/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (full) of the given DCpxTrs could be proven to be BOUNDS(1, n^1).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), 72 ms] (4) CpxRelTRS (5) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] (6) CpxTRS (7) CpxTrsMatchBoundsProof [FINISHED, 4 ms] (8) BOUNDS(1, n^1) ---------------------------------------- (0) Obligation: The Derivational Complexity (full) of the given DCpxTrs could be proven to be BOUNDS(1, n^1). The TRS R consists of the following rules: 1(2(1(1(0(0(1(0(2(1(1(0(2(x1))))))))))))) -> 1(2(2(1(0(0(2(0(1(2(2(0(1(0(0(2(2(x1))))))))))))))))) 2(0(1(2(1(2(1(2(1(0(0(1(0(x1))))))))))))) -> 2(2(0(0(2(1(1(1(0(0(0(1(1(1(2(2(0(x1))))))))))))))))) 2(1(0(0(2(1(1(1(0(1(2(2(0(x1))))))))))))) -> 0(0(2(1(0(0(1(1(2(2(2(2(2(1(2(1(0(x1))))))))))))))))) 2(2(0(0(1(0(2(1(1(1(0(0(1(x1))))))))))))) -> 2(2(1(1(2(0(0(1(2(2(0(2(0(0(2(2(2(x1))))))))))))))))) 2(2(1(0(1(0(1(1(0(0(0(2(0(x1))))))))))))) -> 0(2(2(1(0(0(1(0(0(1(1(1(1(0(0(2(0(x1))))))))))))))))) 2(2(1(1(0(2(0(0(0(2(2(0(0(x1))))))))))))) -> 0(0(1(2(0(0(0(2(2(0(1(2(2(2(2(0(0(x1))))))))))))))))) 2(2(2(1(0(2(0(0(1(0(1(0(2(x1))))))))))))) -> 0(2(2(0(0(2(2(2(1(2(0(0(2(2(0(2(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(0(x_1)) -> 0(encArg(x_1)) encArg(cons_1(x_1)) -> 1(encArg(x_1)) encArg(cons_2(x_1)) -> 2(encArg(x_1)) encode_1(x_1) -> 1(encArg(x_1)) encode_2(x_1) -> 2(encArg(x_1)) encode_0(x_1) -> 0(encArg(x_1)) ---------------------------------------- (2) Obligation: The Runtime Complexity (full) of the given CpxRelTRS could be proven to be BOUNDS(1, n^1). The TRS R consists of the following rules: 1(2(1(1(0(0(1(0(2(1(1(0(2(x1))))))))))))) -> 1(2(2(1(0(0(2(0(1(2(2(0(1(0(0(2(2(x1))))))))))))))))) 2(0(1(2(1(2(1(2(1(0(0(1(0(x1))))))))))))) -> 2(2(0(0(2(1(1(1(0(0(0(1(1(1(2(2(0(x1))))))))))))))))) 2(1(0(0(2(1(1(1(0(1(2(2(0(x1))))))))))))) -> 0(0(2(1(0(0(1(1(2(2(2(2(2(1(2(1(0(x1))))))))))))))))) 2(2(0(0(1(0(2(1(1(1(0(0(1(x1))))))))))))) -> 2(2(1(1(2(0(0(1(2(2(0(2(0(0(2(2(2(x1))))))))))))))))) 2(2(1(0(1(0(1(1(0(0(0(2(0(x1))))))))))))) -> 0(2(2(1(0(0(1(0(0(1(1(1(1(0(0(2(0(x1))))))))))))))))) 2(2(1(1(0(2(0(0(0(2(2(0(0(x1))))))))))))) -> 0(0(1(2(0(0(0(2(2(0(1(2(2(2(2(0(0(x1))))))))))))))))) 2(2(2(1(0(2(0(0(1(0(1(0(2(x1))))))))))))) -> 0(2(2(0(0(2(2(2(1(2(0(0(2(2(0(2(2(x1))))))))))))))))) The (relative) TRS S consists of the following rules: encArg(0(x_1)) -> 0(encArg(x_1)) encArg(cons_1(x_1)) -> 1(encArg(x_1)) encArg(cons_2(x_1)) -> 2(encArg(x_1)) encode_1(x_1) -> 1(encArg(x_1)) encode_2(x_1) -> 2(encArg(x_1)) encode_0(x_1) -> 0(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(1, n^1). The TRS R consists of the following rules: 1(2(1(1(0(0(1(0(2(1(1(0(2(x1))))))))))))) -> 1(2(2(1(0(0(2(0(1(2(2(0(1(0(0(2(2(x1))))))))))))))))) 2(0(1(2(1(2(1(2(1(0(0(1(0(x1))))))))))))) -> 