/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(1, n^2).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, 280 ms] (2) BOUNDS(1, n^2) ---------------------------------------- (0) Obligation: Complexity Int TRS consisting of the following rules: eval_loops_start(v_n, v_x.0, v_y.0) -> Com_1(eval_loops_bb0_in(v_n, v_x.0, v_y.0)) :|: TRUE eval_loops_bb0_in(v_n, v_x.0, v_y.0) -> Com_1(eval_loops_bb1_in(v_n, v_n, v_y.0)) :|: v_n >= 0 eval_loops_bb0_in(v_n, v_x.0, v_y.0) -> Com_1(eval_loops_bb6_in(v_n, v_x.0, v_y.0)) :|: v_n < 0 eval_loops_bb1_in(v_n, v_x.0, v_y.0) -> Com_1(eval_loops_bb2_in(v_n, v_x.0, v_y.0)) :|: v_x.0 >= 0 eval_loops_bb1_in(v_n, v_x.0, v_y.0) -> Com_1(eval_loops_bb6_in(v_n, v_x.0, v_y.0)) :|: v_x.0 < 0 eval_loops_bb2_in(v_n, v_x.0, v_y.0) -> Com_1(eval_loops_bb3_in(v_n, v_x.0, 1)) :|: 1 < v_x.0 eval_loops_bb2_in(v_n, v_x.0, v_y.0) -> Com_1(eval_loops_bb5_in(v_n, v_x.0, v_y.0)) :|: 1 >= v_x.0 eval_loops_bb3_in(v_n, v_x.0, v_y.0) -> Com_1(eval_loops_bb4_in(v_n, v_x.0, v_y.0)) :|: v_y.0 < v_x.0 eval_loops_bb3_in(v_n, v_x.0, v_y.0) -> Com_1(eval_loops_bb5_in(v_n, v_x.0, v_y.0)) :|: v_y.0 >= v_x.0 eval_loops_bb4_in(v_n, v_x.0, v_y.0) -> Com_1(eval_loops_bb3_in(v_n, v_x.0, 2 * v_y.0)) :|: TRUE eval_loops_bb5_in(v_n, v_x.0, v_y.0) -> Com_1(eval_loops_bb1_in(v_n, v_x.0 - 1, v_y.0)) :|: TRUE eval_loops_bb6_in(v_n, v_x.0, v_y.0) -> Com_1(eval_loops_stop(v_n, v_x.0, v_y.0)) :|: TRUE The start-symbols are:[eval_loops_start_3] ---------------------------------------- (1) Koat Proof (FINISHED) YES(?, 34*Ar_0 + 8*Ar_0^2 + 33) Initial complexity problem: 1: T: (Comp: ?, Cost: 1) evalloopsstart(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsbb0in(Ar_0, Ar_1, Ar_2)) (Comp: ?, Cost: 1) evalloopsbb0in(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsbb1in(Ar_0, Ar_0, Ar_2)) [ Ar_0 >= 0 ] (Comp: ?, Cost: 1) evalloopsbb0in(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsbb6in(Ar_0, Ar_1, Ar_2)) [ 0 >= Ar_0 + 1 ] (Comp: ?, Cost: 1) evalloopsbb1in(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsbb2in(Ar_0, Ar_1, Ar_2)) [ Ar_1 >= 0 ] (Comp: ?, Cost: 1) evalloopsbb1in(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsbb6in(Ar_0, Ar_1, Ar_2)) [ 0 >= Ar_1 + 1 ] (Comp: ?, Cost: 1) evalloopsbb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsbb3in(Ar_0, Ar_1, 1)) [ Ar_1 >= 2 ] (Comp: ?, Cost: 1) evalloopsbb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsbb5in(Ar_0, Ar_1, Ar_2)) [ 1 >= Ar_1 ] (Comp: ?, Cost: 1) evalloopsbb3in(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsbb4in(Ar_0, Ar_1, Ar_2)) [ Ar_1 >= Ar_2 + 1 ] (Comp: ?, Cost: 1) evalloopsbb3in(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsbb5in(Ar_0, Ar_1, Ar_2)) [ Ar_2 >= Ar_1 ] (Comp: ?, Cost: 1) evalloopsbb4in(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsbb3in(Ar_0, Ar_1, 2*Ar_2)) (Comp: ?, Cost: 1) evalloopsbb5in(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsbb1in(Ar_0, Ar_1 - 1, Ar_2)) (Comp: ?, Cost: 1) evalloopsbb6in(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsstop(Ar_0, Ar_1, Ar_2)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsstart(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) evalloopsstart(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsbb0in(Ar_0, Ar_1, Ar_2)) (Comp: 1, Cost: 1) evalloopsbb0in(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsbb1in(Ar_0, Ar_0, Ar_2)) [ Ar_0 >= 0 ] (Comp: 1, Cost: 1) evalloopsbb0in(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsbb6in(Ar_0, Ar_1, Ar_2)) [ 0 >= Ar_0 + 1 ] (Comp: ?, Cost: 1) evalloopsbb1in(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsbb2in(Ar_0, Ar_1, Ar_2)) [ Ar_1 >= 0 ] (Comp: ?, Cost: 1) evalloopsbb1in(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsbb6in(Ar_0, Ar_1, Ar_2)) [ 0 >= Ar_1 + 1 ] (Comp: ?, Cost: 1) evalloopsbb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsbb3in(Ar_0, Ar_1, 1)) [ Ar_1 >= 2 ] (Comp: ?, Cost: 1) evalloopsbb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsbb5in(Ar_0, Ar_1, Ar_2)) [ 1 >= Ar_1 ] (Comp: ?, Cost: 1) evalloopsbb3in(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsbb4in(Ar_0, Ar_1, Ar_2)) [ Ar_1 >= Ar_2 + 1 ] (Comp: ?, Cost: 1) evalloopsbb3in(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsbb5in(Ar_0, Ar_1, Ar_2)) [ Ar_2 >= Ar_1 ] (Comp: ?, Cost: 1) evalloopsbb4in(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsbb3in(Ar_0, Ar_1, 2*Ar_2)) (Comp: ?, Cost: 1) evalloopsbb5in(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsbb1in(Ar_0, Ar_1 - 1, Ar_2)) (Comp: ?, Cost: 1) evalloopsbb6in(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsstop(Ar_0, Ar_1, Ar_2)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsstart(Ar_0, Ar_1, Ar_2)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 