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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (full) of the given DCpxTrs could be proven to be BOUNDS(n^1, n^1).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), 54 ms] (4) CpxRelTRS (5) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] (6) CpxTRS (7) CpxTrsMatchBoundsProof [FINISHED, 52 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 (full) 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: 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(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 (full) 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: 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(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: 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: 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: 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 4. The certificate found is represented by the following graph. 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(856,794,[2_1|4]), (856,809,[2_1|4]), (856,827,[2_1|4]), (857,858,[2_1|4]), (858,859,[1_1|4]), (859,860,[1_1|4]), (860,861,[0_1|4]), (861,862,[1_1|4]), (862,863,[2_1|4]), (863,864,[0_1|4]), (864,865,[1_1|4]), (865,866,[2_1|4]), (866,867,[0_1|4]), (867,868,[1_1|4]), (868,773,[2_1|4]), (868,782,[2_1|4]), (868,794,[2_1|4]), (868,809,[2_1|4]), (868,827,[2_1|4]), (869,870,[2_1|4]), (870,871,[1_1|4]), (871,872,[1_1|4]), (872,873,[0_1|4]), (873,874,[1_1|4]), (874,875,[2_1|4]), (875,876,[0_1|4]), (876,877,[1_1|4]), (877,878,[2_1|4]), (878,879,[0_1|4]), (879,880,[1_1|4]), (880,881,[2_1|4]), (881,882,[0_1|4]), (882,883,[1_1|4]), (883,773,[2_1|4]), (883,782,[2_1|4]), (883,794,[2_1|4]), (883,809,[2_1|4]), (883,827,[2_1|4]), (884,885,[2_1|4]), (885,886,[1_1|4]), (886,887,[1_1|4]), (887,888,[0_1|4]), (888,889,[1_1|4]), (889,890,[2_1|4]), (890,891,[0_1|4]), (891,892,[1_1|4]), (892,893,[2_1|4]), (893,894,[0_1|4]), (894,895,[1_1|4]), (895,896,[2_1|4]), (896,897,[0_1|4]), (897,898,[1_1|4]), (898,899,[2_1|4]), (899,900,[0_1|4]), (900,901,[1_1|4]), (901,773,[2_1|4]), (901,782,[2_1|4]), (901,794,[2_1|4]), (901,809,[2_1|4]), (901,827,[2_1|4]), (902,903,[2_1|4]), (903,904,[1_1|4]), (904,905,[1_1|4]), (905,906,[0_1|4]), (906,907,[1_1|4]), (907,908,[2_1|4]), (908,909,[0_1|4]), (909,910,[1_1|4]), (910,911,[2_1|4]), (911,912,[0_1|4]), (912,913,[1_1|4]), (913,914,[2_1|4]), (914,915,[0_1|4]), (915,916,[1_1|4]), (916,917,[2_1|4]), (917,918,[0_1|4]), (918,919,[1_1|4]), (919,920,[2_1|4]), (920,921,[0_1|4]), (921,922,[1_1|4]), (922,773,[2_1|4]), (922,782,[2_1|4]), (922,794,[2_1|4]), (922,809,[2_1|4]), (922,827,[2_1|4]), (923,924,[2_1|4]), (924,925,[1_1|4]), (925,926,[1_1|4]), (926,927,[0_1|4]), (927,928,[1_1|4]), (928,929,[2_1|4]), (929,930,[0_1|4]), (930,931,[1_1|4]), (931,848,[2_1|4]), (931,857,[2_1|4]), (931,869,[2_1|4]), (931,884,[2_1|4]), (931,902,[2_1|4]), (932,933,[2_1|4]), (933,934,[1_1|4]), (934,935,[1_1|4]), (935,936,[0_1|4]), (936,937,[1_1|4]), (937,938,[2_1|4]), (938,939,[0_1|4]), (939,940,[1_1|4]), (940,941,[2_1|4]), (941,942,[0_1|4]), (942,943,[1_1|4]), (943,848,[2_1|4]), (943,857,[2_1|4]), (943,869,[2_1|4]), (943,884,[2_1|4]), (943,902,[2_1|4]), (944,945,[2_1|4]), (945,946,[1_1|4]), (946,947,[1_1|4]), (947,948,[0_1|4]), (948,949,[1_1|4]), (949,950,[2_1|4]), (950,951,[0_1|4]), (951,952,[1_1|4]), (952,953,[2_1|4]), (953,954,[0_1|4]), (954,955,[1_1|4]), (955,956,[2_1|4]), (956,957,[0_1|4]), (957,958,[1_1|4]), (958,848,[2_1|4]), (958,857,[2_1|4]), (958,869,[2_1|4]), (958,884