2(2(0(0(2(1(1(1(0(0(0(1(1(1(2(2(0(x1))))))))))))))))) 2(1(0(0(2(1(1(1(0(1(2(2(0(x1))))))))))))) -> 0(0(2(1(0(0(1(1(2(2(2(2(2(1(2(1(0(x1))))))))))))))))) 2(2(0(0(1(0(2(1(1(1(0(0(1(x1))))))))))))) -> 2(2(1(1(2(0(0(1(2(2(0(2(0(0(2(2(2(x1))))))))))))))))) 2(2(1(0(1(0(1(1(0(0(0(2(0(x1))))))))))))) -> 0(2(2(1(0(0(1(0(0(1(1(1(1(0(0(2(0(x1))))))))))))))))) 2(2(1(1(0(2(0(0(0(2(2(0(0(x1))))))))))))) -> 0(0(1(2(0(0(0(2(2(0(1(2(2(2(2(0(0(x1))))))))))))))))) 2(2(2(1(0(2(0(0(1(0(1(0(2(x1))))))))))))) -> 0(2(2(0(0(2(2(2(1(2(0(0(2(2(0(2(2(x1))))))))))))))))) The (relative) TRS S consists of the following rules: encArg(0(x_1)) -> 0(encArg(x_1)) encArg(cons_1(x_1)) -> 1(encArg(x_1)) encArg(cons_2(x_1)) -> 2(encArg(x_1)) encode_1(x_1) -> 1(encArg(x_1)) encode_2(x_1) -> 2(encArg(x_1)) encode_0(x_1) -> 0(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: 1(2(1(1(0(0(1(0(2(1(1(0(2(x1))))))))))))) -> 1(2(2(1(0(0(2(0(1(2(2(0(1(0(0(2(2(x1))))))))))))))))) 2(0(1(2(1(2(1(2(1(0(0(1(0(x1))))))))))))) -> 2(2(0(0(2(1(1(1(0(0(0(1(1(1(2(2(0(x1))))))))))))))))) 2(1(0(0(2(1(1(1(0(1(2(2(0(x1))))))))))))) -> 0(0(2(1(0(0(1(1(2(2(2(2(2(1(2(1(0(x1))))))))))))))))) 2(2(0(0(1(0(2(1(1(1(0(0(1(x1))))))))))))) -> 2(2(1(1(2(0(0(1(2(2(0(2(0(0(2(2(2(x1))))))))))))))))) 2(2(1(0(1(0(1(1(0(0(0(2(0(x1))))))))))))) -> 0(2(2(1(0(0(1(0(0(1(1(1(1(0(0(2(0(x1))))))))))))))))) 2(2(1(1(0(2(0(0(0(2(2(0(0(x1))))))))))))) -> 0(0(1(2(0(0(0(2(2(0(1(2(2(2(2(0(0(x1))))))))))))))))) 2(2(2(1(0(2(0(0(1(0(1(0(2(x1))))))))))))) -> 0(2(2(0(0(2(2(2(1(2(0(0(2(2(0(2(2(x1))))))))))))))))) encArg(0(x_1)) -> 0(encArg(x_1)) encArg(cons_1(x_1)) -> 1(encArg(x_1)) encArg(cons_2(x_1)) -> 2(encArg(x_1)) encode_1(x_1) -> 1(encArg(x_1)) encode_2(x_1) -> 2(encArg(x_1)) encode_0(x_1) -> 0(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 2. The certificate found is represented by the following graph. "[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, 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] {(35,36,[1_1|0, 2_1|0, encArg_1|0, encode_1_1|0, encode_2_1|0, encode_0_1|0]), (35,37,[0_1|1, 1_1|1, 2_1|1]), (35,38,[1_1|2]), (35,54,[2_1|2]), (35,70,[0_1|2]), (35,86,[2_1|2]), (35,102,[0_1|2]), (35,118,[0_1|2]), (35,134,[0_1|2]), (36,36,[0_1|0, cons_1_1|0, cons_2_1|0]), (37,36,[encArg_1|1]), (37,37,[0_1|1, 1_1|1, 2_1|1]), (37,38,[1_1|2]), (37,54,[2_1|2]), (37,70,[0_1|2]), (37,86,[2_1|2]), (37,102,[0_1|2]), (37,118,[0_1|2]), (37,134,[0_1|2]), (38,39,[2_1|2]), (39,40,[2_1|2]), (40,41,[1_1|2]), (41,42,[0_1|2]), (42,43,[0_1|2]), (43,44,[2_1|2]), (44,45,[0_1|2]), (45,46,[1_1|2]), (46,47,[2_1|2]), (47,48,[2_1|2]), (48,49,[0_1|2]), (49,50,[1_1|2]), (50,51,[0_1|2]), (51,52,[0_1|2]), (52,53,[2_1|2]), (52,86,[2_1|2]), (52,102,[0_1|2]), (52,118,[0_1|2]), (52,134,[0_1|2]), (53,37,[2_1|2]), (53,54,[2_1|2]), (53,86,[2_1|2]), (53,103,[2_1|2]), (53,135,[2_1|2]), (53,70,[0_1|2]), (53,102,[0_1|2]), (53,118,[0_1|2]), (53,134,[0_1|2]), (54,55,[2_1|2]), (55,56,[0_1|2]), (56,57,[0_1|2]), (57,58,[2_1|2]), (58,59,[1_1|2]), (59,60,[1_1|2]), (60,61,[1_1|2]), (61,62,[0_1|2]), (62,63,[0_1|2]), (63,64,[0_1|2]), (64,65,[1_1|2]), (65,66,[1_1|2]), (66,67,[1_1|2]), (67,68,[2_1|2]), (67,86,[2_1|2]), (68,69,[2_1|2]), (68,54,[2_1|2]), (69,37,[0_1|2]), (69,70,[0_1|2]), (69,102,[0_1|2]), (69,118,[0_1|2]), (69,134,[0_1|2]), (70,71,[0_1|2]), (71,72,[2_1|2]), (72,73,[1_1|2]), (73,74,[0_1|2]), (74,75,[0_1|2]), (75,76,[1_1|2]), (76,77,[1_1|2]), (77,78,[2_1|2]), (78,79,[2_1|2]), (79,80,[2_1|2]), (80,81,[2_1|2]), (81,82,[2_1|2]), (82,83,[1_1|2]), (83,84,[2_1|2]), (83,70,[0_1|2]), (84,85,[1_1|2]), (85,37,[0_1|2]), (85,70,[0_1|2]), (85,102,[0_1|2]), (85,118,[0_1|2]), (85,134,[0_1|2]), (85,56,[0_1|2]), (86,87,[2_1|2]), (87,88,[1_1|2]), (88,89,[1_1|2]), (89,90,[2_1|2]), (90,91,[0_1|2]), (91,92,[0_1|2]), (92,93,[1_1|2]), (93,94,[2_1|2]), (94,95,[2_1|2]), (95,96,[0_1|2]), (96,97,[2_1|2]), (97,98,[0_1|2]), (98,99,[0_1|2]), (99,100,[2_1|2]), (99,134,[0_1|2]), (100,101,[2_1|2]), (100,86,[2_1|2]), (100,102,[0_1|2]), (100,118,[0_1|2]), (100,134,[0_1|2]), (101,37,[2_1|2]), (101,38,[2_1|2]), (101,120,[2_1|2]), (101,54,[2_1|2]), (101,70,[0_1|2]), (101,86,[2_1|2]), (101,102,[0_1|2]), (101,118,[0_1|2]), (101,134,[0_1|2]), (102,103,[2_1|2]), (103,104,[2_1|2]), (104,105,[1_1|2]), (105,106,[0_1|2]), (106,107,[0_1|2]), (107,108,[1_1|2]), (108,109,[0_1|2]), (109,110,[0_1|2]), (110,111,[1_1|2]), (111,112,[1_1|2]), (112,113,[1_1|2]), (113,114,[1_1|2]), (114,115,[0_1|2]), (115,116,[0_1|2]), (116,117,[2_1|2]), (116,54,[2_1|2]), (117,37,[0_1|2]), (117,70,[0_1|2]), (117,102,[0_1|2]), (117,118,[0_1|2]), (117,134,[0_1|2]), (118,119,[0_1|2]), (119,120,[1_1|2]), (120,121,[2_1|2]), (121,122,[0_1|2]), (122,123,[0_1|2]), (123,124,[0_1|2]), (124,125,[2_1|2]), (125,126,[2_1|2]), (126,127,[0_1|2]), (127,128,[1_1|2]), (128,129,[2_1|2]), (129,130,[2_1|2]), (130,131,[2_1|2]), (130,86,[2_1|2]), (131,132,[2_1|2]), (132,133,[0_1|2]), (133,37,[0_1|2]), (133,70,[0_1|2]), (133,102,[0_1|2]), (133,118,[0_1|2]), (133,134,[0_1|2]), (133,71,[0_1|2]), (133,119,[0_1|2]), (133,57,[0_1|2]), (133,138,[0_1|2]), (134,135,[2_1|2]), (135,136,[2_1|2]), (136,137,[0_1|2]), (137,138,[0_1|2]), (138,139,[2_1|2]), (139,140,[2_1|2]), (140,141,[2_1|2]), (141,142,[1_1|2]), (142,143,[2_1|2]), (143,144,[0_1|2]), (144,145,[0_1|2]), (145,146,[2_1|2]), (146,147,[2_1|2]), (147,148,[0_1|2]), (148,149,[2_1|2]), (148,86,[2_1|2]), (148,102,[0_1|2]), (148,118,[0_1|2]), (148,134,[0_1|2]), (149,37,[2_1|2]), (149,54,[2_1|2]), (149,86,[2_1|2]), (149,103,[2_1|2]), (149,135,[2_1|2]), (149,70,[0_1|2]), (149,102,[0_1|2]), (149,118,[0_1|2]), (149,134,[0_1|2])}" ---------------------------------------- (8) BOUNDS(1, n^1)