/export/starexec/sandbox2/solver/bin/starexec_run_c_complexity /export/starexec/sandbox2/benchmark/theBenchmark.c /export/starexec/sandbox2/output/output_files -------------------------------------------------------------------------------- WORST_CASE(?, O(n^2)) proof of /export/starexec/sandbox2/output/output_files/bench.koat # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(1, n^2). (0) CpxIntTrs (1) Koat Proof [FINISHED, 74 ms] (2) BOUNDS(1, n^2) ---------------------------------------- (0) Obligation: Complexity Int TRS consisting of the following rules: eval_textbook_ex4_start(v_i.0, v_j.0, v_m, v_n) -> Com_1(eval_textbook_ex4_bb0_in(v_i.0, v_j.0, v_m, v_n)) :|: TRUE eval_textbook_ex4_bb0_in(v_i.0, v_j.0, v_m, v_n) -> Com_1(eval_textbook_ex4_bb1_in(v_i.0, 1, v_m, v_n)) :|: TRUE eval_textbook_ex4_bb1_in(v_i.0, v_j.0, v_m, v_n) -> Com_1(eval_textbook_ex4_bb2_in(1, v_j.0, v_m, v_n)) :|: v_j.0 <= v_n eval_textbook_ex4_bb1_in(v_i.0, v_j.0, v_m, v_n) -> Com_1(eval_textbook_ex4_bb5_in(v_i.0, v_j.0, v_m, v_n)) :|: v_j.0 > v_n eval_textbook_ex4_bb2_in(v_i.0, v_j.0, v_m, v_n) -> Com_1(eval_textbook_ex4_bb3_in(v_i.0, v_j.0, v_m, v_n)) :|: v_i.0 <= v_m eval_textbook_ex4_bb2_in(v_i.0, v_j.0, v_m, v_n) -> Com_1(eval_textbook_ex4_bb4_in(v_i.0, v_j.0, v_m, v_n)) :|: v_i.0 > v_m eval_textbook_ex4_bb3_in(v_i.0, v_j.0, v_m, v_n) -> Com_1(eval_textbook_ex4_bb2_in(v_i.0 + 1, v_j.0, v_m, v_n)) :|: TRUE eval_textbook_ex4_bb4_in(v_i.0, v_j.0, v_m, v_n) -> Com_1(eval_textbook_ex4_bb1_in(v_i.0, v_j.0 + 1, v_m, v_n)) :|: TRUE eval_textbook_ex4_bb5_in(v_i.0, v_j.0, v_m, v_n) -> Com_1(eval_textbook_ex4_stop(v_i.0, v_j.0, v_m, v_n)) :|: TRUE The start-symbols are:[eval_textbook_ex4_start_4] ---------------------------------------- (1) Koat Proof (FINISHED) YES(?, 9*ar_1 + 2*ar_1*ar_3 + 6) Initial complexity problem: 1: T: (Comp: ?, Cost: 1) evaltextbookex4start(ar_0, ar_1, ar_2, ar_3) -> Com_1(evaltextbookex4bb0in(ar_0, ar_1, ar_2, ar_3)) (Comp: ?, Cost: 1) evaltextbookex4bb0in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evaltextbookex4bb1in(1, ar_1, ar_2, ar_3)) (Comp: ?, Cost: 1) evaltextbookex4bb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evaltextbookex4bb2in(ar_0, ar_1, 1, ar_3)) [ ar_1 >= ar_0 ] (Comp: ?, Cost: 1) evaltextbookex4bb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evaltextbookex4bb5in(ar_0, ar_1, ar_2, ar_3)) [ ar_0 >= ar_1 + 1 ] (Comp: ?, Cost: 1) evaltextbookex4bb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evaltextbookex4bb3in(ar_0, ar_1, ar_2, ar_3)) [ ar_3 >= ar_2 ] (Comp: ?, Cost: 1) evaltextbookex4bb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evaltextbookex4bb4in(ar_0, ar_1, ar_2, ar_3)) [ ar_2 >= ar_3 + 1 ] (Comp: ?, Cost: 1) evaltextbookex4bb3in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evaltextbookex4bb2in(ar_0, ar_1, ar_2 + 1, ar_3)) (Comp: ?, Cost: 1) evaltextbookex4bb4in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evaltextbookex4bb1in(ar_0 + 1, ar_1, ar_2, ar_3)) (Comp: ?, Cost: 1) evaltextbookex4bb5in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evaltextbookex4stop(ar_0, ar_1, ar_2, ar_3)) (Comp: 1, Cost: 