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

(0) Obligation:

Clauses:

mult(0, Y, 0).
mult(s(X), Y, Z) :- ','(mult(X, Y, Z1), add(Z1, Y, Z)).
add(0, Y, Y).
add(s(X), Y, s(Z)) :- add(X, Y, Z).

Query: mult(g,g,a)

(1) PrologToPiTRSViaGraphTransformerProof (SOUND transformation)

Transformed Prolog program to (Pi-)TRS.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
2 [label="A: mult(T1, T2, T3)\n\nT1 is ground\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
4 [label="4: mult(T1, T2, T3), SCOPE: 1, CLAUSE: 0 | mult(T1, T2, T3), SCOPE: 1, CLAUSE: 1\n\nT1 is ground\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: true | mult(0, T5, T3), SCOPE: 1, CLAUSE: 1\n\nT5 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: mult(T1, T2, T3), SCOPE: 1, CLAUSE: 1\n\nT1 is ground\nT2 is ground\nX2 is free\nmult(T1, T2, T3) !=? mult(0, X2, 0)", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: mult(0, T5, T3), SCOPE: 1, CLAUSE: 1\n\nT5 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: ','(mult(T9, T10, X12), add(X12, T10, T12))\n\nT9 is ground\nT10 is ground\nX12 is free", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: ','(mult(T9, T10, X12), add(X12, T10, T12)), SCOPE: 2, CLAUSE: 0 | ','(mult(T9, T10, X12), add(X12, T10, T12)), SCOPE: 2, CLAUSE: 1\n\nT9 is ground\nT10 is ground\nX12 is free", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: ','(mult(T9, T10, X12), add(X12, T10, T12)), SCOPE: 2, CLAUSE: 0\n\nT9 is ground\nT10 is ground\nX12 is free", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: ','(mult(T9, T10, X12), add(X12, T10, T12)), SCOPE: 2, CLAUSE: 1\n\nT9 is ground\nT10 is ground\nX12 is free", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: add(0, T17, T12)\n\nT17 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: add(0, T17, T12), SCOPE: 3, CLAUSE: 2 | add(0, T17, T12), SCOPE: 3, CLAUSE: 3\n\nT17 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="38: add(0, T17, T12), SCOPE: 3, CLAUSE: 2\n\nT17 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
39 [label="39: add(0, T17, T12), SCOPE: 3, CLAUSE: 3\n\nT17 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
44 [label="44: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
45 [label="45: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
46 [label="46: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
51 [label="51: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
55 [label="B: ','(','(mult(T29, T30, X40), add(X40, T30, X41)), add(X41, T30, T12))\n\nT29 is ground\nT30 is ground\nX41 is free\nX40 is free", fontsize=16, style=filled, fillcolor=lightyellow];
56 [label="56: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
64 [label="C: mult(T29, T30, X40)\n\nT29 is ground\nT30 is ground\nX40 is free", fontsize=16, style=filled, fillcolor=lightyellow];
65 [label="G: ','(add(T33, T30, X41), add(X41, T30, T12))\n\nT30 is ground\nT33 is ground\nX41 is free", fontsize=16, style=filled, fillcolor=lightyellow];
72 [label="72: mult(T29, T30, X40), SCOPE: 4, CLAUSE: 0 | mult(T29, T30, X40), SCOPE: 4, CLAUSE: 1\n\nT29 is ground\nT30 is ground\nX40 is free", fontsize=16, style=filled, fillcolor=lightyellow];
84 [label="84: mult(T29, T30, X40), SCOPE: 4, CLAUSE: 0\n\nT29 is ground\nT30 is ground\nX40 is free", fontsize=16, style=filled, fillcolor=lightyellow];
85 [label="85: mult(T29, T30, X40), SCOPE: 4, CLAUSE: 1\n\nT29 is ground\nT30 is ground\nX40 is free", fontsize=16, style=filled, fillcolor=lightyellow];
90 [label="90: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
91 [label="91: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
92 [label="92: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
108 [label="D: ','(mult(T45, T46, X63), add(X63, T46, X64))\n\nT45 is ground\nT46 is ground\nX64 is free\nX63 is free", fontsize=16, style=filled, fillcolor=lightyellow];
109 [label="109: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
112 [label="112: mult(T45, T46, X63)\n\nT45 is ground\nT46 is ground\nX63 is free", fontsize=16, style=filled, fillcolor=lightyellow];
113 [label="E: add(T49, T46, X64)\n\nT46 is ground\nT49 is ground\nX64 is free", fontsize=16, style=filled, fillcolor=lightyellow];
116 [label="116: add(T49, T46, X64), SCOPE: 5, CLAUSE: 2 | add(T49, T46, X64), SCOPE: 5, CLAUSE: 3\n\nT46 is ground\nT49 is ground\nX64 is free", fontsize=16, style=filled, fillcolor=lightyellow];
118 [label="118: add(T49, T46, X64), SCOPE: 5, CLAUSE: 2\n\nT46 is ground\nT49 is ground\nX64 is free", fontsize=16, style=filled, fillcolor=lightyellow];
119 [label="119: add(T49, T46, X64), SCOPE: 5, CLAUSE: 3\n\nT46 is ground\nT49 is ground\nX64 is free", fontsize=16, style=filled, fillcolor=lightyellow];
122 [label="122: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
123 [label="123: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
124 [label="124: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
128 [label="128: add(T63, T64, X87)\n\nT63 is ground\nT64 is ground\nX87 is free", fontsize=16, style=filled, fillcolor=lightyellow];
129 [label="129: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
132 [label="132: add(T33, T30, X41)\n\nT30 is ground\nT33 is ground\nX41 is free", fontsize=16, style=filled, fillcolor=lightyellow];
133 [label="F: add(T69, T30, T12)\n\nT30 is ground\nT69 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
134 [label="134: add(T69, T30, T12), SCOPE: 6, CLAUSE: 2 | add(T69, T30, T12), SCOPE: 6, CLAUSE: 3\n\nT30 is ground\nT69 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
135 [label="135: add(T69, T30, T12), SCOPE: 6, CLAUSE: 2\n\nT30 is ground\nT69 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
136 [label="136: add(T69, T30, T12), SCOPE: 6, CLAUSE: 3\n\nT30 is ground\nT69 is ground", 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: add(T85, T86, T88)\n\nT85 is ground\nT86 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
141 [label="141: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
142 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
2 -> 142 [arrowhead = none ];
142 -> 4;

143 [label="EVAL with clause\nmult(0, X2, 0).\nand substitutionT1 -> 0,\nT2 -> T5,\nX2 -> T5,\nT3 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 143 [arrowhead = none ];
143 -> 11;

144 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
4 -> 144 [arrowhead = none ];
144 -> 13;

145 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
11 -> 145 [arrowhead = none ];
145 -> 16;

146 [label="EVAL with clause\nmult(s(X9), X10, X11) :- ','(mult(X9, X10, X12), add(X12, X10, X11)).\nand substitutionX9 -> T9,\nT1 -> s(T9),\nT2 -> T10,\nX10 -> T10,\nT3 -> T12,\nX11 -> T12,\nT11 -> T12", fontsize=14, style = filled, fillcolor = lightblue];
13 -> 146 [arrowhead = none ];
146 -> 22;

147 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
13 -> 147 [arrowhead = none ];
147 -> 24;

148 [label="BACKTRACK\nfor clause: mult(s(X), Y, Z) :- ','(mult(X, Y, Z1), add(Z1, Y, Z))because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
16 -> 148 [arrowhead = none ];
148 -> 18;

149 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
22 -> 149 [arrowhead = none ];
149 -> 26;

150 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
26 -> 150 [arrowhead = none ];
150 -> 28;

151 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
26 -> 151 [arrowhead = none ];
151 -> 29;

