/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(n^2, n^2).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, 375 ms] (2) BOUNDS(1, n^2) (3) Loat Proof [FINISHED, 1037 ms] (4) BOUNDS(n^2, INF) ---------------------------------------- (0) Obligation: Complexity Int TRS consisting of the following rules: evalEx1start(A, B, C, D) -> Com_1(evalEx1entryin(A, B, C, D)) :|: TRUE evalEx1entryin(A, B, C, D) -> Com_1(evalEx1bb6in(0, A, C, D)) :|: TRUE evalEx1bb6in(A, B, C, D) -> Com_1(evalEx1bbin(A, B, C, D)) :|: B >= A + 1 evalEx1bb6in(A, B, C, D) -> Com_1(evalEx1returnin(A, B, C, D)) :|: A >= B evalEx1bbin(A, B, C, D) -> Com_1(evalEx1bb4in(A, B, A + 1, B)) :|: TRUE evalEx1bb4in(A, B, C, D) -> Com_1(evalEx1bb1in(A, B, C, D)) :|: D >= C + 1 evalEx1bb4in(A, B, C, D) -> Com_1(evalEx1bb5in(A, B, C, D)) :|: C >= D evalEx1bb1in(A, B, C, D) -> Com_1(evalEx1bb4in(A, B, C, D - 1)) :|: 0 >= E + 1 evalEx1bb1in(A, B, C, D) -> Com_1(evalEx1bb4in(A, B, C, D - 1)) :|: 0 >= E + 1 && E >= 1 evalEx1bb1in(A, B, C, D) -> Com_1(evalEx1bb4in(A, B, C, D - 1)) :|: E >= 1 evalEx1bb1in(A, B, C, D) -> Com_1(evalEx1bb4in(A, B, C, D)) :|: 0 >= 1 evalEx1bb1in(A, B, C, D) -> Com_1(evalEx1bb4in(A, B, C + 1, D - 1)) :|: 0 >= 1 evalEx1bb1in(A, B, C, D) -> Com_1(evalEx1bb4in(A, B, C + 1, D)) :|: TRUE evalEx1bb5in(A, B, C, D) -> Com_1(evalEx1bb6in(A + 1, D, C, D)) :|: TRUE evalEx1returnin(A, B, C, D) -> Com_1(evalEx1stop(A, B, C, D)) :|: TRUE The start-symbols are:[evalEx1start_4] ---------------------------------------- (1) Koat Proof (FINISHED) YES(?, 22*Ar_0 + 24*Ar_0^2 + 14) Initial complexity problem: 1: T: (Comp: ?, Cost: 1) evalEx1start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1entryin(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: ?, Cost: 1) evalEx1entryin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1bb6in(0, Ar_0, Ar_2, Ar_3)) (Comp: ?, Cost: 1) evalEx1bb6in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1bbin(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_1 >= Ar_0 + 1 ] (Comp: ?, Cost: 1) evalEx1bb6in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1returnin(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= Ar_1 ] (Comp: ?, Cost: 1) evalEx1bbin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1bb4in(Ar_0, Ar_1, Ar_0 + 1, Ar_1)) (Comp: ?, Cost: 1) evalEx1bb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1bb1in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_3 >= Ar_2 + 1 ] (Comp: ?, Cost: 1) evalEx1bb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1bb5in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= Ar_3 ] (Comp: ?, Cost: 1) evalEx1bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1bb4in(Ar_0, Ar_1, Ar_2, Ar_3 - 1)) [ 0 >= E + 1 ] (Comp: ?, Cost: 1) evalEx1bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1bb4in(Ar_0, Ar_1, Ar_2, Ar_3 - 1)) [ 0 >= E + 1 /\ E >= 1 ] (Comp: ?, Cost: 1) evalEx1bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1bb4in(Ar_0, Ar_1, Ar_2, Ar_3 - 1)) [ E >= 1 ] (Comp: ?, Cost: 1) evalEx1bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1bb4in(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 >= 1 ] (Comp: ?, Cost: 1) evalEx1bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1bb4in(Ar_0, Ar_1, Ar_2 + 1, Ar_3 - 1)) [ 0 >= 1 ] (Comp: ?, Cost: 1) evalEx1bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1bb4in(Ar_0, Ar_1, Ar_2 + 1, Ar_3)) (Comp: ?, Cost: 1) evalEx1bb5in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1bb6in(Ar_0 + 1, Ar_3, Ar_2, Ar_3)) (Comp: ?, Cost: 1) evalEx1returnin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1stop(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1start(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 Testing for reachability in the complexity graph removes the following transitions from problem 1: evalEx1bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1bb4in(Ar_0, Ar_1, Ar_2, Ar_3 - 1)) [ 0 >= E + 1 /\ E >= 1 ] evalEx1bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1bb4in(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 >= 1 ] evalEx1bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1bb4in(Ar_0, Ar_1, Ar_2 + 1, Ar_3 - 1)) [ 0 >= 1 ] We thus obtain the following problem: 2: T: (Comp: ?, Cost: 1) evalEx1bb5in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1bb6in(Ar_0 + 1, Ar_3, Ar_2, Ar_3)) (Comp: ?, Cost: 1) evalEx1bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1bb4in(Ar_0, Ar_1, Ar_2 + 1, Ar_3)) (Comp: ?, Cost: 1) evalEx1bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1bb4in(Ar_0, Ar_1, Ar_2, Ar_3 - 1)) [ E >= 1 ] (Comp: ?, Cost: 1) evalEx1bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1bb4in(Ar_0, Ar_1, Ar_2, Ar_3 - 1)) [ 0 >= E + 1 ] (Comp: ?, Cost: 1) evalEx1bb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1bb5in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= Ar_3 ] (Comp: ?, Cost: 1) evalEx1bb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1bb1in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_3 >= Ar_2 + 1 ] (Comp: ?, Cost: 1) evalEx1returnin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1stop(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: ?, Cost: 1) evalEx1bbin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1bb4in(Ar_0, Ar_1, Ar_0 + 1, Ar_1)) (Comp: ?, Cost: 1) evalEx1bb6in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1returnin(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= Ar_1 ] (Comp: ?, Cost: 1) evalEx1bb6in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1bbin(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_1 >= Ar_0 + 1 ] (Comp: ?, Cost: 1) evalEx1entryin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1bb6in(0, Ar_0, Ar_2, Ar_3)) (Comp: ?, Cost: 1) evalEx1start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1entryin(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1start(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 Repeatedly propagating knowledge in problem 2 produces the following problem: 3: T: (Comp: ?, Cost: 1) evalEx1bb5in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1bb6in(Ar_0 + 1, Ar_3, Ar_2, Ar_3)) (Comp: ?, Cost: 1) evalEx1bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1bb4in(Ar_0, Ar_1, Ar_2 + 1, Ar_3)) (Comp: ?, Cost: 1) evalEx1bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1bb4in(Ar_0, Ar_1, Ar_2, Ar_3 - 1)) [ E >= 1 ] (Comp: ?, Cost: 1) evalEx1bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1bb4in(Ar_0, Ar_1, Ar_2, Ar_3 - 1)) [ 0 >= E + 1 ] (Comp: ?, Cost: 1) evalEx1bb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1bb5in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= Ar_3 ] (Comp: ?, Cost: 1) evalEx1bb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1bb1in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_3 >= Ar_2 + 1 ] (Comp: ?, Cost: 1) evalEx1returnin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1stop(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: ?, Cost: 1) evalEx1bbin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1bb4in(Ar_0, Ar_1, Ar_0 + 1, Ar_1)) (Comp: ?, Cost: 1) evalEx1bb6in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1returnin(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= Ar_1 ] (Comp: ?, Cost: 1) evalEx1bb6in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1bbin(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_1 >= Ar_0 + 1 ] (Comp: 1, Cost: 1) evalEx1entryin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1bb6in(0, Ar_0, Ar_2, Ar_3)) (Comp: 1, Cost: 1) evalEx1start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1entryin(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1start(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 A polynomial rank function with Pol(evalEx1bb5in) = 2 Pol(evalEx1bb6in) = 2 Pol(evalEx1bb1in) = 2 Pol(evalEx1bb4in) = 2 Pol(evalEx1returnin) = 1 Pol(evalEx1stop) = 0 Pol(evalEx1bbin) = 2 Pol(evalEx1entryin) = 2 Pol(evalEx1start) = 2 Pol(koat_start) = 2 orients all transitions weakly and the transitions evalEx1returnin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1stop(Ar_0, Ar_1, Ar_2, Ar_3)) evalEx1bb6in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1returnin(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= Ar_1 ] strictly and produces the following problem: 4: T: (Comp: ?, Cost: 1) evalEx1bb5in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1bb6in(Ar_0 + 1, Ar_3, Ar_2, Ar_3)) (Comp: ?, Cost: 1) evalEx1bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1bb4in(Ar_0, Ar_1, Ar_2 + 1, Ar_3)) (Comp: ?, Cost: 1) evalEx1bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1bb4in(Ar_0, Ar_1, Ar_2, Ar_3 - 1)) [ E >= 1 ] (Comp: ?, Cost: 1) evalEx1bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1bb4in(Ar_0, Ar_1, Ar_2, Ar_3 - 1)) [ 0 >= E + 1 ] (Comp: ?, Cost: 1) evalEx1bb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1bb5in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= Ar_3 ] (Comp: ?, Cost: 1) evalEx1bb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1bb1in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_3 >= Ar_2 + 1 ] (Comp: 2, Cost: 1) evalEx1returnin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1stop(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: ?, Cost: 1) evalEx1bbin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1bb4in(Ar_0, Ar_1, Ar_0 + 1, Ar_1)) (Comp: 2, Cost: 1) evalEx1bb6in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1returnin(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= Ar_1 ] (Comp: ?, Cost: 1) evalEx1bb6in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1bbin(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_1 >= Ar_0 + 1 ] (Comp: 1, Cost: 1) evalEx1entryin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1bb6in(0, Ar_0, Ar_2, Ar_3)) (Comp: 1, Cost: 1) evalEx1start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1entryin(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1start(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 A polynomial rank function with Pol(evalEx1bb5in) = -V_1 + V_4 Pol(evalEx1bb6in) = -V_1 + V_2 + 1 Pol(evalEx1bb1in) = -V_1 + V_4 Pol(evalEx1bb4in) = -V_1 + V_4 Pol(evalEx1returnin) = -V_1 + V_2 Pol(evalEx1stop) = -V_1 + V_2 Pol(evalEx1bbin) = -V_1 + V_2 Pol(evalEx1entryin) = V_1 + 1 Pol(evalEx1start) = V_1 + 1 Pol(koat_start) = V_1 + 1 orients all transitions weakly and the transition evalEx1bb6in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1bbin(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_1 >= Ar_0 + 1 ] strictly and produces the following problem: 5: T: (Comp: ?, Cost: 1) evalEx1bb5in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1bb6in(Ar_0 + 1, Ar_3, Ar_2, Ar_3)) (Comp: ?, Cost: 1) evalEx1bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1bb4in(Ar_0, Ar_1, Ar_2 + 1, Ar_3)) (Comp: ?, Cost: 1) evalEx1bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1bb4in(Ar_0, Ar_1, Ar_2, Ar_3 - 1)) [ E >= 1 ] (Comp: ?, Cost: 1) evalEx1bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1bb4in(Ar_0, Ar_1, Ar_2, Ar_3 - 1)) [ 0 >= E + 1 ] (Comp: ?, Cost: 1) evalEx1bb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1bb5in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= Ar_3 ] (Comp: ?, Cost: 1) evalEx1bb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1bb1in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_3 >= Ar_2 + 1 ] (Comp: 2, Cost: 1) evalEx1returnin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1stop(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: ?, Cost: 1) evalEx1bbin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1bb4in(Ar_0, Ar_1, Ar_0 + 1, Ar_1)) (Comp: 2, Cost: 1) evalEx1bb6in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1returnin(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= Ar_1 ] (Comp: Ar_0 + 1, Cost: 1) evalEx1bb6in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1bbin(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_1 >= Ar_0 + 1 ] (Comp: 1, Cost: 1) evalEx1entryin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1bb6in(0, Ar_0, Ar_2, Ar_3)) (Comp: 1, Cost: 1) evalEx1start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1entryin(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1start(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 Repeatedly propagating knowledge in problem 5 produces the following problem: 6: T: (Comp: ?, Cost: 1) evalEx1bb5in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1bb6in(Ar_0 + 1, Ar_3, Ar_2, Ar_3)) (Comp: ?, Cost: 1) evalEx1bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1bb4in(Ar_0, Ar_1, Ar_2 + 1, Ar_3)) (Comp: ?, Cost: 1) evalEx1bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1bb4in(Ar_0, Ar_1, Ar_2, Ar_3 - 1)) [ E >= 1 ] (Comp: ?, Cost: 1) evalEx1bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1bb4in(Ar_0, Ar_1, Ar_2, Ar_3 - 1)) [ 0 >= E + 1 ] (Comp: ?, Cost: 1) evalEx1bb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1bb5in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= Ar_3 ] (Comp: ?, Cost: 1) evalEx1bb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1bb1in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_3 >= Ar_2 + 1 ] (Comp: 2, Cost: 1) evalEx1returnin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1stop(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: Ar_0 + 1, Cost: 1) evalEx1bbin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1bb4in(Ar_0, Ar_1, Ar_0 + 1, Ar_1)) (Comp: 2, Cost: 1) evalEx1bb6in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1returnin(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= Ar_1 ] (Comp: Ar_0 + 1, Cost: 1) evalEx1bb6in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1bbin(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_1 >= Ar_0 + 1 ] (Comp: 1, Cost: 1) evalEx1entryin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1bb6in(0, Ar_0, Ar_2, Ar_3)) (Comp: 1, Cost: 1) evalEx1start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1entryin(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1start(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 A polynomial rank function with Pol(evalEx1bb5in) = 1 Pol(evalEx1bb6in) = 0 Pol(evalEx1bb4in) = 2 Pol(evalEx1bb1in) = 2 and size complexities S("koat_start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1start(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 <= 0 ]", 0-0) = Ar_0 S("koat_start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1start(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 <= 0 ]", 0-1) = Ar_1 S("koat_start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1start(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 <= 0 ]", 0-2) = Ar_2 S("koat_start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1start(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 <= 0 ]", 0-3) = Ar_3 S("evalEx1start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1entryin(Ar_0, Ar_1, Ar_2, Ar_3))", 0-0) = Ar_0 S("evalEx1start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1entryin(Ar_0, Ar_1, Ar_2, Ar_3))", 0-1) = Ar_1 S("evalEx1start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1entryin(Ar_0, Ar_1, Ar_2, Ar_3))", 0-2) = Ar_2 S("evalEx1start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1entryin(Ar_0, Ar_1, Ar_2, Ar_3))", 0-3) = Ar_3 S("evalEx1entryin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1bb6in(0, Ar_0, Ar_2, Ar_3))", 0-0) = 0 S("evalEx1entryin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1bb6in(0, Ar_0, Ar_2, Ar_3))", 0-1) = Ar_0 S("evalEx1entryin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1bb6in(0, Ar_0, Ar_2, Ar_3))", 0-2) = Ar_2 S("evalEx1entryin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1bb6in(0, Ar_0, Ar_2, Ar_3))", 0-3) = Ar_3 S("evalEx1bb6in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1bbin(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_1 >= Ar_0 + 1 ]", 0-0) = ? S("evalEx1bb6in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1bbin(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_1 >= Ar_0 + 1 ]", 0-1) = ? S("evalEx1bb6in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1bbin(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_1 >= Ar_0 + 1 ]", 0-2) = ? S("evalEx1bb6in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1bbin(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_1 >= Ar_0 + 1 ]", 0-3) = ? S("evalEx1bb6in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1returnin(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= Ar_1 ]", 0-0) = ? S("evalEx1bb6in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1returnin(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= Ar_1 ]", 0-1) = ? S("evalEx1bb6in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1returnin(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= Ar_1 ]", 0-2) = ? S("evalEx1bb6in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1returnin(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= Ar_1 ]", 0-3) = ? S("evalEx1bbin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1bb4in(Ar_0, Ar_1, Ar_0 + 1, Ar_1))", 0-0) = ? S("evalEx1bbin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1bb4in(Ar_0, Ar_1, Ar_0 + 1, Ar_1))", 0-1) = ? S("evalEx1bbin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1bb4in(Ar_0, Ar_1, Ar_0 + 1, Ar_1))", 0-2) = ? S("evalEx1bbin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1bb4in(Ar_0, Ar_1, Ar_0 + 1, Ar_1))", 0-3) = ? S("evalEx1returnin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1stop(Ar_0, Ar_1, Ar_2, Ar_3))", 0-0) = ? S("evalEx1returnin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1stop(Ar_0, Ar_1, Ar_2, Ar_3))", 0-1) = ? S("evalEx1returnin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1stop(Ar_0, Ar_1, Ar_2, Ar_3))", 0-2) = ? S("evalEx1returnin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1stop(Ar_0, Ar_1, Ar_2, Ar_3))", 0-3) = ? S("evalEx1bb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1bb1in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_3 >= Ar_2 + 1 ]", 0-0) = ? S("evalEx1bb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1bb1in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_3 >= Ar_2 + 1 ]", 0-1) = ? S("evalEx1bb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1bb1in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_3 >= Ar_2 + 1 ]", 0-2) = ? S("evalEx1bb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1bb1in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_3 >= Ar_2 + 1 ]", 0-3) = ? S("evalEx1bb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1bb5in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= Ar_3 ]", 0-0) = ? S("evalEx1bb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1bb5in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= Ar_3 ]", 0-1) = ? S("evalEx1bb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1bb5in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= Ar_3 ]", 0-2) = ? S("evalEx1bb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1bb5in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= Ar_3 ]", 0-3) = ? S("evalEx1bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1bb4in(Ar_0, Ar_1, Ar_2, Ar_3 - 1)) [ 0 >= E + 1 ]", 0-0) = ? S("evalEx1bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1bb4in(Ar_0, Ar_1, Ar_2, Ar_3 - 1)) [ 0 >= E + 1 ]", 0-1) = ? S("evalEx1bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1bb4in(Ar_0, Ar_1, Ar_2, Ar_3 - 1)) [ 0 >= E + 1 ]", 0-2) = ? S("evalEx1bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1bb4in(Ar_0, Ar_1, Ar_2, Ar_3 - 1)) [ 0 >= E + 1 ]", 0-3) = ? S("evalEx1bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1bb4in(Ar_0, Ar_1, Ar_2, Ar_3 - 1)) [ E >= 1 ]", 0-0) = ? S("evalEx1bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1bb4in(Ar_0, Ar_1, Ar_2, Ar_3 - 1)) [ E >= 1 ]", 0-1) = ? S("evalEx1bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1bb4in(Ar_0, Ar_1, Ar_2, Ar_3 - 1)) [ E >= 1 ]", 0-2) = ? S("evalEx1bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1bb4in(Ar_0, Ar_1, Ar_2, Ar_3 - 1)) [ E >= 1 ]", 0-3) = ? S("evalEx1bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1bb4in(Ar_0, Ar_1, Ar_2 + 1, Ar_3))", 0-0) = ? S("evalEx1bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1bb4in(Ar_0, Ar_1, Ar_2 + 1, Ar_3))", 0-1) = ? S("evalEx1bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1bb4in(Ar_0, Ar_1, Ar_2 + 1, Ar_3))", 0-2) = ? S("evalEx1bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1bb4in(Ar_0, Ar_1, Ar_2 + 1, Ar_3))", 0-3) = ? S("evalEx1bb5in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1bb6in(Ar_0 + 1, Ar_3, Ar_2, Ar_3))", 0-0) = ? S("evalEx1bb5in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1bb6in(Ar_0 + 1, Ar_3, Ar_2, Ar_3))", 0-1) = ? S("evalEx1bb5in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1bb6in(Ar_0 + 1, Ar_3, Ar_2, Ar_3))", 0-2) = ? S("evalEx1bb5in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1bb6in(Ar_0 + 1, Ar_3, Ar_2, Ar_3))", 0-3) = ? orients the transitions evalEx1bb5in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1bb6in(Ar_0 + 1, Ar_3, Ar_2, Ar_3)) evalEx1bb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1bb5in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= Ar_3 ] evalEx1bb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1bb1in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_3 >= Ar_2 + 1 ] evalEx1bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1bb4in(Ar_0, Ar_1, Ar_2 + 1, Ar_3)) evalEx1bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1bb4in(Ar_0, Ar_1, Ar_2, Ar_3 - 1)) [ E >= 1 ] evalEx1bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1bb4in(Ar_0, Ar_1, Ar_2, Ar_3 - 1)) [ 0 >= E + 1 ] weakly and the transitions evalEx1bb5in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1bb6in(Ar_0 + 1, Ar_3, Ar_2, Ar_3)) evalEx1bb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1bb5in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= Ar_3 ] strictly and produces the following problem: 7: T: (Comp: 2*Ar_0 + 2, Cost: 1) evalEx1bb5in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1bb6in(Ar_0 + 1, Ar_3, Ar_2, Ar_3)) (Comp: ?, Cost: 1) evalEx1bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1bb4in(Ar_0, Ar_1, Ar_2 + 1, Ar_3)) (Comp: ?, Cost: 1) evalEx1bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1bb4in(Ar_0, Ar_1, Ar_2, Ar_3 - 1)) [ E >= 1 ] (Comp: ?, Cost: 1) evalEx1bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1bb4in(Ar_0, Ar_1, Ar_2, Ar_3 - 1)) [ 0 >= E + 1 ] (Comp: 2*Ar_0 + 2, Cost: 1) evalEx1bb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1bb5in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= Ar_3 ] (Comp: ?, Cost: 1) evalEx1bb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1bb1in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_3 >= Ar_2 + 1 ] (Comp: 2, Cost: 1) evalEx1returnin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1stop(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: Ar_0 + 1, Cost: 1) evalEx1bbin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1bb4in(Ar_0, Ar_1, Ar_0 + 1, Ar_1)) (Comp: 2, Cost: 1) evalEx1bb6in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1returnin(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= Ar_1 ] (Comp: Ar_0 + 1, Cost: 1) evalEx1bb6in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1bbin(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_1 >= Ar_0 + 1 ] (Comp: 1, Cost: 1) evalEx1entryin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1bb6in(0, Ar_0, Ar_2, Ar_3)) (Comp: 1, Cost: 1) evalEx1start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1entryin(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1start(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 Applied AI with 'oct' on problem 7 to obtain the following invariants: For symbol evalEx1bb1in: X_2 - X_4 >= 0 /\ X_4 - 2 >= 0 /\ X_3 + X_4 - 3 >= 0 /\ -X_3 + X_4 - 1 >= 0 /\ X_2 + X_4 - 4 >= 0 /\ X_1 + X_4 - 2 >= 0 /\ -X_1 + X_4 - 2 >= 0 /\ X_2 - X_3 - 1 >= 0 /\ X_3 - 1 >= 0 /\ X_2 + X_3 - 3 >= 0 /\ X_1 + X_3 - 1 >= 0 /\ -X_1 + X_3 - 1 >= 0 /\ X_2 - 2 >= 0 /\ X_1 + X_2 - 2 >= 0 /\ -X_1 + X_2 - 2 >= 0 /\ X_1 >= 0 For symbol evalEx1bb4in: X_2 - X_4 >= 0 /\ X_4 - 1 >= 0 /\ X_3 + X_4 - 2 >= 0 /\ -X_3 + X_4 >= 0 /\ X_2 + X_4 - 2 >= 0 /\ X_1 + X_4 - 1 >= 0 /\ -X_1 + X_4 - 1 >= 0 /\ X_2 - X_3 >= 0 /\ X_3 - 1 >= 0 /\ X_2 + X_3 - 2 >= 0 /\ X_1 + X_3 - 1 >= 0 /\ -X_1 + X_3 - 1 >= 0 /\ X_2 - 1 >= 0 /\ X_1 + X_2 - 1 >= 0 /\ -X_1 + X_2 - 1 >= 0 /\ X_1 >= 0 For symbol evalEx1bb5in: X_3 - X_4 >= 0 /\ X_2 - X_4 >= 0 /\ X_4 - 1 >= 0 /\ X_3 + X_4 - 2 >= 0 /\ -X_3 + X_4 >= 0 /\ X_2 + X_4 - 2 >= 0 /\ X_1 + X_4 - 1 >= 0 /\ -X_1 + X_4 - 1 >= 0 /\ X_2 - X_3 >= 0 /\ X_3 - 1 >= 0 /\ X_2 + X_3 - 2 >= 0 /\ X_1 + X_3 - 