/export/starexec/sandbox/solver/bin/starexec_run_rcdcRelativeAlsoLower /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files -------------------------------------------------------------------------------- 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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (full) of the given DCpxTrs could be proven to be BOUNDS(n^1, n^1).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), 52 ms] (4) CpxRelTRS (5) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] (6) CpxTRS (7) CpxTrsMatchBoundsProof [FINISHED, 82 ms] (8) BOUNDS(1, n^1) (9) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 5 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(1(1(x1))) -> 0(1(2(3(4(1(x1)))))) 0(1(1(x1))) -> 1(3(1(3(4(0(x1)))))) 0(1(1(x1))) -> 5(1(3(0(3(1(x1)))))) 0(1(4(x1))) -> 0(1(3(4(x1)))) 0(1(4(x1))) -> 1(3(4(2(0(x1))))) 0(1(4(x1))) -> 1(5(3(4(0(x1))))) 0(1(4(x1))) -> 3(1(3(4(0(x1))))) 0(1(4(x1))) -> 1(1(5(3(4(0(x1)))))) 0(1(4(x1))) -> 1(2(3(4(3(0(x1)))))) 0(1(4(x1))) -> 1(3(3(4(4(0(x1)))))) 0(1(4(x1))) -> 1(3(4(5(0(5(x1)))))) 0(1(4(x1))) -> 3(4(5(5(0(1(x1)))))) 1(2(4(x1))) -> 1(2(3(4(1(x1))))) 1(2(4(x1))) -> 5(3(4(1(2(x1))))) 1(2(4(x1))) -> 1(5(2(3(4(3(x1)))))) 1(2(4(x1))) -> 2(3(3(4(5(1(x1)))))) 5(2(1(x1))) -> 1(2(2(3(5(4(x1)))))) 5(2(1(x1))) -> 1(3(2(5(3(4(x1)))))) 5(2(4(x1))) -> 0(5(2(3(4(x1))))) 5(2(4(x1))) -> 5(5(3(4(2(x1))))) 5(2(4(x1))) -> 0(3(4(4(5(2(x1)))))) 5(2(4(x1))) -> 2(2(5(3(4(4(x1)))))) 5(2(4(x1))) -> 2(3(3(4(5(5(x1)))))) 5(2(4(x1))) -> 2(3(4(3(5(3(x1)))))) 5(2(4(x1))) -> 2(5(3(4(0(5(x1)))))) 5(2(4(x1))) -> 3(1(2(5(3(4(x1)))))) 5(2(4(x1))) -> 5(2(3(4(0(3(x1)))))) 0(0(2(4(x1)))) -> 0(4(3(0(2(x1))))) 0(1(1(5(x1)))) -> 0(1(3(5(1(2(x1)))))) 0(1(2(4(x1)))) -> 0(1(2(3(3(4(x1)))))) 0(1(4(5(x1)))) -> 3(4(0(5(1(x1))))) 0(1(4(5(x1)))) -> 3(4(5(3(0(1(x1)))))) 0(4(2(1(x1)))) -> 0(4(1(2(3(4(x1)))))) 0(5(1(4(x1)))) -> 0(0(3(5(4(1(x1)))))) 1(0(1(4(x1)))) -> 3(4(0(1(2(1(x1)))))) 1(1(2(4(x1)))) -> 1(3(4(1(2(x1))))) 1(2(2(4(x1)))) -> 2(2(2(3(4(1(x1)))))) 1(2(4(2(x1)))) -> 1(2(3(4(2(3(x1)))))) 1(5(2(1(x1)))) -> 1(0(3(5(1(2(x1)))))) 1(5(2(4(x1)))) -> 3(2(3(5(1(4(x1)))))) 1(5(2(4(x1)))) -> 5(2(3(3(1(4(x1)))))) 5(0(1(4(x1)))) -> 3(4(0(5(3(1(x1)))))) 5(2(5(1(x1)))) -> 3(5(1(5(2(x1))))) 0(1(1(5(4(x1))))) -> 0(0(5(4(1(1(x1)))))) 1(0(0(2(4(x1))))) -> 0(2(0(4(4(1(x1)))))) 1(0(4(2(1(x1))))) -> 1(2(3(4(1(0(x1)))))) 1(2(5(0(1(x1))))) -> 2(3(5(1(0(1(x1)))))) 1(5(3(0(4(x1))))) -> 5(1(3(4(0(4(x1)))))) 5(0(5(2(4(x1))))) -> 4(0(0(5(5(2(x1)))))) 5(3(2(4(2(x1))))) -> 2(1(5(3(4(2(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(2(x_1)) -> 2(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_1(x_1)) -> 1(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(1(1(x1))) -> 0(1(2(3(4(1(x1)))))) 0(1(1(x1))) -> 1(3(1(3(4(0(x1)))))) 0(1(1(x1))) -> 5(1(3(0(3(1(x1)))))) 0(1(4(x1))) -> 0(1(3(4(x1)))) 0(1(4(x1))) -> 1(3(4(2(0(x1))))) 0(1(4(x1))) -> 1(5(3(4(0(x1))))) 0(1(4(x1))) -> 3(1(3(4(0(x1))))) 0(1(4(x1))) -> 1(1(5(3(4(0(x1)))))) 0(1(4(x1))) -> 1(2(3(4(3(0(x1)))))) 0(1(4(x1))) -> 1(3(3(4(4(0(x1)))))) 0(1(4(x1))) -> 