/export/starexec/sandbox2/solver/bin/starexec_run_rcdcRelativeAlsoLower /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files -------------------------------------------------------------------------------- WORST_CASE(Omega(n^1), O(n^1)) proof of /export/starexec/sandbox2/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (full) of the given DCpxTrs could be proven to be BOUNDS(n^1, n^1).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), 44 ms] (4) CpxRelTRS (5) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] (6) CpxTRS (7) CpxTrsMatchBoundsProof [FINISHED, 408 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))) -> 0(1(2(0(x1)))) 0(0(1(x1))) -> 0(3(1(0(x1)))) 0(0(1(x1))) -> 1(0(4(0(x1)))) 0(0(1(x1))) -> 0(1(3(0(2(x1))))) 0(0(1(x1))) -> 0(1(3(0(4(x1))))) 0(0(1(x1))) -> 0(2(0(1(2(x1))))) 0(0(1(x1))) -> 0(3(0(1(2(x1))))) 0(0(1(x1))) -> 0(3(0(3(1(x1))))) 0(0(1(x1))) -> 0(4(0(4(1(x1))))) 0(0(1(x1))) -> 1(2(0(2(0(x1))))) 0(0(1(x1))) -> 1(2(2(0(0(x1))))) 0(0(1(x1))) -> 0(0(2(2(1(2(x1)))))) 0(0(1(x1))) -> 0(1(2(4(2(0(x1)))))) 0(0(1(x1))) -> 1(2(0(3(0(4(x1)))))) 0(1(1(x1))) -> 0(2(1(1(x1)))) 0(1(1(x1))) -> 0(3(1(1(x1)))) 0(1(1(x1))) -> 1(1(3(0(4(x1))))) 0(1(1(x1))) -> 1(2(0(2(1(x1))))) 0(1(1(x1))) -> 1(0(3(1(2(4(x1)))))) 0(1(1(x1))) -> 1(0(4(2(1(2(x1)))))) 0(1(1(x1))) -> 1(1(2(4(3(0(x1)))))) 0(1(1(x1))) -> 1(2(1(0(4(4(x1)))))) 0(1(1(x1))) -> 1(2(2(1(3(0(x1)))))) 0(5(1(x1))) -> 0(3(1(5(x1)))) 0(5(1(x1))) -> 0(4(5(1(x1)))) 0(5(1(x1))) -> 0(2(3(1(5(x1))))) 0(5(1(x1))) -> 0(3(1(5(2(x1))))) 0(5(1(x1))) -> 0(3(1(2(5(2(x1)))))) 5(0(1(x1))) -> 5(1(2(4(0(x1))))) 5(0(1(x1))) -> 5(0(2(1(2(4(x1)))))) 5(0(1(x1))) -> 5(1(2(3(0(4(x1)))))) 0(0(1(5(x1)))) -> 0(4(1(0(5(x1))))) 0(0(2(1(x1)))) -> 2(0(3(0(2(1(x1)))))) 0(0(2(1(x1)))) -> 2(3(0(2(0(1(x1)))))) 0(1(0(1(x1)))) -> 1(0(2(0(1(x1))))) 0(1(1(1(x1)))) -> 1(1(3(1(0(x1))))) 5(0(1(1(x1)))) -> 1(5(1(2(0(x1))))) 5(3(0(1(x1)))) -> 5(1(2(3(0(x1))))) 5(3(1(5(x1)))) -> 5(3(1(2(5(x1))))) 5(3(2(1(x1)))) -> 1(2(3(5(2(x1))))) 5(4(0(1(x1)))) -> 1(2(5(0(4(x1))))) 0(0(5(1(5(x1))))) -> 1(2(5(5(0(0(x1)))))) 0(5(3(0(1(x1))))) -> 1(0(5(3(0(4(x1)))))) 0(5(3(4(1(x1))))) -> 1(0(3(5(4(5(x1)))))) 0(5(4(0(1(x1))))) -> 0(1(3(0(4(5(x1)))))) 5(4(2(1(1(x1))))) -> 5(4(1(2(1(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(1(x_1)) -> 1(encArg(x_1)) 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_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))) -> 0(1(2(0(x1)))) 0(0(1(x1))) -> 0(3(1(0(x1)))) 0(0(1(x1))) -> 1(0(4(0(x1)))) 0(0(1(x1))) -> 0(1(3(0(2(x1))))) 0(0(1(x1))) -> 