152 [label="EVAL with clause\nmult(0, X17, 0).\nand substitutionT9 -> 0,\nT10 -> T17,\nX17 -> T17,\nX12 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
28 -> 152 [arrowhead = none ];
152 -> 32;

153 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
28 -> 153 [arrowhead = none ];
153 -> 33;

154 [label="EVAL with clause\nmult(s(X37), X38, X39) :- ','(mult(X37, X38, X40), add(X40, X38, X39)).\nand substitutionX37 -> T29,\nT9 -> s(T29),\nT10 -> T30,\nX38 -> T30,\nX12 -> X41,\nX39 -> X41", fontsize=14, style = filled, fillcolor = lightblue];
29 -> 154 [arrowhead = none ];
154 -> 55;

155 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
29 -> 155 [arrowhead = none ];
155 -> 56;

156 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
32 -> 156 [arrowhead = none ];
156 -> 36;

157 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
36 -> 157 [arrowhead = none ];
157 -> 38;

158 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
36 -> 158 [arrowhead = none ];
158 -> 39;

159 [label="EVAL with clause\nadd(0, X24, X24).\nand substitutionT17 -> T24,\nX24 -> T24,\nT12 -> T24", fontsize=14, style = filled, fillcolor = lightblue];
38 -> 159 [arrowhead = none ];
159 -> 44;

160 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
38 -> 160 [arrowhead = none ];
160 -> 45;

161 [label="BACKTRACK\nfor clause: add(s(X), Y, s(Z)) :- add(X, Y, Z)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
39 -> 161 [arrowhead = none ];
161 -> 51;

162 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
44 -> 162 [arrowhead = none ];
162 -> 46;

163 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
55 -> 163 [arrowhead = none ];
163 -> 64;

164 [label="SPLIT 2\nnew knowledge:\nT29 is ground\nT30 is ground\nT33 is ground\nreplacements:X40 -> T33", fontsize=14, style = filled, fillcolor = violet];
55 -> 164 [arrowhead = none ];
164 -> 65;

165 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
64 -> 165 [arrowhead = none ];
165 -> 72;

166 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
65 -> 166 [arrowhead = none ];
166 -> 132;

167 [label="SPLIT 2\nnew knowledge:\nT33 is ground\nT30 is ground\nT69 is ground\nreplacements:X41 -> T69", fontsize=14, style = filled, fillcolor = violet];
65 -> 167 [arrowhead = none ];
167 -> 133;

168 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
72 -> 168 [arrowhead = none ];
168 -> 84;

169 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
72 -> 169 [arrowhead = none ];
169 -> 85;

170 [label="EVAL with clause\nmult(0, X50, 0).\nand substitutionT29 -> 0,\nT30 -> T40,\nX50 -> T40,\nX40 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
84 -> 170 [arrowhead = none ];
170 -> 90;

171 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
84 -> 171 [arrowhead = none ];
171 -> 91;

172 [label="EVAL with clause\nmult(s(X60), X61, X62) :- ','(mult(X60, X61, X63), add(X63, X61, X62)).\nand substitutionX60 -> T45,\nT29 -> s(T45),\nT30 -> T46,\nX61 -> T46,\nX40 -> X64,\nX62 -> X64", fontsize=14, style = filled, fillcolor = lightblue];
85 -> 172 [arrowhead = none ];
172 -> 108;

173 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
85 -> 173 [arrowhead = none ];
173 -> 109;

174 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
90 -> 174 [arrowhead = none ];
174 -> 92;

175 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
108 -> 175 [arrowhead = none ];
175 -> 112;

176 [label="SPLIT 2\nnew knowledge:\nT45 is ground\nT46 is ground\nT49 is ground\nreplacements:X63 -> T49", fontsize=14, style = filled, fillcolor = violet];
108 -> 176 [arrowhead = none ];
176 -> 113;

177 [label="INSTANCE with matching:\nT29 -> T45\nT30 -> T46\nX40 -> X63", fontsize=14, style = filled, fillcolor = lightgrey];
112 -> 177 [arrowhead = none , style=dashed];
177 -> 64[style=dashed];

178 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
113 -> 178 [arrowhead = none ];
178 -> 116;

179 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
116 -> 179 [arrowhead = none ];
179 -> 118;

180 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
116 -> 180 [arrowhead = none ];
180 -> 119;

181 [label="EVAL with clause\nadd(0, X75, X75).\nand substitutionT49 -> 0,\nT46 -> T58,\nX75 -> T58,\nX64 -> T58", fontsize=14, style = filled, fillcolor = lightblue];
118 -> 181 [arrowhead = none ];
181 -> 122;

182 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
118 -> 182 [arrowhead = none ];
182 -> 123;

183 [label="EVAL with clause\nadd(s(X84), X85, s(X86)) :- add(X84, X85, X86).\nand substitutionX84 -> T63,\nT49 -> s(T63),\nT46 -> T64,\nX85 -> T64,\nX86 -> X87,\nX64 -> s(X87)", fontsize=14, style = filled, fillcolor = lightblue];
119 -> 183 [arrowhead = none ];
183 -> 128;

184 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
119 -> 184 [arrowhead = none ];
184 -> 129;

185 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
122 -> 185 [arrowhead = none ];
185 -> 124;

186 [label="INSTANCE with matching:\nT49 -> T63\nT46 -> T64\nX64 -> X87", fontsize=14, style = filled, fillcolor = lightgrey];
128 -> 186 [arrowhead = none , style=dashed];
186 -> 113[style=dashed];

187 [label="INSTANCE with matching:\nT49 -> T33\nT46 -> T30\nX64 -> X41", fontsize=14, style = filled, fillcolor = lightgrey];
132 -> 187 [arrowhead = none , style=dashed];
187 -> 113[style=dashed];

188 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
133 -> 188 [arrowhead = none ];
188 -> 134;

189 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
134 -> 189 [arrowhead = none ];
189 -> 135;

190 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
134 -> 190 [arrowhead = none ];
190 -> 136;

191 [label="EVAL with clause\nadd(0, X100, X100).\nand substitutionT69 -> 0,\nT30 -> T78,\nX100 -> T78,\nT12 -> T78", fontsize=14, style = filled, fillcolor = lightblue];
135 -> 191 [arrowhead = none ];
191 -> 137;

192 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
135 -> 192 [arrowhead = none ];
192 -> 138;

193 [label="EVAL with clause\nadd(s(X107), X108, s(X109)) :- add(X107, X108, X109).\nand substitutionX107 -> T85,\nT69 -> s(T85),\nT30 -> T86,\nX108 -> T86,\nX109 -> T88,\nT12 -> s(T88),\nT87 -> T88", fontsize=14, style = filled, fillcolor = lightblue];
136 -> 193 [arrowhead = none ];
193 -> 140;

194 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
136 -> 194 [arrowhead = none ];
194 -> 141;

195 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
137 -> 195 [arrowhead = none ];
195 -> 139;

196 [label="INSTANCE with matching:\nT69 -> T85\nT30 -> T86\nT12 -> T88", fontsize=14, style = filled, fillcolor = lightgrey];
140 -> 196 [arrowhead = none , style=dashed];
196 -> 133[style=dashed];

}

(2) Obligation:

