/export/starexec/sandbox/solver/bin/starexec_run_complexity /export/starexec/sandbox/benchmark/theBenchmark.koat /export/starexec/sandbox/output/output_files -------------------------------------------------------------------------------- WORST_CASE(Omega(n^2), O(n^2)) proof of /export/starexec/sandbox/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(n^2, n^2).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, 123 ms] (2) BOUNDS(1, n^2) (3) Loat Proof [FINISHED, 827 ms] (4) BOUNDS(n^2, INF) ---------------------------------------- (0) Obligation: Complexity Int TRS consisting of the following rules: eval_ax_start(v__0, v__01, v_3, v_i, v_j, v_n) -> Com_1(eval_ax_bb0_in(v__0, v__01, v_3, v_i, v_j, v_n)) :|: TRUE eval_ax_bb0_in(v__0, v__01, v_3, v_i, v_j, v_n) -> Com_1(eval_ax_0(v__0, v__01, v_3, v_i, v_j, v_n)) :|: TRUE eval_ax_0(v__0, v__01, v_3, v_i, v_j, v_n) -> Com_1(eval_ax_1(v__0, v__01, v_3, v_i, v_j, v_n)) :|: TRUE eval_ax_1(v__0, v__01, v_3, v_i, v_j, v_n) -> Com_1(eval_ax_2(v__0, v__01, v_3, v_i, v_j, v_n)) :|: TRUE eval_ax_2(v__0, v__01, v_3, v_i, v_j, v_n) -> Com_1(eval_ax_3(v__0, v__01, v_3, v_i, v_j, v_n)) :|: TRUE eval_ax_3(v__0, v__01, v_3, v_i, v_j, v_n) -> Com_1(eval_ax_4(v__0, v__01, v_3, v_i, v_j, v_n)) :|: TRUE eval_ax_4(v__0, v__01, v_3, v_i, v_j, v_n) -> Com_1(eval_ax_5(v__0, v__01, v_3, v_i, v_j, v_n)) :|: TRUE eval_ax_5(v__0, v__01, v_3, v_i, v_j, v_n) -> Com_1(eval_ax_6(v__0, v__01, v_3, v_i, v_j, v_n)) :|: TRUE eval_ax_6(v__0, v__01, v_3, v_i, v_j, v_n) -> Com_1(eval_ax_bb1_in(0, v__01, v_3, v_i, v_j, v_n)) :|: TRUE eval_ax_bb1_in(v__0, v__01, v_3, v_i, v_j, v_n) -> Com_1(eval_ax_bb2_in(v__0, 0, v_3, v_i, v_j, v_n)) :|: TRUE eval_ax_bb2_in(v__0, v__01, v_3, v_i, v_j, v_n) -> Com_1(eval_ax_bb3_in(v__0, v__01, v_3, v_i, v_j, v_n)) :|: v__01 < v_n - 1 eval_ax_bb2_in(v__0, v__01, v_3, v_i, v_j, v_n) -> Com_1(eval_ax_bb4_in(v__0, v__01, v_3, v_i, v_j, v_n)) :|: v__01 >= v_n - 1 eval_ax_bb3_in(v__0, v__01, v_3, v_i, v_j, v_n) -> Com_1(eval_ax_bb2_in(v__0, v__01 + 1, v_3, v_i, v_j, v_n)) :|: TRUE eval_ax_bb4_in(v__0, v__01, v_3, v_i, v_j, v_n) -> Com_1(eval_ax_12(v__0, v__01, v__0 + 1, v_i, v_j, v_n)) :|: TRUE eval_ax_12(v__0, v__01, v_3, v_i, v_j, v_n) -> Com_1(eval_ax_13(v__0, v__01, v_3, v_i, v_j, v_n)) :|: TRUE eval_ax_13(v__0, v__01, v_3, v_i, v_j, v_n) -> Com_1(eval_ax_bb1_in(v_3, v__01, v_3, v_i, v_j, v_n)) :|: v__01 >= v_n - 1 && v_3 < v_n - 1 eval_ax_13(v__0, v__01, v_3, v_i, v_j, v_n) -> Com_1(eval_ax_bb5_in(v__0, v__01, v_3, v_i, v_j, v_n)) :|: v__01 < v_n - 1 eval_ax_13(v__0, v__01, v_3, v_i, v_j, v_n) -> Com_1(eval_ax_bb5_in(v__0, v__01, v_3, v_i, v_j, v_n)) :|: v_3 >= v_n - 1 eval_ax_bb5_in(v__0, v__01, v_3, v_i, v_j, v_n) -> Com_1(eval_ax_stop(v__0, v__01, v_3, v_i, v_j, v_n)) :|: TRUE The start-symbols are:[eval_ax_start_6] ---------------------------------------- (1) Koat Proof (FINISHED) YES(?, 15*Ar_2 + 2*Ar_2^2 + 27) Initial complexity problem: 1: T: (Comp: ?, Cost: 1) evalaxstart(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalaxbb0in(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: ?, Cost: 1) evalaxbb0in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalax0(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: ?, Cost: 1) evalax0(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalax1(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: ?, Cost: 1) evalax1(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalax2(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: ?, Cost: 1) evalax2(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalax3(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: ?, Cost: 1) evalax3(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalax4(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: ?, Cost: 1) evalax4(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalax5(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: ?, Cost: 1) evalax5(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalax6(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: ?, Cost: 1) evalax6(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalaxbb1in(0, Ar_1, Ar_2, Ar_3)) (Comp: ?, Cost: 1) evalaxbb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalaxbb2in(Ar_0, 0, Ar_2, Ar_3)) (Comp: ?, Cost: 1) evalaxbb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalaxbb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= Ar_1 + 2 ] (Comp: ?, Cost: 1) evalaxbb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalaxbb4in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_1 + 1 >= Ar_2 ] (Comp: ?, Cost: 1) evalaxbb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalaxbb2in(Ar_0, Ar_1 + 1, Ar_2, Ar_3)) (Comp: ?, Cost: 1) evalaxbb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalax12(Ar_0, Ar_1, Ar_2, Ar_0 + 1)) (Comp: ?, Cost: 1) evalax12(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalax13(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: ?, Cost: 1) evalax13(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalaxbb1in(Ar_3, Ar_1, Ar_2, Ar_3)) [ Ar_1 + 1 >= Ar_2 /\ Ar_2 >= Ar_3 + 2 ] (Comp: ?, Cost: 1) evalax13(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalaxbb5in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= Ar_1 + 2 ] (Comp: ?, Cost: 1) evalax13(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalaxbb5in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_3 + 1 >= Ar_2 ] (Comp: ?, Cost: 1) evalaxbb5in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalaxstop(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalaxstart(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) evalaxstart(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalaxbb0in(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 1) evalaxbb0in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalax0(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 1) evalax0(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalax1(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 1) evalax1(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalax2(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 1) evalax2(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalax3(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 1) evalax3(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalax4(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 1) evalax4(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalax5(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 