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

(0) Obligation:

Clauses:

even(0, true).
even(s(0), false).
even(s(s(X)), B) :- even(X, B).
half(0, 0).
half(s(s(X)), s(Y)) :- half(X, Y).
plus(0, Y, Y).
plus(s(X), Y, s(Z)) :- plus(X, Y, Z).
times(0, Y, 0).
times(s(X), Y, Z) :- ','(even(s(X), B), if(B, s(X), Y, Z)).
if(true, s(X), Y, Z) :- ','(half(s(X), X1), ','(times(X1, Y, Y1), plus(Y1, Y1, Z))).
if(false, s(X), Y, Z) :- ','(times(X, Y, U), plus(Y, U, Z)).

Query: times(g,g,a)

(1) PrologToPiTRSViaGraphTransformerProof (SOUND transformation)

Transformed Prolog program to (Pi-)TRS.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
2 [label="A: times(T1, T2, T3)\n\nT1 is ground\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
6 [label="6: times(T1, T2, T3), SCOPE: 1, CLAUSE: 7 | times(T1, T2, T3), SCOPE: 1, CLAUSE: 8\n\nT1 is ground\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: true | times(0, T5, T3), SCOPE: 1, CLAUSE: 8\n\nT5 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: times(T1, T2, T3), SCOPE: 1, CLAUSE: 8\n\nT1 is ground\nT2 is ground\nX3 is free\ntimes(T1, T2, T3) !=? times(0, X3, 0)", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: times(0, T5, T3), SCOPE: 1, CLAUSE: 8\n\nT5 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: ','(even(s(T9), X13), if(X13, s(T9), T10, T12))\n\nT9 is ground\nT10 is ground\nX13 is free", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: ','(even(s(T9), X13), if(X13, s(T9), T10, T12)), SCOPE: 2, CLAUSE: 0 | ','(even(s(T9), X13), if(X13, s(T9), T10, T12)), SCOPE: 2, CLAUSE: 1 | ','(even(s(T9), X13), if(X13, s(T9), T10, T12)), SCOPE: 2, CLAUSE: 2\n\nT9 is ground\nT10 is ground\nX13 is free", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: ','(even(s(T9), X13), if(X13, s(T9), T10, T12)), SCOPE: 2, CLAUSE: 1 | ','(even(s(T9), X13), if(X13, s(T9), T10, T12)), SCOPE: 2, CLAUSE: 2\n\nT9 is ground\nT10 is ground\nX13 is free", fontsize=16, style=filled, fillcolor=lightyellow];
43 [label="43: ','(even(s(T9), X13), if(X13, s(T9), T10, T12)), SCOPE: 2, CLAUSE: 1\n\nT9 is ground\nT10 is ground\nX13 is free", fontsize=16, style=filled, fillcolor=lightyellow];
46 [label="46: ','(even(s(T9), X13), if(X13, s(T9), T10, T12)), SCOPE: 2, CLAUSE: 2\n\nT9 is ground\nT10 is ground\nX13 is free", fontsize=16, style=filled, fillcolor=lightyellow];
48 [label="48: if(false, s(0), T10, T12)\n\nT10 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
49 [label="49: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
50 [label="50: if(false, s(0), T10, T12), SCOPE: 3, CLAUSE: 9 | if(false, s(0), T10, T12), SCOPE: 3, CLAUSE: 10\n\nT10 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
52 [label="52: if(false, s(0), T10, T12), SCOPE: 3, CLAUSE: 10\n\nT10 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
56 [label="B: ','(times(0, T21, X37), plus(T21, X37, T23))\n\nT21 is ground\nX37 is free", fontsize=16, style=filled, fillcolor=lightyellow];
80 [label="K: times(0, T21, X37)\n\nT21 is ground\nX37 is free", fontsize=16, style=filled, fillcolor=lightyellow];
81 [label="D: plus(T21, T27, T23)\n\nT21 is ground\nT27 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
84 [label="84: times(0, T21, X37), SCOPE: 4, CLAUSE: 7 | times(0, T21, X37), SCOPE: 4, CLAUSE: 8\n\nT21 is ground\nX37 is free", fontsize=16, style=filled, fillcolor=lightyellow];
87 [label="87: times(0, T21, X37), SCOPE: 4, CLAUSE: 7\n\nT21 is ground\nX37 is free", fontsize=16, style=filled, fillcolor=lightyellow];
88 [label="88: times(0, T21, X37), SCOPE: 4, CLAUSE: 8\n\nT21 is ground\nX37 is free", fontsize=16, style=filled, fillcolor=lightyellow];
89 [label="89: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
90 [label="90: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
91 [label="91: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
92 [label="92: plus(T21, T27, T23), SCOPE: 5, CLAUSE: 5 | plus(T21, T27, T23), SCOPE: 5, CLAUSE: 6\n\nT21 is ground\nT27 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
94 [label="94: plus(T21, T27, T23), SCOPE: 5, CLAUSE: 5\n\nT21 is ground\nT27 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
95 [label="95: plus(T21, T27, T23), SCOPE: 5, CLAUSE: 6\n\nT21 is ground\nT27 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
98 [label="98: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
99 [label="99: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
100 [label="100: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
114 [label="114: plus(T47, T48, T50)\n\nT47 is ground\nT48 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
115 [label="115: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
118 [label="C: ','(even(T55, X78), if(X78, s(s(T55)), T10, T12))\n\nT10 is ground\nT55 is ground\nX78 is free", fontsize=16, style=filled, fillcolor=lightyellow];
119 [label="119: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
123 [label="E: even(T55, X78)\n\nT55 is ground\nX78 is free", fontsize=16, style=filled, fillcolor=lightyellow];
124 [label="L: if(T56, s(s(T55)), T10, T12)\n\nT10 is ground\nT55 is ground\nT56 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
137 [label="137: even(T55, X78), SCOPE: 6, CLAUSE: 0 | even(T55, X78), SCOPE: 6, CLAUSE: 1 | even(T55, X78), SCOPE: 6, CLAUSE: 2\n\nT55 is ground\nX78 is free", fontsize=16, style=filled, fillcolor=lightyellow];
138 [label="138: even(T55, X78), SCOPE: 6, CLAUSE: 0\n\nT55 is ground\nX78 is free", fontsize=16, style=filled, fillcolor=lightyellow];
139 [label="139: even(T55, X78), SCOPE: 6, CLAUSE: 1 | even(T55, X78), SCOPE: 6, CLAUSE: 2\n\nT55 is ground\nX78 is free", fontsize=16, style=filled, fillcolor=lightyellow];
143 [label="143: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
144 [label="144: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
145 [label="145: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
148 [label="148: even(T55, X78), SCOPE: 6, CLAUSE: 1\n\nT55 is ground\nX78 is free", fontsize=16, style=filled, fillcolor=lightyellow];
149 [label="149: even(T55, X78), SCOPE: 6, CLAUSE: 2\n\nT55 is ground\nX78 is free", fontsize=16, style=filled, fillcolor=lightyellow];
155 [label="155: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
156 [label="156: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
157 [label="157: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
160 [label="160: even(T59, X87)\n\nT59 is ground\nX87 is free", fontsize=16, style=filled, fillcolor=lightyellow];
162 [label="162: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
169 [label="169: if(T56, s(s(T55)), T10, T12), SCOPE: 7, CLAUSE: 9 | if(T56, s(s(T55)), T10, T12), SCOPE: 7, CLAUSE: 10\n\nT10 is ground\nT55 is ground\nT56 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
172 [label="172: if(T56, s(s(T55)), T10, T12), SCOPE: 7, CLAUSE: 9\n\nT10 is ground\nT55 is ground\nT56 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
173 [label="173: if(T56, s(s(T55)), T10, T12), SCOPE: 7, CLAUSE: 10\n\nT10 is ground\nT55 is ground\nT56 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
174 [label="M: ','(half(s(s(T75)), X112), ','(times(X112, T76, X113), plus(X113, X113, T78)))\n\nT75 is ground\nT76 is ground\nX112 is free\nX113 is free", fontsize=16, style=filled, fillcolor=lightyellow];
175 [label="175: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
179 [label="O: half(s(s(T75)), X112)\n\nT75 is ground\nX112 is free", fontsize=16, style=filled, fillcolor=lightyellow];
180 [label="U: ','(times(T79, T76, X113), plus(X113, X113, T78))\n\nT76 is ground\nT79 is ground\nX113 is free", fontsize=16, style=filled, fillcolor=lightyellow];
185 [label="185: half(s(s(T75)), X112), SCOPE: 8, CLAUSE: 3 | half(s(s(T75)), X112), SCOPE: 8, CLAUSE: 4\n\nT75 is ground\nX112 is free", fontsize=16, style=filled, fillcolor=lightyellow];
186 [label="186: half(s(s(T75)), X112), SCOPE: 8, CLAUSE: 4\n\nT75 is ground\nX112 is free", fontsize=16, style=filled, fillcolor=lightyellow];
198 [label="F: half(T82, X122)\n\nT82 is ground\nX122 is free", fontsize=16, style=filled, fillcolor=lightyellow];
199 [label="199: half(T82, X122), SCOPE: 9, CLAUSE: 3 | half(T82, X122), SCOPE: 9, CLAUSE: 4\n\nT82 is ground\nX122 is free", fontsize=16, style=filled, fillcolor=lightyellow];
200 [label="200: half(T82, X122), SCOPE: 9, CLAUSE: 3\n\nT82 is ground\nX122 is free", fontsize=16, style=filled, fillcolor=lightyellow];
201 [label="201: half(T82, X122), SCOPE: 9, CLAUSE: 4\n\nT82 is ground\nX122 is free", fontsize=16, style=filled, fillcolor=lightyellow];
202 [label="202: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
203 [label="203: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
204 [label="204: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
206 [label="206: half(T85, X131)\n\nT85 is ground\nX131 is free", fontsize=16, style=filled, fillcolor=lightyellow];
207 [label="207: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
212 [label="G: times(T79, T76, X113)\n\nT76 is ground\nT79 is ground\nX113 is free", fontsize=16, style=filled, fillcolor=lightyellow];
213 [label="P: plus(T88, T88, T78)\n\nT88 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
218 [label="218: times(T79, T76, X113), SCOPE: 10, CLAUSE: 7 | times(T79, T76, X113), SCOPE: 10, CLAUSE: 8\n\nT76 is ground\nT79 is ground\nX113 is free", fontsize=16, style=filled, fillcolor=lightyellow];
219 [label="219: times(T79, T76, X113), SCOPE: 10, CLAUSE: 7\n\nT76 is ground\nT79 is ground\nX113 is free", fontsize=16, style=filled, fillcolor=lightyellow];
220 [label="220: times(T79, T76, X113), SCOPE: 10, CLAUSE: 8\n\nT76 is ground\nT79 is ground\nX113 is free", fontsize=16, style=filled, fillcolor=lightyellow];
221 [label="221: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
222 [label="222: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
223 [label="223: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
228 [label="H: ','(even(s(T100), X153), if(X153, s(T100), T101, X154))\n\nT100 is ground\nT101 is ground\nX154 is free\nX153 is free", fontsize=16, style=filled, fillcolor=lightyellow];
229 [label="229: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
230 [label="230: even(s(T100), X153)\n\nT100 is ground\nX153 is free", fontsize=16, style=filled, fillcolor=lightyellow];
231 [label="Q: if(T102, s(T100), T101, X154)\n\nT100 is ground\nT101 is ground\nT102 is ground\nX154 is free", fontsize=16, style=filled, fillcolor=lightyellow];
232 [label="232: if(T102, s(T100), T101, X154), SCOPE: 11, CLAUSE: 9 | if(T102, s(T100), T101, X154), SCOPE: 11, CLAUSE: 10\n\nT100 is ground\nT101 is ground\nT102 is ground\nX154 is free", fontsize=16, style=filled, fillcolor=lightyellow];
233 [label="233: if(T102, s(T100), T101, X154), SCOPE: 11, CLAUSE: 9\n\nT100 is ground\nT101 is ground\nT102 is ground\nX154 is free", fontsize=16, style=filled, fillcolor=lightyellow];
234 [label="234: if(T102, s(T100), T101, X154), SCOPE: 11, CLAUSE: 10\n\nT100 is ground\nT101 is ground\nT102 is ground\nX154 is free", fontsize=16, style=filled, fillcolor=lightyellow];
235 [label="R: ','(half(s(T113), X184), ','(times(X184, T114, X185), plus(X185, X185, X186)))\n\nT113 is ground\nT114 is ground\nX186 is free\nX184 is free\nX185 is free", fontsize=16, style=filled, fillcolor=lightyellow];
236 [label="236: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
237 [label="237: half(s(T113), X184)\n\nT113 is ground\nX184 is free", fontsize=16, style=filled, fillcolor=lightyellow];
238 [label="V: ','(times(T115, T114, X185), plus(X185, X185, X186))\n\nT114 is ground\nT115 is ground\nX186 is free\nX185 is free", fontsize=16, style=filled, fillcolor=lightyellow];
244 [label="244: times(T115, T114, X185)\n\nT114 is ground\nT115 is ground\nX185 is free", fontsize=16, style=filled, fillcolor=lightyellow];
245 [label="T: plus(T118, T118, X186)\n\nT118 is ground\nX186 is free", fontsize=16, style=filled, fillcolor=lightyellow];
256 [label="256: plus(T118, T118, X186), SCOPE: 12, CLAUSE: 5 | plus(T118, T118, X186), SCOPE: 12, CLAUSE: 6\n\nT118 is ground\nX186 is free", fontsize=16, style=filled, fillcolor=lightyellow];
259 [label="259: plus(T118, T118, X186), SCOPE: 12, CLAUSE: 5\n\nT118 is ground\nX186 is free", fontsize=16, style=filled, fillcolor=lightyellow];
260 [label="260: plus(T118, T118, X186), SCOPE: 12, CLAUSE: 6\n\nT118 is ground\nX186 is free", fontsize=16, style=filled, fillcolor=lightyellow];
261 [label="261: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
262 [label="262: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
263 [label="263: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
272 [label="272: plus(T123, s(T123), X209)\n\nT123 is ground\nX209 is free", fontsize=16, style=filled, fillcolor=lightyellow];
273 [label="273: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
278 [label="278: plus(T123, s(T123), X209), SCOPE: 13, CLAUSE: 5 | plus(T123, s(T123), X209), SCOPE: 13, CLAUSE: 6\n\nT123 is ground\nX209 is free", fontsize=16, style=filled, fillcolor=lightyellow];
282 [label="282: plus(T123, s(T123), X209), SCOPE: 13, CLAUSE: 5\n\nT123 is ground\nX209 is free", fontsize=16, style=filled, fillcolor=lightyellow];
284 [label="284: plus(T123, s(T123), X209), SCOPE: 13, CLAUSE: 6\n\nT123 is ground\nX209 is free", fontsize=16, style=filled, fillcolor=lightyellow];
285 [label="285: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
286 [label="286: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
287 [label="287: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
299 [label="299: plus(T126, s(s(T126)), X228)\n\nT126 is ground\nX228 is free", fontsize=16, style=filled, fillcolor=lightyellow];
300 [label="300: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
304 [label="304: plus(T126, s(s(T126)), X228), SCOPE: 14, CLAUSE: 5 | plus(T126, s(s(T126)), X228), SCOPE: 14, CLAUSE: 6\n\nT126 is ground\nX228 is free", fontsize=16, style=filled, fillcolor=lightyellow];
305 [label="305: plus(T126, s(s(T126)), X228), SCOPE: 14, CLAUSE: 5\n\nT126 is ground\nX228 is free", fontsize=16, style=filled, fillcolor=lightyellow];
306 [label="306: plus(T126, s(s(T126)), X228), SCOPE: 14, CLAUSE: 6\n\nT126 is ground\nX228 is free", fontsize=16, style=filled, fillcolor=lightyellow];
311 [label="311: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
312 [label="312: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
313 [label="313: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
316 [label="316: plus(T129, s(s(s(T129))), X247)\n\nT129 is ground\nX247 is free", fontsize=16, style=filled, fillcolor=lightyellow];
317 [label="317: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
328 [label="328: plus(T129, s(s(s(T129))), X247), SCOPE: 15, CLAUSE: 5 | plus(T129, s(s(s(T129))), X247), SCOPE: 15, CLAUSE: 6\n\nT129 is ground\nX247 is free", fontsize=16, style=filled, fillcolor=lightyellow];
329 [label="329: plus(T129, s(s(s(T129))), X247), SCOPE: 15, CLAUSE: 5\n\nT129 is ground\nX247 is free", fontsize=16, style=filled, fillcolor=lightyellow];
330 [label="330: plus(T129, s(s(s(T129))), X247), SCOPE: 15, CLAUSE: 6\n\nT129 is ground\nX247 is free", fontsize=16, style=filled, fillcolor=lightyellow];
331 [label="331: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
332 [label="332: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
333 [label="333: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
341 [label="341: plus(T132, s(s(s(s(T132)))), X266)\n\nT132 is ground\nX266 is free", fontsize=16, style=filled, fillcolor=lightyellow];
342 [label="342: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
352 [label="352: plus(T132, s(s(s(s(T132)))), X266), SCOPE: 16, CLAUSE: 5 | plus(T132, s(s(s(s(T132)))), X266), SCOPE: 16, CLAUSE: 6\n\nT132 is ground\nX266 is free", fontsize=16, style=filled, fillcolor=lightyellow];
353 [label="353: plus(T132, s(s(s(s(T132)))), X266), SCOPE: 16, CLAUSE: 5\n\nT132 is ground\nX266 is free", fontsize=16, style=filled, fillcolor=lightyellow];
354 [label="354: plus(T132, s(s(s(s(T132)))), X266), SCOPE: 16, CLAUSE: 6\n\nT132 is ground\nX266 is free", fontsize=16, style=filled, fillcolor=lightyellow];
357 [label="357: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
358 [label="358: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
359 [label="359: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
367 [label="367: plus(T135, s(s(s(s(s(T135))))), X285)\n\nT135 is ground\nX285 is free", fontsize=16, style=filled, fillcolor=lightyellow];
370 [label="370: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
375 [label="375: plus(T135, s(s(s(s(s(T135))))), X285), SCOPE: 17, CLAUSE: 5 | plus(T135, s(s(s(s(s(T135))))), X285), SCOPE: 17, CLAUSE: 6\n\nT135 is ground\nX285 is free", fontsize=16, style=filled, fillcolor=lightyellow];
379 [label="379: plus(T135, s(s(s(s(s(T135))))), X285), SCOPE: 17, CLAUSE: 5\n\nT135 is ground\nX285 is free", fontsize=16, style=filled, fillcolor=lightyellow];
380 [label="380: plus(T135, s(s(s(s(s(T135))))), X285), SCOPE: 17, CLAUSE: 6\n\nT135 is ground\nX285 is free", fontsize=16, style=filled, fillcolor=lightyellow];
381 [label="381: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
382 [label="382: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
383 [label="383: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
392 [label="392: plus(T138, s(s(s(s(s(s(T138)))))), X304)\n\nT138 is ground\nX304 is free", fontsize=16, style=filled, fillcolor=lightyellow];
393 [label="393: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
400 [label="400: plus(T138, s(s(s(s(s(s(T138)))))), X304), SCOPE: 18, CLAUSE: 5 | plus(T138, s(s(s(s(s(s(T138)))))), X304), SCOPE: 18, CLAUSE: 6\n\nT138 is ground\nX304 is free", fontsize=16, style=filled, fillcolor=lightyellow];
403 [label="403: plus(T138, s(s(s(s(s(s(T138)))))), X304), SCOPE: 18, CLAUSE: 5\n\nT138 is ground\nX304 is free", fontsize=16, style=filled, fillcolor=lightyellow];
404 [label="404: plus(T138, s(s(s(s(s(s(T138)))))), X304), SCOPE: 18, CLAUSE: 6\n\nT138 is ground\nX304 is free", fontsize=16, style=filled, fillcolor=lightyellow];
405 [label="405: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
407 [label="407: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
408 [label="408: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
416 [label="416: plus(T141, s(s(s(s(s(s(s(T141))))))), X323)\n\nT141 is ground\nX323 is free", fontsize=16, style=filled, fillcolor=lightyellow];
417 [label="417: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
423 [label="423: plus(T141, s(s(s(s(s(s(s(T141))))))), X323), SCOPE: 19, CLAUSE: 5 | plus(T141, s(s(s(s(s(s(s(T141))))))), X323), SCOPE: 19, CLAUSE: 6\n\nT141 is ground\nX323 is free", fontsize=16, style=filled, fillcolor=lightyellow];
427 [label="427: plus(T141, s(s(s(s(s(s(s(T141))))))), X323), SCOPE: 19, CLAUSE: 5\n\nT141 is ground\nX323 is free", fontsize=16, style=filled, fillcolor=lightyellow];
428 [label="428: plus(T141, s(s(s(s(s(s(s(T141))))))), X323), SCOPE: 19, CLAUSE: 6\n\nT141 is ground\nX323 is free", fontsize=16, style=filled, fillcolor=lightyellow];
430 [label="430: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
433 [label="433: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
434 [label="434: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
440 [label="440: plus(T144, s(s(s(s(s(s(s(s(T144)))))))), X342)\n\nT144 is ground\nX342 is free", fontsize=16, style=filled, fillcolor=lightyellow];
441 [label="441: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
443 [label="I: plus(T144, s(T145), X342)\n\nT144 is ground\nT145 is ground\nX342 is free", fontsize=16, style=filled, fillcolor=lightyellow];
449 [label="449: plus(T144, s(T145), X342), SCOPE: 20, CLAUSE: 5 | plus(T144, s(T145), X342), SCOPE: 20, CLAUSE: 6\n\nT144 is ground\nT145 is ground\nX342 is free", fontsize=16, style=filled, fillcolor=lightyellow];
453 [label="453: plus(T144, s(T145), X342), SCOPE: 20, CLAUSE: 5\n\nT144 is ground\nT145 is ground\nX342 is free", fontsize=16, style=filled, fillcolor=lightyellow];
454 [label="454: plus(T144, s(T145), X342), SCOPE: 20, CLAUSE: 6\n\nT144 is ground\nT145 is ground\nX342 is free", 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];
462 [label="462: plus(T157, s(T158), X363)\n\nT157 is ground\nT158 is ground\nX363 is free", fontsize=16, style=filled, fillcolor=lightyellow];
465 [label="465: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
477 [label="S: ','(times(T165, T166, X378), plus(T166, X378, X379))\n\nT165 is ground\nT166 is ground\nX379 is free\nX378 is free", fontsize=16, style=filled, fillcolor=lightyellow];
478 [label="478: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
503 [label="503: times(T165, T166, X378)\n\nT165 is ground\nT166 is ground\nX378 is free", fontsize=16, style=filled, fillcolor=lightyellow];
504 [label="J: plus(T166, T169, X379)\n\nT166 is ground\nT169 is ground\nX379 is free", fontsize=16, style=filled, fillcolor=lightyellow];
514 [label="514: plus(T166, T169, X379), SCOPE: 21, CLAUSE: 5 | plus(T166, T169, X379), SCOPE: 21, CLAUSE: 6\n\nT166 is ground\nT169 is ground\nX379 is free", fontsize=16, style=filled, fillcolor=lightyellow];
518 [label="518: plus(T166, T169, X379), SCOPE: 21, CLAUSE: 5\n\nT166 is ground\nT169 is ground\nX379 is free", fontsize=16, style=filled, fillcolor=lightyellow];
520 [label="520: plus(T166, T169, X379), SCOPE: 21, CLAUSE: 6\n\nT166 is ground\nT169 is ground\nX379 is free", fontsize=16, style=filled, fillcolor=lightyellow];
524 [label="524: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
525 [label="525: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
526 [label="526: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
559 [label="559: plus(T183, T184, X402)\n\nT183 is ground\nT184 is ground\nX402 is free", fontsize=16, style=filled, fillcolor=lightyellow];
560 [label="560: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
569 [label="569: plus(T88, T88, T78), SCOPE: 22, CLAUSE: 5 | plus(T88, T88, T78), SCOPE: 22, CLAUSE: 6\n\nT88 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
570 [label="570: plus(T88, T88, T78), SCOPE: 22, CLAUSE: 5\n\nT88 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
571 [label="571: plus(T88, T88, T78), SCOPE: 22, CLAUSE: 6\n\nT88 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
572 [label="572: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
573 [label="573: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
574 [label="574: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
582 [label="582: plus(T191, s(T191), T193)\n\nT191 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
584 [label="584: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
588 [label="588: plus(T191, s(T191), T193), SCOPE: 23, CLAUSE: 5 | plus(T191, s(T191), T193), SCOPE: 23, CLAUSE: 6\n\nT191 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
592 [label="592: plus(T191, s(T191), T193), SCOPE: 23, CLAUSE: 5\n\nT191 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
593 [label="593: plus(T191, s(T191), T193), SCOPE: 23, CLAUSE: 6\n\nT191 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
596 [label="596: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
597 [label="597: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
598 [label="598: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
601 [label="601: plus(T198, s(s(T198)), T200)\n\nT198 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
602 [label="602: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
604 [label="604: plus(T198, s(s(T198)), T200), SCOPE: 24, CLAUSE: 5 | plus(T198, s(s(T198)), T200), SCOPE: 24, CLAUSE: 6\n\nT198 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
607 [label="607: plus(T198, s(s(T198)), T200), SCOPE: 24, CLAUSE: 5\n\nT198 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
608 [label="608: plus(T198, s(s(T198)), T200), SCOPE: 24, CLAUSE: 6\n\nT198 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
609 [label="609: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
611 [label="611: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
613 [label="613: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
615 [label="615: plus(T205, s(s(s(T205))), T207)\n\nT205 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
616 [label="616: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
625 [label="625: plus(T205, s(s(s(T205))), T207), SCOPE: 25, CLAUSE: 5 | plus(T205, s(s(s(T205))), T207), SCOPE: 25, CLAUSE: 6\n\nT205 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
626 [label="626: plus(T205, s(s(s(T205))), T207), SCOPE: 25, CLAUSE: 5\n\nT205 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
627 [label="627: plus(T205, s(s(s(T205))), T207), SCOPE: 25, CLAUSE: 6\n\nT205 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
629 [label="629: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
632 [label="632: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
633 [label="633: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
639 [label="639: plus(T212, s(s(s(s(T212)))), T214)\n\nT212 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
640 [label="640: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
643 [label="643: plus(T212, s(s(s(s(T212)))), T214), SCOPE: 26, CLAUSE: 5 | plus(T212, s(s(s(s(T212)))), T214), SCOPE: 26, CLAUSE: 6\n\nT212 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
644 [label="644: plus(T212, s(s(s(s(T212)))), T214), SCOPE: 26, CLAUSE: 5\n\nT212 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
645 [label="645: plus(T212, s(s(s(s(T212)))), T214), SCOPE: 26, CLAUSE: 6\n\nT212 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
646 [label="646: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
647 [label="647: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
648 [label="648: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
655 [label="655: plus(T219, s(s(s(s(s(T219))))), T221)\n\nT219 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
656 [label="656: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
662 [label="662: plus(T219, s(s(s(s(s(T219))))), T221), SCOPE: 27, CLAUSE: 5 | plus(T219, s(s(s(s(s(T219))))), T221), SCOPE: 27, CLAUSE: 6\n\nT219 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
663 [label="663: plus(T219, s(s(s(s(s(T219))))), T221), SCOPE: 27, CLAUSE: 5\n\nT219 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
664 [label="664: plus(T219, s(s(s(s(s(T219))))), T221), SCOPE: 27, CLAUSE: 6\n\nT219 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
668 [label="668: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
669 [label="669: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
670 [label="670: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
674 [label="674: plus(T226, s(s(s(s(s(s(T226)))))), T228)\n\nT226 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
675 [label="675: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
680 [label="680: plus(T226, s(s(s(s(s(s(T226)))))), T228), SCOPE: 28, CLAUSE: 5 | plus(T226, s(s(s(s(s(s(T226)))))), T228), SCOPE: 28, CLAUSE: 6\n\nT226 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
683 [label="683: plus(T226, s(s(s(s(s(s(T226)))))), T228), SCOPE: 28, CLAUSE: 5\n\nT226 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
684 [label="684: plus(T226, s(s(s(s(s(s(T226)))))), T228), SCOPE: 28, CLAUSE: 6\n\nT226 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
685 [label="685: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
686 [label="686: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
687 [label="687: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
689 [label="689: plus(T233, s(s(s(s(s(s(s(T233))))))), T235)\n\nT233 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
690 [label="690: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
696 [label="696: plus(T233, s(s(s(s(s(s(s(T233))))))), T235), SCOPE: 29, CLAUSE: 5 | plus(T233, s(s(s(s(s(s(s(T233))))))), T235), SCOPE: 29, CLAUSE: 6\n\nT233 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
699 [label="699: plus(T233, s(s(s(s(s(s(s(T233))))))), T235), SCOPE: 29, CLAUSE: 5\n\nT233 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
700 [label="700: plus(T233, s(s(s(s(s(s(s(T233))))))), T235), SCOPE: 29, CLAUSE: 6\n\nT233 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
701 [label="701: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
702 [label="702: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
703 [label="703: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
705 [label="705: plus(T240, s(s(s(s(s(s(s(s(T240)))))))), T242)\n\nT240 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
707 [label="707: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
712 [label="712: plus(T240, s(T243), T242)\n\nT240 is ground\nT243 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
714 [label="N: ','(times(s(T252), T253, X547), plus(T253, X547, T255))\n\nT252 is ground\nT253 is ground\nX547 is free", fontsize=16, style=filled, fillcolor=lightyellow];
715 [label="715: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
721 [label="721: times(s(T252), T253, X547)\n\nT252 is ground\nT253 is ground\nX547 is free", fontsize=16, style=filled, fillcolor=lightyellow];
722 [label="722: plus(T253, T256, T255)\n\nT253 is ground\nT256 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
723 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
2 -> 723 [arrowhead = none ];
723 -> 6;

724 [label="EVAL with clause\ntimes(0, X3, 0).\nand substitutionT1 -> 0,\nT2 -> T5,\nX3 -> T5,\nT3 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
6 -> 724 [arrowhead = none ];
724 -> 10;

725 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
6 -> 725 [arrowhead = none ];
725 -> 14;

726 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
10 -> 726 [arrowhead = none ];
726 -> 17;

727 [label="EVAL with clause\ntimes(s(X10), X11, X12) :- ','(even(s(X10), X13), if(X13, s(X10), X11, X12)).\nand substitutionX10 -> T9,\nT1 -> s(T9),\nT2 -> T10,\nX11 -> T10,\nT3 -> T12,\nX12 -> T12,\nT11 -> T12", fontsize=14, style = filled, fillcolor = lightblue];
14 -> 727 [arrowhead = none ];
727 -> 24;

728 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
14 -> 728 [arrowhead = none ];
728 -> 26;

729 [label="BACKTRACK\nfor clause: times(s(X), Y, Z) :- ','(even(s(X), B), if(B, s(X), Y, Z))because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
17 -> 729 [arrowhead = none ];
729 -> 19;

730 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
24 -> 730 [arrowhead = none ];
730 -> 29;

731 [label="BACKTRACK\nfor clause: even(0, true)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
29 -> 731 [arrowhead = none ];
731 -> 31;

732 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
31 -> 732 [arrowhead = none ];
732 -> 43;

733 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
31 -> 733 [arrowhead = none ];
733 -> 46;

734 [label="EVAL with clause\neven(s(0), false).\nand substitutionT9 -> 0,\nX13 -> false", fontsize=14, style = filled, fillcolor = lightblue];
43 -> 734 [arrowhead = none ];
734 -> 48;

735 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
43 -> 735 [arrowhead = none ];
735 -> 49;

736 [label="EVAL with clause\neven(s(s(X76)), X77) :- even(X76, X77).\nand substitutionX76 -> T55,\nT9 -> s(T55),\nX13 -> X78,\nX77 -> X78", fontsize=14, style = filled, fillcolor = lightblue];
46 -> 736 [arrowhead = none ];
736 -> 118;

737 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
46 -> 737 [arrowhead = none ];
737 -> 119;

738 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
48 -> 738 [arrowhead = none ];
738 -> 50;

739 [label="BACKTRACK\nfor clause: if(true, s(X), Y, Z) :- ','(half(s(X), X1), ','(times(X1, Y, Y1), plus(Y1, Y1, Z)))because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
50 -> 739 [arrowhead = none ];
739 -> 52;

740 [label="ONLY EVAL with clause\nif(false, s(X34), X35, X36) :- ','(times(X34, X35, X37), plus(X35, X37, X36)).\nand substitutionX34 -> 0,\nT10 -> T21,\nX35 -> T21,\nT12 -> T23,\nX36 -> T23,\nT22 -> T23", fontsize=14, style = filled, fillcolor = lightblue];
52 -> 740 [arrowhead = none ];
740 -> 56;

741 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
56 -> 741 [arrowhead = none ];
741 -> 80;

742 [label="SPLIT 2\nnew knowledge:\nT21 is ground\nT27 is ground\nreplacements:X37 -> T27", fontsize=14, style = filled, fillcolor = violet];
56 -> 742 [arrowhead = none ];
742 -> 81;

743 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
80 -> 743 [arrowhead = none ];
743 -> 84;

744 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
81 -> 744 [arrowhead = none ];
744 -> 92;

745 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
84 -> 745 [arrowhead = none ];
745 -> 87;

746 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
84 -> 746 [arrowhead = none ];
746 -> 88;

747 [label="ONLY EVAL with clause\ntimes(0, X48, 0).\nand substitutionT21 -> T33,\nX48 -> T33,\nX37 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
87 -> 747 [arrowhead = none ];
747 -> 89;

748 [label="BACKTRACK\nfor clause: times(s(X), Y, Z) :- ','(even(s(X), B), if(B, s(X), Y, Z))because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
88 -> 748 [arrowhead = none ];
748 -> 91;

749 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
89 -> 749 [arrowhead = none ];
749 -> 90;

750 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
92 -> 750 [arrowhead = none ];
750 -> 94;

751 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
92 -> 751 [arrowhead = none ];
751 -> 95;

752 [label="EVAL with clause\nplus(0, X58, X58).\nand substitutionT21 -> 0,\nT27 -> T40,\nX58 -> T40,\nT23 -> T40", fontsize=14, style = filled, fillcolor = lightblue];
94 -> 752 [arrowhead = none ];
752 -> 98;

753 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
94 -> 753 [arrowhead = none ];
753 -> 99;

754 [label="EVAL with clause\nplus(s(X65), X66, s(X67)) :- plus(X65, X66, X67).\nand substitutionX65 -> T47,\nT21 -> s(T47),\nT27 -> T48,\nX66 -> T48,\nX67 -> T50,\nT23 -> s(T50),\nT49 -> T50", fontsize=14, style = filled, fillcolor = lightblue];
95 -> 754 [arrowhead = none ];
754 -> 114;

755 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
95 -> 755 [arrowhead = none ];
755 -> 115;

756 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
98 -> 756 [arrowhead = none ];
756 -> 100;

757 [label="INSTANCE with matching:\nT21 -> T47\nT27 -> T48\nT23 -> T50", fontsize=14, style = filled, fillcolor = lightgrey];
114 -> 757 [arrowhead = none , style=dashed];
757 -> 81[style=dashed];

758 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
118 -> 758 [arrowhead = none ];
758 -> 123;

759 [label="SPLIT 2\nnew knowledge:\nT55 is ground\nT56 is ground\nreplacements:X78 -> T56", fontsize=14, style = filled, fillcolor = violet];
118 -> 759 [arrowhead = none ];
759 -> 124;

760 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
123 -> 760 [arrowhead = none ];
760 -> 137;

761 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
124 -> 761 [arrowhead = none ];
761 -> 169;

762 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
137 -> 762 [arrowhead = none ];
762 -> 138;

763 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
137 -> 763 [arrowhead = none ];
763 -> 139;

764 [label="EVAL with clause\neven(0, true).\nand substitutionT55 -> 0,\nX78 -> true", fontsize=14, style = filled, fillcolor = lightblue];
138 -> 764 [arrowhead = none ];
764 -> 143;

765 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
138 -> 765 [arrowhead = none ];
765 -> 144;

766 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
139 -> 766 [arrowhead = none ];
766 -> 148;

767 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
139 -> 767 [arrowhead = none ];
767 -> 149;

768 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
143 -> 768 [arrowhead = none ];
768 -> 145;

769 [label="EVAL with clause\neven(s(0), false).\nand substitutionT55 -> s(0),\nX78 -> false", fontsize=14, style = filled, fillcolor = lightblue];
148 -> 769 [arrowhead = none ];
769 -> 155;

770 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
148 -> 770 [arrowhead = none ];
770 -> 156;

771 [label="EVAL with clause\neven(s(s(X85)), X86) :- even(X85, X86).\nand substitutionX85 -> T59,\nT55 -> s(s(T59)),\nX78 -> X87,\nX86 -> X87", fontsize=14, style = filled, fillcolor = lightblue];
149 -> 771 [arrowhead = none ];
771 -> 160;

772 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
149 -> 772 [arrowhead = none ];
772 -> 162;

773 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
155 -> 773 [arrowhead = none ];
773 -> 157;

774 [label="INSTANCE with matching:\nT55 -> T59\nX78 -> X87", fontsize=14, style = filled, fillcolor = lightgrey];
160 -> 774 [arrowhead = none , style=dashed];
774 -> 123[style=dashed];

775 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
169 -> 775 [arrowhead = none ];
775 -> 172;

776 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
169 -> 776 [arrowhead = none ];
776 -> 173;

