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

(0) Obligation:

Clauses:

conf(X) :- ','(del2(X, Z), ','(del(U, Y, Z), conf(Y))).
del2(X, Y) :- ','(del(U, X, Z), del(V, Z, Y)).
del(X, .(X, T), T).
del(X, .(H, T), .(H, T1)) :- del(X, T, T1).
s2l(s(X), .(Y, Xs)) :- s2l(X, Xs).
s2l(0, []).
goal(X) :- ','(s2l(X, XS), conf(XS)).

Query: conf(g)

(1) PrologToPiTRSViaGraphTransformerProof (SOUND transformation)

Transformed Prolog program to (Pi-)TRS.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
3 [label="A: conf(T2)\n\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
6 [label="6: conf(T2), SCOPE: 1, CLAUSE: 0\n\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: ','(del2(T4, X3), ','(del(X4, X5, X3), conf(X5)))\n\nT4 is ground\nX3 is free\nX4 is free\nX5 is free", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: ','(del2(T4, X3), ','(del(X4, X5, X3), conf(X5))), SCOPE: 2, CLAUSE: 1\n\nT4 is ground\nX3 is free\nX4 is free\nX5 is free", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="B: ','(','(del(X23, T11, X24), del(X25, X24, X26)), ','(del(X4, X5, X26), conf(X5)))\n\nT11 is ground\nX4 is free\nX5 is free\nX26 is free\nX23 is free\nX24 is free\nX25 is free", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="C: del(X23, T11, X24)\n\nT11 is ground\nX23 is free\nX24 is free", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="K: ','(del(X25, T17, X26), ','(del(X4, X5, X26), conf(X5)))\n\nT17 is ground\nX4 is free\nX5 is free\nX26 is free\nX25 is free", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: del(X23, T11, X24), SCOPE: 3, CLAUSE: 2 | del(X23, T11, X24), SCOPE: 3, CLAUSE: 3\n\nT11 is ground\nX23 is free\nX24 is free", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: del(X23, T11, X24), SCOPE: 3, CLAUSE: 2\n\nT11 is ground\nX23 is free\nX24 is free", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: del(X23, T11, X24), SCOPE: 3, CLAUSE: 3\n\nT11 is ground\nX23 is free\nX24 is free", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: del(X70, T37, X71)\n\nT37 is ground\nX70 is free\nX71 is free", fontsize=16, style=filled, fillcolor=lightyellow];
39 [label="39: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
43 [label="43: del(X25, T17, X26)\n\nT17 is ground\nX26 is free\nX25 is free", fontsize=16, style=filled, fillcolor=lightyellow];
44 [label="L: ','(del(X4, X5, T47), conf(X5))\n\nT47 is ground\nX4 is free\nX5 is free", fontsize=16, style=filled, fillcolor=lightyellow];
52 [label="D: del(X4, X5, T47)\n\nT47 is ground\nX4 is free\nX5 is free", fontsize=16, style=filled, fillcolor=lightyellow];
53 [label="F: conf(T54)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
55 [label="55: del(X4, X5, T47), SCOPE: 4, CLAUSE: 2 | del(X4, X5, T47), SCOPE: 4, CLAUSE: 3\n\nT47 is ground\nX4 is free\nX5 is free", fontsize=16, style=filled, fillcolor=lightyellow];
58 [label="58: del(X4, X5, T47), SCOPE: 4, CLAUSE: 2\n\nT47 is ground\nX4 is free\nX5 is free", fontsize=16, style=filled, fillcolor=lightyellow];
59 [label="59: del(X4, X5, T47), SCOPE: 4, CLAUSE: 3\n\nT47 is ground\nX4 is free\nX5 is free", fontsize=16, style=filled, fillcolor=lightyellow];
62 [label="62: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
63 [label="63: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
68 [label="68: del(X141, X143, T66)\n\nT66 is ground\nX141 is free\nX143 is free", fontsize=16, style=filled, fillcolor=lightyellow];
69 [label="69: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
73 [label="73: conf(T54), SCOPE: 5, CLAUSE: 0\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
75 [label="G: ','(del2(T74, X164), ','(del(X165, X166, X164), conf(X166)))\n\nX164 is free\nX165 is free\nX166 is free", fontsize=16, style=filled, fillcolor=lightyellow];
77 [label="I: del2(T74, X164)\n\nX164 is free", fontsize=16, style=filled, fillcolor=lightyellow];
79 [label="M: ','(del(X165, X166, T76), conf(X166))\n\nX165 is free\nX166 is free", fontsize=16, style=filled, fillcolor=lightyellow];
82 [label="82: del2(T74, X164), SCOPE: 6, CLAUSE: 1\n\nX164 is free", fontsize=16, style=filled, fillcolor=lightyellow];
88 [label="J: ','(del(X196, T86, X197), del(X198, X197, X199))\n\nX199 is free\nX196 is free\nX197 is free\nX198 is free", fontsize=16, style=filled, fillcolor=lightyellow];
91 [label="91: del(X196, T86, X197)\n\nX196 is free\nX197 is free", fontsize=16, style=filled, fillcolor=lightyellow];
92 [label="E: del(X198, T92, X199)\n\nX199 is free\nX198 is free", fontsize=16, style=filled, fillcolor=lightyellow];
94 [label="94: del(X198, T92, X199), SCOPE: 7, CLAUSE: 2 | del(X198, T92, X199), SCOPE: 7, CLAUSE: 3\n\nX199 is free\nX198 is free", fontsize=16, style=filled, fillcolor=lightyellow];
99 [label="99: del(X198, T92, X199), SCOPE: 7, CLAUSE: 2\n\nX199 is free\nX198 is free", fontsize=16, style=filled, fillcolor=lightyellow];
100 [label="100: del(X198, T92, X199), SCOPE: 7, CLAUSE: 3\n\nX199 is free\nX198 is free", fontsize=16, style=filled, fillcolor=lightyellow];
105 [label="105: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
106 [label="106: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
107 [label="107: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
110 [label="110: del(X249, T117, X250)\n\nX249 is free\nX250 is free", fontsize=16, style=filled, fillcolor=lightyellow];
111 [label="111: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
115 [label="H: del(X165, X166, T76)\n\nX165 is free\nX166 is free", fontsize=16, style=filled, fillcolor=lightyellow];
116 [label="116: conf(T124)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
121 [label="121: del(X165, X166, T76), SCOPE: 8, CLAUSE: 2 | del(X165, X166, T76), SCOPE: 8, CLAUSE: 3\n\nX165 is free\nX166 is free", fontsize=16, style=filled, fillcolor=lightyellow];
124 [label="124: del(X165, X166, T76), SCOPE: 8, CLAUSE: 2\n\nX165 is free\nX166 is free", fontsize=16, style=filled, fillcolor=lightyellow];
125 [label="125: del(X165, X166, T76), SCOPE: 8, CLAUSE: 3\n\nX165 is free\nX166 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];
132 [label="132: del(X308, X310, T137)\n\nX308 is free\nX310 is free", fontsize=16, style=filled, fillcolor=lightyellow];
134 [label="134: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
135 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
3 -> 135 [arrowhead = none ];
135 -> 6;

136 [label="ONLY EVAL with clause\nconf(X2) :- ','(del2(X2, X3), ','(del(X4, X5, X3), conf(X5))).\nand substitutionT2 -> T4,\nX2 -> T4", fontsize=14, style = filled, fillcolor = lightblue];
6 -> 136 [arrowhead = none ];
136 -> 7;

137 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
7 -> 137 [arrowhead = none ];
137 -> 11;

138 [label="ONLY EVAL with clause\ndel2(X21, X22) :- ','(del(X23, X21, X24), del(X25, X24, X22)).\nand substitutionT4 -> T11,\nX21 -> T11,\nX3 -> X26,\nX22 -> X26", fontsize=14, style = filled, fillcolor = lightblue];
11 -> 138 [arrowhead = none ];
138 -> 13;

139 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
13 -> 139 [arrowhead = none ];
139 -> 15;

140 [label="SPLIT 2\nnew knowledge:\nT16 is ground\nT11 is ground\nT17 is ground\nreplacements:X23 -> T16,\nX24 -> T17", fontsize=14, style = filled, fillcolor = violet];
13 -> 140 [arrowhead = none ];
140 -> 17;

141 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
15 -> 141 [arrowhead = none ];
141 -> 22;

142 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
17 -> 142 [arrowhead = none ];
142 -> 43;

143 [label="SPLIT 2\nnew knowledge:\nT46 is ground\nT17 is ground\nT47 is ground\nreplacements:X25 -> T46,\nX26 -> T47", fontsize=14, style = filled, fillcolor = violet];
17 -> 143 [arrowhead = none ];
143 -> 44;

144 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
22 -> 144 [arrowhead = none ];
144 -> 24;

145 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
22 -> 145 [arrowhead = none ];
145 -> 26;

146 [label="EVAL with clause\ndel(X51, .(X51, X52), X52).\nand substitutionX23 -> T30,\nX51 -> T30,\nX53 -> T30,\nX52 -> T31,\nT11 -> .(T30, T31),\nX24 -> T31", fontsize=14, style = filled, fillcolor = lightblue];
24 -> 146 [arrowhead = none ];
146 -> 30;

147 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
24 -> 147 [arrowhead = none ];
147 -> 31;

148 [label="EVAL with clause\ndel(X66, .(X67, X68), .(X67, X69)) :- del(X66, X68, X69).\nand substitutionX23 -> X70,\nX66 -> X70,\nX67 -> T36,\nX68 -> T37,\nT11 -> .(T36, T37),\nX69 -> X71,\nX24 -> .(T36, X71)", fontsize=14, style = filled, fillcolor = lightblue];
26 -> 148 [arrowhead = none ];
148 -> 37;

149 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
26 -> 149 [arrowhead = none ];
149 -> 39;

150 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
30 -> 150 [arrowhead = none ];
150 -> 34;

151 [label="INSTANCE with matching:\nX23 -> X70\nT11 -> T37\nX24 -> X71", fontsize=14, style = filled, fillcolor = lightgrey];
37 -> 151 [arrowhead = none , style=dashed];
151 -> 15[style=dashed];

152 [label="INSTANCE with matching:\nX23 -> X25\nT11 -> T17\nX24 -> X26", fontsize=14, style = filled, fillcolor = lightgrey];
43 -> 152 [arrowhead = none , style=dashed];
152 -> 15[style=dashed];

153 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
44 -> 153 [arrowhead = none ];
153 -> 52;

154 [label="SPLIT 2\nnew knowledge:\nT47 is ground\nreplacements:X4 -> T53,\nX5 -> T54", fontsize=14, style = filled, fillcolor = violet];
44 -> 154 [arrowhead = none ];
154 -> 53;

155 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
52 -> 155 [arrowhead = none ];
155 -> 55;

156 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
53 -> 156 [arrowhead = none ];
156 -> 73;

157 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
55 -> 157 [arrowhead = none ];
157 -> 58;

158 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
55 -> 158 [arrowhead = none ];
158 -> 59;

159 [label="ONLY EVAL with clause\ndel(X119, .(X119, X120), X120).\nand substitutionX4 -> X121,\nX119 -> X121,\nX120 -> T60,\nX5 -> .(X121, T60),\nT47 -> T60,\nX122 -> T60", fontsize=14, style = filled, fillcolor = lightblue];
58 -> 159 [arrowhead = none ];
159 -> 62;

160 [label="EVAL with clause\ndel(X137, .(X138, X139), .(X138, X140)) :- del(X137, X139, X140).\nand substitutionX4 -> X141,\nX137 -> X141,\nX138 -> T65,\nX139 -> X143,\nX5 -> .(T65, X143),\nX142 -> T65,\nX140 -> T66,\nT47 -> .(T65, T66)", fontsize=14, style = filled, fillcolor = lightblue];
59 -> 160 [arrowhead = none ];
160 -> 68;

161 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
59 -> 161 [arrowhead = none ];
161 -> 69;

162 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
62 -> 162 [arrowhead = none ];
162 -> 63;

163 [label="INSTANCE with matching:\nX4 -> X141\nX5 -> X143\nT47 -> T66", fontsize=14, style = filled, fillcolor = lightgrey];
68 -> 163 [arrowhead = none , style=dashed];
163 -> 52[style=dashed];

164 [label="ONLY EVAL with clause\nconf(X163) :- ','(del2(X163, X164), ','(del(X165, X166, X164), conf(X166))).\nand substitutionT54 -> T74,\nX163 -> T74,\nT73 -> T74", fontsize=14, style = filled, fillcolor = lightblue];
73 -> 164 [arrowhead = none ];
164 -> 75;

165 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
75 -> 165 [arrowhead = none ];
165 -> 77;

166 [label="SPLIT 2\nreplacements:X164 -> T76", fontsize=14, style = filled, fillcolor = violet];
75 -> 166 [arrowhead = none ];
166 -> 79;

167 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
77 -> 167 [arrowhead = none ];
167 -> 82;

168 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
79 -> 168 [arrowhead = none ];
168 -> 115;

169 [label="SPLIT 2\nreplacements:X165 -> T123,\nX166 -> T124", fontsize=14, style = filled, fillcolor = violet];
79 -> 169 [arrowhead = none ];
169 -> 116;

170 [label="ONLY EVAL with clause\ndel2(X194, X195) :- ','(del(X196, X194, X197), del(X198, X197, X195)).\nand substitutionT74 -> T86,\nX194 -> T86,\nX164 -> X199,\nX195 -> X199,\nT85 -> T86", fontsize=14, style = filled, fillcolor = lightblue];
82 -> 170 [arrowhead = none ];
170 -> 88;

171 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
88 -> 171 [arrowhead = none ];
171 -> 91;

172 [label="SPLIT 2\nreplacements:X196 -> T91,\nX197 -> T92", fontsize=14, style = filled, fillcolor = violet];
88 -> 172 [arrowhead = none ];
172 -> 92;

173 [label="INSTANCE with matching:\nX198 -> X196\nT92 -> T86\nX199 -> X197", fontsize=14, style = filled, fillcolor = lightgrey];
91 -> 173 [arrowhead = none , style=dashed];
173 -> 92[style=dashed];

174 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
92 -> 174 [arrowhead = none ];
174 -> 94;

175 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
94 -> 175 [arrowhead = none ];
175 -> 99;

176 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
94 -> 176 [arrowhead = none ];
176 -> 100;

177 [label="EVAL with clause\ndel(X230, .(X230, X231), X231).\nand substitutionX198 -> T109,\nX230 -> T109,\nX232 -> T109,\nX231 -> T110,\nT92 -> .(T109, T110),\nX199 -> T110", fontsize=14, style = filled, fillcolor = lightblue];
99 -> 177 [arrowhead = none ];
177 -> 105;

178 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
99 -> 178 [arrowhead = none ];
178 -> 106;

179 [label="EVAL with clause\ndel(X245, .(X246, X247), .(X246, X248)) :- del(X245, X247, X248).\nand substitutionX198 -> X249,\nX245 -> X249,\nX246 -> T115,\nX247 -> T117,\nT92 -> .(T115, T117),\nX248 -> X250,\nX199 -> .(T115, X250),\nT116 -> T117", fontsize=14, style = filled, fillcolor = lightblue];
100 -> 179 [arrowhead = none ];
179 -> 110;

180 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
100 -> 180 [arrowhead = none ];
180 -> 111;

181 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
105 -> 181 [arrowhead = none ];
181 -> 107;

182 [label="INSTANCE with matching:\nX198 -> X249\nT92 -> T117\nX199 -> X250", fontsize=14, style = filled, fillcolor = lightgrey];
110 -> 182 [arrowhead = none , style=dashed];
182 -> 92[style=dashed];