0) koat_start(ar_0, ar_1, ar_2, ar_3) -> Com_1(evaltextbookex4start(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) evaltextbookex4start(ar_0, ar_1, ar_2, ar_3) -> Com_1(evaltextbookex4bb0in(ar_0, ar_1, ar_2, ar_3)) (Comp: 1, Cost: 1) evaltextbookex4bb0in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evaltextbookex4bb1in(1, ar_1, ar_2, ar_3)) (Comp: ?, Cost: 1) evaltextbookex4bb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evaltextbookex4bb2in(ar_0, ar_1, 1, ar_3)) [ ar_1 >= ar_0 ] (Comp: ?, Cost: 1) evaltextbookex4bb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evaltextbookex4bb5in(ar_0, ar_1, ar_2, ar_3)) [ ar_0 >= ar_1 + 1 ] (Comp: ?, Cost: 1) evaltextbookex4bb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evaltextbookex4bb3in(ar_0, ar_1, ar_2, ar_3)) [ ar_3 >= ar_2 ] (Comp: ?, Cost: 1) evaltextbookex4bb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evaltextbookex4bb4in(ar_0, ar_1, ar_2, ar_3)) [ ar_2 >= ar_3 + 1 ] (Comp: ?, Cost: 1) evaltextbookex4bb3in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evaltextbookex4bb2in(ar_0, ar_1, ar_2 + 1, ar_3)) (Comp: ?, Cost: 1) evaltextbookex4bb4in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evaltextbookex4bb1in(ar_0 + 1, ar_1, ar_2, ar_3)) (Comp: ?, Cost: 1) evaltextbookex4bb5in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evaltextbookex4stop(ar_0, ar_1, ar_2, ar_3)) (Comp: 1, Cost: 0) koat_start(ar_0, ar_1, ar_2, ar_3) -> Com_1(evaltextbookex4start(ar_0, ar_1, ar_2, ar_3)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 A polynomial rank function with Pol(evaltextbookex4start) = 2 Pol(evaltextbookex4bb0in) = 2 Pol(evaltextbookex4bb1in) = 2 Pol(evaltextbookex4bb2in) = 2 Pol(evaltextbookex4bb5in) = 1 Pol(evaltextbookex4bb3in) = 2 Pol(evaltextbookex4bb4in) = 2 Pol(evaltextbookex4stop) = 0 Pol(koat_start) = 2 orients all transitions weakly and the transitions evaltextbookex4bb5in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evaltextbookex4stop(ar_0, ar_1, ar_2, ar_3)) evaltextbookex4bb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evaltextbookex4bb5in(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) evaltextbookex4start(ar_0, ar_1, ar_2, ar_3) -> Com_1(evaltextbookex4bb0in(ar_0, ar_1, ar_2, ar_3)) (Comp: 1, Cost: 1) evaltextbookex4bb0in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evaltextbookex4bb1in(1, ar_1, ar_2, ar_3)) (Comp: ?, Cost: 1) evaltextbookex4bb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evaltextbookex4bb2in(ar_0, ar_1, 1, ar_3)) [ ar_1 >= ar_0 ] (Comp: 2, Cost: 1) evaltextbookex4bb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evaltextbookex4bb5in(ar_0, ar_1, ar_2, ar_3)) [ ar_0 >= ar_1 + 1 ] (Comp: ?, Cost: 1) evaltextbookex4bb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evaltextbookex4bb3in(ar_0, ar_1, ar_2, ar_3)) [ ar_3 >= ar_2 ] (Comp: ?, Cost: 1) evaltextbookex4bb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evaltextbookex4bb4in(ar_0, ar_1, ar_2, ar_3)) [ ar_2 >= ar_3 + 1 ] (Comp: ?, Cost: 1) evaltextbookex4bb3in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evaltextbookex4bb2in(ar_0, ar_1, ar_2 + 1, ar_3)) (Comp: ?, Cost: 1) evaltextbookex4bb4in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evaltextbookex4bb1in(ar_0 + 1, ar_1, ar_2, ar_3)) (Comp: 2, Cost: 