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

(0) Obligation:

Clauses:

gcd(X, Y, D) :- ','(le(X, Y), gcd_le(X, Y, D)).
gcd(X, Y, D) :- ','(gt(X, Y), gcd_le(Y, X, D)).
gcd_le(0, Y, Y).
gcd_le(s(X), Y, D) :- ','(add(s(X), Z, Y), gcd(s(X), Z, D)).
gt(s(X), s(Y)) :- gt(X, Y).
gt(s(X), 0).
le(s(X), s(Y)) :- le(X, Y).
le(0, s(Y)).
le(0, 0).
add(s(X), Y, s(Z)) :- add(X, Y, Z).
add(0, X, X).

Query: gcd(g,g,a)

(1) PrologToTRSTransformerProof (SOUND transformation)

Transformed Prolog program to TRS.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
3 [label="3: gcd(T1, T2, T3)\n\nT1 is ground\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
6 [label="6: gcd(T1, T2, T3), SCOPE: 1, CLAUSE: 0 | gcd(T1, T2, T3), SCOPE: 1, CLAUSE: 1\n\nT1 is ground\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: gcd(T1, T2, T3), SCOPE: 1, CLAUSE: 0\n\nT1 is ground\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: gcd(T1, T2, T3), SCOPE: 1, CLAUSE: 1\n\nT1 is ground\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: ','(le(T22, T23), gcd_le(T22, T23, T25))\n\nT22 is ground\nT23 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: le(T22, T23)\n\nT22 is ground\nT23 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: gcd_le(T22, T23, T25)\n\nT22 is ground\nT23 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: le(T22, T23), SCOPE: 2, CLAUSE: 6 | le(T22, T23), SCOPE: 2, CLAUSE: 7 | le(T22, T23), SCOPE: 2, CLAUSE: 8\n\nT22 is ground\nT23 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: le(T22, T23), SCOPE: 2, CLAUSE: 6\n\nT22 is ground\nT23 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: le(T22, T23), SCOPE: 2, CLAUSE: 7 | le(T22, T23), SCOPE: 2, CLAUSE: 8\n\nT22 is ground\nT23 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
39 [label="39: le(T38, T39)\n\nT38 is ground\nT39 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
40 [label="40: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
41 [label="41: le(T22, T23), SCOPE: 2, CLAUSE: 7\n\nT22 is ground\nT23 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
43 [label="43: le(T22, T23), SCOPE: 2, CLAUSE: 8\n\nT22 is ground\nT23 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
46 [label="46: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
50 [label="50: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
51 [label="51: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
54 [label="54: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
56 [label="56: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
58 [label="58: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
63 [label="63: gcd_le(T22, T23, T25), SCOPE: 3, CLAUSE: 2 | gcd_le(T22, T23, T25), SCOPE: 3, CLAUSE: 3\n\nT22 is ground\nT23 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
64 [label="64: gcd_le(T22, T23, T25), SCOPE: 3, CLAUSE: 2\n\nT22 is ground\nT23 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
65 [label="65: gcd_le(T22, T23, T25), SCOPE: 3, CLAUSE: 3\n\nT22 is ground\nT23 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
66 [label="66: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
67 [label="67: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
68 [label="68: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
78 [label="78: ','(add(s(T60), X58, T61), gcd(s(T60), X58, T63))\n\nT60 is ground\nT61 is ground\nX58 is free", fontsize=16, style=filled, fillcolor=lightyellow];
79 [label="79: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
82 [label="82: add(s(T60), X58, T61)\n\nT60 is ground\nT61 is ground\nX58 is free", fontsize=16, style=filled, fillcolor=lightyellow];
84 [label="84: gcd(s(T60), T66, T63)\n\nT60 is ground\nT66 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
85 [label="85: add(s(T60), X58, T61), SCOPE: 4, CLAUSE: 9 | add(s(T60), X58, T61), SCOPE: 4, CLAUSE: 10\n\nT60 is ground\nT61 is ground\nX58 is free", fontsize=16, style=filled, fillcolor=lightyellow];
86 [label="86: add(s(T60), X58, T61), SCOPE: 4, CLAUSE: 9\n\nT60 is ground\nT61 is ground\nX58 is free", fontsize=16, style=filled, fillcolor=lightyellow];
88 [label="88: add(s(T60), X58, T61), SCOPE: 4, CLAUSE: 10\n\nT60 is ground\nT61 is ground\nX58 is free", fontsize=16, style=filled, fillcolor=lightyellow];
91 [label="91: add(T77, X86, T78)\n\nT77 is ground\nT78 is ground\nX86 is free", fontsize=16, style=filled, fillcolor=lightyellow];
92 [label="92: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
94 [label="94: add(T77, X86, T78), SCOPE: 5, CLAUSE: 9 | add(T77, X86, T78), SCOPE: 5, CLAUSE: 10\n\nT77 is ground\nT78 is ground\nX86 is free", fontsize=16, style=filled, fillcolor=lightyellow];
96 [label="96: add(T77, X86, T78), SCOPE: 5, CLAUSE: 9\n\nT77 is ground\nT78 is ground\nX86 is free", fontsize=16, style=filled, fillcolor=lightyellow];
97 [label="97: add(T77, X86, T78), SCOPE: 5, CLAUSE: 10\n\nT77 is ground\nT78 is ground\nX86 is free", fontsize=16, style=filled, fillcolor=lightyellow];
100 [label="100: add(T89, X110, T90)\n\nT89 is ground\nT90 is ground\nX110 is free", fontsize=16, style=filled, fillcolor=lightyellow];
101 [label="101: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
106 [label="106: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
108 [label="108: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
109 [label="109: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
112 [label="112: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
119 [label="119: ','(gt(T108, T109), gcd_le(T109, T108, T111))\n\nT108 is ground\nT109 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
125 [label="125: gt(T108, T109)\n\nT108 is ground\nT109 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
126 [label="126: gcd_le(T109, T108, T111)\n\nT108 is ground\nT109 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
131 [label="131: gt(T108, T109), SCOPE: 6, CLAUSE: 4 | gt(T108, T109), SCOPE: 6, CLAUSE: 5\n\nT108 is ground\nT109 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
136 [label="136: gt(T108, T109), SCOPE: 6, CLAUSE: 4\n\nT108 is ground\nT109 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
137 [label="137: gt(T108, T109), SCOPE: 6, CLAUSE: 5\n\nT108 is ground\nT109 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
142 [label="142: gt(T124, T125)\n\nT124 is ground\nT125 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
143 [label="143: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
149 [label="149: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
150 [label="150: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
151 [label="151: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
152 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
3 -> 152 [arrowhead = none ];
152 -> 6;