1(3(4(5(0(5(x1)))))) 0(1(4(x1))) -> 3(4(5(5(0(1(x1)))))) 1(2(4(x1))) -> 1(2(3(4(1(x1))))) 1(2(4(x1))) -> 5(3(4(1(2(x1))))) 1(2(4(x1))) -> 1(5(2(3(4(3(x1)))))) 1(2(4(x1))) -> 2(3(3(4(5(1(x1)))))) 5(2(1(x1))) -> 1(2(2(3(5(4(x1)))))) 5(2(1(x1))) -> 1(3(2(5(3(4(x1)))))) 5(2(4(x1))) -> 0(5(2(3(4(x1))))) 5(2(4(x1))) -> 5(5(3(4(2(x1))))) 5(2(4(x1))) -> 0(3(4(4(5(2(x1)))))) 5(2(4(x1))) -> 2(2(5(3(4(4(x1)))))) 5(2(4(x1))) -> 2(3(3(4(5(5(x1)))))) 5(2(4(x1))) -> 2(3(4(3(5(3(x1)))))) 5(2(4(x1))) -> 2(5(3(4(0(5(x1)))))) 5(2(4(x1))) -> 3(1(2(5(3(4(x1)))))) 5(2(4(x1))) -> 5(2(3(4(0(3(x1)))))) 0(0(2(4(x1)))) -> 0(4(3(0(2(x1))))) 0(1(1(5(x1)))) -> 0(1(3(5(1(2(x1)))))) 0(1(2(4(x1)))) -> 0(1(2(3(3(4(x1)))))) 0(1(4(5(x1)))) -> 3(4(0(5(1(x1))))) 0(1(4(5(x1)))) -> 3(4(5(3(0(1(x1)))))) 0(4(2(1(x1)))) -> 0(4(1(2(3(4(x1)))))) 0(5(1(4(x1)))) -> 0(0(3(5(4(1(x1)))))) 1(0(1(4(x1)))) -> 3(4(0(1(2(1(x1)))))) 1(1(2(4(x1)))) -> 1(3(4(1(2(x1))))) 1(2(2(4(x1)))) -> 2(2(2(3(4(1(x1)))))) 1(2(4(2(x1)))) -> 1(2(3(4(2(3(x1)))))) 1(5(2(1(x1)))) -> 1(0(3(5(1(2(x1)))))) 1(5(2(4(x1)))) -> 3(2(3(5(1(4(x1)))))) 1(5(2(4(x1)))) -> 5(2(3(3(1(4(x1)))))) 5(0(1(4(x1)))) -> 3(4(0(5(3(1(x1)))))) 5(2(5(1(x1)))) -> 3(5(1(5(2(x1))))) 0(1(1(5(4(x1))))) -> 0(0(5(4(1(1(x1)))))) 1(0(0(2(4(x1))))) -> 0(2(0(4(4(1(x1)))))) 1(0(4(2(1(x1))))) -> 1(2(3(4(1(0(x1)))))) 1(2(5(0(1(x1))))) -> 2(3(5(1(0(1(x1)))))) 1(5(3(0(4(x1))))) -> 5(1(3(4(0(4(x1)))))) 5(0(5(2(4(x1))))) -> 4(0(0(5(5(2(x1)))))) 5(3(2(4(2(x1))))) -> 2(1(5(3(4(2(x1)))))) The (relative) TRS S consists of the following rules: encArg(2(x_1)) -> 2(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_1(x_1)) -> 1(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(1(1(x1))) -> 0(1(2(3(4(1(x1)))))) 0(1(1(x1))) -> 1(3(1(3(4(0(x1)))))) 0(1(1(x1))) -> 5(1(3(0(3(1(x1)))))) 0(1(4(x1))) -> 0(1(3(4(x1)))) 0(1(4(x1))) -> 1(3(4(2(0(x1))))) 0(1(4(x1))) -> 1(5(3(4(0(x1))))) 0(1(4(x1))) -> 3(1(3(4(0(x1))))) 0(1(4(x1))) -> 1(1(5(3(4(0(x1)))))) 0(1(4(x1))) -> 1(2(3(4(3(0(x1)))))) 0(1(4(x1))) -> 1(3(3(4(4(0(x1)))))) 0(1(4(x1))) -> 1(3(4(5(0(5(x1)))))) 0(1(4(x1))) -> 3(4(5(5(0(1(x1)))))) 1(2(4(x1))) -> 1(2(3(4(1(x1))))) 1(2(4(x1))) -> 5(3(4(1(2(x1))))) 1(2(4(x1))) -> 1(5(2(3(4(3(x1)))))) 1(2(4(x1))) -> 2(3(3(4(5(1(x1)))))) 5(2(1(x1))) -> 1(2(2(3(5(4(x1)))))) 5(2(1(x1))) -> 1(3(2(5(3(4(x1)))))) 5(2(4(x1))) -> 0(5(2(3(4(x1))))) 5(2(4(x1))) -> 5(5(3(4(2(x1))))) 5(2(4(x1))) -> 0(3(4(4(5(2(x1)))))) 5(2(4(x1))) -> 2(2(5(3(4(4(x1)))))) 5(2(4(x1))) -> 2(3(3(4(5(5(x1)))))) 5(2(4(x1))) -> 2(3(4(3(5(3(x1)))))) 5(2(4(x1))) -> 2(5(3(4(0(5(x1)))))) 5(2(4(x1))) -> 3(1(2(5(3(4(x1)))))) 5(2(4(x1))) -> 5(2(3(4(0(3(x1)))))) 0(0(2(4(x1)))) -> 0(4(3(0(2(x1))))) 0(1(1(5(x1)))) -> 0(1(3(5(1(2(x1)))))) 0(1(2(4(x1)))) -> 0(1(2(3(3(4(x1)))))) 0(1(4(5(x1)))) -> 3(4(0(5(1(x1))))) 0(1(4(5(x1)))) -> 3(4(5(3(0(1(x1)))))) 0(4(2(1(x1)))) -> 0(4(1(2(3(4(x1)))))) 0(5(1(4(x1)))) -> 0(0(3(5(4(1(x1)))))) 1(0(1(4(x1)))) -> 3(4(0(1(2(1(x1)))))) 1(1(2(4(x1)))) -> 1(3(4(1(2(x1))))) 1(2(2(4(x1)))) -> 2(2(2(3(4(1(x1)))))) 1(2(4(2(x1)))) -> 1(2(3(4(2(3(x1)))))) 1(5(2(1(x1)))) -> 1(0(3(5(1(2(x1)))))) 1(5(2(4(x1)))) -> 