/export/starexec/sandbox2/solver/bin/starexec_run_complexity /export/starexec/sandbox2/benchmark/theBenchmark.koat /export/starexec/sandbox2/output/output_files -------------------------------------------------------------------------------- WORST_CASE(Omega(n^2), 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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(n^2, n^2). (0) CpxIntTrs (1) Koat Proof [FINISHED, 26 ms] (2) BOUNDS(1, n^2) (3) Loat Proof [FINISHED, 534 ms] (4) BOUNDS(n^2, INF) ---------------------------------------- (0) Obligation: Complexity Int TRS consisting of the following rules: evalNestedMultipleDepstart(A, B, C, D, E) -> Com_1(evalNestedMultipleDepentryin(A, B, C, D, E)) :|: TRUE evalNestedMultipleDepentryin(A, B, C, D, E) -> Com_1(evalNestedMultipleDepbb3in(0, B, C, D, E)) :|: TRUE evalNestedMultipleDepbb3in(A, B, C, D, E) -> Com_1(evalNestedMultipleDepbbin(A, B, C, D, E)) :|: B >= A + 1 evalNestedMultipleDepbb3in(A, B, C, D, E) -> Com_1(evalNestedMultipleDepreturnin(A, B, C, D, E)) :|: A >= B evalNestedMultipleDepbbin(A, B, C, D, E) -> Com_1(evalNestedMultipleDepbb2in(A, B, A + 1, 0, E)) :|: TRUE evalNestedMultipleDepbb2in(A, B, C, D, E) -> Com_1(evalNestedMultipleDepbb1in(A, B, C, D, E)) :|: E >= D + 1 evalNestedMultipleDepbb2in(A, B, C, D, E) -> Com_1(evalNestedMultipleDepbb3in(C, B, C, D, E)) :|: D >= E evalNestedMultipleDepbb1in(A, B, C, D, E) -> Com_1(evalNestedMultipleDepbb2in(A, B, C, D + 1, E)) :|: TRUE evalNestedMultipleDepreturnin(A, B, C, D, E) -> Com_1(evalNestedMultipleDepstop(A, B, C, D, E)) :|: TRUE The start-symbols are:[evalNestedMultipleDepstart_5] ---------------------------------------- (1) Koat Proof (FINISHED) YES(?, 5*Ar_1 + 2*Ar_1*Ar_4 + 6) Initial complexity problem: 1: T: (Comp: ?, Cost: 1) evalNestedMultipleDepstart(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalNestedMultipleDepentryin(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) (Comp: ?, Cost: 1) evalNestedMultipleDepentryin(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalNestedMultipleDepbb3in(0, Ar_1, Ar_2, Ar_3, Ar_4)) (Comp: ?, Cost: 1) evalNestedMultipleDepbb3in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalNestedMultipleDepbbin(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) [ Ar_1 >= Ar_0 + 1 ] (Comp: ?, Cost: 1) evalNestedMultipleDepbb3in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalNestedMultipleDepreturnin(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) [ Ar_0 >= Ar_1 ] (Comp: ?, Cost: 1) evalNestedMultipleDepbbin(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalNestedMultipleDepbb2in(Ar_0, Ar_1, Ar_0 + 1, 0, Ar_4)) (Comp: ?, Cost: 1) evalNestedMultipleDepbb2in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalNestedMultipleDepbb1in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) [ Ar_4 >= Ar_3 + 1 ] (Comp: ?, Cost: 1) evalNestedMultipleDepbb2in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalNestedMultipleDepbb3in(Ar_2, Ar_1, Ar_2, Ar_3, Ar_4)) [ Ar_3 >= Ar_4 ] (Comp: ?, Cost: 1) evalNestedMultipleDepbb1in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalNestedMultipleDepbb2in(Ar_0, Ar_1, Ar_2, Ar_3 + 1, Ar_4)) (Comp: ?, Cost: 1) evalNestedMultipleDepreturnin(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalNestedMultipleDepstop(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalNestedMultipleDepstart(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) [ 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) evalNestedMultipleDepstart(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalNestedMultipleDepentryin(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) (Comp: 1, Cost: 1) evalNestedMultipleDepentryin(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalNestedMultipleDepbb3in(0, Ar_1, Ar_2, Ar_3, Ar_4)) (Comp: ?, Cost: 1) evalNestedMultipleDepbb3in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalNestedMultipleDepbbin(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) [ Ar_1 >= Ar_0 + 1 ] (Comp: ?, Cost: 1) evalNestedMultipleDepbb3in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalNestedMultipleDepreturnin(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) [ Ar_0 >= Ar_1 ] (Comp: ?, Cost: 1) evalNestedMultipleDepbbin(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalNestedMultipleDepbb2in(Ar_0, Ar_1, Ar_0 + 1, 0, Ar_4)) (Comp: ?, Cost: 1) evalNestedMultipleDepbb2in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalNestedMultipleDepbb1in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) [ Ar_4 >= Ar_3 + 1 ] (Comp: ?, Cost: 1) evalNestedMultipleDepbb2in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalNestedMultipleDepbb3in(Ar_2, Ar_1, Ar_2, Ar_3, Ar_4)) [ Ar_3 >= Ar_4 ] (Comp: ?, Cost: 1) evalNestedMultipleDepbb1in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalNestedMultipleDepbb2in(Ar_0, Ar_1, Ar_2, Ar_3 + 1, Ar_4)) (Comp: ?, Cost: 1) evalNestedMultipleDepreturnin(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalNestedMultipleDepstop(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalNestedMultipleDepstart(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 A polynomial rank function with Pol(evalNestedMultipleDepstart) = 2 Pol(evalNestedMultipleDepentryin) = 2 Pol(evalNestedMultipleDepbb3in) = 2 Pol(evalNestedMultipleDepbbin) = 2 Pol(evalNestedMultipleDepreturnin) = 1 Pol(evalNestedMultipleDepbb2in) = 