/export/starexec/sandbox/solver/bin/starexec_run_complexity /export/starexec/sandbox/benchmark/theBenchmark.koat /export/starexec/sandbox/output/output_files -------------------------------------------------------------------------------- WORST_CASE(Omega(n^2), O(n^2)) proof of /export/starexec/sandbox/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(n^2, n^2). (0) CpxIntTrs (1) Koat Proof [FINISHED, 135 ms] (2) BOUNDS(1, n^2) (3) Loat Proof [FINISHED, 933 ms] (4) BOUNDS(n^2, INF) ---------------------------------------- (0) Obligation: Complexity Int TRS consisting of the following rules: evalinsertsortstart(A, B, C, D) -> Com_1(evalinsertsortentryin(A, B, C, D)) :|: TRUE evalinsertsortentryin(A, B, C, D) -> Com_1(evalinsertsortbb5in(1, B, C, D)) :|: TRUE evalinsertsortbb5in(A, B, C, D) -> Com_1(evalinsertsortbbin(A, B, C, D)) :|: B >= A + 1 evalinsertsortbb5in(A, B, C, D) -> Com_1(evalinsertsortreturnin(A, B, C, D)) :|: A >= B evalinsertsortbbin(A, B, C, D) -> Com_1(evalinsertsortbb2in(A, B, E, A - 1)) :|: TRUE evalinsertsortbb2in(A, B, C, D) -> Com_1(evalinsertsortbb4in(A, B, C, D)) :|: 0 >= D + 1 evalinsertsortbb2in(A, B, C, D) -> Com_1(evalinsertsortbb3in(A, B, C, D)) :|: D >= 0 evalinsertsortbb3in(A, B, C, D) -> Com_1(evalinsertsortbb1in(A, B, C, D)) :|: E >= C + 1 evalinsertsortbb3in(A, B, C, D) -> Com_1(evalinsertsortbb4in(A, B, C, D)) :|: C >= E evalinsertsortbb1in(A, B, C, D) -> Com_1(evalinsertsortbb2in(A, B, C, D - 1)) :|: TRUE evalinsertsortbb4in(A, B, C, D) -> Com_1(evalinsertsortbb5in(A + 1, B, C, D)) :|: TRUE evalinsertsortreturnin(A, B, C, D) -> Com_1(evalinsertsortstop(A, B, C, D)) :|: TRUE The start-symbols are:[evalinsertsortstart_4] ---------------------------------------- (1) Koat Proof (FINISHED) YES(?, 41*Ar_1 + 6*Ar_1^2 + 6) Initial complexity problem: 1: T: (Comp: ?, Cost: 1) evalinsertsortstart(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalinsertsortentryin(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: ?, Cost: 1) evalinsertsortentryin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalinsertsortbb5in(1, Ar_1, Ar_2, Ar_3)) (Comp: ?, Cost: 1) evalinsertsortbb5in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalinsertsortbbin(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_1 >= Ar_0 + 1 ] (Comp: ?, Cost: 1) evalinsertsortbb5in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalinsertsortreturnin(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= Ar_1 ] (Comp: ?, Cost: 1) evalinsertsortbbin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalinsertsortbb2in(Ar_0, Ar_1, Fresh_0, Ar_0 - 1)) (Comp: ?, Cost: 1) evalinsertsortbb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalinsertsortbb4in(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 >= Ar_3 + 1 ] (Comp: ?, Cost: 1) evalinsertsortbb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalinsertsortbb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_3 >= 0 ] (Comp: ?, Cost: 1) evalinsertsortbb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalinsertsortbb1in(Ar_0, Ar_1, Ar_2, Ar_3)) [ E >= Ar_2 + 1 ] (Comp: ?, Cost: 1) evalinsertsortbb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalinsertsortbb4in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= E ] (Comp: ?, Cost: 1) evalinsertsortbb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalinsertsortbb2in(Ar_0, Ar_1, Ar_2, Ar_3 - 1)) (Comp: ?, Cost: 1) evalinsertsortbb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalinsertsortbb5in(Ar_0 + 1, Ar_1, Ar_2, Ar_3)) (Comp: ?, Cost: 1) evalinsertsortreturnin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalinsertsortstop(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalinsertsortstart(Ar_0, Ar_1, Ar_2, Ar_3)) [ 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) evalinsertsortstart(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalinsertsortentryin(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 1) evalinsertsortentryin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalinsertsortbb5in(1, Ar_1, Ar_2, Ar_3)) (Comp: ?, Cost: 1) evalinsertsortbb5in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalinsertsortbbin(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_1 >= Ar_0 + 1 ] (Comp: ?, Cost: 1) evalinsertsortbb5in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalinsertsortreturnin(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= Ar_1 ] (Comp: ?, Cost: 1) evalinsertsortbbin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalinsertsortbb2in(Ar_0, Ar_1, Fresh_0, Ar_0 - 1)) (Comp: ?, Cost: 1) evalinsertsortbb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalinsertsortbb4in(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 >= Ar_3 + 1 ] (Comp: ?, Cost: 1) evalinsertsortbb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalinsertsortbb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_3 >= 0 ] (Comp: ?, Cost: 1) evalinsertsortbb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalinsertsortbb1in(Ar_0, Ar_1, Ar_2, Ar_3)) [ E >= Ar_2 + 1 ] (Comp: ?, Cost: 1) evalinsertsortbb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalinsertsortbb4in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= E ] (Comp: ?, Cost: 1) evalinsertsortbb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalinsertsortbb2in(Ar_0, Ar_1, Ar_2, Ar_3 - 1)) (Comp: ?, Cost: 1) evalinsertsortbb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalinsertsortbb5in(Ar_0 + 1, Ar_1, Ar_2, Ar_3)) (Comp: ?, Cost: 1) evalinsertsortreturnin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalinsertsortstop(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalinsertsortstart(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 