/export/starexec/sandbox2/solver/bin/starexec_run_complexity /export/starexec/sandbox2/benchmark/theBenchmark.koat /export/starexec/sandbox2/output/output_files -------------------------------------------------------------------------------- WORST_CASE(Omega(n^2), O(n^2)) proof of /export/starexec/sandbox2/benchmark/theBenchmark.koat # AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(n^2, n^2). (0) CpxIntTrs (1) Koat Proof [FINISHED, 132 ms] (2) BOUNDS(1, n^2) (3) Loat Proof [FINISHED, 731 ms] (4) BOUNDS(n^2, INF) ---------------------------------------- (0) Obligation: Complexity Int TRS consisting of the following rules: eval_abc_start(v_3, v_i_0, v_j_0, v_n) -> Com_1(eval_abc_bb0_in(v_3, v_i_0, v_j_0, v_n)) :|: TRUE eval_abc_bb0_in(v_3, v_i_0, v_j_0, v_n) -> Com_1(eval_abc_0(v_3, v_i_0, v_j_0, v_n)) :|: TRUE eval_abc_0(v_3, v_i_0, v_j_0, v_n) -> Com_1(eval_abc_1(v_3, v_i_0, v_j_0, v_n)) :|: TRUE eval_abc_1(v_3, v_i_0, v_j_0, v_n) -> Com_1(eval_abc_2(v_3, v_i_0, v_j_0, v_n)) :|: TRUE eval_abc_2(v_3, v_i_0, v_j_0, v_n) -> Com_1(eval_abc_3(v_3, v_i_0, v_j_0, v_n)) :|: TRUE eval_abc_3(v_3, v_i_0, v_j_0, v_n) -> Com_1(eval_abc_4(v_3, v_i_0, v_j_0, v_n)) :|: TRUE eval_abc_4(v_3, v_i_0, v_j_0, v_n) -> Com_1(eval_abc_bb1_in(v_3, 1, v_j_0, v_n)) :|: TRUE eval_abc_bb1_in(v_3, v_i_0, v_j_0, v_n) -> Com_1(eval_abc_bb2_in(v_3, v_i_0, 1, v_n)) :|: v_i_0 <= v_n eval_abc_bb1_in(v_3, v_i_0, v_j_0, v_n) -> Com_1(eval_abc_bb5_in(v_3, v_i_0, v_j_0, v_n)) :|: v_i_0 > v_n eval_abc_bb2_in(v_3, v_i_0, v_j_0, v_n) -> Com_1(eval_abc_bb3_in(v_3, v_i_0, v_j_0, v_n)) :|: v_j_0 <= v_n eval_abc_bb2_in(v_3, v_i_0, v_j_0, v_n) -> Com_1(eval_abc_bb4_in(v_3, v_i_0, v_j_0, v_n)) :|: v_j_0 > v_n eval_abc_bb3_in(v_3, v_i_0, v_j_0, v_n) -> Com_1(eval_abc_bb2_in(v_3, v_i_0, v_j_0 + 1, v_n)) :|: TRUE eval_abc_bb4_in(v_3, v_i_0, v_j_0, v_n) -> Com_1(eval_abc_8(v_i_0 + 1, v_i_0, v_j_0, v_n)) :|: TRUE eval_abc_8(v_3, v_i_0, v_j_0, v_n) -> Com_1(eval_abc_9(v_3, v_i_0, v_j_0, v_n)) :|: TRUE eval_abc_9(v_3, v_i_0, v_j_0, v_n) -> Com_1(eval_abc_bb1_in(v_3, v_3, v_j_0, v_n)) :|: TRUE eval_abc_bb5_in(v_3, v_i_0, v_j_0, v_n) -> Com_1(eval_abc_stop(v_3, v_i_0, v_j_0, v_n)) :|: TRUE The start-symbols are:[eval_abc_start_4] ---------------------------------------- (1) Koat Proof (FINISHED) YES(?, 21*Ar_1 + 2*Ar_1^2 + 11) 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(1, 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, 1, Ar_3)) [ Ar_1 >= Ar_0 ] (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 + 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_1 >= 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_1 + 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(evalabc8(Ar_0, Ar_1, Ar_2, Ar_0 + 1)) (Comp: ?, Cost: 1) evalabc8(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalabc9(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: ?, Cost: 1) evalabc9(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(1, 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, 1, Ar_3)) [ Ar_1 >= Ar_0 ] (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 + 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_1 >= 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_1 + 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(evalabc8(Ar_0, Ar_1, Ar_2, Ar_0 + 1)) (Comp: ?, Cost: 1) evalabc8(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalabc9(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: ?, Cost: 1) evalabc9(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(evalabc8) = 2 Pol(evalabc9) = 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 + 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(1, 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, 1, Ar_3)) [ Ar_1 >= Ar_0 ] (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 + 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_1 >= 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_1 + 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(evalabc8(Ar_0, Ar_1, Ar_2, Ar_0 + 1)) (Comp: ?, Cost: 