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

(0) Obligation:

Clauses:

reach(X, Y, Edges, Not_Visited) :- member(.(X, .(Y, [])), Edges).
reach(X, Z, Edges, Not_Visited) :- ','(member(.(X, .(Y, [])), Edges), ','(member(Y, Not_Visited), ','(delete(Y, Not_Visited, V1), reach(Y, Z, Edges, V1)))).
member(H, .(H, L)).
member(X, .(H, L)) :- member(X, L).
delete(X, .(X, Y), Y).
delete(X, .(H, T1), .(H, T2)) :- delete(X, T1, T2).

Query: reach(g,g,g,g)

(1) PrologToPiTRSViaGraphTransformerProof (SOUND transformation)

Transformed Prolog program to (Pi-)TRS.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="A: reach(T3, T4, T5, T6)\n\nT3 is ground\nT4 is ground\nT5 is ground\nT6 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
5 [label="5: reach(T3, T4, T5, T6), SCOPE: 1, CLAUSE: 0 | reach(T3, T4, T5, T6), SCOPE: 1, CLAUSE: 1\n\nT3 is ground\nT4 is ground\nT5 is ground\nT6 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: member(.(T11, .(T12, [])), T13) | reach(T11, T12, T13, T14), SCOPE: 1, CLAUSE: 1\n\nT11 is ground\nT12 is ground\nT13 is ground\nT14 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: member(.(T11, .(T12, [])), T13), SCOPE: 2, CLAUSE: 2 | member(.(T11, .(T12, [])), T13), SCOPE: 2, CLAUSE: 3 | ?_2 | reach(T11, T12, T13, T14), SCOPE: 1, CLAUSE: 1\n\nT11 is ground\nT12 is ground\nT13 is ground\nT14 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: member(.(T11, .(T12, [])), T13), SCOPE: 2, CLAUSE: 2\n\nT11 is ground\nT12 is ground\nT13 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: member(.(T11, .(T12, [])), T13), SCOPE: 2, CLAUSE: 3 | ?_2 | reach(T11, T12, T13, T14), SCOPE: 1, CLAUSE: 1\n\nT11 is ground\nT12 is ground\nT13 is ground\nT14 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: member(.(T11, .(T12, [])), T13), SCOPE: 2, CLAUSE: 3\n\nT11 is ground\nT12 is ground\nT13 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: ?_2 | reach(T11, T12, T13, T14), SCOPE: 1, CLAUSE: 1\n\nT11 is ground\nT12 is ground\nT13 is ground\nT14 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="B: member(.(T46, .(T47, [])), T49)\n\nT46 is ground\nT47 is ground\nT49 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: member(.(T46, .(T47, [])), T49), SCOPE: 3, CLAUSE: 2 | member(.(T46, .(T47, [])), T49), SCOPE: 3, CLAUSE: 3\n\nT46 is ground\nT47 is ground\nT49 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: member(.(T46, .(T47, [])), T49), SCOPE: 3, CLAUSE: 2\n\nT46 is ground\nT47 is ground\nT49 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: member(.(T46, .(T47, [])), T49), SCOPE: 3, CLAUSE: 3\n\nT46 is ground\nT47 is ground\nT49 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="38: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
42 [label="42: member(.(T79, .(T80, [])), T82)\n\nT79 is ground\nT80 is ground\nT82 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
43 [label="43: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
49 [label="49: reach(T11, T12, T13, T14), SCOPE: 1, CLAUSE: 1\n\nT11 is ground\nT12 is ground\nT13 is ground\nT14 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
57 [label="C: ','(member(.(T103, .(X81, [])), T105), ','(member(X81, T106), ','(delete(X81, T106, X82), reach(X81, T104, T105, X82))))\n\nT103 is ground\nT104 is ground\nT105 is ground\nT106 is ground\nX81 is free\nX82 is free", fontsize=16, style=filled, fillcolor=lightyellow];
58 [label="D: member(.(T103, .(X81, [])), T105)\n\nT103 is ground\nT105 is ground\nX81 is free", fontsize=16, style=filled, fillcolor=lightyellow];
59 [label="G: ','(member(T113, T106), ','(delete(T113, T106, X82), reach(T113, T104, T105, X82)))\n\nT104 is ground\nT105 is ground\nT106 is ground\nT113 is ground\nX82 is free", fontsize=16, style=filled, fillcolor=lightyellow];
60 [label="60: member(.(T103, .(X81, [])), T105), SCOPE: 4, CLAUSE: 2 | member(.(T103, .(X81, [])), T105), SCOPE: 4, CLAUSE: 3\n\nT103 is ground\nT105 is ground\nX81 is free", fontsize=16, style=filled, fillcolor=lightyellow];
62 [label="62: member(.(T103, .(X81, [])), T105), SCOPE: 4, CLAUSE: 2\n\nT103 is ground\nT105 is ground\nX81 is free", fontsize=16, style=filled, fillcolor=lightyellow];
63 [label="63: member(.(T103, .(X81, [])), T105), SCOPE: 4, CLAUSE: 3\n\nT103 is ground\nT105 is ground\nX81 is free", fontsize=16, style=filled, fillcolor=lightyellow];
67 [label="67: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
68 [label="68: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
69 [label="69: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
71 [label="71: member(.(T141, .(X121, [])), T143)\n\nT141 is ground\nT143 is ground\nX121 is free", fontsize=16, style=filled, fillcolor=lightyellow];
74 [label="74: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
78 [label="E: member(T113, T106)\n\nT106 is ground\nT113 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
79 [label="H: ','(delete(T113, T106, X82), reach(T113, T104, T105, X82))\n\nT104 is ground\nT105 is ground\nT106 is ground\nT113 is ground\nX82 is free", fontsize=16, style=filled, fillcolor=lightyellow];
82 [label="82: member(T113, T106), SCOPE: 5, CLAUSE: 2 | member(T113, T106), SCOPE: 5, CLAUSE: 3\n\nT106 is ground\nT113 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
84 [label="84: member(T113, T106), SCOPE: 5, CLAUSE: 2\n\nT106 is ground\nT113 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
85 [label="85: member(T113, T106), SCOPE: 5, CLAUSE: 3\n\nT106 is ground\nT113 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
86 [label="86: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
87 [label="87: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
89 [label="89: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
92 [label="92: member(T174, T176)\n\nT174 is ground\nT176 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
93 [label="93: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
104 [label="F: delete(T113, T106, X82)\n\nT106 is ground\nT113 is ground\nX82 is free", fontsize=16, style=filled, fillcolor=lightyellow];
105 [label="105: reach(T113, T104, T105, T185)\n\nT104 is ground\nT105 is ground\nT113 is ground\nT185 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
108 [label="108: delete(T113, T106, X82), SCOPE: 6, CLAUSE: 4 | delete(T113, T106, X82), SCOPE: 6, CLAUSE: 5\n\nT106 is ground\nT113 is ground\nX82 is free", fontsize=16, style=filled, fillcolor=lightyellow];
109 [label="109: delete(T113, T106, X82), SCOPE: 6, CLAUSE: 4\n\nT106 is ground\nT113 is ground\nX82 is free", fontsize=16, style=filled, fillcolor=lightyellow];
110 [label="110: delete(T113, T106, X82), SCOPE: 6, CLAUSE: 5\n\nT106 is ground\nT113 is ground\nX82 is free", fontsize=16, style=filled, fillcolor=lightyellow];
113 [label="113: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
114 [label="114: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
115 [label="115: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
119 [label="119: delete(T206, T208, X191)\n\nT206 is ground\nT208 is ground\nX191 is free", fontsize=16, style=filled, fillcolor=lightyellow];
122 [label="122: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
123 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 123 [arrowhead = none ];
123 -> 5;

124 [label="ONLY EVAL with clause\nreach(X5, X6, X7, X8) :- member(.(X5, .(X6, [])), X7).\nand substitutionT3 -> T11,\nX5 -> T11,\nT4 -> T12,\nX6 -> T12,\nT5 -> T13,\nX7 -> T13,\nT6 -> T14,\nX8 -> T14", fontsize=14, style = filled, fillcolor = lightblue];
5 -> 124 [arrowhead = none ];
124 -> 8;

125 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
8 -> 125 [arrowhead = none ];
125 -> 9;

126 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
9 -> 126 [arrowhead = none ];
126 -> 13;

127 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
9 -> 127 [arrowhead = none ];
127 -> 14;

128 [label="EVAL with clause\nmember(X17, .(X17, X18)).\nand substitutionT11 -> T27,\nT12 -> T28,\nX17 -> .(T27, .(T28, [])),\nX18 -> T29,\nT13 -> .(.(T27, .(T28, [])), T29)", fontsize=14, style = filled, fillcolor = lightblue];
13 -> 128 [arrowhead = none ];
128 -> 15;

129 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
13 -> 129 [arrowhead = none ];
129 -> 16;

130 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
14 -> 130 [arrowhead = none ];
130 -> 19;

131 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
14 -> 131 [arrowhead = none ];
131 -> 20;

132 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
15 -> 132 [arrowhead = none ];
132 -> 17;

133 [label="EVAL with clause\nmember(X31, .(X32, X33)) :- member(X31, X33).\nand substitutionT11 -> T46,\nT12 -> T47,\nX31 -> .(T46, .(T47, [])),\nX32 -> T48,\nX33 -> T49,\nT13 -> .(T48, T49)", fontsize=14, style = filled, fillcolor = lightblue];
19 -> 133 [arrowhead = none ];
133 -> 21;

134 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
19 -> 134 [arrowhead = none ];
134 -> 23;

135 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
20 -> 135 [arrowhead = none ];
135 -> 49;

136 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
21 -> 136 [arrowhead = none ];
136 -> 31;

137 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
31 -> 137 [arrowhead = none ];
137 -> 34;

138 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
31 -> 138 [arrowhead = none ];
138 -> 35;

139 [label="EVAL with clause\nmember(X46, .(X46, X47)).\nand substitutionT46 -> T68,\nT47 -> T69,\nX46 -> .(T68, .(T69, [])),\nX47 -> T70,\nT49 -> .(.(T68, .(T69, [])), T70)", fontsize=14, style = filled, fillcolor = lightblue];
34 -> 139 [arrowhead = none ];
139 -> 36;

140 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
34 -> 140 [arrowhead = none ];
140 -> 37;

141 [label="EVAL with clause\nmember(X54, .(X55, X56)) :- member(X54, X56).\nand substitutionT46 -> T79,\nT47 -> T80,\nX54 -> .(T79, .(T80, [])),\nX55 -> T81,\nX56 -> T82,\nT49 -> .(T81, T82)", fontsize=14, style = filled, fillcolor = lightblue];
35 -> 141 [arrowhead = none ];
141 -> 42;

142 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
35 -> 142 [arrowhead = none ];
142 -> 43;

143 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
36 -> 143 [arrowhead = none ];
143 -> 38;

144 [label="INSTANCE with matching:\nT46 -> T79\nT47 -> T80\nT49 -> T82", fontsize=14, style = filled, fillcolor = lightgrey];
42 -> 144 [arrowhead = none , style=dashed];
144 -> 21[style=dashed];

145 [label="ONLY EVAL with clause\nreach(X77, X78, X79, X80) :- ','(member(.(X77, .(X81, [])), X79), ','(member(X81, X80), ','(delete(X81, X80, X82), reach(X81, X78, X79, X82)))).\nand substitutionT11 -> T103,\nX77 -> T103,\nT12 -> T104,\nX78 -> T104,\nT13 -> T105,\nX79 -> T105,\nT14 -> T106,\nX80 -> T106", fontsize=14, style = filled, fillcolor = lightblue];
49 -> 145 [arrowhead = none ];
145 -> 57;

146 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
57 -> 146 [arrowhead = none ];
146 -> 58;

147 [label="SPLIT 2\nnew knowledge:\nT103 is ground\nT113 is ground\nT105 is ground\nreplacements:X81 -> T113", fontsize=14, style = filled, fillcolor = violet];
57 -> 147 [arrowhead = none ];
147 -> 59;

148 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
58 -> 148 [arrowhead = none ];
148 -> 60;

149 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
59 -> 149 [arrowhead = none ];
149 -> 78;

150 [label="SPLIT 2\nnew knowledge:\nT113 is ground\nT106 is ground", fontsize=14, style = filled, fillcolor = violet];
59 -> 150 [arrowhead = none ];
150 -> 79;

151 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
60 -> 151 [arrowhead = none ];
151 -> 62;

152 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
60 -> 152 [arrowhead = none ];
152 -> 63;

153 [label="EVAL with clause\nmember(X107, .(X107, X108)).\nand substitutionT103 -> T132,\nX81 -> T133,\nX107 -> .(T132, .(T133, [])),\nX109 -> T133,\nX108 -> T134,\nT105 -> .(.(T132, .(T133, [])), T134)", fontsize=14, style = filled, fillcolor = lightblue];
62 -> 153 [arrowhead = none ];
153 -> 67;

154 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
62 -> 154 [arrowhead = none ];
154 -> 68;

155 [label="EVAL with clause\nmember(X118, .(X119, X120)) :- member(X118, X120).\nand substitutionT103 -> T141,\nX81 -> X121,\nX118 -> .(T141, .(X121, [])),\nX119 -> T142,\nX120 -> T143,\nT105 -> .(T142, T143)", fontsize=14, style = filled, fillcolor = lightblue];
63 -> 155 [arrowhead = none ];
155 -> 71;

156 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
63 -> 156 [arrowhead = none ];
156 -> 74;

157 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
67 -> 157 [arrowhead = none ];
157 -> 69;

158 [label="INSTANCE with matching:\nT103 -> T141\nX81 -> X121\nT105 -> T143", fontsize=14, style = filled, fillcolor = lightgrey];
71 -> 158 [arrowhead = none , style=dashed];
158 -> 58[style=dashed];

