5.60/2.58 WORST_CASE(Omega(n^1), O(n^1)) 5.60/2.59 proof of /export/starexec/sandbox/benchmark/theBenchmark.koat 5.60/2.59 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 5.60/2.59 5.60/2.59 5.60/2.59 The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(n^1, n^1). 5.60/2.59 5.60/2.59 (0) CpxIntTrs 5.60/2.59 (1) Koat Proof [FINISHED, 306 ms] 5.60/2.59 (2) BOUNDS(1, n^1) 5.60/2.59 (3) Loat Proof [FINISHED, 820 ms] 5.60/2.59 (4) BOUNDS(n^1, INF) 5.60/2.59 5.60/2.59 5.60/2.59 ---------------------------------------- 5.60/2.59 5.60/2.59 (0) 5.60/2.59 Obligation: 5.60/2.59 Complexity Int TRS consisting of the following rules: 5.60/2.59 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 5.60/2.59 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 5.60/2.59 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 5.60/2.59 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 5.60/2.59 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 5.60/2.59 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 5.60/2.59 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 5.60/2.59 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 5.60/2.59 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 5.60/2.59 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 5.60/2.59 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 5.60/2.59 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 5.60/2.59 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 5.60/2.59 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 5.60/2.59 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 5.60/2.59 5.60/2.59 The start-symbols are:[eval_speedDis2_start_5] 5.60/2.59 5.60/2.59 5.60/2.59 ---------------------------------------- 5.60/2.59 5.60/2.59 (1) Koat Proof (FINISHED) 5.60/2.59 YES(?, 2*ar_3 + 4*ar_4 + 2*ar_1 + 13) 5.60/2.59 5.60/2.59 5.60/2.59 5.60/2.59 Initial complexity problem: 5.60/2.59 5.60/2.59 1: T: 5.60/2.59 5.60/2.59 (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)) 5.60/2.59 5.60/2.59 (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)) 5.60/2.59 5.60/2.59 (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)) 5.60/2.59 5.60/2.59 (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)) 5.60/2.59 5.60/2.59 (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)) 5.60/2.59 5.60/2.59 (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)) 5.60/2.59 5.60/2.59 (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)) 5.60/2.59 5.60/2.59 (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)) 5.60/2.59 5.60/2.59 (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 ] 5.60/2.59 5.60/2.59 (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 ] 5.60/2.59 5.60/2.59 (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 ] 5.60/2.59 5.60/2.59 (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 ] 5.60/2.59 5.60/2.59 (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 ] 5.60/2.59 5.60/2.59 (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 ] 5.60/2.59 5.60/2.59 (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)) 5.60/2.59 5.60/2.59 (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 ] 5.60/2.59 5.60/2.59 start location: koat_start 5.60/2.59 5.60/2.59 leaf cost: 0 5.60/2.59 5.60/2.59 5.60/2.59 5.60/2.59 Testing for reachability in the complexity graph removes the following transitions from problem 1: 5.60/2.59 5.60/2.59 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 ] 5.60/2.59 5.60/2.59 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 ] 5.60/2.59 5.60/2.59 We thus obtain the following problem: 5.60/2.59 5.60/2.59 2: T: 5.60/2.59 5.60/2.59 (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)) 5.60/2.59 5.60/2.59 (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 ] 5.60/2.59 5.60/2.59 (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 ] 5.60/2.59 5.60/2.59 (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 ] 5.60/2.59 5.60/2.59 (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 ] 5.60/2.59 5.60/2.59 (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)) 5.60/2.59 5.60/2.59 (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)) 5.60/2.59 5.60/2.59 (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)) 5.60/2.59 5.60/2.59 (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)) 5.60/2.59 5.60/2.59 (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)) 5.60/2.59 5.60/2.59 (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)) 5.60/2.59 5.60/2.59 (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)) 5.60/2.59 5.60/2.59 (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)) 5.60/2.59 5.60/2.59 (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 ] 5.60/2.59 5.60/2.59 start location: koat_start 5.60/2.59 5.60/2.59 leaf cost: 0 5.60/2.59 5.60/2.59 5.60/2.59 5.60/2.59 Repeatedly propagating knowledge in problem 2 produces the following problem: 5.60/2.59 5.60/2.59 3: T: 5.60/2.59 5.60/2.59 (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)) 5.60/2.59 5.60/2.59 (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 ] 5.60/2.59 5.60/2.59 (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 ] 5.60/2.59 5.60/2.59 (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 ] 5.60/2.59 5.60/2.59 (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 ] 5.60/2.59 5.60/2.59 (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)) 5.60/2.59 5.60/2.59 (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)) 5.60/2.59 5.60/2.59 (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)) 5.60/2.59 5.60/2.59 (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)) 5.60/2.59 5.60/2.59 (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)) 5.60/2.59 5.60/2.59 (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)) 5.60/2.59 5.60/2.59 (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)) 5.60/2.59 5.60/2.59 (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)) 5.60/2.59 5.60/2.59 (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 ] 5.60/2.59 5.60/2.59 start location: koat_start 5.60/2.59 5.60/2.59 leaf cost: 0 5.60/2.59 5.60/2.59 5.60/2.59 5.60/2.59 A polynomial rank function with 5.60/2.59 5.60/2.59 Pol(evalspeedDis2bb3in) = 1 5.60/2.59 5.60/2.59 Pol(evalspeedDis2stop) = 0 5.60/2.59 5.60/2.59 