1) evalax5(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalax6(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 1) evalax6(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalaxbb1in(0, Ar_1, Ar_2, Ar_3)) (Comp: ?, Cost: 1) evalaxbb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalaxbb2in(Ar_0, 0, Ar_2, Ar_3)) (Comp: ?, Cost: 1) evalaxbb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalaxbb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= Ar_1 + 2 ] (Comp: ?, Cost: 1) evalaxbb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalaxbb4in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_1 + 1 >= Ar_2 ] (Comp: ?, Cost: 1) evalaxbb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalaxbb2in(Ar_0, Ar_1 + 1, Ar_2, Ar_3)) (Comp: ?, Cost: 1) evalaxbb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalax12(Ar_0, Ar_1, Ar_2, Ar_0 + 1)) (Comp: ?, Cost: 1) evalax12(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalax13(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: ?, Cost: 1) evalax13(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalaxbb1in(Ar_3, Ar_1, Ar_2, Ar_3)) [ Ar_1 + 1 >= Ar_2 /\ Ar_2 >= Ar_3 + 2 ] (Comp: ?, Cost: 1) evalax13(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalaxbb5in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= Ar_1 + 2 ] (Comp: ?, Cost: 1) evalax13(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalaxbb5in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_3 + 1 >= Ar_2 ] (Comp: ?, Cost: 1) evalaxbb5in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalaxstop(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalaxstart(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 A polynomial rank function with Pol(evalaxstart) = 2 Pol(evalaxbb0in) = 2 Pol(evalax0) = 2 Pol(evalax1) = 2 Pol(evalax2) = 2 Pol(evalax3) = 2 Pol(evalax4) = 2 Pol(evalax5) = 2 Pol(evalax6) = 2 Pol(evalaxbb1in) = 2 Pol(evalaxbb2in) = 2 Pol(evalaxbb3in) = 2 Pol(evalaxbb4in) = 2 Pol(evalax12) = 2 Pol(evalax13) = 2 Pol(evalaxbb5in) = 1 Pol(evalaxstop) = 0 Pol(koat_start) = 2 orients all transitions weakly and the transitions evalaxbb5in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalaxstop(Ar_0, Ar_1, Ar_2, Ar_3)) evalax13(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalaxbb5in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_3 + 1 >= Ar_2 ] evalax13(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalaxbb5in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= Ar_1 + 2 ] strictly and produces the following problem: 3: T: (Comp: 1, Cost: 1) evalaxstart(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalaxbb0in(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 1) evalaxbb0in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalax0(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 1) evalax0(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalax1(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 1) evalax1(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalax2(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 1) evalax2(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalax3(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 1) evalax3(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalax4(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 1) evalax4(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalax5(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 1) evalax5(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalax6(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 1) evalax6(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalaxbb1in(0, Ar_1, Ar_2, Ar_3)) (Comp: ?, Cost: 1) evalaxbb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalaxbb2in(Ar_0, 0, Ar_2, Ar_3)) (Comp: ?, Cost: 1) evalaxbb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalaxbb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= Ar_1 + 2 ] (Comp: ?, Cost: 1) evalaxbb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalaxbb4in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_1 + 1 >= Ar_2 ] (Comp: ?, Cost: 1) evalaxbb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalaxbb2in(Ar_0, Ar_1 + 1, Ar_2, Ar_3)) (Comp: ?, Cost: 1) evalaxbb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalax12(Ar_0, Ar_1, Ar_2, Ar_0 + 1)) (Comp: ?, Cost: 1) evalax12(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalax13(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: ?, Cost: 1) evalax13(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalaxbb1in(Ar_3, Ar_1, Ar_2, Ar_3)) [ Ar_1 + 1 >= Ar_2 /\ Ar_2 >= Ar_3 + 2 ] (Comp: 2, Cost: 1) evalax13(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalaxbb5in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= Ar_1 + 2 ] (Comp: 2, Cost: 1) evalax13(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalaxbb5in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_3 + 1 >= Ar_2 ] (Comp: 2, Cost: 1) evalaxbb5in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalaxstop(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalaxstart(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 A polynomial rank function with Pol(evalaxstart) = V_3 Pol(evalaxbb0in) = V_3 Pol(evalax0) = V_3 Pol(evalax1) = V_3 Pol(evalax2) = V_3 Pol(evalax3) = V_3 Pol(evalax4) = V_3 Pol(evalax5) = V_3 Pol(evalax6) = V_3 Pol(evalaxbb1in) = -V_1 + V_3 - 1 Pol(evalaxbb2in) = -V_1 + V_3 - 1 Pol(evalaxbb3in) = -V_1 + V_3 - 1 Pol(evalaxbb4in) = -V_1 + V_3 - 1 Pol(evalax12) = V_3 - V_4 Pol(evalax13) = V_3 - V_4 Pol(evalaxbb5in) = V_3 - V_4 Pol(evalaxstop) = V_3 - V_4 Pol(koat_start) = V_3 orients all transitions weakly and the transition evalax13(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalaxbb1in(Ar_3, Ar_1, Ar_2, Ar_3)) [ Ar_1 + 1 >= Ar_2 /\ Ar_2 >= Ar_3 + 2 ] strictly and produces the following problem: 4: T: (Comp: 1, Cost: 1) evalaxstart(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalaxbb0in(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 1) evalaxbb0in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalax0(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 1) evalax0(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalax1(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 1) evalax1(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalax2(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 1) evalax2(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalax3(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 1) evalax3(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalax4(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 