WORST_CASE(Omega(n^1), O(n^1)) proof of /export/starexec/sandbox/benchmark/theBenchmark.xml # AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (full) of the given DCpxTrs could be proven to be BOUNDS(n^1, n^1). (0) DCpxTrs (1) DerivationalComplexityToRuntimeComplexityProof [BOTH BOUNDS(ID, ID), 0 ms] (2) CpxRelTRS (3) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 56 ms] (4) CpxRelTRS (5) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] (6) CpxTRS (7) CpxTrsMatchBoundsProof [FINISHED, 145 ms] (8) BOUNDS(1, n^1) (9) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] (10) TRS for Loop Detection (11) DecreasingLoopProof [LOWER BOUND(ID), 0 ms] (12) BEST (13) proven lower bound (14) LowerBoundPropagationProof [FINISHED, 0 ms] (15) BOUNDS(n^1, INF) (16) TRS for Loop Detection ---------------------------------------- (0) Obligation: The Derivational Complexity (full) of the given DCpxTrs could be proven to be BOUNDS(n^1, n^1). The TRS R consists of the following rules: 0(0(1(x1))) -> 2(0(3(3(0(1(x1)))))) 0(1(0(x1))) -> 0(1(3(4(0(3(x1)))))) 0(1(0(x1))) -> 2(0(3(0(1(4(x1)))))) 0(1(1(x1))) -> 0(3(1(3(1(x1))))) 0(1(1(x1))) -> 1(3(0(1(4(x1))))) 0(1(1(x1))) -> 0(1(3(1(3(1(x1)))))) 0(1(1(x1))) -> 1(3(2(1(3(0(x1)))))) 0(1(1(x1))) -> 1(3(3(1(4(0(x1)))))) 0(1(1(x1))) -> 3(0(3(1(5(1(x1)))))) 0(1(1(x1))) -> 5(0(3(1(5(1(x1)))))) 0(5(0(x1))) -> 3(0(3(5(0(x1))))) 0(5(0(x1))) -> 3(5(0(0(3(x1))))) 0(5(0(x1))) -> 5(0(3(0(2(x1))))) 0(5(0(x1))) -> 5(0(3(3(0(x1))))) 0(5(0(x1))) -> 4(5(0(3(3(0(x1)))))) 0(5(0(x1))) -> 4(5(0(3(5(0(x1)))))) 0(5(0(x1))) -> 5(3(0(1(3(0(x1)))))) 2(0(0(x1))) -> 0(3(0(3(2(x1))))) 2(0(0(x1))) -> 0(3(3(0(2(3(x1)))))) 2(0(0(x1))) -> 0(3(5(2(0(3(x1)))))) 5(1(0(x1))) -> 3(5(0(1(4(3(x1)))))) 5(1(0(x1))) -> 3(5(1(4(0(3(x1)))))) 5(1(1(x1))) -> 3(1(5(1(x1)))) 5(1(1(x1))) -> 1(3(1(3(5(x1))))) 5(1(1(x1))) -> 1(3(3(3(5(1(x1)))))) 5(1(1(x1))) -> 1(3(5(5(1(4(x1)))))) 0(2(0(1(x1)))) -> 0(2(3(3(0(1(x1)))))) 0(5(1(0(x1)))) -> 0(0(1(3(5(x1))))) 0(5(4(0(x1)))) -> 0(4(5(0(3(x1))))) 2(0(2(0(x1)))) -> 3(0(3(0(2(2(x1)))))) 2(0(4(1(x1)))) -> 2(3(0(1(4(4(x1)))))) 2(0(5(0(x1)))) -> 0(0(3(5(2(x1))))) 2(2(4(1(x1)))) -> 3(2(4(3(2(1(x1)))))) 5(1(0(1(x1)))) -> 0(5(1(4(3(1(x1)))))) 5(1(1(0(x1)))) -> 0(5(1(5(1(x1))))) 5(1(2(0(x1)))) -> 3(1(3(5(0(2(x1)))))) 5(1(5(0(x1)))) -> 5(3(5(0(1(x1))))) 5(2(0(1(x1)))) -> 5(1(0(3(2(x1))))) 5(3(1(1(x1)))) -> 5(3(1(3(1(5(x1)))))) 5(4(1(1(x1)))) -> 5(1(4(1(4(5(x1)))))) 5(5(1(0(x1)))) -> 5(0(5(1(3(x1))))) 5(5(1(1(x1)))) -> 5(1(3(5(0(1(x1)))))) 0(2(4(1(0(x1))))) -> 2(4(0(0(1(3(x1)))))) 0(5(5(1(1(x1))))) -> 5(1(3(5(0(1(x1)))))) 2(2(2(4(1(x1))))) -> 1(2(2(1(4(2(x1)))))) 2(5(0(1(1(x1))))) -> 5(1(2(0(1(3(x1)))))) 5(0(2(4(1(x1))))) -> 5(1(4(0(3(2(x1)))))) 5(2(4(1(0(x1))))) -> 0(2(3(4(5(1(x1)))))) 5(3(0(4(1(x1))))) -> 5(3(0(1(4(1(x1)))))) 5(3(4(1(1(x1))))) -> 1(4(3(5(2(1(x1)))))) S is empty. Rewrite Strategy: FULL ---------------------------------------- (1) DerivationalComplexityToRuntimeComplexityProof (BOTH BOUNDS(ID, ID)) The following rules have been added to S to convert the given derivational complexity problem to a runtime complexity problem: encArg(1(x_1)) -> 1(encArg(x_1)) encArg(3(x_1)) -> 3(encArg(x_1)) encArg(4(x_1)) -> 4(encArg(x_1)) encArg(cons_0(x_1)) -> 0(encArg(x_1)) encArg(cons_2(x_1)) -> 2(encArg(x_1)) encArg(cons_5(x_1)) -> 5(encArg(x_1)) encode_0(x_1) -> 0(encArg(x_1)) encode_1(x_1) -> 1(encArg(x_1)) encode_2(x_1) -> 2(encArg(x_1)) encode_3(x_1) -> 3(encArg(x_1)) encode_4(x_1) -> 4(encArg(x_1)) encode_5(x_1) -> 5(encArg(x_1)) ---------------------------------------- (2) Obligation: The Runtime Complexity (full) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^1). The TRS R consists of the following rules: 0(0(1(x1))) -> 2(0(3(3(0(1(x1)))))) 0(1(0(x1))) -> 0(1(3(4(0(3(x1)))))) 0(1(0(x1))) -> 2(0(3(0(1(4(x1)))))) 0(1(1(x1))) -> 0(3(1(3(1(x1))))) 0(1(1(x1))) -> 1(3(0(1(4(x1))))) 0(1(1(x1))) -> 0(1(3(1(3(1(x1)))))) 0(1(1(x1))) -> 1(3(2(1(3(0(x1)))))) 0(1(1(x1))) -> 1(3(3(1(4(0(x1)))))) 0(1(1(x1))) -> 3(0(3(1(5(1(x1)))))) 0(1(1(x1))) -> 5(0(3(1(5(1(x1)))))) 0(5(0(x1))) -> 3(0(3(5(0(x1))))) 0(5(0(x1))) -> 3(5(0(0(3(x1))))) 0(5(0(x1))) -> 5(0(3(0(2(x1))))) 0(5(0(x1))) -> 5(0(3(3(0(x1))))) 0(5(0(x1))) -> 4(5(0(3(3(0(x1)))))) 0(5(0(x1))) -> 4(5(0(3(5(0(x1)))))) 0(5(0(x1))) -> 5(3(0(1(3(0(x1)))))) 2(0(0(x1))) -> 0(3(0(3(2(x1))))) 2(0(0(x1))) -> 0(3(3(0(2(3(x1)))))) 2(0(0(x1))) -> 0(3(5(2(0(3(x1)))))) 5(1(0(x1))) -> 3(5(0(1(4(3(x1)))))) 5(1(0(x1))) -> 3(5(1(4(0(3(x1)))))) 5(1(1(x1))) -> 3(1(5(1(x1)))) 5(1(1(x1))) -> 1(3(1(3(5(x1))))) 5(1(1(x1))) -> 1(3(3(3(5(1(x1)))))) 5(1(1(x1))) -> 1(3(5(5(1(4(x1)))))) 0(2(0(1(x1)))) -> 0(2(3(3(0(1(x1)))))) 0(5(1(0(x1)))) -> 0(0(1(3(5(x1))))) 0(5(4(0(x1)))) -> 0(4(5(0(3(x1))))) 2(0(2(0(x1)))) -> 3(0(3(0(2(2(x1)))))) 2(0(4(1(x1)))) -> 2(3(0(1(4(4(x1)))))) 2(0(5(0(x1)))) -> 0(0(3(5(2(x1))))) 2(2(4(1(x1)))) -> 3(2(4(3(2(1(x1)))))) 5(1(0(1(x1)))) -> 0(5(1(4(3(1(x1)))))) 5(1(1(0(x1)))) -> 0(5(1(5(1(x1))))) 5(1(2(0(x1)))) -> 3(1(3(5(0(2(x1)))))) 5(1(5(0(x1)))) -> 5(3(5(0(1(x1))))) 5(2(0(1(x1)))) -> 5(1(0(3(2(x1))))) 5(3(1(1(x1)))) -> 5(3(1(3(1(5(x1)))))) 5(4(1(1(x1)))) -> 5(1(4(1(4(5(x1)))))) 5(5(1(0(x1)))) -> 5(0(5(1(3(x1))))) 5(5(1(1(x1)))) -> 5(1(3(5(0(1(x1)))))) 0(2(4(1(0(x1))))) -> 2(4(0(0(1(3(x1)))))) 0(5(5(1(1(x1))))) -> 5(1(3(5(0(1(x1)))))) 2(2(2(4(1(x1))))) -> 1(2(2(1(4(2(x1)))))) 2(5(0(1(1(x1))))) -> 5(1(2(0(1(3(x1)))))) 5(0(2(4(1(x1))))) -> 5(1(4(0(3(2(x1)))))) 5(2(4(1(0(x1))))) -> 0(2(3(4(5(1(x1)))))) 5(3(0(4(1(x1))))) -> 5(3(0(1(4(1(x1)))))) 5(3(4(1(1(x1))))) -> 1(4(3(5(2(1(x1)))))) The (relative) TRS S consists of the following rules: encArg(1(x_1)) -> 1(encArg(x_1)) encArg(3(x_1)) -> 3(encArg(x_1)) encArg(4(x_1)) -> 4(encArg(x_1)) encArg(cons_0(x_1)) -> 0(encArg(x_1)) encArg(cons_2(x_1)) -> 2(encArg(x_1)) encArg(cons_5(x_1)) -> 5(encArg(x_1)) encode_0(x_1) -> 0(encArg(x_1)) encode_1(x_1) -> 1(encArg(x_1)) encode_2(x_1) -> 2(encArg(x_1)) encode_3(x_1) -> 3(encArg(x_1)) encode_4(x_1) -> 4(encArg(x_1)) encode_5(x_1) -> 5(encArg(x_1)) Rewrite Strategy: FULL ---------------------------------------- (3) SInnermostTerminationProof (BOTH CONCRETE BOUNDS(ID, ID)) proved innermost termination of relative rules ---------------------------------------- (4) Obligation: The Runtime Complexity (full) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^1). The TRS R consists of the following rules: 0(0(1(x1))) -> 2(0(3(3(0(1(x1)))))) 0(1(0(x1))) -> 0(1(3(4(0(3(x1)))))) 0(1(0(x1))) -> 2(0(3(0(1(4(x1)))))) 0(1(1(x1))) -> 0(3(1(3(1(x1))))) 0(1(1(x1))) -> 1(3(0(1(4(x1))))) 0(1(1(x1))) -> 0(1(3(1(3(1(x1)))))) 0(1(1(x1))) -> 1(3(2(1(3(0(x1)))))) 0(1(1(x1))) -> 1(3(3(1(4(0(x1)))))) 0(1(1(x1))) -> 3(0(3(1(5(1(x1)))))) 0(1(1(x1))) -> 5(0(3(1(5(1(x1)))))) 0(5(0(x1))) -> 3(0(3(5(0(x1))))) 0(5(0(x1))) -> 3(5(0(0(3(x1))))) 0(5(0(x1))) -> 5(0(3(0(2(x1))))) 0(5(0(x1))) -> 5(0(3(3(0(x1))))) 0(5(0(x1))) -> 4(5(0(3(3(0(x1)))))) 0(5(0(x1))) -> 4(5(0(3(5(0(x1)))))) 0(5(0(x1))) -> 5(3(0(1(3(0(x1)))))) 2(0(0(x1))) -> 0(3(0(3(2(x1))))) 2(0(0(x1))) -> 0(3(3(0(2(3(x1)))))) 2(0(0(x1))) -> 0(3(5(2(0(3(x1)))))) 5(1(0(x1))) -> 3(5(0(1(4(3(x1)))))) 5(1(0(x1))) -> 