5.63/2.98 WORST_CASE(Omega(n^1), O(n^2)) 5.76/2.99 proof of /export/starexec/sandbox/benchmark/theBenchmark.koat 5.76/2.99 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 5.76/2.99 5.76/2.99 5.76/2.99 The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(n^1, n^2). 5.76/2.99 5.76/2.99 (0) CpxIntTrs 5.76/2.99 (1) Koat Proof [FINISHED, 318 ms] 5.76/2.99 (2) BOUNDS(1, n^2) 5.76/2.99 (3) Loat Proof [FINISHED, 611 ms] 5.76/2.99 (4) BOUNDS(n^1, INF) 5.76/2.99 5.76/2.99 5.76/2.99 ---------------------------------------- 5.76/2.99 5.76/2.99 (0) 5.76/2.99 Obligation: 5.76/2.99 Complexity Int TRS consisting of the following rules: 5.76/2.99 eval_start_start(v_n, v_x_0, v_y_0) -> Com_1(eval_start_bb0_in(v_n, v_x_0, v_y_0)) :|: TRUE 5.76/2.99 eval_start_bb0_in(v_n, v_x_0, v_y_0) -> Com_1(eval_start_0(v_n, v_x_0, v_y_0)) :|: TRUE 5.76/2.99 eval_start_0(v_n, v_x_0, v_y_0) -> Com_1(eval_start_1(v_n, v_x_0, v_y_0)) :|: TRUE 5.76/2.99 eval_start_1(v_n, v_x_0, v_y_0) -> Com_1(eval_start_2(v_n, v_x_0, v_y_0)) :|: TRUE 5.76/2.99 eval_start_2(v_n, v_x_0, v_y_0) -> Com_1(eval_start_3(v_n, v_x_0, v_y_0)) :|: TRUE 5.76/2.99 eval_start_3(v_n, v_x_0, v_y_0) -> Com_1(eval_start_4(v_n, v_x_0, v_y_0)) :|: TRUE 5.76/2.99 eval_start_4(v_n, v_x_0, v_y_0) -> Com_1(eval_start_5(v_n, v_x_0, v_y_0)) :|: TRUE 5.76/2.99 eval_start_5(v_n, v_x_0, v_y_0) -> Com_1(eval_start_6(v_n, v_x_0, v_y_0)) :|: TRUE 5.76/2.99 eval_start_6(v_n, v_x_0, v_y_0) -> Com_1(eval_start_bb1_in(v_n, 0, 0)) :|: TRUE 5.76/2.99 eval_start_bb1_in(v_n, v_x_0, v_y_0) -> Com_1(eval_start_bb2_in(v_n, v_x_0, v_y_0)) :|: v_x_0 < v_n 5.76/2.99 eval_start_bb1_in(v_n, v_x_0, v_y_0) -> Com_1(eval_start_bb3_in(v_n, v_x_0, v_y_0)) :|: v_x_0 >= v_n 5.76/2.99 eval_start_bb2_in(v_n, v_x_0, v_y_0) -> Com_1(eval_start_bb1_in(v_n, v_x_0 + 1, v_y_0 + 1)) :|: TRUE 5.76/2.99 eval_start_bb3_in(v_n, v_x_0, v_y_0) -> Com_1(eval_start_bb4_in(v_n, v_x_0, v_y_0)) :|: v_y_0 > 0 5.76/2.99 eval_start_bb3_in(v_n, v_x_0, v_y_0) -> Com_1(eval_start_bb5_in(v_n, v_x_0, v_y_0)) :|: v_y_0 <= 0 5.76/2.99 eval_start_bb4_in(v_n, v_x_0, v_y_0) -> Com_1(eval_start_bb1_in(v_n, v_x_0, v_y_0 - 1)) :|: TRUE 5.76/2.99 eval_start_bb5_in(v_n, v_x_0, v_y_0) -> Com_1(eval_start_stop(v_n, v_x_0, v_y_0)) :|: TRUE 5.76/2.99 5.76/2.99 The start-symbols are:[eval_start_start_3] 5.76/2.99 5.76/2.99 5.76/2.99 ---------------------------------------- 5.76/2.99 5.76/2.99 (1) Koat Proof (FINISHED) 5.76/2.99 YES(?, 9*ar_2^2 + 8*ar_2 + 19) 5.76/2.99 5.76/2.99 5.76/2.99 5.76/2.99 Initial complexity problem: 5.76/2.99 5.76/2.99 1: T: 5.76/2.99 5.76/2.99 (Comp: ?, Cost: 1) evalstartstart(ar_0, ar_1, ar_2) -> Com_1(evalstartbb0in(ar_0, ar_1, ar_2)) 5.76/2.99 5.76/2.99 (Comp: ?, Cost: 1) evalstartbb0in(ar_0, ar_1, ar_2) -> Com_1(evalstart0(ar_0, ar_1, ar_2)) 5.76/2.99 5.76/2.99 (Comp: ?, Cost: 1) evalstart0(ar_0, ar_1, ar_2) -> Com_1(evalstart1(ar_0, ar_1, ar_2)) 5.76/2.99 5.76/2.99 (Comp: ?, Cost: 1) evalstart1(ar_0, ar_1, ar_2) -> Com_1(evalstart2(ar_0, ar_1, ar_2)) 5.76/2.99 5.76/2.99 (Comp: ?, Cost: 1) evalstart2(ar_0, ar_1, ar_2) -> Com_1(evalstart3(ar_0, ar_1, ar_2)) 5.76/2.99 5.76/2.99 (Comp: ?, Cost: 1) evalstart3(ar_0, ar_1, ar_2) -> Com_1(evalstart4(ar_0, ar_1, ar_2)) 5.76/2.99 5.76/2.99 (Comp: ?, Cost: 1) evalstart4(ar_0, ar_1, ar_2) -> Com_1(evalstart5(ar_0, ar_1, ar_2)) 5.76/2.99 5.76/2.99 (Comp: ?, Cost: 1) evalstart5(ar_0, ar_1, ar_2) -> Com_1(evalstart6(ar_0, ar_1, ar_2)) 5.76/2.99 5.76/2.99 (Comp: ?, Cost: 1) evalstart6(ar_0, ar_1, ar_2) -> Com_1(evalstartbb1in(0, 0, ar_2)) 5.76/2.99 5.76/2.99 (Comp: ?, Cost: 1) evalstartbb1in(ar_0, ar_1, ar_2) -> Com_1(evalstartbb2in(ar_0, ar_1, ar_2)) [ ar_2 >= ar_0 + 1 ] 5.76/2.99 5.76/2.99 (Comp: ?, Cost: 1) evalstartbb1in(ar_0, ar_1, ar_2) -> Com_1(evalstartbb3in(ar_0, ar_1, ar_2)) [ ar_0 >= ar_2 ] 5.76/2.99 5.76/2.99 (Comp: ?, Cost: 1) evalstartbb2in(ar_0, ar_1, ar_2) -> Com_1(evalstartbb1in(ar_0 + 1, ar_1 + 1, ar_2)) 5.76/2.99 5.76/2.99 (Comp: ?, Cost: 1) evalstartbb3in(ar_0, ar_1, ar_2) -> Com_1(evalstartbb4in(ar_0, ar_1, ar_2)) [ ar_1 >= 1 ] 5.76/2.99 5.76/2.99 (Comp: ?, Cost: 1) evalstartbb3in(ar_0, ar_1, ar_2) -> Com_1(evalstartbb5in(ar_0, ar_1, ar_2)) [ 0 >= ar_1 ] 5.76/2.99 5.76/2.99 (Comp: ?, Cost: 1) evalstartbb4in(ar_0, ar_1, ar_2) -> Com_1(evalstartbb1in(ar_0, ar_1 - 1, ar_2)) 5.76/2.99 