/export/starexec/sandbox/solver/bin/starexec_run_c_complexity /export/starexec/sandbox/benchmark/theBenchmark.c /export/starexec/sandbox/output/output_files -------------------------------------------------------------------------------- WORST_CASE(?, O(n^2)) proof of /export/starexec/sandbox/output/output_files/bench.koat # AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(1, n^2). (0) CpxIntTrs (1) Koat Proof [FINISHED, 76 ms] (2) BOUNDS(1, n^2) ---------------------------------------- (0) Obligation: Complexity Int TRS consisting of the following rules: eval_ex_paper2_start(v_n, v_x.0, v_y.0) -> Com_1(eval_ex_paper2_bb0_in(v_n, v_x.0, v_y.0)) :|: TRUE eval_ex_paper2_bb0_in(v_n, v_x.0, v_y.0) -> Com_1(eval_ex_paper2_bb1_in(v_n, 1, v_y.0)) :|: TRUE eval_ex_paper2_bb1_in(v_n, v_x.0, v_y.0) -> Com_1(eval_ex_paper2_bb2_in(v_n, v_x.0, v_x.0)) :|: v_x.0 <= v_n eval_ex_paper2_bb1_in(v_n, v_x.0, v_y.0) -> Com_1(eval_ex_paper2_bb5_in(v_n, v_x.0, v_y.0)) :|: v_x.0 > v_n eval_ex_paper2_bb2_in(v_n, v_x.0, v_y.0) -> Com_1(eval_ex_paper2_bb3_in(v_n, v_x.0, v_y.0)) :|: v_y.0 <= v_n eval_ex_paper2_bb2_in(v_n, v_x.0, v_y.0) -> Com_1(eval_ex_paper2_bb4_in(v_n, v_x.0, v_y.0)) :|: v_y.0 > v_n eval_ex_paper2_bb3_in(v_n, v_x.0, v_y.0) -> Com_1(eval_ex_paper2_bb2_in(v_n, v_x.0, v_y.0 + 1)) :|: TRUE eval_ex_paper2_bb4_in(v_n, v_x.0, v_y.0) -> Com_1(eval_ex_paper2_bb1_in(v_n, v_x.0 + 1, v_y.0)) :|: TRUE eval_ex_paper2_bb5_in(v_n, v_x.0, v_y.0) -> Com_1(eval_ex_paper2_stop(v_n, v_x.0, v_y.0)) :|: TRUE The start-symbols are:[eval_ex_paper2_start_3] ---------------------------------------- (1) Koat Proof (FINISHED) YES(?, 27*Ar_1 + 6*Ar_1^2 + 6) Initial complexity problem: 1: T: (Comp: ?, Cost: 1) evalexpaper2start(Ar_0, Ar_1, Ar_2) -> Com_1(evalexpaper2bb0in(Ar_0, Ar_1, Ar_2)) (Comp: ?, Cost: 1) evalexpaper2bb0in(Ar_0, Ar_1, Ar_2) -> Com_1(evalexpaper2bb1in(1, Ar_1, Ar_2)) (Comp: ?, Cost: 1) evalexpaper2bb1in(Ar_0, Ar_1, Ar_2) -> Com_1(evalexpaper2bb2in(Ar_0, Ar_1, Ar_0)) [ Ar_1 >= Ar_0 ] (Comp: ?, Cost: 1) evalexpaper2bb1in(Ar_0, Ar_1, Ar_2) -> Com_1(evalexpaper2bb5in(Ar_0, Ar_1, Ar_2)) [ Ar_0 >= Ar_1 + 1 ] (Comp: ?, Cost: 1) evalexpaper2bb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalexpaper2bb3in(Ar_0, Ar_1, Ar_2)) [ Ar_1 >= Ar_2 ] (Comp: ?, Cost: 1) evalexpaper2bb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalexpaper2bb4in(Ar_0, Ar_1, Ar_2)) [ Ar_2 >= Ar_1 + 1 ] (Comp: ?, Cost: 