/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(1, n^2).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, 276 ms] (2) BOUNDS(1, n^2) ---------------------------------------- (0) Obligation: Complexity Int TRS consisting of the following rules: eval_speed_pldi09_fig1_start(v_n, v_x.0, v_y.0) -> Com_1(eval_speed_pldi09_fig1_bb0_in(v_n, v_x.0, v_y.0)) :|: TRUE eval_speed_pldi09_fig1_bb0_in(v_n, v_x.0, v_y.0) -> Com_1(eval_speed_pldi09_fig1_bb1_in(v_n, 0, 0)) :|: TRUE eval_speed_pldi09_fig1_bb1_in(v_n, v_x.0, v_y.0) -> Com_1(eval_speed_pldi09_fig1_bb2_in(v_n, v_x.0, v_y.0)) :|: v_x.0 < v_n eval_speed_pldi09_fig1_bb1_in(v_n, v_x.0, v_y.0) -> Com_1(eval_speed_pldi09_fig1_bb3_in(v_n, v_x.0, v_y.0)) :|: v_x.0 >= v_n eval_speed_pldi09_fig1_bb2_in(v_n, v_x.0, v_y.0) -> Com_1(eval_speed_pldi09_fig1_bb1_in(v_n, v_x.0 + 1, v_y.0 + 1)) :|: TRUE eval_speed_pldi09_fig1_bb3_in(v_n, v_x.0, v_y.0) -> Com_1(eval_speed_pldi09_fig1_bb4_in(v_n, v_x.0, v_y.0)) :|: v_y.0 > 0 eval_speed_pldi09_fig1_bb3_in(v_n, v_x.0, v_y.0) -> Com_1(eval_speed_pldi09_fig1_bb5_in(v_n, v_x.0, v_y.0)) :|: v_y.0 <= 0 eval_speed_pldi09_fig1_bb4_in(v_n, v_x.0, v_y.0) -> Com_1(eval_speed_pldi09_fig1_bb1_in(v_n, v_x.0, v_y.0 - 1)) :|: TRUE eval_speed_pldi09_fig1_bb5_in(v_n, v_x.0, v_y.0) -> Com_1(eval_speed_pldi09_fig1_stop(v_n, v_x.0, v_y.0)) :|: TRUE The start-symbols are:[eval_speed_pldi09_fig1_start_3] ---------------------------------------- (1) Koat Proof (FINISHED) YES(?, 9*ar_2^2 + 8*ar_2 + 12) Initial complexity problem: 1: T: (Comp: ?, Cost: 1) evalspeedpldi09fig1start(ar_0, ar_1, ar_2) -> Com_1(evalspeedpldi09fig1bb0in(ar_0, ar_1, ar_2)) (Comp: ?, Cost: 1) evalspeedpldi09fig1bb0in(ar_0, ar_1, ar_2) -> Com_1(evalspeedpldi09fig1bb1in(0, 0, ar_2)) (Comp: ?, Cost: 1) evalspeedpldi09fig1bb1in(ar_0, ar_1, ar_2) -> Com_1(evalspeedpldi09fig1bb2in(ar_0, ar_1, ar_2)) [ ar_2 >= ar_0 + 1 ] (Comp: ?, Cost: 1) evalspeedpldi09fig1bb1in(ar_0, ar_1, ar_2) -> Com_1(evalspeedpldi09fig1bb3in(ar_0, ar_1, ar_2)) [ ar_0 >= ar_2 ] (Comp: ?, Cost: 1) evalspeedpldi09fig1bb2in(ar_0, ar_1, ar_2) -> Com_1(evalspeedpldi09fig1bb1in(ar_0 + 1, ar_1 + 1, ar_2)) (Comp: ?, Cost: 1) evalspeedpldi09fig1bb3in(ar_0, ar_1, ar_2) -> Com_1(evalspeedpldi09fig1bb4in(ar_0, ar_1, ar_2)) [ ar_1 >= 1 ] (Comp: ?, Cost: 1) evalspeedpldi09fig1bb3in(ar_0, ar_1, ar_2) -> Com_1(evalspeedpldi09fig1bb5in(ar_0, ar_1, ar_2)) [ 0 >= ar_1 ] (Comp: ?, Cost: 1) evalspeedpldi09fig1bb4in(ar_0, ar_1, ar_2) -> Com_1(evalspeedpldi09fig1bb1in(ar_0, ar_1 - 1, ar_2)) (Comp: ?, Cost: 1) evalspeedpldi09fig1bb5in(ar_0, ar_1, ar_2) -> Com_1(evalspeedpldi09fig1stop(ar_0, ar_1, ar_2)) (Comp: 