/export/starexec/sandbox2/solver/bin/starexec_run_complexity /export/starexec/sandbox2/benchmark/theBenchmark.koat /export/starexec/sandbox2/output/output_files -------------------------------------------------------------------------------- WORST_CASE(Omega(n^1), O(n^1)) proof of /export/starexec/sandbox2/benchmark/theBenchmark.koat # AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(n^1, n^1). (0) CpxIntTrs (1) Koat Proof [FINISHED, 141 ms] (2) BOUNDS(1, n^1) (3) Loat Proof [FINISHED, 628 ms] (4) BOUNDS(n^1, INF) ---------------------------------------- (0) Obligation: Complexity Int TRS consisting of the following rules: eval_wise_start(v__0, v__01, v_x, v_y) -> Com_1(eval_wise_bb0_in(v__0, v__01, v_x, v_y)) :|: TRUE eval_wise_bb0_in(v__0, v__01, v_x, v_y) -> Com_1(eval_wise_0(v__0, v__01, v_x, v_y)) :|: TRUE eval_wise_0(v__0, v__01, v_x, v_y) -> Com_1(eval_wise_1(v__0, v__01, v_x, v_y)) :|: TRUE eval_wise_1(v__0, v__01, v_x, v_y) -> Com_1(eval_wise_2(v__0, v__01, v_x, v_y)) :|: TRUE eval_wise_2(v__0, v__01, v_x, v_y) -> Com_1(eval_wise_bb2_in(v__0, v__01, v_x, v_y)) :|: v_x < 0 eval_wise_2(v__0, v__01, v_x, v_y) -> Com_1(eval_wise_bb2_in(v__0, v__01, v_x, v_y)) :|: v_y < 0 eval_wise_2(v__0, v__01, v_x, v_y) -> Com_1(eval_wise_bb1_in(v_x, v_y, v_x, v_y)) :|: v_x >= 0 && v_y >= 0 eval_wise_bb1_in(v__0, v__01, v_x, v_y) -> Com_1(eval_wise__critedge_in(v__0, v__01, v_x, v_y)) :|: v__0 - v__01 > 2 eval_wise_bb1_in(v__0, v__01, v_x, v_y) -> Com_1(eval_wise__critedge_in(v__0, v__01, v_x, v_y)) :|: v__01 - v__0 > 2 eval_wise_bb1_in(v__0, v__01, v_x, v_y) -> Com_1(eval_wise_bb2_in(v__0, v__01, v_x, v_y)) :|: v__0 - v__01 <= 2 && v__01 - v__0 <= 2 eval_wise__critedge_in(v__0, v__01, v_x, v_y) -> Com_1(eval_wise_bb1_in(v__0 + 1, v__01, v_x, v_y)) :|: v__0 < v__01 eval_wise__critedge_in(v__0, v__01, v_x, v_y) -> Com_1(eval_wise_bb1_in(v__0, v__01, v_x, v_y)) :|: v__0 < v__01 && v__0 >= v__01 eval_wise__critedge_in(v__0, v__01, v_x, v_y) -> Com_1(eval_wise_bb1_in(v__0 + 1, v__01 + 1, v_x, v_y)) :|: v__0 >= v__01 && v__0 < v__01 eval_wise__critedge_in(v__0, v__01, v_x, v_y) -> Com_1(eval_wise_bb1_in(v__0, v__01 + 1, v_x, v_y)) :|: v__0 >= v__01 eval_wise_bb2_in(v__0, v__01, v_x, v_y) -> Com_1(eval_wise_stop(v__0, v__01, v_x, v_y)) :|: TRUE The start-symbols are:[eval_wise_start_4] ---------------------------------------- (1) Koat Proof (FINISHED) YES(?, 8*Ar_0 + 8*Ar_1 + 17) Initial complexity problem: 1: T: (Comp: ?, Cost: 1) evalwisestart(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwisebb0in(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: ?, Cost: 1) evalwisebb0in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwise0(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: ?, Cost: 1) evalwise0(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwise1(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: ?, Cost: 1) evalwise1(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwise2(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: ?, Cost: 1) evalwise2(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwisebb2in(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 >= Ar_0 + 1 ] (Comp: ?, Cost: 1) evalwise2(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwisebb2in(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 >= Ar_1 + 1 ] (Comp: ?, Cost: 1) evalwise2(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwisebb1in(Ar_0, Ar_1, Ar_0, Ar_1)) [ Ar_0 >= 0 /\ Ar_1 >= 0 ] (Comp: ?, Cost: 1) evalwisebb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwisecritedgein(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= Ar_3 + 3 ] (Comp: ?, Cost: 1) evalwisebb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwisecritedgein(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_3 >= Ar_2 + 3 ] (Comp: ?, Cost: 1) evalwisebb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwisebb2in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_3 + 2 >= Ar_2 /\ Ar_2 + 2 >= Ar_3 ] (Comp: ?, Cost: 1) evalwisecritedgein(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwisebb1in(Ar_0, Ar_1, Ar_2 + 1, Ar_3)) [ Ar_3 >= Ar_2 + 1 ] (Comp: ?, Cost: 1) evalwisecritedgein(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwisebb1in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_3 >= Ar_2 + 1 /\ Ar_2 >= Ar_3 ] (Comp: ?, Cost: 1) evalwisecritedgein(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwisebb1in(Ar_0, Ar_1, Ar_2 + 1, Ar_3 + 1)) [ Ar_2 >= Ar_3 /\ Ar_3 >= Ar_2 + 1 ] (Comp: ?, Cost: 1) evalwisecritedgein(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwisebb1in(Ar_0, Ar_1, Ar_2, Ar_3 + 1)) [ Ar_2 >= Ar_3 ] (Comp: ?, Cost: 1) evalwisebb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwisestop(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwisestart(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 Testing for