/export/starexec/sandbox2/solver/bin/starexec_run_c_complexity /export/starexec/sandbox2/benchmark/theBenchmark.c /export/starexec/sandbox2/output/output_files -------------------------------------------------------------------------------- WORST_CASE(?, O(n^1)) proof of /export/starexec/sandbox2/output/output_files/bench.koat # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(1, n^1). (0) CpxIntTrs (1) Koat Proof [FINISHED, 45 ms] (2) BOUNDS(1, n^1) ---------------------------------------- (0) Obligation: Complexity Int TRS consisting of the following rules: eval_easy2_start(v_.0, v_z) -> Com_1(eval_easy2_bb0_in(v_.0, v_z)) :|: TRUE eval_easy2_bb0_in(v_.0, v_z) -> Com_1(eval_easy2_bb1_in(v_z, v_z)) :|: TRUE eval_easy2_bb1_in(v_.0, v_z) -> Com_1(eval_easy2_bb2_in(v_.0, v_z)) :|: v_.0 > 0 eval_easy2_bb1_in(v_.0, v_z) -> Com_1(eval_easy2_bb3_in(v_.0, v_z)) :|: v_.0 <= 0 eval_easy2_bb2_in(v_.0, v_z) -> Com_1(eval_easy2_bb1_in(v_.0 - 1, v_z)) :|: TRUE eval_easy2_bb3_in(v_.0, v_z) -> Com_1(eval_easy2_stop(v_.0, v_z)) :|: TRUE The start-symbols are:[eval_easy2_start_2] ---------------------------------------- (1) Koat Proof (FINISHED) YES(?, 2*ar_1 + 8) Initial complexity problem: 1: T: (Comp: ?, Cost: 1) evaleasy2start(ar_0, ar_1) -> Com_1(evaleasy2bb0in(ar_0, ar_1)) (Comp: ?, Cost: 1) evaleasy2bb0in(ar_0, ar_1) -> Com_1(evaleasy2bb1in(ar_1, ar_1)) (Comp: ?, Cost: 1) evaleasy2bb1in(ar_0, ar_1) -> Com_1(evaleasy2bb2in(ar_0, ar_1)) [ ar_0 >= 1 ] (Comp: ?, Cost: 1) evaleasy2bb1in(ar_0, ar_1) -> Com_1(evaleasy2bb3in(ar_0, ar_1)) [ 0 >= ar_0 ] (Comp: ?, Cost: 1) evaleasy2bb2in(ar_0, ar_1) -> Com_1(evaleasy2bb1in(ar_0 - 1, ar_1)) (Comp: ?, Cost: 1) evaleasy2bb3in(ar_0, ar_1) -> Com_1(evaleasy2stop(ar_0, ar_1)) (Comp: 1, Cost: 0) koat_start(ar_0, ar_1) -> Com_1(evaleasy2start(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) evaleasy2start(ar_0, ar_1) -> Com_1(evaleasy2bb0in(ar_0, ar_1)) (Comp: 1, Cost: 1) evaleasy2bb0in(ar_0, ar_1) -> Com_1(evaleasy2bb1in(ar_1, ar_1)) (Comp: ?, Cost: 1) evaleasy2bb1in(ar_0, ar_1) -> Com_1(evaleasy2bb2in(ar_0, ar_1)) [ ar_0 >= 1 ] (Comp: ?, Cost: 1) evaleasy2bb1in(ar_0, ar_1) -> Com_1(evaleasy2bb3in(ar_0, ar_1)) [ 0 >= ar_0 ] (Comp: ?, Cost: 1) evaleasy2bb2in(ar_0, ar_1) -> Com_1(evaleasy2bb1in(ar_0 - 1, ar_1)) (Comp: ?, Cost: 1) evaleasy2bb3in(ar_0, ar_1) -> Com_1(evaleasy2stop(ar_0, ar_1)) (Comp: 1, Cost: 0) koat_start(ar_0, ar_1) -> Com_1(evaleasy2start(ar_0, ar_1)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 A polynomial rank function with Pol(evaleasy2start) = 2 Pol(evaleasy2bb0in) = 2 Pol(evaleasy2bb1in) = 2 Pol(evaleasy2bb2in) = 2 Pol(evaleasy2bb3in) = 1 Pol(evaleasy2stop) = 0 Pol(koat_start) = 2 orients all transitions