/export/starexec/sandbox/solver/bin/starexec_run_c_complexity /export/starexec/sandbox/benchmark/theBenchmark.c /export/starexec/sandbox/output/output_files -------------------------------------------------------------------------------- WORST_CASE(?, O(n^2)) proof of /export/starexec/sandbox/output/output_files/bench.koat # AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(1, n^2). (0) CpxIntTrs (1) Koat Proof [FINISHED, 89 ms] (2) BOUNDS(1, n^2) ---------------------------------------- (0) Obligation: Complexity Int TRS consisting of the following rules: eval_ax_start(v_.0, v_.01, v_i, v_j, v_n) -> Com_1(eval_ax_bb0_in(v_.0, v_.01, v_i, v_j, v_n)) :|: TRUE eval_ax_bb0_in(v_.0, v_.01, v_i, v_j, v_n) -> Com_1(eval_ax_bb1_in(0, v_.01, v_i, v_j, v_n)) :|: TRUE eval_ax_bb1_in(v_.0, v_.01, v_i, v_j, v_n) -> Com_1(eval_ax_bb2_in(v_.0, 0, v_i, v_j, v_n)) :|: TRUE eval_ax_bb2_in(v_.0, v_.01, v_i, v_j, v_n) -> Com_1(eval_ax_bb3_in(v_.0, v_.01, v_i, v_j, v_n)) :|: v_.01 < v_n - 1 eval_ax_bb2_in(v_.0, v_.01, v_i, v_j, v_n) -> Com_1(eval_ax_bb4_in(v_.0, v_.01, v_i, v_j, v_n)) :|: v_.01 >= v_n - 1 eval_ax_bb3_in(v_.0, v_.01, v_i, v_j, v_n) -> Com_1(eval_ax_bb2_in(v_.0, v_.01 + 1, v_i, v_j, v_n)) :|: TRUE eval_ax_bb4_in(v_.0, v_.01, v_i, v_j, v_n) -> Com_1(eval_ax_bb1_in(v_.0 + 1, v_.01, v_i, v_j, v_n)) :|: v_.01 >= v_n - 1 && v_.0 + 1 < v_n - 1 eval_ax_bb4_in(v_.0, v_.01, v_i, v_j, v_n) -> Com_1(eval_ax_.critedge_in(v_.0, v_.01, v_i, v_j, v_n)) :|: v_.01 < v_n - 1 eval_ax_bb4_in(v_.0, v_.01, v_i, v_j, v_n) -> Com_1(eval_ax_.critedge_in(v_.0, v_.01, v_i, v_j, v_n)) :|: v_.0 + 1 >= v_n - 1 eval_ax_.critedge_in(v_.0, v_.01, v_i, v_j, v_n) -> Com_1(eval_ax_stop(v_.0, v_.01, v_i, v_j, v_n)) :|: TRUE The start-symbols are:[eval_ax_start_5] ---------------------------------------- (1) Koat Proof (FINISHED) YES(?, 7*Ar_2 + 2*Ar_2^2 + 10) Initial complexity problem: 1: T: (Comp: ?, Cost: 1) evalaxstart(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxbb0in(Ar_0, Ar_1, Ar_2)) (Comp: ?, Cost: 1) evalaxbb0in(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxbb1in(0, Ar_1, Ar_2)) (Comp: ?, Cost: 1) evalaxbb1in(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxbb2in(Ar_0, 0, Ar_2)) (Comp: ?, Cost: 1) evalaxbb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxbb3in(Ar_0, Ar_1, Ar_2)) [ Ar_2 >= Ar_1 + 2 ] (Comp: ?, Cost: 1) evalaxbb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxbb4in(Ar_0, Ar_1, Ar_2)) [ Ar_1 + 1 >= Ar_2 ] (Comp: ?, Cost: 1) evalaxbb3in(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxbb2in(Ar_0, Ar_1 + 1, Ar_2)) (Comp: ?, Cost: 1) evalaxbb4in(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxbb1in(Ar_0 + 1, Ar_1, Ar_2)) [ Ar_1 + 1 >= Ar_2 /\ Ar_2 >= Ar_0 + 3 ] (Comp: ?, Cost: 1) evalaxbb4in(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxcritedgein(Ar_0, Ar_1, Ar_2)) [ Ar_2 >= Ar_1 + 2 ] (Comp: ?, Cost: 1) evalaxbb4in(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxcritedgein(Ar_0, Ar_1, Ar_2)) [ Ar_0 + 2 >= Ar_2 ] (Comp: ?, Cost: 1) evalaxcritedgein(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxstop(Ar_0, Ar_1, Ar_2)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxstart(Ar_0, Ar_1, Ar_2)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 Testing for reachability in the complexity