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

(0) Obligation:

Clauses:

parse(Xs, T) :- ','(app(As, .(a, .(s(A, B, C), .(b, Bs))), Xs), ','(app(As, .(s(a, s(A, B, C), b), Bs), Ys), parse(Ys, T))).
parse(Xs, T) :- ','(app(As, .(a, .(s(A, B), .(b, Bs))), Xs), ','(app(As, .(s(a, s(A, B), b), Bs), Ys), parse(Ys, T))).
parse(Xs, T) :- ','(app(As, .(a, .(b, Bs)), Xs), ','(app(As, .(s(a, b), Bs), Ys), parse(Ys, T))).
parse(.(s(A, B), []), s(A, B)).
parse(.(s(A, B, C), []), s(A, B, C)).
app([], X, X).
app(.(X, Xs), Ys, .(X, Zs)) :- app(Xs, Ys, Zs).

Query: parse(g,a)

(1) PrologToPiTRSViaGraphTransformerProof (SOUND transformation)

Transformed Prolog program to (Pi-)TRS.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="A: parse(T1, T2)\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
5 [label="5: parse(T1, T2), SCOPE: 1, CLAUSE: 0 | parse(T1, T2), SCOPE: 1, CLAUSE: 1 | parse(T1, T2), SCOPE: 1, CLAUSE: 2 | parse(T1, T2), SCOPE: 1, CLAUSE: 3 | parse(T1, T2), SCOPE: 1, CLAUSE: 4\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: ','(app(X5, .(a, .(s(X6, X7, X8), .(b, X9))), T5), ','(app(X5, .(s(a, s(X6, X7, X8), b), X9), X10), parse(X10, T7))) | parse(T5, T2), SCOPE: 1, CLAUSE: 1 | parse(T5, T2), SCOPE: 1, CLAUSE: 2 | parse(T5, T2), SCOPE: 1, CLAUSE: 3 | parse(T5, T2), SCOPE: 1, CLAUSE: 4\n\nT5 is ground\nX5 is free\nX6 is free\nX7 is free\nX8 is free\nX9 is free\nX10 is free", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: ','(app(X5, .(a, .(s(X6, X7, X8), .(b, X9))), T5), ','(app(X5, .(s(a, s(X6, X7, X8), b), X9), X10), parse(X10, T7))), SCOPE: 2, CLAUSE: 5 | ','(app(X5, .(a, .(s(X6, X7, X8), .(b, X9))), T5), ','(app(X5, .(s(a, s(X6, X7, X8), b), X9), X10), parse(X10, T7))), SCOPE: 2, CLAUSE: 6 | ?_2 | parse(T5, T2), SCOPE: 1, CLAUSE: 1 | parse(T5, T2), SCOPE: 1, CLAUSE: 2 | parse(T5, T2), SCOPE: 1, CLAUSE: 3 | parse(T5, T2), SCOPE: 1, CLAUSE: 4\n\nT5 is ground\nX5 is free\nX6 is free\nX7 is free\nX8 is free\nX9 is free\nX10 is free", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: ','(app(X5, .(a, .(s(X6, X7, X8), .(b, X9))), T5), ','(app(X5, .(s(a, s(X6, X7, X8), b), X9), X10), parse(X10, T7))), SCOPE: 2, CLAUSE: 5\n\nT5 is ground\nX5 is free\nX6 is free\nX7 is free\nX8 is free\nX9 is free\nX10 is free", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: ','(app(X5, .(a, .(s(X6, X7, X8), .(b, X9))), T5), ','(app(X5, .(s(a, s(X6, X7, X8), b), X9), X10), parse(X10, T7))), SCOPE: 2, CLAUSE: 6 | ?_2 | parse(T5, T2), SCOPE: 1, CLAUSE: 1 | parse(T5, T2), SCOPE: 1, CLAUSE: 2 | parse(T5, T2), SCOPE: 1, CLAUSE: 3 | parse(T5, T2), SCOPE: 1, CLAUSE: 4\n\nT5 is ground\nX5 is free\nX6 is free\nX7 is free\nX8 is free\nX9 is free\nX10 is free", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="B: ','(app([], .(s(a, s(T24, T25, T26), b), T27), X10), parse(X10, T7))\n\nT24 is ground\nT25 is ground\nT26 is ground\nT27 is ground\nX10 is free", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="L: app([], .(s(a, s(T24, T25, T26), b), T27), X10)\n\nT24 is ground\nT25 is ground\nT26 is ground\nT27 is ground\nX10 is free", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: parse(T33, T7)\n\nT33 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: app([], .(s(a, s(T24, T25, T26), b), T27), X10), SCOPE: 3, CLAUSE: 5 | app([], .(s(a, s(T24, T25, T26), b), T27), X10), SCOPE: 3, CLAUSE: 6\n\nT24 is ground\nT25 is ground\nT26 is ground\nT27 is ground\nX10 is free", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: app([], .(s(a, s(T24, T25, T26), b), T27), X10), SCOPE: 3, CLAUSE: 5\n\nT24 is ground\nT25 is ground\nT26 is ground\nT27 is ground\nX10 is free", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: app([], .(s(a, s(T24, T25, T26), b), T27), X10), SCOPE: 3, CLAUSE: 6\n\nT24 is ground\nT25 is ground\nT26 is ground\nT27 is ground\nX10 is free", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="38: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
39 [label="39: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
40 [label="40: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
59 [label="59: ','(app(X5, .(a, .(s(X6, X7, X8), .(b, X9))), T5), ','(app(X5, .(s(a, s(X6, X7, X8), b), X9), X10), parse(X10, T7))), SCOPE: 2, CLAUSE: 6\n\nT5 is ground\nX5 is free\nX6 is free\nX7 is free\nX8 is free\nX9 is free\nX10 is free", fontsize=16, style=filled, fillcolor=lightyellow];
60 [label="60: ?_2 | parse(T5, T2), SCOPE: 1, CLAUSE: 1 | parse(T5, T2), SCOPE: 1, CLAUSE: 2 | parse(T5, T2), SCOPE: 1, CLAUSE: 3 | parse(T5, T2), SCOPE: 1, CLAUSE: 4\n\nT5 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
62 [label="C: ','(app(X109, .(a, .(s(X110, X111, X112), .(b, X113))), T69), ','(app(.(T68, X109), .(s(a, s(X110, X111, X112), b), X113), X10), parse(X10, T7)))\n\nT68 is ground\nT69 is ground\nX10 is free\nX109 is free\nX110 is free\nX111 is free\nX112 is free\nX113 is free", fontsize=16, style=filled, fillcolor=lightyellow];
64 [label="64: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
70 [label="F: app(X109, .(a, .(s(X110, X111, X112), .(b, X113))), T69)\n\nT69 is ground\nX109 is free\nX110 is free\nX111 is free\nX112 is free\nX113 is free", fontsize=16, style=filled, fillcolor=lightyellow];
71 [label="N: ','(app(.(T68, T78), .(s(a, s(T79, T80, T81), b), T82), X10), parse(X10, T7))\n\nT68 is ground\nT78 is ground\nT79 is ground\nT80 is ground\nT81 is ground\nT82 is ground\nX10 is free", fontsize=16, style=filled, fillcolor=lightyellow];
73 [label="73: app(X109, .(a, .(s(X110, X111, X112), .(b, X113))), T69), SCOPE: 4, CLAUSE: 5 | app(X109, .(a, .(s(X110, X111, X112), .(b, X113))), T69), SCOPE: 4, CLAUSE: 6\n\nT69 is ground\nX109 is free\nX110 is free\nX111 is free\nX112 is free\nX113 is free", fontsize=16, style=filled, fillcolor=lightyellow];
74 [label="74: app(X109, .(a, .(s(X110, X111, X112), .(b, X113))), T69), SCOPE: 4, CLAUSE: 5\n\nT69 is ground\nX109 is free\nX110 is free\nX111 is free\nX112 is free\nX113 is free", fontsize=16, style=filled, fillcolor=lightyellow];
75 [label="75: app(X109, .(a, .(s(X110, X111, X112), .(b, X113))), T69), SCOPE: 4, CLAUSE: 6\n\nT69 is ground\nX109 is free\nX110 is free\nX111 is free\nX112 is free\nX113 is free", fontsize=16, style=filled, fillcolor=lightyellow];
90 [label="90: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
91 [label="91: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
92 [label="92: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
95 [label="95: app(X184, .(a, .(s(X185, X186, X187), .(b, X188))), T116)\n\nT116 is ground\nX184 is free\nX185 is free\nX186 is free\nX187 is free\nX188 is free", fontsize=16, style=filled, fillcolor=lightyellow];
96 [label="96: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
111 [label="M: app(.(T68, T78), .(s(a, s(T79, T80, T81), b), T82), X10)\n\nT68 is ground\nT78 is ground\nT79 is ground\nT80 is ground\nT81 is ground\nT82 is ground\nX10 is free", fontsize=16, style=filled, fillcolor=lightyellow];
112 [label="112: parse(T125, T7)\n\nT125 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
113 [label="113: app(.(T68, T78), .(s(a, s(T79, T80, T81), b), T82), X10), SCOPE: 5, CLAUSE: 5 | app(.(T68, T78), .(s(a, s(T79, T80, T81), b), T82), X10), SCOPE: 5, CLAUSE: 6\n\nT68 is ground\nT78 is ground\nT79 is ground\nT80 is ground\nT81 is ground\nT82 is ground\nX10 is free", fontsize=16, style=filled, fillcolor=lightyellow];
114 [label="114: app(.(T68, T78), .(s(a, s(T79, T80, T81), b), T82), X10), SCOPE: 5, CLAUSE: 6\n\nT68 is ground\nT78 is ground\nT79 is ground\nT80 is ground\nT81 is ground\nT82 is ground\nX10 is free", fontsize=16, style=filled, fillcolor=lightyellow];
118 [label="G: app(T147, .(s(a, s(T148, T149, T150), b), T151), X220)\n\nT147 is ground\nT148 is ground\nT149 is ground\nT150 is ground\nT151 is ground\nX220 is free", fontsize=16, style=filled, fillcolor=lightyellow];
121 [label="121: app(T147, .(s(a, s(T148, T149, T150), b), T151), X220), SCOPE: 6, CLAUSE: 5 | app(T147, .(s(a, s(T148, T149, T150), b), T151), X220), SCOPE: 6, CLAUSE: 6\n\nT147 is ground\nT148 is ground\nT149 is ground\nT150 is ground\nT151 is ground\nX220 is free", fontsize=16, style=filled, fillcolor=lightyellow];
122 [label="122: app(T147, .(s(a, s(T148, T149, T150), b), T151), X220), SCOPE: 6, CLAUSE: 5\n\nT147 is ground\nT148 is ground\nT149 is ground\nT150 is ground\nT151 is ground\nX220 is free", fontsize=16, style=filled, fillcolor=lightyellow];
123 [label="123: app(T147, .(s(a, s(T148, T149, T150), b), T151), X220), SCOPE: 6, CLAUSE: 6\n\nT147 is ground\nT148 is ground\nT149 is ground\nT150 is ground\nT151 is ground\nX220 is free", fontsize=16, style=filled, fillcolor=lightyellow];
128 [label="128: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
129 [label="129: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
130 [label="130: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
134 [label="134: app(T193, .(s(a, s(T194, T195, T196), b), T197), X242)\n\nT193 is ground\nT194 is ground\nT195 is ground\nT196 is ground\nT197 is ground\nX242 is free", fontsize=16, style=filled, fillcolor=lightyellow];
135 [label="135: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
144 [label="144: parse(T5, T2), SCOPE: 1, CLAUSE: 1 | parse(T5, T2), SCOPE: 1, CLAUSE: 2 | parse(T5, T2), SCOPE: 1, CLAUSE: 3 | parse(T5, T2), SCOPE: 1, CLAUSE: 4\n\nT5 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
149 [label="149: parse(T5, T2), SCOPE: 1, CLAUSE: 1\n\nT5 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
150 [label="150: parse(T5, T2), SCOPE: 1, CLAUSE: 2 | parse(T5, T2), SCOPE: 1, CLAUSE: 3 | parse(T5, T2), SCOPE: 1, CLAUSE: 4\n\nT5 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
152 [label="D: ','(app(X289, .(a, .(s(X290, X291), .(b, X292))), T230), ','(app(X289, .(s(a, s(X290, X291), b), X292), X293), parse(X293, T232)))\n\nT230 is ground\nX289 is free\nX290 is free\nX291 is free\nX292 is free\nX293 is free", fontsize=16, style=filled, fillcolor=lightyellow];
155 [label="H: app(X289, .(a, .(s(X290, X291), .(b, X292))), T230)\n\nT230 is ground\nX289 is free\nX290 is free\nX291 is free\nX292 is free", fontsize=16, style=filled, fillcolor=lightyellow];
156 [label="O: ','(app(T239, .(s(a, s(T240, T241), b), T242), X293), parse(X293, T232))\n\nT239 is ground\nT240 is ground\nT241 is ground\nT242 is ground\nX293 is free", fontsize=16, style=filled, fillcolor=lightyellow];
159 [label="159: app(X289, .(a, .(s(X290, X291), .(b, X292))), T230), SCOPE: 7, CLAUSE: 5 | app(X289, .(a, .(s(X290, X291), .(b, X292))), T230), SCOPE: 7, CLAUSE: 6\n\nT230 is ground\nX289 is free\nX290 is free\nX291 is free\nX292 is free", fontsize=16, style=filled, fillcolor=lightyellow];
163 [label="163: app(X289, .(a, .(s(X290, X291), .(b, X292))), T230), SCOPE: 7, CLAUSE: 5\n\nT230 is ground\nX289 is free\nX290 is free\nX291 is free\nX292 is free", fontsize=16, style=filled, fillcolor=lightyellow];
164 [label="164: app(X289, .(a, .(s(X290, X291), .(b, X292))), T230), SCOPE: 7, CLAUSE: 6\n\nT230 is ground\nX289 is free\nX290 is free\nX291 is free\nX292 is free", fontsize=16, style=filled, fillcolor=lightyellow];
169 [label="169: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
170 [label="170: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
171 [label="171: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
176 [label="176: app(X353, .(a, .(s(X354, X355), .(b, X356))), T269)\n\nT269 is ground\nX353 is free\nX354 is free\nX355 is free\nX356 is free", fontsize=16, style=filled, fillcolor=lightyellow];
177 [label="177: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
183 [label="I: app(T239, .(s(a, s(T240, T241), b), T242), X293)\n\nT239 is ground\nT240 is ground\nT241 is ground\nT242 is ground\nX293 is free", fontsize=16, style=filled, fillcolor=lightyellow];
184 [label="184: parse(T282, T232)\n\nT282 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
194 [label="194: app(T239, .(s(a, s(T240, T241), b), T242), X293), SCOPE: 8, CLAUSE: 5 | app(T239, .(s(a, s(T240, T241), b), T242), X293), SCOPE: 8, CLAUSE: 6\n\nT239 is ground\nT240 is ground\nT241 is ground\nT242 is ground\nX293 is free", fontsize=16, style=filled, fillcolor=lightyellow];
195 [label="195: app(T239, .(s(a, s(T240, T241), b), T242), X293), SCOPE: 8, CLAUSE: 5\n\nT239 is ground\nT240 is ground\nT241 is ground\nT242 is ground\nX293 is free", fontsize=16, style=filled, fillcolor=lightyellow];
196 [label="196: app(T239, .(s(a, s(T240, T241), b), T242), X293), SCOPE: 8, CLAUSE: 6\n\nT239 is ground\nT240 is ground\nT241 is ground\nT242 is ground\nX293 is free", fontsize=16, style=filled, fillcolor=lightyellow];
197 [label="197: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
198 [label="198: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
199 [label="199: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
205 [label="205: app(T315, .(s(a, s(T316, T317), b), T318), X388)\n\nT315 is ground\nT316 is ground\nT317 is ground\nT318 is ground\nX388 is free", fontsize=16, style=filled, fillcolor=lightyellow];
206 [label="206: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
213 [label="213: parse(T5, T2), SCOPE: 1, CLAUSE: 2\n\nT5 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
214 [label="214: parse(T5, T2), SCOPE: 1, CLAUSE: 3 | parse(T5, T2), SCOPE: 1, CLAUSE: 4\n\nT5 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
218 [label="E: ','(app(X423, .(a, .(b, X424)), T341), ','(app(X423, .(s(a, b), X424), X425), parse(X425, T343)))\n\nT341 is ground\nX423 is free\nX424 is free\nX425 is free", fontsize=16, style=filled, fillcolor=lightyellow];
219 [label="J: app(X423, .(a, .(b, X424)), T341)\n\nT341 is ground\nX423 is free\nX424 is free", fontsize=16, style=filled, fillcolor=lightyellow];
220 [label="P: ','(app(T346, .(s(a, b), T347), X425), parse(X425, T343))\n\nT346 is ground\nT347 is ground\nX425 is free", fontsize=16, style=filled, fillcolor=lightyellow];
221 [label="221: app(X423, .(a, .(b, X424)), T341), SCOPE: 9, CLAUSE: 5 | app(X423, .(a, .(b, X424)), T341), SCOPE: 9, CLAUSE: 6\n\nT341 is ground\nX423 is free\nX424 is free", fontsize=16, style=filled, fillcolor=lightyellow];
222 [label="222: app(X423, .(a, .(b, X424)), T341), SCOPE: 9, CLAUSE: 5\n\nT341 is ground\nX423 is free\nX424 is free", fontsize=16, style=filled, fillcolor=lightyellow];
223 [label="223: app(X423, .(a, .(b, X424)), T341), SCOPE: 9, CLAUSE: 6\n\nT341 is ground\nX423 is free\nX424 is free", fontsize=16, style=filled, fillcolor=lightyellow];
224 [label="224: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
225 [label="225: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
226 [label="226: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
231 [label="231: app(X463, .(a, .(b, X464)), T360)\n\nT360 is ground\nX463 is free\nX464 is free", fontsize=16, style=filled, fillcolor=lightyellow];
232 [label="232: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
238 [label="K: app(T346, .(s(a, b), T347), X425)\n\nT346 is ground\nT347 is ground\nX425 is free", fontsize=16, style=filled, fillcolor=lightyellow];
239 [label="239: parse(T365, T343)\n\nT365 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
240 [label="240: app(T346, .(s(a, b), T347), X425), SCOPE: 10, CLAUSE: 5 | app(T346, .(s(a, b), T347), X425), SCOPE: 10, CLAUSE: 6\n\nT346 is ground\nT347 is ground\nX425 is free", fontsize=16, style=filled, fillcolor=lightyellow];
243 [label="243: app(T346, .(s(a, b), T347), X425), SCOPE: 10, CLAUSE: 5\n\nT346 is ground\nT347 is ground\nX425 is free", fontsize=16, style=filled, fillcolor=lightyellow];
244 [label="244: app(T346, .(s(a, b), T347), X425), SCOPE: 10, CLAUSE: 6\n\nT346 is ground\nT347 is ground\nX425 is free", fontsize=16, style=filled, fillcolor=lightyellow];
245 [label="245: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
246 [label="246: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
247 [label="247: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
250 [label="250: app(T380, .(s(a, b), T381), X492)\n\nT380 is ground\nT381 is ground\nX492 is free", fontsize=16, style=filled, fillcolor=lightyellow];
251 [label="251: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
259 [label="259: parse(T5, T2), SCOPE: 1, CLAUSE: 3\n\nT5 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
260 [label="260: parse(T5, T2), SCOPE: 1, CLAUSE: 4\n\nT5 is ground", 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];
267 [label="267: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
268 [label="268: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
269 [label="269: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
270 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 270 [arrowhead = none ];
270 -> 5;

271 [label="ONLY EVAL with clause\nparse(X3, X4) :- ','(app(X5, .(a, .(s(X6, X7, X8), .(b, X9))), X3), ','(app(X5, .(s(a, s(X6, X7, X8), b), X9), X10), parse(X10, X4))).\nand substitutionT1 -> T5,\nX3 -> T5,\nT2 -> T7,\nX4 -> T7,\nT6 -> T7", fontsize=14, style = filled, fillcolor = lightblue];
5 -> 271 [arrowhead = none ];
271 -> 7;

272 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
7 -> 272 [arrowhead = none ];
272 -> 8;

273 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
8 -> 273 [arrowhead = none ];
273 -> 13;

274 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
8 -> 274 [arrowhead = none ];
274 -> 14;

275 [label="EVAL with clause\napp([], X31, X31).\nand substitutionX5 -> [],\nX6 -> T24,\nX7 -> T25,\nX8 -> T26,\nX9 -> T27,\nX31 -> .(a, .(s(T24, T25, T26), .(b, T27))),\nX32 -> T24,\nX33 -> T25,\nX34 -> T26,\nX35 -> T27,\nT5 -> .(a, .(s(T24, T25, T26), .(b, T27)))", fontsize=14, style = filled, fillcolor = lightblue];
13 -> 275 [arrowhead = none ];
275 -> 16;

276 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
13 -> 276 [arrowhead = none ];
276 -> 21;

277 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
14 -> 277 [arrowhead = none ];
277 -> 59;

278 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
14 -> 278 [arrowhead = none ];
278 -> 60;

279 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
16 -> 279 [arrowhead = none ];
279 -> 25;

280 [label="SPLIT 2\nnew knowledge:\nT24 is ground\nT25 is ground\nT26 is ground\nT27 is ground\nT33 is ground\nreplacements:X10 -> T33", fontsize=14, style = filled, fillcolor = violet];
16 -> 280 [arrowhead = none ];
280 -> 26;

281 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
25 -> 281 [arrowhead = none ];
281 -> 31;

282 [label="INSTANCE with matching:\nT1 -> T33\nT2 -> T7", fontsize=14, style = filled, fillcolor = lightgrey];
26 -> 282 [arrowhead = none , style=dashed];
282 -> 1[style=dashed];

283 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
31 -> 283 [arrowhead = none ];
283 -> 36;

284 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
31 -> 284 [arrowhead = none ];
284 -> 37;

285 [label="ONLY EVAL with clause\napp([], X51, X51).\nand substitutionT24 -> T54,\nT25 -> T55,\nT26 -> T56,\nT27 -> T57,\nX51 -> .(s(a, s(T54, T55, T56), b), T57),\nX10 -> .(s(a, s(T54, T55, T56), b), T57)", fontsize=14, style = filled, fillcolor = lightblue];
36 -> 285 [arrowhead = none ];
285 -> 38;

286 [label="BACKTRACK\nfor clause: app(.(X, Xs), Ys, .(X, Zs)) :- app(Xs, Ys, Zs)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
37 -> 286 [arrowhead = none ];
286 -> 40;

287 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
38 -> 287 [arrowhead = none ];
287 -> 39;

288 [label="EVAL with clause\napp(.(X104, X105), X106, .(X104, X107)) :- app(X105, X106, X107).\nand substitutionX104 -> T68,\nX105 -> X109,\nX5 -> .(T68, X109),\nX6 -> X110,\nX7 -> X111,\nX8 -> X112,\nX9 -> X113,\nX106 -> .(a, .(s(X110, X111, X112), .(b, X113))),\nX108 -> T68,\nX107 -> T69,\nT5 -> .(T68, T69)", fontsize=14, style = filled, fillcolor = lightblue];
59 -> 288 [arrowhead = none ];
288 -> 62;

289 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
59 -> 289 [arrowhead = none ];
289 -> 64;

290 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
60 -> 290 [arrowhead = none ];
290 -> 144;

291 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
62 -> 291 [arrowhead = none ];
291 -> 70;

292 [label="SPLIT 2\nnew knowledge:\nT78 is ground\nT79 is ground\nT80 is ground\nT81 is ground\nT82 is ground\nT69 is ground\nreplacements:X109 -> T78,\nX110 -> T79,\nX111 -> T80,\nX112 -> T81,\nX113 -> T82", fontsize=14, style = filled, fillcolor = violet];
62 -> 292 [arrowhead = none ];
292 -> 71;

293 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
70 -> 293 [arrowhead = none ];
293 -> 73;

294 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
71 -> 294 [arrowhead = none ];
294 -> 111;

295 [label="SPLIT 2\nnew knowledge:\nT68 is ground\nT78 is ground\nT79 is ground\nT80 is ground\nT81 is ground\nT82 is ground\nT125 is ground\nreplacements:X10 -> T125", fontsize=14, style = filled, fillcolor = violet];
71 -> 295 [arrowhead = none ];
295 -> 112;

296 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
73 -> 296 [arrowhead = none ];
296 -> 74;

297 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
73 -> 297 [arrowhead = none ];
297 -> 75;

298 [label="EVAL with clause\napp([], X154, X154).\nand substitutionX109 -> [],\nX110 -> T107,\nX111 -> T108,\nX112 -> T109,\nX113 -> T110,\nX154 -> .(a, .(s(T107, T108, T109), .(b, T110))),\nX155 -> T107,\nX156 -> T108,\nX157 -> T109,\nX158 -> T110,\nT69 -> .(a, .(s(T107, T108, T109), .(b, T110)))", fontsize=14, style = filled, fillcolor = lightblue];
74 -> 298 [arrowhead = none ];
298 -> 90;

299 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
74 -> 299 [arrowhead = none ];
299 -> 91;

300 [label="EVAL with clause\napp(.(X179, X180), X181, .(X179, X182)) :- app(X180, X181, X182).\nand substitutionX179 -> T115,\nX180 -> X184,\nX109 -> .(T115, X184),\nX110 -> X185,\nX111 -> X186,\nX112 -> X187,\nX113 -> X188,\nX181 -> .(a, .(s(X185, X186, X187), .(b, X188))),\nX183 -> T115,\nX182 -> T116,\nT69 -> .(T115, T116)", fontsize=14, style = filled, fillcolor = lightblue];
75 -> 300 [arrowhead = none ];
300 -> 95;

301 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
75 -> 301 [arrowhead = none ];
301 -> 96;

302 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
90 -> 302 [arrowhead = none ];
302 -> 92;

303 [label="INSTANCE with matching:\nX109 -> X184\nX110 -> X185\nX111 -> X186\nX112 -> X187\nX113 -> X188\nT69 -> T116", fontsize=14, style = filled, fillcolor = lightgrey];
95 -> 303 [arrowhead = none , style=dashed];
303 -> 70[style=dashed];

304 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
111 -> 304 [arrowhead = none ];
304 -> 113;

305 [label="INSTANCE with matching:\nT1 -> T125\nT2 -> T7", fontsize=14, style = filled, fillcolor = lightgrey];
112 -> 305 [arrowhead = none , style=dashed];
305 -> 1[style=dashed];

306 [label="BACKTRACK\nfor clause: app([], X, X)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
113 -> 306 [arrowhead = none ];
306 -> 114;

307 [label="ONLY EVAL with clause\napp(.(X216, X217), X218, .(X216, X219)) :- app(X217, X218, X219).\nand substitutionT68 -> T146,\nX216 -> T146,\nT78 -> T147,\nX217 -> T147,\nT79 -> T148,\nT80 -> T149,\nT81 -> T150,\nT82 -> T151,\nX218 -> .(s(a, s(T148, T149, T150), b), T151),\nX219 -> X220,\nX10 -> .(T146, X220)", fontsize=14, style = filled, fillcolor = lightblue];
114 -> 307 [arrowhead = none ];
307 -> 118;

308 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
118 -> 308 [arrowhead = none ];
308 -> 121;

309 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
121 -> 309 [arrowhead = none ];
309 -> 122;

310 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
121 -> 310 [arrowhead = none ];
310 -> 123;

311 [label="EVAL with clause\napp([], X227, X227).\nand substitutionT147 -> [],\nT148 -> T176,\nT149 -> T177,\nT150 -> T178,\nT151 -> T179,\nX227 -> .(s(a, s(T176, T177, T178), b), T179),\nX220 -> .(s(a, s(T176, T177, T178), b), T179)", fontsize=14, style = filled, fillcolor = lightblue];
122 -> 311 [arrowhead = none ];
311 -> 128;

312 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
122 -> 312 [arrowhead = none ];
312 -> 129;

313 [label="EVAL with clause\napp(.(X238, X239), X240, .(X238, X241)) :- app(X239, X240, X241).\nand substitutionX238 -> T192,\nX239 -> T193,\nT147 -> .(T192, T193),\nT148 -> T194,\nT149 -> T195,\nT150 -> T196,\nT151 -> T197,\nX240 -> .(s(a, s(T194, T195, T196), b), T197),\nX241 -> X242,\nX220 -> .(T192, X242)", fontsize=14, style = filled, fillcolor = lightblue];
123 -> 313 [arrowhead = none ];
313 -> 134;

314 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
123 -> 314 [arrowhead = none ];
314 -> 135;

315 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
128 -> 315 [arrowhead = none ];
315 -> 130;

316 [label="INSTANCE with matching:\nT147 -> T193\nT148 -> T194\nT149 -> T195\nT150 -> T196\nT151 -> T197\nX220 -> X242", fontsize=14, style = filled, fillcolor = lightgrey];
134 -> 316 [arrowhead = none , style=dashed];
316 -> 118[style=dashed];

317 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
144 -> 317 [arrowhead = none ];
317 -> 149;

318 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
144 -> 318 [arrowhead = none ];
318 -> 150;

319 [label="ONLY EVAL with clause\nparse(X287, X288) :- ','(app(X289, .(a, .(s(X290, X291), .(b, X292))), X287), ','(app(X289, .(s(a, s(X290, X291), b), X292), X293), parse(X293, X288))).\nand substitutionT5 -> T230,\nX287 -> T230,\nT2 -> T232,\nX288 -> T232,\nT231 -> T232", fontsize=14, style = filled, fillcolor = lightblue];
149 -> 319 [arrowhead = none ];
319 -> 152;

320 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
150 -> 320 [arrowhead = none ];
320 -> 213;

321 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
150 -> 321 [arrowhead = none ];
321 -> 214;

322 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
152 -> 322 [arrowhead = none ];
322 -> 155;

323 [label="SPLIT 2\nnew knowledge:\nT239 is ground\nT240 is ground\nT241 is ground\nT242 is ground\nT230 is ground\nreplacements:X289 -> T239,\nX290 -> T240,\nX291 -> T241,\nX292 -> T242", fontsize=14, style = filled, fillcolor = violet];
152 -> 323 [arrowhead = none ];
323 -> 156;

324 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
155 -> 324 [arrowhead = none ];
324 -> 159;

325 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
156 -> 325 [arrowhead = none ];
325 -> 183;

326 [label="SPLIT 2\nnew knowledge:\nT239 is ground\nT240 is ground\nT241 is ground\nT242 is ground\nT282 is ground\nreplacements:X293 -> T282", fontsize=14, style = filled, fillcolor = violet];
156 -> 326 [arrowhead = none ];
326 -> 184;

327 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
159 -> 327 [arrowhead = none ];
327 -> 163;

328 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
159 -> 328 [arrowhead = none ];
328 -> 164;

329 [label="EVAL with clause\napp([], X326, X326).\nand substitutionX289 -> [],\nX290 -> T261,\nX291 -> T262,\nX292 -> T263,\nX326 -> .(a, .(s(T261, T262), .(b, T263))),\nX327 -> T261,\nX328 -> T262,\nX329 -> T263,\nT230 -> .(a, .(s(T261, T262), .(b, T263)))", fontsize=14, style = filled, fillcolor = lightblue];
163 -> 329 [arrowhead = none ];
329 -> 169;

330 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
163 -> 330 [arrowhead = none ];
330 -> 170;

331 [label="EVAL with clause\napp(.(X348, X349), X350, .(X348, X351)) :- app(X349, X350, X351).\nand substitutionX348 -> T268,\nX349 -> X353,\nX289 -> .(T268, X353),\nX290 -> X354,\nX291 -> X355,\nX292 -> X356,\nX350 -> .(a, .(s(X354, X355), .(b, X356))),\nX352 -> T268,\nX351 -> T269,\nT230 -> .(T268, T269)", fontsize=14, style = filled, fillcolor = lightblue];
164 -> 331 [arrowhead = none ];
331 -> 176;

332 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
164 -> 332 [arrowhead = none ];
332 -> 177;

333 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
169 -> 333 [arrowhead = none ];
333 -> 171;

334 [label="INSTANCE with matching:\nX289 -> X353\nX290 -> X354\nX291 -> X355\nX292 -> X356\nT230 -> T269", fontsize=14, style = filled, fillcolor = lightgrey];
176 -> 334 [arrowhead = none , style=dashed];
334 -> 155[style=dashed];

335 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
183 -> 335 [arrowhead = none ];
335 -> 194;

336 [label="INSTANCE with matching:\nT1 -> T282\nT2 -> T232", fontsize=14, style = filled, fillcolor = lightgrey];
184 -> 336 [arrowhead = none , style=dashed];
336 -> 1[style=dashed];

337 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
194 -> 337 [arrowhead = none ];
337 -> 195;

338 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
194 -> 338 [arrowhead = none ];
338 -> 196;

339 [label="EVAL with clause\napp([], X373, X373).\nand substitutionT239 -> [],\nT240 -> T301,\nT241 -> T302,\nT242 -> T303,\nX373 -> .(s(a, s(T301, T302), b), T303),\nX293 -> .(s(a, s(T301, T302), b), T303)", fontsize=14, style = filled, fillcolor = lightblue];
195 -> 339 [arrowhead = none ];
339 -> 197;

340 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
195 -> 340 [arrowhead = none ];
340 -> 198;

341 [label="EVAL with clause\napp(.(X384, X385), X386, .(X384, X387)) :- app(X385, X386, X387).\nand substitutionX384 -> T314,\nX385 -> T315,\nT239 -> .(T314, T315),\nT240 -> T316,\nT241 -> T317,\nT242 -> T318,\nX386 -> .(s(a, s(T316, T317), b), T318),\nX387 -> X388,\nX293 -> .(T314, X388)", fontsize=14, style = filled, fillcolor = lightblue];
196 -> 341 [arrowhead = none ];
341 -> 205;

342 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
196 -> 342 [arrowhead = none ];
342 -> 206;

343 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
197 -> 343 [arrowhead = none ];
343 -> 199;

344 [label="INSTANCE with matching:\nT239 -> T315\nT240 -> T316\nT241 -> T317\nT242 -> T318\nX293 -> X388", fontsize=14, style = filled, fillcolor = lightgrey];
205 -> 344 [arrowhead = none , style=dashed];
344 -> 183[style=dashed];

345 [label="ONLY EVAL with clause\nparse(X421, X422) :- ','(app(X423, .(a, .(b, X424)), X421), ','(app(X423, .(s(a, b), X424), X425), parse(X425, X422))).\nand substitutionT5 -> T341,\nX421 -> T341,\nT2 -> T343,\nX422 -> T343,\nT342 -> T343", fontsize=14, style = filled, fillcolor = lightblue];
213 -> 345 [arrowhead = none ];
345 -> 218;

346 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
214 -> 346 [arrowhead = none ];
346 -> 259;

347 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
214 -> 347 [arrowhead = none ];
347 -> 260;

348 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
218 -> 348 [arrowhead = none ];
348 -> 219;

349 [label="SPLIT 2\nnew knowledge:\nT346 is ground\nT347 is ground\nT341 is ground\nreplacements:X423 -> T346,\nX424 -> T347", fontsize=14, style = filled, fillcolor = violet];
218 -> 349 [arrowhead = none ];
349 -> 220;

350 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
219 -> 350 [arrowhead = none ];
350 -> 221;

351 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
220 -> 351 [arrowhead = none ];
351 -> 238;

352 [label="SPLIT 2\nnew knowledge:\nT346 is ground\nT347 is ground\nT365 is ground\nreplacements:X425 -> T365", fontsize=14, style = filled, fillcolor = violet];
220 -> 352 [arrowhead = none ];
352 -> 239;

353 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
221 -> 353 [arrowhead = none ];
353 -> 222;

354 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
221 -> 354 [arrowhead = none ];
354 -> 223;

355 [label="EVAL with clause\napp([], X442, X442).\nand substitutionX423 -> [],\nX424 -> T354,\nX442 -> .(a, .(b, T354)),\nX443 -> T354,\nT341 -> .(a, .(b, T354))", fontsize=14, style = filled, fillcolor = lightblue];
222 -> 355 [arrowhead = none ];
355 -> 224;

356 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
222 -> 356 [arrowhead = none ];
356 -> 225;

357 [label="EVAL with clause\napp(.(X458, X459), X460, .(X458, X461)) :- app(X459, X460, X461).\nand substitutionX458 -> T359,\nX459 -> X463,\nX423 -> .(T359, X463),\nX424 -> X464,\nX460 -> .(a, .(b, X464)),\nX462 -> T359,\nX461 -> T360,\nT341 -> .(T359, T360)", fontsize=14, style = filled, fillcolor = lightblue];
223 -> 357 [arrowhead = none ];
357 -> 231;

358 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
223 -> 358 [arrowhead = none ];
358 -> 232;

359 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
224 -> 359 [arrowhead = none ];
359 -> 226;

360 [label="INSTANCE with matching:\nX423 -> X463\nX424 -> X464\nT341 -> T360", fontsize=14, style = filled, fillcolor = lightgrey];
231 -> 360 [arrowhead = none , style=dashed];
360 -> 219[style=dashed];

361 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
238 -> 361 [arrowhead = none ];
361 -> 240;

362 [label="INSTANCE with matching:\nT1 -> T365\nT2 -> T343", fontsize=14, style = filled, fillcolor = lightgrey];
239 -> 362 [arrowhead = none , style=dashed];
362 -> 1[style=dashed];

363 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
240 -> 363 [arrowhead = none ];
363 -> 243;

364 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
240 -> 364 [arrowhead = none ];
364 -> 244;

365 [label="EVAL with clause\napp([], X477, X477).\nand substitutionT346 -> [],\nT347 -> T372,\nX477 -> .(s(a, b), T372),\nX425 -> .(s(a, b), T372)", fontsize=14, style = filled, fillcolor = lightblue];
243 -> 365 [arrowhead = none ];
365 -> 245;

366 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
243 -> 366 [arrowhead = none ];
366 -> 246;

367 [label="EVAL with clause\napp(.(X488, X489), X490, .(X488, X491)) :- app(X489, X490, X491).\nand substitutionX488 -> T379,\nX489 -> T380,\nT346 -> .(T379, T380),\nT347 -> T381,\nX490 -> .(s(a, b), T381),\nX491 -> X492,\nX425 -> .(T379, X492)", fontsize=14, style = filled, fillcolor = lightblue];
244 -> 367 [arrowhead = none ];
367 -> 250;

368 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
244 -> 368 [arrowhead = none ];
368 -> 251;

369 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
245 -> 369 [arrowhead = none ];
369 -> 247;

370 [label="INSTANCE with matching:\nT346 -> T380\nT347 -> T381\nX425 -> X492", fontsize=14, style = filled, fillcolor = lightgrey];
250 -> 370 [arrowhead = none , style=dashed];
370 -> 238[style=dashed];

371 [label="EVAL with clause\nparse(.(s(X511, X512), []), s(X511, X512)).\nand substitutionX511 -> T394,\nX512 -> T395,\nT5 -> .(s(T394, T395), []),\nT2 -> s(T394, T395)", fontsize=14, style = filled, fillcolor = lightblue];
259 -> 371 [arrowhead = none ];
371 -> 261;

372 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
259 -> 372 [arrowhead = none ];
372 -> 262;

373 [label="EVAL with clause\nparse(.(s(X519, X520, X521), []), s(X519, X520, X521)).\nand substitutionX519 -> T402,\nX520 -> T403,\nX521 -> T404,\nT5 -> .(s(T402, T403, T404), []),\nT2 -> s(T402, T403, T404)", fontsize=14, style = filled, fillcolor = lightblue];
260 -> 373 [arrowhead = none ];
373 -> 267;

374 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
260 -> 374 [arrowhead = none ];
374 -> 268;

375 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
261 -> 375 [arrowhead = none ];
375 -> 263;

376 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
267 -> 376 [arrowhead = none ];
376 -> 269;

}

(2) Obligation:

