/export/starexec/sandbox/solver/bin/starexec_run_complexity /export/starexec/sandbox/benchmark/theBenchmark.koat /export/starexec/sandbox/output/output_files -------------------------------------------------------------------------------- WORST_CASE(Omega(n^1), O(n^1)) 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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(n^1, n^1). (0) CpxIntTrs (1) Koat Proof [FINISHED, 307 ms] (2) BOUNDS(1, n^1) (3) Loat Proof [FINISHED, 929 ms] (4) BOUNDS(n^1, INF) ---------------------------------------- (0) Obligation: Complexity Int TRS consisting of the following rules: eval_speedDis2_start(v__0, v__01, v_n, v_x, v_z) -> Com_1(eval_speedDis2_bb0_in(v__0, v__01, v_n, v_x, v_z)) :|: TRUE eval_speedDis2_bb0_in(v__0, v__01, v_n, v_x, v_z) -> Com_1(eval_speedDis2_0(v__0, v__01, v_n, v_x, v_z)) :|: TRUE eval_speedDis2_0(v__0, v__01, v_n, v_x, v_z) -> Com_1(eval_speedDis2_1(v__0, v__01, v_n, v_x, v_z)) :|: TRUE eval_speedDis2_1(v__0, v__01, v_n, v_x, v_z) -> Com_1(eval_speedDis2_2(v__0, v__01, v_n, v_x, v_z)) :|: TRUE eval_speedDis2_2(v__0, v__01, v_n, v_x, v_z) -> Com_1(eval_speedDis2_3(v__0, v__01, v_n, v_x, v_z)) :|: TRUE eval_speedDis2_3(v__0, v__01, v_n, v_x, v_z) -> Com_1(eval_speedDis2_4(v__0, v__01, v_n, v_x, v_z)) :|: TRUE eval_speedDis2_4(v__0, v__01, v_n, v_x, v_z) -> Com_1(eval_speedDis2_5(v__0, v__01, v_n, v_x, v_z)) :|: TRUE eval_speedDis2_5(v__0, v__01, v_n, v_x, v_z) -> Com_1(eval_speedDis2_bb1_in(v_x, v_z, v_n, v_x, v_z)) :|: TRUE eval_speedDis2_bb1_in(v__0, v__01, v_n, v_x, v_z) -> Com_1(eval_speedDis2_bb2_in(v__0, v__01, v_n, v_x, v_z)) :|: v__0 < v_n eval_speedDis2_bb1_in(v__0, v__01, v_n, v_x, v_z) -> Com_1(eval_speedDis2_bb3_in(v__0, v__01, v_n, v_x, v_z)) :|: v__0 >= v_n eval_speedDis2_bb2_in(v__0, v__01, v_n, v_x, v_z) -> Com_1(eval_speedDis2_bb1_in(v__0 + 1, v__01, v_n, v_x, v_z)) :|: v__01 > v__0 eval_speedDis2_bb2_in(v__0, v__01, v_n, v_x, v_z) -> Com_1(eval_speedDis2_bb1_in(v__0, v__01, v_n, v_x, v_z)) :|: v__01 > v__0 && v__01 <= v__0 eval_speedDis2_bb2_in(v__0, v__01, v_n, v_x, v_z) -> Com_1(eval_speedDis2_bb1_in(v__0 + 1, v__01 + 1, v_n, v_x, v_z)) :|: v__01 <= v__0 && v__01 > v__0 eval_speedDis2_bb2_in(v__0, v__01, v_n, v_x, v_z) -> Com_1(eval_speedDis2_bb1_in(v__0, v__01 + 1, v_n, v_x, v_z)) :|: v__01 <= v__0 eval_speedDis2_bb3_in(v__0, v__01, v_n, v_x, v_z) -> Com_1(eval_speedDis2_stop(v__0, v__01, v_n, v_x, v_z)) :|: TRUE The start-symbols are:[eval_speedDis2_start_5] ---------------------------------------- (1) Koat Proof (FINISHED) YES(?, 2*ar_3 + 4*ar_4 + 2*ar_1 + 13) Initial complexity problem: 1: T: (Comp: ?, Cost: 1) evalspeedDis2start(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalspeedDis2bb0in(ar_0, ar_1, ar_2, ar_3, ar_4)) (Comp: ?, Cost: 1) evalspeedDis2bb0in(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalspeedDis20(ar_0, ar_1, ar_2, ar_3, ar_4)) (Comp: ?, Cost: 1) evalspeedDis20(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalspeedDis21(ar_0, ar_1, ar_2, ar_3, ar_4)) (Comp: ?, Cost: 1) evalspeedDis21(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalspeedDis22(ar_0, ar_1, ar_2, ar_3, ar_4)) (Comp: ?, Cost: 1) evalspeedDis22(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalspeedDis23(ar_0, ar_1, ar_2, ar_3, ar_4)) (Comp: ?, Cost: 1) evalspeedDis23(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalspeedDis24(ar_0, ar_1, ar_2, ar_3, ar_4)) (Comp: ?, Cost: 1) evalspeedDis24(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalspeedDis25(ar_0, ar_1, ar_2, ar_3, ar_4)) (Comp: ?, Cost: 1) evalspeedDis25(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalspeedDis2bb1in(ar_1, ar_1, ar_3, ar_3, ar_4)) (Comp: ?, Cost: 1) evalspeedDis2bb1in(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalspeedDis2bb2in(ar_0, ar_1, ar_2, ar_3, ar_4)) [ ar_4 >= ar_0 + 1 ] (Comp: ?, Cost: 1) evalspeedDis2bb1in(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalspeedDis2bb3in(ar_0, ar_1, ar_2, ar_3, ar_4)) [ ar_0 >= ar_4 ] (Comp: ?, Cost: 1) evalspeedDis2bb2in(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalspeedDis2bb1in(ar_0 + 1, ar_1, ar_2, ar_3, ar_4)) [ ar_2 >= ar_0 + 1 ] (Comp: ?, Cost: 1) evalspeedDis2bb2in(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalspeedDis2bb1in(ar_0, ar_1, ar_2, ar_3, ar_4)) [ ar_2 >= ar_0 + 1 /\ ar_0 >= ar_2 ] (Comp: ?, Cost: 1) evalspeedDis2bb2in(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalspeedDis2bb1in(ar_0 + 1, ar_1, ar_2 + 1, ar_3, ar_4)) [ ar_0 >= ar_2 /\ ar_2 >= ar_0 + 1 ] (Comp: ?, Cost: 1) evalspeedDis2bb2in(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalspeedDis2bb1in(ar_0, ar_1, ar_2 + 1, ar_3, ar_4)) [ ar_0 >= ar_2 ] (Comp: ?, Cost: 1) evalspeedDis2bb3in(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalspeedDis2stop(ar_0, ar_1, ar_2, ar_3, ar_4)) (Comp: 1, Cost: 0) koat_start(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalspeedDis2start(ar_0, ar_1, ar_2, ar_3, ar_4)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 Testing for reachability in the complexity graph removes the following transitions from problem 1: evalspeedDis2bb2in(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalspeedDis2bb1in(ar_0, ar_1, ar_2, ar_3, ar_4)) [ ar_2 >= ar_0 + 1 /\ ar_0 >= ar_2 ] evalspeedDis2bb2in(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalspeedDis2bb1in(ar_0 + 1, ar_1, ar_2 + 1, ar_3, ar_4)) [ ar_0 >= ar_2 /\ ar_2 >= ar_0 + 1 ] We thus obtain the following problem: 2: T: (Comp: ?, Cost: 1) evalspeedDis2bb3in(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalspeedDis2stop(ar_0, ar_1, ar_2, ar_3, ar_4)) (Comp: ?, Cost: 1) evalspeedDis2bb2in(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalspeedDis2bb1in(ar_0, ar_1, ar_2 + 1, ar_3, ar_4)) [ ar_0 >= ar_2 ] (Comp: ?, Cost: 1) evalspeedDis2bb2in(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalspeedDis2bb1in(ar_0 + 1, ar_1, ar_2, ar_3, ar_4)) [ ar_2 >= ar_0 + 1 ] (Comp: ?, Cost: 1) evalspeedDis2bb1in(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalspeedDis2bb3in(ar_0, ar_1, ar_2, ar_3, ar_4)) [ ar_0 >= ar_4 ] (Comp: ?, Cost: 1) evalspeedDis2bb1in(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalspeedDis2bb2in(ar_0, ar_1, ar_2, ar_3, ar_4)) [ ar_4 >= ar_0 + 1 ] (Comp: ?, Cost: 1) evalspeedDis25(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalspeedDis2bb1in(ar_1, ar_1, ar_3, ar_3, ar_4)) (Comp: ?, Cost: 1) evalspeedDis24(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalspeedDis25(ar_0, ar_1, ar_2, ar_3, ar_4)) (Comp: ?, Cost: 1) evalspeedDis23(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalspeedDis24(ar_0, ar_1, ar_2, ar_3, ar_4)) (Comp: ?, Cost: 1) evalspeedDis22(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalspeedDis23(ar_0, ar_1, ar_2, ar_3, ar_4)) (Comp: ?, Cost: 1) evalspeedDis21(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalspeedDis22(ar_0, ar_1, ar_2, ar_3, ar_4)) (Comp: ?, Cost: 1) evalspeedDis20(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalspeedDis21(ar_0, ar_1, ar_2, ar_3, ar_4)) (Comp: ?, Cost: 1) evalspeedDis2bb0in(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalspeedDis20(ar_0, ar_1, ar_2, ar_3, ar_4)) (Comp: ?, Cost: 1) evalspeedDis2start(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalspeedDis2bb0in(ar_0, ar_1, ar_2, ar_3, ar_4)) (Comp: 1, Cost: 0) koat_start(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalspeedDis2start(ar_0, ar_1, ar_2, ar_3, ar_4)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 Repeatedly propagating knowledge in problem 2 produces the following problem: 3: T: (Comp: ?, Cost: 