A polynomial rank function with Pol(evalinsertsortstart) = 2 Pol(evalinsertsortentryin) = 2 Pol(evalinsertsortbb5in) = 2 Pol(evalinsertsortbbin) = 2 Pol(evalinsertsortreturnin) = 1 Pol(evalinsertsortbb2in) = 2 Pol(evalinsertsortbb4in) = 2 Pol(evalinsertsortbb3in) = 2 Pol(evalinsertsortbb1in) = 2 Pol(evalinsertsortstop) = 0 Pol(koat_start) = 2 orients all transitions weakly and the transitions evalinsertsortreturnin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalinsertsortstop(Ar_0, Ar_1, Ar_2, Ar_3)) evalinsertsortbb5in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalinsertsortreturnin(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= Ar_1 ] strictly and produces the following problem: 3: T: (Comp: 1, Cost: 1) evalinsertsortstart(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalinsertsortentryin(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 1) evalinsertsortentryin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalinsertsortbb5in(1, Ar_1, Ar_2, Ar_3)) (Comp: ?, Cost: 1) evalinsertsortbb5in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalinsertsortbbin(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_1 >= Ar_0 + 1 ] (Comp: 2, Cost: 1) evalinsertsortbb5in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalinsertsortreturnin(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= Ar_1 ] (Comp: ?, Cost: 1) evalinsertsortbbin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalinsertsortbb2in(Ar_0, Ar_1, Fresh_0, Ar_0 - 1)) (Comp: ?, Cost: 1) evalinsertsortbb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalinsertsortbb4in(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 >= Ar_3 + 1 ] (Comp: ?, Cost: 1) evalinsertsortbb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalinsertsortbb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_3 >= 0 ] (Comp: ?, Cost: 1) evalinsertsortbb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalinsertsortbb1in(Ar_0, Ar_1, Ar_2, Ar_3)) [ E >= Ar_2 + 1 ] (Comp: ?, Cost: 1) evalinsertsortbb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalinsertsortbb4in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= E ] (Comp: ?, Cost: 1) evalinsertsortbb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalinsertsortbb2in(Ar_0, Ar_1, Ar_2, Ar_3 - 1)) (Comp: ?, Cost: 1) evalinsertsortbb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalinsertsortbb5in(Ar_0 + 1, Ar_1, Ar_2, Ar_3)) (Comp: 2, Cost: 1) evalinsertsortreturnin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalinsertsortstop(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalinsertsortstart(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 A polynomial rank function with Pol(evalinsertsortstart) = V_2 Pol(evalinsertsortentryin) = V_2 Pol(evalinsertsortbb5in) = -V_1 + V_2 + 1 Pol(evalinsertsortbbin) = -V_1 + V_2 Pol(evalinsertsortreturnin) = -V_1 + V_2 Pol(evalinsertsortbb2in) = -V_1 + V_2 Pol(evalinsertsortbb4in) = -V_1 + V_2 Pol(evalinsertsortbb3in) = -V_1 + V_2 Pol(evalinsertsortbb1in) = -V_1 + V_2 Pol(evalinsertsortstop) = -V_1 + V_2 Pol(koat_start) = V_2 orients all transitions weakly and the transition evalinsertsortbb5in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalinsertsortbbin(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_1 >= Ar_0 + 1 ] strictly and produces the following problem: 4: T: (Comp: 1, Cost: 1) evalinsertsortstart(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalinsertsortentryin(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 1) evalinsertsortentryin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalinsertsortbb5in(1, Ar_1, Ar_2, Ar_3)) (Comp: Ar_1, Cost: 1) evalinsertsortbb5in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalinsertsortbbin(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_1 >= Ar_0 + 1 ] (Comp: 2, Cost: 1) evalinsertsortbb5in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalinsertsortreturnin(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= Ar_1 ] (Comp: ?, Cost: 1) evalinsertsortbbin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalinsertsortbb2in(Ar_0, Ar_1, Fresh_0, Ar_0 - 1)) (Comp: ?, Cost: 1) evalinsertsortbb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalinsertsortbb4in(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 >= Ar_3 + 1 ] (Comp: ?, Cost: 1) evalinsertsortbb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalinsertsortbb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_3 >= 0 ] (Comp: ?, Cost: 1) evalinsertsortbb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalinsertsortbb1in(Ar_0, Ar_1, Ar_2, Ar_3)) [ E >= Ar_2 + 1 ] (Comp: ?, Cost: 1) evalinsertsortbb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalinsertsortbb4in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= E ] (Comp: ?, Cost: 1) evalinsertsortbb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalinsertsortbb2in(Ar_0, Ar_1, Ar_2, Ar_3 - 1)) (Comp: ?, Cost: 1) evalinsertsortbb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalinsertsortbb5in(Ar_0 + 1, Ar_1, Ar_2, Ar_3)) (Comp: 2, Cost: 1) evalinsertsortreturnin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalinsertsortstop(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalinsertsortstart(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 Repeatedly propagating knowledge in problem 4 produces the following problem: 5: T: (Comp: 1, Cost: 1) evalinsertsortstart(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalinsertsortentryin(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 1) evalinsertsortentryin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalinsertsortbb5in(1, Ar_1, Ar_2, Ar_3)) (Comp: Ar_1, Cost: 