/export/starexec/sandbox2/solver/bin/starexec_run_rcdcRelativeAlsoLower /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files -------------------------------------------------------------------------------- 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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(n^1, n^1).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), 33 ms] (4) CpxRelTRS (5) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] (6) CpxTRS (7) CpxTrsMatchBoundsProof [FINISHED, 507 ms] (8) BOUNDS(1, n^1) (9) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] (10) TRS for Loop Detection (11) DecreasingLoopProof [LOWER BOUND(ID), 1 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)))))))))))))))))))))) 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(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(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(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(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)))))))))))))))))))))) 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(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(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(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(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)))))))))))))))))))))) 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(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(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(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(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)))))))))))))))))))))) 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(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(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(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(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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(3816,3640,[2_1|4]), (3816,3655,[2_1|4]), (3816,3673,[2_1|4]), (3816,3694,[2_1|4]), (3816,3718,[2_1|4]), (3816,3745,[2_1|4]), (3816,3775,[2_1|4]), (3817,3818,[2_1|4]), (3818,3819,[1_1|4]), (3819,3820,[1_1|4]), (3820,3821,[0_1|4]), (3821,3822,[1_1|4]), (3822,3823,[2_1|4]), (3823,3824,[0_1|4]), (3824,3825,[1_1|4]), (3825,3826,[2_1|4]), (3826,3827,[0_1|4]), (3827,3828,[1_1|4]), (3828,3619,[2_1|4]), (3828,3628,[2_1|4]), (3828,3640,[2_1|4]), (3828,3655,[2_1|4]), (3828,3673,[2_1|4]), (3828,3694,[2_1|4]), (3828,3718,[2_1|4]), (3828,3745,[2_1|4]), (3828,3775,[2_1|4]), (3829,3830,[2_1|4]), (3830,3831,[1_1|4]), (3831,3832,[1_1|4]), (3832,3833,[0_1|4]), (3833,3834,[1_1|4]), (3834,3835,[2_1|4]), (3835,3836,[0_1|4]), (3836,3837,[1_1|4]), (3837,3838,[2_1|4]), (3838,3839,[0_1|4]), (3839,3840,[1_1|4]), (3840,3841,[2_1|4]), (3841,3842,[0_1|4]), (3842,3843,[1_1|4]), (3843,3619,[2_1|4]), (3843,3628,[2_1|4]), (3843,3640,[2_1|4]), (3843,3655,[2_1|4]), (3843,3673,[2_1|4]), (3843,3694,[2_1|4]), 