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

parseA_in_ga(.(a, .(s(T24, T25, T26), .(b, T27))), T7) → U1_ga(T24, T25, T26, T27, T7, pB_in_ggggaa(T24, T25, T26, T27, X10, T7))
pB_in_ggggaa(T24, T25, T26, T27, T33, T7) → U12_ggggaa(T24, T25, T26, T27, T33, T7, appL_in_gggga(T24, T25, T26, T27, T33))
appL_in_gggga(T54, T55, T56, T57, .(s(a, s(T54, T55, T56), b), T57)) → appL_out_gggga(T54, T55, T56, T57, .(s(a, s(T54, T55, T56), b), T57))
U12_ggggaa(T24, T25, T26, T27, T33, T7, appL_out_gggga(T24, T25, T26, T27, T33)) → U13_ggggaa(T24, T25, T26, T27, T33, T7, parseA_in_ga(T33, T7))
parseA_in_ga(.(T68, T69), T7) → U2_ga(T68, T69, T7, pC_in_aaaaaggaa(X109, X110, X111, X112, X113, T69, T68, X10, T7))
pC_in_aaaaaggaa(T78, T79, T80, T81, T82, T69, T68, X10, T7) → U14_aaaaaggaa(T78, T79, T80, T81, T82, T69, T68, X10, T7, appF_in_aaaaag(T78, T79, T80, T81, T82, T69))
appF_in_aaaaag([], T107, T108, T109, T110, .(a, .(s(T107, T108, T109), .(b, T110)))) → appF_out_aaaaag([], T107, T108, T109, T110, .(a, .(s(T107, T108, T109), .(b, T110))))
appF_in_aaaaag(.(T115, X184), X185, X186, X187, X188, .(T115, T116)) → U5_aaaaag(T115, X184, X185, X186, X187, X188, T116, appF_in_aaaaag(X184, X185, X186, X187, X188, T116))
U5_aaaaag(T115, X184, X185, X186, X187, X188, T116, appF_out_aaaaag(X184, X185, X186, X187, X188, T116)) → appF_out_aaaaag(.(T115, X184), X185, X186, X187, X188, .(T115, T116))
U14_aaaaaggaa(T78, T79, T80, T81, T82, T69, T68, X10, T7, appF_out_aaaaag(T78, T79, T80, T81, T82, T69)) → U15_aaaaaggaa(T78, T79, T80, T81, T82, T69, T68, X10, T7, pN_in_ggggggaa(T68, T78, T79, T80, T81, T82, X10, T7))
pN_in_ggggggaa(T68, T78, T79, T80, T81, T82, T125, T7) → U16_ggggggaa(T68, T78, T79, T80, T81, T82, T125, T7, appM_in_gggggga(T68, T78, T79, T80, T81, T82, T125))
appM_in_gggggga(T146, T147, T148, T149, T150, T151, .(T146, X220)) → U11_gggggga(T146, T147, T148, T149, T150, T151, X220, appG_in_ggggga(T147, T148, T149, T150, T151, X220))
appG_in_ggggga([], T176, T177, T178, T179, .(s(a, s(T176, T177, T178), b), T179)) → appG_out_ggggga([], T176, T177, T178, T179, .(s(a, s(T176, T177, T178), b), T179))
appG_in_ggggga(.(T192, T193), T194, T195, T196, T197, .(T192, X242)) → U6_ggggga(T192, T193, T194, T195, T196, T197, X242, appG_in_ggggga(T193, T194, T195, T196, T197, X242))
U6_ggggga(T192, T193, T194, T195, T196, T197, X242, appG_out_ggggga(T193, T194, T195, T196, T197, X242)) → appG_out_ggggga(.(T192, T193), T194, T195, T196, T197, .(T192, X242))
U11_gggggga(T146, T147, T148, T149, T150, T151, X220, appG_out_ggggga(T147, T148, T149, T150, T151, X220)) → appM_out_gggggga(T146, T147, T148, T149, T150, T151, .(T146, X220))
U16_ggggggaa(T68, T78, T79, T80, T81, T82, T125, T7, appM_out_gggggga(T68, T78, T79, T80, T81, T82, T125)) → U17_ggggggaa(T68, T78, T79, T80, T81, T82, T125, T7, parseA_in_ga(T125, T7))
parseA_in_ga(T230, T232) → U3_ga(T230, T232, pD_in_aaaagaa(X289, X290, X291, X292, T230, X293, T232))
pD_in_aaaagaa(T239, T240, T241, T242, T230, X293, T232) → U18_aaaagaa(T239, T240, T241, T242, T230, X293, T232, appH_in_aaaag(T239, T240, T241, T242, T230))
appH_in_aaaag([], T261, T262, T263, .(a, .(s(T261, T262), .(b, T263)))) → appH_out_aaaag([], T261, T262, T263, .(a, .(s(T261, T262), .(b, T263))))
appH_in_aaaag(.(T268, X353), X354, X355, X356, .(T268, T269)) → U7_aaaag(T268, X353, X354, X355, X356, T269, appH_in_aaaag(X353, X354, X355, X356, T269))
U7_aaaag(T268, X353, X354, X355, X356, T269, appH_out_aaaag(X353, X354, X355, X356, T269)) → appH_out_aaaag(.(T268, X353), X354, X355, X356, .(T268, T269))
U18_aaaagaa(T239, T240, T241, T242, T230, X293, T232, appH_out_aaaag(T239, T240, T241, T242, T230)) → U19_aaaagaa(T239, T240, T241, T242, T230, X293, T232, pO_in_ggggaa(T239, T240, T241, T242, X293, T232))
pO_in_ggggaa(T239, T240, T241, T242, T282, T232) → U20_ggggaa(T239, T240, T241, T242, T282, T232, appI_in_gggga(T239, T240, T241, T242, T282))
appI_in_gggga([], T301, T302, T303, .(s(a, s(T301, T302), b), T303)) → appI_out_gggga([], T301, T302, T303, .(s(a, s(T301, T302), b), T303))
appI_in_gggga(.(T314, T315), T316, T317, T318, .(T314, X388)) → U8_gggga(T314, T315, T316, T317, T318, X388, appI_in_gggga(T315, T316, T317, T318, X388))
U8_gggga(T314, T315, T316, T317, T318, X388, appI_out_gggga(T315, T316, T317, T318, X388)) → appI_out_gggga(.(T314, T315), T316, T317, T318, .(T314, X388))
U20_ggggaa(T239, T240, T241, T242, T282, T232, appI_out_gggga(T239, T240, T241, T242, T282)) → U21_ggggaa(T239, T240, T241, T242, T282, T232, parseA_in_ga(T282, T232))
parseA_in_ga(T341, T343) → U4_ga(T341, T343, pE_in_aagaa(X423, X424, T341, X425, T343))
pE_in_aagaa(T346, T347, T341, X425, T343) → U22_aagaa(T346, T347, T341, X425, T343, appJ_in_aag(T346, T347, T341))
appJ_in_aag([], T354, .(a, .(b, T354))) → appJ_out_aag([], T354, .(a, .(b, T354)))
appJ_in_aag(.(T359, X463), X464, .(T359, T360)) → U9_aag(T359, X463, X464, T360, appJ_in_aag(X463, X464, T360))
U9_aag(T359, X463, X464, T360, appJ_out_aag(X463, X464, T360)) → appJ_out_aag(.(T359, X463), X464, .(T359, T360))
U22_aagaa(T346, T347, T341, X425, T343, appJ_out_aag(T346, T347, T341)) → U23_aagaa(T346, T347, T341, X425, T343, pP_in_ggaa(T346, T347, X425, T343))
pP_in_ggaa(T346, T347, T365, T343) → U24_ggaa(T346, T347, T365, T343, appK_in_gga(T346, T347, T365))
appK_in_gga([], T372, .(s(a, b), T372)) → appK_out_gga([], T372, .(s(a, b), T372))
appK_in_gga(.(T379, T380), T381, .(T379, X492)) → U10_gga(T379, T380, T381, X492, appK_in_gga(T380, T381, X492))
U10_gga(T379, T380, T381, X492, appK_out_gga(T380, T381, X492)) → appK_out_gga(.(T379, T380), T381, .(T379, X492))
U24_ggaa(T346, T347, T365, T343, appK_out_gga(T346, T347, T365)) → U25_ggaa(T346, T347, T365, T343, parseA_in_ga(T365, T343))
parseA_in_ga(.(s(T394, T395), []), s(T394, T395)) → parseA_out_ga(.(s(T394, T395), []), s(T394, T395))
parseA_in_ga(.(s(T402, T403, T404), []), s(T402, T403, T404)) → parseA_out_ga(.(s(T402, T403, T404), []), s(T402, T403, T404))
U25_ggaa(T346, T347, T365, T343, parseA_out_ga(T365, T343)) → pP_out_ggaa(T346, T347, T365, T343)
U23_aagaa(T346, T347, T341, X425, T343, pP_out_ggaa(T346, T347, X425, T343)) → pE_out_aagaa(T346, T347, T341, X425, T343)
U4_ga(T341, T343, pE_out_aagaa(X423, X424, T341, X425, T343)) → parseA_out_ga(T341, T343)
U21_ggggaa(T239, T240, T241, T242, T282, T232, parseA_out_ga(T282, T232)) → pO_out_ggggaa(T239, T240, T241, T242, T282, T232)
U19_aaaagaa(T239, T240, T241, T242, T230, X293, T232, pO_out_ggggaa(T239, T240, T241, T242, X293, T232)) → pD_out_aaaagaa(T239, T240, T241, T242, T230, X293, T232)
U3_ga(T230, T232, pD_out_aaaagaa(X289, X290, X291, X292, T230, X293, T232)) → parseA_out_ga(T230, T232)
U17_ggggggaa(T68, T78, T79, T80, T81, T82, T125, T7, parseA_out_ga(T125, T7)) → pN_out_ggggggaa(T68, T78, T79, T80, T81, T82, T125, T7)
U15_aaaaaggaa(T78, T79, T80, T81, T82, T69, T68, X10, T7, pN_out_ggggggaa(T68, T78, T79, T80, T81, T82, X10, T7)) → pC_out_aaaaaggaa(T78, T79, T80, T81, T82, T69, T68, X10, T7)
U2_ga(T68, T69, T7, pC_out_aaaaaggaa(X109, X110, X111, X112, X113, T69, T68, X10, T7)) → parseA_out_ga(.(T68, T69), T7)
U13_ggggaa(T24, T25, T26, T27, T33, T7, parseA_out_ga(T33, T7)) → pB_out_ggggaa(T24, T25, T26, T27, T33, T7)
U1_ga(T24, T25, T26, T27, T7, pB_out_ggggaa(T24, T25, T26, T27, X10, T7)) → parseA_out_ga(.(a, .(s(T24, T25, T26), .(b, T27))), T7)

The argument filtering Pi contains the following mapping:
parseA_in_ga(x1, x2)  =  parseA_in_ga(x1)
.(x1, x2)  =  .(x1, x2)
a  =  a
s(x1, x2, x3)  =  s(x1, x2, x3)
b  =  b
U1_ga(x1, x2, x3, x4, x5, x6)  =  U1_ga(x1, x2, x3, x4, x6)
pB_in_ggggaa(x1, x2, x3, x4, x5, x6)  =  pB_in_ggggaa(x1, x2, x3, x4)
U12_ggggaa(x1, x2, x3, x4, x5, x6, x7)  =  U12_ggggaa(x1, x2, x3, x4, x7)
appL_in_gggga(x1, x2, x3, x4, x5)  =  appL_in_gggga(x1, x2, x3, x4)
appL_out_gggga(x1, x2, x3, x4, x5)  =  appL_out_gggga(x1, x2, x3, x4, x5)
U13_ggggaa(x1, x2, x3, x4, x5, x6, x7)  =  U13_ggggaa(x1, x2, x3, x4, x5, x7)
U2_ga(x1, x2, x3, x4)  =  U2_ga(x1, x2, x4)
pC_in_aaaaaggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  pC_in_aaaaaggaa(x6, x7)
U14_aaaaaggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10)  =  U14_aaaaaggaa(x6, x7, x10)
appF_in_aaaaag(x1, x2, x3, x4, x5, x6)  =  appF_in_aaaaag(x6)
appF_out_aaaaag(x1, x2, x3, x4, x5, x6)  =  appF_out_aaaaag(x1, x2, x3, x4, x5, x6)
U5_aaaaag(x1, x2, x3, x4, x5, x6, x7, x8)  =  U5_aaaaag(x1, x7, x8)
U15_aaaaaggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10)  =  U15_aaaaaggaa(x1, x2, x3, x4, x5, x6, x7, x10)
pN_in_ggggggaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  pN_in_ggggggaa(x1, x2, x3, x4, x5, x6)
U16_ggggggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U16_ggggggaa(x1, x2, x3, x4, x5, x6, x9)
appM_in_gggggga(x1, x2, x3, x4, x5, x6, x7)  =  appM_in_gggggga(x1, x2, x3, x4, x5, x6)
U11_gggggga(x1, x2, x3, x4, x5, x6, x7, x8)  =  U11_gggggga(x1, x2, x3, x4, x5, x6, x8)
appG_in_ggggga(x1, x2, x3, x4, x5, x6)  =  appG_in_ggggga(x1, x2, x3, x4, x5)
[]  =  []
appG_out_ggggga(x1, x2, x3, x4, x5, x6)  =  appG_out_ggggga(x1, x2, x3, x4, x5, x6)
U6_ggggga(x1, x2, x3, x4, x5, x6, x7, x8)  =  U6_ggggga(x1, x2, x3, x4, x5, x6, x8)
appM_out_gggggga(x1, x2, x3, x4, x5, x6, x7)  =  appM_out_gggggga(x1, x2, x3, x4, x5, x6, x7)
U17_ggggggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U17_ggggggaa(x1, x2, x3, x4, x5, x6, x7, x9)
U3_ga(x1, x2, x3)  =  U3_ga(x1, x3)
pD_in_aaaagaa(x1, x2, x3, x4, x5, x6, x7)  =  pD_in_aaaagaa(x5)
U18_aaaagaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U18_aaaagaa(x5, x8)
appH_in_aaaag(x1, x2, x3, x4, x5)  =  appH_in_aaaag(x5)
s(x1, x2)  =  s(x1, x2)
appH_out_aaaag(x1, x2, x3, x4, x5)  =  appH_out_aaaag(x1, x2, x3, x4, x5)
U7_aaaag(x1, x2, x3, x4, x5, x6, x7)  =  U7_aaaag(x1, x6, x7)
U19_aaaagaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U19_aaaagaa(x1, x2, x3, x4, x5, x8)
pO_in_ggggaa(x1, x2, x3, x4, x5, x6)  =  pO_in_ggggaa(x1, x2, x3, x4)
U20_ggggaa(x1, x2, x3, x4, x5, x6, x7)  =  U20_ggggaa(x1, x2, x3, x4, x7)
appI_in_gggga(x1, x2, x3, x4, x5)  =  appI_in_gggga(x1, x2, x3, x4)
appI_out_gggga(x1, x2, x3, x4, x5)  =  appI_out_gggga(x1, x2, x3, x4, x5)
U8_gggga(x1, x2, x3, x4, x5, x6, x7)  =  U8_gggga(x1, x2, x3, x4, x5, x7)
U21_ggggaa(x1, x2, x3, x4, x5, x6, x7)  =  U21_ggggaa(x1, x2, x3, x4, x5, x7)
U4_ga(x1, x2, x3)  =  U4_ga(x1, x3)
pE_in_aagaa(x1, x2, x3, x4, x5)  =  pE_in_aagaa(x3)
U22_aagaa(x1, x2, x3, x4, x5, x6)  =  U22_aagaa(x3, x6)
appJ_in_aag(x1, x2, x3)  =  appJ_in_aag(x3)
appJ_out_aag(x1, x2, x3)  =  appJ_out_aag(x1, x2, x3)
U9_aag(x1, x2, x3, x4, x5)  =  U9_aag(x1, x4, x5)
U23_aagaa(x1, x2, x3, x4, x5, x6)  =  U23_aagaa(x1, x2, x3, x6)
pP_in_ggaa(x1, x2, x3, x4)  =  pP_in_ggaa(x1, x2)
U24_ggaa(x1, x2, x3, x4, x5)  =  U24_ggaa(x1, x2, x5)
appK_in_gga(x1, x2, x3)  =  appK_in_gga(x1, x2)
appK_out_gga(x1, x2, x3)  =  appK_out_gga(x1, x2, x3)
U10_gga(x1, x2, x3, x4, x5)  =  U10_gga(x1, x2, x3, x5)
U25_ggaa(x1, x2, x3, x4, x5)  =  U25_ggaa(x1, x2, x3, x5)
parseA_out_ga(x1, x2)  =  parseA_out_ga(x1, x2)
pP_out_ggaa(x1, x2, x3, x4)  =  pP_out_ggaa(x1, x2, x3, x4)
pE_out_aagaa(x1, x2, x3, x4, x5)  =  pE_out_aagaa(x1, x2, x3, x4, x5)
pO_out_ggggaa(x1, x2, x3, x4, x5, x6)  =  pO_out_ggggaa(x1, x2, x3, x4, x5, x6)
pD_out_aaaagaa(x1, x2, x3, x4, x5, x6, x7)  =  pD_out_aaaagaa(x1, x2, x3, x4, x5, x6, x7)
pN_out_ggggggaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  pN_out_ggggggaa(x1, x2, x3, x4, x5, x6, x7, x8)
pC_out_aaaaaggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  pC_out_aaaaaggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)
pB_out_ggggaa(x1, x2, x3, x4, x5, x6)  =  pB_out_ggggaa(x1, x2, x3, x4, x5, x6)

(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:

PARSEA_IN_GA(.(a, .(s(T24, T25, T26), .(b, T27))), T7) → U1_GA(T24, T25, T26, T27, T7, pB_in_ggggaa(T24, T25, T26, T27, X10, T7))
PARSEA_IN_GA(.(a, .(s(T24, T25, T26), .(b, T27))), T7) → PB_IN_GGGGAA(T24, T25, T26, T27, X10, T7)
PB_IN_GGGGAA(T24, T25, T26, T27, T33, T7) → U12_GGGGAA(T24, T25, T26, T27, T33, T7, appL_in_gggga(T24, T25, T26, T27, T33))
PB_IN_GGGGAA(T24, T25, T26, T27, T33, T7) → APPL_IN_GGGGA(T24, T25, T26, T27, T33)
U12_GGGGAA(T24, T25, T26, T27, T33, T7, appL_out_gggga(T24, T25, T26, T27, T33)) → U13_GGGGAA(T24, T25, T26, T27, T33, T7, parseA_in_ga(T33, T7))
U12_GGGGAA(T24, T25, T26, T27, T33, T7, appL_out_gggga(T24, T25, T26, T27, T33)) → PARSEA_IN_GA(T33, T7)
PARSEA_IN_GA(.(T68, T69), T7) → U2_GA(T68, T69, T7, pC_in_aaaaaggaa(X109, X110, X111, X112, X113, T69, T68, X10, T7))
PARSEA_IN_GA(.(T68, T69), T7) → PC_IN_AAAAAGGAA(X109, X110, X111, X112, X113, T69, T68, X10, T7)
PC_IN_AAAAAGGAA(T78, T79, T80, T81, T82, T69, T68, X10, T7) → U14_AAAAAGGAA(T78, T79, T80, T81, T82, T69, T68, X10, T7, appF_in_aaaaag(T78, T79, T80, T81, T82, T69))
PC_IN_AAAAAGGAA(T78, T79, T80, T81, T82, T69, T68, X10, T7) → APPF_IN_AAAAAG(T78, T79, T80, T81, T82, T69)
APPF_IN_AAAAAG(.(T115, X184), X185, X186, X187, X188, .(T115, T116)) → U5_AAAAAG(T115, X184, X185, X186, X187, X188, T116, appF_in_aaaaag(X184, X185, X186, X187, X188, T116))
APPF_IN_AAAAAG(.(T115, X184), X185, X186, X187, X188, .(T115, T116)) → APPF_IN_AAAAAG(X184, X185, X186, X187, X188, T116)
U14_AAAAAGGAA(T78, T79, T80, T81, T82, T69, T68, X10, T7, appF_out_aaaaag(T78, T79, T80, T81, T82, T69)) → U15_AAAAAGGAA(T78, T79, T80, T81, T82, T69, T68, X10, T7, pN_in_ggggggaa(T68, T78, T79, T80, T81, T82, X10, T7))
U14_AAAAAGGAA(T78, T79, T80, T81, T82, T69, T68, X10, T7, appF_out_aaaaag(T78, T79, T80, T81, T82, T69)) → PN_IN_GGGGGGAA(T68, T78, T79, T80, T81, T82, X10, T7)
PN_IN_GGGGGGAA(T68, T78, T79, T80, T81, T82, T125, T7) → U16_GGGGGGAA(T68, T78, T79, T80, T81, T82, T125, T7, appM_in_gggggga(T68, T78, T79, T80, T81, T82, T125))
PN_IN_GGGGGGAA(T68, T78, T79, T80, T81, T82, T125, T7) → APPM_IN_GGGGGGA(T68, T78, T79, T80, T81, T82, T125)
APPM_IN_GGGGGGA(T146, T147, T148, T149, T150, T151, .(T146, X220)) → U11_GGGGGGA(T146, T147, T148, T149, T150, T151, X220, appG_in_ggggga(T147, T148, T149, T150, T151, X220))
APPM_IN_GGGGGGA(T146, T147, T148, T149, T150, T151, .(T146, X220)) → APPG_IN_GGGGGA(T147, T148, T149, T150, T151, X220)
APPG_IN_GGGGGA(.(T192, T193), T194, T195, T196, T197, .(T192, X242)) → U6_GGGGGA(T192, T193, T194, T195, T196, T197, X242, appG_in_ggggga(T193, T194, T195, T196, T197, X242))
APPG_IN_GGGGGA(.(T192, T193), T194, T195, T196, T197, .(T192, X242)) → APPG_IN_GGGGGA(T193, T194, T195, T196, T197, X242)
U16_GGGGGGAA(T68, T78, T79, T80, T81, T82, T125, T7, appM_out_gggggga(T68, T78, T79, T80, T81, T82, T125)) → U17_GGGGGGAA(T68, T78, T79, T80, T81, T82, T125, T7, parseA_in_ga(T125, T7))
U16_GGGGGGAA(T68, T78, T79, T80, T81, T82, T125, T7, appM_out_gggggga(T68, T78, T79, T80, T81, T82, T125)) → PARSEA_IN_GA(T125, T7)
PARSEA_IN_GA(T230, T232) → U3_GA(T230, T232, pD_in_aaaagaa(X289, X290, X291, X292, T230, X293, T232))
PARSEA_IN_GA(T230, T232) → PD_IN_AAAAGAA(X289, X290, X291, X292, T230, X293, T232)
PD_IN_AAAAGAA(T239, T240, T241, T242, T230, X293, T232) → U18_AAAAGAA(T239, T240, T241, T242, T230, X293, T232, appH_in_aaaag(T239, T240, T241, T242, T230))
PD_IN_AAAAGAA(T239, T240, T241, T242, T230, X293, T232) → APPH_IN_AAAAG(T239, T240, T241, T242, T230)
APPH_IN_AAAAG(.(T268, X353), X354, X355, X356, .(T268, T269)) → U7_AAAAG(T268, X353, X354, X355, X356, T269, appH_in_aaaag(X353, X354, X355, X356, T269))
APPH_IN_AAAAG(.(T268, X353), X354, X355, X356, .(T268, T269)) → APPH_IN_AAAAG(X353, X354, X355, X356, T269)
U18_AAAAGAA(T239, T240, T241, T242, T230, X293, T232, appH_out_aaaag(T239, T240, T241, T242, T230)) → U19_AAAAGAA(T239, T240, T241, T242, T230, X293, T232, pO_in_ggggaa(T239, T240, T241, T242, X293, T232))
U18_AAAAGAA(T239, T240, T241, T242, T230, X293, T232, appH_out_aaaag(T239, T240, T241, T242, T230)) → PO_IN_GGGGAA(T239, T240, T241, T242, X293, T232)
PO_IN_GGGGAA(T239, T240, T241, T242, T282, T232) → U20_GGGGAA(T239, T240, T241, T242, T282, T232, appI_in_gggga(T239, T240, T241, T242, T282))
PO_IN_GGGGAA(T239, T240, T241, T242, T282, T232) → APPI_IN_GGGGA(T239, T240, T241, T242, T282)
APPI_IN_GGGGA(.(T314, T315), T316, T317, T318, .(T314, X388)) → U8_GGGGA(T314, T315, T316, T317, T318, X388, appI_in_gggga(T315, T316, T317, T318, X388))
APPI_IN_GGGGA(.(T314, T315), T316, T317, T318, .(T314, X388)) → APPI_IN_GGGGA(T315, T316, T317, T318, X388)
U20_GGGGAA(T239, T240, T241, T242, T282, T232, appI_out_gggga(T239, T240, T241, T242, T282)) → U21_GGGGAA(T239, T240, T241, T242, T282, T232, parseA_in_ga(T282, T232))
U20_GGGGAA(T239, T240, T241, T242, T282, T232, appI_out_gggga(T239, T240, T241, T242, T282)) → PARSEA_IN_GA(T282, T232)
PARSEA_IN_GA(T341, T343) → U4_GA(T341, T343, pE_in_aagaa(X423, X424, T341, X425, T343))
PARSEA_IN_GA(T341, T343) → PE_IN_AAGAA(X423, X424, T341, X425, T343)
PE_IN_AAGAA(T346, T347, T341, X425, T343) → U22_AAGAA(T346, T347, T341, X425, T343, appJ_in_aag(T346, T347, T341))
PE_IN_AAGAA(T346, T347, T341, X425, T343) → APPJ_IN_AAG(T346, T347, T341)
APPJ_IN_AAG(.(T359, X463), X464, .(T359, T360)) → U9_AAG(T359, X463, X464, T360, appJ_in_aag(X463, X464, T360))
APPJ_IN_AAG(.(T359, X463), X464, .(T359, T360)) → APPJ_IN_AAG(X463, X464, T360)
U22_AAGAA(T346, T347, T341, X425, T343, appJ_out_aag(T346, T347, T341)) → U23_AAGAA(T346, T347, T341, X425, T343, pP_in_ggaa(T346, T347, X425, T343))
U22_AAGAA(T346, T347, T341, X425, T343, appJ_out_aag(T346, T347, T341)) → PP_IN_GGAA(T346, T347, X425, T343)
PP_IN_GGAA(T346, T347, T365, T343) → U24_GGAA(T346, T347, T365, T343, appK_in_gga(T346, T347, T365))
PP_IN_GGAA(T346, T347, T365, T343) → APPK_IN_GGA(T346, T347, T365)
APPK_IN_GGA(.(T379, T380), T381, .(T379, X492)) → U10_GGA(T379, T380, T381, X492, appK_in_gga(T380, T381, X492))
APPK_IN_GGA(.(T379, T380), T381, .(T379, X492)) → APPK_IN_GGA(T380, T381, X492)
U24_GGAA(T346, T347, T365, T343, appK_out_gga(T346, T347, T365)) → U25_GGAA(T346, T347, T365, T343, parseA_in_ga(T365, T343))
U24_GGAA(T346, T347, T365, T343, appK_out_gga(T346, T347, T365)) → PARSEA_IN_GA(T365, T343)

The TRS R consists of the following rules:

parseA_in_ga(.(a, .(s(T24, T25, T26), .(b, T27))), T7) → U1_ga(T24, T25, T26, T27, T7, pB_in_ggggaa(T24, T25, T26, T27, X10, T7))
pB_in_ggggaa(T24, T25, T26, T27, T33, T7) → U12_ggggaa(T24, T25, T26, T27, T33, T7, appL_in_gggga(T24, T25, T26, T27, T33))
appL_in_gggga(T54, T55, T56, T57, .(s(a, s(T54, T55, T56), b), T57)) → appL_out_gggga(T54, T55, T56, T57, .(s(a, s(T54, T55, T56), b), T57))
U12_ggggaa(T24, T25, T26, T27, T33, T7, appL_out_gggga(T24, T25, T26, T27, T33)) → U13_ggggaa(T24, T25, T26, T27, T33, T7, parseA_in_ga(T33, T7))
parseA_in_ga(.(T68, T69), T7) → U2_ga(T68, T69, T7, pC_in_aaaaaggaa(X109, X110, X111, X112, X113, T69, T68, X10, T7))
pC_in_aaaaaggaa(T78, T79, T80, T81, T82, T69, T68, X10, T7) → U14_aaaaaggaa(T78, T79, T80, T81, T82, T69, T68, X10, T7, appF_in_aaaaag(T78, T79, T80, T81, T82, T69))
appF_in_aaaaag([], T107, T108, T109, T110, .(a, .(s(T107, T108, T109), .(b, T110)))) → appF_out_aaaaag([], T107, T108, T109, T110, .(a, .(s(T107, T108, T109), .(b, T110))))
appF_in_aaaaag(.(T115, X184), X185, X186, X187, X188, .(T115, T116)) → U5_aaaaag(T115, X184, X185, X186, X187, X188, T116, appF_in_aaaaag(X184, X185, X186, X187, X188, T116))
U5_aaaaag(T115, X184, X185, X186, X187, X188, T116, appF_out_aaaaag(X184, X185, X186, X187, X188, T116)) → appF_out_aaaaag(.(T115, X184), X185, X186, X187, X188, .(T115, T116))
U14_aaaaaggaa(T78, T79, T80, T81, T82, T69, T68, X10, T7, appF_out_aaaaag(T78, T79, T80, T81, T82, T69)) → U15_aaaaaggaa(T78, T79, T80, T81, T82, T69, T68, X10, T7, pN_in_ggggggaa(T68, T78, T79, T80, T81, T82, X10, T7))
pN_in_ggggggaa(T68, T78, T79, T80, T81, T82, T125, T7) → U16_ggggggaa(T68, T78, T79, T80, T81, T82, T125, T7, appM_in_gggggga(T68, T78, T79, T80, T81, T82, T125))
appM_in_gggggga(T146, T147, T148, T149, T150, T151, .(T146, X220)) → U11_gggggga(T146, T147, T148, T149, T150, T151, X220, appG_in_ggggga(T147, T148, T149, T150, T151, X220))
appG_in_ggggga([], T176, T177, T178, T179, .(s(a, s(T176, T177, T178), b), T179)) → appG_out_ggggga([], T176, T177, T178, T179, .(s(a, s(T176, T177, T178), b), T179))
appG_in_ggggga(.(T192, T193), T194, T195, T196, T197, .(T192, X242)) → U6_ggggga(T192, T193, T194, T195, T196, T197, X242, appG_in_ggggga(T193, T194, T195, T196, T197, X242))
U6_ggggga(T192, T193, T194, T195, T196, T197, X242, appG_out_ggggga(T193, T194, T195, T196, T197, X242)) → appG_out_ggggga(.(T192, T193), T194, T195, T196, T197, .(T192, X242))
U11_gggggga(T146, T147, T148, T149, T150, T151, X220, appG_out_ggggga(T147, T148, T149, T150, T151, X220)) → appM_out_gggggga(T146, T147, T148, T149, T150, T151, .(T146, X220))
U16_ggggggaa(T68, T78, T79, T80, T81, T82, T125, T7, appM_out_gggggga(T68, T78, T79, T80, T81, T82, T125)) → U17_ggggggaa(T68, T78, T79, T80, T81, T82, T125, T7, parseA_in_ga(T125, T7))
parseA_in_ga(T230, T232) → U3_ga(T230, T232, pD_in_aaaagaa(X289, X290, X291, X292, T230, X293, T232))
pD_in_aaaagaa(T239, T240, T241, T242, T230, X293, T232) → U18_aaaagaa(T239, T240, T241, T242, T230, X293, T232, appH_in_aaaag(T239, T240, T241, T242, T230))
appH_in_aaaag([], T261, T262, T263, .(a, .(s(T261, T262), .(b, T263)))) → appH_out_aaaag([], T261, T262, T263, .(a, .(s(T261, T262), .(b, T263))))
appH_in_aaaag(.(T268, X353), X354, X355, X356, .(T268, T269)) → U7_aaaag(T268, X353, X354, X355, X356, T269, appH_in_aaaag(X353, X354, X355, X356, T269))
U7_aaaag(T268, X353, X354, X355, X356, T269, appH_out_aaaag(X353, X354, X355, X356, T269)) → appH_out_aaaag(.(T268, X353), X354, X355, X356, .(T268, T269))
U18_aaaagaa(T239, T240, T241, T242, T230, X293, T232, appH_out_aaaag(T239, T240, T241, T242, T230)) → U19_aaaagaa(T239, T240, T241, T242, T230, X293, T232, pO_in_ggggaa(T239, T240, T241, T242, X293, T232))
pO_in_ggggaa(T239, T240, T241, T242, T282, T232) → U20_ggggaa(T239, T240, T241, T242, T282, T232, appI_in_gggga(T239, T240, T241, T242, T282))
appI_in_gggga([], T301, T302, T303, .(s(a, s(T301, T302), b), T303)) → appI_out_gggga([], T301, T302, T303, .(s(a, s(T301, T302), b), T303))
appI_in_gggga(.(T314, T315), T316, T317, T318, .(T314, X388)) → U8_gggga(T314, T315, T316, T317, T318, X388, appI_in_gggga(T315, T316, T317, T318, X388))
U8_gggga(T314, T315, T316, T317, T318, X388, appI_out_gggga(T315, T316, T317, T318, X388)) → appI_out_gggga(.(T314, T315), T316, T317, T318, .(T314, X388))
U20_ggggaa(T239, T240, T241, T242, T282, T232, appI_out_gggga(T239, T240, T241, T242, T282)) → U21_ggggaa(T239, T240, T241, T242, T282, T232, parseA_in_ga(T282, T232))
parseA_in_ga(T341, T343) → U4_ga(T341, T343, pE_in_aagaa(X423, X424, T341, X425, T343))
pE_in_aagaa(T346, T347, T341, X425, T343) → U22_aagaa(T346, T347, T341, X425, T343, appJ_in_aag(T346, T347, T341))
appJ_in_aag([], T354, .(a, .(b, T354))) → appJ_out_aag([], T354, .(a, .(b, T354)))
appJ_in_aag(.(T359, X463), X464, .(T359, T360)) → U9_aag(T359, X463, X464, T360, appJ_in_aag(X463, X464, T360))
U9_aag(T359, X463, X464, T360, appJ_out_aag(X463, X464, T360)) → appJ_out_aag(.(T359, X463), X464, .(T359, T360))
U22_aagaa(T346, T347, T341, X425, T343, appJ_out_aag(T346, T347, T341)) → U23_aagaa(T346, T347, T341, X425, T343, pP_in_ggaa(T346, T347, X425, T343))
pP_in_ggaa(T346, T347, T365, T343) → U24_ggaa(T346, T347, T365, T343, appK_in_gga(T346, T347, T365))
appK_in_gga([], T372, .(s(a, b), T372)) → appK_out_gga([], T372, .(s(a, b), T372))
appK_in_gga(.(T379, T380), T381, .(T379, X492)) → U10_gga(T379, T380, T381, X492, appK_in_gga(T380, T381, X492))
U10_gga(T379, T380, T381, X492, appK_out_gga(T380, T381, X492)) → appK_out_gga(.(T379, T380), T381, .(T379, X492))
U24_ggaa(T346, T347, T365, T343, appK_out_gga(T346, T347, T365)) → U25_ggaa(T346, T347, T365, T343, parseA_in_ga(T365, T343))
parseA_in_ga(.(s(T394, T395), []), s(T394, T395)) → parseA_out_ga(.(s(T394, T395), []), s(T394, T395))
parseA_in_ga(.(s(T402, T403, T404), []), s(T402, T403, T404)) → parseA_out_ga(.(s(T402, T403, T404), []), s(T402, T403, T404))
U25_ggaa(T346, T347, T365, T343, parseA_out_ga(T365, T343)) → pP_out_ggaa(T346, T347, T365, T343)
U23_aagaa(T346, T347, T341, X425, T343, pP_out_ggaa(T346, T347, X425, T343)) → pE_out_aagaa(T346, T347, T341, X425, T343)
U4_ga(T341, T343, pE_out_aagaa(X423, X424, T341, X425, T343)) → parseA_out_ga(T341, T343)
U21_ggggaa(T239, T240, T241, T242, T282, T232, parseA_out_ga(T282, T232)) → pO_out_ggggaa(T239, T240, T241, T242, T282, T232)
U19_aaaagaa(T239, T240, T241, T242, T230, X293, T232, pO_out_ggggaa(T239, T240, T241, T242, X293, T232)) → pD_out_aaaagaa(T239, T240, T241, T242, T230, X293, T232)
U3_ga(T230, T232, pD_out_aaaagaa(X289, X290, X291, X292, T230, X293, T232)) → parseA_out_ga(T230, T232)
U17_ggggggaa(T68, T78, T79, T80, T81, T82, T125, T7, parseA_out_ga(T125, T7)) → pN_out_ggggggaa(T68, T78, T79, T80, T81, T82, T125, T7)
U15_aaaaaggaa(T78, T79, T80, T81, T82, T69, T68, X10, T7, pN_out_ggggggaa(T68, T78, T79, T80, T81, T82, X10, T7)) → pC_out_aaaaaggaa(T78, T79, T80, T81, T82, T69, T68, X10, T7)
U2_ga(T68, T69, T7, pC_out_aaaaaggaa(X109, X110, X111, X112, X113, T69, T68, X10, T7)) → parseA_out_ga(.(T68, T69), T7)
U13_ggggaa(T24, T25, T26, T27, T33, T7, parseA_out_ga(T33, T7)) → pB_out_ggggaa(T24, T25, T26, T27, T33, T7)
U1_ga(T24, T25, T26, T27, T7, pB_out_ggggaa(T24, T25, T26, T27, X10, T7)) → parseA_out_ga(.(a, .(s(T24, T25, T26), .(b, T27))), T7)

The argument filtering Pi contains the following mapping:
parseA_in_ga(x1, x2)  =  parseA_in_ga(x1)
.(x1, x2)  =  .(x1, x2)
a  =  a
s(x1, x2, x3)  =  s(x1, x2, x3)
b  =  b
U1_ga(x1, x2, x3, x4, x5, x6)  =  U1_ga(x1, x2, x3, x4, x6)
pB_in_ggggaa(x1, x2, x3, x4, x5, x6)  =  pB_in_ggggaa(x1, x2, x3, x4)
U12_ggggaa(x1, x2, x3, x4, x5, x6, x7)  =  U12_ggggaa(x1, x2, x3, x4, x7)
appL_in_gggga(x1, x2, x3, x4, x5)  =  appL_in_gggga(x1, x2, x3, x4)
appL_out_gggga(x1, x2, x3, x4, x5)  =  appL_out_gggga(x1, x2, x3, x4, x5)
U13_ggggaa(x1, x2, x3, x4, x5, x6, x7)  =  U13_ggggaa(x1, x2, x3, x4, x5, x7)
U2_ga(x1, x2, x3, x4)  =  U2_ga(x1, x2, x4)
pC_in_aaaaaggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  pC_in_aaaaaggaa(x6, x7)
U14_aaaaaggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10)  =  U14_aaaaaggaa(x6, x7, x10)
appF_in_aaaaag(x1, x2, x3, x4, x5, x6)  =  appF_in_aaaaag(x6)
appF_out_aaaaag(x1, x2, x3, x4, x5, x6)  =  appF_out_aaaaag(x1, x2, x3, x4, x5, x6)
U5_aaaaag(x1, x2, x3, x4, x5, x6, x7, x8)  =  U5_aaaaag(x1, x7, x8)
U15_aaaaaggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10)  =  U15_aaaaaggaa(x1, x2, x3, x4, x5, x6, x7, x10)
pN_in_ggggggaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  pN_in_ggggggaa(x1, x2, x3, x4, x5, x6)
U16_ggggggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U16_ggggggaa(x1, x2, x3, x4, x5, x6, x9)
appM_in_gggggga(x1, x2, x3, x4, x5, x6, x7)  =  appM_in_gggggga(x1, x2, x3, x4, x5, x6)
U11_gggggga(x1, x2, x3, x4, x5, x6, x7, x8)  =  U11_gggggga(x1, x2, x3, x4, x5, x6, x8)
appG_in_ggggga(x1, x2, x3, x4, x5, x6)  =  appG_in_ggggga(x1, x2, x3, x4, x5)
[]  =  []
appG_out_ggggga(x1, x2, x3, x4, x5, x6)  =  appG_out_ggggga(x1, x2, x3, x4, x5, x6)
U6_ggggga(x1, x2, x3, x4, x5, x6, x7, x8)  =  U6_ggggga(x1, x2, x3, x4, x5, x6, x8)
appM_out_gggggga(x1, x2, x3, x4, x5, x6, x7)  =  appM_out_gggggga(x1, x2, x3, x4, x5, x6, x7)
U17_ggggggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U17_ggggggaa(x1, x2, x3, x4, x5, x6, x7, x9)
U3_ga(x1, x2, x3)  =  U3_ga(x1, x3)
pD_in_aaaagaa(x1, x2, x3, x4, x5, x6, x7)  =  pD_in_aaaagaa(x5)
U18_aaaagaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U18_aaaagaa(x5, x8)
appH_in_aaaag(x1, x2, x3, x4, x5)  =  appH_in_aaaag(x5)
s(x1, x2)  =  s(x1, x2)
appH_out_aaaag(x1, x2, x3, x4, x5)  =  appH_out_aaaag(x1, x2, x3, x4, x5)
U7_aaaag(x1, x2, x3, x4, x5, x6, x7)  =  U7_aaaag(x1, x6, x7)
U19_aaaagaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U19_aaaagaa(x1, x2, x3, x4, x5, x8)
pO_in_ggggaa(x1, x2, x3, x4, x5, x6)  =  pO_in_ggggaa(x1, x2, x3, x4)
U20_ggggaa(x1, x2, x3, x4, x5, x6, x7)  =  U20_ggggaa(x1, x2, x3, x4, x7)
appI_in_gggga(x1, x2, x3, x4, x5)  =  appI_in_gggga(x1, x2, x3, x4)
appI_out_gggga(x1, x2, x3, x4, x5)  =  appI_out_gggga(x1, x2, x3, x4, x5)
U8_gggga(x1, x2, x3, x4, x5, x6, x7)  =  U8_gggga(x1, x2, x3, x4, x5, x7)
U21_ggggaa(x1, x2, x3, x4, x5, x6, x7)  =  U21_ggggaa(x1, x2, x3, x4, x5, x7)
U4_ga(x1, x2, x3)  =  U4_ga(x1, x3)
pE_in_aagaa(x1, x2, x3, x4, x5)  =  pE_in_aagaa(x3)
U22_aagaa(x1, x2, x3, x4, x5, x6)  =  U22_aagaa(x3, x6)
appJ_in_aag(x1, x2, x3)  =  appJ_in_aag(x3)
appJ_out_aag(x1, x2, x3)  =  appJ_out_aag(x1, x2, x3)
U9_aag(x1, x2, x3, x4, x5)  =  U9_aag(x1, x4, x5)
U23_aagaa(x1, x2, x3, x4, x5, x6)  =  U23_aagaa(x1, x2, x3, x6)
pP_in_ggaa(x1, x2, x3, x4)  =  pP_in_ggaa(x1, x2)
U24_ggaa(x1, x2, x3, x4, x5)  =  U24_ggaa(x1, x2, x5)
appK_in_gga(x1, x2, x3)  =  appK_in_gga(x1, x2)
appK_out_gga(x1, x2, x3)  =  appK_out_gga(x1, x2, x3)
U10_gga(x1, x2, x3, x4, x5)  =  U10_gga(x1, x2, x3, x5)
U25_ggaa(x1, x2, x3, x4, x5)  =  U25_ggaa(x1, x2, x3, x5)
parseA_out_ga(x1, x2)  =  parseA_out_ga(x1, x2)
pP_out_ggaa(x1, x2, x3, x4)  =  pP_out_ggaa(x1, x2, x3, x4)
pE_out_aagaa(x1, x2, x3, x4, x5)  =  pE_out_aagaa(x1, x2, x3, x4, x5)
pO_out_ggggaa(x1, x2, x3, x4, x5, x6)  =  pO_out_ggggaa(x1, x2, x3, x4, x5, x6)
pD_out_aaaagaa(x1, x2, x3, x4, x5, x6, x7)  =  pD_out_aaaagaa(x1, x2, x3, x4, x5, x6, x7)
pN_out_ggggggaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  pN_out_ggggggaa(x1, x2, x3, x4, x5, x6, x7, x8)
pC_out_aaaaaggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  pC_out_aaaaaggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)
pB_out_ggggaa(x1, x2, x3, x4, x5, x6)  =  pB_out_ggggaa(x1, x2, x3, x4, x5, x6)
PARSEA_IN_GA(x1, x2)  =  PARSEA_IN_GA(x1)
U1_GA(x1, x2, x3, x4, x5, x6)  =  U1_GA(x1, x2, x3, x4, x6)
PB_IN_GGGGAA(x1, x2, x3, x4, x5, x6)  =  PB_IN_GGGGAA(x1, x2, x3, x4)
U12_GGGGAA(x1, x2, x3, x4, x5, x6, x7)  =  U12_GGGGAA(x1, x2, x3, x4, x7)
APPL_IN_GGGGA(x1, x2, x3, x4, x5)  =  APPL_IN_GGGGA(x1, x2, x3, x4)
U13_GGGGAA(x1, x2, x3, x4, x5, x6, x7)  =  U13_GGGGAA(x1, x2, x3, x4, x5, x7)
U2_GA(x1, x2, x3, x4)  =  U2_GA(x1, x2, x4)
PC_IN_AAAAAGGAA(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  PC_IN_AAAAAGGAA(x6, x7)
U14_AAAAAGGAA(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10)  =  U14_AAAAAGGAA(x6, x7, x10)
APPF_IN_AAAAAG(x1, x2, x3, x4, x5, x6)  =  APPF_IN_AAAAAG(x6)
U5_AAAAAG(x1, x2, x3, x4, x5, x6, x7, x8)  =  U5_AAAAAG(x1, x7, x8)
U15_AAAAAGGAA(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10)  =  U15_AAAAAGGAA(x1, x2, x3, x4, x5, x6, x7, x10)
PN_IN_GGGGGGAA(x1, x2, x3, x4, x5, x6, x7, x8)  =  PN_IN_GGGGGGAA(x1, x2, x3, x4, x5, x6)
U16_GGGGGGAA(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U16_GGGGGGAA(x1, x2, x3, x4, x5, x6, x9)
APPM_IN_GGGGGGA(x1, x2, x3, x4, x5, x6, x7)  =  APPM_IN_GGGGGGA(x1, x2, x3, x4, x5, x6)
U11_GGGGGGA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U11_GGGGGGA(x1, x2, x3, x4, x5, x6, x8)
APPG_IN_GGGGGA(x1, x2, x3, x4, x5, x6)  =  APPG_IN_GGGGGA(x1, x2, x3, x4, x5)
U6_GGGGGA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U6_GGGGGA(x1, x2, x3, x4, x5, x6, x8)
U17_GGGGGGAA(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U17_GGGGGGAA(x1, x2, x3, x4, x5, x6, x7, x9)
U3_GA(x1, x2, x3)  =  U3_GA(x1, x3)
PD_IN_AAAAGAA(x1, x2, x3, x4, x5, x6, x7)  =  PD_IN_AAAAGAA(x5)
U18_AAAAGAA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U18_AAAAGAA(x5, x8)
APPH_IN_AAAAG(x1, x2, x3, x4, x5)  =  APPH_IN_AAAAG(x5)
U7_AAAAG(x1, x2, x3, x4, x5, x6, x7)  =  U7_AAAAG(x1, x6, x7)
U19_AAAAGAA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U19_AAAAGAA(x1, x2, x3, x4, x5, x8)
PO_IN_GGGGAA(x1, x2, x3, x4, x5, x6)  =  PO_IN_GGGGAA(x1, x2, x3, x4)
U20_GGGGAA(x1, x2, x3, x4, x5, x6, x7)  =  U20_GGGGAA(x1, x2, x3, x4, x7)
APPI_IN_GGGGA(x1, x2, x3, x4, x5)  =  APPI_IN_GGGGA(x1, x2, x3, x4)
U8_GGGGA(x1, x2, x3, x4, x5, x6, x7)  =  U8_GGGGA(x1, x2, x3, x4, x5, x7)
U21_GGGGAA(x1, x2, x3, x4, x5, x6, x7)  =  U21_GGGGAA(x1, x2, x3, x4, x5, x7)
U4_GA(x1, x2, x3)  =  U4_GA(x1, x3)
PE_IN_AAGAA(x1, x2, x3, x4, x5)  =  PE_IN_AAGAA(x3)
U22_AAGAA(x1, x2, x3, x4, x5, x6)  =  U22_AAGAA(x3, x6)
APPJ_IN_AAG(x1, x2, x3)  =  APPJ_IN_AAG(x3)
U9_AAG(x1, x2, x3, x4, x5)  =  U9_AAG(x1, x4, x5)
U23_AAGAA(x1, x2, x3, x4, x5, x6)  =  U23_AAGAA(x1, x2, x3, x6)
PP_IN_GGAA(x1, x2, x3, x4)  =  PP_IN_GGAA(x1, x2)
U24_GGAA(x1, x2, x3, x4, x5)  =  U24_GGAA(x1, x2, x5)
APPK_IN_GGA(x1, x2, x3)  =  APPK_IN_GGA(x1, x2)
U10_GGA(x1, x2, x3, x4, x5)  =  U10_GGA(x1, x2, x3, x5)
U25_GGAA(x1, x2, x3, x4, x5)  =  U25_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:

PARSEA_IN_GA(.(a, .(s(T24, T25, T26), .(b, T27))), T7) → U1_GA(T24, T25, T26, T27, T7, pB_in_ggggaa(T24, T25, T26, T27, X10, T7))
PARSEA_IN_GA(.(a, .(s(T24, T25, T26), .(b, T27))), T7) → PB_IN_GGGGAA(T24, T25, T26, T27, X10, T7)
PB_IN_GGGGAA(T24, T25, T26, T27, T33, T7) → U12_GGGGAA(T24, T25, T26, T27, T33, T7, appL_in_gggga(T24, T25, T26, T27, T33))
PB_IN_GGGGAA(T24, T25, T26, T27, T33, T7) → APPL_IN_GGGGA(T24, T25, T26, T27, T33)
U12_GGGGAA(T24, T25, T26, T27, T33, T7, appL_out_gggga(T24, T25, T26, T27, T33)) → U13_GGGGAA(T24, T25, T26, T27, T33, T7, parseA_in_ga(T33, T7))
U12_GGGGAA(T24, T25, T26, T27, T33, T7, appL_out_gggga(T24, T25, T26, T27, T33)) → PARSEA_IN_GA(T33, T7)
PARSEA_IN_GA(.(T68, T69), T7) → U2_GA(T68, T69, T7, pC_in_aaaaaggaa(X109, X110, X111, X112, X113, T69, T68, X10, T7))
PARSEA_IN_GA(.(T68, T69), T7) → PC_IN_AAAAAGGAA(X109, X110, X111, X112, X113, T69, T68, X10, T7)
PC_IN_AAAAAGGAA(T78, T79, T80, T81, T82, T69, T68, X10, T7) → U14_AAAAAGGAA(T78, T79, T80, T81, T82, T69, T68, X10, T7, appF_in_aaaaag(T78, T79, T80, T81, T82, T69))
PC_IN_AAAAAGGAA(T78, T79, T80, T81, T82, T69, T68, X10, T7) → APPF_IN_AAAAAG(T78, T79, T80, T81, T82, T69)
APPF_IN_AAAAAG(.(T115, X184), X185, X186, X187, X188, .(T115, T116)) → U5_AAAAAG(T115, X184, X185, X186, X187, X188, T116, appF_in_aaaaag(X184, X185, X186, X187, X188, T116))
APPF_IN_AAAAAG(.(T115, X184), X185, X186, X187, X188, .(T115, T116)) → APPF_IN_AAAAAG(X184, X185, X186, X187, X188, T116)
U14_AAAAAGGAA(T78, T79, T80, T81, T82, T69, T68, X10, T7, appF_out_aaaaag(T78, T79, T80, T81, T82, T69)) → U15_AAAAAGGAA(T78, T79, T80, T81, T82, T69, T68, X10, T7, pN_in_ggggggaa(T68, T78, T79, T80, T81, T82, X10, T7))
U14_AAAAAGGAA(T78, T79, T80, T81, T82, T69, T68, X10, T7, appF_out_aaaaag(T78, T79, T80, T81, T82, T69)) → PN_IN_GGGGGGAA(T68, T78, T79, T80, T81, T82, X10, T7)
PN_IN_GGGGGGAA(T68, T78, T79, T80, T81, T82, T125, T7) → U16_GGGGGGAA(T68, T78, T79, T80, T81, T82, T125, T7, appM_in_gggggga(T68, T78, T79, T80, T81, T82, T125))
PN_IN_GGGGGGAA(T68, T78, T79, T80, T81, T82, T125, T7) → APPM_IN_GGGGGGA(T68, T78, T79, T80, T81, T82, T125)
APPM_IN_GGGGGGA(T146, T147, T148, T149, T150, T151, .(T146, X220)) → U11_GGGGGGA(T146, T147, T148, T149, T150, T151, X220, appG_in_ggggga(T147, T148, T149, T150, T151, X220))
APPM_IN_GGGGGGA(T146, T147, T148, T149, T150, T151, .(T146, X220)) → APPG_IN_GGGGGA(T147, T148, T149, T150, T151, X220)
APPG_IN_GGGGGA(.(T192, T193), T194, T195, T196, T197, .(T192, X242)) → U6_GGGGGA(T192, T193, T194, T195, T196, T197, X242, appG_in_ggggga(T193, T194, T195, T196, T197, X242))
APPG_IN_GGGGGA(.(T192, T193), T194, T195, T196, T197, .(T192, X242)) → APPG_IN_GGGGGA(T193, T194, T195, T196, T197, X242)
U16_GGGGGGAA(T68, T78, T79, T80, T81, T82, T125, T7, appM_out_gggggga(T68, T78, T79, T80, T81, T82, T125)) → U17_GGGGGGAA(T68, T78, T79, T80, T81, T82, T125, T7, parseA_in_ga(T125, T7))
U16_GGGGGGAA(T68, T78, T79, T80, T81, T82, T125, T7, appM_out_gggggga(T68, T78, T79, T80, T81, T82, T125)) → PARSEA_IN_GA(T125, T7)
PARSEA_IN_GA(T230, T232) → U3_GA(T230, T232, pD_in_aaaagaa(X289, X290, X291, X292, T230, X293, T232))
PARSEA_IN_GA(T230, T232) → PD_IN_AAAAGAA(X289, X290, X291, X292, T230, X293, T232)
PD_IN_AAAAGAA(T239, T240, T241, T242, T230, X293, T232) → U18_AAAAGAA(T239, T240, T241, T242, T230, X293, T232, appH_in_aaaag(T239, T240, T241, T242, T230))
PD_IN_AAAAGAA(T239, T240, T241, T242, T230, X293, T232) → APPH_IN_AAAAG(T239, T240, T241, T242, T230)
APPH_IN_AAAAG(.(T268, X353), X354, X355, X356, .(T268, T269)) → U7_AAAAG(T268, X353, X354, X355, X356, T269, appH_in_aaaag(X353, X354, X355, X356, T269))
APPH_IN_AAAAG(.(T268, X353), X354, X355, X356, .(T268, T269)) → APPH_IN_AAAAG(X353, X354, X355, X356, T269)
U18_AAAAGAA(T239, T240, T241, T242, T230, X293, T232, appH_out_aaaag(T239, T240, T241, T242, T230)) → U19_AAAAGAA(T239, T240, T241, T242, T230, X293, T232, pO_in_ggggaa(T239, T240, T241, T242, X293, T232))
U18_AAAAGAA(T239, T240, T241, T242, T230, X293, T232, appH_out_aaaag(T239, T240, T241, T242, T230)) → PO_IN_GGGGAA(T239, T240, T241, T242, X293, T232)
PO_IN_GGGGAA(T239, T240, T241, T242, T282, T232) → U20_GGGGAA(T239, T240, T241, T242, T282, T232, appI_in_gggga(T239, T240, T241, T242, T282))
PO_IN_GGGGAA(T239, T240, T241, T242, T282, T232) → APPI_IN_GGGGA(T239, T240, T241, T242, T282)
APPI_IN_GGGGA(.(T314, T315), T316, T317, T318, .(T314, X388)) → U8_GGGGA(T314, T315, T316, T317, T318, X388, appI_in_gggga(T315, T316, T317, T318, X388))
APPI_IN_GGGGA(.(T314, T315), T316, T317, T318, .(T314, X388)) → APPI_IN_GGGGA(T315, T316, T317, T318, X388)
U20_GGGGAA(T239, T240, T241, T242, T282, T232, appI_out_gggga(T239, T240, T241, T242, T282)) → U21_GGGGAA(T239, T240, T241, T242, T282, T232, parseA_in_ga(T282, T232))
U20_GGGGAA(T239, T240, T241, T242, T282, T232, appI_out_gggga(T239, T240, T241, T242, T282)) → PARSEA_IN_GA(T282, T232)
PARSEA_IN_GA(T341, T343) → U4_GA(T341, T343, pE_in_aagaa(X423, X424, T341, X425, T343))
PARSEA_IN_GA(T341, T343) → PE_IN_AAGAA(X423, X424, T341, X425, T343)
PE_IN_AAGAA(T346, T347, T341, X425, T343) → U22_AAGAA(T346, T347, T341, X425, T343, appJ_in_aag(T346, T347, T341))
PE_IN_AAGAA(T346, T347, T341, X425, T343) → APPJ_IN_AAG(T346, T347, T341)
APPJ_IN_AAG(.(T359, X463), X464, .(T359, T360)) → U9_AAG(T359, X463, X464, T360, appJ_in_aag(X463, X464, T360))
APPJ_IN_AAG(.(T359, X463), X464, .(T359, T360)) → APPJ_IN_AAG(X463, X464, T360)
U22_AAGAA(T346, T347, T341, X425, T343, appJ_out_aag(T346, T347, T341)) → U23_AAGAA(T346, T347, T341, X425, T343, pP_in_ggaa(T346, T347, X425, T343))
U22_AAGAA(T346, T347, T341, X425, T343, appJ_out_aag(T346, T347, T341)) → PP_IN_GGAA(T346, T347, X425, T343)
PP_IN_GGAA(T346, T347, T365, T343) → U24_GGAA(T346, T347, T365, T343, appK_in_gga(T346, T347, T365))
PP_IN_GGAA(T346, T347, T365, T343) → APPK_IN_GGA(T346, T347, T365)
APPK_IN_GGA(.(T379, T380), T381, .(T379, X492)) → U10_GGA(T379, T380, T381, X492, appK_in_gga(T380, T381, X492))
APPK_IN_GGA(.(T379, T380), T381, .(T379, X492)) → APPK_IN_GGA(T380, T381, X492)
U24_GGAA(T346, T347, T365, T343, appK_out_gga(T346, T347, T365)) → U25_GGAA(T346, T347, T365, T343, parseA_in_ga(T365, T343))
U24_GGAA(T346, T347, T365, T343, appK_out_gga(T346, T347, T365)) → PARSEA_IN_GA(T365, T343)

The TRS R consists of the following rules:

parseA_in_ga(.(a, .(s(T24, T25, T26), .(b, T27))), T7) → U1_ga(T24, T25, T26, T27, T7, pB_in_ggggaa(T24, T25, T26, T27, X10, T7))
pB_in_ggggaa(T24, T25, T26, T27, T33, T7) → U12_ggggaa(T24, T25, T26, T27, T33, T7, appL_in_gggga(T24, T25, T26, T27, T33))
appL_in_gggga(T54, T55, T56, T57, .(s(a, s(T54, T55, T56), b), T57)) → appL_out_gggga(T54, T55, T56, T57, .(s(a, s(T54, T55, T56), b), T57))
U12_ggggaa(T24, T25, T26, T27, T33, T7, appL_out_gggga(T24, T25, T26, T27, T33)) → U13_ggggaa(T24, T25, T26, T27, T33, T7, parseA_in_ga(T33, T7))
parseA_in_ga(.(T68, T69), T7) → U2_ga(T68, T69, T7, pC_in_aaaaaggaa(X109, X110, X111, X112, X113, T69, T68, X10, T7))
pC_in_aaaaaggaa(T78, T79, T80, T81, T82, T69, T68, X10, T7) → U14_aaaaaggaa(T78, T79, T80, T81, T82, T69, T68, X10, T7, appF_in_aaaaag(T78, T79, T80, T81, T82, T69))
appF_in_aaaaag([], T107, T108, T109, T110, .(a, .(s(T107, T108, T109), .(b, T110)))) → appF_out_aaaaag([], T107, T108, T109, T110, .(a, .(s(T107, T108, T109), .(b, T110))))
appF_in_aaaaag(.(T115, X184), X185, X186, X187, X188, .(T115, T116)) → U5_aaaaag(T115, X184, X185, X186, X187, X188, T116, appF_in_aaaaag(X184, X185, X186, X187, X188, T116))
U5_aaaaag(T115, X184, X185, X186, X187, X188, T116, appF_out_aaaaag(X184, X185, X186, X187, X188, T116)) → appF_out_aaaaag(.(T115, X184), X185, X186, X187, X188, .(T115, T116))
U14_aaaaaggaa(T78, T79, T80, T81, T82, T69, T68, X10, T7, appF_out_aaaaag(T78, T79, T80, T81, T82, T69)) → U15_aaaaaggaa(T78, T79, T80, T81, T82, T69, T68, X10, T7, pN_in_ggggggaa(T68, T78, T79, T80, T81, T82, X10, T7))
pN_in_ggggggaa(T68, T78, T79, T80, T81, T82, T125, T7) → U16_ggggggaa(T68, T78, T79, T80, T81, T82, T125, T7, appM_in_gggggga(T68, T78, T79, T80, T81, T82, T125))
appM_in_gggggga(T146, T147, T148, T149, T150, T151, .(T146, X220)) → U11_gggggga(T146, T147, T148, T149, T150, T151, X220, appG_in_ggggga(T147, T148, T149, T150, T151, X220))
appG_in_ggggga([], T176, T177, T178, T179, .(s(a, s(T176, T177, T178), b), T179)) → appG_out_ggggga([], T176, T177, T178, T179, .(s(a, s(T176, T177, T178), b), T179))
appG_in_ggggga(.(T192, T193), T194, T195, T196, T197, .(T192, X242)) → U6_ggggga(T192, T193, T194, T195, T196, T197, X242, appG_in_ggggga(T193, T194, T195, T196, T197, X242))
U6_ggggga(T192, T193, T194, T195, T196, T197, X242, appG_out_ggggga(T193, T194, T195, T196, T197, X242)) → appG_out_ggggga(.(T192, T193), T194, T195, T196, T197, .(T192, X242))
U11_gggggga(T146, T147, T148, T149, T150, T151, X220, appG_out_ggggga(T147, T148, T149, T150, T151, X220)) → appM_out_gggggga(T146, T147, T148, T149, T150, T151, .(T146, X220))
U16_ggggggaa(T68, T78, T79, T80, T81, T82, T125, T7, appM_out_gggggga(T68, T78, T79, T80, T81, T82, T125)) → U17_ggggggaa(T68, T78, T79, T80, T81, T82, T125, T7, parseA_in_ga(T125, T7))
parseA_in_ga(T230, T232) → U3_ga(T230, T232, pD_in_aaaagaa(X289, X290, X291, X292, T230, X293, T232))
pD_in_aaaagaa(T239, T240, T241, T242, T230, X293, T232) → U18_aaaagaa(T239, T240, T241, T242, T230, X293, T232, appH_in_aaaag(T239, T240, T241, T242, T230))
appH_in_aaaag([], T261, T262, T263, .(a, .(s(T261, T262), .(b, T263)))) → appH_out_aaaag([], T261, T262, T263, .(a, .(s(T261, T262), .(b, T263))))
appH_in_aaaag(.(T268, X353), X354, X355, X356, .(T268, T269)) → U7_aaaag(T268, X353, X354, X355, X356, T269, appH_in_aaaag(X353, X354, X355, X356, T269))
U7_aaaag(T268, X353, X354, X355, X356, T269, appH_out_aaaag(X353, X354, X355, X356, T269)) → appH_out_aaaag(.(T268, X353), X354, X355, X356, .(T268, T269))
U18_aaaagaa(T239, T240, T241, T242, T230, X293, T232, appH_out_aaaag(T239, T240, T241, T242, T230)) → U19_aaaagaa(T239, T240, T241, T242, T230, X293, T232, pO_in_ggggaa(T239, T240, T241, T242, X293, T232))
pO_in_ggggaa(T239, T240, T241, T242, T282, T232) → U20_ggggaa(T239, T240, T241, T242, T282, T232, appI_in_gggga(T239, T240, T241, T242, T282))
appI_in_gggga([], T301, T302, T303, .(s(a, s(T301, T302), b), T303)) → appI_out_gggga([], T301, T302, T303, .(s(a, s(T301, T302), b), T303))
appI_in_gggga(.(T314, T315), T316, T317, T318, .(T314, X388)) → U8_gggga(T314, T315, T316, T317, T318, X388, appI_in_gggga(T315, T316, T317, T318, X388))
U8_gggga(T314, T315, T316, T317, T318, X388, appI_out_gggga(T315, T316, T317, T318, X388)) → appI_out_gggga(.(T314, T315), T316, T317, T318, .(T314, X388))
U20_ggggaa(T239, T240, T241, T242, T282, T232, appI_out_gggga(T239, T240, T241, T242, T282)) → U21_ggggaa(T239, T240, T241, T242, T282, T232, parseA_in_ga(T282, T232))
parseA_in_ga(T341, T343) → U4_ga(T341, T343, pE_in_aagaa(X423, X424, T341, X425, T343))
pE_in_aagaa(T346, T347, T341, X425, T343) → U22_aagaa(T346, T347, T341, X425, T343, appJ_in_aag(T346, T347, T341))
appJ_in_aag([], T354, .(a, .(b, T354))) → appJ_out_aag([], T354, .(a, .(b, T354)))
appJ_in_aag(.(T359, X463), X464, .(T359, T360)) → U9_aag(T359, X463, X464, T360, appJ_in_aag(X463, X464, T360))
U9_aag(T359, X463, X464, T360, appJ_out_aag(X463, X464, T360)) → appJ_out_aag(.(T359, X463), X464, .(T359, T360))
U22_aagaa(T346, T347, T341, X425, T343, appJ_out_aag(T346, T347, T341)) → U23_aagaa(T346, T347, T341, X425, T343, pP_in_ggaa(T346, T347, X425, T343))
pP_in_ggaa(T346, T347, T365, T343) → U24_ggaa(T346, T347, T365, T343, appK_in_gga(T346, T347, T365))
appK_in_gga([], T372, .(s(a, b), T372)) → appK_out_gga([], T372, .(s(a, b), T372))
appK_in_gga(.(T379, T380), T381, .(T379, X492)) → U10_gga(T379, T380, T381, X492, appK_in_gga(T380, T381, X492))
U10_gga(T379, T380, T381, X492, appK_out_gga(T380, T381, X492)) → appK_out_gga(.(T379, T380), T381, .(T379, X492))
U24_ggaa(T346, T347, T365, T343, appK_out_gga(T346, T347, T365)) → U25_ggaa(T346, T347, T365, T343, parseA_in_ga(T365, T343))
parseA_in_ga(.(s(T394, T395), []), s(T394, T395)) → parseA_out_ga(.(s(T394, T395), []), s(T394, T395))
parseA_in_ga(.(s(T402, T403, T404), []), s(T402, T403, T404)) → parseA_out_ga(.(s(T402, T403, T404), []), s(T402, T403, T404))
U25_ggaa(T346, T347, T365, T343, parseA_out_ga(T365, T343)) → pP_out_ggaa(T346, T347, T365, T343)
U23_aagaa(T346, T347, T341, X425, T343, pP_out_ggaa(T346, T347, X425, T343)) → pE_out_aagaa(T346, T347, T341, X425, T343)
U4_ga(T341, T343, pE_out_aagaa(X423, X424, T341, X425, T343)) → parseA_out_ga(T341, T343)
U21_ggggaa(T239, T240, T241, T242, T282, T232, parseA_out_ga(T282, T232)) → pO_out_ggggaa(T239, T240, T241, T242, T282, T232)
U19_aaaagaa(T239, T240, T241, T242, T230, X293, T232, pO_out_ggggaa(T239, T240, T241, T242, X293, T232)) → pD_out_aaaagaa(T239, T240, T241, T242, T230, X293, T232)
U3_ga(T230, T232, pD_out_aaaagaa(X289, X290, X291, X292, T230, X293, T232)) → parseA_out_ga(T230, T232)
U17_ggggggaa(T68, T78, T79, T80, T81, T82, T125, T7, parseA_out_ga(T125, T7)) → pN_out_ggggggaa(T68, T78, T79, T80, T81, T82, T125, T7)
U15_aaaaaggaa(T78, T79, T80, T81, T82, T69, T68, X10, T7, pN_out_ggggggaa(T68, T78, T79, T80, T81, T82, X10, T7)) → pC_out_aaaaaggaa(T78, T79, T80, T81, T82, T69, T68, X10, T7)
U2_ga(T68, T69, T7, pC_out_aaaaaggaa(X109, X110, X111, X112, X113, T69, T68, X10, T7)) → parseA_out_ga(.(T68, T69), T7)
U13_ggggaa(T24, T25, T26, T27, T33, T7, parseA_out_ga(T33, T7)) → pB_out_ggggaa(T24, T25, T26, T27, T33, T7)
U1_ga(T24, T25, T26, T27, T7, pB_out_ggggaa(T24, T25, T26, T27, X10, T7)) → parseA_out_ga(.(a, .(s(T24, T25, T26), .(b, T27))), T7)

The argument filtering Pi contains the following mapping:
parseA_in_ga(x1, x2)  =  parseA_in_ga(x1)
.(x1, x2)  =  .(x1, x2)
a  =  a
s(x1, x2, x3)  =  s(x1, x2, x3)
b  =  b
U1_ga(x1, x2, x3, x4, x5, x6)  =  U1_ga(x1, x2, x3, x4, x6)
pB_in_ggggaa(x1, x2, x3, x4, x5, x6)  =  pB_in_ggggaa(x1, x2, x3, x4)
U12_ggggaa(x1, x2, x3, x4, x5, x6, x7)  =  U12_ggggaa(x1, x2, x3, x4, x7)
appL_in_gggga(x1, x2, x3, x4, x5)  =  appL_in_gggga(x1, x2, x3, x4)
appL_out_gggga(x1, x2, x3, x4, x5)  =  appL_out_gggga(x1, x2, x3, x4, x5)
U13_ggggaa(x1, x2, x3, x4, x5, x6, x7)  =  U13_ggggaa(x1, x2, x3, x4, x5, x7)
U2_ga(x1, x2, x3, x4)  =  U2_ga(x1, x2, x4)
pC_in_aaaaaggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  pC_in_aaaaaggaa(x6, x7)
U14_aaaaaggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10)  =  U14_aaaaaggaa(x6, x7, x10)
appF_in_aaaaag(x1, x2, x3, x4, x5, x6)  =  appF_in_aaaaag(x6)
appF_out_aaaaag(x1, x2, x3, x4, x5, x6)  =  appF_out_aaaaag(x1, x2, x3, x4, x5, x6)
U5_aaaaag(x1, x2, x3, x4, x5, x6, x7, x8)  =  U5_aaaaag(x1, x7, x8)
U15_aaaaaggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10)  =  U15_aaaaaggaa(x1, x2, x3, x4, x5, x6, x7, x10)
pN_in_ggggggaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  pN_in_ggggggaa(x1, x2, x3, x4, x5, x6)
U16_ggggggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U16_ggggggaa(x1, x2, x3, x4, x5, x6, x9)
appM_in_gggggga(x1, x2, x3, x4, x5, x6, x7)  =  appM_in_gggggga(x1, x2, x3, x4, x5, x6)
U11_gggggga(x1, x2, x3, x4, x5, x6, x7, x8)  =  U11_gggggga(x1, x2, x3, x4, x5, x6, x8)
appG_in_ggggga(x1, x2, x3, x4, x5, x6)  =  appG_in_ggggga(x1, x2, x3, x4, x5)
[]  =  []
appG_out_ggggga(x1, x2, x3, x4, x5, x6)  =  appG_out_ggggga(x1, x2, x3, x4, x5, x6)
U6_ggggga(x1, x2, x3, x4, x5, x6, x7, x8)  =  U6_ggggga(x1, x2, x3, x4, x5, x6, x8)
appM_out_gggggga(x1, x2, x3, x4, x5, x6, x7)  =  appM_out_gggggga(x1, x2, x3, x4, x5, x6, x7)
U17_ggggggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U17_ggggggaa(x1, x2, x3, x4, x5, x6, x7, x9)
U3_ga(x1, x2, x3)  =  U3_ga(x1, x3)
pD_in_aaaagaa(x1, x2, x3, x4, x5, x6, x7)  =  pD_in_aaaagaa(x5)
U18_aaaagaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U18_aaaagaa(x5, x8)
appH_in_aaaag(x1, x2, x3, x4, x5)  =  appH_in_aaaag(x5)
s(x1, x2)  =  s(x1, x2)
appH_out_aaaag(x1, x2, x3, x4, x5)  =  appH_out_aaaag(x1, x2, x3, x4, x5)
U7_aaaag(x1, x2, x3, x4, x5, x6, x7)  =  U7_aaaag(x1, x6, x7)
U19_aaaagaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U19_aaaagaa(x1, x2, x3, x4, x5, x8)
pO_in_ggggaa(x1, x2, x3, x4, x5, x6)  =  pO_in_ggggaa(x1, x2, x3, x4)
U20_ggggaa(x1, x2, x3, x4, x5, x6, x7)  =  U20_ggggaa(x1, x2, x3, x4, x7)
appI_in_gggga(x1, x2, x3, x4, x5)  =  appI_in_gggga(x1, x2, x3, x4)
appI_out_gggga(x1, x2, x3, x4, x5)  =  appI_out_gggga(x1, x2, x3, x4, x5)
U8_gggga(x1, x2, x3, x4, x5, x6, x7)  =  U8_gggga(x1, x2, x3, x4, x5, x7)
U21_ggggaa(x1, x2, x3, x4, x5, x6, x7)  =  U21_ggggaa(x1, x2, x3, x4, x5, x7)
U4_ga(x1, x2, x3)  =  U4_ga(x1, x3)
pE_in_aagaa(x1, x2, x3, x4, x5)  =  pE_in_aagaa(x3)
U22_aagaa(x1, x2, x3, x4, x5, x6)  =  U22_aagaa(x3, x6)
appJ_in_aag(x1, x2, x3)  =  appJ_in_aag(x3)
appJ_out_aag(x1, x2, x3)  =  appJ_out_aag(x1, x2, x3)
U9_aag(x1, x2, x3, x4, x5)  =  U9_aag(x1, x4, x5)
U23_aagaa(x1, x2, x3, x4, x5, x6)  =  U23_aagaa(x1, x2, x3, x6)
pP_in_ggaa(x1, x2, x3, x4)  =  pP_in_ggaa(x1, x2)
U24_ggaa(x1, x2, x3, x4, x5)  =  U24_ggaa(x1, x2, x5)
appK_in_gga(x1, x2, x3)  =  appK_in_gga(x1, x2)
appK_out_gga(x1, x2, x3)  =  appK_out_gga(x1, x2, x3)
U10_gga(x1, x2, x3, x4, x5)  =  U10_gga(x1, x2, x3, x5)
U25_ggaa(x1, x2, x3, x4, x5)  =  U25_ggaa(x1, x2, x3, x5)
parseA_out_ga(x1, x2)  =  parseA_out_ga(x1, x2)
pP_out_ggaa(x1, x2, x3, x4)  =  pP_out_ggaa(x1, x2, x3, x4)
pE_out_aagaa(x1, x2, x3, x4, x5)  =  pE_out_aagaa(x1, x2, x3, x4, x5)
pO_out_ggggaa(x1, x2, x3, x4, x5, x6)  =  pO_out_ggggaa(x1, x2, x3, x4, x5, x6)
pD_out_aaaagaa(x1, x2, x3, x4, x5, x6, x7)  =  pD_out_aaaagaa(x1, x2, x3, x4, x5, x6, x7)
pN_out_ggggggaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  pN_out_ggggggaa(x1, x2, x3, x4, x5, x6, x7, x8)
pC_out_aaaaaggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  pC_out_aaaaaggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)
pB_out_ggggaa(x1, x2, x3, x4, x5, x6)  =  pB_out_ggggaa(x1, x2, x3, x4, x5, x6)
PARSEA_IN_GA(x1, x2)  =  PARSEA_IN_GA(x1)
U1_GA(x1, x2, x3, x4, x5, x6)  =  U1_GA(x1, x2, x3, x4, x6)
PB_IN_GGGGAA(x1, x2, x3, x4, x5, x6)  =  PB_IN_GGGGAA(x1, x2, x3, x4)
U12_GGGGAA(x1, x2, x3, x4, x5, x6, x7)  =  U12_GGGGAA(x1, x2, x3, x4, x7)
APPL_IN_GGGGA(x1, x2, x3, x4, x5)  =  APPL_IN_GGGGA(x1, x2, x3, x4)
U13_GGGGAA(x1, x2, x3, x4, x5, x6, x7)  =  U13_GGGGAA(x1, x2, x3, x4, x5, x7)
U2_GA(x1, x2, x3, x4)  =  U2_GA(x1, x2, x4)
PC_IN_AAAAAGGAA(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  PC_IN_AAAAAGGAA(x6, x7)
U14_AAAAAGGAA(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10)  =  U14_AAAAAGGAA(x6, x7, x10)
APPF_IN_AAAAAG(x1, x2, x3, x4, x5, x6)  =  APPF_IN_AAAAAG(x6)
U5_AAAAAG(x1, x2, x3, x4, x5, x6, x7, x8)  =  U5_AAAAAG(x1, x7, x8)
U15_AAAAAGGAA(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10)  =  U15_AAAAAGGAA(x1, x2, x3, x4, x5, x6, x7, x10)
PN_IN_GGGGGGAA(x1, x2, x3, x4, x5, x6, x7, x8)  =  PN_IN_GGGGGGAA(x1, x2, x3, x4, x5, x6)
U16_GGGGGGAA(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U16_GGGGGGAA(x1, x2, x3, x4, x5, x6, x9)
APPM_IN_GGGGGGA(x1, x2, x3, x4, x5, x6, x7)  =  APPM_IN_GGGGGGA(x1, x2, x3, x4, x5, x6)
U11_GGGGGGA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U11_GGGGGGA(x1, x2, x3, x4, x5, x6, x8)
APPG_IN_GGGGGA(x1, x2, x3, x4, x5, x6)  =  APPG_IN_GGGGGA(x1, x2, x3, x4, x5)
U6_GGGGGA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U6_GGGGGA(x1, x2, x3, x4, x5, x6, x8)
U17_GGGGGGAA(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U17_GGGGGGAA(x1, x2, x3, x4, x5, x6, x7, x9)
U3_GA(x1, x2, x3)  =  U3_GA(x1, x3)
PD_IN_AAAAGAA(x1, x2, x3, x4, x5, x6, x7)  =  PD_IN_AAAAGAA(x5)
U18_AAAAGAA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U18_AAAAGAA(x5, x8)
APPH_IN_AAAAG(x1, x2, x3, x4, x5)  =  APPH_IN_AAAAG(x5)
U7_AAAAG(x1, x2, x3, x4, x5, x6, x7)  =  U7_AAAAG(x1, x6, x7)
U19_AAAAGAA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U19_AAAAGAA(x1, x2, x3, x4, x5, x8)
PO_IN_GGGGAA(x1, x2, x3, x4, x5, x6)  =  PO_IN_GGGGAA(x1, x2, x3, x4)
U20_GGGGAA(x1, x2, x3, x4, x5, x6, x7)  =  U20_GGGGAA(x1, x2, x3, x4, x7)
APPI_IN_GGGGA(x1, x2, x3, x4, x5)  =  APPI_IN_GGGGA(x1, x2, x3, x4)
U8_GGGGA(x1, x2, x3, x4, x5, x6, x7)  =  U8_GGGGA(x1, x2, x3, x4, x5, x7)
U21_GGGGAA(x1, x2, x3, x4, x5, x6, x7)  =  U21_GGGGAA(x1, x2, x3, x4, x5, x7)
U4_GA(x1, x2, x3)  =  U4_GA(x1, x3)
PE_IN_AAGAA(x1, x2, x3, x4, x5)  =  PE_IN_AAGAA(x3)
U22_AAGAA(x1, x2, x3, x4, x5, x6)  =  U22_AAGAA(x3, x6)
APPJ_IN_AAG(x1, x2, x3)  =  APPJ_IN_AAG(x3)
U9_AAG(x1, x2, x3, x4, x5)  =  U9_AAG(x1, x4, x5)
U23_AAGAA(x1, x2, x3, x4, x5, x6)  =  U23_AAGAA(x1, x2, x3, x6)
PP_IN_GGAA(x1, x2, x3, x4)  =  PP_IN_GGAA(x1, x2)
U24_GGAA(x1, x2, x3, x4, x5)  =  U24_GGAA(x1, x2, x5)
APPK_IN_GGA(x1, x2, x3)  =  APPK_IN_GGA(x1, x2)
U10_GGA(x1, x2, x3, x4, x5)  =  U10_GGA(x1, x2, x3, x5)
U25_GGAA(x1, x2, x3, x4, x5)  =  U25_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 7 SCCs with 26 less nodes.

(6) Complex Obligation (AND)

(7) Obligation:

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

APPK_IN_GGA(.(T379, T380), T381, .(T379, X492)) → APPK_IN_GGA(T380, T381, X492)

The TRS R consists of the following rules:

parseA_in_ga(.(a, .(s(T24, T25, T26), .(b, T27))), T7) → U1_ga(T24, T25, T26, T27, T7, pB_in_ggggaa(T24, T25, T26, T27, X10, T7))
pB_in_ggggaa(T24, T25, T26, T27, T33, T7) → U12_ggggaa(T24, T25, T26, T27, T33, T7, appL_in_gggga(T24, T25, T26, T27, T33))
appL_in_gggga(T54, T55, T56, T57, .(s(a, s(T54, T55, T56), b), T57)) → appL_out_gggga(T54, T55, T56, T57, .(s(a, s(T54, T55, T56), b), T57))
U12_ggggaa(T24, T25, T26, T27, T33, T7, appL_out_gggga(T24, T25, T26, T27, T33)) → U13_ggggaa(T24, T25, T26, T27, T33, T7, parseA_in_ga(T33, T7))
parseA_in_ga(.(T68, T69), T7) → U2_ga(T68, T69, T7, pC_in_aaaaaggaa(X109, X110, X111, X112, X113, T69, T68, X10, T7))
pC_in_aaaaaggaa(T78, T79, T80, T81, T82, T69, T68, X10, T7) → U14_aaaaaggaa(T78, T79, T80, T81, T82, T69, T68, X10, T7, appF_in_aaaaag(T78, T79, T80, T81, T82, T69))
appF_in_aaaaag([], T107, T108, T109, T110, .(a, .(s(T107, T108, T109), .(b, T110)))) → appF_out_aaaaag([], T107, T108, T109, T110, .(a, .(s(T107, T108, T109), .(b, T110))))
appF_in_aaaaag(.(T115, X184), X185, X186, X187, X188, .(T115, T116)) → U5_aaaaag(T115, X184, X185, X186, X187, X188, T116, appF_in_aaaaag(X184, X185, X186, X187, X188, T116))
U5_aaaaag(T115, X184, X185, X186, X187, X188, T116, appF_out_aaaaag(X184, X185, X186, X187, X188, T116)) → appF_out_aaaaag(.(T115, X184), X185, X186, X187, X188, .(T115, T116))
U14_aaaaaggaa(T78, T79, T80, T81, T82, T69, T68, X10, T7, appF_out_aaaaag(T78, T79, T80, T81, T82, T69)) → U15_aaaaaggaa(T78, T79, T80, T81, T82, T69, T68, X10, T7, pN_in_ggggggaa(T68, T78, T79, T80, T81, T82, X10, T7))
pN_in_ggggggaa(T68, T78, T79, T80, T81, T82, T125, T7) → U16_ggggggaa(T68, T78, T79, T80, T81, T82, T125, T7, appM_in_gggggga(T68, T78, T79, T80, T81, T82, T125))
appM_in_gggggga(T146, T147, T148, T149, T150, T151, .(T146, X220)) → U11_gggggga(T146, T147, T148, T149, T150, T151, X220, appG_in_ggggga(T147, T148, T149, T150, T151, X220))
appG_in_ggggga([], T176, T177, T178, T179, .(s(a, s(T176, T177, T178), b), T179)) → appG_out_ggggga([], T176, T177, T178, T179, .(s(a, s(T176, T177, T178), b), T179))
appG_in_ggggga(.(T192, T193), T194, T195, T196, T197, .(T192, X242)) → U6_ggggga(T192, T193, T194, T195, T196, T197, X242, appG_in_ggggga(T193, T194, T195, T196, T197, X242))
U6_ggggga(T192, T193, T194, T195, T196, T197, X242, appG_out_ggggga(T193, T194, T195, T196, T197, X242)) → appG_out_ggggga(.(T192, T193), T194, T195, T196, T197, .(T192, X242))
U11_gggggga(T146, T147, T148, T149, T150, T151, X220, appG_out_ggggga(T147, T148, T149, T150, T151, X220)) → appM_out_gggggga(T146, T147, T148, T149, T150, T151, .(T146, X220))
U16_ggggggaa(T68, T78, T79, T80, T81, T82, T125, T7, appM_out_gggggga(T68, T78, T79, T80, T81, T82, T125)) → U17_ggggggaa(T68, T78, T79, T80, T81, T82, T125, T7, parseA_in_ga(T125, T7))
parseA_in_ga(T230, T232) → U3_ga(T230, T232, pD_in_aaaagaa(X289, X290, X291, X292, T230, X293, T232))
pD_in_aaaagaa(T239, T240, T241, T242, T230, X293, T232) → U18_aaaagaa(T239, T240, T241, T242, T230, X293, T232, appH_in_aaaag(T239, T240, T241, T242, T230))
appH_in_aaaag([], T261, T262, T263, .(a, .(s(T261, T262), .(b, T263)))) → appH_out_aaaag([], T261, T262, T263, .(a, .(s(T261, T262), .(b, T263))))
appH_in_aaaag(.(T268, X353), X354, X355, X356, .(T268, T269)) → U7_aaaag(T268, X353, X354, X355, X356, T269, appH_in_aaaag(X353, X354, X355, X356, T269))
U7_aaaag(T268, X353, X354, X355, X356, T269, appH_out_aaaag(X353, X354, X355, X356, T269)) → appH_out_aaaag(.(T268, X353), X354, X355, X356, .(T268, T269))
U18_aaaagaa(T239, T240, T241, T242, T230, X293, T232, appH_out_aaaag(T239, T240, T241, T242, T230)) → U19_aaaagaa(T239, T240, T241, T242, T230, X293, T232, pO_in_ggggaa(T239, T240, T241, T242, X293, T232))
pO_in_ggggaa(T239, T240, T241, T242, T282, T232) → U20_ggggaa(T239, T240, T241, T242, T282, T232, appI_in_gggga(T239, T240, T241, T242, T282))
appI_in_gggga([], T301, T302, T303, .(s(a, s(T301, T302), b), T303)) → appI_out_gggga([], T301, T302, T303, .(s(a, s(T301, T302), b), T303))
appI_in_gggga(.(T314, T315), T316, T317, T318, .(T314, X388)) → U8_gggga(T314, T315, T316, T317, T318, X388, appI_in_gggga(T315, T316, T317, T318, X388))
U8_gggga(T314, T315, T316, T317, T318, X388, appI_out_gggga(T315, T316, T317, T318, X388)) → appI_out_gggga(.(T314, T315), T316, T317, T318, .(T314, X388))
U20_ggggaa(T239, T240, T241, T242, T282, T232, appI_out_gggga(T239, T240, T241, T242, T282)) → U21_ggggaa(T239, T240, T241, T242, T282, T232, parseA_in_ga(T282, T232))
parseA_in_ga(T341, T343) → U4_ga(T341, T343, pE_in_aagaa(X423, X424, T341, X425, T343))
pE_in_aagaa(T346, T347, T341, X425, T343) → U22_aagaa(T346, T347, T341, X425, T343, appJ_in_aag(T346, T347, T341))
appJ_in_aag([], T354, .(a, .(b, T354))) → appJ_out_aag([], T354, .(a, .(b, T354)))
appJ_in_aag(.(T359, X463), X464, .(T359, T360)) → U9_aag(T359, X463, X464, T360, appJ_in_aag(X463, X464, T360))
U9_aag(T359, X463, X464, T360, appJ_out_aag(X463, X464, T360)) → appJ_out_aag(.(T359, X463), X464, .(T359, T360))
U22_aagaa(T346, T347, T341, X425, T343, appJ_out_aag(T346, T347, T341)) → U23_aagaa(T346, T347, T341, X425, T343, pP_in_ggaa(T346, T347, X425, T343))
pP_in_ggaa(T346, T347, T365, T343) → U24_ggaa(T346, T347, T365, T343, appK_in_gga(T346, T347, T365))
appK_in_gga([], T372, .(s(a, b), T372)) → appK_out_gga([], T372, .(s(a, b), T372))
appK_in_gga(.(T379, T380), T381, .(T379, X492)) → U10_gga(T379, T380, T381, X492, appK_in_gga(T380, T381, X492))
U10_gga(T379, T380, T381, X492, appK_out_gga(T380, T381, X492)) → appK_out_gga(.(T379, T380), T381, .(T379, X492))
U24_ggaa(T346, T347, T365, T343, appK_out_gga(T346, T347, T365)) → U25_ggaa(T346, T347, T365, T343, parseA_in_ga(T365, T343))
parseA_in_ga(.(s(T394, T395), []), s(T394, T395)) → parseA_out_ga(.(s(T394, T395), []), s(T394, T395))
parseA_in_ga(.(s(T402, T403, T404), []), s(T402, T403, T404)) → parseA_out_ga(.(s(T402, T403, T404), []), s(T402, T403, T404))
U25_ggaa(T346, T347, T365, T343, parseA_out_ga(T365, T343)) → pP_out_ggaa(T346, T347, T365, T343)
U23_aagaa(T346, T347, T341, X425, T343, pP_out_ggaa(T346, T347, X425, T343)) → pE_out_aagaa(T346, T347, T341, X425, T343)
U4_ga(T341, T343, pE_out_aagaa(X423, X424, T341, X425, T343)) → parseA_out_ga(T341, T343)
U21_ggggaa(T239, T240, T241, T242, T282, T232, parseA_out_ga(T282, T232)) → pO_out_ggggaa(T239, T240, T241, T242, T282, T232)
U19_aaaagaa(T239, T240, T241, T242, T230, X293, T232, pO_out_ggggaa(T239, T240, T241, T242, X293, T232)) → pD_out_aaaagaa(T239, T240, T241, T242, T230, X293, T232)
U3_ga(T230, T232, pD_out_aaaagaa(X289, X290, X291, X292, T230, X293, T232)) → parseA_out_ga(T230, T232)
U17_ggggggaa(T68, T78, T79, T80, T81, T82, T125, T7, parseA_out_ga(T125, T7)) → pN_out_ggggggaa(T68, T78, T79, T80, T81, T82, T125, T7)
U15_aaaaaggaa(T78, T79, T80, T81, T82, T69, T68, X10, T7, pN_out_ggggggaa(T68, T78, T79, T80, T81, T82, X10, T7)) → pC_out_aaaaaggaa(T78, T79, T80, T81, T82, T69, T68, X10, T7)
U2_ga(T68, T69, T7, pC_out_aaaaaggaa(X109, X110, X111, X112, X113, T69, T68, X10, T7)) → parseA_out_ga(.(T68, T69), T7)
U13_ggggaa(T24, T25, T26, T27, T33, T7, parseA_out_ga(T33, T7)) → pB_out_ggggaa(T24, T25, T26, T27, T33, T7)
U1_ga(T24, T25, T26, T27, T7, pB_out_ggggaa(T24, T25, T26, T27, X10, T7)) → parseA_out_ga(.(a, .(s(T24, T25, T26), .(b, T27))), T7)