0(1(3(0(4(x1))))) 0(0(1(x1))) -> 0(2(0(1(2(x1))))) 0(0(1(x1))) -> 0(3(0(1(2(x1))))) 0(0(1(x1))) -> 0(3(0(3(1(x1))))) 0(0(1(x1))) -> 0(4(0(4(1(x1))))) 0(0(1(x1))) -> 1(2(0(2(0(x1))))) 0(0(1(x1))) -> 1(2(2(0(0(x1))))) 0(0(1(x1))) -> 0(0(2(2(1(2(x1)))))) 0(0(1(x1))) -> 0(1(2(4(2(0(x1)))))) 0(0(1(x1))) -> 1(2(0(3(0(4(x1)))))) 0(1(1(x1))) -> 0(2(1(1(x1)))) 0(1(1(x1))) -> 0(3(1(1(x1)))) 0(1(1(x1))) -> 1(1(3(0(4(x1))))) 0(1(1(x1))) -> 1(2(0(2(1(x1))))) 0(1(1(x1))) -> 1(0(3(1(2(4(x1)))))) 0(1(1(x1))) -> 1(0(4(2(1(2(x1)))))) 0(1(1(x1))) -> 1(1(2(4(3(0(x1)))))) 0(1(1(x1))) -> 1(2(1(0(4(4(x1)))))) 0(1(1(x1))) -> 1(2(2(1(3(0(x1)))))) 0(5(1(x1))) -> 0(3(1(5(x1)))) 0(5(1(x1))) -> 0(4(5(1(x1)))) 0(5(1(x1))) -> 0(2(3(1(5(x1))))) 0(5(1(x1))) -> 0(3(1(5(2(x1))))) 0(5(1(x1))) -> 0(3(1(2(5(2(x1)))))) 5(0(1(x1))) -> 5(1(2(4(0(x1))))) 5(0(1(x1))) -> 5(0(2(1(2(4(x1)))))) 5(0(1(x1))) -> 5(1(2(3(0(4(x1)))))) 0(0(1(5(x1)))) -> 0(4(1(0(5(x1))))) 0(0(2(1(x1)))) -> 2(0(3(0(2(1(x1)))))) 0(0(2(1(x1)))) -> 2(3(0(2(0(1(x1)))))) 0(1(0(1(x1)))) -> 1(0(2(0(1(x1))))) 0(1(1(1(x1)))) -> 1(1(3(1(0(x1))))) 5(0(1(1(x1)))) -> 1(5(1(2(0(x1))))) 5(3(0(1(x1)))) -> 5(1(2(3(0(x1))))) 5(3(1(5(x1)))) -> 5(3(1(2(5(x1))))) 5(3(2(1(x1)))) -> 1(2(3(5(2(x1))))) 5(4(0(1(x1)))) -> 1(2(5(0(4(x1))))) 0(0(5(1(5(x1))))) -> 1(2(5(5(0(0(x1)))))) 0(5(3(0(1(x1))))) -> 1(0(5(3(0(4(x1)))))) 0(5(3(4(1(x1))))) -> 1(0(3(5(4(5(x1)))))) 0(5(4(0(1(x1))))) -> 0(1(3(0(4(5(x1)))))) 5(4(2(1(1(x1))))) -> 5(4(1(2(1(2(x1)))))) The (relative) TRS S consists of the following rules: encArg(1(x_1)) -> 1(encArg(x_1)) 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_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))) -> 0(1(2(0(x1)))) 0(0(1(x1))) -> 0(3(1(0(x1)))) 0(0(1(x1))) -> 1(0(4(0(x1)))) 0(0(1(x1))) -> 0(1(3(0(2(x1))))) 0(0(1(x1))) -> 0(1(3(0(4(x1))))) 0(0(1(x1))) -> 0(2(0(1(2(x1))))) 0(0(1(x1))) -> 0(3(0(1(2(x1))))) 0(0(1(x1))) -> 0(3(0(3(1(x1))))) 0(0(1(x1))) -> 0(4(0(4(1(x1))))) 0(0(1(x1))) -> 1(2(0(2(0(x1))))) 0(0(1(x1))) -> 1(2(2(0(0(x1))))) 0(0(1(x1))) -> 0(0(2(2(1(2(x1)))))) 0(0(1(x1))) -> 0(1(2(4(2(0(x1)))))) 0(0(1(x1))) -> 1(2(0(3(0(4(x1)))))) 0(1(1(x1))) -> 0(2(1(1(x1)))) 0(1(1(x1))) -> 0(3(1(1(x1)))) 0(1(1(x1))) -> 1(1(3(0(4(x1))))) 0(1(1(x1))) -> 1(2(0(2(1(x1))))) 0(1(1(x1))) -> 1(0(3(1(2(4(x1)))))) 0(1(1(x1))) -> 1(0(4(2(1(2(x1)))))) 0(1(1(x1))) -> 1(1(2(4(3(0(x1)))))) 0(1(1(x1))) -> 1(2(1(0(4(4(x1)))))) 0(1(1(x1))) -> 1(2(2(1(3(0(x1)))))) 0(5(1(x1))) -> 0(3(1(5(x1)))) 0(5(1(x1))) -> 0(4(5(1(x1)))) 0(5(1(x1))) -> 0(2(3(1(5(x1))))) 0(5(1(x1))) -> 0(3(1(5(2(x1))))) 0(5(1(x1))) -> 