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

multA_in_gga(0, T5, 0) → multA_out_gga(0, T5, 0)
multA_in_gga(s(0), T24, T24) → multA_out_gga(s(0), T24, T24)
multA_in_gga(s(s(T29)), T30, T12) → U1_gga(T29, T30, T12, pB_in_ggaaa(T29, T30, X40, X41, T12))
pB_in_ggaaa(T29, T30, T33, X41, T12) → U5_ggaaa(T29, T30, T33, X41, T12, multC_in_gga(T29, T30, T33))
multC_in_gga(0, T40, 0) → multC_out_gga(0, T40, 0)
multC_in_gga(s(T45), T46, X64) → U2_gga(T45, T46, X64, pD_in_ggaa(T45, T46, X63, X64))
pD_in_ggaa(T45, T46, T49, X64) → U9_ggaa(T45, T46, T49, X64, multC_in_gga(T45, T46, T49))
U9_ggaa(T45, T46, T49, X64, multC_out_gga(T45, T46, T49)) → U10_ggaa(T45, T46, T49, X64, addE_in_gga(T49, T46, X64))
addE_in_gga(0, T58, T58) → addE_out_gga(0, T58, T58)
addE_in_gga(s(T63), T64, s(X87)) → U3_gga(T63, T64, X87, addE_in_gga(T63, T64, X87))
U3_gga(T63, T64, X87, addE_out_gga(T63, T64, X87)) → addE_out_gga(s(T63), T64, s(X87))
U10_ggaa(T45, T46, T49, X64, addE_out_gga(T49, T46, X64)) → pD_out_ggaa(T45, T46, T49, X64)
U2_gga(T45, T46, X64, pD_out_ggaa(T45, T46, X63, X64)) → multC_out_gga(s(T45), T46, X64)
U5_ggaaa(T29, T30, T33, X41, T12, multC_out_gga(T29, T30, T33)) → U6_ggaaa(T29, T30, T33, X41, T12, pG_in_ggaa(T33, T30, X41, T12))
pG_in_ggaa(T33, T30, T69, T12) → U7_ggaa(T33, T30, T69, T12, addE_in_gga(T33, T30, T69))
U7_ggaa(T33, T30, T69, T12, addE_out_gga(T33, T30, T69)) → U8_ggaa(T33, T30, T69, T12, addF_in_gga(T69, T30, T12))
addF_in_gga(0, T78, T78) → addF_out_gga(0, T78, T78)
addF_in_gga(s(T85), T86, s(T88)) → U4_gga(T85, T86, T88, addF_in_gga(T85, T86, T88))
U4_gga(T85, T86, T88, addF_out_gga(T85, T86, T88)) → addF_out_gga(s(T85), T86, s(T88))
U8_ggaa(T33, T30, T69, T12, addF_out_gga(T69, T30, T12)) → pG_out_ggaa(T33, T30, T69, T12)
U6_ggaaa(T29, T30, T33, X41, T12, pG_out_ggaa(T33, T30, X41, T12)) → pB_out_ggaaa(T29, T30, T33, X41, T12)
U1_gga(T29, T30, T12, pB_out_ggaaa(T29, T30, X40, X41, T12)) → multA_out_gga(s(s(T29)), T30, T12)

The argument filtering Pi contains the following mapping:
multA_in_gga(x1, x2, x3)  =  multA_in_gga(x1, x2)
0  =  0
multA_out_gga(x1, x2, x3)  =  multA_out_gga(x1, x2, x3)
s(x1)  =  s(x1)
U1_gga(x1, x2, x3, x4)  =  U1_gga(x1, x2, x4)
pB_in_ggaaa(x1, x2, x3, x4, x5)  =  pB_in_ggaaa(x1, x2)
U5_ggaaa(x1, x2, x3, x4, x5, x6)  =  U5_ggaaa(x1, x2, x6)
multC_in_gga(x1, x2, x3)  =  multC_in_gga(x1, x2)
multC_out_gga(x1, x2, x3)  =  multC_out_gga(x1, x2, x3)
U2_gga(x1, x2, x3, x4)  =  U2_gga(x1, x2, x4)
pD_in_ggaa(x1, x2, x3, x4)  =  pD_in_ggaa(x1, x2)
U9_ggaa(x1, x2, x3, x4, x5)  =  U9_ggaa(x1, x2, x5)
U10_ggaa(x1, x2, x3, x4, x5)  =  U10_ggaa(x1, x2, x3, x5)
addE_in_gga(x1, x2, x3)  =  addE_in_gga(x1, x2)
addE_out_gga(x1, x2, x3)  =  addE_out_gga(x1, x2, x3)
U3_gga(x1, x2, x3, x4)  =  U3_gga(x1, x2, x4)
pD_out_ggaa(x1, x2, x3, x4)  =  pD_out_ggaa(x1, x2, x3, x4)
U6_ggaaa(x1, x2, x3, x4, x5, x6)  =  U6_ggaaa(x1, x2, x3, x6)
pG_in_ggaa(x1, x2, x3, x4)  =  pG_in_ggaa(x1, x2)
U7_ggaa(x1, x2, x3, x4, x5)  =  U7_ggaa(x1, x2, x5)
U8_ggaa(x1, x2, x3, x4, x5)  =  U8_ggaa(x1, x2, x3, x5)
addF_in_gga(x1, x2, x3)  =  addF_in_gga(x1, x2)
addF_out_gga(x1, x2, x3)  =  addF_out_gga(x1, x2, x3)
U4_gga(x1, x2, x3, x4)  =  U4_gga(x1, x2, x4)
pG_out_ggaa(x1, x2, x3, x4)  =  pG_out_ggaa(x1, x2, x3, x4)
pB_out_ggaaa(x1, x2, x3, x4, x5)  =  pB_out_ggaaa(x1, x2, x3, x4, x5)

(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:

MULTA_IN_GGA(s(s(T29)), T30, T12) → U1_GGA(T29, T30, T12, pB_in_ggaaa(T29, T30, X40, X41, T12))
MULTA_IN_GGA(s(s(T29)), T30, T12) → PB_IN_GGAAA(T29, T30, X40, X41, T12)
PB_IN_GGAAA(T29, T30, T33, X41, T12) → U5_GGAAA(T29, T30, T33, X41, T12, multC_in_gga(T29, T30, T33))
PB_IN_GGAAA(T29, T30, T33, X41, T12) → MULTC_IN_GGA(T29, T30, T33)
MULTC_IN_GGA(s(T45), T46, X64) → U2_GGA(T45, T46, X64, pD_in_ggaa(T45, T46, X63, X64))
MULTC_IN_GGA(s(T45), T46, X64) → PD_IN_GGAA(T45, T46, X63, X64)
PD_IN_GGAA(T45, T46, T49, X64) → U9_GGAA(T45, T46, T49, X64, multC_in_gga(T45, T46, T49))
PD_IN_GGAA(T45, T46, T49, X64) → MULTC_IN_GGA(T45, T46, T49)
U9_GGAA(T45, T46, T49, X64, multC_out_gga(T45, T46, T49)) → U10_GGAA(T45, T46, T49, X64, addE_in_gga(T49, T46, X64))
U9_GGAA(T45, T46, T49, X64, multC_out_gga(T45, T46, T49)) → ADDE_IN_GGA(T49, T46, X64)
ADDE_IN_GGA(s(T63), T64, s(X87)) → U3_GGA(T63, T64, X87, addE_in_gga(T63, T64, X87))
ADDE_IN_GGA(s(T63), T64, s(X87)) → ADDE_IN_GGA(T63, T64, X87)
U5_GGAAA(T29, T30, T33, X41, T12, multC_out_gga(T29, T30, T33)) → U6_GGAAA(T29, T30, T33, X41, T12, pG_in_ggaa(T33, T30, X41, T12))
U5_GGAAA(T29, T30, T33, X41, T12, multC_out_gga(T29, T30, T33)) → PG_IN_GGAA(T33, T30, X41, T12)
PG_IN_GGAA(T33, T30, T69, T12) → U7_GGAA(T33, T30, T69, T12, addE_in_gga(T33, T30, T69))
PG_IN_GGAA(T33, T30, T69, T12) → ADDE_IN_GGA(T33, T30, T69)
U7_GGAA(T33, T30, T69, T12, addE_out_gga(T33, T30, T69)) → U8_GGAA(T33, T30, T69, T12, addF_in_gga(T69, T30, T12))
U7_GGAA(T33, T30, T69, T12, addE_out_gga(T33, T30, T69)) → ADDF_IN_GGA(T69, T30, T12)
ADDF_IN_GGA(s(T85), T86, s(T88)) → U4_GGA(T85, T86, T88, addF_in_gga(T85, T86, T88))
ADDF_IN_GGA(s(T85), T86, s(T88)) → ADDF_IN_GGA(T85, T86, T88)