1) evalax4(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalax5(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 1) evalax5(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalax6(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 1) evalax6(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalaxbb1in(0, Ar_1, Ar_2, Ar_3)) (Comp: ?, Cost: 1) evalaxbb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalaxbb2in(Ar_0, 0, Ar_2, Ar_3)) (Comp: ?, Cost: 1) evalaxbb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalaxbb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= Ar_1 + 2 ] (Comp: ?, Cost: 1) evalaxbb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalaxbb4in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_1 + 1 >= Ar_2 ] (Comp: ?, Cost: 1) evalaxbb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalaxbb2in(Ar_0, Ar_1 + 1, Ar_2, Ar_3)) (Comp: ?, Cost: 1) evalaxbb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalax12(Ar_0, Ar_1, Ar_2, Ar_0 + 1)) (Comp: ?, Cost: 1) evalax12(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalax13(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: Ar_2, Cost: 1) evalax13(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalaxbb1in(Ar_3, Ar_1, Ar_2, Ar_3)) [ Ar_1 + 1 >= Ar_2 /\ Ar_2 >= Ar_3 + 2 ] (Comp: 2, Cost: 1) evalax13(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalaxbb5in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= Ar_1 + 2 ] (Comp: 2, Cost: 1) evalax13(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalaxbb5in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_3 + 1 >= Ar_2 ] (Comp: 2, Cost: 1) evalaxbb5in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalaxstop(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalaxstart(Ar_0, Ar_1, Ar_2, Ar_3)) [ 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) evalaxstart(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalaxbb0in(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 1) evalaxbb0in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalax0(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 1) evalax0(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalax1(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 1) evalax1(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalax2(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 1) evalax2(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalax3(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 1) evalax3(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalax4(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 1) evalax4(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalax5(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 1) evalax5(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalax6(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 1) evalax6(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalaxbb1in(0, Ar_1, Ar_2, Ar_3)) (Comp: Ar_2 + 1, Cost: 1) evalaxbb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalaxbb2in(Ar_0, 0, Ar_2, Ar_3)) (Comp: ?, Cost: 1) evalaxbb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalaxbb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= Ar_1 + 2 ] (Comp: ?, Cost: 1) evalaxbb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalaxbb4in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_1 + 1 >= Ar_2 ] (Comp: ?, Cost: 1) evalaxbb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalaxbb2in(Ar_0, Ar_1 + 1, Ar_2, Ar_3)) (Comp: ?, Cost: 1) evalaxbb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalax12(Ar_0, Ar_1, Ar_2, Ar_0 + 1)) (Comp: ?, Cost: 1) evalax12(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalax13(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: Ar_2, Cost: 1) evalax13(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalaxbb1in(Ar_3, Ar_1, Ar_2, Ar_3)) [ Ar_1 + 1 >= Ar_2 /\ Ar_2 >= Ar_3 + 2 ] (Comp: 2, Cost: 1) evalax13(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalaxbb5in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= Ar_1 + 2 ] (Comp: 2, Cost: 1) evalax13(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalaxbb5in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_3 + 1 >= Ar_2 ] (Comp: 2, Cost: 1) evalaxbb5in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalaxstop(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalaxstart(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 A polynomial rank function with Pol(evalaxbb4in) = 2 Pol(evalax12) = 1 Pol(evalaxbb3in) = 3 Pol(evalaxbb2in) = 3 Pol(evalax13) = 0 and size complexities S("koat_start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalaxstart(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(evalaxstart(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(evalaxstart(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(evalaxstart(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 <= 0 ]", 0-3) = Ar_3 S("evalaxbb5in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalaxstop(Ar_0, Ar_1, Ar_2, Ar_3))", 0-0) = ? S("evalaxbb5in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalaxstop(Ar_0, Ar_1, Ar_2, Ar_3))", 0-1) = ? S("evalaxbb5in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalaxstop(Ar_0, Ar_1, Ar_2, Ar_3))", 0-2) = Ar_2 S("evalaxbb5in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalaxstop(Ar_0, Ar_1, Ar_2, Ar_3))", 0-3) = ? S("evalax13(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalaxbb5in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_3 + 1 >= Ar_2 ]", 0-0) = ? S("evalax13(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalaxbb5in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_3 + 1 >= Ar_2 ]", 0-1) = ? S("evalax13(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalaxbb5in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_3 + 1 >= Ar_2 ]", 0-2) = Ar_2 S("evalax13(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalaxbb5in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_3 + 1 >= Ar_2 ]", 0-3) = ? S("evalax13(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalaxbb5in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= Ar_1 + 2 ]", 0-0) = ? S("evalax13(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalaxbb5in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= Ar_1 + 2 ]", 0-1) = ? S("evalax13(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalaxbb5in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= Ar_1 + 2 ]", 0-2) = Ar_2 