777 [label="EVAL with clause\nif(true, s(X109), X110, X111) :- ','(half(s(X109), X112), ','(times(X112, X110, X113), plus(X113, X113, X111))).\nand substitutionT56 -> true,\nT55 -> T75,\nX109 -> s(T75),\nT10 -> T76,\nX110 -> T76,\nT12 -> T78,\nX111 -> T78,\nT77 -> T78", fontsize=14, style = filled, fillcolor = lightblue];
172 -> 777 [arrowhead = none ];
777 -> 174;

778 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
172 -> 778 [arrowhead = none ];
778 -> 175;

779 [label="EVAL with clause\nif(false, s(X544), X545, X546) :- ','(times(X544, X545, X547), plus(X545, X547, X546)).\nand substitutionT56 -> false,\nT55 -> T252,\nX544 -> s(T252),\nT10 -> T253,\nX545 -> T253,\nT12 -> T255,\nX546 -> T255,\nT254 -> T255", fontsize=14, style = filled, fillcolor = lightblue];
173 -> 779 [arrowhead = none ];
779 -> 714;

780 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
173 -> 780 [arrowhead = none ];
780 -> 715;

781 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
174 -> 781 [arrowhead = none ];
781 -> 179;

782 [label="SPLIT 2\nnew knowledge:\nT75 is ground\nT79 is ground\nreplacements:X112 -> T79", fontsize=14, style = filled, fillcolor = violet];
174 -> 782 [arrowhead = none ];
782 -> 180;

783 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
179 -> 783 [arrowhead = none ];
783 -> 185;

784 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
180 -> 784 [arrowhead = none ];
784 -> 212;

785 [label="SPLIT 2\nnew knowledge:\nT79 is ground\nT76 is ground\nT88 is ground\nreplacements:X113 -> T88", fontsize=14, style = filled, fillcolor = violet];
180 -> 785 [arrowhead = none ];
785 -> 213;

786 [label="BACKTRACK\nfor clause: half(0, 0)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
185 -> 786 [arrowhead = none ];
786 -> 186;

787 [label="ONLY EVAL with clause\nhalf(s(s(X120)), s(X121)) :- half(X120, X121).\nand substitutionT75 -> T82,\nX120 -> T82,\nX121 -> X122,\nX112 -> s(X122)", fontsize=14, style = filled, fillcolor = lightblue];
186 -> 787 [arrowhead = none ];
787 -> 198;

788 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
198 -> 788 [arrowhead = none ];
788 -> 199;

789 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
199 -> 789 [arrowhead = none ];
789 -> 200;

790 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
199 -> 790 [arrowhead = none ];
790 -> 201;

791 [label="EVAL with clause\nhalf(0, 0).\nand substitutionT82 -> 0,\nX122 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
200 -> 791 [arrowhead = none ];
791 -> 202;

792 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
200 -> 792 [arrowhead = none ];
792 -> 203;

793 [label="EVAL with clause\nhalf(s(s(X129)), s(X130)) :- half(X129, X130).\nand substitutionX129 -> T85,\nT82 -> s(s(T85)),\nX130 -> X131,\nX122 -> s(X131)", fontsize=14, style = filled, fillcolor = lightblue];
201 -> 793 [arrowhead = none ];
793 -> 206;

794 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
201 -> 794 [arrowhead = none ];
794 -> 207;

795 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
202 -> 795 [arrowhead = none ];
795 -> 204;

796 [label="INSTANCE with matching:\nT82 -> T85\nX122 -> X131", fontsize=14, style = filled, fillcolor = lightgrey];
206 -> 796 [arrowhead = none , style=dashed];
796 -> 198[style=dashed];

797 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
212 -> 797 [arrowhead = none ];
797 -> 218;

798 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
213 -> 798 [arrowhead = none ];
798 -> 569;

799 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
218 -> 799 [arrowhead = none ];
799 -> 219;

800 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
218 -> 800 [arrowhead = none ];
800 -> 220;

801 [label="EVAL with clause\ntimes(0, X140, 0).\nand substitutionT79 -> 0,\nT76 -> T95,\nX140 -> T95,\nX113 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
219 -> 801 [arrowhead = none ];
801 -> 221;

802 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
219 -> 802 [arrowhead = none ];
802 -> 222;

803 [label="EVAL with clause\ntimes(s(X150), X151, X152) :- ','(even(s(X150), X153), if(X153, s(X150), X151, X152)).\nand substitutionX150 -> T100,\nT79 -> s(T100),\nT76 -> T101,\nX151 -> T101,\nX113 -> X154,\nX152 -> X154", fontsize=14, style = filled, fillcolor = lightblue];
220 -> 803 [arrowhead = none ];
803 -> 228;

804 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
220 -> 804 [arrowhead = none ];
804 -> 229;

805 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
221 -> 805 [arrowhead = none ];
805 -> 223;

806 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
228 -> 806 [arrowhead = none ];
806 -> 230;

807 [label="SPLIT 2\nnew knowledge:\nT100 is ground\nT102 is ground\nreplacements:X153 -> T102", fontsize=14, style = filled, fillcolor = violet];
228 -> 807 [arrowhead = none ];
807 -> 231;

808 [label="INSTANCE with matching:\nT55 -> s(T100)\nX78 -> X153", fontsize=14, style = filled, fillcolor = lightgrey];
230 -> 808 [arrowhead = none , style=dashed];
808 -> 123[style=dashed];

809 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
231 -> 809 [arrowhead = none ];
809 -> 232;

810 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
232 -> 810 [arrowhead = none ];
810 -> 233;

811 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
232 -> 811 [arrowhead = none ];
811 -> 234;

812 [label="EVAL with clause\nif(true, s(X181), X182, X183) :- ','(half(s(X181), X184), ','(times(X184, X182, X185), plus(X185, X185, X183))).\nand substitutionT102 -> true,\nT100 -> T113,\nX181 -> T113,\nT101 -> T114,\nX182 -> T114,\nX154 -> X186,\nX183 -> X186", fontsize=14, style = filled, fillcolor = lightblue];
233 -> 812 [arrowhead = none ];
812 -> 235;

813 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
233 -> 813 [arrowhead = none ];
813 -> 236;

814 [label="EVAL with clause\nif(false, s(X375), X376, X377) :- ','(times(X375, X376, X378), plus(X376, X378, X377)).\nand substitutionT102 -> false,\nT100 -> T165,\nX375 -> T165,\nT101 -> T166,\nX376 -> T166,\nX154 -> X379,\nX377 -> X379", fontsize=14, style = filled, fillcolor = lightblue];
234 -> 814 [arrowhead = none ];
814 -> 477;

815 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
234 -> 815 [arrowhead = none ];
815 -> 478;

816 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
235 -> 816 [arrowhead = none ];
816 -> 237;

817 [label="SPLIT 2\nnew knowledge:\nT113 is ground\nT115 is ground\nreplacements:X184 -> T115", fontsize=14, style = filled, fillcolor = violet];
235 -> 817 [arrowhead = none ];
817 -> 238;

818 [label="INSTANCE with matching:\nT82 -> s(T113)\nX122 -> X184", fontsize=14, style = filled, fillcolor = lightgrey];
237 -> 818 [arrowhead = none , style=dashed];
818 -> 198[style=dashed];

819 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
238 -> 819 [arrowhead = none ];
819 -> 244;

820 [label="SPLIT 2\nnew knowledge:\nT115 is ground\nT114 is ground\nT118 is ground\nreplacements:X185 -> T118", fontsize=14, style = filled, fillcolor = violet];
238 -> 820 [arrowhead = none ];
820 -> 245;

821 [label="INSTANCE with matching:\nT79 -> T115\nT76 -> T114\nX113 -> X185", fontsize=14, style = filled, fillcolor = lightgrey];
244 -> 821 [arrowhead = none , style=dashed];
821 -> 212[style=dashed];

822 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
245 -> 822 [arrowhead = none ];
822 -> 256;

823 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
256 -> 823 [arrowhead = none ];
823 -> 259;

824 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
256 -> 824 [arrowhead = none ];
824 -> 260;

825 [label="EVAL with clause\nplus(0, X197, X197).\nand substitutionT118 -> 0,\nX197 -> 0,\nX186 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
259 -> 825 [arrowhead = none ];
825 -> 261;

826 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
259 -> 826 [arrowhead = none ];
826 -> 262;

827 [label="EVAL with clause\nplus(s(X206), X207, s(X208)) :- plus(X206, X207, X208).\nand substitutionX206 -> T123,\nT118 -> s(T123),\nX207 -> s(T123),\nX208 -> X209,\nX186 -> s(X209)", fontsize=14, style = filled, fillcolor = lightblue];
260 -> 827 [arrowhead = none ];
827 -> 272;

828 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
260 -> 828 [arrowhead = none ];
828 -> 273;

829 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
261 -> 829 [arrowhead = none ];
829 -> 263;

830 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
272 -> 830 [arrowhead = none ];
830 -> 278;

831 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
278 -> 831 [arrowhead = none ];
831 -> 282;

832 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
278 -> 832 [arrowhead = none ];
832 -> 284;

833 [label="EVAL with clause\nplus(0, X216, X216).\nand substitutionT123 -> 0,\nX216 -> s(0),\nX209 -> s(0)", fontsize=14, style = filled, fillcolor = lightblue];
282 -> 833 [arrowhead = none ];
833 -> 285;

834 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
282 -> 834 [arrowhead = none ];
834 -> 286;

835 [label="EVAL with clause\nplus(s(X225), X226, s(X227)) :- plus(X225, X226, X227).\nand substitutionX225 -> T126,\nT123 -> s(T126),\nX226 -> s(s(T126)),\nX227 -> X228,\nX209 -> s(X228)", fontsize=14, style = filled, fillcolor = lightblue];
284 -> 835 [arrowhead = none ];
835 -> 299;

836 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
284 -> 836 [arrowhead = none ];
836 -> 300;

837 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
285 -> 837 [arrowhead = none ];
837 -> 287;

838 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
299 -> 838 [arrowhead = none ];
838 -> 304;

839 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
304 -> 839 [arrowhead = none ];
839 -> 305;

840 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
304 -> 840 [arrowhead = none ];
840 -> 306;

841 [label="EVAL with clause\nplus(0, X235, X235).\nand substitutionT126 -> 0,\nX235 -> s(s(0)),\nX228 -> s(s(0))", fontsize=14, style = filled, fillcolor = lightblue];
305 -> 841 [arrowhead = none ];
841 -> 311;

842 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
305 -> 842 [arrowhead = none ];
842 -> 312;

843 [label="EVAL with clause\nplus(s(X244), X245, s(X246)) :- plus(X244, X245, X246).\nand substitutionX244 -> T129,\nT126 -> s(T129),\nX245 -> s(s(s(T129))),\nX246 -> X247,\nX228 -> s(X247)", fontsize=14, style = filled, fillcolor = lightblue];
306 -> 843 [arrowhead = none ];
843 -> 316;

844 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
306 -> 844 [arrowhead = none ];
844 -> 317;

845 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
311 -> 845 [arrowhead = none ];
845 -> 313;

846 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
316 -> 846 [arrowhead = none ];
846 -> 328;

847 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
328 -> 847 [arrowhead = none ];
847 -> 329;

848 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
328 -> 848 [arrowhead = none ];
848 -> 330;

849 [label="EVAL with clause\nplus(0, X254, X254).\nand substitutionT129 -> 0,\nX254 -> s(s(s(0))),\nX247 -> s(s(s(0)))", fontsize=14, style = filled, fillcolor = lightblue];
329 -> 849 [arrowhead = none ];
849 -> 331;

850 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
329 -> 850 [arrowhead = none ];
850 -> 332;

851 [label="EVAL with clause\nplus(s(X263), X264, s(X265)) :- plus(X263, X264, X265).\nand substitutionX263 -> T132,\nT129 -> s(T132),\nX264 -> s(s(s(s(T132)))),\nX265 -> X266,\nX247 -> s(X266)", fontsize=14, style = filled, fillcolor = lightblue];
330 -> 851 [arrowhead = none ];
851 -> 341;

852 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
330 -> 852 [arrowhead = none ];
852 -> 342;

853 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
331 -> 853 [arrowhead = none ];
853 -> 333;

854 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
341 -> 854 [arrowhead = none ];
854 -> 352;

855 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
352 -> 855 [arrowhead = none ];
855 -> 353;

856 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
352 -> 856 [arrowhead = none ];
856 -> 354;

857 [label="EVAL with clause\nplus(0, X273, X273).\nand substitutionT132 -> 0,\nX273 -> s(s(s(s(0)))),\nX266 -> s(s(s(s(0))))", fontsize=14, style = filled, fillcolor = lightblue];
353 -> 857 [arrowhead = none ];
857 -> 357;

858 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
353 -> 858 [arrowhead = none ];
858 -> 358;

859 [label="EVAL with clause\nplus(s(X282), X283, s(X284)) :- plus(X282, X283, X284).\nand substitutionX282 -> T135,\nT132 -> s(T135),\nX283 -> s(s(s(s(s(T135))))),\nX284 -> X285,\nX266 -> s(X285)", fontsize=14, style = filled, fillcolor = lightblue];
354 -> 859 [arrowhead = none ];
859 -> 367;

860 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
354 -> 860 [arrowhead = none ];
860 -> 370;

861 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
357 -> 861 [arrowhead = none ];
861 -> 359;

862 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
367 -> 862 [arrowhead = none ];
862 -> 375;

863 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
375 -> 863 [arrowhead = none ];
863 -> 379;

864 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
375 -> 864 [arrowhead = none ];
864 -> 380;

865 [label="EVAL with clause\nplus(0, X292, X292).\nand substitutionT135 -> 0,\nX292 -> s(s(s(s(s(0))))),\nX285 -> s(s(s(s(s(0)))))", fontsize=14, style = filled, fillcolor = lightblue];
379 -> 865 [arrowhead = none ];
865 -> 381;

866 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
379 -> 866 [arrowhead = none ];
866 -> 382;

867 [label="EVAL with clause\nplus(s(X301), X302, s(X303)) :- plus(X301, X302, X303).\nand substitutionX301 -> T138,\nT135 -> s(T138),\nX302 -> s(s(s(s(s(s(T138)))))),\nX303 -> X304,\nX285 -> s(X304)", fontsize=14, style = filled, fillcolor = lightblue];
380 -> 867 [arrowhead = none ];
867 -> 392;

868 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
380 -> 868 [arrowhead = none ];
868 -> 393;

869 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
381 -> 869 [arrowhead = none ];
869 -> 383;

870 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
392 -> 870 [arrowhead = none ];
870 -> 400;

871 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
400 -> 871 [arrowhead = none ];
871 -> 403;

872 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
400 -> 872 [arrowhead = none ];
872 -> 404;

873 [label="EVAL with clause\nplus(0, X311, X311).\nand substitutionT138 -> 0,\nX311 -> s(s(s(s(s(s(0)))))),\nX304 -> s(s(s(s(s(s(0))))))", fontsize=14, style = filled, fillcolor = lightblue];
403 -> 873 [arrowhead = none ];
873 -> 405;

874 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
403 -> 874 [arrowhead = none ];
874 -> 407;

875 [label="EVAL with clause\nplus(s(X320), X321, s(X322)) :- plus(X320, X321, X322).\nand substitutionX320 -> T141,\nT138 -> s(T141),\nX321 -> s(s(s(s(s(s(s(T141))))))),\nX322 -> X323,\nX304 -> s(X323)", fontsize=14, style = filled, fillcolor = lightblue];
404 -> 875 [arrowhead = none ];
875 -> 416;

876 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
404 -> 876 [arrowhead = none ];
876 -> 417;

877 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
405 -> 877 [arrowhead = none ];
877 -> 408;

878 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
416 -> 878 [arrowhead = none ];
878 -> 423;

879 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
423 -> 879 [arrowhead = none ];
879 -> 427;

880 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
423 -> 880 [arrowhead = none ];
880 -> 428;

881 [label="EVAL with clause\nplus(0, X330, X330).\nand substitutionT141 -> 0,\nX330 -> s(s(s(s(s(s(s(0))))))),\nX323 -> s(s(s(s(s(s(s(0)))))))", fontsize=14, style = filled, fillcolor = lightblue];
427 -> 881 [arrowhead = none ];
881 -> 430;

882 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
427 -> 882 [arrowhead = none ];
882 -> 433;

883 [label="EVAL with clause\nplus(s(X339), X340, s(X341)) :- plus(X339, X340, X341).\nand substitutionX339 -> T144,\nT141 -> s(T144),\nX340 -> s(s(s(s(s(s(s(s(T144)))))))),\nX341 -> X342,\nX323 -> s(X342)", fontsize=14, style = filled, fillcolor = lightblue];
428 -> 883 [arrowhead = none ];
883 -> 440;

884 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
428 -> 884 [arrowhead = none ];
884 -> 441;

885 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
430 -> 885 [arrowhead = none ];
885 -> 434;

886 [label="GENERALIZATION\nT145 <-- s(s(s(s(s(s(s(T144)))))))\n\nNew Knowledge:\nT145 is ground", fontsize=14, style = filled, fillcolor = violet];
440 -> 886 [arrowhead = none ];
886 -> 443;

887 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
443 -> 887 [arrowhead = none ];
887 -> 449;

888 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
449 -> 888 [arrowhead = none ];
888 -> 453;

889 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
449 -> 889 [arrowhead = none ];
889 -> 454;

890 [label="EVAL with clause\nplus(0, X351, X351).\nand substitutionT144 -> 0,\nT145 -> T152,\nX351 -> s(T152),\nX342 -> s(T152)", fontsize=14, style = filled, fillcolor = lightblue];
453 -> 890 [arrowhead = none ];
890 -> 456;

891 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
453 -> 891 [arrowhead = none ];
891 -> 457;

892 [label="EVAL with clause\nplus(s(X360), X361, s(X362)) :- plus(X360, X361, X362).\nand substitutionX360 -> T157,\nT144 -> s(T157),\nT145 -> T158,\nX361 -> s(T158),\nX362 -> X363,\nX342 -> s(X363)", fontsize=14, style = filled, fillcolor = lightblue];
454 -> 892 [arrowhead = none ];
892 -> 462;

893 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
454 -> 893 [arrowhead = none ];
893 -> 465;

894 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
456 -> 894 [arrowhead = none ];
894 -> 458;

895 [label="INSTANCE with matching:\nT144 -> T157\nT145 -> T158\nX342 -> X363", fontsize=14, style = filled, fillcolor = lightgrey];
462 -> 895 [arrowhead = none , style=dashed];
895 -> 443[style=dashed];

896 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
477 -> 896 [arrowhead = none ];
896 -> 503;

897 [label="SPLIT 2\nnew knowledge:\nT165 is ground\nT166 is ground\nT169 is ground\nreplacements:X378 -> T169", fontsize=14, style = filled, fillcolor = violet];
477 -> 897 [arrowhead = none ];
897 -> 504;

898 [label="INSTANCE with matching:\nT79 -> T165\nT76 -> T166\nX113 -> X378", fontsize=14, style = filled, fillcolor = lightgrey];
503 -> 898 [arrowhead = none , style=dashed];
898 -> 212[style=dashed];

899 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
504 -> 899 [arrowhead = none ];
899 -> 514;

900 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
514 -> 900 [arrowhead = none ];
900 -> 518;

901 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
514 -> 901 [arrowhead = none ];
901 -> 520;

902 [label="EVAL with clause\nplus(0, X390, X390).\nand substitutionT166 -> 0,\nT169 -> T178,\nX390 -> T178,\nX379 -> T178", fontsize=14, style = filled, fillcolor = lightblue];
518 -> 902 [arrowhead = none ];
902 -> 524;

903 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
518 -> 903 [arrowhead = none ];
903 -> 525;

904 [label="EVAL with clause\nplus(s(X399), X400, s(X401)) :- plus(X399, X400, X401).\nand substitutionX399 -> T183,\nT166 -> s(T183),\nT169 -> T184,\nX400 -> T184,\nX401 -> X402,\nX379 -> s(X402)", fontsize=14, style = filled, fillcolor = lightblue];
520 -> 904 [arrowhead = none ];
904 -> 559;

905 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
520 -> 905 [arrowhead = none ];
905 -> 560;

906 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
524 -> 906 [arrowhead = none ];
906 -> 526;

907 [label="INSTANCE with matching:\nT166 -> T183\nT169 -> T184\nX379 -> X402", fontsize=14, style = filled, fillcolor = lightgrey];
559 -> 907 [arrowhead = none , style=dashed];
907 -> 504[style=dashed];

908 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
569 -> 908 [arrowhead = none ];
908 -> 570;

909 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
569 -> 909 [arrowhead = none ];
909 -> 571;

910 [label="EVAL with clause\nplus(0, X411, X411).\nand substitutionT88 -> 0,\nX411 -> 0,\nT78 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
570 -> 910 [arrowhead = none ];
910 -> 572;

911 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
570 -> 911 [arrowhead = none ];
911 -> 573;

912 [label="EVAL with clause\nplus(s(X418), X419, s(X420)) :- plus(X418, X419, X420).\nand substitutionX418 -> T191,\nT88 -> s(T191),\nX419 -> s(T191),\nX420 -> T193,\nT78 -> s(T193),\nT192 -> T193", fontsize=14, style = filled, fillcolor = lightblue];
571 -> 912 [arrowhead = none ];
912 -> 582;

913 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
571 -> 913 [arrowhead = none ];
913 -> 584;

914 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
572 -> 914 [arrowhead = none ];
914 -> 574;

915 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
582 -> 915 [arrowhead = none ];
915 -> 588;

916 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
588 -> 916 [arrowhead = none ];
916 -> 592;

917 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
588 -> 917 [arrowhead = none ];
917 -> 593;

918 [label="EVAL with clause\nplus(0, X427, X427).\nand substitutionT191 -> 0,\nX427 -> s(0),\nT193 -> s(0)", fontsize=14, style = filled, fillcolor = lightblue];
592 -> 918 [arrowhead = none ];
918 -> 596;

919 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
592 -> 919 [arrowhead = none ];
919 -> 597;

920 [label="EVAL with clause\nplus(s(X434), X435, s(X436)) :- plus(X434, X435, X436).\nand substitutionX434 -> T198,\nT191 -> s(T198),\nX435 -> s(s(T198)),\nX436 -> T200,\nT193 -> s(T200),\nT199 -> T200", fontsize=14, style = filled, fillcolor = lightblue];
593 -> 920 [arrowhead = none ];
920 -> 601;

921 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
593 -> 921 [arrowhead = none ];
921 -> 602;

922 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
596 -> 922 [arrowhead = none ];
922 -> 598;

923 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
601 -> 923 [arrowhead = none ];
923 -> 604;

924 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
604 -> 924 [arrowhead = none ];
924 -> 607;

925 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
604 -> 925 [arrowhead = none ];
925 -> 608;

926 [label="EVAL with clause\nplus(0, X443, X443).\nand substitutionT198 -> 0,\nX443 -> s(s(0)),\nT200 -> s(s(0))", fontsize=14, style = filled, fillcolor = lightblue];
607 -> 926 [arrowhead = none ];
926 -> 609;

927 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
607 -> 927 [arrowhead = none ];
927 -> 611;

928 [label="EVAL with clause\nplus(s(X450), X451, s(X452)) :- plus(X450, X451, X452).\nand substitutionX450 -> T205,\nT198 -> s(T205),\nX451 -> s(s(s(T205))),\nX452 -> T207,\nT200 -> s(T207),\nT206 -> T207", fontsize=14, style = filled, fillcolor = lightblue];
608 -> 928 [arrowhead = none ];
928 -> 615;

929 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
608 -> 929 [arrowhead = none ];
929 -> 616;

930 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
609 -> 930 [arrowhead = none ];
930 -> 613;

931 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
615 -> 931 [arrowhead = none ];
931 -> 625;

932 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
625 -> 932 [arrowhead = none ];
932 -> 626;

933 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
625 -> 933 [arrowhead = none ];
933 -> 627;

934 [label="EVAL with clause\nplus(0, X459, X459).\nand substitutionT205 -> 0,\nX459 -> s(s(s(0))),\nT207 -> s(s(s(0)))", fontsize=14, style = filled, fillcolor = lightblue];
626 -> 934 [arrowhead = none ];
934 -> 629;

935 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
626 -> 935 [arrowhead = none ];
935 -> 632;

936 [label="EVAL with clause\nplus(s(X466), X467, s(X468)) :- plus(X466, X467, X468).\nand substitutionX466 -> T212,\nT205 -> s(T212),\nX467 -> s(s(s(s(T212)))),\nX468 -> T214,\nT207 -> s(T214),\nT213 -> T214", fontsize=14, style = filled, fillcolor = lightblue];
627 -> 936 [arrowhead = none ];
936 -> 639;

937 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
627 -> 937 [arrowhead = none ];
937 -> 640;

938 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
629 -> 938 [arrowhead = none ];
938 -> 633;

939 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
639 -> 939 [arrowhead = none ];
939 -> 643;

940 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
643 -> 940 [arrowhead = none ];
940 -> 644;

941 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
643 -> 941 [arrowhead = none ];
941 -> 645;

942 [label="EVAL with clause\nplus(0, X475, X475).\nand substitutionT212 -> 0,\nX475 -> s(s(s(s(0)))),\nT214 -> s(s(s(s(0))))", fontsize=14, style = filled, fillcolor = lightblue];
644 -> 942 [arrowhead = none ];
942 -> 646;

943 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
644 -> 943 [arrowhead = none ];
943 -> 647;

944 [label="EVAL with clause\nplus(s(X482), X483, s(X484)) :- plus(X482, X483, X484).\nand substitutionX482 -> T219,\nT212 -> s(T219),\nX483 -> s(s(s(s(s(T219))))),\nX484 -> T221,\nT214 -> s(T221),\nT220 -> T221", fontsize=14, style = filled, fillcolor = lightblue];
645 -> 944 [arrowhead = none ];
944 -> 655;

945 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
645 -> 945 [arrowhead = none ];
945 -> 656;

946 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
646 -> 946 [arrowhead = none ];
946 -> 648;

947 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
655 -> 947 [arrowhead = none ];
947 -> 662;

948 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
662 -> 948 [arrowhead = none ];
948 -> 663;

949 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
662 -> 949 [arrowhead = none ];
949 -> 664;

950 [label="EVAL with clause\nplus(0, X491, X491).\nand substitutionT219 -> 0,\nX491 -> s(s(s(s(s(0))))),\nT221 -> s(s(s(s(s(0)))))", fontsize=14, style = filled, fillcolor = lightblue];
663 -> 950 [arrowhead = none ];
950 -> 668;

951 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
663 -> 951 [arrowhead = none ];
951 -> 669;

952 [label="EVAL with clause\nplus(s(X498), X499, s(X500)) :- plus(X498, X499, X500).\nand substitutionX498 -> T226,\nT219 -> s(T226),\nX499 -> s(s(s(s(s(s(T226)))))),\nX500 -> T228,\nT221 -> s(T228),\nT227 -> T228", fontsize=14, style = filled, fillcolor = lightblue];
664 -> 952 [arrowhead = none ];
952 -> 674;

953 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
664 -> 953 [arrowhead = none ];
953 -> 675;

954 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
668 -> 954 [arrowhead = none ];
954 -> 670;

955 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
674 -> 955 [arrowhead = none ];
955 -> 680;

956 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
680 -> 956 [arrowhead = none ];
956 -> 683;

957 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
680 -> 957 [arrowhead = none ];
957 -> 684;

958 [label="EVAL with clause\nplus(0, X507, X507).\nand substitutionT226 -> 0,\nX507 -> s(s(s(s(s(s(0)))))),\nT228 -> s(s(s(s(s(s(0))))))", fontsize=14, style = filled, fillcolor = lightblue];
683 -> 958 [arrowhead = none ];
958 -> 685;

959 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
683 -> 959 [arrowhead = none ];
959 -> 686;

960 [label="EVAL with clause\nplus(s(X514), X515, s(X516)) :- plus(X514, X515, X516).\nand substitutionX514 -> T233,\nT226 -> s(T233),\nX515 -> s(s(s(s(s(s(s(T233))))))),\nX516 -> T235,\nT228 -> s(T235),\nT234 -> T235", fontsize=14, style = filled, fillcolor = lightblue];
684 -> 960 [arrowhead = none ];
960 -> 689;

961 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
684 -> 961 [arrowhead = none ];
961 -> 690;

962 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
685 -> 962 [arrowhead = none ];
962 -> 687;

963 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
689 -> 963 [arrowhead = none ];
963 -> 696;

964 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
696 -> 964 [arrowhead = none ];
964 -> 699;

965 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
696 -> 965 [arrowhead = none ];
965 -> 700;

966 [label="EVAL with clause\nplus(0, X523, X523).\nand substitutionT233 -> 0,\nX523 -> s(s(s(s(s(s(s(0))))))),\nT235 -> s(s(s(s(s(s(s(0)))))))", fontsize=14, style = filled, fillcolor = lightblue];
699 -> 966 [arrowhead = none ];
966 -> 701;

967 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
699 -> 967 [arrowhead = none ];
967 -> 702;

968 [label="EVAL with clause\nplus(s(X530), X531, s(X532)) :- plus(X530, X531, X532).\nand substitutionX530 -> T240,\nT233 -> s(T240),\nX531 -> s(s(s(s(s(s(s(s(T240)))))))),\nX532 -> T242,\nT235 -> s(T242),\nT241 -> T242", fontsize=14, style = filled, fillcolor = lightblue];
700 -> 968 [arrowhead = none ];
968 -> 705;

969 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
700 -> 969 [arrowhead = none ];
969 -> 707;

970 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
701 -> 970 [arrowhead = none ];
970 -> 703;

971 [label="GENERALIZATION\nT243 <-- s(s(s(s(s(s(s(T240)))))))\n\nNew Knowledge:\nT243 is ground", fontsize=14, style = filled, fillcolor = violet];
705 -> 971 [arrowhead = none ];
971 -> 712;

972 [label="INSTANCE with matching:\nT21 -> T240\nT27 -> s(T243)\nT23 -> T242", fontsize=14, style = filled, fillcolor = lightgrey];
712 -> 972 [arrowhead = none , style=dashed];
972 -> 81[style=dashed];

973 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
714 -> 973 [arrowhead = none ];
973 -> 721;

974 [label="SPLIT 2\nnew knowledge:\nT252 is ground\nT253 is ground\nT256 is ground\nreplacements:X547 -> T256", fontsize=14, style = filled, fillcolor = violet];
714 -> 974 [arrowhead = none ];
974 -> 722;

975 [label="INSTANCE with matching:\nT79 -> s(T252)\nT76 -> T253\nX113 -> X547", fontsize=14, style = filled, fillcolor = lightgrey];
721 -> 975 [arrowhead = none , style=dashed];
975 -> 212[style=dashed];

976 [label="INSTANCE with matching:\nT21 -> T253\nT27 -> T256\nT23 -> T255", fontsize=14, style = filled, fillcolor = lightgrey];
722 -> 976 [arrowhead = none , style=dashed];
976 -> 81[style=dashed];

}

(2) Obligation:

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

timesA_in_gga(0, T5, 0) → timesA_out_gga(0, T5, 0)
timesA_in_gga(s(0), T21, T23) → U1_gga(T21, T23, pB_in_gaa(T21, X37, T23))
pB_in_gaa(T21, T27, T23) → U16_gaa(T21, T27, T23, timesK_in_ga(T21, T27))
timesK_in_ga(T33, 0) → timesK_out_ga(T33, 0)
U16_gaa(T21, T27, T23, timesK_out_ga(T21, T27)) → U17_gaa(T21, T27, T23, plusD_in_gga(T21, T27, T23))
plusD_in_gga(0, T40, T40) → plusD_out_gga(0, T40, T40)
plusD_in_gga(s(T47), T48, s(T50)) → U3_gga(T47, T48, T50, plusD_in_gga(T47, T48, T50))
U3_gga(T47, T48, T50, plusD_out_gga(T47, T48, T50)) → plusD_out_gga(s(T47), T48, s(T50))
U17_gaa(T21, T27, T23, plusD_out_gga(T21, T27, T23)) → pB_out_gaa(T21, T27, T23)
U1_gga(T21, T23, pB_out_gaa(T21, X37, T23)) → timesA_out_gga(s(0), T21, T23)
timesA_in_gga(s(s(T55)), T10, T12) → U2_gga(T55, T10, T12, pC_in_gaga(T55, X78, T10, T12))
pC_in_gaga(T55, T56, T10, T12) → U18_gaga(T55, T56, T10, T12, evenE_in_ga(T55, T56))
evenE_in_ga(0, true) → evenE_out_ga(0, true)
evenE_in_ga(s(0), false) → evenE_out_ga(s(0), false)
evenE_in_ga(s(s(T59)), X87) → U4_ga(T59, X87, evenE_in_ga(T59, X87))
U4_ga(T59, X87, evenE_out_ga(T59, X87)) → evenE_out_ga(s(s(T59)), X87)
U18_gaga(T55, T56, T10, T12, evenE_out_ga(T55, T56)) → U19_gaga(T55, T56, T10, T12, ifL_in_ggga(T56, T55, T10, T12))
ifL_in_ggga(true, T75, T76, T78) → U9_ggga(T75, T76, T78, pM_in_gagaa(T75, X112, T76, X113, T78))
pM_in_gagaa(T75, T79, T76, X113, T78) → U20_gagaa(T75, T79, T76, X113, T78, halfO_in_ga(T75, T79))
halfO_in_ga(T82, s(X122)) → U11_ga(T82, X122, halfF_in_ga(T82, X122))
halfF_in_ga(0, 0) → halfF_out_ga(0, 0)
halfF_in_ga(s(s(T85)), s(X131)) → U5_ga(T85, X131, halfF_in_ga(T85, X131))
U5_ga(T85, X131, halfF_out_ga(T85, X131)) → halfF_out_ga(s(s(T85)), s(X131))
U11_ga(T82, X122, halfF_out_ga(T82, X122)) → halfO_out_ga(T82, s(X122))
U20_gagaa(T75, T79, T76, X113, T78, halfO_out_ga(T75, T79)) → U21_gagaa(T75, T79, T76, X113, T78, pU_in_ggaa(T79, T76, X113, T78))
pU_in_ggaa(T79, T76, T88, T78) → U22_ggaa(T79, T76, T88, T78, timesG_in_gga(T79, T76, T88))
timesG_in_gga(0, T95, 0) → timesG_out_gga(0, T95, 0)
timesG_in_gga(s(T100), T101, X154) → U6_gga(T100, T101, X154, pH_in_gaga(T100, X153, T101, X154))
pH_in_gaga(T100, T102, T101, X154) → U24_gaga(T100, T102, T101, X154, evenE_in_ga(s(T100), T102))
U24_gaga(T100, T102, T101, X154, evenE_out_ga(s(T100), T102)) → U25_gaga(T100, T102, T101, X154, ifQ_in_ggga(T102, T100, T101, X154))
ifQ_in_ggga(true, T113, T114, X186) → U13_ggga(T113, T114, X186, pR_in_gagaa(T113, X184, T114, X185, X186))
pR_in_gagaa(T113, T115, T114, X185, X186) → U26_gagaa(T113, T115, T114, X185, X186, halfF_in_ga(s(T113), T115))
U26_gagaa(T113, T115, T114, X185, X186, halfF_out_ga(s(T113), T115)) → U27_gagaa(T113, T115, T114, X185, X186, pV_in_ggaa(T115, T114, X185, X186))
pV_in_ggaa(T115, T114, T118, X186) → U28_ggaa(T115, T114, T118, X186, timesG_in_gga(T115, T114, T118))
U28_ggaa(T115, T114, T118, X186, timesG_out_gga(T115, T114, T118)) → U29_ggaa(T115, T114, T118, X186, plusT_in_ga(T118, X186))
plusT_in_ga(0, 0) → plusT_out_ga(0, 0)
plusT_in_ga(s(0), s(s(0))) → plusT_out_ga(s(0), s(s(0)))
plusT_in_ga(s(s(0)), s(s(s(s(0))))) → plusT_out_ga(s(s(0)), s(s(s(s(0)))))
plusT_in_ga(s(s(s(0))), s(s(s(s(s(s(0))))))) → plusT_out_ga(s(s(s(0))), s(s(s(s(s(s(0)))))))
plusT_in_ga(s(s(s(s(0)))), s(s(s(s(s(s(s(s(0))))))))) → plusT_out_ga(s(s(s(s(0)))), s(s(s(s(s(s(s(s(0)))))))))
plusT_in_ga(s(s(s(s(s(0))))), s(s(s(s(s(s(s(s(s(s(0))))))))))) → plusT_out_ga(s(s(s(s(s(0))))), s(s(s(s(s(s(s(s(s(s(0)))))))))))
plusT_in_ga(s(s(s(s(s(s(0)))))), s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))) → plusT_out_ga(s(s(s(s(s(s(0)))))), s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))
plusT_in_ga(s(s(s(s(s(s(s(0))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))) → plusT_out_ga(s(s(s(s(s(s(s(0))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))
plusT_in_ga(s(s(s(s(s(s(s(s(T144)))))))), s(s(s(s(s(s(s(s(X342))))))))) → U15_ga(T144, X342, plusI_in_gga(T144, s(s(s(s(s(s(s(T144))))))), X342))
plusI_in_gga(0, T152, s(T152)) → plusI_out_gga(0, T152, s(T152))
plusI_in_gga(s(T157), T158, s(X363)) → U7_gga(T157, T158, X363, plusI_in_gga(T157, T158, X363))
U7_gga(T157, T158, X363, plusI_out_gga(T157, T158, X363)) → plusI_out_gga(s(T157), T158, s(X363))
U15_ga(T144, X342, plusI_out_gga(T144, s(s(s(s(s(s(s(T144))))))), X342)) → plusT_out_ga(s(s(s(s(s(s(s(s(T144)))))))), s(s(s(s(s(s(s(s(X342)))))))))
U29_ggaa(T115, T114, T118, X186, plusT_out_ga(T118, X186)) → pV_out_ggaa(T115, T114, T118, X186)
U27_gagaa(T113, T115, T114, X185, X186, pV_out_ggaa(T115, T114, X185, X186)) → pR_out_gagaa(T113, T115, T114, X185, X186)
U13_ggga(T113, T114, X186, pR_out_gagaa(T113, X184, T114, X185, X186)) → ifQ_out_ggga(true, T113, T114, X186)
ifQ_in_ggga(false, T165, T166, X379) → U14_ggga(T165, T166, X379, pS_in_ggaa(T165, T166, X378, X379))
pS_in_ggaa(T165, T166, T169, X379) → U30_ggaa(T165, T166, T169, X379, timesG_in_gga(T165, T166, T169))
U30_ggaa(T165, T166, T169, X379, timesG_out_gga(T165, T166, T169)) → U31_ggaa(T165, T166, T169, X379, plusJ_in_gga(T166, T169, X379))
plusJ_in_gga(0, T178, T178) → plusJ_out_gga(0, T178, T178)
plusJ_in_gga(s(T183), T184, s(X402)) → U8_gga(T183, T184, X402, plusJ_in_gga(T183, T184, X402))
U8_gga(T183, T184, X402, plusJ_out_gga(T183, T184, X402)) → plusJ_out_gga(s(T183), T184, s(X402))
U31_ggaa(T165, T166, T169, X379, plusJ_out_gga(T166, T169, X379)) → pS_out_ggaa(T165, T166, T169, X379)
U14_ggga(T165, T166, X379, pS_out_ggaa(T165, T166, X378, X379)) → ifQ_out_ggga(false, T165, T166, X379)
U25_gaga(T100, T102, T101, X154, ifQ_out_ggga(T102, T100, T101, X154)) → pH_out_gaga(T100, T102, T101, X154)
U6_gga(T100, T101, X154, pH_out_gaga(T100, X153, T101, X154)) → timesG_out_gga(s(T100), T101, X154)
U22_ggaa(T79, T76, T88, T78, timesG_out_gga(T79, T76, T88)) → U23_ggaa(T79, T76, T88, T78, plusP_in_ga(T88, T78))
plusP_in_ga(0, 0) → plusP_out_ga(0, 0)
plusP_in_ga(s(0), s(s(0))) → plusP_out_ga(s(0), s(s(0)))
plusP_in_ga(s(s(0)), s(s(s(s(0))))) → plusP_out_ga(s(s(0)), s(s(s(s(0)))))
plusP_in_ga(s(s(s(0))), s(s(s(s(s(s(0))))))) → plusP_out_ga(s(s(s(0))), s(s(s(s(s(s(0)))))))
plusP_in_ga(s(s(s(s(0)))), s(s(s(s(s(s(s(s(0))))))))) → plusP_out_ga(s(s(s(s(0)))), s(s(s(s(s(s(s(s(0)))))))))
plusP_in_ga(s(s(s(s(s(0))))), s(s(s(s(s(s(s(s(s(s(0))))))))))) → plusP_out_ga(s(s(s(s(s(0))))), s(s(s(s(s(s(s(s(s(s(0)))))))))))
plusP_in_ga(s(s(s(s(s(s(0)))))), s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))) → plusP_out_ga(s(s(s(s(s(s(0)))))), s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))
plusP_in_ga(s(s(s(s(s(s(s(0))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))) → plusP_out_ga(s(s(s(s(s(s(s(0))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))
plusP_in_ga(s(s(s(s(s(s(s(s(T240)))))))), s(s(s(s(s(s(s(s(T242))))))))) → U12_ga(T240, T242, plusD_in_gga(T240, s(s(s(s(s(s(s(s(T240)))))))), T242))
U12_ga(T240, T242, plusD_out_gga(T240, s(s(s(s(s(s(s(s(T240)))))))), T242)) → plusP_out_ga(s(s(s(s(s(s(s(s(T240)))))))), s(s(s(s(s(s(s(s(T242)))))))))
U23_ggaa(T79, T76, T88, T78, plusP_out_ga(T88, T78)) → pU_out_ggaa(T79, T76, T88, T78)
U21_gagaa(T75, T79, T76, X113, T78, pU_out_ggaa(T79, T76, X113, T78)) → pM_out_gagaa(T75, T79, T76, X113, T78)
U9_ggga(T75, T76, T78, pM_out_gagaa(T75, X112, T76, X113, T78)) → ifL_out_ggga(true, T75, T76, T78)
ifL_in_ggga(false, T252, T253, T255) → U10_ggga(T252, T253, T255, pN_in_ggaa(T252, T253, X547, T255))
pN_in_ggaa(T252, T253, T256, T255) → U32_ggaa(T252, T253, T256, T255, timesG_in_gga(s(T252), T253, T256))
U32_ggaa(T252, T253, T256, T255, timesG_out_gga(s(T252), T253, T256)) → U33_ggaa(T252, T253, T256, T255, plusD_in_gga(T253, T256, T255))
U33_ggaa(T252, T253, T256, T255, plusD_out_gga(T253, T256, T255)) → pN_out_ggaa(T252, T253, T256, T255)
U10_ggga(T252, T253, T255, pN_out_ggaa(T252, T253, X547, T255)) → ifL_out_ggga(false, T252, T253, T255)
U19_gaga(T55, T56, T10, T12, ifL_out_ggga(T56, T55, T10, T12)) → pC_out_gaga(T55, T56, T10, T12)
U2_gga(T55, T10, T12, pC_out_gaga(T55, X78, T10, T12)) → timesA_out_gga(s(s(T55)), T10, T12)