1) evaltextbookex4bb5in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evaltextbookex4stop(ar_0, ar_1, ar_2, ar_3)) (Comp: 1, Cost: 0) koat_start(ar_0, ar_1, ar_2, ar_3) -> Com_1(evaltextbookex4start(ar_0, ar_1, ar_2, ar_3)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 A polynomial rank function with Pol(evaltextbookex4start) = V_2 Pol(evaltextbookex4bb0in) = V_2 Pol(evaltextbookex4bb1in) = -V_1 + V_2 + 1 Pol(evaltextbookex4bb2in) = -V_1 + V_2 Pol(evaltextbookex4bb5in) = -V_1 + V_2 Pol(evaltextbookex4bb3in) = -V_1 + V_2 Pol(evaltextbookex4bb4in) = -V_1 + V_2 Pol(evaltextbookex4stop) = -V_1 + V_2 Pol(koat_start) = V_2 orients all transitions weakly and the transition evaltextbookex4bb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evaltextbookex4bb2in(ar_0, ar_1, 1, ar_3)) [ ar_1 >= ar_0 ] strictly and produces the following problem: 4: T: (Comp: 1, Cost: 1) evaltextbookex4start(ar_0, ar_1, ar_2, ar_3) -> Com_1(evaltextbookex4bb0in(ar_0, ar_1, ar_2, ar_3)) (Comp: 1, Cost: 1) evaltextbookex4bb0in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evaltextbookex4bb1in(1, ar_1, ar_2, ar_3)) (Comp: ar_1, Cost: 1) evaltextbookex4bb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evaltextbookex4bb2in(ar_0, ar_1, 1, ar_3)) [ ar_1 >= ar_0 ] (Comp: 2, Cost: 1) evaltextbookex4bb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evaltextbookex4bb5in(ar_0, ar_1, ar_2, ar_3)) [ ar_0 >= ar_1 + 1 ] (Comp: ?, Cost: 1) evaltextbookex4bb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evaltextbookex4bb3in(ar_0, ar_1, ar_2, ar_3)) [ ar_3 >= ar_2 ] (Comp: ?, Cost: 1) evaltextbookex4bb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evaltextbookex4bb4in(ar_0, ar_1, ar_2, ar_3)) [ ar_2 >= ar_3 + 1 ] (Comp: ?, Cost: 1) evaltextbookex4bb3in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evaltextbookex4bb2in(ar_0, ar_1, ar_2 + 1, ar_3)) (Comp: ?, Cost: 1) evaltextbookex4bb4in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evaltextbookex4bb1in(ar_0 + 1, ar_1, ar_2, ar_3)) (Comp: 2, Cost: 1) evaltextbookex4bb5in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evaltextbookex4stop(ar_0, ar_1, ar_2, ar_3)) (Comp: 1, Cost: 0) koat_start(ar_0, ar_1, ar_2, ar_3) -> Com_1(evaltextbookex4start(ar_0, ar_1, ar_2, ar_3)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 A polynomial rank function with Pol(evaltextbookex4bb4in) = 1 Pol(evaltextbookex4bb1in) = 0 Pol(evaltextbookex4bb3in) = 2 Pol(evaltextbookex4bb2in) = 2 and size complexities S("koat_start(ar_0, ar_1, ar_2, ar_3) -> Com_1(evaltextbookex4start(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(evaltextbookex4start(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(evaltextbookex4start(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(evaltextbookex4start(ar_0, ar_1, ar_2, ar_3)) [ 0 <= 0 ]", 0-3) = ar_3 S("evaltextbookex4bb5in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evaltextbookex4stop(ar_0, ar_1, ar_2, ar_3))", 0-0) = ? S("evaltextbookex4bb5in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evaltextbookex4stop(ar_0, ar_1, ar_2, ar_3))", 0-1) = ar_1 S("evaltextbookex4bb5in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evaltextbookex4stop(ar_0, ar_1, ar_2, ar_3))", 