1, Cost: 0) koat_start(ar_0, ar_1, ar_2) -> Com_1(evalspeedpldi09fig1start(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) evalspeedpldi09fig1start(ar_0, ar_1, ar_2) -> Com_1(evalspeedpldi09fig1bb0in(ar_0, ar_1, ar_2)) (Comp: 1, Cost: 1) evalspeedpldi09fig1bb0in(ar_0, ar_1, ar_2) -> Com_1(evalspeedpldi09fig1bb1in(0, 0, ar_2)) (Comp: ?, Cost: 1) evalspeedpldi09fig1bb1in(ar_0, ar_1, ar_2) -> Com_1(evalspeedpldi09fig1bb2in(ar_0, ar_1, ar_2)) [ ar_2 >= ar_0 + 1 ] (Comp: ?, Cost: 1) evalspeedpldi09fig1bb1in(ar_0, ar_1, ar_2) -> Com_1(evalspeedpldi09fig1bb3in(ar_0, ar_1, ar_2)) [ ar_0 >= ar_2 ] (Comp: ?, Cost: 1) evalspeedpldi09fig1bb2in(ar_0, ar_1, ar_2) -> Com_1(evalspeedpldi09fig1bb1in(ar_0 + 1, ar_1 + 1, ar_2)) (Comp: ?, Cost: 1) evalspeedpldi09fig1bb3in(ar_0, ar_1, ar_2) -> Com_1(evalspeedpldi09fig1bb4in(ar_0, ar_1, ar_2)) [ ar_1 >= 1 ] (Comp: ?, Cost: 1) evalspeedpldi09fig1bb3in(ar_0, ar_1, ar_2) -> Com_1(evalspeedpldi09fig1bb5in(ar_0, ar_1, ar_2)) [ 0 >= ar_1 ] (Comp: ?, Cost: 1) evalspeedpldi09fig1bb4in(ar_0, ar_1, ar_2) -> Com_1(evalspeedpldi09fig1bb1in(ar_0, ar_1 - 1, ar_2)) (Comp: ?, Cost: 1) evalspeedpldi09fig1bb5in(ar_0, ar_1, ar_2) -> Com_1(evalspeedpldi09fig1stop(ar_0, ar_1, ar_2)) (Comp: 1, Cost: 0) koat_start(ar_0, ar_1, ar_2) -> Com_1(evalspeedpldi09fig1start(ar_0, ar_1, ar_2)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 A polynomial rank function with Pol(evalspeedpldi09fig1start) = 2 Pol(evalspeedpldi09fig1bb0in) = 2 Pol(evalspeedpldi09fig1bb1in) = 2 Pol(evalspeedpldi09fig1bb2in) = 2 Pol(evalspeedpldi09fig1bb3in) = 2 Pol(evalspeedpldi09fig1bb4in) = 2 Pol(evalspeedpldi09fig1bb5in) = 1 Pol(evalspeedpldi09fig1stop) = 0 Pol(koat_start) = 2 orients all transitions weakly and the transitions evalspeedpldi09fig1bb5in(ar_0, ar_1, ar_2) -> Com_1(evalspeedpldi09fig1stop(ar_0, ar_1, ar_2)) evalspeedpldi09fig1bb3in(ar_0, ar_1, ar_2) -> Com_1(evalspeedpldi09fig1bb5in(ar_0, ar_1, ar_2)) [ 0 >= ar_1 ] strictly and produces the following problem: 3: T: (Comp: 1, Cost: 1) evalspeedpldi09fig1start(ar_0, ar_1, ar_2) -> Com_1(evalspeedpldi09fig1bb0in(ar_0, ar_1, ar_2)) (Comp: 1, Cost: 1) evalspeedpldi09fig1bb0in(ar_0, ar_1, ar_2) -> Com_1(evalspeedpldi09fig1bb1in(0, 0, ar_2)) (Comp: ?, Cost: 1) evalspeedpldi09fig1bb1in(ar_0, ar_1, ar_2) -> Com_1(evalspeedpldi09fig1bb2in(ar_0, ar_1, ar_2)) [ ar_2 >= ar_0 + 1 ] (Comp: ?, Cost: 1) evalspeedpldi09fig1bb1in(ar_0, ar_1, ar_2) -> Com_1(evalspeedpldi09fig1bb3in(ar_0, ar_1, ar_2)) [ ar_0 >= ar_2 ] (Comp: ?, Cost: 1) evalspeedpldi09fig1bb2in(ar_0, ar_1, ar_2) -> Com_1(evalspeedpldi09fig1bb1in(ar_0 + 1, ar_1 + 1, ar_2)) (Comp: ?, Cost: 1) evalspeedpldi09fig1bb3in(ar_0, ar_1, ar_2) -> Com_1(evalspeedpldi09fig1bb4in(ar_0, ar_1, ar_2)) [ ar_1 >= 1 ] (Comp: 2, Cost: 1) evalspeedpldi09fig1bb3in(ar_0, ar_1, ar_2) -> Com_1(evalspeedpldi09fig1bb5in(ar_0, ar_1, ar_2)) [ 0 >= ar_1 ] (Comp: ?, Cost: 1) evalspeedpldi09fig1bb4in(ar_0, ar_1, ar_2) -> Com_1(evalspeedpldi09fig1bb1in(ar_0, ar_1 - 1, ar_2)) (Comp: 2, Cost: 1) evalspeedpldi09fig1bb5in(ar_0, ar_1, ar_2) -> Com_1(evalspeedpldi09fig1stop(ar_0, ar_1, ar_2)) (Comp: 1, Cost: 0) koat_start(ar_0, ar_1, ar_2) -> Com_1(evalspeedpldi09fig1start(ar_0, ar_1, ar_2)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 A polynomial rank function with Pol(evalspeedpldi09fig1start) = V_3 Pol(evalspeedpldi09fig1bb0in) = V_3 Pol(evalspeedpldi09fig1bb1in) = -V_1 + V_3 Pol(evalspeedpldi09fig1bb2in) = -V_1 + V_3 - 1 Pol(evalspeedpldi09fig1bb3in) = -V_1 + V_3 Pol(evalspeedpldi09fig1bb4in) = -V_1 + V_3 Pol(evalspeedpldi09fig1bb5in) = -V_1 + V_3 Pol(evalspeedpldi09fig1stop) = -V_1 + V_3 Pol(koat_start) = V_3 orients all transitions weakly and the transition evalspeedpldi09fig1bb1in(ar_0, ar_1, ar_2) -> Com_1(evalspeedpldi09fig1bb2in(ar_0, ar_1, ar_2)) [ ar_2 >= ar_0 + 1 ] strictly and produces the following problem: 4: T: (Comp: 1, Cost: 1) evalspeedpldi09fig1start(ar_0, ar_1, ar_2) -> Com_1(evalspeedpldi09fig1bb0in(ar_0, ar_1, ar_2)) (Comp: 1, Cost: 1) evalspeedpldi09fig1bb0in(ar_0, ar_1, ar_2) -> Com_1(evalspeedpldi09fig1bb1in(0, 0, ar_2)) (Comp: ar_2, Cost: 1) evalspeedpldi09fig1bb1in(ar_0, ar_1, ar_2) -> Com_1(evalspeedpldi09fig1bb2in(ar_0, ar_1, ar_2)) [ ar_2 >= ar_0 + 1 ] (Comp: ?, Cost: 1) evalspeedpldi09fig1bb1in(ar_0, ar_1, ar_2) -> Com_1(evalspeedpldi09fig1bb3in(ar_0, ar_1, ar_2)) [ ar_0 >= ar_2 ] (Comp: ?, Cost: 1) evalspeedpldi09fig1bb2in(ar_0, ar_1, ar_2) -> Com_1(evalspeedpldi09fig1bb1in(ar_0 + 1, ar_1 + 1, ar_2)) (Comp: ?, Cost: 1) evalspeedpldi09fig1bb3in(ar_0, ar_1, ar_2) -> Com_1(evalspeedpldi09fig1bb4in(ar_0, ar_1, ar_2)) [ ar_1 >= 1 ] (Comp: 2, Cost: 1) evalspeedpldi09fig1bb3in(ar_0, ar_1, ar_2) -> Com_1(evalspeedpldi09fig1bb5in(ar_0, ar_1, ar_2)) [ 0 >= ar_1 ] (Comp: ?, Cost: 1) evalspeedpldi09fig1bb4in(ar_0, ar_1, ar_2) -> Com_1(evalspeedpldi09fig1bb1in(ar_0, ar_1 - 1, ar_2)) (Comp: 2, Cost: 1) evalspeedpldi09fig1bb5in(ar_0, ar_1, ar_2) -> Com_1(evalspeedpldi09fig1stop(ar_0, ar_1, ar_2)) (Comp: 1, Cost: 0) koat_start(ar_0, ar_1, ar_2) -> Com_1(evalspeedpldi09fig1start(ar_0, ar_1, ar_2)) [ 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) evalspeedpldi09fig1start(ar_0, ar_1, ar_2) -> Com_1(evalspeedpldi09fig1bb0in(ar_0, ar_1, ar_2)) (Comp: 1, Cost: 