1) evalexpaper2bb3in(Ar_0, Ar_1, Ar_2) -> Com_1(evalexpaper2bb2in(Ar_0, Ar_1, Ar_2 + 1)) (Comp: ?, Cost: 1) evalexpaper2bb4in(Ar_0, Ar_1, Ar_2) -> Com_1(evalexpaper2bb1in(Ar_0 + 1, Ar_1, Ar_2)) (Comp: ?, Cost: 1) evalexpaper2bb5in(Ar_0, Ar_1, Ar_2) -> Com_1(evalexpaper2stop(Ar_0, Ar_1, Ar_2)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2) -> Com_1(evalexpaper2start(Ar_0, Ar_1, Ar_2)) [ 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) evalexpaper2start(Ar_0, Ar_1, Ar_2) -> Com_1(evalexpaper2bb0in(Ar_0, Ar_1, Ar_2)) (Comp: 1, Cost: 1) evalexpaper2bb0in(Ar_0, Ar_1, Ar_2) -> Com_1(evalexpaper2bb1in(1, Ar_1, Ar_2)) (Comp: ?, Cost: 1) evalexpaper2bb1in(Ar_0, Ar_1, Ar_2) -> Com_1(evalexpaper2bb2in(Ar_0, Ar_1, Ar_0)) [ Ar_1 >= Ar_0 ] (Comp: ?, Cost: 1) evalexpaper2bb1in(Ar_0, Ar_1, Ar_2) -> Com_1(evalexpaper2bb5in(Ar_0, Ar_1, Ar_2)) [ Ar_0 >= Ar_1 + 1 ] (Comp: ?, Cost: 1) evalexpaper2bb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalexpaper2bb3in(Ar_0, Ar_1, Ar_2)) [ Ar_1 >= Ar_2 ] (Comp: ?, Cost: 1) evalexpaper2bb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalexpaper2bb4in(Ar_0, Ar_1, Ar_2)) [ Ar_2 >= Ar_1 + 1 ] (Comp: ?, Cost: 1) evalexpaper2bb3in(Ar_0, Ar_1, Ar_2) -> Com_1(evalexpaper2bb2in(Ar_0, Ar_1, Ar_2 + 1)) (Comp: ?, Cost: 1) evalexpaper2bb4in(Ar_0, Ar_1, Ar_2) -> Com_1(evalexpaper2bb1in(Ar_0 + 1, Ar_1, Ar_2)) (Comp: ?, Cost: 1) evalexpaper2bb5in(Ar_0, Ar_1, Ar_2) -> Com_1(evalexpaper2stop(Ar_0, Ar_1, Ar_2)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2) -> Com_1(evalexpaper2start(Ar_0, Ar_1, Ar_2)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 A polynomial rank function with Pol(evalexpaper2start) = 2 Pol(evalexpaper2bb0in) = 2 Pol(evalexpaper2bb1in) = 2 Pol(evalexpaper2bb2in) = 2 Pol(evalexpaper2bb5in) = 1 Pol(evalexpaper2bb3in) = 2 Pol(evalexpaper2bb4in) = 2 Pol(evalexpaper2stop) = 0 Pol(koat_start) = 2 orients all transitions weakly and the transitions evalexpaper2bb5in(Ar_0, Ar_1, Ar_2) -> Com_1(evalexpaper2stop(Ar_0, Ar_1, Ar_2)) evalexpaper2bb1in(Ar_0, Ar_1, Ar_2) -> Com_1(evalexpaper2bb5in(Ar_0, Ar_1, Ar_2)) [ Ar_0 >= Ar_1 + 1 ] strictly and produces the following problem: 3: T: (Comp: 1, Cost: 1) evalexpaper2start(Ar_0, Ar_1, Ar_2) -> Com_1(evalexpaper2bb0in(Ar_0, Ar_1, Ar_2)) (Comp: 1, Cost: 1) evalexpaper2bb0in(Ar_0, Ar_1, Ar_2) -> Com_1(evalexpaper2bb1in(1, Ar_1, Ar_2)) (Comp: ?, Cost: 1) evalexpaper2bb1in(Ar_0, Ar_1, Ar_2) -> Com_1(evalexpaper2bb2in(Ar_0, Ar_1, Ar_0)) [ Ar_1 >= Ar_0 ] (Comp: 2, Cost: 1) evalexpaper2bb1in(Ar_0, Ar_1, Ar_2) -> Com_1(evalexpaper2bb5in(Ar_0, Ar_1, Ar_2)) [ Ar_0 >= Ar_1 + 1 ] (Comp: ?, Cost: 1) evalexpaper2bb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalexpaper2bb3in(Ar_0, Ar_1, Ar_2)) [ Ar_1 >= Ar_2 ] (Comp: ?, Cost: 1) evalexpaper2bb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalexpaper2bb4in(Ar_0, Ar_1, Ar_2)) [ Ar_2 >= Ar_1 + 1 ] (Comp: ?, Cost: 1) evalexpaper2bb3in(Ar_0, Ar_1, Ar_2) -> Com_1(evalexpaper2bb2in(Ar_0, Ar_1, Ar_2 + 1)) (Comp: ?, Cost: 1) evalexpaper2bb4in(Ar_0, Ar_1, Ar_2) -> Com_1(evalexpaper2bb1in(Ar_0 + 1, Ar_1, Ar_2)) (Comp: 2, Cost: 1) evalexpaper2bb5in(Ar_0, Ar_1, Ar_2) -> Com_1(evalexpaper2stop(Ar_0, Ar_1, Ar_2)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2) -> Com_1(evalexpaper2start(Ar_0, Ar_1, Ar_2)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 A polynomial rank function with Pol(evalexpaper2start) = V_2 Pol(evalexpaper2bb0in) = V_2 Pol(evalexpaper2bb1in) = -V_1 + V_2 + 1 Pol(evalexpaper2bb2in) = -V_1 + V_2 Pol(evalexpaper2bb5in) = -V_1 + V_2 Pol(evalexpaper2bb3in) = -V_1 + V_2 Pol(evalexpaper2bb4in) = -V_1 + V_2 Pol(evalexpaper2stop) = -V_1 + V_2 Pol(koat_start) = V_2 orients all transitions weakly and the transition evalexpaper2bb1in(Ar_0, Ar_1, Ar_2) -> Com_1(evalexpaper2bb2in(Ar_0, Ar_1, Ar_0)) [ Ar_1 >= Ar_0 ] strictly and produces the following problem: 4: T: (Comp: 1, Cost: 1) evalexpaper2start(Ar_0, Ar_1, Ar_2) -> Com_1(evalexpaper2bb0in(Ar_0, Ar_1, Ar_2)) (Comp: 1, Cost: 1) evalexpaper2bb0in(Ar_0, Ar_1, Ar_2) -> Com_1(evalexpaper2bb1in(1, Ar_1, Ar_2)) (Comp: Ar_1, Cost: 1) evalexpaper2bb1in(Ar_0, Ar_1, Ar_2) -> Com_1(evalexpaper2bb2in(Ar_0, Ar_1, Ar_0)) [ Ar_1 >= Ar_0 ] (Comp: 2, Cost: 1) evalexpaper2bb1in(Ar_0, Ar_1, Ar_2) -> Com_1(evalexpaper2bb5in(Ar_0, Ar_1, Ar_2)) [ Ar_0 >= Ar_1 + 1 ] (Comp: ?, Cost: 1) evalexpaper2bb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalexpaper2bb3in(Ar_0, Ar_1, Ar_2)) [ Ar_1 >= Ar_2 ] (Comp: ?, Cost: 1) evalexpaper2bb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalexpaper2bb4in(Ar_0, Ar_1, Ar_2)) [ Ar_2 >= Ar_1 + 1 ] (Comp: ?, Cost: 1) evalexpaper2bb3in(Ar_0, Ar_1, Ar_2) -> Com_1(evalexpaper2bb2in(Ar_0, Ar_1, Ar_2 + 1)) (Comp: ?, Cost: 1) evalexpaper2bb4in(Ar_0, Ar_1, Ar_2) -> Com_1(evalexpaper2bb1in(Ar_0 + 1, Ar_1, Ar_2)) (Comp: 2, Cost: 