2 Pol(evalNestedMultipleDepbb1in) = 2 Pol(evalNestedMultipleDepstop) = 0 Pol(koat_start) = 2 orients all transitions weakly and the transitions evalNestedMultipleDepreturnin(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalNestedMultipleDepstop(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) evalNestedMultipleDepbb3in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalNestedMultipleDepreturnin(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) [ Ar_0 >= Ar_1 ] strictly and produces the following problem: 3: T: (Comp: 1, Cost: 1) evalNestedMultipleDepstart(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalNestedMultipleDepentryin(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) (Comp: 1, Cost: 1) evalNestedMultipleDepentryin(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalNestedMultipleDepbb3in(0, Ar_1, Ar_2, Ar_3, Ar_4)) (Comp: ?, Cost: 1) evalNestedMultipleDepbb3in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalNestedMultipleDepbbin(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) [ Ar_1 >= Ar_0 + 1 ] (Comp: 2, Cost: 1) evalNestedMultipleDepbb3in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalNestedMultipleDepreturnin(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) [ Ar_0 >= Ar_1 ] (Comp: ?, Cost: 1) evalNestedMultipleDepbbin(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalNestedMultipleDepbb2in(Ar_0, Ar_1, Ar_0 + 1, 0, Ar_4)) (Comp: ?, Cost: 1) evalNestedMultipleDepbb2in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalNestedMultipleDepbb1in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) [ Ar_4 >= Ar_3 + 1 ] (Comp: ?, Cost: 1) evalNestedMultipleDepbb2in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalNestedMultipleDepbb3in(Ar_2, Ar_1, Ar_2, Ar_3, Ar_4)) [ Ar_3 >= Ar_4 ] (Comp: ?, Cost: 1) evalNestedMultipleDepbb1in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalNestedMultipleDepbb2in(Ar_0, Ar_1, Ar_2, Ar_3 + 1, Ar_4)) (Comp: 2, Cost: 1) evalNestedMultipleDepreturnin(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalNestedMultipleDepstop(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalNestedMultipleDepstart(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 A polynomial rank function with Pol(evalNestedMultipleDepstart) = V_2 Pol(evalNestedMultipleDepentryin) = V_2 Pol(evalNestedMultipleDepbb3in) = -V_1 + V_2 Pol(evalNestedMultipleDepbbin) = -V_1 + V_2 - 1 Pol(evalNestedMultipleDepreturnin) = -V_1 + V_2 Pol(evalNestedMultipleDepbb2in) = V_2 - V_3 Pol(evalNestedMultipleDepbb1in) = V_2 - V_3 Pol(evalNestedMultipleDepstop) = -V_1 + V_2 Pol(koat_start) = V_2 orients all transitions weakly and the transition evalNestedMultipleDepbb3in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalNestedMultipleDepbbin(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) [ Ar_1 >= Ar_0 + 1 ] strictly and produces the following problem: 4: T: (Comp: 1, Cost: 1) evalNestedMultipleDepstart(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalNestedMultipleDepentryin(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) (Comp: 1, Cost: 1) evalNestedMultipleDepentryin(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalNestedMultipleDepbb3in(0, Ar_1, Ar_2, Ar_3, Ar_4)) (Comp: Ar_1, Cost: 1) evalNestedMultipleDepbb3in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalNestedMultipleDepbbin(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) [ Ar_1 >= Ar_0 + 1 ] (Comp: 2, Cost: 1) evalNestedMultipleDepbb3in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalNestedMultipleDepreturnin(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) [ Ar_0 >= Ar_1 ] (Comp: ?, Cost: 1) evalNestedMultipleDepbbin(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalNestedMultipleDepbb2in(Ar_0, Ar_1, Ar_0 + 1, 0, Ar_4)) (Comp: ?, Cost: 1) evalNestedMultipleDepbb2in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalNestedMultipleDepbb1in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) [ Ar_4 >= Ar_3 + 1 ] (Comp: ?, Cost: 1) evalNestedMultipleDepbb2in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalNestedMultipleDepbb3in(Ar_2, Ar_1, Ar_2, Ar_3, Ar_4)) [ Ar_3 >= Ar_4 ] (Comp: ?, Cost: 1) evalNestedMultipleDepbb1in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalNestedMultipleDepbb2in(Ar_0, Ar_1, Ar_2, Ar_3 + 1, Ar_4)) (Comp: 2, Cost: 1) evalNestedMultipleDepreturnin(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalNestedMultipleDepstop(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalNestedMultipleDepstart(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) [ 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) evalNestedMultipleDepstart(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalNestedMultipleDepentryin(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) (Comp: 1, Cost: 1) evalNestedMultipleDepentryin(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalNestedMultipleDepbb3in(0, Ar_1, Ar_2, Ar_3, Ar_4)) (Comp: Ar_1, Cost: 1) evalNestedMultipleDepbb3in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalNestedMultipleDepbbin(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) [ Ar_1 >= Ar_0 + 1 ] (Comp: 2, Cost: 1) evalNestedMultipleDepbb3in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalNestedMultipleDepreturnin(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) [ Ar_0 >= Ar_1 ] (Comp: Ar_1, Cost: 