/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^2)) 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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(1, n^2). (0) CpxIntTrs (1) Koat Proof [FINISHED, 49 ms] (2) BOUNDS(1, n^2) ---------------------------------------- (0) Obligation: Complexity Int TRS consisting of the following rules: eval_foo_start(v_.0, v_.01, v_i, v_j, v_n) -> Com_1(eval_foo_bb0_in(v_.0, v_.01, v_i, v_j, v_n)) :|: TRUE eval_foo_bb0_in(v_.0, v_.01, v_i, v_j, v_n) -> Com_1(eval_foo_bb1_in(v_i, v_.01, v_i, v_j, v_n)) :|: TRUE eval_foo_bb1_in(v_.0, v_.01, v_i, v_j, v_n) -> Com_1(eval_foo_bb2_in(v_.0, 0, v_i, v_j, v_n)) :|: v_.0 < v_n eval_foo_bb1_in(v_.0, v_.01, v_i, v_j, v_n) -> Com_1(eval_foo_bb5_in(v_.0, v_.01, v_i, v_j, v_n)) :|: v_.0 >= v_n eval_foo_bb2_in(v_.0, v_.01, v_i, v_j, v_n) -> Com_1(eval_foo_bb3_in(v_.0, v_.01, v_i, v_j, v_n)) :|: v_.01 <= v_.0 eval_foo_bb2_in(v_.0, v_.01, v_i, v_j, v_n) -> Com_1(eval_foo_bb4_in(v_.0, v_.01, v_i, v_j, v_n)) :|: v_.01 > v_.0 eval_foo_bb3_in(v_.0, v_.01, v_i, v_j, v_n) -> Com_1(eval_foo_bb2_in(v_.0, v_.01 + 1, v_i, v_j, v_n)) :|: TRUE eval_foo_bb4_in(v_.0, v_.01, v_i, v_j, v_n) -> Com_1(eval_foo_bb1_in(v_.0 + 1, v_.01, v_i, v_j, v_n)) :|: TRUE eval_foo_bb5_in(v_.0, v_.01, v_i, v_j, v_n) -> Com_1(eval_foo_stop(v_.0, v_.01, v_i, v_j, v_n)) :|: TRUE The start-symbols are:[eval_foo_start_5] ---------------------------------------- (1) Koat Proof (FINISHED) YES(?, 49*ar_1 + 49*ar_2 + 6*ar_1^2 + 12*ar_1*ar_2 + 6*ar_2^2 + 49) Initial complexity problem: 1: T: (Comp: ?, Cost: 1) evalfoostart(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb0in(ar_0, ar_1, ar_2, ar_3)) (Comp: ?, Cost: 1) evalfoobb0in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb1in(ar_1, ar_1, ar_2, ar_3)) (Comp: ?, Cost: 1) evalfoobb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb2in(ar_0, ar_1, ar_2, 0)) [ ar_2 >= ar_0 + 1 ] (Comp: ?, Cost: 1) evalfoobb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb5in(ar_0, ar_1, ar_2, ar_3)) [ ar_0 >= ar_2 ] (Comp: ?, Cost: 1) evalfoobb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb3in(ar_0, ar_1, ar_2, ar_3)) [ ar_0 >= ar_3 ] (Comp: ?, Cost: 1) evalfoobb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb4in(ar_0, ar_1, ar_2, ar_3)) [ ar_3 >= ar_0 + 1 ] (Comp: ?, Cost: 1) evalfoobb3in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb2in(ar_0, ar_1, ar_2, ar_3 + 1)) (Comp: ?, Cost: 1) evalfoobb4in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb1in(ar_0 + 1, ar_1, ar_2, ar_3)) (Comp: ?, Cost: 1) evalfoobb5in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoostop(ar_0, ar_1, ar_2, ar_3)) (Comp: 1, Cost: 0) koat_start(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoostart(ar_0, ar_1, ar_2, ar_3)) [ 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) evalfoostart(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb0in(ar_0, ar_1, ar_2, ar_3)) (Comp: 1, Cost: 1) evalfoobb0in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb1in(ar_1, ar_1, ar_2, ar_3)) (Comp: ?, Cost: 1) evalfoobb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb2in(ar_0, ar_1, ar_2, 0)) [ ar_2 >= ar_0 + 1 ] (Comp: ?, Cost: 1) evalfoobb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb5in(ar_0, ar_1, ar_2, ar_3)) [ ar_0 >= ar_2 ] (Comp: ?, Cost: 1) evalfoobb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb3in(ar_0, ar_1, ar_2, ar_3)) [ ar_0 >= ar_3 ] (Comp: ?, Cost: 1) evalfoobb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb4in(ar_0, ar_1, ar_2, ar_3)) [ ar_3 >= ar_0 + 1 ] (Comp: ?, Cost: 1) evalfoobb3in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb2in(ar_0, ar_1, ar_2, ar_3 + 1)) (Comp: ?, Cost: 1) evalfoobb4in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb1in(ar_0 + 1, ar_1, ar_2, ar_3)) (Comp: ?, Cost: 1) evalfoobb5in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoostop(ar_0, ar_1, ar_2, ar_3)) (Comp: 1, Cost: 0) koat_start(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoostart(ar_0, ar_1, ar_2, ar_3)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 A polynomial rank function with Pol(evalfoostart) = 2 Pol(evalfoobb0in) = 2 Pol(evalfoobb1in) = 2 Pol(evalfoobb2in) = 2 Pol(evalfoobb5in) = 1 Pol(evalfoobb3in) = 2 Pol(evalfoobb4in) = 2 Pol(evalfoostop) = 0 Pol(koat_start) = 2 orients all transitions weakly and the transitions evalfoobb5in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoostop(ar_0, ar_1, ar_2, ar_3)) evalfoobb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb5in(ar_0, ar_1, ar_2, ar_3)) [ ar_0 >= ar_2 ] strictly and produces the following problem: 3: T: (Comp: 1, Cost: 1) evalfoostart(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb0in(ar_0, ar_1, ar_2, ar_3)) (Comp: 1, Cost: 1) evalfoobb0in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb1in(ar_1, ar_1, ar_2, ar_3)) (Comp: ?, Cost: 1) evalfoobb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb2in(ar_0, ar_1, ar_2, 0)) [ ar_2 >= ar_0 + 1 ] (Comp: 2, Cost: 1) evalfoobb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb5in(ar_0, ar_1, ar_2, ar_3)) [ ar_0 >= ar_2 ] (Comp: ?, Cost: 1) evalfoobb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb3in(ar_0, ar_1, ar_2, ar_3)) [ ar_0 >= ar_3 ] (Comp: ?, Cost: 1) evalfoobb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb4in(ar_0, ar_1, ar_2, ar_3)) [ ar_3 >= ar_0 + 1 ] (Comp: ?, Cost: 1) evalfoobb3in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb2in(ar_0, ar_1, ar_2, ar_3 + 1)) (Comp: ?, Cost: 1) evalfoobb4in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb1in(ar_0 + 1, ar_1, ar_2, ar_3)) (Comp: 2, Cost: 1) evalfoobb5in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoostop(ar_0, ar_1, ar_2, ar_3)) (Comp: 1, Cost: 0) koat_start(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoostart(ar_0, ar_1, ar_2, ar_3)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 A polynomial rank function with Pol(evalfoostart) = -V_2 + V_3 + 1 Pol(evalfoobb0in) = -V_2 + V_3 + 1 Pol(evalfoobb1in) = -V_1 + V_3 + 1 Pol(evalfoobb2in) = -V_1 + V_3 Pol(evalfoobb5in) = -V_1 + V_3 Pol(evalfoobb3in) = -V_1 + V_3 Pol(evalfoobb4in) = -V_1 + V_3 Pol(evalfoostop) = -V_1 + V_3 Pol(koat_start) = -V_2 + V_3 + 1 orients all transitions weakly and the transition evalfoobb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb2in(ar_0, ar_1, ar_2, 0)) [ ar_2 >= ar_0 + 1 ] strictly and produces the following problem: 4: T: (Comp: 1, Cost: 1) evalfoostart(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb0in(ar_0, ar_1, ar_2, ar_3)) (Comp: 1, Cost: 