graph removes the following transition from problem 1: evalaxbb4in(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxcritedgein(Ar_0, Ar_1, Ar_2)) [ Ar_2 >= Ar_1 + 2 ] We thus obtain the following problem: 2: T: (Comp: ?, Cost: 1) evalaxcritedgein(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxstop(Ar_0, Ar_1, Ar_2)) (Comp: ?, Cost: 1) evalaxbb4in(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxcritedgein(Ar_0, Ar_1, Ar_2)) [ Ar_0 + 2 >= Ar_2 ] (Comp: ?, Cost: 1) evalaxbb4in(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxbb1in(Ar_0 + 1, Ar_1, Ar_2)) [ Ar_1 + 1 >= Ar_2 /\ Ar_2 >= Ar_0 + 3 ] (Comp: ?, Cost: 1) evalaxbb3in(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxbb2in(Ar_0, Ar_1 + 1, Ar_2)) (Comp: ?, Cost: 1) evalaxbb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxbb4in(Ar_0, Ar_1, Ar_2)) [ Ar_1 + 1 >= Ar_2 ] (Comp: ?, Cost: 1) evalaxbb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxbb3in(Ar_0, Ar_1, Ar_2)) [ Ar_2 >= Ar_1 + 2 ] (Comp: ?, Cost: 1) evalaxbb1in(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxbb2in(Ar_0, 0, Ar_2)) (Comp: ?, Cost: 1) evalaxbb0in(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxbb1in(0, Ar_1, Ar_2)) (Comp: ?, Cost: 1) evalaxstart(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxbb0in(Ar_0, Ar_1, Ar_2)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxstart(Ar_0, Ar_1, Ar_2)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 Repeatedly propagating knowledge in problem 2 produces the following problem: 3: T: (Comp: ?, Cost: 1) evalaxcritedgein(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxstop(Ar_0, Ar_1, Ar_2)) (Comp: ?, Cost: 1) evalaxbb4in(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxcritedgein(Ar_0, Ar_1, Ar_2)) [ Ar_0 + 2 >= Ar_2 ] (Comp: ?, Cost: 1) evalaxbb4in(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxbb1in(Ar_0 + 1, Ar_1, Ar_2)) [ Ar_1 + 1 >= Ar_2 /\ Ar_2 >= Ar_0 + 3 ] (Comp: ?, Cost: 1) evalaxbb3in(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxbb2in(Ar_0, Ar_1 + 1, Ar_2)) (Comp: ?, Cost: 1) evalaxbb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxbb4in(Ar_0, Ar_1, Ar_2)) [ Ar_1 + 1 >= Ar_2 ] (Comp: ?, Cost: 1) evalaxbb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxbb3in(Ar_0, Ar_1, Ar_2)) [ Ar_2 >= Ar_1 + 2 ] (Comp: ?, Cost: 1) evalaxbb1in(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxbb2in(Ar_0, 0, Ar_2)) (Comp: 1, Cost: 1) evalaxbb0in(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxbb1in(0, Ar_1, Ar_2)) (Comp: 1, Cost: 1) evalaxstart(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxbb0in(Ar_0, Ar_1, Ar_2)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxstart(Ar_0, Ar_1, Ar_2)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 A polynomial rank function with Pol(evalaxcritedgein) = 1 Pol(evalaxstop) = 0 Pol(evalaxbb4in) = 2 Pol(evalaxbb1in) = 2 Pol(evalaxbb3in) = 2 Pol(evalaxbb2in) = 2 Pol(evalaxbb0in) = 2 Pol(evalaxstart) = 2 Pol(koat_start) = 2 orients all transitions weakly and the transitions evalaxcritedgein(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxstop(Ar_0, Ar_1, Ar_2)) evalaxbb4in(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxcritedgein(Ar_0, Ar_1, Ar_2)) [ Ar_0 + 2 >= Ar_2 ] strictly and produces the following problem: 