/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(1, n^2). (0) CpxIntTrs (1) Koat Proof [FINISHED, 84 ms] (2) BOUNDS(1, n^2) ---------------------------------------- (0) Obligation: Complexity Int TRS consisting of the following rules: eval_while2_start(v_N, v_i.0, v_j.0) -> Com_1(eval_while2_bb0_in(v_N, v_i.0, v_j.0)) :|: TRUE eval_while2_bb0_in(v_N, v_i.0, v_j.0) -> Com_1(eval_while2_bb1_in(v_N, v_N, v_j.0)) :|: TRUE eval_while2_bb1_in(v_N, v_i.0, v_j.0) -> Com_1(eval_while2_bb2_in(v_N, v_i.0, v_N)) :|: v_i.0 > 0 eval_while2_bb1_in(v_N, v_i.0, v_j.0) -> Com_1(eval_while2_bb5_in(v_N, v_i.0, v_j.0)) :|: v_i.0 <= 0 eval_while2_bb2_in(v_N, v_i.0, v_j.0) -> Com_1(eval_while2_bb3_in(v_N, v_i.0, v_j.0)) :|: v_j.0 > 0 eval_while2_bb2_in(v_N, v_i.0, v_j.0) -> Com_1(eval_while2_bb4_in(v_N, v_i.0, v_j.0)) :|: v_j.0 <= 0 eval_while2_bb3_in(v_N, v_i.0, v_j.0) -> Com_1(eval_while2_bb2_in(v_N, v_i.0, v_j.0 - 1)) :|: TRUE eval_while2_bb4_in(v_N, v_i.0, v_j.0) -> Com_1(eval_while2_bb1_in(v_N, v_i.0 - 1, v_j.0)) :|: TRUE eval_while2_bb5_in(v_N, v_i.0, v_j.0) -> Com_1(eval_while2_stop(v_N, v_i.0, v_j.0)) :|: TRUE The start-symbols are:[eval_while2_start_3] ---------------------------------------- (1) Koat Proof (FINISHED) YES(?, 7*ar_1 + 2*ar_1^2 + 11) Initial complexity problem: 1: T: (Comp: ?, Cost: 1) evalwhile2start(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb0in(ar_0, ar_1, ar_2)) (Comp: ?, Cost: 1) evalwhile2bb0in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb1in(ar_1, ar_1, ar_2)) (Comp: ?, Cost: 1) evalwhile2bb1in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb2in(ar_0, ar_1, ar_1)) [ ar_0 >= 1 ] (Comp: ?, Cost: 1) evalwhile2bb1in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb5in(ar_0, ar_1, ar_2)) [ 0 >= ar_0 ] (Comp: ?, Cost: 1) evalwhile2bb2in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb3in(ar_0, ar_1, ar_2)) [ ar_2 >= 1 ] (Comp: ?, Cost: 1) evalwhile2bb2in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb4in(ar_0, ar_1, ar_2)) [ 0 >= ar_2 ] (Comp: ?, Cost: 1) evalwhile2bb3in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb2in(ar_0, ar_1, ar_2 - 1)) (Comp: ?, Cost: 1) evalwhile2bb4in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb1in(ar_0 - 1, ar_1, ar_2)) (Comp: ?, Cost: 1) evalwhile2bb5in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2stop(ar_0, ar_1, ar_2)) (Comp: 1, Cost: 0) koat_start(ar_0, ar_1, ar_2) -> Com_1(evalwhile2start(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) evalwhile2start(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb0in(ar_0, ar_1, ar_2)) (Comp: 1, Cost: 1) evalwhile2bb0in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb1in(ar_1, ar_1, ar_2)) (Comp: ?, Cost: 1) evalwhile2bb1in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb2in(ar_0, ar_1, ar_1)) [ ar_0 >= 1 ] (Comp: ?, Cost: 1) evalwhile2bb1in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb5in(ar_0, ar_1, ar_2)) [ 0 >= ar_0 ] (Comp: ?, Cost: 1) evalwhile2bb2in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb3in(ar_0, ar_1, ar_2)) [ ar_2 >= 1 ] (Comp: ?, Cost: 1) evalwhile2bb2in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb4in(ar_0, ar_1, ar_2)) [ 0 >= ar_2 ] (Comp: ?, Cost: 1) evalwhile2bb3in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb2in(ar_0, ar_1, ar_2 - 1)) (Comp: ?, Cost: 1) evalwhile2bb4in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb1in(ar_0 - 1, ar_1, ar_2)) (Comp: ?, Cost: 1) evalwhile2bb5in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2stop(ar_0, ar_1, ar_2)) (Comp: 1, Cost: 0) koat_start(ar_0, ar_1, ar_2) -> Com_1(evalwhile2start(ar_0, ar_1, ar_2)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 A polynomial rank function with Pol(evalwhile2start) = 2 Pol(evalwhile2bb0in) = 2 Pol(evalwhile2bb1in) = 2 Pol(evalwhile2bb2in) = 2 Pol(evalwhile2bb5in) = 1 Pol(evalwhile2bb3in) = 