0-2) = ? S("evaltextbookex4bb5in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evaltextbookex4stop(ar_0, ar_1, ar_2, ar_3))", 0-3) = ar_3 S("evaltextbookex4bb4in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evaltextbookex4bb1in(ar_0 + 1, ar_1, ar_2, ar_3))", 0-0) = ? S("evaltextbookex4bb4in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evaltextbookex4bb1in(ar_0 + 1, ar_1, ar_2, ar_3))", 0-1) = ar_1 S("evaltextbookex4bb4in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evaltextbookex4bb1in(ar_0 + 1, ar_1, ar_2, ar_3))", 0-2) = ? S("evaltextbookex4bb4in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evaltextbookex4bb1in(ar_0 + 1, ar_1, ar_2, ar_3))", 0-3) = ar_3 S("evaltextbookex4bb3in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evaltextbookex4bb2in(ar_0, ar_1, ar_2 + 1, ar_3))", 0-0) = ? S("evaltextbookex4bb3in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evaltextbookex4bb2in(ar_0, ar_1, ar_2 + 1, ar_3))", 0-1) = ar_1 S("evaltextbookex4bb3in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evaltextbookex4bb2in(ar_0, ar_1, ar_2 + 1, ar_3))", 0-2) = ? S("evaltextbookex4bb3in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evaltextbookex4bb2in(ar_0, ar_1, ar_2 + 1, ar_3))", 0-3) = ar_3 S("evaltextbookex4bb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evaltextbookex4bb4in(ar_0, ar_1, ar_2, ar_3)) [ ar_2 >= ar_3 + 1 ]", 0-0) = ? S("evaltextbookex4bb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evaltextbookex4bb4in(ar_0, ar_1, ar_2, ar_3)) [ ar_2 >= ar_3 + 1 ]", 0-1) = ar_1 S("evaltextbookex4bb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evaltextbookex4bb4in(ar_0, ar_1, ar_2, ar_3)) [ ar_2 >= ar_3 + 1 ]", 0-2) = ? S("evaltextbookex4bb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evaltextbookex4bb4in(ar_0, ar_1, ar_2, ar_3)) [ ar_2 >= ar_3 + 1 ]", 0-3) = ar_3 S("evaltextbookex4bb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evaltextbookex4bb3in(ar_0, ar_1, ar_2, ar_3)) [ ar_3 >= ar_2 ]", 0-0) = ? S("evaltextbookex4bb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evaltextbookex4bb3in(ar_0, ar_1, ar_2, ar_3)) [ ar_3 >= ar_2 ]", 0-1) = ar_1 S("evaltextbookex4bb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evaltextbookex4bb3in(ar_0, ar_1, ar_2, ar_3)) [ ar_3 >= ar_2 ]", 0-2) = ? S("evaltextbookex4bb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evaltextbookex4bb3in(ar_0, ar_1, ar_2, ar_3)) [ ar_3 >= ar_2 ]", 0-3) = ar_3 S("evaltextbookex4bb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evaltextbookex4bb5in(ar_0, ar_1, ar_2, ar_3)) [ ar_0 >= ar_1 + 1 ]", 0-0) = ? S("evaltextbookex4bb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evaltextbookex4bb5in(ar_0, ar_1, ar_2, ar_3)) [ ar_0 >= ar_1 + 1 ]", 0-1) = ar_1 S("evaltextbookex4bb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evaltextbookex4bb5in(ar_0, ar_1, ar_2, ar_3)) [ ar_0 >= ar_1 + 1 ]", 0-2) = ? S("evaltextbookex4bb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evaltextbookex4bb5in(ar_0, ar_1, ar_2, ar_3)) [ ar_0 >= ar_1 + 1 ]", 0-3) = ar_3 S("evaltextbookex4bb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evaltextbookex4bb2in(ar_0, ar_1, 1, ar_3)) [ ar_1 >= ar_0 ]", 