1 >= 0 /\ -X_1 + X_3 - 1 >= 0 /\ X_2 - 1 >= 0 /\ X_1 + X_2 - 1 >= 0 /\ -X_1 + X_2 - 1 >= 0 /\ X_1 >= 0 For symbol evalEx1bb6in: X_1 >= 0 For symbol evalEx1bbin: X_2 - 1 >= 0 /\ X_1 + X_2 - 1 >= 0 /\ -X_1 + X_2 - 1 >= 0 /\ X_1 >= 0 For symbol evalEx1returnin: X_1 - X_2 >= 0 /\ X_1 >= 0 This yielded the following problem: 8: T: (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1start(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 <= 0 ] (Comp: 1, Cost: 1) evalEx1start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1entryin(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 1) evalEx1entryin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1bb6in(0, Ar_0, Ar_2, Ar_3)) (Comp: Ar_0 + 1, Cost: 1) evalEx1bb6in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1bbin(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= 0 /\ Ar_1 >= Ar_0 + 1 ] (Comp: 2, Cost: 1) evalEx1bb6in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1returnin(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= 0 /\ Ar_0 >= Ar_1 ] (Comp: Ar_0 + 1, Cost: 1) evalEx1bbin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1bb4in(Ar_0, Ar_1, Ar_0 + 1, Ar_1)) [ Ar_1 - 1 >= 0 /\ Ar_0 + Ar_1 - 1 >= 0 /\ -Ar_0 + Ar_1 - 1 >= 0 /\ Ar_0 >= 0 ] (Comp: 2, Cost: 1) evalEx1returnin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1stop(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 - Ar_1 >= 0 /\ Ar_0 >= 0 ] (Comp: ?, Cost: 1) evalEx1bb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1bb1in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_1 - Ar_3 >= 0 /\ Ar_3 - 1 >= 0 /\ Ar_2 + Ar_3 - 2 >= 0 /\ -Ar_2 + Ar_3 >= 0 /\ Ar_1 + Ar_3 - 2 >= 0 /\ Ar_0 + Ar_3 - 1 >= 0 /\ -Ar_0 + Ar_3 - 1 >= 0 /\ Ar_1 - Ar_2 >= 0 /\ Ar_2 - 1 >= 0 /\ Ar_1 + Ar_2 - 2 >= 0 /\ Ar_0 + Ar_2 - 1 >= 0 /\ -Ar_0 + Ar_2 - 1 >= 0 /\ Ar_1 - 1 >= 0 /\ Ar_0 + Ar_1 - 1 >= 0 /\ -Ar_0 + Ar_1 - 1 >= 0 /\ Ar_0 >= 0 /\ Ar_3 >= Ar_2 + 1 ] (Comp: 2*Ar_0 + 2, Cost: 1) evalEx1bb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1bb5in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_1 - Ar_3 >= 0 /\ Ar_3 - 1 >= 0 /\ Ar_2 + Ar_3 - 2 >= 0 /\ -Ar_2 + Ar_3 >= 0 /\ Ar_1 + Ar_3 - 2 >= 0 /\ Ar_0 + Ar_3 - 1 >= 0 /\ -Ar_0 + Ar_3 - 1 >= 0 /\ Ar_1 - Ar_2 >= 0 /\ Ar_2 - 1 >= 0 /\ Ar_1 + Ar_2 - 2 >= 0 /\ Ar_0 + Ar_2 - 1 >= 0 /\ -Ar_0 + Ar_2 - 1 >= 0 /\ Ar_1 - 1 >= 0 /\ Ar_0 + Ar_1 - 1 >= 0 /\ -Ar_0 + Ar_1 - 1 >= 0 /\ Ar_0 >= 0 /\ Ar_2 >= Ar_3 ] (Comp: ?, Cost: 1) evalEx1bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1bb4in(Ar_0, Ar_1, Ar_2, Ar_3 - 1)) [ Ar_1 - Ar_3 >= 0 /\ Ar_3 - 2 >= 0 /\ Ar_2 + Ar_3 - 3 >= 0 /\ -Ar_2 + Ar_3 - 1 >= 0 /\ Ar_1 + Ar_3 - 4 >= 0 /\ Ar_0 + Ar_3 - 2 >= 0 /\ -Ar_0 + Ar_3 - 2 >= 0 /\ Ar_1 - Ar_2 - 1 >= 0 /\ Ar_2 - 1 >= 0 /\ Ar_1 + Ar_2 - 3 >= 0 /\ Ar_0 + Ar_2 - 1 >= 0 /\ -Ar_0 + Ar_2 - 1 >= 0 /\ Ar_1 - 2 >= 0 /\ Ar_0 + Ar_1 - 2 >= 0 /\ -Ar_0 + Ar_1 - 2 >= 0 /\ Ar_0 >= 0 /\ 0 >= E + 1 ] (Comp: ?, Cost: 1) evalEx1bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1bb4in(Ar_0, Ar_1, Ar_2, Ar_3 - 1)) [ Ar_1 - Ar_3 >= 0 /\ Ar_3 - 2 >= 0 /\ Ar_2 + Ar_3 - 3 >= 0 /\ -Ar_2 + Ar_3 - 1 >= 0 /\ Ar_1 + Ar_3 - 4 >= 0 /\ Ar_0 + Ar_3 - 2 >= 0 /\ -Ar_0 + Ar_3 - 2 >= 0 /\ Ar_1 - Ar_2 - 1 >= 0 /\ Ar_2 - 1 >= 0 /\ Ar_1 + Ar_2 - 3 >= 0 /\ Ar_0 + Ar_2 - 1 >= 0 /\ -Ar_0 + Ar_2 - 1 >= 0 /\ Ar_1 - 2 >= 0 /\ Ar_0 + Ar_1 - 2 >= 0 /\ -Ar_0 + Ar_1 - 2 >= 0 /\ Ar_0 >= 0 /\ E >= 1 ] (Comp: ?, Cost: 1) evalEx1bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1bb4in(Ar_0, Ar_1, Ar_2 + 1, Ar_3)) [ Ar_1 - Ar_3 >= 0 /\ Ar_3 - 2 >= 0 /\ Ar_2 + Ar_3 - 3 >= 0 /\ -Ar_2 + Ar_3 - 1 >= 0 /\ Ar_1 + Ar_3 - 4 >= 0 /\ Ar_0 + Ar_3 - 2 >= 0 /\ -Ar_0 + Ar_3 - 2 >= 0 /\ Ar_1 - Ar_2 - 1 >= 0 /\ Ar_2 - 1 >= 0 /\ Ar_1 + Ar_2 - 3 >= 0 /\ Ar_0 + Ar_2 - 1 >= 0 /\ -Ar_0 + Ar_2 - 1 >= 0 /\ Ar_1 - 2 >= 0 /\ Ar_0 + Ar_1 - 2 >= 0 /\ -Ar_0 + Ar_1 - 2 >= 0 /\ Ar_0 >= 0 ] (Comp: 2*Ar_0 + 2, Cost: 1) evalEx1bb5in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1bb6in(Ar_0 + 1, Ar_3, Ar_2, Ar_3)) [ Ar_2 - Ar_3 >= 0 /\ Ar_1 - Ar_3 >= 0 /\ Ar_3 - 1 >= 0 /\ Ar_2 + Ar_3 - 2 >= 0 /\ -Ar_2 + Ar_3 >= 0 /\ Ar_1 + Ar_3 - 2 >= 0 /\ Ar_0 + Ar_3 - 1 >= 0 /\ -Ar_0 + Ar_3 - 1 >= 0 /\ Ar_1 - Ar_2 >= 0 /\ Ar_2 - 1 >= 0 /\ Ar_1 + Ar_2 - 2 >= 0 /\ Ar_0 + Ar_2 - 1 >= 0 /\ -Ar_0 + Ar_2 - 1 >= 0 /\ Ar_1 - 1 >= 0 /\ Ar_0 + Ar_1 - 1 >= 0 /\ -Ar_0 + Ar_1 - 1 >= 0 /\ Ar_0 >= 0 ] start location: koat_start leaf cost: 0 A polynomial rank function with Pol(koat_start) = V_1 Pol(evalEx1start) = V_1 Pol(evalEx1entryin) = V_1 Pol(evalEx1bb6in) = V_2 Pol(evalEx1bbin) = V_2 Pol(evalEx1returnin) = V_2 Pol(evalEx1bb4in) = V_4 Pol(evalEx1stop) = V_2 Pol(evalEx1bb1in) = V_4 Pol(evalEx1bb5in) = V_4 orients all transitions weakly and the transitions evalEx1bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1bb4in(Ar_0, Ar_1, Ar_2, Ar_3 - 1)) [ Ar_1 - Ar_3 >= 0 /\ Ar_3 - 2 >= 0 /\ Ar_2 + Ar_3 - 3 >= 0 /\ -Ar_2 + Ar_3 - 1 >= 0 /\ Ar_1 + Ar_3 - 4 >= 0 /\ Ar_0 + Ar_3 - 2 >= 0 /\ -Ar_0 + Ar_3 - 2 >= 0 /\ Ar_1 - Ar_2 - 1 >= 0 /\ Ar_2 - 1 >= 0 /\ Ar_1 + Ar_2 - 3 >= 0 /\ Ar_0 + Ar_2 - 1 >= 0 /\ -Ar_0 + Ar_2 - 1 >= 0 /\ Ar_1 - 2 >= 0 /\ Ar_0 + Ar_1 - 2 >= 0 /\ -Ar_0 + Ar_1 - 2 >= 0 /\ Ar_0 >= 0 /\ E >= 1 ] evalEx1bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1bb4in(Ar_0, Ar_1, Ar_2, Ar_3 - 1)) [ Ar_1 - Ar_3 >= 0 /\ Ar_3 - 2 >= 0 /\ Ar_2 + Ar_3 - 3 >= 0 /\ -Ar_2 + Ar_3 - 1 >= 0 /\ Ar_1 + Ar_3 - 4 >= 0 /\ Ar_0 + Ar_3 - 2 >= 0 /\ -Ar_0 + Ar_3 - 2 >= 0 /\ Ar_1 - Ar_2 - 1 >= 0 /\ Ar_2 - 1 >= 0 /\ Ar_1 + Ar_2 - 3 >= 0 /\ Ar_0 + Ar_2 - 1 >= 0 /\ -Ar_0 + Ar_2 - 1 >= 0 /\ Ar_1 - 2 >= 0 /\ Ar_0 + Ar_1 - 2 >= 0 /\ -Ar_0 + Ar_1 - 2 >= 0 /\ Ar_0 >= 0 /\ 0 >= E + 1 ] strictly and produces the following problem: 9: T: (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1start(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 <= 0 ] (Comp: 1, Cost: 1) evalEx1start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1entryin(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 1) evalEx1entryin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1bb6in(0, Ar_0, Ar_2, Ar_3)) (Comp: Ar_0 + 1, Cost: 1) evalEx1bb6in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1bbin(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= 0 /\ Ar_1 >= Ar_0 + 1 ] (Comp: 2, Cost: 1) evalEx1bb6in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1returnin(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= 0 /\ Ar_0 >= Ar_1 ] (Comp: Ar_0 + 1, Cost: 1) evalEx1bbin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1bb4in(Ar_0, Ar_1, Ar_0 + 1, Ar_1)) [ Ar_1 - 1 >= 0 /\ Ar_0 + Ar_1 - 1 >= 0 /\ -Ar_0 + Ar_1 - 1 >= 0 /\ Ar_0 >= 0 ] (Comp: 2, Cost: 1) evalEx1returnin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1stop(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 - Ar_1 >= 0 /\ Ar_0 >= 0 ] (Comp: ?, Cost: 1) evalEx1bb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1bb1in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_1 - Ar_3 >= 0 /\ Ar_3 - 1 >= 0 /\ Ar_2 + Ar_3 - 2 >= 0 /\ -Ar_2 + Ar_3 >= 0 /\ Ar_1 + Ar_3 - 2 >= 0 /\ Ar_0 + Ar_3 - 1 >= 0 /\ -Ar_0 + Ar_3 - 1 >= 0 /\ Ar_1 - Ar_2 >= 0 /\ Ar_2 - 1 >= 0 /\ Ar_1 + Ar_2 - 2 >= 0 /\ Ar_0 + Ar_2 - 1 >= 0 /\ -Ar_0 + Ar_2 - 1 >= 0 /\ Ar_1 - 1 >= 0 /\ Ar_0 + Ar_1 - 1 >= 0 /\ -Ar_0 + Ar_1 - 1 >= 0 /\ Ar_0 >= 0 /\ Ar_3 >= Ar_2 + 1 ] (Comp: 2*Ar_0 + 2, Cost: 1) evalEx1bb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1bb5in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_1 - Ar_3 >= 0 /\ Ar_3 - 1 >= 0 /\ Ar_2 + Ar_3 - 2 >= 0 /\ -Ar_2 + Ar_3 >= 0 /\ Ar_1 + Ar_3 - 2 >= 0 /\ Ar_0 + Ar_3 - 1 >= 0 /\ -Ar_0 + Ar_3 - 1 >= 0 /\ Ar_1 - Ar_2 >= 0 /\ Ar_2 - 1 >= 0 /\ Ar_1 + Ar_2 - 2 >= 0 /\ Ar_0 + Ar_2 - 1 >= 0 /\ -Ar_0 + Ar_2 - 1 >= 0 /\ Ar_1 - 1 >= 0 /\ Ar_0 + Ar_1 - 1 >= 0 /\ -Ar_0 + Ar_1 - 1 >= 0 /\ Ar_0 >= 0 /\ Ar_2 >= Ar_3 ] (Comp: Ar_0, Cost: 1) evalEx1bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1bb4in(Ar_0, Ar_1, Ar_2, Ar_3 - 1)) [ Ar_1 - Ar_3 >= 0 /\ Ar_3 - 2 >= 0 /\ Ar_2 + Ar_3 - 3 >= 0 /\ -Ar_2 + Ar_3 - 1 >= 0 /\ Ar_1 + Ar_3 - 4 >= 0 /\ Ar_0 + Ar_3 - 2 >= 0 /\ -Ar_0 + Ar_3 - 2 >= 0 /\ Ar_1 - Ar_2 - 1 >= 0 /\ Ar_2 - 1 >= 0 /\ Ar_1 + Ar_2 - 3 >= 0 /\ Ar_0 + Ar_2 - 1 >= 0 /\ -Ar_0 + Ar_2 - 1 >= 0 /\ Ar_1 - 2 >= 0 /\ Ar_0 + Ar_1 - 2 >= 0 /\ -Ar_0 + Ar_1 - 2 >= 0 /\ Ar_0 >= 0 /\ 0 >= E + 1 ] (Comp: Ar_0, Cost: 1) evalEx1bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1bb4in(Ar_0, Ar_1, Ar_2, Ar_3 - 1)) [ Ar_1 - Ar_3 >= 0 /\ Ar_3 - 2 >= 0 /\ Ar_2 + Ar_3 - 3 >= 0 /\ -Ar_2 + Ar_3 - 1 >= 0 /\ Ar_1 + Ar_3 - 4 >= 0 /\ Ar_0 + Ar_3 - 2 >= 0 /\ -Ar_0 + Ar_3 - 2 >= 0 /\ Ar_1 - Ar_2 - 1 >= 0 /\ Ar_2 - 1 >= 0 /\ Ar_1 + Ar_2 - 3 >= 0 /\ Ar_0 + Ar_2 - 1 >= 0 /\ -Ar_0 + Ar_2 - 1 >= 0 /\ Ar_1 - 2 >= 0 /\ Ar_0 + Ar_1 - 2 >= 0 /\ -Ar_0 + Ar_1 - 2 >= 0 /\ Ar_0 >= 0 /\ E >= 1 ] (Comp: ?, Cost: 1) evalEx1bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1bb4in(Ar_0, Ar_1, Ar_2 + 1, Ar_3)) [ Ar_1 - Ar_3 >= 0 /\ Ar_3 - 2 >= 0 /\ Ar_2 + Ar_3 - 3 >= 0 /\ -Ar_2 + Ar_3 - 1 >= 0 /\ Ar_1 + Ar_3 - 4 >= 0 /\ Ar_0 + Ar_3 - 2 >= 0 /\ -Ar_0 + Ar_3 - 2 >= 0 /\ Ar_1 - Ar_2 - 1 >= 0 /\ Ar_2 - 1 >= 0 /\ Ar_1 + Ar_2 - 3 >= 0 /\ Ar_0 + Ar_2 - 1 >= 0 /\ -Ar_0 + Ar_2 - 1 >= 0 /\ Ar_1 - 2 >= 0 /\ Ar_0 + Ar_1 - 2 >= 0 /\ -Ar_0 + Ar_1 - 2 >= 0 /\ Ar_0 >= 0 ] (Comp: 2*Ar_0 + 2, Cost: 1) evalEx1bb5in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1bb6in(Ar_0 + 1, Ar_3, Ar_2, Ar_3)) [ Ar_2 - Ar_3 >= 0 /\ Ar_1 - Ar_3 >= 0 /\ Ar_3 - 1 >= 0 /\ Ar_2 + Ar_3 - 2 >= 0 /\ -Ar_2 + Ar_3 >= 0 /\ Ar_1 + Ar_3 - 2 >= 0 /\ Ar_0 + Ar_3 - 1 >= 0 /\ -Ar_0 + Ar_3 - 1 >= 0 /\ Ar_1 - Ar_2 >= 0 /\ Ar_2 - 1 >= 0 /\ Ar_1 + Ar_2 - 2 >= 0 /\ Ar_0 + Ar_2 - 1 >= 0 /\ -Ar_0 + Ar_2 - 1 >= 0 /\ Ar_1 - 1 >= 0 /\ Ar_0 + Ar_1 - 1 >= 0 /\ -Ar_0 + Ar_1 - 1 >= 0 /\ Ar_0 >= 0 ] start location: koat_start leaf cost: 0 A polynomial rank function with Pol(evalEx1bb4in) = -2*V_3 + 2*V_4 + 1 Pol(evalEx1bb1in) = -2*V_3 + 2*V_4 and size complexities S("evalEx1bb5in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1bb6in(Ar_0 + 1, Ar_3, Ar_2, Ar_3)) [ Ar_2 - Ar_3 >= 0 /\\ Ar_1 - Ar_3 >= 0 /\\ Ar_3 - 1 >= 0 /\\ Ar_2 + Ar_3 - 2 >= 0 /\\ -Ar_2 + Ar_3 >= 0 /\\ Ar_1 + Ar_3 - 2 >= 0 /\\ Ar_0 + Ar_3 - 1 >= 0 /\\ -Ar_0 + Ar_3 - 1 >= 0 /\\ Ar_1 - Ar_2 >= 0 /\\ Ar_2 - 1 >= 0 /\\ Ar_1 + Ar_2 - 2 >= 0 /\\ Ar_0 + Ar_2 - 1 >= 0 /\\ -Ar_0 + Ar_2 - 1 >= 0 /\\ Ar_1 - 1 >= 0 /\\ Ar_0 + Ar_1 - 1 >= 0 /\\ -Ar_0 + Ar_1 - 1 >= 0 /\\ Ar_0 >= 0 ]", 0-0) = Ar_0 S("evalEx1bb5in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1bb6in(Ar_0 + 1, Ar_3, Ar_2, Ar_3)) [ Ar_2 - Ar_3 >= 0 /\\ Ar_1 - Ar_3 >= 0 /\\ Ar_3 - 1 >= 0 /\\ Ar_2 + Ar_3 - 2 >= 0 /\\ -Ar_2 + Ar_3 >= 0 /\\ Ar_1 + Ar_3 - 2 >= 0 /\\ Ar_0 + Ar_3 - 1 >= 0 /\\ -Ar_0 + Ar_3 - 1 >= 0 /\\ Ar_1 - Ar_2 >= 0 /\\ Ar_2 - 1 >= 0 /\\ Ar_1 + Ar_2 - 2 >= 0 /\\ Ar_0 + Ar_2 - 1 >= 0 /\\ -Ar_0 + Ar_2 - 1 >= 0 /\\ Ar_1 - 1 >= 0 /\\ Ar_0 + Ar_1 - 1 >= 0 /\\ -Ar_0 + Ar_1 - 1 >= 0 /\\ Ar_0 >= 0 ]", 0-1) = Ar_0 S("evalEx1bb5in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1bb6in(Ar_0 + 1, Ar_3, Ar_2, Ar_3)) [ Ar_2 - Ar_3 >= 0 /\\ Ar_1 - Ar_3 >= 0 /\\ Ar_3 - 1 >= 0 /\\ Ar_2 + Ar_3 - 2 >= 0 /\\ -Ar_2 + Ar_3 >= 0 /\\ Ar_1 + Ar_3 - 2 >= 0 /\\ Ar_0 + Ar_3 - 1 >= 0 /\\ -Ar_0 + Ar_3 - 1 >= 0 /\\ Ar_1 - Ar_2 >= 0 /\\ Ar_2 - 1 >= 0 /\\ Ar_1 + Ar_2 - 2 >= 0 /\\ Ar_0 + Ar_2 - 1 >= 0 /\\ -Ar_0 + Ar_2 - 1 >= 0 /\\ Ar_1 - 1 >= 0 /\\ Ar_0 + Ar_1 - 1 >= 0 /\\ -Ar_0 + Ar_1 - 1 >= 0 /\\ Ar_0 >= 0 ]", 0-2) = Ar_0 S("evalEx1bb5in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1bb6in(Ar_0 + 1, Ar_3, Ar_2, Ar_3)) [ Ar_2 - Ar_3 >= 0 /\\ Ar_1 - Ar_3 >= 0 /\\ Ar_3 - 1 >= 0 /\\ Ar_2 + Ar_3 - 2 >= 0 /\\ -Ar_2 + Ar_3 >= 0 /\\ Ar_1 + Ar_3 - 2 >= 0 /\\ Ar_0 + Ar_3 - 1 >= 0 /\\ -Ar_0 + Ar_3 - 1 >= 0 /\\ Ar_1 - Ar_2 >= 0 /\\ Ar_2 - 1 >= 0 /\\ Ar_1 + Ar_2 - 2 >= 0 /\\ Ar_0 + Ar_2 - 1 >= 0 /\\ -Ar_0 + Ar_2 - 1 >= 0 /\\ Ar_1 - 1 >= 0 /\\ Ar_0 + Ar_1 - 1 >= 0 /\\ -Ar_0 + Ar_1 - 1 >= 0 /\\ Ar_0 >= 0 ]", 0-3) = Ar_0 S("evalEx1bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1bb4in(Ar_0, Ar_1, Ar_2 + 1, Ar_3)) [ Ar_1 - Ar_3 >= 0 /\\ Ar_3 - 2 >= 0 /\\ Ar_2 + Ar_3 - 3 >= 0 /\\ -Ar_2 + Ar_3 - 1 >= 0 /\\ Ar_1 + Ar_3 - 4 >= 0 /\\ Ar_0 + Ar_3 - 2 >= 0 /\\ -Ar_0 + Ar_3 - 2 >= 0 /\\ Ar_1 - Ar_2 - 1 >= 0 /\\ Ar_2 - 1 >= 0 /\\ Ar_1 + Ar_2 - 3 >= 0 /\\ Ar_0 + Ar_2 - 1 >= 0 /\\ -Ar_0 + Ar_2 - 1 >= 0 /\\ Ar_1 - 2 >= 0 /\\ Ar_0 + Ar_1 - 2 >= 0 /\\ -Ar_0 + Ar_1 - 2 >= 0 /\\ Ar_0 >= 0 ]", 0-0) = Ar_0 S("evalEx1bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1bb4in(Ar_0, Ar_1, Ar_2 + 1, Ar_3)) [ Ar_1 - Ar_3 >= 0 /\\ Ar_3 - 2 >= 0 /\\ Ar_2 + Ar_3 - 3 >= 0 /\\ -Ar_2 + Ar_3 - 1 >= 0 /\\ Ar_1 + Ar_3 - 4 >= 0 /\\ Ar_0 + Ar_3 - 2 >= 0 /\\ -Ar_0 + Ar_3 - 2 >= 0 /\\ Ar_1 - Ar_2 - 1 >= 0 /\\ Ar_2 - 1 >= 0 /\\ Ar_1 + Ar_2 - 3 >= 0 /\\ Ar_0 + Ar_2 - 1 >= 0 /\\ -Ar_0 + Ar_2 - 1 >= 0 /\\ Ar_1 - 2 >= 0 /\\ Ar_0 + Ar_1 - 2 >= 0 /\\ -Ar_0 + Ar_1 - 2 >= 0 /\\ Ar_0 >= 0 ]", 0-1) = Ar_0 S("evalEx1bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1bb4in(Ar_0, Ar_1, Ar_2 + 1, Ar_3)) [ Ar_1 - Ar_3 >= 0 /\\ Ar_3 - 2 >= 0 /\\ Ar_2 + Ar_3 - 3 >= 0 /\\ -Ar_2 + Ar_3 - 1 >= 0 /\\ Ar_1 + Ar_3 - 4 >= 0 /\\ Ar_0 + Ar_3 - 2 >= 0 /\\ -Ar_0 + Ar_3 - 2 >= 0 /\\ Ar_1 - Ar_2 - 1 >= 0 /\\ Ar_2 - 1 >= 0 /\\ Ar_1 + Ar_2 - 3 >= 0 /\\ Ar_0 + Ar_2 - 1 >= 0 /\\ -Ar_0 + Ar_2 - 1 >= 0 /\\ Ar_1 - 2 >= 0 /\\ Ar_0 + Ar_1 - 2 >= 0 /\\ -Ar_0 + Ar_1 - 2 >= 0 /\\ Ar_0 >= 0 ]", 0-2) = Ar_0 S("evalEx1bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1bb4in(Ar_0, Ar_1, Ar_2 + 1, Ar_3)) [ Ar_1 - Ar_3 >= 0 /\\ Ar_3 - 2 >= 0 /\\ Ar_2 + Ar_3 - 3 >= 0 /\\ -Ar_2 + Ar_3 - 1 >= 0 /\\ Ar_1 + Ar_3 - 4 >= 0 /\\ Ar_0 + Ar_3 - 2 >= 0 /\\ -Ar_0 + Ar_3 - 2 >= 0 /\\ Ar_1 - Ar_2 - 1 >= 0 /\\ Ar_2 - 1 >= 0 /\\ Ar_1 + Ar_2 - 3 >= 0 /\\ Ar_0 + Ar_2 - 1 >= 0 /\\ -Ar_0 + Ar_2 - 1 >= 0 /\\ Ar_1 - 2 >= 0 /\\ Ar_0 + Ar_1 - 2 >= 0 /\\ -Ar_0 + Ar_1 - 2 >= 0 /\\ Ar_0 >= 0 ]", 0-3) = Ar_0 S("evalEx1bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1bb4in(Ar_0, Ar_1, Ar_2, Ar_3 - 1)) [ Ar_1 - Ar_3 >= 0 /\\ Ar_3 - 2 >= 0 /\\ Ar_2 + Ar_3 - 3 >= 0 /\\ -Ar_2 + Ar_3 - 1 >= 0 /\\ Ar_1 + Ar_3 - 4 >= 0 /\\ Ar_0 + Ar_3 - 2 >= 0 /\\ -Ar_0 + Ar_3 - 2 >= 0 /\\ Ar_1 - Ar_2 - 1 >= 0 /\\ Ar_2 - 1 >= 0 /\\ Ar_1 + Ar_2 - 3 >= 0 /\\ Ar_0 + Ar_2 - 1 >= 0 /\\ -Ar_0 + Ar_2 - 1 >= 0 /\\ Ar_1 - 2 >= 0 /\\ Ar_0 + Ar_1 - 2 >= 0 /\\ -Ar_0 + Ar_1 - 2 >= 0 /\\ Ar_0 >= 0 /\\ E >= 1 ]", 0-0) = Ar_0 S("evalEx1bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1bb4in(Ar_0, Ar_1, Ar_2, Ar_3 - 1)) [ Ar_1 - Ar_3 >= 0 /\\ Ar_3 - 2 >= 0 /\\ Ar_2 + Ar_3 - 3 >= 0 /\\ -Ar_2 + Ar_3 - 1 >= 0 /\\ Ar_1 + Ar_3 - 4 >= 0 /\\ Ar_0 + Ar_3 - 2 >= 0 /\\ -Ar_0 + Ar_3 - 2 >= 0 /\\ Ar_1 - Ar_2 - 1 >= 0 /\\ Ar_2 - 1 >= 0 /\\ Ar_1 + Ar_2 - 3 >= 0 /\\ Ar_0 + Ar_2 - 1 >= 0 /\\ -Ar_0 + Ar_2 - 1 >= 0 /\\ Ar_1 - 2 >= 0 /\\ Ar_0 + Ar_1 - 2 >= 0 /\\ -Ar_0 + Ar_1 - 2 >= 0 /\\ Ar_0 >= 0 /\\ E >= 1 ]", 0-1) = Ar_0 S("evalEx1bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1bb4in(Ar_0, Ar_1, Ar_2, Ar_3 - 1)) [ Ar_1 - Ar_3 >= 0 /\\ Ar_3 - 2 >= 0 /\\ Ar_2 + Ar_3 - 3 >= 0 /\\ -Ar_2 + Ar_3 - 1 >= 0 /\\ Ar_1 + Ar_3 - 4 >= 0 /\\ Ar_0 + Ar_3 - 2 >= 0 /\\ -Ar_0 + Ar_3 - 2 >= 0 /\\ Ar_1 - Ar_2 - 1 >= 0 /\\ Ar_2 - 1 >= 0 /\\ Ar_1 + Ar_2 - 3 >= 0 /\\ Ar_0 + Ar_2 - 1 >= 0 /\\ -Ar_0 + Ar_2 - 1 >= 0 /\\ Ar_1 - 2 >= 0 /\\ Ar_0 + Ar_1 - 2 >= 0 /\\ -Ar_0 + Ar_1 - 2 >= 0 /\\ Ar_0 >= 0 /\\ E >= 1 ]", 0-2) = Ar_0 S("evalEx1bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1bb4in(Ar_0, Ar_1, Ar_2, Ar_3 - 1)) [ Ar_1 - Ar_3 >= 0 /\\ Ar_3 - 2 >= 0 /\\ Ar_2 + Ar_3 - 3 >= 0 /\\ -Ar_2 + Ar_3 - 1 >= 0 /\\ Ar_1 + Ar_3 - 4 >= 0 /\\ Ar_0 + Ar_3 - 2 >= 0 /\\ -Ar_0 + Ar_3 - 2 >= 0 /\\ Ar_1 - Ar_2 - 1 >= 0 /\\ Ar_2 - 1 >= 0 /\\ Ar_1 + Ar_2 - 3 >= 0 /\\ Ar_0 + Ar_2 - 1 >= 0 /\\ -Ar_0 + Ar_2 - 1 >= 0 /\\ Ar_1 - 2 >= 0 /\\ Ar_0 + Ar_1 - 2 >= 0 /\\ -Ar_0 + Ar_1 - 2 >= 0 /\\ Ar_0 >= 0 /\\ E >= 1 ]", 0-3) = Ar_0 S("evalEx1bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1bb4in(Ar_0, Ar_1, Ar_2, Ar_3 - 1)) [ Ar_1 - Ar_3 >= 0 /\\ Ar_3 - 2 >= 0 /\\ Ar_2 + Ar_3 - 3 >= 0 /\\ -Ar_2 + Ar_3 - 1 >= 0 /\\ Ar_1 + Ar_3 - 4 >= 0 /\\ Ar_0 + Ar_3 - 2 >= 0 /\\ -Ar_0 + Ar_3 - 2 >= 0 /\\ Ar_1 - Ar_2 - 1 >= 0 /\\ Ar_2 - 1 >= 0 /\\ Ar_1 + Ar_2 - 3 >= 0 /\\ Ar_0 + Ar_2 - 1 >= 0 /\\ -Ar_0 + Ar_2 - 1 >= 0 /\\ Ar_1 - 2 >= 0 /\\ Ar_0 + Ar_1 - 2 >= 0 /\\ -Ar_0 + Ar_1 - 2 >= 0 /\\ Ar_0 >= 0 /\\ 0 >= E + 1 ]", 0-0) = Ar_0 S("evalEx1bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1bb4in(Ar_0, Ar_1, Ar_2, Ar_3 - 1)) [ Ar_1 - Ar_3 >= 0 /\\ Ar_3 - 2 >= 0 /\\ Ar_2 + Ar_3 - 3 >= 0 /\\ -Ar_2 + Ar_3 - 1 >= 0 /\\ Ar_1 + Ar_3 - 4 >= 0 /\\ Ar_0 + Ar_3 - 2 >= 0 /\\ -Ar_0 + Ar_3 - 2 >= 0 /\\ Ar_1 - Ar_2 - 1 >= 0 /\\ Ar_2 - 1 >= 0 /\\ Ar_1 + Ar_2 - 3 >= 0 /\\ Ar_0 + Ar_2 - 1 >= 0 /\\ -Ar_0 + Ar_2 - 1 >= 0 /\\ Ar_1 - 2 >= 0 /\\ Ar_0 + Ar_1 - 2 >= 0 /\\ -Ar_0 + Ar_1 - 2 >= 0 /\\ Ar_0 >= 0 /\\ 0 >= E + 1 ]", 0-1) = Ar_0 S("evalEx1bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1bb4in(Ar_0, Ar_1, Ar_2, Ar_3 - 1)) [ Ar_1 - Ar_3 >= 0 /\\ Ar_3 - 2 >= 0 /\\ Ar_2 + Ar_3 - 3 >= 0 /\\ -Ar_2 + Ar_3 - 1 >= 0 /\\ Ar_1 + Ar_3 - 4 >= 0 /\\ Ar_0 + Ar_3 - 2 >= 0 /\\ -Ar_0 + Ar_3 - 2 >= 0 /\\ Ar_1 - Ar_2 - 1 >= 0 /\\ Ar_2 - 1 >= 0 /\\ Ar_1 + Ar_2 - 3 >= 0 /\\ Ar_0 + Ar_2 - 1 >= 0 /\\ -Ar_0 + Ar_2 - 1 >= 0 /\\ Ar_1 - 2 >= 0 /\\ Ar_0 + Ar_1 - 2 >= 0 /\\ -Ar_0 + Ar_1 - 2 >= 0 /\\ Ar_0 >= 0 /\\ 0 >= E + 1 ]", 0-2) = Ar_0 S("evalEx1bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1bb4in(Ar_0, Ar_1, Ar_2, Ar_3 - 1)) [ Ar_1 - Ar_3 >= 0 /\\ Ar_3 - 2 >= 0 /\\ Ar_2 + Ar_3 - 3 >= 0 /\\ -Ar_2 + Ar_3 - 1 >= 0 /\\ Ar_1 + Ar_3 - 4 >= 0 /\\ Ar_0 + Ar_3 - 2 >= 0 /\\ -Ar_0 + Ar_3 - 2 >= 0 /\\ Ar_1 - Ar_2 - 1 >= 0 /\\ Ar_2 - 1 >= 0 /\\ Ar_1 + Ar_2 - 3 >= 0 /\\ Ar_0 + Ar_2 - 1 >= 0 /\\ -Ar_0 + Ar_2 - 1 >= 0 /\\ Ar_1 - 2 >= 0 /\\ Ar_0 + Ar_1 - 2 >= 0 /\\ -Ar_0 + Ar_1 - 2 >= 0 /\\ Ar_0 >= 0 /\\ 0 >= E + 1 ]", 0-3) = Ar_0 S("evalEx1bb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1bb5in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_1 - Ar_3 >= 0 /\\ Ar_3 - 1 >= 0 /\\ Ar_2 + Ar_3 - 2 >= 0 /\\ -Ar_2 + Ar_3 >= 0 /\\ Ar_1 + Ar_3 - 2 >= 0 /\\ Ar_0 + Ar_3 - 1 >= 0 /\\ -Ar_0 + Ar_3 - 1 >= 0 /\\ Ar_1 - Ar_2 >= 0 /\\ Ar_2 - 1 >= 0 /\\ Ar_1 + Ar_2 - 2 >= 0 /\\ Ar_0 + Ar_2 - 1 >= 0 /\\ -Ar_0 + Ar_2 - 1 >= 0 /\\ Ar_1 - 1 >= 0 /\\ Ar_0 + Ar_1 - 1 >= 0 /\\ -Ar_0 + Ar_1 - 1 >= 0 /\\ Ar_0 >= 0 /\\ Ar_2 >= Ar_3 ]", 0-0) = Ar_0 S("evalEx1bb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1bb5in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_1 - Ar_3 >= 0 /\\ Ar_3 - 1 >= 0 /\\ Ar_2 + Ar_3 - 2 >= 0 /\\ -Ar_2 + Ar_3 >= 0 /\\ Ar_1 + Ar_3 - 2 >= 0 /\\ Ar_0 + Ar_3 - 1 >= 0 /\\ -Ar_0 + Ar_3 - 1 >= 0 /\\ Ar_1 - Ar_2 >= 0 /\\ Ar_2 - 1 >= 0 /\\ Ar_1 + Ar_2 - 2 >= 0 /\\ Ar_0 + Ar_2 - 1 >= 0 /\\ -Ar_0 + Ar_2 - 1 >= 0 /\\ Ar_1 - 1 >= 0 /\\ Ar_0 + Ar_1 - 1 >= 0 /\\ -Ar_0 + Ar_1 - 1 >= 0 /\\ Ar_0 >= 0 /\\ Ar_2 >= Ar_3 ]", 0-1) = Ar_0 S("evalEx1bb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1bb5in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_1 - Ar_3 >= 0 /\\ Ar_3 - 1 >= 0 /\\ Ar_2 + Ar_3 - 2 >= 0 /\\ -Ar_2 + Ar_3 >= 0 /\\ Ar_1 + Ar_3 - 2 >= 0 /\\ Ar_0 + Ar_3 - 1 >= 0 /\\ -Ar_0 + Ar_3 - 1 >= 0 /\\ Ar_1 - Ar_2 >= 0 /\\ Ar_2 - 1 >= 0 /\\ Ar_1 + Ar_2 - 2 >= 0 /\\ Ar_0 + Ar_2 - 1 >= 0 /\\ -Ar_0 + Ar_2 - 1 >= 0 /\\ Ar_1 - 1 >= 0 /\\ Ar_0 + Ar_1 - 1 >= 0 /\\ -Ar_0 + Ar_1 - 1 >= 0 /\\ Ar_0 >= 0 /\\ Ar_2 >= Ar_3 ]", 0-2) = Ar_0 S("evalEx1bb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1bb5in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_1 - Ar_3 >= 0 /\\ Ar_3 - 1 >= 0 /\\ Ar_2 + Ar_3 - 2 >= 0 /\\ -Ar_2 + Ar_3 >= 0 /\\ Ar_1 + Ar_3 - 2 >= 0 /\\ Ar_0 + Ar_3 - 1 >= 0 /\\ -Ar_0 + Ar_3 - 1 >= 0 /\\ Ar_1 - Ar_2 >= 0 /\\ Ar_2 - 1 >= 0 /\\ Ar_1 + Ar_2 - 2 >= 0 /\\ Ar_0 + Ar_2 - 1 >= 0 /\\ -Ar_0 + Ar_2 - 1 >= 0 /\\ Ar_1 - 1 >= 0 /\\ Ar_0 + Ar_1 - 1 >= 0 /\\ -Ar_0 + Ar_1 - 1 >= 0 /\\ Ar_0 >= 0 /\\ Ar_2 >= Ar_3 ]", 0-3) = Ar_0 S("evalEx1bb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1bb1in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_1 - Ar_3 >= 0 /\\ Ar_3 - 1 >= 0 /\\ Ar_2 + Ar_3 - 2 >= 0 /\\ -Ar_2 + Ar_3 >= 0 /\\ Ar_1 + Ar_3 - 2 >= 0 /\\ Ar_0 + Ar_3 - 1 >= 0 /\\ -Ar_0 + Ar_3 - 1 >= 0 /\\ Ar_1 - Ar_2 >= 0 /\\ Ar_2 - 1 >= 0 /\\ Ar_1 + Ar_2 - 2 >= 0 /\\ Ar_0 + Ar_2 - 1 >= 0 /\\ -Ar_0 + Ar_2 - 1 >= 0 /\\ Ar_1 - 1 >= 0 /\\ Ar_0 + Ar_1 - 1 >= 0 /\\ -Ar_0 + Ar_1 - 1 >= 0 /\\ Ar_0 >= 0 /\\ Ar_3 >= Ar_2 + 1 ]", 0-0) = Ar_0 S("evalEx1bb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1bb1in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_1 - Ar_3 >= 0 /\\ Ar_3 - 1 >= 0 /\\ Ar_2 + Ar_3 - 2 >= 0 /\\ -Ar_2 + Ar_3 >= 0 /\\ Ar_1 + Ar_3 - 2 >= 0 /\\ Ar_0 + Ar_3 - 1 >= 0 /\\ -Ar_0 + Ar_3 - 1 >= 0 /\\ Ar_1 - Ar_2 >= 0 /\\ Ar_2 - 1 >= 0 /\\ Ar_1 + Ar_2 - 2 >= 0 /\\ Ar_0 + Ar_2 - 1 >= 0 /\\ -Ar_0 + Ar_2 - 1 >= 0 /\\ Ar_1 - 1 >= 0 /\\ Ar_0 + Ar_1 - 1 >= 0 /\\ -Ar_0 + Ar_1 - 1 >= 0 /\\ Ar_0 >= 0 /\\ Ar_3 >= Ar_2 + 1 ]", 0-1) = Ar_0 S("evalEx1bb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1bb1in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_1 - Ar_3 >= 0 /\\ Ar_3 - 1 >= 0 /\\ Ar_2 + Ar_3 - 2 >= 0 /\\ -Ar_2 + Ar_3 >= 0 /\\ Ar_1 + Ar_3 - 2 >= 0 /\\ Ar_0 + Ar_3 - 1 >= 0 /\\ -Ar_0 + Ar_3 - 1 >= 0 /\\ Ar_1 - Ar_2 >= 0 /\\ Ar_2 - 1 >= 0 /\\ Ar_1 + Ar_2 - 2 >= 0 /\\ Ar_0 + Ar_2 - 1 >= 0 /\\ -Ar_0 + Ar_2 - 1 >= 0 /\\ Ar_1 - 1 >= 0 /\\ Ar_0 + Ar_1 - 1 >= 0 /\\ -Ar_0 + Ar_1 - 1 >= 0 /\\ Ar_0 >= 0 /\\ Ar_3 >= Ar_2 + 1 ]", 0-2) = Ar_0 S("evalEx1bb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1bb1in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_1 - Ar_3 >= 0 /\\ Ar_3 - 1 >= 0 /\\ Ar_2 + Ar_3 - 2 >= 0 /\\ -Ar_2 + Ar_3 >= 0 /\\ Ar_1 + Ar_3 - 2 >= 0 /\\ Ar_0 + Ar_3 - 1 >= 0 /\\ -Ar_0 + Ar_3 - 1 >= 0 /\\ Ar_1 - Ar_2 >= 0 /\\ Ar_2 - 1 >= 0 /\\ Ar_1 + Ar_2 - 2 >= 0 /\\ Ar_0 + Ar_2 - 1 >= 0 /\\ -Ar_0 + Ar_2 - 1 >= 0 /\\ Ar_1 - 1 >= 0 /\\ Ar_0 + Ar_1 - 1 >= 0 /\\ -Ar_0 + Ar_1 - 1 >= 0 /\\ Ar_0 >= 0 /\\ Ar_3 >= Ar_2 + 1 ]", 0-3) = Ar_0 S("evalEx1returnin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1stop(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 - Ar_1 >= 0 /\\ Ar_0 >= 0 ]", 0-0) = Ar_0 S("evalEx1returnin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1stop(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 - Ar_1 >= 0 /\\ Ar_0 >= 0 ]", 0-1) = Ar_0 S("evalEx1returnin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1stop(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 - Ar_1 >= 0 /\\ Ar_0 >= 0 ]", 0-2) = Ar_0 + Ar_2 S("evalEx1returnin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1stop(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 - Ar_1 >= 0 /\\ Ar_0 >= 0 ]", 0-3) = Ar_0 + Ar_3 S("evalEx1bbin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1bb4in(Ar_0, Ar_1, Ar_0 + 1, Ar_1)) [ Ar_1 - 1 >= 0 /\\ Ar_0 + Ar_1 - 1 >= 0 /\\ -Ar_0 + Ar_1 - 1 >= 0 /\\ Ar_0 >= 0 ]", 0-0) = Ar_0 S("evalEx1bbin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1bb4in(Ar_0, Ar_1, Ar_0 + 1, Ar_1)) [ Ar_1 - 1 >= 0 /\\ Ar_0 + Ar_1 - 1 >= 0 /\\ -Ar_0 + Ar_1 - 1 >= 0 /\\ Ar_0 >= 0 ]", 0-1) = Ar_0 S("evalEx1bbin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1bb4in(Ar_0, Ar_1, Ar_0 + 1, Ar_1)) [ Ar_1 - 1 >= 0 /\\ Ar_0 + Ar_1 - 1 >= 0 /\\ -Ar_0 + Ar_1 - 1 >= 0 /\\ Ar_0 >= 0 ]", 0-2) = Ar_0 S("evalEx1bbin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1bb4in(Ar_0, Ar_1, Ar_0 + 1, Ar_1)) [ Ar_1 - 1 >= 0 /\\ Ar_0 + Ar_1 - 1 >= 0 /\\ -Ar_0 + Ar_1 - 1 >= 0 /\\ Ar_0 >= 0 ]", 0-3) = Ar_0 S("evalEx1bb6in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1returnin(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= 0 /\\ Ar_0 >= Ar_1 ]", 0-0) = Ar_0 S("evalEx1bb6in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1returnin(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= 0 /\\ Ar_0 >= Ar_1 ]", 0-1) = Ar_0 S("evalEx1bb6in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1returnin(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= 0 /\\ Ar_0 >= Ar_1 ]", 0-2) = Ar_0 + Ar_2 S("evalEx1bb6in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1returnin(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= 0 /\\ Ar_0 >= Ar_1 ]", 0-3) = Ar_0 + Ar_3 S("evalEx1bb6in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1bbin(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= 0 /\\ Ar_1 >= Ar_0 + 1 ]", 0-0) = Ar_0 S("evalEx1bb6in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1bbin(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= 0 /\\ Ar_1 >= Ar_0 + 1 ]", 0-1) = Ar_0 S("evalEx1bb6in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1bbin(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= 0 /\\ Ar_1 >= Ar_0 + 1 ]", 0-2) = Ar_0 + Ar_2 S("evalEx1bb6in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1bbin(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= 