2 Pol(evalwhile2bb4in) = 2 Pol(evalwhile2stop) = 0 Pol(koat_start) = 2 orients all transitions weakly and the transitions evalwhile2bb5in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2stop(ar_0, ar_1, ar_2)) evalwhile2bb1in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb5in(ar_0, ar_1, ar_2)) [ 0 >= ar_0 ] strictly and produces the following problem: 3: T: (Comp: 1, Cost: 1) evalwhile2start(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb0in(ar_0, ar_1, ar_2)) (Comp: 1, Cost: 1) evalwhile2bb0in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb1in(ar_1, ar_1, ar_2)) (Comp: ?, Cost: 1) evalwhile2bb1in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb2in(ar_0, ar_1, ar_1)) [ ar_0 >= 1 ] (Comp: 2, Cost: 1) evalwhile2bb1in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb5in(ar_0, ar_1, ar_2)) [ 0 >= ar_0 ] (Comp: ?, Cost: 1) evalwhile2bb2in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb3in(ar_0, ar_1, ar_2)) [ ar_2 >= 1 ] (Comp: ?, Cost: 1) evalwhile2bb2in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb4in(ar_0, ar_1, ar_2)) [ 0 >= ar_2 ] (Comp: ?, Cost: 1) evalwhile2bb3in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb2in(ar_0, ar_1, ar_2 - 1)) (Comp: ?, Cost: 1) evalwhile2bb4in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb1in(ar_0 - 1, ar_1, ar_2)) (Comp: 2, Cost: 1) evalwhile2bb5in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2stop(ar_0, ar_1, ar_2)) (Comp: 1, Cost: 0) koat_start(ar_0, ar_1, ar_2) -> Com_1(evalwhile2start(ar_0, ar_1, ar_2)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 A polynomial rank function with Pol(evalwhile2start) = V_2 + 1 Pol(evalwhile2bb0in) = V_2 + 1 Pol(evalwhile2bb1in) = V_1 + 1 Pol(evalwhile2bb2in) = V_1 Pol(evalwhile2bb5in) = V_1 Pol(evalwhile2bb3in) = V_1 Pol(evalwhile2bb4in) = V_1 Pol(evalwhile2stop) = V_1 Pol(koat_start) = V_2 + 1 orients all transitions weakly and the transition evalwhile2bb1in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb2in(ar_0, ar_1, ar_1)) [ ar_0 >= 1 ] strictly and produces the following problem: 4: T: (Comp: 1, Cost: 1) evalwhile2start(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb0in(ar_0, ar_1, ar_2)) (Comp: 1, Cost: 1) evalwhile2bb0in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb1in(ar_1, ar_1, ar_2)) (Comp: ar_1 + 1, Cost: 1) evalwhile2bb1in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb2in(ar_0, ar_1, ar_1)) [ ar_0 >= 1 ] (Comp: 2, Cost: 1) evalwhile2bb1in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb5in(ar_0, ar_1, ar_2)) [ 0 >= ar_0 ] (Comp: ?, Cost: 1) evalwhile2bb2in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb3in(ar_0, ar_1, ar_2)) [ ar_2 >= 1 ] (Comp: ?, Cost: 1) evalwhile2bb2in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb4in(ar_0, ar_1, ar_2)) [ 0 >= ar_2 ] (Comp: ?, Cost: 1) evalwhile2bb3in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb2in(ar_0, ar_1, ar_2 - 1)) (Comp: ?, Cost: 1) evalwhile2bb4in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb1in(ar_0 - 1, ar_1, ar_2)) (Comp: 2, Cost: 1) evalwhile2bb5in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2stop(ar_0, ar_1, ar_2)) (Comp: 1, Cost: 0) koat_start(ar_0, ar_1, ar_2) -> Com_1(evalwhile2start(ar_0, ar_1, ar_2)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 A polynomial rank function with Pol(evalwhile2bb4in) = 1 Pol(evalwhile2bb1in) = 0 Pol(evalwhile2bb3in) = 2 Pol(evalwhile2bb2in) = 2 and size complexities S("koat_start(ar_0, ar_1, ar_2) -> Com_1(evalwhile2start(ar_0, ar_1, ar_2)) [ 0 <= 0 ]", 0-0) = ar_0 S("koat_start(ar_0, ar_1, ar_2) -> Com_1(evalwhile2start(ar_0, ar_1, ar_2)) [ 0 <= 0 ]", 0-1) = ar_1 S("koat_start(ar_0, ar_1, ar_2) -> Com_1(evalwhile2start(ar_0, ar_1, ar_2)) [ 0 <= 0 ]", 0-2) = ar_2 S("evalwhile2bb5in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2stop(ar_0, ar_1, ar_2))", 