3(5(1(4(0(3(x1)))))) 5(1(1(x1))) -> 3(1(5(1(x1)))) 5(1(1(x1))) -> 1(3(1(3(5(x1))))) 5(1(1(x1))) -> 1(3(3(3(5(1(x1)))))) 5(1(1(x1))) -> 1(3(5(5(1(4(x1)))))) 0(2(0(1(x1)))) -> 0(2(3(3(0(1(x1)))))) 0(5(1(0(x1)))) -> 0(0(1(3(5(x1))))) 0(5(4(0(x1)))) -> 0(4(5(0(3(x1))))) 2(0(2(0(x1)))) -> 3(0(3(0(2(2(x1)))))) 2(0(4(1(x1)))) -> 2(3(0(1(4(4(x1)))))) 2(0(5(0(x1)))) -> 0(0(3(5(2(x1))))) 2(2(4(1(x1)))) -> 3(2(4(3(2(1(x1)))))) 5(1(0(1(x1)))) -> 0(5(1(4(3(1(x1)))))) 5(1(1(0(x1)))) -> 0(5(1(5(1(x1))))) 5(1(2(0(x1)))) -> 3(1(3(5(0(2(x1)))))) 5(1(5(0(x1)))) -> 5(3(5(0(1(x1))))) 5(2(0(1(x1)))) -> 5(1(0(3(2(x1))))) 5(3(1(1(x1)))) -> 5(3(1(3(1(5(x1)))))) 5(4(1(1(x1)))) -> 5(1(4(1(4(5(x1)))))) 5(5(1(0(x1)))) -> 5(0(5(1(3(x1))))) 5(5(1(1(x1)))) -> 5(1(3(5(0(1(x1)))))) 0(2(4(1(0(x1))))) -> 2(4(0(0(1(3(x1)))))) 0(5(5(1(1(x1))))) -> 5(1(3(5(0(1(x1)))))) 2(2(2(4(1(x1))))) -> 1(2(2(1(4(2(x1)))))) 2(5(0(1(1(x1))))) -> 5(1(2(0(1(3(x1)))))) 5(0(2(4(1(x1))))) -> 5(1(4(0(3(2(x1)))))) 5(2(4(1(0(x1))))) -> 0(2(3(4(5(1(x1)))))) 5(3(0(4(1(x1))))) -> 5(3(0(1(4(1(x1)))))) 5(3(4(1(1(x1))))) -> 1(4(3(5(2(1(x1)))))) The (relative) TRS S consists of the following rules: encArg(1(x_1)) -> 1(encArg(x_1)) encArg(3(x_1)) -> 3(encArg(x_1)) encArg(4(x_1)) -> 4(encArg(x_1)) encArg(cons_0(x_1)) -> 0(encArg(x_1)) encArg(cons_2(x_1)) -> 2(encArg(x_1)) encArg(cons_5(x_1)) -> 5(encArg(x_1)) encode_0(x_1) -> 0(encArg(x_1)) encode_1(x_1) -> 1(encArg(x_1)) encode_2(x_1) -> 2(encArg(x_1)) encode_3(x_1) -> 3(encArg(x_1)) encode_4(x_1) -> 4(encArg(x_1)) encode_5(x_1) -> 5(encArg(x_1)) Rewrite Strategy: FULL ---------------------------------------- (5) RelTrsToTrsProof (UPPER BOUND(ID)) transformed relative TRS to TRS ---------------------------------------- (6) Obligation: The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^1). The TRS R consists of the following rules: 0(0(1(x1))) -> 2(0(3(3(0(1(x1)))))) 0(1(0(x1))) -> 0(1(3(4(0(3(x1)))))) 0(1(0(x1))) -> 2(0(3(0(1(4(x1)))))) 0(1(1(x1))) -> 0(3(1(3(1(x1))))) 0(1(1(x1))) -> 1(3(0(1(4(x1))))) 0(1(1(x1))) -> 0(1(3(1(3(1(x1)))))) 0(1(1(x1))) -> 1(3(2(1(3(0(x1)))))) 0(1(1(x1))) -> 1(3(3(1(4(0(x1)))))) 0(1(1(x1))) -> 3(0(3(1(5(1(x1)))))) 0(1(1(x1))) -> 5(0(3(1(5(1(x1)))))) 0(5(0(x1))) -> 3(0(3(5(0(x1))))) 0(5(0(x1))) -> 3(5(0(0(3(x1))))) 0(5(0(x1))) -> 5(0(3(0(2(x1))))) 0(5(0(x1))) -> 5(0(3(3(0(x1))))) 0(5(0(x1))) -> 4(5(0(3(3(0(x1)))))) 0(5(0(x1))) -> 4(5(0(3(5(0(x1)))))) 0(5(0(x1))) -> 5(3(0(1(3(0(x1)))))) 2(0(0(x1))) -> 0(3(0(3(2(x1))))) 2(0(0(x1))) -> 0(3(3(0(2(3(x1)))))) 2(0(0(x1))) -> 0(3(5(2(0(3(x1)))))) 5(1(0(x1))) -> 3(5(0(1(4(3(x1)))))) 5(1(0(x1))) -> 3(5(1(4(0(3(x1)))))) 5(1(1(x1))) -> 3(1(5(1(x1)))) 5(1(1(x1))) -> 1(3(1(3(5(x1))))) 5(1(1(x1))) -> 1(3(3(3(5(1(x1)))))) 5(1(1(x1))) -> 1(3(5(5(1(4(x1)))))) 0(2(0(1(x1)))) -> 0(2(3(3(0(1(x1)))))) 0(5(1(0(x1)))) -> 0(0(1(3(5(x1))))) 0(5(4(0(x1)))) -> 0(4(5(0(3(x1))))) 2(0(2(0(x1)))) -> 3(0(3(0(2(2(x1)))))) 2(0(4(1(x1)))) -> 2(3(0(1(4(4(x1)))))) 2(0(5(0(x1)))) -> 0(0(3(5(2(x1))))) 2(2(4(1(x1)))) -> 3(2(4(3(2(1(x1)))))) 5(1(0(1(x1)))) -> 0(5(1(4(3(1(x1)))))) 5(1(1(0(x1)))) -> 0(5(1(5(1(x1))))) 5(1(2(0(x1)))) -> 3(1(3(5(0(2(x1)))))) 5(1(5(0(x1)))) -> 5(3(5(0(1(x1))))) 5(2(0(1(x1)))) -> 5(1(0(3(2(x1))))) 5(3(1(1(x1)))) -> 5(3(1(3(1(5(x1)))))) 5(4(1(1(x1)))) -> 5(1(4(1(4(5(x1)))))) 5(5(1(0(x1)))) -> 5(0(5(1(3(x1))))) 5(5(1(1(x1)))) -> 5(1(3(5(0(1(x1)))))) 0(2(4(1(0(x1))))) -> 2(4(0(0(1(3(x1)))))) 0(5(5(1(1(x1))))) -> 5(1(3(5(0(1(x1)))))) 2(2(2(4(1(x1))))) -> 1(2(2(1(4(2(x1)))))) 2(5(0(1(1(x1))))) -> 5(1(2(0(1(3(x1)))))) 5(0(2(4(1(x1))))) -> 5(1(4(0(3(2(x1)))))) 