A polynomial rank function with Pol(evalloopsstart) = 2 Pol(evalloopsbb0in) = 2 Pol(evalloopsbb1in) = 2 Pol(evalloopsbb6in) = 1 Pol(evalloopsbb2in) = 2 Pol(evalloopsbb3in) = 2 Pol(evalloopsbb5in) = 2 Pol(evalloopsbb4in) = 2 Pol(evalloopsstop) = 0 Pol(koat_start) = 2 orients all transitions weakly and the transitions evalloopsbb6in(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsstop(Ar_0, Ar_1, Ar_2)) evalloopsbb1in(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsbb6in(Ar_0, Ar_1, Ar_2)) [ 0 >= Ar_1 + 1 ] strictly and produces the following problem: 3: T: (Comp: 1, Cost: 1) evalloopsstart(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsbb0in(Ar_0, Ar_1, Ar_2)) (Comp: 1, Cost: 1) evalloopsbb0in(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsbb1in(Ar_0, Ar_0, Ar_2)) [ Ar_0 >= 0 ] (Comp: 1, Cost: 1) evalloopsbb0in(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsbb6in(Ar_0, Ar_1, Ar_2)) [ 0 >= Ar_0 + 1 ] (Comp: ?, Cost: 1) evalloopsbb1in(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsbb2in(Ar_0, Ar_1, Ar_2)) [ Ar_1 >= 0 ] (Comp: 2, Cost: 1) evalloopsbb1in(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsbb6in(Ar_0, Ar_1, Ar_2)) [ 0 >= Ar_1 + 1 ] (Comp: ?, Cost: 1) evalloopsbb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsbb3in(Ar_0, Ar_1, 1)) [ Ar_1 >= 2 ] (Comp: ?, Cost: 1) evalloopsbb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsbb5in(Ar_0, Ar_1, Ar_2)) [ 1 >= Ar_1 ] (Comp: ?, Cost: 1) evalloopsbb3in(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsbb4in(Ar_0, Ar_1, Ar_2)) [ Ar_1 >= Ar_2 + 1 ] (Comp: ?, Cost: 1) evalloopsbb3in(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsbb5in(Ar_0, Ar_1, Ar_2)) [ Ar_2 >= Ar_1 ] (Comp: ?, Cost: 1) evalloopsbb4in(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsbb3in(Ar_0, Ar_1, 2*Ar_2)) (Comp: ?, Cost: 1) evalloopsbb5in(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsbb1in(Ar_0, Ar_1 - 1, Ar_2)) (Comp: 2, Cost: 1) evalloopsbb6in(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsstop(Ar_0, Ar_1, Ar_2)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsstart(Ar_0, Ar_1, Ar_2)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 A polynomial rank function with Pol(evalloopsbb5in) = 2*V_2 Pol(evalloopsbb1in) = 2*V_2 + 2 Pol(evalloopsbb4in) = 2*V_2 Pol(evalloopsbb3in) = 2*V_2 Pol(evalloopsbb2in) = 2*V_2 + 1 and size complexities S("koat_start(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsstart(Ar_0, Ar_1, Ar_2)) [ 0 <= 0 ]", 0-0) = Ar_0 S("koat_start(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsstart(Ar_0, Ar_1, Ar_2)) [ 0 <= 0 ]", 0-1) = Ar_1 S("koat_start(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsstart(Ar_0, Ar_1, Ar_2)) [ 0 <= 0 ]", 0-2) = Ar_2 S("evalloopsbb6in(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsstop(Ar_0, Ar_1, Ar_2))", 0-0) = Ar_0 S("evalloopsbb6in(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsstop(Ar_0, Ar_1, Ar_2))", 0-1) = ? S("evalloopsbb6in(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsstop(Ar_0, Ar_1, Ar_2))", 0-2) = ? S("evalloopsbb5in(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsbb1in(Ar_0, Ar_1 - 1, Ar_2))", 0-0) = Ar_0 S("evalloopsbb5in(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsbb1in(Ar_0, Ar_1 - 1, Ar_2))", 0-1) = ? S("evalloopsbb5in(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsbb1in(Ar_0, Ar_1 - 1, Ar_2))", 0-2) = ? S("evalloopsbb4in(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsbb3in(Ar_0, Ar_1, 2*Ar_2))", 0-0) = Ar_0 S("evalloopsbb4in(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsbb3in(Ar_0, Ar_1, 2*Ar_2))", 0-1) = ? S("evalloopsbb4in(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsbb3in(Ar_0, Ar_1, 2*Ar_2))", 0-2) = ? S("evalloopsbb3in(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsbb5in(Ar_0, Ar_1, Ar_2)) [ Ar_2 >= Ar_1 ]", 0-0) = Ar_0 S("evalloopsbb3in(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsbb5in(Ar_0, Ar_1, Ar_2)) [ Ar_2 >= Ar_1 ]", 0-1) = ? S("evalloopsbb3in(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsbb5in(Ar_0, Ar_1, Ar_2)) [ Ar_2 >= Ar_1 ]", 0-2) = ? S("evalloopsbb3in(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsbb4in(Ar_0, Ar_1, Ar_2)) [ Ar_1 >= Ar_2 + 1 ]", 0-0) = Ar_0 S("evalloopsbb3in(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsbb4in(Ar_0, Ar_1, Ar_2)) [ Ar_1 >= Ar_2 + 1 ]", 0-1) = ? S("evalloopsbb3in(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsbb4in(Ar_0, Ar_1, Ar_2)) [ Ar_1 >= Ar_2 + 1 ]", 0-2) = ? S("evalloopsbb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsbb5in(Ar_0, Ar_1, Ar_2)) [ 