Left Termination proof of /tmp/tmpndTWGr/fib

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

0 Prolog

↳1 PrologToPiTRSViaGraphTransformerProof (⇒, 207 ms)

↳2 PiTRS

↳3 DependencyPairsProof (⇔, 495 ms)

↳4 PiDP

↳5 DependencyGraphProof (⇔, 0 ms)

↳6 AND

    ↳7 PiDP

        ↳8 UsableRulesProof (⇔, 0 ms)

        ↳9 PiDP

        ↳10 PiDPToQDPProof (⇒, 22 ms)

        ↳11 QDP

        ↳12 QDPSizeChangeProof (⇔, 0 ms)

        ↳13 YES

    ↳14 PiDP

        ↳15 UsableRulesProof (⇔, 0 ms)

        ↳16 PiDP

        ↳17 PiDPToQDPProof (⇒, 0 ms)

        ↳18 QDP

        ↳19 QDPSizeChangeProof (⇔, 0 ms)

        ↳20 YES

    ↳21 PiDP

        ↳22 UsableRulesProof (⇔, 0 ms)

        ↳23 PiDP

        ↳24 PiDPToQDPProof (⇒, 0 ms)

        ↳25 QDP

        ↳26 QDPSizeChangeProof (⇔, 0 ms)

        ↳27 YES

    ↳28 PiDP

        ↳29 UsableRulesProof (⇔, 0 ms)

        ↳30 PiDP

        ↳31 PiDPToQDPProof (⇒, 0 ms)

        ↳32 QDP

        ↳33 QDPSizeChangeProof (⇔, 0 ms)

        ↳34 YES

    ↳35 PiDP

        ↳36 UsableRulesProof (⇔, 0 ms)

        ↳37 PiDP

        ↳38 PiDPToQDPProof (⇒, 0 ms)

        ↳39 QDP

        ↳40 QDPSizeChangeProof (⇔, 0 ms)

        ↳41 YES

(0) Obligation:

Clauses:

add(0, 0, 0).
add(s(X), Y, s(N)) :- add(X, Y, N).
add(X, s(Y), s(N)) :- add(X, Y, N).
fib(0, 0).
fib(s(0), s(0)).
fib(s(s(X)), N) :- ','(fib(s(X), N1), ','(fib(X, N2), add(N1, N2, N))).

Query: fib(g,a)

(1) PrologToPiTRSViaGraphTransformerProof (SOUND transformation)

Transformed Prolog program to (Pi-)TRS.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
3 [label="A: fib(T1, T2)\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
6 [label="6: fib(T1, T2), SCOPE: 1, CLAUSE: 3 | fib(T1, T2), SCOPE: 1, CLAUSE: 4 | fib(T1, T2), SCOPE: 1, CLAUSE: 5\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: true | fib(0, T2), SCOPE: 1, CLAUSE: 4 | fib(0, T2), SCOPE: 1, CLAUSE: 5\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: fib(T1, T2), SCOPE: 1, CLAUSE: 4 | fib(T1, T2), SCOPE: 1, CLAUSE: 5\n\nT1 is ground\nfib(T1, T2) !=? fib(0, 0)", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: fib(0, T2), SCOPE: 1, CLAUSE: 4 | fib(0, T2), SCOPE: 1, CLAUSE: 5\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: fib(0, T2), SCOPE: 1, CLAUSE: 5\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: true | fib(s(0), T2), SCOPE: 1, CLAUSE: 5\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: fib(T1, T2), SCOPE: 1, CLAUSE: 5\n\nT1 is ground\nfib(T1, T2) !=? fib(0, 0)\nfib(T1, T2) !=? fib(s(0), s(0))", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: fib(s(0), T2), SCOPE: 1, CLAUSE: 5\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
39 [label="39: ','(fib(s(T5), X9), ','(fib(T5, X10), add(X9, X10, T7)))\n\nT5 is ground\nX9 is free\nX10 is free", fontsize=16, style=filled, fillcolor=lightyellow];
41 [label="41: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
43 [label="43: ','(fib(s(T5), X9), ','(fib(T5, X10), add(X9, X10, T7))), SCOPE: 2, CLAUSE: 3 | ','(fib(s(T5), X9), ','(fib(T5, X10), add(X9, X10, T7))), SCOPE: 2, CLAUSE: 4 | ','(fib(s(T5), X9), ','(fib(T5, X10), add(X9, X10, T7))), SCOPE: 2, CLAUSE: 5\n\nT5 is ground\nX9 is free\nX10 is free", fontsize=16, style=filled, fillcolor=lightyellow];
46 [label="46: ','(fib(s(T5), X9), ','(fib(T5, X10), add(X9, X10, T7))), SCOPE: 2, CLAUSE: 4 | ','(fib(s(T5), X9), ','(fib(T5, X10), add(X9, X10, T7))), SCOPE: 2, CLAUSE: 5\n\nT5 is ground\nX9 is free\nX10 is free", fontsize=16, style=filled, fillcolor=lightyellow];
47 [label="47: ','(fib(s(T5), X9), ','(fib(T5, X10), add(X9, X10, T7))), SCOPE: 2, CLAUSE: 4\n\nT5 is ground\nX9 is free\nX10 is free", fontsize=16, style=filled, fillcolor=lightyellow];
49 [label="49: ','(fib(s(T5), X9), ','(fib(T5, X10), add(X9, X10, T7))), SCOPE: 2, CLAUSE: 5\n\nT5 is ground\nX9 is free\nX10 is free", fontsize=16, style=filled, fillcolor=lightyellow];
51 [label="B: ','(fib(0, X10), add(s(0), X10, T7))\n\nX10 is free", fontsize=16, style=filled, fillcolor=lightyellow];
52 [label="52: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
57 [label="K: fib(0, X10)\n\nX10 is free", fontsize=16, style=filled, fillcolor=lightyellow];
58 [label="E: add(s(0), T8, T7)\n\nT8 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
61 [label="61: fib(0, X10), SCOPE: 3, CLAUSE: 3 | fib(0, X10), SCOPE: 3, CLAUSE: 4 | fib(0, X10), SCOPE: 3, CLAUSE: 5\n\nX10 is free", fontsize=16, style=filled, fillcolor=lightyellow];
67 [label="67: fib(0, X10), SCOPE: 3, CLAUSE: 3\n\nX10 is free", fontsize=16, style=filled, fillcolor=lightyellow];
68 [label="68: fib(0, X10), SCOPE: 3, CLAUSE: 4 | fib(0, X10), SCOPE: 3, CLAUSE: 5\n\nX10 is free", fontsize=16, style=filled, fillcolor=lightyellow];
71 [label="71: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
73 [label="73: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
77 [label="77: fib(0, X10), SCOPE: 3, CLAUSE: 5\n\nX10 is free", fontsize=16, style=filled, fillcolor=lightyellow];
78 [label="78: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
86 [label="86: add(s(0), T8, T7), SCOPE: 4, CLAUSE: 0 | add(s(0), T8, T7), SCOPE: 4, CLAUSE: 1 | add(s(0), T8, T7), SCOPE: 4, CLAUSE: 2\n\nT8 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
87 [label="87: add(s(0), T8, T7), SCOPE: 4, CLAUSE: 1 | add(s(0), T8, T7), SCOPE: 4, CLAUSE: 2\n\nT8 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
92 [label="92: add(s(0), T8, T7), SCOPE: 4, CLAUSE: 1\n\nT8 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
93 [label="93: add(s(0), T8, T7), SCOPE: 4, CLAUSE: 2\n\nT8 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
98 [label="D: add(0, T17, T19)\n\nT17 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
101 [label="101: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
103 [label="103: add(0, T17, T19), SCOPE: 5, CLAUSE: 0 | add(0, T17, T19), SCOPE: 5, CLAUSE: 1 | add(0, T17, T19), SCOPE: 5, CLAUSE: 2\n\nT17 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
104 [label="104: add(0, T17, T19), SCOPE: 5, CLAUSE: 0\n\nT17 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
105 [label="105: add(0, T17, T19), SCOPE: 5, CLAUSE: 1 | add(0, T17, T19), SCOPE: 5, CLAUSE: 2\n\nT17 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
107 [label="107: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
109 [label="109: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
110 [label="110: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
112 [label="112: add(0, T17, T19), SCOPE: 5, CLAUSE: 2\n\nT17 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
122 [label="122: add(0, T24, T26)\n\nT24 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
125 [label="125: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
129 [label="129: add(s(0), T31, T33)\n\nT31 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
130 [label="130: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
141 [label="C: ','(','(fib(s(T36), X59), ','(fib(T36, X60), add(X59, X60, X61))), ','(fib(s(T36), X10), add(X61, X10, T7)))\n\nT36 is ground\nX10 is free\nX61 is free\nX59 is free\nX60 is free", fontsize=16, style=filled, fillcolor=lightyellow];
143 [label="143: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
148 [label="F: fib(s(T36), X59)\n\nT36 is ground\nX59 is free", fontsize=16, style=filled, fillcolor=lightyellow];
150 [label="L: ','(','(fib(T36, X60), add(T37, X60, X61)), ','(fib(s(T36), X10), add(X61, X10, T7)))\n\nT36 is ground\nT37 is ground\nX10 is free\nX61 is free\nX60 is free", fontsize=16, style=filled, fillcolor=lightyellow];
152 [label="152: fib(s(T36), X59), SCOPE: 6, CLAUSE: 3 | fib(s(T36), X59), SCOPE: 6, CLAUSE: 4 | fib(s(T36), X59), SCOPE: 6, CLAUSE: 5\n\nT36 is ground\nX59 is free", fontsize=16, style=filled, fillcolor=lightyellow];
154 [label="154: fib(s(T36), X59), SCOPE: 6, CLAUSE: 4 | fib(s(T36), X59), SCOPE: 6, CLAUSE: 5\n\nT36 is ground\nX59 is free", fontsize=16, style=filled, fillcolor=lightyellow];
157 [label="157: fib(s(T36), X59), SCOPE: 6, CLAUSE: 4\n\nT36 is ground\nX59 is free", fontsize=16, style=filled, fillcolor=lightyellow];
158 [label="158: fib(s(T36), X59), SCOPE: 6, CLAUSE: 5\n\nT36 is ground\nX59 is free", fontsize=16, style=filled, fillcolor=lightyellow];
160 [label="160: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
163 [label="163: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
164 [label="164: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
170 [label="G: ','(fib(s(T40), X72), ','(fib(T40, X73), add(X72, X73, X74)))\n\nT40 is ground\nX74 is free\nX72 is free\nX73 is free", fontsize=16, style=filled, fillcolor=lightyellow];
172 [label="172: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
177 [label="177: fib(s(T40), X72)\n\nT40 is ground\nX72 is free", fontsize=16, style=filled, fillcolor=lightyellow];
179 [label="N: ','(fib(T40, X73), add(T41, X73, X74))\n\nT40 is ground\nT41 is ground\nX74 is free\nX73 is free", fontsize=16, style=filled, fillcolor=lightyellow];
201 [label="I: fib(T40, X73)\n\nT40 is ground\nX73 is free", fontsize=16, style=filled, fillcolor=lightyellow];
202 [label="H: add(T41, T42, X74)\n\nT41 is ground\nT42 is ground\nX74 is free", fontsize=16, style=filled, fillcolor=lightyellow];
206 [label="206: fib(T40, X73), SCOPE: 7, CLAUSE: 3 | fib(T40, X73), SCOPE: 7, CLAUSE: 4 | fib(T40, X73), SCOPE: 7, CLAUSE: 5\n\nT40 is ground\nX73 is free", fontsize=16, style=filled, fillcolor=lightyellow];
207 [label="207: fib(T40, X73), SCOPE: 7, CLAUSE: 3\n\nT40 is ground\nX73 is free", fontsize=16, style=filled, fillcolor=lightyellow];
208 [label="208: fib(T40, X73), SCOPE: 7, CLAUSE: 4 | fib(T40, X73), SCOPE: 7, CLAUSE: 5\n\nT40 is ground\nX73 is free", fontsize=16, style=filled, fillcolor=lightyellow];
209 [label="209: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
210 [label="210: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
211 [label="211: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
212 [label="212: fib(T40, X73), SCOPE: 7, CLAUSE: 4\n\nT40 is ground\nX73 is free", fontsize=16, style=filled, fillcolor=lightyellow];
213 [label="213: fib(T40, X73), SCOPE: 7, CLAUSE: 5\n\nT40 is ground\nX73 is free", fontsize=16, style=filled, fillcolor=lightyellow];
214 [label="214: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
215 [label="215: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
216 [label="216: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
227 [label="227: ','(fib(s(T45), X85), ','(fib(T45, X86), add(X85, X86, X87)))\n\nT45 is ground\nX87 is free\nX85 is free\nX86 is free", fontsize=16, style=filled, fillcolor=lightyellow];
228 [label="228: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
231 [label="231: add(T41, T42, X74), SCOPE: 8, CLAUSE: 0 | add(T41, T42, X74), SCOPE: 8, CLAUSE: 1 | add(T41, T42, X74), SCOPE: 8, CLAUSE: 2\n\nT41 is ground\nT42 is ground\nX74 is free", fontsize=16, style=filled, fillcolor=lightyellow];
232 [label="232: add(T41, T42, X74), SCOPE: 8, CLAUSE: 0\n\nT41 is ground\nT42 is ground\nX74 is free", fontsize=16, style=filled, fillcolor=lightyellow];
233 [label="233: add(T41, T42, X74), SCOPE: 8, CLAUSE: 1 | add(T41, T42, X74), SCOPE: 8, CLAUSE: 2\n\nT41 is ground\nT42 is ground\nX74 is free", fontsize=16, style=filled, fillcolor=lightyellow];
234 [label="234: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
235 [label="235: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
236 [label="236: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
243 [label="243: add(T41, T42, X74), SCOPE: 8, CLAUSE: 1\n\nT41 is ground\nT42 is ground\nX74 is free", fontsize=16, style=filled, fillcolor=lightyellow];
244 [label="244: add(T41, T42, X74), SCOPE: 8, CLAUSE: 2\n\nT41 is ground\nT42 is ground\nX74 is free", fontsize=16, style=filled, fillcolor=lightyellow];
251 [label="251: add(T54, T55, X107)\n\nT54 is ground\nT55 is ground\nX107 is free", fontsize=16, style=filled, fillcolor=lightyellow];
252 [label="252: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
257 [label="257: add(T60, T61, X119)\n\nT60 is ground\nT61 is ground\nX119 is free", fontsize=16, style=filled, fillcolor=lightyellow];
258 [label="258: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
261 [label="261: fib(T36, X60)\n\nT36 is ground\nX60 is free", fontsize=16, style=filled, fillcolor=lightyellow];
262 [label="M: ','(add(T37, T62, X61), ','(fib(s(T36), X10), add(X61, X10, T7)))\n\nT36 is ground\nT37 is ground\nT62 is ground\nX10 is free\nX61 is free", fontsize=16, style=filled, fillcolor=lightyellow];
263 [label="263: add(T37, T62, X61)\n\nT37 is ground\nT62 is ground\nX61 is free", fontsize=16, style=filled, fillcolor=lightyellow];
264 [label="O: ','(fib(s(T36), X10), add(T63, X10, T7))\n\nT36 is ground\nT63 is ground\nX10 is free", fontsize=16, style=filled, fillcolor=lightyellow];
265 [label="265: fib(s(T36), X10)\n\nT36 is ground\nX10 is free", fontsize=16, style=filled, fillcolor=lightyellow];
266 [label="J: add(T63, T64, T7)\n\nT63 is ground\nT64 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
267 [label="267: add(T63, T64, T7), SCOPE: 9, CLAUSE: 0 | add(T63, T64, T7), SCOPE: 9, CLAUSE: 1 | add(T63, T64, T7), SCOPE: 9, CLAUSE: 2\n\nT63 is ground\nT64 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
268 [label="268: add(T63, T64, T7), SCOPE: 9, CLAUSE: 0\n\nT63 is ground\nT64 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
269 [label="269: add(T63, T64, T7), SCOPE: 9, CLAUSE: 1 | add(T63, T64, T7), SCOPE: 9, CLAUSE: 2\n\nT63 is ground\nT64 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
270 [label="270: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
271 [label="271: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
272 [label="272: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
273 [label="273: add(T63, T64, T7), SCOPE: 9, CLAUSE: 1\n\nT63 is ground\nT64 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
274 [label="274: add(T63, T64, T7), SCOPE: 9, CLAUSE: 2\n\nT63 is ground\nT64 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
275 [label="275: add(T77, T78, T80)\n\nT77 is ground\nT78 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
276 [label="276: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
277 [label="277: add(T87, T88, T90)\n\nT87 is ground\nT88 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
278 [label="278: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
279 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
3 -> 279 [arrowhead = none ];
279 -> 6;

