/export/starexec/sandbox2/solver/bin/starexec_run_rcdcRelativeAlsoLower /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files -------------------------------------------------------------------------------- WORST_CASE(?, O(n^1)) proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml # AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(1, n^1). (0) DCpxTrs (1) DerivationalComplexityToRuntimeComplexityProof [BOTH BOUNDS(ID, ID), 0 ms] (2) CpxRelTRS (3) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 35 ms] (4) CpxRelTRS (5) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] (6) CpxTRS (7) CpxTrsMatchBoundsProof [FINISHED, 16 ms] (8) BOUNDS(1, n^1) ---------------------------------------- (0) Obligation: The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(1, n^1). The TRS R consists of the following rules: a(a(a(b(x1)))) -> b(a(b(a(x1)))) b(b(a(x1))) -> a(a(a(b(x1)))) S is empty. Rewrite Strategy: INNERMOST ---------------------------------------- (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(cons_a(x_1)) -> a(encArg(x_1)) encArg(cons_b(x_1)) -> b(encArg(x_1)) encode_a(x_1) -> a(encArg(x_1)) encode_b(x_1) -> b(encArg(x_1)) ---------------------------------------- (2) Obligation: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(1, n^1). The TRS R consists of the following rules: a(a(a(b(x1)))) -> b(a(b(a(x1)))) b(b(a(x1))) -> a(a(a(b(x1)))) The (relative) TRS S consists of the following rules: encArg(cons_a(x_1)) -> a(encArg(x_1)) encArg(cons_b(x_1)) -> b(encArg(x_1)) encode_a(x_1) -> a(encArg(x_1)) encode_b(x_1) -> b(encArg(x_1)) Rewrite Strategy: INNERMOST ---------------------------------------- (3) SInnermostTerminationProof (BOTH CONCRETE BOUNDS(ID, ID)) proved innermost termination of relative rules ---------------------------------------- (4) Obligation: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(1, n^1). The TRS R consists of the following rules: a(a(a(b(x1)))) -> b(a(b(a(x1)))) b(b(a(x1))) -> a(a(a(b(x1)))) The (relative) TRS S consists of the following rules: encArg(cons_a(x_1)) -> a(encArg(x_1)) encArg(cons_b(x_1)) -> b(encArg(x_1)) encode_a(x_1) -> a(encArg(x_1)) encode_b(x_1) -> b(encArg(x_1)) Rewrite Strategy: INNERMOST ---------------------------------------- (5) RelTrsToTrsProof (UPPER BOUND(ID)) transformed relative TRS to TRS ---------------------------------------- (6) Obligation: The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^1). The TRS R consists of the following rules: a(a(a(b(x1)))) -> b(a(b(a(x1)))) b(b(a(x1))) -> a(a(a(b(x1)))) encArg(cons_a(x_1)) -> a(encArg(x_1)) encArg(cons_b(x_1)) -> b(encArg(x_1)) encode_a(x_1) -> a(encArg(x_1)) encode_b(x_1) -> b(encArg(x_1)) S is empty. Rewrite Strategy: INNERMOST ---------------------------------------- (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 7. The certificate found is represented by the following