/export/starexec/sandbox2/solver/bin/starexec_run_complexity /export/starexec/sandbox2/benchmark/theBenchmark.koat /export/starexec/sandbox2/output/output_files -------------------------------------------------------------------------------- WORST_CASE(Omega(n^2), O(n^2)) proof of /export/starexec/sandbox2/benchmark/theBenchmark.koat # AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(n^2, n^2). (0) CpxIntTrs (1) Koat Proof [FINISHED, 118 ms] (2) BOUNDS(1, n^2) (3) Loat Proof [FINISHED, 726 ms] (4) BOUNDS(n^2, INF) ---------------------------------------- (0) Obligation: Complexity Int TRS consisting of the following rules: eval_while2_start(v_3, v_N, v_i_0, v_j_0) -> Com_1(eval_while2_bb0_in(v_3, v_N, v_i_0, v_j_0)) :|: TRUE eval_while2_bb0_in(v_3, v_N, v_i_0, v_j_0) -> Com_1(eval_while2_0(v_3, v_N, v_i_0, v_j_0)) :|: TRUE eval_while2_0(v_3, v_N, v_i_0, v_j_0) -> Com_1(eval_while2_1(v_3, v_N, v_i_0, v_j_0)) :|: TRUE eval_while2_1(v_3, v_N, v_i_0, v_j_0) -> Com_1(eval_while2_2(v_3, v_N, v_i_0, v_j_0)) :|: TRUE eval_while2_2(v_3, v_N, v_i_0, v_j_0) -> Com_1(eval_while2_3(v_3, v_N, v_i_0, v_j_0)) :|: TRUE eval_while2_3(v_3, v_N, v_i_0, v_j_0) -> Com_1(eval_while2_4(v_3, v_N, v_i_0, v_j_0)) :|: TRUE eval_while2_4(v_3, v_N, v_i_0, v_j_0) -> Com_1(eval_while2_bb1_in(v_3, v_N, v_N, v_j_0)) :|: TRUE eval_while2_bb1_in(v_3, v_N, v_i_0, v_j_0) -> Com_1(eval_while2_bb2_in(v_3, v_N, v_i_0, v_N)) :|: v_i_0 > 0 eval_while2_bb1_in(v_3, v_N, v_i_0, v_j_0) -> Com_1(eval_while2_bb5_in(v_3, v_N, v_i_0, v_j_0)) :|: v_i_0 <= 0 eval_while2_bb2_in(v_3, v_N, v_i_0, v_j_0) -> Com_1(eval_while2_bb3_in(v_3, v_N, v_i_0, v_j_0)) :|: v_j_0 > 0 eval_while2_bb2_in(v_3, v_N, v_i_0, v_j_0) -> Com_1(eval_while2_bb4_in(v_3, v_N, v_i_0, v_j_0)) :|: v_j_0 <= 0 eval_while2_bb3_in(v_3, v_N, v_i_0, v_j_0) -> Com_1(eval_while2_bb2_in(v_3, v_N, v_i_0, v_j_0 - 1)) :|: TRUE eval_while2_bb4_in(v_3, v_N, v_i_0, v_j_0) -> Com_1(eval_while2_8(v_i_0 - 1, v_N, v_i_0, v_j_0)) :|: TRUE eval_while2_8(v_3, v_N, v_i_0, v_j_0) -> Com_1(eval_while2_9(v_3, v_N, v_i_0, v_j_0)) :|: TRUE eval_while2_9(v_3, v_N, v_i_0, v_j_0) -> Com_1(eval_while2_bb1_in(v_3, v_N, v_3, v_j_0)) :|: TRUE eval_while2_bb5_in(v_3, v_N, v_i_0, v_j_0) -> Com_1(eval_while2_stop(v_3, v_N, v_i_0, v_j_0)) :|: TRUE The start-symbols are:[eval_while2_start_4] ---------------------------------------- (1) Koat Proof (FINISHED) YES(?, 17*Ar_1 + 2*Ar_1^2 + 11) Initial complexity problem: 1: T: (Comp: ?, Cost: 1) evalwhile2start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile2bb0in(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: ?, Cost: 1) evalwhile2bb0in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile20(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: ?, Cost: 1) evalwhile20(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile21(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: ?, Cost: 1) evalwhile21(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile22(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: ?, Cost: 1) evalwhile22(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile23(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: ?, Cost: 1) evalwhile23(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile24(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: ?, Cost: 1) evalwhile24(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile2bb1in(Ar_1, Ar_1, Ar_2, Ar_3)) (Comp: ?, Cost: 1) evalwhile2bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile2bb2in(Ar_0, Ar_1, Ar_1, Ar_3)) [ Ar_0 >= 1 ] (Comp: ?, Cost: 1) evalwhile2bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile2bb5in(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 >= Ar_0 ] (Comp: ?, Cost: 1) evalwhile2bb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile2bb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= 1 ] (Comp: ?, Cost: 1) evalwhile2bb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile2bb4in(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 >= Ar_2 ] (Comp: ?, Cost: 1) evalwhile2bb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile2bb2in(Ar_0, Ar_1, Ar_2 - 1, Ar_3)) (Comp: ?, Cost: 1) evalwhile2bb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile28(Ar_0, Ar_1, Ar_2, Ar_0 - 1)) (Comp: ?, Cost: 1) evalwhile28(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile29(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: ?, Cost: 1) evalwhile29(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile2bb1in(Ar_3, Ar_1, Ar_2, Ar_3)) (Comp: ?, Cost: 1) evalwhile2bb5in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile2stop(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile2start(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) evalwhile2start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile2bb0in(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 1) evalwhile2bb0in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile20(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 1) evalwhile20(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile21(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 1) evalwhile21(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile22(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 1) evalwhile22(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile23(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 1) evalwhile23(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile24(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 