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

(0) Obligation:

Clauses:

dis(or(B1, B2)) :- ','(con(B1), dis(B2)).
dis(B) :- con(B).
con(and(B1, B2)) :- ','(dis(B1), con(B2)).
con(B) :- bool(B).
bool(0).
bool(1).

Query: dis(g)

(1) PrologToPiTRSViaGraphTransformerProof (SOUND transformation)

Transformed Prolog program to (Pi-)TRS.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
3 [label="A: dis(T1)\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
6 [label="6: dis(T1), SCOPE: 1, CLAUSE: 0 | dis(T1), SCOPE: 1, CLAUSE: 1\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: ','(con(T4), dis(T5)) | dis(or(T4, T5)), SCOPE: 1, CLAUSE: 1\n\nT4 is ground\nT5 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: dis(T1), SCOPE: 1, CLAUSE: 1\n\nT1 is ground\nX3 is free\nX4 is free\ndis(T1) !=? dis(or(X3, X4))", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: ','(con(T4), dis(T5)), SCOPE: 2, CLAUSE: 2 | ','(con(T4), dis(T5)), SCOPE: 2, CLAUSE: 3 | ?_2 | dis(or(T4, T5)), SCOPE: 1, CLAUSE: 1\n\nT4 is ground\nT5 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: ','(con(T4), dis(T5)), SCOPE: 2, CLAUSE: 2\n\nT4 is ground\nT5 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: ','(con(T4), dis(T5)), SCOPE: 2, CLAUSE: 3 | ?_2 | dis(or(T4, T5)), SCOPE: 1, CLAUSE: 1\n\nT4 is ground\nT5 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="B: ','(','(dis(T14), con(T15)), dis(T5))\n\nT5 is ground\nT14 is ground\nT15 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: dis(T14)\n\nT14 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="E: ','(con(T15), dis(T5))\n\nT5 is ground\nT15 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
39 [label="C: con(T15)\n\nT15 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
40 [label="40: dis(T5)\n\nT5 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
45 [label="45: con(T15), SCOPE: 3, CLAUSE: 2 | con(T15), SCOPE: 3, CLAUSE: 3\n\nT15 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
57 [label="57: con(T15), SCOPE: 3, CLAUSE: 2\n\nT15 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
59 [label="59: con(T15), SCOPE: 3, CLAUSE: 3\n\nT15 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
67 [label="D: ','(dis(T32), con(T33))\n\nT32 is ground\nT33 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
70 [label="70: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
75 [label="75: dis(T32)\n\nT32 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
76 [label="76: con(T33)\n\nT33 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
90 [label="90: bool(T42)\n\nT42 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
91 [label="91: bool(T42), SCOPE: 4, CLAUSE: 4 | bool(T42), SCOPE: 4, CLAUSE: 5\n\nT42 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
93 [label="93: bool(T42), SCOPE: 4, CLAUSE: 4\n\nT42 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
94 [label="94: bool(T42), SCOPE: 4, CLAUSE: 5\n\nT42 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
95 [label="95: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
96 [label="96: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
97 [label="97: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
100 [label="100: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
101 [label="101: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
102 [label="102: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
115 [label="115: ','(con(T4), dis(T5)), SCOPE: 2, CLAUSE: 3\n\nT4 is ground\nT5 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
116 [label="116: ?_2 | dis(or(T4, T5)), SCOPE: 1, CLAUSE: 1\n\nT4 is ground\nT5 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
117 [label="117: ','(bool(T53), dis(T5))\n\nT5 is ground\nT53 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
118 [label="118: ','(bool(T53), dis(T5)), SCOPE: 5, CLAUSE: 4 | ','(bool(T53), dis(T5)), SCOPE: 5, CLAUSE: 5\n\nT5 is ground\nT53 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
119 [label="119: ','(bool(T53), dis(T5)), SCOPE: 5, CLAUSE: 4\n\nT5 is ground\nT53 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
120 [label="120: ','(bool(T53), dis(T5)), SCOPE: 5, CLAUSE: 5\n\nT5 is ground\nT53 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
121 [label="121: dis(T5)\n\nT5 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
122 [label="122: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
123 [label="123: dis(T5)\n\nT5 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
124 [label="124: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
125 [label="125: dis(or(T4, T5)), SCOPE: 1, CLAUSE: 1\n\nT4 is ground\nT5 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
126 [label="126: con(or(T64, T65))\n\nT64 is ground\nT65 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
127 [label="127: con(T67)\n\nT67 is ground\nX3 is free\nX4 is free\ndis(T67) !=? dis(or(X3, X4))", fontsize=16, style=filled, fillcolor=lightyellow];
128 [label="128: con(T67), SCOPE: 6, CLAUSE: 2 | con(T67), SCOPE: 6, CLAUSE: 3\n\nT67 is ground\nX3 is free\nX4 is free\ndis(T67) !=? dis(or(X3, X4))", fontsize=16, style=filled, fillcolor=lightyellow];
129 [label="129: con(T67), SCOPE: 6, CLAUSE: 2\n\nT67 is ground\nX3 is free\nX4 is free\ndis(T67) !=? dis(or(X3, X4))", fontsize=16, style=filled, fillcolor=lightyellow];
130 [label="130: con(T67), SCOPE: 6, CLAUSE: 3\n\nT67 is ground\nX3 is free\nX4 is free\ndis(T67) !=? dis(or(X3, X4))", fontsize=16, style=filled, fillcolor=lightyellow];
131 [label="131: ','(dis(T76), con(T77))\n\nT76 is ground\nT77 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
132 [label="132: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
133 [label="133: bool(T82)\n\nT82 is ground\nX3 is free\nX4 is free\ndis(T82) !=? dis(or(X3, X4))", fontsize=16, style=filled, fillcolor=lightyellow];
134 [label="134: bool(T82), SCOPE: 7, CLAUSE: 4 | bool(T82), SCOPE: 7, CLAUSE: 5\n\nT82 is ground\nX3 is free\nX4 is free\ndis(T82) !=? dis(or(X3, X4))", fontsize=16, style=filled, fillcolor=lightyellow];
135 [label="135: bool(T82), SCOPE: 7, CLAUSE: 4\n\nT82 is ground\nX3 is free\nX4 is free\ndis(T82) !=? dis(or(X3, X4))", fontsize=16, style=filled, fillcolor=lightyellow];
136 [label="136: bool(T82), SCOPE: 7, CLAUSE: 5\n\nT82 is ground\nX3 is free\nX4 is free\ndis(T82) !=? dis(or(X3, X4))", fontsize=16, style=filled, fillcolor=lightyellow];
137 [label="137: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
138 [label="138: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
139 [label="139: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
140 [label="140: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
141 [label="141: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
142 [label="142: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
143 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
3 -> 143 [arrowhead = none ];
143 -> 6;