153 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
6 -> 153 [arrowhead = none ];
153 -> 9;

154 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
6 -> 154 [arrowhead = none ];
154 -> 10;

155 [label="ONLY EVAL with clause\ngcd(X17, X18, X19) :- ','(le(X17, X18), gcd_le(X17, X18, X19)).\nand substitutionT1 -> T22,\nX17 -> T22,\nT2 -> T23,\nX18 -> T23,\nT3 -> T25,\nX19 -> T25,\nT24 -> T25", fontsize=14, style = filled, fillcolor = lightblue];
9 -> 155 [arrowhead = none ];
155 -> 20;

156 [label="ONLY EVAL with clause\ngcd(X133, X134, X135) :- ','(gt(X133, X134), gcd_le(X134, X133, X135)).\nand substitutionT1 -> T108,\nX133 -> T108,\nT2 -> T109,\nX134 -> T109,\nT3 -> T111,\nX135 -> T111,\nT110 -> T111", fontsize=14, style = filled, fillcolor = lightblue];
10 -> 156 [arrowhead = none ];
156 -> 119;

157 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
20 -> 157 [arrowhead = none ];
157 -> 24;

158 [label="SPLIT 2\nnew knowledge:\nT22 is ground\nT23 is ground", fontsize=14, style = filled, fillcolor = violet];
20 -> 158 [arrowhead = none ];
158 -> 26;

159 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
24 -> 159 [arrowhead = none ];
159 -> 30;

160 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
26 -> 160 [arrowhead = none ];
160 -> 63;

161 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
30 -> 161 [arrowhead = none ];
161 -> 34;

162 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
30 -> 162 [arrowhead = none ];
162 -> 36;

163 [label="EVAL with clause\nle(s(X32), s(X33)) :- le(X32, X33).\nand substitutionX32 -> T38,\nT22 -> s(T38),\nX33 -> T39,\nT23 -> s(T39)", fontsize=14, style = filled, fillcolor = lightblue];
34 -> 163 [arrowhead = none ];
163 -> 39;

164 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
34 -> 164 [arrowhead = none ];
164 -> 40;

165 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
36 -> 165 [arrowhead = none ];
165 -> 41;

166 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
36 -> 166 [arrowhead = none ];
166 -> 43;

167 [label="INSTANCE with matching:\nT22 -> T38\nT23 -> T39", fontsize=14, style = filled, fillcolor = lightgrey];
39 -> 167 [arrowhead = none , style=dashed];
167 -> 24[style=dashed];

168 [label="EVAL with clause\nle(0, s(X40)).\nand substitutionT22 -> 0,\nX40 -> T46,\nT23 -> s(T46)", fontsize=14, style = filled, fillcolor = lightblue];
41 -> 168 [arrowhead = none ];
168 -> 46;

169 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
41 -> 169 [arrowhead = none ];
169 -> 50;

170 [label="EVAL with clause\nle(0, 0).\nand substitutionT22 -> 0,\nT23 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
43 -> 170 [arrowhead = none ];
170 -> 54;

171 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
43 -> 171 [arrowhead = none ];
171 -> 56;