1) evalabc8(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalabc9(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: ?, Cost: 1) evalabc9(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 Pol(evalabcbb0in) = V_2 Pol(evalabc0) = V_2 Pol(evalabc1) = V_2 Pol(evalabc2) = V_2 Pol(evalabc3) = V_2 Pol(evalabc4) = V_2 Pol(evalabcbb1in) = -V_1 + V_2 + 1 Pol(evalabcbb2in) = -V_1 + V_2 Pol(evalabcbb5in) = -V_1 + V_2 + 1 Pol(evalabcbb3in) = -V_1 + V_2 Pol(evalabcbb4in) = -V_1 + V_2 Pol(evalabc8) = V_2 - V_4 + 1 Pol(evalabc9) = V_2 - V_4 + 1 Pol(evalabcstop) = -V_1 + V_2 + 1 Pol(koat_start) = V_2 orients all transitions weakly and the transition evalabcbb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalabcbb2in(Ar_0, Ar_1, 1, Ar_3)) [ Ar_1 >= Ar_0 ] 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(1, Ar_1, Ar_2, Ar_3)) (Comp: Ar_1, Cost: 1) evalabcbb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalabcbb2in(Ar_0, Ar_1, 1, Ar_3)) [ Ar_1 >= Ar_0 ] (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 + 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_1 >= 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_1 + 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(evalabc8(Ar_0, Ar_1, Ar_2, Ar_0 + 1)) (Comp: ?, Cost: 1) evalabc8(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalabc9(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: ?, Cost: 1) evalabc9(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(evalabc8) = 2 Pol(evalabcbb3in) = 4 Pol(evalabcbb2in) = 4 Pol(evalabc9) = 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("evalabc9(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalabcbb1in(Ar_3, Ar_1, Ar_2, Ar_3))", 0-0) = ? S("evalabc9(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalabcbb1in(Ar_3, Ar_1, Ar_2, Ar_3))", 0-1) = Ar_1 S("evalabc9(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalabcbb1in(Ar_3, Ar_1, Ar_2, Ar_3))", 0-2) = ? S("evalabc9(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalabcbb1in(Ar_3, Ar_1, Ar_2, Ar_3))", 0-3) = ? S("evalabc8(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalabc9(Ar_0, Ar_1, Ar_2, Ar_3))", 0-0) = ? S("evalabc8(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalabc9(Ar_0, Ar_1, Ar_2, Ar_3))", 0-1) = Ar_1 S("evalabc8(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalabc9(Ar_0, Ar_1, Ar_2, Ar_3))", 0-2) = ? S("evalabc8(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalabc9(Ar_0, Ar_1, Ar_2, Ar_3))", 0-3) = ? S("evalabcbb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalabc8(Ar_0, Ar_1, Ar_2, Ar_0 + 1))", 0-0) = ? S("evalabcbb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalabc8(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(evalabc8(Ar_0, Ar_1, Ar_2, Ar_0 + 1))", 0-2) = ? S("evalabcbb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalabc8(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_1 + 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_1 + 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_1 + 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_1 + 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_1 >= 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_1 >= 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_1 >= 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_1 >= 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 + 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 + 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 + 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 + 1 ]", 0-3) = ? S("evalabcbb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalabcbb2in(Ar_0, Ar_1, 1, Ar_3)) [ Ar_1 >= Ar_0 ]", 0-0) = ? S("evalabcbb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalabcbb2in(Ar_0, Ar_1, 1, Ar_3)) [ Ar_1 >= Ar_0 ]", 0-1) = Ar_1 S("evalabcbb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalabcbb2in(Ar_0, Ar_1, 1, Ar_3)) [ Ar_1 >= Ar_0 ]", 0-2) = 1 S("evalabcbb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalabcbb2in(Ar_0, Ar_1, 1, Ar_3)) [ Ar_1 >= Ar_0 ]", 0-3) = ? S("evalabc4(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalabcbb1in(1, Ar_1, Ar_2, Ar_3))", 