0-0) = ? S("evaltextbookex4bb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evaltextbookex4bb2in(ar_0, ar_1, 1, ar_3)) [ ar_1 >= ar_0 ]", 0-1) = ar_1 S("evaltextbookex4bb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evaltextbookex4bb2in(ar_0, ar_1, 1, ar_3)) [ ar_1 >= ar_0 ]", 0-2) = 1 S("evaltextbookex4bb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evaltextbookex4bb2in(ar_0, ar_1, 1, ar_3)) [ ar_1 >= ar_0 ]", 0-3) = ar_3 S("evaltextbookex4bb0in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evaltextbookex4bb1in(1, ar_1, ar_2, ar_3))", 0-0) = 1 S("evaltextbookex4bb0in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evaltextbookex4bb1in(1, ar_1, ar_2, ar_3))", 0-1) = ar_1 S("evaltextbookex4bb0in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evaltextbookex4bb1in(1, ar_1, ar_2, ar_3))", 0-2) = ar_2 S("evaltextbookex4bb0in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evaltextbookex4bb1in(1, ar_1, ar_2, ar_3))", 0-3) = ar_3 S("evaltextbookex4start(ar_0, ar_1, ar_2, ar_3) -> Com_1(evaltextbookex4bb0in(ar_0, ar_1, ar_2, ar_3))", 0-0) = ar_0 S("evaltextbookex4start(ar_0, ar_1, ar_2, ar_3) -> Com_1(evaltextbookex4bb0in(ar_0, ar_1, ar_2, ar_3))", 0-1) = ar_1 S("evaltextbookex4start(ar_0, ar_1, ar_2, ar_3) -> Com_1(evaltextbookex4bb0in(ar_0, ar_1, ar_2, ar_3))", 0-2) = ar_2 S("evaltextbookex4start(ar_0, ar_1, ar_2, ar_3) -> Com_1(evaltextbookex4bb0in(ar_0, ar_1, ar_2, ar_3))", 0-3) = ar_3 orients the transitions evaltextbookex4bb4in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evaltextbookex4bb1in(ar_0 + 1, ar_1, ar_2, ar_3)) evaltextbookex4bb3in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evaltextbookex4bb2in(ar_0, ar_1, ar_2 + 1, ar_3)) evaltextbookex4bb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evaltextbookex4bb4in(ar_0, ar_1, ar_2, ar_3)) [ ar_2 >= ar_3 + 1 ] evaltextbookex4bb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evaltextbookex4bb3in(ar_0, ar_1, ar_2, ar_3)) [ ar_3 >= ar_2 ] weakly and the transitions evaltextbookex4bb4in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evaltextbookex4bb1in(ar_0 + 1, ar_1, ar_2, ar_3)) evaltextbookex4bb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evaltextbookex4bb4in(ar_0, ar_1, ar_2, ar_3)) [ ar_2 >= ar_3 + 1 ] strictly and produces the following problem: 5: T: (Comp: 1, Cost: 1) evaltextbookex4start(ar_0, ar_1, ar_2, ar_3) -> Com_1(evaltextbookex4bb0in(ar_0, ar_1, ar_2, ar_3)) (Comp: 1, Cost: 1) evaltextbookex4bb0in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evaltextbookex4bb1in(1, ar_1, ar_2, ar_3)) (Comp: ar_1, Cost: 1) evaltextbookex4bb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evaltextbookex4bb2in(ar_0, ar_1, 1, ar_3)) [ ar_1 >= ar_0 ] (Comp: 2, Cost: 1) evaltextbookex4bb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evaltextbookex4bb5in(ar_0, ar_1, ar_2, ar_3)) [ ar_0 >= ar_1 + 1 ] (Comp: ?, Cost: 1) evaltextbookex4bb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evaltextbookex4bb3in(ar_0, ar_1, ar_2, ar_3)) [ ar_3 >= ar_2 ] (Comp: 2*ar_1, Cost: 1) evaltextbookex4bb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evaltextbookex4bb4in(ar_0, ar_1, ar_2, ar_3)) [ ar_2 >= ar_3 + 1 ] (Comp: ?, Cost: 