/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(n^2, n^2).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, 11 ms] (2) BOUNDS(1, n^2) (3) Loat Proof [FINISHED, 428 ms] (4) BOUNDS(n^2, INF) ---------------------------------------- (0) Obligation: Complexity Int TRS consisting of the following rules: evalrealselectstart(A, B, C) -> Com_1(evalrealselectentryin(A, B, C)) :|: TRUE evalrealselectentryin(A, B, C) -> Com_1(evalrealselectbb6in(0, B, C)) :|: TRUE evalrealselectbb6in(A, B, C) -> Com_1(evalrealselectbbin(A, B, C)) :|: B >= 2 + A evalrealselectbb6in(A, B, C) -> Com_1(evalrealselectreturnin(A, B, C)) :|: A + 1 >= B evalrealselectbbin(A, B, C) -> Com_1(evalrealselectbb4in(A, B, A + 1)) :|: TRUE evalrealselectbb4in(A, B, C) -> Com_1(evalrealselectbb1in(A, B, C)) :|: B >= C + 1 evalrealselectbb4in(A, B, C) -> Com_1(evalrealselectbb5in(A, B, C)) :|: C >= B evalrealselectbb1in(A, B, C) -> Com_1(evalrealselectbb4in(A, B, C + 1)) :|: D >= E + 1 evalrealselectbb1in(A, B, C) -> Com_1(evalrealselectbb4in(A, B, C + 1)) :|: E >= D evalrealselectbb5in(A, B, C) -> Com_1(evalrealselectbb6in(A + 1, B, C)) :|: TRUE evalrealselectreturnin(A, B, C) -> Com_1(evalrealselectstop(A, B, C)) :|: TRUE The start-symbols are:[evalrealselectstart_3] ---------------------------------------- (1) Koat Proof (FINISHED) YES(?, 72*Ar_1 + 9*Ar_1^2 + 69) Initial complexity problem: 1: T: (Comp: ?, Cost: 1) evalrealselectstart(Ar_0, Ar_1, Ar_2) -> Com_1(evalrealselectentryin(Ar_0, Ar_1, Ar_2)) (Comp: ?, Cost: 1) evalrealselectentryin(Ar_0, Ar_1, Ar_2) -> Com_1(evalrealselectbb6in(0, Ar_1, Ar_2)) (Comp: ?, Cost: 1) evalrealselectbb6in(Ar_0, Ar_1, Ar_2) -> Com_1(evalrealselectbbin(Ar_0, Ar_1, Ar_2)) [ Ar_1 >= Ar_0 + 2 ] (Comp: ?, Cost: 1) evalrealselectbb6in(Ar_0, Ar_1, Ar_2) -> Com_1(evalrealselectreturnin(Ar_0, Ar_1, Ar_2)) [ Ar_0 + 1 >= Ar_1 ] (Comp: ?, Cost: 1) evalrealselectbbin(Ar_0, Ar_1, Ar_2) -> Com_1(evalrealselectbb4in(Ar_0, Ar_1, Ar_0 + 1)) (Comp: ?, Cost: 1) evalrealselectbb4in(Ar_0, Ar_1, Ar_2) -> Com_1(evalrealselectbb1in(Ar_0, Ar_1, Ar_2)) [ Ar_1 >= Ar_2 + 1 ] (Comp: ?, Cost: 1) evalrealselectbb4in(Ar_0, Ar_1, Ar_2) -> Com_1(evalrealselectbb5in(Ar_0, Ar_1, Ar_2)) [ Ar_2 >= Ar_1 ] (Comp: ?, Cost: 1) evalrealselectbb1in(Ar_0, Ar_1, Ar_2) -> Com_1(evalrealselectbb4in(Ar_0, Ar_1, Ar_2 + 1)) [ D >= E + 1 ] (Comp: ?, Cost: 1) evalrealselectbb1in(Ar_0, Ar_1, Ar_2) -> Com_1(evalrealselectbb4in(Ar_0, Ar_1, Ar_2 + 1)) [ D >= E ] (Comp: ?, Cost: 1) evalrealselectbb5in(Ar_0, Ar_1, Ar_2) -> Com_1(evalrealselectbb6in(Ar_0 + 1, Ar_1, Ar_2)) (Comp: ?, Cost: 1) evalrealselectreturnin(Ar_0, Ar_1, Ar_2) -> Com_1(evalrealselectstop(Ar_0, Ar_1, Ar_2)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2) -> Com_1(evalrealselectstart(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) evalrealselectstart(Ar_0, Ar_1, Ar_2) -> Com_1(evalrealselectentryin(Ar_0, Ar_1, Ar_2)) (Comp: 1, Cost: 1) evalrealselectentryin(Ar_0, Ar_1, Ar_2) -> Com_1(evalrealselectbb6in(0, Ar_1, Ar_2)) (Comp: ?, Cost: 1) evalrealselectbb6in(Ar_0, Ar_1, Ar_2) -> Com_1(evalrealselectbbin(Ar_0, Ar_1, Ar_2)) [ Ar_1 >= Ar_0 + 2 ] (Comp: ?, Cost: 1) evalrealselectbb6in(Ar_0, Ar_1, Ar_2) -> Com_1(evalrealselectreturnin(Ar_0, Ar_1, Ar_2)) [ Ar_0 + 1 >= Ar_1 ] (Comp: ?, Cost: 1) evalrealselectbbin(Ar_0, Ar_1, Ar_2) -> Com_1(evalrealselectbb4in(Ar_0, Ar_1, Ar_0 + 1)) (Comp: ?, Cost: 1) evalrealselectbb4in(Ar_0, Ar_1, Ar_2) -> Com_1(evalrealselectbb1in(Ar_0, Ar_1, Ar_2)) [ Ar_1 >= Ar_2 + 1 ] (Comp: ?, Cost: 1) evalrealselectbb4in(Ar_0, Ar_1, Ar_2) -> Com_1(evalrealselectbb5in(Ar_0, Ar_1, Ar_2)) [ Ar_2 >= Ar_1 ] (Comp: ?, Cost: 1) evalrealselectbb1in(Ar_0, Ar_1, Ar_2) -> Com_1(evalrealselectbb4in(Ar_0, Ar_1, Ar_2 + 1)) [ D >= E + 1 ] (Comp: ?, Cost: 