5.76/2.99 (Comp: ?, Cost: 1) evalstartbb5in(ar_0, ar_1, ar_2) -> Com_1(evalstartstop(ar_0, ar_1, ar_2)) 5.76/2.99 5.76/2.99 (Comp: 1, Cost: 0) koat_start(ar_0, ar_1, ar_2) -> Com_1(evalstartstart(ar_0, ar_1, ar_2)) [ 0 <= 0 ] 5.76/2.99 5.76/2.99 start location: koat_start 5.76/2.99 5.76/2.99 leaf cost: 0 5.76/2.99 5.76/2.99 5.76/2.99 5.76/2.99 Repeatedly propagating knowledge in problem 1 produces the following problem: 5.76/2.99 5.76/2.99 2: T: 5.76/2.99 5.76/2.99 (Comp: 1, Cost: 1) evalstartstart(ar_0, ar_1, ar_2) -> Com_1(evalstartbb0in(ar_0, ar_1, ar_2)) 5.76/2.99 5.76/2.99 (Comp: 1, Cost: 1) evalstartbb0in(ar_0, ar_1, ar_2) -> Com_1(evalstart0(ar_0, ar_1, ar_2)) 5.76/2.99 5.76/2.99 (Comp: 1, Cost: 1) evalstart0(ar_0, ar_1, ar_2) -> Com_1(evalstart1(ar_0, ar_1, ar_2)) 5.76/2.99 5.76/2.99 (Comp: 1, Cost: 1) evalstart1(ar_0, ar_1, ar_2) -> Com_1(evalstart2(ar_0, ar_1, ar_2)) 5.76/2.99 5.76/2.99 (Comp: 1, Cost: 1) evalstart2(ar_0, ar_1, ar_2) -> Com_1(evalstart3(ar_0, ar_1, ar_2)) 5.76/2.99 5.76/2.99 (Comp: 1, Cost: 1) evalstart3(ar_0, ar_1, ar_2) -> Com_1(evalstart4(ar_0, ar_1, ar_2)) 5.76/2.99 5.76/2.99 (Comp: 1, Cost: 1) evalstart4(ar_0, ar_1, ar_2) -> Com_1(evalstart5(ar_0, ar_1, ar_2)) 5.76/2.99 5.76/2.99 (Comp: 1, Cost: 1) evalstart5(ar_0, ar_1, ar_2) -> Com_1(evalstart6(ar_0, ar_1, ar_2)) 5.76/2.99 5.76/2.99 (Comp: 1, Cost: 1) evalstart6(ar_0, ar_1, ar_2) -> Com_1(evalstartbb1in(0, 0, ar_2)) 5.76/2.99 5.76/2.99 (Comp: ?, Cost: 1) evalstartbb1in(ar_0, ar_1, ar_2) -> Com_1(evalstartbb2in(ar_0, ar_1, ar_2)) [ ar_2 >= ar_0 + 1 ] 5.76/2.99 5.76/2.99 (Comp: ?, Cost: 1) evalstartbb1in(ar_0, ar_1, ar_2) -> Com_1(evalstartbb3in(ar_0, ar_1, ar_2)) [ ar_0 >= ar_2 ] 5.76/2.99 5.76/2.99 (Comp: ?, Cost: 1) evalstartbb2in(ar_0, ar_1, ar_2) -> Com_1(evalstartbb1in(ar_0 + 1, ar_1 + 1, ar_2)) 5.76/2.99 5.76/2.99 (Comp: ?, Cost: 1) evalstartbb3in(ar_0, ar_1, ar_2) -> Com_1(evalstartbb4in(ar_0, ar_1, ar_2)) [ ar_1 >= 1 ] 5.76/2.99 5.76/2.99 (Comp: ?, Cost: 1) evalstartbb3in(ar_0, ar_1, ar_2) -> Com_1(evalstartbb5in(ar_0, ar_1, ar_2)) [ 0 >= ar_1 ] 5.76/2.99 5.76/2.99 (Comp: ?, Cost: 1) evalstartbb4in(ar_0, ar_1, ar_2) -> Com_1(evalstartbb1in(ar_0, ar_1 - 1, ar_2)) 5.76/2.99 5.76/2.99 (Comp: ?, Cost: 1) evalstartbb5in(ar_0, ar_1, ar_2) -> Com_1(evalstartstop(ar_0, ar_1, ar_2)) 5.76/2.99 5.76/2.99 (Comp: 1, Cost: 0) koat_start(ar_0, ar_1, ar_2) -> Com_1(evalstartstart(ar_0, ar_1, ar_2)) [ 0 <= 0 ] 5.76/2.99 5.76/2.99 start location: koat_start 5.76/2.99 5.76/2.99 leaf cost: 0 5.76/2.99 5.76/2.99 5.76/2.99 5.76/2.99 A polynomial rank function with 5.76/2.99 5.76/2.99 Pol(evalstartstart) = 2 5.76/2.99 5.76/2.99 Pol(evalstartbb0in) = 2 5.76/2.99 5.76/2.99 Pol(evalstart0) = 2 5.76/2.99 5.76/2.99 Pol(evalstart1) = 2 5.76/2.99 5.76/2.99 Pol(evalstart2) = 2 5.76/2.99 5.76/2.99 Pol(evalstart3) = 2 5.76/2.99 5.76/2.99 Pol(evalstart4) = 2 5.76/2.99 5.76/2.99 Pol(evalstart5) = 2 5.76/2.99 5.76/2.99 Pol(evalstart6) = 2 5.76/2.99 5.76/2.99 Pol(evalstartbb1in) = 2 5.76/2.99 5.76/2.99 Pol(evalstartbb2in) = 2 5.76/2.99 5.76/2.99 Pol(evalstartbb3in) = 2 5.76/2.99 5.76/2.99 Pol(evalstartbb4in) = 2 5.76/2.99 5.76/2.99 Pol(evalstartbb5in) = 1 5.76/2.99 5.76/2.99 Pol(evalstartstop) = 0 5.76/2.99 5.76/2.99 Pol(koat_start) = 2 5.76/2.99 5.76/2.99 orients all transitions weakly and the transitions 5.76/2.99 5.76/2.99 evalstartbb5in(ar_0, ar_1, ar_2) -> Com_1(evalstartstop(ar_0, ar_1, ar_2)) 5.76/2.99 5.76/2.99 evalstartbb3in(ar_0, ar_1, ar_2) -> Com_1(evalstartbb5in(ar_0, ar_1, ar_2)) [ 0 >= ar_1 ] 5.76/2.99 5.76/2.99 strictly and produces the following problem: 5.76/2.99 5.76/2.99 3: T: 5.76/2.99 5.76/2.99 (Comp: 1, Cost: 1) evalstartstart(ar_0, ar_1, ar_2) -> Com_1(evalstartbb0in(ar_0, ar_1, ar_2)) 5.76/2.99 5.76/2.99 (Comp: 1, Cost: 1) evalstartbb0in(ar_0, ar_1, ar_2) -> Com_1(evalstart0(ar_0, ar_1, ar_2)) 5.76/2.99 5.76/2.99 (Comp: 1, Cost: 1) evalstart0(ar_0, ar_1, ar_2) -> Com_1(evalstart1(ar_0, ar_1, ar_2)) 5.76/2.99 5.76/2.99 (Comp: 1, Cost: 1) evalstart1(ar_0, ar_1, ar_2) -> Com_1(evalstart2(ar_0, ar_1, ar_2)) 5.76/2.99 5.76/2.99 (Comp: 1, Cost: 1) evalstart2(ar_0, ar_1, ar_2) -> Com_1(evalstart3(ar_0, ar_1, ar_2)) 5.76/2.99 5.76/2.99 (Comp: 1, Cost: 1) evalstart3(ar_0, ar_1, ar_2) -> Com_1(evalstart4(ar_0, ar_1, ar_2)) 5.76/2.99 5.76/2.99 (Comp: 1, Cost: 