/export/starexec/sandbox/solver/bin/starexec_run_rcdcRelativeAlsoLower /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files -------------------------------------------------------------------------------- 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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(n^1, n^1).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), 86 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: 0(1(2(1(x1)))) -> 1(2(1(1(0(1(2(0(1(2(x1)))))))))) 0(1(2(1(x1)))) -> 1(2(1(1(0(1(2(0(1(2(0(1(2(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(1(x_1)) -> 1(encArg(x_1)) encArg(2(x_1)) -> 2(encArg(x_1)) encArg(cons_0(x_1)) -> 0(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)) ---------------------------------------- (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: 0(1(2(1(x1)))) -> 1(2(1(1(0(1(2(0(1(2(x1)))))))))) 0(1(2(1(x1)))) -> 1(2(1(1(0(1(2(0(1(2(0(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(cons_0(x_1)) -> 0(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)) 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: 0(1(2(1(x1)))) -> 1(2(1(1(0(1(2(0(1(2(x1)))))))))) 0(1(2(1(x1)))) -> 1(2(1(1(0(1(2(0(1(2(0(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(cons_0(x_1)) -> 0(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)) 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: 0(1(2(1(x1)))) -> 1(2(1(1(0(1(2(0(1(2(x1)))))))))) 0(1(2(1(x1)))) -> 1(2(1(1(0(1(2(0(1(2(0(1(2(x1))))))))))))) encArg(1(x_1)) -> 1(encArg(x_1)) encArg(2(x_1)) -> 2(encArg(x_1)) encArg(cons_0(x_1)) -> 0(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)) 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. "[7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 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, 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, 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] {(7,8,[0_1|0, encArg_1|0, encode_0_1|0, encode_1_1|0, encode_2_1|0]), (7,9,[1_1|1]), (7,18,[1_1|1]), (7,30,[1_1|1, 2_1|1, 0_1|1]), (7,31,[1_1|2]), (7,40,[1_1|2]), (8,8,[1_1|0, 2_1|0, cons_0_1|0]), (9,10,[2_1|1]), (10,11,[1_1|1]), (11,12,[1_1|1]), (12,13,[0_1|1]), (12,52,[1_1|2]), (12,61,[1_1|2]), (13,14,[1_1|1]), (14,15,[2_1|1]), (15,16,[0_1|1]), (15,9,[1_1|1]), (15,18,[1_1|1]), (16,17,[1_1|1]), (17,8,[2_1|1]), (18,19,[2_1|1]), (19,20,[1_1|1]), (20,21,[1_1|1]), (21,22,[0_1|1]), (21,73,[1_1|2]), (21,82,[1_1|2]), (22,23,[1_1|1]), (23,24,[2_1|1]), (24,25,[0_1|1]), (24,52,[1_1|2]), (24,61,[1_1|2]), (25,26,[1_1|1]), (26,27,[2_1|1]), (27,28,[0_1|1]), (27,9,[1_1|1]), (27,18,[1_1|1]), (28,29,[1_1|1]), (29,8,[2_1|1]), (30,8,[encArg_1|1]), (30,30,[1_1|1, 2_1|1, 0_1|1]), (30,31,[1_1|2]), (30,40,[1_1|2]), (31,32,[2_1|2]), (32,33,[1_1|2]), (33,34,[1_1|2]), (34,35,[0_1|2]), (34,94,[1_1|3]), (34,103,[1_1|3]), (35,36,[1_1|2]), (36,37,[2_1|2]), (37,38,[0_1|2]), (37,31,[1_1|2]), (37,40,[1_1|2]), (37,94,[1_1|3]), (37,103,[1_1|3]), (38,39,[1_1|2]), (39,30,[2_1|2]), (39,31,[2_1|2]), (39,40,[2_1|2]), (39,33,[2_1|2]), (39,42,[2_1|2]), (40,41,[2_1|2]), (41,42,[1_1|2]), (42,43,[1_1|2]), (43,44,[0_1|2]), (43,94,[1_1|3]), (43,103,[1_1|3]), (44,45,[1_1|2]), (45,46,[2_1|2]), (46,47,[0_1|2]), (46,94,[1_1|3]), (46,103,[1_1|3]), (47,48,[1_1|2]), (48,49,[2_1|2]), (49,50,[0_1|2]), (49,31,[1_1|2]), (49,40,[1_1|2]), (49,94,[1_1|3]), (49,103,[1_1|3]), (50,51,[1_1|2]), (51,30,[2_1|2]), (51,31,[2_1|2]), (51,40,[2_1|2]), (51,33,[2_1|2]), (51,42,[2_1|2]), (52,53,[2_1|2]), (53,54,[1_1|2]), (54,55,[1_1|2]), (55,56,[0_1|2]), (56,57,[1_1|2]), (57,58,[2_1|2]), (58,59,[0_1|2]), (59,60,[1_1|2]), (60,9,[2_1|2]), (60,18,[2_1|2]), (61,62,[2_1|2]), (62,63,[1_1|2]), (63,64,[1_1|2]), (64,65,[0_1|2]), (65,66,[1_1|2]), (66,67,[2_1|2]), (67,68,[0_1|2]), (68