3(2(3(5(1(4(x1)))))) 1(5(2(4(x1)))) -> 5(2(3(3(1(4(x1)))))) 5(0(1(4(x1)))) -> 3(4(0(5(3(1(x1)))))) 5(2(5(1(x1)))) -> 3(5(1(5(2(x1))))) 0(1(1(5(4(x1))))) -> 0(0(5(4(1(1(x1)))))) 1(0(0(2(4(x1))))) -> 0(2(0(4(4(1(x1)))))) 1(0(4(2(1(x1))))) -> 1(2(3(4(1(0(x1)))))) 1(2(5(0(1(x1))))) -> 2(3(5(1(0(1(x1)))))) 1(5(3(0(4(x1))))) -> 5(1(3(4(0(4(x1)))))) 5(0(5(2(4(x1))))) -> 4(0(0(5(5(2(x1)))))) 5(3(2(4(2(x1))))) -> 2(1(5(3(4(2(x1)))))) The (relative) TRS S consists of the following rules: encArg(2(x_1)) -> 2(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_1(x_1)) -> 1(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(1(1(x1))) -> 0(1(2(3(4(1(x1)))))) 0(1(1(x1))) -> 1(3(1(3(4(0(x1)))))) 0(1(1(x1))) -> 5(1(3(0(3(1(x1)))))) 0(1(4(x1))) -> 0(1(3(4(x1)))) 0(1(4(x1))) -> 1(3(4(2(0(x1))))) 0(1(4(x1))) -> 1(5(3(4(0(x1))))) 0(1(4(x1))) -> 3(1(3(4(0(x1))))) 0(1(4(x1))) -> 1(1(5(3(4(0(x1)))))) 0(1(4(x1))) -> 1(2(3(4(3(0(x1)))))) 0(1(4(x1))) -> 1(3(3(4(4(0(x1)))))) 0(1(4(x1))) -> 1(3(4(5(0(5(x1)))))) 0(1(4(x1))) -> 3(4(5(5(0(1(x1)))))) 1(2(4(x1))) -> 1(2(3(4(1(x1))))) 1(2(4(x1))) -> 5(3(4(1(2(x1))))) 1(2(4(x1))) -> 1(5(2(3(4(3(x1)))))) 1(2(4(x1))) -> 2(3(3(4(5(1(x1)))))) 5(2(1(x1))) -> 1(2(2(3(5(4(x1)))))) 5(2(1(x1))) -> 1(3(2(5(3(4(x1)))))) 5(2(4(x1))) -> 0(5(2(3(4(x1))))) 5(2(4(x1))) -> 5(5(3(4(2(x1))))) 5(2(4(x1))) -> 0(3(4(4(5(2(x1)))))) 5(2(4(x1))) -> 2(2(5(3(4(4(x1)))))) 5(2(4(x1))) -> 2(3(3(4(5(5(x1)))))) 5(2(4(x1))) -> 2(3(4(3(5(3(x1)))))) 5(2(4(x1))) -> 2(5(3(4(0(5(x1)))))) 5(2(4(x1))) -> 3(1(2(5(3(4(x1)))))) 5(2(4(x1))) -> 5(2(3(4(0(3(x1)))))) 0(0(2(4(x1)))) -> 0(4(3(0(2(x1))))) 0(1(1(5(x1)))) -> 0(1(3(5(1(2(x1)))))) 0(1(2(4(x1)))) -> 0(1(2(3(3(4(x1)))))) 0(1(4(5(x1)))) -> 3(4(0(5(1(x1))))) 0(1(4(5(x1)))) -> 3(4(5(3(0(1(x1)))))) 0(4(2(1(x1)))) -> 0(4(1(2(3(4(x1)))))) 0(5(1(4(x1)))) -> 0(0(3(5(4(1(x1)))))) 1(0(1(4(x1)))) -> 3(4(0(1(2(1(x1)))))) 1(1(2(4(x1)))) -> 1(3(4(1(2(x1))))) 1(2(2(4(x1)))) -> 2(2(2(3(4(1(x1)))))) 1(2(4(2(x1)))) -> 1(2(3(4(2(3(x1)))))) 1(5(2(1(x1)))) -> 1(0(3(5(1(2(x1)))))) 1(5(2(4(x1)))) -> 3(2(3(5(1(4(x1)))))) 1(5(2(4(x1)))) -> 5(2(3(3(1(4(x1)))))) 5(0(1(4(x1)))) -> 3(4(0(5(3(1(x1)))))) 5(2(5(1(x1)))) -> 3(5(1(5(2(x1))))) 0(1(1(5(4(x1))))) -> 0(0(5(4(1(1(x1)))))) 1(0(0(2(4(x1))))) -> 0(2(0(4(4(1(x1)))))) 1(0(4(2(1(x1))))) -> 1(2(3(4(1(0(x1)))))) 1(2(5(0(1(x1))))) -> 2(3(5(1(0(1(x1)))))) 1(5(3(0(4(x1))))) -> 5(1(3(4(0(4(x1)))))) 5(0(5(2(4(x1))))) -> 4(0(0(5(5(2(x1)))))) 5(3(2(4(2(x1))))) -> 2(1(5(3(4(2(x1)))))) encArg(2(x_1)) -> 2(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_1(x_1)) -> 1(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 3. The certificate found is represented by the following graph. 