reachability in the complexity graph removes the following transitions from problem 1: evalwisecritedgein(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwisebb1in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_3 >= Ar_2 + 1 /\ Ar_2 >= Ar_3 ] evalwisecritedgein(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwisebb1in(Ar_0, Ar_1, Ar_2 + 1, Ar_3 + 1)) [ Ar_2 >= Ar_3 /\ Ar_3 >= Ar_2 + 1 ] We thus obtain the following problem: 2: T: (Comp: ?, Cost: 1) evalwisecritedgein(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwisebb1in(Ar_0, Ar_1, Ar_2 + 1, Ar_3)) [ Ar_3 >= Ar_2 + 1 ] (Comp: ?, Cost: 1) evalwisecritedgein(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwisebb1in(Ar_0, Ar_1, Ar_2, Ar_3 + 1)) [ Ar_2 >= Ar_3 ] (Comp: ?, Cost: 1) evalwisebb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwisebb2in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_3 + 2 >= Ar_2 /\ Ar_2 + 2 >= Ar_3 ] (Comp: ?, Cost: 1) evalwisebb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwisecritedgein(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_3 >= Ar_2 + 3 ] (Comp: ?, Cost: 1) evalwisebb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwisecritedgein(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= Ar_3 + 3 ] (Comp: ?, Cost: 1) evalwisebb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwisestop(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: ?, Cost: 1) evalwise2(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwisebb1in(Ar_0, Ar_1, Ar_0, Ar_1)) [ Ar_0 >= 0 /\ Ar_1 >= 0 ] (Comp: ?, Cost: 1) evalwise2(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwisebb2in(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 >= Ar_1 + 1 ] (Comp: ?, Cost: 1) evalwise2(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwisebb2in(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 >= Ar_0 + 1 ] (Comp: ?, Cost: 1) evalwise1(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwise2(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: ?, Cost: 1) evalwise0(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwise1(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: ?, Cost: 1) evalwisebb0in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwise0(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: ?, Cost: 1) evalwisestart(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwisebb0in(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwisestart(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 Repeatedly propagating knowledge in problem 2 produces the following problem: 3: T: (Comp: ?, Cost: 1) evalwisecritedgein(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwisebb1in(Ar_0, Ar_1, Ar_2 + 1, Ar_3)) [ Ar_3 >= Ar_2 + 1 ] (Comp: ?, Cost: 1) evalwisecritedgein(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwisebb1in(Ar_0, Ar_1, Ar_2, Ar_3 + 1)) [ Ar_2 >= Ar_3 ] (Comp: ?, Cost: 1) evalwisebb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwisebb2in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_3 + 2 >= Ar_2 /\ Ar_2 + 2 >= Ar_3 ] (Comp: ?, Cost: 1) evalwisebb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwisecritedgein(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_3 >= Ar_2 + 3 ] (Comp: ?, Cost: 1) evalwisebb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwisecritedgein(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= Ar_3 + 3 ] (Comp: ?, Cost: 1) evalwisebb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwisestop(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 1) evalwise2(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwisebb1in(Ar_0, Ar_1, Ar_0, Ar_1)) [ Ar_0 >= 0 /\ Ar_1 >= 0 ] (Comp: 1, Cost: 1) evalwise2(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwisebb2in(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 >= Ar_1 + 1 ] (Comp: 1, Cost: 1) evalwise2(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwisebb2in(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 >= Ar_0 + 1 ] (Comp: 1, Cost: 1) evalwise1(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwise2(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 1) evalwise0(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwise1(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 1) evalwisebb0in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwise0(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 1) evalwisestart(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwisebb0in(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwisestart(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 A polynomial rank function with Pol(evalwisecritedgein) = 2 Pol(evalwisebb1in) = 2 Pol(evalwisebb2in) = 1 Pol(evalwisestop) = 0 Pol(evalwise2) = 2 Pol(evalwise1) = 2 Pol(evalwise0) = 2 Pol(evalwisebb0in) = 2 Pol(evalwisestart) = 2 Pol(koat_start) = 2 orients all transitions weakly