1 >= Ar_1 ]", 0-0) = Ar_0 S("evalloopsbb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsbb5in(Ar_0, Ar_1, Ar_2)) [ 1 >= Ar_1 ]", 0-1) = ? S("evalloopsbb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsbb5in(Ar_0, Ar_1, Ar_2)) [ 1 >= Ar_1 ]", 0-2) = ? S("evalloopsbb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsbb3in(Ar_0, Ar_1, 1)) [ Ar_1 >= 2 ]", 0-0) = Ar_0 S("evalloopsbb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsbb3in(Ar_0, Ar_1, 1)) [ Ar_1 >= 2 ]", 0-1) = ? S("evalloopsbb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsbb3in(Ar_0, Ar_1, 1)) [ Ar_1 >= 2 ]", 0-2) = 1 S("evalloopsbb1in(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsbb6in(Ar_0, Ar_1, Ar_2)) [ 0 >= Ar_1 + 1 ]", 0-0) = Ar_0 S("evalloopsbb1in(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsbb6in(Ar_0, Ar_1, Ar_2)) [ 0 >= Ar_1 + 1 ]", 0-1) = ? S("evalloopsbb1in(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsbb6in(Ar_0, Ar_1, Ar_2)) [ 0 >= Ar_1 + 1 ]", 0-2) = ? S("evalloopsbb1in(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsbb2in(Ar_0, Ar_1, Ar_2)) [ Ar_1 >= 0 ]", 0-0) = Ar_0 S("evalloopsbb1in(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsbb2in(Ar_0, Ar_1, Ar_2)) [ Ar_1 >= 0 ]", 0-1) = ? S("evalloopsbb1in(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsbb2in(Ar_0, Ar_1, Ar_2)) [ Ar_1 >= 0 ]", 0-2) = ? S("evalloopsbb0in(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsbb6in(Ar_0, Ar_1, Ar_2)) [ 0 >= Ar_0 + 1 ]", 0-0) = Ar_0 S("evalloopsbb0in(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsbb6in(Ar_0, Ar_1, Ar_2)) [ 0 >= Ar_0 + 1 ]", 0-1) = Ar_1 S("evalloopsbb0in(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsbb6in(Ar_0, Ar_1, Ar_2)) [ 0 >= Ar_0 + 1 ]", 0-2) = Ar_2 S("evalloopsbb0in(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsbb1in(Ar_0, Ar_0, Ar_2)) [ Ar_0 >= 0 ]", 0-0) = Ar_0 S("evalloopsbb0in(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsbb1in(Ar_0, Ar_0, Ar_2)) [ Ar_0 >= 0 ]", 0-1) = Ar_0 S("evalloopsbb0in(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsbb1in(Ar_0, Ar_0, Ar_2)) [ Ar_0 >= 0 ]", 0-2) = Ar_2 S("evalloopsstart(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsbb0in(Ar_0, Ar_1, Ar_2))", 0-0) = Ar_0 S("evalloopsstart(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsbb0in(Ar_0, Ar_1, Ar_2))", 0-1) = Ar_1 S("evalloopsstart(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsbb0in(Ar_0, Ar_1, Ar_2))", 0-2) = Ar_2 orients the transitions evalloopsbb5in(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsbb1in(Ar_0, Ar_1 - 1, Ar_2)) evalloopsbb4in(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsbb3in(Ar_0, Ar_1, 2*Ar_2)) evalloopsbb3in(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsbb5in(Ar_0, Ar_1, Ar_2)) [ Ar_2 >= Ar_1 ] evalloopsbb3in(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsbb4in(Ar_0, Ar_1, Ar_2)) [ Ar_1 >= Ar_2 + 1 ] evalloopsbb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsbb5in(Ar_0, Ar_1, Ar_2)) [ 1 >= Ar_1 ] evalloopsbb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsbb3in(Ar_0, Ar_1, 1)) [ Ar_1 >= 2 ] evalloopsbb1in(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsbb2in(Ar_0, Ar_1, Ar_2)) [ Ar_1 >= 0 ] weakly and the transitions evalloopsbb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsbb3in(Ar_0, Ar_1, 1)) [ Ar_1 >= 2 ] evalloopsbb1in(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsbb2in(Ar_0, Ar_1, Ar_2)) [ Ar_1 >= 0 ] strictly and produces the following problem: 4: T: (Comp: 1, Cost: 1) evalloopsstart(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsbb0in(Ar_0, Ar_1, Ar_2)) (Comp: 1, Cost: 1) evalloopsbb0in(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsbb1in(Ar_0, Ar_0, Ar_2)) [ Ar_0 >= 0 ] (Comp: 1, Cost: 1) evalloopsbb0in(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsbb6in(Ar_0, Ar_1, Ar_2)) [ 0 >= Ar_0 + 1 ] (Comp: 2*Ar_0 + 2, Cost: 1) evalloopsbb1in(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsbb2in(Ar_0, Ar_1, Ar_2)) [ Ar_1 >= 0 ] (Comp: 2, Cost: 1) evalloopsbb1in(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsbb6in(Ar_0, Ar_1, Ar_2)) [ 0 >= Ar_1 + 1 ] (Comp: 2*Ar_0 + 2, Cost: 1) evalloopsbb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsbb3in(Ar_0, Ar_1, 1)) [ Ar_1 >= 2 ] (Comp: ?, Cost: 1) evalloopsbb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsbb5in(Ar_0, Ar_1, Ar_2)) [ 1 >= Ar_1 ] (Comp: ?, Cost: 1) evalloopsbb3in(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsbb4in(Ar_0, Ar_1, Ar_2)) [ Ar_1 >= Ar_2 + 1 ] (Comp: ?, Cost: 1) evalloopsbb3in(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsbb5in(Ar_0, Ar_1, Ar_2)) [ Ar_2 >= Ar_1 ] (Comp: ?, Cost: 