/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(n^1, n^1).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), 36 ms] (4) CpxRelTRS (5) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] (6) CpxTRS (7) CpxTrsMatchBoundsProof [FINISHED, 56 ms] (8) BOUNDS(1, n^1) (9) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 5 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))))))))))))) 0(1(2(1(x1)))) -> 1(2(1(1(0(1(2(0(1(2(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(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(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))))))))))))) 0(1(2(1(x1)))) -> 1(2(1(1(0(1(2(0(1(2(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(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(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))))))))))))) 0(1(2(1(x1)))) -> 1(2(1(1(0(1(2(0(1(2(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(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(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))))))))))))) 0(1(2(1(x1)))) -> 1(2(1(1(0(1(2(0(1(2(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(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(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. 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(860,786,[2_1|4]), (860,798,[2_1|4]), (860,813,[2_1|4]), (860,831,[2_1|4]), (861,862,[2_1|4]), (862,863,[1_1|4]), (863,864,[1_1|4]), (864,865,[0_1|4]), (865,866,[1_1|4]), (866,867,[2_1|4]), (867,868,[0_1|4]), (868,869,[1_1|4]), (869,870,[2_1|4]), (870,871,[0_1|4]), (871,872,[1_1|4]), (872,777,[2_1|4]), (872,786,[2_1|4]), (872,798,[2_1|4]), (872,813,[2_1|4]), (872,831,[2_1|4]), (873,874,[2_1|4]), (874,875,[1_1|4]), (875,876,[1_1|4]), (876,877,[0_1|4]), (877,878,[1_1|4]), (878,879,[2_1|4]), (879,880,[0_1|4]), (880,881,[1_1|4]), (881,882,[2_1|4]), (882,883,[0_1|4]), (883,884,[1_1|4]), (884,885,[2_1|4]), (885,886,[0_1|4]), (886,887,[1_1|4]), (887,777,[2_1|4]), (887,786,[2_1|4]), (887,798,[2_1|4]), (887,813,[2_1|4]), (887,831,[2_1|4]), (888,889,[2_1|4]), (889,890,[1_1|4]), (890,891,[1_1|4]), (891,892,[0_1|4]), (892,893,[1_1|4]), (893,894,[2_1|4]), (894,895,[0_1|4]), (895,896,[1_1|4]), (896,897,[2_1|4]), (897,898,[0_1|4]), (898,899,[1_1|4]), (899,900,[2_1|4]), (900,901,[0_1|4]), (901,902,[1_1|4]), (902,903,[2_1|4]), (903,904,[0_1|4]), (904,905,[1_1|4]), (905,777,[2_1|4]), (905,786,[2_1|4]), (905,798,[2_1|4]), (905,813,[2_1|4]), (905,831,[2_1|4]), (906,907,[2_1|4]), (907,908,[1_1|4]), (908,909,[1_1|4]), (909,910,[0_1|4]), (910,911,[1_1|4]), (911,912,[2_1|4]), (912,913,[0_1|4]), (913,914,[1_1|4]), (914,915,[2_1|4]), (915,916,[0_1|4]), (916,917,[1_1|4]), (917,918,[2_1|4]), (918,919,[0_1|4]), (919,920,[1_1|4]), (920,921,[2_1|4]), (921,922,[0_1|4]), (922,923,[1_1|4]), (923,924,[2_1|4]), (924,925,[0_1|4]), (925,926,[1_1|4]), (926,777,[2_1|4]), (926,786,[2_1|4]), (926,798,[2_1|4]), (926,813,[2_1|4]), (926,831,[2_1|4]), (927,928,[2_1|4]), (928,929,[1_1|4]), (929,930,[1_1|4]), (930,931,[0_1|4]), (931,932,[1_1|4]), (932,933,[2_1|4]), (933,934,[0_1|4]), (934,935,[1_1|4]), (935,852,[2_1|4]), (935,861,[2_1|4]), (935,873,[2_1|4]), (935,888,[2_1|4]), (935,906,[2_1|4]), (936,937,[2_1|4]), (937,938,[1_1|4]), (938,939,[1_1|4]), (939,940,[0_1|4]), (940,941,[1_1|4]), (941,942,[2_1|4]), (942,943,[0_1|4]), (943,944,[1_1|4]), (944,945,[2_1|4]), (945,946,[0_1|4]), (946,947,[1_1|4]), (947,852,[2_1|4]), (947,861,[2_1|4]), (947,873,[2_1|4]), (947,888,[2_1|4]), (947,906,[2_1|4]), (948,949,[2_1|4]), (949,950,[1_1|4]), (950,951,[1_1|4]), (951,952,[0_1|4]), (952,953,[1_1|4]), (953,954,[2_1|4]), (954,955,[0_1|4]), (955,956,[1_1|4]), (956,957,[2_1|4]), (957,958,[0_1|4]), (958,959,[1_1|4]), (959,960,[2_1|4]), (960,961,[0_1|4]), (961,962,[1_1|4]), (962,852,[2_1|4]), (962,861,[2_1|4]), (962,873,[2_1|4]), (962,888,[2_1|