/export/starexec/sandbox/solver/bin/starexec_run_complexity /export/starexec/sandbox/benchmark/theBenchmark.koat /export/starexec/sandbox/output/output_files -------------------------------------------------------------------------------- WORST_CASE(Omega(n^1), O(n^1)) 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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(n^1, n^1). (0) CpxIntTrs (1) Koat Proof [FINISHED, 274 ms] (2) BOUNDS(1, n^1) (3) Loat Proof [FINISHED, 395 ms] (4) BOUNDS(n^1, INF) ---------------------------------------- (0) Obligation: Complexity Int TRS consisting of the following rules: start(A, B, C, D, E, F) -> Com_1(stop(A, B, C, D, E, F)) :|: 0 >= A && B >= C && B <= C && D >= A && D <= A && E >= F && E <= F start(A, B, C, D, E, F) -> Com_1(lbl6(A, B, C, D, E, F)) :|: A >= 1 && A >= C && B >= C && B <= C && D >= A && D <= A && E >= F && E <= F start(A, B, C, D, E, F) -> Com_1(lbl121(A, B, C, D, B - D, F)) :|: A >= 1 && C >= A + 1 && B >= C && B <= C && D >= A && D <= A && E >= F && E <= F lbl6(A, B, C, D, E, F) -> Com_1(stop(A, B, C, D, E, F)) :|: A >= 1 && A >= C && E >= F && E <= F && D >= A && D <= A && B >= C && B <= C lbl111(A, B, C, D, E, F) -> Com_1(cut(A, B, C, D, E, F)) :|: C >= A + 1 && A >= 2 && E >= 0 && E <= 0 && D >= A && D <= A && B >= C && B <= C lbl111(A, B, C, D, E, F) -> Com_1(lbl111(A, B, C, D, E - 1, F)) :|: E >= 1 && A >= E + 1 && E >= 0 && C >= E + A + 1 && A >= E + 2 && D >= A && D <= A && B >= C && B <= C lbl111(A, B, C, D, E, F) -> Com_1(lbl121(A, B, C, D, E - D, F)) :|: E >= 1 && E >= A && E >= 0 && C >= E + A + 1 && A >= E + 2 && D >= A && D <= A && B >= C && B <= C lbl121(A, B, C, D, E, F) -> Com_1(cut(A, B, C, D, E, F)) :|: C >= A + 1 && A >= 1 && C >= A && E >= 0 && E <= 0 && D >= A && D <= A && B >= C && B <= C lbl121(A, B, C, D, E, F) -> Com_1(lbl111(A, B, C, D, E - 1, F)) :|: E >= 1 && A >= E + 1 && C >= A + 1 && A >= 1 && E >= 0 && C >= E + A && D >= A && D <= A && B >= C && B <= C lbl121(A, B, C, D, E, F) -> Com_1(lbl121(A, B, C, D, E - D, F)) :|: E >= 1 && E >= A && C >= A + 1 && A >= 1 && E >= 0 && C >= E + A && D >= A && D <= A && B >= C && B <= C cut(A, B, C, D, E, F) -> Com_1(stop(A, B, C, D, E, F)) :|: A >= 1 && C >= A + 1 && E >= 0 && E <= 0 && D >= A && D <= A && B >= C && B <= C start0(A, B, C, D, E, F) -> Com_1(start(A, C, C, A, F, F)) :|: TRUE The start-symbols are:[start0_6] ---------------------------------------- (1) Koat Proof (FINISHED) YES(?, 8*Ar_2 + 25) 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_3, Ar_4, Ar_5)) [ 0 >= Ar_0 /\ Ar_1 = Ar_2 /\ Ar_3 = Ar_0 /\ Ar_4 = Ar_5 ] (Comp: ?, Cost: 1) start(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(lbl6(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5)) [ Ar_0 >= 1 /\ Ar_0 >= Ar_2 /\ Ar_1 = Ar_2 /\ Ar_3 = Ar_0 /\ Ar_4 = Ar_5 ] (Comp: ?, Cost: 1) start(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(lbl121(Ar_0, Ar_1, Ar_2, Ar_3, Ar_1 - Ar_3, Ar_5)) [ Ar_0 >= 1 /\ Ar_2 >= Ar_0 + 1 /\ Ar_1 = Ar_2 /\ Ar_3 = Ar_0 /\ Ar_4 = Ar_5 ] (Comp: ?, Cost: 1) lbl6(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_0 >= Ar_2 /\ Ar_4 = Ar_5 /\ Ar_3 = Ar_0 /\ Ar_1 = Ar_2 ] (Comp: ?, Cost: 1) lbl111(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(cut(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5)) [ Ar_2 >= Ar_0 + 1 /\ Ar_0 >= 2 /\ Ar_4 = 0 /\ Ar_3 = Ar_0 /\ Ar_1 = Ar_2 ] (Comp: ?, Cost: 1) lbl111(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(lbl111(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4 - 1, Ar_5)) [ Ar_4 >= 1 /\ Ar_0 >= Ar_4 + 1 /\ Ar_4 >= 0 /\ Ar_2 >= Ar_4 + Ar_0 + 1 /\ Ar_0 >= Ar_4 + 2 /\ Ar_3 = Ar_0 /\ Ar_1 = Ar_2 ] (Comp: ?, Cost: 1) lbl111(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(lbl121(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4 - Ar_3, Ar_5)) [ Ar_4 >= 1 /\ Ar_4 >= Ar_0 /\ Ar_4 >= 0 /\ Ar_2 >= Ar_4 + Ar_0 + 1 /\ Ar_0 >= Ar_4 + 2 /\ Ar_3 = Ar_0 /\ Ar_1 = Ar_2 ] (Comp: ?, Cost: 1) lbl121(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(cut(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5)) [ Ar_2 >= Ar_0 + 1 /\ Ar_0 >= 1 /\ Ar_2 >= Ar_0 /\ Ar_4 = 0 /\ Ar_3 = Ar_0 /\ Ar_1 = Ar_2 ] (Comp: ?, Cost: 1) lbl121(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(lbl111(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4 - 1, Ar_5)) [ Ar_4 >= 1 /\ Ar_0 >= Ar_4 + 1 /\ Ar_2 >= Ar_0 + 1 /\ Ar_0 >= 1 /\ Ar_4 >= 0 /\ Ar_2 >= Ar_4 + Ar_0 /\ Ar_3 = Ar_0 /\ Ar_1 = Ar_2 ] (Comp: ?, Cost: 