/export/starexec/sandbox2/solver/bin/starexec_run_standard /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files -------------------------------------------------------------------------------- YES proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty Outermost Termination of the given OTRS could be proven: (0) OTRS (1) Raffelsieper-Zantema-Transformation [SOUND, 0 ms] (2) QTRS (3) AAECC Innermost [EQUIVALENT, 2 ms] (4) QTRS (5) DependencyPairsProof [EQUIVALENT, 0 ms] (6) QDP (7) DependencyGraphProof [EQUIVALENT, 0 ms] (8) QDP (9) UsableRulesProof [EQUIVALENT, 0 ms] (10) QDP (11) QReductionProof [EQUIVALENT, 0 ms] (12) QDP (13) RFCMatchBoundsDPProof [EQUIVALENT, 23 ms] (14) YES ---------------------------------------- (0) Obligation: Term rewrite system R: The TRS R consists of the following rules: a -> f(a) f(f(f(f(f(x))))) -> b Outermost Strategy. ---------------------------------------- (1) Raffelsieper-Zantema-Transformation (SOUND) We applied the Raffelsieper-Zantema transformation to transform the outermost TRS to a standard TRS. ---------------------------------------- (2) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: down(a) -> up(f(a)) down(f(f(f(f(f(x)))))) -> up(b) top(up(x)) -> top(down(x)) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) f_flat(up(x_1)) -> up(f(x_1)) Q is empty. ---------------------------------------- (3) AAECC Innermost (EQUIVALENT) We have applied [NOC,AAECCNOC] to switch to innermost. The TRS R 1 is down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) f_flat(up(x_1)) -> up(f(x_1)) down(a) -> up(f(a)) down(f(f(f(f(f(x)))))) -> up(b) The TRS R 2 is top(up(x)) -> top(down(x)) The signature Sigma is {top_1} ---------------------------------------- (4) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: down(a) -> up(f(a)) down(f(f(f(f(f(x)))))) -> up(b) top(up(x)) -> top(down(x)) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(f(f(f(f(f(x0)))))) top(up(x0)) down(f(a)) down(f(b)) down(f(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(fresh_constant))))) f_flat(up(x0)) ---------------------------------------- (5) DependencyPairsProof (EQUIVALENT) Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. ---------------------------------------- (6) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(x)) -> TOP(down(x)) TOP(up(x)) -> DOWN(x) DOWN(f(a)) -> F_FLAT(down(a)) DOWN(f(a)) -> DOWN(a) DOWN(f(b)) -> F_FLAT(down(b)) DOWN(f(b)) -> DOWN(b) DOWN(f(fresh_constant)) -> F_FLAT(down(fresh_constant)) DOWN(f(fresh_constant)) -> DOWN(fresh_constant) DOWN(f(f(a))) -> F_FLAT(down(f(a))) DOWN(f(f(a))) -> DOWN(f(a)) DOWN(f(f(b))) -> F_FLAT(down(f(b))) DOWN(f(f(b))) -> DOWN(f(b)) DOWN(f(f(fresh_constant))) -> F_FLAT(down(f(fresh_constant))) DOWN(f(f(fresh_constant))) -> DOWN(f(fresh_constant)) DOWN(f(f(f(a)))) -> F_FLAT(down(f(f(a)))) DOWN(f(f(f(a)))) -> DOWN(f(f(a))) DOWN(f(f(f(b)))) -> F_FLAT(down(f(f(b)))) DOWN(f(f(f(b)))) -> DOWN(f(f(b))) DOWN(f(f(f(fresh_constant)))) -> F_FLAT(down(f(f(fresh_constant)))) DOWN(f(f(f(fresh_constant)))) -> DOWN(f(f(fresh_constant))) DOWN(f(f(f(f(a))))) -> F_FLAT(down(f(f(f(a))))) DOWN(f(f(f(f(a))))) -> DOWN(f(f(f(a)))) DOWN(f(f(f(f(b))))) -> F_FLAT(down(f(f(f(b))))) DOWN(f(f(f(f(b))))) -> DOWN(f(f(f(b)))) DOWN(f(f(f(f(fresh_constant))))) -> F_FLAT(down(f(f(f(fresh_constant))))) DOWN(f(f(f(f(fresh_constant))))) -> DOWN(f(f(f(fresh_constant)))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(f(f(f(f(f(x)))))) -> up(b) top(up(x)) -> top(down(x)) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(f(f(f(f(f(x0)))))) top(up(x0)) down(f(a)) down(f(b)) down(f(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(fresh_constant))))) f_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (7) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 