144 [label="EVAL with clause\ndis(or(X3, X4)) :- ','(con(X3), dis(X4)).\nand substitutionX3 -> T4,\nX4 -> T5,\nT1 -> or(T4, T5)", fontsize=14, style = filled, fillcolor = lightblue];
6 -> 144 [arrowhead = none ];
144 -> 10;

145 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
6 -> 145 [arrowhead = none ];
145 -> 12;

146 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
10 -> 146 [arrowhead = none ];
146 -> 16;

147 [label="ONLY EVAL with clause\ndis(X67) :- con(X67).\nand substitutionT1 -> T67,\nX67 -> T67", fontsize=14, style = filled, fillcolor = lightblue];
12 -> 147 [arrowhead = none ];
147 -> 127;

148 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
16 -> 148 [arrowhead = none ];
148 -> 19;

149 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
16 -> 149 [arrowhead = none ];
149 -> 20;

150 [label="EVAL with clause\ncon(and(X13, X14)) :- ','(dis(X13), con(X14)).\nand substitutionX13 -> T14,\nX14 -> T15,\nT4 -> and(T14, T15)", fontsize=14, style = filled, fillcolor = lightblue];
19 -> 150 [arrowhead = none ];
150 -> 23;

151 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
19 -> 151 [arrowhead = none ];
151 -> 24;

152 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
20 -> 152 [arrowhead = none ];
152 -> 115;

153 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
20 -> 153 [arrowhead = none ];
153 -> 116;

154 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
23 -> 154 [arrowhead = none ];
154 -> 27;

155 [label="SPLIT 2\nnew knowledge:\nT14 is ground", fontsize=14, style = filled, fillcolor = violet];
23 -> 155 [arrowhead = none ];
155 -> 30;

156 [label="INSTANCE with matching:\nT1 -> T14", fontsize=14, style = filled, fillcolor = lightgrey];
27 -> 156 [arrowhead = none , style=dashed];
156 -> 3[style=dashed];

157 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
30 -> 157 [arrowhead = none ];
157 -> 39;

158 [label="SPLIT 2\nnew knowledge:\nT15 is ground", fontsize=14, style = filled, fillcolor = violet];
30 -> 158 [arrowhead = none ];
158 -> 40;

159 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
39 -> 159 [arrowhead = none ];
159 -> 45;

160 [label="INSTANCE with matching:\nT1 -> T5", fontsize=14, style = filled, fillcolor = lightgrey];
40 -> 160 [arrowhead = none , style=dashed];
160 -> 3[style=dashed];

161 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
45 -> 161 [arrowhead = none ];
161 -> 57;

162 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
45 -> 162 [arrowhead = none ];
162 -> 59;

163 [label="EVAL with clause\ncon(and(X31, X32)) :- ','(dis(X31), con(X32)).\nand substitutionX31 -> T32,\nX32 -> T33,\nT15 -> and(T32, T33)", fontsize=14, style = filled, fillcolor = lightblue];
57 -> 163 [arrowhead = none ];
163 -> 67;

164 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
57 -> 164 [arrowhead = none ];
164 -> 70;

165 [label="ONLY EVAL with clause\ncon(X41) :- bool(X41).\nand substitutionT15 -> T42,\nX41 -> T42", fontsize=14, style = filled, fillcolor = lightblue];
59 -> 165 [arrowhead = none ];
165 -> 90;

166 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
67 -> 166 [arrowhead = none ];
166 -> 75;

167 [label="SPLIT 2\nnew knowledge:\nT32 is ground", fontsize=14, style = filled, fillcolor = violet];
67 -> 167 [arrowhead = none ];
167 -> 76;

168 [label="INSTANCE with matching:\nT1 -> T32", fontsize=14, style = filled, fillcolor = lightgrey];
75 -> 168 [arrowhead = none , style=dashed];
168 -> 3[style=dashed];