280 [label="EVAL with clause\nfib(0, 0).\nand substitutionT1 -> 0,\nT2 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
6 -> 280 [arrowhead = none ];
280 -> 9;

281 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
6 -> 281 [arrowhead = none ];
281 -> 11;

282 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
9 -> 282 [arrowhead = none ];
282 -> 15;

283 [label="EVAL with clause\nfib(s(0), s(0)).\nand substitutionT1 -> s(0),\nT2 -> s(0)", fontsize=14, style = filled, fillcolor = lightblue];
11 -> 283 [arrowhead = none ];
283 -> 28;

284 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
11 -> 284 [arrowhead = none ];
284 -> 30;

285 [label="BACKTRACK\nfor clause: fib(s(0), s(0))because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
15 -> 285 [arrowhead = none ];
285 -> 19;

286 [label="BACKTRACK\nfor clause: fib(s(s(X)), N) :- ','(fib(s(X), N1), ','(fib(X, N2), add(N1, N2, N)))because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
19 -> 286 [arrowhead = none ];
286 -> 23;

287 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
28 -> 287 [arrowhead = none ];
287 -> 32;

288 [label="EVAL with clause\nfib(s(s(X7)), X8) :- ','(fib(s(X7), X9), ','(fib(X7, X10), add(X9, X10, X8))).\nand substitutionX7 -> T5,\nT1 -> s(s(T5)),\nT2 -> T7,\nX8 -> T7,\nT6 -> T7", fontsize=14, style = filled, fillcolor = lightblue];
30 -> 288 [arrowhead = none ];
288 -> 39;

289 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
30 -> 289 [arrowhead = none ];
289 -> 41;

290 [label="BACKTRACK\nfor clause: fib(s(s(X)), N) :- ','(fib(s(X), N1), ','(fib(X, N2), add(N1, N2, N)))because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
32 -> 290 [arrowhead = none ];
290 -> 34;

291 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
39 -> 291 [arrowhead = none ];
291 -> 43;

292 [label="BACKTRACK\nfor clause: fib(0, 0)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
43 -> 292 [arrowhead = none ];
292 -> 46;

293 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
46 -> 293 [arrowhead = none ];
293 -> 47;

294 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
46 -> 294 [arrowhead = none ];
294 -> 49;

295 [label="EVAL with clause\nfib(s(0), s(0)).\nand substitutionT5 -> 0,\nX9 -> s(0)", fontsize=14, style = filled, fillcolor = lightblue];
47 -> 295 [arrowhead = none ];
295 -> 51;

296 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
47 -> 296 [arrowhead = none ];
296 -> 52;

297 [label="EVAL with clause\nfib(s(s(X57)), X58) :- ','(fib(s(X57), X59), ','(fib(X57, X60), add(X59, X60, X58))).\nand substitutionX57 -> T36,\nT5 -> s(T36),\nX9 -> X61,\nX58 -> X61", fontsize=14, style = filled, fillcolor = lightblue];
49 -> 297 [arrowhead = none ];
297 -> 141;

298 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
49 -> 298 [arrowhead = none ];
298 -> 143;

299 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
51 -> 299 [arrowhead = none ];
299 -> 57;

300 [label="SPLIT 2\nnew knowledge:\nT8 is ground\nreplacements:X10 -> T8", fontsize=14, style = filled, fillcolor = violet];
51 -> 300 [arrowhead = none ];
300 -> 58;

301 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
57 -> 301 [arrowhead = none ];
301 -> 61;

302 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
58 -> 302 [arrowhead = none ];
302 -> 86;

303 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
61 -> 303 [arrowhead = none ];
303 -> 67;

304 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
61 -> 304 [arrowhead = none ];
304 -> 68;

305 [label="ONLY EVAL with clause\nfib(0, 0).\nand substitutionX10 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
67 -> 305 [arrowhead = none ];
305 -> 71;

306 [label="BACKTRACK\nfor clause: fib(s(0), s(0))because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
68 -> 306 [arrowhead = none ];
306 -> 77;

307 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
71 -> 307 [arrowhead = none ];
307 -> 73;

308 [label="BACKTRACK\nfor clause: fib(s(s(X)), N) :- ','(fib(s(X), N1), ','(fib(X, N2), add(N1, N2, N)))because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
77 -> 308 [arrowhead = none ];
308 -> 78;

309 [label="BACKTRACK\nfor clause: add(0, 0, 0)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
86 -> 309 [arrowhead = none ];
309 -> 87;

310 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
87 -> 310 [arrowhead = none ];
310 -> 92;

311 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
87 -> 311 [arrowhead = none ];
311 -> 93;

312 [label="EVAL with clause\nadd(s(X25), X26, s(X27)) :- add(X25, X26, X27).\nand substitutionX25 -> 0,\nT8 -> T17,\nX26 -> T17,\nX27 -> T19,\nT7 -> s(T19),\nT18 -> T19", fontsize=14, style = filled, fillcolor = lightblue];
92 -> 312 [arrowhead = none ];
312 -> 98;

313 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
92 -> 313 [arrowhead = none ];
313 -> 101;

314 [label="EVAL with clause\nadd(X46, s(X47), s(X48)) :- add(X46, X47, X48).\nand substitutionX46 -> s(0),\nX47 -> T31,\nT8 -> s(T31),\nX48 -> T33,\nT7 -> s(T33),\nT32 -> T33", fontsize=14, style = filled, fillcolor = lightblue];
93 -> 314 [arrowhead = none ];
314 -> 129;

315 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
93 -> 315 [arrowhead = none ];
315 -> 130;

316 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
98 -> 316 [arrowhead = none ];
316 -> 103;

317 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
103 -> 317 [arrowhead = none ];
317 -> 104;

318 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
103 -> 318 [arrowhead = none ];
318 -> 105;

319 [label="EVAL with clause\nadd(0, 0, 0).\nand substitutionT17 -> 0,\nT19 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
104 -> 319 [arrowhead = none ];
319 -> 107;

320 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
104 -> 320 [arrowhead = none ];
320 -> 109;

321 [label="BACKTRACK\nfor clause: add(s(X), Y, s(N)) :- add(X, Y, N)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
105 -> 321 [arrowhead = none ];
321 -> 112;

322 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
107 -> 322 [arrowhead = none ];
322 -> 110;

323 [label="EVAL with clause\nadd(X37, s(X38), s(X39)) :- add(X37, X38, X39).\nand substitutionX37 -> 0,\nX38 -> T24,\nT17 -> s(T24),\nX39 -> T26,\nT19 -> s(T26),\nT25 -> T26", fontsize=14, style = filled, fillcolor = lightblue];
112 -> 323 [arrowhead = none ];
323 -> 122;

324 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
112 -> 324 [arrowhead = none ];
324 -> 125;

325 [label="INSTANCE with matching:\nT17 -> T24\nT19 -> T26", fontsize=14, style = filled, fillcolor = lightgrey];
122 -> 325 [arrowhead = none , style=dashed];
325 -> 98[style=dashed];

326 [label="INSTANCE with matching:\nT8 -> T31\nT7 -> T33", fontsize=14, style = filled, fillcolor = lightgrey];
129 -> 326 [arrowhead = none , style=dashed];
326 -> 58[style=dashed];

327 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
141 -> 327 [arrowhead = none ];
327 -> 148;

328 [label="SPLIT 2\nnew knowledge:\nT36 is ground\nT37 is ground\nreplacements:X59 -> T37", fontsize=14, style = filled, fillcolor = violet];
141 -> 328 [arrowhead = none ];
328 -> 150;

329 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
148 -> 329 [arrowhead = none ];
329 -> 152;

330 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
150 -> 330 [arrowhead = none ];
330 -> 261;

331 [label="SPLIT 2\nnew knowledge:\nT36 is ground\nT62 is ground\nreplacements:X60 -> T62", fontsize=14, style = filled, fillcolor = violet];
150 -> 331 [arrowhead = none ];
331 -> 262;

332 [label="BACKTRACK\nfor clause: fib(0, 0)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
152 -> 332 [arrowhead = none ];
332 -> 154;

333 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
154 -> 333 [arrowhead = none ];
333 -> 157;

334 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
154 -> 334 [arrowhead = none ];
334 -> 158;

335 [label="EVAL with clause\nfib(s(0), s(0)).\nand substitutionT36 -> 0,\nX59 -> s(0)", fontsize=14, style = filled, fillcolor = lightblue];
157 -> 335 [arrowhead = none ];
335 -> 160;

336 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
157 -> 336 [arrowhead = none ];
336 -> 163;

337 [label="EVAL with clause\nfib(s(s(X70)), X71) :- ','(fib(s(X70), X72), ','(fib(X70, X73), add(X72, X73, X71))).\nand substitutionX70 -> T40,\nT36 -> s(T40),\nX59 -> X74,\nX71 -> X74", fontsize=14, style = filled, fillcolor = lightblue];
158 -> 337 [arrowhead = none ];
337 -> 170;

338 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
158 -> 338 [arrowhead = none ];
338 -> 172;

339 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
160 -> 339 [arrowhead = none ];
339 -> 164;

340 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
170 -> 340 [arrowhead = none ];
340 -> 177;

341 [label="SPLIT 2\nnew knowledge:\nT40 is ground\nT41 is ground\nreplacements:X72 -> T41", fontsize=14, style = filled, fillcolor = violet];
170 -> 341 [arrowhead = none ];
341 -> 179;

342 [label="INSTANCE with matching:\nT36 -> T40\nX59 -> X72", fontsize=14, style = filled, fillcolor = lightgrey];
177 -> 342 [arrowhead = none , style=dashed];
342 -> 148[style=dashed];

343 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
179 -> 343 [arrowhead = none ];
343 -> 201;

344 [label="SPLIT 2\nnew knowledge:\nT40 is ground\nT42 is ground\nreplacements:X73 -> T42", fontsize=14, style = filled, fillcolor = violet];
179 -> 344 [arrowhead = none ];
344 -> 202;

345 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
201 -> 345 [arrowhead = none ];
345 -> 206;

346 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
202 -> 346 [arrowhead = none ];
346 -> 231;

347 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
206 -> 347 [arrowhead = none ];
347 -> 207;

348 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
206 -> 348 [arrowhead = none ];
348 -> 208;

349 [label="EVAL with clause\nfib(0, 0).\nand substitutionT40 -> 0,\nX73 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
207 -> 349 [arrowhead = none ];
349 -> 209;

350 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
207 -> 350 [arrowhead = none ];
350 -> 210;

351 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
208 -> 351 [arrowhead = none ];
351 -> 212;

352 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
208 -> 352 [arrowhead = none ];
352 -> 213;

353 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
209 -> 353 [arrowhead = none ];
353 -> 211;

354 [label="EVAL with clause\nfib(s(0), s(0)).\nand substitutionT40 -> s(0),\nX73 -> s(0)", fontsize=14, style = filled, fillcolor = lightblue];
212 -> 354 [arrowhead = none ];
354 -> 214;

355 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
212 -> 355 [arrowhead = none ];
355 -> 215;

356 [label="EVAL with clause\nfib(s(s(X83)), X84) :- ','(fib(s(X83), X85), ','(fib(X83, X86), add(X85, X86, X84))).\nand substitutionX83 -> T45,\nT40 -> s(s(T45)),\nX73 -> X87,\nX84 -> X87", fontsize=14, style = filled, fillcolor = lightblue];
213 -> 356 [arrowhead = none ];
356 -> 227;

357 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
213 -> 357 [arrowhead = none ];
357 -> 228;

358 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
214 -> 358 [arrowhead = none ];
358 -> 216;

359 [label="INSTANCE with matching:\nT40 -> T45\nX72 -> X85\nX73 -> X86\nX74 -> X87", fontsize=14, style = filled, fillcolor = lightgrey];
227 -> 359 [arrowhead = none , style=dashed];
359 -> 170[style=dashed];

360 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
231 -> 360 [arrowhead = none ];
360 -> 232;

361 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
231 -> 361 [arrowhead = none ];
361 -> 233;

362 [label="EVAL with clause\nadd(0, 0, 0).\nand substitutionT41 -> 0,\nT42 -> 0,\nX74 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
232 -> 362 [arrowhead = none ];
362 -> 234;

363 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
232 -> 363 [arrowhead = none ];
363 -> 235;

364 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
233 -> 364 [arrowhead = none ];
364 -> 243;

365 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
233 -> 365 [arrowhead = none ];
365 -> 244;

366 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
234 -> 366 [arrowhead = none ];
366 -> 236;

367 [label="EVAL with clause\nadd(s(X104), X105, s(X106)) :- add(X104, X105, X106).\nand substitutionX104 -> T54,\nT41 -> s(T54),\nT42 -> T55,\nX105 -> T55,\nX106 -> X107,\nX74 -> s(X107)", fontsize=14, style = filled, fillcolor = lightblue];
243 -> 367 [arrowhead = none ];
367 -> 251;

368 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
243 -> 368 [arrowhead = none ];
368 -> 252;

369 [label="EVAL with clause\nadd(X116, s(X117), s(X118)) :- add(X116, X117, X118).\nand substitutionT41 -> T60,\nX116 -> T60,\nX117 -> T61,\nT42 -> s(T61),\nX118 -> X119,\nX74 -> s(X119)", fontsize=14, style = filled, fillcolor = lightblue];
244 -> 369 [arrowhead = none ];
369 -> 257;

370 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
244 -> 370 [arrowhead = none ];
370 -> 258;

371 [label="INSTANCE with matching:\nT41 -> T54\nT42 -> T55\nX74 -> X107", fontsize=14, style = filled, fillcolor = lightgrey];
251 -> 371 [arrowhead = none , style=dashed];
371 -> 202[style=dashed];

372 [label="INSTANCE with matching:\nT41 -> T60\nT42 -> T61\nX74 -> X119", fontsize=14, style = filled, fillcolor = lightgrey];
257 -> 372 [arrowhead = none , style=dashed];
372 -> 202[style=dashed];

373 [label="INSTANCE with matching:\nT40 -> T36\nX73 -> X60", fontsize=14, style = filled, fillcolor = lightgrey];
261 -> 373 [arrowhead = none , style=dashed];
373 -> 201[style=dashed];

374 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
262 -> 374 [arrowhead = none ];
374 -> 263;

375 [label="SPLIT 2\nnew knowledge:\nT37 is ground\nT62 is ground\nT63 is ground\nreplacements:X61 -> T63", fontsize=14, style = filled, fillcolor = violet];
262 -> 375 [arrowhead = none ];
375 -> 264;

376 [label="INSTANCE with matching:\nT41 -> T37\nT42 -> T62\nX74 -> X61", fontsize=14, style = filled, fillcolor = lightgrey];
263 -> 376 [arrowhead = none , style=dashed];
376 -> 202[style=dashed];

377 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
264 -> 377 [arrowhead = none ];
377 -> 265;

378 [label="SPLIT 2\nnew knowledge:\nT36 is ground\nT64 is ground\nreplacements:X10 -> T64", fontsize=14, style = filled, fillcolor = violet];
264 -> 378 [arrowhead = none ];
378 -> 266;

379 [label="INSTANCE with matching:\nX59 -> X10", fontsize=14, style = filled, fillcolor = lightgrey];
265 -> 379 [arrowhead = none , style=dashed];
379 -> 148[style=dashed];

380 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
266 -> 380 [arrowhead = none ];
380 -> 267;

381 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
267 -> 381 [arrowhead = none ];
381 -> 268;

382 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
267 -> 382 [arrowhead = none ];
382 -> 269;

383 [label="EVAL with clause\nadd(0, 0, 0).\nand substitutionT63 -> 0,\nT64 -> 0,\nT7 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
268 -> 383 [arrowhead = none ];
383 -> 270;

384 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
268 -> 384 [arrowhead = none ];
384 -> 271;

385 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
269 -> 385 [arrowhead = none ];
385 -> 273;

386 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
269 -> 386 [arrowhead = none ];
386 -> 274;