0 /\\ Ar_1 >= Ar_0 + 1 ]", 0-3) = Ar_0 + Ar_3 S("evalEx1entryin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1bb6in(0, Ar_0, Ar_2, Ar_3))", 0-0) = 0 S("evalEx1entryin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1bb6in(0, Ar_0, Ar_2, Ar_3))", 0-1) = Ar_0 S("evalEx1entryin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1bb6in(0, Ar_0, Ar_2, Ar_3))", 0-2) = Ar_2 S("evalEx1entryin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1bb6in(0, Ar_0, Ar_2, Ar_3))", 0-3) = Ar_3 S("evalEx1start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1entryin(Ar_0, Ar_1, Ar_2, Ar_3))", 0-0) = Ar_0 S("evalEx1start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1entryin(Ar_0, Ar_1, Ar_2, Ar_3))", 0-1) = Ar_1 S("evalEx1start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1entryin(Ar_0, Ar_1, Ar_2, Ar_3))", 0-2) = Ar_2 S("evalEx1start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1entryin(Ar_0, Ar_1, Ar_2, Ar_3))", 0-3) = Ar_3 S("koat_start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1start(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 <= 0 ]", 0-0) = Ar_0 S("koat_start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1start(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 <= 0 ]", 0-1) = Ar_1 S("koat_start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1start(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 <= 0 ]", 0-2) = Ar_2 S("koat_start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1start(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 <= 0 ]", 0-3) = Ar_3 orients the transitions evalEx1bb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1bb1in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_1 - Ar_3 >= 0 /\ Ar_3 - 1 >= 0 /\ Ar_2 + Ar_3 - 2 >= 0 /\ -Ar_2 + Ar_3 >= 0 /\ Ar_1 + Ar_3 - 2 >= 0 /\ Ar_0 + Ar_3 - 1 >= 0 /\ -Ar_0 + Ar_3 - 1 >= 0 /\ Ar_1 - Ar_2 >= 0 /\ Ar_2 - 1 >= 0 /\ Ar_1 + Ar_2 - 2 >= 0 /\ Ar_0 + Ar_2 - 1 >= 0 /\ -Ar_0 + Ar_2 - 1 >= 0 /\ Ar_1 - 1 >= 0 /\ Ar_0 + Ar_1 - 1 >= 0 /\ -Ar_0 + Ar_1 - 1 >= 0 /\ Ar_0 >= 0 /\ Ar_3 >= Ar_2 + 1 ] evalEx1bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1bb4in(Ar_0, Ar_1, Ar_2 + 1, Ar_3)) [ Ar_1 - Ar_3 >= 0 /\ Ar_3 - 2 >= 0 /\ Ar_2 + Ar_3 - 3 >= 0 /\ -Ar_2 + Ar_3 - 1 >= 0 /\ Ar_1 + Ar_3 - 4 >= 0 /\ Ar_0 + Ar_3 - 2 >= 0 /\ -Ar_0 + Ar_3 - 2 >= 0 /\ Ar_1 - Ar_2 - 1 >= 0 /\ Ar_2 - 1 >= 0 /\ Ar_1 + Ar_2 - 3 >= 0 /\ Ar_0 + Ar_2 - 1 >= 0 /\ -Ar_0 + Ar_2 - 1 >= 0 /\ Ar_1 - 2 >= 0 /\ Ar_0 + Ar_1 - 2 >= 0 /\ -Ar_0 + Ar_1 - 2 >= 0 /\ Ar_0 >= 0 ] weakly and the transitions evalEx1bb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1bb1in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_1 - Ar_3 >= 0 /\ Ar_3 - 1 >= 0 /\ Ar_2 + Ar_3 - 2 >= 0 /\ -Ar_2 + Ar_3 >= 0 /\ Ar_1 + Ar_3 - 2 >= 0 /\ Ar_0 + Ar_3 - 1 >= 0 /\ -Ar_0 + Ar_3 - 1 >= 0 /\ Ar_1 - Ar_2 >= 0 /\ Ar_2 - 1 >= 0 /\ Ar_1 + Ar_2 - 2 >= 0 /\ Ar_0 + Ar_2 - 1 >= 0 /\ -Ar_0 + Ar_2 - 1 >= 0 /\ Ar_1 - 1 >= 0 /\ Ar_0 + Ar_1 - 1 >= 0 /\ -Ar_0 + Ar_1 - 1 >= 0 /\ Ar_0 >= 0 /\ Ar_3 >= Ar_2 + 1 ] evalEx1bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1bb4in(Ar_0, Ar_1, Ar_2 + 1, Ar_3)) [ Ar_1 - Ar_3 >= 0 /\ Ar_3 - 2 >= 0 /\ Ar_2 + Ar_3 - 3 >= 0 /\ -Ar_2 + Ar_3 - 1 >= 0 /\ Ar_1 + Ar_3 - 4 >= 0 /\ Ar_0 + Ar_3 - 2 >= 0 /\ -Ar_0 + Ar_3 - 2 >= 0 /\ Ar_1 - Ar_2 - 1 >= 0 /\ Ar_2 - 1 >= 0 /\ Ar_1 + Ar_2 - 3 >= 0 /\ Ar_0 + Ar_2 - 1 >= 0 /\ -Ar_0 + Ar_2 - 1 >= 0 /\ Ar_1 - 2 >= 0 /\ Ar_0 + Ar_1 - 2 >= 0 /\ -Ar_0 + Ar_1 - 2 >= 0 /\ Ar_0 >= 0 ] strictly and produces the following problem: 10: T: (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1start(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 <= 0 ] (Comp: 1, Cost: 1) evalEx1start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1entryin(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 1) evalEx1entryin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1bb6in(0, Ar_0, Ar_2, Ar_3)) (Comp: Ar_0 + 1, Cost: 1) evalEx1bb6in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1bbin(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= 0 /\ Ar_1 >= Ar_0 + 1 ] (Comp: 2, Cost: 1) evalEx1bb6in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1returnin(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= 0 /\ Ar_0 >= Ar_1 ] (Comp: Ar_0 + 1, Cost: 1) evalEx1bbin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1bb4in(Ar_0, Ar_1, Ar_0 + 1, Ar_1)) [ Ar_1 - 1 >= 0 /\ Ar_0 + Ar_1 - 1 >= 0 /\ -Ar_0 + Ar_1 - 1 >= 0 /\ Ar_0 >= 0 ] (Comp: 2, Cost: 1) evalEx1returnin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1stop(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 - Ar_1 >= 0 /\ Ar_0 >= 0 ] (Comp: 12*Ar_0^2 + 7*Ar_0 + 1, Cost: 1) evalEx1bb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1bb1in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_1 - Ar_3 >= 0 /\ Ar_3 - 1 >= 0 /\ Ar_2 + Ar_3 - 2 >= 0 /\ -Ar_2 + Ar_3 >= 0 /\ Ar_1 + Ar_3 - 2 >= 0 /\ Ar_0 + Ar_3 - 1 >= 0 /\ -Ar_0 + Ar_3 - 1 >= 0 /\ Ar_1 - Ar_2 >= 0 /\ Ar_2 - 1 >= 0 /\ Ar_1 + Ar_2 - 2 >= 0 /\ Ar_0 + Ar_2 - 1 >= 0 /\ -Ar_0 + Ar_2 - 1 >= 0 /\ Ar_1 - 1 >= 0 /\ Ar_0 + Ar_1 - 1 >= 0 /\ -Ar_0 + Ar_1 - 1 >= 0 /\ Ar_0 >= 0 /\ Ar_3 >= Ar_2 + 1 ] (Comp: 2*Ar_0 + 2, Cost: 1) evalEx1bb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1bb5in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_1 - Ar_3 >= 0 /\ Ar_3 - 1 >= 0 /\ Ar_2 + Ar_3 - 2 >= 0 /\ -Ar_2 + Ar_3 >= 0 /\ Ar_1 + Ar_3 - 2 >= 0 /\ Ar_0 + Ar_3 - 1 >= 0 /\ -Ar_0 + Ar_3 - 1 >= 0 /\ Ar_1 - Ar_2 >= 0 /\ Ar_2 - 1 >= 0 /\ Ar_1 + Ar_2 - 2 >= 0 /\ Ar_0 + Ar_2 - 1 >= 0 /\ -Ar_0 + Ar_2 - 1 >= 0 /\ Ar_1 - 1 >= 0 /\ Ar_0 + Ar_1 - 1 >= 0 /\ -Ar_0 + Ar_1 - 1 >= 0 /\ Ar_0 >= 0 /\ Ar_2 >= Ar_3 ] (Comp: Ar_0, Cost: 1) evalEx1bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1bb4in(Ar_0, Ar_1, Ar_2, Ar_3 - 1)) [ Ar_1 - Ar_3 >= 0 /\ Ar_3 - 2 >= 0 /\ Ar_2 + Ar_3 - 3 >= 0 /\ -Ar_2 + Ar_3 - 1 >= 0 /\ Ar_1 + Ar_3 - 4 >= 0 /\ Ar_0 + Ar_3 - 2 >= 0 /\ -Ar_0 + Ar_3 - 2 >= 0 /\ Ar_1 - Ar_2 - 1 >= 0 /\ Ar_2 - 1 >= 0 /\ Ar_1 + Ar_2 - 3 >= 0 /\ Ar_0 + Ar_2 - 1 >= 0 /\ -Ar_0 + Ar_2 - 1 >= 0 /\ Ar_1 - 2 >= 0 /\ Ar_0 + Ar_1 - 2 >= 0 /\ -Ar_0 + Ar_1 - 2 >= 0 /\ Ar_0 >= 0 /\ 0 >= E + 1 ] (Comp: Ar_0, Cost: 1) evalEx1bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1bb4in(Ar_0, Ar_1, Ar_2, Ar_3 - 1)) [ Ar_1 - Ar_3 >= 0 /\ Ar_3 - 2 >= 0 /\ Ar_2 + Ar_3 - 3 >= 0 /\ -Ar_2 + Ar_3 - 1 >= 0 /\ Ar_1 + Ar_3 - 4 >= 0 /\ Ar_0 + Ar_3 - 2 >= 0 /\ -Ar_0 + Ar_3 - 2 >= 0 /\ Ar_1 - Ar_2 - 1 >= 0 /\ Ar_2 - 1 >= 0 /\ Ar_1 + Ar_2 - 3 >= 0 /\ Ar_0 + Ar_2 - 1 >= 0 /\ -Ar_0 + Ar_2 - 1 >= 0 /\ Ar_1 - 2 >= 0 /\ Ar_0 + Ar_1 - 2 >= 0 /\ -Ar_0 + Ar_1 - 2 >= 0 /\ Ar_0 >= 0 /\ E >= 1 ] (Comp: 12*Ar_0^2 + 7*Ar_0 + 1, Cost: 1) evalEx1bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1bb4in(Ar_0, Ar_1, Ar_2 + 1, Ar_3)) [ Ar_1 - Ar_3 >= 0 /\ Ar_3 - 2 >= 0 /\ Ar_2 + Ar_3 - 3 >= 0 /\ -Ar_2 + Ar_3 - 1 >= 0 /\ Ar_1 + Ar_3 - 4 >= 0 /\ Ar_0 + Ar_3 - 2 >= 0 /\ -Ar_0 + Ar_3 - 2 >= 0 /\ Ar_1 - Ar_2 - 1 >= 0 /\ Ar_2 - 1 >= 0 /\ Ar_1 + Ar_2 - 3 >= 0 /\ Ar_0 + Ar_2 - 1 >= 0 /\ -Ar_0 + Ar_2 - 1 >= 0 /\ Ar_1 - 2 >= 0 /\ Ar_0 + Ar_1 - 2 >= 0 /\ -Ar_0 + Ar_1 - 2 >= 0 /\ Ar_0 >= 0 ] (Comp: 2*Ar_0 + 2, Cost: 1) evalEx1bb5in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalEx1bb6in(Ar_0 + 1, Ar_3, Ar_2, Ar_3)) [ Ar_2 - Ar_3 >= 0 /\ Ar_1 - Ar_3 >= 0 /\ Ar_3 - 1 >= 0 /\ Ar_2 + Ar_3 - 2 >= 0 /\ -Ar_2 + Ar_3 >= 0 /\ Ar_1 + Ar_3 - 2 >= 0 /\ Ar_0 + Ar_3 - 1 >= 0 /\ -Ar_0 + Ar_3 - 1 >= 0 /\ Ar_1 - Ar_2 >= 0 /\ Ar_2 - 1 >= 0 /\ Ar_1 + Ar_2 - 2 >= 0 /\ Ar_0 + Ar_2 - 1 >= 0 /\ -Ar_0 + Ar_2 - 1 >= 0 /\ Ar_1 - 1 >= 0 /\ Ar_0 + Ar_1 - 1 >= 0 /\ -Ar_0 + Ar_1 - 1 >= 0 /\ Ar_0 >= 0 ] start location: koat_start leaf cost: 0 Complexity upper bound 22*Ar_0 + 24*Ar_0^2 + 14 Time: 0.437 sec (SMT: 0.340 sec) ---------------------------------------- (2) BOUNDS(1, n^2) ---------------------------------------- (3) Loat Proof (FINISHED) ### Pre-processing the ITS problem ### Initial linear ITS problem Start location: evalEx1start 0: evalEx1start -> evalEx1entryin : [], cost: 1 1: evalEx1entryin -> evalEx1bb6in : A'=0, B'=A, [], cost: 1 2: evalEx1bb6in -> evalEx1bbin : [ B>=1+A ], cost: 1 3: evalEx1bb6in -> evalEx1returnin : [ A>=B ], cost: 1 4: evalEx1bbin -> evalEx1bb4in : C'=1+A, D'=B, [], cost: 1 5: evalEx1bb4in -> evalEx1bb1in : [ D>=1+C ], cost: 1 6: evalEx1bb4in -> evalEx1bb5in : [ C>=D ], cost: 1 7: evalEx1bb1in -> evalEx1bb4in : D'=-1+D, [ 0>=1+free ], cost: 1 8: evalEx1bb1in -> evalEx1bb4in : D'=-1+D, [ 0>=1+free_1 && free_1>=1 ], cost: 1 9: evalEx1bb1in -> evalEx1bb4in : D'=-1+D, [ free_2>=1 ], cost: 1 10: evalEx1bb1in -> evalEx1bb4in : [ 0>=1 ], cost: 1 11: evalEx1bb1in -> evalEx1bb4in : C'=1+C, D'=-1+D, [ 0>=1 ], cost: 1 12: evalEx1bb1in -> evalEx1bb4in : C'=1+C, [], cost: 1 13: evalEx1bb5in -> evalEx1bb6in : A'=1+A, B'=D, [], cost: 1 14: evalEx1returnin -> evalEx1stop : [], cost: 1 Checking for constant complexity: The following rule is satisfiable with cost >= 1, yielding constant complexity: 0: evalEx1start -> evalEx1entryin : [], cost: 1 Removed unreachable and leaf rules: Start location: evalEx1start 0: evalEx1start -> evalEx1entryin : [], cost: 1 1: evalEx1entryin -> evalEx1bb6in : A'=0, B'=A, [], cost: 1 2: evalEx1bb6in -> evalEx1bbin : [ B>=1+A ], cost: 1 4: evalEx1bbin -> evalEx1bb4in : C'=1+A, D'=B, [], cost: 1 5: evalEx1bb4in -> evalEx1bb1in : [ D>=1+C ], cost: 1 6: evalEx1bb4in -> evalEx1bb5in : [ C>=D ], cost: 1 7: evalEx1bb1in -> evalEx1bb4in : D'=-1+D, [ 0>=1+free ], cost: 1 8: evalEx1bb1in -> evalEx1bb4in : D'=-1+D, [ 0>=1+free_1 && free_1>=1 ], cost: 1 9: evalEx1bb1in -> evalEx1bb4in : D'=-1+D, [ free_2>=1 ], cost: 1 10: evalEx1bb1in -> evalEx1bb4in : [ 0>=1 ], cost: 1 11: evalEx1bb1in -> evalEx1bb4in : C'=1+C, D'=-1+D, [ 0>=1 ], cost: 1 12: evalEx1bb1in -> evalEx1bb4in : C'=1+C, [], cost: 1 13: evalEx1bb5in -> evalEx1bb6in : A'=1+A, B'=D, [], cost: 1 Removed rules with unsatisfiable guard: Start location: evalEx1start 0: evalEx1start -> evalEx1entryin : [], cost: 1 1: evalEx1entryin -> evalEx1bb6in : A'=0, B'=A, [], cost: 1 2: evalEx1bb6in -> evalEx1bbin : [ B>=1+A ], cost: 1 4: evalEx1bbin -> evalEx1bb4in : C'=1+A, D'=B, [], cost: 1 5: evalEx1bb4in -> evalEx1bb1in : [ D>=1+C ], cost: 1 6: evalEx1bb4in -> evalEx1bb5in : [ C>=D ], cost: 1 7: evalEx1bb1in -> evalEx1bb4in : D'=-1+D, [ 0>=1+free ], cost: 1 9: evalEx1bb1in -> evalEx1bb4in : D'=-1+D, [ free_2>=1 ], cost: 1 12: evalEx1bb1in -> evalEx1bb4in : C'=1+C, [], cost: 1 13: evalEx1bb5in -> evalEx1bb6in : A'=1+A, B'=D, [], cost: 1 Simplified all rules, resulting in: Start location: evalEx1start 0: evalEx1start -> evalEx1entryin : [], cost: 1 1: evalEx1entryin -> evalEx1bb6in : A'=0, B'=A, [], cost: 1 2: evalEx1bb6in -> evalEx1bbin : [ B>=1+A ], cost: 1 4: evalEx1bbin -> evalEx1bb4in : C'=1+A, D'=B, [], cost: 1 5: evalEx1bb4in -> evalEx1bb1in : [ D>=1+C ], cost: 1 6: evalEx1bb4in -> evalEx1bb5in : [ C>=D ], cost: 1 9: evalEx1bb1in -> evalEx1bb4in : D'=-1+D, [], cost: 1 12: evalEx1bb1in -> evalEx1bb4in : C'=1+C, [], cost: 1 13: evalEx1bb5in -> evalEx1bb6in : A'=1+A, B'=D, [], cost: 1 ### Simplification by acceleration and chaining ### Eliminated locations (on linear paths): Start location: evalEx1start 15: evalEx1start -> evalEx1bb6in : A'=0, B'=A, [], cost: 2 16: evalEx1bb6in -> evalEx1bb4in : C'=1+A, D'=B, [ B>=1+A ], cost: 2 5: evalEx1bb4in -> evalEx1bb1in : [ D>=1+C ], cost: 1 17: evalEx1bb4in -> evalEx1bb6in : A'=1+A, B'=D, [ C>=D ], cost: 2 9: evalEx1bb1in -> evalEx1bb4in : D'=-1+D, [], cost: 1 12: evalEx1bb1in -> evalEx1bb4in : C'=1+C, [], cost: 1 Eliminated locations (on tree-shaped paths): Start location: evalEx1start 15: evalEx1start -> evalEx1bb6in : A'=0, B'=A, [], cost: 2 16: evalEx1bb6in -> evalEx1bb4in : C'=1+A, D'=B, [ B>=1+A ], cost: 2 17: evalEx1bb4in -> evalEx1bb6in : A'=1+A, B'=D, [ C>=D ], cost: 2 18: evalEx1bb4in -> evalEx1bb4in : D'=-1+D, [ D>=1+C ], cost: 2 19: evalEx1bb4in -> evalEx1bb4in : C'=1+C, [ D>=1+C ], cost: 2 Accelerating simple loops of location 4. Accelerating the following rules: 18: evalEx1bb4in -> evalEx1bb4in : D'=-1+D, [ D>=1+C ], cost: 2 19: evalEx1bb4in -> evalEx1bb4in : C'=1+C, [ D>=1+C ], cost: 2 Accelerated rule 18 with metering function -C+D, yielding the new rule 20. Accelerated rule 19 with metering function -C+D, yielding the new rule 21. Removing the simple loops: 18 19. Accelerated all simple loops using metering functions (where possible): Start location: evalEx1start 15: evalEx1start -> evalEx1bb6in : A'=0, B'=A, [], cost: 2 16: evalEx1bb6in -> evalEx1bb4in : C'=1+A, D'=B, [ B>=1+A ], cost: 2 17: evalEx1bb4in -> evalEx1bb6in : A'=1+A, B'=D, [ C>=D ], cost: 2 20: evalEx1bb4in -> evalEx1bb4in : D'=C, [ D>=1+C ], cost: -2*C+2*D 21: evalEx1bb4in -> evalEx1bb4in : C'=D, [ D>=1+C ], cost: -2*C+2*D Chained accelerated rules (with incoming rules): Start location: evalEx1start 15: evalEx1start -> evalEx1bb6in : A'=0, B'=A, [], cost: 2 16: evalEx1bb6in -> evalEx1bb4in : C'=1+A, D'=B, [ B>=1+A ], cost: 2 22: evalEx1bb6in -> evalEx1bb4in : C'=1+A, D'=1+A, [ B>=2+A ], cost: -2*A+2*B 23: evalEx1bb6in -> evalEx1bb4in : C'=B, D'=B, [ B>=2+A ], cost: -2*A+2*B 17: evalEx1bb4in -> evalEx1bb6in : A'=1+A, B'=D, [ C>=D ], cost: 2 Eliminated locations (on tree-shaped paths): Start location: evalEx1start 15: evalEx1start -> evalEx1bb6in : A'=0, B'=A, [], cost: 2 24: evalEx1bb6in -> evalEx1bb6in : A'=1+A, B'=B, C'=1+A, D'=B, [ B>=1+A && 1+A>=B ], cost: 4 25: evalEx1bb6in -> evalEx1bb6in : A'=1+A, B'=1+A, C'=1+A, D'=1+A, [ B>=2+A ], cost: 2-2*A+2*B 26: evalEx1bb6in -> evalEx1bb6in : A'=1+A, B'=B, C'=B, D'=B, [ B>=2+A ], cost: 2-2*A+2*B Accelerating simple loops of location 2. Simplified some of the simple loops (and removed duplicate rules). Accelerating the following rules: 24: evalEx1bb6in -> evalEx1bb6in : A'=1+A, C'=1+A, D'=B, [ 1+A-B==0 ], cost: 4 25: evalEx1bb6in -> evalEx1bb6in : A'=1+A, B'=1+A, C'=1+A, D'=1+A, [ B>=2+A ], cost: 2-2*A+2*B 26: evalEx1bb6in -> evalEx1bb6in : A'=1+A, C'=B, D'=B, [ B>=2+A ], cost: 2-2*A+2*B Accelerated rule 24 with metering function -A+B, yielding the new rule 27. Found no metering function for rule 25. Accelerated rule 26 with metering function -1-A+B, yielding the new rule 28. Removing the simple loops: 24 26. Accelerated all simple loops using metering functions (where possible): Start location: evalEx1start 15: evalEx1start -> evalEx1bb6in : A'=0, B'=A, [], cost: 2 25: evalEx1bb6in -> evalEx1bb6in : A'=1+A, B'=1+A, C'=1+A, D'=1+A, [ B>=2+A ], cost: 2-2*A+2*B 27: evalEx1bb6in -> evalEx1bb6in : A'=B, C'=B, D'=B, [ 1+A-B==0 ], cost: -4*A+4*B 28: evalEx1bb6in -> evalEx1bb6in : A'=-1+B, C'=B, D'=B, [ B>=2+A ], cost: -3-(1+A-B)^2-3*A+2*A*(1+A-B)-2*(1+A-B)*B+3*B Chained accelerated rules (with incoming rules): Start location: evalEx1start 15: evalEx1start -> evalEx1bb6in : A'=0, B'=A, [], cost: 2 29: evalEx1start -> evalEx1bb6in : A'=1, B'=1, C'=1, D'=1, [ A>=2 ], cost: 4+2*A 30: evalEx1start -> evalEx1bb6in : B'=A, C'=A, D'=A, [ 1-A==0 ], cost: 2+4*A 31: evalEx1start -> evalEx1bb6in : A'=-1+A, B'=A, C'=A, D'=A, [ A>=2 ], cost: -1-(-1+A)^2+2*(-1+A)*A+3*A Removed unreachable locations (and leaf rules with constant cost): Start location: evalEx1start 29: evalEx1start -> evalEx1bb6in : A'=1, B'=1, C'=1, D'=1, [ A>=2 ], cost: 4+2*A 30: evalEx1start -> evalEx1bb6in : B'=A, C'=A, D'=A, [ 1-A==0 ], cost: 2+4*A 31: evalEx1start -> evalEx1bb6in : A'=-1+A, B'=A, C'=A, D'=A, [ A>=2 ], cost: -1-(-1+A)^2+2*(-1+A)*A+3*A ### Computing asymptotic complexity ### Fully simplified ITS problem Start location: evalEx1start 29: evalEx1start -> evalEx1bb6in : A'=1, B'=1, C'=1, D'=1, [ A>=2 ], cost: 4+2*A 30: evalEx1start -> evalEx1bb6in : B'=A, C'=A, D'=A, [ 1-A==0 ], cost: 2+4*A 31: evalEx1start -> evalEx1bb6in : A'=-1+A, B'=A, C'=A, D'=A, [ A>=2 ], cost: -1-(-1+A)^2+2*(-1+A)*A+3*A Computing asymptotic complexity for rule 29 Solved the limit problem by the following transformations: Created initial limit problem: 4+2*A (+), -1+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 4+2*n has complexity: Poly(n^1) Found new complexity Poly(n^1). Computing asymptotic complexity for rule 31 Solved the limit problem by the following transformations: Created initial limit problem: -1+A (+/+!), -2+3*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 -2+3*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: -2+3*n+n^2 Rule cost: -1-(-1+A)^2+2*(-1+A)*A+3*A Rule guard: [ A>=2 ] WORST_CASE(Omega(n^2),?) ---------------------------------------- (4) BOUNDS(n^2, INF)