183 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
115 -> 183 [arrowhead = none ];
183 -> 121;

184 [label="INSTANCE with matching:\nT54 -> T124", fontsize=14, style = filled, fillcolor = lightgrey];
116 -> 184 [arrowhead = none , style=dashed];
184 -> 53[style=dashed];

185 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
121 -> 185 [arrowhead = none ];
185 -> 124;

186 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
121 -> 186 [arrowhead = none ];
186 -> 125;

187 [label="ONLY EVAL with clause\ndel(X286, .(X286, X287), X287).\nand substitutionX165 -> X288,\nX286 -> X288,\nX287 -> T130,\nX166 -> .(X288, T130),\nT76 -> T130,\nX289 -> T130", fontsize=14, style = filled, fillcolor = lightblue];
124 -> 187 [arrowhead = none ];
187 -> 128;

188 [label="EVAL with clause\ndel(X304, .(X305, X306), .(X305, X307)) :- del(X304, X306, X307).\nand substitutionX165 -> X308,\nX304 -> X308,\nX305 -> T135,\nX306 -> X310,\nX166 -> .(T135, X310),\nX309 -> T135,\nX307 -> T137,\nT76 -> .(T135, T137),\nT136 -> T137", fontsize=14, style = filled, fillcolor = lightblue];
125 -> 188 [arrowhead = none ];
188 -> 132;

189 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
125 -> 189 [arrowhead = none ];
189 -> 134;

190 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
128 -> 190 [arrowhead = none ];
190 -> 129;

191 [label="INSTANCE with matching:\nX165 -> X308\nX166 -> X310\nT76 -> T137", fontsize=14, style = filled, fillcolor = lightgrey];
132 -> 191 [arrowhead = none , style=dashed];
191 -> 115[style=dashed];

}

(2) Obligation:

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

confA_in_g(T11) → U1_g(T11, pB_in_agaaaaa(X23, T11, X24, X25, X26, X4, X5))
pB_in_agaaaaa(T16, T11, T17, X25, X26, X4, X5) → U8_agaaaaa(T16, T11, T17, X25, X26, X4, X5, delC_in_aga(T16, T11, T17))
delC_in_aga(T30, .(T30, T31), T31) → delC_out_aga(T30, .(T30, T31), T31)
delC_in_aga(X70, .(T36, T37), .(T36, X71)) → U2_aga(X70, T36, T37, X71, delC_in_aga(X70, T37, X71))
U2_aga(X70, T36, T37, X71, delC_out_aga(X70, T37, X71)) → delC_out_aga(X70, .(T36, T37), .(T36, X71))
U8_agaaaaa(T16, T11, T17, X25, X26, X4, X5, delC_out_aga(T16, T11, T17)) → U9_agaaaaa(T16, T11, T17, X25, X26, X4, X5, pK_in_agaaa(X25, T17, X26, X4, X5))
pK_in_agaaa(T46, T17, T47, X4, X5) → U10_agaaa(T46, T17, T47, X4, X5, delC_in_aga(T46, T17, T47))
U10_agaaa(T46, T17, T47, X4, X5, delC_out_aga(T46, T17, T47)) → U11_agaaa(T46, T17, T47, X4, X5, pL_in_aag(X4, X5, T47))
pL_in_aag(T53, T54, T47) → U12_aag(T53, T54, T47, delD_in_aag(T53, T54, T47))
delD_in_aag(X121, .(X121, T60), T60) → delD_out_aag(X121, .(X121, T60), T60)
delD_in_aag(X141, .(T65, X143), .(T65, T66)) → U3_aag(X141, T65, X143, T66, delD_in_aag(X141, X143, T66))
U3_aag(X141, T65, X143, T66, delD_out_aag(X141, X143, T66)) → delD_out_aag(X141, .(T65, X143), .(T65, T66))
U12_aag(T53, T54, T47, delD_out_aag(T53, T54, T47)) → U13_aag(T53, T54, T47, confF_in_g(T54))
confF_in_g(T74) → U5_g(T74, pG_in_gaaa(T74, X164, X165, X166))
pG_in_gaaa(T74, T76, X165, X166) → U14_gaaa(T74, T76, X165, X166, del2I_in_ga(T74, T76))
del2I_in_ga(T86, X199) → U7_ga(T86, X199, pJ_in_agaaa(X196, T86, X197, X198, X199))
pJ_in_agaaa(T91, T86, T92, X198, X199) → U18_agaaa(T91, T86, T92, X198, X199, delE_in_aga(T91, T86, T92))
delE_in_aga(T109, .(T109, T110), T110) → delE_out_aga(T109, .(T109, T110), T110)
delE_in_aga(X249, .(T115, T117), .(T115, X250)) → U4_aga(X249, T115, T117, X250, delE_in_aga(X249, T117, X250))
U4_aga(X249, T115, T117, X250, delE_out_aga(X249, T117, X250)) → delE_out_aga(X249, .(T115, T117), .(T115, X250))
U18_agaaa(T91, T86, T92, X198, X199, delE_out_aga(T91, T86, T92)) → U19_agaaa(T91, T86, T92, X198, X199, delE_in_aga(X198, T92, X199))
U19_agaaa(T91, T86, T92, X198, X199, delE_out_aga(X198, T92, X199)) → pJ_out_agaaa(T91, T86, T92, X198, X199)
U7_ga(T86, X199, pJ_out_agaaa(X196, T86, X197, X198, X199)) → del2I_out_ga(T86, X199)
U14_gaaa(T74, T76, X165, X166, del2I_out_ga(T74, T76)) → U15_gaaa(T74, T76, X165, X166, pM_in_aag(X165, X166, T76))
pM_in_aag(T123, T124, T76) → U16_aag(T123, T124, T76, delH_in_aag(T123, T124, T76))
delH_in_aag(X288, .(X288, T130), T130) → delH_out_aag(X288, .(X288, T130), T130)
delH_in_aag(X308, .(T135, X310), .(T135, T137)) → U6_aag(X308, T135, X310, T137, delH_in_aag(X308, X310, T137))
U6_aag(X308, T135, X310, T137, delH_out_aag(X308, X310, T137)) → delH_out_aag(X308, .(T135, X310), .(T135, T137))
U16_aag(T123, T124, T76, delH_out_aag(T123, T124, T76)) → U17_aag(T123, T124, T76, confF_in_g(T124))
U17_aag(T123, T124, T76, confF_out_g(T124)) → pM_out_aag(T123, T124, T76)
U15_gaaa(T74, T76, X165, X166, pM_out_aag(X165, X166, T76)) → pG_out_gaaa(T74, T76, X165, X166)
U5_g(T74, pG_out_gaaa(T74, X164, X165, X166)) → confF_out_g(T74)
U13_aag(T53, T54, T47, confF_out_g(T54)) → pL_out_aag(T53, T54, T47)
U11_agaaa(T46, T17, T47, X4, X5, pL_out_aag(X4, X5, T47)) → pK_out_agaaa(T46, T17, T47, X4, X5)
U9_agaaaaa(T16, T11, T17, X25, X26, X4, X5, pK_out_agaaa(X25, T17, X26, X4, X5)) → pB_out_agaaaaa(T16, T11, T17, X25, X26, X4, X5)
U1_g(T11, pB_out_agaaaaa(X23, T11, X24, X25, X26, X4, X5)) → confA_out_g(T11)

The argument filtering Pi contains the following mapping:
confA_in_g(x1)  =  confA_in_g(x1)
U1_g(x1, x2)  =  U1_g(x1, x2)
pB_in_agaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  pB_in_agaaaaa(x2)
U8_agaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U8_agaaaaa(x2, x8)
delC_in_aga(x1, x2, x3)  =  delC_in_aga(x2)
.(x1, x2)  =  .(x2)
delC_out_aga(x1, x2, x3)  =  delC_out_aga(x2, x3)
U2_aga(x1, x2, x3, x4, x5)  =  U2_aga(x3, x5)
U9_agaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U9_agaaaaa(x2, x3, x8)
pK_in_agaaa(x1, x2, x3, x4, x5)  =  pK_in_agaaa(x2)
U10_agaaa(x1, x2, x3, x4, x5, x6)  =  U10_agaaa(x2, x6)
U11_agaaa(x1, x2, x3, x4, x5, x6)  =  U11_agaaa(x2, x3, x6)
pL_in_aag(x1, x2, x3)  =  pL_in_aag(x3)
U12_aag(x1, x2, x3, x4)  =  U12_aag(x3, x4)
delD_in_aag(x1, x2, x3)  =  delD_in_aag(x3)
delD_out_aag(x1, x2, x3)  =  delD_out_aag(x2, x3)
U3_aag(x1, x2, x3, x4, x5)  =  U3_aag(x4, x5)
U13_aag(x1, x2, x3, x4)  =  U13_aag(x2, x3, x4)
confF_in_g(x1)  =  confF_in_g(x1)
U5_g(x1, x2)  =  U5_g(x1, x2)
pG_in_gaaa(x1, x2, x3, x4)  =  pG_in_gaaa(x1)
U14_gaaa(x1, x2, x3, x4, x5)  =  U14_gaaa(x1, x5)
del2I_in_ga(x1, x2)  =  del2I_in_ga(x1)
U7_ga(x1, x2, x3)  =  U7_ga(x1, x3)
pJ_in_agaaa(x1, x2, x3, x4, x5)  =  pJ_in_agaaa(x2)
U18_agaaa(x1, x2, x3, x4, x5, x6)  =  U18_agaaa(x2, x6)
delE_in_aga(x1, x2, x3)  =  delE_in_aga(x2)
delE_out_aga(x1, x2, x3)  =  delE_out_aga(x2, x3)
U4_aga(x1, x2, x3, x4, x5)  =  U4_aga(x3, x5)
U19_agaaa(x1, x2, x3, x4, x5, x6)  =  U19_agaaa(x2, x3, x6)
pJ_out_agaaa(x1, x2, x3, x4, x5)  =  pJ_out_agaaa(x2, x3, x5)
del2I_out_ga(x1, x2)  =  del2I_out_ga(x1, x2)
U15_gaaa(x1, x2, x3, x4, x5)  =  U15_gaaa(x1, x2, x5)
pM_in_aag(x1, x2, x3)  =  pM_in_aag(x3)
U16_aag(x1, x2, x3, x4)  =  U16_aag(x3, x4)
delH_in_aag(x1, x2, x3)  =  delH_in_aag(x3)
delH_out_aag(x1, x2, x3)  =  delH_out_aag(x2, x3)
U6_aag(x1, x2, x3, x4, x5)  =  U6_aag(x4, x5)
U17_aag(x1, x2, x3, x4)  =  U17_aag(x2, x3, x4)
confF_out_g(x1)  =  confF_out_g(x1)
pM_out_aag(x1, x2, x3)  =  pM_out_aag(x2, x3)
pG_out_gaaa(x1, x2, x3, x4)  =  pG_out_gaaa(x1, x2, x4)
pL_out_aag(x1, x2, x3)  =  pL_out_aag(x2, x3)
pK_out_agaaa(x1, x2, x3, x4, x5)  =  pK_out_agaaa(x2, x3, x5)
pB_out_agaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  pB_out_agaaaaa(x2, x3, x5, x7)
confA_out_g(x1)  =  confA_out_g(x1)

(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:

CONFA_IN_G(T11) → U1_G(T11, pB_in_agaaaaa(X23, T11, X24, X25, X26, X4, X5))
CONFA_IN_G(T11) → PB_IN_AGAAAAA(X23, T11, X24, X25, X26, X4, X5)
PB_IN_AGAAAAA(T16, T11, T17, X25, X26, X4, X5) → U8_AGAAAAA(T16, T11, T17, X25, X26, X4, X5, delC_in_aga(T16, T11, T17))
PB_IN_AGAAAAA(T16, T11, T17, X25, X26, X4, X5) → DELC_IN_AGA(T16, T11, T17)
DELC_IN_AGA(X70, .(T36, T37), .(T36, X71)) → U2_AGA(X70, T36, T37, X71, delC_in_aga(X70, T37, X71))
DELC_IN_AGA(X70, .(T36, T37), .(T36, X71)) → DELC_IN_AGA(X70, T37, X71)
U8_AGAAAAA(T16, T11, T17, X25, X26, X4, X5, delC_out_aga(T16, T11, T17)) → U9_AGAAAAA(T16, T11, T17, X25, X26, X4, X5, pK_in_agaaa(X25, T17, X26, X4, X5))
U8_AGAAAAA(T16, T11, T17, X25, X26, X4, X5, delC_out_aga(T16, T11, T17)) → PK_IN_AGAAA(X25, T17, X26, X4, X5)
PK_IN_AGAAA(T46, T17, T47, X4, X5) → U10_AGAAA(T46, T17, T47, X4, X5, delC_in_aga(T46, T17, T47))
PK_IN_AGAAA(T46, T17, T47, X4, X5) → DELC_IN_AGA(T46, T17, T47)
U10_AGAAA(T46, T17, T47, X4, X5, delC_out_aga(T46, T17, T47)) → U11_AGAAA(T46, T17, T47, X4, X5, pL_in_aag(X4, X5, T47))
U10_AGAAA(T46, T17, T47, X4, X5, delC_out_aga(T46, T17, T47)) → PL_IN_AAG(X4, X5, T47)
PL_IN_AAG(T53, T54, T47) → U12_AAG(T53, T54, T47, delD_in_aag(T53, T54, T47))
PL_IN_AAG(T53, T54, T47) → DELD_IN_AAG(T53, T54, T47)
DELD_IN_AAG(X141, .(T65, X143), .(T65, T66)) → U3_AAG(X141, T65, X143, T66, delD_in_aag(X141, X143, T66))
DELD_IN_AAG(X141, .(T65, X143), .(T65, T66)) → DELD_IN_AAG(X141, X143, T66)
U12_AAG(T53, T54, T47, delD_out_aag(T53, T54, T47)) → U13_AAG(T53, T54, T47, confF_in_g(T54))
U12_AAG(T53, T54, T47, delD_out_aag(T53, T54, T47)) → CONFF_IN_G(T54)
CONFF_IN_G(T74) → U5_G(T74, pG_in_gaaa(T74, X164, X165, X166))
CONFF_IN_G(T74) → PG_IN_GAAA(T74, X164, X165, X166)
PG_IN_GAAA(T74, T76, X165, X166) → U14_GAAA(T74, T76, X165, X166, del2I_in_ga(T74, T76))
PG_IN_GAAA(T74, T76, X165, X166) → DEL2I_IN_GA(T74, T76)
DEL2I_IN_GA(T86, X199) → U7_GA(T86, X199, pJ_in_agaaa(X196, T86, X197, X198, X199))
DEL2I_IN_GA(T86, X199) → PJ_IN_AGAAA(X196, T86, X197, X198, X199)
PJ_IN_AGAAA(T91, T86, T92, X198, X199) → U18_AGAAA(T91, T86, T92, X198, X199, delE_in_aga(T91, T86, T92))
PJ_IN_AGAAA(T91, T86, T92, X198, X199) → DELE_IN_AGA(T91, T86, T92)
DELE_IN_AGA(X249, .(T115, T117), .(T115, X250)) → U4_AGA(X249, T115, T117, X250, delE_in_aga(X249, T117, X250))
DELE_IN_AGA(X249, .(T115, T117), .(T115, X250)) → DELE_IN_AGA(X249, T117, X250)
U18_AGAAA(T91, T86, T92, X198, X199, delE_out_aga(T91, T86, T92)) → U19_AGAAA(T91, T86, T92, X198, X199, delE_in_aga(X198, T92, X199))
U18_AGAAA(T91, T86, T92, X198, X199, delE_out_aga(T91, T86, T92)) → DELE_IN_AGA(X198, T92, X199)
U14_GAAA(T74, T76, X165, X166, del2I_out_ga(T74, T76)) → U15_GAAA(T74, T76, X165, X166, pM_in_aag(X165, X166, T76))
U14_GAAA(T74, T76, X165, X166, del2I_out_ga(T74, T76)) → PM_IN_AAG(X165, X166, T76)
PM_IN_AAG(T123, T124, T76) → U16_AAG(T123, T124, T76, delH_in_aag(T123, T124, T76))
PM_IN_AAG(T123, T124, T76) → DELH_IN_AAG(T123, T124, T76)
DELH_IN_AAG(X308, .(T135, X310), .(T135, T137)) → U6_AAG(X308, T135, X310, T137, delH_in_aag(X308, X310, T137))
DELH_IN_AAG(X308, .(T135, X310), .(T135, T137)) → DELH_IN_AAG(X308, X310, T137)
U16_AAG(T123, T124, T76, delH_out_aag(T123, T124, T76)) → U17_AAG(T123, T124, T76, confF_in_g(T124))
U16_AAG(T123, T124, T76, delH_out_aag(T123, T124, T76)) → CONFF_IN_G(T124)