The TRS R consists of the following rules:

multA_in_gga(0, T5, 0) → multA_out_gga(0, T5, 0)
multA_in_gga(s(0), T24, T24) → multA_out_gga(s(0), T24, T24)
multA_in_gga(s(s(T29)), T30, T12) → U1_gga(T29, T30, T12, pB_in_ggaaa(T29, T30, X40, X41, T12))
pB_in_ggaaa(T29, T30, T33, X41, T12) → U5_ggaaa(T29, T30, T33, X41, T12, multC_in_gga(T29, T30, T33))
multC_in_gga(0, T40, 0) → multC_out_gga(0, T40, 0)
multC_in_gga(s(T45), T46, X64) → U2_gga(T45, T46, X64, pD_in_ggaa(T45, T46, X63, X64))
pD_in_ggaa(T45, T46, T49, X64) → U9_ggaa(T45, T46, T49, X64, multC_in_gga(T45, T46, T49))
U9_ggaa(T45, T46, T49, X64, multC_out_gga(T45, T46, T49)) → U10_ggaa(T45, T46, T49, X64, addE_in_gga(T49, T46, X64))
addE_in_gga(0, T58, T58) → addE_out_gga(0, T58, T58)
addE_in_gga(s(T63), T64, s(X87)) → U3_gga(T63, T64, X87, addE_in_gga(T63, T64, X87))
U3_gga(T63, T64, X87, addE_out_gga(T63, T64, X87)) → addE_out_gga(s(T63), T64, s(X87))
U10_ggaa(T45, T46, T49, X64, addE_out_gga(T49, T46, X64)) → pD_out_ggaa(T45, T46, T49, X64)
U2_gga(T45, T46, X64, pD_out_ggaa(T45, T46, X63, X64)) → multC_out_gga(s(T45), T46, X64)
U5_ggaaa(T29, T30, T33, X41, T12, multC_out_gga(T29, T30, T33)) → U6_ggaaa(T29, T30, T33, X41, T12, pG_in_ggaa(T33, T30, X41, T12))
pG_in_ggaa(T33, T30, T69, T12) → U7_ggaa(T33, T30, T69, T12, addE_in_gga(T33, T30, T69))
U7_ggaa(T33, T30, T69, T12, addE_out_gga(T33, T30, T69)) → U8_ggaa(T33, T30, T69, T12, addF_in_gga(T69, T30, T12))
addF_in_gga(0, T78, T78) → addF_out_gga(0, T78, T78)
addF_in_gga(s(T85), T86, s(T88)) → U4_gga(T85, T86, T88, addF_in_gga(T85, T86, T88))
U4_gga(T85, T86, T88, addF_out_gga(T85, T86, T88)) → addF_out_gga(s(T85), T86, s(T88))
U8_ggaa(T33, T30, T69, T12, addF_out_gga(T69, T30, T12)) → pG_out_ggaa(T33, T30, T69, T12)
U6_ggaaa(T29, T30, T33, X41, T12, pG_out_ggaa(T33, T30, X41, T12)) → pB_out_ggaaa(T29, T30, T33, X41, T12)
U1_gga(T29, T30, T12, pB_out_ggaaa(T29, T30, X40, X41, T12)) → multA_out_gga(s(s(T29)), T30, T12)

The argument filtering Pi contains the following mapping:
multA_in_gga(x1, x2, x3)  =  multA_in_gga(x1, x2)
0  =  0
multA_out_gga(x1, x2, x3)  =  multA_out_gga(x1, x2, x3)
s(x1)  =  s(x1)
U1_gga(x1, x2, x3, x4)  =  U1_gga(x1, x2, x4)
pB_in_ggaaa(x1, x2, x3, x4, x5)  =  pB_in_ggaaa(x1, x2)
U5_ggaaa(x1, x2, x3, x4, x5, x6)  =  U5_ggaaa(x1, x2, x6)
multC_in_gga(x1, x2, x3)  =  multC_in_gga(x1, x2)
multC_out_gga(x1, x2, x3)  =  multC_out_gga(x1, x2, x3)
U2_gga(x1, x2, x3, x4)  =  U2_gga(x1, x2, x4)
pD_in_ggaa(x1, x2, x3, x4)  =  pD_in_ggaa(x1, x2)
U9_ggaa(x1, x2, x3, x4, x5)  =  U9_ggaa(x1, x2, x5)
U10_ggaa(x1, x2, x3, x4, x5)  =  U10_ggaa(x1, x2, x3, x5)
addE_in_gga(x1, x2, x3)  =  addE_in_gga(x1, x2)
addE_out_gga(x1, x2, x3)  =  addE_out_gga(x1, x2, x3)
U3_gga(x1, x2, x3, x4)  =  U3_gga(x1, x2, x4)
pD_out_ggaa(x1, x2, x3, x4)  =  pD_out_ggaa(x1, x2, x3, x4)
U6_ggaaa(x1, x2, x3, x4, x5, x6)  =  U6_ggaaa(x1, x2, x3, x6)
pG_in_ggaa(x1, x2, x3, x4)  =  pG_in_ggaa(x1, x2)
U7_ggaa(x1, x2, x3, x4, x5)  =  U7_ggaa(x1, x2, x5)
U8_ggaa(x1, x2, x3, x4, x5)  =  U8_ggaa(x1, x2, x3, x5)
addF_in_gga(x1, x2, x3)  =  addF_in_gga(x1, x2)
addF_out_gga(x1, x2, x3)  =  addF_out_gga(x1, x2, x3)
U4_gga(x1, x2, x3, x4)  =  U4_gga(x1, x2, x4)
pG_out_ggaa(x1, x2, x3, x4)  =  pG_out_ggaa(x1, x2, x3, x4)
pB_out_ggaaa(x1, x2, x3, x4, x5)  =  pB_out_ggaaa(x1, x2, x3, x4, x5)
MULTA_IN_GGA(x1, x2, x3)  =  MULTA_IN_GGA(x1, x2)
U1_GGA(x1, x2, x3, x4)  =  U1_GGA(x1, x2, x4)
PB_IN_GGAAA(x1, x2, x3, x4, x5)  =  PB_IN_GGAAA(x1, x2)
U5_GGAAA(x1, x2, x3, x4, x5, x6)  =  U5_GGAAA(x1, x2, x6)
MULTC_IN_GGA(x1, x2, x3)  =  MULTC_IN_GGA(x1, x2)
U2_GGA(x1, x2, x3, x4)  =  U2_GGA(x1, x2, x4)
PD_IN_GGAA(x1, x2, x3, x4)  =  PD_IN_GGAA(x1, x2)
U9_GGAA(x1, x2, x3, x4, x5)  =  U9_GGAA(x1, x2, x5)
U10_GGAA(x1, x2, x3, x4, x5)  =  U10_GGAA(x1, x2, x3, x5)
ADDE_IN_GGA(x1, x2, x3)  =  ADDE_IN_GGA(x1, x2)
U3_GGA(x1, x2, x3, x4)  =  U3_GGA(x1, x2, x4)
U6_GGAAA(x1, x2, x3, x4, x5, x6)  =  U6_GGAAA(x1, x2, x3, x6)
PG_IN_GGAA(x1, x2, x3, x4)  =  PG_IN_GGAA(x1, x2)
U7_GGAA(x1, x2, x3, x4, x5)  =  U7_GGAA(x1, x2, x5)
U8_GGAA(x1, x2, x3, x4, x5)  =  U8_GGAA(x1, x2, x3, x5)
ADDF_IN_GGA(x1, x2, x3)  =  ADDF_IN_GGA(x1, x2)
U4_GGA(x1, x2, x3, x4)  =  U4_GGA(x1, x2, x4)