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265, 266, 267, 268, 269, 270, 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, 576, 577, 578, 579, 580, 581, 582] {(57,58,[0_1|0, 1_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]), (57,59,[1_1|1]), (57,63,[5_1|1]), (57,67,[1_1|1]), (57,72,[2_1|1]), (57,77,[1_1|1]), (57,82,[2_1|1]), (57,87,[0_1|1]), (57,91,[5_1|1]), (57,95,[0_1|1]), (57,100,[2_1|1]), (57,105,[2_1|1]), (57,110,[2_1|1]), (57,115,[2_1|1]), (57,120,[3_1|1]), (57,125,[5_1|1]), (57,130,[2_1|1]), 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(333,426,[1_1|3]), (334,335,[3_1|2]), (335,336,[4_1|2]), (336,337,[3_1|2]), (337,338,[5_1|2]), (337,368,[2_1|2]), (338,135,[3_1|2]), (338,363,[3_1|2]), (339,340,[5_1|2]), (340,341,[3_1|2]), (341,342,[4_1|2]), (342,343,[0_1|2]), (342,224,[0_1|2]), (343,135,[5_1|2]), (343,363,[5_1|2, 4_1|2]), (343,301,[1_1|2]), (343,306,[1_1|2]), (343,311,[0_1|2]), (343,315,[5_1|2]), (343,319,[0_1|2]), (343,324,[2_1|2]), (343,329,[2_1|2]), (343,334,[2_1|2]), (343,339,[2_1|2]), (343,344,[3_1|2]), (343,349,[5_1|2]), (343,354,[3_1|2]), (343,358,[3_1|2]), (343,368,[2_1|2]), (343,421,[1_1|3]), (343,426,[1_1|3]), (344,345,[1_1|2]), (345,346,[2_1|2]), (346,347,[5_1|2]), (347,348,[3_1|2]), (348,135,[4_1|2]), (348,363,[4_1|2]), (349,350,[2_1|2]), (350,351,[3_1|2]), (351,352,[4_1|2]), (352,353,[0_1|2]), (353,135,[3_1|2]), (353,363,[3_1|2]), (354,355,[5_1|2]), (355,356,[1_1|2]), (355,281,[1_1|2]), (355,286,[3_1|2]), (355,291,[5_1|2]), (355,568,[1_1|3]), (355,573,[3_1|3]), (355,578,[5_1|3]), (356,357,[5_1|2]), (356,301,[1_1|2]), (356,306,[1_1|2]), (356,311,[0_1|2]), (356,315,[5_1|2]), (356,319,[0_1|2]), (356,324,[2_1|2]), (356,329,[2_1|2]), (356,334,[2_1|2]), (356,339,[2_1|2]), (356,344,[3_1|2]), (356,349,[5_1|2]), (356,354,[3_1|2]), (356,511,[1_1|3]), (356,516,[1_1|3]), (356,521,[3_1|3]), (356,525,[0_1|3]), (356,529,[5_1|3]), (356,533,[0_1|3]), (356,538,[2_1|3]), (356,543,[2_1|3]), (356,548,[2_1|3]), (356,553,[2_1|3]), (356,558,[3_1|3]), (356,563,[5_1|3]), (357,135,[2_1|2]), (357,141,[2_1|2]), (357,164,[2_1|2]), (357,168,[2_1|2]), (357,176,[2_1|2]), (357,181,[2_1|2]), (357,186,[2_1|2]), (357,191,[2_1|2]), (357,229,[2_1|2]), (357,237,[2_1|2]), (357,247,[2_1|2]), (357,272,[2_1|2]), (357,277,[2_1|2]), (357,281,[2_1|2]), (357,301,[2_1|2]), (357,306,[2_1|2]), (357,147,[2_1|2]), (357,297,[2_1|2]), (358,359,[4_1|2]), (359,360,[0_1|2]), (360,361,[5_1|2]), (361,362,[3_1|2]), (362,135,[1_1|2]), (362,363,[1_1|2]), (362,229,[1_1|2]), (362,233,[5_1|2]), (362,237,[1_1|2]), (362,242,[2_1|2]), (362,247,[1_1|2]), (362,252,[2_1|2]), (362,257,[2_1|2]), (362,262,[3_1|2]), (362,267,[0_1|2]), (362,272,[1_1|2]), (362,277,[1_1|2]), (362,281,[1_1|2]), (362,286,[3_1|2]), (362,291,[5_1|2]), (362,296,[5_1|2]), (363,364,[0_1|2]), (364,365,[0_1|2]), (365,366,[5_1|2]), (366,367,[5_1|2]), (366,301,[1_1|2]), (366,306,[1_1|2]), (366,311,[0_1|2]), (366,315,[5_1|2]), (366,319,[0_1|2]), (366,324,[2_1|2]), (366,329,[2_1|2]), (366,334,[2_1|2]), (366,339,[2_1|2]), (366,344,[3_1|2]), (366,349,[5_1|2]), (366,354,[3_1|2]), (366,511,[1_1|3]), (366,516,[1_1|3]), (366,521,[3_1|3]), (366,525,[0_1|3]), (366,529,[5_1|3]), (366,533,[0_1|3]), (366,538,[2_1|3]), (366,543,[2_1|3]), (366,548,[2_1|3]), (366,553,[2_1|3]), (366,558,[3_1|3]), (366,563,[5_1|3]), (367,135,[2_1|2]), (367,363,[2_1|2]), (368,369,[1_1|2]), (369,370,[5_1|2]), (370,371,[3_1|2]), (371,372,[4_1|2]), (372,135,[2_1|2]), (372,242,[2_1|2]), (372,252,[2_1|2]), (372,257,[2_1|2]), (372,324,[2_1|2]), (372,329