4.66/2.24 WORST_CASE(Omega(n^1), O(n^1)) 4.66/2.25 proof of /export/starexec/sandbox/benchmark/theBenchmark.koat 4.66/2.25 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 4.66/2.25 4.66/2.25 4.66/2.25 The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(n^1, n^1). 4.66/2.25 4.66/2.25 (0) CpxIntTrs 4.66/2.25 (1) Koat Proof [FINISHED, 120 ms] 4.66/2.25 (2) BOUNDS(1, n^1) 4.66/2.25 (3) Loat Proof [FINISHED, 626 ms] 4.66/2.25 (4) BOUNDS(n^1, INF) 4.66/2.25 4.66/2.25 4.66/2.25 ---------------------------------------- 4.66/2.25 4.66/2.25 (0) 4.66/2.25 Obligation: 4.66/2.25 Complexity Int TRS consisting of the following rules: 4.66/2.25 evalSimpleSingle2start(A, B, C, D) -> Com_1(evalSimpleSingle2entryin(A, B, C, D)) :|: TRUE 4.66/2.25 evalSimpleSingle2entryin(A, B, C, D) -> Com_1(evalSimpleSingle2bb4in(0, 0, C, D)) :|: TRUE 4.66/2.25 evalSimpleSingle2bb4in(A, B, C, D) -> Com_1(evalSimpleSingle2bbin(A, B, C, D)) :|: 0 >= E + 1 4.66/2.25 evalSimpleSingle2bb4in(A, B, C, D) -> Com_1(evalSimpleSingle2bbin(A, B, C, D)) :|: E >= 1 4.66/2.25 evalSimpleSingle2bb4in(A, B, C, D) -> Com_1(evalSimpleSingle2returnin(A, B, C, D)) :|: TRUE 4.66/2.25 evalSimpleSingle2bbin(A, B, C, D) -> Com_1(evalSimpleSingle2bb1in(A, B, C, D)) :|: C >= B + 1 4.66/2.25 evalSimpleSingle2bbin(A, B, C, D) -> Com_1(evalSimpleSingle2bb2in(A, B, C, D)) :|: B >= C 4.66/2.25 evalSimpleSingle2bb1in(A, B, C, D) -> Com_1(evalSimpleSingle2bb4in(A + 1, B + 1, C, D)) :|: TRUE 4.66/2.25 evalSimpleSingle2bb2in(A, B, C, D) -> Com_1(evalSimpleSingle2bb3in(A, B, C, D)) :|: D >= A + 1 4.66/2.25 evalSimpleSingle2bb2in(A, B, C, D) -> Com_1(evalSimpleSingle2returnin(A, B, C, D)) :|: A >= D 4.66/2.25 evalSimpleSingle2bb3in(A, B, C, D) -> Com_1(evalSimpleSingle2bb4in(A + 1, B + 1, C, D)) :|: TRUE 4.66/2.25 evalSimpleSingle2returnin(A, B, C, D) -> Com_1(evalSimpleSingle2stop(A, B, C, D)) :|: TRUE 4.66/2.25 4.66/2.25 The start-symbols are:[evalSimpleSingle2start_4] 4.66/2.25 4.66/2.25 4.66/2.25 ---------------------------------------- 4.66/2.25 4.66/2.25 (1) Koat Proof (FINISHED) 4.66/2.25 YES(?, 6*ar_3 + 6*ar_2 + 24) 4.66/2.25 4.66/2.25 4.66/2.25 4.66/2.25 Initial complexity problem: 4.66/2.25 4.66/2.25 1: T: 4.66/2.25 4.66/2.25 (Comp: ?, Cost: 1) evalSimpleSingle2start(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalSimpleSingle2entryin(ar_0, ar_1, ar_2, ar_3)) 4.66/2.25 4.66/2.25 (Comp: ?, Cost: 1) evalSimpleSingle2entryin(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalSimpleSingle2bb4in(0, 0, ar_2, ar_3)) 4.66/2.25 4.66/2.25 (Comp: ?, Cost: 1) evalSimpleSingle2bb4in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalSimpleSingle2bbin(ar_0, ar_1, ar_2, ar_3)) [ 0 >= e + 1 ] 4.86/2.25 4.86/2.25 (Comp: ?, Cost: 1) evalSimpleSingle2bb4in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalSimpleSingle2bbin(ar_0, ar_1, ar_2, ar_3)) [ e >= 1 ] 4.86/2.25 4.86/2.25 (Comp: ?, Cost: 1) evalSimpleSingle2bb4in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalSimpleSingle2returnin(ar_0, ar_1, ar_2, ar_3)) 4.86/2.25 4.86/2.25 (Comp: ?, Cost: 1) evalSimpleSingle2bbin(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalSimpleSingle2bb1in(ar_0, ar_1, ar_2, ar_3)) [ ar_2 >= ar_1 + 1 ] 4.86/2.25 4.86/2.25 (Comp: ?, Cost: 1) evalSimpleSingle2bbin(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalSimpleSingle2bb2in(ar_0, ar_1, ar_2, ar_3)) [ ar_1 >= ar_2 ] 4.86/2.25 4.86/2.25 (Comp: ?, Cost: 1) evalSimpleSingle2bb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalSimpleSingle2bb4in(ar_0 + 1, ar_1 + 1, ar_2, ar_3)) 4.86/2.25 4.86/2.25 (Comp: ?, Cost: 1) evalSimpleSingle2bb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalSimpleSingle2bb3in(ar_0, ar_1, ar_2, ar_3)) [ ar_3 >= ar_0 + 1 ] 4.86/2.25 4.86/2.25 (Comp: ?, Cost: 