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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (full) of the given DCpxTrs could be proven to be BOUNDS(n^1, n^1).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), 71 ms] (4) CpxRelTRS (5) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] (6) CpxTRS (7) CpxTrsMatchBoundsProof [FINISHED, 491 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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(3908,3777,[2_1|4]), (3909,3910,[2_1|4]), (3910,3911,[1_1|4]), (3911,3912,[1_1|4]), (3912,3913,[0_1|4]), (3913,3914,[1_1|4]), (3914,3915,[2_1|4]), (3915,3916,[0_1|4]), (3916,3917,[1_1|4]), (3917,3918,[2_1|4]), (3918,3919,[0_1|4]), (3919,3920,[1_1|4]), (3920,3921,[2_1|4]), (3921,3922,[0_1|4]), (3922,3923,[1_1|4]), (3923,3924,[2_1|4]), (3924,3925,[0_1|4]), (3925,3926,[1_1|4]), (3926,3927,[2_1|4]), (3927,3928,[0_1|4]), (3928,3929,[1_1|4]), (3929,3930,[2_1|4]), (3930,3931,[0_1|4]), (3931,3932,[1_1|4]), (3932,3933,[2_1|4]), (3933,3934,[0_1|4]), (3934,3935,[1_1|4]), (3935,3621,[2_1|4]), (3935,3630,[2_1|4]), (3935,3642,[2_1|4]), (3935,3657,[2_1|4]), (3935,3675,[2_1|4]), (3935,3696,[2_1|4]), (3935,3720,[2_1|4]), (3935,3747,[2_1|4]), (3935,3777,[2_1|4]), (3936,3937,[2_1|4]), (3937,3938,[1_1|4]), (3938,3939,[1_1|4]), (3939,3940,[0_1|4]), (3940,3941,[1_1|4]), (3941,3942,[2_1|4]), (3942,3943,[0_1|4]), (3943,3944,[1_1|4]), (3944,3945,[2_1|4]), (3945,3946,[0_1|4]), (3946,3947,[1_1|4]), (3947,3948,[2_1|4]), (3948,3949,[0_1|4]), (3949,3950,[1_1|4]), (3950,3951,[2_1|4]), (3951,3952,[0_1|4]), (3952,3953,[1_1|4]), (3953,3954,[2_1|4]), (3954,3955,[0_1|4]), (3955,3956,[1_1|4]), (3956,3957,[2_1|4]), (3957,3958,[0_1|4]), (3958,3959,[1_1|4]), (3959,3960,[2_1|4]), (3960,3961,[0_1|4]), (3961,3962,[1_1|4]), (3962,3963,[2_1|4]), (3963,3964,[0_1|4]), (3964,3965,[1_1|4]), (3965,3621,[2_1|4]), (3965,3630,[2_1|4]), (3965,3642,[2_1|4]), (3965,3657,[2_1|4]), (3965,3675,[2_1|4]), (3965,3696,[2_1|4]), (3965,3720,[2_1|4]), (3965,3747,[2_1|4]), (3965,3777,[2_1|4]), (3966,3967,[2_1|4]), (3967,3968,[1_1|4]), (3968,3969,[1_1|4]), (3969,3970,[0_1|4]), (3970,3971,[1_1|4]), (3971,3972,[2_1|4]), (3972,3973,[0_1|4]), (3973,3974,[1_1|4]), (3974,3975,[2_1|4]), (3975,3976,[0_1|4]), (3976,3977,[1_1|4]), (3977,3978,[2_1|4]), (3978,3979,[0_1|4]), (3979,3980,[1_1|4]), (3980,3981,[2_1|4]), (3981,3982,[0_1|4]), (3982,3983,[1_1|4]), (3983,3984,[2_1|4]), (3984,3985,[0_1|4]), (3985,3986,[1_1|4]), (3986,3987,[2_1|4]), (3987,3988,[0_1|4]), (3988,3989,[1_1|4]), (3989,3990,[2_1|4]), (3990,3991,[0_1|4]), (3991,3992,[1_1|4]), (3992,3993,[2_1|4]), (3993,3994,[0_1|4]), (3994,3995,[1_1|4]), (3995,3996,[2_1|4]), (3996,3997,[0_1|4]), (3997,3998,[1_1|4]), (3998,3621,[2_1|4]), (3998,3630,[2_1|4]), (3998,3642,[2_1|4]), (3998,3657,[2_1|4]), (3998,3675,[2_1|4]), (3998