1) evalstart4(ar_0, ar_1, ar_2) -> Com_1(evalstart5(ar_0, ar_1, ar_2)) 5.76/2.99 5.76/2.99 (Comp: 1, Cost: 1) evalstart5(ar_0, ar_1, ar_2) -> Com_1(evalstart6(ar_0, ar_1, ar_2)) 5.76/2.99 5.76/2.99 (Comp: 1, Cost: 1) evalstart6(ar_0, ar_1, ar_2) -> Com_1(evalstartbb1in(0, 0, ar_2)) 5.76/2.99 5.76/2.99 (Comp: ?, Cost: 1) evalstartbb1in(ar_0, ar_1, ar_2) -> Com_1(evalstartbb2in(ar_0, ar_1, ar_2)) [ ar_2 >= ar_0 + 1 ] 5.76/2.99 5.76/2.99 (Comp: ?, Cost: 1) evalstartbb1in(ar_0, ar_1, ar_2) -> Com_1(evalstartbb3in(ar_0, ar_1, ar_2)) [ ar_0 >= ar_2 ] 5.76/2.99 5.76/2.99 (Comp: ?, Cost: 1) evalstartbb2in(ar_0, ar_1, ar_2) -> Com_1(evalstartbb1in(ar_0 + 1, ar_1 + 1, ar_2)) 5.76/2.99 5.76/2.99 (Comp: ?, Cost: 1) evalstartbb3in(ar_0, ar_1, ar_2) -> Com_1(evalstartbb4in(ar_0, ar_1, ar_2)) [ ar_1 >= 1 ] 5.76/2.99 5.76/2.99 (Comp: 2, Cost: 1) evalstartbb3in(ar_0, ar_1, ar_2) -> Com_1(evalstartbb5in(ar_0, ar_1, ar_2)) [ 0 >= ar_1 ] 5.76/2.99 5.76/2.99 (Comp: ?, Cost: 1) evalstartbb4in(ar_0, ar_1, ar_2) -> Com_1(evalstartbb1in(ar_0, ar_1 - 1, ar_2)) 5.76/2.99 5.76/2.99 (Comp: 2, Cost: 1) evalstartbb5in(ar_0, ar_1, ar_2) -> Com_1(evalstartstop(ar_0, ar_1, ar_2)) 5.76/2.99 5.76/2.99 (Comp: 1, Cost: 0) koat_start(ar_0, ar_1, ar_2) -> Com_1(evalstartstart(ar_0, ar_1, ar_2)) [ 0 <= 0 ] 5.76/2.99 5.76/2.99 start location: koat_start 5.76/2.99 5.76/2.99 leaf cost: 0 5.76/2.99 5.76/2.99 5.76/2.99 5.76/2.99 A polynomial rank function with 5.76/2.99 5.76/2.99 Pol(evalstartstart) = V_3 5.76/2.99 5.76/2.99 Pol(evalstartbb0in) = V_3 5.76/2.99 5.76/2.99 Pol(evalstart0) = V_3 5.76/2.99 5.76/2.99 Pol(evalstart1) = V_3 5.76/2.99 5.76/2.99 Pol(evalstart2) = V_3 5.76/2.99 5.76/2.99 Pol(evalstart3) = V_3 5.76/2.99 5.76/2.99 Pol(evalstart4) = V_3 5.76/2.99 5.76/2.99 Pol(evalstart5) = V_3 5.76/2.99 5.76/2.99 Pol(evalstart6) = V_3 5.76/2.99 5.76/2.99 Pol(evalstartbb1in) = -V_1 + V_3 5.76/2.99 5.76/2.99 Pol(evalstartbb2in) = -V_1 + V_3 - 1 5.76/2.99 5.76/2.99 Pol(evalstartbb3in) = -V_1 + V_3 5.76/2.99 5.76/2.99 Pol(evalstartbb4in) = -V_1 + V_3 5.76/2.99 5.76/2.99 Pol(evalstartbb5in) = -V_1 + V_3 5.76/2.99 5.76/2.99 Pol(evalstartstop) = -V_1 + V_3 5.76/2.99 5.76/2.99 Pol(koat_start) = V_3 5.76/2.99 5.76/2.99 orients all transitions weakly and the transition 5.76/2.99 5.76/2.99 evalstartbb1in(ar_0, ar_1, ar_2) -> Com_1(evalstartbb2in(ar_0, ar_1, ar_2)) [ ar_2 >= ar_0 + 1 ] 5.76/2.99 5.76/2.99 strictly and produces the following problem: 5.76/2.99 5.76/2.99 4: T: 5.76/2.99 5.76/2.99 (Comp: 1, Cost: 1) evalstartstart(ar_0, ar_1, ar_2) -> Com_1(evalstartbb0in(ar_0, ar_1, ar_2)) 5.76/2.99 5.76/2.99 (Comp: 1, Cost: 1) evalstartbb0in(ar_0, ar_1, ar_2) -> Com_1(evalstart0(ar_0, ar_1, ar_2)) 5.76/2.99 5.76/2.99 (Comp: 1, Cost: 1) evalstart0(ar_0, ar_1, ar_2) -> Com_1(evalstart1(ar_0, ar_1, ar_2)) 5.76/2.99 5.76/2.99 (Comp: 1, Cost: 1) evalstart1(ar_0, ar_1, ar_2) -> Com_1(evalstart2(ar_0, ar_1, ar_2)) 5.76/2.99 5.76/2.99 (Comp: 1, Cost: 1) evalstart2(ar_0, ar_1, ar_2) -> Com_1(evalstart3(ar_0, ar_1, ar_2)) 5.76/2.99 5.76/2.99 (Comp: 1, Cost: 1) evalstart3(ar_0, ar_1, ar_2) -> Com_1(evalstart4(ar_0, ar_1, ar_2)) 5.76/2.99 5.76/2.99 (Comp: 1, Cost: 1) evalstart4(ar_0, ar_1, ar_2) -> Com_1(evalstart5(ar_0, ar_1, ar_2)) 5.76/2.99 5.76/2.99 (Comp: 1, Cost: 1) evalstart5(ar_0, ar_1, ar_2) -> Com_1(evalstart6(ar_0, ar_1, ar_2)) 5.76/2.99 5.76/2.99 (Comp: 1, Cost: 1) evalstart6(ar_0, ar_1, ar_2) -> Com_1(evalstartbb1in(0, 0, ar_2)) 5.76/2.99 5.76/2.99 (Comp: ar_2, Cost: 1) evalstartbb1in(ar_0, ar_1, ar_2) -> Com_1(evalstartbb2in(ar_0, ar_1, ar_2)) [ ar_2 >= ar_0 + 1 ] 5.76/2.99 5.76/2.99 (Comp: ?, Cost: 1) evalstartbb1in(ar_0, ar_1, ar_2) -> Com_1(evalstartbb3in(ar_0, ar_1, ar_2)) [ ar_0 >= ar_2 ] 5.76/2.99 5.76/2.99 (Comp: ?, Cost: 1) evalstartbb2in(ar_0, ar_1, ar_2) -> Com_1(evalstartbb1in(ar_0 + 1, ar_1 + 1, ar_2)) 5.76/2.99 5.76/2.99 (Comp: ?, Cost: 1) evalstartbb3in(ar_0, ar_1, ar_2) -> Com_1(evalstartbb4in(ar_0, ar_1, ar_2)) [ ar_1 >= 1 ] 5.76/2.99 5.76/2.99 (Comp: 2, Cost: 1) evalstartbb3in(ar_0, ar_1, ar_2) -> Com_1(evalstartbb5in(ar_0, ar_1, ar_2)) [ 0 >= ar_1 ] 5.76/2.99 5.76/2.99 (Comp: ?, Cost: 1) evalstartbb4in(ar_0, ar_1, ar_2) -> Com_1(evalstartbb1in(ar_0, ar_1 - 1, ar_2)) 5.76/2.99 5.76/2.99 (Comp: 2, Cost: 1) evalstartbb5in(ar_0, ar_1, ar_2) -> Com_1(evalstartstop(ar_0, ar_1, ar_2)) 