1) evalloopsbb4in(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsbb3in(Ar_0, Ar_1, 2*Ar_2)) (Comp: ?, Cost: 1) evalloopsbb5in(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsbb1in(Ar_0, Ar_1 - 1, Ar_2)) (Comp: 2, Cost: 1) evalloopsbb6in(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsstop(Ar_0, Ar_1, Ar_2)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsstart(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) evalloopsstart(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsbb0in(Ar_0, Ar_1, Ar_2)) (Comp: 1, Cost: 1) evalloopsbb0in(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsbb1in(Ar_0, Ar_0, Ar_2)) [ Ar_0 >= 0 ] (Comp: 1, Cost: 1) evalloopsbb0in(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsbb6in(Ar_0, Ar_1, Ar_2)) [ 0 >= Ar_0 + 1 ] (Comp: 2*Ar_0 + 2, Cost: 1) evalloopsbb1in(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsbb2in(Ar_0, Ar_1, Ar_2)) [ Ar_1 >= 0 ] (Comp: 2, Cost: 1) evalloopsbb1in(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsbb6in(Ar_0, Ar_1, Ar_2)) [ 0 >= Ar_1 + 1 ] (Comp: 2*Ar_0 + 2, Cost: 1) evalloopsbb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsbb3in(Ar_0, Ar_1, 1)) [ Ar_1 >= 2 ] (Comp: 2*Ar_0 + 2, Cost: 1) evalloopsbb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsbb5in(Ar_0, Ar_1, Ar_2)) [ 1 >= Ar_1 ] (Comp: ?, Cost: 1) evalloopsbb3in(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsbb4in(Ar_0, Ar_1, Ar_2)) [ Ar_1 >= Ar_2 + 1 ] (Comp: ?, Cost: 1) evalloopsbb3in(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsbb5in(Ar_0, Ar_1, Ar_2)) [ Ar_2 >= Ar_1 ] (Comp: ?, Cost: 1) evalloopsbb4in(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsbb3in(Ar_0, Ar_1, 2*Ar_2)) (Comp: ?, Cost: 1) evalloopsbb5in(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsbb1in(Ar_0, Ar_1 - 1, Ar_2)) (Comp: 2, Cost: 1) evalloopsbb6in(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsstop(Ar_0, Ar_1, Ar_2)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsstart(Ar_0, Ar_1, Ar_2)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 A polynomial rank function with Pol(evalloopsbb5in) = 1 Pol(evalloopsbb1in) = 0 Pol(evalloopsbb4in) = 2 Pol(evalloopsbb3in) = 2 and size complexities S("koat_start(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsstart(Ar_0, Ar_1, Ar_2)) [ 0 <= 0 ]", 0-0) = Ar_0 S("koat_start(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsstart(Ar_0, Ar_1, Ar_2)) [ 0 <= 0 ]", 0-1) = Ar_1 S("koat_start(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsstart(Ar_0, Ar_1, Ar_2)) [ 0 <= 0 ]", 0-2) = Ar_2 S("evalloopsbb6in(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsstop(Ar_0, Ar_1, Ar_2))", 0-0) = Ar_0 S("evalloopsbb6in(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsstop(Ar_0, Ar_1, Ar_2))", 0-1) = ? S("evalloopsbb6in(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsstop(Ar_0, Ar_1, Ar_2))", 0-2) = ? S("evalloopsbb5in(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsbb1in(Ar_0, Ar_1 - 1, Ar_2))", 0-0) = Ar_0 S("evalloopsbb5in(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsbb1in(Ar_0, Ar_1 - 1, Ar_2))", 0-1) = ? S("evalloopsbb5in(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsbb1in(Ar_0, Ar_1 - 1, Ar_2))", 0-2) = ? S("evalloopsbb4in(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsbb3in(Ar_0, Ar_1, 2*Ar_2))", 0-0) = Ar_0 S("evalloopsbb4in(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsbb3in(Ar_0, Ar_1, 2*Ar_2))", 0-1) = ? S("evalloopsbb4in(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsbb3in(Ar_0, Ar_1, 2*Ar_2))", 0-2) = ? S("evalloopsbb3in(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsbb5in(Ar_0, Ar_1, Ar_2)) [ Ar_2 >= Ar_1 ]", 0-0) = Ar_0 S("evalloopsbb3in(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsbb5in(Ar_0, Ar_1, Ar_2)) [ Ar_2 >= Ar_1 ]", 0-1) = ? S("evalloopsbb3in(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsbb5in(Ar_0, Ar_1, Ar_2)) [ Ar_2 >= Ar_1 ]", 0-2) = ? S("evalloopsbb3in(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsbb4in(Ar_0, Ar_1, Ar_2)) [ Ar_1 >= Ar_2 + 1 ]", 0-0) = Ar_0 S("evalloopsbb3in(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsbb4in(Ar_0, Ar_1, Ar_2)) [ Ar_1 >= Ar_2 + 1 ]", 0-1) = ? S("evalloopsbb3in(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsbb4in(Ar_0, Ar_1, Ar_2)) [ Ar_1 >= Ar_2 + 1 ]", 0-2) = ? S("evalloopsbb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsbb5in(Ar_0, Ar_1, Ar_2)) [ 1 >= Ar_1 ]", 0-0) = Ar_0 S("evalloopsbb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsbb5in(Ar_0, Ar_1, Ar_2)) [ 1 >= Ar_1 ]", 0-1) = ? S("evalloopsbb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsbb5in(Ar_0, Ar_1, Ar_2)) [ 1 >= Ar_1 ]", 