S("evalax13(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalaxbb5in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= Ar_1 + 2 ]", 0-3) = ? S("evalax13(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalaxbb1in(Ar_3, Ar_1, Ar_2, Ar_3)) [ Ar_1 + 1 >= Ar_2 /\\ Ar_2 >= Ar_3 + 2 ]", 0-0) = ? S("evalax13(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalaxbb1in(Ar_3, Ar_1, Ar_2, Ar_3)) [ Ar_1 + 1 >= Ar_2 /\\ Ar_2 >= Ar_3 + 2 ]", 0-1) = ? S("evalax13(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalaxbb1in(Ar_3, Ar_1, Ar_2, Ar_3)) [ Ar_1 + 1 >= Ar_2 /\\ Ar_2 >= Ar_3 + 2 ]", 0-2) = Ar_2 S("evalax13(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalaxbb1in(Ar_3, Ar_1, Ar_2, Ar_3)) [ Ar_1 + 1 >= Ar_2 /\\ Ar_2 >= Ar_3 + 2 ]", 0-3) = ? S("evalax12(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalax13(Ar_0, Ar_1, Ar_2, Ar_3))", 0-0) = ? S("evalax12(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalax13(Ar_0, Ar_1, Ar_2, Ar_3))", 0-1) = ? S("evalax12(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalax13(Ar_0, Ar_1, Ar_2, Ar_3))", 0-2) = Ar_2 S("evalax12(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalax13(Ar_0, Ar_1, Ar_2, Ar_3))", 0-3) = ? S("evalaxbb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalax12(Ar_0, Ar_1, Ar_2, Ar_0 + 1))", 0-0) = ? S("evalaxbb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalax12(Ar_0, Ar_1, Ar_2, Ar_0 + 1))", 0-1) = ? S("evalaxbb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalax12(Ar_0, Ar_1, Ar_2, Ar_0 + 1))", 0-2) = Ar_2 S("evalaxbb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalax12(Ar_0, Ar_1, Ar_2, Ar_0 + 1))", 0-3) = ? S("evalaxbb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalaxbb2in(Ar_0, Ar_1 + 1, Ar_2, Ar_3))", 0-0) = ? S("evalaxbb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalaxbb2in(Ar_0, Ar_1 + 1, Ar_2, Ar_3))", 0-1) = ? S("evalaxbb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalaxbb2in(Ar_0, Ar_1 + 1, Ar_2, Ar_3))", 0-2) = Ar_2 S("evalaxbb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalaxbb2in(Ar_0, Ar_1 + 1, Ar_2, Ar_3))", 0-3) = ? S("evalaxbb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalaxbb4in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_1 + 1 >= Ar_2 ]", 0-0) = ? S("evalaxbb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalaxbb4in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_1 + 1 >= Ar_2 ]", 0-1) = ? S("evalaxbb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalaxbb4in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_1 + 1 >= Ar_2 ]", 0-2) = Ar_2 S("evalaxbb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalaxbb4in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_1 + 1 >= Ar_2 ]", 0-3) = ? S("evalaxbb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalaxbb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= Ar_1 + 2 ]", 0-0) = ? S("evalaxbb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalaxbb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= Ar_1 + 2 ]", 0-1) = ? S("evalaxbb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalaxbb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= Ar_1 + 2 ]", 0-2) = Ar_2 S("evalaxbb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalaxbb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= Ar_1 + 2 ]", 0-3) = ? S("evalaxbb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalaxbb2in(Ar_0, 0, Ar_2, Ar_3))", 0-0) = ? S("evalaxbb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalaxbb2in(Ar_0, 0, Ar_2, Ar_3))", 0-1) = 0 S("evalaxbb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalaxbb2in(Ar_0, 0, Ar_2, Ar_3))", 0-2) = Ar_2 S("evalaxbb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalaxbb2in(Ar_0, 0, Ar_2, Ar_3))", 0-3) = ? S("evalax6(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalaxbb1in(0, Ar_1, Ar_2, Ar_3))", 0-0) = 0 S("evalax6(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalaxbb1in(0, Ar_1, Ar_2, Ar_3))", 0-1) = Ar_1 S("evalax6(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalaxbb1in(0, Ar_1, Ar_2, Ar_3))", 0-2) = Ar_2 S("evalax6(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalaxbb1in(0, Ar_1, Ar_2, Ar_3))", 0-3) = Ar_3 S("evalax5(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalax6(Ar_0, Ar_1, Ar_2, Ar_3))", 0-0) = Ar_0 S("evalax5(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalax6(Ar_0, Ar_1, Ar_2, Ar_3))", 0-1) = Ar_1 S("evalax5(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalax6(Ar_0, Ar_1, Ar_2, Ar_3))", 0-2) = Ar_2 S("evalax5(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalax6(Ar_0, Ar_1, Ar_2, Ar_3))", 0-3) = Ar_3 S("evalax4(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalax5(Ar_0, Ar_1, Ar_2, Ar_3))", 0-0) = Ar_0 S("evalax4(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalax5(Ar_0, Ar_1, Ar_2, Ar_3))", 0-1) = Ar_1 S("evalax4(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalax5(Ar_0, Ar_1, Ar_2, Ar_3))", 0-2) = Ar_2 S("evalax4(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalax5(Ar_0, Ar_1, Ar_2, Ar_3))", 0-3) = Ar_3 S("evalax3(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalax4(Ar_0, Ar_1, Ar_2, Ar_3))", 0-0) = Ar_0 S("evalax3(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalax4(Ar_0, Ar_1, Ar_2, Ar_3))", 0-1) = Ar_1 S("evalax3(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalax4(Ar_0, Ar_1, Ar_2, Ar_3))", 0-2) = Ar_2 S("evalax3(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalax4(Ar_0, Ar_1, Ar_2, Ar_3))", 0-3) = Ar_3 S("evalax2(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalax3(Ar_0, Ar_1, Ar_2, Ar_3))", 0-0) = Ar_0 S("evalax2(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalax3(Ar_0, Ar_1, Ar_2, Ar_3))", 0-1) = Ar_1 S("evalax2(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalax3(Ar_0, Ar_1, Ar_2, Ar_3))", 0-2) = Ar_2 S("evalax2(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalax3(Ar_0, Ar_1, Ar_2, Ar_3))", 0-3) = Ar_3 S("evalax1(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalax2(Ar_0, Ar_1, Ar_2, Ar_3))", 0-0) = Ar_0 S("evalax1(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalax2(Ar_0, Ar_1, Ar_2, Ar_3))", 0-1) = Ar_1 S("evalax1(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalax2(Ar_0, Ar_1, Ar_2, Ar_3))", 0-2) = Ar_2 S("evalax1(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalax2(Ar_0, Ar_1, Ar_2, Ar_3))", 0-3) = Ar_3 S("evalax0(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalax1(Ar_0, Ar_1, Ar_2, Ar_3))", 0-0) = Ar_0 S("evalax0(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalax1(Ar_0, Ar_1, Ar_2, Ar_3))", 0-1) = Ar_1 