/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty


The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(1, max(3, 3 + 4 * Arg_0) + nat(8 * Arg_0)).

(0) CpxIntTrs
(1) Koat2 Proof [FINISHED, 154 ms]
(2) BOUNDS(1, max(3, 3 + 4 * Arg_0) + nat(8 * Arg_0))


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(0)
Obligation:
Complexity Int TRS consisting of the following rules:
start(A) -> Com_1(a(A)) :|: A >= 1
start(A) -> Com_1(a(A)) :|: A >= 2
start(A) -> Com_1(a(A)) :|: A >= 4
a(A) -> Com_1(a(B)) :|: 2 * B >= 2 && A >= 2 * B && A <= 2 * B
a(A) -> Com_1(a(B)) :|: 2 * B >= 1 && A >= 2 * B + 1 && A <= 2 * B + 1

The start-symbols are:[start_1]


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(1) Koat2 Proof (FINISHED)
YES( ?, 3+2*(2*max([0, Arg_0])+max([0, Arg_0])+max([0, Arg_0])+max([0, Arg_0])+max([0, Arg_0])) {O(n)})



Initial Complexity Problem:

  Start:  start  

  Program_Vars:  Arg_0  

  Temp_Vars:  B  

  Locations:  a, start  

  Transitions:

  a(Arg_0) -> a(B):|:2 <= (2)*B && Arg_0 <= (2)*B && (2)*B <= Arg_0

  a(Arg_0) -> a(B):|:1 <= (2)*B && Arg_0 <= (2)*B+1 && (2)*B+1 <= Arg_0

  start(Arg_0) -> a(Arg_0):|:1 <= Arg_0

  start(Arg_0) -> a(Arg_0):|:2 <= Arg_0

  start(Arg_0) -> a(Arg_0):|:4 <= Arg_0  



Timebounds: 

  Overall timebound: 3+2*(2*max([0, Arg_0])+max([0, Arg_0])+max([0, Arg_0])+max([0, Arg_0])+max([0, Arg_0])) {O(n)}

  3: a->a: 2*max([0, Arg_0])+max([0, Arg_0])+max([0, Arg_0])+max([0, Arg_0])+max([0, Arg_0]) {O(n)}

  4: a->a: 2*max([0, Arg_0])+max([0, Arg_0])+max([0, Arg_0])+max([0, Arg_0])+max([0, Arg_0]) {O(n)}

  0: start->a: 1 {O(1)}

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

  2: start->a: 1 {O(1)}



Costbounds:

  Overall costbound: 3+2*(2*max([0, Arg_0])+max([0, Arg_0])+max([0, Arg_0])+max([0, Arg_0])+max([0, Arg_0])) {O(n)}

  3: a->a: 2*max([0, Arg_0])+max([0, Arg_0])+max([0, Arg_0])+max([0, Arg_0])+max([0, Arg_0]) {O(n)}

  4: a->a: 2*max([0, Arg_0])+max([0, Arg_0])+max([0, Arg_0])+max([0, Arg_0])+max([0, Arg_0]) {O(n)}

  0: start->a: 1 {O(1)}

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

  2: start->a: 1 {O(1)}



Sizebounds:

`Lower:

  3: a->a, Arg_0: 1 {O(1)}

  4: a->a, Arg_0: 1 {O(1)}

  0: start->a, Arg_0: 1 {O(1)}

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

  2: start->a, Arg_0: 4 {O(1)}

`Upper:

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

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

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


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(2)
BOUNDS(1, max(3, 3 + 4 * Arg_0) + nat(8 * Arg_0))