387 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
270 -> 387 [arrowhead = none ];
387 -> 272;

388 [label="EVAL with clause\nadd(s(X132), X133, s(X134)) :- add(X132, X133, X134).\nand substitutionX132 -> T77,\nT63 -> s(T77),\nT64 -> T78,\nX133 -> T78,\nX134 -> T80,\nT7 -> s(T80),\nT79 -> T80", fontsize=14, style = filled, fillcolor = lightblue];
273 -> 388 [arrowhead = none ];
388 -> 275;

389 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
273 -> 389 [arrowhead = none ];
389 -> 276;

390 [label="EVAL with clause\nadd(X141, s(X142), s(X143)) :- add(X141, X142, X143).\nand substitutionT63 -> T87,\nX141 -> T87,\nX142 -> T88,\nT64 -> s(T88),\nX143 -> T90,\nT7 -> s(T90),\nT89 -> T90", fontsize=14, style = filled, fillcolor = lightblue];
274 -> 390 [arrowhead = none ];
390 -> 277;

391 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
274 -> 391 [arrowhead = none ];
391 -> 278;

392 [label="INSTANCE with matching:\nT63 -> T77\nT64 -> T78\nT7 -> T80", fontsize=14, style = filled, fillcolor = lightgrey];
275 -> 392 [arrowhead = none , style=dashed];
392 -> 266[style=dashed];

393 [label="INSTANCE with matching:\nT63 -> T87\nT64 -> T88\nT7 -> T90", fontsize=14, style = filled, fillcolor = lightgrey];
277 -> 393 [arrowhead = none , style=dashed];
393 -> 266[style=dashed];

}

(2) Obligation:

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

fibA_in_ga(0, 0) → fibA_out_ga(0, 0)
fibA_in_ga(s(0), s(0)) → fibA_out_ga(s(0), s(0))
fibA_in_ga(s(s(0)), T7) → U1_ga(T7, pB_in_aa(X10, T7))
pB_in_aa(T8, T7) → U12_aa(T8, T7, fibK_in_a(T8))
fibK_in_a(0) → fibK_out_a(0)
U12_aa(T8, T7, fibK_out_a(T8)) → U13_aa(T8, T7, addE_in_ga(T8, T7))
addE_in_ga(T17, s(T19)) → U4_ga(T17, T19, addD_in_ga(T17, T19))
addD_in_ga(0, 0) → addD_out_ga(0, 0)
addD_in_ga(s(T24), s(T26)) → U3_ga(T24, T26, addD_in_ga(T24, T26))
U3_ga(T24, T26, addD_out_ga(T24, T26)) → addD_out_ga(s(T24), s(T26))
U4_ga(T17, T19, addD_out_ga(T17, T19)) → addE_out_ga(T17, s(T19))
addE_in_ga(s(T31), s(T33)) → U5_ga(T31, T33, addE_in_ga(T31, T33))
U5_ga(T31, T33, addE_out_ga(T31, T33)) → addE_out_ga(s(T31), s(T33))
U13_aa(T8, T7, addE_out_ga(T8, T7)) → pB_out_aa(T8, T7)
U1_ga(T7, pB_out_aa(X10, T7)) → fibA_out_ga(s(s(0)), T7)
fibA_in_ga(s(s(s(T36))), T7) → U2_ga(T36, T7, pC_in_gaaaaa(T36, X59, X60, X61, X10, T7))
pC_in_gaaaaa(T36, T37, X60, X61, X10, T7) → U14_gaaaaa(T36, T37, X60, X61, X10, T7, fibF_in_ga(T36, T37))
fibF_in_ga(0, s(0)) → fibF_out_ga(0, s(0))
fibF_in_ga(s(T40), X74) → U6_ga(T40, X74, pG_in_gaaa(T40, X72, X73, X74))
pG_in_gaaa(T40, T41, X73, X74) → U18_gaaa(T40, T41, X73, X74, fibF_in_ga(T40, T41))
U18_gaaa(T40, T41, X73, X74, fibF_out_ga(T40, T41)) → U19_gaaa(T40, T41, X73, X74, pN_in_gaga(T40, X73, T41, X74))
pN_in_gaga(T40, T42, T41, X74) → U20_gaga(T40, T42, T41, X74, fibI_in_ga(T40, T42))
fibI_in_ga(0, 0) → fibI_out_ga(0, 0)
fibI_in_ga(s(0), s(0)) → fibI_out_ga(s(0), s(0))
fibI_in_ga(s(s(T45)), X87) → U9_ga(T45, X87, pG_in_gaaa(T45, X85, X86, X87))
U9_ga(T45, X87, pG_out_gaaa(T45, X85, X86, X87)) → fibI_out_ga(s(s(T45)), X87)
U20_gaga(T40, T42, T41, X74, fibI_out_ga(T40, T42)) → U21_gaga(T40, T42, T41, X74, addH_in_gga(T41, T42, X74))
addH_in_gga(0, 0, 0) → addH_out_gga(0, 0, 0)
addH_in_gga(s(T54), T55, s(X107)) → U7_gga(T54, T55, X107, addH_in_gga(T54, T55, X107))
addH_in_gga(T60, s(T61), s(X119)) → U8_gga(T60, T61, X119, addH_in_gga(T60, T61, X119))
U8_gga(T60, T61, X119, addH_out_gga(T60, T61, X119)) → addH_out_gga(T60, s(T61), s(X119))
U7_gga(T54, T55, X107, addH_out_gga(T54, T55, X107)) → addH_out_gga(s(T54), T55, s(X107))
U21_gaga(T40, T42, T41, X74, addH_out_gga(T41, T42, X74)) → pN_out_gaga(T40, T42, T41, X74)
U19_gaaa(T40, T41, X73, X74, pN_out_gaga(T40, X73, T41, X74)) → pG_out_gaaa(T40, T41, X73, X74)
U6_ga(T40, X74, pG_out_gaaa(T40, X72, X73, X74)) → fibF_out_ga(s(T40), X74)
U14_gaaaaa(T36, T37, X60, X61, X10, T7, fibF_out_ga(T36, T37)) → U15_gaaaaa(T36, T37, X60, X61, X10, T7, pL_in_gagaaa(T36, X60, T37, X61, X10, T7))
pL_in_gagaaa(T36, T62, T37, X61, X10, T7) → U16_gagaaa(T36, T62, T37, X61, X10, T7, fibI_in_ga(T36, T62))
U16_gagaaa(T36, T62, T37, X61, X10, T7, fibI_out_ga(T36, T62)) → U17_gagaaa(T36, T62, T37, X61, X10, T7, pM_in_ggagaa(T37, T62, X61, T36, X10, T7))
pM_in_ggagaa(T37, T62, T63, T36, X10, T7) → U22_ggagaa(T37, T62, T63, T36, X10, T7, addH_in_gga(T37, T62, T63))
U22_ggagaa(T37, T62, T63, T36, X10, T7, addH_out_gga(T37, T62, T63)) → U23_ggagaa(T37, T62, T63, T36, X10, T7, pO_in_gaga(T36, X10, T63, T7))
pO_in_gaga(T36, T64, T63, T7) → U24_gaga(T36, T64, T63, T7, fibF_in_ga(T36, T64))
U24_gaga(T36, T64, T63, T7, fibF_out_ga(T36, T64)) → U25_gaga(T36, T64, T63, T7, addJ_in_gga(T63, T64, T7))
addJ_in_gga(0, 0, 0) → addJ_out_gga(0, 0, 0)
addJ_in_gga(s(T77), T78, s(T80)) → U10_gga(T77, T78, T80, addJ_in_gga(T77, T78, T80))
addJ_in_gga(T87, s(T88), s(T90)) → U11_gga(T87, T88, T90, addJ_in_gga(T87, T88, T90))
U11_gga(T87, T88, T90, addJ_out_gga(T87, T88, T90)) → addJ_out_gga(T87, s(T88), s(T90))
U10_gga(T77, T78, T80, addJ_out_gga(T77, T78, T80)) → addJ_out_gga(s(T77), T78, s(T80))
U25_gaga(T36, T64, T63, T7, addJ_out_gga(T63, T64, T7)) → pO_out_gaga(T36, T64, T63, T7)
U23_ggagaa(T37, T62, T63, T36, X10, T7, pO_out_gaga(T36, X10, T63, T7)) → pM_out_ggagaa(T37, T62, T63, T36, X10, T7)
U17_gagaaa(T36, T62, T37, X61, X10, T7, pM_out_ggagaa(T37, T62, X61, T36, X10, T7)) → pL_out_gagaaa(T36, T62, T37, X61, X10, T7)
U15_gaaaaa(T36, T37, X60, X61, X10, T7, pL_out_gagaaa(T36, X60, T37, X61, X10, T7)) → pC_out_gaaaaa(T36, T37, X60, X61, X10, T7)
U2_ga(T36, T7, pC_out_gaaaaa(T36, X59, X60, X61, X10, T7)) → fibA_out_ga(s(s(s(T36))), T7)

(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:

FIBA_IN_GA(s(s(0)), T7) → U1_GA(T7, pB_in_aa(X10, T7))
FIBA_IN_GA(s(s(0)), T7) → PB_IN_AA(X10, T7)
PB_IN_AA(T8, T7) → U12_AA(T8, T7, fibK_in_a(T8))
PB_IN_AA(T8, T7) → FIBK_IN_A(T8)
U12_AA(T8, T7, fibK_out_a(T8)) → U13_AA(T8, T7, addE_in_ga(T8, T7))
U12_AA(T8, T7, fibK_out_a(T8)) → ADDE_IN_GA(T8, T7)
ADDE_IN_GA(T17, s(T19)) → U4_GA(T17, T19, addD_in_ga(T17, T19))
ADDE_IN_GA(T17, s(T19)) → ADDD_IN_GA(T17, T19)
ADDD_IN_GA(s(T24), s(T26)) → U3_GA(T24, T26, addD_in_ga(T24, T26))
ADDD_IN_GA(s(T24), s(T26)) → ADDD_IN_GA(T24, T26)
ADDE_IN_GA(s(T31), s(T33)) → U5_GA(T31, T33, addE_in_ga(T31, T33))
ADDE_IN_GA(s(T31), s(T33)) → ADDE_IN_GA(T31, T33)
FIBA_IN_GA(s(s(s(T36))), T7) → U2_GA(T36, T7, pC_in_gaaaaa(T36, X59, X60, X61, X10, T7))
FIBA_IN_GA(s(s(s(T36))), T7) → PC_IN_GAAAAA(T36, X59, X60, X61, X10, T7)
PC_IN_GAAAAA(T36, T37, X60, X61, X10, T7) → U14_GAAAAA(T36, T37, X60, X61, X10, T7, fibF_in_ga(T36, T37))
PC_IN_GAAAAA(T36, T37, X60, X61, X10, T7) → FIBF_IN_GA(T36, T37)
FIBF_IN_GA(s(T40), X74) → U6_GA(T40, X74, pG_in_gaaa(T40, X72, X73, X74))
FIBF_IN_GA(s(T40), X74) → PG_IN_GAAA(T40, X72, X73, X74)
PG_IN_GAAA(T40, T41, X73, X74) → U18_GAAA(T40, T41, X73, X74, fibF_in_ga(T40, T41))
PG_IN_GAAA(T40, T41, X73, X74) → FIBF_IN_GA(T40, T41)
U18_GAAA(T40, T41, X73, X74, fibF_out_ga(T40, T41)) → U19_GAAA(T40, T41, X73, X74, pN_in_gaga(T40, X73, T41, X74))
U18_GAAA(T40, T41, X73, X74, fibF_out_ga(T40, T41)) → PN_IN_GAGA(T40, X73, T41, X74)
PN_IN_GAGA(T40, T42, T41, X74) → U20_GAGA(T40, T42, T41, X74, fibI_in_ga(T40, T42))
PN_IN_GAGA(T40, T42, T41, X74) → FIBI_IN_GA(T40, T42)
FIBI_IN_GA(s(s(T45)), X87) → U9_GA(T45, X87, pG_in_gaaa(T45, X85, X86, X87))
FIBI_IN_GA(s(s(T45)), X87) → PG_IN_GAAA(T45, X85, X86, X87)
U20_GAGA(T40, T42, T41, X74, fibI_out_ga(T40, T42)) → U21_GAGA(T40, T42, T41, X74, addH_in_gga(T41, T42, X74))
U20_GAGA(T40, T42, T41, X74, fibI_out_ga(T40, T42)) → ADDH_IN_GGA(T41, T42, X74)
ADDH_IN_GGA(s(T54), T55, s(X107)) → U7_GGA(T54, T55, X107, addH_in_gga(T54, T55, X107))
ADDH_IN_GGA(s(T54), T55, s(X107)) → ADDH_IN_GGA(T54, T55, X107)
ADDH_IN_GGA(T60, s(T61), s(X119)) → U8_GGA(T60, T61, X119, addH_in_gga(T60, T61, X119))
ADDH_IN_GGA(T60, s(T61), s(X119)) → ADDH_IN_GGA(T60, T61, X119)
U14_GAAAAA(T36, T37, X60, X61, X10, T7, fibF_out_ga(T36, T37)) → U15_GAAAAA(T36, T37, X60, X61, X10, T7, pL_in_gagaaa(T36, X60, T37, X61, X10, T7))
U14_GAAAAA(T36, T37, X60, X61, X10, T7, fibF_out_ga(T36, T37)) → PL_IN_GAGAAA(T36, X60, T37, X61, X10, T7)
PL_IN_GAGAAA(T36, T62, T37, X61, X10, T7) → U16_GAGAAA(T36, T62, T37, X61, X10, T7, fibI_in_ga(T36, T62))
PL_IN_GAGAAA(T36, T62, T37, X61, X10, T7) → FIBI_IN_GA(T36, T62)
U16_GAGAAA(T36, T62, T37, X61, X10, T7, fibI_out_ga(T36, T62)) → U17_GAGAAA(T36, T62, T37, X61, X10, T7, pM_in_ggagaa(T37, T62, X61, T36, X10, T7))
U16_GAGAAA(T36, T62, T37, X61, X10, T7, fibI_out_ga(T36, T62)) → PM_IN_GGAGAA(T37, T62, X61, T36, X10, T7)
PM_IN_GGAGAA(T37, T62, T63, T36, X10, T7) → U22_GGAGAA(T37, T62, T63, T36, X10, T7, addH_in_gga(T37, T62, T63))
PM_IN_GGAGAA(T37, T62, T63, T36, X10, T7) → ADDH_IN_GGA(T37, T62, T63)
U22_GGAGAA(T37, T62, T63, T36, X10, T7, addH_out_gga(T37, T62, T63)) → U23_GGAGAA(T37, T62, T63, T36, X10, T7, pO_in_gaga(T36, X10, T63, T7))
U22_GGAGAA(T37, T62, T63, T36, X10, T7, addH_out_gga(T37, T62, T63)) → PO_IN_GAGA(T36, X10, T63, T7)
PO_IN_GAGA(T36, T64, T63, T7) → U24_GAGA(T36, T64, T63, T7, fibF_in_ga(T36, T64))
PO_IN_GAGA(T36, T64, T63, T7) → FIBF_IN_GA(T36, T64)
U24_GAGA(T36, T64, T63, T7, fibF_out_ga(T36, T64)) → U25_GAGA(T36, T64, T63, T7, addJ_in_gga(T63, T64, T7))
U24_GAGA(T36, T64, T63, T7, fibF_out_ga(T36, T64)) → ADDJ_IN_GGA(T63, T64, T7)
ADDJ_IN_GGA(s(T77), T78, s(T80)) → U10_GGA(T77, T78, T80, addJ_in_gga(T77, T78, T80))
ADDJ_IN_GGA(s(T77), T78, s(T80)) → ADDJ_IN_GGA(T77, T78, T80)
ADDJ_IN_GGA(T87, s(T88), s(T90)) → U11_GGA(T87, T88, T90, addJ_in_gga(T87, T88, T90))
ADDJ_IN_GGA(T87, s(T88), s(T90)) → ADDJ_IN_GGA(T87, T88, T90)