1) evalSimpleSingle2bb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalSimpleSingle2returnin(ar_0, ar_1, ar_2, ar_3)) [ ar_0 >= ar_3 ] 4.86/2.25 4.86/2.25 (Comp: ?, Cost: 1) evalSimpleSingle2bb3in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalSimpleSingle2bb4in(ar_0 + 1, ar_1 + 1, ar_2, ar_3)) 4.86/2.25 4.86/2.25 (Comp: ?, Cost: 1) evalSimpleSingle2returnin(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalSimpleSingle2stop(ar_0, ar_1, ar_2, ar_3)) 4.86/2.25 4.86/2.25 (Comp: 1, Cost: 0) koat_start(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalSimpleSingle2start(ar_0, ar_1, ar_2, ar_3)) [ 0 <= 0 ] 4.86/2.25 4.86/2.25 start location: koat_start 4.86/2.25 4.86/2.25 leaf cost: 0 4.86/2.25 4.86/2.25 4.86/2.25 4.86/2.25 Repeatedly propagating knowledge in problem 1 produces the following problem: 4.86/2.25 4.86/2.25 2: T: 4.86/2.25 4.86/2.25 (Comp: 1, Cost: 1) evalSimpleSingle2start(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalSimpleSingle2entryin(ar_0, ar_1, ar_2, ar_3)) 4.86/2.25 4.86/2.25 (Comp: 1, Cost: 1) evalSimpleSingle2entryin(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalSimpleSingle2bb4in(0, 0, ar_2, ar_3)) 4.86/2.25 4.86/2.25 (Comp: ?, Cost: 1) evalSimpleSingle2bb4in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalSimpleSingle2bbin(ar_0, ar_1, ar_2, ar_3)) [ 0 >= e + 1 ] 4.86/2.25 4.86/2.25 (Comp: ?, Cost: 1) evalSimpleSingle2bb4in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalSimpleSingle2bbin(ar_0, ar_1, ar_2, ar_3)) [ e >= 1 ] 4.86/2.25 4.86/2.25 (Comp: ?, Cost: 1) evalSimpleSingle2bb4in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalSimpleSingle2returnin(ar_0, ar_1, ar_2, ar_3)) 4.86/2.25 4.86/2.25 (Comp: ?, Cost: 1) evalSimpleSingle2bbin(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalSimpleSingle2bb1in(ar_0, ar_1, ar_2, ar_3)) [ ar_2 >= ar_1 + 1 ] 4.86/2.25 4.86/2.25 (Comp: ?, Cost: 1) evalSimpleSingle2bbin(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalSimpleSingle2bb2in(ar_0, ar_1, ar_2, ar_3)) [ ar_1 >= ar_2 ] 4.86/2.25 4.86/2.25 (Comp: ?, Cost: 1) evalSimpleSingle2bb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalSimpleSingle2bb4in(ar_0 + 1, ar_1 + 1, ar_2, ar_3)) 4.86/2.25 4.86/2.25 (Comp: ?, Cost: 1) evalSimpleSingle2bb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalSimpleSingle2bb3in(ar_0, ar_1, ar_2, ar_3)) [ ar_3 >= ar_0 + 1 ] 4.86/2.25 4.86/2.25 (Comp: ?, Cost: 1) evalSimpleSingle2bb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalSimpleSingle2returnin(ar_0, ar_1, ar_2, ar_3)) [ ar_0 >= ar_3 ] 4.86/2.25 4.86/2.25 (Comp: ?, Cost: 1) evalSimpleSingle2bb3in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalSimpleSingle2bb4in(ar_0 + 1, ar_1 + 1, ar_2, ar_3)) 4.86/2.25 4.86/2.25 (Comp: ?, Cost: 1) evalSimpleSingle2returnin(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalSimpleSingle2stop(ar_0, ar_1, ar_2, ar_3)) 4.86/2.25 4.86/2.25 (Comp: 1, Cost: 0) koat_start(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalSimpleSingle2start(ar_0, ar_1, ar_2, ar_3)) [ 0 <= 0 ] 4.86/2.25 4.86/2.25 start location: koat_start 4.86/2.25 4.86/2.25 leaf cost: 0 4.86/2.25 4.86/2.25 4.86/2.25 4.86/2.25 A polynomial rank function with 4.86/2.25 4.86/2.25 Pol(evalSimpleSingle2start) = 2 4.86/2.25 4.86/2.25 Pol(evalSimpleSingle2entryin) = 2 4.86/2.25 4.86/2.25 Pol(evalSimpleSingle2bb4in) = 2 4.86/2.25 4.86/2.25 Pol(evalSimpleSingle2bbin) = 2 4.86/2.25 4.86/2.25 Pol(evalSimpleSingle2returnin) = 1 4.86/2.25 4.86/2.25 Pol(evalSimpleSingle2bb1in) = 2 4.86/2.25 4.86/2.25 Pol(evalSimpleSingle2bb2in) = 2 4.86/2.25 4.86/2.25 Pol(evalSimpleSingle2bb3in) = 2 4.86/2.25 4.86/2.25 Pol(evalSimpleSingle2stop) = 0 4.86/2.25 4.86/2.25 Pol(koat_start) = 2 4.86/2.25 4.86/2.25 orients all transitions weakly and the transitions 4.86/2.25 4.86/2.25 