The TRS R consists of the following rules:

confA_in_g(T11) → U1_g(T11, pB_in_agaaaaa(X23, T11, X24, X25, X26, X4, X5))
pB_in_agaaaaa(T16, T11, T17, X25, X26, X4, X5) → U8_agaaaaa(T16, T11, T17, X25, X26, X4, X5, delC_in_aga(T16, T11, T17))
delC_in_aga(T30, .(T30, T31), T31) → delC_out_aga(T30, .(T30, T31), T31)
delC_in_aga(X70, .(T36, T37), .(T36, X71)) → U2_aga(X70, T36, T37, X71, delC_in_aga(X70, T37, X71))
U2_aga(X70, T36, T37, X71, delC_out_aga(X70, T37, X71)) → delC_out_aga(X70, .(T36, T37), .(T36, X71))
U8_agaaaaa(T16, T11, T17, X25, X26, X4, X5, delC_out_aga(T16, T11, T17)) → U9_agaaaaa(T16, T11, T17, X25, X26, X4, X5, pK_in_agaaa(X25, T17, X26, X4, X5))
pK_in_agaaa(T46, T17, T47, X4, X5) → U10_agaaa(T46, T17, T47, X4, X5, delC_in_aga(T46, T17, T47))
U10_agaaa(T46, T17, T47, X4, X5, delC_out_aga(T46, T17, T47)) → U11_agaaa(T46, T17, T47, X4, X5, pL_in_aag(X4, X5, T47))
pL_in_aag(T53, T54, T47) → U12_aag(T53, T54, T47, delD_in_aag(T53, T54, T47))
delD_in_aag(X121, .(X121, T60), T60) → delD_out_aag(X121, .(X121, T60), T60)
delD_in_aag(X141, .(T65, X143), .(T65, T66)) → U3_aag(X141, T65, X143, T66, delD_in_aag(X141, X143, T66))
U3_aag(X141, T65, X143, T66, delD_out_aag(X141, X143, T66)) → delD_out_aag(X141, .(T65, X143), .(T65, T66))
U12_aag(T53, T54, T47, delD_out_aag(T53, T54, T47)) → U13_aag(T53, T54, T47, confF_in_g(T54))
confF_in_g(T74) → U5_g(T74, pG_in_gaaa(T74, X164, X165, X166))
pG_in_gaaa(T74, T76, X165, X166) → U14_gaaa(T74, T76, X165, X166, del2I_in_ga(T74, T76))
del2I_in_ga(T86, X199) → U7_ga(T86, X199, pJ_in_agaaa(X196, T86, X197, X198, X199))
pJ_in_agaaa(T91, T86, T92, X198, X199) → U18_agaaa(T91, T86, T92, X198, X199, delE_in_aga(T91, T86, T92))
delE_in_aga(T109, .(T109, T110), T110) → delE_out_aga(T109, .(T109, T110), T110)
delE_in_aga(X249, .(T115, T117), .(T115, X250)) → U4_aga(X249, T115, T117, X250, delE_in_aga(X249, T117, X250))
U4_aga(X249, T115, T117, X250, delE_out_aga(X249, T117, X250)) → delE_out_aga(X249, .(T115, T117), .(T115, X250))
U18_agaaa(T91, T86, T92, X198, X199, delE_out_aga(T91, T86, T92)) → U19_agaaa(T91, T86, T92, X198, X199, delE_in_aga(X198, T92, X199))
U19_agaaa(T91, T86, T92, X198, X199, delE_out_aga(X198, T92, X199)) → pJ_out_agaaa(T91, T86, T92, X198, X199)
U7_ga(T86, X199, pJ_out_agaaa(X196, T86, X197, X198, X199)) → del2I_out_ga(T86, X199)
U14_gaaa(T74, T76, X165, X166, del2I_out_ga(T74, T76)) → U15_gaaa(T74, T76, X165, X166, pM_in_aag(X165, X166, T76))
pM_in_aag(T123, T124, T76) → U16_aag(T123, T124, T76, delH_in_aag(T123, T124, T76))
delH_in_aag(X288, .(X288, T130), T130) → delH_out_aag(X288, .(X288, T130), T130)
delH_in_aag(X308, .(T135, X310), .(T135, T137)) → U6_aag(X308, T135, X310, T137, delH_in_aag(X308, X310, T137))
U6_aag(X308, T135, X310, T137, delH_out_aag(X308, X310, T137)) → delH_out_aag(X308, .(T135, X310), .(T135, T137))
U16_aag(T123, T124, T76, delH_out_aag(T123, T124, T76)) → U17_aag(T123, T124, T76, confF_in_g(T124))
U17_aag(T123, T124, T76, confF_out_g(T124)) → pM_out_aag(T123, T124, T76)
U15_gaaa(T74, T76, X165, X166, pM_out_aag(X165, X166, T76)) → pG_out_gaaa(T74, T76, X165, X166)
U5_g(T74, pG_out_gaaa(T74, X164, X165, X166)) → confF_out_g(T74)
U13_aag(T53, T54, T47, confF_out_g(T54)) → pL_out_aag(T53, T54, T47)
U11_agaaa(T46, T17, T47, X4, X5, pL_out_aag(X4, X5, T47)) → pK_out_agaaa(T46, T17, T47, X4, X5)
U9_agaaaaa(T16, T11, T17, X25, X26, X4, X5, pK_out_agaaa(X25, T17, X26, X4, X5)) → pB_out_agaaaaa(T16, T11, T17, X25, X26, X4, X5)
U1_g(T11, pB_out_agaaaaa(X23, T11, X24, X25, X26, X4, X5)) → confA_out_g(T11)

The argument filtering Pi contains the following mapping:
confA_in_g(x1)  =  confA_in_g(x1)
U1_g(x1, x2)  =  U1_g(x1, x2)
pB_in_agaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  pB_in_agaaaaa(x2)
U8_agaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U8_agaaaaa(x2, x8)
delC_in_aga(x1, x2, x3)  =  delC_in_aga(x2)
.(x1, x2)  =  .(x2)
delC_out_aga(x1, x2, x3)  =  delC_out_aga(x2, x3)
U2_aga(x1, x2, x3, x4, x5)  =  U2_aga(x3, x5)
U9_agaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U9_agaaaaa(x2, x3, x8)
pK_in_agaaa(x1, x2, x3, x4, x5)  =  pK_in_agaaa(x2)
U10_agaaa(x1, x2, x3, x4, x5, x6)  =  U10_agaaa(x2, x6)
U11_agaaa(x1, x2, x3, x4, x5, x6)  =  U11_agaaa(x2, x3, x6)
pL_in_aag(x1, x2, x3)  =  pL_in_aag(x3)
U12_aag(x1, x2, x3, x4)  =  U12_aag(x3, x4)
delD_in_aag(x1, x2, x3)  =  delD_in_aag(x3)
delD_out_aag(x1, x2, x3)  =  delD_out_aag(x2, x3)
U3_aag(x1, x2, x3, x4, x5)  =  U3_aag(x4, x5)
U13_aag(x1, x2, x3, x4)  =  U13_aag(x2, x3, x4)
confF_in_g(x1)  =  confF_in_g(x1)
U5_g(x1, x2)  =  U5_g(x1, x2)
pG_in_gaaa(x1, x2, x3, x4)  =  pG_in_gaaa(x1)
U14_gaaa(x1, x2, x3, x4, x5)  =  U14_gaaa(x1, x5)
del2I_in_ga(x1, x2)  =  del2I_in_ga(x1)
U7_ga(x1, x2, x3)  =  U7_ga(x1, x3)
pJ_in_agaaa(x1, x2, x3, x4, x5)  =  pJ_in_agaaa(x2)
U18_agaaa(x1, x2, x3, x4, x5, x6)  =  U18_agaaa(x2, x6)
delE_in_aga(x1, x2, x3)  =  delE_in_aga(x2)
delE_out_aga(x1, x2, x3)  =  delE_out_aga(x2, x3)
U4_aga(x1, x2, x3, x4, x5)  =  U4_aga(x3, x5)
U19_agaaa(x1, x2, x3, x4, x5, x6)  =  U19_agaaa(x2, x3, x6)
pJ_out_agaaa(x1, x2, x3, x4, x5)  =  pJ_out_agaaa(x2, x3, x5)
del2I_out_ga(x1, x2)  =  del2I_out_ga(x1, x2)
U15_gaaa(x1, x2, x3, x4, x5)  =  U15_gaaa(x1, x2, x5)
pM_in_aag(x1, x2, x3)  =  pM_in_aag(x3)
U16_aag(x1, x2, x3, x4)  =  U16_aag(x3, x4)
delH_in_aag(x1, x2, x3)  =  delH_in_aag(x3)
delH_out_aag(x1, x2, x3)  =  delH_out_aag(x2, x3)
U6_aag(x1, x2, x3, x4, x5)  =  U6_aag(x4, x5)
U17_aag(x1, x2, x3, x4)  =  U17_aag(x2, x3, x4)
confF_out_g(x1)  =  confF_out_g(x1)
pM_out_aag(x1, x2, x3)  =  pM_out_aag(x2, x3)
pG_out_gaaa(x1, x2, x3, x4)  =  pG_out_gaaa(x1, x2, x4)
pL_out_aag(x1, x2, x3)  =  pL_out_aag(x2, x3)
pK_out_agaaa(x1, x2, x3, x4, x5)  =  pK_out_agaaa(x2, x3, x5)
pB_out_agaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  pB_out_agaaaaa(x2, x3, x5, x7)
confA_out_g(x1)  =  confA_out_g(x1)
CONFA_IN_G(x1)  =  CONFA_IN_G(x1)
U1_G(x1, x2)  =  U1_G(x1, x2)
PB_IN_AGAAAAA(x1, x2, x3, x4, x5, x6, x7)  =  PB_IN_AGAAAAA(x2)
U8_AGAAAAA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U8_AGAAAAA(x2, x8)
DELC_IN_AGA(x1, x2, x3)  =  DELC_IN_AGA(x2)
U2_AGA(x1, x2, x3, x4, x5)  =  U2_AGA(x3, x5)
U9_AGAAAAA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U9_AGAAAAA(x2, x3, x8)
PK_IN_AGAAA(x1, x2, x3, x4, x5)  =  PK_IN_AGAAA(x2)
U10_AGAAA(x1, x2, x3, x4, x5, x6)  =  U10_AGAAA(x2, x6)
U11_AGAAA(x1, x2, x3, x4, x5, x6)  =  U11_AGAAA(x2, x3, x6)
PL_IN_AAG(x1, x2, x3)  =  PL_IN_AAG(x3)
U12_AAG(x1, x2, x3, x4)  =  U12_AAG(x3, x4)
DELD_IN_AAG(x1, x2, x3)  =  DELD_IN_AAG(x3)
U3_AAG(x1, x2, x3, x4, x5)  =  U3_AAG(x4, x5)
U13_AAG(x1, x2, x3, x4)  =  U13_AAG(x2, x3, x4)
CONFF_IN_G(x1)  =  CONFF_IN_G(x1)
U5_G(x1, x2)  =  U5_G(x1, x2)
PG_IN_GAAA(x1, x2, x3, x4)  =  PG_IN_GAAA(x1)
U14_GAAA(x1, x2, x3, x4, x5)  =  U14_GAAA(x1, x5)
DEL2I_IN_GA(x1, x2)  =  DEL2I_IN_GA(x1)
U7_GA(x1, x2, x3)  =  U7_GA(x1, x3)
PJ_IN_AGAAA(x1, x2, x3, x4, x5)  =  PJ_IN_AGAAA(x2)
U18_AGAAA(x1, x2, x3, x4, x5, x6)  =  U18_AGAAA(x2, x6)
DELE_IN_AGA(x1, x2, x3)  =  DELE_IN_AGA(x2)
U4_AGA(x1, x2, x3, x4, x5)  =  U4_AGA(x3, x5)
U19_AGAAA(x1, x2, x3, x4, x5, x6)  =  U19_AGAAA(x2, x3, x6)
U15_GAAA(x1, x2, x3, x4, x5)  =  U15_GAAA(x1, x2, x5)
PM_IN_AAG(x1, x2, x3)  =  PM_IN_AAG(x3)
U16_AAG(x1, x2, x3, x4)  =  U16_AAG(x3, x4)
DELH_IN_AAG(x1, x2, x3)  =  DELH_IN_AAG(x3)
U6_AAG(x1, x2, x3, x4, x5)  =  U6_AAG(x4, x5)
U17_AAG(x1, x2, x3, x4)  =  U17_AAG(x2, x3, x4)

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

(4) Obligation:

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

CONFA_IN_G(T11) → U1_G(T11, pB_in_agaaaaa(X23, T11, X24, X25, X26, X4, X5))
CONFA_IN_G(T11) → PB_IN_AGAAAAA(X23, T11, X24, X25, X26, X4, X5)
PB_IN_AGAAAAA(T16, T11, T17, X25, X26, X4, X5) → U8_AGAAAAA(T16, T11, T17, X25, X26, X4, X5, delC_in_aga(T16, T11, T17))
PB_IN_AGAAAAA(T16, T11, T17, X25, X26, X4, X5) → DELC_IN_AGA(T16, T11, T17)
DELC_IN_AGA(X70, .(T36, T37), .(T36, X71)) → U2_AGA(X70, T36, T37, X71, delC_in_aga(X70, T37, X71))
DELC_IN_AGA(X70, .(T36, T37), .(T36, X71)) → DELC_IN_AGA(X70, T37, X71)
U8_AGAAAAA(T16, T11, T17, X25, X26, X4, X5, delC_out_aga(T16, T11, T17)) → U9_AGAAAAA(T16, T11, T17, X25, X26, X4, X5, pK_in_agaaa(X25, T17, X26, X4, X5))
U8_AGAAAAA(T16, T11, T17, X25, X26, X4, X5, delC_out_aga(T16, T11, T17)) → PK_IN_AGAAA(X25, T17, X26, X4, X5)
PK_IN_AGAAA(T46, T17, T47, X4, X5) → U10_AGAAA(T46, T17, T47, X4, X5, delC_in_aga(T46, T17, T47))
PK_IN_AGAAA(T46, T17, T47, X4, X5) → DELC_IN_AGA(T46, T17, T47)
U10_AGAAA(T46, T17, T47, X4, X5, delC_out_aga(T46, T17, T47)) → U11_AGAAA(T46, T17, T47, X4, X5, pL_in_aag(X4, X5, T47))
U10_AGAAA(T46, T17, T47, X4, X5, delC_out_aga(T46, T17, T47)) → PL_IN_AAG(X4, X5, T47)
PL_IN_AAG(T53, T54, T47) → U12_AAG(T53, T54, T47, delD_in_aag(T53, T54, T47))
PL_IN_AAG(T53, T54, T47) → DELD_IN_AAG(T53, T54, T47)
DELD_IN_AAG(X141, .(T65, X143), .(T65, T66)) → U3_AAG(X141, T65, X143, T66, delD_in_aag(X141, X143, T66))
DELD_IN_AAG(X141, .(T65, X143), .(T65, T66)) → DELD_IN_AAG(X141, X143, T66)
U12_AAG(T53, T54, T47, delD_out_aag(T53, T54, T47)) → U13_AAG(T53, T54, T47, confF_in_g(T54))
U12_AAG(T53, T54, T47, delD_out_aag(T53, T54, T47)) → CONFF_IN_G(T54)
CONFF_IN_G(T74) → U5_G(T74, pG_in_gaaa(T74, X164, X165, X166))
CONFF_IN_G(T74) → PG_IN_GAAA(T74, X164, X165, X166)
PG_IN_GAAA(T74, T76, X165, X166) → U14_GAAA(T74, T76, X165, X166, del2I_in_ga(T74, T76))
PG_IN_GAAA(T74, T76, X165, X166) → DEL2I_IN_GA(T74, T76)
DEL2I_IN_GA(T86, X199) → U7_GA(T86, X199, pJ_in_agaaa(X196, T86, X197, X198, X199))
DEL2I_IN_GA(T86, X199) → PJ_IN_AGAAA(X196, T86, X197, X198, X199)
PJ_IN_AGAAA(T91, T86, T92, X198, X199) → U18_AGAAA(T91, T86, T92, X198, X199, delE_in_aga(T91, T86, T92))
PJ_IN_AGAAA(T91, T86, T92, X198, X199) → DELE_IN_AGA(T91, T86, T92)
DELE_IN_AGA(X249, .(T115, T117), .(T115, X250)) → U4_AGA(X249, T115, T117, X250, delE_in_aga(X249, T117, X250))
DELE_IN_AGA(X249, .(T115, T117), .(T115, X250)) → DELE_IN_AGA(X249, T117, X250)
U18_AGAAA(T91, T86, T92, X198, X199, delE_out_aga(T91, T86, T92)) → U19_AGAAA(T91, T86, T92, X198, X199, delE_in_aga(X198, T92, X199))
U18_AGAAA(T91, T86, T92, X198, X199, delE_out_aga(T91, T86, T92)) → DELE_IN_AGA(X198, T92, X199)
U14_GAAA(T74, T76, X165, X166, del2I_out_ga(T74, T76)) → U15_GAAA(T74, T76, X165, X166, pM_in_aag(X165, X166, T76))
U14_GAAA(T74, T76, X165, X166, del2I_out_ga(T74, T76)) → PM_IN_AAG(X165, X166, T76)
PM_IN_AAG(T123, T124, T76) → U16_AAG(T123, T124, T76, delH_in_aag(T123, T124, T76))
PM_IN_AAG(T123, T124, T76) → DELH_IN_AAG(T123, T124, T76)
DELH_IN_AAG(X308, .(T135, X310), .(T135, T137)) → U6_AAG(X308, T135, X310, T137, delH_in_aag(X308, X310, T137))
DELH_IN_AAG(X308, .(T135, X310), .(T135, T137)) → DELH_IN_AAG(X308, X310, T137)
U16_AAG(T123, T124, T76, delH_out_aag(T123, T124, T76)) → U17_AAG(T123, T124, T76, confF_in_g(T124))
U16_AAG(T123, T124, T76, delH_out_aag(T123, T124, T76)) → CONFF_IN_G(T124)

The TRS R consists of the following rules:

confA_in_g(T11) → U1_g(T11, pB_in_agaaaaa(X23, T11, X24, X25, X26, X4, X5))
pB_in_agaaaaa(T16, T11, T17, X25, X26, X4, X5) → U8_agaaaaa(T16, T11, T17, X25, X26, X4, X5, delC_in_aga(T16, T11, T17))
delC_in_aga(T30, .(T30, T31), T31) → delC_out_aga(T30, .(T30, T31), T31)
delC_in_aga(X70, .(T36, T37), .(T36, X71)) → U2_aga(X70, T36, T37, X71, delC_in_aga(X70, T37, X71))
U2_aga(X70, T36, T37, X71, delC_out_aga(X70, T37, X71)) → delC_out_aga(X70, .(T36, T37), .(T36, X71))
U8_agaaaaa(T16, T11, T17, X25, X26, X4, X5, delC_out_aga(T16, T11, T17)) → U9_agaaaaa(T16, T11, T17, X25, X26, X4, X5, pK_in_agaaa(X25, T17, X26, X4, X5))
pK_in_agaaa(T46, T17, T47, X4, X5) → U10_agaaa(T46, T17, T47, X4, X5, delC_in_aga(T46, T17, T47))
U10_agaaa(T46, T17, T47, X4, X5, delC_out_aga(T46, T17, T47)) → U11_agaaa(T46, T17, T47, X4, X5, pL_in_aag(X4, X5, T47))
pL_in_aag(T53, T54, T47) → U12_aag(T53, T54, T47, delD_in_aag(T53, T54, T47))
delD_in_aag(X121, .(X121, T60), T60) → delD_out_aag(X121, .(X121, T60), T60)
delD_in_aag(X141, .(T65, X143), .(T65, T66)) → U3_aag(X141, T65, X143, T66, delD_in_aag(X141, X143, T66))
U3_aag(X141, T65, X143, T66, delD_out_aag(X141, X143, T66)) → delD_out_aag(X141, .(T65, X143), .(T65, T66))
U12_aag(T53, T54, T47, delD_out_aag(T53, T54, T47)) → U13_aag(T53, T54, T47, confF_in_g(T54))
confF_in_g(T74) → U5_g(T74, pG_in_gaaa(T74, X164, X165, X166))
pG_in_gaaa(T74, T76, X165, X166) → U14_gaaa(T74, T76, X165, X166, del2I_in_ga(T74, T76))
del2I_in_ga(T86, X199) → U7_ga(T86, X199, pJ_in_agaaa(X196, T86, X197, X198, X199))
pJ_in_agaaa(T91, T86, T92, X198, X199) → U18_agaaa(T91, T86, T92, X198, X199, delE_in_aga(T91, T86, T92))
delE_in_aga(T109, .(T109, T110), T110) → delE_out_aga(T109, .(T109, T110), T110)
delE_in_aga(X249, .(T115, T117), .(T115, X250)) → U4_aga(X249, T115, T117, X250, delE_in_aga(X249, T117, X250))
U4_aga(X249, T115, T117, X250, delE_out_aga(X249, T117, X250)) → delE_out_aga(X249, .(T115, T117), .(T115, X250))
U18_agaaa(T91, T86, T92, X198, X199, delE_out_aga(T91, T86, T92)) → U19_agaaa(T91, T86, T92, X198, X199, delE_in_aga(X198, T92, X199))
U19_agaaa(T91, T86, T92, X198, X199, delE_out_aga(X198, T92, X199)) → pJ_out_agaaa(T91, T86, T92, X198, X199)
U7_ga(T86, X199, pJ_out_agaaa(X196, T86, X197, X198, X199)) → del2I_out_ga(T86, X199)
U14_gaaa(T74, T76, X165, X166, del2I_out_ga(T74, T76)) → U15_gaaa(T74, T76, X165, X166, pM_in_aag(X165, X166, T76))
pM_in_aag(T123, T124, T76) → U16_aag(T123, T124, T76, delH_in_aag(T123, T124, T76))
delH_in_aag(X288, .(X288, T130), T130) → delH_out_aag(X288, .(X288, T130), T130)
delH_in_aag(X308, .(T135, X310), .(T135, T137)) → U6_aag(X308, T135, X310, T137, delH_in_aag(X308, X310, T137))
U6_aag(X308, T135, X310, T137, delH_out_aag(X308, X310, T137)) → delH_out_aag(X308, .(T135, X310), .(T135, T137))
U16_aag(T123, T124, T76, delH_out_aag(T123, T124, T76)) → U17_aag(T123, T124, T76, confF_in_g(T124))
U17_aag(T123, T124, T76, confF_out_g(T124)) → pM_out_aag(T123, T124, T76)
U15_gaaa(T74, T76, X165, X166, pM_out_aag(X165, X166, T76)) → pG_out_gaaa(T74, T76, X165, X166)
U5_g(T74, pG_out_gaaa(T74, X164, X165, X166)) → confF_out_g(T74)
U13_aag(T53, T54, T47, confF_out_g(T54)) → pL_out_aag(T53, T54, T47)
U11_agaaa(T46, T17, T47, X4, X5, pL_out_aag(X4, X5, T47)) → pK_out_agaaa(T46, T17, T47, X4, X5)
U9_agaaaaa(T16, T11, T17, X25, X26, X4, X5, pK_out_agaaa(X25, T17, X26, X4, X5)) → pB_out_agaaaaa(T16, T11, T17, X25, X26, X4, X5)
U1_g(T11, pB_out_agaaaaa(X23, T11, X24, X25, X26, X4, X5)) → confA_out_g(T11)

The argument filtering Pi contains the following mapping:
confA_in_g(x1)  =  confA_in_g(x1)
U1_g(x1, x2)  =  U1_g(x1, x2)
pB_in_agaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  pB_in_agaaaaa(x2)
U8_agaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U8_agaaaaa(x2, x8)
delC_in_aga(x1, x2, x3)  =  delC_in_aga(x2)
.(x1, x2)  =  .(x2)
delC_out_aga(x1, x2, x3)  =  delC_out_aga(x2, x3)
U2_aga(x1, x2, x3, x4, x5)  =  U2_aga(x3, x5)
U9_agaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U9_agaaaaa(x2, x3, x8)
pK_in_agaaa(x1, x2, x3, x4, x5)  =  pK_in_agaaa(x2)
U10_agaaa(x1, x2, x3, x4, x5, x6)  =  U10_agaaa(x2, x6)
U11_agaaa(x1, x2, x3, x4, x5, x6)  =  U11_agaaa(x2, x3, x6)
pL_in_aag(x1, x2, x3)  =  pL_in_aag(x3)
U12_aag(x1, x2, x3, x4)  =  U12_aag(x3, x4)
delD_in_aag(x1, x2, x3)  =  delD_in_aag(x3)
delD_out_aag(x1, x2, x3)  =  delD_out_aag(x2, x3)
U3_aag(x1, x2, x3, x4, x5)  =  U3_aag(x4, x5)
U13_aag(x1, x2, x3, x4)  =  U13_aag(x2, x3, x4)
confF_in_g(x1)  =  confF_in_g(x1)
U5_g(x1, x2)  =  U5_g(x1, x2)
pG_in_gaaa(x1, x2, x3, x4)  =  pG_in_gaaa(x1)
U14_gaaa(x1, x2, x3, x4, x5)  =  U14_gaaa(x1, x5)
del2I_in_ga(x1, x2)  =  del2I_in_ga(x1)
U7_ga(x1, x2, x3)  =  U7_ga(x1, x3)
pJ_in_agaaa(x1, x2, x3, x4, x5)  =  pJ_in_agaaa(x2)
U18_agaaa(x1, x2, x3, x4, x5, x6)  =  U18_agaaa(x2, x6)
delE_in_aga(x1, x2, x3)  =  delE_in_aga(x2)
delE_out_aga(x1, x2, x3)  =  delE_out_aga(x2, x3)
U4_aga(x1, x2, x3, x4, x5)  =  U4_aga(x3, x5)
U19_agaaa(x1, x2, x3, x4, x5, x6)  =  U19_agaaa(x2, x3, x6)
pJ_out_agaaa(x1, x2, x3, x4, x5)  =  pJ_out_agaaa(x2, x3, x5)
del2I_out_ga(x1, x2)  =  del2I_out_ga(x1, x2)
U15_gaaa(x1, x2, x3, x4, x5)  =  U15_gaaa(x1, x2, x5)
pM_in_aag(x1, x2, x3)  =  pM_in_aag(x3)
U16_aag(x1, x2, x3, x4)  =  U16_aag(x3, x4)
delH_in_aag(x1, x2, x3)  =  delH_in_aag(x3)
delH_out_aag(x1, x2, x3)  =  delH_out_aag(x2, x3)
U6_aag(x1, x2, x3, x4, x5)  =  U6_aag(x4, x5)
U17_aag(x1, x2, x3, x4)  =  U17_aag(x2, x3, x4)
confF_out_g(x1)  =  confF_out_g(x1)
pM_out_aag(x1, x2, x3)  =  pM_out_aag(x2, x3)
pG_out_gaaa(x1, x2, x3, x4)  =  pG_out_gaaa(x1, x2, x4)
pL_out_aag(x1, x2, x3)  =  pL_out_aag(x2, x3)
pK_out_agaaa(x1, x2, x3, x4, x5)  =  pK_out_agaaa(x2, x3, x5)
pB_out_agaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  pB_out_agaaaaa(x2, x3, x5, x7)
confA_out_g(x1)  =  confA_out_g(x1)
CONFA_IN_G(x1)  =  CONFA_IN_G(x1)
U1_G(x1, x2)  =  U1_G(x1, x2)
PB_IN_AGAAAAA(x1, x2, x3, x4, x5, x6, x7)  =  PB_IN_AGAAAAA(x2)
U8_AGAAAAA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U8_AGAAAAA(x2, x8)
DELC_IN_AGA(x1, x2, x3)  =  DELC_IN_AGA(x2)
U2_AGA(x1, x2, x3, x4, x5)  =  U2_AGA(x3, x5)
U9_AGAAAAA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U9_AGAAAAA(x2, x3, x8)
PK_IN_AGAAA(x1, x2, x3, x4, x5)  =  PK_IN_AGAAA(x2)
U10_AGAAA(x1, x2, x3, x4, x5, x6)  =  U10_AGAAA(x2, x6)
U11_AGAAA(x1, x2, x3, x4, x5, x6)  =  U11_AGAAA(x2, x3, x6)
PL_IN_AAG(x1, x2, x3)  =  PL_IN_AAG(x3)
U12_AAG(x1, x2, x3, x4)  =  U12_AAG(x3, x4)
DELD_IN_AAG(x1, x2, x3)  =  DELD_IN_AAG(x3)
U3_AAG(x1, x2, x3, x4, x5)  =  U3_AAG(x4, x5)
U13_AAG(x1, x2, x3, x4)  =  U13_AAG(x2, x3, x4)
CONFF_IN_G(x1)  =  CONFF_IN_G(x1)
U5_G(x1, x2)  =  U5_G(x1, x2)
PG_IN_GAAA(x1, x2, x3, x4)  =  PG_IN_GAAA(x1)
U14_GAAA(x1, x2, x3, x4, x5)  =  U14_GAAA(x1, x5)
DEL2I_IN_GA(x1, x2)  =  DEL2I_IN_GA(x1)
U7_GA(x1, x2, x3)  =  U7_GA(x1, x3)
PJ_IN_AGAAA(x1, x2, x3, x4, x5)  =  PJ_IN_AGAAA(x2)
U18_AGAAA(x1, x2, x3, x4, x5, x6)  =  U18_AGAAA(x2, x6)
DELE_IN_AGA(x1, x2, x3)  =  DELE_IN_AGA(x2)
U4_AGA(x1, x2, x3, x4, x5)  =  U4_AGA(x3, x5)
U19_AGAAA(x1, x2, x3, x4, x5, x6)  =  U19_AGAAA(x2, x3, x6)
U15_GAAA(x1, x2, x3, x4, x5)  =  U15_GAAA(x1, x2, x5)
PM_IN_AAG(x1, x2, x3)  =  PM_IN_AAG(x3)
U16_AAG(x1, x2, x3, x4)  =  U16_AAG(x3, x4)
DELH_IN_AAG(x1, x2, x3)  =  DELH_IN_AAG(x3)
U6_AAG(x1, x2, x3, x4, x5)  =  U6_AAG(x4, x5)
U17_AAG(x1, x2, x3, x4)  =  U17_AAG(x2, x3, x4)

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

(5) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 5 SCCs with 29 less nodes.