1) evalspeedDis2bb3in(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalspeedDis2stop(ar_0, ar_1, ar_2, ar_3, ar_4)) (Comp: ?, Cost: 1) evalspeedDis2bb2in(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalspeedDis2bb1in(ar_0, ar_1, ar_2 + 1, ar_3, ar_4)) [ ar_0 >= ar_2 ] (Comp: ?, Cost: 1) evalspeedDis2bb2in(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalspeedDis2bb1in(ar_0 + 1, ar_1, ar_2, ar_3, ar_4)) [ ar_2 >= ar_0 + 1 ] (Comp: ?, Cost: 1) evalspeedDis2bb1in(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalspeedDis2bb3in(ar_0, ar_1, ar_2, ar_3, ar_4)) [ ar_0 >= ar_4 ] (Comp: ?, Cost: 1) evalspeedDis2bb1in(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalspeedDis2bb2in(ar_0, ar_1, ar_2, ar_3, ar_4)) [ ar_4 >= ar_0 + 1 ] (Comp: 1, Cost: 1) evalspeedDis25(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalspeedDis2bb1in(ar_1, ar_1, ar_3, ar_3, ar_4)) (Comp: 1, Cost: 1) evalspeedDis24(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalspeedDis25(ar_0, ar_1, ar_2, ar_3, ar_4)) (Comp: 1, Cost: 1) evalspeedDis23(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalspeedDis24(ar_0, ar_1, ar_2, ar_3, ar_4)) (Comp: 1, Cost: 1) evalspeedDis22(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalspeedDis23(ar_0, ar_1, ar_2, ar_3, ar_4)) (Comp: 1, Cost: 1) evalspeedDis21(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalspeedDis22(ar_0, ar_1, ar_2, ar_3, ar_4)) (Comp: 1, Cost: 1) evalspeedDis20(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalspeedDis21(ar_0, ar_1, ar_2, ar_3, ar_4)) (Comp: 1, Cost: 1) evalspeedDis2bb0in(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalspeedDis20(ar_0, ar_1, ar_2, ar_3, ar_4)) (Comp: 1, Cost: 1) evalspeedDis2start(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalspeedDis2bb0in(ar_0, ar_1, ar_2, ar_3, ar_4)) (Comp: 1, Cost: 0) koat_start(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalspeedDis2start(ar_0, ar_1, ar_2, ar_3, ar_4)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 A polynomial rank function with Pol(evalspeedDis2bb3in) = 1 Pol(evalspeedDis2stop) = 0 Pol(evalspeedDis2bb2in) = 2 Pol(evalspeedDis2bb1in) = 2 Pol(evalspeedDis25) = 2 Pol(evalspeedDis24) = 2 Pol(evalspeedDis23) = 2 Pol(evalspeedDis22) = 2 Pol(evalspeedDis21) = 2 Pol(evalspeedDis20) = 2 Pol(evalspeedDis2bb0in) = 2 Pol(evalspeedDis2start) = 2 Pol(koat_start) = 2 orients all transitions weakly and the transitions evalspeedDis2bb3in(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalspeedDis2stop(ar_0, ar_1, ar_2, ar_3, ar_4)) evalspeedDis2bb1in(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalspeedDis2bb3in(ar_0, ar_1, ar_2, ar_3, ar_4)) [ ar_0 >= ar_4 ] strictly and produces the following problem: 4: T: (Comp: 2, Cost: 1) evalspeedDis2bb3in(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalspeedDis2stop(ar_0, ar_1, ar_2, ar_3, ar_4)) (Comp: ?, Cost: 1) evalspeedDis2bb2in(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalspeedDis2bb1in(ar_0, ar_1, ar_2 + 1, ar_3, ar_4)) [ ar_0 >= ar_2 ] (Comp: ?, Cost: 1) evalspeedDis2bb2in(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalspeedDis2bb1in(ar_0 + 1, ar_1, ar_2, ar_3, ar_4)) [ ar_2 >= ar_0 + 1 ] (Comp: 2, Cost: 1) evalspeedDis2bb1in(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalspeedDis2bb3in(ar_0, ar_1, ar_2, ar_3, ar_4)) [ ar_0 >= ar_4 ] (Comp: ?, Cost: 1) evalspeedDis2bb1in(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalspeedDis2bb2in(ar_0, ar_1, ar_2, ar_3, ar_4)) [ ar_4 >= ar_0 + 1 ] (Comp: 1, Cost: 1) evalspeedDis25(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalspeedDis2bb1in(ar_1, ar_1, ar_3, ar_3, ar_4)) (Comp: 1, Cost: 1) evalspeedDis24(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalspeedDis25(ar_0, ar_1, ar_2, ar_3, ar_4)) (Comp: 1, Cost: 