Pol(evalspeedDis2bb2in) = 2 5.60/2.59 5.60/2.59 Pol(evalspeedDis2bb1in) = 2 5.60/2.59 5.60/2.59 Pol(evalspeedDis25) = 2 5.60/2.59 5.60/2.59 Pol(evalspeedDis24) = 2 5.60/2.59 5.60/2.59 Pol(evalspeedDis23) = 2 5.60/2.59 5.60/2.59 Pol(evalspeedDis22) = 2 5.60/2.59 5.60/2.59 Pol(evalspeedDis21) = 2 5.60/2.59 5.60/2.59 Pol(evalspeedDis20) = 2 5.60/2.59 5.60/2.59 Pol(evalspeedDis2bb0in) = 2 5.60/2.59 5.60/2.59 Pol(evalspeedDis2start) = 2 5.60/2.59 5.60/2.59 Pol(koat_start) = 2 5.60/2.59 5.60/2.59 orients all transitions weakly and the transitions 5.60/2.59 5.60/2.59 evalspeedDis2bb3in(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalspeedDis2stop(ar_0, ar_1, ar_2, ar_3, ar_4)) 5.60/2.59 5.60/2.59 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 ] 5.60/2.59 5.60/2.59 strictly and produces the following problem: 5.60/2.59 5.60/2.59 4: T: 5.60/2.59 5.60/2.59 (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)) 5.60/2.59 5.60/2.59 (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 ] 5.60/2.59 5.60/2.59 (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 ] 5.60/2.59 5.60/2.59 (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 ] 5.60/2.59 5.60/2.59 (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 ] 5.60/2.59 5.60/2.59 (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)) 5.60/2.59 5.60/2.59 (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)) 5.60/2.59 5.60/2.59 (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)) 5.60/2.59 5.60/2.59 (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)) 5.60/2.59 5.60/2.59 (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)) 5.60/2.59 5.60/2.59 (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)) 5.60/2.59 5.60/2.59 (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)) 5.60/2.59 5.60/2.59 (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)) 5.60/2.59 5.60/2.59 (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 ] 5.60/2.59 5.60/2.59 start location: koat_start 5.60/2.59 5.60/2.59 leaf cost: 0 5.60/2.59 5.60/2.59 5.60/2.59 5.60/2.59 Applied AI with 'oct' on problem 4 to obtain the following invariants: 5.60/2.59 5.60/2.59 For symbol evalspeedDis2bb1in: X_3 - X_4 >= 0 /\ X_1 - X_2 >= 0 5.60/2.59 5.60/2.59 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 5.60/2.59 5.60/2.59 For symbol evalspeedDis2bb3in: X_1 - X_5 >= 0 /\ X_3 - X_4 >= 0 /\ X_1 - X_2 >= 0 5.60/2.59 5.60/2.59 5.60/2.59 5.60/2.59 5.60/2.59 5.60/2.59 This yielded the following problem: 5.60/2.59 5.60/2.59 5: T: 5.60/2.59 5.60/2.59 (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 ] 5.60/2.59 5.60/2.59 (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)) 5.60/2.59 5.60/2.59 (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)) 5.60/2.59 5.60/2.59 (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)) 5.60/2.59 5.60/2.59 (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)) 5.60/2.59 5.60/2.59 (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)) 5.60/2.59 5.60/2.59 (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)) 5.60/2.59 5.60/2.59 (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)) 5.60/2.59 