0-0) = ? S("evalwhile2bb5in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2stop(ar_0, ar_1, ar_2))", 0-1) = ar_1 S("evalwhile2bb5in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2stop(ar_0, ar_1, ar_2))", 0-2) = ? S("evalwhile2bb4in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb1in(ar_0 - 1, ar_1, ar_2))", 0-0) = ? S("evalwhile2bb4in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb1in(ar_0 - 1, ar_1, ar_2))", 0-1) = ar_1 S("evalwhile2bb4in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb1in(ar_0 - 1, ar_1, ar_2))", 0-2) = ? S("evalwhile2bb3in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb2in(ar_0, ar_1, ar_2 - 1))", 0-0) = ? S("evalwhile2bb3in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb2in(ar_0, ar_1, ar_2 - 1))", 0-1) = ar_1 S("evalwhile2bb3in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb2in(ar_0, ar_1, ar_2 - 1))", 0-2) = ? S("evalwhile2bb2in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb4in(ar_0, ar_1, ar_2)) [ 0 >= ar_2 ]", 0-0) = ? S("evalwhile2bb2in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb4in(ar_0, ar_1, ar_2)) [ 0 >= ar_2 ]", 0-1) = ar_1 S("evalwhile2bb2in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb4in(ar_0, ar_1, ar_2)) [ 0 >= ar_2 ]", 0-2) = ? S("evalwhile2bb2in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb3in(ar_0, ar_1, ar_2)) [ ar_2 >= 1 ]", 0-0) = ? S("evalwhile2bb2in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb3in(ar_0, ar_1, ar_2)) [ ar_2 >= 1 ]", 0-1) = ar_1 S("evalwhile2bb2in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb3in(ar_0, ar_1, ar_2)) [ ar_2 >= 1 ]", 0-2) = ? S("evalwhile2bb1in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb5in(ar_0, ar_1, ar_2)) [ 0 >= ar_0 ]", 0-0) = ? S("evalwhile2bb1in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb5in(ar_0, ar_1, ar_2)) [ 0 >= ar_0 ]", 0-1) = ar_1 S("evalwhile2bb1in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb5in(ar_0, ar_1, ar_2)) [ 0 >= ar_0 ]", 0-2) = ? S("evalwhile2bb1in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb2in(ar_0, ar_1, ar_1)) [ ar_0 >= 1 ]", 0-0) = ? S("evalwhile2bb1in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb2in(ar_0, ar_1, ar_1)) [ ar_0 >= 1 ]", 0-1) = ar_1 S("evalwhile2bb1in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb2in(ar_0, ar_1, ar_1)) [ ar_0 >= 1 ]", 0-2) = ar_1 S("evalwhile2bb0in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb1in(ar_1, ar_1, ar_2))", 0-0) = ar_1 S("evalwhile2bb0in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb1in(ar_1, ar_1, ar_2))", 0-1) = ar_1 S("evalwhile2bb0in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb1in(ar_1, ar_1, ar_2))", 0-2) = ar_2 S("evalwhile2start(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb0in(ar_0, ar_1, ar_2))", 0-0) = ar_0 S("evalwhile2start(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb0in(ar_0, ar_1, ar_2))", 0-1) = ar_1 S("evalwhile2start(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb0in(ar_0, ar_1, ar_2))", 0-2) = ar_2 orients the transitions evalwhile2bb4in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb1in(ar_0 - 1, ar_1, ar_2)) evalwhile2bb3in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb2in(ar_0, ar_1, ar_2 - 1)) evalwhile2bb2in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb4in(ar_0, ar_1, ar_2)) [ 0 >= ar_2 ] evalwhile2bb2in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb3in(ar_0, ar_1, ar_2)) [ ar_2 >= 1 ] weakly and the transitions evalwhile2bb4in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb1in(ar_0 - 1, ar_1, ar_2)) evalwhile2bb2in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb4in(ar_0, ar_1, ar_2)) [ 0 >= ar_2 ] strictly and produces the following problem: 5: T: (Comp: 1, Cost: 1) evalwhile2start(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb0in(ar_0, ar_1, ar_2)) (Comp: 1, Cost: 