Left Termination of the query pattern
maxsort(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))).

Query: maxsort(g,a)

(1) PrologToPiTRSViaGraphTransformerProof (SOUND transformation)

Transformed Prolog program to (Pi-)TRS.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="A: maxsort(T1, T2)\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
5 [label="5: maxsort(T1, T2), SCOPE: 1, CLAUSE: 15 | maxsort(T1, T2), SCOPE: 1, CLAUSE: 16\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: true | maxsort([], T2), SCOPE: 1, CLAUSE: 16\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: maxsort(T1, T2), SCOPE: 1, CLAUSE: 16\n\nT1 is ground\nmaxsort(T1, T2) !=? maxsort([], [])", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: maxsort([], T2), SCOPE: 1, CLAUSE: 16\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: ','(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];
24 [label="24: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: ','(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];
28 [label="28: ','(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];
30 [label="30: ','(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];
31 [label="31: ','(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];
34 [label="B: ','(del(T17, .(T17, []), X14), maxsort(X14, T18))\n\nT17 is ground\nX14 is free", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="K: del(T17, .(T17, []), X14)\n\nT17 is ground\nX14 is free", fontsize=16, style=filled, fillcolor=lightyellow];
39 [label="39: maxsort(T21, T18)\n\nT21 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
45 [label="45: 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];
47 [label="47: 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];
52 [label="52: del(T17, .(T17, []), X14), SCOPE: 3, CLAUSE: 5\n\nT17 is ground\nX14 is free", fontsize=16, style=filled, fillcolor=lightyellow];
54 [label="54: del(T17, .(T17, []), X14), SCOPE: 3, CLAUSE: 6\n\nT17 is ground\nX14 is free", fontsize=16, style=filled, fillcolor=lightyellow];
55 [label="55: eq(T29, T29)\n\nT29 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
56 [label="56: eq(T29, T29), SCOPE: 4, CLAUSE: 0 | eq(T29, T29), SCOPE: 4, CLAUSE: 1\n\nT29 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
57 [label="57: eq(T29, T29), SCOPE: 4, CLAUSE: 0\n\nT29 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
58 [label="58: eq(T29, T29), SCOPE: 4, CLAUSE: 1\n\nT29 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
59 [label="59: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
60 [label="60: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
61 [label="61: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
62 [label="62: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
63 [label="63: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
64 [label="64: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
67 [label="67: ','(neq(T33, T33), del(T33, [], X55))\n\nT33 is ground\nX55 is free", fontsize=16, style=filled, fillcolor=lightyellow];
68 [label="68: ','(neq(T33, T33), del(T33, [], X55)), SCOPE: 5, CLAUSE: 2 | ','(neq(T33, T33), del(T33, [], X55)), SCOPE: 5, CLAUSE: 3\n\nT33 is ground\nX55 is free", fontsize=16, style=filled, fillcolor=lightyellow];
69 [label="69: ','(neq(T33, T33), del(T33, [], X55)), SCOPE: 5, CLAUSE: 3\n\nT33 is ground\nX55 is free", fontsize=16, style=filled, fillcolor=lightyellow];
70 [label="70: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
89 [label="89: ','(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];
90 [label="90: ','(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];
131 [label="131: ','(','(ge(T51, T52), max(.(T51, T53), T55)), ','(del(T55, .(T51, .(T52, T53)), X14), maxsort(X14, T56)))\n\nT51 is ground\nT52 is ground\nT53 is ground\nX14 is free", fontsize=16, style=filled, fillcolor=lightyellow];
132 [label="132: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
136 [label="136: ','(','(ge(T51, T52), max(.(T51, T53), T55)), ','(del(T55, .(T51, .(T52, T53)), X14), maxsort(X14, T56))), SCOPE: 6, CLAUSE: 7 | ','(','(ge(T51, T52), max(.(T51, T53), T55)), ','(del(T55, .(T51, .(T52, T53)), X14), maxsort(X14, T56))), SCOPE: 6, CLAUSE: 8 | ','(','(ge(T51, T52), max(.(T51, T53), T55)), ','(del(T55, .(T51, .(T52, T53)), X14), maxsort(X14, T56))), SCOPE: 6, CLAUSE: 9\n\nT51 is ground\nT52 is ground\nT53 is ground\nX14 is free", fontsize=16, style=filled, fillcolor=lightyellow];
139 [label="139: ','(','(ge(T51, T52), max(.(T51, T53), T55)), ','(del(T55, .(T51, .(T52, T53)), X14), maxsort(X14, T56))), SCOPE: 6, CLAUSE: 7\n\nT51 is ground\nT52 is ground\nT53 is ground\nX14 is free", fontsize=16, style=filled, fillcolor=lightyellow];
140 [label="140: ','(','(ge(T51, T52), max(.(T51, T53), T55)), ','(del(T55, .(T51, .(T52, T53)), X14), maxsort(X14, T56))), SCOPE: 6, CLAUSE: 8 | ','(','(ge(T51, T52), max(.(T51, T53), T55)), ','(del(T55, .(T51, .(T52, T53)), X14), maxsort(X14, T56))), SCOPE: 6, CLAUSE: 9\n\nT51 is ground\nT52 is ground\nT53 is ground\nX14 is free", fontsize=16, style=filled, fillcolor=lightyellow];
142 [label="C: ','(max(.(t, T53), T55), ','(del(T55, .(t, .(t, T53)), X14), maxsort(X14, T56)))\n\nT53 is ground\nX14 is free", fontsize=16, style=filled, fillcolor=lightyellow];
144 [label="144: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
148 [label="G: max(.(t, T53), T55)\n\nT53 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
149 [label="P: ','(del(T57, .(t, .(t, T53)), X14), maxsort(X14, T58))\n\nT53 is ground\nT57 is ground\nX14 is free", fontsize=16, style=filled, fillcolor=lightyellow];
150 [label="150: max(.(t, T53), T55), SCOPE: 7, CLAUSE: 11 | max(.(t, T53), T55), SCOPE: 7, CLAUSE: 12 | max(.(t, T53), T55), SCOPE: 7, CLAUSE: 13 | max(.(t, T53), T55), SCOPE: 7, CLAUSE: 14\n\nT53 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
151 [label="151: max(.(t, T53), T55), SCOPE: 7, CLAUSE: 12 | max(.(t, T53), T55), SCOPE: 7, CLAUSE: 13 | max(.(t, T53), T55), SCOPE: 7, CLAUSE: 14\n\nT53 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
152 [label="152: max(.(t, T53), T55), SCOPE: 7, CLAUSE: 12\n\nT53 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
153 [label="153: max(.(t, T53), T55), SCOPE: 7, CLAUSE: 13 | max(.(t, T53), T55), SCOPE: 7, CLAUSE: 14\n\nT53 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
158 [label="158: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
161 [label="161: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
180 [label="180: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
183 [label="183: max(.(t, T53), T55), SCOPE: 7, CLAUSE: 13\n\nT53 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
184 [label="184: max(.(t, T53), T55), SCOPE: 7, CLAUSE: 14\n\nT53 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
189 [label="189: ','(ge(t, T71), max(.(t, T72), T74))\n\nT71 is ground\nT72 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
191 [label="191: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
195 [label="195: ','(ge(t, T71), max(.(t, T72), T74)), SCOPE: 8, CLAUSE: 7 | ','(ge(t, T71), max(.(t, T72), T74)), SCOPE: 8, CLAUSE: 8 | ','(ge(t, T71), max(.(t, T72), T74)), SCOPE: 8, CLAUSE: 9\n\nT71 is ground\nT72 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
196 [label="196: ','(ge(t, T71), max(.(t, T72), T74)), SCOPE: 8, CLAUSE: 7\n\nT71 is ground\nT72 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
197 [label="197: ','(ge(t, T71), max(.(t, T72), T74)), SCOPE: 8, CLAUSE: 8 | ','(ge(t, T71), max(.(t, T72), T74)), SCOPE: 8, CLAUSE: 9\n\nT71 is ground\nT72 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
198 [label="198: max(.(t, T72), T74)\n\nT72 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
199 [label="199: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
200 [label="200: ','(ge(t, T71), max(.(t, T72), T74)), SCOPE: 8, CLAUSE: 8\n\nT71 is ground\nT72 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
201 [label="201: ','(ge(t, T71), max(.(t, T72), T74)), SCOPE: 8, CLAUSE: 9\n\nT71 is ground\nT72 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
202 [label="202: max(.(t, T72), T74)\n\nT72 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
221 [label="221: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
227 [label="227: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
228 [label="228: ','(gt(T81, t), max(.(T81, T82), T84))\n\nT81 is ground\nT82 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
229 [label="229: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
230 [label="230: ','(gt(T81, t), max(.(T81, T82), T84)), SCOPE: 9, CLAUSE: 10\n\nT81 is ground\nT82 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
231 [label="231: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
264 [label="L: del(T57, .(t, .(t, T53)), X14)\n\nT53 is ground\nT57 is ground\nX14 is free", fontsize=16, style=filled, fillcolor=lightyellow];
265 [label="265: maxsort(T87, T58)\n\nT87 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
270 [label="270: del(T57, .(t, .(t, T53)), X14), SCOPE: 10, CLAUSE: 4 | del(T57, .(t, .(t, T53)), X14), SCOPE: 10, CLAUSE: 5 | del(T57, .(t, .(t, T53)), X14), SCOPE: 10, CLAUSE: 6\n\nT53 is ground\nT57 is ground\nX14 is free", fontsize=16, style=filled, fillcolor=lightyellow];
272 [label="272: del(T57, .(t, .(t, T53)), X14), SCOPE: 10, CLAUSE: 5 | del(T57, .(t, .(t, T53)), X14), SCOPE: 10, CLAUSE: 6\n\nT53 is ground\nT57 is ground\nX14 is free", fontsize=16, style=filled, fillcolor=lightyellow];
286 [label="286: del(T57, .(t, .(t, T53)), X14), SCOPE: 10, CLAUSE: 5\n\nT53 is ground\nT57 is ground\nX14 is free", fontsize=16, style=filled, fillcolor=lightyellow];
287 [label="287: del(T57, .(t, .(t, T53)), X14), SCOPE: 10, CLAUSE: 6\n\nT53 is ground\nT57 is ground\nX14 is free", fontsize=16, style=filled, fillcolor=lightyellow];
290 [label="290: eq(T99, t)\n\nT99 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
291 [label="291: eq(T99, t), SCOPE: 11, CLAUSE: 0 | eq(T99, t), SCOPE: 11, CLAUSE: 1\n\nT99 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
292 [label="292: eq(T99, t), SCOPE: 11, CLAUSE: 0\n\nT99 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
293 [label="293: eq(T99, t), SCOPE: 11, CLAUSE: 1\n\nT99 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
294 [label="294: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
296 [label="296: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
298 [label="298: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
299 [label="299: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
319 [label="319: ','(neq(T107, t), del(T107, .(t, T108), X150))\n\nT107 is ground\nT108 is ground\nX150 is free", fontsize=16, style=filled, fillcolor=lightyellow];
322 [label="322: ','(neq(T107, t), del(T107, .(t, T108), X150)), SCOPE: 12, CLAUSE: 2 | ','(neq(T107, t), del(T107, .(t, T108), X150)), SCOPE: 12, CLAUSE: 3\n\nT107 is ground\nT108 is ground\nX150 is free", fontsize=16, style=filled, fillcolor=lightyellow];
324 [label="324: ','(neq(T107, t), del(T107, .(t, T108), X150)), SCOPE: 12, CLAUSE: 3\n\nT107 is ground\nT108 is ground\nX150 is free", fontsize=16, style=filled, fillcolor=lightyellow];
326 [label="326: del(f, .(t, T108), X150)\n\nT108 is ground\nX150 is free", fontsize=16, style=filled, fillcolor=lightyellow];
327 [label="327: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
332 [label="332: del(f, .(t, T108), X150), SCOPE: 13, CLAUSE: 4 | del(f, .(t, T108), X150), SCOPE: 13, CLAUSE: 5 | del(f, .(t, T108), X150), SCOPE: 13, CLAUSE: 6\n\nT108 is ground\nX150 is free", fontsize=16, style=filled, fillcolor=lightyellow];
333 [label="333: del(f, .(t, T108), X150), SCOPE: 13, CLAUSE: 5 | del(f, .(t, T108), X150), SCOPE: 13, CLAUSE: 6\n\nT108 is ground\nX150 is free", fontsize=16, style=filled, fillcolor=lightyellow];
344 [label="344: del(f, .(t, T108), X150), SCOPE: 13, CLAUSE: 5\n\nT108 is ground\nX150 is free", fontsize=16, style=filled, fillcolor=lightyellow];
345 [label="345: del(f, .(t, T108), X150), SCOPE: 13, CLAUSE: 6\n\nT108 is ground\nX150 is free", fontsize=16, style=filled, fillcolor=lightyellow];
349 [label="349: eq(f, t)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
350 [label="350: eq(f, t), SCOPE: 14, CLAUSE: 0 | eq(f, t), SCOPE: 14, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
351 [label="351: eq(f, t), SCOPE: 14, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
352 [label="352: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
355 [label="355: ','(neq(f, t), del(f, T118, X187))\n\nT118 is ground\nX187 is free", fontsize=16, style=filled, fillcolor=lightyellow];
356 [label="356: ','(neq(f, t), del(f, T118, X187)), SCOPE: 15, CLAUSE: 2 | ','(neq(f, t), del(f, T118, X187)), SCOPE: 15, CLAUSE: 3\n\nT118 is ground\nX187 is free", fontsize=16, style=filled, fillcolor=lightyellow];
357 [label="357: ','(neq(f, t), del(f, T118, X187)), SCOPE: 15, CLAUSE: 3\n\nT118 is ground\nX187 is free", fontsize=16, style=filled, fillcolor=lightyellow];
358 [label="H: del(f, T118, X187)\n\nT118 is ground\nX187 is free", fontsize=16, style=filled, fillcolor=lightyellow];
361 [label="361: del(f, T118, X187), SCOPE: 16, CLAUSE: 4 | del(f, T118, X187), SCOPE: 16, CLAUSE: 5 | del(f, T118, X187), SCOPE: 16, CLAUSE: 6\n\nT118 is ground\nX187 is free", fontsize=16, style=filled, fillcolor=lightyellow];
364 [label="364: del(f, T118, X187), SCOPE: 16, CLAUSE: 4\n\nT118 is ground\nX187 is free", fontsize=16, style=filled, fillcolor=lightyellow];
365 [label="365: del(f, T118, X187), SCOPE: 16, CLAUSE: 5 | del(f, T118, X187), SCOPE: 16, CLAUSE: 6\n\nT118 is ground\nX187 is free", fontsize=16, style=filled, fillcolor=lightyellow];
368 [label="368: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
369 [label="369: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
370 [label="370: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
374 [label="374: del(f, T118, X187), SCOPE: 16, CLAUSE: 5\n\nT118 is ground\nX187 is free", fontsize=16, style=filled, fillcolor=lightyellow];
375 [label="375: del(f, T118, X187), SCOPE: 16, CLAUSE: 6\n\nT118 is ground\nX187 is free", fontsize=16, style=filled, fillcolor=lightyellow];
378 [label="378: eq(f, T127)\n\nT127 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
379 [label="379: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
380 [label="380: eq(f, T127), SCOPE: 17, CLAUSE: 0 | eq(f, T127), SCOPE: 17, CLAUSE: 1\n\nT127 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
381 [label="381: eq(f, T127), SCOPE: 17, CLAUSE: 1\n\nT127 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
382 [label="382: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
383 [label="383: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
384 [label="384: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
390 [label="390: ','(neq(f, T133), del(f, T134, X228))\n\nT133 is ground\nT134 is ground\nX228 is free", fontsize=16, style=filled, fillcolor=lightyellow];
391 [label="391: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
394 [label="394: ','(neq(f, T133), del(f, T134, X228)), SCOPE: 18, CLAUSE: 2 | ','(neq(f, T133), del(f, T134, X228)), SCOPE: 18, CLAUSE: 3\n\nT133 is ground\nT134 is ground\nX228 is free", fontsize=16, style=filled, fillcolor=lightyellow];
395 [label="395: ','(neq(f, T133), del(f, T134, X228)), SCOPE: 18, CLAUSE: 3\n\nT133 is ground\nT134 is ground\nX228 is free", fontsize=16, style=filled, fillcolor=lightyellow];
396 [label="396: del(f, T134, X228)\n\nT134 is ground\nX228 is free", fontsize=16, style=filled, fillcolor=lightyellow];
397 [label="397: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
398 [label="398: ','(','(ge(T51, T52), max(.(T51, T53), T55)), ','(del(T55, .(T51, .(T52, T53)), X14), maxsort(X14, T56))), SCOPE: 6, CLAUSE: 8\n\nT51 is ground\nT52 is ground\nT53 is ground\nX14 is free", fontsize=16, style=filled, fillcolor=lightyellow];
399 [label="399: ','(','(ge(T51, T52), max(.(T51, T53), T55)), ','(del(T55, .(T51, .(T52, T53)), X14), maxsort(X14, T56))), SCOPE: 6, CLAUSE: 9\n\nT51 is ground\nT52 is ground\nT53 is ground\nX14 is free", fontsize=16, style=filled, fillcolor=lightyellow];
400 [label="D: ','(max(.(t, T53), T55), ','(del(T55, .(t, .(f, T53)), X14), maxsort(X14, T56)))\n\nT53 is ground\nX14 is free", fontsize=16, style=filled, fillcolor=lightyellow];
401 [label="401: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
407 [label="407: max(.(t, T53), T55)\n\nT53 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
408 [label="Q: ','(del(T135, .(t, .(f, T53)), X14), maxsort(X14, T136))\n\nT53 is ground\nT135 is ground\nX14 is free", fontsize=16, style=filled, fillcolor=lightyellow];
409 [label="M: del(T135, .(t, .(f, T53)), X14)\n\nT53 is ground\nT135 is ground\nX14 is free", fontsize=16, style=filled, fillcolor=lightyellow];
410 [label="410: maxsort(T139, T136)\n\nT139 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
413 [label="413: del(T135, .(t, .(f, T53)), X14), SCOPE: 19, CLAUSE: 4 | del(T135, .(t, .(f, T53)), X14), SCOPE: 19, CLAUSE: 5 | del(T135, .(t, .(f, T53)), X14), SCOPE: 19, CLAUSE: 6\n\nT53 is ground\nT135 is ground\nX14 is free", fontsize=16, style=filled, fillcolor=lightyellow];
414 [label="414: del(T135, .(t, .(f, T53)), X14), SCOPE: 19, CLAUSE: 5 | del(T135, .(t, .(f, T53)), X14), SCOPE: 19, CLAUSE: 6\n\nT53 is ground\nT135 is ground\nX14 is free", fontsize=16, style=filled, fillcolor=lightyellow];
417 [label="417: del(T135, .(t, .(f, T53)), X14), SCOPE: 19, CLAUSE: 5\n\nT53 is ground\nT135 is ground\nX14 is free", fontsize=16, style=filled, fillcolor=lightyellow];
418 [label="418: del(T135, .(t, .(f, T53)), X14), SCOPE: 19, CLAUSE: 6\n\nT53 is ground\nT135 is ground\nX14 is free", fontsize=16, style=filled, fillcolor=lightyellow];
421 [label="421: eq(T151, t)\n\nT151 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
422 [label="422: eq(T151, t), SCOPE: 20, CLAUSE: 0 | eq(T151, t), SCOPE: 20, CLAUSE: 1\n\nT151 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
423 [label="423: eq(T151, t), SCOPE: 20, CLAUSE: 0\n\nT151 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
424 [label="424: eq(T151, t), SCOPE: 20, CLAUSE: 1\n\nT151 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
425 [label="425: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
426 [label="426: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
427 [label="427: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
428 [label="428: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
435 [label="435: ','(neq(T159, t), del(T159, .(f, T160), X269))\n\nT159 is ground\nT160 is ground\nX269 is free", fontsize=16, style=filled, fillcolor=lightyellow];
436 [label="436: ','(neq(T159, t), del(T159, .(f, T160), X269)), SCOPE: 21, CLAUSE: 2 | ','(neq(T159, t), del(T159, .(f, T160), X269)), SCOPE: 21, CLAUSE: 3\n\nT159 is ground\nT160 is ground\nX269 is free", fontsize=16, style=filled, fillcolor=lightyellow];
437 [label="437: ','(neq(T159, t), del(T159, .(f, T160), X269)), SCOPE: 21, CLAUSE: 3\n\nT159 is ground\nT160 is ground\nX269 is free", fontsize=16, style=filled, fillcolor=lightyellow];
438 [label="438: del(f, .(f, T160), X269)\n\nT160 is ground\nX269 is free", fontsize=16, style=filled, fillcolor=lightyellow];
439 [label="439: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
446 [label="E: ','(max(.(f, T53), T55), ','(del(T55, .(f, .(f, T53)), X14), maxsort(X14, T56)))\n\nT53 is ground\nX14 is free", fontsize=16, style=filled, fillcolor=lightyellow];
447 [label="447: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
450 [label="I: max(.(f, T53), T55)\n\nT53 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
451 [label="R: ','(del(T163, .(f, .(f, T53)), X14), maxsort(X14, T164))\n\nT53 is ground\nT163 is ground\nX14 is free", fontsize=16, style=filled, fillcolor=lightyellow];
452 [label="452: max(.(f, T53), T55), SCOPE: 22, CLAUSE: 11 | max(.(f, T53), T55), SCOPE: 22, CLAUSE: 12 | max(.(f, T53), T55), SCOPE: 22, CLAUSE: 13 | max(.(f, T53), T55), SCOPE: 22, CLAUSE: 14\n\nT53 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
453 [label="453: max(.(f, T53), T55), SCOPE: 22, CLAUSE: 12 | max(.(f, T53), T55), SCOPE: 22, CLAUSE: 13 | max(.(f, T53), T55), SCOPE: 22, CLAUSE: 14\n\nT53 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
454 [label="454: max(.(f, T53), T55), SCOPE: 22, CLAUSE: 12\n\nT53 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
455 [label="455: max(.(f, T53), T55), SCOPE: 22, CLAUSE: 13 | max(.(f, T53), T55), SCOPE: 22, CLAUSE: 14\n\nT53 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
456 [label="456: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
457 [label="457: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
458 [label="458: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
459 [label="459: max(.(f, T53), T55), SCOPE: 22, CLAUSE: 13\n\nT53 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
460 [label="460: max(.(f, T53), T55), SCOPE: 22, CLAUSE: 14\n\nT53 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
461 [label="461: ','(ge(f, T177), max(.(f, T178), T180))\n\nT177 is ground\nT178 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
462 [label="462: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
463 [label="463: ','(ge(f, T177), max(.(f, T178), T180)), SCOPE: 23, CLAUSE: 7 | ','(ge(f, T177), max(.(f, T178), T180)), SCOPE: 23, CLAUSE: 8 | ','(ge(f, T177), max(.(f, T178), T180)), SCOPE: 23, CLAUSE: 9\n\nT177 is ground\nT178 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
464 [label="464: ','(ge(f, T177), max(.(f, T178), T180)), SCOPE: 23, CLAUSE: 8 | ','(ge(f, T177), max(.(f, T178), T180)), SCOPE: 23, CLAUSE: 9\n\nT177 is ground\nT178 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
465 [label="465: ','(ge(f, T177), max(.(f, T178), T180)), SCOPE: 23, CLAUSE: 9\n\nT177 is ground\nT178 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
466 [label="466: max(.(f, T178), T180)\n\nT178 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
467 [label="467: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
468 [label="468: ','(gt(T187, f), max(.(T187, T188), T190))\n\nT187 is ground\nT188 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
469 [label="469: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
470 [label="470: ','(gt(T187, f), max(.(T187, T188), T190)), SCOPE: 24, CLAUSE: 10\n\nT187 is ground\nT188 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
471 [label="471: max(.(t, T188), T190)\n\nT188 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
472 [label="472: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
479 [label="N: del(T163, .(f, .(f, T53)), X14)\n\nT53 is ground\nT163 is ground\nX14 is free", fontsize=16, style=filled, fillcolor=lightyellow];
480 [label="480: maxsort(T193, T164)\n\nT193 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
481 [label="481: del(T163, .(f, .(f, T53)), X14), SCOPE: 25, CLAUSE: 4 | del(T163, .(f, .(f, T53)), X14), SCOPE: 25, CLAUSE: 5 | del(T163, .(f, .(f, T53)), X14), SCOPE: 25, CLAUSE: 6\n\nT53 is ground\nT163 is ground\nX14 is free", fontsize=16, style=filled, fillcolor=lightyellow];
483 [label="483: del(T163, .(f, .(f, T53)), X14), SCOPE: 25, CLAUSE: 5 | del(T163, .(f, .(f, T53)), X14), SCOPE: 25, CLAUSE: 6\n\nT53 is ground\nT163 is ground\nX14 is free", fontsize=16, style=filled, fillcolor=lightyellow];
487 [label="487: del(T163, .(f, .(f, T53)), X14), SCOPE: 25, CLAUSE: 5\n\nT53 is ground\nT163 is ground\nX14 is free", fontsize=16, style=filled, fillcolor=lightyellow];
488 [label="488: del(T163, .(f, .(f, T53)), X14), SCOPE: 25, CLAUSE: 6\n\nT53 is ground\nT163 is ground\nX14 is free", fontsize=16, style=filled, fillcolor=lightyellow];
490 [label="490: eq(T205, f)\n\nT205 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
491 [label="491: eq(T205, f), SCOPE: 26, CLAUSE: 0 | eq(T205, f), SCOPE: 26, CLAUSE: 1\n\nT205 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
492 [label="492: eq(T205, f), SCOPE: 26, CLAUSE: 1\n\nT205 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
493 [label="493: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
494 [label="494: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
495 [label="495: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
496 [label="496: ','(neq(T213, f), del(T213, .(f, T214), X347))\n\nT213 is ground\nT214 is ground\nX347 is free", fontsize=16, style=filled, fillcolor=lightyellow];
497 [label="497: ','(neq(T213, f), del(T213, .(f, T214), X347)), SCOPE: 27, CLAUSE: 2 | ','(neq(T213, f), del(T213, .(f, T214), X347)), SCOPE: 27, CLAUSE: 3\n\nT213 is ground\nT214 is ground\nX347 is free", fontsize=16, style=filled, fillcolor=lightyellow];
498 [label="498: ','(neq(T213, f), del(T213, .(f, T214), X347)), SCOPE: 27, CLAUSE: 2\n\nT213 is ground\nT214 is ground\nX347 is free", fontsize=16, style=filled, fillcolor=lightyellow];
499 [label="499: ','(neq(T213, f), del(T213, .(f, T214), X347)), SCOPE: 27, CLAUSE: 3\n\nT213 is ground\nT214 is ground\nX347 is free", fontsize=16, style=filled, fillcolor=lightyellow];
500 [label="500: del(t, .(f, T214), X347)\n\nT214 is ground\nX347 is free", fontsize=16, style=filled, fillcolor=lightyellow];
501 [label="501: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
508 [label="508: del(t, .(f, T214), X347), SCOPE: 28, CLAUSE: 4 | del(t, .(f, T214), X347), SCOPE: 28, CLAUSE: 5 | del(t, .(f, T214), X347), SCOPE: 28, CLAUSE: 6\n\nT214 is ground\nX347 is free", fontsize=16, style=filled, fillcolor=lightyellow];
509 [label="509: del(t, .(f, T214), X347), SCOPE: 28, CLAUSE: 5 | del(t, .(f, T214), X347), SCOPE: 28, CLAUSE: 6\n\nT214 is ground\nX347 is free", fontsize=16, style=filled, fillcolor=lightyellow];
510 [label="510: del(t, .(f, T214), X347), SCOPE: 28, CLAUSE: 5\n\nT214 is ground\nX347 is free", fontsize=16, style=filled, fillcolor=lightyellow];
511 [label="511: del(t, .(f, T214), X347), SCOPE: 28, CLAUSE: 6\n\nT214 is ground\nX347 is free", fontsize=16, style=filled, fillcolor=lightyellow];
512 [label="512: eq(t, f)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
513 [label="513: eq(t, f), SCOPE: 29, CLAUSE: 0 | eq(t, f), SCOPE: 29, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
514 [label="514: eq(t, f), SCOPE: 29, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
515 [label="515: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
516 [label="516: ','(neq(t, f), del(t, T224, X384))\n\nT224 is ground\nX384 is free", fontsize=16, style=filled, fillcolor=lightyellow];
517 [label="517: ','(neq(t, f), del(t, T224, X384)), SCOPE: 30, CLAUSE: 2 | ','(neq(t, f), del(t, T224, X384)), SCOPE: 30, CLAUSE: 3\n\nT224 is ground\nX384 is free", fontsize=16, style=filled, fillcolor=lightyellow];
518 [label="518: ','(neq(t, f), del(t, T224, X384)), SCOPE: 30, CLAUSE: 2\n\nT224 is ground\nX384 is free", fontsize=16, style=filled, fillcolor=lightyellow];
519 [label="519: ','(neq(t, f), del(t, T224, X384)), SCOPE: 30, CLAUSE: 3\n\nT224 is ground\nX384 is free", fontsize=16, style=filled, fillcolor=lightyellow];
522 [label="J: del(t, T224, X384)\n\nT224 is ground\nX384 is free", fontsize=16, style=filled, fillcolor=lightyellow];
532 [label="532: del(t, T224, X384), SCOPE: 31, CLAUSE: 4 | del(t, T224, X384), SCOPE: 31, CLAUSE: 5 | del(t, T224, X384), SCOPE: 31, CLAUSE: 6\n\nT224 is ground\nX384 is free", fontsize=16, style=filled, fillcolor=lightyellow];
533 [label="533: del(t, T224, X384), SCOPE: 31, CLAUSE: 4\n\nT224 is ground\nX384 is free", fontsize=16, style=filled, fillcolor=lightyellow];
534 [label="534: del(t, T224, X384), SCOPE: 31, CLAUSE: 5 | del(t, T224, X384), SCOPE: 31, CLAUSE: 6\n\nT224 is ground\nX384 is free", fontsize=16, style=filled, fillcolor=lightyellow];
535 [label="535: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
536 [label="536: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
537 [label="537: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
541 [label="541: del(t, T224, X384), SCOPE: 31, CLAUSE: 5\n\nT224 is ground\nX384 is free", fontsize=16, style=filled, fillcolor=lightyellow];
542 [label="542: del(t, T224, X384), SCOPE: 31, CLAUSE: 6\n\nT224 is ground\nX384 is free", fontsize=16, style=filled, fillcolor=lightyellow];
547 [label="547: eq(t, T233)\n\nT233 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
548 [label="548: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
549 [label="549: eq(t, T233), SCOPE: 32, CLAUSE: 0 | eq(t, T233), SCOPE: 32, CLAUSE: 1\n\nT233 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
550 [label="550: eq(t, T233), SCOPE: 32, CLAUSE: 0\n\nT233 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
551 [label="551: eq(t, T233), SCOPE: 32, CLAUSE: 1\n\nT233 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
552 [label="552: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
553 [label="553: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
554 [label="554: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
555 [label="555: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
556 [label="556: ','(neq(t, T239), del(t, T240, X427))\n\nT239 is ground\nT240 is ground\nX427 is free", fontsize=16, style=filled, fillcolor=lightyellow];
557 [label="557: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
558 [label="558: ','(neq(t, T239), del(t, T240, X427)), SCOPE: 33, CLAUSE: 2 | ','(neq(t, T239), del(t, T240, X427)), SCOPE: 33, CLAUSE: 3\n\nT239 is ground\nT240 is ground\nX427 is free", fontsize=16, style=filled, fillcolor=lightyellow];
559 [label="559: ','(neq(t, T239), del(t, T240, X427)), SCOPE: 33, CLAUSE: 2\n\nT239 is ground\nT240 is ground\nX427 is free", fontsize=16, style=filled, fillcolor=lightyellow];
560 [label="560: ','(neq(t, T239), del(t, T240, X427)), SCOPE: 33, CLAUSE: 3\n\nT239 is ground\nT240 is ground\nX427 is free", fontsize=16, style=filled, fillcolor=lightyellow];
561 [label="561: del(t, T240, X427)\n\nT240 is ground\nX427 is free", fontsize=16, style=filled, fillcolor=lightyellow];
562 [label="562: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
563 [label="563: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
564 [label="564: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
565 [label="565: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
569 [label="569: ','(','(gt(T250, T249), max(.(T250, T251), T253)), ','(del(T253, .(T249, .(T250, T251)), X14), maxsort(X14, T254)))\n\nT249 is ground\nT250 is ground\nT251 is ground\nX14 is free", fontsize=16, style=filled, fillcolor=lightyellow];
570 [label="570: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
571 [label="571: ','(','(gt(T250, T249), max(.(T250, T251), T253)), ','(del(T253, .(T249, .(T250, T251)), X14), maxsort(X14, T254))), SCOPE: 34, CLAUSE: 10\n\nT249 is ground\nT250 is ground\nT251 is ground\nX14 is free", fontsize=16, style=filled, fillcolor=lightyellow];
572 [label="F: ','(max(.(t, T251), T253), ','(del(T253, .(f, .(t, T251)), X14), maxsort(X14, T254)))\n\nT251 is ground\nX14 is free", fontsize=16, style=filled, fillcolor=lightyellow];
573 [label="573: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
579 [label="579: max(.(t, T251), T253)\n\nT251 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
580 [label="S: ','(del(T255, .(f, .(t, T251)), X14), maxsort(X14, T256))\n\nT251 is ground\nT255 is ground\nX14 is free", fontsize=16, style=filled, fillcolor=lightyellow];
581 [label="O: del(T255, .(f, .(t, T251)), X14)\n\nT251 is ground\nT255 is ground\nX14 is free", fontsize=16, style=filled, fillcolor=lightyellow];
582 [label="582: maxsort(T259, T256)\n\nT259 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
655 [label="655: del(T255, .(f, .(t, T251)), X14), SCOPE: 35, CLAUSE: 4 | del(T255, .(f, .(t, T251)), X14), SCOPE: 35, CLAUSE: 5 | del(T255, .(f, .(t, T251)), X14), SCOPE: 35, CLAUSE: 6\n\nT251 is ground\nT255 is ground\nX14 is free", fontsize=16, style=filled, fillcolor=lightyellow];
662 [label="662: del(T255, .(f, .(t, T251)), X14), SCOPE: 35, CLAUSE: 5 | del(T255, .(f, .(t, T251)), X14), SCOPE: 35, CLAUSE: 6\n\nT251 is ground\nT255 is ground\nX14 is free", fontsize=16, style=filled, fillcolor=lightyellow];
663 [label="663: del(T255, .(f, .(t, T251)), X14), SCOPE: 35, CLAUSE: 5\n\nT251 is ground\nT255 is ground\nX14 is free", fontsize=16, style=filled, fillcolor=lightyellow];
664 [label="664: del(T255, .(f, .(t, T251)), X14), SCOPE: 35, CLAUSE: 6\n\nT251 is ground\nT255 is ground\nX14 is free", fontsize=16, style=filled, fillcolor=lightyellow];
668 [label="668: eq(T271, f)\n\nT271 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
669 [label="669: eq(T271, f), SCOPE: 36, CLAUSE: 0 | eq(T271, f), SCOPE: 36, CLAUSE: 1\n\nT271 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
670 [label="670: eq(T271, f), SCOPE: 36, CLAUSE: 1\n\nT271 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
671 [label="671: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
672 [label="672: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
673 [label="673: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
677 [label="677: ','(neq(T279, f), del(T279, .(t, T280), X480))\n\nT279 is ground\nT280 is ground\nX480 is free", fontsize=16, style=filled, fillcolor=lightyellow];
678 [label="678: ','(neq(T279, f), del(T279, .(t, T280), X480)), SCOPE: 37, CLAUSE: 2 | ','(neq(T279, f), del(T279, .(t, T280), X480)), SCOPE: 37, CLAUSE: 3\n\nT279 is ground\nT280 is ground\nX480 is free", fontsize=16, style=filled, fillcolor=lightyellow];
679 [label="679: ','(neq(T279, f), del(T279, .(t, T280), X480)), SCOPE: 37, CLAUSE: 2\n\nT279 is ground\nT280 is ground\nX480 is free", fontsize=16, style=filled, fillcolor=lightyellow];
680 [label="680: ','(neq(T279, f), del(T279, .(t, T280), X480)), SCOPE: 37, CLAUSE: 3\n\nT279 is ground\nT280 is ground\nX480 is free", fontsize=16, style=filled, fillcolor=lightyellow];
681 [label="681: del(t, .(t, T280), X480)\n\nT280 is ground\nX480 is free", fontsize=16, style=filled, fillcolor=lightyellow];
682 [label="682: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
684 [label="684: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
685 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 685 [arrowhead = none ];
685 -> 5;

686 [label="EVAL with clause\nmaxsort([], []).\nand substitutionT1 -> [],\nT2 -> []", fontsize=14, style = filled, fillcolor = lightblue];
5 -> 686 [arrowhead = none ];
686 -> 10;

687 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
5 -> 687 [arrowhead = none ];
687 -> 12;

688 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
10 -> 688 [arrowhead = none ];
688 -> 14;

689 [label="EVAL with clause\nmaxsort(.(X10, X11), .(X12, X13)) :- ','(max(.(X10, X11), X12), ','(del(X12, .(X10, X11), X14), maxsort(X14, X13))).\nand substitutionX10 -> T7,\nX11 -> T8,\nT1 -> .(T7, T8),\nX12 -> T11,\nX13 -> T12,\nT2 -> .(T11, T12),\nT9 -> T11,\nT10 -> T12", fontsize=14, style = filled, fillcolor = lightblue];
12 -> 689 [arrowhead = none ];
689 -> 22;

690 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
12 -> 690 [arrowhead = none ];
690 -> 24;

691 [label="BACKTRACK\nfor clause: maxsort(.(X, XS), .(Y, YS)) :- ','(max(.(X, XS), Y), ','(del(Y, .(X, XS), ZS), maxsort(ZS, YS)))because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
14 -> 691 [arrowhead = none ];
691 -> 18;

692 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
22 -> 692 [arrowhead = none ];
692 -> 26;

693 [label="BACKTRACK\nfor clause: max([], f)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
26 -> 693 [arrowhead = none ];
693 -> 28;

694 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
28 -> 694 [arrowhead = none ];
694 -> 30;

695 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
28 -> 695 [arrowhead = none ];
695 -> 31;

696 [label="EVAL with clause\nmax(.(X19, []), X19).\nand substitutionT7 -> T17,\nX19 -> T17,\nT8 -> [],\nT11 -> T17,\nT12 -> T18", fontsize=14, style = filled, fillcolor = lightblue];
30 -> 696 [arrowhead = none ];
696 -> 34;

697 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
30 -> 697 [arrowhead = none ];
697 -> 36;

698 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
31 -> 698 [arrowhead = none ];
698 -> 89;

699 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
31 -> 699 [arrowhead = none ];
699 -> 90;

700 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
34 -> 700 [arrowhead = none ];
700 -> 38;

701 [label="SPLIT 2\nnew knowledge:\nT17 is ground\nT21 is ground\nreplacements:X14 -> T21", fontsize=14, style = filled, fillcolor = violet];
34 -> 701 [arrowhead = none ];
701 -> 39;

702 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
38 -> 702 [arrowhead = none ];
702 -> 45;

703 [label="INSTANCE with matching:\nT1 -> T21\nT2 -> T18", fontsize=14, style = filled, fillcolor = lightgrey];
39 -> 703 [arrowhead = none , style=dashed];
703 -> 1[style=dashed];

704 [label="BACKTRACK\nfor clause: del(X1, [], [])because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
45 -> 704 [arrowhead = none ];
704 -> 47;

705 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
47 -> 705 [arrowhead = none ];
705 -> 52;

706 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
47 -> 706 [arrowhead = none ];
706 -> 54;

707 [label="ONLY EVAL with clause\ndel(X37, .(X38, X39), X39) :- eq(X37, X38).\nand substitutionT17 -> T29,\nX37 -> T29,\nX38 -> T29,\nX39 -> [],\nX14 -> []", fontsize=14, style = filled, fillcolor = lightblue];
52 -> 707 [arrowhead = none ];
707 -> 55;

708 [label="ONLY EVAL with clause\ndel(X51, .(X52, X53), .(X52, X54)) :- ','(neq(X51, X52), del(X51, X53, X54)).\nand substitutionT17 -> T33,\nX51 -> T33,\nX52 -> T33,\nX53 -> [],\nX54 -> X55,\nX14 -> .(T33, X55)", fontsize=14, style = filled, fillcolor = lightblue];
54 -> 708 [arrowhead = none ];
708 -> 67;

709 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
55 -> 709 [arrowhead = none ];
709 -> 56;

710 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
56 -> 710 [arrowhead = none ];
710 -> 57;

711 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
56 -> 711 [arrowhead = none ];
711 -> 58;

712 [label="EVAL with clause\neq(t, t).\nand substitutionT29 -> t", fontsize=14, style = filled, fillcolor = lightblue];
57 -> 712 [arrowhead = none ];
712 -> 59;

713 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
57 -> 713 [arrowhead = none ];
713 -> 60;

714 [label="EVAL with clause\neq(f, f).\nand substitutionT29 -> f", fontsize=14, style = filled, fillcolor = lightblue];
58 -> 714 [arrowhead = none ];
714 -> 62;

715 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
58 -> 715 [arrowhead = none ];
715 -> 63;

716 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
59 -> 716 [arrowhead = none ];
716 -> 61;

717 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
62 -> 717 [arrowhead = none ];
717 -> 64;

718 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
67 -> 718 [arrowhead = none ];
718 -> 68;

719 [label="BACKTRACK\nfor clause: neq(t, f)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
68 -> 719 [arrowhead = none ];
719 -> 69;

720 [label="BACKTRACK\nfor clause: neq(f, t)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
69 -> 720 [arrowhead = none ];
720 -> 70;

721 [label="EVAL with clause\nmax(.(X73, .(X74, X75)), X76) :- ','(ge(X73, X74), max(.(X73, X75), X76)).\nand substitutionT7 -> T51,\nX73 -> T51,\nX74 -> T52,\nX75 -> T53,\nT8 -> .(T52, T53),\nT11 -> T55,\nX76 -> T55,\nT54 -> T55,\nT12 -> T56", fontsize=14, style = filled, fillcolor = lightblue];
89 -> 721 [arrowhead = none ];
721 -> 131;

722 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
89 -> 722 [arrowhead = none ];
722 -> 132;

723 [label="EVAL with clause\nmax(.(X440, .(X441, X442)), X443) :- ','(gt(X441, X440), max(.(X441, X442), X443)).\nand substitutionT7 -> T249,\nX440 -> T249,\nX441 -> T250,\nX442 -> T251,\nT8 -> .(T250, T251),\nT11 -> T253,\nX443 -> T253,\nT252 -> T253,\nT12 -> T254", fontsize=14, style = filled, fillcolor = lightblue];
90 -> 723 [arrowhead = none ];
723 -> 569;

724 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
90 -> 724 [arrowhead = none ];
724 -> 570;

725 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
131 -> 725 [arrowhead = none ];
725 -> 136;

726 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
136 -> 726 [arrowhead = none ];
726 -> 139;

727 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
136 -> 727 [arrowhead = none ];
727 -> 140;

728 [label="EVAL with clause\nge(t, t).\nand substitutionT51 -> t,\nT52 -> t", fontsize=14, style = filled, fillcolor = lightblue];
139 -> 728 [arrowhead = none ];
728 -> 142;

729 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
139 -> 729 [arrowhead = none ];
729 -> 144;

730 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
140 -> 730 [arrowhead = none ];
730 -> 398;

731 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
140 -> 731 [arrowhead = none ];
731 -> 399;

732 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
142 -> 732 [arrowhead = none ];
732 -> 148;

733 [label="SPLIT 2\nnew knowledge:\nT53 is ground\nT57 is ground\nreplacements:T55 -> T57,\nT56 -> T58", fontsize=14, style = filled, fillcolor = violet];
142 -> 733 [arrowhead = none ];
733 -> 149;

734 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
148 -> 734 [arrowhead = none ];
734 -> 150;

735 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
149 -> 735 [arrowhead = none ];
735 -> 264;

736 [label="SPLIT 2\nnew knowledge:\nT57 is ground\nT53 is ground\nT87 is ground\nreplacements:X14 -> T87", fontsize=14, style = filled, fillcolor = violet];
149 -> 736 [arrowhead = none ];
736 -> 265;

737 [label="BACKTRACK\nfor clause: max([], f)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
150 -> 737 [arrowhead = none ];
737 -> 151;

738 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
151 -> 738 [arrowhead = none ];
738 -> 152;

739 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
151 -> 739 [arrowhead = none ];
739 -> 153;

740 [label="EVAL with clause\nmax(.(X81, []), X81).\nand substitutionX81 -> t,\nT53 -> [],\nT55 -> t", fontsize=14, style = filled, fillcolor = lightblue];
152 -> 740 [arrowhead = none ];
740 -> 158;

741 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
152 -> 741 [arrowhead = none ];
741 -> 161;

742 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
153 -> 742 [arrowhead = none ];
742 -> 183;

743 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
153 -> 743 [arrowhead = none ];
743 -> 184;

744 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
158 -> 744 [arrowhead = none ];
744 -> 180;

745 [label="EVAL with clause\nmax(.(X98, .(X99, X100)), X101) :- ','(ge(X98, X99), max(.(X98, X100), X101)).\nand substitutionX98 -> t,\nX99 -> T71,\nX100 -> T72,\nT53 -> .(T71, T72),\nT55 -> T74,\nX101 -> T74,\nT73 -> T74", fontsize=14, style = filled, fillcolor = lightblue];
183 -> 745 [arrowhead = none ];
745 -> 189;

746 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
183 -> 746 [arrowhead = none ];
746 -> 191;

747 [label="EVAL with clause\nmax(.(X110, .(X111, X112)), X113) :- ','(gt(X111, X110), max(.(X111, X112), X113)).\nand substitutionX110 -> t,\nX111 -> T81,\nX112 -> T82,\nT53 -> .(T81, T82),\nT55 -> T84,\nX113 -> T84,\nT83 -> T84", fontsize=14, style = filled, fillcolor = lightblue];
184 -> 747 [arrowhead = none ];
747 -> 228;

748 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
184 -> 748 [arrowhead = none ];
748 -> 229;

749 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
189 -> 749 [arrowhead = none ];
749 -> 195;

750 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
195 -> 750 [arrowhead = none ];
750 -> 196;

751 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
195 -> 751 [arrowhead = none ];
751 -> 197;

752 [label="EVAL with clause\nge(t, t).\nand substitutionT71 -> t", fontsize=14, style = filled, fillcolor = lightblue];
196 -> 752 [arrowhead = none ];
752 -> 198;

753 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
196 -> 753 [arrowhead = none ];
753 -> 199;

754 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
197 -> 754 [arrowhead = none ];
754 -> 200;

755 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
197 -> 755 [arrowhead = none ];
755 -> 201;

756 [label="INSTANCE with matching:\nT53 -> T72\nT55 -> T74", fontsize=14, style = filled, fillcolor = lightgrey];
198 -> 756 [arrowhead = none , style=dashed];
756 -> 148[style=dashed];

757 [label="EVAL with clause\nge(t, f).\nand substitutionT71 -> f", fontsize=14, style = filled, fillcolor = lightblue];
200 -> 757 [arrowhead = none ];
757 -> 202;

758 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
200 -> 758 [arrowhead = none ];
758 -> 221;

759 [label="BACKTRACK\nfor clause: ge(f, f)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
201 -> 759 [arrowhead = none ];
759 -> 227;

760 [label="INSTANCE with matching:\nT53 -> T72\nT55 -> T74", fontsize=14, style = filled, fillcolor = lightgrey];
202 -> 760 [arrowhead = none , style=dashed];
760 -> 148[style=dashed];

761 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
228 -> 761 [arrowhead = none ];
761 -> 230;

762 [label="BACKTRACK\nfor clause: gt(t, f)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
230 -> 762 [arrowhead = none ];
762 -> 231;

763 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
264 -> 763 [arrowhead = none ];
763 -> 270;

764 [label="INSTANCE with matching:\nT1 -> T87\nT2 -> T58", fontsize=14, style = filled, fillcolor = lightgrey];
265 -> 764 [arrowhead = none , style=dashed];
764 -> 1[style=dashed];

765 [label="BACKTRACK\nfor clause: del(X1, [], [])because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
270 -> 765 [arrowhead = none ];
765 -> 272;

766 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
272 -> 766 [arrowhead = none ];
766 -> 286;

767 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
272 -> 767 [arrowhead = none ];
767 -> 287;

768 [label="ONLY EVAL with clause\ndel(X131, .(X132, X133), X133) :- eq(X131, X132).\nand substitutionT57 -> T99,\nX131 -> T99,\nX132 -> t,\nT53 -> T100,\nX133 -> .(t, T100),\nX14 -> .(t, T100)", fontsize=14, style = filled, fillcolor = lightblue];
286 -> 768 [arrowhead = none ];
768 -> 290;

769 [label="ONLY EVAL with clause\ndel(X146, .(X147, X148), .(X147, X149)) :- ','(neq(X146, X147), del(X146, X148, X149)).\nand substitutionT57 -> T107,\nX146 -> T107,\nX147 -> t,\nT53 -> T108,\nX148 -> .(t, T108),\nX149 -> X150,\nX14 -> .(t, X150)", fontsize=14, style = filled, fillcolor = lightblue];
287 -> 769 [arrowhead = none ];
769 -> 319;

770 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
290 -> 770 [arrowhead = none ];
770 -> 291;

771 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
291 -> 771 [arrowhead = none ];
771 -> 292;

772 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
291 -> 772 [arrowhead = none ];
772 -> 293;