1) evalwhile24(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile2bb1in(Ar_1, Ar_1, Ar_2, Ar_3)) (Comp: ?, Cost: 1) evalwhile2bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile2bb2in(Ar_0, Ar_1, Ar_1, Ar_3)) [ Ar_0 >= 1 ] (Comp: ?, Cost: 1) evalwhile2bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile2bb5in(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 >= Ar_0 ] (Comp: ?, Cost: 1) evalwhile2bb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile2bb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= 1 ] (Comp: ?, Cost: 1) evalwhile2bb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile2bb4in(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 >= Ar_2 ] (Comp: ?, Cost: 1) evalwhile2bb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile2bb2in(Ar_0, Ar_1, Ar_2 - 1, Ar_3)) (Comp: ?, Cost: 1) evalwhile2bb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile28(Ar_0, Ar_1, Ar_2, Ar_0 - 1)) (Comp: ?, Cost: 1) evalwhile28(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile29(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: ?, Cost: 1) evalwhile29(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile2bb1in(Ar_3, Ar_1, Ar_2, Ar_3)) (Comp: ?, Cost: 1) evalwhile2bb5in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile2stop(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile2start(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 A polynomial rank function with Pol(evalwhile2start) = 2 Pol(evalwhile2bb0in) = 2 Pol(evalwhile20) = 2 Pol(evalwhile21) = 2 Pol(evalwhile22) = 2 Pol(evalwhile23) = 2 Pol(evalwhile24) = 2 Pol(evalwhile2bb1in) = 2 Pol(evalwhile2bb2in) = 2 Pol(evalwhile2bb5in) = 1 Pol(evalwhile2bb3in) = 2 Pol(evalwhile2bb4in) = 2 Pol(evalwhile28) = 2 Pol(evalwhile29) = 2 Pol(evalwhile2stop) = 0 Pol(koat_start) = 2 orients all transitions weakly and the transitions evalwhile2bb5in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile2stop(Ar_0, Ar_1, Ar_2, Ar_3)) evalwhile2bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile2bb5in(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 >= Ar_0 ] strictly and produces the following problem: 3: T: (Comp: 1, Cost: 1) evalwhile2start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile2bb0in(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 1) evalwhile2bb0in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile20(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 1) evalwhile20(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile21(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 1) evalwhile21(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile22(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 1) evalwhile22(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile23(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 1) evalwhile23(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile24(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 1) evalwhile24(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile2bb1in(Ar_1, Ar_1, Ar_2, Ar_3)) (Comp: ?, Cost: 1) evalwhile2bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile2bb2in(Ar_0, Ar_1, Ar_1, Ar_3)) [ Ar_0 >= 1 ] (Comp: 2, Cost: 1) evalwhile2bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile2bb5in(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 >= Ar_0 ] (Comp: ?, Cost: 1) evalwhile2bb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile2bb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= 1 ] (Comp: ?, Cost: 1) evalwhile2bb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile2bb4in(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 >= Ar_2 ] (Comp: ?, Cost: 1) evalwhile2bb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile2bb2in(Ar_0, Ar_1, Ar_2 - 1, Ar_3)) (Comp: ?, Cost: 1) evalwhile2bb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile28(Ar_0, Ar_1, Ar_2, Ar_0 - 1)) (Comp: ?, Cost: 1) evalwhile28(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile29(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: ?, Cost: 1) evalwhile29(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile2bb1in(Ar_3, Ar_1, Ar_2, Ar_3)) (Comp: 2, Cost: 1) evalwhile2bb5in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile2stop(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile2start(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 A polynomial rank function with Pol(evalwhile2start) = V_2 Pol(evalwhile2bb0in) = V_2 Pol(evalwhile20) = V_2 Pol(evalwhile21) = V_2 Pol(evalwhile22) = V_2 Pol(evalwhile23) = V_2 Pol(evalwhile24) = V_2 Pol(evalwhile2bb1in) = V_1 Pol(evalwhile2bb2in) = V_1 - 1 Pol(evalwhile2bb5in) = V_1 Pol(evalwhile2bb3in) = V_1 - 1 Pol(evalwhile2bb4in) = V_1 - 1 Pol(evalwhile28) = V_4 Pol(evalwhile29) = V_4 Pol(evalwhile2stop) = V_1 Pol(koat_start) = V_2 orients all transitions weakly and the transition evalwhile2bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile2bb2in(Ar_0, Ar_1, Ar_1, Ar_3)) [ Ar_0 >= 1 ] strictly and produces the following problem: 4: T: (Comp: 1, Cost: 1) evalwhile2start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile2bb0in(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 1) evalwhile2bb0in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile20(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 1) evalwhile20(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile21(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 