The argument filtering Pi contains the following mapping:
parseA_in_ga(x1, x2)  =  parseA_in_ga(x1)
.(x1, x2)  =  .(x1, x2)
a  =  a
s(x1, x2, x3)  =  s(x1, x2, x3)
b  =  b
U1_ga(x1, x2, x3, x4, x5, x6)  =  U1_ga(x1, x2, x3, x4, x6)
pB_in_ggggaa(x1, x2, x3, x4, x5, x6)  =  pB_in_ggggaa(x1, x2, x3, x4)
U12_ggggaa(x1, x2, x3, x4, x5, x6, x7)  =  U12_ggggaa(x1, x2, x3, x4, x7)
appL_in_gggga(x1, x2, x3, x4, x5)  =  appL_in_gggga(x1, x2, x3, x4)
appL_out_gggga(x1, x2, x3, x4, x5)  =  appL_out_gggga(x1, x2, x3, x4, x5)
U13_ggggaa(x1, x2, x3, x4, x5, x6, x7)  =  U13_ggggaa(x1, x2, x3, x4, x5, x7)
U2_ga(x1, x2, x3, x4)  =  U2_ga(x1, x2, x4)
pC_in_aaaaaggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  pC_in_aaaaaggaa(x6, x7)
U14_aaaaaggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10)  =  U14_aaaaaggaa(x6, x7, x10)
appF_in_aaaaag(x1, x2, x3, x4, x5, x6)  =  appF_in_aaaaag(x6)
appF_out_aaaaag(x1, x2, x3, x4, x5, x6)  =  appF_out_aaaaag(x1, x2, x3, x4, x5, x6)
U5_aaaaag(x1, x2, x3, x4, x5, x6, x7, x8)  =  U5_aaaaag(x1, x7, x8)
U15_aaaaaggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10)  =  U15_aaaaaggaa(x1, x2, x3, x4, x5, x6, x7, x10)
pN_in_ggggggaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  pN_in_ggggggaa(x1, x2, x3, x4, x5, x6)
U16_ggggggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U16_ggggggaa(x1, x2, x3, x4, x5, x6, x9)
appM_in_gggggga(x1, x2, x3, x4, x5, x6, x7)  =  appM_in_gggggga(x1, x2, x3, x4, x5, x6)
U11_gggggga(x1, x2, x3, x4, x5, x6, x7, x8)  =  U11_gggggga(x1, x2, x3, x4, x5, x6, x8)
appG_in_ggggga(x1, x2, x3, x4, x5, x6)  =  appG_in_ggggga(x1, x2, x3, x4, x5)
[]  =  []
appG_out_ggggga(x1, x2, x3, x4, x5, x6)  =  appG_out_ggggga(x1, x2, x3, x4, x5, x6)
U6_ggggga(x1, x2, x3, x4, x5, x6, x7, x8)  =  U6_ggggga(x1, x2, x3, x4, x5, x6, x8)
appM_out_gggggga(x1, x2, x3, x4, x5, x6, x7)  =  appM_out_gggggga(x1, x2, x3, x4, x5, x6, x7)
U17_ggggggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U17_ggggggaa(x1, x2, x3, x4, x5, x6, x7, x9)
U3_ga(x1, x2, x3)  =  U3_ga(x1, x3)
pD_in_aaaagaa(x1, x2, x3, x4, x5, x6, x7)  =  pD_in_aaaagaa(x5)
U18_aaaagaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U18_aaaagaa(x5, x8)
appH_in_aaaag(x1, x2, x3, x4, x5)  =  appH_in_aaaag(x5)
s(x1, x2)  =  s(x1, x2)
appH_out_aaaag(x1, x2, x3, x4, x5)  =  appH_out_aaaag(x1, x2, x3, x4, x5)
U7_aaaag(x1, x2, x3, x4, x5, x6, x7)  =  U7_aaaag(x1, x6, x7)
U19_aaaagaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U19_aaaagaa(x1, x2, x3, x4, x5, x8)
pO_in_ggggaa(x1, x2, x3, x4, x5, x6)  =  pO_in_ggggaa(x1, x2, x3, x4)
U20_ggggaa(x1, x2, x3, x4, x5, x6, x7)  =  U20_ggggaa(x1, x2, x3, x4, x7)
appI_in_gggga(x1, x2, x3, x4, x5)  =  appI_in_gggga(x1, x2, x3, x4)
appI_out_gggga(x1, x2, x3, x4, x5)  =  appI_out_gggga(x1, x2, x3, x4, x5)
U8_gggga(x1, x2, x3, x4, x5, x6, x7)  =  U8_gggga(x1, x2, x3, x4, x5, x7)
U21_ggggaa(x1, x2, x3, x4, x5, x6, x7)  =  U21_ggggaa(x1, x2, x3, x4, x5, x7)
U4_ga(x1, x2, x3)  =  U4_ga(x1, x3)
pE_in_aagaa(x1, x2, x3, x4, x5)  =  pE_in_aagaa(x3)
U22_aagaa(x1, x2, x3, x4, x5, x6)  =  U22_aagaa(x3, x6)
appJ_in_aag(x1, x2, x3)  =  appJ_in_aag(x3)
appJ_out_aag(x1, x2, x3)  =  appJ_out_aag(x1, x2, x3)
U9_aag(x1, x2, x3, x4, x5)  =  U9_aag(x1, x4, x5)
U23_aagaa(x1, x2, x3, x4, x5, x6)  =  U23_aagaa(x1, x2, x3, x6)
pP_in_ggaa(x1, x2, x3, x4)  =  pP_in_ggaa(x1, x2)
U24_ggaa(x1, x2, x3, x4, x5)  =  U24_ggaa(x1, x2, x5)
appK_in_gga(x1, x2, x3)  =  appK_in_gga(x1, x2)
appK_out_gga(x1, x2, x3)  =  appK_out_gga(x1, x2, x3)
U10_gga(x1, x2, x3, x4, x5)  =  U10_gga(x1, x2, x3, x5)
U25_ggaa(x1, x2, x3, x4, x5)  =  U25_ggaa(x1, x2, x3, x5)
parseA_out_ga(x1, x2)  =  parseA_out_ga(x1, x2)
pP_out_ggaa(x1, x2, x3, x4)  =  pP_out_ggaa(x1, x2, x3, x4)
pE_out_aagaa(x1, x2, x3, x4, x5)  =  pE_out_aagaa(x1, x2, x3, x4, x5)
pO_out_ggggaa(x1, x2, x3, x4, x5, x6)  =  pO_out_ggggaa(x1, x2, x3, x4, x5, x6)
pD_out_aaaagaa(x1, x2, x3, x4, x5, x6, x7)  =  pD_out_aaaagaa(x1, x2, x3, x4, x5, x6, x7)
pN_out_ggggggaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  pN_out_ggggggaa(x1, x2, x3, x4, x5, x6, x7, x8)
pC_out_aaaaaggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  pC_out_aaaaaggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)
pB_out_ggggaa(x1, x2, x3, x4, x5, x6)  =  pB_out_ggggaa(x1, x2, x3, x4, x5, x6)
APPK_IN_GGA(x1, x2, x3)  =  APPK_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:

APPK_IN_GGA(.(T379, T380), T381, .(T379, X492)) → APPK_IN_GGA(T380, T381, X492)

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

APPK_IN_GGA(.(T379, T380), T381) → APPK_IN_GGA(T380, T381)

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:

• APPK_IN_GGA(.(T379, T380), T381) → APPK_IN_GGA(T380, T381)
  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:

APPJ_IN_AAG(.(T359, X463), X464, .(T359, T360)) → APPJ_IN_AAG(X463, X464, T360)

The TRS R consists of the following rules:

parseA_in_ga(.(a, .(s(T24, T25, T26), .(b, T27))), T7) → U1_ga(T24, T25, T26, T27, T7, pB_in_ggggaa(T24, T25, T26, T27, X10, T7))
pB_in_ggggaa(T24, T25, T26, T27, T33, T7) → U12_ggggaa(T24, T25, T26, T27, T33, T7, appL_in_gggga(T24, T25, T26, T27, T33))
appL_in_gggga(T54, T55, T56, T57, .(s(a, s(T54, T55, T56), b), T57)) → appL_out_gggga(T54, T55, T56, T57, .(s(a, s(T54, T55, T56), b), T57))
U12_ggggaa(T24, T25, T26, T27, T33, T7, appL_out_gggga(T24, T25, T26, T27, T33)) → U13_ggggaa(T24, T25, T26, T27, T33, T7, parseA_in_ga(T33, T7))
parseA_in_ga(.(T68, T69), T7) → U2_ga(T68, T69, T7, pC_in_aaaaaggaa(X109, X110, X111, X112, X113, T69, T68, X10, T7))
pC_in_aaaaaggaa(T78, T79, T80, T81, T82, T69, T68, X10, T7) → U14_aaaaaggaa(T78, T79, T80, T81, T82, T69, T68, X10, T7, appF_in_aaaaag(T78, T79, T80, T81, T82, T69))
appF_in_aaaaag([], T107, T108, T109, T110, .(a, .(s(T107, T108, T109), .(b, T110)))) → appF_out_aaaaag([], T107, T108, T109, T110, .(a, .(s(T107, T108, T109), .(b, T110))))
appF_in_aaaaag(.(T115, X184), X185, X186, X187, X188, .(T115, T116)) → U5_aaaaag(T115, X184, X185, X186, X187, X188, T116, appF_in_aaaaag(X184, X185, X186, X187, X188, T116))
U5_aaaaag(T115, X184, X185, X186, X187, X188, T116, appF_out_aaaaag(X184, X185, X186, X187, X188, T116)) → appF_out_aaaaag(.(T115, X184), X185, X186, X187, X188, .(T115, T116))
U14_aaaaaggaa(T78, T79, T80, T81, T82, T69, T68, X10, T7, appF_out_aaaaag(T78, T79, T80, T81, T82, T69)) → U15_aaaaaggaa(T78, T79, T80, T81, T82, T69, T68, X10, T7, pN_in_ggggggaa(T68, T78, T79, T80, T81, T82, X10, T7))
pN_in_ggggggaa(T68, T78, T79, T80, T81, T82, T125, T7) → U16_ggggggaa(T68, T78, T79, T80, T81, T82, T125, T7, appM_in_gggggga(T68, T78, T79, T80, T81, T82, T125))
appM_in_gggggga(T146, T147, T148, T149, T150, T151, .(T146, X220)) → U11_gggggga(T146, T147, T148, T149, T150, T151, X220, appG_in_ggggga(T147, T148, T149, T150, T151, X220))
appG_in_ggggga([], T176, T177, T178, T179, .(s(a, s(T176, T177, T178), b), T179)) → appG_out_ggggga([], T176, T177, T178, T179, .(s(a, s(T176, T177, T178), b), T179))
appG_in_ggggga(.(T192, T193), T194, T195, T196, T197, .(T192, X242)) → U6_ggggga(T192, T193, T194, T195, T196, T197, X242, appG_in_ggggga(T193, T194, T195, T196, T197, X242))
U6_ggggga(T192, T193, T194, T195, T196, T197, X242, appG_out_ggggga(T193, T194, T195, T196, T197, X242)) → appG_out_ggggga(.(T192, T193), T194, T195, T196, T197, .(T192, X242))
U11_gggggga(T146, T147, T148, T149, T150, T151, X220, appG_out_ggggga(T147, T148, T149, T150, T151, X220)) → appM_out_gggggga(T146, T147, T148, T149, T150, T151, .(T146, X220))
U16_ggggggaa(T68, T78, T79, T80, T81, T82, T125, T7, appM_out_gggggga(T68, T78, T79, T80, T81, T82, T125)) → U17_ggggggaa(T68, T78, T79, T80, T81, T82, T125, T7, parseA_in_ga(T125, T7))
parseA_in_ga(T230, T232) → U3_ga(T230, T232, pD_in_aaaagaa(X289, X290, X291, X292, T230, X293, T232))
pD_in_aaaagaa(T239, T240, T241, T242, T230, X293, T232) → U18_aaaagaa(T239, T240, T241, T242, T230, X293, T232, appH_in_aaaag(T239, T240, T241, T242, T230))
appH_in_aaaag([], T261, T262, T263, .(a, .(s(T261, T262), .(b, T263)))) → appH_out_aaaag([], T261, T262, T263, .(a, .(s(T261, T262), .(b, T263))))
appH_in_aaaag(.(T268, X353), X354, X355, X356, .(T268, T269)) → U7_aaaag(T268, X353, X354, X355, X356, T269, appH_in_aaaag(X353, X354, X355, X356, T269))
U7_aaaag(T268, X353, X354, X355, X356, T269, appH_out_aaaag(X353, X354, X355, X356, T269)) → appH_out_aaaag(.(T268, X353), X354, X355, X356, .(T268, T269))
U18_aaaagaa(T239, T240, T241, T242, T230, X293, T232, appH_out_aaaag(T239, T240, T241, T242, T230)) → U19_aaaagaa(T239, T240, T241, T242, T230, X293, T232, pO_in_ggggaa(T239, T240, T241, T242, X293, T232))
pO_in_ggggaa(T239, T240, T241, T242, T282, T232) → U20_ggggaa(T239, T240, T241, T242, T282, T232, appI_in_gggga(T239, T240, T241, T242, T282))
appI_in_gggga([], T301, T302, T303, .(s(a, s(T301, T302), b), T303)) → appI_out_gggga([], T301, T302, T303, .(s(a, s(T301, T302), b), T303))
appI_in_gggga(.(T314, T315), T316, T317, T318, .(T314, X388)) → U8_gggga(T314, T315, T316, T317, T318, X388, appI_in_gggga(T315, T316, T317, T318, X388))
U8_gggga(T314, T315, T316, T317, T318, X388, appI_out_gggga(T315, T316, T317, T318, X388)) → appI_out_gggga(.(T314, T315), T316, T317, T318, .(T314, X388))
U20_ggggaa(T239, T240, T241, T242, T282, T232, appI_out_gggga(T239, T240, T241, T242, T282)) → U21_ggggaa(T239, T240, T241, T242, T282, T232, parseA_in_ga(T282, T232))
parseA_in_ga(T341, T343) → U4_ga(T341, T343, pE_in_aagaa(X423, X424, T341, X425, T343))
pE_in_aagaa(T346, T347, T341, X425, T343) → U22_aagaa(T346, T347, T341, X425, T343, appJ_in_aag(T346, T347, T341))
appJ_in_aag([], T354, .(a, .(b, T354))) → appJ_out_aag([], T354, .(a, .(b, T354)))
appJ_in_aag(.(T359, X463), X464, .(T359, T360)) → U9_aag(T359, X463, X464, T360, appJ_in_aag(X463, X464, T360))
U9_aag(T359, X463, X464, T360, appJ_out_aag(X463, X464, T360)) → appJ_out_aag(.(T359, X463), X464, .(T359, T360))
U22_aagaa(T346, T347, T341, X425, T343, appJ_out_aag(T346, T347, T341)) → U23_aagaa(T346, T347, T341, X425, T343, pP_in_ggaa(T346, T347, X425, T343))
pP_in_ggaa(T346, T347, T365, T343) → U24_ggaa(T346, T347, T365, T343, appK_in_gga(T346, T347, T365))
appK_in_gga([], T372, .(s(a, b), T372)) → appK_out_gga([], T372, .(s(a, b), T372))
appK_in_gga(.(T379, T380), T381, .(T379, X492)) → U10_gga(T379, T380, T381, X492, appK_in_gga(T380, T381, X492))
U10_gga(T379, T380, T381, X492, appK_out_gga(T380, T381, X492)) → appK_out_gga(.(T379, T380), T381, .(T379, X492))
U24_ggaa(T346, T347, T365, T343, appK_out_gga(T346, T347, T365)) → U25_ggaa(T346, T347, T365, T343, parseA_in_ga(T365, T343))
parseA_in_ga(.(s(T394, T395), []), s(T394, T395)) → parseA_out_ga(.(s(T394, T395), []), s(T394, T395))
parseA_in_ga(.(s(T402, T403, T404), []), s(T402, T403, T404)) → parseA_out_ga(.(s(T402, T403, T404), []), s(T402, T403, T404))
U25_ggaa(T346, T347, T365, T343, parseA_out_ga(T365, T343)) → pP_out_ggaa(T346, T347, T365, T343)
U23_aagaa(T346, T347, T341, X425, T343, pP_out_ggaa(T346, T347, X425, T343)) → pE_out_aagaa(T346, T347, T341, X425, T343)
U4_ga(T341, T343, pE_out_aagaa(X423, X424, T341, X425, T343)) → parseA_out_ga(T341, T343)
U21_ggggaa(T239, T240, T241, T242, T282, T232, parseA_out_ga(T282, T232)) → pO_out_ggggaa(T239, T240, T241, T242, T282, T232)
U19_aaaagaa(T239, T240, T241, T242, T230, X293, T232, pO_out_ggggaa(T239, T240, T241, T242, X293, T232)) → pD_out_aaaagaa(T239, T240, T241, T242, T230, X293, T232)
U3_ga(T230, T232, pD_out_aaaagaa(X289, X290, X291, X292, T230, X293, T232)) → parseA_out_ga(T230, T232)
U17_ggggggaa(T68, T78, T79, T80, T81, T82, T125, T7, parseA_out_ga(T125, T7)) → pN_out_ggggggaa(T68, T78, T79, T80, T81, T82, T125, T7)
U15_aaaaaggaa(T78, T79, T80, T81, T82, T69, T68, X10, T7, pN_out_ggggggaa(T68, T78, T79, T80, T81, T82, X10, T7)) → pC_out_aaaaaggaa(T78, T79, T80, T81, T82, T69, T68, X10, T7)
U2_ga(T68, T69, T7, pC_out_aaaaaggaa(X109, X110, X111, X112, X113, T69, T68, X10, T7)) → parseA_out_ga(.(T68, T69), T7)
U13_ggggaa(T24, T25, T26, T27, T33, T7, parseA_out_ga(T33, T7)) → pB_out_ggggaa(T24, T25, T26, T27, T33, T7)
U1_ga(T24, T25, T26, T27, T7, pB_out_ggggaa(T24, T25, T26, T27, X10, T7)) → parseA_out_ga(.(a, .(s(T24, T25, T26), .(b, T27))), T7)

The argument filtering Pi contains the following mapping:
parseA_in_ga(x1, x2)  =  parseA_in_ga(x1)
.(x1, x2)  =  .(x1, x2)
a  =  a
s(x1, x2, x3)  =  s(x1, x2, x3)
b  =  b
U1_ga(x1, x2, x3, x4, x5, x6)  =  U1_ga(x1, x2, x3, x4, x6)
pB_in_ggggaa(x1, x2, x3, x4, x5, x6)  =  pB_in_ggggaa(x1, x2, x3, x4)
U12_ggggaa(x1, x2, x3, x4, x5, x6, x7)  =  U12_ggggaa(x1, x2, x3, x4, x7)
appL_in_gggga(x1, x2, x3, x4, x5)  =  appL_in_gggga(x1, x2, x3, x4)
appL_out_gggga(x1, x2, x3, x4, x5)  =  appL_out_gggga(x1, x2, x3, x4, x5)
U13_ggggaa(x1, x2, x3, x4, x5, x6, x7)  =  U13_ggggaa(x1, x2, x3, x4, x5, x7)
U2_ga(x1, x2, x3, x4)  =  U2_ga(x1, x2, x4)
pC_in_aaaaaggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  pC_in_aaaaaggaa(x6, x7)
U14_aaaaaggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10)  =  U14_aaaaaggaa(x6, x7, x10)
appF_in_aaaaag(x1, x2, x3, x4, x5, x6)  =  appF_in_aaaaag(x6)
appF_out_aaaaag(x1, x2, x3, x4, x5, x6)  =  appF_out_aaaaag(x1, x2, x3, x4, x5, x6)
U5_aaaaag(x1, x2, x3, x4, x5, x6, x7, x8)  =  U5_aaaaag(x1, x7, x8)
U15_aaaaaggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10)  =  U15_aaaaaggaa(x1, x2, x3, x4, x5, x6, x7, x10)
pN_in_ggggggaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  pN_in_ggggggaa(x1, x2, x3, x4, x5, x6)
U16_ggggggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U16_ggggggaa(x1, x2, x3, x4, x5, x6, x9)
appM_in_gggggga(x1, x2, x3, x4, x5, x6, x7)  =  appM_in_gggggga(x1, x2, x3, x4, x5, x6)
U11_gggggga(x1, x2, x3, x4, x5, x6, x7, x8)  =  U11_gggggga(x1, x2, x3, x4, x5, x6, x8)
appG_in_ggggga(x1, x2, x3, x4, x5, x6)  =  appG_in_ggggga(x1, x2, x3, x4, x5)
[]  =  []
appG_out_ggggga(x1, x2, x3, x4, x5, x6)  =  appG_out_ggggga(x1, x2, x3, x4, x5, x6)
U6_ggggga(x1, x2, x3, x4, x5, x6, x7, x8)  =  U6_ggggga(x1, x2, x3, x4, x5, x6, x8)
appM_out_gggggga(x1, x2, x3, x4, x5, x6, x7)  =  appM_out_gggggga(x1, x2, x3, x4, x5, x6, x7)
U17_ggggggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U17_ggggggaa(x1, x2, x3, x4, x5, x6, x7, x9)
U3_ga(x1, x2, x3)  =  U3_ga(x1, x3)
pD_in_aaaagaa(x1, x2, x3, x4, x5, x6, x7)  =  pD_in_aaaagaa(x5)
U18_aaaagaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U18_aaaagaa(x5, x8)
appH_in_aaaag(x1, x2, x3, x4, x5)  =  appH_in_aaaag(x5)
s(x1, x2)  =  s(x1, x2)
appH_out_aaaag(x1, x2, x3, x4, x5)  =  appH_out_aaaag(x1, x2, x3, x4, x5)
U7_aaaag(x1, x2, x3, x4, x5, x6, x7)  =  U7_aaaag(x1, x6, x7)
U19_aaaagaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U19_aaaagaa(x1, x2, x3, x4, x5, x8)
pO_in_ggggaa(x1, x2, x3, x4, x5, x6)  =  pO_in_ggggaa(x1, x2, x3, x4)
U20_ggggaa(x1, x2, x3, x4, x5, x6, x7)  =  U20_ggggaa(x1, x2, x3, x4, x7)
appI_in_gggga(x1, x2, x3, x4, x5)  =  appI_in_gggga(x1, x2, x3, x4)
appI_out_gggga(x1, x2, x3, x4, x5)  =  appI_out_gggga(x1, x2, x3, x4, x5)
U8_gggga(x1, x2, x3, x4, x5, x6, x7)  =  U8_gggga(x1, x2, x3, x4, x5, x7)
U21_ggggaa(x1, x2, x3, x4, x5, x6, x7)  =  U21_ggggaa(x1, x2, x3, x4, x5, x7)
U4_ga(x1, x2, x3)  =  U4_ga(x1, x3)
pE_in_aagaa(x1, x2, x3, x4, x5)  =  pE_in_aagaa(x3)
U22_aagaa(x1, x2, x3, x4, x5, x6)  =  U22_aagaa(x3, x6)
appJ_in_aag(x1, x2, x3)  =  appJ_in_aag(x3)
appJ_out_aag(x1, x2, x3)  =  appJ_out_aag(x1, x2, x3)
U9_aag(x1, x2, x3, x4, x5)  =  U9_aag(x1, x4, x5)
U23_aagaa(x1, x2, x3, x4, x5, x6)  =  U23_aagaa(x1, x2, x3, x6)
pP_in_ggaa(x1, x2, x3, x4)  =  pP_in_ggaa(x1, x2)
U24_ggaa(x1, x2, x3, x4, x5)  =  U24_ggaa(x1, x2, x5)
appK_in_gga(x1, x2, x3)  =  appK_in_gga(x1, x2)
appK_out_gga(x1, x2, x3)  =  appK_out_gga(x1, x2, x3)
U10_gga(x1, x2, x3, x4, x5)  =  U10_gga(x1, x2, x3, x5)
U25_ggaa(x1, x2, x3, x4, x5)  =  U25_ggaa(x1, x2, x3, x5)
parseA_out_ga(x1, x2)  =  parseA_out_ga(x1, x2)
pP_out_ggaa(x1, x2, x3, x4)  =  pP_out_ggaa(x1, x2, x3, x4)
pE_out_aagaa(x1, x2, x3, x4, x5)  =  pE_out_aagaa(x1, x2, x3, x4, x5)
pO_out_ggggaa(x1, x2, x3, x4, x5, x6)  =  pO_out_ggggaa(x1, x2, x3, x4, x5, x6)
pD_out_aaaagaa(x1, x2, x3, x4, x5, x6, x7)  =  pD_out_aaaagaa(x1, x2, x3, x4, x5, x6, x7)
pN_out_ggggggaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  pN_out_ggggggaa(x1, x2, x3, x4, x5, x6, x7, x8)
pC_out_aaaaaggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  pC_out_aaaaaggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)
pB_out_ggggaa(x1, x2, x3, x4, x5, x6)  =  pB_out_ggggaa(x1, x2, x3, x4, x5, x6)
APPJ_IN_AAG(x1, x2, x3)  =  APPJ_IN_AAG(x3)

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:

APPJ_IN_AAG(.(T359, X463), X464, .(T359, T360)) → APPJ_IN_AAG(X463, X464, T360)

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

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:

APPJ_IN_AAG(.(T359, T360)) → APPJ_IN_AAG(T360)

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:

• APPJ_IN_AAG(.(T359, T360)) → APPJ_IN_AAG(T360)
  The graph contains the following edges 1 > 1

(20) YES

(21) Obligation:

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

APPI_IN_GGGGA(.(T314, T315), T316, T317, T318, .(T314, X388)) → APPI_IN_GGGGA(T315, T316, T317, T318, X388)

The TRS R consists of the following rules:

parseA_in_ga(.(a, .(s(T24, T25, T26), .(b, T27))), T7) → U1_ga(T24, T25, T26, T27, T7, pB_in_ggggaa(T24, T25, T26, T27, X10, T7))
pB_in_ggggaa(T24, T25, T26, T27, T33, T7) → U12_ggggaa(T24, T25, T26, T27, T33, T7, appL_in_gggga(T24, T25, T26, T27, T33))
appL_in_gggga(T54, T55, T56, T57, .(s(a, s(T54, T55, T56), b), T57)) → appL_out_gggga(T54, T55, T56, T57, .(s(a, s(T54, T55, T56), b), T57))
U12_ggggaa(T24, T25, T26, T27, T33, T7, appL_out_gggga(T24, T25, T26, T27, T33)) → U13_ggggaa(T24, T25, T26, T27, T33, T7, parseA_in_ga(T33, T7))
parseA_in_ga(.(T68, T69), T7) → U2_ga(T68, T69, T7, pC_in_aaaaaggaa(X109, X110, X111, X112, X113, T69, T68, X10, T7))
pC_in_aaaaaggaa(T78, T79, T80, T81, T82, T69, T68, X10, T7) → U14_aaaaaggaa(T78, T79, T80, T81, T82, T69, T68, X10, T7, appF_in_aaaaag(T78, T79, T80, T81, T82, T69))
appF_in_aaaaag([], T107, T108, T109, T110, .(a, .(s(T107, T108, T109), .(b, T110)))) → appF_out_aaaaag([], T107, T108, T109, T110, .(a, .(s(T107, T108, T109), .(b, T110))))
appF_in_aaaaag(.(T115, X184), X185, X186, X187, X188, .(T115, T116)) → U5_aaaaag(T115, X184, X185, X186, X187, X188, T116, appF_in_aaaaag(X184, X185, X186, X187, X188, T116))
U5_aaaaag(T115, X184, X185, X186, X187, X188, T116, appF_out_aaaaag(X184, X185, X186, X187, X188, T116)) → appF_out_aaaaag(.(T115, X184), X185, X186, X187, X188, .(T115, T116))
U14_aaaaaggaa(T78, T79, T80, T81, T82, T69, T68, X10, T7, appF_out_aaaaag(T78, T79, T80, T81, T82, T69)) → U15_aaaaaggaa(T78, T79, T80, T81, T82, T69, T68, X10, T7, pN_in_ggggggaa(T68, T78, T79, T80, T81, T82, X10, T7))
pN_in_ggggggaa(T68, T78, T79, T80, T81, T82, T125, T7) → U16_ggggggaa(T68, T78, T79, T80, T81, T82, T125, T7, appM_in_gggggga(T68, T78, T79, T80, T81, T82, T125))
appM_in_gggggga(T146, T147, T148, T149, T150, T151, .(T146, X220)) → U11_gggggga(T146, T147, T148, T149, T150, T151, X220, appG_in_ggggga(T147, T148, T149, T150, T151, X220))
appG_in_ggggga([], T176, T177, T178, T179, .(s(a, s(T176, T177, T178), b), T179)) → appG_out_ggggga([], T176, T177, T178, T179, .(s(a, s(T176, T177, T178), b), T179))
appG_in_ggggga(.(T192, T193), T194, T195, T196, T197, .(T192, X242)) → U6_ggggga(T192, T193, T194, T195, T196, T197, X242, appG_in_ggggga(T193, T194, T195, T196, T197, X242))
U6_ggggga(T192, T193, T194, T195, T196, T197, X242, appG_out_ggggga(T193, T194, T195, T196, T197, X242)) → appG_out_ggggga(.(T192, T193), T194, T195, T196, T197, .(T192, X242))
U11_gggggga(T146, T147, T148, T149, T150, T151, X220, appG_out_ggggga(T147, T148, T149, T150, T151, X220)) → appM_out_gggggga(T146, T147, T148, T149, T150, T151, .(T146, X220))
U16_ggggggaa(T68, T78, T79, T80, T81, T82, T125, T7, appM_out_gggggga(T68, T78, T79, T80, T81, T82, T125)) → U17_ggggggaa(T68, T78, T79, T80, T81, T82, T125, T7, parseA_in_ga(T125, T7))
parseA_in_ga(T230, T232) → U3_ga(T230, T232, pD_in_aaaagaa(X289, X290, X291, X292, T230, X293, T232))
pD_in_aaaagaa(T239, T240, T241, T242, T230, X293, T232) → U18_aaaagaa(T239, T240, T241, T242, T230, X293, T232, appH_in_aaaag(T239, T240, T241, T242, T230))
appH_in_aaaag([], T261, T262, T263, .(a, .(s(T261, T262), .(b, T263)))) → appH_out_aaaag([], T261, T262, T263, .(a, .(s(T261, T262), .(b, T263))))
appH_in_aaaag(.(T268, X353), X354, X355, X356, .(T268, T269)) → U7_aaaag(T268, X353, X354, X355, X356, T269, appH_in_aaaag(X353, X354, X355, X356, T269))
U7_aaaag(T268, X353, X354, X355, X356, T269, appH_out_aaaag(X353, X354, X355, X356, T269)) → appH_out_aaaag(.(T268, X353), X354, X355, X356, .(T268, T269))
U18_aaaagaa(T239, T240, T241, T242, T230, X293, T232, appH_out_aaaag(T239, T240, T241, T242, T230)) → U19_aaaagaa(T239, T240, T241, T242, T230, X293, T232, pO_in_ggggaa(T239, T240, T241, T242, X293, T232))
pO_in_ggggaa(T239, T240, T241, T242, T282, T232) → U20_ggggaa(T239, T240, T241, T242, T282, T232, appI_in_gggga(T239, T240, T241, T242, T282))
appI_in_gggga([], T301, T302, T303, .(s(a, s(T301, T302), b), T303)) → appI_out_gggga([], T301, T302, T303, .(s(a, s(T301, T302), b), T303))
appI_in_gggga(.(T314, T315), T316, T317, T318, .(T314, X388)) → U8_gggga(T314, T315, T316, T317, T318, X388, appI_in_gggga(T315, T316, T317, T318, X388))
U8_gggga(T314, T315, T316, T317, T318, X388, appI_out_gggga(T315, T316, T317, T318, X388)) → appI_out_gggga(.(T314, T315), T316, T317, T318, .(T314, X388))
U20_ggggaa(T239, T240, T241, T242, T282, T232, appI_out_gggga(T239, T240, T241, T242, T282)) → U21_ggggaa(T239, T240, T241, T242, T282, T232, parseA_in_ga(T282, T232))
parseA_in_ga(T341, T343) → U4_ga(T341, T343, pE_in_aagaa(X423, X424, T341, X425, T343))
pE_in_aagaa(T346, T347, T341, X425, T343) → U22_aagaa(T346, T347, T341, X425, T343, appJ_in_aag(T346, T347, T341))
appJ_in_aag([], T354, .(a, .(b, T354))) → appJ_out_aag([], T354, .(a, .(b, T354)))
appJ_in_aag(.(T359, X463), X464, .(T359, T360)) → U9_aag(T359, X463, X464, T360, appJ_in_aag(X463, X464, T360))
U9_aag(T359, X463, X464, T360, appJ_out_aag(X463, X464, T360)) → appJ_out_aag(.(T359, X463), X464, .(T359, T360))
U22_aagaa(T346, T347, T341, X425, T343, appJ_out_aag(T346, T347, T341)) → U23_aagaa(T346, T347, T341, X425, T343, pP_in_ggaa(T346, T347, X425, T343))
pP_in_ggaa(T346, T347, T365, T343) → U24_ggaa(T346, T347, T365, T343, appK_in_gga(T346, T347, T365))
appK_in_gga([], T372, .(s(a, b), T372)) → appK_out_gga([], T372, .(s(a, b), T372))
appK_in_gga(.(T379, T380), T381, .(T379, X492)) → U10_gga(T379, T380, T381, X492, appK_in_gga(T380, T381, X492))
U10_gga(T379, T380, T381, X492, appK_out_gga(T380, T381, X492)) → appK_out_gga(.(T379, T380), T381, .(T379, X492))
U24_ggaa(T346, T347, T365, T343, appK_out_gga(T346, T347, T365)) → U25_ggaa(T346, T347, T365, T343, parseA_in_ga(T365, T343))
parseA_in_ga(.(s(T394, T395), []), s(T394, T395)) → parseA_out_ga(.(s(T394, T395), []), s(T394, T395))
parseA_in_ga(.(s(T402, T403, T404), []), s(T402, T403, T404)) → parseA_out_ga(.(s(T402, T403, T404), []), s(T402, T403, T404))
U25_ggaa(T346, T347, T365, T343, parseA_out_ga(T365, T343)) → pP_out_ggaa(T346, T347, T365, T343)
U23_aagaa(T346, T347, T341, X425, T343, pP_out_ggaa(T346, T347, X425, T343)) → pE_out_aagaa(T346, T347, T341, X425, T343)
U4_ga(T341, T343, pE_out_aagaa(X423, X424, T341, X425, T343)) → parseA_out_ga(T341, T343)
U21_ggggaa(T239, T240, T241, T242, T282, T232, parseA_out_ga(T282, T232)) → pO_out_ggggaa(T239, T240, T241, T242, T282, T232)
U19_aaaagaa(T239, T240, T241, T242, T230, X293, T232, pO_out_ggggaa(T239, T240, T241, T242, X293, T232)) → pD_out_aaaagaa(T239, T240, T241, T242, T230, X293, T232)
U3_ga(T230, T232, pD_out_aaaagaa(X289, X290, X291, X292, T230, X293, T232)) → parseA_out_ga(T230, T232)
U17_ggggggaa(T68, T78, T79, T80, T81, T82, T125, T7, parseA_out_ga(T125, T7)) → pN_out_ggggggaa(T68, T78, T79, T80, T81, T82, T125, T7)
U15_aaaaaggaa(T78, T79, T80, T81, T82, T69, T68, X10, T7, pN_out_ggggggaa(T68, T78, T79, T80, T81, T82, X10, T7)) → pC_out_aaaaaggaa(T78, T79, T80, T81, T82, T69, T68, X10, T7)
U2_ga(T68, T69, T7, pC_out_aaaaaggaa(X109, X110, X111, X112, X113, T69, T68, X10, T7)) → parseA_out_ga(.(T68, T69), T7)
U13_ggggaa(T24, T25, T26, T27, T33, T7, parseA_out_ga(T33, T7)) → pB_out_ggggaa(T24, T25, T26, T27, T33, T7)
U1_ga(T24, T25, T26, T27, T7, pB_out_ggggaa(T24, T25, T26, T27, X10, T7)) → parseA_out_ga(.(a, .(s(T24, T25, T26), .(b, T27))), T7)

The argument filtering Pi contains the following mapping:
parseA_in_ga(x1, x2)  =  parseA_in_ga(x1)
.(x1, x2)  =  .(x1, x2)
a  =  a
s(x1, x2, x3)  =  s(x1, x2, x3)
b  =  b
U1_ga(x1, x2, x3, x4, x5, x6)  =  U1_ga(x1, x2, x3, x4, x6)
pB_in_ggggaa(x1, x2, x3, x4, x5, x6)  =  pB_in_ggggaa(x1, x2, x3, x4)
U12_ggggaa(x1, x2, x3, x4, x5, x6, x7)  =  U12_ggggaa(x1, x2, x3, x4, x7)
appL_in_gggga(x1, x2, x3, x4, x5)  =  appL_in_gggga(x1, x2, x3, x4)
appL_out_gggga(x1, x2, x3, x4, x5)  =  appL_out_gggga(x1, x2, x3, x4, x5)
U13_ggggaa(x1, x2, x3, x4, x5, x6, x7)  =  U13_ggggaa(x1, x2, x3, x4, x5, x7)
U2_ga(x1, x2, x3, x4)  =  U2_ga(x1, x2, x4)
pC_in_aaaaaggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  pC_in_aaaaaggaa(x6, x7)
U14_aaaaaggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10)  =  U14_aaaaaggaa(x6, x7, x10)
appF_in_aaaaag(x1, x2, x3, x4, x5, x6)  =  appF_in_aaaaag(x6)
appF_out_aaaaag(x1, x2, x3, x4, x5, x6)  =  appF_out_aaaaag(x1, x2, x3, x4, x5, x6)
U5_aaaaag(x1, x2, x3, x4, x5, x6, x7, x8)  =  U5_aaaaag(x1, x7, x8)
U15_aaaaaggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10)  =  U15_aaaaaggaa(x1, x2, x3, x4, x5, x6, x7, x10)
pN_in_ggggggaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  pN_in_ggggggaa(x1, x2, x3, x4, x5, x6)
U16_ggggggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U16_ggggggaa(x1, x2, x3, x4, x5, x6, x9)
appM_in_gggggga(x1, x2, x3, x4, x5, x6, x7)  =  appM_in_gggggga(x1, x2, x3, x4, x5, x6)
U11_gggggga(x1, x2, x3, x4, x5, x6, x7, x8)  =  U11_gggggga(x1, x2, x3, x4, x5, x6, x8)
appG_in_ggggga(x1, x2, x3, x4, x5, x6)  =  appG_in_ggggga(x1, x2, x3, x4, x5)
[]  =  []
appG_out_ggggga(x1, x2, x3, x4, x5, x6)  =  appG_out_ggggga(x1, x2, x3, x4, x5, x6)
U6_ggggga(x1, x2, x3, x4, x5, x6, x7, x8)  =  U6_ggggga(x1, x2, x3, x4, x5, x6, x8)
appM_out_gggggga(x1, x2, x3, x4, x5, x6, x7)  =  appM_out_gggggga(x1, x2, x3, x4, x5, x6, x7)
U17_ggggggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U17_ggggggaa(x1, x2, x3, x4, x5, x6, x7, x9)
U3_ga(x1, x2, x3)  =  U3_ga(x1, x3)
pD_in_aaaagaa(x1, x2, x3, x4, x5, x6, x7)  =  pD_in_aaaagaa(x5)
U18_aaaagaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U18_aaaagaa(x5, x8)
appH_in_aaaag(x1, x2, x3, x4, x5)  =  appH_in_aaaag(x5)
s(x1, x2)  =  s(x1, x2)
appH_out_aaaag(x1, x2, x3, x4, x5)  =  appH_out_aaaag(x1, x2, x3, x4, x5)
U7_aaaag(x1, x2, x3, x4, x5, x6, x7)  =  U7_aaaag(x1, x6, x7)
U19_aaaagaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U19_aaaagaa(x1, x2, x3, x4, x5, x8)
pO_in_ggggaa(x1, x2, x3, x4, x5, x6)  =  pO_in_ggggaa(x1, x2, x3, x4)
U20_ggggaa(x1, x2, x3, x4, x5, x6, x7)  =  U20_ggggaa(x1, x2, x3, x4, x7)
appI_in_gggga(x1, x2, x3, x4, x5)  =  appI_in_gggga(x1, x2, x3, x4)
appI_out_gggga(x1, x2, x3, x4, x5)  =  appI_out_gggga(x1, x2, x3, x4, x5)
U8_gggga(x1, x2, x3, x4, x5, x6, x7)  =  U8_gggga(x1, x2, x3, x4, x5, x7)
U21_ggggaa(x1, x2, x3, x4, x5, x6, x7)  =  U21_ggggaa(x1, x2, x3, x4, x5, x7)
U4_ga(x1, x2, x3)  =  U4_ga(x1, x3)
pE_in_aagaa(x1, x2, x3, x4, x5)  =  pE_in_aagaa(x3)
U22_aagaa(x1, x2, x3, x4, x5, x6)  =  U22_aagaa(x3, x6)
appJ_in_aag(x1, x2, x3)  =  appJ_in_aag(x3)
appJ_out_aag(x1, x2, x3)  =  appJ_out_aag(x1, x2, x3)
U9_aag(x1, x2, x3, x4, x5)  =  U9_aag(x1, x4, x5)
U23_aagaa(x1, x2, x3, x4, x5, x6)  =  U23_aagaa(x1, x2, x3, x6)
pP_in_ggaa(x1, x2, x3, x4)  =  pP_in_ggaa(x1, x2)
U24_ggaa(x1, x2, x3, x4, x5)  =  U24_ggaa(x1, x2, x5)
appK_in_gga(x1, x2, x3)  =  appK_in_gga(x1, x2)
appK_out_gga(x1, x2, x3)  =  appK_out_gga(x1, x2, x3)
U10_gga(x1, x2, x3, x4, x5)  =  U10_gga(x1, x2, x3, x5)
U25_ggaa(x1, x2, x3, x4, x5)  =  U25_ggaa(x1, x2, x3, x5)
parseA_out_ga(x1, x2)  =  parseA_out_ga(x1, x2)
pP_out_ggaa(x1, x2, x3, x4)  =  pP_out_ggaa(x1, x2, x3, x4)
pE_out_aagaa(x1, x2, x3, x4, x5)  =  pE_out_aagaa(x1, x2, x3, x4, x5)
pO_out_ggggaa(x1, x2, x3, x4, x5, x6)  =  pO_out_ggggaa(x1, x2, x3, x4, x5, x6)
pD_out_aaaagaa(x1, x2, x3, x4, x5, x6, x7)  =  pD_out_aaaagaa(x1, x2, x3, x4, x5, x6, x7)
pN_out_ggggggaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  pN_out_ggggggaa(x1, x2, x3, x4, x5, x6, x7, x8)
pC_out_aaaaaggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  pC_out_aaaaaggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)
pB_out_ggggaa(x1, x2, x3, x4, x5, x6)  =  pB_out_ggggaa(x1, x2, x3, x4, x5, x6)
APPI_IN_GGGGA(x1, x2, x3, x4, x5)  =  APPI_IN_GGGGA(x1, x2, x3, x4)