5.60/2.59 (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)) 5.60/2.59 5.60/2.59 (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 ] 5.60/2.59 5.60/2.59 (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 ] 5.60/2.59 5.60/2.59 (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 ] 5.60/2.59 5.60/2.59 (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 ] 5.60/2.59 5.60/2.59 (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 ] 5.60/2.59 5.60/2.59 start location: koat_start 5.60/2.59 5.60/2.59 leaf cost: 0 5.60/2.59 5.60/2.59 5.60/2.59 5.60/2.59 A polynomial rank function with 5.60/2.59 5.60/2.59 Pol(koat_start) = -V_2 + V_5 5.60/2.59 5.60/2.59 Pol(evalspeedDis2start) = -V_2 + V_5 5.60/2.59 5.60/2.59 Pol(evalspeedDis2bb0in) = -V_2 + V_5 5.60/2.59 5.60/2.59 Pol(evalspeedDis20) = -V_2 + V_5 5.60/2.59 5.60/2.59 Pol(evalspeedDis21) = -V_2 + V_5 5.60/2.59 5.60/2.59 Pol(evalspeedDis22) = -V_2 + V_5 5.60/2.59 5.60/2.59 Pol(evalspeedDis23) = -V_2 + V_5 5.60/2.59 5.60/2.59 Pol(evalspeedDis24) = -V_2 + V_5 5.60/2.59 5.60/2.59 Pol(evalspeedDis25) = -V_2 + V_5 5.60/2.59 5.60/2.59 Pol(evalspeedDis2bb1in) = -V_1 + V_5 5.60/2.59 5.60/2.59 Pol(evalspeedDis2bb2in) = -V_1 + V_5 5.60/2.59 5.60/2.59 Pol(evalspeedDis2bb3in) = -V_1 + V_5 5.60/2.59 5.60/2.59 Pol(evalspeedDis2stop) = -V_1 + V_5 5.60/2.59 5.60/2.59 orients all transitions weakly and the transition 5.60/2.59 5.60/2.59 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 ] 5.60/2.59 5.60/2.59 strictly and produces the following problem: 5.60/2.60 5.60/2.60 6: T: 5.60/2.60 5.60/2.60 (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 ] 5.60/2.60 5.60/2.60 (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)) 5.60/2.60 5.60/2.60 (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)) 5.60/2.60 5.60/2.60 (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)) 5.60/2.60 5.60/2.60 (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)) 5.60/2.60 5.60/2.60 (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)) 5.60/2.60 5.60/2.60 (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)) 5.60/2.60 5.60/2.60 (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)) 5.60/2.60 5.60/2.60 (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)) 5.60/2.60 5.60/2.60 (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 ] 5.60/2.60 5.60/2.60 (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 ] 5.60/2.60 5.60/2.60 (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 ] 5.60/2.60 5.60/2.60 (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 ] 5.60/2.60 5.60/2.60 (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 ] 5.60/2.60 5.60/2.60 start location: koat_start 5.60/2.60 5.60/2.60 leaf cost: 0 5.60/2.60 5.60/2.60 5.60/2.60 5.60/2.60 A polynomial rank function with 5.60/2.60 5.60/2.60 Pol(koat_start) = -V_4 + V_5 5.60/2.60 5.60/2.60 Pol(evalspeedDis2start) = -V_4 + V_5 5.60/2.60 5.60/2.60 Pol(evalspeedDis2bb0in) = -V_4 + V_5 5.60/2.60 5.60/2.60 Pol(evalspeedDis20) = -V_4 + V_5 5.60/2.60 5.60/2.60 Pol(evalspeedDis21) = -V_4 + V_5 5.60/2.60 5.60/2.60 Pol(evalspeedDis22) = -V_4 + V_5 5.60/2.60 5.60/2.60 