S("evalax0(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalax1(Ar_0, Ar_1, Ar_2, Ar_3))", 0-2) = Ar_2 S("evalax0(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalax1(Ar_0, Ar_1, Ar_2, Ar_3))", 0-3) = Ar_3 S("evalaxbb0in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalax0(Ar_0, Ar_1, Ar_2, Ar_3))", 0-0) = Ar_0 S("evalaxbb0in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalax0(Ar_0, Ar_1, Ar_2, Ar_3))", 0-1) = Ar_1 S("evalaxbb0in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalax0(Ar_0, Ar_1, Ar_2, Ar_3))", 0-2) = Ar_2 S("evalaxbb0in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalax0(Ar_0, Ar_1, Ar_2, Ar_3))", 0-3) = Ar_3 S("evalaxstart(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalaxbb0in(Ar_0, Ar_1, Ar_2, Ar_3))", 0-0) = Ar_0 S("evalaxstart(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalaxbb0in(Ar_0, Ar_1, Ar_2, Ar_3))", 0-1) = Ar_1 S("evalaxstart(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalaxbb0in(Ar_0, Ar_1, Ar_2, Ar_3))", 0-2) = Ar_2 S("evalaxstart(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalaxbb0in(Ar_0, Ar_1, Ar_2, Ar_3))", 0-3) = Ar_3 orients the transitions evalaxbb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalax12(Ar_0, Ar_1, Ar_2, Ar_0 + 1)) evalaxbb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalaxbb2in(Ar_0, Ar_1 + 1, Ar_2, Ar_3)) evalaxbb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalaxbb4in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_1 + 1 >= Ar_2 ] evalaxbb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalaxbb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= Ar_1 + 2 ] evalax12(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalax13(Ar_0, Ar_1, Ar_2, Ar_3)) weakly and the transitions evalaxbb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalax12(Ar_0, Ar_1, Ar_2, Ar_0 + 1)) evalaxbb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalaxbb4in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_1 + 1 >= Ar_2 ] evalax12(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalax13(Ar_0, Ar_1, Ar_2, Ar_3)) strictly and produces the following problem: 6: T: (Comp: 1, Cost: 1) evalaxstart(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalaxbb0in(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 1) evalaxbb0in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalax0(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 1) evalax0(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalax1(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 1) evalax1(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalax2(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 1) evalax2(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalax3(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 1) evalax3(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalax4(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 1) evalax4(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalax5(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 1) evalax5(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalax6(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 1) evalax6(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalaxbb1in(0, Ar_1, Ar_2, Ar_3)) (Comp: Ar_2 + 1, Cost: 1) evalaxbb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalaxbb2in(Ar_0, 0, Ar_2, Ar_3)) (Comp: ?, Cost: 1) evalaxbb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalaxbb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= Ar_1 + 2 ] (Comp: 3*Ar_2 + 3, Cost: 1) evalaxbb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalaxbb4in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_1 + 1 >= Ar_2 ] (Comp: ?, Cost: 1) evalaxbb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalaxbb2in(Ar_0, Ar_1 + 1, Ar_2, Ar_3)) (Comp: 3*Ar_2 + 3, Cost: 1) evalaxbb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalax12(Ar_0, Ar_1, Ar_2, Ar_0 + 1)) (Comp: 3*Ar_2 + 3, Cost: 1) evalax12(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalax13(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: Ar_2, Cost: 1) evalax13(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalaxbb1in(Ar_3, Ar_1, Ar_2, Ar_3)) [ Ar_1 + 1 >= Ar_2 /\ Ar_2 >= Ar_3 + 2 ] (Comp: 2, Cost: 1) evalax13(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalaxbb5in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= Ar_1 + 2 ] (Comp: 2, Cost: 1) evalax13(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalaxbb5in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_3 + 1 >= Ar_2 ] (Comp: 2, Cost: 1) evalaxbb5in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalaxstop(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalaxstart(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 A polynomial rank function with Pol(evalaxbb3in) = -V_2 + V_3 Pol(evalaxbb2in) = -V_2 + V_3 + 1 and size complexities S("koat_start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalaxstart(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(evalaxstart(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(evalaxstart(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(evalaxstart(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 <= 0 ]", 0-3) = Ar_3 S("evalaxbb5in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalaxstop(Ar_0, Ar_1, Ar_2, Ar_3))", 0-0) = 3*Ar_2 + 2187 S("evalaxbb5in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalaxstop(Ar_0, Ar_1, Ar_2, Ar_3))", 0-1) = ? S("evalaxbb5in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalaxstop(Ar_0, Ar_1, Ar_2, Ar_3))", 0-2) = Ar_2 S("evalaxbb5in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalaxstop(Ar_0, Ar_1, Ar_2, Ar_3))", 0-3) = 3*Ar_2 + 243 S("evalax13(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalaxbb5in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_3 + 1 >= Ar_2 ]", 0-0) = 3*Ar_2 + 729 S("evalax13(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalaxbb5in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_3 + 1 >= Ar_2 ]", 0-1) = ? S("evalax13(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalaxbb5in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_3 + 1 >= Ar_2 ]", 0-2) = Ar_2 S("evalax13(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalaxbb5in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_3 + 1 >= Ar_2 ]", 0-3) = 3*Ar_2 + 81 S("evalax13(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalaxbb5in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= Ar_1 + 2 ]", 0-0) = 3*Ar_2 + 729 