,[2_1|2]), (372,334,[2_1|2]), (372,339,[2_1|2]), (372,368,[2_1|2]), (373,374,[2_1|2]), (374,375,[2_1|2]), (375,376,[3_1|2]), (376,377,[5_1|2]), (377,131,[4_1|2]), (378,379,[3_1|2]), (379,380,[2_1|2]), (380,381,[5_1|2]), (381,382,[3_1|2]), (382,131,[4_1|2]), (383,384,[1_1|3]), (384,385,[2_1|3]), (385,386,[3_1|3]), (386,387,[4_1|3]), (387,177,[1_1|3]), (388,389,[3_1|3]), (389,390,[1_1|3]), (390,391,[3_1|3]), (391,392,[4_1|3]), (392,177,[0_1|3]), (393,394,[1_1|3]), (394,395,[3_1|3]), (395,396,[0_1|3]), (396,397,[3_1|3]), (397,177,[1_1|3]), (398,399,[1_1|3]), (399,400,[3_1|3]), (400,401,[5_1|3]), (401,402,[1_1|3]), (402,178,[2_1|3]), (403,404,[2_1|3]), (404,405,[3_1|3]), (405,406,[4_1|3]), (406,363,[1_1|3]), (407,408,[3_1|3]), (408,409,[4_1|3]), (409,410,[1_1|3]), (410,363,[2_1|3]), (411,412,[5_1|3]), (412,413,[2_1|3]), (413,414,[3_1|3]), (414,415,[4_1|3]), (415,363,[3_1|3]), (416,417,[3_1|3]), (417,418,[3_1|3]), (418,419,[4_1|3]), (419,420,[5_1|3]), (420,363,[1_1|3]), (421,422,[2_1|3]), (422,423,[2_1|3]), (423,424,[3_1|3]), (424,425,[5_1|3]), (425,369,[4_1|3]), (426,427,[3_1|3]), (427,428,[2_1|3]), (428,429,[5_1|3]), (429,430,[3_1|3]), (430,369,[4_1|3]), (431,432,[4_1|3]), (432,433,[0_1|3]), (433,434,[5_1|3]), (434,435,[3_1|3]), (435,363,[1_1|3]), (436,437,[1_1|3]), (437,438,[2_1|3]), (438,439,[3_1|3]), (439,440,[4_1|3]), (440,141,[1_1|3]), (440,164,[1_1|3]), (440,168,[1_1|3]), (440,176,[1_1|3]), (440,181,[1_1|3]), (440,186,[1_1|3]), (440,191,[1_1|3]), (440,229,[1_1|3]), (440,237,[1_1|3]), (440,247,[1_1|3]), (440,272,[1_1|3]), (440,277,[1_1|3]), (440,281,[1_1|3]), (440,301,[1_1|3]), (440,306,[1_1|3]), (440,147,[1_1|3]), (440,297,[1_1|3]), (440,177,[1_1|3]), (441,442,[3_1|3]), (442,443,[1_1|3]), (443,444,[3_1|3]), (444,445,[4_1|3]), (445,141,[0_1|3]), (445,164,[0_1|3]), (445,168,[0_1|3]), (445,176,[0_1|3]), (445,181,[0_1|3]), (445,186,[0_1|3]), (445,191,[0_1|3]), (445,229,[0_1|3]), (445,237,[0_1|3]), (445,247,[0_1|3]), (445,272,[0_1|3]), (445,277,[0_1|3]), (445,281,[0_1|3]), (445,301,[0_1|3]), (445,306,[0_1|3]), (445,147,[0_1|3]), (445,297,[0_1|3]), (445,177,[0_1|3]), (446,447,[1_1|3]), (447,448,[3_1|3]), (448,449,[0_1|3]), (449,450,[3_1|3]), (450,141,[1_1|3]), (450,164,[1_1|3]), (450,168,[1_1|3]), (450,176,[1_1|3]), (450,181,[1_1|3]), (450,186,[1_1|3]), (450,191,[1_1|3]), (450,229,[1_1|3]), (450,237,[1_1|3]), (450,247,[1_1|3]), (450,272,[1_1|3]), (450,277,[1_1|3]), (450,281,[1_1|3]), (450,301,[1_1|3]), (450,306,[1_1|3]), (450,147,[1_1|3]), (450,297,[1_1|3]), (450,177,[1_1|3]), (451,452,[1_1|3]), (452,453,[3_1|3]), (453,454,[5_1|3]), (454,455,[1_1|3]), (455,169,[2_1|3]), (455,238,[2_1|3]), (455,178,[2_1|3]), (456,457,[1_1|3]), (457,458,[3_1|3]), (458,363,[4_1|3]), (459,460,[3_1|3]), (460,461,[4_1|3]), (461,462,[2_1|3]), (462,363,[0_1|3]), (463,464,[5_1|3]), (464,465,[3_1|3]), (465,466,[4_1|3]), (466,363,[0_1|3]), (467,468,[1_1|3]), (468,469,[3_1|3]), (469,470,[4_1|3]), (470,363,[0_1|3]), (471,472,[1_1|3]), (472,473,[5_1|3]), (473,474,[3_1|3]), (474,475,[4_1|3]), (475,363,[0_1|3]), (476,477,[2_1|3]), (477,478,[3_1|3]), (478,479,[4_1|3]), (479,480,[