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

(4) Obligation:

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

MULTA_IN_GGA(s(s(T29)), T30, T12) → U1_GGA(T29, T30, T12, pB_in_ggaaa(T29, T30, X40, X41, T12))
MULTA_IN_GGA(s(s(T29)), T30, T12) → PB_IN_GGAAA(T29, T30, X40, X41, T12)
PB_IN_GGAAA(T29, T30, T33, X41, T12) → U5_GGAAA(T29, T30, T33, X41, T12, multC_in_gga(T29, T30, T33))
PB_IN_GGAAA(T29, T30, T33, X41, T12) → MULTC_IN_GGA(T29, T30, T33)
MULTC_IN_GGA(s(T45), T46, X64) → U2_GGA(T45, T46, X64, pD_in_ggaa(T45, T46, X63, X64))
MULTC_IN_GGA(s(T45), T46, X64) → PD_IN_GGAA(T45, T46, X63, X64)
PD_IN_GGAA(T45, T46, T49, X64) → U9_GGAA(T45, T46, T49, X64, multC_in_gga(T45, T46, T49))
PD_IN_GGAA(T45, T46, T49, X64) → MULTC_IN_GGA(T45, T46, T49)
U9_GGAA(T45, T46, T49, X64, multC_out_gga(T45, T46, T49)) → U10_GGAA(T45, T46, T49, X64, addE_in_gga(T49, T46, X64))
U9_GGAA(T45, T46, T49, X64, multC_out_gga(T45, T46, T49)) → ADDE_IN_GGA(T49, T46, X64)
ADDE_IN_GGA(s(T63), T64, s(X87)) → U3_GGA(T63, T64, X87, addE_in_gga(T63, T64, X87))
ADDE_IN_GGA(s(T63), T64, s(X87)) → ADDE_IN_GGA(T63, T64, X87)
U5_GGAAA(T29, T30, T33, X41, T12, multC_out_gga(T29, T30, T33)) → U6_GGAAA(T29, T30, T33, X41, T12, pG_in_ggaa(T33, T30, X41, T12))
U5_GGAAA(T29, T30, T33, X41, T12, multC_out_gga(T29, T30, T33)) → PG_IN_GGAA(T33, T30, X41, T12)
PG_IN_GGAA(T33, T30, T69, T12) → U7_GGAA(T33, T30, T69, T12, addE_in_gga(T33, T30, T69))
PG_IN_GGAA(T33, T30, T69, T12) → ADDE_IN_GGA(T33, T30, T69)
U7_GGAA(T33, T30, T69, T12, addE_out_gga(T33, T30, T69)) → U8_GGAA(T33, T30, T69, T12, addF_in_gga(T69, T30, T12))
U7_GGAA(T33, T30, T69, T12, addE_out_gga(T33, T30, T69)) → ADDF_IN_GGA(T69, T30, T12)
ADDF_IN_GGA(s(T85), T86, s(T88)) → U4_GGA(T85, T86, T88, addF_in_gga(T85, T86, T88))
ADDF_IN_GGA(s(T85), T86, s(T88)) → ADDF_IN_GGA(T85, T86, T88)

The TRS R consists of the following rules:

multA_in_gga(0, T5, 0) → multA_out_gga(0, T5, 0)
multA_in_gga(s(0), T24, T24) → multA_out_gga(s(0), T24, T24)
multA_in_gga(s(s(T29)), T30, T12) → U1_gga(T29, T30, T12, pB_in_ggaaa(T29, T30, X40, X41, T12))
pB_in_ggaaa(T29, T30, T33, X41, T12) → U5_ggaaa(T29, T30, T33, X41, T12, multC_in_gga(T29, T30, T33))
multC_in_gga(0, T40, 0) → multC_out_gga(0, T40, 0)
multC_in_gga(s(T45), T46, X64) → U2_gga(T45, T46, X64, pD_in_ggaa(T45, T46, X63, X64))
pD_in_ggaa(T45, T46, T49, X64) → U9_ggaa(T45, T46, T49, X64, multC_in_gga(T45, T46, T49))
U9_ggaa(T45, T46, T49, X64, multC_out_gga(T45, T46, T49)) → U10_ggaa(T45, T46, T49, X64, addE_in_gga(T49, T46, X64))
addE_in_gga(0, T58, T58) → addE_out_gga(0, T58, T58)
addE_in_gga(s(T63), T64, s(X87)) → U3_gga(T63, T64, X87, addE_in_gga(T63, T64, X87))
U3_gga(T63, T64, X87, addE_out_gga(T63, T64, X87)) → addE_out_gga(s(T63), T64, s(X87))
U10_ggaa(T45, T46, T49, X64, addE_out_gga(T49, T46, X64)) → pD_out_ggaa(T45, T46, T49, X64)
U2_gga(T45, T46, X64, pD_out_ggaa(T45, T46, X63, X64)) → multC_out_gga(s(T45), T46, X64)
U5_ggaaa(T29, T30, T33, X41, T12, multC_out_gga(T29, T30, T33)) → U6_ggaaa(T29, T30, T33, X41, T12, pG_in_ggaa(T33, T30, X41, T12))
pG_in_ggaa(T33, T30, T69, T12) → U7_ggaa(T33, T30, T69, T12, addE_in_gga(T33, T30, T69))
U7_ggaa(T33, T30, T69, T12, addE_out_gga(T33, T30, T69)) → U8_ggaa(T33, T30, T69, T12, addF_in_gga(T69, T30, T12))
addF_in_gga(0, T78, T78) → addF_out_gga(0, T78, T78)
addF_in_gga(s(T85), T86, s(T88)) → U4_gga(T85, T86, T88, addF_in_gga(T85, T86, T88))
U4_gga(T85, T86, T88, addF_out_gga(T85, T86, T88)) → addF_out_gga(s(T85), T86, s(T88))
U8_ggaa(T33, T30, T69, T12, addF_out_gga(T69, T30, T12)) → pG_out_ggaa(T33, T30, T69, T12)
U6_ggaaa(T29, T30, T33, X41, T12, pG_out_ggaa(T33, T30, X41, T12)) → pB_out_ggaaa(T29, T30, T33, X41, T12)
U1_gga(T29, T30, T12, pB_out_ggaaa(T29, T30, X40, X41, T12)) → multA_out_gga(s(s(T29)), T30, T12)