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(3880,3881,[0_1|4]), (3881,3882,[1_1|4]), (3882,3619,[2_1|4]), (3882,3628,[2_1|4]), (3882,3640,[2_1|4]), (3882,3655,[2_1|4]), (3882,3673,[2_1|4]), (3882,3694,[2_1|4]), (3882,3718,[2_1|4]), (3882,3745,[2_1|4]), (3882,3775,[2_1|4]), (3883,3884,[2_1|4]), (3884,3885,[1_1|4]), (3885,3886,[1_1|4]), (3886,3887,[0_1|4]), (3887,3888,[1_1|4]), (3888,3889,[2_1|4]), (3889,3890,[0_1|4]), (3890,3891,[1_1|4]), (3891,3892,[2_1|4]), (3892,3893,[0_1|4]), (3893,3894,[1_1|4]), (3894,3895,[2_1|4]), (3895,3896,[0_1|4]), (3896,3897,[1_1|4]), (3897,3898,[2_1|4]), (3898,3899,[0_1|4]), (3899,3900,[1_1|4]), (3900,3901,[2_1|4]), (3901,3902,[0_1|4]), (3902,3903,[1_1|4]), (3903,3904,[2_1|4]), (3904,3905,[0_1|4]), (3905,3906,[1_1|4]), (3906,3619,[2_1|4]), (3906,3628,[2_1|4]), (3906,3640,[2_1|4]), (3906,3655,[2_1|4]), (3906,3673,[2_1|4]), (3906,3694,[2_1|4]), (3906,3718,[2_1|4]), (3906,3745,[2_1|4]), (3906,3775,[2_1|4]), (3907,3908,[2_1|4]), (3908,3909,[1_1|4]), (3909,3910,[1_1|4]), (3910,3911,[0_1|4]), (3911,3912,[1_1|4]), (3912,3913,[2_1|4]), (3913,3914,[0_1|4]), (3914,3915,[1_1|4]), (3915,3916,[2_1|4]), (3916,3917,[0_1|4]), (3917,3918,[1_1|4]), (3918,3919,[2_1|4]), (3919,3920,[0_1|4]), (3920,3921,[1_1|4]), (3921,3922,[2_1|4]), (3922,3923,[0_1|4]), (3923,3924,[1_1|4]), (3924,3925,[2_1|4]), (3925,3926,[0_1|4]), (3926,3927,[1_1|4]), (3927,3928,[2_1|4]), (3928,3929,[0_1|4]), (3929,3930,[1_1|4]), (3930,3931,[2_1|4]), (3931,3932,[0_1|4]), (3932,3933,[1_1|4]), (3933,3619,[2_1|4]), (3933,3628,[2_1|4]), (3933,3640,[2_1|4]), (3933,3655,[2_1|4]), (3933,3673,[2_1|4]), (3933,3694,[2_1|4]), (3933,3718,[2_1|4]), (3933,3745,[2_1|4]), (3933,3775,[2_1|4]), (3934,3935,[2_1|4]), (3935,3936,[1_1|4]), (3936,3937,[1_1|4]), (3937,3938,[0_1|4]), (3938,3939,[1_1|4]), (3939,3940,[2_1|4]), (3940,3941,[0_1|4]), (3941,3942,[1_1|4]), (3942,3943,[2_1|4]), (3943,3944,[0_1|4]), (3944,3945,[1_1|4]), (3945,3946,[2_1|4]), (3946,3947,[0_1|4]), (3947,3948,[1_1|4]), (3948,3949,[2_1|4]), (3949,3950,[0_1|4]), (3950,3951,[1_1|4]), (3951,3952,[2_1|4]), (3952,3953,[0_1|4]), (3953,3954,[1_1|4]), (3954,3955,[2_1|4]), (3955,3956,[0_1|4]), (3956,3957,[1_1|4]), (3957,3958,[2_1|4]), (3958,3959,[0_1|4]), (3959,3960,[1_1|4]), (3960,3961,[2_1|4]), (3961,3962,[0_1|4]), (3962,3963,[1_1|4]), (3963,3619,[2_1|4]), (3963,3628,[2_1|4]), (3963,3640,[2_1|4]), (3963,3655,[2_1|4]), (3963,3673,[2_1|4]), (3963,3694,[2_1|4]), (3963,3718,[2_1|4]), (3963,3745,[2_1|4]), (3963,3775,[2_1|4]), (3964,3965,[2_1|4]), (3965,3966,[1_1|4]), (3966,3967,[1_1|4]), (3967,3968,[0_1|4]), (3968,3969,[1_1|4]), (3969,3970,[2_1|4]), (3970,3971,[0_1|4]), (3971,3972,[1_1|4]), (3972,3973,[2_1|4]), (3973,3974,[0_1|4]), (3974,3975,[1_1|4]), (3975,3976,[2_1|4]), (3976,3977,[0_1|4]), (3977,3978,[1_1|4]), (3978,3979,[2_1|4]), (3979,3980,[0_1|4]), (3980,3981,[1_1|4]), (3981,3982,[2_1|4]), (3982,3983,[0_1|4]), (3983,3984,[1_1|4]), (3984,3985,[2_1|4]), (3985,3986,[0_1|4]), (3986,3987,[1_1|4]), (3987,3988,[2_1|4]), (3988,3989,[0_1|4]), (3989,3990,[1_1|4]), (3990,3991,[2_1|4]), (3991,3992,[0_1|4]), (3992,3993,[1_1|4]), (3993,3994,[2_1|4]), (3994,3995,[0_1|4]), (3995,3996,[1_1|4]), (3996,3619,[2_1|4]), (3996,3628,[2_1|4]), (3996,3640,[2_1|4]), (3996,3655,[2_1|4]), (3996,3673,[2_1|4]), (3996,3694,[2_1|4]), (3996,3718,[2_1|4]), (3996,3745,[2_1|4]), (3996,3775,[2_1|4]), (3997,3998,[2_1|4]), (3998,3999,[1_1|4]), (3999,4000,[1_1|4