1) evalwhile2bb0in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb1in(ar_1, ar_1, ar_2)) (Comp: ar_1 + 1, Cost: 1) evalwhile2bb1in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb2in(ar_0, ar_1, ar_1)) [ ar_0 >= 1 ] (Comp: 2, Cost: 1) evalwhile2bb1in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb5in(ar_0, ar_1, ar_2)) [ 0 >= ar_0 ] (Comp: ?, Cost: 1) evalwhile2bb2in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb3in(ar_0, ar_1, ar_2)) [ ar_2 >= 1 ] (Comp: 2*ar_1 + 2, Cost: 1) evalwhile2bb2in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb4in(ar_0, ar_1, ar_2)) [ 0 >= ar_2 ] (Comp: ?, Cost: 1) evalwhile2bb3in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb2in(ar_0, ar_1, ar_2 - 1)) (Comp: 2*ar_1 + 2, Cost: 1) evalwhile2bb4in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb1in(ar_0 - 1, ar_1, ar_2)) (Comp: 2, Cost: 1) evalwhile2bb5in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2stop(ar_0, ar_1, ar_2)) (Comp: 1, Cost: 0) koat_start(ar_0, ar_1, ar_2) -> Com_1(evalwhile2start(ar_0, ar_1, ar_2)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 A polynomial rank function with Pol(evalwhile2bb3in) = V_3 - 1 Pol(evalwhile2bb2in) = V_3 and size complexities S("koat_start(ar_0, ar_1, ar_2) -> Com_1(evalwhile2start(ar_0, ar_1, ar_2)) [ 0 <= 0 ]", 0-0) = ar_0 S("koat_start(ar_0, ar_1, ar_2) -> Com_1(evalwhile2start(ar_0, ar_1, ar_2)) [ 0 <= 0 ]", 0-1) = ar_1 S("koat_start(ar_0, ar_1, ar_2) -> Com_1(evalwhile2start(ar_0, ar_1, ar_2)) [ 0 <= 0 ]", 0-2) = ar_2 S("evalwhile2bb5in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2stop(ar_0, ar_1, ar_2))", 0-0) = 3*ar_1 + 162 S("evalwhile2bb5in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2stop(ar_0, ar_1, ar_2))", 0-1) = ar_1 S("evalwhile2bb5in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2stop(ar_0, ar_1, ar_2))", 0-2) = ? S("evalwhile2bb4in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb1in(ar_0 - 1, ar_1, ar_2))", 0-0) = 3*ar_1 + 18 S("evalwhile2bb4in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb1in(ar_0 - 1, ar_1, ar_2))", 0-1) = ar_1 S("evalwhile2bb4in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb1in(ar_0 - 1, ar_1, ar_2))", 0-2) = ? S("evalwhile2bb3in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb2in(ar_0, ar_1, ar_2 - 1))", 0-0) = 3*ar_1 + 18 S("evalwhile2bb3in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb2in(ar_0, ar_1, ar_2 - 1))", 0-1) = ar_1 S("evalwhile2bb3in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb2in(ar_0, ar_1, ar_2 - 1))", 0-2) = ? S("evalwhile2bb2in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb4in(ar_0, ar_1, ar_2)) [ 0 >= ar_2 ]", 0-0) = 3*ar_1 + 18 S("evalwhile2bb2in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb4in(ar_0, ar_1, ar_2)) [ 0 >= ar_2 ]", 0-1) = ar_1 S("evalwhile2bb2in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb4in(ar_0, ar_1, ar_2)) [ 0 >= ar_2 ]", 0-2) = ? S("evalwhile2bb2in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb3in(ar_0, ar_1, ar_2)) [ ar_2 >= 1 ]", 0-0) = 3*ar_1 + 18 S("evalwhile2bb2in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb3in(ar_0, ar_1, ar_2)) [ ar_2 >= 1 ]", 0-1) = ar_1 S("evalwhile2bb2in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb3in(ar_0, ar_1, ar_2)) [ ar_2 >= 1 ]", 0-2) = ? S("evalwhile2bb1in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb5in(ar_0, ar_1, ar_2)) [ 0 >= ar_0 ]", 0-0) = 3*ar_1 + 54 S("evalwhile2bb1in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb5in(ar_0, ar_1, ar_2)) [ 0 >= ar_0 ]", 0-1) = ar_1 S("evalwhile2bb1in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb5in(ar_0, ar_1, ar_2)) [ 0 >= ar_0 ]", 0-2) = ? S("evalwhile2bb1in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb2in(ar_0, ar_1, ar_1)) [ ar_0 >= 1 ]", 0-0) = 3*ar_1 + 18 S("evalwhile2bb1in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb2in(ar_0, ar_1, ar_1)) [ ar_0 >= 1 ]", 0-1) = ar_1 S("evalwhile2bb1in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb2in(ar_0, ar_1, ar_1)) [ ar_0 >= 1 ]", 0-2) = ar_1 S("evalwhile2bb0in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb1in(ar_1, ar_1, ar_2))", 0-0) = ar_1 S("evalwhile2bb0in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb1in(ar_1, ar_1, ar_2))", 0-1) = ar_1 S("evalwhile2bb0in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb1in(ar_1, ar_1, ar_2))", 0-2) = ar_2 S("evalwhile2start(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb0in(ar_0, ar_1, ar_2))", 0-0) = ar_0 S("evalwhile2start(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb0in(ar_0, ar_1, ar_2))", 0-1) = ar_1 S("evalwhile2start(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb0in(ar_0, ar_1, ar_2))", 0-2) = ar_2 orients the transitions evalwhile2bb3in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb2in(ar_0, ar_1, ar_2 - 1)) evalwhile2bb2in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb3in(ar_0, ar_1, ar_2)) [ ar_2 >= 1 ] weakly and the transition evalwhile2bb2in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb3in(ar_0, ar_1, ar_2)) [ ar_2 >= 1 ] strictly and produces the following problem: 6: T: (Comp: 1, Cost: 1) evalwhile2start(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb0in(ar_0, ar_1, ar_2)) (Comp: 1, Cost: 1) evalwhile2bb0in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb1in(ar_1, ar_1, ar_2)) (Comp: ar_1 + 1, Cost: 1) evalwhile2bb1in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb2in(ar_0, ar_1, ar_1)) [ ar_0 >= 1 ] (Comp: 2, Cost: 1) evalwhile2bb1in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb5in(ar_0, ar_1, ar_2)) [ 0 >= ar_0 ] (Comp: ar_1^2 + ar_1, Cost: 1) evalwhile2bb2in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb3in(ar_0, ar_1, ar_2)) [ ar_2 >= 1 ] (Comp: 2*ar_1 + 2, Cost: 1) evalwhile2bb2in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb4in(ar_0, ar_1, ar_2)) [ 0 >= ar_2 ] (Comp: ?, Cost: 1) evalwhile2bb3in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb2in(ar_0, ar_1, ar_2 - 1)) (Comp: 2*ar_1 + 2, Cost: 1) evalwhile2bb4in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb1in(ar_0 - 1, ar_1, ar_2)) (Comp: 2, Cost: 1) evalwhile2bb5in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2stop(ar_0, ar_1, ar_2)) (Comp: 1, Cost: 0) koat_start(ar_0, ar_1, ar_2) -> Com_1(evalwhile2start(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) evalwhile2start(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb0in(ar_0, ar_1, ar_2)) (Comp: 1, Cost: 1) evalwhile2bb0in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb1in(ar_1, ar_1, ar_2)) (Comp: ar_1 + 1, Cost: 1) evalwhile2bb1in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb2in(ar_0, ar_1, ar_1)) [ ar_0 >= 1 ] (Comp: 2, Cost: 1) evalwhile2bb1in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb5in(ar_0, ar_1, ar_2)) [ 0 >= ar_0 ] (Comp: ar_1^2 + ar_1, Cost: 1) evalwhile2bb2in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb3in(ar_0, ar_1, ar_2)) [ ar_2 >= 1 ] (Comp: 2*ar_1 + 2, Cost: 1) evalwhile2bb2in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb4in(ar_0, ar_1, ar_2)) [ 0 >= ar_2 ] (Comp: ar_1^2 + ar_1, Cost: 1) evalwhile2bb3in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb2in(ar_0, ar_1, ar_2 - 1)) (Comp: 2*ar_1 + 2, Cost: 1) evalwhile2bb4in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb1in(ar_0 - 1, ar_1, ar_2)) (Comp: 2, Cost: 1) evalwhile2bb5in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2stop(ar_0, ar_1, ar_2)) (Comp: 1, Cost: 0) koat_start(ar_0, ar_1, ar_2) -> Com_1(evalwhile2start(ar_0, ar_1, ar_2)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 Complexity upper bound 7*ar_1 + 2*ar_1^2 + 11 Time: 0.080 sec (SMT: 0.071 sec) ---------------------------------------- (2) BOUNDS(1, n^2)