1) evalwhile21(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile22(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 1) evalwhile22(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile23(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 1) evalwhile23(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile24(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 1) evalwhile24(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile2bb1in(Ar_1, Ar_1, Ar_2, Ar_3)) (Comp: Ar_1, Cost: 1) evalwhile2bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile2bb2in(Ar_0, Ar_1, Ar_1, Ar_3)) [ Ar_0 >= 1 ] (Comp: 2, Cost: 1) evalwhile2bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile2bb5in(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 >= Ar_0 ] (Comp: ?, Cost: 1) evalwhile2bb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile2bb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= 1 ] (Comp: ?, Cost: 1) evalwhile2bb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile2bb4in(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 >= Ar_2 ] (Comp: ?, Cost: 1) evalwhile2bb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile2bb2in(Ar_0, Ar_1, Ar_2 - 1, Ar_3)) (Comp: ?, Cost: 1) evalwhile2bb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile28(Ar_0, Ar_1, Ar_2, Ar_0 - 1)) (Comp: ?, Cost: 1) evalwhile28(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile29(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: ?, Cost: 1) evalwhile29(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile2bb1in(Ar_3, Ar_1, Ar_2, Ar_3)) (Comp: 2, Cost: 1) evalwhile2bb5in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile2stop(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile2start(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 A polynomial rank function with Pol(evalwhile2bb4in) = 3 Pol(evalwhile28) = 2 Pol(evalwhile2bb3in) = 4 Pol(evalwhile2bb2in) = 4 Pol(evalwhile29) = 1 Pol(evalwhile2bb1in) = 0 and size complexities S("koat_start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile2start(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(evalwhile2start(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(evalwhile2start(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(evalwhile2start(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 <= 0 ]", 0-3) = Ar_3 S("evalwhile2bb5in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile2stop(Ar_0, Ar_1, Ar_2, Ar_3))", 0-0) = ? S("evalwhile2bb5in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile2stop(Ar_0, Ar_1, Ar_2, Ar_3))", 0-1) = Ar_1 S("evalwhile2bb5in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile2stop(Ar_0, Ar_1, Ar_2, Ar_3))", 0-2) = ? S("evalwhile2bb5in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile2stop(Ar_0, Ar_1, Ar_2, Ar_3))", 0-3) = ? S("evalwhile29(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile2bb1in(Ar_3, Ar_1, Ar_2, Ar_3))", 0-0) = ? S("evalwhile29(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile2bb1in(Ar_3, Ar_1, Ar_2, Ar_3))", 0-1) = Ar_1 S("evalwhile29(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile2bb1in(Ar_3, Ar_1, Ar_2, Ar_3))", 0-2) = ? S("evalwhile29(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile2bb1in(Ar_3, Ar_1, Ar_2, Ar_3))", 0-3) = ? S("evalwhile28(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile29(Ar_0, Ar_1, Ar_2, Ar_3))", 0-0) = ? S("evalwhile28(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile29(Ar_0, Ar_1, Ar_2, Ar_3))", 0-1) = Ar_1 S("evalwhile28(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile29(Ar_0, Ar_1, Ar_2, Ar_3))", 0-2) = ? S("evalwhile28(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile29(Ar_0, Ar_1, Ar_2, Ar_3))", 0-3) = ? S("evalwhile2bb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile28(Ar_0, Ar_1, Ar_2, Ar_0 - 1))", 0-0) = ? S("evalwhile2bb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile28(Ar_0, Ar_1, Ar_2, Ar_0 - 1))", 0-1) = Ar_1 S("evalwhile2bb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile28(Ar_0, Ar_1, Ar_2, Ar_0 - 1))", 0-2) = ? S("evalwhile2bb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile28(Ar_0, Ar_1, Ar_2, Ar_0 - 1))", 0-3) = ? S("evalwhile2bb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile2bb2in(Ar_0, Ar_1, Ar_2 - 1, Ar_3))", 0-0) = ? S("evalwhile2bb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile2bb2in(Ar_0, Ar_1, Ar_2 - 1, Ar_3))", 0-1) = Ar_1 S("evalwhile2bb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile2bb2in(Ar_0, Ar_1, Ar_2 - 1, Ar_3))", 0-2) = ? S("evalwhile2bb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile2bb2in(Ar_0, Ar_1, Ar_2 - 1, Ar_3))", 0-3) = ? S("evalwhile2bb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile2bb4in(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 >= Ar_2 ]", 0-0) = ? S("evalwhile2bb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile2bb4in(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 >= Ar_2 ]", 0-1) = Ar_1 S("evalwhile2bb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile2bb4in(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 >= Ar_2 ]", 0-2) = ? S("evalwhile2bb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile2bb4in(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 >= Ar_2 ]", 0-3) = ? S("evalwhile2bb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile2bb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= 1 ]", 0-0) = ? S("evalwhile2bb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile2bb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= 