Pol(evalspeedDis23) = -V_4 + V_5 5.60/2.60 5.60/2.60 Pol(evalspeedDis24) = -V_4 + V_5 5.60/2.60 5.60/2.60 Pol(evalspeedDis25) = -V_4 + V_5 5.60/2.60 5.60/2.60 Pol(evalspeedDis2bb1in) = -V_3 + V_5 5.60/2.60 5.60/2.60 Pol(evalspeedDis2bb2in) = -V_3 + V_5 5.60/2.60 5.60/2.60 Pol(evalspeedDis2bb3in) = -V_3 + V_5 5.60/2.60 5.60/2.60 Pol(evalspeedDis2stop) = -V_3 + V_5 5.60/2.60 5.60/2.60 orients all transitions weakly and the transition 5.60/2.60 5.60/2.60 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 ] 5.60/2.60 5.60/2.60 strictly and produces the following problem: 5.60/2.60 5.60/2.60 7: T: 5.60/2.60 5.60/2.60 (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 ] 5.60/2.60 5.60/2.60 (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)) 5.60/2.60 5.60/2.60 (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)) 5.60/2.60 5.60/2.60 (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)) 5.60/2.60 5.60/2.60 (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)) 5.60/2.60 5.60/2.60 (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)) 5.60/2.60 5.60/2.60 (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)) 5.60/2.60 5.60/2.60 (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)) 5.60/2.60 5.60/2.60 (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)) 5.60/2.60 5.60/2.60 (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 ] 5.60/2.60 5.60/2.60 (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 ] 5.60/2.60 5.60/2.60 (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 ] 5.60/2.60 5.60/2.60 (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 ] 5.60/2.60 5.60/2.60 (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 ] 5.60/2.60 5.60/2.60 start location: koat_start 5.60/2.60 5.60/2.60 leaf cost: 0 5.60/2.60 5.60/2.60 5.60/2.60 5.60/2.60 Repeatedly propagating knowledge in problem 7 produces the following problem: 5.60/2.60 5.60/2.60 8: T: 5.60/2.60 5.60/2.60 (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 ] 5.60/2.60 5.60/2.60 (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)) 5.60/2.60 5.60/2.60 (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)) 5.60/2.60 5.60/2.60 (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)) 5.60/2.60 5.60/2.60 (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)) 5.60/2.60 5.60/2.60 (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)) 5.60/2.60 5.60/2.60 (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)) 5.60/2.60 5.60/2.60 (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)) 5.60/2.60 5.60/2.60 (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)) 5.60/2.60 5.60/2.60 (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 ] 5.60/2.60 5.60/2.60 (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 ] 5.60/2.60 5.60/2.60 (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 ] 5.60/2.60 5.60/2.60 (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 ] 5.60/2.60 5.60/2.60 (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 ] 5.60/2.60 5.60/2.60 start location: koat_start 5.60/2.60 5.60/2.60 leaf cost: 0 5.60/2.60 5.60/2.60 5.60/2.60 5.60/2.60 Complexity upper bound 2*ar_3 + 4*ar_4 + 