4: T: (Comp: 2, Cost: 1) evalaxcritedgein(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxstop(Ar_0, Ar_1, Ar_2)) (Comp: 2, Cost: 1) evalaxbb4in(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxcritedgein(Ar_0, Ar_1, Ar_2)) [ Ar_0 + 2 >= Ar_2 ] (Comp: ?, Cost: 1) evalaxbb4in(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxbb1in(Ar_0 + 1, Ar_1, Ar_2)) [ Ar_1 + 1 >= Ar_2 /\ Ar_2 >= Ar_0 + 3 ] (Comp: ?, Cost: 1) evalaxbb3in(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxbb2in(Ar_0, Ar_1 + 1, Ar_2)) (Comp: ?, Cost: 1) evalaxbb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxbb4in(Ar_0, Ar_1, Ar_2)) [ Ar_1 + 1 >= Ar_2 ] (Comp: ?, Cost: 1) evalaxbb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxbb3in(Ar_0, Ar_1, Ar_2)) [ Ar_2 >= Ar_1 + 2 ] (Comp: ?, Cost: 1) evalaxbb1in(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxbb2in(Ar_0, 0, Ar_2)) (Comp: 1, Cost: 1) evalaxbb0in(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxbb1in(0, Ar_1, Ar_2)) (Comp: 1, Cost: 1) evalaxstart(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxbb0in(Ar_0, Ar_1, Ar_2)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxstart(Ar_0, Ar_1, Ar_2)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 A polynomial rank function with Pol(evalaxcritedgein) = -V_1 + V_3 Pol(evalaxstop) = -V_1 + V_3 Pol(evalaxbb4in) = -V_1 + V_3 Pol(evalaxbb1in) = -V_1 + V_3 Pol(evalaxbb3in) = -V_1 + V_3 Pol(evalaxbb2in) = -V_1 + V_3 Pol(evalaxbb0in) = V_3 Pol(evalaxstart) = V_3 Pol(koat_start) = V_3 orients all transitions weakly and the transition evalaxbb4in(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxbb1in(Ar_0 + 1, Ar_1, Ar_2)) [ Ar_1 + 1 >= Ar_2 /\ Ar_2 >= Ar_0 + 3 ] strictly and produces the following problem: 5: T: (Comp: 2, Cost: 1) evalaxcritedgein(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxstop(Ar_0, Ar_1, Ar_2)) (Comp: 2, Cost: 1) evalaxbb4in(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxcritedgein(Ar_0, Ar_1, Ar_2)) [ Ar_0 + 2 >= Ar_2 ] (Comp: Ar_2, Cost: 1) evalaxbb4in(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxbb1in(Ar_0 + 1, Ar_1, Ar_2)) [ Ar_1 + 1 >= Ar_2 /\ Ar_2 >= Ar_0 + 3 ] (Comp: ?, Cost: 1) evalaxbb3in(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxbb2in(Ar_0, Ar_1 + 1, Ar_2)) (Comp: ?, Cost: 1) evalaxbb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxbb4in(Ar_0, Ar_1, Ar_2)) [ Ar_1 + 1 >= Ar_2 ] (Comp: ?, Cost: 1) evalaxbb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxbb3in(Ar_0, Ar_1, Ar_2)) [ Ar_2 >= Ar_1 + 2 ] (Comp: ?, Cost: 1) evalaxbb1in(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxbb2in(Ar_0, 0, Ar_2)) (Comp: 1, Cost: 1) evalaxbb0in(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxbb1in(0, Ar_1, Ar_2)) (Comp: 1, Cost: 1) evalaxstart(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxbb0in(Ar_0, Ar_1, Ar_2)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxstart(Ar_0, Ar_1, Ar_2)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 Repeatedly propagating knowledge in problem 5 produces the following problem: 6: T: (Comp: 2, Cost: 1) evalaxcritedgein(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxstop(Ar_0, Ar_1, Ar_2)) (Comp: 2, Cost: 1) evalaxbb4in(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxcritedgein(Ar_0, Ar_1, Ar_2)) [ Ar_0 + 2 >= Ar_2 ] (Comp: Ar_2, Cost: 1) evalaxbb4in(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxbb1in(Ar_0 + 1, Ar_1, Ar_2)) [ Ar_1 + 1 >= Ar_2 /\ Ar_2 >= Ar_0 + 3 ] (Comp: ?, Cost: 1) evalaxbb3in(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxbb2in(Ar_0, Ar_1 + 1, Ar_2)) (Comp: ?, Cost: 1) evalaxbb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxbb4in(Ar_0, Ar_1, Ar_2)) [ Ar_1 + 1 >= Ar_2 ] (Comp: ?, Cost: 1) evalaxbb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxbb3in(Ar_0, Ar_1, Ar_2)) [ Ar_2 >= Ar_1 + 2 ] (Comp: Ar_2 + 1, Cost: 1) evalaxbb1in(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxbb2in(Ar_0, 0, Ar_2)) (Comp: 1, Cost: 1) evalaxbb0in(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxbb1in(0, Ar_1, Ar_2)) (Comp: 1, Cost: 1) evalaxstart(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxbb0in(Ar_0, Ar_1, Ar_2)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxstart(Ar_0, Ar_1, Ar_2)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 A polynomial rank function with Pol(evalaxbb3in) = 1 Pol(evalaxbb2in) = 1 Pol(evalaxbb4in) = 0 and size complexities S("koat_start(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxstart(Ar_0, Ar_1, Ar_2)) [ 0 <= 0 ]", 0-0) = Ar_0 S("koat_start(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxstart(Ar_0, Ar_1, Ar_2)) [ 0 <= 0 ]", 0-1) = Ar_1 S("koat_start(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxstart(Ar_0, Ar_1, Ar_2)) [ 0 <= 0 ]", 0-2) = Ar_2 S("evalaxstart(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxbb0in(Ar_0, Ar_1, Ar_2))", 0-0) = Ar_0 S("evalaxstart(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxbb0in(Ar_0, Ar_1, Ar_2))", 0-1) = Ar_1 S("evalaxstart(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxbb0in(Ar_0, Ar_1, Ar_2))", 0-2) = Ar_2 S("evalaxbb0in(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxbb1in(0, Ar_1, Ar_2))", 0-0) = 0 S("evalaxbb0in(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxbb1in(0, Ar_1, Ar_2))", 0-1) = Ar_1 S("evalaxbb0in(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxbb1in(0, Ar_1, Ar_2))", 0-2) = Ar_2 S("evalaxbb1in(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxbb2in(Ar_0, 0, Ar_2))", 0-0) = Ar_2 S("evalaxbb1in(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxbb2in(Ar_0, 0, Ar_2))", 0-1) = 0 S("evalaxbb1in(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxbb2in(Ar_0, 0, Ar_2))", 0-2) = Ar_2 S("evalaxbb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxbb3in(Ar_0, Ar_1, Ar_2)) [ Ar_2 >= Ar_1 + 2 ]", 0-0) = Ar_2 S("evalaxbb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxbb3in(Ar_0, Ar_1, Ar_2)) [ Ar_2 >= Ar_1 + 2 ]", 0-1) = ? S("evalaxbb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxbb3in(Ar_0, Ar_1, Ar_2)) [ Ar_2 >= Ar_1 + 2 ]", 0-2) = Ar_2 S("evalaxbb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxbb4in(Ar_0, Ar_1, Ar_2)) [ Ar_1 + 1 >= Ar_2 ]", 0-0) = Ar_2 S("evalaxbb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxbb4in(Ar_0, Ar_1, Ar_2)) [ Ar_1 + 1 >= Ar_2 ]", 0-1) = ? S("evalaxbb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxbb4in(Ar_0, Ar_1, Ar_2)) [ Ar_1 + 1 >= Ar_2 ]", 0-2) = Ar_2 S("evalaxbb3in(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxbb2in(Ar_0, Ar_1 + 1, Ar_2))", 0-0) = Ar_2 