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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(n^1, n^1).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), 36 ms] (4) CpxRelTRS (5) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] (6) CpxTRS (7) CpxTrsMatchBoundsProof [FINISHED, 0 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 (innermost) of the given DCpxTrs could be proven to be BOUNDS(n^1, n^1). The TRS R consists of the following rules: c(b(a(a(x1)))) -> a(a(b(a(a(b(c(b(a(c(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(b(x_1)) -> b(encArg(x_1)) encArg(a(x_1)) -> a(encArg(x_1)) encArg(cons_c(x_1)) -> c(encArg(x_1)) encode_c(x_1) -> c(encArg(x_1)) encode_b(x_1) -> b(encArg(x_1)) encode_a(x_1) -> a(encArg(x_1)) ---------------------------------------- (2) Obligation: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^1). The TRS R consists of the following rules: c(b(a(a(x1)))) -> a(a(b(a(a(b(c(b(a(c(x1)))))))))) The (relative) TRS S consists of the following rules: encArg(b(x_1)) -> b(encArg(x_1)) encArg(a(x_1)) -> a(encArg(x_1)) encArg(cons_c(x_1)) -> c(encArg(x_1)) encode_c(x_1) -> c(encArg(x_1)) encode_b(x_1) -> b(encArg(x_1)) encode_a(x_1) -> a(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(n^1, n^1). The TRS R consists of the following rules: c(b(a(a(x1)))) -> a(a(b(a(a(b(c(b(a(c(x1)))))))))) The (relative) TRS S consists of the following rules: encArg(b(x_1)) -> b(encArg(x_1)) encArg(a(x_1)) -> a(encArg(x_1)) encArg(cons_c(x_1)) -> c(encArg(x_1)) encode_c(x_1) -> c(encArg(x_1)) encode_b(x_1) -> b(encArg(x_1)) encode_a(x_1) -> a(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: c(b(a(a(x1)))) -> a(a(b(a(a(b(c(b(a(c(x1)))))))))) encArg(b(x_1)) -> b(encArg(x_1)) encArg(a(x_1)) -> a(encArg(x_1)) encArg(cons_c(x_1)) -> c(encArg(x_1)) encode_c(x_1) -> c(encArg(x_1)) encode_b(x_1) -> b(encArg(x_1)) encode_a(x_1) -> a(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 4. The certificate found is represented by the following graph. "[5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 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, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 90, 91, 92, 93, 94, 95, 96, 97, 98] {(5,6,[c_1|0, encArg_1|0, encode_c_1|0, encode_b_1|0, encode_a_1|0]), (5,7,[a_1|1]), (5,16,[b_1|1, a_1|1, c_1|1]), (5,22,[a_1|2]), (6,6,[b_1|0, a_1|0, cons_c_1|0]), (7,8,[a_1|1]), (8,9,[b_1|1]), (9,10,[a_1|1]), (10,11,[a_1|1]), (11,12,[b_1|1]), (12,13,[c_1|1]), (12,31,[a_1|2]), (13,14,[b_1|1]), (14,15,[a_1|1]), (15,6,[c_1|1]), (15,7,[a_1|1]), (16,6,[encArg_1|1]), (16,16,[b_1|1, a_1|1, c_1|1]), (16,22,[a_1|2]), (22,23,[a_1|2]), (23,24,[b_1|2]), (24,25,[a_1|2]), (25,26,[a_1|2]), (26,27,[b_1|2]), (27,28,[c_1|2]), (27,40,[a_1|3]), (28,29,[b_1|2]), (29,30,[a_1|2]), (30,16,[c_1|2]), (30,22,[c_1|2, a_1|2]), (30,23,[c_1|2]), (30,49,[a_1|3]), (31,32,[a_1|2]), (32,33,[b_1|2]), (33,34,[a_1|2]), (34,35,[a_1|2]), (35,36,[b_1|2]), (36,37,[c_1|2]), (37,38,[b_1|2]), (38,39,[a_1|2]), (39,7,[c_1|2]), (40,41,[a_1|3]), (41,42,[b_1|3]), (42,43,[a_1|3]), (43,44,[a_1|3]), (44,45,[b_1|3]), (45,46,[c_1|3]), (46,47,[b_1|3]), (47,48,[a_1|3]), (48,22,[c_1|3]), (48,49,[c_1|3]), (49,50,[a_1|3]), (50,51,[b_1|3]), (51,52,[a_1|3]), (52,53,[a_1|3]), (53,54,[b