773 [label="EVAL with clause\neq(t, t).\nand substitutionT99 -> t", fontsize=14, style = filled, fillcolor = lightblue];
292 -> 773 [arrowhead = none ];
773 -> 294;

774 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
292 -> 774 [arrowhead = none ];
774 -> 296;

775 [label="BACKTRACK\nfor clause: eq(f, f)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
293 -> 775 [arrowhead = none ];
775 -> 299;

776 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
294 -> 776 [arrowhead = none ];
776 -> 298;

777 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
319 -> 777 [arrowhead = none ];
777 -> 322;

778 [label="BACKTRACK\nfor clause: neq(t, f)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
322 -> 778 [arrowhead = none ];
778 -> 324;

779 [label="EVAL with clause\nneq(f, t).\nand substitutionT107 -> f", fontsize=14, style = filled, fillcolor = lightblue];
324 -> 779 [arrowhead = none ];
779 -> 326;

780 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
324 -> 780 [arrowhead = none ];
780 -> 327;

781 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
326 -> 781 [arrowhead = none ];
781 -> 332;

782 [label="BACKTRACK\nfor clause: del(X1, [], [])because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
332 -> 782 [arrowhead = none ];
782 -> 333;

783 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
333 -> 783 [arrowhead = none ];
783 -> 344;

784 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
333 -> 784 [arrowhead = none ];
784 -> 345;

785 [label="ONLY EVAL with clause\ndel(X168, .(X169, X170), X170) :- eq(X168, X169).\nand substitutionX168 -> f,\nX169 -> t,\nT108 -> T115,\nX170 -> T115,\nX150 -> T115", fontsize=14, style = filled, fillcolor = lightblue];
344 -> 785 [arrowhead = none ];
785 -> 349;

786 [label="ONLY EVAL with clause\ndel(X183, .(X184, X185), .(X184, X186)) :- ','(neq(X183, X184), del(X183, X185, X186)).\nand substitutionX183 -> f,\nX184 -> t,\nT108 -> T118,\nX185 -> T118,\nX186 -> X187,\nX150 -> .(t, X187)", fontsize=14, style = filled, fillcolor = lightblue];
345 -> 786 [arrowhead = none ];
786 -> 355;

787 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
349 -> 787 [arrowhead = none ];
787 -> 350;

788 [label="BACKTRACK\nfor clause: eq(t, t)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
350 -> 788 [arrowhead = none ];
788 -> 351;

789 [label="BACKTRACK\nfor clause: eq(f, f)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
351 -> 789 [arrowhead = none ];
789 -> 352;

790 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
355 -> 790 [arrowhead = none ];
790 -> 356;

791 [label="BACKTRACK\nfor clause: neq(t, f)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
356 -> 791 [arrowhead = none ];
791 -> 357;

792 [label="ONLY EVAL with clause\nneq(f, t).\nand substitution", fontsize=14, style = filled, fillcolor = lightblue];
357 -> 792 [arrowhead = none ];
792 -> 358;

793 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
358 -> 793 [arrowhead = none ];
793 -> 361;

794 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
361 -> 794 [arrowhead = none ];
794 -> 364;

795 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
361 -> 795 [arrowhead = none ];
795 -> 365;

796 [label="EVAL with clause\ndel(X198, [], []).\nand substitutionX198 -> f,\nT118 -> [],\nX187 -> []", fontsize=14, style = filled, fillcolor = lightblue];
364 -> 796 [arrowhead = none ];
796 -> 368;

797 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
364 -> 797 [arrowhead = none ];
797 -> 369;

798 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
365 -> 798 [arrowhead = none ];
798 -> 374;

799 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
365 -> 799 [arrowhead = none ];
799 -> 375;

800 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
368 -> 800 [arrowhead = none ];
800 -> 370;

801 [label="EVAL with clause\ndel(X211, .(X212, X213), X213) :- eq(X211, X212).\nand substitutionX211 -> f,\nX212 -> T127,\nX213 -> T128,\nT118 -> .(T127, T128),\nX187 -> T128", fontsize=14, style = filled, fillcolor = lightblue];
374 -> 801 [arrowhead = none ];
801 -> 378;

802 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
374 -> 802 [arrowhead = none ];
802 -> 379;

803 [label="EVAL with clause\ndel(X224, .(X225, X226), .(X225, X227)) :- ','(neq(X224, X225), del(X224, X226, X227)).\nand substitutionX224 -> f,\nX225 -> T133,\nX226 -> T134,\nT118 -> .(T133, T134),\nX227 -> X228,\nX187 -> .(T133, X228)", fontsize=14, style = filled, fillcolor = lightblue];
375 -> 803 [arrowhead = none ];
803 -> 390;

804 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
375 -> 804 [arrowhead = none ];
804 -> 391;

805 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
378 -> 805 [arrowhead = none ];
805 -> 380;

806 [label="BACKTRACK\nfor clause: eq(t, t)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
380 -> 806 [arrowhead = none ];
806 -> 381;

807 [label="EVAL with clause\neq(f, f).\nand substitutionT127 -> f", fontsize=14, style = filled, fillcolor = lightblue];
381 -> 807 [arrowhead = none ];
807 -> 382;

808 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
381 -> 808 [arrowhead = none ];
808 -> 383;

809 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
382 -> 809 [arrowhead = none ];
809 -> 384;

810 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
390 -> 810 [arrowhead = none ];
810 -> 394;

811 [label="BACKTRACK\nfor clause: neq(t, f)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
394 -> 811 [arrowhead = none ];
811 -> 395;

812 [label="EVAL with clause\nneq(f, t).\nand substitutionT133 -> t", fontsize=14, style = filled, fillcolor = lightblue];
395 -> 812 [arrowhead = none ];
812 -> 396;

813 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
395 -> 813 [arrowhead = none ];
813 -> 397;

814 [label="INSTANCE with matching:\nT118 -> T134\nX187 -> X228", fontsize=14, style = filled, fillcolor = lightgrey];
396 -> 814 [arrowhead = none , style=dashed];
814 -> 358[style=dashed];

815 [label="EVAL with clause\nge(t, f).\nand substitutionT51 -> t,\nT52 -> f", fontsize=14, style = filled, fillcolor = lightblue];
398 -> 815 [arrowhead = none ];
815 -> 400;

816 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
398 -> 816 [arrowhead = none ];
816 -> 401;

817 [label="EVAL with clause\nge(f, f).\nand substitutionT51 -> f,\nT52 -> f", fontsize=14, style = filled, fillcolor = lightblue];
399 -> 817 [arrowhead = none ];
817 -> 446;

818 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
399 -> 818 [arrowhead = none ];
818 -> 447;

819 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
400 -> 819 [arrowhead = none ];
819 -> 407;

820 [label="SPLIT 2\nnew knowledge:\nT53 is ground\nT135 is ground\nreplacements:T55 -> T135,\nT56 -> T136", fontsize=14, style = filled, fillcolor = violet];
400 -> 820 [arrowhead = none ];
820 -> 408;

821 [label="INSTANCE", fontsize=14, style = filled, fillcolor = lightgrey];
407 -> 821 [arrowhead = none , style=dashed];
821 -> 148[style=dashed];

822 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
408 -> 822 [arrowhead = none ];
822 -> 409;

823 [label="SPLIT 2\nnew knowledge:\nT135 is ground\nT53 is ground\nT139 is ground\nreplacements:X14 -> T139", fontsize=14, style = filled, fillcolor = violet];
408 -> 823 [arrowhead = none ];
823 -> 410;

824 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
409 -> 824 [arrowhead = none ];
824 -> 413;

825 [label="INSTANCE with matching:\nT1 -> T139\nT2 -> T136", fontsize=14, style = filled, fillcolor = lightgrey];
410 -> 825 [arrowhead = none , style=dashed];
825 -> 1[style=dashed];

826 [label="BACKTRACK\nfor clause: del(X1, [], [])because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
413 -> 826 [arrowhead = none ];
826 -> 414;

827 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
414 -> 827 [arrowhead = none ];
827 -> 417;

828 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
414 -> 828 [arrowhead = none ];
828 -> 418;

829 [label="ONLY EVAL with clause\ndel(X250, .(X251, X252), X252) :- eq(X250, X251).\nand substitutionT135 -> T151,\nX250 -> T151,\nX251 -> t,\nT53 -> T152,\nX252 -> .(f, T152),\nX14 -> .(f, T152)", fontsize=14, style = filled, fillcolor = lightblue];
417 -> 829 [arrowhead = none ];
829 -> 421;

830 [label="ONLY EVAL with clause\ndel(X265, .(X266, X267), .(X266, X268)) :- ','(neq(X265, X266), del(X265, X267, X268)).\nand substitutionT135 -> T159,\nX265 -> T159,\nX266 -> t,\nT53 -> T160,\nX267 -> .(f, T160),\nX268 -> X269,\nX14 -> .(t, X269)", fontsize=14, style = filled, fillcolor = lightblue];
418 -> 830 [arrowhead = none ];
830 -> 435;

831 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
421 -> 831 [arrowhead = none ];
831 -> 422;

832 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
422 -> 832 [arrowhead = none ];
832 -> 423;

833 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
422 -> 833 [arrowhead = none ];
833 -> 424;

834 [label="EVAL with clause\neq(t, t).\nand substitutionT151 -> t", fontsize=14, style = filled, fillcolor = lightblue];
423 -> 834 [arrowhead = none ];
834 -> 425;

835 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
423 -> 835 [arrowhead = none ];
835 -> 426;

836 [label="BACKTRACK\nfor clause: eq(f, f)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
424 -> 836 [arrowhead = none ];
836 -> 428;

837 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
425 -> 837 [arrowhead = none ];
837 -> 427;

838 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
435 -> 838 [arrowhead = none ];
838 -> 436;

839 [label="BACKTRACK\nfor clause: neq(t, f)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
436 -> 839 [arrowhead = none ];
839 -> 437;

840 [label="EVAL with clause\nneq(f, t).\nand substitutionT159 -> f", fontsize=14, style = filled, fillcolor = lightblue];
437 -> 840 [arrowhead = none ];
840 -> 438;

841 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
437 -> 841 [arrowhead = none ];
841 -> 439;

842 [label="INSTANCE with matching:\nT118 -> .(f, T160)\nX187 -> X269", fontsize=14, style = filled, fillcolor = lightgrey];
438 -> 842 [arrowhead = none , style=dashed];
842 -> 358[style=dashed];

843 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
446 -> 843 [arrowhead = none ];
843 -> 450;

844 [label="SPLIT 2\nnew knowledge:\nT53 is ground\nT163 is ground\nreplacements:T55 -> T163,\nT56 -> T164", fontsize=14, style = filled, fillcolor = violet];
446 -> 844 [arrowhead = none ];
844 -> 451;

845 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
450 -> 845 [arrowhead = none ];
845 -> 452;

846 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
451 -> 846 [arrowhead = none ];
846 -> 479;

847 [label="SPLIT 2\nnew knowledge:\nT163 is ground\nT53 is ground\nT193 is ground\nreplacements:X14 -> T193", fontsize=14, style = filled, fillcolor = violet];
451 -> 847 [arrowhead = none ];
847 -> 480;

848 [label="BACKTRACK\nfor clause: max([], f)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
452 -> 848 [arrowhead = none ];
848 -> 453;

849 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
453 -> 849 [arrowhead = none ];
849 -> 454;

850 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
453 -> 850 [arrowhead = none ];
850 -> 455;

851 [label="EVAL with clause\nmax(.(X278, []), X278).\nand substitutionX278 -> f,\nT53 -> [],\nT55 -> f", fontsize=14, style = filled, fillcolor = lightblue];
454 -> 851 [arrowhead = none ];
851 -> 456;

852 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
454 -> 852 [arrowhead = none ];
852 -> 457;

853 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
455 -> 853 [arrowhead = none ];
853 -> 459;

854 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
455 -> 854 [arrowhead = none ];
854 -> 460;

855 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
456 -> 855 [arrowhead = none ];
855 -> 458;

856 [label="EVAL with clause\nmax(.(X295, .(X296, X297)), X298) :- ','(ge(X295, X296), max(.(X295, X297), X298)).\nand substitutionX295 -> f,\nX296 -> T177,\nX297 -> T178,\nT53 -> .(T177, T178),\nT55 -> T180,\nX298 -> T180,\nT179 -> T180", fontsize=14, style = filled, fillcolor = lightblue];
459 -> 856 [arrowhead = none ];
856 -> 461;

857 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
459 -> 857 [arrowhead = none ];
857 -> 462;

858 [label="EVAL with clause\nmax(.(X307, .(X308, X309)), X310) :- ','(gt(X308, X307), max(.(X308, X309), X310)).\nand substitutionX307 -> f,\nX308 -> T187,\nX309 -> T188,\nT53 -> .(T187, T188),\nT55 -> T190,\nX310 -> T190,\nT189 -> T190", fontsize=14, style = filled, fillcolor = lightblue];
460 -> 858 [arrowhead = none ];
858 -> 468;

859 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
460 -> 859 [arrowhead = none ];
859 -> 469;

860 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
461 -> 860 [arrowhead = none ];
860 -> 463;

861 [label="BACKTRACK\nfor clause: ge(t, t)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
463 -> 861 [arrowhead = none ];
861 -> 464;

862 [label="BACKTRACK\nfor clause: ge(t, f)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
464 -> 862 [arrowhead = none ];
862 -> 465;

863 [label="EVAL with clause\nge(f, f).\nand substitutionT177 -> f", fontsize=14, style = filled, fillcolor = lightblue];
465 -> 863 [arrowhead = none ];
863 -> 466;

864 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
465 -> 864 [arrowhead = none ];
864 -> 467;

865 [label="INSTANCE with matching:\nT53 -> T178\nT55 -> T180", fontsize=14, style = filled, fillcolor = lightgrey];
466 -> 865 [arrowhead = none , style=dashed];
865 -> 450[style=dashed];

866 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
468 -> 866 [arrowhead = none ];
866 -> 470;

867 [label="EVAL with clause\ngt(t, f).\nand substitutionT187 -> t", fontsize=14, style = filled, fillcolor = lightblue];
470 -> 867 [arrowhead = none ];
867 -> 471;

868 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
470 -> 868 [arrowhead = none ];
868 -> 472;

869 [label="INSTANCE with matching:\nT53 -> T188\nT55 -> T190", fontsize=14, style = filled, fillcolor = lightgrey];
471 -> 869 [arrowhead = none , style=dashed];
869 -> 148[style=dashed];

870 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
479 -> 870 [arrowhead = none ];
870 -> 481;

871 [label="INSTANCE with matching:\nT1 -> T193\nT2 -> T164", fontsize=14, style = filled, fillcolor = lightgrey];
480 -> 871 [arrowhead = none , style=dashed];
871 -> 1[style=dashed];

872 [label="BACKTRACK\nfor clause: del(X1, [], [])because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
481 -> 872 [arrowhead = none ];
872 -> 483;

873 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
483 -> 873 [arrowhead = none ];
873 -> 487;

874 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
483 -> 874 [arrowhead = none ];
874 -> 488;

875 [label="ONLY EVAL with clause\ndel(X328, .(X329, X330), X330) :- eq(X328, X329).\nand substitutionT163 -> T205,\nX328 -> T205,\nX329 -> f,\nT53 -> T206,\nX330 -> .(f, T206),\nX14 -> .(f, T206)", fontsize=14, style = filled, fillcolor = lightblue];
487 -> 875 [arrowhead = none ];
875 -> 490;

876 [label="ONLY EVAL with clause\ndel(X343, .(X344, X345), .(X344, X346)) :- ','(neq(X343, X344), del(X343, X345, X346)).\nand substitutionT163 -> T213,\nX343 -> T213,\nX344 -> f,\nT53 -> T214,\nX345 -> .(f, T214),\nX346 -> X347,\nX14 -> .(f, X347)", fontsize=14, style = filled, fillcolor = lightblue];
488 -> 876 [arrowhead = none ];
876 -> 496;

877 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
490 -> 877 [arrowhead = none ];
877 -> 491;

878 [label="BACKTRACK\nfor clause: eq(t, t)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
491 -> 878 [arrowhead = none ];
878 -> 492;

879 [label="EVAL with clause\neq(f, f).\nand substitutionT205 -> f", fontsize=14, style = filled, fillcolor = lightblue];
492 -> 879 [arrowhead = none ];
879 -> 493;

880 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
492 -> 880 [arrowhead = none ];
880 -> 494;

881 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
493 -> 881 [arrowhead = none ];
881 -> 495;

882 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
496 -> 882 [arrowhead = none ];
882 -> 497;

883 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
497 -> 883 [arrowhead = none ];
883 -> 498;

884 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
497 -> 884 [arrowhead = none ];
884 -> 499;

885 [label="EVAL with clause\nneq(t, f).\nand substitutionT213 -> t", fontsize=14, style = filled, fillcolor = lightblue];
498 -> 885 [arrowhead = none ];
885 -> 500;

886 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
498 -> 886 [arrowhead = none ];
886 -> 501;

887 [label="BACKTRACK\nfor clause: neq(f, t)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
499 -> 887 [arrowhead = none ];
887 -> 565;

888 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
500 -> 888 [arrowhead = none ];
888 -> 508;

889 [label="BACKTRACK\nfor clause: del(X1, [], [])because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
508 -> 889 [arrowhead = none ];
889 -> 509;

890 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
509 -> 890 [arrowhead = none ];
890 -> 510;

891 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
509 -> 891 [arrowhead = none ];
891 -> 511;

892 [label="ONLY EVAL with clause\ndel(X365, .(X366, X367), X367) :- eq(X365, X366).\nand substitutionX365 -> t,\nX366 -> f,\nT214 -> T221,\nX367 -> T221,\nX347 -> T221", fontsize=14, style = filled, fillcolor = lightblue];
510 -> 892 [arrowhead = none ];
892 -> 512;

893 [label="ONLY EVAL with clause\ndel(X380, .(X381, X382), .(X381, X383)) :- ','(neq(X380, X381), del(X380, X382, X383)).\nand substitutionX380 -> t,\nX381 -> f,\nT214 -> T224,\nX382 -> T224,\nX383 -> X384,\nX347 -> .(f, X384)", fontsize=14, style = filled, fillcolor = lightblue];
511 -> 893 [arrowhead = none ];
893 -> 516;

894 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
512 -> 894 [arrowhead = none ];
894 -> 513;

895 [label="BACKTRACK\nfor clause: eq(t, t)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
513 -> 895 [arrowhead = none ];
895 -> 514;

896 [label="BACKTRACK\nfor clause: eq(f, f)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
514 -> 896 [arrowhead = none ];
896 -> 515;

897 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
516 -> 897 [arrowhead = none ];
897 -> 517;

898 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
517 -> 898 [arrowhead = none ];
898 -> 518;

899 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
517 -> 899 [arrowhead = none ];
899 -> 519;

900 [label="ONLY EVAL with clause\nneq(t, f).\nand substitution", fontsize=14, style = filled, fillcolor = lightblue];
518 -> 900 [arrowhead = none ];
900 -> 522;

901 [label="BACKTRACK\nfor clause: neq(f, t)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
519 -> 901 [arrowhead = none ];
901 -> 564;

902 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
522 -> 902 [arrowhead = none ];
902 -> 532;

903 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
532 -> 903 [arrowhead = none ];
903 -> 533;

904 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
532 -> 904 [arrowhead = none ];
904 -> 534;

905 [label="EVAL with clause\ndel(X397, [], []).\nand substitutionX397 -> t,\nT224 -> [],\nX384 -> []", fontsize=14, style = filled, fillcolor = lightblue];
533 -> 905 [arrowhead = none ];
905 -> 535;

906 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
533 -> 906 [arrowhead = none ];
906 -> 536;

907 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
534 -> 907 [arrowhead = none ];
907 -> 541;

908 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
534 -> 908 [arrowhead = none ];
908 -> 542;

909 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
535 -> 909 [arrowhead = none ];
909 -> 537;

910 [label="EVAL with clause\ndel(X410, .(X411, X412), X412) :- eq(X410, X411).\nand substitutionX410 -> t,\nX411 -> T233,\nX412 -> T234,\nT224 -> .(T233, T234),\nX384 -> T234", fontsize=14, style = filled, fillcolor = lightblue];
541 -> 910 [arrowhead = none ];
910 -> 547;

911 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
541 -> 911 [arrowhead = none ];
911 -> 548;

912 [label="EVAL with clause\ndel(X423, .(X424, X425), .(X424, X426)) :- ','(neq(X423, X424), del(X423, X425, X426)).\nand substitutionX423 -> t,\nX424 -> T239,\nX425 -> T240,\nT224 -> .(T239, T240),\nX426 -> X427,\nX384 -> .(T239, X427)", fontsize=14, style = filled, fillcolor = lightblue];
542 -> 912 [arrowhead = none ];
912 -> 556;

913 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
542 -> 913 [arrowhead = none ];
913 -> 557;

914 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
547 -> 914 [arrowhead = none ];
914 -> 549;

915 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
549 -> 915 [arrowhead = none ];
915 -> 550;

916 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
549 -> 916 [arrowhead = none ];
916 -> 551;

917 [label="EVAL with clause\neq(t, t).\nand substitutionT233 -> t", fontsize=14, style = filled, fillcolor = lightblue];
550 -> 917 [arrowhead = none ];
917 -> 552;

918 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
550 -> 918 [arrowhead = none ];
918 -> 553;

919 [label="BACKTRACK\nfor clause: eq(f, f)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
551 -> 919 [arrowhead = none ];
919 -> 555;

920 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
552 -> 920 [arrowhead = none ];
920 -> 554;

921 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
556 -> 921 [arrowhead = none ];
921 -> 558;

922 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
558 -> 922 [arrowhead = none ];
922 -> 559;

923 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
558 -> 923 [arrowhead = none ];
923 -> 560;

924 [label="EVAL with clause\nneq(t, f).\nand substitutionT239 -> f", fontsize=14, style = filled, fillcolor = lightblue];
559 -> 924 [arrowhead = none ];
924 -> 561;

925 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
559 -> 925 [arrowhead = none ];
925 -> 562;

926 [label="BACKTRACK\nfor clause: neq(f, t)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
560 -> 926 [arrowhead = none ];
926 -> 563;

927 [label="INSTANCE with matching:\nT224 -> T240\nX384 -> X427", fontsize=14, style = filled, fillcolor = lightgrey];
561 -> 927 [arrowhead = none , style=dashed];
927 -> 522[style=dashed];

928 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
569 -> 928 [arrowhead = none ];
928 -> 571;

929 [label="EVAL with clause\ngt(t, f).\nand substitutionT250 -> t,\nT249 -> f", fontsize=14, style = filled, fillcolor = lightblue];
571 -> 929 [arrowhead = none ];
929 -> 572;

930 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
571 -> 930 [arrowhead = none ];
930 -> 573;

931 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
572 -> 931 [arrowhead = none ];
931 -> 579;

932 [label="SPLIT 2\nnew knowledge:\nT251 is ground\nT255 is ground\nreplacements:T253 -> T255,\nT254 -> T256", fontsize=14, style = filled, fillcolor = violet];
572 -> 932 [arrowhead = none ];
932 -> 580;

933 [label="INSTANCE with matching:\nT53 -> T251\nT55 -> T253", fontsize=14, style = filled, fillcolor = lightgrey];
579 -> 933 [arrowhead = none , style=dashed];
933 -> 148[style=dashed];

934 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
580 -> 934 [arrowhead = none ];
934 -> 581;

935 [label="SPLIT 2\nnew knowledge:\nT255 is ground\nT251 is ground\nT259 is ground\nreplacements:X14 -> T259", fontsize=14, style = filled, fillcolor = violet];
580 -> 935 [arrowhead = none ];
935 -> 582;

936 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
581 -> 936 [arrowhead = none ];
936 -> 655;

937 [label="INSTANCE with matching:\nT1 -> T259\nT2 -> T256", fontsize=14, style = filled, fillcolor = lightgrey];
582 -> 937 [arrowhead = none , style=dashed];
937 -> 1[style=dashed];

938 [label="BACKTRACK\nfor clause: del(X1, [], [])because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
655 -> 938 [arrowhead = none ];
938 -> 662;

939 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
662 -> 939 [arrowhead = none ];
939 -> 663;

940 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
662 -> 940 [arrowhead = none ];
940 -> 664;

941 [label="ONLY EVAL with clause\ndel(X461, .(X462, X463), X463) :- eq(X461, X462).\nand substitutionT255 -> T271,\nX461 -> T271,\nX462 -> f,\nT251 -> T272,\nX463 -> .(t, T272),\nX14 -> .(t, T272)", fontsize=14, style = filled, fillcolor = lightblue];
663 -> 941 [arrowhead = none ];
941 -> 668;

942 [label="ONLY EVAL with clause\ndel(X476, .(X477, X478), .(X477, X479)) :- ','(neq(X476, X477), del(X476, X478, X479)).\nand substitutionT255 -> T279,\nX476 -> T279,\nX477 -> f,\nT251 -> T280,\nX478 -> .(t, T280),\nX479 -> X480,\nX14 -> .(f, X480)", fontsize=14, style = filled, fillcolor = lightblue];
664 -> 942 [arrowhead = none ];
942 -> 677;

943 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
668 -> 943 [arrowhead = none ];
943 -> 669;

944 [label="BACKTRACK\nfor clause: eq(t, t)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
669 -> 944 [arrowhead = none ];
944 -> 670;

945 [label="EVAL with clause\neq(f, f).\nand substitutionT271 -> f", fontsize=14, style = filled, fillcolor = lightblue];
670 -> 945 [arrowhead = none ];
945 -> 671;

946 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
670 -> 946 [arrowhead = none ];
946 -> 672;

947 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
671 -> 947 [arrowhead = none ];
947 -> 673;

948 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
677 -> 948 [arrowhead = none ];
948 -> 678;

949 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
678 -> 949 [arrowhead = none ];
949 -> 679;

950 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
678 -> 950 [arrowhead = none ];
950 -> 680;

951 [label="EVAL with clause\nneq(t, f).\nand substitutionT279 -> t", fontsize=14, style = filled, fillcolor = lightblue];
679 -> 951 [arrowhead = none ];
951 -> 681;

952 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
679 -> 952 [arrowhead = none ];
952 -> 682;

953 [label="BACKTRACK\nfor clause: neq(f, t)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
680 -> 953 [arrowhead = none ];
953 -> 684;

954 [label="INSTANCE with matching:\nT224 -> .(t, T280)\nX384 -> X480", fontsize=14, style = filled, fillcolor = lightgrey];
681 -> 954 [arrowhead = none , style=dashed];
954 -> 522[style=dashed];

}

(2) Obligation:

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

maxsortA_in_ga([], []) → maxsortA_out_ga([], [])
maxsortA_in_ga(.(T17, []), .(T17, T18)) → U1_ga(T17, T18, pB_in_gaa(T17, X14, T18))
pB_in_gaa(T17, T21, T18) → U16_gaa(T17, T21, T18, delK_in_ga(T17, T21))
delK_in_ga(t, []) → delK_out_ga(t, [])
delK_in_ga(f, []) → delK_out_ga(f, [])
U16_gaa(T17, T21, T18, delK_out_ga(T17, T21)) → U17_gaa(T17, T21, T18, maxsortA_in_ga(T21, T18))
maxsortA_in_ga(.(t, .(t, T53)), .(T55, T56)) → U2_ga(T53, T55, T56, pC_in_gaaa(T53, T55, X14, T56))
pC_in_gaaa(T53, T57, X14, T58) → U18_gaaa(T53, T57, X14, T58, maxG_in_ga(T53, T57))
maxG_in_ga([], t) → maxG_out_ga([], t)
maxG_in_ga(.(t, T72), T74) → U6_ga(T72, T74, maxG_in_ga(T72, T74))
maxG_in_ga(.(f, T72), T74) → U7_ga(T72, T74, maxG_in_ga(T72, T74))
U7_ga(T72, T74, maxG_out_ga(T72, T74)) → maxG_out_ga(.(f, T72), T74)
U6_ga(T72, T74, maxG_out_ga(T72, T74)) → maxG_out_ga(.(t, T72), T74)
U18_gaaa(T53, T57, X14, T58, maxG_out_ga(T53, T57)) → U19_gaaa(T53, T57, X14, T58, pP_in_ggaa(T57, T53, X14, T58))
pP_in_ggaa(T57, T53, T87, T58) → U20_ggaa(T57, T53, T87, T58, delL_in_gga(T57, T53, T87))
delL_in_gga(t, T100, .(t, T100)) → delL_out_gga(t, T100, .(t, T100))
delL_in_gga(f, T118, .(t, .(t, X187))) → U12_gga(T118, X187, delH_in_ga(T118, X187))
delH_in_ga([], []) → delH_out_ga([], [])
delH_in_ga(.(f, T128), T128) → delH_out_ga(.(f, T128), T128)
delH_in_ga(.(t, T134), .(t, X228)) → U8_ga(T134, X228, delH_in_ga(T134, X228))
U8_ga(T134, X228, delH_out_ga(T134, X228)) → delH_out_ga(.(t, T134), .(t, X228))
U12_gga(T118, X187, delH_out_ga(T118, X187)) → delL_out_gga(f, T118, .(t, .(t, X187)))
U20_ggaa(T57, T53, T87, T58, delL_out_gga(T57, T53, T87)) → U21_ggaa(T57, T53, T87, T58, maxsortA_in_ga(T87, T58))
maxsortA_in_ga(.(t, .(f, T53)), .(T55, T56)) → U3_ga(T53, T55, T56, pD_in_gaaa(T53, T55, X14, T56))
pD_in_gaaa(T53, T135, X14, T136) → U22_gaaa(T53, T135, X14, T136, maxG_in_ga(T53, T135))
U22_gaaa(T53, T135, X14, T136, maxG_out_ga(T53, T135)) → U23_gaaa(T53, T135, X14, T136, pQ_in_ggaa(T135, T53, X14, T136))
pQ_in_ggaa(T135, T53, T139, T136) → U24_ggaa(T135, T53, T139, T136, delM_in_gga(T135, T53, T139))
delM_in_gga(t, T152, .(f, T152)) → delM_out_gga(t, T152, .(f, T152))
delM_in_gga(f, T160, .(t, X269)) → U13_gga(T160, X269, delH_in_ga(.(f, T160), X269))
U13_gga(T160, X269, delH_out_ga(.(f, T160), X269)) → delM_out_gga(f, T160, .(t, X269))
U24_ggaa(T135, T53, T139, T136, delM_out_gga(T135, T53, T139)) → U25_ggaa(T135, T53, T139, T136, maxsortA_in_ga(T139, T136))
maxsortA_in_ga(.(f, .(f, T53)), .(T55, T56)) → U4_ga(T53, T55, T56, pE_in_gaaa(T53, T55, X14, T56))
pE_in_gaaa(T53, T163, X14, T164) → U26_gaaa(T53, T163, X14, T164, maxI_in_ga(T53, T163))
maxI_in_ga([], f) → maxI_out_ga([], f)
maxI_in_ga(.(f, T178), T180) → U9_ga(T178, T180, maxI_in_ga(T178, T180))
maxI_in_ga(.(t, T188), T190) → U10_ga(T188, T190, maxG_in_ga(T188, T190))
U10_ga(T188, T190, maxG_out_ga(T188, T190)) → maxI_out_ga(.(t, T188), T190)
U9_ga(T178, T180, maxI_out_ga(T178, T180)) → maxI_out_ga(.(f, T178), T180)
U26_gaaa(T53, T163, X14, T164, maxI_out_ga(T53, T163)) → U27_gaaa(T53, T163, X14, T164, pR_in_ggaa(T163, T53, X14, T164))
pR_in_ggaa(T163, T53, T193, T164) → U28_ggaa(T163, T53, T193, T164, delN_in_gga(T163, T53, T193))
delN_in_gga(f, T206, .(f, T206)) → delN_out_gga(f, T206, .(f, T206))
delN_in_gga(t, T224, .(f, .(f, X384))) → U14_gga(T224, X384, delJ_in_ga(T224, X384))
delJ_in_ga([], []) → delJ_out_ga([], [])
delJ_in_ga(.(t, T234), T234) → delJ_out_ga(.(t, T234), T234)
delJ_in_ga(.(f, T240), .(f, X427)) → U11_ga(T240, X427, delJ_in_ga(T240, X427))
U11_ga(T240, X427, delJ_out_ga(T240, X427)) → delJ_out_ga(.(f, T240), .(f, X427))
U14_gga(T224, X384, delJ_out_ga(T224, X384)) → delN_out_gga(t, T224, .(f, .(f, X384)))
U28_ggaa(T163, T53, T193, T164, delN_out_gga(T163, T53, T193)) → U29_ggaa(T163, T53, T193, T164, maxsortA_in_ga(T193, T164))
maxsortA_in_ga(.(f, .(t, T251)), .(T253, T254)) → U5_ga(T251, T253, T254, pF_in_gaaa(T251, T253, X14, T254))
pF_in_gaaa(T251, T255, X14, T256) → U30_gaaa(T251, T255, X14, T256, maxG_in_ga(T251, T255))
U30_gaaa(T251, T255, X14, T256, maxG_out_ga(T251, T255)) → U31_gaaa(T251, T255, X14, T256, pS_in_ggaa(T255, T251, X14, T256))
pS_in_ggaa(T255, T251, T259, T256) → U32_ggaa(T255, T251, T259, T256, delO_in_gga(T255, T251, T259))
delO_in_gga(f, T272, .(t, T272)) → delO_out_gga(f, T272, .(t, T272))
delO_in_gga(t, T280, .(f, X480)) → U15_gga(T280, X480, delJ_in_ga(.(t, T280), X480))
U15_gga(T280, X480, delJ_out_ga(.(t, T280), X480)) → delO_out_gga(t, T280, .(f, X480))
U32_ggaa(T255, T251, T259, T256, delO_out_gga(T255, T251, T259)) → U33_ggaa(T255, T251, T259, T256, maxsortA_in_ga(T259, T256))
U33_ggaa(T255, T251, T259, T256, maxsortA_out_ga(T259, T256)) → pS_out_ggaa(T255, T251, T259, T256)
U31_gaaa(T251, T255, X14, T256, pS_out_ggaa(T255, T251, X14, T256)) → pF_out_gaaa(T251, T255, X14, T256)
U5_ga(T251, T253, T254, pF_out_gaaa(T251, T253, X14, T254)) → maxsortA_out_ga(.(f, .(t, T251)), .(T253, T254))
U29_ggaa(T163, T53, T193, T164, maxsortA_out_ga(T193, T164)) → pR_out_ggaa(T163, T53, T193, T164)
U27_gaaa(T53, T163, X14, T164, pR_out_ggaa(T163, T53, X14, T164)) → pE_out_gaaa(T53, T163, X14, T164)
U4_ga(T53, T55, T56, pE_out_gaaa(T53, T55, X14, T56)) → maxsortA_out_ga(.(f, .(f, T53)), .(T55, T56))
U25_ggaa(T135, T53, T139, T136, maxsortA_out_ga(T139, T136)) → pQ_out_ggaa(T135, T53, T139, T136)
U23_gaaa(T53, T135, X14, T136, pQ_out_ggaa(T135, T53, X14, T136)) → pD_out_gaaa(T53, T135, X14, T136)
U3_ga(T53, T55, T56, pD_out_gaaa(T53, T55, X14, T56)) → maxsortA_out_ga(.(t, .(f, T53)), .(T55, T56))
U21_ggaa(T57, T53, T87, T58, maxsortA_out_ga(T87, T58)) → pP_out_ggaa(T57, T53, T87, T58)
U19_gaaa(T53, T57, X14, T58, pP_out_ggaa(T57, T53, X14, T58)) → pC_out_gaaa(T53, T57, X14, T58)
U2_ga(T53, T55, T56, pC_out_gaaa(T53, T55, X14, T56)) → maxsortA_out_ga(.(t, .(t, T53)), .(T55, T56))
U17_gaa(T17, T21, T18, maxsortA_out_ga(T21, T18)) → pB_out_gaa(T17, T21, T18)
U1_ga(T17, T18, pB_out_gaa(T17, X14, T18)) → maxsortA_out_ga(.(T17, []), .(T17, T18))

The argument filtering Pi contains the following mapping:
maxsortA_in_ga(x1, x2)  =  maxsortA_in_ga(x1)
[]  =  []
maxsortA_out_ga(x1, x2)  =  maxsortA_out_ga(x1, x2)
.(x1, x2)  =  .(x1, x2)
U1_ga(x1, x2, x3)  =  U1_ga(x1, x3)
pB_in_gaa(x1, x2, x3)  =  pB_in_gaa(x1)
U16_gaa(x1, x2, x3, x4)  =  U16_gaa(x1, x4)
delK_in_ga(x1, x2)  =  delK_in_ga(x1)
t  =  t
delK_out_ga(x1, x2)  =  delK_out_ga(x1, x2)
f  =  f
U17_gaa(x1, x2, x3, x4)  =  U17_gaa(x1, x2, x4)
U2_ga(x1, x2, x3, x4)  =  U2_ga(x1, x4)
pC_in_gaaa(x1, x2, x3, x4)  =  pC_in_gaaa(x1)
U18_gaaa(x1, x2, x3, x4, x5)  =  U18_gaaa(x1, x5)
maxG_in_ga(x1, x2)  =  maxG_in_ga(x1)
maxG_out_ga(x1, x2)  =  maxG_out_ga(x1, x2)
U6_ga(x1, x2, x3)  =  U6_ga(x1, x3)
U7_ga(x1, x2, x3)  =  U7_ga(x1, x3)
U19_gaaa(x1, x2, x3, x4, x5)  =  U19_gaaa(x1, x2, x5)
pP_in_ggaa(x1, x2, x3, x4)  =  pP_in_ggaa(x1, x2)
U20_ggaa(x1, x2, x3, x4, x5)  =  U20_ggaa(x1, x2, x5)
delL_in_gga(x1, x2, x3)  =  delL_in_gga(x1, x2)
delL_out_gga(x1, x2, x3)  =  delL_out_gga(x1, x2, x3)
U12_gga(x1, x2, x3)  =  U12_gga(x1, x3)
delH_in_ga(x1, x2)  =  delH_in_ga(x1)
delH_out_ga(x1, x2)  =  delH_out_ga(x1, x2)
U8_ga(x1, x2, x3)  =  U8_ga(x1, x3)
U21_ggaa(x1, x2, x3, x4, x5)  =  U21_ggaa(x1, x2, x3, x5)
U3_ga(x1, x2, x3, x4)  =  U3_ga(x1, x4)
pD_in_gaaa(x1, x2, x3, x4)  =  pD_in_gaaa(x1)
U22_gaaa(x1, x2, x3, x4, x5)  =  U22_gaaa(x1, x5)
U23_gaaa(x1, x2, x3, x4, x5)  =  U23_gaaa(x1, x2, x5)
pQ_in_ggaa(x1, x2, x3, x4)  =  pQ_in_ggaa(x1, x2)
U24_ggaa(x1, x2, x3, x4, x5)  =  U24_ggaa(x1, x2, x5)
delM_in_gga(x1, x2, x3)  =  delM_in_gga(x1, x2)
delM_out_gga(x1, x2, x3)  =  delM_out_gga(x1, x2, x3)
U13_gga(x1, x2, x3)  =  U13_gga(x1, x3)
U25_ggaa(x1, x2, x3, x4, x5)  =  U25_ggaa(x1, x2, x3, x5)
U4_ga(x1, x2, x3, x4)  =  U4_ga(x1, x4)
pE_in_gaaa(x1, x2, x3, x4)  =  pE_in_gaaa(x1)
U26_gaaa(x1, x2, x3, x4, x5)  =  U26_gaaa(x1, x5)
maxI_in_ga(x1, x2)  =  maxI_in_ga(x1)
maxI_out_ga(x1, x2)  =  maxI_out_ga(x1, x2)
U9_ga(x1, x2, x3)  =  U9_ga(x1, x3)
U10_ga(x1, x2, x3)  =  U10_ga(x1, x3)
U27_gaaa(x1, x2, x3, x4, x5)  =  U27_gaaa(x1, x2, x5)
pR_in_ggaa(x1, x2, x3, x4)  =  pR_in_ggaa(x1, x2)
U28_ggaa(x1, x2, x3, x4, x5)  =  U28_ggaa(x1, x2, x5)
delN_in_gga(x1, x2, x3)  =  delN_in_gga(x1, x2)
delN_out_gga(x1, x2, x3)  =  delN_out_gga(x1, x2, x3)
U14_gga(x1, x2, x3)  =  U14_gga(x1, x3)
delJ_in_ga(x1, x2)  =  delJ_in_ga(x1)
delJ_out_ga(x1, x2)  =  delJ_out_ga(x1, x2)
U11_ga(x1, x2, x3)  =  U11_ga(x1, x3)
U29_ggaa(x1, x2, x3, x4, x5)  =  U29_ggaa(x1, x2, x3, x5)
U5_ga(x1, x2, x3, x4)  =  U5_ga(x1, x4)
pF_in_gaaa(x1, x2, x3, x4)  =  pF_in_gaaa(x1)
U30_gaaa(x1, x2, x3, x4, x5)  =  U30_gaaa(x1, x5)
U31_gaaa(x1, x2, x3, x4, x5)  =  U31_gaaa(x1, x2, x5)
pS_in_ggaa(x1, x2, x3, x4)  =  pS_in_ggaa(x1, x2)
U32_ggaa(x1, x2, x3, x4, x5)  =  U32_ggaa(x1, x2, x5)
delO_in_gga(x1, x2, x3)  =  delO_in_gga(x1, x2)
delO_out_gga(x1, x2, x3)  =  delO_out_gga(x1, x2, x3)
U15_gga(x1, x2, x3)  =  U15_gga(x1, x3)
U33_ggaa(x1, x2, x3, x4, x5)  =  U33_ggaa(x1, x2, x3, x5)
pS_out_ggaa(x1, x2, x3, x4)  =  pS_out_ggaa(x1, x2, x3, x4)
pF_out_gaaa(x1, x2, x3, x4)  =  pF_out_gaaa(x1, x2, x3, x4)
pR_out_ggaa(x1, x2, x3, x4)  =  pR_out_ggaa(x1, x2, x3, x4)
pE_out_gaaa(x1, x2, x3, x4)  =  pE_out_gaaa(x1, x2, x3, x4)
pQ_out_ggaa(x1, x2, x3, x4)  =  pQ_out_ggaa(x1, x2, x3, x4)
pD_out_gaaa(x1, x2, x3, x4)  =  pD_out_gaaa(x1, x2, x3, x4)
pP_out_ggaa(x1, x2, x3, x4)  =  pP_out_ggaa(x1, x2, x3, x4)
pC_out_gaaa(x1, x2, x3, x4)  =  pC_out_gaaa(x1, x2, x3, x4)
pB_out_gaa(x1, x2, x3)  =  pB_out_gaa(x1, x2, x3)

(3) 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:

MAXSORTA_IN_GA(.(T17, []), .(T17, T18)) → U1_GA(T17, T18, pB_in_gaa(T17, X14, T18))
MAXSORTA_IN_GA(.(T17, []), .(T17, T18)) → PB_IN_GAA(T17, X14, T18)
PB_IN_GAA(T17, T21, T18) → U16_GAA(T17, T21, T18, delK_in_ga(T17, T21))
PB_IN_GAA(T17, T21, T18) → DELK_IN_GA(T17, T21)
U16_GAA(T17, T21, T18, delK_out_ga(T17, T21)) → U17_GAA(T17, T21, T18, maxsortA_in_ga(T21, T18))
U16_GAA(T17, T21, T18, delK_out_ga(T17, T21)) → MAXSORTA_IN_GA(T21, T18)
MAXSORTA_IN_GA(.(t, .(t, T53)), .(T55, T56)) → U2_GA(T53, T55, T56, pC_in_gaaa(T53, T55, X14, T56))
MAXSORTA_IN_GA(.(t, .(t, T53)), .(T55, T56)) → PC_IN_GAAA(T53, T55, X14, T56)
PC_IN_GAAA(T53, T57, X14, T58) → U18_GAAA(T53, T57, X14, T58, maxG_in_ga(T53, T57))
PC_IN_GAAA(T53, T57, X14, T58) → MAXG_IN_GA(T53, T57)
MAXG_IN_GA(.(t, T72), T74) → U6_GA(T72, T74, maxG_in_ga(T72, T74))
MAXG_IN_GA(.(t, T72), T74) → MAXG_IN_GA(T72, T74)
MAXG_IN_GA(.(f, T72), T74) → U7_GA(T72, T74, maxG_in_ga(T72, T74))
MAXG_IN_GA(.(f, T72), T74) → MAXG_IN_GA(T72, T74)
U18_GAAA(T53, T57, X14, T58, maxG_out_ga(T53, T57)) → U19_GAAA(T53, T57, X14, T58, pP_in_ggaa(T57, T53, X14, T58))
U18_GAAA(T53, T57, X14, T58, maxG_out_ga(T53, T57)) → PP_IN_GGAA(T57, T53, X14, T58)
PP_IN_GGAA(T57, T53, T87, T58) → U20_GGAA(T57, T53, T87, T58, delL_in_gga(T57, T53, T87))
PP_IN_GGAA(T57, T53, T87, T58) → DELL_IN_GGA(T57, T53, T87)
DELL_IN_GGA(f, T118, .(t, .(t, X187))) → U12_GGA(T118, X187, delH_in_ga(T118, X187))
DELL_IN_GGA(f, T118, .(t, .(t, X187))) → DELH_IN_GA(T118, X187)
DELH_IN_GA(.(t, T134), .(t, X228)) → U8_GA(T134, X228, delH_in_ga(T134, X228))
DELH_IN_GA(.(t, T134), .(t, X228)) → DELH_IN_GA(T134, X228)
U20_GGAA(T57, T53, T87, T58, delL_out_gga(T57, T53, T87)) → U21_GGAA(T57, T53, T87, T58, maxsortA_in_ga(T87, T58))
U20_GGAA(T57, T53, T87, T58, delL_out_gga(T57, T53, T87)) → MAXSORTA_IN_GA(T87, T58)
MAXSORTA_IN_GA(.(t, .(f, T53)), .(T55, T56)) → U3_GA(T53, T55, T56, pD_in_gaaa(T53, T55, X14, T56))
MAXSORTA_IN_GA(.(t, .(f, T53)), .(T55, T56)) → PD_IN_GAAA(T53, T55, X14, T56)
PD_IN_GAAA(T53, T135, X14, T136) → U22_GAAA(T53, T135, X14, T136, maxG_in_ga(T53, T135))
PD_IN_GAAA(T53, T135, X14, T136) → MAXG_IN_GA(T53, T135)
U22_GAAA(T53, T135, X14, T136, maxG_out_ga(T53, T135)) → U23_GAAA(T53, T135, X14, T136, pQ_in_ggaa(T135, T53, X14, T136))
U22_GAAA(T53, T135, X14, T136, maxG_out_ga(T53, T135)) → PQ_IN_GGAA(T135, T53, X14, T136)
PQ_IN_GGAA(T135, T53, T139, T136) → U24_GGAA(T135, T53, T139, T136, delM_in_gga(T135, T53, T139))
PQ_IN_GGAA(T135, T53, T139, T136) → DELM_IN_GGA(T135, T53, T139)
DELM_IN_GGA(f, T160, .(t, X269)) → U13_GGA(T160, X269, delH_in_ga(.(f, T160), X269))
DELM_IN_GGA(f, T160, .(t, X269)) → DELH_IN_GA(.(f, T160), X269)
U24_GGAA(T135, T53, T139, T136, delM_out_gga(T135, T53, T139)) → U25_GGAA(T135, T53, T139, T136, maxsortA_in_ga(T139, T136))
U24_GGAA(T135, T53, T139, T136, delM_out_gga(T135, T53, T139)) → MAXSORTA_IN_GA(T139, T136)
MAXSORTA_IN_GA(.(f, .(f, T53)), .(T55, T56)) → U4_GA(T53, T55, T56, pE_in_gaaa(T53, T55, X14, T56))
MAXSORTA_IN_GA(.(f, .(f, T53)), .(T55, T56)) → PE_IN_GAAA(T53, T55, X14, T56)
PE_IN_GAAA(T53, T163, X14, T164) → U26_GAAA(T53, T163, X14, T164, maxI_in_ga(T53, T163))
PE_IN_GAAA(T53, T163, X14, T164) → MAXI_IN_GA(T53, T163)
MAXI_IN_GA(.(f, T178), T180) → U9_GA(T178, T180, maxI_in_ga(T178, T180))
MAXI_IN_GA(.(f, T178), T180) → MAXI_IN_GA(T178, T180)
MAXI_IN_GA(.(t, T188), T190) → U10_GA(T188, T190, maxG_in_ga(T188, T190))
MAXI_IN_GA(.(t, T188), T190) → MAXG_IN_GA(T188, T190)
U26_GAAA(T53, T163, X14, T164, maxI_out_ga(T53, T163)) → U27_GAAA(T53, T163, X14, T164, pR_in_ggaa(T163, T53, X14, T164))
U26_GAAA(T53, T163, X14, T164, maxI_out_ga(T53, T163)) → PR_IN_GGAA(T163, T53, X14, T164)
PR_IN_GGAA(T163, T53, T193, T164) → U28_GGAA(T163, T53, T193, T164, delN_in_gga(T163, T53, T193))
PR_IN_GGAA(T163, T53, T193, T164) → DELN_IN_GGA(T163, T53, T193)
DELN_IN_GGA(t, T224, .(f, .(f, X384))) → U14_GGA(T224, X384, delJ_in_ga(T224, X384))
DELN_IN_GGA(t, T224, .(f, .(f, X384))) → DELJ_IN_GA(T224, X384)
DELJ_IN_GA(.(f, T240), .(f, X427)) → U11_GA(T240, X427, delJ_in_ga(T240, X427))
DELJ_IN_GA(.(f, T240), .(f, X427)) → DELJ_IN_GA(T240, X427)
U28_GGAA(T163, T53, T193, T164, delN_out_gga(T163, T53, T193)) → U29_GGAA(T163, T53, T193, T164, maxsortA_in_ga(T193, T164))
U28_GGAA(T163, T53, T193, T164, delN_out_gga(T163, T53, T193)) → MAXSORTA_IN_GA(T193, T164)
MAXSORTA_IN_GA(.(f, .(t, T251)), .(T253, T254)) → U5_GA(T251, T253, T254, pF_in_gaaa(T251, T253, X14, T254))
MAXSORTA_IN_GA(.(f, .(t, T251)), .(T253, T254)) → PF_IN_GAAA(T251, T253, X14, T254)
PF_IN_GAAA(T251, T255, X14, T256) → U30_GAAA(T251, T255, X14, T256, maxG_in_ga(T251, T255))
PF_IN_GAAA(T251, T255, X14, T256) → MAXG_IN_GA(T251, T255)
U30_GAAA(T251, T255, X14, T256, maxG_out_ga(T251, T255)) → U31_GAAA(T251, T255, X14, T256, pS_in_ggaa(T255, T251, X14, T256))
U30_GAAA(T251, T255, X14, T256, maxG_out_ga(T251, T255)) → PS_IN_GGAA(T255, T251, X14, T256)
PS_IN_GGAA(T255, T251, T259, T256) → U32_GGAA(T255, T251, T259, T256, delO_in_gga(T255, T251, T259))
PS_IN_GGAA(T255, T251, T259, T256) → DELO_IN_GGA(T255, T251, T259)
DELO_IN_GGA(t, T280, .(f, X480)) → U15_GGA(T280, X480, delJ_in_ga(.(t, T280), X480))
DELO_IN_GGA(t, T280, .(f, X480)) → DELJ_IN_GA(.(t, T280), X480)
U32_GGAA(T255, T251, T259, T256, delO_out_gga(T255, T251, T259)) → U33_GGAA(T255, T251, T259, T256, maxsortA_in_ga(T259, T256))
U32_GGAA(T255, T251, T259, T256, delO_out_gga(T255, T251, T259)) → MAXSORTA_IN_GA(T259, T256)