5(2(4(1(0(x1))))) -> 0(2(3(4(5(1(x1)))))) 5(3(0(4(1(x1))))) -> 5(3(0(1(4(1(x1)))))) 5(3(4(1(1(x1))))) -> 1(4(3(5(2(1(x1)))))) encArg(1(x_1)) -> 1(encArg(x_1)) encArg(3(x_1)) -> 3(encArg(x_1)) encArg(4(x_1)) -> 4(encArg(x_1)) encArg(cons_0(x_1)) -> 0(encArg(x_1)) encArg(cons_2(x_1)) -> 2(encArg(x_1)) encArg(cons_5(x_1)) -> 5(encArg(x_1)) encode_0(x_1) -> 0(encArg(x_1)) encode_1(x_1) -> 1(encArg(x_1)) encode_2(x_1) -> 2(encArg(x_1)) encode_3(x_1) -> 3(encArg(x_1)) encode_4(x_1) -> 4(encArg(x_1)) encode_5(x_1) -> 5(encArg(x_1)) S is empty. Rewrite Strategy: FULL ---------------------------------------- (7) CpxTrsMatchBoundsProof (FINISHED) A linear upper bound on the runtime complexity of the TRS R could be shown with a Match Bound [MATCHBOUNDS1,MATCHBOUNDS2] of 4. The certificate found is represented by the following graph. 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271, 272, 273, 274, 275, 276, 277, 278, 279, 280, 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, 409, 410, 411, 412, 413, 414, 415, 416, 417, 418, 419, 420, 421, 422, 423, 424, 425, 426, 427, 428, 429, 430, 431, 432, 433, 434, 435, 436, 437, 438, 439, 440, 441, 442, 443, 444, 445, 446, 447, 448, 449, 450, 451, 452, 453, 454, 455, 456, 457, 458, 459, 460, 461, 462, 463, 464, 465, 466, 467, 468, 469, 470, 471, 472, 473, 474, 475, 476, 477, 478, 479, 480, 481, 482, 483, 484, 485, 486, 487, 488, 489, 490, 491, 492, 493, 494, 495, 496, 497, 498, 499, 500, 501, 502, 503, 504, 505, 506, 507, 508, 509, 510, 511, 512, 513, 514, 515, 516, 517, 518, 519, 520, 521, 522, 523, 524, 525, 526, 527, 528, 529, 530, 531, 532, 533, 534, 535, 536, 537, 538, 539, 540, 541, 542, 543, 544, 545, 546, 547, 548, 549, 550, 551, 552, 553, 554, 555, 556, 557, 558, 559, 560, 561, 562, 563, 564, 565, 566, 567, 568, 569, 570, 571, 572, 573, 574, 575] {(65,66,[0_1|0, 2_1|0, 5_1|0, encArg_1|0, encode_0_1|0, encode_1_1|0, encode_2_1|0, encode_3_1|0, encode_4_1|0, encode_5_1|0]), (65,67,[0_1|1]), (65,71,[1_1|1]), (65,75,[0_1|1]), (65,80,[1_1|1]), (65,85,[1_1|1]), (65,90,[3_1|1]), (65,95,[5_1|1]), (65,100,[3_1|1]), (65,103,[1_1|1]), (65,107,[1_1|1]), (65,112,[1_1|1]), (65,117,[5_1|1]), (65,122,[1_1|1]), (65,127,[5_1|1]), (65,132,[1_1|1, 3_1|1, 4_1|1, 0_1|1, 2_1|1, 5_1|1]), (65,133,[2_1|2]), (65,138,[0_1|2]), 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(418,157,[1_1|3]), (418,487,[1_1|3]), (418,491,[1_1|3]), (418,496,[1_1|3]), (419,420,[0_1|3]), (420,421,[3_1|3]), (421,422,[1_1|3]), (422,423,[5_1|3]), (423,152,[1_1|3]), (423,161,[1_1|3]), (423,166,[1_1|3]), (423,268,[1_1|3]), (423,296,[1_1|3]), (423,300,[1_1|3]), (423,305,[1_1|3]), (423,342,[1_1|3]), (423,139,[1_1|3]), (423,157,[1_1|3]), (423,487,[1_1|3]), (423,491,[1_1|3]), (423,496,[1_1|3]), (424,425,[0_1|3]), (425,426,[3_1|3]), (426,427,[3_1|3]), (427,428,[0_1|3]), (428,139,[1_1|3]), (428,157,[1_1|3]), (428,214,[1_1|3]), (428,487,[1_1|3]), (428,491,[1_1|3]), (428,496,[1_1|3]), (428,544,[1_1|3]), (429,430,[0_1|3]), (430,431,[3_1|3]), (431,432,[5_1|3]), (432,177,[0_1|3]), (432,190,[0_1|3]), (432,194,[0_1|3]), (432,353,[0_1|3]), (433,434,[5_1|3]), (434,435,[0_1|3]), (435,436,[0_1|3]), (436,177,[3_1|3]), (436,190,[3_1|3]), (436,194,[3_1|3]), (436,353,[3_1|3]), (437,438,[0_1|3]), (438,439,[3_1|3]), (439,440,[0_1|3]), (440,177,[2_1|3]), (440,190,[2_1|3]), (440,194,[2_1|3]), (440,353,[2_1|3]), (441,442,[0_1|3]), (442,443,[3_1|3]), (443,444,[3_1|3]), (444,177,[0_1|3]), (444,190,[0_1|3]), (444,194,[0_1|3]), (444,353,[0_1|3]), (445,446,[5_1|3]), (446,447,[0_1|3]), (447,448,[3_1|3]), (448,449,[3_1|3]), (449,177,[0_1|3]