/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(n^2, n^2).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, 96 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_abc_start(v_4, v_i_0, v_j_0, v_n) -> Com_1(eval_abc_bb0_in(v_4, v_i_0, v_j_0, v_n)) :|: TRUE eval_abc_bb0_in(v_4, v_i_0, v_j_0, v_n) -> Com_1(eval_abc_0(v_4, v_i_0, v_j_0, v_n)) :|: TRUE eval_abc_0(v_4, v_i_0, v_j_0, v_n) -> Com_1(eval_abc_1(v_4, v_i_0, v_j_0, v_n)) :|: TRUE eval_abc_1(v_4, v_i_0, v_j_0, v_n) -> Com_1(eval_abc_2(v_4, v_i_0, v_j_0, v_n)) :|: TRUE eval_abc_2(v_4, v_i_0, v_j_0, v_n) -> Com_1(eval_abc_3(v_4, v_i_0, v_j_0, v_n)) :|: TRUE eval_abc_3(v_4, v_i_0, v_j_0, v_n) -> Com_1(eval_abc_4(v_4, v_i_0, v_j_0, v_n)) :|: TRUE eval_abc_4(v_4, v_i_0, v_j_0, v_n) -> Com_1(eval_abc_bb1_in(v_4, 0, v_j_0, v_n)) :|: TRUE eval_abc_bb1_in(v_4, v_i_0, v_j_0, v_n) -> Com_1(eval_abc_bb2_in(v_4, v_i_0, 0, v_n)) :|: v_i_0 <= v_n - 1 eval_abc_bb1_in(v_4, v_i_0, v_j_0, v_n) -> Com_1(eval_abc_bb5_in(v_4, v_i_0, v_j_0, v_n)) :|: v_i_0 > v_n - 1 eval_abc_bb2_in(v_4, v_i_0, v_j_0, v_n) -> Com_1(eval_abc_bb3_in(v_4, v_i_0, v_j_0, v_n)) :|: v_j_0 <= v_i_0 eval_abc_bb2_in(v_4, v_i_0, v_j_0, v_n) -> Com_1(eval_abc_bb4_in(v_4, v_i_0, v_j_0, v_n)) :|: v_j_0 > v_i_0 eval_abc_bb3_in(v_4, v_i_0, v_j_0, v_n) -> Com_1(eval_abc_bb2_in(v_4, v_i_0, v_j_0 + 1, v_n)) :|: TRUE eval_abc_bb4_in(v_4, v_i_0, v_j_0, v_n) -> Com_1(eval_abc_10(v_i_0 + 1, v_i_0, v_j_0, v_n)) :|: TRUE eval_abc_10(v_4, v_i_0, v_j_0, v_n) -> Com_1(eval_abc_11(v_4, v_i_0, v_j_0, v_n)) :|: TRUE eval_abc_11(v_4, v_i_0, v_j_0, v_n) -> Com_1(eval_abc_bb1_in(v_4, v_4, v_j_0, v_n)) :|: TRUE eval_abc_bb5_in(v_4, v_i_0, v_j_0, v_n) -> Com_1(eval_abc_stop(v_4, v_i_0, v_j_0, v_n)) :|: TRUE The start-symbols are:[eval_abc_start_4] ---------------------------------------- (1) Koat Proof (FINISHED) YES(?, 155*ar_1 + 8*ar_1^2 + 158) Initial complexity problem: 1: T: (Comp: ?, Cost: 1) evalabcstart(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabcbb0in(ar_0, ar_1, ar_2, ar_3)) (Comp: ?, Cost: 1) evalabcbb0in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabc0(ar_0, ar_1, ar_2, ar_3)) (Comp: ?, Cost: 1) evalabc0(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabc1(ar_0, ar_1, ar_2, ar_3)) (Comp: ?, Cost: 1) evalabc1(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabc2(ar_0, ar_1, ar_2, ar_3)) (Comp: ?, Cost: 1) evalabc2(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabc3(ar_0, ar_1, ar_2, ar_3)) (Comp: ?, Cost: 1) evalabc3(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabc4(ar_0, ar_1, ar_2, ar_3)) (Comp: ?, Cost: 1) evalabc4(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabcbb1in(0, ar_1, ar_2, ar_3)) (Comp: ?, Cost: 1) evalabcbb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabcbb2in(ar_0, ar_1, 0, ar_3)) [ ar_1 >= ar_0 + 1 ] (Comp: ?, Cost: 1) evalabcbb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabcbb5in(ar_0, ar_1, ar_2, ar_3)) [ ar_0 >= ar_1 ] (Comp: ?, Cost: 1) evalabcbb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabcbb3in(ar_0, ar_1, ar_2, ar_3)) [ ar_0 >= ar_2 ] (Comp: ?, Cost: 1) evalabcbb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabcbb4in(ar_0, ar_1, ar_2, ar_3)) [ ar_2 >= ar_0 + 1 ] (Comp: ?, Cost: 1) evalabcbb3in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabcbb2in(ar_0, ar_1, ar_2 + 1, ar_3)) (Comp: ?, Cost: 1) evalabcbb4in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabc10(ar_0, ar_1, ar_2, ar_0 + 1)) (Comp: ?, Cost: 1) evalabc10(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabc11(ar_0, ar_1, ar_2, ar_3)) (Comp: ?, Cost: 1) evalabc11(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabcbb1in(ar_3, ar_1, ar_2, ar_3)) (Comp: ?, Cost: 1) evalabcbb5in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabcstop(ar_0, ar_1, ar_2, ar_3)) (Comp: 1, Cost: 0) koat_start(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabcstart(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) evalabcstart(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabcbb0in(ar_0, ar_1, ar_2, ar_3)) (Comp: 1, Cost: 1) evalabcbb0in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabc0(ar_0, ar_1, ar_2, ar_3)) (Comp: 1, Cost: 1) evalabc0(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabc1(ar_0, ar_1, ar_2, ar_3)) (Comp: 1, Cost: 1) evalabc1(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabc2(ar_0, ar_1, ar_2, ar_3)) (Comp: 1, Cost: 1) evalabc2(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabc3(ar_0, ar_1, ar_2, ar_3)) (Comp: 1, Cost: 1) evalabc3(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabc4(ar_0, ar_1, ar_2, ar_3)) (Comp: 1, Cost: 1) evalabc4(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabcbb1in(0, ar_1, ar_2, ar_3)) (Comp: ?, Cost: 1) evalabcbb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabcbb2in(ar_0, ar_1, 0, ar_3)) [ ar_1 >= ar_0 + 1 ] (Comp: ?, Cost: 1) evalabcbb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabcbb5in(ar_0, ar_1, ar_2, ar_3)) [ ar_0 >= ar_1 ] (Comp: ?, Cost: 1) evalabcbb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabcbb3in(ar_0, ar_1, ar_2, ar_3)) [ ar_0 >= ar_2 ] (Comp: ?, Cost: 1) evalabcbb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabcbb4in(ar_0, ar_1, ar_2, ar_3)) [ ar_2 >= ar_0 + 1 ] (Comp: ?, Cost: 1) evalabcbb3in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabcbb2in(ar_0, ar_1, ar_2 + 1, ar_3)) (Comp: ?, Cost: 1) evalabcbb4in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabc10(ar_0, ar_1, ar_2, ar_0 + 1)) (Comp: ?, Cost: 1) evalabc10(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabc11(ar_0, ar_1, ar_2, ar_3)) (Comp: ?, Cost: 1) evalabc11(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabcbb1in(ar_3, ar_1, ar_2, ar_3)) (Comp: ?, Cost: 1) evalabcbb5in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabcstop(ar_0, ar_1, ar_2, ar_3)) (Comp: 1, Cost: 0) koat_start(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabcstart(ar_0, ar_1, ar_2, ar_3)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 A polynomial rank function with Pol(evalabcstart) = 2 Pol(evalabcbb0in) = 2 Pol(evalabc0) = 2 Pol(evalabc1) = 2 Pol(evalabc2) = 2 Pol(evalabc3) = 2 Pol(evalabc4) = 2 Pol(evalabcbb1in) = 2 Pol(evalabcbb2in) = 2 Pol(evalabcbb5in) = 1 Pol(evalabcbb3in) = 2 Pol(evalabcbb4in) = 2 Pol(evalabc10) = 2 Pol(evalabc11) = 2 Pol(evalabcstop) = 0 Pol(koat_start) = 2 orients all transitions weakly and the transitions evalabcbb5in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabcstop(ar_0, ar_1, ar_2, ar_3)) evalabcbb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabcbb5in(ar_0, ar_1, ar_2, ar_3)) [ ar_0 >= ar_1 ] strictly and produces the following problem: 3: T: (Comp: 1, Cost: 1) evalabcstart(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabcbb0in(ar_0, ar_1, ar_2, ar_3)) (Comp: 1, Cost: 1) evalabcbb0in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabc0(ar_0, ar_1, ar_2, ar_3)) (Comp: 1, Cost: 1) evalabc0(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabc1(ar_0, ar_1, ar_2, ar_3)) (Comp: 1, Cost: 1) evalabc1(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabc2(ar_0, ar_1, ar_2, ar_3)) (Comp: 1, Cost: 1) evalabc2(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabc3(ar_0, ar_1, ar_2, ar_3)) (Comp: 1, Cost: 1) evalabc3(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabc4(ar_0, ar_1, ar_2, ar_3)) (Comp: 1, Cost: 1) evalabc4(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabcbb1in(0, ar_1, ar_2, ar_3)) (Comp: ?, Cost: 1) evalabcbb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabcbb2in(ar_0, ar_1, 0, ar_3)) [ ar_1 >= ar_0 + 1 ] (Comp: 2, Cost: 1) evalabcbb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabcbb5in(ar_0, ar_1, ar_2, ar_3)) [ ar_0 >= ar_1 ] (Comp: ?, Cost: 1) evalabcbb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabcbb3in(ar_0, ar_1, ar_2, ar_3)) [ ar_0 >= ar_2 ] (Comp: ?, Cost: 1) evalabcbb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabcbb4in(ar_0, ar_1, ar_2, ar_3)) [ ar_2 >= ar_0 + 1 ] (Comp: ?, Cost: 1) evalabcbb3in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabcbb2in(ar_0, ar_1, ar_2 + 1, ar_3)) (Comp: ?, Cost: 1) evalabcbb4in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabc10(ar_0, ar_1, ar_2, ar_0 + 1)) (Comp: ?, Cost: 1) evalabc10(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabc11(ar_0, ar_1, ar_2, ar_3)) (Comp: ?, Cost: 1) evalabc11(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabcbb1in(ar_3, ar_1, ar_2, ar_3)) (Comp: 2, Cost: 1) evalabcbb5in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabcstop(ar_0, ar_1, ar_2, ar_3)) (Comp: 1, Cost: 0) koat_start(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabcstart(ar_0, ar_1, ar_2, ar_3)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 A polynomial rank function with Pol(evalabcstart) = V_2 + 1 Pol(evalabcbb0in) = V_2 + 1 Pol(evalabc0) = V_2 + 1 Pol(evalabc1) = V_2 + 1 Pol(evalabc2) = V_2 + 1 Pol(evalabc3) = V_2 + 1 Pol(evalabc4) = V_2 + 1 Pol(evalabcbb1in) = -V_1 + V_2 + 1 Pol(evalabcbb2in) = -V_1 + V_2 Pol(evalabcbb5in) = -V_1 + V_2 Pol(evalabcbb3in) = -V_1 + V_2 Pol(evalabcbb4in) = -V_1 + V_2 Pol(evalabc10) = V_2 - V_4 + 1 Pol(evalabc11) = V_2 - V_4 + 1 Pol(evalabcstop) = -V_1 + V_2 Pol(koat_start) = V_2 + 1 orients all transitions weakly and the transition evalabcbb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabcbb2in(ar_0, ar_1, 0, ar_3)) [ ar_1 >= ar_0 + 1 ] strictly and produces the following problem: 4: T: (Comp: 1, Cost: 1) evalabcstart(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabcbb0in(ar_0, ar_1, ar_2, ar_3)) (Comp: 1, Cost: 1) evalabcbb0in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabc0(ar_0, ar_1, ar_2, ar_3)) (Comp: 1, Cost: 1) evalabc0(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabc1(ar_0, ar_1, ar_2, ar_3)) (Comp: 1, Cost: 1) evalabc1(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabc2(ar_0, ar_1, ar_2, ar_3)) (Comp: 1, Cost: 1) evalabc2(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabc3(ar_0, ar_1, ar_2, ar_3)) (Comp: 1, Cost: 1) evalabc3(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabc4(ar_0, ar_1, ar_2, ar_3)) (Comp: 1, Cost: 1) evalabc4(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabcbb1in(0, ar_1, ar_2, ar_3)) (Comp: ar_1 + 1, Cost: 1) evalabcbb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabcbb2in(ar_0, ar_1, 0, ar_3)) [ ar_1 >= ar_0 + 1 ] (Comp: 2, Cost: 1) evalabcbb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabcbb5in(ar_0, ar_1, ar_2, ar_3)) [ ar_0 >= ar_1 ] (Comp: ?, Cost: 1) evalabcbb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabcbb3in(ar_0, ar_1, ar_2, ar_3)) [ ar_0 >= ar_2 ] (Comp: ?, Cost: 1) evalabcbb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabcbb4in(ar_0, ar_1, ar_2, ar_3)) [ ar_2 >= ar_0 + 1 ] (Comp: ?, Cost: 1) evalabcbb3in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabcbb2in(ar_0, ar_1, ar_2 + 1, ar_3)) (Comp: ?, Cost: 1) evalabcbb4in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabc10(ar_0, ar_1, ar_2, ar_0 + 1)) (Comp: ?, Cost: 1) evalabc10(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabc11(ar_0, ar_1, ar_2, ar_3)) (Comp: ?, Cost: 1) evalabc11(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabcbb1in(ar_3, ar_1, ar_2, ar_3)) (Comp: 2, Cost: 1) evalabcbb5in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabcstop(ar_0, ar_1, ar_2, ar_3)) (Comp: 1, Cost: 0) koat_start(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabcstart(ar_0, ar_1, ar_2, ar_3)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 A polynomial rank function with Pol(evalabcbb4in) = 3 Pol(evalabc10) = 2 Pol(evalabcbb3in) = 4 Pol(evalabcbb2in) = 4 Pol(evalabc11) = 1 Pol(evalabcbb1in) = 0 and size complexities S("koat_start(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabcstart(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(evalabcstart(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(evalabcstart(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(evalabcstart(ar_0, ar_1, ar_2, ar_3)) [ 0 <= 0 ]", 0-3) = ar_3 S("evalabcbb5in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabcstop(ar_0, ar_1, ar_2, ar_3))", 0-0) = ? S("evalabcbb5in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabcstop(ar_0, ar_1, ar_2, ar_3))", 0-1) = ar_1 S("evalabcbb5in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabcstop(ar_0, ar_1, ar_2, ar_3))", 0-2) = ? S("evalabcbb5in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabcstop(ar_0, ar_1, ar_2, ar_3))", 0-3) = ? S("evalabc11(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabcbb1in(ar_3, ar_1, ar_2, ar_3))", 0-0) = ? S("evalabc11(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabcbb1in(ar_3, ar_1, ar_2, ar_3))", 0-1) = ar_1 S("evalabc11(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabcbb1in(ar_3, ar_1, ar_2, ar_3))", 0-2) = ? S("evalabc11(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabcbb1in(ar_3, ar_1, ar_2, ar_3))", 0-3) = ? S("evalabc10(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabc11(ar_0, ar_1, ar_2, ar_3))", 0-0) = ? S("evalabc10(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabc11(ar_0, ar_1, ar_2, ar_3))", 0-1) = ar_1 S("evalabc10(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabc11(ar_0, ar_1, ar_2, ar_3))", 0-2) = ? S("evalabc10(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabc11(ar_0, ar_1, ar_2, ar_3))", 0-3) = ? S("evalabcbb4in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabc10(ar_0, ar_1, ar_2, ar_0 + 1))", 0-0) = ? S("evalabcbb4in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabc10(ar_0, ar_1, ar_2, ar_0 + 1))", 0-1) = ar_1 S("evalabcbb4in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabc10(ar_0, ar_1, ar_2, ar_0 + 1))", 0-2) = ? S("evalabcbb4in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabc10(ar_0, ar_1, ar_2, ar_0 + 1))", 0-3) = ? S("evalabcbb3in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabcbb2in(ar_0, ar_1, ar_2 + 1, ar_3))", 0-0) = ? S("evalabcbb3in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabcbb2in(ar_0, ar_1, ar_2 + 1, ar_3))", 0-1) = ar_1 S("evalabcbb3in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabcbb2in(ar_0, ar_1, ar_2 + 1, ar_3))", 0-2) = ? S("evalabcbb3in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabcbb2in(ar_0, ar_1, ar_2 + 1, ar_3))", 0-3) = ? S("evalabcbb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabcbb4in(ar_0, ar_1, ar_2, ar_3)) [ ar_2 >= ar_0 + 1 ]", 0-0) = ? S("evalabcbb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabcbb4in(ar_0, ar_1, ar_2, ar_3)) [ ar_2 >= ar_0 + 1 ]", 0-1) = ar_1 S("evalabcbb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabcbb4in(ar_0, ar_1, ar_2, ar_3)) [ ar_2 >= ar_0 + 1 ]", 0-2) = ? S("evalabcbb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabcbb4in(ar_0, ar_1, ar_2, ar_3)) [ ar_2 >= ar_0 + 1 ]", 0-3) = ? S("evalabcbb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabcbb3in(ar_0, ar_1, ar_2, ar_3)) [ ar_0 >= ar_2 ]", 0-0) = ? S("evalabcbb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabcbb3in(ar_0, ar_1, ar_2, ar_3)) [ ar_0 >= ar_2 ]", 0-1) = ar_1 S("evalabcbb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabcbb3in(ar_0, ar_1, ar_2, ar_3)) [ ar_0 >= ar_2 ]", 0-2) = ? S("evalabcbb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabcbb3in(ar_0, ar_1, ar_2, ar_3)) [ ar_0 >= ar_2 ]", 0-3) = ? S("evalabcbb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabcbb5in(ar_0, ar_1, ar_2, ar_3)) [ ar_0 >= ar_1 ]", 0-0) = ? S("evalabcbb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabcbb5in(ar_0, ar_1, ar_2, ar_3)) [ ar_0 >= ar_1 ]", 0-1) = ar_1 S("evalabcbb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabcbb5in(ar_0, ar_1, ar_2, ar_3)) [ ar_0 >= ar_1 ]", 0-2) = ? S("evalabcbb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabcbb5in(ar_0, ar_1, ar_2, ar_3)) [ ar_0 >= ar_1 ]", 0-3) = ? S("evalabcbb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabcbb2in(ar_0, ar_1, 0, ar_3)) [ ar_1 >= ar_0 + 1 ]", 0-0) = ? S("evalabcbb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabcbb2in(ar_0, ar_1, 0, ar_3)) [ ar_1 >= ar_0 + 1 ]", 0-1) = ar_1 S("evalabcbb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabcbb2in(ar_0, ar_1, 0, ar_3)) [ ar_1 >= ar_0 + 1 ]", 0-2) = 0 S("evalabcbb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabcbb2in(ar_0, ar_1, 0, ar_3)) [ ar_1 >= ar_0 + 1 ]", 0-3) = ? S("evalabc4(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabcbb1in(0, ar_1, ar_2, ar_3))", 0-0) = 0 S("evalabc4(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabcbb1in(0, ar_1, ar_2, ar_3))", 0-1) = ar_1 S("evalabc4(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabcbb1in(0, ar_1, ar_2, ar_3))", 0-2) = ar_2 S("evalabc4(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabcbb1in(0, ar_1, ar_2, ar_3))", 0-3) = ar_3 S("evalabc3(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabc4(ar_0, ar_1, ar_2, ar_3))", 0-0) = ar_0 S("evalabc3(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabc4(ar_0, ar_1, ar_2, ar_3))", 0-1) = ar_1 S("evalabc3(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabc4(ar_0, ar_1, ar_2, ar_3))", 0-2) = ar_2 S("evalabc3(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabc4(ar_0, ar_1, ar_2, ar_3))", 0-3) = ar_3 S("evalabc2(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabc3(ar_0, ar_1, ar_2, ar_3))", 0-0) = ar_0 S("evalabc2(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabc3(ar_0, ar_1, ar_2, ar_3))", 0-1) = ar_1 S("evalabc2(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabc3(ar_0, ar_1, ar_2, ar_3))", 0-2) = ar_2 S("evalabc2(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabc3(ar_0, ar_1, ar_2, ar_3))", 0-3) = ar_3 S("evalabc1(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabc2(ar_0, ar_1, ar_2, ar_3))", 0-0) = ar_0 S("evalabc1(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabc2(ar_0, ar_1, ar_2, ar_3))", 0-1) = ar_1 S("evalabc1(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabc2(ar_0, ar_1, ar_2, ar_3))", 0-2) = ar_2 S("evalabc1(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabc2(ar_0, ar_1, ar_2, ar_3))", 0-3) = ar_3 S("evalabc0(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabc1(ar_0, ar_1, ar_2, ar_3))", 0-0) = ar_0 S("evalabc0(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabc1(ar_0, ar_1, ar_2, ar_3))", 0-1) = ar_1 S("evalabc0(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabc1(ar_0, ar_1, ar_2, ar_3))", 0-2) = ar_2 S("evalabc0(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabc1(ar_0, ar_1, ar_2, ar_3))", 0-3) = ar_3 S("evalabcbb0in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabc0(ar_0, ar_1, ar_2, ar_3))", 0-0) = ar_0 S("evalabcbb0in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabc0(ar_0, ar_1, ar_2, ar_3))", 0-1) = ar_1 S("evalabcbb0in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabc0(ar_0, ar_1, ar_2, ar_3))", 0-2) = ar_2 S("evalabcbb0in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabc0(ar_0, ar_1, ar_2, ar_3))", 0-3) = ar_3 S("evalabcstart(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabcbb0in(ar_0, ar_1, ar_2, ar_3))", 0-0) = ar_0 S("evalabcstart(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabcbb0in(ar_0, ar_1, ar_2, ar_3))", 0-1) = ar_1 S("evalabcstart(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabcbb0in(ar_0, ar_1, ar_2, ar_3))", 0-2) = ar_2 S("evalabcstart(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabcbb0in(ar_0, ar_1, ar_2, ar_3))", 0-3) = ar_3 orients the transitions evalabcbb4in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabc10(ar_0, ar_1, ar_2, ar_0 + 1)) evalabcbb3in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabcbb2in(ar_0, ar_1, ar_2 + 1, ar_3)) evalabcbb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabcbb4in(ar_0, ar_1, ar_2, ar_3)) [ ar_2 >= ar_0 + 1 ] evalabcbb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabcbb3in(ar_0, ar_1, ar_2, ar_3)) [ ar_0 >= ar_2 ] evalabc11(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabcbb1in(ar_3, ar_1, ar_2, ar_3)) evalabc10(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabc11(ar_0, ar_1, ar_2, ar_3)) weakly and the transitions evalabcbb4in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabc10(ar_0, ar_1, ar_2, ar_0 + 1)) evalabcbb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabcbb4in(ar_0, ar_1, ar_2, ar_3)) [ ar_2 >= ar_0 + 1 ] evalabc11(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabcbb1in(ar_3, ar_1, ar_2, ar_3)) evalabc10(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabc11(ar_0, ar_1, ar_2, ar_3)) strictly and produces the following problem: 5: T: (Comp: 1, Cost: 1) evalabcstart(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabcbb0in(ar_0, ar_1, ar_2, ar_3)) (Comp: 1, Cost: 1) evalabcbb0in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabc0(ar_0, ar_1, ar_2, ar_3)) (Comp: 1, Cost: 1) evalabc0(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabc1(ar_0, ar_1, ar_2, ar_3)) (Comp: 1, Cost: 1) evalabc1(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabc2(ar_0, ar_1, ar_2, ar_3)) (Comp: 1, Cost: 1) evalabc2(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabc3(ar_0, ar_1, ar_2, ar_3)) (Comp: 1, Cost: 1) evalabc3(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabc4(ar_0, ar_1, ar_2, ar_3)) (Comp: 1, Cost: 1) evalabc4(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabcbb1in(0, ar_1, ar_2, ar_3)) (Comp: ar_1 + 1, Cost: 1) evalabcbb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabcbb2in(ar_0, ar_1, 0, ar_3)) [ ar_1 >= ar_0 + 1 ] (Comp: 2, Cost: 1) evalabcbb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabcbb5in(ar_0, ar_1, ar_2, ar_3)) [ ar_0 >= ar_1 ] (Comp: ?, Cost: 1) evalabcbb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabcbb3in(ar_0, ar_1, ar_2, ar_3)) [ ar_0 >= ar_2 ] (Comp: 4*ar_1 + 4, Cost: 1) evalabcbb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabcbb4in(ar_0, ar_1, ar_2, ar_3)) [ ar_2 >= ar_0 + 1 ] (Comp: ?, Cost: 1) evalabcbb3in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabcbb2in(ar_0, ar_1, ar_2 + 1, ar_3)) (Comp: 4*ar_1 + 4, Cost: 1) evalabcbb4in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabc10(ar_0, ar_1, ar_2, ar_0 + 1)) (Comp: 4*ar_1 + 4, Cost: 1) evalabc10(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabc11(ar_0, ar_1, ar_2, ar_3)) (Comp: 4*ar_1 + 4, Cost: 1) evalabc11(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabcbb1in(ar_3, ar_1, ar_2, ar_3)) (Comp: 2, Cost: 1) evalabcbb5in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabcstop(ar_0, ar_1, ar_2, ar_3)) (Comp: 1, Cost: 0) koat_start(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabcstart(ar_0, ar_1, ar_2, ar_3)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 A polynomial rank function with Pol(evalabcbb3in) = V_1 - V_3 Pol(evalabcbb2in) = V_1 - V_3 + 1 and size complexities S("koat_start(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabcstart(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(evalabcstart(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(evalabcstart(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(evalabcstart(ar_0, ar_1, ar_2, ar_3)) [ 0 <= 0 ]", 0-3) = ar_3 S("evalabcbb5in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabcstop(ar_0, ar_1, ar_2, ar_3))", 0-0) = 4*ar_1 + 1024 S("evalabcbb5in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabcstop(ar_0, ar_1, ar_2, ar_3))", 0-1) = ar_1 S("evalabcbb5in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabcstop(ar_0, ar_1, ar_2, ar_3))", 0-2) = ? S("evalabcbb5in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabcstop(ar_0, ar_1, ar_2, ar_3))", 0-3) = 4*ar_1 + 4*ar_3 + 4096 S("evalabc11(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabcbb1in(ar_3, ar_1, ar_2, ar_3))", 0-0) = 4*ar_1 + 64 S("evalabc11(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabcbb1in(ar_3, ar_1, ar_2, ar_3))", 0-1) = ar_1 S("evalabc11(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabcbb1in(ar_3, ar_1, ar_2, ar_3))", 0-2) = ? S("evalabc11(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabcbb1in(ar_3, ar_1, ar_2, ar_3))", 0-3) = 4*ar_1 + 256 S("evalabc10(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabc11(ar_0, ar_1, ar_2, ar_3))", 0-0) = 4*ar_1 + 1024 S("evalabc10(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabc11(ar_0, ar_1, ar_2, ar_3))", 0-1) = ar_1 S("evalabc10(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabc11(ar_0, ar_1, ar_2, ar_3))", 0-2) = ? S("evalabc10(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabc11(ar_0, ar_1, ar_2, ar_3))", 0-3) = 4*ar_1 + 64 S("evalabcbb4in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabc10(ar_0, ar_1, ar_2, ar_0 + 1))", 0-0) = 4*ar_1 + 256 S("evalabcbb4in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabc10(ar_0, ar_1, ar_2, ar_0 + 1))", 0-1) = ar_1 S("evalabcbb4in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabc10(ar_0, ar_1, ar_2, ar_0 + 1))", 0-2) = ? S("evalabcbb4in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabc10(ar_0, ar_1, ar_2, ar_0 + 1))", 0-3) = 4*ar_1 + 64 S("evalabcbb3in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabcbb2in(ar_0, ar_1, ar_2 + 1, ar_3))", 0-0) = 4*ar_1 + 64 S("evalabcbb3in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabcbb2in(ar_0, ar_1, ar_2 + 1, ar_3))", 0-1) = ar_1 S("evalabcbb3in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabcbb2in(ar_0, ar_1, ar_2 + 1, ar_3))", 0-2) = ? S("evalabcbb3in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabcbb2in(ar_0, ar_1, ar_2 + 1, ar_3))", 