169 [label="INSTANCE with matching:\nT15 -> T33", fontsize=14, style = filled, fillcolor = lightgrey];
76 -> 169 [arrowhead = none , style=dashed];
169 -> 39[style=dashed];

170 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
90 -> 170 [arrowhead = none ];
170 -> 91;

171 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
91 -> 171 [arrowhead = none ];
171 -> 93;

172 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
91 -> 172 [arrowhead = none ];
172 -> 94;

173 [label="EVAL with clause\nbool(0).\nand substitutionT42 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
93 -> 173 [arrowhead = none ];
173 -> 95;

174 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
93 -> 174 [arrowhead = none ];
174 -> 96;

175 [label="EVAL with clause\nbool(1).\nand substitutionT42 -> 1", fontsize=14, style = filled, fillcolor = lightblue];
94 -> 175 [arrowhead = none ];
175 -> 100;

176 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
94 -> 176 [arrowhead = none ];
176 -> 101;

177 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
95 -> 177 [arrowhead = none ];
177 -> 97;

178 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
100 -> 178 [arrowhead = none ];
178 -> 102;

179 [label="ONLY EVAL with clause\ncon(X52) :- bool(X52).\nand substitutionT4 -> T53,\nX52 -> T53", fontsize=14, style = filled, fillcolor = lightblue];
115 -> 179 [arrowhead = none ];
179 -> 117;

180 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
116 -> 180 [arrowhead = none ];
180 -> 125;

181 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
117 -> 181 [arrowhead = none ];
181 -> 118;

182 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
118 -> 182 [arrowhead = none ];
182 -> 119;

183 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
118 -> 183 [arrowhead = none ];
183 -> 120;

184 [label="EVAL with clause\nbool(0).\nand substitutionT53 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
119 -> 184 [arrowhead = none ];
184 -> 121;

185 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
119 -> 185 [arrowhead = none ];
185 -> 122;

186 [label="EVAL with clause\nbool(1).\nand substitutionT53 -> 1", fontsize=14, style = filled, fillcolor = lightblue];
120 -> 186 [arrowhead = none ];
186 -> 123;

187 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
120 -> 187 [arrowhead = none ];
187 -> 124;

188 [label="INSTANCE with matching:\nT1 -> T5", fontsize=14, style = filled, fillcolor = lightgrey];
121 -> 188 [arrowhead = none , style=dashed];
188 -> 3[style=dashed];

189 [label="INSTANCE with matching:\nT1 -> T5", fontsize=14, style = filled, fillcolor = lightgrey];
123 -> 189 [arrowhead = none , style=dashed];
189 -> 3[style=dashed];

190 [label="ONLY EVAL with clause\ndis(X63) :- con(X63).\nand substitutionT4 -> T64,\nT5 -> T65,\nX63 -> or(T64, T65)", fontsize=14, style = filled, fillcolor = lightblue];
125 -> 190 [arrowhead = none ];
190 -> 126;

191 [label="INSTANCE with matching:\nT15 -> or(T64, T65)", fontsize=14, style = filled, fillcolor = lightgrey];
126 -> 191 [arrowhead = none , style=dashed];
191 -> 39[style=dashed];

192 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
127 -> 192 [arrowhead = none ];
192 -> 128;

193 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
128 -> 193 [arrowhead = none ];
193 -> 129;

194 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
128 -> 194 [arrowhead = none ];
194 -> 130;

195 [label="EVAL with clause\ncon(and(X76, X77)) :- ','(dis(X76), con(X77)).\nand substitutionX76 -> T76,\nX77 -> T77,\nT67 -> and(T76, T77)", fontsize=14, style = filled, fillcolor = lightblue];
129 -> 195 [arrowhead = none ];
195 -> 131;

196 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
129 -> 196 [arrowhead = none ];
196 -> 132;

197 [label="ONLY EVAL with clause\ncon(X82) :- bool(X82).\nand substitutionT67 -> T82,\nX82 -> T82", fontsize=14, style = filled, fillcolor = lightblue];
130 -> 197 [arrowhead = none ];
197 -> 133;

198 [label="INSTANCE with matching:\nT32 -> T76\nT33 -> T77", fontsize=14, style = filled, fillcolor = lightgrey];
131 -> 198 [arrowhead = none , style=dashed];
198 -> 67[style=dashed];

199 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
133 -> 199 [arrowhead = none ];
199 -> 134;

200 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
134 -> 200 [arrowhead = none ];
200 -> 135;

201 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
134 -> 201 [arrowhead = none ];
201 -> 136;

202 [label="EVAL with clause\nbool(0).\nand substitutionT82 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
135 -> 202 [arrowhead = none ];
202 -> 137;

203 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
135 -> 203 [arrowhead = none ];
203 -> 138;

204 [label="EVAL with clause\nbool(1).\nand substitutionT82 -> 1", fontsize=14, style = filled, fillcolor = lightblue];
136 -> 204 [arrowhead = none ];
204 -> 140;

205 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
136 -> 205 [arrowhead = none ];
205 -> 141;

206 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
137 -> 206 [arrowhead = none ];
206 -> 139;

207 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
140 -> 207 [arrowhead = none ];
207 -> 142;

}

(2) Obligation:

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

disA_in_g(or(and(T14, T15), T5)) → U1_g(T14, T15, T5, pB_in_ggg(T14, T15, T5))
pB_in_ggg(T14, T15, T5) → U7_ggg(T14, T15, T5, disA_in_g(T14))
disA_in_g(or(0, T5)) → U2_g(T5, disA_in_g(T5))
disA_in_g(or(1, T5)) → U3_g(T5, disA_in_g(T5))
disA_in_g(or(T64, T65)) → U4_g(T64, T65, conC_in_g(or(T64, T65)))
conC_in_g(and(T32, T33)) → U6_g(T32, T33, pD_in_gg(T32, T33))
pD_in_gg(T32, T33) → U11_gg(T32, T33, disA_in_g(T32))
disA_in_g(and(T76, T77)) → U5_g(T76, T77, pD_in_gg(T76, T77))
U5_g(T76, T77, pD_out_gg(T76, T77)) → disA_out_g(and(T76, T77))
disA_in_g(0) → disA_out_g(0)
disA_in_g(1) → disA_out_g(1)
U11_gg(T32, T33, disA_out_g(T32)) → U12_gg(T32, T33, conC_in_g(T33))
conC_in_g(0) → conC_out_g(0)
conC_in_g(1) → conC_out_g(1)
U12_gg(T32, T33, conC_out_g(T33)) → pD_out_gg(T32, T33)
U6_g(T32, T33, pD_out_gg(T32, T33)) → conC_out_g(and(T32, T33))
U4_g(T64, T65, conC_out_g(or(T64, T65))) → disA_out_g(or(T64, T65))
U3_g(T5, disA_out_g(T5)) → disA_out_g(or(1, T5))
U2_g(T5, disA_out_g(T5)) → disA_out_g(or(0, T5))
U7_ggg(T14, T15, T5, disA_out_g(T14)) → U8_ggg(T14, T15, T5, pE_in_gg(T15, T5))
pE_in_gg(T15, T5) → U9_gg(T15, T5, conC_in_g(T15))
U9_gg(T15, T5, conC_out_g(T15)) → U10_gg(T15, T5, disA_in_g(T5))
U10_gg(T15, T5, disA_out_g(T5)) → pE_out_gg(T15, T5)
U8_ggg(T14, T15, T5, pE_out_gg(T15, T5)) → pB_out_ggg(T14, T15, T5)
U1_g(T14, T15, T5, pB_out_ggg(T14, T15, T5)) → disA_out_g(or(and(T14, T15), T5))

Pi is empty.

(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:

DISA_IN_G(or(and(T14, T15), T5)) → U1_G(T14, T15, T5, pB_in_ggg(T14, T15, T5))
DISA_IN_G(or(and(T14, T15), T5)) → PB_IN_GGG(T14, T15, T5)
PB_IN_GGG(T14, T15, T5) → U7_GGG(T14, T15, T5, disA_in_g(T14))
PB_IN_GGG(T14, T15, T5) → DISA_IN_G(T14)
DISA_IN_G(or(0, T5)) → U2_G(T5, disA_in_g(T5))
DISA_IN_G(or(0, T5)) → DISA_IN_G(T5)
DISA_IN_G(or(1, T5)) → U3_G(T5, disA_in_g(T5))
DISA_IN_G(or(1, T5)) → DISA_IN_G(T5)
DISA_IN_G(or(T64, T65)) → U4_G(T64, T65, conC_in_g(or(T64, T65)))
DISA_IN_G(or(T64, T65)) → CONC_IN_G(or(T64, T65))
CONC_IN_G(and(T32, T33)) → U6_G(T32, T33, pD_in_gg(T32, T33))
CONC_IN_G(and(T32, T33)) → PD_IN_GG(T32, T33)
PD_IN_GG(T32, T33) → U11_GG(T32, T33, disA_in_g(T32))
PD_IN_GG(T32, T33) → DISA_IN_G(T32)
DISA_IN_G(and(T76, T77)) → U5_G(T76, T77, pD_in_gg(T76, T77))
DISA_IN_G(and(T76, T77)) → PD_IN_GG(T76, T77)
U11_GG(T32, T33, disA_out_g(T32)) → U12_GG(T32, T33, conC_in_g(T33))
U11_GG(T32, T33, disA_out_g(T32)) → CONC_IN_G(T33)
U7_GGG(T14, T15, T5, disA_out_g(T14)) → U8_GGG(T14, T15, T5, pE_in_gg(T15, T5))
U7_GGG(T14, T15, T5, disA_out_g(T14)) → PE_IN_GG(T15, T5)
PE_IN_GG(T15, T5) → U9_GG(T15, T5, conC_in_g(T15))
PE_IN_GG(T15, T5) → CONC_IN_G(T15)
U9_GG(T15, T5, conC_out_g(T15)) → U10_GG(T15, T5, disA_in_g(T5))
U9_GG(T15, T5, conC_out_g(T15)) → DISA_IN_G(T5)