The argument filtering Pi contains the following mapping:
multA_in_gga(x1, x2, x3)  =  multA_in_gga(x1, x2)
0  =  0
multA_out_gga(x1, x2, x3)  =  multA_out_gga(x1, x2, x3)
s(x1)  =  s(x1)
U1_gga(x1, x2, x3, x4)  =  U1_gga(x1, x2, x4)
pB_in_ggaaa(x1, x2, x3, x4, x5)  =  pB_in_ggaaa(x1, x2)
U5_ggaaa(x1, x2, x3, x4, x5, x6)  =  U5_ggaaa(x1, x2, x6)
multC_in_gga(x1, x2, x3)  =  multC_in_gga(x1, x2)
multC_out_gga(x1, x2, x3)  =  multC_out_gga(x1, x2, x3)
U2_gga(x1, x2, x3, x4)  =  U2_gga(x1, x2, x4)
pD_in_ggaa(x1, x2, x3, x4)  =  pD_in_ggaa(x1, x2)
U9_ggaa(x1, x2, x3, x4, x5)  =  U9_ggaa(x1, x2, x5)
U10_ggaa(x1, x2, x3, x4, x5)  =  U10_ggaa(x1, x2, x3, x5)
addE_in_gga(x1, x2, x3)  =  addE_in_gga(x1, x2)
addE_out_gga(x1, x2, x3)  =  addE_out_gga(x1, x2, x3)
U3_gga(x1, x2, x3, x4)  =  U3_gga(x1, x2, x4)
pD_out_ggaa(x1, x2, x3, x4)  =  pD_out_ggaa(x1, x2, x3, x4)
U6_ggaaa(x1, x2, x3, x4, x5, x6)  =  U6_ggaaa(x1, x2, x3, x6)
pG_in_ggaa(x1, x2, x3, x4)  =  pG_in_ggaa(x1, x2)
U7_ggaa(x1, x2, x3, x4, x5)  =  U7_ggaa(x1, x2, x5)
U8_ggaa(x1, x2, x3, x4, x5)  =  U8_ggaa(x1, x2, x3, x5)
addF_in_gga(x1, x2, x3)  =  addF_in_gga(x1, x2)
addF_out_gga(x1, x2, x3)  =  addF_out_gga(x1, x2, x3)
U4_gga(x1, x2, x3, x4)  =  U4_gga(x1, x2, x4)
pG_out_ggaa(x1, x2, x3, x4)  =  pG_out_ggaa(x1, x2, x3, x4)
pB_out_ggaaa(x1, x2, x3, x4, x5)  =  pB_out_ggaaa(x1, x2, x3, x4, x5)
MULTA_IN_GGA(x1, x2, x3)  =  MULTA_IN_GGA(x1, x2)
U1_GGA(x1, x2, x3, x4)  =  U1_GGA(x1, x2, x4)
PB_IN_GGAAA(x1, x2, x3, x4, x5)  =  PB_IN_GGAAA(x1, x2)
U5_GGAAA(x1, x2, x3, x4, x5, x6)  =  U5_GGAAA(x1, x2, x6)
MULTC_IN_GGA(x1, x2, x3)  =  MULTC_IN_GGA(x1, x2)
U2_GGA(x1, x2, x3, x4)  =  U2_GGA(x1, x2, x4)
PD_IN_GGAA(x1, x2, x3, x4)  =  PD_IN_GGAA(x1, x2)
U9_GGAA(x1, x2, x3, x4, x5)  =  U9_GGAA(x1, x2, x5)
U10_GGAA(x1, x2, x3, x4, x5)  =  U10_GGAA(x1, x2, x3, x5)
ADDE_IN_GGA(x1, x2, x3)  =  ADDE_IN_GGA(x1, x2)
U3_GGA(x1, x2, x3, x4)  =  U3_GGA(x1, x2, x4)
U6_GGAAA(x1, x2, x3, x4, x5, x6)  =  U6_GGAAA(x1, x2, x3, x6)
PG_IN_GGAA(x1, x2, x3, x4)  =  PG_IN_GGAA(x1, x2)
U7_GGAA(x1, x2, x3, x4, x5)  =  U7_GGAA(x1, x2, x5)
U8_GGAA(x1, x2, x3, x4, x5)  =  U8_GGAA(x1, x2, x3, x5)
ADDF_IN_GGA(x1, x2, x3)  =  ADDF_IN_GGA(x1, x2)
U4_GGA(x1, x2, x3, x4)  =  U4_GGA(x1, x2, x4)

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

(5) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 3 SCCs with 16 less nodes.

(6) Complex Obligation (AND)

(7) Obligation:

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

ADDF_IN_GGA(s(T85), T86, s(T88)) → ADDF_IN_GGA(T85, T86, T88)

The TRS R consists of the following rules:

multA_in_gga(0, T5, 0) → multA_out_gga(0, T5, 0)
multA_in_gga(s(0), T24, T24) → multA_out_gga(s(0), T24, T24)
multA_in_gga(s(s(T29)), T30, T12) → U1_gga(T29, T30, T12, pB_in_ggaaa(T29, T30, X40, X41, T12))
pB_in_ggaaa(T29, T30, T33, X41, T12) → U5_ggaaa(T29, T30, T33, X41, T12, multC_in_gga(T29, T30, T33))
multC_in_gga(0, T40, 0) → multC_out_gga(0, T40, 0)
multC_in_gga(s(T45), T46, X64) → U2_gga(T45, T46, X64, pD_in_ggaa(T45, T46, X63, X64))
pD_in_ggaa(T45, T46, T49, X64) → U9_ggaa(T45, T46, T49, X64, multC_in_gga(T45, T46, T49))
U9_ggaa(T45, T46, T49, X64, multC_out_gga(T45, T46, T49)) → U10_ggaa(T45, T46, T49, X64, addE_in_gga(T49, T46, X64))
addE_in_gga(0, T58, T58) → addE_out_gga(0, T58, T58)
addE_in_gga(s(T63), T64, s(X87)) → U3_gga(T63, T64, X87, addE_in_gga(T63, T64, X87))
U3_gga(T63, T64, X87, addE_out_gga(T63, T64, X87)) → addE_out_gga(s(T63), T64, s(X87))
U10_ggaa(T45, T46, T49, X64, addE_out_gga(T49, T46, X64)) → pD_out_ggaa(T45, T46, T49, X64)
U2_gga(T45, T46, X64, pD_out_ggaa(T45, T46, X63, X64)) → multC_out_gga(s(T45), T46, X64)
U5_ggaaa(T29, T30, T33, X41, T12, multC_out_gga(T29, T30, T33)) → U6_ggaaa(T29, T30, T33, X41, T12, pG_in_ggaa(T33, T30, X41, T12))
pG_in_ggaa(T33, T30, T69, T12) → U7_ggaa(T33, T30, T69, T12, addE_in_gga(T33, T30, T69))
U7_ggaa(T33, T30, T69, T12, addE_out_gga(T33, T30, T69)) → U8_ggaa(T33, T30, T69, T12, addF_in_gga(T69, T30, T12))
addF_in_gga(0, T78, T78) → addF_out_gga(0, T78, T78)
addF_in_gga(s(T85), T86, s(T88)) → U4_gga(T85, T86, T88, addF_in_gga(T85, T86, T88))
U4_gga(T85, T86, T88, addF_out_gga(T85, T86, T88)) → addF_out_gga(s(T85), T86, s(T88))
U8_ggaa(T33, T30, T69, T12, addF_out_gga(T69, T30, T12)) → pG_out_ggaa(T33, T30, T69, T12)
U6_ggaaa(T29, T30, T33, X41, T12, pG_out_ggaa(T33, T30, X41, T12)) → pB_out_ggaaa(T29, T30, T33, X41, T12)
U1_gga(T29, T30, T12, pB_out_ggaaa(T29, T30, X40, X41, T12)) → multA_out_gga(s(s(T29)), T30, T12)