S("evalaxbb3in(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxbb2in(Ar_0, Ar_1 + 1, Ar_2))", 0-1) = ? S("evalaxbb3in(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxbb2in(Ar_0, Ar_1 + 1, Ar_2))", 0-2) = Ar_2 S("evalaxbb4in(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxbb1in(Ar_0 + 1, Ar_1, Ar_2)) [ Ar_1 + 1 >= Ar_2 /\\ Ar_2 >= Ar_0 + 3 ]", 0-0) = Ar_2 S("evalaxbb4in(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxbb1in(Ar_0 + 1, Ar_1, Ar_2)) [ Ar_1 + 1 >= Ar_2 /\\ Ar_2 >= Ar_0 + 3 ]", 0-1) = ? S("evalaxbb4in(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxbb1in(Ar_0 + 1, Ar_1, Ar_2)) [ Ar_1 + 1 >= Ar_2 /\\ Ar_2 >= Ar_0 + 3 ]", 0-2) = Ar_2 S("evalaxbb4in(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxcritedgein(Ar_0, Ar_1, Ar_2)) [ Ar_0 + 2 >= Ar_2 ]", 0-0) = Ar_2 S("evalaxbb4in(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxcritedgein(Ar_0, Ar_1, Ar_2)) [ Ar_0 + 2 >= Ar_2 ]", 0-1) = ? S("evalaxbb4in(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxcritedgein(Ar_0, Ar_1, Ar_2)) [ Ar_0 + 2 >= Ar_2 ]", 0-2) = Ar_2 S("evalaxcritedgein(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxstop(Ar_0, Ar_1, Ar_2))", 0-0) = Ar_2 S("evalaxcritedgein(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxstop(Ar_0, Ar_1, Ar_2))", 0-1) = ? S("evalaxcritedgein(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxstop(Ar_0, Ar_1, Ar_2))", 0-2) = Ar_2 orients the transitions evalaxbb3in(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxbb2in(Ar_0, Ar_1 + 1, Ar_2)) evalaxbb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxbb4in(Ar_0, Ar_1, Ar_2)) [ Ar_1 + 1 >= Ar_2 ] evalaxbb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxbb3in(Ar_0, Ar_1, Ar_2)) [ Ar_2 >= Ar_1 + 2 ] weakly and the transition evalaxbb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxbb4in(Ar_0, Ar_1, Ar_2)) [ Ar_1 + 1 >= Ar_2 ] strictly and produces the following problem: 7: T: (Comp: 2, Cost: 1) evalaxcritedgein(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxstop(Ar_0, Ar_1, Ar_2)) (Comp: 2, Cost: 1) evalaxbb4in(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxcritedgein(Ar_0, Ar_1, Ar_2)) [ Ar_0 + 2 >= Ar_2 ] (Comp: Ar_2, Cost: 1) evalaxbb4in(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxbb1in(Ar_0 + 1, Ar_1, Ar_2)) [ Ar_1 + 1 >= Ar_2 /\ Ar_2 >= Ar_0 + 3 ] (Comp: ?, Cost: 1) evalaxbb3in(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxbb2in(Ar_0, Ar_1 + 1, Ar_2)) (Comp: Ar_2 + 1, Cost: 1) evalaxbb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxbb4in(Ar_0, Ar_1, Ar_2)) [ Ar_1 + 1 >= Ar_2 ] (Comp: ?, Cost: 1) evalaxbb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxbb3in(Ar_0, Ar_1, Ar_2)) [ Ar_2 >= Ar_1 + 2 ] (Comp: Ar_2 + 1, Cost: 1) evalaxbb1in(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxbb2in(Ar_0, 0, Ar_2)) (Comp: 1, Cost: 1) evalaxbb0in(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxbb1in(0, Ar_1, Ar_2)) (Comp: 1, Cost: 1) evalaxstart(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxbb0in(Ar_0, Ar_1, Ar_2)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxstart(Ar_0, Ar_1, Ar_2)) [ 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) -> Com_1(evalaxstart(Ar_0, Ar_1, Ar_2)) [ 0 <= 0 ]", 0-0) = Ar_0 