25 less nodes. ---------------------------------------- (8) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(x)) -> TOP(down(x)) The TRS R consists of the following rules: down(a) -> up(f(a)) down(f(f(f(f(f(x)))))) -> up(b) top(up(x)) -> top(down(x)) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(f(f(f(f(f(x0)))))) top(up(x0)) down(f(a)) down(f(b)) down(f(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(fresh_constant))))) f_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (9) UsableRulesProof (EQUIVALENT) As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R. ---------------------------------------- (10) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(x)) -> TOP(down(x)) The TRS R consists of the following rules: down(a) -> up(f(a)) down(f(f(f(f(f(x)))))) -> up(b) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(f(f(f(f(f(x0)))))) top(up(x0)) down(f(a)) down(f(b)) down(f(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(fresh_constant))))) f_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (11) QReductionProof (EQUIVALENT) We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. top(up(x0)) ---------------------------------------- (12) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(x)) -> TOP(down(x)) The TRS R consists of the following rules: down(a) -> up(f(a)) down(f(f(f(f(f(x)))))) -> up(b) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(f(f(f(f(f(x0)))))) down(f(a)) down(f(b)) down(f(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(fresh_constant))))) f_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (13) RFCMatchBoundsDPProof (EQUIVALENT) Finiteness of the DP problem can be shown by a matchbound of 9. As the DP problem is minimal we only have to initialize the certificate graph by the rules of P: TOP(up(x)) -> TOP(down(x)) To find matches we regarded all rules of R and P: down(a) -> up(f(a)) down(f(f(f(f(f(x)))))) -> up(b) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) f_flat(up(x_1)) -> up(f(x_1)) TOP(up(x)) -> TOP(down(x)) The certificate found is represented by the following graph. The certificate consists of the following enumerated nodes: 277, 278, 281, 282, 283, 284, 285, 286, 287, 288, 289, 290, 291, 292, 293, 294, 295, 296, 297, 298, 299, 300, 301, 302, 303, 304, 305, 306, 307, 308, 309, 310, 311, 312, 313, 314, 315, 316, 317, 318, 319, 320, 321, 322, 323, 324, 325, 326, 327, 328, 329, 330, 331, 332, 333, 334, 335, 336, 337, 338, 339, 340, 341, 342, 343, 344, 345, 346, 347, 348, 349, 350, 351, 352, 353, 354, 355, 356, 357, 358, 359, 360, 361, 362, 363, 364, 365, 366, 367, 368, 369, 370, 371, 372, 373, 374, 375, 376, 377, 378, 379, 380, 381, 382, 383, 384, 385, 386, 387, 388, 389, 390, 391, 392, 393, 394, 395, 396, 397, 398, 399, 400, 401, 402, 403, 404, 405, 406, 407, 408, 414, 416, 417, 419, 420, 421, 422 Node 277 is start node and node 278 is final node. Those nodes are connected through the following edges: * 277 to 281 labelled TOP_1(0)* 277 to 298 labelled TOP_1(1)* 277 to 323 labelled TOP_1(2)* 277 to 330 labelled TOP_1(3)* 277 to 376 labelled TOP_1(4)* 277 to 419 labelled