,69,[1_1|2]), (69,70,[2_1|2]), (70,71,[0_1|2]), (71,72,[1_1|2]), (72,9,[2_1|2]), (72,18,[2_1|2]), (73,74,[2_1|2]), (74,75,[1_1|2]), (75,76,[1_1|2]), (76,77,[0_1|2]), (77,78,[1_1|2]), (78,79,[2_1|2]), (79,80,[0_1|2]), (80,81,[1_1|2]), (81,52,[2_1|2]), (81,61,[2_1|2]), (82,83,[2_1|2]), (83,84,[1_1|2]), (84,85,[1_1|2]), (85,86,[0_1|2]), (86,87,[1_1|2]), (87,88,[2_1|2]), (88,89,[0_1|2]), (89,90,[1_1|2]), (90,91,[2_1|2]), (91,92,[0_1|2]), (92,93,[1_1|2]), (93,52,[2_1|2]), (93,61,[2_1|2]), (94,95,[2_1|3]), (95,96,[1_1|3]), (96,97,[1_1|3]), (97,98,[0_1|3]), (97,141,[1_1|4]), (97,150,[1_1|4]), (98,99,[1_1|3]), (99,100,[2_1|3]), (100,101,[0_1|3]), (100,115,[1_1|4]), (100,124,[1_1|4]), (101,102,[1_1|3]), (102,31,[2_1|3]), (102,40,[2_1|3]), (102,94,[2_1|3]), (102,103,[2_1|3]), (102,34,[2_1|3]), (102,43,[2_1|3]), (103,104,[2_1|3]), (104,105,[1_1|3]), (105,106,[1_1|3]), (106,107,[0_1|3]), (106,162,[1_1|4]), (106,171,[1_1|4]), (107,108,[1_1|3]), (108,109,[2_1|3]), (109,110,[0_1|3]), (109,141,[1_1|4]), (109,150,[1_1|4]), (110,111,[1_1|3]), (111,112,[2_1|3]), (112,113,[0_1|3]), (112,115,[1_1|4]), (112,124,[1_1|4]), (113,114,[1_1|3]), (114,31,[2_1|3]), (114,40,[2_1|3]), (114,94,[2_1|3]), (114,103,[2_1|3]), (114,34,[2_1|3]), (114,43,[2_1|3]), (115,116,[2_1|4]), (116,117,[1_1|4]), (117,118,[1_1|4]), (118,119,[0_1|4]), (119,120,[1_1|4]), (120,121,[2_1|4]), (121,122,[0_1|4]), (122,123,[1_1|4]), (123,94,[2_1|4]), (123,103,[2_1|4]), (124,125,[2_1|4]), (125,126,[1_1|4]), (126,127,[1_1|4]), (127,128,[0_1|4]), (128,129,[1_1|4]), (129,130,[2_1|4]), (130,131,[0_1|4]), (131,132,[1_1|4]), (132,133,[2_1|4]), (133,134,[0_1|4]), (134,135,[1_1|4]), (135,94,[2_1|4]), (135,103,[2_1|4]), (141,142,[2_1|4]), (142,143,[1_1|4]), (143,144,[1_1|4]), (144,145,[0_1|4]), (145,146,[1_1|4]), (146,147,[2_1|4]), (147,148,[0_1|4]), (148,149,[1_1|4]), (149,115,[2_1|4]), (149,124,[2_1|4]), (150,151,[2_1|4]), (151,152,[1_1|4]), (152,153,[1_1|4]), (153,154,[0_1|4]), (154,155,[1_1|4]), (155,156,[2_1|4]), (156,157,[0_1|4]), (157,158,[1_1|4]), (158,159,[2_1|4]), (159,160,[0_1|4]), (160,161,[1_1|4]), (161,115,[2_1|4]), (161,124,[2_1|4]), (162,163,[2_1|4]), (163,164,[1_1|4]), (164,165,[1_1|4]), (165,166,[0_1|4]), (166,167,[1_1|4]), (167,168,[2_1|4]), (168,169,[0_1|4]), (169,170,[1_1|4]), (170,141,[2_1|4]), (170,150,[2_1|4]), (171,172,[2_1|4]), (172,173,[1_1|4]), (173,174,[1_1|4]), (174,175,[0_1|4]), (175,176,[1_1|4]), (176,177,[2_1|4]), (177,178,[0_1|4]), (178,179,[1_1|4]), (179,180,[2_1|4]), (180,181,[0_1|4]), (181,182,[1_1|4]), (182,141,[2_1|4]), (182,150,[2_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: 0(1(2(1(x1)))) -> 1(2(1(1(0(1(2(0(1(2(x1)))))))))) 0(1(2(1(x1)))) -> 1(2(1(1(0(1(2(0(1(2(0(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(cons_0(x_1)) -> 0(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)) 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 0(1(2(1(x1)))) ->^+ 1(2(1(1(0(1(2(0(1(2(x1)))))))))) gives rise to a decreasing loop by considering the right hand sides subterm at position [0,0,0,0,0,0,0]. The pumping substitution is [x1 / 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 (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^1). The TRS R consists of the following rules: 0(1(2(1(x1)))) -> 1(2(1(1(0(1(2(0(1(2(x1)))))))))) 0(1(2(1(x1)))) -> 1(2(1(1(0(1(2(0(1(2(0(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(cons_0(x_1)) -> 0(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)) 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: 0(1(2(1(x1)))) -> 1(2(1(1(0(1(2(0(1(2(x1)))))))))) 0(1(2(1(x1)))) -> 1(2(1(1(0(1(2(0(1(2(0(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(cons_0(x_1)) -> 0(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)) Rewrite Strategy: INNERMOST