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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (full) of the given DCpxTrs could be proven to be BOUNDS(n^1, n^1).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), 138 ms] (4) CpxRelTRS (5) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] (6) CpxTRS (7) CpxTrsMatchBoundsProof [FINISHED, 331 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. "[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, 640, 641, 642, 643] {(63,64,[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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(614,217,[4_1|4]), (614,222,[4_1|4]), (614,227,[4_1|4]), (614,232,[4_1|4]), (614,236,[4_1|4]), (614,259,[4_1|4]), (614,264,[4_1|4]), (614,288,[4_1|4]), (614,300,[4_1|4]), (614,304,[4_1|4]), (615,616,[2_1|4]), (616,617,[0_1|4]), (617,618,[2_1|4]), (618,541,[1_1|4]), (618,559,[1_1|4]), (618,574,[1_1|4]), (618,200,[1_1|4]), (618,218,[1_1|4]), (618,233,[1_1|4]), (618,386,[1_1|4]), (618,404,[1_1|4]), (618,117,[1_1|4]), (618,124,[1_1|4]), (618,151,[1_1|4]), (618,155,[1_1|4]), (618,169,[1_1|4]), (618,188,[1_1|4]), (618,199,[1_1|4]), (618,203,[1_1|4]), (618,207,[1_1|4]), (618,212,[1_1|4]), (618,217,[1_1|4]), (618,222,[1_1|4]), (618,227,[1_1|4]), (618,232,[1_1|4]), (618,236,[1_1|4]), (618,259,[1_1|4]), (618,264,[1_1|4]), (618,288,[1_1|4]), (618,300,[1_1|4]), (618,304,[1_1|4]), (619,620,[0_1|4]), (620,621,[3_1|4]), (621,622,[1_1|4]), (622,623,[2_1|4]), (623,541,[4_1|4]), (623,559,[4_1|4]), (623,574,[4_1|4]), (623,200,[4_1|4]), (623,218,[4_1|4]), (623,233,[4_1|4]), (623,386,[4_1|4]), (623,404,[4_1|4]), (623,117,[4_1|4]), (623,124,[4_1|4]), (623,151,[4_1|4]), (623,155,[4_1|4]), (623,169,[4_1|4]), (623,188,[4_1|4]), (623,199,[4_1|4]), (623,203,[4_1|4]), (623,207,[4_1|4]), (623,212,[4_1|4]), (623,217,[4_1|4]), (623,222,[4_1|4]), (623,227,[4_1|4]), (623,232,[4_1|4]), (623,236,[4_1|4]), (623,259,[4_1|4]), (623,264,[4_1|4]), (623,288,[4_1|4]), (623,300,[4_1|4]), (623,304,[4_1|4]), (624,625,[0_1|4]), (625,626,[4_1|4]), (626,627,[2_1|4]), (627,628,[1_1|4]), (628,541,[2_1|4]), (628,559,[2_1|4]), (628,574,[2_1|4]), (628,200,[2_1|4]), (628,218,[2_1|4]), (628,233,[2_1|4]), (628,386,[2_1|4]), (628,404,[2_1|4]), (628,117,[2_1|4]), (628,124,[2_1|4]), (628,151,[2_1|4]), (628,155,[2_1|4]), (628,169,[2_1|4]), (628,188,[2_1|4]), (628,199,[2_1|4]), (628,203,[2_1|4]), (628,207,[2_1|4]), (628,212,[2_1|4]), (628,217,[2_1|4]), (628,222,[2_1|4]), (628,227,[2_1|4]), (628,232,[2_1|4]), (628,236,[2_1|4]), (628,259,[2_1|4]), (628,264,[2_1|4]), (628,288,[2_1|4]), (628,300,[2_1|4]), (628,304,[2_1|4]), (629,630,[1_1|4]), (630,631,[2_1|4]), (631,632,[4_1|4]), (632,633,[3_1|4]), (633,541,[0_1|4]), (633,559,[0_1|4]), (633,574,[0_1|4]), (633,200,[0_1|4]), (633,218,[0_1|4]), (633,233,[0_1|4]), (633,386,[0_1|4]), (633,404,[0_1|4]), (633,117,[0_1|4]), (633,124,[0_1|4, 1_1|2]), (633,151,[0_1|4, 1_1|2]), (633,155,[0_1|4, 1_1|2]), (633,169,[0_1|4, 1_1|2]), (633,188,[0_1|4, 1_1|2]), (633,199,[0_1|4, 1_1|2]), (633,203,[0_1|4, 1_1|2]), (633,207,[0_1|4, 1_1|2]), (633,212,[0_1|4, 1_1|2]), (633,217,[0_1|4, 1_1|2]), (633,222,[0_1|4, 1_1|2]), (633,227,[0_1|4, 1_1|2]), (633,232,[0_1|4, 1_1|2]), (633,236,[0_1|4, 1_1|2]), (633,259,[0_1|4, 1_1|2]), (633,264,[0_1|4, 