159 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
78 -> 159 [arrowhead = none ];
159 -> 82;

160 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
79 -> 160 [arrowhead = none ];
160 -> 104;

161 [label="SPLIT 2\nnew knowledge:\nT113 is ground\nT106 is ground\nT185 is ground\nreplacements:X82 -> T185", fontsize=14, style = filled, fillcolor = violet];
79 -> 161 [arrowhead = none ];
161 -> 105;

162 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
82 -> 162 [arrowhead = none ];
162 -> 84;

163 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
82 -> 163 [arrowhead = none ];
163 -> 85;

164 [label="EVAL with clause\nmember(X144, .(X144, X145)).\nand substitutionT113 -> T166,\nX144 -> T166,\nX145 -> T167,\nT106 -> .(T166, T167)", fontsize=14, style = filled, fillcolor = lightblue];
84 -> 164 [arrowhead = none ];
164 -> 86;

165 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
84 -> 165 [arrowhead = none ];
165 -> 87;

166 [label="EVAL with clause\nmember(X152, .(X153, X154)) :- member(X152, X154).\nand substitutionT113 -> T174,\nX152 -> T174,\nX153 -> T175,\nX154 -> T176,\nT106 -> .(T175, T176)", fontsize=14, style = filled, fillcolor = lightblue];
85 -> 166 [arrowhead = none ];
166 -> 92;

167 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
85 -> 167 [arrowhead = none ];
167 -> 93;

168 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
86 -> 168 [arrowhead = none ];
168 -> 89;

169 [label="INSTANCE with matching:\nT113 -> T174\nT106 -> T176", fontsize=14, style = filled, fillcolor = lightgrey];
92 -> 169 [arrowhead = none , style=dashed];
169 -> 78[style=dashed];

170 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
104 -> 170 [arrowhead = none ];
170 -> 108;

171 [label="INSTANCE with matching:\nT3 -> T113\nT4 -> T104\nT5 -> T105\nT6 -> T185", fontsize=14, style = filled, fillcolor = lightgrey];
105 -> 171 [arrowhead = none , style=dashed];
171 -> 1[style=dashed];

172 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
108 -> 172 [arrowhead = none ];
172 -> 109;

173 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
108 -> 173 [arrowhead = none ];
173 -> 110;

174 [label="EVAL with clause\ndelete(X175, .(X175, X176), X176).\nand substitutionT113 -> T198,\nX175 -> T198,\nX176 -> T199,\nT106 -> .(T198, T199),\nX82 -> T199", fontsize=14, style = filled, fillcolor = lightblue];
109 -> 174 [arrowhead = none ];
174 -> 113;

175 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
109 -> 175 [arrowhead = none ];
175 -> 114;

176 [label="EVAL with clause\ndelete(X187, .(X188, X189), .(X188, X190)) :- delete(X187, X189, X190).\nand substitutionT113 -> T206,\nX187 -> T206,\nX188 -> T207,\nX189 -> T208,\nT106 -> .(T207, T208),\nX190 -> X191,\nX82 -> .(T207, X191)", fontsize=14, style = filled, fillcolor = lightblue];
110 -> 176 [arrowhead = none ];
176 -> 119;

177 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
110 -> 177 [arrowhead = none ];
177 -> 122;

178 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
113 -> 178 [arrowhead = none ];
178 -> 115;

179 [label="INSTANCE with matching:\nT113 -> T206\nT106 -> T208\nX82 -> X191", fontsize=14, style = filled, fillcolor = lightgrey];
119 -> 179 [arrowhead = none , style=dashed];
179 -> 104[style=dashed];

}

(2) Obligation:

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

reachA_in_gggg(T27, T28, .(.(T27, .(T28, [])), T29), T14) → reachA_out_gggg(T27, T28, .(.(T27, .(T28, [])), T29), T14)
reachA_in_gggg(T46, T47, .(T48, T49), T14) → U1_gggg(T46, T47, T48, T49, T14, memberB_in_ggg(T46, T47, T49))
memberB_in_ggg(T68, T69, .(.(T68, .(T69, [])), T70)) → memberB_out_ggg(T68, T69, .(.(T68, .(T69, [])), T70))
memberB_in_ggg(T79, T80, .(T81, T82)) → U3_ggg(T79, T80, T81, T82, memberB_in_ggg(T79, T80, T82))
U3_ggg(T79, T80, T81, T82, memberB_out_ggg(T79, T80, T82)) → memberB_out_ggg(T79, T80, .(T81, T82))
U1_gggg(T46, T47, T48, T49, T14, memberB_out_ggg(T46, T47, T49)) → reachA_out_gggg(T46, T47, .(T48, T49), T14)
reachA_in_gggg(T103, T104, T105, T106) → U2_gggg(T103, T104, T105, T106, pC_in_gaggag(T103, X81, T105, T106, X82, T104))
pC_in_gaggag(T103, T113, T105, T106, X82, T104) → U7_gaggag(T103, T113, T105, T106, X82, T104, memberD_in_gag(T103, T113, T105))
memberD_in_gag(T132, T133, .(.(T132, .(T133, [])), T134)) → memberD_out_gag(T132, T133, .(.(T132, .(T133, [])), T134))
memberD_in_gag(T141, X121, .(T142, T143)) → U4_gag(T141, X121, T142, T143, memberD_in_gag(T141, X121, T143))
U4_gag(T141, X121, T142, T143, memberD_out_gag(T141, X121, T143)) → memberD_out_gag(T141, X121, .(T142, T143))
U7_gaggag(T103, T113, T105, T106, X82, T104, memberD_out_gag(T103, T113, T105)) → U8_gaggag(T103, T113, T105, T106, X82, T104, pG_in_ggagg(T113, T106, X82, T104, T105))
pG_in_ggagg(T113, T106, X82, T104, T105) → U9_ggagg(T113, T106, X82, T104, T105, memberE_in_gg(T113, T106))
memberE_in_gg(T166, .(T166, T167)) → memberE_out_gg(T166, .(T166, T167))
memberE_in_gg(T174, .(T175, T176)) → U5_gg(T174, T175, T176, memberE_in_gg(T174, T176))
U5_gg(T174, T175, T176, memberE_out_gg(T174, T176)) → memberE_out_gg(T174, .(T175, T176))
U9_ggagg(T113, T106, X82, T104, T105, memberE_out_gg(T113, T106)) → U10_ggagg(T113, T106, X82, T104, T105, pH_in_ggagg(T113, T106, X82, T104, T105))
pH_in_ggagg(T113, T106, T185, T104, T105) → U11_ggagg(T113, T106, T185, T104, T105, deleteF_in_gga(T113, T106, T185))
deleteF_in_gga(T198, .(T198, T199), T199) → deleteF_out_gga(T198, .(T198, T199), T199)
deleteF_in_gga(T206, .(T207, T208), .(T207, X191)) → U6_gga(T206, T207, T208, X191, deleteF_in_gga(T206, T208, X191))
U6_gga(T206, T207, T208, X191, deleteF_out_gga(T206, T208, X191)) → deleteF_out_gga(T206, .(T207, T208), .(T207, X191))
U11_ggagg(T113, T106, T185, T104, T105, deleteF_out_gga(T113, T106, T185)) → U12_ggagg(T113, T106, T185, T104, T105, reachA_in_gggg(T113, T104, T105, T185))
U12_ggagg(T113, T106, T185, T104, T105, reachA_out_gggg(T113, T104, T105, T185)) → pH_out_ggagg(T113, T106, T185, T104, T105)
U10_ggagg(T113, T106, X82, T104, T105, pH_out_ggagg(T113, T106, X82, T104, T105)) → pG_out_ggagg(T113, T106, X82, T104, T105)
U8_gaggag(T103, T113, T105, T106, X82, T104, pG_out_ggagg(T113, T106, X82, T104, T105)) → pC_out_gaggag(T103, T113, T105, T106, X82, T104)
U2_gggg(T103, T104, T105, T106, pC_out_gaggag(T103, X81, T105, T106, X82, T104)) → reachA_out_gggg(T103, T104, T105, T106)

The argument filtering Pi contains the following mapping:
reachA_in_gggg(x1, x2, x3, x4)  =  reachA_in_gggg(x1, x2, x3, x4)
.(x1, x2)  =  .(x1, x2)
[]  =  []
reachA_out_gggg(x1, x2, x3, x4)  =  reachA_out_gggg(x1, x2, x3, x4)
U1_gggg(x1, x2, x3, x4, x5, x6)  =  U1_gggg(x1, x2, x3, x4, x5, x6)
memberB_in_ggg(x1, x2, x3)  =  memberB_in_ggg(x1, x2, x3)
memberB_out_ggg(x1, x2, x3)  =  memberB_out_ggg(x1, x2, x3)
U3_ggg(x1, x2, x3, x4, x5)  =  U3_ggg(x1, x2, x3, x4, x5)
U2_gggg(x1, x2, x3, x4, x5)  =  U2_gggg(x1, x2, x3, x4, x5)
pC_in_gaggag(x1, x2, x3, x4, x5, x6)  =  pC_in_gaggag(x1, x3, x4, x6)
U7_gaggag(x1, x2, x3, x4, x5, x6, x7)  =  U7_gaggag(x1, x3, x4, x6, x7)
memberD_in_gag(x1, x2, x3)  =  memberD_in_gag(x1, x3)
memberD_out_gag(x1, x2, x3)  =  memberD_out_gag(x1, x2, x3)
U4_gag(x1, x2, x3, x4, x5)  =  U4_gag(x1, x3, x4, x5)
U8_gaggag(x1, x2, x3, x4, x5, x6, x7)  =  U8_gaggag(x1, x2, x3, x4, x6, x7)
pG_in_ggagg(x1, x2, x3, x4, x5)  =  pG_in_ggagg(x1, x2, x4, x5)
U9_ggagg(x1, x2, x3, x4, x5, x6)  =  U9_ggagg(x1, x2, x4, x5, x6)
memberE_in_gg(x1, x2)  =  memberE_in_gg(x1, x2)
memberE_out_gg(x1, x2)  =  memberE_out_gg(x1, x2)
U5_gg(x1, x2, x3, x4)  =  U5_gg(x1, x2, x3, x4)
U10_ggagg(x1, x2, x3, x4, x5, x6)  =  U10_ggagg(x1, x2, x4, x5, x6)
pH_in_ggagg(x1, x2, x3, x4, x5)  =  pH_in_ggagg(x1, x2, x4, x5)
U11_ggagg(x1, x2, x3, x4, x5, x6)  =  U11_ggagg(x1, x2, x4, x5, x6)
deleteF_in_gga(x1, x2, x3)  =  deleteF_in_gga(x1, x2)
deleteF_out_gga(x1, x2, x3)  =  deleteF_out_gga(x1, x2, x3)
U6_gga(x1, x2, x3, x4, x5)  =  U6_gga(x1, x2, x3, x5)
U12_ggagg(x1, x2, x3, x4, x5, x6)  =  U12_ggagg(x1, x2, x3, x4, x5, x6)
pH_out_ggagg(x1, x2, x3, x4, x5)  =  pH_out_ggagg(x1, x2, x3, x4, x5)
pG_out_ggagg(x1, x2, x3, x4, x5)  =  pG_out_ggagg(x1, x2, x3, x4, x5)
pC_out_gaggag(x1, x2, x3, x4, x5, x6)  =  pC_out_gaggag(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:

REACHA_IN_GGGG(T46, T47, .(T48, T49), T14) → U1_GGGG(T46, T47, T48, T49, T14, memberB_in_ggg(T46, T47, T49))
REACHA_IN_GGGG(T46, T47, .(T48, T49), T14) → MEMBERB_IN_GGG(T46, T47, T49)
MEMBERB_IN_GGG(T79, T80, .(T81, T82)) → U3_GGG(T79, T80, T81, T82, memberB_in_ggg(T79, T80, T82))
MEMBERB_IN_GGG(T79, T80, .(T81, T82)) → MEMBERB_IN_GGG(T79, T80, T82)
REACHA_IN_GGGG(T103, T104, T105, T106) → U2_GGGG(T103, T104, T105, T106, pC_in_gaggag(T103, X81, T105, T106, X82, T104))
REACHA_IN_GGGG(T103, T104, T105, T106) → PC_IN_GAGGAG(T103, X81, T105, T106, X82, T104)
PC_IN_GAGGAG(T103, T113, T105, T106, X82, T104) → U7_GAGGAG(T103, T113, T105, T106, X82, T104, memberD_in_gag(T103, T113, T105))
PC_IN_GAGGAG(T103, T113, T105, T106, X82, T104) → MEMBERD_IN_GAG(T103, T113, T105)
MEMBERD_IN_GAG(T141, X121, .(T142, T143)) → U4_GAG(T141, X121, T142, T143, memberD_in_gag(T141, X121, T143))
MEMBERD_IN_GAG(T141, X121, .(T142, T143)) → MEMBERD_IN_GAG(T141, X121, T143)
U7_GAGGAG(T103, T113, T105, T106, X82, T104, memberD_out_gag(T103, T113, T105)) → U8_GAGGAG(T103, T113, T105, T106, X82, T104, pG_in_ggagg(T113, T106, X82, T104, T105))
U7_GAGGAG(T103, T113, T105, T106, X82, T104, memberD_out_gag(T103, T113, T105)) → PG_IN_GGAGG(T113, T106, X82, T104, T105)
PG_IN_GGAGG(T113, T106, X82, T104, T105) → U9_GGAGG(T113, T106, X82, T104, T105, memberE_in_gg(T113, T106))
PG_IN_GGAGG(T113, T106, X82, T104, T105) → MEMBERE_IN_GG(T113, T106)
MEMBERE_IN_GG(T174, .(T175, T176)) → U5_GG(T174, T175, T176, memberE_in_gg(T174, T176))
MEMBERE_IN_GG(T174, .(T175, T176)) → MEMBERE_IN_GG(T174, T176)
U9_GGAGG(T113, T106, X82, T104, T105, memberE_out_gg(T113, T106)) → U10_GGAGG(T113, T106, X82, T104, T105, pH_in_ggagg(T113, T106, X82, T104, T105))
U9_GGAGG(T113, T106, X82, T104, T105, memberE_out_gg(T113, T106)) → PH_IN_GGAGG(T113, T106, X82, T104, T105)
PH_IN_GGAGG(T113, T106, T185, T104, T105) → U11_GGAGG(T113, T106, T185, T104, T105, deleteF_in_gga(T113, T106, T185))
PH_IN_GGAGG(T113, T106, T185, T104, T105) → DELETEF_IN_GGA(T113, T106, T185)
DELETEF_IN_GGA(T206, .(T207, T208), .(T207, X191)) → U6_GGA(T206, T207, T208, X191, deleteF_in_gga(T206, T208, X191))
DELETEF_IN_GGA(T206, .(T207, T208), .(T207, X191)) → DELETEF_IN_GGA(T206, T208, X191)
U11_GGAGG(T113, T106, T185, T104, T105, deleteF_out_gga(T113, T106, T185)) → U12_GGAGG(T113, T106, T185, T104, T105, reachA_in_gggg(T113, T104, T105, T185))
U11_GGAGG(T113, T106, T185, T104, T105, deleteF_out_gga(T113, T106, T185)) → REACHA_IN_GGGG(T113, T104, T105, T185)