), (449,190,[0_1|3]), (449,194,[0_1|3]), (449,353,[0_1|3]), (450,451,[5_1|3]), (451,452,[0_1|3]), (452,453,[3_1|3]), (453,454,[5_1|3]), (454,177,[0_1|3]), (454,190,[0_1|3]), (454,194,[0_1|3]), (454,353,[0_1|3]), (455,456,[3_1|3]), (456,457,[0_1|3]), (457,458,[1_1|3]), (458,459,[3_1|3]), (459,177,[0_1|3]), (459,190,[0_1|3]), (459,194,[0_1|3]), (459,353,[0_1|3]), (460,461,[0_1|3]), (461,462,[1_1|3]), (462,463,[3_1|3]), (463,325,[5_1|3]), (463,288,[5_1|3]), (463,310,[5_1|3]), (463,479,[5_1|3]), (463,542,[5_1|3]), (463,529,[5_1|3]), (463,556,[5_1|3]), (464,465,[1_1|3]), (465,466,[3_1|3]), (466,467,[5_1|3]), (467,468,[0_1|3]), (468,134,[2_1|3]), (468,144,[2_1|3]), (468,362,[2_1|3]), (468,572,[2_1|3]), (469,470,[5_1|3]), (470,471,[0_1|3]), (471,472,[1_1|3]), (472,473,[4_1|3]), (473,138,[3_1|3]), (473,148,[3_1|3]), (473,156,[3_1|3]), (473,212,[3_1|3]), (473,216,[3_1|3]), (473,225,[3_1|3]), (473,235,[3_1|3]), (473,239,[3_1|3]), (473,244,[3_1|3]), (473,259,[3_1|3]), (473,288,[3_1|3]), (473,310,[3_1|3]), (473,327,[3_1|3]), (473,479,[3_1|3]), (473,213,[3_1|3]), (473,260,[3_1|3]), (473,542,[3_1|3]), (473,543,[3_1|3]), (474,475,[5_1|3]), (475,476,[1_1|3]), (476,477,[4_1|3]), (477,478,[0_1|3]), (478,138,[3_1|3]), (478,148,[3_1|3]), (478,156,[3_1|3]), (478,212,[3_1|3]), (478,216,[3_1|3]), (478,225,[3_1|3]), (478,235,[3_1|3]), (478,239,[3_1|3]), (478,244,[3_1|3]), (478,259,[3_1|3]), (478,288,[3_1|3]), (478,310,[3_1|3]), (478,327,[3_1|3]), (478,479,[3_1|3]), (478,213,[3_1|3]), (478,260,[3_1|3]), (478,542,[3_1|3]), (478,543,[3_1|3]), (479,480,[5_1|3]), (480,481,[1_1|3]), (481,482,[4_1|3]), (482,483,[3_1|3]), (483,139,[1_1|3]), (483,157,[1_1|3]), (483,214,[1_1|3]), (483,487,[1_1|3]), (483,491,[1_1|3]), (483,496,[1_1|3]), (483,544,[1_1|3]), (484,485,[1_1|3]), (485,486,[5_1|3]), (486,152,[1_1|3]), (486,161,[1_1|3]), (486,166,[1_1|3]), (486,268,[1_1|3]), (486,296,[1_1|3]), (486,300,[1_1|3]), (486,305,[1_1|3]), (486,342,[1_1|3]), (486,487,[1_1|3]), (486,491,[1_1|3]), (486,496,[1_1|3]), (486,139,[1_1|3]), (486,157,[1_1|3]), (487,488,[3_1|3]), (488,489,[1_1|3]), (489,490,[3_1|3]), (490,152,[5_1|3]), (490,161,[5_1|3]), (490,166,[5_1|3]), (490,268,[5_1|3]), (490,296,[5_1|3]), (490,300,[5_1|3]), (490,305,[5_1|3]), (490,342,[5_1|3]), (490,487,[5_1|3]), (490,491,[5_1|3]), (490,496,[5_1|3]), (490,139,[5_1|3]), (490,157,[5_1|3]), (491,492,[3_1|3]), (492,493,[3_1|3]), (493,494,[3_1|3]), (494,495,[5_1|3]), (495,152,[1_1|3]), (495,161,[1_1|3]), (495,166,[1_1|3]), (495,268,[1_1|3]), (495,296,[1_1|3]), (495,300,[1_1|3]), (495,305,[1_1|3]), (495,342,[1_1|3]), (495,487,[1_1|3]), (495,491,[1_1|3]), (495,496,[1_1|3]), (495,139,[1_1|3]), (495,157,[1_1|3]), (496,497,[3_1|3]), (497,498,[5_1|3]), (498,499,[5_1|3]), (499,500,[1_1|3]), (500,152,[4_1|3]), (500,161,[4_1|3]), (500,166,[4_1|3]), (500,268,[4_1|3]), (500,296,[4_1|3]), (500,300,[4_1|3]), (500,305,[4_1|3]), (500,342,[4_1|3]), (500,487,[4_1|3]), (500,491,[4_1|3]), (500,496,[4_1|3]), (500,139,[4_1|3]), (500,157,[4_1|3]), (501,502,[3_1|3]), (502,503,[5_1|3]), (503,504,[0_1|3]), (504,177,[1_1|3]), (504,190,[1_1|3]), (504,194,[1_1|3]), (504,353,[1_1|3]), (505,506,[2_1|3]), (506,507,[3_1|3]), (507,508,[3_1|3]), (508,509,[0_1|3]), (509,139,[1_1|3]), (509,157,[1_1|3]), (509,214,[1_1|3]), (509,487,[1_1|3]), (509,491,[1_1|3]), (509,496,[1_1|3]), (509,544,[1_1|3]), (510,511,[3_1|3]), (511,512,[0_1|3]), (512,513,[3_1|3]), (513,213,[2_1|3]), (513,260,[2_1|3]), (513,543,[2_1|3]), (513,479,[2_1|3]), (514,515,[3_1|3]), (515,516,[3_1|3]), (516,517,[0_1|3]), (517,518,[2_1|3]), (518,213,[3_1|3]), (518,260,[3