1_1|2]), (633,288,[0_1|4]), (633,300,[0_1|4]), (633,304,[0_1|4]), (633,118,[0_1|2]), (633,121,[0_1|2]), (633,127,[0_1|2]), (633,131,[0_1|2]), (633,135,[0_1|2]), (633,139,[0_1|2]), (633,143,[0_1|2]), (633,147,[0_1|2]), (633,159,[0_1|2]), (633,164,[0_1|2]), (633,174,[0_1|2]), (633,178,[2_1|2]), (633,183,[2_1|2]), (633,193,[0_1|2]), (633,196,[0_1|2]), (633,240,[0_1|2]), (633,243,[0_1|2]), (633,246,[0_1|2]), (633,250,[0_1|2]), (633,254,[0_1|2]), (633,269,[0_1|2]), (633,313,[0_1|3]), (633,316,[0_1|3]), (633,319,[1_1|3]), (633,322,[0_1|3]), (633,326,[0_1|3]), (633,330,[0_1|3]), (633,334,[0_1|3]), (633,338,[0_1|3]), (633,342,[0_1|3]), (633,346,[1_1|3]), (633,350,[1_1|3]), (633,354,[0_1|3]), (633,359,[0_1|3]), (633,364,[1_1|3]), (633,369,[2_1|3]), (633,374,[2_1|3]), (633,379,[0_1|3]), (633,382,[0_1|3]), (633,385,[1_1|3]), (633,389,[1_1|3]), (633,393,[1_1|3]), (633,398,[1_1|3]), (633,403,[1_1|3]), (633,408,[1_1|3]), (633,413,[1_1|3]), (633,418,[0_1|3]), (633,421,[0_1|3]), (633,424,[0_1|3]), (633,428,[0_1|3]), (633,432,[0_1|3]), (634,635,[2_1|4]), (635,636,[1_1|4]), (636,637,[0_1|4]), (637,638,[4_1|4]), (638,541,[4_1|4]), (638,559,[4_1|4]), (638,574,[4_1|4]), (638,200,[4_1|4]), (638,218,[4_1|4]), (638,233,[4_1|4]), (638,386,[4_1|4]), (638,404,[4_1|4]), (638,117,[4_1|4]), (638,124,[4_1|4]), (638,151,[4_1|4]), (638,155,[4_1|4]), (638,169,[4_1|4]), (638,188,[4_1|4]), (638,199,[4_1|4]), (638,203,[4_1|4]), (638,207,[4_1|4]), (638,212,[4_1|4]), (638,217,[4_1|4]), (638,222,[4_1|4]), (638,227,[4_1|4]), (638,232,[4_1|4]), (638,236,[4_1|4]), (638,259,[4_1|4]), (638,264,[4_1|4]), (638,288,[4_1|4]), (638,300,[4_1|4]), (638,304,[4_1|4]), (639,640,[2_1|4]), (640,641,[2_1|4]), (641,642,[1_1|4]), (642,643,[3_1|4]), (643,541,[0_1|4]), (643,559,[0_1|4]), (643,574,[0_1|4]), (643,200,[0_1|4]), (643,218,[0_1|4]), (643,233,[0_1|4]), (643,386,[0_1|4]), (643,404,[0_1|4]), (643,117,[0_1|4]), (643,124,[0_1|4, 1_1|2]), (643,151,[0_1|4, 1_1|2]), (643,155,[0_1|4, 1_1|2]), (643,169,[0_1|4, 1_1|2]), (643,188,[0_1|4, 1_1|2]), (643,199,[0_1|4, 1_1|2]), (643,203,[0_1|4, 1_1|2]), (643,207,[0_1|4, 1_1|2]), (643,212,[0_1|4, 1_1|2]), (643,217,[0_1|4, 1_1|2]), (643,222,[0_1|4, 1_1|2]), (643,227,[0_1|4, 1_1|2]), (643,232,[0_1|4, 1_1|2]), (643,236,[0_1|4, 1_1|2]), (643,259,[0_1|4, 1_1|2]), (643,264,[0_1|4, 1_1|2]), (643,288,[0_1|4]), (643,300,[0_1|4]), (643,304,[0_1|4]), (643,118,[0_1|2]), (643,121,[0_1|2]), (643,127,[0_1|2]), (643,131,[0_1|2]), (643,135,[0_1|2]), (643,139,[0_1|2]), (643,143,[0_1|2]), (643,147,[0_1|2]), (643,159,[0_1|2]), (643,164,[0_1|2]), (643,174,[0_1|2]), (643,178,[2_1|2]), (643,183,[2_1|2]), (643,193,[0_1|2]), (643,196,[0_1|2]), (643,240,[0_1|2]), (643,243,[0_1|2]), (643,246,[0_1|2]), (643,250,[0_1|2]), (643,254,[0_1|2]), (643,269,[0_1|2]), (643,313,[0_1|3]), (643,316,[0_1|3]), (643,319,[1_1|3]), (643,322,[0_1|3]), (643,326,[0_1|3]), (643,330,[0_1|3]), (643,334,[0_1|3]), (643,338,[0_1|3]), (643,342,[0_1|3]), (643,346,[1_1|3]), (643,350,[1_1|3]), (643,354,[0_1|3]), (643,359,[0_1|3]), (643,364,[1_1|3]), (643,369,[2_1|3]), (643,374,[2_1|3]), (643,379,[0_1|3]), (643,382,[0_1|3]), (643,385,[1_1|3]), (643,389,[1_1|3]), (643,393,[1_1|3]), (643,398,[1_1|3]), (643,403,[1_1|3]), (643,408,[1_1|3]), (643,413,[1_1|3]), (643,418,[0_1|3]), (643,421,[0_1|3]), (643,424,[0_1|3]), (643,428,[0_1|3]), (643,432,[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