0-3) = 4*ar_1 + 4*ar_3 + 4096 S("evalabcbb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabcbb4in(ar_0, ar_1, ar_2, ar_3)) [ ar_2 >= ar_0 + 1 ]", 0-0) = 4*ar_1 + 64 S("evalabcbb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabcbb4in(ar_0, ar_1, ar_2, ar_3)) [ ar_2 >= ar_0 + 1 ]", 0-1) = ar_1 S("evalabcbb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabcbb4in(ar_0, ar_1, ar_2, ar_3)) [ ar_2 >= ar_0 + 1 ]", 0-2) = ? S("evalabcbb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabcbb4in(ar_0, ar_1, ar_2, ar_3)) [ ar_2 >= ar_0 + 1 ]", 0-3) = 4*ar_1 + 4*ar_3 + 16384 S("evalabcbb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabcbb3in(ar_0, ar_1, ar_2, ar_3)) [ ar_0 >= ar_2 ]", 0-0) = 4*ar_1 + 64 S("evalabcbb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabcbb3in(ar_0, ar_1, ar_2, ar_3)) [ ar_0 >= ar_2 ]", 0-1) = ar_1 S("evalabcbb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabcbb3in(ar_0, ar_1, ar_2, ar_3)) [ ar_0 >= ar_2 ]", 0-2) = ? S("evalabcbb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabcbb3in(ar_0, ar_1, ar_2, ar_3)) [ ar_0 >= ar_2 ]", 0-3) = 4*ar_1 + 4*ar_3 + 4096 S("evalabcbb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabcbb5in(ar_0, ar_1, ar_2, ar_3)) [ ar_0 >= ar_1 ]", 0-0) = 4*ar_1 + 256 S("evalabcbb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabcbb5in(ar_0, ar_1, ar_2, ar_3)) [ ar_0 >= ar_1 ]", 0-1) = ar_1 S("evalabcbb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabcbb5in(ar_0, ar_1, ar_2, ar_3)) [ ar_0 >= ar_1 ]", 0-2) = ? S("evalabcbb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabcbb5in(ar_0, ar_1, ar_2, ar_3)) [ ar_0 >= ar_1 ]", 0-3) = 4*ar_1 + 4*ar_3 + 1024 S("evalabcbb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabcbb2in(ar_0, ar_1, 0, ar_3)) [ ar_1 >= ar_0 + 1 ]", 0-0) = 4*ar_1 + 64 S("evalabcbb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabcbb2in(ar_0, ar_1, 0, ar_3)) [ ar_1 >= ar_0 + 1 ]", 0-1) = ar_1 S("evalabcbb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabcbb2in(ar_0, ar_1, 0, ar_3)) [ ar_1 >= ar_0 + 1 ]", 0-2) = 0 S("evalabcbb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabcbb2in(ar_0, ar_1, 0, ar_3)) [ ar_1 >= ar_0 + 1 ]", 0-3) = 4*ar_1 + 4*ar_3 + 1024 S("evalabc4(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabcbb1in(0, ar_1, ar_2, ar_3))", 0-0) = 0 S("evalabc4(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabcbb1in(0, ar_1, ar_2, ar_3))", 0-1) = ar_1 S("evalabc4(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabcbb1in(0, ar_1, ar_2, ar_3))", 0-2) = ar_2 S("evalabc4(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabcbb1in(0, ar_1, ar_2, ar_3))", 0-3) = ar_3 S("evalabc3(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabc4(ar_0, ar_1, ar_2, ar_3))", 0-0) = ar_0 S("evalabc3(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabc4(ar_0, ar_1, ar_2, ar_3))", 0-1) = ar_1 S("evalabc3(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabc4(ar_0, ar_1, ar_2, ar_3))", 0-2) = ar_2 S("evalabc3(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabc4(ar_0, ar_1, ar_2, ar_3))", 0-3) = ar_3 S("evalabc2(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabc3(ar_0, ar_1, ar_2, ar_3))", 0-0) = ar_0 S("evalabc2(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabc3(ar_0, ar_1, ar_2, ar_3))", 0-1) = ar_1 S("evalabc2(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabc3(ar_0, ar_1, ar_2, ar_3))", 0-2) = ar_2 S("evalabc2(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabc3(ar_0, ar_1, ar_2, ar_3))", 0-3) = ar_3 S("evalabc1(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabc2(ar_0, ar_1, ar_2, ar_3))", 0-0) = ar_0 S("evalabc1(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabc2(ar_0, ar_1, ar_2, ar_3))", 0-1) = ar_1 S("evalabc1(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabc2(ar_0, ar_1, ar_2, ar_3))", 0-2) = ar_2 S("evalabc1(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabc2(ar_0, ar_1, ar_2, ar_3))", 0-3) = ar_3 S("evalabc0(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabc1(ar_0, ar_1, ar_2, ar_3))", 0-0) = ar_0 S("evalabc0(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabc1(ar_0, ar_1, ar_2, ar_3))", 0-1) = ar_1 S("evalabc0(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabc1(ar_0, ar_1, ar_2, ar_3))", 0-2) = ar_2 S("evalabc0(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabc1(ar_0, ar_1, ar_2, ar_3))", 0-3) = ar_3 S("evalabcbb0in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabc0(ar_0, ar_1, ar_2, ar_3))", 0-0) = ar_0 S("evalabcbb0in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabc0(ar_0, ar_1, ar_2, ar_3))", 0-1) = ar_1 S("evalabcbb0in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabc0(ar_0, ar_1, ar_2, ar_3))", 0-2) = ar_2 S("evalabcbb0in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabc0(ar_0, ar_1, ar_2, ar_3))", 0-3) = ar_3 S("evalabcstart(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabcbb0in(ar_0, ar_1, ar_2, ar_3))", 0-0) = ar_0 S("evalabcstart(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabcbb0in(ar_0, ar_1, ar_2, ar_3))", 0-1) = ar_1 