/export/starexec/sandbox/solver/bin/starexec_run_complexity /export/starexec/sandbox/benchmark/theBenchmark.koat /export/starexec/sandbox/output/output_files -------------------------------------------------------------------------------- WORST_CASE(Omega(n^2), O(n^2)) proof of /export/starexec/sandbox/benchmark/theBenchmark.koat # AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(n^2, n^2). (0) CpxIntTrs (1) Koat Proof [FINISHED, 138 ms] (2) BOUNDS(1, n^2) (3) Loat Proof [FINISHED, 1026 ms] (4) BOUNDS(n^2, INF) ---------------------------------------- (0) Obligation: Complexity Int TRS consisting of the following rules: f0(A, B, C, D, E, F, G) -> Com_1(f10(H, 0, C, D, E, F, G)) :|: TRUE f10(A, B, C, D, E, F, G) -> Com_1(f10(A, B + 1, C, D, E, F, G)) :|: C >= B + 1 f18(A, B, C, D, E, F, G) -> Com_1(f21(A, B, C, D, E, 0, G)) :|: D >= 2 + E f21(A, B, C, D, E, F, G) -> Com_1(f21(A, B, C, D, E, F + 1, G)) :|: D >= E + 2 + F f21(A, B, C, D, E, F, G) -> Com_1(f21(A, B, C, D, E, F + 1, H)) :|: D >= E + 2 + F f32(A, B, C, D, E, F, G) -> Com_1(f32(A, B, C, D, E + 1, F, G)) :|: D >= 2 + E f32(A, B, C, D, E, F, G) -> Com_1(f41(A, B, C, D, E, F, G)) :|: E + 1 >= D f21(A, B, C, D, E, F, G) -> Com_1(f18(A, B, C, D, E + 1, F, G)) :|: F + E + 1 >= D f18(A, B, C, D, E, F, G) -> Com_1(f32(A, B, C, D, 0, F, G)) :|: E + 1 >= D f10(A, B, C, D, E, F, G) -> Com_1(f18(A, B, C, C, 0, F, G)) :|: B >= C The start-symbols are:[f0_7] ---------------------------------------- (1) Koat Proof (FINISHED) YES(?, 8*Ar_2 + 24*Ar_2^2 + 10) Initial complexity problem: 1: T: (Comp: ?, Cost: 1) f0(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5, Ar_6) -> Com_1(f10(Fresh_1, 0, Ar_2, Ar_3, Ar_4, Ar_5, Ar_6)) (Comp: ?, Cost: 1) f10(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5, Ar_6) -> Com_1(f10(Ar_0, Ar_1 + 1, Ar_2, Ar_3, Ar_4, Ar_5, Ar_6)) [ Ar_2 >= Ar_1 + 1 ] (Comp: ?, Cost: 1) f18(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5, Ar_6) -> Com_1(f21(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, 0, Ar_6)) [ Ar_3 >= Ar_4 + 2 ] (Comp: ?, Cost: 1) f21(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5, Ar_6) -> Com_1(f21(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5 + 1, Ar_6)) [ Ar_3 >= Ar_4 + Ar_5 + 2 ] (Comp: ?, Cost: 1) f21(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5, Ar_6) -> Com_1(f21(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5 + 1, Fresh_0)) [ Ar_3 >= Ar_4 + Ar_5 + 2 ] (Comp: ?, Cost: 1) f32(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5, Ar_6) -> Com_1(f32(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4 + 1, Ar_5, Ar_6)) [ Ar_3 >= Ar_4 + 2 ] (Comp: ?, Cost: 1) f32(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5, Ar_6) -> Com_1(f41(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5, Ar_6)) [ Ar_4 + 1 >= Ar_3 ] (Comp: ?, Cost: 1) f21(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5, Ar_6) -> Com_1(f18(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4 + 1, Ar_5, Ar_6)) [ Ar_5 + Ar_4 + 1 >= Ar_3 ] (Comp: ?, Cost: 1) f18(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5, Ar_6) -> Com_1(f32(Ar_0, Ar_1, Ar_2, Ar_3, 0, Ar_5, Ar_6)) [ Ar_4 + 1 >= Ar_3 ] (Comp: ?, Cost: 1) f10(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5, Ar_6) -> Com_1(f18(Ar_0, Ar_1, Ar_2, Ar_2, 0, Ar_5, Ar_6)) [ Ar_1 >= Ar_2 ] (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5, Ar_6) -> Com_1(f0(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5, Ar_6)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 Slicing away variables that do not contribute to conditions from problem 1 leaves variables [Ar_1, Ar_2, Ar_3, Ar_4, Ar_5]. We thus obtain the following problem: 2: T: (Comp: 1, Cost: 0) koat_start(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f0(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5)) [ 0 <= 0 ] (Comp: ?, Cost: 1) f10(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f18(Ar_1, Ar_2, Ar_2, 0, Ar_5)) [ Ar_1 >= Ar_2 ] (Comp: ?, Cost: 1) f18(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f32(Ar_1, Ar_2, Ar_3, 0, Ar_5)) [ Ar_4 + 1 >= Ar_3 ] (Comp: ?, Cost: 1) f21(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f18(Ar_1, Ar_2, Ar_3, Ar_4 + 1, Ar_5)) [ Ar_5 + Ar_4 + 1 >= Ar_3 ] (Comp: ?, Cost: 1) f32(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f41(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5)) [ Ar_4 + 1 >= Ar_3 ] (Comp: ?, Cost: 1) f32(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f32(Ar_1, Ar_2, Ar_3, Ar_4 + 1, Ar_5)) [ Ar_3 >= Ar_4 + 2 ] (Comp: ?, Cost: 1) f21(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f21(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5 + 1)) [ Ar_3 >= Ar_4 + Ar_5 + 2 ] (Comp: ?, Cost: 1) f21(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f21(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5 + 1)) [ Ar_3 >= Ar_4 + Ar_5 + 2 ] (Comp: ?, Cost: 1) f18(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f21(Ar_1, Ar_2, Ar_3, Ar_4, 0)) [ Ar_3 >= Ar_4 + 2 ] (Comp: ?, Cost: 1) f10(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f10(Ar_1 + 1, Ar_2, Ar_3, Ar_4, Ar_5)) [ Ar_2 >= Ar_1 + 1 ] (Comp: ?, Cost: 1) f0(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f10(0, Ar_2, Ar_3, Ar_4, Ar_5)) start location: koat_start leaf cost: 0 Repeatedly propagating knowledge in problem 2 produces the following problem: 3: T: (Comp: 1, Cost: 0) koat_start(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f0(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5)) [ 0 <= 0 ] (Comp: ?, Cost: 1) f10(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f18(Ar_1, Ar_2, Ar_2, 0, Ar_5)) [ Ar_1 >= Ar_2 ] (Comp: ?, Cost: 1) f18(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f32(Ar_1, Ar_2, Ar_3, 0, Ar_5)) [ Ar_4 + 1 >= Ar_3 ] (Comp: ?, Cost: 1) f21(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f18(Ar_1, Ar_2, Ar_3, Ar_4 + 1, Ar_5)) [ Ar_5 + Ar_4 + 1 >= Ar_3 ] (Comp: ?, Cost: 1) f32(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f41(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5)) [ Ar_4 + 1 >= Ar_3 ] (Comp: ?, Cost: 1) f32(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f32(Ar_1, Ar_2, Ar_3, Ar_4 + 1, Ar_5)) [ Ar_3 >= Ar_4 + 2 ] (Comp: ?, Cost: 1) f21(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f21(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5 + 1)) [ Ar_3 >= Ar_4 + Ar_5 + 2 ] (Comp: ?, Cost: 1) f21(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f21(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5 + 1)) [ Ar_3 >= Ar_4 + Ar_5 + 2 ] (Comp: ?, Cost: 1) f18(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f21(Ar_1, Ar_2, Ar_3, Ar_4, 0)) [ Ar_3 >= Ar_4 + 2 ] (Comp: ?, Cost: 1) f10(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f10(Ar_1 + 1, Ar_2, Ar_3, Ar_4, Ar_5)) [ Ar_2 >= Ar_1 + 1 ] (Comp: 1, Cost: 1) f0(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f10(0, Ar_2, Ar_3, Ar_4, Ar_5)) start location: koat_start leaf cost: 0 A polynomial rank function with Pol(koat_start) = 3 Pol(f0) = 3 Pol(f10) = 3 Pol(f18) = 2 Pol(f32) = 1 Pol(f21) = 2 Pol(f41) = 0 orients all transitions weakly and the transitions f32(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f41(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5)) [ Ar_4 + 1 >= Ar_3 ] f18(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f32(Ar_1, Ar_2, Ar_3, 0, Ar_5)) [ Ar_4 + 1 >= Ar_3 ] f10(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f18(Ar_1, Ar_2, Ar_2, 0, Ar_5)) [ Ar_1 >= Ar_2 ] strictly and produces the following problem: 4: T: (Comp: 1, Cost: 0) koat_start(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f0(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5)) [ 0 <= 0 ] (Comp: 3, Cost: 1) f10(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f18(Ar_1, Ar_2, Ar_2, 0, Ar_5)) [ Ar_1 >= Ar_2 ] (Comp: 3, Cost: 1) f18(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f32(Ar_1, Ar_2, Ar_3, 0, Ar_5)) [ Ar_4 + 1 >= Ar_3 ] (Comp: ?, Cost: 1) f21(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f18(Ar_1, Ar_2, Ar_3, Ar_4 + 1, Ar_5)) [ Ar_5 + Ar_4 + 1 >= Ar_3 ] (Comp: 3, Cost: 1) f32(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f41(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5)) [ Ar_4 + 1 >= Ar_3 ] (Comp: ?, Cost: 1) f32(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f32(Ar_1, Ar_2, Ar_3, Ar_4 + 1, Ar_5)) [ Ar_3 >= Ar_4 + 2 ] (Comp: ?, Cost: 1) f21(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f21(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5 + 1)) [ Ar_3 >= Ar_4 + Ar_5 + 2 ] (Comp: ?, Cost: 1) f21(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f21(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5 + 1)) [ Ar_3 >= Ar_4 + Ar_5 + 2 ] (Comp: ?, Cost: 1) f18(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f21(Ar_1, Ar_2, Ar_3, Ar_4, 0)) [ Ar_3 >= Ar_4 + 2 ] (Comp: ?, Cost: 1) f10(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f10(Ar_1 + 1, Ar_2, Ar_3, Ar_4, Ar_5)) [ Ar_2 >= Ar_1 + 1 ] (Comp: 1, Cost: 1) f0(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f10(0, Ar_2, Ar_3, Ar_4, Ar_5)) start location: koat_start leaf cost: 0 A polynomial rank function with Pol(koat_start) = V_2 Pol(f0) = V_2 Pol(f10) = V_2 Pol(f18) = V_3 Pol(f32) = V_3 - V_4 Pol(f21) = V_3 Pol(f41) = V_3 - V_4 orients all transitions weakly and the transition f32(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f32(Ar_1, Ar_2, Ar_3, Ar_4 + 1, Ar_5)) [ Ar_3 >= Ar_4 + 2 ] strictly and produces the following problem: 5: T: (Comp: 1, Cost: 0) koat_start(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f0(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5)) [ 0 <= 0 ] (Comp: 3, Cost: 1) f10(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f18(Ar_1, Ar_2, Ar_2, 0, Ar_5)) [ Ar_1 >= Ar_2 ] (Comp: 3, Cost: 1) f18(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f32(Ar_1, Ar_2, Ar_3, 0, Ar_5)) [ Ar_4 + 1 >= Ar_3 ] (Comp: ?, Cost: 1) f21(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f18(Ar_1, Ar_2, Ar_3, Ar_4 + 1, Ar_5)) [ Ar_5 + Ar_4 + 1 >= Ar_3 ] (Comp: 3, Cost: 1) f32(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f41(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5)) [ Ar_4 + 1 >= Ar_3 ] (Comp: Ar_2, Cost: 1) f32(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f32(Ar_1, Ar_2, Ar_3, Ar_4 + 1, Ar_5)) [ Ar_3 >= Ar_4 + 2 ] (Comp: ?, Cost: 1) f21(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f21(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5 + 1)) [ Ar_3 >= Ar_4 + Ar_5 + 2 ] (Comp: ?, Cost: 1) f21(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f21(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5 + 1)) [ Ar_3 >= Ar_4 + Ar_5 + 2 ] (Comp: ?, Cost: 1) f18(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f21(Ar_1, Ar_2, Ar_3, Ar_4, 0)) [ Ar_3 >= Ar_4 + 2 ] (Comp: ?, Cost: 1) f10(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f10(Ar_1 + 1, Ar_2, Ar_3, Ar_4, Ar_5)) [ Ar_2 >= Ar_1 + 1 ] (Comp: 1, Cost: 1) f0(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f10(0, Ar_2, Ar_3, Ar_4, Ar_5)) start location: koat_start leaf cost: 0 A polynomial rank function with Pol(koat_start) = V_2 Pol(f0) = V_2 Pol(f10) = -V_1 + V_2 Pol(f18) = -V_1 + V_2 Pol(f32) = -V_1 + V_2 Pol(f21) = -V_1 + V_2 Pol(f41) = -V_1 + V_2 orients all transitions weakly and the transition f10(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f10(Ar_1 + 1, Ar_2, Ar_3, Ar_4, Ar_5)) [ Ar_2 >= Ar_1 + 1 ] strictly and produces the following problem: 6: T: (Comp: 1, Cost: 0) koat_start(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f0(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5)) [ 0 <= 0 ] (Comp: 3, Cost: 1) f10(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f18(Ar_1, Ar_2, Ar_2, 0, Ar_5)) [ Ar_1 >= Ar_2 ] (Comp: 3, Cost: 1) f18(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f32(Ar_1, Ar_2, Ar_3, 0, Ar_5)) [ Ar_4 + 1 >= Ar_3 ] (Comp: ?, Cost: 1) f21(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f18(Ar_1, Ar_2, Ar_3, Ar_4 + 1, Ar_5)) [ Ar_5 + Ar_4 + 1 >= Ar_3 ] (Comp: 3, Cost: 1) f32(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f41(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5)) [ Ar_4 + 1 >= Ar_3 ] (Comp: Ar_2, Cost: 1) f32(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f32(Ar_1, Ar_2, Ar_3, Ar_4 + 