/export/starexec/sandbox2/solver/bin/starexec_run_complexity /export/starexec/sandbox2/benchmark/theBenchmark.koat /export/starexec/sandbox2/output/output_files -------------------------------------------------------------------------------- WORST_CASE(Omega(n^2), O(n^2)) proof of /export/starexec/sandbox2/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(n^2, n^2).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, 200 ms] (2) BOUNDS(1, n^2) (3) Loat Proof [FINISHED, 595 ms] (4) BOUNDS(n^2, INF) ---------------------------------------- (0) Obligation: Complexity Int TRS consisting of the following rules: start(A, B, C, D, E, F) -> Com_1(stop(A, B, C, F, E, F)) :|: 0 >= A && B >= C && B <= C && D >= E && D <= E && F >= A && F <= A start(A, B, C, D, E, F) -> Com_1(lbl62(A, F - 1, C, F, E, F)) :|: A >= 1 && B >= C && B <= C && D >= E && D <= E && F >= A && F <= A lbl72(A, B, C, D, E, F) -> Com_1(stop(A, B, C, D, E, F)) :|: A >= 1 && D >= 0 && D <= 0 && B >= 0 && B <= 0 && F >= A && F <= A lbl72(A, B, C, D, E, F) -> Com_1(lbl72(A, F, C, D - 1, E, F)) :|: D >= 1 && 0 >= A && D >= 0 && A >= D + 1 && B >= 0 && B <= 0 && F >= A && F <= A lbl72(A, B, C, D, E, F) -> Com_1(lbl62(A, F - 1, C, D, E, F)) :|: A >= 1 && D >= 1 && D >= 0 && A >= D + 1 && B >= 0 && B <= 0 && F >= A && F <= A lbl62(A, B, C, D, E, F) -> Com_1(lbl72(A, B, C, D - 1, E, F)) :|: A >= D && A >= 1 && D >= 1 && B >= 0 && B <= 0 && F >= A && F <= A lbl62(A, B, C, D, E, F) -> Com_1(lbl62(A, B - 1, C, D, E, F)) :|: B >= 1 && A >= D && A >= B + 1 && B >= 0 && D >= 1 && F >= A && F <= A start0(A, B, C, D, E, F) -> Com_1(start(A, C, C, E, E, A)) :|: TRUE The start-symbols are:[start0_6] ---------------------------------------- (1) Koat Proof (FINISHED) YES(?, 7*Ar_0 + 2*Ar_0^2 + 5) Initial complexity problem: 1: T: (Comp: ?, Cost: 1) start(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(stop(Ar_0, Ar_1, Ar_2, Ar_5, Ar_4, Ar_5)) [ 0 >= Ar_0 /\ Ar_1 = Ar_2 /\ Ar_3 = Ar_4 /\ Ar_5 = Ar_0 ] (Comp: ?, Cost: 1) start(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(lbl62(Ar_0, Ar_5 - 1, Ar_2, Ar_5, Ar_4, Ar_5)) [ Ar_0 >= 1 /\ Ar_1 = Ar_2 /\ Ar_3 = Ar_4 /\ Ar_5 = Ar_0 ] (Comp: ?, Cost: 1) lbl72(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(stop(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5)) [ Ar_0 >= 1 /\ Ar_3 = 0 /\ Ar_1 = 0 /\ Ar_5 = Ar_0 ] (Comp: ?, Cost: 1) lbl72(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(lbl72(Ar_0, Ar_5, Ar_2, Ar_3 - 1, Ar_4, Ar_5)) [ Ar_3 >= 1 /\ 0 >= Ar_0 /\ Ar_3 >= 0 /\ Ar_0 >= Ar_3 + 1 /\ Ar_1 = 0 /\ Ar_5 = Ar_0 ] (Comp: ?, Cost: 1) lbl72(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(lbl62(Ar_0, Ar_5 - 1, Ar_2, Ar_3, Ar_4, Ar_5)) [ Ar_0 >= 1 /\ Ar_3 >= 1 /\ Ar_3 >= 0 /\ Ar_0 >= Ar_3 + 1 /\ Ar_1 = 0 /\ Ar_5 = Ar_0 ] (Comp: ?, Cost: 1) lbl62(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(lbl72(Ar_0, Ar_1, Ar_2, Ar_3 - 1, Ar_4, Ar_5)) [ Ar_0 >= Ar_3 /\ Ar_0 >= 1 /\ Ar_3 >= 1 /\ Ar_1 = 0 /\ Ar_5 = Ar_0 ] (Comp: ?, Cost: 1) lbl62(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(lbl62(Ar_0, Ar_1 - 1, Ar_2, Ar_3, Ar_4, Ar_5)) [ Ar_1 >= 1 /\ Ar_0 >= Ar_3 /\ Ar_0 >= Ar_1 + 1 /\ Ar_1 >= 0 /\ Ar_3 >= 1 /\ Ar_5 = Ar_0 ] (Comp: ?, Cost: 1) start0(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(start(Ar_0, Ar_2, Ar_2, Ar_4, Ar_4, Ar_0)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(start0(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 Testing for reachability in the complexity graph removes the following transition from problem 1: lbl72(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(lbl72(Ar_0, Ar_5, Ar_2, Ar_3 - 1, Ar_4, Ar_5)) [ Ar_3 >= 1 /\ 0 >= Ar_0 /\ Ar_3 >= 0 /\ Ar_0 >= Ar_3 + 1 /\ Ar_1 = 0 /\ Ar_5 = Ar_0 ] We thus obtain the following problem: 2: T: (Comp: ?, Cost: 1) lbl72(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(lbl62(Ar_0, Ar_5 - 1, Ar_2, Ar_3, Ar_4, Ar_5)) [ Ar_0 >= 1 /\ Ar_3 >= 1 /\ Ar_3 >= 0 /\ Ar_0 >= Ar_3 + 1 /\ Ar_1 = 0 /\ Ar_5 = Ar_0 ] (Comp: ?, Cost: 1) lbl72(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(stop(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5)) [ Ar_0 >= 1 /\ Ar_3 = 0 /\ Ar_1 = 0 /\ Ar_5 = Ar_0 ] (Comp: ?, Cost: 1) lbl62(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(lbl62(Ar_0, Ar_1 - 1, Ar_2, Ar_3, Ar_4, Ar_5)) [ Ar_1 >= 1 /\ Ar_0 >= Ar_3 /\ Ar_0 >= Ar_1 + 1 /\ Ar_1 >= 0 /\ Ar_3 >= 1 /\ Ar_5 = Ar_0 ] (Comp: ?, Cost: 1) lbl62(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(lbl72(Ar_0, Ar_1, Ar_2, Ar_3 - 1, Ar_4, Ar_5)) [ Ar_0 >= Ar_3 /\ Ar_0 >= 1 /\ Ar_3 >= 1 /\ Ar_1 = 0 /\ Ar_5 = Ar_0 ] (Comp: ?, Cost: 1) start(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(lbl62(Ar_0, Ar_5 - 1, Ar_2, Ar_5, Ar_4, Ar_5)) [ Ar_0 >= 1 /\ Ar_1 = Ar_2 /\ Ar_3 = Ar_4 /\ Ar_5 = Ar_0 ] (Comp: ?, Cost: 1) start(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(stop(Ar_0, Ar_1, Ar_2, Ar_5, Ar_4, Ar_5)) [ 0 >= Ar_0 /\ Ar_1 = Ar_2 /\ Ar_3 = Ar_4 /\ Ar_5 = Ar_0 ] (Comp: ?, Cost: 1) start0(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(start(Ar_0, Ar_2, Ar_2, Ar_4, Ar_4, Ar_0)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(start0(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 Repeatedly propagating knowledge in problem 2 produces the following problem: 3: T: (Comp: ?, Cost: 1) lbl72(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(lbl62(Ar_0, Ar_5 - 1, Ar_2, Ar_3, Ar_4, Ar_5)) [ Ar_0 >= 1 /\ Ar_3 >= 1 /\ Ar_3 >= 0 /\ Ar_0 >= Ar_3 + 1 /\ Ar_1 = 0 /\ Ar_5 = Ar_0 ] (Comp: ?, Cost: 1) lbl72(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(stop(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5)) [ Ar_0 >= 1 /\ Ar_3 = 0 /\ Ar_1 = 0 /\ Ar_5 = Ar_0 ] (Comp: ?, Cost: 1) lbl62(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(lbl62(Ar_0, Ar_1 - 1, Ar_2, Ar_3, Ar_4, Ar_5)) [ Ar_1 >= 1 /\ Ar_0 >= Ar_3 /\ Ar_0 >= Ar_1 + 1 /\ Ar_1 >= 0 /\ Ar_3 >= 1 /\ Ar_5 = Ar_0 ] (Comp: ?, Cost: 1) lbl62(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(lbl72(Ar_0, Ar_1, Ar_2, Ar_3 - 1, Ar_4, Ar_5)) [ Ar_0 >= Ar_3 /\ Ar_0 >= 1 /\ Ar_3 >= 1 /\ Ar_1 = 0 /\ Ar_5 = Ar_0 ] (Comp: 1, Cost: 1) start(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(lbl62(Ar_0, Ar_5 - 1, Ar_2, Ar_5, Ar_4, Ar_5)) [ Ar_0 >= 1 /\ Ar_1 = Ar_2 /\ Ar_3 = Ar_4 /\ Ar_5 = Ar_0 ] (Comp: 1, Cost: 1) start(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(stop(Ar_0, Ar_1, Ar_2, Ar_5, Ar_4, Ar_5)) [ 0 >= Ar_0 /\ Ar_1 = Ar_2 /\ Ar_3 = Ar_4 /\ Ar_5 = Ar_0 ] (Comp: 1, Cost: 1) start0(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(start(Ar_0, Ar_2, Ar_2, Ar_4, Ar_4, Ar_0)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(start0(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 A polynomial rank function with Pol(lbl72) = 1 Pol(lbl62) = 1 Pol(stop) = 0 Pol(start) = 1 Pol(start0) = 1 Pol(koat_start) = 1 orients all transitions weakly and the transition lbl72(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(stop(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5)) [ Ar_0 >= 1 /\ Ar_3 = 0 /\ Ar_1 = 0 /\ Ar_5 = Ar_0 ] strictly and produces the following problem: 4: T: (Comp: ?, Cost: 1) lbl72(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(lbl62(Ar_0, Ar_5 - 1, Ar_2, Ar_3, Ar_4, Ar_5)) [ Ar_0 >= 1 /\ Ar_3 >= 1 /\ Ar_3 >= 0 /\ Ar_0 >= Ar_3 + 1 /\ Ar_1 = 0 /\ Ar_5 = Ar_0 ] (Comp: 1, Cost: 1) lbl72(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(stop(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5)) [ Ar_0 >= 1 /\ Ar_3 = 0 /\ Ar_1 = 0 /\ Ar_5 = Ar_0 ] (Comp: ?, Cost: 1) lbl62(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(lbl62(Ar_0, Ar_1 - 1, Ar_2, Ar_3, Ar_4, Ar_5)) [ Ar_1 >= 1 /\ Ar_0 >= Ar_3 /\ Ar_0 >= Ar_1 + 1 /\ Ar_1 >= 0 /\ Ar_3 >= 1 /\ Ar_5 = Ar_0 ] (Comp: ?, Cost: 1) lbl62(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(lbl72(Ar_0, Ar_1, Ar_2, Ar_3 - 1, Ar_4, Ar_5)) [ Ar_0 >= Ar_3 /\ Ar_0 >= 1 /\ Ar_3 >= 1 /\ Ar_1 = 0 /\ Ar_5 = Ar_0 ] (Comp: 1, Cost: 1) start(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(lbl62(Ar_0, Ar_5 - 1, Ar_2, Ar_5, Ar_4, Ar_5)) [ Ar_0 >= 1 /\ Ar_1 = Ar_2 /\ Ar_3 = Ar_4 /\ Ar_5 = Ar_0 ] (Comp: 1, Cost: 1) start(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(stop(Ar_0, Ar_1, Ar_2, Ar_5, Ar_4, Ar_5)) [ 0 >= Ar_0 /\ Ar_1 = Ar_2 /\ Ar_3 = Ar_4 /\ Ar_5 = Ar_0 ] (Comp: 1, Cost: 1) start0(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(start(Ar_0, Ar_2, Ar_2, Ar_4, Ar_4, Ar_0)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(start0(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 A polynomial rank function with Pol(lbl72) = 2*V_4 + 1 Pol(lbl62) = 2*V_4 Pol(stop) = 2*V_1 + 2*V_4 - 2*V_6 Pol(start) = 2*V_1 Pol(start0) = 2*V_1 Pol(koat_start) = 2*V_1 orients all transitions weakly and the transitions lbl72(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(lbl62(Ar_0, Ar_5 - 1, Ar_2, Ar_3, Ar_4, Ar_5)) [ Ar_0 >= 1 /\ Ar_3 >= 1 /\ Ar_3 >= 0 /\ Ar_0 >= Ar_3 + 1 /\ Ar_1 = 0 /\ Ar_5 = Ar_0 ] lbl62(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(lbl72(Ar_0, Ar_1, Ar_2, Ar_3 - 1, Ar_4, Ar_5)) [ Ar_0 >= Ar_3 /\ Ar_0 >= 1 /\ Ar_3 >= 1 /\ Ar_1 = 0 /\ Ar_5 = Ar_0 ] strictly and produces the following problem: 5: T: (Comp: 2*Ar_0, Cost: 1) lbl72(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(lbl62(Ar_0, Ar_5 - 1, Ar_2, Ar_3, Ar_4, Ar_5)) [ Ar_0 >= 1 /\ Ar_3 >= 1 /\ Ar_3 >= 0 /\ Ar_0 >= Ar_3 + 1 /\ Ar_1 = 0 /\ Ar_5 = Ar_0 ] (Comp: 1, Cost: 1) lbl72(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(stop(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5)) [ Ar_0 >= 1 /\ Ar_3 = 0 /\ Ar_1 = 0 /\ Ar_5 = Ar_0 ] (Comp: ?, Cost: 1) lbl62(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(lbl62(Ar_0, Ar_1 - 1, Ar_2, Ar_3, Ar_4, Ar_5)) [ Ar_1 >= 1 /\ Ar_0 >= Ar_3 /\ Ar_0 >= Ar_1 + 1 /\ Ar_1 >= 0 /\ Ar_3 >= 1 /\ Ar_5 = Ar_0 ] (Comp: 2*Ar_0, Cost: 1) lbl62(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(lbl72(Ar_0, Ar_1, Ar_2, Ar_3 - 1, Ar_4, Ar_5)) [ Ar_0 >= Ar_3 /\ Ar_0 >= 1 /\ Ar_3 >= 1 /\ Ar_1 = 0 /\ Ar_5 = Ar_0 ] (Comp: 1, Cost: 1) start(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(lbl62(Ar_0, Ar_5 - 1, Ar_2, Ar_5, Ar_4, Ar_5)) [ Ar_0 >= 1 /\ Ar_1 = Ar_2 /\ Ar_3 = Ar_4 /\ Ar_5 = Ar_0 ] (Comp: 1, Cost: 1) start(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(stop(Ar_0, Ar_1, Ar_2, Ar_5, Ar_4, Ar_5)) [ 0 >= Ar_0 /\ Ar_1 = Ar_2 /\ Ar_3 = Ar_4 /\ Ar_5 = Ar_0 ] (Comp: 1, Cost: 1) start0(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(start(Ar_0, Ar_2, Ar_2, Ar_4, Ar_4, Ar_0)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(start0(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 A polynomial rank function with Pol(lbl62) = V_2 + 1 and size