1) evalrealselectbb1in(Ar_0, Ar_1, Ar_2) -> Com_1(evalrealselectbb4in(Ar_0, Ar_1, Ar_2 + 1)) [ D >= E ] (Comp: ?, Cost: 1) evalrealselectbb5in(Ar_0, Ar_1, Ar_2) -> Com_1(evalrealselectbb6in(Ar_0 + 1, Ar_1, Ar_2)) (Comp: ?, Cost: 1) evalrealselectreturnin(Ar_0, Ar_1, Ar_2) -> Com_1(evalrealselectstop(Ar_0, Ar_1, Ar_2)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2) -> Com_1(evalrealselectstart(Ar_0, Ar_1, Ar_2)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 A polynomial rank function with Pol(evalrealselectstart) = 2 Pol(evalrealselectentryin) = 2 Pol(evalrealselectbb6in) = 2 Pol(evalrealselectbbin) = 2 Pol(evalrealselectreturnin) = 1 Pol(evalrealselectbb4in) = 2 Pol(evalrealselectbb1in) = 2 Pol(evalrealselectbb5in) = 2 Pol(evalrealselectstop) = 0 Pol(koat_start) = 2 orients all transitions weakly and the transitions evalrealselectreturnin(Ar_0, Ar_1, Ar_2) -> Com_1(evalrealselectstop(Ar_0, Ar_1, Ar_2)) evalrealselectbb6in(Ar_0, Ar_1, Ar_2) -> Com_1(evalrealselectreturnin(Ar_0, Ar_1, Ar_2)) [ Ar_0 + 1 >= Ar_1 ] strictly and produces the following problem: 3: T: (Comp: 1, Cost: 1) evalrealselectstart(Ar_0, Ar_1, Ar_2) -> Com_1(evalrealselectentryin(Ar_0, Ar_1, Ar_2)) (Comp: 1, Cost: 1) evalrealselectentryin(Ar_0, Ar_1, Ar_2) -> Com_1(evalrealselectbb6in(0, Ar_1, Ar_2)) (Comp: ?, Cost: 1) evalrealselectbb6in(Ar_0, Ar_1, Ar_2) -> Com_1(evalrealselectbbin(Ar_0, Ar_1, Ar_2)) [ Ar_1 >= Ar_0 + 2 ] (Comp: 2, Cost: 1) evalrealselectbb6in(Ar_0, Ar_1, Ar_2) -> Com_1(evalrealselectreturnin(Ar_0, Ar_1, Ar_2)) [ Ar_0 + 1 >= Ar_1 ] (Comp: ?, Cost: 1) evalrealselectbbin(Ar_0, Ar_1, Ar_2) -> Com_1(evalrealselectbb4in(Ar_0, Ar_1, Ar_0 + 1)) (Comp: ?, Cost: 1) evalrealselectbb4in(Ar_0, Ar_1, Ar_2) -> Com_1(evalrealselectbb1in(Ar_0, Ar_1, Ar_2)) [ Ar_1 >= Ar_2 + 1 ] (Comp: ?, Cost: 1) evalrealselectbb4in(Ar_0, Ar_1, Ar_2) -> Com_1(evalrealselectbb5in(Ar_0, Ar_1, Ar_2)) [ Ar_2 >= Ar_1 ] (Comp: ?, Cost: 1) evalrealselectbb1in(Ar_0, Ar_1, Ar_2) -> Com_1(evalrealselectbb4in(Ar_0, Ar_1, Ar_2 + 1)) [ D >= E + 1 ] (Comp: ?, Cost: 1) evalrealselectbb1in(Ar_0, Ar_1, Ar_2) -> Com_1(evalrealselectbb4in(Ar_0, Ar_1, Ar_2 + 1)) [ D >= E ] (Comp: ?, Cost: 1) evalrealselectbb5in(Ar_0, Ar_1, Ar_2) -> Com_1(evalrealselectbb6in(Ar_0 + 1, Ar_1, Ar_2)) (Comp: 2, Cost: 1) evalrealselectreturnin(Ar_0, Ar_1, Ar_2) -> Com_1(evalrealselectstop(Ar_0, Ar_1, Ar_2)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2) -> Com_1(evalrealselectstart(Ar_0, Ar_1, Ar_2)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 A polynomial rank function with Pol(evalrealselectstart) = V_2 + 1 Pol(evalrealselectentryin) = V_2 + 1 Pol(evalrealselectbb6in) = -V_1 + V_2 + 1 Pol(evalrealselectbbin) = -V_1 + V_2 Pol(evalrealselectreturnin) = -V_1 + V_2 Pol(evalrealselectbb4in) = -V_1 + V_2 Pol(evalrealselectbb1in) = -V_1 + V_2 Pol(evalrealselectbb5in) = -V_1 + V_2 Pol(evalrealselectstop) = -V_1 + V_2 Pol(koat_start) = V_2 + 1 orients all transitions weakly and the transition evalrealselectbb6in(Ar_0, Ar_1, Ar_2) -> Com_1(evalrealselectbbin(Ar_0, Ar_1, Ar_2)) [ Ar_1 >= Ar_0 + 2 ] strictly and produces the following problem: 4: T: (Comp: 1, Cost: 1) evalrealselectstart(Ar_0, Ar_1, Ar_2) -> Com_1(evalrealselectentryin(Ar_0, Ar_1, Ar_2)) (Comp: 1, Cost: 1) evalrealselectentryin(Ar_0, Ar_1, Ar_2) -> Com_1(evalrealselectbb6in(0, Ar_1, Ar_2)) (Comp: Ar_1 + 1, Cost: 1) evalrealselectbb6in(Ar_0, Ar_1, Ar_2) -> Com_1(evalrealselectbbin(Ar_0, Ar_1, Ar_2)) [ Ar_1 >= Ar_0 + 2 ] (Comp: 2, Cost: 1) evalrealselectbb6in(Ar_0, Ar_1, Ar_2) -> Com_1(evalrealselectreturnin(Ar_0, Ar_1, Ar_2)) [ Ar_0 + 1 >= Ar_1 ] (Comp: ?, Cost: 1) evalrealselectbbin(Ar_0, Ar_1, Ar_2) -> Com_1(evalrealselectbb4in(Ar_0, Ar_1, Ar_0 + 1)) (Comp: ?, Cost: 