The TRS R consists of the following rules:

fibA_in_ga(0, 0) → fibA_out_ga(0, 0)
fibA_in_ga(s(0), s(0)) → fibA_out_ga(s(0), s(0))
fibA_in_ga(s(s(0)), T7) → U1_ga(T7, pB_in_aa(X10, T7))
pB_in_aa(T8, T7) → U12_aa(T8, T7, fibK_in_a(T8))
fibK_in_a(0) → fibK_out_a(0)
U12_aa(T8, T7, fibK_out_a(T8)) → U13_aa(T8, T7, addE_in_ga(T8, T7))
addE_in_ga(T17, s(T19)) → U4_ga(T17, T19, addD_in_ga(T17, T19))
addD_in_ga(0, 0) → addD_out_ga(0, 0)
addD_in_ga(s(T24), s(T26)) → U3_ga(T24, T26, addD_in_ga(T24, T26))
U3_ga(T24, T26, addD_out_ga(T24, T26)) → addD_out_ga(s(T24), s(T26))
U4_ga(T17, T19, addD_out_ga(T17, T19)) → addE_out_ga(T17, s(T19))
addE_in_ga(s(T31), s(T33)) → U5_ga(T31, T33, addE_in_ga(T31, T33))
U5_ga(T31, T33, addE_out_ga(T31, T33)) → addE_out_ga(s(T31), s(T33))
U13_aa(T8, T7, addE_out_ga(T8, T7)) → pB_out_aa(T8, T7)
U1_ga(T7, pB_out_aa(X10, T7)) → fibA_out_ga(s(s(0)), T7)
fibA_in_ga(s(s(s(T36))), T7) → U2_ga(T36, T7, pC_in_gaaaaa(T36, X59, X60, X61, X10, T7))
pC_in_gaaaaa(T36, T37, X60, X61, X10, T7) → U14_gaaaaa(T36, T37, X60, X61, X10, T7, fibF_in_ga(T36, T37))
fibF_in_ga(0, s(0)) → fibF_out_ga(0, s(0))
fibF_in_ga(s(T40), X74) → U6_ga(T40, X74, pG_in_gaaa(T40, X72, X73, X74))
pG_in_gaaa(T40, T41, X73, X74) → U18_gaaa(T40, T41, X73, X74, fibF_in_ga(T40, T41))
U18_gaaa(T40, T41, X73, X74, fibF_out_ga(T40, T41)) → U19_gaaa(T40, T41, X73, X74, pN_in_gaga(T40, X73, T41, X74))
pN_in_gaga(T40, T42, T41, X74) → U20_gaga(T40, T42, T41, X74, fibI_in_ga(T40, T42))
fibI_in_ga(0, 0) → fibI_out_ga(0, 0)
fibI_in_ga(s(0), s(0)) → fibI_out_ga(s(0), s(0))
fibI_in_ga(s(s(T45)), X87) → U9_ga(T45, X87, pG_in_gaaa(T45, X85, X86, X87))
U9_ga(T45, X87, pG_out_gaaa(T45, X85, X86, X87)) → fibI_out_ga(s(s(T45)), X87)
U20_gaga(T40, T42, T41, X74, fibI_out_ga(T40, T42)) → U21_gaga(T40, T42, T41, X74, addH_in_gga(T41, T42, X74))
addH_in_gga(0, 0, 0) → addH_out_gga(0, 0, 0)
addH_in_gga(s(T54), T55, s(X107)) → U7_gga(T54, T55, X107, addH_in_gga(T54, T55, X107))
addH_in_gga(T60, s(T61), s(X119)) → U8_gga(T60, T61, X119, addH_in_gga(T60, T61, X119))
U8_gga(T60, T61, X119, addH_out_gga(T60, T61, X119)) → addH_out_gga(T60, s(T61), s(X119))
U7_gga(T54, T55, X107, addH_out_gga(T54, T55, X107)) → addH_out_gga(s(T54), T55, s(X107))
U21_gaga(T40, T42, T41, X74, addH_out_gga(T41, T42, X74)) → pN_out_gaga(T40, T42, T41, X74)
U19_gaaa(T40, T41, X73, X74, pN_out_gaga(T40, X73, T41, X74)) → pG_out_gaaa(T40, T41, X73, X74)
U6_ga(T40, X74, pG_out_gaaa(T40, X72, X73, X74)) → fibF_out_ga(s(T40), X74)
U14_gaaaaa(T36, T37, X60, X61, X10, T7, fibF_out_ga(T36, T37)) → U15_gaaaaa(T36, T37, X60, X61, X10, T7, pL_in_gagaaa(T36, X60, T37, X61, X10, T7))
pL_in_gagaaa(T36, T62, T37, X61, X10, T7) → U16_gagaaa(T36, T62, T37, X61, X10, T7, fibI_in_ga(T36, T62))
U16_gagaaa(T36, T62, T37, X61, X10, T7, fibI_out_ga(T36, T62)) → U17_gagaaa(T36, T62, T37, X61, X10, T7, pM_in_ggagaa(T37, T62, X61, T36, X10, T7))
pM_in_ggagaa(T37, T62, T63, T36, X10, T7) → U22_ggagaa(T37, T62, T63, T36, X10, T7, addH_in_gga(T37, T62, T63))
U22_ggagaa(T37, T62, T63, T36, X10, T7, addH_out_gga(T37, T62, T63)) → U23_ggagaa(T37, T62, T63, T36, X10, T7, pO_in_gaga(T36, X10, T63, T7))
pO_in_gaga(T36, T64, T63, T7) → U24_gaga(T36, T64, T63, T7, fibF_in_ga(T36, T64))
U24_gaga(T36, T64, T63, T7, fibF_out_ga(T36, T64)) → U25_gaga(T36, T64, T63, T7, addJ_in_gga(T63, T64, T7))
addJ_in_gga(0, 0, 0) → addJ_out_gga(0, 0, 0)
addJ_in_gga(s(T77), T78, s(T80)) → U10_gga(T77, T78, T80, addJ_in_gga(T77, T78, T80))
addJ_in_gga(T87, s(T88), s(T90)) → U11_gga(T87, T88, T90, addJ_in_gga(T87, T88, T90))
U11_gga(T87, T88, T90, addJ_out_gga(T87, T88, T90)) → addJ_out_gga(T87, s(T88), s(T90))
U10_gga(T77, T78, T80, addJ_out_gga(T77, T78, T80)) → addJ_out_gga(s(T77), T78, s(T80))
U25_gaga(T36, T64, T63, T7, addJ_out_gga(T63, T64, T7)) → pO_out_gaga(T36, T64, T63, T7)
U23_ggagaa(T37, T62, T63, T36, X10, T7, pO_out_gaga(T36, X10, T63, T7)) → pM_out_ggagaa(T37, T62, T63, T36, X10, T7)
U17_gagaaa(T36, T62, T37, X61, X10, T7, pM_out_ggagaa(T37, T62, X61, T36, X10, T7)) → pL_out_gagaaa(T36, T62, T37, X61, X10, T7)
U15_gaaaaa(T36, T37, X60, X61, X10, T7, pL_out_gagaaa(T36, X60, T37, X61, X10, T7)) → pC_out_gaaaaa(T36, T37, X60, X61, X10, T7)
U2_ga(T36, T7, pC_out_gaaaaa(T36, X59, X60, X61, X10, T7)) → fibA_out_ga(s(s(s(T36))), T7)

The argument filtering Pi contains the following mapping:
fibA_in_ga(x1, x2) = fibA_in_ga(x1)
0 = 0
fibA_out_ga(x1, x2) = fibA_out_ga(x1, x2)
s(x1) = s(x1)
U1_ga(x1, x2) = U1_ga(x2)
pB_in_aa(x1, x2) = pB_in_aa
U12_aa(x1, x2, x3) = U12_aa(x3)
fibK_in_a(x1) = fibK_in_a
fibK_out_a(x1) = fibK_out_a(x1)
U13_aa(x1, x2, x3) = U13_aa(x1, x3)
addE_in_ga(x1, x2) = addE_in_ga(x1)
U4_ga(x1, x2, x3) = U4_ga(x1, x3)
addD_in_ga(x1, x2) = addD_in_ga(x1)
addD_out_ga(x1, x2) = addD_out_ga(x1, x2)
U3_ga(x1, x2, x3) = U3_ga(x1, x3)
addE_out_ga(x1, x2) = addE_out_ga(x1, x2)
U5_ga(x1, x2, x3) = U5_ga(x1, x3)
pB_out_aa(x1, x2) = pB_out_aa(x1, x2)
U2_ga(x1, x2, x3) = U2_ga(x1, x3)
pC_in_gaaaaa(x1, x2, x3, x4, x5, x6) = pC_in_gaaaaa(x1)
U14_gaaaaa(x1, x2, x3, x4, x5, x6, x7) = U14_gaaaaa(x1, x7)
fibF_in_ga(x1, x2) = fibF_in_ga(x1)
fibF_out_ga(x1, x2) = fibF_out_ga(x1, x2)
U6_ga(x1, x2, x3) = U6_ga(x1, x3)
pG_in_gaaa(x1, x2, x3, x4) = pG_in_gaaa(x1)
U18_gaaa(x1, x2, x3, x4, x5) = U18_gaaa(x1, x5)
U19_gaaa(x1, x2, x3, x4, x5) = U19_gaaa(x1, x2, x5)
pN_in_gaga(x1, x2, x3, x4) = pN_in_gaga(x1, x3)
U20_gaga(x1, x2, x3, x4, x5) = U20_gaga(x1, x3, x5)
fibI_in_ga(x1, x2) = fibI_in_ga(x1)
fibI_out_ga(x1, x2) = fibI_out_ga(x1, x2)
U9_ga(x1, x2, x3) = U9_ga(x1, x3)
pG_out_gaaa(x1, x2, x3, x4) = pG_out_gaaa(x1, x2, x3, x4)
U21_gaga(x1, x2, x3, x4, x5) = U21_gaga(x1, x2, x3, x5)
addH_in_gga(x1, x2, x3) = addH_in_gga(x1, x2)
addH_out_gga(x1, x2, x3) = addH_out_gga(x1, x2, x3)
U7_gga(x1, x2, x3, x4) = U7_gga(x1, x2, x4)
U8_gga(x1, x2, x3, x4) = U8_gga(x1, x2, x4)
pN_out_gaga(x1, x2, x3, x4) = pN_out_gaga(x1, x2, x3, x4)
U15_gaaaaa(x1, x2, x3, x4, x5, x6, x7) = U15_gaaaaa(x1, x2, x7)
pL_in_gagaaa(x1, x2, x3, x4, x5, x6) = pL_in_gagaaa(x1, x3)
U16_gagaaa(x1, x2, x3, x4, x5, x6, x7) = U16_gagaaa(x1, x3, x7)
U17_gagaaa(x1, x2, x3, x4, x5, x6, x7) = U17_gagaaa(x1, x2, x3, x7)
pM_in_ggagaa(x1, x2, x3, x4, x5, x6) = pM_in_ggagaa(x1, x2, x4)
U22_ggagaa(x1, x2, x3, x4, x5, x6, x7) = U22_ggagaa(x1, x2, x4, x7)
U23_ggagaa(x1, x2, x3, x4, x5, x6, x7) = U23_ggagaa(x1, x2, x3, x4, x7)
pO_in_gaga(x1, x2, x3, x4) = pO_in_gaga(x1, x3)
U24_gaga(x1, x2, x3, x4, x5) = U24_gaga(x1, x3, x5)
U25_gaga(x1, x2, x3, x4, x5) = U25_gaga(x1, x2, x3, x5)
addJ_in_gga(x1, x2, x3) = addJ_in_gga(x1, x2)
addJ_out_gga(x1, x2, x3) = addJ_out_gga(x1, x2, x3)
U10_gga(x1, x2, x3, x4) = U10_gga(x1, x2, x4)
U11_gga(x1, x2, x3, x4) = U11_gga(x1, x2, x4)
pO_out_gaga(x1, x2, x3, x4) = pO_out_gaga(x1, x2, x3, x4)
pM_out_ggagaa(x1, x2, x3, x4, x5, x6) = pM_out_ggagaa(x1, x2, x3, x4, x5, x6)
pL_out_gagaaa(x1, x2, x3, x4, x5, x6) = pL_out_gagaaa(x1, x2, x3, x4, x5, x6)
pC_out_gaaaaa(x1, x2, x3, x4, x5, x6) = pC_out_gaaaaa(x1, x2, x3, x4, x5, x6)
FIBA_IN_GA(x1, x2) = FIBA_IN_GA(x1)
U1_GA(x1, x2) = U1_GA(x2)
PB_IN_AA(x1, x2) = PB_IN_AA
U12_AA(x1, x2, x3) = U12_AA(x3)
FIBK_IN_A(x1) = FIBK_IN_A
U13_AA(x1, x2, x3) = U13_AA(x1, x3)
ADDE_IN_GA(x1, x2) = ADDE_IN_GA(x1)
U4_GA(x1, x2, x3) = U4_GA(x1, x3)
ADDD_IN_GA(x1, x2) = ADDD_IN_GA(x1)
U3_GA(x1, x2, x3) = U3_GA(x1, x3)
U5_GA(x1, x2, x3) = U5_GA(x1, x3)
U2_GA(x1, x2, x3) = U2_GA(x1, x3)
PC_IN_GAAAAA(x1, x2, x3, x4, x5, x6) = PC_IN_GAAAAA(x1)
U14_GAAAAA(x1, x2, x3, x4, x5, x6, x7) = U14_GAAAAA(x1, x7)
FIBF_IN_GA(x1, x2) = FIBF_IN_GA(x1)
U6_GA(x1, x2, x3) = U6_GA(x1, x3)
PG_IN_GAAA(x1, x2, x3, x4) = PG_IN_GAAA(x1)
U18_GAAA(x1, x2, x3, x4, x5) = U18_GAAA(x1, x5)
U19_GAAA(x1, x2, x3, x4, x5) = U19_GAAA(x1, x2, x5)
PN_IN_GAGA(x1, x2, x3, x4) = PN_IN_GAGA(x1, x3)
U20_GAGA(x1, x2, x3, x4, x5) = U20_GAGA(x1, x3, x5)
FIBI_IN_GA(x1, x2) = FIBI_IN_GA(x1)
U9_GA(x1, x2, x3) = U9_GA(x1, x3)
U21_GAGA(x1, x2, x3, x4, x5) = U21_GAGA(x1, x2, x3, x5)
ADDH_IN_GGA(x1, x2, x3) = ADDH_IN_GGA(x1, x2)
U7_GGA(x1, x2, x3, x4) = U7_GGA(x1, x2, x4)
U8_GGA(x1, x2, x3, x4) = U8_GGA(x1, x2, x4)
U15_GAAAAA(x1, x2, x3, x4, x5, x6, x7) = U15_GAAAAA(x1, x2, x7)
PL_IN_GAGAAA(x1, x2, x3, x4, x5, x6) = PL_IN_GAGAAA(x1, x3)
U16_GAGAAA(x1, x2, x3, x4, x5, x6, x7) = U16_GAGAAA(x1, x3, x7)
U17_GAGAAA(x1, x2, x3, x4, x5, x6, x7) = U17_GAGAAA(x1, x2, x3, x7)
PM_IN_GGAGAA(x1, x2, x3, x4, x5, x6) = PM_IN_GGAGAA(x1, x2, x4)
U22_GGAGAA(x1, x2, x3, x4, x5, x6, x7) = U22_GGAGAA(x1, x2, x4, x7)
U23_GGAGAA(x1, x2, x3, x4, x5, x6, x7) = U23_GGAGAA(x1, x2, x3, x4, x7)
PO_IN_GAGA(x1, x2, x3, x4) = PO_IN_GAGA(x1, x3)
U24_GAGA(x1, x2, x3, x4, x5) = U24_GAGA(x1, x3, x5)
U25_GAGA(x1, x2, x3, x4, x5) = U25_GAGA(x1, x2, x3, x5)
ADDJ_IN_GGA(x1, x2, x3) = ADDJ_IN_GGA(x1, x2)
U10_GGA(x1, x2, x3, x4) = U10_GGA(x1, x2, x4)
U11_GGA(x1, x2, x3, x4) = U11_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:

FIBA_IN_GA(s(s(0)), T7) → U1_GA(T7, pB_in_aa(X10, T7))
FIBA_IN_GA(s(s(0)), T7) → PB_IN_AA(X10, T7)
PB_IN_AA(T8, T7) → U12_AA(T8, T7, fibK_in_a(T8))
PB_IN_AA(T8, T7) → FIBK_IN_A(T8)
U12_AA(T8, T7, fibK_out_a(T8)) → U13_AA(T8, T7, addE_in_ga(T8, T7))
U12_AA(T8, T7, fibK_out_a(T8)) → ADDE_IN_GA(T8, T7)
ADDE_IN_GA(T17, s(T19)) → U4_GA(T17, T19, addD_in_ga(T17, T19))
ADDE_IN_GA(T17, s(T19)) → ADDD_IN_GA(T17, T19)
ADDD_IN_GA(s(T24), s(T26)) → U3_GA(T24, T26, addD_in_ga(T24, T26))
ADDD_IN_GA(s(T24), s(T26)) → ADDD_IN_GA(T24, T26)
ADDE_IN_GA(s(T31), s(T33)) → U5_GA(T31, T33, addE_in_ga(T31, T33))
ADDE_IN_GA(s(T31), s(T33)) → ADDE_IN_GA(T31, T33)
FIBA_IN_GA(s(s(s(T36))), T7) → U2_GA(T36, T7, pC_in_gaaaaa(T36, X59, X60, X61, X10, T7))
FIBA_IN_GA(s(s(s(T36))), T7) → PC_IN_GAAAAA(T36, X59, X60, X61, X10, T7)
PC_IN_GAAAAA(T36, T37, X60, X61, X10, T7) → U14_GAAAAA(T36, T37, X60, X61, X10, T7, fibF_in_ga(T36, T37))
PC_IN_GAAAAA(T36, T37, X60, X61, X10, T7) → FIBF_IN_GA(T36, T37)
FIBF_IN_GA(s(T40), X74) → U6_GA(T40, X74, pG_in_gaaa(T40, X72, X73, X74))
FIBF_IN_GA(s(T40), X74) → PG_IN_GAAA(T40, X72, X73, X74)
PG_IN_GAAA(T40, T41, X73, X74) → U18_GAAA(T40, T41, X73, X74, fibF_in_ga(T40, T41))
PG_IN_GAAA(T40, T41, X73, X74) → FIBF_IN_GA(T40, T41)
U18_GAAA(T40, T41, X73, X74, fibF_out_ga(T40, T41)) → U19_GAAA(T40, T41, X73, X74, pN_in_gaga(T40, X73, T41, X74))
U18_GAAA(T40, T41, X73, X74, fibF_out_ga(T40, T41)) → PN_IN_GAGA(T40, X73, T41, X74)
PN_IN_GAGA(T40, T42, T41, X74) → U20_GAGA(T40, T42, T41, X74, fibI_in_ga(T40, T42))
PN_IN_GAGA(T40, T42, T41, X74) → FIBI_IN_GA(T40, T42)
FIBI_IN_GA(s(s(T45)), X87) → U9_GA(T45, X87, pG_in_gaaa(T45, X85, X86, X87))
FIBI_IN_GA(s(s(T45)), X87) → PG_IN_GAAA(T45, X85, X86, X87)
U20_GAGA(T40, T42, T41, X74, fibI_out_ga(T40, T42)) → U21_GAGA(T40, T42, T41, X74, addH_in_gga(T41, T42, X74))
U20_GAGA(T40, T42, T41, X74, fibI_out_ga(T40, T42)) → ADDH_IN_GGA(T41, T42, X74)
ADDH_IN_GGA(s(T54), T55, s(X107)) → U7_GGA(T54, T55, X107, addH_in_gga(T54, T55, X107))
ADDH_IN_GGA(s(T54), T55, s(X107)) → ADDH_IN_GGA(T54, T55, X107)
ADDH_IN_GGA(T60, s(T61), s(X119)) → U8_GGA(T60, T61, X119, addH_in_gga(T60, T61, X119))
ADDH_IN_GGA(T60, s(T61), s(X119)) → ADDH_IN_GGA(T60, T61, X119)
U14_GAAAAA(T36, T37, X60, X61, X10, T7, fibF_out_ga(T36, T37)) → U15_GAAAAA(T36, T37, X60, X61, X10, T7, pL_in_gagaaa(T36, X60, T37, X61, X10, T7))
U14_GAAAAA(T36, T37, X60, X61, X10, T7, fibF_out_ga(T36, T37)) → PL_IN_GAGAAA(T36, X60, T37, X61, X10, T7)
PL_IN_GAGAAA(T36, T62, T37, X61, X10, T7) → U16_GAGAAA(T36, T62, T37, X61, X10, T7, fibI_in_ga(T36, T62))
PL_IN_GAGAAA(T36, T62, T37, X61, X10, T7) → FIBI_IN_GA(T36, T62)
U16_GAGAAA(T36, T62, T37, X61, X10, T7, fibI_out_ga(T36, T62)) → U17_GAGAAA(T36, T62, T37, X61, X10, T7, pM_in_ggagaa(T37, T62, X61, T36, X10, T7))
U16_GAGAAA(T36, T62, T37, X61, X10, T7, fibI_out_ga(T36, T62)) → PM_IN_GGAGAA(T37, T62, X61, T36, X10, T7)
PM_IN_GGAGAA(T37, T62, T63, T36, X10, T7) → U22_GGAGAA(T37, T62, T63, T36, X10, T7, addH_in_gga(T37, T62, T63))
PM_IN_GGAGAA(T37, T62, T63, T36, X10, T7) → ADDH_IN_GGA(T37, T62, T63)
U22_GGAGAA(T37, T62, T63, T36, X10, T7, addH_out_gga(T37, T62, T63)) → U23_GGAGAA(T37, T62, T63, T36, X10, T7, pO_in_gaga(T36, X10, T63, T7))
U22_GGAGAA(T37, T62, T63, T36, X10, T7, addH_out_gga(T37, T62, T63)) → PO_IN_GAGA(T36, X10, T63, T7)
PO_IN_GAGA(T36, T64, T63, T7) → U24_GAGA(T36, T64, T63, T7, fibF_in_ga(T36, T64))
PO_IN_GAGA(T36, T64, T63, T7) → FIBF_IN_GA(T36, T64)
U24_GAGA(T36, T64, T63, T7, fibF_out_ga(T36, T64)) → U25_GAGA(T36, T64, T63, T7, addJ_in_gga(T63, T64, T7))
U24_GAGA(T36, T64, T63, T7, fibF_out_ga(T36, T64)) → ADDJ_IN_GGA(T63, T64, T7)
ADDJ_IN_GGA(s(T77), T78, s(T80)) → U10_GGA(T77, T78, T80, addJ_in_gga(T77, T78, T80))
ADDJ_IN_GGA(s(T77), T78, s(T80)) → ADDJ_IN_GGA(T77, T78, T80)
ADDJ_IN_GGA(T87, s(T88), s(T90)) → U11_GGA(T87, T88, T90, addJ_in_gga(T87, T88, T90))
ADDJ_IN_GGA(T87, s(T88), s(T90)) → ADDJ_IN_GGA(T87, T88, T90)

The TRS R consists of the following rules:

fibA_in_ga(0, 0) → fibA_out_ga(0, 0)
fibA_in_ga(s(0), s(0)) → fibA_out_ga(s(0), s(0))
fibA_in_ga(s(s(0)), T7) → U1_ga(T7, pB_in_aa(X10, T7))
pB_in_aa(T8, T7) → U12_aa(T8, T7, fibK_in_a(T8))
fibK_in_a(0) → fibK_out_a(0)
U12_aa(T8, T7, fibK_out_a(T8)) → U13_aa(T8, T7, addE_in_ga(T8, T7))
addE_in_ga(T17, s(T19)) → U4_ga(T17, T19, addD_in_ga(T17, T19))
addD_in_ga(0, 0) → addD_out_ga(0, 0)
addD_in_ga(s(T24), s(T26)) → U3_ga(T24, T26, addD_in_ga(T24, T26))
U3_ga(T24, T26, addD_out_ga(T24, T26)) → addD_out_ga(s(T24), s(T26))
U4_ga(T17, T19, addD_out_ga(T17, T19)) → addE_out_ga(T17, s(T19))
addE_in_ga(s(T31), s(T33)) → U5_ga(T31, T33, addE_in_ga(T31, T33))
U5_ga(T31, T33, addE_out_ga(T31, T33)) → addE_out_ga(s(T31), s(T33))
U13_aa(T8, T7, addE_out_ga(T8, T7)) → pB_out_aa(T8, T7)
U1_ga(T7, pB_out_aa(X10, T7)) → fibA_out_ga(s(s(0)), T7)
fibA_in_ga(s(s(s(T36))), T7) → U2_ga(T36, T7, pC_in_gaaaaa(T36, X59, X60, X61, X10, T7))
pC_in_gaaaaa(T36, T37, X60, X61, X10, T7) → U14_gaaaaa(T36, T37, X60, X61, X10, T7, fibF_in_ga(T36, T37))
fibF_in_ga(0, s(0)) → fibF_out_ga(0, s(0))
fibF_in_ga(s(T40), X74) → U6_ga(T40, X74, pG_in_gaaa(T40, X72, X73, X74))
pG_in_gaaa(T40, T41, X73, X74) → U18_gaaa(T40, T41, X73, X74, fibF_in_ga(T40, T41))
U18_gaaa(T40, T41, X73, X74, fibF_out_ga(T40, T41)) → U19_gaaa(T40, T41, X73, X74, pN_in_gaga(T40, X73, T41, X74))
pN_in_gaga(T40, T42, T41, X74) → U20_gaga(T40, T42, T41, X74, fibI_in_ga(T40, T42))
fibI_in_ga(0, 0) → fibI_out_ga(0, 0)
fibI_in_ga(s(0), s(0)) → fibI_out_ga(s(0), s(0))
fibI_in_ga(s(s(T45)), X87) → U9_ga(T45, X87, pG_in_gaaa(T45, X85, X86, X87))
U9_ga(T45, X87, pG_out_gaaa(T45, X85, X86, X87)) → fibI_out_ga(s(s(T45)), X87)
U20_gaga(T40, T42, T41, X74, fibI_out_ga(T40, T42)) → U21_gaga(T40, T42, T41, X74, addH_in_gga(T41, T42, X74))
addH_in_gga(0, 0, 0) → addH_out_gga(0, 0, 0)
addH_in_gga(s(T54), T55, s(X107)) → U7_gga(T54, T55, X107, addH_in_gga(T54, T55, X107))
addH_in_gga(T60, s(T61), s(X119)) → U8_gga(T60, T61, X119, addH_in_gga(T60, T61, X119))
U8_gga(T60, T61, X119, addH_out_gga(T60, T61, X119)) → addH_out_gga(T60, s(T61), s(X119))
U7_gga(T54, T55, X107, addH_out_gga(T54, T55, X107)) → addH_out_gga(s(T54), T55, s(X107))
U21_gaga(T40, T42, T41, X74, addH_out_gga(T41, T42, X74)) → pN_out_gaga(T40, T42, T41, X74)
U19_gaaa(T40, T41, X73, X74, pN_out_gaga(T40, X73, T41, X74)) → pG_out_gaaa(T40, T41, X73, X74)
U6_ga(T40, X74, pG_out_gaaa(T40, X72, X73, X74)) → fibF_out_ga(s(T40), X74)
U14_gaaaaa(T36, T37, X60, X61, X10, T7, fibF_out_ga(T36, T37)) → U15_gaaaaa(T36, T37, X60, X61, X10, T7, pL_in_gagaaa(T36, X60, T37, X61, X10, T7))
pL_in_gagaaa(T36, T62, T37, X61, X10, T7) → U16_gagaaa(T36, T62, T37, X61, X10, T7, fibI_in_ga(T36, T62))
U16_gagaaa(T36, T62, T37, X61, X10, T7, fibI_out_ga(T36, T62)) → U17_gagaaa(T36, T62, T37, X61, X10, T7, pM_in_ggagaa(T37, T62, X61, T36, X10, T7))
pM_in_ggagaa(T37, T62, T63, T36, X10, T7) → U22_ggagaa(T37, T62, T63, T36, X10, T7, addH_in_gga(T37, T62, T63))
U22_ggagaa(T37, T62, T63, T36, X10, T7, addH_out_gga(T37, T62, T63)) → U23_ggagaa(T37, T62, T63, T36, X10, T7, pO_in_gaga(T36, X10, T63, T7))
pO_in_gaga(T36, T64, T63, T7) → U24_gaga(T36, T64, T63, T7, fibF_in_ga(T36, T64))
U24_gaga(T36, T64, T63, T7, fibF_out_ga(T36, T64)) → U25_gaga(T36, T64, T63, T7, addJ_in_gga(T63, T64, T7))
addJ_in_gga(0, 0, 0) → addJ_out_gga(0, 0, 0)
addJ_in_gga(s(T77), T78, s(T80)) → U10_gga(T77, T78, T80, addJ_in_gga(T77, T78, T80))
addJ_in_gga(T87, s(T88), s(T90)) → U11_gga(T87, T88, T90, addJ_in_gga(T87, T88, T90))
U11_gga(T87, T88, T90, addJ_out_gga(T87, T88, T90)) → addJ_out_gga(T87, s(T88), s(T90))
U10_gga(T77, T78, T80, addJ_out_gga(T77, T78, T80)) → addJ_out_gga(s(T77), T78, s(T80))
U25_gaga(T36, T64, T63, T7, addJ_out_gga(T63, T64, T7)) → pO_out_gaga(T36, T64, T63, T7)
U23_ggagaa(T37, T62, T63, T36, X10, T7, pO_out_gaga(T36, X10, T63, T7)) → pM_out_ggagaa(T37, T62, T63, T36, X10, T7)
U17_gagaaa(T36, T62, T37, X61, X10, T7, pM_out_ggagaa(T37, T62, X61, T36, X10, T7)) → pL_out_gagaaa(T36, T62, T37, X61, X10, T7)
U15_gaaaaa(T36, T37, X60, X61, X10, T7, pL_out_gagaaa(T36, X60, T37, X61, X10, T7)) → pC_out_gaaaaa(T36, T37, X60, X61, X10, T7)
U2_ga(T36, T7, pC_out_gaaaaa(T36, X59, X60, X61, X10, T7)) → fibA_out_ga(s(s(s(T36))), T7)

(5) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 5 SCCs with 38 less nodes.