The argument filtering Pi contains the following mapping:
multA_in_gga(x1, x2, x3)  =  multA_in_gga(x1, x2)
0  =  0
multA_out_gga(x1, x2, x3)  =  multA_out_gga(x1, x2, x3)
s(x1)  =  s(x1)
U1_gga(x1, x2, x3, x4)  =  U1_gga(x1, x2, x4)
pB_in_ggaaa(x1, x2, x3, x4, x5)  =  pB_in_ggaaa(x1, x2)
U5_ggaaa(x1, x2, x3, x4, x5, x6)  =  U5_ggaaa(x1, x2, x6)
multC_in_gga(x1, x2, x3)  =  multC_in_gga(x1, x2)
multC_out_gga(x1, x2, x3)  =  multC_out_gga(x1, x2, x3)
U2_gga(x1, x2, x3, x4)  =  U2_gga(x1, x2, x4)
pD_in_ggaa(x1, x2, x3, x4)  =  pD_in_ggaa(x1, x2)
U9_ggaa(x1, x2, x3, x4, x5)  =  U9_ggaa(x1, x2, x5)
U10_ggaa(x1, x2, x3, x4, x5)  =  U10_ggaa(x1, x2, x3, x5)
addE_in_gga(x1, x2, x3)  =  addE_in_gga(x1, x2)
addE_out_gga(x1, x2, x3)  =  addE_out_gga(x1, x2, x3)
U3_gga(x1, x2, x3, x4)  =  U3_gga(x1, x2, x4)
pD_out_ggaa(x1, x2, x3, x4)  =  pD_out_ggaa(x1, x2, x3, x4)
U6_ggaaa(x1, x2, x3, x4, x5, x6)  =  U6_ggaaa(x1, x2, x3, x6)
pG_in_ggaa(x1, x2, x3, x4)  =  pG_in_ggaa(x1, x2)
U7_ggaa(x1, x2, x3, x4, x5)  =  U7_ggaa(x1, x2, x5)
U8_ggaa(x1, x2, x3, x4, x5)  =  U8_ggaa(x1, x2, x3, x5)
addF_in_gga(x1, x2, x3)  =  addF_in_gga(x1, x2)
addF_out_gga(x1, x2, x3)  =  addF_out_gga(x1, x2, x3)
U4_gga(x1, x2, x3, x4)  =  U4_gga(x1, x2, x4)
pG_out_ggaa(x1, x2, x3, x4)  =  pG_out_ggaa(x1, x2, x3, x4)
pB_out_ggaaa(x1, x2, x3, x4, x5)  =  pB_out_ggaaa(x1, x2, x3, x4, x5)
ADDF_IN_GGA(x1, x2, x3)  =  ADDF_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:

ADDF_IN_GGA(s(T85), T86, s(T88)) → ADDF_IN_GGA(T85, T86, T88)

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

ADDF_IN_GGA(s(T85), T86) → ADDF_IN_GGA(T85, T86)

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:

• ADDF_IN_GGA(s(T85), T86) → ADDF_IN_GGA(T85, T86)
  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:

ADDE_IN_GGA(s(T63), T64, s(X87)) → ADDE_IN_GGA(T63, T64, X87)

The TRS R consists of the following rules:

multA_in_gga(0, T5, 0) → multA_out_gga(0, T5, 0)
multA_in_gga(s(0), T24, T24) → multA_out_gga(s(0), T24, T24)
multA_in_gga(s(s(T29)), T30, T12) → U1_gga(T29, T30, T12, pB_in_ggaaa(T29, T30, X40, X41, T12))
pB_in_ggaaa(T29, T30, T33, X41, T12) → U5_ggaaa(T29, T30, T33, X41, T12, multC_in_gga(T29, T30, T33))
multC_in_gga(0, T40, 0) → multC_out_gga(0, T40, 0)
multC_in_gga(s(T45), T46, X64) → U2_gga(T45, T46, X64, pD_in_ggaa(T45, T46, X63, X64))
pD_in_ggaa(T45, T46, T49, X64) → U9_ggaa(T45, T46, T49, X64, multC_in_gga(T45, T46, T49))
U9_ggaa(T45, T46, T49, X64, multC_out_gga(T45, T46, T49)) → U10_ggaa(T45, T46, T49, X64, addE_in_gga(T49, T46, X64))
addE_in_gga(0, T58, T58) → addE_out_gga(0, T58, T58)
addE_in_gga(s(T63), T64, s(X87)) → U3_gga(T63, T64, X87, addE_in_gga(T63, T64, X87))
U3_gga(T63, T64, X87, addE_out_gga(T63, T64, X87)) → addE_out_gga(s(T63), T64, s(X87))
U10_ggaa(T45, T46, T49, X64, addE_out_gga(T49, T46, X64)) → pD_out_ggaa(T45, T46, T49, X64)
U2_gga(T45, T46, X64, pD_out_ggaa(T45, T46, X63, X64)) → multC_out_gga(s(T45), T46, X64)
U5_ggaaa(T29, T30, T33, X41, T12, multC_out_gga(T29, T30, T33)) → U6_ggaaa(T29, T30, T33, X41, T12, pG_in_ggaa(T33, T30, X41, T12))
pG_in_ggaa(T33, T30, T69, T12) → U7_ggaa(T33, T30, T69, T12, addE_in_gga(T33, T30, T69))
U7_ggaa(T33, T30, T69, T12, addE_out_gga(T33, T30, T69)) → U8_ggaa(T33, T30, T69, T12, addF_in_gga(T69, T30, T12))
addF_in_gga(0, T78, T78) → addF_out_gga(0, T78, T78)
addF_in_gga(s(T85), T86, s(T88)) → U4_gga(T85, T86, T88, addF_in_gga(T85, T86, T88))
U4_gga(T85, T86, T88, addF_out_gga(T85, T86, T88)) → addF_out_gga(s(T85), T86, s(T88))
U8_ggaa(T33, T30, T69, T12, addF_out_gga(T69, T30, T12)) → pG_out_ggaa(T33, T30, T69, T12)
U6_ggaaa(T29, T30, T33, X41, T12, pG_out_ggaa(T33, T30, X41, T12)) → pB_out_ggaaa(T29, T30, T33, X41, T12)
U1_gga(T29, T30, T12, pB_out_ggaaa(T29, T30, X40, X41, T12)) → multA_out_gga(s(s(T29)), T30, T12)

The argument filtering Pi contains the following mapping:
multA_in_gga(x1, x2, x3)  =  multA_in_gga(x1, x2)
0  =  0
multA_out_gga(x1, x2, x3)  =  multA_out_gga(x1, x2, x3)
s(x1)  =  s(x1)
U1_gga(x1, x2, x3, x4)  =  U1_gga(x1, x2, x4)
pB_in_ggaaa(x1, x2, x3, x4, x5)  =  pB_in_ggaaa(x1, x2)
U5_ggaaa(x1, x2, x3, x4, x5, x6)  =  U5_ggaaa(x1, x2, x6)
multC_in_gga(x1, x2, x3)  =  multC_in_gga(x1, x2)
multC_out_gga(x1, x2, x3)  =  multC_out_gga(x1, x2, x3)
U2_gga(x1, x2, x3, x4)  =  U2_gga(x1, x2, x4)
pD_in_ggaa(x1, x2, x3, x4)  =  pD_in_ggaa(x1, x2)
U9_ggaa(x1, x2, x3, x4, x5)  =  U9_ggaa(x1, x2, x5)
U10_ggaa(x1, x2, x3, x4, x5)  =  U10_ggaa(x1, x2, x3, x5)
addE_in_gga(x1, x2, x3)  =  addE_in_gga(x1, x2)
addE_out_gga(x1, x2, x3)  =  addE_out_gga(x1, x2, x3)
U3_gga(x1, x2, x3, x4)  =  U3_gga(x1, x2, x4)
pD_out_ggaa(x1, x2, x3, x4)  =  pD_out_ggaa(x1, x2, x3, x4)
U6_ggaaa(x1, x2, x3, x4, x5, x6)  =  U6_ggaaa(x1, x2, x3, x6)
pG_in_ggaa(x1, x2, x3, x4)  =  pG_in_ggaa(x1, x2)
U7_ggaa(x1, x2, x3, x4, x5)  =  U7_ggaa(x1, x2, x5)
U8_ggaa(x1, x2, x3, x4, x5)  =  U8_ggaa(x1, x2, x3, x5)
addF_in_gga(x1, x2, x3)  =  addF_in_gga(x1, x2)
addF_out_gga(x1, x2, x3)  =  addF_out_gga(x1, x2, x3)
U4_gga(x1, x2, x3, x4)  =  U4_gga(x1, x2, x4)
pG_out_ggaa(x1, x2, x3, x4)  =  pG_out_ggaa(x1, x2, x3, x4)
pB_out_ggaaa(x1, x2, x3, x4, x5)  =  pB_out_ggaaa(x1, x2, x3, x4, x5)
ADDE_IN_GGA(x1, x2, x3)  =  ADDE_IN_GGA(x1, x2)