0(3(1(2(5(2(x1)))))) 5(0(1(x1))) -> 5(1(2(4(0(x1))))) 5(0(1(x1))) -> 5(0(2(1(2(4(x1)))))) 5(0(1(x1))) -> 5(1(2(3(0(4(x1)))))) 0(0(1(5(x1)))) -> 0(4(1(0(5(x1))))) 0(0(2(1(x1)))) -> 2(0(3(0(2(1(x1)))))) 0(0(2(1(x1)))) -> 2(3(0(2(0(1(x1)))))) 0(1(0(1(x1)))) -> 1(0(2(0(1(x1))))) 0(1(1(1(x1)))) -> 1(1(3(1(0(x1))))) 5(0(1(1(x1)))) -> 1(5(1(2(0(x1))))) 5(3(0(1(x1)))) -> 5(1(2(3(0(x1))))) 5(3(1(5(x1)))) -> 5(3(1(2(5(x1))))) 5(3(2(1(x1)))) -> 1(2(3(5(2(x1))))) 5(4(0(1(x1)))) -> 1(2(5(0(4(x1))))) 0(0(5(1(5(x1))))) -> 1(2(5(5(0(0(x1)))))) 0(5(3(0(1(x1))))) -> 1(0(5(3(0(4(x1)))))) 0(5(3(4(1(x1))))) -> 1(0(3(5(4(5(x1)))))) 0(5(4(0(1(x1))))) -> 0(1(3(0(4(5(x1)))))) 5(4(2(1(1(x1))))) -> 5(4(1(2(1(2(x1)))))) The (relative) TRS S consists of the following rules: encArg(1(x_1)) -> 1(encArg(x_1)) 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_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))) -> 0(1(2(0(x1)))) 0(0(1(x1))) -> 0(3(1(0(x1)))) 0(0(1(x1))) -> 1(0(4(0(x1)))) 0(0(1(x1))) -> 0(1(3(0(2(x1))))) 0(0(1(x1))) -> 0(1(3(0(4(x1))))) 0(0(1(x1))) -> 0(2(0(1(2(x1))))) 0(0(1(x1))) -> 0(3(0(1(2(x1))))) 0(0(1(x1))) -> 0(3(0(3(1(x1))))) 0(0(1(x1))) -> 0(4(0(4(1(x1))))) 0(0(1(x1))) -> 1(2(0(2(0(x1))))) 0(0(1(x1))) -> 1(2(2(0(0(x1))))) 0(0(1(x1))) -> 0(0(2(2(1(2(x1)))))) 0(0(1(x1))) -> 0(1(2(4(2(0(x1)))))) 0(0(1(x1))) -> 1(2(0(3(0(4(x1)))))) 0(1(1(x1))) -> 0(2(1(1(x1)))) 0(1(1(x1))) -> 0(3(1(1(x1)))) 0(1(1(x1))) -> 1(1(3(0(4(x1))))) 0(1(1(x1))) -> 1(2(0(2(1(x1))))) 0(1(1(x1))) -> 1(0(3(1(2(4(x1)))))) 0(1(1(x1))) -> 1(0(4(2(1(2(x1)))))) 0(1(1(x1))) -> 1(1(2(4(3(0(x1)))))) 0(1(1(x1))) -> 1(2(1(0(4(4(x1)))))) 0(1(1(x1))) -> 1(2(2(1(3(0(x1)))))) 0(5(1(x1))) -> 0(3(1(5(x1)))) 0(5(1(x1))) -> 0(4(5(1(x1)))) 0(5(1(x1))) -> 0(2(3(1(5(x1))))) 0(5(1(x1))) -> 0(3(1(5(2(x1))))) 0(5(1(x1))) -> 0(3(1(2(5(2(x1)))))) 5(0(1(x1))) -> 5(1(2(4(0(x1))))) 5(0(1(x1))) -> 5(0(2(1(2(4(x1)))))) 5(0(1(x1))) -> 5(1(2(3(0(4(x1)))))) 0(0(1(5(x1)))) -> 0(4(1(0(5(x1))))) 0(0(2(1(x1)))) -> 2(0(3(0(2(1(x1)))))) 0(0(2(1(x1)))) -> 2(3(0(2(0(1(x1)))))) 0(1(0(1(x1)))) -> 1(0(2(0(1(x1))))) 0(1(1(1(x1)))) -> 1(1(3(1(0(x1))))) 5(0(1(1(x1)))) -> 1(5(1(2(0(x1))))) 5(3(0(1(x1)))) -> 5(1(2(3(0(x1))))) 5(3(1(5(x1)))) -> 5(3(1(2(5(x1))))) 5(3(2(1(x1)))) -> 1(2(3(5(2(x1))))) 5(4(0(1(x1)))) -> 1(2(5(0(4(x1))))) 0(0(5(1(5(x1))))) -> 1(2(5(5(0(0(x1)))))) 0(5(3(0(1(x1))))) -> 1(0(5(3(0(4(x1)))))) 0(5(3(4(1(x1))))) -> 1(0(3(5(4(5(x1)))))) 0(5(4(0(1(x1))))) -> 0(1(3(0(4(5(x1)))))) 5(4(2(1(1(x1))))) -> 5(4(1(2(1(2(x1)))))) encArg(1(x_1)) -> 1(encArg(x_1)) 