1) evalNestedMultipleDepbbin(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalNestedMultipleDepbb2in(Ar_0, Ar_1, Ar_0 + 1, 0, Ar_4)) (Comp: ?, Cost: 1) evalNestedMultipleDepbb2in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalNestedMultipleDepbb1in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) [ Ar_4 >= Ar_3 + 1 ] (Comp: ?, Cost: 1) evalNestedMultipleDepbb2in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalNestedMultipleDepbb3in(Ar_2, Ar_1, Ar_2, Ar_3, Ar_4)) [ Ar_3 >= Ar_4 ] (Comp: ?, Cost: 1) evalNestedMultipleDepbb1in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalNestedMultipleDepbb2in(Ar_0, Ar_1, Ar_2, Ar_3 + 1, Ar_4)) (Comp: 2, Cost: 1) evalNestedMultipleDepreturnin(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalNestedMultipleDepstop(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalNestedMultipleDepstart(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 A polynomial rank function with Pol(evalNestedMultipleDepbb2in) = 1 Pol(evalNestedMultipleDepbb3in) = 0 Pol(evalNestedMultipleDepbb1in) = 1 and size complexities S("koat_start(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalNestedMultipleDepstart(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) [ 0 <= 0 ]", 0-0) = Ar_0 S("koat_start(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalNestedMultipleDepstart(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) [ 0 <= 0 ]", 0-1) = Ar_1 S("koat_start(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalNestedMultipleDepstart(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) [ 0 <= 0 ]", 0-2) = Ar_2 S("koat_start(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalNestedMultipleDepstart(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) [ 0 <= 0 ]", 0-3) = Ar_3 S("koat_start(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalNestedMultipleDepstart(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) [ 0 <= 0 ]", 0-4) = Ar_4 S("evalNestedMultipleDepreturnin(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalNestedMultipleDepstop(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4))", 0-0) = Ar_1 S("evalNestedMultipleDepreturnin(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalNestedMultipleDepstop(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4))", 0-1) = Ar_1 S("evalNestedMultipleDepreturnin(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalNestedMultipleDepstop(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4))", 0-2) = Ar_1 + Ar_2 S("evalNestedMultipleDepreturnin(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalNestedMultipleDepstop(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4))", 0-3) = ? S("evalNestedMultipleDepreturnin(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalNestedMultipleDepstop(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4))", 0-4) = Ar_4 S("evalNestedMultipleDepbb1in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalNestedMultipleDepbb2in(Ar_0, Ar_1, Ar_2, Ar_3 + 1, Ar_4))", 0-0) = Ar_1 S("evalNestedMultipleDepbb1in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalNestedMultipleDepbb2in(Ar_0, Ar_1, Ar_2, Ar_3 + 1, Ar_4))", 0-1) = Ar_1 S("evalNestedMultipleDepbb1in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalNestedMultipleDepbb2in(Ar_0, Ar_1, Ar_2, Ar_3 + 1, Ar_4))", 0-2) = Ar_1 S("evalNestedMultipleDepbb1in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalNestedMultipleDepbb2in(Ar_0, Ar_1, Ar_2, Ar_3 + 1, Ar_4))", 0-3) = ? S("evalNestedMultipleDepbb1in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalNestedMultipleDepbb2in(Ar_0, Ar_1, Ar_2, Ar_3 + 1, Ar_4))", 0-4) = Ar_4 S("evalNestedMultipleDepbb2in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalNestedMultipleDepbb3in(Ar_2, Ar_1, Ar_2, Ar_3, Ar_4)) [ Ar_3 >= Ar_4 ]", 0-0) = Ar_1 S("evalNestedMultipleDepbb2in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalNestedMultipleDepbb3in(Ar_2, Ar_1, Ar_2, Ar_3, Ar_4)) [ Ar_3 >= Ar_4 ]", 0-1) = Ar_1 S("evalNestedMultipleDepbb2in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalNestedMultipleDepbb3in(Ar_2, Ar_1, Ar_2, Ar_3, Ar_4)) [ Ar_3 >= Ar_4 ]", 0-2) = Ar_1 S("evalNestedMultipleDepbb2in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalNestedMultipleDepbb3in(Ar_2, Ar_1, Ar_2, Ar_3, Ar_4)) [ Ar_3 >= Ar_4 ]", 0-3) = ? S("evalNestedMultipleDepbb2in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalNestedMultipleDepbb3in(Ar_2, Ar_1, Ar_2, Ar_3, Ar_4)) [ Ar_3 >= Ar_4 ]", 0-4) = Ar_4 S("evalNestedMultipleDepbb2in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalNestedMultipleDepbb1in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) [ Ar_4 >= Ar_3 + 1 ]", 0-0) = Ar_1 S("evalNestedMultipleDepbb2in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalNestedMultipleDepbb1in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) [ Ar_4 >= Ar_3 + 1 ]", 0-1) = Ar_1 S("evalNestedMultipleDepbb2in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalNestedMultipleDepbb1in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) [ Ar_4 >= Ar_3 + 1 ]", 0-2) = Ar_1 S("evalNestedMultipleDepbb2in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalNestedMultipleDepbb1in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) [ Ar_4 >= Ar_3 + 1 ]", 0-3) = ? S("evalNestedMultipleDepbb2in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalNestedMultipleDepbb1in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) [ Ar_4 >= Ar_3 + 1 ]", 0-4) = Ar_4 S("evalNestedMultipleDepbbin(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalNestedMultipleDepbb2in(Ar_0, Ar_1, Ar_0 + 1, 0, Ar_4))", 0-0) = Ar_1 S("evalNestedMultipleDepbbin(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalNestedMultipleDepbb2in(Ar_0, Ar_1, Ar_0 + 1, 0, Ar_4))", 0-1) = Ar_1 S("evalNestedMultipleDepbbin(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalNestedMultipleDepbb2in(Ar_0, Ar_1, Ar_0 + 1, 0, Ar_4))", 0-2) = Ar_1 S("evalNestedMultipleDepbbin(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalNestedMultipleDepbb2in(Ar_0, Ar_1, Ar_0 + 1, 0, Ar_4))", 0-3) = 0 S("evalNestedMultipleDepbbin(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalNestedMultipleDepbb2in(Ar_0, Ar_1, Ar_0 + 1, 0, Ar_4))", 0-4) = Ar_4 S("evalNestedMultipleDepbb3in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalNestedMultipleDepreturnin(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) [ Ar_0 >= Ar_1 ]", 0-0) = Ar_1 S("evalNestedMultipleDepbb3in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalNestedMultipleDepreturnin(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) [ Ar_0 >= Ar_1 ]", 0-1) = Ar_1 S("evalNestedMultipleDepbb3in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalNestedMultipleDepreturnin(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) [ Ar_0 >= Ar_1 ]", 0-2) = Ar_1 + Ar_2 S("evalNestedMultipleDepbb3in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalNestedMultipleDepreturnin(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) [ Ar_0 >= Ar_1 ]", 0-3) = ? S("evalNestedMultipleDepbb3in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalNestedMultipleDepreturnin(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) [ Ar_0 >= Ar_1 ]", 0-4) = Ar_4 S("evalNestedMultipleDepbb3in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalNestedMultipleDepbbin(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) [ Ar_1 >= Ar_0 + 1 ]", 0-0) = Ar_1 S("evalNestedMultipleDepbb3in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalNestedMultipleDepbbin(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) [ Ar_1 >= Ar_0 + 1 ]", 0-1) = Ar_1 S("evalNestedMultipleDepbb3in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalNestedMultipleDepbbin(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) [ Ar_1 >= Ar_0 + 1 ]", 0-2) = Ar_1 + Ar_2 S("evalNestedMultipleDepbb3in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalNestedMultipleDepbbin(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) [ Ar_1 >= Ar_0 + 1 ]", 0-3) = ? S("evalNestedMultipleDepbb3in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalNestedMultipleDepbbin(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) [ Ar_1 >= Ar_0 + 1 ]", 0-4) = Ar_4 S("evalNestedMultipleDepentryin(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalNestedMultipleDepbb3in(0, Ar_1, Ar_2, Ar_3, Ar_4))", 0-0) = 0 S("evalNestedMultipleDepentryin(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalNestedMultipleDepbb3in(0, Ar_1, Ar_2, Ar_3, Ar_4))", 0-1) = Ar_1 S("evalNestedMultipleDepentryin(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalNestedMultipleDepbb3in(0, Ar_1, Ar_2, Ar_3, Ar_4))", 0-2) = Ar_2 S("evalNestedMultipleDepentryin(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalNestedMultipleDepbb3in(0, Ar_1, Ar_2, Ar_3, Ar_4))", 0-3) = Ar_3 S("evalNestedMultipleDepentryin(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalNestedMultipleDepbb3in(0, Ar_1, Ar_2, Ar_3, Ar_4))", 0-4) = Ar_4 S("evalNestedMultipleDepstart(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalNestedMultipleDepentryin(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4))", 0-0) = Ar_0 S("evalNestedMultipleDepstart(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalNestedMultipleDepentryin(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4))", 0-1) = Ar_1 S("evalNestedMultipleDepstart(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalNestedMultipleDepentryin(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4))", 0-2) = Ar_2 S("evalNestedMultipleDepstart(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalNestedMultipleDepentryin(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4))", 0-3) = Ar_3 S("evalNestedMultipleDepstart(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalNestedMultipleDepentryin(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4))", 0-4) = Ar_4 orients the transitions evalNestedMultipleDepbb2in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalNestedMultipleDepbb3in(Ar_2, Ar_1, Ar_2, Ar_3, Ar_4)) [ Ar_3 >= Ar_4 ] evalNestedMultipleDepbb2in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalNestedMultipleDepbb1in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) [ Ar_4 >= Ar_3 + 1 ] evalNestedMultipleDepbb1in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalNestedMultipleDepbb2in(Ar_0, Ar_1, Ar_2, Ar_3 + 1, Ar_4)) weakly and the transition evalNestedMultipleDepbb2in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalNestedMultipleDepbb3in(Ar_2, Ar_1, Ar_2, Ar_3, Ar_4)) [ Ar_3 >= Ar_4 ] strictly and produces the following problem: 6: T: (Comp: 1, Cost: 1) evalNestedMultipleDepstart(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalNestedMultipleDepentryin(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) (Comp: 