/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (full) of the given DCpxTrs could be proven to be BOUNDS(n^1, n^1).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), 55 ms] (4) CpxRelTRS (5) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] (6) CpxTRS (7) CpxTrsMatchBoundsProof [FINISHED, 459 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)))))))))))))))))))))) 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: 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)))))))))))))))))))))) 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: 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)))))))))))))))))))))) 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: 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)))))))))))))))))))))) 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: 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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(3879,3880,[1_1|4]), (3880,3881,[2_1|4]), (3881,3882,[0_1|4]), (3882,3883,[1_1|4]), (3883,3884,[2_1|4]), (3884,3885,[0_1|4]), (3885,3886,[1_1|4]), (3886,3623,[2_1|4]), (3886,3632,[2_1|4]), (3886,3644,[2_1|4]), (3886,3659,[2_1|4]), (3886,3677,[2_1|4]), (3886,3698,[2_1|4]), (3886,3722,[2_1|4]), (3886,3749,[2_1|4]), (3886,3779,[2_1|4]), (3887,3888,[2_1|4]), (3888,3889,[1_1|4]), (3889,3890,[1_1|4]), (3890,3891,[0_1|4]), (3891,3892,[1_1|4]), (3892,3893,[2_1|4]), (3893,3894,[0_1|4]), (3894,3895,[1_1|4]), (3895,3896,[2_1|4]), (3896,3897,[0_1|4]), (3897,3898,[1_1|4]), (3898,3899,[2_1|4]), (3899,3900,[0_1|4]), (3900,3901,[1_1|4]), (3901,3902,[2_1|4]), (3902,3903,[0_1|4]), (3903,3904,[1_1|4]), (3904,3905,[2_1|4]), (3905,3906,[0_1|4]), (3906,3907,[1_1|4]), (3907,3908,[2_1|4]), (3908,3909,[0_1|4]), (3909,3910,[1_1|4]), (3910,3623,[2_1|4]), (3910,3632,[2_1|4]), (3910,3644,[2_1|4]), (3910,3659,[2_1|4]), (3910,3677,[2_1|4]), (3910,3698,[2_1|4]), (3910,3722,[2_1|4]), (3910,3749,[2_1|4]), (3910,3779,[2_1|4]), (3911,3912,[2_1|4]), (3912,3913,[1_1|4]), (3913,3914,[1_1|4]), (3914,3915,[0_1|4]), (3915,3916,[1_1|4]), (3916,3917,[2_1|4]), (3917,3918,[0_1|4]), (3918,3919,[1_1|4]), (3919,3920,[2_1|4]), (3920,3921,[0_1|4]), (3921,3922,[1_1|4]), (3922,3923,[2_1|4]), (3923,3924,[0_1|4]), (3924,3925,[1_1|4]), (3925,3926,[2_1|4]), (3926,3927,[0_1|4]), (3927,3928,[1_1|4]), (3928,3929,[2_1|4]), (3929,3930,[0_1|4]), (3930,3931,[1_1|4]), (3931,3932,[2_1|4]), (3932,3933,[0_1|4]), (3933,3934,[1_1|4]), (3934,3935,[2_1|4]), (3935,3936,[0_1|4]), (3936,3937,[1_1|4]), (3937,3623,[2_1|4]), (3937,3632,[2_1|4]), (3937,3644,[2_1|4