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_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. "[59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126, 127, 128, 129, 130, 131, 132, 133, 134, 135, 136, 137, 138, 139, 140, 141, 142, 143, 144, 145, 146, 147, 148, 149, 150, 151, 152, 153, 154, 155, 156, 157, 158, 159, 160, 161, 162, 163, 164, 165, 166, 167, 168, 169, 170, 171, 172, 173, 174, 175, 176, 177, 178, 179, 180, 181, 182, 183, 184, 185, 186, 187, 188, 189, 190, 191, 192, 193, 194, 195, 196, 197, 198, 199, 200, 201, 202, 203, 204, 205, 206, 207, 208, 209, 210, 211, 212, 213, 214, 215, 216, 217, 218, 219, 220, 221, 222, 223, 224, 225, 226, 227, 228, 229, 230, 231, 232, 233, 234, 235, 236, 237, 238, 239, 240, 241, 242, 243, 244, 245, 246, 247, 248, 249, 250, 251, 252, 253, 254, 255, 256, 257, 258, 259, 260, 261, 262, 263, 264, 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, 583, 584, 585, 586, 587, 588, 589, 590, 591, 592, 593, 594, 595, 596, 597, 598, 599, 600, 601, 602, 603, 604, 605, 606, 607, 608, 609, 610, 611, 612, 613, 614, 615, 616, 617, 618, 619, 620, 621, 622, 623, 624, 625, 626, 627, 628, 629, 630, 631, 632, 633, 634, 635, 636, 637, 638, 639] {(59,60,[0_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]), 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(610,255,[4_1|4]), (610,260,[4_1|4]), (610,284,[4_1|4]), (610,296,[4_1|4]), (610,300,[4_1|4]), (611,612,[2_1|4]), (612,613,[0_1|4]), (613,614,[2_1|4]), (614,537,[1_1|4]), (614,555,[1_1|4]), (614,570,[1_1|4]), (614,196,[1_1|4]), (614,214,[1_1|4]), (614,229,[1_1|4]), (614,382,[1_1|4]), (614,400,[1_1|4]), (614,113,[1_1|4]), (614,120,[1_1|4]), (614,147,[1_1|4]), (614,151,[1_1|4]), (614,165,[1_1|4]), (614,184,[1_1|4]), (614,195,[1_1|4]), (614,199,[1_1|4]), (614,203,[1_1|4]), (614,208,[1_1|4]), (614,213,[1_1|4]), (614,218,[1_1|4]), (614,223,[1_1|4]), (614,228,[1_1|4]), (614,232,[1_1|4]), (614,255,[1_1|4]), (614,260,[1_1|4]), (614,284,[1_1|4]), (614,296,[1_1|4]), (614,300,[1_1|4]), (615,616,[0_1|4]), (616,617,[3_1|4]), (617,618,[1_1|4]), (618,619,[2_1|4]), (619,537,[4_1|4]), (619,555,[4_1|4]), (619,570,[4_1|4]), (619,196,[4_1|4]), (619,214,[4_1|4]), (619,229,[4_1|4]), (619,382,[4_1|4]), (619,400,[4_1|4]), (619,113,[4_1|4]), (619,120,[4_1|4]), (619,147,[4_1|4]), (619,151,[4_1|4]), (619,165,[4_1|4]), (619,184,[4_1|4]), (619,195,[4_1|4]), (619,199,[4_1|4]), (619,203,[4_1|4]), (619,208,[4_1|4]), (619,213,[4_1|4]), (619,218,[4_1|4]), (619,223,[4_1|4]), (619,228,[4_1|4]), (619,232,[4_1|4]), (619,255,[4_1|4]), (619,260,[4_1|4]), (619,284,[4_1|4