S("koat_start(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxstart(Ar_0, Ar_1, Ar_2)) [ 0 <= 0 ]", 0-1) = Ar_1 S("koat_start(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxstart(Ar_0, Ar_1, Ar_2)) [ 0 <= 0 ]", 0-2) = Ar_2 S("evalaxstart(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxbb0in(Ar_0, Ar_1, Ar_2))", 0-0) = Ar_0 S("evalaxstart(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxbb0in(Ar_0, Ar_1, Ar_2))", 0-1) = Ar_1 S("evalaxstart(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxbb0in(Ar_0, Ar_1, Ar_2))", 0-2) = Ar_2 S("evalaxbb0in(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxbb1in(0, Ar_1, Ar_2))", 0-0) = 0 S("evalaxbb0in(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxbb1in(0, Ar_1, Ar_2))", 0-1) = Ar_1 S("evalaxbb0in(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxbb1in(0, Ar_1, Ar_2))", 0-2) = Ar_2 S("evalaxbb1in(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxbb2in(Ar_0, 0, Ar_2))", 0-0) = Ar_2 S("evalaxbb1in(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxbb2in(Ar_0, 0, Ar_2))", 0-1) = 0 S("evalaxbb1in(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxbb2in(Ar_0, 0, Ar_2))", 0-2) = Ar_2 S("evalaxbb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxbb3in(Ar_0, Ar_1, Ar_2)) [ Ar_2 >= Ar_1 + 2 ]", 0-0) = Ar_2 S("evalaxbb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxbb3in(Ar_0, Ar_1, Ar_2)) [ Ar_2 >= Ar_1 + 2 ]", 0-1) = ? S("evalaxbb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxbb3in(Ar_0, Ar_1, Ar_2)) [ Ar_2 >= Ar_1 + 2 ]", 0-2) = Ar_2 S("evalaxbb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxbb4in(Ar_0, Ar_1, Ar_2)) [ Ar_1 + 1 >= Ar_2 ]", 0-0) = Ar_2 S("evalaxbb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxbb4in(Ar_0, Ar_1, Ar_2)) [ Ar_1 + 1 >= Ar_2 ]", 0-1) = ? S("evalaxbb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxbb4in(Ar_0, Ar_1, Ar_2)) [ Ar_1 + 1 >= Ar_2 ]", 0-2) = Ar_2 S("evalaxbb3in(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxbb2in(Ar_0, Ar_1 + 1, Ar_2))", 0-0) = Ar_2 S("evalaxbb3in(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxbb2in(Ar_0, Ar_1 + 1, Ar_2))", 0-1) = ? S("evalaxbb3in(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxbb2in(Ar_0, Ar_1 + 1, Ar_2))", 0-2) = Ar_2 S("evalaxbb4in(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxbb1in(Ar_0 + 1, Ar_1, Ar_2)) [ Ar_1 + 1 >= Ar_2 /\\ Ar_2 >= Ar_0 + 3 ]", 0-0) = Ar_2 S("evalaxbb4in(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxbb1in(Ar_0 + 1, Ar_1, Ar_2)) [ Ar_1 + 1 >= Ar_2 /\\ Ar_2 >= Ar_0 + 3 ]", 0-1) = ? S("evalaxbb4in(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxbb1in(Ar_0 + 1, Ar_1, Ar_2)) [ Ar_1 + 1 >= Ar_2 /\\ Ar_2 >= Ar_0 + 3 ]", 0-2) = Ar_2 S("evalaxbb4in(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxcritedgein(Ar_0, Ar_1, Ar_2)) [ Ar_0 + 2 >= Ar_2 ]", 0-0) = Ar_2 S("evalaxbb4in(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxcritedgein(Ar_0, Ar_1, Ar_2)) [ Ar_0 + 2 >= Ar_2 ]", 0-1) = ? S("evalaxbb4in(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxcritedgein(Ar_0, Ar_1, Ar_2)) [ Ar_0 + 2 >= Ar_2 ]", 0-2) = Ar_2 S("evalaxcritedgein(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxstop(Ar_0, Ar_1, Ar_2))", 0-0) = Ar_2 S("evalaxcritedgein(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxstop(Ar_0, Ar_1, Ar_2))", 