evalSimpleSingle2returnin(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalSimpleSingle2stop(ar_0, ar_1, ar_2, ar_3)) 4.86/2.25 4.86/2.25 evalSimpleSingle2bb4in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalSimpleSingle2returnin(ar_0, ar_1, ar_2, ar_3)) 4.86/2.25 4.86/2.25 evalSimpleSingle2bb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalSimpleSingle2returnin(ar_0, ar_1, ar_2, ar_3)) [ ar_0 >= ar_3 ] 4.86/2.25 4.86/2.25 strictly and produces the following problem: 4.86/2.25 4.86/2.25 3: T: 4.86/2.25 4.86/2.25 (Comp: 1, Cost: 1) evalSimpleSingle2start(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalSimpleSingle2entryin(ar_0, ar_1, ar_2, ar_3)) 4.86/2.25 4.86/2.25 (Comp: 1, Cost: 1) evalSimpleSingle2entryin(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalSimpleSingle2bb4in(0, 0, ar_2, ar_3)) 4.86/2.25 4.86/2.25 (Comp: ?, Cost: 1) evalSimpleSingle2bb4in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalSimpleSingle2bbin(ar_0, ar_1, ar_2, ar_3)) [ 0 >= e + 1 ] 4.86/2.25 4.86/2.25 (Comp: ?, Cost: 1) evalSimpleSingle2bb4in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalSimpleSingle2bbin(ar_0, ar_1, ar_2, ar_3)) [ e >= 1 ] 4.86/2.25 4.86/2.25 (Comp: 2, Cost: 1) evalSimpleSingle2bb4in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalSimpleSingle2returnin(ar_0, ar_1, ar_2, ar_3)) 4.86/2.25 4.86/2.25 (Comp: ?, Cost: 1) evalSimpleSingle2bbin(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalSimpleSingle2bb1in(ar_0, ar_1, ar_2, ar_3)) [ ar_2 >= ar_1 + 1 ] 4.86/2.25 4.86/2.25 (Comp: ?, Cost: 1) evalSimpleSingle2bbin(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalSimpleSingle2bb2in(ar_0, ar_1, ar_2, ar_3)) [ ar_1 >= ar_2 ] 4.86/2.25 4.86/2.25 (Comp: ?, Cost: 1) evalSimpleSingle2bb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalSimpleSingle2bb4in(ar_0 + 1, ar_1 + 1, ar_2, ar_3)) 4.86/2.25 4.86/2.25 (Comp: ?, Cost: 1) evalSimpleSingle2bb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalSimpleSingle2bb3in(ar_0, ar_1, ar_2, ar_3)) [ ar_3 >= ar_0 + 1 ] 4.86/2.25 4.86/2.25 (Comp: 2, Cost: 1) evalSimpleSingle2bb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalSimpleSingle2returnin(ar_0, ar_1, ar_2, ar_3)) [ ar_0 >= ar_3 ] 4.86/2.25 4.86/2.25 (Comp: ?, Cost: 1) evalSimpleSingle2bb3in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalSimpleSingle2bb4in(ar_0 + 1, ar_1 + 1, ar_2, ar_3)) 4.86/2.25 4.86/2.25 (Comp: 2, Cost: 1) evalSimpleSingle2returnin(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalSimpleSingle2stop(ar_0, ar_1, ar_2, ar_3)) 4.86/2.25 4.86/2.25 (Comp: 1, Cost: 0) koat_start(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalSimpleSingle2start(ar_0, ar_1, ar_2, ar_3)) [ 0 <= 0 ] 4.86/2.25 4.86/2.25 start location: koat_start 4.86/2.25 4.86/2.25 leaf cost: 0 4.86/2.25 4.86/2.25 4.86/2.25 4.86/2.25 A polynomial rank function with 4.86/2.25 4.86/2.25 Pol(evalSimpleSingle2start) = V_3 + 1 4.86/2.25 4.86/2.25 Pol(evalSimpleSingle2entryin) = V_3 + 1 4.86/2.25 4.86/2.25 Pol(evalSimpleSingle2bb4in) = -V_2 + V_3 + 1 4.86/2.25 4.86/2.25 Pol(evalSimpleSingle2bbin) = -V_2 + V_3 + 1 4.86/2.25 4.86/2.25 Pol(evalSimpleSingle2returnin) = -V_2 + V_3 4.86/2.25 4.86/2.25 Pol(evalSimpleSingle2bb1in) = -V_2 + V_3 4.86/2.25 4.86/2.25 Pol(evalSimpleSingle2bb2in) = -V_2 + V_3 + 1 4.86/2.25 4.86/2.25 Pol(evalSimpleSingle2bb3in) = -V_2 + V_3 4.86/2.25 4.86/2.25 Pol(evalSimpleSingle2stop) = -V_2 + V_3 4.86/2.25 4.86/2.25 Pol(koat_start) = V_3 + 1 4.86/2.25 4.86/2.25 orients all transitions weakly and the transition 4.86/2.25 4.86/2.25 evalSimpleSingle2bbin(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalSimpleSingle2bb1in(ar_0, ar_1, ar_2, ar_3)) [ ar_2 >= ar_1 + 1 ] 4.86/2.25 4.86/2.25 strictly and produces the following problem: 4.86/2.25 4.86/2.25 4: T: 4.86/2.25 4.86/2.25 (Comp: 1, Cost: 1) evalSimpleSingle2start(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalSimpleSingle2entryin(ar_0, ar_1, ar_2, ar_3)) 4.86/2.25 4.86/2.25 (Comp: 1, Cost: 1) evalSimpleSingle2entryin(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalSimpleSingle2bb4in(0, 0, ar_2, ar_3)) 4.86/2.25 4.86/2.25 (Comp: ?, Cost: 1) evalSimpleSingle2bb4in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalSimpleSingle2bbin(ar_0, ar_1, ar_2, ar_3)) [ 0 >= e + 1 ] 4.86/2.25 4.86/2.25 (Comp: ?, Cost: 1) evalSimpleSingle2bb4in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalSimpleSingle2bbin(ar_0, ar_1, ar_2, ar_3)) [ e >= 1 ] 4.86/2.25 4.86/2.25 (Comp: 2, Cost: 1) evalSimpleSingle2bb4in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalSimpleSingle2returnin(ar_0, ar_1, ar_2, ar_3)) 4.86/2.25 4.86/2.25 (Comp: ar_2 + 1, Cost: 1) evalSimpleSingle2bbin(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalSimpleSingle2bb1in(ar_0, ar_1, ar_2, ar_3)) [ ar_2 >= ar_1 + 1 ] 4.86/2.25 4.86/2.25 (Comp: ?, Cost: 1) evalSimpleSingle2bbin(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalSimpleSingle2bb2in(ar_0, ar_1, ar_2, ar_3)) [ ar_1 >= ar_2 ] 4.86/2.25 4.86/2.25 (Comp: ?, Cost: 1) evalSimpleSingle2bb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalSimpleSingle2bb4in(ar_0 + 1, ar_1 + 1, ar_2, ar_3)) 4.86/2.25 4.86/2.25 (Comp: ?, Cost: 1) evalSimpleSingle2bb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalSimpleSingle2bb3in(ar_0, ar_1, ar_2, ar_3)) [ ar_3 >= ar_0 + 1 ] 4.86/2.25 4.86/2.25 (Comp: 2, Cost: 1) evalSimpleSingle2bb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalSimpleSingle2returnin(ar_0, ar_1, ar_2, ar_3)) [ ar_0 >= ar_3 ] 4.86/2.25 4.86/2.25 (Comp: ?, Cost: 1) evalSimpleSingle2bb3in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalSimpleSingle2bb4in(ar_0 + 1, ar_1 + 1, ar_2, ar_3)) 4.86/2.25 4.86/2.25 (Comp: 2, Cost: 1) evalSimpleSingle2returnin(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalSimpleSingle2stop(ar_0, ar_1, ar_2, ar_3)) 4.86/2.25 4.86/2.25 (Comp: 1, Cost: 0) koat_start(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalSimpleSingle2start(ar_0, ar_1, ar_2, ar_3)) [ 0 <= 0 ] 4.86/2.25 4.86/2.25 start location: koat_start 4.86/2.25 4.86/2.25 leaf cost: 0 4.86/2.25 4.86/2.25 4.86/2.25 4.86/2.25 Repeatedly propagating knowledge in problem 4 produces the following problem: 4.86/2.25 4.86/2.25 5: T: 4.86/2.25 4.86/2.25 (Comp: 1, Cost: 1) evalSimpleSingle2start(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalSimpleSingle2entryin(ar_0, ar_1, ar_2, ar_3)) 4.86/2.25 4.86/2.25 (Comp: 1, Cost: 1) evalSimpleSingle2entryin(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalSimpleSingle2bb4in(0, 0, ar_2, ar_3)) 4.86/2.25 4.86/2.25 (Comp: ?, Cost: 1) evalSimpleSingle2bb4in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalSimpleSingle2bbin(ar_0, ar_1, ar_2, ar_3)) [ 0 >= e + 1 ] 4.86/2.25 4.86/2.25 (Comp: ?, Cost: 1) evalSimpleSingle2bb4in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalSimpleSingle2bbin(ar_0, ar_1, ar_2, ar_3)) [ e >= 1 ] 4.86/2.25 4.86/2.25 (Comp: 2, Cost: 1) evalSimpleSingle2bb4in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalSimpleSingle2returnin(ar_0, ar_1, ar_2, ar_3)) 4.86/2.25 4.86/2.25 (Comp: ar_2 + 1, Cost: 1) evalSimpleSingle2bbin(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalSimpleSingle2bb1in(ar_0, ar_1, ar_2, ar_3)) [ ar_2 >= ar_1 + 1 ] 4.86/2.25 4.86/2.25 (Comp: ?, Cost: 1) evalSimpleSingle2bbin(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalSimpleSingle2bb2in(ar_0, ar_1, ar_2, ar_3)) [ ar_1 >= ar_2 ] 4.86/2.25 4.86/2.25 (Comp: ar_2 + 1, Cost: 1) evalSimpleSingle2bb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalSimpleSingle2bb4in(ar_0 + 1, ar_1 + 1, ar_2, ar_3)) 4.86/2.25 4.86/2.25 (Comp: ?, Cost: 1) evalSimpleSingle2bb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalSimpleSingle2bb3in(ar_0, ar_1, ar_2, ar_3)) [ ar_3 >= ar_0 + 1 ] 4.86/2.25 4.86/2.25 (Comp: 2, Cost: 1) evalSimpleSingle2bb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalSimpleSingle2returnin(ar_0, ar_1, ar_2, ar_3)) [ ar_0 >= ar_3 ] 4.86/2.25 4.86/2.25 (Comp: ?, Cost: 1) evalSimpleSingle2bb3in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalSimpleSingle2bb4in(ar_0 + 1, ar_1 + 1, ar_2, ar_3)) 4.86/2.25 4.86/2.25 (Comp: 2, Cost: 1) evalSimpleSingle2returnin(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalSimpleSingle2stop(ar_0, ar_1, ar_2, ar_3)) 4.86/2.25 4.86/2.25 (Comp: 1, Cost: 0) koat_start(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalSimpleSingle2start(ar_0, ar_1, ar_2, ar_3)) [ 0 <= 0 ] 4.86/2.25 4.86/2.25 start location: koat_start 4.86/2.25 4.86/2.25 leaf cost: 0 4.86/2.25 4.86/2.25 4.86/2.25 4.86/2.25 A polynomial rank function with 4.86/2.25 4.86/2.25 Pol(evalSimpleSingle2start) = V_4 + 1 4.86/2.25 4.86/2.25 Pol(evalSimpleSingle2entryin) = V_4 + 1 4.86/2.25 4.86/2.25 Pol(evalSimpleSingle2bb4in) = -V_1 + V_4 + 1 4.86/2.25 4.86/2.25 Pol(evalSimpleSingle2bbin) = -V_1 + V_4 + 1 4.86/2.25 4.86/2.25 Pol(evalSimpleSingle2returnin) = -V_1 + V_4 + 1 4.86/2.25 4.86/2.25 Pol(evalSimpleSingle2bb1in) = -V_1 + V_4 4.86/2.25 4.86/2.25 Pol(evalSimpleSingle2bb2in) = -V_1 + V_4 + 1 4.86/2.25 4.86/2.25 Pol(evalSimpleSingle2bb3in) = -V_1 + V_4 4.86/2.25 4.86/2.25 Pol(evalSimpleSingle2stop) = -V_1 + V_4 + 1 4.86/2.25 4.86/2.25 Pol(koat_start) = V_4 + 1 4.86/2.25 4.86/2.25 orients all transitions weakly and the transition 4.86/2.25 4.86/2.25 evalSimpleSingle2bb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalSimpleSingle2bb3in(ar_0, ar_1, ar_2, ar_3)) [ ar_3 >= ar_0 + 1 ] 4.86/2.25 4.86/2.25 strictly and produces the following problem: 4.86/2.25 4.86/2.25 6: T: 4.86/2.25 4.86/2.25 (Comp: 1, Cost: 1) evalSimpleSingle2start(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalSimpleSingle2entryin(ar_0, ar_1, ar_2, ar_3)) 4.86/2.25 4.86/2.25 (Comp: 1, Cost: 1) evalSimpleSingle2entryin(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalSimpleSingle2bb4in(0, 0, ar_2, ar_3)) 4.86/2.25 4.86/2.25 (Comp: ?, Cost: 1) evalSimpleSingle2bb4in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalSimpleSingle2bbin(ar_0, ar_1, ar_2, ar_3)) [ 0 >= e + 1 ] 4.86/2.25 4.86/2.25 (Comp: ?, Cost: 1) evalSimpleSingle2bb4in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalSimpleSingle2bbin(ar_0, ar_1, ar_2, ar_3)) [ e >= 1 ] 4.86/2.25 4.86/2.25 (Comp: 2, Cost: 1) evalSimpleSingle2bb4in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalSimpleSingle2returnin(ar_0, ar_1, ar_2, ar_3)) 4.86/2.25 4.86/2.25 (Comp: ar_2 + 1, Cost: 1) evalSimpleSingle2bbin(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalSimpleSingle2bb1in(ar_0, ar_1, ar_2, ar_3)) [ ar_2 >= ar_1 + 1 ] 4.86/2.25 4.86/2.25 (Comp: ?, Cost: 1) evalSimpleSingle2bbin(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalSimpleSingle2bb2in(ar_0, ar_1, ar_2, ar_3)) [ ar_1 >= ar_2 ] 4.86/2.25 4.86/2.25 (Comp: ar_2 + 1, Cost: 1) evalSimpleSingle2bb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalSimpleSingle2bb4in(ar_0 + 1, ar_1 + 1, ar_2, ar_3)) 4.86/2.25 4.86/2.25 (Comp: ar_3 + 1, Cost: 1) evalSimpleSingle2bb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalSimpleSingle2bb3in(ar_0, ar_1, ar_2, ar_3)) [ ar_3 >= ar_0 + 1 ] 4.86/2.25 4.86/2.25 (Comp: 2, Cost: 1) evalSimpleSingle2bb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalSimpleSingle2returnin(ar_0, ar_1, ar_2, ar_3)) [ ar_0 >= ar_3 ] 4.86/2.25 4.86/2.25 (Comp: ?, Cost: 1) evalSimpleSingle2bb3in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalSimpleSingle2bb4in(ar_0 + 1, ar_1 + 