172 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
46 -> 172 [arrowhead = none ];
172 -> 51;

173 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
54 -> 173 [arrowhead = none ];
173 -> 58;

174 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
63 -> 174 [arrowhead = none ];
174 -> 64;

175 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
63 -> 175 [arrowhead = none ];
175 -> 65;

176 [label="EVAL with clause\ngcd_le(0, X47, X47).\nand substitutionT22 -> 0,\nT23 -> T53,\nX47 -> T53,\nT25 -> T53", fontsize=14, style = filled, fillcolor = lightblue];
64 -> 176 [arrowhead = none ];
176 -> 66;

177 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
64 -> 177 [arrowhead = none ];
177 -> 67;

178 [label="EVAL with clause\ngcd_le(s(X55), X56, X57) :- ','(add(s(X55), X58, X56), gcd(s(X55), X58, X57)).\nand substitutionX55 -> T60,\nT22 -> s(T60),\nT23 -> T61,\nX56 -> T61,\nT25 -> T63,\nX57 -> T63,\nT62 -> T63", fontsize=14, style = filled, fillcolor = lightblue];
65 -> 178 [arrowhead = none ];
178 -> 78;

179 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
65 -> 179 [arrowhead = none ];
179 -> 79;

180 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
66 -> 180 [arrowhead = none ];
180 -> 68;

181 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
78 -> 181 [arrowhead = none ];
181 -> 82;

182 [label="SPLIT 2\nnew knowledge:\nT60 is ground\nT66 is ground\nT61 is ground\nreplacements:X58 -> T66", fontsize=14, style = filled, fillcolor = violet];
78 -> 182 [arrowhead = none ];
182 -> 84;

183 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
82 -> 183 [arrowhead = none ];
183 -> 85;

184 [label="INSTANCE with matching:\nT1 -> s(T60)\nT2 -> T66\nT3 -> T63", fontsize=14, style = filled, fillcolor = lightgrey];
84 -> 184 [arrowhead = none , style=dashed];
184 -> 3[style=dashed];

185 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
85 -> 185 [arrowhead = none ];
185 -> 86;

186 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
85 -> 186 [arrowhead = none ];
186 -> 88;

187 [label="EVAL with clause\nadd(s(X83), X84, s(X85)) :- add(X83, X84, X85).\nand substitutionT60 -> T77,\nX83 -> T77,\nX58 -> X86,\nX84 -> X86,\nX85 -> T78,\nT61 -> s(T78)", fontsize=14, style = filled, fillcolor = lightblue];
86 -> 187 [arrowhead = none ];
187 -> 91;

188 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
86 -> 188 [arrowhead = none ];
188 -> 92;

189 [label="BACKTRACK\nfor clause: add(0, X, X)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
88 -> 189 [arrowhead = none ];
189 -> 112;

190 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
91 -> 190 [arrowhead = none ];
190 -> 94;

191 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
94 -> 191 [arrowhead = none ];
191 -> 96;

192 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
94 -> 192 [arrowhead = none ];
192 -> 97;

193 [label="EVAL with clause\nadd(s(X107), X108, s(X109)) :- add(X107, X108, X109).\nand substitutionX107 -> T89,\nT77 -> s(T89),\nX86 -> X110,\nX108 -> X110,\nX109 -> T90,\nT78 -> s(T90)", fontsize=14, style = filled, fillcolor = lightblue];
96 -> 193 [arrowhead = none ];
193 -> 100;

194 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
96 -> 194 [arrowhead = none ];
194 -> 101;

195 [label="EVAL with clause\nadd(0, X119, X119).\nand substitutionT77 -> 0,\nX86 -> T95,\nX119 -> T95,\nT78 -> T95,\nX120 -> T95", fontsize=14, style = filled, fillcolor = lightblue];
97 -> 195 [arrowhead = none ];
195 -> 106;

196 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
97 -> 196 [arrowhead = none ];
196 -> 108;

197 [label="INSTANCE with matching:\nT77 -> T89\nX86 -> X110\nT78 -> T90", fontsize=14, style = filled, fillcolor = lightgrey];
100 -> 197 [arrowhead = none , style=dashed];
197 -> 91[style=dashed];

198 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
106 -> 198 [arrowhead = none ];
198 -> 109;

199 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
119 -> 199 [arrowhead = none ];
199 -> 125;

200 [label="SPLIT 2\nnew knowledge:\nT108 is ground\nT109 is ground", fontsize=14, style = filled, fillcolor = violet];
119 -> 200 [arrowhead = none ];
200 -> 126;

201 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
125 -> 201 [arrowhead = none ];
201 -> 131;

202 [label="INSTANCE with matching:\nT22 -> T109\nT23 -> T108\nT25 -> T111", fontsize=14, style = filled, fillcolor = lightgrey];
126 -> 202 [arrowhead = none , style=dashed];
202 -> 26[style=dashed];