0-1) = ? S("evalaxcritedgein(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxstop(Ar_0, Ar_1, Ar_2))", 0-2) = Ar_2 orients the transitions evalaxbb3in(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxbb2in(Ar_0, Ar_1 + 1, Ar_2)) evalaxbb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxbb3in(Ar_0, Ar_1, Ar_2)) [ Ar_2 >= Ar_1 + 2 ] weakly and the transition evalaxbb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxbb3in(Ar_0, Ar_1, Ar_2)) [ Ar_2 >= Ar_1 + 2 ] strictly and produces the following problem: 8: T: (Comp: 2, Cost: 1) evalaxcritedgein(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxstop(Ar_0, Ar_1, Ar_2)) (Comp: 2, Cost: 1) evalaxbb4in(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxcritedgein(Ar_0, Ar_1, Ar_2)) [ Ar_0 + 2 >= Ar_2 ] (Comp: Ar_2, Cost: 1) evalaxbb4in(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxbb1in(Ar_0 + 1, Ar_1, Ar_2)) [ Ar_1 + 1 >= Ar_2 /\ Ar_2 >= Ar_0 + 3 ] (Comp: ?, Cost: 1) evalaxbb3in(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxbb2in(Ar_0, Ar_1 + 1, Ar_2)) (Comp: Ar_2 + 1, Cost: 1) evalaxbb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxbb4in(Ar_0, Ar_1, Ar_2)) [ Ar_1 + 1 >= Ar_2 ] (Comp: Ar_2^2 + 2*Ar_2 + 1, Cost: 1) evalaxbb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxbb3in(Ar_0, Ar_1, Ar_2)) [ Ar_2 >= Ar_1 + 2 ] (Comp: Ar_2 + 1, Cost: 1) evalaxbb1in(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxbb2in(Ar_0, 0, Ar_2)) (Comp: 1, Cost: 1) evalaxbb0in(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxbb1in(0, Ar_1, Ar_2)) (Comp: 1, Cost: 1) evalaxstart(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxbb0in(Ar_0, Ar_1, Ar_2)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxstart(Ar_0, Ar_1, Ar_2)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 Repeatedly propagating knowledge in problem 8 produces the following problem: 9: T: (Comp: 2, Cost: 1) evalaxcritedgein(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxstop(Ar_0, Ar_1, Ar_2)) (Comp: 2, Cost: 1) evalaxbb4in(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxcritedgein(Ar_0, Ar_1, Ar_2)) [ Ar_0 + 2 >= Ar_2 ] (Comp: Ar_2, Cost: 1) evalaxbb4in(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxbb1in(Ar_0 + 1, Ar_1, Ar_2)) [ Ar_1 + 1 >= Ar_2 /\ Ar_2 >= Ar_0 + 3 ] (Comp: Ar_2^2 + 2*Ar_2 + 1, Cost: 1) evalaxbb3in(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxbb2in(Ar_0, Ar_1 + 1, Ar_2)) (Comp: Ar_2 + 1, Cost: 1) evalaxbb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxbb4in(Ar_0, Ar_1, Ar_2)) [ Ar_1 + 1 >= Ar_2 ] (Comp: Ar_2^2 + 2*Ar_2 + 1, Cost: 1) evalaxbb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxbb3in(Ar_0, Ar_1, Ar_2)) [ Ar_2 >= Ar_1 + 2 ] (Comp: Ar_2 + 1, Cost: 1) evalaxbb1in(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxbb2in(Ar_0, 0, Ar_2)) (Comp: 1, Cost: 1) evalaxbb0in(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxbb1in(0, Ar_1, Ar_2)) (Comp: 1, Cost: 1) evalaxstart(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxbb0in(Ar_0, Ar_1, Ar_2)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxstart(Ar_0, Ar_1, Ar_2)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 Complexity upper bound 7*Ar_2 + 2*Ar_2^2 + 10 Time: 0.076 sec (SMT: 0.060 sec) ---------------------------------------- (2) BOUNDS(1, n^2)