5.76/2.99 5.76/2.99 (Comp: 1, Cost: 0) koat_start(ar_0, ar_1, ar_2) -> Com_1(evalstartstart(ar_0, ar_1, ar_2)) [ 0 <= 0 ] 5.76/2.99 5.76/2.99 start location: koat_start 5.76/2.99 5.76/2.99 leaf cost: 0 5.76/2.99 5.76/2.99 5.76/2.99 5.76/2.99 Repeatedly propagating knowledge in problem 4 produces the following problem: 5.76/2.99 5.76/2.99 5: T: 5.76/2.99 5.76/2.99 (Comp: 1, Cost: 1) evalstartstart(ar_0, ar_1, ar_2) -> Com_1(evalstartbb0in(ar_0, ar_1, ar_2)) 5.76/2.99 5.76/2.99 (Comp: 1, Cost: 1) evalstartbb0in(ar_0, ar_1, ar_2) -> Com_1(evalstart0(ar_0, ar_1, ar_2)) 5.76/2.99 5.76/2.99 (Comp: 1, Cost: 1) evalstart0(ar_0, ar_1, ar_2) -> Com_1(evalstart1(ar_0, ar_1, ar_2)) 5.76/2.99 5.76/2.99 (Comp: 1, Cost: 1) evalstart1(ar_0, ar_1, ar_2) -> Com_1(evalstart2(ar_0, ar_1, ar_2)) 5.76/2.99 5.76/2.99 (Comp: 1, Cost: 1) evalstart2(ar_0, ar_1, ar_2) -> Com_1(evalstart3(ar_0, ar_1, ar_2)) 5.76/2.99 5.76/2.99 (Comp: 1, Cost: 1) evalstart3(ar_0, ar_1, ar_2) -> Com_1(evalstart4(ar_0, ar_1, ar_2)) 5.76/2.99 5.76/2.99 (Comp: 1, Cost: 1) evalstart4(ar_0, ar_1, ar_2) -> Com_1(evalstart5(ar_0, ar_1, ar_2)) 5.76/2.99 5.76/2.99 (Comp: 1, Cost: 1) evalstart5(ar_0, ar_1, ar_2) -> Com_1(evalstart6(ar_0, ar_1, ar_2)) 5.76/2.99 5.76/2.99 (Comp: 1, Cost: 1) evalstart6(ar_0, ar_1, ar_2) -> Com_1(evalstartbb1in(0, 0, ar_2)) 5.76/2.99 5.76/2.99 (Comp: ar_2, Cost: 1) evalstartbb1in(ar_0, ar_1, ar_2) -> Com_1(evalstartbb2in(ar_0, ar_1, ar_2)) [ ar_2 >= ar_0 + 1 ] 5.76/2.99 5.76/2.99 (Comp: ?, Cost: 1) evalstartbb1in(ar_0, ar_1, ar_2) -> Com_1(evalstartbb3in(ar_0, ar_1, ar_2)) [ ar_0 >= ar_2 ] 5.76/2.99 5.76/2.99 (Comp: ar_2, Cost: 1) evalstartbb2in(ar_0, ar_1, ar_2) -> Com_1(evalstartbb1in(ar_0 + 1, ar_1 + 1, ar_2)) 5.76/2.99 5.76/2.99 (Comp: ?, Cost: 1) evalstartbb3in(ar_0, ar_1, ar_2) -> Com_1(evalstartbb4in(ar_0, ar_1, ar_2)) [ ar_1 >= 1 ] 5.76/2.99 5.76/2.99 (Comp: 2, Cost: 1) evalstartbb3in(ar_0, ar_1, ar_2) -> Com_1(evalstartbb5in(ar_0, ar_1, ar_2)) [ 0 >= ar_1 ] 5.76/2.99 5.76/2.99 (Comp: ?, Cost: 1) evalstartbb4in(ar_0, ar_1, ar_2) -> Com_1(evalstartbb1in(ar_0, ar_1 - 1, ar_2)) 5.76/2.99 5.76/2.99 (Comp: 2, Cost: 1) evalstartbb5in(ar_0, ar_1, ar_2) -> Com_1(evalstartstop(ar_0, ar_1, ar_2)) 5.76/2.99 5.76/2.99 (Comp: 1, Cost: 0) koat_start(ar_0, ar_1, ar_2) -> Com_1(evalstartstart(ar_0, ar_1, ar_2)) [ 0 <= 0 ] 5.76/2.99 5.76/2.99 start location: koat_start 5.76/2.99 5.76/2.99 leaf cost: 0 5.76/2.99 5.76/2.99 5.76/2.99 5.76/2.99 Applied AI with 'oct' on problem 5 to obtain the following invariants: 5.76/2.99 5.76/2.99 For symbol evalstartbb1in: X_1 - X_2 >= 0 /\ X_2 >= 0 /\ X_1 + X_2 >= 0 /\ X_1 >= 0 5.76/2.99 5.76/2.99 For symbol evalstartbb2in: 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 5.76/2.99 5.76/2.99 For symbol evalstartbb3in: X_1 - X_3 >= 0 /\ X_1 - X_2 >= 0 /\ X_2 >= 0 /\ X_1 + X_2 >= 0 /\ X_1 >= 0 5.76/2.99 5.76/2.99 For symbol evalstartbb4in: X_1 - X_3 >= 0 /\ X_1 - X_2 >= 0 /\ X_2 - 1 >= 0 /\ X_1 + X_2 - 2 >= 0 /\ X_1 - 1 >= 0 5.76/2.99 5.76/2.99 For symbol evalstartbb5in: X_1 - X_3 >= 0 /\ -X_2 >= 0 /\ X_1 - X_2 >= 0 /\ X_2 >= 0 /\ X_1 + X_2 >= 0 /\ X_1 >= 0 5.76/2.99 5.76/2.99 5.76/2.99 5.76/2.99 5.76/2.99 5.76/2.99 This yielded the following problem: 5.76/2.99 5.76/2.99 6: T: 5.76/2.99 5.76/2.99 (Comp: 1, Cost: 0) koat_start(ar_0, ar_1, ar_2) -> Com_1(evalstartstart(ar_0, ar_1, ar_2)) [ 0 <= 0 ] 5.76/2.99 5.76/2.99 (Comp: 2, Cost: 1) evalstartbb5in(ar_0, ar_1, ar_2) -> Com_1(evalstartstop(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 ] 5.76/2.99 5.76/2.99 (Comp: ?, Cost: 1) evalstartbb4in(ar_0, ar_1, ar_2) -> Com_1(evalstartbb1in(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 ] 5.76/2.99 5.76/2.99 (Comp: 2, Cost: 1) evalstartbb3in(ar_0, ar_1, ar_2) -> Com_1(evalstartbb5in(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 ] 5.76/2.99 5.76/2.99 (Comp: ?, Cost: 1) evalstartbb3in(ar_0, ar_1, ar_2) -> Com_1(evalstartbb4in(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 ] 5.76/2.99 5.76/2.99 (Comp: ar_2, Cost: 1) evalstartbb2in(ar_0, ar_1, ar_2) -> Com_1(evalstartbb1in(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 ] 5.76/2.99 5.76/2.99 (Comp: ?, Cost: 1) evalstartbb1in(ar_0, ar_1, ar_2) -> Com_1(evalstartbb3in(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 ] 5.76/2.99 5.76/2.99 (Comp: ar_2, Cost: 