1) evaltextbookex4bb3in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evaltextbookex4bb2in(ar_0, ar_1, ar_2 + 1, ar_3)) (Comp: 2*ar_1, Cost: 1) evaltextbookex4bb4in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evaltextbookex4bb1in(ar_0 + 1, ar_1, ar_2, ar_3)) (Comp: 2, Cost: 1) evaltextbookex4bb5in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evaltextbookex4stop(ar_0, ar_1, ar_2, ar_3)) (Comp: 1, Cost: 0) koat_start(ar_0, ar_1, ar_2, ar_3) -> Com_1(evaltextbookex4start(ar_0, ar_1, ar_2, ar_3)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 A polynomial rank function with Pol(evaltextbookex4bb3in) = -V_3 + V_4 Pol(evaltextbookex4bb2in) = -V_3 + V_4 + 1 and size complexities S("koat_start(ar_0, ar_1, ar_2, ar_3) -> Com_1(evaltextbookex4start(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(evaltextbookex4start(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(evaltextbookex4start(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(evaltextbookex4start(ar_0, ar_1, ar_2, ar_3)) [ 0 <= 0 ]", 0-3) = ar_3 S("evaltextbookex4bb5in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evaltextbookex4stop(ar_0, ar_1, ar_2, ar_3))", 0-0) = 2*ar_1 + 20 S("evaltextbookex4bb5in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evaltextbookex4stop(ar_0, ar_1, ar_2, ar_3))", 0-1) = ar_1 S("evaltextbookex4bb5in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evaltextbookex4stop(ar_0, ar_1, ar_2, ar_3))", 0-2) = ? S("evaltextbookex4bb5in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evaltextbookex4stop(ar_0, ar_1, ar_2, ar_3))", 0-3) = ar_3 S("evaltextbookex4bb4in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evaltextbookex4bb1in(ar_0 + 1, ar_1, ar_2, ar_3))", 0-0) = 2*ar_1 + 4 S("evaltextbookex4bb4in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evaltextbookex4bb1in(ar_0 + 1, ar_1, ar_2, ar_3))", 0-1) = ar_1 S("evaltextbookex4bb4in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evaltextbookex4bb1in(ar_0 + 1, ar_1, ar_2, ar_3))", 0-2) = ? S("evaltextbookex4bb4in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evaltextbookex4bb1in(ar_0 + 1, ar_1, ar_2, ar_3))", 0-3) = ar_3 S("evaltextbookex4bb3in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evaltextbookex4bb2in(ar_0, ar_1, ar_2 + 1, ar_3))", 0-0) = 2*ar_1 + 4 S("evaltextbookex4bb3in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evaltextbookex4bb2in(ar_0, ar_1, ar_2 + 1, ar_3))", 0-1) = ar_1 S("evaltextbookex4bb3in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evaltextbookex4bb2in(ar_0, ar_1, ar_2 + 1, ar_3))", 0-2) = ? S("evaltextbookex4bb3in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evaltextbookex4bb2in(ar_0, ar_1, ar_2 + 1, ar_3))", 0-3) = ar_3 S("evaltextbookex4bb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evaltextbookex4bb4in(ar_0, ar_1, ar_2, ar_3)) [ ar_2 >= ar_3 + 1 ]", 0-0) = 2*ar_1 + 4 S("evaltextbookex4bb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evaltextbookex4bb4in(ar_0, ar_1, ar_2, ar_3)) [ ar_2 >= ar_3 + 1 ]", 0-1) = ar_1 S("evaltextbookex4bb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evaltextbookex4bb4in(ar_0, ar_1, ar_2, ar_3)) [ ar_2 >= ar_3 + 1 ]", 0-2) = ? S("evaltextbookex4bb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evaltextbookex4bb4in(ar_0, ar_1, ar_2, ar_3)) [ ar_2 >= ar_3 + 1 ]", 0-3) = ar_3 S("evaltextbookex4bb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evaltextbookex4bb3in(ar_0, ar_1, ar_2, ar_3)) [ ar_3 >= ar_2 ]", 0-0) = 2*ar_1 + 4 S("evaltextbookex4bb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evaltextbookex4bb3in(ar_0, ar_1, ar_2, ar_3)) [ ar_3 >= ar_2 ]", 0-1) = ar_1 S("evaltextbookex4bb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evaltextbookex4bb3in(ar_0, ar_1, ar_2, ar_3)) [ ar_3 >= ar_2 ]", 0-2) = ? S("evaltextbookex4bb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evaltextbookex4bb3in(ar_0, ar_1, ar_2, ar_3)) [ ar_3 >= ar_2 ]", 0-3) = ar_3 S("evaltextbookex4bb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evaltextbookex4bb5in(ar_0, ar_1, ar_2, ar_3)) [ ar_0 >= ar_1 + 1 ]", 0-0) = 2*ar_1 + 10 S("evaltextbookex4bb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evaltextbookex4bb5in(ar_0, ar_1, ar_2, ar_3)) [ ar_0 >= ar_1 + 1 ]", 0-1) = ar_1 S("evaltextbookex4bb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evaltextbookex4bb5in(ar_0, ar_1, ar_2, ar_3)) [ ar_0 >= ar_1 + 1 ]", 0-2) = ? S("evaltextbookex4bb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evaltextbookex4bb5in(ar_0, ar_1, ar_2, ar_3)) [ ar_0 >= ar_1 + 1 ]", 0-3) = ar_3 S("evaltextbookex4bb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evaltextbookex4bb2in(ar_0, ar_1, 1, ar_3)) [ ar_1 >= ar_0 ]", 0-0) = 2*ar_1 + 4 S("evaltextbookex4bb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evaltextbookex4bb2in(ar_0, ar_1, 1, ar_3)) [ ar_1 >= ar_0 ]", 0-1) = ar_1 S("evaltextbookex4bb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evaltextbookex4bb2in(ar_0, ar_1, 1, ar_3)) [ ar_1 >= ar_0 ]", 0-2) = 1 S("evaltextbookex4bb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evaltextbookex4bb2in(ar_0, ar_1, 1, ar_3)) [ ar_1 >= ar_0 ]", 0-3) = ar_3 S("evaltextbookex4bb0in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evaltextbookex4bb1in(1, ar_1, ar_2, ar_3))", 0-0) = 1 S("evaltextbookex4bb0in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evaltextbookex4bb1in(1, ar_1, ar_2, ar_3))", 0-1) = ar_1 S("evaltextbookex4bb0in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evaltextbookex4bb1in(1, ar_1, ar_2, ar_3))", 0-2) = ar_2 S("evaltextbookex4bb0in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evaltextbookex4bb1in(1, ar_1, ar_2, ar_3))", 0-3) = ar_3 S("evaltextbookex4start(ar_0, ar_1, ar_2, ar_3) -> Com_1(evaltextbookex4bb0in(ar_0, ar_1, ar_2, ar_3))", 0-0) = ar_0 S("evaltextbookex4start(ar_0, ar_1, ar_2, ar_3) -> Com_1(evaltextbookex4bb0in(ar_0, ar_1, ar_2, ar_3))", 0-1) = ar_1 S("evaltextbookex4start(ar_0, ar_1, ar_2, ar_3) -> Com_1(evaltextbookex4bb0in(ar_0, ar_1, ar_2, ar_3))", 0-2) = ar_2 S("evaltextbookex4start(ar_0, ar_1, ar_2, ar_3) -> Com_1(evaltextbookex4bb0in(ar_0, ar_1, ar_2, ar_3))", 0-3) = ar_3 orients the transitions evaltextbookex4bb3in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evaltextbookex4bb2in(ar_0, ar_1, ar_2 + 1, ar_3)) evaltextbookex4bb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evaltextbookex4bb3in(ar_0, ar_1, ar_2, ar_3)) [ ar_3 >= ar_2 ] weakly and the transition