(6) Complex Obligation (AND)

(7) Obligation:

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

ADDJ_IN_GGA(T87, s(T88), s(T90)) → ADDJ_IN_GGA(T87, T88, T90)
ADDJ_IN_GGA(s(T77), T78, s(T80)) → ADDJ_IN_GGA(T77, T78, T80)

The TRS R consists of the following rules:

fibA_in_ga(0, 0) → fibA_out_ga(0, 0)
fibA_in_ga(s(0), s(0)) → fibA_out_ga(s(0), s(0))
fibA_in_ga(s(s(0)), T7) → U1_ga(T7, pB_in_aa(X10, T7))
pB_in_aa(T8, T7) → U12_aa(T8, T7, fibK_in_a(T8))
fibK_in_a(0) → fibK_out_a(0)
U12_aa(T8, T7, fibK_out_a(T8)) → U13_aa(T8, T7, addE_in_ga(T8, T7))
addE_in_ga(T17, s(T19)) → U4_ga(T17, T19, addD_in_ga(T17, T19))
addD_in_ga(0, 0) → addD_out_ga(0, 0)
addD_in_ga(s(T24), s(T26)) → U3_ga(T24, T26, addD_in_ga(T24, T26))
U3_ga(T24, T26, addD_out_ga(T24, T26)) → addD_out_ga(s(T24), s(T26))
U4_ga(T17, T19, addD_out_ga(T17, T19)) → addE_out_ga(T17, s(T19))
addE_in_ga(s(T31), s(T33)) → U5_ga(T31, T33, addE_in_ga(T31, T33))
U5_ga(T31, T33, addE_out_ga(T31, T33)) → addE_out_ga(s(T31), s(T33))
U13_aa(T8, T7, addE_out_ga(T8, T7)) → pB_out_aa(T8, T7)
U1_ga(T7, pB_out_aa(X10, T7)) → fibA_out_ga(s(s(0)), T7)
fibA_in_ga(s(s(s(T36))), T7) → U2_ga(T36, T7, pC_in_gaaaaa(T36, X59, X60, X61, X10, T7))
pC_in_gaaaaa(T36, T37, X60, X61, X10, T7) → U14_gaaaaa(T36, T37, X60, X61, X10, T7, fibF_in_ga(T36, T37))
fibF_in_ga(0, s(0)) → fibF_out_ga(0, s(0))
fibF_in_ga(s(T40), X74) → U6_ga(T40, X74, pG_in_gaaa(T40, X72, X73, X74))
pG_in_gaaa(T40, T41, X73, X74) → U18_gaaa(T40, T41, X73, X74, fibF_in_ga(T40, T41))
U18_gaaa(T40, T41, X73, X74, fibF_out_ga(T40, T41)) → U19_gaaa(T40, T41, X73, X74, pN_in_gaga(T40, X73, T41, X74))
pN_in_gaga(T40, T42, T41, X74) → U20_gaga(T40, T42, T41, X74, fibI_in_ga(T40, T42))
fibI_in_ga(0, 0) → fibI_out_ga(0, 0)
fibI_in_ga(s(0), s(0)) → fibI_out_ga(s(0), s(0))
fibI_in_ga(s(s(T45)), X87) → U9_ga(T45, X87, pG_in_gaaa(T45, X85, X86, X87))
U9_ga(T45, X87, pG_out_gaaa(T45, X85, X86, X87)) → fibI_out_ga(s(s(T45)), X87)
U20_gaga(T40, T42, T41, X74, fibI_out_ga(T40, T42)) → U21_gaga(T40, T42, T41, X74, addH_in_gga(T41, T42, X74))
addH_in_gga(0, 0, 0) → addH_out_gga(0, 0, 0)
addH_in_gga(s(T54), T55, s(X107)) → U7_gga(T54, T55, X107, addH_in_gga(T54, T55, X107))
addH_in_gga(T60, s(T61), s(X119)) → U8_gga(T60, T61, X119, addH_in_gga(T60, T61, X119))
U8_gga(T60, T61, X119, addH_out_gga(T60, T61, X119)) → addH_out_gga(T60, s(T61), s(X119))
U7_gga(T54, T55, X107, addH_out_gga(T54, T55, X107)) → addH_out_gga(s(T54), T55, s(X107))
U21_gaga(T40, T42, T41, X74, addH_out_gga(T41, T42, X74)) → pN_out_gaga(T40, T42, T41, X74)
U19_gaaa(T40, T41, X73, X74, pN_out_gaga(T40, X73, T41, X74)) → pG_out_gaaa(T40, T41, X73, X74)
U6_ga(T40, X74, pG_out_gaaa(T40, X72, X73, X74)) → fibF_out_ga(s(T40), X74)
U14_gaaaaa(T36, T37, X60, X61, X10, T7, fibF_out_ga(T36, T37)) → U15_gaaaaa(T36, T37, X60, X61, X10, T7, pL_in_gagaaa(T36, X60, T37, X61, X10, T7))
pL_in_gagaaa(T36, T62, T37, X61, X10, T7) → U16_gagaaa(T36, T62, T37, X61, X10, T7, fibI_in_ga(T36, T62))
U16_gagaaa(T36, T62, T37, X61, X10, T7, fibI_out_ga(T36, T62)) → U17_gagaaa(T36, T62, T37, X61, X10, T7, pM_in_ggagaa(T37, T62, X61, T36, X10, T7))
pM_in_ggagaa(T37, T62, T63, T36, X10, T7) → U22_ggagaa(T37, T62, T63, T36, X10, T7, addH_in_gga(T37, T62, T63))
U22_ggagaa(T37, T62, T63, T36, X10, T7, addH_out_gga(T37, T62, T63)) → U23_ggagaa(T37, T62, T63, T36, X10, T7, pO_in_gaga(T36, X10, T63, T7))
pO_in_gaga(T36, T64, T63, T7) → U24_gaga(T36, T64, T63, T7, fibF_in_ga(T36, T64))
U24_gaga(T36, T64, T63, T7, fibF_out_ga(T36, T64)) → U25_gaga(T36, T64, T63, T7, addJ_in_gga(T63, T64, T7))
addJ_in_gga(0, 0, 0) → addJ_out_gga(0, 0, 0)
addJ_in_gga(s(T77), T78, s(T80)) → U10_gga(T77, T78, T80, addJ_in_gga(T77, T78, T80))
addJ_in_gga(T87, s(T88), s(T90)) → U11_gga(T87, T88, T90, addJ_in_gga(T87, T88, T90))
U11_gga(T87, T88, T90, addJ_out_gga(T87, T88, T90)) → addJ_out_gga(T87, s(T88), s(T90))
U10_gga(T77, T78, T80, addJ_out_gga(T77, T78, T80)) → addJ_out_gga(s(T77), T78, s(T80))
U25_gaga(T36, T64, T63, T7, addJ_out_gga(T63, T64, T7)) → pO_out_gaga(T36, T64, T63, T7)
U23_ggagaa(T37, T62, T63, T36, X10, T7, pO_out_gaga(T36, X10, T63, T7)) → pM_out_ggagaa(T37, T62, T63, T36, X10, T7)
U17_gagaaa(T36, T62, T37, X61, X10, T7, pM_out_ggagaa(T37, T62, X61, T36, X10, T7)) → pL_out_gagaaa(T36, T62, T37, X61, X10, T7)
U15_gaaaaa(T36, T37, X60, X61, X10, T7, pL_out_gagaaa(T36, X60, T37, X61, X10, T7)) → pC_out_gaaaaa(T36, T37, X60, X61, X10, T7)
U2_ga(T36, T7, pC_out_gaaaaa(T36, X59, X60, X61, X10, T7)) → fibA_out_ga(s(s(s(T36))), T7)

(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:

ADDJ_IN_GGA(T87, s(T88), s(T90)) → ADDJ_IN_GGA(T87, T88, T90)
ADDJ_IN_GGA(s(T77), T78, s(T80)) → ADDJ_IN_GGA(T77, T78, T80)

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

ADDJ_IN_GGA(T87, s(T88)) → ADDJ_IN_GGA(T87, T88)
ADDJ_IN_GGA(s(T77), T78) → ADDJ_IN_GGA(T77, T78)

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:

ADDJ_IN_GGA(T87, s(T88)) → ADDJ_IN_GGA(T87, T88)
The graph contains the following edges 1 >= 1, 2 > 2
ADDJ_IN_GGA(s(T77), T78) → ADDJ_IN_GGA(T77, T78)
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:

ADDH_IN_GGA(T60, s(T61), s(X119)) → ADDH_IN_GGA(T60, T61, X119)
ADDH_IN_GGA(s(T54), T55, s(X107)) → ADDH_IN_GGA(T54, T55, X107)

The TRS R consists of the following rules:

fibA_in_ga(0, 0) → fibA_out_ga(0, 0)
fibA_in_ga(s(0), s(0)) → fibA_out_ga(s(0), s(0))
fibA_in_ga(s(s(0)), T7) → U1_ga(T7, pB_in_aa(X10, T7))
pB_in_aa(T8, T7) → U12_aa(T8, T7, fibK_in_a(T8))
fibK_in_a(0) → fibK_out_a(0)
U12_aa(T8, T7, fibK_out_a(T8)) → U13_aa(T8, T7, addE_in_ga(T8, T7))
addE_in_ga(T17, s(T19)) → U4_ga(T17, T19, addD_in_ga(T17, T19))
addD_in_ga(0, 0) → addD_out_ga(0, 0)
addD_in_ga(s(T24), s(T26)) → U3_ga(T24, T26, addD_in_ga(T24, T26))
U3_ga(T24, T26, addD_out_ga(T24, T26)) → addD_out_ga(s(T24), s(T26))
U4_ga(T17, T19, addD_out_ga(T17, T19)) → addE_out_ga(T17, s(T19))
addE_in_ga(s(T31), s(T33)) → U5_ga(T31, T33, addE_in_ga(T31, T33))
U5_ga(T31, T33, addE_out_ga(T31, T33)) → addE_out_ga(s(T31), s(T33))
U13_aa(T8, T7, addE_out_ga(T8, T7)) → pB_out_aa(T8, T7)
U1_ga(T7, pB_out_aa(X10, T7)) → fibA_out_ga(s(s(0)), T7)
fibA_in_ga(s(s(s(T36))), T7) → U2_ga(T36, T7, pC_in_gaaaaa(T36, X59, X60, X61, X10, T7))
pC_in_gaaaaa(T36, T37, X60, X61, X10, T7) → U14_gaaaaa(T36, T37, X60, X61, X10, T7, fibF_in_ga(T36, T37))
fibF_in_ga(0, s(0)) → fibF_out_ga(0, s(0))
fibF_in_ga(s(T40), X74) → U6_ga(T40, X74, pG_in_gaaa(T40, X72, X73, X74))
pG_in_gaaa(T40, T41, X73, X74) → U18_gaaa(T40, T41, X73, X74, fibF_in_ga(T40, T41))
U18_gaaa(T40, T41, X73, X74, fibF_out_ga(T40, T41)) → U19_gaaa(T40, T41, X73, X74, pN_in_gaga(T40, X73, T41, X74))
pN_in_gaga(T40, T42, T41, X74) → U20_gaga(T40, T42, T41, X74, fibI_in_ga(T40, T42))
fibI_in_ga(0, 0) → fibI_out_ga(0, 0)
fibI_in_ga(s(0), s(0)) → fibI_out_ga(s(0), s(0))
fibI_in_ga(s(s(T45)), X87) → U9_ga(T45, X87, pG_in_gaaa(T45, X85, X86, X87))
U9_ga(T45, X87, pG_out_gaaa(T45, X85, X86, X87)) → fibI_out_ga(s(s(T45)), X87)
U20_gaga(T40, T42, T41, X74, fibI_out_ga(T40, T42)) → U21_gaga(T40, T42, T41, X74, addH_in_gga(T41, T42, X74))
addH_in_gga(0, 0, 0) → addH_out_gga(0, 0, 0)
addH_in_gga(s(T54), T55, s(X107)) → U7_gga(T54, T55, X107, addH_in_gga(T54, T55, X107))
addH_in_gga(T60, s(T61), s(X119)) → U8_gga(T60, T61, X119, addH_in_gga(T60, T61, X119))
U8_gga(T60, T61, X119, addH_out_gga(T60, T61, X119)) → addH_out_gga(T60, s(T61), s(X119))
U7_gga(T54, T55, X107, addH_out_gga(T54, T55, X107)) → addH_out_gga(s(T54), T55, s(X107))
U21_gaga(T40, T42, T41, X74, addH_out_gga(T41, T42, X74)) → pN_out_gaga(T40, T42, T41, X74)
U19_gaaa(T40, T41, X73, X74, pN_out_gaga(T40, X73, T41, X74)) → pG_out_gaaa(T40, T41, X73, X74)
U6_ga(T40, X74, pG_out_gaaa(T40, X72, X73, X74)) → fibF_out_ga(s(T40), X74)
U14_gaaaaa(T36, T37, X60, X61, X10, T7, fibF_out_ga(T36, T37)) → U15_gaaaaa(T36, T37, X60, X61, X10, T7, pL_in_gagaaa(T36, X60, T37, X61, X10, T7))
pL_in_gagaaa(T36, T62, T37, X61, X10, T7) → U16_gagaaa(T36, T62, T37, X61, X10, T7, fibI_in_ga(T36, T62))
U16_gagaaa(T36, T62, T37, X61, X10, T7, fibI_out_ga(T36, T62)) → U17_gagaaa(T36, T62, T37, X61, X10, T7, pM_in_ggagaa(T37, T62, X61, T36, X10, T7))
pM_in_ggagaa(T37, T62, T63, T36, X10, T7) → U22_ggagaa(T37, T62, T63, T36, X10, T7, addH_in_gga(T37, T62, T63))
U22_ggagaa(T37, T62, T63, T36, X10, T7, addH_out_gga(T37, T62, T63)) → U23_ggagaa(T37, T62, T63, T36, X10, T7, pO_in_gaga(T36, X10, T63, T7))
pO_in_gaga(T36, T64, T63, T7) → U24_gaga(T36, T64, T63, T7, fibF_in_ga(T36, T64))
U24_gaga(T36, T64, T63, T7, fibF_out_ga(T36, T64)) → U25_gaga(T36, T64, T63, T7, addJ_in_gga(T63, T64, T7))
addJ_in_gga(0, 0, 0) → addJ_out_gga(0, 0, 0)
addJ_in_gga(s(T77), T78, s(T80)) → U10_gga(T77, T78, T80, addJ_in_gga(T77, T78, T80))
addJ_in_gga(T87, s(T88), s(T90)) → U11_gga(T87, T88, T90, addJ_in_gga(T87, T88, T90))
U11_gga(T87, T88, T90, addJ_out_gga(T87, T88, T90)) → addJ_out_gga(T87, s(T88), s(T90))
U10_gga(T77, T78, T80, addJ_out_gga(T77, T78, T80)) → addJ_out_gga(s(T77), T78, s(T80))
U25_gaga(T36, T64, T63, T7, addJ_out_gga(T63, T64, T7)) → pO_out_gaga(T36, T64, T63, T7)
U23_ggagaa(T37, T62, T63, T36, X10, T7, pO_out_gaga(T36, X10, T63, T7)) → pM_out_ggagaa(T37, T62, T63, T36, X10, T7)
U17_gagaaa(T36, T62, T37, X61, X10, T7, pM_out_ggagaa(T37, T62, X61, T36, X10, T7)) → pL_out_gagaaa(T36, T62, T37, X61, X10, T7)
U15_gaaaaa(T36, T37, X60, X61, X10, T7, pL_out_gagaaa(T36, X60, T37, X61, X10, T7)) → pC_out_gaaaaa(T36, T37, X60, X61, X10, T7)
U2_ga(T36, T7, pC_out_gaaaaa(T36, X59, X60, X61, X10, T7)) → fibA_out_ga(s(s(s(T36))), T7)

(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:

ADDH_IN_GGA(T60, s(T61), s(X119)) → ADDH_IN_GGA(T60, T61, X119)
ADDH_IN_GGA(s(T54), T55, s(X107)) → ADDH_IN_GGA(T54, T55, X107)

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

ADDH_IN_GGA(T60, s(T61)) → ADDH_IN_GGA(T60, T61)
ADDH_IN_GGA(s(T54), T55) → ADDH_IN_GGA(T54, T55)

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:

ADDH_IN_GGA(T60, s(T61)) → ADDH_IN_GGA(T60, T61)
The graph contains the following edges 1 >= 1, 2 > 2
ADDH_IN_GGA(s(T54), T55) → ADDH_IN_GGA(T54, T55)
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:

FIBF_IN_GA(s(T40), X74) → PG_IN_GAAA(T40, X72, X73, X74)
PG_IN_GAAA(T40, T41, X73, X74) → U18_GAAA(T40, T41, X73, X74, fibF_in_ga(T40, T41))
U18_GAAA(T40, T41, X73, X74, fibF_out_ga(T40, T41)) → PN_IN_GAGA(T40, X73, T41, X74)
PN_IN_GAGA(T40, T42, T41, X74) → FIBI_IN_GA(T40, T42)
FIBI_IN_GA(s(s(T45)), X87) → PG_IN_GAAA(T45, X85, X86, X87)
PG_IN_GAAA(T40, T41, X73, X74) → FIBF_IN_GA(T40, T41)