_1|3]), (518,543,[3_1|3]), (518,479,[3_1|3]), (519,520,[3_1|3]), (520,521,[5_1|3]), (521,522,[2_1|3]), (522,523,[0_1|3]), (523,213,[3_1|3]), (523,260,[3_1|3]), (523,543,[3_1|3]), (523,479,[3_1|3]), (524,525,[1_1|2]), (525,526,[3_1|2]), (526,527,[5_1|2]), (527,528,[0_1|2]), (527,138,[0_1|2]), (527,143,[2_1|2]), (527,148,[0_1|2]), (527,152,[1_1|2]), (527,156,[0_1|2]), (527,161,[1_1|2]), (527,166,[1_1|2]), (527,171,[3_1|2]), (527,176,[5_1|2]), (527,381,[0_1|3]), (527,386,[2_1|3]), (527,391,[0_1|3]), (527,395,[1_1|3]), (527,399,[0_1|3]), (527,404,[1_1|3]), (527,409,[1_1|3]), (527,414,[3_1|3]), (527,419,[5_1|3]), (528,132,[1_1|2]), (528,152,[1_1|2]), (528,161,[1_1|2]), (528,166,[1_1|2]), (528,268,[1_1|2]), (528,296,[1_1|2]), (528,300,[1_1|2]), (528,305,[1_1|2]), (528,342,[1_1|2]), (529,530,[0_1|3]), (530,531,[5_1|3]), (531,532,[1_1|3]), (532,325,[3_1|3]), (532,288,[3_1|3]), (532,310,[3_1|3]), (532,479,[3_1|3]), (532,542,[3_1|3]), (533,534,[0_1|3]), (534,535,[3_1|3]), (535,536,[3_1|3]), (536,537,[0_1|3]), (537,234,[1_1|3]), (538,539,[1_1|3]), (539,540,[0_1|3]), (540,541,[3_1|3]), (541,139,[2_1|3]), (541,157,[2_1|3]), (541,214,[2_1|3]), (541,487,[2_1|3]), (541,491,[2_1|3]), (541,496,[2_1|3]), (541,544,[2_1|3]), (542,543,[0_1|3]), (543,544,[1_1|3]), (544,545,[3_1|3]), (545,288,[5_1|3]), (545,310,[5_1|3]), (545,479,[5_1|3]), (545,542,[5_1|3]), (545,529,[5_1|3]), (545,556,[5_1|3]), (546,547,[0_1|4]), (547,548,[3_1|4]), (548,549,[3_1|4]), (549,550,[0_1|4]), (550,462,[1_1|4]), (551,552,[3_1|3]), (552,553,[1_1|3]), (553,554,[3_1|3]), (554,555,[1_1|3]), (555,296,[5_1|3]), (555,300,[5_1|3]), (555,305,[5_1|3]), (555,487,[5_1|3]), (555,491,[5_1|3]), (555,496,[5_1|3]), (556,557,[1_1|3]), (557,558,[3_1|3]), (558,559,[5_1|3]), (559,560,[0_1|3]), (560,296,[1_1|3]), (560,300,[1_1|3]), (560,305,[1_1|3]), (560,487,[1_1|3]), (560,491,[1_1|3]), (560,496,[1_1|3]), (561,562,[5_1|4]), (562,563,[0_1|4]), (563,564,[1_1|4]), (564,565,[4_1|4]), (565,540,[3_1|4]), (566,567,[5_1|4]), (567,568,[1_1|4]), (568,569,[4_1|4]), (569,570,[0_1|4]), (570,540,[3_1|4]), (571,572,[0_1|4]), (572,573,[3_1|4]), (573,574,[3_1|4]), (574,575,[0_1|4]), (575,544,[1_1|4])}" ---------------------------------------- (8) BOUNDS(1, n^1) ---------------------------------------- (9) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) Transformed a relative TRS into a decreasing-loop problem. ---------------------------------------- (10) Obligation: Analyzing the following TRS for decreasing loops: The Runtime Complexity (full) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^1). The TRS R consists of the following rules: 0(0(1(x1))) -> 2(0(3(3(0(1(x1)))))) 0(1(0(x1))) -> 0(1(3(4(0(3(x1)))))) 0(1(0(x1))) -> 2(0(3(0(1(4(x1)))))) 0(1(1(x1))) -> 0(3(1(3(1(x1))))) 0(1(1(x1))) -> 1(3(0(1(4(x1))))) 0(1(1(x1))) -> 0(1(3(1(3(1(x1)))))) 0(1(1(x1))) -> 1(3(2(1(3(0(x1)))))) 0(1(1(x1))) -> 1(3(3(1(4(0(x1)))))) 0(1(1(x1))) -> 3(0(3(1(5(1(x1)))))) 0(1(1(x1))) -> 5(0(3(1(5(1(x1)))))) 0(5(0(x1))) -> 3(0(3(5(0(x1))))) 0(5(0(x1))) -> 3(5(0(0(3(x1))))) 0(5(0(x1))) -> 5(0(3(0(2(x1))))) 0(5(0(x1))) -> 5(0(3(3(0(x1))))) 0(5(0(x1))) -> 4(5(0(3(3(0(x1)))))) 0(5(0(x1))) -> 4(5(0(3(5(0(x1)))))) 0(5(0(x1))) -> 5(3(0(1(3(0(x1)))))) 2(0(0(x1))) -> 0(3(0(3(2(x1))))) 2(0(0(x1))) -> 0(3(3(0(2(3(x1)))))) 2(0(0(x1))) -> 0(3(5(2(0(3(x1)))))) 5(1(0(x1))) -> 3(5(0(1(4(3(x1)))))) 5(1(0(x1))) -> 3(5(1(4(0(3(x1)))))) 5(1(1(x1))) -> 3(1(5(1(x1)))) 5(1(1(x1))) -> 1(3(1(3(5(x1))))) 5(1(1(x1))) -> 1(3(3(3(5(1(x1)))))) 5(1(1(x1))) -> 1(3(5(5(1(4(x1)))))) 0(2(0(1(x1)))) -> 0(2(3(3(0(1(x1)))))) 0(5(1(0(x1)))) -> 