1 ]", 0-1) = Ar_1 S("evalwhile2bb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile2bb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= 1 ]", 0-2) = ? S("evalwhile2bb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile2bb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= 1 ]", 0-3) = ? S("evalwhile2bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile2bb5in(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 >= Ar_0 ]", 0-0) = ? S("evalwhile2bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile2bb5in(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 >= Ar_0 ]", 0-1) = Ar_1 S("evalwhile2bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile2bb5in(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 >= Ar_0 ]", 0-2) = ? S("evalwhile2bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile2bb5in(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 >= Ar_0 ]", 0-3) = ? S("evalwhile2bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile2bb2in(Ar_0, Ar_1, Ar_1, Ar_3)) [ Ar_0 >= 1 ]", 0-0) = ? S("evalwhile2bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile2bb2in(Ar_0, Ar_1, Ar_1, Ar_3)) [ Ar_0 >= 1 ]", 0-1) = Ar_1 S("evalwhile2bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile2bb2in(Ar_0, Ar_1, Ar_1, Ar_3)) [ Ar_0 >= 1 ]", 0-2) = Ar_1 S("evalwhile2bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile2bb2in(Ar_0, Ar_1, Ar_1, Ar_3)) [ Ar_0 >= 1 ]", 0-3) = ? S("evalwhile24(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile2bb1in(Ar_1, Ar_1, Ar_2, Ar_3))", 0-0) = Ar_1 S("evalwhile24(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile2bb1in(Ar_1, Ar_1, Ar_2, Ar_3))", 0-1) = Ar_1 S("evalwhile24(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile2bb1in(Ar_1, Ar_1, Ar_2, Ar_3))", 0-2) = Ar_2 S("evalwhile24(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile2bb1in(Ar_1, Ar_1, Ar_2, Ar_3))", 0-3) = Ar_3 S("evalwhile23(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile24(Ar_0, Ar_1, Ar_2, Ar_3))", 0-0) = Ar_0 S("evalwhile23(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile24(Ar_0, Ar_1, Ar_2, Ar_3))", 0-1) = Ar_1 S("evalwhile23(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile24(Ar_0, Ar_1, Ar_2, Ar_3))", 0-2) = Ar_2 S("evalwhile23(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile24(Ar_0, Ar_1, Ar_2, Ar_3))", 0-3) = Ar_3 S("evalwhile22(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile23(Ar_0, Ar_1, Ar_2, Ar_3))", 0-0) = Ar_0 S("evalwhile22(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile23(Ar_0, Ar_1, Ar_2, Ar_3))", 0-1) = Ar_1 S("evalwhile22(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile23(Ar_0, Ar_1, Ar_2, Ar_3))", 0-2) = Ar_2 S("evalwhile22(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile23(Ar_0, Ar_1, Ar_2, Ar_3))", 0-3) = Ar_3 S("evalwhile21(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile22(Ar_0, Ar_1, Ar_2, Ar_3))", 0-0) = Ar_0 S("evalwhile21(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile22(Ar_0, Ar_1, Ar_2, Ar_3))", 0-1) = Ar_1 S("evalwhile21(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile22(Ar_0, Ar_1, Ar_2, Ar_3))", 0-2) = Ar_2 S("evalwhile21(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile22(Ar_0, Ar_1, Ar_2, Ar_3))", 0-3) = Ar_3 S("evalwhile20(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile21(Ar_0, Ar_1, Ar_2, Ar_3))", 0-0) = Ar_0 S("evalwhile20(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile21(Ar_0, Ar_1, Ar_2, Ar_3))", 0-1) = Ar_1 S("evalwhile20(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile21(Ar_0, Ar_1, Ar_2, Ar_3))", 0-2) = Ar_2 S("evalwhile20(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile21(Ar_0, Ar_1, Ar_2, Ar_3))", 0-3) = Ar_3 S("evalwhile2bb0in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile20(Ar_0, Ar_1, Ar_2, Ar_3))", 0-0) = Ar_0 S("evalwhile2bb0in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile20(Ar_0, Ar_1, Ar_2, Ar_3))", 0-1) = Ar_1 S("evalwhile2bb0in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile20(Ar_0, Ar_1, Ar_2, Ar_3))", 0-2) = Ar_2 S("evalwhile2bb0in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile20(Ar_0, Ar_1, Ar_2, Ar_3))", 0-3) = Ar_3 S("evalwhile2start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile2bb0in(Ar_0, Ar_1, Ar_2, Ar_3))", 0-0) = Ar_0 S("evalwhile2start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile2bb0in(Ar_0, Ar_1, Ar_2, Ar_3))", 0-1) = Ar_1 S("evalwhile2start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile2bb0in(Ar_0, Ar_1, Ar_2, Ar_3))", 0-2) = Ar_2 S("evalwhile2start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile2bb0in(Ar_0, Ar_1, Ar_2, Ar_3))", 0-3) = Ar_3 orients the transitions evalwhile2bb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile28(Ar_0, Ar_1, Ar_2, Ar_0 - 1)) evalwhile2bb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile2bb2in(Ar_0, Ar_1, Ar_2 - 1, Ar_3)) evalwhile2bb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile2bb4in(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 >= Ar_2 ] evalwhile2bb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile2bb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= 1 ] evalwhile29(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile2bb1in(Ar_3, Ar_1, Ar_2, Ar_3)) evalwhile28(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile29(Ar_0, Ar_1, Ar_2, Ar_3)) weakly and the transitions evalwhile2bb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile28(Ar_0, Ar_1, Ar_2, Ar_0 - 1)) evalwhile2bb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile2bb4in(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 >= Ar_2 ] evalwhile29(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile2bb1in(Ar_3, Ar_1, Ar_2, Ar_3)) evalwhile28(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile29(Ar_0, Ar_1, Ar_2, Ar_3)) strictly and produces the following problem: 5: T: (Comp: 1, Cost: 1) evalwhile2start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile2bb0in(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 1) evalwhile2bb0in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile20(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 1) evalwhile20(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile21(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 1) evalwhile21(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile22(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 1) evalwhile22(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile23(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 1) evalwhile23(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile24(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 1) evalwhile24(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile2bb1in(Ar_1, Ar_1, Ar_2, Ar_3)) (Comp: Ar_1, Cost: 1) evalwhile2bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile2bb2in(Ar_0, Ar_1, Ar_1, Ar_3)) [ Ar_0 >= 1 ] (Comp: 2, Cost: 1) evalwhile2bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile2bb5in(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 >= Ar_0 ] (Comp: ?, Cost: 1) evalwhile2bb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile2bb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= 1 ] (Comp: 4*Ar_1, Cost: 1) evalwhile2bb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile2bb4in(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 >= Ar_2 ] (Comp: ?, Cost: 1) evalwhile2bb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile2bb2in(Ar_0, Ar_1, Ar_2 - 1, Ar_3)) (Comp: 4*Ar_1, Cost: 1) evalwhile2bb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile28(Ar_0, Ar_1, Ar_2, Ar_0 - 1)) (Comp: 4*Ar_1, Cost: 1) evalwhile28(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile29(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 4*Ar_1, Cost: 1) evalwhile29(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile2bb1in(Ar_3, Ar_1, Ar_2, Ar_3)) (Comp: 2, Cost: 1) evalwhile2bb5in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile2stop(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile2start(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 A polynomial rank function with Pol(evalwhile2bb3in) = V_3 - 1 Pol(evalwhile2bb2in) = V_3 and size complexities S("koat_start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile2start(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(evalwhile2start(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(evalwhile2start(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(evalwhile2start(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 <= 0 ]", 0-3) = Ar_3 S("evalwhile2bb5in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile2stop(Ar_0, Ar_1, Ar_2, Ar_3))", 0-0) = 5*Ar_1 S("evalwhile2bb5in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile2stop(Ar_0, Ar_1, Ar_2, Ar_3))", 0-1) = Ar_1 S("evalwhile2bb5in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile2stop(Ar_0, Ar_1, Ar_2, Ar_3))", 0-2) = ? S("evalwhile2bb5in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile2stop(Ar_0, Ar_1, Ar_2, Ar_3))", 0-3) = 5*Ar_1 + 5*Ar_3 S("evalwhile29(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile2bb1in(Ar_3, Ar_1, Ar_2, Ar_3))", 0-0) = 5*Ar_1 S("evalwhile29(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile2bb1in(Ar_3, Ar_1, Ar_2, Ar_3))", 0-1) = Ar_1 S("evalwhile29(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile2bb1in(Ar_3, Ar_1, Ar_2, Ar_3))", 0-2) = ? S("evalwhile29(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile2bb1in(Ar_3, Ar_1, Ar_2, Ar_3))", 0-3) = 5*Ar_1 S("evalwhile28(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile29(Ar_0, Ar_1, Ar_2, Ar_3))", 0-0) = 5*Ar_1 S("evalwhile28(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile29(Ar_0, Ar_1, Ar_2, Ar_3))", 0-1) = Ar_1 S("evalwhile28(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile29(Ar_0, Ar_1, Ar_2, Ar_3))", 0-2) = ? S("evalwhile28(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile29(Ar_0, Ar_1, Ar_2, Ar_3))", 0-3) = 5*Ar_1 S("evalwhile2bb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile28(Ar_0, Ar_1, Ar_2, Ar_0 - 1))", 0-0) = 5*Ar_1 S("evalwhile2bb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile28(Ar_0, Ar_1, Ar_2, Ar_0 - 1))", 0-1) = Ar_1 S("evalwhile2bb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile28(Ar_0, Ar_1, Ar_2, Ar_0 - 1))", 0-2) = ? S("evalwhile2bb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile28(Ar_0, Ar_1, Ar_2, Ar_0 - 1))", 0-3) = 5*Ar_1 S("evalwhile2bb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile2bb2in(Ar_0, Ar_1, Ar_2 - 1, Ar_3))", 0-0) = 5*Ar_1 S("evalwhile2bb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile2bb2in(Ar_0, Ar_1, Ar_2 - 1, Ar_3))", 0-1) = Ar_1 S("evalwhile2bb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile2bb2in(Ar_0, Ar_1, Ar_2 - 1, Ar_3))", 0-2) = ? S("evalwhile2bb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile2bb2in(Ar_0, Ar_1, Ar_2 - 1, Ar_3))", 0-3) = 5*Ar_1 + 5*Ar_3 