,3696,[2_1|4]), (3998,3720,[2_1|4]), (3998,3747,[2_1|4]), (3998,3777,[2_1|4]), (3999,4000,[2_1|4]), (4000,4001,[1_1|4]), (4001,4002,[1_1|4]), (4002,4003,[0_1|4]), (4003,4004,[1_1|4]), (4004,4005,[2_1|4]), (4005,4006,[0_1|4]), (4006,4007,[1_1|4]), (4007,3810,[2_1|4]), (4007,3819,[2_1|4]), (4007,3831,[2_1|4]), (4007,3846,[2_1|4]), (4007,3864,[2_1|4]), (4007,3885,[2_1|4]), (4007,3909,[2_1|4]), (4007,3936,[2_1|4]), (4007,3966,[2_1|4]), (4008,4009,[2_1|4]), (4009,4010,[1_1|4]), (4010,4011,[1_1|4]), (4011,4012,[0_1|4]), (4012,4013,[1_1|4]), (4013,4014,[2_1|4]), (4014,4015,[0_1|4]), (4015,4016,[1_1|4]), (4016,4017,[2_1|4]), (4017,4018,[0_1|4]), (4018,4019,[1_1|4]), (4019,3810,[2_1|4]), (4019,3819,[2_1|4]), (4019,3831,[2_1|4]), (4019,3846,[2_1|4]), (4019,3864,[2_1|4]), (4019,3885,[2_1|4]), (4019,3909,[2_1|4]), (4019,3936,[2_1|4]), (4019,3966,[2_1|4]), (4020,4021,[2_1|4]), (4021,4022,[1_1|4]), (4022,4023,[1_1|4]), (4023,4024,[0_1|4]), (4024,4025,[1_1|4]), (4025,4026,[2_1|4]), (4026,4027,[0_1|4]), (4027,4028,[1_1|4]), (4028,4029,[2_1|4]), (4029,4030,[0_1|4]), (4030,4031,[1_1|4]), (4031,4032,[2_1|4]), (4032,4033,[0_1|4]), (4033,4034,[1_1|4]), (4034,3810,[2_1|4]), (4034,3819,[2_1|4]), (4034,3831,[2_1|4]), (4034,3846,[2_1|4]), (4034,3864,[2_1|4]), (4034,3885,[2_1|4]), (4034,3909,[2_1|4]), (4034,3936,[2_1|4]), (4034,3966,[2_1|4]), (4035,4036,[2_1|4]), (4036,4037,[1_1|4]), (4037,4038,[1_1|4]), (4038,4039,[0_1|4]), (4039,4040,[1_1|4]), (4040,4041,[2_1|4]), (4041,4042,[0_1|4]), (4042,4043,[1_1|4]), (4043,4044,[2_1|4]), (4044,4045,[0_1|4]), (4045,4046,[1_1|4]), (4046,4047,[2_1|4]), (4047,4048,[0_1|4]), (4048,4049,[1_1|4]), (4049,4050,[2_1|4]), (4050,4051,[0_1|4]), (4051,4052,[1_1|4]), (4052,3810,[2_1|4]), (4052,3819,[2_1|4]), (4052,3831,[2_1|4]), (4052,3846,[2_1|4]), (4052,3864,[2_1|4]), (4052,3885,[2_1|4]), (4052,3909,[2_1|4]), (4052,3936,[2_1|4]), (4052,3966,[2_1|4]), (4053,4054,[2_1|4]), (4054,4055,[1_1|4]), (4055,4056,[1_1|4]), (4056,4057,[0_1|4]), (4057,4058,[1_1|4]), (4058,4059,[2_1|4]), (4059,4060,[0_1|4]), (4060,4061,[1_1|4]), (4061,4062,[2_1|4]), (4062,4063,[0_1|4]), (4063,4064,[1_1|4]), (4064,4065,[2_1|4]), (4065,4066,[0_1|4]), (4066,4067,[1_1|4]), (4067,4068,[2_1|4]), (4068,4069,[0_1|4]), (4069,4070,[1_1|4]), (4070,4071,[2_1|4]), (4071,4072,[0_1|4]), (4072,4073,[1_1|4]), (4073,3810,[2_1|4]), (4073,3819,[2_1|4]), (4073,3831,[2_1|4]), (4073,3846,[2_1|4]), (4073,3864,[2_1|4]), (4073,3885,[2_1|4]), (4073,3909,[2_1|4]), (4073,3936,[2_1|4]), (4073,3966,[2_1|4]), (4074,4075,[2_1|4]), (4075,4076,[1_1|4]), (4076,4077,[1_1|4]), (4077,4078,[0_1|4]), (4078,4079,[1_1|4]), (4079,4080,[2_1|4]), (4080,4081,[0_1|4]), (4081,4082,[1_1|4]), (4082,4083,[2_1|4]), (4083,4084,[0_1|4]), (4084,4085,[1_1|4]), (4085,4086,[2_1|4]), (4086,4087,[0_1|4]), (4087,4088,[1_1|4]), (4088,4089,[2_