0(0(1(3(5(x1))))) 0(5(4(0(x1)))) -> 0(4(5(0(3(x1))))) 2(0(2(0(x1)))) -> 3(0(3(0(2(2(x1)))))) 2(0(4(1(x1)))) -> 2(3(0(1(4(4(x1)))))) 2(0(5(0(x1)))) -> 0(0(3(5(2(x1))))) 2(2(4(1(x1)))) -> 3(2(4(3(2(1(x1)))))) 5(1(0(1(x1)))) -> 0(5(1(4(3(1(x1)))))) 5(1(1(0(x1)))) -> 0(5(1(5(1(x1))))) 5(1(2(0(x1)))) -> 3(1(3(5(0(2(x1)))))) 5(1(5(0(x1)))) -> 5(3(5(0(1(x1))))) 5(2(0(1(x1)))) -> 5(1(0(3(2(x1))))) 5(3(1(1(x1)))) -> 5(3(1(3(1(5(x1)))))) 5(4(1(1(x1)))) -> 5(1(4(1(4(5(x1)))))) 5(5(1(0(x1)))) -> 5(0(5(1(3(x1))))) 5(5(1(1(x1)))) -> 5(1(3(5(0(1(x1)))))) 0(2(4(1(0(x1))))) -> 2(4(0(0(1(3(x1)))))) 0(5(5(1(1(x1))))) -> 5(1(3(5(0(1(x1)))))) 2(2(2(4(1(x1))))) -> 1(2(2(1(4(2(x1)))))) 2(5(0(1(1(x1))))) -> 5(1(2(0(1(3(x1)))))) 5(0(2(4(1(x1))))) -> 5(1(4(0(3(2(x1)))))) 5(2(4(1(0(x1))))) -> 0(2(3(4(5(1(x1)))))) 5(3(0(4(1(x1))))) -> 5(3(0(1(4(1(x1)))))) 5(3(4(1(1(x1))))) -> 1(4(3(5(2(1(x1)))))) The (relative) TRS S consists of the following rules: encArg(1(x_1)) -> 1(encArg(x_1)) encArg(3(x_1)) -> 3(encArg(x_1)) encArg(4(x_1)) -> 4(encArg(x_1)) encArg(cons_0(x_1)) -> 0(encArg(x_1)) encArg(cons_2(x_1)) -> 2(encArg(x_1)) encArg(cons_5(x_1)) -> 5(encArg(x_1)) encode_0(x_1) -> 0(encArg(x_1)) encode_1(x_1) -> 1(encArg(x_1)) encode_2(x_1) -> 2(encArg(x_1)) encode_3(x_1) -> 3(encArg(x_1)) encode_4(x_1) -> 4(encArg(x_1)) encode_5(x_1) -> 5(encArg(x_1)) Rewrite Strategy: FULL ---------------------------------------- (11) DecreasingLoopProof (LOWER BOUND(ID)) The following loop(s) give(s) rise to the lower bound Omega(n^1): The rewrite sequence 5(1(1(x1))) ->^+ 1(3(3(3(5(1(x1)))))) gives rise to a decreasing loop by considering the right hand sides subterm at position [0,0,0,0]. The pumping substitution is [x1 / 1(x1)]. The result substitution is [ ]. ---------------------------------------- (12) Complex Obligation (BEST) ---------------------------------------- (13) Obligation: Proved the lower bound n^1 for the following obligation: The Runtime Complexity (full) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^1). The TRS R consists of the following rules: 0(0(1(x1))) -> 2(0(3(3(0(1(x1)))))) 0(1(0(x1))) -> 0(1(3(4(0(3(x1)))))) 0(1(0(x1))) -> 2(0(3(0(1(4(x1)))))) 0(1(1(x1))) -> 0(3(1(3(1(x1))))) 0(1(1(x1))) -> 1(3(0(1(4(x1))))) 0(1(1(x1))) -> 0(1(3(1(3(1(x1)))))) 0(1(1(x1))) -> 1(3(2(1(3(0(x1)))))) 0(1(1(x1))) -> 1(3(3(1(4(0(x1)))))) 0(1(1(x1))) -> 3(0(3(1(5(1(x1)))))) 0(1(1(x1))) -> 5(0(3(1(5(1(x1)))))) 0(5(0(x1))) -> 3(0(3(5(0(x1))))) 0(5(0(x1))) -> 3(5(0(0(3(x1))))) 0(5(0(x1))) -> 5(0(3(0(2(x1))))) 0(5(0(x1))) -> 5(0(3(3(0(x1))))) 0(5(0(x1))) -> 4(5(0(3(3(0(x1)))))) 0(5(0(x1))) -> 4(5(0(3(5(0(x1)))))) 0(5(0(x1))) -> 5(3(0(1(3(0(x1)))))) 2(0(0(x1))) -> 0(3(0(3(2(x1))))) 2(0(0(x1))) -> 0(3(3(0(2(3(x1)))))) 2(0(0(x1))) -> 0(3(5(2(0(3(x1)))))) 5(1(0(x1))) -> 3(5(0(1(4(3(x1)))))) 5(1(0(x1))) -> 3(5(1(4(0(3(x1)))))) 5(1(1(x1))) -> 3(1(5(1(x1)))) 5(1(1(x1))) -> 1(3(1(3(5(x1))))) 5(1(1(x1))) -> 1(3(3(3(5(1(x1)))))) 5(1(1(x1))) -> 1(3(5(5(1(4(x1)))))) 0(2(0(1(x1)))) -> 0(2(3(3(0(1(x1)))))) 0(5(1(0(x1)))) -> 0(0(1(3(5(x1))))) 0(5(4(0(x1)))) -> 0(4(5(0(3(x1))))) 2(0(2(0(x1)))) -> 3(0(3(0(2(2(x1)))))) 2(0(4(1(x1)))) -> 2(3(0(1(4(4(x1)))))) 2(0(5(0(x1)))) -> 0(0(3(5(2(x1))))) 2(2(4(1(x1)))) -> 3(2(4(3(2(1(x1)))))) 5(1(0(1(x1)))) -> 0(5(1(4(3(1(x1)))))) 5(1(1(0(x1)))) -> 0(5(1(5(1(x1))))) 5(1(2(0(x1)))) -> 3(1(3(5(0(2(x1)))))) 5(1(5(0(x1)))) -> 5(3(5(0(1(x1))))) 5(2(0(1(x1)))) -> 5(1(0(3(2(x1))))) 5(3(1(1(x1)))) -> 5(3(1(3(1(5(x1)))))) 5(4(1(1(x1)))) -> 5(1(4(1(4(5(x1)))))) 5(5(1(0(x1)))) -> 5(0(5(1(3(x1))))) 5(5(1(1(x1)))) -> 