The TRS R consists of the following rules:

reachA_in_gggg(T27, T28, .(.(T27, .(T28, [])), T29), T14) → reachA_out_gggg(T27, T28, .(.(T27, .(T28, [])), T29), T14)
reachA_in_gggg(T46, T47, .(T48, T49), T14) → U1_gggg(T46, T47, T48, T49, T14, memberB_in_ggg(T46, T47, T49))
memberB_in_ggg(T68, T69, .(.(T68, .(T69, [])), T70)) → memberB_out_ggg(T68, T69, .(.(T68, .(T69, [])), T70))
memberB_in_ggg(T79, T80, .(T81, T82)) → U3_ggg(T79, T80, T81, T82, memberB_in_ggg(T79, T80, T82))
U3_ggg(T79, T80, T81, T82, memberB_out_ggg(T79, T80, T82)) → memberB_out_ggg(T79, T80, .(T81, T82))
U1_gggg(T46, T47, T48, T49, T14, memberB_out_ggg(T46, T47, T49)) → reachA_out_gggg(T46, T47, .(T48, T49), T14)
reachA_in_gggg(T103, T104, T105, T106) → U2_gggg(T103, T104, T105, T106, pC_in_gaggag(T103, X81, T105, T106, X82, T104))
pC_in_gaggag(T103, T113, T105, T106, X82, T104) → U7_gaggag(T103, T113, T105, T106, X82, T104, memberD_in_gag(T103, T113, T105))
memberD_in_gag(T132, T133, .(.(T132, .(T133, [])), T134)) → memberD_out_gag(T132, T133, .(.(T132, .(T133, [])), T134))
memberD_in_gag(T141, X121, .(T142, T143)) → U4_gag(T141, X121, T142, T143, memberD_in_gag(T141, X121, T143))
U4_gag(T141, X121, T142, T143, memberD_out_gag(T141, X121, T143)) → memberD_out_gag(T141, X121, .(T142, T143))
U7_gaggag(T103, T113, T105, T106, X82, T104, memberD_out_gag(T103, T113, T105)) → U8_gaggag(T103, T113, T105, T106, X82, T104, pG_in_ggagg(T113, T106, X82, T104, T105))
pG_in_ggagg(T113, T106, X82, T104, T105) → U9_ggagg(T113, T106, X82, T104, T105, memberE_in_gg(T113, T106))
memberE_in_gg(T166, .(T166, T167)) → memberE_out_gg(T166, .(T166, T167))
memberE_in_gg(T174, .(T175, T176)) → U5_gg(T174, T175, T176, memberE_in_gg(T174, T176))
U5_gg(T174, T175, T176, memberE_out_gg(T174, T176)) → memberE_out_gg(T174, .(T175, T176))
U9_ggagg(T113, T106, X82, T104, T105, memberE_out_gg(T113, T106)) → U10_ggagg(T113, T106, X82, T104, T105, pH_in_ggagg(T113, T106, X82, T104, T105))
pH_in_ggagg(T113, T106, T185, T104, T105) → U11_ggagg(T113, T106, T185, T104, T105, deleteF_in_gga(T113, T106, T185))
deleteF_in_gga(T198, .(T198, T199), T199) → deleteF_out_gga(T198, .(T198, T199), T199)
deleteF_in_gga(T206, .(T207, T208), .(T207, X191)) → U6_gga(T206, T207, T208, X191, deleteF_in_gga(T206, T208, X191))
U6_gga(T206, T207, T208, X191, deleteF_out_gga(T206, T208, X191)) → deleteF_out_gga(T206, .(T207, T208), .(T207, X191))
U11_ggagg(T113, T106, T185, T104, T105, deleteF_out_gga(T113, T106, T185)) → U12_ggagg(T113, T106, T185, T104, T105, reachA_in_gggg(T113, T104, T105, T185))
U12_ggagg(T113, T106, T185, T104, T105, reachA_out_gggg(T113, T104, T105, T185)) → pH_out_ggagg(T113, T106, T185, T104, T105)
U10_ggagg(T113, T106, X82, T104, T105, pH_out_ggagg(T113, T106, X82, T104, T105)) → pG_out_ggagg(T113, T106, X82, T104, T105)
U8_gaggag(T103, T113, T105, T106, X82, T104, pG_out_ggagg(T113, T106, X82, T104, T105)) → pC_out_gaggag(T103, T113, T105, T106, X82, T104)
U2_gggg(T103, T104, T105, T106, pC_out_gaggag(T103, X81, T105, T106, X82, T104)) → reachA_out_gggg(T103, T104, T105, T106)

The argument filtering Pi contains the following mapping:
reachA_in_gggg(x1, x2, x3, x4)  =  reachA_in_gggg(x1, x2, x3, x4)
.(x1, x2)  =  .(x1, x2)
[]  =  []
reachA_out_gggg(x1, x2, x3, x4)  =  reachA_out_gggg(x1, x2, x3, x4)
U1_gggg(x1, x2, x3, x4, x5, x6)  =  U1_gggg(x1, x2, x3, x4, x5, x6)
memberB_in_ggg(x1, x2, x3)  =  memberB_in_ggg(x1, x2, x3)
memberB_out_ggg(x1, x2, x3)  =  memberB_out_ggg(x1, x2, x3)
U3_ggg(x1, x2, x3, x4, x5)  =  U3_ggg(x1, x2, x3, x4, x5)
U2_gggg(x1, x2, x3, x4, x5)  =  U2_gggg(x1, x2, x3, x4, x5)
pC_in_gaggag(x1, x2, x3, x4, x5, x6)  =  pC_in_gaggag(x1, x3, x4, x6)
U7_gaggag(x1, x2, x3, x4, x5, x6, x7)  =  U7_gaggag(x1, x3, x4, x6, x7)
memberD_in_gag(x1, x2, x3)  =  memberD_in_gag(x1, x3)
memberD_out_gag(x1, x2, x3)  =  memberD_out_gag(x1, x2, x3)
U4_gag(x1, x2, x3, x4, x5)  =  U4_gag(x1, x3, x4, x5)
U8_gaggag(x1, x2, x3, x4, x5, x6, x7)  =  U8_gaggag(x1, x2, x3, x4, x6, x7)
pG_in_ggagg(x1, x2, x3, x4, x5)  =  pG_in_ggagg(x1, x2, x4, x5)
U9_ggagg(x1, x2, x3, x4, x5, x6)  =  U9_ggagg(x1, x2, x4, x5, x6)
memberE_in_gg(x1, x2)  =  memberE_in_gg(x1, x2)
memberE_out_gg(x1, x2)  =  memberE_out_gg(x1, x2)
U5_gg(x1, x2, x3, x4)  =  U5_gg(x1, x2, x3, x4)
U10_ggagg(x1, x2, x3, x4, x5, x6)  =  U10_ggagg(x1, x2, x4, x5, x6)
pH_in_ggagg(x1, x2, x3, x4, x5)  =  pH_in_ggagg(x1, x2, x4, x5)
U11_ggagg(x1, x2, x3, x4, x5, x6)  =  U11_ggagg(x1, x2, x4, x5, x6)
deleteF_in_gga(x1, x2, x3)  =  deleteF_in_gga(x1, x2)
deleteF_out_gga(x1, x2, x3)  =  deleteF_out_gga(x1, x2, x3)
U6_gga(x1, x2, x3, x4, x5)  =  U6_gga(x1, x2, x3, x5)
U12_ggagg(x1, x2, x3, x4, x5, x6)  =  U12_ggagg(x1, x2, x3, x4, x5, x6)
pH_out_ggagg(x1, x2, x3, x4, x5)  =  pH_out_ggagg(x1, x2, x3, x4, x5)
pG_out_ggagg(x1, x2, x3, x4, x5)  =  pG_out_ggagg(x1, x2, x3, x4, x5)
pC_out_gaggag(x1, x2, x3, x4, x5, x6)  =  pC_out_gaggag(x1, x2, x3, x4, x5, x6)
REACHA_IN_GGGG(x1, x2, x3, x4)  =  REACHA_IN_GGGG(x1, x2, x3, x4)
U1_GGGG(x1, x2, x3, x4, x5, x6)  =  U1_GGGG(x1, x2, x3, x4, x5, x6)
MEMBERB_IN_GGG(x1, x2, x3)  =  MEMBERB_IN_GGG(x1, x2, x3)
U3_GGG(x1, x2, x3, x4, x5)  =  U3_GGG(x1, x2, x3, x4, x5)
U2_GGGG(x1, x2, x3, x4, x5)  =  U2_GGGG(x1, x2, x3, x4, x5)
PC_IN_GAGGAG(x1, x2, x3, x4, x5, x6)  =  PC_IN_GAGGAG(x1, x3, x4, x6)
U7_GAGGAG(x1, x2, x3, x4, x5, x6, x7)  =  U7_GAGGAG(x1, x3, x4, x6, x7)
MEMBERD_IN_GAG(x1, x2, x3)  =  MEMBERD_IN_GAG(x1, x3)
U4_GAG(x1, x2, x3, x4, x5)  =  U4_GAG(x1, x3, x4, x5)
U8_GAGGAG(x1, x2, x3, x4, x5, x6, x7)  =  U8_GAGGAG(x1, x2, x3, x4, x6, x7)
PG_IN_GGAGG(x1, x2, x3, x4, x5)  =  PG_IN_GGAGG(x1, x2, x4, x5)
U9_GGAGG(x1, x2, x3, x4, x5, x6)  =  U9_GGAGG(x1, x2, x4, x5, x6)
MEMBERE_IN_GG(x1, x2)  =  MEMBERE_IN_GG(x1, x2)
U5_GG(x1, x2, x3, x4)  =  U5_GG(x1, x2, x3, x4)
U10_GGAGG(x1, x2, x3, x4, x5, x6)  =  U10_GGAGG(x1, x2, x4, x5, x6)
PH_IN_GGAGG(x1, x2, x3, x4, x5)  =  PH_IN_GGAGG(x1, x2, x4, x5)
U11_GGAGG(x1, x2, x3, x4, x5, x6)  =  U11_GGAGG(x1, x2, x4, x5, x6)
DELETEF_IN_GGA(x1, x2, x3)  =  DELETEF_IN_GGA(x1, x2)
U6_GGA(x1, x2, x3, x4, x5)  =  U6_GGA(x1, x2, x3, x5)
U12_GGAGG(x1, x2, x3, x4, x5, x6)  =  U12_GGAGG(x1, x2, x3, x4, x5, x6)

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

(4) Obligation:

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

REACHA_IN_GGGG(T46, T47, .(T48, T49), T14) → U1_GGGG(T46, T47, T48, T49, T14, memberB_in_ggg(T46, T47, T49))
REACHA_IN_GGGG(T46, T47, .(T48, T49), T14) → MEMBERB_IN_GGG(T46, T47, T49)
MEMBERB_IN_GGG(T79, T80, .(T81, T82)) → U3_GGG(T79, T80, T81, T82, memberB_in_ggg(T79, T80, T82))
MEMBERB_IN_GGG(T79, T80, .(T81, T82)) → MEMBERB_IN_GGG(T79, T80, T82)
REACHA_IN_GGGG(T103, T104, T105, T106) → U2_GGGG(T103, T104, T105, T106, pC_in_gaggag(T103, X81, T105, T106, X82, T104))
REACHA_IN_GGGG(T103, T104, T105, T106) → PC_IN_GAGGAG(T103, X81, T105, T106, X82, T104)
PC_IN_GAGGAG(T103, T113, T105, T106, X82, T104) → U7_GAGGAG(T103, T113, T105, T106, X82, T104, memberD_in_gag(T103, T113, T105))
PC_IN_GAGGAG(T103, T113, T105, T106, X82, T104) → MEMBERD_IN_GAG(T103, T113, T105)
MEMBERD_IN_GAG(T141, X121, .(T142, T143)) → U4_GAG(T141, X121, T142, T143, memberD_in_gag(T141, X121, T143))
MEMBERD_IN_GAG(T141, X121, .(T142, T143)) → MEMBERD_IN_GAG(T141, X121, T143)
U7_GAGGAG(T103, T113, T105, T106, X82, T104, memberD_out_gag(T103, T113, T105)) → U8_GAGGAG(T103, T113, T105, T106, X82, T104, pG_in_ggagg(T113, T106, X82, T104, T105))
U7_GAGGAG(T103, T113, T105, T106, X82, T104, memberD_out_gag(T103, T113, T105)) → PG_IN_GGAGG(T113, T106, X82, T104, T105)
PG_IN_GGAGG(T113, T106, X82, T104, T105) → U9_GGAGG(T113, T106, X82, T104, T105, memberE_in_gg(T113, T106))
PG_IN_GGAGG(T113, T106, X82, T104, T105) → MEMBERE_IN_GG(T113, T106)
MEMBERE_IN_GG(T174, .(T175, T176)) → U5_GG(T174, T175, T176, memberE_in_gg(T174, T176))
MEMBERE_IN_GG(T174, .(T175, T176)) → MEMBERE_IN_GG(T174, T176)
U9_GGAGG(T113, T106, X82, T104, T105, memberE_out_gg(T113, T106)) → U10_GGAGG(T113, T106, X82, T104, T105, pH_in_ggagg(T113, T106, X82, T104, T105))
U9_GGAGG(T113, T106, X82, T104, T105, memberE_out_gg(T113, T106)) → PH_IN_GGAGG(T113, T106, X82, T104, T105)
PH_IN_GGAGG(T113, T106, T185, T104, T105) → U11_GGAGG(T113, T106, T185, T104, T105, deleteF_in_gga(T113, T106, T185))
PH_IN_GGAGG(T113, T106, T185, T104, T105) → DELETEF_IN_GGA(T113, T106, T185)
DELETEF_IN_GGA(T206, .(T207, T208), .(T207, X191)) → U6_GGA(T206, T207, T208, X191, deleteF_in_gga(T206, T208, X191))
DELETEF_IN_GGA(T206, .(T207, T208), .(T207, X191)) → DELETEF_IN_GGA(T206, T208, X191)
U11_GGAGG(T113, T106, T185, T104, T105, deleteF_out_gga(T113, T106, T185)) → U12_GGAGG(T113, T106, T185, T104, T105, reachA_in_gggg(T113, T104, T105, T185))
U11_GGAGG(T113, T106, T185, T104, T105, deleteF_out_gga(T113, T106, T185)) → REACHA_IN_GGGG(T113, T104, T105, T185)

The TRS R consists of the following rules:

reachA_in_gggg(T27, T28, .(.(T27, .(T28, [])), T29), T14) → reachA_out_gggg(T27, T28, .(.(T27, .(T28, [])), T29), T14)
reachA_in_gggg(T46, T47, .(T48, T49), T14) → U1_gggg(T46, T47, T48, T49, T14, memberB_in_ggg(T46, T47, T49))
memberB_in_ggg(T68, T69, .(.(T68, .(T69, [])), T70)) → memberB_out_ggg(T68, T69, .(.(T68, .(T69, [])), T70))
memberB_in_ggg(T79, T80, .(T81, T82)) → U3_ggg(T79, T80, T81, T82, memberB_in_ggg(T79, T80, T82))
U3_ggg(T79, T80, T81, T82, memberB_out_ggg(T79, T80, T82)) → memberB_out_ggg(T79, T80, .(T81, T82))
U1_gggg(T46, T47, T48, T49, T14, memberB_out_ggg(T46, T47, T49)) → reachA_out_gggg(T46, T47, .(T48, T49), T14)
reachA_in_gggg(T103, T104, T105, T106) → U2_gggg(T103, T104, T105, T106, pC_in_gaggag(T103, X81, T105, T106, X82, T104))
pC_in_gaggag(T103, T113, T105, T106, X82, T104) → U7_gaggag(T103, T113, T105, T106, X82, T104, memberD_in_gag(T103, T113, T105))
memberD_in_gag(T132, T133, .(.(T132, .(T133, [])), T134)) → memberD_out_gag(T132, T133, .(.(T132, .(T133, [])), T134))
memberD_in_gag(T141, X121, .(T142, T143)) → U4_gag(T141, X121, T142, T143, memberD_in_gag(T141, X121, T143))
U4_gag(T141, X121, T142, T143, memberD_out_gag(T141, X121, T143)) → memberD_out_gag(T141, X121, .(T142, T143))
U7_gaggag(T103, T113, T105, T106, X82, T104, memberD_out_gag(T103, T113, T105)) → U8_gaggag(T103, T113, T105, T106, X82, T104, pG_in_ggagg(T113, T106, X82, T104, T105))
pG_in_ggagg(T113, T106, X82, T104, T105) → U9_ggagg(T113, T106, X82, T104, T105, memberE_in_gg(T113, T106))
memberE_in_gg(T166, .(T166, T167)) → memberE_out_gg(T166, .(T166, T167))
memberE_in_gg(T174, .(T175, T176)) → U5_gg(T174, T175, T176, memberE_in_gg(T174, T176))
U5_gg(T174, T175, T176, memberE_out_gg(T174, T176)) → memberE_out_gg(T174, .(T175, T176))
U9_ggagg(T113, T106, X82, T104, T105, memberE_out_gg(T113, T106)) → U10_ggagg(T113, T106, X82, T104, T105, pH_in_ggagg(T113, T106, X82, T104, T105))
pH_in_ggagg(T113, T106, T185, T104, T105) → U11_ggagg(T113, T106, T185, T104, T105, deleteF_in_gga(T113, T106, T185))
deleteF_in_gga(T198, .(T198, T199), T199) → deleteF_out_gga(T198, .(T198, T199), T199)
deleteF_in_gga(T206, .(T207, T208), .(T207, X191)) → U6_gga(T206, T207, T208, X191, deleteF_in_gga(T206, T208, X191))
U6_gga(T206, T207, T208, X191, deleteF_out_gga(T206, T208, X191)) → deleteF_out_gga(T206, .(T207, T208), .(T207, X191))
U11_ggagg(T113, T106, T185, T104, T105, deleteF_out_gga(T113, T106, T185)) → U12_ggagg(T113, T106, T185, T104, T105, reachA_in_gggg(T113, T104, T105, T185))
U12_ggagg(T113, T106, T185, T104, T105, reachA_out_gggg(T113, T104, T105, T185)) → pH_out_ggagg(T113, T106, T185, T104, T105)
U10_ggagg(T113, T106, X82, T104, T105, pH_out_ggagg(T113, T106, X82, T104, T105)) → pG_out_ggagg(T113, T106, X82, T104, T105)
U8_gaggag(T103, T113, T105, T106, X82, T104, pG_out_ggagg(T113, T106, X82, T104, T105)) → pC_out_gaggag(T103, T113, T105, T106, X82, T104)
U2_gggg(T103, T104, T105, T106, pC_out_gaggag(T103, X81, T105, T106, X82, T104)) → reachA_out_gggg(T103, T104, T105, T106)

The argument filtering Pi contains the following mapping:
reachA_in_gggg(x1, x2, x3, x4)  =  reachA_in_gggg(x1, x2, x3, x4)
.(x1, x2)  =  .(x1, x2)
[]  =  []
reachA_out_gggg(x1, x2, x3, x4)  =  reachA_out_gggg(x1, x2, x3, x4)
U1_gggg(x1, x2, x3, x4, x5, x6)  =  U1_gggg(x1, x2, x3, x4, x5, x6)
memberB_in_ggg(x1, x2, x3)  =  memberB_in_ggg(x1, x2, x3)
memberB_out_ggg(x1, x2, x3)  =  memberB_out_ggg(x1, x2, x3)
U3_ggg(x1, x2, x3, x4, x5)  =  U3_ggg(x1, x2, x3, x4, x5)
U2_gggg(x1, x2, x3, x4, x5)  =  U2_gggg(x1, x2, x3, x4, x5)
pC_in_gaggag(x1, x2, x3, x4, x5, x6)  =  pC_in_gaggag(x1, x3, x4, x6)
U7_gaggag(x1, x2, x3, x4, x5, x6, x7)  =  U7_gaggag(x1, x3, x4, x6, x7)
memberD_in_gag(x1, x2, x3)  =  memberD_in_gag(x1, x3)
memberD_out_gag(x1, x2, x3)  =  memberD_out_gag(x1, x2, x3)
U4_gag(x1, x2, x3, x4, x5)  =  U4_gag(x1, x3, x4, x5)
U8_gaggag(x1, x2, x3, x4, x5, x6, x7)  =  U8_gaggag(x1, x2, x3, x4, x6, x7)
pG_in_ggagg(x1, x2, x3, x4, x5)  =  pG_in_ggagg(x1, x2, x4, x5)
U9_ggagg(x1, x2, x3, x4, x5, x6)  =  U9_ggagg(x1, x2, x4, x5, x6)
memberE_in_gg(x1, x2)  =  memberE_in_gg(x1, x2)
memberE_out_gg(x1, x2)  =  memberE_out_gg(x1, x2)
U5_gg(x1, x2, x3, x4)  =  U5_gg(x1, x2, x3, x4)
U10_ggagg(x1, x2, x3, x4, x5, x6)  =  U10_ggagg(x1, x2, x4, x5, x6)
pH_in_ggagg(x1, x2, x3, x4, x5)  =  pH_in_ggagg(x1, x2, x4, x5)
U11_ggagg(x1, x2, x3, x4, x5, x6)  =  U11_ggagg(x1, x2, x4, x5, x6)
deleteF_in_gga(x1, x2, x3)  =  deleteF_in_gga(x1, x2)
deleteF_out_gga(x1, x2, x3)  =  deleteF_out_gga(x1, x2, x3)
U6_gga(x1, x2, x3, x4, x5)  =  U6_gga(x1, x2, x3, x5)
U12_ggagg(x1, x2, x3, x4, x5, x6)  =  U12_ggagg(x1, x2, x3, x4, x5, x6)
pH_out_ggagg(x1, x2, x3, x4, x5)  =  pH_out_ggagg(x1, x2, x3, x4, x5)
pG_out_ggagg(x1, x2, x3, x4, x5)  =  pG_out_ggagg(x1, x2, x3, x4, x5)
pC_out_gaggag(x1, x2, x3, x4, x5, x6)  =  pC_out_gaggag(x1, x2, x3, x4, x5, x6)
REACHA_IN_GGGG(x1, x2, x3, x4)  =  REACHA_IN_GGGG(x1, x2, x3, x4)
U1_GGGG(x1, x2, x3, x4, x5, x6)  =  U1_GGGG(x1, x2, x3, x4, x5, x6)
MEMBERB_IN_GGG(x1, x2, x3)  =  MEMBERB_IN_GGG(x1, x2, x3)
U3_GGG(x1, x2, x3, x4, x5)  =  U3_GGG(x1, x2, x3, x4, x5)
U2_GGGG(x1, x2, x3, x4, x5)  =  U2_GGGG(x1, x2, x3, x4, x5)
PC_IN_GAGGAG(x1, x2, x3, x4, x5, x6)  =  PC_IN_GAGGAG(x1, x3, x4, x6)
U7_GAGGAG(x1, x2, x3, x4, x5, x6, x7)  =  U7_GAGGAG(x1, x3, x4, x6, x7)
MEMBERD_IN_GAG(x1, x2, x3)  =  MEMBERD_IN_GAG(x1, x3)
U4_GAG(x1, x2, x3, x4, x5)  =  U4_GAG(x1, x3, x4, x5)
U8_GAGGAG(x1, x2, x3, x4, x5, x6, x7)  =  U8_GAGGAG(x1, x2, x3, x4, x6, x7)
PG_IN_GGAGG(x1, x2, x3, x4, x5)  =  PG_IN_GGAGG(x1, x2, x4, x5)
U9_GGAGG(x1, x2, x3, x4, x5, x6)  =  U9_GGAGG(x1, x2, x4, x5, x6)
MEMBERE_IN_GG(x1, x2)  =  MEMBERE_IN_GG(x1, x2)
U5_GG(x1, x2, x3, x4)  =  U5_GG(x1, x2, x3, x4)
U10_GGAGG(x1, x2, x3, x4, x5, x6)  =  U10_GGAGG(x1, x2, x4, x5, x6)
PH_IN_GGAGG(x1, x2, x3, x4, x5)  =  PH_IN_GGAGG(x1, x2, x4, x5)
U11_GGAGG(x1, x2, x3, x4, x5, x6)  =  U11_GGAGG(x1, x2, x4, x5, x6)
DELETEF_IN_GGA(x1, x2, x3)  =  DELETEF_IN_GGA(x1, x2)
U6_GGA(x1, x2, x3, x4, x5)  =  U6_GGA(x1, x2, x3, x5)
U12_GGAGG(x1, x2, x3, x4, x5, x6)  =  U12_GGAGG(x1, x2, x3, x4, x5, x6)

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

(5) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 5 SCCs with 13 less nodes.