1, Cost: 1) evalNestedMultipleDepentryin(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalNestedMultipleDepbb3in(0, Ar_1, Ar_2, Ar_3, Ar_4)) (Comp: Ar_1, Cost: 1) evalNestedMultipleDepbb3in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalNestedMultipleDepbbin(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) [ Ar_1 >= Ar_0 + 1 ] (Comp: 2, Cost: 1) evalNestedMultipleDepbb3in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalNestedMultipleDepreturnin(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) [ Ar_0 >= Ar_1 ] (Comp: Ar_1, Cost: 1) evalNestedMultipleDepbbin(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalNestedMultipleDepbb2in(Ar_0, Ar_1, Ar_0 + 1, 0, Ar_4)) (Comp: ?, Cost: 1) evalNestedMultipleDepbb2in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalNestedMultipleDepbb1in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) [ Ar_4 >= Ar_3 + 1 ] (Comp: Ar_1, Cost: 1) evalNestedMultipleDepbb2in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalNestedMultipleDepbb3in(Ar_2, Ar_1, Ar_2, Ar_3, Ar_4)) [ Ar_3 >= Ar_4 ] (Comp: ?, Cost: 1) evalNestedMultipleDepbb1in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalNestedMultipleDepbb2in(Ar_0, Ar_1, Ar_2, Ar_3 + 1, Ar_4)) (Comp: 2, Cost: 1) evalNestedMultipleDepreturnin(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalNestedMultipleDepstop(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalNestedMultipleDepstart(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 A polynomial rank function with Pol(evalNestedMultipleDepbb2in) = -V_4 + V_5 + 1 Pol(evalNestedMultipleDepbb1in) = -V_4 + V_5 and size complexities S("koat_start(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalNestedMultipleDepstart(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) [ 0 <= 0 ]", 0-0) = Ar_0 S("koat_start(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalNestedMultipleDepstart(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) [ 0 <= 0 ]", 0-1) = Ar_1 S("koat_start(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalNestedMultipleDepstart(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) [ 0 <= 0 ]", 0-2) = Ar_2 S("koat_start(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalNestedMultipleDepstart(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) [ 0 <= 0 ]", 0-3) = Ar_3 S("koat_start(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalNestedMultipleDepstart(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) [ 0 <= 0 ]", 0-4) = Ar_4 S("evalNestedMultipleDepreturnin(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalNestedMultipleDepstop(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4))", 0-0) = Ar_1 S("evalNestedMultipleDepreturnin(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalNestedMultipleDepstop(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4))", 0-1) = Ar_1 S("evalNestedMultipleDepreturnin(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalNestedMultipleDepstop(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4))", 0-2) = Ar_1 + Ar_2 S("evalNestedMultipleDepreturnin(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalNestedMultipleDepstop(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4))", 0-3) = ? S("evalNestedMultipleDepreturnin(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalNestedMultipleDepstop(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4))", 0-4) = Ar_4 S("evalNestedMultipleDepbb1in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalNestedMultipleDepbb2in(Ar_0, Ar_1, Ar_2, Ar_3 + 1, Ar_4))", 0-0) = Ar_1 S("evalNestedMultipleDepbb1in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalNestedMultipleDepbb2in(Ar_0, Ar_1, Ar_2, Ar_3 + 1, Ar_4))", 0-1) = Ar_1 S("evalNestedMultipleDepbb1in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalNestedMultipleDepbb2in(Ar_0, Ar_1, Ar_2, Ar_3 + 1, Ar_4))", 0-2) = Ar_1 S("evalNestedMultipleDepbb1in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalNestedMultipleDepbb2in(Ar_0, Ar_1, Ar_2, Ar_3 + 1, Ar_4))", 0-3) = ? S("evalNestedMultipleDepbb1in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalNestedMultipleDepbb2in(Ar_0, Ar_1, Ar_2, Ar_3 + 1, Ar_4))", 0-4) = Ar_4 S("evalNestedMultipleDepbb2in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalNestedMultipleDepbb3in(Ar_2, Ar_1, Ar_2, Ar_3, Ar_4)) [ Ar_3 >= Ar_4 ]", 0-0) = Ar_1 S("evalNestedMultipleDepbb2in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalNestedMultipleDepbb3in(Ar_2, Ar_1, Ar_2, Ar_3, Ar_4)) [ Ar_3 >= Ar_4 ]", 0-1) = Ar_1 S("evalNestedMultipleDepbb2in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalNestedMultipleDepbb3in(Ar_2, Ar_1, Ar_2, Ar_3, Ar_4)) [ Ar_3 >= Ar_4 ]", 0-2) = Ar_1 S("evalNestedMultipleDepbb2in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalNestedMultipleDepbb3in(Ar_2, Ar_1, Ar_2, Ar_3, Ar_4)) [ Ar_3 >= Ar_4 ]", 0-3) = ? S("evalNestedMultipleDepbb2in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalNestedMultipleDepbb3in(Ar_2, Ar_1, Ar_2, Ar_3, Ar_4)) [ Ar_3 >= Ar_4 ]", 0-4) = Ar_4 S("evalNestedMultipleDepbb2in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalNestedMultipleDepbb1in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) [ Ar_4 >= Ar_3 + 1 ]", 0-0) = Ar_1 S("evalNestedMultipleDepbb2in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalNestedMultipleDepbb1in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) [ Ar_4 >= Ar_3 + 1 ]", 0-1) = Ar_1 S("evalNestedMultipleDepbb2in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalNestedMultipleDepbb1in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) [ Ar_4 >= Ar_3 + 1 ]", 0-2) = Ar_1 S("evalNestedMultipleDepbb2in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalNestedMultipleDepbb1in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) [ Ar_4 >= Ar_3 + 1 ]", 0-3) = ? S("evalNestedMultipleDepbb2in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalNestedMultipleDepbb1in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) [ Ar_4 >= Ar_3 + 1 ]", 0-4) = Ar_4 S("evalNestedMultipleDepbbin(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalNestedMultipleDepbb2in(Ar_0, Ar_1, Ar_0 + 1, 0, Ar_4))", 0-0) = Ar_1 S("evalNestedMultipleDepbbin(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalNestedMultipleDepbb2in(Ar_0, Ar_1, Ar_0 + 1, 0, Ar_4))", 0-1) = Ar_1 S("evalNestedMultipleDepbbin(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalNestedMultipleDepbb2in(Ar_0, Ar_1, Ar_0 + 1, 0, Ar_4))", 0-2) = Ar_1 S("evalNestedMultipleDepbbin(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalNestedMultipleDepbb2in(Ar_0, Ar_1, Ar_0 + 1, 0, Ar_4))", 0-3) = 0 S("evalNestedMultipleDepbbin(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalNestedMultipleDepbb2in(Ar_0, Ar_1, Ar_0 + 1, 0, Ar_4))", 0-4) = Ar_4 S("evalNestedMultipleDepbb3in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalNestedMultipleDepreturnin(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) [ Ar_0 >= Ar_1 ]", 0-0) = Ar_1 S("evalNestedMultipleDepbb3in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalNestedMultipleDepreturnin(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) [ Ar_0 >= Ar_1 ]", 0-1) = Ar_1 S("evalNestedMultipleDepbb3in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalNestedMultipleDepreturnin(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) [ Ar_0 >= Ar_1 ]", 0-2) = Ar_1 + Ar_2 S("evalNestedMultipleDepbb3in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalNestedMultipleDepreturnin(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) [ Ar_0 >= Ar_1 ]", 0-3) = ? S("evalNestedMultipleDepbb3in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalNestedMultipleDepreturnin(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) [ Ar_0 >= Ar_1 ]", 0-4) = Ar_4 S("evalNestedMultipleDepbb3in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalNestedMultipleDepbbin(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) [ Ar_1 >= Ar_0 + 1 ]", 0-0) = Ar_1 S("evalNestedMultipleDepbb3in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalNestedMultipleDepbbin(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) [ Ar_1 >= Ar_0 + 1 ]", 0-1) = Ar_1 S("evalNestedMultipleDepbb3in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalNestedMultipleDepbbin(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) [ Ar_1 >= Ar_0 + 1 ]", 0-2) = Ar_1 + Ar_2 S("evalNestedMultipleDepbb3in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalNestedMultipleDepbbin(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) [ Ar_1 >= Ar_0 + 1 ]", 0-3) = ? S("evalNestedMultipleDepbb3in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalNestedMultipleDepbbin(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) [ Ar_1 >= Ar_0 + 1 ]", 0-4) = Ar_4 S("evalNestedMultipleDepentryin(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalNestedMultipleDepbb3in(0, Ar_1, Ar_2, Ar_3, Ar_4))", 0-0) = 0 S("evalNestedMultipleDepentryin(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalNestedMultipleDepbb3in(0, Ar_1, Ar_2, Ar_3, Ar_4))", 0-1) = Ar_1 S("evalNestedMultipleDepentryin(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalNestedMultipleDepbb3in(0, Ar_1, Ar_2, Ar_3, Ar_4))", 0-2) = Ar_2 S("evalNestedMultipleDepentryin(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalNestedMultipleDepbb3in(0, Ar_1, Ar_2, Ar_3, Ar_4))", 0-3) = Ar_3 S("evalNestedMultipleDepentryin(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalNestedMultipleDepbb3in(0, Ar_1, Ar_2, Ar_3, Ar_4))", 0-4) = Ar_4 S("evalNestedMultipleDepstart(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalNestedMultipleDepentryin(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4))", 0-0) = Ar_0 S("evalNestedMultipleDepstart(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalNestedMultipleDepentryin(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4))", 0-1) = Ar_1 S("evalNestedMultipleDepstart(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalNestedMultipleDepentryin(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4))", 0-2) = Ar_2 S("evalNestedMultipleDepstart(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalNestedMultipleDepentryin(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4))", 0-3) = Ar_3 S("evalNestedMultipleDepstart(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalNestedMultipleDepentryin(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4))", 0-4) = Ar_4 orients the transitions evalNestedMultipleDepbb2in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalNestedMultipleDepbb1in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) [ Ar_4 >= Ar_3 + 1 ] evalNestedMultipleDepbb1in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalNestedMultipleDepbb2in(Ar_0, Ar_1, Ar_2, Ar_3 + 1, Ar_4)) weakly