and the transitions evalwisebb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwisestop(Ar_0, Ar_1, Ar_2, Ar_3)) evalwisebb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwisebb2in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_3 + 2 >= Ar_2 /\ Ar_2 + 2 >= Ar_3 ] strictly and produces the following problem: 4: T: (Comp: ?, Cost: 1) evalwisecritedgein(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwisebb1in(Ar_0, Ar_1, Ar_2 + 1, Ar_3)) [ Ar_3 >= Ar_2 + 1 ] (Comp: ?, Cost: 1) evalwisecritedgein(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwisebb1in(Ar_0, Ar_1, Ar_2, Ar_3 + 1)) [ Ar_2 >= Ar_3 ] (Comp: 2, Cost: 1) evalwisebb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwisebb2in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_3 + 2 >= Ar_2 /\ Ar_2 + 2 >= Ar_3 ] (Comp: ?, Cost: 1) evalwisebb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwisecritedgein(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_3 >= Ar_2 + 3 ] (Comp: ?, Cost: 1) evalwisebb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwisecritedgein(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= Ar_3 + 3 ] (Comp: 2, Cost: 1) evalwisebb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwisestop(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 1) evalwise2(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwisebb1in(Ar_0, Ar_1, Ar_0, Ar_1)) [ Ar_0 >= 0 /\ Ar_1 >= 0 ] (Comp: 1, Cost: 1) evalwise2(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwisebb2in(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 >= Ar_1 + 1 ] (Comp: 1, Cost: 1) evalwise2(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwisebb2in(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 >= Ar_0 + 1 ] (Comp: 1, Cost: 1) evalwise1(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwise2(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 1) evalwise0(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwise1(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 1) evalwisebb0in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwise0(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 1) evalwisestart(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwisebb0in(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwisestart(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 A polynomial rank function with Pol(evalwisebb1in) = -2*V_3 + 2*V_4 + 1 Pol(evalwisecritedgein) = -2*V_3 + 2*V_4 and size complexities S("koat_start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwisestart(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(evalwisestart(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(evalwisestart(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(evalwisestart(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 <= 0 ]", 0-3) = Ar_3 S("evalwisestart(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwisebb0in(Ar_0, Ar_1, Ar_2, Ar_3))", 0-0) = Ar_0 S("evalwisestart(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwisebb0in(Ar_0, Ar_1, Ar_2, Ar_3))", 0-1) = Ar_1 S("evalwisestart(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwisebb0in(Ar_0, Ar_1, Ar_2, Ar_3))", 0-2) = Ar_2 S("evalwisestart(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwisebb0in(Ar_0, Ar_1, Ar_2, Ar_3))", 0-3) = Ar_3 S("evalwisebb0in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwise0(Ar_0, Ar_1, Ar_2, Ar_3))", 0-0) = Ar_0 S("evalwisebb0in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwise0(Ar_0, Ar_1, Ar_2, Ar_3))", 0-1) = Ar_1 S("evalwisebb0in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwise0(Ar_0, Ar_1, Ar_2, Ar_3))", 0-2) = Ar_2 S("evalwisebb0in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwise0(Ar_0, Ar_1, Ar_2, Ar_3))", 0-3) = Ar_3 S("evalwise0(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwise1(Ar_0, Ar_1, Ar_2, Ar_3))", 0-0) = Ar_0 S("evalwise0(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwise1(Ar_0, Ar_1, Ar_2, Ar_3))", 0-1) = Ar_1 S("evalwise0(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwise1(Ar_0, Ar_1, Ar_2, Ar_3))", 0-2) = Ar_2 S("evalwise0(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwise1(Ar_0, Ar_1, Ar_2, Ar_3))", 0-3) = Ar_3 S("evalwise1(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwise2(Ar_0, Ar_1, Ar_2, Ar_3))", 0-0) = Ar_0 S("evalwise1(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwise2(Ar_0, Ar_1, Ar_2, Ar_3))", 0-1) = Ar_1 S("evalwise1(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwise2(Ar_0, Ar_1, Ar_2, Ar_3))", 0-2) = Ar_2 S("evalwise1(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwise2(Ar_0, Ar_1, Ar_2, Ar_3))", 0-3) = Ar_3 S("evalwise2(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwisebb2in(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 >= Ar_0 + 1 ]", 0-0) = Ar_0 