]), (4000,4001,[0_1|4]), (4001,4002,[1_1|4]), (4002,4003,[2_1|4]), (4003,4004,[0_1|4]), (4004,4005,[1_1|4]), (4005,3808,[2_1|4]), (4005,3817,[2_1|4]), (4005,3829,[2_1|4]), (4005,3844,[2_1|4]), (4005,3862,[2_1|4]), (4005,3883,[2_1|4]), (4005,3907,[2_1|4]), (4005,3934,[2_1|4]), (4005,3964,[2_1|4]), (4006,4007,[2_1|4]), (4007,4008,[1_1|4]), (4008,4009,[1_1|4]), (4009,4010,[0_1|4]), (4010,4011,[1_1|4]), (4011,4012,[2_1|4]), (4012,4013,[0_1|4]), (4013,4014,[1_1|4]), (4014,4015,[2_1|4]), (4015,4016,[0_1|4]), (4016,4017,[1_1|4]), (4017,3808,[2_1|4]), (4017,3817,[2_1|4]), (4017,3829,[2_1|4]), (4017,3844,[2_1|4]), (4017,3862,[2_1|4]), (4017,3883,[2_1|4]), (4017,3907,[2_1|4]), (4017,3934,[2_1|4]), (4017,3964,[2_1|4]), (4018,4019,[2_1|4]), (4019,4020,[1_1|4]), (4020,4021,[1_1|4]), (4021,4022,[0_1|4]), (4022,4023,[1_1|4]), (4023,4024,[2_1|4]), (4024,4025,[0_1|4]), (4025,4026,[1_1|4]), (4026,4027,[2_1|4]), (4027,4028,[0_1|4]), (4028,4029,[1_1|4]), (4029,4030,[2_1|4]), (4030,4031,[0_1|4]), (4031,4032,[1_1|4]), (4032,3808,[2_1|4]), (4032,3817,[2_1|4]), (4032,3829,[2_1|4]), (4032,3844,[2_1|4]), (4032,3862,[2_1|4]), (4032,3883,[2_1|4]), (4032,3907,[2_1|4]), (4032,3934,[2_1|4]), (4032,3964,[2_1|4]), (4033,4034,[2_1|4]), (4034,4035,[1_1|4]), (4035,4036,[1_1|4]), (4036,4037,[0_1|4]), (4037,4038,[1_1|4]), (4038,4039,[2_1|4]), (4039,4040,[0_1|4]), (4040,4041,[1_1|4]), (4041,4042,[2_1|4]), (4042,4043,[0_1|4]), (4043,4044,[1_1|4]), (4044,4045,[2_1|4]), (4045,4046,[0_1|4]), (4046,4047,[1_1|4]), (4047,4048,[2_1|4]), (4048,4049,[0_1|4]), (4049,4050,[1_1|4]), (4050,3808,[2_1|4]), (4050,3817,[2_1|4]), (4050,3829,[2_1|4]), (4050,3844,[2_1|4]), (4050,3862,[2_1|4]), (4050,3883,[2_1|4]), (4050,3907,[2_1|4]), (4050,3934,[2_1|4]), (4050,3964,[2_1|4]), (4051,4052,[2_1|4]), (4052,4053,[1_1|4]), (4053,4054,[1_1|4]), (4054,4055,[0_1|4]), (4055,4056,[1_1|4]), (4056,4057,[2_1|4]), (4057,4058,[0_1|4]), (4058,4059,[1_1|4]), (4059,4060,[2_1|4]), (4060,4061,[0_1|4]), (4061,4062,[1_1|4]), (4062,4063,[2_1|4]), (4063,4064,[0_1|4]), (4064,4065,[1_1|4]), (4065,4066,[2_1|4]), (4066,4067,[0_1|4]), (4067,4068,[1_1|4]), (4068,4069,[2_1|4]), (4069,4070,[0_1|4]), (4070,4071,[1_1|4]), (4071,3808,[2_1|4]), (4071,3817,[2_1|4]), (4071,3829,[2_1|4]), (4071,3844,[2_1|4]), (4071,3862,[2_1|4]), (4071,3883,[2_1|4]), (4071,3907,[2_1|4]), (4071,3934,[2_1|4]), (4071,3964,[2_1|4]), (4072,4073,[2_1|4]), (4073,4074,[1_1|4]), (4074,4075,[1_1|4]), (4075,4076,[0_1|4]), (4076,4077,[1_1|4]), (4077,4078,[2_1|4]), (4078,4079,[0_1|4]), (4079,4080,[1_1|4]), (4080,4081,[2_1|4]), (4081,4082,[0_1|4]), (4082,4083,[1_1|4]), (4083,4084,[2_1|4]), (4084,4085,[0_1|4]), (4085,4086,[1_1|4]), (4086,4087,[2_1|4]), (4087,4088,[0_1|4]), (4088,4089,[1_1|4]), (4089,4090,[2_1|4]), (4090,4091,[0_1|4]), (4091,4092,[1_1|4]), (4092,4093,[2_1|4]), (4093,4094,[0_1|4]), (4094,4095,[1_1|4]), (4095,3808,[2_1|4]), (4095,3817,[2_1|4]), (4095,3829,[2_1|4]), (4095,3844,[2_1|4]), (4095,3862,[2_1|4]), (4095,3883,[2_1|4]), (4095,3907,[2_1|4]), (4095,3934,[2_1|4]), (4095,3964,[2_1|4]), (4096,4097,[2_1|4]), (4097,4098,[1_1|4]), (4098,4099,[1_1|4]), (4099,4100,[0_1|4]), (4100,4101,[1_1|4]), (4101,4102,[2_1|4]), (4102,4103,[0_1|4