evaltextbookex4bb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evaltextbookex4bb3in(ar_0, ar_1, ar_2, ar_3)) [ ar_3 >= ar_2 ] strictly and produces the following problem: 6: T: (Comp: 1, Cost: 1) evaltextbookex4start(ar_0, ar_1, ar_2, ar_3) -> Com_1(evaltextbookex4bb0in(ar_0, ar_1, ar_2, ar_3)) (Comp: 1, Cost: 1) evaltextbookex4bb0in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evaltextbookex4bb1in(1, ar_1, ar_2, ar_3)) (Comp: ar_1, Cost: 1) evaltextbookex4bb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evaltextbookex4bb2in(ar_0, ar_1, 1, ar_3)) [ ar_1 >= ar_0 ] (Comp: 2, Cost: 1) evaltextbookex4bb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evaltextbookex4bb5in(ar_0, ar_1, ar_2, ar_3)) [ ar_0 >= ar_1 + 1 ] (Comp: ar_1*ar_3 + 2*ar_1, Cost: 1) evaltextbookex4bb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evaltextbookex4bb3in(ar_0, ar_1, ar_2, ar_3)) [ ar_3 >= ar_2 ] (Comp: 2*ar_1, Cost: 1) evaltextbookex4bb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evaltextbookex4bb4in(ar_0, ar_1, ar_2, ar_3)) [ ar_2 >= ar_3 + 1 ] (Comp: ?, Cost: 1) evaltextbookex4bb3in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evaltextbookex4bb2in(ar_0, ar_1, ar_2 + 1, ar_3)) (Comp: 2*ar_1, Cost: 1) evaltextbookex4bb4in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evaltextbookex4bb1in(ar_0 + 1, ar_1, ar_2, ar_3)) (Comp: 2, Cost: 1) evaltextbookex4bb5in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evaltextbookex4stop(ar_0, ar_1, ar_2, ar_3)) (Comp: 1, Cost: 0) koat_start(ar_0, ar_1, ar_2, ar_3) -> Com_1(evaltextbookex4start(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) evaltextbookex4start(ar_0, ar_1, ar_2, ar_3) -> Com_1(evaltextbookex4bb0in(ar_0, ar_1, ar_2, ar_3)) (Comp: 1, Cost: 1) evaltextbookex4bb0in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evaltextbookex4bb1in(1, ar_1, ar_2, ar_3)) (Comp: ar_1, Cost: 1) evaltextbookex4bb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evaltextbookex4bb2in(ar_0, ar_1, 1, ar_3)) [ ar_1 >= ar_0 ] (Comp: 2, Cost: 1) evaltextbookex4bb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evaltextbookex4bb5in(ar_0, ar_1, ar_2, ar_3)) [ ar_0 >= ar_1 + 1 ] (Comp: ar_1*ar_3 + 2*ar_1, Cost: 1) evaltextbookex4bb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evaltextbookex4bb3in(ar_0, ar_1, ar_2, ar_3)) [ ar_3 >= ar_2 ] (Comp: 2*ar_1, Cost: 1) evaltextbookex4bb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evaltextbookex4bb4in(ar_0, ar_1, ar_2, ar_3)) [ ar_2 >= ar_3 + 1 ] (Comp: ar_1*ar_3 + 2*ar_1, Cost: 1) evaltextbookex4bb3in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evaltextbookex4bb2in(ar_0, ar_1, ar_2 + 1, ar_3)) (Comp: 2*ar_1, Cost: 1) evaltextbookex4bb4in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evaltextbookex4bb1in(ar_0 + 1, ar_1, ar_2, ar_3)) (Comp: 2, Cost: 1) evaltextbookex4bb5in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evaltextbookex4stop(ar_0, ar_1, ar_2, ar_3)) (Comp: 1, Cost: 0) koat_start(ar_0, ar_1, ar_2, ar_3) -> Com_1(evaltextbookex4start(ar_0, ar_1, ar_2, ar_3)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 Complexity upper bound 9*ar_1 + 2*ar_1*ar_3 + 6 Time: 0.097 sec (SMT: 0.082 sec) ---------------------------------------- (2) BOUNDS(1, n^2)