1) evalfoobb0in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb1in(ar_1, ar_1, ar_2, ar_3)) (Comp: ar_1 + ar_2 + 1, Cost: 1) evalfoobb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb2in(ar_0, ar_1, ar_2, 0)) [ ar_2 >= ar_0 + 1 ] (Comp: 2, Cost: 1) evalfoobb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb5in(ar_0, ar_1, ar_2, ar_3)) [ ar_0 >= ar_2 ] (Comp: ?, Cost: 1) evalfoobb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb3in(ar_0, ar_1, ar_2, ar_3)) [ ar_0 >= ar_3 ] (Comp: ?, Cost: 1) evalfoobb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb4in(ar_0, ar_1, ar_2, ar_3)) [ ar_3 >= ar_0 + 1 ] (Comp: ?, Cost: 1) evalfoobb3in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb2in(ar_0, ar_1, ar_2, ar_3 + 1)) (Comp: ?, Cost: 1) evalfoobb4in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb1in(ar_0 + 1, ar_1, ar_2, ar_3)) (Comp: 2, Cost: 1) evalfoobb5in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoostop(ar_0, ar_1, ar_2, ar_3)) (Comp: 1, Cost: 0) koat_start(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoostart(ar_0, ar_1, ar_2, ar_3)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 A polynomial rank function with Pol(evalfoobb4in) = 1 Pol(evalfoobb1in) = 0 Pol(evalfoobb3in) = 2 Pol(evalfoobb2in) = 2 and size complexities S("koat_start(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoostart(ar_0, ar_1, ar_2, ar_3)) [ 0 <= 0 ]", 0-0) = ar_0 S("koat_start(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoostart(ar_0, ar_1, ar_2, ar_3)) [ 0 <= 0 ]", 0-1) = ar_1 S("koat_start(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoostart(ar_0, ar_1, ar_2, ar_3)) [ 0 <= 0 ]", 0-2) = ar_2 S("koat_start(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoostart(ar_0, ar_1, ar_2, ar_3)) [ 0 <= 0 ]", 0-3) = ar_3 S("evalfoobb5in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoostop(ar_0, ar_1, ar_2, ar_3))", 0-0) = ? S("evalfoobb5in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoostop(ar_0, ar_1, ar_2, ar_3))", 0-1) = ar_1 S("evalfoobb5in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoostop(ar_0, ar_1, ar_2, ar_3))", 0-2) = ar_2 S("evalfoobb5in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoostop(ar_0, ar_1, ar_2, ar_3))", 0-3) = ? S("evalfoobb4in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb1in(ar_0 + 1, ar_1, ar_2, ar_3))", 0-0) = ? S("evalfoobb4in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb1in(ar_0 + 1, ar_1, ar_2, ar_3))", 0-1) = ar_1 S("evalfoobb4in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb1in(ar_0 + 1, ar_1, ar_2, ar_3))", 0-2) = ar_2 S("evalfoobb4in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb1in(ar_0 + 1, ar_1, ar_2, ar_3))", 0-3) = ? S("evalfoobb3in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb2in(ar_0, ar_1, ar_2, ar_3 + 1))", 0-0) = ? S("evalfoobb3in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb2in(ar_0, ar_1, ar_2, ar_3 + 1))", 0-1) = ar_1 S("evalfoobb3in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb2in(ar_0, ar_1, ar_2, ar_3 + 1))", 0-2) = ar_2 S("evalfoobb3in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb2in(ar_0, ar_1, ar_2, ar_3 + 1))", 0-3) = ? S("evalfoobb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb4in(ar_0, ar_1, ar_2, ar_3)) [ ar_3 >= ar_0 + 1 ]", 0-0) = ? S("evalfoobb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb4in(ar_0, ar_1, ar_2, ar_3)) [ ar_3 >= ar_0 + 1 ]", 0-1) = ar_1 S("evalfoobb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb4in(ar_0, ar_1, ar_2, ar_3)) [ ar_3 >= ar_0 + 1 ]", 0-2) = ar_2 S("evalfoobb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb4in(ar_0, ar_1, ar_2, ar_3)) [ ar_3 >= ar_0 + 1 ]", 0-3) = ? S("evalfoobb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb3in(ar_0, ar_1, ar_2, ar_3)) [ ar_0 >= ar_3 ]", 0-0) = ? S("evalfoobb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb3in(ar_0, ar_1, ar_2, ar_3)) [ ar_0 >= ar_3 ]", 0-1) = ar_1 S("evalfoobb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb3in(ar_0, ar_1, ar_2, ar_3)) [ ar_0 >= ar_3 ]", 0-2) = ar_2 S("evalfoobb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb3in(ar_0, ar_1, ar_2, ar_3)) [ ar_0 >= ar_3 ]", 0-3) = ? S("evalfoobb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb5in(ar_0, ar_1, ar_2, ar_3)) [ ar_0 >= ar_2 ]", 0-0) = ? S("evalfoobb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb5in(ar_0, ar_1, ar_2, ar_3)) [ ar_0 >= ar_2 ]", 0-1) = ar_1 S("evalfoobb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb5in(ar_0, ar_1, ar_2, ar_3)) [ ar_0 >= ar_2 ]", 0-2) = ar_2 S("evalfoobb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb5in(ar_0, ar_1, ar_2, ar_3)) [ ar_0 >= ar_2 ]", 0-3) = ? S("evalfoobb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb2in(ar_0, ar_1, ar_2, 0)) [ ar_2 >= ar_0 + 1 ]", 0-0) = ? S("evalfoobb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb2in(ar_0, ar_1, ar_2, 0)) [ ar_2 >= ar_0 + 1 ]", 0-1) = ar_1 S("evalfoobb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb2in(ar_0, ar_1, ar_2, 0)) [ ar_2 >= ar_0 + 1 ]", 0-2) = ar_2 S("evalfoobb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb2in(ar_0, ar_1, ar_2, 0)) [ ar_2 >= ar_0 + 1 ]", 0-3) = 0 S("evalfoobb0in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb1in(ar_1, ar_1, ar_2, ar_3))", 0-0) = ar_1 S("evalfoobb0in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb1in(ar_1, ar_1, ar_2, ar_3))", 0-1) = ar_1 S("evalfoobb0in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb1in(ar_1, ar_1, ar_2, ar_3))", 0-2) = ar_2 S("evalfoobb0in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb1in(ar_1, ar_1, ar_2, ar_3))", 0-3) = ar_3 S("evalfoostart(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb0in(ar_0, ar_1, ar_2, ar_3))", 0-0) = ar_0 S("evalfoostart(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb0in(ar_0, ar_1, ar_2, ar_3))", 0-1) = ar_1 S("evalfoostart(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb0in(ar_0, ar_1, ar_2, ar_3))", 0-2) = ar_2 S("evalfoostart(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb0in(ar_0, ar_1, ar_2, ar_3))", 0-3) = ar_3 orients the transitions evalfoobb4in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb1in(ar_0 + 1, ar_1, ar_2, ar_3)) evalfoobb3in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb2in(ar_0, ar_1, ar_2, ar_3 + 1)) evalfoobb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb4in(ar_0, ar_1, ar_2, ar_3)) [ ar_3 >= ar_0 + 1 ] evalfoobb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb3in(ar_0, ar_1, ar_2, ar_3)) [ ar_0 >= ar_3 ] weakly and the transitions evalfoobb4in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb1in(ar_0 + 