The TRS R consists of the following rules:

disA_in_g(or(and(T14, T15), T5)) → U1_g(T14, T15, T5, pB_in_ggg(T14, T15, T5))
pB_in_ggg(T14, T15, T5) → U7_ggg(T14, T15, T5, disA_in_g(T14))
disA_in_g(or(0, T5)) → U2_g(T5, disA_in_g(T5))
disA_in_g(or(1, T5)) → U3_g(T5, disA_in_g(T5))
disA_in_g(or(T64, T65)) → U4_g(T64, T65, conC_in_g(or(T64, T65)))
conC_in_g(and(T32, T33)) → U6_g(T32, T33, pD_in_gg(T32, T33))
pD_in_gg(T32, T33) → U11_gg(T32, T33, disA_in_g(T32))
disA_in_g(and(T76, T77)) → U5_g(T76, T77, pD_in_gg(T76, T77))
U5_g(T76, T77, pD_out_gg(T76, T77)) → disA_out_g(and(T76, T77))
disA_in_g(0) → disA_out_g(0)
disA_in_g(1) → disA_out_g(1)
U11_gg(T32, T33, disA_out_g(T32)) → U12_gg(T32, T33, conC_in_g(T33))
conC_in_g(0) → conC_out_g(0)
conC_in_g(1) → conC_out_g(1)
U12_gg(T32, T33, conC_out_g(T33)) → pD_out_gg(T32, T33)
U6_g(T32, T33, pD_out_gg(T32, T33)) → conC_out_g(and(T32, T33))
U4_g(T64, T65, conC_out_g(or(T64, T65))) → disA_out_g(or(T64, T65))
U3_g(T5, disA_out_g(T5)) → disA_out_g(or(1, T5))
U2_g(T5, disA_out_g(T5)) → disA_out_g(or(0, T5))
U7_ggg(T14, T15, T5, disA_out_g(T14)) → U8_ggg(T14, T15, T5, pE_in_gg(T15, T5))
pE_in_gg(T15, T5) → U9_gg(T15, T5, conC_in_g(T15))
U9_gg(T15, T5, conC_out_g(T15)) → U10_gg(T15, T5, disA_in_g(T5))
U10_gg(T15, T5, disA_out_g(T5)) → pE_out_gg(T15, T5)
U8_ggg(T14, T15, T5, pE_out_gg(T15, T5)) → pB_out_ggg(T14, T15, T5)
U1_g(T14, T15, T5, pB_out_ggg(T14, T15, T5)) → disA_out_g(or(and(T14, T15), T5))

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

(4) Obligation:

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

DISA_IN_G(or(and(T14, T15), T5)) → U1_G(T14, T15, T5, pB_in_ggg(T14, T15, T5))
DISA_IN_G(or(and(T14, T15), T5)) → PB_IN_GGG(T14, T15, T5)
PB_IN_GGG(T14, T15, T5) → U7_GGG(T14, T15, T5, disA_in_g(T14))
PB_IN_GGG(T14, T15, T5) → DISA_IN_G(T14)
DISA_IN_G(or(0, T5)) → U2_G(T5, disA_in_g(T5))
DISA_IN_G(or(0, T5)) → DISA_IN_G(T5)
DISA_IN_G(or(1, T5)) → U3_G(T5, disA_in_g(T5))
DISA_IN_G(or(1, T5)) → DISA_IN_G(T5)
DISA_IN_G(or(T64, T65)) → U4_G(T64, T65, conC_in_g(or(T64, T65)))
DISA_IN_G(or(T64, T65)) → CONC_IN_G(or(T64, T65))
CONC_IN_G(and(T32, T33)) → U6_G(T32, T33, pD_in_gg(T32, T33))
CONC_IN_G(and(T32, T33)) → PD_IN_GG(T32, T33)
PD_IN_GG(T32, T33) → U11_GG(T32, T33, disA_in_g(T32))
PD_IN_GG(T32, T33) → DISA_IN_G(T32)
DISA_IN_G(and(T76, T77)) → U5_G(T76, T77, pD_in_gg(T76, T77))
DISA_IN_G(and(T76, T77)) → PD_IN_GG(T76, T77)
U11_GG(T32, T33, disA_out_g(T32)) → U12_GG(T32, T33, conC_in_g(T33))
U11_GG(T32, T33, disA_out_g(T32)) → CONC_IN_G(T33)
U7_GGG(T14, T15, T5, disA_out_g(T14)) → U8_GGG(T14, T15, T5, pE_in_gg(T15, T5))
U7_GGG(T14, T15, T5, disA_out_g(T14)) → PE_IN_GG(T15, T5)
PE_IN_GG(T15, T5) → U9_GG(T15, T5, conC_in_g(T15))
PE_IN_GG(T15, T5) → CONC_IN_G(T15)
U9_GG(T15, T5, conC_out_g(T15)) → U10_GG(T15, T5, disA_in_g(T5))
U9_GG(T15, T5, conC_out_g(T15)) → DISA_IN_G(T5)