0-2) = ? S("evalloopsbb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsbb3in(Ar_0, Ar_1, 1)) [ Ar_1 >= 2 ]", 0-0) = Ar_0 S("evalloopsbb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsbb3in(Ar_0, Ar_1, 1)) [ Ar_1 >= 2 ]", 0-1) = ? S("evalloopsbb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsbb3in(Ar_0, Ar_1, 1)) [ Ar_1 >= 2 ]", 0-2) = 1 S("evalloopsbb1in(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsbb6in(Ar_0, Ar_1, Ar_2)) [ 0 >= Ar_1 + 1 ]", 0-0) = Ar_0 S("evalloopsbb1in(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsbb6in(Ar_0, Ar_1, Ar_2)) [ 0 >= Ar_1 + 1 ]", 0-1) = ? S("evalloopsbb1in(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsbb6in(Ar_0, Ar_1, Ar_2)) [ 0 >= Ar_1 + 1 ]", 0-2) = ? S("evalloopsbb1in(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsbb2in(Ar_0, Ar_1, Ar_2)) [ Ar_1 >= 0 ]", 0-0) = Ar_0 S("evalloopsbb1in(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsbb2in(Ar_0, Ar_1, Ar_2)) [ Ar_1 >= 0 ]", 0-1) = ? S("evalloopsbb1in(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsbb2in(Ar_0, Ar_1, Ar_2)) [ Ar_1 >= 0 ]", 0-2) = ? S("evalloopsbb0in(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsbb6in(Ar_0, Ar_1, Ar_2)) [ 0 >= Ar_0 + 1 ]", 0-0) = Ar_0 S("evalloopsbb0in(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsbb6in(Ar_0, Ar_1, Ar_2)) [ 0 >= Ar_0 + 1 ]", 0-1) = Ar_1 S("evalloopsbb0in(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsbb6in(Ar_0, Ar_1, Ar_2)) [ 0 >= Ar_0 + 1 ]", 0-2) = Ar_2 S("evalloopsbb0in(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsbb1in(Ar_0, Ar_0, Ar_2)) [ Ar_0 >= 0 ]", 0-0) = Ar_0 S("evalloopsbb0in(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsbb1in(Ar_0, Ar_0, Ar_2)) [ Ar_0 >= 0 ]", 0-1) = Ar_0 S("evalloopsbb0in(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsbb1in(Ar_0, Ar_0, Ar_2)) [ Ar_0 >= 0 ]", 0-2) = Ar_2 S("evalloopsstart(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsbb0in(Ar_0, Ar_1, Ar_2))", 0-0) = Ar_0 S("evalloopsstart(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsbb0in(Ar_0, Ar_1, Ar_2))", 0-1) = Ar_1 S("evalloopsstart(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsbb0in(Ar_0, Ar_1, Ar_2))", 0-2) = Ar_2 orients the transitions evalloopsbb5in(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsbb1in(Ar_0, Ar_1 - 1, Ar_2)) evalloopsbb4in(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsbb3in(Ar_0, Ar_1, 2*Ar_2)) evalloopsbb3in(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsbb5in(Ar_0, Ar_1, Ar_2)) [ Ar_2 >= Ar_1 ] evalloopsbb3in(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsbb4in(Ar_0, Ar_1, Ar_2)) [ Ar_1 >= Ar_2 + 1 ] weakly and the transitions evalloopsbb5in(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsbb1in(Ar_0, Ar_1 - 1, Ar_2)) evalloopsbb3in(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsbb5in(Ar_0, Ar_1, Ar_2)) [ Ar_2 >= Ar_1 ] strictly and produces the following problem: 6: T: (Comp: 1, Cost: 1) evalloopsstart(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsbb0in(Ar_0, Ar_1, Ar_2)) (Comp: 1, Cost: 1) evalloopsbb0in(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsbb1in(Ar_0, Ar_0, Ar_2)) [ Ar_0 >= 0 ] (Comp: 1, Cost: 1) evalloopsbb0in(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsbb6in(Ar_0, Ar_1, Ar_2)) [ 0 >= Ar_0 + 1 ] (Comp: 2*Ar_0 + 2, Cost: 1) evalloopsbb1in(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsbb2in(Ar_0, Ar_1, Ar_2)) [ Ar_1 >= 0 ] (Comp: 2, Cost: 1) evalloopsbb1in(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsbb6in(Ar_0, Ar_1, Ar_2)) [ 0 >= Ar_1 + 1 ] (Comp: 2*Ar_0 + 2, Cost: 1) evalloopsbb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsbb3in(Ar_0, Ar_1, 1)) [ Ar_1 >= 2 ] (Comp: 2*Ar_0 + 2, Cost: 1) evalloopsbb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsbb5in(Ar_0, Ar_1, Ar_2)) [ 1 >= Ar_1 ] (Comp: ?, Cost: 1) evalloopsbb3in(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsbb4in(Ar_0, Ar_1, Ar_2)) [ Ar_1 >= Ar_2 + 1 ] (Comp: 6*Ar_0 + 6, Cost: 1) evalloopsbb3in(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsbb5in(Ar_0, Ar_1, Ar_2)) [ Ar_2 >= Ar_1 ] (Comp: ?, Cost: 1) evalloopsbb4in(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsbb3in(Ar_0, Ar_1, 2*Ar_2)) (Comp: 6*Ar_0 + 6, Cost: 1) evalloopsbb5in(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsbb1in(Ar_0, Ar_1 - 1, Ar_2)) (Comp: 2, Cost: 1) evalloopsbb6in(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsstop(Ar_0, Ar_1, Ar_2)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsstart(Ar_0, Ar_1, Ar_2)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 Applied AI with 'oct' on problem 6 to obtain the following invariants: For symbol evalloopsbb1in: X_1 - X_2 >= 0 /\ X_1 >= 0 For symbol evalloopsbb2in: X_1 - X_2 >= 0 /\ X_2 >= 0 /\ X_1 + X_2 >= 0 /\ X_1 >= 0 For symbol evalloopsbb3in: X_3 - 1 >= 0 /\ X_2 + X_3 - 3 >= 0 /\ X_1 + X_3 - 3 >= 0 /\ X_1 - X_2 >= 0 /\ X_2 - 2 >= 0 /\ X_1 + X_2 - 4 >= 0 /\ X_1 - 2 >= 0 For symbol evalloopsbb4in: X_2 - X_3 - 1 >= 0 /\ X_1 - X_3 - 1 >= 0 /\ X_3 - 1 >= 0 /\ X_2 + X_3 - 3 >= 0 /\ X_1 + X_3 - 3 >= 0 /\ X_1 - X_2 >= 0 /\ X_2 - 2 >= 0 /\ X_1 + X_2 - 4 >= 0 /\ X_1 - 2 >= 0 For symbol evalloopsbb5in: X_1 - X_2 >= 0 /\ X_2 >= 0 /\ X_1 + X_2 >= 0 /\ X_1 >= 0 This yielded the following problem: 7: T: (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsstart(Ar_0, Ar_1, Ar_2)) [ 0 <= 0 ] (Comp: 2, Cost: 1) evalloopsbb6in(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsstop(Ar_0, Ar_1, Ar_2)) (Comp: 6*Ar_0 + 6, Cost: 1) evalloopsbb5in(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsbb1in(Ar_0, Ar_1 - 1, Ar_2)) [ Ar_0 - Ar_1 >= 0 /\ Ar_1 >= 0 /\ Ar_0 + Ar_1 >= 0 /\ Ar_0 >= 0 ] (Comp: ?, Cost: 1) evalloopsbb4in(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsbb3in(Ar_0, Ar_1, 2*Ar_2)) [ Ar_1 - Ar_2 - 1 >= 0 /\ Ar_0 - Ar_2 - 1 >= 0 /\ Ar_2 - 1 >= 0 /\ Ar_1 + Ar_2 - 3 >= 0 /\ Ar_0 + Ar_2 - 3 >= 0 /\ Ar_0 - Ar_1 >= 0 /\ Ar_1 - 2 >= 0 /\ Ar_0 + Ar_1 - 4 >= 0 /\ Ar_0 - 2 >= 0 ] (Comp: 6*Ar_0 + 6, Cost: 1) evalloopsbb3in(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsbb5in(Ar_0, Ar_1, Ar_2)) [ Ar_2 - 1 >= 0 /\ Ar_1 + Ar_2 - 3 >= 0 /\ Ar_0 + Ar_2 - 3 >= 0 /\ Ar_0 - Ar_1 >= 0 /\ Ar_1 - 2 >= 0 /\ Ar_0 + Ar_1 - 4 >= 0 /\ Ar_0 - 2 >= 0 /\ Ar_2 >= Ar_1 ] (Comp: ?, Cost: 1) evalloopsbb3in(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsbb4in(Ar_0, Ar_1, Ar_2)) [ Ar_2 - 1 >= 0 /\ Ar_1 + Ar_2 - 3 >= 0 /\ Ar_0 + Ar_2 - 3 >= 0 /\ Ar_0 - Ar_1 >= 0 /\ Ar_1 - 2 >= 0 /\ Ar_0 + Ar_1 - 4 >= 0 /\ Ar_0 - 2 >= 0 /\ Ar_1 >= Ar_2 + 1 ] (Comp: 2*Ar_0 + 2, Cost: 1) evalloopsbb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsbb5in(Ar_0, Ar_1, Ar_2)) [ Ar_0 - Ar_1 >= 0 /\ Ar_1 >= 0 /\ Ar_0 + Ar_1 >= 0 /\ Ar_0 >= 0 /\ 1 >= Ar_1 ] (Comp: 2*Ar_0 + 2, Cost: 1) evalloopsbb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsbb3in(Ar_0, Ar_1, 1)) [ Ar_0 - Ar_1 >= 0 /\ Ar_1 >= 0 /\ Ar_0 + Ar_1 >= 0 /\ Ar_0 >= 0 /\ Ar_1 >= 2 ] (Comp: 2, Cost: 1) evalloopsbb1in(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsbb6in(Ar_0, Ar_1, Ar_2)) [ Ar_0 - Ar_1 >= 0 /\ Ar_0 >= 0 /\ 0 >= Ar_1 + 1 ] (Comp: 2*Ar_0 + 2, Cost: 1) evalloopsbb1in(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsbb2in(Ar_0, Ar_1, Ar_2)) [ Ar_0 - Ar_1 >= 0 /\ Ar_0 >= 0 /\ Ar_1 >= 0 ] (Comp: 1, Cost: 1) evalloopsbb0in(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsbb6in(Ar_0, Ar_1, Ar_2)) [ 0 >= Ar_0 + 1 ] (Comp: 1, Cost: 1) evalloopsbb0in(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsbb1in(Ar_0, Ar_0, Ar_2)) [ Ar_0 >= 0 ] (Comp: 1, Cost: 1) evalloopsstart(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsbb0in(Ar_0, Ar_1, Ar_2)) start location: koat_start leaf cost: 0 A polynomial rank function with Pol(evalloopsbb4in) = 2*V_1 - 2*V_3 - 1 Pol(evalloopsbb3in) = 2*V_1 - 2*V_3 and size complexities S("evalloopsstart(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsbb0in(Ar_0, Ar_1, Ar_2))", 0-0) = Ar_0 S("evalloopsstart(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsbb0in(Ar_0, Ar_1, Ar_2))", 0-1) = Ar_1 S("evalloopsstart(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsbb0in(Ar_0, Ar_1, Ar_2))", 0-2) = Ar_2 S("evalloopsbb0in(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsbb1in(Ar_0, Ar_0, Ar_2)) [ Ar_0 >= 0 ]", 0-0) = Ar_0 S("evalloopsbb0in(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsbb1in(Ar_0, Ar_0, Ar_2)) [ Ar_0 >= 0 ]", 0-1) = Ar_0 S("evalloopsbb0in(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsbb1in(Ar_0, Ar_0, Ar_2)) [ Ar_0 >= 0 ]", 0-2) = Ar_2 S("evalloopsbb0in(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsbb6in(Ar_0, Ar_1, Ar_2)) [ 0 >= Ar_0 + 1 ]", 0-0) = Ar_0 S("evalloopsbb0in(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsbb6in(Ar_0, Ar_1, Ar_2)) [ 0 >= Ar_0 + 1 ]", 0-1) = Ar_1 S("evalloopsbb0in(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsbb6in(Ar_0, Ar_1, Ar_2)) [ 0 >= Ar_0 + 1 ]", 0-2) = Ar_2 S("evalloopsbb1in(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsbb2in(Ar_0, Ar_1, Ar_2)) [ Ar_0 - Ar_1 >= 0 /\\ Ar_0 >= 0 /\\ Ar_1 >= 0 ]", 0-0) = Ar_0 S("evalloopsbb1in(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsbb2in(Ar_0, Ar_1, Ar_2)) [ Ar_0 - Ar_1 >= 0 /\\ Ar_0 >= 0 /\\ Ar_1 >= 0 ]", 0-1) = Ar_0 + 2 S("evalloopsbb1in(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsbb2in(Ar_0, Ar_1, Ar_2)) [ Ar_0 - Ar_1 >= 0 /\\ Ar_0 >= 0 /\\ Ar_1 >= 0 ]", 0-2) = 2*Ar_0 + 2*Ar_2 + 16 S("evalloopsbb1in(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsbb6in(Ar_0, Ar_1, Ar_2)) [ Ar_0 - Ar_1 >= 0 /\\ Ar_0 >= 0 /\\ 0 >= Ar_1 + 1 ]", 0-0) = Ar_0 S("evalloopsbb1in(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsbb6in(Ar_0, Ar_1, Ar_2)) [ Ar_0 - Ar_1 >= 0 /\\ Ar_0 >= 0 /\\ 0 >= Ar_1 + 1 ]", 0-1) = Ar_0 + 2 S("evalloopsbb1in(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsbb6in(Ar_0, Ar_1, Ar_2)) [ Ar_0 - Ar_1 >= 0 /\\ Ar_0 >= 0 /\\ 0 >= Ar_1 + 1 ]", 0-2) = 2*Ar_0 + 2*Ar_2 + 32 S("evalloopsbb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsbb3in(Ar_0, Ar_1, 1)) [ Ar_0 - Ar_1 >= 0 /\\ Ar_1 >= 0 /\\ Ar_0 + Ar_1 >= 0 /\\ Ar_0 >= 0 /\\ Ar_1 >= 2 ]", 0-0) = Ar_0 S("evalloopsbb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsbb3in(Ar_0, Ar_1, 1)) [ Ar_0 - Ar_1 >= 0 /\\ Ar_1 >= 0 /\\ Ar_0 + Ar_1 >= 0 /\\ Ar_0 >= 0 /\\ Ar_1 >= 2 ]", 0-1) = Ar_0 + 2 S("evalloopsbb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsbb3in(Ar_0, Ar_1, 1)) [ Ar_0 - Ar_1 >= 0 /\\ Ar_1 >= 0 /\\ Ar_0 + Ar_1 >= 0 /\\ Ar_0 >= 0 /\\ Ar_1 >= 2 ]", 0-2) = 1 S("evalloopsbb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsbb5in(Ar_0, Ar_1, Ar_2)) [ Ar_0 - Ar_1 >= 0 /\\ Ar_1 >= 0 /\\ Ar_0 + Ar_1 >= 0 /\\ Ar_0 >= 0 /\\ 1 >= Ar_1 ]", 0-0) = Ar_0 S("evalloopsbb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsbb5in(Ar_0, Ar_1, Ar_2)) [ Ar_0 - Ar_1 >= 0 /\\ Ar_1 >= 0 /\\ Ar_0 + Ar_1 >= 0 /\\ Ar_0 >= 0 /\\ 1 >= Ar_1 ]", 0-1) = 1 S("evalloopsbb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsbb5in(Ar_0, Ar_1, Ar_2)) [ Ar_0 - Ar_1 >= 0 /\\ Ar_1 >= 0 /\\ Ar_0 + Ar_1 >= 0 /\\ Ar_0 >= 0 /\\ 1 >= Ar_1 ]", 0-2) = 2*Ar_0 + 2*Ar_2 + 16 S("evalloopsbb3in(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsbb4in(Ar_0, Ar_1, Ar_2)) [ Ar_2 - 1 >= 0 /\\ Ar_1 + Ar_2 - 3 >= 0 /\\ Ar_0 + Ar_2 - 3 >= 0 /\\ Ar_0 - Ar_1 >= 0 /\\ Ar_1 - 2 >= 0 /\\ Ar_0 + Ar_1 - 4 >= 0 /\\ Ar_0 - 2 >= 0 /\\ Ar_1 >= Ar_2 + 1 ]", 0-0) = Ar_0 S("evalloopsbb3in(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsbb4in(Ar_0, Ar_1, Ar_2)) [ Ar_2 - 1 >= 0 /\\ Ar_1 + Ar_2 - 3 >= 0 /\\ Ar_0 + Ar_2 - 3 >= 0 /\\ Ar_0 - Ar_1 >= 0 /\\ Ar_1 - 2 >= 0 /\\ Ar_0 + Ar_1 - 4 >= 0 /\\ Ar_0 - 2 >= 0 /\\ Ar_1 >= Ar_2 + 1 ]", 0-1) = Ar_0 + 2 S("evalloopsbb3in(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsbb4in(Ar_0, Ar_1, Ar_2)) [ Ar_2 - 1 >= 0 /\\ Ar_1 + Ar_2 - 3 >= 0 /\\ Ar_0 + Ar_2 - 3 >= 0 /\\ Ar_0 - Ar_1 >= 0 /\\ Ar_1 - 2 >= 0 /\\ Ar_0 + Ar_1 - 4 >= 0 /\\ Ar_0 - 2 >= 0 /\\ Ar_1 >= Ar_2 + 1 ]", 0-2) = 2*Ar_0 + 10 S("evalloopsbb3in(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsbb5in(Ar_0, Ar_1, Ar_2)) [ Ar_2 - 1 >= 0 /\\ Ar_1 + Ar_2 - 3 >= 0 /\\ Ar_0 + Ar_2 - 3 >= 0 /\\ Ar_0 - Ar_1 >= 0 /\\ Ar_1 - 2 >= 0 /\\ Ar_0 + Ar_1 - 4 >= 0 /\\ Ar_0 - 2 >= 0 /\\ Ar_2 >= Ar_1 ]", 0-0) = Ar_0 S("evalloopsbb3in(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsbb5in(Ar_0, Ar_1, Ar_2)) [ Ar_2 - 1 >= 0 /\\ Ar_1 + Ar_2 - 3 >= 0 /\\ Ar_0 + Ar_2 - 3 >= 0 /\\ Ar_0 - Ar_1 >= 0 /\\ Ar_1 - 2 >= 0 /\\ Ar_0 + Ar_1 - 4 >= 0 /\\ Ar_0 - 2 >= 0 /\\ Ar_2 >= Ar_1 ]", 0-1) = Ar_0 + 2 S("evalloopsbb3in(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsbb5in(Ar_0, Ar_1, Ar_2)) [ Ar_2 - 1 >= 0 /\\ Ar_1 + Ar_2 - 3 >= 0 /\\ Ar_0 + Ar_2 - 3 >= 0 /\\ Ar_0 - Ar_1 >= 0 /\\ Ar_1 - 2 >= 0 /\\ Ar_0 + Ar_1 - 4 >= 0 /\\ Ar_0 - 2 >= 0 /\\ Ar_2 >= Ar_1 ]", 0-2) = 2*Ar_0 + 8 S("evalloopsbb4in(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsbb3in(Ar_0, Ar_1, 2*Ar_2)) [ Ar_1 - Ar_2 - 1 >= 0 /\\ Ar_0 - Ar_2 - 1 >= 0 /\\ Ar_2 - 1 >= 0 /\\ Ar_1 + Ar_2 - 3 >= 0 /\\ Ar_0 + Ar_2 - 3 >= 0 /\\ Ar_0 - Ar_1 >= 0 /\\ Ar_1 - 2 >= 0 /\\ Ar_0 + Ar_1 - 4 >= 0 /\\ Ar_0 - 2 >= 0 ]", 0-0) = Ar_0 S("evalloopsbb4in(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsbb3in(Ar_0, Ar_1, 2*Ar_2)) [ Ar_1 - Ar_2 - 1 >= 0 /\\ Ar_0 - Ar_2 - 1 >= 0 /\\ Ar_2 - 1 >= 0 /\\ Ar_1 + Ar_2 - 3 >= 0 /\\ Ar_0 + Ar_2 - 3 >= 0 /\\ Ar_0 - Ar_1 >= 0 /\\ Ar_1 - 2 >= 0 /\\ Ar_0 + Ar_1 - 4 >= 0 /\\ Ar_0 - 2 >= 0 ]", 0-1) = Ar_0 + 2 S("evalloopsbb4in(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsbb3in(Ar_0, Ar_1, 2*Ar_2)) [ Ar_1 - Ar_2 - 1 >= 0 /\\ Ar_0 - Ar_2 - 1 >= 0 /\\ Ar_2 - 1 >= 0 /\\ Ar_1 + Ar_2 - 3 >= 0 /\\ Ar_0 + Ar_2 - 3 >= 0 /\\ Ar_0 - Ar_1 >= 0 /\\ Ar_1 - 2 >= 0 /\\ Ar_0 + Ar_1 - 4 >= 0 /\\ Ar_0 - 2 >= 0 ]", 0-2) = 2*Ar_0 + 4 S("evalloopsbb5in(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsbb1in(Ar_0, Ar_1 - 1, Ar_2)) [ Ar_0 - Ar_1 >= 0 /\\ Ar_1 >= 0 /\\ Ar_0 + Ar_1 >= 0 /\\ Ar_0 >= 0 ]", 0-0) = Ar_0 S("evalloopsbb5in(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsbb1in(Ar_0, Ar_1 - 1, Ar_2)) [ Ar_0 - Ar_1 >= 0 /\\ Ar_1 >= 0 /\\ Ar_0 + Ar_1 >= 0 /\\ Ar_0 >= 0 ]", 0-1) = Ar_0 + 2 S("evalloopsbb5in(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsbb1in(Ar_0, Ar_1 - 1, Ar_2)) [ Ar_0 - Ar_1 >= 0 /\\ Ar_1 >= 0 /\\ Ar_0 + Ar_1 >= 0 /\\ Ar_0 >= 0 ]", 0-2) = 2*Ar_0 + 2*Ar_2 + 16 S("evalloopsbb6in(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsstop(Ar_0, Ar_1, Ar_2))", 0-0) = Ar_0 S("evalloopsbb6in(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsstop(Ar_0, Ar_1, Ar_2))", 0-1) = Ar_0 + Ar_1 + 2 S("evalloopsbb6in(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsstop(Ar_0, Ar_1, Ar_2))", 0-2) = 2*Ar_0 + 2*Ar_2 + 64 S("koat_start(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsstart(Ar_0, Ar_1, Ar_2)) [ 0 <= 0 ]", 0-0) = Ar_0 S("koat_start(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsstart(Ar_0, Ar_1, Ar_2)) [ 0 <= 0 ]", 0-1) = Ar_1 S("koat_start(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsstart(Ar_0, Ar_1, Ar_2)) [ 0 <= 0 ]", 0-2) = Ar_2 orients the transitions evalloopsbb4in(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsbb3in(Ar_0, Ar_1, 2*Ar_2)) [ Ar_1 - Ar_2 - 1 >= 0 /\ Ar_0 - Ar_2 - 1 >= 0 /\ Ar_2 - 1 >= 0 /\ Ar_1 + Ar_2 - 3 >= 0 /\ Ar_0 + Ar_2 - 3 >= 0 /\ Ar_0 - Ar_1 >= 0 /\ Ar_1 - 2 >= 0 /\ Ar_0 + Ar_1 - 4 >= 0 /\ Ar_0 - 2 >= 0 ] evalloopsbb3in(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsbb4in(Ar_0, Ar_1, Ar_2)) [ Ar_2 - 1 >= 0 /\ Ar_1 + Ar_2 - 3 >= 0 /\ Ar_0 + Ar_2 - 3 >= 0 /\ Ar_0 - Ar_1 >= 0 /\ Ar_1 - 2 >= 0 /\ Ar_0 + Ar_1 - 4 >= 0 /\ Ar_0 - 2 >= 0 /\ Ar_1 >= Ar_2 + 1 ] weakly and the transitions evalloopsbb4in(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsbb3in(Ar_0, Ar_1, 2*Ar_2)) [ Ar_1 - Ar_2 - 1 >= 0 /\ Ar_0 - Ar_2 - 1 >= 0 /\ Ar_2 - 1 >= 0 /\ Ar_1 + Ar_2 - 3 >= 0 /\ Ar_0 + Ar_2 - 3 >= 0 /\ Ar_0 - Ar_1 >= 0 /\ Ar_1 - 2 >= 0 /\ Ar_0 + Ar_1 - 4 >= 0 /\ Ar_0 - 2 >= 0 ] evalloopsbb3in(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsbb4in(Ar_0, Ar_1, Ar_2)) [ Ar_2 - 1 >= 0 /\ Ar_1 + Ar_2 - 3 >= 0 /\ Ar_0 + Ar_2 - 3 >= 0 /\ Ar_0 - Ar_1 >= 0 /\ Ar_1 - 2 >= 0 /\ Ar_0 + Ar_1 - 4 >= 0 /\ Ar_0 - 2 >= 0 /\ Ar_1 >= Ar_2 + 1 ] strictly and produces the following problem: 8: T: (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsstart(Ar_0, Ar_1, Ar_2)) [ 0 <= 0 ] (Comp: 2, Cost: 1) evalloopsbb6in(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsstop(Ar_0, Ar_1, Ar_2)) (Comp: 6*Ar_0 + 6, Cost: 1) evalloopsbb5in(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsbb1in(Ar_0, Ar_1 - 1, Ar_2)) [ Ar_0 - Ar_1 >= 0 /\ Ar_1 >= 0 /\ Ar_0 + Ar_1 >= 0 /\ Ar_0 >= 0 ] (Comp: 4*Ar_0^2 + 8*Ar_0 + 4, Cost: 1) evalloopsbb4in(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsbb3in(Ar_0, Ar_1, 2*Ar_2)) [ Ar_1 - Ar_2 - 1 >= 0 /\ Ar_0 - Ar_2 - 1 >= 0 /\ Ar_2 - 1 >= 0 /\ Ar_1 + Ar_2 - 3 >= 0 /\ Ar_0 + Ar_2 - 3 >= 0 /\ Ar_0 - Ar_1 >= 0 /\ Ar_1 - 2 >= 0 /\ Ar_0 + Ar_1 - 4 >= 0 /\ Ar_0 - 2 >= 0 ] (Comp: 6*Ar_0 + 6, Cost: 1) evalloopsbb3in(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsbb5in(Ar_0, Ar_1, Ar_2)) [ Ar_2 - 1 >= 0 /\ Ar_1 + Ar_2 - 3 >= 0 /\ Ar_0 + Ar_2 - 3 >= 0 /\ Ar_0 - Ar_1 >= 0 /\ Ar_1 - 2 >= 0 /\ Ar_0 + Ar_1 - 4 >= 0 /\ Ar_0 - 2 >= 0 /\ Ar_2 >= Ar_1 ] (Comp: 4*Ar_0^2 + 8*Ar_0 + 4, Cost: 1) evalloopsbb3in(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsbb4in(Ar_0, Ar_1, Ar_2)) [ Ar_2 - 1 >= 0 /\ Ar_1 + Ar_2 - 3 >= 0 /\ Ar_0 + Ar_2 - 3 >= 0 /\ Ar_0 - Ar_1 >= 0 /\ Ar_1 - 2 >= 0 /\ Ar_0 + Ar_1 - 4 >= 0 /\ Ar_0 - 2 >= 0 /\ Ar_1 >= Ar_2 + 1 ] (Comp: 2*Ar_0 + 2, Cost: 1) evalloopsbb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsbb5in(Ar_0, Ar_1, Ar_2)) [ Ar_0 - Ar_1 >= 0 /\ Ar_1 >= 0 /\ Ar_0 + Ar_1 >= 0 /\ Ar_0 >= 0 /\ 1 >= Ar_1 ] (Comp: 2*Ar_0 + 2, Cost: 1) evalloopsbb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsbb3in(Ar_0, Ar_1, 1)) [ Ar_0 - Ar_1 >= 0 /\ Ar_1 >= 0 /\ Ar_0 + Ar_1 >= 0 /\ Ar_0 >= 0 /\ Ar_1 >= 2 ] (Comp: 2, Cost: 1) evalloopsbb1in(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsbb6in(Ar_0, Ar_1, Ar_2)) [ Ar_0 - Ar_1 >= 0 /\ Ar_0 >= 0 /\ 0 >= Ar_1 + 1 ] (Comp: 2*Ar_0 + 2, Cost: 1) evalloopsbb1in(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsbb2in(Ar_0, Ar_1, Ar_2)) [ Ar_0 - Ar_1 >= 0 /\ Ar_0 >= 0 /\ Ar_1 >= 0 ] (Comp: 1, Cost: 1) evalloopsbb0in(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsbb6in(Ar_0, Ar_1, Ar_2)) [ 0 >= Ar_0 + 1 ] (Comp: 1, Cost: 1) evalloopsbb0in(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsbb1in(Ar_0, Ar_0, Ar_2)) [ Ar_0 >= 0 ] (Comp: 1, Cost: 1) evalloopsstart(Ar_0, Ar_1, Ar_2) -> Com_1(evalloopsbb0in(Ar_0, Ar_1, Ar_2)) start location: koat_start leaf cost: 0 Complexity upper bound 34*Ar_0 + 8*Ar_0^2 + 33 Time: 0.231 sec (SMT: 0.190 sec) ---------------------------------------- (2) BOUNDS(1, n^2)