1) lbl121(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(lbl121(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4 - Ar_3, Ar_5)) [ Ar_4 >= 1 /\ Ar_4 >= Ar_0 /\ Ar_2 >= Ar_0 + 1 /\ Ar_0 >= 1 /\ Ar_4 >= 0 /\ Ar_2 >= Ar_4 + Ar_0 /\ Ar_3 = Ar_0 /\ Ar_1 = Ar_2 ] (Comp: ?, Cost: 1) cut(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_2 >= Ar_0 + 1 /\ Ar_4 = 0 /\ Ar_3 = Ar_0 /\ Ar_1 = Ar_2 ] (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_0, Ar_5, Ar_5)) (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: lbl111(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(lbl121(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4 - Ar_3, Ar_5)) [ Ar_4 >= 1 /\ Ar_4 >= Ar_0 /\ Ar_4 >= 0 /\ Ar_2 >= Ar_4 + Ar_0 + 1 /\ Ar_0 >= Ar_4 + 2 /\ Ar_3 = Ar_0 /\ Ar_1 = Ar_2 ] We thus obtain the following problem: 2: T: (Comp: ?, Cost: 1) cut(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_2 >= Ar_0 + 1 /\ Ar_4 = 0 /\ Ar_3 = Ar_0 /\ Ar_1 = Ar_2 ] (Comp: ?, Cost: 1) lbl121(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(cut(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5)) [ Ar_2 >= Ar_0 + 1 /\ Ar_0 >= 1 /\ Ar_2 >= Ar_0 /\ Ar_4 = 0 /\ Ar_3 = Ar_0 /\ Ar_1 = Ar_2 ] (Comp: ?, Cost: 1) lbl111(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(lbl111(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4 - 1, Ar_5)) [ Ar_4 >= 1 /\ Ar_0 >= Ar_4 + 1 /\ Ar_4 >= 0 /\ Ar_2 >= Ar_4 + Ar_0 + 1 /\ Ar_0 >= Ar_4 + 2 /\ Ar_3 = Ar_0 /\ Ar_1 = Ar_2 ] (Comp: ?, Cost: 1) lbl111(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(cut(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5)) [ Ar_2 >= Ar_0 + 1 /\ Ar_0 >= 2 /\ Ar_4 = 0 /\ Ar_3 = Ar_0 /\ Ar_1 = Ar_2 ] (Comp: ?, Cost: 1) lbl121(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(lbl121(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4 - Ar_3, Ar_5)) [ Ar_4 >= 1 /\ Ar_4 >= Ar_0 /\ Ar_2 >= Ar_0 + 1 /\ Ar_0 >= 1 /\ Ar_4 >= 0 /\ Ar_2 >= Ar_4 + Ar_0 /\ Ar_3 = Ar_0 /\ Ar_1 = Ar_2 ] (Comp: ?, Cost: 1) lbl121(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(lbl111(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4 - 1, Ar_5)) [ Ar_4 >= 1 /\ Ar_0 >= Ar_4 + 1 /\ Ar_2 >= Ar_0 + 1 /\ Ar_0 >= 1 /\ Ar_4 >= 0 /\ Ar_2 >= Ar_4 + Ar_0 /\ Ar_3 = Ar_0 /\ Ar_1 = Ar_2 ] (Comp: ?, Cost: 1) lbl6(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_0 >= Ar_2 /\ Ar_4 = Ar_5 /\ Ar_3 = Ar_0 /\ Ar_1 = Ar_2 ] (Comp: ?, Cost: 1) start(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(lbl121(Ar_0, Ar_1, Ar_2, Ar_3, Ar_1 - Ar_3, Ar_5)) [ Ar_0 >= 1 /\ Ar_2 >= Ar_0 + 1 /\ Ar_1 = Ar_2 /\ Ar_3 = Ar_0 /\ Ar_4 = Ar_5 ] (Comp: ?, Cost: 1) start(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(lbl6(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5)) [ Ar_0 >= 1 /\ Ar_0 >= Ar_2 /\ Ar_1 = Ar_2 /\ Ar_3 = Ar_0 /\ Ar_4 = Ar_5 ] (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_3, Ar_4, Ar_5)) [ 0 >= Ar_0 /\ Ar_1 = Ar_2 /\ Ar_3 = Ar_0 /\ Ar_4 = Ar_5 ] (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_0, Ar_5, Ar_5)) (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) cut(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_2 >= Ar_0 + 1 /\ Ar_4 = 0 /\ Ar_3 = Ar_0 /\ Ar_1 = Ar_2 ] (Comp: ?, Cost: 1) lbl121(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(cut(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5)) [ Ar_2 >= Ar_0 + 1 /\ Ar_0 >= 1 /\ Ar_2 >= Ar_0 /\ Ar_4 = 0 /\ Ar_3 = Ar_0 /\ Ar_1 = Ar_2 ] (Comp: ?, Cost: 1) lbl111(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(lbl111(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4 - 1, Ar_5)) [ Ar_4 >= 1 /\ Ar_0 >= Ar_4 + 1 /\ Ar_4 >= 0 /\ Ar_2 >= Ar_4 + Ar_0 + 1 /\ Ar_0 >= Ar_4 + 2 /\ Ar_3 = Ar_0 /\ Ar_1 = Ar_2 ] (Comp: ?, Cost: 1) lbl111(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(cut(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5)) [ Ar_2 >= Ar_0 + 1 /\ Ar_0 >= 2 /\ Ar_4 = 0 /\ Ar_3 = Ar_0 /\ Ar_1 = Ar_2 ] (Comp: ?, Cost: 1) lbl121(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(lbl121(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4 - Ar_3, Ar_5)) [ Ar_4 >= 1 /\ Ar_4 >= Ar_0 /\ Ar_2 >= Ar_0 + 1 /\ Ar_0 >= 1 /\ Ar_4 >= 0 /\ Ar_2 >= Ar_4 + Ar_0 /\ Ar_3 = Ar_0 /\ Ar_1 = Ar_2 ] (Comp: ?, Cost: 1) lbl121(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(lbl111(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4 - 1, Ar_5)) [ Ar_4 >= 1 /\ Ar_0 >= Ar_4 + 1 /\ Ar_2 >= Ar_0 + 1 /\ Ar_0 >= 1 /\ Ar_4 >= 0 /\ Ar_2 >= Ar_4 + Ar_0 /\ Ar_3 = Ar_0 /\ Ar_1 = Ar_2 ] (Comp: 1, Cost: 1) lbl6(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_0 >= Ar_2 /\ Ar_4 = Ar_5 /\ Ar_3 = Ar_0 /\ Ar_1 = Ar_2 ] (Comp: 1, Cost: 1) start(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(lbl121(Ar_0, Ar_1, Ar_2, Ar_3, Ar_1 - Ar_3, Ar_5)) [ Ar_0 >= 1 /\ Ar_2 >= Ar_0 + 1 /\ Ar_1 = Ar_2 /\ Ar_3 = Ar_0 /\ Ar_4 = Ar_5 ] (Comp: 1, Cost: 1) start(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(lbl6(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5)) [ Ar_0 >= 1 /\ Ar_0 >= Ar_2 /\ Ar_1 = Ar_2 /\ Ar_3 = Ar_0 /\ Ar_4 = Ar_5 ] (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_3, Ar_4, Ar_5)) [ 0 >= Ar_0 /\ Ar_1 = Ar_2 /\ Ar_3 = Ar_0 /\ Ar_4 = Ar_5 ] (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_0, Ar_5, Ar_5)) (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(cut) = 1 Pol(stop) = 0 Pol(lbl121) = 3 Pol(lbl111) = 2 Pol(lbl6) = 0 Pol(start) = 3 Pol(start0) = 3 Pol(koat_start) = 3 orients all transitions weakly and the transitions lbl121(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(lbl111(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4 - 1, Ar_5)) [ Ar_4 >= 1 /\ Ar_0 >= Ar_4 + 1 /\ Ar_2 >= Ar_0 + 1 /\ Ar_0 >= 1 /\ Ar_4 >= 0 /\ Ar_2 >= Ar_4 + Ar_0 /\ Ar_3 = Ar_0 /\ Ar_1 = Ar_2 ] lbl121(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(cut(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5)) [ Ar_2 >= Ar_0 + 1 /\ Ar_0 >= 1 /\ Ar_2 >= Ar_0 /\ Ar_4 = 0 /\ Ar_3 = Ar_0 /\ Ar_1 = Ar_2 ] lbl111(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(cut(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5)) [ Ar_2 >= Ar_0 + 1 /\ Ar_0 >= 2 /\ Ar_4 = 0 /\ Ar_3 = Ar_0 /\ Ar_1 = Ar_2 ] cut(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_2 >= Ar_0 + 1 /\ Ar_4 = 0 /\ Ar_3 = Ar_0 /\ Ar_1 = Ar_2 ] strictly and produces the following problem: 4: T: (Comp: 3, Cost: 1) cut(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_2 >= Ar_0 + 1 /\ Ar_4 = 0 /\ Ar_3 = Ar_0 /\ Ar_1 = Ar_2 ] (Comp: 3, Cost: 1) lbl121(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(cut(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5)) [ Ar_2 >= Ar_0 + 1 /\ Ar_0 >= 1 /\ Ar_2 >= Ar_0 /\ Ar_4 = 0 /\ Ar_3 = Ar_0 /\ Ar_1 = Ar_2 ] (Comp: ?, Cost: 1) lbl111(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(lbl111(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4 - 1, Ar_5)) [ Ar_4 >= 1 /\ Ar_0 >= Ar_4 + 1 /\ Ar_4 >= 0 /\ Ar_2 >= Ar_4 + Ar_0 + 1 /\ Ar_0 >= Ar_4 + 2 /\ Ar_3 = Ar_0 /\ Ar_1 = Ar_2 ] (Comp: 3, Cost: 1) lbl111(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(cut(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5)) [ Ar_2 >= Ar_0 + 1 /\ Ar_0 >= 2 /\ Ar_4 = 0 /\ Ar_3 = Ar_0 /\ Ar_1 = Ar_2 ] (Comp: ?, Cost: 1) lbl121(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(lbl121(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4 - Ar_3, Ar_5)) [ Ar_4 >= 1 /\ Ar_4 >= Ar_0 /\ Ar_2 >= Ar_0 + 1 /\ Ar_0 >= 1 /\ Ar_4 >= 0 /\ Ar_2 >= Ar_4 + Ar_0 /\ Ar_3 = Ar_0 /\ Ar_1 = Ar_2 ] (Comp: 3, Cost: 1) lbl121(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(lbl111(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4 - 1, Ar_5)) [ Ar_4 >= 1 /\ Ar_0 >= Ar_4 + 1 /\ Ar_2 >= Ar_0 + 1 /\ Ar_0 >= 1 /\ Ar_4 >= 0 /\ Ar_2 >= Ar_4 + Ar_0 /\ Ar_3 = Ar_0 /\ Ar_1 = Ar_2 ] (Comp: 1, Cost: 1) lbl6(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_0 >= Ar_2 /\ Ar_4 = Ar_5 /\ Ar_3 = Ar_0 /\ Ar_1 = Ar_2 ] (Comp: 1, Cost: 1) start(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(lbl121(Ar_0, Ar_1, Ar_2, Ar_3, Ar_1 - Ar_3, Ar_5)) [ Ar_0 >= 1 /\ Ar_2 >= Ar_0 + 1 /\ Ar_1 = Ar_2 /\ Ar_3 = Ar_0 /\ Ar_4 = Ar_5 ] (Comp: 1, Cost: 1) start(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(lbl6(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5)) [ Ar_0 >= 1 /\ Ar_0 >= Ar_2 /\ Ar_1 = Ar_2 /\ Ar_3 = Ar_0 /\ Ar_4 = Ar_5 ] (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_3, Ar_4, Ar_5)) [ 0 >= Ar_0 /\ Ar_1 = Ar_2 /\ Ar_3 = Ar_0 /\ Ar_4 = Ar_5 ] (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_0, Ar_5, Ar_5)) (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(lbl121) = V_5 + 1 Pol(lbl111) = V_5 + 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_0, Ar_5, Ar_5))", 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_0, Ar_5, Ar_5))", 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_0, Ar_5, Ar_5))", 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_0, Ar_5, Ar_5))", 0-3) = 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_0, Ar_5, Ar_5))", 