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:

APPI_IN_GGGGA(.(T314, T315), T316, T317, T318, .(T314, X388)) → APPI_IN_GGGGA(T315, T316, T317, T318, X388)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
APPI_IN_GGGGA(x1, x2, x3, x4, x5)  =  APPI_IN_GGGGA(x1, x2, x3, x4)

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:

APPI_IN_GGGGA(.(T314, T315), T316, T317, T318) → APPI_IN_GGGGA(T315, T316, T317, T318)

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:

• APPI_IN_GGGGA(.(T314, T315), T316, T317, T318) → APPI_IN_GGGGA(T315, T316, T317, T318)
  The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3, 4 >= 4

(27) YES

(28) Obligation:

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

APPH_IN_AAAAG(.(T268, X353), X354, X355, X356, .(T268, T269)) → APPH_IN_AAAAG(X353, X354, X355, X356, T269)

The TRS R consists of the following rules:

parseA_in_ga(.(a, .(s(T24, T25, T26), .(b, T27))), T7) → U1_ga(T24, T25, T26, T27, T7, pB_in_ggggaa(T24, T25, T26, T27, X10, T7))
pB_in_ggggaa(T24, T25, T26, T27, T33, T7) → U12_ggggaa(T24, T25, T26, T27, T33, T7, appL_in_gggga(T24, T25, T26, T27, T33))
appL_in_gggga(T54, T55, T56, T57, .(s(a, s(T54, T55, T56), b), T57)) → appL_out_gggga(T54, T55, T56, T57, .(s(a, s(T54, T55, T56), b), T57))
U12_ggggaa(T24, T25, T26, T27, T33, T7, appL_out_gggga(T24, T25, T26, T27, T33)) → U13_ggggaa(T24, T25, T26, T27, T33, T7, parseA_in_ga(T33, T7))
parseA_in_ga(.(T68, T69), T7) → U2_ga(T68, T69, T7, pC_in_aaaaaggaa(X109, X110, X111, X112, X113, T69, T68, X10, T7))
pC_in_aaaaaggaa(T78, T79, T80, T81, T82, T69, T68, X10, T7) → U14_aaaaaggaa(T78, T79, T80, T81, T82, T69, T68, X10, T7, appF_in_aaaaag(T78, T79, T80, T81, T82, T69))
appF_in_aaaaag([], T107, T108, T109, T110, .(a, .(s(T107, T108, T109), .(b, T110)))) → appF_out_aaaaag([], T107, T108, T109, T110, .(a, .(s(T107, T108, T109), .(b, T110))))
appF_in_aaaaag(.(T115, X184), X185, X186, X187, X188, .(T115, T116)) → U5_aaaaag(T115, X184, X185, X186, X187, X188, T116, appF_in_aaaaag(X184, X185, X186, X187, X188, T116))
U5_aaaaag(T115, X184, X185, X186, X187, X188, T116, appF_out_aaaaag(X184, X185, X186, X187, X188, T116)) → appF_out_aaaaag(.(T115, X184), X185, X186, X187, X188, .(T115, T116))
U14_aaaaaggaa(T78, T79, T80, T81, T82, T69, T68, X10, T7, appF_out_aaaaag(T78, T79, T80, T81, T82, T69)) → U15_aaaaaggaa(T78, T79, T80, T81, T82, T69, T68, X10, T7, pN_in_ggggggaa(T68, T78, T79, T80, T81, T82, X10, T7))
pN_in_ggggggaa(T68, T78, T79, T80, T81, T82, T125, T7) → U16_ggggggaa(T68, T78, T79, T80, T81, T82, T125, T7, appM_in_gggggga(T68, T78, T79, T80, T81, T82, T125))
appM_in_gggggga(T146, T147, T148, T149, T150, T151, .(T146, X220)) → U11_gggggga(T146, T147, T148, T149, T150, T151, X220, appG_in_ggggga(T147, T148, T149, T150, T151, X220))
appG_in_ggggga([], T176, T177, T178, T179, .(s(a, s(T176, T177, T178), b), T179)) → appG_out_ggggga([], T176, T177, T178, T179, .(s(a, s(T176, T177, T178), b), T179))
appG_in_ggggga(.(T192, T193), T194, T195, T196, T197, .(T192, X242)) → U6_ggggga(T192, T193, T194, T195, T196, T197, X242, appG_in_ggggga(T193, T194, T195, T196, T197, X242))
U6_ggggga(T192, T193, T194, T195, T196, T197, X242, appG_out_ggggga(T193, T194, T195, T196, T197, X242)) → appG_out_ggggga(.(T192, T193), T194, T195, T196, T197, .(T192, X242))
U11_gggggga(T146, T147, T148, T149, T150, T151, X220, appG_out_ggggga(T147, T148, T149, T150, T151, X220)) → appM_out_gggggga(T146, T147, T148, T149, T150, T151, .(T146, X220))
U16_ggggggaa(T68, T78, T79, T80, T81, T82, T125, T7, appM_out_gggggga(T68, T78, T79, T80, T81, T82, T125)) → U17_ggggggaa(T68, T78, T79, T80, T81, T82, T125, T7, parseA_in_ga(T125, T7))
parseA_in_ga(T230, T232) → U3_ga(T230, T232, pD_in_aaaagaa(X289, X290, X291, X292, T230, X293, T232))
pD_in_aaaagaa(T239, T240, T241, T242, T230, X293, T232) → U18_aaaagaa(T239, T240, T241, T242, T230, X293, T232, appH_in_aaaag(T239, T240, T241, T242, T230))
appH_in_aaaag([], T261, T262, T263, .(a, .(s(T261, T262), .(b, T263)))) → appH_out_aaaag([], T261, T262, T263, .(a, .(s(T261, T262), .(b, T263))))
appH_in_aaaag(.(T268, X353), X354, X355, X356, .(T268, T269)) → U7_aaaag(T268, X353, X354, X355, X356, T269, appH_in_aaaag(X353, X354, X355, X356, T269))
U7_aaaag(T268, X353, X354, X355, X356, T269, appH_out_aaaag(X353, X354, X355, X356, T269)) → appH_out_aaaag(.(T268, X353), X354, X355, X356, .(T268, T269))
U18_aaaagaa(T239, T240, T241, T242, T230, X293, T232, appH_out_aaaag(T239, T240, T241, T242, T230)) → U19_aaaagaa(T239, T240, T241, T242, T230, X293, T232, pO_in_ggggaa(T239, T240, T241, T242, X293, T232))
pO_in_ggggaa(T239, T240, T241, T242, T282, T232) → U20_ggggaa(T239, T240, T241, T242, T282, T232, appI_in_gggga(T239, T240, T241, T242, T282))
appI_in_gggga([], T301, T302, T303, .(s(a, s(T301, T302), b), T303)) → appI_out_gggga([], T301, T302, T303, .(s(a, s(T301, T302), b), T303))
appI_in_gggga(.(T314, T315), T316, T317, T318, .(T314, X388)) → U8_gggga(T314, T315, T316, T317, T318, X388, appI_in_gggga(T315, T316, T317, T318, X388))
U8_gggga(T314, T315, T316, T317, T318, X388, appI_out_gggga(T315, T316, T317, T318, X388)) → appI_out_gggga(.(T314, T315), T316, T317, T318, .(T314, X388))
U20_ggggaa(T239, T240, T241, T242, T282, T232, appI_out_gggga(T239, T240, T241, T242, T282)) → U21_ggggaa(T239, T240, T241, T242, T282, T232, parseA_in_ga(T282, T232))
parseA_in_ga(T341, T343) → U4_ga(T341, T343, pE_in_aagaa(X423, X424, T341, X425, T343))
pE_in_aagaa(T346, T347, T341, X425, T343) → U22_aagaa(T346, T347, T341, X425, T343, appJ_in_aag(T346, T347, T341))
appJ_in_aag([], T354, .(a, .(b, T354))) → appJ_out_aag([], T354, .(a, .(b, T354)))
appJ_in_aag(.(T359, X463), X464, .(T359, T360)) → U9_aag(T359, X463, X464, T360, appJ_in_aag(X463, X464, T360))
U9_aag(T359, X463, X464, T360, appJ_out_aag(X463, X464, T360)) → appJ_out_aag(.(T359, X463), X464, .(T359, T360))
U22_aagaa(T346, T347, T341, X425, T343, appJ_out_aag(T346, T347, T341)) → U23_aagaa(T346, T347, T341, X425, T343, pP_in_ggaa(T346, T347, X425, T343))
pP_in_ggaa(T346, T347, T365, T343) → U24_ggaa(T346, T347, T365, T343, appK_in_gga(T346, T347, T365))
appK_in_gga([], T372, .(s(a, b), T372)) → appK_out_gga([], T372, .(s(a, b), T372))
appK_in_gga(.(T379, T380), T381, .(T379, X492)) → U10_gga(T379, T380, T381, X492, appK_in_gga(T380, T381, X492))
U10_gga(T379, T380, T381, X492, appK_out_gga(T380, T381, X492)) → appK_out_gga(.(T379, T380), T381, .(T379, X492))
U24_ggaa(T346, T347, T365, T343, appK_out_gga(T346, T347, T365)) → U25_ggaa(T346, T347, T365, T343, parseA_in_ga(T365, T343))
parseA_in_ga(.(s(T394, T395), []), s(T394, T395)) → parseA_out_ga(.(s(T394, T395), []), s(T394, T395))
parseA_in_ga(.(s(T402, T403, T404), []), s(T402, T403, T404)) → parseA_out_ga(.(s(T402, T403, T404), []), s(T402, T403, T404))
U25_ggaa(T346, T347, T365, T343, parseA_out_ga(T365, T343)) → pP_out_ggaa(T346, T347, T365, T343)
U23_aagaa(T346, T347, T341, X425, T343, pP_out_ggaa(T346, T347, X425, T343)) → pE_out_aagaa(T346, T347, T341, X425, T343)
U4_ga(T341, T343, pE_out_aagaa(X423, X424, T341, X425, T343)) → parseA_out_ga(T341, T343)
U21_ggggaa(T239, T240, T241, T242, T282, T232, parseA_out_ga(T282, T232)) → pO_out_ggggaa(T239, T240, T241, T242, T282, T232)
U19_aaaagaa(T239, T240, T241, T242, T230, X293, T232, pO_out_ggggaa(T239, T240, T241, T242, X293, T232)) → pD_out_aaaagaa(T239, T240, T241, T242, T230, X293, T232)
U3_ga(T230, T232, pD_out_aaaagaa(X289, X290, X291, X292, T230, X293, T232)) → parseA_out_ga(T230, T232)
U17_ggggggaa(T68, T78, T79, T80, T81, T82, T125, T7, parseA_out_ga(T125, T7)) → pN_out_ggggggaa(T68, T78, T79, T80, T81, T82, T125, T7)
U15_aaaaaggaa(T78, T79, T80, T81, T82, T69, T68, X10, T7, pN_out_ggggggaa(T68, T78, T79, T80, T81, T82, X10, T7)) → pC_out_aaaaaggaa(T78, T79, T80, T81, T82, T69, T68, X10, T7)
U2_ga(T68, T69, T7, pC_out_aaaaaggaa(X109, X110, X111, X112, X113, T69, T68, X10, T7)) → parseA_out_ga(.(T68, T69), T7)
U13_ggggaa(T24, T25, T26, T27, T33, T7, parseA_out_ga(T33, T7)) → pB_out_ggggaa(T24, T25, T26, T27, T33, T7)
U1_ga(T24, T25, T26, T27, T7, pB_out_ggggaa(T24, T25, T26, T27, X10, T7)) → parseA_out_ga(.(a, .(s(T24, T25, T26), .(b, T27))), T7)

The argument filtering Pi contains the following mapping:
parseA_in_ga(x1, x2)  =  parseA_in_ga(x1)
.(x1, x2)  =  .(x1, x2)
a  =  a
s(x1, x2, x3)  =  s(x1, x2, x3)
b  =  b
U1_ga(x1, x2, x3, x4, x5, x6)  =  U1_ga(x1, x2, x3, x4, x6)
pB_in_ggggaa(x1, x2, x3, x4, x5, x6)  =  pB_in_ggggaa(x1, x2, x3, x4)
U12_ggggaa(x1, x2, x3, x4, x5, x6, x7)  =  U12_ggggaa(x1, x2, x3, x4, x7)
appL_in_gggga(x1, x2, x3, x4, x5)  =  appL_in_gggga(x1, x2, x3, x4)
appL_out_gggga(x1, x2, x3, x4, x5)  =  appL_out_gggga(x1, x2, x3, x4, x5)
U13_ggggaa(x1, x2, x3, x4, x5, x6, x7)  =  U13_ggggaa(x1, x2, x3, x4, x5, x7)
U2_ga(x1, x2, x3, x4)  =  U2_ga(x1, x2, x4)
pC_in_aaaaaggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  pC_in_aaaaaggaa(x6, x7)
U14_aaaaaggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10)  =  U14_aaaaaggaa(x6, x7, x10)
appF_in_aaaaag(x1, x2, x3, x4, x5, x6)  =  appF_in_aaaaag(x6)
appF_out_aaaaag(x1, x2, x3, x4, x5, x6)  =  appF_out_aaaaag(x1, x2, x3, x4, x5, x6)
U5_aaaaag(x1, x2, x3, x4, x5, x6, x7, x8)  =  U5_aaaaag(x1, x7, x8)
U15_aaaaaggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10)  =  U15_aaaaaggaa(x1, x2, x3, x4, x5, x6, x7, x10)
pN_in_ggggggaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  pN_in_ggggggaa(x1, x2, x3, x4, x5, x6)
U16_ggggggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U16_ggggggaa(x1, x2, x3, x4, x5, x6, x9)
appM_in_gggggga(x1, x2, x3, x4, x5, x6, x7)  =  appM_in_gggggga(x1, x2, x3, x4, x5, x6)
U11_gggggga(x1, x2, x3, x4, x5, x6, x7, x8)  =  U11_gggggga(x1, x2, x3, x4, x5, x6, x8)
appG_in_ggggga(x1, x2, x3, x4, x5, x6)  =  appG_in_ggggga(x1, x2, x3, x4, x5)
[]  =  []
appG_out_ggggga(x1, x2, x3, x4, x5, x6)  =  appG_out_ggggga(x1, x2, x3, x4, x5, x6)
U6_ggggga(x1, x2, x3, x4, x5, x6, x7, x8)  =  U6_ggggga(x1, x2, x3, x4, x5, x6, x8)
appM_out_gggggga(x1, x2, x3, x4, x5, x6, x7)  =  appM_out_gggggga(x1, x2, x3, x4, x5, x6, x7)
U17_ggggggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U17_ggggggaa(x1, x2, x3, x4, x5, x6, x7, x9)
U3_ga(x1, x2, x3)  =  U3_ga(x1, x3)
pD_in_aaaagaa(x1, x2, x3, x4, x5, x6, x7)  =  pD_in_aaaagaa(x5)
U18_aaaagaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U18_aaaagaa(x5, x8)
appH_in_aaaag(x1, x2, x3, x4, x5)  =  appH_in_aaaag(x5)
s(x1, x2)  =  s(x1, x2)
appH_out_aaaag(x1, x2, x3, x4, x5)  =  appH_out_aaaag(x1, x2, x3, x4, x5)
U7_aaaag(x1, x2, x3, x4, x5, x6, x7)  =  U7_aaaag(x1, x6, x7)
U19_aaaagaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U19_aaaagaa(x1, x2, x3, x4, x5, x8)
pO_in_ggggaa(x1, x2, x3, x4, x5, x6)  =  pO_in_ggggaa(x1, x2, x3, x4)
U20_ggggaa(x1, x2, x3, x4, x5, x6, x7)  =  U20_ggggaa(x1, x2, x3, x4, x7)
appI_in_gggga(x1, x2, x3, x4, x5)  =  appI_in_gggga(x1, x2, x3, x4)
appI_out_gggga(x1, x2, x3, x4, x5)  =  appI_out_gggga(x1, x2, x3, x4, x5)
U8_gggga(x1, x2, x3, x4, x5, x6, x7)  =  U8_gggga(x1, x2, x3, x4, x5, x7)
U21_ggggaa(x1, x2, x3, x4, x5, x6, x7)  =  U21_ggggaa(x1, x2, x3, x4, x5, x7)
U4_ga(x1, x2, x3)  =  U4_ga(x1, x3)
pE_in_aagaa(x1, x2, x3, x4, x5)  =  pE_in_aagaa(x3)
U22_aagaa(x1, x2, x3, x4, x5, x6)  =  U22_aagaa(x3, x6)
appJ_in_aag(x1, x2, x3)  =  appJ_in_aag(x3)
appJ_out_aag(x1, x2, x3)  =  appJ_out_aag(x1, x2, x3)
U9_aag(x1, x2, x3, x4, x5)  =  U9_aag(x1, x4, x5)
U23_aagaa(x1, x2, x3, x4, x5, x6)  =  U23_aagaa(x1, x2, x3, x6)
pP_in_ggaa(x1, x2, x3, x4)  =  pP_in_ggaa(x1, x2)
U24_ggaa(x1, x2, x3, x4, x5)  =  U24_ggaa(x1, x2, x5)
appK_in_gga(x1, x2, x3)  =  appK_in_gga(x1, x2)
appK_out_gga(x1, x2, x3)  =  appK_out_gga(x1, x2, x3)
U10_gga(x1, x2, x3, x4, x5)  =  U10_gga(x1, x2, x3, x5)
U25_ggaa(x1, x2, x3, x4, x5)  =  U25_ggaa(x1, x2, x3, x5)
parseA_out_ga(x1, x2)  =  parseA_out_ga(x1, x2)
pP_out_ggaa(x1, x2, x3, x4)  =  pP_out_ggaa(x1, x2, x3, x4)
pE_out_aagaa(x1, x2, x3, x4, x5)  =  pE_out_aagaa(x1, x2, x3, x4, x5)
pO_out_ggggaa(x1, x2, x3, x4, x5, x6)  =  pO_out_ggggaa(x1, x2, x3, x4, x5, x6)
pD_out_aaaagaa(x1, x2, x3, x4, x5, x6, x7)  =  pD_out_aaaagaa(x1, x2, x3, x4, x5, x6, x7)
pN_out_ggggggaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  pN_out_ggggggaa(x1, x2, x3, x4, x5, x6, x7, x8)
pC_out_aaaaaggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  pC_out_aaaaaggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)
pB_out_ggggaa(x1, x2, x3, x4, x5, x6)  =  pB_out_ggggaa(x1, x2, x3, x4, x5, x6)
APPH_IN_AAAAG(x1, x2, x3, x4, x5)  =  APPH_IN_AAAAG(x5)

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:

APPH_IN_AAAAG(.(T268, X353), X354, X355, X356, .(T268, T269)) → APPH_IN_AAAAG(X353, X354, X355, X356, T269)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
APPH_IN_AAAAG(x1, x2, x3, x4, x5)  =  APPH_IN_AAAAG(x5)

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:

APPH_IN_AAAAG(.(T268, T269)) → APPH_IN_AAAAG(T269)

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:

• APPH_IN_AAAAG(.(T268, T269)) → APPH_IN_AAAAG(T269)
  The graph contains the following edges 1 > 1

(34) YES

(35) Obligation:

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

APPG_IN_GGGGGA(.(T192, T193), T194, T195, T196, T197, .(T192, X242)) → APPG_IN_GGGGGA(T193, T194, T195, T196, T197, X242)

The TRS R consists of the following rules:

parseA_in_ga(.(a, .(s(T24, T25, T26), .(b, T27))), T7) → U1_ga(T24, T25, T26, T27, T7, pB_in_ggggaa(T24, T25, T26, T27, X10, T7))
pB_in_ggggaa(T24, T25, T26, T27, T33, T7) → U12_ggggaa(T24, T25, T26, T27, T33, T7, appL_in_gggga(T24, T25, T26, T27, T33))
appL_in_gggga(T54, T55, T56, T57, .(s(a, s(T54, T55, T56), b), T57)) → appL_out_gggga(T54, T55, T56, T57, .(s(a, s(T54, T55, T56), b), T57))
U12_ggggaa(T24, T25, T26, T27, T33, T7, appL_out_gggga(T24, T25, T26, T27, T33)) → U13_ggggaa(T24, T25, T26, T27, T33, T7, parseA_in_ga(T33, T7))
parseA_in_ga(.(T68, T69), T7) → U2_ga(T68, T69, T7, pC_in_aaaaaggaa(X109, X110, X111, X112, X113, T69, T68, X10, T7))
pC_in_aaaaaggaa(T78, T79, T80, T81, T82, T69, T68, X10, T7) → U14_aaaaaggaa(T78, T79, T80, T81, T82, T69, T68, X10, T7, appF_in_aaaaag(T78, T79, T80, T81, T82, T69))
appF_in_aaaaag([], T107, T108, T109, T110, .(a, .(s(T107, T108, T109), .(b, T110)))) → appF_out_aaaaag([], T107, T108, T109, T110, .(a, .(s(T107, T108, T109), .(b, T110))))
appF_in_aaaaag(.(T115, X184), X185, X186, X187, X188, .(T115, T116)) → U5_aaaaag(T115, X184, X185, X186, X187, X188, T116, appF_in_aaaaag(X184, X185, X186, X187, X188, T116))
U5_aaaaag(T115, X184, X185, X186, X187, X188, T116, appF_out_aaaaag(X184, X185, X186, X187, X188, T116)) → appF_out_aaaaag(.(T115, X184), X185, X186, X187, X188, .(T115, T116))
U14_aaaaaggaa(T78, T79, T80, T81, T82, T69, T68, X10, T7, appF_out_aaaaag(T78, T79, T80, T81, T82, T69)) → U15_aaaaaggaa(T78, T79, T80, T81, T82, T69, T68, X10, T7, pN_in_ggggggaa(T68, T78, T79, T80, T81, T82, X10, T7))
pN_in_ggggggaa(T68, T78, T79, T80, T81, T82, T125, T7) → U16_ggggggaa(T68, T78, T79, T80, T81, T82, T125, T7, appM_in_gggggga(T68, T78, T79, T80, T81, T82, T125))
appM_in_gggggga(T146, T147, T148, T149, T150, T151, .(T146, X220)) → U11_gggggga(T146, T147, T148, T149, T150, T151, X220, appG_in_ggggga(T147, T148, T149, T150, T151, X220))
appG_in_ggggga([], T176, T177, T178, T179, .(s(a, s(T176, T177, T178), b), T179)) → appG_out_ggggga([], T176, T177, T178, T179, .(s(a, s(T176, T177, T178), b), T179))
appG_in_ggggga(.(T192, T193), T194, T195, T196, T197, .(T192, X242)) → U6_ggggga(T192, T193, T194, T195, T196, T197, X242, appG_in_ggggga(T193, T194, T195, T196, T197, X242))
U6_ggggga(T192, T193, T194, T195, T196, T197, X242, appG_out_ggggga(T193, T194, T195, T196, T197, X242)) → appG_out_ggggga(.(T192, T193), T194, T195, T196, T197, .(T192, X242))
U11_gggggga(T146, T147, T148, T149, T150, T151, X220, appG_out_ggggga(T147, T148, T149, T150, T151, X220)) → appM_out_gggggga(T146, T147, T148, T149, T150, T151, .(T146, X220))
U16_ggggggaa(T68, T78, T79, T80, T81, T82, T125, T7, appM_out_gggggga(T68, T78, T79, T80, T81, T82, T125)) → U17_ggggggaa(T68, T78, T79, T80, T81, T82, T125, T7, parseA_in_ga(T125, T7))
parseA_in_ga(T230, T232) → U3_ga(T230, T232, pD_in_aaaagaa(X289, X290, X291, X292, T230, X293, T232))
pD_in_aaaagaa(T239, T240, T241, T242, T230, X293, T232) → U18_aaaagaa(T239, T240, T241, T242, T230, X293, T232, appH_in_aaaag(T239, T240, T241, T242, T230))
appH_in_aaaag([], T261, T262, T263, .(a, .(s(T261, T262), .(b, T263)))) → appH_out_aaaag([], T261, T262, T263, .(a, .(s(T261, T262), .(b, T263))))
appH_in_aaaag(.(T268, X353), X354, X355, X356, .(T268, T269)) → U7_aaaag(T268, X353, X354, X355, X356, T269, appH_in_aaaag(X353, X354, X355, X356, T269))
U7_aaaag(T268, X353, X354, X355, X356, T269, appH_out_aaaag(X353, X354, X355, X356, T269)) → appH_out_aaaag(.(T268, X353), X354, X355, X356, .(T268, T269))
U18_aaaagaa(T239, T240, T241, T242, T230, X293, T232, appH_out_aaaag(T239, T240, T241, T242, T230)) → U19_aaaagaa(T239, T240, T241, T242, T230, X293, T232, pO_in_ggggaa(T239, T240, T241, T242, X293, T232))
pO_in_ggggaa(T239, T240, T241, T242, T282, T232) → U20_ggggaa(T239, T240, T241, T242, T282, T232, appI_in_gggga(T239, T240, T241, T242, T282))
appI_in_gggga([], T301, T302, T303, .(s(a, s(T301, T302), b), T303)) → appI_out_gggga([], T301, T302, T303, .(s(a, s(T301, T302), b), T303))
appI_in_gggga(.(T314, T315), T316, T317, T318, .(T314, X388)) → U8_gggga(T314, T315, T316, T317, T318, X388, appI_in_gggga(T315, T316, T317, T318, X388))
U8_gggga(T314, T315, T316, T317, T318, X388, appI_out_gggga(T315, T316, T317, T318, X388)) → appI_out_gggga(.(T314, T315), T316, T317, T318, .(T314, X388))
U20_ggggaa(T239, T240, T241, T242, T282, T232, appI_out_gggga(T239, T240, T241, T242, T282)) → U21_ggggaa(T239, T240, T241, T242, T282, T232, parseA_in_ga(T282, T232))
parseA_in_ga(T341, T343) → U4_ga(T341, T343, pE_in_aagaa(X423, X424, T341, X425, T343))
pE_in_aagaa(T346, T347, T341, X425, T343) → U22_aagaa(T346, T347, T341, X425, T343, appJ_in_aag(T346, T347, T341))
appJ_in_aag([], T354, .(a, .(b, T354))) → appJ_out_aag([], T354, .(a, .(b, T354)))
appJ_in_aag(.(T359, X463), X464, .(T359, T360)) → U9_aag(T359, X463, X464, T360, appJ_in_aag(X463, X464, T360))
U9_aag(T359, X463, X464, T360, appJ_out_aag(X463, X464, T360)) → appJ_out_aag(.(T359, X463), X464, .(T359, T360))
U22_aagaa(T346, T347, T341, X425, T343, appJ_out_aag(T346, T347, T341)) → U23_aagaa(T346, T347, T341, X425, T343, pP_in_ggaa(T346, T347, X425, T343))
pP_in_ggaa(T346, T347, T365, T343) → U24_ggaa(T346, T347, T365, T343, appK_in_gga(T346, T347, T365))
appK_in_gga([], T372, .(s(a, b), T372)) → appK_out_gga([], T372, .(s(a, b), T372))
appK_in_gga(.(T379, T380), T381, .(T379, X492)) → U10_gga(T379, T380, T381, X492, appK_in_gga(T380, T381, X492))
U10_gga(T379, T380, T381, X492, appK_out_gga(T380, T381, X492)) → appK_out_gga(.(T379, T380), T381, .(T379, X492))
U24_ggaa(T346, T347, T365, T343, appK_out_gga(T346, T347, T365)) → U25_ggaa(T346, T347, T365, T343, parseA_in_ga(T365, T343))
parseA_in_ga(.(s(T394, T395), []), s(T394, T395)) → parseA_out_ga(.(s(T394, T395), []), s(T394, T395))
parseA_in_ga(.(s(T402, T403, T404), []), s(T402, T403, T404)) → parseA_out_ga(.(s(T402, T403, T404), []), s(T402, T403, T404))
U25_ggaa(T346, T347, T365, T343, parseA_out_ga(T365, T343)) → pP_out_ggaa(T346, T347, T365, T343)
U23_aagaa(T346, T347, T341, X425, T343, pP_out_ggaa(T346, T347, X425, T343)) → pE_out_aagaa(T346, T347, T341, X425, T343)
U4_ga(T341, T343, pE_out_aagaa(X423, X424, T341, X425, T343)) → parseA_out_ga(T341, T343)
U21_ggggaa(T239, T240, T241, T242, T282, T232, parseA_out_ga(T282, T232)) → pO_out_ggggaa(T239, T240, T241, T242, T282, T232)
U19_aaaagaa(T239, T240, T241, T242, T230, X293, T232, pO_out_ggggaa(T239, T240, T241, T242, X293, T232)) → pD_out_aaaagaa(T239, T240, T241, T242, T230, X293, T232)
U3_ga(T230, T232, pD_out_aaaagaa(X289, X290, X291, X292, T230, X293, T232)) → parseA_out_ga(T230, T232)
U17_ggggggaa(T68, T78, T79, T80, T81, T82, T125, T7, parseA_out_ga(T125, T7)) → pN_out_ggggggaa(T68, T78, T79, T80, T81, T82, T125, T7)
U15_aaaaaggaa(T78, T79, T80, T81, T82, T69, T68, X10, T7, pN_out_ggggggaa(T68, T78, T79, T80, T81, T82, X10, T7)) → pC_out_aaaaaggaa(T78, T79, T80, T81, T82, T69, T68, X10, T7)
U2_ga(T68, T69, T7, pC_out_aaaaaggaa(X109, X110, X111, X112, X113, T69, T68, X10, T7)) → parseA_out_ga(.(T68, T69), T7)
U13_ggggaa(T24, T25, T26, T27, T33, T7, parseA_out_ga(T33, T7)) → pB_out_ggggaa(T24, T25, T26, T27, T33, T7)
U1_ga(T24, T25, T26, T27, T7, pB_out_ggggaa(T24, T25, T26, T27, X10, T7)) → parseA_out_ga(.(a, .(s(T24, T25, T26), .(b, T27))), T7)

The argument filtering Pi contains the following mapping:
parseA_in_ga(x1, x2)  =  parseA_in_ga(x1)
.(x1, x2)  =  .(x1, x2)
a  =  a
s(x1, x2, x3)  =  s(x1, x2, x3)
b  =  b
U1_ga(x1, x2, x3, x4, x5, x6)  =  U1_ga(x1, x2, x3, x4, x6)
pB_in_ggggaa(x1, x2, x3, x4, x5, x6)  =  pB_in_ggggaa(x1, x2, x3, x4)
U12_ggggaa(x1, x2, x3, x4, x5, x6, x7)  =  U12_ggggaa(x1, x2, x3, x4, x7)
appL_in_gggga(x1, x2, x3, x4, x5)  =  appL_in_gggga(x1, x2, x3, x4)
appL_out_gggga(x1, x2, x3, x4, x5)  =  appL_out_gggga(x1, x2, x3, x4, x5)
U13_ggggaa(x1, x2, x3, x4, x5, x6, x7)  =  U13_ggggaa(x1, x2, x3, x4, x5, x7)
U2_ga(x1, x2, x3, x4)  =  U2_ga(x1, x2, x4)
pC_in_aaaaaggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  pC_in_aaaaaggaa(x6, x7)
U14_aaaaaggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10)  =  U14_aaaaaggaa(x6, x7, x10)
appF_in_aaaaag(x1, x2, x3, x4, x5, x6)  =  appF_in_aaaaag(x6)
appF_out_aaaaag(x1, x2, x3, x4, x5, x6)  =  appF_out_aaaaag(x1, x2, x3, x4, x5, x6)
U5_aaaaag(x1, x2, x3, x4, x5, x6, x7, x8)  =  U5_aaaaag(x1, x7, x8)
U15_aaaaaggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10)  =  U15_aaaaaggaa(x1, x2, x3, x4, x5, x6, x7, x10)
pN_in_ggggggaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  pN_in_ggggggaa(x1, x2, x3, x4, x5, x6)
U16_ggggggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U16_ggggggaa(x1, x2, x3, x4, x5, x6, x9)
appM_in_gggggga(x1, x2, x3, x4, x5, x6, x7)  =  appM_in_gggggga(x1, x2, x3, x4, x5, x6)
U11_gggggga(x1, x2, x3, x4, x5, x6, x7, x8)  =  U11_gggggga(x1, x2, x3, x4, x5, x6, x8)
appG_in_ggggga(x1, x2, x3, x4, x5, x6)  =  appG_in_ggggga(x1, x2, x3, x4, x5)
[]  =  []
appG_out_ggggga(x1, x2, x3, x4, x5, x6)  =  appG_out_ggggga(x1, x2, x3, x4, x5, x6)
U6_ggggga(x1, x2, x3, x4, x5, x6, x7, x8)  =  U6_ggggga(x1, x2, x3, x4, x5, x6, x8)
appM_out_gggggga(x1, x2, x3, x4, x5, x6, x7)  =  appM_out_gggggga(x1, x2, x3, x4, x5, x6, x7)
U17_ggggggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U17_ggggggaa(x1, x2, x3, x4, x5, x6, x7, x9)
U3_ga(x1, x2, x3)  =  U3_ga(x1, x3)
pD_in_aaaagaa(x1, x2, x3, x4, x5, x6, x7)  =  pD_in_aaaagaa(x5)
U18_aaaagaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U18_aaaagaa(x5, x8)
appH_in_aaaag(x1, x2, x3, x4, x5)  =  appH_in_aaaag(x5)
s(x1, x2)  =  s(x1, x2)
appH_out_aaaag(x1, x2, x3, x4, x5)  =  appH_out_aaaag(x1, x2, x3, x4, x5)
U7_aaaag(x1, x2, x3, x4, x5, x6, x7)  =  U7_aaaag(x1, x6, x7)
U19_aaaagaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U19_aaaagaa(x1, x2, x3, x4, x5, x8)
pO_in_ggggaa(x1, x2, x3, x4, x5, x6)  =  pO_in_ggggaa(x1, x2, x3, x4)
U20_ggggaa(x1, x2, x3, x4, x5, x6, x7)  =  U20_ggggaa(x1, x2, x3, x4, x7)
appI_in_gggga(x1, x2, x3, x4, x5)  =  appI_in_gggga(x1, x2, x3, x4)
appI_out_gggga(x1, x2, x3, x4, x5)  =  appI_out_gggga(x1, x2, x3, x4, x5)
U8_gggga(x1, x2, x3, x4, x5, x6, x7)  =  U8_gggga(x1, x2, x3, x4, x5, x7)
U21_ggggaa(x1, x2, x3, x4, x5, x6, x7)  =  U21_ggggaa(x1, x2, x3, x4, x5, x7)
U4_ga(x1, x2, x3)  =  U4_ga(x1, x3)
pE_in_aagaa(x1, x2, x3, x4, x5)  =  pE_in_aagaa(x3)
U22_aagaa(x1, x2, x3, x4, x5, x6)  =  U22_aagaa(x3, x6)
appJ_in_aag(x1, x2, x3)  =  appJ_in_aag(x3)
appJ_out_aag(x1, x2, x3)  =  appJ_out_aag(x1, x2, x3)
U9_aag(x1, x2, x3, x4, x5)  =  U9_aag(x1, x4, x5)
U23_aagaa(x1, x2, x3, x4, x5, x6)  =  U23_aagaa(x1, x2, x3, x6)
pP_in_ggaa(x1, x2, x3, x4)  =  pP_in_ggaa(x1, x2)
U24_ggaa(x1, x2, x3, x4, x5)  =  U24_ggaa(x1, x2, x5)
appK_in_gga(x1, x2, x3)  =  appK_in_gga(x1, x2)
appK_out_gga(x1, x2, x3)  =  appK_out_gga(x1, x2, x3)
U10_gga(x1, x2, x3, x4, x5)  =  U10_gga(x1, x2, x3, x5)
U25_ggaa(x1, x2, x3, x4, x5)  =  U25_ggaa(x1, x2, x3, x5)
parseA_out_ga(x1, x2)  =  parseA_out_ga(x1, x2)
pP_out_ggaa(x1, x2, x3, x4)  =  pP_out_ggaa(x1, x2, x3, x4)
pE_out_aagaa(x1, x2, x3, x4, x5)  =  pE_out_aagaa(x1, x2, x3, x4, x5)
pO_out_ggggaa(x1, x2, x3, x4, x5, x6)  =  pO_out_ggggaa(x1, x2, x3, x4, x5, x6)
pD_out_aaaagaa(x1, x2, x3, x4, x5, x6, x7)  =  pD_out_aaaagaa(x1, x2, x3, x4, x5, x6, x7)
pN_out_ggggggaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  pN_out_ggggggaa(x1, x2, x3, x4, x5, x6, x7, x8)
pC_out_aaaaaggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  pC_out_aaaaaggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)
pB_out_ggggaa(x1, x2, x3, x4, x5, x6)  =  pB_out_ggggaa(x1, x2, x3, x4, x5, x6)
APPG_IN_GGGGGA(x1, x2, x3, x4, x5, x6)  =  APPG_IN_GGGGGA(x1, x2, x3, x4, x5)

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

(36) UsableRulesProof (EQUIVALENT transformation)

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

(37) Obligation:

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

APPG_IN_GGGGGA(.(T192, T193), T194, T195, T196, T197, .(T192, X242)) → APPG_IN_GGGGGA(T193, T194, T195, T196, T197, X242)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
APPG_IN_GGGGGA(x1, x2, x3, x4, x5, x6)  =  APPG_IN_GGGGGA(x1, x2, x3, x4, x5)

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

(38) PiDPToQDPProof (SOUND transformation)

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

(39) Obligation:

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

APPG_IN_GGGGGA(.(T192, T193), T194, T195, T196, T197) → APPG_IN_GGGGGA(T193, T194, T195, T196, T197)

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

(40) 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:

• APPG_IN_GGGGGA(.(T192, T193), T194, T195, T196, T197) → APPG_IN_GGGGGA(T193, T194, T195, T196, T197)
  The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3, 4 >= 4, 5 >= 5

(41) YES

(42) Obligation:

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

APPF_IN_AAAAAG(.(T115, X184), X185, X186, X187, X188, .(T115, T116)) → APPF_IN_AAAAAG(X184, X185, X186, X187, X188, T116)

The TRS R consists of the following rules:

parseA_in_ga(.(a, .(s(T24, T25, T26), .(b, T27))), T7) → U1_ga(T24, T25, T26, T27, T7, pB_in_ggggaa(T24, T25, T26, T27, X10, T7))
pB_in_ggggaa(T24, T25, T26, T27, T33, T7) → U12_ggggaa(T24, T25, T26, T27, T33, T7, appL_in_gggga(T24, T25, T26, T27, T33))
appL_in_gggga(T54, T55, T56, T57, .(s(a, s(T54, T55, T56), b), T57)) → appL_out_gggga(T54, T55, T56, T57, .(s(a, s(T54, T55, T56), b), T57))
U12_ggggaa(T24, T25, T26, T27, T33, T7, appL_out_gggga(T24, T25, T26, T27, T33)) → U13_ggggaa(T24, T25, T26, T27, T33, T7, parseA_in_ga(T33, T7))
parseA_in_ga(.(T68, T69), T7) → U2_ga(T68, T69, T7, pC_in_aaaaaggaa(X109, X110, X111, X112, X113, T69, T68, X10, T7))
pC_in_aaaaaggaa(T78, T79, T80, T81, T82, T69, T68, X10, T7) → U14_aaaaaggaa(T78, T79, T80, T81, T82, T69, T68, X10, T7, appF_in_aaaaag(T78, T79, T80, T81, T82, T69))
appF_in_aaaaag([], T107, T108, T109, T110, .(a, .(s(T107, T108, T109), .(b, T110)))) → appF_out_aaaaag([], T107, T108, T109, T110, .(a, .(s(T107, T108, T109), .(b, T110))))
appF_in_aaaaag(.(T115, X184), X185, X186, X187, X188, .(T115, T116)) → U5_aaaaag(T115, X184, X185, X186, X187, X188, T116, appF_in_aaaaag(X184, X185, X186, X187, X188, T116))
U5_aaaaag(T115, X184, X185, X186, X187, X188, T116, appF_out_aaaaag(X184, X185, X186, X187, X188, T116)) → appF_out_aaaaag(.(T115, X184), X185, X186, X187, X188, .(T115, T116))
U14_aaaaaggaa(T78, T79, T80, T81, T82, T69, T68, X10, T7, appF_out_aaaaag(T78, T79, T80, T81, T82, T69)) → U15_aaaaaggaa(T78, T79, T80, T81, T82, T69, T68, X10, T7, pN_in_ggggggaa(T68, T78, T79, T80, T81, T82, X10, T7))
pN_in_ggggggaa(T68, T78, T79, T80, T81, T82, T125, T7) → U16_ggggggaa(T68, T78, T79, T80, T81, T82, T125, T7, appM_in_gggggga(T68, T78, T79, T80, T81, T82, T125))
appM_in_gggggga(T146, T147, T148, T149, T150, T151, .(T146, X220)) → U11_gggggga(T146, T147, T148, T149, T150, T151, X220, appG_in_ggggga(T147, T148, T149, T150, T151, X220))
appG_in_ggggga([], T176, T177, T178, T179, .(s(a, s(T176, T177, T178), b), T179)) → appG_out_ggggga([], T176, T177, T178, T179, .(s(a, s(T176, T177, T178), b), T179))
appG_in_ggggga(.(T192, T193), T194, T195, T196, T197, .(T192, X242)) → U6_ggggga(T192, T193, T194, T195, T196, T197, X242, appG_in_ggggga(T193, T194, T195, T196, T197, X242))
U6_ggggga(T192, T193, T194, T195, T196, T197, X242, appG_out_ggggga(T193, T194, T195, T196, T197, X242)) → appG_out_ggggga(.(T192, T193), T194, T195, T196, T197, .(T192, X242))
U11_gggggga(T146, T147, T148, T149, T150, T151, X220, appG_out_ggggga(T147, T148, T149, T150, T151, X220)) → appM_out_gggggga(T146, T147, T148, T149, T150, T151, .(T146, X220))
U16_ggggggaa(T68, T78, T79, T80, T81, T82, T125, T7, appM_out_gggggga(T68, T78, T79, T80, T81, T82, T125)) → U17_ggggggaa(T68, T78, T79, T80, T81, T82, T125, T7, parseA_in_ga(T125, T7))
parseA_in_ga(T230, T232) → U3_ga(T230, T232, pD_in_aaaagaa(X289, X290, X291, X292, T230, X293, T232))
pD_in_aaaagaa(T239, T240, T241, T242, T230, X293, T232) → U18_aaaagaa(T239, T240, T241, T242, T230, X293, T232, appH_in_aaaag(T239, T240, T241, T242, T230))
appH_in_aaaag([], T261, T262, T263, .(a, .(s(T261, T262), .(b, T263)))) → appH_out_aaaag([], T261, T262, T263, .(a, .(s(T261, T262), .(b, T263))))
appH_in_aaaag(.(T268, X353), X354, X355, X356, .(T268, T269)) → U7_aaaag(T268, X353, X354, X355, X356, T269, appH_in_aaaag(X353, X354, X355, X356, T269))
U7_aaaag(T268, X353, X354, X355, X356, T269, appH_out_aaaag(X353, X354, X355, X356, T269)) → appH_out_aaaag(.(T268, X353), X354, X355, X356, .(T268, T269))
U18_aaaagaa(T239, T240, T241, T242, T230, X293, T232, appH_out_aaaag(T239, T240, T241, T242, T230)) → U19_aaaagaa(T239, T240, T241, T242, T230, X293, T232, pO_in_ggggaa(T239, T240, T241, T242, X293, T232))
pO_in_ggggaa(T239, T240, T241, T242, T282, T232) → U20_ggggaa(T239, T240, T241, T242, T282, T232, appI_in_gggga(T239, T240, T241, T242, T282))
appI_in_gggga([], T301, T302, T303, .(s(a, s(T301, T302), b), T303)) → appI_out_gggga([], T301, T302, T303, .(s(a, s(T301, T302), b), T303))
appI_in_gggga(.(T314, T315), T316, T317, T318, .(T314, X388)) → U8_gggga(T314, T315, T316, T317, T318, X388, appI_in_gggga(T315, T316, T317, T318, X388))
U8_gggga(T314, T315, T316, T317, T318, X388, appI_out_gggga(T315, T316, T317, T318, X388)) → appI_out_gggga(.(T314, T315), T316, T317, T318, .(T314, X388))
U20_ggggaa(T239, T240, T241, T242, T282, T232, appI_out_gggga(T239, T240, T241, T242, T282)) → U21_ggggaa(T239, T240, T241, T242, T282, T232, parseA_in_ga(T282, T232))
parseA_in_ga(T341, T343) → U4_ga(T341, T343, pE_in_aagaa(X423, X424, T341, X425, T343))
pE_in_aagaa(T346, T347, T341, X425, T343) → U22_aagaa(T346, T347, T341, X425, T343, appJ_in_aag(T346, T347, T341))
appJ_in_aag([], T354, .(a, .(b, T354))) → appJ_out_aag([], T354, .(a, .(b, T354)))
appJ_in_aag(.(T359, X463), X464, .(T359, T360)) → U9_aag(T359, X463, X464, T360, appJ_in_aag(X463, X464, T360))
U9_aag(T359, X463, X464, T360, appJ_out_aag(X463, X464, T360)) → appJ_out_aag(.(T359, X463), X464, .(T359, T360))
U22_aagaa(T346, T347, T341, X425, T343, appJ_out_aag(T346, T347, T341)) → U23_aagaa(T346, T347, T341, X425, T343, pP_in_ggaa(T346, T347, X425, T343))
pP_in_ggaa(T346, T347, T365, T343) → U24_ggaa(T346, T347, T365, T343, appK_in_gga(T346, T347, T365))
appK_in_gga([], T372, .(s(a, b), T372)) → appK_out_gga([], T372, .(s(a, b), T372))
appK_in_gga(.(T379, T380), T381, .(T379, X492)) → U10_gga(T379, T380, T381, X492, appK_in_gga(T380, T381, X492))
U10_gga(T379, T380, T381, X492, appK_out_gga(T380, T381, X492)) → appK_out_gga(.(T379, T380), T381, .(T379, X492))
U24_ggaa(T346, T347, T365, T343, appK_out_gga(T346, T347, T365)) → U25_ggaa(T346, T347, T365, T343, parseA_in_ga(T365, T343))
parseA_in_ga(.(s(T394, T395), []), s(T394, T395)) → parseA_out_ga(.(s(T394, T395), []), s(T394, T395))
parseA_in_ga(.(s(T402, T403, T404), []), s(T402, T403, T404)) → parseA_out_ga(.(s(T402, T403, T404), []), s(T402, T403, T404))
U25_ggaa(T346, T347, T365, T343, parseA_out_ga(T365, T343)) → pP_out_ggaa(T346, T347, T365, T343)
U23_aagaa(T346, T347, T341, X425, T343, pP_out_ggaa(T346, T347, X425, T343)) → pE_out_aagaa(T346, T347, T341, X425, T343)
U4_ga(T341, T343, pE_out_aagaa(X423, X424, T341, X425, T343)) → parseA_out_ga(T341, T343)
U21_ggggaa(T239, T240, T241, T242, T282, T232, parseA_out_ga(T282, T232)) → pO_out_ggggaa(T239, T240, T241, T242, T282, T232)
U19_aaaagaa(T239, T240, T241, T242, T230, X293, T232, pO_out_ggggaa(T239, T240, T241, T242, X293, T232)) → pD_out_aaaagaa(T239, T240, T241, T242, T230, X293, T232)
U3_ga(T230, T232, pD_out_aaaagaa(X289, X290, X291, X292, T230, X293, T232)) → parseA_out_ga(T230, T232)
U17_ggggggaa(T68, T78, T79, T80, T81, T82, T125, T7, parseA_out_ga(T125, T7)) → pN_out_ggggggaa(T68, T78, T79, T80, T81, T82, T125, T7)
U15_aaaaaggaa(T78, T79, T80, T81, T82, T69, T68, X10, T7, pN_out_ggggggaa(T68, T78, T79, T80, T81, T82, X10, T7)) → pC_out_aaaaaggaa(T78, T79, T80, T81, T82, T69, T68, X10, T7)
U2_ga(T68, T69, T7, pC_out_aaaaaggaa(X109, X110, X111, X112, X113, T69, T68, X10, T7)) → parseA_out_ga(.(T68, T69), T7)
U13_ggggaa(T24, T25, T26, T27, T33, T7, parseA_out_ga(T33, T7)) → pB_out_ggggaa(T24, T25, T26, T27, T33, T7)
U1_ga(T24, T25, T26, T27, T7, pB_out_ggggaa(T24, T25, T26, T27, X10, T7)) → parseA_out_ga(.(a, .(s(T24, T25, T26), .(b, T27))), T7)