1, Ar_5)) [ Ar_3 >= Ar_4 + 2 ] (Comp: ?, Cost: 1) f21(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f21(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5 + 1)) [ Ar_3 >= Ar_4 + Ar_5 + 2 ] (Comp: ?, Cost: 1) f21(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f21(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5 + 1)) [ Ar_3 >= Ar_4 + Ar_5 + 2 ] (Comp: ?, Cost: 1) f18(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f21(Ar_1, Ar_2, Ar_3, Ar_4, 0)) [ Ar_3 >= Ar_4 + 2 ] (Comp: Ar_2, Cost: 1) f10(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f10(Ar_1 + 1, Ar_2, Ar_3, Ar_4, Ar_5)) [ Ar_2 >= Ar_1 + 1 ] (Comp: 1, Cost: 1) f0(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f10(0, Ar_2, Ar_3, Ar_4, Ar_5)) start location: koat_start leaf cost: 0 A polynomial rank function with Pol(f21) = V_3 - V_4 - 1 Pol(f18) = V_3 - V_4 and size complexities S("f0(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f10(0, Ar_2, Ar_3, Ar_4, Ar_5))", 0-0) = 0 S("f0(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f10(0, Ar_2, Ar_3, Ar_4, Ar_5))", 0-1) = Ar_2 S("f0(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f10(0, Ar_2, Ar_3, Ar_4, Ar_5))", 0-2) = Ar_3 S("f0(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f10(0, Ar_2, Ar_3, Ar_4, Ar_5))", 0-3) = Ar_4 S("f0(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f10(0, Ar_2, Ar_3, Ar_4, Ar_5))", 0-4) = Ar_5 S("f10(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f10(Ar_1 + 1, Ar_2, Ar_3, Ar_4, Ar_5)) [ Ar_2 >= Ar_1 + 1 ]", 0-0) = Ar_2 S("f10(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f10(Ar_1 + 1, Ar_2, Ar_3, Ar_4, Ar_5)) [ Ar_2 >= Ar_1 + 1 ]", 0-1) = Ar_2 S("f10(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f10(Ar_1 + 1, Ar_2, Ar_3, Ar_4, Ar_5)) [ Ar_2 >= Ar_1 + 1 ]", 0-2) = Ar_3 S("f10(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f10(Ar_1 + 1, Ar_2, Ar_3, Ar_4, Ar_5)) [ Ar_2 >= Ar_1 + 1 ]", 0-3) = Ar_4 S("f10(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f10(Ar_1 + 1, Ar_2, Ar_3, Ar_4, Ar_5)) [ Ar_2 >= Ar_1 + 1 ]", 0-4) = Ar_5 S("f18(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f21(Ar_1, Ar_2, Ar_3, Ar_4, 0)) [ Ar_3 >= Ar_4 + 2 ]", 0-0) = Ar_2 S("f18(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f21(Ar_1, Ar_2, Ar_3, Ar_4, 0)) [ Ar_3 >= Ar_4 + 2 ]", 0-1) = Ar_2 S("f18(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f21(Ar_1, Ar_2, Ar_3, Ar_4, 0)) [ Ar_3 >= Ar_4 + 2 ]", 0-2) = Ar_2 S("f18(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f21(Ar_1, Ar_2, Ar_3, Ar_4, 0)) [ Ar_3 >= Ar_4 + 2 ]", 0-3) = ? S("f18(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f21(Ar_1, Ar_2, Ar_3, Ar_4, 0)) [ Ar_3 >= Ar_4 + 2 ]", 0-4) = 0 S("f21(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f21(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5 + 1)) [ Ar_3 >= Ar_4 + Ar_5 + 2 ]", 0-0) = Ar_2 S("f21(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f21(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5 + 1)) [ Ar_3 >= Ar_4 + Ar_5 + 2 ]", 0-1) = Ar_2 S("f21(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f21(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5 + 1)) [ Ar_3 >= Ar_4 + Ar_5 + 2 ]", 0-2) = Ar_2 S("f21(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f21(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5 + 1)) [ Ar_3 >= Ar_4 + Ar_5 + 2 ]", 0-3) = ? S("f21(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f21(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5 + 1)) [ Ar_3 >= Ar_4 + Ar_5 + 2 ]", 0-4) = ? S("f21(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f21(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5 + 1)) [ Ar_3 >= Ar_4 + Ar_5 + 2 ]", 0-0) = Ar_2 S("f21(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f21(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5 + 1)) [ Ar_3 >= Ar_4 + Ar_5 + 2 ]", 0-1) = Ar_2 S("f21(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f21(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5 + 1)) [ Ar_3 >= Ar_4 + Ar_5 + 2 ]", 0-2) = Ar_2 S("f21(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f21(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5 + 1)) [ Ar_3 >= Ar_4 + Ar_5 + 2 ]", 0-3) = ? S("f21(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f21(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5 + 1)) [ Ar_3 >= Ar_4 + Ar_5 + 2 ]", 0-4) = ? S("f32(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f32(Ar_1, Ar_2, Ar_3, Ar_4 + 1, Ar_5)) [ Ar_3 >= Ar_4 + 2 ]", 0-0) = Ar_2 S("f32(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f32(Ar_1, Ar_2, Ar_3, Ar_4 + 1, Ar_5)) [ Ar_3 >= Ar_4 + 2 ]", 0-1) = Ar_2 S("f32(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f32(Ar_1, Ar_2, Ar_3, Ar_4 + 1, Ar_5)) [ Ar_3 >= Ar_4 + 2 ]", 0-2) = Ar_2 S("f32(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f32(Ar_1, Ar_2, Ar_3, Ar_4 + 1, Ar_5)) [ Ar_3 >= Ar_4 + 2 ]", 0-3) = Ar_2 S("f32(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f32(Ar_1, Ar_2, Ar_3, Ar_4 + 1, Ar_5)) [ Ar_3 >= Ar_4 + 2 ]", 0-4) = ? S("f32(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f41(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5)) [ Ar_4 + 1 >= Ar_3 ]", 0-0) = Ar_2 S("f32(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f41(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5)) [ Ar_4 + 1 >= Ar_3 ]", 0-1) = Ar_2 S("f32(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f41(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5)) [ Ar_4 + 1 >= Ar_3 ]", 0-2) = Ar_2 S("f32(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f41(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5)) [ Ar_4 + 1 >= Ar_3 ]", 0-3) = Ar_2 S("f32(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f41(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5)) [ Ar_4 + 1 >= Ar_3 ]", 0-4) = ? S("f21(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f18(Ar_1, Ar_2, Ar_3, Ar_4 + 1, Ar_5)) [ Ar_5 + Ar_4 + 1 >= Ar_3 ]", 0-0) = Ar_2 S("f21(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f18(Ar_1, Ar_2, Ar_3, Ar_4 + 1, Ar_5)) [ Ar_5 + Ar_4 + 1 >= Ar_3 ]", 0-1) = Ar_2 S("f21(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f18(Ar_1, Ar_2, Ar_3, Ar_4 + 1, Ar_5)) [ Ar_5 + Ar_4 + 1 >= Ar_3 ]", 0-2) = Ar_2 S("f21(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f18(Ar_1, Ar_2, Ar_3, Ar_4 + 1, Ar_5)) [ Ar_5 + Ar_4 + 1 >= Ar_3 ]", 0-3) = ? S("f21(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f18(Ar_1, Ar_2, Ar_3, Ar_4 + 1, Ar_5)) [ Ar_5 + Ar_4 + 1 >= Ar_3 ]", 0-4) = ? S("f18(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f32(Ar_1, Ar_2, Ar_3, 0, Ar_5)) [ Ar_4 + 1 >= Ar_3 ]", 0-0) = Ar_2 S("f18(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f32(Ar_1, Ar_2, Ar_3, 0, Ar_5)) [ Ar_4 + 1 >= Ar_3 ]", 0-1) = Ar_2 S("f18(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f32(Ar_1, Ar_2, Ar_3, 0, Ar_5)) [ Ar_4 + 1 >= Ar_3 ]", 0-2) = Ar_2 S("f18(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f32(Ar_1, Ar_2, Ar_3, 0, Ar_5)) [ Ar_4 + 1 >= Ar_3 ]", 0-3) = 0 S("f18(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f32(Ar_1, Ar_2, Ar_3, 0, Ar_5)) [ Ar_4 + 1 >= Ar_3 ]", 0-4) = ? S("f10(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f18(Ar_1, Ar_2, Ar_2, 0, Ar_5)) [ Ar_1 >= Ar_2 ]", 0-0) = Ar_2 S("f10(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f18(Ar_1, Ar_2, Ar_2, 0, Ar_5)) [ Ar_1 >= Ar_2 ]", 0-1) = Ar_2 S("f10(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f18(Ar_1, Ar_2, Ar_2, 0, Ar_5)) [ Ar_1 >= Ar_2 ]", 0-2) = Ar_2 S("f10(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f18(Ar_1, Ar_2, Ar_2, 0, Ar_5)) [ Ar_1 >= Ar_2 ]", 0-3) = 0 S("f10(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f18(Ar_1, Ar_2, Ar_2, 0, Ar_5)) [ Ar_1 >= Ar_2 ]", 0-4) = Ar_5 S("koat_start(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f0(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5)) [ 0 <= 0 ]", 0-0) = Ar_1 S("koat_start(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f0(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5)) [ 0 <= 0 ]", 0-1) = Ar_2 S("koat_start(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f0(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5)) [ 0 <= 0 ]", 0-2) = Ar_3 S("koat_start(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f0(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5)) [ 0 <= 0 ]", 0-3) = Ar_4 S("koat_start(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f0(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5)) [ 0 <= 0 ]", 0-4) = Ar_5 orients the transitions f21(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f21(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5 + 1)) [ Ar_3 >= Ar_4 + Ar_5 + 2 ] f21(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f18(Ar_1, Ar_2, Ar_3, Ar_4 + 1, Ar_5)) [ Ar_5 + Ar_4 + 1 >= Ar_3 ] f18(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f21(Ar_1, Ar_2, Ar_3, Ar_4, 0)) [ Ar_3 >= Ar_4 + 2 ] weakly and the transition f18(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f21(Ar_1, Ar_2, Ar_3, Ar_4, 0)) [ Ar_3 >= Ar_4 + 2 ] strictly and produces the following problem: 7: T: (Comp: 1, Cost: 0) koat_start(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f0(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5)) [ 0 <= 0 ] (Comp: 3, Cost: 1) f10(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f18(Ar_1, Ar_2, Ar_2, 0, Ar_5)) [ Ar_1 >= Ar_2 ] (Comp: 3, Cost: 1) f18(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f32(Ar_1, Ar_2, Ar_3, 0, Ar_5)) [ Ar_4 + 1 >= Ar_3 ] (Comp: ?, Cost: 1) f21(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f18(Ar_1, Ar_2, Ar_3, Ar_4 + 1, Ar_5)) [ Ar_5 + Ar_4 + 1 >= Ar_3 ] (Comp: 3, Cost: 1) f32(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f41(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5)) [ Ar_4 + 1 >= Ar_3 ] (Comp: Ar_2, Cost: 1) f32(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f32(Ar_1, Ar_2, Ar_3, Ar_4 + 1, Ar_5)) [ Ar_3 >= Ar_4 + 2 ] (Comp: ?, Cost: 1) f21(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f21(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5 + 1)) [ Ar_3 >= Ar_4 + Ar_5 + 2 ] (Comp: ?, Cost: 1) f21(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f21(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5 + 1)) [ Ar_3 >= Ar_4 + Ar_5 + 2 ] (Comp: 3*Ar_2, Cost: 1) f18(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f21(Ar_1, Ar_2, Ar_3, Ar_4, 0)) [ Ar_3 >= Ar_4 + 2 ] (Comp: Ar_2, Cost: 1) f10(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f10(Ar_1 + 1, Ar_2, Ar_3, Ar_4, Ar_5)) [ Ar_2 >= Ar_1 + 1 ] (Comp: 1, Cost: 1) f0(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f10(0, Ar_2, Ar_3, Ar_4, Ar_5)) start location: koat_start leaf cost: 0 A polynomial rank function with Pol(f21) = 1 Pol(f18) = 0 and size complexities S("f0(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f10(0, Ar_2, Ar_3, Ar_4, Ar_5))", 0-0) = 0 S("f0(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f10(0, Ar_2, Ar_3, Ar_4, Ar_5))", 0-1) = Ar_2 S("f0(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f10(0, Ar_2, Ar_3, Ar_4, Ar_5))", 0-2) = Ar_3 S("f0(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f10(0, Ar_2, Ar_3, Ar_4, Ar_5))", 0-3) = Ar_4 S("f0(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f10(0, Ar_2, Ar_3, Ar_4, Ar_5))", 0-4) = Ar_5 S("f10(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f10(Ar_1 + 1, Ar_2, Ar_3, Ar_4, Ar_5)) [ Ar_2 >= Ar_1 + 1 ]", 0-0) = Ar_2 S("f10(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f10(Ar_1 + 1, Ar_2, Ar_3, Ar_4, Ar_5)) [ Ar_2 >= Ar_1 + 1 ]", 0-1) = Ar_2 S("f10(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f10(Ar_1 + 1, Ar_2, Ar_3, Ar_4, Ar_5)) [ Ar_2 >= Ar_1 + 1 ]", 0-2) = Ar_3 S("f10(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f10(Ar_1 + 1, Ar_2, Ar_3, Ar_4, Ar_5)) [ Ar_2 >= Ar_1 + 1 ]", 0-3) = Ar_4 S("f10(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f10(Ar_1 + 1, Ar_2, Ar_3, Ar_4, Ar_5)) [ Ar_2 >= Ar_1 + 1 ]", 0-4) = Ar_5 S("f18(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f21(Ar_1, Ar_2, Ar_3, Ar_4, 0)) [ Ar_3 >= Ar_4 + 2 ]", 0-0) = Ar_2 S("f18(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f21(Ar_1, Ar_2, Ar_3, Ar_4, 0)) [ Ar_3 >= Ar_4 + 2 ]", 0-1) = Ar_2 S("f18(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f21(Ar_1, Ar_2, Ar_3, Ar_4, 0)) [ Ar_3 >= Ar_4 + 2 ]", 0-2) = Ar_2 S("f18(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f21(Ar_1, Ar_2, Ar_3, Ar_4, 0)) [ Ar_3 >= Ar_4 + 2 ]", 0-3) = ? S("f18(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f21(Ar_1, Ar_2, Ar_3, Ar_4, 0)) [ Ar_3 >= Ar_4 + 2 ]", 0-4) = 0 S("f21(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f21(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5 + 1)) [ Ar_3 >= Ar_4 + Ar_5 + 2 ]", 0-0) = Ar_2 S("f21(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f21(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5 + 1)) [ Ar_3 >= Ar_4 + Ar_5 + 2 ]", 0-1) = Ar_2 S("f21(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f21(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5 + 1)) [ Ar_3 >= Ar_4 + Ar_5 + 2 ]", 0-2) = Ar_2 S("f21(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f21(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5 + 1)) [ Ar_3 >= Ar_4 + Ar_5 + 2 ]", 0-3) = ? S("f21(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f21(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5 + 1)) [ Ar_3 >= Ar_4 + Ar_5 + 2 ]", 0-4) = ? S("f21(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f21(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5 + 1)) [ Ar_3 >= Ar_4 + Ar_5 + 2 ]", 0-0) = Ar_2 S("f21(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f21(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5 + 1)) [ Ar_3 >= Ar_4 + Ar_5 + 2 ]", 0-1) = Ar_2 S("f21(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f21(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5 + 1)) [ Ar_3 >= Ar_4 + Ar_5 + 2 ]", 0-2) = Ar_2 S("f21(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f21(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5 + 1)) [ Ar_3 >= Ar_4 + Ar_5 + 2 ]", 0-3) = ? S("f21(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f21(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5 + 1)) [ Ar_3 >= Ar_4 + Ar_5 + 2 ]", 0-4) = ? S("f32(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f32(Ar_1, Ar_2, Ar_3, Ar_4 + 1, Ar_5)) [ Ar_3 >= Ar_4 + 2 ]", 0-0) = Ar_2 S("f32(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f32(Ar_1, Ar_2, Ar_3, Ar_4 + 1, Ar_5)) [ Ar_3 >= Ar_4 + 2 ]", 0-1) = Ar_2 S("f32(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f32(Ar_1, Ar_2, Ar_3, Ar_4 + 1, Ar_5)) [ Ar_3 >= Ar_4 + 2 ]", 0-2) = Ar_2 S("f32(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f32(Ar_1, Ar_2, Ar_3, Ar_4 + 1, Ar_5)) [ Ar_3 >= Ar_4 + 2 ]", 0-3) = Ar_2 S("f32(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f32(Ar_1, Ar_2, Ar_3, Ar_4 + 1, Ar_5)) [ Ar_3 >= Ar_4 + 2 ]", 0-4) = ? S("f32(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f41(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5)) [ Ar_4 + 1 >= Ar_3 ]", 0-0) = Ar_2 S("f32(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f41(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5)) [ Ar_4 + 1 >= Ar_3 ]", 0-1) = Ar_2 S("f32(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f41(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5)) [ Ar_4 + 1 >= Ar_3 ]", 0-2) = Ar_2 S("f32(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f41(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5)) [ Ar_4 + 1 >= Ar_3 ]", 0-3) = Ar_2 S("f32(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f41(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5)) [ Ar_4 + 1 >= Ar_3 ]", 0-4) = ? S("f21(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f18(Ar_1, Ar_2, Ar_3, Ar_4 + 1, Ar_5)) [ Ar_5 + Ar_4 + 1 >= Ar_3 ]", 0-0) = Ar_2 S("f21(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f18(Ar_1, Ar_2, Ar_3, Ar_4 + 1, Ar_5)) [ Ar_5 + Ar_4 + 1 >= Ar_3 ]", 0-1) = Ar_2 S("f21(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f18(Ar_1, Ar_2, Ar_3, Ar_4 + 1, Ar_5)) [ Ar_5 + Ar_4 + 1 >= Ar_3 ]", 0-2) = Ar_2 S("f21(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f18(Ar_1, Ar_2, Ar_3, Ar_4 + 1, Ar_5)) [ Ar_5 + Ar_4 + 1 >= Ar_3 ]", 0-3) = ? S("f21(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f18(Ar_1, Ar_2, Ar_3, Ar_4 + 1, Ar_5)) [ Ar_5 + Ar_4 + 1 >= Ar_3 ]", 0-4) = ? S("f18(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f32(Ar_1, Ar_2, Ar_3, 0, Ar_5)) [ Ar_4 + 1 >= Ar_3 ]", 0-0) = Ar_2 S("f18(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f32(Ar_1, Ar_2, Ar_3, 0, Ar_5)) [ Ar_4 + 1 >= Ar_3 ]", 0-1) = Ar_2 S("f18(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f32(Ar_1, Ar_2, Ar_3, 0, Ar_5)) [ Ar_4 + 1 >= Ar_3 ]", 0-2) = Ar_2 S("f18(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f32(Ar_1, Ar_2, Ar_3, 0, Ar_5)) [ Ar_4 + 1 >= Ar_3 ]", 0-3) = 0 S("f18(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f32(Ar_1, Ar_2, Ar_3, 0, Ar_5)) [ Ar_4 + 1 >= Ar_3 ]", 0-4) = ? S("f10(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f18(Ar_1, Ar_2, Ar_2, 0, Ar_5)) [ Ar_1 >= Ar_2 ]", 0-0) = Ar_2 S("f10(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f18(Ar_1, Ar_2, Ar_2, 0, Ar_5)) [ Ar_1 >= Ar_2 ]", 0-1) = Ar_2 S("f10(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f18(Ar_1, Ar_2, Ar_2, 0, Ar_5)) [ Ar_1 >= Ar_2 ]", 0-2) = Ar_2 S("f10(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f18(Ar_1, Ar_2, Ar_2, 0, Ar_5)) [ Ar_1 >= Ar_2 ]", 0-3) = 0 S("f10(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f18(Ar_1, Ar_2, Ar_2, 0, Ar_5)) [ Ar_1 >= Ar_2 ]", 0-4) = Ar_5 S("koat_start(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f0(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5)) [ 0 <= 0 ]", 0-0) = Ar_1 S("koat_start(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f0(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5)) [ 0 <= 0 ]", 0-1) = Ar_2 S("koat_start(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f0(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5)) [ 0 <= 0 ]", 0-2) = Ar_3 S("koat_start(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f0(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5)) [ 0 <= 0 ]", 0-3) = Ar_4 S("koat_start(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f0(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5)) [ 0 <= 0 ]", 0-4) = Ar_5 orients the transitions f21(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f21(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5 + 1)) [ Ar_3 >= Ar_4 + Ar_5 + 2 ] f21(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f18(Ar_1, Ar_2, Ar_3, Ar_4 + 1, Ar_5)) [ Ar_5 + Ar_4 + 1 >= Ar_3 ] weakly and the transition f21(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f18(Ar_1, Ar_2, Ar_3, Ar_4 + 1, Ar_5)) [ Ar_5 + Ar_4 + 1 >= Ar_3 ] strictly and produces the following problem: 8: T: (Comp: 1, Cost: 0) koat_start(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f0(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5)) [ 0 <= 0 ] (Comp: 3, Cost: 1) f10(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f18(Ar_1, Ar_2, Ar_2, 0, Ar_5)) [ Ar_1 >= Ar_2 ] (Comp: 3, Cost: 1) f18(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f32(Ar_1, Ar_2, Ar_3, 0, Ar_5)) [ Ar_4 + 1 >= Ar_3 ] (Comp: 3*Ar_2, Cost: 1) f21(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f18(Ar_1, Ar_2, Ar_3, Ar_4 + 1, Ar_5)) [ Ar_5 + Ar_4 + 1 >= Ar_3 ] (Comp: 3, Cost: 1) f32(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f41(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5)) [ Ar_4 + 1 >= Ar_3 ] (Comp: Ar_2, Cost: 1) f32(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f32(Ar_1, Ar_2, Ar_3, Ar_4 + 1, Ar_5)) [ Ar_3 >= Ar_4 + 2 ] (Comp: ?, Cost: 1) f21(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f21(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5 + 1)) [ Ar_3 >= Ar_4 + Ar_5 + 2 ] (Comp: ?, Cost: 1) f21(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f21(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5 + 1)) [ Ar_3 >= Ar_4 + Ar_5 + 2 ] (Comp: 3*Ar_2, Cost: 1) f18(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f21(Ar_1, Ar_2, Ar_3, Ar_4, 0)) [ Ar_3 >= Ar_4 + 2 ] (Comp: Ar_2, Cost: 1) f10(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f10(Ar_1 + 1, Ar_2, Ar_3, Ar_4, Ar_5)) [ Ar_2 >= Ar_1 + 1 ] (Comp: 1, Cost: 1) f0(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f10(0, Ar_2, Ar_3, Ar_4, Ar_5)) start location: koat_start leaf cost: 0 A polynomial rank function with Pol(f21) = V_3 - V_4 - V_5 and size complexities S("f0(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f10(0, Ar_2, Ar_3, Ar_4, Ar_5))", 0-0) = 0 S("f0(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f10(0, Ar_2, Ar_3, Ar_4, Ar_5))", 0-1) = Ar_2 S("f0(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f10(0, Ar_2, Ar_3, Ar_4, Ar_5))", 0-2) = Ar_3 S("f0(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f10(0, Ar_2, Ar_3, Ar_4, Ar_5))", 0-3) = Ar_4 S("f0(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f10(0, Ar_2, Ar_3, Ar_4, Ar_5))", 0-4) = Ar_5 S("f10(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f10(Ar_1 + 1, Ar_2, Ar_3, Ar_4, Ar_5)) [ Ar_2 >= Ar_1 + 1 ]", 0-0) = Ar_2 S("f10(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f10(Ar_1 + 1, Ar_2, Ar_3, Ar_4, Ar_5)) [ Ar_2 >= Ar_1 + 1 ]", 0-1) = Ar_2 S("f10(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f10(Ar_1 + 1, Ar_2, Ar_3, Ar_4, Ar_5)) [ Ar_2 >= Ar_1 + 1 ]", 0-2) = Ar_3 S("f10(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f10(Ar_1 + 1, Ar_2, Ar_3, Ar_4, Ar_5)) [ Ar_2 >= Ar_1 + 1 ]", 0-3) = Ar_4 S("f10(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f10(Ar_1 + 1, Ar_2, Ar_3, Ar_4, Ar_5)) [ Ar_2 >= Ar_1 + 1 ]", 0-4) = Ar_5 S("f18(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f21(Ar_1, Ar_2, Ar_3, Ar_4, 0)) [ Ar_3 >= Ar_4 + 2 ]", 0-0) = Ar_2 S("f18(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f21(Ar_1, Ar_2, Ar_3, Ar_4, 0)) [ Ar_3 >= Ar_4 + 2 ]", 0-1) = Ar_2 S("f18(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f21(Ar_1, Ar_2, Ar_3, Ar_4, 0)) [ Ar_3 >= Ar_4 + 2 ]", 0-2) = Ar_2 S("f18(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f21(Ar_1, Ar_2, Ar_3, Ar_4, 0)) [ Ar_3 >= Ar_4 + 2 ]", 0-3) = 3*Ar_2 S("f18(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f21(Ar_1, Ar_2, Ar_3, Ar_4, 0)) [ Ar_3 >= Ar_4 + 2 ]", 0-4) = 0 S("f21(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f21(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5 + 1)) [ Ar_3 >= Ar_4 + Ar_5 + 2 ]", 0-0) = Ar_2 S("f21(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f21(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5 + 1)) [ Ar_3 >= Ar_4 + Ar_5 + 2 ]", 0-1) = Ar_2 S("f21(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f21(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5 + 1)) [ Ar_3 >= Ar_4 + Ar_5 + 2 ]", 0-2) = Ar_2 S("f21(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f21(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5 + 1)) [ Ar_3 >= Ar_4 + Ar_5 + 2 ]", 0-3) = 3*Ar_2 S("f21(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f21(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5 + 1)) [ Ar_3 >= Ar_4 + Ar_5 + 2 ]", 0-4) = ? S("f21(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f21(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5 + 1)) [ Ar_3 >= Ar_4 + Ar_5 + 2 ]", 0-0) = Ar_2 S("f21(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f21(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5 + 1)) [ Ar_3 >= Ar_4 + Ar_5 + 2 ]", 0-1) = Ar_2 S("f21(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f21(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5 + 1)) [ Ar_3 >= Ar_4 + Ar_5 + 2 ]", 0-2) = Ar_2 S("f21(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f21(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5 + 1)) [ Ar_3 >= Ar_4 + Ar_5 + 2 ]", 0-3) = 3*Ar_2 S("f21(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f21(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5 + 1)) [ Ar_3 >= Ar_4 + Ar_5 + 2 ]", 0-4) = ? S("f32(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f32(Ar_1, Ar_2, Ar_3, Ar_4 + 1, Ar_5)) [ Ar_3 >= Ar_4 + 2 ]", 0-0) = Ar_2 S("f32(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f32(Ar_1, Ar_2, Ar_3, Ar_4 + 1, Ar_5)) [ Ar_3 >= Ar_4 + 2 ]", 0-1) = Ar_2 S("f32(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f32(Ar_1, Ar_2, Ar_3, Ar_4 + 1, Ar_5)) [ Ar_3 >= Ar_4 + 2 ]", 0-2) = Ar_2 S("f32(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f32(Ar_1, Ar_2, Ar_3, Ar_4 + 1, Ar_5)) [ Ar_3 >= Ar_4 + 2 ]", 0-3) = Ar_2 