1) evalstartbb1in(ar_0, ar_1, ar_2) -> Com_1(evalstartbb2in(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 ] 5.76/2.99 5.76/2.99 (Comp: 1, Cost: 1) evalstart6(ar_0, ar_1, ar_2) -> Com_1(evalstartbb1in(0, 0, ar_2)) 5.76/2.99 5.76/2.99 (Comp: 1, Cost: 1) evalstart5(ar_0, ar_1, ar_2) -> Com_1(evalstart6(ar_0, ar_1, ar_2)) 5.76/2.99 5.76/2.99 (Comp: 1, Cost: 1) evalstart4(ar_0, ar_1, ar_2) -> Com_1(evalstart5(ar_0, ar_1, ar_2)) 5.76/2.99 5.76/2.99 (Comp: 1, Cost: 1) evalstart3(ar_0, ar_1, ar_2) -> Com_1(evalstart4(ar_0, ar_1, ar_2)) 5.76/2.99 5.76/2.99 (Comp: 1, Cost: 1) evalstart2(ar_0, ar_1, ar_2) -> Com_1(evalstart3(ar_0, ar_1, ar_2)) 5.76/2.99 5.76/2.99 (Comp: 1, Cost: 1) evalstart1(ar_0, ar_1, ar_2) -> Com_1(evalstart2(ar_0, ar_1, ar_2)) 5.76/2.99 5.76/2.99 (Comp: 1, Cost: 1) evalstart0(ar_0, ar_1, ar_2) -> Com_1(evalstart1(ar_0, ar_1, ar_2)) 5.76/2.99 5.76/2.99 (Comp: 1, Cost: 1) evalstartbb0in(ar_0, ar_1, ar_2) -> Com_1(evalstart0(ar_0, ar_1, ar_2)) 5.76/2.99 5.76/2.99 (Comp: 1, Cost: 1) evalstartstart(ar_0, ar_1, ar_2) -> Com_1(evalstartbb0in(ar_0, ar_1, ar_2)) 5.76/2.99 5.76/2.99 start location: koat_start 5.76/2.99 5.76/2.99 leaf cost: 0 5.76/2.99 5.76/2.99 5.76/2.99 5.76/2.99 A polynomial rank function with 5.76/2.99 5.76/2.99 Pol(evalstartbb4in) = 3*V_2 5.76/2.99 5.76/2.99 Pol(evalstartbb1in) = 3*V_2 + 2 5.76/2.99 5.76/2.99 Pol(evalstartbb3in) = 3*V_2 + 1 5.76/2.99 5.76/2.99 and size complexities 5.76/2.99 5.76/2.99 S("evalstartstart(ar_0, ar_1, ar_2) -> Com_1(evalstartbb0in(ar_0, ar_1, ar_2))", 0-0) = ar_0 5.76/2.99 5.76/2.99 S("evalstartstart(ar_0, ar_1, ar_2) -> Com_1(evalstartbb0in(ar_0, ar_1, ar_2))", 0-1) = ar_1 5.76/2.99 5.76/2.99 S("evalstartstart(ar_0, ar_1, ar_2) -> Com_1(evalstartbb0in(ar_0, ar_1, ar_2))", 0-2) = ar_2 5.76/2.99 5.76/2.99 S("evalstartbb0in(ar_0, ar_1, ar_2) -> Com_1(evalstart0(ar_0, ar_1, ar_2))", 0-0) = ar_0 5.76/2.99 5.76/2.99 S("evalstartbb0in(ar_0, ar_1, ar_2) -> Com_1(evalstart0(ar_0, ar_1, ar_2))", 0-1) = ar_1 5.76/2.99 5.76/2.99 S("evalstartbb0in(ar_0, ar_1, ar_2) -> Com_1(evalstart0(ar_0, ar_1, ar_2))", 0-2) = ar_2 5.76/2.99 5.76/2.99 S("evalstart0(ar_0, ar_1, ar_2) -> Com_1(evalstart1(ar_0, ar_1, ar_2))", 0-0) = ar_0 5.76/2.99 5.76/2.99 S("evalstart0(ar_0, ar_1, ar_2) -> Com_1(evalstart1(ar_0, ar_1, ar_2))", 0-1) = ar_1 5.76/2.99 5.76/2.99 S("evalstart0(ar_0, ar_1, ar_2) -> Com_1(evalstart1(ar_0, ar_1, ar_2))", 0-2) = ar_2 5.76/2.99 5.76/2.99 S("evalstart1(ar_0, ar_1, ar_2) -> Com_1(evalstart2(ar_0, ar_1, ar_2))", 0-0) = ar_0 5.76/2.99 5.76/2.99 S("evalstart1(ar_0, ar_1, ar_2) -> Com_1(evalstart2(ar_0, ar_1, ar_2))", 0-1) = ar_1 5.76/2.99 5.76/2.99 S("evalstart1(ar_0, ar_1, ar_2) -> Com_1(evalstart2(ar_0, ar_1, ar_2))", 0-2) = ar_2 5.76/2.99 5.76/2.99 S("evalstart2(ar_0, ar_1, ar_2) -> Com_1(evalstart3(ar_0, ar_1, ar_2))", 0-0) = ar_0 5.76/2.99 5.76/2.99 S("evalstart2(ar_0, ar_1, ar_2) -> Com_1(evalstart3(ar_0, ar_1, ar_2))", 0-1) = ar_1 5.76/2.99 5.76/2.99 S("evalstart2(ar_0, ar_1, ar_2) -> Com_1(evalstart3(ar_0, ar_1, ar_2))", 0-2) = ar_2 5.76/2.99 5.76/2.99 S("evalstart3(ar_0, ar_1, ar_2) -> Com_1(evalstart4(ar_0, ar_1, ar_2))", 0-0) = ar_0 5.76/2.99 5.76/2.99 S("evalstart3(ar_0, ar_1, ar_2) -> Com_1(evalstart4(ar_0, ar_1, ar_2))", 0-1) = ar_1 5.76/2.99 5.76/2.99 S("evalstart3(ar_0, ar_1, ar_2) -> Com_1(evalstart4(ar_0, ar_1, ar_2))", 0-2) = ar_2 5.76/2.99 5.76/2.99 S("evalstart4(ar_0, ar_1, ar_2) -> Com_1(evalstart5(ar_0, ar_1, ar_2))", 0-0) = ar_0 5.76/2.99 5.76/2.99 S("evalstart4(ar_0, ar_1, ar_2) -> Com_1(evalstart5(ar_0, ar_1, ar_2))", 0-1) = ar_1 5.76/2.99 5.76/2.99 S("evalstart4(ar_0, ar_1, ar_2) -> Com_1(evalstart5(ar_0, ar_1, ar_2))", 0-2) = ar_2 5.76/2.99 5.76/2.99 S("evalstart5(ar_0, ar_1, ar_2) -> Com_1(evalstart6(ar_0, ar_1, ar_2))", 0-0) = ar_0 5.76/2.99 5.76/2.99 S("evalstart5(ar_0, ar_1, ar_2) -> Com_1(evalstart6(ar_0, ar_1, ar_2))", 0-1) = ar_1 5.76/2.99 5.76/2.99 S("evalstart5(ar_0, ar_1, ar_2) -> Com_1(evalstart6(ar_0, ar_1, ar_2))", 0-2) = ar_2 5.76/2.99 5.76/2.99 S("evalstart6(ar_0, ar_1, ar_2) -> Com_1(evalstartbb1in(0, 0, ar_2))", 0-0) = 0 5.76/2.99 5.76/2.99 S("evalstart6(ar_0, ar_1, ar_2) -> Com_1(evalstartbb1in(0, 0, ar_2))", 0-1) = 0 5.76/2.99 5.76/2.99 S("evalstart6(ar_0, ar_1, ar_2) -> Com_1(evalstartbb1in(0, 0, ar_2))", 0-2) = ar_2 5.76/2.99 