1) evalinsertsortbb5in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalinsertsortbbin(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_1 >= Ar_0 + 1 ] (Comp: 2, Cost: 1) evalinsertsortbb5in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalinsertsortreturnin(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= Ar_1 ] (Comp: Ar_1, Cost: 1) evalinsertsortbbin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalinsertsortbb2in(Ar_0, Ar_1, Fresh_0, Ar_0 - 1)) (Comp: ?, Cost: 1) evalinsertsortbb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalinsertsortbb4in(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 >= Ar_3 + 1 ] (Comp: ?, Cost: 1) evalinsertsortbb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalinsertsortbb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_3 >= 0 ] (Comp: ?, Cost: 1) evalinsertsortbb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalinsertsortbb1in(Ar_0, Ar_1, Ar_2, Ar_3)) [ E >= Ar_2 + 1 ] (Comp: ?, Cost: 1) evalinsertsortbb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalinsertsortbb4in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= E ] (Comp: ?, Cost: 1) evalinsertsortbb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalinsertsortbb2in(Ar_0, Ar_1, Ar_2, Ar_3 - 1)) (Comp: ?, Cost: 1) evalinsertsortbb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalinsertsortbb5in(Ar_0 + 1, Ar_1, Ar_2, Ar_3)) (Comp: 2, Cost: 1) evalinsertsortreturnin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalinsertsortstop(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalinsertsortstart(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 A polynomial rank function with Pol(evalinsertsortbb4in) = 1 Pol(evalinsertsortbb5in) = 0 Pol(evalinsertsortbb3in) = 2 Pol(evalinsertsortbb1in) = 2 Pol(evalinsertsortbb2in) = 2 and size complexities S("koat_start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalinsertsortstart(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(evalinsertsortstart(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(evalinsertsortstart(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(evalinsertsortstart(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 <= 0 ]", 0-3) = Ar_3 S("evalinsertsortreturnin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalinsertsortstop(Ar_0, Ar_1, Ar_2, Ar_3))", 0-0) = ? S("evalinsertsortreturnin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalinsertsortstop(Ar_0, Ar_1, Ar_2, Ar_3))", 0-1) = Ar_1 S("evalinsertsortreturnin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalinsertsortstop(Ar_0, Ar_1, Ar_2, Ar_3))", 0-2) = ? S("evalinsertsortreturnin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalinsertsortstop(Ar_0, Ar_1, Ar_2, Ar_3))", 0-3) = ? S("evalinsertsortbb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalinsertsortbb5in(Ar_0 + 1, Ar_1, Ar_2, Ar_3))", 0-0) = ? S("evalinsertsortbb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalinsertsortbb5in(Ar_0 + 1, Ar_1, Ar_2, Ar_3))", 0-1) = Ar_1 S("evalinsertsortbb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalinsertsortbb5in(Ar_0 + 1, Ar_1, Ar_2, Ar_3))", 0-2) = ? S("evalinsertsortbb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalinsertsortbb5in(Ar_0 + 1, Ar_1, Ar_2, Ar_3))", 0-3) = ? S("evalinsertsortbb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalinsertsortbb2in(Ar_0, Ar_1, Ar_2, Ar_3 - 1))", 0-0) = ? S("evalinsertsortbb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalinsertsortbb2in(Ar_0, Ar_1, Ar_2, Ar_3 - 1))", 0-1) = Ar_1 S("evalinsertsortbb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalinsertsortbb2in(Ar_0, Ar_1, Ar_2, Ar_3 - 1))", 0-2) = ? S("evalinsertsortbb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalinsertsortbb2in(Ar_0, Ar_1, Ar_2, Ar_3 - 1))", 0-3) = ? S("evalinsertsortbb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalinsertsortbb4in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= E ]", 0-0) = ? S("evalinsertsortbb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalinsertsortbb4in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= E ]", 0-1) = Ar_1 S("evalinsertsortbb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalinsertsortbb4in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= E ]", 0-2) = ? S("evalinsertsortbb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalinsertsortbb4in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= E ]", 0-3) = ? S("evalinsertsortbb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalinsertsortbb1in(Ar_0, Ar_1, Ar_2, Ar_3)) [ E >= Ar_2 + 1 ]", 0-0) = ? S("evalinsertsortbb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalinsertsortbb1in(Ar_0, Ar_1, Ar_2, Ar_3)) [ E >= Ar_2 + 1 ]", 0-1) = Ar_1 S("evalinsertsortbb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalinsertsortbb1in(Ar_0, Ar_1, Ar_2, Ar_3)) [ E >= Ar_2 + 1 ]", 0-2) = ? S("evalinsertsortbb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalinsertsortbb1in(Ar_0, Ar_1, Ar_2, Ar_3)) [ E >= Ar_2 + 1 ]", 0-3) = ? S("evalinsertsortbb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalinsertsortbb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_3 >= 0 ]", 0-0) = ? S("evalinsertsortbb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalinsertsortbb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_3 >= 0 ]", 0-1) = Ar_1 S("evalinsertsortbb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalinsertsortbb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_3 >= 0 ]", 0-2) = ? S("evalinsertsortbb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalinsertsortbb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_3 >= 0 ]", 0-3) = ? S("evalinsertsortbb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalinsertsortbb4in(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 >= Ar_3 + 1 ]", 0-0) = ? S("evalinsertsortbb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalinsertsortbb4in(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 >= Ar_3 + 1 ]", 0-1) = Ar_1 S("evalinsertsortbb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalinsertsortbb4in(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 >= Ar_3 + 1 ]", 0-2) = ? S("evalinsertsortbb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalinsertsortbb4in(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 >= Ar_3 + 1 ]", 0-3) = ? S("evalinsertsortbbin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalinsertsortbb2in(Ar_0, Ar_1, Fresh_0, Ar_0 - 1))", 0-0) = ? S("evalinsertsortbbin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalinsertsortbb2in(Ar_0, Ar_1, Fresh_0, Ar_0 - 1))", 0-1) = Ar_1 S("evalinsertsortbbin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalinsertsortbb2in(Ar_0, Ar_1, Fresh_0, Ar_0 - 1))", 0-2) = ? S("evalinsertsortbbin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalinsertsortbb2in(Ar_0, Ar_1, Fresh_0, Ar_0 - 1))", 0-3) = ? S("evalinsertsortbb5in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalinsertsortreturnin(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= Ar_1 ]", 0-0) = ? S("evalinsertsortbb5in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalinsertsortreturnin(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= Ar_1 ]", 0-1) = Ar_1 S("evalinsertsortbb5in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalinsertsortreturnin(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= Ar_1 ]", 0-2) = ? S("evalinsertsortbb5in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalinsertsortreturnin(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= Ar_1 ]", 0-3) = ? S("evalinsertsortbb5in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalinsertsortbbin(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_1 >= Ar_0 + 1 ]", 0-0) = ? S("evalinsertsortbb5in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalinsertsortbbin(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_1 >= Ar_0 + 1 ]", 0-1) = Ar_1 S("evalinsertsortbb5in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalinsertsortbbin(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_1 >= Ar_0 + 1 ]", 0-2) = ? S("evalinsertsortbb5in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalinsertsortbbin(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_1 >= Ar_0 + 1 ]", 0-3) = ? S("evalinsertsortentryin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalinsertsortbb5in(1, Ar_1, Ar_2, Ar_3))", 0-0) = 1 S("evalinsertsortentryin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalinsertsortbb5in(1, Ar_1, Ar_2, Ar_3))", 0-1) = Ar_1 S("evalinsertsortentryin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalinsertsortbb5in(1, Ar_1, Ar_2, Ar_3))", 0-2) = Ar_2 S("evalinsertsortentryin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalinsertsortbb5in(1, Ar_1, Ar_2, Ar_3))", 0-3) = Ar_3 S("evalinsertsortstart(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalinsertsortentryin(Ar_0, Ar_1, Ar_2, Ar_3))", 0-0) = Ar_0 S("evalinsertsortstart(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalinsertsortentryin(Ar_0, Ar_1, Ar_2, Ar_3))", 0-1) = Ar_1 S("evalinsertsortstart(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalinsertsortentryin(Ar_0, Ar_1, Ar_2, Ar_3))", 0-2) = Ar_2 S("evalinsertsortstart(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalinsertsortentryin(Ar_0, Ar_1, Ar_2, Ar_3))", 0-3) = Ar_3 orients the transitions evalinsertsortbb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalinsertsortbb5in(Ar_0 + 1, Ar_1, Ar_2, Ar_3)) evalinsertsortbb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalinsertsortbb4in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= E ] evalinsertsortbb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalinsertsortbb1in(Ar_0, Ar_1, Ar_2, Ar_3)) [ E >= Ar_2 + 1 ] evalinsertsortbb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalinsertsortbb4in(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 >= Ar_3 + 1 ] evalinsertsortbb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalinsertsortbb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_3 >= 0 ] evalinsertsortbb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalinsertsortbb2in(Ar_0, Ar_1, Ar_2, Ar_3 - 1)) weakly and the transitions evalinsertsortbb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalinsertsortbb5in(Ar_0 + 1, Ar_1, Ar_2, Ar_3)) evalinsertsortbb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalinsertsortbb4in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= E ] evalinsertsortbb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalinsertsortbb4in(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 >= Ar_3 + 1 ] strictly and produces the following problem: 6: T: (Comp: 1, Cost: 1) evalinsertsortstart(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalinsertsortentryin(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 1) evalinsertsortentryin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalinsertsortbb5in(1, Ar_1, Ar_2, Ar_3)) (Comp: Ar_1, Cost: 1) evalinsertsortbb5in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalinsertsortbbin(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_1 >= Ar_0 + 1 ] (Comp: 2, Cost: 1) evalinsertsortbb5in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalinsertsortreturnin(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= Ar_1 ] (Comp: Ar_1, Cost: 1) evalinsertsortbbin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalinsertsortbb2in(Ar_0, Ar_1, Fresh_0, Ar_0 - 1)) (Comp: 