0-0) = 1 S("evalabc4(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalabcbb1in(1, Ar_1, Ar_2, Ar_3))", 0-1) = Ar_1 S("evalabc4(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalabcbb1in(1, Ar_1, Ar_2, Ar_3))", 0-2) = Ar_2 S("evalabc4(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalabcbb1in(1, 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(evalabc8(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_1 + 1 ] evalabcbb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalabcbb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_1 >= Ar_2 ] evalabc9(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalabcbb1in(Ar_3, Ar_1, Ar_2, Ar_3)) evalabc8(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalabc9(Ar_0, Ar_1, Ar_2, Ar_3)) weakly and the transitions evalabcbb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalabc8(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_1 + 1 ] evalabc9(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalabcbb1in(Ar_3, Ar_1, Ar_2, Ar_3)) evalabc8(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalabc9(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(1, Ar_1, Ar_2, Ar_3)) (Comp: Ar_1, Cost: 1) evalabcbb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalabcbb2in(Ar_0, Ar_1, 1, Ar_3)) [ Ar_1 >= Ar_0 ] (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 + 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_1 >= Ar_2 ] (Comp: 4*Ar_1, 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_1 + 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, Cost: 1) evalabcbb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalabc8(Ar_0, Ar_1, Ar_2, Ar_0 + 1)) (Comp: 4*Ar_1, Cost: 1) evalabc8(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalabc9(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 4*Ar_1, Cost: 1) evalabc9(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_2 - V_3 Pol(evalabcbb2in) = V_2 - 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 + 272 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 + 1024 S("evalabc9(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalabcbb1in(Ar_3, Ar_1, Ar_2, Ar_3))", 0-0) = 4*Ar_1 + 16 S("evalabc9(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalabcbb1in(Ar_3, Ar_1, Ar_2, Ar_3))", 0-1) = Ar_1 S("evalabc9(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalabcbb1in(Ar_3, Ar_1, Ar_2, Ar_3))", 0-2) = ? S("evalabc9(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalabcbb1in(Ar_3, Ar_1, Ar_2, Ar_3))", 0-3) = 4*Ar_1 + 64 S("evalabc8(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalabc9(Ar_0, Ar_1, Ar_2, Ar_3))", 0-0) = 4*Ar_1 + 256 S("evalabc8(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalabc9(Ar_0, Ar_1, Ar_2, Ar_3))", 0-1) = Ar_1 S("evalabc8(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalabc9(Ar_0, Ar_1, Ar_2, Ar_3))", 0-2) = ? S("evalabc8(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalabc9(Ar_0, Ar_1, Ar_2, Ar_3))", 0-3) = 4*Ar_1 + 16 S("evalabcbb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalabc8(Ar_0, Ar_1, Ar_2, Ar_0 + 1))", 0-0) = 4*Ar_1 + 64 S("evalabcbb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalabc8(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(evalabc8(Ar_0, Ar_1, Ar_2, Ar_0 + 1))", 0-2) = ? S("evalabcbb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalabc8(Ar_0, Ar_1, Ar_2, Ar_0 + 1))", 0-3) = 4*Ar_1 + 16 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 + 16 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 + 1024 S("evalabcbb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalabcbb4in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= Ar_1 + 1 ]", 0-0) = 4*Ar_1 + 16 S("evalabcbb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalabcbb4in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= Ar_1 + 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_1 + 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_1 + 1 ]", 0-3) = 4*Ar_1 + 4*Ar_3 + 4096 S("evalabcbb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalabcbb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_1 >= Ar_2 ]", 0-0) = 4*Ar_1 + 16 S("evalabcbb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalabcbb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_1 >= 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_1 >= 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_1 >= Ar_2 ]", 0-3) = 4*Ar_1 + 4*Ar_3 + 1024 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 + 1 ]", 0-0) = 4*Ar_1 + 68 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 + 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 + 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 + 1 ]", 0-3) = 4*Ar_1 + 4*Ar_3 + 256 S("evalabcbb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalabcbb2in(Ar_0, Ar_1, 1, Ar_3)) [ Ar_1 >= Ar_0 ]", 0-0) = 4*Ar_1 + 16 S("evalabcbb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalabcbb2in(Ar_0, Ar_1, 1, Ar_3)) [ Ar_1 >= Ar_0 ]", 0-1) = Ar_1 S("evalabcbb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalabcbb2in(Ar_0, Ar_1, 1, Ar_3)) [ Ar_1 >= Ar_0 ]", 0-2) = 1 S("evalabcbb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalabcbb2in(Ar_0, Ar_1, 1, Ar_3)) [ Ar_1 >= Ar_0 ]", 0-3) = 4*Ar_1 + 4*Ar_3 + 256 S("evalabc4(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalabcbb1in(1, Ar_1, Ar_2, Ar_3))", 0-0) = 1 S("evalabc4(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalabcbb1in(1, Ar_1, Ar_2, Ar_3))", 0-1) = Ar_1 S("evalabc4(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalabcbb1in(1, Ar_1, Ar_2, Ar_3))", 0-2) = Ar_2 S("evalabc4(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalabcbb1in(1, 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_1 >= 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_1 >= 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(1, Ar_1, Ar_2, Ar_3)) (Comp: Ar_1, Cost: 1) evalabcbb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalabcbb2in(Ar_0, Ar_1, 1, Ar_3)) [ Ar_1 >= Ar_0 ] (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 + 1 ] (Comp: Ar_1^2 + 2*Ar_1, Cost: 1) evalabcbb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalabcbb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_1 >= Ar_2 ] (Comp: 4*Ar_1, 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_1 + 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, Cost: 1) evalabcbb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalabc8(Ar_0, Ar_1, Ar_2, Ar_0 + 1)) (Comp: 4*Ar_1, Cost: 1) evalabc8(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalabc9(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 4*Ar_1, Cost: 1) evalabc9(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(1, Ar_1, Ar_2, Ar_3)) (Comp: Ar_1, Cost: 1) evalabcbb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalabcbb2in(Ar_0, Ar_1, 1, Ar_3)) [ Ar_1 >= Ar_0 ] (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 + 1 ] (Comp: Ar_1^2 + 2*Ar_1, Cost: 1) evalabcbb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalabcbb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_1 >= Ar_2 ] (Comp: 4*Ar_1, 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_1 + 1 ] (Comp: Ar_1^2 + 2*Ar_1, 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, Cost: 1) evalabcbb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalabc8(Ar_0, Ar_1, Ar_2, Ar_0 + 1)) (Comp: 4*Ar_1, Cost: 1) evalabc8(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalabc9(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 4*Ar_1, Cost: 1) evalabc9(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 21*Ar_1 + 2*Ar_1^2 + 11 Time: 0.128 sec (SMT: 0.089 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'=1, [], cost: 1 7: evalabcbb1in -> evalabcbb2in : C'=1, [ B>=A ], cost: 1 8: evalabcbb1in -> evalabcbb5in : [ A>=1+B ], cost: 1 9: evalabcbb2in -> evalabcbb3in : [ B>=C ], cost: 1 10: evalabcbb2in -> evalabcbb4in : [ C>=1+B ], cost: 1 11: evalabcbb3in -> evalabcbb2in : C'=1+C, [], cost: 1 12: evalabcbb4in -> evalabc8 : D'=1+A, [], cost: 1 13: evalabc8 -> evalabc9 : [], cost: 1 14: evalabc9 -> evalabcbb1in : A'=D, [], cost: 1 15: evalabcbb5in -> evalabcstop : [], cost: 1 Checking for constant complexity: The