5.76/2.99 S("evalstartbb1in(ar_0, ar_1, ar_2) -> Com_1(evalstartbb2in(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 5.76/2.99 5.76/2.99 S("evalstartbb1in(ar_0, ar_1, ar_2) -> Com_1(evalstartbb2in(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 5.76/2.99 5.76/2.99 S("evalstartbb1in(ar_0, ar_1, ar_2) -> Com_1(evalstartbb2in(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 5.76/2.99 5.76/2.99 S("evalstartbb1in(ar_0, ar_1, ar_2) -> Com_1(evalstartbb3in(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 5.76/2.99 5.76/2.99 S("evalstartbb1in(ar_0, ar_1, ar_2) -> Com_1(evalstartbb3in(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 5.76/2.99 5.76/2.99 S("evalstartbb1in(ar_0, ar_1, ar_2) -> Com_1(evalstartbb3in(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 5.76/2.99 5.76/2.99 S("evalstartbb2in(ar_0, ar_1, ar_2) -> Com_1(evalstartbb1in(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 5.76/2.99 5.76/2.99 S("evalstartbb2in(ar_0, ar_1, ar_2) -> Com_1(evalstartbb1in(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 5.76/2.99 5.76/2.99 S("evalstartbb2in(ar_0, ar_1, ar_2) -> Com_1(evalstartbb1in(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 5.76/2.99 5.76/2.99 S("evalstartbb3in(ar_0, ar_1, ar_2) -> Com_1(evalstartbb4in(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 5.76/2.99 5.76/2.99 S("evalstartbb3in(ar_0, ar_1, ar_2) -> Com_1(evalstartbb4in(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 5.76/2.99 5.76/2.99 S("evalstartbb3in(ar_0, ar_1, ar_2) -> Com_1(evalstartbb4in(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 5.76/2.99 5.76/2.99 S("evalstartbb3in(ar_0, ar_1, ar_2) -> Com_1(evalstartbb5in(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 5.76/2.99 5.76/2.99 S("evalstartbb3in(ar_0, ar_1, ar_2) -> Com_1(evalstartbb5in(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 5.76/2.99 5.76/2.99 S("evalstartbb3in(ar_0, ar_1, ar_2) -> Com_1(evalstartbb5in(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 5.76/2.99 5.76/2.99 S("evalstartbb4in(ar_0, ar_1, ar_2) -> Com_1(evalstartbb1in(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 5.76/2.99 5.76/2.99 S("evalstartbb4in(ar_0, ar_1, ar_2) -> Com_1(evalstartbb1in(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 5.76/2.99 5.76/2.99 S("evalstartbb4in(ar_0, ar_1, ar_2) -> Com_1(evalstartbb1in(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 5.76/2.99 5.76/2.99 S("evalstartbb5in(ar_0, ar_1, ar_2) -> Com_1(evalstartstop(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 5.76/2.99 5.76/2.99 S("evalstartbb5in(ar_0, ar_1, ar_2) -> Com_1(evalstartstop(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 5.76/2.99 5.76/2.99 S("evalstartbb5in(ar_0, ar_1, ar_2) -> Com_1(evalstartstop(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 5.76/2.99 5.76/2.99 S("koat_start(ar_0, ar_1, ar_2) -> Com_1(evalstartstart(ar_0, ar_1, ar_2)) [ 0 <= 0 ]", 0-0) = ar_0 5.76/2.99 5.76/2.99 S("koat_start(ar_0, ar_1, ar_2) -> Com_1(evalstartstart(ar_0, ar_1, ar_2)) [ 0 <= 0 ]", 0-1) = ar_1 5.76/2.99 5.76/2.99 S("koat_start(ar_0, ar_1, ar_2) -> Com_1(evalstartstart(ar_0, ar_1, ar_2)) [ 0 <= 0 ]", 0-2) = ar_2 5.76/2.99 5.76/2.99 orients the transitions 5.76/2.99 5.76/2.99 evalstartbb4in(ar_0, ar_1, ar_2) -> Com_1(evalstartbb1in(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 ] 5.76/2.99 5.76/2.99 evalstartbb3in(ar_0, ar_1, ar_2) -> Com_1(evalstartbb4in(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 ] 5.76/2.99 5.76/2.99 evalstartbb1in(ar_0, ar_1, ar_2) -> Com_1(evalstartbb3in(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 ] 5.76/2.99 5.76/2.99 weakly and the transitions 5.76/2.99 5.76/2.99 evalstartbb4in(ar_0, ar_1, ar_2) -> Com_1(evalstartbb1in(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 ] 5.76/2.99 5.76/2.99 evalstartbb3in(ar_0, ar_1, ar_2) -> Com_1(evalstartbb4in(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 ] 5.76/2.99 5.76/2.99 evalstartbb1in(ar_0, ar_1, ar_2) -> Com_1(evalstartbb3in(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 ] 5.76/2.99 5.76/2.99 strictly and produces the following problem: 5.76/2.99 5.76/2.99 7: T: 5.76/2.99 5.76/2.99 (Comp: 1, Cost: 0) koat_start(ar_0, ar_1, ar_2) -> Com_1(evalstartstart(ar_0, ar_1, ar_2)) [ 0 <= 0 ] 5.76/2.99 5.76/2.99 (Comp: 2, Cost: 