(6) Complex Obligation (AND)

(7) Obligation:

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

DELH_IN_AAG(X308, .(T135, X310), .(T135, T137)) → DELH_IN_AAG(X308, X310, T137)

The TRS R consists of the following rules:

confA_in_g(T11) → U1_g(T11, pB_in_agaaaaa(X23, T11, X24, X25, X26, X4, X5))
pB_in_agaaaaa(T16, T11, T17, X25, X26, X4, X5) → U8_agaaaaa(T16, T11, T17, X25, X26, X4, X5, delC_in_aga(T16, T11, T17))
delC_in_aga(T30, .(T30, T31), T31) → delC_out_aga(T30, .(T30, T31), T31)
delC_in_aga(X70, .(T36, T37), .(T36, X71)) → U2_aga(X70, T36, T37, X71, delC_in_aga(X70, T37, X71))
U2_aga(X70, T36, T37, X71, delC_out_aga(X70, T37, X71)) → delC_out_aga(X70, .(T36, T37), .(T36, X71))
U8_agaaaaa(T16, T11, T17, X25, X26, X4, X5, delC_out_aga(T16, T11, T17)) → U9_agaaaaa(T16, T11, T17, X25, X26, X4, X5, pK_in_agaaa(X25, T17, X26, X4, X5))
pK_in_agaaa(T46, T17, T47, X4, X5) → U10_agaaa(T46, T17, T47, X4, X5, delC_in_aga(T46, T17, T47))
U10_agaaa(T46, T17, T47, X4, X5, delC_out_aga(T46, T17, T47)) → U11_agaaa(T46, T17, T47, X4, X5, pL_in_aag(X4, X5, T47))
pL_in_aag(T53, T54, T47) → U12_aag(T53, T54, T47, delD_in_aag(T53, T54, T47))
delD_in_aag(X121, .(X121, T60), T60) → delD_out_aag(X121, .(X121, T60), T60)
delD_in_aag(X141, .(T65, X143), .(T65, T66)) → U3_aag(X141, T65, X143, T66, delD_in_aag(X141, X143, T66))
U3_aag(X141, T65, X143, T66, delD_out_aag(X141, X143, T66)) → delD_out_aag(X141, .(T65, X143), .(T65, T66))
U12_aag(T53, T54, T47, delD_out_aag(T53, T54, T47)) → U13_aag(T53, T54, T47, confF_in_g(T54))
confF_in_g(T74) → U5_g(T74, pG_in_gaaa(T74, X164, X165, X166))
pG_in_gaaa(T74, T76, X165, X166) → U14_gaaa(T74, T76, X165, X166, del2I_in_ga(T74, T76))
del2I_in_ga(T86, X199) → U7_ga(T86, X199, pJ_in_agaaa(X196, T86, X197, X198, X199))
pJ_in_agaaa(T91, T86, T92, X198, X199) → U18_agaaa(T91, T86, T92, X198, X199, delE_in_aga(T91, T86, T92))
delE_in_aga(T109, .(T109, T110), T110) → delE_out_aga(T109, .(T109, T110), T110)
delE_in_aga(X249, .(T115, T117), .(T115, X250)) → U4_aga(X249, T115, T117, X250, delE_in_aga(X249, T117, X250))
U4_aga(X249, T115, T117, X250, delE_out_aga(X249, T117, X250)) → delE_out_aga(X249, .(T115, T117), .(T115, X250))
U18_agaaa(T91, T86, T92, X198, X199, delE_out_aga(T91, T86, T92)) → U19_agaaa(T91, T86, T92, X198, X199, delE_in_aga(X198, T92, X199))
U19_agaaa(T91, T86, T92, X198, X199, delE_out_aga(X198, T92, X199)) → pJ_out_agaaa(T91, T86, T92, X198, X199)
U7_ga(T86, X199, pJ_out_agaaa(X196, T86, X197, X198, X199)) → del2I_out_ga(T86, X199)
U14_gaaa(T74, T76, X165, X166, del2I_out_ga(T74, T76)) → U15_gaaa(T74, T76, X165, X166, pM_in_aag(X165, X166, T76))
pM_in_aag(T123, T124, T76) → U16_aag(T123, T124, T76, delH_in_aag(T123, T124, T76))
delH_in_aag(X288, .(X288, T130), T130) → delH_out_aag(X288, .(X288, T130), T130)
delH_in_aag(X308, .(T135, X310), .(T135, T137)) → U6_aag(X308, T135, X310, T137, delH_in_aag(X308, X310, T137))
U6_aag(X308, T135, X310, T137, delH_out_aag(X308, X310, T137)) → delH_out_aag(X308, .(T135, X310), .(T135, T137))
U16_aag(T123, T124, T76, delH_out_aag(T123, T124, T76)) → U17_aag(T123, T124, T76, confF_in_g(T124))
U17_aag(T123, T124, T76, confF_out_g(T124)) → pM_out_aag(T123, T124, T76)
U15_gaaa(T74, T76, X165, X166, pM_out_aag(X165, X166, T76)) → pG_out_gaaa(T74, T76, X165, X166)
U5_g(T74, pG_out_gaaa(T74, X164, X165, X166)) → confF_out_g(T74)
U13_aag(T53, T54, T47, confF_out_g(T54)) → pL_out_aag(T53, T54, T47)
U11_agaaa(T46, T17, T47, X4, X5, pL_out_aag(X4, X5, T47)) → pK_out_agaaa(T46, T17, T47, X4, X5)
U9_agaaaaa(T16, T11, T17, X25, X26, X4, X5, pK_out_agaaa(X25, T17, X26, X4, X5)) → pB_out_agaaaaa(T16, T11, T17, X25, X26, X4, X5)
U1_g(T11, pB_out_agaaaaa(X23, T11, X24, X25, X26, X4, X5)) → confA_out_g(T11)

The argument filtering Pi contains the following mapping:
confA_in_g(x1)  =  confA_in_g(x1)
U1_g(x1, x2)  =  U1_g(x1, x2)
pB_in_agaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  pB_in_agaaaaa(x2)
U8_agaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U8_agaaaaa(x2, x8)
delC_in_aga(x1, x2, x3)  =  delC_in_aga(x2)
.(x1, x2)  =  .(x2)
delC_out_aga(x1, x2, x3)  =  delC_out_aga(x2, x3)
U2_aga(x1, x2, x3, x4, x5)  =  U2_aga(x3, x5)
U9_agaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U9_agaaaaa(x2, x3, x8)
pK_in_agaaa(x1, x2, x3, x4, x5)  =  pK_in_agaaa(x2)
U10_agaaa(x1, x2, x3, x4, x5, x6)  =  U10_agaaa(x2, x6)
U11_agaaa(x1, x2, x3, x4, x5, x6)  =  U11_agaaa(x2, x3, x6)
pL_in_aag(x1, x2, x3)  =  pL_in_aag(x3)
U12_aag(x1, x2, x3, x4)  =  U12_aag(x3, x4)
delD_in_aag(x1, x2, x3)  =  delD_in_aag(x3)
delD_out_aag(x1, x2, x3)  =  delD_out_aag(x2, x3)
U3_aag(x1, x2, x3, x4, x5)  =  U3_aag(x4, x5)
U13_aag(x1, x2, x3, x4)  =  U13_aag(x2, x3, x4)
confF_in_g(x1)  =  confF_in_g(x1)
U5_g(x1, x2)  =  U5_g(x1, x2)
pG_in_gaaa(x1, x2, x3, x4)  =  pG_in_gaaa(x1)
U14_gaaa(x1, x2, x3, x4, x5)  =  U14_gaaa(x1, x5)
del2I_in_ga(x1, x2)  =  del2I_in_ga(x1)
U7_ga(x1, x2, x3)  =  U7_ga(x1, x3)
pJ_in_agaaa(x1, x2, x3, x4, x5)  =  pJ_in_agaaa(x2)
U18_agaaa(x1, x2, x3, x4, x5, x6)  =  U18_agaaa(x2, x6)
delE_in_aga(x1, x2, x3)  =  delE_in_aga(x2)
delE_out_aga(x1, x2, x3)  =  delE_out_aga(x2, x3)
U4_aga(x1, x2, x3, x4, x5)  =  U4_aga(x3, x5)
U19_agaaa(x1, x2, x3, x4, x5, x6)  =  U19_agaaa(x2, x3, x6)
pJ_out_agaaa(x1, x2, x3, x4, x5)  =  pJ_out_agaaa(x2, x3, x5)
del2I_out_ga(x1, x2)  =  del2I_out_ga(x1, x2)
U15_gaaa(x1, x2, x3, x4, x5)  =  U15_gaaa(x1, x2, x5)
pM_in_aag(x1, x2, x3)  =  pM_in_aag(x3)
U16_aag(x1, x2, x3, x4)  =  U16_aag(x3, x4)
delH_in_aag(x1, x2, x3)  =  delH_in_aag(x3)
delH_out_aag(x1, x2, x3)  =  delH_out_aag(x2, x3)
U6_aag(x1, x2, x3, x4, x5)  =  U6_aag(x4, x5)
U17_aag(x1, x2, x3, x4)  =  U17_aag(x2, x3, x4)
confF_out_g(x1)  =  confF_out_g(x1)
pM_out_aag(x1, x2, x3)  =  pM_out_aag(x2, x3)
pG_out_gaaa(x1, x2, x3, x4)  =  pG_out_gaaa(x1, x2, x4)
pL_out_aag(x1, x2, x3)  =  pL_out_aag(x2, x3)
pK_out_agaaa(x1, x2, x3, x4, x5)  =  pK_out_agaaa(x2, x3, x5)
pB_out_agaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  pB_out_agaaaaa(x2, x3, x5, x7)
confA_out_g(x1)  =  confA_out_g(x1)
DELH_IN_AAG(x1, x2, x3)  =  DELH_IN_AAG(x3)

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:

DELH_IN_AAG(X308, .(T135, X310), .(T135, T137)) → DELH_IN_AAG(X308, X310, T137)

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

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:

DELH_IN_AAG(.(T137)) → DELH_IN_AAG(T137)

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:

• DELH_IN_AAG(.(T137)) → DELH_IN_AAG(T137)
  The graph contains the following edges 1 > 1

(13) YES

(14) Obligation:

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

DELE_IN_AGA(X249, .(T115, T117), .(T115, X250)) → DELE_IN_AGA(X249, T117, X250)

The TRS R consists of the following rules:

confA_in_g(T11) → U1_g(T11, pB_in_agaaaaa(X23, T11, X24, X25, X26, X4, X5))
pB_in_agaaaaa(T16, T11, T17, X25, X26, X4, X5) → U8_agaaaaa(T16, T11, T17, X25, X26, X4, X5, delC_in_aga(T16, T11, T17))
delC_in_aga(T30, .(T30, T31), T31) → delC_out_aga(T30, .(T30, T31), T31)
delC_in_aga(X70, .(T36, T37), .(T36, X71)) → U2_aga(X70, T36, T37, X71, delC_in_aga(X70, T37, X71))
U2_aga(X70, T36, T37, X71, delC_out_aga(X70, T37, X71)) → delC_out_aga(X70, .(T36, T37), .(T36, X71))
U8_agaaaaa(T16, T11, T17, X25, X26, X4, X5, delC_out_aga(T16, T11, T17)) → U9_agaaaaa(T16, T11, T17, X25, X26, X4, X5, pK_in_agaaa(X25, T17, X26, X4, X5))
pK_in_agaaa(T46, T17, T47, X4, X5) → U10_agaaa(T46, T17, T47, X4, X5, delC_in_aga(T46, T17, T47))
U10_agaaa(T46, T17, T47, X4, X5, delC_out_aga(T46, T17, T47)) → U11_agaaa(T46, T17, T47, X4, X5, pL_in_aag(X4, X5, T47))
pL_in_aag(T53, T54, T47) → U12_aag(T53, T54, T47, delD_in_aag(T53, T54, T47))
delD_in_aag(X121, .(X121, T60), T60) → delD_out_aag(X121, .(X121, T60), T60)
delD_in_aag(X141, .(T65, X143), .(T65, T66)) → U3_aag(X141, T65, X143, T66, delD_in_aag(X141, X143, T66))
U3_aag(X141, T65, X143, T66, delD_out_aag(X141, X143, T66)) → delD_out_aag(X141, .(T65, X143), .(T65, T66))
U12_aag(T53, T54, T47, delD_out_aag(T53, T54, T47)) → U13_aag(T53, T54, T47, confF_in_g(T54))
confF_in_g(T74) → U5_g(T74, pG_in_gaaa(T74, X164, X165, X166))
pG_in_gaaa(T74, T76, X165, X166) → U14_gaaa(T74, T76, X165, X166, del2I_in_ga(T74, T76))
del2I_in_ga(T86, X199) → U7_ga(T86, X199, pJ_in_agaaa(X196, T86, X197, X198, X199))
pJ_in_agaaa(T91, T86, T92, X198, X199) → U18_agaaa(T91, T86, T92, X198, X199, delE_in_aga(T91, T86, T92))
delE_in_aga(T109, .(T109, T110), T110) → delE_out_aga(T109, .(T109, T110), T110)
delE_in_aga(X249, .(T115, T117), .(T115, X250)) → U4_aga(X249, T115, T117, X250, delE_in_aga(X249, T117, X250))
U4_aga(X249, T115, T117, X250, delE_out_aga(X249, T117, X250)) → delE_out_aga(X249, .(T115, T117), .(T115, X250))
U18_agaaa(T91, T86, T92, X198, X199, delE_out_aga(T91, T86, T92)) → U19_agaaa(T91, T86, T92, X198, X199, delE_in_aga(X198, T92, X199))
U19_agaaa(T91, T86, T92, X198, X199, delE_out_aga(X198, T92, X199)) → pJ_out_agaaa(T91, T86, T92, X198, X199)
U7_ga(T86, X199, pJ_out_agaaa(X196, T86, X197, X198, X199)) → del2I_out_ga(T86, X199)
U14_gaaa(T74, T76, X165, X166, del2I_out_ga(T74, T76)) → U15_gaaa(T74, T76, X165, X166, pM_in_aag(X165, X166, T76))
pM_in_aag(T123, T124, T76) → U16_aag(T123, T124, T76, delH_in_aag(T123, T124, T76))
delH_in_aag(X288, .(X288, T130), T130) → delH_out_aag(X288, .(X288, T130), T130)
delH_in_aag(X308, .(T135, X310), .(T135, T137)) → U6_aag(X308, T135, X310, T137, delH_in_aag(X308, X310, T137))
U6_aag(X308, T135, X310, T137, delH_out_aag(X308, X310, T137)) → delH_out_aag(X308, .(T135, X310), .(T135, T137))
U16_aag(T123, T124, T76, delH_out_aag(T123, T124, T76)) → U17_aag(T123, T124, T76, confF_in_g(T124))
U17_aag(T123, T124, T76, confF_out_g(T124)) → pM_out_aag(T123, T124, T76)
U15_gaaa(T74, T76, X165, X166, pM_out_aag(X165, X166, T76)) → pG_out_gaaa(T74, T76, X165, X166)
U5_g(T74, pG_out_gaaa(T74, X164, X165, X166)) → confF_out_g(T74)
U13_aag(T53, T54, T47, confF_out_g(T54)) → pL_out_aag(T53, T54, T47)
U11_agaaa(T46, T17, T47, X4, X5, pL_out_aag(X4, X5, T47)) → pK_out_agaaa(T46, T17, T47, X4, X5)
U9_agaaaaa(T16, T11, T17, X25, X26, X4, X5, pK_out_agaaa(X25, T17, X26, X4, X5)) → pB_out_agaaaaa(T16, T11, T17, X25, X26, X4, X5)
U1_g(T11, pB_out_agaaaaa(X23, T11, X24, X25, X26, X4, X5)) → confA_out_g(T11)