S("evalax13(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalaxbb5in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= Ar_1 + 2 ]", 0-1) = ? S("evalax13(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalaxbb5in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= Ar_1 + 2 ]", 0-2) = Ar_2 S("evalax13(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalaxbb5in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= Ar_1 + 2 ]", 0-3) = 3*Ar_2 + 81 S("evalax13(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalaxbb1in(Ar_3, Ar_1, Ar_2, Ar_3)) [ Ar_1 + 1 >= Ar_2 /\\ Ar_2 >= Ar_3 + 2 ]", 0-0) = 3*Ar_2 + 27 S("evalax13(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalaxbb1in(Ar_3, Ar_1, Ar_2, Ar_3)) [ Ar_1 + 1 >= Ar_2 /\\ Ar_2 >= Ar_3 + 2 ]", 0-1) = ? S("evalax13(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalaxbb1in(Ar_3, Ar_1, Ar_2, Ar_3)) [ Ar_1 + 1 >= Ar_2 /\\ Ar_2 >= Ar_3 + 2 ]", 0-2) = Ar_2 S("evalax13(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalaxbb1in(Ar_3, Ar_1, Ar_2, Ar_3)) [ Ar_1 + 1 >= Ar_2 /\\ Ar_2 >= Ar_3 + 2 ]", 0-3) = 3*Ar_2 + 81 S("evalax12(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalax13(Ar_0, Ar_1, Ar_2, Ar_3))", 0-0) = 3*Ar_2 + 243 S("evalax12(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalax13(Ar_0, Ar_1, Ar_2, Ar_3))", 0-1) = ? S("evalax12(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalax13(Ar_0, Ar_1, Ar_2, Ar_3))", 0-2) = Ar_2 S("evalax12(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalax13(Ar_0, Ar_1, Ar_2, Ar_3))", 0-3) = 3*Ar_2 + 27 S("evalaxbb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalax12(Ar_0, Ar_1, Ar_2, Ar_0 + 1))", 0-0) = 3*Ar_2 + 81 S("evalaxbb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalax12(Ar_0, Ar_1, Ar_2, Ar_0 + 1))", 0-1) = ? S("evalaxbb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalax12(Ar_0, Ar_1, Ar_2, Ar_0 + 1))", 0-2) = Ar_2 S("evalaxbb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalax12(Ar_0, Ar_1, Ar_2, Ar_0 + 1))", 0-3) = 3*Ar_2 + 27 S("evalaxbb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalaxbb2in(Ar_0, Ar_1 + 1, Ar_2, Ar_3))", 0-0) = 3*Ar_2 + 27 S("evalaxbb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalaxbb2in(Ar_0, Ar_1 + 1, Ar_2, Ar_3))", 0-1) = ? S("evalaxbb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalaxbb2in(Ar_0, Ar_1 + 1, Ar_2, Ar_3))", 0-2) = Ar_2 S("evalaxbb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalaxbb2in(Ar_0, Ar_1 + 1, Ar_2, Ar_3))", 0-3) = 3*Ar_2 + 3*Ar_3 + 729 S("evalaxbb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalaxbb4in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_1 + 1 >= Ar_2 ]", 0-0) = 3*Ar_2 + 27 S("evalaxbb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalaxbb4in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_1 + 1 >= Ar_2 ]", 0-1) = ? S("evalaxbb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalaxbb4in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_1 + 1 >= Ar_2 ]", 0-2) = Ar_2 S("evalaxbb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalaxbb4in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_1 + 1 >= Ar_2 ]", 0-3) = 3*Ar_2 + 3*Ar_3 + 2187 S("evalaxbb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalaxbb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= Ar_1 + 2 ]", 0-0) = 3*Ar_2 + 27 S("evalaxbb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalaxbb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= Ar_1 + 2 ]", 0-1) = ? S("evalaxbb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalaxbb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= Ar_1 + 2 ]", 0-2) = Ar_2 S("evalaxbb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalaxbb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= Ar_1 + 2 ]", 0-3) = 3*Ar_2 + 3*Ar_3 + 729 S("evalaxbb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalaxbb2in(Ar_0, 0, Ar_2, Ar_3))", 0-0) = 3*Ar_2 + 27 S("evalaxbb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalaxbb2in(Ar_0, 0, Ar_2, Ar_3))", 0-1) = 0 S("evalaxbb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalaxbb2in(Ar_0, 0, Ar_2, Ar_3))", 0-2) = Ar_2 S("evalaxbb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalaxbb2in(Ar_0, 0, Ar_2, Ar_3))", 0-3) = 3*Ar_2 + 3*Ar_3 + 243 S("evalax6(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalaxbb1in(0, Ar_1, Ar_2, Ar_3))", 0-0) = 0 S("evalax6(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalaxbb1in(0, Ar_1, Ar_2, Ar_3))", 0-1) = Ar_1 S("evalax6(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalaxbb1in(0, Ar_1, Ar_2, Ar_3))", 0-2) = Ar_2 S("evalax6(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalaxbb1in(0, Ar_1, Ar_2, Ar_3))", 0-3) = Ar_3 S("evalax5(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalax6(Ar_0, Ar_1, Ar_2, Ar_3))", 0-0) = Ar_0 S("evalax5(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalax6(Ar_0, Ar_1, Ar_2, Ar_3))", 0-1) = Ar_1 S("evalax5(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalax6(Ar_0, Ar_1, Ar_2, Ar_3))", 0-2) = Ar_2 S("evalax5(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalax6(Ar_0, Ar_1, Ar_2, Ar_3))", 0-3) = Ar_3 S("evalax4(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalax5(Ar_0, Ar_1, Ar_2, Ar_3))", 0-0) = Ar_0 S("evalax4(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalax5(Ar_0, Ar_1, Ar_2, Ar_3))", 0-1) = Ar_1 S("evalax4(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalax5(Ar_0, Ar_1, Ar_2, Ar_3))", 0-2) = Ar_2 S("evalax4(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalax5(Ar_0, Ar_1, Ar_2, Ar_3))", 0-3) = Ar_3 S("evalax3(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalax4(Ar_0, Ar_1, Ar_2, Ar_3))", 0-0) = Ar_0 S("evalax3(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalax4(Ar_0, Ar_1, Ar_2, Ar_3))", 0-1) = Ar_1 S("evalax3(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalax4(Ar_0, Ar_1, Ar_2, Ar_3))", 0-2) = Ar_2 S("evalax3(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalax4(Ar_0, Ar_1, Ar_2, Ar_3))", 0-3) = Ar_3 S("evalax2(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalax3(Ar_0, Ar_1, Ar_2, Ar_3))", 0-0) = Ar_0 S("evalax2(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalax3(Ar_0, Ar_1, Ar_2, Ar_3))", 0-1) = Ar_1 S("evalax2(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalax3(Ar_0, Ar_1, Ar_2, Ar_3))", 0-2) = Ar_2 S("evalax2(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalax3(Ar_0, Ar_1, Ar_2, Ar_3))", 0-3) = Ar_3 S("evalax1(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalax2(Ar_0, Ar_1, Ar_2, Ar_3))", 0-0) = Ar_0 S("evalax1(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalax2(Ar_0, Ar_1, Ar_2, Ar_3))", 