/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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(2 + 101 * Arg_1, 103)).

(0) CpxIntTrs
(1) Koat2 Proof [FINISHED, 349 ms]
(2) BOUNDS(1, max(2 + 101 * Arg_1, 103))


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(0)
Obligation:
Complexity Int TRS consisting of the following rules:
f300(A, B, C, D, E) -> Com_1(f300(1 + A, B, C, D, E)) :|: 100 >= A && B >= 1
f300(A, B, C, D, E) -> Com_1(f3(A, B, 0, 0, 0)) :|: A >= 101
f2(A, B, C, D, E) -> Com_1(f300(1, B, C, D, E)) :|: B >= 1
f2(A, B, C, D, E) -> Com_1(f3(0, B, 0, 0, 0)) :|: 0 >= B

The start-symbols are:[f2_5]


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(1) Koat2 Proof (FINISHED)
YES( ?, max([103, 2+101*Arg_1]) {O(n)})



Initial Complexity Problem:

  Start:  f2  

  Program_Vars:  Arg_0, Arg_1, Arg_2, Arg_3, Arg_4  

  Temp_Vars:    

  Locations:  f2, f3, f300  

  Transitions:

  f2(Arg_0,Arg_1,Arg_2,Arg_3,Arg_4) -> f3(0,Arg_1,0,0,0):|:Arg_1 <= 0

  f2(Arg_0,Arg_1,Arg_2,Arg_3,Arg_4) -> f300(1,Arg_1,Arg_2,Arg_3,Arg_4):|:1 <= Arg_1

  f300(Arg_0,Arg_1,Arg_2,Arg_3,Arg_4) -> f3(Arg_0,Arg_1,0,0,0):|:1 <= Arg_1 && 2 <= Arg_0+Arg_1 && 1 <= Arg_0 && 101 <= Arg_0

  f300(Arg_0,Arg_1,Arg_2,Arg_3,Arg_4) -> f300(1+Arg_0,Arg_1,Arg_2,Arg_3,Arg_4):|:1 <= Arg_1 && 2 <= Arg_0+Arg_1 && 1 <= Arg_0 && Arg_0 <= 100 && 1 <= Arg_1  



Timebounds: 

  Overall timebound: max([103, 2+101*Arg_1]) {O(n)}

  2: f2->f300: 1 {O(1)}

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

  0: f300->f300: max([100, -1+101*Arg_1]) {O(n)}

  1: f300->f3: 1 {O(1)}



Costbounds:

  Overall costbound: max([103, 2+101*Arg_1]) {O(n)}

  2: f2->f300: 1 {O(1)}

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

  0: f300->f300: max([100, -1+101*Arg_1]) {O(n)}

  1: f300->f3: 1 {O(1)}



Sizebounds:

`Lower:

  2: f2->f300, Arg_0: 1 {O(1)}

  2: f2->f300, Arg_1: 1 {O(1)}

  2: f2->f300, Arg_2: Arg_2 {O(n)}

  2: f2->f300, Arg_3: Arg_3 {O(n)}

  2: f2->f300, Arg_4: Arg_4 {O(n)}

  3: f2->f3, Arg_0: 0 {O(1)}

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

  3: f2->f3, Arg_2: 0 {O(1)}

  3: f2->f3, Arg_3: 0 {O(1)}

  3: f2->f3, Arg_4: 0 {O(1)}

  0: f300->f300, Arg_0: 2 {O(1)}

  0: f300->f300, Arg_1: 1 {O(1)}

  0: f300->f300, Arg_2: Arg_2 {O(n)}

  0: f300->f300, Arg_3: Arg_3 {O(n)}

  0: f300->f300, Arg_4: Arg_4 {O(n)}

  1: f300->f3, Arg_0: 101 {O(1)}

  1: f300->f3, Arg_1: 1 {O(1)}

  1: f300->f3, Arg_2: 0 {O(1)}

  1: f300->f3, Arg_3: 0 {O(1)}

  1: f300->f3, Arg_4: 0 {O(1)}

`Upper:

  2: f2->f300, Arg_0: 1 {O(1)}

  2: f2->f300, Arg_1: Arg_1 {O(n)}

  2: f2->f300, Arg_2: Arg_2 {O(n)}

  2: f2->f300, Arg_3: Arg_3 {O(n)}

  2: f2->f300, Arg_4: Arg_4 {O(n)}

  3: f2->f3, Arg_0: 0 {O(1)}

  3: f2->f3, Arg_1: 0 {O(1)}

  3: f2->f3, Arg_2: 0 {O(1)}

  3: f2->f3, Arg_3: 0 {O(1)}

  3: f2->f3, Arg_4: 0 {O(1)}

  0: f300->f300, Arg_0: 101 {O(1)}

  0: f300->f300, Arg_1: Arg_1 {O(n)}

  0: f300->f300, Arg_2: Arg_2 {O(n)}

  0: f300->f300, Arg_3: Arg_3 {O(n)}

  0: f300->f300, Arg_4: Arg_4 {O(n)}

  1: f300->f3, Arg_0: 101 {O(1)}

  1: f300->f3, Arg_1: Arg_1 {O(n)}

  1: f300->f3, Arg_2: 0 {O(1)}

  1: f300->f3, Arg_3: 0 {O(1)}

  1: f300->f3, Arg_4: 0 {O(1)}


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(2)
BOUNDS(1, max(2 + 101 * Arg_1, 103))