4]), (962,906,[2_1|4]), (963,964,[2_1|4]), (964,965,[1_1|4]), (965,966,[1_1|4]), (966,967,[0_1|4]), (967,968,[1_1|4]), (968,969,[2_1|4]), (969,970,[0_1|4]), (970,971,[1_1|4]), (971,972,[2_1|4]), (972,973,[0_1|4]), (973,974,[1_1|4]), (974,975,[2_1|4]), (975,976,[0_1|4]), (976,977,[1_1|4]), (977,978,[2_1|4]), (978,979,[0_1|4]), (979,980,[1_1|4]), (980,852,[2_1|4]), (980,861,[2_1|4]), (980,873,[2_1|4]), (980,888,[2_1|4]), (980,906,[2_1|4]), (981,982,[2_1|4]), (982,983,[1_1|4]), (983,984,[1_1|4]), (984,985,[0_1|4]), (985,986,[1_1|4]), (986,987,[2_1|4]), (987,988,[0_1|4]), (988,989,[1_1|4]), (989,990,[2_1|4]), (990,991,[0_1|4]), (991,992,[1_1|4]), (992,993,[2_1|4]), (993,994,[0_1|4]), (994,995,[1_1|4]), (995,996,[2_1|4]), (996,997,[0_1|4]), (997,998,[1_1|4]), (998,999,[2_1|4]), (999,1000,[0_1|4]), (1000,1001,[1_1|4]), (1001,852,[2_1|4]), (1001,861,[2_1|4]), (1001,873,[2_1|4]), (1001,888,[2_1|4]), (1001,906,[2_1|4]), (1002,1003,[2_1|4]), (1003,1004,[1_1|4]), (1004,1005,[1_1|4]), (1005,1006,[0_1|4]), (1006,1007,[1_1|4]), (1007,1008,[2_1|4]), (1008,1009,[0_1|4]), (1009,1010,[1_1|4]), (1010,927,[2_1|4]), (1010,936,[2_1|4]), (1010,948,[2_1|4]), (1010,963,[2_1|4]), (1010,981,[2_1|4]), (1011,1012,[2_1|4]), (1012,1013,[1_1|4]), (1013,1014,[1_1|4]), (1014,1015,[0_1|4]), (1015,1016,[1_1|4]), (1016,1017,[2_1|4]), (1017,1018,[0_1|4]), (1018,1019,[1_1|4]), (1019,1020,[2_1|4]), (1020,1021,[0_1|4]), (1021,1022,[1_1|4]), (1022,927,[2_1|4]), (1022,936,[2_1|4]), (1022,948,[2_1|4]), (1022,963,[2_1|4]), (1022,981,[2_1|4]), (1023,1024,[2_1|4]), (1024,1025,[1_1|4]), (1025,1026,[1_1|4]), (1026,1027,[0_1|4]), (1027,1028,[1_1|4]), (1028,1029,[2_1|4]), (1029,1030,[0_1|4]), (1030,1031,[1_1|4]), (1031,1032,[2_1|4]), (1032,1033,[0_1|4]), (1033,1034,[1_1|4]), (1034,1035,[2_1|4]), (1035,1036,[0_1|4]), (1036,1037,[1_1|4]), (1037,927,[2_1|4]), (1037,936,[2_1|4]), (1037,948,[2_1|4]), (1037,963,[2_1|4]), (1037,981,[2_1|4]), (1038,1039,[2_1|4]), (1039,1040,[1_1|4]), (1040,1041,[1_1|4]), (1041,1042,[0_1|4]), (1042,1043,[1_1|4]), (1043,1044,[2_1|4]), (1044,1045,[0_1|4]), (1045,1046,[1_1|4]), (1046,1047,[2_1|4]), (1047,1048,[0_1|4]), (1048,1049,[1_1|4]), (1049,1050,[2_1|4]), (1050,1051,[0_1|4]), (1051,1052,[1_1|4]), (1052,1053,[2_1|4]), (1053,1054,[0_1|4]), (1054,1055,[1_1|4]), (1055,927,[2_1|4]), (1055,936,[2_1|4]), (1055,948,[2_1|4]), (1055,963,[2_1|4]), (1055,981,[2_1|4]), (1056,1057,[2_1|4]), (1057,1058,[1_1|4]), (1058,1059,[1_1|4]), (1059,1060,[0_1|4]), (1060,1061,[1_1|4]), (1061,1062,[2_1|4]), (1062,1063,[0_1|4]), (1063,1064,[1_1|4]), (1064,1065,[2_1|4]), (1065,1066,[0_1|4]), (1066,1067,[1_1|4]), (1067,1068,[2_1|4]), (1068,1069,[0_1|4]), (1069,1070,[1_1|4]), (1070,1071,[2_1|4]), (1071,1072,[0_1|4]), (1072,1073,[1_1|4]), (1073,1074,[2_1|4]), (1074,1075,[0_1|4]), (1075,1076,[1_1|4]), (1076,927,[2_1|4]), (1076,936,[2_1|4]), (1076,948,[2_1|4]), (1076,963,[2_1|4]), (1076,981,[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))))))))))))) 0(1(2(1(x1)))) -> 1(2(1(1(0(1(2(0(1(2(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(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(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(0(1(2(0(1(2(0(1(2(x1))))))))))))))))))) gives rise to a decreasing loop by considering the right hand sides subterm at position [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]. 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))))))))))))) 0(1(2(1(x1)))) -> 1(2(1(1(0(1(2(0(1(2(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(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(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))))))))))))) 0(1(2(1(x1)))) -> 1(2(1(1(0(1(2(0(1(2(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(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(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