1) evalspeedpldi09fig1bb0in(ar_0, ar_1, ar_2) -> Com_1(evalspeedpldi09fig1bb1in(0, 0, ar_2)) (Comp: ar_2, Cost: 1) evalspeedpldi09fig1bb1in(ar_0, ar_1, ar_2) -> Com_1(evalspeedpldi09fig1bb2in(ar_0, ar_1, ar_2)) [ ar_2 >= ar_0 + 1 ] (Comp: ?, Cost: 1) evalspeedpldi09fig1bb1in(ar_0, ar_1, ar_2) -> Com_1(evalspeedpldi09fig1bb3in(ar_0, ar_1, ar_2)) [ ar_0 >= ar_2 ] (Comp: ar_2, Cost: 1) evalspeedpldi09fig1bb2in(ar_0, ar_1, ar_2) -> Com_1(evalspeedpldi09fig1bb1in(ar_0 + 1, ar_1 + 1, ar_2)) (Comp: ?, Cost: 1) evalspeedpldi09fig1bb3in(ar_0, ar_1, ar_2) -> Com_1(evalspeedpldi09fig1bb4in(ar_0, ar_1, ar_2)) [ ar_1 >= 1 ] (Comp: 2, Cost: 1) evalspeedpldi09fig1bb3in(ar_0, ar_1, ar_2) -> Com_1(evalspeedpldi09fig1bb5in(ar_0, ar_1, ar_2)) [ 0 >= ar_1 ] (Comp: ?, Cost: 1) evalspeedpldi09fig1bb4in(ar_0, ar_1, ar_2) -> Com_1(evalspeedpldi09fig1bb1in(ar_0, ar_1 - 1, ar_2)) (Comp: 2, Cost: 1) evalspeedpldi09fig1bb5in(ar_0, ar_1, ar_2) -> Com_1(evalspeedpldi09fig1stop(ar_0, ar_1, ar_2)) (Comp: 1, Cost: 0) koat_start(ar_0, ar_1, ar_2) -> Com_1(evalspeedpldi09fig1start(ar_0, ar_1, ar_2)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 Applied AI with 'oct' on problem 5 to obtain the following invariants: For symbol evalspeedpldi09fig1bb1in: X_1 - X_2 >= 0 /\ X_2 >= 0 /\ X_1 + X_2 >= 0 /\ X_1 >= 0 For symbol evalspeedpldi09fig1bb2in: X_3 - 1 >= 0 /\ X_2 + X_3 - 1 >= 0 /\ -X_2 + X_3 - 1 >= 0 /\ X_1 + X_3 - 1 >= 0 /\ -X_1 + X_3 - 1 >= 0 /\ X_1 - X_2 >= 0 /\ X_2 >= 0 /\ X_1 + X_2 >= 0 /\ X_1 >= 0 For symbol evalspeedpldi09fig1bb3in: X_1 - X_3 >= 0 /\ X_1 - X_2 >= 0 /\ X_2 >= 0 /\ X_1 + X_2 >= 0 /\ X_1 >= 0 For symbol evalspeedpldi09fig1bb4in: X_1 - X_3 >= 0 /\ X_1 - X_2 >= 0 /\ X_2 - 1 >= 0 /\ X_1 + X_2 - 2 >= 0 /\ X_1 - 1 >= 0 For symbol evalspeedpldi09fig1bb5in: X_1 - X_3 >= 0 /\ -X_2 >= 0 /\ X_1 - X_2 >= 0 /\ X_2 >= 0 /\ X_1 + X_2 >= 0 /\ X_1 >= 0 This yielded the following problem: 6: T: (Comp: 1, Cost: 0) koat_start(ar_0, ar_1, ar_2) -> Com_1(evalspeedpldi09fig1start(ar_0, ar_1, ar_2)) [ 0 <= 0 ] (Comp: 2, Cost: 1) evalspeedpldi09fig1bb5in(ar_0, ar_1, ar_2) -> Com_1(evalspeedpldi09fig1stop(ar_0, ar_1, ar_2)) [ ar_0 - ar_2 >= 0 /\ -ar_1 >= 0 /\ ar_0 - ar_1 >= 0 /\ ar_1 >= 0 /\ ar_0 + ar_1 >= 0 /\ ar_0 >= 0 ] (Comp: ?, Cost: 1) evalspeedpldi09fig1bb4in(ar_0, ar_1, ar_2) -> Com_1(evalspeedpldi09fig1bb1in(ar_0, ar_1 - 1, ar_2)) [ ar_0 - ar_2 >= 0 /\ ar_0 - ar_1 >= 0 /\ ar_1 - 1 >= 0 /\ ar_0 + ar_1 - 2 >= 0 /\ ar_0 - 1 >= 0 ] (Comp: 2, Cost: 1) evalspeedpldi09fig1bb3in(ar_0, ar_1, ar_2) -> Com_1(evalspeedpldi09fig1bb5in(ar_0, ar_1, ar_2)) [ ar_0 - ar_2 >= 0 /\ ar_0 - ar_1 >= 0 /\ ar_1 >= 0 /\ ar_0 + ar_1 >= 0 /\ ar_0 >= 