The argument filtering Pi contains the following mapping:
timesA_in_gga(x1, x2, x3)  =  timesA_in_gga(x1, x2)
0  =  0
timesA_out_gga(x1, x2, x3)  =  timesA_out_gga(x1, x2, x3)
s(x1)  =  s(x1)
U1_gga(x1, x2, x3)  =  U1_gga(x1, x3)
pB_in_gaa(x1, x2, x3)  =  pB_in_gaa(x1)
U16_gaa(x1, x2, x3, x4)  =  U16_gaa(x1, x4)
timesK_in_ga(x1, x2)  =  timesK_in_ga(x1)
timesK_out_ga(x1, x2)  =  timesK_out_ga(x1, x2)
U17_gaa(x1, x2, x3, x4)  =  U17_gaa(x1, x2, x4)
plusD_in_gga(x1, x2, x3)  =  plusD_in_gga(x1, x2)
plusD_out_gga(x1, x2, x3)  =  plusD_out_gga(x1, x2, x3)
U3_gga(x1, x2, x3, x4)  =  U3_gga(x1, x2, x4)
pB_out_gaa(x1, x2, x3)  =  pB_out_gaa(x1, x2, x3)
U2_gga(x1, x2, x3, x4)  =  U2_gga(x1, x2, x4)
pC_in_gaga(x1, x2, x3, x4)  =  pC_in_gaga(x1, x3)
U18_gaga(x1, x2, x3, x4, x5)  =  U18_gaga(x1, x3, x5)
evenE_in_ga(x1, x2)  =  evenE_in_ga(x1)
evenE_out_ga(x1, x2)  =  evenE_out_ga(x1, x2)
U4_ga(x1, x2, x3)  =  U4_ga(x1, x3)
U19_gaga(x1, x2, x3, x4, x5)  =  U19_gaga(x1, x2, x3, x5)
ifL_in_ggga(x1, x2, x3, x4)  =  ifL_in_ggga(x1, x2, x3)
true  =  true
U9_ggga(x1, x2, x3, x4)  =  U9_ggga(x1, x2, x4)
pM_in_gagaa(x1, x2, x3, x4, x5)  =  pM_in_gagaa(x1, x3)
U20_gagaa(x1, x2, x3, x4, x5, x6)  =  U20_gagaa(x1, x3, x6)
halfO_in_ga(x1, x2)  =  halfO_in_ga(x1)
U11_ga(x1, x2, x3)  =  U11_ga(x1, x3)
halfF_in_ga(x1, x2)  =  halfF_in_ga(x1)
halfF_out_ga(x1, x2)  =  halfF_out_ga(x1, x2)
U5_ga(x1, x2, x3)  =  U5_ga(x1, x3)
halfO_out_ga(x1, x2)  =  halfO_out_ga(x1, x2)
U21_gagaa(x1, x2, x3, x4, x5, x6)  =  U21_gagaa(x1, x2, x3, x6)
pU_in_ggaa(x1, x2, x3, x4)  =  pU_in_ggaa(x1, x2)
U22_ggaa(x1, x2, x3, x4, x5)  =  U22_ggaa(x1, x2, x5)
timesG_in_gga(x1, x2, x3)  =  timesG_in_gga(x1, x2)
timesG_out_gga(x1, x2, x3)  =  timesG_out_gga(x1, x2, x3)
U6_gga(x1, x2, x3, x4)  =  U6_gga(x1, x2, x4)
pH_in_gaga(x1, x2, x3, x4)  =  pH_in_gaga(x1, x3)
U24_gaga(x1, x2, x3, x4, x5)  =  U24_gaga(x1, x3, x5)
U25_gaga(x1, x2, x3, x4, x5)  =  U25_gaga(x1, x2, x3, x5)
ifQ_in_ggga(x1, x2, x3, x4)  =  ifQ_in_ggga(x1, x2, x3)
U13_ggga(x1, x2, x3, x4)  =  U13_ggga(x1, x2, x4)
pR_in_gagaa(x1, x2, x3, x4, x5)  =  pR_in_gagaa(x1, x3)
U26_gagaa(x1, x2, x3, x4, x5, x6)  =  U26_gagaa(x1, x3, x6)
U27_gagaa(x1, x2, x3, x4, x5, x6)  =  U27_gagaa(x1, x2, x3, x6)
pV_in_ggaa(x1, x2, x3, x4)  =  pV_in_ggaa(x1, x2)
U28_ggaa(x1, x2, x3, x4, x5)  =  U28_ggaa(x1, x2, x5)
U29_ggaa(x1, x2, x3, x4, x5)  =  U29_ggaa(x1, x2, x3, x5)
plusT_in_ga(x1, x2)  =  plusT_in_ga(x1)
plusT_out_ga(x1, x2)  =  plusT_out_ga(x1, x2)
U15_ga(x1, x2, x3)  =  U15_ga(x1, x3)
plusI_in_gga(x1, x2, x3)  =  plusI_in_gga(x1, x2)
plusI_out_gga(x1, x2, x3)  =  plusI_out_gga(x1, x2, x3)
U7_gga(x1, x2, x3, x4)  =  U7_gga(x1, x2, x4)
pV_out_ggaa(x1, x2, x3, x4)  =  pV_out_ggaa(x1, x2, x3, x4)
pR_out_gagaa(x1, x2, x3, x4, x5)  =  pR_out_gagaa(x1, x2, x3, x4, x5)
ifQ_out_ggga(x1, x2, x3, x4)  =  ifQ_out_ggga(x1, x2, x3, x4)
false  =  false
U14_ggga(x1, x2, x3, x4)  =  U14_ggga(x1, x2, x4)
pS_in_ggaa(x1, x2, x3, x4)  =  pS_in_ggaa(x1, x2)
U30_ggaa(x1, x2, x3, x4, x5)  =  U30_ggaa(x1, x2, x5)
U31_ggaa(x1, x2, x3, x4, x5)  =  U31_ggaa(x1, x2, x3, x5)
plusJ_in_gga(x1, x2, x3)  =  plusJ_in_gga(x1, x2)
plusJ_out_gga(x1, x2, x3)  =  plusJ_out_gga(x1, x2, x3)
U8_gga(x1, x2, x3, x4)  =  U8_gga(x1, x2, x4)
pS_out_ggaa(x1, x2, x3, x4)  =  pS_out_ggaa(x1, x2, x3, x4)
pH_out_gaga(x1, x2, x3, x4)  =  pH_out_gaga(x1, x2, x3, x4)
U23_ggaa(x1, x2, x3, x4, x5)  =  U23_ggaa(x1, x2, x3, x5)
plusP_in_ga(x1, x2)  =  plusP_in_ga(x1)
plusP_out_ga(x1, x2)  =  plusP_out_ga(x1, x2)
U12_ga(x1, x2, x3)  =  U12_ga(x1, x3)
pU_out_ggaa(x1, x2, x3, x4)  =  pU_out_ggaa(x1, x2, x3, x4)
pM_out_gagaa(x1, x2, x3, x4, x5)  =  pM_out_gagaa(x1, x2, x3, x4, x5)
ifL_out_ggga(x1, x2, x3, x4)  =  ifL_out_ggga(x1, x2, x3, x4)
U10_ggga(x1, x2, x3, x4)  =  U10_ggga(x1, x2, x4)
pN_in_ggaa(x1, x2, x3, x4)  =  pN_in_ggaa(x1, x2)
U32_ggaa(x1, x2, x3, x4, x5)  =  U32_ggaa(x1, x2, x5)
U33_ggaa(x1, x2, x3, x4, x5)  =  U33_ggaa(x1, x2, x3, x5)
pN_out_ggaa(x1, x2, x3, x4)  =  pN_out_ggaa(x1, x2, x3, x4)
pC_out_gaga(x1, x2, x3, x4)  =  pC_out_gaga(x1, x2, x3, x4)

(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:

TIMESA_IN_GGA(s(0), T21, T23) → U1_GGA(T21, T23, pB_in_gaa(T21, X37, T23))
TIMESA_IN_GGA(s(0), T21, T23) → PB_IN_GAA(T21, X37, T23)
PB_IN_GAA(T21, T27, T23) → U16_GAA(T21, T27, T23, timesK_in_ga(T21, T27))
PB_IN_GAA(T21, T27, T23) → TIMESK_IN_GA(T21, T27)
U16_GAA(T21, T27, T23, timesK_out_ga(T21, T27)) → U17_GAA(T21, T27, T23, plusD_in_gga(T21, T27, T23))
U16_GAA(T21, T27, T23, timesK_out_ga(T21, T27)) → PLUSD_IN_GGA(T21, T27, T23)
PLUSD_IN_GGA(s(T47), T48, s(T50)) → U3_GGA(T47, T48, T50, plusD_in_gga(T47, T48, T50))
PLUSD_IN_GGA(s(T47), T48, s(T50)) → PLUSD_IN_GGA(T47, T48, T50)
TIMESA_IN_GGA(s(s(T55)), T10, T12) → U2_GGA(T55, T10, T12, pC_in_gaga(T55, X78, T10, T12))
TIMESA_IN_GGA(s(s(T55)), T10, T12) → PC_IN_GAGA(T55, X78, T10, T12)
PC_IN_GAGA(T55, T56, T10, T12) → U18_GAGA(T55, T56, T10, T12, evenE_in_ga(T55, T56))
PC_IN_GAGA(T55, T56, T10, T12) → EVENE_IN_GA(T55, T56)
EVENE_IN_GA(s(s(T59)), X87) → U4_GA(T59, X87, evenE_in_ga(T59, X87))
EVENE_IN_GA(s(s(T59)), X87) → EVENE_IN_GA(T59, X87)
U18_GAGA(T55, T56, T10, T12, evenE_out_ga(T55, T56)) → U19_GAGA(T55, T56, T10, T12, ifL_in_ggga(T56, T55, T10, T12))
U18_GAGA(T55, T56, T10, T12, evenE_out_ga(T55, T56)) → IFL_IN_GGGA(T56, T55, T10, T12)
IFL_IN_GGGA(true, T75, T76, T78) → U9_GGGA(T75, T76, T78, pM_in_gagaa(T75, X112, T76, X113, T78))
IFL_IN_GGGA(true, T75, T76, T78) → PM_IN_GAGAA(T75, X112, T76, X113, T78)
PM_IN_GAGAA(T75, T79, T76, X113, T78) → U20_GAGAA(T75, T79, T76, X113, T78, halfO_in_ga(T75, T79))
PM_IN_GAGAA(T75, T79, T76, X113, T78) → HALFO_IN_GA(T75, T79)
HALFO_IN_GA(T82, s(X122)) → U11_GA(T82, X122, halfF_in_ga(T82, X122))
HALFO_IN_GA(T82, s(X122)) → HALFF_IN_GA(T82, X122)
HALFF_IN_GA(s(s(T85)), s(X131)) → U5_GA(T85, X131, halfF_in_ga(T85, X131))
HALFF_IN_GA(s(s(T85)), s(X131)) → HALFF_IN_GA(T85, X131)
U20_GAGAA(T75, T79, T76, X113, T78, halfO_out_ga(T75, T79)) → U21_GAGAA(T75, T79, T76, X113, T78, pU_in_ggaa(T79, T76, X113, T78))
U20_GAGAA(T75, T79, T76, X113, T78, halfO_out_ga(T75, T79)) → PU_IN_GGAA(T79, T76, X113, T78)
PU_IN_GGAA(T79, T76, T88, T78) → U22_GGAA(T79, T76, T88, T78, timesG_in_gga(T79, T76, T88))
PU_IN_GGAA(T79, T76, T88, T78) → TIMESG_IN_GGA(T79, T76, T88)
TIMESG_IN_GGA(s(T100), T101, X154) → U6_GGA(T100, T101, X154, pH_in_gaga(T100, X153, T101, X154))
TIMESG_IN_GGA(s(T100), T101, X154) → PH_IN_GAGA(T100, X153, T101, X154)
PH_IN_GAGA(T100, T102, T101, X154) → U24_GAGA(T100, T102, T101, X154, evenE_in_ga(s(T100), T102))
PH_IN_GAGA(T100, T102, T101, X154) → EVENE_IN_GA(s(T100), T102)
U24_GAGA(T100, T102, T101, X154, evenE_out_ga(s(T100), T102)) → U25_GAGA(T100, T102, T101, X154, ifQ_in_ggga(T102, T100, T101, X154))
U24_GAGA(T100, T102, T101, X154, evenE_out_ga(s(T100), T102)) → IFQ_IN_GGGA(T102, T100, T101, X154)
IFQ_IN_GGGA(true, T113, T114, X186) → U13_GGGA(T113, T114, X186, pR_in_gagaa(T113, X184, T114, X185, X186))
IFQ_IN_GGGA(true, T113, T114, X186) → PR_IN_GAGAA(T113, X184, T114, X185, X186)
PR_IN_GAGAA(T113, T115, T114, X185, X186) → U26_GAGAA(T113, T115, T114, X185, X186, halfF_in_ga(s(T113), T115))
PR_IN_GAGAA(T113, T115, T114, X185, X186) → HALFF_IN_GA(s(T113), T115)
U26_GAGAA(T113, T115, T114, X185, X186, halfF_out_ga(s(T113), T115)) → U27_GAGAA(T113, T115, T114, X185, X186, pV_in_ggaa(T115, T114, X185, X186))
U26_GAGAA(T113, T115, T114, X185, X186, halfF_out_ga(s(T113), T115)) → PV_IN_GGAA(T115, T114, X185, X186)
PV_IN_GGAA(T115, T114, T118, X186) → U28_GGAA(T115, T114, T118, X186, timesG_in_gga(T115, T114, T118))
PV_IN_GGAA(T115, T114, T118, X186) → TIMESG_IN_GGA(T115, T114, T118)
U28_GGAA(T115, T114, T118, X186, timesG_out_gga(T115, T114, T118)) → U29_GGAA(T115, T114, T118, X186, plusT_in_ga(T118, X186))
U28_GGAA(T115, T114, T118, X186, timesG_out_gga(T115, T114, T118)) → PLUST_IN_GA(T118, X186)
PLUST_IN_GA(s(s(s(s(s(s(s(s(T144)))))))), s(s(s(s(s(s(s(s(X342))))))))) → U15_GA(T144, X342, plusI_in_gga(T144, s(s(s(s(s(s(s(T144))))))), X342))
PLUST_IN_GA(s(s(s(s(s(s(s(s(T144)))))))), s(s(s(s(s(s(s(s(X342))))))))) → PLUSI_IN_GGA(T144, s(s(s(s(s(s(s(T144))))))), X342)
PLUSI_IN_GGA(s(T157), T158, s(X363)) → U7_GGA(T157, T158, X363, plusI_in_gga(T157, T158, X363))
PLUSI_IN_GGA(s(T157), T158, s(X363)) → PLUSI_IN_GGA(T157, T158, X363)
IFQ_IN_GGGA(false, T165, T166, X379) → U14_GGGA(T165, T166, X379, pS_in_ggaa(T165, T166, X378, X379))
IFQ_IN_GGGA(false, T165, T166, X379) → PS_IN_GGAA(T165, T166, X378, X379)
PS_IN_GGAA(T165, T166, T169, X379) → U30_GGAA(T165, T166, T169, X379, timesG_in_gga(T165, T166, T169))
PS_IN_GGAA(T165, T166, T169, X379) → TIMESG_IN_GGA(T165, T166, T169)
U30_GGAA(T165, T166, T169, X379, timesG_out_gga(T165, T166, T169)) → U31_GGAA(T165, T166, T169, X379, plusJ_in_gga(T166, T169, X379))
U30_GGAA(T165, T166, T169, X379, timesG_out_gga(T165, T166, T169)) → PLUSJ_IN_GGA(T166, T169, X379)
PLUSJ_IN_GGA(s(T183), T184, s(X402)) → U8_GGA(T183, T184, X402, plusJ_in_gga(T183, T184, X402))
PLUSJ_IN_GGA(s(T183), T184, s(X402)) → PLUSJ_IN_GGA(T183, T184, X402)
U22_GGAA(T79, T76, T88, T78, timesG_out_gga(T79, T76, T88)) → U23_GGAA(T79, T76, T88, T78, plusP_in_ga(T88, T78))
U22_GGAA(T79, T76, T88, T78, timesG_out_gga(T79, T76, T88)) → PLUSP_IN_GA(T88, T78)
PLUSP_IN_GA(s(s(s(s(s(s(s(s(T240)))))))), s(s(s(s(s(s(s(s(T242))))))))) → U12_GA(T240, T242, plusD_in_gga(T240, s(s(s(s(s(s(s(s(T240)))))))), T242))
PLUSP_IN_GA(s(s(s(s(s(s(s(s(T240)))))))), s(s(s(s(s(s(s(s(T242))))))))) → PLUSD_IN_GGA(T240, s(s(s(s(s(s(s(s(T240)))))))), T242)
IFL_IN_GGGA(false, T252, T253, T255) → U10_GGGA(T252, T253, T255, pN_in_ggaa(T252, T253, X547, T255))
IFL_IN_GGGA(false, T252, T253, T255) → PN_IN_GGAA(T252, T253, X547, T255)
PN_IN_GGAA(T252, T253, T256, T255) → U32_GGAA(T252, T253, T256, T255, timesG_in_gga(s(T252), T253, T256))
PN_IN_GGAA(T252, T253, T256, T255) → TIMESG_IN_GGA(s(T252), T253, T256)
U32_GGAA(T252, T253, T256, T255, timesG_out_gga(s(T252), T253, T256)) → U33_GGAA(T252, T253, T256, T255, plusD_in_gga(T253, T256, T255))
U32_GGAA(T252, T253, T256, T255, timesG_out_gga(s(T252), T253, T256)) → PLUSD_IN_GGA(T253, T256, T255)

The TRS R consists of the following rules:

timesA_in_gga(0, T5, 0) → timesA_out_gga(0, T5, 0)
timesA_in_gga(s(0), T21, T23) → U1_gga(T21, T23, pB_in_gaa(T21, X37, T23))
pB_in_gaa(T21, T27, T23) → U16_gaa(T21, T27, T23, timesK_in_ga(T21, T27))
timesK_in_ga(T33, 0) → timesK_out_ga(T33, 0)
U16_gaa(T21, T27, T23, timesK_out_ga(T21, T27)) → U17_gaa(T21, T27, T23, plusD_in_gga(T21, T27, T23))
plusD_in_gga(0, T40, T40) → plusD_out_gga(0, T40, T40)
plusD_in_gga(s(T47), T48, s(T50)) → U3_gga(T47, T48, T50, plusD_in_gga(T47, T48, T50))
U3_gga(T47, T48, T50, plusD_out_gga(T47, T48, T50)) → plusD_out_gga(s(T47), T48, s(T50))
U17_gaa(T21, T27, T23, plusD_out_gga(T21, T27, T23)) → pB_out_gaa(T21, T27, T23)
U1_gga(T21, T23, pB_out_gaa(T21, X37, T23)) → timesA_out_gga(s(0), T21, T23)
timesA_in_gga(s(s(T55)), T10, T12) → U2_gga(T55, T10, T12, pC_in_gaga(T55, X78, T10, T12))
pC_in_gaga(T55, T56, T10, T12) → U18_gaga(T55, T56, T10, T12, evenE_in_ga(T55, T56))
evenE_in_ga(0, true) → evenE_out_ga(0, true)
evenE_in_ga(s(0), false) → evenE_out_ga(s(0), false)
evenE_in_ga(s(s(T59)), X87) → U4_ga(T59, X87, evenE_in_ga(T59, X87))
U4_ga(T59, X87, evenE_out_ga(T59, X87)) → evenE_out_ga(s(s(T59)), X87)
U18_gaga(T55, T56, T10, T12, evenE_out_ga(T55, T56)) → U19_gaga(T55, T56, T10, T12, ifL_in_ggga(T56, T55, T10, T12))
ifL_in_ggga(true, T75, T76, T78) → U9_ggga(T75, T76, T78, pM_in_gagaa(T75, X112, T76, X113, T78))
pM_in_gagaa(T75, T79, T76, X113, T78) → U20_gagaa(T75, T79, T76, X113, T78, halfO_in_ga(T75, T79))
halfO_in_ga(T82, s(X122)) → U11_ga(T82, X122, halfF_in_ga(T82, X122))
halfF_in_ga(0, 0) → halfF_out_ga(0, 0)
halfF_in_ga(s(s(T85)), s(X131)) → U5_ga(T85, X131, halfF_in_ga(T85, X131))
U5_ga(T85, X131, halfF_out_ga(T85, X131)) → halfF_out_ga(s(s(T85)), s(X131))
U11_ga(T82, X122, halfF_out_ga(T82, X122)) → halfO_out_ga(T82, s(X122))
U20_gagaa(T75, T79, T76, X113, T78, halfO_out_ga(T75, T79)) → U21_gagaa(T75, T79, T76, X113, T78, pU_in_ggaa(T79, T76, X113, T78))
pU_in_ggaa(T79, T76, T88, T78) → U22_ggaa(T79, T76, T88, T78, timesG_in_gga(T79, T76, T88))
timesG_in_gga(0, T95, 0) → timesG_out_gga(0, T95, 0)
timesG_in_gga(s(T100), T101, X154) → U6_gga(T100, T101, X154, pH_in_gaga(T100, X153, T101, X154))
pH_in_gaga(T100, T102, T101, X154) → U24_gaga(T100, T102, T101, X154, evenE_in_ga(s(T100), T102))
U24_gaga(T100, T102, T101, X154, evenE_out_ga(s(T100), T102)) → U25_gaga(T100, T102, T101, X154, ifQ_in_ggga(T102, T100, T101, X154))
ifQ_in_ggga(true, T113, T114, X186) → U13_ggga(T113, T114, X186, pR_in_gagaa(T113, X184, T114, X185, X186))
pR_in_gagaa(T113, T115, T114, X185, X186) → U26_gagaa(T113, T115, T114, X185, X186, halfF_in_ga(s(T113), T115))
U26_gagaa(T113, T115, T114, X185, X186, halfF_out_ga(s(T113), T115)) → U27_gagaa(T113, T115, T114, X185, X186, pV_in_ggaa(T115, T114, X185, X186))
pV_in_ggaa(T115, T114, T118, X186) → U28_ggaa(T115, T114, T118, X186, timesG_in_gga(T115, T114, T118))
U28_ggaa(T115, T114, T118, X186, timesG_out_gga(T115, T114, T118)) → U29_ggaa(T115, T114, T118, X186, plusT_in_ga(T118, X186))
plusT_in_ga(0, 0) → plusT_out_ga(0, 0)
plusT_in_ga(s(0), s(s(0))) → plusT_out_ga(s(0), s(s(0)))
plusT_in_ga(s(s(0)), s(s(s(s(0))))) → plusT_out_ga(s(s(0)), s(s(s(s(0)))))
plusT_in_ga(s(s(s(0))), s(s(s(s(s(s(0))))))) → plusT_out_ga(s(s(s(0))), s(s(s(s(s(s(0)))))))
plusT_in_ga(s(s(s(s(0)))), s(s(s(s(s(s(s(s(0))))))))) → plusT_out_ga(s(s(s(s(0)))), s(s(s(s(s(s(s(s(0)))))))))
plusT_in_ga(s(s(s(s(s(0))))), s(s(s(s(s(s(s(s(s(s(0))))))))))) → plusT_out_ga(s(s(s(s(s(0))))), s(s(s(s(s(s(s(s(s(s(0)))))))))))
plusT_in_ga(s(s(s(s(s(s(0)))))), s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))) → plusT_out_ga(s(s(s(s(s(s(0)))))), s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))
plusT_in_ga(s(s(s(s(s(s(s(0))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))) → plusT_out_ga(s(s(s(s(s(s(s(0))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))
plusT_in_ga(s(s(s(s(s(s(s(s(T144)))))))), s(s(s(s(s(s(s(s(X342))))))))) → U15_ga(T144, X342, plusI_in_gga(T144, s(s(s(s(s(s(s(T144))))))), X342))
plusI_in_gga(0, T152, s(T152)) → plusI_out_gga(0, T152, s(T152))
plusI_in_gga(s(T157), T158, s(X363)) → U7_gga(T157, T158, X363, plusI_in_gga(T157, T158, X363))
U7_gga(T157, T158, X363, plusI_out_gga(T157, T158, X363)) → plusI_out_gga(s(T157), T158, s(X363))
U15_ga(T144, X342, plusI_out_gga(T144, s(s(s(s(s(s(s(T144))))))), X342)) → plusT_out_ga(s(s(s(s(s(s(s(s(T144)))))))), s(s(s(s(s(s(s(s(X342)))))))))
U29_ggaa(T115, T114, T118, X186, plusT_out_ga(T118, X186)) → pV_out_ggaa(T115, T114, T118, X186)
U27_gagaa(T113, T115, T114, X185, X186, pV_out_ggaa(T115, T114, X185, X186)) → pR_out_gagaa(T113, T115, T114, X185, X186)
U13_ggga(T113, T114, X186, pR_out_gagaa(T113, X184, T114, X185, X186)) → ifQ_out_ggga(true, T113, T114, X186)
ifQ_in_ggga(false, T165, T166, X379) → U14_ggga(T165, T166, X379, pS_in_ggaa(T165, T166, X378, X379))
pS_in_ggaa(T165, T166, T169, X379) → U30_ggaa(T165, T166, T169, X379, timesG_in_gga(T165, T166, T169))
U30_ggaa(T165, T166, T169, X379, timesG_out_gga(T165, T166, T169)) → U31_ggaa(T165, T166, T169, X379, plusJ_in_gga(T166, T169, X379))
plusJ_in_gga(0, T178, T178) → plusJ_out_gga(0, T178, T178)
plusJ_in_gga(s(T183), T184, s(X402)) → U8_gga(T183, T184, X402, plusJ_in_gga(T183, T184, X402))
U8_gga(T183, T184, X402, plusJ_out_gga(T183, T184, X402)) → plusJ_out_gga(s(T183), T184, s(X402))
U31_ggaa(T165, T166, T169, X379, plusJ_out_gga(T166, T169, X379)) → pS_out_ggaa(T165, T166, T169, X379)
U14_ggga(T165, T166, X379, pS_out_ggaa(T165, T166, X378, X379)) → ifQ_out_ggga(false, T165, T166, X379)
U25_gaga(T100, T102, T101, X154, ifQ_out_ggga(T102, T100, T101, X154)) → pH_out_gaga(T100, T102, T101, X154)
U6_gga(T100, T101, X154, pH_out_gaga(T100, X153, T101, X154)) → timesG_out_gga(s(T100), T101, X154)
U22_ggaa(T79, T76, T88, T78, timesG_out_gga(T79, T76, T88)) → U23_ggaa(T79, T76, T88, T78, plusP_in_ga(T88, T78))
plusP_in_ga(0, 0) → plusP_out_ga(0, 0)
plusP_in_ga(s(0), s(s(0))) → plusP_out_ga(s(0), s(s(0)))
plusP_in_ga(s(s(0)), s(s(s(s(0))))) → plusP_out_ga(s(s(0)), s(s(s(s(0)))))
plusP_in_ga(s(s(s(0))), s(s(s(s(s(s(0))))))) → plusP_out_ga(s(s(s(0))), s(s(s(s(s(s(0)))))))
plusP_in_ga(s(s(s(s(0)))), s(s(s(s(s(s(s(s(0))))))))) → plusP_out_ga(s(s(s(s(0)))), s(s(s(s(s(s(s(s(0)))))))))
plusP_in_ga(s(s(s(s(s(0))))), s(s(s(s(s(s(s(s(s(s(0))))))))))) → plusP_out_ga(s(s(s(s(s(0))))), s(s(s(s(s(s(s(s(s(s(0)))))))))))
plusP_in_ga(s(s(s(s(s(s(0)))))), s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))) → plusP_out_ga(s(s(s(s(s(s(0)))))), s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))
plusP_in_ga(s(s(s(s(s(s(s(0))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))) → plusP_out_ga(s(s(s(s(s(s(s(0))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))
plusP_in_ga(s(s(s(s(s(s(s(s(T240)))))))), s(s(s(s(s(s(s(s(T242))))))))) → U12_ga(T240, T242, plusD_in_gga(T240, s(s(s(s(s(s(s(s(T240)))))))), T242))
U12_ga(T240, T242, plusD_out_gga(T240, s(s(s(s(s(s(s(s(T240)))))))), T242)) → plusP_out_ga(s(s(s(s(s(s(s(s(T240)))))))), s(s(s(s(s(s(s(s(T242)))))))))
U23_ggaa(T79, T76, T88, T78, plusP_out_ga(T88, T78)) → pU_out_ggaa(T79, T76, T88, T78)
U21_gagaa(T75, T79, T76, X113, T78, pU_out_ggaa(T79, T76, X113, T78)) → pM_out_gagaa(T75, T79, T76, X113, T78)
U9_ggga(T75, T76, T78, pM_out_gagaa(T75, X112, T76, X113, T78)) → ifL_out_ggga(true, T75, T76, T78)
ifL_in_ggga(false, T252, T253, T255) → U10_ggga(T252, T253, T255, pN_in_ggaa(T252, T253, X547, T255))
pN_in_ggaa(T252, T253, T256, T255) → U32_ggaa(T252, T253, T256, T255, timesG_in_gga(s(T252), T253, T256))
U32_ggaa(T252, T253, T256, T255, timesG_out_gga(s(T252), T253, T256)) → U33_ggaa(T252, T253, T256, T255, plusD_in_gga(T253, T256, T255))
U33_ggaa(T252, T253, T256, T255, plusD_out_gga(T253, T256, T255)) → pN_out_ggaa(T252, T253, T256, T255)
U10_ggga(T252, T253, T255, pN_out_ggaa(T252, T253, X547, T255)) → ifL_out_ggga(false, T252, T253, T255)
U19_gaga(T55, T56, T10, T12, ifL_out_ggga(T56, T55, T10, T12)) → pC_out_gaga(T55, T56, T10, T12)
U2_gga(T55, T10, T12, pC_out_gaga(T55, X78, T10, T12)) → timesA_out_gga(s(s(T55)), T10, T12)