S("evalwise2(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwisebb2in(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 >= Ar_0 + 1 ]", 0-1) = Ar_1 S("evalwise2(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwisebb2in(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 >= Ar_0 + 1 ]", 0-2) = Ar_2 S("evalwise2(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwisebb2in(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 >= Ar_0 + 1 ]", 0-3) = Ar_3 S("evalwise2(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwisebb2in(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 >= Ar_1 + 1 ]", 0-0) = Ar_0 S("evalwise2(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwisebb2in(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 >= Ar_1 + 1 ]", 0-1) = Ar_1 S("evalwise2(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwisebb2in(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 >= Ar_1 + 1 ]", 0-2) = Ar_2 S("evalwise2(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwisebb2in(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 >= Ar_1 + 1 ]", 0-3) = Ar_3 S("evalwise2(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwisebb1in(Ar_0, Ar_1, Ar_0, Ar_1)) [ Ar_0 >= 0 /\\ Ar_1 >= 0 ]", 0-0) = Ar_0 S("evalwise2(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwisebb1in(Ar_0, Ar_1, Ar_0, Ar_1)) [ Ar_0 >= 0 /\\ Ar_1 >= 0 ]", 0-1) = Ar_1 S("evalwise2(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwisebb1in(Ar_0, Ar_1, Ar_0, Ar_1)) [ Ar_0 >= 0 /\\ Ar_1 >= 0 ]", 0-2) = Ar_0 S("evalwise2(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwisebb1in(Ar_0, Ar_1, Ar_0, Ar_1)) [ Ar_0 >= 0 /\\ Ar_1 >= 0 ]", 0-3) = Ar_1 S("evalwisebb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwisestop(Ar_0, Ar_1, Ar_2, Ar_3))", 0-0) = Ar_0 S("evalwisebb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwisestop(Ar_0, Ar_1, Ar_2, Ar_3))", 0-1) = Ar_1 S("evalwisebb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwisestop(Ar_0, Ar_1, Ar_2, Ar_3))", 0-2) = ? S("evalwisebb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwisestop(Ar_0, Ar_1, Ar_2, Ar_3))", 0-3) = ? S("evalwisebb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwisecritedgein(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= Ar_3 + 3 ]", 0-0) = Ar_0 S("evalwisebb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwisecritedgein(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= Ar_3 + 3 ]", 0-1) = Ar_1 S("evalwisebb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwisecritedgein(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= Ar_3 + 3 ]", 0-2) = Ar_0 S("evalwisebb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwisecritedgein(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= Ar_3 + 3 ]", 0-3) = ? S("evalwisebb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwisecritedgein(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_3 >= Ar_2 + 3 ]", 0-0) = Ar_0 S("evalwisebb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwisecritedgein(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_3 >= Ar_2 + 3 ]", 0-1) = Ar_1 S("evalwisebb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwisecritedgein(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_3 >= Ar_2 + 3 ]", 0-2) = ? S("evalwisebb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwisecritedgein(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_3 >= Ar_2 + 3 ]", 0-3) = Ar_1 S("evalwisebb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwisebb2in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_3 + 2 >= Ar_2 /\\ Ar_2 + 2 >= Ar_3 ]", 0-0) = Ar_0 S("evalwisebb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwisebb2in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_3 + 2 >= Ar_2 /\\ Ar_2 + 2 >= Ar_3 ]", 0-1) = Ar_1 S("evalwisebb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwisebb2in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_3 + 2 >= Ar_2 /\\ Ar_2 + 2 >= Ar_3 ]", 0-2) = ? S("evalwisebb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwisebb2in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_3 + 2 >= Ar_2 /\\ Ar_2 + 2 >= Ar_3 ]", 0-3) = ? S("evalwisecritedgein(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwisebb1in(Ar_0, Ar_1, Ar_2, Ar_3 + 1)) [ Ar_2 >= Ar_3 ]", 0-0) = Ar_0 S("evalwisecritedgein(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwisebb1in(Ar_0, Ar_1, Ar_2, Ar_3 + 1)) [ Ar_2 >= Ar_3 ]", 0-1) = Ar_1 S("evalwisecritedgein(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwisebb1in(Ar_0, Ar_1, Ar_2, Ar_3 + 1)) [ Ar_2 >= Ar_3 ]", 0-2) = Ar_0 S("evalwisecritedgein(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwisebb1in(Ar_0, Ar_1, Ar_2, Ar_3 + 1)) [ Ar_2 >= Ar_3 ]", 0-3) = ? S("evalwisecritedgein(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwisebb1in(Ar_0, Ar_1, Ar_2 + 1, Ar_3)) [ Ar_3 >= Ar_2 + 1 ]", 0-0) = Ar_0 