1) evalrealselectbb4in(Ar_0, Ar_1, Ar_2) -> Com_1(evalrealselectbb1in(Ar_0, Ar_1, Ar_2)) [ Ar_1 >= Ar_2 + 1 ] (Comp: ?, Cost: 1) evalrealselectbb4in(Ar_0, Ar_1, Ar_2) -> Com_1(evalrealselectbb5in(Ar_0, Ar_1, Ar_2)) [ Ar_2 >= Ar_1 ] (Comp: ?, Cost: 1) evalrealselectbb1in(Ar_0, Ar_1, Ar_2) -> Com_1(evalrealselectbb4in(Ar_0, Ar_1, Ar_2 + 1)) [ D >= E + 1 ] (Comp: ?, Cost: 1) evalrealselectbb1in(Ar_0, Ar_1, Ar_2) -> Com_1(evalrealselectbb4in(Ar_0, Ar_1, Ar_2 + 1)) [ D >= E ] (Comp: ?, Cost: 1) evalrealselectbb5in(Ar_0, Ar_1, Ar_2) -> Com_1(evalrealselectbb6in(Ar_0 + 1, Ar_1, Ar_2)) (Comp: 2, Cost: 1) evalrealselectreturnin(Ar_0, Ar_1, Ar_2) -> Com_1(evalrealselectstop(Ar_0, Ar_1, Ar_2)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2) -> Com_1(evalrealselectstart(Ar_0, Ar_1, Ar_2)) [ 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) evalrealselectstart(Ar_0, Ar_1, Ar_2) -> Com_1(evalrealselectentryin(Ar_0, Ar_1, Ar_2)) (Comp: 1, Cost: 1) evalrealselectentryin(Ar_0, Ar_1, Ar_2) -> Com_1(evalrealselectbb6in(0, Ar_1, Ar_2)) (Comp: Ar_1 + 1, Cost: 1) evalrealselectbb6in(Ar_0, Ar_1, Ar_2) -> Com_1(evalrealselectbbin(Ar_0, Ar_1, Ar_2)) [ Ar_1 >= Ar_0 + 2 ] (Comp: 2, Cost: 1) evalrealselectbb6in(Ar_0, Ar_1, Ar_2) -> Com_1(evalrealselectreturnin(Ar_0, Ar_1, Ar_2)) [ Ar_0 + 1 >= Ar_1 ] (Comp: Ar_1 + 1, Cost: 1) evalrealselectbbin(Ar_0, Ar_1, Ar_2) -> Com_1(evalrealselectbb4in(Ar_0, Ar_1, Ar_0 + 1)) (Comp: ?, Cost: 1) evalrealselectbb4in(Ar_0, Ar_1, Ar_2) -> Com_1(evalrealselectbb1in(Ar_0, Ar_1, Ar_2)) [ Ar_1 >= Ar_2 + 1 ] (Comp: ?, Cost: 1) evalrealselectbb4in(Ar_0, Ar_1, Ar_2) -> Com_1(evalrealselectbb5in(Ar_0, Ar_1, Ar_2)) [ Ar_2 >= Ar_1 ] (Comp: ?, Cost: 1) evalrealselectbb1in(Ar_0, Ar_1, Ar_2) -> Com_1(evalrealselectbb4in(Ar_0, Ar_1, Ar_2 + 1)) [ D >= E + 1 ] (Comp: ?, Cost: 1) evalrealselectbb1in(Ar_0, Ar_1, Ar_2) -> Com_1(evalrealselectbb4in(Ar_0, Ar_1, Ar_2 + 1)) [ D >= E ] (Comp: ?, Cost: 1) evalrealselectbb5in(Ar_0, Ar_1, Ar_2) -> Com_1(evalrealselectbb6in(Ar_0 + 1, Ar_1, Ar_2)) (Comp: 2, Cost: 1) evalrealselectreturnin(Ar_0, Ar_1, Ar_2) -> Com_1(evalrealselectstop(Ar_0, Ar_1, Ar_2)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2) -> Com_1(evalrealselectstart(Ar_0, Ar_1, Ar_2)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 A polynomial rank function with Pol(evalrealselectbb5in) = 1 Pol(evalrealselectbb6in) = 0 Pol(evalrealselectbb4in) = 2 Pol(evalrealselectbb1in) = 2 and size complexities S("koat_start(Ar_0, Ar_1, Ar_2) -> Com_1(evalrealselectstart(Ar_0, Ar_1, Ar_2)) [ 0 <= 0 ]", 0-0) = Ar_0 S("koat_start(Ar_0, Ar_1, Ar_2) -> Com_1(evalrealselectstart(Ar_0, Ar_1, Ar_2)) [ 0 <= 0 ]", 0-1) = Ar_1 S("koat_start(Ar_0, Ar_1, Ar_2) -> Com_1(evalrealselectstart(Ar_0, Ar_1, Ar_2)) [ 0 <= 0 ]", 0-2) = Ar_2 S("evalrealselectreturnin(Ar_0, Ar_1, Ar_2) -> Com_1(evalrealselectstop(Ar_0, Ar_1, Ar_2))", 0-0) = ? S("evalrealselectreturnin(Ar_0, Ar_1, Ar_2) -> Com_1(evalrealselectstop(Ar_0, Ar_1, Ar_2))", 0-1) = Ar_1 S("evalrealselectreturnin(Ar_0, Ar_1, Ar_2) -> Com_1(evalrealselectstop(Ar_0, Ar_1, Ar_2))", 0-2) = ? S("evalrealselectbb5in(Ar_0, Ar_1, Ar_2) -> Com_1(evalrealselectbb6in(Ar_0 + 1, Ar_1, Ar_2))", 0-0) = ? S("evalrealselectbb5in(Ar_0, Ar_1, Ar_2) -> Com_1(evalrealselectbb6in(Ar_0 + 1, Ar_1, Ar_2))", 0-1) = Ar_1 S("evalrealselectbb5in(Ar_0, Ar_1, Ar_2) -> Com_1(evalrealselectbb6in(Ar_0 + 1, Ar_1, Ar_2))", 0-2) = ? S("evalrealselectbb1in(Ar_0, Ar_1, Ar_2) -> Com_1(evalrealselectbb4in(Ar_0, Ar_1, Ar_2 + 1)) [ D >= E ]", 0-0) = ? S("evalrealselectbb1in(Ar_0, Ar_1, Ar_2) -> Com_1(evalrealselectbb4in(Ar_0, Ar_1, Ar_2 + 1)) [ D >= E ]", 0-1) = Ar_1 S("evalrealselectbb1in(Ar_0, Ar_1, Ar_2) -> Com_1(evalrealselectbb4in(Ar_0, Ar_1, Ar_2 + 1)) [ D >= E ]", 