(6) Complex Obligation (AND)

(7) Obligation:

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

DELETEF_IN_GGA(T206, .(T207, T208), .(T207, X191)) → DELETEF_IN_GGA(T206, T208, X191)

The TRS R consists of the following rules:

reachA_in_gggg(T27, T28, .(.(T27, .(T28, [])), T29), T14) → reachA_out_gggg(T27, T28, .(.(T27, .(T28, [])), T29), T14)
reachA_in_gggg(T46, T47, .(T48, T49), T14) → U1_gggg(T46, T47, T48, T49, T14, memberB_in_ggg(T46, T47, T49))
memberB_in_ggg(T68, T69, .(.(T68, .(T69, [])), T70)) → memberB_out_ggg(T68, T69, .(.(T68, .(T69, [])), T70))
memberB_in_ggg(T79, T80, .(T81, T82)) → U3_ggg(T79, T80, T81, T82, memberB_in_ggg(T79, T80, T82))
U3_ggg(T79, T80, T81, T82, memberB_out_ggg(T79, T80, T82)) → memberB_out_ggg(T79, T80, .(T81, T82))
U1_gggg(T46, T47, T48, T49, T14, memberB_out_ggg(T46, T47, T49)) → reachA_out_gggg(T46, T47, .(T48, T49), T14)
reachA_in_gggg(T103, T104, T105, T106) → U2_gggg(T103, T104, T105, T106, pC_in_gaggag(T103, X81, T105, T106, X82, T104))
pC_in_gaggag(T103, T113, T105, T106, X82, T104) → U7_gaggag(T103, T113, T105, T106, X82, T104, memberD_in_gag(T103, T113, T105))
memberD_in_gag(T132, T133, .(.(T132, .(T133, [])), T134)) → memberD_out_gag(T132, T133, .(.(T132, .(T133, [])), T134))
memberD_in_gag(T141, X121, .(T142, T143)) → U4_gag(T141, X121, T142, T143, memberD_in_gag(T141, X121, T143))
U4_gag(T141, X121, T142, T143, memberD_out_gag(T141, X121, T143)) → memberD_out_gag(T141, X121, .(T142, T143))
U7_gaggag(T103, T113, T105, T106, X82, T104, memberD_out_gag(T103, T113, T105)) → U8_gaggag(T103, T113, T105, T106, X82, T104, pG_in_ggagg(T113, T106, X82, T104, T105))
pG_in_ggagg(T113, T106, X82, T104, T105) → U9_ggagg(T113, T106, X82, T104, T105, memberE_in_gg(T113, T106))
memberE_in_gg(T166, .(T166, T167)) → memberE_out_gg(T166, .(T166, T167))
memberE_in_gg(T174, .(T175, T176)) → U5_gg(T174, T175, T176, memberE_in_gg(T174, T176))
U5_gg(T174, T175, T176, memberE_out_gg(T174, T176)) → memberE_out_gg(T174, .(T175, T176))
U9_ggagg(T113, T106, X82, T104, T105, memberE_out_gg(T113, T106)) → U10_ggagg(T113, T106, X82, T104, T105, pH_in_ggagg(T113, T106, X82, T104, T105))
pH_in_ggagg(T113, T106, T185, T104, T105) → U11_ggagg(T113, T106, T185, T104, T105, deleteF_in_gga(T113, T106, T185))
deleteF_in_gga(T198, .(T198, T199), T199) → deleteF_out_gga(T198, .(T198, T199), T199)
deleteF_in_gga(T206, .(T207, T208), .(T207, X191)) → U6_gga(T206, T207, T208, X191, deleteF_in_gga(T206, T208, X191))
U6_gga(T206, T207, T208, X191, deleteF_out_gga(T206, T208, X191)) → deleteF_out_gga(T206, .(T207, T208), .(T207, X191))
U11_ggagg(T113, T106, T185, T104, T105, deleteF_out_gga(T113, T106, T185)) → U12_ggagg(T113, T106, T185, T104, T105, reachA_in_gggg(T113, T104, T105, T185))
U12_ggagg(T113, T106, T185, T104, T105, reachA_out_gggg(T113, T104, T105, T185)) → pH_out_ggagg(T113, T106, T185, T104, T105)
U10_ggagg(T113, T106, X82, T104, T105, pH_out_ggagg(T113, T106, X82, T104, T105)) → pG_out_ggagg(T113, T106, X82, T104, T105)
U8_gaggag(T103, T113, T105, T106, X82, T104, pG_out_ggagg(T113, T106, X82, T104, T105)) → pC_out_gaggag(T103, T113, T105, T106, X82, T104)
U2_gggg(T103, T104, T105, T106, pC_out_gaggag(T103, X81, T105, T106, X82, T104)) → reachA_out_gggg(T103, T104, T105, T106)

The argument filtering Pi contains the following mapping:
reachA_in_gggg(x1, x2, x3, x4)  =  reachA_in_gggg(x1, x2, x3, x4)
.(x1, x2)  =  .(x1, x2)
[]  =  []
reachA_out_gggg(x1, x2, x3, x4)  =  reachA_out_gggg(x1, x2, x3, x4)
U1_gggg(x1, x2, x3, x4, x5, x6)  =  U1_gggg(x1, x2, x3, x4, x5, x6)
memberB_in_ggg(x1, x2, x3)  =  memberB_in_ggg(x1, x2, x3)
memberB_out_ggg(x1, x2, x3)  =  memberB_out_ggg(x1, x2, x3)
U3_ggg(x1, x2, x3, x4, x5)  =  U3_ggg(x1, x2, x3, x4, x5)
U2_gggg(x1, x2, x3, x4, x5)  =  U2_gggg(x1, x2, x3, x4, x5)
pC_in_gaggag(x1, x2, x3, x4, x5, x6)  =  pC_in_gaggag(x1, x3, x4, x6)
U7_gaggag(x1, x2, x3, x4, x5, x6, x7)  =  U7_gaggag(x1, x3, x4, x6, x7)
memberD_in_gag(x1, x2, x3)  =  memberD_in_gag(x1, x3)
memberD_out_gag(x1, x2, x3)  =  memberD_out_gag(x1, x2, x3)
U4_gag(x1, x2, x3, x4, x5)  =  U4_gag(x1, x3, x4, x5)
U8_gaggag(x1, x2, x3, x4, x5, x6, x7)  =  U8_gaggag(x1, x2, x3, x4, x6, x7)
pG_in_ggagg(x1, x2, x3, x4, x5)  =  pG_in_ggagg(x1, x2, x4, x5)
U9_ggagg(x1, x2, x3, x4, x5, x6)  =  U9_ggagg(x1, x2, x4, x5, x6)
memberE_in_gg(x1, x2)  =  memberE_in_gg(x1, x2)
memberE_out_gg(x1, x2)  =  memberE_out_gg(x1, x2)
U5_gg(x1, x2, x3, x4)  =  U5_gg(x1, x2, x3, x4)
U10_ggagg(x1, x2, x3, x4, x5, x6)  =  U10_ggagg(x1, x2, x4, x5, x6)
pH_in_ggagg(x1, x2, x3, x4, x5)  =  pH_in_ggagg(x1, x2, x4, x5)
U11_ggagg(x1, x2, x3, x4, x5, x6)  =  U11_ggagg(x1, x2, x4, x5, x6)
deleteF_in_gga(x1, x2, x3)  =  deleteF_in_gga(x1, x2)
deleteF_out_gga(x1, x2, x3)  =  deleteF_out_gga(x1, x2, x3)
U6_gga(x1, x2, x3, x4, x5)  =  U6_gga(x1, x2, x3, x5)
U12_ggagg(x1, x2, x3, x4, x5, x6)  =  U12_ggagg(x1, x2, x3, x4, x5, x6)
pH_out_ggagg(x1, x2, x3, x4, x5)  =  pH_out_ggagg(x1, x2, x3, x4, x5)
pG_out_ggagg(x1, x2, x3, x4, x5)  =  pG_out_ggagg(x1, x2, x3, x4, x5)
pC_out_gaggag(x1, x2, x3, x4, x5, x6)  =  pC_out_gaggag(x1, x2, x3, x4, x5, x6)
DELETEF_IN_GGA(x1, x2, x3)  =  DELETEF_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:

DELETEF_IN_GGA(T206, .(T207, T208), .(T207, X191)) → DELETEF_IN_GGA(T206, T208, X191)

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

DELETEF_IN_GGA(T206, .(T207, T208)) → DELETEF_IN_GGA(T206, T208)

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:

• DELETEF_IN_GGA(T206, .(T207, T208)) → DELETEF_IN_GGA(T206, T208)
  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:

MEMBERE_IN_GG(T174, .(T175, T176)) → MEMBERE_IN_GG(T174, T176)

The TRS R consists of the following rules:

reachA_in_gggg(T27, T28, .(.(T27, .(T28, [])), T29), T14) → reachA_out_gggg(T27, T28, .(.(T27, .(T28, [])), T29), T14)
reachA_in_gggg(T46, T47, .(T48, T49), T14) → U1_gggg(T46, T47, T48, T49, T14, memberB_in_ggg(T46, T47, T49))
memberB_in_ggg(T68, T69, .(.(T68, .(T69, [])), T70)) → memberB_out_ggg(T68, T69, .(.(T68, .(T69, [])), T70))
memberB_in_ggg(T79, T80, .(T81, T82)) → U3_ggg(T79, T80, T81, T82, memberB_in_ggg(T79, T80, T82))
U3_ggg(T79, T80, T81, T82, memberB_out_ggg(T79, T80, T82)) → memberB_out_ggg(T79, T80, .(T81, T82))
U1_gggg(T46, T47, T48, T49, T14, memberB_out_ggg(T46, T47, T49)) → reachA_out_gggg(T46, T47, .(T48, T49), T14)
reachA_in_gggg(T103, T104, T105, T106) → U2_gggg(T103, T104, T105, T106, pC_in_gaggag(T103, X81, T105, T106, X82, T104))
pC_in_gaggag(T103, T113, T105, T106, X82, T104) → U7_gaggag(T103, T113, T105, T106, X82, T104, memberD_in_gag(T103, T113, T105))
memberD_in_gag(T132, T133, .(.(T132, .(T133, [])), T134)) → memberD_out_gag(T132, T133, .(.(T132, .(T133, [])), T134))
memberD_in_gag(T141, X121, .(T142, T143)) → U4_gag(T141, X121, T142, T143, memberD_in_gag(T141, X121, T143))
U4_gag(T141, X121, T142, T143, memberD_out_gag(T141, X121, T143)) → memberD_out_gag(T141, X121, .(T142, T143))
U7_gaggag(T103, T113, T105, T106, X82, T104, memberD_out_gag(T103, T113, T105)) → U8_gaggag(T103, T113, T105, T106, X82, T104, pG_in_ggagg(T113, T106, X82, T104, T105))
pG_in_ggagg(T113, T106, X82, T104, T105) → U9_ggagg(T113, T106, X82, T104, T105, memberE_in_gg(T113, T106))
memberE_in_gg(T166, .(T166, T167)) → memberE_out_gg(T166, .(T166, T167))
memberE_in_gg(T174, .(T175, T176)) → U5_gg(T174, T175, T176, memberE_in_gg(T174, T176))
U5_gg(T174, T175, T176, memberE_out_gg(T174, T176)) → memberE_out_gg(T174, .(T175, T176))
U9_ggagg(T113, T106, X82, T104, T105, memberE_out_gg(T113, T106)) → U10_ggagg(T113, T106, X82, T104, T105, pH_in_ggagg(T113, T106, X82, T104, T105))
pH_in_ggagg(T113, T106, T185, T104, T105) → U11_ggagg(T113, T106, T185, T104, T105, deleteF_in_gga(T113, T106, T185))
deleteF_in_gga(T198, .(T198, T199), T199) → deleteF_out_gga(T198, .(T198, T199), T199)
deleteF_in_gga(T206, .(T207, T208), .(T207, X191)) → U6_gga(T206, T207, T208, X191, deleteF_in_gga(T206, T208, X191))
U6_gga(T206, T207, T208, X191, deleteF_out_gga(T206, T208, X191)) → deleteF_out_gga(T206, .(T207, T208), .(T207, X191))
U11_ggagg(T113, T106, T185, T104, T105, deleteF_out_gga(T113, T106, T185)) → U12_ggagg(T113, T106, T185, T104, T105, reachA_in_gggg(T113, T104, T105, T185))
U12_ggagg(T113, T106, T185, T104, T105, reachA_out_gggg(T113, T104, T105, T185)) → pH_out_ggagg(T113, T106, T185, T104, T105)
U10_ggagg(T113, T106, X82, T104, T105, pH_out_ggagg(T113, T106, X82, T104, T105)) → pG_out_ggagg(T113, T106, X82, T104, T105)
U8_gaggag(T103, T113, T105, T106, X82, T104, pG_out_ggagg(T113, T106, X82, T104, T105)) → pC_out_gaggag(T103, T113, T105, T106, X82, T104)
U2_gggg(T103, T104, T105, T106, pC_out_gaggag(T103, X81, T105, T106, X82, T104)) → reachA_out_gggg(T103, T104, T105, T106)

The argument filtering Pi contains the following mapping:
reachA_in_gggg(x1, x2, x3, x4)  =  reachA_in_gggg(x1, x2, x3, x4)
.(x1, x2)  =  .(x1, x2)
[]  =  []
reachA_out_gggg(x1, x2, x3, x4)  =  reachA_out_gggg(x1, x2, x3, x4)
U1_gggg(x1, x2, x3, x4, x5, x6)  =  U1_gggg(x1, x2, x3, x4, x5, x6)
memberB_in_ggg(x1, x2, x3)  =  memberB_in_ggg(x1, x2, x3)
memberB_out_ggg(x1, x2, x3)  =  memberB_out_ggg(x1, x2, x3)
U3_ggg(x1, x2, x3, x4, x5)  =  U3_ggg(x1, x2, x3, x4, x5)
U2_gggg(x1, x2, x3, x4, x5)  =  U2_gggg(x1, x2, x3, x4, x5)
pC_in_gaggag(x1, x2, x3, x4, x5, x6)  =  pC_in_gaggag(x1, x3, x4, x6)
U7_gaggag(x1, x2, x3, x4, x5, x6, x7)  =  U7_gaggag(x1, x3, x4, x6, x7)
memberD_in_gag(x1, x2, x3)  =  memberD_in_gag(x1, x3)
memberD_out_gag(x1, x2, x3)  =  memberD_out_gag(x1, x2, x3)
U4_gag(x1, x2, x3, x4, x5)  =  U4_gag(x1, x3, x4, x5)
U8_gaggag(x1, x2, x3, x4, x5, x6, x7)  =  U8_gaggag(x1, x2, x3, x4, x6, x7)
pG_in_ggagg(x1, x2, x3, x4, x5)  =  pG_in_ggagg(x1, x2, x4, x5)
U9_ggagg(x1, x2, x3, x4, x5, x6)  =  U9_ggagg(x1, x2, x4, x5, x6)
memberE_in_gg(x1, x2)  =  memberE_in_gg(x1, x2)
memberE_out_gg(x1, x2)  =  memberE_out_gg(x1, x2)
U5_gg(x1, x2, x3, x4)  =  U5_gg(x1, x2, x3, x4)
U10_ggagg(x1, x2, x3, x4, x5, x6)  =  U10_ggagg(x1, x2, x4, x5, x6)
pH_in_ggagg(x1, x2, x3, x4, x5)  =  pH_in_ggagg(x1, x2, x4, x5)
U11_ggagg(x1, x2, x3, x4, x5, x6)  =  U11_ggagg(x1, x2, x4, x5, x6)
deleteF_in_gga(x1, x2, x3)  =  deleteF_in_gga(x1, x2)
deleteF_out_gga(x1, x2, x3)  =  deleteF_out_gga(x1, x2, x3)
U6_gga(x1, x2, x3, x4, x5)  =  U6_gga(x1, x2, x3, x5)
U12_ggagg(x1, x2, x3, x4, x5, x6)  =  U12_ggagg(x1, x2, x3, x4, x5, x6)
pH_out_ggagg(x1, x2, x3, x4, x5)  =  pH_out_ggagg(x1, x2, x3, x4, x5)
pG_out_ggagg(x1, x2, x3, x4, x5)  =  pG_out_ggagg(x1, x2, x3, x4, x5)
pC_out_gaggag(x1, x2, x3, x4, x5, x6)  =  pC_out_gaggag(x1, x2, x3, x4, x5, x6)
MEMBERE_IN_GG(x1, x2)  =  MEMBERE_IN_GG(x1, x2)