]), (619,296,[4_1|4]), (619,300,[4_1|4]), (620,621,[0_1|4]), (621,622,[4_1|4]), (622,623,[2_1|4]), (623,624,[1_1|4]), (624,537,[2_1|4]), (624,555,[2_1|4]), (624,570,[2_1|4]), (624,196,[2_1|4]), (624,214,[2_1|4]), (624,229,[2_1|4]), (624,382,[2_1|4]), (624,400,[2_1|4]), (624,113,[2_1|4]), (624,120,[2_1|4]), (624,147,[2_1|4]), (624,151,[2_1|4]), (624,165,[2_1|4]), (624,184,[2_1|4]), (624,195,[2_1|4]), (624,199,[2_1|4]), (624,203,[2_1|4]), (624,208,[2_1|4]), (624,213,[2_1|4]), (624,218,[2_1|4]), (624,223,[2_1|4]), (624,228,[2_1|4]), (624,232,[2_1|4]), (624,255,[2_1|4]), (624,260,[2_1|4]), (624,284,[2_1|4]), (624,296,[2_1|4]), (624,300,[2_1|4]), (625,626,[1_1|4]), (626,627,[2_1|4]), (627,628,[4_1|4]), (628,629,[3_1|4]), (629,537,[0_1|4]), (629,555,[0_1|4]), (629,570,[0_1|4]), (629,196,[0_1|4]), (629,214,[0_1|4]), (629,229,[0_1|4]), (629,382,[0_1|4]), (629,400,[0_1|4]), (629,113,[0_1|4]), (629,120,[0_1|4, 1_1|2]), (629,147,[0_1|4, 1_1|2]), (629,151,[0_1|4, 1_1|2]), (629,165,[0_1|4, 1_1|2]), (629,184,[0_1|4, 1_1|2]), (629,195,[0_1|4, 1_1|2]), (629,199,[0_1|4, 1_1|2]), (629,203,[0_1|4, 1_1|2]), (629,208,[0_1|4, 1_1|2]), (629,213,[0_1|4, 1_1|2]), (629,218,[0_1|4, 1_1|2]), (629,223,[0_1|4, 1_1|2]), (629,228,[0_1|4, 1_1|2]), (629,232,[0_1|4, 1_1|2]), (629,255,[0_1|4, 1_1|2]), (629,260,[0_1|4, 1_1|2]), (629,284,[0_1|4]), (629,296,[0_1|4]), (629,300,[0_1|4]), (629,114,[0_1|2]), (629,117,[0_1|2]), (629,123,[0_1|2]), (629,127,[0_1|2]), (629,131,[0_1|2]), (629,135,[0_1|2]), (629,139,[0_1|2]), (629,143,[0_1|2]), (629,155,[0_1|2]), (629,160,[0_1|2]), (629,170,[0_1|2]), (629,174,[2_1|2]), (629,179,[2_1|2]), (629,189,[0_1|2]), (629,192,[0_1|2]), (629,236,[0_1|2]), (629,239,[0_1|2]), (629,242,[0_1|2]), (629,246,[0_1|2]), (629,250,[0_1|2]), (629,265,[0_1|2]), (629,309,[0_1|3]), (629,312,[0_1|3]), (629,315,[1_1|3]), (629,318,[0_1|3]), (629,322,[0_1|3]), (629,326,[0_1|3]), (629,330,[0_1|3]), (629,334,[0_1|3]), (629,338,[0_1|3]), (629,342,[1_1|3]), (629,346,[1_1|3]), (629,350,[0_1|3]), (629,355,[0_1|3]), (629,360,[1_1|3]), (629,365,[2_1|3]), (629,370,[2_1|3]), (629,375,[0_1|3]), (629,378,[0_1|3]), (629,381,[1_1|3]), (629,385,[1_1|3]), (629,389,[1_1|3]), (629,394,[1_1|3]), (629,399,[1_1|3]), (629,404,[1_1|3]), (629,409,[1_1|3]), (629,414,[0_1|3]), (629,417,[0_1|3]), (629,420,[0_1|3]), (629,424,[0_1|3]), (629,428,[0_1|3]), (630,631,[2_1|4]), (631,632,[1_1|4]), (632,633,[0_1|4]), (633,634,[4_1|4]), (634,537,[4_1|4]), (634,555,[4_1|4]), (634,570,[4_1|4]), (634,196,[4_1|4]), (634,214,[4_1|4]), (634,229,[4_1|4]), (634,382,[4_1|4]), (634,400