TOP_1(5)* 277 to 422 labelled TOP_1(6)* 278 to 278 labelled #_1(0)* 281 to 278 labelled down_1(0)* 281 to 282 labelled up_1(1)* 281 to 284 labelled f_flat_1(1)* 281 to 286 labelled f_flat_1(1)* 281 to 289 labelled f_flat_1(1)* 281 to 293 labelled f_flat_1(1)* 281 to 310 labelled up_1(2)* 282 to 283 labelled f_1(1)* 282 to 278 labelled b(1)* 283 to 278 labelled a(1)* 284 to 285 labelled down_1(1)* 284 to 299 labelled up_1(2)* 285 to 278 labelled a(1), b(1), fresh_constant(1)* 286 to 287 labelled down_1(1)* 286 to 301 labelled f_flat_1(2)* 286 to 318 labelled up_1(3)* 287 to 288 labelled f_1(1)* 288 to 278 labelled a(1), b(1), fresh_constant(1)* 289 to 290 labelled down_1(1)* 289 to 303 labelled f_flat_1(2)* 289 to 327 labelled up_1(3)* 290 to 291 labelled f_1(1)* 291 to 292 labelled f_1(1)* 292 to 278 labelled a(1), b(1), fresh_constant(1)* 293 to 294 labelled down_1(1)* 293 to 306 labelled f_flat_1(2)* 293 to 331 labelled up_1(3)* 294 to 295 labelled f_1(1)* 295 to 296 labelled f_1(1)* 296 to 297 labelled f_1(1)* 297 to 278 labelled a(1), b(1), fresh_constant(1)* 298 to 282 labelled down_1(1)* 298 to 301 labelled f_flat_1(2)* 298 to 310 labelled down_1(1)* 298 to 318 labelled up_1(3)* 298 to 303 labelled f_flat_1(2)* 298 to 327 labelled up_1(3)* 298 to 306 labelled f_flat_1(2)* 298 to 331 labelled up_1(3)* 298 to 336 labelled f_flat_1(2)* 298 to 353 labelled up_1(2)* 299 to 300 labelled f_1(2)* 300 to 278 labelled a(2)* 301 to 302 labelled down_1(2)* 301 to 311 labelled up_1(3)* 302 to 278 labelled a(2), b(2), fresh_constant(2)* 303 to 304 labelled down_1(2)* 303 to 313 labelled f_flat_1(3)* 303 to 324 labelled up_1(4)* 304 to 305 labelled f_1(2)* 305 to 278 labelled a(2), b(2), fresh_constant(2)* 306 to 307 labelled down_1(2)* 306 to 315 labelled f_flat_1(3)* 306 to 329 labelled up_1(4)* 307 to 308 labelled f_1(2)* 308 to 309 labelled f_1(2)* 309 to 278 labelled a(2), b(2), fresh_constant(2)* 310 to 299 labelled f_1(2)* 310 to 318 labelled f_1(2)* 310 to 327 labelled f_1(2)* 310 to 331 labelled f_1(2)* 311 to 312 labelled f_1(3)* 312 to 278 labelled a(3)* 313 to 314 labelled down_1(3)* 313 to 319 labelled up_1(4)* 314 to 278 labelled a(3), b(3), fresh_constant(3)* 315 to 316 labelled down_1(3)* 315 to 321 labelled f_flat_1(4)* 315 to 328 labelled up_1(5)* 316 to 317 labelled f_1(3)* 317 to 278 labelled a(3), b(3), fresh_constant(3)* 318 to 311 labelled f_1(3)* 319 to 320 labelled f_1(4)* 320 to 278 labelled a(4)* 321 to 322 labelled down_1(4)* 321 to 325 labelled up_1(5)* 322 to 278 labelled a(4), b(4), fresh_constant(4)* 323 to 318 labelled down_1(2)* 323 to 315 labelled f_flat_1(3)* 323 to 327 labelled down_1(2)* 323 to 329 labelled up_1(4)* 323 to 332 labelled f_flat_1(3)* 323 to 331 labelled down_1(2)* 323 to 354 labelled f_flat_1(3)* 323 to 353 labelled down_1(2)* 323 to 370 labelled up_1(3)* 324 to 319 labelled f_1(4)* 325 to 326 labelled f_1(5)* 326 to 278 labelled a(5)* 327 to 324 labelled f_1(3)* 328 to 325 labelled f_1(5)* 329 to 328 labelled f_1(4)* 329 to 366 labelled f_1(4)* 329 to 369 labelled f_1(4)* 330 to 329 labelled down_1(3)* 330 to 344 labelled f_flat_1(4)* 330 to 369 labelled up_1(5)* 330 to 371 labelled f_flat_1(4)* 330 to 370 labelled down_1(3)* 330 to 384 labelled up_1(4)* 331 to 329 labelled f_1(3)* 332 to 333 labelled down_1(3)* 332 to 341 labelled f_flat_1(4)* 332 to 366 labelled up_1(5)* 333 to 334 labelled f_1(3)* 334 to 335 labelled f_1(3)* 335 to 278 labelled a(3)* 