1, ar_2, ar_3)) 4.86/2.25 4.86/2.25 (Comp: 2, Cost: 1) evalSimpleSingle2returnin(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalSimpleSingle2stop(ar_0, ar_1, ar_2, ar_3)) 4.86/2.25 4.86/2.25 (Comp: 1, Cost: 0) koat_start(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalSimpleSingle2start(ar_0, ar_1, ar_2, ar_3)) [ 0 <= 0 ] 4.86/2.25 4.86/2.25 start location: koat_start 4.86/2.25 4.86/2.25 leaf cost: 0 4.86/2.25 4.86/2.25 4.86/2.25 4.86/2.25 Repeatedly propagating knowledge in problem 6 produces the following problem: 4.86/2.25 4.86/2.25 7: T: 4.86/2.25 4.86/2.25 (Comp: 1, Cost: 1) evalSimpleSingle2start(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalSimpleSingle2entryin(ar_0, ar_1, ar_2, ar_3)) 4.86/2.25 4.86/2.25 (Comp: 1, Cost: 1) evalSimpleSingle2entryin(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalSimpleSingle2bb4in(0, 0, ar_2, ar_3)) 4.86/2.25 4.86/2.25 (Comp: ar_3 + ar_2 + 3, Cost: 1) evalSimpleSingle2bb4in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalSimpleSingle2bbin(ar_0, ar_1, ar_2, ar_3)) [ 0 >= e + 1 ] 4.86/2.25 4.86/2.25 (Comp: ar_3 + ar_2 + 3, Cost: 1) evalSimpleSingle2bb4in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalSimpleSingle2bbin(ar_0, ar_1, ar_2, ar_3)) [ e >= 1 ] 4.86/2.25 4.86/2.25 (Comp: 2, Cost: 1) evalSimpleSingle2bb4in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalSimpleSingle2returnin(ar_0, ar_1, ar_2, ar_3)) 4.86/2.25 4.86/2.25 (Comp: ar_2 + 1, Cost: 1) evalSimpleSingle2bbin(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalSimpleSingle2bb1in(ar_0, ar_1, ar_2, ar_3)) [ ar_2 >= ar_1 + 1 ] 4.86/2.25 4.86/2.25 (Comp: 2*ar_3 + 2*ar_2 + 6, Cost: 1) evalSimpleSingle2bbin(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalSimpleSingle2bb2in(ar_0, ar_1, ar_2, ar_3)) [ ar_1 >= ar_2 ] 4.86/2.25 4.86/2.25 (Comp: ar_2 + 1, Cost: 1) evalSimpleSingle2bb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalSimpleSingle2bb4in(ar_0 + 1, ar_1 + 1, ar_2, ar_3)) 4.86/2.25 4.86/2.25 (Comp: ar_3 + 1, Cost: 1) evalSimpleSingle2bb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalSimpleSingle2bb3in(ar_0, ar_1, ar_2, ar_3)) [ ar_3 >= ar_0 + 1 ] 4.86/2.25 4.86/2.25 (Comp: 2, Cost: 1) evalSimpleSingle2bb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalSimpleSingle2returnin(ar_0, ar_1, ar_2, ar_3)) [ ar_0 >= ar_3 ] 4.86/2.25 4.86/2.25 (Comp: ar_3 + 1, Cost: 1) evalSimpleSingle2bb3in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalSimpleSingle2bb4in(ar_0 + 1, ar_1 + 1, ar_2, ar_3)) 4.86/2.25 4.86/2.25 (Comp: 2, Cost: 1) evalSimpleSingle2returnin(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalSimpleSingle2stop(ar_0, ar_1, ar_2, ar_3)) 4.86/2.25 4.86/2.25 (Comp: 1, Cost: 0) koat_start(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalSimpleSingle2start(ar_0, ar_1, ar_2, ar_3)) [ 0 <= 0 ] 4.86/2.25 4.86/2.25 start location: koat_start 4.86/2.25 4.86/2.25 leaf cost: 0 4.86/2.25 4.86/2.25 4.86/2.25 4.86/2.25 Complexity upper bound 6*ar_3 + 6*ar_2 + 24 4.86/2.25 4.86/2.25 4.86/2.25 4.86/2.25 Time: 0.143 sec (SMT: 0.122 sec) 4.86/2.25 4.86/2.25 4.86/2.25 ---------------------------------------- 4.86/2.25 4.86/2.25 (2) 4.86/2.25 BOUNDS(1, n^1) 4.86/2.25 4.86/2.25 ---------------------------------------- 4.86/2.25 4.86/2.25 (3) Loat Proof (FINISHED) 4.86/2.25 4.86/2.25 4.86/2.25 ### Pre-processing the ITS problem ### 4.86/2.25 4.86/2.25 4.86/2.25 4.86/2.25 Initial linear ITS problem 4.86/2.25 4.86/2.25 Start location: evalSimpleSingle2start 4.86/2.25 4.86/2.25 0: evalSimpleSingle2start -> evalSimpleSingle2entryin : [], cost: 1 4.86/2.25 4.86/2.25 1: evalSimpleSingle2entryin -> evalSimpleSingle2bb4in : A'=0, B'=0, [], cost: 1 4.86/2.25 4.86/2.25 2: evalSimpleSingle2bb4in -> evalSimpleSingle2bbin : [ 0>=1+free ], cost: 1 4.86/2.25 