The TRS R consists of the following rules:

disA_in_g(or(and(T14, T15), T5)) → U1_g(T14, T15, T5, pB_in_ggg(T14, T15, T5))
pB_in_ggg(T14, T15, T5) → U7_ggg(T14, T15, T5, disA_in_g(T14))
disA_in_g(or(0, T5)) → U2_g(T5, disA_in_g(T5))
disA_in_g(or(1, T5)) → U3_g(T5, disA_in_g(T5))
disA_in_g(or(T64, T65)) → U4_g(T64, T65, conC_in_g(or(T64, T65)))
conC_in_g(and(T32, T33)) → U6_g(T32, T33, pD_in_gg(T32, T33))
pD_in_gg(T32, T33) → U11_gg(T32, T33, disA_in_g(T32))
disA_in_g(and(T76, T77)) → U5_g(T76, T77, pD_in_gg(T76, T77))
U5_g(T76, T77, pD_out_gg(T76, T77)) → disA_out_g(and(T76, T77))
disA_in_g(0) → disA_out_g(0)
disA_in_g(1) → disA_out_g(1)
U11_gg(T32, T33, disA_out_g(T32)) → U12_gg(T32, T33, conC_in_g(T33))
conC_in_g(0) → conC_out_g(0)
conC_in_g(1) → conC_out_g(1)
U12_gg(T32, T33, conC_out_g(T33)) → pD_out_gg(T32, T33)
U6_g(T32, T33, pD_out_gg(T32, T33)) → conC_out_g(and(T32, T33))
U4_g(T64, T65, conC_out_g(or(T64, T65))) → disA_out_g(or(T64, T65))
U3_g(T5, disA_out_g(T5)) → disA_out_g(or(1, T5))
U2_g(T5, disA_out_g(T5)) → disA_out_g(or(0, T5))
U7_ggg(T14, T15, T5, disA_out_g(T14)) → U8_ggg(T14, T15, T5, pE_in_gg(T15, T5))
pE_in_gg(T15, T5) → U9_gg(T15, T5, conC_in_g(T15))
U9_gg(T15, T5, conC_out_g(T15)) → U10_gg(T15, T5, disA_in_g(T5))
U10_gg(T15, T5, disA_out_g(T5)) → pE_out_gg(T15, T5)
U8_ggg(T14, T15, T5, pE_out_gg(T15, T5)) → pB_out_ggg(T14, T15, T5)
U1_g(T14, T15, T5, pB_out_ggg(T14, T15, T5)) → disA_out_g(or(and(T14, T15), T5))

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

(5) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 10 less nodes.

(6) Obligation:

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

DISA_IN_G(or(and(T14, T15), T5)) → PB_IN_GGG(T14, T15, T5)
PB_IN_GGG(T14, T15, T5) → U7_GGG(T14, T15, T5, disA_in_g(T14))
U7_GGG(T14, T15, T5, disA_out_g(T14)) → PE_IN_GG(T15, T5)
PE_IN_GG(T15, T5) → U9_GG(T15, T5, conC_in_g(T15))
U9_GG(T15, T5, conC_out_g(T15)) → DISA_IN_G(T5)
DISA_IN_G(or(0, T5)) → DISA_IN_G(T5)
DISA_IN_G(or(1, T5)) → DISA_IN_G(T5)
DISA_IN_G(and(T76, T77)) → PD_IN_GG(T76, T77)
PD_IN_GG(T32, T33) → U11_GG(T32, T33, disA_in_g(T32))
U11_GG(T32, T33, disA_out_g(T32)) → CONC_IN_G(T33)
CONC_IN_G(and(T32, T33)) → PD_IN_GG(T32, T33)
PD_IN_GG(T32, T33) → DISA_IN_G(T32)
PE_IN_GG(T15, T5) → CONC_IN_G(T15)
PB_IN_GGG(T14, T15, T5) → DISA_IN_G(T14)

The TRS R consists of the following rules:

disA_in_g(or(and(T14, T15), T5)) → U1_g(T14, T15, T5, pB_in_ggg(T14, T15, T5))
pB_in_ggg(T14, T15, T5) → U7_ggg(T14, T15, T5, disA_in_g(T14))
disA_in_g(or(0, T5)) → U2_g(T5, disA_in_g(T5))
disA_in_g(or(1, T5)) → U3_g(T5, disA_in_g(T5))
disA_in_g(or(T64, T65)) → U4_g(T64, T65, conC_in_g(or(T64, T65)))
conC_in_g(and(T32, T33)) → U6_g(T32, T33, pD_in_gg(T32, T33))
pD_in_gg(T32, T33) → U11_gg(T32, T33, disA_in_g(T32))
disA_in_g(and(T76, T77)) → U5_g(T76, T77, pD_in_gg(T76, T77))
U5_g(T76, T77, pD_out_gg(T76, T77)) → disA_out_g(and(T76, T77))
disA_in_g(0) → disA_out_g(0)
disA_in_g(1) → disA_out_g(1)
U11_gg(T32, T33, disA_out_g(T32)) → U12_gg(T32, T33, conC_in_g(T33))
conC_in_g(0) → conC_out_g(0)
conC_in_g(1) → conC_out_g(1)
U12_gg(T32, T33, conC_out_g(T33)) → pD_out_gg(T32, T33)
U6_g(T32, T33, pD_out_gg(T32, T33)) → conC_out_g(and(T32, T33))
U4_g(T64, T65, conC_out_g(or(T64, T65))) → disA_out_g(or(T64, T65))
U3_g(T5, disA_out_g(T5)) → disA_out_g(or(1, T5))
U2_g(T5, disA_out_g(T5)) → disA_out_g(or(0, T5))
U7_ggg(T14, T15, T5, disA_out_g(T14)) → U8_ggg(T14, T15, T5, pE_in_gg(T15, T5))
pE_in_gg(T15, T5) → U9_gg(T15, T5, conC_in_g(T15))
U9_gg(T15, T5, conC_out_g(T15)) → U10_gg(T15, T5, disA_in_g(T5))
U10_gg(T15, T5, disA_out_g(T5)) → pE_out_gg(T15, T5)
U8_ggg(T14, T15, T5, pE_out_gg(T15, T5)) → pB_out_ggg(T14, T15, T5)
U1_g(T14, T15, T5, pB_out_ggg(T14, T15, T5)) → disA_out_g(or(and(T14, T15), T5))