203 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
131 -> 203 [arrowhead = none ];
203 -> 136;

204 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
131 -> 204 [arrowhead = none ];
204 -> 137;

205 [label="EVAL with clause\ngt(s(X148), s(X149)) :- gt(X148, X149).\nand substitutionX148 -> T124,\nT108 -> s(T124),\nX149 -> T125,\nT109 -> s(T125)", fontsize=14, style = filled, fillcolor = lightblue];
136 -> 205 [arrowhead = none ];
205 -> 142;

206 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
136 -> 206 [arrowhead = none ];
206 -> 143;

207 [label="EVAL with clause\ngt(s(X154), 0).\nand substitutionX154 -> T130,\nT108 -> s(T130),\nT109 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
137 -> 207 [arrowhead = none ];
207 -> 149;

208 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
137 -> 208 [arrowhead = none ];
208 -> 150;

209 [label="INSTANCE with matching:\nT108 -> T124\nT109 -> T125", fontsize=14, style = filled, fillcolor = lightgrey];
142 -> 209 [arrowhead = none , style=dashed];
209 -> 125[style=dashed];

210 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
149 -> 210 [arrowhead = none ];
210 -> 151;

}

(2) Obligation:

Q restricted rewrite system:
The TRS R consists of the following rules:

f3_in(T22, T23) → U1(f20_in(T22, T23), T22, T23)
U1(f20_out1(T25), T22, T23) → f3_out1(T25)
f3_in(T108, T109) → U2(f119_in(T108, T109), T108, T109)
U2(f119_out1(T111), T108, T109) → f3_out1(T111)
f24_in(s(T38), s(T39)) → U3(f24_in(T38, T39), s(T38), s(T39))
U3(f24_out1, s(T38), s(T39)) → f24_out1
f24_in(0, s(T46)) → f24_out1
f24_in(0, 0) → f24_out1
f91_in(s(T89), s(T90)) → U4(f91_in(T89, T90), s(T89), s(T90))
U4(f91_out1(X110), s(T89), s(T90)) → f91_out1(X110)
f91_in(0, T95) → f91_out1(T95)
f26_in(0, T53) → f26_out1(T53)
f26_in(s(T60), T61) → U5(f78_in(T60, T61), s(T60), T61)
U5(f78_out1(X58, T63), s(T60), T61) → f26_out1(T63)
f125_in(s(T124), s(T125)) → U6(f125_in(T124, T125), s(T124), s(T125))
U6(f125_out1, s(T124), s(T125)) → f125_out1
f125_in(s(T130), 0) → f125_out1
f82_in(T77, s(T78)) → U7(f91_in(T77, T78), T77, s(T78))
U7(f91_out1(X86), T77, s(T78)) → f82_out1(X86)
f20_in(T22, T23) → U8(f24_in(T22, T23), T22, T23)
U8(f24_out1, T22, T23) → U9(f26_in(T22, T23), T22, T23)
U9(f26_out1(T25), T22, T23) → f20_out1(T25)
f78_in(T60, T61) → U10(f82_in(T60, T61), T60, T61)
U10(f82_out1(T66), T60, T61) → U11(f3_in(s(T60), T66), T60, T61, T66)
U11(f3_out1(T63), T60, T61, T66) → f78_out1(T66, T63)
f119_in(T108, T109) → U12(f125_in(T108, T109), T108, T109)
U12(f125_out1, T108, T109) → U13(f26_in(T109, T108), T108, T109)
U13(f26_out1(T111), T108, T109) → f119_out1(T111)

Q is empty.

(3) DependencyPairsProof (EQUIVALENT transformation)

Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.

(4) Obligation:

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

F3_IN(T22, T23) → U1¹(f20_in(T22, T23), T22, T23)
F3_IN(T22, T23) → F20_IN(T22, T23)
F3_IN(T108, T109) → U2¹(f119_in(T108, T109), T108, T109)
F3_IN(T108, T109) → F119_IN(T108, T109)
F24_IN(s(T38), s(T39)) → U3¹(f24_in(T38, T39), s(T38), s(T39))
F24_IN(s(T38), s(T39)) → F24_IN(T38, T39)
F91_IN(s(T89), s(T90)) → U4¹(f91_in(T89, T90), s(T89), s(T90))
F91_IN(s(T89), s(T90)) → F91_IN(T89, T90)
F26_IN(s(T60), T61) → U5¹(f78_in(T60, T61), s(T60), T61)
F26_IN(s(T60), T61) → F78_IN(T60, T61)
F125_IN(s(T124), s(T125)) → U6¹(f125_in(T124, T125), s(T124), s(T125))
F125_IN(s(T124), s(T125)) → F125_IN(T124, T125)
F82_IN(T77, s(T78)) → U7¹(f91_in(T77, T78), T77, s(T78))
F82_IN(T77, s(T78)) → F91_IN(T77, T78)
F20_IN(T22, T23) → U8¹(f24_in(T22, T23), T22, T23)
F20_IN(T22, T23) → F24_IN(T22, T23)
U8¹(f24_out1, T22, T23) → U9¹(f26_in(T22, T23), T22, T23)
U8¹(f24_out1, T22, T23) → F26_IN(T22, T23)
F78_IN(T60, T61) → U10¹(f82_in(T60, T61), T60, T61)
F78_IN(T60, T61) → F82_IN(T60, T61)
U10¹(f82_out1(T66), T60, T61) → U11¹(f3_in(s(T60), T66), T60, T61, T66)
U10¹(f82_out1(T66), T60, T61) → F3_IN(s(T60), T66)
F119_IN(T108, T109) → U12¹(f125_in(T108, T109), T108, T109)
F119_IN(T108, T109) → F125_IN(T108, T109)
U12¹(f125_out1, T108, T109) → U13¹(f26_in(T109, T108), T108, T109)
U12¹(f125_out1, T108, T109) → F26_IN(T109, T108)

