/export/starexec/sandbox/solver/bin/starexec_run_c_complexity /export/starexec/sandbox/benchmark/theBenchmark.c /export/starexec/sandbox/output/output_files -------------------------------------------------------------------------------- WORST_CASE(?, O(n^1)) proof of /export/starexec/sandbox/output/output_files/bench.koat # AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(1, n^1). (0) CpxIntTrs (1) Koat Proof [FINISHED, 46 ms] (2) BOUNDS(1, n^1) ---------------------------------------- (0) Obligation: Complexity Int TRS consisting of the following rules: eval_textbook_ex1_start(v_a, v_b, v_i.0) -> Com_1(eval_textbook_ex1_bb0_in(v_a, v_b, v_i.0)) :|: TRUE eval_textbook_ex1_bb0_in(v_a, v_b, v_i.0) -> Com_1(eval_textbook_ex1_bb1_in(v_a, v_b, v_a)) :|: TRUE eval_textbook_ex1_bb1_in(v_a, v_b, v_i.0) -> Com_1(eval_textbook_ex1_bb2_in(v_a, v_b, v_i.0)) :|: v_i.0 <= v_b eval_textbook_ex1_bb1_in(v_a, v_b, v_i.0) -> Com_1(eval_textbook_ex1_bb3_in(v_a, v_b, v_i.0)) :|: v_i.0 > v_b eval_textbook_ex1_bb2_in(v_a, v_b, v_i.0) -> Com_1(eval_textbook_ex1_bb1_in(v_a, v_b, v_i.0 + 1)) :|: TRUE eval_textbook_ex1_bb3_in(v_a, v_b, v_i.0) -> Com_1(eval_textbook_ex1_stop(v_a, v_b, v_i.0)) :|: TRUE The start-symbols are:[eval_textbook_ex1_start_3] ---------------------------------------- (1) Koat Proof (FINISHED) YES(?, 2*Ar_1 + 2*Ar_2 + 8) Initial complexity problem: 1: T: (Comp: ?, Cost: 1) evaltextbookex1start(Ar_0, Ar_1, Ar_2) -> Com_1(evaltextbookex1bb0in(Ar_0, Ar_1, Ar_2)) (Comp: ?, Cost: 1) evaltextbookex1bb0in(Ar_0, Ar_1, Ar_2) -> Com_1(evaltextbookex1bb1in(Ar_1, Ar_1, Ar_2)) (Comp: ?, Cost: 1) evaltextbookex1bb1in(Ar_0, Ar_1, Ar_2) -> Com_1(evaltextbookex1bb2in(Ar_0, Ar_1, Ar_2)) [ Ar_2 >= Ar_0 ] (Comp: ?, Cost: 1) evaltextbookex1bb1in(Ar_0, Ar_1, Ar_2) -> Com_1(evaltextbookex1bb3in(Ar_0, Ar_1, Ar_2)) [ Ar_0 >= Ar_2 + 1 ] (Comp: ?, Cost: 1) evaltextbookex1bb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evaltextbookex1bb1in(Ar_0 + 1, Ar_1, Ar_2)) (Comp: ?, Cost: 1) evaltextbookex1bb3in(Ar_0, Ar_1, Ar_2) -> Com_1(evaltextbookex1stop(Ar_0, Ar_1, Ar_2)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2) -> Com_1(evaltextbookex1start(Ar_0, Ar_1, Ar_2)) [ 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) evaltextbookex1start(Ar_0, Ar_1, Ar_2) -> Com_1(evaltextbookex1bb0in(Ar_0, Ar_1, Ar_2)) (Comp: 1, Cost: 1) evaltextbookex1bb0in(Ar_0, Ar_1, Ar_2) -> Com_1(evaltextbookex1bb1in(Ar_1, Ar_1, Ar_2)) (Comp: ?, Cost: 1) evaltextbookex1bb1in(Ar_0, Ar_1, Ar_2) -> Com_1(evaltextbookex1bb2in(Ar_0, Ar_1, Ar_2)) [ Ar_2 >= Ar_0 ] (Comp: ?, Cost: 1) evaltextbookex1bb1in(Ar_0, Ar_1, Ar_2) -> Com_1(evaltextbookex1bb3in(Ar_0, Ar_1, Ar_2)) [ Ar_0 >= Ar_2 + 1 ] (Comp: ?, Cost: 1) evaltextbookex1bb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evaltextbookex1bb1in(Ar_0 + 1, Ar_1, Ar_2)) (Comp: ?, Cost: 1) evaltextbookex1bb3in(Ar_0, Ar_1, Ar_2) -> Com_1(evaltextbookex1stop(Ar_0, Ar_1, Ar_2)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2) -> Com_1(evaltextbookex1start(Ar_0, Ar_1, Ar_2)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 A polynomial rank function with Pol(evaltextbookex1start) = 2 Pol(evaltextbookex1bb0in) = 2 Pol(evaltextbookex1bb1in) = 2 Pol(evaltextbookex1bb2in) = 2 Pol(evaltextbookex1bb3in) = 1 Pol(evaltextbookex1stop) = 0 Pol(koat_start) = 2 orients all transitions weakly and the transitions evaltextbookex1bb3in(Ar_0, Ar_1, Ar_2) -> Com_1(evaltextbookex1stop(Ar_0, Ar_1, Ar_2)) evaltextbookex1bb1in(Ar_0, Ar_1, Ar_2) -> Com_1(evaltextbookex1bb3in(Ar_0, Ar_1, Ar_2)) [ Ar_0 >= Ar_2 + 1 ] strictly and produces the following problem: 3: T: (Comp: 1, Cost: 1) evaltextbookex1start(Ar_0, Ar_1, Ar_2) -> Com_1(evaltextbookex1bb0in(Ar_0, Ar_1, Ar_2)) (Comp: 1, Cost: 1) evaltextbookex1bb0in(Ar_0, Ar_1, Ar_2) -> Com_1(evaltextbookex1bb1in(Ar_1, Ar_1, Ar_2)) (Comp: ?, Cost: 1) evaltextbookex1bb1in(Ar_0, Ar_1, Ar_2) -> Com_1(evaltextbookex1bb2in(Ar_0, Ar_1, Ar_2)) [ Ar_2 >= Ar_0 ] (Comp: 2, Cost: 1) evaltextbookex1bb1in(Ar_0, Ar_1, Ar_2) -> Com_1(evaltextbookex1bb3in(Ar_0, Ar_1, Ar_2)) [ Ar_0 >= Ar_2 + 1 ] (Comp: ?, Cost: 1) evaltextbookex1bb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evaltextbookex1bb1in(Ar_0 + 1, Ar_1, Ar_2)) (Comp: 2, Cost: 1) evaltextbookex1bb3in(Ar_0, Ar_1, Ar_2) -> Com_1(evaltextbookex1stop(Ar_0, Ar_1, Ar_2)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2) -> Com_1(evaltextbookex1start(Ar_0, Ar_1, Ar_2)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 A polynomial rank function with Pol(evaltextbookex1start) = -V_2 + V_3 + 1 Pol(evaltextbookex1bb0in) = -V_2 + V_3 + 1 Pol(evaltextbookex1bb1in) = -V_1 + V_3 + 1 Pol(evaltextbookex1bb2in) = -V_1 + V_3 Pol(evaltextbookex1bb3in) = -V_1 + V_3 Pol(evaltextbookex1stop) = -V_1 + V_3 Pol(koat_start) = -V_2 + V_3 + 1 orients all transitions weakly and the transition evaltextbookex1bb1in(Ar_0, Ar_1, Ar_2) -> Com_1(evaltextbookex1bb2in(Ar_0, Ar_1, Ar_2)) [ Ar_2 >= Ar_0 ] strictly and produces the following problem: 4: T: (Comp: 1, Cost: 1) evaltextbookex1start(Ar_0, Ar_1, Ar_2) -> Com_1(evaltextbookex1bb0in(Ar_0, Ar_1, Ar_2)) (Comp: 1, Cost: 1) evaltextbookex1bb0in(Ar_0, Ar_1, Ar_2) -> Com_1(evaltextbookex1bb1in(Ar_1, Ar_1, Ar_2)) (Comp: Ar_1 + Ar_2 + 1, Cost: 1) evaltextbookex1bb1in(Ar_0, Ar_1, Ar_2) -> Com_1(evaltextbookex1bb2in(Ar_0, Ar_1, Ar_2)) [ Ar_2 >= Ar_0 ] (Comp: 2, Cost: 1) evaltextbookex1bb1in(Ar_0, Ar_1, Ar_2) -> Com_1(evaltextbookex1bb3in(Ar_0, Ar_1, Ar_2)) [ Ar_0 >= Ar_2 + 1 ] (Comp: ?, Cost: 1) evaltextbookex1bb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evaltextbookex1bb1in(Ar_0 + 1, Ar_1, Ar_2)) (Comp: 2, Cost: 1) evaltextbookex1bb3in(Ar_0, Ar_1, Ar_2) -> Com_1(evaltextbookex1stop(Ar_0, Ar_1, Ar_2)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2) -> Com_1(evaltextbookex1start(Ar_0, Ar_1, Ar_2)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 Repeatedly propagating knowledge in problem 4 produces the following problem: 5: T: (Comp: 1, Cost: 1) evaltextbookex1start(Ar_0, Ar_1, Ar_2) -> Com_1(evaltextbookex1bb0in(Ar_0, Ar_1, Ar_2)) (Comp: 1, Cost: 1) evaltextbookex1bb0in(Ar_0, Ar_1, Ar_2) -> Com_1(evaltextbookex1bb1in(Ar_1, Ar_1, Ar_2)) (Comp: Ar_1 + Ar_2 + 1, Cost: 1) evaltextbookex1bb1in(Ar_0, Ar_1, Ar_2) -> Com_1(evaltextbookex1bb2in(Ar_0, Ar_1, Ar_2)) [ Ar_2 >= Ar_0 ] (Comp: 2, Cost: 1) evaltextbookex1bb1in(Ar_0, Ar_1, Ar_2) -> Com_1(evaltextbookex1bb3in(Ar_0, Ar_1, Ar_2)) [ Ar_0 >= Ar_2 + 1 ] (Comp: Ar_1 + Ar_2 + 1, Cost: 1) evaltextbookex1bb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evaltextbookex1bb1in(Ar_0 + 1, Ar_1, Ar_2)) (Comp: 2, Cost: 1) evaltextbookex1bb3in(Ar_0, Ar_1, Ar_2) -> Com_1(evaltextbookex1stop(Ar_0, Ar_1, Ar_2)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2) -> Com_1(evaltextbookex1start(Ar_0, Ar_1, Ar_2)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 Complexity upper bound 2*Ar_1 + 2*Ar_2 + 8 Time: 0.037 sec (SMT: 0.030 sec) ---------------------------------------- (2) BOUNDS(1, n^1)