The TRS R consists of the following rules:

maxsortA_in_ga([], []) → maxsortA_out_ga([], [])
maxsortA_in_ga(.(T17, []), .(T17, T18)) → U1_ga(T17, T18, pB_in_gaa(T17, X14, T18))
pB_in_gaa(T17, T21, T18) → U16_gaa(T17, T21, T18, delK_in_ga(T17, T21))
delK_in_ga(t, []) → delK_out_ga(t, [])
delK_in_ga(f, []) → delK_out_ga(f, [])
U16_gaa(T17, T21, T18, delK_out_ga(T17, T21)) → U17_gaa(T17, T21, T18, maxsortA_in_ga(T21, T18))
maxsortA_in_ga(.(t, .(t, T53)), .(T55, T56)) → U2_ga(T53, T55, T56, pC_in_gaaa(T53, T55, X14, T56))
pC_in_gaaa(T53, T57, X14, T58) → U18_gaaa(T53, T57, X14, T58, maxG_in_ga(T53, T57))
maxG_in_ga([], t) → maxG_out_ga([], t)
maxG_in_ga(.(t, T72), T74) → U6_ga(T72, T74, maxG_in_ga(T72, T74))
maxG_in_ga(.(f, T72), T74) → U7_ga(T72, T74, maxG_in_ga(T72, T74))
U7_ga(T72, T74, maxG_out_ga(T72, T74)) → maxG_out_ga(.(f, T72), T74)
U6_ga(T72, T74, maxG_out_ga(T72, T74)) → maxG_out_ga(.(t, T72), T74)
U18_gaaa(T53, T57, X14, T58, maxG_out_ga(T53, T57)) → U19_gaaa(T53, T57, X14, T58, pP_in_ggaa(T57, T53, X14, T58))
pP_in_ggaa(T57, T53, T87, T58) → U20_ggaa(T57, T53, T87, T58, delL_in_gga(T57, T53, T87))
delL_in_gga(t, T100, .(t, T100)) → delL_out_gga(t, T100, .(t, T100))
delL_in_gga(f, T118, .(t, .(t, X187))) → U12_gga(T118, X187, delH_in_ga(T118, X187))
delH_in_ga([], []) → delH_out_ga([], [])
delH_in_ga(.(f, T128), T128) → delH_out_ga(.(f, T128), T128)
delH_in_ga(.(t, T134), .(t, X228)) → U8_ga(T134, X228, delH_in_ga(T134, X228))
U8_ga(T134, X228, delH_out_ga(T134, X228)) → delH_out_ga(.(t, T134), .(t, X228))
U12_gga(T118, X187, delH_out_ga(T118, X187)) → delL_out_gga(f, T118, .(t, .(t, X187)))
U20_ggaa(T57, T53, T87, T58, delL_out_gga(T57, T53, T87)) → U21_ggaa(T57, T53, T87, T58, maxsortA_in_ga(T87, T58))
maxsortA_in_ga(.(t, .(f, T53)), .(T55, T56)) → U3_ga(T53, T55, T56, pD_in_gaaa(T53, T55, X14, T56))
pD_in_gaaa(T53, T135, X14, T136) → U22_gaaa(T53, T135, X14, T136, maxG_in_ga(T53, T135))
U22_gaaa(T53, T135, X14, T136, maxG_out_ga(T53, T135)) → U23_gaaa(T53, T135, X14, T136, pQ_in_ggaa(T135, T53, X14, T136))
pQ_in_ggaa(T135, T53, T139, T136) → U24_ggaa(T135, T53, T139, T136, delM_in_gga(T135, T53, T139))
delM_in_gga(t, T152, .(f, T152)) → delM_out_gga(t, T152, .(f, T152))
delM_in_gga(f, T160, .(t, X269)) → U13_gga(T160, X269, delH_in_ga(.(f, T160), X269))
U13_gga(T160, X269, delH_out_ga(.(f, T160), X269)) → delM_out_gga(f, T160, .(t, X269))
U24_ggaa(T135, T53, T139, T136, delM_out_gga(T135, T53, T139)) → U25_ggaa(T135, T53, T139, T136, maxsortA_in_ga(T139, T136))
maxsortA_in_ga(.(f, .(f, T53)), .(T55, T56)) → U4_ga(T53, T55, T56, pE_in_gaaa(T53, T55, X14, T56))
pE_in_gaaa(T53, T163, X14, T164) → U26_gaaa(T53, T163, X14, T164, maxI_in_ga(T53, T163))
maxI_in_ga([], f) → maxI_out_ga([], f)
maxI_in_ga(.(f, T178), T180) → U9_ga(T178, T180, maxI_in_ga(T178, T180))
maxI_in_ga(.(t, T188), T190) → U10_ga(T188, T190, maxG_in_ga(T188, T190))
U10_ga(T188, T190, maxG_out_ga(T188, T190)) → maxI_out_ga(.(t, T188), T190)
U9_ga(T178, T180, maxI_out_ga(T178, T180)) → maxI_out_ga(.(f, T178), T180)
U26_gaaa(T53, T163, X14, T164, maxI_out_ga(T53, T163)) → U27_gaaa(T53, T163, X14, T164, pR_in_ggaa(T163, T53, X14, T164))
pR_in_ggaa(T163, T53, T193, T164) → U28_ggaa(T163, T53, T193, T164, delN_in_gga(T163, T53, T193))
delN_in_gga(f, T206, .(f, T206)) → delN_out_gga(f, T206, .(f, T206))
delN_in_gga(t, T224, .(f, .(f, X384))) → U14_gga(T224, X384, delJ_in_ga(T224, X384))
delJ_in_ga([], []) → delJ_out_ga([], [])
delJ_in_ga(.(t, T234), T234) → delJ_out_ga(.(t, T234), T234)
delJ_in_ga(.(f, T240), .(f, X427)) → U11_ga(T240, X427, delJ_in_ga(T240, X427))
U11_ga(T240, X427, delJ_out_ga(T240, X427)) → delJ_out_ga(.(f, T240), .(f, X427))
U14_gga(T224, X384, delJ_out_ga(T224, X384)) → delN_out_gga(t, T224, .(f, .(f, X384)))
U28_ggaa(T163, T53, T193, T164, delN_out_gga(T163, T53, T193)) → U29_ggaa(T163, T53, T193, T164, maxsortA_in_ga(T193, T164))
maxsortA_in_ga(.(f, .(t, T251)), .(T253, T254)) → U5_ga(T251, T253, T254, pF_in_gaaa(T251, T253, X14, T254))
pF_in_gaaa(T251, T255, X14, T256) → U30_gaaa(T251, T255, X14, T256, maxG_in_ga(T251, T255))
U30_gaaa(T251, T255, X14, T256, maxG_out_ga(T251, T255)) → U31_gaaa(T251, T255, X14, T256, pS_in_ggaa(T255, T251, X14, T256))
pS_in_ggaa(T255, T251, T259, T256) → U32_ggaa(T255, T251, T259, T256, delO_in_gga(T255, T251, T259))
delO_in_gga(f, T272, .(t, T272)) → delO_out_gga(f, T272, .(t, T272))
delO_in_gga(t, T280, .(f, X480)) → U15_gga(T280, X480, delJ_in_ga(.(t, T280), X480))
U15_gga(T280, X480, delJ_out_ga(.(t, T280), X480)) → delO_out_gga(t, T280, .(f, X480))
U32_ggaa(T255, T251, T259, T256, delO_out_gga(T255, T251, T259)) → U33_ggaa(T255, T251, T259, T256, maxsortA_in_ga(T259, T256))
U33_ggaa(T255, T251, T259, T256, maxsortA_out_ga(T259, T256)) → pS_out_ggaa(T255, T251, T259, T256)
U31_gaaa(T251, T255, X14, T256, pS_out_ggaa(T255, T251, X14, T256)) → pF_out_gaaa(T251, T255, X14, T256)
U5_ga(T251, T253, T254, pF_out_gaaa(T251, T253, X14, T254)) → maxsortA_out_ga(.(f, .(t, T251)), .(T253, T254))
U29_ggaa(T163, T53, T193, T164, maxsortA_out_ga(T193, T164)) → pR_out_ggaa(T163, T53, T193, T164)
U27_gaaa(T53, T163, X14, T164, pR_out_ggaa(T163, T53, X14, T164)) → pE_out_gaaa(T53, T163, X14, T164)
U4_ga(T53, T55, T56, pE_out_gaaa(T53, T55, X14, T56)) → maxsortA_out_ga(.(f, .(f, T53)), .(T55, T56))
U25_ggaa(T135, T53, T139, T136, maxsortA_out_ga(T139, T136)) → pQ_out_ggaa(T135, T53, T139, T136)
U23_gaaa(T53, T135, X14, T136, pQ_out_ggaa(T135, T53, X14, T136)) → pD_out_gaaa(T53, T135, X14, T136)
U3_ga(T53, T55, T56, pD_out_gaaa(T53, T55, X14, T56)) → maxsortA_out_ga(.(t, .(f, T53)), .(T55, T56))
U21_ggaa(T57, T53, T87, T58, maxsortA_out_ga(T87, T58)) → pP_out_ggaa(T57, T53, T87, T58)
U19_gaaa(T53, T57, X14, T58, pP_out_ggaa(T57, T53, X14, T58)) → pC_out_gaaa(T53, T57, X14, T58)
U2_ga(T53, T55, T56, pC_out_gaaa(T53, T55, X14, T56)) → maxsortA_out_ga(.(t, .(t, T53)), .(T55, T56))
U17_gaa(T17, T21, T18, maxsortA_out_ga(T21, T18)) → pB_out_gaa(T17, T21, T18)
U1_ga(T17, T18, pB_out_gaa(T17, X14, T18)) → maxsortA_out_ga(.(T17, []), .(T17, T18))

The argument filtering Pi contains the following mapping:
maxsortA_in_ga(x1, x2)  =  maxsortA_in_ga(x1)
[]  =  []
maxsortA_out_ga(x1, x2)  =  maxsortA_out_ga(x1, x2)
.(x1, x2)  =  .(x1, x2)
U1_ga(x1, x2, x3)  =  U1_ga(x1, x3)
pB_in_gaa(x1, x2, x3)  =  pB_in_gaa(x1)
U16_gaa(x1, x2, x3, x4)  =  U16_gaa(x1, x4)
delK_in_ga(x1, x2)  =  delK_in_ga(x1)
t  =  t
delK_out_ga(x1, x2)  =  delK_out_ga(x1, x2)
f  =  f
U17_gaa(x1, x2, x3, x4)  =  U17_gaa(x1, x2, x4)
U2_ga(x1, x2, x3, x4)  =  U2_ga(x1, x4)
pC_in_gaaa(x1, x2, x3, x4)  =  pC_in_gaaa(x1)
U18_gaaa(x1, x2, x3, x4, x5)  =  U18_gaaa(x1, x5)
maxG_in_ga(x1, x2)  =  maxG_in_ga(x1)
maxG_out_ga(x1, x2)  =  maxG_out_ga(x1, x2)
U6_ga(x1, x2, x3)  =  U6_ga(x1, x3)
U7_ga(x1, x2, x3)  =  U7_ga(x1, x3)
U19_gaaa(x1, x2, x3, x4, x5)  =  U19_gaaa(x1, x2, x5)
pP_in_ggaa(x1, x2, x3, x4)  =  pP_in_ggaa(x1, x2)
U20_ggaa(x1, x2, x3, x4, x5)  =  U20_ggaa(x1, x2, x5)
delL_in_gga(x1, x2, x3)  =  delL_in_gga(x1, x2)
delL_out_gga(x1, x2, x3)  =  delL_out_gga(x1, x2, x3)
U12_gga(x1, x2, x3)  =  U12_gga(x1, x3)
delH_in_ga(x1, x2)  =  delH_in_ga(x1)
delH_out_ga(x1, x2)  =  delH_out_ga(x1, x2)
U8_ga(x1, x2, x3)  =  U8_ga(x1, x3)
U21_ggaa(x1, x2, x3, x4, x5)  =  U21_ggaa(x1, x2, x3, x5)
U3_ga(x1, x2, x3, x4)  =  U3_ga(x1, x4)
pD_in_gaaa(x1, x2, x3, x4)  =  pD_in_gaaa(x1)
U22_gaaa(x1, x2, x3, x4, x5)  =  U22_gaaa(x1, x5)
U23_gaaa(x1, x2, x3, x4, x5)  =  U23_gaaa(x1, x2, x5)
pQ_in_ggaa(x1, x2, x3, x4)  =  pQ_in_ggaa(x1, x2)
U24_ggaa(x1, x2, x3, x4, x5)  =  U24_ggaa(x1, x2, x5)
delM_in_gga(x1, x2, x3)  =  delM_in_gga(x1, x2)
delM_out_gga(x1, x2, x3)  =  delM_out_gga(x1, x2, x3)
U13_gga(x1, x2, x3)  =  U13_gga(x1, x3)
U25_ggaa(x1, x2, x3, x4, x5)  =  U25_ggaa(x1, x2, x3, x5)
U4_ga(x1, x2, x3, x4)  =  U4_ga(x1, x4)
pE_in_gaaa(x1, x2, x3, x4)  =  pE_in_gaaa(x1)
U26_gaaa(x1, x2, x3, x4, x5)  =  U26_gaaa(x1, x5)
maxI_in_ga(x1, x2)  =  maxI_in_ga(x1)
maxI_out_ga(x1, x2)  =  maxI_out_ga(x1, x2)
U9_ga(x1, x2, x3)  =  U9_ga(x1, x3)
U10_ga(x1, x2, x3)  =  U10_ga(x1, x3)
U27_gaaa(x1, x2, x3, x4, x5)  =  U27_gaaa(x1, x2, x5)
pR_in_ggaa(x1, x2, x3, x4)  =  pR_in_ggaa(x1, x2)
U28_ggaa(x1, x2, x3, x4, x5)  =  U28_ggaa(x1, x2, x5)
delN_in_gga(x1, x2, x3)  =  delN_in_gga(x1, x2)
delN_out_gga(x1, x2, x3)  =  delN_out_gga(x1, x2, x3)
U14_gga(x1, x2, x3)  =  U14_gga(x1, x3)
delJ_in_ga(x1, x2)  =  delJ_in_ga(x1)
delJ_out_ga(x1, x2)  =  delJ_out_ga(x1, x2)
U11_ga(x1, x2, x3)  =  U11_ga(x1, x3)
U29_ggaa(x1, x2, x3, x4, x5)  =  U29_ggaa(x1, x2, x3, x5)
U5_ga(x1, x2, x3, x4)  =  U5_ga(x1, x4)
pF_in_gaaa(x1, x2, x3, x4)  =  pF_in_gaaa(x1)
U30_gaaa(x1, x2, x3, x4, x5)  =  U30_gaaa(x1, x5)
U31_gaaa(x1, x2, x3, x4, x5)  =  U31_gaaa(x1, x2, x5)
pS_in_ggaa(x1, x2, x3, x4)  =  pS_in_ggaa(x1, x2)
U32_ggaa(x1, x2, x3, x4, x5)  =  U32_ggaa(x1, x2, x5)
delO_in_gga(x1, x2, x3)  =  delO_in_gga(x1, x2)
delO_out_gga(x1, x2, x3)  =  delO_out_gga(x1, x2, x3)
U15_gga(x1, x2, x3)  =  U15_gga(x1, x3)
U33_ggaa(x1, x2, x3, x4, x5)  =  U33_ggaa(x1, x2, x3, x5)
pS_out_ggaa(x1, x2, x3, x4)  =  pS_out_ggaa(x1, x2, x3, x4)
pF_out_gaaa(x1, x2, x3, x4)  =  pF_out_gaaa(x1, x2, x3, x4)
pR_out_ggaa(x1, x2, x3, x4)  =  pR_out_ggaa(x1, x2, x3, x4)
pE_out_gaaa(x1, x2, x3, x4)  =  pE_out_gaaa(x1, x2, x3, x4)
pQ_out_ggaa(x1, x2, x3, x4)  =  pQ_out_ggaa(x1, x2, x3, x4)
pD_out_gaaa(x1, x2, x3, x4)  =  pD_out_gaaa(x1, x2, x3, x4)
pP_out_ggaa(x1, x2, x3, x4)  =  pP_out_ggaa(x1, x2, x3, x4)
pC_out_gaaa(x1, x2, x3, x4)  =  pC_out_gaaa(x1, x2, x3, x4)
pB_out_gaa(x1, x2, x3)  =  pB_out_gaa(x1, x2, x3)
MAXSORTA_IN_GA(x1, x2)  =  MAXSORTA_IN_GA(x1)
U1_GA(x1, x2, x3)  =  U1_GA(x1, x3)
PB_IN_GAA(x1, x2, x3)  =  PB_IN_GAA(x1)
U16_GAA(x1, x2, x3, x4)  =  U16_GAA(x1, x4)
DELK_IN_GA(x1, x2)  =  DELK_IN_GA(x1)
U17_GAA(x1, x2, x3, x4)  =  U17_GAA(x1, x2, x4)
U2_GA(x1, x2, x3, x4)  =  U2_GA(x1, x4)
PC_IN_GAAA(x1, x2, x3, x4)  =  PC_IN_GAAA(x1)
U18_GAAA(x1, x2, x3, x4, x5)  =  U18_GAAA(x1, x5)
MAXG_IN_GA(x1, x2)  =  MAXG_IN_GA(x1)
U6_GA(x1, x2, x3)  =  U6_GA(x1, x3)
U7_GA(x1, x2, x3)  =  U7_GA(x1, x3)
U19_GAAA(x1, x2, x3, x4, x5)  =  U19_GAAA(x1, x2, x5)
PP_IN_GGAA(x1, x2, x3, x4)  =  PP_IN_GGAA(x1, x2)
U20_GGAA(x1, x2, x3, x4, x5)  =  U20_GGAA(x1, x2, x5)
DELL_IN_GGA(x1, x2, x3)  =  DELL_IN_GGA(x1, x2)
U12_GGA(x1, x2, x3)  =  U12_GGA(x1, x3)
DELH_IN_GA(x1, x2)  =  DELH_IN_GA(x1)
U8_GA(x1, x2, x3)  =  U8_GA(x1, x3)
U21_GGAA(x1, x2, x3, x4, x5)  =  U21_GGAA(x1, x2, x3, x5)
U3_GA(x1, x2, x3, x4)  =  U3_GA(x1, x4)
PD_IN_GAAA(x1, x2, x3, x4)  =  PD_IN_GAAA(x1)
U22_GAAA(x1, x2, x3, x4, x5)  =  U22_GAAA(x1, x5)
U23_GAAA(x1, x2, x3, x4, x5)  =  U23_GAAA(x1, x2, x5)
PQ_IN_GGAA(x1, x2, x3, x4)  =  PQ_IN_GGAA(x1, x2)
U24_GGAA(x1, x2, x3, x4, x5)  =  U24_GGAA(x1, x2, x5)
DELM_IN_GGA(x1, x2, x3)  =  DELM_IN_GGA(x1, x2)
U13_GGA(x1, x2, x3)  =  U13_GGA(x1, x3)
U25_GGAA(x1, x2, x3, x4, x5)  =  U25_GGAA(x1, x2, x3, x5)
U4_GA(x1, x2, x3, x4)  =  U4_GA(x1, x4)
PE_IN_GAAA(x1, x2, x3, x4)  =  PE_IN_GAAA(x1)
U26_GAAA(x1, x2, x3, x4, x5)  =  U26_GAAA(x1, x5)
MAXI_IN_GA(x1, x2)  =  MAXI_IN_GA(x1)
U9_GA(x1, x2, x3)  =  U9_GA(x1, x3)
U10_GA(x1, x2, x3)  =  U10_GA(x1, x3)
U27_GAAA(x1, x2, x3, x4, x5)  =  U27_GAAA(x1, x2, x5)
PR_IN_GGAA(x1, x2, x3, x4)  =  PR_IN_GGAA(x1, x2)
U28_GGAA(x1, x2, x3, x4, x5)  =  U28_GGAA(x1, x2, x5)
DELN_IN_GGA(x1, x2, x3)  =  DELN_IN_GGA(x1, x2)
U14_GGA(x1, x2, x3)  =  U14_GGA(x1, x3)
DELJ_IN_GA(x1, x2)  =  DELJ_IN_GA(x1)
U11_GA(x1, x2, x3)  =  U11_GA(x1, x3)
U29_GGAA(x1, x2, x3, x4, x5)  =  U29_GGAA(x1, x2, x3, x5)
U5_GA(x1, x2, x3, x4)  =  U5_GA(x1, x4)
PF_IN_GAAA(x1, x2, x3, x4)  =  PF_IN_GAAA(x1)
U30_GAAA(x1, x2, x3, x4, x5)  =  U30_GAAA(x1, x5)
U31_GAAA(x1, x2, x3, x4, x5)  =  U31_GAAA(x1, x2, x5)
PS_IN_GGAA(x1, x2, x3, x4)  =  PS_IN_GGAA(x1, x2)
U32_GGAA(x1, x2, x3, x4, x5)  =  U32_GGAA(x1, x2, x5)
DELO_IN_GGA(x1, x2, x3)  =  DELO_IN_GGA(x1, x2)
U15_GGA(x1, x2, x3)  =  U15_GGA(x1, x3)
U33_GGAA(x1, x2, x3, x4, x5)  =  U33_GGAA(x1, x2, x3, x5)

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

(4) Obligation:

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

MAXSORTA_IN_GA(.(T17, []), .(T17, T18)) → U1_GA(T17, T18, pB_in_gaa(T17, X14, T18))
MAXSORTA_IN_GA(.(T17, []), .(T17, T18)) → PB_IN_GAA(T17, X14, T18)
PB_IN_GAA(T17, T21, T18) → U16_GAA(T17, T21, T18, delK_in_ga(T17, T21))
PB_IN_GAA(T17, T21, T18) → DELK_IN_GA(T17, T21)
U16_GAA(T17, T21, T18, delK_out_ga(T17, T21)) → U17_GAA(T17, T21, T18, maxsortA_in_ga(T21, T18))
U16_GAA(T17, T21, T18, delK_out_ga(T17, T21)) → MAXSORTA_IN_GA(T21, T18)
MAXSORTA_IN_GA(.(t, .(t, T53)), .(T55, T56)) → U2_GA(T53, T55, T56, pC_in_gaaa(T53, T55, X14, T56))
MAXSORTA_IN_GA(.(t, .(t, T53)), .(T55, T56)) → PC_IN_GAAA(T53, T55, X14, T56)
PC_IN_GAAA(T53, T57, X14, T58) → U18_GAAA(T53, T57, X14, T58, maxG_in_ga(T53, T57))
PC_IN_GAAA(T53, T57, X14, T58) → MAXG_IN_GA(T53, T57)
MAXG_IN_GA(.(t, T72), T74) → U6_GA(T72, T74, maxG_in_ga(T72, T74))
MAXG_IN_GA(.(t, T72), T74) → MAXG_IN_GA(T72, T74)
MAXG_IN_GA(.(f, T72), T74) → U7_GA(T72, T74, maxG_in_ga(T72, T74))
MAXG_IN_GA(.(f, T72), T74) → MAXG_IN_GA(T72, T74)
U18_GAAA(T53, T57, X14, T58, maxG_out_ga(T53, T57)) → U19_GAAA(T53, T57, X14, T58, pP_in_ggaa(T57, T53, X14, T58))
U18_GAAA(T53, T57, X14, T58, maxG_out_ga(T53, T57)) → PP_IN_GGAA(T57, T53, X14, T58)
PP_IN_GGAA(T57, T53, T87, T58) → U20_GGAA(T57, T53, T87, T58, delL_in_gga(T57, T53, T87))
PP_IN_GGAA(T57, T53, T87, T58) → DELL_IN_GGA(T57, T53, T87)
DELL_IN_GGA(f, T118, .(t, .(t, X187))) → U12_GGA(T118, X187, delH_in_ga(T118, X187))
DELL_IN_GGA(f, T118, .(t, .(t, X187))) → DELH_IN_GA(T118, X187)
DELH_IN_GA(.(t, T134), .(t, X228)) → U8_GA(T134, X228, delH_in_ga(T134, X228))
DELH_IN_GA(.(t, T134), .(t, X228)) → DELH_IN_GA(T134, X228)
U20_GGAA(T57, T53, T87, T58, delL_out_gga(T57, T53, T87)) → U21_GGAA(T57, T53, T87, T58, maxsortA_in_ga(T87, T58))
U20_GGAA(T57, T53, T87, T58, delL_out_gga(T57, T53, T87)) → MAXSORTA_IN_GA(T87, T58)
MAXSORTA_IN_GA(.(t, .(f, T53)), .(T55, T56)) → U3_GA(T53, T55, T56, pD_in_gaaa(T53, T55, X14, T56))
MAXSORTA_IN_GA(.(t, .(f, T53)), .(T55, T56)) → PD_IN_GAAA(T53, T55, X14, T56)
PD_IN_GAAA(T53, T135, X14, T136) → U22_GAAA(T53, T135, X14, T136, maxG_in_ga(T53, T135))
PD_IN_GAAA(T53, T135, X14, T136) → MAXG_IN_GA(T53, T135)
U22_GAAA(T53, T135, X14, T136, maxG_out_ga(T53, T135)) → U23_GAAA(T53, T135, X14, T136, pQ_in_ggaa(T135, T53, X14, T136))
U22_GAAA(T53, T135, X14, T136, maxG_out_ga(T53, T135)) → PQ_IN_GGAA(T135, T53, X14, T136)
PQ_IN_GGAA(T135, T53, T139, T136) → U24_GGAA(T135, T53, T139, T136, delM_in_gga(T135, T53, T139))
PQ_IN_GGAA(T135, T53, T139, T136) → DELM_IN_GGA(T135, T53, T139)
DELM_IN_GGA(f, T160, .(t, X269)) → U13_GGA(T160, X269, delH_in_ga(.(f, T160), X269))
DELM_IN_GGA(f, T160, .(t, X269)) → DELH_IN_GA(.(f, T160), X269)
U24_GGAA(T135, T53, T139, T136, delM_out_gga(T135, T53, T139)) → U25_GGAA(T135, T53, T139, T136, maxsortA_in_ga(T139, T136))
U24_GGAA(T135, T53, T139, T136, delM_out_gga(T135, T53, T139)) → MAXSORTA_IN_GA(T139, T136)
MAXSORTA_IN_GA(.(f, .(f, T53)), .(T55, T56)) → U4_GA(T53, T55, T56, pE_in_gaaa(T53, T55, X14, T56))
MAXSORTA_IN_GA(.(f, .(f, T53)), .(T55, T56)) → PE_IN_GAAA(T53, T55, X14, T56)
PE_IN_GAAA(T53, T163, X14, T164) → U26_GAAA(T53, T163, X14, T164, maxI_in_ga(T53, T163))
PE_IN_GAAA(T53, T163, X14, T164) → MAXI_IN_GA(T53, T163)
MAXI_IN_GA(.(f, T178), T180) → U9_GA(T178, T180, maxI_in_ga(T178, T180))
MAXI_IN_GA(.(f, T178), T180) → MAXI_IN_GA(T178, T180)
MAXI_IN_GA(.(t, T188), T190) → U10_GA(T188, T190, maxG_in_ga(T188, T190))
MAXI_IN_GA(.(t, T188), T190) → MAXG_IN_GA(T188, T190)
U26_GAAA(T53, T163, X14, T164, maxI_out_ga(T53, T163)) → U27_GAAA(T53, T163, X14, T164, pR_in_ggaa(T163, T53, X14, T164))
U26_GAAA(T53, T163, X14, T164, maxI_out_ga(T53, T163)) → PR_IN_GGAA(T163, T53, X14, T164)
PR_IN_GGAA(T163, T53, T193, T164) → U28_GGAA(T163, T53, T193, T164, delN_in_gga(T163, T53, T193))
PR_IN_GGAA(T163, T53, T193, T164) → DELN_IN_GGA(T163, T53, T193)
DELN_IN_GGA(t, T224, .(f, .(f, X384))) → U14_GGA(T224, X384, delJ_in_ga(T224, X384))
DELN_IN_GGA(t, T224, .(f, .(f, X384))) → DELJ_IN_GA(T224, X384)
DELJ_IN_GA(.(f, T240), .(f, X427)) → U11_GA(T240, X427, delJ_in_ga(T240, X427))
DELJ_IN_GA(.(f, T240), .(f, X427)) → DELJ_IN_GA(T240, X427)
U28_GGAA(T163, T53, T193, T164, delN_out_gga(T163, T53, T193)) → U29_GGAA(T163, T53, T193, T164, maxsortA_in_ga(T193, T164))
U28_GGAA(T163, T53, T193, T164, delN_out_gga(T163, T53, T193)) → MAXSORTA_IN_GA(T193, T164)
MAXSORTA_IN_GA(.(f, .(t, T251)), .(T253, T254)) → U5_GA(T251, T253, T254, pF_in_gaaa(T251, T253, X14, T254))
MAXSORTA_IN_GA(.(f, .(t, T251)), .(T253, T254)) → PF_IN_GAAA(T251, T253, X14, T254)
PF_IN_GAAA(T251, T255, X14, T256) → U30_GAAA(T251, T255, X14, T256, maxG_in_ga(T251, T255))
PF_IN_GAAA(T251, T255, X14, T256) → MAXG_IN_GA(T251, T255)
U30_GAAA(T251, T255, X14, T256, maxG_out_ga(T251, T255)) → U31_GAAA(T251, T255, X14, T256, pS_in_ggaa(T255, T251, X14, T256))
U30_GAAA(T251, T255, X14, T256, maxG_out_ga(T251, T255)) → PS_IN_GGAA(T255, T251, X14, T256)
PS_IN_GGAA(T255, T251, T259, T256) → U32_GGAA(T255, T251, T259, T256, delO_in_gga(T255, T251, T259))
PS_IN_GGAA(T255, T251, T259, T256) → DELO_IN_GGA(T255, T251, T259)
DELO_IN_GGA(t, T280, .(f, X480)) → U15_GGA(T280, X480, delJ_in_ga(.(t, T280), X480))
DELO_IN_GGA(t, T280, .(f, X480)) → DELJ_IN_GA(.(t, T280), X480)
U32_GGAA(T255, T251, T259, T256, delO_out_gga(T255, T251, T259)) → U33_GGAA(T255, T251, T259, T256, maxsortA_in_ga(T259, T256))
U32_GGAA(T255, T251, T259, T256, delO_out_gga(T255, T251, T259)) → MAXSORTA_IN_GA(T259, T256)

The TRS R consists of the following rules:

maxsortA_in_ga([], []) → maxsortA_out_ga([], [])
maxsortA_in_ga(.(T17, []), .(T17, T18)) → U1_ga(T17, T18, pB_in_gaa(T17, X14, T18))
pB_in_gaa(T17, T21, T18) → U16_gaa(T17, T21, T18, delK_in_ga(T17, T21))
delK_in_ga(t, []) → delK_out_ga(t, [])
delK_in_ga(f, []) → delK_out_ga(f, [])
U16_gaa(T17, T21, T18, delK_out_ga(T17, T21)) → U17_gaa(T17, T21, T18, maxsortA_in_ga(T21, T18))
maxsortA_in_ga(.(t, .(t, T53)), .(T55, T56)) → U2_ga(T53, T55, T56, pC_in_gaaa(T53, T55, X14, T56))
pC_in_gaaa(T53, T57, X14, T58) → U18_gaaa(T53, T57, X14, T58, maxG_in_ga(T53, T57))
maxG_in_ga([], t) → maxG_out_ga([], t)
maxG_in_ga(.(t, T72), T74) → U6_ga(T72, T74, maxG_in_ga(T72, T74))
maxG_in_ga(.(f, T72), T74) → U7_ga(T72, T74, maxG_in_ga(T72, T74))
U7_ga(T72, T74, maxG_out_ga(T72, T74)) → maxG_out_ga(.(f, T72), T74)
U6_ga(T72, T74, maxG_out_ga(T72, T74)) → maxG_out_ga(.(t, T72), T74)
U18_gaaa(T53, T57, X14, T58, maxG_out_ga(T53, T57)) → U19_gaaa(T53, T57, X14, T58, pP_in_ggaa(T57, T53, X14, T58))
pP_in_ggaa(T57, T53, T87, T58) → U20_ggaa(T57, T53, T87, T58, delL_in_gga(T57, T53, T87))
delL_in_gga(t, T100, .(t, T100)) → delL_out_gga(t, T100, .(t, T100))
delL_in_gga(f, T118, .(t, .(t, X187))) → U12_gga(T118, X187, delH_in_ga(T118, X187))
delH_in_ga([], []) → delH_out_ga([], [])
delH_in_ga(.(f, T128), T128) → delH_out_ga(.(f, T128), T128)
delH_in_ga(.(t, T134), .(t, X228)) → U8_ga(T134, X228, delH_in_ga(T134, X228))
U8_ga(T134, X228, delH_out_ga(T134, X228)) → delH_out_ga(.(t, T134), .(t, X228))
U12_gga(T118, X187, delH_out_ga(T118, X187)) → delL_out_gga(f, T118, .(t, .(t, X187)))
U20_ggaa(T57, T53, T87, T58, delL_out_gga(T57, T53, T87)) → U21_ggaa(T57, T53, T87, T58, maxsortA_in_ga(T87, T58))
maxsortA_in_ga(.(t, .(f, T53)), .(T55, T56)) → U3_ga(T53, T55, T56, pD_in_gaaa(T53, T55, X14, T56))
pD_in_gaaa(T53, T135, X14, T136) → U22_gaaa(T53, T135, X14, T136, maxG_in_ga(T53, T135))
U22_gaaa(T53, T135, X14, T136, maxG_out_ga(T53, T135)) → U23_gaaa(T53, T135, X14, T136, pQ_in_ggaa(T135, T53, X14, T136))
pQ_in_ggaa(T135, T53, T139, T136) → U24_ggaa(T135, T53, T139, T136, delM_in_gga(T135, T53, T139))
delM_in_gga(t, T152, .(f, T152)) → delM_out_gga(t, T152, .(f, T152))
delM_in_gga(f, T160, .(t, X269)) → U13_gga(T160, X269, delH_in_ga(.(f, T160), X269))
U13_gga(T160, X269, delH_out_ga(.(f, T160), X269)) → delM_out_gga(f, T160, .(t, X269))
U24_ggaa(T135, T53, T139, T136, delM_out_gga(T135, T53, T139)) → U25_ggaa(T135, T53, T139, T136, maxsortA_in_ga(T139, T136))
maxsortA_in_ga(.(f, .(f, T53)), .(T55, T56)) → U4_ga(T53, T55, T56, pE_in_gaaa(T53, T55, X14, T56))
pE_in_gaaa(T53, T163, X14, T164) → U26_gaaa(T53, T163, X14, T164, maxI_in_ga(T53, T163))
maxI_in_ga([], f) → maxI_out_ga([], f)
maxI_in_ga(.(f, T178), T180) → U9_ga(T178, T180, maxI_in_ga(T178, T180))
maxI_in_ga(.(t, T188), T190) → U10_ga(T188, T190, maxG_in_ga(T188, T190))
U10_ga(T188, T190, maxG_out_ga(T188, T190)) → maxI_out_ga(.(t, T188), T190)
U9_ga(T178, T180, maxI_out_ga(T178, T180)) → maxI_out_ga(.(f, T178), T180)
U26_gaaa(T53, T163, X14, T164, maxI_out_ga(T53, T163)) → U27_gaaa(T53, T163, X14, T164, pR_in_ggaa(T163, T53, X14, T164))
pR_in_ggaa(T163, T53, T193, T164) → U28_ggaa(T163, T53, T193, T164, delN_in_gga(T163, T53, T193))
delN_in_gga(f, T206, .(f, T206)) → delN_out_gga(f, T206, .(f, T206))
delN_in_gga(t, T224, .(f, .(f, X384))) → U14_gga(T224, X384, delJ_in_ga(T224, X384))
delJ_in_ga([], []) → delJ_out_ga([], [])
delJ_in_ga(.(t, T234), T234) → delJ_out_ga(.(t, T234), T234)
delJ_in_ga(.(f, T240), .(f, X427)) → U11_ga(T240, X427, delJ_in_ga(T240, X427))
U11_ga(T240, X427, delJ_out_ga(T240, X427)) → delJ_out_ga(.(f, T240), .(f, X427))
U14_gga(T224, X384, delJ_out_ga(T224, X384)) → delN_out_gga(t, T224, .(f, .(f, X384)))
U28_ggaa(T163, T53, T193, T164, delN_out_gga(T163, T53, T193)) → U29_ggaa(T163, T53, T193, T164, maxsortA_in_ga(T193, T164))
maxsortA_in_ga(.(f, .(t, T251)), .(T253, T254)) → U5_ga(T251, T253, T254, pF_in_gaaa(T251, T253, X14, T254))
pF_in_gaaa(T251, T255, X14, T256) → U30_gaaa(T251, T255, X14, T256, maxG_in_ga(T251, T255))
U30_gaaa(T251, T255, X14, T256, maxG_out_ga(T251, T255)) → U31_gaaa(T251, T255, X14, T256, pS_in_ggaa(T255, T251, X14, T256))
pS_in_ggaa(T255, T251, T259, T256) → U32_ggaa(T255, T251, T259, T256, delO_in_gga(T255, T251, T259))
delO_in_gga(f, T272, .(t, T272)) → delO_out_gga(f, T272, .(t, T272))
delO_in_gga(t, T280, .(f, X480)) → U15_gga(T280, X480, delJ_in_ga(.(t, T280), X480))
U15_gga(T280, X480, delJ_out_ga(.(t, T280), X480)) → delO_out_gga(t, T280, .(f, X480))
U32_ggaa(T255, T251, T259, T256, delO_out_gga(T255, T251, T259)) → U33_ggaa(T255, T251, T259, T256, maxsortA_in_ga(T259, T256))
U33_ggaa(T255, T251, T259, T256, maxsortA_out_ga(T259, T256)) → pS_out_ggaa(T255, T251, T259, T256)
U31_gaaa(T251, T255, X14, T256, pS_out_ggaa(T255, T251, X14, T256)) → pF_out_gaaa(T251, T255, X14, T256)
U5_ga(T251, T253, T254, pF_out_gaaa(T251, T253, X14, T254)) → maxsortA_out_ga(.(f, .(t, T251)), .(T253, T254))
U29_ggaa(T163, T53, T193, T164, maxsortA_out_ga(T193, T164)) → pR_out_ggaa(T163, T53, T193, T164)
U27_gaaa(T53, T163, X14, T164, pR_out_ggaa(T163, T53, X14, T164)) → pE_out_gaaa(T53, T163, X14, T164)
U4_ga(T53, T55, T56, pE_out_gaaa(T53, T55, X14, T56)) → maxsortA_out_ga(.(f, .(f, T53)), .(T55, T56))
U25_ggaa(T135, T53, T139, T136, maxsortA_out_ga(T139, T136)) → pQ_out_ggaa(T135, T53, T139, T136)
U23_gaaa(T53, T135, X14, T136, pQ_out_ggaa(T135, T53, X14, T136)) → pD_out_gaaa(T53, T135, X14, T136)
U3_ga(T53, T55, T56, pD_out_gaaa(T53, T55, X14, T56)) → maxsortA_out_ga(.(t, .(f, T53)), .(T55, T56))
U21_ggaa(T57, T53, T87, T58, maxsortA_out_ga(T87, T58)) → pP_out_ggaa(T57, T53, T87, T58)
U19_gaaa(T53, T57, X14, T58, pP_out_ggaa(T57, T53, X14, T58)) → pC_out_gaaa(T53, T57, X14, T58)
U2_ga(T53, T55, T56, pC_out_gaaa(T53, T55, X14, T56)) → maxsortA_out_ga(.(t, .(t, T53)), .(T55, T56))
U17_gaa(T17, T21, T18, maxsortA_out_ga(T21, T18)) → pB_out_gaa(T17, T21, T18)
U1_ga(T17, T18, pB_out_gaa(T17, X14, T18)) → maxsortA_out_ga(.(T17, []), .(T17, T18))

The argument filtering Pi contains the following mapping:
maxsortA_in_ga(x1, x2)  =  maxsortA_in_ga(x1)
[]  =  []
maxsortA_out_ga(x1, x2)  =  maxsortA_out_ga(x1, x2)
.(x1, x2)  =  .(x1, x2)
U1_ga(x1, x2, x3)  =  U1_ga(x1, x3)
pB_in_gaa(x1, x2, x3)  =  pB_in_gaa(x1)
U16_gaa(x1, x2, x3, x4)  =  U16_gaa(x1, x4)
delK_in_ga(x1, x2)  =  delK_in_ga(x1)
t  =  t
delK_out_ga(x1, x2)  =  delK_out_ga(x1, x2)
f  =  f
U17_gaa(x1, x2, x3, x4)  =  U17_gaa(x1, x2, x4)
U2_ga(x1, x2, x3, x4)  =  U2_ga(x1, x4)
pC_in_gaaa(x1, x2, x3, x4)  =  pC_in_gaaa(x1)
U18_gaaa(x1, x2, x3, x4, x5)  =  U18_gaaa(x1, x5)
maxG_in_ga(x1, x2)  =  maxG_in_ga(x1)
maxG_out_ga(x1, x2)  =  maxG_out_ga(x1, x2)
U6_ga(x1, x2, x3)  =  U6_ga(x1, x3)
U7_ga(x1, x2, x3)  =  U7_ga(x1, x3)
U19_gaaa(x1, x2, x3, x4, x5)  =  U19_gaaa(x1, x2, x5)
pP_in_ggaa(x1, x2, x3, x4)  =  pP_in_ggaa(x1, x2)
U20_ggaa(x1, x2, x3, x4, x5)  =  U20_ggaa(x1, x2, x5)
delL_in_gga(x1, x2, x3)  =  delL_in_gga(x1, x2)
delL_out_gga(x1, x2, x3)  =  delL_out_gga(x1, x2, x3)
U12_gga(x1, x2, x3)  =  U12_gga(x1, x3)
delH_in_ga(x1, x2)  =  delH_in_ga(x1)
delH_out_ga(x1, x2)  =  delH_out_ga(x1, x2)
U8_ga(x1, x2, x3)  =  U8_ga(x1, x3)
U21_ggaa(x1, x2, x3, x4, x5)  =  U21_ggaa(x1, x2, x3, x5)
U3_ga(x1, x2, x3, x4)  =  U3_ga(x1, x4)
pD_in_gaaa(x1, x2, x3, x4)  =  pD_in_gaaa(x1)
U22_gaaa(x1, x2, x3, x4, x5)  =  U22_gaaa(x1, x5)
U23_gaaa(x1, x2, x3, x4, x5)  =  U23_gaaa(x1, x2, x5)
pQ_in_ggaa(x1, x2, x3, x4)  =  pQ_in_ggaa(x1, x2)
U24_ggaa(x1, x2, x3, x4, x5)  =  U24_ggaa(x1, x2, x5)
delM_in_gga(x1, x2, x3)  =  delM_in_gga(x1, x2)
delM_out_gga(x1, x2, x3)  =  delM_out_gga(x1, x2, x3)
U13_gga(x1, x2, x3)  =  U13_gga(x1, x3)
U25_ggaa(x1, x2, x3, x4, x5)  =  U25_ggaa(x1, x2, x3, x5)
U4_ga(x1, x2, x3, x4)  =  U4_ga(x1, x4)
pE_in_gaaa(x1, x2, x3, x4)  =  pE_in_gaaa(x1)
U26_gaaa(x1, x2, x3, x4, x5)  =  U26_gaaa(x1, x5)
maxI_in_ga(x1, x2)  =  maxI_in_ga(x1)
maxI_out_ga(x1, x2)  =  maxI_out_ga(x1, x2)
U9_ga(x1, x2, x3)  =  U9_ga(x1, x3)
U10_ga(x1, x2, x3)  =  U10_ga(x1, x3)
U27_gaaa(x1, x2, x3, x4, x5)  =  U27_gaaa(x1, x2, x5)
pR_in_ggaa(x1, x2, x3, x4)  =  pR_in_ggaa(x1, x2)
U28_ggaa(x1, x2, x3, x4, x5)  =  U28_ggaa(x1, x2, x5)
delN_in_gga(x1, x2, x3)  =  delN_in_gga(x1, x2)
delN_out_gga(x1, x2, x3)  =  delN_out_gga(x1, x2, x3)
U14_gga(x1, x2, x3)  =  U14_gga(x1, x3)
delJ_in_ga(x1, x2)  =  delJ_in_ga(x1)
delJ_out_ga(x1, x2)  =  delJ_out_ga(x1, x2)
U11_ga(x1, x2, x3)  =  U11_ga(x1, x3)
U29_ggaa(x1, x2, x3, x4, x5)  =  U29_ggaa(x1, x2, x3, x5)
U5_ga(x1, x2, x3, x4)  =  U5_ga(x1, x4)
pF_in_gaaa(x1, x2, x3, x4)  =  pF_in_gaaa(x1)
U30_gaaa(x1, x2, x3, x4, x5)  =  U30_gaaa(x1, x5)
U31_gaaa(x1, x2, x3, x4, x5)  =  U31_gaaa(x1, x2, x5)
pS_in_ggaa(x1, x2, x3, x4)  =  pS_in_ggaa(x1, x2)
U32_ggaa(x1, x2, x3, x4, x5)  =  U32_ggaa(x1, x2, x5)
delO_in_gga(x1, x2, x3)  =  delO_in_gga(x1, x2)
delO_out_gga(x1, x2, x3)  =  delO_out_gga(x1, x2, x3)
U15_gga(x1, x2, x3)  =  U15_gga(x1, x3)
U33_ggaa(x1, x2, x3, x4, x5)  =  U33_ggaa(x1, x2, x3, x5)
pS_out_ggaa(x1, x2, x3, x4)  =  pS_out_ggaa(x1, x2, x3, x4)
pF_out_gaaa(x1, x2, x3, x4)  =  pF_out_gaaa(x1, x2, x3, x4)
pR_out_ggaa(x1, x2, x3, x4)  =  pR_out_ggaa(x1, x2, x3, x4)
pE_out_gaaa(x1, x2, x3, x4)  =  pE_out_gaaa(x1, x2, x3, x4)
pQ_out_ggaa(x1, x2, x3, x4)  =  pQ_out_ggaa(x1, x2, x3, x4)
pD_out_gaaa(x1, x2, x3, x4)  =  pD_out_gaaa(x1, x2, x3, x4)
pP_out_ggaa(x1, x2, x3, x4)  =  pP_out_ggaa(x1, x2, x3, x4)
pC_out_gaaa(x1, x2, x3, x4)  =  pC_out_gaaa(x1, x2, x3, x4)
pB_out_gaa(x1, x2, x3)  =  pB_out_gaa(x1, x2, x3)
MAXSORTA_IN_GA(x1, x2)  =  MAXSORTA_IN_GA(x1)
U1_GA(x1, x2, x3)  =  U1_GA(x1, x3)
PB_IN_GAA(x1, x2, x3)  =  PB_IN_GAA(x1)
U16_GAA(x1, x2, x3, x4)  =  U16_GAA(x1, x4)
DELK_IN_GA(x1, x2)  =  DELK_IN_GA(x1)
U17_GAA(x1, x2, x3, x4)  =  U17_GAA(x1, x2, x4)
U2_GA(x1, x2, x3, x4)  =  U2_GA(x1, x4)
PC_IN_GAAA(x1, x2, x3, x4)  =  PC_IN_GAAA(x1)
U18_GAAA(x1, x2, x3, x4, x5)  =  U18_GAAA(x1, x5)
MAXG_IN_GA(x1, x2)  =  MAXG_IN_GA(x1)
U6_GA(x1, x2, x3)  =  U6_GA(x1, x3)
U7_GA(x1, x2, x3)  =  U7_GA(x1, x3)
U19_GAAA(x1, x2, x3, x4, x5)  =  U19_GAAA(x1, x2, x5)
PP_IN_GGAA(x1, x2, x3, x4)  =  PP_IN_GGAA(x1, x2)
U20_GGAA(x1, x2, x3, x4, x5)  =  U20_GGAA(x1, x2, x5)
DELL_IN_GGA(x1, x2, x3)  =  DELL_IN_GGA(x1, x2)
U12_GGA(x1, x2, x3)  =  U12_GGA(x1, x3)
DELH_IN_GA(x1, x2)  =  DELH_IN_GA(x1)
U8_GA(x1, x2, x3)  =  U8_GA(x1, x3)
U21_GGAA(x1, x2, x3, x4, x5)  =  U21_GGAA(x1, x2, x3, x5)
U3_GA(x1, x2, x3, x4)  =  U3_GA(x1, x4)
PD_IN_GAAA(x1, x2, x3, x4)  =  PD_IN_GAAA(x1)
U22_GAAA(x1, x2, x3, x4, x5)  =  U22_GAAA(x1, x5)
U23_GAAA(x1, x2, x3, x4, x5)  =  U23_GAAA(x1, x2, x5)
PQ_IN_GGAA(x1, x2, x3, x4)  =  PQ_IN_GGAA(x1, x2)
U24_GGAA(x1, x2, x3, x4, x5)  =  U24_GGAA(x1, x2, x5)
DELM_IN_GGA(x1, x2, x3)  =  DELM_IN_GGA(x1, x2)
U13_GGA(x1, x2, x3)  =  U13_GGA(x1, x3)
U25_GGAA(x1, x2, x3, x4, x5)  =  U25_GGAA(x1, x2, x3, x5)
U4_GA(x1, x2, x3, x4)  =  U4_GA(x1, x4)
PE_IN_GAAA(x1, x2, x3, x4)  =  PE_IN_GAAA(x1)
U26_GAAA(x1, x2, x3, x4, x5)  =  U26_GAAA(x1, x5)
MAXI_IN_GA(x1, x2)  =  MAXI_IN_GA(x1)
U9_GA(x1, x2, x3)  =  U9_GA(x1, x3)
U10_GA(x1, x2, x3)  =  U10_GA(x1, x3)
U27_GAAA(x1, x2, x3, x4, x5)  =  U27_GAAA(x1, x2, x5)
PR_IN_GGAA(x1, x2, x3, x4)  =  PR_IN_GGAA(x1, x2)
U28_GGAA(x1, x2, x3, x4, x5)  =  U28_GGAA(x1, x2, x5)
DELN_IN_GGA(x1, x2, x3)  =  DELN_IN_GGA(x1, x2)
U14_GGA(x1, x2, x3)  =  U14_GGA(x1, x3)
DELJ_IN_GA(x1, x2)  =  DELJ_IN_GA(x1)
U11_GA(x1, x2, x3)  =  U11_GA(x1, x3)
U29_GGAA(x1, x2, x3, x4, x5)  =  U29_GGAA(x1, x2, x3, x5)
U5_GA(x1, x2, x3, x4)  =  U5_GA(x1, x4)
PF_IN_GAAA(x1, x2, x3, x4)  =  PF_IN_GAAA(x1)
U30_GAAA(x1, x2, x3, x4, x5)  =  U30_GAAA(x1, x5)
U31_GAAA(x1, x2, x3, x4, x5)  =  U31_GAAA(x1, x2, x5)
PS_IN_GGAA(x1, x2, x3, x4)  =  PS_IN_GGAA(x1, x2)
U32_GGAA(x1, x2, x3, x4, x5)  =  U32_GGAA(x1, x2, x5)
DELO_IN_GGA(x1, x2, x3)  =  DELO_IN_GGA(x1, x2)
U15_GGA(x1, x2, x3)  =  U15_GGA(x1, x3)
U33_GGAA(x1, x2, x3, x4, x5)  =  U33_GGAA(x1, x2, x3, x5)