1) evalexpaper2bb5in(Ar_0, Ar_1, Ar_2) -> Com_1(evalexpaper2stop(Ar_0, Ar_1, Ar_2)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2) -> Com_1(evalexpaper2start(Ar_0, Ar_1, Ar_2)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 A polynomial rank function with Pol(evalexpaper2bb4in) = 1 Pol(evalexpaper2bb1in) = 0 Pol(evalexpaper2bb3in) = 2 Pol(evalexpaper2bb2in) = 2 and size complexities S("koat_start(Ar_0, Ar_1, Ar_2) -> Com_1(evalexpaper2start(Ar_0, Ar_1, Ar_2)) [ 0 <= 0 ]", 0-0) = Ar_0 S("koat_start(Ar_0, Ar_1, Ar_2) -> Com_1(evalexpaper2start(Ar_0, Ar_1, Ar_2)) [ 0 <= 0 ]", 0-1) = Ar_1 S("koat_start(Ar_0, Ar_1, Ar_2) -> Com_1(evalexpaper2start(Ar_0, Ar_1, Ar_2)) [ 0 <= 0 ]", 0-2) = Ar_2 S("evalexpaper2bb5in(Ar_0, Ar_1, Ar_2) -> Com_1(evalexpaper2stop(Ar_0, Ar_1, Ar_2))", 0-0) = ? S("evalexpaper2bb5in(Ar_0, Ar_1, Ar_2) -> Com_1(evalexpaper2stop(Ar_0, Ar_1, Ar_2))", 0-1) = Ar_1 S("evalexpaper2bb5in(Ar_0, Ar_1, Ar_2) -> Com_1(evalexpaper2stop(Ar_0, Ar_1, Ar_2))", 0-2) = ? S("evalexpaper2bb4in(Ar_0, Ar_1, Ar_2) -> Com_1(evalexpaper2bb1in(Ar_0 + 1, Ar_1, Ar_2))", 0-0) = ? S("evalexpaper2bb4in(Ar_0, Ar_1, Ar_2) -> Com_1(evalexpaper2bb1in(Ar_0 + 1, Ar_1, Ar_2))", 0-1) = Ar_1 S("evalexpaper2bb4in(Ar_0, Ar_1, Ar_2) -> Com_1(evalexpaper2bb1in(Ar_0 + 1, Ar_1, Ar_2))", 0-2) = ? S("evalexpaper2bb3in(Ar_0, Ar_1, Ar_2) -> Com_1(evalexpaper2bb2in(Ar_0, Ar_1, Ar_2 + 1))", 0-0) = ? S("evalexpaper2bb3in(Ar_0, Ar_1, Ar_2) -> Com_1(evalexpaper2bb2in(Ar_0, Ar_1, Ar_2 + 1))", 0-1) = Ar_1 S("evalexpaper2bb3in(Ar_0, Ar_1, Ar_2) -> Com_1(evalexpaper2bb2in(Ar_0, Ar_1, Ar_2 + 1))", 0-2) = ? S("evalexpaper2bb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalexpaper2bb4in(Ar_0, Ar_1, Ar_2)) [ Ar_2 >= Ar_1 + 1 ]", 0-0) = ? S("evalexpaper2bb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalexpaper2bb4in(Ar_0, Ar_1, Ar_2)) [ Ar_2 >= Ar_1 + 1 ]", 0-1) = Ar_1 S("evalexpaper2bb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalexpaper2bb4in(Ar_0, Ar_1, Ar_2)) [ Ar_2 >= Ar_1 + 1 ]", 0-2) = ? S("evalexpaper2bb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalexpaper2bb3in(Ar_0, Ar_1, Ar_2)) [ Ar_1 >= Ar_2 ]", 0-0) = ? S("evalexpaper2bb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalexpaper2bb3in(Ar_0, Ar_1, Ar_2)) [ Ar_1 >= Ar_2 ]", 0-1) = Ar_1 S("evalexpaper2bb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalexpaper2bb3in(Ar_0, Ar_1, Ar_2)) [ Ar_1 >= Ar_2 ]", 0-2) = ? S("evalexpaper2bb1in(Ar_0, Ar_1, Ar_2) -> Com_1(evalexpaper2bb5in(Ar_0, Ar_1, Ar_2)) [ Ar_0 >= Ar_1 + 1 ]", 0-0) = ? S("evalexpaper2bb1in(Ar_0, Ar_1, Ar_2) -> Com_1(evalexpaper2bb5in(Ar_0, Ar_1, Ar_2)) [ Ar_0 >= Ar_1 + 1 ]", 0-1) = Ar_1 S("evalexpaper2bb1in(Ar_0, Ar_1, Ar_2) -> Com_1(evalexpaper2bb5in(Ar_0, Ar_1, Ar_2)) [ Ar_0 >= Ar_1 + 1 ]", 0-2) = ? S("evalexpaper2bb1in(Ar_0, Ar_1, Ar_2) -> Com_1(evalexpaper2bb2in(Ar_0, Ar_1, Ar_0)) [ Ar_1 >= Ar_0 ]", 0-0) = ? S("evalexpaper2bb1in(Ar_0, Ar_1, Ar_2) -> Com_1(evalexpaper2bb2in(Ar_0, Ar_1, Ar_0)) [ Ar_1 >= Ar_0 ]", 0-1) = Ar_1 S("evalexpaper2bb1in(Ar_0, Ar_1, Ar_2) -> Com_1(evalexpaper2bb2in(Ar_0, Ar_1, Ar_0)) [ Ar_1 >= Ar_0 ]", 0-2) = ? S("evalexpaper2bb0in(Ar_0, Ar_1, Ar_2) -> Com_1(evalexpaper2bb1in(1, Ar_1, Ar_2))", 0-0) = 1 S("evalexpaper2bb0in(Ar_0, Ar_1, Ar_2) -> Com_1(evalexpaper2bb1in(1, Ar_1, Ar_2))", 0-1) = Ar_1 S("evalexpaper2bb0in(Ar_0, Ar_1, Ar_2) -> Com_1(evalexpaper2bb1in(1, Ar_1, Ar_2))", 0-2) = Ar_2 S("evalexpaper2start(Ar_0, Ar_1, Ar_2) -> Com_1(evalexpaper2bb0in(Ar_0, Ar_1, Ar_2))", 0-0) = Ar_0 S("evalexpaper2start(Ar_0, Ar_1, Ar_2) -> Com_1(evalexpaper2bb0in(Ar_0, Ar_1, Ar_2))", 0-1) = Ar_1 S("evalexpaper2start(Ar_0, Ar_1, Ar_2) -> Com_1(evalexpaper2bb0in(Ar_0, Ar_1, Ar_2))", 0-2) = Ar_2 orients the transitions evalexpaper2bb4in(Ar_0, Ar_1, Ar_2) -> Com_1(evalexpaper2bb1in(Ar_0 + 1, Ar_1, Ar_2)) evalexpaper2bb3in(Ar_0, Ar_1, Ar_2) -> Com_1(evalexpaper2bb2in(Ar_0, Ar_1, Ar_2 + 1)) evalexpaper2bb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalexpaper2bb4in(Ar_0, Ar_1, Ar_2)) [ Ar_2 >= Ar_1 + 1 ] evalexpaper2bb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalexpaper2bb3in(Ar_0, Ar_1, Ar_2)) [ Ar_1 >= Ar_2 ] weakly and the transitions evalexpaper2bb4in(Ar_0, Ar_1, Ar_2) -> Com_1(evalexpaper2bb1in(Ar_0 + 1, Ar_1, Ar_2)) evalexpaper2bb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalexpaper2bb4in(Ar_0, Ar_1, Ar_2)) [ Ar_2 >= Ar_1 + 1 ] strictly and produces the following problem: 5: T: (Comp: 1, Cost: 1) evalexpaper2start(Ar_0, Ar_1, Ar_2) -> Com_1(evalexpaper2bb0in(Ar_0, Ar_1, Ar_2)) (Comp: 1, Cost: 1) evalexpaper2bb0in(Ar_0, Ar_1, Ar_2) -> Com_1(evalexpaper2bb1in(1, Ar_1, Ar_2)) (Comp: Ar_1, Cost: 1) evalexpaper2bb1in(Ar_0, Ar_1, Ar_2) -> Com_1(evalexpaper2bb2in(Ar_0, Ar_1, Ar_0)) [ Ar_1 >= Ar_0 ] (Comp: 2, Cost: 1) evalexpaper2bb1in(Ar_0, Ar_1, Ar_2) -> Com_1(evalexpaper2bb5in(Ar_0, Ar_1, Ar_2)) [ Ar_0 >= Ar_1 + 1 ] (Comp: ?, Cost: 