1) evalspeedDis23(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalspeedDis24(ar_0, ar_1, ar_2, ar_3, ar_4)) (Comp: 1, Cost: 1) evalspeedDis22(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalspeedDis23(ar_0, ar_1, ar_2, ar_3, ar_4)) (Comp: 1, Cost: 1) evalspeedDis21(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalspeedDis22(ar_0, ar_1, ar_2, ar_3, ar_4)) (Comp: 1, Cost: 1) evalspeedDis20(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalspeedDis21(ar_0, ar_1, ar_2, ar_3, ar_4)) (Comp: 1, Cost: 1) evalspeedDis2bb0in(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalspeedDis20(ar_0, ar_1, ar_2, ar_3, ar_4)) (Comp: 1, Cost: 1) evalspeedDis2start(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalspeedDis2bb0in(ar_0, ar_1, ar_2, ar_3, ar_4)) (Comp: 1, Cost: 0) koat_start(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalspeedDis2start(ar_0, ar_1, ar_2, ar_3, ar_4)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 Applied AI with 'oct' on problem 4 to obtain the following invariants: For symbol evalspeedDis2bb1in: X_3 - X_4 >= 0 /\ X_1 - X_2 >= 0 For symbol evalspeedDis2bb2in: -X_2 + X_5 - 1 >= 0 /\ -X_1 + X_5 - 1 >= 0 /\ X_3 - X_4 >= 0 /\ X_1 - X_2 >= 0 For symbol evalspeedDis2bb3in: X_1 - X_5 >= 0 /\ X_3 - X_4 >= 0 /\ X_1 - X_2 >= 0 This yielded the following problem: 5: T: (Comp: 1, Cost: 0) koat_start(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalspeedDis2start(ar_0, ar_1, ar_2, ar_3, ar_4)) [ 0 <= 0 ] (Comp: 1, Cost: 1) evalspeedDis2start(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalspeedDis2bb0in(ar_0, ar_1, ar_2, ar_3, ar_4)) (Comp: 1, Cost: 1) evalspeedDis2bb0in(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalspeedDis20(ar_0, ar_1, ar_2, ar_3, ar_4)) (Comp: 1, Cost: 1) evalspeedDis20(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalspeedDis21(ar_0, ar_1, ar_2, ar_3, ar_4)) (Comp: 1, Cost: 1) evalspeedDis21(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalspeedDis22(ar_0, ar_1, ar_2, ar_3, ar_4)) (Comp: 1, Cost: 1) evalspeedDis22(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalspeedDis23(ar_0, ar_1, ar_2, ar_3, ar_4)) (Comp: 1, Cost: 1) evalspeedDis23(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalspeedDis24(ar_0, ar_1, ar_2, ar_3, ar_4)) (Comp: 1, Cost: 1) evalspeedDis24(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalspeedDis25(ar_0, ar_1, ar_2, ar_3, ar_4)) (Comp: 1, Cost: 1) evalspeedDis25(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalspeedDis2bb1in(ar_1, ar_1, ar_3, ar_3, ar_4)) (Comp: ?, Cost: 1) evalspeedDis2bb1in(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalspeedDis2bb2in(ar_0, ar_1, ar_2, ar_3, ar_4)) [ ar_2 - ar_3 >= 0 /\ ar_0 - ar_1 >= 0 /\ ar_4 >= ar_0 + 1 ] (Comp: 2, Cost: 1) evalspeedDis2bb1in(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalspeedDis2bb3in(ar_0, ar_1, ar_2, ar_3, ar_4)) [ ar_2 - ar_3 >= 0 /\ ar_0 - ar_1 >= 0 /\ ar_0 >= ar_4 ] (Comp: ?, Cost: 1) evalspeedDis2bb2in(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalspeedDis2bb1in(ar_0 + 1, ar_1, ar_2, ar_3, ar_4)) [ -ar_1 + ar_4 - 1 >= 0 /\ -ar_0 + ar_4 - 1 >= 0 /\ ar_2 - ar_3 >= 0 /\ ar_0 - ar_1 >= 0 /\ ar_2 >= ar_0 + 1 ] (Comp: ?, Cost: 1) evalspeedDis2bb2in(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalspeedDis2bb1in(ar_0, ar_1, ar_2 + 1, ar_3, ar_4)) [ -ar_1 + ar_4 - 1 >= 0 /\ -ar_0 + ar_4 - 1 >= 0 /\ ar_2 - ar_3 >= 0 /\ ar_0 - ar_1 >= 0 /\ ar_0 >= ar_2 ] (Comp: 2, Cost: 1) evalspeedDis2bb3in(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalspeedDis2stop(ar_0, ar_1, ar_2, ar_3, ar_4)) [ ar_0 - ar_4 >= 0 /\ ar_2 - ar_3 >= 0 /\ ar_0 - ar_1 >= 0 ] start location: koat_start leaf cost: 0 A polynomial rank function with Pol(koat_start) = -V_2 + V_5 Pol(evalspeedDis2start) = -V_2 + V_5 Pol(evalspeedDis2bb0in) = -V_2 + V_5 Pol(evalspeedDis20) = -V_2 + V_5 