2*ar_1 + 13 5.60/2.60 5.60/2.60 5.60/2.60 5.60/2.60 Time: 0.329 sec (SMT: 0.250 sec) 5.60/2.60 5.60/2.60 5.60/2.60 ---------------------------------------- 5.60/2.60 5.60/2.60 (2) 5.60/2.60 BOUNDS(1, n^1) 5.60/2.60 5.60/2.60 ---------------------------------------- 5.60/2.60 5.60/2.60 (3) Loat Proof (FINISHED) 5.60/2.60 5.60/2.60 5.60/2.60 ### Pre-processing the ITS problem ### 5.60/2.60 5.60/2.60 5.60/2.60 5.60/2.60 Initial linear ITS problem 5.60/2.60 5.60/2.60 Start location: evalspeedDis2start 5.60/2.60 5.60/2.60 0: evalspeedDis2start -> evalspeedDis2bb0in : [], cost: 1 5.60/2.60 5.60/2.60 1: evalspeedDis2bb0in -> evalspeedDis20 : [], cost: 1 5.60/2.60 5.60/2.60 2: evalspeedDis20 -> evalspeedDis21 : [], cost: 1 5.60/2.60 5.60/2.60 3: evalspeedDis21 -> evalspeedDis22 : [], cost: 1 5.60/2.60 5.60/2.60 4: evalspeedDis22 -> evalspeedDis23 : [], cost: 1 5.60/2.60 5.60/2.60 5: evalspeedDis23 -> evalspeedDis24 : [], cost: 1 5.60/2.60 5.60/2.60 6: evalspeedDis24 -> evalspeedDis25 : [], cost: 1 5.60/2.60 5.60/2.60 7: evalspeedDis25 -> evalspeedDis2bb1in : A'=B, C'=D, [], cost: 1 5.60/2.60 5.60/2.60 8: evalspeedDis2bb1in -> evalspeedDis2bb2in : [ E>=1+A ], cost: 1 5.60/2.60 5.60/2.60 9: evalspeedDis2bb1in -> evalspeedDis2bb3in : [ A>=E ], cost: 1 5.60/2.60 5.60/2.60 10: evalspeedDis2bb2in -> evalspeedDis2bb1in : A'=1+A, [ C>=1+A ], cost: 1 5.60/2.60 5.60/2.60 11: evalspeedDis2bb2in -> evalspeedDis2bb1in : [ C>=1+A && A>=C ], cost: 1 5.60/2.60 5.60/2.60 12: evalspeedDis2bb2in -> evalspeedDis2bb1in : A'=1+A, C'=1+C, [ A>=C && C>=1+A ], cost: 1 5.60/2.60 5.60/2.60 13: evalspeedDis2bb2in -> evalspeedDis2bb1in : C'=1+C, [ A>=C ], cost: 1 5.60/2.60 5.60/2.60 14: evalspeedDis2bb3in -> evalspeedDis2stop : [], cost: 1 5.60/2.60 5.60/2.60 5.60/2.60 5.60/2.60 Removed unreachable and leaf rules: 5.60/2.60 5.60/2.60 Start location: evalspeedDis2start 5.60/2.60 5.60/2.60 0: evalspeedDis2start -> evalspeedDis2bb0in : [], cost: 1 5.60/2.60 5.60/2.60 1: evalspeedDis2bb0in -> evalspeedDis20 : [], cost: 1 5.60/2.60 5.60/2.60 2: evalspeedDis20 -> evalspeedDis21 : [], cost: 1 5.60/2.60 5.60/2.60 3: evalspeedDis21 -> evalspeedDis22 : [], cost: 1 5.60/2.60 5.60/2.60 4: evalspeedDis22 -> evalspeedDis23 : [], cost: 1 5.60/2.60 5.60/2.60 5: evalspeedDis23 -> evalspeedDis24 : [], cost: 1 5.60/2.60 5.60/2.60 6: evalspeedDis24 -> evalspeedDis25 : [], cost: 1 5.60/2.60 5.60/2.60 7: evalspeedDis25 -> evalspeedDis2bb1in : A'=B, C'=D, [], cost: 1 5.60/2.60 5.60/2.60 8: evalspeedDis2bb1in -> evalspeedDis2bb2in : [ E>=1+A ], cost: 1 5.60/2.60 5.60/2.60 10: evalspeedDis2bb2in -> evalspeedDis2bb1in : A'=1+A, [ C>=1+A ], cost: 1 5.60/2.60 5.60/2.60 11: evalspeedDis2bb2in -> evalspeedDis2bb1in : [ C>=1+A && A>=C ], cost: 1 5.60/2.60 5.60/2.60 12: evalspeedDis2bb2in -> evalspeedDis2bb1in : A'=1+A, C'=1+C, [ A>=C && C>=1+A ], cost: 1 5.60/2.60 5.60/2.60 13: evalspeedDis2bb2in -> evalspeedDis2bb1in : C'=1+C, [ A>=C ], cost: 1 5.60/2.60 5.60/2.60 5.60/2.60 5.60/2.60 Removed rules with unsatisfiable guard: 5.60/2.60 5.60/2.60 Start location: evalspeedDis2start 5.60/2.60 5.60/2.60 0: evalspeedDis2start -> evalspeedDis2bb0in : [], cost: 1 5.60/2.60 5.60/2.60 1: evalspeedDis2bb0in -> evalspeedDis20 : [], cost: 1 5.60/2.60 5.60/2.60 2: evalspeedDis20 -> evalspeedDis21 : [], cost: 1 5.60/2.60 5.60/2.60 3: evalspeedDis21 -> evalspeedDis22 : [], cost: 1 5.60/2.60 5.60/2.60 4: evalspeedDis22 -> evalspeedDis23 : [], cost: 1 5.60/2.60 5.60/2.60 5: evalspeedDis23 -> evalspeedDis24 : [], cost: 1 5.60/2.60 5.60/2.60 6: evalspeedDis24 -> evalspeedDis25 : [], cost: 1 5.60/2.60 5.60/2.60 7: evalspeedDis25 -> evalspeedDis2bb1in : A'=B, C'=D, [], cost: 1 5.60/2.60 5.60/2.60 8: evalspeedDis2bb1in -> evalspeedDis2bb2in : [ E>=1+A ], cost: 1 5.60/2.60 5.60/2.60 10: evalspeedDis2bb2in -> evalspeedDis2bb1in : A'=1+A, [ C>=1+A ], cost: 1 5.60/2.60 5.60/2.60 13: evalspeedDis2bb2in -> evalspeedDis2bb1in : C'=1+C, [ A>=C ], cost: 1 5.60/2.60 5.60/2.60 5.60/2.60 5.60/2.60 ### Simplification by acceleration and chaining ### 5.60/2.60 5.60/2.60 5.60/2.60 5.60/2.60 Eliminated locations (on linear paths): 5.60/2.60 5.60/2.60 Start location: evalspeedDis2start 5.60/2.60 5.60/2.60 21: evalspeedDis2start -> evalspeedDis2bb1in : A'=B, C'=D, [], cost: 8 5.60/2.60 5.60/2.60 8: evalspeedDis2bb1in -> evalspeedDis2bb2in : [ E>=1+A ], cost: 1 5.60/2.60 5.60/2.60 10: evalspeedDis2bb2in -> evalspeedDis2bb1in : A'=1+A, [ C>=1+A ], cost: 1 5.60/2.60 5.60/2.60 13: evalspeedDis2bb2in -> evalspeedDis2bb1in : C'=1+C, [ A>=C ], cost: 1 5.60/2.60 5.60/2.60 5.60/2.60 5.60/2.60 Eliminated locations (on tree-shaped paths): 5.60/2.60 5.60/2.60 Start location: evalspeedDis2start 5.60/2.60 5.60/2.60 21: evalspeedDis2start -> evalspeedDis2bb1in : A'=B, C'=D, [], cost: 8 5.60/2.60 5.60/2.60 22: evalspeedDis2bb1in -> evalspeedDis2bb1in : A'=1+A, [ E>=1+A && C>=1+A ], cost: 2 5.60/2.60 5.60/2.60 23: evalspeedDis2bb1in -> evalspeedDis2bb1in : C'=1+C, [ E>=1+A && A>=C ], cost: 2 5.60/2.60 5.60/2.60 5.60/2.60 5.60/2.60 Accelerating simple loops of location 8. 5.60/2.60 5.60/2.60 Accelerating the following rules: 5.60/2.60 5.60/2.60 22: evalspeedDis2bb1in -> evalspeedDis2bb1in : A'=1+A, [ E>=1+A && C>=1+A ], cost: 2 5.60/2.60 5.60/2.60 23: evalspeedDis2bb1in -> evalspeedDis2bb1in : C'=1+C, [ E>=1+A && A>=C ], cost: 2 5.60/2.60 5.60/2.60 5.60/2.60 5.60/2.60 Accelerated rule 22 with backward acceleration, yielding the new rule 24. 5.60/2.60 5.60/2.60 Accelerated rule 22 with backward acceleration, yielding the new rule 25. 5.60/2.60 5.60/2.60 Accelerated rule 23 with metering function 1-C+A, yielding the new rule 26. 5.60/2.60 5.60/2.60 Removing the simple loops: 22 23. 5.60/2.60 5.60/2.60 5.60/2.60 5.60/2.60 Accelerated all simple loops using metering functions (where possible): 5.60/2.60 5.60/2.60 Start location: evalspeedDis2start 5.60/2.60 5.60/2.60 21: evalspeedDis2start -> evalspeedDis2bb1in : A'=B, C'=D, [], cost: 8 5.60/2.60 5.60/2.60 24: evalspeedDis2bb1in -> evalspeedDis2bb1in : A'=E, [ E>=1+A && C>=1+A && C>=E ], cost: -2*A+2*E 5.60/2.60 5.60/2.60 25: evalspeedDis2bb1in -> evalspeedDis2bb1in : A'=C, [ E>=1+A && C>=1+A && E>=C ], cost: 2*C-2*A 5.60/2.60 5.60/2.60 26: evalspeedDis2bb1in -> evalspeedDis2bb1in : C'=1+A, [ E>=1+A && A>=C ], cost: 2-2*C+2*A 