The argument filtering Pi contains the following mapping:
confA_in_g(x1)  =  confA_in_g(x1)
U1_g(x1, x2)  =  U1_g(x1, x2)
pB_in_agaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  pB_in_agaaaaa(x2)
U8_agaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U8_agaaaaa(x2, x8)
delC_in_aga(x1, x2, x3)  =  delC_in_aga(x2)
.(x1, x2)  =  .(x2)
delC_out_aga(x1, x2, x3)  =  delC_out_aga(x2, x3)
U2_aga(x1, x2, x3, x4, x5)  =  U2_aga(x3, x5)
U9_agaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U9_agaaaaa(x2, x3, x8)
pK_in_agaaa(x1, x2, x3, x4, x5)  =  pK_in_agaaa(x2)
U10_agaaa(x1, x2, x3, x4, x5, x6)  =  U10_agaaa(x2, x6)
U11_agaaa(x1, x2, x3, x4, x5, x6)  =  U11_agaaa(x2, x3, x6)
pL_in_aag(x1, x2, x3)  =  pL_in_aag(x3)
U12_aag(x1, x2, x3, x4)  =  U12_aag(x3, x4)
delD_in_aag(x1, x2, x3)  =  delD_in_aag(x3)
delD_out_aag(x1, x2, x3)  =  delD_out_aag(x2, x3)
U3_aag(x1, x2, x3, x4, x5)  =  U3_aag(x4, x5)
U13_aag(x1, x2, x3, x4)  =  U13_aag(x2, x3, x4)
confF_in_g(x1)  =  confF_in_g(x1)
U5_g(x1, x2)  =  U5_g(x1, x2)
pG_in_gaaa(x1, x2, x3, x4)  =  pG_in_gaaa(x1)
U14_gaaa(x1, x2, x3, x4, x5)  =  U14_gaaa(x1, x5)
del2I_in_ga(x1, x2)  =  del2I_in_ga(x1)
U7_ga(x1, x2, x3)  =  U7_ga(x1, x3)
pJ_in_agaaa(x1, x2, x3, x4, x5)  =  pJ_in_agaaa(x2)
U18_agaaa(x1, x2, x3, x4, x5, x6)  =  U18_agaaa(x2, x6)
delE_in_aga(x1, x2, x3)  =  delE_in_aga(x2)
delE_out_aga(x1, x2, x3)  =  delE_out_aga(x2, x3)
U4_aga(x1, x2, x3, x4, x5)  =  U4_aga(x3, x5)
U19_agaaa(x1, x2, x3, x4, x5, x6)  =  U19_agaaa(x2, x3, x6)
pJ_out_agaaa(x1, x2, x3, x4, x5)  =  pJ_out_agaaa(x2, x3, x5)
del2I_out_ga(x1, x2)  =  del2I_out_ga(x1, x2)
U15_gaaa(x1, x2, x3, x4, x5)  =  U15_gaaa(x1, x2, x5)
pM_in_aag(x1, x2, x3)  =  pM_in_aag(x3)
U16_aag(x1, x2, x3, x4)  =  U16_aag(x3, x4)
delH_in_aag(x1, x2, x3)  =  delH_in_aag(x3)
delH_out_aag(x1, x2, x3)  =  delH_out_aag(x2, x3)
U6_aag(x1, x2, x3, x4, x5)  =  U6_aag(x4, x5)
U17_aag(x1, x2, x3, x4)  =  U17_aag(x2, x3, x4)
confF_out_g(x1)  =  confF_out_g(x1)
pM_out_aag(x1, x2, x3)  =  pM_out_aag(x2, x3)
pG_out_gaaa(x1, x2, x3, x4)  =  pG_out_gaaa(x1, x2, x4)
pL_out_aag(x1, x2, x3)  =  pL_out_aag(x2, x3)
pK_out_agaaa(x1, x2, x3, x4, x5)  =  pK_out_agaaa(x2, x3, x5)
pB_out_agaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  pB_out_agaaaaa(x2, x3, x5, x7)
confA_out_g(x1)  =  confA_out_g(x1)
DELE_IN_AGA(x1, x2, x3)  =  DELE_IN_AGA(x2)

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

(15) UsableRulesProof (EQUIVALENT transformation)

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

(16) Obligation:

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

DELE_IN_AGA(X249, .(T115, T117), .(T115, X250)) → DELE_IN_AGA(X249, T117, X250)

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

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

(17) PiDPToQDPProof (SOUND transformation)

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

(18) Obligation:

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

DELE_IN_AGA(.(T117)) → DELE_IN_AGA(T117)

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:

• DELE_IN_AGA(.(T117)) → DELE_IN_AGA(T117)
  The graph contains the following edges 1 > 1

(20) YES

(21) Obligation:

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

CONFF_IN_G(T74) → PG_IN_GAAA(T74, X164, X165, X166)
PG_IN_GAAA(T74, T76, X165, X166) → U14_GAAA(T74, T76, X165, X166, del2I_in_ga(T74, T76))
U14_GAAA(T74, T76, X165, X166, del2I_out_ga(T74, T76)) → PM_IN_AAG(X165, X166, T76)
PM_IN_AAG(T123, T124, T76) → U16_AAG(T123, T124, T76, delH_in_aag(T123, T124, T76))
U16_AAG(T123, T124, T76, delH_out_aag(T123, T124, T76)) → CONFF_IN_G(T124)

The TRS R consists of the following rules:

confA_in_g(T11) → U1_g(T11, pB_in_agaaaaa(X23, T11, X24, X25, X26, X4, X5))
pB_in_agaaaaa(T16, T11, T17, X25, X26, X4, X5) → U8_agaaaaa(T16, T11, T17, X25, X26, X4, X5, delC_in_aga(T16, T11, T17))
delC_in_aga(T30, .(T30, T31), T31) → delC_out_aga(T30, .(T30, T31), T31)
delC_in_aga(X70, .(T36, T37), .(T36, X71)) → U2_aga(X70, T36, T37, X71, delC_in_aga(X70, T37, X71))
U2_aga(X70, T36, T37, X71, delC_out_aga(X70, T37, X71)) → delC_out_aga(X70, .(T36, T37), .(T36, X71))
U8_agaaaaa(T16, T11, T17, X25, X26, X4, X5, delC_out_aga(T16, T11, T17)) → U9_agaaaaa(T16, T11, T17, X25, X26, X4, X5, pK_in_agaaa(X25, T17, X26, X4, X5))
pK_in_agaaa(T46, T17, T47, X4, X5) → U10_agaaa(T46, T17, T47, X4, X5, delC_in_aga(T46, T17, T47))
U10_agaaa(T46, T17, T47, X4, X5, delC_out_aga(T46, T17, T47)) → U11_agaaa(T46, T17, T47, X4, X5, pL_in_aag(X4, X5, T47))
pL_in_aag(T53, T54, T47) → U12_aag(T53, T54, T47, delD_in_aag(T53, T54, T47))
delD_in_aag(X121, .(X121, T60), T60) → delD_out_aag(X121, .(X121, T60), T60)
delD_in_aag(X141, .(T65, X143), .(T65, T66)) → U3_aag(X141, T65, X143, T66, delD_in_aag(X141, X143, T66))
U3_aag(X141, T65, X143, T66, delD_out_aag(X141, X143, T66)) → delD_out_aag(X141, .(T65, X143), .(T65, T66))
U12_aag(T53, T54, T47, delD_out_aag(T53, T54, T47)) → U13_aag(T53, T54, T47, confF_in_g(T54))
confF_in_g(T74) → U5_g(T74, pG_in_gaaa(T74, X164, X165, X166))
pG_in_gaaa(T74, T76, X165, X166) → U14_gaaa(T74, T76, X165, X166, del2I_in_ga(T74, T76))
del2I_in_ga(T86, X199) → U7_ga(T86, X199, pJ_in_agaaa(X196, T86, X197, X198, X199))
pJ_in_agaaa(T91, T86, T92, X198, X199) → U18_agaaa(T91, T86, T92, X198, X199, delE_in_aga(T91, T86, T92))
delE_in_aga(T109, .(T109, T110), T110) → delE_out_aga(T109, .(T109, T110), T110)
delE_in_aga(X249, .(T115, T117), .(T115, X250)) → U4_aga(X249, T115, T117, X250, delE_in_aga(X249, T117, X250))
U4_aga(X249, T115, T117, X250, delE_out_aga(X249, T117, X250)) → delE_out_aga(X249, .(T115, T117), .(T115, X250))
U18_agaaa(T91, T86, T92, X198, X199, delE_out_aga(T91, T86, T92)) → U19_agaaa(T91, T86, T92, X198, X199, delE_in_aga(X198, T92, X199))
U19_agaaa(T91, T86, T92, X198, X199, delE_out_aga(X198, T92, X199)) → pJ_out_agaaa(T91, T86, T92, X198, X199)
U7_ga(T86, X199, pJ_out_agaaa(X196, T86, X197, X198, X199)) → del2I_out_ga(T86, X199)
U14_gaaa(T74, T76, X165, X166, del2I_out_ga(T74, T76)) → U15_gaaa(T74, T76, X165, X166, pM_in_aag(X165, X166, T76))
pM_in_aag(T123, T124, T76) → U16_aag(T123, T124, T76, delH_in_aag(T123, T124, T76))
delH_in_aag(X288, .(X288, T130), T130) → delH_out_aag(X288, .(X288, T130), T130)
delH_in_aag(X308, .(T135, X310), .(T135, T137)) → U6_aag(X308, T135, X310, T137, delH_in_aag(X308, X310, T137))
U6_aag(X308, T135, X310, T137, delH_out_aag(X308, X310, T137)) → delH_out_aag(X308, .(T135, X310), .(T135, T137))
U16_aag(T123, T124, T76, delH_out_aag(T123, T124, T76)) → U17_aag(T123, T124, T76, confF_in_g(T124))
U17_aag(T123, T124, T76, confF_out_g(T124)) → pM_out_aag(T123, T124, T76)
U15_gaaa(T74, T76, X165, X166, pM_out_aag(X165, X166, T76)) → pG_out_gaaa(T74, T76, X165, X166)
U5_g(T74, pG_out_gaaa(T74, X164, X165, X166)) → confF_out_g(T74)
U13_aag(T53, T54, T47, confF_out_g(T54)) → pL_out_aag(T53, T54, T47)
U11_agaaa(T46, T17, T47, X4, X5, pL_out_aag(X4, X5, T47)) → pK_out_agaaa(T46, T17, T47, X4, X5)
U9_agaaaaa(T16, T11, T17, X25, X26, X4, X5, pK_out_agaaa(X25, T17, X26, X4, X5)) → pB_out_agaaaaa(T16, T11, T17, X25, X26, X4, X5)
U1_g(T11, pB_out_agaaaaa(X23, T11, X24, X25, X26, X4, X5)) → confA_out_g(T11)

The argument filtering Pi contains the following mapping:
confA_in_g(x1)  =  confA_in_g(x1)
U1_g(x1, x2)  =  U1_g(x1, x2)
pB_in_agaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  pB_in_agaaaaa(x2)
U8_agaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U8_agaaaaa(x2, x8)
delC_in_aga(x1, x2, x3)  =  delC_in_aga(x2)
.(x1, x2)  =  .(x2)
delC_out_aga(x1, x2, x3)  =  delC_out_aga(x2, x3)
U2_aga(x1, x2, x3, x4, x5)  =  U2_aga(x3, x5)
U9_agaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U9_agaaaaa(x2, x3, x8)
pK_in_agaaa(x1, x2, x3, x4, x5)  =  pK_in_agaaa(x2)
U10_agaaa(x1, x2, x3, x4, x5, x6)  =  U10_agaaa(x2, x6)
U11_agaaa(x1, x2, x3, x4, x5, x6)  =  U11_agaaa(x2, x3, x6)
pL_in_aag(x1, x2, x3)  =  pL_in_aag(x3)
U12_aag(x1, x2, x3, x4)  =  U12_aag(x3, x4)
delD_in_aag(x1, x2, x3)  =  delD_in_aag(x3)
delD_out_aag(x1, x2, x3)  =  delD_out_aag(x2, x3)
U3_aag(x1, x2, x3, x4, x5)  =  U3_aag(x4, x5)
U13_aag(x1, x2, x3, x4)  =  U13_aag(x2, x3, x4)
confF_in_g(x1)  =  confF_in_g(x1)
U5_g(x1, x2)  =  U5_g(x1, x2)
pG_in_gaaa(x1, x2, x3, x4)  =  pG_in_gaaa(x1)
U14_gaaa(x1, x2, x3, x4, x5)  =  U14_gaaa(x1, x5)
del2I_in_ga(x1, x2)  =  del2I_in_ga(x1)
U7_ga(x1, x2, x3)  =  U7_ga(x1, x3)
pJ_in_agaaa(x1, x2, x3, x4, x5)  =  pJ_in_agaaa(x2)
U18_agaaa(x1, x2, x3, x4, x5, x6)  =  U18_agaaa(x2, x6)
delE_in_aga(x1, x2, x3)  =  delE_in_aga(x2)
delE_out_aga(x1, x2, x3)  =  delE_out_aga(x2, x3)
U4_aga(x1, x2, x3, x4, x5)  =  U4_aga(x3, x5)
U19_agaaa(x1, x2, x3, x4, x5, x6)  =  U19_agaaa(x2, x3, x6)
pJ_out_agaaa(x1, x2, x3, x4, x5)  =  pJ_out_agaaa(x2, x3, x5)
del2I_out_ga(x1, x2)  =  del2I_out_ga(x1, x2)
U15_gaaa(x1, x2, x3, x4, x5)  =  U15_gaaa(x1, x2, x5)
pM_in_aag(x1, x2, x3)  =  pM_in_aag(x3)
U16_aag(x1, x2, x3, x4)  =  U16_aag(x3, x4)
delH_in_aag(x1, x2, x3)  =  delH_in_aag(x3)
delH_out_aag(x1, x2, x3)  =  delH_out_aag(x2, x3)
U6_aag(x1, x2, x3, x4, x5)  =  U6_aag(x4, x5)
U17_aag(x1, x2, x3, x4)  =  U17_aag(x2, x3, x4)
confF_out_g(x1)  =  confF_out_g(x1)
pM_out_aag(x1, x2, x3)  =  pM_out_aag(x2, x3)
pG_out_gaaa(x1, x2, x3, x4)  =  pG_out_gaaa(x1, x2, x4)
pL_out_aag(x1, x2, x3)  =  pL_out_aag(x2, x3)
pK_out_agaaa(x1, x2, x3, x4, x5)  =  pK_out_agaaa(x2, x3, x5)
pB_out_agaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  pB_out_agaaaaa(x2, x3, x5, x7)
confA_out_g(x1)  =  confA_out_g(x1)
CONFF_IN_G(x1)  =  CONFF_IN_G(x1)
PG_IN_GAAA(x1, x2, x3, x4)  =  PG_IN_GAAA(x1)
U14_GAAA(x1, x2, x3, x4, x5)  =  U14_GAAA(x1, x5)
PM_IN_AAG(x1, x2, x3)  =  PM_IN_AAG(x3)
U16_AAG(x1, x2, x3, x4)  =  U16_AAG(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:

CONFF_IN_G(T74) → PG_IN_GAAA(T74, X164, X165, X166)
PG_IN_GAAA(T74, T76, X165, X166) → U14_GAAA(T74, T76, X165, X166, del2I_in_ga(T74, T76))
U14_GAAA(T74, T76, X165, X166, del2I_out_ga(T74, T76)) → PM_IN_AAG(X165, X166, T76)
PM_IN_AAG(T123, T124, T76) → U16_AAG(T123, T124, T76, delH_in_aag(T123, T124, T76))
U16_AAG(T123, T124, T76, delH_out_aag(T123, T124, T76)) → CONFF_IN_G(T124)

The TRS R consists of the following rules:

del2I_in_ga(T86, X199) → U7_ga(T86, X199, pJ_in_agaaa(X196, T86, X197, X198, X199))
delH_in_aag(X288, .(X288, T130), T130) → delH_out_aag(X288, .(X288, T130), T130)
delH_in_aag(X308, .(T135, X310), .(T135, T137)) → U6_aag(X308, T135, X310, T137, delH_in_aag(X308, X310, T137))
U7_ga(T86, X199, pJ_out_agaaa(X196, T86, X197, X198, X199)) → del2I_out_ga(T86, X199)
U6_aag(X308, T135, X310, T137, delH_out_aag(X308, X310, T137)) → delH_out_aag(X308, .(T135, X310), .(T135, T137))
pJ_in_agaaa(T91, T86, T92, X198, X199) → U18_agaaa(T91, T86, T92, X198, X199, delE_in_aga(T91, T86, T92))
U18_agaaa(T91, T86, T92, X198, X199, delE_out_aga(T91, T86, T92)) → U19_agaaa(T91, T86, T92, X198, X199, delE_in_aga(X198, T92, X199))
delE_in_aga(T109, .(T109, T110), T110) → delE_out_aga(T109, .(T109, T110), T110)
delE_in_aga(X249, .(T115, T117), .(T115, X250)) → U4_aga(X249, T115, T117, X250, delE_in_aga(X249, T117, X250))
U19_agaaa(T91, T86, T92, X198, X199, delE_out_aga(X198, T92, X199)) → pJ_out_agaaa(T91, T86, T92, X198, X199)
U4_aga(X249, T115, T117, X250, delE_out_aga(X249, T117, X250)) → delE_out_aga(X249, .(T115, T117), .(T115, X250))

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x2)
del2I_in_ga(x1, x2)  =  del2I_in_ga(x1)
U7_ga(x1, x2, x3)  =  U7_ga(x1, x3)
pJ_in_agaaa(x1, x2, x3, x4, x5)  =  pJ_in_agaaa(x2)
U18_agaaa(x1, x2, x3, x4, x5, x6)  =  U18_agaaa(x2, x6)
delE_in_aga(x1, x2, x3)  =  delE_in_aga(x2)
delE_out_aga(x1, x2, x3)  =  delE_out_aga(x2, x3)
U4_aga(x1, x2, x3, x4, x5)  =  U4_aga(x3, x5)
U19_agaaa(x1, x2, x3, x4, x5, x6)  =  U19_agaaa(x2, x3, x6)
pJ_out_agaaa(x1, x2, x3, x4, x5)  =  pJ_out_agaaa(x2, x3, x5)
del2I_out_ga(x1, x2)  =  del2I_out_ga(x1, x2)
delH_in_aag(x1, x2, x3)  =  delH_in_aag(x3)
delH_out_aag(x1, x2, x3)  =  delH_out_aag(x2, x3)
U6_aag(x1, x2, x3, x4, x5)  =  U6_aag(x4, x5)
CONFF_IN_G(x1)  =  CONFF_IN_G(x1)
PG_IN_GAAA(x1, x2, x3, x4)  =  PG_IN_GAAA(x1)
U14_GAAA(x1, x2, x3, x4, x5)  =  U14_GAAA(x1, x5)
PM_IN_AAG(x1, x2, x3)  =  PM_IN_AAG(x3)
U16_AAG(x1, x2, x3, x4)  =  U16_AAG(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:

CONFF_IN_G(T74) → PG_IN_GAAA(T74)
PG_IN_GAAA(T74) → U14_GAAA(T74, del2I_in_ga(T74))
U14_GAAA(T74, del2I_out_ga(T74, T76)) → PM_IN_AAG(T76)
PM_IN_AAG(T76) → U16_AAG(T76, delH_in_aag(T76))
U16_AAG(T76, delH_out_aag(T124, T76)) → CONFF_IN_G(T124)

The TRS R consists of the following rules:

del2I_in_ga(T86) → U7_ga(T86, pJ_in_agaaa(T86))
delH_in_aag(T130) → delH_out_aag(.(T130), T130)
delH_in_aag(.(T137)) → U6_aag(T137, delH_in_aag(T137))
U7_ga(T86, pJ_out_agaaa(T86, X197, X199)) → del2I_out_ga(T86, X199)
U6_aag(T137, delH_out_aag(X310, T137)) → delH_out_aag(.(X310), .(T137))
pJ_in_agaaa(T86) → U18_agaaa(T86, delE_in_aga(T86))
U18_agaaa(T86, delE_out_aga(T86, T92)) → U19_agaaa(T86, T92, delE_in_aga(T92))
delE_in_aga(.(T110)) → delE_out_aga(.(T110), T110)
delE_in_aga(.(T117)) → U4_aga(T117, delE_in_aga(T117))
U19_agaaa(T86, T92, delE_out_aga(T92, X199)) → pJ_out_agaaa(T86, T92, X199)
U4_aga(T117, delE_out_aga(T117, X250)) → delE_out_aga(.(T117), .(X250))

The set Q consists of the following terms:

del2I_in_ga(x0)
delH_in_aag(x0)
U7_ga(x0, x1)
U6_aag(x0, x1)
pJ_in_agaaa(x0)
U18_agaaa(x0, x1)
delE_in_aga(x0)
U19_agaaa(x0, x1, x2)
U4_aga(x0, x1)

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

(26) QDPOrderProof (EQUIVALENT transformation)

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

The following pairs can be oriented strictly and are deleted.

U14_GAAA(T74, del2I_out_ga(T74, T76)) → PM_IN_AAG(T76)

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

POL( U14_GAAA(x₁, x₂) ) = max{0, 2x₂ - 1}

POL( del2I_in_ga(x₁) ) = x₁

POL( U7_ga(x₁, x₂) ) = x₂

POL( pJ_in_agaaa(x₁) ) = x₁

POL( U16_AAG(x₁, x₂) ) = x₂

POL( delH_in_aag(x₁) ) = 2x₁ + 2

POL( delH_out_aag(x₁, x₂) ) = 2x₁

POL( .(x₁) ) = x₁ + 1

POL( U6_aag(x₁, x₂) ) = x₂ + 2

POL( U18_agaaa(x₁, x₂) ) = max{0, x₂ - 1}

POL( delE_in_aga(x₁) ) = x₁ + 1

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

POL( del2I_out_ga(x₁, x₂) ) = x₂ + 2

POL( U19_agaaa(x₁, ..., x₃) ) = x₃

POL( delE_out_aga(x₁, x₂) ) = x₂ + 2

POL( U4_aga(x₁, x₂) ) = x₂ + 1

POL( CONFF_IN_G(x₁) ) = 2x₁

POL( PG_IN_GAAA(x₁) ) = 2x₁

POL( PM_IN_AAG(x₁) ) = 2x₁ + 2

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

del2I_in_ga(T86) → U7_ga(T86, pJ_in_agaaa(T86))
delH_in_aag(T130) → delH_out_aag(.(T130), T130)
delH_in_aag(.(T137)) → U6_aag(T137, delH_in_aag(T137))
pJ_in_agaaa(T86) → U18_agaaa(T86, delE_in_aga(T86))
U7_ga(T86, pJ_out_agaaa(T86, X197, X199)) → del2I_out_ga(T86, X199)
U6_aag(T137, delH_out_aag(X310, T137)) → delH_out_aag(.(X310), .(T137))
delE_in_aga(.(T110)) → delE_out_aga(.(T110), T110)
delE_in_aga(.(T117)) → U4_aga(T117, delE_in_aga(T117))
U18_agaaa(T86, delE_out_aga(T86, T92)) → U19_agaaa(T86, T92, delE_in_aga(T92))
U19_agaaa(T86, T92, delE_out_aga(T92, X199)) → pJ_out_agaaa(T86, T92, X199)
U4_aga(T117, delE_out_aga(T117, X250)) → delE_out_aga(.(T117), .(X250))

(27) Obligation:

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

CONFF_IN_G(T74) → PG_IN_GAAA(T74)
PG_IN_GAAA(T74) → U14_GAAA(T74, del2I_in_ga(T74))
PM_IN_AAG(T76) → U16_AAG(T76, delH_in_aag(T76))
U16_AAG(T76, delH_out_aag(T124, T76)) → CONFF_IN_G(T124)

The TRS R consists of the following rules:

del2I_in_ga(T86) → U7_ga(T86, pJ_in_agaaa(T86))
delH_in_aag(T130) → delH_out_aag(.(T130), T130)
delH_in_aag(.(T137)) → U6_aag(T137, delH_in_aag(T137))
U7_ga(T86, pJ_out_agaaa(T86, X197, X199)) → del2I_out_ga(T86, X199)
U6_aag(T137, delH_out_aag(X310, T137)) → delH_out_aag(.(X310), .(T137))
pJ_in_agaaa(T86) → U18_agaaa(T86, delE_in_aga(T86))
U18_agaaa(T86, delE_out_aga(T86, T92)) → U19_agaaa(T86, T92, delE_in_aga(T92))
delE_in_aga(.(T110)) → delE_out_aga(.(T110), T110)
delE_in_aga(.(T117)) → U4_aga(T117, delE_in_aga(T117))
U19_agaaa(T86, T92, delE_out_aga(T92, X199)) → pJ_out_agaaa(T86, T92, X199)
U4_aga(T117, delE_out_aga(T117, X250)) → delE_out_aga(.(T117), .(X250))

The set Q consists of the following terms:

del2I_in_ga(x0)
delH_in_aag(x0)
U7_ga(x0, x1)
U6_aag(x0, x1)
pJ_in_agaaa(x0)
U18_agaaa(x0, x1)
delE_in_aga(x0)
U19_agaaa(x0, x1, x2)
U4_aga(x0, x1)

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

(28) DependencyGraphProof (EQUIVALENT transformation)

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

(29) TRUE

(30) Obligation:

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

DELD_IN_AAG(X141, .(T65, X143), .(T65, T66)) → DELD_IN_AAG(X141, X143, T66)

The TRS R consists of the following rules:

confA_in_g(T11) → U1_g(T11, pB_in_agaaaaa(X23, T11, X24, X25, X26, X4, X5))
pB_in_agaaaaa(T16, T11, T17, X25, X26, X4, X5) → U8_agaaaaa(T16, T11, T17, X25, X26, X4, X5, delC_in_aga(T16, T11, T17))
delC_in_aga(T30, .(T30, T31), T31) → delC_out_aga(T30, .(T30, T31), T31)
delC_in_aga(X70, .(T36, T37), .(T36, X71)) → U2_aga(X70, T36, T37, X71, delC_in_aga(X70, T37, X71))
U2_aga(X70, T36, T37, X71, delC_out_aga(X70, T37, X71)) → delC_out_aga(X70, .(T36, T37), .(T36, X71))
U8_agaaaaa(T16, T11, T17, X25, X26, X4, X5, delC_out_aga(T16, T11, T17)) → U9_agaaaaa(T16, T11, T17, X25, X26, X4, X5, pK_in_agaaa(X25, T17, X26, X4, X5))
pK_in_agaaa(T46, T17, T47, X4, X5) → U10_agaaa(T46, T17, T47, X4, X5, delC_in_aga(T46, T17, T47))
U10_agaaa(T46, T17, T47, X4, X5, delC_out_aga(T46, T17, T47)) → U11_agaaa(T46, T17, T47, X4, X5, pL_in_aag(X4, X5, T47))
pL_in_aag(T53, T54, T47) → U12_aag(T53, T54, T47, delD_in_aag(T53, T54, T47))
delD_in_aag(X121, .(X121, T60), T60) → delD_out_aag(X121, .(X121, T60), T60)
delD_in_aag(X141, .(T65, X143), .(T65, T66)) → U3_aag(X141, T65, X143, T66, delD_in_aag(X141, X143, T66))
U3_aag(X141, T65, X143, T66, delD_out_aag(X141, X143, T66)) → delD_out_aag(X141, .(T65, X143), .(T65, T66))
U12_aag(T53, T54, T47, delD_out_aag(T53, T54, T47)) → U13_aag(T53, T54, T47, confF_in_g(T54))
confF_in_g(T74) → U5_g(T74, pG_in_gaaa(T74, X164, X165, X166))
pG_in_gaaa(T74, T76, X165, X166) → U14_gaaa(T74, T76, X165, X166, del2I_in_ga(T74, T76))
del2I_in_ga(T86, X199) → U7_ga(T86, X199, pJ_in_agaaa(X196, T86, X197, X198, X199))
pJ_in_agaaa(T91, T86, T92, X198, X199) → U18_agaaa(T91, T86, T92, X198, X199, delE_in_aga(T91, T86, T92))
delE_in_aga(T109, .(T109, T110), T110) → delE_out_aga(T109, .(T109, T110), T110)
delE_in_aga(X249, .(T115, T117), .(T115, X250)) → U4_aga(X249, T115, T117, X250, delE_in_aga(X249, T117, X250))
U4_aga(X249, T115, T117, X250, delE_out_aga(X249, T117, X250)) → delE_out_aga(X249, .(T115, T117), .(T115, X250))
U18_agaaa(T91, T86, T92, X198, X199, delE_out_aga(T91, T86, T92)) → U19_agaaa(T91, T86, T92, X198, X199, delE_in_aga(X198, T92, X199))
U19_agaaa(T91, T86, T92, X198, X199, delE_out_aga(X198, T92, X199)) → pJ_out_agaaa(T91, T86, T92, X198, X199)
U7_ga(T86, X199, pJ_out_agaaa(X196, T86, X197, X198, X199)) → del2I_out_ga(T86, X199)
U14_gaaa(T74, T76, X165, X166, del2I_out_ga(T74, T76)) → U15_gaaa(T74, T76, X165, X166, pM_in_aag(X165, X166, T76))
pM_in_aag(T123, T124, T76) → U16_aag(T123, T124, T76, delH_in_aag(T123, T124, T76))
delH_in_aag(X288, .(X288, T130), T130) → delH_out_aag(X288, .(X288, T130), T130)
delH_in_aag(X308, .(T135, X310), .(T135, T137)) → U6_aag(X308, T135, X310, T137, delH_in_aag(X308, X310, T137))
U6_aag(X308, T135, X310, T137, delH_out_aag(X308, X310, T137)) → delH_out_aag(X308, .(T135, X310), .(T135, T137))
U16_aag(T123, T124, T76, delH_out_aag(T123, T124, T76)) → U17_aag(T123, T124, T76, confF_in_g(T124))
U17_aag(T123, T124, T76, confF_out_g(T124)) → pM_out_aag(T123, T124, T76)
U15_gaaa(T74, T76, X165, X166, pM_out_aag(X165, X166, T76)) → pG_out_gaaa(T74, T76, X165, X166)
U5_g(T74, pG_out_gaaa(T74, X164, X165, X166)) → confF_out_g(T74)
U13_aag(T53, T54, T47, confF_out_g(T54)) → pL_out_aag(T53, T54, T47)
U11_agaaa(T46, T17, T47, X4, X5, pL_out_aag(X4, X5, T47)) → pK_out_agaaa(T46, T17, T47, X4, X5)
U9_agaaaaa(T16, T11, T17, X25, X26, X4, X5, pK_out_agaaa(X25, T17, X26, X4, X5)) → pB_out_agaaaaa(T16, T11, T17, X25, X26, X4, X5)
U1_g(T11, pB_out_agaaaaa(X23, T11, X24, X25, X26, X4, X5)) → confA_out_g(T11)