0-1) = Ar_1 S("evalax1(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalax2(Ar_0, Ar_1, Ar_2, Ar_3))", 0-2) = Ar_2 S("evalax1(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalax2(Ar_0, Ar_1, Ar_2, Ar_3))", 0-3) = Ar_3 S("evalax0(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalax1(Ar_0, Ar_1, Ar_2, Ar_3))", 0-0) = Ar_0 S("evalax0(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalax1(Ar_0, Ar_1, Ar_2, Ar_3))", 0-1) = Ar_1 S("evalax0(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalax1(Ar_0, Ar_1, Ar_2, Ar_3))", 0-2) = Ar_2 S("evalax0(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalax1(Ar_0, Ar_1, Ar_2, Ar_3))", 0-3) = Ar_3 S("evalaxbb0in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalax0(Ar_0, Ar_1, Ar_2, Ar_3))", 0-0) = Ar_0 S("evalaxbb0in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalax0(Ar_0, Ar_1, Ar_2, Ar_3))", 0-1) = Ar_1 S("evalaxbb0in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalax0(Ar_0, Ar_1, Ar_2, Ar_3))", 0-2) = Ar_2 S("evalaxbb0in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalax0(Ar_0, Ar_1, Ar_2, Ar_3))", 0-3) = Ar_3 S("evalaxstart(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalaxbb0in(Ar_0, Ar_1, Ar_2, Ar_3))", 0-0) = Ar_0 S("evalaxstart(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalaxbb0in(Ar_0, Ar_1, Ar_2, Ar_3))", 0-1) = Ar_1 S("evalaxstart(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalaxbb0in(Ar_0, Ar_1, Ar_2, Ar_3))", 0-2) = Ar_2 S("evalaxstart(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalaxbb0in(Ar_0, Ar_1, Ar_2, Ar_3))", 0-3) = Ar_3 orients the transitions evalaxbb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalaxbb2in(Ar_0, Ar_1 + 1, Ar_2, Ar_3)) evalaxbb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalaxbb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= Ar_1 + 2 ] weakly and the transition evalaxbb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalaxbb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= Ar_1 + 2 ] strictly and produces the following problem: 7: T: (Comp: 1, Cost: 1) evalaxstart(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalaxbb0in(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 1) evalaxbb0in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalax0(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 1) evalax0(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalax1(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 1) evalax1(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalax2(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 1) evalax2(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalax3(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 1) evalax3(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalax4(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 1) evalax4(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalax5(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 1) evalax5(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalax6(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 1) evalax6(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalaxbb1in(0, Ar_1, Ar_2, Ar_3)) (Comp: Ar_2 + 1, Cost: 1) evalaxbb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalaxbb2in(Ar_0, 0, Ar_2, Ar_3)) (Comp: Ar_2^2 + 2*Ar_2 + 1, Cost: 1) evalaxbb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalaxbb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= Ar_1 + 2 ] (Comp: 3*Ar_2 + 3, Cost: 1) evalaxbb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalaxbb4in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_1 + 1 >= Ar_2 ] (Comp: ?, Cost: 1) evalaxbb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalaxbb2in(Ar_0, Ar_1 + 1, Ar_2, Ar_3)) (Comp: 3*Ar_2 + 3, Cost: 1) evalaxbb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalax12(Ar_0, Ar_1, Ar_2, Ar_0 + 1)) (Comp: 3*Ar_2 + 3, Cost: 1) evalax12(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalax13(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: Ar_2, Cost: 1) evalax13(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalaxbb1in(Ar_3, Ar_1, Ar_2, Ar_3)) [ Ar_1 + 1 >= Ar_2 /\ Ar_2 >= Ar_3 + 2 ] (Comp: 2, Cost: 1) evalax13(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalaxbb5in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= Ar_1 + 2 ] (Comp: 2, Cost: 1) evalax13(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalaxbb5in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_3 + 1 >= Ar_2 ] (Comp: 2, Cost: 1) evalaxbb5in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalaxstop(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalaxstart(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 Repeatedly propagating knowledge in problem 7 produces the following problem: 8: T: (Comp: 1, Cost: 1) evalaxstart(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalaxbb0in(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 1) evalaxbb0in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalax0(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 1) evalax0(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalax1(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 1) evalax1(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalax2(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 1) evalax2(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalax3(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 1) evalax3(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalax4(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 1) evalax4(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalax5(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 1) evalax5(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalax6(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 1) evalax6(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalaxbb1in(0, Ar_1, Ar_2, Ar_3)) (Comp: Ar_2 + 1, Cost: 1) evalaxbb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalaxbb2in(Ar_0, 0, Ar_2, Ar_3)) (Comp: Ar_2^2 + 2*Ar_2 + 1, Cost: 1) evalaxbb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalaxbb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= Ar_1 + 2 ] (Comp: 3*Ar_2 + 3, Cost: 1) evalaxbb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalaxbb4in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_1 + 1 >= Ar_2 ] (Comp: Ar_2^2 + 2*Ar_2 + 1, Cost: 1) evalaxbb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalaxbb2in(Ar_0, Ar_1 + 1, Ar_2, Ar_3)) (Comp: 3*Ar_2 + 3, Cost: 