The argument filtering Pi contains the following mapping:
timesA_in_gga(x1, x2, x3)  =  timesA_in_gga(x1, x2)
0  =  0
timesA_out_gga(x1, x2, x3)  =  timesA_out_gga(x1, x2, x3)
s(x1)  =  s(x1)
U1_gga(x1, x2, x3)  =  U1_gga(x1, x3)
pB_in_gaa(x1, x2, x3)  =  pB_in_gaa(x1)
U16_gaa(x1, x2, x3, x4)  =  U16_gaa(x1, x4)
timesK_in_ga(x1, x2)  =  timesK_in_ga(x1)
timesK_out_ga(x1, x2)  =  timesK_out_ga(x1, x2)
U17_gaa(x1, x2, x3, x4)  =  U17_gaa(x1, x2, x4)
plusD_in_gga(x1, x2, x3)  =  plusD_in_gga(x1, x2)
plusD_out_gga(x1, x2, x3)  =  plusD_out_gga(x1, x2, x3)
U3_gga(x1, x2, x3, x4)  =  U3_gga(x1, x2, x4)
pB_out_gaa(x1, x2, x3)  =  pB_out_gaa(x1, x2, x3)
U2_gga(x1, x2, x3, x4)  =  U2_gga(x1, x2, x4)
pC_in_gaga(x1, x2, x3, x4)  =  pC_in_gaga(x1, x3)
U18_gaga(x1, x2, x3, x4, x5)  =  U18_gaga(x1, x3, x5)
evenE_in_ga(x1, x2)  =  evenE_in_ga(x1)
evenE_out_ga(x1, x2)  =  evenE_out_ga(x1, x2)
U4_ga(x1, x2, x3)  =  U4_ga(x1, x3)
U19_gaga(x1, x2, x3, x4, x5)  =  U19_gaga(x1, x2, x3, x5)
ifL_in_ggga(x1, x2, x3, x4)  =  ifL_in_ggga(x1, x2, x3)
true  =  true
U9_ggga(x1, x2, x3, x4)  =  U9_ggga(x1, x2, x4)
pM_in_gagaa(x1, x2, x3, x4, x5)  =  pM_in_gagaa(x1, x3)
U20_gagaa(x1, x2, x3, x4, x5, x6)  =  U20_gagaa(x1, x3, x6)
halfO_in_ga(x1, x2)  =  halfO_in_ga(x1)
U11_ga(x1, x2, x3)  =  U11_ga(x1, x3)
halfF_in_ga(x1, x2)  =  halfF_in_ga(x1)
halfF_out_ga(x1, x2)  =  halfF_out_ga(x1, x2)
U5_ga(x1, x2, x3)  =  U5_ga(x1, x3)
halfO_out_ga(x1, x2)  =  halfO_out_ga(x1, x2)
U21_gagaa(x1, x2, x3, x4, x5, x6)  =  U21_gagaa(x1, x2, x3, x6)
pU_in_ggaa(x1, x2, x3, x4)  =  pU_in_ggaa(x1, x2)
U22_ggaa(x1, x2, x3, x4, x5)  =  U22_ggaa(x1, x2, x5)
timesG_in_gga(x1, x2, x3)  =  timesG_in_gga(x1, x2)
timesG_out_gga(x1, x2, x3)  =  timesG_out_gga(x1, x2, x3)
U6_gga(x1, x2, x3, x4)  =  U6_gga(x1, x2, x4)
pH_in_gaga(x1, x2, x3, x4)  =  pH_in_gaga(x1, x3)
U24_gaga(x1, x2, x3, x4, x5)  =  U24_gaga(x1, x3, x5)
U25_gaga(x1, x2, x3, x4, x5)  =  U25_gaga(x1, x2, x3, x5)
ifQ_in_ggga(x1, x2, x3, x4)  =  ifQ_in_ggga(x1, x2, x3)
U13_ggga(x1, x2, x3, x4)  =  U13_ggga(x1, x2, x4)
pR_in_gagaa(x1, x2, x3, x4, x5)  =  pR_in_gagaa(x1, x3)
U26_gagaa(x1, x2, x3, x4, x5, x6)  =  U26_gagaa(x1, x3, x6)
U27_gagaa(x1, x2, x3, x4, x5, x6)  =  U27_gagaa(x1, x2, x3, x6)
pV_in_ggaa(x1, x2, x3, x4)  =  pV_in_ggaa(x1, x2)
U28_ggaa(x1, x2, x3, x4, x5)  =  U28_ggaa(x1, x2, x5)
U29_ggaa(x1, x2, x3, x4, x5)  =  U29_ggaa(x1, x2, x3, x5)
plusT_in_ga(x1, x2)  =  plusT_in_ga(x1)
plusT_out_ga(x1, x2)  =  plusT_out_ga(x1, x2)
U15_ga(x1, x2, x3)  =  U15_ga(x1, x3)
plusI_in_gga(x1, x2, x3)  =  plusI_in_gga(x1, x2)
plusI_out_gga(x1, x2, x3)  =  plusI_out_gga(x1, x2, x3)
U7_gga(x1, x2, x3, x4)  =  U7_gga(x1, x2, x4)
pV_out_ggaa(x1, x2, x3, x4)  =  pV_out_ggaa(x1, x2, x3, x4)
pR_out_gagaa(x1, x2, x3, x4, x5)  =  pR_out_gagaa(x1, x2, x3, x4, x5)
ifQ_out_ggga(x1, x2, x3, x4)  =  ifQ_out_ggga(x1, x2, x3, x4)
false  =  false
U14_ggga(x1, x2, x3, x4)  =  U14_ggga(x1, x2, x4)
pS_in_ggaa(x1, x2, x3, x4)  =  pS_in_ggaa(x1, x2)
U30_ggaa(x1, x2, x3, x4, x5)  =  U30_ggaa(x1, x2, x5)
U31_ggaa(x1, x2, x3, x4, x5)  =  U31_ggaa(x1, x2, x3, x5)
plusJ_in_gga(x1, x2, x3)  =  plusJ_in_gga(x1, x2)
plusJ_out_gga(x1, x2, x3)  =  plusJ_out_gga(x1, x2, x3)
U8_gga(x1, x2, x3, x4)  =  U8_gga(x1, x2, x4)
pS_out_ggaa(x1, x2, x3, x4)  =  pS_out_ggaa(x1, x2, x3, x4)
pH_out_gaga(x1, x2, x3, x4)  =  pH_out_gaga(x1, x2, x3, x4)
U23_ggaa(x1, x2, x3, x4, x5)  =  U23_ggaa(x1, x2, x3, x5)
plusP_in_ga(x1, x2)  =  plusP_in_ga(x1)
plusP_out_ga(x1, x2)  =  plusP_out_ga(x1, x2)
U12_ga(x1, x2, x3)  =  U12_ga(x1, x3)
pU_out_ggaa(x1, x2, x3, x4)  =  pU_out_ggaa(x1, x2, x3, x4)
pM_out_gagaa(x1, x2, x3, x4, x5)  =  pM_out_gagaa(x1, x2, x3, x4, x5)
ifL_out_ggga(x1, x2, x3, x4)  =  ifL_out_ggga(x1, x2, x3, x4)
U10_ggga(x1, x2, x3, x4)  =  U10_ggga(x1, x2, x4)
pN_in_ggaa(x1, x2, x3, x4)  =  pN_in_ggaa(x1, x2)
U32_ggaa(x1, x2, x3, x4, x5)  =  U32_ggaa(x1, x2, x5)
U33_ggaa(x1, x2, x3, x4, x5)  =  U33_ggaa(x1, x2, x3, x5)
pN_out_ggaa(x1, x2, x3, x4)  =  pN_out_ggaa(x1, x2, x3, x4)
pC_out_gaga(x1, x2, x3, x4)  =  pC_out_gaga(x1, x2, x3, x4)
TIMESA_IN_GGA(x1, x2, x3)  =  TIMESA_IN_GGA(x1, x2)
U1_GGA(x1, x2, x3)  =  U1_GGA(x1, x3)
PB_IN_GAA(x1, x2, x3)  =  PB_IN_GAA(x1)
U16_GAA(x1, x2, x3, x4)  =  U16_GAA(x1, x4)
TIMESK_IN_GA(x1, x2)  =  TIMESK_IN_GA(x1)
U17_GAA(x1, x2, x3, x4)  =  U17_GAA(x1, x2, x4)
PLUSD_IN_GGA(x1, x2, x3)  =  PLUSD_IN_GGA(x1, x2)
U3_GGA(x1, x2, x3, x4)  =  U3_GGA(x1, x2, x4)
U2_GGA(x1, x2, x3, x4)  =  U2_GGA(x1, x2, x4)
PC_IN_GAGA(x1, x2, x3, x4)  =  PC_IN_GAGA(x1, x3)
U18_GAGA(x1, x2, x3, x4, x5)  =  U18_GAGA(x1, x3, x5)
EVENE_IN_GA(x1, x2)  =  EVENE_IN_GA(x1)
U4_GA(x1, x2, x3)  =  U4_GA(x1, x3)
U19_GAGA(x1, x2, x3, x4, x5)  =  U19_GAGA(x1, x2, x3, x5)
IFL_IN_GGGA(x1, x2, x3, x4)  =  IFL_IN_GGGA(x1, x2, x3)
U9_GGGA(x1, x2, x3, x4)  =  U9_GGGA(x1, x2, x4)
PM_IN_GAGAA(x1, x2, x3, x4, x5)  =  PM_IN_GAGAA(x1, x3)
U20_GAGAA(x1, x2, x3, x4, x5, x6)  =  U20_GAGAA(x1, x3, x6)
HALFO_IN_GA(x1, x2)  =  HALFO_IN_GA(x1)
U11_GA(x1, x2, x3)  =  U11_GA(x1, x3)
HALFF_IN_GA(x1, x2)  =  HALFF_IN_GA(x1)
U5_GA(x1, x2, x3)  =  U5_GA(x1, x3)
U21_GAGAA(x1, x2, x3, x4, x5, x6)  =  U21_GAGAA(x1, x2, x3, x6)
PU_IN_GGAA(x1, x2, x3, x4)  =  PU_IN_GGAA(x1, x2)
U22_GGAA(x1, x2, x3, x4, x5)  =  U22_GGAA(x1, x2, x5)
TIMESG_IN_GGA(x1, x2, x3)  =  TIMESG_IN_GGA(x1, x2)
U6_GGA(x1, x2, x3, x4)  =  U6_GGA(x1, x2, x4)
PH_IN_GAGA(x1, x2, x3, x4)  =  PH_IN_GAGA(x1, x3)
U24_GAGA(x1, x2, x3, x4, x5)  =  U24_GAGA(x1, x3, x5)
U25_GAGA(x1, x2, x3, x4, x5)  =  U25_GAGA(x1, x2, x3, x5)
IFQ_IN_GGGA(x1, x2, x3, x4)  =  IFQ_IN_GGGA(x1, x2, x3)
U13_GGGA(x1, x2, x3, x4)  =  U13_GGGA(x1, x2, x4)
PR_IN_GAGAA(x1, x2, x3, x4, x5)  =  PR_IN_GAGAA(x1, x3)
U26_GAGAA(x1, x2, x3, x4, x5, x6)  =  U26_GAGAA(x1, x3, x6)
U27_GAGAA(x1, x2, x3, x4, x5, x6)  =  U27_GAGAA(x1, x2, x3, x6)
PV_IN_GGAA(x1, x2, x3, x4)  =  PV_IN_GGAA(x1, x2)
U28_GGAA(x1, x2, x3, x4, x5)  =  U28_GGAA(x1, x2, x5)
U29_GGAA(x1, x2, x3, x4, x5)  =  U29_GGAA(x1, x2, x3, x5)
PLUST_IN_GA(x1, x2)  =  PLUST_IN_GA(x1)
U15_GA(x1, x2, x3)  =  U15_GA(x1, x3)
PLUSI_IN_GGA(x1, x2, x3)  =  PLUSI_IN_GGA(x1, x2)
U7_GGA(x1, x2, x3, x4)  =  U7_GGA(x1, x2, x4)
U14_GGGA(x1, x2, x3, x4)  =  U14_GGGA(x1, x2, x4)
PS_IN_GGAA(x1, x2, x3, x4)  =  PS_IN_GGAA(x1, x2)
U30_GGAA(x1, x2, x3, x4, x5)  =  U30_GGAA(x1, x2, x5)
U31_GGAA(x1, x2, x3, x4, x5)  =  U31_GGAA(x1, x2, x3, x5)
PLUSJ_IN_GGA(x1, x2, x3)  =  PLUSJ_IN_GGA(x1, x2)
U8_GGA(x1, x2, x3, x4)  =  U8_GGA(x1, x2, x4)
U23_GGAA(x1, x2, x3, x4, x5)  =  U23_GGAA(x1, x2, x3, x5)
PLUSP_IN_GA(x1, x2)  =  PLUSP_IN_GA(x1)
U12_GA(x1, x2, x3)  =  U12_GA(x1, x3)
U10_GGGA(x1, x2, x3, x4)  =  U10_GGGA(x1, x2, x4)
PN_IN_GGAA(x1, x2, x3, x4)  =  PN_IN_GGAA(x1, x2)
U32_GGAA(x1, x2, x3, x4, x5)  =  U32_GGAA(x1, x2, x5)
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:

TIMESA_IN_GGA(s(0), T21, T23) → U1_GGA(T21, T23, pB_in_gaa(T21, X37, T23))
TIMESA_IN_GGA(s(0), T21, T23) → PB_IN_GAA(T21, X37, T23)
PB_IN_GAA(T21, T27, T23) → U16_GAA(T21, T27, T23, timesK_in_ga(T21, T27))
PB_IN_GAA(T21, T27, T23) → TIMESK_IN_GA(T21, T27)
U16_GAA(T21, T27, T23, timesK_out_ga(T21, T27)) → U17_GAA(T21, T27, T23, plusD_in_gga(T21, T27, T23))
U16_GAA(T21, T27, T23, timesK_out_ga(T21, T27)) → PLUSD_IN_GGA(T21, T27, T23)
PLUSD_IN_GGA(s(T47), T48, s(T50)) → U3_GGA(T47, T48, T50, plusD_in_gga(T47, T48, T50))
PLUSD_IN_GGA(s(T47), T48, s(T50)) → PLUSD_IN_GGA(T47, T48, T50)
TIMESA_IN_GGA(s(s(T55)), T10, T12) → U2_GGA(T55, T10, T12, pC_in_gaga(T55, X78, T10, T12))
TIMESA_IN_GGA(s(s(T55)), T10, T12) → PC_IN_GAGA(T55, X78, T10, T12)
PC_IN_GAGA(T55, T56, T10, T12) → U18_GAGA(T55, T56, T10, T12, evenE_in_ga(T55, T56))
PC_IN_GAGA(T55, T56, T10, T12) → EVENE_IN_GA(T55, T56)
EVENE_IN_GA(s(s(T59)), X87) → U4_GA(T59, X87, evenE_in_ga(T59, X87))
EVENE_IN_GA(s(s(T59)), X87) → EVENE_IN_GA(T59, X87)
U18_GAGA(T55, T56, T10, T12, evenE_out_ga(T55, T56)) → U19_GAGA(T55, T56, T10, T12, ifL_in_ggga(T56, T55, T10, T12))
U18_GAGA(T55, T56, T10, T12, evenE_out_ga(T55, T56)) → IFL_IN_GGGA(T56, T55, T10, T12)
IFL_IN_GGGA(true, T75, T76, T78) → U9_GGGA(T75, T76, T78, pM_in_gagaa(T75, X112, T76, X113, T78))
IFL_IN_GGGA(true, T75, T76, T78) → PM_IN_GAGAA(T75, X112, T76, X113, T78)
PM_IN_GAGAA(T75, T79, T76, X113, T78) → U20_GAGAA(T75, T79, T76, X113, T78, halfO_in_ga(T75, T79))
PM_IN_GAGAA(T75, T79, T76, X113, T78) → HALFO_IN_GA(T75, T79)
HALFO_IN_GA(T82, s(X122)) → U11_GA(T82, X122, halfF_in_ga(T82, X122))
HALFO_IN_GA(T82, s(X122)) → HALFF_IN_GA(T82, X122)
HALFF_IN_GA(s(s(T85)), s(X131)) → U5_GA(T85, X131, halfF_in_ga(T85, X131))
HALFF_IN_GA(s(s(T85)), s(X131)) → HALFF_IN_GA(T85, X131)
U20_GAGAA(T75, T79, T76, X113, T78, halfO_out_ga(T75, T79)) → U21_GAGAA(T75, T79, T76, X113, T78, pU_in_ggaa(T79, T76, X113, T78))
U20_GAGAA(T75, T79, T76, X113, T78, halfO_out_ga(T75, T79)) → PU_IN_GGAA(T79, T76, X113, T78)
PU_IN_GGAA(T79, T76, T88, T78) → U22_GGAA(T79, T76, T88, T78, timesG_in_gga(T79, T76, T88))
PU_IN_GGAA(T79, T76, T88, T78) → TIMESG_IN_GGA(T79, T76, T88)
TIMESG_IN_GGA(s(T100), T101, X154) → U6_GGA(T100, T101, X154, pH_in_gaga(T100, X153, T101, X154))
TIMESG_IN_GGA(s(T100), T101, X154) → PH_IN_GAGA(T100, X153, T101, X154)
PH_IN_GAGA(T100, T102, T101, X154) → U24_GAGA(T100, T102, T101, X154, evenE_in_ga(s(T100), T102))
PH_IN_GAGA(T100, T102, T101, X154) → EVENE_IN_GA(s(T100), T102)
U24_GAGA(T100, T102, T101, X154, evenE_out_ga(s(T100), T102)) → U25_GAGA(T100, T102, T101, X154, ifQ_in_ggga(T102, T100, T101, X154))
U24_GAGA(T100, T102, T101, X154, evenE_out_ga(s(T100), T102)) → IFQ_IN_GGGA(T102, T100, T101, X154)
IFQ_IN_GGGA(true, T113, T114, X186) → U13_GGGA(T113, T114, X186, pR_in_gagaa(T113, X184, T114, X185, X186))
IFQ_IN_GGGA(true, T113, T114, X186) → PR_IN_GAGAA(T113, X184, T114, X185, X186)
PR_IN_GAGAA(T113, T115, T114, X185, X186) → U26_GAGAA(T113, T115, T114, X185, X186, halfF_in_ga(s(T113), T115))
PR_IN_GAGAA(T113, T115, T114, X185, X186) → HALFF_IN_GA(s(T113), T115)
U26_GAGAA(T113, T115, T114, X185, X186, halfF_out_ga(s(T113), T115)) → U27_GAGAA(T113, T115, T114, X185, X186, pV_in_ggaa(T115, T114, X185, X186))
U26_GAGAA(T113, T115, T114, X185, X186, halfF_out_ga(s(T113), T115)) → PV_IN_GGAA(T115, T114, X185, X186)
PV_IN_GGAA(T115, T114, T118, X186) → U28_GGAA(T115, T114, T118, X186, timesG_in_gga(T115, T114, T118))
PV_IN_GGAA(T115, T114, T118, X186) → TIMESG_IN_GGA(T115, T114, T118)
U28_GGAA(T115, T114, T118, X186, timesG_out_gga(T115, T114, T118)) → U29_GGAA(T115, T114, T118, X186, plusT_in_ga(T118, X186))
U28_GGAA(T115, T114, T118, X186, timesG_out_gga(T115, T114, T118)) → PLUST_IN_GA(T118, X186)
PLUST_IN_GA(s(s(s(s(s(s(s(s(T144)))))))), s(s(s(s(s(s(s(s(X342))))))))) → U15_GA(T144, X342, plusI_in_gga(T144, s(s(s(s(s(s(s(T144))))))), X342))
PLUST_IN_GA(s(s(s(s(s(s(s(s(T144)))))))), s(s(s(s(s(s(s(s(X342))))))))) → PLUSI_IN_GGA(T144, s(s(s(s(s(s(s(T144))))))), X342)
PLUSI_IN_GGA(s(T157), T158, s(X363)) → U7_GGA(T157, T158, X363, plusI_in_gga(T157, T158, X363))
PLUSI_IN_GGA(s(T157), T158, s(X363)) → PLUSI_IN_GGA(T157, T158, X363)
IFQ_IN_GGGA(false, T165, T166, X379) → U14_GGGA(T165, T166, X379, pS_in_ggaa(T165, T166, X378, X379))
IFQ_IN_GGGA(false, T165, T166, X379) → PS_IN_GGAA(T165, T166, X378, X379)
PS_IN_GGAA(T165, T166, T169, X379) → U30_GGAA(T165, T166, T169, X379, timesG_in_gga(T165, T166, T169))
PS_IN_GGAA(T165, T166, T169, X379) → TIMESG_IN_GGA(T165, T166, T169)
U30_GGAA(T165, T166, T169, X379, timesG_out_gga(T165, T166, T169)) → U31_GGAA(T165, T166, T169, X379, plusJ_in_gga(T166, T169, X379))
U30_GGAA(T165, T166, T169, X379, timesG_out_gga(T165, T166, T169)) → PLUSJ_IN_GGA(T166, T169, X379)
PLUSJ_IN_GGA(s(T183), T184, s(X402)) → U8_GGA(T183, T184, X402, plusJ_in_gga(T183, T184, X402))
PLUSJ_IN_GGA(s(T183), T184, s(X402)) → PLUSJ_IN_GGA(T183, T184, X402)
U22_GGAA(T79, T76, T88, T78, timesG_out_gga(T79, T76, T88)) → U23_GGAA(T79, T76, T88, T78, plusP_in_ga(T88, T78))
U22_GGAA(T79, T76, T88, T78, timesG_out_gga(T79, T76, T88)) → PLUSP_IN_GA(T88, T78)
PLUSP_IN_GA(s(s(s(s(s(s(s(s(T240)))))))), s(s(s(s(s(s(s(s(T242))))))))) → U12_GA(T240, T242, plusD_in_gga(T240, s(s(s(s(s(s(s(s(T240)))))))), T242))
PLUSP_IN_GA(s(s(s(s(s(s(s(s(T240)))))))), s(s(s(s(s(s(s(s(T242))))))))) → PLUSD_IN_GGA(T240, s(s(s(s(s(s(s(s(T240)))))))), T242)
IFL_IN_GGGA(false, T252, T253, T255) → U10_GGGA(T252, T253, T255, pN_in_ggaa(T252, T253, X547, T255))
IFL_IN_GGGA(false, T252, T253, T255) → PN_IN_GGAA(T252, T253, X547, T255)
PN_IN_GGAA(T252, T253, T256, T255) → U32_GGAA(T252, T253, T256, T255, timesG_in_gga(s(T252), T253, T256))
PN_IN_GGAA(T252, T253, T256, T255) → TIMESG_IN_GGA(s(T252), T253, T256)
U32_GGAA(T252, T253, T256, T255, timesG_out_gga(s(T252), T253, T256)) → U33_GGAA(T252, T253, T256, T255, plusD_in_gga(T253, T256, T255))
U32_GGAA(T252, T253, T256, T255, timesG_out_gga(s(T252), T253, T256)) → PLUSD_IN_GGA(T253, T256, T255)

The TRS R consists of the following rules:

timesA_in_gga(0, T5, 0) → timesA_out_gga(0, T5, 0)
timesA_in_gga(s(0), T21, T23) → U1_gga(T21, T23, pB_in_gaa(T21, X37, T23))
pB_in_gaa(T21, T27, T23) → U16_gaa(T21, T27, T23, timesK_in_ga(T21, T27))
timesK_in_ga(T33, 0) → timesK_out_ga(T33, 0)
U16_gaa(T21, T27, T23, timesK_out_ga(T21, T27)) → U17_gaa(T21, T27, T23, plusD_in_gga(T21, T27, T23))
plusD_in_gga(0, T40, T40) → plusD_out_gga(0, T40, T40)
plusD_in_gga(s(T47), T48, s(T50)) → U3_gga(T47, T48, T50, plusD_in_gga(T47, T48, T50))
U3_gga(T47, T48, T50, plusD_out_gga(T47, T48, T50)) → plusD_out_gga(s(T47), T48, s(T50))
U17_gaa(T21, T27, T23, plusD_out_gga(T21, T27, T23)) → pB_out_gaa(T21, T27, T23)
U1_gga(T21, T23, pB_out_gaa(T21, X37, T23)) → timesA_out_gga(s(0), T21, T23)
timesA_in_gga(s(s(T55)), T10, T12) → U2_gga(T55, T10, T12, pC_in_gaga(T55, X78, T10, T12))
pC_in_gaga(T55, T56, T10, T12) → U18_gaga(T55, T56, T10, T12, evenE_in_ga(T55, T56))
evenE_in_ga(0, true) → evenE_out_ga(0, true)
evenE_in_ga(s(0), false) → evenE_out_ga(s(0), false)
evenE_in_ga(s(s(T59)), X87) → U4_ga(T59, X87, evenE_in_ga(T59, X87))
U4_ga(T59, X87, evenE_out_ga(T59, X87)) → evenE_out_ga(s(s(T59)), X87)
U18_gaga(T55, T56, T10, T12, evenE_out_ga(T55, T56)) → U19_gaga(T55, T56, T10, T12, ifL_in_ggga(T56, T55, T10, T12))
ifL_in_ggga(true, T75, T76, T78) → U9_ggga(T75, T76, T78, pM_in_gagaa(T75, X112, T76, X113, T78))
pM_in_gagaa(T75, T79, T76, X113, T78) → U20_gagaa(T75, T79, T76, X113, T78, halfO_in_ga(T75, T79))
halfO_in_ga(T82, s(X122)) → U11_ga(T82, X122, halfF_in_ga(T82, X122))
halfF_in_ga(0, 0) → halfF_out_ga(0, 0)
halfF_in_ga(s(s(T85)), s(X131)) → U5_ga(T85, X131, halfF_in_ga(T85, X131))
U5_ga(T85, X131, halfF_out_ga(T85, X131)) → halfF_out_ga(s(s(T85)), s(X131))
U11_ga(T82, X122, halfF_out_ga(T82, X122)) → halfO_out_ga(T82, s(X122))
U20_gagaa(T75, T79, T76, X113, T78, halfO_out_ga(T75, T79)) → U21_gagaa(T75, T79, T76, X113, T78, pU_in_ggaa(T79, T76, X113, T78))
pU_in_ggaa(T79, T76, T88, T78) → U22_ggaa(T79, T76, T88, T78, timesG_in_gga(T79, T76, T88))
timesG_in_gga(0, T95, 0) → timesG_out_gga(0, T95, 0)
timesG_in_gga(s(T100), T101, X154) → U6_gga(T100, T101, X154, pH_in_gaga(T100, X153, T101, X154))
pH_in_gaga(T100, T102, T101, X154) → U24_gaga(T100, T102, T101, X154, evenE_in_ga(s(T100), T102))
U24_gaga(T100, T102, T101, X154, evenE_out_ga(s(T100), T102)) → U25_gaga(T100, T102, T101, X154, ifQ_in_ggga(T102, T100, T101, X154))
ifQ_in_ggga(true, T113, T114, X186) → U13_ggga(T113, T114, X186, pR_in_gagaa(T113, X184, T114, X185, X186))
pR_in_gagaa(T113, T115, T114, X185, X186) → U26_gagaa(T113, T115, T114, X185, X186, halfF_in_ga(s(T113), T115))
U26_gagaa(T113, T115, T114, X185, X186, halfF_out_ga(s(T113), T115)) → U27_gagaa(T113, T115, T114, X185, X186, pV_in_ggaa(T115, T114, X185, X186))
pV_in_ggaa(T115, T114, T118, X186) → U28_ggaa(T115, T114, T118, X186, timesG_in_gga(T115, T114, T118))
U28_ggaa(T115, T114, T118, X186, timesG_out_gga(T115, T114, T118)) → U29_ggaa(T115, T114, T118, X186, plusT_in_ga(T118, X186))
plusT_in_ga(0, 0) → plusT_out_ga(0, 0)
plusT_in_ga(s(0), s(s(0))) → plusT_out_ga(s(0), s(s(0)))
plusT_in_ga(s(s(0)), s(s(s(s(0))))) → plusT_out_ga(s(s(0)), s(s(s(s(0)))))
plusT_in_ga(s(s(s(0))), s(s(s(s(s(s(0))))))) → plusT_out_ga(s(s(s(0))), s(s(s(s(s(s(0)))))))
plusT_in_ga(s(s(s(s(0)))), s(s(s(s(s(s(s(s(0))))))))) → plusT_out_ga(s(s(s(s(0)))), s(s(s(s(s(s(s(s(0)))))))))
plusT_in_ga(s(s(s(s(s(0))))), s(s(s(s(s(s(s(s(s(s(0))))))))))) → plusT_out_ga(s(s(s(s(s(0))))), s(s(s(s(s(s(s(s(s(s(0)))))))))))
plusT_in_ga(s(s(s(s(s(s(0)))))), s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))) → plusT_out_ga(s(s(s(s(s(s(0)))))), s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))
plusT_in_ga(s(s(s(s(s(s(s(0))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))) → plusT_out_ga(s(s(s(s(s(s(s(0))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))
plusT_in_ga(s(s(s(s(s(s(s(s(T144)))))))), s(s(s(s(s(s(s(s(X342))))))))) → U15_ga(T144, X342, plusI_in_gga(T144, s(s(s(s(s(s(s(T144))))))), X342))
plusI_in_gga(0, T152, s(T152)) → plusI_out_gga(0, T152, s(T152))
plusI_in_gga(s(T157), T158, s(X363)) → U7_gga(T157, T158, X363, plusI_in_gga(T157, T158, X363))
U7_gga(T157, T158, X363, plusI_out_gga(T157, T158, X363)) → plusI_out_gga(s(T157), T158, s(X363))
U15_ga(T144, X342, plusI_out_gga(T144, s(s(s(s(s(s(s(T144))))))), X342)) → plusT_out_ga(s(s(s(s(s(s(s(s(T144)))))))), s(s(s(s(s(s(s(s(X342)))))))))
U29_ggaa(T115, T114, T118, X186, plusT_out_ga(T118, X186)) → pV_out_ggaa(T115, T114, T118, X186)
U27_gagaa(T113, T115, T114, X185, X186, pV_out_ggaa(T115, T114, X185, X186)) → pR_out_gagaa(T113, T115, T114, X185, X186)
U13_ggga(T113, T114, X186, pR_out_gagaa(T113, X184, T114, X185, X186)) → ifQ_out_ggga(true, T113, T114, X186)
ifQ_in_ggga(false, T165, T166, X379) → U14_ggga(T165, T166, X379, pS_in_ggaa(T165, T166, X378, X379))
pS_in_ggaa(T165, T166, T169, X379) → U30_ggaa(T165, T166, T169, X379, timesG_in_gga(T165, T166, T169))
U30_ggaa(T165, T166, T169, X379, timesG_out_gga(T165, T166, T169)) → U31_ggaa(T165, T166, T169, X379, plusJ_in_gga(T166, T169, X379))
plusJ_in_gga(0, T178, T178) → plusJ_out_gga(0, T178, T178)
plusJ_in_gga(s(T183), T184, s(X402)) → U8_gga(T183, T184, X402, plusJ_in_gga(T183, T184, X402))
U8_gga(T183, T184, X402, plusJ_out_gga(T183, T184, X402)) → plusJ_out_gga(s(T183), T184, s(X402))
U31_ggaa(T165, T166, T169, X379, plusJ_out_gga(T166, T169, X379)) → pS_out_ggaa(T165, T166, T169, X379)
U14_ggga(T165, T166, X379, pS_out_ggaa(T165, T166, X378, X379)) → ifQ_out_ggga(false, T165, T166, X379)
U25_gaga(T100, T102, T101, X154, ifQ_out_ggga(T102, T100, T101, X154)) → pH_out_gaga(T100, T102, T101, X154)
U6_gga(T100, T101, X154, pH_out_gaga(T100, X153, T101, X154)) → timesG_out_gga(s(T100), T101, X154)
U22_ggaa(T79, T76, T88, T78, timesG_out_gga(T79, T76, T88)) → U23_ggaa(T79, T76, T88, T78, plusP_in_ga(T88, T78))
plusP_in_ga(0, 0) → plusP_out_ga(0, 0)
plusP_in_ga(s(0), s(s(0))) → plusP_out_ga(s(0), s(s(0)))
plusP_in_ga(s(s(0)), s(s(s(s(0))))) → plusP_out_ga(s(s(0)), s(s(s(s(0)))))
plusP_in_ga(s(s(s(0))), s(s(s(s(s(s(0))))))) → plusP_out_ga(s(s(s(0))), s(s(s(s(s(s(0)))))))
plusP_in_ga(s(s(s(s(0)))), s(s(s(s(s(s(s(s(0))))))))) → plusP_out_ga(s(s(s(s(0)))), s(s(s(s(s(s(s(s(0)))))))))
plusP_in_ga(s(s(s(s(s(0))))), s(s(s(s(s(s(s(s(s(s(0))))))))))) → plusP_out_ga(s(s(s(s(s(0))))), s(s(s(s(s(s(s(s(s(s(0)))))))))))
plusP_in_ga(s(s(s(s(s(s(0)))))), s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))) → plusP_out_ga(s(s(s(s(s(s(0)))))), s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))
plusP_in_ga(s(s(s(s(s(s(s(0))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))) → plusP_out_ga(s(s(s(s(s(s(s(0))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))
plusP_in_ga(s(s(s(s(s(s(s(s(T240)))))))), s(s(s(s(s(s(s(s(T242))))))))) → U12_ga(T240, T242, plusD_in_gga(T240, s(s(s(s(s(s(s(s(T240)))))))), T242))
U12_ga(T240, T242, plusD_out_gga(T240, s(s(s(s(s(s(s(s(T240)))))))), T242)) → plusP_out_ga(s(s(s(s(s(s(s(s(T240)))))))), s(s(s(s(s(s(s(s(T242)))))))))
U23_ggaa(T79, T76, T88, T78, plusP_out_ga(T88, T78)) → pU_out_ggaa(T79, T76, T88, T78)
U21_gagaa(T75, T79, T76, X113, T78, pU_out_ggaa(T79, T76, X113, T78)) → pM_out_gagaa(T75, T79, T76, X113, T78)
U9_ggga(T75, T76, T78, pM_out_gagaa(T75, X112, T76, X113, T78)) → ifL_out_ggga(true, T75, T76, T78)
ifL_in_ggga(false, T252, T253, T255) → U10_ggga(T252, T253, T255, pN_in_ggaa(T252, T253, X547, T255))
pN_in_ggaa(T252, T253, T256, T255) → U32_ggaa(T252, T253, T256, T255, timesG_in_gga(s(T252), T253, T256))
U32_ggaa(T252, T253, T256, T255, timesG_out_gga(s(T252), T253, T256)) → U33_ggaa(T252, T253, T256, T255, plusD_in_gga(T253, T256, T255))
U33_ggaa(T252, T253, T256, T255, plusD_out_gga(T253, T256, T255)) → pN_out_ggaa(T252, T253, T256, T255)
U10_ggga(T252, T253, T255, pN_out_ggaa(T252, T253, X547, T255)) → ifL_out_ggga(false, T252, T253, T255)
U19_gaga(T55, T56, T10, T12, ifL_out_ggga(T56, T55, T10, T12)) → pC_out_gaga(T55, T56, T10, T12)
U2_gga(T55, T10, T12, pC_out_gaga(T55, X78, T10, T12)) → timesA_out_gga(s(s(T55)), T10, T12)