S("evalwisecritedgein(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwisebb1in(Ar_0, Ar_1, Ar_2 + 1, Ar_3)) [ Ar_3 >= Ar_2 + 1 ]", 0-1) = Ar_1 S("evalwisecritedgein(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwisebb1in(Ar_0, Ar_1, Ar_2 + 1, Ar_3)) [ Ar_3 >= Ar_2 + 1 ]", 0-2) = ? S("evalwisecritedgein(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwisebb1in(Ar_0, Ar_1, Ar_2 + 1, Ar_3)) [ Ar_3 >= Ar_2 + 1 ]", 0-3) = Ar_1 orients the transitions evalwisebb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwisecritedgein(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_3 >= Ar_2 + 3 ] evalwisecritedgein(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwisebb1in(Ar_0, Ar_1, Ar_2 + 1, Ar_3)) [ Ar_3 >= Ar_2 + 1 ] weakly and the transitions evalwisecritedgein(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwisebb1in(Ar_0, Ar_1, Ar_2 + 1, Ar_3)) [ Ar_3 >= Ar_2 + 1 ] evalwisebb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwisecritedgein(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_3 >= Ar_2 + 3 ] strictly and produces the following problem: 5: T: (Comp: 2*Ar_0 + 2*Ar_1 + 1, Cost: 1) evalwisecritedgein(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwisebb1in(Ar_0, Ar_1, Ar_2 + 1, Ar_3)) [ Ar_3 >= Ar_2 + 1 ] (Comp: ?, Cost: 1) evalwisecritedgein(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwisebb1in(Ar_0, Ar_1, Ar_2, Ar_3 + 1)) [ Ar_2 >= Ar_3 ] (Comp: 2, Cost: 1) evalwisebb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwisebb2in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_3 + 2 >= Ar_2 /\ Ar_2 + 2 >= Ar_3 ] (Comp: 2*Ar_0 + 2*Ar_1 + 1, Cost: 1) evalwisebb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwisecritedgein(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_3 >= Ar_2 + 3 ] (Comp: ?, Cost: 1) evalwisebb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwisecritedgein(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= Ar_3 + 3 ] (Comp: 2, Cost: 1) evalwisebb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwisestop(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 1) evalwise2(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwisebb1in(Ar_0, Ar_1, Ar_0, Ar_1)) [ Ar_0 >= 0 /\ Ar_1 >= 0 ] (Comp: 1, Cost: 1) evalwise2(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwisebb2in(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 >= Ar_1 + 1 ] (Comp: 1, Cost: 1) evalwise2(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwisebb2in(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 >= Ar_0 + 1 ] (Comp: 1, Cost: 1) evalwise1(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwise2(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 1) evalwise0(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwise1(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 1) evalwisebb0in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwise0(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 1) evalwisestart(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwisebb0in(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwisestart(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 A polynomial rank function with Pol(evalwisecritedgein) = 2*V_3 - 2*V_4 + 1 Pol(evalwisebb1in) = 2*V_3 - 2*V_4 + 2 and size complexities S("koat_start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwisestart(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(evalwisestart(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(evalwisestart(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(evalwisestart(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 <= 0 ]", 0-3) = Ar_3 S("evalwisestart(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwisebb0in(Ar_0, Ar_1, Ar_2, Ar_3))", 0-0) = Ar_0 S("evalwisestart(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwisebb0in(Ar_0, Ar_1, Ar_2, Ar_3))", 0-1) = Ar_1 S("evalwisestart(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwisebb0in(Ar_0, Ar_1, Ar_2, Ar_3))", 0-2) = Ar_2 S("evalwisestart(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwisebb0in(Ar_0, Ar_1, Ar_2, Ar_3))", 0-3) = Ar_3 S("evalwisebb0in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwise0(Ar_0, Ar_1, Ar_2, Ar_3))", 0-0) = Ar_0 S("evalwisebb0in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwise0(Ar_0, Ar_1, Ar_2, Ar_3))", 0-1) = Ar_1 S("evalwisebb0in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwise0(Ar_0, Ar_1, Ar_2, Ar_3))", 0-2) = Ar_2 S("evalwisebb0in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwise0(Ar_0, Ar_1, Ar_2, Ar_3))", 0-3) = Ar_3 S("evalwise0(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwise1(Ar_0, Ar_1, Ar_2, Ar_3))", 0-0) = Ar_0 S("evalwise0(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwise1(Ar_0, Ar_1, Ar_2, Ar_3))", 0-1) = Ar_1 S("evalwise0(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwise1(Ar_0, Ar_1, Ar_2, Ar_3))", 0-2) = Ar_2 