1|4]), (4089,4090,[0_1|4]), (4090,4091,[1_1|4]), (4091,4092,[2_1|4]), (4092,4093,[0_1|4]), (4093,4094,[1_1|4]), (4094,4095,[2_1|4]), (4095,4096,[0_1|4]), (4096,4097,[1_1|4]), (4097,3810,[2_1|4]), (4097,3819,[2_1|4]), (4097,3831,[2_1|4]), (4097,3846,[2_1|4]), (4097,3864,[2_1|4]), (4097,3885,[2_1|4]), (4097,3909,[2_1|4]), (4097,3936,[2_1|4]), (4097,3966,[2_1|4]), (4098,4099,[2_1|4]), (4099,4100,[1_1|4]), (4100,4101,[1_1|4]), (4101,4102,[0_1|4]), (4102,4103,[1_1|4]), (4103,4104,[2_1|4]), (4104,4105,[0_1|4]), (4105,4106,[1_1|4]), (4106,4107,[2_1|4]), (4107,4108,[0_1|4]), (4108,4109,[1_1|4]), (4109,4110,[2_1|4]), (4110,4111,[0_1|4]), (4111,4112,[1_1|4]), (4112,4113,[2_1|4]), (4113,4114,[0_1|4]), (4114,4115,[1_1|4]), (4115,4116,[2_1|4]), (4116,4117,[0_1|4]), (4117,4118,[1_1|4]), (4118,4119,[2_1|4]), (4119,4120,[0_1|4]), (4120,4121,[1_1|4]), (4121,4122,[2_1|4]), (4122,4123,[0_1|4]), (4123,4124,[1_1|4]), (4124,3810,[2_1|4]), (4124,3819,[2_1|4]), (4124,3831,[2_1|4]), (4124,3846,[2_1|4]), (4124,3864,[2_1|4]), (4124,3885,[2_1|4]), (4124,3909,[2_1|4]), (4124,3936,[2_1|4]), (4124,3966,[2_1|4]), (4125,4126,[2_1|4]), (4126,4127,[1_1|4]), (4127,4128,[1_1|4]), (4128,4129,[0_1|4]), (4129,4130,[1_1|4]), (4130,4131,[2_1|4]), (4131,4132,[0_1|4]), (4132,4133,[1_1|4]), (4133,4134,[2_1|4]), (4134,4135,[0_1|4]), (4135,4136,[1_1|4]), (4136,4137,[2_1|4]), (4137,4138,[0_1|4]), (4138,4139,[1_1|4]), (4139,4140,[2_1|4]), (4140,4141,[0_1|4]), (4141,4142,[1_1|4]), (4142,4143,[2_1|4]), (4143,4144,[0_1|4]), (4144,4145,[1_1|4]), (4145,4146,[2_1|4]), (4146,4147,[0_1|4]), (4147,4148,[1_1|4]), (4148,4149,[2_1|4]), (4149,4150,[0_1|4]), (4150,4151,[1_1|4]), (4151,4152,[2_1|4]), (4152,4153,[0_1|4]), (4153,4154,[1_1|4]), (4154,3810,[2_1|4]), (4154,3819,[2_1|4]), (4154,3831,[2_1|4]), (4154,3846,[2_1|4]), (4154,3864,[2_1|4]), (4154,3885,[2_1|4]), (4154,3909,[2_1|4]), (4154,3936,[2_1|4]), (4154,3966,[2_1|4]), (4155,4156,[2_1|4]), (4156,4157,[1_1|4]), (4157,4158,[1_1|4]), (4158,4159,[0_1|4]), (4159,4160,[1_1|4]), (4160,4161,[2_1|4]), (4161,4162,[0_1|4]), (4162,4163,[1_1|4]), (4163,4164,[2_1|4]), (4164,4165,[0_1|4]), (4165,4166,[1_1|4]), (4166,4167,[2_1|4]), (4167,4168,[0_1|4]), (4168,4169,[1_1|4]), (4169,4170,[2_1|4]), (4170,4171,[0_1|4]), (4171,4172,[1_1|4]), (4172,4173,[2_1|4]), (4173,4174,[0_1|4]), (4174,4175,[1_1|4]), (4175,4176,[2_1|4]), (4176,4177,[0_1|4]), (4177,4178,[1_1|4]), (4178,4179,[2_1|4]), (4179,4180,[0_1|4]), (4180,4181,[1_1|4]), (4181,4182,[2_1|4]), (4182,4183,[0_1|4]), (4183,4184,[1_1|4]), (4184,4185,[2_1|4]), (4185,4186,[0_1|4]), (4186,4187,[1_1|4]), (4187,3810,[2_1|4]), (4187,3819,[2_1|4]), (4187,3831,[2_1|4]), (4187,3846,[2_1|4]), (4187,3864,[2_1|4]), (4187,3885,[2_1|4]), (4187,3909,[2_1|4]), (4187,3936,[2_1|4]), (4187,3966,[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