3_1|3]), (480,363,[0_1|3]), (481,482,[3_1|3]), (482,483,[3_1|3]), (483,484,[4_1|3]), (484,485,[4_1|3]), (485,363,[0_1|3]), (486,487,[3_1|3]), (487,488,[4_1|3]), (488,489,[5_1|3]), (489,490,[0_1|3]), (490,363,[5_1|3]), (491,492,[4_1|3]), (492,493,[5_1|3]), (493,494,[5_1|3]), (494,495,[0_1|3]), (495,363,[1_1|3]), (496,497,[0_1|3]), (497,498,[3_1|3]), (498,499,[5_1|3]), (499,500,[4_1|3]), (500,363,[1_1|3]), (501,502,[4_1|3]), (502,503,[0_1|3]), (503,504,[1_1|3]), (504,505,[2_1|3]), (505,363,[1_1|3]), (506,507,[4_1|3]), (507,508,[1_1|3]), (508,509,[2_1|3]), (509,510,[3_1|3]), (510,369,[4_1|3]), (511,512,[2_1|3]), (512,513,[2_1|3]), (513,514,[3_1|3]), (514,515,[5_1|3]), (515,141,[4_1|3]), (515,164,[4_1|3]), (515,168,[4_1|3]), (515,176,[4_1|3]), (515,181,[4_1|3]), (515,186,[4_1|3]), (515,191,[4_1|3]), (515,229,[4_1|3]), (515,237,[4_1|3]), (515,247,[4_1|3]), (515,272,[4_1|3]), (515,277,[4_1|3]), (515,281,[4_1|3]), (515,301,[4_1|3]), (515,306,[4_1|3]), (515,177,[4_1|3]), (516,517,[3_1|3]), (517,518,[2_1|3]), (518,519,[5_1|3]), (519,520,[3_1|3]), (520,141,[4_1|3]), (520,164,[4_1|3]), (520,168,[4_1|3]), (520,176,[4_1|3]), (520,181,[4_1|3]), (520,186,[4_1|3]), (520,191,[4_1|3]), (520,229,[4_1|3]), (520,237,[4_1|3]), (520,247,[4_1|3]), (520,272,[4_1|3]), (520,277,[4_1|3]), (520,281,[4_1|3]), (520,301,[4_1|3]), (520,306,[4_1|3]), (520,177,[4_1|3]), (521,522,[5_1|3]), (522,523,[1_1|3]), (523,524,[5_1|3]), (524,147,[2_1|3]), (524,297,[2_1|3]), (525,526,[5_1|3]), (526,527,[2_1|3]), (527,528,[3_1|3]), (528,363,[4_1|3]), (529,530,[5_1|3]), (530,531,[3_1|3]), (531,532,[4_1|3]), (532,363,[2_1|3]), (533,534,[3_1|3]), (534,535,[4_1|3]), (535,536,[4_1|3]), (536,537,[5_1|3]), (537,363,[2_1|3]), (538,539,[2_1|3]), (539,540,[5_1|3]), (540,541,[3_1|3]), (541,542,[4_1|3]), (542,363,[4_1|3]), (543,544,[3_1|3]), (544,545,[3_1|3]), (545,546,[4_1|3]), (546,547,[5_1|3]), (547,363,[5_1|3]), (548,549,[3_1|3]), (549,550,[4_1|3]), (550,551,[3_1|3]), (551,552,[5_1|3]), (552,363,[3_1|3]), (553,554,[5_1|3]), (554,555,[3_1|3]), (555,556,[4_1|3]), (556,557,[0_1|3]), (557,363,[5_1|3]), (558,559,[1_1|3]), (559,560,[2_1|3]), (560,561,[5_1|3]), (561,562,[3_1|3]), (562,363,[4_1|3]), (563,564,[2_1|3]), (564,565,[3_1|3]), (565,566,[4_1|3]), (566,567,[0_1|3]), (567,363,[3_1|3]), (568,569,[0_1|3]), (569,570,[3_1|3]), (570,571,[5_1|3]), (571,572,[1_1|3]), (572,141,[2_1|3]), (572,164,[2_1|3]), (572,168,[2_1|3]), (572,176,[2_1|3]), (572,181,[2_1|3]), (572,186,[2_1|3]), (572,191,[2_1|3]), (572,229,[2_1|3]), (572,237,[2_1|3]), (572,247,[2_1|3]), (572,272,[2_1|3]), (572,277,[2_1|3]), (572,281,[2_1|3]), (572,301,[2_1|3]), (572,306,[2_1|3]), (572,177,[2_1|3]), (573,574,[2_1|3]), (574,575,[3_1|3]), (575,576,[5_1|3]), (576,577,[1_1|3]), (577,363,[4_1|3]), (578,579,[2_1|3]), (579,580,[3_1|3]), (580,581,[3_1|3]), (581,582,[1_1|3]), (582,363,[4_1|3])}" ---------------------------------------- (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(1(1(x1))) -> 0(1(2(3(4(1(x1)))))) 0(1(1(x1))) -> 1(3(1(3(4(0(x1)))))) 0(1(1(x1))) -> 5(1(3(0(3(1(x1)))))) 0(1(4(x1))) -> 0(1(3(4(x1)))) 0(1(4(x1))) -> 1(3(4(2(0(x1))))) 