S("evalwhile2bb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile2bb4in(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 >= Ar_2 ]", 0-0) = 5*Ar_1 S("evalwhile2bb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile2bb4in(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 >= Ar_2 ]", 0-1) = Ar_1 S("evalwhile2bb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile2bb4in(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 >= Ar_2 ]", 0-2) = ? S("evalwhile2bb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile2bb4in(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 >= Ar_2 ]", 0-3) = 5*Ar_1 + 5*Ar_3 S("evalwhile2bb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile2bb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= 1 ]", 0-0) = 5*Ar_1 S("evalwhile2bb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile2bb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= 1 ]", 0-1) = Ar_1 S("evalwhile2bb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile2bb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= 1 ]", 0-2) = ? S("evalwhile2bb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile2bb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= 1 ]", 0-3) = 5*Ar_1 + 5*Ar_3 S("evalwhile2bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile2bb5in(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 >= Ar_0 ]", 0-0) = 5*Ar_1 S("evalwhile2bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile2bb5in(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 >= Ar_0 ]", 0-1) = Ar_1 S("evalwhile2bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile2bb5in(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 >= Ar_0 ]", 0-2) = ? S("evalwhile2bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile2bb5in(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 >= Ar_0 ]", 0-3) = 5*Ar_1 + 5*Ar_3 S("evalwhile2bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile2bb2in(Ar_0, Ar_1, Ar_1, Ar_3)) [ Ar_0 >= 1 ]", 0-0) = 5*Ar_1 S("evalwhile2bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile2bb2in(Ar_0, Ar_1, Ar_1, Ar_3)) [ Ar_0 >= 1 ]", 0-1) = Ar_1 S("evalwhile2bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile2bb2in(Ar_0, Ar_1, Ar_1, Ar_3)) [ Ar_0 >= 1 ]", 0-2) = Ar_1 S("evalwhile2bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile2bb2in(Ar_0, Ar_1, Ar_1, Ar_3)) [ Ar_0 >= 1 ]", 0-3) = 5*Ar_1 + 5*Ar_3 S("evalwhile24(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile2bb1in(Ar_1, Ar_1, Ar_2, Ar_3))", 0-0) = Ar_1 S("evalwhile24(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile2bb1in(Ar_1, Ar_1, Ar_2, Ar_3))", 0-1) = Ar_1 S("evalwhile24(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile2bb1in(Ar_1, Ar_1, Ar_2, Ar_3))", 0-2) = Ar_2 S("evalwhile24(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile2bb1in(Ar_1, Ar_1, Ar_2, Ar_3))", 0-3) = Ar_3 S("evalwhile23(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile24(Ar_0, Ar_1, Ar_2, Ar_3))", 0-0) = Ar_0 S("evalwhile23(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile24(Ar_0, Ar_1, Ar_2, Ar_3))", 0-1) = Ar_1 S("evalwhile23(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile24(Ar_0, Ar_1, Ar_2, Ar_3))", 0-2) = Ar_2 S("evalwhile23(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile24(Ar_0, Ar_1, Ar_2, Ar_3))", 0-3) = Ar_3 S("evalwhile22(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile23(Ar_0, Ar_1, Ar_2, Ar_3))", 0-0) = Ar_0 S("evalwhile22(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile23(Ar_0, Ar_1, Ar_2, Ar_3))", 0-1) = Ar_1 S("evalwhile22(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile23(Ar_0, Ar_1, Ar_2, Ar_3))", 0-2) = Ar_2 S("evalwhile22(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile23(Ar_0, Ar_1, Ar_2, Ar_3))", 0-3) = Ar_3 S("evalwhile21(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile22(Ar_0, Ar_1, Ar_2, Ar_3))", 0-0) = Ar_0 S("evalwhile21(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile22(Ar_0, Ar_1, Ar_2, Ar_3))", 0-1) = Ar_1 S("evalwhile21(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile22(Ar_0, Ar_1, Ar_2, Ar_3))", 0-2) = Ar_2 S("evalwhile21(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile22(Ar_0, Ar_1, Ar_2, Ar_3))", 0-3) = Ar_3 S("evalwhile20(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile21(Ar_0, Ar_1, Ar_2, Ar_3))", 0-0) = Ar_0 S("evalwhile20(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile21(Ar_0, Ar_1, Ar_2, Ar_3))", 0-1) = Ar_1 S("evalwhile20(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile21(Ar_0, Ar_1, Ar_2, Ar_3))", 0-2) = Ar_2 S("evalwhile20(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile21(Ar_0, Ar_1, Ar_2, Ar_3))", 0-3) = Ar_3 S("evalwhile2bb0in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile20(Ar_0, Ar_1, Ar_2, Ar_3))", 0-0) = Ar_0 S("evalwhile2bb0in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile20(Ar_0, Ar_1, Ar_2, Ar_3))", 0-1) = Ar_1 S("evalwhile2bb0in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile20(Ar_0, Ar_1, Ar_2, Ar_3))", 0-2) = Ar_2 S("evalwhile2bb0in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile20(Ar_0, Ar_1, Ar_2, Ar_3))", 0-3) = Ar_3 S("evalwhile2start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile2bb0in(Ar_0, Ar_1, Ar_2, Ar_3))", 0-0) = Ar_0 S("evalwhile2start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile2bb0in(Ar_0, Ar_1, Ar_2, Ar_3))", 0-1) = Ar_1 S("evalwhile2start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile2bb0in(Ar_0, Ar_1, Ar_2, Ar_3))", 