,[2_1|4]), (958,902,[2_1|4]), (959,960,[2_1|4]), (960,961,[1_1|4]), (961,962,[1_1|4]), (962,963,[0_1|4]), (963,964,[1_1|4]), (964,965,[2_1|4]), (965,966,[0_1|4]), (966,967,[1_1|4]), (967,968,[2_1|4]), (968,969,[0_1|4]), (969,970,[1_1|4]), (970,971,[2_1|4]), (971,972,[0_1|4]), (972,973,[1_1|4]), (973,974,[2_1|4]), (974,975,[0_1|4]), (975,976,[1_1|4]), (976,848,[2_1|4]), (976,857,[2_1|4]), (976,869,[2_1|4]), (976,884,[2_1|4]), (976,902,[2_1|4]), (977,978,[2_1|4]), (978,979,[1_1|4]), (979,980,[1_1|4]), (980,981,[0_1|4]), (981,982,[1_1|4]), (982,983,[2_1|4]), (983,984,[0_1|4]), (984,985,[1_1|4]), (985,986,[2_1|4]), (986,987,[0_1|4]), (987,988,[1_1|4]), (988,989,[2_1|4]), (989,990,[0_1|4]), (990,991,[1_1|4]), (991,992,[2_1|4]), (992,993,[0_1|4]), (993,994,[1_1|4]), (994,995,[2_1|4]), (995,996,[0_1|4]), (996,997,[1_1|4]), (997,848,[2_1|4]), (997,857,[2_1|4]), (997,869,[2_1|4]), (997,884,[2_1|4]), (997,902,[2_1|4]), (998,999,[2_1|4]), (999,1000,[1_1|4]), (1000,1001,[1_1|4]), (1001,1002,[0_1|4]), (1002,1003,[1_1|4]), (1003,1004,[2_1|4]), (1004,1005,[0_1|4]), (1005,1006,[1_1|4]), (1006,923,[2_1|4]), (1006,932,[2_1|4]), (1006,944,[2_1|4]), (1006,959,[2_1|4]), (1006,977,[2_1|4]), (1007,1008,[2_1|4]), (1008,1009,[1_1|4]), (1009,1010,[1_1|4]), (1010,1011,[0_1|4]), (1011,1012,[1_1|4]), (1012,1013,[2_1|4]), (1013,1014,[0_1|4]), (1014,1015,[1_1|4]), (1015,1016,[2_1|4]), (1016,1017,[0_1|4]), (1017,1018,[1_1|4]), (1018,923,[2_1|4]), (1018,932,[2_1|4]), (1018,944,[2_1|4]), (1018,959,[2_1|4]), (1018,977,[2_1|4]), (1019,1020,[2_1|4]), (1020,1021,[1_1|4]), (1021,1022,[1_1|4]), (1022,1023,[0_1|4]), (1023,1024,[1_1|4]), (1024,1025,[2_1|4]), (1025,1026,[0_1|4]), (1026,1027,[1_1|4]), (1027,1028,[2_1|4]), (1028,1029,[0_1|4]), (1029,1030,[1_1|4]), (1030,1031,[2_1|4]), (1031,1032,[0_1|4]), (1032,1033,[1_1|4]), (1033,923,[2_1|4]), (1033,932,[2_1|4]), (1033,944,[2_1|4]), (1033,959,[2_1|4]), (1033,977,[2_1|4]), (1034,1035,[2_1|4]), (1035,1036,[1_1|4]), (1036,1037,[1_1|4]), (1037,1038,[0_1|4]), (1038,1039,[1_1|4]), (1039,1040,[2_1|4]), (1040,1041,[0_1|4]), (1041,1042,[1_1|4]), (1042,1043,[2_1|4]), (1043,1044,[0_1|4]), (1044,1045,[1_1|4]), (1045,1046,[2_1|4]), (1046,1047,[0_1|4]), (1047,1048,[1_1|4]), (1048,1049,[2_1|4]), (1049,1050,[0_1|4]), (1050,1051,[1_1|4]), (1051,923,[2_1|4]), (1051,932,[2_1|4]), (1051,944,[2_1|4]), (1051,959,[2_1|4]), (1051,977,[2_1|4]), (1052,1053,[2_1|4]), (1053,1054,[1_1|4]), (1054,1055,[1_1|4]), (1055,1056,[0_1|4]), (1056,1057,[1_1|4]), (1057,1058,[2_1|4]), (1058,1059,[0_1|4]), (1059,1060,[1_1|4]), (1060,1061,[2_1|4]), (1061,1062,[0_1|4]), (1062,1063,[1_1|4]), (1063,1064,[2_1|4]), (1064,1065,[0_1|4]), (1065,1066,[1_1|4]), (1066,1067,[2_1|4]), (1067,1068,[0_1|4]), (1068,1069,[1_1|4]), (1069,1070,[2_1|4]), (1070,1071,[0_1|4]), (1071,1072,[1_1|4]), (1072,923,[2_1|4]), (1072,932,[2_1|4]), (1072,944,[2_1|4]), (1072,959,[2_1|4]), (1072,977,[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 (full) 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: FULL ---------------------------------------- (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 (full) 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: FULL ---------------------------------------- (14) LowerBoundPropagationProof (FINISHED) Propagated lower bound. ---------------------------------------- (15) BOUNDS(n^1, INF) ---------------------------------------- (16) Obligation: Analyzing the following TRS for decreasing loops: The Runtime Complexity (full) 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: FULL