S("evalabcstart(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabcbb0in(ar_0, ar_1, ar_2, ar_3))", 0-2) = ar_2 S("evalabcstart(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabcbb0in(ar_0, ar_1, ar_2, ar_3))", 0-3) = ar_3 orients the transitions evalabcbb3in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabcbb2in(ar_0, ar_1, ar_2 + 1, ar_3)) evalabcbb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabcbb3in(ar_0, ar_1, ar_2, ar_3)) [ ar_0 >= ar_2 ] weakly and the transition evalabcbb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabcbb3in(ar_0, ar_1, ar_2, ar_3)) [ ar_0 >= ar_2 ] strictly and produces the following problem: 6: T: (Comp: 1, Cost: 1) evalabcstart(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabcbb0in(ar_0, ar_1, ar_2, ar_3)) (Comp: 1, Cost: 1) evalabcbb0in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabc0(ar_0, ar_1, ar_2, ar_3)) (Comp: 1, Cost: 1) evalabc0(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabc1(ar_0, ar_1, ar_2, ar_3)) (Comp: 1, Cost: 1) evalabc1(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabc2(ar_0, ar_1, ar_2, ar_3)) (Comp: 1, Cost: 1) evalabc2(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabc3(ar_0, ar_1, ar_2, ar_3)) (Comp: 1, Cost: 1) evalabc3(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabc4(ar_0, ar_1, ar_2, ar_3)) (Comp: 1, Cost: 1) evalabc4(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabcbb1in(0, ar_1, ar_2, ar_3)) (Comp: ar_1 + 1, Cost: 1) evalabcbb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabcbb2in(ar_0, ar_1, 0, ar_3)) [ ar_1 >= ar_0 + 1 ] (Comp: 2, Cost: 1) evalabcbb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabcbb5in(ar_0, ar_1, ar_2, ar_3)) [ ar_0 >= ar_1 ] (Comp: 4*ar_1^2 + 69*ar_1 + 65, Cost: 1) evalabcbb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabcbb3in(ar_0, ar_1, ar_2, ar_3)) [ ar_0 >= ar_2 ] (Comp: 4*ar_1 + 4, Cost: 1) evalabcbb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabcbb4in(ar_0, ar_1, ar_2, ar_3)) [ ar_2 >= ar_0 + 1 ] (Comp: ?, Cost: 1) evalabcbb3in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabcbb2in(ar_0, ar_1, ar_2 + 1, ar_3)) (Comp: 4*ar_1 + 4, Cost: 1) evalabcbb4in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabc10(ar_0, ar_1, ar_2, ar_0 + 1)) (Comp: 4*ar_1 + 4, Cost: 1) evalabc10(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabc11(ar_0, ar_1, ar_2, ar_3)) (Comp: 4*ar_1 + 4, Cost: 1) evalabc11(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabcbb1in(ar_3, ar_1, ar_2, ar_3)) (Comp: 2, Cost: 1) evalabcbb5in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabcstop(ar_0, ar_1, ar_2, ar_3)) (Comp: 1, Cost: 0) koat_start(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabcstart(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) evalabcstart(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabcbb0in(ar_0, ar_1, ar_2, ar_3)) (Comp: 1, Cost: 1) evalabcbb0in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabc0(ar_0, ar_1, ar_2, ar_3)) (Comp: 1, Cost: 1) evalabc0(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabc1(ar_0, ar_1, ar_2, ar_3)) (Comp: 1, Cost: 1) evalabc1(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabc2(ar_0, ar_1, ar_2, ar_3)) (Comp: 1, Cost: 1) evalabc2(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabc3(ar_0, ar_1, ar_2, ar_3)) (Comp: 1, Cost: 1) evalabc3(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabc4(ar_0, ar_1, ar_2, ar_3)) (Comp: 1, Cost: 1) evalabc4(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabcbb1in(0, ar_1, ar_2, ar_3)) (Comp: ar_1 + 1, Cost: 1) evalabcbb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabcbb2in(ar_0, ar_1, 0, ar_3)) [ ar_1 >= ar_0 + 1 ] (Comp: 2, Cost: 1) evalabcbb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabcbb5in(ar_0, ar_1, ar_2, ar_3)) [ ar_0 >= ar_1 ] (Comp: 4*ar_1^2 + 69*ar_1 + 65, Cost: 1) evalabcbb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabcbb3in(ar_0, ar_1, ar_2, ar_3)) [ ar_0 >= ar_2 ] (Comp: 4*ar_1 + 4, Cost: 1) evalabcbb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabcbb4in(ar_0, ar_1, ar_2, ar_3)) [ ar_2 >= ar_0 + 1 ] (Comp: 4*ar_1^2 + 69*ar_1 + 65, Cost: 1) evalabcbb3in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabcbb2in(ar_0, ar_1, ar_2 + 1, ar_3)) (Comp: 4*ar_1 + 4, Cost: 1) evalabcbb4in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabc10(ar_0, ar_1, ar_2, ar_0 + 1)) (Comp: 4*ar_1 + 4, Cost: 1) evalabc10(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabc11(ar_0, ar_1, ar_2, ar_3)) (Comp: 4*ar_1 + 4, Cost: 1) evalabc11(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabcbb1in(ar_3, ar_1, ar_2, ar_3)) (Comp: 2, Cost: 1) evalabcbb5in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabcstop(ar_0, ar_1, ar_2, ar_3)) (Comp: 1, Cost: 0) koat_start(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalabcstart(ar_0, ar_1, ar_2, ar_3)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 Complexity upper bound 155*ar_1 + 8*ar_1^2 + 158 Time: 0.158 sec (SMT: 0.118 sec) ---------------------------------------- (2) BOUNDS(1, n^2) ---------------------------------------- (3) Loat