The argument filtering Pi contains the following mapping:
timesA_in_gga(x1, x2, x3)  =  timesA_in_gga(x1, x2)
0  =  0
timesA_out_gga(x1, x2, x3)  =  timesA_out_gga(x1, x2, x3)
s(x1)  =  s(x1)
U1_gga(x1, x2, x3)  =  U1_gga(x1, x3)
pB_in_gaa(x1, x2, x3)  =  pB_in_gaa(x1)
U16_gaa(x1, x2, x3, x4)  =  U16_gaa(x1, x4)
timesK_in_ga(x1, x2)  =  timesK_in_ga(x1)
timesK_out_ga(x1, x2)  =  timesK_out_ga(x1, x2)
U17_gaa(x1, x2, x3, x4)  =  U17_gaa(x1, x2, x4)
plusD_in_gga(x1, x2, x3)  =  plusD_in_gga(x1, x2)
plusD_out_gga(x1, x2, x3)  =  plusD_out_gga(x1, x2, x3)
U3_gga(x1, x2, x3, x4)  =  U3_gga(x1, x2, x4)
pB_out_gaa(x1, x2, x3)  =  pB_out_gaa(x1, x2, x3)
U2_gga(x1, x2, x3, x4)  =  U2_gga(x1, x2, x4)
pC_in_gaga(x1, x2, x3, x4)  =  pC_in_gaga(x1, x3)
U18_gaga(x1, x2, x3, x4, x5)  =  U18_gaga(x1, x3, x5)
evenE_in_ga(x1, x2)  =  evenE_in_ga(x1)
evenE_out_ga(x1, x2)  =  evenE_out_ga(x1, x2)
U4_ga(x1, x2, x3)  =  U4_ga(x1, x3)
U19_gaga(x1, x2, x3, x4, x5)  =  U19_gaga(x1, x2, x3, x5)
ifL_in_ggga(x1, x2, x3, x4)  =  ifL_in_ggga(x1, x2, x3)
true  =  true
U9_ggga(x1, x2, x3, x4)  =  U9_ggga(x1, x2, x4)
pM_in_gagaa(x1, x2, x3, x4, x5)  =  pM_in_gagaa(x1, x3)
U20_gagaa(x1, x2, x3, x4, x5, x6)  =  U20_gagaa(x1, x3, x6)
halfO_in_ga(x1, x2)  =  halfO_in_ga(x1)
U11_ga(x1, x2, x3)  =  U11_ga(x1, x3)
halfF_in_ga(x1, x2)  =  halfF_in_ga(x1)
halfF_out_ga(x1, x2)  =  halfF_out_ga(x1, x2)
U5_ga(x1, x2, x3)  =  U5_ga(x1, x3)
halfO_out_ga(x1, x2)  =  halfO_out_ga(x1, x2)
U21_gagaa(x1, x2, x3, x4, x5, x6)  =  U21_gagaa(x1, x2, x3, x6)
pU_in_ggaa(x1, x2, x3, x4)  =  pU_in_ggaa(x1, x2)
U22_ggaa(x1, x2, x3, x4, x5)  =  U22_ggaa(x1, x2, x5)
timesG_in_gga(x1, x2, x3)  =  timesG_in_gga(x1, x2)
timesG_out_gga(x1, x2, x3)  =  timesG_out_gga(x1, x2, x3)
U6_gga(x1, x2, x3, x4)  =  U6_gga(x1, x2, x4)
pH_in_gaga(x1, x2, x3, x4)  =  pH_in_gaga(x1, x3)
U24_gaga(x1, x2, x3, x4, x5)  =  U24_gaga(x1, x3, x5)
U25_gaga(x1, x2, x3, x4, x5)  =  U25_gaga(x1, x2, x3, x5)
ifQ_in_ggga(x1, x2, x3, x4)  =  ifQ_in_ggga(x1, x2, x3)
U13_ggga(x1, x2, x3, x4)  =  U13_ggga(x1, x2, x4)
pR_in_gagaa(x1, x2, x3, x4, x5)  =  pR_in_gagaa(x1, x3)
U26_gagaa(x1, x2, x3, x4, x5, x6)  =  U26_gagaa(x1, x3, x6)
U27_gagaa(x1, x2, x3, x4, x5, x6)  =  U27_gagaa(x1, x2, x3, x6)
pV_in_ggaa(x1, x2, x3, x4)  =  pV_in_ggaa(x1, x2)
U28_ggaa(x1, x2, x3, x4, x5)  =  U28_ggaa(x1, x2, x5)
U29_ggaa(x1, x2, x3, x4, x5)  =  U29_ggaa(x1, x2, x3, x5)
plusT_in_ga(x1, x2)  =  plusT_in_ga(x1)
plusT_out_ga(x1, x2)  =  plusT_out_ga(x1, x2)
U15_ga(x1, x2, x3)  =  U15_ga(x1, x3)
plusI_in_gga(x1, x2, x3)  =  plusI_in_gga(x1, x2)
plusI_out_gga(x1, x2, x3)  =  plusI_out_gga(x1, x2, x3)
U7_gga(x1, x2, x3, x4)  =  U7_gga(x1, x2, x4)
pV_out_ggaa(x1, x2, x3, x4)  =  pV_out_ggaa(x1, x2, x3, x4)
pR_out_gagaa(x1, x2, x3, x4, x5)  =  pR_out_gagaa(x1, x2, x3, x4, x5)
ifQ_out_ggga(x1, x2, x3, x4)  =  ifQ_out_ggga(x1, x2, x3, x4)
false  =  false
U14_ggga(x1, x2, x3, x4)  =  U14_ggga(x1, x2, x4)
pS_in_ggaa(x1, x2, x3, x4)  =  pS_in_ggaa(x1, x2)
U30_ggaa(x1, x2, x3, x4, x5)  =  U30_ggaa(x1, x2, x5)
U31_ggaa(x1, x2, x3, x4, x5)  =  U31_ggaa(x1, x2, x3, x5)
plusJ_in_gga(x1, x2, x3)  =  plusJ_in_gga(x1, x2)
plusJ_out_gga(x1, x2, x3)  =  plusJ_out_gga(x1, x2, x3)
U8_gga(x1, x2, x3, x4)  =  U8_gga(x1, x2, x4)
pS_out_ggaa(x1, x2, x3, x4)  =  pS_out_ggaa(x1, x2, x3, x4)
pH_out_gaga(x1, x2, x3, x4)  =  pH_out_gaga(x1, x2, x3, x4)
U23_ggaa(x1, x2, x3, x4, x5)  =  U23_ggaa(x1, x2, x3, x5)
plusP_in_ga(x1, x2)  =  plusP_in_ga(x1)
plusP_out_ga(x1, x2)  =  plusP_out_ga(x1, x2)
U12_ga(x1, x2, x3)  =  U12_ga(x1, x3)
pU_out_ggaa(x1, x2, x3, x4)  =  pU_out_ggaa(x1, x2, x3, x4)
pM_out_gagaa(x1, x2, x3, x4, x5)  =  pM_out_gagaa(x1, x2, x3, x4, x5)
ifL_out_ggga(x1, x2, x3, x4)  =  ifL_out_ggga(x1, x2, x3, x4)
U10_ggga(x1, x2, x3, x4)  =  U10_ggga(x1, x2, x4)
pN_in_ggaa(x1, x2, x3, x4)  =  pN_in_ggaa(x1, x2)
U32_ggaa(x1, x2, x3, x4, x5)  =  U32_ggaa(x1, x2, x5)
U33_ggaa(x1, x2, x3, x4, x5)  =  U33_ggaa(x1, x2, x3, x5)
pN_out_ggaa(x1, x2, x3, x4)  =  pN_out_ggaa(x1, x2, x3, x4)
pC_out_gaga(x1, x2, x3, x4)  =  pC_out_gaga(x1, x2, x3, x4)
TIMESA_IN_GGA(x1, x2, x3)  =  TIMESA_IN_GGA(x1, x2)
U1_GGA(x1, x2, x3)  =  U1_GGA(x1, x3)
PB_IN_GAA(x1, x2, x3)  =  PB_IN_GAA(x1)
U16_GAA(x1, x2, x3, x4)  =  U16_GAA(x1, x4)
TIMESK_IN_GA(x1, x2)  =  TIMESK_IN_GA(x1)
U17_GAA(x1, x2, x3, x4)  =  U17_GAA(x1, x2, x4)
PLUSD_IN_GGA(x1, x2, x3)  =  PLUSD_IN_GGA(x1, x2)
U3_GGA(x1, x2, x3, x4)  =  U3_GGA(x1, x2, x4)
U2_GGA(x1, x2, x3, x4)  =  U2_GGA(x1, x2, x4)
PC_IN_GAGA(x1, x2, x3, x4)  =  PC_IN_GAGA(x1, x3)
U18_GAGA(x1, x2, x3, x4, x5)  =  U18_GAGA(x1, x3, x5)
EVENE_IN_GA(x1, x2)  =  EVENE_IN_GA(x1)
U4_GA(x1, x2, x3)  =  U4_GA(x1, x3)
U19_GAGA(x1, x2, x3, x4, x5)  =  U19_GAGA(x1, x2, x3, x5)
IFL_IN_GGGA(x1, x2, x3, x4)  =  IFL_IN_GGGA(x1, x2, x3)
U9_GGGA(x1, x2, x3, x4)  =  U9_GGGA(x1, x2, x4)
PM_IN_GAGAA(x1, x2, x3, x4, x5)  =  PM_IN_GAGAA(x1, x3)
U20_GAGAA(x1, x2, x3, x4, x5, x6)  =  U20_GAGAA(x1, x3, x6)
HALFO_IN_GA(x1, x2)  =  HALFO_IN_GA(x1)
U11_GA(x1, x2, x3)  =  U11_GA(x1, x3)
HALFF_IN_GA(x1, x2)  =  HALFF_IN_GA(x1)
U5_GA(x1, x2, x3)  =  U5_GA(x1, x3)
U21_GAGAA(x1, x2, x3, x4, x5, x6)  =  U21_GAGAA(x1, x2, x3, x6)
PU_IN_GGAA(x1, x2, x3, x4)  =  PU_IN_GGAA(x1, x2)
U22_GGAA(x1, x2, x3, x4, x5)  =  U22_GGAA(x1, x2, x5)
TIMESG_IN_GGA(x1, x2, x3)  =  TIMESG_IN_GGA(x1, x2)
U6_GGA(x1, x2, x3, x4)  =  U6_GGA(x1, x2, x4)
PH_IN_GAGA(x1, x2, x3, x4)  =  PH_IN_GAGA(x1, x3)
U24_GAGA(x1, x2, x3, x4, x5)  =  U24_GAGA(x1, x3, x5)
U25_GAGA(x1, x2, x3, x4, x5)  =  U25_GAGA(x1, x2, x3, x5)
IFQ_IN_GGGA(x1, x2, x3, x4)  =  IFQ_IN_GGGA(x1, x2, x3)
U13_GGGA(x1, x2, x3, x4)  =  U13_GGGA(x1, x2, x4)
PR_IN_GAGAA(x1, x2, x3, x4, x5)  =  PR_IN_GAGAA(x1, x3)
U26_GAGAA(x1, x2, x3, x4, x5, x6)  =  U26_GAGAA(x1, x3, x6)
U27_GAGAA(x1, x2, x3, x4, x5, x6)  =  U27_GAGAA(x1, x2, x3, x6)
PV_IN_GGAA(x1, x2, x3, x4)  =  PV_IN_GGAA(x1, x2)
U28_GGAA(x1, x2, x3, x4, x5)  =  U28_GGAA(x1, x2, x5)
U29_GGAA(x1, x2, x3, x4, x5)  =  U29_GGAA(x1, x2, x3, x5)
PLUST_IN_GA(x1, x2)  =  PLUST_IN_GA(x1)
U15_GA(x1, x2, x3)  =  U15_GA(x1, x3)
PLUSI_IN_GGA(x1, x2, x3)  =  PLUSI_IN_GGA(x1, x2)
U7_GGA(x1, x2, x3, x4)  =  U7_GGA(x1, x2, x4)
U14_GGGA(x1, x2, x3, x4)  =  U14_GGGA(x1, x2, x4)
PS_IN_GGAA(x1, x2, x3, x4)  =  PS_IN_GGAA(x1, x2)
U30_GGAA(x1, x2, x3, x4, x5)  =  U30_GGAA(x1, x2, x5)
U31_GGAA(x1, x2, x3, x4, x5)  =  U31_GGAA(x1, x2, x3, x5)
PLUSJ_IN_GGA(x1, x2, x3)  =  PLUSJ_IN_GGA(x1, x2)
U8_GGA(x1, x2, x3, x4)  =  U8_GGA(x1, x2, x4)
U23_GGAA(x1, x2, x3, x4, x5)  =  U23_GGAA(x1, x2, x3, x5)
PLUSP_IN_GA(x1, x2)  =  PLUSP_IN_GA(x1)
U12_GA(x1, x2, x3)  =  U12_GA(x1, x3)
U10_GGGA(x1, x2, x3, x4)  =  U10_GGGA(x1, x2, x4)
PN_IN_GGAA(x1, x2, x3, x4)  =  PN_IN_GGAA(x1, x2)
U32_GGAA(x1, x2, x3, x4, x5)  =  U32_GGAA(x1, x2, x5)
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 6 SCCs with 52 less nodes.

(6) Complex Obligation (AND)

(7) Obligation:

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

PLUSJ_IN_GGA(s(T183), T184, s(X402)) → PLUSJ_IN_GGA(T183, T184, X402)

The TRS R consists of the following rules:

timesA_in_gga(0, T5, 0) → timesA_out_gga(0, T5, 0)
timesA_in_gga(s(0), T21, T23) → U1_gga(T21, T23, pB_in_gaa(T21, X37, T23))
pB_in_gaa(T21, T27, T23) → U16_gaa(T21, T27, T23, timesK_in_ga(T21, T27))
timesK_in_ga(T33, 0) → timesK_out_ga(T33, 0)
U16_gaa(T21, T27, T23, timesK_out_ga(T21, T27)) → U17_gaa(T21, T27, T23, plusD_in_gga(T21, T27, T23))
plusD_in_gga(0, T40, T40) → plusD_out_gga(0, T40, T40)
plusD_in_gga(s(T47), T48, s(T50)) → U3_gga(T47, T48, T50, plusD_in_gga(T47, T48, T50))
U3_gga(T47, T48, T50, plusD_out_gga(T47, T48, T50)) → plusD_out_gga(s(T47), T48, s(T50))
U17_gaa(T21, T27, T23, plusD_out_gga(T21, T27, T23)) → pB_out_gaa(T21, T27, T23)
U1_gga(T21, T23, pB_out_gaa(T21, X37, T23)) → timesA_out_gga(s(0), T21, T23)
timesA_in_gga(s(s(T55)), T10, T12) → U2_gga(T55, T10, T12, pC_in_gaga(T55, X78, T10, T12))
pC_in_gaga(T55, T56, T10, T12) → U18_gaga(T55, T56, T10, T12, evenE_in_ga(T55, T56))
evenE_in_ga(0, true) → evenE_out_ga(0, true)
evenE_in_ga(s(0), false) → evenE_out_ga(s(0), false)
evenE_in_ga(s(s(T59)), X87) → U4_ga(T59, X87, evenE_in_ga(T59, X87))
U4_ga(T59, X87, evenE_out_ga(T59, X87)) → evenE_out_ga(s(s(T59)), X87)
U18_gaga(T55, T56, T10, T12, evenE_out_ga(T55, T56)) → U19_gaga(T55, T56, T10, T12, ifL_in_ggga(T56, T55, T10, T12))
ifL_in_ggga(true, T75, T76, T78) → U9_ggga(T75, T76, T78, pM_in_gagaa(T75, X112, T76, X113, T78))
pM_in_gagaa(T75, T79, T76, X113, T78) → U20_gagaa(T75, T79, T76, X113, T78, halfO_in_ga(T75, T79))
halfO_in_ga(T82, s(X122)) → U11_ga(T82, X122, halfF_in_ga(T82, X122))
halfF_in_ga(0, 0) → halfF_out_ga(0, 0)
halfF_in_ga(s(s(T85)), s(X131)) → U5_ga(T85, X131, halfF_in_ga(T85, X131))
U5_ga(T85, X131, halfF_out_ga(T85, X131)) → halfF_out_ga(s(s(T85)), s(X131))
U11_ga(T82, X122, halfF_out_ga(T82, X122)) → halfO_out_ga(T82, s(X122))
U20_gagaa(T75, T79, T76, X113, T78, halfO_out_ga(T75, T79)) → U21_gagaa(T75, T79, T76, X113, T78, pU_in_ggaa(T79, T76, X113, T78))
pU_in_ggaa(T79, T76, T88, T78) → U22_ggaa(T79, T76, T88, T78, timesG_in_gga(T79, T76, T88))
timesG_in_gga(0, T95, 0) → timesG_out_gga(0, T95, 0)
timesG_in_gga(s(T100), T101, X154) → U6_gga(T100, T101, X154, pH_in_gaga(T100, X153, T101, X154))
pH_in_gaga(T100, T102, T101, X154) → U24_gaga(T100, T102, T101, X154, evenE_in_ga(s(T100), T102))
U24_gaga(T100, T102, T101, X154, evenE_out_ga(s(T100), T102)) → U25_gaga(T100, T102, T101, X154, ifQ_in_ggga(T102, T100, T101, X154))
ifQ_in_ggga(true, T113, T114, X186) → U13_ggga(T113, T114, X186, pR_in_gagaa(T113, X184, T114, X185, X186))
pR_in_gagaa(T113, T115, T114, X185, X186) → U26_gagaa(T113, T115, T114, X185, X186, halfF_in_ga(s(T113), T115))
U26_gagaa(T113, T115, T114, X185, X186, halfF_out_ga(s(T113), T115)) → U27_gagaa(T113, T115, T114, X185, X186, pV_in_ggaa(T115, T114, X185, X186))
pV_in_ggaa(T115, T114, T118, X186) → U28_ggaa(T115, T114, T118, X186, timesG_in_gga(T115, T114, T118))
U28_ggaa(T115, T114, T118, X186, timesG_out_gga(T115, T114, T118)) → U29_ggaa(T115, T114, T118, X186, plusT_in_ga(T118, X186))
plusT_in_ga(0, 0) → plusT_out_ga(0, 0)
plusT_in_ga(s(0), s(s(0))) → plusT_out_ga(s(0), s(s(0)))
plusT_in_ga(s(s(0)), s(s(s(s(0))))) → plusT_out_ga(s(s(0)), s(s(s(s(0)))))
plusT_in_ga(s(s(s(0))), s(s(s(s(s(s(0))))))) → plusT_out_ga(s(s(s(0))), s(s(s(s(s(s(0)))))))
plusT_in_ga(s(s(s(s(0)))), s(s(s(s(s(s(s(s(0))))))))) → plusT_out_ga(s(s(s(s(0)))), s(s(s(s(s(s(s(s(0)))))))))
plusT_in_ga(s(s(s(s(s(0))))), s(s(s(s(s(s(s(s(s(s(0))))))))))) → plusT_out_ga(s(s(s(s(s(0))))), s(s(s(s(s(s(s(s(s(s(0)))))))))))
plusT_in_ga(s(s(s(s(s(s(0)))))), s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))) → plusT_out_ga(s(s(s(s(s(s(0)))))), s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))
plusT_in_ga(s(s(s(s(s(s(s(0))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))) → plusT_out_ga(s(s(s(s(s(s(s(0))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))
plusT_in_ga(s(s(s(s(s(s(s(s(T144)))))))), s(s(s(s(s(s(s(s(X342))))))))) → U15_ga(T144, X342, plusI_in_gga(T144, s(s(s(s(s(s(s(T144))))))), X342))
plusI_in_gga(0, T152, s(T152)) → plusI_out_gga(0, T152, s(T152))
plusI_in_gga(s(T157), T158, s(X363)) → U7_gga(T157, T158, X363, plusI_in_gga(T157, T158, X363))
U7_gga(T157, T158, X363, plusI_out_gga(T157, T158, X363)) → plusI_out_gga(s(T157), T158, s(X363))
U15_ga(T144, X342, plusI_out_gga(T144, s(s(s(s(s(s(s(T144))))))), X342)) → plusT_out_ga(s(s(s(s(s(s(s(s(T144)))))))), s(s(s(s(s(s(s(s(X342)))))))))
U29_ggaa(T115, T114, T118, X186, plusT_out_ga(T118, X186)) → pV_out_ggaa(T115, T114, T118, X186)
U27_gagaa(T113, T115, T114, X185, X186, pV_out_ggaa(T115, T114, X185, X186)) → pR_out_gagaa(T113, T115, T114, X185, X186)
U13_ggga(T113, T114, X186, pR_out_gagaa(T113, X184, T114, X185, X186)) → ifQ_out_ggga(true, T113, T114, X186)
ifQ_in_ggga(false, T165, T166, X379) → U14_ggga(T165, T166, X379, pS_in_ggaa(T165, T166, X378, X379))
pS_in_ggaa(T165, T166, T169, X379) → U30_ggaa(T165, T166, T169, X379, timesG_in_gga(T165, T166, T169))
U30_ggaa(T165, T166, T169, X379, timesG_out_gga(T165, T166, T169)) → U31_ggaa(T165, T166, T169, X379, plusJ_in_gga(T166, T169, X379))
plusJ_in_gga(0, T178, T178) → plusJ_out_gga(0, T178, T178)
plusJ_in_gga(s(T183), T184, s(X402)) → U8_gga(T183, T184, X402, plusJ_in_gga(T183, T184, X402))
U8_gga(T183, T184, X402, plusJ_out_gga(T183, T184, X402)) → plusJ_out_gga(s(T183), T184, s(X402))
U31_ggaa(T165, T166, T169, X379, plusJ_out_gga(T166, T169, X379)) → pS_out_ggaa(T165, T166, T169, X379)
U14_ggga(T165, T166, X379, pS_out_ggaa(T165, T166, X378, X379)) → ifQ_out_ggga(false, T165, T166, X379)
U25_gaga(T100, T102, T101, X154, ifQ_out_ggga(T102, T100, T101, X154)) → pH_out_gaga(T100, T102, T101, X154)
U6_gga(T100, T101, X154, pH_out_gaga(T100, X153, T101, X154)) → timesG_out_gga(s(T100), T101, X154)
U22_ggaa(T79, T76, T88, T78, timesG_out_gga(T79, T76, T88)) → U23_ggaa(T79, T76, T88, T78, plusP_in_ga(T88, T78))
plusP_in_ga(0, 0) → plusP_out_ga(0, 0)
plusP_in_ga(s(0), s(s(0))) → plusP_out_ga(s(0), s(s(0)))
plusP_in_ga(s(s(0)), s(s(s(s(0))))) → plusP_out_ga(s(s(0)), s(s(s(s(0)))))
plusP_in_ga(s(s(s(0))), s(s(s(s(s(s(0))))))) → plusP_out_ga(s(s(s(0))), s(s(s(s(s(s(0)))))))
plusP_in_ga(s(s(s(s(0)))), s(s(s(s(s(s(s(s(0))))))))) → plusP_out_ga(s(s(s(s(0)))), s(s(s(s(s(s(s(s(0)))))))))
plusP_in_ga(s(s(s(s(s(0))))), s(s(s(s(s(s(s(s(s(s(0))))))))))) → plusP_out_ga(s(s(s(s(s(0))))), s(s(s(s(s(s(s(s(s(s(0)))))))))))
plusP_in_ga(s(s(s(s(s(s(0)))))), s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))) → plusP_out_ga(s(s(s(s(s(s(0)))))), s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))
plusP_in_ga(s(s(s(s(s(s(s(0))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))) → plusP_out_ga(s(s(s(s(s(s(s(0))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))
plusP_in_ga(s(s(s(s(s(s(s(s(T240)))))))), s(s(s(s(s(s(s(s(T242))))))))) → U12_ga(T240, T242, plusD_in_gga(T240, s(s(s(s(s(s(s(s(T240)))))))), T242))
U12_ga(T240, T242, plusD_out_gga(T240, s(s(s(s(s(s(s(s(T240)))))))), T242)) → plusP_out_ga(s(s(s(s(s(s(s(s(T240)))))))), s(s(s(s(s(s(s(s(T242)))))))))
U23_ggaa(T79, T76, T88, T78, plusP_out_ga(T88, T78)) → pU_out_ggaa(T79, T76, T88, T78)
U21_gagaa(T75, T79, T76, X113, T78, pU_out_ggaa(T79, T76, X113, T78)) → pM_out_gagaa(T75, T79, T76, X113, T78)
U9_ggga(T75, T76, T78, pM_out_gagaa(T75, X112, T76, X113, T78)) → ifL_out_ggga(true, T75, T76, T78)
ifL_in_ggga(false, T252, T253, T255) → U10_ggga(T252, T253, T255, pN_in_ggaa(T252, T253, X547, T255))
pN_in_ggaa(T252, T253, T256, T255) → U32_ggaa(T252, T253, T256, T255, timesG_in_gga(s(T252), T253, T256))
U32_ggaa(T252, T253, T256, T255, timesG_out_gga(s(T252), T253, T256)) → U33_ggaa(T252, T253, T256, T255, plusD_in_gga(T253, T256, T255))
U33_ggaa(T252, T253, T256, T255, plusD_out_gga(T253, T256, T255)) → pN_out_ggaa(T252, T253, T256, T255)
U10_ggga(T252, T253, T255, pN_out_ggaa(T252, T253, X547, T255)) → ifL_out_ggga(false, T252, T253, T255)
U19_gaga(T55, T56, T10, T12, ifL_out_ggga(T56, T55, T10, T12)) → pC_out_gaga(T55, T56, T10, T12)
U2_gga(T55, T10, T12, pC_out_gaga(T55, X78, T10, T12)) → timesA_out_gga(s(s(T55)), T10, T12)

The argument filtering Pi contains the following mapping:
timesA_in_gga(x1, x2, x3)  =  timesA_in_gga(x1, x2)
0  =  0
timesA_out_gga(x1, x2, x3)  =  timesA_out_gga(x1, x2, x3)
s(x1)  =  s(x1)
U1_gga(x1, x2, x3)  =  U1_gga(x1, x3)
pB_in_gaa(x1, x2, x3)  =  pB_in_gaa(x1)
U16_gaa(x1, x2, x3, x4)  =  U16_gaa(x1, x4)
timesK_in_ga(x1, x2)  =  timesK_in_ga(x1)
timesK_out_ga(x1, x2)  =  timesK_out_ga(x1, x2)
U17_gaa(x1, x2, x3, x4)  =  U17_gaa(x1, x2, x4)
plusD_in_gga(x1, x2, x3)  =  plusD_in_gga(x1, x2)
plusD_out_gga(x1, x2, x3)  =  plusD_out_gga(x1, x2, x3)
U3_gga(x1, x2, x3, x4)  =  U3_gga(x1, x2, x4)
pB_out_gaa(x1, x2, x3)  =  pB_out_gaa(x1, x2, x3)
U2_gga(x1, x2, x3, x4)  =  U2_gga(x1, x2, x4)
pC_in_gaga(x1, x2, x3, x4)  =  pC_in_gaga(x1, x3)
U18_gaga(x1, x2, x3, x4, x5)  =  U18_gaga(x1, x3, x5)
evenE_in_ga(x1, x2)  =  evenE_in_ga(x1)
evenE_out_ga(x1, x2)  =  evenE_out_ga(x1, x2)
U4_ga(x1, x2, x3)  =  U4_ga(x1, x3)
U19_gaga(x1, x2, x3, x4, x5)  =  U19_gaga(x1, x2, x3, x5)
ifL_in_ggga(x1, x2, x3, x4)  =  ifL_in_ggga(x1, x2, x3)
true  =  true
U9_ggga(x1, x2, x3, x4)  =  U9_ggga(x1, x2, x4)
pM_in_gagaa(x1, x2, x3, x4, x5)  =  pM_in_gagaa(x1, x3)
U20_gagaa(x1, x2, x3, x4, x5, x6)  =  U20_gagaa(x1, x3, x6)
halfO_in_ga(x1, x2)  =  halfO_in_ga(x1)
U11_ga(x1, x2, x3)  =  U11_ga(x1, x3)
halfF_in_ga(x1, x2)  =  halfF_in_ga(x1)
halfF_out_ga(x1, x2)  =  halfF_out_ga(x1, x2)
U5_ga(x1, x2, x3)  =  U5_ga(x1, x3)
halfO_out_ga(x1, x2)  =  halfO_out_ga(x1, x2)
U21_gagaa(x1, x2, x3, x4, x5, x6)  =  U21_gagaa(x1, x2, x3, x6)
pU_in_ggaa(x1, x2, x3, x4)  =  pU_in_ggaa(x1, x2)
U22_ggaa(x1, x2, x3, x4, x5)  =  U22_ggaa(x1, x2, x5)
timesG_in_gga(x1, x2, x3)  =  timesG_in_gga(x1, x2)
timesG_out_gga(x1, x2, x3)  =  timesG_out_gga(x1, x2, x3)
U6_gga(x1, x2, x3, x4)  =  U6_gga(x1, x2, x4)
pH_in_gaga(x1, x2, x3, x4)  =  pH_in_gaga(x1, x3)
U24_gaga(x1, x2, x3, x4, x5)  =  U24_gaga(x1, x3, x5)
U25_gaga(x1, x2, x3, x4, x5)  =  U25_gaga(x1, x2, x3, x5)
ifQ_in_ggga(x1, x2, x3, x4)  =  ifQ_in_ggga(x1, x2, x3)
U13_ggga(x1, x2, x3, x4)  =  U13_ggga(x1, x2, x4)
pR_in_gagaa(x1, x2, x3, x4, x5)  =  pR_in_gagaa(x1, x3)
U26_gagaa(x1, x2, x3, x4, x5, x6)  =  U26_gagaa(x1, x3, x6)
U27_gagaa(x1, x2, x3, x4, x5, x6)  =  U27_gagaa(x1, x2, x3, x6)
pV_in_ggaa(x1, x2, x3, x4)  =  pV_in_ggaa(x1, x2)
U28_ggaa(x1, x2, x3, x4, x5)  =  U28_ggaa(x1, x2, x5)
U29_ggaa(x1, x2, x3, x4, x5)  =  U29_ggaa(x1, x2, x3, x5)
plusT_in_ga(x1, x2)  =  plusT_in_ga(x1)
plusT_out_ga(x1, x2)  =  plusT_out_ga(x1, x2)
U15_ga(x1, x2, x3)  =  U15_ga(x1, x3)
plusI_in_gga(x1, x2, x3)  =  plusI_in_gga(x1, x2)
plusI_out_gga(x1, x2, x3)  =  plusI_out_gga(x1, x2, x3)
U7_gga(x1, x2, x3, x4)  =  U7_gga(x1, x2, x4)
pV_out_ggaa(x1, x2, x3, x4)  =  pV_out_ggaa(x1, x2, x3, x4)
pR_out_gagaa(x1, x2, x3, x4, x5)  =  pR_out_gagaa(x1, x2, x3, x4, x5)
ifQ_out_ggga(x1, x2, x3, x4)  =  ifQ_out_ggga(x1, x2, x3, x4)
false  =  false
U14_ggga(x1, x2, x3, x4)  =  U14_ggga(x1, x2, x4)
pS_in_ggaa(x1, x2, x3, x4)  =  pS_in_ggaa(x1, x2)
U30_ggaa(x1, x2, x3, x4, x5)  =  U30_ggaa(x1, x2, x5)
U31_ggaa(x1, x2, x3, x4, x5)  =  U31_ggaa(x1, x2, x3, x5)
plusJ_in_gga(x1, x2, x3)  =  plusJ_in_gga(x1, x2)
plusJ_out_gga(x1, x2, x3)  =  plusJ_out_gga(x1, x2, x3)
U8_gga(x1, x2, x3, x4)  =  U8_gga(x1, x2, x4)
pS_out_ggaa(x1, x2, x3, x4)  =  pS_out_ggaa(x1, x2, x3, x4)
pH_out_gaga(x1, x2, x3, x4)  =  pH_out_gaga(x1, x2, x3, x4)
U23_ggaa(x1, x2, x3, x4, x5)  =  U23_ggaa(x1, x2, x3, x5)
plusP_in_ga(x1, x2)  =  plusP_in_ga(x1)
plusP_out_ga(x1, x2)  =  plusP_out_ga(x1, x2)
U12_ga(x1, x2, x3)  =  U12_ga(x1, x3)
pU_out_ggaa(x1, x2, x3, x4)  =  pU_out_ggaa(x1, x2, x3, x4)
pM_out_gagaa(x1, x2, x3, x4, x5)  =  pM_out_gagaa(x1, x2, x3, x4, x5)
ifL_out_ggga(x1, x2, x3, x4)  =  ifL_out_ggga(x1, x2, x3, x4)
U10_ggga(x1, x2, x3, x4)  =  U10_ggga(x1, x2, x4)
pN_in_ggaa(x1, x2, x3, x4)  =  pN_in_ggaa(x1, x2)
U32_ggaa(x1, x2, x3, x4, x5)  =  U32_ggaa(x1, x2, x5)
U33_ggaa(x1, x2, x3, x4, x5)  =  U33_ggaa(x1, x2, x3, x5)
pN_out_ggaa(x1, x2, x3, x4)  =  pN_out_ggaa(x1, x2, x3, x4)
pC_out_gaga(x1, x2, x3, x4)  =  pC_out_gaga(x1, x2, x3, x4)
PLUSJ_IN_GGA(x1, x2, x3)  =  PLUSJ_IN_GGA(x1, x2)

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:

PLUSJ_IN_GGA(s(T183), T184, s(X402)) → PLUSJ_IN_GGA(T183, T184, X402)

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

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:

PLUSJ_IN_GGA(s(T183), T184) → PLUSJ_IN_GGA(T183, T184)

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:

• PLUSJ_IN_GGA(s(T183), T184) → PLUSJ_IN_GGA(T183, T184)
  The graph contains the following edges 1 > 1, 2 >= 2

(13) YES

(14) Obligation:

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

PLUSI_IN_GGA(s(T157), T158, s(X363)) → PLUSI_IN_GGA(T157, T158, X363)

The TRS R consists of the following rules:

timesA_in_gga(0, T5, 0) → timesA_out_gga(0, T5, 0)
timesA_in_gga(s(0), T21, T23) → U1_gga(T21, T23, pB_in_gaa(T21, X37, T23))
pB_in_gaa(T21, T27, T23) → U16_gaa(T21, T27, T23, timesK_in_ga(T21, T27))
timesK_in_ga(T33, 0) → timesK_out_ga(T33, 0)
U16_gaa(T21, T27, T23, timesK_out_ga(T21, T27)) → U17_gaa(T21, T27, T23, plusD_in_gga(T21, T27, T23))
plusD_in_gga(0, T40, T40) → plusD_out_gga(0, T40, T40)
plusD_in_gga(s(T47), T48, s(T50)) → U3_gga(T47, T48, T50, plusD_in_gga(T47, T48, T50))
U3_gga(T47, T48, T50, plusD_out_gga(T47, T48, T50)) → plusD_out_gga(s(T47), T48, s(T50))
U17_gaa(T21, T27, T23, plusD_out_gga(T21, T27, T23)) → pB_out_gaa(T21, T27, T23)
U1_gga(T21, T23, pB_out_gaa(T21, X37, T23)) → timesA_out_gga(s(0), T21, T23)
timesA_in_gga(s(s(T55)), T10, T12) → U2_gga(T55, T10, T12, pC_in_gaga(T55, X78, T10, T12))
pC_in_gaga(T55, T56, T10, T12) → U18_gaga(T55, T56, T10, T12, evenE_in_ga(T55, T56))
evenE_in_ga(0, true) → evenE_out_ga(0, true)
evenE_in_ga(s(0), false) → evenE_out_ga(s(0), false)
evenE_in_ga(s(s(T59)), X87) → U4_ga(T59, X87, evenE_in_ga(T59, X87))
U4_ga(T59, X87, evenE_out_ga(T59, X87)) → evenE_out_ga(s(s(T59)), X87)
U18_gaga(T55, T56, T10, T12, evenE_out_ga(T55, T56)) → U19_gaga(T55, T56, T10, T12, ifL_in_ggga(T56, T55, T10, T12))
ifL_in_ggga(true, T75, T76, T78) → U9_ggga(T75, T76, T78, pM_in_gagaa(T75, X112, T76, X113, T78))
pM_in_gagaa(T75, T79, T76, X113, T78) → U20_gagaa(T75, T79, T76, X113, T78, halfO_in_ga(T75, T79))
halfO_in_ga(T82, s(X122)) → U11_ga(T82, X122, halfF_in_ga(T82, X122))
halfF_in_ga(0, 0) → halfF_out_ga(0, 0)
halfF_in_ga(s(s(T85)), s(X131)) → U5_ga(T85, X131, halfF_in_ga(T85, X131))
U5_ga(T85, X131, halfF_out_ga(T85, X131)) → halfF_out_ga(s(s(T85)), s(X131))
U11_ga(T82, X122, halfF_out_ga(T82, X122)) → halfO_out_ga(T82, s(X122))
U20_gagaa(T75, T79, T76, X113, T78, halfO_out_ga(T75, T79)) → U21_gagaa(T75, T79, T76, X113, T78, pU_in_ggaa(T79, T76, X113, T78))
pU_in_ggaa(T79, T76, T88, T78) → U22_ggaa(T79, T76, T88, T78, timesG_in_gga(T79, T76, T88))
timesG_in_gga(0, T95, 0) → timesG_out_gga(0, T95, 0)
timesG_in_gga(s(T100), T101, X154) → U6_gga(T100, T101, X154, pH_in_gaga(T100, X153, T101, X154))
pH_in_gaga(T100, T102, T101, X154) → U24_gaga(T100, T102, T101, X154, evenE_in_ga(s(T100), T102))
U24_gaga(T100, T102, T101, X154, evenE_out_ga(s(T100), T102)) → U25_gaga(T100, T102, T101, X154, ifQ_in_ggga(T102, T100, T101, X154))
ifQ_in_ggga(true, T113, T114, X186) → U13_ggga(T113, T114, X186, pR_in_gagaa(T113, X184, T114, X185, X186))
pR_in_gagaa(T113, T115, T114, X185, X186) → U26_gagaa(T113, T115, T114, X185, X186, halfF_in_ga(s(T113), T115))
U26_gagaa(T113, T115, T114, X185, X186, halfF_out_ga(s(T113), T115)) → U27_gagaa(T113, T115, T114, X185, X186, pV_in_ggaa(T115, T114, X185, X186))
pV_in_ggaa(T115, T114, T118, X186) → U28_ggaa(T115, T114, T118, X186, timesG_in_gga(T115, T114, T118))
U28_ggaa(T115, T114, T118, X186, timesG_out_gga(T115, T114, T118)) → U29_ggaa(T115, T114, T118, X186, plusT_in_ga(T118, X186))
plusT_in_ga(0, 0) → plusT_out_ga(0, 0)
plusT_in_ga(s(0), s(s(0))) → plusT_out_ga(s(0), s(s(0)))
plusT_in_ga(s(s(0)), s(s(s(s(0))))) → plusT_out_ga(s(s(0)), s(s(s(s(0)))))
plusT_in_ga(s(s(s(0))), s(s(s(s(s(s(0))))))) → plusT_out_ga(s(s(s(0))), s(s(s(s(s(s(0)))))))
plusT_in_ga(s(s(s(s(0)))), s(s(s(s(s(s(s(s(0))))))))) → plusT_out_ga(s(s(s(s(0)))), s(s(s(s(s(s(s(s(0)))))))))
plusT_in_ga(s(s(s(s(s(0))))), s(s(s(s(s(s(s(s(s(s(0))))))))))) → plusT_out_ga(s(s(s(s(s(0))))), s(s(s(s(s(s(s(s(s(s(0)))))))))))
plusT_in_ga(s(s(s(s(s(s(0)))))), s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))) → plusT_out_ga(s(s(s(s(s(s(0)))))), s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))
plusT_in_ga(s(s(s(s(s(s(s(0))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))) → plusT_out_ga(s(s(s(s(s(s(s(0))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))
plusT_in_ga(s(s(s(s(s(s(s(s(T144)))))))), s(s(s(s(s(s(s(s(X342))))))))) → U15_ga(T144, X342, plusI_in_gga(T144, s(s(s(s(s(s(s(T144))))))), X342))
plusI_in_gga(0, T152, s(T152)) → plusI_out_gga(0, T152, s(T152))
plusI_in_gga(s(T157), T158, s(X363)) → U7_gga(T157, T158, X363, plusI_in_gga(T157, T158, X363))
U7_gga(T157, T158, X363, plusI_out_gga(T157, T158, X363)) → plusI_out_gga(s(T157), T158, s(X363))
U15_ga(T144, X342, plusI_out_gga(T144, s(s(s(s(s(s(s(T144))))))), X342)) → plusT_out_ga(s(s(s(s(s(s(s(s(T144)))))))), s(s(s(s(s(s(s(s(X342)))))))))
U29_ggaa(T115, T114, T118, X186, plusT_out_ga(T118, X186)) → pV_out_ggaa(T115, T114, T118, X186)
U27_gagaa(T113, T115, T114, X185, X186, pV_out_ggaa(T115, T114, X185, X186)) → pR_out_gagaa(T113, T115, T114, X185, X186)
U13_ggga(T113, T114, X186, pR_out_gagaa(T113, X184, T114, X185, X186)) → ifQ_out_ggga(true, T113, T114, X186)
ifQ_in_ggga(false, T165, T166, X379) → U14_ggga(T165, T166, X379, pS_in_ggaa(T165, T166, X378, X379))
pS_in_ggaa(T165, T166, T169, X379) → U30_ggaa(T165, T166, T169, X379, timesG_in_gga(T165, T166, T169))
U30_ggaa(T165, T166, T169, X379, timesG_out_gga(T165, T166, T169)) → U31_ggaa(T165, T166, T169, X379, plusJ_in_gga(T166, T169, X379))
plusJ_in_gga(0, T178, T178) → plusJ_out_gga(0, T178, T178)
plusJ_in_gga(s(T183), T184, s(X402)) → U8_gga(T183, T184, X402, plusJ_in_gga(T183, T184, X402))
U8_gga(T183, T184, X402, plusJ_out_gga(T183, T184, X402)) → plusJ_out_gga(s(T183), T184, s(X402))
U31_ggaa(T165, T166, T169, X379, plusJ_out_gga(T166, T169, X379)) → pS_out_ggaa(T165, T166, T169, X379)
U14_ggga(T165, T166, X379, pS_out_ggaa(T165, T166, X378, X379)) → ifQ_out_ggga(false, T165, T166, X379)
U25_gaga(T100, T102, T101, X154, ifQ_out_ggga(T102, T100, T101, X154)) → pH_out_gaga(T100, T102, T101, X154)
U6_gga(T100, T101, X154, pH_out_gaga(T100, X153, T101, X154)) → timesG_out_gga(s(T100), T101, X154)
U22_ggaa(T79, T76, T88, T78, timesG_out_gga(T79, T76, T88)) → U23_ggaa(T79, T76, T88, T78, plusP_in_ga(T88, T78))
plusP_in_ga(0, 0) → plusP_out_ga(0, 0)
plusP_in_ga(s(0), s(s(0))) → plusP_out_ga(s(0), s(s(0)))
plusP_in_ga(s(s(0)), s(s(s(s(0))))) → plusP_out_ga(s(s(0)), s(s(s(s(0)))))
plusP_in_ga(s(s(s(0))), s(s(s(s(s(s(0))))))) → plusP_out_ga(s(s(s(0))), s(s(s(s(s(s(0)))))))
plusP_in_ga(s(s(s(s(0)))), s(s(s(s(s(s(s(s(0))))))))) → plusP_out_ga(s(s(s(s(0)))), s(s(s(s(s(s(s(s(0)))))))))
plusP_in_ga(s(s(s(s(s(0))))), s(s(s(s(s(s(s(s(s(s(0))))))))))) → plusP_out_ga(s(s(s(s(s(0))))), s(s(s(s(s(s(s(s(s(s(0)))))))))))
plusP_in_ga(s(s(s(s(s(s(0)))))), s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))) → plusP_out_ga(s(s(s(s(s(s(0)))))), s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))
plusP_in_ga(s(s(s(s(s(s(s(0))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))) → plusP_out_ga(s(s(s(s(s(s(s(0))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))
plusP_in_ga(s(s(s(s(s(s(s(s(T240)))))))), s(s(s(s(s(s(s(s(T242))))))))) → U12_ga(T240, T242, plusD_in_gga(T240, s(s(s(s(s(s(s(s(T240)))))))), T242))
U12_ga(T240, T242, plusD_out_gga(T240, s(s(s(s(s(s(s(s(T240)))))))), T242)) → plusP_out_ga(s(s(s(s(s(s(s(s(T240)))))))), s(s(s(s(s(s(s(s(T242)))))))))
U23_ggaa(T79, T76, T88, T78, plusP_out_ga(T88, T78)) → pU_out_ggaa(T79, T76, T88, T78)
U21_gagaa(T75, T79, T76, X113, T78, pU_out_ggaa(T79, T76, X113, T78)) → pM_out_gagaa(T75, T79, T76, X113, T78)
U9_ggga(T75, T76, T78, pM_out_gagaa(T75, X112, T76, X113, T78)) → ifL_out_ggga(true, T75, T76, T78)
ifL_in_ggga(false, T252, T253, T255) → U10_ggga(T252, T253, T255, pN_in_ggaa(T252, T253, X547, T255))
pN_in_ggaa(T252, T253, T256, T255) → U32_ggaa(T252, T253, T256, T255, timesG_in_gga(s(T252), T253, T256))
U32_ggaa(T252, T253, T256, T255, timesG_out_gga(s(T252), T253, T256)) → U33_ggaa(T252, T253, T256, T255, plusD_in_gga(T253, T256, T255))
U33_ggaa(T252, T253, T256, T255, plusD_out_gga(T253, T256, T255)) → pN_out_ggaa(T252, T253, T256, T255)
U10_ggga(T252, T253, T255, pN_out_ggaa(T252, T253, X547, T255)) → ifL_out_ggga(false, T252, T253, T255)
U19_gaga(T55, T56, T10, T12, ifL_out_ggga(T56, T55, T10, T12)) → pC_out_gaga(T55, T56, T10, T12)
U2_gga(T55, T10, T12, pC_out_gaga(T55, X78, T10, T12)) → timesA_out_gga(s(s(T55)), T10, T12)

