/export/starexec/sandbox2/solver/bin/starexec_run_complexity /export/starexec/sandbox2/benchmark/theBenchmark.koat /export/starexec/sandbox2/output/output_files -------------------------------------------------------------------------------- WORST_CASE(?, O(n^2)) 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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(1, n^2). (0) CpxIntTrs (1) Koat Proof [FINISHED, 128 ms] (2) BOUNDS(1, n^2) ---------------------------------------- (0) Obligation: Complexity Int TRS consisting of the following rules: evalspeedpldi2start(A, B, C) -> Com_1(evalspeedpldi2entryin(A, B, C)) :|: TRUE evalspeedpldi2entryin(A, B, C) -> Com_1(evalspeedpldi2bb5in(B, 0, A)) :|: A >= 0 && B >= 1 evalspeedpldi2entryin(A, B, C) -> Com_1(evalspeedpldi2returnin(A, B, C)) :|: 0 >= A + 1 evalspeedpldi2entryin(A, B, C) -> Com_1(evalspeedpldi2returnin(A, B, C)) :|: 0 >= B evalspeedpldi2bb5in(A, B, C) -> Com_1(evalspeedpldi2bb2in(A, B, C)) :|: C >= 1 evalspeedpldi2bb5in(A, B, C) -> Com_1(evalspeedpldi2returnin(A, B, C)) :|: 0 >= C evalspeedpldi2bb2in(A, B, C) -> Com_1(evalspeedpldi2bb3in(A, B, C)) :|: A >= B + 1 evalspeedpldi2bb2in(A, B, C) -> Com_1(evalspeedpldi2bb5in(A, 0, C)) :|: B >= A evalspeedpldi2bb3in(A, B, C) -> Com_1(evalspeedpldi2bb5in(A, B + 1, C - 1)) :|: TRUE evalspeedpldi2returnin(A, B, C) -> Com_1(evalspeedpldi2stop(A, B, C)) :|: TRUE The start-symbols are:[evalspeedpldi2start_3] ---------------------------------------- (1) Koat Proof (FINISHED) YES(?, 17*Ar_0 + 18*Ar_0^2 + 18*Ar_0*Ar_1 + 8*Ar_1 + 11) Initial complexity problem: 1: T: (Comp: ?, Cost: 1) evalspeedpldi2start(Ar_0, Ar_1, Ar_2) -> Com_1(evalspeedpldi2entryin(Ar_0, Ar_1, Ar_2)) (Comp: ?, Cost: 1) evalspeedpldi2entryin(Ar_0, Ar_1, Ar_2) -> Com_1(evalspeedpldi2bb5in(Ar_1, 0, Ar_0)) [ Ar_0 >= 0 /\ Ar_1 >= 1 ] (Comp: ?, Cost: 1) evalspeedpldi2entryin(Ar_0, Ar_1, Ar_2) -> Com_1(evalspeedpldi2returnin(Ar_0, Ar_1, Ar_2)) [ 0 >= Ar_0 + 1 ] (Comp: ?, Cost: 1) evalspeedpldi2entryin(Ar_0, Ar_1, Ar_2) -> Com_1(evalspeedpldi2returnin(Ar_0, Ar_1, Ar_2)) [ 0 >= Ar_1 ] (Comp: ?, Cost: 1) evalspeedpldi2bb5in(Ar_0, Ar_1, Ar_2) -> Com_1(evalspeedpldi2bb2in(Ar_0, Ar_1, Ar_2)) [ Ar_2 >= 1 ] (Comp: ?, Cost: 1) evalspeedpldi2bb5in(Ar_0, Ar_1, Ar_2) -> Com_1(evalspeedpldi2returnin(Ar_0, Ar_1, Ar_2)) [ 0 >= Ar_2 ] (Comp: ?, Cost: 1) evalspeedpldi2bb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalspeedpldi2bb3in(Ar_0, Ar_1, Ar_2)) [ Ar_0 >= Ar_1 + 1 ] (Comp: ?, Cost: 1) evalspeedpldi2bb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalspeedpldi2bb5in(Ar_0, 0, Ar_2)) [ Ar_1 >= Ar_0 ] (Comp: ?, Cost: 1) evalspeedpldi2bb3in(Ar_0, Ar_1, Ar_2) -> Com_1(evalspeedpldi2bb5in(Ar_0, Ar_1 + 1, Ar_2 - 1)) (Comp: ?, Cost: 1) evalspeedpldi2returnin(Ar_0, Ar_1, Ar_2) -> Com_1(evalspeedpldi2stop(Ar_0, Ar_1, Ar_2)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2) -> Com_1(evalspeedpldi2start(Ar_0, Ar_1, Ar_2)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 Repeatedly propagating knowledge in problem 1 produces the following problem: 2: T: (Comp: 1, Cost: 1) evalspeedpldi2start(Ar_0, Ar_1, Ar_2) -> Com_1(evalspeedpldi2entryin(Ar_0, Ar_1, Ar_2)) (Comp: 1, Cost: 1) evalspeedpldi2entryin(Ar_0, Ar_1, Ar_2) -> Com_1(evalspeedpldi2bb5in(Ar_1, 0, Ar_0)) [ Ar_0 >= 0 /\ Ar_1 >= 1 ] (Comp: 1, Cost: 1) evalspeedpldi2entryin(Ar_0, Ar_1, Ar_2) -> Com_1(evalspeedpldi2returnin(Ar_0, Ar_1, Ar_2)) [ 0 >= Ar_0 + 1 ] (Comp: 1, Cost: 1) evalspeedpldi2entryin(Ar_0, Ar_1, Ar_2) -> Com_1(evalspeedpldi2returnin(Ar_0, Ar_1, Ar_2)) [ 0 >= Ar_1 ] (Comp: ?, Cost: 1) evalspeedpldi2bb5in(Ar_0, Ar_1, Ar_2) -> Com_1(evalspeedpldi2bb2in(Ar_0, Ar_1, Ar_2)) [ Ar_2 >= 1 ] (Comp: ?, Cost: 1) evalspeedpldi2bb5in(Ar_0, Ar_1, Ar_2) -> Com_1(evalspeedpldi2returnin(Ar_0, Ar_1, Ar_2)) [ 0 >= Ar_2 ] (Comp: ?, Cost: 1) evalspeedpldi2bb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalspeedpldi2bb3in(Ar_0, Ar_1, Ar_2)) [ Ar_0 >= Ar_1 + 1 ] (Comp: ?, Cost: 1) evalspeedpldi2bb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalspeedpldi2bb5in(Ar_0, 0, Ar_2)) [ Ar_1 >= Ar_0 ] (Comp: ?, Cost: 1) evalspeedpldi2bb3in(Ar_0, Ar_1, Ar_2) -> Com_1(evalspeedpldi2bb5in(Ar_0, Ar_1 + 1, Ar_2 - 1)) (Comp: ?, Cost: 1) evalspeedpldi2returnin(Ar_0, Ar_1, Ar_2) -> Com_1(evalspeedpldi2stop(Ar_0, Ar_1, Ar_2)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2) -> Com_1(evalspeedpldi2start(Ar_0, Ar_1, Ar_2)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 A polynomial rank function with Pol(evalspeedpldi2start) = 2 Pol(evalspeedpldi2entryin) = 2 Pol(evalspeedpldi2bb5in) = 2 Pol(evalspeedpldi2returnin) = 1 Pol(evalspeedpldi2bb2in) = 2 Pol(evalspeedpldi2bb3in) = 2 Pol(evalspeedpldi2stop) = 0 Pol(koat_start) = 2 orients all transitions weakly and the transitions evalspeedpldi2returnin(Ar_0, Ar_1, Ar_2) -> Com_1(evalspeedpldi2stop(Ar_0, Ar_1, Ar_2)) evalspeedpldi2bb5in(Ar_0, Ar_1, Ar_2) -> Com_1(evalspeedpldi2returnin(Ar_0, Ar_1, Ar_2)) [ 0 >= Ar_2 ] strictly and produces the following problem: 3: T: (Comp: 1, Cost: 1) evalspeedpldi2start(Ar_0, Ar_1, Ar_2) -> Com_1(evalspeedpldi2entryin(Ar_0, Ar_1, Ar_2)) (Comp: 1, Cost: 1) evalspeedpldi2entryin(Ar_0, Ar_1, Ar_2) -> Com_1(evalspeedpldi2bb5in(Ar_1, 0, Ar_0)) [ Ar_0 >= 0 /\ Ar_1 >= 1 ] (Comp: 1, Cost: 1) evalspeedpldi2entryin(Ar_0, Ar_1, Ar_2) -> Com_1(evalspeedpldi2returnin(Ar_0, Ar_1, Ar_2)) [ 0 >= Ar_0 + 1 ] (Comp: 1, Cost: 1) evalspeedpldi2entryin(Ar_0, Ar_1, Ar_2) -> Com_1(evalspeedpldi2returnin(Ar_0, Ar_1, Ar_2)) [ 0 >= Ar_1 ] (Comp: ?, Cost: 1) evalspeedpldi2bb5in(Ar_0, Ar_1, Ar_2) -> Com_1(evalspeedpldi2bb2in(Ar_0, Ar_1, Ar_2)) [ Ar_2 >= 1 ] (Comp: 2, Cost: 1) evalspeedpldi2bb5in(Ar_0, Ar_1, Ar_2) -> Com_1(evalspeedpldi2returnin(Ar_0, Ar_1, Ar_2)) [ 0 >= Ar_2 ] (Comp: ?, Cost: 1) evalspeedpldi2bb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalspeedpldi2bb3in(Ar_0, Ar_1, Ar_2)) [ Ar_0 >= Ar_1 + 1 ] (Comp: ?, Cost: 1) evalspeedpldi2bb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalspeedpldi2bb5in(Ar_0, 0, Ar_2)) [ Ar_1 >= Ar_0 ] (Comp: ?, Cost: 1) evalspeedpldi2bb3in(Ar_0, Ar_1, Ar_2) -> Com_1(evalspeedpldi2bb5in(Ar_0, Ar_1 + 1, Ar_2 - 1)) (Comp: 2, Cost: 1) evalspeedpldi2returnin(Ar_0, Ar_1, Ar_2) -> Com_1(evalspeedpldi2stop(Ar_0, Ar_1, Ar_2)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2) -> Com_1(evalspeedpldi2start(Ar_0, Ar_1, Ar_2)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 Applied AI with 'oct' on problem 3 to obtain the following invariants: For symbol evalspeedpldi2bb2in: X_3 - 1 >= 0 /\ X_2 + X_3 - 1 >= 0 /\ X_1 + X_3 - 2 >= 0 /\ X_2 >= 0 /\ X_1 + X_2 - 1 >= 0 /\ X_1 - 1 >= 0 For symbol evalspeedpldi2bb3in: X_3 - 1 >= 0 /\ X_2 + X_3 - 1 >= 0 /\ X_1 + X_3 - 2 >= 0 /\ X_1 - X_2 - 1 >= 0 /\ X_2 >= 0 /\ X_1 + X_2 - 1 >= 0 /\ X_1 - 1 >= 0 For symbol evalspeedpldi2bb5in: X_3 >= 0 /\ X_2 + X_3 >= 0 /\ X_1 + X_3 - 1 >= 0 /\ X_2 >= 0 /\ X_1 + X_2 - 1 >= 0 /\ X_1 - 1 >= 0 This yielded the following problem: 4: T: (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2) -> Com_1(evalspeedpldi2start(Ar_0, Ar_1, Ar_2)) [ 0 <= 0 ] (Comp: 2, Cost: 1) evalspeedpldi2returnin(Ar_0, Ar_1, Ar_2) -> Com_1(evalspeedpldi2stop(Ar_0, Ar_1, Ar_2)) (Comp: ?, Cost: 1) evalspeedpldi2bb3in(Ar_0, Ar_1, Ar_2) -> Com_1(evalspeedpldi2bb5in(Ar_0, Ar_1 + 1, Ar_2 - 1)) [ Ar_2 - 1 >= 0 /\ Ar_1 + Ar_2 - 1 >= 0 /\ Ar_0 + Ar_2 - 2 >= 0 /\ Ar_0 - Ar_1 - 1 >= 0 /\ Ar_1 >= 0 /\ Ar_0 + Ar_1 - 1 >= 0 /\ Ar_0 - 1 >= 0 ] (Comp: ?, Cost: 1) evalspeedpldi2bb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalspeedpldi2bb5in(Ar_0, 0, Ar_2)) [ Ar_2 - 1 >= 0 /\ Ar_1 + Ar_2 - 1 >= 0 /\ Ar_0 + Ar_2 - 2 >= 0 /\ Ar_1 >= 0 /\ Ar_0 + Ar_1 - 1 >= 0 /\ Ar_0 - 1 >= 0 /\ Ar_1 >= Ar_0 ] (Comp: ?, Cost: 1) evalspeedpldi2bb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalspeedpldi2bb3in(Ar_0, Ar_1, Ar_2)) [ Ar_2 - 1 >= 0 /\ Ar_1 + Ar_2 - 1 >= 0 /\ Ar_0 + Ar_2 - 2 >= 0 /\ Ar_1 >= 0 /\ Ar_0 + Ar_1 - 1 >= 0 /\ Ar_0 - 1 >= 0 /\ Ar_0 >= Ar_1 + 1 ] (Comp: 2, Cost: 1) evalspeedpldi2bb5in(Ar_0, Ar_1, Ar_2) -> Com_1(evalspeedpldi2returnin(Ar_0, Ar_1, Ar_2)) [ Ar_2 >= 0 /\ Ar_1 + Ar_2 >= 0 /\ Ar_0 + Ar_2 - 1 >= 0 /\ Ar_1 >= 0 /\ Ar_0 + Ar_1 - 1 >= 0 /\ Ar_0 - 1 >= 0 /\ 0 >= Ar_2 ] (Comp: ?, Cost: 1) evalspeedpldi2bb5in(Ar_0, Ar_1, Ar_2) -> Com_1(evalspeedpldi2bb2in(Ar_0, Ar_1, Ar_2)) [ Ar_2 >= 0 /\ Ar_1 + Ar_2 >= 0 /\ Ar_0 + Ar_2 - 1 >= 0 /\ Ar_1 >= 0 /\ Ar_0 + Ar_1 - 1 >= 0 /\ Ar_0 - 1 >= 0 /\ Ar_2 >= 1 ] (Comp: 1, Cost: 1) evalspeedpldi2entryin(Ar_0, Ar_1, Ar_2) -> Com_1(evalspeedpldi2returnin(Ar_0, Ar_1, Ar_2)) [ 0 >= Ar_1 ] (Comp: 1, Cost: 1) evalspeedpldi2entryin(Ar_0, Ar_1, Ar_2) -> Com_1(evalspeedpldi2returnin(Ar_0, Ar_1, Ar_2)) [ 0 >= Ar_0 + 1 ] (Comp: 1, Cost: 1) evalspeedpldi2entryin(Ar_0, Ar_1, Ar_2) -> Com_1(evalspeedpldi2bb5in(Ar_1, 0, Ar_0)) [ Ar_0 >= 0 /\ Ar_1 >= 1 ] (Comp: 1, Cost: 1) evalspeedpldi2start(Ar_0, Ar_1, Ar_2) -> Com_1(evalspeedpldi2entryin(Ar_0, Ar_1, Ar_2)) start location: koat_start leaf cost: 0 A polynomial rank function with Pol(koat_start) = 2*V_1 + 2*V_2 Pol(evalspeedpldi2start) = 2*V_1 + 2*V_2 Pol(evalspeedpldi2returnin) = 2*V_1 + 2*V_2 Pol(evalspeedpldi2stop) = 2*V_1 + 2*V_2 Pol(evalspeedpldi2bb3in) = 2*V_1 + 2*V_2 + 2*V_3 Pol(evalspeedpldi2bb5in) = 2*V_1 + 2*V_2 + 2*V_3 Pol(evalspeedpldi2bb2in) = 2*V_1 + 2*V_2 + 2*V_3 Pol(evalspeedpldi2entryin) = 2*V_1 + 2*V_2 orients all transitions weakly and the transition evalspeedpldi2bb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalspeedpldi2bb5in(Ar_0, 0, Ar_2)) [ Ar_2 - 1 >= 0 /\ Ar_1 + Ar_2 - 1 >= 0 /\ Ar_0 + Ar_2 - 2 >= 0 /\ Ar_1 >= 0 /\ Ar_0 + Ar_1 - 1 >= 0 /\ Ar_0 - 1 >= 0 /\ Ar_1 >= Ar_0 ] strictly and produces the following problem: 5: T: (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2) -> Com_1(evalspeedpldi2start(Ar_0, Ar_1, Ar_2)) [ 0 <= 0 ] (Comp: 2, Cost: 1) evalspeedpldi2returnin(Ar_0, Ar_1, Ar_2) -> Com_1(evalspeedpldi2stop(Ar_0, Ar_1, Ar_2)) (Comp: ?, Cost: 1) evalspeedpldi2bb3in(Ar_0, Ar_1, Ar_2) -> Com_1(evalspeedpldi2bb5in(Ar_0, Ar_1 + 1, Ar_2 - 1)) [ Ar_2 - 1 >= 0 /\ Ar_1 + Ar_2 - 1 >= 0 /\ Ar_0 + Ar_2 - 2 >= 0 /\ Ar_0 - Ar_1 - 1 >= 0 /\ Ar_1 >= 0 /\ Ar_0 + Ar_1 - 1 >= 0 /\ Ar_0 - 1 >= 0 ] (Comp: 2*Ar_0 + 2*Ar_1, Cost: 1) evalspeedpldi2bb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalspeedpldi2bb5in(Ar_0, 0, Ar_2)) [ Ar_2 - 1 >= 0 /\ Ar_1 + Ar_2 - 1 >= 0 /\ Ar_0 + Ar_2 - 2 >= 0 /\ Ar_1 >= 0 /\ Ar_0 + Ar_1 - 1 >= 0 /\ Ar_0 - 1 >= 0 /\ Ar_1 >= Ar_0 ] (Comp: ?, Cost: 1) evalspeedpldi2bb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalspeedpldi2bb3in(Ar_0, Ar_1, Ar_2)) [ Ar_2 - 1 >= 0 /\ Ar_1 + Ar_2 - 1 >= 0 /\ Ar_0 + Ar_2 - 2 >= 0 /\ Ar_1 >= 0 /\ Ar_0 + Ar_1 - 1 >= 0 /\ Ar_0 - 1 >= 0 /\ Ar_0 >= Ar_1 + 1 ] (Comp: 2, Cost: 1) evalspeedpldi2bb5in(Ar_0, Ar_1, Ar_2) -> Com_1(evalspeedpldi2returnin(Ar_0, Ar_1, Ar_2)) [ Ar_2 >= 0 /\ Ar_1 + Ar_2 >= 0 /\ Ar_0 + Ar_2 - 1 >= 0 /\ Ar_1 >= 0 /\ Ar_0 + Ar_1 - 1 >= 0 /\ Ar_0 - 1 >= 0 /\ 0 >= Ar_2 ] (Comp: ?, Cost: 1) evalspeedpldi2bb5in(Ar_0, Ar_1, Ar_2) -> Com_1(evalspeedpldi2bb2in(Ar_0, Ar_1, Ar_2)) [ Ar_2 >= 0 /\ Ar_1 + Ar_2 >= 0 /\ Ar_0 + Ar_2 - 1 >= 0 /\ Ar_1 >= 0 /\ Ar_0 + Ar_1 - 1 >= 0 /\ Ar_0 - 1 >= 0 /\ Ar_2 >= 1 ] (Comp: 1, Cost: 1) evalspeedpldi2entryin(Ar_0, Ar_1, Ar_2) -> Com_1(evalspeedpldi2returnin(Ar_0, Ar_1, Ar_2)) [ 0 >= Ar_1 ] (Comp: 1, Cost: 1) evalspeedpldi2entryin(Ar_0, Ar_1, Ar_2) -> Com_1(evalspeedpldi2returnin(Ar_0, Ar_1, Ar_2)) [ 0 >= Ar_0 + 1 ] (Comp: 1, Cost: 1) evalspeedpldi2entryin(Ar_0, Ar_1, Ar_2) -> Com_1(evalspeedpldi2bb5in(Ar_1, 0, Ar_0)) [ Ar_0 >= 0 /\ Ar_1 >= 1 ] (Comp: 1, Cost: 1) evalspeedpldi2start(Ar_0, Ar_1, Ar_2) -> Com_1(evalspeedpldi2entryin(Ar_0, Ar_1, Ar_2)) start location: koat_start leaf cost: 0 A polynomial rank function with Pol(evalspeedpldi2bb5in) = 3*V_3 + 1 Pol(evalspeedpldi2bb2in) = 3*V_3 Pol(evalspeedpldi2bb3in) = 3*V_3 - 1 and size complexities S("evalspeedpldi2start(Ar_0, Ar_1, Ar_2) -> Com_1(evalspeedpldi2entryin(Ar_0, Ar_1, Ar_2))", 0-0) = Ar_0 S("evalspeedpldi2start(Ar_0, Ar_1, Ar_2) -> Com_1(evalspeedpldi2entryin(Ar_0, Ar_1, Ar_2))", 0-1) = Ar_1 S("evalspeedpldi2start(Ar_0, Ar_1, Ar_2) -> Com_1(evalspeedpldi2entryin(Ar_0, Ar_1, Ar_2))", 0-2) = Ar_2 S("evalspeedpldi2entryin(Ar_0, Ar_1, Ar_2) -> Com_1(evalspeedpldi2bb5in(Ar_1, 0, Ar_0)) [ Ar_0 >= 0 /\\ Ar_1 >= 1 ]", 0-0) = Ar_1 S("evalspeedpldi2entryin(Ar_0, Ar_1, Ar_2) -> Com_1(evalspeedpldi2bb5in(Ar_1, 0, Ar_0)) [ Ar_0 >= 0 /\\ Ar_1 >= 1 ]", 0-1) = 0 S("evalspeedpldi2entryin(Ar_0, Ar_1, Ar_2) -> Com_1(evalspeedpldi2bb5in(Ar_1, 0, Ar_0)) [ Ar_0 >= 0 /\\ Ar_1 >= 1 ]", 0-2) = Ar_0 S("evalspeedpldi2entryin(Ar_0, Ar_1, Ar_2) -> Com_1(evalspeedpldi2returnin(Ar_0, Ar_1, Ar_2)) [ 0 >= Ar_0 + 1 ]", 0-0) = Ar_0 S("evalspeedpldi2entryin(Ar_0, Ar_1, Ar_2) -> Com_1(evalspeedpldi2returnin(Ar_0, Ar_1, Ar_2)) [ 0 >= Ar_0 + 1 ]", 0-1) = Ar_1 S("evalspeedpldi2entryin(Ar_0, Ar_1, Ar_2) -> Com_1(evalspeedpldi2returnin(Ar_0, Ar_1, Ar_2)) [ 0 >= Ar_0 + 1 ]", 0-2) = Ar_2 S("evalspeedpldi2entryin(Ar_0, Ar_1, Ar_2) -> Com_1(evalspeedpldi2returnin(Ar_0, Ar_1, Ar_2)) [ 0 >= Ar_1 ]", 0-0) = Ar_0 S("evalspeedpldi2entryin(Ar_0, Ar_1, Ar_2) -> Com_1(evalspeedpldi2returnin(Ar_0, Ar_1, Ar_2)) [ 0 >= Ar_1 ]", 0-1) = Ar_1 S("evalspeedpldi2entryin(Ar_0, Ar_1, Ar_2) -> Com_1(evalspeedpldi2returnin(Ar_0, Ar_1, Ar_2)) [ 0 >= Ar_1 ]", 0-2) = Ar_2 S("evalspeedpldi2bb5in(Ar_0, Ar_1, Ar_2) -> Com_1(evalspeedpldi2bb2in(Ar_0, Ar_1, Ar_2)) [ Ar_2 >= 0 /\\ Ar_1 + Ar_2 >= 0 /\\ Ar_0 + Ar_2 - 1 >= 0 /\\ Ar_1 >= 0 /\\ Ar_0 + Ar_1 - 1 >= 0 /\\ Ar_0 - 1 >= 0 /\\ Ar_2 >= 1 ]", 0-0) = Ar_1 S("evalspeedpldi2bb5in(Ar_0, Ar_1, Ar_2) -> Com_1(evalspeedpldi2bb2in(Ar_0, Ar_1, Ar_2)) [ Ar_2 >= 0 /\\ Ar_1 + Ar_2 >= 0 /\\ Ar_0 + Ar_2 - 1 >= 0 /\\ Ar_1 >= 0 /\\ Ar_0 + Ar_1 - 1 >= 0 /\\ Ar_0 - 1 >= 0 /\\ Ar_2 >= 1 ]", 0-1) = Ar_1 S("evalspeedpldi2bb5in(Ar_0, Ar_1, Ar_2) -> Com_1(evalspeedpldi2bb2in(Ar_0, Ar_1, Ar_2)) [ Ar_2 >= 0 /\\ Ar_1 + Ar_2 >= 0 /\\ Ar_0 + Ar_2 - 1 >= 0 /\\ Ar_1 >= 0 /\\ Ar_0 + Ar_1 - 1 >= 0 /\\ Ar_0 - 1 >= 0 /\\ Ar_2 >= 1 ]", 0-2) = Ar_0 S("evalspeedpldi2bb5in(Ar_0, Ar_1, Ar_2) -> Com_1(evalspeedpldi2returnin(Ar_0, Ar_1, Ar_2)) [ Ar_2 >= 0 /\\ Ar_1 + Ar_2 >= 0 /\\ Ar_0 + Ar_2 - 1 >= 0 /\\ Ar_1 >= 0 /\\ Ar_0 + Ar_1 - 1 >= 0 /\\ Ar_0 - 1 >= 0 /\\ 0 >= Ar_2 ]", 0-0) = Ar_1 S("evalspeedpldi2bb5in(Ar_0, Ar_1, Ar_2) -> Com_1(evalspeedpldi2returnin(Ar_0, Ar_1, Ar_2)) [ Ar_2 >= 0 /\\ Ar_1 + Ar_2 >= 0 /\\ Ar_0 + Ar_2 - 1 >= 0 /\\ Ar_1 >= 0 /\\ Ar_0 + Ar_1 - 1 >= 0 /\\ Ar_0 - 1 >= 0 /\\ 0 >= Ar_2 ]", 0-1) = Ar_1 S("evalspeedpldi2bb5in(Ar_0, Ar_1, Ar_2) -> Com_1(evalspeedpldi2returnin(Ar_0, Ar_1, Ar_2)) [ Ar_2 >= 0 /\\ Ar_1 + Ar_2 >= 0 /\\ Ar_0 + Ar_2 - 1 >= 0 /\\ Ar_1 >= 0 /\\ Ar_0 + Ar_1 - 1 >= 0 /\\ Ar_0 - 1 >= 0 /\\ 0 >= Ar_2 ]", 0-2) = 0 S("evalspeedpldi2bb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalspeedpldi2bb3in(Ar_0, Ar_1, Ar_2)) [ Ar_2 - 1 >= 0 /\\ Ar_1 + Ar_2 - 1 >= 0 /\\ Ar_0 + Ar_2 - 2 >= 0 /\\ Ar_1 >= 0 /\\ Ar_0 + Ar_1 - 1 >= 0 /\\ Ar_0 - 1 >= 0 /\\ Ar_0 >= Ar_1 + 1 ]", 0-0) = Ar_1 S("evalspeedpldi2bb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalspeedpldi2bb3in(Ar_0, Ar_1, Ar_2)) [ Ar_2 - 1 >= 0 /\\ Ar_1 + Ar_2 - 1 >= 0 /\\ Ar_0 + Ar_2 - 2 >= 0 /\\ Ar_1 >= 0 /\\ Ar_0 + Ar_1 - 1 >= 0 /\\ Ar_0 - 1 >= 0 /\\ Ar_0 >= Ar_1 + 1 ]", 0-1) = Ar_1 S("evalspeedpldi2bb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalspeedpldi2bb3in(Ar_0, Ar_1, Ar_2)) [ Ar_2 - 1 >= 0 /\\ Ar_1 + Ar_2 - 1 >= 0 /\\ Ar_0 + Ar_2 - 2 >= 0 /\\ Ar_1 >= 0 /\\ Ar_0 + Ar_1 - 1 >= 0 /\\ Ar_0 - 1 >= 0 /\\ Ar_0 >= Ar_1 + 1 ]", 0-2) = Ar_0 S("evalspeedpldi2bb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalspeedpldi2bb5in(Ar_0, 0, Ar_2)) [ Ar_2 - 1 >= 0 /\\ Ar_1 + Ar_2 - 1 >= 0 /\\ Ar_0 + Ar_2 - 2 >= 0 /\\ Ar_1 >= 0 /\\ Ar_0 + Ar_1 - 1 >= 0 /\\ Ar_0 - 1 >= 0 /\\ Ar_1 >= Ar_0 ]", 0-0) = Ar_1 S("evalspeedpldi2bb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalspeedpldi2bb5in(Ar_0, 0, Ar_2)) [ Ar_2 - 1 >= 0 /\\ Ar_1 + Ar_2 - 1 >= 0 /\\ Ar_0 + Ar_2 - 2 >= 0 /\\ Ar_1 >= 0 /\\ Ar_0 + Ar_1 - 1 >= 0 /\\ Ar_0 - 1 >= 0 /\\ Ar_1 >= Ar_0 ]", 0-1) = 0 S("evalspeedpldi2bb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalspeedpldi2bb5in(Ar_0, 0, Ar_2)) [ Ar_2 - 1 >= 0 /\\ Ar_1 + Ar_2 - 1 >= 0 /\\ Ar_0 + Ar_2 - 2 >= 0 /\\ Ar_1 >= 0 /\\ Ar_0 + Ar_1 - 1 >= 0 /\\ Ar_0 - 1 >= 0 /\\ Ar_1 >= Ar_0 ]", 0-2) = Ar_0 S("evalspeedpldi2bb3in(Ar_0, Ar_1, Ar_2) -> Com_1(evalspeedpldi2bb5in(Ar_0, Ar_1 + 1, Ar_2 - 1)) [ Ar_2 - 1 >= 0 /\\ Ar_1 + Ar_2 - 1 >= 0 /\\ Ar_0 + Ar_2 - 2 >= 0 /\\ Ar_0 - Ar_1 - 1 >= 0 /\\ Ar_1 >= 0 /\\ Ar_0 + Ar_1 - 1 >= 0 /\\ Ar_0 - 1 >= 0 ]", 0-0) = Ar_1 S("evalspeedpldi2bb3in(Ar_0, Ar_1, Ar_2) -> Com_1(evalspeedpldi2bb5in(Ar_0, Ar_1 + 1, Ar_2 - 1)) [ Ar_2 - 1 >= 0 /\\ Ar_1 + Ar_2 - 1 >= 0 /\\ Ar_0 + Ar_2 - 2 >= 0 /\\ Ar_0 - Ar_1 - 1 >= 0 /\\ Ar_1 >= 0 /\\ Ar_0 + Ar_1 - 1 >= 0 /\\ Ar_0 - 1 >= 0 ]", 0-1) = Ar_1 S("evalspeedpldi2bb3in(Ar_0, Ar_1, Ar_2) -> Com_1(evalspeedpldi2bb5in(Ar_0, Ar_1 + 1, Ar_2 - 1)) [ Ar_2 - 1 >= 0 /\\ Ar_1 + Ar_2 - 1 >= 0 /\\ Ar_0 + Ar_2 - 2 >= 0 /\\ Ar_0 - Ar_1 - 1 >= 0 /\\ Ar_1 >= 0 /\\ Ar_0 + Ar_1 - 1 >= 0 /\\ Ar_0 - 1 >= 0 ]", 0-2) = Ar_0 S("evalspeedpldi2returnin(Ar_0, Ar_1, Ar_2) -> Com_1(evalspeedpldi2stop(Ar_0, Ar_1, Ar_2))", 0-0) = Ar_0 + Ar_1 S("evalspeedpldi2returnin(Ar_0, Ar_1, Ar_2) -> Com_1(evalspeedpldi2stop(Ar_0, Ar_1, Ar_2))", 0-1) = Ar_1 S("evalspeedpldi2returnin(Ar_0, Ar_1, Ar_2) -> Com_1(evalspeedpldi2stop(Ar_0, Ar_1, Ar_2))", 0-2) = Ar_2 S("koat_start(Ar_0, Ar_1, Ar_2) -> Com_1(evalspeedpldi2start(Ar_0, Ar_1, Ar_2)) [ 0 <= 0 ]", 0-0) = Ar_0 S("koat_start(Ar_0, Ar_1, Ar_2) -> Com_1(evalspeedpldi2start(Ar_0, Ar_1, Ar_2)) [ 0 <= 0 ]", 0-1) = Ar_1 S("koat_start(Ar_0, Ar_1, Ar_2) -> Com_1(evalspeedpldi2start(Ar_0, Ar_1, Ar_2)) [ 0 <= 0 ]", 0-2) = Ar_2 orients the transitions evalspeedpldi2bb5in(Ar_0, Ar_1, Ar_2) -> Com_1(evalspeedpldi2bb2in(Ar_0, Ar_1, Ar_2)) [ Ar_2 >= 0 /\ Ar_1 + Ar_2 >= 0 /\ Ar_0 + Ar_2 - 1 >= 0 /\ Ar_1 >= 0 /\ Ar_0 + Ar_1 - 1 >= 0 /\ Ar_0 - 1 >= 0 /\ Ar_2 >= 1 ] evalspeedpldi2bb3in(Ar_0, Ar_1, Ar_2) -> Com_1(evalspeedpldi2bb5in(Ar_0, Ar_1 + 1, Ar_2 - 1)) [ Ar_2 - 1 >= 0 /\ Ar_1 + Ar_2 - 1 >= 0 /\ Ar_0 + Ar_2 - 2 >= 0 /\ Ar_0 - Ar_1 - 1 >= 0 /\ Ar_1 >= 0 /\ Ar_0 + Ar_1 - 1 >= 0 /\ Ar_0 - 1 >= 0 ] evalspeedpldi2bb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalspeedpldi2bb3in(Ar_0, Ar_1, Ar_2)) [ Ar_2 - 1 >= 0 /\ Ar_1 + Ar_2 - 1 >= 0 /\ Ar_0 + Ar_2 - 2 >= 0 /\ Ar_1 >= 0 /\ Ar_0 + Ar_1 - 1 >= 0 /\ Ar_0 - 1 >= 0 /\ Ar_0 >= Ar_1 + 1 ] weakly and the transitions evalspeedpldi2bb5in(Ar_0, Ar_1, Ar_2) -> Com_1(evalspeedpldi2bb2in(Ar_0, Ar_1, Ar_2)) [ Ar_2 >= 0 /\ Ar_1 + Ar_2 >= 0 /\ Ar_0 + Ar_2 - 1 >= 0 /\ Ar_1 >= 0 /\ Ar_0 + Ar_1 - 1 >= 0 /\ Ar_0 - 1 >= 0 /\ Ar_2 >= 1 ] evalspeedpldi2bb3in(Ar_0, Ar_1, Ar_2) -> Com_1(evalspeedpldi2bb5in(Ar_0, Ar_1 + 1, Ar_2 - 1)) [ Ar_2 - 1 >= 0 /\ Ar_1 + Ar_2 - 1 >= 0 /\ Ar_0 + Ar_2 - 