1, ar_1, ar_2, ar_3)) evalfoobb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb4in(ar_0, ar_1, ar_2, ar_3)) [ ar_3 >= ar_0 + 1 ] strictly and produces the following problem: 5: T: (Comp: 1, Cost: 1) evalfoostart(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb0in(ar_0, ar_1, ar_2, ar_3)) (Comp: 1, Cost: 1) evalfoobb0in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb1in(ar_1, ar_1, ar_2, ar_3)) (Comp: ar_1 + ar_2 + 1, Cost: 1) evalfoobb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb2in(ar_0, ar_1, ar_2, 0)) [ ar_2 >= ar_0 + 1 ] (Comp: 2, Cost: 1) evalfoobb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb5in(ar_0, ar_1, ar_2, ar_3)) [ ar_0 >= ar_2 ] (Comp: ?, Cost: 1) evalfoobb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb3in(ar_0, ar_1, ar_2, ar_3)) [ ar_0 >= ar_3 ] (Comp: 2*ar_1 + 2*ar_2 + 2, Cost: 1) evalfoobb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb4in(ar_0, ar_1, ar_2, ar_3)) [ ar_3 >= ar_0 + 1 ] (Comp: ?, Cost: 1) evalfoobb3in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb2in(ar_0, ar_1, ar_2, ar_3 + 1)) (Comp: 2*ar_1 + 2*ar_2 + 2, Cost: 1) evalfoobb4in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb1in(ar_0 + 1, ar_1, ar_2, ar_3)) (Comp: 2, Cost: 1) evalfoobb5in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoostop(ar_0, ar_1, ar_2, ar_3)) (Comp: 1, Cost: 0) koat_start(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoostart(ar_0, ar_1, ar_2, ar_3)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 A polynomial rank function with Pol(evalfoobb3in) = V_1 - V_4 Pol(evalfoobb2in) = V_1 - V_4 + 1 and size complexities S("koat_start(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoostart(ar_0, ar_1, ar_2, ar_3)) [ 0 <= 0 ]", 0-0) = ar_0 S("koat_start(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoostart(ar_0, ar_1, ar_2, ar_3)) [ 0 <= 0 ]", 0-1) = ar_1 S("koat_start(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoostart(ar_0, ar_1, ar_2, ar_3)) [ 0 <= 0 ]", 0-2) = ar_2 S("koat_start(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoostart(ar_0, ar_1, ar_2, ar_3)) [ 0 <= 0 ]", 0-3) = ar_3 S("evalfoobb5in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoostop(ar_0, ar_1, ar_2, ar_3))", 0-0) = 3*ar_1 + 3*ar_2 + 162 S("evalfoobb5in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoostop(ar_0, ar_1, ar_2, ar_3))", 0-1) = ar_1 S("evalfoobb5in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoostop(ar_0, ar_1, ar_2, ar_3))", 0-2) = ar_2 S("evalfoobb5in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoostop(ar_0, ar_1, ar_2, ar_3))", 0-3) = ? S("evalfoobb4in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb1in(ar_0 + 1, ar_1, ar_2, ar_3))", 0-0) = 3*ar_1 + 3*ar_2 + 18 S("evalfoobb4in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb1in(ar_0 + 1, ar_1, ar_2, ar_3))", 0-1) = ar_1 S("evalfoobb4in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb1in(ar_0 + 1, ar_1, ar_2, ar_3))", 0-2) = ar_2 S("evalfoobb4in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb1in(ar_0 + 1, ar_1, ar_2, ar_3))", 0-3) = ? S("evalfoobb3in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb2in(ar_0, ar_1, ar_2, ar_3 + 1))", 0-0) = 3*ar_1 + 3*ar_2 + 18 S("evalfoobb3in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb2in(ar_0, ar_1, ar_2, ar_3 + 1))", 0-1) = ar_1 