The argument filtering Pi contains the following mapping:
parseA_in_ga(x1, x2)  =  parseA_in_ga(x1)
.(x1, x2)  =  .(x1, x2)
a  =  a
s(x1, x2, x3)  =  s(x1, x2, x3)
b  =  b
U1_ga(x1, x2, x3, x4, x5, x6)  =  U1_ga(x1, x2, x3, x4, x6)
pB_in_ggggaa(x1, x2, x3, x4, x5, x6)  =  pB_in_ggggaa(x1, x2, x3, x4)
U12_ggggaa(x1, x2, x3, x4, x5, x6, x7)  =  U12_ggggaa(x1, x2, x3, x4, x7)
appL_in_gggga(x1, x2, x3, x4, x5)  =  appL_in_gggga(x1, x2, x3, x4)
appL_out_gggga(x1, x2, x3, x4, x5)  =  appL_out_gggga(x1, x2, x3, x4, x5)
U13_ggggaa(x1, x2, x3, x4, x5, x6, x7)  =  U13_ggggaa(x1, x2, x3, x4, x5, x7)
U2_ga(x1, x2, x3, x4)  =  U2_ga(x1, x2, x4)
pC_in_aaaaaggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  pC_in_aaaaaggaa(x6, x7)
U14_aaaaaggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10)  =  U14_aaaaaggaa(x6, x7, x10)
appF_in_aaaaag(x1, x2, x3, x4, x5, x6)  =  appF_in_aaaaag(x6)
appF_out_aaaaag(x1, x2, x3, x4, x5, x6)  =  appF_out_aaaaag(x1, x2, x3, x4, x5, x6)
U5_aaaaag(x1, x2, x3, x4, x5, x6, x7, x8)  =  U5_aaaaag(x1, x7, x8)
U15_aaaaaggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10)  =  U15_aaaaaggaa(x1, x2, x3, x4, x5, x6, x7, x10)
pN_in_ggggggaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  pN_in_ggggggaa(x1, x2, x3, x4, x5, x6)
U16_ggggggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U16_ggggggaa(x1, x2, x3, x4, x5, x6, x9)
appM_in_gggggga(x1, x2, x3, x4, x5, x6, x7)  =  appM_in_gggggga(x1, x2, x3, x4, x5, x6)
U11_gggggga(x1, x2, x3, x4, x5, x6, x7, x8)  =  U11_gggggga(x1, x2, x3, x4, x5, x6, x8)
appG_in_ggggga(x1, x2, x3, x4, x5, x6)  =  appG_in_ggggga(x1, x2, x3, x4, x5)
[]  =  []
appG_out_ggggga(x1, x2, x3, x4, x5, x6)  =  appG_out_ggggga(x1, x2, x3, x4, x5, x6)
U6_ggggga(x1, x2, x3, x4, x5, x6, x7, x8)  =  U6_ggggga(x1, x2, x3, x4, x5, x6, x8)
appM_out_gggggga(x1, x2, x3, x4, x5, x6, x7)  =  appM_out_gggggga(x1, x2, x3, x4, x5, x6, x7)
U17_ggggggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U17_ggggggaa(x1, x2, x3, x4, x5, x6, x7, x9)
U3_ga(x1, x2, x3)  =  U3_ga(x1, x3)
pD_in_aaaagaa(x1, x2, x3, x4, x5, x6, x7)  =  pD_in_aaaagaa(x5)
U18_aaaagaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U18_aaaagaa(x5, x8)
appH_in_aaaag(x1, x2, x3, x4, x5)  =  appH_in_aaaag(x5)
s(x1, x2)  =  s(x1, x2)
appH_out_aaaag(x1, x2, x3, x4, x5)  =  appH_out_aaaag(x1, x2, x3, x4, x5)
U7_aaaag(x1, x2, x3, x4, x5, x6, x7)  =  U7_aaaag(x1, x6, x7)
U19_aaaagaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U19_aaaagaa(x1, x2, x3, x4, x5, x8)
pO_in_ggggaa(x1, x2, x3, x4, x5, x6)  =  pO_in_ggggaa(x1, x2, x3, x4)
U20_ggggaa(x1, x2, x3, x4, x5, x6, x7)  =  U20_ggggaa(x1, x2, x3, x4, x7)
appI_in_gggga(x1, x2, x3, x4, x5)  =  appI_in_gggga(x1, x2, x3, x4)
appI_out_gggga(x1, x2, x3, x4, x5)  =  appI_out_gggga(x1, x2, x3, x4, x5)
U8_gggga(x1, x2, x3, x4, x5, x6, x7)  =  U8_gggga(x1, x2, x3, x4, x5, x7)
U21_ggggaa(x1, x2, x3, x4, x5, x6, x7)  =  U21_ggggaa(x1, x2, x3, x4, x5, x7)
U4_ga(x1, x2, x3)  =  U4_ga(x1, x3)
pE_in_aagaa(x1, x2, x3, x4, x5)  =  pE_in_aagaa(x3)
U22_aagaa(x1, x2, x3, x4, x5, x6)  =  U22_aagaa(x3, x6)
appJ_in_aag(x1, x2, x3)  =  appJ_in_aag(x3)
appJ_out_aag(x1, x2, x3)  =  appJ_out_aag(x1, x2, x3)
U9_aag(x1, x2, x3, x4, x5)  =  U9_aag(x1, x4, x5)
U23_aagaa(x1, x2, x3, x4, x5, x6)  =  U23_aagaa(x1, x2, x3, x6)
pP_in_ggaa(x1, x2, x3, x4)  =  pP_in_ggaa(x1, x2)
U24_ggaa(x1, x2, x3, x4, x5)  =  U24_ggaa(x1, x2, x5)
appK_in_gga(x1, x2, x3)  =  appK_in_gga(x1, x2)
appK_out_gga(x1, x2, x3)  =  appK_out_gga(x1, x2, x3)
U10_gga(x1, x2, x3, x4, x5)  =  U10_gga(x1, x2, x3, x5)
U25_ggaa(x1, x2, x3, x4, x5)  =  U25_ggaa(x1, x2, x3, x5)
parseA_out_ga(x1, x2)  =  parseA_out_ga(x1, x2)
pP_out_ggaa(x1, x2, x3, x4)  =  pP_out_ggaa(x1, x2, x3, x4)
pE_out_aagaa(x1, x2, x3, x4, x5)  =  pE_out_aagaa(x1, x2, x3, x4, x5)
pO_out_ggggaa(x1, x2, x3, x4, x5, x6)  =  pO_out_ggggaa(x1, x2, x3, x4, x5, x6)
pD_out_aaaagaa(x1, x2, x3, x4, x5, x6, x7)  =  pD_out_aaaagaa(x1, x2, x3, x4, x5, x6, x7)
pN_out_ggggggaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  pN_out_ggggggaa(x1, x2, x3, x4, x5, x6, x7, x8)
pC_out_aaaaaggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  pC_out_aaaaaggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)
pB_out_ggggaa(x1, x2, x3, x4, x5, x6)  =  pB_out_ggggaa(x1, x2, x3, x4, x5, x6)
APPF_IN_AAAAAG(x1, x2, x3, x4, x5, x6)  =  APPF_IN_AAAAAG(x6)

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

(43) UsableRulesProof (EQUIVALENT transformation)

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

(44) Obligation:

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

APPF_IN_AAAAAG(.(T115, X184), X185, X186, X187, X188, .(T115, T116)) → APPF_IN_AAAAAG(X184, X185, X186, X187, X188, T116)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
APPF_IN_AAAAAG(x1, x2, x3, x4, x5, x6)  =  APPF_IN_AAAAAG(x6)

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

(45) PiDPToQDPProof (SOUND transformation)

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

(46) Obligation:

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

APPF_IN_AAAAAG(.(T115, T116)) → APPF_IN_AAAAAG(T116)

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

(47) 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:

• APPF_IN_AAAAAG(.(T115, T116)) → APPF_IN_AAAAAG(T116)
  The graph contains the following edges 1 > 1

(48) YES

(49) Obligation:

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

PARSEA_IN_GA(.(a, .(s(T24, T25, T26), .(b, T27))), T7) → PB_IN_GGGGAA(T24, T25, T26, T27, X10, T7)
PB_IN_GGGGAA(T24, T25, T26, T27, T33, T7) → U12_GGGGAA(T24, T25, T26, T27, T33, T7, appL_in_gggga(T24, T25, T26, T27, T33))
U12_GGGGAA(T24, T25, T26, T27, T33, T7, appL_out_gggga(T24, T25, T26, T27, T33)) → PARSEA_IN_GA(T33, T7)
PARSEA_IN_GA(.(T68, T69), T7) → PC_IN_AAAAAGGAA(X109, X110, X111, X112, X113, T69, T68, X10, T7)
PC_IN_AAAAAGGAA(T78, T79, T80, T81, T82, T69, T68, X10, T7) → U14_AAAAAGGAA(T78, T79, T80, T81, T82, T69, T68, X10, T7, appF_in_aaaaag(T78, T79, T80, T81, T82, T69))
U14_AAAAAGGAA(T78, T79, T80, T81, T82, T69, T68, X10, T7, appF_out_aaaaag(T78, T79, T80, T81, T82, T69)) → PN_IN_GGGGGGAA(T68, T78, T79, T80, T81, T82, X10, T7)
PN_IN_GGGGGGAA(T68, T78, T79, T80, T81, T82, T125, T7) → U16_GGGGGGAA(T68, T78, T79, T80, T81, T82, T125, T7, appM_in_gggggga(T68, T78, T79, T80, T81, T82, T125))
U16_GGGGGGAA(T68, T78, T79, T80, T81, T82, T125, T7, appM_out_gggggga(T68, T78, T79, T80, T81, T82, T125)) → PARSEA_IN_GA(T125, T7)
PARSEA_IN_GA(T230, T232) → PD_IN_AAAAGAA(X289, X290, X291, X292, T230, X293, T232)
PD_IN_AAAAGAA(T239, T240, T241, T242, T230, X293, T232) → U18_AAAAGAA(T239, T240, T241, T242, T230, X293, T232, appH_in_aaaag(T239, T240, T241, T242, T230))
U18_AAAAGAA(T239, T240, T241, T242, T230, X293, T232, appH_out_aaaag(T239, T240, T241, T242, T230)) → PO_IN_GGGGAA(T239, T240, T241, T242, X293, T232)
PO_IN_GGGGAA(T239, T240, T241, T242, T282, T232) → U20_GGGGAA(T239, T240, T241, T242, T282, T232, appI_in_gggga(T239, T240, T241, T242, T282))
U20_GGGGAA(T239, T240, T241, T242, T282, T232, appI_out_gggga(T239, T240, T241, T242, T282)) → PARSEA_IN_GA(T282, T232)
PARSEA_IN_GA(T341, T343) → PE_IN_AAGAA(X423, X424, T341, X425, T343)
PE_IN_AAGAA(T346, T347, T341, X425, T343) → U22_AAGAA(T346, T347, T341, X425, T343, appJ_in_aag(T346, T347, T341))
U22_AAGAA(T346, T347, T341, X425, T343, appJ_out_aag(T346, T347, T341)) → PP_IN_GGAA(T346, T347, X425, T343)
PP_IN_GGAA(T346, T347, T365, T343) → U24_GGAA(T346, T347, T365, T343, appK_in_gga(T346, T347, T365))
U24_GGAA(T346, T347, T365, T343, appK_out_gga(T346, T347, T365)) → PARSEA_IN_GA(T365, T343)

The TRS R consists of the following rules:

parseA_in_ga(.(a, .(s(T24, T25, T26), .(b, T27))), T7) → U1_ga(T24, T25, T26, T27, T7, pB_in_ggggaa(T24, T25, T26, T27, X10, T7))
pB_in_ggggaa(T24, T25, T26, T27, T33, T7) → U12_ggggaa(T24, T25, T26, T27, T33, T7, appL_in_gggga(T24, T25, T26, T27, T33))
appL_in_gggga(T54, T55, T56, T57, .(s(a, s(T54, T55, T56), b), T57)) → appL_out_gggga(T54, T55, T56, T57, .(s(a, s(T54, T55, T56), b), T57))
U12_ggggaa(T24, T25, T26, T27, T33, T7, appL_out_gggga(T24, T25, T26, T27, T33)) → U13_ggggaa(T24, T25, T26, T27, T33, T7, parseA_in_ga(T33, T7))
parseA_in_ga(.(T68, T69), T7) → U2_ga(T68, T69, T7, pC_in_aaaaaggaa(X109, X110, X111, X112, X113, T69, T68, X10, T7))
pC_in_aaaaaggaa(T78, T79, T80, T81, T82, T69, T68, X10, T7) → U14_aaaaaggaa(T78, T79, T80, T81, T82, T69, T68, X10, T7, appF_in_aaaaag(T78, T79, T80, T81, T82, T69))
appF_in_aaaaag([], T107, T108, T109, T110, .(a, .(s(T107, T108, T109), .(b, T110)))) → appF_out_aaaaag([], T107, T108, T109, T110, .(a, .(s(T107, T108, T109), .(b, T110))))
appF_in_aaaaag(.(T115, X184), X185, X186, X187, X188, .(T115, T116)) → U5_aaaaag(T115, X184, X185, X186, X187, X188, T116, appF_in_aaaaag(X184, X185, X186, X187, X188, T116))
U5_aaaaag(T115, X184, X185, X186, X187, X188, T116, appF_out_aaaaag(X184, X185, X186, X187, X188, T116)) → appF_out_aaaaag(.(T115, X184), X185, X186, X187, X188, .(T115, T116))
U14_aaaaaggaa(T78, T79, T80, T81, T82, T69, T68, X10, T7, appF_out_aaaaag(T78, T79, T80, T81, T82, T69)) → U15_aaaaaggaa(T78, T79, T80, T81, T82, T69, T68, X10, T7, pN_in_ggggggaa(T68, T78, T79, T80, T81, T82, X10, T7))
pN_in_ggggggaa(T68, T78, T79, T80, T81, T82, T125, T7) → U16_ggggggaa(T68, T78, T79, T80, T81, T82, T125, T7, appM_in_gggggga(T68, T78, T79, T80, T81, T82, T125))
appM_in_gggggga(T146, T147, T148, T149, T150, T151, .(T146, X220)) → U11_gggggga(T146, T147, T148, T149, T150, T151, X220, appG_in_ggggga(T147, T148, T149, T150, T151, X220))
appG_in_ggggga([], T176, T177, T178, T179, .(s(a, s(T176, T177, T178), b), T179)) → appG_out_ggggga([], T176, T177, T178, T179, .(s(a, s(T176, T177, T178), b), T179))
appG_in_ggggga(.(T192, T193), T194, T195, T196, T197, .(T192, X242)) → U6_ggggga(T192, T193, T194, T195, T196, T197, X242, appG_in_ggggga(T193, T194, T195, T196, T197, X242))
U6_ggggga(T192, T193, T194, T195, T196, T197, X242, appG_out_ggggga(T193, T194, T195, T196, T197, X242)) → appG_out_ggggga(.(T192, T193), T194, T195, T196, T197, .(T192, X242))
U11_gggggga(T146, T147, T148, T149, T150, T151, X220, appG_out_ggggga(T147, T148, T149, T150, T151, X220)) → appM_out_gggggga(T146, T147, T148, T149, T150, T151, .(T146, X220))
U16_ggggggaa(T68, T78, T79, T80, T81, T82, T125, T7, appM_out_gggggga(T68, T78, T79, T80, T81, T82, T125)) → U17_ggggggaa(T68, T78, T79, T80, T81, T82, T125, T7, parseA_in_ga(T125, T7))
parseA_in_ga(T230, T232) → U3_ga(T230, T232, pD_in_aaaagaa(X289, X290, X291, X292, T230, X293, T232))
pD_in_aaaagaa(T239, T240, T241, T242, T230, X293, T232) → U18_aaaagaa(T239, T240, T241, T242, T230, X293, T232, appH_in_aaaag(T239, T240, T241, T242, T230))
appH_in_aaaag([], T261, T262, T263, .(a, .(s(T261, T262), .(b, T263)))) → appH_out_aaaag([], T261, T262, T263, .(a, .(s(T261, T262), .(b, T263))))
appH_in_aaaag(.(T268, X353), X354, X355, X356, .(T268, T269)) → U7_aaaag(T268, X353, X354, X355, X356, T269, appH_in_aaaag(X353, X354, X355, X356, T269))
U7_aaaag(T268, X353, X354, X355, X356, T269, appH_out_aaaag(X353, X354, X355, X356, T269)) → appH_out_aaaag(.(T268, X353), X354, X355, X356, .(T268, T269))
U18_aaaagaa(T239, T240, T241, T242, T230, X293, T232, appH_out_aaaag(T239, T240, T241, T242, T230)) → U19_aaaagaa(T239, T240, T241, T242, T230, X293, T232, pO_in_ggggaa(T239, T240, T241, T242, X293, T232))
pO_in_ggggaa(T239, T240, T241, T242, T282, T232) → U20_ggggaa(T239, T240, T241, T242, T282, T232, appI_in_gggga(T239, T240, T241, T242, T282))
appI_in_gggga([], T301, T302, T303, .(s(a, s(T301, T302), b), T303)) → appI_out_gggga([], T301, T302, T303, .(s(a, s(T301, T302), b), T303))
appI_in_gggga(.(T314, T315), T316, T317, T318, .(T314, X388)) → U8_gggga(T314, T315, T316, T317, T318, X388, appI_in_gggga(T315, T316, T317, T318, X388))
U8_gggga(T314, T315, T316, T317, T318, X388, appI_out_gggga(T315, T316, T317, T318, X388)) → appI_out_gggga(.(T314, T315), T316, T317, T318, .(T314, X388))
U20_ggggaa(T239, T240, T241, T242, T282, T232, appI_out_gggga(T239, T240, T241, T242, T282)) → U21_ggggaa(T239, T240, T241, T242, T282, T232, parseA_in_ga(T282, T232))
parseA_in_ga(T341, T343) → U4_ga(T341, T343, pE_in_aagaa(X423, X424, T341, X425, T343))
pE_in_aagaa(T346, T347, T341, X425, T343) → U22_aagaa(T346, T347, T341, X425, T343, appJ_in_aag(T346, T347, T341))
appJ_in_aag([], T354, .(a, .(b, T354))) → appJ_out_aag([], T354, .(a, .(b, T354)))
appJ_in_aag(.(T359, X463), X464, .(T359, T360)) → U9_aag(T359, X463, X464, T360, appJ_in_aag(X463, X464, T360))
U9_aag(T359, X463, X464, T360, appJ_out_aag(X463, X464, T360)) → appJ_out_aag(.(T359, X463), X464, .(T359, T360))
U22_aagaa(T346, T347, T341, X425, T343, appJ_out_aag(T346, T347, T341)) → U23_aagaa(T346, T347, T341, X425, T343, pP_in_ggaa(T346, T347, X425, T343))
pP_in_ggaa(T346, T347, T365, T343) → U24_ggaa(T346, T347, T365, T343, appK_in_gga(T346, T347, T365))
appK_in_gga([], T372, .(s(a, b), T372)) → appK_out_gga([], T372, .(s(a, b), T372))
appK_in_gga(.(T379, T380), T381, .(T379, X492)) → U10_gga(T379, T380, T381, X492, appK_in_gga(T380, T381, X492))
U10_gga(T379, T380, T381, X492, appK_out_gga(T380, T381, X492)) → appK_out_gga(.(T379, T380), T381, .(T379, X492))
U24_ggaa(T346, T347, T365, T343, appK_out_gga(T346, T347, T365)) → U25_ggaa(T346, T347, T365, T343, parseA_in_ga(T365, T343))
parseA_in_ga(.(s(T394, T395), []), s(T394, T395)) → parseA_out_ga(.(s(T394, T395), []), s(T394, T395))
parseA_in_ga(.(s(T402, T403, T404), []), s(T402, T403, T404)) → parseA_out_ga(.(s(T402, T403, T404), []), s(T402, T403, T404))
U25_ggaa(T346, T347, T365, T343, parseA_out_ga(T365, T343)) → pP_out_ggaa(T346, T347, T365, T343)
U23_aagaa(T346, T347, T341, X425, T343, pP_out_ggaa(T346, T347, X425, T343)) → pE_out_aagaa(T346, T347, T341, X425, T343)
U4_ga(T341, T343, pE_out_aagaa(X423, X424, T341, X425, T343)) → parseA_out_ga(T341, T343)
U21_ggggaa(T239, T240, T241, T242, T282, T232, parseA_out_ga(T282, T232)) → pO_out_ggggaa(T239, T240, T241, T242, T282, T232)
U19_aaaagaa(T239, T240, T241, T242, T230, X293, T232, pO_out_ggggaa(T239, T240, T241, T242, X293, T232)) → pD_out_aaaagaa(T239, T240, T241, T242, T230, X293, T232)
U3_ga(T230, T232, pD_out_aaaagaa(X289, X290, X291, X292, T230, X293, T232)) → parseA_out_ga(T230, T232)
U17_ggggggaa(T68, T78, T79, T80, T81, T82, T125, T7, parseA_out_ga(T125, T7)) → pN_out_ggggggaa(T68, T78, T79, T80, T81, T82, T125, T7)
U15_aaaaaggaa(T78, T79, T80, T81, T82, T69, T68, X10, T7, pN_out_ggggggaa(T68, T78, T79, T80, T81, T82, X10, T7)) → pC_out_aaaaaggaa(T78, T79, T80, T81, T82, T69, T68, X10, T7)
U2_ga(T68, T69, T7, pC_out_aaaaaggaa(X109, X110, X111, X112, X113, T69, T68, X10, T7)) → parseA_out_ga(.(T68, T69), T7)
U13_ggggaa(T24, T25, T26, T27, T33, T7, parseA_out_ga(T33, T7)) → pB_out_ggggaa(T24, T25, T26, T27, T33, T7)
U1_ga(T24, T25, T26, T27, T7, pB_out_ggggaa(T24, T25, T26, T27, X10, T7)) → parseA_out_ga(.(a, .(s(T24, T25, T26), .(b, T27))), T7)

The argument filtering Pi contains the following mapping:
parseA_in_ga(x1, x2)  =  parseA_in_ga(x1)
.(x1, x2)  =  .(x1, x2)
a  =  a
s(x1, x2, x3)  =  s(x1, x2, x3)
b  =  b
U1_ga(x1, x2, x3, x4, x5, x6)  =  U1_ga(x1, x2, x3, x4, x6)
pB_in_ggggaa(x1, x2, x3, x4, x5, x6)  =  pB_in_ggggaa(x1, x2, x3, x4)
U12_ggggaa(x1, x2, x3, x4, x5, x6, x7)  =  U12_ggggaa(x1, x2, x3, x4, x7)
appL_in_gggga(x1, x2, x3, x4, x5)  =  appL_in_gggga(x1, x2, x3, x4)
appL_out_gggga(x1, x2, x3, x4, x5)  =  appL_out_gggga(x1, x2, x3, x4, x5)
U13_ggggaa(x1, x2, x3, x4, x5, x6, x7)  =  U13_ggggaa(x1, x2, x3, x4, x5, x7)
U2_ga(x1, x2, x3, x4)  =  U2_ga(x1, x2, x4)
pC_in_aaaaaggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  pC_in_aaaaaggaa(x6, x7)
U14_aaaaaggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10)  =  U14_aaaaaggaa(x6, x7, x10)
appF_in_aaaaag(x1, x2, x3, x4, x5, x6)  =  appF_in_aaaaag(x6)
appF_out_aaaaag(x1, x2, x3, x4, x5, x6)  =  appF_out_aaaaag(x1, x2, x3, x4, x5, x6)
U5_aaaaag(x1, x2, x3, x4, x5, x6, x7, x8)  =  U5_aaaaag(x1, x7, x8)
U15_aaaaaggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10)  =  U15_aaaaaggaa(x1, x2, x3, x4, x5, x6, x7, x10)
pN_in_ggggggaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  pN_in_ggggggaa(x1, x2, x3, x4, x5, x6)
U16_ggggggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U16_ggggggaa(x1, x2, x3, x4, x5, x6, x9)
appM_in_gggggga(x1, x2, x3, x4, x5, x6, x7)  =  appM_in_gggggga(x1, x2, x3, x4, x5, x6)
U11_gggggga(x1, x2, x3, x4, x5, x6, x7, x8)  =  U11_gggggga(x1, x2, x3, x4, x5, x6, x8)
appG_in_ggggga(x1, x2, x3, x4, x5, x6)  =  appG_in_ggggga(x1, x2, x3, x4, x5)
[]  =  []
appG_out_ggggga(x1, x2, x3, x4, x5, x6)  =  appG_out_ggggga(x1, x2, x3, x4, x5, x6)
U6_ggggga(x1, x2, x3, x4, x5, x6, x7, x8)  =  U6_ggggga(x1, x2, x3, x4, x5, x6, x8)
appM_out_gggggga(x1, x2, x3, x4, x5, x6, x7)  =  appM_out_gggggga(x1, x2, x3, x4, x5, x6, x7)
U17_ggggggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U17_ggggggaa(x1, x2, x3, x4, x5, x6, x7, x9)
U3_ga(x1, x2, x3)  =  U3_ga(x1, x3)
pD_in_aaaagaa(x1, x2, x3, x4, x5, x6, x7)  =  pD_in_aaaagaa(x5)
U18_aaaagaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U18_aaaagaa(x5, x8)
appH_in_aaaag(x1, x2, x3, x4, x5)  =  appH_in_aaaag(x5)
s(x1, x2)  =  s(x1, x2)
appH_out_aaaag(x1, x2, x3, x4, x5)  =  appH_out_aaaag(x1, x2, x3, x4, x5)
U7_aaaag(x1, x2, x3, x4, x5, x6, x7)  =  U7_aaaag(x1, x6, x7)
U19_aaaagaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U19_aaaagaa(x1, x2, x3, x4, x5, x8)
pO_in_ggggaa(x1, x2, x3, x4, x5, x6)  =  pO_in_ggggaa(x1, x2, x3, x4)
U20_ggggaa(x1, x2, x3, x4, x5, x6, x7)  =  U20_ggggaa(x1, x2, x3, x4, x7)
appI_in_gggga(x1, x2, x3, x4, x5)  =  appI_in_gggga(x1, x2, x3, x4)
appI_out_gggga(x1, x2, x3, x4, x5)  =  appI_out_gggga(x1, x2, x3, x4, x5)
U8_gggga(x1, x2, x3, x4, x5, x6, x7)  =  U8_gggga(x1, x2, x3, x4, x5, x7)
U21_ggggaa(x1, x2, x3, x4, x5, x6, x7)  =  U21_ggggaa(x1, x2, x3, x4, x5, x7)
U4_ga(x1, x2, x3)  =  U4_ga(x1, x3)
pE_in_aagaa(x1, x2, x3, x4, x5)  =  pE_in_aagaa(x3)
U22_aagaa(x1, x2, x3, x4, x5, x6)  =  U22_aagaa(x3, x6)
appJ_in_aag(x1, x2, x3)  =  appJ_in_aag(x3)
appJ_out_aag(x1, x2, x3)  =  appJ_out_aag(x1, x2, x3)
U9_aag(x1, x2, x3, x4, x5)  =  U9_aag(x1, x4, x5)
U23_aagaa(x1, x2, x3, x4, x5, x6)  =  U23_aagaa(x1, x2, x3, x6)
pP_in_ggaa(x1, x2, x3, x4)  =  pP_in_ggaa(x1, x2)
U24_ggaa(x1, x2, x3, x4, x5)  =  U24_ggaa(x1, x2, x5)
appK_in_gga(x1, x2, x3)  =  appK_in_gga(x1, x2)
appK_out_gga(x1, x2, x3)  =  appK_out_gga(x1, x2, x3)
U10_gga(x1, x2, x3, x4, x5)  =  U10_gga(x1, x2, x3, x5)
U25_ggaa(x1, x2, x3, x4, x5)  =  U25_ggaa(x1, x2, x3, x5)
parseA_out_ga(x1, x2)  =  parseA_out_ga(x1, x2)
pP_out_ggaa(x1, x2, x3, x4)  =  pP_out_ggaa(x1, x2, x3, x4)
pE_out_aagaa(x1, x2, x3, x4, x5)  =  pE_out_aagaa(x1, x2, x3, x4, x5)
pO_out_ggggaa(x1, x2, x3, x4, x5, x6)  =  pO_out_ggggaa(x1, x2, x3, x4, x5, x6)
pD_out_aaaagaa(x1, x2, x3, x4, x5, x6, x7)  =  pD_out_aaaagaa(x1, x2, x3, x4, x5, x6, x7)
pN_out_ggggggaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  pN_out_ggggggaa(x1, x2, x3, x4, x5, x6, x7, x8)
pC_out_aaaaaggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  pC_out_aaaaaggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)
pB_out_ggggaa(x1, x2, x3, x4, x5, x6)  =  pB_out_ggggaa(x1, x2, x3, x4, x5, x6)
PARSEA_IN_GA(x1, x2)  =  PARSEA_IN_GA(x1)
PB_IN_GGGGAA(x1, x2, x3, x4, x5, x6)  =  PB_IN_GGGGAA(x1, x2, x3, x4)
U12_GGGGAA(x1, x2, x3, x4, x5, x6, x7)  =  U12_GGGGAA(x1, x2, x3, x4, x7)
PC_IN_AAAAAGGAA(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  PC_IN_AAAAAGGAA(x6, x7)
U14_AAAAAGGAA(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10)  =  U14_AAAAAGGAA(x6, x7, x10)
PN_IN_GGGGGGAA(x1, x2, x3, x4, x5, x6, x7, x8)  =  PN_IN_GGGGGGAA(x1, x2, x3, x4, x5, x6)
U16_GGGGGGAA(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U16_GGGGGGAA(x1, x2, x3, x4, x5, x6, x9)
PD_IN_AAAAGAA(x1, x2, x3, x4, x5, x6, x7)  =  PD_IN_AAAAGAA(x5)
U18_AAAAGAA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U18_AAAAGAA(x5, x8)
PO_IN_GGGGAA(x1, x2, x3, x4, x5, x6)  =  PO_IN_GGGGAA(x1, x2, x3, x4)
U20_GGGGAA(x1, x2, x3, x4, x5, x6, x7)  =  U20_GGGGAA(x1, x2, x3, x4, x7)
PE_IN_AAGAA(x1, x2, x3, x4, x5)  =  PE_IN_AAGAA(x3)
U22_AAGAA(x1, x2, x3, x4, x5, x6)  =  U22_AAGAA(x3, x6)
PP_IN_GGAA(x1, x2, x3, x4)  =  PP_IN_GGAA(x1, x2)
U24_GGAA(x1, x2, x3, x4, x5)  =  U24_GGAA(x1, x2, x5)

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

(50) UsableRulesProof (EQUIVALENT transformation)

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

(51) Obligation:

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

PARSEA_IN_GA(.(a, .(s(T24, T25, T26), .(b, T27))), T7) → PB_IN_GGGGAA(T24, T25, T26, T27, X10, T7)
PB_IN_GGGGAA(T24, T25, T26, T27, T33, T7) → U12_GGGGAA(T24, T25, T26, T27, T33, T7, appL_in_gggga(T24, T25, T26, T27, T33))
U12_GGGGAA(T24, T25, T26, T27, T33, T7, appL_out_gggga(T24, T25, T26, T27, T33)) → PARSEA_IN_GA(T33, T7)
PARSEA_IN_GA(.(T68, T69), T7) → PC_IN_AAAAAGGAA(X109, X110, X111, X112, X113, T69, T68, X10, T7)
PC_IN_AAAAAGGAA(T78, T79, T80, T81, T82, T69, T68, X10, T7) → U14_AAAAAGGAA(T78, T79, T80, T81, T82, T69, T68, X10, T7, appF_in_aaaaag(T78, T79, T80, T81, T82, T69))
U14_AAAAAGGAA(T78, T79, T80, T81, T82, T69, T68, X10, T7, appF_out_aaaaag(T78, T79, T80, T81, T82, T69)) → PN_IN_GGGGGGAA(T68, T78, T79, T80, T81, T82, X10, T7)
PN_IN_GGGGGGAA(T68, T78, T79, T80, T81, T82, T125, T7) → U16_GGGGGGAA(T68, T78, T79, T80, T81, T82, T125, T7, appM_in_gggggga(T68, T78, T79, T80, T81, T82, T125))
U16_GGGGGGAA(T68, T78, T79, T80, T81, T82, T125, T7, appM_out_gggggga(T68, T78, T79, T80, T81, T82, T125)) → PARSEA_IN_GA(T125, T7)
PARSEA_IN_GA(T230, T232) → PD_IN_AAAAGAA(X289, X290, X291, X292, T230, X293, T232)
PD_IN_AAAAGAA(T239, T240, T241, T242, T230, X293, T232) → U18_AAAAGAA(T239, T240, T241, T242, T230, X293, T232, appH_in_aaaag(T239, T240, T241, T242, T230))
U18_AAAAGAA(T239, T240, T241, T242, T230, X293, T232, appH_out_aaaag(T239, T240, T241, T242, T230)) → PO_IN_GGGGAA(T239, T240, T241, T242, X293, T232)
PO_IN_GGGGAA(T239, T240, T241, T242, T282, T232) → U20_GGGGAA(T239, T240, T241, T242, T282, T232, appI_in_gggga(T239, T240, T241, T242, T282))
U20_GGGGAA(T239, T240, T241, T242, T282, T232, appI_out_gggga(T239, T240, T241, T242, T282)) → PARSEA_IN_GA(T282, T232)
PARSEA_IN_GA(T341, T343) → PE_IN_AAGAA(X423, X424, T341, X425, T343)
PE_IN_AAGAA(T346, T347, T341, X425, T343) → U22_AAGAA(T346, T347, T341, X425, T343, appJ_in_aag(T346, T347, T341))
U22_AAGAA(T346, T347, T341, X425, T343, appJ_out_aag(T346, T347, T341)) → PP_IN_GGAA(T346, T347, X425, T343)
PP_IN_GGAA(T346, T347, T365, T343) → U24_GGAA(T346, T347, T365, T343, appK_in_gga(T346, T347, T365))
U24_GGAA(T346, T347, T365, T343, appK_out_gga(T346, T347, T365)) → PARSEA_IN_GA(T365, T343)

The TRS R consists of the following rules:

appL_in_gggga(T54, T55, T56, T57, .(s(a, s(T54, T55, T56), b), T57)) → appL_out_gggga(T54, T55, T56, T57, .(s(a, s(T54, T55, T56), b), T57))
appF_in_aaaaag([], T107, T108, T109, T110, .(a, .(s(T107, T108, T109), .(b, T110)))) → appF_out_aaaaag([], T107, T108, T109, T110, .(a, .(s(T107, T108, T109), .(b, T110))))
appF_in_aaaaag(.(T115, X184), X185, X186, X187, X188, .(T115, T116)) → U5_aaaaag(T115, X184, X185, X186, X187, X188, T116, appF_in_aaaaag(X184, X185, X186, X187, X188, T116))
appM_in_gggggga(T146, T147, T148, T149, T150, T151, .(T146, X220)) → U11_gggggga(T146, T147, T148, T149, T150, T151, X220, appG_in_ggggga(T147, T148, T149, T150, T151, X220))
appH_in_aaaag([], T261, T262, T263, .(a, .(s(T261, T262), .(b, T263)))) → appH_out_aaaag([], T261, T262, T263, .(a, .(s(T261, T262), .(b, T263))))
appH_in_aaaag(.(T268, X353), X354, X355, X356, .(T268, T269)) → U7_aaaag(T268, X353, X354, X355, X356, T269, appH_in_aaaag(X353, X354, X355, X356, T269))
appI_in_gggga([], T301, T302, T303, .(s(a, s(T301, T302), b), T303)) → appI_out_gggga([], T301, T302, T303, .(s(a, s(T301, T302), b), T303))
appI_in_gggga(.(T314, T315), T316, T317, T318, .(T314, X388)) → U8_gggga(T314, T315, T316, T317, T318, X388, appI_in_gggga(T315, T316, T317, T318, X388))
appJ_in_aag([], T354, .(a, .(b, T354))) → appJ_out_aag([], T354, .(a, .(b, T354)))
appJ_in_aag(.(T359, X463), X464, .(T359, T360)) → U9_aag(T359, X463, X464, T360, appJ_in_aag(X463, X464, T360))
appK_in_gga([], T372, .(s(a, b), T372)) → appK_out_gga([], T372, .(s(a, b), T372))
appK_in_gga(.(T379, T380), T381, .(T379, X492)) → U10_gga(T379, T380, T381, X492, appK_in_gga(T380, T381, X492))
U5_aaaaag(T115, X184, X185, X186, X187, X188, T116, appF_out_aaaaag(X184, X185, X186, X187, X188, T116)) → appF_out_aaaaag(.(T115, X184), X185, X186, X187, X188, .(T115, T116))
U11_gggggga(T146, T147, T148, T149, T150, T151, X220, appG_out_ggggga(T147, T148, T149, T150, T151, X220)) → appM_out_gggggga(T146, T147, T148, T149, T150, T151, .(T146, X220))
U7_aaaag(T268, X353, X354, X355, X356, T269, appH_out_aaaag(X353, X354, X355, X356, T269)) → appH_out_aaaag(.(T268, X353), X354, X355, X356, .(T268, T269))
U8_gggga(T314, T315, T316, T317, T318, X388, appI_out_gggga(T315, T316, T317, T318, X388)) → appI_out_gggga(.(T314, T315), T316, T317, T318, .(T314, X388))
U9_aag(T359, X463, X464, T360, appJ_out_aag(X463, X464, T360)) → appJ_out_aag(.(T359, X463), X464, .(T359, T360))
U10_gga(T379, T380, T381, X492, appK_out_gga(T380, T381, X492)) → appK_out_gga(.(T379, T380), T381, .(T379, X492))
appG_in_ggggga([], T176, T177, T178, T179, .(s(a, s(T176, T177, T178), b), T179)) → appG_out_ggggga([], T176, T177, T178, T179, .(s(a, s(T176, T177, T178), b), T179))
appG_in_ggggga(.(T192, T193), T194, T195, T196, T197, .(T192, X242)) → U6_ggggga(T192, T193, T194, T195, T196, T197, X242, appG_in_ggggga(T193, T194, T195, T196, T197, X242))
U6_ggggga(T192, T193, T194, T195, T196, T197, X242, appG_out_ggggga(T193, T194, T195, T196, T197, X242)) → appG_out_ggggga(.(T192, T193), T194, T195, T196, T197, .(T192, X242))