4.86/2.25 3: evalSimpleSingle2bb4in -> evalSimpleSingle2bbin : [ free_1>=1 ], cost: 1 4.86/2.25 4.86/2.25 4: evalSimpleSingle2bb4in -> evalSimpleSingle2returnin : [], cost: 1 4.86/2.25 4.86/2.25 5: evalSimpleSingle2bbin -> evalSimpleSingle2bb1in : [ C>=1+B ], cost: 1 4.86/2.25 4.86/2.25 6: evalSimpleSingle2bbin -> evalSimpleSingle2bb2in : [ B>=C ], cost: 1 4.86/2.25 4.86/2.25 7: evalSimpleSingle2bb1in -> evalSimpleSingle2bb4in : A'=1+A, B'=1+B, [], cost: 1 4.86/2.25 4.86/2.25 8: evalSimpleSingle2bb2in -> evalSimpleSingle2bb3in : [ D>=1+A ], cost: 1 4.86/2.25 4.86/2.25 9: evalSimpleSingle2bb2in -> evalSimpleSingle2returnin : [ A>=D ], cost: 1 4.86/2.25 4.86/2.25 10: evalSimpleSingle2bb3in -> evalSimpleSingle2bb4in : A'=1+A, B'=1+B, [], cost: 1 4.86/2.25 4.86/2.25 11: evalSimpleSingle2returnin -> evalSimpleSingle2stop : [], cost: 1 4.86/2.25 4.86/2.25 4.86/2.25 4.86/2.25 Removed unreachable and leaf rules: 4.86/2.25 4.86/2.25 Start location: evalSimpleSingle2start 4.86/2.25 4.86/2.25 0: evalSimpleSingle2start -> evalSimpleSingle2entryin : [], cost: 1 4.86/2.25 4.86/2.25 1: evalSimpleSingle2entryin -> evalSimpleSingle2bb4in : A'=0, B'=0, [], cost: 1 4.86/2.25 4.86/2.25 2: evalSimpleSingle2bb4in -> evalSimpleSingle2bbin : [ 0>=1+free ], cost: 1 4.86/2.25 4.86/2.25 3: evalSimpleSingle2bb4in -> evalSimpleSingle2bbin : [ free_1>=1 ], cost: 1 4.86/2.25 4.86/2.25 5: evalSimpleSingle2bbin -> evalSimpleSingle2bb1in : [ C>=1+B ], cost: 1 4.86/2.25 4.86/2.25 6: evalSimpleSingle2bbin -> evalSimpleSingle2bb2in : [ B>=C ], cost: 1 4.86/2.25 4.86/2.25 7: evalSimpleSingle2bb1in -> evalSimpleSingle2bb4in : A'=1+A, B'=1+B, [], cost: 1 4.86/2.25 4.86/2.25 8: evalSimpleSingle2bb2in -> evalSimpleSingle2bb3in : [ D>=1+A ], cost: 1 4.86/2.25 4.86/2.25 10: evalSimpleSingle2bb3in -> evalSimpleSingle2bb4in : A'=1+A, B'=1+B, [], cost: 1 4.86/2.25 4.86/2.25 4.86/2.25 4.86/2.25 Simplified all rules, resulting in: 4.86/2.25 4.86/2.25 Start location: evalSimpleSingle2start 4.86/2.25 4.86/2.25 0: evalSimpleSingle2start -> evalSimpleSingle2entryin : [], cost: 1 4.86/2.25 4.86/2.25 1: evalSimpleSingle2entryin -> evalSimpleSingle2bb4in : A'=0, B'=0, [], cost: 1 4.86/2.25 4.86/2.25 3: evalSimpleSingle2bb4in -> evalSimpleSingle2bbin : [], cost: 1 4.86/2.25 4.86/2.25 5: evalSimpleSingle2bbin -> evalSimpleSingle2bb1in : [ C>=1+B ], cost: 1 4.86/2.25 4.86/2.25 6: evalSimpleSingle2bbin -> evalSimpleSingle2bb2in : [ B>=C ], cost: 1 4.86/2.25 4.86/2.25 7: evalSimpleSingle2bb1in -> evalSimpleSingle2bb4in : A'=1+A, B'=1+B, [], cost: 1 4.86/2.25 4.86/2.25 8: evalSimpleSingle2bb2in -> evalSimpleSingle2bb3in : [ D>=1+A ], cost: 1 4.86/2.25 4.86/2.25 10: evalSimpleSingle2bb3in -> evalSimpleSingle2bb4in : A'=1+A, B'=1+B, [], cost: 1 4.86/2.25 4.86/2.25 4.86/2.25 4.86/2.25 ### Simplification by acceleration and chaining ### 4.86/2.25 4.86/2.25 4.86/2.25 4.86/2.25 Eliminated locations (on linear paths): 4.86/2.25 4.86/2.25 Start location: evalSimpleSingle2start 4.86/2.25 4.86/2.25 12: evalSimpleSingle2start -> evalSimpleSingle2bb4in : A'=0, B'=0, [], cost: 2 4.86/2.25 4.86/2.25 3: evalSimpleSingle2bb4in -> evalSimpleSingle2bbin : [], cost: 1 4.86/2.25 4.86/2.25 13: evalSimpleSingle2bbin -> evalSimpleSingle2bb4in : A'=1+A, B'=1+B, [ C>=1+B ], cost: 2 4.86/2.25 4.86/2.25 15: evalSimpleSingle2bbin -> evalSimpleSingle2bb4in : A'=1+A, B'=1+B, [ B>=C && D>=1+A ], cost: 3 4.86/2.25 4.86/2.25 4.86/2.25 4.86/2.25 Eliminated locations (on tree-shaped paths): 4.86/2.25 4.86/2.25 Start location: evalSimpleSingle2start 4.86/2.25 4.86/2.25 12: evalSimpleSingle2start -> evalSimpleSingle2bb4in : A'=0, B'=0, [], cost: 2 4.86/2.25 4.86/2.25 16: evalSimpleSingle2bb4in -> evalSimpleSingle2bb4in : A'=1+A, B'=1+B, [ C>=1+B ], cost: 3 4.86/2.25 4.86/2.25 17: evalSimpleSingle2bb4in -> evalSimpleSingle2bb4in : A'=1+A, B'=1+B, [ B>=C && D>=1+A ], cost: 4 4.86/2.25 4.86/2.25 4.86/2.25 4.86/2.25 Accelerating simple loops of location 2. 