The argument filtering Pi contains the following mapping:
timesA_in_gga(x1, x2, x3)  =  timesA_in_gga(x1, x2)
0  =  0
timesA_out_gga(x1, x2, x3)  =  timesA_out_gga(x1, x2, x3)
s(x1)  =  s(x1)
U1_gga(x1, x2, x3)  =  U1_gga(x1, x3)
pB_in_gaa(x1, x2, x3)  =  pB_in_gaa(x1)
U16_gaa(x1, x2, x3, x4)  =  U16_gaa(x1, x4)
timesK_in_ga(x1, x2)  =  timesK_in_ga(x1)
timesK_out_ga(x1, x2)  =  timesK_out_ga(x1, x2)
U17_gaa(x1, x2, x3, x4)  =  U17_gaa(x1, x2, x4)
plusD_in_gga(x1, x2, x3)  =  plusD_in_gga(x1, x2)
plusD_out_gga(x1, x2, x3)  =  plusD_out_gga(x1, x2, x3)
U3_gga(x1, x2, x3, x4)  =  U3_gga(x1, x2, x4)
pB_out_gaa(x1, x2, x3)  =  pB_out_gaa(x1, x2, x3)
U2_gga(x1, x2, x3, x4)  =  U2_gga(x1, x2, x4)
pC_in_gaga(x1, x2, x3, x4)  =  pC_in_gaga(x1, x3)
U18_gaga(x1, x2, x3, x4, x5)  =  U18_gaga(x1, x3, x5)
evenE_in_ga(x1, x2)  =  evenE_in_ga(x1)
evenE_out_ga(x1, x2)  =  evenE_out_ga(x1, x2)
U4_ga(x1, x2, x3)  =  U4_ga(x1, x3)
U19_gaga(x1, x2, x3, x4, x5)  =  U19_gaga(x1, x2, x3, x5)
ifL_in_ggga(x1, x2, x3, x4)  =  ifL_in_ggga(x1, x2, x3)
true  =  true
U9_ggga(x1, x2, x3, x4)  =  U9_ggga(x1, x2, x4)
pM_in_gagaa(x1, x2, x3, x4, x5)  =  pM_in_gagaa(x1, x3)
U20_gagaa(x1, x2, x3, x4, x5, x6)  =  U20_gagaa(x1, x3, x6)
halfO_in_ga(x1, x2)  =  halfO_in_ga(x1)
U11_ga(x1, x2, x3)  =  U11_ga(x1, x3)
halfF_in_ga(x1, x2)  =  halfF_in_ga(x1)
halfF_out_ga(x1, x2)  =  halfF_out_ga(x1, x2)
U5_ga(x1, x2, x3)  =  U5_ga(x1, x3)
halfO_out_ga(x1, x2)  =  halfO_out_ga(x1, x2)
U21_gagaa(x1, x2, x3, x4, x5, x6)  =  U21_gagaa(x1, x2, x3, x6)
pU_in_ggaa(x1, x2, x3, x4)  =  pU_in_ggaa(x1, x2)
U22_ggaa(x1, x2, x3, x4, x5)  =  U22_ggaa(x1, x2, x5)
timesG_in_gga(x1, x2, x3)  =  timesG_in_gga(x1, x2)
timesG_out_gga(x1, x2, x3)  =  timesG_out_gga(x1, x2, x3)
U6_gga(x1, x2, x3, x4)  =  U6_gga(x1, x2, x4)
pH_in_gaga(x1, x2, x3, x4)  =  pH_in_gaga(x1, x3)
U24_gaga(x1, x2, x3, x4, x5)  =  U24_gaga(x1, x3, x5)
U25_gaga(x1, x2, x3, x4, x5)  =  U25_gaga(x1, x2, x3, x5)
ifQ_in_ggga(x1, x2, x3, x4)  =  ifQ_in_ggga(x1, x2, x3)
U13_ggga(x1, x2, x3, x4)  =  U13_ggga(x1, x2, x4)
pR_in_gagaa(x1, x2, x3, x4, x5)  =  pR_in_gagaa(x1, x3)
U26_gagaa(x1, x2, x3, x4, x5, x6)  =  U26_gagaa(x1, x3, x6)
U27_gagaa(x1, x2, x3, x4, x5, x6)  =  U27_gagaa(x1, x2, x3, x6)
pV_in_ggaa(x1, x2, x3, x4)  =  pV_in_ggaa(x1, x2)
U28_ggaa(x1, x2, x3, x4, x5)  =  U28_ggaa(x1, x2, x5)
U29_ggaa(x1, x2, x3, x4, x5)  =  U29_ggaa(x1, x2, x3, x5)
plusT_in_ga(x1, x2)  =  plusT_in_ga(x1)
plusT_out_ga(x1, x2)  =  plusT_out_ga(x1, x2)
U15_ga(x1, x2, x3)  =  U15_ga(x1, x3)
plusI_in_gga(x1, x2, x3)  =  plusI_in_gga(x1, x2)
plusI_out_gga(x1, x2, x3)  =  plusI_out_gga(x1, x2, x3)
U7_gga(x1, x2, x3, x4)  =  U7_gga(x1, x2, x4)
pV_out_ggaa(x1, x2, x3, x4)  =  pV_out_ggaa(x1, x2, x3, x4)
pR_out_gagaa(x1, x2, x3, x4, x5)  =  pR_out_gagaa(x1, x2, x3, x4, x5)
ifQ_out_ggga(x1, x2, x3, x4)  =  ifQ_out_ggga(x1, x2, x3, x4)
false  =  false
U14_ggga(x1, x2, x3, x4)  =  U14_ggga(x1, x2, x4)
pS_in_ggaa(x1, x2, x3, x4)  =  pS_in_ggaa(x1, x2)
U30_ggaa(x1, x2, x3, x4, x5)  =  U30_ggaa(x1, x2, x5)
U31_ggaa(x1, x2, x3, x4, x5)  =  U31_ggaa(x1, x2, x3, x5)
plusJ_in_gga(x1, x2, x3)  =  plusJ_in_gga(x1, x2)
plusJ_out_gga(x1, x2, x3)  =  plusJ_out_gga(x1, x2, x3)
U8_gga(x1, x2, x3, x4)  =  U8_gga(x1, x2, x4)
pS_out_ggaa(x1, x2, x3, x4)  =  pS_out_ggaa(x1, x2, x3, x4)
pH_out_gaga(x1, x2, x3, x4)  =  pH_out_gaga(x1, x2, x3, x4)
U23_ggaa(x1, x2, x3, x4, x5)  =  U23_ggaa(x1, x2, x3, x5)
plusP_in_ga(x1, x2)  =  plusP_in_ga(x1)
plusP_out_ga(x1, x2)  =  plusP_out_ga(x1, x2)
U12_ga(x1, x2, x3)  =  U12_ga(x1, x3)
pU_out_ggaa(x1, x2, x3, x4)  =  pU_out_ggaa(x1, x2, x3, x4)
pM_out_gagaa(x1, x2, x3, x4, x5)  =  pM_out_gagaa(x1, x2, x3, x4, x5)
ifL_out_ggga(x1, x2, x3, x4)  =  ifL_out_ggga(x1, x2, x3, x4)
U10_ggga(x1, x2, x3, x4)  =  U10_ggga(x1, x2, x4)
pN_in_ggaa(x1, x2, x3, x4)  =  pN_in_ggaa(x1, x2)
U32_ggaa(x1, x2, x3, x4, x5)  =  U32_ggaa(x1, x2, x5)
U33_ggaa(x1, x2, x3, x4, x5)  =  U33_ggaa(x1, x2, x3, x5)
pN_out_ggaa(x1, x2, x3, x4)  =  pN_out_ggaa(x1, x2, x3, x4)
pC_out_gaga(x1, x2, x3, x4)  =  pC_out_gaga(x1, x2, x3, x4)
PLUSI_IN_GGA(x1, x2, x3)  =  PLUSI_IN_GGA(x1, x2)

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:

PLUSI_IN_GGA(s(T157), T158, s(X363)) → PLUSI_IN_GGA(T157, T158, X363)

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

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:

PLUSI_IN_GGA(s(T157), T158) → PLUSI_IN_GGA(T157, T158)

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:

• PLUSI_IN_GGA(s(T157), T158) → PLUSI_IN_GGA(T157, T158)
  The graph contains the following edges 1 > 1, 2 >= 2

(20) YES

(21) Obligation:

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

HALFF_IN_GA(s(s(T85)), s(X131)) → HALFF_IN_GA(T85, X131)

The TRS R consists of the following rules:

timesA_in_gga(0, T5, 0) → timesA_out_gga(0, T5, 0)
timesA_in_gga(s(0), T21, T23) → U1_gga(T21, T23, pB_in_gaa(T21, X37, T23))
pB_in_gaa(T21, T27, T23) → U16_gaa(T21, T27, T23, timesK_in_ga(T21, T27))
timesK_in_ga(T33, 0) → timesK_out_ga(T33, 0)
U16_gaa(T21, T27, T23, timesK_out_ga(T21, T27)) → U17_gaa(T21, T27, T23, plusD_in_gga(T21, T27, T23))
plusD_in_gga(0, T40, T40) → plusD_out_gga(0, T40, T40)
plusD_in_gga(s(T47), T48, s(T50)) → U3_gga(T47, T48, T50, plusD_in_gga(T47, T48, T50))
U3_gga(T47, T48, T50, plusD_out_gga(T47, T48, T50)) → plusD_out_gga(s(T47), T48, s(T50))
U17_gaa(T21, T27, T23, plusD_out_gga(T21, T27, T23)) → pB_out_gaa(T21, T27, T23)
U1_gga(T21, T23, pB_out_gaa(T21, X37, T23)) → timesA_out_gga(s(0), T21, T23)
timesA_in_gga(s(s(T55)), T10, T12) → U2_gga(T55, T10, T12, pC_in_gaga(T55, X78, T10, T12))
pC_in_gaga(T55, T56, T10, T12) → U18_gaga(T55, T56, T10, T12, evenE_in_ga(T55, T56))
evenE_in_ga(0, true) → evenE_out_ga(0, true)
evenE_in_ga(s(0), false) → evenE_out_ga(s(0), false)
evenE_in_ga(s(s(T59)), X87) → U4_ga(T59, X87, evenE_in_ga(T59, X87))
U4_ga(T59, X87, evenE_out_ga(T59, X87)) → evenE_out_ga(s(s(T59)), X87)
U18_gaga(T55, T56, T10, T12, evenE_out_ga(T55, T56)) → U19_gaga(T55, T56, T10, T12, ifL_in_ggga(T56, T55, T10, T12))
ifL_in_ggga(true, T75, T76, T78) → U9_ggga(T75, T76, T78, pM_in_gagaa(T75, X112, T76, X113, T78))
pM_in_gagaa(T75, T79, T76, X113, T78) → U20_gagaa(T75, T79, T76, X113, T78, halfO_in_ga(T75, T79))
halfO_in_ga(T82, s(X122)) → U11_ga(T82, X122, halfF_in_ga(T82, X122))
halfF_in_ga(0, 0) → halfF_out_ga(0, 0)
halfF_in_ga(s(s(T85)), s(X131)) → U5_ga(T85, X131, halfF_in_ga(T85, X131))
U5_ga(T85, X131, halfF_out_ga(T85, X131)) → halfF_out_ga(s(s(T85)), s(X131))
U11_ga(T82, X122, halfF_out_ga(T82, X122)) → halfO_out_ga(T82, s(X122))
U20_gagaa(T75, T79, T76, X113, T78, halfO_out_ga(T75, T79)) → U21_gagaa(T75, T79, T76, X113, T78, pU_in_ggaa(T79, T76, X113, T78))
pU_in_ggaa(T79, T76, T88, T78) → U22_ggaa(T79, T76, T88, T78, timesG_in_gga(T79, T76, T88))
timesG_in_gga(0, T95, 0) → timesG_out_gga(0, T95, 0)
timesG_in_gga(s(T100), T101, X154) → U6_gga(T100, T101, X154, pH_in_gaga(T100, X153, T101, X154))
pH_in_gaga(T100, T102, T101, X154) → U24_gaga(T100, T102, T101, X154, evenE_in_ga(s(T100), T102))
U24_gaga(T100, T102, T101, X154, evenE_out_ga(s(T100), T102)) → U25_gaga(T100, T102, T101, X154, ifQ_in_ggga(T102, T100, T101, X154))
ifQ_in_ggga(true, T113, T114, X186) → U13_ggga(T113, T114, X186, pR_in_gagaa(T113, X184, T114, X185, X186))
pR_in_gagaa(T113, T115, T114, X185, X186) → U26_gagaa(T113, T115, T114, X185, X186, halfF_in_ga(s(T113), T115))
U26_gagaa(T113, T115, T114, X185, X186, halfF_out_ga(s(T113), T115)) → U27_gagaa(T113, T115, T114, X185, X186, pV_in_ggaa(T115, T114, X185, X186))
pV_in_ggaa(T115, T114, T118, X186) → U28_ggaa(T115, T114, T118, X186, timesG_in_gga(T115, T114, T118))
U28_ggaa(T115, T114, T118, X186, timesG_out_gga(T115, T114, T118)) → U29_ggaa(T115, T114, T118, X186, plusT_in_ga(T118, X186))
plusT_in_ga(0, 0) → plusT_out_ga(0, 0)
plusT_in_ga(s(0), s(s(0))) → plusT_out_ga(s(0), s(s(0)))
plusT_in_ga(s(s(0)), s(s(s(s(0))))) → plusT_out_ga(s(s(0)), s(s(s(s(0)))))
plusT_in_ga(s(s(s(0))), s(s(s(s(s(s(0))))))) → plusT_out_ga(s(s(s(0))), s(s(s(s(s(s(0)))))))
plusT_in_ga(s(s(s(s(0)))), s(s(s(s(s(s(s(s(0))))))))) → plusT_out_ga(s(s(s(s(0)))), s(s(s(s(s(s(s(s(0)))))))))
plusT_in_ga(s(s(s(s(s(0))))), s(s(s(s(s(s(s(s(s(s(0))))))))))) → plusT_out_ga(s(s(s(s(s(0))))), s(s(s(s(s(s(s(s(s(s(0)))))))))))
plusT_in_ga(s(s(s(s(s(s(0)))))), s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))) → plusT_out_ga(s(s(s(s(s(s(0)))))), s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))
plusT_in_ga(s(s(s(s(s(s(s(0))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))) → plusT_out_ga(s(s(s(s(s(s(s(0))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))
plusT_in_ga(s(s(s(s(s(s(s(s(T144)))))))), s(s(s(s(s(s(s(s(X342))))))))) → U15_ga(T144, X342, plusI_in_gga(T144, s(s(s(s(s(s(s(T144))))))), X342))
plusI_in_gga(0, T152, s(T152)) → plusI_out_gga(0, T152, s(T152))
plusI_in_gga(s(T157), T158, s(X363)) → U7_gga(T157, T158, X363, plusI_in_gga(T157, T158, X363))
U7_gga(T157, T158, X363, plusI_out_gga(T157, T158, X363)) → plusI_out_gga(s(T157), T158, s(X363))
U15_ga(T144, X342, plusI_out_gga(T144, s(s(s(s(s(s(s(T144))))))), X342)) → plusT_out_ga(s(s(s(s(s(s(s(s(T144)))))))), s(s(s(s(s(s(s(s(X342)))))))))
U29_ggaa(T115, T114, T118, X186, plusT_out_ga(T118, X186)) → pV_out_ggaa(T115, T114, T118, X186)
U27_gagaa(T113, T115, T114, X185, X186, pV_out_ggaa(T115, T114, X185, X186)) → pR_out_gagaa(T113, T115, T114, X185, X186)
U13_ggga(T113, T114, X186, pR_out_gagaa(T113, X184, T114, X185, X186)) → ifQ_out_ggga(true, T113, T114, X186)
ifQ_in_ggga(false, T165, T166, X379) → U14_ggga(T165, T166, X379, pS_in_ggaa(T165, T166, X378, X379))
pS_in_ggaa(T165, T166, T169, X379) → U30_ggaa(T165, T166, T169, X379, timesG_in_gga(T165, T166, T169))
U30_ggaa(T165, T166, T169, X379, timesG_out_gga(T165, T166, T169)) → U31_ggaa(T165, T166, T169, X379, plusJ_in_gga(T166, T169, X379))
plusJ_in_gga(0, T178, T178) → plusJ_out_gga(0, T178, T178)
plusJ_in_gga(s(T183), T184, s(X402)) → U8_gga(T183, T184, X402, plusJ_in_gga(T183, T184, X402))
U8_gga(T183, T184, X402, plusJ_out_gga(T183, T184, X402)) → plusJ_out_gga(s(T183), T184, s(X402))
U31_ggaa(T165, T166, T169, X379, plusJ_out_gga(T166, T169, X379)) → pS_out_ggaa(T165, T166, T169, X379)
U14_ggga(T165, T166, X379, pS_out_ggaa(T165, T166, X378, X379)) → ifQ_out_ggga(false, T165, T166, X379)
U25_gaga(T100, T102, T101, X154, ifQ_out_ggga(T102, T100, T101, X154)) → pH_out_gaga(T100, T102, T101, X154)
U6_gga(T100, T101, X154, pH_out_gaga(T100, X153, T101, X154)) → timesG_out_gga(s(T100), T101, X154)
U22_ggaa(T79, T76, T88, T78, timesG_out_gga(T79, T76, T88)) → U23_ggaa(T79, T76, T88, T78, plusP_in_ga(T88, T78))
plusP_in_ga(0, 0) → plusP_out_ga(0, 0)
plusP_in_ga(s(0), s(s(0))) → plusP_out_ga(s(0), s(s(0)))
plusP_in_ga(s(s(0)), s(s(s(s(0))))) → plusP_out_ga(s(s(0)), s(s(s(s(0)))))
plusP_in_ga(s(s(s(0))), s(s(s(s(s(s(0))))))) → plusP_out_ga(s(s(s(0))), s(s(s(s(s(s(0)))))))
plusP_in_ga(s(s(s(s(0)))), s(s(s(s(s(s(s(s(0))))))))) → plusP_out_ga(s(s(s(s(0)))), s(s(s(s(s(s(s(s(0)))))))))
plusP_in_ga(s(s(s(s(s(0))))), s(s(s(s(s(s(s(s(s(s(0))))))))))) → plusP_out_ga(s(s(s(s(s(0))))), s(s(s(s(s(s(s(s(s(s(0)))))))))))
plusP_in_ga(s(s(s(s(s(s(0)))))), s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))) → plusP_out_ga(s(s(s(s(s(s(0)))))), s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))
plusP_in_ga(s(s(s(s(s(s(s(0))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))) → plusP_out_ga(s(s(s(s(s(s(s(0))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))
plusP_in_ga(s(s(s(s(s(s(s(s(T240)))))))), s(s(s(s(s(s(s(s(T242))))))))) → U12_ga(T240, T242, plusD_in_gga(T240, s(s(s(s(s(s(s(s(T240)))))))), T242))
U12_ga(T240, T242, plusD_out_gga(T240, s(s(s(s(s(s(s(s(T240)))))))), T242)) → plusP_out_ga(s(s(s(s(s(s(s(s(T240)))))))), s(s(s(s(s(s(s(s(T242)))))))))
U23_ggaa(T79, T76, T88, T78, plusP_out_ga(T88, T78)) → pU_out_ggaa(T79, T76, T88, T78)
U21_gagaa(T75, T79, T76, X113, T78, pU_out_ggaa(T79, T76, X113, T78)) → pM_out_gagaa(T75, T79, T76, X113, T78)
U9_ggga(T75, T76, T78, pM_out_gagaa(T75, X112, T76, X113, T78)) → ifL_out_ggga(true, T75, T76, T78)
ifL_in_ggga(false, T252, T253, T255) → U10_ggga(T252, T253, T255, pN_in_ggaa(T252, T253, X547, T255))
pN_in_ggaa(T252, T253, T256, T255) → U32_ggaa(T252, T253, T256, T255, timesG_in_gga(s(T252), T253, T256))
U32_ggaa(T252, T253, T256, T255, timesG_out_gga(s(T252), T253, T256)) → U33_ggaa(T252, T253, T256, T255, plusD_in_gga(T253, T256, T255))
U33_ggaa(T252, T253, T256, T255, plusD_out_gga(T253, T256, T255)) → pN_out_ggaa(T252, T253, T256, T255)
U10_ggga(T252, T253, T255, pN_out_ggaa(T252, T253, X547, T255)) → ifL_out_ggga(false, T252, T253, T255)
U19_gaga(T55, T56, T10, T12, ifL_out_ggga(T56, T55, T10, T12)) → pC_out_gaga(T55, T56, T10, T12)
U2_gga(T55, T10, T12, pC_out_gaga(T55, X78, T10, T12)) → timesA_out_gga(s(s(T55)), T10, T12)

The argument filtering Pi contains the following mapping:
timesA_in_gga(x1, x2, x3)  =  timesA_in_gga(x1, x2)
0  =  0
timesA_out_gga(x1, x2, x3)  =  timesA_out_gga(x1, x2, x3)
s(x1)  =  s(x1)
U1_gga(x1, x2, x3)  =  U1_gga(x1, x3)
pB_in_gaa(x1, x2, x3)  =  pB_in_gaa(x1)
U16_gaa(x1, x2, x3, x4)  =  U16_gaa(x1, x4)
timesK_in_ga(x1, x2)  =  timesK_in_ga(x1)
timesK_out_ga(x1, x2)  =  timesK_out_ga(x1, x2)
U17_gaa(x1, x2, x3, x4)  =  U17_gaa(x1, x2, x4)
plusD_in_gga(x1, x2, x3)  =  plusD_in_gga(x1, x2)
plusD_out_gga(x1, x2, x3)  =  plusD_out_gga(x1, x2, x3)
U3_gga(x1, x2, x3, x4)  =  U3_gga(x1, x2, x4)
pB_out_gaa(x1, x2, x3)  =  pB_out_gaa(x1, x2, x3)
U2_gga(x1, x2, x3, x4)  =  U2_gga(x1, x2, x4)
pC_in_gaga(x1, x2, x3, x4)  =  pC_in_gaga(x1, x3)
U18_gaga(x1, x2, x3, x4, x5)  =  U18_gaga(x1, x3, x5)
evenE_in_ga(x1, x2)  =  evenE_in_ga(x1)
evenE_out_ga(x1, x2)  =  evenE_out_ga(x1, x2)
U4_ga(x1, x2, x3)  =  U4_ga(x1, x3)
U19_gaga(x1, x2, x3, x4, x5)  =  U19_gaga(x1, x2, x3, x5)
ifL_in_ggga(x1, x2, x3, x4)  =  ifL_in_ggga(x1, x2, x3)
true  =  true
U9_ggga(x1, x2, x3, x4)  =  U9_ggga(x1, x2, x4)
pM_in_gagaa(x1, x2, x3, x4, x5)  =  pM_in_gagaa(x1, x3)
U20_gagaa(x1, x2, x3, x4, x5, x6)  =  U20_gagaa(x1, x3, x6)
halfO_in_ga(x1, x2)  =  halfO_in_ga(x1)
U11_ga(x1, x2, x3)  =  U11_ga(x1, x3)
halfF_in_ga(x1, x2)  =  halfF_in_ga(x1)
halfF_out_ga(x1, x2)  =  halfF_out_ga(x1, x2)
U5_ga(x1, x2, x3)  =  U5_ga(x1, x3)
halfO_out_ga(x1, x2)  =  halfO_out_ga(x1, x2)
U21_gagaa(x1, x2, x3, x4, x5, x6)  =  U21_gagaa(x1, x2, x3, x6)
pU_in_ggaa(x1, x2, x3, x4)  =  pU_in_ggaa(x1, x2)
U22_ggaa(x1, x2, x3, x4, x5)  =  U22_ggaa(x1, x2, x5)
timesG_in_gga(x1, x2, x3)  =  timesG_in_gga(x1, x2)
timesG_out_gga(x1, x2, x3)  =  timesG_out_gga(x1, x2, x3)
U6_gga(x1, x2, x3, x4)  =  U6_gga(x1, x2, x4)
pH_in_gaga(x1, x2, x3, x4)  =  pH_in_gaga(x1, x3)
U24_gaga(x1, x2, x3, x4, x5)  =  U24_gaga(x1, x3, x5)
U25_gaga(x1, x2, x3, x4, x5)  =  U25_gaga(x1, x2, x3, x5)
ifQ_in_ggga(x1, x2, x3, x4)  =  ifQ_in_ggga(x1, x2, x3)
U13_ggga(x1, x2, x3, x4)  =  U13_ggga(x1, x2, x4)
pR_in_gagaa(x1, x2, x3, x4, x5)  =  pR_in_gagaa(x1, x3)
U26_gagaa(x1, x2, x3, x4, x5, x6)  =  U26_gagaa(x1, x3, x6)
U27_gagaa(x1, x2, x3, x4, x5, x6)  =  U27_gagaa(x1, x2, x3, x6)
pV_in_ggaa(x1, x2, x3, x4)  =  pV_in_ggaa(x1, x2)
U28_ggaa(x1, x2, x3, x4, x5)  =  U28_ggaa(x1, x2, x5)
U29_ggaa(x1, x2, x3, x4, x5)  =  U29_ggaa(x1, x2, x3, x5)
plusT_in_ga(x1, x2)  =  plusT_in_ga(x1)
plusT_out_ga(x1, x2)  =  plusT_out_ga(x1, x2)
U15_ga(x1, x2, x3)  =  U15_ga(x1, x3)
plusI_in_gga(x1, x2, x3)  =  plusI_in_gga(x1, x2)
plusI_out_gga(x1, x2, x3)  =  plusI_out_gga(x1, x2, x3)
U7_gga(x1, x2, x3, x4)  =  U7_gga(x1, x2, x4)
pV_out_ggaa(x1, x2, x3, x4)  =  pV_out_ggaa(x1, x2, x3, x4)
pR_out_gagaa(x1, x2, x3, x4, x5)  =  pR_out_gagaa(x1, x2, x3, x4, x5)
ifQ_out_ggga(x1, x2, x3, x4)  =  ifQ_out_ggga(x1, x2, x3, x4)
false  =  false
U14_ggga(x1, x2, x3, x4)  =  U14_ggga(x1, x2, x4)
pS_in_ggaa(x1, x2, x3, x4)  =  pS_in_ggaa(x1, x2)
U30_ggaa(x1, x2, x3, x4, x5)  =  U30_ggaa(x1, x2, x5)
U31_ggaa(x1, x2, x3, x4, x5)  =  U31_ggaa(x1, x2, x3, x5)
plusJ_in_gga(x1, x2, x3)  =  plusJ_in_gga(x1, x2)
plusJ_out_gga(x1, x2, x3)  =  plusJ_out_gga(x1, x2, x3)
U8_gga(x1, x2, x3, x4)  =  U8_gga(x1, x2, x4)
pS_out_ggaa(x1, x2, x3, x4)  =  pS_out_ggaa(x1, x2, x3, x4)
pH_out_gaga(x1, x2, x3, x4)  =  pH_out_gaga(x1, x2, x3, x4)
U23_ggaa(x1, x2, x3, x4, x5)  =  U23_ggaa(x1, x2, x3, x5)
plusP_in_ga(x1, x2)  =  plusP_in_ga(x1)
plusP_out_ga(x1, x2)  =  plusP_out_ga(x1, x2)
U12_ga(x1, x2, x3)  =  U12_ga(x1, x3)
pU_out_ggaa(x1, x2, x3, x4)  =  pU_out_ggaa(x1, x2, x3, x4)
pM_out_gagaa(x1, x2, x3, x4, x5)  =  pM_out_gagaa(x1, x2, x3, x4, x5)
ifL_out_ggga(x1, x2, x3, x4)  =  ifL_out_ggga(x1, x2, x3, x4)
U10_ggga(x1, x2, x3, x4)  =  U10_ggga(x1, x2, x4)
pN_in_ggaa(x1, x2, x3, x4)  =  pN_in_ggaa(x1, x2)
U32_ggaa(x1, x2, x3, x4, x5)  =  U32_ggaa(x1, x2, x5)
U33_ggaa(x1, x2, x3, x4, x5)  =  U33_ggaa(x1, x2, x3, x5)
pN_out_ggaa(x1, x2, x3, x4)  =  pN_out_ggaa(x1, x2, x3, x4)
pC_out_gaga(x1, x2, x3, x4)  =  pC_out_gaga(x1, x2, x3, x4)
HALFF_IN_GA(x1, x2)  =  HALFF_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:

HALFF_IN_GA(s(s(T85)), s(X131)) → HALFF_IN_GA(T85, X131)

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

HALFF_IN_GA(s(s(T85))) → HALFF_IN_GA(T85)

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:

• HALFF_IN_GA(s(s(T85))) → HALFF_IN_GA(T85)
  The graph contains the following edges 1 > 1

(27) YES

(28) Obligation:

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

EVENE_IN_GA(s(s(T59)), X87) → EVENE_IN_GA(T59, X87)