following rule is satisfiable with cost >= 1, yielding constant complexity: 0: evalabcstart -> evalabcbb0in : [], 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'=1, [], cost: 1 7: evalabcbb1in -> evalabcbb2in : C'=1, [ B>=A ], cost: 1 9: evalabcbb2in -> evalabcbb3in : [ B>=C ], cost: 1 10: evalabcbb2in -> evalabcbb4in : [ C>=1+B ], cost: 1 11: evalabcbb3in -> evalabcbb2in : C'=1+C, [], cost: 1 12: evalabcbb4in -> evalabc8 : D'=1+A, [], cost: 1 13: evalabc8 -> evalabc9 : [], cost: 1 14: evalabc9 -> evalabcbb1in : A'=D, [], cost: 1 ### Simplification by acceleration and chaining ### Eliminated locations (on linear paths): Start location: evalabcstart 21: evalabcstart -> evalabcbb1in : A'=1, [], cost: 7 7: evalabcbb1in -> evalabcbb2in : C'=1, [ B>=A ], cost: 1 22: evalabcbb2in -> evalabcbb2in : C'=1+C, [ B>=C ], cost: 2 25: evalabcbb2in -> evalabcbb1in : A'=1+A, D'=1+A, [ C>=1+B ], cost: 4 Accelerating simple loops of location 8. Accelerating the following rules: 22: evalabcbb2in -> evalabcbb2in : C'=1+C, [ B>=C ], cost: 2 Accelerated rule 22 with metering function 1-C+B, 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'=1, [], cost: 7 7: evalabcbb1in -> evalabcbb2in : C'=1, [ B>=A ], cost: 1 25: evalabcbb2in -> evalabcbb1in : A'=1+A, D'=1+A, [ C>=1+B ], cost: 4 26: evalabcbb2in -> evalabcbb2in : C'=1+B, [ B>=C ], cost: 2-2*C+2*B Chained accelerated rules (with incoming rules): Start location: evalabcstart 21: evalabcstart -> evalabcbb1in : A'=1, [], cost: 7 7: evalabcbb1in -> evalabcbb2in : C'=1, [ B>=A ], cost: 1 27: evalabcbb1in -> evalabcbb2in : C'=1+B, [ B>=A && B>=1 ], cost: 1+2*B 25: evalabcbb2in -> evalabcbb1in : A'=1+A, D'=1+A, [ C>=1+B ], cost: 4 Eliminated locations (on tree-shaped paths): Start location: evalabcstart 21: evalabcstart -> evalabcbb1in : A'=1, [], cost: 7 28: evalabcbb1in -> evalabcbb1in : A'=1+A, C'=1, D'=1+A, [ B>=A && 1>=1+B ], cost: 5 29: evalabcbb1in -> evalabcbb1in : A'=1+A, C'=1+B, D'=1+A, [ B>=A && B>=1 ], cost: 5+2*B Accelerating simple loops of location 7. Accelerating the following rules: 28: evalabcbb1in -> evalabcbb1in : A'=1+A, C'=1, D'=1+A, [ B>=A && 1>=1+B ], cost: 5 29: evalabcbb1in -> evalabcbb1in : A'=1+A, C'=1+B, D'=1+A, [ B>=A && B>=1 ], cost: 5+2*B Accelerated rule 28 with metering function 1-A+B, yielding the new rule 30. Accelerated rule 29 with metering function 1-A+B, yielding the new rule 31. Removing the simple loops: 28 29. Accelerated all simple loops using metering functions (where possible): Start location: evalabcstart 21: evalabcstart -> evalabcbb1in : A'=1, [], cost: 7 30: evalabcbb1in -> evalabcbb1in : A'=1+B, C'=1, D'=1+B, [ B>=A && 1>=1+B ], cost: 5-5*A+5*B 31: evalabcbb1in -> evalabcbb1in : A'=1+B, C'=1+B, D'=1+B, [ B>=A && B>=1 ], cost: 5-5*A-2*(-1+A-B)*B+5*B Chained accelerated rules (with incoming rules): Start location: evalabcstart 21: evalabcstart -> evalabcbb1in : A'=1, [], cost: 7 32: evalabcstart -> evalabcbb1in : A'=1+B, C'=1+B, D'=1+B, [ B>=1 ], cost: 7+5*B+2*B^2 Removed unreachable locations (and leaf rules with constant cost): Start location: evalabcstart 32: evalabcstart -> evalabcbb1in : A'=1+B, C'=1+B, D'=1+B, [ B>=1 ], cost: 7+5*B+2*B^2 ### Computing asymptotic complexity ### Fully simplified ITS problem Start location: evalabcstart 32: evalabcstart -> evalabcbb1in : A'=1+B, C'=1+B, D'=1+B, [ B>=1 ], cost: 7+5*B+2*B^2 Computing asymptotic complexity for rule 32 Solved the limit problem by the following transformations: Created initial limit problem: 7+5*B+2*B^2 (+), B (+/+!) [not solved] removing all constraints (solved by SMT) resulting limit problem: [solved] applying transformation rule (C) using substitution {B==n} resulting limit problem: [solved] Solution: B / n Resulting cost 7+5*n+2*n^2 has complexity: Poly(n^2) Found new complexity Poly(n^2). Obtained the following overall complexity (w.r.t. the length of the input n): Complexity: Poly(n^2) Cpx degree: 2 Solved cost: 7+5*n+2*n^2 Rule cost: 7+5*B+2*B^2 Rule guard: [ B>=1 ] WORST_CASE(Omega(n^2),?) ---------------------------------------- (4) BOUNDS(n^2, INF)