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

(7) UsableRulesProof (EQUIVALENT transformation)

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

(8) Obligation:

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

DISA_IN_G(or(and(T14, T15), T5)) → PB_IN_GGG(T14, T15, T5)
PB_IN_GGG(T14, T15, T5) → U7_GGG(T14, T15, T5, disA_in_g(T14))
U7_GGG(T14, T15, T5, disA_out_g(T14)) → PE_IN_GG(T15, T5)
PE_IN_GG(T15, T5) → U9_GG(T15, T5, conC_in_g(T15))
U9_GG(T15, T5, conC_out_g(T15)) → DISA_IN_G(T5)
DISA_IN_G(or(0, T5)) → DISA_IN_G(T5)
DISA_IN_G(or(1, T5)) → DISA_IN_G(T5)
DISA_IN_G(and(T76, T77)) → PD_IN_GG(T76, T77)
PD_IN_GG(T32, T33) → U11_GG(T32, T33, disA_in_g(T32))
U11_GG(T32, T33, disA_out_g(T32)) → CONC_IN_G(T33)
CONC_IN_G(and(T32, T33)) → PD_IN_GG(T32, T33)
PD_IN_GG(T32, T33) → DISA_IN_G(T32)
PE_IN_GG(T15, T5) → CONC_IN_G(T15)
PB_IN_GGG(T14, T15, T5) → DISA_IN_G(T14)

The TRS R consists of the following rules:

disA_in_g(or(and(T14, T15), T5)) → U1_g(T14, T15, T5, pB_in_ggg(T14, T15, T5))
disA_in_g(or(0, T5)) → U2_g(T5, disA_in_g(T5))
disA_in_g(or(1, T5)) → U3_g(T5, disA_in_g(T5))
disA_in_g(or(T64, T65)) → U4_g(T64, T65, conC_in_g(or(T64, T65)))
disA_in_g(and(T76, T77)) → U5_g(T76, T77, pD_in_gg(T76, T77))
disA_in_g(0) → disA_out_g(0)
disA_in_g(1) → disA_out_g(1)
conC_in_g(and(T32, T33)) → U6_g(T32, T33, pD_in_gg(T32, T33))
conC_in_g(0) → conC_out_g(0)
conC_in_g(1) → conC_out_g(1)
U1_g(T14, T15, T5, pB_out_ggg(T14, T15, T5)) → disA_out_g(or(and(T14, T15), T5))
U2_g(T5, disA_out_g(T5)) → disA_out_g(or(0, T5))
U3_g(T5, disA_out_g(T5)) → disA_out_g(or(1, T5))
U5_g(T76, T77, pD_out_gg(T76, T77)) → disA_out_g(and(T76, T77))
U6_g(T32, T33, pD_out_gg(T32, T33)) → conC_out_g(and(T32, T33))
pB_in_ggg(T14, T15, T5) → U7_ggg(T14, T15, T5, disA_in_g(T14))
pD_in_gg(T32, T33) → U11_gg(T32, T33, disA_in_g(T32))
U7_ggg(T14, T15, T5, disA_out_g(T14)) → U8_ggg(T14, T15, T5, pE_in_gg(T15, T5))
U11_gg(T32, T33, disA_out_g(T32)) → U12_gg(T32, T33, conC_in_g(T33))
U8_ggg(T14, T15, T5, pE_out_gg(T15, T5)) → pB_out_ggg(T14, T15, T5)
U12_gg(T32, T33, conC_out_g(T33)) → pD_out_gg(T32, T33)
pE_in_gg(T15, T5) → U9_gg(T15, T5, conC_in_g(T15))
U9_gg(T15, T5, conC_out_g(T15)) → U10_gg(T15, T5, disA_in_g(T5))
U10_gg(T15, T5, disA_out_g(T5)) → pE_out_gg(T15, T5)

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

(9) PiDPToQDPProof (EQUIVALENT transformation)

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

(10) Obligation:

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

DISA_IN_G(or(and(T14, T15), T5)) → PB_IN_GGG(T14, T15, T5)
PB_IN_GGG(T14, T15, T5) → U7_GGG(T14, T15, T5, disA_in_g(T14))
U7_GGG(T14, T15, T5, disA_out_g(T14)) → PE_IN_GG(T15, T5)
PE_IN_GG(T15, T5) → U9_GG(T15, T5, conC_in_g(T15))
U9_GG(T15, T5, conC_out_g(T15)) → DISA_IN_G(T5)
DISA_IN_G(or(0, T5)) → DISA_IN_G(T5)
DISA_IN_G(or(1, T5)) → DISA_IN_G(T5)
DISA_IN_G(and(T76, T77)) → PD_IN_GG(T76, T77)
PD_IN_GG(T32, T33) → U11_GG(T32, T33, disA_in_g(T32))
U11_GG(T32, T33, disA_out_g(T32)) → CONC_IN_G(T33)
CONC_IN_G(and(T32, T33)) → PD_IN_GG(T32, T33)
PD_IN_GG(T32, T33) → DISA_IN_G(T32)
PE_IN_GG(T15, T5) → CONC_IN_G(T15)
PB_IN_GGG(T14, T15, T5) → DISA_IN_G(T14)