2*Ar_1, Cost: 1) evalinsertsortbb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalinsertsortbb4in(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 >= Ar_3 + 1 ] (Comp: ?, Cost: 1) evalinsertsortbb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalinsertsortbb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_3 >= 0 ] (Comp: ?, Cost: 1) evalinsertsortbb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalinsertsortbb1in(Ar_0, Ar_1, Ar_2, Ar_3)) [ E >= Ar_2 + 1 ] (Comp: 2*Ar_1, Cost: 1) evalinsertsortbb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalinsertsortbb4in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= E ] (Comp: ?, Cost: 1) evalinsertsortbb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalinsertsortbb2in(Ar_0, Ar_1, Ar_2, Ar_3 - 1)) (Comp: 2*Ar_1, Cost: 1) evalinsertsortbb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalinsertsortbb5in(Ar_0 + 1, Ar_1, Ar_2, Ar_3)) (Comp: 2, Cost: 1) evalinsertsortreturnin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalinsertsortstop(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalinsertsortstart(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 A polynomial rank function with Pol(evalinsertsortbb3in) = V_4 Pol(evalinsertsortbb1in) = V_4 Pol(evalinsertsortbb2in) = V_4 + 1 and size complexities S("koat_start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalinsertsortstart(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(evalinsertsortstart(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(evalinsertsortstart(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(evalinsertsortstart(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 <= 0 ]", 0-3) = Ar_3 S("evalinsertsortreturnin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalinsertsortstop(Ar_0, Ar_1, Ar_2, Ar_3))", 0-0) = 2*Ar_1 + 20 S("evalinsertsortreturnin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalinsertsortstop(Ar_0, Ar_1, Ar_2, Ar_3))", 0-1) = Ar_1 S("evalinsertsortreturnin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalinsertsortstop(Ar_0, Ar_1, Ar_2, Ar_3))", 0-2) = ? S("evalinsertsortreturnin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalinsertsortstop(Ar_0, Ar_1, Ar_2, Ar_3))", 0-3) = ? S("evalinsertsortbb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalinsertsortbb5in(Ar_0 + 1, Ar_1, Ar_2, Ar_3))", 0-0) = 2*Ar_1 + 4 S("evalinsertsortbb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalinsertsortbb5in(Ar_0 + 1, Ar_1, Ar_2, Ar_3))", 0-1) = Ar_1 S("evalinsertsortbb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalinsertsortbb5in(Ar_0 + 1, Ar_1, Ar_2, Ar_3))", 0-2) = ? S("evalinsertsortbb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalinsertsortbb5in(Ar_0 + 1, Ar_1, Ar_2, Ar_3))", 0-3) = ? S("evalinsertsortbb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalinsertsortbb2in(Ar_0, Ar_1, Ar_2, Ar_3 - 1))", 0-0) = 2*Ar_1 + 4 S("evalinsertsortbb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalinsertsortbb2in(Ar_0, Ar_1, Ar_2, Ar_3 - 1))", 0-1) = Ar_1 S("evalinsertsortbb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalinsertsortbb2in(Ar_0, Ar_1, Ar_2, Ar_3 - 1))", 0-2) = ? S("evalinsertsortbb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalinsertsortbb2in(Ar_0, Ar_1, Ar_2, Ar_3 - 1))", 0-3) = ? S("evalinsertsortbb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalinsertsortbb4in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= E ]", 0-0) = 2*Ar_1 + 4 S("evalinsertsortbb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalinsertsortbb4in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= E ]", 0-1) = Ar_1 S("evalinsertsortbb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalinsertsortbb4in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= E ]", 0-2) = ? S("evalinsertsortbb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalinsertsortbb4in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= E ]", 0-3) = ? S("evalinsertsortbb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalinsertsortbb1in(Ar_0, Ar_1, Ar_2, Ar_3)) [ E >= Ar_2 + 1 ]", 0-0) = 2*Ar_1 + 4 S("evalinsertsortbb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalinsertsortbb1in(Ar_0, Ar_1, Ar_2, Ar_3)) [ E >= Ar_2 + 1 ]", 0-1) = Ar_1 S("evalinsertsortbb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalinsertsortbb1in(Ar_0, Ar_1, Ar_2, Ar_3)) [ E >= Ar_2 + 1 ]", 0-2) = ? S("evalinsertsortbb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalinsertsortbb1in(Ar_0, Ar_1, Ar_2, Ar_3)) [ E >= Ar_2 + 1 ]", 0-3) = ? S("evalinsertsortbb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalinsertsortbb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_3 >= 0 ]", 0-0) = 2*Ar_1 + 4 S("evalinsertsortbb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalinsertsortbb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_3 >= 0 ]", 0-1) = Ar_1 S("evalinsertsortbb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalinsertsortbb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_3 >= 0 ]", 0-2) = ? S("evalinsertsortbb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalinsertsortbb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_3 >= 0 ]", 0-3) = ? S("evalinsertsortbb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalinsertsortbb4in(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 >= Ar_3 + 1 ]", 0-0) = 2*Ar_1 + 4 S("evalinsertsortbb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalinsertsortbb4in(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 >= Ar_3 + 1 ]", 0-1) = Ar_1 S("evalinsertsortbb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalinsertsortbb4in(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 >= Ar_3 + 1 ]", 0-2) = ? S("evalinsertsortbb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalinsertsortbb4in(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 >= Ar_3 + 1 ]", 0-3) = ? S("evalinsertsortbbin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalinsertsortbb2in(Ar_0, Ar_1, Fresh_0, Ar_0 - 1))", 0-0) = 2*Ar_1 + 4 S("evalinsertsortbbin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalinsertsortbb2in(Ar_0, Ar_1, Fresh_0, Ar_0 - 1))", 0-1) = Ar_1 S("evalinsertsortbbin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalinsertsortbb2in(Ar_0, Ar_1, Fresh_0, Ar_0 - 1))", 0-2) = ? S("evalinsertsortbbin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalinsertsortbb2in(Ar_0, Ar_1, Fresh_0, Ar_0 - 1))", 0-3) = 2*Ar_1 + 10 S("evalinsertsortbb5in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalinsertsortreturnin(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= Ar_1 ]", 0-0) = 2*Ar_1 + 10 S("evalinsertsortbb5in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalinsertsortreturnin(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= Ar_1 ]", 0-1) = Ar_1 S("evalinsertsortbb5in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalinsertsortreturnin(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= Ar_1 ]", 0-2) = ? S("evalinsertsortbb5in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalinsertsortreturnin(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= Ar_1 ]", 0-3) = ? S("evalinsertsortbb5in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalinsertsortbbin(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_1 >= Ar_0 + 1 ]", 0-0) = 2*Ar_1 + 4 S("evalinsertsortbb5in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalinsertsortbbin(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_1 >= Ar_0 + 1 ]", 0-1) = Ar_1 S("evalinsertsortbb5in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalinsertsortbbin(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_1 >= Ar_0 + 1 ]", 0-2) = ? S("evalinsertsortbb5in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalinsertsortbbin(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_1 >= Ar_0 + 1 ]", 0-3) = ? S("evalinsertsortentryin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalinsertsortbb5in(1, Ar_1, Ar_2, Ar_3))", 0-0) = 1 S("evalinsertsortentryin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalinsertsortbb5in(1, Ar_1, Ar_2, Ar_3))", 0-1) = Ar_1 S("evalinsertsortentryin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalinsertsortbb5in(1, Ar_1, Ar_2, Ar_3))", 0-2) = Ar_2 S("evalinsertsortentryin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalinsertsortbb5in(1, Ar_1, Ar_2, Ar_3))", 0-3) = Ar_3 S("evalinsertsortstart(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalinsertsortentryin(Ar_0, Ar_1, Ar_2, Ar_3))", 0-0) = Ar_0 S("evalinsertsortstart(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalinsertsortentryin(Ar_0, Ar_1, Ar_2, Ar_3))", 0-1) = Ar_1 S("evalinsertsortstart(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalinsertsortentryin(Ar_0, Ar_1, Ar_2, Ar_3))", 0-2) = Ar_2 S("evalinsertsortstart(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalinsertsortentryin(Ar_0, Ar_1, Ar_2, Ar_3))", 0-3) = Ar_3 orients the transitions evalinsertsortbb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalinsertsortbb1in(Ar_0, Ar_1, Ar_2, Ar_3)) [ E >= Ar_2 + 1 ] evalinsertsortbb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalinsertsortbb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_3 >= 0 ] evalinsertsortbb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalinsertsortbb2in(Ar_0, Ar_1, Ar_2, Ar_3 - 1)) weakly and the transition evalinsertsortbb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalinsertsortbb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_3 >= 0 ] strictly and produces the following problem: 7: T: (Comp: 1, Cost: 1) evalinsertsortstart(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalinsertsortentryin(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 1) evalinsertsortentryin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalinsertsortbb5in(1, Ar_1, Ar_2, Ar_3)) (Comp: Ar_1, Cost: 1) evalinsertsortbb5in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalinsertsortbbin(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_1 >= Ar_0 + 1 ] (Comp: 2, Cost: 1) evalinsertsortbb5in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalinsertsortreturnin(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= Ar_1 ] (Comp: Ar_1, Cost: 1) evalinsertsortbbin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalinsertsortbb2in(Ar_0, Ar_1, Fresh_0, Ar_0 - 1)) (Comp: 2*Ar_1, Cost: 1) evalinsertsortbb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalinsertsortbb4in(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 >= Ar_3 + 1 ] (Comp: 2*Ar_1^2 + 11*Ar_1, Cost: 1) evalinsertsortbb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalinsertsortbb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_3 >= 0 ] (Comp: ?, Cost: 1) evalinsertsortbb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalinsertsortbb1in(Ar_0, Ar_1, Ar_2, Ar_3)) [ E >= Ar_2 + 1 ] (Comp: 2*Ar_1, Cost: 1) evalinsertsortbb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalinsertsortbb4in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= E ] (Comp: ?, Cost: 1) evalinsertsortbb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalinsertsortbb2in(Ar_0, Ar_1, Ar_2, Ar_3 - 1)) (Comp: 2*Ar_1, Cost: 1) evalinsertsortbb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalinsertsortbb5in(Ar_0 + 1, Ar_1, Ar_2, Ar_3)) (Comp: 2, Cost: 1) evalinsertsortreturnin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalinsertsortstop(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalinsertsortstart(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 