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
a  =  a
s(x1, x2, x3)  =  s(x1, x2, x3)
b  =  b
appL_in_gggga(x1, x2, x3, x4, x5)  =  appL_in_gggga(x1, x2, x3, x4)
appL_out_gggga(x1, x2, x3, x4, x5)  =  appL_out_gggga(x1, x2, x3, x4, x5)
appF_in_aaaaag(x1, x2, x3, x4, x5, x6)  =  appF_in_aaaaag(x6)
appF_out_aaaaag(x1, x2, x3, x4, x5, x6)  =  appF_out_aaaaag(x1, x2, x3, x4, x5, x6)
U5_aaaaag(x1, x2, x3, x4, x5, x6, x7, x8)  =  U5_aaaaag(x1, x7, x8)
appM_in_gggggga(x1, x2, x3, x4, x5, x6, x7)  =  appM_in_gggggga(x1, x2, x3, x4, x5, x6)
U11_gggggga(x1, x2, x3, x4, x5, x6, x7, x8)  =  U11_gggggga(x1, x2, x3, x4, x5, x6, x8)
appG_in_ggggga(x1, x2, x3, x4, x5, x6)  =  appG_in_ggggga(x1, x2, x3, x4, x5)
[]  =  []
appG_out_ggggga(x1, x2, x3, x4, x5, x6)  =  appG_out_ggggga(x1, x2, x3, x4, x5, x6)
U6_ggggga(x1, x2, x3, x4, x5, x6, x7, x8)  =  U6_ggggga(x1, x2, x3, x4, x5, x6, x8)
appM_out_gggggga(x1, x2, x3, x4, x5, x6, x7)  =  appM_out_gggggga(x1, x2, x3, x4, x5, x6, x7)
appH_in_aaaag(x1, x2, x3, x4, x5)  =  appH_in_aaaag(x5)
s(x1, x2)  =  s(x1, x2)
appH_out_aaaag(x1, x2, x3, x4, x5)  =  appH_out_aaaag(x1, x2, x3, x4, x5)
U7_aaaag(x1, x2, x3, x4, x5, x6, x7)  =  U7_aaaag(x1, x6, x7)
appI_in_gggga(x1, x2, x3, x4, x5)  =  appI_in_gggga(x1, x2, x3, x4)
appI_out_gggga(x1, x2, x3, x4, x5)  =  appI_out_gggga(x1, x2, x3, x4, x5)
U8_gggga(x1, x2, x3, x4, x5, x6, x7)  =  U8_gggga(x1, x2, x3, x4, x5, x7)
appJ_in_aag(x1, x2, x3)  =  appJ_in_aag(x3)
appJ_out_aag(x1, x2, x3)  =  appJ_out_aag(x1, x2, x3)
U9_aag(x1, x2, x3, x4, x5)  =  U9_aag(x1, x4, x5)
appK_in_gga(x1, x2, x3)  =  appK_in_gga(x1, x2)
appK_out_gga(x1, x2, x3)  =  appK_out_gga(x1, x2, x3)
U10_gga(x1, x2, x3, x4, x5)  =  U10_gga(x1, x2, x3, x5)
PARSEA_IN_GA(x1, x2)  =  PARSEA_IN_GA(x1)
PB_IN_GGGGAA(x1, x2, x3, x4, x5, x6)  =  PB_IN_GGGGAA(x1, x2, x3, x4)
U12_GGGGAA(x1, x2, x3, x4, x5, x6, x7)  =  U12_GGGGAA(x1, x2, x3, x4, x7)
PC_IN_AAAAAGGAA(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  PC_IN_AAAAAGGAA(x6, x7)
U14_AAAAAGGAA(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10)  =  U14_AAAAAGGAA(x6, x7, x10)
PN_IN_GGGGGGAA(x1, x2, x3, x4, x5, x6, x7, x8)  =  PN_IN_GGGGGGAA(x1, x2, x3, x4, x5, x6)
U16_GGGGGGAA(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U16_GGGGGGAA(x1, x2, x3, x4, x5, x6, x9)
PD_IN_AAAAGAA(x1, x2, x3, x4, x5, x6, x7)  =  PD_IN_AAAAGAA(x5)
U18_AAAAGAA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U18_AAAAGAA(x5, x8)
PO_IN_GGGGAA(x1, x2, x3, x4, x5, x6)  =  PO_IN_GGGGAA(x1, x2, x3, x4)
U20_GGGGAA(x1, x2, x3, x4, x5, x6, x7)  =  U20_GGGGAA(x1, x2, x3, x4, x7)
PE_IN_AAGAA(x1, x2, x3, x4, x5)  =  PE_IN_AAGAA(x3)
U22_AAGAA(x1, x2, x3, x4, x5, x6)  =  U22_AAGAA(x3, x6)
PP_IN_GGAA(x1, x2, x3, x4)  =  PP_IN_GGAA(x1, x2)
U24_GGAA(x1, x2, x3, x4, x5)  =  U24_GGAA(x1, x2, x5)

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

(52) PiDPToQDPProof (SOUND transformation)

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

(53) Obligation:

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

PARSEA_IN_GA(.(a, .(s(T24, T25, T26), .(b, T27)))) → PB_IN_GGGGAA(T24, T25, T26, T27)
PB_IN_GGGGAA(T24, T25, T26, T27) → U12_GGGGAA(T24, T25, T26, T27, appL_in_gggga(T24, T25, T26, T27))
U12_GGGGAA(T24, T25, T26, T27, appL_out_gggga(T24, T25, T26, T27, T33)) → PARSEA_IN_GA(T33)
PARSEA_IN_GA(.(T68, T69)) → PC_IN_AAAAAGGAA(T69, T68)
PC_IN_AAAAAGGAA(T69, T68) → U14_AAAAAGGAA(T69, T68, appF_in_aaaaag(T69))
U14_AAAAAGGAA(T69, T68, appF_out_aaaaag(T78, T79, T80, T81, T82, T69)) → PN_IN_GGGGGGAA(T68, T78, T79, T80, T81, T82)
PN_IN_GGGGGGAA(T68, T78, T79, T80, T81, T82) → U16_GGGGGGAA(T68, T78, T79, T80, T81, T82, appM_in_gggggga(T68, T78, T79, T80, T81, T82))
U16_GGGGGGAA(T68, T78, T79, T80, T81, T82, appM_out_gggggga(T68, T78, T79, T80, T81, T82, T125)) → PARSEA_IN_GA(T125)
PARSEA_IN_GA(T230) → PD_IN_AAAAGAA(T230)
PD_IN_AAAAGAA(T230) → U18_AAAAGAA(T230, appH_in_aaaag(T230))
U18_AAAAGAA(T230, appH_out_aaaag(T239, T240, T241, T242, T230)) → PO_IN_GGGGAA(T239, T240, T241, T242)
PO_IN_GGGGAA(T239, T240, T241, T242) → U20_GGGGAA(T239, T240, T241, T242, appI_in_gggga(T239, T240, T241, T242))
U20_GGGGAA(T239, T240, T241, T242, appI_out_gggga(T239, T240, T241, T242, T282)) → PARSEA_IN_GA(T282)
PARSEA_IN_GA(T341) → PE_IN_AAGAA(T341)
PE_IN_AAGAA(T341) → U22_AAGAA(T341, appJ_in_aag(T341))
U22_AAGAA(T341, appJ_out_aag(T346, T347, T341)) → PP_IN_GGAA(T346, T347)
PP_IN_GGAA(T346, T347) → U24_GGAA(T346, T347, appK_in_gga(T346, T347))
U24_GGAA(T346, T347, appK_out_gga(T346, T347, T365)) → PARSEA_IN_GA(T365)

The TRS R consists of the following rules:

appL_in_gggga(T54, T55, T56, T57) → appL_out_gggga(T54, T55, T56, T57, .(s(a, s(T54, T55, T56), b), T57))
appF_in_aaaaag(.(a, .(s(T107, T108, T109), .(b, T110)))) → appF_out_aaaaag([], T107, T108, T109, T110, .(a, .(s(T107, T108, T109), .(b, T110))))
appF_in_aaaaag(.(T115, T116)) → U5_aaaaag(T115, T116, appF_in_aaaaag(T116))
appM_in_gggggga(T146, T147, T148, T149, T150, T151) → U11_gggggga(T146, T147, T148, T149, T150, T151, appG_in_ggggga(T147, T148, T149, T150, T151))
appH_in_aaaag(.(a, .(s(T261, T262), .(b, T263)))) → appH_out_aaaag([], T261, T262, T263, .(a, .(s(T261, T262), .(b, T263))))
appH_in_aaaag(.(T268, T269)) → U7_aaaag(T268, T269, appH_in_aaaag(T269))
appI_in_gggga([], T301, T302, T303) → appI_out_gggga([], T301, T302, T303, .(s(a, s(T301, T302), b), T303))
appI_in_gggga(.(T314, T315), T316, T317, T318) → U8_gggga(T314, T315, T316, T317, T318, appI_in_gggga(T315, T316, T317, T318))
appJ_in_aag(.(a, .(b, T354))) → appJ_out_aag([], T354, .(a, .(b, T354)))
appJ_in_aag(.(T359, T360)) → U9_aag(T359, T360, appJ_in_aag(T360))
appK_in_gga([], T372) → appK_out_gga([], T372, .(s(a, b), T372))
appK_in_gga(.(T379, T380), T381) → U10_gga(T379, T380, T381, appK_in_gga(T380, T381))
U5_aaaaag(T115, T116, appF_out_aaaaag(X184, X185, X186, X187, X188, T116)) → appF_out_aaaaag(.(T115, X184), X185, X186, X187, X188, .(T115, T116))
U11_gggggga(T146, T147, T148, T149, T150, T151, appG_out_ggggga(T147, T148, T149, T150, T151, X220)) → appM_out_gggggga(T146, T147, T148, T149, T150, T151, .(T146, X220))
U7_aaaag(T268, T269, appH_out_aaaag(X353, X354, X355, X356, T269)) → appH_out_aaaag(.(T268, X353), X354, X355, X356, .(T268, T269))
U8_gggga(T314, T315, T316, T317, T318, appI_out_gggga(T315, T316, T317, T318, X388)) → appI_out_gggga(.(T314, T315), T316, T317, T318, .(T314, X388))
U9_aag(T359, T360, appJ_out_aag(X463, X464, T360)) → appJ_out_aag(.(T359, X463), X464, .(T359, T360))
U10_gga(T379, T380, T381, appK_out_gga(T380, T381, X492)) → appK_out_gga(.(T379, T380), T381, .(T379, X492))
appG_in_ggggga([], T176, T177, T178, T179) → appG_out_ggggga([], T176, T177, T178, T179, .(s(a, s(T176, T177, T178), b), T179))
appG_in_ggggga(.(T192, T193), T194, T195, T196, T197) → U6_ggggga(T192, T193, T194, T195, T196, T197, appG_in_ggggga(T193, T194, T195, T196, T197))
U6_ggggga(T192, T193, T194, T195, T196, T197, appG_out_ggggga(T193, T194, T195, T196, T197, X242)) → appG_out_ggggga(.(T192, T193), T194, T195, T196, T197, .(T192, X242))

The set Q consists of the following terms:

appL_in_gggga(x0, x1, x2, x3)
appF_in_aaaaag(x0)
appM_in_gggggga(x0, x1, x2, x3, x4, x5)
appH_in_aaaag(x0)
appI_in_gggga(x0, x1, x2, x3)
appJ_in_aag(x0)
appK_in_gga(x0, x1)
U5_aaaaag(x0, x1, x2)
U11_gggggga(x0, x1, x2, x3, x4, x5, x6)
U7_aaaag(x0, x1, x2)
U8_gggga(x0, x1, x2, x3, x4, x5)
U9_aag(x0, x1, x2)
U10_gga(x0, x1, x2, x3)
appG_in_ggggga(x0, x1, x2, x3, x4)
U6_ggggga(x0, x1, x2, x3, x4, x5, x6)

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

(54) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule PB_IN_GGGGAA(T24, T25, T26, T27) → U12_GGGGAA(T24, T25, T26, T27, appL_in_gggga(T24, T25, T26, T27)) at position [4] we obtained the following new rules [LPAR04]:

PB_IN_GGGGAA(T24, T25, T26, T27) → U12_GGGGAA(T24, T25, T26, T27, appL_out_gggga(T24, T25, T26, T27, .(s(a, s(T24, T25, T26), b), T27)))

(55) Obligation:

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

PARSEA_IN_GA(.(a, .(s(T24, T25, T26), .(b, T27)))) → PB_IN_GGGGAA(T24, T25, T26, T27)
U12_GGGGAA(T24, T25, T26, T27, appL_out_gggga(T24, T25, T26, T27, T33)) → PARSEA_IN_GA(T33)
PARSEA_IN_GA(.(T68, T69)) → PC_IN_AAAAAGGAA(T69, T68)
PC_IN_AAAAAGGAA(T69, T68) → U14_AAAAAGGAA(T69, T68, appF_in_aaaaag(T69))
U14_AAAAAGGAA(T69, T68, appF_out_aaaaag(T78, T79, T80, T81, T82, T69)) → PN_IN_GGGGGGAA(T68, T78, T79, T80, T81, T82)
PN_IN_GGGGGGAA(T68, T78, T79, T80, T81, T82) → U16_GGGGGGAA(T68, T78, T79, T80, T81, T82, appM_in_gggggga(T68, T78, T79, T80, T81, T82))
U16_GGGGGGAA(T68, T78, T79, T80, T81, T82, appM_out_gggggga(T68, T78, T79, T80, T81, T82, T125)) → PARSEA_IN_GA(T125)
PARSEA_IN_GA(T230) → PD_IN_AAAAGAA(T230)
PD_IN_AAAAGAA(T230) → U18_AAAAGAA(T230, appH_in_aaaag(T230))
U18_AAAAGAA(T230, appH_out_aaaag(T239, T240, T241, T242, T230)) → PO_IN_GGGGAA(T239, T240, T241, T242)
PO_IN_GGGGAA(T239, T240, T241, T242) → U20_GGGGAA(T239, T240, T241, T242, appI_in_gggga(T239, T240, T241, T242))
U20_GGGGAA(T239, T240, T241, T242, appI_out_gggga(T239, T240, T241, T242, T282)) → PARSEA_IN_GA(T282)
PARSEA_IN_GA(T341) → PE_IN_AAGAA(T341)
PE_IN_AAGAA(T341) → U22_AAGAA(T341, appJ_in_aag(T341))
U22_AAGAA(T341, appJ_out_aag(T346, T347, T341)) → PP_IN_GGAA(T346, T347)
PP_IN_GGAA(T346, T347) → U24_GGAA(T346, T347, appK_in_gga(T346, T347))
U24_GGAA(T346, T347, appK_out_gga(T346, T347, T365)) → PARSEA_IN_GA(T365)
PB_IN_GGGGAA(T24, T25, T26, T27) → U12_GGGGAA(T24, T25, T26, T27, appL_out_gggga(T24, T25, T26, T27, .(s(a, s(T24, T25, T26), b), T27)))

The TRS R consists of the following rules:

appL_in_gggga(T54, T55, T56, T57) → appL_out_gggga(T54, T55, T56, T57, .(s(a, s(T54, T55, T56), b), T57))
appF_in_aaaaag(.(a, .(s(T107, T108, T109), .(b, T110)))) → appF_out_aaaaag([], T107, T108, T109, T110, .(a, .(s(T107, T108, T109), .(b, T110))))
appF_in_aaaaag(.(T115, T116)) → U5_aaaaag(T115, T116, appF_in_aaaaag(T116))
appM_in_gggggga(T146, T147, T148, T149, T150, T151) → U11_gggggga(T146, T147, T148, T149, T150, T151, appG_in_ggggga(T147, T148, T149, T150, T151))
appH_in_aaaag(.(a, .(s(T261, T262), .(b, T263)))) → appH_out_aaaag([], T261, T262, T263, .(a, .(s(T261, T262), .(b, T263))))
appH_in_aaaag(.(T268, T269)) → U7_aaaag(T268, T269, appH_in_aaaag(T269))
appI_in_gggga([], T301, T302, T303) → appI_out_gggga([], T301, T302, T303, .(s(a, s(T301, T302), b), T303))
appI_in_gggga(.(T314, T315), T316, T317, T318) → U8_gggga(T314, T315, T316, T317, T318, appI_in_gggga(T315, T316, T317, T318))
appJ_in_aag(.(a, .(b, T354))) → appJ_out_aag([], T354, .(a, .(b, T354)))
appJ_in_aag(.(T359, T360)) → U9_aag(T359, T360, appJ_in_aag(T360))
appK_in_gga([], T372) → appK_out_gga([], T372, .(s(a, b), T372))
appK_in_gga(.(T379, T380), T381) → U10_gga(T379, T380, T381, appK_in_gga(T380, T381))
U5_aaaaag(T115, T116, appF_out_aaaaag(X184, X185, X186, X187, X188, T116)) → appF_out_aaaaag(.(T115, X184), X185, X186, X187, X188, .(T115, T116))
U11_gggggga(T146, T147, T148, T149, T150, T151, appG_out_ggggga(T147, T148, T149, T150, T151, X220)) → appM_out_gggggga(T146, T147, T148, T149, T150, T151, .(T146, X220))
U7_aaaag(T268, T269, appH_out_aaaag(X353, X354, X355, X356, T269)) → appH_out_aaaag(.(T268, X353), X354, X355, X356, .(T268, T269))
U8_gggga(T314, T315, T316, T317, T318, appI_out_gggga(T315, T316, T317, T318, X388)) → appI_out_gggga(.(T314, T315), T316, T317, T318, .(T314, X388))
U9_aag(T359, T360, appJ_out_aag(X463, X464, T360)) → appJ_out_aag(.(T359, X463), X464, .(T359, T360))
U10_gga(T379, T380, T381, appK_out_gga(T380, T381, X492)) → appK_out_gga(.(T379, T380), T381, .(T379, X492))
appG_in_ggggga([], T176, T177, T178, T179) → appG_out_ggggga([], T176, T177, T178, T179, .(s(a, s(T176, T177, T178), b), T179))
appG_in_ggggga(.(T192, T193), T194, T195, T196, T197) → U6_ggggga(T192, T193, T194, T195, T196, T197, appG_in_ggggga(T193, T194, T195, T196, T197))
U6_ggggga(T192, T193, T194, T195, T196, T197, appG_out_ggggga(T193, T194, T195, T196, T197, X242)) → appG_out_ggggga(.(T192, T193), T194, T195, T196, T197, .(T192, X242))

The set Q consists of the following terms:

appL_in_gggga(x0, x1, x2, x3)
appF_in_aaaaag(x0)
appM_in_gggggga(x0, x1, x2, x3, x4, x5)
appH_in_aaaag(x0)
appI_in_gggga(x0, x1, x2, x3)
appJ_in_aag(x0)
appK_in_gga(x0, x1)
U5_aaaaag(x0, x1, x2)
U11_gggggga(x0, x1, x2, x3, x4, x5, x6)
U7_aaaag(x0, x1, x2)
U8_gggga(x0, x1, x2, x3, x4, x5)
U9_aag(x0, x1, x2)
U10_gga(x0, x1, x2, x3)
appG_in_ggggga(x0, x1, x2, x3, x4)
U6_ggggga(x0, x1, x2, x3, x4, x5, x6)

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

(56) UsableRulesProof (EQUIVALENT transformation)

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

(57) Obligation:

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

PARSEA_IN_GA(.(a, .(s(T24, T25, T26), .(b, T27)))) → PB_IN_GGGGAA(T24, T25, T26, T27)
U12_GGGGAA(T24, T25, T26, T27, appL_out_gggga(T24, T25, T26, T27, T33)) → PARSEA_IN_GA(T33)
PARSEA_IN_GA(.(T68, T69)) → PC_IN_AAAAAGGAA(T69, T68)
PC_IN_AAAAAGGAA(T69, T68) → U14_AAAAAGGAA(T69, T68, appF_in_aaaaag(T69))
U14_AAAAAGGAA(T69, T68, appF_out_aaaaag(T78, T79, T80, T81, T82, T69)) → PN_IN_GGGGGGAA(T68, T78, T79, T80, T81, T82)
PN_IN_GGGGGGAA(T68, T78, T79, T80, T81, T82) → U16_GGGGGGAA(T68, T78, T79, T80, T81, T82, appM_in_gggggga(T68, T78, T79, T80, T81, T82))
U16_GGGGGGAA(T68, T78, T79, T80, T81, T82, appM_out_gggggga(T68, T78, T79, T80, T81, T82, T125)) → PARSEA_IN_GA(T125)
PARSEA_IN_GA(T230) → PD_IN_AAAAGAA(T230)
PD_IN_AAAAGAA(T230) → U18_AAAAGAA(T230, appH_in_aaaag(T230))
U18_AAAAGAA(T230, appH_out_aaaag(T239, T240, T241, T242, T230)) → PO_IN_GGGGAA(T239, T240, T241, T242)
PO_IN_GGGGAA(T239, T240, T241, T242) → U20_GGGGAA(T239, T240, T241, T242, appI_in_gggga(T239, T240, T241, T242))
U20_GGGGAA(T239, T240, T241, T242, appI_out_gggga(T239, T240, T241, T242, T282)) → PARSEA_IN_GA(T282)
PARSEA_IN_GA(T341) → PE_IN_AAGAA(T341)
PE_IN_AAGAA(T341) → U22_AAGAA(T341, appJ_in_aag(T341))
U22_AAGAA(T341, appJ_out_aag(T346, T347, T341)) → PP_IN_GGAA(T346, T347)
PP_IN_GGAA(T346, T347) → U24_GGAA(T346, T347, appK_in_gga(T346, T347))
U24_GGAA(T346, T347, appK_out_gga(T346, T347, T365)) → PARSEA_IN_GA(T365)
PB_IN_GGGGAA(T24, T25, T26, T27) → U12_GGGGAA(T24, T25, T26, T27, appL_out_gggga(T24, T25, T26, T27, .(s(a, s(T24, T25, T26), b), T27)))

The TRS R consists of the following rules:

appK_in_gga([], T372) → appK_out_gga([], T372, .(s(a, b), T372))
appK_in_gga(.(T379, T380), T381) → U10_gga(T379, T380, T381, appK_in_gga(T380, T381))
U10_gga(T379, T380, T381, appK_out_gga(T380, T381, X492)) → appK_out_gga(.(T379, T380), T381, .(T379, X492))
appJ_in_aag(.(a, .(b, T354))) → appJ_out_aag([], T354, .(a, .(b, T354)))
appJ_in_aag(.(T359, T360)) → U9_aag(T359, T360, appJ_in_aag(T360))
U9_aag(T359, T360, appJ_out_aag(X463, X464, T360)) → appJ_out_aag(.(T359, X463), X464, .(T359, T360))
appI_in_gggga([], T301, T302, T303) → appI_out_gggga([], T301, T302, T303, .(s(a, s(T301, T302), b), T303))
appI_in_gggga(.(T314, T315), T316, T317, T318) → U8_gggga(T314, T315, T316, T317, T318, appI_in_gggga(T315, T316, T317, T318))
U8_gggga(T314, T315, T316, T317, T318, appI_out_gggga(T315, T316, T317, T318, X388)) → appI_out_gggga(.(T314, T315), T316, T317, T318, .(T314, X388))
appH_in_aaaag(.(a, .(s(T261, T262), .(b, T263)))) → appH_out_aaaag([], T261, T262, T263, .(a, .(s(T261, T262), .(b, T263))))
appH_in_aaaag(.(T268, T269)) → U7_aaaag(T268, T269, appH_in_aaaag(T269))
U7_aaaag(T268, T269, appH_out_aaaag(X353, X354, X355, X356, T269)) → appH_out_aaaag(.(T268, X353), X354, X355, X356, .(T268, T269))
appM_in_gggggga(T146, T147, T148, T149, T150, T151) → U11_gggggga(T146, T147, T148, T149, T150, T151, appG_in_ggggga(T147, T148, T149, T150, T151))
appG_in_ggggga([], T176, T177, T178, T179) → appG_out_ggggga([], T176, T177, T178, T179, .(s(a, s(T176, T177, T178), b), T179))
appG_in_ggggga(.(T192, T193), T194, T195, T196, T197) → U6_ggggga(T192, T193, T194, T195, T196, T197, appG_in_ggggga(T193, T194, T195, T196, T197))
U11_gggggga(T146, T147, T148, T149, T150, T151, appG_out_ggggga(T147, T148, T149, T150, T151, X220)) → appM_out_gggggga(T146, T147, T148, T149, T150, T151, .(T146, X220))
U6_ggggga(T192, T193, T194, T195, T196, T197, appG_out_ggggga(T193, T194, T195, T196, T197, X242)) → appG_out_ggggga(.(T192, T193), T194, T195, T196, T197, .(T192, X242))
appF_in_aaaaag(.(a, .(s(T107, T108, T109), .(b, T110)))) → appF_out_aaaaag([], T107, T108, T109, T110, .(a, .(s(T107, T108, T109), .(b, T110))))
appF_in_aaaaag(.(T115, T116)) → U5_aaaaag(T115, T116, appF_in_aaaaag(T116))
U5_aaaaag(T115, T116, appF_out_aaaaag(X184, X185, X186, X187, X188, T116)) → appF_out_aaaaag(.(T115, X184), X185, X186, X187, X188, .(T115, T116))

The set Q consists of the following terms:

appL_in_gggga(x0, x1, x2, x3)
appF_in_aaaaag(x0)
appM_in_gggggga(x0, x1, x2, x3, x4, x5)
appH_in_aaaag(x0)
appI_in_gggga(x0, x1, x2, x3)
appJ_in_aag(x0)
appK_in_gga(x0, x1)
U5_aaaaag(x0, x1, x2)
U11_gggggga(x0, x1, x2, x3, x4, x5, x6)
U7_aaaag(x0, x1, x2)
U8_gggga(x0, x1, x2, x3, x4, x5)
U9_aag(x0, x1, x2)
U10_gga(x0, x1, x2, x3)
appG_in_ggggga(x0, x1, x2, x3, x4)
U6_ggggga(x0, x1, x2, x3, x4, x5, x6)

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

(58) QReductionProof (EQUIVALENT transformation)

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

appL_in_gggga(x0, x1, x2, x3)

(59) Obligation:

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

PARSEA_IN_GA(.(a, .(s(T24, T25, T26), .(b, T27)))) → PB_IN_GGGGAA(T24, T25, T26, T27)
U12_GGGGAA(T24, T25, T26, T27, appL_out_gggga(T24, T25, T26, T27, T33)) → PARSEA_IN_GA(T33)
PARSEA_IN_GA(.(T68, T69)) → PC_IN_AAAAAGGAA(T69, T68)
PC_IN_AAAAAGGAA(T69, T68) → U14_AAAAAGGAA(T69, T68, appF_in_aaaaag(T69))
U14_AAAAAGGAA(T69, T68, appF_out_aaaaag(T78, T79, T80, T81, T82, T69)) → PN_IN_GGGGGGAA(T68, T78, T79, T80, T81, T82)
PN_IN_GGGGGGAA(T68, T78, T79, T80, T81, T82) → U16_GGGGGGAA(T68, T78, T79, T80, T81, T82, appM_in_gggggga(T68, T78, T79, T80, T81, T82))
U16_GGGGGGAA(T68, T78, T79, T80, T81, T82, appM_out_gggggga(T68, T78, T79, T80, T81, T82, T125)) → PARSEA_IN_GA(T125)
PARSEA_IN_GA(T230) → PD_IN_AAAAGAA(T230)
PD_IN_AAAAGAA(T230) → U18_AAAAGAA(T230, appH_in_aaaag(T230))
U18_AAAAGAA(T230, appH_out_aaaag(T239, T240, T241, T242, T230)) → PO_IN_GGGGAA(T239, T240, T241, T242)
PO_IN_GGGGAA(T239, T240, T241, T242) → U20_GGGGAA(T239, T240, T241, T242, appI_in_gggga(T239, T240, T241, T242))
U20_GGGGAA(T239, T240, T241, T242, appI_out_gggga(T239, T240, T241, T242, T282)) → PARSEA_IN_GA(T282)
PARSEA_IN_GA(T341) → PE_IN_AAGAA(T341)
PE_IN_AAGAA(T341) → U22_AAGAA(T341, appJ_in_aag(T341))
U22_AAGAA(T341, appJ_out_aag(T346, T347, T341)) → PP_IN_GGAA(T346, T347)
PP_IN_GGAA(T346, T347) → U24_GGAA(T346, T347, appK_in_gga(T346, T347))
U24_GGAA(T346, T347, appK_out_gga(T346, T347, T365)) → PARSEA_IN_GA(T365)
PB_IN_GGGGAA(T24, T25, T26, T27) → U12_GGGGAA(T24, T25, T26, T27, appL_out_gggga(T24, T25, T26, T27, .(s(a, s(T24, T25, T26), b), T27)))

The TRS R consists of the following rules:

appK_in_gga([], T372) → appK_out_gga([], T372, .(s(a, b), T372))
appK_in_gga(.(T379, T380), T381) → U10_gga(T379, T380, T381, appK_in_gga(T380, T381))
U10_gga(T379, T380, T381, appK_out_gga(T380, T381, X492)) → appK_out_gga(.(T379, T380), T381, .(T379, X492))
appJ_in_aag(.(a, .(b, T354))) → appJ_out_aag([], T354, .(a, .(b, T354)))
appJ_in_aag(.(T359, T360)) → U9_aag(T359, T360, appJ_in_aag(T360))
U9_aag(T359, T360, appJ_out_aag(X463, X464, T360)) → appJ_out_aag(.(T359, X463), X464, .(T359, T360))
appI_in_gggga([], T301, T302, T303) → appI_out_gggga([], T301, T302, T303, .(s(a, s(T301, T302), b), T303))
appI_in_gggga(.(T314, T315), T316, T317, T318) → U8_gggga(T314, T315, T316, T317, T318, appI_in_gggga(T315, T316, T317, T318))
U8_gggga(T314, T315, T316, T317, T318, appI_out_gggga(T315, T316, T317, T318, X388)) → appI_out_gggga(.(T314, T315), T316, T317, T318, .(T314, X388))
appH_in_aaaag(.(a, .(s(T261, T262), .(b, T263)))) → appH_out_aaaag([], T261, T262, T263, .(a, .(s(T261, T262), .(b, T263))))
appH_in_aaaag(.(T268, T269)) → U7_aaaag(T268, T269, appH_in_aaaag(T269))
U7_aaaag(T268, T269, appH_out_aaaag(X353, X354, X355, X356, T269)) → appH_out_aaaag(.(T268, X353), X354, X355, X356, .(T268, T269))
appM_in_gggggga(T146, T147, T148, T149, T150, T151) → U11_gggggga(T146, T147, T148, T149, T150, T151, appG_in_ggggga(T147, T148, T149, T150, T151))
appG_in_ggggga([], T176, T177, T178, T179) → appG_out_ggggga([], T176, T177, T178, T179, .(s(a, s(T176, T177, T178), b), T179))
appG_in_ggggga(.(T192, T193), T194, T195, T196, T197) → U6_ggggga(T192, T193, T194, T195, T196, T197, appG_in_ggggga(T193, T194, T195, T196, T197))
U11_gggggga(T146, T147, T148, T149, T150, T151, appG_out_ggggga(T147, T148, T149, T150, T151, X220)) → appM_out_gggggga(T146, T147, T148, T149, T150, T151, .(T146, X220))
U6_ggggga(T192, T193, T194, T195, T196, T197, appG_out_ggggga(T193, T194, T195, T196, T197, X242)) → appG_out_ggggga(.(T192, T193), T194, T195, T196, T197, .(T192, X242))
appF_in_aaaaag(.(a, .(s(T107, T108, T109), .(b, T110)))) → appF_out_aaaaag([], T107, T108, T109, T110, .(a, .(s(T107, T108, T109), .(b, T110))))
appF_in_aaaaag(.(T115, T116)) → U5_aaaaag(T115, T116, appF_in_aaaaag(T116))
U5_aaaaag(T115, T116, appF_out_aaaaag(X184, X185, X186, X187, X188, T116)) → appF_out_aaaaag(.(T115, X184), X185, X186, X187, X188, .(T115, T116))

The set Q consists of the following terms:

appF_in_aaaaag(x0)
appM_in_gggggga(x0, x1, x2, x3, x4, x5)
appH_in_aaaag(x0)
appI_in_gggga(x0, x1, x2, x3)
appJ_in_aag(x0)
appK_in_gga(x0, x1)
U5_aaaaag(x0, x1, x2)
U11_gggggga(x0, x1, x2, x3, x4, x5, x6)
U7_aaaag(x0, x1, x2)
U8_gggga(x0, x1, x2, x3, x4, x5)
U9_aag(x0, x1, x2)
U10_gga(x0, x1, x2, x3)
appG_in_ggggga(x0, x1, x2, x3, x4)
U6_ggggga(x0, x1, x2, x3, x4, x5, x6)

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

(60) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule PN_IN_GGGGGGAA(T68, T78, T79, T80, T81, T82) → U16_GGGGGGAA(T68, T78, T79, T80, T81, T82, appM_in_gggggga(T68, T78, T79, T80, T81, T82)) at position [6] we obtained the following new rules [LPAR04]:

PN_IN_GGGGGGAA(T68, T78, T79, T80, T81, T82) → U16_GGGGGGAA(T68, T78, T79, T80, T81, T82, U11_gggggga(T68, T78, T79, T80, T81, T82, appG_in_ggggga(T78, T79, T80, T81, T82)))

(61) Obligation:

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

PARSEA_IN_GA(.(a, .(s(T24, T25, T26), .(b, T27)))) → PB_IN_GGGGAA(T24, T25, T26, T27)
U12_GGGGAA(T24, T25, T26, T27, appL_out_gggga(T24, T25, T26, T27, T33)) → PARSEA_IN_GA(T33)
PARSEA_IN_GA(.(T68, T69)) → PC_IN_AAAAAGGAA(T69, T68)
PC_IN_AAAAAGGAA(T69, T68) → U14_AAAAAGGAA(T69, T68, appF_in_aaaaag(T69))
U14_AAAAAGGAA(T69, T68, appF_out_aaaaag(T78, T79, T80, T81, T82, T69)) → PN_IN_GGGGGGAA(T68, T78, T79, T80, T81, T82)
U16_GGGGGGAA(T68, T78, T79, T80, T81, T82, appM_out_gggggga(T68, T78, T79, T80, T81, T82, T125)) → PARSEA_IN_GA(T125)
PARSEA_IN_GA(T230) → PD_IN_AAAAGAA(T230)
PD_IN_AAAAGAA(T230) → U18_AAAAGAA(T230, appH_in_aaaag(T230))
U18_AAAAGAA(T230, appH_out_aaaag(T239, T240, T241, T242, T230)) → PO_IN_GGGGAA(T239, T240, T241, T242)
PO_IN_GGGGAA(T239, T240, T241, T242) → U20_GGGGAA(T239, T240, T241, T242, appI_in_gggga(T239, T240, T241, T242))
U20_GGGGAA(T239, T240, T241, T242, appI_out_gggga(T239, T240, T241, T242, T282)) → PARSEA_IN_GA(T282)
PARSEA_IN_GA(T341) → PE_IN_AAGAA(T341)
PE_IN_AAGAA(T341) → U22_AAGAA(T341, appJ_in_aag(T341))
U22_AAGAA(T341, appJ_out_aag(T346, T347, T341)) → PP_IN_GGAA(T346, T347)
PP_IN_GGAA(T346, T347) → U24_GGAA(T346, T347, appK_in_gga(T346, T347))
U24_GGAA(T346, T347, appK_out_gga(T346, T347, T365)) → PARSEA_IN_GA(T365)
PB_IN_GGGGAA(T24, T25, T26, T27) → U12_GGGGAA(T24, T25, T26, T27, appL_out_gggga(T24, T25, T26, T27, .(s(a, s(T24, T25, T26), b), T27)))
PN_IN_GGGGGGAA(T68, T78, T79, T80, T81, T82) → U16_GGGGGGAA(T68, T78, T79, T80, T81, T82, U11_gggggga(T68, T78, T79, T80, T81, T82, appG_in_ggggga(T78, T79, T80, T81, T82)))

The TRS R consists of the following rules:

appK_in_gga([], T372) → appK_out_gga([], T372, .(s(a, b), T372))
appK_in_gga(.(T379, T380), T381) → U10_gga(T379, T380, T381, appK_in_gga(T380, T381))
U10_gga(T379, T380, T381, appK_out_gga(T380, T381, X492)) → appK_out_gga(.(T379, T380), T381, .(T379, X492))
appJ_in_aag(.(a, .(b, T354))) → appJ_out_aag([], T354, .(a, .(b, T354)))
appJ_in_aag(.(T359, T360)) → U9_aag(T359, T360, appJ_in_aag(T360))
U9_aag(T359, T360, appJ_out_aag(X463, X464, T360)) → appJ_out_aag(.(T359, X463), X464, .(T359, T360))
appI_in_gggga([], T301, T302, T303) → appI_out_gggga([], T301, T302, T303, .(s(a, s(T301, T302), b), T303))
appI_in_gggga(.(T314, T315), T316, T317, T318) → U8_gggga(T314, T315, T316, T317, T318, appI_in_gggga(T315, T316, T317, T318))
U8_gggga(T314, T315, T316, T317, T318, appI_out_gggga(T315, T316, T317, T318, X388)) → appI_out_gggga(.(T314, T315), T316, T317, T318, .(T314, X388))
appH_in_aaaag(.(a, .(s(T261, T262), .(b, T263)))) → appH_out_aaaag([], T261, T262, T263, .(a, .(s(T261, T262), .(b, T263))))
appH_in_aaaag(.(T268, T269)) → U7_aaaag(T268, T269, appH_in_aaaag(T269))
U7_aaaag(T268, T269, appH_out_aaaag(X353, X354, X355, X356, T269)) → appH_out_aaaag(.(T268, X353), X354, X355, X356, .(T268, T269))
appM_in_gggggga(T146, T147, T148, T149, T150, T151) → U11_gggggga(T146, T147, T148, T149, T150, T151, appG_in_ggggga(T147, T148, T149, T150, T151))
appG_in_ggggga([], T176, T177, T178, T179) → appG_out_ggggga([], T176, T177, T178, T179, .(s(a, s(T176, T177, T178), b), T179))
appG_in_ggggga(.(T192, T193), T194, T195, T196, T197) → U6_ggggga(T192, T193, T194, T195, T196, T197, appG_in_ggggga(T193, T194, T195, T196, T197))
U11_gggggga(T146, T147, T148, T149, T150, T151, appG_out_ggggga(T147, T148, T149, T150, T151, X220)) → appM_out_gggggga(T146, T147, T148, T149, T150, T151, .(T146, X220))
U6_ggggga(T192, T193, T194, T195, T196, T197, appG_out_ggggga(T193, T194, T195, T196, T197, X242)) → appG_out_ggggga(.(T192, T193), T194, T195, T196, T197, .(T192, X242))
appF_in_aaaaag(.(a, .(s(T107, T108, T109), .(b, T110)))) → appF_out_aaaaag([], T107, T108, T109, T110, .(a, .(s(T107, T108, T109), .(b, T110))))
appF_in_aaaaag(.(T115, T116)) → U5_aaaaag(T115, T116, appF_in_aaaaag(T116))
U5_aaaaag(T115, T116, appF_out_aaaaag(X184, X185, X186, X187, X188, T116)) → appF_out_aaaaag(.(T115, X184), X185, X186, X187, X188, .(T115, T116))

The set Q consists of the following terms:

appF_in_aaaaag(x0)
appM_in_gggggga(x0, x1, x2, x3, x4, x5)
appH_in_aaaag(x0)
appI_in_gggga(x0, x1, x2, x3)
appJ_in_aag(x0)
appK_in_gga(x0, x1)
U5_aaaaag(x0, x1, x2)
U11_gggggga(x0, x1, x2, x3, x4, x5, x6)
U7_aaaag(x0, x1, x2)
U8_gggga(x0, x1, x2, x3, x4, x5)
U9_aag(x0, x1, x2)
U10_gga(x0, x1, x2, x3)
appG_in_ggggga(x0, x1, x2, x3, x4)
U6_ggggga(x0, x1, x2, x3, x4, x5, x6)

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

(62) UsableRulesProof (EQUIVALENT transformation)

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

(63) Obligation:

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

PARSEA_IN_GA(.(a, .(s(T24, T25, T26), .(b, T27)))) → PB_IN_GGGGAA(T24, T25, T26, T27)
U12_GGGGAA(T24, T25, T26, T27, appL_out_gggga(T24, T25, T26, T27, T33)) → PARSEA_IN_GA(T33)
PARSEA_IN_GA(.(T68, T69)) → PC_IN_AAAAAGGAA(T69, T68)
PC_IN_AAAAAGGAA(T69, T68) → U14_AAAAAGGAA(T69, T68, appF_in_aaaaag(T69))
U14_AAAAAGGAA(T69, T68, appF_out_aaaaag(T78, T79, T80, T81, T82, T69)) → PN_IN_GGGGGGAA(T68, T78, T79, T80, T81, T82)
U16_GGGGGGAA(T68, T78, T79, T80, T81, T82, appM_out_gggggga(T68, T78, T79, T80, T81, T82, T125)) → PARSEA_IN_GA(T125)
PARSEA_IN_GA(T230) → PD_IN_AAAAGAA(T230)
PD_IN_AAAAGAA(T230) → U18_AAAAGAA(T230, appH_in_aaaag(T230))
U18_AAAAGAA(T230, appH_out_aaaag(T239, T240, T241, T242, T230)) → PO_IN_GGGGAA(T239, T240, T241, T242)
PO_IN_GGGGAA(T239, T240, T241, T242) → U20_GGGGAA(T239, T240, T241, T242, appI_in_gggga(T239, T240, T241, T242))
U20_GGGGAA(T239, T240, T241, T242, appI_out_gggga(T239, T240, T241, T242, T282)) → PARSEA_IN_GA(T282)
PARSEA_IN_GA(T341) → PE_IN_AAGAA(T341)
PE_IN_AAGAA(T341) → U22_AAGAA(T341, appJ_in_aag(T341))
U22_AAGAA(T341, appJ_out_aag(T346, T347, T341)) → PP_IN_GGAA(T346, T347)
PP_IN_GGAA(T346, T347) → U24_GGAA(T346, T347, appK_in_gga(T346, T347))
U24_GGAA(T346, T347, appK_out_gga(T346, T347, T365)) → PARSEA_IN_GA(T365)
PB_IN_GGGGAA(T24, T25, T26, T27) → U12_GGGGAA(T24, T25, T26, T27, appL_out_gggga(T24, T25, T26, T27, .(s(a, s(T24, T25, T26), b), T27)))
PN_IN_GGGGGGAA(T68, T78, T79, T80, T81, T82) → U16_GGGGGGAA(T68, T78, T79, T80, T81, T82, U11_gggggga(T68, T78, T79, T80, T81, T82, appG_in_ggggga(T78, T79, T80, T81, T82)))

