/export/starexec/sandbox/solver/bin/starexec_run_complexity /export/starexec/sandbox/benchmark/theBenchmark.koat /export/starexec/sandbox/output/output_files -------------------------------------------------------------------------------- WORST_CASE(?, O(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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(1, 1). (0) CpxIntTrs (1) Koat Proof [FINISHED, 17 ms] (2) BOUNDS(1, 1) ---------------------------------------- (0) Obligation: Complexity Int TRS consisting of the following rules: evaleasy1start(A, B) -> Com_1(evaleasy1entryin(A, B)) :|: TRUE evaleasy1entryin(A, B) -> Com_1(evaleasy1bb3in(0, B)) :|: TRUE evaleasy1bb3in(A, B) -> Com_1(evaleasy1bbin(A, B)) :|: 39 >= A evaleasy1bb3in(A, B) -> Com_1(evaleasy1returnin(A, B)) :|: A >= 40 evaleasy1bbin(A, B) -> Com_1(evaleasy1bb1in(A, B)) :|: B >= 0 && B <= 0 evaleasy1bbin(A, B) -> Com_1(evaleasy1bb2in(A, B)) :|: 0 >= B + 1 evaleasy1bbin(A, B) -> Com_1(evaleasy1bb2in(A, B)) :|: B >= 1 evaleasy1bb1in(A, B) -> Com_1(evaleasy1bb3in(A + 1, B)) :|: TRUE evaleasy1bb2in(A, B) -> Com_1(evaleasy1bb3in(A + 2, B)) :|: TRUE evaleasy1returnin(A, B) -> Com_1(evaleasy1stop(A, B)) :|: TRUE The start-symbols are:[evaleasy1start_2] ---------------------------------------- (1) Koat Proof (FINISHED) YES(?, 286) Initial complexity problem: 1: T: (Comp: ?, Cost: 1) evaleasy1start(Ar_0, Ar_1) -> Com_1(evaleasy1entryin(Ar_0, Ar_1)) (Comp: ?, Cost: 1) evaleasy1entryin(Ar_0, Ar_1) -> Com_1(evaleasy1bb3in(0, Ar_1)) (Comp: ?, Cost: 1) evaleasy1bb3in(Ar_0, Ar_1) -> Com_1(evaleasy1bbin(Ar_0, Ar_1)) [ 39 >= Ar_0 ] (Comp: ?, Cost: 1) evaleasy1bb3in(Ar_0, Ar_1) -> Com_1(evaleasy1returnin(Ar_0, Ar_1)) [ Ar_0 >= 40 ] (Comp: ?, Cost: 1) evaleasy1bbin(Ar_0, Ar_1) -> Com_1(evaleasy1bb1in(Ar_0, Ar_1)) [ Ar_1 = 0 ] (Comp: ?, Cost: 1) evaleasy1bbin(Ar_0, Ar_1) -> Com_1(evaleasy1bb2in(Ar_0, Ar_1)) [ 0 >= Ar_1 + 1 ] (Comp: ?, Cost: 1) evaleasy1bbin(Ar_0, Ar_1) -> Com_1(evaleasy1bb2in(Ar_0, Ar_1)) [ Ar_1 >= 1 ] (Comp: ?, Cost: 1) evaleasy1bb1in(Ar_0, Ar_1) -> Com_1(evaleasy1bb3in(Ar_0 + 1, Ar_1)) (Comp: ?, Cost: 1) evaleasy1bb2in(Ar_0, Ar_1) -> Com_1(evaleasy1bb3in(Ar_0 + 2, Ar_1)) (Comp: ?, Cost: 1) evaleasy1returnin(Ar_0, Ar_1) -> Com_1(evaleasy1stop(Ar_0, Ar_1)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1) -> Com_1(evaleasy1start(Ar_0, Ar_1)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 Repeatedly propagating knowledge in problem 1 produces the following problem: 2: T: (Comp: 1, Cost: 1) evaleasy1start(Ar_0, Ar_1) -> Com_1(evaleasy1entryin(Ar_0, Ar_1)) (Comp: 1, Cost: 1) evaleasy1entryin(Ar_0, Ar_1) -> Com_1(evaleasy1bb3in(0, Ar_1)) (Comp: ?, Cost: 1) evaleasy1bb3in(Ar_0, Ar_1) -> Com_1(evaleasy1bbin(Ar_0, Ar_1)) [ 39 >= Ar_0 ] (Comp: ?, Cost: 1) evaleasy1bb3in(Ar_0, Ar_1) -> Com_1(evaleasy1returnin(Ar_0, Ar_1)) [ Ar_0 >= 40 ] (Comp: ?, Cost: 1) evaleasy1bbin(Ar_0, Ar_1) -> Com_1(evaleasy1bb1in(Ar_0, Ar_1)) [ Ar_1 = 0 ] (Comp: ?, Cost: 1) evaleasy1bbin(Ar_0, Ar_1) -> Com_1(evaleasy1bb2in(Ar_0, Ar_1)) [ 0 >= Ar_1 + 1 ] (Comp: ?, Cost: 1) evaleasy1bbin(Ar_0, Ar_1) -> Com_1(evaleasy1bb2in(Ar_0, Ar_1)) [ Ar_1 >= 1 ] (Comp: ?, Cost: 1) evaleasy1bb1in(Ar_0, Ar_1) -> Com_1(evaleasy1bb3in(Ar_0 + 1, Ar_1)) (Comp: ?, Cost: 1) evaleasy1bb2in(Ar_0, Ar_1) -> Com_1(evaleasy1bb3in(Ar_0 + 2, Ar_1)) (Comp: ?, Cost: 1) evaleasy1returnin(Ar_0, Ar_1) -> Com_1(evaleasy1stop(Ar_0, Ar_1)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1) -> Com_1(evaleasy1start(Ar_0, Ar_1)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 A polynomial rank function with Pol(evaleasy1start) = 2 Pol(evaleasy1entryin) = 2 Pol(evaleasy1bb3in) = 2 Pol(evaleasy1bbin) = 2 Pol(evaleasy1returnin) = 1 Pol(evaleasy1bb1in) = 2 Pol(evaleasy1bb2in) = 2 Pol(evaleasy1stop) = 0 Pol(koat_start) = 2 orients all transitions weakly and the transitions evaleasy1returnin(Ar_0, Ar_1) -> Com_1(evaleasy1stop(Ar_0, Ar_1)) evaleasy1bb3in(Ar_0, Ar_1) -> Com_1(evaleasy1returnin(Ar_0, Ar_1)) [ Ar_0 >= 40 ] strictly and produces the following problem: 3: T: (Comp: 1, Cost: 1) evaleasy1start(Ar_0, Ar_1) -> Com_1(evaleasy1entryin(Ar_0, Ar_1)) (Comp: 1, Cost: 1) evaleasy1entryin(Ar_0, Ar_1) -> Com_1(evaleasy1bb3in(0, Ar_1)) (Comp: ?, Cost: 1) evaleasy1bb3in(Ar_0, Ar_1) -> Com_1(evaleasy1bbin(Ar_0, Ar_1)) [ 39 >= Ar_0 ] (Comp: 2, Cost: 1) evaleasy1bb3in(Ar_0, Ar_1) -> Com_1(evaleasy1returnin(Ar_0, Ar_1)) [ Ar_0 >= 40 ] (Comp: ?, Cost: 1) evaleasy1bbin(Ar_0, Ar_1) -> Com_1(evaleasy1bb1in(Ar_0, Ar_1)) [ Ar_1 = 0 ] (Comp: ?, Cost: 1) evaleasy1bbin(Ar_0, Ar_1) -> Com_1(evaleasy1bb2in(Ar_0, Ar_1)) [ 0 >= Ar_1 + 1 ] (Comp: ?, Cost: 1) evaleasy1bbin(Ar_0, Ar_1) -> Com_1(evaleasy1bb2in(Ar_0, Ar_1)) [ Ar_1 >= 1 ] (Comp: ?, Cost: 1) evaleasy1bb1in(Ar_0, Ar_1) -> Com_1(evaleasy1bb3in(Ar_0 + 1, Ar_1)) (Comp: ?, Cost: 1) evaleasy1bb2in(Ar_0, Ar_1) -> Com_1(evaleasy1bb3in(Ar_0 + 2, Ar_1)) (Comp: 2, Cost: 1) evaleasy1returnin(Ar_0, Ar_1) -> Com_1(evaleasy1stop(Ar_0, Ar_1)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1) -> Com_1(evaleasy1start(Ar_0, Ar_1)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 A polynomial rank function with Pol(evaleasy1start) = 40 Pol(evaleasy1entryin) = 40 Pol(evaleasy1bb3in) = -V_1 + 40 Pol(evaleasy1bbin) = -V_1 + 39 