S("evalwise0(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwise1(Ar_0, Ar_1, Ar_2, Ar_3))", 0-3) = Ar_3 S("evalwise1(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwise2(Ar_0, Ar_1, Ar_2, Ar_3))", 0-0) = Ar_0 S("evalwise1(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwise2(Ar_0, Ar_1, Ar_2, Ar_3))", 0-1) = Ar_1 S("evalwise1(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwise2(Ar_0, Ar_1, Ar_2, Ar_3))", 0-2) = Ar_2 S("evalwise1(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwise2(Ar_0, Ar_1, Ar_2, Ar_3))", 0-3) = Ar_3 S("evalwise2(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwisebb2in(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 >= Ar_0 + 1 ]", 0-0) = Ar_0 S("evalwise2(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwisebb2in(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 >= Ar_0 + 1 ]", 0-1) = Ar_1 S("evalwise2(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwisebb2in(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 >= Ar_0 + 1 ]", 0-2) = Ar_2 S("evalwise2(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwisebb2in(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 >= Ar_0 + 1 ]", 0-3) = Ar_3 S("evalwise2(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwisebb2in(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 >= Ar_1 + 1 ]", 0-0) = Ar_0 S("evalwise2(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwisebb2in(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 >= Ar_1 + 1 ]", 0-1) = Ar_1 S("evalwise2(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwisebb2in(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 >= Ar_1 + 1 ]", 0-2) = Ar_2 S("evalwise2(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwisebb2in(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 >= Ar_1 + 1 ]", 0-3) = Ar_3 S("evalwise2(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwisebb1in(Ar_0, Ar_1, Ar_0, Ar_1)) [ Ar_0 >= 0 /\\ Ar_1 >= 0 ]", 0-0) = Ar_0 S("evalwise2(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwisebb1in(Ar_0, Ar_1, Ar_0, Ar_1)) [ Ar_0 >= 0 /\\ Ar_1 >= 0 ]", 0-1) = Ar_1 S("evalwise2(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwisebb1in(Ar_0, Ar_1, Ar_0, Ar_1)) [ Ar_0 >= 0 /\\ Ar_1 >= 0 ]", 0-2) = Ar_0 S("evalwise2(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwisebb1in(Ar_0, Ar_1, Ar_0, Ar_1)) [ Ar_0 >= 0 /\\ Ar_1 >= 0 ]", 0-3) = Ar_1 S("evalwisebb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwisestop(Ar_0, Ar_1, Ar_2, Ar_3))", 0-0) = Ar_0 S("evalwisebb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwisestop(Ar_0, Ar_1, Ar_2, Ar_3))", 0-1) = Ar_1 S("evalwisebb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwisestop(Ar_0, Ar_1, Ar_2, Ar_3))", 0-2) = 3*Ar_0 + 3*Ar_1 + 3*Ar_2 + 81 S("evalwisebb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwisestop(Ar_0, Ar_1, Ar_2, Ar_3))", 0-3) = ? S("evalwisebb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwisecritedgein(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= Ar_3 + 3 ]", 0-0) = Ar_0 S("evalwisebb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwisecritedgein(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= Ar_3 + 3 ]", 0-1) = Ar_1 S("evalwisebb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwisecritedgein(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= Ar_3 + 3 ]", 0-2) = Ar_0 S("evalwisebb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwisecritedgein(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= Ar_3 + 3 ]", 0-3) = ? S("evalwisebb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwisecritedgein(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_3 >= Ar_2 + 3 ]", 0-0) = Ar_0 S("evalwisebb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwisecritedgein(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_3 >= Ar_2 + 3 ]", 0-1) = Ar_1 S("evalwisebb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwisecritedgein(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_3 >= Ar_2 + 3 ]", 0-2) = 3*Ar_0 + 3*Ar_1 + 9 S("evalwisebb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwisecritedgein(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_3 >= Ar_2 + 3 ]", 0-3) = Ar_1 S("evalwisebb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwisebb2in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_3 + 2 >= Ar_2 /\\ Ar_2 + 2 >= Ar_3 ]", 0-0) = Ar_0 S("evalwisebb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwisebb2in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_3 + 2 >= Ar_2 /\\ Ar_2 + 2 >= Ar_3 ]", 0-1) = Ar_1 S("evalwisebb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwisebb2in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_3 + 2 >= Ar_2 /\\ Ar_2 + 2 >= Ar_3 ]", 0-2) = 3*Ar_0 + 3*Ar_1 + 27 S("evalwisebb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwisebb2in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_3 + 2 >= Ar_2 /\\ Ar_2 + 2 >= Ar_3 ]", 0-3) = ? S("evalwisecritedgein(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwisebb1in(Ar_0, Ar_1, Ar_2, Ar_3 + 1)) [ Ar_2 >= Ar_3 ]", 0-0) = Ar_0 S("evalwisecritedgein(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwisebb1in(Ar_0, Ar_1, Ar_2, Ar_3 + 1)) [ Ar_2 >= Ar_3 ]", 0-1) = Ar_1 S("evalwisecritedgein(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwisebb1in(Ar_0, Ar_1, Ar_2, Ar_3 + 1)) [ Ar_2 >= Ar_3 ]", 0-2) = Ar_0 S("evalwisecritedgein(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwisebb1in(Ar_0, Ar_1, Ar_2, Ar_3 + 1)) [ Ar_2 >= Ar_3 ]", 0-3) = ? S("evalwisecritedgein(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwisebb1in(Ar_0, Ar_1, Ar_2 + 1, Ar_3)) [ Ar_3 >= Ar_2 + 1 ]", 0-0) = Ar_0 S("evalwisecritedgein(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwisebb1in(Ar_0, Ar_1, Ar_2 + 1, Ar_3)) [ Ar_3 >= Ar_2 + 1 ]", 0-1) = Ar_1 S("evalwisecritedgein(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwisebb1in(Ar_0, Ar_1, Ar_2 + 1, Ar_3)) [ Ar_3 >= Ar_2 + 1 ]", 0-2) = 3*Ar_0 + 3*Ar_1 + 9 S("evalwisecritedgein(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwisebb1in(Ar_0, Ar_1, Ar_2 + 1, Ar_3)) [ Ar_3 >= Ar_2 + 1 ]", 0-3) = Ar_1 orients the transitions evalwisecritedgein(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwisebb1in(Ar_0, Ar_1, Ar_2, Ar_3 + 1)) [ Ar_2 >= Ar_3 ] evalwisebb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwisecritedgein(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= Ar_3 + 3 ] weakly and the transitions evalwisecritedgein(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwisebb1in(Ar_0, Ar_1, Ar_2, Ar_3 + 1)) [ Ar_2 >= Ar_3 ] evalwisebb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwisecritedgein(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= Ar_3 + 3 ] strictly and produces the following problem: 6: T: (Comp: 2*Ar_0 + 2*Ar_1 + 1, Cost: 1) evalwisecritedgein(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwisebb1in(Ar_0, Ar_1, Ar_2 + 1, Ar_3)) [ Ar_3 >= Ar_2 + 1 ] (Comp: 2*Ar_0 + 2*Ar_1 + 2, Cost: 1) evalwisecritedgein(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwisebb1in(Ar_0, Ar_1, Ar_2, Ar_3 + 1)) [ Ar_2 >= Ar_3 ] (Comp: 2, Cost: 1) evalwisebb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwisebb2in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_3 + 2 >= Ar_2 /\ Ar_2 + 2 >= Ar_3 ] (Comp: 2*Ar_0 + 2*Ar_1 + 1, Cost: 1) evalwisebb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwisecritedgein(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_3 >= Ar_2 + 3 ] (Comp: 2*Ar_0 + 2*Ar_1 + 2, Cost: 1) evalwisebb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwisecritedgein(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= Ar_3 + 3 ] (Comp: 2, Cost: 1) evalwisebb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwisestop(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 1) evalwise2(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwisebb1in(Ar_0, Ar_1, Ar_0, Ar_1)) [ Ar_0 >= 0 /\ Ar_1 >= 0 ] (Comp: 1, Cost: 1) evalwise2(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwisebb2in(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 >= Ar_1 + 1 ] (Comp: 1, Cost: 1) evalwise2(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwisebb2in(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 >= Ar_0 + 1 ] (Comp: 1, Cost: 1) evalwise1(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwise2(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 1) evalwise0(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwise1(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 1) evalwisebb0in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwise0(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 1) evalwisestart(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwisebb0in(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalwisestart(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 Complexity upper bound 8*Ar_0 + 8*Ar_1 + 17 Time: 0.123 sec (SMT: 0.090 sec) ---------------------------------------- (2) BOUNDS(1, n^1) ---------------------------------------- (3) Loat Proof (FINISHED) ### Pre-processing the ITS problem ### Initial linear ITS problem Start location: evalwisestart 0: evalwisestart -> evalwisebb0in : [], cost: 1 1: evalwisebb0in -> evalwise0 : [], cost: 1 2: evalwise0 -> evalwise1 : [], cost: 1 3: evalwise1 -> evalwise2 : [], cost: 1 4: evalwise2 -> evalwisebb2in : [ 0>=1+A ], cost: 1 5: evalwise2 -> evalwisebb2in : [ 0>=1+B ], cost: 1 6: evalwise2 -> evalwisebb1in : C'=A, D'=B, [ A>=0 && B>=0 ], cost: 1 7: evalwisebb1in -> evalwisecritedgein : [ C>=3+D ], cost: 1 8: evalwisebb1in -> evalwisecritedgein : [ D>=3+C ], cost: 1 9: evalwisebb1in -> evalwisebb2in : [ 2+D>=C && 2+C>=D ], cost: 1 10: evalwisecritedgein -> evalwisebb1in : C'=1+C, [ D>=1+C ], cost: 1 11: evalwisecritedgein -> evalwisebb1in : [ D>=1+C && C>=D ], cost: 1 12: evalwisecritedgein -> evalwisebb1in : C'=1+C, D'=1+D, [ C>=D && D>=1+C ], cost: 1 13: evalwisecritedgein -> evalwisebb1in : D'=1+D, [ C>=D ], cost: 1 14: evalwisebb2in -> evalwisestop : [], cost: 1 Checking for constant complexity: The following rule is satisfiable with cost >= 1, yielding constant complexity: 0: evalwisestart -> evalwisebb0in : [], cost: 1 Removed unreachable and leaf rules: Start location: evalwisestart 0: evalwisestart -> evalwisebb0in : [], cost: 1 1: evalwisebb0in -> evalwise0 : [], cost: 1 2: evalwise0 -> evalwise1 : [], cost: 1 3: evalwise1 -> evalwise2 : [], cost: 1 6: evalwise2 -> evalwisebb1in : C'=A, D'=B, [ A>=0 && B>=0 ], cost: 1 7: evalwisebb1in -> evalwisecritedgein : [ C>=3+D ], cost: 1 8: evalwisebb1in -> evalwisecritedgein : [ D>=3+C ], cost: 1 10: evalwisecritedgein -> evalwisebb1in : C'=1+C, [ D>=1+C ], cost: 1 11: evalwisecritedgein -> evalwisebb1in : [ D>=1+C && C>=D ], cost: 1 12: evalwisecritedgein -> evalwisebb1in : C'=1+C, D'=1+D, [ C>=D && D>=1+C ], cost: 1 13: evalwisecritedgein -> evalwisebb1in : D'=1+D, [ C>=D ], cost: 1 Removed rules with unsatisfiable guard: Start location: evalwisestart 0: evalwisestart -> evalwisebb0in : [], cost: 1 1: evalwisebb0in -> evalwise0 : [], cost: 1 2: evalwise0 -> evalwise1 : [], cost: 1 3: evalwise1 -> evalwise2 : [], cost: 1 6: evalwise2 -> evalwisebb1in : C'=A, D'=B, [ A>=0 && B>=0 ], cost: 1 7: evalwisebb1in -> evalwisecritedgein : [ C>=3+D ], cost: 1 8: evalwisebb1in -> evalwisecritedgein : [ D>=3+C ], cost: 1 10: evalwisecritedgein -> evalwisebb1in : C'=1+C, [ D>=1+C ], cost: 1 13: evalwisecritedgein -> evalwisebb1in : D'=1+D, [ C>=D ], cost: 1 ### Simplification by acceleration and chaining ### Eliminated locations (on linear paths): Start location: evalwisestart 18: evalwisestart -> evalwisebb1in : C'=A, D'=B, [ A>=0 && B>=0 ], cost: 5 7: evalwisebb1in -> evalwisecritedgein : [ C>=3+D ], cost: 1 8: evalwisebb1in -> evalwisecritedgein : [ D>=3+C ], cost: 1 10: evalwisecritedgein -> evalwisebb1in : C'=1+C, [ D>=1+C ], cost: 1 13: evalwisecritedgein -> evalwisebb1in : D'=1+D, [ C>=D ], cost: 1 Eliminated locations (on tree-shaped paths): Start location: evalwisestart 18: evalwisestart -> evalwisebb1in : C'=A, D'=B, [ A>=0 && B>=0 ], cost: 5 19: evalwisebb1in -> evalwisebb1in : D'=1+D, [ C>=3+D ], cost: 2 20: evalwisebb1in -> evalwisebb1in : C'=1+C, [ D>=3+C ], cost: 2 Accelerating simple loops of location 5. Accelerating the following rules: 19: evalwisebb1in -> evalwisebb1in : D'=1+D, [ C>=3+D ], cost: 2 20: evalwisebb1in -> evalwisebb1in : C'=1+C, [ D>=3+C ], cost: 2 Accelerated rule 19 with metering function -2+C-D, yielding the new rule 21. Accelerated rule 20 with metering function -2-C+D, yielding the new rule 22. Removing the simple loops: 19 20. Accelerated all simple loops using metering functions (where possible): Start location: evalwisestart 18: evalwisestart -> evalwisebb1in : C'=A, D'=B, [ A>=0 && B>=0 ], cost: 5 21: evalwisebb1in -> evalwisebb1in : D'=-2+C, [ C>=3+D ], cost: -4+2*C-2*D 22: evalwisebb1in -> evalwisebb1in : C'=-2+D, [ D>=3+C ], cost: -4-2*C+2*D Chained accelerated rules (with incoming rules): Start location: evalwisestart 18: evalwisestart -> evalwisebb1in : C'=A, D'=B, [ A>=0 && B>=0 ], cost: 5 23: evalwisestart -> evalwisebb1in : C'=A, D'=-2+A, [ A>=0 && B>=0 && A>=3+B ], cost: 1+2*A-2*B 24: evalwisestart -> evalwisebb1in : C'=-2+B, D'=B, [ A>=0 && B>=0 && B>=3+A ], cost: 1-2*A+2*B Removed unreachable locations (and leaf rules with constant cost): Start location: evalwisestart 23: evalwisestart -> evalwisebb1in : C'=A, D'=-2+A, [ A>=0 && B>=0 && A>=3+B ], cost: 1+2*A-2*B 24: evalwisestart -> evalwisebb1in : C'=-2+B, D'=B, [ A>=0 && B>=0 && B>=3+A ], cost: 1-2*A+2*B ### Computing asymptotic complexity ### Fully simplified ITS problem Start location: evalwisestart 23: evalwisestart -> evalwisebb1in : C'=A, D'=-2+A, [ A>=0 && B>=0 && A>=3+B ], cost: 1+2*A-2*B 24: evalwisestart -> evalwisebb1in : C'=-2+B, D'=B, [ A>=0 && B>=0 && B>=3+A ], cost: 1-2*A+2*B Computing asymptotic complexity for rule 23 Simplified the guard: 23: evalwisestart -> evalwisebb1in : C'=A, D'=-2+A, [ B>=0 && A>=3+B ], cost: 1+2*A-2*B Solved the limit problem by the following transformations: Created initial limit problem: 1+2*A-2*B (+), 1+B (+/+!), -2+A-B (+/+!) [not solved] removing all constraints (solved by SMT) resulting limit problem: [solved] applying transformation rule (C) using substitution {A==n,B==0} resulting limit problem: [solved] Solution: A / n B / 0 Resulting cost 1+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: 1+2*n Rule cost: 1+2*A-2*B Rule guard: [ B>=0 && A>=3+B ] WORST_CASE(Omega(n^1),?) ---------------------------------------- (4) BOUNDS(n^1, INF)