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


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WORST_CASE(?, O(n^1))
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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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(1, nat(1 + 2 * Arg_2) + max(1, 2 + Arg_0 + -1 * Arg_1)).

(0) CpxIntTrs
(1) Koat2 Proof [FINISHED, 215 ms]
(2) BOUNDS(1, nat(1 + 2 * Arg_2) + max(1, 2 + Arg_0 + -1 * Arg_1))


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(0)
Obligation:
Complexity Int TRS consisting of the following rules:
eval0(A, B, C) -> Com_1(eval1(A, B, C)) :|: A >= 1
eval1(A, B, C) -> Com_1(eval1(A, B + A, C)) :|: A >= B + 1 && C >= A + 1 && A >= 1
eval1(A, B, C) -> Com_1(eval1(A, B, B - A)) :|: A >= B + 1 && C >= A + 1 && A >= 1

The start-symbols are:[eval0_3]


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(1) Koat2 Proof (FINISHED)
YES( ?, max([0, 1+2*Arg_2])+max([1, 2+Arg_0-Arg_1]) {O(n)})



Initial Complexity Problem:

  Start:  eval0  

  Program_Vars:  Arg_0, Arg_1, Arg_2  

  Temp_Vars:    

  Locations:  eval0, eval1  

  Transitions:

  0: eval0->eval1

  1: eval1->eval1

  2: eval1->eval1  



Timebounds: 

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

  0: eval0->eval1: 1 {O(1)}

  1: eval1->eval1: max([0, 1+Arg_0-Arg_1]) {O(n)}

  2: eval1->eval1: max([0, 1+2*Arg_2]) {O(n)}



Costbounds:

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

  0: eval0->eval1: 1 {O(1)}

  1: eval1->eval1: max([0, 1+Arg_0-Arg_1]) {O(n)}

  2: eval1->eval1: max([0, 1+2*Arg_2]) {O(n)}



Sizebounds:

`Lower:

  0: eval0->eval1, Arg_0: 1 {O(1)}

  0: eval0->eval1, Arg_1: Arg_1 {O(n)}

  0: eval0->eval1, Arg_2: Arg_2 {O(n)}

  1: eval1->eval1, Arg_0: 1 {O(1)}

  1: eval1->eval1, Arg_1: Arg_1 {O(n)}

  1: eval1->eval1, Arg_2: 2 {O(1)}

  2: eval1->eval1, Arg_0: 1 {O(1)}

  2: eval1->eval1, Arg_1: Arg_1 {O(n)}

  2: eval1->eval1, Arg_2: Arg_1-Arg_0 {O(n)}

`Upper:

  0: eval0->eval1, Arg_0: Arg_0 {O(n)}

  0: eval0->eval1, Arg_1: Arg_1 {O(n)}

  0: eval0->eval1, Arg_2: Arg_2 {O(n)}

  1: eval1->eval1, Arg_0: Arg_0 {O(n)}

  1: eval1->eval1, Arg_1: 2*Arg_0 {O(n)}

  1: eval1->eval1, Arg_2: Arg_2 {O(n)}

  2: eval1->eval1, Arg_0: Arg_0 {O(n)}

  2: eval1->eval1, Arg_1: max([Arg_1, 2*Arg_0]) {O(n)}

  2: eval1->eval1, Arg_2: -1 {O(1)}


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(2)
BOUNDS(1, nat(1 + 2 * Arg_2) + max(1, 2 + Arg_0 + -1 * Arg_1))