Pol(evaleasy1returnin) = -V_1 Pol(evaleasy1bb1in) = -V_1 + 39 Pol(evaleasy1bb2in) = -V_1 + 39 Pol(evaleasy1stop) = -V_1 Pol(koat_start) = 40 orients all transitions weakly and the transition evaleasy1bb3in(Ar_0, Ar_1) -> Com_1(evaleasy1bbin(Ar_0, Ar_1)) [ 39 >= Ar_0 ] strictly and produces the following problem: 4: T: (Comp: 1, Cost: 1) evaleasy1start(Ar_0, Ar_1) -> Com_1(evaleasy1entryin(Ar_0, Ar_1)) (Comp: 1, Cost: 1) evaleasy1entryin(Ar_0, Ar_1) -> Com_1(evaleasy1bb3in(0, Ar_1)) (Comp: 40, Cost: 1) evaleasy1bb3in(Ar_0, Ar_1) -> Com_1(evaleasy1bbin(Ar_0, Ar_1)) [ 39 >= Ar_0 ] (Comp: 2, Cost: 1) evaleasy1bb3in(Ar_0, Ar_1) -> Com_1(evaleasy1returnin(Ar_0, Ar_1)) [ Ar_0 >= 40 ] (Comp: ?, Cost: 1) evaleasy1bbin(Ar_0, Ar_1) -> Com_1(evaleasy1bb1in(Ar_0, Ar_1)) [ Ar_1 = 0 ] (Comp: ?, Cost: 1) evaleasy1bbin(Ar_0, Ar_1) -> Com_1(evaleasy1bb2in(Ar_0, Ar_1)) [ 0 >= Ar_1 + 1 ] (Comp: ?, Cost: 1) evaleasy1bbin(Ar_0, Ar_1) -> Com_1(evaleasy1bb2in(Ar_0, Ar_1)) [ Ar_1 >= 1 ] (Comp: ?, Cost: 1) evaleasy1bb1in(Ar_0, Ar_1) -> Com_1(evaleasy1bb3in(Ar_0 + 1, Ar_1)) (Comp: ?, Cost: 1) evaleasy1bb2in(Ar_0, Ar_1) -> Com_1(evaleasy1bb3in(Ar_0 + 2, Ar_1)) (Comp: 2, Cost: 1) evaleasy1returnin(Ar_0, Ar_1) -> Com_1(evaleasy1stop(Ar_0, Ar_1)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1) -> Com_1(evaleasy1start(Ar_0, Ar_1)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 Repeatedly propagating knowledge in problem 4 produces the following problem: 5: T: (Comp: 1, Cost: 1) evaleasy1start(Ar_0, Ar_1) -> Com_1(evaleasy1entryin(Ar_0, Ar_1)) (Comp: 1, Cost: 1) evaleasy1entryin(Ar_0, Ar_1) -> Com_1(evaleasy1bb3in(0, Ar_1)) (Comp: 40, Cost: 1) evaleasy1bb3in(Ar_0, Ar_1) -> Com_1(evaleasy1bbin(Ar_0, Ar_1)) [ 39 >= Ar_0 ] (Comp: 2, Cost: 1) evaleasy1bb3in(Ar_0, Ar_1) -> Com_1(evaleasy1returnin(Ar_0, Ar_1)) [ Ar_0 >= 40 ] (Comp: 40, Cost: 1) evaleasy1bbin(Ar_0, Ar_1) -> Com_1(evaleasy1bb1in(Ar_0, Ar_1)) [ Ar_1 = 0 ] (Comp: 40, Cost: 1) evaleasy1bbin(Ar_0, Ar_1) -> Com_1(evaleasy1bb2in(Ar_0, Ar_1)) [ 0 >= Ar_1 + 1 ] (Comp: 40, Cost: 1) evaleasy1bbin(Ar_0, Ar_1) -> Com_1(evaleasy1bb2in(Ar_0, Ar_1)) [ Ar_1 >= 1 ] (Comp: 40, Cost: 1) evaleasy1bb1in(Ar_0, Ar_1) -> Com_1(evaleasy1bb3in(Ar_0 + 1, Ar_1)) (Comp: 80, Cost: 1) evaleasy1bb2in(Ar_0, Ar_1) -> Com_1(evaleasy1bb3in(Ar_0 + 2, Ar_1)) (Comp: 2, Cost: 1) evaleasy1returnin(Ar_0, Ar_1) -> Com_1(evaleasy1stop(Ar_0, Ar_1)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1) -> Com_1(evaleasy1start(Ar_0, Ar_1)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 Complexity upper bound 286 Time: 0.057 sec (SMT: 0.051 sec) ---------------------------------------- (2) BOUNDS(1, 1)