,[4_1|4]), (634,113,[4_1|4]), (634,120,[4_1|4]), (634,147,[4_1|4]), (634,151,[4_1|4]), (634,165,[4_1|4]), (634,184,[4_1|4]), (634,195,[4_1|4]), (634,199,[4_1|4]), (634,203,[4_1|4]), (634,208,[4_1|4]), (634,213,[4_1|4]), (634,218,[4_1|4]), (634,223,[4_1|4]), (634,228,[4_1|4]), (634,232,[4_1|4]), (634,255,[4_1|4]), (634,260,[4_1|4]), (634,284,[4_1|4]), (634,296,[4_1|4]), (634,300,[4_1|4]), (635,636,[2_1|4]), (636,637,[2_1|4]), (637,638,[1_1|4]), (638,639,[3_1|4]), (639,537,[0_1|4]), (639,555,[0_1|4]), (639,570,[0_1|4]), (639,196,[0_1|4]), (639,214,[0_1|4]), (639,229,[0_1|4]), (639,382,[0_1|4]), (639,400,[0_1|4]), (639,113,[0_1|4]), (639,120,[0_1|4, 1_1|2]), (639,147,[0_1|4, 1_1|2]), (639,151,[0_1|4, 1_1|2]), (639,165,[0_1|4, 1_1|2]), (639,184,[0_1|4, 1_1|2]), (639,195,[0_1|4, 1_1|2]), (639,199,[0_1|4, 1_1|2]), (639,203,[0_1|4, 1_1|2]), (639,208,[0_1|4, 1_1|2]), (639,213,[0_1|4, 1_1|2]), (639,218,[0_1|4, 1_1|2]), (639,223,[0_1|4, 1_1|2]), (639,228,[0_1|4, 1_1|2]), (639,232,[0_1|4, 1_1|2]), (639,255,[0_1|4, 1_1|2]), (639,260,[0_1|4, 1_1|2]), (639,284,[0_1|4]), (639,296,[0_1|4]), (639,300,[0_1|4]), (639,114,[0_1|2]), (639,117,[0_1|2]), (639,123,[0_1|2]), (639,127,[0_1|2]), (639,131,[0_1|2]), (639,135,[0_1|2]), (639,139,[0_1|2]), (639,143,[0_1|2]), (639,155,[0_1|2]), (639,160,[0_1|2]), (639,170,[0_1|2]), (639,174,[2_1|2]), (639,179,[2_1|2]), (639,189,[0_1|2]), (639,192,[0_1|2]), (639,236,[0_1|2]), (639,239,[0_1|2]), (639,242,[0_1|2]), (639,246,[0_1|2]), (639,250,[0_1|2]), (639,265,[0_1|2]), (639,309,[0_1|3]), (639,312,[0_1|3]), (639,315,[1_1|3]), (639,318,[0_1|3]), (639,322,[0_1|3]), (639,326,[0_1|3]), (639,330,[0_1|3]), (639,334,[0_1|3]), (639,338,[0_1|3]), (639,342,[1_1|3]), (639,346,[1_1|3]), (639,350,[0_1|3]), (639,355,[0_1|3]), (639,360,[1_1|3]), (639,365,[2_1|3]), (639,370,[2_1|3]), (639,375,[0_1|3]), (639,378,[0_1|3]), (639,381,[1_1|3]), (639,385,[1_1|3]), (639,389,[1_1|3]), (639,394,[1_1|3]), (639,399,[1_1|3]), (639,404,[1_1|3]), (639,409,[1_1|3]), (639,414,[0_1|3]), (639,417,[0_1|3]), (639,420,[0_1|3]), (639,424,[0_1|3]), (639,428,[0_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(0(1(x1))) -> 0(1(2(0(x1)))) 0(0(1(x1))) -> 0(3(1(0(x1)))) 0(0(1(x1))) -> 1(0(4(0(x1)))) 0(0(1(x1))) -> 0(1(3(0(2(x1))))) 0(0(1(x1))) -> 0(1(3(0(4(x1))))) 0(0(1(x1))) -> 0(2(0(1(2(x1))))) 0(0(1(x1))) -> 0(3(0(1(2(x1))))) 0(0(1(x1))) -> 0(3(0(3(1(x1))))) 0(0(1(x1))) -> 0(4(0(4(1(x1))))) 0(0(1(x1))) -> 1(2(0(2(0(x1))))) 0(0(1(x1))) -> 1(2(2(0(0(x1))))) 0(0(1(x1))) -> 0(0(2(2(1(2(x1)))))) 0(0(1(x1))) -> 0(1(2(4(2(0(x1)))))) 