1) evalexpaper2bb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalexpaper2bb3in(Ar_0, Ar_1, Ar_2)) [ Ar_1 >= Ar_2 ] (Comp: 2*Ar_1, Cost: 1) evalexpaper2bb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalexpaper2bb4in(Ar_0, Ar_1, Ar_2)) [ Ar_2 >= Ar_1 + 1 ] (Comp: ?, Cost: 1) evalexpaper2bb3in(Ar_0, Ar_1, Ar_2) -> Com_1(evalexpaper2bb2in(Ar_0, Ar_1, Ar_2 + 1)) (Comp: 2*Ar_1, Cost: 1) evalexpaper2bb4in(Ar_0, Ar_1, Ar_2) -> Com_1(evalexpaper2bb1in(Ar_0 + 1, Ar_1, Ar_2)) (Comp: 2, Cost: 1) evalexpaper2bb5in(Ar_0, Ar_1, Ar_2) -> Com_1(evalexpaper2stop(Ar_0, Ar_1, Ar_2)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2) -> Com_1(evalexpaper2start(Ar_0, Ar_1, Ar_2)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 A polynomial rank function with Pol(evalexpaper2bb3in) = V_2 - V_3 Pol(evalexpaper2bb2in) = V_2 - V_3 + 1 and size complexities S("koat_start(Ar_0, Ar_1, Ar_2) -> Com_1(evalexpaper2start(Ar_0, Ar_1, Ar_2)) [ 0 <= 0 ]", 0-0) = Ar_0 S("koat_start(Ar_0, Ar_1, Ar_2) -> Com_1(evalexpaper2start(Ar_0, Ar_1, Ar_2)) [ 0 <= 0 ]", 0-1) = Ar_1 S("koat_start(Ar_0, Ar_1, Ar_2) -> Com_1(evalexpaper2start(Ar_0, Ar_1, Ar_2)) [ 0 <= 0 ]", 0-2) = Ar_2 S("evalexpaper2bb5in(Ar_0, Ar_1, Ar_2) -> Com_1(evalexpaper2stop(Ar_0, Ar_1, Ar_2))", 0-0) = 2*Ar_1 + 20 S("evalexpaper2bb5in(Ar_0, Ar_1, Ar_2) -> Com_1(evalexpaper2stop(Ar_0, Ar_1, Ar_2))", 0-1) = Ar_1 S("evalexpaper2bb5in(Ar_0, Ar_1, Ar_2) -> Com_1(evalexpaper2stop(Ar_0, Ar_1, Ar_2))", 0-2) = ? S("evalexpaper2bb4in(Ar_0, Ar_1, Ar_2) -> Com_1(evalexpaper2bb1in(Ar_0 + 1, Ar_1, Ar_2))", 0-0) = 2*Ar_1 + 4 S("evalexpaper2bb4in(Ar_0, Ar_1, Ar_2) -> Com_1(evalexpaper2bb1in(Ar_0 + 1, Ar_1, Ar_2))", 0-1) = Ar_1 S("evalexpaper2bb4in(Ar_0, Ar_1, Ar_2) -> Com_1(evalexpaper2bb1in(Ar_0 + 1, Ar_1, Ar_2))", 0-2) = ? S("evalexpaper2bb3in(Ar_0, Ar_1, Ar_2) -> Com_1(evalexpaper2bb2in(Ar_0, Ar_1, Ar_2 + 1))", 0-0) = 2*Ar_1 + 4 S("evalexpaper2bb3in(Ar_0, Ar_1, Ar_2) -> Com_1(evalexpaper2bb2in(Ar_0, Ar_1, Ar_2 + 1))", 0-1) = Ar_1 S("evalexpaper2bb3in(Ar_0, Ar_1, Ar_2) -> Com_1(evalexpaper2bb2in(Ar_0, Ar_1, Ar_2 + 1))", 0-2) = ? S("evalexpaper2bb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalexpaper2bb4in(Ar_0, Ar_1, Ar_2)) [ Ar_2 >= Ar_1 + 1 ]", 0-0) = 2*Ar_1 + 4 S("evalexpaper2bb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalexpaper2bb4in(Ar_0, Ar_1, Ar_2)) [ Ar_2 >= Ar_1 + 1 ]", 0-1) = Ar_1 S("evalexpaper2bb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalexpaper2bb4in(Ar_0, Ar_1, Ar_2)) [ Ar_2 >= Ar_1 + 1 ]", 0-2) = ? S("evalexpaper2bb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalexpaper2bb3in(Ar_0, Ar_1, Ar_2)) [ Ar_1 >= Ar_2 ]", 0-0) = 2*Ar_1 + 4 S("evalexpaper2bb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalexpaper2bb3in(Ar_0, Ar_1, Ar_2)) [ Ar_1 >= Ar_2 ]", 0-1) = Ar_1 S("evalexpaper2bb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalexpaper2bb3in(Ar_0, Ar_1, Ar_2)) [ Ar_1 >= Ar_2 ]", 0-2) = ? S("evalexpaper2bb1in(Ar_0, Ar_1, Ar_2) -> Com_1(evalexpaper2bb5in(Ar_0, Ar_1, Ar_2)) [ Ar_0 >= Ar_1 + 1 ]", 0-0) = 2*Ar_1 + 10 S("evalexpaper2bb1in(Ar_0, Ar_1, Ar_2) -> Com_1(evalexpaper2bb5in(Ar_0, Ar_1, Ar_2)) [ Ar_0 >= Ar_1 + 1 ]", 0-1) = Ar_1 S("evalexpaper2bb1in(Ar_0, Ar_1, Ar_2) -> Com_1(evalexpaper2bb5in(Ar_0, Ar_1, Ar_2)) [ Ar_0 >= Ar_1 + 1 ]", 0-2) = ? S("evalexpaper2bb1in(Ar_0, Ar_1, Ar_2) -> Com_1(evalexpaper2bb2in(Ar_0, Ar_1, Ar_0)) [ Ar_1 >= Ar_0 ]", 0-0) = 2*Ar_1 + 4 S("evalexpaper2bb1in(Ar_0, Ar_1, Ar_2) -> Com_1(evalexpaper2bb2in(Ar_0, Ar_1, Ar_0)) [ Ar_1 >= Ar_0 ]", 0-1) = Ar_1 S("evalexpaper2bb1in(Ar_0, Ar_1, Ar_2) -> Com_1(evalexpaper2bb2in(Ar_0, Ar_1, Ar_0)) [ Ar_1 >= Ar_0 ]", 0-2) = 2*Ar_1 + 10 S("evalexpaper2bb0in(Ar_0, Ar_1, Ar_2) -> Com_1(evalexpaper2bb1in(1, Ar_1, Ar_2))", 0-0) = 1 S("evalexpaper2bb0in(Ar_0, Ar_1, Ar_2) -> Com_1(evalexpaper2bb1in(1, Ar_1, Ar_2))", 0-1) = Ar_1 S("evalexpaper2bb0in(Ar_0, Ar_1, Ar_2) -> Com_1(evalexpaper2bb1in(1, Ar_1, Ar_2))", 0-2) = Ar_2 S("evalexpaper2start(Ar_0, Ar_1, Ar_2) -> Com_1(evalexpaper2bb0in(Ar_0, Ar_1, Ar_2))", 0-0) = Ar_0 S("evalexpaper2start(Ar_0, Ar_1, Ar_2) -> Com_1(evalexpaper2bb0in(Ar_0, Ar_1, Ar_2))", 0-1) = Ar_1 S("evalexpaper2start(Ar_0, Ar_1, Ar_2) -> Com_1(evalexpaper2bb0in(Ar_0, Ar_1, Ar_2))", 0-2) = Ar_2 orients the transitions evalexpaper2bb3in(Ar_0, Ar_1, Ar_2) -> Com_1(evalexpaper2bb2in(Ar_0, Ar_1, Ar_2 + 1)) evalexpaper2bb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalexpaper2bb3in(Ar_0, Ar_1, Ar_2)) [ Ar_1 >= Ar_2 ] weakly and the transition evalexpaper2bb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalexpaper2bb3in(Ar_0, Ar_1, Ar_2)) [ Ar_1 >= Ar_2 ] strictly and produces the following problem: 6: T: (Comp: 1, Cost: 1) evalexpaper2start(Ar_0, Ar_1, Ar_2) -> Com_1(evalexpaper2bb0in(Ar_0, Ar_1, Ar_2)) (Comp: 1, Cost: 1) evalexpaper2bb0in(Ar_0, Ar_1, Ar_2) -> Com_1(evalexpaper2bb1in(1, Ar_1, Ar_2)) (Comp: Ar_1, Cost: 