/export/starexec/sandbox2/solver/bin/starexec_run_complexity /export/starexec/sandbox2/benchmark/theBenchmark.koat /export/starexec/sandbox2/output/output_files


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WORST_CASE(?, O(n^1))
proof of /export/starexec/sandbox2/benchmark/theBenchmark.koat
# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty


The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(1, max(3, 3 + Arg_0)).

(0) CpxIntTrs
(1) Koat2 Proof [FINISHED, 236 ms]
(2) BOUNDS(1, max(3, 3 + Arg_0))


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(0)
Obligation:
Complexity Int TRS consisting of the following rules:
f2(A, B, C) -> Com_1(f2(-(B) + A, 1 + B, C)) :|: A >= 1
f3(A, B, C) -> Com_1(f2(A, B, C)) :|: B >= 1
f2(A, B, C) -> Com_1(f4(A, B, D)) :|: 0 >= A
f3(A, B, C) -> Com_1(f4(A, B, D)) :|: 0 >= B

The start-symbols are:[f3_3]


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(1) Koat2 Proof (FINISHED)
YES( ?, max([3, 3+Arg_0]) {O(n)})



Initial Complexity Problem:

  Start:  f3  

  Program_Vars:  Arg_0, Arg_1, Arg_2  

  Temp_Vars:  D  

  Locations:  f2, f3, f4  

  Transitions:

  f2(Arg_0,Arg_1,Arg_2) -> f2(-Arg_1+Arg_0,1+Arg_1,Arg_2):|:1 <= Arg_1 && 1 <= Arg_0

  f2(Arg_0,Arg_1,Arg_2) -> f4(Arg_0,Arg_1,D):|:1 <= Arg_1 && Arg_0 <= 0

  f3(Arg_0,Arg_1,Arg_2) -> f2(Arg_0,Arg_1,Arg_2):|:1 <= Arg_1

  f3(Arg_0,Arg_1,Arg_2) -> f4(Arg_0,Arg_1,D):|:Arg_1 <= 0  



Timebounds: 

  Overall timebound: max([3, 3+Arg_0]) {O(n)}

  0: f2->f2: max([0, Arg_0]) {O(n)}

  2: f2->f4: 1 {O(1)}

  1: f3->f2: 1 {O(1)}

  3: f3->f4: 1 {O(1)}



Costbounds:

  Overall costbound: max([3, 3+Arg_0]) {O(n)}

  0: f2->f2: max([0, Arg_0]) {O(n)}

  2: f2->f4: 1 {O(1)}

  1: f3->f2: 1 {O(1)}

  3: f3->f4: 1 {O(1)}



Sizebounds:

`Lower:

  0: f2->f2, Arg_1: 2 {O(1)}

  0: f2->f2, Arg_2: Arg_2 {O(n)}

  2: f2->f4, Arg_1: 1 {O(1)}

  1: f3->f2, Arg_0: Arg_0 {O(n)}

  1: f3->f2, Arg_1: 1 {O(1)}

  1: f3->f2, Arg_2: Arg_2 {O(n)}

  3: f3->f4, Arg_0: Arg_0 {O(n)}

  3: f3->f4, Arg_1: Arg_1 {O(n)}

`Upper:

  0: f2->f2, Arg_0: Arg_0 {O(n)}

  0: f2->f2, Arg_1: Arg_1+max([0, Arg_0]) {O(n)}

  0: f2->f2, Arg_2: Arg_2 {O(n)}

  2: f2->f4, Arg_0: 0 {O(1)}

  2: f2->f4, Arg_1: max([Arg_1, Arg_1+max([0, Arg_0])]) {O(n)}

  1: f3->f2, Arg_0: Arg_0 {O(n)}

  1: f3->f2, Arg_1: Arg_1 {O(n)}

  1: f3->f2, Arg_2: Arg_2 {O(n)}

  3: f3->f4, Arg_0: Arg_0 {O(n)}

  3: f3->f4, Arg_1: 0 {O(1)}


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(2)
BOUNDS(1, max(3, 3 + Arg_0))