The TRS R consists of the following rules:

f3_in(T22, T23) → U1(f20_in(T22, T23), T22, T23)
U1(f20_out1(T25), T22, T23) → f3_out1(T25)
f3_in(T108, T109) → U2(f119_in(T108, T109), T108, T109)
U2(f119_out1(T111), T108, T109) → f3_out1(T111)
f24_in(s(T38), s(T39)) → U3(f24_in(T38, T39), s(T38), s(T39))
U3(f24_out1, s(T38), s(T39)) → f24_out1
f24_in(0, s(T46)) → f24_out1
f24_in(0, 0) → f24_out1
f91_in(s(T89), s(T90)) → U4(f91_in(T89, T90), s(T89), s(T90))
U4(f91_out1(X110), s(T89), s(T90)) → f91_out1(X110)
f91_in(0, T95) → f91_out1(T95)
f26_in(0, T53) → f26_out1(T53)
f26_in(s(T60), T61) → U5(f78_in(T60, T61), s(T60), T61)
U5(f78_out1(X58, T63), s(T60), T61) → f26_out1(T63)
f125_in(s(T124), s(T125)) → U6(f125_in(T124, T125), s(T124), s(T125))
U6(f125_out1, s(T124), s(T125)) → f125_out1
f125_in(s(T130), 0) → f125_out1
f82_in(T77, s(T78)) → U7(f91_in(T77, T78), T77, s(T78))
U7(f91_out1(X86), T77, s(T78)) → f82_out1(X86)
f20_in(T22, T23) → U8(f24_in(T22, T23), T22, T23)
U8(f24_out1, T22, T23) → U9(f26_in(T22, T23), T22, T23)
U9(f26_out1(T25), T22, T23) → f20_out1(T25)
f78_in(T60, T61) → U10(f82_in(T60, T61), T60, T61)
U10(f82_out1(T66), T60, T61) → U11(f3_in(s(T60), T66), T60, T61, T66)
U11(f3_out1(T63), T60, T61, T66) → f78_out1(T66, T63)
f119_in(T108, T109) → U12(f125_in(T108, T109), T108, T109)
U12(f125_out1, T108, T109) → U13(f26_in(T109, T108), T108, T109)
U13(f26_out1(T111), T108, T109) → f119_out1(T111)

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

(5) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 4 SCCs with 14 less nodes.

(6) Complex Obligation (AND)

(7) Obligation:

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

F125_IN(s(T124), s(T125)) → F125_IN(T124, T125)

The TRS R consists of the following rules:

f3_in(T22, T23) → U1(f20_in(T22, T23), T22, T23)
U1(f20_out1(T25), T22, T23) → f3_out1(T25)
f3_in(T108, T109) → U2(f119_in(T108, T109), T108, T109)
U2(f119_out1(T111), T108, T109) → f3_out1(T111)
f24_in(s(T38), s(T39)) → U3(f24_in(T38, T39), s(T38), s(T39))
U3(f24_out1, s(T38), s(T39)) → f24_out1
f24_in(0, s(T46)) → f24_out1
f24_in(0, 0) → f24_out1
f91_in(s(T89), s(T90)) → U4(f91_in(T89, T90), s(T89), s(T90))
U4(f91_out1(X110), s(T89), s(T90)) → f91_out1(X110)
f91_in(0, T95) → f91_out1(T95)
f26_in(0, T53) → f26_out1(T53)
f26_in(s(T60), T61) → U5(f78_in(T60, T61), s(T60), T61)
U5(f78_out1(X58, T63), s(T60), T61) → f26_out1(T63)
f125_in(s(T124), s(T125)) → U6(f125_in(T124, T125), s(T124), s(T125))
U6(f125_out1, s(T124), s(T125)) → f125_out1
f125_in(s(T130), 0) → f125_out1
f82_in(T77, s(T78)) → U7(f91_in(T77, T78), T77, s(T78))
U7(f91_out1(X86), T77, s(T78)) → f82_out1(X86)
f20_in(T22, T23) → U8(f24_in(T22, T23), T22, T23)
U8(f24_out1, T22, T23) → U9(f26_in(T22, T23), T22, T23)
U9(f26_out1(T25), T22, T23) → f20_out1(T25)
f78_in(T60, T61) → U10(f82_in(T60, T61), T60, T61)
U10(f82_out1(T66), T60, T61) → U11(f3_in(s(T60), T66), T60, T61, T66)
U11(f3_out1(T63), T60, T61, T66) → f78_out1(T66, T63)
f119_in(T108, T109) → U12(f125_in(T108, T109), T108, T109)
U12(f125_out1, T108, T109) → U13(f26_in(T109, T108), T108, T109)
U13(f26_out1(T111), T108, T109) → f119_out1(T111)