The TRS R consists of the following rules:

disA_in_g(or(and(T14, T15), T5)) → U1_g(T14, T15, T5, pB_in_ggg(T14, T15, T5))
disA_in_g(or(0, T5)) → U2_g(T5, disA_in_g(T5))
disA_in_g(or(1, T5)) → U3_g(T5, disA_in_g(T5))
disA_in_g(or(T64, T65)) → U4_g(T64, T65, conC_in_g(or(T64, T65)))
disA_in_g(and(T76, T77)) → U5_g(T76, T77, pD_in_gg(T76, T77))
disA_in_g(0) → disA_out_g(0)
disA_in_g(1) → disA_out_g(1)
conC_in_g(and(T32, T33)) → U6_g(T32, T33, pD_in_gg(T32, T33))
conC_in_g(0) → conC_out_g(0)
conC_in_g(1) → conC_out_g(1)
U1_g(T14, T15, T5, pB_out_ggg(T14, T15, T5)) → disA_out_g(or(and(T14, T15), T5))
U2_g(T5, disA_out_g(T5)) → disA_out_g(or(0, T5))
U3_g(T5, disA_out_g(T5)) → disA_out_g(or(1, T5))
U5_g(T76, T77, pD_out_gg(T76, T77)) → disA_out_g(and(T76, T77))
U6_g(T32, T33, pD_out_gg(T32, T33)) → conC_out_g(and(T32, T33))
pB_in_ggg(T14, T15, T5) → U7_ggg(T14, T15, T5, disA_in_g(T14))
pD_in_gg(T32, T33) → U11_gg(T32, T33, disA_in_g(T32))
U7_ggg(T14, T15, T5, disA_out_g(T14)) → U8_ggg(T14, T15, T5, pE_in_gg(T15, T5))
U11_gg(T32, T33, disA_out_g(T32)) → U12_gg(T32, T33, conC_in_g(T33))
U8_ggg(T14, T15, T5, pE_out_gg(T15, T5)) → pB_out_ggg(T14, T15, T5)
U12_gg(T32, T33, conC_out_g(T33)) → pD_out_gg(T32, T33)
pE_in_gg(T15, T5) → U9_gg(T15, T5, conC_in_g(T15))
U9_gg(T15, T5, conC_out_g(T15)) → U10_gg(T15, T5, disA_in_g(T5))
U10_gg(T15, T5, disA_out_g(T5)) → pE_out_gg(T15, T5)

The set Q consists of the following terms:

disA_in_g(x0)
conC_in_g(x0)
U1_g(x0, x1, x2, x3)
U2_g(x0, x1)
U3_g(x0, x1)
U5_g(x0, x1, x2)
U6_g(x0, x1, x2)
pB_in_ggg(x0, x1, x2)
pD_in_gg(x0, x1)
U7_ggg(x0, x1, x2, x3)
U11_gg(x0, x1, x2)
U8_ggg(x0, x1, x2, x3)
U12_gg(x0, x1, x2)
pE_in_gg(x0, x1)
U9_gg(x0, x1, x2)
U10_gg(x0, x1, x2)

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

(11) 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:

• PB_IN_GGG(T14, T15, T5) → DISA_IN_G(T14)
  The graph contains the following edges 1 >= 1

• PB_IN_GGG(T14, T15, T5) → U7_GGG(T14, T15, T5, disA_in_g(T14))
  The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 3

• U7_GGG(T14, T15, T5, disA_out_g(T14)) → PE_IN_GG(T15, T5)
  The graph contains the following edges 2 >= 1, 3 >= 2

• DISA_IN_G(or(and(T14, T15), T5)) → PB_IN_GGG(T14, T15, T5)
  The graph contains the following edges 1 > 1, 1 > 2, 1 > 3

• U9_GG(T15, T5, conC_out_g(T15)) → DISA_IN_G(T5)
  The graph contains the following edges 2 >= 1

• PD_IN_GG(T32, T33) → DISA_IN_G(T32)
  The graph contains the following edges 1 >= 1

• PE_IN_GG(T15, T5) → U9_GG(T15, T5, conC_in_g(T15))
  The graph contains the following edges 1 >= 1, 2 >= 2

• PE_IN_GG(T15, T5) → CONC_IN_G(T15)
  The graph contains the following edges 1 >= 1

• DISA_IN_G(and(T76, T77)) → PD_IN_GG(T76, T77)
  The graph contains the following edges 1 > 1, 1 > 2

• PD_IN_GG(T32, T33) → U11_GG(T32, T33, disA_in_g(T32))
  The graph contains the following edges 1 >= 1, 2 >= 2

• U11_GG(T32, T33, disA_out_g(T32)) → CONC_IN_G(T33)
  The graph contains the following edges 2 >= 1

• CONC_IN_G(and(T32, T33)) → PD_IN_GG(T32, T33)
  The graph contains the following edges 1 > 1, 1 > 2

• DISA_IN_G(or(0, T5)) → DISA_IN_G(T5)
  The graph contains the following edges 1 > 1

• DISA_IN_G(or(1, T5)) → DISA_IN_G(T5)
  The graph contains the following edges 1 > 1

(12) YES