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

(15) UsableRulesProof (EQUIVALENT transformation)

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

(16) Obligation:

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

MEMBERE_IN_GG(T174, .(T175, T176)) → MEMBERE_IN_GG(T174, T176)

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

(17) PiDPToQDPProof (EQUIVALENT 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:

MEMBERE_IN_GG(T174, .(T175, T176)) → MEMBERE_IN_GG(T174, T176)

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:

• MEMBERE_IN_GG(T174, .(T175, T176)) → MEMBERE_IN_GG(T174, T176)
  The graph contains the following edges 1 >= 1, 2 > 2

(20) YES

(21) Obligation:

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

MEMBERD_IN_GAG(T141, X121, .(T142, T143)) → MEMBERD_IN_GAG(T141, X121, T143)

The TRS R consists of the following rules:

reachA_in_gggg(T27, T28, .(.(T27, .(T28, [])), T29), T14) → reachA_out_gggg(T27, T28, .(.(T27, .(T28, [])), T29), T14)
reachA_in_gggg(T46, T47, .(T48, T49), T14) → U1_gggg(T46, T47, T48, T49, T14, memberB_in_ggg(T46, T47, T49))
memberB_in_ggg(T68, T69, .(.(T68, .(T69, [])), T70)) → memberB_out_ggg(T68, T69, .(.(T68, .(T69, [])), T70))
memberB_in_ggg(T79, T80, .(T81, T82)) → U3_ggg(T79, T80, T81, T82, memberB_in_ggg(T79, T80, T82))
U3_ggg(T79, T80, T81, T82, memberB_out_ggg(T79, T80, T82)) → memberB_out_ggg(T79, T80, .(T81, T82))
U1_gggg(T46, T47, T48, T49, T14, memberB_out_ggg(T46, T47, T49)) → reachA_out_gggg(T46, T47, .(T48, T49), T14)
reachA_in_gggg(T103, T104, T105, T106) → U2_gggg(T103, T104, T105, T106, pC_in_gaggag(T103, X81, T105, T106, X82, T104))
pC_in_gaggag(T103, T113, T105, T106, X82, T104) → U7_gaggag(T103, T113, T105, T106, X82, T104, memberD_in_gag(T103, T113, T105))
memberD_in_gag(T132, T133, .(.(T132, .(T133, [])), T134)) → memberD_out_gag(T132, T133, .(.(T132, .(T133, [])), T134))
memberD_in_gag(T141, X121, .(T142, T143)) → U4_gag(T141, X121, T142, T143, memberD_in_gag(T141, X121, T143))
U4_gag(T141, X121, T142, T143, memberD_out_gag(T141, X121, T143)) → memberD_out_gag(T141, X121, .(T142, T143))
U7_gaggag(T103, T113, T105, T106, X82, T104, memberD_out_gag(T103, T113, T105)) → U8_gaggag(T103, T113, T105, T106, X82, T104, pG_in_ggagg(T113, T106, X82, T104, T105))
pG_in_ggagg(T113, T106, X82, T104, T105) → U9_ggagg(T113, T106, X82, T104, T105, memberE_in_gg(T113, T106))
memberE_in_gg(T166, .(T166, T167)) → memberE_out_gg(T166, .(T166, T167))
memberE_in_gg(T174, .(T175, T176)) → U5_gg(T174, T175, T176, memberE_in_gg(T174, T176))
U5_gg(T174, T175, T176, memberE_out_gg(T174, T176)) → memberE_out_gg(T174, .(T175, T176))
U9_ggagg(T113, T106, X82, T104, T105, memberE_out_gg(T113, T106)) → U10_ggagg(T113, T106, X82, T104, T105, pH_in_ggagg(T113, T106, X82, T104, T105))
pH_in_ggagg(T113, T106, T185, T104, T105) → U11_ggagg(T113, T106, T185, T104, T105, deleteF_in_gga(T113, T106, T185))
deleteF_in_gga(T198, .(T198, T199), T199) → deleteF_out_gga(T198, .(T198, T199), T199)
deleteF_in_gga(T206, .(T207, T208), .(T207, X191)) → U6_gga(T206, T207, T208, X191, deleteF_in_gga(T206, T208, X191))
U6_gga(T206, T207, T208, X191, deleteF_out_gga(T206, T208, X191)) → deleteF_out_gga(T206, .(T207, T208), .(T207, X191))
U11_ggagg(T113, T106, T185, T104, T105, deleteF_out_gga(T113, T106, T185)) → U12_ggagg(T113, T106, T185, T104, T105, reachA_in_gggg(T113, T104, T105, T185))
U12_ggagg(T113, T106, T185, T104, T105, reachA_out_gggg(T113, T104, T105, T185)) → pH_out_ggagg(T113, T106, T185, T104, T105)
U10_ggagg(T113, T106, X82, T104, T105, pH_out_ggagg(T113, T106, X82, T104, T105)) → pG_out_ggagg(T113, T106, X82, T104, T105)
U8_gaggag(T103, T113, T105, T106, X82, T104, pG_out_ggagg(T113, T106, X82, T104, T105)) → pC_out_gaggag(T103, T113, T105, T106, X82, T104)
U2_gggg(T103, T104, T105, T106, pC_out_gaggag(T103, X81, T105, T106, X82, T104)) → reachA_out_gggg(T103, T104, T105, T106)

The argument filtering Pi contains the following mapping:
reachA_in_gggg(x1, x2, x3, x4)  =  reachA_in_gggg(x1, x2, x3, x4)
.(x1, x2)  =  .(x1, x2)
[]  =  []
reachA_out_gggg(x1, x2, x3, x4)  =  reachA_out_gggg(x1, x2, x3, x4)
U1_gggg(x1, x2, x3, x4, x5, x6)  =  U1_gggg(x1, x2, x3, x4, x5, x6)
memberB_in_ggg(x1, x2, x3)  =  memberB_in_ggg(x1, x2, x3)
memberB_out_ggg(x1, x2, x3)  =  memberB_out_ggg(x1, x2, x3)
U3_ggg(x1, x2, x3, x4, x5)  =  U3_ggg(x1, x2, x3, x4, x5)
U2_gggg(x1, x2, x3, x4, x5)  =  U2_gggg(x1, x2, x3, x4, x5)
pC_in_gaggag(x1, x2, x3, x4, x5, x6)  =  pC_in_gaggag(x1, x3, x4, x6)
U7_gaggag(x1, x2, x3, x4, x5, x6, x7)  =  U7_gaggag(x1, x3, x4, x6, x7)
memberD_in_gag(x1, x2, x3)  =  memberD_in_gag(x1, x3)
memberD_out_gag(x1, x2, x3)  =  memberD_out_gag(x1, x2, x3)
U4_gag(x1, x2, x3, x4, x5)  =  U4_gag(x1, x3, x4, x5)
U8_gaggag(x1, x2, x3, x4, x5, x6, x7)  =  U8_gaggag(x1, x2, x3, x4, x6, x7)
pG_in_ggagg(x1, x2, x3, x4, x5)  =  pG_in_ggagg(x1, x2, x4, x5)
U9_ggagg(x1, x2, x3, x4, x5, x6)  =  U9_ggagg(x1, x2, x4, x5, x6)
memberE_in_gg(x1, x2)  =  memberE_in_gg(x1, x2)
memberE_out_gg(x1, x2)  =  memberE_out_gg(x1, x2)
U5_gg(x1, x2, x3, x4)  =  U5_gg(x1, x2, x3, x4)
U10_ggagg(x1, x2, x3, x4, x5, x6)  =  U10_ggagg(x1, x2, x4, x5, x6)
pH_in_ggagg(x1, x2, x3, x4, x5)  =  pH_in_ggagg(x1, x2, x4, x5)
U11_ggagg(x1, x2, x3, x4, x5, x6)  =  U11_ggagg(x1, x2, x4, x5, x6)
deleteF_in_gga(x1, x2, x3)  =  deleteF_in_gga(x1, x2)
deleteF_out_gga(x1, x2, x3)  =  deleteF_out_gga(x1, x2, x3)
U6_gga(x1, x2, x3, x4, x5)  =  U6_gga(x1, x2, x3, x5)
U12_ggagg(x1, x2, x3, x4, x5, x6)  =  U12_ggagg(x1, x2, x3, x4, x5, x6)
pH_out_ggagg(x1, x2, x3, x4, x5)  =  pH_out_ggagg(x1, x2, x3, x4, x5)
pG_out_ggagg(x1, x2, x3, x4, x5)  =  pG_out_ggagg(x1, x2, x3, x4, x5)
pC_out_gaggag(x1, x2, x3, x4, x5, x6)  =  pC_out_gaggag(x1, x2, x3, x4, x5, x6)
MEMBERD_IN_GAG(x1, x2, x3)  =  MEMBERD_IN_GAG(x1, x3)

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:

MEMBERD_IN_GAG(T141, X121, .(T142, T143)) → MEMBERD_IN_GAG(T141, X121, T143)

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

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:

MEMBERD_IN_GAG(T141, .(T142, T143)) → MEMBERD_IN_GAG(T141, T143)

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:

• MEMBERD_IN_GAG(T141, .(T142, T143)) → MEMBERD_IN_GAG(T141, T143)
  The graph contains the following edges 1 >= 1, 2 > 2

(27) YES

(28) Obligation:

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

MEMBERB_IN_GGG(T79, T80, .(T81, T82)) → MEMBERB_IN_GGG(T79, T80, T82)

The TRS R consists of the following rules:

reachA_in_gggg(T27, T28, .(.(T27, .(T28, [])), T29), T14) → reachA_out_gggg(T27, T28, .(.(T27, .(T28, [])), T29), T14)
reachA_in_gggg(T46, T47, .(T48, T49), T14) → U1_gggg(T46, T47, T48, T49, T14, memberB_in_ggg(T46, T47, T49))
memberB_in_ggg(T68, T69, .(.(T68, .(T69, [])), T70)) → memberB_out_ggg(T68, T69, .(.(T68, .(T69, [])), T70))
memberB_in_ggg(T79, T80, .(T81, T82)) → U3_ggg(T79, T80, T81, T82, memberB_in_ggg(T79, T80, T82))
U3_ggg(T79, T80, T81, T82, memberB_out_ggg(T79, T80, T82)) → memberB_out_ggg(T79, T80, .(T81, T82))
U1_gggg(T46, T47, T48, T49, T14, memberB_out_ggg(T46, T47, T49)) → reachA_out_gggg(T46, T47, .(T48, T49), T14)
reachA_in_gggg(T103, T104, T105, T106) → U2_gggg(T103, T104, T105, T106, pC_in_gaggag(T103, X81, T105, T106, X82, T104))
pC_in_gaggag(T103, T113, T105, T106, X82, T104) → U7_gaggag(T103, T113, T105, T106, X82, T104, memberD_in_gag(T103, T113, T105))
memberD_in_gag(T132, T133, .(.(T132, .(T133, [])), T134)) → memberD_out_gag(T132, T133, .(.(T132, .(T133, [])), T134))
memberD_in_gag(T141, X121, .(T142, T143)) → U4_gag(T141, X121, T142, T143, memberD_in_gag(T141, X121, T143))
U4_gag(T141, X121, T142, T143, memberD_out_gag(T141, X121, T143)) → memberD_out_gag(T141, X121, .(T142, T143))
U7_gaggag(T103, T113, T105, T106, X82, T104, memberD_out_gag(T103, T113, T105)) → U8_gaggag(T103, T113, T105, T106, X82, T104, pG_in_ggagg(T113, T106, X82, T104, T105))
pG_in_ggagg(T113, T106, X82, T104, T105) → U9_ggagg(T113, T106, X82, T104, T105, memberE_in_gg(T113, T106))
memberE_in_gg(T166, .(T166, T167)) → memberE_out_gg(T166, .(T166, T167))
memberE_in_gg(T174, .(T175, T176)) → U5_gg(T174, T175, T176, memberE_in_gg(T174, T176))
U5_gg(T174, T175, T176, memberE_out_gg(T174, T176)) → memberE_out_gg(T174, .(T175, T176))
U9_ggagg(T113, T106, X82, T104, T105, memberE_out_gg(T113, T106)) → U10_ggagg(T113, T106, X82, T104, T105, pH_in_ggagg(T113, T106, X82, T104, T105))
pH_in_ggagg(T113, T106, T185, T104, T105) → U11_ggagg(T113, T106, T185, T104, T105, deleteF_in_gga(T113, T106, T185))
deleteF_in_gga(T198, .(T198, T199), T199) → deleteF_out_gga(T198, .(T198, T199), T199)
deleteF_in_gga(T206, .(T207, T208), .(T207, X191)) → U6_gga(T206, T207, T208, X191, deleteF_in_gga(T206, T208, X191))
U6_gga(T206, T207, T208, X191, deleteF_out_gga(T206, T208, X191)) → deleteF_out_gga(T206, .(T207, T208), .(T207, X191))
U11_ggagg(T113, T106, T185, T104, T105, deleteF_out_gga(T113, T106, T185)) → U12_ggagg(T113, T106, T185, T104, T105, reachA_in_gggg(T113, T104, T105, T185))
U12_ggagg(T113, T106, T185, T104, T105, reachA_out_gggg(T113, T104, T105, T185)) → pH_out_ggagg(T113, T106, T185, T104, T105)
U10_ggagg(T113, T106, X82, T104, T105, pH_out_ggagg(T113, T106, X82, T104, T105)) → pG_out_ggagg(T113, T106, X82, T104, T105)
U8_gaggag(T103, T113, T105, T106, X82, T104, pG_out_ggagg(T113, T106, X82, T104, T105)) → pC_out_gaggag(T103, T113, T105, T106, X82, T104)
U2_gggg(T103, T104, T105, T106, pC_out_gaggag(T103, X81, T105, T106, X82, T104)) → reachA_out_gggg(T103, T104, T105, T106)