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

(5) DependencyGraphProof (EQUIVALENT transformation)

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

(6) Complex Obligation (AND)

(7) Obligation:

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

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

The TRS R consists of the following rules:

maxsortA_in_ga([], []) → maxsortA_out_ga([], [])
maxsortA_in_ga(.(T17, []), .(T17, T18)) → U1_ga(T17, T18, pB_in_gaa(T17, X14, T18))
pB_in_gaa(T17, T21, T18) → U16_gaa(T17, T21, T18, delK_in_ga(T17, T21))
delK_in_ga(t, []) → delK_out_ga(t, [])
delK_in_ga(f, []) → delK_out_ga(f, [])
U16_gaa(T17, T21, T18, delK_out_ga(T17, T21)) → U17_gaa(T17, T21, T18, maxsortA_in_ga(T21, T18))
maxsortA_in_ga(.(t, .(t, T53)), .(T55, T56)) → U2_ga(T53, T55, T56, pC_in_gaaa(T53, T55, X14, T56))
pC_in_gaaa(T53, T57, X14, T58) → U18_gaaa(T53, T57, X14, T58, maxG_in_ga(T53, T57))
maxG_in_ga([], t) → maxG_out_ga([], t)
maxG_in_ga(.(t, T72), T74) → U6_ga(T72, T74, maxG_in_ga(T72, T74))
maxG_in_ga(.(f, T72), T74) → U7_ga(T72, T74, maxG_in_ga(T72, T74))
U7_ga(T72, T74, maxG_out_ga(T72, T74)) → maxG_out_ga(.(f, T72), T74)
U6_ga(T72, T74, maxG_out_ga(T72, T74)) → maxG_out_ga(.(t, T72), T74)
U18_gaaa(T53, T57, X14, T58, maxG_out_ga(T53, T57)) → U19_gaaa(T53, T57, X14, T58, pP_in_ggaa(T57, T53, X14, T58))
pP_in_ggaa(T57, T53, T87, T58) → U20_ggaa(T57, T53, T87, T58, delL_in_gga(T57, T53, T87))
delL_in_gga(t, T100, .(t, T100)) → delL_out_gga(t, T100, .(t, T100))
delL_in_gga(f, T118, .(t, .(t, X187))) → U12_gga(T118, X187, delH_in_ga(T118, X187))
delH_in_ga([], []) → delH_out_ga([], [])
delH_in_ga(.(f, T128), T128) → delH_out_ga(.(f, T128), T128)
delH_in_ga(.(t, T134), .(t, X228)) → U8_ga(T134, X228, delH_in_ga(T134, X228))
U8_ga(T134, X228, delH_out_ga(T134, X228)) → delH_out_ga(.(t, T134), .(t, X228))
U12_gga(T118, X187, delH_out_ga(T118, X187)) → delL_out_gga(f, T118, .(t, .(t, X187)))
U20_ggaa(T57, T53, T87, T58, delL_out_gga(T57, T53, T87)) → U21_ggaa(T57, T53, T87, T58, maxsortA_in_ga(T87, T58))
maxsortA_in_ga(.(t, .(f, T53)), .(T55, T56)) → U3_ga(T53, T55, T56, pD_in_gaaa(T53, T55, X14, T56))
pD_in_gaaa(T53, T135, X14, T136) → U22_gaaa(T53, T135, X14, T136, maxG_in_ga(T53, T135))
U22_gaaa(T53, T135, X14, T136, maxG_out_ga(T53, T135)) → U23_gaaa(T53, T135, X14, T136, pQ_in_ggaa(T135, T53, X14, T136))
pQ_in_ggaa(T135, T53, T139, T136) → U24_ggaa(T135, T53, T139, T136, delM_in_gga(T135, T53, T139))
delM_in_gga(t, T152, .(f, T152)) → delM_out_gga(t, T152, .(f, T152))
delM_in_gga(f, T160, .(t, X269)) → U13_gga(T160, X269, delH_in_ga(.(f, T160), X269))
U13_gga(T160, X269, delH_out_ga(.(f, T160), X269)) → delM_out_gga(f, T160, .(t, X269))
U24_ggaa(T135, T53, T139, T136, delM_out_gga(T135, T53, T139)) → U25_ggaa(T135, T53, T139, T136, maxsortA_in_ga(T139, T136))
maxsortA_in_ga(.(f, .(f, T53)), .(T55, T56)) → U4_ga(T53, T55, T56, pE_in_gaaa(T53, T55, X14, T56))
pE_in_gaaa(T53, T163, X14, T164) → U26_gaaa(T53, T163, X14, T164, maxI_in_ga(T53, T163))
maxI_in_ga([], f) → maxI_out_ga([], f)
maxI_in_ga(.(f, T178), T180) → U9_ga(T178, T180, maxI_in_ga(T178, T180))
maxI_in_ga(.(t, T188), T190) → U10_ga(T188, T190, maxG_in_ga(T188, T190))
U10_ga(T188, T190, maxG_out_ga(T188, T190)) → maxI_out_ga(.(t, T188), T190)
U9_ga(T178, T180, maxI_out_ga(T178, T180)) → maxI_out_ga(.(f, T178), T180)
U26_gaaa(T53, T163, X14, T164, maxI_out_ga(T53, T163)) → U27_gaaa(T53, T163, X14, T164, pR_in_ggaa(T163, T53, X14, T164))
pR_in_ggaa(T163, T53, T193, T164) → U28_ggaa(T163, T53, T193, T164, delN_in_gga(T163, T53, T193))
delN_in_gga(f, T206, .(f, T206)) → delN_out_gga(f, T206, .(f, T206))
delN_in_gga(t, T224, .(f, .(f, X384))) → U14_gga(T224, X384, delJ_in_ga(T224, X384))
delJ_in_ga([], []) → delJ_out_ga([], [])
delJ_in_ga(.(t, T234), T234) → delJ_out_ga(.(t, T234), T234)
delJ_in_ga(.(f, T240), .(f, X427)) → U11_ga(T240, X427, delJ_in_ga(T240, X427))
U11_ga(T240, X427, delJ_out_ga(T240, X427)) → delJ_out_ga(.(f, T240), .(f, X427))
U14_gga(T224, X384, delJ_out_ga(T224, X384)) → delN_out_gga(t, T224, .(f, .(f, X384)))
U28_ggaa(T163, T53, T193, T164, delN_out_gga(T163, T53, T193)) → U29_ggaa(T163, T53, T193, T164, maxsortA_in_ga(T193, T164))
maxsortA_in_ga(.(f, .(t, T251)), .(T253, T254)) → U5_ga(T251, T253, T254, pF_in_gaaa(T251, T253, X14, T254))
pF_in_gaaa(T251, T255, X14, T256) → U30_gaaa(T251, T255, X14, T256, maxG_in_ga(T251, T255))
U30_gaaa(T251, T255, X14, T256, maxG_out_ga(T251, T255)) → U31_gaaa(T251, T255, X14, T256, pS_in_ggaa(T255, T251, X14, T256))
pS_in_ggaa(T255, T251, T259, T256) → U32_ggaa(T255, T251, T259, T256, delO_in_gga(T255, T251, T259))
delO_in_gga(f, T272, .(t, T272)) → delO_out_gga(f, T272, .(t, T272))
delO_in_gga(t, T280, .(f, X480)) → U15_gga(T280, X480, delJ_in_ga(.(t, T280), X480))
U15_gga(T280, X480, delJ_out_ga(.(t, T280), X480)) → delO_out_gga(t, T280, .(f, X480))
U32_ggaa(T255, T251, T259, T256, delO_out_gga(T255, T251, T259)) → U33_ggaa(T255, T251, T259, T256, maxsortA_in_ga(T259, T256))
U33_ggaa(T255, T251, T259, T256, maxsortA_out_ga(T259, T256)) → pS_out_ggaa(T255, T251, T259, T256)
U31_gaaa(T251, T255, X14, T256, pS_out_ggaa(T255, T251, X14, T256)) → pF_out_gaaa(T251, T255, X14, T256)
U5_ga(T251, T253, T254, pF_out_gaaa(T251, T253, X14, T254)) → maxsortA_out_ga(.(f, .(t, T251)), .(T253, T254))
U29_ggaa(T163, T53, T193, T164, maxsortA_out_ga(T193, T164)) → pR_out_ggaa(T163, T53, T193, T164)
U27_gaaa(T53, T163, X14, T164, pR_out_ggaa(T163, T53, X14, T164)) → pE_out_gaaa(T53, T163, X14, T164)
U4_ga(T53, T55, T56, pE_out_gaaa(T53, T55, X14, T56)) → maxsortA_out_ga(.(f, .(f, T53)), .(T55, T56))
U25_ggaa(T135, T53, T139, T136, maxsortA_out_ga(T139, T136)) → pQ_out_ggaa(T135, T53, T139, T136)
U23_gaaa(T53, T135, X14, T136, pQ_out_ggaa(T135, T53, X14, T136)) → pD_out_gaaa(T53, T135, X14, T136)
U3_ga(T53, T55, T56, pD_out_gaaa(T53, T55, X14, T56)) → maxsortA_out_ga(.(t, .(f, T53)), .(T55, T56))
U21_ggaa(T57, T53, T87, T58, maxsortA_out_ga(T87, T58)) → pP_out_ggaa(T57, T53, T87, T58)
U19_gaaa(T53, T57, X14, T58, pP_out_ggaa(T57, T53, X14, T58)) → pC_out_gaaa(T53, T57, X14, T58)
U2_ga(T53, T55, T56, pC_out_gaaa(T53, T55, X14, T56)) → maxsortA_out_ga(.(t, .(t, T53)), .(T55, T56))
U17_gaa(T17, T21, T18, maxsortA_out_ga(T21, T18)) → pB_out_gaa(T17, T21, T18)
U1_ga(T17, T18, pB_out_gaa(T17, X14, T18)) → maxsortA_out_ga(.(T17, []), .(T17, T18))

The argument filtering Pi contains the following mapping:
maxsortA_in_ga(x1, x2)  =  maxsortA_in_ga(x1)
[]  =  []
maxsortA_out_ga(x1, x2)  =  maxsortA_out_ga(x1, x2)
.(x1, x2)  =  .(x1, x2)
U1_ga(x1, x2, x3)  =  U1_ga(x1, x3)
pB_in_gaa(x1, x2, x3)  =  pB_in_gaa(x1)
U16_gaa(x1, x2, x3, x4)  =  U16_gaa(x1, x4)
delK_in_ga(x1, x2)  =  delK_in_ga(x1)
t  =  t
delK_out_ga(x1, x2)  =  delK_out_ga(x1, x2)
f  =  f
U17_gaa(x1, x2, x3, x4)  =  U17_gaa(x1, x2, x4)
U2_ga(x1, x2, x3, x4)  =  U2_ga(x1, x4)
pC_in_gaaa(x1, x2, x3, x4)  =  pC_in_gaaa(x1)
U18_gaaa(x1, x2, x3, x4, x5)  =  U18_gaaa(x1, x5)
maxG_in_ga(x1, x2)  =  maxG_in_ga(x1)
maxG_out_ga(x1, x2)  =  maxG_out_ga(x1, x2)
U6_ga(x1, x2, x3)  =  U6_ga(x1, x3)
U7_ga(x1, x2, x3)  =  U7_ga(x1, x3)
U19_gaaa(x1, x2, x3, x4, x5)  =  U19_gaaa(x1, x2, x5)
pP_in_ggaa(x1, x2, x3, x4)  =  pP_in_ggaa(x1, x2)
U20_ggaa(x1, x2, x3, x4, x5)  =  U20_ggaa(x1, x2, x5)
delL_in_gga(x1, x2, x3)  =  delL_in_gga(x1, x2)
delL_out_gga(x1, x2, x3)  =  delL_out_gga(x1, x2, x3)
U12_gga(x1, x2, x3)  =  U12_gga(x1, x3)
delH_in_ga(x1, x2)  =  delH_in_ga(x1)
delH_out_ga(x1, x2)  =  delH_out_ga(x1, x2)
U8_ga(x1, x2, x3)  =  U8_ga(x1, x3)
U21_ggaa(x1, x2, x3, x4, x5)  =  U21_ggaa(x1, x2, x3, x5)
U3_ga(x1, x2, x3, x4)  =  U3_ga(x1, x4)
pD_in_gaaa(x1, x2, x3, x4)  =  pD_in_gaaa(x1)
U22_gaaa(x1, x2, x3, x4, x5)  =  U22_gaaa(x1, x5)
U23_gaaa(x1, x2, x3, x4, x5)  =  U23_gaaa(x1, x2, x5)
pQ_in_ggaa(x1, x2, x3, x4)  =  pQ_in_ggaa(x1, x2)
U24_ggaa(x1, x2, x3, x4, x5)  =  U24_ggaa(x1, x2, x5)
delM_in_gga(x1, x2, x3)  =  delM_in_gga(x1, x2)
delM_out_gga(x1, x2, x3)  =  delM_out_gga(x1, x2, x3)
U13_gga(x1, x2, x3)  =  U13_gga(x1, x3)
U25_ggaa(x1, x2, x3, x4, x5)  =  U25_ggaa(x1, x2, x3, x5)
U4_ga(x1, x2, x3, x4)  =  U4_ga(x1, x4)
pE_in_gaaa(x1, x2, x3, x4)  =  pE_in_gaaa(x1)
U26_gaaa(x1, x2, x3, x4, x5)  =  U26_gaaa(x1, x5)
maxI_in_ga(x1, x2)  =  maxI_in_ga(x1)
maxI_out_ga(x1, x2)  =  maxI_out_ga(x1, x2)
U9_ga(x1, x2, x3)  =  U9_ga(x1, x3)
U10_ga(x1, x2, x3)  =  U10_ga(x1, x3)
U27_gaaa(x1, x2, x3, x4, x5)  =  U27_gaaa(x1, x2, x5)
pR_in_ggaa(x1, x2, x3, x4)  =  pR_in_ggaa(x1, x2)
U28_ggaa(x1, x2, x3, x4, x5)  =  U28_ggaa(x1, x2, x5)
delN_in_gga(x1, x2, x3)  =  delN_in_gga(x1, x2)
delN_out_gga(x1, x2, x3)  =  delN_out_gga(x1, x2, x3)
U14_gga(x1, x2, x3)  =  U14_gga(x1, x3)
delJ_in_ga(x1, x2)  =  delJ_in_ga(x1)
delJ_out_ga(x1, x2)  =  delJ_out_ga(x1, x2)
U11_ga(x1, x2, x3)  =  U11_ga(x1, x3)
U29_ggaa(x1, x2, x3, x4, x5)  =  U29_ggaa(x1, x2, x3, x5)
U5_ga(x1, x2, x3, x4)  =  U5_ga(x1, x4)
pF_in_gaaa(x1, x2, x3, x4)  =  pF_in_gaaa(x1)
U30_gaaa(x1, x2, x3, x4, x5)  =  U30_gaaa(x1, x5)
U31_gaaa(x1, x2, x3, x4, x5)  =  U31_gaaa(x1, x2, x5)
pS_in_ggaa(x1, x2, x3, x4)  =  pS_in_ggaa(x1, x2)
U32_ggaa(x1, x2, x3, x4, x5)  =  U32_ggaa(x1, x2, x5)
delO_in_gga(x1, x2, x3)  =  delO_in_gga(x1, x2)
delO_out_gga(x1, x2, x3)  =  delO_out_gga(x1, x2, x3)
U15_gga(x1, x2, x3)  =  U15_gga(x1, x3)
U33_ggaa(x1, x2, x3, x4, x5)  =  U33_ggaa(x1, x2, x3, x5)
pS_out_ggaa(x1, x2, x3, x4)  =  pS_out_ggaa(x1, x2, x3, x4)
pF_out_gaaa(x1, x2, x3, x4)  =  pF_out_gaaa(x1, x2, x3, x4)
pR_out_ggaa(x1, x2, x3, x4)  =  pR_out_ggaa(x1, x2, x3, x4)
pE_out_gaaa(x1, x2, x3, x4)  =  pE_out_gaaa(x1, x2, x3, x4)
pQ_out_ggaa(x1, x2, x3, x4)  =  pQ_out_ggaa(x1, x2, x3, x4)
pD_out_gaaa(x1, x2, x3, x4)  =  pD_out_gaaa(x1, x2, x3, x4)
pP_out_ggaa(x1, x2, x3, x4)  =  pP_out_ggaa(x1, x2, x3, x4)
pC_out_gaaa(x1, x2, x3, x4)  =  pC_out_gaaa(x1, x2, x3, x4)
pB_out_gaa(x1, x2, x3)  =  pB_out_gaa(x1, x2, x3)
DELJ_IN_GA(x1, x2)  =  DELJ_IN_GA(x1)

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

(8) UsableRulesProof (EQUIVALENT transformation)

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

(9) Obligation:

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

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

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

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

(10) PiDPToQDPProof (SOUND transformation)

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

(11) Obligation:

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

DELJ_IN_GA(.(f, T240)) → DELJ_IN_GA(T240)

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

(12) QDPSizeChangeProof (EQUIVALENT transformation)

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

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

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

(13) YES

(14) Obligation:

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

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

The TRS R consists of the following rules:

maxsortA_in_ga([], []) → maxsortA_out_ga([], [])
maxsortA_in_ga(.(T17, []), .(T17, T18)) → U1_ga(T17, T18, pB_in_gaa(T17, X14, T18))
pB_in_gaa(T17, T21, T18) → U16_gaa(T17, T21, T18, delK_in_ga(T17, T21))
delK_in_ga(t, []) → delK_out_ga(t, [])
delK_in_ga(f, []) → delK_out_ga(f, [])
U16_gaa(T17, T21, T18, delK_out_ga(T17, T21)) → U17_gaa(T17, T21, T18, maxsortA_in_ga(T21, T18))
maxsortA_in_ga(.(t, .(t, T53)), .(T55, T56)) → U2_ga(T53, T55, T56, pC_in_gaaa(T53, T55, X14, T56))
pC_in_gaaa(T53, T57, X14, T58) → U18_gaaa(T53, T57, X14, T58, maxG_in_ga(T53, T57))
maxG_in_ga([], t) → maxG_out_ga([], t)
maxG_in_ga(.(t, T72), T74) → U6_ga(T72, T74, maxG_in_ga(T72, T74))
maxG_in_ga(.(f, T72), T74) → U7_ga(T72, T74, maxG_in_ga(T72, T74))
U7_ga(T72, T74, maxG_out_ga(T72, T74)) → maxG_out_ga(.(f, T72), T74)
U6_ga(T72, T74, maxG_out_ga(T72, T74)) → maxG_out_ga(.(t, T72), T74)
U18_gaaa(T53, T57, X14, T58, maxG_out_ga(T53, T57)) → U19_gaaa(T53, T57, X14, T58, pP_in_ggaa(T57, T53, X14, T58))
pP_in_ggaa(T57, T53, T87, T58) → U20_ggaa(T57, T53, T87, T58, delL_in_gga(T57, T53, T87))
delL_in_gga(t, T100, .(t, T100)) → delL_out_gga(t, T100, .(t, T100))
delL_in_gga(f, T118, .(t, .(t, X187))) → U12_gga(T118, X187, delH_in_ga(T118, X187))
delH_in_ga([], []) → delH_out_ga([], [])
delH_in_ga(.(f, T128), T128) → delH_out_ga(.(f, T128), T128)
delH_in_ga(.(t, T134), .(t, X228)) → U8_ga(T134, X228, delH_in_ga(T134, X228))
U8_ga(T134, X228, delH_out_ga(T134, X228)) → delH_out_ga(.(t, T134), .(t, X228))
U12_gga(T118, X187, delH_out_ga(T118, X187)) → delL_out_gga(f, T118, .(t, .(t, X187)))
U20_ggaa(T57, T53, T87, T58, delL_out_gga(T57, T53, T87)) → U21_ggaa(T57, T53, T87, T58, maxsortA_in_ga(T87, T58))
maxsortA_in_ga(.(t, .(f, T53)), .(T55, T56)) → U3_ga(T53, T55, T56, pD_in_gaaa(T53, T55, X14, T56))
pD_in_gaaa(T53, T135, X14, T136) → U22_gaaa(T53, T135, X14, T136, maxG_in_ga(T53, T135))
U22_gaaa(T53, T135, X14, T136, maxG_out_ga(T53, T135)) → U23_gaaa(T53, T135, X14, T136, pQ_in_ggaa(T135, T53, X14, T136))
pQ_in_ggaa(T135, T53, T139, T136) → U24_ggaa(T135, T53, T139, T136, delM_in_gga(T135, T53, T139))
delM_in_gga(t, T152, .(f, T152)) → delM_out_gga(t, T152, .(f, T152))
delM_in_gga(f, T160, .(t, X269)) → U13_gga(T160, X269, delH_in_ga(.(f, T160), X269))
U13_gga(T160, X269, delH_out_ga(.(f, T160), X269)) → delM_out_gga(f, T160, .(t, X269))
U24_ggaa(T135, T53, T139, T136, delM_out_gga(T135, T53, T139)) → U25_ggaa(T135, T53, T139, T136, maxsortA_in_ga(T139, T136))
maxsortA_in_ga(.(f, .(f, T53)), .(T55, T56)) → U4_ga(T53, T55, T56, pE_in_gaaa(T53, T55, X14, T56))
pE_in_gaaa(T53, T163, X14, T164) → U26_gaaa(T53, T163, X14, T164, maxI_in_ga(T53, T163))
maxI_in_ga([], f) → maxI_out_ga([], f)
maxI_in_ga(.(f, T178), T180) → U9_ga(T178, T180, maxI_in_ga(T178, T180))
maxI_in_ga(.(t, T188), T190) → U10_ga(T188, T190, maxG_in_ga(T188, T190))
U10_ga(T188, T190, maxG_out_ga(T188, T190)) → maxI_out_ga(.(t, T188), T190)
U9_ga(T178, T180, maxI_out_ga(T178, T180)) → maxI_out_ga(.(f, T178), T180)
U26_gaaa(T53, T163, X14, T164, maxI_out_ga(T53, T163)) → U27_gaaa(T53, T163, X14, T164, pR_in_ggaa(T163, T53, X14, T164))
pR_in_ggaa(T163, T53, T193, T164) → U28_ggaa(T163, T53, T193, T164, delN_in_gga(T163, T53, T193))
delN_in_gga(f, T206, .(f, T206)) → delN_out_gga(f, T206, .(f, T206))
delN_in_gga(t, T224, .(f, .(f, X384))) → U14_gga(T224, X384, delJ_in_ga(T224, X384))
delJ_in_ga([], []) → delJ_out_ga([], [])
delJ_in_ga(.(t, T234), T234) → delJ_out_ga(.(t, T234), T234)
delJ_in_ga(.(f, T240), .(f, X427)) → U11_ga(T240, X427, delJ_in_ga(T240, X427))
U11_ga(T240, X427, delJ_out_ga(T240, X427)) → delJ_out_ga(.(f, T240), .(f, X427))
U14_gga(T224, X384, delJ_out_ga(T224, X384)) → delN_out_gga(t, T224, .(f, .(f, X384)))
U28_ggaa(T163, T53, T193, T164, delN_out_gga(T163, T53, T193)) → U29_ggaa(T163, T53, T193, T164, maxsortA_in_ga(T193, T164))
maxsortA_in_ga(.(f, .(t, T251)), .(T253, T254)) → U5_ga(T251, T253, T254, pF_in_gaaa(T251, T253, X14, T254))
pF_in_gaaa(T251, T255, X14, T256) → U30_gaaa(T251, T255, X14, T256, maxG_in_ga(T251, T255))
U30_gaaa(T251, T255, X14, T256, maxG_out_ga(T251, T255)) → U31_gaaa(T251, T255, X14, T256, pS_in_ggaa(T255, T251, X14, T256))
pS_in_ggaa(T255, T251, T259, T256) → U32_ggaa(T255, T251, T259, T256, delO_in_gga(T255, T251, T259))
delO_in_gga(f, T272, .(t, T272)) → delO_out_gga(f, T272, .(t, T272))
delO_in_gga(t, T280, .(f, X480)) → U15_gga(T280, X480, delJ_in_ga(.(t, T280), X480))
U15_gga(T280, X480, delJ_out_ga(.(t, T280), X480)) → delO_out_gga(t, T280, .(f, X480))
U32_ggaa(T255, T251, T259, T256, delO_out_gga(T255, T251, T259)) → U33_ggaa(T255, T251, T259, T256, maxsortA_in_ga(T259, T256))
U33_ggaa(T255, T251, T259, T256, maxsortA_out_ga(T259, T256)) → pS_out_ggaa(T255, T251, T259, T256)
U31_gaaa(T251, T255, X14, T256, pS_out_ggaa(T255, T251, X14, T256)) → pF_out_gaaa(T251, T255, X14, T256)
U5_ga(T251, T253, T254, pF_out_gaaa(T251, T253, X14, T254)) → maxsortA_out_ga(.(f, .(t, T251)), .(T253, T254))
U29_ggaa(T163, T53, T193, T164, maxsortA_out_ga(T193, T164)) → pR_out_ggaa(T163, T53, T193, T164)
U27_gaaa(T53, T163, X14, T164, pR_out_ggaa(T163, T53, X14, T164)) → pE_out_gaaa(T53, T163, X14, T164)
U4_ga(T53, T55, T56, pE_out_gaaa(T53, T55, X14, T56)) → maxsortA_out_ga(.(f, .(f, T53)), .(T55, T56))
U25_ggaa(T135, T53, T139, T136, maxsortA_out_ga(T139, T136)) → pQ_out_ggaa(T135, T53, T139, T136)
U23_gaaa(T53, T135, X14, T136, pQ_out_ggaa(T135, T53, X14, T136)) → pD_out_gaaa(T53, T135, X14, T136)
U3_ga(T53, T55, T56, pD_out_gaaa(T53, T55, X14, T56)) → maxsortA_out_ga(.(t, .(f, T53)), .(T55, T56))
U21_ggaa(T57, T53, T87, T58, maxsortA_out_ga(T87, T58)) → pP_out_ggaa(T57, T53, T87, T58)
U19_gaaa(T53, T57, X14, T58, pP_out_ggaa(T57, T53, X14, T58)) → pC_out_gaaa(T53, T57, X14, T58)
U2_ga(T53, T55, T56, pC_out_gaaa(T53, T55, X14, T56)) → maxsortA_out_ga(.(t, .(t, T53)), .(T55, T56))
U17_gaa(T17, T21, T18, maxsortA_out_ga(T21, T18)) → pB_out_gaa(T17, T21, T18)
U1_ga(T17, T18, pB_out_gaa(T17, X14, T18)) → maxsortA_out_ga(.(T17, []), .(T17, T18))

The argument filtering Pi contains the following mapping:
maxsortA_in_ga(x1, x2)  =  maxsortA_in_ga(x1)
[]  =  []
maxsortA_out_ga(x1, x2)  =  maxsortA_out_ga(x1, x2)
.(x1, x2)  =  .(x1, x2)
U1_ga(x1, x2, x3)  =  U1_ga(x1, x3)
pB_in_gaa(x1, x2, x3)  =  pB_in_gaa(x1)
U16_gaa(x1, x2, x3, x4)  =  U16_gaa(x1, x4)
delK_in_ga(x1, x2)  =  delK_in_ga(x1)
t  =  t
delK_out_ga(x1, x2)  =  delK_out_ga(x1, x2)
f  =  f
U17_gaa(x1, x2, x3, x4)  =  U17_gaa(x1, x2, x4)
U2_ga(x1, x2, x3, x4)  =  U2_ga(x1, x4)
pC_in_gaaa(x1, x2, x3, x4)  =  pC_in_gaaa(x1)
U18_gaaa(x1, x2, x3, x4, x5)  =  U18_gaaa(x1, x5)
maxG_in_ga(x1, x2)  =  maxG_in_ga(x1)
maxG_out_ga(x1, x2)  =  maxG_out_ga(x1, x2)
U6_ga(x1, x2, x3)  =  U6_ga(x1, x3)
U7_ga(x1, x2, x3)  =  U7_ga(x1, x3)
U19_gaaa(x1, x2, x3, x4, x5)  =  U19_gaaa(x1, x2, x5)
pP_in_ggaa(x1, x2, x3, x4)  =  pP_in_ggaa(x1, x2)
U20_ggaa(x1, x2, x3, x4, x5)  =  U20_ggaa(x1, x2, x5)
delL_in_gga(x1, x2, x3)  =  delL_in_gga(x1, x2)
delL_out_gga(x1, x2, x3)  =  delL_out_gga(x1, x2, x3)
U12_gga(x1, x2, x3)  =  U12_gga(x1, x3)
delH_in_ga(x1, x2)  =  delH_in_ga(x1)
delH_out_ga(x1, x2)  =  delH_out_ga(x1, x2)
U8_ga(x1, x2, x3)  =  U8_ga(x1, x3)
U21_ggaa(x1, x2, x3, x4, x5)  =  U21_ggaa(x1, x2, x3, x5)
U3_ga(x1, x2, x3, x4)  =  U3_ga(x1, x4)
pD_in_gaaa(x1, x2, x3, x4)  =  pD_in_gaaa(x1)
U22_gaaa(x1, x2, x3, x4, x5)  =  U22_gaaa(x1, x5)
U23_gaaa(x1, x2, x3, x4, x5)  =  U23_gaaa(x1, x2, x5)
pQ_in_ggaa(x1, x2, x3, x4)  =  pQ_in_ggaa(x1, x2)
U24_ggaa(x1, x2, x3, x4, x5)  =  U24_ggaa(x1, x2, x5)
delM_in_gga(x1, x2, x3)  =  delM_in_gga(x1, x2)
delM_out_gga(x1, x2, x3)  =  delM_out_gga(x1, x2, x3)
U13_gga(x1, x2, x3)  =  U13_gga(x1, x3)
U25_ggaa(x1, x2, x3, x4, x5)  =  U25_ggaa(x1, x2, x3, x5)
U4_ga(x1, x2, x3, x4)  =  U4_ga(x1, x4)
pE_in_gaaa(x1, x2, x3, x4)  =  pE_in_gaaa(x1)
U26_gaaa(x1, x2, x3, x4, x5)  =  U26_gaaa(x1, x5)
maxI_in_ga(x1, x2)  =  maxI_in_ga(x1)
maxI_out_ga(x1, x2)  =  maxI_out_ga(x1, x2)
U9_ga(x1, x2, x3)  =  U9_ga(x1, x3)
U10_ga(x1, x2, x3)  =  U10_ga(x1, x3)
U27_gaaa(x1, x2, x3, x4, x5)  =  U27_gaaa(x1, x2, x5)
pR_in_ggaa(x1, x2, x3, x4)  =  pR_in_ggaa(x1, x2)
U28_ggaa(x1, x2, x3, x4, x5)  =  U28_ggaa(x1, x2, x5)
delN_in_gga(x1, x2, x3)  =  delN_in_gga(x1, x2)
delN_out_gga(x1, x2, x3)  =  delN_out_gga(x1, x2, x3)
U14_gga(x1, x2, x3)  =  U14_gga(x1, x3)
delJ_in_ga(x1, x2)  =  delJ_in_ga(x1)
delJ_out_ga(x1, x2)  =  delJ_out_ga(x1, x2)
U11_ga(x1, x2, x3)  =  U11_ga(x1, x3)
U29_ggaa(x1, x2, x3, x4, x5)  =  U29_ggaa(x1, x2, x3, x5)
U5_ga(x1, x2, x3, x4)  =  U5_ga(x1, x4)
pF_in_gaaa(x1, x2, x3, x4)  =  pF_in_gaaa(x1)
U30_gaaa(x1, x2, x3, x4, x5)  =  U30_gaaa(x1, x5)
U31_gaaa(x1, x2, x3, x4, x5)  =  U31_gaaa(x1, x2, x5)
pS_in_ggaa(x1, x2, x3, x4)  =  pS_in_ggaa(x1, x2)
U32_ggaa(x1, x2, x3, x4, x5)  =  U32_ggaa(x1, x2, x5)
delO_in_gga(x1, x2, x3)  =  delO_in_gga(x1, x2)
delO_out_gga(x1, x2, x3)  =  delO_out_gga(x1, x2, x3)
U15_gga(x1, x2, x3)  =  U15_gga(x1, x3)
U33_ggaa(x1, x2, x3, x4, x5)  =  U33_ggaa(x1, x2, x3, x5)
pS_out_ggaa(x1, x2, x3, x4)  =  pS_out_ggaa(x1, x2, x3, x4)
pF_out_gaaa(x1, x2, x3, x4)  =  pF_out_gaaa(x1, x2, x3, x4)
pR_out_ggaa(x1, x2, x3, x4)  =  pR_out_ggaa(x1, x2, x3, x4)
pE_out_gaaa(x1, x2, x3, x4)  =  pE_out_gaaa(x1, x2, x3, x4)
pQ_out_ggaa(x1, x2, x3, x4)  =  pQ_out_ggaa(x1, x2, x3, x4)
pD_out_gaaa(x1, x2, x3, x4)  =  pD_out_gaaa(x1, x2, x3, x4)
pP_out_ggaa(x1, x2, x3, x4)  =  pP_out_ggaa(x1, x2, x3, x4)
pC_out_gaaa(x1, x2, x3, x4)  =  pC_out_gaaa(x1, x2, x3, x4)
pB_out_gaa(x1, x2, x3)  =  pB_out_gaa(x1, x2, x3)
DELH_IN_GA(x1, x2)  =  DELH_IN_GA(x1)

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

(15) UsableRulesProof (EQUIVALENT transformation)

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

(16) Obligation:

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

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

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

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

(17) PiDPToQDPProof (SOUND transformation)

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

(18) Obligation:

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

DELH_IN_GA(.(t, T134)) → DELH_IN_GA(T134)

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

(19) QDPSizeChangeProof (EQUIVALENT transformation)

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

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

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

(20) YES

(21) Obligation:

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

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

The TRS R consists of the following rules:

maxsortA_in_ga([], []) → maxsortA_out_ga([], [])
maxsortA_in_ga(.(T17, []), .(T17, T18)) → U1_ga(T17, T18, pB_in_gaa(T17, X14, T18))
pB_in_gaa(T17, T21, T18) → U16_gaa(T17, T21, T18, delK_in_ga(T17, T21))
delK_in_ga(t, []) → delK_out_ga(t, [])
delK_in_ga(f, []) → delK_out_ga(f, [])
U16_gaa(T17, T21, T18, delK_out_ga(T17, T21)) → U17_gaa(T17, T21, T18, maxsortA_in_ga(T21, T18))
maxsortA_in_ga(.(t, .(t, T53)), .(T55, T56)) → U2_ga(T53, T55, T56, pC_in_gaaa(T53, T55, X14, T56))
pC_in_gaaa(T53, T57, X14, T58) → U18_gaaa(T53, T57, X14, T58, maxG_in_ga(T53, T57))
maxG_in_ga([], t) → maxG_out_ga([], t)
maxG_in_ga(.(t, T72), T74) → U6_ga(T72, T74, maxG_in_ga(T72, T74))
maxG_in_ga(.(f, T72), T74) → U7_ga(T72, T74, maxG_in_ga(T72, T74))
U7_ga(T72, T74, maxG_out_ga(T72, T74)) → maxG_out_ga(.(f, T72), T74)
U6_ga(T72, T74, maxG_out_ga(T72, T74)) → maxG_out_ga(.(t, T72), T74)
U18_gaaa(T53, T57, X14, T58, maxG_out_ga(T53, T57)) → U19_gaaa(T53, T57, X14, T58, pP_in_ggaa(T57, T53, X14, T58))
pP_in_ggaa(T57, T53, T87, T58) → U20_ggaa(T57, T53, T87, T58, delL_in_gga(T57, T53, T87))
delL_in_gga(t, T100, .(t, T100)) → delL_out_gga(t, T100, .(t, T100))
delL_in_gga(f, T118, .(t, .(t, X187))) → U12_gga(T118, X187, delH_in_ga(T118, X187))
delH_in_ga([], []) → delH_out_ga([], [])
delH_in_ga(.(f, T128), T128) → delH_out_ga(.(f, T128), T128)
delH_in_ga(.(t, T134), .(t, X228)) → U8_ga(T134, X228, delH_in_ga(T134, X228))
U8_ga(T134, X228, delH_out_ga(T134, X228)) → delH_out_ga(.(t, T134), .(t, X228))
U12_gga(T118, X187, delH_out_ga(T118, X187)) → delL_out_gga(f, T118, .(t, .(t, X187)))
U20_ggaa(T57, T53, T87, T58, delL_out_gga(T57, T53, T87)) → U21_ggaa(T57, T53, T87, T58, maxsortA_in_ga(T87, T58))
maxsortA_in_ga(.(t, .(f, T53)), .(T55, T56)) → U3_ga(T53, T55, T56, pD_in_gaaa(T53, T55, X14, T56))
pD_in_gaaa(T53, T135, X14, T136) → U22_gaaa(T53, T135, X14, T136, maxG_in_ga(T53, T135))
U22_gaaa(T53, T135, X14, T136, maxG_out_ga(T53, T135)) → U23_gaaa(T53, T135, X14, T136, pQ_in_ggaa(T135, T53, X14, T136))
pQ_in_ggaa(T135, T53, T139, T136) → U24_ggaa(T135, T53, T139, T136, delM_in_gga(T135, T53, T139))
delM_in_gga(t, T152, .(f, T152)) → delM_out_gga(t, T152, .(f, T152))
delM_in_gga(f, T160, .(t, X269)) → U13_gga(T160, X269, delH_in_ga(.(f, T160), X269))
U13_gga(T160, X269, delH_out_ga(.(f, T160), X269)) → delM_out_gga(f, T160, .(t, X269))
U24_ggaa(T135, T53, T139, T136, delM_out_gga(T135, T53, T139)) → U25_ggaa(T135, T53, T139, T136, maxsortA_in_ga(T139, T136))
maxsortA_in_ga(.(f, .(f, T53)), .(T55, T56)) → U4_ga(T53, T55, T56, pE_in_gaaa(T53, T55, X14, T56))
pE_in_gaaa(T53, T163, X14, T164) → U26_gaaa(T53, T163, X14, T164, maxI_in_ga(T53, T163))
maxI_in_ga([], f) → maxI_out_ga([], f)
maxI_in_ga(.(f, T178), T180) → U9_ga(T178, T180, maxI_in_ga(T178, T180))
maxI_in_ga(.(t, T188), T190) → U10_ga(T188, T190, maxG_in_ga(T188, T190))
U10_ga(T188, T190, maxG_out_ga(T188, T190)) → maxI_out_ga(.(t, T188), T190)
U9_ga(T178, T180, maxI_out_ga(T178, T180)) → maxI_out_ga(.(f, T178), T180)
U26_gaaa(T53, T163, X14, T164, maxI_out_ga(T53, T163)) → U27_gaaa(T53, T163, X14, T164, pR_in_ggaa(T163, T53, X14, T164))
pR_in_ggaa(T163, T53, T193, T164) → U28_ggaa(T163, T53, T193, T164, delN_in_gga(T163, T53, T193))
delN_in_gga(f, T206, .(f, T206)) → delN_out_gga(f, T206, .(f, T206))
delN_in_gga(t, T224, .(f, .(f, X384))) → U14_gga(T224, X384, delJ_in_ga(T224, X384))
delJ_in_ga([], []) → delJ_out_ga([], [])
delJ_in_ga(.(t, T234), T234) → delJ_out_ga(.(t, T234), T234)
delJ_in_ga(.(f, T240), .(f, X427)) → U11_ga(T240, X427, delJ_in_ga(T240, X427))
U11_ga(T240, X427, delJ_out_ga(T240, X427)) → delJ_out_ga(.(f, T240), .(f, X427))
U14_gga(T224, X384, delJ_out_ga(T224, X384)) → delN_out_gga(t, T224, .(f, .(f, X384)))
U28_ggaa(T163, T53, T193, T164, delN_out_gga(T163, T53, T193)) → U29_ggaa(T163, T53, T193, T164, maxsortA_in_ga(T193, T164))
maxsortA_in_ga(.(f, .(t, T251)), .(T253, T254)) → U5_ga(T251, T253, T254, pF_in_gaaa(T251, T253, X14, T254))
pF_in_gaaa(T251, T255, X14, T256) → U30_gaaa(T251, T255, X14, T256, maxG_in_ga(T251, T255))
U30_gaaa(T251, T255, X14, T256, maxG_out_ga(T251, T255)) → U31_gaaa(T251, T255, X14, T256, pS_in_ggaa(T255, T251, X14, T256))
pS_in_ggaa(T255, T251, T259, T256) → U32_ggaa(T255, T251, T259, T256, delO_in_gga(T255, T251, T259))
delO_in_gga(f, T272, .(t, T272)) → delO_out_gga(f, T272, .(t, T272))
delO_in_gga(t, T280, .(f, X480)) → U15_gga(T280, X480, delJ_in_ga(.(t, T280), X480))
U15_gga(T280, X480, delJ_out_ga(.(t, T280), X480)) → delO_out_gga(t, T280, .(f, X480))
U32_ggaa(T255, T251, T259, T256, delO_out_gga(T255, T251, T259)) → U33_ggaa(T255, T251, T259, T256, maxsortA_in_ga(T259, T256))
U33_ggaa(T255, T251, T259, T256, maxsortA_out_ga(T259, T256)) → pS_out_ggaa(T255, T251, T259, T256)
U31_gaaa(T251, T255, X14, T256, pS_out_ggaa(T255, T251, X14, T256)) → pF_out_gaaa(T251, T255, X14, T256)
U5_ga(T251, T253, T254, pF_out_gaaa(T251, T253, X14, T254)) → maxsortA_out_ga(.(f, .(t, T251)), .(T253, T254))
U29_ggaa(T163, T53, T193, T164, maxsortA_out_ga(T193, T164)) → pR_out_ggaa(T163, T53, T193, T164)
U27_gaaa(T53, T163, X14, T164, pR_out_ggaa(T163, T53, X14, T164)) → pE_out_gaaa(T53, T163, X14, T164)
U4_ga(T53, T55, T56, pE_out_gaaa(T53, T55, X14, T56)) → maxsortA_out_ga(.(f, .(f, T53)), .(T55, T56))
U25_ggaa(T135, T53, T139, T136, maxsortA_out_ga(T139, T136)) → pQ_out_ggaa(T135, T53, T139, T136)
U23_gaaa(T53, T135, X14, T136, pQ_out_ggaa(T135, T53, X14, T136)) → pD_out_gaaa(T53, T135, X14, T136)
U3_ga(T53, T55, T56, pD_out_gaaa(T53, T55, X14, T56)) → maxsortA_out_ga(.(t, .(f, T53)), .(T55, T56))
U21_ggaa(T57, T53, T87, T58, maxsortA_out_ga(T87, T58)) → pP_out_ggaa(T57, T53, T87, T58)
U19_gaaa(T53, T57, X14, T58, pP_out_ggaa(T57, T53, X14, T58)) → pC_out_gaaa(T53, T57, X14, T58)
U2_ga(T53, T55, T56, pC_out_gaaa(T53, T55, X14, T56)) → maxsortA_out_ga(.(t, .(t, T53)), .(T55, T56))
U17_gaa(T17, T21, T18, maxsortA_out_ga(T21, T18)) → pB_out_gaa(T17, T21, T18)
U1_ga(T17, T18, pB_out_gaa(T17, X14, T18)) → maxsortA_out_ga(.(T17, []), .(T17, T18))