Repeatedly propagating knowledge in problem 7 produces the following problem: 8: T: (Comp: 1, Cost: 1) evalinsertsortstart(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalinsertsortentryin(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 1) evalinsertsortentryin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalinsertsortbb5in(1, Ar_1, Ar_2, Ar_3)) (Comp: Ar_1, Cost: 1) evalinsertsortbb5in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalinsertsortbbin(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_1 >= Ar_0 + 1 ] (Comp: 2, Cost: 1) evalinsertsortbb5in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalinsertsortreturnin(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= Ar_1 ] (Comp: Ar_1, Cost: 1) evalinsertsortbbin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalinsertsortbb2in(Ar_0, Ar_1, Fresh_0, Ar_0 - 1)) (Comp: 2*Ar_1, Cost: 1) evalinsertsortbb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalinsertsortbb4in(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 >= Ar_3 + 1 ] (Comp: 2*Ar_1^2 + 11*Ar_1, Cost: 1) evalinsertsortbb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalinsertsortbb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_3 >= 0 ] (Comp: 2*Ar_1^2 + 11*Ar_1, Cost: 1) evalinsertsortbb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalinsertsortbb1in(Ar_0, Ar_1, Ar_2, Ar_3)) [ E >= Ar_2 + 1 ] (Comp: 2*Ar_1, Cost: 1) evalinsertsortbb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalinsertsortbb4in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= E ] (Comp: 2*Ar_1^2 + 11*Ar_1, Cost: 1) evalinsertsortbb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalinsertsortbb2in(Ar_0, Ar_1, Ar_2, Ar_3 - 1)) (Comp: 2*Ar_1, Cost: 1) evalinsertsortbb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalinsertsortbb5in(Ar_0 + 1, Ar_1, Ar_2, Ar_3)) (Comp: 2, Cost: 1) evalinsertsortreturnin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalinsertsortstop(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalinsertsortstart(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 Complexity upper bound 41*Ar_1 + 6*Ar_1^2 + 6 Time: 0.116 sec (SMT: 0.087 sec) ---------------------------------------- (2) BOUNDS(1, n^2) ---------------------------------------- (3) Loat Proof (FINISHED) ### Pre-processing the ITS problem ### Initial linear ITS problem Start location: evalinsertsortstart 0: evalinsertsortstart -> evalinsertsortentryin : [], cost: 1 1: evalinsertsortentryin -> evalinsertsortbb5in : A'=1, [], cost: 1 2: evalinsertsortbb5in -> evalinsertsortbbin : [ B>=1+A ], cost: 1 3: evalinsertsortbb5in -> evalinsertsortreturnin : [ A>=B ], cost: 1 4: evalinsertsortbbin -> evalinsertsortbb2in : C'=free, D'=-1+A, [], cost: 1 5: evalinsertsortbb2in -> evalinsertsortbb4in : [ 0>=1+D ], cost: 1 6: evalinsertsortbb2in -> evalinsertsortbb3in : [ D>=0 ], cost: 1 7: evalinsertsortbb3in -> evalinsertsortbb1in : [ free_1>=1+C ], cost: 1 8: evalinsertsortbb3in -> evalinsertsortbb4in : [ C>=free_2 ], cost: 1 9: evalinsertsortbb1in -> evalinsertsortbb2in : D'=-1+D, [], cost: 1 10: evalinsertsortbb4in -> evalinsertsortbb5in : A'=1+A, [], cost: 1 11: evalinsertsortreturnin -> evalinsertsortstop : [], cost: 1 Checking for constant complexity: The following rule is satisfiable with cost >= 1, yielding constant complexity: 0: evalinsertsortstart -> evalinsertsortentryin : [], cost: 1 Removed unreachable and leaf rules: Start location: evalinsertsortstart 0: evalinsertsortstart -> evalinsertsortentryin : [], cost: 1 1: evalinsertsortentryin -> evalinsertsortbb5in : A'=1, [], cost: 1 2: evalinsertsortbb5in -> evalinsertsortbbin : [ B>=1+A ], cost: 1 4: evalinsertsortbbin -> evalinsertsortbb2in : C'=free, D'=-1+A, [], cost: 1 5: evalinsertsortbb2in -> evalinsertsortbb4in : [ 0>=1+D ], cost: 1 6: evalinsertsortbb2in -> evalinsertsortbb3in : [ D>=0 ], cost: 1 7: evalinsertsortbb3in -> evalinsertsortbb1in : [ free_1>=1+C ], cost: 1 8: evalinsertsortbb3in -> evalinsertsortbb4in : [ C>=free_2 ], cost: 1 9: evalinsertsortbb1in -> evalinsertsortbb2in : D'=-1+D, [], cost: 1 10: evalinsertsortbb4in -> evalinsertsortbb5in : A'=1+A, [], cost: 1 Simplified all rules, resulting in: Start location: evalinsertsortstart 0: evalinsertsortstart -> evalinsertsortentryin : [], cost: 1 1: evalinsertsortentryin -> evalinsertsortbb5in : A'=1, [], cost: 1 2: evalinsertsortbb5in -> evalinsertsortbbin : [ B>=1+A ], cost: 1 4: evalinsertsortbbin -> evalinsertsortbb2in : C'=free, D'=-1+A, [], cost: 1 5: evalinsertsortbb2in -> evalinsertsortbb4in : [ 0>=1+D ], cost: 1 6: evalinsertsortbb2in -> evalinsertsortbb3in : [ D>=0 ], cost: 1 7: evalinsertsortbb3in -> evalinsertsortbb1in : [], cost: 1 8: evalinsertsortbb3in -> evalinsertsortbb4in : [], cost: 1 9: evalinsertsortbb1in -> evalinsertsortbb2in : D'=-1+D, [], cost: 1 10: evalinsertsortbb4in -> evalinsertsortbb5in : A'=1+A, [], cost: 1 ### Simplification by acceleration and chaining ### Eliminated locations (on linear paths): Start location: evalinsertsortstart 12: evalinsertsortstart -> evalinsertsortbb5in : A'=1, [], cost: 2 13: evalinsertsortbb5in -> evalinsertsortbb2in : C'=free, D'=-1+A, [ B>=1+A ], cost: 2 5: evalinsertsortbb2in -> evalinsertsortbb4in : [ 0>=1+D ], cost: 1 6: evalinsertsortbb2in -> evalinsertsortbb3in : [ D>=0 ], cost: 1 8: evalinsertsortbb3in -> evalinsertsortbb4in : [], cost: 1 14: evalinsertsortbb3in -> evalinsertsortbb2in : D'=-1+D, [], cost: 2 10: evalinsertsortbb4in -> evalinsertsortbb5in : A'=1+A, [], cost: 1 Eliminated locations (on tree-shaped paths): Start location: evalinsertsortstart 