The TRS R consists of the following rules:

fibA_in_ga(0, 0) → fibA_out_ga(0, 0)
fibA_in_ga(s(0), s(0)) → fibA_out_ga(s(0), s(0))
fibA_in_ga(s(s(0)), T7) → U1_ga(T7, pB_in_aa(X10, T7))
pB_in_aa(T8, T7) → U12_aa(T8, T7, fibK_in_a(T8))
fibK_in_a(0) → fibK_out_a(0)
U12_aa(T8, T7, fibK_out_a(T8)) → U13_aa(T8, T7, addE_in_ga(T8, T7))
addE_in_ga(T17, s(T19)) → U4_ga(T17, T19, addD_in_ga(T17, T19))
addD_in_ga(0, 0) → addD_out_ga(0, 0)
addD_in_ga(s(T24), s(T26)) → U3_ga(T24, T26, addD_in_ga(T24, T26))
U3_ga(T24, T26, addD_out_ga(T24, T26)) → addD_out_ga(s(T24), s(T26))
U4_ga(T17, T19, addD_out_ga(T17, T19)) → addE_out_ga(T17, s(T19))
addE_in_ga(s(T31), s(T33)) → U5_ga(T31, T33, addE_in_ga(T31, T33))
U5_ga(T31, T33, addE_out_ga(T31, T33)) → addE_out_ga(s(T31), s(T33))
U13_aa(T8, T7, addE_out_ga(T8, T7)) → pB_out_aa(T8, T7)
U1_ga(T7, pB_out_aa(X10, T7)) → fibA_out_ga(s(s(0)), T7)
fibA_in_ga(s(s(s(T36))), T7) → U2_ga(T36, T7, pC_in_gaaaaa(T36, X59, X60, X61, X10, T7))
pC_in_gaaaaa(T36, T37, X60, X61, X10, T7) → U14_gaaaaa(T36, T37, X60, X61, X10, T7, fibF_in_ga(T36, T37))
fibF_in_ga(0, s(0)) → fibF_out_ga(0, s(0))
fibF_in_ga(s(T40), X74) → U6_ga(T40, X74, pG_in_gaaa(T40, X72, X73, X74))
pG_in_gaaa(T40, T41, X73, X74) → U18_gaaa(T40, T41, X73, X74, fibF_in_ga(T40, T41))
U18_gaaa(T40, T41, X73, X74, fibF_out_ga(T40, T41)) → U19_gaaa(T40, T41, X73, X74, pN_in_gaga(T40, X73, T41, X74))
pN_in_gaga(T40, T42, T41, X74) → U20_gaga(T40, T42, T41, X74, fibI_in_ga(T40, T42))
fibI_in_ga(0, 0) → fibI_out_ga(0, 0)
fibI_in_ga(s(0), s(0)) → fibI_out_ga(s(0), s(0))
fibI_in_ga(s(s(T45)), X87) → U9_ga(T45, X87, pG_in_gaaa(T45, X85, X86, X87))
U9_ga(T45, X87, pG_out_gaaa(T45, X85, X86, X87)) → fibI_out_ga(s(s(T45)), X87)
U20_gaga(T40, T42, T41, X74, fibI_out_ga(T40, T42)) → U21_gaga(T40, T42, T41, X74, addH_in_gga(T41, T42, X74))
addH_in_gga(0, 0, 0) → addH_out_gga(0, 0, 0)
addH_in_gga(s(T54), T55, s(X107)) → U7_gga(T54, T55, X107, addH_in_gga(T54, T55, X107))
addH_in_gga(T60, s(T61), s(X119)) → U8_gga(T60, T61, X119, addH_in_gga(T60, T61, X119))
U8_gga(T60, T61, X119, addH_out_gga(T60, T61, X119)) → addH_out_gga(T60, s(T61), s(X119))
U7_gga(T54, T55, X107, addH_out_gga(T54, T55, X107)) → addH_out_gga(s(T54), T55, s(X107))
U21_gaga(T40, T42, T41, X74, addH_out_gga(T41, T42, X74)) → pN_out_gaga(T40, T42, T41, X74)
U19_gaaa(T40, T41, X73, X74, pN_out_gaga(T40, X73, T41, X74)) → pG_out_gaaa(T40, T41, X73, X74)
U6_ga(T40, X74, pG_out_gaaa(T40, X72, X73, X74)) → fibF_out_ga(s(T40), X74)
U14_gaaaaa(T36, T37, X60, X61, X10, T7, fibF_out_ga(T36, T37)) → U15_gaaaaa(T36, T37, X60, X61, X10, T7, pL_in_gagaaa(T36, X60, T37, X61, X10, T7))
pL_in_gagaaa(T36, T62, T37, X61, X10, T7) → U16_gagaaa(T36, T62, T37, X61, X10, T7, fibI_in_ga(T36, T62))
U16_gagaaa(T36, T62, T37, X61, X10, T7, fibI_out_ga(T36, T62)) → U17_gagaaa(T36, T62, T37, X61, X10, T7, pM_in_ggagaa(T37, T62, X61, T36, X10, T7))
pM_in_ggagaa(T37, T62, T63, T36, X10, T7) → U22_ggagaa(T37, T62, T63, T36, X10, T7, addH_in_gga(T37, T62, T63))
U22_ggagaa(T37, T62, T63, T36, X10, T7, addH_out_gga(T37, T62, T63)) → U23_ggagaa(T37, T62, T63, T36, X10, T7, pO_in_gaga(T36, X10, T63, T7))
pO_in_gaga(T36, T64, T63, T7) → U24_gaga(T36, T64, T63, T7, fibF_in_ga(T36, T64))
U24_gaga(T36, T64, T63, T7, fibF_out_ga(T36, T64)) → U25_gaga(T36, T64, T63, T7, addJ_in_gga(T63, T64, T7))
addJ_in_gga(0, 0, 0) → addJ_out_gga(0, 0, 0)
addJ_in_gga(s(T77), T78, s(T80)) → U10_gga(T77, T78, T80, addJ_in_gga(T77, T78, T80))
addJ_in_gga(T87, s(T88), s(T90)) → U11_gga(T87, T88, T90, addJ_in_gga(T87, T88, T90))
U11_gga(T87, T88, T90, addJ_out_gga(T87, T88, T90)) → addJ_out_gga(T87, s(T88), s(T90))
U10_gga(T77, T78, T80, addJ_out_gga(T77, T78, T80)) → addJ_out_gga(s(T77), T78, s(T80))
U25_gaga(T36, T64, T63, T7, addJ_out_gga(T63, T64, T7)) → pO_out_gaga(T36, T64, T63, T7)
U23_ggagaa(T37, T62, T63, T36, X10, T7, pO_out_gaga(T36, X10, T63, T7)) → pM_out_ggagaa(T37, T62, T63, T36, X10, T7)
U17_gagaaa(T36, T62, T37, X61, X10, T7, pM_out_ggagaa(T37, T62, X61, T36, X10, T7)) → pL_out_gagaaa(T36, T62, T37, X61, X10, T7)
U15_gaaaaa(T36, T37, X60, X61, X10, T7, pL_out_gagaaa(T36, X60, T37, X61, X10, T7)) → pC_out_gaaaaa(T36, T37, X60, X61, X10, T7)
U2_ga(T36, T7, pC_out_gaaaaa(T36, X59, X60, X61, X10, T7)) → fibA_out_ga(s(s(s(T36))), T7)

(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:

FIBF_IN_GA(s(T40), X74) → PG_IN_GAAA(T40, X72, X73, X74)
PG_IN_GAAA(T40, T41, X73, X74) → U18_GAAA(T40, T41, X73, X74, fibF_in_ga(T40, T41))
U18_GAAA(T40, T41, X73, X74, fibF_out_ga(T40, T41)) → PN_IN_GAGA(T40, X73, T41, X74)
PN_IN_GAGA(T40, T42, T41, X74) → FIBI_IN_GA(T40, T42)
FIBI_IN_GA(s(s(T45)), X87) → PG_IN_GAAA(T45, X85, X86, X87)
PG_IN_GAAA(T40, T41, X73, X74) → FIBF_IN_GA(T40, T41)

The TRS R consists of the following rules:

fibF_in_ga(0, s(0)) → fibF_out_ga(0, s(0))
fibF_in_ga(s(T40), X74) → U6_ga(T40, X74, pG_in_gaaa(T40, X72, X73, X74))
U6_ga(T40, X74, pG_out_gaaa(T40, X72, X73, X74)) → fibF_out_ga(s(T40), X74)
pG_in_gaaa(T40, T41, X73, X74) → U18_gaaa(T40, T41, X73, X74, fibF_in_ga(T40, T41))
U18_gaaa(T40, T41, X73, X74, fibF_out_ga(T40, T41)) → U19_gaaa(T40, T41, X73, X74, pN_in_gaga(T40, X73, T41, X74))
U19_gaaa(T40, T41, X73, X74, pN_out_gaga(T40, X73, T41, X74)) → pG_out_gaaa(T40, T41, X73, X74)
pN_in_gaga(T40, T42, T41, X74) → U20_gaga(T40, T42, T41, X74, fibI_in_ga(T40, T42))
U20_gaga(T40, T42, T41, X74, fibI_out_ga(T40, T42)) → U21_gaga(T40, T42, T41, X74, addH_in_gga(T41, T42, X74))
fibI_in_ga(0, 0) → fibI_out_ga(0, 0)
fibI_in_ga(s(0), s(0)) → fibI_out_ga(s(0), s(0))
fibI_in_ga(s(s(T45)), X87) → U9_ga(T45, X87, pG_in_gaaa(T45, X85, X86, X87))
U21_gaga(T40, T42, T41, X74, addH_out_gga(T41, T42, X74)) → pN_out_gaga(T40, T42, T41, X74)
U9_ga(T45, X87, pG_out_gaaa(T45, X85, X86, X87)) → fibI_out_ga(s(s(T45)), X87)
addH_in_gga(0, 0, 0) → addH_out_gga(0, 0, 0)
addH_in_gga(s(T54), T55, s(X107)) → U7_gga(T54, T55, X107, addH_in_gga(T54, T55, X107))
addH_in_gga(T60, s(T61), s(X119)) → U8_gga(T60, T61, X119, addH_in_gga(T60, T61, X119))
U7_gga(T54, T55, X107, addH_out_gga(T54, T55, X107)) → addH_out_gga(s(T54), T55, s(X107))
U8_gga(T60, T61, X119, addH_out_gga(T60, T61, X119)) → addH_out_gga(T60, s(T61), s(X119))

The argument filtering Pi contains the following mapping:
0 = 0
s(x1) = s(x1)
fibF_in_ga(x1, x2) = fibF_in_ga(x1)
fibF_out_ga(x1, x2) = fibF_out_ga(x1, x2)
U6_ga(x1, x2, x3) = U6_ga(x1, x3)
pG_in_gaaa(x1, x2, x3, x4) = pG_in_gaaa(x1)
U18_gaaa(x1, x2, x3, x4, x5) = U18_gaaa(x1, x5)
U19_gaaa(x1, x2, x3, x4, x5) = U19_gaaa(x1, x2, x5)
pN_in_gaga(x1, x2, x3, x4) = pN_in_gaga(x1, x3)
U20_gaga(x1, x2, x3, x4, x5) = U20_gaga(x1, x3, x5)
fibI_in_ga(x1, x2) = fibI_in_ga(x1)
fibI_out_ga(x1, x2) = fibI_out_ga(x1, x2)
U9_ga(x1, x2, x3) = U9_ga(x1, x3)
pG_out_gaaa(x1, x2, x3, x4) = pG_out_gaaa(x1, x2, x3, x4)
U21_gaga(x1, x2, x3, x4, x5) = U21_gaga(x1, x2, x3, x5)
addH_in_gga(x1, x2, x3) = addH_in_gga(x1, x2)
addH_out_gga(x1, x2, x3) = addH_out_gga(x1, x2, x3)
U7_gga(x1, x2, x3, x4) = U7_gga(x1, x2, x4)
U8_gga(x1, x2, x3, x4) = U8_gga(x1, x2, x4)
pN_out_gaga(x1, x2, x3, x4) = pN_out_gaga(x1, x2, x3, x4)
FIBF_IN_GA(x1, x2) = FIBF_IN_GA(x1)
PG_IN_GAAA(x1, x2, x3, x4) = PG_IN_GAAA(x1)
U18_GAAA(x1, x2, x3, x4, x5) = U18_GAAA(x1, x5)
PN_IN_GAGA(x1, x2, x3, x4) = PN_IN_GAGA(x1, x3)
FIBI_IN_GA(x1, x2) = FIBI_IN_GA(x1)

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:

FIBF_IN_GA(s(T40)) → PG_IN_GAAA(T40)
PG_IN_GAAA(T40) → U18_GAAA(T40, fibF_in_ga(T40))
U18_GAAA(T40, fibF_out_ga(T40, T41)) → PN_IN_GAGA(T40, T41)
PN_IN_GAGA(T40, T41) → FIBI_IN_GA(T40)
FIBI_IN_GA(s(s(T45))) → PG_IN_GAAA(T45)
PG_IN_GAAA(T40) → FIBF_IN_GA(T40)

The TRS R consists of the following rules:

fibF_in_ga(0) → fibF_out_ga(0, s(0))
fibF_in_ga(s(T40)) → U6_ga(T40, pG_in_gaaa(T40))
U6_ga(T40, pG_out_gaaa(T40, X72, X73, X74)) → fibF_out_ga(s(T40), X74)
pG_in_gaaa(T40) → U18_gaaa(T40, fibF_in_ga(T40))
U18_gaaa(T40, fibF_out_ga(T40, T41)) → U19_gaaa(T40, T41, pN_in_gaga(T40, T41))
U19_gaaa(T40, T41, pN_out_gaga(T40, X73, T41, X74)) → pG_out_gaaa(T40, T41, X73, X74)
pN_in_gaga(T40, T41) → U20_gaga(T40, T41, fibI_in_ga(T40))
U20_gaga(T40, T41, fibI_out_ga(T40, T42)) → U21_gaga(T40, T42, T41, addH_in_gga(T41, T42))
fibI_in_ga(0) → fibI_out_ga(0, 0)
fibI_in_ga(s(0)) → fibI_out_ga(s(0), s(0))
fibI_in_ga(s(s(T45))) → U9_ga(T45, pG_in_gaaa(T45))
U21_gaga(T40, T42, T41, addH_out_gga(T41, T42, X74)) → pN_out_gaga(T40, T42, T41, X74)
U9_ga(T45, pG_out_gaaa(T45, X85, X86, X87)) → fibI_out_ga(s(s(T45)), X87)
addH_in_gga(0, 0) → addH_out_gga(0, 0, 0)
addH_in_gga(s(T54), T55) → U7_gga(T54, T55, addH_in_gga(T54, T55))
addH_in_gga(T60, s(T61)) → U8_gga(T60, T61, addH_in_gga(T60, T61))
U7_gga(T54, T55, addH_out_gga(T54, T55, X107)) → addH_out_gga(s(T54), T55, s(X107))
U8_gga(T60, T61, addH_out_gga(T60, T61, X119)) → addH_out_gga(T60, s(T61), s(X119))

The set Q consists of the following terms:

fibF_in_ga(x0)
U6_ga(x0, x1)
pG_in_gaaa(x0)
U18_gaaa(x0, x1)
U19_gaaa(x0, x1, x2)
pN_in_gaga(x0, x1)
U20_gaga(x0, x1, x2)
fibI_in_ga(x0)
U21_gaga(x0, x1, x2, x3)
U9_ga(x0, x1)
addH_in_gga(x0, x1)
U7_gga(x0, x1, x2)
U8_gga(x0, x1, x2)

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:

PG_IN_GAAA(T40) → FIBF_IN_GA(T40)
The graph contains the following edges 1 >= 1
PG_IN_GAAA(T40) → U18_GAAA(T40, fibF_in_ga(T40))
The graph contains the following edges 1 >= 1
U18_GAAA(T40, fibF_out_ga(T40, T41)) → PN_IN_GAGA(T40, T41)
The graph contains the following edges 1 >= 1, 2 > 1, 2 > 2
PN_IN_GAGA(T40, T41) → FIBI_IN_GA(T40)
The graph contains the following edges 1 >= 1
FIBI_IN_GA(s(s(T45))) → PG_IN_GAAA(T45)
The graph contains the following edges 1 > 1
FIBF_IN_GA(s(T40)) → PG_IN_GAAA(T40)
The graph contains the following edges 1 > 1

(27) YES

(28) Obligation:

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

ADDD_IN_GA(s(T24), s(T26)) → ADDD_IN_GA(T24, T26)