0-2) = Ar_2 S("evalwhile2start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile2bb0in(Ar_0, Ar_1, Ar_2, Ar_3))", 0-3) = Ar_3 orients the transitions evalwhile2bb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile2bb2in(Ar_0, Ar_1, Ar_2 - 1, Ar_3)) evalwhile2bb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile2bb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= 1 ] weakly and the transition evalwhile2bb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile2bb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= 1 ] strictly and produces the following problem: 6: T: (Comp: 1, Cost: 1) evalwhile2start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile2bb0in(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 1) evalwhile2bb0in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile20(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 1) evalwhile20(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile21(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 1) evalwhile21(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile22(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 1) evalwhile22(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile23(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 1) evalwhile23(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile24(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 1) evalwhile24(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile2bb1in(Ar_1, Ar_1, Ar_2, Ar_3)) (Comp: Ar_1, Cost: 1) evalwhile2bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile2bb2in(Ar_0, Ar_1, Ar_1, Ar_3)) [ Ar_0 >= 1 ] (Comp: 2, Cost: 1) evalwhile2bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile2bb5in(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 >= Ar_0 ] (Comp: Ar_1^2, Cost: 1) evalwhile2bb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile2bb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= 1 ] (Comp: 4*Ar_1, Cost: 1) evalwhile2bb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile2bb4in(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 >= Ar_2 ] (Comp: ?, Cost: 1) evalwhile2bb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile2bb2in(Ar_0, Ar_1, Ar_2 - 1, Ar_3)) (Comp: 4*Ar_1, Cost: 1) evalwhile2bb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile28(Ar_0, Ar_1, Ar_2, Ar_0 - 1)) (Comp: 4*Ar_1, Cost: 1) evalwhile28(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile29(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 4*Ar_1, Cost: 1) evalwhile29(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile2bb1in(Ar_3, Ar_1, Ar_2, Ar_3)) (Comp: 2, Cost: 1) evalwhile2bb5in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile2stop(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile2start(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) evalwhile2start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile2bb0in(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 1) evalwhile2bb0in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile20(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 1) evalwhile20(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile21(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 1) evalwhile21(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile22(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 1) evalwhile22(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile23(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 1) evalwhile23(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile24(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 1) evalwhile24(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile2bb1in(Ar_1, Ar_1, Ar_2, Ar_3)) (Comp: Ar_1, Cost: 1) evalwhile2bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile2bb2in(Ar_0, Ar_1, Ar_1, Ar_3)) [ Ar_0 >= 1 ] (Comp: 2, Cost: 1) evalwhile2bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile2bb5in(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 >= Ar_0 ] (Comp: Ar_1^2, Cost: 1) evalwhile2bb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile2bb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= 1 ] (Comp: 4*Ar_1, Cost: 1) evalwhile2bb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile2bb4in(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 >= Ar_2 ] (Comp: Ar_1^2, Cost: 1) evalwhile2bb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile2bb2in(Ar_0, Ar_1, Ar_2 - 1, Ar_3)) (Comp: 4*Ar_1, Cost: 1) evalwhile2bb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile28(Ar_0, Ar_1, Ar_2, Ar_0 - 1)) (Comp: 4*Ar_1, Cost: 1) evalwhile28(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile29(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 4*Ar_1, Cost: 1) evalwhile29(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile2bb1in(Ar_3, Ar_1, Ar_2, Ar_3)) (Comp: 2, Cost: 1) evalwhile2bb5in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile2stop(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwhile2start(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 Complexity upper bound 17*Ar_1 + 2*Ar_1^2 + 11 Time: 0.130 sec (SMT: 0.094 sec) ---------------------------------------- (2) BOUNDS(1, n^2) ---------------------------------------- (3) Loat Proof (FINISHED) ### Pre-processing the ITS problem ### Initial linear ITS problem Start location: evalwhile2start 0: evalwhile2start -> evalwhile2bb0in : [], cost: 1 1: evalwhile2bb0in -> evalwhile20 : [], cost: 1 2: evalwhile20 -> evalwhile21 : [], cost: 1 3: evalwhile21 -> evalwhile22 : [], cost: 1 4: evalwhile22 -> evalwhile23 : [], cost: 1 5: evalwhile23 -> evalwhile24 : [], cost: 1 6: evalwhile24 -> evalwhile2bb1in : A'=B, [], cost: 1 7: evalwhile2bb1in -> evalwhile2bb2in : C'=B, [ A>=1 ], cost: 1 8: evalwhile2bb1in -> evalwhile2bb5in : [ 0>=A ], cost: 1 9: evalwhile2bb2in -> evalwhile2bb3in : [ C>=1 ], cost: 1 10: evalwhile2bb2in -> evalwhile2bb4in : [ 0>=C ], cost: 1 11: evalwhile2bb3in -> evalwhile2bb2in : C'=-1+C, [], cost: 1 12: evalwhile2bb4in -> evalwhile28 : D'=-1+A, [], cost: 1 13: evalwhile28 -> evalwhile29 : [], cost: 1 14: evalwhile29 -> evalwhile2bb1in : A'=D, [], cost: 1 15: evalwhile2bb5in -> evalwhile2stop : [], cost: 1 Checking for constant complexity: The following rule is satisfiable with cost >= 1, yielding constant complexity: 0: evalwhile2start -> evalwhile2bb0in : [], cost: 1 Removed unreachable and leaf rules: Start location: evalwhile2start 0: evalwhile2start -> evalwhile2bb0in : [], cost: 1 1: evalwhile2bb0in -> evalwhile20 : [], cost: 1 2: evalwhile20 -> evalwhile21 : [], cost: 1 3: evalwhile21 -> evalwhile22 : [], cost: 1 4: evalwhile22 -> evalwhile23 : [], cost: 1 5: evalwhile23 -> evalwhile24 : [], cost: 1 6: evalwhile24 -> evalwhile2bb1in : A'=B, [], cost: 1 7: evalwhile2bb1in -> evalwhile2bb2in : C'=B, [ A>=1 ], cost: 1 9: evalwhile2bb2in -> evalwhile2bb3in : [ C>=1 ], cost: 1 10: evalwhile2bb2in -> evalwhile2bb4in : [ 0>=C ], cost: 1 11: evalwhile2bb3in -> evalwhile2bb2in : C'=-1+C, [], cost: 1 12: evalwhile2bb4in -> evalwhile28 : D'=-1+A, [], cost: 1 13: evalwhile28 -> evalwhile29 : [], cost: 1 14: evalwhile29 -> evalwhile2bb1in : A'=D, [], cost: 1 ### Simplification by acceleration and chaining ### Eliminated locations (on linear paths): Start location: evalwhile2start 21: evalwhile2start -> evalwhile2bb1in : A'=B, [], cost: 7 7: evalwhile2bb1in -> evalwhile2bb2in : C'=B, [ A>=1 ], cost: 1 22: evalwhile2bb2in -> evalwhile2bb2in : C'=-1+C, [ C>=1 ], cost: 2 25: evalwhile2bb2in -> evalwhile2bb1in : A'=-1+A, D'=-1+A, [ 0>=C ], cost: 4 Accelerating simple loops of location 8. Accelerating the following rules: 22: evalwhile2bb2in -> evalwhile2bb2in : C'=-1+C, [ C>=1 ], cost: 2 Accelerated rule 22 with metering function C, yielding the new rule 26. Removing the simple loops: 22. Accelerated all simple loops using metering functions (where possible): Start location: evalwhile2start 21: evalwhile2start -> evalwhile2bb1in : A'=B, [], cost: 7 7: evalwhile2bb1in -> evalwhile2bb2in : C'=B, [ A>=1 ], cost: 1 25: evalwhile2bb2in -> evalwhile2bb1in : A'=-1+A, D'=-1+A, [ 0>=C ], cost: 4 26: evalwhile2bb2in -> evalwhile2bb2in : C'=0, [ C>=1 ], cost: 2*C Chained accelerated rules (with incoming rules): Start location: evalwhile2start 21: evalwhile2start -> evalwhile2bb1in : A'=B, [], cost: 7 7: evalwhile2bb1in -> evalwhile2bb2in : C'=B, [ A>=1 ], cost: 1 27: evalwhile2bb1in -> evalwhile2bb2in : C'=0, [ A>=1 && B>=1 ], cost: 1+2*B 25: evalwhile2bb2in -> evalwhile2bb1in : A'=-1+A, D'=-1+A, [ 0>=C ], cost: 4 Eliminated locations (on tree-shaped paths): Start location: evalwhile2start 21: evalwhile2start -> evalwhile2bb1in : A'=B, [], cost: 7 28: evalwhile2bb1in -> evalwhile2bb1in : A'=-1+A, C'=B, D'=-1+A, [ A>=1 && 0>=B ], cost: 5 29: evalwhile2bb1in -> evalwhile2bb1in : A'=-1+A, C'=0, D'=-1+A, [ A>=1 && B>=1 ], cost: 5+2*B Accelerating simple loops of location 7. Accelerating the following rules: 28: evalwhile2bb1in -> evalwhile2bb1in : A'=-1+A, C'=B, D'=-1+A, [ A>=1 && 0>=B ], cost: 5 29: evalwhile2bb1in -> evalwhile2bb1in : A'=-1+A, C'=0, D'=-1+A, [ A>=1 && B>=1 ], cost: 5+2*B Accelerated rule 28 with metering function A, yielding the new rule 30. Accelerated rule 29 with metering function A, yielding the new rule 31. Removing the simple loops: 28 29. Accelerated all simple loops using metering functions (where possible): Start location: evalwhile2start 21: evalwhile2start -> evalwhile2bb1in : A'=B, [], cost: 7 30: evalwhile2bb1in -> evalwhile2bb1in : A'=0, C'=B, D'=0, [ A>=1 && 0>=B ], cost: 5*A 31: evalwhile2bb1in -> evalwhile2bb1in : A'=0, C'=0, D'=0, [ A>=1 && B>=1 ], cost: 2*A*B+5*A Chained accelerated rules (with incoming rules): Start location: evalwhile2start 21: evalwhile2start -> evalwhile2bb1in : A'=B, [], cost: 7 32: evalwhile2start -> evalwhile2bb1in : A'=0, C'=0, D'=0, [ B>=1 ], cost: 7+5*B+2*B^2 Removed unreachable locations (and leaf rules with constant cost): Start location: evalwhile2start 32: evalwhile2start -> evalwhile2bb1in : A'=0, C'=0, D'=0, [ B>=1 ], cost: 7+5*B+2*B^2 ### Computing asymptotic complexity ### Fully simplified ITS problem Start location: evalwhile2start 32: evalwhile2start -> evalwhile2bb1in : A'=0, C'=0, D'=0, [ B>=1 ], cost: 7+5*B+2*B^2 Computing asymptotic complexity for rule 32 Solved the limit problem by the following transformations: Created initial limit problem: 7+5*B+2*B^2 (+), B (+/+!) [not solved] removing all constraints (solved by SMT) resulting limit problem: [solved] applying transformation rule (C) using substitution {B==n} resulting limit problem: [solved] Solution: B / n Resulting cost 7+5*n+2*n^2 has complexity: Poly(n^2) Found new complexity Poly(n^2). Obtained the following overall complexity (w.r.t. the length of the input n): Complexity: Poly(n^2) Cpx degree: 2 Solved cost: 7+5*n+2*n^2 Rule cost: 7+5*B+2*B^2 Rule guard: [ B>=1 ] WORST_CASE(Omega(n^2),?) ---------------------------------------- (4) BOUNDS(n^2, INF)