0(0(1(x1))) -> 1(2(0(3(0(4(x1)))))) 0(1(1(x1))) -> 0(2(1(1(x1)))) 0(1(1(x1))) -> 0(3(1(1(x1)))) 0(1(1(x1))) -> 1(1(3(0(4(x1))))) 0(1(1(x1))) -> 1(2(0(2(1(x1))))) 0(1(1(x1))) -> 1(0(3(1(2(4(x1)))))) 0(1(1(x1))) -> 1(0(4(2(1(2(x1)))))) 0(1(1(x1))) -> 1(1(2(4(3(0(x1)))))) 0(1(1(x1))) -> 1(2(1(0(4(4(x1)))))) 0(1(1(x1))) -> 1(2(2(1(3(0(x1)))))) 0(5(1(x1))) -> 0(3(1(5(x1)))) 0(5(1(x1))) -> 0(4(5(1(x1)))) 0(5(1(x1))) -> 0(2(3(1(5(x1))))) 0(5(1(x1))) -> 0(3(1(5(2(x1))))) 0(5(1(x1))) -> 0(3(1(2(5(2(x1)))))) 5(0(1(x1))) -> 5(1(2(4(0(x1))))) 5(0(1(x1))) -> 5(0(2(1(2(4(x1)))))) 5(0(1(x1))) -> 5(1(2(3(0(4(x1)))))) 0(0(1(5(x1)))) -> 0(4(1(0(5(x1))))) 0(0(2(1(x1)))) -> 2(0(3(0(2(1(x1)))))) 0(0(2(1(x1)))) -> 2(3(0(2(0(1(x1)))))) 0(1(0(1(x1)))) -> 1(0(2(0(1(x1))))) 0(1(1(1(x1)))) -> 1(1(3(1(0(x1))))) 5(0(1(1(x1)))) -> 1(5(1(2(0(x1))))) 5(3(0(1(x1)))) -> 5(1(2(3(0(x1))))) 5(3(1(5(x1)))) -> 5(3(1(2(5(x1))))) 5(3(2(1(x1)))) -> 1(2(3(5(2(x1))))) 5(4(0(1(x1)))) -> 1(2(5(0(4(x1))))) 0(0(5(1(5(x1))))) -> 1(2(5(5(0(0(x1)))))) 0(5(3(0(1(x1))))) -> 1(0(5(3(0(4(x1)))))) 0(5(3(4(1(x1))))) -> 1(0(3(5(4(5(x1)))))) 0(5(4(0(1(x1))))) -> 0(1(3(0(4(5(x1)))))) 5(4(2(1(1(x1))))) -> 5(4(1(2(1(2(x1)))))) The (relative) TRS S consists of the following rules: encArg(1(x_1)) -> 1(encArg(x_1)) 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_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 0(1(1(x1))) ->^+ 1(2(2(1(3(0(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 / 1(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))) -> 0(1(2(0(x1)))) 0(0(1(x1))) -> 0(3(1(0(x1)))) 0(0(1(x1))) -> 1(0(4(0(x1)))) 0(0(1(x1))) -> 0(1(3(0(2(x1))))) 0(0(1(x1))) -> 0(1(3(0(4(x1))))) 0(0(1(x1))) -> 0(2(0(1(2(x1))))) 0(0(1(x1))) -> 0(3(0(1(2(x1))))) 0(0(1(x1))) -> 0(3(0(3(1(x1))))) 0(0(1(x1))) -> 0(4(0(4(1(x1))))) 0(0(1(x1))) -> 1(2(0(2(0(x1))))) 0(0(1(x1))) -> 1(2(2(0(0(x1))))) 0(0(1(x1))) -> 0(0(2(2(1(2(x1)))))) 0(0(1(x1))) -> 0(1(2(4(2(0(x1)))))) 0(0(1(x1))) -> 1(2(0(3(0(4(x1)))))) 0(1(1(x1))) -> 0(2(1(1(x1)))) 0(1(1(x1))) -> 0(3(1(1(x1)))) 0(1(1(x1))) -> 1(1(3(0(4(x1))))) 0(1(1(x1))) -> 1(2(0(2(1(x1))))) 0(1(1(x1))) -> 1(0(3(1(2(4(x1)))))) 0(1(1(x1))) -> 1(0(4(2(1(2(x1)))))) 0(1(1(x1))) -> 1(1(2(4(3(0(x1)))))) 0(1(1(x1))) -> 1(2(1(0(4(4(x1)))))) 0(1(1(x1))) -> 1(2(2(1(3(0(x1)))))) 0(5(1(x1))) -> 0(3(1(5(x1)))) 0(5(1(x1))) -> 0(4(5(1(x1)))) 0(5(1(x1))) -> 0(2(3(1(5(x1))))) 0(5(1(x1))) -> 0(3(1(5(2(x1))))) 0(5(1(x1))) -> 0(3(1(2(5(2(x1)))))) 5(0(1(x1))) -> 5(1(2(4(0(x1))))) 5(0(1(x1))) -> 5(0(2(1(2(4(x1)))))) 5(0(1(x1))) -> 5(1(2(3(0(4(x1)))))) 0(0(1(5(x1)))) -> 