0 /\ 0 >= ar_1 ] (Comp: ?, Cost: 1) evalspeedpldi09fig1bb3in(ar_0, ar_1, ar_2) -> Com_1(evalspeedpldi09fig1bb4in(ar_0, ar_1, ar_2)) [ ar_0 - ar_2 >= 0 /\ ar_0 - ar_1 >= 0 /\ ar_1 >= 0 /\ ar_0 + ar_1 >= 0 /\ ar_0 >= 0 /\ ar_1 >= 1 ] (Comp: ar_2, Cost: 1) evalspeedpldi09fig1bb2in(ar_0, ar_1, ar_2) -> Com_1(evalspeedpldi09fig1bb1in(ar_0 + 1, ar_1 + 1, ar_2)) [ ar_2 - 1 >= 0 /\ ar_1 + ar_2 - 1 >= 0 /\ -ar_1 + ar_2 - 1 >= 0 /\ ar_0 + ar_2 - 1 >= 0 /\ -ar_0 + ar_2 - 1 >= 0 /\ ar_0 - ar_1 >= 0 /\ ar_1 >= 0 /\ ar_0 + ar_1 >= 0 /\ ar_0 >= 0 ] (Comp: ?, Cost: 1) evalspeedpldi09fig1bb1in(ar_0, ar_1, ar_2) -> Com_1(evalspeedpldi09fig1bb3in(ar_0, ar_1, ar_2)) [ ar_0 - ar_1 >= 0 /\ ar_1 >= 0 /\ ar_0 + ar_1 >= 0 /\ ar_0 >= 0 /\ ar_0 >= ar_2 ] (Comp: ar_2, Cost: 1) evalspeedpldi09fig1bb1in(ar_0, ar_1, ar_2) -> Com_1(evalspeedpldi09fig1bb2in(ar_0, ar_1, ar_2)) [ ar_0 - ar_1 >= 0 /\ ar_1 >= 0 /\ ar_0 + ar_1 >= 0 /\ ar_0 >= 0 /\ ar_2 >= ar_0 + 1 ] (Comp: 1, Cost: 1) evalspeedpldi09fig1bb0in(ar_0, ar_1, ar_2) -> Com_1(evalspeedpldi09fig1bb1in(0, 0, ar_2)) (Comp: 1, Cost: 1) evalspeedpldi09fig1start(ar_0, ar_1, ar_2) -> Com_1(evalspeedpldi09fig1bb0in(ar_0, ar_1, ar_2)) start location: koat_start leaf cost: 0 A polynomial rank function with Pol(evalspeedpldi09fig1bb4in) = 3*V_2 Pol(evalspeedpldi09fig1bb1in) = 3*V_2 + 2 Pol(evalspeedpldi09fig1bb3in) = 3*V_2 + 1 and size complexities S("evalspeedpldi09fig1start(ar_0, ar_1, ar_2) -> Com_1(evalspeedpldi09fig1bb0in(ar_0, ar_1, ar_2))", 0-0) = ar_0 S("evalspeedpldi09fig1start(ar_0, ar_1, ar_2) -> Com_1(evalspeedpldi09fig1bb0in(ar_0, ar_1, ar_2))", 0-1) = ar_1 S("evalspeedpldi09fig1start(ar_0, ar_1, ar_2) -> Com_1(evalspeedpldi09fig1bb0in(ar_0, ar_1, ar_2))", 0-2) = ar_2 S("evalspeedpldi09fig1bb0in(ar_0, ar_1, ar_2) -> Com_1(evalspeedpldi09fig1bb1in(0, 0, ar_2))", 0-0) = 0 S("evalspeedpldi09fig1bb0in(ar_0, ar_1, ar_2) -> Com_1(evalspeedpldi09fig1bb1in(0, 0, ar_2))", 0-1) = 0 S("evalspeedpldi09fig1bb0in(ar_0, ar_1, ar_2) -> Com_1(evalspeedpldi09fig1bb1in(0, 0, ar_2))", 0-2) = ar_2 S("evalspeedpldi09fig1bb1in(ar_0, ar_1, ar_2) -> Com_1(evalspeedpldi09fig1bb2in(ar_0, ar_1, ar_2)) [ ar_0 - ar_1 >= 0 /\\ ar_1 >= 0 /\\ ar_0 + ar_1 >= 0 /\\ ar_0 >= 0 /\\ ar_2 >= ar_0 + 1 ]", 0-0) = ar_2 S("evalspeedpldi09fig1bb1in(ar_0, ar_1, ar_2) -> Com_1(evalspeedpldi09fig1bb2in(ar_0, ar_1, ar_2)) [ ar_0 - ar_1 >= 0 /\\ ar_1 >= 0 /\\ ar_0 + ar_1 >= 0 /\\ ar_0 >= 0 /\\ ar_2 >= ar_0 + 1 ]", 0-1) = ar_2 S("evalspeedpldi09fig1bb1in(ar_0, ar_1, ar_2) -> Com_1(evalspeedpldi09fig1bb2in(ar_0, ar_1, ar_2)) [ ar_0 - ar_1 >= 0 /\\ ar_1 >= 0 /\\ ar_0 + ar_1 >= 0 /\\ ar_0 >= 0 /\\ ar_2 >= ar_0 + 1 ]", 0-2) = ar_2 S("evalspeedpldi09fig1bb1in(ar_0, ar_1, ar_2) -> Com_1(evalspeedpldi09fig1bb3in(ar_0, ar_1, ar_2)) [ ar_0 - ar_1 >= 0 /\\ ar_1 >= 0 /\\ ar_0 + ar_1 >= 0 /\\ ar_0 >= 0 /\\ ar_0 >= ar_2 ]", 0-0) = ar_2 S("evalspeedpldi09fig1bb1in(ar_0, ar_1, ar_2) -> Com_1(evalspeedpldi09fig1bb3in(ar_0, ar_1, ar_2)) [ ar_0 - ar_1 >= 0 /\\ ar_1 >= 0 /\\ ar_0 + ar_1 >= 0 /\\ ar_0 >= 0 /\\ ar_0 >= ar_2 ]", 0-1) = ar_2 S("evalspeedpldi09fig1bb1in(ar_0, ar_1, ar_2) -> Com_1(evalspeedpldi09fig1bb3in(ar_0, ar_1, ar_2)) [ ar_0 - ar_1 >= 0 /\\ ar_1 >= 0 /\\ ar_0 + ar_1 >= 0 /\\ ar_0 >= 0 /\\ ar_0 >= ar_2 ]", 0-2) = ar_2 S("evalspeedpldi09fig1bb2in(ar_0, ar_1, ar_2) -> Com_1(evalspeedpldi09fig1bb1in(ar_0 + 1, ar_1 + 1, ar_2)) [ ar_2 - 1 >= 0 /\\ ar_1 + ar_2 - 1 >= 0 /\\ -ar_1 + ar_2 - 1 >= 0 /\\ ar_0 + ar_2 - 1 >= 0 /\\ -ar_0 + ar_2 - 1 >= 0 /\\ ar_0 - ar_1 >= 0 /\\ ar_1 >= 0 /\\ ar_0 + ar_1 >= 0 /\\ ar_0 >= 0 ]", 0-0) = ar_2 S("evalspeedpldi09fig1bb2in(ar_0, ar_1, ar_2) -> Com_1(evalspeedpldi09fig1bb1in(ar_0 + 1, ar_1 + 1, ar_2)) [ ar_2 - 1 >= 0 /\\ ar_1 + ar_2 - 1 >= 0 /\\ -ar_1 + ar_2 - 1 >= 0 /\\ ar_0 + ar_2 - 1 >= 0 /\\ -ar_0 + ar_2 - 1 >= 0 /\\ ar_0 - ar_1 >= 0 /\\ ar_1 >= 0 /\\ ar_0 + ar_1 >= 0 /\\ ar_0 >= 0 ]", 0-1) = ar_2 S("evalspeedpldi09fig1bb2in(ar_0, ar_1, ar_2) -> Com_1(evalspeedpldi09fig1bb1in(ar_0 + 1, ar_1 + 1, ar_2)) [ ar_2 - 1 >= 0 /\\ ar_1 + ar_2 - 1 >= 0 /\\ -ar_1 + ar_2 - 1 >= 0 /\\ ar_0 + ar_2 - 1 >= 0 /\\ -ar_0 + ar_2 - 1 >= 0 /\\ ar_0 - ar_1 >= 0 /\\ ar_1 >= 0 /\\ ar_0 + ar_1 >= 0 /\\ ar_0 >= 0 ]", 0-2) = ar_2 S("evalspeedpldi09fig1bb3in(ar_0, ar_1, ar_2) -> Com_1(evalspeedpldi09fig1bb4in(ar_0, ar_1, ar_2)) [ ar_0 - ar_2 >= 0 /\\ ar_0 - ar_1 >= 0 /\\ ar_1 >= 0 /\\ ar_0 + ar_1 >= 0 /\\ ar_0 >= 0 /\\ ar_1 >= 1 ]", 0-0) = ar_2 S("evalspeedpldi09fig1bb3in(ar_0, ar_1, ar_2) -> Com_1(evalspeedpldi09fig1bb4in(ar_0, ar_1, ar_2)) [ ar_0 - ar_2 >= 0 /\\ ar_0 - ar_1 >= 0 /\\ ar_1 >= 0 /\\ ar_0 + ar_1 >= 0 /\\ ar_0 >= 0 /\\ ar_1 >= 1 ]", 0-1) = ar_2 S("evalspeedpldi09fig1bb3in(ar_0, ar_1, ar_2) -> Com_1(evalspeedpldi09fig1bb4in(ar_0, ar_1, ar_2)) [ ar_0 - ar_2 >= 0 /\\ ar_0 - ar_1 >= 0 /\\ ar_1 >= 0 /\\ ar_0 + ar_1 >= 0 /\\ ar_0 >= 0 /\\ ar_1 >= 1 ]", 0-2) = ar_2 S("evalspeedpldi09fig1bb3in(ar_0, ar_1, ar_2) -> Com_1(evalspeedpldi09fig1bb5in(ar_0, ar_1, ar_2)) [ ar_0 - ar_2 >= 0 /\\ ar_0 - ar_1 >= 0 /\\ ar_1 >= 0 /\\ ar_0 + ar_1 >= 0 /\\ ar_0 >= 0 /\\ 0 >= ar_1 ]", 0-0) = ar_2 S("evalspeedpldi09fig1bb3in(ar_0, ar_1, ar_2) -> Com_1(evalspeedpldi09fig1bb5in(ar_0, ar_1, ar_2)) [ ar_0 - ar_2 >= 0 /\\ ar_0 - ar_1 >= 0 /\\ ar_1 >= 0 /\\ ar_0 + ar_1 >= 0 /\\ ar_0 >= 0 /\\ 0 >= ar_1 ]", 