4.86/2.25 4.86/2.25 Accelerating the following rules: 4.86/2.25 4.86/2.25 16: evalSimpleSingle2bb4in -> evalSimpleSingle2bb4in : A'=1+A, B'=1+B, [ C>=1+B ], cost: 3 4.86/2.25 4.86/2.25 17: evalSimpleSingle2bb4in -> evalSimpleSingle2bb4in : A'=1+A, B'=1+B, [ B>=C && D>=1+A ], cost: 4 4.86/2.25 4.86/2.25 4.86/2.25 4.86/2.25 Accelerated rule 16 with metering function C-B, yielding the new rule 18. 4.86/2.25 4.86/2.25 Accelerated rule 17 with metering function D-A, yielding the new rule 19. 4.86/2.25 4.86/2.25 Removing the simple loops: 16 17. 4.86/2.25 4.86/2.25 4.86/2.25 4.86/2.25 Accelerated all simple loops using metering functions (where possible): 4.86/2.25 4.86/2.25 Start location: evalSimpleSingle2start 4.86/2.25 4.86/2.25 12: evalSimpleSingle2start -> evalSimpleSingle2bb4in : A'=0, B'=0, [], cost: 2 4.86/2.25 4.86/2.25 18: evalSimpleSingle2bb4in -> evalSimpleSingle2bb4in : A'=C+A-B, B'=C, [ C>=1+B ], cost: 3*C-3*B 4.86/2.25 4.86/2.25 19: evalSimpleSingle2bb4in -> evalSimpleSingle2bb4in : A'=D, B'=D-A+B, [ B>=C && D>=1+A ], cost: 4*D-4*A 4.86/2.25 4.86/2.25 4.86/2.25 4.86/2.25 Chained accelerated rules (with incoming rules): 4.86/2.25 4.86/2.25 Start location: evalSimpleSingle2start 4.86/2.25 4.86/2.25 12: evalSimpleSingle2start -> evalSimpleSingle2bb4in : A'=0, B'=0, [], cost: 2 4.86/2.25 4.86/2.25 20: evalSimpleSingle2start -> evalSimpleSingle2bb4in : A'=C, B'=C, [ C>=1 ], cost: 2+3*C 4.86/2.25 4.86/2.25 21: evalSimpleSingle2start -> evalSimpleSingle2bb4in : A'=D, B'=D, [ 0>=C && D>=1 ], cost: 2+4*D 4.86/2.25 4.86/2.25 4.86/2.25 4.86/2.25 Removed unreachable locations (and leaf rules with constant cost): 4.86/2.25 4.86/2.25 Start location: evalSimpleSingle2start 4.86/2.25 4.86/2.25 20: evalSimpleSingle2start -> evalSimpleSingle2bb4in : A'=C, B'=C, [ C>=1 ], cost: 2+3*C 4.86/2.25 4.86/2.25 21: evalSimpleSingle2start -> evalSimpleSingle2bb4in : A'=D, B'=D, [ 0>=C && D>=1 ], cost: 2+4*D 4.86/2.25 4.86/2.25 4.86/2.25 4.86/2.25 ### Computing asymptotic complexity ### 4.86/2.25 4.86/2.25 4.86/2.25 4.86/2.25 Fully simplified ITS problem 4.86/2.25 4.86/2.25 Start location: evalSimpleSingle2start 4.86/2.25 4.86/2.25 20: evalSimpleSingle2start -> evalSimpleSingle2bb4in : A'=C, B'=C, [ C>=1 ], cost: 2+3*C 4.86/2.25 4.86/2.25 21: evalSimpleSingle2start -> evalSimpleSingle2bb4in : A'=D, B'=D, [ 0>=C && D>=1 ], cost: 2+4*D 4.86/2.25 4.86/2.25 4.86/2.25 4.86/2.25 Computing asymptotic complexity for rule 20 4.86/2.25 4.86/2.25 Solved the limit problem by the following transformations: 4.86/2.25 4.86/2.25 Created initial limit problem: 4.86/2.25 4.86/2.25 2+3*C (+), C (+/+!) [not solved] 4.86/2.25 4.86/2.25 4.86/2.25 4.86/2.25 removing all constraints (solved by SMT) 4.86/2.25 4.86/2.25 resulting limit problem: [solved] 4.86/2.25 4.86/2.25 4.86/2.25 4.86/2.25 applying transformation rule (C) using substitution {C==n} 4.86/2.25 4.86/2.25 resulting limit problem: 4.86/2.25 4.86/2.25 [solved] 4.86/2.25 4.86/2.25 4.86/2.25 4.86/2.25 Solution: 4.86/2.25 4.86/2.25 C / n 4.86/2.25 4.86/2.25 Resulting cost 2+3*n has complexity: Poly(n^1) 4.86/2.25 4.86/2.25 4.86/2.25 4.86/2.25 Found new complexity Poly(n^1). 4.86/2.25 4.86/2.25 4.86/2.25 4.86/2.25 Obtained the following overall complexity (w.r.t. the length of the input n): 4.86/2.25 4.86/2.25 Complexity: Poly(n^1) 4.86/2.25 4.86/2.25 Cpx degree: 1 4.86/2.25 4.86/2.25 Solved cost: 2+3*n 4.86/2.25 4.86/2.25 Rule cost: 2+3*C 4.86/2.25 4.86/2.25 Rule guard: [ C>=1 ] 4.86/2.25 4.86/2.25 4.86/2.25 4.86/2.25 WORST_CASE(Omega(n^1),?) 4.86/2.25 4.86/2.25 4.86/2.25 ---------------------------------------- 4.86/2.25 4.86/2.25 (4) 4.86/2.25 BOUNDS(n^1, INF) 4.86/2.26 EOF