The argument filtering Pi contains the following mapping:
maxsortA_in_ga(x1, x2)  =  maxsortA_in_ga(x1)
[]  =  []
maxsortA_out_ga(x1, x2)  =  maxsortA_out_ga(x1, x2)
.(x1, x2)  =  .(x1, x2)
U1_ga(x1, x2, x3)  =  U1_ga(x1, x3)
pB_in_gaa(x1, x2, x3)  =  pB_in_gaa(x1)
U16_gaa(x1, x2, x3, x4)  =  U16_gaa(x1, x4)
delK_in_ga(x1, x2)  =  delK_in_ga(x1)
t  =  t
delK_out_ga(x1, x2)  =  delK_out_ga(x1, x2)
f  =  f
U17_gaa(x1, x2, x3, x4)  =  U17_gaa(x1, x2, x4)
U2_ga(x1, x2, x3, x4)  =  U2_ga(x1, x4)
pC_in_gaaa(x1, x2, x3, x4)  =  pC_in_gaaa(x1)
U18_gaaa(x1, x2, x3, x4, x5)  =  U18_gaaa(x1, x5)
maxG_in_ga(x1, x2)  =  maxG_in_ga(x1)
maxG_out_ga(x1, x2)  =  maxG_out_ga(x1, x2)
U6_ga(x1, x2, x3)  =  U6_ga(x1, x3)
U7_ga(x1, x2, x3)  =  U7_ga(x1, x3)
U19_gaaa(x1, x2, x3, x4, x5)  =  U19_gaaa(x1, x2, x5)
pP_in_ggaa(x1, x2, x3, x4)  =  pP_in_ggaa(x1, x2)
U20_ggaa(x1, x2, x3, x4, x5)  =  U20_ggaa(x1, x2, x5)
delL_in_gga(x1, x2, x3)  =  delL_in_gga(x1, x2)
delL_out_gga(x1, x2, x3)  =  delL_out_gga(x1, x2, x3)
U12_gga(x1, x2, x3)  =  U12_gga(x1, x3)
delH_in_ga(x1, x2)  =  delH_in_ga(x1)
delH_out_ga(x1, x2)  =  delH_out_ga(x1, x2)
U8_ga(x1, x2, x3)  =  U8_ga(x1, x3)
U21_ggaa(x1, x2, x3, x4, x5)  =  U21_ggaa(x1, x2, x3, x5)
U3_ga(x1, x2, x3, x4)  =  U3_ga(x1, x4)
pD_in_gaaa(x1, x2, x3, x4)  =  pD_in_gaaa(x1)
U22_gaaa(x1, x2, x3, x4, x5)  =  U22_gaaa(x1, x5)
U23_gaaa(x1, x2, x3, x4, x5)  =  U23_gaaa(x1, x2, x5)
pQ_in_ggaa(x1, x2, x3, x4)  =  pQ_in_ggaa(x1, x2)
U24_ggaa(x1, x2, x3, x4, x5)  =  U24_ggaa(x1, x2, x5)
delM_in_gga(x1, x2, x3)  =  delM_in_gga(x1, x2)
delM_out_gga(x1, x2, x3)  =  delM_out_gga(x1, x2, x3)
U13_gga(x1, x2, x3)  =  U13_gga(x1, x3)
U25_ggaa(x1, x2, x3, x4, x5)  =  U25_ggaa(x1, x2, x3, x5)
U4_ga(x1, x2, x3, x4)  =  U4_ga(x1, x4)
pE_in_gaaa(x1, x2, x3, x4)  =  pE_in_gaaa(x1)
U26_gaaa(x1, x2, x3, x4, x5)  =  U26_gaaa(x1, x5)
maxI_in_ga(x1, x2)  =  maxI_in_ga(x1)
maxI_out_ga(x1, x2)  =  maxI_out_ga(x1, x2)
U9_ga(x1, x2, x3)  =  U9_ga(x1, x3)
U10_ga(x1, x2, x3)  =  U10_ga(x1, x3)
U27_gaaa(x1, x2, x3, x4, x5)  =  U27_gaaa(x1, x2, x5)
pR_in_ggaa(x1, x2, x3, x4)  =  pR_in_ggaa(x1, x2)
U28_ggaa(x1, x2, x3, x4, x5)  =  U28_ggaa(x1, x2, x5)
delN_in_gga(x1, x2, x3)  =  delN_in_gga(x1, x2)
delN_out_gga(x1, x2, x3)  =  delN_out_gga(x1, x2, x3)
U14_gga(x1, x2, x3)  =  U14_gga(x1, x3)
delJ_in_ga(x1, x2)  =  delJ_in_ga(x1)
delJ_out_ga(x1, x2)  =  delJ_out_ga(x1, x2)
U11_ga(x1, x2, x3)  =  U11_ga(x1, x3)
U29_ggaa(x1, x2, x3, x4, x5)  =  U29_ggaa(x1, x2, x3, x5)
U5_ga(x1, x2, x3, x4)  =  U5_ga(x1, x4)
pF_in_gaaa(x1, x2, x3, x4)  =  pF_in_gaaa(x1)
U30_gaaa(x1, x2, x3, x4, x5)  =  U30_gaaa(x1, x5)
U31_gaaa(x1, x2, x3, x4, x5)  =  U31_gaaa(x1, x2, x5)
pS_in_ggaa(x1, x2, x3, x4)  =  pS_in_ggaa(x1, x2)
U32_ggaa(x1, x2, x3, x4, x5)  =  U32_ggaa(x1, x2, x5)
delO_in_gga(x1, x2, x3)  =  delO_in_gga(x1, x2)
delO_out_gga(x1, x2, x3)  =  delO_out_gga(x1, x2, x3)
U15_gga(x1, x2, x3)  =  U15_gga(x1, x3)
U33_ggaa(x1, x2, x3, x4, x5)  =  U33_ggaa(x1, x2, x3, x5)
pS_out_ggaa(x1, x2, x3, x4)  =  pS_out_ggaa(x1, x2, x3, x4)
pF_out_gaaa(x1, x2, x3, x4)  =  pF_out_gaaa(x1, x2, x3, x4)
pR_out_ggaa(x1, x2, x3, x4)  =  pR_out_ggaa(x1, x2, x3, x4)
pE_out_gaaa(x1, x2, x3, x4)  =  pE_out_gaaa(x1, x2, x3, x4)
pQ_out_ggaa(x1, x2, x3, x4)  =  pQ_out_ggaa(x1, x2, x3, x4)
pD_out_gaaa(x1, x2, x3, x4)  =  pD_out_gaaa(x1, x2, x3, x4)
pP_out_ggaa(x1, x2, x3, x4)  =  pP_out_ggaa(x1, x2, x3, x4)
pC_out_gaaa(x1, x2, x3, x4)  =  pC_out_gaaa(x1, x2, x3, x4)
pB_out_gaa(x1, x2, x3)  =  pB_out_gaa(x1, x2, x3)
MAXG_IN_GA(x1, x2)  =  MAXG_IN_GA(x1)

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

(22) UsableRulesProof (EQUIVALENT transformation)

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

(23) Obligation:

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

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

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

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

(24) PiDPToQDPProof (SOUND transformation)

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

(25) Obligation:

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

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

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

(26) QDPSizeChangeProof (EQUIVALENT transformation)

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

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

• MAXG_IN_GA(.(f, T72)) → MAXG_IN_GA(T72)
  The graph contains the following edges 1 > 1

• MAXG_IN_GA(.(t, T72)) → MAXG_IN_GA(T72)
  The graph contains the following edges 1 > 1

(27) YES

(28) Obligation:

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

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

The TRS R consists of the following rules:

maxsortA_in_ga([], []) → maxsortA_out_ga([], [])
maxsortA_in_ga(.(T17, []), .(T17, T18)) → U1_ga(T17, T18, pB_in_gaa(T17, X14, T18))
pB_in_gaa(T17, T21, T18) → U16_gaa(T17, T21, T18, delK_in_ga(T17, T21))
delK_in_ga(t, []) → delK_out_ga(t, [])
delK_in_ga(f, []) → delK_out_ga(f, [])
U16_gaa(T17, T21, T18, delK_out_ga(T17, T21)) → U17_gaa(T17, T21, T18, maxsortA_in_ga(T21, T18))
maxsortA_in_ga(.(t, .(t, T53)), .(T55, T56)) → U2_ga(T53, T55, T56, pC_in_gaaa(T53, T55, X14, T56))
pC_in_gaaa(T53, T57, X14, T58) → U18_gaaa(T53, T57, X14, T58, maxG_in_ga(T53, T57))
maxG_in_ga([], t) → maxG_out_ga([], t)
maxG_in_ga(.(t, T72), T74) → U6_ga(T72, T74, maxG_in_ga(T72, T74))
maxG_in_ga(.(f, T72), T74) → U7_ga(T72, T74, maxG_in_ga(T72, T74))
U7_ga(T72, T74, maxG_out_ga(T72, T74)) → maxG_out_ga(.(f, T72), T74)
U6_ga(T72, T74, maxG_out_ga(T72, T74)) → maxG_out_ga(.(t, T72), T74)
U18_gaaa(T53, T57, X14, T58, maxG_out_ga(T53, T57)) → U19_gaaa(T53, T57, X14, T58, pP_in_ggaa(T57, T53, X14, T58))
pP_in_ggaa(T57, T53, T87, T58) → U20_ggaa(T57, T53, T87, T58, delL_in_gga(T57, T53, T87))
delL_in_gga(t, T100, .(t, T100)) → delL_out_gga(t, T100, .(t, T100))
delL_in_gga(f, T118, .(t, .(t, X187))) → U12_gga(T118, X187, delH_in_ga(T118, X187))
delH_in_ga([], []) → delH_out_ga([], [])
delH_in_ga(.(f, T128), T128) → delH_out_ga(.(f, T128), T128)
delH_in_ga(.(t, T134), .(t, X228)) → U8_ga(T134, X228, delH_in_ga(T134, X228))
U8_ga(T134, X228, delH_out_ga(T134, X228)) → delH_out_ga(.(t, T134), .(t, X228))
U12_gga(T118, X187, delH_out_ga(T118, X187)) → delL_out_gga(f, T118, .(t, .(t, X187)))
U20_ggaa(T57, T53, T87, T58, delL_out_gga(T57, T53, T87)) → U21_ggaa(T57, T53, T87, T58, maxsortA_in_ga(T87, T58))
maxsortA_in_ga(.(t, .(f, T53)), .(T55, T56)) → U3_ga(T53, T55, T56, pD_in_gaaa(T53, T55, X14, T56))
pD_in_gaaa(T53, T135, X14, T136) → U22_gaaa(T53, T135, X14, T136, maxG_in_ga(T53, T135))
U22_gaaa(T53, T135, X14, T136, maxG_out_ga(T53, T135)) → U23_gaaa(T53, T135, X14, T136, pQ_in_ggaa(T135, T53, X14, T136))
pQ_in_ggaa(T135, T53, T139, T136) → U24_ggaa(T135, T53, T139, T136, delM_in_gga(T135, T53, T139))
delM_in_gga(t, T152, .(f, T152)) → delM_out_gga(t, T152, .(f, T152))
delM_in_gga(f, T160, .(t, X269)) → U13_gga(T160, X269, delH_in_ga(.(f, T160), X269))
U13_gga(T160, X269, delH_out_ga(.(f, T160), X269)) → delM_out_gga(f, T160, .(t, X269))
U24_ggaa(T135, T53, T139, T136, delM_out_gga(T135, T53, T139)) → U25_ggaa(T135, T53, T139, T136, maxsortA_in_ga(T139, T136))
maxsortA_in_ga(.(f, .(f, T53)), .(T55, T56)) → U4_ga(T53, T55, T56, pE_in_gaaa(T53, T55, X14, T56))
pE_in_gaaa(T53, T163, X14, T164) → U26_gaaa(T53, T163, X14, T164, maxI_in_ga(T53, T163))
maxI_in_ga([], f) → maxI_out_ga([], f)
maxI_in_ga(.(f, T178), T180) → U9_ga(T178, T180, maxI_in_ga(T178, T180))
maxI_in_ga(.(t, T188), T190) → U10_ga(T188, T190, maxG_in_ga(T188, T190))
U10_ga(T188, T190, maxG_out_ga(T188, T190)) → maxI_out_ga(.(t, T188), T190)
U9_ga(T178, T180, maxI_out_ga(T178, T180)) → maxI_out_ga(.(f, T178), T180)
U26_gaaa(T53, T163, X14, T164, maxI_out_ga(T53, T163)) → U27_gaaa(T53, T163, X14, T164, pR_in_ggaa(T163, T53, X14, T164))
pR_in_ggaa(T163, T53, T193, T164) → U28_ggaa(T163, T53, T193, T164, delN_in_gga(T163, T53, T193))
delN_in_gga(f, T206, .(f, T206)) → delN_out_gga(f, T206, .(f, T206))
delN_in_gga(t, T224, .(f, .(f, X384))) → U14_gga(T224, X384, delJ_in_ga(T224, X384))
delJ_in_ga([], []) → delJ_out_ga([], [])
delJ_in_ga(.(t, T234), T234) → delJ_out_ga(.(t, T234), T234)
delJ_in_ga(.(f, T240), .(f, X427)) → U11_ga(T240, X427, delJ_in_ga(T240, X427))
U11_ga(T240, X427, delJ_out_ga(T240, X427)) → delJ_out_ga(.(f, T240), .(f, X427))
U14_gga(T224, X384, delJ_out_ga(T224, X384)) → delN_out_gga(t, T224, .(f, .(f, X384)))
U28_ggaa(T163, T53, T193, T164, delN_out_gga(T163, T53, T193)) → U29_ggaa(T163, T53, T193, T164, maxsortA_in_ga(T193, T164))
maxsortA_in_ga(.(f, .(t, T251)), .(T253, T254)) → U5_ga(T251, T253, T254, pF_in_gaaa(T251, T253, X14, T254))
pF_in_gaaa(T251, T255, X14, T256) → U30_gaaa(T251, T255, X14, T256, maxG_in_ga(T251, T255))
U30_gaaa(T251, T255, X14, T256, maxG_out_ga(T251, T255)) → U31_gaaa(T251, T255, X14, T256, pS_in_ggaa(T255, T251, X14, T256))
pS_in_ggaa(T255, T251, T259, T256) → U32_ggaa(T255, T251, T259, T256, delO_in_gga(T255, T251, T259))
delO_in_gga(f, T272, .(t, T272)) → delO_out_gga(f, T272, .(t, T272))
delO_in_gga(t, T280, .(f, X480)) → U15_gga(T280, X480, delJ_in_ga(.(t, T280), X480))
U15_gga(T280, X480, delJ_out_ga(.(t, T280), X480)) → delO_out_gga(t, T280, .(f, X480))
U32_ggaa(T255, T251, T259, T256, delO_out_gga(T255, T251, T259)) → U33_ggaa(T255, T251, T259, T256, maxsortA_in_ga(T259, T256))
U33_ggaa(T255, T251, T259, T256, maxsortA_out_ga(T259, T256)) → pS_out_ggaa(T255, T251, T259, T256)
U31_gaaa(T251, T255, X14, T256, pS_out_ggaa(T255, T251, X14, T256)) → pF_out_gaaa(T251, T255, X14, T256)
U5_ga(T251, T253, T254, pF_out_gaaa(T251, T253, X14, T254)) → maxsortA_out_ga(.(f, .(t, T251)), .(T253, T254))
U29_ggaa(T163, T53, T193, T164, maxsortA_out_ga(T193, T164)) → pR_out_ggaa(T163, T53, T193, T164)
U27_gaaa(T53, T163, X14, T164, pR_out_ggaa(T163, T53, X14, T164)) → pE_out_gaaa(T53, T163, X14, T164)
U4_ga(T53, T55, T56, pE_out_gaaa(T53, T55, X14, T56)) → maxsortA_out_ga(.(f, .(f, T53)), .(T55, T56))
U25_ggaa(T135, T53, T139, T136, maxsortA_out_ga(T139, T136)) → pQ_out_ggaa(T135, T53, T139, T136)
U23_gaaa(T53, T135, X14, T136, pQ_out_ggaa(T135, T53, X14, T136)) → pD_out_gaaa(T53, T135, X14, T136)
U3_ga(T53, T55, T56, pD_out_gaaa(T53, T55, X14, T56)) → maxsortA_out_ga(.(t, .(f, T53)), .(T55, T56))
U21_ggaa(T57, T53, T87, T58, maxsortA_out_ga(T87, T58)) → pP_out_ggaa(T57, T53, T87, T58)
U19_gaaa(T53, T57, X14, T58, pP_out_ggaa(T57, T53, X14, T58)) → pC_out_gaaa(T53, T57, X14, T58)
U2_ga(T53, T55, T56, pC_out_gaaa(T53, T55, X14, T56)) → maxsortA_out_ga(.(t, .(t, T53)), .(T55, T56))
U17_gaa(T17, T21, T18, maxsortA_out_ga(T21, T18)) → pB_out_gaa(T17, T21, T18)
U1_ga(T17, T18, pB_out_gaa(T17, X14, T18)) → maxsortA_out_ga(.(T17, []), .(T17, T18))

The argument filtering Pi contains the following mapping:
maxsortA_in_ga(x1, x2)  =  maxsortA_in_ga(x1)
[]  =  []
maxsortA_out_ga(x1, x2)  =  maxsortA_out_ga(x1, x2)
.(x1, x2)  =  .(x1, x2)
U1_ga(x1, x2, x3)  =  U1_ga(x1, x3)
pB_in_gaa(x1, x2, x3)  =  pB_in_gaa(x1)
U16_gaa(x1, x2, x3, x4)  =  U16_gaa(x1, x4)
delK_in_ga(x1, x2)  =  delK_in_ga(x1)
t  =  t
delK_out_ga(x1, x2)  =  delK_out_ga(x1, x2)
f  =  f
U17_gaa(x1, x2, x3, x4)  =  U17_gaa(x1, x2, x4)
U2_ga(x1, x2, x3, x4)  =  U2_ga(x1, x4)
pC_in_gaaa(x1, x2, x3, x4)  =  pC_in_gaaa(x1)
U18_gaaa(x1, x2, x3, x4, x5)  =  U18_gaaa(x1, x5)
maxG_in_ga(x1, x2)  =  maxG_in_ga(x1)
maxG_out_ga(x1, x2)  =  maxG_out_ga(x1, x2)
U6_ga(x1, x2, x3)  =  U6_ga(x1, x3)
U7_ga(x1, x2, x3)  =  U7_ga(x1, x3)
U19_gaaa(x1, x2, x3, x4, x5)  =  U19_gaaa(x1, x2, x5)
pP_in_ggaa(x1, x2, x3, x4)  =  pP_in_ggaa(x1, x2)
U20_ggaa(x1, x2, x3, x4, x5)  =  U20_ggaa(x1, x2, x5)
delL_in_gga(x1, x2, x3)  =  delL_in_gga(x1, x2)
delL_out_gga(x1, x2, x3)  =  delL_out_gga(x1, x2, x3)
U12_gga(x1, x2, x3)  =  U12_gga(x1, x3)
delH_in_ga(x1, x2)  =  delH_in_ga(x1)
delH_out_ga(x1, x2)  =  delH_out_ga(x1, x2)
U8_ga(x1, x2, x3)  =  U8_ga(x1, x3)
U21_ggaa(x1, x2, x3, x4, x5)  =  U21_ggaa(x1, x2, x3, x5)
U3_ga(x1, x2, x3, x4)  =  U3_ga(x1, x4)
pD_in_gaaa(x1, x2, x3, x4)  =  pD_in_gaaa(x1)
U22_gaaa(x1, x2, x3, x4, x5)  =  U22_gaaa(x1, x5)
U23_gaaa(x1, x2, x3, x4, x5)  =  U23_gaaa(x1, x2, x5)
pQ_in_ggaa(x1, x2, x3, x4)  =  pQ_in_ggaa(x1, x2)
U24_ggaa(x1, x2, x3, x4, x5)  =  U24_ggaa(x1, x2, x5)
delM_in_gga(x1, x2, x3)  =  delM_in_gga(x1, x2)
delM_out_gga(x1, x2, x3)  =  delM_out_gga(x1, x2, x3)
U13_gga(x1, x2, x3)  =  U13_gga(x1, x3)
U25_ggaa(x1, x2, x3, x4, x5)  =  U25_ggaa(x1, x2, x3, x5)
U4_ga(x1, x2, x3, x4)  =  U4_ga(x1, x4)
pE_in_gaaa(x1, x2, x3, x4)  =  pE_in_gaaa(x1)
U26_gaaa(x1, x2, x3, x4, x5)  =  U26_gaaa(x1, x5)
maxI_in_ga(x1, x2)  =  maxI_in_ga(x1)
maxI_out_ga(x1, x2)  =  maxI_out_ga(x1, x2)
U9_ga(x1, x2, x3)  =  U9_ga(x1, x3)
U10_ga(x1, x2, x3)  =  U10_ga(x1, x3)
U27_gaaa(x1, x2, x3, x4, x5)  =  U27_gaaa(x1, x2, x5)
pR_in_ggaa(x1, x2, x3, x4)  =  pR_in_ggaa(x1, x2)
U28_ggaa(x1, x2, x3, x4, x5)  =  U28_ggaa(x1, x2, x5)
delN_in_gga(x1, x2, x3)  =  delN_in_gga(x1, x2)
delN_out_gga(x1, x2, x3)  =  delN_out_gga(x1, x2, x3)
U14_gga(x1, x2, x3)  =  U14_gga(x1, x3)
delJ_in_ga(x1, x2)  =  delJ_in_ga(x1)
delJ_out_ga(x1, x2)  =  delJ_out_ga(x1, x2)
U11_ga(x1, x2, x3)  =  U11_ga(x1, x3)
U29_ggaa(x1, x2, x3, x4, x5)  =  U29_ggaa(x1, x2, x3, x5)
U5_ga(x1, x2, x3, x4)  =  U5_ga(x1, x4)
pF_in_gaaa(x1, x2, x3, x4)  =  pF_in_gaaa(x1)
U30_gaaa(x1, x2, x3, x4, x5)  =  U30_gaaa(x1, x5)
U31_gaaa(x1, x2, x3, x4, x5)  =  U31_gaaa(x1, x2, x5)
pS_in_ggaa(x1, x2, x3, x4)  =  pS_in_ggaa(x1, x2)
U32_ggaa(x1, x2, x3, x4, x5)  =  U32_ggaa(x1, x2, x5)
delO_in_gga(x1, x2, x3)  =  delO_in_gga(x1, x2)
delO_out_gga(x1, x2, x3)  =  delO_out_gga(x1, x2, x3)
U15_gga(x1, x2, x3)  =  U15_gga(x1, x3)
U33_ggaa(x1, x2, x3, x4, x5)  =  U33_ggaa(x1, x2, x3, x5)
pS_out_ggaa(x1, x2, x3, x4)  =  pS_out_ggaa(x1, x2, x3, x4)
pF_out_gaaa(x1, x2, x3, x4)  =  pF_out_gaaa(x1, x2, x3, x4)
pR_out_ggaa(x1, x2, x3, x4)  =  pR_out_ggaa(x1, x2, x3, x4)
pE_out_gaaa(x1, x2, x3, x4)  =  pE_out_gaaa(x1, x2, x3, x4)
pQ_out_ggaa(x1, x2, x3, x4)  =  pQ_out_ggaa(x1, x2, x3, x4)
pD_out_gaaa(x1, x2, x3, x4)  =  pD_out_gaaa(x1, x2, x3, x4)
pP_out_ggaa(x1, x2, x3, x4)  =  pP_out_ggaa(x1, x2, x3, x4)
pC_out_gaaa(x1, x2, x3, x4)  =  pC_out_gaaa(x1, x2, x3, x4)
pB_out_gaa(x1, x2, x3)  =  pB_out_gaa(x1, x2, x3)
MAXI_IN_GA(x1, x2)  =  MAXI_IN_GA(x1)

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

(29) UsableRulesProof (EQUIVALENT transformation)

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

(30) Obligation:

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

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

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

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

(31) PiDPToQDPProof (SOUND transformation)

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

(32) Obligation:

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

MAXI_IN_GA(.(f, T178)) → MAXI_IN_GA(T178)

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

(33) QDPSizeChangeProof (EQUIVALENT transformation)

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

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

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

(34) YES

(35) Obligation:

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

MAXSORTA_IN_GA(.(T17, []), .(T17, T18)) → PB_IN_GAA(T17, X14, T18)
PB_IN_GAA(T17, T21, T18) → U16_GAA(T17, T21, T18, delK_in_ga(T17, T21))
U16_GAA(T17, T21, T18, delK_out_ga(T17, T21)) → MAXSORTA_IN_GA(T21, T18)
MAXSORTA_IN_GA(.(t, .(t, T53)), .(T55, T56)) → PC_IN_GAAA(T53, T55, X14, T56)
PC_IN_GAAA(T53, T57, X14, T58) → U18_GAAA(T53, T57, X14, T58, maxG_in_ga(T53, T57))
U18_GAAA(T53, T57, X14, T58, maxG_out_ga(T53, T57)) → PP_IN_GGAA(T57, T53, X14, T58)
PP_IN_GGAA(T57, T53, T87, T58) → U20_GGAA(T57, T53, T87, T58, delL_in_gga(T57, T53, T87))
U20_GGAA(T57, T53, T87, T58, delL_out_gga(T57, T53, T87)) → MAXSORTA_IN_GA(T87, T58)
MAXSORTA_IN_GA(.(t, .(f, T53)), .(T55, T56)) → PD_IN_GAAA(T53, T55, X14, T56)
PD_IN_GAAA(T53, T135, X14, T136) → U22_GAAA(T53, T135, X14, T136, maxG_in_ga(T53, T135))
U22_GAAA(T53, T135, X14, T136, maxG_out_ga(T53, T135)) → PQ_IN_GGAA(T135, T53, X14, T136)
PQ_IN_GGAA(T135, T53, T139, T136) → U24_GGAA(T135, T53, T139, T136, delM_in_gga(T135, T53, T139))
U24_GGAA(T135, T53, T139, T136, delM_out_gga(T135, T53, T139)) → MAXSORTA_IN_GA(T139, T136)
MAXSORTA_IN_GA(.(f, .(f, T53)), .(T55, T56)) → PE_IN_GAAA(T53, T55, X14, T56)
PE_IN_GAAA(T53, T163, X14, T164) → U26_GAAA(T53, T163, X14, T164, maxI_in_ga(T53, T163))
U26_GAAA(T53, T163, X14, T164, maxI_out_ga(T53, T163)) → PR_IN_GGAA(T163, T53, X14, T164)
PR_IN_GGAA(T163, T53, T193, T164) → U28_GGAA(T163, T53, T193, T164, delN_in_gga(T163, T53, T193))
U28_GGAA(T163, T53, T193, T164, delN_out_gga(T163, T53, T193)) → MAXSORTA_IN_GA(T193, T164)
MAXSORTA_IN_GA(.(f, .(t, T251)), .(T253, T254)) → PF_IN_GAAA(T251, T253, X14, T254)
PF_IN_GAAA(T251, T255, X14, T256) → U30_GAAA(T251, T255, X14, T256, maxG_in_ga(T251, T255))
U30_GAAA(T251, T255, X14, T256, maxG_out_ga(T251, T255)) → PS_IN_GGAA(T255, T251, X14, T256)
PS_IN_GGAA(T255, T251, T259, T256) → U32_GGAA(T255, T251, T259, T256, delO_in_gga(T255, T251, T259))
U32_GGAA(T255, T251, T259, T256, delO_out_gga(T255, T251, T259)) → MAXSORTA_IN_GA(T259, T256)

The TRS R consists of the following rules:

maxsortA_in_ga([], []) → maxsortA_out_ga([], [])
maxsortA_in_ga(.(T17, []), .(T17, T18)) → U1_ga(T17, T18, pB_in_gaa(T17, X14, T18))
pB_in_gaa(T17, T21, T18) → U16_gaa(T17, T21, T18, delK_in_ga(T17, T21))
delK_in_ga(t, []) → delK_out_ga(t, [])
delK_in_ga(f, []) → delK_out_ga(f, [])
U16_gaa(T17, T21, T18, delK_out_ga(T17, T21)) → U17_gaa(T17, T21, T18, maxsortA_in_ga(T21, T18))
maxsortA_in_ga(.(t, .(t, T53)), .(T55, T56)) → U2_ga(T53, T55, T56, pC_in_gaaa(T53, T55, X14, T56))
pC_in_gaaa(T53, T57, X14, T58) → U18_gaaa(T53, T57, X14, T58, maxG_in_ga(T53, T57))
maxG_in_ga([], t) → maxG_out_ga([], t)
maxG_in_ga(.(t, T72), T74) → U6_ga(T72, T74, maxG_in_ga(T72, T74))
maxG_in_ga(.(f, T72), T74) → U7_ga(T72, T74, maxG_in_ga(T72, T74))
U7_ga(T72, T74, maxG_out_ga(T72, T74)) → maxG_out_ga(.(f, T72), T74)
U6_ga(T72, T74, maxG_out_ga(T72, T74)) → maxG_out_ga(.(t, T72), T74)
U18_gaaa(T53, T57, X14, T58, maxG_out_ga(T53, T57)) → U19_gaaa(T53, T57, X14, T58, pP_in_ggaa(T57, T53, X14, T58))
pP_in_ggaa(T57, T53, T87, T58) → U20_ggaa(T57, T53, T87, T58, delL_in_gga(T57, T53, T87))
delL_in_gga(t, T100, .(t, T100)) → delL_out_gga(t, T100, .(t, T100))
delL_in_gga(f, T118, .(t, .(t, X187))) → U12_gga(T118, X187, delH_in_ga(T118, X187))
delH_in_ga([], []) → delH_out_ga([], [])
delH_in_ga(.(f, T128), T128) → delH_out_ga(.(f, T128), T128)
delH_in_ga(.(t, T134), .(t, X228)) → U8_ga(T134, X228, delH_in_ga(T134, X228))
U8_ga(T134, X228, delH_out_ga(T134, X228)) → delH_out_ga(.(t, T134), .(t, X228))
U12_gga(T118, X187, delH_out_ga(T118, X187)) → delL_out_gga(f, T118, .(t, .(t, X187)))
U20_ggaa(T57, T53, T87, T58, delL_out_gga(T57, T53, T87)) → U21_ggaa(T57, T53, T87, T58, maxsortA_in_ga(T87, T58))
maxsortA_in_ga(.(t, .(f, T53)), .(T55, T56)) → U3_ga(T53, T55, T56, pD_in_gaaa(T53, T55, X14, T56))
pD_in_gaaa(T53, T135, X14, T136) → U22_gaaa(T53, T135, X14, T136, maxG_in_ga(T53, T135))
U22_gaaa(T53, T135, X14, T136, maxG_out_ga(T53, T135)) → U23_gaaa(T53, T135, X14, T136, pQ_in_ggaa(T135, T53, X14, T136))
pQ_in_ggaa(T135, T53, T139, T136) → U24_ggaa(T135, T53, T139, T136, delM_in_gga(T135, T53, T139))
delM_in_gga(t, T152, .(f, T152)) → delM_out_gga(t, T152, .(f, T152))
delM_in_gga(f, T160, .(t, X269)) → U13_gga(T160, X269, delH_in_ga(.(f, T160), X269))
U13_gga(T160, X269, delH_out_ga(.(f, T160), X269)) → delM_out_gga(f, T160, .(t, X269))
U24_ggaa(T135, T53, T139, T136, delM_out_gga(T135, T53, T139)) → U25_ggaa(T135, T53, T139, T136, maxsortA_in_ga(T139, T136))
maxsortA_in_ga(.(f, .(f, T53)), .(T55, T56)) → U4_ga(T53, T55, T56, pE_in_gaaa(T53, T55, X14, T56))
pE_in_gaaa(T53, T163, X14, T164) → U26_gaaa(T53, T163, X14, T164, maxI_in_ga(T53, T163))
maxI_in_ga([], f) → maxI_out_ga([], f)
maxI_in_ga(.(f, T178), T180) → U9_ga(T178, T180, maxI_in_ga(T178, T180))
maxI_in_ga(.(t, T188), T190) → U10_ga(T188, T190, maxG_in_ga(T188, T190))
U10_ga(T188, T190, maxG_out_ga(T188, T190)) → maxI_out_ga(.(t, T188), T190)
U9_ga(T178, T180, maxI_out_ga(T178, T180)) → maxI_out_ga(.(f, T178), T180)
U26_gaaa(T53, T163, X14, T164, maxI_out_ga(T53, T163)) → U27_gaaa(T53, T163, X14, T164, pR_in_ggaa(T163, T53, X14, T164))
pR_in_ggaa(T163, T53, T193, T164) → U28_ggaa(T163, T53, T193, T164, delN_in_gga(T163, T53, T193))
delN_in_gga(f, T206, .(f, T206)) → delN_out_gga(f, T206, .(f, T206))
delN_in_gga(t, T224, .(f, .(f, X384))) → U14_gga(T224, X384, delJ_in_ga(T224, X384))
delJ_in_ga([], []) → delJ_out_ga([], [])
delJ_in_ga(.(t, T234), T234) → delJ_out_ga(.(t, T234), T234)
delJ_in_ga(.(f, T240), .(f, X427)) → U11_ga(T240, X427, delJ_in_ga(T240, X427))
U11_ga(T240, X427, delJ_out_ga(T240, X427)) → delJ_out_ga(.(f, T240), .(f, X427))
U14_gga(T224, X384, delJ_out_ga(T224, X384)) → delN_out_gga(t, T224, .(f, .(f, X384)))
U28_ggaa(T163, T53, T193, T164, delN_out_gga(T163, T53, T193)) → U29_ggaa(T163, T53, T193, T164, maxsortA_in_ga(T193, T164))
maxsortA_in_ga(.(f, .(t, T251)), .(T253, T254)) → U5_ga(T251, T253, T254, pF_in_gaaa(T251, T253, X14, T254))
pF_in_gaaa(T251, T255, X14, T256) → U30_gaaa(T251, T255, X14, T256, maxG_in_ga(T251, T255))
U30_gaaa(T251, T255, X14, T256, maxG_out_ga(T251, T255)) → U31_gaaa(T251, T255, X14, T256, pS_in_ggaa(T255, T251, X14, T256))
pS_in_ggaa(T255, T251, T259, T256) → U32_ggaa(T255, T251, T259, T256, delO_in_gga(T255, T251, T259))
delO_in_gga(f, T272, .(t, T272)) → delO_out_gga(f, T272, .(t, T272))
delO_in_gga(t, T280, .(f, X480)) → U15_gga(T280, X480, delJ_in_ga(.(t, T280), X480))
U15_gga(T280, X480, delJ_out_ga(.(t, T280), X480)) → delO_out_gga(t, T280, .(f, X480))
U32_ggaa(T255, T251, T259, T256, delO_out_gga(T255, T251, T259)) → U33_ggaa(T255, T251, T259, T256, maxsortA_in_ga(T259, T256))
U33_ggaa(T255, T251, T259, T256, maxsortA_out_ga(T259, T256)) → pS_out_ggaa(T255, T251, T259, T256)
U31_gaaa(T251, T255, X14, T256, pS_out_ggaa(T255, T251, X14, T256)) → pF_out_gaaa(T251, T255, X14, T256)
U5_ga(T251, T253, T254, pF_out_gaaa(T251, T253, X14, T254)) → maxsortA_out_ga(.(f, .(t, T251)), .(T253, T254))
U29_ggaa(T163, T53, T193, T164, maxsortA_out_ga(T193, T164)) → pR_out_ggaa(T163, T53, T193, T164)
U27_gaaa(T53, T163, X14, T164, pR_out_ggaa(T163, T53, X14, T164)) → pE_out_gaaa(T53, T163, X14, T164)
U4_ga(T53, T55, T56, pE_out_gaaa(T53, T55, X14, T56)) → maxsortA_out_ga(.(f, .(f, T53)), .(T55, T56))
U25_ggaa(T135, T53, T139, T136, maxsortA_out_ga(T139, T136)) → pQ_out_ggaa(T135, T53, T139, T136)
U23_gaaa(T53, T135, X14, T136, pQ_out_ggaa(T135, T53, X14, T136)) → pD_out_gaaa(T53, T135, X14, T136)
U3_ga(T53, T55, T56, pD_out_gaaa(T53, T55, X14, T56)) → maxsortA_out_ga(.(t, .(f, T53)), .(T55, T56))
U21_ggaa(T57, T53, T87, T58, maxsortA_out_ga(T87, T58)) → pP_out_ggaa(T57, T53, T87, T58)
U19_gaaa(T53, T57, X14, T58, pP_out_ggaa(T57, T53, X14, T58)) → pC_out_gaaa(T53, T57, X14, T58)
U2_ga(T53, T55, T56, pC_out_gaaa(T53, T55, X14, T56)) → maxsortA_out_ga(.(t, .(t, T53)), .(T55, T56))
U17_gaa(T17, T21, T18, maxsortA_out_ga(T21, T18)) → pB_out_gaa(T17, T21, T18)
U1_ga(T17, T18, pB_out_gaa(T17, X14, T18)) → maxsortA_out_ga(.(T17, []), .(T17, T18))

The argument filtering Pi contains the following mapping:
maxsortA_in_ga(x1, x2)  =  maxsortA_in_ga(x1)
[]  =  []
maxsortA_out_ga(x1, x2)  =  maxsortA_out_ga(x1, x2)
.(x1, x2)  =  .(x1, x2)
U1_ga(x1, x2, x3)  =  U1_ga(x1, x3)
pB_in_gaa(x1, x2, x3)  =  pB_in_gaa(x1)
U16_gaa(x1, x2, x3, x4)  =  U16_gaa(x1, x4)
delK_in_ga(x1, x2)  =  delK_in_ga(x1)
t  =  t
delK_out_ga(x1, x2)  =  delK_out_ga(x1, x2)
f  =  f
U17_gaa(x1, x2, x3, x4)  =  U17_gaa(x1, x2, x4)
U2_ga(x1, x2, x3, x4)  =  U2_ga(x1, x4)
pC_in_gaaa(x1, x2, x3, x4)  =  pC_in_gaaa(x1)
U18_gaaa(x1, x2, x3, x4, x5)  =  U18_gaaa(x1, x5)
maxG_in_ga(x1, x2)  =  maxG_in_ga(x1)
maxG_out_ga(x1, x2)  =  maxG_out_ga(x1, x2)
U6_ga(x1, x2, x3)  =  U6_ga(x1, x3)
U7_ga(x1, x2, x3)  =  U7_ga(x1, x3)
U19_gaaa(x1, x2, x3, x4, x5)  =  U19_gaaa(x1, x2, x5)
pP_in_ggaa(x1, x2, x3, x4)  =  pP_in_ggaa(x1, x2)
U20_ggaa(x1, x2, x3, x4, x5)  =  U20_ggaa(x1, x2, x5)
delL_in_gga(x1, x2, x3)  =  delL_in_gga(x1, x2)
delL_out_gga(x1, x2, x3)  =  delL_out_gga(x1, x2, x3)
U12_gga(x1, x2, x3)  =  U12_gga(x1, x3)
delH_in_ga(x1, x2)  =  delH_in_ga(x1)
delH_out_ga(x1, x2)  =  delH_out_ga(x1, x2)
U8_ga(x1, x2, x3)  =  U8_ga(x1, x3)
U21_ggaa(x1, x2, x3, x4, x5)  =  U21_ggaa(x1, x2, x3, x5)
U3_ga(x1, x2, x3, x4)  =  U3_ga(x1, x4)
pD_in_gaaa(x1, x2, x3, x4)  =  pD_in_gaaa(x1)
U22_gaaa(x1, x2, x3, x4, x5)  =  U22_gaaa(x1, x5)
U23_gaaa(x1, x2, x3, x4, x5)  =  U23_gaaa(x1, x2, x5)
pQ_in_ggaa(x1, x2, x3, x4)  =  pQ_in_ggaa(x1, x2)
U24_ggaa(x1, x2, x3, x4, x5)  =  U24_ggaa(x1, x2, x5)
delM_in_gga(x1, x2, x3)  =  delM_in_gga(x1, x2)
delM_out_gga(x1, x2, x3)  =  delM_out_gga(x1, x2, x3)
U13_gga(x1, x2, x3)  =  U13_gga(x1, x3)
U25_ggaa(x1, x2, x3, x4, x5)  =  U25_ggaa(x1, x2, x3, x5)
U4_ga(x1, x2, x3, x4)  =  U4_ga(x1, x4)
pE_in_gaaa(x1, x2, x3, x4)  =  pE_in_gaaa(x1)
U26_gaaa(x1, x2, x3, x4, x5)  =  U26_gaaa(x1, x5)
maxI_in_ga(x1, x2)  =  maxI_in_ga(x1)
maxI_out_ga(x1, x2)  =  maxI_out_ga(x1, x2)
U9_ga(x1, x2, x3)  =  U9_ga(x1, x3)
U10_ga(x1, x2, x3)  =  U10_ga(x1, x3)
U27_gaaa(x1, x2, x3, x4, x5)  =  U27_gaaa(x1, x2, x5)
pR_in_ggaa(x1, x2, x3, x4)  =  pR_in_ggaa(x1, x2)
U28_ggaa(x1, x2, x3, x4, x5)  =  U28_ggaa(x1, x2, x5)
delN_in_gga(x1, x2, x3)  =  delN_in_gga(x1, x2)
delN_out_gga(x1, x2, x3)  =  delN_out_gga(x1, x2, x3)
U14_gga(x1, x2, x3)  =  U14_gga(x1, x3)
delJ_in_ga(x1, x2)  =  delJ_in_ga(x1)
delJ_out_ga(x1, x2)  =  delJ_out_ga(x1, x2)
U11_ga(x1, x2, x3)  =  U11_ga(x1, x3)
U29_ggaa(x1, x2, x3, x4, x5)  =  U29_ggaa(x1, x2, x3, x5)
U5_ga(x1, x2, x3, x4)  =  U5_ga(x1, x4)
pF_in_gaaa(x1, x2, x3, x4)  =  pF_in_gaaa(x1)
U30_gaaa(x1, x2, x3, x4, x5)  =  U30_gaaa(x1, x5)
U31_gaaa(x1, x2, x3, x4, x5)  =  U31_gaaa(x1, x2, x5)
pS_in_ggaa(x1, x2, x3, x4)  =  pS_in_ggaa(x1, x2)
U32_ggaa(x1, x2, x3, x4, x5)  =  U32_ggaa(x1, x2, x5)
delO_in_gga(x1, x2, x3)  =  delO_in_gga(x1, x2)
delO_out_gga(x1, x2, x3)  =  delO_out_gga(x1, x2, x3)
U15_gga(x1, x2, x3)  =  U15_gga(x1, x3)
U33_ggaa(x1, x2, x3, x4, x5)  =  U33_ggaa(x1, x2, x3, x5)
pS_out_ggaa(x1, x2, x3, x4)  =  pS_out_ggaa(x1, x2, x3, x4)
pF_out_gaaa(x1, x2, x3, x4)  =  pF_out_gaaa(x1, x2, x3, x4)
pR_out_ggaa(x1, x2, x3, x4)  =  pR_out_ggaa(x1, x2, x3, x4)
pE_out_gaaa(x1, x2, x3, x4)  =  pE_out_gaaa(x1, x2, x3, x4)
pQ_out_ggaa(x1, x2, x3, x4)  =  pQ_out_ggaa(x1, x2, x3, x4)
pD_out_gaaa(x1, x2, x3, x4)  =  pD_out_gaaa(x1, x2, x3, x4)
pP_out_ggaa(x1, x2, x3, x4)  =  pP_out_ggaa(x1, x2, x3, x4)
pC_out_gaaa(x1, x2, x3, x4)  =  pC_out_gaaa(x1, x2, x3, x4)
pB_out_gaa(x1, x2, x3)  =  pB_out_gaa(x1, x2, x3)
MAXSORTA_IN_GA(x1, x2)  =  MAXSORTA_IN_GA(x1)
PB_IN_GAA(x1, x2, x3)  =  PB_IN_GAA(x1)
U16_GAA(x1, x2, x3, x4)  =  U16_GAA(x1, x4)
PC_IN_GAAA(x1, x2, x3, x4)  =  PC_IN_GAAA(x1)
U18_GAAA(x1, x2, x3, x4, x5)  =  U18_GAAA(x1, x5)
PP_IN_GGAA(x1, x2, x3, x4)  =  PP_IN_GGAA(x1, x2)
U20_GGAA(x1, x2, x3, x4, x5)  =  U20_GGAA(x1, x2, x5)
PD_IN_GAAA(x1, x2, x3, x4)  =  PD_IN_GAAA(x1)
U22_GAAA(x1, x2, x3, x4, x5)  =  U22_GAAA(x1, x5)
PQ_IN_GGAA(x1, x2, x3, x4)  =  PQ_IN_GGAA(x1, x2)
U24_GGAA(x1, x2, x3, x4, x5)  =  U24_GGAA(x1, x2, x5)
PE_IN_GAAA(x1, x2, x3, x4)  =  PE_IN_GAAA(x1)
U26_GAAA(x1, x2, x3, x4, x5)  =  U26_GAAA(x1, x5)
PR_IN_GGAA(x1, x2, x3, x4)  =  PR_IN_GGAA(x1, x2)
U28_GGAA(x1, x2, x3, x4, x5)  =  U28_GGAA(x1, x2, x5)
PF_IN_GAAA(x1, x2, x3, x4)  =  PF_IN_GAAA(x1)
U30_GAAA(x1, x2, x3, x4, x5)  =  U30_GAAA(x1, x5)
PS_IN_GGAA(x1, x2, x3, x4)  =  PS_IN_GGAA(x1, x2)
U32_GGAA(x1, x2, x3, x4, x5)  =  U32_GGAA(x1, x2, x5)

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

(36) UsableRulesProof (EQUIVALENT transformation)

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

(37) Obligation:

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

MAXSORTA_IN_GA(.(T17, []), .(T17, T18)) → PB_IN_GAA(T17, X14, T18)
PB_IN_GAA(T17, T21, T18) → U16_GAA(T17, T21, T18, delK_in_ga(T17, T21))
U16_GAA(T17, T21, T18, delK_out_ga(T17, T21)) → MAXSORTA_IN_GA(T21, T18)
MAXSORTA_IN_GA(.(t, .(t, T53)), .(T55, T56)) → PC_IN_GAAA(T53, T55, X14, T56)
PC_IN_GAAA(T53, T57, X14, T58) → U18_GAAA(T53, T57, X14, T58, maxG_in_ga(T53, T57))
U18_GAAA(T53, T57, X14, T58, maxG_out_ga(T53, T57)) → PP_IN_GGAA(T57, T53, X14, T58)
PP_IN_GGAA(T57, T53, T87, T58) → U20_GGAA(T57, T53, T87, T58, delL_in_gga(T57, T53, T87))
U20_GGAA(T57, T53, T87, T58, delL_out_gga(T57, T53, T87)) → MAXSORTA_IN_GA(T87, T58)
MAXSORTA_IN_GA(.(t, .(f, T53)), .(T55, T56)) → PD_IN_GAAA(T53, T55, X14, T56)
PD_IN_GAAA(T53, T135, X14, T136) → U22_GAAA(T53, T135, X14, T136, maxG_in_ga(T53, T135))
U22_GAAA(T53, T135, X14, T136, maxG_out_ga(T53, T135)) → PQ_IN_GGAA(T135, T53, X14, T136)
PQ_IN_GGAA(T135, T53, T139, T136) → U24_GGAA(T135, T53, T139, T136, delM_in_gga(T135, T53, T139))
U24_GGAA(T135, T53, T139, T136, delM_out_gga(T135, T53, T139)) → MAXSORTA_IN_GA(T139, T136)
MAXSORTA_IN_GA(.(f, .(f, T53)), .(T55, T56)) → PE_IN_GAAA(T53, T55, X14, T56)
PE_IN_GAAA(T53, T163, X14, T164) → U26_GAAA(T53, T163, X14, T164, maxI_in_ga(T53, T163))
U26_GAAA(T53, T163, X14, T164, maxI_out_ga(T53, T163)) → PR_IN_GGAA(T163, T53, X14, T164)
PR_IN_GGAA(T163, T53, T193, T164) → U28_GGAA(T163, T53, T193, T164, delN_in_gga(T163, T53, T193))
U28_GGAA(T163, T53, T193, T164, delN_out_gga(T163, T53, T193)) → MAXSORTA_IN_GA(T193, T164)
MAXSORTA_IN_GA(.(f, .(t, T251)), .(T253, T254)) → PF_IN_GAAA(T251, T253, X14, T254)
PF_IN_GAAA(T251, T255, X14, T256) → U30_GAAA(T251, T255, X14, T256, maxG_in_ga(T251, T255))
U30_GAAA(T251, T255, X14, T256, maxG_out_ga(T251, T255)) → PS_IN_GGAA(T255, T251, X14, T256)
PS_IN_GGAA(T255, T251, T259, T256) → U32_GGAA(T255, T251, T259, T256, delO_in_gga(T255, T251, T259))
U32_GGAA(T255, T251, T259, T256, delO_out_gga(T255, T251, T259)) → MAXSORTA_IN_GA(T259, T256)

The TRS R consists of the following rules:

delK_in_ga(t, []) → delK_out_ga(t, [])
delK_in_ga(f, []) → delK_out_ga(f, [])
maxG_in_ga([], t) → maxG_out_ga([], t)
maxG_in_ga(.(t, T72), T74) → U6_ga(T72, T74, maxG_in_ga(T72, T74))
maxG_in_ga(.(f, T72), T74) → U7_ga(T72, T74, maxG_in_ga(T72, T74))
delL_in_gga(t, T100, .(t, T100)) → delL_out_gga(t, T100, .(t, T100))
delL_in_gga(f, T118, .(t, .(t, X187))) → U12_gga(T118, X187, delH_in_ga(T118, X187))
delM_in_gga(t, T152, .(f, T152)) → delM_out_gga(t, T152, .(f, T152))
delM_in_gga(f, T160, .(t, X269)) → U13_gga(T160, X269, delH_in_ga(.(f, T160), X269))
maxI_in_ga([], f) → maxI_out_ga([], f)
maxI_in_ga(.(f, T178), T180) → U9_ga(T178, T180, maxI_in_ga(T178, T180))
maxI_in_ga(.(t, T188), T190) → U10_ga(T188, T190, maxG_in_ga(T188, T190))
delN_in_gga(f, T206, .(f, T206)) → delN_out_gga(f, T206, .(f, T206))
delN_in_gga(t, T224, .(f, .(f, X384))) → U14_gga(T224, X384, delJ_in_ga(T224, X384))
delO_in_gga(f, T272, .(t, T272)) → delO_out_gga(f, T272, .(t, T272))
delO_in_gga(t, T280, .(f, X480)) → U15_gga(T280, X480, delJ_in_ga(.(t, T280), X480))
U6_ga(T72, T74, maxG_out_ga(T72, T74)) → maxG_out_ga(.(t, T72), T74)
U7_ga(T72, T74, maxG_out_ga(T72, T74)) → maxG_out_ga(.(f, T72), T74)
U12_gga(T118, X187, delH_out_ga(T118, X187)) → delL_out_gga(f, T118, .(t, .(t, X187)))
U13_gga(T160, X269, delH_out_ga(.(f, T160), X269)) → delM_out_gga(f, T160, .(t, X269))
U9_ga(T178, T180, maxI_out_ga(T178, T180)) → maxI_out_ga(.(f, T178), T180)
U10_ga(T188, T190, maxG_out_ga(T188, T190)) → maxI_out_ga(.(t, T188), T190)
U14_gga(T224, X384, delJ_out_ga(T224, X384)) → delN_out_gga(t, T224, .(f, .(f, X384)))
U15_gga(T280, X480, delJ_out_ga(.(t, T280), X480)) → delO_out_gga(t, T280, .(f, X480))
delH_in_ga([], []) → delH_out_ga([], [])
delH_in_ga(.(f, T128), T128) → delH_out_ga(.(f, T128), T128)
delH_in_ga(.(t, T134), .(t, X228)) → U8_ga(T134, X228, delH_in_ga(T134, X228))
delJ_in_ga([], []) → delJ_out_ga([], [])
delJ_in_ga(.(t, T234), T234) → delJ_out_ga(.(t, T234), T234)
delJ_in_ga(.(f, T240), .(f, X427)) → U11_ga(T240, X427, delJ_in_ga(T240, X427))
U8_ga(T134, X228, delH_out_ga(T134, X228)) → delH_out_ga(.(t, T134), .(t, X228))
U11_ga(T240, X427, delJ_out_ga(T240, X427)) → delJ_out_ga(.(f, T240), .(f, X427))