graph. "[33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68] {(33,34,[a_1|0, b_1|0, encArg_1|0, encode_a_1|0, encode_b_1|0]), (33,35,[a_1|1, b_1|1]), (33,36,[b_1|2]), (33,39,[a_1|2]), (33,42,[b_1|3]), (34,34,[cons_a_1|0, cons_b_1|0]), (35,34,[encArg_1|1]), (35,35,[a_1|1, b_1|1]), (35,36,[b_1|2]), (35,39,[a_1|2]), (35,42,[b_1|3]), (36,37,[a_1|2]), (37,38,[b_1|2]), (37,45,[a_1|3]), (37,48,[b_1|4]), (38,35,[a_1|2]), (38,36,[a_1|2, b_1|2]), (38,42,[a_1|2]), (39,40,[a_1|2]), (40,41,[a_1|2]), (41,35,[b_1|2]), (41,39,[b_1|2, a_1|2]), (41,37,[b_1|2]), (41,45,[a_1|3]), (41,43,[b_1|2]), (41,42,[b_1|3]), (41,48,[b_1|4]), (42,43,[a_1|3]), (43,44,[b_1|3]), (43,45,[a_1|3]), (43,51,[a_1|4]), (43,48,[b_1|4]), (43,54,[b_1|5]), (44,35,[a_1|3]), (44,39,[a_1|3]), (44,37,[a_1|3]), (44,36,[b_1|2]), (44,43,[a_1|3]), (44,42,[a_1|3, b_1|3]), (44,48,[a_1|3]), (45,46,[a_1|3]), (46,47,[a_1|3]), (47,37,[b_1|3]), (47,43,[b_1|3]), (47,35,[b_1|3]), (47,36,[b_1|3]), (47,45,[a_1|3]), (47,42,[b_1|3]), (47,39,[b_1|3, a_1|2]), (47,51,[a_1|4]), (47,48,[b_1|4, b_1|3]), (47,49,[b_1|3]), (47,54,[b_1|5]), (47,55,[b_1|3]), (48,49,[a_1|4]), (49,50,[b_1|4]), (49,45,[a_1|3]), (49,51,[a_1|4]), (49,48,[b_1|4]), (49,54,[b_1|5]), (50,37,[a_1|4]), (50,43,[a_1|4]), (50,35,[a_1|4]), (50,36,[a_1|4, b_1|2]), (50,42,[a_1|4, b_1|3]), (50,39,[a_1|4]), (50,48,[a_1|4]), (50,49,[a_1|4]), (50,54,[a_1|4]), (50,55,[a_1|4]), (51,52,[a_1|4]), (52,53,[a_1|4]), (53,35,[b_1|4]), (53,39,[b_1|4, a_1|2]), (53,37,[b_1|4]), (53,43,[b_1|4]), (53,49,[b_1|4]), (53,42,[b_1|4, b_1|3]), (53,48,[b_1|4]), (53,45,[a_1|3]), (53,51,[a_1|4]), (53,57,[a_1|5]), (53,36,[b_1|4]), (53,54,[b_1|5, b_1|4]), (53,60,[b_1|6, b_1|4]), (53,55,[b_1|4]), (54,55,[a_1|5]), (55,56,[b_1|5]), (55,51,[a_1|4]), (55,45,[a_1|3]), (55,54,[b_1|5]), (55,48,[b_1|4]), (56,35,[a_1|5]), (56,39,[a_1|5]), (56,37,[a_1|5]), (56,43,[a_1|5]), (56,49,[a_1|5]), (56,42,[a_1|5, b_1|3]), (56,48,[a_1|5]), (56,36,[b_1|2, a_1|5]), (56,54,[a_1|5]), (56,60,[a_1|5]), (56,55,[a_1|5]), (57,58,[a_1|5]), (58,59,[a_1|5]), (59,49,[b_1|5]), (59,37,[b_1|5]), (59,43,[b_1|5]), (59,35,[b_1|5]), (59,36,[b_1|5]), (59,42,[b_1|5, b_1|3]), (59,39,[b_1|5, a_1|2]), (59,48,[b_1|5, b_1|4]), (59,57,[a_1|5]), (59,55,[b_1|5]), (59,54,[b_1|5]), (59,45,[a_1|3]), (59,51,[a_1|4]), (59,63,[a_1|6]), (59,60,[b_1|6, b_1|5]), (59,66,[b_1|7]), (60,61,[a_1|6]), (61,62,[b_1|6]), (61,45,[a_1|3]), (61,51,[a_1|4]), (61,48,[b_1|4]), (61,54,[b_1|5]), (62,49,[a_1|6]), (62,37,[a_1|6]), (62,43,[a_1|6]), (62,35,[a_1|6]), (62,36,[a_1|6, b_1|2]), (62,42,[a_1|6, b_1|3]), (62,39,[a_1|6]), (62,48,[a_1|6]), (62,55,[a_1|6]), (62,54,[a_1|6]), (62,60,[a_1|6]), (62,66,[a_1|6]), (63,64,[a_1|6]), (64,65,[a_1|6]), (65,55,[b_1|6]), (65,35,[b_1|6]), (65,39,[b_1|6, a_1|2]), (65,37,[b_1|6]), (65,43,[b_1|6]), (65,49,[b_1|6]), (65,42,[b_1|6, b_1|3]), (65,48,[b_1|6, b_1|4]), (65,36,[b_1|6]), (65,54,[b_1|6, b_1|5]), (65,60,[b_1|6]), (65,63,[a_1|6]), (65,57,[a_1|5]), (65,45,[a_1|3]), (65,51,[a_1|4]), (65,66,[b_1|7]), (66,67,[a_1|7]), (67,68,[b_1|7]), (67,51,[a_1|4]), (67,45,[a_1|3]), (67,54,[b_1|5]), (67,48,[b_1|4]), (68,55,[a_1|7]), (68,35,[a_1|7]), (68,39,[a_1|7]), (68,37,[a_1|7]), (68,43,[a_1|7]), (68,49,[a_1|7]), (68,42,[a_1|7, b_1|3]), (68,48,[a_1|7]), (68,36,[a_1|7, b_1|2]), (68,54,[a_1|7]), (68,60,[a_1|7]), (68,66,[a_1|7])}" ---------------------------------------- (8) BOUNDS(1, n^1)