]), (4103,4104,[1_1|4]), (4104,4105,[2_1|4]), (4105,4106,[0_1|4]), (4106,4107,[1_1|4]), (4107,4108,[2_1|4]), (4108,4109,[0_1|4]), (4109,4110,[1_1|4]), (4110,4111,[2_1|4]), (4111,4112,[0_1|4]), (4112,4113,[1_1|4]), (4113,4114,[2_1|4]), (4114,4115,[0_1|4]), (4115,4116,[1_1|4]), (4116,4117,[2_1|4]), (4117,4118,[0_1|4]), (4118,4119,[1_1|4]), (4119,4120,[2_1|4]), (4120,4121,[0_1|4]), (4121,4122,[1_1|4]), (4122,3808,[2_1|4]), (4122,3817,[2_1|4]), (4122,3829,[2_1|4]), (4122,3844,[2_1|4]), (4122,3862,[2_1|4]), (4122,3883,[2_1|4]), (4122,3907,[2_1|4]), (4122,3934,[2_1|4]), (4122,3964,[2_1|4]), (4123,4124,[2_1|4]), (4124,4125,[1_1|4]), (4125,4126,[1_1|4]), (4126,4127,[0_1|4]), (4127,4128,[1_1|4]), (4128,4129,[2_1|4]), (4129,4130,[0_1|4]), (4130,4131,[1_1|4]), (4131,4132,[2_1|4]), (4132,4133,[0_1|4]), (4133,4134,[1_1|4]), (4134,4135,[2_1|4]), (4135,4136,[0_1|4]), (4136,4137,[1_1|4]), (4137,4138,[2_1|4]), (4138,4139,[0_1|4]), (4139,4140,[1_1|4]), (4140,4141,[2_1|4]), (4141,4142,[0_1|4]), (4142,4143,[1_1|4]), (4143,4144,[2_1|4]), (4144,4145,[0_1|4]), (4145,4146,[1_1|4]), (4146,4147,[2_1|4]), (4147,4148,[0_1|4]), (4148,4149,[1_1|4]), (4149,4150,[2_1|4]), (4150,4151,[0_1|4]), (4151,4152,[1_1|4]), (4152,3808,[2_1|4]), (4152,3817,[2_1|4]), (4152,3829,[2_1|4]), (4152,3844,[2_1|4]), (4152,3862,[2_1|4]), (4152,3883,[2_1|4]), (4152,3907,[2_1|4]), (4152,3934,[2_1|4]), (4152,3964,[2_1|4]), (4153,4154,[2_1|4]), (4154,4155,[1_1|4]), (4155,4156,[1_1|4]), (4156,4157,[0_1|4]), (4157,4158,[1_1|4]), (4158,4159,[2_1|4]), (4159,4160,[0_1|4]), (4160,4161,[1_1|4]), (4161,4162,[2_1|4]), (4162,4163,[0_1|4]), (4163,4164,[1_1|4]), (4164,4165,[2_1|4]), (4165,4166,[0_1|4]), (4166,4167,[1_1|4]), (4167,4168,[2_1|4]), (4168,4169,[0_1|4]), (4169,4170,[1_1|4]), (4170,4171,[2_1|4]), (4171,4172,[0_1|4]), (4172,4173,[1_1|4]), (4173,4174,[2_1|4]), (4174,4175,[0_1|4]), (4175,4176,[1_1|4]), (4176,4177,[2_1|4]), (4177,4178,[0_1|4]), (4178,4179,[1_1|4]), (4179,4180,[2_1|4]), (4180,4181,[0_1|4]), (4181,4182,[1_1|4]), (4182,4183,[2_1|4]), (4183,4184,[0_1|4]), (4184,4185,[1_1|4]), (4185,3808,[2_1|4]), (4185,3817,[2_1|4]), (4185,3829,[2_1|4]), (4185,3844,[2_1|4]), (4185,3862,[2_1|4]), (4185,3883,[2_1|4]), (4185,3907,[2_1|4]), (4185,3934,[2_1|4]), (4185,3964,[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)))))))))))))))))))))) 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(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(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(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(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)))))))))))))))))))))) 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(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(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(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(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)))))))))))))))))))))) 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(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(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(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(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