5.60/2.60 5.60/2.60 5.60/2.60 5.60/2.60 Chained accelerated rules (with incoming rules): 5.60/2.60 5.60/2.60 Start location: evalspeedDis2start 5.60/2.60 5.60/2.60 21: evalspeedDis2start -> evalspeedDis2bb1in : A'=B, C'=D, [], cost: 8 5.60/2.60 5.60/2.60 27: evalspeedDis2start -> evalspeedDis2bb1in : A'=E, C'=D, [ E>=1+B && D>=1+B && D>=E ], cost: 8+2*E-2*B 5.60/2.60 5.60/2.60 28: evalspeedDis2start -> evalspeedDis2bb1in : A'=D, C'=D, [ E>=1+B && D>=1+B && E>=D ], cost: 8+2*D-2*B 5.60/2.60 5.60/2.60 29: evalspeedDis2start -> evalspeedDis2bb1in : A'=B, C'=1+B, [ E>=1+B && B>=D ], cost: 10-2*D+2*B 5.60/2.60 5.60/2.60 5.60/2.60 5.60/2.60 Removed unreachable locations (and leaf rules with constant cost): 5.60/2.60 5.60/2.60 Start location: evalspeedDis2start 5.60/2.60 5.60/2.60 27: evalspeedDis2start -> evalspeedDis2bb1in : A'=E, C'=D, [ E>=1+B && D>=1+B && D>=E ], cost: 8+2*E-2*B 5.60/2.60 5.60/2.60 28: evalspeedDis2start -> evalspeedDis2bb1in : A'=D, C'=D, [ E>=1+B && D>=1+B && E>=D ], cost: 8+2*D-2*B 5.60/2.60 5.60/2.60 29: evalspeedDis2start -> evalspeedDis2bb1in : A'=B, C'=1+B, [ E>=1+B && B>=D ], cost: 10-2*D+2*B 5.60/2.60 5.60/2.60 5.60/2.60 5.60/2.60 ### Computing asymptotic complexity ### 5.60/2.60 5.60/2.60 5.60/2.60 5.60/2.60 Fully simplified ITS problem 5.60/2.60 5.60/2.60 Start location: evalspeedDis2start 5.60/2.60 5.60/2.60 27: evalspeedDis2start -> evalspeedDis2bb1in : A'=E, C'=D, [ E>=1+B && D>=1+B && D>=E ], cost: 8+2*E-2*B 5.60/2.60 5.60/2.60 28: evalspeedDis2start -> evalspeedDis2bb1in : A'=D, C'=D, [ E>=1+B && D>=1+B && E>=D ], cost: 8+2*D-2*B 5.60/2.60 5.60/2.60 29: evalspeedDis2start -> evalspeedDis2bb1in : A'=B, C'=1+B, [ E>=1+B && B>=D ], cost: 10-2*D+2*B 5.60/2.60 5.60/2.60 5.60/2.60 5.60/2.60 Computing asymptotic complexity for rule 27 5.60/2.60 5.60/2.60 Solved the limit problem by the following transformations: 5.60/2.60 5.60/2.60 Created initial limit problem: 5.60/2.60 5.60/2.60 1+D-E (+/+!), D-B (+/+!), 8+2*E-2*B (+), E-B (+/+!) [not solved] 5.60/2.60 5.60/2.60 5.60/2.60 5.60/2.60 removing all constraints (solved by SMT) 5.60/2.60 5.60/2.60 resulting limit problem: [solved] 5.60/2.60 5.60/2.60 5.60/2.60 5.60/2.60 applying transformation rule (C) using substitution {D==n,E==0,B==-n} 5.60/2.60 5.60/2.60 resulting limit problem: 5.60/2.60 5.60/2.60 [solved] 5.60/2.60 5.60/2.60 5.60/2.60 5.60/2.60 Solution: 5.60/2.60 5.60/2.60 D / n 5.60/2.60 5.60/2.60 E / 0 5.60/2.60 5.60/2.60 B / -n 5.60/2.60 5.60/2.60 Resulting cost 8+2*n has complexity: Poly(n^1) 5.60/2.60 5.60/2.60 5.60/2.60 5.60/2.60 Found new complexity Poly(n^1). 5.60/2.60 5.60/2.60 5.60/2.60 5.60/2.60 Obtained the following overall complexity (w.r.t. the length of the input n): 5.60/2.60 5.60/2.60 Complexity: Poly(n^1) 5.60/2.60 5.60/2.60 Cpx degree: 1 5.60/2.60 5.60/2.60 Solved cost: 8+2*n 5.60/2.60 5.60/2.60 Rule cost: 8+2*E-2*B 5.60/2.60 5.60/2.60 Rule guard: [ E>=1+B && D>=1+B && D>=E ] 5.60/2.60 5.60/2.60 5.60/2.60 5.60/2.60 WORST_CASE(Omega(n^1),?) 5.60/2.60 5.60/2.60 5.60/2.60 ---------------------------------------- 5.60/2.60 5.60/2.60 (4) 5.60/2.60 BOUNDS(n^1, INF) 5.60/2.61 EOF