0(4(1(0(5(x1))))) 0(0(2(1(x1)))) -> 2(0(3(0(2(1(x1)))))) 0(0(2(1(x1)))) -> 2(3(0(2(0(1(x1)))))) 0(1(0(1(x1)))) -> 1(0(2(0(1(x1))))) 0(1(1(1(x1)))) -> 1(1(3(1(0(x1))))) 5(0(1(1(x1)))) -> 1(5(1(2(0(x1))))) 5(3(0(1(x1)))) -> 5(1(2(3(0(x1))))) 5(3(1(5(x1)))) -> 5(3(1(2(5(x1))))) 5(3(2(1(x1)))) -> 1(2(3(5(2(x1))))) 5(4(0(1(x1)))) -> 1(2(5(0(4(x1))))) 0(0(5(1(5(x1))))) -> 1(2(5(5(0(0(x1)))))) 0(5(3(0(1(x1))))) -> 1(0(5(3(0(4(x1)))))) 0(5(3(4(1(x1))))) -> 1(0(3(5(4(5(x1)))))) 0(5(4(0(1(x1))))) -> 0(1(3(0(4(5(x1)))))) 5(4(2(1(1(x1))))) -> 5(4(1(2(1(2(x1)))))) The (relative) TRS S consists of the following rules: encArg(1(x_1)) -> 1(encArg(x_1)) 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_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))) -> 0(1(2(0(x1)))) 0(0(1(x1))) -> 0(3(1(0(x1)))) 0(0(1(x1))) -> 1(0(4(0(x1)))) 0(0(1(x1))) -> 0(1(3(0(2(x1))))) 0(0(1(x1))) -> 0(1(3(0(4(x1))))) 0(0(1(x1))) -> 0(2(0(1(2(x1))))) 0(0(1(x1))) -> 0(3(0(1(2(x1))))) 0(0(1(x1))) -> 0(3(0(3(1(x1))))) 0(0(1(x1))) -> 0(4(0(4(1(x1))))) 0(0(1(x1))) -> 1(2(0(2(0(x1))))) 0(0(1(x1))) -> 1(2(2(0(0(x1))))) 0(0(1(x1))) -> 0(0(2(2(1(2(x1)))))) 0(0(1(x1))) -> 0(1(2(4(2(0(x1)))))) 0(0(1(x1))) -> 1(2(0(3(0(4(x1)))))) 0(1(1(x1))) -> 0(2(1(1(x1)))) 0(1(1(x1))) -> 0(3(1(1(x1)))) 0(1(1(x1))) -> 1(1(3(0(4(x1))))) 0(1(1(x1))) -> 1(2(0(2(1(x1))))) 0(1(1(x1))) -> 1(0(3(1(2(4(x1)))))) 0(1(1(x1))) -> 1(0(4(2(1(2(x1)))))) 0(1(1(x1))) -> 1(1(2(4(3(0(x1)))))) 0(1(1(x1))) -> 1(2(1(0(4(4(x1)))))) 0(1(1(x1))) -> 1(2(2(1(3(0(x1)))))) 0(5(1(x1))) -> 0(3(1(5(x1)))) 0(5(1(x1))) -> 0(4(5(1(x1)))) 0(5(1(x1))) -> 0(2(3(1(5(x1))))) 0(5(1(x1))) -> 0(3(1(5(2(x1))))) 0(5(1(x1))) -> 0(3(1(2(5(2(x1)))))) 5(0(1(x1))) -> 5(1(2(4(0(x1))))) 5(0(1(x1))) -> 5(0(2(1(2(4(x1)))))) 5(0(1(x1))) -> 5(1(2(3(0(4(x1)))))) 0(0(1(5(x1)))) -> 0(4(1(0(5(x1))))) 0(0(2(1(x1)))) -> 2(0(3(0(2(1(x1)))))) 0(0(2(1(x1)))) -> 2(3(0(2(0(1(x1)))))) 0(1(0(1(x1)))) -> 1(0(2(0(1(x1))))) 0(1(1(1(x1)))) -> 1(1(3(1(0(x1))))) 5(0(1(1(x1)))) -> 1(5(1(2(0(x1))))) 5(3(0(1(x1)))) -> 5(1(2(3(0(x1))))) 5(3(1(5(x1)))) -> 5(3(1(2(5(x1))))) 5(3(2(1(x1)))) -> 1(2(3(5(2(x1))))) 5(4(0(1(x1)))) -> 1(2(5(0(4(x1))))) 0(0(5(1(5(x1))))) -> 1(2(5(5(0(0(x1)))))) 0(5(3(0(1(x1))))) -> 1(0(5(3(0(4(x1)))))) 0(5(3(4(1(x1))))) -> 1(0(3(5(4(5(x1)))))) 0(5(4(0(1(x1))))) -> 0(1(3(0(4(5(x1)))))) 5(4(2(1(1(x1))))) -> 5(4(1(2(1(2(x1)))))) The (relative) TRS S consists of the following rules: encArg(1(x_1)) -> 1(encArg(x_1)) 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_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