5(1(3(5(0(1(x1)))))) 0(2(4(1(0(x1))))) -> 2(4(0(0(1(3(x1)))))) 0(5(5(1(1(x1))))) -> 5(1(3(5(0(1(x1)))))) 2(2(2(4(1(x1))))) -> 1(2(2(1(4(2(x1)))))) 2(5(0(1(1(x1))))) -> 5(1(2(0(1(3(x1)))))) 5(0(2(4(1(x1))))) -> 5(1(4(0(3(2(x1)))))) 5(2(4(1(0(x1))))) -> 0(2(3(4(5(1(x1)))))) 5(3(0(4(1(x1))))) -> 5(3(0(1(4(1(x1)))))) 5(3(4(1(1(x1))))) -> 1(4(3(5(2(1(x1)))))) The (relative) TRS S consists of the following rules: encArg(1(x_1)) -> 1(encArg(x_1)) encArg(3(x_1)) -> 3(encArg(x_1)) encArg(4(x_1)) -> 4(encArg(x_1)) encArg(cons_0(x_1)) -> 0(encArg(x_1)) encArg(cons_2(x_1)) -> 2(encArg(x_1)) encArg(cons_5(x_1)) -> 5(encArg(x_1)) encode_0(x_1) -> 0(encArg(x_1)) encode_1(x_1) -> 1(encArg(x_1)) encode_2(x_1) -> 2(encArg(x_1)) encode_3(x_1) -> 3(encArg(x_1)) encode_4(x_1) -> 4(encArg(x_1)) encode_5(x_1) -> 5(encArg(x_1)) Rewrite Strategy: FULL ---------------------------------------- (14) LowerBoundPropagationProof (FINISHED) Propagated lower bound. ---------------------------------------- (15) BOUNDS(n^1, INF) ---------------------------------------- (16) Obligation: Analyzing the following TRS for decreasing loops: The Runtime Complexity (full) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^1). The TRS R consists of the following rules: 0(0(1(x1))) -> 2(0(3(3(0(1(x1)))))) 0(1(0(x1))) -> 0(1(3(4(0(3(x1)))))) 0(1(0(x1))) -> 2(0(3(0(1(4(x1)))))) 0(1(1(x1))) -> 0(3(1(3(1(x1))))) 0(1(1(x1))) -> 1(3(0(1(4(x1))))) 0(1(1(x1))) -> 0(1(3(1(3(1(x1)))))) 0(1(1(x1))) -> 1(3(2(1(3(0(x1)))))) 0(1(1(x1))) -> 1(3(3(1(4(0(x1)))))) 0(1(1(x1))) -> 3(0(3(1(5(1(x1)))))) 0(1(1(x1))) -> 5(0(3(1(5(1(x1)))))) 0(5(0(x1))) -> 3(0(3(5(0(x1))))) 0(5(0(x1))) -> 3(5(0(0(3(x1))))) 0(5(0(x1))) -> 5(0(3(0(2(x1))))) 0(5(0(x1))) -> 5(0(3(3(0(x1))))) 0(5(0(x1))) -> 4(5(0(3(3(0(x1)))))) 0(5(0(x1))) -> 4(5(0(3(5(0(x1)))))) 0(5(0(x1))) -> 5(3(0(1(3(0(x1)))))) 2(0(0(x1))) -> 0(3(0(3(2(x1))))) 2(0(0(x1))) -> 0(3(3(0(2(3(x1)))))) 2(0(0(x1))) -> 0(3(5(2(0(3(x1)))))) 5(1(0(x1))) -> 3(5(0(1(4(3(x1)))))) 5(1(0(x1))) -> 3(5(1(4(0(3(x1)))))) 5(1(1(x1))) -> 3(1(5(1(x1)))) 5(1(1(x1))) -> 1(3(1(3(5(x1))))) 5(1(1(x1))) -> 1(3(3(3(5(1(x1)))))) 5(1(1(x1))) -> 1(3(5(5(1(4(x1)))))) 0(2(0(1(x1)))) -> 0(2(3(3(0(1(x1)))))) 0(5(1(0(x1)))) -> 0(0(1(3(5(x1))))) 0(5(4(0(x1)))) -> 0(4(5(0(3(x1))))) 2(0(2(0(x1)))) -> 3(0(3(0(2(2(x1)))))) 2(0(4(1(x1)))) -> 2(3(0(1(4(4(x1)))))) 2(0(5(0(x1)))) -> 0(0(3(5(2(x1))))) 2(2(4(1(x1)))) -> 3(2(4(3(2(1(x1)))))) 5(1(0(1(x1)))) -> 0(5(1(4(3(1(x1)))))) 5(1(1(0(x1)))) -> 0(5(1(5(1(x1))))) 5(1(2(0(x1)))) -> 3(1(3(5(0(2(x1)))))) 5(1(5(0(x1)))) -> 5(3(5(0(1(x1))))) 5(2(0(1(x1)))) -> 5(1(0(3(2(x1))))) 5(3(1(1(x1)))) -> 5(3(1(3(1(5(x1)))))) 5(4(1(1(x1)))) -> 5(1(4(1(4(5(x1)))))) 5(5(1(0(x1)))) -> 5(0(5(1(3(x1))))) 5(5(1(1(x1)))) -> 5(1(3(5(0(1(x1)))))) 0(2(4(1(0(x1))))) -> 2(4(0(0(1(3(x1)))))) 0(5(5(1(1(x1))))) -> 5(1(3(5(0(1(x1)))))) 2(2(2(4(1(x1))))) -> 1(2(2(1(4(2(x1)))))) 2(5(0(1(1(x1))))) -> 5(1(2(0(1(3(x1)))))) 5(0(2(4(1(x1))))) -> 5(1(4(0(3(2(x1)))))) 5(2(4(1(0(x1))))) -> 0(2(3(4(5(1(x1)))))) 5(3(0(4(1(x1))))) -> 5(3(0(1(4(1(x1)))))) 5(3(4(1(1(x1))))) -> 1(4(3(5(2(1(x1)))))) The (relative) TRS S consists of the following rules: encArg(1(x_1)) -> 1(encArg(x_1)) encArg(3(x_1)) -> 3(encArg(x_1)) encArg(4(x_1)) -> 4(encArg(x_1)) encArg(cons_0(x_1)) -> 0(encArg(x_1)) encArg(cons_2(x_1)) -> 2(encArg(x_1)) encArg(cons_5(x_1)) -> 5(encArg(x_1)) encode_0(x_1) -> 0(encArg(x_1)) encode_1(x_1) -> 1(encArg(x_1)) encode_2(x_1) -> 2(encArg(x_1)) encode_3(x_1) -> 3(encArg(x_1)) encode_4(x_1) -> 4(encArg(x_1)) encode_5(x_1) -> 5(encArg(x_1)) Rewrite Strategy: FULL