The TRS R consists of the following rules:

appG_in_ggggga([], T176, T177, T178, T179) → appG_out_ggggga([], T176, T177, T178, T179, .(s(a, s(T176, T177, T178), b), T179))
appG_in_ggggga(.(T192, T193), T194, T195, T196, T197) → U6_ggggga(T192, T193, T194, T195, T196, T197, appG_in_ggggga(T193, T194, T195, T196, T197))
U11_gggggga(T146, T147, T148, T149, T150, T151, appG_out_ggggga(T147, T148, T149, T150, T151, X220)) → appM_out_gggggga(T146, T147, T148, T149, T150, T151, .(T146, X220))
U6_ggggga(T192, T193, T194, T195, T196, T197, appG_out_ggggga(T193, T194, T195, T196, T197, X242)) → appG_out_ggggga(.(T192, T193), T194, T195, T196, T197, .(T192, X242))
appK_in_gga([], T372) → appK_out_gga([], T372, .(s(a, b), T372))
appK_in_gga(.(T379, T380), T381) → U10_gga(T379, T380, T381, appK_in_gga(T380, T381))
U10_gga(T379, T380, T381, appK_out_gga(T380, T381, X492)) → appK_out_gga(.(T379, T380), T381, .(T379, X492))
appJ_in_aag(.(a, .(b, T354))) → appJ_out_aag([], T354, .(a, .(b, T354)))
appJ_in_aag(.(T359, T360)) → U9_aag(T359, T360, appJ_in_aag(T360))
U9_aag(T359, T360, appJ_out_aag(X463, X464, T360)) → appJ_out_aag(.(T359, X463), X464, .(T359, T360))
appI_in_gggga([], T301, T302, T303) → appI_out_gggga([], T301, T302, T303, .(s(a, s(T301, T302), b), T303))
appI_in_gggga(.(T314, T315), T316, T317, T318) → U8_gggga(T314, T315, T316, T317, T318, appI_in_gggga(T315, T316, T317, T318))
U8_gggga(T314, T315, T316, T317, T318, appI_out_gggga(T315, T316, T317, T318, X388)) → appI_out_gggga(.(T314, T315), T316, T317, T318, .(T314, X388))
appH_in_aaaag(.(a, .(s(T261, T262), .(b, T263)))) → appH_out_aaaag([], T261, T262, T263, .(a, .(s(T261, T262), .(b, T263))))
appH_in_aaaag(.(T268, T269)) → U7_aaaag(T268, T269, appH_in_aaaag(T269))
U7_aaaag(T268, T269, appH_out_aaaag(X353, X354, X355, X356, T269)) → appH_out_aaaag(.(T268, X353), X354, X355, X356, .(T268, T269))
appF_in_aaaaag(.(a, .(s(T107, T108, T109), .(b, T110)))) → appF_out_aaaaag([], T107, T108, T109, T110, .(a, .(s(T107, T108, T109), .(b, T110))))
appF_in_aaaaag(.(T115, T116)) → U5_aaaaag(T115, T116, appF_in_aaaaag(T116))
U5_aaaaag(T115, T116, appF_out_aaaaag(X184, X185, X186, X187, X188, T116)) → appF_out_aaaaag(.(T115, X184), X185, X186, X187, X188, .(T115, T116))

The set Q consists of the following terms:

appF_in_aaaaag(x0)
appM_in_gggggga(x0, x1, x2, x3, x4, x5)
appH_in_aaaag(x0)
appI_in_gggga(x0, x1, x2, x3)
appJ_in_aag(x0)
appK_in_gga(x0, x1)
U5_aaaaag(x0, x1, x2)
U11_gggggga(x0, x1, x2, x3, x4, x5, x6)
U7_aaaag(x0, x1, x2)
U8_gggga(x0, x1, x2, x3, x4, x5)
U9_aag(x0, x1, x2)
U10_gga(x0, x1, x2, x3)
appG_in_ggggga(x0, x1, x2, x3, x4)
U6_ggggga(x0, x1, x2, x3, x4, x5, x6)

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

(64) QReductionProof (EQUIVALENT transformation)

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

appM_in_gggggga(x0, x1, x2, x3, x4, x5)

(65) Obligation:

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

PARSEA_IN_GA(.(a, .(s(T24, T25, T26), .(b, T27)))) → PB_IN_GGGGAA(T24, T25, T26, T27)
U12_GGGGAA(T24, T25, T26, T27, appL_out_gggga(T24, T25, T26, T27, T33)) → PARSEA_IN_GA(T33)
PARSEA_IN_GA(.(T68, T69)) → PC_IN_AAAAAGGAA(T69, T68)
PC_IN_AAAAAGGAA(T69, T68) → U14_AAAAAGGAA(T69, T68, appF_in_aaaaag(T69))
U14_AAAAAGGAA(T69, T68, appF_out_aaaaag(T78, T79, T80, T81, T82, T69)) → PN_IN_GGGGGGAA(T68, T78, T79, T80, T81, T82)
U16_GGGGGGAA(T68, T78, T79, T80, T81, T82, appM_out_gggggga(T68, T78, T79, T80, T81, T82, T125)) → PARSEA_IN_GA(T125)
PARSEA_IN_GA(T230) → PD_IN_AAAAGAA(T230)
PD_IN_AAAAGAA(T230) → U18_AAAAGAA(T230, appH_in_aaaag(T230))
U18_AAAAGAA(T230, appH_out_aaaag(T239, T240, T241, T242, T230)) → PO_IN_GGGGAA(T239, T240, T241, T242)
PO_IN_GGGGAA(T239, T240, T241, T242) → U20_GGGGAA(T239, T240, T241, T242, appI_in_gggga(T239, T240, T241, T242))
U20_GGGGAA(T239, T240, T241, T242, appI_out_gggga(T239, T240, T241, T242, T282)) → PARSEA_IN_GA(T282)
PARSEA_IN_GA(T341) → PE_IN_AAGAA(T341)
PE_IN_AAGAA(T341) → U22_AAGAA(T341, appJ_in_aag(T341))
U22_AAGAA(T341, appJ_out_aag(T346, T347, T341)) → PP_IN_GGAA(T346, T347)
PP_IN_GGAA(T346, T347) → U24_GGAA(T346, T347, appK_in_gga(T346, T347))
U24_GGAA(T346, T347, appK_out_gga(T346, T347, T365)) → PARSEA_IN_GA(T365)
PB_IN_GGGGAA(T24, T25, T26, T27) → U12_GGGGAA(T24, T25, T26, T27, appL_out_gggga(T24, T25, T26, T27, .(s(a, s(T24, T25, T26), b), T27)))
PN_IN_GGGGGGAA(T68, T78, T79, T80, T81, T82) → U16_GGGGGGAA(T68, T78, T79, T80, T81, T82, U11_gggggga(T68, T78, T79, T80, T81, T82, appG_in_ggggga(T78, T79, T80, T81, T82)))

The TRS R consists of the following rules:

appG_in_ggggga([], T176, T177, T178, T179) → appG_out_ggggga([], T176, T177, T178, T179, .(s(a, s(T176, T177, T178), b), T179))
appG_in_ggggga(.(T192, T193), T194, T195, T196, T197) → U6_ggggga(T192, T193, T194, T195, T196, T197, appG_in_ggggga(T193, T194, T195, T196, T197))
U11_gggggga(T146, T147, T148, T149, T150, T151, appG_out_ggggga(T147, T148, T149, T150, T151, X220)) → appM_out_gggggga(T146, T147, T148, T149, T150, T151, .(T146, X220))
U6_ggggga(T192, T193, T194, T195, T196, T197, appG_out_ggggga(T193, T194, T195, T196, T197, X242)) → appG_out_ggggga(.(T192, T193), T194, T195, T196, T197, .(T192, X242))
appK_in_gga([], T372) → appK_out_gga([], T372, .(s(a, b), T372))
appK_in_gga(.(T379, T380), T381) → U10_gga(T379, T380, T381, appK_in_gga(T380, T381))
U10_gga(T379, T380, T381, appK_out_gga(T380, T381, X492)) → appK_out_gga(.(T379, T380), T381, .(T379, X492))
appJ_in_aag(.(a, .(b, T354))) → appJ_out_aag([], T354, .(a, .(b, T354)))
appJ_in_aag(.(T359, T360)) → U9_aag(T359, T360, appJ_in_aag(T360))
U9_aag(T359, T360, appJ_out_aag(X463, X464, T360)) → appJ_out_aag(.(T359, X463), X464, .(T359, T360))
appI_in_gggga([], T301, T302, T303) → appI_out_gggga([], T301, T302, T303, .(s(a, s(T301, T302), b), T303))
appI_in_gggga(.(T314, T315), T316, T317, T318) → U8_gggga(T314, T315, T316, T317, T318, appI_in_gggga(T315, T316, T317, T318))
U8_gggga(T314, T315, T316, T317, T318, appI_out_gggga(T315, T316, T317, T318, X388)) → appI_out_gggga(.(T314, T315), T316, T317, T318, .(T314, X388))
appH_in_aaaag(.(a, .(s(T261, T262), .(b, T263)))) → appH_out_aaaag([], T261, T262, T263, .(a, .(s(T261, T262), .(b, T263))))
appH_in_aaaag(.(T268, T269)) → U7_aaaag(T268, T269, appH_in_aaaag(T269))
U7_aaaag(T268, T269, appH_out_aaaag(X353, X354, X355, X356, T269)) → appH_out_aaaag(.(T268, X353), X354, X355, X356, .(T268, T269))
appF_in_aaaaag(.(a, .(s(T107, T108, T109), .(b, T110)))) → appF_out_aaaaag([], T107, T108, T109, T110, .(a, .(s(T107, T108, T109), .(b, T110))))
appF_in_aaaaag(.(T115, T116)) → U5_aaaaag(T115, T116, appF_in_aaaaag(T116))
U5_aaaaag(T115, T116, appF_out_aaaaag(X184, X185, X186, X187, X188, T116)) → appF_out_aaaaag(.(T115, X184), X185, X186, X187, X188, .(T115, T116))

The set Q consists of the following terms:

appF_in_aaaaag(x0)
appH_in_aaaag(x0)
appI_in_gggga(x0, x1, x2, x3)
appJ_in_aag(x0)
appK_in_gga(x0, x1)
U5_aaaaag(x0, x1, x2)
U11_gggggga(x0, x1, x2, x3, x4, x5, x6)
U7_aaaag(x0, x1, x2)
U8_gggga(x0, x1, x2, x3, x4, x5)
U9_aag(x0, x1, x2)
U10_gga(x0, x1, x2, x3)
appG_in_ggggga(x0, x1, x2, x3, x4)
U6_ggggga(x0, x1, x2, x3, x4, x5, x6)

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

(66) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule U12_GGGGAA(T24, T25, T26, T27, appL_out_gggga(T24, T25, T26, T27, T33)) → PARSEA_IN_GA(T33) we obtained the following new rules [LPAR04]:

U12_GGGGAA(z0, z1, z2, z3, appL_out_gggga(z0, z1, z2, z3, .(s(a, s(z0, z1, z2), b), z3))) → PARSEA_IN_GA(.(s(a, s(z0, z1, z2), b), z3))

(67) Obligation:

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

PARSEA_IN_GA(.(a, .(s(T24, T25, T26), .(b, T27)))) → PB_IN_GGGGAA(T24, T25, T26, T27)
PARSEA_IN_GA(.(T68, T69)) → PC_IN_AAAAAGGAA(T69, T68)
PC_IN_AAAAAGGAA(T69, T68) → U14_AAAAAGGAA(T69, T68, appF_in_aaaaag(T69))
U14_AAAAAGGAA(T69, T68, appF_out_aaaaag(T78, T79, T80, T81, T82, T69)) → PN_IN_GGGGGGAA(T68, T78, T79, T80, T81, T82)
U16_GGGGGGAA(T68, T78, T79, T80, T81, T82, appM_out_gggggga(T68, T78, T79, T80, T81, T82, T125)) → PARSEA_IN_GA(T125)
PARSEA_IN_GA(T230) → PD_IN_AAAAGAA(T230)
PD_IN_AAAAGAA(T230) → U18_AAAAGAA(T230, appH_in_aaaag(T230))
U18_AAAAGAA(T230, appH_out_aaaag(T239, T240, T241, T242, T230)) → PO_IN_GGGGAA(T239, T240, T241, T242)
PO_IN_GGGGAA(T239, T240, T241, T242) → U20_GGGGAA(T239, T240, T241, T242, appI_in_gggga(T239, T240, T241, T242))
U20_GGGGAA(T239, T240, T241, T242, appI_out_gggga(T239, T240, T241, T242, T282)) → PARSEA_IN_GA(T282)
PARSEA_IN_GA(T341) → PE_IN_AAGAA(T341)
PE_IN_AAGAA(T341) → U22_AAGAA(T341, appJ_in_aag(T341))
U22_AAGAA(T341, appJ_out_aag(T346, T347, T341)) → PP_IN_GGAA(T346, T347)
PP_IN_GGAA(T346, T347) → U24_GGAA(T346, T347, appK_in_gga(T346, T347))
U24_GGAA(T346, T347, appK_out_gga(T346, T347, T365)) → PARSEA_IN_GA(T365)
PB_IN_GGGGAA(T24, T25, T26, T27) → U12_GGGGAA(T24, T25, T26, T27, appL_out_gggga(T24, T25, T26, T27, .(s(a, s(T24, T25, T26), b), T27)))
PN_IN_GGGGGGAA(T68, T78, T79, T80, T81, T82) → U16_GGGGGGAA(T68, T78, T79, T80, T81, T82, U11_gggggga(T68, T78, T79, T80, T81, T82, appG_in_ggggga(T78, T79, T80, T81, T82)))
U12_GGGGAA(z0, z1, z2, z3, appL_out_gggga(z0, z1, z2, z3, .(s(a, s(z0, z1, z2), b), z3))) → PARSEA_IN_GA(.(s(a, s(z0, z1, z2), b), z3))

The TRS R consists of the following rules:

appG_in_ggggga([], T176, T177, T178, T179) → appG_out_ggggga([], T176, T177, T178, T179, .(s(a, s(T176, T177, T178), b), T179))
appG_in_ggggga(.(T192, T193), T194, T195, T196, T197) → U6_ggggga(T192, T193, T194, T195, T196, T197, appG_in_ggggga(T193, T194, T195, T196, T197))
U11_gggggga(T146, T147, T148, T149, T150, T151, appG_out_ggggga(T147, T148, T149, T150, T151, X220)) → appM_out_gggggga(T146, T147, T148, T149, T150, T151, .(T146, X220))
U6_ggggga(T192, T193, T194, T195, T196, T197, appG_out_ggggga(T193, T194, T195, T196, T197, X242)) → appG_out_ggggga(.(T192, T193), T194, T195, T196, T197, .(T192, X242))
appK_in_gga([], T372) → appK_out_gga([], T372, .(s(a, b), T372))
appK_in_gga(.(T379, T380), T381) → U10_gga(T379, T380, T381, appK_in_gga(T380, T381))
U10_gga(T379, T380, T381, appK_out_gga(T380, T381, X492)) → appK_out_gga(.(T379, T380), T381, .(T379, X492))
appJ_in_aag(.(a, .(b, T354))) → appJ_out_aag([], T354, .(a, .(b, T354)))
appJ_in_aag(.(T359, T360)) → U9_aag(T359, T360, appJ_in_aag(T360))
U9_aag(T359, T360, appJ_out_aag(X463, X464, T360)) → appJ_out_aag(.(T359, X463), X464, .(T359, T360))
appI_in_gggga([], T301, T302, T303) → appI_out_gggga([], T301, T302, T303, .(s(a, s(T301, T302), b), T303))
appI_in_gggga(.(T314, T315), T316, T317, T318) → U8_gggga(T314, T315, T316, T317, T318, appI_in_gggga(T315, T316, T317, T318))
U8_gggga(T314, T315, T316, T317, T318, appI_out_gggga(T315, T316, T317, T318, X388)) → appI_out_gggga(.(T314, T315), T316, T317, T318, .(T314, X388))
appH_in_aaaag(.(a, .(s(T261, T262), .(b, T263)))) → appH_out_aaaag([], T261, T262, T263, .(a, .(s(T261, T262), .(b, T263))))
appH_in_aaaag(.(T268, T269)) → U7_aaaag(T268, T269, appH_in_aaaag(T269))
U7_aaaag(T268, T269, appH_out_aaaag(X353, X354, X355, X356, T269)) → appH_out_aaaag(.(T268, X353), X354, X355, X356, .(T268, T269))
appF_in_aaaaag(.(a, .(s(T107, T108, T109), .(b, T110)))) → appF_out_aaaaag([], T107, T108, T109, T110, .(a, .(s(T107, T108, T109), .(b, T110))))
appF_in_aaaaag(.(T115, T116)) → U5_aaaaag(T115, T116, appF_in_aaaaag(T116))
U5_aaaaag(T115, T116, appF_out_aaaaag(X184, X185, X186, X187, X188, T116)) → appF_out_aaaaag(.(T115, X184), X185, X186, X187, X188, .(T115, T116))

The set Q consists of the following terms:

appF_in_aaaaag(x0)
appH_in_aaaag(x0)
appI_in_gggga(x0, x1, x2, x3)
appJ_in_aag(x0)
appK_in_gga(x0, x1)
U5_aaaaag(x0, x1, x2)
U11_gggggga(x0, x1, x2, x3, x4, x5, x6)
U7_aaaag(x0, x1, x2)
U8_gggga(x0, x1, x2, x3, x4, x5)
U9_aag(x0, x1, x2)
U10_gga(x0, x1, x2, x3)
appG_in_ggggga(x0, x1, x2, x3, x4)
U6_ggggga(x0, x1, x2, x3, x4, x5, x6)

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

(68) QDPOrderProof (EQUIVALENT transformation)

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

The following pairs can be oriented strictly and are deleted.

PARSEA_IN_GA(.(a, .(s(T24, T25, T26), .(b, T27)))) → PB_IN_GGGGAA(T24, T25, T26, T27)
PC_IN_AAAAAGGAA(T69, T68) → U14_AAAAAGGAA(T69, T68, appF_in_aaaaag(T69))
PP_IN_GGAA(T346, T347) → U24_GGAA(T346, T347, appK_in_gga(T346, T347))

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

POL(.(x₁, x₂)) = 1 + x₂   
POL(PARSEA_IN_GA(x₁)) = x₁   
POL(PB_IN_GGGGAA(x₁, x₂, x₃, x₄)) = 1 + x₄   
POL(PC_IN_AAAAAGGAA(x₁, x₂)) = 1 + x₁   
POL(PD_IN_AAAAGAA(x₁)) = x₁   
POL(PE_IN_AAGAA(x₁)) = x₁   
POL(PN_IN_GGGGGGAA(x₁, x₂, x₃, x₄, x₅, x₆)) = 1 + x₂ + x₆   
POL(PO_IN_GGGGAA(x₁, x₂, x₃, x₄)) = x₁ + x₄   
POL(PP_IN_GGAA(x₁, x₂)) = 1 + x₁ + x₂   
POL(U10_gga(x₁, x₂, x₃, x₄)) = 1 + x₄   
POL(U11_gggggga(x₁, x₂, x₃, x₄, x₅, x₆, x₇)) = x₇   
POL(U12_GGGGAA(x₁, x₂, x₃, x₄, x₅)) = 1 + x₅   
POL(U14_AAAAAGGAA(x₁, x₂, x₃)) = x₃   
POL(U16_GGGGGGAA(x₁, x₂, x₃, x₄, x₅, x₆, x₇)) = x₇   
POL(U18_AAAAGAA(x₁, x₂)) = x₂   
POL(U20_GGGGAA(x₁, x₂, x₃, x₄, x₅)) = x₅   
POL(U22_AAGAA(x₁, x₂)) = x₂   
POL(U24_GGAA(x₁, x₂, x₃)) = x₃   
POL(U5_aaaaag(x₁, x₂, x₃)) = 1 + x₃   
POL(U6_ggggga(x₁, x₂, x₃, x₄, x₅, x₆, x₇)) = 1 + x₇   
POL(U7_aaaag(x₁, x₂, x₃)) = 1 + x₃   
POL(U8_gggga(x₁, x₂, x₃, x₄, x₅, x₆)) = 1 + x₆   
POL(U9_aag(x₁, x₂, x₃)) = 1 + x₃   
POL([]) = 1   
POL(a) = 0   
POL(appF_in_aaaaag(x₁)) = x₁   
POL(appF_out_aaaaag(x₁, x₂, x₃, x₄, x₅, x₆)) = 1 + x₁ + x₅   
POL(appG_in_ggggga(x₁, x₂, x₃, x₄, x₅)) = 1 + x₁ + x₅   
POL(appG_out_ggggga(x₁, x₂, x₃, x₄, x₅, x₆)) = 1 + x₆   
POL(appH_in_aaaag(x₁)) = x₁   
POL(appH_out_aaaag(x₁, x₂, x₃, x₄, x₅)) = x₁ + x₄   
POL(appI_in_gggga(x₁, x₂, x₃, x₄)) = x₁ + x₄   
POL(appI_out_gggga(x₁, x₂, x₃, x₄, x₅)) = x₅   
POL(appJ_in_aag(x₁)) = x₁   
POL(appJ_out_aag(x₁, x₂, x₃)) = 1 + x₁ + x₂   
POL(appK_in_gga(x₁, x₂)) = x₁ + x₂   
POL(appK_out_gga(x₁, x₂, x₃)) = x₃   
POL(appL_out_gggga(x₁, x₂, x₃, x₄, x₅)) = x₄   
POL(appM_out_gggggga(x₁, x₂, x₃, x₄, x₅, x₆, x₇)) = x₇   
POL(b) = 0   
POL(s(x₁, x₂)) = 0   
POL(s(x₁, x₂, x₃)) = 0   

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

appF_in_aaaaag(.(a, .(s(T107, T108, T109), .(b, T110)))) → appF_out_aaaaag([], T107, T108, T109, T110, .(a, .(s(T107, T108, T109), .(b, T110))))
appF_in_aaaaag(.(T115, T116)) → U5_aaaaag(T115, T116, appF_in_aaaaag(T116))
appH_in_aaaag(.(a, .(s(T261, T262), .(b, T263)))) → appH_out_aaaag([], T261, T262, T263, .(a, .(s(T261, T262), .(b, T263))))
appH_in_aaaag(.(T268, T269)) → U7_aaaag(T268, T269, appH_in_aaaag(T269))
appI_in_gggga([], T301, T302, T303) → appI_out_gggga([], T301, T302, T303, .(s(a, s(T301, T302), b), T303))
appI_in_gggga(.(T314, T315), T316, T317, T318) → U8_gggga(T314, T315, T316, T317, T318, appI_in_gggga(T315, T316, T317, T318))
appJ_in_aag(.(a, .(b, T354))) → appJ_out_aag([], T354, .(a, .(b, T354)))
appJ_in_aag(.(T359, T360)) → U9_aag(T359, T360, appJ_in_aag(T360))
appK_in_gga([], T372) → appK_out_gga([], T372, .(s(a, b), T372))
appK_in_gga(.(T379, T380), T381) → U10_gga(T379, T380, T381, appK_in_gga(T380, T381))
appG_in_ggggga([], T176, T177, T178, T179) → appG_out_ggggga([], T176, T177, T178, T179, .(s(a, s(T176, T177, T178), b), T179))
appG_in_ggggga(.(T192, T193), T194, T195, T196, T197) → U6_ggggga(T192, T193, T194, T195, T196, T197, appG_in_ggggga(T193, T194, T195, T196, T197))
U11_gggggga(T146, T147, T148, T149, T150, T151, appG_out_ggggga(T147, T148, T149, T150, T151, X220)) → appM_out_gggggga(T146, T147, T148, T149, T150, T151, .(T146, X220))
U6_ggggga(T192, T193, T194, T195, T196, T197, appG_out_ggggga(T193, T194, T195, T196, T197, X242)) → appG_out_ggggga(.(T192, T193), T194, T195, T196, T197, .(T192, X242))
U10_gga(T379, T380, T381, appK_out_gga(T380, T381, X492)) → appK_out_gga(.(T379, T380), T381, .(T379, X492))
U9_aag(T359, T360, appJ_out_aag(X463, X464, T360)) → appJ_out_aag(.(T359, X463), X464, .(T359, T360))
U8_gggga(T314, T315, T316, T317, T318, appI_out_gggga(T315, T316, T317, T318, X388)) → appI_out_gggga(.(T314, T315), T316, T317, T318, .(T314, X388))
U7_aaaag(T268, T269, appH_out_aaaag(X353, X354, X355, X356, T269)) → appH_out_aaaag(.(T268, X353), X354, X355, X356, .(T268, T269))
U5_aaaaag(T115, T116, appF_out_aaaaag(X184, X185, X186, X187, X188, T116)) → appF_out_aaaaag(.(T115, X184), X185, X186, X187, X188, .(T115, T116))

(69) Obligation:

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

PARSEA_IN_GA(.(T68, T69)) → PC_IN_AAAAAGGAA(T69, T68)
U14_AAAAAGGAA(T69, T68, appF_out_aaaaag(T78, T79, T80, T81, T82, T69)) → PN_IN_GGGGGGAA(T68, T78, T79, T80, T81, T82)
U16_GGGGGGAA(T68, T78, T79, T80, T81, T82, appM_out_gggggga(T68, T78, T79, T80, T81, T82, T125)) → PARSEA_IN_GA(T125)
PARSEA_IN_GA(T230) → PD_IN_AAAAGAA(T230)
PD_IN_AAAAGAA(T230) → U18_AAAAGAA(T230, appH_in_aaaag(T230))
U18_AAAAGAA(T230, appH_out_aaaag(T239, T240, T241, T242, T230)) → PO_IN_GGGGAA(T239, T240, T241, T242)
PO_IN_GGGGAA(T239, T240, T241, T242) → U20_GGGGAA(T239, T240, T241, T242, appI_in_gggga(T239, T240, T241, T242))
U20_GGGGAA(T239, T240, T241, T242, appI_out_gggga(T239, T240, T241, T242, T282)) → PARSEA_IN_GA(T282)
PARSEA_IN_GA(T341) → PE_IN_AAGAA(T341)
PE_IN_AAGAA(T341) → U22_AAGAA(T341, appJ_in_aag(T341))
U22_AAGAA(T341, appJ_out_aag(T346, T347, T341)) → PP_IN_GGAA(T346, T347)
U24_GGAA(T346, T347, appK_out_gga(T346, T347, T365)) → PARSEA_IN_GA(T365)
PB_IN_GGGGAA(T24, T25, T26, T27) → U12_GGGGAA(T24, T25, T26, T27, appL_out_gggga(T24, T25, T26, T27, .(s(a, s(T24, T25, T26), b), T27)))
PN_IN_GGGGGGAA(T68, T78, T79, T80, T81, T82) → U16_GGGGGGAA(T68, T78, T79, T80, T81, T82, U11_gggggga(T68, T78, T79, T80, T81, T82, appG_in_ggggga(T78, T79, T80, T81, T82)))
U12_GGGGAA(z0, z1, z2, z3, appL_out_gggga(z0, z1, z2, z3, .(s(a, s(z0, z1, z2), b), z3))) → PARSEA_IN_GA(.(s(a, s(z0, z1, z2), b), z3))

The TRS R consists of the following rules:

appG_in_ggggga([], T176, T177, T178, T179) → appG_out_ggggga([], T176, T177, T178, T179, .(s(a, s(T176, T177, T178), b), T179))
appG_in_ggggga(.(T192, T193), T194, T195, T196, T197) → U6_ggggga(T192, T193, T194, T195, T196, T197, appG_in_ggggga(T193, T194, T195, T196, T197))
U11_gggggga(T146, T147, T148, T149, T150, T151, appG_out_ggggga(T147, T148, T149, T150, T151, X220)) → appM_out_gggggga(T146, T147, T148, T149, T150, T151, .(T146, X220))
U6_ggggga(T192, T193, T194, T195, T196, T197, appG_out_ggggga(T193, T194, T195, T196, T197, X242)) → appG_out_ggggga(.(T192, T193), T194, T195, T196, T197, .(T192, X242))
appK_in_gga([], T372) → appK_out_gga([], T372, .(s(a, b), T372))
appK_in_gga(.(T379, T380), T381) → U10_gga(T379, T380, T381, appK_in_gga(T380, T381))
U10_gga(T379, T380, T381, appK_out_gga(T380, T381, X492)) → appK_out_gga(.(T379, T380), T381, .(T379, X492))
appJ_in_aag(.(a, .(b, T354))) → appJ_out_aag([], T354, .(a, .(b, T354)))
appJ_in_aag(.(T359, T360)) → U9_aag(T359, T360, appJ_in_aag(T360))
U9_aag(T359, T360, appJ_out_aag(X463, X464, T360)) → appJ_out_aag(.(T359, X463), X464, .(T359, T360))
appI_in_gggga([], T301, T302, T303) → appI_out_gggga([], T301, T302, T303, .(s(a, s(T301, T302), b), T303))
appI_in_gggga(.(T314, T315), T316, T317, T318) → U8_gggga(T314, T315, T316, T317, T318, appI_in_gggga(T315, T316, T317, T318))
U8_gggga(T314, T315, T316, T317, T318, appI_out_gggga(T315, T316, T317, T318, X388)) → appI_out_gggga(.(T314, T315), T316, T317, T318, .(T314, X388))
appH_in_aaaag(.(a, .(s(T261, T262), .(b, T263)))) → appH_out_aaaag([], T261, T262, T263, .(a, .(s(T261, T262), .(b, T263))))
appH_in_aaaag(.(T268, T269)) → U7_aaaag(T268, T269, appH_in_aaaag(T269))
U7_aaaag(T268, T269, appH_out_aaaag(X353, X354, X355, X356, T269)) → appH_out_aaaag(.(T268, X353), X354, X355, X356, .(T268, T269))
appF_in_aaaaag(.(a, .(s(T107, T108, T109), .(b, T110)))) → appF_out_aaaaag([], T107, T108, T109, T110, .(a, .(s(T107, T108, T109), .(b, T110))))
appF_in_aaaaag(.(T115, T116)) → U5_aaaaag(T115, T116, appF_in_aaaaag(T116))
U5_aaaaag(T115, T116, appF_out_aaaaag(X184, X185, X186, X187, X188, T116)) → appF_out_aaaaag(.(T115, X184), X185, X186, X187, X188, .(T115, T116))

The set Q consists of the following terms:

appF_in_aaaaag(x0)
appH_in_aaaag(x0)
appI_in_gggga(x0, x1, x2, x3)
appJ_in_aag(x0)
appK_in_gga(x0, x1)
U5_aaaaag(x0, x1, x2)
U11_gggggga(x0, x1, x2, x3, x4, x5, x6)
U7_aaaag(x0, x1, x2)
U8_gggga(x0, x1, x2, x3, x4, x5)
U9_aag(x0, x1, x2)
U10_gga(x0, x1, x2, x3)
appG_in_ggggga(x0, x1, x2, x3, x4)
U6_ggggga(x0, x1, x2, x3, x4, x5, x6)

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

(70) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 10 less nodes.

(71) Obligation:

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

PARSEA_IN_GA(T230) → PD_IN_AAAAGAA(T230)
PD_IN_AAAAGAA(T230) → U18_AAAAGAA(T230, appH_in_aaaag(T230))
U18_AAAAGAA(T230, appH_out_aaaag(T239, T240, T241, T242, T230)) → PO_IN_GGGGAA(T239, T240, T241, T242)
PO_IN_GGGGAA(T239, T240, T241, T242) → U20_GGGGAA(T239, T240, T241, T242, appI_in_gggga(T239, T240, T241, T242))
U20_GGGGAA(T239, T240, T241, T242, appI_out_gggga(T239, T240, T241, T242, T282)) → PARSEA_IN_GA(T282)

The TRS R consists of the following rules:

appG_in_ggggga([], T176, T177, T178, T179) → appG_out_ggggga([], T176, T177, T178, T179, .(s(a, s(T176, T177, T178), b), T179))
appG_in_ggggga(.(T192, T193), T194, T195, T196, T197) → U6_ggggga(T192, T193, T194, T195, T196, T197, appG_in_ggggga(T193, T194, T195, T196, T197))
U11_gggggga(T146, T147, T148, T149, T150, T151, appG_out_ggggga(T147, T148, T149, T150, T151, X220)) → appM_out_gggggga(T146, T147, T148, T149, T150, T151, .(T146, X220))
U6_ggggga(T192, T193, T194, T195, T196, T197, appG_out_ggggga(T193, T194, T195, T196, T197, X242)) → appG_out_ggggga(.(T192, T193), T194, T195, T196, T197, .(T192, X242))
appK_in_gga([], T372) → appK_out_gga([], T372, .(s(a, b), T372))
appK_in_gga(.(T379, T380), T381) → U10_gga(T379, T380, T381, appK_in_gga(T380, T381))
U10_gga(T379, T380, T381, appK_out_gga(T380, T381, X492)) → appK_out_gga(.(T379, T380), T381, .(T379, X492))
appJ_in_aag(.(a, .(b, T354))) → appJ_out_aag([], T354, .(a, .(b, T354)))
appJ_in_aag(.(T359, T360)) → U9_aag(T359, T360, appJ_in_aag(T360))
U9_aag(T359, T360, appJ_out_aag(X463, X464, T360)) → appJ_out_aag(.(T359, X463), X464, .(T359, T360))
appI_in_gggga([], T301, T302, T303) → appI_out_gggga([], T301, T302, T303, .(s(a, s(T301, T302), b), T303))
appI_in_gggga(.(T314, T315), T316, T317, T318) → U8_gggga(T314, T315, T316, T317, T318, appI_in_gggga(T315, T316, T317, T318))
U8_gggga(T314, T315, T316, T317, T318, appI_out_gggga(T315, T316, T317, T318, X388)) → appI_out_gggga(.(T314, T315), T316, T317, T318, .(T314, X388))
appH_in_aaaag(.(a, .(s(T261, T262), .(b, T263)))) → appH_out_aaaag([], T261, T262, T263, .(a, .(s(T261, T262), .(b, T263))))
appH_in_aaaag(.(T268, T269)) → U7_aaaag(T268, T269, appH_in_aaaag(T269))
U7_aaaag(T268, T269, appH_out_aaaag(X353, X354, X355, X356, T269)) → appH_out_aaaag(.(T268, X353), X354, X355, X356, .(T268, T269))
appF_in_aaaaag(.(a, .(s(T107, T108, T109), .(b, T110)))) → appF_out_aaaaag([], T107, T108, T109, T110, .(a, .(s(T107, T108, T109), .(b, T110))))
appF_in_aaaaag(.(T115, T116)) → U5_aaaaag(T115, T116, appF_in_aaaaag(T116))
U5_aaaaag(T115, T116, appF_out_aaaaag(X184, X185, X186, X187, X188, T116)) → appF_out_aaaaag(.(T115, X184), X185, X186, X187, X188, .(T115, T116))

The set Q consists of the following terms:

appF_in_aaaaag(x0)
appH_in_aaaag(x0)
appI_in_gggga(x0, x1, x2, x3)
appJ_in_aag(x0)
appK_in_gga(x0, x1)
U5_aaaaag(x0, x1, x2)
U11_gggggga(x0, x1, x2, x3, x4, x5, x6)
U7_aaaag(x0, x1, x2)
U8_gggga(x0, x1, x2, x3, x4, x5)
U9_aag(x0, x1, x2)
U10_gga(x0, x1, x2, x3)
appG_in_ggggga(x0, x1, x2, x3, x4)
U6_ggggga(x0, x1, x2, x3, x4, x5, x6)

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

(72) UsableRulesProof (EQUIVALENT transformation)

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

(73) Obligation:

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

PARSEA_IN_GA(T230) → PD_IN_AAAAGAA(T230)
PD_IN_AAAAGAA(T230) → U18_AAAAGAA(T230, appH_in_aaaag(T230))
U18_AAAAGAA(T230, appH_out_aaaag(T239, T240, T241, T242, T230)) → PO_IN_GGGGAA(T239, T240, T241, T242)
PO_IN_GGGGAA(T239, T240, T241, T242) → U20_GGGGAA(T239, T240, T241, T242, appI_in_gggga(T239, T240, T241, T242))
U20_GGGGAA(T239, T240, T241, T242, appI_out_gggga(T239, T240, T241, T242, T282)) → PARSEA_IN_GA(T282)

The TRS R consists of the following rules:

appI_in_gggga([], T301, T302, T303) → appI_out_gggga([], T301, T302, T303, .(s(a, s(T301, T302), b), T303))
appI_in_gggga(.(T314, T315), T316, T317, T318) → U8_gggga(T314, T315, T316, T317, T318, appI_in_gggga(T315, T316, T317, T318))
U8_gggga(T314, T315, T316, T317, T318, appI_out_gggga(T315, T316, T317, T318, X388)) → appI_out_gggga(.(T314, T315), T316, T317, T318, .(T314, X388))
appH_in_aaaag(.(a, .(s(T261, T262), .(b, T263)))) → appH_out_aaaag([], T261, T262, T263, .(a, .(s(T261, T262), .(b, T263))))
appH_in_aaaag(.(T268, T269)) → U7_aaaag(T268, T269, appH_in_aaaag(T269))
U7_aaaag(T268, T269, appH_out_aaaag(X353, X354, X355, X356, T269)) → appH_out_aaaag(.(T268, X353), X354, X355, X356, .(T268, T269))

The set Q consists of the following terms:

appF_in_aaaaag(x0)
appH_in_aaaag(x0)
appI_in_gggga(x0, x1, x2, x3)
appJ_in_aag(x0)
appK_in_gga(x0, x1)
U5_aaaaag(x0, x1, x2)
U11_gggggga(x0, x1, x2, x3, x4, x5, x6)
U7_aaaag(x0, x1, x2)
U8_gggga(x0, x1, x2, x3, x4, x5)
U9_aag(x0, x1, x2)
U10_gga(x0, x1, x2, x3)
appG_in_ggggga(x0, x1, x2, x3, x4)
U6_ggggga(x0, x1, x2, x3, x4, x5, x6)

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

(74) QReductionProof (EQUIVALENT transformation)

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

appF_in_aaaaag(x0)
appJ_in_aag(x0)
appK_in_gga(x0, x1)
U5_aaaaag(x0, x1, x2)
U11_gggggga(x0, x1, x2, x3, x4, x5, x6)
U9_aag(x0, x1, x2)
U10_gga(x0, x1, x2, x3)
appG_in_ggggga(x0, x1, x2, x3, x4)
U6_ggggga(x0, x1, x2, x3, x4, x5, x6)

(75) Obligation:

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

PARSEA_IN_GA(T230) → PD_IN_AAAAGAA(T230)
PD_IN_AAAAGAA(T230) → U18_AAAAGAA(T230, appH_in_aaaag(T230))
U18_AAAAGAA(T230, appH_out_aaaag(T239, T240, T241, T242, T230)) → PO_IN_GGGGAA(T239, T240, T241, T242)
PO_IN_GGGGAA(T239, T240, T241, T242) → U20_GGGGAA(T239, T240, T241, T242, appI_in_gggga(T239, T240, T241, T242))
U20_GGGGAA(T239, T240, T241, T242, appI_out_gggga(T239, T240, T241, T242, T282)) → PARSEA_IN_GA(T282)

The TRS R consists of the following rules:

appI_in_gggga([], T301, T302, T303) → appI_out_gggga([], T301, T302, T303, .(s(a, s(T301, T302), b), T303))
appI_in_gggga(.(T314, T315), T316, T317, T318) → U8_gggga(T314, T315, T316, T317, T318, appI_in_gggga(T315, T316, T317, T318))
U8_gggga(T314, T315, T316, T317, T318, appI_out_gggga(T315, T316, T317, T318, X388)) → appI_out_gggga(.(T314, T315), T316, T317, T318, .(T314, X388))
appH_in_aaaag(.(a, .(s(T261, T262), .(b, T263)))) → appH_out_aaaag([], T261, T262, T263, .(a, .(s(T261, T262), .(b, T263))))
appH_in_aaaag(.(T268, T269)) → U7_aaaag(T268, T269, appH_in_aaaag(T269))
U7_aaaag(T268, T269, appH_out_aaaag(X353, X354, X355, X356, T269)) → appH_out_aaaag(.(T268, X353), X354, X355, X356, .(T268, T269))

The set Q consists of the following terms:

appH_in_aaaag(x0)
appI_in_gggga(x0, x1, x2, x3)
U7_aaaag(x0, x1, x2)
U8_gggga(x0, x1, x2, x3, x4, x5)

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

(76) QDPOrderProof (EQUIVALENT transformation)

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

The following pairs can be oriented strictly and are deleted.

PO_IN_GGGGAA(T239, T240, T241, T242) → U20_GGGGAA(T239, T240, T241, T242, appI_in_gggga(T239, T240, T241, T242))

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

POL(.(x₁, x₂)) = 1 + x₂   
POL(PARSEA_IN_GA(x₁)) = x₁   
POL(PD_IN_AAAAGAA(x₁)) = x₁   
POL(PO_IN_GGGGAA(x₁, x₂, x₃, x₄)) = 1 + x₁ + x₄   
POL(U18_AAAAGAA(x₁, x₂)) = x₂   
POL(U20_GGGGAA(x₁, x₂, x₃, x₄, x₅)) = x₅   
POL(U7_aaaag(x₁, x₂, x₃)) = 1 + x₃   
POL(U8_gggga(x₁, x₂, x₃, x₄, x₅, x₆)) = 1 + x₆   
POL([]) = 1   
POL(a) = 0   
POL(appH_in_aaaag(x₁)) = x₁   
POL(appH_out_aaaag(x₁, x₂, x₃, x₄, x₅)) = 1 + x₁ + x₄   
POL(appI_in_gggga(x₁, x₂, x₃, x₄)) = x₁ + x₄   
POL(appI_out_gggga(x₁, x₂, x₃, x₄, x₅)) = x₅   
POL(b) = 0   
POL(s(x₁, x₂)) = 0   
POL(s(x₁, x₂, x₃)) = 0   

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

appH_in_aaaag(.(a, .(s(T261, T262), .(b, T263)))) → appH_out_aaaag([], T261, T262, T263, .(a, .(s(T261, T262), .(b, T263))))
appH_in_aaaag(.(T268, T269)) → U7_aaaag(T268, T269, appH_in_aaaag(T269))
appI_in_gggga([], T301, T302, T303) → appI_out_gggga([], T301, T302, T303, .(s(a, s(T301, T302), b), T303))
appI_in_gggga(.(T314, T315), T316, T317, T318) → U8_gggga(T314, T315, T316, T317, T318, appI_in_gggga(T315, T316, T317, T318))
U8_gggga(T314, T315, T316, T317, T318, appI_out_gggga(T315, T316, T317, T318, X388)) → appI_out_gggga(.(T314, T315), T316, T317, T318, .(T314, X388))
U7_aaaag(T268, T269, appH_out_aaaag(X353, X354, X355, X356, T269)) → appH_out_aaaag(.(T268, X353), X354, X355, X356, .(T268, T269))

(77) Obligation:

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

PARSEA_IN_GA(T230) → PD_IN_AAAAGAA(T230)
PD_IN_AAAAGAA(T230) → U18_AAAAGAA(T230, appH_in_aaaag(T230))
U18_AAAAGAA(T230, appH_out_aaaag(T239, T240, T241, T242, T230)) → PO_IN_GGGGAA(T239, T240, T241, T242)
U20_GGGGAA(T239, T240, T241, T242, appI_out_gggga(T239, T240, T241, T242, T282)) → PARSEA_IN_GA(T282)

The TRS R consists of the following rules:

appI_in_gggga([], T301, T302, T303) → appI_out_gggga([], T301, T302, T303, .(s(a, s(T301, T302), b), T303))
appI_in_gggga(.(T314, T315), T316, T317, T318) → U8_gggga(T314, T315, T316, T317, T318, appI_in_gggga(T315, T316, T317, T318))
U8_gggga(T314, T315, T316, T317, T318, appI_out_gggga(T315, T316, T317, T318, X388)) → appI_out_gggga(.(T314, T315), T316, T317, T318, .(T314, X388))
appH_in_aaaag(.(a, .(s(T261, T262), .(b, T263)))) → appH_out_aaaag([], T261, T262, T263, .(a, .(s(T261, T262), .(b, T263))))
appH_in_aaaag(.(T268, T269)) → U7_aaaag(T268, T269, appH_in_aaaag(T269))
U7_aaaag(T268, T269, appH_out_aaaag(X353, X354, X355, X356, T269)) → appH_out_aaaag(.(T268, X353), X354, X355, X356, .(T268, T269))

The set Q consists of the following terms:

appH_in_aaaag(x0)
appI_in_gggga(x0, x1, x2, x3)
U7_aaaag(x0, x1, x2)
U8_gggga(x0, x1, x2, x3, x4, x5)

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

(78) DependencyGraphProof (EQUIVALENT transformation)

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

(79) TRUE