Proof (FINISHED) ### Pre-processing the ITS problem ### Initial linear ITS problem Start location: evalabcstart 0: evalabcstart -> evalabcbb0in : [], cost: 1 1: evalabcbb0in -> evalabc0 : [], cost: 1 2: evalabc0 -> evalabc1 : [], cost: 1 3: evalabc1 -> evalabc2 : [], cost: 1 4: evalabc2 -> evalabc3 : [], cost: 1 5: evalabc3 -> evalabc4 : [], cost: 1 6: evalabc4 -> evalabcbb1in : A'=0, [], cost: 1 7: evalabcbb1in -> evalabcbb2in : C'=0, [ B>=1+A ], cost: 1 8: evalabcbb1in -> evalabcbb5in : [ A>=B ], cost: 1 9: evalabcbb2in -> evalabcbb3in : [ A>=C ], cost: 1 10: evalabcbb2in -> evalabcbb4in : [ C>=1+A ], cost: 1 11: evalabcbb3in -> evalabcbb2in : C'=1+C, [], cost: 1 12: evalabcbb4in -> evalabc10 : D'=1+A, [], cost: 1 13: evalabc10 -> evalabc11 : [], cost: 1 14: evalabc11 -> evalabcbb1in : A'=D, [], cost: 1 15: evalabcbb5in -> evalabcstop : [], cost: 1 Removed unreachable and leaf rules: Start location: evalabcstart 0: evalabcstart -> evalabcbb0in : [], cost: 1 1: evalabcbb0in -> evalabc0 : [], cost: 1 2: evalabc0 -> evalabc1 : [], cost: 1 3: evalabc1 -> evalabc2 : [], cost: 1 4: evalabc2 -> evalabc3 : [], cost: 1 5: evalabc3 -> evalabc4 : [], cost: 1 6: evalabc4 -> evalabcbb1in : A'=0, [], cost: 1 7: evalabcbb1in -> evalabcbb2in : C'=0, [ B>=1+A ], cost: 1 9: evalabcbb2in -> evalabcbb3in : [ A>=C ], cost: 1 10: evalabcbb2in -> evalabcbb4in : [ C>=1+A ], cost: 1 11: evalabcbb3in -> evalabcbb2in : C'=1+C, [], cost: 1 12: evalabcbb4in -> evalabc10 : D'=1+A, [], cost: 1 13: evalabc10 -> evalabc11 : [], cost: 1 14: evalabc11 -> evalabcbb1in : A'=D, [], cost: 1 ### Simplification by acceleration and chaining ### Eliminated locations (on linear paths): Start location: evalabcstart 21: evalabcstart -> evalabcbb1in : A'=0, [], cost: 7 7: evalabcbb1in -> evalabcbb2in : C'=0, [ B>=1+A ], cost: 1 22: evalabcbb2in -> evalabcbb2in : C'=1+C, [ A>=C ], cost: 2 25: evalabcbb2in -> evalabcbb1in : A'=1+A, D'=1+A, [ C>=1+A ], cost: 4 Accelerating simple loops of location 8. Accelerating the following rules: 22: evalabcbb2in -> evalabcbb2in : C'=1+C, [ A>=C ], cost: 2 Accelerated rule 22 with metering function 1-C+A, yielding the new rule 26. Removing the simple loops: 22. Accelerated all simple loops using metering functions (where possible): Start location: evalabcstart 21: evalabcstart -> evalabcbb1in : A'=0, [], cost: 7 7: evalabcbb1in -> evalabcbb2in : C'=0, [ B>=1+A ], cost: 1 25: evalabcbb2in -> evalabcbb1in : A'=1+A, D'=1+A, [ C>=1+A ], cost: 4 26: evalabcbb2in -> evalabcbb2in : C'=1+A, [ A>=C ], cost: 2-2*C+2*A Chained accelerated rules (with incoming rules): Start location: evalabcstart 21: evalabcstart -> evalabcbb1in : A'=0, [], cost: 7 7: evalabcbb1in -> evalabcbb2in : C'=0, [ B>=1+A ], cost: 1 27: evalabcbb1in -> evalabcbb2in : C'=1+A, [ B>=1+A && A>=0 ], cost: 3+2*A 25: evalabcbb2in -> evalabcbb1in : A'=1+A, D'=1+A, [ C>=1+A ], cost: 4 Eliminated locations (on tree-shaped paths): Start location: evalabcstart 21: evalabcstart -> evalabcbb1in : A'=0, [], cost: 7 28: evalabcbb1in -> evalabcbb1in : A'=1+A, C'=0, D'=1+A, [ B>=1+A && 0>=1+A ], cost: 5 29: evalabcbb1in -> evalabcbb1in : A'=1+A, C'=1+A, D'=1+A, [ B>=1+A && A>=0 ], cost: 7+2*A Accelerating simple loops of location 7. Accelerating the following rules: 28: evalabcbb1in -> evalabcbb1in : A'=1+A, C'=0, D'=1+A, [ B>=1+A && 0>=1+A ], cost: 5 29: evalabcbb1in -> evalabcbb1in : A'=1+A, C'=1+A, D'=1+A, [ B>=1+A && A>=0 ], cost: 7+2*A Accelerated rule 28 with backward acceleration, yielding the new rule 30. Accelerated rule 28 with backward acceleration, yielding the new rule 31. Accelerated rule 29 with metering function -A+B, yielding the new rule 32. Removing the simple loops: 28 29. Accelerated all simple loops using metering functions (where possible): Start location: evalabcstart 21: evalabcstart -> evalabcbb1in : A'=0, [], cost: 7 30: evalabcbb1in -> evalabcbb1in : A'=B, C'=0, D'=B, [ B>=1+A && 0>=1+A && 0>=B ], cost: -5*A+5*B 31: evalabcbb1in -> evalabcbb1in : A'=0, C'=0, D'=0, [ B>=1+A && 0>=1+A && B>=0 ], cost: -5*A 32: evalabcbb1in -> evalabcbb1in : A'=B, C'=B, D'=B, [ B>=1+A && A>=0 ], cost: (A-B)^2-6*A-2*A*(A-B)+6*B Chained accelerated rules (with incoming rules): Start location: evalabcstart 21: evalabcstart -> evalabcbb1in : A'=0, [], cost: 7 33: evalabcstart -> evalabcbb1in : A'=B, C'=B, D'=B, [ B>=1 ], cost: 7+B^2+6*B Removed unreachable locations (and leaf rules with constant cost): Start location: evalabcstart 33: evalabcstart -> evalabcbb1in : A'=B, C'=B, D'=B, [ B>=1 ], cost: 7+B^2+6*B ### Computing asymptotic complexity ### Fully simplified ITS problem Start location: evalabcstart 33: evalabcstart -> evalabcbb1in : A'=B, C'=B, D'=B, [ B>=1 ], cost: 7+B^2+6*B Computing asymptotic complexity for rule 33 Solved the limit problem by the following transformations: Created initial limit problem: 7+B^2+6*B (+), 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+n^2+6*n 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+n^2+6*n Rule cost: 7+B^2+6*B Rule guard: [ B>=1 ] WORST_CASE(Omega(n^2),?) ---------------------------------------- (4) BOUNDS(n^2, INF)