0(1(4(x1))) -> 1(5(3(4(0(x1))))) 0(1(4(x1))) -> 3(1(3(4(0(x1))))) 0(1(4(x1))) -> 1(1(5(3(4(0(x1)))))) 0(1(4(x1))) -> 1(2(3(4(3(0(x1)))))) 0(1(4(x1))) -> 1(3(3(4(4(0(x1)))))) 0(1(4(x1))) -> 1(3(4(5(0(5(x1)))))) 0(1(4(x1))) -> 3(4(5(5(0(1(x1)))))) 1(2(4(x1))) -> 1(2(3(4(1(x1))))) 1(2(4(x1))) -> 5(3(4(1(2(x1))))) 1(2(4(x1))) -> 1(5(2(3(4(3(x1)))))) 1(2(4(x1))) -> 2(3(3(4(5(1(x1)))))) 5(2(1(x1))) -> 1(2(2(3(5(4(x1)))))) 5(2(1(x1))) -> 1(3(2(5(3(4(x1)))))) 5(2(4(x1))) -> 0(5(2(3(4(x1))))) 5(2(4(x1))) -> 5(5(3(4(2(x1))))) 5(2(4(x1))) -> 0(3(4(4(5(2(x1)))))) 5(2(4(x1))) -> 2(2(5(3(4(4(x1)))))) 5(2(4(x1))) -> 2(3(3(4(5(5(x1)))))) 5(2(4(x1))) -> 2(3(4(3(5(3(x1)))))) 5(2(4(x1))) -> 2(5(3(4(0(5(x1)))))) 5(2(4(x1))) -> 3(1(2(5(3(4(x1)))))) 5(2(4(x1))) -> 5(2(3(4(0(3(x1)))))) 0(0(2(4(x1)))) -> 0(4(3(0(2(x1))))) 0(1(1(5(x1)))) -> 0(1(3(5(1(2(x1)))))) 0(1(2(4(x1)))) -> 0(1(2(3(3(4(x1)))))) 0(1(4(5(x1)))) -> 3(4(0(5(1(x1))))) 0(1(4(5(x1)))) -> 3(4(5(3(0(1(x1)))))) 0(4(2(1(x1)))) -> 0(4(1(2(3(4(x1)))))) 0(5(1(4(x1)))) -> 0(0(3(5(4(1(x1)))))) 1(0(1(4(x1)))) -> 3(4(0(1(2(1(x1)))))) 1(1(2(4(x1)))) -> 1(3(4(1(2(x1))))) 1(2(2(4(x1)))) -> 2(2(2(3(4(1(x1)))))) 1(2(4(2(x1)))) -> 1(2(3(4(2(3(x1)))))) 1(5(2(1(x1)))) -> 1(0(3(5(1(2(x1)))))) 1(5(2(4(x1)))) -> 3(2(3(5(1(4(x1)))))) 1(5(2(4(x1)))) -> 5(2(3(3(1(4(x1)))))) 5(0(1(4(x1)))) -> 3(4(0(5(3(1(x1)))))) 5(2(5(1(x1)))) -> 3(5(1(5(2(x1))))) 0(1(1(5(4(x1))))) -> 0(0(5(4(1(1(x1)))))) 1(0(0(2(4(x1))))) -> 0(2(0(4(4(1(x1)))))) 1(0(4(2(1(x1))))) -> 1(2(3(4(1(0(x1)))))) 1(2(5(0(1(x1))))) -> 2(3(5(1(0(1(x1)))))) 1(5(3(0(4(x1))))) -> 5(1(3(4(0(4(x1)))))) 5(0(5(2(4(x1))))) -> 4(0(0(5(5(2(x1)))))) 5(3(2(4(2(x1))))) -> 2(1(5(3(4(2(x1)))))) The (relative) TRS S consists of the following rules: encArg(2(x_1)) -> 2(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_1(x_1)) -> 1(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 1(2(4(x1))) ->^+ 2(3(3(4(5(1(x1)))))) gives rise to a decreasing loop by considering the right hand sides subterm at position [0,0,0,0,0]. The pumping substitution is [x1 / 2(4(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(1(1(x1))) -> 0(1(2(3(4(1(x1)))))) 0(1(1(x1))) -> 1(3(1(3(4(0(x1)))))) 0(1(1(x1))) -> 5(1(3(0(3(1(x1)))))) 0(1(4(x1))) -> 0(1(3(4(x1)))) 0(1(4(x1))) -> 1(3(4(2(0(x1))))) 0(1(4(x1))) -> 1(5(3(4(0(x1))))) 0(1(4(x1))) -> 3(1(3(4(0(x1))))) 0(1(4(x1))) -> 1(1(5(3(4(0(x1)))))) 0(1(4(x1))) -> 1(2(3(4(3(0(x1)))))) 0(1(4(x1))) -> 1(3(3(4(4(0(x1)))))) 0(1(4(x1))) -> 1(3(4(5(0(5(x1)))))) 0(1(4(x1))) -> 3(4(5(5(0(1(x1)))))) 1(2(4(x1))) -> 1(2(3(4(1(x1))))) 1(2(4(x1))) -> 5(3(4(1(2(x1))))) 1(2(4(x1))) -> 1(5(2(3(4(3(x1)))))) 1(2(4(x1))) -> 2(3(3(4(5(1(x1)))))) 5(2(1(x1))) -> 1(2(2(3(5(4(x1)))))) 5(2(1(x1))) -> 1(3(2(5(3(4(x1)))))) 5(2(4(x1))) -> 0(5(2(3(4(x1))))) 5(2(4(x1))) -> 5(5(3(4(2(x1))))) 5(2(4(x1))) -> 0(3(4(4(5(2(x1)))))) 5(2(4(x1))) -> 2(2(5(3(4(4(x1)))))) 5(2(4(x1))) -> 2(3(3(4(5(5(x1)))))) 5(2(4(x1))) -> 2(3(4(3(5(3(x1)))))) 5(2(4(x1))) -> 2(5(3(4(0(5(x1)))))) 5(2(4(x1))) -> 3(1(2(5(3(4(x1)))))) 5(2(4(x1))) -> 5(2(3(4(0(3(x1)))))) 0(0(2(4(x1)))) -> 0(4(3(0(2(x1))))) 