The argument filtering Pi contains the following mapping:
confA_in_g(x1)  =  confA_in_g(x1)
U1_g(x1, x2)  =  U1_g(x1, x2)
pB_in_agaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  pB_in_agaaaaa(x2)
U8_agaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U8_agaaaaa(x2, x8)
delC_in_aga(x1, x2, x3)  =  delC_in_aga(x2)
.(x1, x2)  =  .(x2)
delC_out_aga(x1, x2, x3)  =  delC_out_aga(x2, x3)
U2_aga(x1, x2, x3, x4, x5)  =  U2_aga(x3, x5)
U9_agaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U9_agaaaaa(x2, x3, x8)
pK_in_agaaa(x1, x2, x3, x4, x5)  =  pK_in_agaaa(x2)
U10_agaaa(x1, x2, x3, x4, x5, x6)  =  U10_agaaa(x2, x6)
U11_agaaa(x1, x2, x3, x4, x5, x6)  =  U11_agaaa(x2, x3, x6)
pL_in_aag(x1, x2, x3)  =  pL_in_aag(x3)
U12_aag(x1, x2, x3, x4)  =  U12_aag(x3, x4)
delD_in_aag(x1, x2, x3)  =  delD_in_aag(x3)
delD_out_aag(x1, x2, x3)  =  delD_out_aag(x2, x3)
U3_aag(x1, x2, x3, x4, x5)  =  U3_aag(x4, x5)
U13_aag(x1, x2, x3, x4)  =  U13_aag(x2, x3, x4)
confF_in_g(x1)  =  confF_in_g(x1)
U5_g(x1, x2)  =  U5_g(x1, x2)
pG_in_gaaa(x1, x2, x3, x4)  =  pG_in_gaaa(x1)
U14_gaaa(x1, x2, x3, x4, x5)  =  U14_gaaa(x1, x5)
del2I_in_ga(x1, x2)  =  del2I_in_ga(x1)
U7_ga(x1, x2, x3)  =  U7_ga(x1, x3)
pJ_in_agaaa(x1, x2, x3, x4, x5)  =  pJ_in_agaaa(x2)
U18_agaaa(x1, x2, x3, x4, x5, x6)  =  U18_agaaa(x2, x6)
delE_in_aga(x1, x2, x3)  =  delE_in_aga(x2)
delE_out_aga(x1, x2, x3)  =  delE_out_aga(x2, x3)
U4_aga(x1, x2, x3, x4, x5)  =  U4_aga(x3, x5)
U19_agaaa(x1, x2, x3, x4, x5, x6)  =  U19_agaaa(x2, x3, x6)
pJ_out_agaaa(x1, x2, x3, x4, x5)  =  pJ_out_agaaa(x2, x3, x5)
del2I_out_ga(x1, x2)  =  del2I_out_ga(x1, x2)
U15_gaaa(x1, x2, x3, x4, x5)  =  U15_gaaa(x1, x2, x5)
pM_in_aag(x1, x2, x3)  =  pM_in_aag(x3)
U16_aag(x1, x2, x3, x4)  =  U16_aag(x3, x4)
delH_in_aag(x1, x2, x3)  =  delH_in_aag(x3)
delH_out_aag(x1, x2, x3)  =  delH_out_aag(x2, x3)
U6_aag(x1, x2, x3, x4, x5)  =  U6_aag(x4, x5)
U17_aag(x1, x2, x3, x4)  =  U17_aag(x2, x3, x4)
confF_out_g(x1)  =  confF_out_g(x1)
pM_out_aag(x1, x2, x3)  =  pM_out_aag(x2, x3)
pG_out_gaaa(x1, x2, x3, x4)  =  pG_out_gaaa(x1, x2, x4)
pL_out_aag(x1, x2, x3)  =  pL_out_aag(x2, x3)
pK_out_agaaa(x1, x2, x3, x4, x5)  =  pK_out_agaaa(x2, x3, x5)
pB_out_agaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  pB_out_agaaaaa(x2, x3, x5, x7)
confA_out_g(x1)  =  confA_out_g(x1)
DELD_IN_AAG(x1, x2, x3)  =  DELD_IN_AAG(x3)

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

(31) UsableRulesProof (EQUIVALENT transformation)

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

(32) Obligation:

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

DELD_IN_AAG(X141, .(T65, X143), .(T65, T66)) → DELD_IN_AAG(X141, X143, T66)

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

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

(33) PiDPToQDPProof (SOUND transformation)

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

(34) Obligation:

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

DELD_IN_AAG(.(T66)) → DELD_IN_AAG(T66)

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

(35) QDPSizeChangeProof (EQUIVALENT transformation)

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

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

• DELD_IN_AAG(.(T66)) → DELD_IN_AAG(T66)
  The graph contains the following edges 1 > 1

(36) YES

(37) Obligation:

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

DELC_IN_AGA(X70, .(T36, T37), .(T36, X71)) → DELC_IN_AGA(X70, T37, X71)

The TRS R consists of the following rules:

confA_in_g(T11) → U1_g(T11, pB_in_agaaaaa(X23, T11, X24, X25, X26, X4, X5))
pB_in_agaaaaa(T16, T11, T17, X25, X26, X4, X5) → U8_agaaaaa(T16, T11, T17, X25, X26, X4, X5, delC_in_aga(T16, T11, T17))
delC_in_aga(T30, .(T30, T31), T31) → delC_out_aga(T30, .(T30, T31), T31)
delC_in_aga(X70, .(T36, T37), .(T36, X71)) → U2_aga(X70, T36, T37, X71, delC_in_aga(X70, T37, X71))
U2_aga(X70, T36, T37, X71, delC_out_aga(X70, T37, X71)) → delC_out_aga(X70, .(T36, T37), .(T36, X71))
U8_agaaaaa(T16, T11, T17, X25, X26, X4, X5, delC_out_aga(T16, T11, T17)) → U9_agaaaaa(T16, T11, T17, X25, X26, X4, X5, pK_in_agaaa(X25, T17, X26, X4, X5))
pK_in_agaaa(T46, T17, T47, X4, X5) → U10_agaaa(T46, T17, T47, X4, X5, delC_in_aga(T46, T17, T47))
U10_agaaa(T46, T17, T47, X4, X5, delC_out_aga(T46, T17, T47)) → U11_agaaa(T46, T17, T47, X4, X5, pL_in_aag(X4, X5, T47))
pL_in_aag(T53, T54, T47) → U12_aag(T53, T54, T47, delD_in_aag(T53, T54, T47))
delD_in_aag(X121, .(X121, T60), T60) → delD_out_aag(X121, .(X121, T60), T60)
delD_in_aag(X141, .(T65, X143), .(T65, T66)) → U3_aag(X141, T65, X143, T66, delD_in_aag(X141, X143, T66))
U3_aag(X141, T65, X143, T66, delD_out_aag(X141, X143, T66)) → delD_out_aag(X141, .(T65, X143), .(T65, T66))
U12_aag(T53, T54, T47, delD_out_aag(T53, T54, T47)) → U13_aag(T53, T54, T47, confF_in_g(T54))
confF_in_g(T74) → U5_g(T74, pG_in_gaaa(T74, X164, X165, X166))
pG_in_gaaa(T74, T76, X165, X166) → U14_gaaa(T74, T76, X165, X166, del2I_in_ga(T74, T76))
del2I_in_ga(T86, X199) → U7_ga(T86, X199, pJ_in_agaaa(X196, T86, X197, X198, X199))
pJ_in_agaaa(T91, T86, T92, X198, X199) → U18_agaaa(T91, T86, T92, X198, X199, delE_in_aga(T91, T86, T92))
delE_in_aga(T109, .(T109, T110), T110) → delE_out_aga(T109, .(T109, T110), T110)
delE_in_aga(X249, .(T115, T117), .(T115, X250)) → U4_aga(X249, T115, T117, X250, delE_in_aga(X249, T117, X250))
U4_aga(X249, T115, T117, X250, delE_out_aga(X249, T117, X250)) → delE_out_aga(X249, .(T115, T117), .(T115, X250))
U18_agaaa(T91, T86, T92, X198, X199, delE_out_aga(T91, T86, T92)) → U19_agaaa(T91, T86, T92, X198, X199, delE_in_aga(X198, T92, X199))
U19_agaaa(T91, T86, T92, X198, X199, delE_out_aga(X198, T92, X199)) → pJ_out_agaaa(T91, T86, T92, X198, X199)
U7_ga(T86, X199, pJ_out_agaaa(X196, T86, X197, X198, X199)) → del2I_out_ga(T86, X199)
U14_gaaa(T74, T76, X165, X166, del2I_out_ga(T74, T76)) → U15_gaaa(T74, T76, X165, X166, pM_in_aag(X165, X166, T76))
pM_in_aag(T123, T124, T76) → U16_aag(T123, T124, T76, delH_in_aag(T123, T124, T76))
delH_in_aag(X288, .(X288, T130), T130) → delH_out_aag(X288, .(X288, T130), T130)
delH_in_aag(X308, .(T135, X310), .(T135, T137)) → U6_aag(X308, T135, X310, T137, delH_in_aag(X308, X310, T137))
U6_aag(X308, T135, X310, T137, delH_out_aag(X308, X310, T137)) → delH_out_aag(X308, .(T135, X310), .(T135, T137))
U16_aag(T123, T124, T76, delH_out_aag(T123, T124, T76)) → U17_aag(T123, T124, T76, confF_in_g(T124))
U17_aag(T123, T124, T76, confF_out_g(T124)) → pM_out_aag(T123, T124, T76)
U15_gaaa(T74, T76, X165, X166, pM_out_aag(X165, X166, T76)) → pG_out_gaaa(T74, T76, X165, X166)
U5_g(T74, pG_out_gaaa(T74, X164, X165, X166)) → confF_out_g(T74)
U13_aag(T53, T54, T47, confF_out_g(T54)) → pL_out_aag(T53, T54, T47)
U11_agaaa(T46, T17, T47, X4, X5, pL_out_aag(X4, X5, T47)) → pK_out_agaaa(T46, T17, T47, X4, X5)
U9_agaaaaa(T16, T11, T17, X25, X26, X4, X5, pK_out_agaaa(X25, T17, X26, X4, X5)) → pB_out_agaaaaa(T16, T11, T17, X25, X26, X4, X5)
U1_g(T11, pB_out_agaaaaa(X23, T11, X24, X25, X26, X4, X5)) → confA_out_g(T11)

The argument filtering Pi contains the following mapping:
confA_in_g(x1)  =  confA_in_g(x1)
U1_g(x1, x2)  =  U1_g(x1, x2)
pB_in_agaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  pB_in_agaaaaa(x2)
U8_agaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U8_agaaaaa(x2, x8)
delC_in_aga(x1, x2, x3)  =  delC_in_aga(x2)
.(x1, x2)  =  .(x2)
delC_out_aga(x1, x2, x3)  =  delC_out_aga(x2, x3)
U2_aga(x1, x2, x3, x4, x5)  =  U2_aga(x3, x5)
U9_agaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U9_agaaaaa(x2, x3, x8)
pK_in_agaaa(x1, x2, x3, x4, x5)  =  pK_in_agaaa(x2)
U10_agaaa(x1, x2, x3, x4, x5, x6)  =  U10_agaaa(x2, x6)
U11_agaaa(x1, x2, x3, x4, x5, x6)  =  U11_agaaa(x2, x3, x6)
pL_in_aag(x1, x2, x3)  =  pL_in_aag(x3)
U12_aag(x1, x2, x3, x4)  =  U12_aag(x3, x4)
delD_in_aag(x1, x2, x3)  =  delD_in_aag(x3)
delD_out_aag(x1, x2, x3)  =  delD_out_aag(x2, x3)
U3_aag(x1, x2, x3, x4, x5)  =  U3_aag(x4, x5)
U13_aag(x1, x2, x3, x4)  =  U13_aag(x2, x3, x4)
confF_in_g(x1)  =  confF_in_g(x1)
U5_g(x1, x2)  =  U5_g(x1, x2)
pG_in_gaaa(x1, x2, x3, x4)  =  pG_in_gaaa(x1)
U14_gaaa(x1, x2, x3, x4, x5)  =  U14_gaaa(x1, x5)
del2I_in_ga(x1, x2)  =  del2I_in_ga(x1)
U7_ga(x1, x2, x3)  =  U7_ga(x1, x3)
pJ_in_agaaa(x1, x2, x3, x4, x5)  =  pJ_in_agaaa(x2)
U18_agaaa(x1, x2, x3, x4, x5, x6)  =  U18_agaaa(x2, x6)
delE_in_aga(x1, x2, x3)  =  delE_in_aga(x2)
delE_out_aga(x1, x2, x3)  =  delE_out_aga(x2, x3)
U4_aga(x1, x2, x3, x4, x5)  =  U4_aga(x3, x5)
U19_agaaa(x1, x2, x3, x4, x5, x6)  =  U19_agaaa(x2, x3, x6)
pJ_out_agaaa(x1, x2, x3, x4, x5)  =  pJ_out_agaaa(x2, x3, x5)
del2I_out_ga(x1, x2)  =  del2I_out_ga(x1, x2)
U15_gaaa(x1, x2, x3, x4, x5)  =  U15_gaaa(x1, x2, x5)
pM_in_aag(x1, x2, x3)  =  pM_in_aag(x3)
U16_aag(x1, x2, x3, x4)  =  U16_aag(x3, x4)
delH_in_aag(x1, x2, x3)  =  delH_in_aag(x3)
delH_out_aag(x1, x2, x3)  =  delH_out_aag(x2, x3)
U6_aag(x1, x2, x3, x4, x5)  =  U6_aag(x4, x5)
U17_aag(x1, x2, x3, x4)  =  U17_aag(x2, x3, x4)
confF_out_g(x1)  =  confF_out_g(x1)
pM_out_aag(x1, x2, x3)  =  pM_out_aag(x2, x3)
pG_out_gaaa(x1, x2, x3, x4)  =  pG_out_gaaa(x1, x2, x4)
pL_out_aag(x1, x2, x3)  =  pL_out_aag(x2, x3)
pK_out_agaaa(x1, x2, x3, x4, x5)  =  pK_out_agaaa(x2, x3, x5)
pB_out_agaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  pB_out_agaaaaa(x2, x3, x5, x7)
confA_out_g(x1)  =  confA_out_g(x1)
DELC_IN_AGA(x1, x2, x3)  =  DELC_IN_AGA(x2)

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

(38) UsableRulesProof (EQUIVALENT transformation)

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

(39) Obligation:

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

DELC_IN_AGA(X70, .(T36, T37), .(T36, X71)) → DELC_IN_AGA(X70, T37, X71)

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

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

(40) PiDPToQDPProof (SOUND transformation)

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

(41) Obligation:

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

DELC_IN_AGA(.(T37)) → DELC_IN_AGA(T37)

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

(42) 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:

• DELC_IN_AGA(.(T37)) → DELC_IN_AGA(T37)
  The graph contains the following edges 1 > 1

(43) YES