]), (3937,3659,[2_1|4]), (3937,3677,[2_1|4]), (3937,3698,[2_1|4]), (3937,3722,[2_1|4]), (3937,3749,[2_1|4]), (3937,3779,[2_1|4]), (3938,3939,[2_1|4]), (3939,3940,[1_1|4]), (3940,3941,[1_1|4]), (3941,3942,[0_1|4]), (3942,3943,[1_1|4]), (3943,3944,[2_1|4]), (3944,3945,[0_1|4]), (3945,3946,[1_1|4]), (3946,3947,[2_1|4]), (3947,3948,[0_1|4]), (3948,3949,[1_1|4]), (3949,3950,[2_1|4]), (3950,3951,[0_1|4]), (3951,3952,[1_1|4]), (3952,3953,[2_1|4]), (3953,3954,[0_1|4]), (3954,3955,[1_1|4]), (3955,3956,[2_1|4]), (3956,3957,[0_1|4]), (3957,3958,[1_1|4]), (3958,3959,[2_1|4]), (3959,3960,[0_1|4]), (3960,3961,[1_1|4]), (3961,3962,[2_1|4]), (3962,3963,[0_1|4]), (3963,3964,[1_1|4]), (3964,3965,[2_1|4]), (3965,3966,[0_1|4]), (3966,3967,[1_1|4]), (3967,3623,[2_1|4]), (3967,3632,[2_1|4]), (3967,3644,[2_1|4]), (3967,3659,[2_1|4]), (3967,3677,[2_1|4]), (3967,3698,[2_1|4]), (3967,3722,[2_1|4]), (3967,3749,[2_1|4]), (3967,3779,[2_1|4]), (3968,3969,[2_1|4]), (3969,3970,[1_1|4]), (3970,3971,[1_1|4]), (3971,3972,[0_1|4]), (3972,3973,[1_1|4]), (3973,3974,[2_1|4]), (3974,3975,[0_1|4]), (3975,3976,[1_1|4]), (3976,3977,[2_1|4]), (3977,3978,[0_1|4]), (3978,3979,[1_1|4]), (3979,3980,[2_1|4]), (3980,3981,[0_1|4]), (3981,3982,[1_1|4]), (3982,3983,[2_1|4]), (3983,3984,[0_1|4]), (3984,3985,[1_1|4]), (3985,3986,[2_1|4]), (3986,3987,[0_1|4]), (3987,3988,[1_1|4]), (3988,3989,[2_1|4]), (3989,3990,[0_1|4]), (3990,3991,[1_1|4]), (3991,3992,[2_1|4]), (3992,3993,[0_1|4]), (3993,3994,[1_1|4]), (3994,3995,[2_1|4]), (3995,3996,[0_1|4]), (3996,3997,[1_1|4]), (3997,3998,[2_1|4]), (3998,3999,[0_1|4]), (3999,4000,[1_1|4]), (4000,3623,[2_1|4]), (4000,3632,[2_1|4]), (4000,3644,[2_1|4]), (4000,3659,[2_1|4]), (4000,3677,[2_1|4]), (4000,3698,[2_1|4]), (4000,3722,[2_1|4]), (4000,3749,[2_1|4]), (4000,3779,[2_1|4]), (4001,4002,[2_1|4]), (4002,4003,[1_1|4]), (4003,4004,[1_1|4]), (4004,4005,[0_1|4]), (4005,4006,[1_1|4]), (4006,4007,[2_1|4]), (4007,4008,[0_1|4]), (4008,4009,[1_1|4]), (4009,3812,[2_1|4]), (4009,3821,[2_1|4]), (4009,3833,[2_1|4]), (4009,3848,[2_1|4]), (4009,3866,[2_1|4]), (4009,3887,[2_1|4]), (4009,3911,[2_1|4]), (4009,3938,[2_1|4]), (4009,3968,[2_1|4]), (4010,4011,[2_1|4]), (4011,4012,[1_1|4]), (4012,4013,[1_1|4]), (4013,4014,[0_1|4]), (4014,4015,[1_1|4]), (4015,4016,[2_1|4]), (4016,4017,[0_1|4]), (4017,4018,[1_1|4]), (4018,4019,[2_1|4]), (4019,4020,[0_1|4]), (4020,4021,[1_1|4]), (4021,3812,[2_1|4]), (4021,3821,[2_1|4]), (4021,3833,[2_1|4]), (4021,3848,[2_1|4]), (4021,3866,[2_1|4]), (4021,3887,[2_1|4]), (4021,3911,[2_1|4]), (4021,3938,[2_1|4]), (4021,3968