2 >= 0 /\ Ar_0 - Ar_1 - 1 >= 0 /\ Ar_1 >= 0 /\ Ar_0 + Ar_1 - 1 >= 0 /\ Ar_0 - 1 >= 0 ] evalspeedpldi2bb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalspeedpldi2bb3in(Ar_0, Ar_1, Ar_2)) [ Ar_2 - 1 >= 0 /\ Ar_1 + Ar_2 - 1 >= 0 /\ Ar_0 + Ar_2 - 2 >= 0 /\ Ar_1 >= 0 /\ Ar_0 + Ar_1 - 1 >= 0 /\ Ar_0 - 1 >= 0 /\ Ar_0 >= Ar_1 + 1 ] strictly and produces the following problem: 6: T: (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2) -> Com_1(evalspeedpldi2start(Ar_0, Ar_1, Ar_2)) [ 0 <= 0 ] (Comp: 2, Cost: 1) evalspeedpldi2returnin(Ar_0, Ar_1, Ar_2) -> Com_1(evalspeedpldi2stop(Ar_0, Ar_1, Ar_2)) (Comp: 5*Ar_0 + 6*Ar_0^2 + 6*Ar_0*Ar_1 + 2*Ar_1 + 1, Cost: 1) evalspeedpldi2bb3in(Ar_0, Ar_1, Ar_2) -> Com_1(evalspeedpldi2bb5in(Ar_0, Ar_1 + 1, Ar_2 - 1)) [ Ar_2 - 1 >= 0 /\ Ar_1 + Ar_2 - 1 >= 0 /\ Ar_0 + Ar_2 - 2 >= 0 /\ Ar_0 - Ar_1 - 1 >= 0 /\ Ar_1 >= 0 /\ Ar_0 + Ar_1 - 1 >= 0 /\ Ar_0 - 1 >= 0 ] (Comp: 2*Ar_0 + 2*Ar_1, Cost: 1) evalspeedpldi2bb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalspeedpldi2bb5in(Ar_0, 0, Ar_2)) [ Ar_2 - 1 >= 0 /\ Ar_1 + Ar_2 - 1 >= 0 /\ Ar_0 + Ar_2 - 2 >= 0 /\ Ar_1 >= 0 /\ Ar_0 + Ar_1 - 1 >= 0 /\ Ar_0 - 1 >= 0 /\ Ar_1 >= Ar_0 ] (Comp: 5*Ar_0 + 6*Ar_0^2 + 6*Ar_0*Ar_1 + 2*Ar_1 + 1, Cost: 1) evalspeedpldi2bb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalspeedpldi2bb3in(Ar_0, Ar_1, Ar_2)) [ Ar_2 - 1 >= 0 /\ Ar_1 + Ar_2 - 1 >= 0 /\ Ar_0 + Ar_2 - 2 >= 0 /\ Ar_1 >= 0 /\ Ar_0 + Ar_1 - 1 >= 0 /\ Ar_0 - 1 >= 0 /\ Ar_0 >= Ar_1 + 1 ] (Comp: 2, Cost: 1) evalspeedpldi2bb5in(Ar_0, Ar_1, Ar_2) -> Com_1(evalspeedpldi2returnin(Ar_0, Ar_1, Ar_2)) [ Ar_2 >= 0 /\ Ar_1 + Ar_2 >= 0 /\ Ar_0 + Ar_2 - 1 >= 0 /\ Ar_1 >= 0 /\ Ar_0 + Ar_1 - 1 >= 0 /\ Ar_0 - 1 >= 0 /\ 0 >= Ar_2 ] (Comp: 5*Ar_0 + 6*Ar_0^2 + 6*Ar_0*Ar_1 + 2*Ar_1 + 1, Cost: 1) evalspeedpldi2bb5in(Ar_0, Ar_1, Ar_2) -> Com_1(evalspeedpldi2bb2in(Ar_0, Ar_1, Ar_2)) [ Ar_2 >= 0 /\ Ar_1 + Ar_2 >= 0 /\ Ar_0 + Ar_2 - 1 >= 0 /\ Ar_1 >= 0 /\ Ar_0 + Ar_1 - 1 >= 0 /\ Ar_0 - 1 >= 0 /\ Ar_2 >= 1 ] (Comp: 1, Cost: 1) evalspeedpldi2entryin(Ar_0, Ar_1, Ar_2) -> Com_1(evalspeedpldi2returnin(Ar_0, Ar_1, Ar_2)) [ 0 >= Ar_1 ] (Comp: 1, Cost: 1) evalspeedpldi2entryin(Ar_0, Ar_1, Ar_2) -> Com_1(evalspeedpldi2returnin(Ar_0, Ar_1, Ar_2)) [ 0 >= Ar_0 + 1 ] (Comp: 1, Cost: 1) evalspeedpldi2entryin(Ar_0, Ar_1, Ar_2) -> Com_1(evalspeedpldi2bb5in(Ar_1, 0, Ar_0)) [ Ar_0 >= 0 /\ Ar_1 >= 1 ] (Comp: 1, Cost: 1) evalspeedpldi2start(Ar_0, Ar_1, Ar_2) -> Com_1(evalspeedpldi2entryin(Ar_0, Ar_1, Ar_2)) start location: koat_start leaf cost: 0 Complexity upper bound 17*Ar_0 + 18*Ar_0^2 + 18*Ar_0*Ar_1 + 8*Ar_1 + 11 Time: 0.195 sec (SMT: 0.163 sec) ---------------------------------------- (2) BOUNDS(1, n^2)