0-2) = ? S("evalrealselectbb1in(Ar_0, Ar_1, Ar_2) -> Com_1(evalrealselectbb4in(Ar_0, Ar_1, Ar_2 + 1)) [ D >= E + 1 ]", 0-0) = ? S("evalrealselectbb1in(Ar_0, Ar_1, Ar_2) -> Com_1(evalrealselectbb4in(Ar_0, Ar_1, Ar_2 + 1)) [ D >= E + 1 ]", 0-1) = Ar_1 S("evalrealselectbb1in(Ar_0, Ar_1, Ar_2) -> Com_1(evalrealselectbb4in(Ar_0, Ar_1, Ar_2 + 1)) [ D >= E + 1 ]", 0-2) = ? S("evalrealselectbb4in(Ar_0, Ar_1, Ar_2) -> Com_1(evalrealselectbb5in(Ar_0, Ar_1, Ar_2)) [ Ar_2 >= Ar_1 ]", 0-0) = ? S("evalrealselectbb4in(Ar_0, Ar_1, Ar_2) -> Com_1(evalrealselectbb5in(Ar_0, Ar_1, Ar_2)) [ Ar_2 >= Ar_1 ]", 0-1) = Ar_1 S("evalrealselectbb4in(Ar_0, Ar_1, Ar_2) -> Com_1(evalrealselectbb5in(Ar_0, Ar_1, Ar_2)) [ Ar_2 >= Ar_1 ]", 0-2) = ? S("evalrealselectbb4in(Ar_0, Ar_1, Ar_2) -> Com_1(evalrealselectbb1in(Ar_0, Ar_1, Ar_2)) [ Ar_1 >= Ar_2 + 1 ]", 0-0) = ? S("evalrealselectbb4in(Ar_0, Ar_1, Ar_2) -> Com_1(evalrealselectbb1in(Ar_0, Ar_1, Ar_2)) [ Ar_1 >= Ar_2 + 1 ]", 0-1) = Ar_1 S("evalrealselectbb4in(Ar_0, Ar_1, Ar_2) -> Com_1(evalrealselectbb1in(Ar_0, Ar_1, Ar_2)) [ Ar_1 >= Ar_2 + 1 ]", 0-2) = ? S("evalrealselectbbin(Ar_0, Ar_1, Ar_2) -> Com_1(evalrealselectbb4in(Ar_0, Ar_1, Ar_0 + 1))", 0-0) = ? S("evalrealselectbbin(Ar_0, Ar_1, Ar_2) -> Com_1(evalrealselectbb4in(Ar_0, Ar_1, Ar_0 + 1))", 0-1) = Ar_1 S("evalrealselectbbin(Ar_0, Ar_1, Ar_2) -> Com_1(evalrealselectbb4in(Ar_0, Ar_1, Ar_0 + 1))", 0-2) = ? S("evalrealselectbb6in(Ar_0, Ar_1, Ar_2) -> Com_1(evalrealselectreturnin(Ar_0, Ar_1, Ar_2)) [ Ar_0 + 1 >= Ar_1 ]", 0-0) = ? S("evalrealselectbb6in(Ar_0, Ar_1, Ar_2) -> Com_1(evalrealselectreturnin(Ar_0, Ar_1, Ar_2)) [ Ar_0 + 1 >= Ar_1 ]", 0-1) = Ar_1 S("evalrealselectbb6in(Ar_0, Ar_1, Ar_2) -> Com_1(evalrealselectreturnin(Ar_0, Ar_1, Ar_2)) [ Ar_0 + 1 >= Ar_1 ]", 0-2) = ? S("evalrealselectbb6in(Ar_0, Ar_1, Ar_2) -> Com_1(evalrealselectbbin(Ar_0, Ar_1, Ar_2)) [ Ar_1 >= Ar_0 + 2 ]", 0-0) = ? S("evalrealselectbb6in(Ar_0, Ar_1, Ar_2) -> Com_1(evalrealselectbbin(Ar_0, Ar_1, Ar_2)) [ Ar_1 >= Ar_0 + 2 ]", 0-1) = Ar_1 S("evalrealselectbb6in(Ar_0, Ar_1, Ar_2) -> Com_1(evalrealselectbbin(Ar_0, Ar_1, Ar_2)) [ Ar_1 >= Ar_0 + 2 ]", 0-2) = ? S("evalrealselectentryin(Ar_0, Ar_1, Ar_2) -> Com_1(evalrealselectbb6in(0, Ar_1, Ar_2))", 0-0) = 0 S("evalrealselectentryin(Ar_0, Ar_1, Ar_2) -> Com_1(evalrealselectbb6in(0, Ar_1, Ar_2))", 0-1) = Ar_1 S("evalrealselectentryin(Ar_0, Ar_1, Ar_2) -> Com_1(evalrealselectbb6in(0, Ar_1, Ar_2))", 0-2) = Ar_2 S("evalrealselectstart(Ar_0, Ar_1, Ar_2) -> Com_1(evalrealselectentryin(Ar_0, Ar_1, Ar_2))", 0-0) = Ar_0 S("evalrealselectstart(Ar_0, Ar_1, Ar_2) -> Com_1(evalrealselectentryin(Ar_0, Ar_1, Ar_2))", 0-1) = Ar_1 S("evalrealselectstart(Ar_0, Ar_1, Ar_2) -> Com_1(evalrealselectentryin(Ar_0, Ar_1, Ar_2))", 0-2) = Ar_2 orients the transitions evalrealselectbb5in(Ar_0, Ar_1, Ar_2) -> Com_1(evalrealselectbb6in(Ar_0 + 1, Ar_1, Ar_2)) evalrealselectbb4in(Ar_0, Ar_1, Ar_2) -> Com_1(evalrealselectbb5in(Ar_0, Ar_1, Ar_2)) [ Ar_2 >= Ar_1 ] evalrealselectbb4in(Ar_0, Ar_1, Ar_2) -> Com_1(evalrealselectbb1in(Ar_0, Ar_1, Ar_2)) [ Ar_1 >= Ar_2 + 1 ] evalrealselectbb1in(Ar_0, Ar_1, Ar_2) -> Com_1(evalrealselectbb4in(Ar_0, Ar_1, Ar_2 + 1)) [ D >= E + 1 ] evalrealselectbb1in(Ar_0, Ar_1, Ar_2) -> Com_1(evalrealselectbb4in(Ar_0, Ar_1, Ar_2 + 1)) [ D >= E ] weakly and the transitions evalrealselectbb5in(Ar_0, Ar_1, Ar_2) -> Com_1(evalrealselectbb6in(Ar_0 + 1, Ar_1, Ar_2)) evalrealselectbb4in(Ar_0, Ar_1, Ar_2) -> Com_1(evalrealselectbb5in(Ar_0, Ar_1, Ar_2)) [ Ar_2 >= Ar_1 ] strictly and produces the following problem: 6: T: (Comp: 1, Cost: 1) evalrealselectstart(Ar_0, Ar_1, Ar_2) -> Com_1(evalrealselectentryin(Ar_0, Ar_1, Ar_2)) (Comp: 1, Cost: 1) evalrealselectentryin(Ar_0, Ar_1, Ar_2) -> Com_1(evalrealselectbb6in(0, Ar_1, Ar_2)) (Comp: Ar_1 + 