and the transition evalNestedMultipleDepbb2in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalNestedMultipleDepbb1in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) [ Ar_4 >= Ar_3 + 1 ] strictly and produces the following problem: 7: T: (Comp: 1, Cost: 1) evalNestedMultipleDepstart(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalNestedMultipleDepentryin(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) (Comp: 1, Cost: 1) evalNestedMultipleDepentryin(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalNestedMultipleDepbb3in(0, Ar_1, Ar_2, Ar_3, Ar_4)) (Comp: Ar_1, Cost: 1) evalNestedMultipleDepbb3in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalNestedMultipleDepbbin(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) [ Ar_1 >= Ar_0 + 1 ] (Comp: 2, Cost: 1) evalNestedMultipleDepbb3in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalNestedMultipleDepreturnin(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) [ Ar_0 >= Ar_1 ] (Comp: Ar_1, Cost: 1) evalNestedMultipleDepbbin(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalNestedMultipleDepbb2in(Ar_0, Ar_1, Ar_0 + 1, 0, Ar_4)) (Comp: Ar_1*Ar_4 + Ar_1, Cost: 1) evalNestedMultipleDepbb2in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalNestedMultipleDepbb1in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) [ Ar_4 >= Ar_3 + 1 ] (Comp: Ar_1, Cost: 1) evalNestedMultipleDepbb2in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalNestedMultipleDepbb3in(Ar_2, Ar_1, Ar_2, Ar_3, Ar_4)) [ Ar_3 >= Ar_4 ] (Comp: ?, Cost: 1) evalNestedMultipleDepbb1in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalNestedMultipleDepbb2in(Ar_0, Ar_1, Ar_2, Ar_3 + 1, Ar_4)) (Comp: 2, Cost: 1) evalNestedMultipleDepreturnin(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalNestedMultipleDepstop(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalNestedMultipleDepstart(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) [ 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) evalNestedMultipleDepstart(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalNestedMultipleDepentryin(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) (Comp: 1, Cost: 1) evalNestedMultipleDepentryin(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalNestedMultipleDepbb3in(0, Ar_1, Ar_2, Ar_3, Ar_4)) (Comp: Ar_1, Cost: 1) evalNestedMultipleDepbb3in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalNestedMultipleDepbbin(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) [ Ar_1 >= Ar_0 + 1 ] (Comp: 2, Cost: 1) evalNestedMultipleDepbb3in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalNestedMultipleDepreturnin(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) [ Ar_0 >= Ar_1 ] (Comp: Ar_1, Cost: 1) evalNestedMultipleDepbbin(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalNestedMultipleDepbb2in(Ar_0, Ar_1, Ar_0 + 1, 0, Ar_4)) (Comp: Ar_1*Ar_4 + Ar_1, Cost: 1) evalNestedMultipleDepbb2in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalNestedMultipleDepbb1in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) [ Ar_4 >= Ar_3 + 1 ] (Comp: Ar_1, Cost: 1) evalNestedMultipleDepbb2in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalNestedMultipleDepbb3in(Ar_2, Ar_1, Ar_2, Ar_3, Ar_4)) [ Ar_3 >= Ar_4 ] (Comp: Ar_1*Ar_4 + Ar_1, Cost: 1) evalNestedMultipleDepbb1in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalNestedMultipleDepbb2in(Ar_0, Ar_1, Ar_2, Ar_3 + 1, Ar_4)) (Comp: 2, Cost: 1) evalNestedMultipleDepreturnin(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalNestedMultipleDepstop(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalNestedMultipleDepstart(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 Complexity upper bound 5*Ar_1 + 2*Ar_1*Ar_4 + 6 Time: 0.097 sec (SMT: 0.072 sec) ---------------------------------------- (2) BOUNDS(1, n^2) ---------------------------------------- (3) Loat Proof (FINISHED) ### Pre-processing the ITS problem ### Initial linear ITS problem Start location: evalNestedMultipleDepstart 0: evalNestedMultipleDepstart -> evalNestedMultipleDepentryin : [], cost: 1 1: evalNestedMultipleDepentryin -> evalNestedMultipleDepbb3in : A'=0, [], cost: 1 2: evalNestedMultipleDepbb3in -> evalNestedMultipleDepbbin : [ B>=1+A ], cost: 1 3: evalNestedMultipleDepbb3in -> evalNestedMultipleDepreturnin : [ A>=B ], cost: 1 4: evalNestedMultipleDepbbin -> evalNestedMultipleDepbb2in : C'=1+A, D'=0, [], cost: 1 5: evalNestedMultipleDepbb2in -> evalNestedMultipleDepbb1in : [ E>=1+D ], cost: 1 6: evalNestedMultipleDepbb2in -> evalNestedMultipleDepbb3in : A'=C, [ D>=E ], cost: 1 7: evalNestedMultipleDepbb1in -> evalNestedMultipleDepbb2in : D'=1+D, [], cost: 1 8: evalNestedMultipleDepreturnin -> evalNestedMultipleDepstop : [], cost: 1 Checking for constant complexity: The following rule is satisfiable with cost >= 1, yielding constant complexity: 0: evalNestedMultipleDepstart -> evalNestedMultipleDepentryin : [], cost: 1 Removed unreachable and leaf rules: Start location: evalNestedMultipleDepstart 0: evalNestedMultipleDepstart -> evalNestedMultipleDepentryin : [], cost: 1 1: evalNestedMultipleDepentryin -> evalNestedMultipleDepbb3in : A'=0, [], cost: 1 2: evalNestedMultipleDepbb3in -> evalNestedMultipleDepbbin : [ B>=1+A ], cost: 1 4: evalNestedMultipleDepbbin -> evalNestedMultipleDepbb2in : C'=1+A, D'=0, [], cost: 1 5: evalNestedMultipleDepbb2in -> evalNestedMultipleDepbb1in : [ E>=1+D ], cost: 1 6: evalNestedMultipleDepbb2in -> evalNestedMultipleDepbb3in : A'=C, [ D>=E ], cost: 1 7: evalNestedMultipleDepbb1in -> evalNestedMultipleDepbb2in : D'=1+D, [], cost: 1 ### Simplification by acceleration and chaining ### Eliminated locations (on linear paths): Start location: evalNestedMultipleDepstart 9: evalNestedMultipleDepstart -> evalNestedMultipleDepbb3in : A'=0, [], cost: 2 10: evalNestedMultipleDepbb3in -> evalNestedMultipleDepbb2in : C'=1+A, D'=0, [ B>=1+A ], cost: 2 6: evalNestedMultipleDepbb2in -> evalNestedMultipleDepbb3in : A'=C, [ D>=E ], cost: 1 11: evalNestedMultipleDepbb2in -> evalNestedMultipleDepbb2in : D'=1+D, [ E>=1+D ], cost: 2 Accelerating simple loops of location 4. Accelerating the following rules: 11: evalNestedMultipleDepbb2in -> evalNestedMultipleDepbb2in : D'=1+D, [ E>=1+D ], cost: 2 Accelerated rule 11 with metering function -D+E, yielding the new rule 12. Removing the simple loops: 11. Accelerated all simple loops using metering functions (where possible): Start location: evalNestedMultipleDepstart 9: evalNestedMultipleDepstart -> evalNestedMultipleDepbb3in : A'=0, [], cost: 2 10: evalNestedMultipleDepbb3in -> evalNestedMultipleDepbb2in : C'=1+A, D'=0, [ B>=1+A ], cost: 2 6: evalNestedMultipleDepbb2in -> evalNestedMultipleDepbb3in : A'=C, [ D>=E ], cost: 1 12: evalNestedMultipleDepbb2in -> evalNestedMultipleDepbb2in : D'=E, [ E>=1+D ], cost: -2*D+2*E Chained accelerated rules (with incoming rules): Start location: evalNestedMultipleDepstart 9: evalNestedMultipleDepstart -> evalNestedMultipleDepbb3in : A'=0, [], cost: 2 10: evalNestedMultipleDepbb3in -> evalNestedMultipleDepbb2in : C'=1+A, D'=0, [ B>=1+A ], cost: 2 13: evalNestedMultipleDepbb3in -> evalNestedMultipleDepbb2in : C'=1+A, D'=E, [ B>=1+A && E>=1 ], cost: 2+2*E 6: evalNestedMultipleDepbb2in -> evalNestedMultipleDepbb3in : A'=C, [ D>=E ], cost: 1 Eliminated locations (on tree-shaped paths): Start location: evalNestedMultipleDepstart 9: evalNestedMultipleDepstart -> evalNestedMultipleDepbb3in : A'=0, [], cost: 2 14: evalNestedMultipleDepbb3in -> evalNestedMultipleDepbb3in : A'=1+A, C'=1+A, D'=0, [ B>=1+A && 0>=E ], cost: 3 15: evalNestedMultipleDepbb3in -> evalNestedMultipleDepbb3in : A'=1+A, C'=1+A, D'=E, [ B>=1+A && E>=1 ], cost: 3+2*E Accelerating simple loops of location 2. Accelerating the following rules: 14: evalNestedMultipleDepbb3in -> evalNestedMultipleDepbb3in : A'=1+A, C'=1+A, D'=0, [ B>=1+A && 0>=E ], cost: 3 15: evalNestedMultipleDepbb3in -> evalNestedMultipleDepbb3in : A'=1+A, C'=1+A, D'=E, [ B>=1+A && E>=1 ], cost: 3+2*E Accelerated rule 14 with metering function -A+B, yielding the new rule 16. Accelerated rule 15 with metering function -A+B, yielding the new rule 17. Removing the simple loops: 14 15. Accelerated all simple loops using metering functions (where possible): Start location: evalNestedMultipleDepstart 9: evalNestedMultipleDepstart -> evalNestedMultipleDepbb3in : A'=0, [], cost: 2 16: evalNestedMultipleDepbb3in -> evalNestedMultipleDepbb3in : A'=B, C'=B, D'=0, [ B>=1+A && 0>=E ], cost: -3*A+3*B 17: evalNestedMultipleDepbb3in -> evalNestedMultipleDepbb3in : A'=B, C'=B, D'=E, [ B>=1+A && E>=1 ], cost: -2*(A-B)*E-3*A+3*B Chained accelerated rules (with incoming rules): Start location: evalNestedMultipleDepstart 9: evalNestedMultipleDepstart -> evalNestedMultipleDepbb3in : A'=0, [], cost: 2 18: evalNestedMultipleDepstart -> evalNestedMultipleDepbb3in : A'=B, C'=B, D'=0, [ B>=1 && 0>=E ], cost: 2+3*B 19: evalNestedMultipleDepstart -> evalNestedMultipleDepbb3in : A'=B, C'=B, D'=E, [ B>=1 && E>=1 ], cost: 2+2*E*B+3*B Removed unreachable locations (and leaf rules with constant cost): Start location: evalNestedMultipleDepstart 18: evalNestedMultipleDepstart -> evalNestedMultipleDepbb3in : A'=B, C'=B, D'=0, [ B>=1 && 0>=E ], cost: 2+3*B 19: evalNestedMultipleDepstart -> evalNestedMultipleDepbb3in : A'=B, C'=B, D'=E, [ B>=1 && E>=1 ], cost: 2+2*E*B+3*B ### Computing asymptotic complexity ### Fully simplified ITS problem Start location: evalNestedMultipleDepstart 18: evalNestedMultipleDepstart -> evalNestedMultipleDepbb3in : A'=B, C'=B, D'=0, [ B>=1 && 0>=E ], cost: 2+3*B 19: evalNestedMultipleDepstart -> evalNestedMultipleDepbb3in : A'=B, C'=B, D'=E, [ B>=1 && E>=1 ], cost: 2+2*E*B+3*B Computing asymptotic complexity for rule 18 Solved the limit problem by the following transformations: Created initial limit problem: 2+3*B (+), 1-E (+/+!), B (+/+!) [not solved] removing all constraints (solved by SMT) resulting limit problem: [solved] applying transformation rule (C) using substitution {E==-n,B==n} resulting limit problem: [solved] Solution: E / -n B / n Resulting cost 2+3*n has complexity: Poly(n^1) Found new complexity Poly(n^1). Computing asymptotic complexity for rule 19 Solved the limit problem by the following transformations: Created initial limit problem: E (+/+!), B (+/+!), 2+2*E*B+3*B (+) [not solved] removing all constraints (solved by SMT) resulting limit problem: [solved] applying transformation rule (C) using substitution {E==n,B==n} resulting limit problem: [solved] Solution: E / n B / n Resulting cost 2+3*n+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+3*n+2*n^2 Rule cost: 2+2*E*B+3*B Rule guard: [ B>=1 && E>=1 ] WORST_CASE(Omega(n^2),?) ---------------------------------------- (4) BOUNDS(n^2, INF)