1) evalexpaper2bb1in(Ar_0, Ar_1, Ar_2) -> Com_1(evalexpaper2bb2in(Ar_0, Ar_1, Ar_0)) [ Ar_1 >= Ar_0 ] (Comp: 2, Cost: 1) evalexpaper2bb1in(Ar_0, Ar_1, Ar_2) -> Com_1(evalexpaper2bb5in(Ar_0, Ar_1, Ar_2)) [ Ar_0 >= Ar_1 + 1 ] (Comp: 3*Ar_1^2 + 11*Ar_1, Cost: 1) evalexpaper2bb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalexpaper2bb3in(Ar_0, Ar_1, Ar_2)) [ Ar_1 >= Ar_2 ] (Comp: 2*Ar_1, Cost: 1) evalexpaper2bb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalexpaper2bb4in(Ar_0, Ar_1, Ar_2)) [ Ar_2 >= Ar_1 + 1 ] (Comp: ?, Cost: 1) evalexpaper2bb3in(Ar_0, Ar_1, Ar_2) -> Com_1(evalexpaper2bb2in(Ar_0, Ar_1, Ar_2 + 1)) (Comp: 2*Ar_1, Cost: 1) evalexpaper2bb4in(Ar_0, Ar_1, Ar_2) -> Com_1(evalexpaper2bb1in(Ar_0 + 1, Ar_1, Ar_2)) (Comp: 2, Cost: 1) evalexpaper2bb5in(Ar_0, Ar_1, Ar_2) -> Com_1(evalexpaper2stop(Ar_0, Ar_1, Ar_2)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2) -> Com_1(evalexpaper2start(Ar_0, Ar_1, Ar_2)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 Repeatedly propagating knowledge in problem 6 produces the following problem: 7: T: (Comp: 1, Cost: 1) evalexpaper2start(Ar_0, Ar_1, Ar_2) -> Com_1(evalexpaper2bb0in(Ar_0, Ar_1, Ar_2)) (Comp: 1, Cost: 1) evalexpaper2bb0in(Ar_0, Ar_1, Ar_2) -> Com_1(evalexpaper2bb1in(1, Ar_1, Ar_2)) (Comp: Ar_1, Cost: 1) evalexpaper2bb1in(Ar_0, Ar_1, Ar_2) -> Com_1(evalexpaper2bb2in(Ar_0, Ar_1, Ar_0)) [ Ar_1 >= Ar_0 ] (Comp: 2, Cost: 1) evalexpaper2bb1in(Ar_0, Ar_1, Ar_2) -> Com_1(evalexpaper2bb5in(Ar_0, Ar_1, Ar_2)) [ Ar_0 >= Ar_1 + 1 ] (Comp: 3*Ar_1^2 + 11*Ar_1, Cost: 1) evalexpaper2bb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalexpaper2bb3in(Ar_0, Ar_1, Ar_2)) [ Ar_1 >= Ar_2 ] (Comp: 2*Ar_1, Cost: 1) evalexpaper2bb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalexpaper2bb4in(Ar_0, Ar_1, Ar_2)) [ Ar_2 >= Ar_1 + 1 ] (Comp: 3*Ar_1^2 + 11*Ar_1, Cost: 1) evalexpaper2bb3in(Ar_0, Ar_1, Ar_2) -> Com_1(evalexpaper2bb2in(Ar_0, Ar_1, Ar_2 + 1)) (Comp: 2*Ar_1, Cost: 1) evalexpaper2bb4in(Ar_0, Ar_1, Ar_2) -> Com_1(evalexpaper2bb1in(Ar_0 + 1, Ar_1, Ar_2)) (Comp: 2, Cost: 1) evalexpaper2bb5in(Ar_0, Ar_1, Ar_2) -> Com_1(evalexpaper2stop(Ar_0, Ar_1, Ar_2)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2) -> Com_1(evalexpaper2start(Ar_0, Ar_1, Ar_2)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 Complexity upper bound 27*Ar_1 + 6*Ar_1^2 + 6 Time: 0.068 sec (SMT: 0.055 sec) ---------------------------------------- (2) BOUNDS(1, n^2)