The TRS R consists of the following rules:

fibA_in_ga(0, 0) → fibA_out_ga(0, 0)
fibA_in_ga(s(0), s(0)) → fibA_out_ga(s(0), s(0))
fibA_in_ga(s(s(0)), T7) → U1_ga(T7, pB_in_aa(X10, T7))
pB_in_aa(T8, T7) → U12_aa(T8, T7, fibK_in_a(T8))
fibK_in_a(0) → fibK_out_a(0)
U12_aa(T8, T7, fibK_out_a(T8)) → U13_aa(T8, T7, addE_in_ga(T8, T7))
addE_in_ga(T17, s(T19)) → U4_ga(T17, T19, addD_in_ga(T17, T19))
addD_in_ga(0, 0) → addD_out_ga(0, 0)
addD_in_ga(s(T24), s(T26)) → U3_ga(T24, T26, addD_in_ga(T24, T26))
U3_ga(T24, T26, addD_out_ga(T24, T26)) → addD_out_ga(s(T24), s(T26))
U4_ga(T17, T19, addD_out_ga(T17, T19)) → addE_out_ga(T17, s(T19))
addE_in_ga(s(T31), s(T33)) → U5_ga(T31, T33, addE_in_ga(T31, T33))
U5_ga(T31, T33, addE_out_ga(T31, T33)) → addE_out_ga(s(T31), s(T33))
U13_aa(T8, T7, addE_out_ga(T8, T7)) → pB_out_aa(T8, T7)
U1_ga(T7, pB_out_aa(X10, T7)) → fibA_out_ga(s(s(0)), T7)
fibA_in_ga(s(s(s(T36))), T7) → U2_ga(T36, T7, pC_in_gaaaaa(T36, X59, X60, X61, X10, T7))
pC_in_gaaaaa(T36, T37, X60, X61, X10, T7) → U14_gaaaaa(T36, T37, X60, X61, X10, T7, fibF_in_ga(T36, T37))
fibF_in_ga(0, s(0)) → fibF_out_ga(0, s(0))
fibF_in_ga(s(T40), X74) → U6_ga(T40, X74, pG_in_gaaa(T40, X72, X73, X74))
pG_in_gaaa(T40, T41, X73, X74) → U18_gaaa(T40, T41, X73, X74, fibF_in_ga(T40, T41))
U18_gaaa(T40, T41, X73, X74, fibF_out_ga(T40, T41)) → U19_gaaa(T40, T41, X73, X74, pN_in_gaga(T40, X73, T41, X74))
pN_in_gaga(T40, T42, T41, X74) → U20_gaga(T40, T42, T41, X74, fibI_in_ga(T40, T42))
fibI_in_ga(0, 0) → fibI_out_ga(0, 0)
fibI_in_ga(s(0), s(0)) → fibI_out_ga(s(0), s(0))
fibI_in_ga(s(s(T45)), X87) → U9_ga(T45, X87, pG_in_gaaa(T45, X85, X86, X87))
U9_ga(T45, X87, pG_out_gaaa(T45, X85, X86, X87)) → fibI_out_ga(s(s(T45)), X87)
U20_gaga(T40, T42, T41, X74, fibI_out_ga(T40, T42)) → U21_gaga(T40, T42, T41, X74, addH_in_gga(T41, T42, X74))
addH_in_gga(0, 0, 0) → addH_out_gga(0, 0, 0)
addH_in_gga(s(T54), T55, s(X107)) → U7_gga(T54, T55, X107, addH_in_gga(T54, T55, X107))
addH_in_gga(T60, s(T61), s(X119)) → U8_gga(T60, T61, X119, addH_in_gga(T60, T61, X119))
U8_gga(T60, T61, X119, addH_out_gga(T60, T61, X119)) → addH_out_gga(T60, s(T61), s(X119))
U7_gga(T54, T55, X107, addH_out_gga(T54, T55, X107)) → addH_out_gga(s(T54), T55, s(X107))
U21_gaga(T40, T42, T41, X74, addH_out_gga(T41, T42, X74)) → pN_out_gaga(T40, T42, T41, X74)
U19_gaaa(T40, T41, X73, X74, pN_out_gaga(T40, X73, T41, X74)) → pG_out_gaaa(T40, T41, X73, X74)
U6_ga(T40, X74, pG_out_gaaa(T40, X72, X73, X74)) → fibF_out_ga(s(T40), X74)
U14_gaaaaa(T36, T37, X60, X61, X10, T7, fibF_out_ga(T36, T37)) → U15_gaaaaa(T36, T37, X60, X61, X10, T7, pL_in_gagaaa(T36, X60, T37, X61, X10, T7))
pL_in_gagaaa(T36, T62, T37, X61, X10, T7) → U16_gagaaa(T36, T62, T37, X61, X10, T7, fibI_in_ga(T36, T62))
U16_gagaaa(T36, T62, T37, X61, X10, T7, fibI_out_ga(T36, T62)) → U17_gagaaa(T36, T62, T37, X61, X10, T7, pM_in_ggagaa(T37, T62, X61, T36, X10, T7))
pM_in_ggagaa(T37, T62, T63, T36, X10, T7) → U22_ggagaa(T37, T62, T63, T36, X10, T7, addH_in_gga(T37, T62, T63))
U22_ggagaa(T37, T62, T63, T36, X10, T7, addH_out_gga(T37, T62, T63)) → U23_ggagaa(T37, T62, T63, T36, X10, T7, pO_in_gaga(T36, X10, T63, T7))
pO_in_gaga(T36, T64, T63, T7) → U24_gaga(T36, T64, T63, T7, fibF_in_ga(T36, T64))
U24_gaga(T36, T64, T63, T7, fibF_out_ga(T36, T64)) → U25_gaga(T36, T64, T63, T7, addJ_in_gga(T63, T64, T7))
addJ_in_gga(0, 0, 0) → addJ_out_gga(0, 0, 0)
addJ_in_gga(s(T77), T78, s(T80)) → U10_gga(T77, T78, T80, addJ_in_gga(T77, T78, T80))
addJ_in_gga(T87, s(T88), s(T90)) → U11_gga(T87, T88, T90, addJ_in_gga(T87, T88, T90))
U11_gga(T87, T88, T90, addJ_out_gga(T87, T88, T90)) → addJ_out_gga(T87, s(T88), s(T90))
U10_gga(T77, T78, T80, addJ_out_gga(T77, T78, T80)) → addJ_out_gga(s(T77), T78, s(T80))
U25_gaga(T36, T64, T63, T7, addJ_out_gga(T63, T64, T7)) → pO_out_gaga(T36, T64, T63, T7)
U23_ggagaa(T37, T62, T63, T36, X10, T7, pO_out_gaga(T36, X10, T63, T7)) → pM_out_ggagaa(T37, T62, T63, T36, X10, T7)
U17_gagaaa(T36, T62, T37, X61, X10, T7, pM_out_ggagaa(T37, T62, X61, T36, X10, T7)) → pL_out_gagaaa(T36, T62, T37, X61, X10, T7)
U15_gaaaaa(T36, T37, X60, X61, X10, T7, pL_out_gagaaa(T36, X60, T37, X61, X10, T7)) → pC_out_gaaaaa(T36, T37, X60, X61, X10, T7)
U2_ga(T36, T7, pC_out_gaaaaa(T36, X59, X60, X61, X10, T7)) → fibA_out_ga(s(s(s(T36))), T7)

(29) UsableRulesProof (EQUIVALENT transformation)

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

(30) Obligation:

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

ADDD_IN_GA(s(T24), s(T26)) → ADDD_IN_GA(T24, T26)

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

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

(31) PiDPToQDPProof (SOUND transformation)

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

(32) Obligation:

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

ADDD_IN_GA(s(T24)) → ADDD_IN_GA(T24)

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

(33) QDPSizeChangeProof (EQUIVALENT transformation)

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

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

ADDD_IN_GA(s(T24)) → ADDD_IN_GA(T24)
The graph contains the following edges 1 > 1

(34) YES

(35) Obligation:

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

ADDE_IN_GA(s(T31), s(T33)) → ADDE_IN_GA(T31, T33)

The TRS R consists of the following rules:

fibA_in_ga(0, 0) → fibA_out_ga(0, 0)
fibA_in_ga(s(0), s(0)) → fibA_out_ga(s(0), s(0))
fibA_in_ga(s(s(0)), T7) → U1_ga(T7, pB_in_aa(X10, T7))
pB_in_aa(T8, T7) → U12_aa(T8, T7, fibK_in_a(T8))
fibK_in_a(0) → fibK_out_a(0)
U12_aa(T8, T7, fibK_out_a(T8)) → U13_aa(T8, T7, addE_in_ga(T8, T7))
addE_in_ga(T17, s(T19)) → U4_ga(T17, T19, addD_in_ga(T17, T19))
addD_in_ga(0, 0) → addD_out_ga(0, 0)
addD_in_ga(s(T24), s(T26)) → U3_ga(T24, T26, addD_in_ga(T24, T26))
U3_ga(T24, T26, addD_out_ga(T24, T26)) → addD_out_ga(s(T24), s(T26))
U4_ga(T17, T19, addD_out_ga(T17, T19)) → addE_out_ga(T17, s(T19))
addE_in_ga(s(T31), s(T33)) → U5_ga(T31, T33, addE_in_ga(T31, T33))
U5_ga(T31, T33, addE_out_ga(T31, T33)) → addE_out_ga(s(T31), s(T33))
U13_aa(T8, T7, addE_out_ga(T8, T7)) → pB_out_aa(T8, T7)
U1_ga(T7, pB_out_aa(X10, T7)) → fibA_out_ga(s(s(0)), T7)
fibA_in_ga(s(s(s(T36))), T7) → U2_ga(T36, T7, pC_in_gaaaaa(T36, X59, X60, X61, X10, T7))
pC_in_gaaaaa(T36, T37, X60, X61, X10, T7) → U14_gaaaaa(T36, T37, X60, X61, X10, T7, fibF_in_ga(T36, T37))
fibF_in_ga(0, s(0)) → fibF_out_ga(0, s(0))
fibF_in_ga(s(T40), X74) → U6_ga(T40, X74, pG_in_gaaa(T40, X72, X73, X74))
pG_in_gaaa(T40, T41, X73, X74) → U18_gaaa(T40, T41, X73, X74, fibF_in_ga(T40, T41))
U18_gaaa(T40, T41, X73, X74, fibF_out_ga(T40, T41)) → U19_gaaa(T40, T41, X73, X74, pN_in_gaga(T40, X73, T41, X74))
pN_in_gaga(T40, T42, T41, X74) → U20_gaga(T40, T42, T41, X74, fibI_in_ga(T40, T42))
fibI_in_ga(0, 0) → fibI_out_ga(0, 0)
fibI_in_ga(s(0), s(0)) → fibI_out_ga(s(0), s(0))
fibI_in_ga(s(s(T45)), X87) → U9_ga(T45, X87, pG_in_gaaa(T45, X85, X86, X87))
U9_ga(T45, X87, pG_out_gaaa(T45, X85, X86, X87)) → fibI_out_ga(s(s(T45)), X87)
U20_gaga(T40, T42, T41, X74, fibI_out_ga(T40, T42)) → U21_gaga(T40, T42, T41, X74, addH_in_gga(T41, T42, X74))
addH_in_gga(0, 0, 0) → addH_out_gga(0, 0, 0)
addH_in_gga(s(T54), T55, s(X107)) → U7_gga(T54, T55, X107, addH_in_gga(T54, T55, X107))
addH_in_gga(T60, s(T61), s(X119)) → U8_gga(T60, T61, X119, addH_in_gga(T60, T61, X119))
U8_gga(T60, T61, X119, addH_out_gga(T60, T61, X119)) → addH_out_gga(T60, s(T61), s(X119))
U7_gga(T54, T55, X107, addH_out_gga(T54, T55, X107)) → addH_out_gga(s(T54), T55, s(X107))
U21_gaga(T40, T42, T41, X74, addH_out_gga(T41, T42, X74)) → pN_out_gaga(T40, T42, T41, X74)
U19_gaaa(T40, T41, X73, X74, pN_out_gaga(T40, X73, T41, X74)) → pG_out_gaaa(T40, T41, X73, X74)
U6_ga(T40, X74, pG_out_gaaa(T40, X72, X73, X74)) → fibF_out_ga(s(T40), X74)
U14_gaaaaa(T36, T37, X60, X61, X10, T7, fibF_out_ga(T36, T37)) → U15_gaaaaa(T36, T37, X60, X61, X10, T7, pL_in_gagaaa(T36, X60, T37, X61, X10, T7))
pL_in_gagaaa(T36, T62, T37, X61, X10, T7) → U16_gagaaa(T36, T62, T37, X61, X10, T7, fibI_in_ga(T36, T62))
U16_gagaaa(T36, T62, T37, X61, X10, T7, fibI_out_ga(T36, T62)) → U17_gagaaa(T36, T62, T37, X61, X10, T7, pM_in_ggagaa(T37, T62, X61, T36, X10, T7))
pM_in_ggagaa(T37, T62, T63, T36, X10, T7) → U22_ggagaa(T37, T62, T63, T36, X10, T7, addH_in_gga(T37, T62, T63))
U22_ggagaa(T37, T62, T63, T36, X10, T7, addH_out_gga(T37, T62, T63)) → U23_ggagaa(T37, T62, T63, T36, X10, T7, pO_in_gaga(T36, X10, T63, T7))
pO_in_gaga(T36, T64, T63, T7) → U24_gaga(T36, T64, T63, T7, fibF_in_ga(T36, T64))
U24_gaga(T36, T64, T63, T7, fibF_out_ga(T36, T64)) → U25_gaga(T36, T64, T63, T7, addJ_in_gga(T63, T64, T7))
addJ_in_gga(0, 0, 0) → addJ_out_gga(0, 0, 0)
addJ_in_gga(s(T77), T78, s(T80)) → U10_gga(T77, T78, T80, addJ_in_gga(T77, T78, T80))
addJ_in_gga(T87, s(T88), s(T90)) → U11_gga(T87, T88, T90, addJ_in_gga(T87, T88, T90))
U11_gga(T87, T88, T90, addJ_out_gga(T87, T88, T90)) → addJ_out_gga(T87, s(T88), s(T90))
U10_gga(T77, T78, T80, addJ_out_gga(T77, T78, T80)) → addJ_out_gga(s(T77), T78, s(T80))
U25_gaga(T36, T64, T63, T7, addJ_out_gga(T63, T64, T7)) → pO_out_gaga(T36, T64, T63, T7)
U23_ggagaa(T37, T62, T63, T36, X10, T7, pO_out_gaga(T36, X10, T63, T7)) → pM_out_ggagaa(T37, T62, T63, T36, X10, T7)
U17_gagaaa(T36, T62, T37, X61, X10, T7, pM_out_ggagaa(T37, T62, X61, T36, X10, T7)) → pL_out_gagaaa(T36, T62, T37, X61, X10, T7)
U15_gaaaaa(T36, T37, X60, X61, X10, T7, pL_out_gagaaa(T36, X60, T37, X61, X10, T7)) → pC_out_gaaaaa(T36, T37, X60, X61, X10, T7)
U2_ga(T36, T7, pC_out_gaaaaa(T36, X59, X60, X61, X10, T7)) → fibA_out_ga(s(s(s(T36))), T7)

(36) UsableRulesProof (EQUIVALENT transformation)

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

(37) Obligation:

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

ADDE_IN_GA(s(T31), s(T33)) → ADDE_IN_GA(T31, T33)

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

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

(38) PiDPToQDPProof (SOUND transformation)

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

(39) Obligation:

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

ADDE_IN_GA(s(T31)) → ADDE_IN_GA(T31)

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

(40) 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_GA(s(T31)) → ADDE_IN_GA(T31)
The graph contains the following edges 1 > 1

(0) Obligation:

(1) PrologToPiTRSViaGraphTransformerProof (SOUND transformation)

(2) Obligation:

(3) DependencyPairsProof (EQUIVALENT transformation)

(4) Obligation:

(5) DependencyGraphProof (EQUIVALENT transformation)

(6) Complex Obligation (AND)

(7) Obligation:

(8) UsableRulesProof (EQUIVALENT transformation)

(9) Obligation:

(10) PiDPToQDPProof (SOUND transformation)

(11) Obligation:

(12) QDPSizeChangeProof (EQUIVALENT transformation)

(13) YES

(14) Obligation:

(15) UsableRulesProof (EQUIVALENT transformation)

(16) Obligation:

(17) PiDPToQDPProof (SOUND transformation)

(18) Obligation:

(19) QDPSizeChangeProof (EQUIVALENT transformation)

(20) YES

(21) Obligation:

(22) UsableRulesProof (EQUIVALENT transformation)

(23) Obligation:

(24) PiDPToQDPProof (SOUND transformation)

(25) Obligation:

(26) QDPSizeChangeProof (EQUIVALENT transformation)

(27) YES

(28) Obligation:

(29) UsableRulesProof (EQUIVALENT transformation)

(30) Obligation:

(31) PiDPToQDPProof (SOUND transformation)

(32) Obligation:

(33) QDPSizeChangeProof (EQUIVALENT transformation)

(34) YES

(35) Obligation:

(36) UsableRulesProof (EQUIVALENT transformation)

(37) Obligation:

(38) PiDPToQDPProof (SOUND transformation)

(39) Obligation:

(40) QDPSizeChangeProof (EQUIVALENT transformation)

(41) YES