1) evalstartbb5in(ar_0, ar_1, ar_2) -> Com_1(evalstartstop(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 ] 5.76/2.99 5.76/2.99 (Comp: 3*ar_2^2 + 2*ar_2 + 2, Cost: 1) evalstartbb4in(ar_0, ar_1, ar_2) -> Com_1(evalstartbb1in(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 ] 5.76/2.99 5.76/2.99 (Comp: 2, Cost: 1) evalstartbb3in(ar_0, ar_1, ar_2) -> Com_1(evalstartbb5in(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 ] 5.76/2.99 5.76/2.99 (Comp: 3*ar_2^2 + 2*ar_2 + 2, Cost: 1) evalstartbb3in(ar_0, ar_1, ar_2) -> Com_1(evalstartbb4in(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 ] 5.76/2.99 5.76/2.99 (Comp: ar_2, Cost: 1) evalstartbb2in(ar_0, ar_1, ar_2) -> Com_1(evalstartbb1in(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 ] 5.76/2.99 5.76/2.99 (Comp: 3*ar_2^2 + 2*ar_2 + 2, Cost: 1) evalstartbb1in(ar_0, ar_1, ar_2) -> Com_1(evalstartbb3in(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 ] 5.76/2.99 5.76/2.99 (Comp: ar_2, Cost: 1) evalstartbb1in(ar_0, ar_1, ar_2) -> Com_1(evalstartbb2in(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 ] 5.76/2.99 5.76/2.99 (Comp: 1, Cost: 1) evalstart6(ar_0, ar_1, ar_2) -> Com_1(evalstartbb1in(0, 0, ar_2)) 5.76/2.99 5.76/2.99 (Comp: 1, Cost: 1) evalstart5(ar_0, ar_1, ar_2) -> Com_1(evalstart6(ar_0, ar_1, ar_2)) 5.76/2.99 5.76/2.99 (Comp: 1, Cost: 1) evalstart4(ar_0, ar_1, ar_2) -> Com_1(evalstart5(ar_0, ar_1, ar_2)) 5.76/2.99 5.76/2.99 (Comp: 1, Cost: 1) evalstart3(ar_0, ar_1, ar_2) -> Com_1(evalstart4(ar_0, ar_1, ar_2)) 5.76/2.99 5.76/2.99 (Comp: 1, Cost: 1) evalstart2(ar_0, ar_1, ar_2) -> Com_1(evalstart3(ar_0, ar_1, ar_2)) 5.76/2.99 5.76/2.99 (Comp: 1, Cost: 1) evalstart1(ar_0, ar_1, ar_2) -> Com_1(evalstart2(ar_0, ar_1, ar_2)) 5.76/2.99 5.76/2.99 (Comp: 1, Cost: 1) evalstart0(ar_0, ar_1, ar_2) -> Com_1(evalstart1(ar_0, ar_1, ar_2)) 5.76/2.99 5.76/2.99 (Comp: 1, Cost: 1) evalstartbb0in(ar_0, ar_1, ar_2) -> Com_1(evalstart0(ar_0, ar_1, ar_2)) 5.76/2.99 5.76/2.99 (Comp: 1, Cost: 1) evalstartstart(ar_0, ar_1, ar_2) -> Com_1(evalstartbb0in(ar_0, ar_1, ar_2)) 5.76/2.99 5.76/2.99 start location: koat_start 5.76/2.99 5.76/2.99 leaf cost: 0 5.76/2.99 5.76/2.99 5.76/2.99 5.76/2.99 Complexity upper bound 9*ar_2^2 + 8*ar_2 + 19 5.76/2.99 5.76/2.99 5.76/2.99 5.76/2.99 Time: 0.331 sec (SMT: 0.284 sec) 5.76/2.99 5.76/2.99 5.76/2.99 ---------------------------------------- 5.76/2.99 5.76/2.99 (2) 5.76/2.99 BOUNDS(1, n^2) 5.76/2.99 5.76/2.99 ---------------------------------------- 5.76/2.99 5.76/2.99 (3) Loat Proof (FINISHED) 5.76/2.99 5.76/2.99 5.76/2.99 ### Pre-processing the ITS problem ### 5.76/2.99 5.76/2.99 5.76/2.99 5.76/2.99 Initial linear ITS problem 5.76/2.99 5.76/2.99 Start location: evalstartstart 5.76/2.99 5.76/2.99 0: evalstartstart -> evalstartbb0in : [], cost: 1 5.76/2.99 5.76/2.99 1: evalstartbb0in -> evalstart0 : [], cost: 1 5.76/2.99 5.76/2.99 2: evalstart0 -> evalstart1 : [], cost: 1 5.76/2.99 5.76/2.99 3: evalstart1 -> evalstart2 : [], cost: 1 5.76/2.99 5.76/2.99 4: evalstart2 -> evalstart3 : [], cost: 1 5.76/2.99 5.76/2.99 5: evalstart3 -> evalstart4 : [], cost: 1 5.76/2.99 5.76/2.99 6: evalstart4 -> evalstart5 : [], cost: 1 5.76/2.99 5.76/2.99 7: evalstart5 -> evalstart6 : [], cost: 1 5.76/2.99 5.76/2.99 8: evalstart6 -> evalstartbb1in : A'=0, B'=0, [], cost: 1 5.76/2.99 5.76/2.99 9: evalstartbb1in -> evalstartbb2in : [ C>=1+A ], cost: 1 5.76/2.99 5.76/2.99 10: evalstartbb1in -> evalstartbb3in : [ A>=C ], cost: 1 5.76/2.99 5.76/2.99 11: evalstartbb2in -> evalstartbb1in : A'=1+A, B'=1+B, [], cost: 1 5.76/2.99 5.76/2.99 12: evalstartbb3in -> evalstartbb4in : [ B>=1 ], cost: 1 5.76/2.99 5.76/2.99 13: evalstartbb3in -> evalstartbb5in : [ 0>=B ], cost: 1 5.76/2.99 5.76/2.99 14: evalstartbb4in -> evalstartbb1in : B'=-1+B, [], cost: 1 5.76/2.99 5.76/2.99 15: evalstartbb5in -> evalstartstop : [], cost: 1 5.76/2.99 5.76/2.99 5.76/2.99 5.76/2.99 Removed unreachable and leaf rules: 5.76/2.99 5.76/2.99 Start location: evalstartstart 5.76/2.99 5.76/2.99 0: evalstartstart -> evalstartbb0in : [], cost: 1 5.76/2.99 5.76/2.99 1: evalstartbb0in -> evalstart0 : [], cost: 1 5.76/2.99 5.76/2.99 2: evalstart0 -> evalstart1 : [], cost: 1 5.76/2.99 5.76/2.99 3: evalstart1 -> evalstart2 : [], cost: 1 5.76/2.99 5.76/2.99 4: evalstart2 -> evalstart3 : [], cost: 1 5.76/2.99 5.76/2.99 5: evalstart3 -> evalstart4 : [], cost: 1 5.76/2.99 5.76/2.99 6: evalstart4 -> evalstart5 : [], cost: 1 5.76/2.99 5.76/2.99 7: evalstart5 -> evalstart6 : [], cost: 1 5.76/2.99 5.76/2.99 8: evalstart6 -> evalstartbb1in : A'=0, B'=0, [], cost: 1 5.76/2.99 5.76/2.99 9: evalstartbb1in -> evalstartbb2in : [ C>=1+A ], cost: 1 5.76/2.99 5.76/2.99 10: evalstartbb1in -> evalstartbb3in : [ A>=C ], cost: 1 5.76/2.99 5.76/2.99 11: evalstartbb2in -> evalstartbb1in : A'=1+A, B'=1+B, [], cost: 1 5.76/2.99 5.76/2.99 12: evalstartbb3in -> evalstartbb4in : [ B>=1 ], cost: 1 5.76/2.99 5.76/2.99 14: evalstartbb4in -> evalstartbb1in : B'=-1+B, [], cost: 1 5.76/2.99 5.76/2.99 5.76/2.99 5.76/2.99 ### Simplification by acceleration and chaining ### 5.76/2.99 5.76/2.99 5.76/2.99 5.76/2.99 Eliminated locations (on linear paths): 5.76/2.99 5.76/2.99 Start location: evalstartstart 5.76/2.99 5.76/2.99 23: evalstartstart -> evalstartbb1in : A'=0, B'=0, [], cost: 9 5.76/2.99 5.76/2.99 24: evalstartbb1in -> evalstartbb1in : A'=1+A, B'=1+B, [ C>=1+A ], cost: 2 5.76/2.99 5.76/2.99 26: evalstartbb1in -> evalstartbb1in : B'=-1+B, [ A>=C && B>=1 ], cost: 3 5.76/2.99 5.76/2.99 5.76/2.99 5.76/2.99 Accelerating simple loops of location 9. 5.76/2.99 5.76/2.99 Accelerating the following rules: 5.76/2.99 5.76/2.99 24: evalstartbb1in -> evalstartbb1in : A'=1+A, B'=1+B, [ C>=1+A ], cost: 2 5.76/2.99 5.76/2.99 26: evalstartbb1in -> evalstartbb1in : B'=-1+B, [ A>=C && B>=1 ], cost: 3 5.76/2.99 5.76/2.99 5.76/2.99 5.76/2.99 Accelerated rule 24 with metering function C-A, yielding the new rule 27. 5.76/2.99 5.76/2.99 Accelerated rule 26 with metering function B, yielding the new rule 28. 5.76/2.99 5.76/2.99 Removing the simple loops: 24 26. 5.76/2.99 5.76/2.99 5.76/2.99 5.76/2.99 Accelerated all simple loops using metering functions (where possible): 5.76/2.99 5.76/2.99 Start location: evalstartstart 5.76/2.99 5.76/2.99 23: evalstartstart -> evalstartbb1in : A'=0, B'=0, [], cost: 9 5.76/2.99 5.76/2.99 27: evalstartbb1in -> evalstartbb1in : A'=C, B'=C-A+B, [ C>=1+A ], cost: 2*C-2*A 5.76/2.99 5.76/2.99 28: evalstartbb1in -> evalstartbb1in : B'=0, [ A>=C && B>=1 ], cost: 3*B 5.76/2.99 5.76/2.99 5.76/2.99 5.76/2.99 Chained accelerated rules (with incoming rules): 5.76/2.99 5.76/2.99 Start location: evalstartstart 5.76/2.99 5.76/2.99 23: evalstartstart -> evalstartbb1in : A'=0, B'=0, [], cost: 9 5.76/2.99 5.76/2.99 29: evalstartstart -> evalstartbb1in : A'=C, B'=C, [ C>=1 ], cost: 9+2*C 5.76/2.99 5.76/2.99 5.76/2.99 5.76/2.99 Removed unreachable locations (and leaf rules with constant cost): 5.76/2.99 5.76/2.99 Start location: evalstartstart 5.76/2.99 5.76/2.99 29: evalstartstart -> evalstartbb1in : A'=C, B'=C, [ C>=1 ], cost: 9+2*C 5.76/2.99 5.76/2.99 5.76/2.99 5.76/2.99 ### Computing asymptotic complexity ### 5.76/2.99 5.76/2.99 5.76/2.99 5.76/2.99 Fully simplified ITS problem 5.76/2.99 5.76/2.99 Start location: evalstartstart 5.76/2.99 5.76/2.99 29: evalstartstart -> evalstartbb1in : A'=C, B'=C, [ C>=1 ], cost: 9+2*C 5.76/2.99 5.76/2.99 5.76/2.99 5.76/2.99 Computing asymptotic complexity for rule 29 5.76/2.99 5.76/2.99 Solved the limit problem by the following transformations: 5.76/2.99 5.76/2.99 Created initial limit problem: 5.76/2.99 5.76/2.99 9+2*C (+), C (+/+!) [not solved] 5.76/2.99 5.76/2.99 5.76/2.99 5.76/2.99 removing all constraints (solved by SMT) 5.76/2.99 5.76/2.99 resulting limit problem: [solved] 5.76/2.99 5.76/2.99 5.76/2.99 5.76/2.99 applying transformation rule (C) using substitution {C==n} 5.76/2.99 5.76/2.99 resulting limit problem: 5.76/2.99 5.76/2.99 [solved] 5.76/2.99 5.76/2.99 5.76/2.99 5.76/2.99 Solution: 5.76/2.99 5.76/2.99 C / n 5.76/2.99 5.76/2.99 Resulting cost 9+2*n has complexity: Poly(n^1) 5.76/2.99 5.76/2.99 5.76/2.99 5.76/2.99 Found new complexity Poly(n^1). 5.76/2.99 5.76/2.99 5.76/2.99 5.76/2.99 Obtained the following overall complexity (w.r.t. the length of the input n): 5.76/2.99 5.76/2.99 Complexity: Poly(n^1) 5.76/2.99 5.76/2.99 Cpx degree: 1 5.76/2.99 5.76/2.99 Solved cost: 9+2*n 5.76/2.99 5.76/2.99 Rule cost: 9+2*C 5.76/2.99 5.76/2.99 Rule guard: [ C>=1 ] 5.76/2.99 5.76/2.99 5.76/2.99 5.76/2.99 WORST_CASE(Omega(n^1),?) 5.76/2.99 5.76/2.99 5.76/2.99 ---------------------------------------- 5.76/2.99 5.76/2.99 (4) 5.76/2.99 BOUNDS(n^1, INF) 5.76/3.03 EOF