The TRS R consists of the following rules:

timesA_in_gga(0, T5, 0) → timesA_out_gga(0, T5, 0)
timesA_in_gga(s(0), T21, T23) → U1_gga(T21, T23, pB_in_gaa(T21, X37, T23))
pB_in_gaa(T21, T27, T23) → U16_gaa(T21, T27, T23, timesK_in_ga(T21, T27))
timesK_in_ga(T33, 0) → timesK_out_ga(T33, 0)
U16_gaa(T21, T27, T23, timesK_out_ga(T21, T27)) → U17_gaa(T21, T27, T23, plusD_in_gga(T21, T27, T23))
plusD_in_gga(0, T40, T40) → plusD_out_gga(0, T40, T40)
plusD_in_gga(s(T47), T48, s(T50)) → U3_gga(T47, T48, T50, plusD_in_gga(T47, T48, T50))
U3_gga(T47, T48, T50, plusD_out_gga(T47, T48, T50)) → plusD_out_gga(s(T47), T48, s(T50))
U17_gaa(T21, T27, T23, plusD_out_gga(T21, T27, T23)) → pB_out_gaa(T21, T27, T23)
U1_gga(T21, T23, pB_out_gaa(T21, X37, T23)) → timesA_out_gga(s(0), T21, T23)
timesA_in_gga(s(s(T55)), T10, T12) → U2_gga(T55, T10, T12, pC_in_gaga(T55, X78, T10, T12))
pC_in_gaga(T55, T56, T10, T12) → U18_gaga(T55, T56, T10, T12, evenE_in_ga(T55, T56))
evenE_in_ga(0, true) → evenE_out_ga(0, true)
evenE_in_ga(s(0), false) → evenE_out_ga(s(0), false)
evenE_in_ga(s(s(T59)), X87) → U4_ga(T59, X87, evenE_in_ga(T59, X87))
U4_ga(T59, X87, evenE_out_ga(T59, X87)) → evenE_out_ga(s(s(T59)), X87)
U18_gaga(T55, T56, T10, T12, evenE_out_ga(T55, T56)) → U19_gaga(T55, T56, T10, T12, ifL_in_ggga(T56, T55, T10, T12))
ifL_in_ggga(true, T75, T76, T78) → U9_ggga(T75, T76, T78, pM_in_gagaa(T75, X112, T76, X113, T78))
pM_in_gagaa(T75, T79, T76, X113, T78) → U20_gagaa(T75, T79, T76, X113, T78, halfO_in_ga(T75, T79))
halfO_in_ga(T82, s(X122)) → U11_ga(T82, X122, halfF_in_ga(T82, X122))
halfF_in_ga(0, 0) → halfF_out_ga(0, 0)
halfF_in_ga(s(s(T85)), s(X131)) → U5_ga(T85, X131, halfF_in_ga(T85, X131))
U5_ga(T85, X131, halfF_out_ga(T85, X131)) → halfF_out_ga(s(s(T85)), s(X131))
U11_ga(T82, X122, halfF_out_ga(T82, X122)) → halfO_out_ga(T82, s(X122))
U20_gagaa(T75, T79, T76, X113, T78, halfO_out_ga(T75, T79)) → U21_gagaa(T75, T79, T76, X113, T78, pU_in_ggaa(T79, T76, X113, T78))
pU_in_ggaa(T79, T76, T88, T78) → U22_ggaa(T79, T76, T88, T78, timesG_in_gga(T79, T76, T88))
timesG_in_gga(0, T95, 0) → timesG_out_gga(0, T95, 0)
timesG_in_gga(s(T100), T101, X154) → U6_gga(T100, T101, X154, pH_in_gaga(T100, X153, T101, X154))
pH_in_gaga(T100, T102, T101, X154) → U24_gaga(T100, T102, T101, X154, evenE_in_ga(s(T100), T102))
U24_gaga(T100, T102, T101, X154, evenE_out_ga(s(T100), T102)) → U25_gaga(T100, T102, T101, X154, ifQ_in_ggga(T102, T100, T101, X154))
ifQ_in_ggga(true, T113, T114, X186) → U13_ggga(T113, T114, X186, pR_in_gagaa(T113, X184, T114, X185, X186))
pR_in_gagaa(T113, T115, T114, X185, X186) → U26_gagaa(T113, T115, T114, X185, X186, halfF_in_ga(s(T113), T115))
U26_gagaa(T113, T115, T114, X185, X186, halfF_out_ga(s(T113), T115)) → U27_gagaa(T113, T115, T114, X185, X186, pV_in_ggaa(T115, T114, X185, X186))
pV_in_ggaa(T115, T114, T118, X186) → U28_ggaa(T115, T114, T118, X186, timesG_in_gga(T115, T114, T118))
U28_ggaa(T115, T114, T118, X186, timesG_out_gga(T115, T114, T118)) → U29_ggaa(T115, T114, T118, X186, plusT_in_ga(T118, X186))
plusT_in_ga(0, 0) → plusT_out_ga(0, 0)
plusT_in_ga(s(0), s(s(0))) → plusT_out_ga(s(0), s(s(0)))
plusT_in_ga(s(s(0)), s(s(s(s(0))))) → plusT_out_ga(s(s(0)), s(s(s(s(0)))))
plusT_in_ga(s(s(s(0))), s(s(s(s(s(s(0))))))) → plusT_out_ga(s(s(s(0))), s(s(s(s(s(s(0)))))))
plusT_in_ga(s(s(s(s(0)))), s(s(s(s(s(s(s(s(0))))))))) → plusT_out_ga(s(s(s(s(0)))), s(s(s(s(s(s(s(s(0)))))))))
plusT_in_ga(s(s(s(s(s(0))))), s(s(s(s(s(s(s(s(s(s(0))))))))))) → plusT_out_ga(s(s(s(s(s(0))))), s(s(s(s(s(s(s(s(s(s(0)))))))))))
plusT_in_ga(s(s(s(s(s(s(0)))))), s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))) → plusT_out_ga(s(s(s(s(s(s(0)))))), s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))
plusT_in_ga(s(s(s(s(s(s(s(0))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))) → plusT_out_ga(s(s(s(s(s(s(s(0))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))
plusT_in_ga(s(s(s(s(s(s(s(s(T144)))))))), s(s(s(s(s(s(s(s(X342))))))))) → U15_ga(T144, X342, plusI_in_gga(T144, s(s(s(s(s(s(s(T144))))))), X342))
plusI_in_gga(0, T152, s(T152)) → plusI_out_gga(0, T152, s(T152))
plusI_in_gga(s(T157), T158, s(X363)) → U7_gga(T157, T158, X363, plusI_in_gga(T157, T158, X363))
U7_gga(T157, T158, X363, plusI_out_gga(T157, T158, X363)) → plusI_out_gga(s(T157), T158, s(X363))
U15_ga(T144, X342, plusI_out_gga(T144, s(s(s(s(s(s(s(T144))))))), X342)) → plusT_out_ga(s(s(s(s(s(s(s(s(T144)))))))), s(s(s(s(s(s(s(s(X342)))))))))
U29_ggaa(T115, T114, T118, X186, plusT_out_ga(T118, X186)) → pV_out_ggaa(T115, T114, T118, X186)
U27_gagaa(T113, T115, T114, X185, X186, pV_out_ggaa(T115, T114, X185, X186)) → pR_out_gagaa(T113, T115, T114, X185, X186)
U13_ggga(T113, T114, X186, pR_out_gagaa(T113, X184, T114, X185, X186)) → ifQ_out_ggga(true, T113, T114, X186)
ifQ_in_ggga(false, T165, T166, X379) → U14_ggga(T165, T166, X379, pS_in_ggaa(T165, T166, X378, X379))
pS_in_ggaa(T165, T166, T169, X379) → U30_ggaa(T165, T166, T169, X379, timesG_in_gga(T165, T166, T169))
U30_ggaa(T165, T166, T169, X379, timesG_out_gga(T165, T166, T169)) → U31_ggaa(T165, T166, T169, X379, plusJ_in_gga(T166, T169, X379))
plusJ_in_gga(0, T178, T178) → plusJ_out_gga(0, T178, T178)
plusJ_in_gga(s(T183), T184, s(X402)) → U8_gga(T183, T184, X402, plusJ_in_gga(T183, T184, X402))
U8_gga(T183, T184, X402, plusJ_out_gga(T183, T184, X402)) → plusJ_out_gga(s(T183), T184, s(X402))
U31_ggaa(T165, T166, T169, X379, plusJ_out_gga(T166, T169, X379)) → pS_out_ggaa(T165, T166, T169, X379)
U14_ggga(T165, T166, X379, pS_out_ggaa(T165, T166, X378, X379)) → ifQ_out_ggga(false, T165, T166, X379)
U25_gaga(T100, T102, T101, X154, ifQ_out_ggga(T102, T100, T101, X154)) → pH_out_gaga(T100, T102, T101, X154)
U6_gga(T100, T101, X154, pH_out_gaga(T100, X153, T101, X154)) → timesG_out_gga(s(T100), T101, X154)
U22_ggaa(T79, T76, T88, T78, timesG_out_gga(T79, T76, T88)) → U23_ggaa(T79, T76, T88, T78, plusP_in_ga(T88, T78))
plusP_in_ga(0, 0) → plusP_out_ga(0, 0)
plusP_in_ga(s(0), s(s(0))) → plusP_out_ga(s(0), s(s(0)))
plusP_in_ga(s(s(0)), s(s(s(s(0))))) → plusP_out_ga(s(s(0)), s(s(s(s(0)))))
plusP_in_ga(s(s(s(0))), s(s(s(s(s(s(0))))))) → plusP_out_ga(s(s(s(0))), s(s(s(s(s(s(0)))))))
plusP_in_ga(s(s(s(s(0)))), s(s(s(s(s(s(s(s(0))))))))) → plusP_out_ga(s(s(s(s(0)))), s(s(s(s(s(s(s(s(0)))))))))
plusP_in_ga(s(s(s(s(s(0))))), s(s(s(s(s(s(s(s(s(s(0))))))))))) → plusP_out_ga(s(s(s(s(s(0))))), s(s(s(s(s(s(s(s(s(s(0)))))))))))
plusP_in_ga(s(s(s(s(s(s(0)))))), s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))) → plusP_out_ga(s(s(s(s(s(s(0)))))), s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))
plusP_in_ga(s(s(s(s(s(s(s(0))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))) → plusP_out_ga(s(s(s(s(s(s(s(0))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))
plusP_in_ga(s(s(s(s(s(s(s(s(T240)))))))), s(s(s(s(s(s(s(s(T242))))))))) → U12_ga(T240, T242, plusD_in_gga(T240, s(s(s(s(s(s(s(s(T240)))))))), T242))
U12_ga(T240, T242, plusD_out_gga(T240, s(s(s(s(s(s(s(s(T240)))))))), T242)) → plusP_out_ga(s(s(s(s(s(s(s(s(T240)))))))), s(s(s(s(s(s(s(s(T242)))))))))
U23_ggaa(T79, T76, T88, T78, plusP_out_ga(T88, T78)) → pU_out_ggaa(T79, T76, T88, T78)
U21_gagaa(T75, T79, T76, X113, T78, pU_out_ggaa(T79, T76, X113, T78)) → pM_out_gagaa(T75, T79, T76, X113, T78)
U9_ggga(T75, T76, T78, pM_out_gagaa(T75, X112, T76, X113, T78)) → ifL_out_ggga(true, T75, T76, T78)
ifL_in_ggga(false, T252, T253, T255) → U10_ggga(T252, T253, T255, pN_in_ggaa(T252, T253, X547, T255))
pN_in_ggaa(T252, T253, T256, T255) → U32_ggaa(T252, T253, T256, T255, timesG_in_gga(s(T252), T253, T256))
U32_ggaa(T252, T253, T256, T255, timesG_out_gga(s(T252), T253, T256)) → U33_ggaa(T252, T253, T256, T255, plusD_in_gga(T253, T256, T255))
U33_ggaa(T252, T253, T256, T255, plusD_out_gga(T253, T256, T255)) → pN_out_ggaa(T252, T253, T256, T255)
U10_ggga(T252, T253, T255, pN_out_ggaa(T252, T253, X547, T255)) → ifL_out_ggga(false, T252, T253, T255)
U19_gaga(T55, T56, T10, T12, ifL_out_ggga(T56, T55, T10, T12)) → pC_out_gaga(T55, T56, T10, T12)
U2_gga(T55, T10, T12, pC_out_gaga(T55, X78, T10, T12)) → timesA_out_gga(s(s(T55)), T10, T12)

The argument filtering Pi contains the following mapping:
timesA_in_gga(x1, x2, x3)  =  timesA_in_gga(x1, x2)
0  =  0
timesA_out_gga(x1, x2, x3)  =  timesA_out_gga(x1, x2, x3)
s(x1)  =  s(x1)
U1_gga(x1, x2, x3)  =  U1_gga(x1, x3)
pB_in_gaa(x1, x2, x3)  =  pB_in_gaa(x1)
U16_gaa(x1, x2, x3, x4)  =  U16_gaa(x1, x4)
timesK_in_ga(x1, x2)  =  timesK_in_ga(x1)
timesK_out_ga(x1, x2)  =  timesK_out_ga(x1, x2)
U17_gaa(x1, x2, x3, x4)  =  U17_gaa(x1, x2, x4)
plusD_in_gga(x1, x2, x3)  =  plusD_in_gga(x1, x2)
plusD_out_gga(x1, x2, x3)  =  plusD_out_gga(x1, x2, x3)
U3_gga(x1, x2, x3, x4)  =  U3_gga(x1, x2, x4)
pB_out_gaa(x1, x2, x3)  =  pB_out_gaa(x1, x2, x3)
U2_gga(x1, x2, x3, x4)  =  U2_gga(x1, x2, x4)
pC_in_gaga(x1, x2, x3, x4)  =  pC_in_gaga(x1, x3)
U18_gaga(x1, x2, x3, x4, x5)  =  U18_gaga(x1, x3, x5)
evenE_in_ga(x1, x2)  =  evenE_in_ga(x1)
evenE_out_ga(x1, x2)  =  evenE_out_ga(x1, x2)
U4_ga(x1, x2, x3)  =  U4_ga(x1, x3)
U19_gaga(x1, x2, x3, x4, x5)  =  U19_gaga(x1, x2, x3, x5)
ifL_in_ggga(x1, x2, x3, x4)  =  ifL_in_ggga(x1, x2, x3)
true  =  true
U9_ggga(x1, x2, x3, x4)  =  U9_ggga(x1, x2, x4)
pM_in_gagaa(x1, x2, x3, x4, x5)  =  pM_in_gagaa(x1, x3)
U20_gagaa(x1, x2, x3, x4, x5, x6)  =  U20_gagaa(x1, x3, x6)
halfO_in_ga(x1, x2)  =  halfO_in_ga(x1)
U11_ga(x1, x2, x3)  =  U11_ga(x1, x3)
halfF_in_ga(x1, x2)  =  halfF_in_ga(x1)
halfF_out_ga(x1, x2)  =  halfF_out_ga(x1, x2)
U5_ga(x1, x2, x3)  =  U5_ga(x1, x3)
halfO_out_ga(x1, x2)  =  halfO_out_ga(x1, x2)
U21_gagaa(x1, x2, x3, x4, x5, x6)  =  U21_gagaa(x1, x2, x3, x6)
pU_in_ggaa(x1, x2, x3, x4)  =  pU_in_ggaa(x1, x2)
U22_ggaa(x1, x2, x3, x4, x5)  =  U22_ggaa(x1, x2, x5)
timesG_in_gga(x1, x2, x3)  =  timesG_in_gga(x1, x2)
timesG_out_gga(x1, x2, x3)  =  timesG_out_gga(x1, x2, x3)
U6_gga(x1, x2, x3, x4)  =  U6_gga(x1, x2, x4)
pH_in_gaga(x1, x2, x3, x4)  =  pH_in_gaga(x1, x3)
U24_gaga(x1, x2, x3, x4, x5)  =  U24_gaga(x1, x3, x5)
U25_gaga(x1, x2, x3, x4, x5)  =  U25_gaga(x1, x2, x3, x5)
ifQ_in_ggga(x1, x2, x3, x4)  =  ifQ_in_ggga(x1, x2, x3)
U13_ggga(x1, x2, x3, x4)  =  U13_ggga(x1, x2, x4)
pR_in_gagaa(x1, x2, x3, x4, x5)  =  pR_in_gagaa(x1, x3)
U26_gagaa(x1, x2, x3, x4, x5, x6)  =  U26_gagaa(x1, x3, x6)
U27_gagaa(x1, x2, x3, x4, x5, x6)  =  U27_gagaa(x1, x2, x3, x6)
pV_in_ggaa(x1, x2, x3, x4)  =  pV_in_ggaa(x1, x2)
U28_ggaa(x1, x2, x3, x4, x5)  =  U28_ggaa(x1, x2, x5)
U29_ggaa(x1, x2, x3, x4, x5)  =  U29_ggaa(x1, x2, x3, x5)
plusT_in_ga(x1, x2)  =  plusT_in_ga(x1)
plusT_out_ga(x1, x2)  =  plusT_out_ga(x1, x2)
U15_ga(x1, x2, x3)  =  U15_ga(x1, x3)
plusI_in_gga(x1, x2, x3)  =  plusI_in_gga(x1, x2)
plusI_out_gga(x1, x2, x3)  =  plusI_out_gga(x1, x2, x3)
U7_gga(x1, x2, x3, x4)  =  U7_gga(x1, x2, x4)
pV_out_ggaa(x1, x2, x3, x4)  =  pV_out_ggaa(x1, x2, x3, x4)
pR_out_gagaa(x1, x2, x3, x4, x5)  =  pR_out_gagaa(x1, x2, x3, x4, x5)
ifQ_out_ggga(x1, x2, x3, x4)  =  ifQ_out_ggga(x1, x2, x3, x4)
false  =  false
U14_ggga(x1, x2, x3, x4)  =  U14_ggga(x1, x2, x4)
pS_in_ggaa(x1, x2, x3, x4)  =  pS_in_ggaa(x1, x2)
U30_ggaa(x1, x2, x3, x4, x5)  =  U30_ggaa(x1, x2, x5)
U31_ggaa(x1, x2, x3, x4, x5)  =  U31_ggaa(x1, x2, x3, x5)
plusJ_in_gga(x1, x2, x3)  =  plusJ_in_gga(x1, x2)
plusJ_out_gga(x1, x2, x3)  =  plusJ_out_gga(x1, x2, x3)
U8_gga(x1, x2, x3, x4)  =  U8_gga(x1, x2, x4)
pS_out_ggaa(x1, x2, x3, x4)  =  pS_out_ggaa(x1, x2, x3, x4)
pH_out_gaga(x1, x2, x3, x4)  =  pH_out_gaga(x1, x2, x3, x4)
U23_ggaa(x1, x2, x3, x4, x5)  =  U23_ggaa(x1, x2, x3, x5)
plusP_in_ga(x1, x2)  =  plusP_in_ga(x1)
plusP_out_ga(x1, x2)  =  plusP_out_ga(x1, x2)
U12_ga(x1, x2, x3)  =  U12_ga(x1, x3)
pU_out_ggaa(x1, x2, x3, x4)  =  pU_out_ggaa(x1, x2, x3, x4)
pM_out_gagaa(x1, x2, x3, x4, x5)  =  pM_out_gagaa(x1, x2, x3, x4, x5)
ifL_out_ggga(x1, x2, x3, x4)  =  ifL_out_ggga(x1, x2, x3, x4)
U10_ggga(x1, x2, x3, x4)  =  U10_ggga(x1, x2, x4)
pN_in_ggaa(x1, x2, x3, x4)  =  pN_in_ggaa(x1, x2)
U32_ggaa(x1, x2, x3, x4, x5)  =  U32_ggaa(x1, x2, x5)
U33_ggaa(x1, x2, x3, x4, x5)  =  U33_ggaa(x1, x2, x3, x5)
pN_out_ggaa(x1, x2, x3, x4)  =  pN_out_ggaa(x1, x2, x3, x4)
pC_out_gaga(x1, x2, x3, x4)  =  pC_out_gaga(x1, x2, x3, x4)
EVENE_IN_GA(x1, x2)  =  EVENE_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:

EVENE_IN_GA(s(s(T59)), X87) → EVENE_IN_GA(T59, X87)

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

EVENE_IN_GA(s(s(T59))) → EVENE_IN_GA(T59)

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:

• EVENE_IN_GA(s(s(T59))) → EVENE_IN_GA(T59)
  The graph contains the following edges 1 > 1

(34) YES

(35) Obligation:

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

PH_IN_GAGA(T100, T102, T101, X154) → U24_GAGA(T100, T102, T101, X154, evenE_in_ga(s(T100), T102))
U24_GAGA(T100, T102, T101, X154, evenE_out_ga(s(T100), T102)) → IFQ_IN_GGGA(T102, T100, T101, X154)
IFQ_IN_GGGA(true, T113, T114, X186) → PR_IN_GAGAA(T113, X184, T114, X185, X186)
PR_IN_GAGAA(T113, T115, T114, X185, X186) → U26_GAGAA(T113, T115, T114, X185, X186, halfF_in_ga(s(T113), T115))
U26_GAGAA(T113, T115, T114, X185, X186, halfF_out_ga(s(T113), T115)) → PV_IN_GGAA(T115, T114, X185, X186)
PV_IN_GGAA(T115, T114, T118, X186) → TIMESG_IN_GGA(T115, T114, T118)
TIMESG_IN_GGA(s(T100), T101, X154) → PH_IN_GAGA(T100, X153, T101, X154)
IFQ_IN_GGGA(false, T165, T166, X379) → PS_IN_GGAA(T165, T166, X378, X379)
PS_IN_GGAA(T165, T166, T169, X379) → TIMESG_IN_GGA(T165, T166, T169)

The TRS R consists of the following rules:

timesA_in_gga(0, T5, 0) → timesA_out_gga(0, T5, 0)
timesA_in_gga(s(0), T21, T23) → U1_gga(T21, T23, pB_in_gaa(T21, X37, T23))
pB_in_gaa(T21, T27, T23) → U16_gaa(T21, T27, T23, timesK_in_ga(T21, T27))
timesK_in_ga(T33, 0) → timesK_out_ga(T33, 0)
U16_gaa(T21, T27, T23, timesK_out_ga(T21, T27)) → U17_gaa(T21, T27, T23, plusD_in_gga(T21, T27, T23))
plusD_in_gga(0, T40, T40) → plusD_out_gga(0, T40, T40)
plusD_in_gga(s(T47), T48, s(T50)) → U3_gga(T47, T48, T50, plusD_in_gga(T47, T48, T50))
U3_gga(T47, T48, T50, plusD_out_gga(T47, T48, T50)) → plusD_out_gga(s(T47), T48, s(T50))
U17_gaa(T21, T27, T23, plusD_out_gga(T21, T27, T23)) → pB_out_gaa(T21, T27, T23)
U1_gga(T21, T23, pB_out_gaa(T21, X37, T23)) → timesA_out_gga(s(0), T21, T23)
timesA_in_gga(s(s(T55)), T10, T12) → U2_gga(T55, T10, T12, pC_in_gaga(T55, X78, T10, T12))
pC_in_gaga(T55, T56, T10, T12) → U18_gaga(T55, T56, T10, T12, evenE_in_ga(T55, T56))
evenE_in_ga(0, true) → evenE_out_ga(0, true)
evenE_in_ga(s(0), false) → evenE_out_ga(s(0), false)
evenE_in_ga(s(s(T59)), X87) → U4_ga(T59, X87, evenE_in_ga(T59, X87))
U4_ga(T59, X87, evenE_out_ga(T59, X87)) → evenE_out_ga(s(s(T59)), X87)
U18_gaga(T55, T56, T10, T12, evenE_out_ga(T55, T56)) → U19_gaga(T55, T56, T10, T12, ifL_in_ggga(T56, T55, T10, T12))
ifL_in_ggga(true, T75, T76, T78) → U9_ggga(T75, T76, T78, pM_in_gagaa(T75, X112, T76, X113, T78))
pM_in_gagaa(T75, T79, T76, X113, T78) → U20_gagaa(T75, T79, T76, X113, T78, halfO_in_ga(T75, T79))
halfO_in_ga(T82, s(X122)) → U11_ga(T82, X122, halfF_in_ga(T82, X122))
halfF_in_ga(0, 0) → halfF_out_ga(0, 0)
halfF_in_ga(s(s(T85)), s(X131)) → U5_ga(T85, X131, halfF_in_ga(T85, X131))
U5_ga(T85, X131, halfF_out_ga(T85, X131)) → halfF_out_ga(s(s(T85)), s(X131))
U11_ga(T82, X122, halfF_out_ga(T82, X122)) → halfO_out_ga(T82, s(X122))
U20_gagaa(T75, T79, T76, X113, T78, halfO_out_ga(T75, T79)) → U21_gagaa(T75, T79, T76, X113, T78, pU_in_ggaa(T79, T76, X113, T78))
pU_in_ggaa(T79, T76, T88, T78) → U22_ggaa(T79, T76, T88, T78, timesG_in_gga(T79, T76, T88))
timesG_in_gga(0, T95, 0) → timesG_out_gga(0, T95, 0)
timesG_in_gga(s(T100), T101, X154) → U6_gga(T100, T101, X154, pH_in_gaga(T100, X153, T101, X154))
pH_in_gaga(T100, T102, T101, X154) → U24_gaga(T100, T102, T101, X154, evenE_in_ga(s(T100), T102))
U24_gaga(T100, T102, T101, X154, evenE_out_ga(s(T100), T102)) → U25_gaga(T100, T102, T101, X154, ifQ_in_ggga(T102, T100, T101, X154))
ifQ_in_ggga(true, T113, T114, X186) → U13_ggga(T113, T114, X186, pR_in_gagaa(T113, X184, T114, X185, X186))
pR_in_gagaa(T113, T115, T114, X185, X186) → U26_gagaa(T113, T115, T114, X185, X186, halfF_in_ga(s(T113), T115))
U26_gagaa(T113, T115, T114, X185, X186, halfF_out_ga(s(T113), T115)) → U27_gagaa(T113, T115, T114, X185, X186, pV_in_ggaa(T115, T114, X185, X186))
pV_in_ggaa(T115, T114, T118, X186) → U28_ggaa(T115, T114, T118, X186, timesG_in_gga(T115, T114, T118))
U28_ggaa(T115, T114, T118, X186, timesG_out_gga(T115, T114, T118)) → U29_ggaa(T115, T114, T118, X186, plusT_in_ga(T118, X186))
plusT_in_ga(0, 0) → plusT_out_ga(0, 0)
plusT_in_ga(s(0), s(s(0))) → plusT_out_ga(s(0), s(s(0)))
plusT_in_ga(s(s(0)), s(s(s(s(0))))) → plusT_out_ga(s(s(0)), s(s(s(s(0)))))
plusT_in_ga(s(s(s(0))), s(s(s(s(s(s(0))))))) → plusT_out_ga(s(s(s(0))), s(s(s(s(s(s(0)))))))
plusT_in_ga(s(s(s(s(0)))), s(s(s(s(s(s(s(s(0))))))))) → plusT_out_ga(s(s(s(s(0)))), s(s(s(s(s(s(s(s(0)))))))))
plusT_in_ga(s(s(s(s(s(0))))), s(s(s(s(s(s(s(s(s(s(0))))))))))) → plusT_out_ga(s(s(s(s(s(0))))), s(s(s(s(s(s(s(s(s(s(0)))))))))))
plusT_in_ga(s(s(s(s(s(s(0)))))), s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))) → plusT_out_ga(s(s(s(s(s(s(0)))))), s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))
plusT_in_ga(s(s(s(s(s(s(s(0))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))) → plusT_out_ga(s(s(s(s(s(s(s(0))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))
plusT_in_ga(s(s(s(s(s(s(s(s(T144)))))))), s(s(s(s(s(s(s(s(X342))))))))) → U15_ga(T144, X342, plusI_in_gga(T144, s(s(s(s(s(s(s(T144))))))), X342))
plusI_in_gga(0, T152, s(T152)) → plusI_out_gga(0, T152, s(T152))
plusI_in_gga(s(T157), T158, s(X363)) → U7_gga(T157, T158, X363, plusI_in_gga(T157, T158, X363))
U7_gga(T157, T158, X363, plusI_out_gga(T157, T158, X363)) → plusI_out_gga(s(T157), T158, s(X363))
U15_ga(T144, X342, plusI_out_gga(T144, s(s(s(s(s(s(s(T144))))))), X342)) → plusT_out_ga(s(s(s(s(s(s(s(s(T144)))))))), s(s(s(s(s(s(s(s(X342)))))))))
U29_ggaa(T115, T114, T118, X186, plusT_out_ga(T118, X186)) → pV_out_ggaa(T115, T114, T118, X186)
U27_gagaa(T113, T115, T114, X185, X186, pV_out_ggaa(T115, T114, X185, X186)) → pR_out_gagaa(T113, T115, T114, X185, X186)
U13_ggga(T113, T114, X186, pR_out_gagaa(T113, X184, T114, X185, X186)) → ifQ_out_ggga(true, T113, T114, X186)
ifQ_in_ggga(false, T165, T166, X379) → U14_ggga(T165, T166, X379, pS_in_ggaa(T165, T166, X378, X379))
pS_in_ggaa(T165, T166, T169, X379) → U30_ggaa(T165, T166, T169, X379, timesG_in_gga(T165, T166, T169))
U30_ggaa(T165, T166, T169, X379, timesG_out_gga(T165, T166, T169)) → U31_ggaa(T165, T166, T169, X379, plusJ_in_gga(T166, T169, X379))
plusJ_in_gga(0, T178, T178) → plusJ_out_gga(0, T178, T178)
plusJ_in_gga(s(T183), T184, s(X402)) → U8_gga(T183, T184, X402, plusJ_in_gga(T183, T184, X402))
U8_gga(T183, T184, X402, plusJ_out_gga(T183, T184, X402)) → plusJ_out_gga(s(T183), T184, s(X402))
U31_ggaa(T165, T166, T169, X379, plusJ_out_gga(T166, T169, X379)) → pS_out_ggaa(T165, T166, T169, X379)
U14_ggga(T165, T166, X379, pS_out_ggaa(T165, T166, X378, X379)) → ifQ_out_ggga(false, T165, T166, X379)
U25_gaga(T100, T102, T101, X154, ifQ_out_ggga(T102, T100, T101, X154)) → pH_out_gaga(T100, T102, T101, X154)
U6_gga(T100, T101, X154, pH_out_gaga(T100, X153, T101, X154)) → timesG_out_gga(s(T100), T101, X154)
U22_ggaa(T79, T76, T88, T78, timesG_out_gga(T79, T76, T88)) → U23_ggaa(T79, T76, T88, T78, plusP_in_ga(T88, T78))
plusP_in_ga(0, 0) → plusP_out_ga(0, 0)
plusP_in_ga(s(0), s(s(0))) → plusP_out_ga(s(0), s(s(0)))
plusP_in_ga(s(s(0)), s(s(s(s(0))))) → plusP_out_ga(s(s(0)), s(s(s(s(0)))))
plusP_in_ga(s(s(s(0))), s(s(s(s(s(s(0))))))) → plusP_out_ga(s(s(s(0))), s(s(s(s(s(s(0)))))))
plusP_in_ga(s(s(s(s(0)))), s(s(s(s(s(s(s(s(0))))))))) → plusP_out_ga(s(s(s(s(0)))), s(s(s(s(s(s(s(s(0)))))))))
plusP_in_ga(s(s(s(s(s(0))))), s(s(s(s(s(s(s(s(s(s(0))))))))))) → plusP_out_ga(s(s(s(s(s(0))))), s(s(s(s(s(s(s(s(s(s(0)))))))))))
plusP_in_ga(s(s(s(s(s(s(0)))))), s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))) → plusP_out_ga(s(s(s(s(s(s(0)))))), s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))
plusP_in_ga(s(s(s(s(s(s(s(0))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))) → plusP_out_ga(s(s(s(s(s(s(s(0))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))
plusP_in_ga(s(s(s(s(s(s(s(s(T240)))))))), s(s(s(s(s(s(s(s(T242))))))))) → U12_ga(T240, T242, plusD_in_gga(T240, s(s(s(s(s(s(s(s(T240)))))))), T242))
U12_ga(T240, T242, plusD_out_gga(T240, s(s(s(s(s(s(s(s(T240)))))))), T242)) → plusP_out_ga(s(s(s(s(s(s(s(s(T240)))))))), s(s(s(s(s(s(s(s(T242)))))))))
U23_ggaa(T79, T76, T88, T78, plusP_out_ga(T88, T78)) → pU_out_ggaa(T79, T76, T88, T78)
U21_gagaa(T75, T79, T76, X113, T78, pU_out_ggaa(T79, T76, X113, T78)) → pM_out_gagaa(T75, T79, T76, X113, T78)
U9_ggga(T75, T76, T78, pM_out_gagaa(T75, X112, T76, X113, T78)) → ifL_out_ggga(true, T75, T76, T78)
ifL_in_ggga(false, T252, T253, T255) → U10_ggga(T252, T253, T255, pN_in_ggaa(T252, T253, X547, T255))
pN_in_ggaa(T252, T253, T256, T255) → U32_ggaa(T252, T253, T256, T255, timesG_in_gga(s(T252), T253, T256))
U32_ggaa(T252, T253, T256, T255, timesG_out_gga(s(T252), T253, T256)) → U33_ggaa(T252, T253, T256, T255, plusD_in_gga(T253, T256, T255))
U33_ggaa(T252, T253, T256, T255, plusD_out_gga(T253, T256, T255)) → pN_out_ggaa(T252, T253, T256, T255)
U10_ggga(T252, T253, T255, pN_out_ggaa(T252, T253, X547, T255)) → ifL_out_ggga(false, T252, T253, T255)
U19_gaga(T55, T56, T10, T12, ifL_out_ggga(T56, T55, T10, T12)) → pC_out_gaga(T55, T56, T10, T12)
U2_gga(T55, T10, T12, pC_out_gaga(T55, X78, T10, T12)) → timesA_out_gga(s(s(T55)), T10, T12)

The argument filtering Pi contains the following mapping:
timesA_in_gga(x1, x2, x3)  =  timesA_in_gga(x1, x2)
0  =  0
timesA_out_gga(x1, x2, x3)  =  timesA_out_gga(x1, x2, x3)
s(x1)  =  s(x1)
U1_gga(x1, x2, x3)  =  U1_gga(x1, x3)
pB_in_gaa(x1, x2, x3)  =  pB_in_gaa(x1)
U16_gaa(x1, x2, x3, x4)  =  U16_gaa(x1, x4)
timesK_in_ga(x1, x2)  =  timesK_in_ga(x1)
timesK_out_ga(x1, x2)  =  timesK_out_ga(x1, x2)
U17_gaa(x1, x2, x3, x4)  =  U17_gaa(x1, x2, x4)
plusD_in_gga(x1, x2, x3)  =  plusD_in_gga(x1, x2)
plusD_out_gga(x1, x2, x3)  =  plusD_out_gga(x1, x2, x3)
U3_gga(x1, x2, x3, x4)  =  U3_gga(x1, x2, x4)
pB_out_gaa(x1, x2, x3)  =  pB_out_gaa(x1, x2, x3)
U2_gga(x1, x2, x3, x4)  =  U2_gga(x1, x2, x4)
pC_in_gaga(x1, x2, x3, x4)  =  pC_in_gaga(x1, x3)
U18_gaga(x1, x2, x3, x4, x5)  =  U18_gaga(x1, x3, x5)
evenE_in_ga(x1, x2)  =  evenE_in_ga(x1)
evenE_out_ga(x1, x2)  =  evenE_out_ga(x1, x2)
U4_ga(x1, x2, x3)  =  U4_ga(x1, x3)
U19_gaga(x1, x2, x3, x4, x5)  =  U19_gaga(x1, x2, x3, x5)
ifL_in_ggga(x1, x2, x3, x4)  =  ifL_in_ggga(x1, x2, x3)
true  =  true
U9_ggga(x1, x2, x3, x4)  =  U9_ggga(x1, x2, x4)
pM_in_gagaa(x1, x2, x3, x4, x5)  =  pM_in_gagaa(x1, x3)
U20_gagaa(x1, x2, x3, x4, x5, x6)  =  U20_gagaa(x1, x3, x6)
halfO_in_ga(x1, x2)  =  halfO_in_ga(x1)
U11_ga(x1, x2, x3)  =  U11_ga(x1, x3)
halfF_in_ga(x1, x2)  =  halfF_in_ga(x1)
halfF_out_ga(x1, x2)  =  halfF_out_ga(x1, x2)
U5_ga(x1, x2, x3)  =  U5_ga(x1, x3)
halfO_out_ga(x1, x2)  =  halfO_out_ga(x1, x2)
U21_gagaa(x1, x2, x3, x4, x5, x6)  =  U21_gagaa(x1, x2, x3, x6)
pU_in_ggaa(x1, x2, x3, x4)  =  pU_in_ggaa(x1, x2)
U22_ggaa(x1, x2, x3, x4, x5)  =  U22_ggaa(x1, x2, x5)
timesG_in_gga(x1, x2, x3)  =  timesG_in_gga(x1, x2)
timesG_out_gga(x1, x2, x3)  =  timesG_out_gga(x1, x2, x3)
U6_gga(x1, x2, x3, x4)  =  U6_gga(x1, x2, x4)
pH_in_gaga(x1, x2, x3, x4)  =  pH_in_gaga(x1, x3)
U24_gaga(x1, x2, x3, x4, x5)  =  U24_gaga(x1, x3, x5)
U25_gaga(x1, x2, x3, x4, x5)  =  U25_gaga(x1, x2, x3, x5)
ifQ_in_ggga(x1, x2, x3, x4)  =  ifQ_in_ggga(x1, x2, x3)
U13_ggga(x1, x2, x3, x4)  =  U13_ggga(x1, x2, x4)
pR_in_gagaa(x1, x2, x3, x4, x5)  =  pR_in_gagaa(x1, x3)
U26_gagaa(x1, x2, x3, x4, x5, x6)  =  U26_gagaa(x1, x3, x6)
U27_gagaa(x1, x2, x3, x4, x5, x6)  =  U27_gagaa(x1, x2, x3, x6)
pV_in_ggaa(x1, x2, x3, x4)  =  pV_in_ggaa(x1, x2)
U28_ggaa(x1, x2, x3, x4, x5)  =  U28_ggaa(x1, x2, x5)
U29_ggaa(x1, x2, x3, x4, x5)  =  U29_ggaa(x1, x2, x3, x5)
plusT_in_ga(x1, x2)  =  plusT_in_ga(x1)
plusT_out_ga(x1, x2)  =  plusT_out_ga(x1, x2)
U15_ga(x1, x2, x3)  =  U15_ga(x1, x3)
plusI_in_gga(x1, x2, x3)  =  plusI_in_gga(x1, x2)
plusI_out_gga(x1, x2, x3)  =  plusI_out_gga(x1, x2, x3)
U7_gga(x1, x2, x3, x4)  =  U7_gga(x1, x2, x4)
pV_out_ggaa(x1, x2, x3, x4)  =  pV_out_ggaa(x1, x2, x3, x4)
pR_out_gagaa(x1, x2, x3, x4, x5)  =  pR_out_gagaa(x1, x2, x3, x4, x5)
ifQ_out_ggga(x1, x2, x3, x4)  =  ifQ_out_ggga(x1, x2, x3, x4)
false  =  false
U14_ggga(x1, x2, x3, x4)  =  U14_ggga(x1, x2, x4)
pS_in_ggaa(x1, x2, x3, x4)  =  pS_in_ggaa(x1, x2)
U30_ggaa(x1, x2, x3, x4, x5)  =  U30_ggaa(x1, x2, x5)
U31_ggaa(x1, x2, x3, x4, x5)  =  U31_ggaa(x1, x2, x3, x5)
plusJ_in_gga(x1, x2, x3)  =  plusJ_in_gga(x1, x2)
plusJ_out_gga(x1, x2, x3)  =  plusJ_out_gga(x1, x2, x3)
U8_gga(x1, x2, x3, x4)  =  U8_gga(x1, x2, x4)
pS_out_ggaa(x1, x2, x3, x4)  =  pS_out_ggaa(x1, x2, x3, x4)
pH_out_gaga(x1, x2, x3, x4)  =  pH_out_gaga(x1, x2, x3, x4)
U23_ggaa(x1, x2, x3, x4, x5)  =  U23_ggaa(x1, x2, x3, x5)
plusP_in_ga(x1, x2)  =  plusP_in_ga(x1)
plusP_out_ga(x1, x2)  =  plusP_out_ga(x1, x2)
U12_ga(x1, x2, x3)  =  U12_ga(x1, x3)
pU_out_ggaa(x1, x2, x3, x4)  =  pU_out_ggaa(x1, x2, x3, x4)
pM_out_gagaa(x1, x2, x3, x4, x5)  =  pM_out_gagaa(x1, x2, x3, x4, x5)
ifL_out_ggga(x1, x2, x3, x4)  =  ifL_out_ggga(x1, x2, x3, x4)
U10_ggga(x1, x2, x3, x4)  =  U10_ggga(x1, x2, x4)
pN_in_ggaa(x1, x2, x3, x4)  =  pN_in_ggaa(x1, x2)
U32_ggaa(x1, x2, x3, x4, x5)  =  U32_ggaa(x1, x2, x5)
U33_ggaa(x1, x2, x3, x4, x5)  =  U33_ggaa(x1, x2, x3, x5)
pN_out_ggaa(x1, x2, x3, x4)  =  pN_out_ggaa(x1, x2, x3, x4)
pC_out_gaga(x1, x2, x3, x4)  =  pC_out_gaga(x1, x2, x3, x4)
TIMESG_IN_GGA(x1, x2, x3)  =  TIMESG_IN_GGA(x1, x2)
PH_IN_GAGA(x1, x2, x3, x4)  =  PH_IN_GAGA(x1, x3)
U24_GAGA(x1, x2, x3, x4, x5)  =  U24_GAGA(x1, x3, x5)
IFQ_IN_GGGA(x1, x2, x3, x4)  =  IFQ_IN_GGGA(x1, x2, x3)
PR_IN_GAGAA(x1, x2, x3, x4, x5)  =  PR_IN_GAGAA(x1, x3)
U26_GAGAA(x1, x2, x3, x4, x5, x6)  =  U26_GAGAA(x1, x3, x6)
PV_IN_GGAA(x1, x2, x3, x4)  =  PV_IN_GGAA(x1, x2)
PS_IN_GGAA(x1, x2, x3, x4)  =  PS_IN_GGAA(x1, x2)