0-1) = 0 S("evalspeedpldi09fig1bb3in(ar_0, ar_1, ar_2) -> Com_1(evalspeedpldi09fig1bb5in(ar_0, ar_1, ar_2)) [ ar_0 - ar_2 >= 0 /\\ ar_0 - ar_1 >= 0 /\\ ar_1 >= 0 /\\ ar_0 + ar_1 >= 0 /\\ ar_0 >= 0 /\\ 0 >= ar_1 ]", 0-2) = ar_2 S("evalspeedpldi09fig1bb4in(ar_0, ar_1, ar_2) -> Com_1(evalspeedpldi09fig1bb1in(ar_0, ar_1 - 1, ar_2)) [ ar_0 - ar_2 >= 0 /\\ ar_0 - ar_1 >= 0 /\\ ar_1 - 1 >= 0 /\\ ar_0 + ar_1 - 2 >= 0 /\\ ar_0 - 1 >= 0 ]", 0-0) = ar_2 S("evalspeedpldi09fig1bb4in(ar_0, ar_1, ar_2) -> Com_1(evalspeedpldi09fig1bb1in(ar_0, ar_1 - 1, ar_2)) [ ar_0 - ar_2 >= 0 /\\ ar_0 - ar_1 >= 0 /\\ ar_1 - 1 >= 0 /\\ ar_0 + ar_1 - 2 >= 0 /\\ ar_0 - 1 >= 0 ]", 0-1) = ar_2 S("evalspeedpldi09fig1bb4in(ar_0, ar_1, ar_2) -> Com_1(evalspeedpldi09fig1bb1in(ar_0, ar_1 - 1, ar_2)) [ ar_0 - ar_2 >= 0 /\\ ar_0 - ar_1 >= 0 /\\ ar_1 - 1 >= 0 /\\ ar_0 + ar_1 - 2 >= 0 /\\ ar_0 - 1 >= 0 ]", 0-2) = ar_2 S("evalspeedpldi09fig1bb5in(ar_0, ar_1, ar_2) -> Com_1(evalspeedpldi09fig1stop(ar_0, ar_1, ar_2)) [ ar_0 - ar_2 >= 0 /\\ -ar_1 >= 0 /\\ ar_0 - ar_1 >= 0 /\\ ar_1 >= 0 /\\ ar_0 + ar_1 >= 0 /\\ ar_0 >= 0 ]", 0-0) = ar_2 S("evalspeedpldi09fig1bb5in(ar_0, ar_1, ar_2) -> Com_1(evalspeedpldi09fig1stop(ar_0, ar_1, ar_2)) [ ar_0 - ar_2 >= 0 /\\ -ar_1 >= 0 /\\ ar_0 - ar_1 >= 0 /\\ ar_1 >= 0 /\\ ar_0 + ar_1 >= 0 /\\ ar_0 >= 0 ]", 0-1) = 0 S("evalspeedpldi09fig1bb5in(ar_0, ar_1, ar_2) -> Com_1(evalspeedpldi09fig1stop(ar_0, ar_1, ar_2)) [ ar_0 - ar_2 >= 0 /\\ -ar_1 >= 0 /\\ ar_0 - ar_1 >= 0 /\\ ar_1 >= 0 /\\ ar_0 + ar_1 >= 0 /\\ ar_0 >= 0 ]", 0-2) = ar_2 S("koat_start(ar_0, ar_1, ar_2) -> Com_1(evalspeedpldi09fig1start(ar_0, ar_1, ar_2)) [ 0 <= 0 ]", 0-0) = ar_0 S("koat_start(ar_0, ar_1, ar_2) -> Com_1(evalspeedpldi09fig1start(ar_0, ar_1, ar_2)) [ 0 <= 0 ]", 0-1) = ar_1 S("koat_start(ar_0, ar_1, ar_2) -> Com_1(evalspeedpldi09fig1start(ar_0, ar_1, ar_2)) [ 0 <= 0 ]", 0-2) = ar_2 orients the transitions evalspeedpldi09fig1bb4in(ar_0, ar_1, ar_2) -> Com_1(evalspeedpldi09fig1bb1in(ar_0, ar_1 - 1, ar_2)) [ ar_0 - ar_2 >= 0 /\ ar_0 - ar_1 >= 0 /\ ar_1 - 1 >= 0 /\ ar_0 + ar_1 - 2 >= 0 /\ ar_0 - 1 >= 0 ] evalspeedpldi09fig1bb3in(ar_0, ar_1, ar_2) -> Com_1(evalspeedpldi09fig1bb4in(ar_0, ar_1, ar_2)) [ ar_0 - ar_2 >= 0 /\ ar_0 - ar_1 >= 0 /\ ar_1 >= 0 /\ ar_0 + ar_1 >= 0 /\ ar_0 >= 0 /\ ar_1 >= 1 ] evalspeedpldi09fig1bb1in(ar_0, ar_1, ar_2) -> Com_1(evalspeedpldi09fig1bb3in(ar_0, ar_1, ar_2)) [ ar_0 - ar_1 >= 0 /\ ar_1 >= 0 /\ ar_0 + ar_1 >= 0 /\ ar_0 >= 0 /\ ar_0 >= ar_2 ] weakly and the transitions evalspeedpldi09fig1bb4in(ar_0, ar_1, ar_2) -> Com_1(evalspeedpldi09fig1bb1in(ar_0, ar_1 - 1, ar_2)) [ ar_0 - ar_2 >= 0 /\ ar_0 - ar_1 >= 0 /\ ar_1 - 1 >= 0 /\ ar_0 + ar_1 - 2 >= 0 /\ ar_0 - 1 >= 0 ] evalspeedpldi09fig1bb3in(ar_0, ar_1, ar_2) -> Com_1(evalspeedpldi09fig1bb4in(ar_0, ar_1, ar_2)) [ ar_0 - ar_2 >= 0 /\ ar_0 - ar_1 >= 0 /\ ar_1 >= 0 /\ ar_0 + ar_1 >= 0 /\ ar_0 >= 0 /\ ar_1 >= 1 ] evalspeedpldi09fig1bb1in(ar_0, ar_1, ar_2) -> Com_1(evalspeedpldi09fig1bb3in(ar_0, ar_1, ar_2)) [ ar_0 - ar_1 >= 0 /\ ar_1 >= 0 /\ ar_0 + ar_1 >= 0 /\ ar_0 >= 0 /\ ar_0 >= ar_2 ] strictly and produces the following problem: 7: T: (Comp: 1, Cost: 0) koat_start(ar_0, ar_1, ar_2) -> Com_1(evalspeedpldi09fig1start(ar_0, ar_1, ar_2)) [ 0 <= 0 ] (Comp: 2, Cost: 1) evalspeedpldi09fig1bb5in(ar_0, ar_1, ar_2) -> Com_1(evalspeedpldi09fig1stop(ar_0, ar_1, ar_2)) [ ar_0 - ar_2 >= 0 /\ -ar_1 >= 0 /\ ar_0 - ar_1 >= 0 /\ ar_1 >= 0 /\ ar_0 + ar_1 >= 0 /\ ar_0 >= 0 ] (Comp: 3*ar_2^2 + 2*ar_2 + 2, Cost: 1) evalspeedpldi09fig1bb4in(ar_0, ar_1, ar_2) -> Com_1(evalspeedpldi09fig1bb1in(ar_0, ar_1 - 1, ar_2)) [ ar_0 - ar_2 >= 0 /\ ar_0 - ar_1 >= 0 /\ ar_1 - 1 >= 0 /\ ar_0 + ar_1 - 2 >= 0 /\ ar_0 - 1 >= 0 ] (Comp: 2, Cost: 1) evalspeedpldi09fig1bb3in(ar_0, ar_1, ar_2) -> Com_1(evalspeedpldi09fig1bb5in(ar_0, ar_1, ar_2)) [ ar_0 - ar_2 >= 0 /\ ar_0 - ar_1 >= 0 /\ ar_1 >= 0 /\ ar_0 + ar_1 >= 0 /\ ar_0 >= 0 /\ 0 >= ar_1 ] (Comp: 3*ar_2^2 + 2*ar_2 + 2, Cost: 1) evalspeedpldi09fig1bb3in(ar_0, ar_1, ar_2) -> Com_1(evalspeedpldi09fig1bb4in(ar_0, ar_1, ar_2)) [ ar_0 - ar_2 >= 0 /\ ar_0 - ar_1 >= 0 /\ ar_1 >= 0 /\ ar_0 + ar_1 >= 0 /\ ar_0 >= 0 /\ ar_1 >= 1 ] (Comp: ar_2, Cost: 1) evalspeedpldi09fig1bb2in(ar_0, ar_1, ar_2) -> Com_1(evalspeedpldi09fig1bb1in(ar_0 + 1, ar_1 + 1, ar_2)) [ ar_2 - 1 >= 0 /\ ar_1 + ar_2 - 1 >= 0 /\ -ar_1 + ar_2 - 1 >= 0 /\ ar_0 + ar_2 - 1 >= 0 /\ -ar_0 + ar_2 - 1 >= 0 /\ ar_0 - ar_1 >= 0 /\ ar_1 >= 0 /\ ar_0 + ar_1 >= 0 /\ ar_0 >= 0 ] (Comp: 3*ar_2^2 + 2*ar_2 + 2, Cost: 1) evalspeedpldi09fig1bb1in(ar_0, ar_1, ar_2) -> Com_1(evalspeedpldi09fig1bb3in(ar_0, ar_1, ar_2)) [ ar_0 - ar_1 >= 0 /\ ar_1 >= 0 /\ ar_0 + ar_1 >= 0 /\ ar_0 >= 0 /\ ar_0 >= ar_2 ] (Comp: ar_2, Cost: 1) evalspeedpldi09fig1bb1in(ar_0, ar_1, ar_2) -> Com_1(evalspeedpldi09fig1bb2in(ar_0, ar_1, ar_2)) [ ar_0 - ar_1 >= 0 /\ ar_1 >= 0 /\ ar_0 + ar_1 >= 0 /\ ar_0 >= 0 /\ ar_2 >= ar_0 + 1 ] (Comp: 1, Cost: 1) evalspeedpldi09fig1bb0in(ar_0, ar_1, ar_2) -> Com_1(evalspeedpldi09fig1bb1in(0, 0, ar_2)) (Comp: 1, Cost: 1) evalspeedpldi09fig1start(ar_0, ar_1, ar_2) -> Com_1(evalspeedpldi09fig1bb0in(ar_0, ar_1, ar_2)) start location: koat_start leaf cost: 0 Complexity upper bound 9*ar_2^2 + 8*ar_2 + 12 Time: 0.252 sec (SMT: 0.222 sec) ---------------------------------------- (2) BOUNDS(1, n^2)