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

(8) UsableRulesProof (EQUIVALENT transformation)

We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R.

(9) Obligation:

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

F125_IN(s(T124), s(T125)) → F125_IN(T124, T125)

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

(10) 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:

• F125_IN(s(T124), s(T125)) → F125_IN(T124, T125)
  The graph contains the following edges 1 > 1, 2 > 2

(11) YES

(12) Obligation:

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

F91_IN(s(T89), s(T90)) → F91_IN(T89, T90)

The TRS R consists of the following rules:

f3_in(T22, T23) → U1(f20_in(T22, T23), T22, T23)
U1(f20_out1(T25), T22, T23) → f3_out1(T25)
f3_in(T108, T109) → U2(f119_in(T108, T109), T108, T109)
U2(f119_out1(T111), T108, T109) → f3_out1(T111)
f24_in(s(T38), s(T39)) → U3(f24_in(T38, T39), s(T38), s(T39))
U3(f24_out1, s(T38), s(T39)) → f24_out1
f24_in(0, s(T46)) → f24_out1
f24_in(0, 0) → f24_out1
f91_in(s(T89), s(T90)) → U4(f91_in(T89, T90), s(T89), s(T90))
U4(f91_out1(X110), s(T89), s(T90)) → f91_out1(X110)
f91_in(0, T95) → f91_out1(T95)
f26_in(0, T53) → f26_out1(T53)
f26_in(s(T60), T61) → U5(f78_in(T60, T61), s(T60), T61)
U5(f78_out1(X58, T63), s(T60), T61) → f26_out1(T63)
f125_in(s(T124), s(T125)) → U6(f125_in(T124, T125), s(T124), s(T125))
U6(f125_out1, s(T124), s(T125)) → f125_out1
f125_in(s(T130), 0) → f125_out1
f82_in(T77, s(T78)) → U7(f91_in(T77, T78), T77, s(T78))
U7(f91_out1(X86), T77, s(T78)) → f82_out1(X86)
f20_in(T22, T23) → U8(f24_in(T22, T23), T22, T23)
U8(f24_out1, T22, T23) → U9(f26_in(T22, T23), T22, T23)
U9(f26_out1(T25), T22, T23) → f20_out1(T25)
f78_in(T60, T61) → U10(f82_in(T60, T61), T60, T61)
U10(f82_out1(T66), T60, T61) → U11(f3_in(s(T60), T66), T60, T61, T66)
U11(f3_out1(T63), T60, T61, T66) → f78_out1(T66, T63)
f119_in(T108, T109) → U12(f125_in(T108, T109), T108, T109)
U12(f125_out1, T108, T109) → U13(f26_in(T109, T108), T108, T109)
U13(f26_out1(T111), T108, T109) → f119_out1(T111)

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

(13) UsableRulesProof (EQUIVALENT transformation)

We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R.

(14) Obligation:

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

F91_IN(s(T89), s(T90)) → F91_IN(T89, T90)

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

(15) 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:

• F91_IN(s(T89), s(T90)) → F91_IN(T89, T90)
  The graph contains the following edges 1 > 1, 2 > 2

(16) YES

(17) Obligation:

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

F24_IN(s(T38), s(T39)) → F24_IN(T38, T39)

The TRS R consists of the following rules:

f3_in(T22, T23) → U1(f20_in(T22, T23), T22, T23)
U1(f20_out1(T25), T22, T23) → f3_out1(T25)
f3_in(T108, T109) → U2(f119_in(T108, T109), T108, T109)
U2(f119_out1(T111), T108, T109) → f3_out1(T111)
f24_in(s(T38), s(T39)) → U3(f24_in(T38, T39), s(T38), s(T39))
U3(f24_out1, s(T38), s(T39)) → f24_out1
f24_in(0, s(T46)) → f24_out1
f24_in(0, 0) → f24_out1
f91_in(s(T89), s(T90)) → U4(f91_in(T89, T90), s(T89), s(T90))
U4(f91_out1(X110), s(T89), s(T90)) → f91_out1(X110)
f91_in(0, T95) → f91_out1(T95)
f26_in(0, T53) → f26_out1(T53)
f26_in(s(T60), T61) → U5(f78_in(T60, T61), s(T60), T61)
U5(f78_out1(X58, T63), s(T60), T61) → f26_out1(T63)
f125_in(s(T124), s(T125)) → U6(f125_in(T124, T125), s(T124), s(T125))
U6(f125_out1, s(T124), s(T125)) → f125_out1
f125_in(s(T130), 0) → f125_out1
f82_in(T77, s(T78)) → U7(f91_in(T77, T78), T77, s(T78))
U7(f91_out1(X86), T77, s(T78)) → f82_out1(X86)
f20_in(T22, T23) → U8(f24_in(T22, T23), T22, T23)
U8(f24_out1, T22, T23) → U9(f26_in(T22, T23), T22, T23)
U9(f26_out1(T25), T22, T23) → f20_out1(T25)
f78_in(T60, T61) → U10(f82_in(T60, T61), T60, T61)
U10(f82_out1(T66), T60, T61) → U11(f3_in(s(T60), T66), T60, T61, T66)
U11(f3_out1(T63), T60, T61, T66) → f78_out1(T66, T63)
f119_in(T108, T109) → U12(f125_in(T108, T109), T108, T109)
U12(f125_out1, T108, T109) → U13(f26_in(T109, T108), T108, T109)
U13(f26_out1(T111), T108, T109) → f119_out1(T111)

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

(18) UsableRulesProof (EQUIVALENT transformation)

We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R.

(19) Obligation:

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

F24_IN(s(T38), s(T39)) → F24_IN(T38, T39)

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

(20) 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:

• F24_IN(s(T38), s(T39)) → F24_IN(T38, T39)
  The graph contains the following edges 1 > 1, 2 > 2

(21) YES

(22) Obligation:

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

F3_IN(T22, T23) → F20_IN(T22, T23)
F20_IN(T22, T23) → U8¹(f24_in(T22, T23), T22, T23)
U8¹(f24_out1, T22, T23) → F26_IN(T22, T23)
F26_IN(s(T60), T61) → F78_IN(T60, T61)
F78_IN(T60, T61) → U10¹(f82_in(T60, T61), T60, T61)
U10¹(f82_out1(T66), T60, T61) → F3_IN(s(T60), T66)
F3_IN(T108, T109) → F119_IN(T108, T109)
F119_IN(T108, T109) → U12¹(f125_in(T108, T109), T108, T109)
U12¹(f125_out1, T108, T109) → F26_IN(T109, T108)

The TRS R consists of the following rules:

f3_in(T22, T23) → U1(f20_in(T22, T23), T22, T23)
U1(f20_out1(T25), T22, T23) → f3_out1(T25)
f3_in(T108, T109) → U2(f119_in(T108, T109), T108, T109)
U2(f119_out1(T111), T108, T109) → f3_out1(T111)
f24_in(s(T38), s(T39)) → U3(f24_in(T38, T39), s(T38), s(T39))
U3(f24_out1, s(T38), s(T39)) → f24_out1
f24_in(0, s(T46)) → f24_out1
f24_in(0, 0) → f24_out1
f91_in(s(T89), s(T90)) → U4(f91_in(T89, T90), s(T89), s(T90))
U4(f91_out1(X110), s(T89), s(T90)) → f91_out1(X110)
f91_in(0, T95) → f91_out1(T95)
f26_in(0, T53) → f26_out1(T53)
f26_in(s(T60), T61) → U5(f78_in(T60, T61), s(T60), T61)
U5(f78_out1(X58, T63), s(T60), T61) → f26_out1(T63)
f125_in(s(T124), s(T125)) → U6(f125_in(T124, T125), s(T124), s(T125))
U6(f125_out1, s(T124), s(T125)) → f125_out1
f125_in(s(T130), 0) → f125_out1
f82_in(T77, s(T78)) → U7(f91_in(T77, T78), T77, s(T78))
U7(f91_out1(X86), T77, s(T78)) → f82_out1(X86)
f20_in(T22, T23) → U8(f24_in(T22, T23), T22, T23)
U8(f24_out1, T22, T23) → U9(f26_in(T22, T23), T22, T23)
U9(f26_out1(T25), T22, T23) → f20_out1(T25)
f78_in(T60, T61) → U10(f82_in(T60, T61), T60, T61)
U10(f82_out1(T66), T60, T61) → U11(f3_in(s(T60), T66), T60, T61, T66)
U11(f3_out1(T63), T60, T61, T66) → f78_out1(T66, T63)
f119_in(T108, T109) → U12(f125_in(T108, T109), T108, T109)
U12(f125_out1, T108, T109) → U13(f26_in(T109, T108), T108, T109)
U13(f26_out1(T111), T108, T109) → f119_out1(T111)

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

(23) QDPOrderProof (EQUIVALENT transformation)

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

The following pairs can be oriented strictly and are deleted.