1, Cost: 1) evalrealselectbb6in(Ar_0, Ar_1, Ar_2) -> Com_1(evalrealselectbbin(Ar_0, Ar_1, Ar_2)) [ Ar_1 >= Ar_0 + 2 ] (Comp: 2, Cost: 1) evalrealselectbb6in(Ar_0, Ar_1, Ar_2) -> Com_1(evalrealselectreturnin(Ar_0, Ar_1, Ar_2)) [ Ar_0 + 1 >= Ar_1 ] (Comp: Ar_1 + 1, Cost: 1) evalrealselectbbin(Ar_0, Ar_1, Ar_2) -> Com_1(evalrealselectbb4in(Ar_0, Ar_1, Ar_0 + 1)) (Comp: ?, Cost: 1) evalrealselectbb4in(Ar_0, Ar_1, Ar_2) -> Com_1(evalrealselectbb1in(Ar_0, Ar_1, Ar_2)) [ Ar_1 >= Ar_2 + 1 ] (Comp: 2*Ar_1 + 2, Cost: 1) evalrealselectbb4in(Ar_0, Ar_1, Ar_2) -> Com_1(evalrealselectbb5in(Ar_0, Ar_1, Ar_2)) [ Ar_2 >= Ar_1 ] (Comp: ?, Cost: 1) evalrealselectbb1in(Ar_0, Ar_1, Ar_2) -> Com_1(evalrealselectbb4in(Ar_0, Ar_1, Ar_2 + 1)) [ D >= E + 1 ] (Comp: ?, Cost: 1) evalrealselectbb1in(Ar_0, Ar_1, Ar_2) -> Com_1(evalrealselectbb4in(Ar_0, Ar_1, Ar_2 + 1)) [ D >= E ] (Comp: 2*Ar_1 + 2, Cost: 1) evalrealselectbb5in(Ar_0, Ar_1, Ar_2) -> Com_1(evalrealselectbb6in(Ar_0 + 1, Ar_1, Ar_2)) (Comp: 2, Cost: 1) evalrealselectreturnin(Ar_0, Ar_1, Ar_2) -> Com_1(evalrealselectstop(Ar_0, Ar_1, Ar_2)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2) -> Com_1(evalrealselectstart(Ar_0, Ar_1, Ar_2)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 A polynomial rank function with Pol(evalrealselectbb4in) = V_2 - V_3 + 1 Pol(evalrealselectbb1in) = V_2 - V_3 and size complexities S("koat_start(Ar_0, Ar_1, Ar_2) -> Com_1(evalrealselectstart(Ar_0, Ar_1, Ar_2)) [ 0 <= 0 ]", 0-0) = Ar_0 S("koat_start(Ar_0, Ar_1, Ar_2) -> Com_1(evalrealselectstart(Ar_0, Ar_1, Ar_2)) [ 0 <= 0 ]", 0-1) = Ar_1 S("koat_start(Ar_0, Ar_1, Ar_2) -> Com_1(evalrealselectstart(Ar_0, Ar_1, Ar_2)) [ 0 <= 0 ]", 0-2) = Ar_2 S("evalrealselectreturnin(Ar_0, Ar_1, Ar_2) -> Com_1(evalrealselectstop(Ar_0, Ar_1, Ar_2))", 0-0) = 2*Ar_1 + 32 S("evalrealselectreturnin(Ar_0, Ar_1, Ar_2) -> Com_1(evalrealselectstop(Ar_0, Ar_1, Ar_2))", 0-1) = Ar_1 S("evalrealselectreturnin(Ar_0, Ar_1, Ar_2) -> Com_1(evalrealselectstop(Ar_0, Ar_1, Ar_2))", 0-2) = ? S("evalrealselectbb5in(Ar_0, Ar_1, Ar_2) -> Com_1(evalrealselectbb6in(Ar_0 + 1, Ar_1, Ar_2))", 0-0) = 2*Ar_1 + 8 S("evalrealselectbb5in(Ar_0, Ar_1, Ar_2) -> Com_1(evalrealselectbb6in(Ar_0 + 1, Ar_1, Ar_2))", 0-1) = Ar_1 S("evalrealselectbb5in(Ar_0, Ar_1, Ar_2) -> Com_1(evalrealselectbb6in(Ar_0 + 1, Ar_1, Ar_2))", 0-2) = ? S("evalrealselectbb1in(Ar_0, Ar_1, Ar_2) -> Com_1(evalrealselectbb4in(Ar_0, Ar_1, Ar_2 + 1)) [ D >= E ]", 0-0) = 2*Ar_1 + 8 S("evalrealselectbb1in(Ar_0, Ar_1, Ar_2) -> Com_1(evalrealselectbb4in(Ar_0, Ar_1, Ar_2 + 1)) [ D >= E ]", 0-1) = Ar_1 S("evalrealselectbb1in(Ar_0, Ar_1, Ar_2) -> Com_1(evalrealselectbb4in(Ar_0, Ar_1, Ar_2 + 1)) [ D >= E ]", 0-2) = ? S("evalrealselectbb1in(Ar_0, Ar_1, Ar_2) -> Com_1(evalrealselectbb4in(Ar_0, Ar_1, Ar_2 + 1)) [ D >= E + 1 ]", 0-0) = 2*Ar_1 + 8 S("evalrealselectbb1in(Ar_0, Ar_1, Ar_2) -> Com_1(evalrealselectbb4in(Ar_0, Ar_1, Ar_2 + 1)) [ D >= E + 1 ]", 0-1) = Ar_1 S("evalrealselectbb1in(Ar_0, Ar_1, Ar_2) -> Com_1(evalrealselectbb4in(Ar_0, Ar_1, Ar_2 + 1)) [ D >= E + 1 ]", 0-2) = ? S("evalrealselectbb4in(Ar_0, Ar_1, Ar_2) -> Com_1(evalrealselectbb5in(Ar_0, Ar_1, Ar_2)) [ Ar_2 >= Ar_1 ]", 0-0) = 2*Ar_1 + 8 S("evalrealselectbb4in(Ar_0, Ar_1, Ar_2) -> Com_1(evalrealselectbb5in(Ar_0, Ar_1, Ar_2)) [ Ar_2 >= Ar_1 ]", 0-1) = Ar_1 S("evalrealselectbb4in(Ar_0, Ar_1, Ar_2) -> Com_1(evalrealselectbb5in(Ar_0, Ar_1, Ar_2)) [ Ar_2 >= Ar_1 ]", 0-2) = ? S("evalrealselectbb4in(Ar_0, Ar_1, Ar_2) -> Com_1(evalrealselectbb1in(Ar_0, Ar_1, Ar_2)) [ Ar_1 >= Ar_2 + 1 ]", 0-0) = 2*Ar_1 + 8 S("evalrealselectbb4in(Ar_0, Ar_1, Ar_2) -> Com_1(evalrealselectbb1in(Ar_0, Ar_1, Ar_2)) [ Ar_1 >= Ar_2 + 1 ]", 0-1) = Ar_1 S("evalrealselectbb4in(Ar_0, Ar_1, Ar_2) -> Com_1(evalrealselectbb1in(Ar_0, Ar_1, Ar_2)) [ Ar_1 >= Ar_2 + 1 ]", 0-2) = ? S("evalrealselectbbin(Ar_0, Ar_1, Ar_2) -> Com_1(evalrealselectbb4in(Ar_0, Ar_1, Ar_0 + 1))", 0-0) = 2*Ar_1 + 8 S("evalrealselectbbin(Ar_0, Ar_1, Ar_2) -> Com_1(evalrealselectbb4in(Ar_0, Ar_1, Ar_0 + 1))", 0-1) = Ar_1 S("evalrealselectbbin(Ar_0, Ar_1, Ar_2) -> Com_1(evalrealselectbb4in(Ar_0, Ar_1, Ar_0 + 1))", 0-2) = 2*Ar_1 + 18 S("evalrealselectbb6in(Ar_0, Ar_1, Ar_2) -> Com_1(evalrealselectreturnin(Ar_0, Ar_1, Ar_2)) [ Ar_0 + 1 >= Ar_1 ]", 0-0) = 2*Ar_1 + 16 S("evalrealselectbb6in(Ar_0, Ar_1, Ar_2) -> Com_1(evalrealselectreturnin(Ar_0, Ar_1, Ar_2)) [ Ar_0 + 1 >= Ar_1 ]", 0-1) = Ar_1 S("evalrealselectbb6in(Ar_0, Ar_1, Ar_2) -> Com_1(evalrealselectreturnin(Ar_0, Ar_1, Ar_2)) [ Ar_0 + 1 >= Ar_1 ]", 0-2) = ? S("evalrealselectbb6in(Ar_0, Ar_1, Ar_2) -> Com_1(evalrealselectbbin(Ar_0, Ar_1, Ar_2)) [ Ar_1 >= Ar_0 + 2 ]", 0-0) = 2*Ar_1 + 8 S("evalrealselectbb6in(Ar_0, Ar_1, Ar_2) -> Com_1(evalrealselectbbin(Ar_0, Ar_1, Ar_2)) [ Ar_1 >= Ar_0 + 2 ]", 0-1) = Ar_1 S("evalrealselectbb6in(Ar_0, Ar_1, Ar_2) -> Com_1(evalrealselectbbin(Ar_0, Ar_1, Ar_2)) [ Ar_1 >= Ar_0 + 2 ]", 0-2) = ? S("evalrealselectentryin(Ar_0, Ar_1, Ar_2) -> Com_1(evalrealselectbb6in(0, Ar_1, Ar_2))", 0-0) = 0 S("evalrealselectentryin(Ar_0, Ar_1, Ar_2) -> Com_1(evalrealselectbb6in(0, Ar_1, Ar_2))", 0-1) = Ar_1 S("evalrealselectentryin(Ar_0, Ar_1, Ar_2) -> Com_1(evalrealselectbb6in(0, Ar_1, Ar_2))", 0-2) = Ar_2 S("evalrealselectstart(Ar_0, Ar_1, Ar_2) -> Com_1(evalrealselectentryin(Ar_0, Ar_1, Ar_2))", 0-0) = Ar_0 S("evalrealselectstart(Ar_0, Ar_1, Ar_2) -> Com_1(evalrealselectentryin(Ar_0, Ar_1, Ar_2))", 0-1) = Ar_1 S("evalrealselectstart(Ar_0, Ar_1, Ar_2) -> Com_1(evalrealselectentryin(Ar_0, Ar_1, Ar_2))", 0-2) = Ar_2 orients the transitions evalrealselectbb4in(Ar_0, Ar_1, Ar_2) -> Com_1(evalrealselectbb1in(Ar_0, Ar_1, Ar_2)) [ Ar_1 >= Ar_2 + 1 ] evalrealselectbb1in(Ar_0, Ar_1, Ar_2) -> Com_1(evalrealselectbb4in(Ar_0, Ar_1, Ar_2 + 1)) [ D >= E + 1 ] evalrealselectbb1in(Ar_0, Ar_1, Ar_2) -> Com_1(evalrealselectbb4in(Ar_0, Ar_1, Ar_2 + 1)) [ D >= E ] weakly and the transition evalrealselectbb4in(Ar_0, Ar_1, Ar_2) -> Com_1(evalrealselectbb1in(Ar_0, Ar_1, Ar_2)) [ Ar_1 >= Ar_2 + 1 ] strictly and produces the following problem: 7: T: (Comp: 1, Cost: 1) evalrealselectstart(Ar_0, Ar_1, Ar_2) -> Com_1(evalrealselectentryin(Ar_0, Ar_1, Ar_2)) (Comp: 1, Cost: 1) evalrealselectentryin(Ar_0, Ar_1, Ar_2) -> Com_1(evalrealselectbb6in(0, Ar_1, Ar_2)) (Comp: Ar_1 + 1, Cost: 1) evalrealselectbb6in(Ar_0, Ar_1, Ar_2) -> Com_1(evalrealselectbbin(Ar_0, Ar_1, Ar_2)) [ Ar_1 >= Ar_0 + 2 ] (Comp: 2, Cost: 1) evalrealselectbb6in(Ar_0, Ar_1, Ar_2) -> Com_1(evalrealselectreturnin(Ar_0, Ar_1, Ar_2)) [ Ar_0 + 1 >= Ar_1 ] (Comp: Ar_1 + 1, Cost: 1) evalrealselectbbin(Ar_0, Ar_1, Ar_2) -> Com_1(evalrealselectbb4in(Ar_0, Ar_1, Ar_0 + 1)) (Comp: 3*Ar_1^2 + 22*Ar_1 + 19, Cost: 1) evalrealselectbb4in(Ar_0, Ar_1, Ar_2) -> Com_1(evalrealselectbb1in(Ar_0, Ar_1, Ar_2)) [ Ar_1 >= Ar_2 + 1 ] (Comp: 2*Ar_1 + 2, Cost: 1) evalrealselectbb4in(Ar_0, Ar_1, Ar_2) -> Com_1(evalrealselectbb5in(Ar_0, Ar_1, Ar_2)) [ Ar_2 >= Ar_1 ] (Comp: ?, Cost: 1) evalrealselectbb1in(Ar_0, Ar_1, Ar_2) -> Com_1(evalrealselectbb4in(Ar_0, Ar_1, Ar_2 + 1)) [ D >= E + 1 ] (Comp: ?, Cost: 1) evalrealselectbb1in(Ar_0, Ar_1, Ar_2) -> Com_1(evalrealselectbb4in(Ar_0, Ar_1, Ar_2 + 1)) [ D >= E ] (Comp: 2*Ar_1 + 2, Cost: 1) evalrealselectbb5in(Ar_0, Ar_1, Ar_2) -> Com_1(evalrealselectbb6in(Ar_0 + 1, Ar_1, Ar_2)) (Comp: 2, Cost: 1) evalrealselectreturnin(Ar_0, Ar_1, Ar_2) -> Com_1(evalrealselectstop(Ar_0, Ar_1, Ar_2)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2) -> Com_1(evalrealselectstart(Ar_0, Ar_1, Ar_2)) [ 