,[2_1|4]), (4022,4023,[2_1|4]), (4023,4024,[1_1|4]), (4024,4025,[1_1|4]), (4025,4026,[0_1|4]), (4026,4027,[1_1|4]), (4027,4028,[2_1|4]), (4028,4029,[0_1|4]), (4029,4030,[1_1|4]), (4030,4031,[2_1|4]), (4031,4032,[0_1|4]), (4032,4033,[1_1|4]), (4033,4034,[2_1|4]), (4034,4035,[0_1|4]), (4035,4036,[1_1|4]), (4036,3812,[2_1|4]), (4036,3821,[2_1|4]), (4036,3833,[2_1|4]), (4036,3848,[2_1|4]), (4036,3866,[2_1|4]), (4036,3887,[2_1|4]), (4036,3911,[2_1|4]), (4036,3938,[2_1|4]), (4036,3968,[2_1|4]), (4037,4038,[2_1|4]), (4038,4039,[1_1|4]), (4039,4040,[1_1|4]), (4040,4041,[0_1|4]), (4041,4042,[1_1|4]), (4042,4043,[2_1|4]), (4043,4044,[0_1|4]), (4044,4045,[1_1|4]), (4045,4046,[2_1|4]), (4046,4047,[0_1|4]), (4047,4048,[1_1|4]), (4048,4049,[2_1|4]), (4049,4050,[0_1|4]), (4050,4051,[1_1|4]), (4051,4052,[2_1|4]), (4052,4053,[0_1|4]), (4053,4054,[1_1|4]), (4054,3812,[2_1|4]), (4054,3821,[2_1|4]), (4054,3833,[2_1|4]), (4054,3848,[2_1|4]), (4054,3866,[2_1|4]), (4054,3887,[2_1|4]), (4054,3911,[2_1|4]), (4054,3938,[2_1|4]), (4054,3968,[2_1|4]), (4055,4056,[2_1|4]), (4056,4057,[1_1|4]), (4057,4058,[1_1|4]), (4058,4059,[0_1|4]), (4059,4060,[1_1|4]), (4060,4061,[2_1|4]), (4061,4062,[0_1|4]), (4062,4063,[1_1|4]), (4063,4064,[2_1|4]), (4064,4065,[0_1|4]), (4065,4066,[1_1|4]), (4066,4067,[2_1|4]), (4067,4068,[0_1|4]), (4068,4069,[1_1|4]), (4069,4070,[2_1|4]), (4070,4071,[0_1|4]), (4071,4072,[1_1|4]), (4072,4073,[2_1|4]), (4073,4074,[0_1|4]), (4074,4075,[1_1|4]), (4075,3812,[2_1|4]), (4075,3821,[2_1|4]), (4075,3833,[2_1|4]), (4075,3848,[2_1|4]), (4075,3866,[2_1|4]), (4075,3887,[2_1|4]), (4075,3911,[2_1|4]), (4075,3938,[2_1|4]), (4075,3968,[2_1|4]), (4076,4077,[2_1|4]), (4077,4078,[1_1|4]), (4078,4079,[1_1|4]), (4079,4080,[0_1|4]), (4080,4081,[1_1|4]), (4081,4082,[2_1|4]), (4082,4083,[0_1|4]), (4083,4084,[1_1|4]), (4084,4085,[2_1|4]), (4085,4086,[0_1|4]), (4086,4087,[1_1|4]), (4087,4088,[2_1|4]), (4088,4089,[0_1|4]), (4089,4090,[1_1|4]), (4090,4091,[2_1|4]), (4091,4092,[0_1|4]), (4092,4093,[1_1|4]), (4093,4094,[2_1|4]), (4094,4095,[0_1|4]), (4095,4096,[1_1|4]), (4096,4097,[2_1|4]), (4097,4098,[0_1|4]), (4098,4099,[1_1|4]), (4099,3812,[2_1|4]), (4099,3821,[2_1|4]), (4099,3833,[2_1|4]), (4099,3848,[2_1|4]), (4099,3866,[2_1|4]), (4099,3887,[2_1|4]), (4099,3911,[2_1|4]), (4099,3938,[2_1|4]), (4099,3968,[2_1|4]), (4100,4101,[2_1|4]), (4101,4102,[1_1|4]), (4102,4103,[1_1|4]), (4103,4104,[0_1|4]), (4104,4105,[1_1|4]), (4105,4106,[2_1|4]), (4106,4107,[0_1|4]), (4107,4108,[1_1|4]), (4108,4109,[2_1|4]), (4109,4110,[0_1|4]), (4110,4111,[1_1|4]), (4111,4112,[2_1|4]), (4112,4113,[0_1|4]), (4113,4114,[1_1|4]), (4114,4115,[2_1|4]), (4115,4116,[0_1|4]), (4116,4117,[1_1|4]), (4117,4118,[2_1|4]), (4118,4119,[0_1|4]), (4119,4120,[1_1|4]), (4120,4121,[2_1|4]), (4121,4122,[0_1|4]), (4122,4123,[1_1|4]), (4123,4124,[2_1|4]), (4124,4125,[0_1|4]), (4125,4126,[1_1|4]), (4126,3812,[2_1|4]), (4126,3821,[2_1|4]), (4126,3833,[2_1|4]), (4126,3848,[2_1|4]), (4126,3866,[2_1|4]), (4126,3887,[2_1|4]), (4126,3911,[2_1|4]), (4126,3938,[2_1|4]), (4126,3968,[2_1|4]), (4127,4128,[2_1|4]), (4128,4129,[1_1|4]), (4129,4130,[1_1|4]), (4130,4131,[0_1|4]), (4131,4132,[1_1|4]), (4132,4133,[2_1|4]), (4133,4134,[0_1|4]), (4134,4135,[1_1|4]), (4135,4136,[2_1|4]), (4136,4137,[0_1|4]), (4137,4138,[1_1|4]), (4138,4139,[2_1|4]), (4139,4140,[0_1|4]), (4140,4141,[1_1|4]), (4141,4142,[2_1|4]), (4142,4143,[0_1|4]), (4143,4144,[1_1|4]), (4144,4145,[2_1|4]), (4145,4146,[0_1|4]), (4146,4147,[1_1|4]), (4147,4148,[2_1|4]), (4148,4149,[0_1|4]), (4149,4150,[1_1|4]), (4150,4151,[2_1|4]), (4151,4152,[0_1|4]), (4152,4153,[1_1|4]), (4153,4154,[2_1|4]), (4154,4155,[0_1|4]), (4155,4156,[1_1|4]), (4156,3812,[2_1|4]), (4156,3821,[2_1|4]), (4156,3833,[2_1|4]), (4156,3848,[2_1|4]), (4156,3866,[2_1|4]), (4156,3887,[2_1|4]), (4156,3911,[2_1|4]), (4156,3938,[2_1|4]), (4156,3968,[2_1|4]), (4157,4158,[2_1|4]), (4158,4159,[1_1|4]), (4159,4160,[1_1|4]), (4160,4161,[0_1|4]), (4161,4162,[1_1|4]), (4162,4163,[2_1|4]), (4163,4164,[0_1|4]), (4164,4165,[1_1|4]), (4165,4166,[2_1|4]), (4166,4167,[0_1|4]), (4167,4168,[1_1|4]), (4168,4169,[2_1|4]), (4169,4170,[0_1|4]), (4170,4171,[1_1|4]), (4171,4172,[2_1|4]), (4172,4173,[0_1|4]), (4173,4174,[1_1|4]), (4174,4175,[2_1|4]), (4175,4176,[0_1|4]), (4176,4177,[1_1|4]), (4177,4178,[2_1|4]), (4178,4179,[0_1|4]), (4179,4180,[1_1|4]), (4180,4181,[2_1|4]), (4181,4182,[0_1|4]), (4182,4183,[1_1|4]), (4183,4184,[2_1|4]), (4184,4185,[0_1|4]), (4185,4186,[1_1|4]), (4186,4187,[2_1|4]), (4187,4188,[0_1|4]), (4188,4189,[1_1|4]), (4189,3812,[2_1|4]), (4189,3821,[2_1|4]), (4189,3833,[2_1|4]), (4189,3848,[2_1|4]), (4189,3866,[2_1|4]), (4189,3887,[2_1|4]), (4189,3911,[2_1|4]), (4189,3938,[2_1|4]), (4189,3968,[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)))))))))))))))))))))) 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: 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)))))))))))))))))))))) 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: 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)))))))))))))))))))))) 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: FULL