1) evalaxbb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalax12(Ar_0, Ar_1, Ar_2, Ar_0 + 1)) (Comp: 3*Ar_2 + 3, Cost: 1) evalax12(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalax13(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: Ar_2, Cost: 1) evalax13(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalaxbb1in(Ar_3, Ar_1, Ar_2, Ar_3)) [ Ar_1 + 1 >= Ar_2 /\ Ar_2 >= Ar_3 + 2 ] (Comp: 2, Cost: 1) evalax13(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalaxbb5in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= Ar_1 + 2 ] (Comp: 2, Cost: 1) evalax13(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalaxbb5in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_3 + 1 >= Ar_2 ] (Comp: 2, Cost: 1) evalaxbb5in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalaxstop(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalaxstart(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 Complexity upper bound 15*Ar_2 + 2*Ar_2^2 + 27 Time: 0.162 sec (SMT: 0.109 sec) ---------------------------------------- (2) BOUNDS(1, n^2) ---------------------------------------- (3) Loat Proof (FINISHED) ### Pre-processing the ITS problem ### Initial linear ITS problem Start location: evalaxstart 0: evalaxstart -> evalaxbb0in : [], cost: 1 1: evalaxbb0in -> evalax0 : [], cost: 1 2: evalax0 -> evalax1 : [], cost: 1 3: evalax1 -> evalax2 : [], cost: 1 4: evalax2 -> evalax3 : [], cost: 1 5: evalax3 -> evalax4 : [], cost: 1 6: evalax4 -> evalax5 : [], cost: 1 7: evalax5 -> evalax6 : [], cost: 1 8: evalax6 -> evalaxbb1in : A'=0, [], cost: 1 9: evalaxbb1in -> evalaxbb2in : B'=0, [], cost: 1 10: evalaxbb2in -> evalaxbb3in : [ C>=2+B ], cost: 1 11: evalaxbb2in -> evalaxbb4in : [ 1+B>=C ], cost: 1 12: evalaxbb3in -> evalaxbb2in : B'=1+B, [], cost: 1 13: evalaxbb4in -> evalax12 : D'=1+A, [], cost: 1 14: evalax12 -> evalax13 : [], cost: 1 15: evalax13 -> evalaxbb1in : A'=D, [ 1+B>=C && C>=2+D ], cost: 1 16: evalax13 -> evalaxbb5in : [ C>=2+B ], cost: 1 17: evalax13 -> evalaxbb5in : [ 1+D>=C ], cost: 1 18: evalaxbb5in -> evalaxstop : [], cost: 1 Checking for constant complexity: The following rule is satisfiable with cost >= 1, yielding constant complexity: 0: evalaxstart -> evalaxbb0in : [], cost: 1 Removed unreachable and leaf rules: Start location: evalaxstart 0: evalaxstart -> evalaxbb0in : [], cost: 1 1: evalaxbb0in -> evalax0 : [], cost: 1 2: evalax0 -> evalax1 : [], cost: 1 3: evalax1 -> evalax2 : [], cost: 1 4: evalax2 -> evalax3 : [], cost: 1 5: evalax3 -> evalax4 : [], cost: 1 6: evalax4 -> evalax5 : [], cost: 1 7: evalax5 -> evalax6 : [], cost: 1 8: evalax6 -> evalaxbb1in : A'=0, [], cost: 1 9: evalaxbb1in -> evalaxbb2in : B'=0, [], cost: 1 10: evalaxbb2in -> evalaxbb3in : [ C>=2+B ], cost: 1 11: evalaxbb2in -> evalaxbb4in : [ 1+B>=C ], cost: 1 12: evalaxbb3in -> evalaxbb2in : B'=1+B, [], cost: 1 13: evalaxbb4in -> evalax12 : D'=1+A, [], cost: 1 14: evalax12 -> evalax13 : [], cost: 1 15: evalax13 -> evalaxbb1in : A'=D, [ 1+B>=C && C>=2+D ], cost: 1 ### Simplification by acceleration and chaining ### Eliminated locations (on linear paths): Start location: evalaxstart 26: evalaxstart -> evalaxbb1in : A'=0, [], cost: 9 9: evalaxbb1in -> evalaxbb2in : B'=0, [], cost: 1 27: evalaxbb2in -> evalaxbb2in : B'=1+B, [ C>=2+B ], cost: 2 30: evalaxbb2in -> evalaxbb1in : A'=1+A, D'=1+A, [ 1+B>=C && C>=3+A ], cost: 4 Accelerating simple loops of location 10. Accelerating the following rules: 27: evalaxbb2in -> evalaxbb2in : B'=1+B, [ C>=2+B ], cost: 2 Accelerated rule 27 with metering function -1+C-B, yielding the new rule 31. Removing the simple loops: 27. Accelerated all simple loops using metering functions (where possible): Start location: evalaxstart 26: evalaxstart -> evalaxbb1in : A'=0, [], cost: 9 9: evalaxbb1in -> evalaxbb2in : B'=0, [], cost: 1 30: evalaxbb2in -> evalaxbb1in : A'=1+A, D'=1+A, [ 1+B>=C && C>=3+A ], cost: 4 31: evalaxbb2in -> evalaxbb2in : B'=-1+C, [ C>=2+B ], cost: -2+2*C-2*B Chained accelerated rules (with incoming rules): Start location: evalaxstart 26: evalaxstart -> evalaxbb1in : A'=0, [], cost: 9 9: evalaxbb1in -> evalaxbb2in : B'=0, [], cost: 1 32: evalaxbb1in -> evalaxbb2in : B'=-1+C, [ C>=2 ], cost: -1+2*C 30: evalaxbb2in -> evalaxbb1in : A'=1+A, D'=1+A, [ 1+B>=C && C>=3+A ], cost: 4 Eliminated locations (on tree-shaped paths): Start location: evalaxstart 26: evalaxstart -> evalaxbb1in : A'=0, [], cost: 9 33: evalaxbb1in -> evalaxbb1in : A'=1+A, B'=0, D'=1+A, [ 1>=C && C>=3+A ], cost: 5 34: evalaxbb1in -> evalaxbb1in : A'=1+A, B'=-1+C, D'=1+A, [ C>=2 && C>=3+A ], cost: 3+2*C Accelerating simple loops of location 9. Accelerating the following rules: 33: evalaxbb1in -> evalaxbb1in : A'=1+A, B'=0, D'=1+A, [ 1>=C && C>=3+A ], cost: 5 34: evalaxbb1in -> evalaxbb1in : A'=1+A, B'=-1+C, D'=1+A, [ C>=2 && C>=3+A ], cost: 3+2*C Accelerated rule 33 with metering function -2+C-A, yielding the new rule 35. Accelerated rule 34 with metering function -2+C-A, yielding the new rule 36. Removing the simple loops: 33 34. Accelerated all simple loops using metering functions (where possible): Start location: evalaxstart 26: evalaxstart -> evalaxbb1in : A'=0, [], cost: 9 35: evalaxbb1in -> evalaxbb1in : A'=-2+C, B'=0, D'=-2+C, [ 1>=C && C>=3+A ], cost: -10+5*C-5*A 36: evalaxbb1in -> evalaxbb1in : A'=-2+C, B'=-1+C, D'=-2+C, [ C>=2 && C>=3+A ], cost: -6+3*C-3*A+2*C*(-2+C-A) Chained accelerated rules (with incoming rules): Start location: evalaxstart 26: evalaxstart -> evalaxbb1in : A'=0, [], cost: 9 37: evalaxstart -> evalaxbb1in : A'=-2+C, B'=-1+C, D'=-2+C, [ C>=3 ], cost: 3+3*C+2*(-2+C)*C Removed unreachable locations (and leaf rules with constant cost): Start location: evalaxstart 37: evalaxstart -> evalaxbb1in : A'=-2+C, B'=-1+C, D'=-2+C, [ C>=3 ], cost: 3+3*C+2*(-2+C)*C ### Computing asymptotic complexity ### Fully simplified ITS problem Start location: evalaxstart 37: evalaxstart -> evalaxbb1in : A'=-2+C, B'=-1+C, D'=-2+C, [ C>=3 ], cost: 3+3*C+2*(-2+C)*C Computing asymptotic complexity for rule 37 Solved the limit problem by the following transformations: Created initial limit problem: -2+C (+/+!), 3-C+2*C^2 (+) [not solved] removing all constraints (solved by SMT) resulting limit problem: [solved] applying transformation rule (C) using substitution {C==n} resulting limit problem: [solved] Solution: C / n Resulting cost 3-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: 3-n+2*n^2 Rule cost: 3+3*C+2*(-2+C)*C Rule guard: [ C>=3 ] WORST_CASE(Omega(n^2),?) ---------------------------------------- (4) BOUNDS(n^2, INF)