F78_IN(T60, T61) → U10¹(f82_in(T60, T61), T60, T61)

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

POL(0) = 0   
POL(F119_IN(x₁, x₂)) = x₁ + x₂   
POL(F20_IN(x₁, x₂)) = x₁ + x₂   
POL(F26_IN(x₁, x₂)) = x₁ + x₂   
POL(F3_IN(x₁, x₂)) = x₁ + x₂   
POL(F78_IN(x₁, x₂)) = 1 + x₁ + x₂   
POL(U10¹(x₁, x₂, x₃)) = x₁ + x₂   
POL(U12¹(x₁, x₂, x₃)) = x₂ + x₃   
POL(U3(x₁, x₂, x₃)) = 0   
POL(U4(x₁, x₂, x₃)) = x₁   
POL(U6(x₁, x₂, x₃)) = 0   
POL(U7(x₁, x₂, x₃)) = 1 + x₁   
POL(U8¹(x₁, x₂, x₃)) = x₂ + x₃   
POL(f125_in(x₁, x₂)) = 0   
POL(f125_out1) = 0   
POL(f24_in(x₁, x₂)) = 0   
POL(f24_out1) = 0   
POL(f82_in(x₁, x₂)) = x₂   
POL(f82_out1(x₁)) = 1 + x₁   
POL(f91_in(x₁, x₂)) = x₂   
POL(f91_out1(x₁)) = x₁   
POL(s(x₁)) = 1 + x₁   

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

f82_in(T77, s(T78)) → U7(f91_in(T77, T78), T77, s(T78))
f91_in(s(T89), s(T90)) → U4(f91_in(T89, T90), s(T89), s(T90))
f91_in(0, T95) → f91_out1(T95)
U7(f91_out1(X86), T77, s(T78)) → f82_out1(X86)
U4(f91_out1(X110), s(T89), s(T90)) → f91_out1(X110)

(24) Obligation:

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

F3_IN(T22, T23) → F20_IN(T22, T23)
F20_IN(T22, T23) → U8¹(f24_in(T22, T23), T22, T23)
U8¹(f24_out1, T22, T23) → F26_IN(T22, T23)
F26_IN(s(T60), T61) → F78_IN(T60, T61)
U10¹(f82_out1(T66), T60, T61) → F3_IN(s(T60), T66)
F3_IN(T108, T109) → F119_IN(T108, T109)
F119_IN(T108, T109) → U12¹(f125_in(T108, T109), T108, T109)
U12¹(f125_out1, T108, T109) → F26_IN(T109, T108)

The TRS R consists of the following rules:

f3_in(T22, T23) → U1(f20_in(T22, T23), T22, T23)
U1(f20_out1(T25), T22, T23) → f3_out1(T25)
f3_in(T108, T109) → U2(f119_in(T108, T109), T108, T109)
U2(f119_out1(T111), T108, T109) → f3_out1(T111)
f24_in(s(T38), s(T39)) → U3(f24_in(T38, T39), s(T38), s(T39))
U3(f24_out1, s(T38), s(T39)) → f24_out1
f24_in(0, s(T46)) → f24_out1
f24_in(0, 0) → f24_out1
f91_in(s(T89), s(T90)) → U4(f91_in(T89, T90), s(T89), s(T90))
U4(f91_out1(X110), s(T89), s(T90)) → f91_out1(X110)
f91_in(0, T95) → f91_out1(T95)
f26_in(0, T53) → f26_out1(T53)
f26_in(s(T60), T61) → U5(f78_in(T60, T61), s(T60), T61)
U5(f78_out1(X58, T63), s(T60), T61) → f26_out1(T63)
f125_in(s(T124), s(T125)) → U6(f125_in(T124, T125), s(T124), s(T125))
U6(f125_out1, s(T124), s(T125)) → f125_out1
f125_in(s(T130), 0) → f125_out1
f82_in(T77, s(T78)) → U7(f91_in(T77, T78), T77, s(T78))
U7(f91_out1(X86), T77, s(T78)) → f82_out1(X86)
f20_in(T22, T23) → U8(f24_in(T22, T23), T22, T23)
U8(f24_out1, T22, T23) → U9(f26_in(T22, T23), T22, T23)
U9(f26_out1(T25), T22, T23) → f20_out1(T25)
f78_in(T60, T61) → U10(f82_in(T60, T61), T60, T61)
U10(f82_out1(T66), T60, T61) → U11(f3_in(s(T60), T66), T60, T61, T66)
U11(f3_out1(T63), T60, T61, T66) → f78_out1(T66, T63)
f119_in(T108, T109) → U12(f125_in(T108, T109), T108, T109)
U12(f125_out1, T108, T109) → U13(f26_in(T109, T108), T108, T109)
U13(f26_out1(T111), T108, T109) → f119_out1(T111)

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

(25) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 8 less nodes.

(26) TRUE