The argument filtering Pi contains the following mapping:
[]  =  []
.(x1, x2)  =  .(x1, x2)
delK_in_ga(x1, x2)  =  delK_in_ga(x1)
t  =  t
delK_out_ga(x1, x2)  =  delK_out_ga(x1, x2)
f  =  f
maxG_in_ga(x1, x2)  =  maxG_in_ga(x1)
maxG_out_ga(x1, x2)  =  maxG_out_ga(x1, x2)
U6_ga(x1, x2, x3)  =  U6_ga(x1, x3)
U7_ga(x1, x2, x3)  =  U7_ga(x1, x3)
delL_in_gga(x1, x2, x3)  =  delL_in_gga(x1, x2)
delL_out_gga(x1, x2, x3)  =  delL_out_gga(x1, x2, x3)
U12_gga(x1, x2, x3)  =  U12_gga(x1, x3)
delH_in_ga(x1, x2)  =  delH_in_ga(x1)
delH_out_ga(x1, x2)  =  delH_out_ga(x1, x2)
U8_ga(x1, x2, x3)  =  U8_ga(x1, x3)
delM_in_gga(x1, x2, x3)  =  delM_in_gga(x1, x2)
delM_out_gga(x1, x2, x3)  =  delM_out_gga(x1, x2, x3)
U13_gga(x1, x2, x3)  =  U13_gga(x1, x3)
maxI_in_ga(x1, x2)  =  maxI_in_ga(x1)
maxI_out_ga(x1, x2)  =  maxI_out_ga(x1, x2)
U9_ga(x1, x2, x3)  =  U9_ga(x1, x3)
U10_ga(x1, x2, x3)  =  U10_ga(x1, x3)
delN_in_gga(x1, x2, x3)  =  delN_in_gga(x1, x2)
delN_out_gga(x1, x2, x3)  =  delN_out_gga(x1, x2, x3)
U14_gga(x1, x2, x3)  =  U14_gga(x1, x3)
delJ_in_ga(x1, x2)  =  delJ_in_ga(x1)
delJ_out_ga(x1, x2)  =  delJ_out_ga(x1, x2)
U11_ga(x1, x2, x3)  =  U11_ga(x1, x3)
delO_in_gga(x1, x2, x3)  =  delO_in_gga(x1, x2)
delO_out_gga(x1, x2, x3)  =  delO_out_gga(x1, x2, x3)
U15_gga(x1, x2, x3)  =  U15_gga(x1, x3)
MAXSORTA_IN_GA(x1, x2)  =  MAXSORTA_IN_GA(x1)
PB_IN_GAA(x1, x2, x3)  =  PB_IN_GAA(x1)
U16_GAA(x1, x2, x3, x4)  =  U16_GAA(x1, x4)
PC_IN_GAAA(x1, x2, x3, x4)  =  PC_IN_GAAA(x1)
U18_GAAA(x1, x2, x3, x4, x5)  =  U18_GAAA(x1, x5)
PP_IN_GGAA(x1, x2, x3, x4)  =  PP_IN_GGAA(x1, x2)
U20_GGAA(x1, x2, x3, x4, x5)  =  U20_GGAA(x1, x2, x5)
PD_IN_GAAA(x1, x2, x3, x4)  =  PD_IN_GAAA(x1)
U22_GAAA(x1, x2, x3, x4, x5)  =  U22_GAAA(x1, x5)
PQ_IN_GGAA(x1, x2, x3, x4)  =  PQ_IN_GGAA(x1, x2)
U24_GGAA(x1, x2, x3, x4, x5)  =  U24_GGAA(x1, x2, x5)
PE_IN_GAAA(x1, x2, x3, x4)  =  PE_IN_GAAA(x1)
U26_GAAA(x1, x2, x3, x4, x5)  =  U26_GAAA(x1, x5)
PR_IN_GGAA(x1, x2, x3, x4)  =  PR_IN_GGAA(x1, x2)
U28_GGAA(x1, x2, x3, x4, x5)  =  U28_GGAA(x1, x2, x5)
PF_IN_GAAA(x1, x2, x3, x4)  =  PF_IN_GAAA(x1)
U30_GAAA(x1, x2, x3, x4, x5)  =  U30_GAAA(x1, x5)
PS_IN_GGAA(x1, x2, x3, x4)  =  PS_IN_GGAA(x1, x2)
U32_GGAA(x1, x2, x3, x4, x5)  =  U32_GGAA(x1, x2, x5)

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

(38) PiDPToQDPProof (SOUND transformation)

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

(39) Obligation:

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

MAXSORTA_IN_GA(.(T17, [])) → PB_IN_GAA(T17)
PB_IN_GAA(T17) → U16_GAA(T17, delK_in_ga(T17))
U16_GAA(T17, delK_out_ga(T17, T21)) → MAXSORTA_IN_GA(T21)
MAXSORTA_IN_GA(.(t, .(t, T53))) → PC_IN_GAAA(T53)
PC_IN_GAAA(T53) → U18_GAAA(T53, maxG_in_ga(T53))
U18_GAAA(T53, maxG_out_ga(T53, T57)) → PP_IN_GGAA(T57, T53)
PP_IN_GGAA(T57, T53) → U20_GGAA(T57, T53, delL_in_gga(T57, T53))
U20_GGAA(T57, T53, delL_out_gga(T57, T53, T87)) → MAXSORTA_IN_GA(T87)
MAXSORTA_IN_GA(.(t, .(f, T53))) → PD_IN_GAAA(T53)
PD_IN_GAAA(T53) → U22_GAAA(T53, maxG_in_ga(T53))
U22_GAAA(T53, maxG_out_ga(T53, T135)) → PQ_IN_GGAA(T135, T53)
PQ_IN_GGAA(T135, T53) → U24_GGAA(T135, T53, delM_in_gga(T135, T53))
U24_GGAA(T135, T53, delM_out_gga(T135, T53, T139)) → MAXSORTA_IN_GA(T139)
MAXSORTA_IN_GA(.(f, .(f, T53))) → PE_IN_GAAA(T53)
PE_IN_GAAA(T53) → U26_GAAA(T53, maxI_in_ga(T53))
U26_GAAA(T53, maxI_out_ga(T53, T163)) → PR_IN_GGAA(T163, T53)
PR_IN_GGAA(T163, T53) → U28_GGAA(T163, T53, delN_in_gga(T163, T53))
U28_GGAA(T163, T53, delN_out_gga(T163, T53, T193)) → MAXSORTA_IN_GA(T193)
MAXSORTA_IN_GA(.(f, .(t, T251))) → PF_IN_GAAA(T251)
PF_IN_GAAA(T251) → U30_GAAA(T251, maxG_in_ga(T251))
U30_GAAA(T251, maxG_out_ga(T251, T255)) → PS_IN_GGAA(T255, T251)
PS_IN_GGAA(T255, T251) → U32_GGAA(T255, T251, delO_in_gga(T255, T251))
U32_GGAA(T255, T251, delO_out_gga(T255, T251, T259)) → MAXSORTA_IN_GA(T259)

The TRS R consists of the following rules:

delK_in_ga(t) → delK_out_ga(t, [])
delK_in_ga(f) → delK_out_ga(f, [])
maxG_in_ga([]) → maxG_out_ga([], t)
maxG_in_ga(.(t, T72)) → U6_ga(T72, maxG_in_ga(T72))
maxG_in_ga(.(f, T72)) → U7_ga(T72, maxG_in_ga(T72))
delL_in_gga(t, T100) → delL_out_gga(t, T100, .(t, T100))
delL_in_gga(f, T118) → U12_gga(T118, delH_in_ga(T118))
delM_in_gga(t, T152) → delM_out_gga(t, T152, .(f, T152))
delM_in_gga(f, T160) → U13_gga(T160, delH_in_ga(.(f, T160)))
maxI_in_ga([]) → maxI_out_ga([], f)
maxI_in_ga(.(f, T178)) → U9_ga(T178, maxI_in_ga(T178))
maxI_in_ga(.(t, T188)) → U10_ga(T188, maxG_in_ga(T188))
delN_in_gga(f, T206) → delN_out_gga(f, T206, .(f, T206))
delN_in_gga(t, T224) → U14_gga(T224, delJ_in_ga(T224))
delO_in_gga(f, T272) → delO_out_gga(f, T272, .(t, T272))
delO_in_gga(t, T280) → U15_gga(T280, delJ_in_ga(.(t, T280)))
U6_ga(T72, maxG_out_ga(T72, T74)) → maxG_out_ga(.(t, T72), T74)
U7_ga(T72, maxG_out_ga(T72, T74)) → maxG_out_ga(.(f, T72), T74)
U12_gga(T118, delH_out_ga(T118, X187)) → delL_out_gga(f, T118, .(t, .(t, X187)))
U13_gga(T160, delH_out_ga(.(f, T160), X269)) → delM_out_gga(f, T160, .(t, X269))
U9_ga(T178, maxI_out_ga(T178, T180)) → maxI_out_ga(.(f, T178), T180)
U10_ga(T188, maxG_out_ga(T188, T190)) → maxI_out_ga(.(t, T188), T190)
U14_gga(T224, delJ_out_ga(T224, X384)) → delN_out_gga(t, T224, .(f, .(f, X384)))
U15_gga(T280, delJ_out_ga(.(t, T280), X480)) → delO_out_gga(t, T280, .(f, X480))
delH_in_ga([]) → delH_out_ga([], [])
delH_in_ga(.(f, T128)) → delH_out_ga(.(f, T128), T128)
delH_in_ga(.(t, T134)) → U8_ga(T134, delH_in_ga(T134))
delJ_in_ga([]) → delJ_out_ga([], [])
delJ_in_ga(.(t, T234)) → delJ_out_ga(.(t, T234), T234)
delJ_in_ga(.(f, T240)) → U11_ga(T240, delJ_in_ga(T240))
U8_ga(T134, delH_out_ga(T134, X228)) → delH_out_ga(.(t, T134), .(t, X228))
U11_ga(T240, delJ_out_ga(T240, X427)) → delJ_out_ga(.(f, T240), .(f, X427))

The set Q consists of the following terms:

delK_in_ga(x0)
maxG_in_ga(x0)
delL_in_gga(x0, x1)
delM_in_gga(x0, x1)
maxI_in_ga(x0)
delN_in_gga(x0, x1)
delO_in_gga(x0, x1)
U6_ga(x0, x1)
U7_ga(x0, x1)
U12_gga(x0, x1)
U13_gga(x0, x1)
U9_ga(x0, x1)
U10_ga(x0, x1)
U14_gga(x0, x1)
U15_gga(x0, x1)
delH_in_ga(x0)
delJ_in_ga(x0)
U8_ga(x0, x1)
U11_ga(x0, x1)

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

(40) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule PB_IN_GAA(T17) → U16_GAA(T17, delK_in_ga(T17)) at position [1] we obtained the following new rules [LPAR04]:

PB_IN_GAA(t) → U16_GAA(t, delK_out_ga(t, []))
PB_IN_GAA(f) → U16_GAA(f, delK_out_ga(f, []))

(41) Obligation:

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

MAXSORTA_IN_GA(.(T17, [])) → PB_IN_GAA(T17)
U16_GAA(T17, delK_out_ga(T17, T21)) → MAXSORTA_IN_GA(T21)
MAXSORTA_IN_GA(.(t, .(t, T53))) → PC_IN_GAAA(T53)
PC_IN_GAAA(T53) → U18_GAAA(T53, maxG_in_ga(T53))
U18_GAAA(T53, maxG_out_ga(T53, T57)) → PP_IN_GGAA(T57, T53)
PP_IN_GGAA(T57, T53) → U20_GGAA(T57, T53, delL_in_gga(T57, T53))
U20_GGAA(T57, T53, delL_out_gga(T57, T53, T87)) → MAXSORTA_IN_GA(T87)
MAXSORTA_IN_GA(.(t, .(f, T53))) → PD_IN_GAAA(T53)
PD_IN_GAAA(T53) → U22_GAAA(T53, maxG_in_ga(T53))
U22_GAAA(T53, maxG_out_ga(T53, T135)) → PQ_IN_GGAA(T135, T53)
PQ_IN_GGAA(T135, T53) → U24_GGAA(T135, T53, delM_in_gga(T135, T53))
U24_GGAA(T135, T53, delM_out_gga(T135, T53, T139)) → MAXSORTA_IN_GA(T139)
MAXSORTA_IN_GA(.(f, .(f, T53))) → PE_IN_GAAA(T53)
PE_IN_GAAA(T53) → U26_GAAA(T53, maxI_in_ga(T53))
U26_GAAA(T53, maxI_out_ga(T53, T163)) → PR_IN_GGAA(T163, T53)
PR_IN_GGAA(T163, T53) → U28_GGAA(T163, T53, delN_in_gga(T163, T53))
U28_GGAA(T163, T53, delN_out_gga(T163, T53, T193)) → MAXSORTA_IN_GA(T193)
MAXSORTA_IN_GA(.(f, .(t, T251))) → PF_IN_GAAA(T251)
PF_IN_GAAA(T251) → U30_GAAA(T251, maxG_in_ga(T251))
U30_GAAA(T251, maxG_out_ga(T251, T255)) → PS_IN_GGAA(T255, T251)
PS_IN_GGAA(T255, T251) → U32_GGAA(T255, T251, delO_in_gga(T255, T251))
U32_GGAA(T255, T251, delO_out_gga(T255, T251, T259)) → MAXSORTA_IN_GA(T259)
PB_IN_GAA(t) → U16_GAA(t, delK_out_ga(t, []))
PB_IN_GAA(f) → U16_GAA(f, delK_out_ga(f, []))

The TRS R consists of the following rules:

delK_in_ga(t) → delK_out_ga(t, [])
delK_in_ga(f) → delK_out_ga(f, [])
maxG_in_ga([]) → maxG_out_ga([], t)
maxG_in_ga(.(t, T72)) → U6_ga(T72, maxG_in_ga(T72))
maxG_in_ga(.(f, T72)) → U7_ga(T72, maxG_in_ga(T72))
delL_in_gga(t, T100) → delL_out_gga(t, T100, .(t, T100))
delL_in_gga(f, T118) → U12_gga(T118, delH_in_ga(T118))
delM_in_gga(t, T152) → delM_out_gga(t, T152, .(f, T152))
delM_in_gga(f, T160) → U13_gga(T160, delH_in_ga(.(f, T160)))
maxI_in_ga([]) → maxI_out_ga([], f)
maxI_in_ga(.(f, T178)) → U9_ga(T178, maxI_in_ga(T178))
maxI_in_ga(.(t, T188)) → U10_ga(T188, maxG_in_ga(T188))
delN_in_gga(f, T206) → delN_out_gga(f, T206, .(f, T206))
delN_in_gga(t, T224) → U14_gga(T224, delJ_in_ga(T224))
delO_in_gga(f, T272) → delO_out_gga(f, T272, .(t, T272))
delO_in_gga(t, T280) → U15_gga(T280, delJ_in_ga(.(t, T280)))
U6_ga(T72, maxG_out_ga(T72, T74)) → maxG_out_ga(.(t, T72), T74)
U7_ga(T72, maxG_out_ga(T72, T74)) → maxG_out_ga(.(f, T72), T74)
U12_gga(T118, delH_out_ga(T118, X187)) → delL_out_gga(f, T118, .(t, .(t, X187)))
U13_gga(T160, delH_out_ga(.(f, T160), X269)) → delM_out_gga(f, T160, .(t, X269))
U9_ga(T178, maxI_out_ga(T178, T180)) → maxI_out_ga(.(f, T178), T180)
U10_ga(T188, maxG_out_ga(T188, T190)) → maxI_out_ga(.(t, T188), T190)
U14_gga(T224, delJ_out_ga(T224, X384)) → delN_out_gga(t, T224, .(f, .(f, X384)))
U15_gga(T280, delJ_out_ga(.(t, T280), X480)) → delO_out_gga(t, T280, .(f, X480))
delH_in_ga([]) → delH_out_ga([], [])
delH_in_ga(.(f, T128)) → delH_out_ga(.(f, T128), T128)
delH_in_ga(.(t, T134)) → U8_ga(T134, delH_in_ga(T134))
delJ_in_ga([]) → delJ_out_ga([], [])
delJ_in_ga(.(t, T234)) → delJ_out_ga(.(t, T234), T234)
delJ_in_ga(.(f, T240)) → U11_ga(T240, delJ_in_ga(T240))
U8_ga(T134, delH_out_ga(T134, X228)) → delH_out_ga(.(t, T134), .(t, X228))
U11_ga(T240, delJ_out_ga(T240, X427)) → delJ_out_ga(.(f, T240), .(f, X427))

The set Q consists of the following terms:

delK_in_ga(x0)
maxG_in_ga(x0)
delL_in_gga(x0, x1)
delM_in_gga(x0, x1)
maxI_in_ga(x0)
delN_in_gga(x0, x1)
delO_in_gga(x0, x1)
U6_ga(x0, x1)
U7_ga(x0, x1)
U12_gga(x0, x1)
U13_gga(x0, x1)
U9_ga(x0, x1)
U10_ga(x0, x1)
U14_gga(x0, x1)
U15_gga(x0, x1)
delH_in_ga(x0)
delJ_in_ga(x0)
U8_ga(x0, x1)
U11_ga(x0, x1)

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

(42) UsableRulesProof (EQUIVALENT transformation)

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

(43) Obligation:

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

MAXSORTA_IN_GA(.(T17, [])) → PB_IN_GAA(T17)
U16_GAA(T17, delK_out_ga(T17, T21)) → MAXSORTA_IN_GA(T21)
MAXSORTA_IN_GA(.(t, .(t, T53))) → PC_IN_GAAA(T53)
PC_IN_GAAA(T53) → U18_GAAA(T53, maxG_in_ga(T53))
U18_GAAA(T53, maxG_out_ga(T53, T57)) → PP_IN_GGAA(T57, T53)
PP_IN_GGAA(T57, T53) → U20_GGAA(T57, T53, delL_in_gga(T57, T53))
U20_GGAA(T57, T53, delL_out_gga(T57, T53, T87)) → MAXSORTA_IN_GA(T87)
MAXSORTA_IN_GA(.(t, .(f, T53))) → PD_IN_GAAA(T53)
PD_IN_GAAA(T53) → U22_GAAA(T53, maxG_in_ga(T53))
U22_GAAA(T53, maxG_out_ga(T53, T135)) → PQ_IN_GGAA(T135, T53)
PQ_IN_GGAA(T135, T53) → U24_GGAA(T135, T53, delM_in_gga(T135, T53))
U24_GGAA(T135, T53, delM_out_gga(T135, T53, T139)) → MAXSORTA_IN_GA(T139)
MAXSORTA_IN_GA(.(f, .(f, T53))) → PE_IN_GAAA(T53)
PE_IN_GAAA(T53) → U26_GAAA(T53, maxI_in_ga(T53))
U26_GAAA(T53, maxI_out_ga(T53, T163)) → PR_IN_GGAA(T163, T53)
PR_IN_GGAA(T163, T53) → U28_GGAA(T163, T53, delN_in_gga(T163, T53))
U28_GGAA(T163, T53, delN_out_gga(T163, T53, T193)) → MAXSORTA_IN_GA(T193)
MAXSORTA_IN_GA(.(f, .(t, T251))) → PF_IN_GAAA(T251)
PF_IN_GAAA(T251) → U30_GAAA(T251, maxG_in_ga(T251))
U30_GAAA(T251, maxG_out_ga(T251, T255)) → PS_IN_GGAA(T255, T251)
PS_IN_GGAA(T255, T251) → U32_GGAA(T255, T251, delO_in_gga(T255, T251))
U32_GGAA(T255, T251, delO_out_gga(T255, T251, T259)) → MAXSORTA_IN_GA(T259)
PB_IN_GAA(t) → U16_GAA(t, delK_out_ga(t, []))
PB_IN_GAA(f) → U16_GAA(f, delK_out_ga(f, []))

The TRS R consists of the following rules:

delO_in_gga(f, T272) → delO_out_gga(f, T272, .(t, T272))
delO_in_gga(t, T280) → U15_gga(T280, delJ_in_ga(.(t, T280)))
delJ_in_ga(.(t, T234)) → delJ_out_ga(.(t, T234), T234)
U15_gga(T280, delJ_out_ga(.(t, T280), X480)) → delO_out_gga(t, T280, .(f, X480))
maxG_in_ga([]) → maxG_out_ga([], t)
maxG_in_ga(.(t, T72)) → U6_ga(T72, maxG_in_ga(T72))
maxG_in_ga(.(f, T72)) → U7_ga(T72, maxG_in_ga(T72))
U7_ga(T72, maxG_out_ga(T72, T74)) → maxG_out_ga(.(f, T72), T74)
U6_ga(T72, maxG_out_ga(T72, T74)) → maxG_out_ga(.(t, T72), T74)
delN_in_gga(f, T206) → delN_out_gga(f, T206, .(f, T206))
delN_in_gga(t, T224) → U14_gga(T224, delJ_in_ga(T224))
delJ_in_ga([]) → delJ_out_ga([], [])
delJ_in_ga(.(f, T240)) → U11_ga(T240, delJ_in_ga(T240))
U14_gga(T224, delJ_out_ga(T224, X384)) → delN_out_gga(t, T224, .(f, .(f, X384)))
U11_ga(T240, delJ_out_ga(T240, X427)) → delJ_out_ga(.(f, T240), .(f, X427))
maxI_in_ga([]) → maxI_out_ga([], f)
maxI_in_ga(.(f, T178)) → U9_ga(T178, maxI_in_ga(T178))
maxI_in_ga(.(t, T188)) → U10_ga(T188, maxG_in_ga(T188))
U10_ga(T188, maxG_out_ga(T188, T190)) → maxI_out_ga(.(t, T188), T190)
U9_ga(T178, maxI_out_ga(T178, T180)) → maxI_out_ga(.(f, T178), T180)
delM_in_gga(t, T152) → delM_out_gga(t, T152, .(f, T152))
delM_in_gga(f, T160) → U13_gga(T160, delH_in_ga(.(f, T160)))
delH_in_ga(.(f, T128)) → delH_out_ga(.(f, T128), T128)
U13_gga(T160, delH_out_ga(.(f, T160), X269)) → delM_out_gga(f, T160, .(t, X269))
delL_in_gga(t, T100) → delL_out_gga(t, T100, .(t, T100))
delL_in_gga(f, T118) → U12_gga(T118, delH_in_ga(T118))
delH_in_ga([]) → delH_out_ga([], [])
delH_in_ga(.(t, T134)) → U8_ga(T134, delH_in_ga(T134))
U12_gga(T118, delH_out_ga(T118, X187)) → delL_out_gga(f, T118, .(t, .(t, X187)))
U8_ga(T134, delH_out_ga(T134, X228)) → delH_out_ga(.(t, T134), .(t, X228))

The set Q consists of the following terms:

delK_in_ga(x0)
maxG_in_ga(x0)
delL_in_gga(x0, x1)
delM_in_gga(x0, x1)
maxI_in_ga(x0)
delN_in_gga(x0, x1)
delO_in_gga(x0, x1)
U6_ga(x0, x1)
U7_ga(x0, x1)
U12_gga(x0, x1)
U13_gga(x0, x1)
U9_ga(x0, x1)
U10_ga(x0, x1)
U14_gga(x0, x1)
U15_gga(x0, x1)
delH_in_ga(x0)
delJ_in_ga(x0)
U8_ga(x0, x1)
U11_ga(x0, x1)

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

(44) QReductionProof (EQUIVALENT transformation)

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

delK_in_ga(x0)

(45) Obligation:

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

MAXSORTA_IN_GA(.(T17, [])) → PB_IN_GAA(T17)
U16_GAA(T17, delK_out_ga(T17, T21)) → MAXSORTA_IN_GA(T21)
MAXSORTA_IN_GA(.(t, .(t, T53))) → PC_IN_GAAA(T53)
PC_IN_GAAA(T53) → U18_GAAA(T53, maxG_in_ga(T53))
U18_GAAA(T53, maxG_out_ga(T53, T57)) → PP_IN_GGAA(T57, T53)
PP_IN_GGAA(T57, T53) → U20_GGAA(T57, T53, delL_in_gga(T57, T53))
U20_GGAA(T57, T53, delL_out_gga(T57, T53, T87)) → MAXSORTA_IN_GA(T87)
MAXSORTA_IN_GA(.(t, .(f, T53))) → PD_IN_GAAA(T53)
PD_IN_GAAA(T53) → U22_GAAA(T53, maxG_in_ga(T53))
U22_GAAA(T53, maxG_out_ga(T53, T135)) → PQ_IN_GGAA(T135, T53)
PQ_IN_GGAA(T135, T53) → U24_GGAA(T135, T53, delM_in_gga(T135, T53))
U24_GGAA(T135, T53, delM_out_gga(T135, T53, T139)) → MAXSORTA_IN_GA(T139)
MAXSORTA_IN_GA(.(f, .(f, T53))) → PE_IN_GAAA(T53)
PE_IN_GAAA(T53) → U26_GAAA(T53, maxI_in_ga(T53))
U26_GAAA(T53, maxI_out_ga(T53, T163)) → PR_IN_GGAA(T163, T53)
PR_IN_GGAA(T163, T53) → U28_GGAA(T163, T53, delN_in_gga(T163, T53))
U28_GGAA(T163, T53, delN_out_gga(T163, T53, T193)) → MAXSORTA_IN_GA(T193)
MAXSORTA_IN_GA(.(f, .(t, T251))) → PF_IN_GAAA(T251)
PF_IN_GAAA(T251) → U30_GAAA(T251, maxG_in_ga(T251))
U30_GAAA(T251, maxG_out_ga(T251, T255)) → PS_IN_GGAA(T255, T251)
PS_IN_GGAA(T255, T251) → U32_GGAA(T255, T251, delO_in_gga(T255, T251))
U32_GGAA(T255, T251, delO_out_gga(T255, T251, T259)) → MAXSORTA_IN_GA(T259)
PB_IN_GAA(t) → U16_GAA(t, delK_out_ga(t, []))
PB_IN_GAA(f) → U16_GAA(f, delK_out_ga(f, []))

The TRS R consists of the following rules:

delO_in_gga(f, T272) → delO_out_gga(f, T272, .(t, T272))
delO_in_gga(t, T280) → U15_gga(T280, delJ_in_ga(.(t, T280)))
delJ_in_ga(.(t, T234)) → delJ_out_ga(.(t, T234), T234)
U15_gga(T280, delJ_out_ga(.(t, T280), X480)) → delO_out_gga(t, T280, .(f, X480))
maxG_in_ga([]) → maxG_out_ga([], t)
maxG_in_ga(.(t, T72)) → U6_ga(T72, maxG_in_ga(T72))
maxG_in_ga(.(f, T72)) → U7_ga(T72, maxG_in_ga(T72))
U7_ga(T72, maxG_out_ga(T72, T74)) → maxG_out_ga(.(f, T72), T74)
U6_ga(T72, maxG_out_ga(T72, T74)) → maxG_out_ga(.(t, T72), T74)
delN_in_gga(f, T206) → delN_out_gga(f, T206, .(f, T206))
delN_in_gga(t, T224) → U14_gga(T224, delJ_in_ga(T224))
delJ_in_ga([]) → delJ_out_ga([], [])
delJ_in_ga(.(f, T240)) → U11_ga(T240, delJ_in_ga(T240))
U14_gga(T224, delJ_out_ga(T224, X384)) → delN_out_gga(t, T224, .(f, .(f, X384)))
U11_ga(T240, delJ_out_ga(T240, X427)) → delJ_out_ga(.(f, T240), .(f, X427))
maxI_in_ga([]) → maxI_out_ga([], f)
maxI_in_ga(.(f, T178)) → U9_ga(T178, maxI_in_ga(T178))
maxI_in_ga(.(t, T188)) → U10_ga(T188, maxG_in_ga(T188))
U10_ga(T188, maxG_out_ga(T188, T190)) → maxI_out_ga(.(t, T188), T190)
U9_ga(T178, maxI_out_ga(T178, T180)) → maxI_out_ga(.(f, T178), T180)
delM_in_gga(t, T152) → delM_out_gga(t, T152, .(f, T152))
delM_in_gga(f, T160) → U13_gga(T160, delH_in_ga(.(f, T160)))
delH_in_ga(.(f, T128)) → delH_out_ga(.(f, T128), T128)
U13_gga(T160, delH_out_ga(.(f, T160), X269)) → delM_out_gga(f, T160, .(t, X269))
delL_in_gga(t, T100) → delL_out_gga(t, T100, .(t, T100))
delL_in_gga(f, T118) → U12_gga(T118, delH_in_ga(T118))
delH_in_ga([]) → delH_out_ga([], [])
delH_in_ga(.(t, T134)) → U8_ga(T134, delH_in_ga(T134))
U12_gga(T118, delH_out_ga(T118, X187)) → delL_out_gga(f, T118, .(t, .(t, X187)))
U8_ga(T134, delH_out_ga(T134, X228)) → delH_out_ga(.(t, T134), .(t, X228))

The set Q consists of the following terms:

maxG_in_ga(x0)
delL_in_gga(x0, x1)
delM_in_gga(x0, x1)
maxI_in_ga(x0)
delN_in_gga(x0, x1)
delO_in_gga(x0, x1)
U6_ga(x0, x1)
U7_ga(x0, x1)
U12_gga(x0, x1)
U13_gga(x0, x1)
U9_ga(x0, x1)
U10_ga(x0, x1)
U14_gga(x0, x1)
U15_gga(x0, x1)
delH_in_ga(x0)
delJ_in_ga(x0)
U8_ga(x0, x1)
U11_ga(x0, x1)

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

(46) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule U16_GAA(T17, delK_out_ga(T17, T21)) → MAXSORTA_IN_GA(T21) we obtained the following new rules [LPAR04]:

U16_GAA(t, delK_out_ga(t, [])) → MAXSORTA_IN_GA([])
U16_GAA(f, delK_out_ga(f, [])) → MAXSORTA_IN_GA([])

(47) Obligation:

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

MAXSORTA_IN_GA(.(T17, [])) → PB_IN_GAA(T17)
MAXSORTA_IN_GA(.(t, .(t, T53))) → PC_IN_GAAA(T53)
PC_IN_GAAA(T53) → U18_GAAA(T53, maxG_in_ga(T53))
U18_GAAA(T53, maxG_out_ga(T53, T57)) → PP_IN_GGAA(T57, T53)
PP_IN_GGAA(T57, T53) → U20_GGAA(T57, T53, delL_in_gga(T57, T53))
U20_GGAA(T57, T53, delL_out_gga(T57, T53, T87)) → MAXSORTA_IN_GA(T87)
MAXSORTA_IN_GA(.(t, .(f, T53))) → PD_IN_GAAA(T53)
PD_IN_GAAA(T53) → U22_GAAA(T53, maxG_in_ga(T53))
U22_GAAA(T53, maxG_out_ga(T53, T135)) → PQ_IN_GGAA(T135, T53)
PQ_IN_GGAA(T135, T53) → U24_GGAA(T135, T53, delM_in_gga(T135, T53))
U24_GGAA(T135, T53, delM_out_gga(T135, T53, T139)) → MAXSORTA_IN_GA(T139)
MAXSORTA_IN_GA(.(f, .(f, T53))) → PE_IN_GAAA(T53)
PE_IN_GAAA(T53) → U26_GAAA(T53, maxI_in_ga(T53))
U26_GAAA(T53, maxI_out_ga(T53, T163)) → PR_IN_GGAA(T163, T53)
PR_IN_GGAA(T163, T53) → U28_GGAA(T163, T53, delN_in_gga(T163, T53))
U28_GGAA(T163, T53, delN_out_gga(T163, T53, T193)) → MAXSORTA_IN_GA(T193)
MAXSORTA_IN_GA(.(f, .(t, T251))) → PF_IN_GAAA(T251)
PF_IN_GAAA(T251) → U30_GAAA(T251, maxG_in_ga(T251))
U30_GAAA(T251, maxG_out_ga(T251, T255)) → PS_IN_GGAA(T255, T251)
PS_IN_GGAA(T255, T251) → U32_GGAA(T255, T251, delO_in_gga(T255, T251))
U32_GGAA(T255, T251, delO_out_gga(T255, T251, T259)) → MAXSORTA_IN_GA(T259)
PB_IN_GAA(t) → U16_GAA(t, delK_out_ga(t, []))
PB_IN_GAA(f) → U16_GAA(f, delK_out_ga(f, []))
U16_GAA(t, delK_out_ga(t, [])) → MAXSORTA_IN_GA([])
U16_GAA(f, delK_out_ga(f, [])) → MAXSORTA_IN_GA([])

The TRS R consists of the following rules:

delO_in_gga(f, T272) → delO_out_gga(f, T272, .(t, T272))
delO_in_gga(t, T280) → U15_gga(T280, delJ_in_ga(.(t, T280)))
delJ_in_ga(.(t, T234)) → delJ_out_ga(.(t, T234), T234)
U15_gga(T280, delJ_out_ga(.(t, T280), X480)) → delO_out_gga(t, T280, .(f, X480))
maxG_in_ga([]) → maxG_out_ga([], t)
maxG_in_ga(.(t, T72)) → U6_ga(T72, maxG_in_ga(T72))
maxG_in_ga(.(f, T72)) → U7_ga(T72, maxG_in_ga(T72))
U7_ga(T72, maxG_out_ga(T72, T74)) → maxG_out_ga(.(f, T72), T74)
U6_ga(T72, maxG_out_ga(T72, T74)) → maxG_out_ga(.(t, T72), T74)
delN_in_gga(f, T206) → delN_out_gga(f, T206, .(f, T206))
delN_in_gga(t, T224) → U14_gga(T224, delJ_in_ga(T224))
delJ_in_ga([]) → delJ_out_ga([], [])
delJ_in_ga(.(f, T240)) → U11_ga(T240, delJ_in_ga(T240))
U14_gga(T224, delJ_out_ga(T224, X384)) → delN_out_gga(t, T224, .(f, .(f, X384)))
U11_ga(T240, delJ_out_ga(T240, X427)) → delJ_out_ga(.(f, T240), .(f, X427))
maxI_in_ga([]) → maxI_out_ga([], f)
maxI_in_ga(.(f, T178)) → U9_ga(T178, maxI_in_ga(T178))
maxI_in_ga(.(t, T188)) → U10_ga(T188, maxG_in_ga(T188))
U10_ga(T188, maxG_out_ga(T188, T190)) → maxI_out_ga(.(t, T188), T190)
U9_ga(T178, maxI_out_ga(T178, T180)) → maxI_out_ga(.(f, T178), T180)
delM_in_gga(t, T152) → delM_out_gga(t, T152, .(f, T152))
delM_in_gga(f, T160) → U13_gga(T160, delH_in_ga(.(f, T160)))
delH_in_ga(.(f, T128)) → delH_out_ga(.(f, T128), T128)
U13_gga(T160, delH_out_ga(.(f, T160), X269)) → delM_out_gga(f, T160, .(t, X269))
delL_in_gga(t, T100) → delL_out_gga(t, T100, .(t, T100))
delL_in_gga(f, T118) → U12_gga(T118, delH_in_ga(T118))
delH_in_ga([]) → delH_out_ga([], [])
delH_in_ga(.(t, T134)) → U8_ga(T134, delH_in_ga(T134))
U12_gga(T118, delH_out_ga(T118, X187)) → delL_out_gga(f, T118, .(t, .(t, X187)))
U8_ga(T134, delH_out_ga(T134, X228)) → delH_out_ga(.(t, T134), .(t, X228))

The set Q consists of the following terms:

maxG_in_ga(x0)
delL_in_gga(x0, x1)
delM_in_gga(x0, x1)
maxI_in_ga(x0)
delN_in_gga(x0, x1)
delO_in_gga(x0, x1)
U6_ga(x0, x1)
U7_ga(x0, x1)
U12_gga(x0, x1)
U13_gga(x0, x1)
U9_ga(x0, x1)
U10_ga(x0, x1)
U14_gga(x0, x1)
U15_gga(x0, x1)
delH_in_ga(x0)
delJ_in_ga(x0)
U8_ga(x0, x1)
U11_ga(x0, x1)

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

(48) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 5 less nodes.

(49) Obligation:

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

MAXSORTA_IN_GA(.(t, .(t, T53))) → PC_IN_GAAA(T53)
PC_IN_GAAA(T53) → U18_GAAA(T53, maxG_in_ga(T53))
U18_GAAA(T53, maxG_out_ga(T53, T57)) → PP_IN_GGAA(T57, T53)
PP_IN_GGAA(T57, T53) → U20_GGAA(T57, T53, delL_in_gga(T57, T53))
U20_GGAA(T57, T53, delL_out_gga(T57, T53, T87)) → MAXSORTA_IN_GA(T87)
MAXSORTA_IN_GA(.(t, .(f, T53))) → PD_IN_GAAA(T53)
PD_IN_GAAA(T53) → U22_GAAA(T53, maxG_in_ga(T53))
U22_GAAA(T53, maxG_out_ga(T53, T135)) → PQ_IN_GGAA(T135, T53)
PQ_IN_GGAA(T135, T53) → U24_GGAA(T135, T53, delM_in_gga(T135, T53))
U24_GGAA(T135, T53, delM_out_gga(T135, T53, T139)) → MAXSORTA_IN_GA(T139)
MAXSORTA_IN_GA(.(f, .(f, T53))) → PE_IN_GAAA(T53)
PE_IN_GAAA(T53) → U26_GAAA(T53, maxI_in_ga(T53))
U26_GAAA(T53, maxI_out_ga(T53, T163)) → PR_IN_GGAA(T163, T53)
PR_IN_GGAA(T163, T53) → U28_GGAA(T163, T53, delN_in_gga(T163, T53))
U28_GGAA(T163, T53, delN_out_gga(T163, T53, T193)) → MAXSORTA_IN_GA(T193)
MAXSORTA_IN_GA(.(f, .(t, T251))) → PF_IN_GAAA(T251)
PF_IN_GAAA(T251) → U30_GAAA(T251, maxG_in_ga(T251))
U30_GAAA(T251, maxG_out_ga(T251, T255)) → PS_IN_GGAA(T255, T251)
PS_IN_GGAA(T255, T251) → U32_GGAA(T255, T251, delO_in_gga(T255, T251))
U32_GGAA(T255, T251, delO_out_gga(T255, T251, T259)) → MAXSORTA_IN_GA(T259)

The TRS R consists of the following rules:

delO_in_gga(f, T272) → delO_out_gga(f, T272, .(t, T272))
delO_in_gga(t, T280) → U15_gga(T280, delJ_in_ga(.(t, T280)))
delJ_in_ga(.(t, T234)) → delJ_out_ga(.(t, T234), T234)
U15_gga(T280, delJ_out_ga(.(t, T280), X480)) → delO_out_gga(t, T280, .(f, X480))
maxG_in_ga([]) → maxG_out_ga([], t)
maxG_in_ga(.(t, T72)) → U6_ga(T72, maxG_in_ga(T72))
maxG_in_ga(.(f, T72)) → U7_ga(T72, maxG_in_ga(T72))
U7_ga(T72, maxG_out_ga(T72, T74)) → maxG_out_ga(.(f, T72), T74)
U6_ga(T72, maxG_out_ga(T72, T74)) → maxG_out_ga(.(t, T72), T74)
delN_in_gga(f, T206) → delN_out_gga(f, T206, .(f, T206))
delN_in_gga(t, T224) → U14_gga(T224, delJ_in_ga(T224))
delJ_in_ga([]) → delJ_out_ga([], [])
delJ_in_ga(.(f, T240)) → U11_ga(T240, delJ_in_ga(T240))
U14_gga(T224, delJ_out_ga(T224, X384)) → delN_out_gga(t, T224, .(f, .(f, X384)))
U11_ga(T240, delJ_out_ga(T240, X427)) → delJ_out_ga(.(f, T240), .(f, X427))
maxI_in_ga([]) → maxI_out_ga([], f)
maxI_in_ga(.(f, T178)) → U9_ga(T178, maxI_in_ga(T178))
maxI_in_ga(.(t, T188)) → U10_ga(T188, maxG_in_ga(T188))
U10_ga(T188, maxG_out_ga(T188, T190)) → maxI_out_ga(.(t, T188), T190)
U9_ga(T178, maxI_out_ga(T178, T180)) → maxI_out_ga(.(f, T178), T180)
delM_in_gga(t, T152) → delM_out_gga(t, T152, .(f, T152))
delM_in_gga(f, T160) → U13_gga(T160, delH_in_ga(.(f, T160)))
delH_in_ga(.(f, T128)) → delH_out_ga(.(f, T128), T128)
U13_gga(T160, delH_out_ga(.(f, T160), X269)) → delM_out_gga(f, T160, .(t, X269))
delL_in_gga(t, T100) → delL_out_gga(t, T100, .(t, T100))
delL_in_gga(f, T118) → U12_gga(T118, delH_in_ga(T118))
delH_in_ga([]) → delH_out_ga([], [])
delH_in_ga(.(t, T134)) → U8_ga(T134, delH_in_ga(T134))
U12_gga(T118, delH_out_ga(T118, X187)) → delL_out_gga(f, T118, .(t, .(t, X187)))
U8_ga(T134, delH_out_ga(T134, X228)) → delH_out_ga(.(t, T134), .(t, X228))

The set Q consists of the following terms:

maxG_in_ga(x0)
delL_in_gga(x0, x1)
delM_in_gga(x0, x1)
maxI_in_ga(x0)
delN_in_gga(x0, x1)
delO_in_gga(x0, x1)
U6_ga(x0, x1)
U7_ga(x0, x1)
U12_gga(x0, x1)
U13_gga(x0, x1)
U9_ga(x0, x1)
U10_ga(x0, x1)
U14_gga(x0, x1)
U15_gga(x0, x1)
delH_in_ga(x0)
delJ_in_ga(x0)
U8_ga(x0, x1)
U11_ga(x0, x1)

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

(50) QDPOrderProof (EQUIVALENT transformation)

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

The following pairs can be oriented strictly and are deleted.

PQ_IN_GGAA(T135, T53) → U24_GGAA(T135, T53, delM_in_gga(T135, T53))

The remaining pairs can at least be oriented weakly.
Used ordering: Matrix interpretation [MATRO] to (N^2, +, *, >=, >) :

POL(MAXSORTA_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 \ \ 0 /

POL(PC_IN_GAAA(x1)) =

0

+

[ 0, 1 ]

·

x1

POL(U18_GAAA(x1, x2)) =

0

+

[ 0, 1 ]

·

x1

+

[ 0, 0 ]

·

x2

POL(maxG_in_ga(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(maxG_out_ga(x1, x2)) =

/ 1 \ \ 1 /

+

/ 1 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

POL(PP_IN_GGAA(x1, x2)) =

0

+

[ 0, 0 ]

·

x1

+

[ 0, 1 ]

·

x2

POL(U20_GGAA(x1, x2, x3)) =

0

+

[ 0, 0 ]

·

x1

+

[ 0, 0 ]

·

x2

+

[ 0, 1 ]

·

x3

POL(delL_in_gga(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 1 0 \ \ 0 1 /

·

x2

POL(delL_out_gga(x1, x2, x3)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

+

/ 0 0 \ \ 1 0 /

·

x3

POL(f) =

/ 1 \ \ 0 /

POL(PD_IN_GAAA(x1)) =

1

+

[ 0, 1 ]

·

x1

POL(U22_GAAA(x1, x2)) =

1

+

[ 0, 1 ]

·

x1

+

[ 0, 0 ]

·

x2

POL(PQ_IN_GGAA(x1, x2)) =

1

+

[ 0, 0 ]

·

x1

+

[ 0, 1 ]

·

x2

POL(U24_GGAA(x1, x2, x3)) =

0

+

[ 0, 0 ]

·

x1

+

[ 0, 0 ]

·

x2

+

[ 1, 0 ]

·

x3

POL(delM_in_gga(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 1 \ \ 0 1 /

·

x2

POL(delM_out_gga(x1, x2, x3)) =

/ 0 \ \ 0 /

+

/ 0 1 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

+

/ 1 0 \ \ 0 0 /

·

x3

POL(PE_IN_GAAA(x1)) =

1

+

[ 0, 1 ]

·

x1

POL(U26_GAAA(x1, x2)) =

1

+

[ 0, 1 ]

·

x1

+

[ 0, 0 ]

·

x2

POL(maxI_in_ga(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(maxI_out_ga(x1, x2)) =

/ 1 \ \ 1 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

POL(PR_IN_GGAA(x1, x2)) =

1

+

[ 0, 0 ]

·

x1

+

[ 0, 1 ]

·

x2

POL(U28_GGAA(x1, x2, x3)) =

0

+

[ 0, 0 ]

·

x1

+

[ 0, 0 ]

·

x2

+

[ 0, 1 ]

·

x3

POL(delN_in_gga(x1, x2)) =

/ 0 \ \ 1 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 1 /

·

x2

POL(delN_out_gga(x1, x2, x3)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 1 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

+

/ 0 0 \ \ 1 0 /

·

x3

POL(PF_IN_GAAA(x1)) =

0

+

[ 0, 1 ]

·

x1

POL(U30_GAAA(x1, x2)) =

0

+

[ 0, 1 ]

·

x1

+

[ 0, 0 ]

·

x2

POL(PS_IN_GGAA(x1, x2)) =

0

+

[ 0, 0 ]

·

x1

+

[ 0, 1 ]

·

x2

POL(U32_GGAA(x1, x2, x3)) =

0

+

[ 0, 0 ]

·

x1

+

[ 0, 0 ]

·

x2

+

[ 1, 0 ]

·

x3

POL(delO_in_gga(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 1 \ \ 1 1 /

·

x2

POL(delO_out_gga(x1, x2, x3)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

+

/ 1 0 \ \ 0 0 /

·

x3

POL([]) =

/ 0 \ \ 0 /

POL(U6_ga(x1, x2)) =

/ 1 \ \ 1 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 1 0 \ \ 0 0 /

·

x2

POL(U7_ga(x1, x2)) =

/ 1 \ \ 1 /

+

/ 0 1 \ \ 1 0 /

·

x1

+

/ 0 1 \ \ 0 0 /

·

x2

POL(U12_gga(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 1 /

·

x2

POL(delH_in_ga(x1)) =

/ 0 \ \ 0 /

+

/ 1 0 \ \ 0 1 /

·

x1

POL(U13_gga(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 1 /

·

x1

+

/ 1 0 \ \ 0 0 /

·

x2

POL(U9_ga(x1, x2)) =

/ 0 \ \ 0 /

+

/ 1 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

POL(U10_ga(x1, x2)) =

/ 1 \ \ 0 /

+

/ 1 1 \ \ 1 1 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

POL(U14_gga(x1, x2)) =

/ 0 \ \ 1 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 1 0 /

·

x2

POL(delJ_in_ga(x1)) =

/ 0 \ \ 0 /

+

/ 0 1 \ \ 0 0 /

·

x1

POL(U15_gga(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 1 0 /

·

x1

+

/ 1 0 \ \ 0 0 /

·

x2

POL(delJ_out_ga(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 1 \ \ 0 0 /

·

x2

POL(U11_ga(x1, x2)) =

/ 1 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 1 0 \ \ 0 0 /

·

x2

POL(delH_out_ga(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 1 \ \ 0 1 /

·

x2

POL(U8_ga(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 1 \ \ 0 1 /

·

x2

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

delL_in_gga(t, T100) → delL_out_gga(t, T100, .(t, T100))
delL_in_gga(f, T118) → U12_gga(T118, delH_in_ga(T118))
delM_in_gga(t, T152) → delM_out_gga(t, T152, .(f, T152))
delM_in_gga(f, T160) → U13_gga(T160, delH_in_ga(.(f, T160)))
delN_in_gga(f, T206) → delN_out_gga(f, T206, .(f, T206))
delN_in_gga(t, T224) → U14_gga(T224, delJ_in_ga(T224))
delO_in_gga(f, T272) → delO_out_gga(f, T272, .(t, T272))
delO_in_gga(t, T280) → U15_gga(T280, delJ_in_ga(.(t, T280)))
delJ_in_ga(.(t, T234)) → delJ_out_ga(.(t, T234), T234)
U15_gga(T280, delJ_out_ga(.(t, T280), X480)) → delO_out_gga(t, T280, .(f, X480))
delJ_in_ga([]) → delJ_out_ga([], [])
delJ_in_ga(.(f, T240)) → U11_ga(T240, delJ_in_ga(T240))
U14_gga(T224, delJ_out_ga(T224, X384)) → delN_out_gga(t, T224, .(f, .(f, X384)))
U11_ga(T240, delJ_out_ga(T240, X427)) → delJ_out_ga(.(f, T240), .(f, X427))
delH_in_ga(.(f, T128)) → delH_out_ga(.(f, T128), T128)
U13_gga(T160, delH_out_ga(.(f, T160), X269)) → delM_out_gga(f, T160, .(t, X269))
delH_in_ga([]) → delH_out_ga([], [])
delH_in_ga(.(t, T134)) → U8_ga(T134, delH_in_ga(T134))
U12_gga(T118, delH_out_ga(T118, X187)) → delL_out_gga(f, T118, .(t, .(t, X187)))
U8_ga(T134, delH_out_ga(T134, X228)) → delH_out_ga(.(t, T134), .(t, X228))

(51) Obligation:

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

MAXSORTA_IN_GA(.(t, .(t, T53))) → PC_IN_GAAA(T53)
PC_IN_GAAA(T53) → U18_GAAA(T53, maxG_in_ga(T53))
U18_GAAA(T53, maxG_out_ga(T53, T57)) → PP_IN_GGAA(T57, T53)
PP_IN_GGAA(T57, T53) → U20_GGAA(T57, T53, delL_in_gga(T57, T53))
U20_GGAA(T57, T53, delL_out_gga(T57, T53, T87)) → MAXSORTA_IN_GA(T87)
MAXSORTA_IN_GA(.(t, .(f, T53))) → PD_IN_GAAA(T53)
PD_IN_GAAA(T53) → U22_GAAA(T53, maxG_in_ga(T53))
U22_GAAA(T53, maxG_out_ga(T53, T135)) → PQ_IN_GGAA(T135, T53)
U24_GGAA(T135, T53, delM_out_gga(T135, T53, T139)) → MAXSORTA_IN_GA(T139)
MAXSORTA_IN_GA(.(f, .(f, T53))) → PE_IN_GAAA(T53)
PE_IN_GAAA(T53) → U26_GAAA(T53, maxI_in_ga(T53))
U26_GAAA(T53, maxI_out_ga(T53, T163)) → PR_IN_GGAA(T163, T53)
PR_IN_GGAA(T163, T53) → U28_GGAA(T163, T53, delN_in_gga(T163, T53))
U28_GGAA(T163, T53, delN_out_gga(T163, T53, T193)) → MAXSORTA_IN_GA(T193)
MAXSORTA_IN_GA(.(f, .(t, T251))) → PF_IN_GAAA(T251)
PF_IN_GAAA(T251) → U30_GAAA(T251, maxG_in_ga(T251))
U30_GAAA(T251, maxG_out_ga(T251, T255)) → PS_IN_GGAA(T255, T251)
PS_IN_GGAA(T255, T251) → U32_GGAA(T255, T251, delO_in_gga(T255, T251))
U32_GGAA(T255, T251, delO_out_gga(T255, T251, T259)) → MAXSORTA_IN_GA(T259)

The TRS R consists of the following rules:

delO_in_gga(f, T272) → delO_out_gga(f, T272, .(t, T272))
delO_in_gga(t, T280) → U15_gga(T280, delJ_in_ga(.(t, T280)))
delJ_in_ga(.(t, T234)) → delJ_out_ga(.(t, T234), T234)
U15_gga(T280, delJ_out_ga(.(t, T280), X480)) → delO_out_gga(t, T280, .(f, X480))
maxG_in_ga([]) → maxG_out_ga([], t)
maxG_in_ga(.(t, T72)) → U6_ga(T72, maxG_in_ga(T72))
maxG_in_ga(.(f, T72)) → U7_ga(T72, maxG_in_ga(T72))
U7_ga(T72, maxG_out_ga(T72, T74)) → maxG_out_ga(.(f, T72), T74)
U6_ga(T72, maxG_out_ga(T72, T74)) → maxG_out_ga(.(t, T72), T74)
delN_in_gga(f, T206) → delN_out_gga(f, T206, .(f, T206))
delN_in_gga(t, T224) → U14_gga(T224, delJ_in_ga(T224))
delJ_in_ga([]) → delJ_out_ga([], [])
delJ_in_ga(.(f, T240)) → U11_ga(T240, delJ_in_ga(T240))
U14_gga(T224, delJ_out_ga(T224, X384)) → delN_out_gga(t, T224, .(f, .(f, X384)))
U11_ga(T240, delJ_out_ga(T240, X427)) → delJ_out_ga(.(f, T240), .(f, X427))
maxI_in_ga([]) → maxI_out_ga([], f)
maxI_in_ga(.(f, T178)) → U9_ga(T178, maxI_in_ga(T178))
maxI_in_ga(.(t, T188)) → U10_ga(T188, maxG_in_ga(T188))
U10_ga(T188, maxG_out_ga(T188, T190)) → maxI_out_ga(.(t, T188), T190)
U9_ga(T178, maxI_out_ga(T178, T180)) → maxI_out_ga(.(f, T178), T180)
delM_in_gga(t, T152) → delM_out_gga(t, T152, .(f, T152))
delM_in_gga(f, T160) → U13_gga(T160, delH_in_ga(.(f, T160)))
delH_in_ga(.(f, T128)) → delH_out_ga(.(f, T128), T128)
U13_gga(T160, delH_out_ga(.(f, T160), X269)) → delM_out_gga(f, T160, .(t, X269))
delL_in_gga(t, T100) → delL_out_gga(t, T100, .(t, T100))
delL_in_gga(f, T118) → U12_gga(T118, delH_in_ga(T118))
delH_in_ga([]) → delH_out_ga([], [])
delH_in_ga(.(t, T134)) → U8_ga(T134, delH_in_ga(T134))
U12_gga(T118, delH_out_ga(T118, X187)) → delL_out_gga(f, T118, .(t, .(t, X187)))
U8_ga(T134, delH_out_ga(T134, X228)) → delH_out_ga(.(t, T134), .(t, X228))

The set Q consists of the following terms:

maxG_in_ga(x0)
delL_in_gga(x0, x1)
delM_in_gga(x0, x1)
maxI_in_ga(x0)
delN_in_gga(x0, x1)
delO_in_gga(x0, x1)
U6_ga(x0, x1)
U7_ga(x0, x1)
U12_gga(x0, x1)
U13_gga(x0, x1)
U9_ga(x0, x1)
U10_ga(x0, x1)
U14_gga(x0, x1)
U15_gga(x0, x1)
delH_in_ga(x0)
delJ_in_ga(x0)
U8_ga(x0, x1)
U11_ga(x0, x1)

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

(52) DependencyGraphProof (EQUIVALENT transformation)

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

(53) Obligation:

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

PC_IN_GAAA(T53) → U18_GAAA(T53, maxG_in_ga(T53))
U18_GAAA(T53, maxG_out_ga(T53, T57)) → PP_IN_GGAA(T57, T53)
PP_IN_GGAA(T57, T53) → U20_GGAA(T57, T53, delL_in_gga(T57, T53))
U20_GGAA(T57, T53, delL_out_gga(T57, T53, T87)) → MAXSORTA_IN_GA(T87)
MAXSORTA_IN_GA(.(t, .(t, T53))) → PC_IN_GAAA(T53)
MAXSORTA_IN_GA(.(f, .(f, T53))) → PE_IN_GAAA(T53)
PE_IN_GAAA(T53) → U26_GAAA(T53, maxI_in_ga(T53))
U26_GAAA(T53, maxI_out_ga(T53, T163)) → PR_IN_GGAA(T163, T53)
PR_IN_GGAA(T163, T53) → U28_GGAA(T163, T53, delN_in_gga(T163, T53))
U28_GGAA(T163, T53, delN_out_gga(T163, T53, T193)) → MAXSORTA_IN_GA(T193)
MAXSORTA_IN_GA(.(f, .(t, T251))) → PF_IN_GAAA(T251)
PF_IN_GAAA(T251) → U30_GAAA(T251, maxG_in_ga(T251))
U30_GAAA(T251, maxG_out_ga(T251, T255)) → PS_IN_GGAA(T255, T251)
PS_IN_GGAA(T255, T251) → U32_GGAA(T255, T251, delO_in_gga(T255, T251))
U32_GGAA(T255, T251, delO_out_gga(T255, T251, T259)) → MAXSORTA_IN_GA(T259)

The TRS R consists of the following rules:

delO_in_gga(f, T272) → delO_out_gga(f, T272, .(t, T272))
delO_in_gga(t, T280) → U15_gga(T280, delJ_in_ga(.(t, T280)))
delJ_in_ga(.(t, T234)) → delJ_out_ga(.(t, T234), T234)
U15_gga(T280, delJ_out_ga(.(t, T280), X480)) → delO_out_gga(t, T280, .(f, X480))
maxG_in_ga([]) → maxG_out_ga([], t)
maxG_in_ga(.(t, T72)) → U6_ga(T72, maxG_in_ga(T72))
maxG_in_ga(.(f, T72)) → U7_ga(T72, maxG_in_ga(T72))
U7_ga(T72, maxG_out_ga(T72, T74)) → maxG_out_ga(.(f, T72), T74)
U6_ga(T72, maxG_out_ga(T72, T74)) → maxG_out_ga(.(t, T72), T74)
delN_in_gga(f, T206) → delN_out_gga(f, T206, .(f, T206))
delN_in_gga(t, T224) → U14_gga(T224, delJ_in_ga(T224))
delJ_in_ga([]) → delJ_out_ga([], [])
delJ_in_ga(.(f, T240)) → U11_ga(T240, delJ_in_ga(T240))
U14_gga(T224, delJ_out_ga(T224, X384)) → delN_out_gga(t, T224, .(f, .(f, X384)))
U11_ga(T240, delJ_out_ga(T240, X427)) → delJ_out_ga(.(f, T240), .(f, X427))
maxI_in_ga([]) → maxI_out_ga([], f)
maxI_in_ga(.(f, T178)) → U9_ga(T178, maxI_in_ga(T178))
maxI_in_ga(.(t, T188)) → U10_ga(T188, maxG_in_ga(T188))
U10_ga(T188, maxG_out_ga(T188, T190)) → maxI_out_ga(.(t, T188), T190)
U9_ga(T178, maxI_out_ga(T178, T180)) → maxI_out_ga(.(f, T178), T180)
delM_in_gga(t, T152) → delM_out_gga(t, T152, .(f, T152))
delM_in_gga(f, T160) → U13_gga(T160, delH_in_ga(.(f, T160)))
delH_in_ga(.(f, T128)) → delH_out_ga(.(f, T128), T128)
U13_gga(T160, delH_out_ga(.(f, T160), X269)) → delM_out_gga(f, T160, .(t, X269))
delL_in_gga(t, T100) → delL_out_gga(t, T100, .(t, T100))
delL_in_gga(f, T118) → U12_gga(T118, delH_in_ga(T118))
delH_in_ga([]) → delH_out_ga([], [])
delH_in_ga(.(t, T134)) → U8_ga(T134, delH_in_ga(T134))
U12_gga(T118, delH_out_ga(T118, X187)) → delL_out_gga(f, T118, .(t, .(t, X187)))
U8_ga(T134, delH_out_ga(T134, X228)) → delH_out_ga(.(t, T134), .(t, X228))

The set Q consists of the following terms:

maxG_in_ga(x0)
delL_in_gga(x0, x1)
delM_in_gga(x0, x1)
maxI_in_ga(x0)
delN_in_gga(x0, x1)
delO_in_gga(x0, x1)
U6_ga(x0, x1)
U7_ga(x0, x1)
U12_gga(x0, x1)
U13_gga(x0, x1)
U9_ga(x0, x1)
U10_ga(x0, x1)
U14_gga(x0, x1)
U15_gga(x0, x1)
delH_in_ga(x0)
delJ_in_ga(x0)
U8_ga(x0, x1)
U11_ga(x0, x1)

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

(54) UsableRulesProof (EQUIVALENT transformation)

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

(55) Obligation:

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

PC_IN_GAAA(T53) → U18_GAAA(T53, maxG_in_ga(T53))
U18_GAAA(T53, maxG_out_ga(T53, T57)) → PP_IN_GGAA(T57, T53)
PP_IN_GGAA(T57, T53) → U20_GGAA(T57, T53, delL_in_gga(T57, T53))
U20_GGAA(T57, T53, delL_out_gga(T57, T53, T87)) → MAXSORTA_IN_GA(T87)
MAXSORTA_IN_GA(.(t, .(t, T53))) → PC_IN_GAAA(T53)
MAXSORTA_IN_GA(.(f, .(f, T53))) → PE_IN_GAAA(T53)
PE_IN_GAAA(T53) → U26_GAAA(T53, maxI_in_ga(T53))
U26_GAAA(T53, maxI_out_ga(T53, T163)) → PR_IN_GGAA(T163, T53)
PR_IN_GGAA(T163, T53) → U28_GGAA(T163, T53, delN_in_gga(T163, T53))
U28_GGAA(T163, T53, delN_out_gga(T163, T53, T193)) → MAXSORTA_IN_GA(T193)
MAXSORTA_IN_GA(.(f, .(t, T251))) → PF_IN_GAAA(T251)
PF_IN_GAAA(T251) → U30_GAAA(T251, maxG_in_ga(T251))
U30_GAAA(T251, maxG_out_ga(T251, T255)) → PS_IN_GGAA(T255, T251)
PS_IN_GGAA(T255, T251) → U32_GGAA(T255, T251, delO_in_gga(T255, T251))
U32_GGAA(T255, T251, delO_out_gga(T255, T251, T259)) → MAXSORTA_IN_GA(T259)

The TRS R consists of the following rules:

delO_in_gga(f, T272) → delO_out_gga(f, T272, .(t, T272))
delO_in_gga(t, T280) → U15_gga(T280, delJ_in_ga(.(t, T280)))
delJ_in_ga(.(t, T234)) → delJ_out_ga(.(t, T234), T234)
U15_gga(T280, delJ_out_ga(.(t, T280), X480)) → delO_out_gga(t, T280, .(f, X480))
maxG_in_ga([]) → maxG_out_ga([], t)
maxG_in_ga(.(t, T72)) → U6_ga(T72, maxG_in_ga(T72))
maxG_in_ga(.(f, T72)) → U7_ga(T72, maxG_in_ga(T72))
U7_ga(T72, maxG_out_ga(T72, T74)) → maxG_out_ga(.(f, T72), T74)
U6_ga(T72, maxG_out_ga(T72, T74)) → maxG_out_ga(.(t, T72), T74)
delN_in_gga(f, T206) → delN_out_gga(f, T206, .(f, T206))
delN_in_gga(t, T224) → U14_gga(T224, delJ_in_ga(T224))
delJ_in_ga([]) → delJ_out_ga([], [])
delJ_in_ga(.(f, T240)) → U11_ga(T240, delJ_in_ga(T240))
U14_gga(T224, delJ_out_ga(T224, X384)) → delN_out_gga(t, T224, .(f, .(f, X384)))
U11_ga(T240, delJ_out_ga(T240, X427)) → delJ_out_ga(.(f, T240), .(f, X427))
maxI_in_ga([]) → maxI_out_ga([], f)
maxI_in_ga(.(f, T178)) → U9_ga(T178, maxI_in_ga(T178))
maxI_in_ga(.(t, T188)) → U10_ga(T188, maxG_in_ga(T188))
U10_ga(T188, maxG_out_ga(T188, T190)) → maxI_out_ga(.(t, T188), T190)
U9_ga(T178, maxI_out_ga(T178, T180)) → maxI_out_ga(.(f, T178), T180)
delL_in_gga(t, T100) → delL_out_gga(t, T100, .(t, T100))
delL_in_gga(f, T118) → U12_gga(T118, delH_in_ga(T118))
delH_in_ga(.(f, T128)) → delH_out_ga(.(f, T128), T128)
delH_in_ga([]) → delH_out_ga([], [])
delH_in_ga(.(t, T134)) → U8_ga(T134, delH_in_ga(T134))
U12_gga(T118, delH_out_ga(T118, X187)) → delL_out_gga(f, T118, .(t, .(t, X187)))
U8_ga(T134, delH_out_ga(T134, X228)) → delH_out_ga(.(t, T134), .(t, X228))

The set Q consists of the following terms:

maxG_in_ga(x0)
delL_in_gga(x0, x1)
delM_in_gga(x0, x1)
maxI_in_ga(x0)
delN_in_gga(x0, x1)
delO_in_gga(x0, x1)
U6_ga(x0, x1)
U7_ga(x0, x1)
U12_gga(x0, x1)
U13_gga(x0, x1)
U9_ga(x0, x1)
U10_ga(x0, x1)
U14_gga(x0, x1)
U15_gga(x0, x1)
delH_in_ga(x0)
delJ_in_ga(x0)
U8_ga(x0, x1)
U11_ga(x0, x1)

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

(56) QReductionProof (EQUIVALENT transformation)

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

delM_in_gga(x0, x1)
U13_gga(x0, x1)

(57) Obligation:

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

PC_IN_GAAA(T53) → U18_GAAA(T53, maxG_in_ga(T53))
U18_GAAA(T53, maxG_out_ga(T53, T57)) → PP_IN_GGAA(T57, T53)
PP_IN_GGAA(T57, T53) → U20_GGAA(T57, T53, delL_in_gga(T57, T53))
U20_GGAA(T57, T53, delL_out_gga(T57, T53, T87)) → MAXSORTA_IN_GA(T87)
MAXSORTA_IN_GA(.(t, .(t, T53))) → PC_IN_GAAA(T53)
MAXSORTA_IN_GA(.(f, .(f, T53))) → PE_IN_GAAA(T53)
PE_IN_GAAA(T53) → U26_GAAA(T53, maxI_in_ga(T53))
U26_GAAA(T53, maxI_out_ga(T53, T163)) → PR_IN_GGAA(T163, T53)
PR_IN_GGAA(T163, T53) → U28_GGAA(T163, T53, delN_in_gga(T163, T53))
U28_GGAA(T163, T53, delN_out_gga(T163, T53, T193)) → MAXSORTA_IN_GA(T193)
MAXSORTA_IN_GA(.(f, .(t, T251))) → PF_IN_GAAA(T251)
PF_IN_GAAA(T251) → U30_GAAA(T251, maxG_in_ga(T251))
U30_GAAA(T251, maxG_out_ga(T251, T255)) → PS_IN_GGAA(T255, T251)
PS_IN_GGAA(T255, T251) → U32_GGAA(T255, T251, delO_in_gga(T255, T251))
U32_GGAA(T255, T251, delO_out_gga(T255, T251, T259)) → MAXSORTA_IN_GA(T259)

The TRS R consists of the following rules:

delO_in_gga(f, T272) → delO_out_gga(f, T272, .(t, T272))
delO_in_gga(t, T280) → U15_gga(T280, delJ_in_ga(.(t, T280)))
delJ_in_ga(.(t, T234)) → delJ_out_ga(.(t, T234), T234)
U15_gga(T280, delJ_out_ga(.(t, T280), X480)) → delO_out_gga(t, T280, .(f, X480))
maxG_in_ga([]) → maxG_out_ga([], t)
maxG_in_ga(.(t, T72)) → U6_ga(T72, maxG_in_ga(T72))
maxG_in_ga(.(f, T72)) → U7_ga(T72, maxG_in_ga(T72))
U7_ga(T72, maxG_out_ga(T72, T74)) → maxG_out_ga(.(f, T72), T74)
U6_ga(T72, maxG_out_ga(T72, T74)) → maxG_out_ga(.(t, T72), T74)
delN_in_gga(f, T206) → delN_out_gga(f, T206, .(f, T206))
delN_in_gga(t, T224) → U14_gga(T224, delJ_in_ga(T224))
delJ_in_ga([]) → delJ_out_ga([], [])
delJ_in_ga(.(f, T240)) → U11_ga(T240, delJ_in_ga(T240))
U14_gga(T224, delJ_out_ga(T224, X384)) → delN_out_gga(t, T224, .(f, .(f, X384)))
U11_ga(T240, delJ_out_ga(T240, X427)) → delJ_out_ga(.(f, T240), .(f, X427))
maxI_in_ga([]) → maxI_out_ga([], f)
maxI_in_ga(.(f, T178)) → U9_ga(T178, maxI_in_ga(T178))
maxI_in_ga(.(t, T188)) → U10_ga(T188, maxG_in_ga(T188))
U10_ga(T188, maxG_out_ga(T188, T190)) → maxI_out_ga(.(t, T188), T190)
U9_ga(T178, maxI_out_ga(T178, T180)) → maxI_out_ga(.(f, T178), T180)
delL_in_gga(t, T100) → delL_out_gga(t, T100, .(t, T100))
delL_in_gga(f, T118) → U12_gga(T118, delH_in_ga(T118))
delH_in_ga(.(f, T128)) → delH_out_ga(.(f, T128), T128)
delH_in_ga([]) → delH_out_ga([], [])
delH_in_ga(.(t, T134)) → U8_ga(T134, delH_in_ga(T134))
U12_gga(T118, delH_out_ga(T118, X187)) → delL_out_gga(f, T118, .(t, .(t, X187)))
U8_ga(T134, delH_out_ga(T134, X228)) → delH_out_ga(.(t, T134), .(t, X228))

The set Q consists of the following terms:

maxG_in_ga(x0)
delL_in_gga(x0, x1)
maxI_in_ga(x0)
delN_in_gga(x0, x1)
delO_in_gga(x0, x1)
U6_ga(x0, x1)
U7_ga(x0, x1)
U12_gga(x0, x1)
U9_ga(x0, x1)
U10_ga(x0, x1)
U14_gga(x0, x1)
U15_gga(x0, x1)
delH_in_ga(x0)
delJ_in_ga(x0)
U8_ga(x0, x1)
U11_ga(x0, x1)

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

(58) QDPQMonotonicMRRProof (EQUIVALENT transformation)

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

Strictly oriented rules of the TRS R:

delL_in_gga(f, T118) → U12_gga(T118, delH_in_ga(T118))

Used ordering: Polynomial interpretation [POLO]:

POL(.(x₁, x₂)) = 0   
POL(MAXSORTA_IN_GA(x₁)) = 0   
POL(PC_IN_GAAA(x₁)) = 0   
POL(PE_IN_GAAA(x₁)) = 0   
POL(PF_IN_GAAA(x₁)) = 0   
POL(PP_IN_GGAA(x₁, x₂)) = 2·x₁   
POL(PR_IN_GGAA(x₁, x₂)) = 0   
POL(PS_IN_GGAA(x₁, x₂)) = 0   
POL(U10_ga(x₁, x₂)) = x₂   
POL(U11_ga(x₁, x₂)) = 0   
POL(U12_gga(x₁, x₂)) = 0   
POL(U14_gga(x₁, x₂)) = 0   
POL(U15_gga(x₁, x₂)) = 0   
POL(U18_GAAA(x₁, x₂)) = 2·x₂   
POL(U20_GGAA(x₁, x₂, x₃)) = x₃   
POL(U26_GAAA(x₁, x₂)) = 0   
POL(U28_GGAA(x₁, x₂, x₃)) = 0   
POL(U30_GAAA(x₁, x₂)) = 0   
POL(U32_GGAA(x₁, x₂, x₃)) = 0   
POL(U6_ga(x₁, x₂)) = x₂   
POL(U7_ga(x₁, x₂)) = 2·x₂   
POL(U8_ga(x₁, x₂)) = 2   
POL(U9_ga(x₁, x₂)) = 0   
POL([]) = 0   
POL(delH_in_ga(x₁)) = 2   
POL(delH_out_ga(x₁, x₂)) = 1   
POL(delJ_in_ga(x₁)) = 0   
POL(delJ_out_ga(x₁, x₂)) = 0   
POL(delL_in_gga(x₁, x₂)) = x₁   
POL(delL_out_gga(x₁, x₂, x₃)) = 0   
POL(delN_in_gga(x₁, x₂)) = 2·x₁   
POL(delN_out_gga(x₁, x₂, x₃)) = x₁   
POL(delO_in_gga(x₁, x₂)) = x₁   
POL(delO_out_gga(x₁, x₂, x₃)) = 0   
POL(f) = 2   
POL(maxG_in_ga(x₁)) = 0   
POL(maxG_out_ga(x₁, x₂)) = 2·x₂   
POL(maxI_in_ga(x₁)) = 0   
POL(maxI_out_ga(x₁, x₂)) = 0   
POL(t) = 0   

(59) Obligation:

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

PC_IN_GAAA(T53) → U18_GAAA(T53, maxG_in_ga(T53))
U18_GAAA(T53, maxG_out_ga(T53, T57)) → PP_IN_GGAA(T57, T53)
PP_IN_GGAA(T57, T53) → U20_GGAA(T57, T53, delL_in_gga(T57, T53))
U20_GGAA(T57, T53, delL_out_gga(T57, T53, T87)) → MAXSORTA_IN_GA(T87)
MAXSORTA_IN_GA(.(t, .(t, T53))) → PC_IN_GAAA(T53)
MAXSORTA_IN_GA(.(f, .(f, T53))) → PE_IN_GAAA(T53)
PE_IN_GAAA(T53) → U26_GAAA(T53, maxI_in_ga(T53))
U26_GAAA(T53, maxI_out_ga(T53, T163)) → PR_IN_GGAA(T163, T53)
PR_IN_GGAA(T163, T53) → U28_GGAA(T163, T53, delN_in_gga(T163, T53))
U28_GGAA(T163, T53, delN_out_gga(T163, T53, T193)) → MAXSORTA_IN_GA(T193)
MAXSORTA_IN_GA(.(f, .(t, T251))) → PF_IN_GAAA(T251)
PF_IN_GAAA(T251) → U30_GAAA(T251, maxG_in_ga(T251))
U30_GAAA(T251, maxG_out_ga(T251, T255)) → PS_IN_GGAA(T255, T251)
PS_IN_GGAA(T255, T251) → U32_GGAA(T255, T251, delO_in_gga(T255, T251))
U32_GGAA(T255, T251, delO_out_gga(T255, T251, T259)) → MAXSORTA_IN_GA(T259)

The TRS R consists of the following rules:

delO_in_gga(f, T272) → delO_out_gga(f, T272, .(t, T272))
delO_in_gga(t, T280) → U15_gga(T280, delJ_in_ga(.(t, T280)))
delJ_in_ga(.(t, T234)) → delJ_out_ga(.(t, T234), T234)
U15_gga(T280, delJ_out_ga(.(t, T280), X480)) → delO_out_gga(t, T280, .(f, X480))
maxG_in_ga([]) → maxG_out_ga([], t)
maxG_in_ga(.(t, T72)) → U6_ga(T72, maxG_in_ga(T72))
maxG_in_ga(.(f, T72)) → U7_ga(T72, maxG_in_ga(T72))
U7_ga(T72, maxG_out_ga(T72, T74)) → maxG_out_ga(.(f, T72), T74)
U6_ga(T72, maxG_out_ga(T72, T74)) → maxG_out_ga(.(t, T72), T74)
delN_in_gga(f, T206) → delN_out_gga(f, T206, .(f, T206))
delN_in_gga(t, T224) → U14_gga(T224, delJ_in_ga(T224))
delJ_in_ga([]) → delJ_out_ga([], [])
delJ_in_ga(.(f, T240)) → U11_ga(T240, delJ_in_ga(T240))
U14_gga(T224, delJ_out_ga(T224, X384)) → delN_out_gga(t, T224, .(f, .(f, X384)))
U11_ga(T240, delJ_out_ga(T240, X427)) → delJ_out_ga(.(f, T240), .(f, X427))
maxI_in_ga([]) → maxI_out_ga([], f)
maxI_in_ga(.(f, T178)) → U9_ga(T178, maxI_in_ga(T178))
maxI_in_ga(.(t, T188)) → U10_ga(T188, maxG_in_ga(T188))
U10_ga(T188, maxG_out_ga(T188, T190)) → maxI_out_ga(.(t, T188), T190)
U9_ga(T178, maxI_out_ga(T178, T180)) → maxI_out_ga(.(f, T178), T180)
delL_in_gga(t, T100) → delL_out_gga(t, T100, .(t, T100))
delH_in_ga(.(f, T128)) → delH_out_ga(.(f, T128), T128)
delH_in_ga([]) → delH_out_ga([], [])
delH_in_ga(.(t, T134)) → U8_ga(T134, delH_in_ga(T134))
U12_gga(T118, delH_out_ga(T118, X187)) → delL_out_gga(f, T118, .(t, .(t, X187)))
U8_ga(T134, delH_out_ga(T134, X228)) → delH_out_ga(.(t, T134), .(t, X228))

The set Q consists of the following terms:

maxG_in_ga(x0)
delL_in_gga(x0, x1)
maxI_in_ga(x0)
delN_in_gga(x0, x1)
delO_in_gga(x0, x1)
U6_ga(x0, x1)
U7_ga(x0, x1)
U12_gga(x0, x1)
U9_ga(x0, x1)
U10_ga(x0, x1)
U14_gga(x0, x1)
U15_gga(x0, x1)
delH_in_ga(x0)
delJ_in_ga(x0)
U8_ga(x0, x1)
U11_ga(x0, x1)

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

(60) UsableRulesProof (EQUIVALENT transformation)

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

(61) Obligation:

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

PC_IN_GAAA(T53) → U18_GAAA(T53, maxG_in_ga(T53))
U18_GAAA(T53, maxG_out_ga(T53, T57)) → PP_IN_GGAA(T57, T53)
PP_IN_GGAA(T57, T53) → U20_GGAA(T57, T53, delL_in_gga(T57, T53))
U20_GGAA(T57, T53, delL_out_gga(T57, T53, T87)) → MAXSORTA_IN_GA(T87)
MAXSORTA_IN_GA(.(t, .(t, T53))) → PC_IN_GAAA(T53)
MAXSORTA_IN_GA(.(f, .(f, T53))) → PE_IN_GAAA(T53)
PE_IN_GAAA(T53) → U26_GAAA(T53, maxI_in_ga(T53))
U26_GAAA(T53, maxI_out_ga(T53, T163)) → PR_IN_GGAA(T163, T53)
PR_IN_GGAA(T163, T53) → U28_GGAA(T163, T53, delN_in_gga(T163, T53))
U28_GGAA(T163, T53, delN_out_gga(T163, T53, T193)) → MAXSORTA_IN_GA(T193)
MAXSORTA_IN_GA(.(f, .(t, T251))) → PF_IN_GAAA(T251)
PF_IN_GAAA(T251) → U30_GAAA(T251, maxG_in_ga(T251))
U30_GAAA(T251, maxG_out_ga(T251, T255)) → PS_IN_GGAA(T255, T251)
PS_IN_GGAA(T255, T251) → U32_GGAA(T255, T251, delO_in_gga(T255, T251))
U32_GGAA(T255, T251, delO_out_gga(T255, T251, T259)) → MAXSORTA_IN_GA(T259)

The TRS R consists of the following rules:

delO_in_gga(f, T272) → delO_out_gga(f, T272, .(t, T272))
delO_in_gga(t, T280) → U15_gga(T280, delJ_in_ga(.(t, T280)))
delJ_in_ga(.(t, T234)) → delJ_out_ga(.(t, T234), T234)
U15_gga(T280, delJ_out_ga(.(t, T280), X480)) → delO_out_gga(t, T280, .(f, X480))
maxG_in_ga([]) → maxG_out_ga([], t)
maxG_in_ga(.(t, T72)) → U6_ga(T72, maxG_in_ga(T72))
maxG_in_ga(.(f, T72)) → U7_ga(T72, maxG_in_ga(T72))
U7_ga(T72, maxG_out_ga(T72, T74)) → maxG_out_ga(.(f, T72), T74)
U6_ga(T72, maxG_out_ga(T72, T74)) → maxG_out_ga(.(t, T72), T74)
delN_in_gga(f, T206) → delN_out_gga(f, T206, .(f, T206))
delN_in_gga(t, T224) → U14_gga(T224, delJ_in_ga(T224))
delJ_in_ga([]) → delJ_out_ga([], [])
delJ_in_ga(.(f, T240)) → U11_ga(T240, delJ_in_ga(T240))
U14_gga(T224, delJ_out_ga(T224, X384)) → delN_out_gga(t, T224, .(f, .(f, X384)))
U11_ga(T240, delJ_out_ga(T240, X427)) → delJ_out_ga(.(f, T240), .(f, X427))
maxI_in_ga([]) → maxI_out_ga([], f)
maxI_in_ga(.(f, T178)) → U9_ga(T178, maxI_in_ga(T178))
maxI_in_ga(.(t, T188)) → U10_ga(T188, maxG_in_ga(T188))
U10_ga(T188, maxG_out_ga(T188, T190)) → maxI_out_ga(.(t, T188), T190)
U9_ga(T178, maxI_out_ga(T178, T180)) → maxI_out_ga(.(f, T178), T180)
delL_in_gga(t, T100) → delL_out_gga(t, T100, .(t, T100))

The set Q consists of the following terms:

maxG_in_ga(x0)
delL_in_gga(x0, x1)
maxI_in_ga(x0)
delN_in_gga(x0, x1)
delO_in_gga(x0, x1)
U6_ga(x0, x1)
U7_ga(x0, x1)
U12_gga(x0, x1)
U9_ga(x0, x1)
U10_ga(x0, x1)
U14_gga(x0, x1)
U15_gga(x0, x1)
delH_in_ga(x0)
delJ_in_ga(x0)
U8_ga(x0, x1)
U11_ga(x0, x1)

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

(62) QReductionProof (EQUIVALENT transformation)

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

U12_gga(x0, x1)
delH_in_ga(x0)
U8_ga(x0, x1)

(63) Obligation:

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

PC_IN_GAAA(T53) → U18_GAAA(T53, maxG_in_ga(T53))
U18_GAAA(T53, maxG_out_ga(T53, T57)) → PP_IN_GGAA(T57, T53)
PP_IN_GGAA(T57, T53) → U20_GGAA(T57, T53, delL_in_gga(T57, T53))
U20_GGAA(T57, T53, delL_out_gga(T57, T53, T87)) → MAXSORTA_IN_GA(T87)
MAXSORTA_IN_GA(.(t, .(t, T53))) → PC_IN_GAAA(T53)
MAXSORTA_IN_GA(.(f, .(f, T53))) → PE_IN_GAAA(T53)
PE_IN_GAAA(T53) → U26_GAAA(T53, maxI_in_ga(T53))
U26_GAAA(T53, maxI_out_ga(T53, T163)) → PR_IN_GGAA(T163, T53)
PR_IN_GGAA(T163, T53) → U28_GGAA(T163, T53, delN_in_gga(T163, T53))
U28_GGAA(T163, T53, delN_out_gga(T163, T53, T193)) → MAXSORTA_IN_GA(T193)
MAXSORTA_IN_GA(.(f, .(t, T251))) → PF_IN_GAAA(T251)
PF_IN_GAAA(T251) → U30_GAAA(T251, maxG_in_ga(T251))
U30_GAAA(T251, maxG_out_ga(T251, T255)) → PS_IN_GGAA(T255, T251)
PS_IN_GGAA(T255, T251) → U32_GGAA(T255, T251, delO_in_gga(T255, T251))
U32_GGAA(T255, T251, delO_out_gga(T255, T251, T259)) → MAXSORTA_IN_GA(T259)

The TRS R consists of the following rules:

delO_in_gga(f, T272) → delO_out_gga(f, T272, .(t, T272))
delO_in_gga(t, T280) → U15_gga(T280, delJ_in_ga(.(t, T280)))
delJ_in_ga(.(t, T234)) → delJ_out_ga(.(t, T234), T234)
U15_gga(T280, delJ_out_ga(.(t, T280), X480)) → delO_out_gga(t, T280, .(f, X480))
maxG_in_ga([]) → maxG_out_ga([], t)
maxG_in_ga(.(t, T72)) → U6_ga(T72, maxG_in_ga(T72))
maxG_in_ga(.(f, T72)) → U7_ga(T72, maxG_in_ga(T72))
U7_ga(T72, maxG_out_ga(T72, T74)) → maxG_out_ga(.(f, T72), T74)
U6_ga(T72, maxG_out_ga(T72, T74)) → maxG_out_ga(.(t, T72), T74)
delN_in_gga(f, T206) → delN_out_gga(f, T206, .(f, T206))
delN_in_gga(t, T224) → U14_gga(T224, delJ_in_ga(T224))
delJ_in_ga([]) → delJ_out_ga([], [])
delJ_in_ga(.(f, T240)) → U11_ga(T240, delJ_in_ga(T240))
U14_gga(T224, delJ_out_ga(T224, X384)) → delN_out_gga(t, T224, .(f, .(f, X384)))
U11_ga(T240, delJ_out_ga(T240, X427)) → delJ_out_ga(.(f, T240), .(f, X427))
maxI_in_ga([]) → maxI_out_ga([], f)
maxI_in_ga(.(f, T178)) → U9_ga(T178, maxI_in_ga(T178))
maxI_in_ga(.(t, T188)) → U10_ga(T188, maxG_in_ga(T188))
U10_ga(T188, maxG_out_ga(T188, T190)) → maxI_out_ga(.(t, T188), T190)
U9_ga(T178, maxI_out_ga(T178, T180)) → maxI_out_ga(.(f, T178), T180)
delL_in_gga(t, T100) → delL_out_gga(t, T100, .(t, T100))

The set Q consists of the following terms:

maxG_in_ga(x0)
delL_in_gga(x0, x1)
maxI_in_ga(x0)
delN_in_gga(x0, x1)
delO_in_gga(x0, x1)
U6_ga(x0, x1)
U7_ga(x0, x1)
U9_ga(x0, x1)
U10_ga(x0, x1)
U14_gga(x0, x1)
U15_gga(x0, x1)
delJ_in_ga(x0)
U11_ga(x0, x1)

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

(64) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule PP_IN_GGAA(T57, T53) → U20_GGAA(T57, T53, delL_in_gga(T57, T53)) at position [2] we obtained the following new rules [LPAR04]:

PP_IN_GGAA(t, x0) → U20_GGAA(t, x0, delL_out_gga(t, x0, .(t, x0)))

(65) Obligation:

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

PC_IN_GAAA(T53) → U18_GAAA(T53, maxG_in_ga(T53))
U18_GAAA(T53, maxG_out_ga(T53, T57)) → PP_IN_GGAA(T57, T53)
U20_GGAA(T57, T53, delL_out_gga(T57, T53, T87)) → MAXSORTA_IN_GA(T87)
MAXSORTA_IN_GA(.(t, .(t, T53))) → PC_IN_GAAA(T53)
MAXSORTA_IN_GA(.(f, .(f, T53))) → PE_IN_GAAA(T53)
PE_IN_GAAA(T53) → U26_GAAA(T53, maxI_in_ga(T53))
U26_GAAA(T53, maxI_out_ga(T53, T163)) → PR_IN_GGAA(T163, T53)
PR_IN_GGAA(T163, T53) → U28_GGAA(T163, T53, delN_in_gga(T163, T53))
U28_GGAA(T163, T53, delN_out_gga(T163, T53, T193)) → MAXSORTA_IN_GA(T193)
MAXSORTA_IN_GA(.(f, .(t, T251))) → PF_IN_GAAA(T251)
PF_IN_GAAA(T251) → U30_GAAA(T251, maxG_in_ga(T251))
U30_GAAA(T251, maxG_out_ga(T251, T255)) → PS_IN_GGAA(T255, T251)
PS_IN_GGAA(T255, T251) → U32_GGAA(T255, T251, delO_in_gga(T255, T251))
U32_GGAA(T255, T251, delO_out_gga(T255, T251, T259)) → MAXSORTA_IN_GA(T259)
PP_IN_GGAA(t, x0) → U20_GGAA(t, x0, delL_out_gga(t, x0, .(t, x0)))

The TRS R consists of the following rules:

delO_in_gga(f, T272) → delO_out_gga(f, T272, .(t, T272))
delO_in_gga(t, T280) → U15_gga(T280, delJ_in_ga(.(t, T280)))
delJ_in_ga(.(t, T234)) → delJ_out_ga(.(t, T234), T234)
U15_gga(T280, delJ_out_ga(.(t, T280), X480)) → delO_out_gga(t, T280, .(f, X480))
maxG_in_ga([]) → maxG_out_ga([], t)
maxG_in_ga(.(t, T72)) → U6_ga(T72, maxG_in_ga(T72))
maxG_in_ga(.(f, T72)) → U7_ga(T72, maxG_in_ga(T72))
U7_ga(T72, maxG_out_ga(T72, T74)) → maxG_out_ga(.(f, T72), T74)
U6_ga(T72, maxG_out_ga(T72, T74)) → maxG_out_ga(.(t, T72), T74)
delN_in_gga(f, T206) → delN_out_gga(f, T206, .(f, T206))
delN_in_gga(t, T224) → U14_gga(T224, delJ_in_ga(T224))
delJ_in_ga([]) → delJ_out_ga([], [])
delJ_in_ga(.(f, T240)) → U11_ga(T240, delJ_in_ga(T240))
U14_gga(T224, delJ_out_ga(T224, X384)) → delN_out_gga(t, T224, .(f, .(f, X384)))
U11_ga(T240, delJ_out_ga(T240, X427)) → delJ_out_ga(.(f, T240), .(f, X427))
maxI_in_ga([]) → maxI_out_ga([], f)
maxI_in_ga(.(f, T178)) → U9_ga(T178, maxI_in_ga(T178))
maxI_in_ga(.(t, T188)) → U10_ga(T188, maxG_in_ga(T188))
U10_ga(T188, maxG_out_ga(T188, T190)) → maxI_out_ga(.(t, T188), T190)
U9_ga(T178, maxI_out_ga(T178, T180)) → maxI_out_ga(.(f, T178), T180)
delL_in_gga(t, T100) → delL_out_gga(t, T100, .(t, T100))

The set Q consists of the following terms:

maxG_in_ga(x0)
delL_in_gga(x0, x1)
maxI_in_ga(x0)
delN_in_gga(x0, x1)
delO_in_gga(x0, x1)
U6_ga(x0, x1)
U7_ga(x0, x1)
U9_ga(x0, x1)
U10_ga(x0, x1)
U14_gga(x0, x1)
U15_gga(x0, x1)
delJ_in_ga(x0)
U11_ga(x0, x1)

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

(66) UsableRulesProof (EQUIVALENT transformation)

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

(67) Obligation:

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

PC_IN_GAAA(T53) → U18_GAAA(T53, maxG_in_ga(T53))
U18_GAAA(T53, maxG_out_ga(T53, T57)) → PP_IN_GGAA(T57, T53)
U20_GGAA(T57, T53, delL_out_gga(T57, T53, T87)) → MAXSORTA_IN_GA(T87)
MAXSORTA_IN_GA(.(t, .(t, T53))) → PC_IN_GAAA(T53)
MAXSORTA_IN_GA(.(f, .(f, T53))) → PE_IN_GAAA(T53)
PE_IN_GAAA(T53) → U26_GAAA(T53, maxI_in_ga(T53))
U26_GAAA(T53, maxI_out_ga(T53, T163)) → PR_IN_GGAA(T163, T53)
PR_IN_GGAA(T163, T53) → U28_GGAA(T163, T53, delN_in_gga(T163, T53))
U28_GGAA(T163, T53, delN_out_gga(T163, T53, T193)) → MAXSORTA_IN_GA(T193)
MAXSORTA_IN_GA(.(f, .(t, T251))) → PF_IN_GAAA(T251)
PF_IN_GAAA(T251) → U30_GAAA(T251, maxG_in_ga(T251))
U30_GAAA(T251, maxG_out_ga(T251, T255)) → PS_IN_GGAA(T255, T251)
PS_IN_GGAA(T255, T251) → U32_GGAA(T255, T251, delO_in_gga(T255, T251))
U32_GGAA(T255, T251, delO_out_gga(T255, T251, T259)) → MAXSORTA_IN_GA(T259)
PP_IN_GGAA(t, x0) → U20_GGAA(t, x0, delL_out_gga(t, x0, .(t, x0)))

The TRS R consists of the following rules:

delO_in_gga(f, T272) → delO_out_gga(f, T272, .(t, T272))
delO_in_gga(t, T280) → U15_gga(T280, delJ_in_ga(.(t, T280)))
delJ_in_ga(.(t, T234)) → delJ_out_ga(.(t, T234), T234)
U15_gga(T280, delJ_out_ga(.(t, T280), X480)) → delO_out_gga(t, T280, .(f, X480))
maxG_in_ga([]) → maxG_out_ga([], t)
maxG_in_ga(.(t, T72)) → U6_ga(T72, maxG_in_ga(T72))
maxG_in_ga(.(f, T72)) → U7_ga(T72, maxG_in_ga(T72))
U7_ga(T72, maxG_out_ga(T72, T74)) → maxG_out_ga(.(f, T72), T74)
U6_ga(T72, maxG_out_ga(T72, T74)) → maxG_out_ga(.(t, T72), T74)
delN_in_gga(f, T206) → delN_out_gga(f, T206, .(f, T206))
delN_in_gga(t, T224) → U14_gga(T224, delJ_in_ga(T224))
delJ_in_ga([]) → delJ_out_ga([], [])
delJ_in_ga(.(f, T240)) → U11_ga(T240, delJ_in_ga(T240))
U14_gga(T224, delJ_out_ga(T224, X384)) → delN_out_gga(t, T224, .(f, .(f, X384)))
U11_ga(T240, delJ_out_ga(T240, X427)) → delJ_out_ga(.(f, T240), .(f, X427))
maxI_in_ga([]) → maxI_out_ga([], f)
maxI_in_ga(.(f, T178)) → U9_ga(T178, maxI_in_ga(T178))
maxI_in_ga(.(t, T188)) → U10_ga(T188, maxG_in_ga(T188))
U10_ga(T188, maxG_out_ga(T188, T190)) → maxI_out_ga(.(t, T188), T190)
U9_ga(T178, maxI_out_ga(T178, T180)) → maxI_out_ga(.(f, T178), T180)

The set Q consists of the following terms:

maxG_in_ga(x0)
delL_in_gga(x0, x1)
maxI_in_ga(x0)
delN_in_gga(x0, x1)
delO_in_gga(x0, x1)
U6_ga(x0, x1)
U7_ga(x0, x1)
U9_ga(x0, x1)
U10_ga(x0, x1)
U14_gga(x0, x1)
U15_gga(x0, x1)
delJ_in_ga(x0)
U11_ga(x0, x1)

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

(68) QReductionProof (EQUIVALENT transformation)

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

delL_in_gga(x0, x1)

(69) Obligation:

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

PC_IN_GAAA(T53) → U18_GAAA(T53, maxG_in_ga(T53))
U18_GAAA(T53, maxG_out_ga(T53, T57)) → PP_IN_GGAA(T57, T53)
U20_GGAA(T57, T53, delL_out_gga(T57, T53, T87)) → MAXSORTA_IN_GA(T87)
MAXSORTA_IN_GA(.(t, .(t, T53))) → PC_IN_GAAA(T53)
MAXSORTA_IN_GA(.(f, .(f, T53))) → PE_IN_GAAA(T53)
PE_IN_GAAA(T53) → U26_GAAA(T53, maxI_in_ga(T53))
U26_GAAA(T53, maxI_out_ga(T53, T163)) → PR_IN_GGAA(T163, T53)
PR_IN_GGAA(T163, T53) → U28_GGAA(T163, T53, delN_in_gga(T163, T53))
U28_GGAA(T163, T53, delN_out_gga(T163, T53, T193)) → MAXSORTA_IN_GA(T193)
MAXSORTA_IN_GA(.(f, .(t, T251))) → PF_IN_GAAA(T251)
PF_IN_GAAA(T251) → U30_GAAA(T251, maxG_in_ga(T251))
U30_GAAA(T251, maxG_out_ga(T251, T255)) → PS_IN_GGAA(T255, T251)
PS_IN_GGAA(T255, T251) → U32_GGAA(T255, T251, delO_in_gga(T255, T251))
U32_GGAA(T255, T251, delO_out_gga(T255, T251, T259)) → MAXSORTA_IN_GA(T259)
PP_IN_GGAA(t, x0) → U20_GGAA(t, x0, delL_out_gga(t, x0, .(t, x0)))

The TRS R consists of the following rules:

delO_in_gga(f, T272) → delO_out_gga(f, T272, .(t, T272))
delO_in_gga(t, T280) → U15_gga(T280, delJ_in_ga(.(t, T280)))
delJ_in_ga(.(t, T234)) → delJ_out_ga(.(t, T234), T234)
U15_gga(T280, delJ_out_ga(.(t, T280), X480)) → delO_out_gga(t, T280, .(f, X480))
maxG_in_ga([]) → maxG_out_ga([], t)
maxG_in_ga(.(t, T72)) → U6_ga(T72, maxG_in_ga(T72))
maxG_in_ga(.(f, T72)) → U7_ga(T72, maxG_in_ga(T72))
U7_ga(T72, maxG_out_ga(T72, T74)) → maxG_out_ga(.(f, T72), T74)
U6_ga(T72, maxG_out_ga(T72, T74)) → maxG_out_ga(.(t, T72), T74)
delN_in_gga(f, T206) → delN_out_gga(f, T206, .(f, T206))
delN_in_gga(t, T224) → U14_gga(T224, delJ_in_ga(T224))
delJ_in_ga([]) → delJ_out_ga([], [])
delJ_in_ga(.(f, T240)) → U11_ga(T240, delJ_in_ga(T240))
U14_gga(T224, delJ_out_ga(T224, X384)) → delN_out_gga(t, T224, .(f, .(f, X384)))
U11_ga(T240, delJ_out_ga(T240, X427)) → delJ_out_ga(.(f, T240), .(f, X427))
maxI_in_ga([]) → maxI_out_ga([], f)
maxI_in_ga(.(f, T178)) → U9_ga(T178, maxI_in_ga(T178))
maxI_in_ga(.(t, T188)) → U10_ga(T188, maxG_in_ga(T188))
U10_ga(T188, maxG_out_ga(T188, T190)) → maxI_out_ga(.(t, T188), T190)
U9_ga(T178, maxI_out_ga(T178, T180)) → maxI_out_ga(.(f, T178), T180)

The set Q consists of the following terms:

maxG_in_ga(x0)
maxI_in_ga(x0)
delN_in_gga(x0, x1)
delO_in_gga(x0, x1)
U6_ga(x0, x1)
U7_ga(x0, x1)
U9_ga(x0, x1)
U10_ga(x0, x1)
U14_gga(x0, x1)
U15_gga(x0, x1)
delJ_in_ga(x0)
U11_ga(x0, x1)

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

(70) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule U20_GGAA(T57, T53, delL_out_gga(T57, T53, T87)) → MAXSORTA_IN_GA(T87) we obtained the following new rules [LPAR04]:

U20_GGAA(t, z0, delL_out_gga(t, z0, .(t, z0))) → MAXSORTA_IN_GA(.(t, z0))

(71) Obligation:

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

PC_IN_GAAA(T53) → U18_GAAA(T53, maxG_in_ga(T53))
U18_GAAA(T53, maxG_out_ga(T53, T57)) → PP_IN_GGAA(T57, T53)
MAXSORTA_IN_GA(.(t, .(t, T53))) → PC_IN_GAAA(T53)
MAXSORTA_IN_GA(.(f, .(f, T53))) → PE_IN_GAAA(T53)
PE_IN_GAAA(T53) → U26_GAAA(T53, maxI_in_ga(T53))
U26_GAAA(T53, maxI_out_ga(T53, T163)) → PR_IN_GGAA(T163, T53)
PR_IN_GGAA(T163, T53) → U28_GGAA(T163, T53, delN_in_gga(T163, T53))
U28_GGAA(T163, T53, delN_out_gga(T163, T53, T193)) → MAXSORTA_IN_GA(T193)
MAXSORTA_IN_GA(.(f, .(t, T251))) → PF_IN_GAAA(T251)
PF_IN_GAAA(T251) → U30_GAAA(T251, maxG_in_ga(T251))
U30_GAAA(T251, maxG_out_ga(T251, T255)) → PS_IN_GGAA(T255, T251)
PS_IN_GGAA(T255, T251) → U32_GGAA(T255, T251, delO_in_gga(T255, T251))
U32_GGAA(T255, T251, delO_out_gga(T255, T251, T259)) → MAXSORTA_IN_GA(T259)
PP_IN_GGAA(t, x0) → U20_GGAA(t, x0, delL_out_gga(t, x0, .(t, x0)))
U20_GGAA(t, z0, delL_out_gga(t, z0, .(t, z0))) → MAXSORTA_IN_GA(.(t, z0))

The TRS R consists of the following rules:

delO_in_gga(f, T272) → delO_out_gga(f, T272, .(t, T272))
delO_in_gga(t, T280) → U15_gga(T280, delJ_in_ga(.(t, T280)))
delJ_in_ga(.(t, T234)) → delJ_out_ga(.(t, T234), T234)
U15_gga(T280, delJ_out_ga(.(t, T280), X480)) → delO_out_gga(t, T280, .(f, X480))
maxG_in_ga([]) → maxG_out_ga([], t)
maxG_in_ga(.(t, T72)) → U6_ga(T72, maxG_in_ga(T72))
maxG_in_ga(.(f, T72)) → U7_ga(T72, maxG_in_ga(T72))
U7_ga(T72, maxG_out_ga(T72, T74)) → maxG_out_ga(.(f, T72), T74)
U6_ga(T72, maxG_out_ga(T72, T74)) → maxG_out_ga(.(t, T72), T74)
delN_in_gga(f, T206) → delN_out_gga(f, T206, .(f, T206))
delN_in_gga(t, T224) → U14_gga(T224, delJ_in_ga(T224))
delJ_in_ga([]) → delJ_out_ga([], [])
delJ_in_ga(.(f, T240)) → U11_ga(T240, delJ_in_ga(T240))
U14_gga(T224, delJ_out_ga(T224, X384)) → delN_out_gga(t, T224, .(f, .(f, X384)))
U11_ga(T240, delJ_out_ga(T240, X427)) → delJ_out_ga(.(f, T240), .(f, X427))
maxI_in_ga([]) → maxI_out_ga([], f)
maxI_in_ga(.(f, T178)) → U9_ga(T178, maxI_in_ga(T178))
maxI_in_ga(.(t, T188)) → U10_ga(T188, maxG_in_ga(T188))
U10_ga(T188, maxG_out_ga(T188, T190)) → maxI_out_ga(.(t, T188), T190)
U9_ga(T178, maxI_out_ga(T178, T180)) → maxI_out_ga(.(f, T178), T180)

The set Q consists of the following terms:

maxG_in_ga(x0)
maxI_in_ga(x0)
delN_in_gga(x0, x1)
delO_in_gga(x0, x1)
U6_ga(x0, x1)
U7_ga(x0, x1)
U9_ga(x0, x1)
U10_ga(x0, x1)
U14_gga(x0, x1)
U15_gga(x0, x1)
delJ_in_ga(x0)
U11_ga(x0, x1)

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

(72) DependencyGraphProof (EQUIVALENT transformation)

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

(73) Complex Obligation (AND)

(74) Obligation:

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

U18_GAAA(T53, maxG_out_ga(T53, T57)) → PP_IN_GGAA(T57, T53)
PP_IN_GGAA(t, x0) → U20_GGAA(t, x0, delL_out_gga(t, x0, .(t, x0)))
U20_GGAA(t, z0, delL_out_gga(t, z0, .(t, z0))) → MAXSORTA_IN_GA(.(t, z0))
MAXSORTA_IN_GA(.(t, .(t, T53))) → PC_IN_GAAA(T53)
PC_IN_GAAA(T53) → U18_GAAA(T53, maxG_in_ga(T53))

The TRS R consists of the following rules:

delO_in_gga(f, T272) → delO_out_gga(f, T272, .(t, T272))
delO_in_gga(t, T280) → U15_gga(T280, delJ_in_ga(.(t, T280)))
delJ_in_ga(.(t, T234)) → delJ_out_ga(.(t, T234), T234)
U15_gga(T280, delJ_out_ga(.(t, T280), X480)) → delO_out_gga(t, T280, .(f, X480))
maxG_in_ga([]) → maxG_out_ga([], t)
maxG_in_ga(.(t, T72)) → U6_ga(T72, maxG_in_ga(T72))
maxG_in_ga(.(f, T72)) → U7_ga(T72, maxG_in_ga(T72))
U7_ga(T72, maxG_out_ga(T72, T74)) → maxG_out_ga(.(f, T72), T74)
U6_ga(T72, maxG_out_ga(T72, T74)) → maxG_out_ga(.(t, T72), T74)
delN_in_gga(f, T206) → delN_out_gga(f, T206, .(f, T206))
delN_in_gga(t, T224) → U14_gga(T224, delJ_in_ga(T224))
delJ_in_ga([]) → delJ_out_ga([], [])
delJ_in_ga(.(f, T240)) → U11_ga(T240, delJ_in_ga(T240))
U14_gga(T224, delJ_out_ga(T224, X384)) → delN_out_gga(t, T224, .(f, .(f, X384)))
U11_ga(T240, delJ_out_ga(T240, X427)) → delJ_out_ga(.(f, T240), .(f, X427))
maxI_in_ga([]) → maxI_out_ga([], f)
maxI_in_ga(.(f, T178)) → U9_ga(T178, maxI_in_ga(T178))
maxI_in_ga(.(t, T188)) → U10_ga(T188, maxG_in_ga(T188))
U10_ga(T188, maxG_out_ga(T188, T190)) → maxI_out_ga(.(t, T188), T190)
U9_ga(T178, maxI_out_ga(T178, T180)) → maxI_out_ga(.(f, T178), T180)

The set Q consists of the following terms:

maxG_in_ga(x0)
maxI_in_ga(x0)
delN_in_gga(x0, x1)
delO_in_gga(x0, x1)
U6_ga(x0, x1)
U7_ga(x0, x1)
U9_ga(x0, x1)
U10_ga(x0, x1)
U14_gga(x0, x1)
U15_gga(x0, x1)
delJ_in_ga(x0)
U11_ga(x0, x1)

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

(75) 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.