weakly and the transitions evaleasy2bb3in(ar_0, ar_1) -> Com_1(evaleasy2stop(ar_0, ar_1)) evaleasy2bb1in(ar_0, ar_1) -> Com_1(evaleasy2bb3in(ar_0, ar_1)) [ 0 >= ar_0 ] strictly and produces the following problem: 3: T: (Comp: 1, Cost: 1) evaleasy2start(ar_0, ar_1) -> Com_1(evaleasy2bb0in(ar_0, ar_1)) (Comp: 1, Cost: 1) evaleasy2bb0in(ar_0, ar_1) -> Com_1(evaleasy2bb1in(ar_1, ar_1)) (Comp: ?, Cost: 1) evaleasy2bb1in(ar_0, ar_1) -> Com_1(evaleasy2bb2in(ar_0, ar_1)) [ ar_0 >= 1 ] (Comp: 2, Cost: 1) evaleasy2bb1in(ar_0, ar_1) -> Com_1(evaleasy2bb3in(ar_0, ar_1)) [ 0 >= ar_0 ] (Comp: ?, Cost: 1) evaleasy2bb2in(ar_0, ar_1) -> Com_1(evaleasy2bb1in(ar_0 - 1, ar_1)) (Comp: 2, Cost: 1) evaleasy2bb3in(ar_0, ar_1) -> Com_1(evaleasy2stop(ar_0, ar_1)) (Comp: 1, Cost: 0) koat_start(ar_0, ar_1) -> Com_1(evaleasy2start(ar_0, ar_1)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 A polynomial rank function with Pol(evaleasy2start) = V_2 + 1 Pol(evaleasy2bb0in) = V_2 + 1 Pol(evaleasy2bb1in) = V_1 + 1 Pol(evaleasy2bb2in) = V_1 Pol(evaleasy2bb3in) = V_1 Pol(evaleasy2stop) = V_1 Pol(koat_start) = V_2 + 1 orients all transitions weakly and the transition evaleasy2bb1in(ar_0, ar_1) -> Com_1(evaleasy2bb2in(ar_0, ar_1)) [ ar_0 >= 1 ] strictly and produces the following problem: 4: T: (Comp: 1, Cost: 1) evaleasy2start(ar_0, ar_1) -> Com_1(evaleasy2bb0in(ar_0, ar_1)) (Comp: 1, Cost: 1) evaleasy2bb0in(ar_0, ar_1) -> Com_1(evaleasy2bb1in(ar_1, ar_1)) (Comp: ar_1 + 1, Cost: 1) evaleasy2bb1in(ar_0, ar_1) -> Com_1(evaleasy2bb2in(ar_0, ar_1)) [ ar_0 >= 1 ] (Comp: 2, Cost: 1) evaleasy2bb1in(ar_0, ar_1) -> Com_1(evaleasy2bb3in(ar_0, ar_1)) [ 0 >= ar_0 ] (Comp: ?, Cost: 1) evaleasy2bb2in(ar_0, ar_1) -> Com_1(evaleasy2bb1in(ar_0 - 1, ar_1)) (Comp: 2, Cost: 1) evaleasy2bb3in(ar_0, ar_1) -> Com_1(evaleasy2stop(ar_0, ar_1)) (Comp: 1, Cost: 0) koat_start(ar_0, ar_1) -> Com_1(evaleasy2start(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) evaleasy2start(ar_0, ar_1) -> Com_1(evaleasy2bb0in(ar_0, ar_1)) (Comp: 1, Cost: 1) evaleasy2bb0in(ar_0, ar_1) -> Com_1(evaleasy2bb1in(ar_1, ar_1)) (Comp: ar_1 + 1, Cost: 1) evaleasy2bb1in(ar_0, ar_1) -> Com_1(evaleasy2bb2in(ar_0, ar_1)) [ ar_0 >= 1 ] (Comp: 2, Cost: 1) evaleasy2bb1in(ar_0, ar_1) -> Com_1(evaleasy2bb3in(ar_0, ar_1)) [ 0 >= ar_0 ] (Comp: ar_1 + 1, Cost: 1) evaleasy2bb2in(ar_0, ar_1) -> Com_1(evaleasy2bb1in(ar_0 - 1, ar_1)) (Comp: 2, Cost: 1) evaleasy2bb3in(ar_0, ar_1) -> Com_1(evaleasy2stop(ar_0, ar_1)) (Comp: 1, Cost: 0) koat_start(ar_0, ar_1) -> Com_1(evaleasy2start(ar_0, ar_1)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 Complexity upper bound 2*ar_1 + 8 Time: 0.034 sec (SMT: 0.031 sec) ---------------------------------------- (2) BOUNDS(1, n^1)