Pol(evalspeedDis21) = -V_2 + V_5 Pol(evalspeedDis22) = -V_2 + V_5 Pol(evalspeedDis23) = -V_2 + V_5 Pol(evalspeedDis24) = -V_2 + V_5 Pol(evalspeedDis25) = -V_2 + V_5 Pol(evalspeedDis2bb1in) = -V_1 + V_5 Pol(evalspeedDis2bb2in) = -V_1 + V_5 Pol(evalspeedDis2bb3in) = -V_1 + V_5 Pol(evalspeedDis2stop) = -V_1 + V_5 orients all transitions weakly and the transition evalspeedDis2bb2in(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalspeedDis2bb1in(ar_0 + 1, ar_1, ar_2, ar_3, ar_4)) [ -ar_1 + ar_4 - 1 >= 0 /\ -ar_0 + ar_4 - 1 >= 0 /\ ar_2 - ar_3 >= 0 /\ ar_0 - ar_1 >= 0 /\ ar_2 >= ar_0 + 1 ] strictly and produces the following problem: 6: T: (Comp: 1, Cost: 0) koat_start(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalspeedDis2start(ar_0, ar_1, ar_2, ar_3, ar_4)) [ 0 <= 0 ] (Comp: 1, Cost: 1) evalspeedDis2start(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalspeedDis2bb0in(ar_0, ar_1, ar_2, ar_3, ar_4)) (Comp: 1, Cost: 1) evalspeedDis2bb0in(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalspeedDis20(ar_0, ar_1, ar_2, ar_3, ar_4)) (Comp: 1, Cost: 1) evalspeedDis20(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalspeedDis21(ar_0, ar_1, ar_2, ar_3, ar_4)) (Comp: 1, Cost: 1) evalspeedDis21(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalspeedDis22(ar_0, ar_1, ar_2, ar_3, ar_4)) (Comp: 1, Cost: 1) evalspeedDis22(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalspeedDis23(ar_0, ar_1, ar_2, ar_3, ar_4)) (Comp: 1, Cost: 1) evalspeedDis23(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalspeedDis24(ar_0, ar_1, ar_2, ar_3, ar_4)) (Comp: 1, Cost: 1) evalspeedDis24(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalspeedDis25(ar_0, ar_1, ar_2, ar_3, ar_4)) (Comp: 1, Cost: 1) evalspeedDis25(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalspeedDis2bb1in(ar_1, ar_1, ar_3, ar_3, ar_4)) (Comp: ?, Cost: 1) evalspeedDis2bb1in(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalspeedDis2bb2in(ar_0, ar_1, ar_2, ar_3, ar_4)) [ ar_2 - ar_3 >= 0 /\ ar_0 - ar_1 >= 0 /\ ar_4 >= ar_0 + 1 ] (Comp: 2, Cost: 1) evalspeedDis2bb1in(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalspeedDis2bb3in(ar_0, ar_1, ar_2, ar_3, ar_4)) [ ar_2 - ar_3 >= 0 /\ ar_0 - ar_1 >= 0 /\ ar_0 >= ar_4 ] (Comp: ar_1 + ar_4, Cost: 1) evalspeedDis2bb2in(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalspeedDis2bb1in(ar_0 + 1, ar_1, ar_2, ar_3, ar_4)) [ -ar_1 + ar_4 - 1 >= 0 /\ -ar_0 + ar_4 - 1 >= 0 /\ ar_2 - ar_3 >= 0 /\ ar_0 - ar_1 >= 0 /\ ar_2 >= ar_0 + 1 ] (Comp: ?, Cost: 1) evalspeedDis2bb2in(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalspeedDis2bb1in(ar_0, ar_1, ar_2 + 1, ar_3, ar_4)) [ -ar_1 + ar_4 - 1 >= 0 /\ -ar_0 + ar_4 - 1 >= 0 /\ ar_2 - ar_3 >= 0 /\ ar_0 - ar_1 >= 0 /\ ar_0 >= ar_2 ] (Comp: 2, Cost: 1) evalspeedDis2bb3in(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalspeedDis2stop(ar_0, ar_1, ar_2, ar_3, ar_4)) [ ar_0 - ar_4 >= 0 /\ ar_2 - ar_3 >= 0 /\ ar_0 - ar_1 >= 0 ] start location: koat_start leaf cost: 0 A polynomial rank function with Pol(koat_start) = -V_4 + V_5 Pol(evalspeedDis2start) = -V_4 + V_5 Pol(evalspeedDis2bb0in) = -V_4 + V_5 Pol(evalspeedDis20) = -V_4 + V_5 Pol(evalspeedDis21) = -V_4 + V_5 Pol(evalspeedDis22) = -V_4 + V_5 Pol(evalspeedDis23) = -V_4 + V_5 Pol(evalspeedDis24) = -V_4 + V_5 Pol(evalspeedDis25) = -V_4 + V_5 Pol(evalspeedDis2bb1in) = -V_3 + V_5 Pol(evalspeedDis2bb2in) = -V_3 + V_5 Pol(evalspeedDis2bb3in) = -V_3 + V_5 Pol(evalspeedDis2stop) = -V_3 + V_5 orients all transitions weakly and the transition evalspeedDis2bb2in(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalspeedDis2bb1in(ar_0, ar_1, ar_2 + 1, ar_3, ar_4)) [ -ar_1 + ar_4 - 1 >= 0 /\ -ar_0 + ar_4 - 1 >= 0 /\ ar_2 - ar_3 >= 0 /\ ar_0 - ar_1 >= 0 /\ ar_0 >= ar_2 ] strictly and produces the following problem: 7: T: (Comp: 1, Cost: 0) koat_start(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalspeedDis2start(ar_0, ar_1, ar_2, ar_3, ar_4)) [ 0 <= 0 ] (Comp: 1, Cost: 1) evalspeedDis2start(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalspeedDis2bb0in(ar_0, ar_1, ar_2, ar_3, ar_4)) (Comp: 1, Cost: 1) evalspeedDis2bb0in(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalspeedDis20(ar_0, ar_1, ar_2, ar_3, ar_4)) (Comp: 1, Cost: 1) evalspeedDis20(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalspeedDis21(ar_0, ar_1, ar_2, ar_3, ar_4)) (Comp: 1, Cost: 1) evalspeedDis21(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalspeedDis22(ar_0, ar_1, ar_2, ar_3, ar_4)) (Comp: 1, Cost: 1) evalspeedDis22(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalspeedDis23(ar_0, ar_1, ar_2, ar_3, ar_4)) (Comp: 1, Cost: 1) evalspeedDis23(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalspeedDis24(ar_0, ar_1, ar_2, ar_3, ar_4)) (Comp: 1, Cost: 1) evalspeedDis24(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalspeedDis25(ar_0, ar_1, ar_2, ar_3, ar_4)) (Comp: 1, Cost: 1) evalspeedDis25(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalspeedDis2bb1in(ar_1, ar_1, ar_3, ar_3, ar_4)) (Comp: ?, Cost: 1) evalspeedDis2bb1in(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalspeedDis2bb2in(ar_0, ar_1, ar_2, ar_3, ar_4)) [ ar_2 - ar_3 >= 0 /\ ar_0 - ar_1 >= 0 /\ ar_4 >= ar_0 + 1 ] (Comp: 2, Cost: 1) evalspeedDis2bb1in(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalspeedDis2bb3in(ar_0, ar_1, ar_2, ar_3, ar_4)) [ ar_2 - ar_3 >= 0 /\ ar_0 - ar_1 >= 0 /\ ar_0 >= ar_4 ] (Comp: ar_1 + ar_4, Cost: 1) evalspeedDis2bb2in(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalspeedDis2bb1in(ar_0 + 1, ar_1, ar_2, ar_3, ar_4)) [ -ar_1 + ar_4 - 1 >= 0 /\ -ar_0 + ar_4 - 1 >= 0 /\ ar_2 - ar_3 >= 0 /\ ar_0 - ar_1 >= 0 /\ ar_2 >= ar_0 + 1 ] (Comp: ar_3 + ar_4, Cost: 1) evalspeedDis2bb2in(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalspeedDis2bb1in(ar_0, ar_1, ar_2 + 1, ar_3, ar_4)) [ -ar_1 + ar_4 - 1 >= 0 /\ -ar_0 + ar_4 - 1 >= 0 /\ ar_2 - ar_3 >= 0 /\ ar_0 - ar_1 >= 0 /\ ar_0 >= ar_2 ] (Comp: 2, Cost: 1) evalspeedDis2bb3in(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalspeedDis2stop(ar_0, ar_1, ar_2, ar_3, ar_4)) [ ar_0 - ar_4 >= 0 /\ ar_2 - ar_3 >= 0 /\ ar_0 - ar_1 >= 0 ] start location: koat_start leaf cost: 0 Repeatedly propagating knowledge in problem 7 produces the following problem: 8: T: (Comp: 1, Cost: 0) koat_start(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalspeedDis2start(ar_0, ar_1, ar_2, ar_3, ar_4)) [ 0 <= 0 ] (Comp: 1, Cost: 1) evalspeedDis2start(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalspeedDis2bb0in(ar_0, ar_1, ar_2, ar_3, ar_4)) (Comp: 1, Cost: 1) evalspeedDis2bb0in(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalspeedDis20(ar_0, ar_1, ar_2, ar_3, ar_4)) (Comp: 1, Cost: 1) evalspeedDis20(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalspeedDis21(ar_0, ar_1, ar_2, ar_3, ar_4)) (Comp: 1, Cost: 1) evalspeedDis21(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalspeedDis22(ar_0, ar_1, ar_2, ar_3, ar_4)) (Comp: 1, Cost: 1) evalspeedDis22(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalspeedDis23(ar_0, ar_1, ar_2, ar_3, ar_4)) (Comp: 1, Cost: 1) evalspeedDis23(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalspeedDis24(ar_0, ar_1, ar_2, ar_3, ar_4)) (Comp: 1, Cost: 1) evalspeedDis24(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalspeedDis25(ar_0, ar_1, ar_2, ar_3, ar_4)) (Comp: 1, Cost: 1) evalspeedDis25(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalspeedDis2bb1in(ar_1, ar_1, ar_3, ar_3, ar_4)) (Comp: ar_3 + 2*ar_4 + ar_1 + 1, Cost: 1) evalspeedDis2bb1in(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalspeedDis2bb2in(ar_0, ar_1, ar_2, ar_3, ar_4)) [ ar_2 - ar_3 >= 0 /\ ar_0 - ar_1 >= 0 /\ ar_4 >= ar_0 + 1 ] (Comp: 2, Cost: 1) evalspeedDis2bb1in(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalspeedDis2bb3in(ar_0, ar_1, ar_2, ar_3, ar_4)) [ ar_2 - ar_3 >= 0 /\ ar_0 - ar_1 >= 0 /\ ar_0 >= ar_4 ] (Comp: ar_1 + ar_4, Cost: 1) evalspeedDis2bb2in(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalspeedDis2bb1in(ar_0 + 1, ar_1, ar_2, ar_3, ar_4)) [ -ar_1 + ar_4 - 1 >= 0 /\ -ar_0 + ar_4 - 1 >= 0 /\ ar_2 - ar_3 >= 0 /\ ar_0 - ar_1 >= 0 /\ ar_2 >= ar_0 + 1 ] (Comp: ar_3 + ar_4, Cost: 1) evalspeedDis2bb2in(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalspeedDis2bb1in(ar_0, ar_1, ar_2 + 1, ar_3, ar_4)) [ -ar_1 + ar_4 - 1 >= 0 /\ -ar_0 + ar_4 - 1 >= 0 /\ ar_2 - ar_3 >= 0 /\ ar_0 - ar_1 >= 0 /\ ar_0 >= ar_2 ] (Comp: 2, Cost: 1) evalspeedDis2bb3in(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalspeedDis2stop(ar_0, ar_1, ar_2, ar_3, ar_4)) [ ar_0 - ar_4 >= 0 /\ ar_2 - ar_3 >= 0 /\ ar_0 - ar_1 >= 0 ] start location: koat_start leaf cost: 0 Complexity upper bound 2*ar_3 + 4*ar_4 + 2*ar_1 + 13 Time: 0.321 sec (SMT: 0.243 sec) ---------------------------------------- (2) BOUNDS(1, n^1) ---------------------------------------- (3) Loat Proof (FINISHED) ### Pre-processing the ITS problem ### Initial linear ITS problem Start location: evalspeedDis2start 0: evalspeedDis2start -> evalspeedDis2bb0in : [], cost: 1 1: evalspeedDis2bb0in -> evalspeedDis20 : [], cost: 1 2: evalspeedDis20 -> evalspeedDis21 : [], cost: 1 3: evalspeedDis21 -> evalspeedDis22 : [], cost: 1 4: evalspeedDis22 -> evalspeedDis23 : [], cost: 1 5: evalspeedDis23 -> evalspeedDis24 : [], cost: 1 6: evalspeedDis24 -> evalspeedDis25 : [], cost: 1 7: evalspeedDis25 -> evalspeedDis2bb1in : A'=B, C'=D, [], cost: 1 8: evalspeedDis2bb1in -> evalspeedDis2bb2in : [ E>=1+A ], cost: 1 9: evalspeedDis2bb1in -> evalspeedDis2bb3in : [ A>=E ], cost: 1 10: evalspeedDis2bb2in -> evalspeedDis2bb1in : A'=1+A, [ C>=1+A ], cost: 1 11: evalspeedDis2bb2in -> evalspeedDis2bb1in : [ C>=1+A && A>=C ], cost: 1 12: evalspeedDis2bb2in -> evalspeedDis2bb1in : A'=1+A, C'=1+C, [ A>=C && C>=1+A ], cost: 1 13: evalspeedDis2bb2in -> evalspeedDis2bb1in : C'=1+C, [ A>=C ], cost: 1 14: evalspeedDis2bb3in -> evalspeedDis2stop : [], cost: 1 Removed unreachable and leaf rules: Start location: evalspeedDis2start 0: evalspeedDis2start -> evalspeedDis2bb0in : [], cost: 1 1: evalspeedDis2bb0in -> evalspeedDis20 : [], cost: 1 2: evalspeedDis20 -> evalspeedDis21 : [], cost: 1 3: evalspeedDis21 -> evalspeedDis22 : [], cost: 1 4: evalspeedDis22 -> evalspeedDis23 : [], cost: 1 5: evalspeedDis23 -> evalspeedDis24 : [], cost: 1 6: evalspeedDis24 -> evalspeedDis25 : [], cost: 1 7: evalspeedDis25 -> evalspeedDis2bb1in : A'=B, C'=D, [], cost: 1 8: evalspeedDis2bb1in -> evalspeedDis2bb2in : [ E>=1+A ], cost: 1 10: evalspeedDis2bb2in -> evalspeedDis2bb1in : A'=1+A, [ C>=1+A ], cost: 1 11: evalspeedDis2bb2in -> evalspeedDis2bb1in : [ C>=1+A && A>=C ], cost: 1 12: evalspeedDis2bb2in -> evalspeedDis2bb1in : A'=1+A, C'=1+C, [ A>=C && C>=1+A ], cost: 1 13: evalspeedDis2bb2in -> evalspeedDis2bb1in : C'=1+C, [ A>=C ], cost: 1 Removed rules with unsatisfiable guard: Start location: evalspeedDis2start 0: evalspeedDis2start -> evalspeedDis2bb0in : [], cost: 1 1: evalspeedDis2bb0in -> evalspeedDis20 : [], cost: 1 2: evalspeedDis20 -> evalspeedDis21 : [], cost: 1 3: evalspeedDis21 -> evalspeedDis22 : [], cost: 1 4: evalspeedDis22 -> evalspeedDis23 : [], cost: 1 5: evalspeedDis23 -> evalspeedDis24 : [], cost: 1 6: evalspeedDis24 -> evalspeedDis25 : [], cost: 1 7: evalspeedDis25 -> evalspeedDis2bb1in : A'=B, C'=D, [], cost: 1 8: evalspeedDis2bb1in -> evalspeedDis2bb2in : [ E>=1+A ], cost: 1 10: evalspeedDis2bb2in -> evalspeedDis2bb1in : A'=1+A, [ C>=1+A ], cost: 1 13: evalspeedDis2bb2in -> evalspeedDis2bb1in : C'=1+C, [ A>=C ], cost: 1 ### Simplification by acceleration and chaining ### Eliminated locations (on linear paths): Start location: evalspeedDis2start 21: evalspeedDis2start -> evalspeedDis2bb1in : A'=B, C'=D, [], cost: 8 8: evalspeedDis2bb1in -> evalspeedDis2bb2in : [ E>=1+A ], cost: 1 10: evalspeedDis2bb2in -> evalspeedDis2bb1in : A'=1+A, [ C>=1+A ], cost: 1 13: evalspeedDis2bb2in -> evalspeedDis2bb1in : C'=1+C, [ A>=C ], cost: 1 Eliminated locations (on tree-shaped paths): Start location: evalspeedDis2start 21: evalspeedDis2start -> evalspeedDis2bb1in : A'=B, C'=D, [], cost: 8 22: evalspeedDis2bb1in -> evalspeedDis2bb1in : A'=1+A, [ E>=1+A && C>=1+A ], cost: 2 23: evalspeedDis2bb1in -> evalspeedDis2bb1in : C'=1+C, [ E>=1+A && A>=C ], cost: 2 Accelerating simple loops of location 8. Accelerating the following rules: 22: evalspeedDis2bb1in -> evalspeedDis2bb1in : A'=1+A, [ E>=1+A && C>=1+A ], cost: 2 23: evalspeedDis2bb1in -> evalspeedDis2bb1in : C'=1+C, [ E>=1+A && A>=C ], cost: 2 Accelerated rule 22 with backward acceleration, yielding the new rule 24. Accelerated rule 22 with backward acceleration, yielding the new rule 25. Accelerated rule 23 with metering function 1-C+A, yielding the new rule 26. Removing the simple loops: 22 23. Accelerated all simple loops using metering functions (where possible): Start location: evalspeedDis2start 21: evalspeedDis2start -> evalspeedDis2bb1in : A'=B, C'=D, [], cost: 8 24: evalspeedDis2bb1in -> evalspeedDis2bb1in : A'=E, [ E>=1+A && C>=1+A && C>=E ], cost: -2*A+2*E 25: evalspeedDis2bb1in -> evalspeedDis2bb1in : A'=C, [ E>=1+A && C>=1+A && E>=C ], cost: 2*C-2*A 26: evalspeedDis2bb1in -> evalspeedDis2bb1in : C'=1+A, [ E>=1+A && A>=C ], cost: 2-2*C+2*A Chained accelerated rules (with incoming rules): Start location: evalspeedDis2start 21: evalspeedDis2start -> evalspeedDis2bb1in : A'=B, C'=D, [], cost: 8 27: evalspeedDis2start -> evalspeedDis2bb1in : A'=E, C'=D, [ E>=1+B && D>=1+B && D>=E ], cost: 8+2*E-2*B 28: evalspeedDis2start -> evalspeedDis2bb1in : A'=D, C'=D, [ E>=1+B && D>=1+B && E>=D ], cost: 8+2*D-2*B 29: evalspeedDis2start -> evalspeedDis2bb1in : A'=B, C'=1+B, [ E>=1+B && B>=D ], cost: 10-2*D+2*B Removed unreachable locations (and leaf rules with constant cost): Start location: evalspeedDis2start 27: evalspeedDis2start -> evalspeedDis2bb1in : A'=E, C'=D, [ E>=1+B && D>=1+B && D>=E ], cost: 8+2*E-2*B 28: evalspeedDis2start -> evalspeedDis2bb1in : A'=D, C'=D, [ E>=1+B && D>=1+B && E>=D ], cost: 8+2*D-2*B 29: evalspeedDis2start -> evalspeedDis2bb1in : A'=B, C'=1+B, [ E>=1+B && B>=D ], cost: 10-2*D+2*B ### Computing asymptotic complexity ### Fully simplified ITS problem Start location: evalspeedDis2start 27: evalspeedDis2start -> evalspeedDis2bb1in : A'=E, C'=D, [ E>=1+B && D>=1+B && D>=E ], cost: 8+2*E-2*B 28: evalspeedDis2start -> evalspeedDis2bb1in : A'=D, C'=D, [ E>=1+B && D>=1+B && E>=D ], cost: 8+2*D-2*B 29: evalspeedDis2start -> evalspeedDis2bb1in : A'=B, C'=1+B, [ E>=1+B && B>=D ], cost: 10-2*D+2*B Computing asymptotic complexity for rule 27 Solved the limit problem by the following transformations: Created initial limit problem: 1+D-E (+/+!), D-B (+/+!), 8+2*E-2*B (+), E-B (+/+!) [not solved] removing all constraints (solved by SMT) resulting limit problem: [solved] applying transformation rule (C) using substitution {D==n,E==0,B==-n} resulting limit problem: [solved] Solution: D / n E / 0 B / -n Resulting cost 8+2*n has complexity: Poly(n^1) Found new complexity Poly(n^1). Obtained the following overall complexity (w.r.t. the length of the input n): Complexity: Poly(n^1) Cpx degree: 1 Solved cost: 8+2*n Rule cost: 8+2*E-2*B Rule guard: [ E>=1+B && D>=1+B && D>=E ] WORST_CASE(Omega(n^1),?) ---------------------------------------- (4) BOUNDS(n^1, INF)