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) evalrealselectstart(Ar_0, Ar_1, Ar_2) -> Com_1(evalrealselectentryin(Ar_0, Ar_1, Ar_2)) (Comp: 1, Cost: 1) evalrealselectentryin(Ar_0, Ar_1, Ar_2) -> Com_1(evalrealselectbb6in(0, Ar_1, Ar_2)) (Comp: Ar_1 + 1, Cost: 1) evalrealselectbb6in(Ar_0, Ar_1, Ar_2) -> Com_1(evalrealselectbbin(Ar_0, Ar_1, Ar_2)) [ Ar_1 >= Ar_0 + 2 ] (Comp: 2, Cost: 1) evalrealselectbb6in(Ar_0, Ar_1, Ar_2) -> Com_1(evalrealselectreturnin(Ar_0, Ar_1, Ar_2)) [ Ar_0 + 1 >= Ar_1 ] (Comp: Ar_1 + 1, Cost: 1) evalrealselectbbin(Ar_0, Ar_1, Ar_2) -> Com_1(evalrealselectbb4in(Ar_0, Ar_1, Ar_0 + 1)) (Comp: 3*Ar_1^2 + 22*Ar_1 + 19, Cost: 1) evalrealselectbb4in(Ar_0, Ar_1, Ar_2) -> Com_1(evalrealselectbb1in(Ar_0, Ar_1, Ar_2)) [ Ar_1 >= Ar_2 + 1 ] (Comp: 2*Ar_1 + 2, Cost: 1) evalrealselectbb4in(Ar_0, Ar_1, Ar_2) -> Com_1(evalrealselectbb5in(Ar_0, Ar_1, Ar_2)) [ Ar_2 >= Ar_1 ] (Comp: 3*Ar_1^2 + 22*Ar_1 + 19, Cost: 1) evalrealselectbb1in(Ar_0, Ar_1, Ar_2) -> Com_1(evalrealselectbb4in(Ar_0, Ar_1, Ar_2 + 1)) [ D >= E + 1 ] (Comp: 3*Ar_1^2 + 22*Ar_1 + 19, Cost: 1) evalrealselectbb1in(Ar_0, Ar_1, Ar_2) -> Com_1(evalrealselectbb4in(Ar_0, Ar_1, Ar_2 + 1)) [ D >= E ] (Comp: 2*Ar_1 + 2, Cost: 1) evalrealselectbb5in(Ar_0, Ar_1, Ar_2) -> Com_1(evalrealselectbb6in(Ar_0 + 1, Ar_1, Ar_2)) (Comp: 2, Cost: 1) evalrealselectreturnin(Ar_0, Ar_1, Ar_2) -> Com_1(evalrealselectstop(Ar_0, Ar_1, Ar_2)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2) -> Com_1(evalrealselectstart(Ar_0, Ar_1, Ar_2)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 Complexity upper bound 72*Ar_1 + 9*Ar_1^2 + 69 Time: 0.095 sec (SMT: 0.079 sec) ---------------------------------------- (2) BOUNDS(1, n^2) ---------------------------------------- (3) Loat Proof (FINISHED) ### Pre-processing the ITS problem ### Initial linear ITS problem Start location: evalrealselectstart 0: evalrealselectstart -> evalrealselectentryin : [], cost: 1 1: evalrealselectentryin -> evalrealselectbb6in : A'=0, [], cost: 1 2: evalrealselectbb6in -> evalrealselectbbin : [ B>=2+A ], cost: 1 3: evalrealselectbb6in -> evalrealselectreturnin : [ 1+A>=B ], cost: 1 4: evalrealselectbbin -> evalrealselectbb4in : C'=1+A, [], cost: 1 5: evalrealselectbb4in -> evalrealselectbb1in : [ B>=1+C ], cost: 1 6: evalrealselectbb4in -> evalrealselectbb5in : [ C>=B ], cost: 1 7: evalrealselectbb1in -> evalrealselectbb4in : C'=1+C, [ free>=1+free_1 ], cost: 1 8: evalrealselectbb1in -> evalrealselectbb4in : C'=1+C, [ free_2>=free_3 ], cost: 1 9: evalrealselectbb5in -> evalrealselectbb6in : A'=1+A, [], cost: 1 10: evalrealselectreturnin -> evalrealselectstop : [], cost: 1 Checking for constant complexity: The following rule is satisfiable with cost >= 1, yielding constant complexity: 0: evalrealselectstart -> evalrealselectentryin : [], cost: 1 Removed unreachable and leaf rules: Start location: evalrealselectstart 0: evalrealselectstart -> evalrealselectentryin : [], cost: 1 1: evalrealselectentryin -> evalrealselectbb6in : A'=0, [], cost: 1 2: evalrealselectbb6in -> evalrealselectbbin : [ B>=2+A ], cost: 1 4: evalrealselectbbin -> evalrealselectbb4in : C'=1+A, [], cost: 1 5: evalrealselectbb4in -> evalrealselectbb1in : [ B>=1+C ], cost: 1 6: evalrealselectbb4in -> evalrealselectbb5in : [ C>=B ], cost: 1 7: evalrealselectbb1in -> evalrealselectbb4in : C'=1+C, [ free>=1+free_1 ], cost: 1 8: evalrealselectbb1in -> evalrealselectbb4in : C'=1+C, [ free_2>=free_3 ], cost: 1 9: evalrealselectbb5in -> evalrealselectbb6in : A'=1+A, [], cost: 1 Simplified all rules, resulting in: Start location: evalrealselectstart 0: evalrealselectstart -> evalrealselectentryin : [], cost: 1 1: evalrealselectentryin -> evalrealselectbb6in : A'=0, [], cost: 1 2: evalrealselectbb6in -> evalrealselectbbin : [ B>=2+A ], cost: 1 4: evalrealselectbbin -> evalrealselectbb4in : C'=1+A, [], cost: 1 5: evalrealselectbb4in -> evalrealselectbb1in : [ B>=1+C ], cost: 1 6: evalrealselectbb4in -> evalrealselectbb5in : [ C>=B ], cost: 1 8: evalrealselectbb1in -> evalrealselectbb4in : C'=1+C, [], cost: 1 9: evalrealselectbb5in -> evalrealselectbb6in : A'=1+A, [], cost: 1 ### Simplification by acceleration and chaining ### Eliminated locations (on linear paths): Start location: evalrealselectstart 11: evalrealselectstart -> evalrealselectbb6in : A'=0, [], cost: 2 12: evalrealselectbb6in -> evalrealselectbb4in : C'=1+A, [ B>=2+A ], cost: 2 13: evalrealselectbb4in -> evalrealselectbb4in : C'=1+C, [ B>=1+C ], cost: 2 14: evalrealselectbb4in -> evalrealselectbb6in : A'=1+A, [ C>=B ], cost: 2 Accelerating simple loops of location 4. Accelerating the following rules: 13: evalrealselectbb4in -> evalrealselectbb4in : C'=1+C, [ B>=1+C ], cost: 2 Accelerated rule 13 with metering function -C+B, yielding the new rule 15. Removing the simple loops: 13. Accelerated all simple loops using metering functions (where possible): Start location: evalrealselectstart 11: evalrealselectstart -> evalrealselectbb6in : A'=0, [], cost: 2 12: evalrealselectbb6in -> evalrealselectbb4in : C'=1+A, [ B>=2+A ], cost: 2 14: evalrealselectbb4in -> evalrealselectbb6in : A'=1+A, [ C>=B ], cost: 2 15: evalrealselectbb4in -> evalrealselectbb4in : C'=B, [ B>=1+C ], cost: -2*C+2*B Chained accelerated rules (with incoming rules): Start location: evalrealselectstart 11: evalrealselectstart -> evalrealselectbb6in : A'=0, [], cost: 2 12: evalrealselectbb6in -> evalrealselectbb4in : C'=1+A, [ B>=2+A ], cost: 2 16: evalrealselectbb6in -> evalrealselectbb4in : C'=B, [ B>=2+A ], cost: -2*A+2*B 14: evalrealselectbb4in -> evalrealselectbb6in : A'=1+A, [ C>=B ], cost: 2 Eliminated locations (on tree-shaped paths): Start location: evalrealselectstart 11: evalrealselectstart -> evalrealselectbb6in : A'=0, [], cost: 2 17: evalrealselectbb6in -> evalrealselectbb6in : A'=1+A, C'=B, [ B>=2+A ], cost: 2-2*A+2*B Accelerating simple loops of location 2. Accelerating the following rules: 17: evalrealselectbb6in -> evalrealselectbb6in : A'=1+A, C'=B, [ B>=2+A ], cost: 2-2*A+2*B Accelerated rule 17 with metering function -1-A+B, yielding the new rule 18. Removing the simple loops: 17. Accelerated all simple loops using metering functions (where possible): Start location: evalrealselectstart 11: evalrealselectstart -> evalrealselectbb6in : A'=0, [], cost: 2 18: evalrealselectbb6in -> evalrealselectbb6in : A'=-1+B, C'=B, [ B>=2+A ], cost: -3-(1+A-B)^2-3*A+2*A*(1+A-B)-2*(1+A-B)*B+3*B Chained accelerated rules (with incoming rules): Start location: evalrealselectstart 11: evalrealselectstart -> evalrealselectbb6in : A'=0, [], cost: 2 19: evalrealselectstart -> evalrealselectbb6in : A'=-1+B, C'=B, [ B>=2 ], cost: -1+2*(-1+B)*B+3*B-(-1+B)^2 Removed unreachable locations (and leaf rules with constant cost): Start location: evalrealselectstart 19: evalrealselectstart -> evalrealselectbb6in : A'=-1+B, C'=B, [ B>=2 ], cost: -1+2*(-1+B)*B+3*B-(-1+B)^2 ### Computing asymptotic complexity ### Fully simplified ITS problem Start location: evalrealselectstart 19: evalrealselectstart -> evalrealselectbb6in : A'=-1+B, C'=B, [ B>=2 ], cost: -1+2*(-1+B)*B+3*B-(-1+B)^2 Computing asymptotic complexity for rule 19 Solved the limit problem by the following transformations: Created initial limit problem: -2+3*B+B^2 (+), -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 -2+n^2+3*n 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+n^2+3*n Rule cost: -1+2*(-1+B)*B+3*B-(-1+B)^2 Rule guard: [ B>=2 ] WORST_CASE(Omega(n^2),?) ---------------------------------------- (4) BOUNDS(n^2, INF)