0(1(1(5(x1)))) -> 0(1(3(5(1(2(x1)))))) 0(1(2(4(x1)))) -> 0(1(2(3(3(4(x1)))))) 0(1(4(5(x1)))) -> 3(4(0(5(1(x1))))) 0(1(4(5(x1)))) -> 3(4(5(3(0(1(x1)))))) 0(4(2(1(x1)))) -> 0(4(1(2(3(4(x1)))))) 0(5(1(4(x1)))) -> 0(0(3(5(4(1(x1)))))) 1(0(1(4(x1)))) -> 3(4(0(1(2(1(x1)))))) 1(1(2(4(x1)))) -> 1(3(4(1(2(x1))))) 1(2(2(4(x1)))) -> 2(2(2(3(4(1(x1)))))) 1(2(4(2(x1)))) -> 1(2(3(4(2(3(x1)))))) 1(5(2(1(x1)))) -> 1(0(3(5(1(2(x1)))))) 1(5(2(4(x1)))) -> 3(2(3(5(1(4(x1)))))) 1(5(2(4(x1)))) -> 5(2(3(3(1(4(x1)))))) 5(0(1(4(x1)))) -> 3(4(0(5(3(1(x1)))))) 5(2(5(1(x1)))) -> 3(5(1(5(2(x1))))) 0(1(1(5(4(x1))))) -> 0(0(5(4(1(1(x1)))))) 1(0(0(2(4(x1))))) -> 0(2(0(4(4(1(x1)))))) 1(0(4(2(1(x1))))) -> 1(2(3(4(1(0(x1)))))) 1(2(5(0(1(x1))))) -> 2(3(5(1(0(1(x1)))))) 1(5(3(0(4(x1))))) -> 5(1(3(4(0(4(x1)))))) 5(0(5(2(4(x1))))) -> 4(0(0(5(5(2(x1)))))) 5(3(2(4(2(x1))))) -> 2(1(5(3(4(2(x1)))))) The (relative) TRS S consists of the following rules: encArg(2(x_1)) -> 2(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_1(x_1)) -> 1(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(1(1(x1))) -> 0(1(2(3(4(1(x1)))))) 0(1(1(x1))) -> 1(3(1(3(4(0(x1)))))) 0(1(1(x1))) -> 5(1(3(0(3(1(x1)))))) 0(1(4(x1))) -> 0(1(3(4(x1)))) 0(1(4(x1))) -> 1(3(4(2(0(x1))))) 0(1(4(x1))) -> 1(5(3(4(0(x1))))) 0(1(4(x1))) -> 3(1(3(4(0(x1))))) 0(1(4(x1))) -> 1(1(5(3(4(0(x1)))))) 0(1(4(x1))) -> 1(2(3(4(3(0(x1)))))) 0(1(4(x1))) -> 1(3(3(4(4(0(x1)))))) 0(1(4(x1))) -> 1(3(4(5(0(5(x1)))))) 0(1(4(x1))) -> 3(4(5(5(0(1(x1)))))) 1(2(4(x1))) -> 1(2(3(4(1(x1))))) 1(2(4(x1))) -> 5(3(4(1(2(x1))))) 1(2(4(x1))) -> 1(5(2(3(4(3(x1)))))) 1(2(4(x1))) -> 2(3(3(4(5(1(x1)))))) 5(2(1(x1))) -> 1(2(2(3(5(4(x1)))))) 5(2(1(x1))) -> 1(3(2(5(3(4(x1)))))) 5(2(4(x1))) -> 0(5(2(3(4(x1))))) 5(2(4(x1))) -> 5(5(3(4(2(x1))))) 5(2(4(x1))) -> 0(3(4(4(5(2(x1)))))) 5(2(4(x1))) -> 2(2(5(3(4(4(x1)))))) 5(2(4(x1))) -> 2(3(3(4(5(5(x1)))))) 5(2(4(x1))) -> 2(3(4(3(5(3(x1)))))) 5(2(4(x1))) -> 2(5(3(4(0(5(x1)))))) 5(2(4(x1))) -> 3(1(2(5(3(4(x1)))))) 5(2(4(x1))) -> 5(2(3(4(0(3(x1)))))) 0(0(2(4(x1)))) -> 0(4(3(0(2(x1))))) 0(1(1(5(x1)))) -> 0(1(3(5(1(2(x1)))))) 0(1(2(4(x1)))) -> 0(1(2(3(3(4(x1)))))) 0(1(4(5(x1)))) -> 3(4(0(5(1(x1))))) 0(1(4(5(x1)))) -> 3(4(5(3(0(1(x1)))))) 0(4(2(1(x1)))) -> 0(4(1(2(3(4(x1)))))) 0(5(1(4(x1)))) -> 0(0(3(5(4(1(x1)))))) 1(0(1(4(x1)))) -> 3(4(0(1(2(1(x1)))))) 1(1(2(4(x1)))) -> 1(3(4(1(2(x1))))) 1(2(2(4(x1)))) -> 2(2(2(3(4(1(x1)))))) 1(2(4(2(x1)))) -> 1(2(3(4(2(3(x1)))))) 1(5(2(1(x1)))) -> 1(0(3(5(1(2(x1)))))) 1(5(2(4(x1)))) -> 3(2(3(5(1(4(x1)))))) 1(5(2(4(x1)))) -> 5(2(3(3(1(4(x1)))))) 5(0(1(4(x1)))) -> 3(4(0(5(3(1(x1)))))) 5(2(5(1(x1)))) -> 3(5(1(5(2(x1))))) 0(1(1(5(4(x1))))) -> 0(0(5(4(1(1(x1)))))) 1(0(0(2(4(x1))))) -> 0(2(0(4(4(1(x1)))))) 1(0(4(2(1(x1))))) -> 1(2(3(4(1(0(x1)))))) 1(2(5(0(1(x1))))) -> 2(3(5(1(0(1(x1)))))) 1(5(3(0(4(x1))))) -> 5(1(3(4(0(4(x1)))))) 5(0(5(2(4(x1))))) -> 4(0(0(5(5(2(x1)))))) 5(3(2(4(2(x1))))) -> 2(1(5(3(4(2(x1)))))) The (relative) TRS S consists of the following rules: encArg(2(x_1)) -> 2(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_1(x_1)) -> 1(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