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:

PH_IN_GAGA(T100, T102, T101, X154) → U24_GAGA(T100, T102, T101, X154, evenE_in_ga(s(T100), T102))
U24_GAGA(T100, T102, T101, X154, evenE_out_ga(s(T100), T102)) → IFQ_IN_GGGA(T102, T100, T101, X154)
IFQ_IN_GGGA(true, T113, T114, X186) → PR_IN_GAGAA(T113, X184, T114, X185, X186)
PR_IN_GAGAA(T113, T115, T114, X185, X186) → U26_GAGAA(T113, T115, T114, X185, X186, halfF_in_ga(s(T113), T115))
U26_GAGAA(T113, T115, T114, X185, X186, halfF_out_ga(s(T113), T115)) → PV_IN_GGAA(T115, T114, X185, X186)
PV_IN_GGAA(T115, T114, T118, X186) → TIMESG_IN_GGA(T115, T114, T118)
TIMESG_IN_GGA(s(T100), T101, X154) → PH_IN_GAGA(T100, X153, T101, X154)
IFQ_IN_GGGA(false, T165, T166, X379) → PS_IN_GGAA(T165, T166, X378, X379)
PS_IN_GGAA(T165, T166, T169, X379) → TIMESG_IN_GGA(T165, T166, T169)

The TRS R consists of the following rules:

evenE_in_ga(s(0), false) → evenE_out_ga(s(0), false)
evenE_in_ga(s(s(T59)), X87) → U4_ga(T59, X87, evenE_in_ga(T59, X87))
halfF_in_ga(s(s(T85)), s(X131)) → U5_ga(T85, X131, halfF_in_ga(T85, X131))
U4_ga(T59, X87, evenE_out_ga(T59, X87)) → evenE_out_ga(s(s(T59)), X87)
U5_ga(T85, X131, halfF_out_ga(T85, X131)) → halfF_out_ga(s(s(T85)), s(X131))
evenE_in_ga(0, true) → evenE_out_ga(0, true)
halfF_in_ga(0, 0) → halfF_out_ga(0, 0)

The argument filtering Pi contains the following mapping:
0  =  0
s(x1)  =  s(x1)
evenE_in_ga(x1, x2)  =  evenE_in_ga(x1)
evenE_out_ga(x1, x2)  =  evenE_out_ga(x1, x2)
U4_ga(x1, x2, x3)  =  U4_ga(x1, x3)
true  =  true
halfF_in_ga(x1, x2)  =  halfF_in_ga(x1)
halfF_out_ga(x1, x2)  =  halfF_out_ga(x1, x2)
U5_ga(x1, x2, x3)  =  U5_ga(x1, x3)
false  =  false
TIMESG_IN_GGA(x1, x2, x3)  =  TIMESG_IN_GGA(x1, x2)
PH_IN_GAGA(x1, x2, x3, x4)  =  PH_IN_GAGA(x1, x3)
U24_GAGA(x1, x2, x3, x4, x5)  =  U24_GAGA(x1, x3, x5)
IFQ_IN_GGGA(x1, x2, x3, x4)  =  IFQ_IN_GGGA(x1, x2, x3)
PR_IN_GAGAA(x1, x2, x3, x4, x5)  =  PR_IN_GAGAA(x1, x3)
U26_GAGAA(x1, x2, x3, x4, x5, x6)  =  U26_GAGAA(x1, x3, x6)
PV_IN_GGAA(x1, x2, x3, x4)  =  PV_IN_GGAA(x1, x2)
PS_IN_GGAA(x1, x2, x3, x4)  =  PS_IN_GGAA(x1, x2)

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:

PH_IN_GAGA(T100, T101) → U24_GAGA(T100, T101, evenE_in_ga(s(T100)))
U24_GAGA(T100, T101, evenE_out_ga(s(T100), T102)) → IFQ_IN_GGGA(T102, T100, T101)
IFQ_IN_GGGA(true, T113, T114) → PR_IN_GAGAA(T113, T114)
PR_IN_GAGAA(T113, T114) → U26_GAGAA(T113, T114, halfF_in_ga(s(T113)))
U26_GAGAA(T113, T114, halfF_out_ga(s(T113), T115)) → PV_IN_GGAA(T115, T114)
PV_IN_GGAA(T115, T114) → TIMESG_IN_GGA(T115, T114)
TIMESG_IN_GGA(s(T100), T101) → PH_IN_GAGA(T100, T101)
IFQ_IN_GGGA(false, T165, T166) → PS_IN_GGAA(T165, T166)
PS_IN_GGAA(T165, T166) → TIMESG_IN_GGA(T165, T166)

The TRS R consists of the following rules:

evenE_in_ga(s(0)) → evenE_out_ga(s(0), false)
evenE_in_ga(s(s(T59))) → U4_ga(T59, evenE_in_ga(T59))
halfF_in_ga(s(s(T85))) → U5_ga(T85, halfF_in_ga(T85))
U4_ga(T59, evenE_out_ga(T59, X87)) → evenE_out_ga(s(s(T59)), X87)
U5_ga(T85, halfF_out_ga(T85, X131)) → halfF_out_ga(s(s(T85)), s(X131))
evenE_in_ga(0) → evenE_out_ga(0, true)
halfF_in_ga(0) → halfF_out_ga(0, 0)

The set Q consists of the following terms:

evenE_in_ga(x0)
halfF_in_ga(x0)
U4_ga(x0, x1)
U5_ga(x0, x1)

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

(40) QDPOrderProof (EQUIVALENT transformation)

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

The following pairs can be oriented strictly and are deleted.

PS_IN_GGAA(T165, T166) → TIMESG_IN_GGA(T165, T166)

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

POL(0) = 0   
POL(IFQ_IN_GGGA(x₁, x₂, x₃)) = 1 + x₂   
POL(PH_IN_GAGA(x₁, x₂)) = 1 + x₁   
POL(PR_IN_GAGAA(x₁, x₂)) = 1 + x₁   
POL(PS_IN_GGAA(x₁, x₂)) = 1 + x₁   
POL(PV_IN_GGAA(x₁, x₂)) = x₁   
POL(TIMESG_IN_GGA(x₁, x₂)) = x₁   
POL(U24_GAGA(x₁, x₂, x₃)) = 1 + x₁   
POL(U26_GAGAA(x₁, x₂, x₃)) = x₃   
POL(U4_ga(x₁, x₂)) = 0   
POL(U5_ga(x₁, x₂)) = 1 + x₂   
POL(evenE_in_ga(x₁)) = 0   
POL(evenE_out_ga(x₁, x₂)) = 0   
POL(false) = 0   
POL(halfF_in_ga(x₁)) = x₁   
POL(halfF_out_ga(x₁, x₂)) = x₂   
POL(s(x₁)) = 1 + x₁   
POL(true) = 0   

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

halfF_in_ga(s(s(T85))) → U5_ga(T85, halfF_in_ga(T85))
halfF_in_ga(0) → halfF_out_ga(0, 0)
U5_ga(T85, halfF_out_ga(T85, X131)) → halfF_out_ga(s(s(T85)), s(X131))

(41) Obligation:

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

PH_IN_GAGA(T100, T101) → U24_GAGA(T100, T101, evenE_in_ga(s(T100)))
U24_GAGA(T100, T101, evenE_out_ga(s(T100), T102)) → IFQ_IN_GGGA(T102, T100, T101)
IFQ_IN_GGGA(true, T113, T114) → PR_IN_GAGAA(T113, T114)
PR_IN_GAGAA(T113, T114) → U26_GAGAA(T113, T114, halfF_in_ga(s(T113)))
U26_GAGAA(T113, T114, halfF_out_ga(s(T113), T115)) → PV_IN_GGAA(T115, T114)
PV_IN_GGAA(T115, T114) → TIMESG_IN_GGA(T115, T114)
TIMESG_IN_GGA(s(T100), T101) → PH_IN_GAGA(T100, T101)
IFQ_IN_GGGA(false, T165, T166) → PS_IN_GGAA(T165, T166)

The TRS R consists of the following rules:

evenE_in_ga(s(0)) → evenE_out_ga(s(0), false)
evenE_in_ga(s(s(T59))) → U4_ga(T59, evenE_in_ga(T59))
halfF_in_ga(s(s(T85))) → U5_ga(T85, halfF_in_ga(T85))
U4_ga(T59, evenE_out_ga(T59, X87)) → evenE_out_ga(s(s(T59)), X87)
U5_ga(T85, halfF_out_ga(T85, X131)) → halfF_out_ga(s(s(T85)), s(X131))
evenE_in_ga(0) → evenE_out_ga(0, true)
halfF_in_ga(0) → halfF_out_ga(0, 0)

The set Q consists of the following terms:

evenE_in_ga(x0)
halfF_in_ga(x0)
U4_ga(x0, x1)
U5_ga(x0, x1)

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

(42) DependencyGraphProof (EQUIVALENT transformation)

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

(43) Obligation:

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

U24_GAGA(T100, T101, evenE_out_ga(s(T100), T102)) → IFQ_IN_GGGA(T102, T100, T101)
IFQ_IN_GGGA(true, T113, T114) → PR_IN_GAGAA(T113, T114)
PR_IN_GAGAA(T113, T114) → U26_GAGAA(T113, T114, halfF_in_ga(s(T113)))
U26_GAGAA(T113, T114, halfF_out_ga(s(T113), T115)) → PV_IN_GGAA(T115, T114)
PV_IN_GGAA(T115, T114) → TIMESG_IN_GGA(T115, T114)
TIMESG_IN_GGA(s(T100), T101) → PH_IN_GAGA(T100, T101)
PH_IN_GAGA(T100, T101) → U24_GAGA(T100, T101, evenE_in_ga(s(T100)))

The TRS R consists of the following rules:

evenE_in_ga(s(0)) → evenE_out_ga(s(0), false)
evenE_in_ga(s(s(T59))) → U4_ga(T59, evenE_in_ga(T59))
halfF_in_ga(s(s(T85))) → U5_ga(T85, halfF_in_ga(T85))
U4_ga(T59, evenE_out_ga(T59, X87)) → evenE_out_ga(s(s(T59)), X87)
U5_ga(T85, halfF_out_ga(T85, X131)) → halfF_out_ga(s(s(T85)), s(X131))
evenE_in_ga(0) → evenE_out_ga(0, true)
halfF_in_ga(0) → halfF_out_ga(0, 0)

The set Q consists of the following terms:

evenE_in_ga(x0)
halfF_in_ga(x0)
U4_ga(x0, x1)
U5_ga(x0, x1)

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

(44) 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:

evenE_in_ga(s(0)) → evenE_out_ga(s(0), false)

Used ordering: Polynomial interpretation [POLO]:

POL(0) = 0   
POL(IFQ_IN_GGGA(x₁, x₂, x₃)) = x₁   
POL(PH_IN_GAGA(x₁, x₂)) = 1   
POL(PR_IN_GAGAA(x₁, x₂)) = 1   
POL(PV_IN_GGAA(x₁, x₂)) = 1   
POL(TIMESG_IN_GGA(x₁, x₂)) = 1   
POL(U24_GAGA(x₁, x₂, x₃)) = x₃   
POL(U26_GAGAA(x₁, x₂, x₃)) = 1   
POL(U4_ga(x₁, x₂)) = x₂   
POL(U5_ga(x₁, x₂)) = 0   
POL(evenE_in_ga(x₁)) = 1   
POL(evenE_out_ga(x₁, x₂)) = x₂   
POL(false) = 0   
POL(halfF_in_ga(x₁)) = 0   
POL(halfF_out_ga(x₁, x₂)) = 0   
POL(s(x₁)) = 0   
POL(true) = 1   

(45) Obligation:

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

U24_GAGA(T100, T101, evenE_out_ga(s(T100), T102)) → IFQ_IN_GGGA(T102, T100, T101)
IFQ_IN_GGGA(true, T113, T114) → PR_IN_GAGAA(T113, T114)
PR_IN_GAGAA(T113, T114) → U26_GAGAA(T113, T114, halfF_in_ga(s(T113)))
U26_GAGAA(T113, T114, halfF_out_ga(s(T113), T115)) → PV_IN_GGAA(T115, T114)
PV_IN_GGAA(T115, T114) → TIMESG_IN_GGA(T115, T114)
TIMESG_IN_GGA(s(T100), T101) → PH_IN_GAGA(T100, T101)
PH_IN_GAGA(T100, T101) → U24_GAGA(T100, T101, evenE_in_ga(s(T100)))

The TRS R consists of the following rules:

evenE_in_ga(s(s(T59))) → U4_ga(T59, evenE_in_ga(T59))
halfF_in_ga(s(s(T85))) → U5_ga(T85, halfF_in_ga(T85))
U4_ga(T59, evenE_out_ga(T59, X87)) → evenE_out_ga(s(s(T59)), X87)
U5_ga(T85, halfF_out_ga(T85, X131)) → halfF_out_ga(s(s(T85)), s(X131))
evenE_in_ga(0) → evenE_out_ga(0, true)
halfF_in_ga(0) → halfF_out_ga(0, 0)

The set Q consists of the following terms:

evenE_in_ga(x0)
halfF_in_ga(x0)
U4_ga(x0, x1)
U5_ga(x0, x1)

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

(46) QDPOrderProof (EQUIVALENT transformation)

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

The following pairs can be oriented strictly and are deleted.

U24_GAGA(T100, T101, evenE_out_ga(s(T100), T102)) → IFQ_IN_GGGA(T102, T100, T101)
TIMESG_IN_GGA(s(T100), T101) → PH_IN_GAGA(T100, T101)

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

POL( U26_GAGAA(x₁, ..., x₃) ) = x₂ + x₃

POL( halfF_in_ga(x₁) ) = max{0, x₁ - 2}

POL( s(x₁) ) = 2x₁ + 2

POL( U5_ga(x₁, x₂) ) = 2x₁ + 2x₂ + 2

POL( U24_GAGA(x₁, ..., x₃) ) = 2x₁ + x₂ + 1

POL( evenE_in_ga(x₁) ) = 1

POL( U4_ga(x₁, x₂) ) = x₁ + 2

POL( 0 ) = 0

POL( evenE_out_ga(x₁, x₂) ) = 2x₁ + 2

POL( true ) = 0

POL( halfF_out_ga(x₁, x₂) ) = x₂

POL( IFQ_IN_GGGA(x₁, ..., x₃) ) = 2x₂ + x₃

POL( PR_IN_GAGAA(x₁, x₂) ) = 2x₁ + x₂

POL( PV_IN_GGAA(x₁, x₂) ) = x₁ + x₂

POL( TIMESG_IN_GGA(x₁, x₂) ) = x₁ + x₂

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

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

halfF_in_ga(s(s(T85))) → U5_ga(T85, halfF_in_ga(T85))
halfF_in_ga(0) → halfF_out_ga(0, 0)
U5_ga(T85, halfF_out_ga(T85, X131)) → halfF_out_ga(s(s(T85)), s(X131))

(47) Obligation:

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

IFQ_IN_GGGA(true, T113, T114) → PR_IN_GAGAA(T113, T114)
PR_IN_GAGAA(T113, T114) → U26_GAGAA(T113, T114, halfF_in_ga(s(T113)))
U26_GAGAA(T113, T114, halfF_out_ga(s(T113), T115)) → PV_IN_GGAA(T115, T114)
PV_IN_GGAA(T115, T114) → TIMESG_IN_GGA(T115, T114)
PH_IN_GAGA(T100, T101) → U24_GAGA(T100, T101, evenE_in_ga(s(T100)))

The TRS R consists of the following rules:

evenE_in_ga(s(s(T59))) → U4_ga(T59, evenE_in_ga(T59))
halfF_in_ga(s(s(T85))) → U5_ga(T85, halfF_in_ga(T85))
U4_ga(T59, evenE_out_ga(T59, X87)) → evenE_out_ga(s(s(T59)), X87)
U5_ga(T85, halfF_out_ga(T85, X131)) → halfF_out_ga(s(s(T85)), s(X131))
evenE_in_ga(0) → evenE_out_ga(0, true)
halfF_in_ga(0) → halfF_out_ga(0, 0)

The set Q consists of the following terms:

evenE_in_ga(x0)
halfF_in_ga(x0)
U4_ga(x0, x1)
U5_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 0 SCCs with 5 less nodes.

(49) TRUE

(50) Obligation:

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

PLUSD_IN_GGA(s(T47), T48, s(T50)) → PLUSD_IN_GGA(T47, T48, T50)

The TRS R consists of the following rules:

timesA_in_gga(0, T5, 0) → timesA_out_gga(0, T5, 0)
timesA_in_gga(s(0), T21, T23) → U1_gga(T21, T23, pB_in_gaa(T21, X37, T23))
pB_in_gaa(T21, T27, T23) → U16_gaa(T21, T27, T23, timesK_in_ga(T21, T27))
timesK_in_ga(T33, 0) → timesK_out_ga(T33, 0)
U16_gaa(T21, T27, T23, timesK_out_ga(T21, T27)) → U17_gaa(T21, T27, T23, plusD_in_gga(T21, T27, T23))
plusD_in_gga(0, T40, T40) → plusD_out_gga(0, T40, T40)
plusD_in_gga(s(T47), T48, s(T50)) → U3_gga(T47, T48, T50, plusD_in_gga(T47, T48, T50))
U3_gga(T47, T48, T50, plusD_out_gga(T47, T48, T50)) → plusD_out_gga(s(T47), T48, s(T50))
U17_gaa(T21, T27, T23, plusD_out_gga(T21, T27, T23)) → pB_out_gaa(T21, T27, T23)
U1_gga(T21, T23, pB_out_gaa(T21, X37, T23)) → timesA_out_gga(s(0), T21, T23)
timesA_in_gga(s(s(T55)), T10, T12) → U2_gga(T55, T10, T12, pC_in_gaga(T55, X78, T10, T12))
pC_in_gaga(T55, T56, T10, T12) → U18_gaga(T55, T56, T10, T12, evenE_in_ga(T55, T56))
evenE_in_ga(0, true) → evenE_out_ga(0, true)
evenE_in_ga(s(0), false) → evenE_out_ga(s(0), false)
evenE_in_ga(s(s(T59)), X87) → U4_ga(T59, X87, evenE_in_ga(T59, X87))
U4_ga(T59, X87, evenE_out_ga(T59, X87)) → evenE_out_ga(s(s(T59)), X87)
U18_gaga(T55, T56, T10, T12, evenE_out_ga(T55, T56)) → U19_gaga(T55, T56, T10, T12, ifL_in_ggga(T56, T55, T10, T12))
ifL_in_ggga(true, T75, T76, T78) → U9_ggga(T75, T76, T78, pM_in_gagaa(T75, X112, T76, X113, T78))
pM_in_gagaa(T75, T79, T76, X113, T78) → U20_gagaa(T75, T79, T76, X113, T78, halfO_in_ga(T75, T79))
halfO_in_ga(T82, s(X122)) → U11_ga(T82, X122, halfF_in_ga(T82, X122))
halfF_in_ga(0, 0) → halfF_out_ga(0, 0)
halfF_in_ga(s(s(T85)), s(X131)) → U5_ga(T85, X131, halfF_in_ga(T85, X131))
U5_ga(T85, X131, halfF_out_ga(T85, X131)) → halfF_out_ga(s(s(T85)), s(X131))
U11_ga(T82, X122, halfF_out_ga(T82, X122)) → halfO_out_ga(T82, s(X122))
U20_gagaa(T75, T79, T76, X113, T78, halfO_out_ga(T75, T79)) → U21_gagaa(T75, T79, T76, X113, T78, pU_in_ggaa(T79, T76, X113, T78))
pU_in_ggaa(T79, T76, T88, T78) → U22_ggaa(T79, T76, T88, T78, timesG_in_gga(T79, T76, T88))
timesG_in_gga(0, T95, 0) → timesG_out_gga(0, T95, 0)
timesG_in_gga(s(T100), T101, X154) → U6_gga(T100, T101, X154, pH_in_gaga(T100, X153, T101, X154))
pH_in_gaga(T100, T102, T101, X154) → U24_gaga(T100, T102, T101, X154, evenE_in_ga(s(T100), T102))
U24_gaga(T100, T102, T101, X154, evenE_out_ga(s(T100), T102)) → U25_gaga(T100, T102, T101, X154, ifQ_in_ggga(T102, T100, T101, X154))
ifQ_in_ggga(true, T113, T114, X186) → U13_ggga(T113, T114, X186, pR_in_gagaa(T113, X184, T114, X185, X186))
pR_in_gagaa(T113, T115, T114, X185, X186) → U26_gagaa(T113, T115, T114, X185, X186, halfF_in_ga(s(T113), T115))
U26_gagaa(T113, T115, T114, X185, X186, halfF_out_ga(s(T113), T115)) → U27_gagaa(T113, T115, T114, X185, X186, pV_in_ggaa(T115, T114, X185, X186))
pV_in_ggaa(T115, T114, T118, X186) → U28_ggaa(T115, T114, T118, X186, timesG_in_gga(T115, T114, T118))
U28_ggaa(T115, T114, T118, X186, timesG_out_gga(T115, T114, T118)) → U29_ggaa(T115, T114, T118, X186, plusT_in_ga(T118, X186))
plusT_in_ga(0, 0) → plusT_out_ga(0, 0)
plusT_in_ga(s(0), s(s(0))) → plusT_out_ga(s(0), s(s(0)))
plusT_in_ga(s(s(0)), s(s(s(s(0))))) → plusT_out_ga(s(s(0)), s(s(s(s(0)))))
plusT_in_ga(s(s(s(0))), s(s(s(s(s(s(0))))))) → plusT_out_ga(s(s(s(0))), s(s(s(s(s(s(0)))))))
plusT_in_ga(s(s(s(s(0)))), s(s(s(s(s(s(s(s(0))))))))) → plusT_out_ga(s(s(s(s(0)))), s(s(s(s(s(s(s(s(0)))))))))
plusT_in_ga(s(s(s(s(s(0))))), s(s(s(s(s(s(s(s(s(s(0))))))))))) → plusT_out_ga(s(s(s(s(s(0))))), s(s(s(s(s(s(s(s(s(s(0)))))))))))
plusT_in_ga(s(s(s(s(s(s(0)))))), s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))) → plusT_out_ga(s(s(s(s(s(s(0)))))), s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))
plusT_in_ga(s(s(s(s(s(s(s(0))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))) → plusT_out_ga(s(s(s(s(s(s(s(0))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))
plusT_in_ga(s(s(s(s(s(s(s(s(T144)))))))), s(s(s(s(s(s(s(s(X342))))))))) → U15_ga(T144, X342, plusI_in_gga(T144, s(s(s(s(s(s(s(T144))))))), X342))
plusI_in_gga(0, T152, s(T152)) → plusI_out_gga(0, T152, s(T152))
plusI_in_gga(s(T157), T158, s(X363)) → U7_gga(T157, T158, X363, plusI_in_gga(T157, T158, X363))
U7_gga(T157, T158, X363, plusI_out_gga(T157, T158, X363)) → plusI_out_gga(s(T157), T158, s(X363))
U15_ga(T144, X342, plusI_out_gga(T144, s(s(s(s(s(s(s(T144))))))), X342)) → plusT_out_ga(s(s(s(s(s(s(s(s(T144)))))))), s(s(s(s(s(s(s(s(X342)))))))))
U29_ggaa(T115, T114, T118, X186, plusT_out_ga(T118, X186)) → pV_out_ggaa(T115, T114, T118, X186)
U27_gagaa(T113, T115, T114, X185, X186, pV_out_ggaa(T115, T114, X185, X186)) → pR_out_gagaa(T113, T115, T114, X185, X186)
U13_ggga(T113, T114, X186, pR_out_gagaa(T113, X184, T114, X185, X186)) → ifQ_out_ggga(true, T113, T114, X186)
ifQ_in_ggga(false, T165, T166, X379) → U14_ggga(T165, T166, X379, pS_in_ggaa(T165, T166, X378, X379))
pS_in_ggaa(T165, T166, T169, X379) → U30_ggaa(T165, T166, T169, X379, timesG_in_gga(T165, T166, T169))
U30_ggaa(T165, T166, T169, X379, timesG_out_gga(T165, T166, T169)) → U31_ggaa(T165, T166, T169, X379, plusJ_in_gga(T166, T169, X379))
plusJ_in_gga(0, T178, T178) → plusJ_out_gga(0, T178, T178)
plusJ_in_gga(s(T183), T184, s(X402)) → U8_gga(T183, T184, X402, plusJ_in_gga(T183, T184, X402))
U8_gga(T183, T184, X402, plusJ_out_gga(T183, T184, X402)) → plusJ_out_gga(s(T183), T184, s(X402))
U31_ggaa(T165, T166, T169, X379, plusJ_out_gga(T166, T169, X379)) → pS_out_ggaa(T165, T166, T169, X379)
U14_ggga(T165, T166, X379, pS_out_ggaa(T165, T166, X378, X379)) → ifQ_out_ggga(false, T165, T166, X379)
U25_gaga(T100, T102, T101, X154, ifQ_out_ggga(T102, T100, T101, X154)) → pH_out_gaga(T100, T102, T101, X154)
U6_gga(T100, T101, X154, pH_out_gaga(T100, X153, T101, X154)) → timesG_out_gga(s(T100), T101, X154)
U22_ggaa(T79, T76, T88, T78, timesG_out_gga(T79, T76, T88)) → U23_ggaa(T79, T76, T88, T78, plusP_in_ga(T88, T78))
plusP_in_ga(0, 0) → plusP_out_ga(0, 0)
plusP_in_ga(s(0), s(s(0))) → plusP_out_ga(s(0), s(s(0)))
plusP_in_ga(s(s(0)), s(s(s(s(0))))) → plusP_out_ga(s(s(0)), s(s(s(s(0)))))
plusP_in_ga(s(s(s(0))), s(s(s(s(s(s(0))))))) → plusP_out_ga(s(s(s(0))), s(s(s(s(s(s(0)))))))
plusP_in_ga(s(s(s(s(0)))), s(s(s(s(s(s(s(s(0))))))))) → plusP_out_ga(s(s(s(s(0)))), s(s(s(s(s(s(s(s(0)))))))))
plusP_in_ga(s(s(s(s(s(0))))), s(s(s(s(s(s(s(s(s(s(0))))))))))) → plusP_out_ga(s(s(s(s(s(0))))), s(s(s(s(s(s(s(s(s(s(0)))))))))))
plusP_in_ga(s(s(s(s(s(s(0)))))), s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))) → plusP_out_ga(s(s(s(s(s(s(0)))))), s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))
plusP_in_ga(s(s(s(s(s(s(s(0))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))) → plusP_out_ga(s(s(s(s(s(s(s(0))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))
plusP_in_ga(s(s(s(s(s(s(s(s(T240)))))))), s(s(s(s(s(s(s(s(T242))))))))) → U12_ga(T240, T242, plusD_in_gga(T240, s(s(s(s(s(s(s(s(T240)))))))), T242))
U12_ga(T240, T242, plusD_out_gga(T240, s(s(s(s(s(s(s(s(T240)))))))), T242)) → plusP_out_ga(s(s(s(s(s(s(s(s(T240)))))))), s(s(s(s(s(s(s(s(T242)))))))))
U23_ggaa(T79, T76, T88, T78, plusP_out_ga(T88, T78)) → pU_out_ggaa(T79, T76, T88, T78)
U21_gagaa(T75, T79, T76, X113, T78, pU_out_ggaa(T79, T76, X113, T78)) → pM_out_gagaa(T75, T79, T76, X113, T78)
U9_ggga(T75, T76, T78, pM_out_gagaa(T75, X112, T76, X113, T78)) → ifL_out_ggga(true, T75, T76, T78)
ifL_in_ggga(false, T252, T253, T255) → U10_ggga(T252, T253, T255, pN_in_ggaa(T252, T253, X547, T255))
pN_in_ggaa(T252, T253, T256, T255) → U32_ggaa(T252, T253, T256, T255, timesG_in_gga(s(T252), T253, T256))
U32_ggaa(T252, T253, T256, T255, timesG_out_gga(s(T252), T253, T256)) → U33_ggaa(T252, T253, T256, T255, plusD_in_gga(T253, T256, T255))
U33_ggaa(T252, T253, T256, T255, plusD_out_gga(T253, T256, T255)) → pN_out_ggaa(T252, T253, T256, T255)
U10_ggga(T252, T253, T255, pN_out_ggaa(T252, T253, X547, T255)) → ifL_out_ggga(false, T252, T253, T255)
U19_gaga(T55, T56, T10, T12, ifL_out_ggga(T56, T55, T10, T12)) → pC_out_gaga(T55, T56, T10, T12)
U2_gga(T55, T10, T12, pC_out_gaga(T55, X78, T10, T12)) → timesA_out_gga(s(s(T55)), T10, T12)

The argument filtering Pi contains the following mapping:
timesA_in_gga(x1, x2, x3)  =  timesA_in_gga(x1, x2)
0  =  0
timesA_out_gga(x1, x2, x3)  =  timesA_out_gga(x1, x2, x3)
s(x1)  =  s(x1)
U1_gga(x1, x2, x3)  =  U1_gga(x1, x3)
pB_in_gaa(x1, x2, x3)  =  pB_in_gaa(x1)
U16_gaa(x1, x2, x3, x4)  =  U16_gaa(x1, x4)
timesK_in_ga(x1, x2)  =  timesK_in_ga(x1)
timesK_out_ga(x1, x2)  =  timesK_out_ga(x1, x2)
U17_gaa(x1, x2, x3, x4)  =  U17_gaa(x1, x2, x4)
plusD_in_gga(x1, x2, x3)  =  plusD_in_gga(x1, x2)
plusD_out_gga(x1, x2, x3)  =  plusD_out_gga(x1, x2, x3)
U3_gga(x1, x2, x3, x4)  =  U3_gga(x1, x2, x4)
pB_out_gaa(x1, x2, x3)  =  pB_out_gaa(x1, x2, x3)
U2_gga(x1, x2, x3, x4)  =  U2_gga(x1, x2, x4)
pC_in_gaga(x1, x2, x3, x4)  =  pC_in_gaga(x1, x3)
U18_gaga(x1, x2, x3, x4, x5)  =  U18_gaga(x1, x3, x5)
evenE_in_ga(x1, x2)  =  evenE_in_ga(x1)
evenE_out_ga(x1, x2)  =  evenE_out_ga(x1, x2)
U4_ga(x1, x2, x3)  =  U4_ga(x1, x3)
U19_gaga(x1, x2, x3, x4, x5)  =  U19_gaga(x1, x2, x3, x5)
ifL_in_ggga(x1, x2, x3, x4)  =  ifL_in_ggga(x1, x2, x3)
true  =  true
U9_ggga(x1, x2, x3, x4)  =  U9_ggga(x1, x2, x4)
pM_in_gagaa(x1, x2, x3, x4, x5)  =  pM_in_gagaa(x1, x3)
U20_gagaa(x1, x2, x3, x4, x5, x6)  =  U20_gagaa(x1, x3, x6)
halfO_in_ga(x1, x2)  =  halfO_in_ga(x1)
U11_ga(x1, x2, x3)  =  U11_ga(x1, x3)
halfF_in_ga(x1, x2)  =  halfF_in_ga(x1)
halfF_out_ga(x1, x2)  =  halfF_out_ga(x1, x2)
U5_ga(x1, x2, x3)  =  U5_ga(x1, x3)
halfO_out_ga(x1, x2)  =  halfO_out_ga(x1, x2)
U21_gagaa(x1, x2, x3, x4, x5, x6)  =  U21_gagaa(x1, x2, x3, x6)
pU_in_ggaa(x1, x2, x3, x4)  =  pU_in_ggaa(x1, x2)
U22_ggaa(x1, x2, x3, x4, x5)  =  U22_ggaa(x1, x2, x5)
timesG_in_gga(x1, x2, x3)  =  timesG_in_gga(x1, x2)
timesG_out_gga(x1, x2, x3)  =  timesG_out_gga(x1, x2, x3)
U6_gga(x1, x2, x3, x4)  =  U6_gga(x1, x2, x4)
pH_in_gaga(x1, x2, x3, x4)  =  pH_in_gaga(x1, x3)
U24_gaga(x1, x2, x3, x4, x5)  =  U24_gaga(x1, x3, x5)
U25_gaga(x1, x2, x3, x4, x5)  =  U25_gaga(x1, x2, x3, x5)
ifQ_in_ggga(x1, x2, x3, x4)  =  ifQ_in_ggga(x1, x2, x3)
U13_ggga(x1, x2, x3, x4)  =  U13_ggga(x1, x2, x4)
pR_in_gagaa(x1, x2, x3, x4, x5)  =  pR_in_gagaa(x1, x3)
U26_gagaa(x1, x2, x3, x4, x5, x6)  =  U26_gagaa(x1, x3, x6)
U27_gagaa(x1, x2, x3, x4, x5, x6)  =  U27_gagaa(x1, x2, x3, x6)
pV_in_ggaa(x1, x2, x3, x4)  =  pV_in_ggaa(x1, x2)
U28_ggaa(x1, x2, x3, x4, x5)  =  U28_ggaa(x1, x2, x5)
U29_ggaa(x1, x2, x3, x4, x5)  =  U29_ggaa(x1, x2, x3, x5)
plusT_in_ga(x1, x2)  =  plusT_in_ga(x1)
plusT_out_ga(x1, x2)  =  plusT_out_ga(x1, x2)
U15_ga(x1, x2, x3)  =  U15_ga(x1, x3)
plusI_in_gga(x1, x2, x3)  =  plusI_in_gga(x1, x2)
plusI_out_gga(x1, x2, x3)  =  plusI_out_gga(x1, x2, x3)
U7_gga(x1, x2, x3, x4)  =  U7_gga(x1, x2, x4)
pV_out_ggaa(x1, x2, x3, x4)  =  pV_out_ggaa(x1, x2, x3, x4)
pR_out_gagaa(x1, x2, x3, x4, x5)  =  pR_out_gagaa(x1, x2, x3, x4, x5)
ifQ_out_ggga(x1, x2, x3, x4)  =  ifQ_out_ggga(x1, x2, x3, x4)
false  =  false
U14_ggga(x1, x2, x3, x4)  =  U14_ggga(x1, x2, x4)
pS_in_ggaa(x1, x2, x3, x4)  =  pS_in_ggaa(x1, x2)
U30_ggaa(x1, x2, x3, x4, x5)  =  U30_ggaa(x1, x2, x5)
U31_ggaa(x1, x2, x3, x4, x5)  =  U31_ggaa(x1, x2, x3, x5)
plusJ_in_gga(x1, x2, x3)  =  plusJ_in_gga(x1, x2)
plusJ_out_gga(x1, x2, x3)  =  plusJ_out_gga(x1, x2, x3)
U8_gga(x1, x2, x3, x4)  =  U8_gga(x1, x2, x4)
pS_out_ggaa(x1, x2, x3, x4)  =  pS_out_ggaa(x1, x2, x3, x4)
pH_out_gaga(x1, x2, x3, x4)  =  pH_out_gaga(x1, x2, x3, x4)
U23_ggaa(x1, x2, x3, x4, x5)  =  U23_ggaa(x1, x2, x3, x5)
plusP_in_ga(x1, x2)  =  plusP_in_ga(x1)
plusP_out_ga(x1, x2)  =  plusP_out_ga(x1, x2)
U12_ga(x1, x2, x3)  =  U12_ga(x1, x3)
pU_out_ggaa(x1, x2, x3, x4)  =  pU_out_ggaa(x1, x2, x3, x4)
pM_out_gagaa(x1, x2, x3, x4, x5)  =  pM_out_gagaa(x1, x2, x3, x4, x5)
ifL_out_ggga(x1, x2, x3, x4)  =  ifL_out_ggga(x1, x2, x3, x4)
U10_ggga(x1, x2, x3, x4)  =  U10_ggga(x1, x2, x4)
pN_in_ggaa(x1, x2, x3, x4)  =  pN_in_ggaa(x1, x2)
U32_ggaa(x1, x2, x3, x4, x5)  =  U32_ggaa(x1, x2, x5)
U33_ggaa(x1, x2, x3, x4, x5)  =  U33_ggaa(x1, x2, x3, x5)
pN_out_ggaa(x1, x2, x3, x4)  =  pN_out_ggaa(x1, x2, x3, x4)
pC_out_gaga(x1, x2, x3, x4)  =  pC_out_gaga(x1, x2, x3, x4)
PLUSD_IN_GGA(x1, x2, x3)  =  PLUSD_IN_GGA(x1, x2)

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

(51) UsableRulesProof (EQUIVALENT transformation)

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

(52) Obligation:

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

PLUSD_IN_GGA(s(T47), T48, s(T50)) → PLUSD_IN_GGA(T47, T48, T50)

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

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

(53) PiDPToQDPProof (SOUND transformation)

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

(54) Obligation:

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

PLUSD_IN_GGA(s(T47), T48) → PLUSD_IN_GGA(T47, T48)

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

(55) 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:

• PLUSD_IN_GGA(s(T47), T48) → PLUSD_IN_GGA(T47, T48)
  The graph contains the following edges 1 > 1, 2 >= 2

(56) YES