_1|3]), (54,55,[c_1|3]), (54,67,[a_1|4]), (55,56,[b_1|3]), (56,57,[a_1|3]), (57,26,[c_1|3]), (57,58,[a_1|3]), (58,59,[a_1|3]), (59,60,[b_1|3]), (60,61,[a_1|3]), (61,62,[a_1|3]), (62,63,[b_1|3]), (63,64,[c_1|3]), (63,90,[a_1|4]), (64,65,[b_1|3]), (65,66,[a_1|3]), (66,41,[c_1|3]), (66,76,[a_1|4]), (67,68,[a_1|4]), (68,69,[b_1|4]), (69,70,[a_1|4]), (70,71,[a_1|4]), (71,72,[b_1|4]), (72,73,[c_1|4]), (73,74,[b_1|4]), (74,75,[a_1|4]), (75,58,[c_1|4]), (76,77,[a_1|4]), (77,78,[b_1|4]), (78,79,[a_1|4]), (79,80,[a_1|4]), (80,81,[b_1|4]), (81,82,[c_1|4]), (82,83,[b_1|4]), (83,84,[a_1|4]), (84,44,[c_1|4]), (90,91,[a_1|4]), (91,92,[b_1|4]), (92,93,[a_1|4]), (93,94,[a_1|4]), (94,95,[b_1|4]), (95,96,[c_1|4]), (96,97,[b_1|4]), (97,98,[a_1|4]), (98,76,[c_1|4])}" ---------------------------------------- (8) BOUNDS(1, n^1) ---------------------------------------- (9) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) Transformed a relative TRS into a decreasing-loop problem. ---------------------------------------- (10) Obligation: Analyzing the following TRS for decreasing loops: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^1). The TRS R consists of the following rules: c(b(a(a(x1)))) -> a(a(b(a(a(b(c(b(a(c(x1)))))))))) The (relative) TRS S consists of the following rules: encArg(b(x_1)) -> b(encArg(x_1)) encArg(a(x_1)) -> a(encArg(x_1)) encArg(cons_c(x_1)) -> c(encArg(x_1)) encode_c(x_1) -> c(encArg(x_1)) encode_b(x_1) -> b(encArg(x_1)) encode_a(x_1) -> a(encArg(x_1)) Rewrite Strategy: INNERMOST ---------------------------------------- (11) DecreasingLoopProof (LOWER BOUND(ID)) The following loop(s) give(s) rise to the lower bound Omega(n^1): The rewrite sequence c(b(a(a(x1)))) ->^+ a(a(b(a(a(b(c(b(a(c(x1)))))))))) gives rise to a decreasing loop by considering the right hand sides subterm at position [0,0,0,0,0,0,0,0,0]. The pumping substitution is [x1 / b(a(a(x1)))]. The result substitution is [ ]. ---------------------------------------- (12) Complex Obligation (BEST) ---------------------------------------- (13) Obligation: Proved the lower bound n^1 for the following obligation: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^1). The TRS R consists of the following rules: c(b(a(a(x1)))) -> a(a(b(a(a(b(c(b(a(c(x1)))))))))) The (relative) TRS S consists of the following rules: encArg(b(x_1)) -> b(encArg(x_1)) encArg(a(x_1)) -> a(encArg(x_1)) encArg(cons_c(x_1)) -> c(encArg(x_1)) encode_c(x_1) -> c(encArg(x_1)) encode_b(x_1) -> b(encArg(x_1)) encode_a(x_1) -> a(encArg(x_1)) Rewrite Strategy: INNERMOST ---------------------------------------- (14) LowerBoundPropagationProof (FINISHED) Propagated lower bound. ---------------------------------------- (15) BOUNDS(n^1, INF) ---------------------------------------- (16) Obligation: Analyzing the following TRS for decreasing loops: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^1). The TRS R consists of the following rules: c(b(a(a(x1)))) -> a(a(b(a(a(b(c(b(a(c(x1)))))))))) The (relative) TRS S consists of the following rules: encArg(b(x_1)) -> b(encArg(x_1)) encArg(a(x_1)) -> a(encArg(x_1)) encArg(cons_c(x_1)) -> c(encArg(x_1)) encode_c(x_1) -> c(encArg(x_1)) encode_b(x_1) -> b(encArg(x_1)) encode_a(x_1) -> a(encArg(x_1)) Rewrite Strategy: INNERMOST