The argument filtering Pi contains the following mapping:
reachA_in_gggg(x1, x2, x3, x4)  =  reachA_in_gggg(x1, x2, x3, x4)
.(x1, x2)  =  .(x1, x2)
[]  =  []
reachA_out_gggg(x1, x2, x3, x4)  =  reachA_out_gggg(x1, x2, x3, x4)
U1_gggg(x1, x2, x3, x4, x5, x6)  =  U1_gggg(x1, x2, x3, x4, x5, x6)
memberB_in_ggg(x1, x2, x3)  =  memberB_in_ggg(x1, x2, x3)
memberB_out_ggg(x1, x2, x3)  =  memberB_out_ggg(x1, x2, x3)
U3_ggg(x1, x2, x3, x4, x5)  =  U3_ggg(x1, x2, x3, x4, x5)
U2_gggg(x1, x2, x3, x4, x5)  =  U2_gggg(x1, x2, x3, x4, x5)
pC_in_gaggag(x1, x2, x3, x4, x5, x6)  =  pC_in_gaggag(x1, x3, x4, x6)
U7_gaggag(x1, x2, x3, x4, x5, x6, x7)  =  U7_gaggag(x1, x3, x4, x6, x7)
memberD_in_gag(x1, x2, x3)  =  memberD_in_gag(x1, x3)
memberD_out_gag(x1, x2, x3)  =  memberD_out_gag(x1, x2, x3)
U4_gag(x1, x2, x3, x4, x5)  =  U4_gag(x1, x3, x4, x5)
U8_gaggag(x1, x2, x3, x4, x5, x6, x7)  =  U8_gaggag(x1, x2, x3, x4, x6, x7)
pG_in_ggagg(x1, x2, x3, x4, x5)  =  pG_in_ggagg(x1, x2, x4, x5)
U9_ggagg(x1, x2, x3, x4, x5, x6)  =  U9_ggagg(x1, x2, x4, x5, x6)
memberE_in_gg(x1, x2)  =  memberE_in_gg(x1, x2)
memberE_out_gg(x1, x2)  =  memberE_out_gg(x1, x2)
U5_gg(x1, x2, x3, x4)  =  U5_gg(x1, x2, x3, x4)
U10_ggagg(x1, x2, x3, x4, x5, x6)  =  U10_ggagg(x1, x2, x4, x5, x6)
pH_in_ggagg(x1, x2, x3, x4, x5)  =  pH_in_ggagg(x1, x2, x4, x5)
U11_ggagg(x1, x2, x3, x4, x5, x6)  =  U11_ggagg(x1, x2, x4, x5, x6)
deleteF_in_gga(x1, x2, x3)  =  deleteF_in_gga(x1, x2)
deleteF_out_gga(x1, x2, x3)  =  deleteF_out_gga(x1, x2, x3)
U6_gga(x1, x2, x3, x4, x5)  =  U6_gga(x1, x2, x3, x5)
U12_ggagg(x1, x2, x3, x4, x5, x6)  =  U12_ggagg(x1, x2, x3, x4, x5, x6)
pH_out_ggagg(x1, x2, x3, x4, x5)  =  pH_out_ggagg(x1, x2, x3, x4, x5)
pG_out_ggagg(x1, x2, x3, x4, x5)  =  pG_out_ggagg(x1, x2, x3, x4, x5)
pC_out_gaggag(x1, x2, x3, x4, x5, x6)  =  pC_out_gaggag(x1, x2, x3, x4, x5, x6)
MEMBERB_IN_GGG(x1, x2, x3)  =  MEMBERB_IN_GGG(x1, x2, x3)

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:

MEMBERB_IN_GGG(T79, T80, .(T81, T82)) → MEMBERB_IN_GGG(T79, T80, T82)

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

(31) PiDPToQDPProof (EQUIVALENT 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:

MEMBERB_IN_GGG(T79, T80, .(T81, T82)) → MEMBERB_IN_GGG(T79, T80, T82)

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:

• MEMBERB_IN_GGG(T79, T80, .(T81, T82)) → MEMBERB_IN_GGG(T79, T80, T82)
  The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3

(34) YES

(35) Obligation:

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

REACHA_IN_GGGG(T103, T104, T105, T106) → PC_IN_GAGGAG(T103, X81, T105, T106, X82, T104)
PC_IN_GAGGAG(T103, T113, T105, T106, X82, T104) → U7_GAGGAG(T103, T113, T105, T106, X82, T104, memberD_in_gag(T103, T113, T105))
U7_GAGGAG(T103, T113, T105, T106, X82, T104, memberD_out_gag(T103, T113, T105)) → PG_IN_GGAGG(T113, T106, X82, T104, T105)
PG_IN_GGAGG(T113, T106, X82, T104, T105) → U9_GGAGG(T113, T106, X82, T104, T105, memberE_in_gg(T113, T106))
U9_GGAGG(T113, T106, X82, T104, T105, memberE_out_gg(T113, T106)) → PH_IN_GGAGG(T113, T106, X82, T104, T105)
PH_IN_GGAGG(T113, T106, T185, T104, T105) → U11_GGAGG(T113, T106, T185, T104, T105, deleteF_in_gga(T113, T106, T185))
U11_GGAGG(T113, T106, T185, T104, T105, deleteF_out_gga(T113, T106, T185)) → REACHA_IN_GGGG(T113, T104, T105, T185)

The TRS R consists of the following rules:

reachA_in_gggg(T27, T28, .(.(T27, .(T28, [])), T29), T14) → reachA_out_gggg(T27, T28, .(.(T27, .(T28, [])), T29), T14)
reachA_in_gggg(T46, T47, .(T48, T49), T14) → U1_gggg(T46, T47, T48, T49, T14, memberB_in_ggg(T46, T47, T49))
memberB_in_ggg(T68, T69, .(.(T68, .(T69, [])), T70)) → memberB_out_ggg(T68, T69, .(.(T68, .(T69, [])), T70))
memberB_in_ggg(T79, T80, .(T81, T82)) → U3_ggg(T79, T80, T81, T82, memberB_in_ggg(T79, T80, T82))
U3_ggg(T79, T80, T81, T82, memberB_out_ggg(T79, T80, T82)) → memberB_out_ggg(T79, T80, .(T81, T82))
U1_gggg(T46, T47, T48, T49, T14, memberB_out_ggg(T46, T47, T49)) → reachA_out_gggg(T46, T47, .(T48, T49), T14)
reachA_in_gggg(T103, T104, T105, T106) → U2_gggg(T103, T104, T105, T106, pC_in_gaggag(T103, X81, T105, T106, X82, T104))
pC_in_gaggag(T103, T113, T105, T106, X82, T104) → U7_gaggag(T103, T113, T105, T106, X82, T104, memberD_in_gag(T103, T113, T105))
memberD_in_gag(T132, T133, .(.(T132, .(T133, [])), T134)) → memberD_out_gag(T132, T133, .(.(T132, .(T133, [])), T134))
memberD_in_gag(T141, X121, .(T142, T143)) → U4_gag(T141, X121, T142, T143, memberD_in_gag(T141, X121, T143))
U4_gag(T141, X121, T142, T143, memberD_out_gag(T141, X121, T143)) → memberD_out_gag(T141, X121, .(T142, T143))
U7_gaggag(T103, T113, T105, T106, X82, T104, memberD_out_gag(T103, T113, T105)) → U8_gaggag(T103, T113, T105, T106, X82, T104, pG_in_ggagg(T113, T106, X82, T104, T105))
pG_in_ggagg(T113, T106, X82, T104, T105) → U9_ggagg(T113, T106, X82, T104, T105, memberE_in_gg(T113, T106))
memberE_in_gg(T166, .(T166, T167)) → memberE_out_gg(T166, .(T166, T167))
memberE_in_gg(T174, .(T175, T176)) → U5_gg(T174, T175, T176, memberE_in_gg(T174, T176))
U5_gg(T174, T175, T176, memberE_out_gg(T174, T176)) → memberE_out_gg(T174, .(T175, T176))
U9_ggagg(T113, T106, X82, T104, T105, memberE_out_gg(T113, T106)) → U10_ggagg(T113, T106, X82, T104, T105, pH_in_ggagg(T113, T106, X82, T104, T105))
pH_in_ggagg(T113, T106, T185, T104, T105) → U11_ggagg(T113, T106, T185, T104, T105, deleteF_in_gga(T113, T106, T185))
deleteF_in_gga(T198, .(T198, T199), T199) → deleteF_out_gga(T198, .(T198, T199), T199)
deleteF_in_gga(T206, .(T207, T208), .(T207, X191)) → U6_gga(T206, T207, T208, X191, deleteF_in_gga(T206, T208, X191))
U6_gga(T206, T207, T208, X191, deleteF_out_gga(T206, T208, X191)) → deleteF_out_gga(T206, .(T207, T208), .(T207, X191))
U11_ggagg(T113, T106, T185, T104, T105, deleteF_out_gga(T113, T106, T185)) → U12_ggagg(T113, T106, T185, T104, T105, reachA_in_gggg(T113, T104, T105, T185))
U12_ggagg(T113, T106, T185, T104, T105, reachA_out_gggg(T113, T104, T105, T185)) → pH_out_ggagg(T113, T106, T185, T104, T105)
U10_ggagg(T113, T106, X82, T104, T105, pH_out_ggagg(T113, T106, X82, T104, T105)) → pG_out_ggagg(T113, T106, X82, T104, T105)
U8_gaggag(T103, T113, T105, T106, X82, T104, pG_out_ggagg(T113, T106, X82, T104, T105)) → pC_out_gaggag(T103, T113, T105, T106, X82, T104)
U2_gggg(T103, T104, T105, T106, pC_out_gaggag(T103, X81, T105, T106, X82, T104)) → reachA_out_gggg(T103, T104, T105, T106)

The argument filtering Pi contains the following mapping:
reachA_in_gggg(x1, x2, x3, x4)  =  reachA_in_gggg(x1, x2, x3, x4)
.(x1, x2)  =  .(x1, x2)
[]  =  []
reachA_out_gggg(x1, x2, x3, x4)  =  reachA_out_gggg(x1, x2, x3, x4)
U1_gggg(x1, x2, x3, x4, x5, x6)  =  U1_gggg(x1, x2, x3, x4, x5, x6)
memberB_in_ggg(x1, x2, x3)  =  memberB_in_ggg(x1, x2, x3)
memberB_out_ggg(x1, x2, x3)  =  memberB_out_ggg(x1, x2, x3)
U3_ggg(x1, x2, x3, x4, x5)  =  U3_ggg(x1, x2, x3, x4, x5)
U2_gggg(x1, x2, x3, x4, x5)  =  U2_gggg(x1, x2, x3, x4, x5)
pC_in_gaggag(x1, x2, x3, x4, x5, x6)  =  pC_in_gaggag(x1, x3, x4, x6)
U7_gaggag(x1, x2, x3, x4, x5, x6, x7)  =  U7_gaggag(x1, x3, x4, x6, x7)
memberD_in_gag(x1, x2, x3)  =  memberD_in_gag(x1, x3)
memberD_out_gag(x1, x2, x3)  =  memberD_out_gag(x1, x2, x3)
U4_gag(x1, x2, x3, x4, x5)  =  U4_gag(x1, x3, x4, x5)
U8_gaggag(x1, x2, x3, x4, x5, x6, x7)  =  U8_gaggag(x1, x2, x3, x4, x6, x7)
pG_in_ggagg(x1, x2, x3, x4, x5)  =  pG_in_ggagg(x1, x2, x4, x5)
U9_ggagg(x1, x2, x3, x4, x5, x6)  =  U9_ggagg(x1, x2, x4, x5, x6)
memberE_in_gg(x1, x2)  =  memberE_in_gg(x1, x2)
memberE_out_gg(x1, x2)  =  memberE_out_gg(x1, x2)
U5_gg(x1, x2, x3, x4)  =  U5_gg(x1, x2, x3, x4)
U10_ggagg(x1, x2, x3, x4, x5, x6)  =  U10_ggagg(x1, x2, x4, x5, x6)
pH_in_ggagg(x1, x2, x3, x4, x5)  =  pH_in_ggagg(x1, x2, x4, x5)
U11_ggagg(x1, x2, x3, x4, x5, x6)  =  U11_ggagg(x1, x2, x4, x5, x6)
deleteF_in_gga(x1, x2, x3)  =  deleteF_in_gga(x1, x2)
deleteF_out_gga(x1, x2, x3)  =  deleteF_out_gga(x1, x2, x3)
U6_gga(x1, x2, x3, x4, x5)  =  U6_gga(x1, x2, x3, x5)
U12_ggagg(x1, x2, x3, x4, x5, x6)  =  U12_ggagg(x1, x2, x3, x4, x5, x6)
pH_out_ggagg(x1, x2, x3, x4, x5)  =  pH_out_ggagg(x1, x2, x3, x4, x5)
pG_out_ggagg(x1, x2, x3, x4, x5)  =  pG_out_ggagg(x1, x2, x3, x4, x5)
pC_out_gaggag(x1, x2, x3, x4, x5, x6)  =  pC_out_gaggag(x1, x2, x3, x4, x5, x6)
REACHA_IN_GGGG(x1, x2, x3, x4)  =  REACHA_IN_GGGG(x1, x2, x3, x4)
PC_IN_GAGGAG(x1, x2, x3, x4, x5, x6)  =  PC_IN_GAGGAG(x1, x3, x4, x6)
U7_GAGGAG(x1, x2, x3, x4, x5, x6, x7)  =  U7_GAGGAG(x1, x3, x4, x6, x7)
PG_IN_GGAGG(x1, x2, x3, x4, x5)  =  PG_IN_GGAGG(x1, x2, x4, x5)
U9_GGAGG(x1, x2, x3, x4, x5, x6)  =  U9_GGAGG(x1, x2, x4, x5, x6)
PH_IN_GGAGG(x1, x2, x3, x4, x5)  =  PH_IN_GGAGG(x1, x2, x4, x5)
U11_GGAGG(x1, x2, x3, x4, x5, x6)  =  U11_GGAGG(x1, x2, x4, x5, x6)