complexities S("koat_start(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(start0(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5)) [ 0 <= 0 ]", 0-0) = Ar_0 S("koat_start(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(start0(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5)) [ 0 <= 0 ]", 0-1) = Ar_1 S("koat_start(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(start0(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5)) [ 0 <= 0 ]", 0-2) = Ar_2 S("koat_start(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(start0(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5)) [ 0 <= 0 ]", 0-3) = Ar_3 S("koat_start(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(start0(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5)) [ 0 <= 0 ]", 0-4) = Ar_4 S("koat_start(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(start0(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5)) [ 0 <= 0 ]", 0-5) = Ar_5 S("start0(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(start(Ar_0, Ar_2, Ar_2, Ar_4, Ar_4, Ar_0))", 0-0) = Ar_0 S("start0(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(start(Ar_0, Ar_2, Ar_2, Ar_4, Ar_4, Ar_0))", 0-1) = Ar_2 S("start0(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(start(Ar_0, Ar_2, Ar_2, Ar_4, Ar_4, Ar_0))", 0-2) = Ar_2 S("start0(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(start(Ar_0, Ar_2, Ar_2, Ar_4, Ar_4, Ar_0))", 0-3) = Ar_4 S("start0(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(start(Ar_0, Ar_2, Ar_2, Ar_4, Ar_4, Ar_0))", 0-4) = Ar_4 S("start0(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(start(Ar_0, Ar_2, Ar_2, Ar_4, Ar_4, Ar_0))", 0-5) = Ar_0 S("start(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(stop(Ar_0, Ar_1, Ar_2, Ar_5, Ar_4, Ar_5)) [ 0 >= Ar_0 /\\ Ar_1 = Ar_2 /\\ Ar_3 = Ar_4 /\\ Ar_5 = Ar_0 ]", 0-0) = Ar_0 S("start(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(stop(Ar_0, Ar_1, Ar_2, Ar_5, Ar_4, Ar_5)) [ 0 >= Ar_0 /\\ Ar_1 = Ar_2 /\\ Ar_3 = Ar_4 /\\ Ar_5 = Ar_0 ]", 0-1) = Ar_2 S("start(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(stop(Ar_0, Ar_1, Ar_2, Ar_5, Ar_4, Ar_5)) [ 0 >= Ar_0 /\\ Ar_1 = Ar_2 /\\ Ar_3 = Ar_4 /\\ Ar_5 = Ar_0 ]", 0-2) = Ar_2 S("start(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(stop(Ar_0, Ar_1, Ar_2, Ar_5, Ar_4, Ar_5)) [ 0 >= Ar_0 /\\ Ar_1 = Ar_2 /\\ Ar_3 = Ar_4 /\\ Ar_5 = Ar_0 ]", 0-3) = Ar_0 S("start(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(stop(Ar_0, Ar_1, Ar_2, Ar_5, Ar_4, Ar_5)) [ 0 >= Ar_0 /\\ Ar_1 = Ar_2 /\\ Ar_3 = Ar_4 /\\ Ar_5 = Ar_0 ]", 0-4) = Ar_4 S("start(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(stop(Ar_0, Ar_1, Ar_2, Ar_5, Ar_4, Ar_5)) [ 0 >= Ar_0 /\\ Ar_1 = Ar_2 /\\ Ar_3 = Ar_4 /\\ Ar_5 = Ar_0 ]", 0-5) = Ar_0 S("start(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(lbl62(Ar_0, Ar_5 - 1, Ar_2, Ar_5, Ar_4, Ar_5)) [ Ar_0 >= 1 /\\ Ar_1 = Ar_2 /\\ Ar_3 = Ar_4 /\\ Ar_5 = Ar_0 ]", 0-0) = Ar_0 S("start(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(lbl62(Ar_0, Ar_5 - 1, Ar_2, Ar_5, Ar_4, Ar_5)) [ Ar_0 >= 1 /\\ Ar_1 = Ar_2 /\\ Ar_3 = Ar_4 /\\ Ar_5 = Ar_0 ]", 0-1) = Ar_0 S("start(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(lbl62(Ar_0, Ar_5 - 1, Ar_2, Ar_5, Ar_4, Ar_5)) [ Ar_0 >= 1 /\\ Ar_1 = Ar_2 /\\ Ar_3 = Ar_4 /\\ Ar_5 = Ar_0 ]", 0-2) = Ar_2 S("start(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(lbl62(Ar_0, Ar_5 - 1, Ar_2, Ar_5, Ar_4, Ar_5)) [ Ar_0 >= 1 /\\ Ar_1 = Ar_2 /\\ Ar_3 = Ar_4 /\\ Ar_5 = Ar_0 ]", 0-3) = Ar_0 S("start(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(lbl62(Ar_0, Ar_5 - 1, Ar_2, Ar_5, Ar_4, Ar_5)) [ Ar_0 >= 1 /\\ Ar_1 = Ar_2 /\\ Ar_3 = Ar_4 /\\ Ar_5 = Ar_0 ]", 0-4) = Ar_4 S("start(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(lbl62(Ar_0, Ar_5 - 1, Ar_2, Ar_5, Ar_4, Ar_5)) [ Ar_0 >= 1 /\\ Ar_1 = Ar_2 /\\ Ar_3 = Ar_4 /\\ Ar_5 = Ar_0 ]", 0-5) = Ar_0 S("lbl62(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(lbl72(Ar_0, Ar_1, Ar_2, Ar_3 - 1, Ar_4, Ar_5)) [ Ar_0 >= Ar_3 /\\ Ar_0 >= 1 /\\ Ar_3 >= 1 /\\ Ar_1 = 0 /\\ Ar_5 = Ar_0 ]", 0-0) = Ar_0 S("lbl62(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(lbl72(Ar_0, Ar_1, Ar_2, Ar_3 - 1, Ar_4, Ar_5)) [ Ar_0 >= Ar_3 /\\ Ar_0 >= 1 /\\ Ar_3 >= 1 /\\ Ar_1 = 0 /\\ Ar_5 = Ar_0 ]", 0-1) = 0 S("lbl62(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(lbl72(Ar_0, Ar_1, Ar_2, Ar_3 - 1, Ar_4, Ar_5)) [ Ar_0 >= Ar_3 /\\ Ar_0 >= 1 /\\ Ar_3 >= 1 /\\ Ar_1 = 0 /\\ Ar_5 = Ar_0 ]", 0-2) = Ar_2 S("lbl62(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(lbl72(Ar_0, Ar_1, Ar_2, Ar_3 - 1, Ar_4, Ar_5)) [ Ar_0 >= Ar_3 /\\ Ar_0 >= 1 /\\ Ar_3 >= 1 /\\ Ar_1 = 0 /\\ Ar_5 = Ar_0 ]", 0-3) = Ar_0 S("lbl62(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(lbl72(Ar_0, Ar_1, Ar_2, Ar_3 - 1, Ar_4, Ar_5)) [ Ar_0 >= Ar_3 /\\ Ar_0 >= 1 /\\ Ar_3 >= 1 /\\ Ar_1 = 0 /\\ Ar_5 = Ar_0 ]", 0-4) = Ar_4 S("lbl62(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(lbl72(Ar_0, Ar_1, Ar_2, Ar_3 - 1, Ar_4, Ar_5)) [ Ar_0 >= Ar_3 /\\ Ar_0 >= 1 /\\ Ar_3 >= 1 /\\ Ar_1 = 0 /\\ Ar_5 = Ar_0 ]", 0-5) = Ar_0 S("lbl62(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(lbl62(Ar_0, Ar_1 - 1, Ar_2, Ar_3, Ar_4, Ar_5)) [ Ar_1 >= 1 /\\ Ar_0 >= Ar_3 /\\ Ar_0 >= Ar_1 + 1 /\\ Ar_1 >= 0 /\\ Ar_3 >= 1 /\\ Ar_5 = Ar_0 ]", 0-0) = Ar_0 S("lbl62(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(lbl62(Ar_0, Ar_1 - 1, Ar_2, Ar_3, Ar_4, Ar_5)) [ Ar_1 >= 1 /\\ Ar_0 >= Ar_3 /\\ Ar_0 >= Ar_1 + 1 /\\ Ar_1 >= 0 /\\ Ar_3 >= 1 /\\ Ar_5 = Ar_0 ]", 0-1) = Ar_0 S("lbl62(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(lbl62(Ar_0, Ar_1 - 1, Ar_2, Ar_3, Ar_4, Ar_5)) [ Ar_1 >= 1 /\\ Ar_0 >= Ar_3 /\\ Ar_0 >= Ar_1 + 1 /\\ Ar_1 >= 0 /\\ Ar_3 >= 1 /\\ Ar_5 = Ar_0 ]", 0-2) = Ar_2 S("lbl62(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(lbl62(Ar_0, Ar_1 - 1, Ar_2, Ar_3, Ar_4, Ar_5)) [ Ar_1 >= 1 /\\ Ar_0 >= Ar_3 /\\ Ar_0 >= Ar_1 + 1 /\\ Ar_1 >= 0 /\\ Ar_3 >= 1 /\\ Ar_5 = Ar_0 ]", 0-3) = Ar_0 S("lbl62(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(lbl62(Ar_0, Ar_1 - 1, Ar_2, Ar_3, Ar_4, Ar_5)) [ Ar_1 >= 1 /\\ Ar_0 >= Ar_3 /\\ Ar_0 >= Ar_1 + 1 /\\ Ar_1 >= 0 /\\ Ar_3 >= 1 /\\ Ar_5 = Ar_0 ]", 0-4) = Ar_4 S("lbl62(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(lbl62(Ar_0, Ar_1 - 1, Ar_2, Ar_3, Ar_4, Ar_5)) [ Ar_1 >= 1 /\\ Ar_0 >= Ar_3 /\\ Ar_0 >= Ar_1 + 1 /\\ Ar_1 >= 0 /\\ Ar_3 >= 1 /\\ Ar_5 = Ar_0 ]", 0-5) = Ar_0 S("lbl72(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(stop(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5)) [ Ar_0 >= 1 /\\ Ar_3 = 0 /\\ Ar_1 = 0 /\\ Ar_5 = Ar_0 ]", 0-0) = Ar_0 S("lbl72(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(stop(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5)) [ Ar_0 >= 1 /\\ Ar_3 = 0 /\\ Ar_1 = 0 /\\ Ar_5 = Ar_0 ]", 0-1) = 0 S("lbl72(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(stop(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5)) [ Ar_0 >= 1 /\\ Ar_3 = 0 /\\ Ar_1 = 0 /\\ Ar_5 = Ar_0 ]", 0-2) = Ar_2 S("lbl72(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(stop(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5)) [ Ar_0 >= 1 /\\ Ar_3 = 0 /\\ Ar_1 = 0 /\\ Ar_5 = Ar_0 ]", 0-3) = 0 S("lbl72(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(stop(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5)) [ Ar_0 >= 1 /\\ Ar_3 = 0 /\\ Ar_1 = 0 /\\ Ar_5 = Ar_0 ]", 0-4) = Ar_4 S("lbl72(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(stop(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5)) [ Ar_0 >= 1 /\\ Ar_3 = 0 /\\ Ar_1 = 0 /\\ Ar_5 = Ar_0 ]", 0-5) = Ar_0 S("lbl72(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(lbl62(Ar_0, Ar_5 - 1, Ar_2, Ar_3, Ar_4, Ar_5)) [ Ar_0 >= 1 /\\ Ar_3 >= 1 /\\ Ar_3 >= 0 /\\ Ar_0 >= Ar_3 + 1 /\\ Ar_1 = 0 /\\ Ar_5 = Ar_0 ]", 0-0) = Ar_0 S("lbl72(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(lbl62(Ar_0, Ar_5 - 1, Ar_2, Ar_3, Ar_4, Ar_5)) [ Ar_0 >= 1 /\\ Ar_3 >= 1 /\\ Ar_3 >= 0 /\\ Ar_0 >= Ar_3 + 1 /\\ Ar_1 = 0 /\\ Ar_5 = Ar_0 ]", 0-1) = Ar_0 S("lbl72(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(lbl62(Ar_0, Ar_5 - 1, Ar_2, Ar_3, Ar_4, Ar_5)) [ Ar_0 >= 1 /\\ Ar_3 >= 1 /\\ Ar_3 >= 0 /\\ Ar_0 >= Ar_3 + 1 /\\ Ar_1 = 0 /\\ Ar_5 = Ar_0 ]", 0-2) = Ar_2 S("lbl72(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(lbl62(Ar_0, Ar_5 - 1, Ar_2, Ar_3, Ar_4, Ar_5)) [ Ar_0 >= 1 /\\ Ar_3 >= 1 /\\ Ar_3 >= 0 /\\ Ar_0 >= Ar_3 + 1 /\\ Ar_1 = 0 /\\ Ar_5 = Ar_0 ]", 0-3) = Ar_0 S("lbl72(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(lbl62(Ar_0, Ar_5 - 1, Ar_2, Ar_3, Ar_4, Ar_5)) [ Ar_0 >= 1 /\\ Ar_3 >= 1 /\\ Ar_3 >= 0 /\\ Ar_0 >= Ar_3 + 1 /\\ Ar_1 = 0 /\\ Ar_5 = Ar_0 ]", 0-4) = Ar_4 S("lbl72(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(lbl62(Ar_0, Ar_5 - 1, Ar_2, Ar_3, Ar_4, Ar_5)) [ Ar_0 >= 1 /\\ Ar_3 >= 1 /\\ Ar_3 >= 0 /\\ Ar_0 >= Ar_3 + 1 /\\ Ar_1 = 0 /\\ Ar_5 = Ar_0 ]", 0-5) = Ar_0 orients the transitions lbl62(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(lbl62(Ar_0, Ar_1 - 1, Ar_2, Ar_3, Ar_4, Ar_5)) [ Ar_1 >= 1 /\ Ar_0 >= Ar_3 /\ Ar_0 >= Ar_1 + 1 /\ Ar_1 >= 0 /\ Ar_3 >= 1 /\ Ar_5 = Ar_0 ] weakly and the transition lbl62(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(lbl62(Ar_0, Ar_1 - 1, Ar_2, Ar_3, Ar_4, Ar_5)) [ Ar_1 >= 1 /\ Ar_0 >= Ar_3 /\ Ar_0 >= Ar_1 + 1 /\ Ar_1 >= 0 /\ Ar_3 >= 1 /\ Ar_5 = Ar_0 ] strictly and produces the following problem: 6: T: (Comp: 2*Ar_0, Cost: 1) lbl72(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(lbl62(Ar_0, Ar_5 - 1, Ar_2, Ar_3, Ar_4, Ar_5)) [ Ar_0 >= 1 /\ Ar_3 >= 1 /\ Ar_3 >= 0 /\ Ar_0 >= Ar_3 + 1 /\ Ar_1 = 0 /\ Ar_5 = Ar_0 ] (Comp: 1, Cost: 1) lbl72(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(stop(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5)) [ Ar_0 >= 1 /\ Ar_3 = 0 /\ Ar_1 = 0 /\ Ar_5 = Ar_0 ] (Comp: 2*Ar_0^2 + 3*Ar_0 + 1, Cost: 1) lbl62(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(lbl62(Ar_0, Ar_1 - 1, Ar_2, Ar_3, Ar_4, Ar_5)) [ Ar_1 >= 1 /\ Ar_0 >= Ar_3 /\ Ar_0 >= Ar_1 + 1 /\ Ar_1 >= 0 /\ Ar_3 >= 1 /\ Ar_5 = Ar_0 ] (Comp: 2*Ar_0, Cost: 1) lbl62(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(lbl72(Ar_0, Ar_1, Ar_2, Ar_3 - 1, Ar_4, Ar_5)) [ Ar_0 >= Ar_3 /\ Ar_0 >= 1 /\ Ar_3 >= 1 /\ Ar_1 = 0 /\ Ar_5 = Ar_0 ] (Comp: 1, Cost: 1) start(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(lbl62(Ar_0, Ar_5 - 1, Ar_2, Ar_5, Ar_4, Ar_5)) [ Ar_0 >= 1 /\ Ar_1 = Ar_2 /\ Ar_3 = Ar_4 /\ Ar_5 = Ar_0 ] (Comp: 1, Cost: 1) start(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(stop(Ar_0, Ar_1, Ar_2, Ar_5, Ar_4, Ar_5)) [ 0 >= Ar_0 /\ Ar_1 = Ar_2 /\ Ar_3 = Ar_4 /\ Ar_5 = Ar_0 ] (Comp: 1, Cost: 