12: evalinsertsortstart -> evalinsertsortbb5in : A'=1, [], cost: 2 13: evalinsertsortbb5in -> evalinsertsortbb2in : C'=free, D'=-1+A, [ B>=1+A ], cost: 2 16: evalinsertsortbb2in -> evalinsertsortbb2in : D'=-1+D, [ D>=0 ], cost: 3 17: evalinsertsortbb2in -> evalinsertsortbb5in : A'=1+A, [ 0>=1+D ], cost: 2 18: evalinsertsortbb2in -> evalinsertsortbb5in : A'=1+A, [ D>=0 ], cost: 3 Accelerating simple loops of location 4. Accelerating the following rules: 16: evalinsertsortbb2in -> evalinsertsortbb2in : D'=-1+D, [ D>=0 ], cost: 3 Accelerated rule 16 with metering function 1+D, yielding the new rule 19. Removing the simple loops: 16. Accelerated all simple loops using metering functions (where possible): Start location: evalinsertsortstart 12: evalinsertsortstart -> evalinsertsortbb5in : A'=1, [], cost: 2 13: evalinsertsortbb5in -> evalinsertsortbb2in : C'=free, D'=-1+A, [ B>=1+A ], cost: 2 17: evalinsertsortbb2in -> evalinsertsortbb5in : A'=1+A, [ 0>=1+D ], cost: 2 18: evalinsertsortbb2in -> evalinsertsortbb5in : A'=1+A, [ D>=0 ], cost: 3 19: evalinsertsortbb2in -> evalinsertsortbb2in : D'=-1, [ D>=0 ], cost: 3+3*D Chained accelerated rules (with incoming rules): Start location: evalinsertsortstart 12: evalinsertsortstart -> evalinsertsortbb5in : A'=1, [], cost: 2 13: evalinsertsortbb5in -> evalinsertsortbb2in : C'=free, D'=-1+A, [ B>=1+A ], cost: 2 20: evalinsertsortbb5in -> evalinsertsortbb2in : C'=free, D'=-1, [ B>=1+A && -1+A>=0 ], cost: 2+3*A 17: evalinsertsortbb2in -> evalinsertsortbb5in : A'=1+A, [ 0>=1+D ], cost: 2 18: evalinsertsortbb2in -> evalinsertsortbb5in : A'=1+A, [ D>=0 ], cost: 3 Eliminated locations (on tree-shaped paths): Start location: evalinsertsortstart 12: evalinsertsortstart -> evalinsertsortbb5in : A'=1, [], cost: 2 21: evalinsertsortbb5in -> evalinsertsortbb5in : A'=1+A, C'=free, D'=-1+A, [ B>=1+A && 0>=A ], cost: 4 22: evalinsertsortbb5in -> evalinsertsortbb5in : A'=1+A, C'=free, D'=-1+A, [ B>=1+A && -1+A>=0 ], cost: 5 23: evalinsertsortbb5in -> evalinsertsortbb5in : A'=1+A, C'=free, D'=-1, [ B>=1+A && -1+A>=0 ], cost: 4+3*A 24: evalinsertsortbb5in -> [11] : [ B>=1+A && -1+A>=0 ], cost: 2+3*A Accelerating simple loops of location 2. Accelerating the following rules: 21: evalinsertsortbb5in -> evalinsertsortbb5in : A'=1+A, C'=free, D'=-1+A, [ B>=1+A && 0>=A ], cost: 4 22: evalinsertsortbb5in -> evalinsertsortbb5in : A'=1+A, C'=free, D'=-1+A, [ B>=1+A && -1+A>=0 ], cost: 5 23: evalinsertsortbb5in -> evalinsertsortbb5in : A'=1+A, C'=free, D'=-1, [ B>=1+A && -1+A>=0 ], cost: 4+3*A Found no metering function for rule 21. Accelerated rule 22 with metering function -A+B, yielding the new rule 25. Accelerated rule 23 with metering function -A+B, yielding the new rule 26. Removing the simple loops: 22 23. Accelerated all simple loops using metering functions (where possible): Start location: evalinsertsortstart 12: evalinsertsortstart -> evalinsertsortbb5in : A'=1, [], cost: 2 21: evalinsertsortbb5in -> evalinsertsortbb5in : A'=1+A, C'=free, D'=-1+A, [ B>=1+A && 0>=A ], cost: 4 24: evalinsertsortbb5in -> [11] : [ B>=1+A && -1+A>=0 ], cost: 2+3*A 25: evalinsertsortbb5in -> evalinsertsortbb5in : A'=B, C'=free, D'=-2+B, [ B>=1+A && -1+A>=0 ], cost: -5*A+5*B 26: evalinsertsortbb5in -> evalinsertsortbb5in : A'=B, C'=free, D'=-1, [ B>=1+A && -1+A>=0 ], cost: 3/2*(A-B)^2-5/2*A-3*A*(A-B)+5/2*B Chained accelerated rules (with incoming rules): Start location: evalinsertsortstart 12: evalinsertsortstart -> evalinsertsortbb5in : A'=1, [], cost: 2 27: evalinsertsortstart -> evalinsertsortbb5in : A'=B, C'=free, D'=-2+B, [ B>=2 ], cost: -3+5*B 28: evalinsertsortstart -> evalinsertsortbb5in : A'=B, C'=free, D'=-1, [ B>=2 ], cost: -7/2+11/2*B+3/2*(-1+B)^2 24: evalinsertsortbb5in -> [11] : [ B>=1+A && -1+A>=0 ], cost: 2+3*A Eliminated locations (on tree-shaped paths): Start location: evalinsertsortstart 29: evalinsertsortstart -> [11] : A'=1, [ B>=2 ], cost: 7 30: evalinsertsortstart -> [13] : [ B>=2 ], cost: -3+5*B 31: evalinsertsortstart -> [13] : [ B>=2 ], cost: -7/2+11/2*B+3/2*(-1+B)^2 Applied pruning (of leafs and parallel rules): Start location: evalinsertsortstart 30: evalinsertsortstart -> [13] : [ B>=2 ], cost: -3+5*B 31: evalinsertsortstart -> [13] : [ B>=2 ], cost: -7/2+11/2*B+3/2*(-1+B)^2 ### Computing asymptotic complexity ### Fully simplified ITS problem Start location: evalinsertsortstart 30: evalinsertsortstart -> [13] : [ B>=2 ], cost: -3+5*B 31: evalinsertsortstart -> [13] : [ B>=2 ], cost: -7/2+11/2*B+3/2*(-1+B)^2 Computing asymptotic complexity for rule 30 Solved the limit problem by the following transformations: Created initial limit problem: -3+5*B (+), -1+B (+/+!) [not solved] removing all constraints (solved by SMT) resulting limit problem: [solved] applying transformation rule (C) using substitution {B==n} resulting limit problem: [solved] Solution: B / n Resulting cost -3+5*n has complexity: Poly(n^1) Found new complexity Poly(n^1). Computing asymptotic complexity for rule 31 Solved the limit problem by the following transformations: Created initial limit problem: -1+B (+/+!), -2+5/2*B+3/2*B^2 (+) [not solved] removing all constraints (solved by SMT) resulting limit problem: [solved] applying transformation rule (C) using substitution {B==n} resulting limit problem: [solved] Solution: B / n Resulting cost -2+5/2*n+3/2*n^2 has complexity: Poly(n^2) Found new complexity Poly(n^2). Obtained the following overall complexity (w.r.t. the length of the input n): Complexity: Poly(n^2) Cpx degree: 2 Solved cost: -2+5/2*n+3/2*n^2 Rule cost: -7/2+11/2*B+3/2*(-1+B)^2 Rule guard: [ B>=2 ] WORST_CASE(Omega(n^2),?) ---------------------------------------- (4) BOUNDS(n^2, INF)