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:

REACHA_IN_GGGG(T103, T104, T105, T106) → PC_IN_GAGGAG(T103, X81, T105, T106, X82, T104)
PC_IN_GAGGAG(T103, T113, T105, T106, X82, T104) → U7_GAGGAG(T103, T113, T105, T106, X82, T104, memberD_in_gag(T103, T113, T105))
U7_GAGGAG(T103, T113, T105, T106, X82, T104, memberD_out_gag(T103, T113, T105)) → PG_IN_GGAGG(T113, T106, X82, T104, T105)
PG_IN_GGAGG(T113, T106, X82, T104, T105) → U9_GGAGG(T113, T106, X82, T104, T105, memberE_in_gg(T113, T106))
U9_GGAGG(T113, T106, X82, T104, T105, memberE_out_gg(T113, T106)) → PH_IN_GGAGG(T113, T106, X82, T104, T105)
PH_IN_GGAGG(T113, T106, T185, T104, T105) → U11_GGAGG(T113, T106, T185, T104, T105, deleteF_in_gga(T113, T106, T185))
U11_GGAGG(T113, T106, T185, T104, T105, deleteF_out_gga(T113, T106, T185)) → REACHA_IN_GGGG(T113, T104, T105, T185)

The TRS R consists of the following rules:

memberD_in_gag(T132, T133, .(.(T132, .(T133, [])), T134)) → memberD_out_gag(T132, T133, .(.(T132, .(T133, [])), T134))
memberD_in_gag(T141, X121, .(T142, T143)) → U4_gag(T141, X121, T142, T143, memberD_in_gag(T141, X121, T143))
memberE_in_gg(T166, .(T166, T167)) → memberE_out_gg(T166, .(T166, T167))
memberE_in_gg(T174, .(T175, T176)) → U5_gg(T174, T175, T176, memberE_in_gg(T174, T176))
deleteF_in_gga(T198, .(T198, T199), T199) → deleteF_out_gga(T198, .(T198, T199), T199)
deleteF_in_gga(T206, .(T207, T208), .(T207, X191)) → U6_gga(T206, T207, T208, X191, deleteF_in_gga(T206, T208, X191))
U4_gag(T141, X121, T142, T143, memberD_out_gag(T141, X121, T143)) → memberD_out_gag(T141, X121, .(T142, T143))
U5_gg(T174, T175, T176, memberE_out_gg(T174, T176)) → memberE_out_gg(T174, .(T175, T176))
U6_gga(T206, T207, T208, X191, deleteF_out_gga(T206, T208, X191)) → deleteF_out_gga(T206, .(T207, T208), .(T207, X191))

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
[]  =  []
memberD_in_gag(x1, x2, x3)  =  memberD_in_gag(x1, x3)
memberD_out_gag(x1, x2, x3)  =  memberD_out_gag(x1, x2, x3)
U4_gag(x1, x2, x3, x4, x5)  =  U4_gag(x1, x3, x4, x5)
memberE_in_gg(x1, x2)  =  memberE_in_gg(x1, x2)
memberE_out_gg(x1, x2)  =  memberE_out_gg(x1, x2)
U5_gg(x1, x2, x3, x4)  =  U5_gg(x1, x2, x3, x4)
deleteF_in_gga(x1, x2, x3)  =  deleteF_in_gga(x1, x2)
deleteF_out_gga(x1, x2, x3)  =  deleteF_out_gga(x1, x2, x3)
U6_gga(x1, x2, x3, x4, x5)  =  U6_gga(x1, x2, x3, x5)
REACHA_IN_GGGG(x1, x2, x3, x4)  =  REACHA_IN_GGGG(x1, x2, x3, x4)
PC_IN_GAGGAG(x1, x2, x3, x4, x5, x6)  =  PC_IN_GAGGAG(x1, x3, x4, x6)
U7_GAGGAG(x1, x2, x3, x4, x5, x6, x7)  =  U7_GAGGAG(x1, x3, x4, x6, x7)
PG_IN_GGAGG(x1, x2, x3, x4, x5)  =  PG_IN_GGAGG(x1, x2, x4, x5)
U9_GGAGG(x1, x2, x3, x4, x5, x6)  =  U9_GGAGG(x1, x2, x4, x5, x6)
PH_IN_GGAGG(x1, x2, x3, x4, x5)  =  PH_IN_GGAGG(x1, x2, x4, x5)
U11_GGAGG(x1, x2, x3, x4, x5, x6)  =  U11_GGAGG(x1, x2, x4, x5, x6)

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:

REACHA_IN_GGGG(T103, T104, T105, T106) → PC_IN_GAGGAG(T103, T105, T106, T104)
PC_IN_GAGGAG(T103, T105, T106, T104) → U7_GAGGAG(T103, T105, T106, T104, memberD_in_gag(T103, T105))
U7_GAGGAG(T103, T105, T106, T104, memberD_out_gag(T103, T113, T105)) → PG_IN_GGAGG(T113, T106, T104, T105)
PG_IN_GGAGG(T113, T106, T104, T105) → U9_GGAGG(T113, T106, T104, T105, memberE_in_gg(T113, T106))
U9_GGAGG(T113, T106, T104, T105, memberE_out_gg(T113, T106)) → PH_IN_GGAGG(T113, T106, T104, T105)
PH_IN_GGAGG(T113, T106, T104, T105) → U11_GGAGG(T113, T106, T104, T105, deleteF_in_gga(T113, T106))
U11_GGAGG(T113, T106, T104, T105, deleteF_out_gga(T113, T106, T185)) → REACHA_IN_GGGG(T113, T104, T105, T185)

The TRS R consists of the following rules:

memberD_in_gag(T132, .(.(T132, .(T133, [])), T134)) → memberD_out_gag(T132, T133, .(.(T132, .(T133, [])), T134))
memberD_in_gag(T141, .(T142, T143)) → U4_gag(T141, T142, T143, memberD_in_gag(T141, T143))
memberE_in_gg(T166, .(T166, T167)) → memberE_out_gg(T166, .(T166, T167))
memberE_in_gg(T174, .(T175, T176)) → U5_gg(T174, T175, T176, memberE_in_gg(T174, T176))
deleteF_in_gga(T198, .(T198, T199)) → deleteF_out_gga(T198, .(T198, T199), T199)
deleteF_in_gga(T206, .(T207, T208)) → U6_gga(T206, T207, T208, deleteF_in_gga(T206, T208))
U4_gag(T141, T142, T143, memberD_out_gag(T141, X121, T143)) → memberD_out_gag(T141, X121, .(T142, T143))
U5_gg(T174, T175, T176, memberE_out_gg(T174, T176)) → memberE_out_gg(T174, .(T175, T176))
U6_gga(T206, T207, T208, deleteF_out_gga(T206, T208, X191)) → deleteF_out_gga(T206, .(T207, T208), .(T207, X191))

The set Q consists of the following terms:

memberD_in_gag(x0, x1)
memberE_in_gg(x0, x1)
deleteF_in_gga(x0, x1)
U4_gag(x0, x1, x2, x3)
U5_gg(x0, x1, x2, x3)
U6_gga(x0, x1, x2, x3)

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

(40) QDPOrderProof (EQUIVALENT transformation)

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

The following pairs can be oriented strictly and are deleted.

PG_IN_GGAGG(T113, T106, T104, T105) → U9_GGAGG(T113, T106, T104, T105, memberE_in_gg(T113, T106))

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

POL(.(x₁, x₂)) = 1 + x₁ + x₂   
POL(PC_IN_GAGGAG(x₁, x₂, x₃, x₄)) = 1 + x₁ + x₃   
POL(PG_IN_GGAGG(x₁, x₂, x₃, x₄)) = 1 + x₂   
POL(PH_IN_GGAGG(x₁, x₂, x₃, x₄)) = x₂   
POL(REACHA_IN_GGGG(x₁, x₂, x₃, x₄)) = 1 + x₁ + x₄   
POL(U11_GGAGG(x₁, x₂, x₃, x₄, x₅)) = x₅   
POL(U4_gag(x₁, x₂, x₃, x₄)) = x₃   
POL(U5_gg(x₁, x₂, x₃, x₄)) = 0   
POL(U6_gga(x₁, x₂, x₃, x₄)) = 1 + x₂ + x₄   
POL(U7_GAGGAG(x₁, x₂, x₃, x₄, x₅)) = 1 + x₃   
POL(U9_GGAGG(x₁, x₂, x₃, x₄, x₅)) = x₂   
POL([]) = 0   
POL(deleteF_in_gga(x₁, x₂)) = x₂   
POL(deleteF_out_gga(x₁, x₂, x₃)) = 1 + x₁ + x₃   
POL(memberD_in_gag(x₁, x₂)) = x₁ + x₂   
POL(memberD_out_gag(x₁, x₂, x₃)) = 0   
POL(memberE_in_gg(x₁, x₂)) = 0   
POL(memberE_out_gg(x₁, x₂)) = 0   

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

deleteF_in_gga(T198, .(T198, T199)) → deleteF_out_gga(T198, .(T198, T199), T199)
deleteF_in_gga(T206, .(T207, T208)) → U6_gga(T206, T207, T208, deleteF_in_gga(T206, T208))
U6_gga(T206, T207, T208, deleteF_out_gga(T206, T208, X191)) → deleteF_out_gga(T206, .(T207, T208), .(T207, X191))

(41) Obligation:

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

REACHA_IN_GGGG(T103, T104, T105, T106) → PC_IN_GAGGAG(T103, T105, T106, T104)
PC_IN_GAGGAG(T103, T105, T106, T104) → U7_GAGGAG(T103, T105, T106, T104, memberD_in_gag(T103, T105))
U7_GAGGAG(T103, T105, T106, T104, memberD_out_gag(T103, T113, T105)) → PG_IN_GGAGG(T113, T106, T104, T105)
U9_GGAGG(T113, T106, T104, T105, memberE_out_gg(T113, T106)) → PH_IN_GGAGG(T113, T106, T104, T105)
PH_IN_GGAGG(T113, T106, T104, T105) → U11_GGAGG(T113, T106, T104, T105, deleteF_in_gga(T113, T106))
U11_GGAGG(T113, T106, T104, T105, deleteF_out_gga(T113, T106, T185)) → REACHA_IN_GGGG(T113, T104, T105, T185)

The TRS R consists of the following rules:

memberD_in_gag(T132, .(.(T132, .(T133, [])), T134)) → memberD_out_gag(T132, T133, .(.(T132, .(T133, [])), T134))
memberD_in_gag(T141, .(T142, T143)) → U4_gag(T141, T142, T143, memberD_in_gag(T141, T143))
memberE_in_gg(T166, .(T166, T167)) → memberE_out_gg(T166, .(T166, T167))
memberE_in_gg(T174, .(T175, T176)) → U5_gg(T174, T175, T176, memberE_in_gg(T174, T176))
deleteF_in_gga(T198, .(T198, T199)) → deleteF_out_gga(T198, .(T198, T199), T199)
deleteF_in_gga(T206, .(T207, T208)) → U6_gga(T206, T207, T208, deleteF_in_gga(T206, T208))
U4_gag(T141, T142, T143, memberD_out_gag(T141, X121, T143)) → memberD_out_gag(T141, X121, .(T142, T143))
U5_gg(T174, T175, T176, memberE_out_gg(T174, T176)) → memberE_out_gg(T174, .(T175, T176))
U6_gga(T206, T207, T208, deleteF_out_gga(T206, T208, X191)) → deleteF_out_gga(T206, .(T207, T208), .(T207, X191))

The set Q consists of the following terms:

memberD_in_gag(x0, x1)
memberE_in_gg(x0, x1)
deleteF_in_gga(x0, x1)
U4_gag(x0, x1, x2, x3)
U5_gg(x0, x1, x2, x3)
U6_gga(x0, x1, x2, x3)

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

(42) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 6 less nodes.

(43) TRUE