1) start0(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(start(Ar_0, Ar_2, Ar_2, Ar_4, Ar_4, Ar_0)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(start0(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 Complexity upper bound 7*Ar_0 + 2*Ar_0^2 + 5 Time: 0.182 sec (SMT: 0.151 sec) ---------------------------------------- (2) BOUNDS(1, n^2) ---------------------------------------- (3) Loat Proof (FINISHED) ### Pre-processing the ITS problem ### Initial linear ITS problem Start location: start0 0: start -> stop : D'=F, [ 0>=A && B==C && D==E && F==A ], cost: 1 1: start -> lbl62 : B'=-1+F, D'=F, [ A>=1 && B==C && D==E && F==A ], cost: 1 2: lbl72 -> stop : [ A>=1 && D==0 && B==0 && F==A ], cost: 1 3: lbl72 -> lbl72 : B'=F, D'=-1+D, [ D>=1 && 0>=A && D>=0 && A>=1+D && B==0 && F==A ], cost: 1 4: lbl72 -> lbl62 : B'=-1+F, [ A>=1 && D>=1 && D>=0 && A>=1+D && B==0 && F==A ], cost: 1 5: lbl62 -> lbl72 : D'=-1+D, [ A>=D && A>=1 && D>=1 && B==0 && F==A ], cost: 1 6: lbl62 -> lbl62 : B'=-1+B, [ B>=1 && A>=D && A>=1+B && B>=0 && D>=1 && F==A ], cost: 1 7: start0 -> start : B'=C, D'=E, F'=A, [], cost: 1 Checking for constant complexity: The following rule is satisfiable with cost >= 1, yielding constant complexity: 7: start0 -> start : B'=C, D'=E, F'=A, [], cost: 1 Removed unreachable and leaf rules: Start location: start0 1: start -> lbl62 : B'=-1+F, D'=F, [ A>=1 && B==C && D==E && F==A ], cost: 1 3: lbl72 -> lbl72 : B'=F, D'=-1+D, [ D>=1 && 0>=A && D>=0 && A>=1+D && B==0 && F==A ], cost: 1 4: lbl72 -> lbl62 : B'=-1+F, [ A>=1 && D>=1 && D>=0 && A>=1+D && B==0 && F==A ], cost: 1 5: lbl62 -> lbl72 : D'=-1+D, [ A>=D && A>=1 && D>=1 && B==0 && F==A ], cost: 1 6: lbl62 -> lbl62 : B'=-1+B, [ B>=1 && A>=D && A>=1+B && B>=0 && D>=1 && F==A ], cost: 1 7: start0 -> start : B'=C, D'=E, F'=A, [], cost: 1 Removed rules with unsatisfiable guard: Start location: start0 1: start -> lbl62 : B'=-1+F, D'=F, [ A>=1 && B==C && D==E && F==A ], cost: 1 4: lbl72 -> lbl62 : B'=-1+F, [ A>=1 && D>=1 && D>=0 && A>=1+D && B==0 && F==A ], cost: 1 5: lbl62 -> lbl72 : D'=-1+D, [ A>=D && A>=1 && D>=1 && B==0 && F==A ], cost: 1 6: lbl62 -> lbl62 : B'=-1+B, [ B>=1 && A>=D && A>=1+B && B>=0 && D>=1 && F==A ], cost: 1 7: start0 -> start : B'=C, D'=E, F'=A, [], cost: 1 Simplified all rules, resulting in: Start location: start0 1: start -> lbl62 : B'=-1+F, D'=F, [ A>=1 && B==C && D==E && F==A ], cost: 1 4: lbl72 -> lbl62 : B'=-1+F, [ D>=1 && A>=1+D && B==0 && F==A ], cost: 1 5: lbl62 -> lbl72 : D'=-1+D, [ A>=D && A>=1 && D>=1 && B==0 && F==A ], cost: 1 6: lbl62 -> lbl62 : B'=-1+B, [ B>=1 && A>=D && A>=1+B && D>=1 && F==A ], cost: 1 7: start0 -> start : B'=C, D'=E, F'=A, [], cost: 1 ### Simplification by acceleration and chaining ### Accelerating simple loops of location 2. Accelerating the following rules: 6: lbl62 -> lbl62 : B'=-1+B, [ B>=1 && A>=D && A>=1+B && D>=1 && F==A ], cost: 1 Accelerated rule 6 with metering function B, yielding the new rule 8. Removing the simple loops: 6. Accelerated all simple loops using metering functions (where possible): Start location: start0 1: start -> lbl62 : B'=-1+F, D'=F, [ A>=1 && B==C && D==E && F==A ], cost: 1 4: lbl72 -> lbl62 : B'=-1+F, [ D>=1 && A>=1+D && B==0 && F==A ], cost: 1 5: lbl62 -> lbl72 : D'=-1+D, [ A>=D && A>=1 && D>=1 && B==0 && F==A ], cost: 1 8: lbl62 -> lbl62 : B'=0, [ B>=1 && A>=D && A>=1+B && D>=1 && F==A ], cost: B 7: start0 -> start : B'=C, D'=E, F'=A, [], cost: 1 Chained accelerated rules (with incoming rules): Start location: start0 1: start -> lbl62 : B'=-1+F, D'=F, [ A>=1 && B==C && D==E && F==A ], cost: 1 9: start -> lbl62 : B'=0, D'=F, [ A>=1 && B==C && D==E && F==A && -1+F>=1 ], cost: F 4: lbl72 -> lbl62 : B'=-1+F, [ D>=1 && A>=1+D && B==0 && F==A ], cost: 1 10: lbl72 -> lbl62 : B'=0, [ D>=1 && A>=1+D && B==0 && F==A && -1+F>=1 ], cost: F 5: lbl62 -> lbl72 : D'=-1+D, [ A>=D && A>=1 && D>=1 && B==0 && F==A ], cost: 1 7: start0 -> start : B'=C, D'=E, F'=A, [], cost: 1 Eliminated locations (on tree-shaped paths): Start location: start0 13: lbl62 -> lbl62 : B'=-1+F, D'=-1+D, [ A>=D && A>=1 && B==0 && F==A && -1+D>=1 ], cost: 2 14: lbl62 -> lbl62 : B'=0, D'=-1+D, [ A>=D && A>=1 && B==0 && F==A && -1+D>=1 && -1+F>=1 ], cost: 1+F 11: start0 -> lbl62 : B'=-1+A, D'=A, F'=A, [ A>=1 ], cost: 2 12: start0 -> lbl62 : B'=0, D'=A, F'=A, [ -1+A>=1 ], cost: 1+A Accelerating simple loops of location 2. Accelerating the following rules: 13: lbl62 -> lbl62 : B'=-1+F, D'=-1+D, [ A>=D && A>=1 && B==0 && F==A && -1+D>=1 ], cost: 2 14: lbl62 -> lbl62 : B'=0, D'=-1+D, [ A>=D && A>=1 && B==0 && F==A && -1+D>=1 && -1+F>=1 ], cost: 1+F Accelerated rule 13 with NONTERM (after strengthening guard), yielding the new rule 15. Accelerated rule 14 with metering function -1+D, yielding the new rule 16. Removing the simple loops: 14. Accelerated all simple loops using metering functions (where possible): Start location: start0 13: lbl62 -> lbl62 : B'=-1+F, D'=-1+D, [ A>=D && A>=1 && B==0 && F==A && -1+D>=1 ], cost: 2 15: lbl62 -> [6] : [ A>=D && A>=1 && B==0 && F==A && -1+D>=1 && -1+F==0 ], cost: NONTERM 16: lbl62 -> lbl62 : B'=0, D'=1, [ A>=D && A>=1 && B==0 && F==A && -1+D>=1 && -1+F>=1 ], cost: -1+F*(-1+D)+D 11: start0 -> lbl62 : B'=-1+A, D'=A, F'=A, [ A>=1 ], cost: 2 12: start0 -> lbl62 : B'=0, D'=A, F'=A, [ -1+A>=1 ], cost: 1+A Chained accelerated rules (with incoming rules): Start location: start0 11: start0 -> lbl62 : B'=-1+A, D'=A, F'=A, [ A>=1 ], cost: 2 12: start0 -> lbl62 : B'=0, D'=A, F'=A, [ -1+A>=1 ], cost: 1+A 17: start0 -> lbl62 : B'=-1+A, D'=-1+A, F'=A, [ -1+A>=1 ], cost: 3+A 18: start0 -> lbl62 : B'=0, D'=1, F'=A, [ -1+A>=1 ], cost: (-1+A)*A+2*A Removed unreachable locations (and leaf rules with constant cost): Start location: start0 12: start0 -> lbl62 : B'=0, D'=A, F'=A, [ -1+A>=1 ], cost: 1+A 17: start0 -> lbl62 : B'=-1+A, D'=-1+A, F'=A, [ -1+A>=1 ], cost: 3+A 18: start0 -> lbl62 : B'=0, D'=1, F'=A, [ -1+A>=1 ], cost: (-1+A)*A+2*A ### Computing asymptotic complexity ### Fully simplified ITS problem Start location: start0 17: start0 -> lbl62 : B'=-1+A, D'=-1+A, F'=A, [ -1+A>=1 ], cost: 3+A 18: start0 -> lbl62 : B'=0, D'=1, F'=A, [ -1+A>=1 ], cost: (-1+A)*A+2*A Computing asymptotic complexity for rule 17 Solved the limit problem by the following transformations: Created initial limit problem: -1+A (+/+!), 3+A (+) [not solved] removing all constraints (solved by SMT) resulting limit problem: [solved] applying transformation rule (C) using substitution {A==n} resulting limit problem: [solved] Solution: A / n Resulting cost 3+n has complexity: Poly(n^1) Found new complexity Poly(n^1). Computing asymptotic complexity for rule 18 Solved the limit problem by the following transformations: Created initial limit problem: -1+A (+/+!), A+A^2 (+) [not solved] removing all constraints (solved by SMT) resulting limit problem: [solved] applying transformation rule (C) using substitution {A==n} resulting limit problem: [solved] Solution: A / n Resulting cost n+n^2 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: n+n^2 Rule cost: (-1+A)*A+2*A Rule guard: [ -1+A>=1 ] WORST_CASE(Omega(n^2),?) ---------------------------------------- (4) BOUNDS(n^2, INF)