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

(15) UsableRulesProof (EQUIVALENT transformation)

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

(16) Obligation:

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

ADDE_IN_GGA(s(T63), T64, s(X87)) → ADDE_IN_GGA(T63, T64, X87)

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

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

(17) PiDPToQDPProof (SOUND transformation)

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

(18) Obligation:

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

ADDE_IN_GGA(s(T63), T64) → ADDE_IN_GGA(T63, T64)

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:

• ADDE_IN_GGA(s(T63), T64) → ADDE_IN_GGA(T63, T64)
  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:

MULTC_IN_GGA(s(T45), T46, X64) → PD_IN_GGAA(T45, T46, X63, X64)
PD_IN_GGAA(T45, T46, T49, X64) → MULTC_IN_GGA(T45, T46, T49)

The TRS R consists of the following rules:

multA_in_gga(0, T5, 0) → multA_out_gga(0, T5, 0)
multA_in_gga(s(0), T24, T24) → multA_out_gga(s(0), T24, T24)
multA_in_gga(s(s(T29)), T30, T12) → U1_gga(T29, T30, T12, pB_in_ggaaa(T29, T30, X40, X41, T12))
pB_in_ggaaa(T29, T30, T33, X41, T12) → U5_ggaaa(T29, T30, T33, X41, T12, multC_in_gga(T29, T30, T33))
multC_in_gga(0, T40, 0) → multC_out_gga(0, T40, 0)
multC_in_gga(s(T45), T46, X64) → U2_gga(T45, T46, X64, pD_in_ggaa(T45, T46, X63, X64))
pD_in_ggaa(T45, T46, T49, X64) → U9_ggaa(T45, T46, T49, X64, multC_in_gga(T45, T46, T49))
U9_ggaa(T45, T46, T49, X64, multC_out_gga(T45, T46, T49)) → U10_ggaa(T45, T46, T49, X64, addE_in_gga(T49, T46, X64))
addE_in_gga(0, T58, T58) → addE_out_gga(0, T58, T58)
addE_in_gga(s(T63), T64, s(X87)) → U3_gga(T63, T64, X87, addE_in_gga(T63, T64, X87))
U3_gga(T63, T64, X87, addE_out_gga(T63, T64, X87)) → addE_out_gga(s(T63), T64, s(X87))
U10_ggaa(T45, T46, T49, X64, addE_out_gga(T49, T46, X64)) → pD_out_ggaa(T45, T46, T49, X64)
U2_gga(T45, T46, X64, pD_out_ggaa(T45, T46, X63, X64)) → multC_out_gga(s(T45), T46, X64)
U5_ggaaa(T29, T30, T33, X41, T12, multC_out_gga(T29, T30, T33)) → U6_ggaaa(T29, T30, T33, X41, T12, pG_in_ggaa(T33, T30, X41, T12))
pG_in_ggaa(T33, T30, T69, T12) → U7_ggaa(T33, T30, T69, T12, addE_in_gga(T33, T30, T69))
U7_ggaa(T33, T30, T69, T12, addE_out_gga(T33, T30, T69)) → U8_ggaa(T33, T30, T69, T12, addF_in_gga(T69, T30, T12))
addF_in_gga(0, T78, T78) → addF_out_gga(0, T78, T78)
addF_in_gga(s(T85), T86, s(T88)) → U4_gga(T85, T86, T88, addF_in_gga(T85, T86, T88))
U4_gga(T85, T86, T88, addF_out_gga(T85, T86, T88)) → addF_out_gga(s(T85), T86, s(T88))
U8_ggaa(T33, T30, T69, T12, addF_out_gga(T69, T30, T12)) → pG_out_ggaa(T33, T30, T69, T12)
U6_ggaaa(T29, T30, T33, X41, T12, pG_out_ggaa(T33, T30, X41, T12)) → pB_out_ggaaa(T29, T30, T33, X41, T12)
U1_gga(T29, T30, T12, pB_out_ggaaa(T29, T30, X40, X41, T12)) → multA_out_gga(s(s(T29)), T30, T12)

The argument filtering Pi contains the following mapping:
multA_in_gga(x1, x2, x3)  =  multA_in_gga(x1, x2)
0  =  0
multA_out_gga(x1, x2, x3)  =  multA_out_gga(x1, x2, x3)
s(x1)  =  s(x1)
U1_gga(x1, x2, x3, x4)  =  U1_gga(x1, x2, x4)
pB_in_ggaaa(x1, x2, x3, x4, x5)  =  pB_in_ggaaa(x1, x2)
U5_ggaaa(x1, x2, x3, x4, x5, x6)  =  U5_ggaaa(x1, x2, x6)
multC_in_gga(x1, x2, x3)  =  multC_in_gga(x1, x2)
multC_out_gga(x1, x2, x3)  =  multC_out_gga(x1, x2, x3)
U2_gga(x1, x2, x3, x4)  =  U2_gga(x1, x2, x4)
pD_in_ggaa(x1, x2, x3, x4)  =  pD_in_ggaa(x1, x2)
U9_ggaa(x1, x2, x3, x4, x5)  =  U9_ggaa(x1, x2, x5)
U10_ggaa(x1, x2, x3, x4, x5)  =  U10_ggaa(x1, x2, x3, x5)
addE_in_gga(x1, x2, x3)  =  addE_in_gga(x1, x2)
addE_out_gga(x1, x2, x3)  =  addE_out_gga(x1, x2, x3)
U3_gga(x1, x2, x3, x4)  =  U3_gga(x1, x2, x4)
pD_out_ggaa(x1, x2, x3, x4)  =  pD_out_ggaa(x1, x2, x3, x4)
U6_ggaaa(x1, x2, x3, x4, x5, x6)  =  U6_ggaaa(x1, x2, x3, x6)
pG_in_ggaa(x1, x2, x3, x4)  =  pG_in_ggaa(x1, x2)
U7_ggaa(x1, x2, x3, x4, x5)  =  U7_ggaa(x1, x2, x5)
U8_ggaa(x1, x2, x3, x4, x5)  =  U8_ggaa(x1, x2, x3, x5)
addF_in_gga(x1, x2, x3)  =  addF_in_gga(x1, x2)
addF_out_gga(x1, x2, x3)  =  addF_out_gga(x1, x2, x3)
U4_gga(x1, x2, x3, x4)  =  U4_gga(x1, x2, x4)
pG_out_ggaa(x1, x2, x3, x4)  =  pG_out_ggaa(x1, x2, x3, x4)
pB_out_ggaaa(x1, x2, x3, x4, x5)  =  pB_out_ggaaa(x1, x2, x3, x4, x5)
MULTC_IN_GGA(x1, x2, x3)  =  MULTC_IN_GGA(x1, x2)
PD_IN_GGAA(x1, x2, x3, x4)  =  PD_IN_GGAA(x1, x2)

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:

MULTC_IN_GGA(s(T45), T46, X64) → PD_IN_GGAA(T45, T46, X63, X64)
PD_IN_GGAA(T45, T46, T49, X64) → MULTC_IN_GGA(T45, T46, T49)

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

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:

MULTC_IN_GGA(s(T45), T46) → PD_IN_GGAA(T45, T46)
PD_IN_GGAA(T45, T46) → MULTC_IN_GGA(T45, T46)

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:

• PD_IN_GGAA(T45, T46) → MULTC_IN_GGA(T45, T46)
  The graph contains the following edges 1 >= 1, 2 >= 2

• MULTC_IN_GGA(s(T45), T46) → PD_IN_GGAA(T45, T46)
  The graph contains the following edges 1 > 1, 2 >= 2

(27) YES