(76) Obligation:

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

U18_GAAA(T53, maxG_out_ga(T53, T57)) → PP_IN_GGAA(T57, T53)
PP_IN_GGAA(t, x0) → U20_GGAA(t, x0, delL_out_gga(t, x0, .(t, x0)))
U20_GGAA(t, z0, delL_out_gga(t, z0, .(t, z0))) → MAXSORTA_IN_GA(.(t, z0))
MAXSORTA_IN_GA(.(t, .(t, T53))) → PC_IN_GAAA(T53)
PC_IN_GAAA(T53) → U18_GAAA(T53, maxG_in_ga(T53))

The TRS R consists of the following rules:

maxG_in_ga([]) → maxG_out_ga([], t)
maxG_in_ga(.(t, T72)) → U6_ga(T72, maxG_in_ga(T72))
maxG_in_ga(.(f, T72)) → U7_ga(T72, maxG_in_ga(T72))
U7_ga(T72, maxG_out_ga(T72, T74)) → maxG_out_ga(.(f, T72), T74)
U6_ga(T72, maxG_out_ga(T72, T74)) → maxG_out_ga(.(t, T72), T74)

The set Q consists of the following terms:

maxG_in_ga(x0)
maxI_in_ga(x0)
delN_in_gga(x0, x1)
delO_in_gga(x0, x1)
U6_ga(x0, x1)
U7_ga(x0, x1)
U9_ga(x0, x1)
U10_ga(x0, x1)
U14_gga(x0, x1)
U15_gga(x0, x1)
delJ_in_ga(x0)
U11_ga(x0, x1)

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

(77) 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].

maxI_in_ga(x0)
delN_in_gga(x0, x1)
delO_in_gga(x0, x1)
U9_ga(x0, x1)
U10_ga(x0, x1)
U14_gga(x0, x1)
U15_gga(x0, x1)
delJ_in_ga(x0)
U11_ga(x0, x1)

(78) Obligation:

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

U18_GAAA(T53, maxG_out_ga(T53, T57)) → PP_IN_GGAA(T57, T53)
PP_IN_GGAA(t, x0) → U20_GGAA(t, x0, delL_out_gga(t, x0, .(t, x0)))
U20_GGAA(t, z0, delL_out_gga(t, z0, .(t, z0))) → MAXSORTA_IN_GA(.(t, z0))
MAXSORTA_IN_GA(.(t, .(t, T53))) → PC_IN_GAAA(T53)
PC_IN_GAAA(T53) → U18_GAAA(T53, maxG_in_ga(T53))

The TRS R consists of the following rules:

maxG_in_ga([]) → maxG_out_ga([], t)
maxG_in_ga(.(t, T72)) → U6_ga(T72, maxG_in_ga(T72))
maxG_in_ga(.(f, T72)) → U7_ga(T72, maxG_in_ga(T72))
U7_ga(T72, maxG_out_ga(T72, T74)) → maxG_out_ga(.(f, T72), T74)
U6_ga(T72, maxG_out_ga(T72, T74)) → maxG_out_ga(.(t, T72), T74)

The set Q consists of the following terms:

maxG_in_ga(x0)
U6_ga(x0, x1)
U7_ga(x0, x1)

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

(79) QDPOrderProof (EQUIVALENT transformation)

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

The following pairs can be oriented strictly and are deleted.

MAXSORTA_IN_GA(.(t, .(t, T53))) → PC_IN_GAAA(T53)

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

POL(.(x₁, x₂)) = x₁ + x₂   
POL(MAXSORTA_IN_GA(x₁)) = x₁   
POL(PC_IN_GAAA(x₁)) = 1 + x₁   
POL(PP_IN_GGAA(x₁, x₂)) = 1 + x₂   
POL(U18_GAAA(x₁, x₂)) = 1 + x₁   
POL(U20_GGAA(x₁, x₂, x₃)) = x₃   
POL(U6_ga(x₁, x₂)) = 0   
POL(U7_ga(x₁, x₂)) = 0   
POL([]) = 0   
POL(delL_out_gga(x₁, x₂, x₃)) = x₃   
POL(f) = 0   
POL(maxG_in_ga(x₁)) = 0   
POL(maxG_out_ga(x₁, x₂)) = 0   
POL(t) = 1   

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
none

(80) Obligation:

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

U18_GAAA(T53, maxG_out_ga(T53, T57)) → PP_IN_GGAA(T57, T53)
PP_IN_GGAA(t, x0) → U20_GGAA(t, x0, delL_out_gga(t, x0, .(t, x0)))
U20_GGAA(t, z0, delL_out_gga(t, z0, .(t, z0))) → MAXSORTA_IN_GA(.(t, z0))
PC_IN_GAAA(T53) → U18_GAAA(T53, maxG_in_ga(T53))

The TRS R consists of the following rules:

maxG_in_ga([]) → maxG_out_ga([], t)
maxG_in_ga(.(t, T72)) → U6_ga(T72, maxG_in_ga(T72))
maxG_in_ga(.(f, T72)) → U7_ga(T72, maxG_in_ga(T72))
U7_ga(T72, maxG_out_ga(T72, T74)) → maxG_out_ga(.(f, T72), T74)
U6_ga(T72, maxG_out_ga(T72, T74)) → maxG_out_ga(.(t, T72), T74)

The set Q consists of the following terms:

maxG_in_ga(x0)
U6_ga(x0, x1)
U7_ga(x0, x1)

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

(81) DependencyGraphProof (EQUIVALENT transformation)

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

(82) TRUE

(83) Obligation:

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

MAXSORTA_IN_GA(.(f, .(f, T53))) → PE_IN_GAAA(T53)
PE_IN_GAAA(T53) → U26_GAAA(T53, maxI_in_ga(T53))
U26_GAAA(T53, maxI_out_ga(T53, T163)) → PR_IN_GGAA(T163, T53)
PR_IN_GGAA(T163, T53) → U28_GGAA(T163, T53, delN_in_gga(T163, T53))
U28_GGAA(T163, T53, delN_out_gga(T163, T53, T193)) → MAXSORTA_IN_GA(T193)
MAXSORTA_IN_GA(.(f, .(t, T251))) → PF_IN_GAAA(T251)
PF_IN_GAAA(T251) → U30_GAAA(T251, maxG_in_ga(T251))
U30_GAAA(T251, maxG_out_ga(T251, T255)) → PS_IN_GGAA(T255, T251)
PS_IN_GGAA(T255, T251) → U32_GGAA(T255, T251, delO_in_gga(T255, T251))
U32_GGAA(T255, T251, delO_out_gga(T255, T251, T259)) → MAXSORTA_IN_GA(T259)

The TRS R consists of the following rules:

delO_in_gga(f, T272) → delO_out_gga(f, T272, .(t, T272))
delO_in_gga(t, T280) → U15_gga(T280, delJ_in_ga(.(t, T280)))
delJ_in_ga(.(t, T234)) → delJ_out_ga(.(t, T234), T234)
U15_gga(T280, delJ_out_ga(.(t, T280), X480)) → delO_out_gga(t, T280, .(f, X480))
maxG_in_ga([]) → maxG_out_ga([], t)
maxG_in_ga(.(t, T72)) → U6_ga(T72, maxG_in_ga(T72))
maxG_in_ga(.(f, T72)) → U7_ga(T72, maxG_in_ga(T72))
U7_ga(T72, maxG_out_ga(T72, T74)) → maxG_out_ga(.(f, T72), T74)
U6_ga(T72, maxG_out_ga(T72, T74)) → maxG_out_ga(.(t, T72), T74)
delN_in_gga(f, T206) → delN_out_gga(f, T206, .(f, T206))
delN_in_gga(t, T224) → U14_gga(T224, delJ_in_ga(T224))
delJ_in_ga([]) → delJ_out_ga([], [])
delJ_in_ga(.(f, T240)) → U11_ga(T240, delJ_in_ga(T240))
U14_gga(T224, delJ_out_ga(T224, X384)) → delN_out_gga(t, T224, .(f, .(f, X384)))
U11_ga(T240, delJ_out_ga(T240, X427)) → delJ_out_ga(.(f, T240), .(f, X427))
maxI_in_ga([]) → maxI_out_ga([], f)
maxI_in_ga(.(f, T178)) → U9_ga(T178, maxI_in_ga(T178))
maxI_in_ga(.(t, T188)) → U10_ga(T188, maxG_in_ga(T188))
U10_ga(T188, maxG_out_ga(T188, T190)) → maxI_out_ga(.(t, T188), T190)
U9_ga(T178, maxI_out_ga(T178, T180)) → maxI_out_ga(.(f, T178), T180)

The set Q consists of the following terms:

maxG_in_ga(x0)
maxI_in_ga(x0)
delN_in_gga(x0, x1)
delO_in_gga(x0, x1)
U6_ga(x0, x1)
U7_ga(x0, x1)
U9_ga(x0, x1)
U10_ga(x0, x1)
U14_gga(x0, x1)
U15_gga(x0, x1)
delJ_in_ga(x0)
U11_ga(x0, x1)

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

(84) QDPQMonotonicMRRProof (EQUIVALENT transformation)

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

Strictly oriented rules of the TRS R:

delO_in_gga(f, T272) → delO_out_gga(f, T272, .(t, T272))

Used ordering: Polynomial interpretation [POLO]:

POL(.(x₁, x₂)) = 2·x₂   
POL(MAXSORTA_IN_GA(x₁)) = 0   
POL(PE_IN_GAAA(x₁)) = 0   
POL(PF_IN_GAAA(x₁)) = 0   
POL(PR_IN_GGAA(x₁, x₂)) = 0   
POL(PS_IN_GGAA(x₁, x₂)) = x₁   
POL(U10_ga(x₁, x₂)) = 0   
POL(U11_ga(x₁, x₂)) = 0   
POL(U14_gga(x₁, x₂)) = 0   
POL(U15_gga(x₁, x₂)) = 0   
POL(U26_GAAA(x₁, x₂)) = 0   
POL(U28_GGAA(x₁, x₂, x₃)) = 0   
POL(U30_GAAA(x₁, x₂)) = 2·x₂   
POL(U32_GGAA(x₁, x₂, x₃)) = x₃   
POL(U6_ga(x₁, x₂)) = x₂   
POL(U7_ga(x₁, x₂)) = 2·x₂   
POL(U9_ga(x₁, x₂)) = 0   
POL([]) = 0   
POL(delJ_in_ga(x₁)) = 0   
POL(delJ_out_ga(x₁, x₂)) = 0   
POL(delN_in_gga(x₁, x₂)) = 0   
POL(delN_out_gga(x₁, x₂, x₃)) = 0   
POL(delO_in_gga(x₁, x₂)) = x₁   
POL(delO_out_gga(x₁, x₂, x₃)) = 0   
POL(f) = 2   
POL(maxG_in_ga(x₁)) = 0   
POL(maxG_out_ga(x₁, x₂)) = x₂   
POL(maxI_in_ga(x₁)) = 2·x₁   
POL(maxI_out_ga(x₁, x₂)) = 0   
POL(t) = 0   

(85) Obligation:

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

MAXSORTA_IN_GA(.(f, .(f, T53))) → PE_IN_GAAA(T53)
PE_IN_GAAA(T53) → U26_GAAA(T53, maxI_in_ga(T53))
U26_GAAA(T53, maxI_out_ga(T53, T163)) → PR_IN_GGAA(T163, T53)
PR_IN_GGAA(T163, T53) → U28_GGAA(T163, T53, delN_in_gga(T163, T53))
U28_GGAA(T163, T53, delN_out_gga(T163, T53, T193)) → MAXSORTA_IN_GA(T193)
MAXSORTA_IN_GA(.(f, .(t, T251))) → PF_IN_GAAA(T251)
PF_IN_GAAA(T251) → U30_GAAA(T251, maxG_in_ga(T251))
U30_GAAA(T251, maxG_out_ga(T251, T255)) → PS_IN_GGAA(T255, T251)
PS_IN_GGAA(T255, T251) → U32_GGAA(T255, T251, delO_in_gga(T255, T251))
U32_GGAA(T255, T251, delO_out_gga(T255, T251, T259)) → MAXSORTA_IN_GA(T259)

The TRS R consists of the following rules:

delO_in_gga(t, T280) → U15_gga(T280, delJ_in_ga(.(t, T280)))
delJ_in_ga(.(t, T234)) → delJ_out_ga(.(t, T234), T234)
U15_gga(T280, delJ_out_ga(.(t, T280), X480)) → delO_out_gga(t, T280, .(f, X480))
maxG_in_ga([]) → maxG_out_ga([], t)
maxG_in_ga(.(t, T72)) → U6_ga(T72, maxG_in_ga(T72))
maxG_in_ga(.(f, T72)) → U7_ga(T72, maxG_in_ga(T72))
U7_ga(T72, maxG_out_ga(T72, T74)) → maxG_out_ga(.(f, T72), T74)
U6_ga(T72, maxG_out_ga(T72, T74)) → maxG_out_ga(.(t, T72), T74)
delN_in_gga(f, T206) → delN_out_gga(f, T206, .(f, T206))
delN_in_gga(t, T224) → U14_gga(T224, delJ_in_ga(T224))
delJ_in_ga([]) → delJ_out_ga([], [])
delJ_in_ga(.(f, T240)) → U11_ga(T240, delJ_in_ga(T240))
U14_gga(T224, delJ_out_ga(T224, X384)) → delN_out_gga(t, T224, .(f, .(f, X384)))
U11_ga(T240, delJ_out_ga(T240, X427)) → delJ_out_ga(.(f, T240), .(f, X427))
maxI_in_ga([]) → maxI_out_ga([], f)
maxI_in_ga(.(f, T178)) → U9_ga(T178, maxI_in_ga(T178))
maxI_in_ga(.(t, T188)) → U10_ga(T188, maxG_in_ga(T188))
U10_ga(T188, maxG_out_ga(T188, T190)) → maxI_out_ga(.(t, T188), T190)
U9_ga(T178, maxI_out_ga(T178, T180)) → maxI_out_ga(.(f, T178), T180)

The set Q consists of the following terms:

maxG_in_ga(x0)
maxI_in_ga(x0)
delN_in_gga(x0, x1)
delO_in_gga(x0, x1)
U6_ga(x0, x1)
U7_ga(x0, x1)
U9_ga(x0, x1)
U10_ga(x0, x1)
U14_gga(x0, x1)
U15_gga(x0, x1)
delJ_in_ga(x0)
U11_ga(x0, x1)

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

(86) QDPOrderProof (EQUIVALENT transformation)

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

The following pairs can be oriented strictly and are deleted.

U32_GGAA(T255, T251, delO_out_gga(T255, T251, T259)) → MAXSORTA_IN_GA(T259)

The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation:

POL( U26_GAAA(x₁, x₂) ) = 2x₁

POL( maxI_in_ga(x₁) ) = 0

POL( [] ) = 0

POL( maxI_out_ga(x₁, x₂) ) = max{0, 2x₁ + x₂ - 2}

POL( f ) = 0

POL( .(x₁, x₂) ) = x₁ + 2x₂

POL( U9_ga(x₁, x₂) ) = 2x₁ + 2

POL( t ) = 1

POL( U10_ga(x₁, x₂) ) = x₁ + 2

POL( maxG_in_ga(x₁) ) = 1

POL( U28_GGAA(x₁, ..., x₃) ) = x₃

POL( delN_in_gga(x₁, x₂) ) = 2x₂

POL( delN_out_gga(x₁, ..., x₃) ) = x₃

POL( U14_gga(x₁, x₂) ) = 2x₂

POL( delJ_in_ga(x₁) ) = x₁

POL( U30_GAAA(x₁, x₂) ) = 2x₁ + 2x₂

POL( maxG_out_ga(x₁, x₂) ) = x₂

POL( U6_ga(x₁, x₂) ) = x₂

POL( U7_ga(x₁, x₂) ) = x₂

POL( U32_GGAA(x₁, ..., x₃) ) = max{0, x₃ - 1}

POL( delO_in_gga(x₁, x₂) ) = 2x₁ + 2x₂ + 1

POL( U15_gga(x₁, x₂) ) = x₂ + 2

POL( delJ_out_ga(x₁, x₂) ) = 2x₂

POL( delO_out_gga(x₁, ..., x₃) ) = x₃ + 2

POL( U11_ga(x₁, x₂) ) = 2x₂

POL( MAXSORTA_IN_GA(x₁) ) = x₁

POL( PE_IN_GAAA(x₁) ) = 2x₁

POL( PR_IN_GGAA(x₁, x₂) ) = 2x₂

POL( PF_IN_GAAA(x₁) ) = 2x₁ + 2

POL( PS_IN_GGAA(x₁, x₂) ) = 2x₁ + 2x₂

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

delN_in_gga(f, T206) → delN_out_gga(f, T206, .(f, T206))
delN_in_gga(t, T224) → U14_gga(T224, delJ_in_ga(T224))
maxG_in_ga([]) → maxG_out_ga([], t)
maxG_in_ga(.(t, T72)) → U6_ga(T72, maxG_in_ga(T72))
maxG_in_ga(.(f, T72)) → U7_ga(T72, maxG_in_ga(T72))
delO_in_gga(t, T280) → U15_gga(T280, delJ_in_ga(.(t, T280)))
delJ_in_ga(.(t, T234)) → delJ_out_ga(.(t, T234), T234)
U15_gga(T280, delJ_out_ga(.(t, T280), X480)) → delO_out_gga(t, T280, .(f, X480))
delJ_in_ga([]) → delJ_out_ga([], [])
delJ_in_ga(.(f, T240)) → U11_ga(T240, delJ_in_ga(T240))
U14_gga(T224, delJ_out_ga(T224, X384)) → delN_out_gga(t, T224, .(f, .(f, X384)))
U11_ga(T240, delJ_out_ga(T240, X427)) → delJ_out_ga(.(f, T240), .(f, X427))
U7_ga(T72, maxG_out_ga(T72, T74)) → maxG_out_ga(.(f, T72), T74)
U6_ga(T72, maxG_out_ga(T72, T74)) → maxG_out_ga(.(t, T72), T74)

(87) Obligation:

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

MAXSORTA_IN_GA(.(f, .(f, T53))) → PE_IN_GAAA(T53)
PE_IN_GAAA(T53) → U26_GAAA(T53, maxI_in_ga(T53))
U26_GAAA(T53, maxI_out_ga(T53, T163)) → PR_IN_GGAA(T163, T53)
PR_IN_GGAA(T163, T53) → U28_GGAA(T163, T53, delN_in_gga(T163, T53))
U28_GGAA(T163, T53, delN_out_gga(T163, T53, T193)) → MAXSORTA_IN_GA(T193)
MAXSORTA_IN_GA(.(f, .(t, T251))) → PF_IN_GAAA(T251)
PF_IN_GAAA(T251) → U30_GAAA(T251, maxG_in_ga(T251))
U30_GAAA(T251, maxG_out_ga(T251, T255)) → PS_IN_GGAA(T255, T251)
PS_IN_GGAA(T255, T251) → U32_GGAA(T255, T251, delO_in_gga(T255, T251))

The TRS R consists of the following rules:

delO_in_gga(t, T280) → U15_gga(T280, delJ_in_ga(.(t, T280)))
delJ_in_ga(.(t, T234)) → delJ_out_ga(.(t, T234), T234)
U15_gga(T280, delJ_out_ga(.(t, T280), X480)) → delO_out_gga(t, T280, .(f, X480))
maxG_in_ga([]) → maxG_out_ga([], t)
maxG_in_ga(.(t, T72)) → U6_ga(T72, maxG_in_ga(T72))
maxG_in_ga(.(f, T72)) → U7_ga(T72, maxG_in_ga(T72))
U7_ga(T72, maxG_out_ga(T72, T74)) → maxG_out_ga(.(f, T72), T74)
U6_ga(T72, maxG_out_ga(T72, T74)) → maxG_out_ga(.(t, T72), T74)
delN_in_gga(f, T206) → delN_out_gga(f, T206, .(f, T206))
delN_in_gga(t, T224) → U14_gga(T224, delJ_in_ga(T224))
delJ_in_ga([]) → delJ_out_ga([], [])
delJ_in_ga(.(f, T240)) → U11_ga(T240, delJ_in_ga(T240))
U14_gga(T224, delJ_out_ga(T224, X384)) → delN_out_gga(t, T224, .(f, .(f, X384)))
U11_ga(T240, delJ_out_ga(T240, X427)) → delJ_out_ga(.(f, T240), .(f, X427))
maxI_in_ga([]) → maxI_out_ga([], f)
maxI_in_ga(.(f, T178)) → U9_ga(T178, maxI_in_ga(T178))
maxI_in_ga(.(t, T188)) → U10_ga(T188, maxG_in_ga(T188))
U10_ga(T188, maxG_out_ga(T188, T190)) → maxI_out_ga(.(t, T188), T190)
U9_ga(T178, maxI_out_ga(T178, T180)) → maxI_out_ga(.(f, T178), T180)

The set Q consists of the following terms:

maxG_in_ga(x0)
maxI_in_ga(x0)
delN_in_gga(x0, x1)
delO_in_gga(x0, x1)
U6_ga(x0, x1)
U7_ga(x0, x1)
U9_ga(x0, x1)
U10_ga(x0, x1)
U14_gga(x0, x1)
U15_gga(x0, x1)
delJ_in_ga(x0)
U11_ga(x0, x1)

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

(88) DependencyGraphProof (EQUIVALENT transformation)

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

(89) Obligation:

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

PE_IN_GAAA(T53) → U26_GAAA(T53, maxI_in_ga(T53))
U26_GAAA(T53, maxI_out_ga(T53, T163)) → PR_IN_GGAA(T163, T53)
PR_IN_GGAA(T163, T53) → U28_GGAA(T163, T53, delN_in_gga(T163, T53))
U28_GGAA(T163, T53, delN_out_gga(T163, T53, T193)) → MAXSORTA_IN_GA(T193)
MAXSORTA_IN_GA(.(f, .(f, T53))) → PE_IN_GAAA(T53)

The TRS R consists of the following rules:

delO_in_gga(t, T280) → U15_gga(T280, delJ_in_ga(.(t, T280)))
delJ_in_ga(.(t, T234)) → delJ_out_ga(.(t, T234), T234)
U15_gga(T280, delJ_out_ga(.(t, T280), X480)) → delO_out_gga(t, T280, .(f, X480))
maxG_in_ga([]) → maxG_out_ga([], t)
maxG_in_ga(.(t, T72)) → U6_ga(T72, maxG_in_ga(T72))
maxG_in_ga(.(f, T72)) → U7_ga(T72, maxG_in_ga(T72))
U7_ga(T72, maxG_out_ga(T72, T74)) → maxG_out_ga(.(f, T72), T74)
U6_ga(T72, maxG_out_ga(T72, T74)) → maxG_out_ga(.(t, T72), T74)
delN_in_gga(f, T206) → delN_out_gga(f, T206, .(f, T206))
delN_in_gga(t, T224) → U14_gga(T224, delJ_in_ga(T224))
delJ_in_ga([]) → delJ_out_ga([], [])
delJ_in_ga(.(f, T240)) → U11_ga(T240, delJ_in_ga(T240))
U14_gga(T224, delJ_out_ga(T224, X384)) → delN_out_gga(t, T224, .(f, .(f, X384)))
U11_ga(T240, delJ_out_ga(T240, X427)) → delJ_out_ga(.(f, T240), .(f, X427))
maxI_in_ga([]) → maxI_out_ga([], f)
maxI_in_ga(.(f, T178)) → U9_ga(T178, maxI_in_ga(T178))
maxI_in_ga(.(t, T188)) → U10_ga(T188, maxG_in_ga(T188))
U10_ga(T188, maxG_out_ga(T188, T190)) → maxI_out_ga(.(t, T188), T190)
U9_ga(T178, maxI_out_ga(T178, T180)) → maxI_out_ga(.(f, T178), T180)

The set Q consists of the following terms:

maxG_in_ga(x0)
maxI_in_ga(x0)
delN_in_gga(x0, x1)
delO_in_gga(x0, x1)
U6_ga(x0, x1)
U7_ga(x0, x1)
U9_ga(x0, x1)
U10_ga(x0, x1)
U14_gga(x0, x1)
U15_gga(x0, x1)
delJ_in_ga(x0)
U11_ga(x0, x1)

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

(90) 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.

(91) Obligation:

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

PE_IN_GAAA(T53) → U26_GAAA(T53, maxI_in_ga(T53))
U26_GAAA(T53, maxI_out_ga(T53, T163)) → PR_IN_GGAA(T163, T53)
PR_IN_GGAA(T163, T53) → U28_GGAA(T163, T53, delN_in_gga(T163, T53))
U28_GGAA(T163, T53, delN_out_gga(T163, T53, T193)) → MAXSORTA_IN_GA(T193)
MAXSORTA_IN_GA(.(f, .(f, T53))) → PE_IN_GAAA(T53)

The TRS R consists of the following rules:

delN_in_gga(f, T206) → delN_out_gga(f, T206, .(f, T206))
delN_in_gga(t, T224) → U14_gga(T224, delJ_in_ga(T224))
delJ_in_ga(.(t, T234)) → delJ_out_ga(.(t, T234), T234)
delJ_in_ga([]) → delJ_out_ga([], [])
delJ_in_ga(.(f, T240)) → U11_ga(T240, delJ_in_ga(T240))
U14_gga(T224, delJ_out_ga(T224, X384)) → delN_out_gga(t, T224, .(f, .(f, X384)))
U11_ga(T240, delJ_out_ga(T240, X427)) → delJ_out_ga(.(f, T240), .(f, X427))
maxI_in_ga([]) → maxI_out_ga([], f)
maxI_in_ga(.(f, T178)) → U9_ga(T178, maxI_in_ga(T178))
maxI_in_ga(.(t, T188)) → U10_ga(T188, maxG_in_ga(T188))
maxG_in_ga([]) → maxG_out_ga([], t)
maxG_in_ga(.(t, T72)) → U6_ga(T72, maxG_in_ga(T72))
maxG_in_ga(.(f, T72)) → U7_ga(T72, maxG_in_ga(T72))
U10_ga(T188, maxG_out_ga(T188, T190)) → maxI_out_ga(.(t, T188), T190)
U7_ga(T72, maxG_out_ga(T72, T74)) → maxG_out_ga(.(f, T72), T74)
U6_ga(T72, maxG_out_ga(T72, T74)) → maxG_out_ga(.(t, T72), T74)
U9_ga(T178, maxI_out_ga(T178, T180)) → maxI_out_ga(.(f, T178), T180)

The set Q consists of the following terms:

maxG_in_ga(x0)
maxI_in_ga(x0)
delN_in_gga(x0, x1)
delO_in_gga(x0, x1)
U6_ga(x0, x1)
U7_ga(x0, x1)
U9_ga(x0, x1)
U10_ga(x0, x1)
U14_gga(x0, x1)
U15_gga(x0, x1)
delJ_in_ga(x0)
U11_ga(x0, x1)

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

(92) 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].

delO_in_gga(x0, x1)
U15_gga(x0, x1)

(93) Obligation:

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

PE_IN_GAAA(T53) → U26_GAAA(T53, maxI_in_ga(T53))
U26_GAAA(T53, maxI_out_ga(T53, T163)) → PR_IN_GGAA(T163, T53)
PR_IN_GGAA(T163, T53) → U28_GGAA(T163, T53, delN_in_gga(T163, T53))
U28_GGAA(T163, T53, delN_out_gga(T163, T53, T193)) → MAXSORTA_IN_GA(T193)
MAXSORTA_IN_GA(.(f, .(f, T53))) → PE_IN_GAAA(T53)

The TRS R consists of the following rules:

delN_in_gga(f, T206) → delN_out_gga(f, T206, .(f, T206))
delN_in_gga(t, T224) → U14_gga(T224, delJ_in_ga(T224))
delJ_in_ga(.(t, T234)) → delJ_out_ga(.(t, T234), T234)
delJ_in_ga([]) → delJ_out_ga([], [])
delJ_in_ga(.(f, T240)) → U11_ga(T240, delJ_in_ga(T240))
U14_gga(T224, delJ_out_ga(T224, X384)) → delN_out_gga(t, T224, .(f, .(f, X384)))
U11_ga(T240, delJ_out_ga(T240, X427)) → delJ_out_ga(.(f, T240), .(f, X427))
maxI_in_ga([]) → maxI_out_ga([], f)
maxI_in_ga(.(f, T178)) → U9_ga(T178, maxI_in_ga(T178))
maxI_in_ga(.(t, T188)) → U10_ga(T188, maxG_in_ga(T188))
maxG_in_ga([]) → maxG_out_ga([], t)
maxG_in_ga(.(t, T72)) → U6_ga(T72, maxG_in_ga(T72))
maxG_in_ga(.(f, T72)) → U7_ga(T72, maxG_in_ga(T72))
U10_ga(T188, maxG_out_ga(T188, T190)) → maxI_out_ga(.(t, T188), T190)
U7_ga(T72, maxG_out_ga(T72, T74)) → maxG_out_ga(.(f, T72), T74)
U6_ga(T72, maxG_out_ga(T72, T74)) → maxG_out_ga(.(t, T72), T74)
U9_ga(T178, maxI_out_ga(T178, T180)) → maxI_out_ga(.(f, T178), T180)

The set Q consists of the following terms:

maxG_in_ga(x0)
maxI_in_ga(x0)
delN_in_gga(x0, x1)
U6_ga(x0, x1)
U7_ga(x0, x1)
U9_ga(x0, x1)
U10_ga(x0, x1)
U14_gga(x0, x1)
delJ_in_ga(x0)
U11_ga(x0, x1)

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

(94) QDPQMonotonicMRRProof (EQUIVALENT transformation)

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

Strictly oriented rules of the TRS R:

delJ_in_ga(.(t, T234)) → delJ_out_ga(.(t, T234), T234)

Used ordering: Polynomial interpretation [POLO]:

POL(.(x₁, x₂)) = x₁ + x₂   
POL(MAXSORTA_IN_GA(x₁)) = x₁   
POL(PE_IN_GAAA(x₁)) = x₁   
POL(PR_IN_GGAA(x₁, x₂)) = x₂   
POL(U10_ga(x₁, x₂)) = 0   
POL(U11_ga(x₁, x₂)) = x₂   
POL(U14_gga(x₁, x₂)) = x₂   
POL(U26_GAAA(x₁, x₂)) = x₁   
POL(U28_GGAA(x₁, x₂, x₃)) = x₃   
POL(U6_ga(x₁, x₂)) = 0   
POL(U7_ga(x₁, x₂)) = 0   
POL(U9_ga(x₁, x₂)) = 0   
POL([]) = 0   
POL(delJ_in_ga(x₁)) = x₁   
POL(delJ_out_ga(x₁, x₂)) = x₂   
POL(delN_in_gga(x₁, x₂)) = x₂   
POL(delN_out_gga(x₁, x₂, x₃)) = x₃   
POL(f) = 0   
POL(maxG_in_ga(x₁)) = 0   
POL(maxG_out_ga(x₁, x₂)) = 0   
POL(maxI_in_ga(x₁)) = 0   
POL(maxI_out_ga(x₁, x₂)) = 0   
POL(t) = 1   

(95) Obligation:

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

PE_IN_GAAA(T53) → U26_GAAA(T53, maxI_in_ga(T53))
U26_GAAA(T53, maxI_out_ga(T53, T163)) → PR_IN_GGAA(T163, T53)
PR_IN_GGAA(T163, T53) → U28_GGAA(T163, T53, delN_in_gga(T163, T53))
U28_GGAA(T163, T53, delN_out_gga(T163, T53, T193)) → MAXSORTA_IN_GA(T193)
MAXSORTA_IN_GA(.(f, .(f, T53))) → PE_IN_GAAA(T53)

The TRS R consists of the following rules:

delN_in_gga(f, T206) → delN_out_gga(f, T206, .(f, T206))
delN_in_gga(t, T224) → U14_gga(T224, delJ_in_ga(T224))
delJ_in_ga([]) → delJ_out_ga([], [])
delJ_in_ga(.(f, T240)) → U11_ga(T240, delJ_in_ga(T240))
U14_gga(T224, delJ_out_ga(T224, X384)) → delN_out_gga(t, T224, .(f, .(f, X384)))
U11_ga(T240, delJ_out_ga(T240, X427)) → delJ_out_ga(.(f, T240), .(f, X427))
maxI_in_ga([]) → maxI_out_ga([], f)
maxI_in_ga(.(f, T178)) → U9_ga(T178, maxI_in_ga(T178))
maxI_in_ga(.(t, T188)) → U10_ga(T188, maxG_in_ga(T188))
maxG_in_ga([]) → maxG_out_ga([], t)
maxG_in_ga(.(t, T72)) → U6_ga(T72, maxG_in_ga(T72))
maxG_in_ga(.(f, T72)) → U7_ga(T72, maxG_in_ga(T72))
U10_ga(T188, maxG_out_ga(T188, T190)) → maxI_out_ga(.(t, T188), T190)
U7_ga(T72, maxG_out_ga(T72, T74)) → maxG_out_ga(.(f, T72), T74)
U6_ga(T72, maxG_out_ga(T72, T74)) → maxG_out_ga(.(t, T72), T74)
U9_ga(T178, maxI_out_ga(T178, T180)) → maxI_out_ga(.(f, T178), T180)

The set Q consists of the following terms:

maxG_in_ga(x0)
maxI_in_ga(x0)
delN_in_gga(x0, x1)
U6_ga(x0, x1)
U7_ga(x0, x1)
U9_ga(x0, x1)
U10_ga(x0, x1)
U14_gga(x0, x1)
delJ_in_ga(x0)
U11_ga(x0, x1)

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

(96) QDPQMonotonicMRRProof (EQUIVALENT transformation)

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

Strictly oriented rules of the TRS R:

delN_in_gga(f, T206) → delN_out_gga(f, T206, .(f, T206))

Used ordering: Polynomial interpretation [POLO]:

POL(.(x₁, x₂)) = 1 + x₂   
POL(MAXSORTA_IN_GA(x₁)) = x₁   
POL(PE_IN_GAAA(x₁)) = 2 + x₁   
POL(PR_IN_GGAA(x₁, x₂)) = 2 + x₂   
POL(U10_ga(x₁, x₂)) = x₁   
POL(U11_ga(x₁, x₂)) = 1 + x₂   
POL(U14_gga(x₁, x₂)) = 2 + x₂   
POL(U26_GAAA(x₁, x₂)) = 2 + x₁   
POL(U28_GGAA(x₁, x₂, x₃)) = x₃   
POL(U6_ga(x₁, x₂)) = 0   
POL(U7_ga(x₁, x₂)) = 0   
POL(U9_ga(x₁, x₂)) = 1   
POL([]) = 0   
POL(delJ_in_ga(x₁)) = x₁   
POL(delJ_out_ga(x₁, x₂)) = x₂   
POL(delN_in_gga(x₁, x₂)) = 2 + x₂   
POL(delN_out_gga(x₁, x₂, x₃)) = x₃   
POL(f) = 0   
POL(maxG_in_ga(x₁)) = 0   
POL(maxG_out_ga(x₁, x₂)) = 0   
POL(maxI_in_ga(x₁)) = 1 + x₁   
POL(maxI_out_ga(x₁, x₂)) = 0   
POL(t) = 0   

(97) Obligation:

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

PE_IN_GAAA(T53) → U26_GAAA(T53, maxI_in_ga(T53))
U26_GAAA(T53, maxI_out_ga(T53, T163)) → PR_IN_GGAA(T163, T53)
PR_IN_GGAA(T163, T53) → U28_GGAA(T163, T53, delN_in_gga(T163, T53))
U28_GGAA(T163, T53, delN_out_gga(T163, T53, T193)) → MAXSORTA_IN_GA(T193)
MAXSORTA_IN_GA(.(f, .(f, T53))) → PE_IN_GAAA(T53)

The TRS R consists of the following rules:

delN_in_gga(t, T224) → U14_gga(T224, delJ_in_ga(T224))
delJ_in_ga([]) → delJ_out_ga([], [])
delJ_in_ga(.(f, T240)) → U11_ga(T240, delJ_in_ga(T240))
U14_gga(T224, delJ_out_ga(T224, X384)) → delN_out_gga(t, T224, .(f, .(f, X384)))
U11_ga(T240, delJ_out_ga(T240, X427)) → delJ_out_ga(.(f, T240), .(f, X427))
maxI_in_ga([]) → maxI_out_ga([], f)
maxI_in_ga(.(f, T178)) → U9_ga(T178, maxI_in_ga(T178))
maxI_in_ga(.(t, T188)) → U10_ga(T188, maxG_in_ga(T188))
maxG_in_ga([]) → maxG_out_ga([], t)
maxG_in_ga(.(t, T72)) → U6_ga(T72, maxG_in_ga(T72))
maxG_in_ga(.(f, T72)) → U7_ga(T72, maxG_in_ga(T72))
U10_ga(T188, maxG_out_ga(T188, T190)) → maxI_out_ga(.(t, T188), T190)
U7_ga(T72, maxG_out_ga(T72, T74)) → maxG_out_ga(.(f, T72), T74)
U6_ga(T72, maxG_out_ga(T72, T74)) → maxG_out_ga(.(t, T72), T74)
U9_ga(T178, maxI_out_ga(T178, T180)) → maxI_out_ga(.(f, T178), T180)

The set Q consists of the following terms:

maxG_in_ga(x0)
maxI_in_ga(x0)
delN_in_gga(x0, x1)
U6_ga(x0, x1)
U7_ga(x0, x1)
U9_ga(x0, x1)
U10_ga(x0, x1)
U14_gga(x0, x1)
delJ_in_ga(x0)
U11_ga(x0, x1)

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

(98) QDPOrderProof (EQUIVALENT transformation)

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

The following pairs can be oriented strictly and are deleted.

U26_GAAA(T53, maxI_out_ga(T53, T163)) → PR_IN_GGAA(T163, T53)

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

POL(.(x₁, x₂)) = x₁ + x₂   
POL(MAXSORTA_IN_GA(x₁)) = x₁   
POL(PE_IN_GAAA(x₁)) = x₁   
POL(PR_IN_GGAA(x₁, x₂)) = x₁   
POL(U10_ga(x₁, x₂)) = 1 + x₂   
POL(U11_ga(x₁, x₂)) = x₂   
POL(U14_gga(x₁, x₂)) = x₂   
POL(U26_GAAA(x₁, x₂)) = x₂   
POL(U28_GGAA(x₁, x₂, x₃)) = x₃   
POL(U6_ga(x₁, x₂)) = 1 + x₂   
POL(U7_ga(x₁, x₂)) = x₂   
POL(U9_ga(x₁, x₂)) = x₂   
POL([]) = 1   
POL(delJ_in_ga(x₁)) = 1   
POL(delJ_out_ga(x₁, x₂)) = x₂   
POL(delN_in_gga(x₁, x₂)) = x₁   
POL(delN_out_gga(x₁, x₂, x₃)) = x₃   
POL(f) = 0   
POL(maxG_in_ga(x₁)) = x₁   
POL(maxG_out_ga(x₁, x₂)) = x₂   
POL(maxI_in_ga(x₁)) = x₁   
POL(maxI_out_ga(x₁, x₂)) = 1 + x₂   
POL(t) = 1   

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

maxI_in_ga([]) → maxI_out_ga([], f)
maxI_in_ga(.(f, T178)) → U9_ga(T178, maxI_in_ga(T178))
maxI_in_ga(.(t, T188)) → U10_ga(T188, maxG_in_ga(T188))
delN_in_gga(t, T224) → U14_gga(T224, delJ_in_ga(T224))
delJ_in_ga([]) → delJ_out_ga([], [])
delJ_in_ga(.(f, T240)) → U11_ga(T240, delJ_in_ga(T240))
U14_gga(T224, delJ_out_ga(T224, X384)) → delN_out_gga(t, T224, .(f, .(f, X384)))
U11_ga(T240, delJ_out_ga(T240, X427)) → delJ_out_ga(.(f, T240), .(f, X427))
U9_ga(T178, maxI_out_ga(T178, T180)) → maxI_out_ga(.(f, T178), T180)
maxG_in_ga([]) → maxG_out_ga([], t)
maxG_in_ga(.(t, T72)) → U6_ga(T72, maxG_in_ga(T72))
maxG_in_ga(.(f, T72)) → U7_ga(T72, maxG_in_ga(T72))
U10_ga(T188, maxG_out_ga(T188, T190)) → maxI_out_ga(.(t, T188), T190)
U7_ga(T72, maxG_out_ga(T72, T74)) → maxG_out_ga(.(f, T72), T74)
U6_ga(T72, maxG_out_ga(T72, T74)) → maxG_out_ga(.(t, T72), T74)

(99) Obligation:

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

PE_IN_GAAA(T53) → U26_GAAA(T53, maxI_in_ga(T53))
PR_IN_GGAA(T163, T53) → U28_GGAA(T163, T53, delN_in_gga(T163, T53))
U28_GGAA(T163, T53, delN_out_gga(T163, T53, T193)) → MAXSORTA_IN_GA(T193)
MAXSORTA_IN_GA(.(f, .(f, T53))) → PE_IN_GAAA(T53)

The TRS R consists of the following rules:

delN_in_gga(t, T224) → U14_gga(T224, delJ_in_ga(T224))
delJ_in_ga([]) → delJ_out_ga([], [])
delJ_in_ga(.(f, T240)) → U11_ga(T240, delJ_in_ga(T240))
U14_gga(T224, delJ_out_ga(T224, X384)) → delN_out_gga(t, T224, .(f, .(f, X384)))
U11_ga(T240, delJ_out_ga(T240, X427)) → delJ_out_ga(.(f, T240), .(f, X427))
maxI_in_ga([]) → maxI_out_ga([], f)
maxI_in_ga(.(f, T178)) → U9_ga(T178, maxI_in_ga(T178))
maxI_in_ga(.(t, T188)) → U10_ga(T188, maxG_in_ga(T188))
maxG_in_ga([]) → maxG_out_ga([], t)
maxG_in_ga(.(t, T72)) → U6_ga(T72, maxG_in_ga(T72))
maxG_in_ga(.(f, T72)) → U7_ga(T72, maxG_in_ga(T72))
U10_ga(T188, maxG_out_ga(T188, T190)) → maxI_out_ga(.(t, T188), T190)
U7_ga(T72, maxG_out_ga(T72, T74)) → maxG_out_ga(.(f, T72), T74)
U6_ga(T72, maxG_out_ga(T72, T74)) → maxG_out_ga(.(t, T72), T74)
U9_ga(T178, maxI_out_ga(T178, T180)) → maxI_out_ga(.(f, T178), T180)

The set Q consists of the following terms:

maxG_in_ga(x0)
maxI_in_ga(x0)
delN_in_gga(x0, x1)
U6_ga(x0, x1)
U7_ga(x0, x1)
U9_ga(x0, x1)
U10_ga(x0, x1)
U14_gga(x0, x1)
delJ_in_ga(x0)
U11_ga(x0, x1)

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

(100) DependencyGraphProof (EQUIVALENT transformation)

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

(101) TRUE