S("f32(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f32(Ar_1, Ar_2, Ar_3, Ar_4 + 1, Ar_5)) [ Ar_3 >= Ar_4 + 2 ]", 0-4) = ? S("f32(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f41(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5)) [ Ar_4 + 1 >= Ar_3 ]", 0-0) = Ar_2 S("f32(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f41(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5)) [ Ar_4 + 1 >= Ar_3 ]", 0-1) = Ar_2 S("f32(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f41(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5)) [ Ar_4 + 1 >= Ar_3 ]", 0-2) = Ar_2 S("f32(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f41(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5)) [ Ar_4 + 1 >= Ar_3 ]", 0-3) = Ar_2 S("f32(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f41(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5)) [ Ar_4 + 1 >= Ar_3 ]", 0-4) = ? S("f21(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f18(Ar_1, Ar_2, Ar_3, Ar_4 + 1, Ar_5)) [ Ar_5 + Ar_4 + 1 >= Ar_3 ]", 0-0) = Ar_2 S("f21(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f18(Ar_1, Ar_2, Ar_3, Ar_4 + 1, Ar_5)) [ Ar_5 + Ar_4 + 1 >= Ar_3 ]", 0-1) = Ar_2 S("f21(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f18(Ar_1, Ar_2, Ar_3, Ar_4 + 1, Ar_5)) [ Ar_5 + Ar_4 + 1 >= Ar_3 ]", 0-2) = Ar_2 S("f21(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f18(Ar_1, Ar_2, Ar_3, Ar_4 + 1, Ar_5)) [ Ar_5 + Ar_4 + 1 >= Ar_3 ]", 0-3) = 3*Ar_2 S("f21(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f18(Ar_1, Ar_2, Ar_3, Ar_4 + 1, Ar_5)) [ Ar_5 + Ar_4 + 1 >= Ar_3 ]", 0-4) = ? S("f18(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f32(Ar_1, Ar_2, Ar_3, 0, Ar_5)) [ Ar_4 + 1 >= Ar_3 ]", 0-0) = Ar_2 S("f18(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f32(Ar_1, Ar_2, Ar_3, 0, Ar_5)) [ Ar_4 + 1 >= Ar_3 ]", 0-1) = Ar_2 S("f18(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f32(Ar_1, Ar_2, Ar_3, 0, Ar_5)) [ Ar_4 + 1 >= Ar_3 ]", 0-2) = Ar_2 S("f18(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f32(Ar_1, Ar_2, Ar_3, 0, Ar_5)) [ Ar_4 + 1 >= Ar_3 ]", 0-3) = 0 S("f18(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f32(Ar_1, Ar_2, Ar_3, 0, Ar_5)) [ Ar_4 + 1 >= Ar_3 ]", 0-4) = ? S("f10(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f18(Ar_1, Ar_2, Ar_2, 0, Ar_5)) [ Ar_1 >= Ar_2 ]", 0-0) = Ar_2 S("f10(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f18(Ar_1, Ar_2, Ar_2, 0, Ar_5)) [ Ar_1 >= Ar_2 ]", 0-1) = Ar_2 S("f10(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f18(Ar_1, Ar_2, Ar_2, 0, Ar_5)) [ Ar_1 >= Ar_2 ]", 0-2) = Ar_2 S("f10(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f18(Ar_1, Ar_2, Ar_2, 0, Ar_5)) [ Ar_1 >= Ar_2 ]", 0-3) = 0 S("f10(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f18(Ar_1, Ar_2, Ar_2, 0, Ar_5)) [ Ar_1 >= Ar_2 ]", 0-4) = Ar_5 S("koat_start(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f0(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5)) [ 0 <= 0 ]", 0-0) = Ar_1 S("koat_start(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f0(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5)) [ 0 <= 0 ]", 0-1) = Ar_2 S("koat_start(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f0(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5)) [ 0 <= 0 ]", 0-2) = Ar_3 S("koat_start(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f0(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5)) [ 0 <= 0 ]", 0-3) = Ar_4 S("koat_start(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f0(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5)) [ 0 <= 0 ]", 0-4) = Ar_5 orients the transitions f21(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f21(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5 + 1)) [ Ar_3 >= Ar_4 + Ar_5 + 2 ] weakly and the transition f21(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f21(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5 + 1)) [ Ar_3 >= Ar_4 + Ar_5 + 2 ] strictly and produces the following problem: 9: T: (Comp: 1, Cost: 0) koat_start(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f0(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5)) [ 0 <= 0 ] (Comp: 3, Cost: 1) f10(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f18(Ar_1, Ar_2, Ar_2, 0, Ar_5)) [ Ar_1 >= Ar_2 ] (Comp: 3, Cost: 1) f18(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f32(Ar_1, Ar_2, Ar_3, 0, Ar_5)) [ Ar_4 + 1 >= Ar_3 ] (Comp: 3*Ar_2, Cost: 1) f21(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f18(Ar_1, Ar_2, Ar_3, Ar_4 + 1, Ar_5)) [ Ar_5 + Ar_4 + 1 >= Ar_3 ] (Comp: 3, Cost: 1) f32(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f41(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5)) [ Ar_4 + 1 >= Ar_3 ] (Comp: Ar_2, Cost: 1) f32(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f32(Ar_1, Ar_2, Ar_3, Ar_4 + 1, Ar_5)) [ Ar_3 >= Ar_4 + 2 ] (Comp: 12*Ar_2^2, Cost: 1) f21(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f21(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5 + 1)) [ Ar_3 >= Ar_4 + Ar_5 + 2 ] (Comp: 12*Ar_2^2, Cost: 1) f21(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f21(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5 + 1)) [ Ar_3 >= Ar_4 + Ar_5 + 2 ] (Comp: 3*Ar_2, Cost: 1) f18(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f21(Ar_1, Ar_2, Ar_3, Ar_4, 0)) [ Ar_3 >= Ar_4 + 2 ] (Comp: Ar_2, Cost: 1) f10(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f10(Ar_1 + 1, Ar_2, Ar_3, Ar_4, Ar_5)) [ Ar_2 >= Ar_1 + 1 ] (Comp: 1, Cost: 1) f0(Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f10(0, Ar_2, Ar_3, Ar_4, Ar_5)) start location: koat_start leaf cost: 0 Complexity upper bound 8*Ar_2 + 24*Ar_2^2 + 10 Time: 0.168 sec (SMT: 0.121 sec) ---------------------------------------- (2) BOUNDS(1, n^2) ---------------------------------------- (3) Loat Proof (FINISHED) ### Pre-processing the ITS problem ### Initial linear ITS problem Start location: f0 0: f0 -> f10 : A'=free, B'=0, [], cost: 1 1: f10 -> f10 : B'=1+B, [ C>=1+B ], cost: 1 9: f10 -> f18 : D'=C, E'=0, [ B>=C ], cost: 1 2: f18 -> f21 : F'=0, [ D>=2+E ], cost: 1 8: f18 -> f32 : E'=0, [ 1+E>=D ], cost: 1 3: f21 -> f21 : F'=1+F, [ D>=2+F+E ], cost: 1 4: f21 -> f21 : F'=1+F, G'=free_1, [ D>=2+F+E ], cost: 1 7: f21 -> f18 : E'=1+E, [ 1+F+E>=D ], cost: 1 5: f32 -> f32 : E'=1+E, [ D>=2+E ], cost: 1 6: f32 -> f41 : [ 1+E>=D ], cost: 1 Checking for constant complexity: The following rule is satisfiable with cost >= 1, yielding constant complexity: 0: f0 -> f10 : A'=free, B'=0, [], cost: 1 Removed unreachable and leaf rules: Start location: f0 0: f0 -> f10 : A'=free, B'=0, [], cost: 1 1: f10 -> f10 : B'=1+B, [ C>=1+B ], cost: 1 9: f10 -> f18 : D'=C, E'=0, [ B>=C ], cost: 1 2: f18 -> f21 : F'=0, [ D>=2+E ], cost: 1 8: f18 -> f32 : E'=0, [ 1+E>=D ], cost: 1 3: f21 -> f21 : F'=1+F, [ D>=2+F+E ], cost: 1 4: f21 -> f21 : F'=1+F, G'=free_1, [ D>=2+F+E ], cost: 1 7: f21 -> f18 : E'=1+E, [ 1+F+E>=D ], cost: 1 5: f32 -> f32 : E'=1+E, [ D>=2+E ], cost: 1 ### Simplification by acceleration and chaining ### Accelerating simple loops of location 1. Accelerating the following rules: 1: f10 -> f10 : B'=1+B, [ C>=1+B ], cost: 1 Accelerated rule 1 with metering function C-B, yielding the new rule 10. Removing the simple loops: 1. Accelerating simple loops of location 3. Accelerating the following rules: 3: f21 -> f21 : F'=1+F, [ D>=2+F+E ], cost: 1 4: f21 -> f21 : F'=1+F, G'=free_1, [ D>=2+F+E ], cost: 1 Accelerated rule 3 with metering function -1-F+D-E, yielding the new rule 11. Accelerated rule 4 with metering function -1-F+D-E, yielding the new rule 12. Removing the simple loops: 3 4. Accelerating simple loops of location 4. Accelerating the following rules: 5: f32 -> f32 : E'=1+E, [ D>=2+E ], cost: 1 Accelerated rule 5 with metering function -1+D-E, yielding the new rule 13. Removing the simple loops: 5. Accelerated all simple loops using metering functions (where possible): Start location: f0 0: f0 -> f10 : A'=free, B'=0, [], cost: 1 9: f10 -> f18 : D'=C, E'=0, [ B>=C ], cost: 1 10: f10 -> f10 : B'=C, [ C>=1+B ], cost: C-B 2: f18 -> f21 : F'=0, [ D>=2+E ], cost: 1 8: f18 -> f32 : E'=0, [ 1+E>=D ], cost: 1 7: f21 -> f18 : E'=1+E, [ 1+F+E>=D ], cost: 1 11: f21 -> f21 : F'=-1+D-E, [ D>=2+F+E ], cost: -1-F+D-E 12: f21 -> f21 : F'=-1+D-E, G'=free_1, [ D>=2+F+E ], cost: -1-F+D-E 13: f32 -> f32 : E'=-1+D, [ D>=2+E ], cost: -1+D-E Chained accelerated rules (with incoming rules): Start location: f0 0: f0 -> f10 : A'=free, B'=0, [], cost: 1 14: f0 -> f10 : A'=free, B'=C, [ C>=1 ], cost: 1+C 9: f10 -> f18 : D'=C, E'=0, [ B>=C ], cost: 1 2: f18 -> f21 : F'=0, [ D>=2+E ], cost: 1 8: f18 -> f32 : E'=0, [ 1+E>=D ], cost: 1 15: f18 -> f21 : F'=-1+D-E, [ D>=2+E ], cost: D-E 16: f18 -> f21 : F'=-1+D-E, G'=free_1, [ D>=2+E ], cost: D-E 17: f18 -> f32 : E'=-1+D, [ 1+E>=D && D>=2 ], cost: D 7: f21 -> f18 : E'=1+E, [ 1+F+E>=D ], cost: 1 Removed unreachable locations (and leaf rules with constant cost): Start location: f0 0: f0 -> f10 : A'=free, B'=0, [], cost: 1 14: f0 -> f10 : A'=free, B'=C, [ C>=1 ], cost: 1+C 9: f10 -> f18 : D'=C, E'=0, [ B>=C ], cost: 1 2: f18 -> f21 : F'=0, [ D>=2+E ], cost: 1 15: f18 -> f21 : F'=-1+D-E, [ D>=2+E ], cost: D-E 16: f18 -> f21 : F'=-1+D-E, G'=free_1, [ D>=2+E ], cost: D-E 17: f18 -> f32 : E'=-1+D, [ 1+E>=D && D>=2 ], cost: D 7: f21 -> f18 : E'=1+E, [ 1+F+E>=D ], cost: 1 Eliminated locations (on tree-shaped paths): Start location: f0 18: f0 -> f18 : A'=free, B'=0, D'=C, E'=0, [ 0>=C ], cost: 2 19: f0 -> f18 : A'=free, B'=C, D'=C, E'=0, [ C>=1 ], cost: 2+C 17: f18 -> f32 : E'=-1+D, [ 1+E>=D && D>=2 ], cost: D 20: f18 -> f18 : E'=1+E, F'=-1+D-E, [ D>=2+E ], cost: 1+D-E 21: f18 -> f18 : E'=1+E, F'=-1+D-E, G'=free_1, [ D>=2+E ], cost: 1+D-E Accelerating simple loops of location 2. Accelerating the following rules: 20: f18 -> f18 : E'=1+E, F'=-1+D-E, [ D>=2+E ], cost: 1+D-E 21: f18 -> f18 : E'=1+E, F'=-1+D-E, G'=free_1, [ D>=2+E ], cost: 1+D-E Accelerated rule 20 with metering function -1+D-E, yielding the new rule 22. Accelerated rule 21 with metering function -1+D-E, yielding the new rule 23. Removing the simple loops: 20 21. Accelerated all simple loops using metering functions (where possible): Start location: f0 18: f0 -> f18 : A'=free, B'=0, D'=C, E'=0, [ 0>=C ], cost: 2 19: f0 -> f18 : A'=free, B'=C, D'=C, E'=0, [ C>=1 ], cost: 2+C 17: f18 -> f32 : E'=-1+D, [ 1+E>=D && D>=2 ], cost: D 22: f18 -> f18 : E'=-1+D, F'=1, [ D>=2+E ], cost: -3/2+D*(-1+D-E)-E*(-1+D-E)+3/2*D-1/2*(-1+D-E)^2-3/2*E 23: f18 -> f18 : E'=-1+D, F'=1, G'=free_1, [ D>=2+E ], cost: -3/2+D*(-1+D-E)-E*(-1+D-E)+3/2*D-1/2*(-1+D-E)^2-3/2*E Chained accelerated rules (with incoming rules): Start location: f0 18: f0 -> f18 : A'=free, B'=0, D'=C, E'=0, [ 0>=C ], cost: 2 19: f0 -> f18 : A'=free, B'=C, D'=C, E'=0, [ C>=1 ], cost: 2+C 24: f0 -> f18 : A'=free, B'=C, D'=C, E'=-1+C, F'=1, [ C>=2 ], cost: 1/2+5/2*C-1/2*(-1+C)^2+C*(-1+C) 25: f0 -> f18 : A'=free, B'=C, D'=C, E'=-1+C, F'=1, G'=free_1, [ C>=2 ], cost: 1/2+5/2*C-1/2*(-1+C)^2+C*(-1+C) 17: f18 -> f32 : E'=-1+D, [ 1+E>=D && D>=2 ], cost: D Eliminated locations (on tree-shaped paths): Start location: f0 26: f0 -> f32 : A'=free, B'=C, D'=C, E'=-1+C, F'=1, [ C>=2 ], cost: 1/2+7/2*C-1/2*(-1+C)^2+C*(-1+C) 27: f0 -> f32 : A'=free, B'=C, D'=C, E'=-1+C, F'=1, G'=free_1, [ C>=2 ], cost: 1/2+7/2*C-1/2*(-1+C)^2+C*(-1+C) 28: f0 -> [10] : [ C>=1 ], cost: 2+C ### Computing asymptotic complexity ### Fully simplified ITS problem Start location: f0 27: f0 -> f32 : A'=free, B'=C, D'=C, E'=-1+C, F'=1, G'=free_1, [ C>=2 ], cost: 1/2+7/2*C-1/2*(-1+C)^2+C*(-1+C) 28: f0 -> [10] : [ C>=1 ], cost: 2+C Computing asymptotic complexity for rule 27 Solved the limit problem by the following transformations: Created initial limit problem: 7/2*C+1/2*C^2 (+), -1+C (+/+!) [not solved] removing all constraints (solved by SMT) resulting limit problem: [solved] applying transformation rule (C) using substitution {C==n} resulting limit problem: [solved] Solution: C / n Resulting cost 1/2*n^2+7/2*n has complexity: Poly(n^2) Found new complexity Poly(n^2). Obtained the following overall complexity (w.r.t. the length of the input n): Complexity: Poly(n^2) Cpx degree: 2 Solved cost: 1/2*n^2+7/2*n Rule cost: 1/2+7/2*C-1/2*(-1+C)^2+C*(-1+C) Rule guard: [ C>=2 ] WORST_CASE(Omega(n^2),?) ---------------------------------------- (4) BOUNDS(n^2, INF)