/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(1, max(1, 2 + Arg_0 + -1 * Arg_1)).

(0) CpxIntTrs
(1) Koat2 Proof [FINISHED, 147 ms]
(2) BOUNDS(1, max(1, 2 + Arg_0 + -1 * Arg_1))


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(0)
Obligation:
Complexity Int TRS consisting of the following rules:
eval(A, B, C) -> Com_1(eval(A, B + C, C)) :|: A >= B && C >= 1
start(A, B, C) -> Com_1(eval(A, B, C)) :|: TRUE

The start-symbols are:[start_3]


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(1) Koat2 Proof (FINISHED)
YES( ?, max([1, 2+Arg_0-Arg_1]) {O(n)})



Initial Complexity Problem:

  Start:  start  

  Program_Vars:  Arg_0, Arg_1, Arg_2  

  Temp_Vars:    

  Locations:  eval, start  

  Transitions:

  0: eval->eval

  1: start->eval  



Timebounds: 

  Overall timebound: max([1, 2+Arg_0-Arg_1]) {O(n)}

  0: eval->eval: max([0, 1+Arg_0-Arg_1]) {O(n)}

  1: start->eval: 1 {O(1)}



Costbounds:

  Overall costbound: max([1, 2+Arg_0-Arg_1]) {O(n)}

  0: eval->eval: max([0, 1+Arg_0-Arg_1]) {O(n)}

  1: start->eval: 1 {O(1)}



Sizebounds:

`Lower:

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

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

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

  1: start->eval, Arg_0: Arg_0 {O(n)}

  1: start->eval, Arg_1: Arg_1 {O(n)}

  1: start->eval, Arg_2: Arg_2 {O(n)}

`Upper:

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

  0: eval->eval, Arg_1: max([Arg_2, max([Arg_2, Arg_1])])+max([0, Arg_2*(1+Arg_0-Arg_1)]) {O(n^2)}

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

  1: start->eval, Arg_0: Arg_0 {O(n)}

  1: start->eval, Arg_1: Arg_1 {O(n)}

  1: start->eval, Arg_2: Arg_2 {O(n)}


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(2)
BOUNDS(1, max(1, 2 + Arg_0 + -1 * Arg_1))