0-4) = 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_0, Ar_5, Ar_5))", 0-5) = Ar_5 S("start(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)) [ 0 >= Ar_0 /\\ Ar_1 = Ar_2 /\\ Ar_3 = Ar_0 /\\ Ar_4 = Ar_5 ]", 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_3, Ar_4, Ar_5)) [ 0 >= Ar_0 /\\ Ar_1 = Ar_2 /\\ Ar_3 = Ar_0 /\\ Ar_4 = Ar_5 ]", 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_3, Ar_4, Ar_5)) [ 0 >= Ar_0 /\\ Ar_1 = Ar_2 /\\ Ar_3 = Ar_0 /\\ Ar_4 = Ar_5 ]", 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_3, Ar_4, Ar_5)) [ 0 >= Ar_0 /\\ Ar_1 = Ar_2 /\\ Ar_3 = Ar_0 /\\ Ar_4 = Ar_5 ]", 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_3, Ar_4, Ar_5)) [ 0 >= Ar_0 /\\ Ar_1 = Ar_2 /\\ Ar_3 = Ar_0 /\\ Ar_4 = Ar_5 ]", 0-4) = Ar_5 S("start(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)) [ 0 >= Ar_0 /\\ Ar_1 = Ar_2 /\\ Ar_3 = Ar_0 /\\ Ar_4 = Ar_5 ]", 0-5) = Ar_5 S("start(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(lbl6(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5)) [ Ar_0 >= 1 /\\ Ar_0 >= Ar_2 /\\ Ar_1 = Ar_2 /\\ Ar_3 = Ar_0 /\\ Ar_4 = Ar_5 ]", 0-0) = Ar_0 S("start(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(lbl6(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5)) [ Ar_0 >= 1 /\\ Ar_0 >= Ar_2 /\\ Ar_1 = Ar_2 /\\ Ar_3 = Ar_0 /\\ Ar_4 = Ar_5 ]", 0-1) = Ar_2 S("start(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(lbl6(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5)) [ Ar_0 >= 1 /\\ Ar_0 >= Ar_2 /\\ Ar_1 = Ar_2 /\\ Ar_3 = Ar_0 /\\ Ar_4 = Ar_5 ]", 0-2) = Ar_2 S("start(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(lbl6(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5)) [ Ar_0 >= 1 /\\ Ar_0 >= Ar_2 /\\ Ar_1 = Ar_2 /\\ Ar_3 = Ar_0 /\\ Ar_4 = Ar_5 ]", 0-3) = Ar_0 S("start(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(lbl6(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5)) [ Ar_0 >= 1 /\\ Ar_0 >= Ar_2 /\\ Ar_1 = Ar_2 /\\ Ar_3 = Ar_0 /\\ Ar_4 = Ar_5 ]", 0-4) = Ar_5 S("start(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(lbl6(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5)) [ Ar_0 >= 1 /\\ Ar_0 >= Ar_2 /\\ Ar_1 = Ar_2 /\\ Ar_3 = Ar_0 /\\ Ar_4 = Ar_5 ]", 0-5) = Ar_5 S("start(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(lbl121(Ar_0, Ar_1, Ar_2, Ar_3, Ar_1 - Ar_3, Ar_5)) [ Ar_0 >= 1 /\\ Ar_2 >= Ar_0 + 1 /\\ Ar_1 = Ar_2 /\\ Ar_3 = Ar_0 /\\ Ar_4 = Ar_5 ]", 0-0) = Ar_0 S("start(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(lbl121(Ar_0, Ar_1, Ar_2, Ar_3, Ar_1 - Ar_3, Ar_5)) [ Ar_0 >= 1 /\\ Ar_2 >= Ar_0 + 1 /\\ Ar_1 = Ar_2 /\\ Ar_3 = Ar_0 /\\ Ar_4 = Ar_5 ]", 0-1) = Ar_2 S("start(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(lbl121(Ar_0, Ar_1, Ar_2, Ar_3, Ar_1 - Ar_3, Ar_5)) [ Ar_0 >= 1 /\\ Ar_2 >= Ar_0 + 1 /\\ Ar_1 = Ar_2 /\\ Ar_3 = Ar_0 /\\ Ar_4 = Ar_5 ]", 0-2) = Ar_2 S("start(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(lbl121(Ar_0, Ar_1, Ar_2, Ar_3, Ar_1 - Ar_3, Ar_5)) [ Ar_0 >= 1 /\\ Ar_2 >= Ar_0 + 1 /\\ Ar_1 = Ar_2 /\\ Ar_3 = Ar_0 /\\ Ar_4 = Ar_5 ]", 0-3) = Ar_0 S("start(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(lbl121(Ar_0, Ar_1, Ar_2, Ar_3, Ar_1 - Ar_3, Ar_5)) [ Ar_0 >= 1 /\\ Ar_2 >= Ar_0 + 1 /\\ Ar_1 = Ar_2 /\\ Ar_3 = Ar_0 /\\ Ar_4 = Ar_5 ]", 0-4) = Ar_2 S("start(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(lbl121(Ar_0, Ar_1, Ar_2, Ar_3, Ar_1 - Ar_3, Ar_5)) [ Ar_0 >= 1 /\\ Ar_2 >= Ar_0 + 1 /\\ Ar_1 = Ar_2 /\\ Ar_3 = Ar_0 /\\ Ar_4 = Ar_5 ]", 0-5) = Ar_5 S("lbl6(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_0 >= Ar_2 /\\ Ar_4 = Ar_5 /\\ Ar_3 = Ar_0 /\\ Ar_1 = Ar_2 ]", 0-0) = Ar_0 S("lbl6(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_0 >= Ar_2 /\\ Ar_4 = Ar_5 /\\ Ar_3 = Ar_0 /\\ Ar_1 = Ar_2 ]", 0-1) = Ar_2 S("lbl6(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_0 >= Ar_2 /\\ Ar_4 = Ar_5 /\\ Ar_3 = Ar_0 /\\ Ar_1 = Ar_2 ]", 0-2) = Ar_2 S("lbl6(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_0 >= Ar_2 /\\ Ar_4 = Ar_5 /\\ Ar_3 = Ar_0 /\\ Ar_1 = Ar_2 ]", 0-3) = Ar_0 S("lbl6(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_0 >= Ar_2 /\\ Ar_4 = Ar_5 /\\ Ar_3 = Ar_0 /\\ Ar_1 = Ar_2 ]", 0-4) = Ar_5 S("lbl6(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_0 >= Ar_2 /\\ Ar_4 = Ar_5 /\\ Ar_3 = Ar_0 /\\ Ar_1 = Ar_2 ]", 0-5) = Ar_5 S("lbl121(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(lbl111(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4 - 1, Ar_5)) [ Ar_4 >= 1 /\\ Ar_0 >= Ar_4 + 1 /\\ Ar_2 >= Ar_0 + 1 /\\ Ar_0 >= 1 /\\ Ar_4 >= 0 /\\ Ar_2 >= Ar_4 + Ar_0 /\\ Ar_3 = Ar_0 /\\ Ar_1 = Ar_2 ]", 0-0) = Ar_0 S("lbl121(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(lbl111(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4 - 1, Ar_5)) [ Ar_4 >= 1 /\\ Ar_0 >= Ar_4 + 1 /\\ Ar_2 >= Ar_0 + 1 /\\ Ar_0 >= 1 /\\ Ar_4 >= 0 /\\ Ar_2 >= Ar_4 + Ar_0 /\\ Ar_3 = Ar_0 /\\ Ar_1 = Ar_2 ]", 0-1) = Ar_2 S("lbl121(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(lbl111(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4 - 1, Ar_5)) [ Ar_4 >= 1 /\\ Ar_0 >= Ar_4 + 1 /\\ Ar_2 >= Ar_0 + 1 /\\ Ar_0 >= 1 /\\ Ar_4 >= 0 /\\ Ar_2 >= Ar_4 + Ar_0 /\\ Ar_3 = Ar_0 /\\ Ar_1 = Ar_2 ]", 0-2) = Ar_2 S("lbl121(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(lbl111(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4 - 1, Ar_5)) [ Ar_4 >= 1 /\\ Ar_0 >= Ar_4 + 1 /\\ Ar_2 >= Ar_0 + 1 /\\ Ar_0 >= 1 /\\ Ar_4 >= 0 /\\ Ar_2 >= Ar_4 + Ar_0 /\\ Ar_3 = Ar_0 /\\ Ar_1 = Ar_2 ]", 0-3) = Ar_0 S("lbl121(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(lbl111(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4 - 1, Ar_5)) [ Ar_4 >= 1 /\\ Ar_0 >= Ar_4 + 1 /\\ Ar_2 >= Ar_0 + 1 /\\ Ar_0 >= 1 /\\ Ar_4 >= 0 /\\ Ar_2 >= Ar_4 + Ar_0 /\\ Ar_3 = Ar_0 /\\ Ar_1 = Ar_2 ]", 0-4) = Ar_2 S("lbl121(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(lbl111(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4 - 1, Ar_5)) [ Ar_4 >= 1 /\\ Ar_0 >= Ar_4 + 1 /\\ Ar_2 >= Ar_0 + 1 /\\ Ar_0 >= 1 /\\ Ar_4 >= 0 /\\ Ar_2 >= Ar_4 + Ar_0 /\\ Ar_3 = Ar_0 /\\ Ar_1 = Ar_2 ]", 0-5) = Ar_5 S("lbl121(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(lbl121(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4 - Ar_3, Ar_5)) [ Ar_4 >= 1 /\\ Ar_4 >= Ar_0 /\\ Ar_2 >= Ar_0 + 1 /\\ Ar_0 >= 1 /\\ Ar_4 >= 0 /\\ Ar_2 >= Ar_4 + Ar_0 /\\ Ar_3 = Ar_0 /\\ Ar_1 = Ar_2 ]", 0-0) = Ar_0 S("lbl121(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(lbl121(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4 - Ar_3, Ar_5)) [ Ar_4 >= 1 /\\ Ar_4 >= Ar_0 /\\ Ar_2 >= Ar_0 + 1 /\\ Ar_0 >= 1 /\\ Ar_4 >= 0 /\\ Ar_2 >= Ar_4 + Ar_0 /\\ Ar_3 = Ar_0 /\\ Ar_1 = Ar_2 ]", 0-1) = Ar_2 S("lbl121(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(lbl121(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4 - Ar_3, Ar_5)) [ Ar_4 >= 1 /\\ Ar_4 >= Ar_0 /\\ Ar_2 >= Ar_0 + 1 /\\ Ar_0 >= 1 /\\ Ar_4 >= 0 /\\ Ar_2 >= Ar_4 + Ar_0 /\\ Ar_3 = Ar_0 /\\ Ar_1 = Ar_2 ]", 0-2) = Ar_2 S("lbl121(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(lbl121(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4 - Ar_3, Ar_5)) [ Ar_4 >= 1 /\\ Ar_4 >= Ar_0 /\\ Ar_2 >= Ar_0 + 1 /\\ Ar_0 >= 1 /\\ Ar_4 >= 0 /\\ Ar_2 >= Ar_4 + Ar_0 /\\ Ar_3 = Ar_0 /\\ Ar_1 = Ar_2 ]", 0-3) = Ar_0 S("lbl121(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(lbl121(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4 - Ar_3, Ar_5)) [ Ar_4 >= 1 /\\ Ar_4 >= Ar_0 /\\ Ar_2 >= Ar_0 + 1 /\\ Ar_0 >= 1 /\\ Ar_4 >= 0 /\\ Ar_2 >= Ar_4 + Ar_0 /\\ Ar_3 = Ar_0 /\\ Ar_1 = Ar_2 ]", 0-4) = Ar_2 S("lbl121(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(lbl121(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4 - Ar_3, Ar_5)) [ Ar_4 >= 1 /\\ Ar_4 >= Ar_0 /\\ Ar_2 >= Ar_0 + 1 /\\ Ar_0 >= 1 /\\ Ar_4 >= 0 /\\ Ar_2 >= Ar_4 + Ar_0 /\\ Ar_3 = Ar_0 /\\ Ar_1 = Ar_2 ]", 0-5) = Ar_5 S("lbl111(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(cut(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5)) [ Ar_2 >= Ar_0 + 1 /\\ Ar_0 >= 2 /\\ Ar_4 = 0 /\\ Ar_3 = Ar_0 /\\ Ar_1 = Ar_2 ]", 0-0) = Ar_0 S("lbl111(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(cut(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5)) [ Ar_2 >= Ar_0 + 1 /\\ Ar_0 >= 2 /\\ Ar_4 = 0 /\\ Ar_3 = Ar_0 /\\ Ar_1 = Ar_2 ]", 0-1) = Ar_2 S("lbl111(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(cut(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5)) [ Ar_2 >= Ar_0 + 1 /\\ Ar_0 >= 2 /\\ Ar_4 = 0 /\\ Ar_3 = Ar_0 /\\ Ar_1 = Ar_2 ]", 0-2) = Ar_2 S("lbl111(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(cut(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5)) [ Ar_2 >= Ar_0 + 1 /\\ Ar_0 >= 2 /\\ Ar_4 = 0 /\\ Ar_3 = Ar_0 /\\ Ar_1 = Ar_2 ]", 0-3) = Ar_0 S("lbl111(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(cut(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5)) [ Ar_2 >= Ar_0 + 1 /\\ Ar_0 >= 2 /\\ Ar_4 = 0 /\\ Ar_3 = Ar_0 /\\ Ar_1 = Ar_2 ]", 0-4) = 0 S("lbl111(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(cut(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5)) [ Ar_2 >= Ar_0 + 1 /\\ Ar_0 >= 2 /\\ Ar_4 = 0 /\\ Ar_3 = Ar_0 /\\ Ar_1 = Ar_2 ]", 0-5) = Ar_5 S("lbl111(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(lbl111(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4 - 1, Ar_5)) [ Ar_4 >= 1 /\\ Ar_0 >= Ar_4 + 1 /\\ Ar_4 >= 0 /\\ Ar_2 >= Ar_4 + Ar_0 + 1 /\\ Ar_0 >= Ar_4 + 2 /\\ Ar_3 = Ar_0 /\\ Ar_1 = Ar_2 ]", 0-0) = Ar_0 S("lbl111(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(lbl111(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4 - 1, Ar_5)) [ Ar_4 >= 1 /\\ Ar_0 >= Ar_4 + 1 /\\ Ar_4 >= 0 /\\ Ar_2 >= Ar_4 + Ar_0 + 1 /\\ Ar_0 >= Ar_4 + 2 /\\ Ar_3 = Ar_0 /\\ Ar_1 = Ar_2 ]", 0-1) = Ar_2 S("lbl111(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(lbl111(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4 - 1, Ar_5)) [ Ar_4 >= 1 /\\ Ar_0 >= Ar_4 + 1 /\\ Ar_4 >= 0 /\\ Ar_2 >= Ar_4 + Ar_0 + 1 /\\ Ar_0 >= Ar_4 + 2 /\\ Ar_3 = Ar_0 /\\ Ar_1 = Ar_2 ]", 0-2) = Ar_2 S("lbl111(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(lbl111(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4 - 1, Ar_5)) [ Ar_4 >= 1 /\\ Ar_0 >= Ar_4 + 1 /\\ Ar_4 >= 0 /\\ Ar_2 >= Ar_4 + Ar_0 + 1 /\\ Ar_0 >= Ar_4 + 2 /\\ Ar_3 = Ar_0 /\\ Ar_1 = Ar_2 ]", 0-3) = Ar_0 S("lbl111(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(lbl111(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4 - 1, Ar_5)) [ Ar_4 >= 1 /\\ Ar_0 >= Ar_4 + 1 /\\ Ar_4 >= 0 /\\ Ar_2 >= Ar_4 + Ar_0 + 1 /\\ Ar_0 >= Ar_4 + 2 /\\ Ar_3 = Ar_0 /\\ Ar_1 = Ar_2 ]", 0-4) = Ar_2 S("lbl111(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(lbl111(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4 - 1, Ar_5)) [ Ar_4 >= 1 /\\ Ar_0 >= Ar_4 + 1 /\\ Ar_4 >= 0 /\\ Ar_2 >= Ar_4 + Ar_0 + 1 /\\ Ar_0 >= Ar_4 + 2 /\\ Ar_3 = Ar_0 /\\ Ar_1 = Ar_2 ]", 0-5) = Ar_5 S("lbl121(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(cut(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5)) [ Ar_2 >= Ar_0 + 1 /\\ Ar_0 >= 1 /\\ Ar_2 >= Ar_0 /\\ Ar_4 = 0 /\\ Ar_3 = Ar_0 /\\ Ar_1 = Ar_2 ]", 0-0) = Ar_0 S("lbl121(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(cut(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5)) [ Ar_2 >= Ar_0 + 1 /\\ Ar_0 >= 1 /\\ Ar_2 >= Ar_0 /\\ Ar_4 = 0 /\\ Ar_3 = Ar_0 /\\ Ar_1 = Ar_2 ]", 0-1) = Ar_2 S("lbl121(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(cut(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5)) [ Ar_2 >= Ar_0 + 1 /\\ Ar_0 >= 1 /\\ Ar_2 >= Ar_0 /\\ Ar_4 = 0 /\\ Ar_3 = Ar_0 /\\ Ar_1 = Ar_2 ]", 0-2) = Ar_2 S("lbl121(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(cut(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5)) [ Ar_2 >= Ar_0 + 1 /\\ Ar_0 >= 1 /\\ Ar_2 >= Ar_0 /\\ Ar_4 = 0 /\\ Ar_3 = Ar_0 /\\ Ar_1 = Ar_2 ]", 0-3) = Ar_0 S("lbl121(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(cut(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5)) [ Ar_2 >= Ar_0 + 1 /\\ Ar_0 >= 1 /\\ Ar_2 >= Ar_0 /\\ Ar_4 = 0 /\\ Ar_3 = Ar_0 /\\ Ar_1 = Ar_2 ]", 0-4) = 0 S("lbl121(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(cut(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5)) [ Ar_2 >= Ar_0 + 1 /\\ Ar_0 >= 1 /\\ Ar_2 >= Ar_0 /\\ Ar_4 = 0 /\\ Ar_3 = Ar_0 /\\ Ar_1 = Ar_2 ]", 0-5) = Ar_5 S("cut(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_2 >= Ar_0 + 1 /\\ Ar_4 = 0 /\\ Ar_3 = Ar_0 /\\ Ar_1 = Ar_2 ]", 0-0) = Ar_0 S("cut(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_2 >= Ar_0 + 1 /\\ Ar_4 = 0 /\\ Ar_3 = Ar_0 /\\ Ar_1 = Ar_2 ]", 0-1) = Ar_2 S("cut(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_2 >= Ar_0 + 1 /\\ Ar_4 = 0 /\\ Ar_3 = Ar_0 /\\ Ar_1 = Ar_2 ]", 0-2) = Ar_2 S("cut(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_2 >= Ar_0 + 1 /\\ Ar_4 = 0 /\\ Ar_3 = Ar_0 /\\ Ar_1 = Ar_2 ]", 0-3) = Ar_0 S("cut(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_2 >= Ar_0 + 1 /\\ Ar_4 = 0 /\\ Ar_3 = Ar_0 /\\ Ar_1 = Ar_2 ]", 0-4) = 0 S("cut(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_2 >= Ar_0 + 1 /\\ Ar_4 = 0 /\\ Ar_3 = Ar_0 /\\ Ar_1 = Ar_2 ]", 0-5) = Ar_5 orients the transitions lbl121(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(lbl121(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4 - Ar_3, Ar_5)) [ Ar_4 >= 1 /\ Ar_4 >= Ar_0 /\ Ar_2 >= Ar_0 + 1 /\ Ar_0 >= 1 /\ Ar_4 >= 0 /\ Ar_2 >= Ar_4 + Ar_0 /\ Ar_3 = Ar_0 /\ Ar_1 = Ar_2 ] lbl111(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(lbl111(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4 - 1, Ar_5)) [ Ar_4 >= 1 /\ Ar_0 >= Ar_4 + 1 /\ Ar_4 >= 0 /\ Ar_2 >= Ar_4 + Ar_0 + 1 /\ Ar_0 >= Ar_4 + 2 /\ Ar_3 = Ar_0 /\ Ar_1 = Ar_2 ] weakly and the transitions lbl121(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(lbl121(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4 - Ar_3, Ar_5)) [ Ar_4 >= 1 /\ Ar_4 >= Ar_0 /\ Ar_2 >= Ar_0 + 1 /\ Ar_0 >= 1 /\ Ar_4 >= 0 /\ Ar_2 >= Ar_4 + Ar_0 /\ Ar_3 = Ar_0 /\ Ar_1 = Ar_2 ] lbl111(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(lbl111(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4 - 1, Ar_5)) [ Ar_4 >= 1 /\ Ar_0 >= Ar_4 + 1 /\ Ar_4 >= 0 /\ Ar_2 >= Ar_4 + Ar_0 + 1 /\ Ar_0 >= Ar_4 + 2 /\ Ar_3 = Ar_0 /\ Ar_1 = Ar_2 ] strictly and produces the following problem: 5: T: (Comp: 3, Cost: 1) cut(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_2 >= Ar_0 + 1 /\ Ar_4 = 0 /\ Ar_3 = Ar_0 /\ Ar_1 = Ar_2 ] (Comp: 3, Cost: 1) lbl121(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(cut(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5)) [ Ar_2 >= Ar_0 + 1 /\ Ar_0 >= 1 /\ Ar_2 >= Ar_0 /\ Ar_4 = 0 /\ Ar_3 = Ar_0 /\ Ar_1 = Ar_2 ] (Comp: 4*Ar_2 + 4, Cost: 1) lbl111(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(lbl111(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4 - 1, Ar_5)) [ Ar_4 >= 1 /\ Ar_0 >= Ar_4 + 1 /\ Ar_4 >= 0 /\ Ar_2 >= Ar_4 + Ar_0 + 1 /\ Ar_0 >= Ar_4 + 2 /\ Ar_3 = Ar_0 /\ Ar_1 = Ar_2 ] (Comp: 3, Cost: 1) lbl111(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(cut(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5)) [ Ar_2 >= Ar_0 + 1 /\ Ar_0 >= 2 /\ Ar_4 = 0 /\ Ar_3 = Ar_0 /\ Ar_1 = Ar_2 ] (Comp: 4*Ar_2 + 4, Cost: 1) lbl121(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(lbl121(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4 - Ar_3, Ar_5)) [ Ar_4 >= 1 /\ Ar_4 >= Ar_0 /\ Ar_2 >= Ar_0 + 1 /\ Ar_0 >= 1 /\ Ar_4 >= 0 /\ Ar_2 >= Ar_4 + Ar_0 /\ Ar_3 = Ar_0 /\ Ar_1 = Ar_2 ] (Comp: 3, Cost: 1) lbl121(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(lbl111(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4 - 1, Ar_5)) [ Ar_4 >= 1 /\ Ar_0 >= Ar_4 + 1 /\ Ar_2 >= Ar_0 + 1 /\ Ar_0 >= 1 /\ Ar_4 >= 0 /\ Ar_2 >= Ar_4 + Ar_0 /\ Ar_3 = Ar_0 /\ Ar_1 = Ar_2 ] (Comp: 1, Cost: 1) lbl6(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_0 >= Ar_2 /\ Ar_4 = Ar_5 /\ Ar_3 = Ar_0 /\ Ar_1 = Ar_2 ] (Comp: 1, Cost: 1) start(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(lbl121(Ar_0, Ar_1, Ar_2, Ar_3, Ar_1 - Ar_3, Ar_5)) [ Ar_0 >= 1 /\ Ar_2 >= Ar_0 + 1 /\ Ar_1 = Ar_2 /\ Ar_3 = Ar_0 /\ Ar_4 = Ar_5 ] (Comp: 1, Cost: 1) start(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(lbl6(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5)) [ Ar_0 >= 1 /\ Ar_0 >= Ar_2 /\ Ar_1 = Ar_2 /\ Ar_3 = Ar_0 /\ Ar_4 = Ar_5 ] (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_3, Ar_4, Ar_5)) [ 0 >= Ar_0 /\ Ar_1 = Ar_2 /\ Ar_3 = Ar_0 /\ Ar_4 = Ar_5 ] (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_0, Ar_5, Ar_5)) (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 8*Ar_2 + 25 Time: 0.246 sec (SMT: 0.194 sec) ---------------------------------------- (2) BOUNDS(1, n^1) ---------------------------------------- (3) Loat Proof (FINISHED) ### Pre-processing the ITS problem ### Initial linear ITS problem Start location: start0 0: start -> stop : [ 0>=A && B==C && D==A && E==F ], cost: 1 1: start -> lbl6 : [ A>=1 && A>=C && B==C && D==A && E==F ], cost: 1 2: start -> lbl121 : E'=-D+B, [ A>=1 && C>=1+A && B==C && D==A && E==F ], cost: 1 3: lbl6 -> stop : [ A>=1 && A>=C && E==F && D==A && B==C ], cost: 1 4: lbl111 -> cut : [ C>=1+A && A>=2 && E==0 && D==A && B==C ], cost: 1 5: lbl111 -> lbl111 : E'=-1+E, [ E>=1 && A>=1+E && E>=0 && C>=1+A+E && A>=2+E && D==A && B==C ], cost: 1 6: lbl111 -> lbl121 : E'=-D+E, [ E>=1 && E>=A && E>=0 && C>=1+A+E && A>=2+E && D==A && B==C ], cost: 1 7: lbl121 -> cut : [ C>=1+A && A>=1 && C>=A && E==0 && D==A && B==C ], cost: 1 8: lbl121 -> lbl111 : E'=-1+E, [ E>=1 && A>=1+E && C>=1+A && A>=1 && E>=0 && C>=A+E && D==A && B==C ], cost: 1 9: lbl121 -> lbl121 : E'=-D+E, [ E>=1 && E>=A && C>=1+A && A>=1 && E>=0 && C>=A+E && D==A && B==C ], cost: 1 10: cut -> stop : [ A>=1 && C>=1+A && E==0 && D==A && B==C ], cost: 1 11: start0 -> start : B'=C, D'=A, E'=F, [], cost: 1 Checking for constant complexity: The following rule is satisfiable with cost >= 1, yielding constant complexity: 11: start0 -> start : B'=C, D'=A, E'=F, [], cost: 1 Removed unreachable and leaf rules: Start location: start0 2: start -> lbl121 : E'=-D+B, [ A>=1 && C>=1+A && B==C && D==A && E==F ], cost: 1 5: lbl111 -> lbl111 : E'=-1+E, [ E>=1 && A>=1+E && E>=0 && C>=1+A+E && A>=2+E && D==A && B==C ], cost: 1 6: lbl111 -> lbl121 : E'=-D+E, [ E>=1 && E>=A && E>=0 && C>=1+A+E && A>=2+E && D==A && B==C ], cost: 1 8: lbl121 -> lbl111 : E'=-1+E, [ E>=1 && A>=1+E && C>=1+A && A>=1 && E>=0 && C>=A+E && D==A && B==C ], cost: 1 9: lbl121 -> lbl121 : E'=-D+E, [ E>=1 && E>=A && C>=1+A && A>=1 && E>=0 && C>=A+E && D==A && B==C ], cost: 1 11: start0 -> start : B'=C, D'=A, E'=F, [], cost: 1 Removed rules with unsatisfiable guard: Start location: start0 2: start -> lbl121 : E'=-D+B, [ A>=1 && C>=1+A && B==C && D==A && E==F ], cost: 1 5: lbl111 -> lbl111 : E'=-1+E, [ E>=1 && A>=1+E && E>=0 && C>=1+A+E && A>=2+E && D==A && B==C ], cost: 1 8: lbl121 -> lbl111 : E'=-1+E, [ E>=1 && A>=1+E && C>=1+A && A>=1 && E>=0 && C>=A+E && D==A && B==C ], cost: 1 9: lbl121 -> lbl121 : E'=-D+E, [ E>=1 && E>=A && C>=1+A && A>=1 && E>=0 && C>=A+E && D==A && B==C ], cost: 1 11: start0 -> start : B'=C, D'=A, E'=F, [], cost: 1 Simplified all rules, resulting in: Start location: start0 2: start -> lbl121 : E'=-D+B, [ A>=1 && C>=1+A && B==C && D==A && E==F ], cost: 1 5: lbl111 -> lbl111 : E'=-1+E, [ E>=1 && C>=1+A+E && A>=2+E && D==A && B==C ], cost: 1 8: lbl121 -> lbl111 : E'=-1+E, [ E>=1 && A>=1+E && C>=1+A && C>=A+E && D==A && B==C ], cost: 1 9: lbl121 -> lbl121 : E'=-D+E, [ E>=A && C>=1+A && A>=1 && C>=A+E && D==A && B==C ], cost: 1 11: start0 -> start : B'=C, D'=A, E'=F, [], cost: 1 ### Simplification by acceleration and chaining ### Accelerating simple loops of location 2. Accelerating the following rules: 5: lbl111 -> lbl111 : E'=-1+E, [ E>=1 && C>=1+A+E && A>=2+E && D==A && B==C ], cost: 1 Accelerated rule 5 with metering function E, yielding the new rule 12. Removing the simple loops: 5. Accelerating simple loops of location 3. Accelerating the following rules: 9: lbl121 -> lbl121 : E'=-D+E, [ E>=A && C>=1+A && A>=1 && C>=A+E && D==A && B==C ], cost: 1 Accelerated rule 9 with metering function -D+A, yielding the new rule 13. Removing the simple loops: 9. Accelerated all simple loops using metering functions (where possible): Start location: start0 2: start -> lbl121 : E'=-D+B, [ A>=1 && C>=1+A && B==C && D==A && E==F ], cost: 1 12: lbl111 -> lbl111 : E'=0, [ E>=1 && C>=1+A+E && A>=2+E && D==A && B==C ], cost: E 8: lbl121 -> lbl111 : E'=-1+E, [ E>=1 && A>=1+E && C>=1+A && C>=A+E && D==A && B==C ], cost: 1 13: lbl121 -> lbl121 : E'=D*(D-A)+E, [ E>=A && C>=1+A && A>=1 && C>=A+E && D==A && B==C && -D+A>=1 ], cost: -D+A 11: start0 -> start : B'=C, D'=A, E'=F, [], cost: 1 Chained accelerated rules (with incoming rules): Start location: start0 2: start -> lbl121 : E'=-D+B, [ A>=1 && C>=1+A && B==C && D==A && E==F ], cost: 1 8: lbl121 -> lbl111 : E'=-1+E, [ E>=1 && A>=1+E && C>=1+A && C>=A+E && D==A && B==C ], cost: 1 14: lbl121 -> lbl111 : E'=0, [ A>=1+E && C>=1+A && C>=A+E && D==A && B==C && -1+E>=1 ], cost: E 11: start0 -> start : B'=C, D'=A, E'=F, [], cost: 1 Removed unreachable locations (and leaf rules with constant cost): Start location: start0 2: start -> lbl121 : E'=-D+B, [ A>=1 && C>=1+A && B==C && D==A && E==F ], cost: 1 14: lbl121 -> lbl111 : E'=0, [ A>=1+E && C>=1+A && C>=A+E && D==A && B==C && -1+E>=1 ], cost: E 11: start0 -> start : B'=C, D'=A, E'=F, [], cost: 1 Eliminated locations (on linear paths): Start location: start0 16: start0 -> lbl111 : B'=C, D'=A, E'=0, [ A>=1 && A>=1+C-A && -1+C-A>=1 ], cost: 2+C-A ### Computing asymptotic complexity ### Fully simplified ITS problem Start location: start0 16: start0 -> lbl111 : B'=C, D'=A, E'=0, [ A>=1 && A>=1+C-A && -1+C-A>=1 ], cost: 2+C-A Computing asymptotic complexity for rule 16 Simplified the guard: 16: start0 -> lbl111 : B'=C, D'=A, E'=0, [ A>=1+C-A && -1+C-A>=1 ], cost: 2+C-A Solved the limit problem by the following transformations: Created initial limit problem: 2+C-A (+), -C+2*A (+/+!), -1+C-A (+/+!) [not solved] removing all constraints (solved by SMT) resulting limit problem: [solved] applying transformation rule (C) using substitution {C==3*n,A==2*n} resulting limit problem: [solved] Solution: C / 3*n A / 2*n Resulting cost 2+n has complexity: Poly(n^1) Found new complexity Poly(n^1). Obtained the following overall complexity (w.r.t. the length of the input n): Complexity: Poly(n^1) Cpx degree: 1 Solved cost: 2+n Rule cost: 2+C-A Rule guard: [ A>=1+C-A && -1+C-A>=1 ] WORST_CASE(Omega(n^1),?) ---------------------------------------- (4) BOUNDS(n^1, INF)