336 to 337 labelled down_1(2)* 336 to 332 labelled f_flat_1(3)* 336 to 329 labelled up_1(4)* 337 to 338 labelled f_1(2)* 338 to 339 labelled f_1(2)* 339 to 340 labelled f_1(2)* 340 to 278 labelled a(2)* 341 to 342 labelled down_1(4)* 341 to 348 labelled f_flat_1(5)* 341 to 363 labelled up_1(6)* 342 to 343 labelled f_1(4)* 343 to 278 labelled a(4)* 344 to 345 labelled down_1(4)* 344 to 350 labelled f_flat_1(5)* 344 to 368 labelled up_1(6)* 345 to 346 labelled f_1(4)* 346 to 347 labelled f_1(4)* 347 to 278 labelled a(4)* 348 to 349 labelled down_1(5)* 348 to 359 labelled up_1(6)* 349 to 278 labelled a(5)* 350 to 351 labelled down_1(5)* 350 to 361 labelled f_flat_1(6)* 350 to 367 labelled up_1(7)* 351 to 352 labelled f_1(5)* 352 to 278 labelled a(5)* 353 to 326 labelled b(2)* 353 to 359 labelled b(2)* 353 to 367 labelled b(2)* 353 to 404 labelled b(2)* 354 to 355 labelled down_1(3)* 354 to 344 labelled f_flat_1(4)* 354 to 369 labelled up_1(5)* 355 to 356 labelled f_1(3)* 356 to 357 labelled f_1(3)* 357 to 358 labelled f_1(3)* 358 to 278 labelled a(3)* 359 to 360 labelled f_1(6)* 360 to 278 labelled a(6)* 361 to 362 labelled down_1(6)* 361 to 364 labelled up_1(7)* 362 to 278 labelled a(6)* 363 to 359 labelled f_1(6)* 364 to 365 labelled f_1(7)* 365 to 278 labelled a(7)* 366 to 363 labelled f_1(5)* 367 to 364 labelled f_1(7)* 368 to 367 labelled f_1(6)* 369 to 368 labelled f_1(5)* 369 to 407 labelled f_1(5)* 370 to 360 labelled b(3)* 370 to 364 labelled b(3)* 370 to 401 labelled b(3)* 371 to 372 labelled down_1(4)* 371 to 377 labelled f_flat_1(5)* 371 to 407 labelled up_1(6)* 372 to 373 labelled f_1(4)* 373 to 374 labelled f_1(4)* 374 to 375 labelled f_1(4)* 375 to 278 labelled a(4)* 376 to 369 labelled down_1(4)* 376 to 385 labelled f_flat_1(5)* 376 to 384 labelled down_1(4)* 376 to 417 labelled up_1(5)* 376 to 420 labelled up_1(6)* 377 to 378 labelled down_1(5)* 377 to 381 labelled f_flat_1(6)* 377 to 404 labelled up_1(7)* 378 to 379 labelled f_1(5)* 379 to 380 labelled f_1(5)* 380 to 278 labelled a(5)* 381 to 382 labelled down_1(6)* 381 to 390 labelled f_flat_1(7)* 381 to 401 labelled up_1(8)* 382 to 383 labelled f_1(6)* 383 to 278 labelled a(6)* 384 to 365 labelled b(4)* 384 to 396 labelled b(4)* 385 to 386 labelled down_1(5)* 385 to 392 labelled f_flat_1(6)* 385 to 416 labelled up_1(7)* 386 to 387 labelled f_1(5)* 387 to 388 labelled f_1(5)* 388 to 389 labelled f_1(5)* 389 to 278 labelled a(5)* 390 to 391 labelled down_1(7)* 390 to 396 labelled up_1(8)* 391 to 278 labelled a(7)* 392 to 393 labelled down_1(6)* 392 to 398 labelled f_flat_1(7)* 392 to 414 labelled up_1(8)* 393 to 394 labelled f_1(6)* 394 to 395 labelled f_1(6)* 395 to 278 labelled a(6)* 396 to 397 labelled f_1(8)* 397 to 278 labelled a(8)* 398 to 399 labelled down_1(7)* 398 to 402 labelled f_flat_1(8)* 398 to 408 labelled up_1(9)* 399 to 400 labelled f_1(7)* 400 to 278 labelled a(7)* 401 to 396 labelled f_1(8)* 402 to 403 labelled down_1(8)* 402 to 405 labelled up_1(9)* 403 to 278 labelled a(8)* 404 to 401 labelled f_1(7)* 405 to 406 labelled f_1(9)* 406 to 278 labelled a(9)* 407 to 404 labelled f_1(6)* 408 to 405 labelled f_1(9)* 414 to 408 labelled f_1(8)* 416 to 414 labelled f_1(7)* 417 to 397 labelled b(5)* 419 to 417 labelled down_1(5)* 419 to 420 labelled down_1(5)* 419 to 421 labelled up_1(6)* 420 to 416 labelled f_1(6)* 421 to 406 labelled b(6)* 422 to 421 labelled down_1(6) ---------------------------------------- (14) YES