S("evalfoobb3in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb2in(ar_0, ar_1, ar_2, ar_3 + 1))", 0-2) = ar_2 S("evalfoobb3in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb2in(ar_0, ar_1, ar_2, ar_3 + 1))", 0-3) = ? S("evalfoobb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb4in(ar_0, ar_1, ar_2, ar_3)) [ ar_3 >= ar_0 + 1 ]", 0-0) = 3*ar_1 + 3*ar_2 + 18 S("evalfoobb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb4in(ar_0, ar_1, ar_2, ar_3)) [ ar_3 >= ar_0 + 1 ]", 0-1) = ar_1 S("evalfoobb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb4in(ar_0, ar_1, ar_2, ar_3)) [ ar_3 >= ar_0 + 1 ]", 0-2) = ar_2 S("evalfoobb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb4in(ar_0, ar_1, ar_2, ar_3)) [ ar_3 >= ar_0 + 1 ]", 0-3) = ? S("evalfoobb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb3in(ar_0, ar_1, ar_2, ar_3)) [ ar_0 >= ar_3 ]", 0-0) = 3*ar_1 + 3*ar_2 + 18 S("evalfoobb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb3in(ar_0, ar_1, ar_2, ar_3)) [ ar_0 >= ar_3 ]", 0-1) = ar_1 S("evalfoobb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb3in(ar_0, ar_1, ar_2, ar_3)) [ ar_0 >= ar_3 ]", 0-2) = ar_2 S("evalfoobb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb3in(ar_0, ar_1, ar_2, ar_3)) [ ar_0 >= ar_3 ]", 0-3) = ? S("evalfoobb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb5in(ar_0, ar_1, ar_2, ar_3)) [ ar_0 >= ar_2 ]", 0-0) = 3*ar_1 + 3*ar_2 + 54 S("evalfoobb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb5in(ar_0, ar_1, ar_2, ar_3)) [ ar_0 >= ar_2 ]", 0-1) = ar_1 S("evalfoobb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb5in(ar_0, ar_1, ar_2, ar_3)) [ ar_0 >= ar_2 ]", 0-2) = ar_2 S("evalfoobb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb5in(ar_0, ar_1, ar_2, ar_3)) [ ar_0 >= ar_2 ]", 0-3) = ? S("evalfoobb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb2in(ar_0, ar_1, ar_2, 0)) [ ar_2 >= ar_0 + 1 ]", 0-0) = 3*ar_1 + 3*ar_2 + 18 S("evalfoobb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb2in(ar_0, ar_1, ar_2, 0)) [ ar_2 >= ar_0 + 1 ]", 0-1) = ar_1 S("evalfoobb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb2in(ar_0, ar_1, ar_2, 0)) [ ar_2 >= ar_0 + 1 ]", 0-2) = ar_2 S("evalfoobb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb2in(ar_0, ar_1, ar_2, 0)) [ ar_2 >= ar_0 + 1 ]", 0-3) = 0 S("evalfoobb0in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb1in(ar_1, ar_1, ar_2, ar_3))", 0-0) = ar_1 S("evalfoobb0in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb1in(ar_1, ar_1, ar_2, ar_3))", 0-1) = ar_1 S("evalfoobb0in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb1in(ar_1, ar_1, ar_2, ar_3))", 0-2) = ar_2 S("evalfoobb0in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb1in(ar_1, ar_1, ar_2, ar_3))", 0-3) = ar_3 S("evalfoostart(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb0in(ar_0, ar_1, ar_2, ar_3))", 0-0) = ar_0 S("evalfoostart(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb0in(ar_0, ar_1, ar_2, ar_3))", 0-1) = ar_1 S("evalfoostart(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb0in(ar_0, ar_1, ar_2, ar_3))", 0-2) = ar_2 S("evalfoostart(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb0in(ar_0, ar_1, ar_2, ar_3))", 0-3) = ar_3 orients the transitions evalfoobb3in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb2in(ar_0, ar_1, ar_2, ar_3 + 1)) evalfoobb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb3in(ar_0, ar_1, ar_2, ar_3)) [ ar_0 >= ar_3 ] weakly and the transition evalfoobb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb3in(ar_0, ar_1, ar_2, ar_3)) [ ar_0 >= ar_3 ] strictly and produces the following problem: 6: T: (Comp: 1, Cost: 1) evalfoostart(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb0in(ar_0, ar_1, ar_2, ar_3)) (Comp: 1, Cost: 1) evalfoobb0in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb1in(ar_1, ar_1, ar_2, ar_3)) (Comp: ar_1 + ar_2 + 1, Cost: 1) evalfoobb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb2in(ar_0, ar_1, ar_2, 0)) [ ar_2 >= ar_0 + 1 ] (Comp: 2, Cost: 1) evalfoobb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb5in(ar_0, ar_1, ar_2, ar_3)) [ ar_0 >= ar_2 ] (Comp: 3*ar_1^2 + 6*ar_1*ar_2 + 3*ar_2^2 + 22*ar_1 + 22*ar_2 + 19, Cost: 1) evalfoobb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb3in(ar_0, ar_1, ar_2, ar_3)) [ ar_0 >= ar_3 ] (Comp: 2*ar_1 + 2*ar_2 + 2, Cost: 1) evalfoobb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb4in(ar_0, ar_1, ar_2, ar_3)) [ ar_3 >= ar_0 + 1 ] (Comp: ?, Cost: 1) evalfoobb3in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb2in(ar_0, ar_1, ar_2, ar_3 + 1)) (Comp: 2*ar_1 + 2*ar_2 + 2, Cost: 1) evalfoobb4in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb1in(ar_0 + 1, ar_1, ar_2, ar_3)) (Comp: 2, Cost: 1) evalfoobb5in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoostop(ar_0, ar_1, ar_2, ar_3)) (Comp: 1, Cost: 0) koat_start(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoostart(ar_0, ar_1, ar_2, ar_3)) [ 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) evalfoostart(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb0in(ar_0, ar_1, ar_2, ar_3)) (Comp: 1, Cost: 1) evalfoobb0in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb1in(ar_1, ar_1, ar_2, ar_3)) (Comp: ar_1 + ar_2 + 1, Cost: 1) evalfoobb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb2in(ar_0, ar_1, ar_2, 0)) [ ar_2 >= ar_0 + 1 ] (Comp: 2, Cost: 1) evalfoobb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb5in(ar_0, ar_1, ar_2, ar_3)) [ ar_0 >= ar_2 ] (Comp: 3*ar_1^2 + 6*ar_1*ar_2 + 3*ar_2^2 + 22*ar_1 + 22*ar_2 + 19, Cost: 1) evalfoobb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb3in(ar_0, ar_1, ar_2, ar_3)) [ ar_0 >= ar_3 ] (Comp: 2*ar_1 + 2*ar_2 + 2, Cost: 1) evalfoobb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb4in(ar_0, ar_1, ar_2, ar_3)) [ ar_3 >= ar_0 + 1 ] (Comp: 3*ar_1^2 + 6*ar_1*ar_2 + 3*ar_2^2 + 22*ar_1 + 22*ar_2 + 19, Cost: 1) evalfoobb3in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb2in(ar_0, ar_1, ar_2, ar_3 + 1)) (Comp: 2*ar_1 + 2*ar_2 + 2, Cost: 1) evalfoobb4in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb1in(ar_0 + 1, ar_1, ar_2, ar_3)) (Comp: 2, Cost: 1) evalfoobb5in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoostop(ar_0, ar_1, ar_2, ar_3)) (Comp: 1, Cost: 0) koat_start(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoostart(ar_0, ar_1, ar_2, ar_3)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 Complexity upper bound 49*ar_1 + 49*ar_2 + 6*ar_1^2 + 12*ar_1*ar_2 + 6*ar_2^2 + 49 Time: 0.097 sec (SMT: 0.081 sec) ---------------------------------------- (2) BOUNDS(1, n^2)