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


--------------------------------------------------------------------------------


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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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_4) + nat(max(3, 1 + 2 * Arg_1) + max(-2 * Arg_2, -2)) + nat(1 + Arg_4) + nat(1 + Arg_1)).

(0) CpxIntTrs
(1) Koat2 Proof [FINISHED, 1228 ms]
(2) BOUNDS(1, max(1, 2 + Arg_4) + nat(max(3, 1 + 2 * Arg_1) + max(-2 * Arg_2, -2)) + nat(1 + Arg_4) + nat(1 + Arg_1))


----------------------------------------

(0)
Obligation:
Complexity Int TRS consisting of the following rules:
start(A, B, C, D, E, F) -> Com_1(m1(A, B, C, D, E, F)) :|: A >= 0 && B + A + 2 >= 2 * C && B >= A + 1 && 2 * C >= B + A && D >= 0 && E + 1 >= C && E + 1 <= C && F >= A && F <= A
m1(A, B, C, D, E, F) -> Com_1(m1(A, B, H, D, E, G)) :|: B >= 1 && D >= 0 && A >= E + 1 && B + 1 >= G && C + 1 >= H && H >= 1 + C && F + 1 >= G && G >= 1 + F
m1(A, B, C, D, E, F) -> Com_1(m1(H, B, C, D, E, G)) :|: B >= 1 && D >= 0 && B >= F && E + 1 >= H && C >= B + 1 && F + 1 >= G && G >= 1 + F && A + 1 >= H && H >= 1 + A
m1(A, B, C, D, E, F) -> Com_1(m1(A, B, H, D, E, G)) :|: B >= 1 && D >= 0 && B >= F && B + 1 >= H && E >= A && F + 1 >= G && G >= 1 + F && C + 1 >= H && H >= 1 + C
m1(A, B, C, D, E, F) -> Com_1(m1(H, B, C, D, E, G)) :|: B >= 1 && D >= 0 && B >= F && B >= C && E + 1 >= H && A + 1 >= H && H >= 1 + A && F + 1 >= G && G >= 1 + F

The start-symbols are:[start_6]


----------------------------------------

(1) Koat2 Proof (FINISHED)
YES( ?, 1+max([0, 1+Arg_4])+max([0, 1+max([2, 2*Arg_1])+max([-2, -2*Arg_2])])+max([0, 1+Arg_4])+max([0, 1+Arg_1]) {O(n)})



Initial Complexity Problem:

  Start:  start  

  Program_Vars:  Arg_0, Arg_1, Arg_2, Arg_3, Arg_4, Arg_5  

  Temp_Vars:    

  Locations:  m1, start  

  Transitions:

  1: m1->m1

  2: m1->m1

  3: m1->m1

  4: m1->m1

  0: start->m1  



Timebounds: 

  Overall timebound: 1+max([0, 1+Arg_4])+max([0, 1+max([2, 2*Arg_1])+max([-2, -2*Arg_2])])+max([0, 1+Arg_4])+max([0, 1+Arg_1]) {O(n)}

  1: m1->m1: max([0, 1+Arg_1]) {O(n)}

  2: m1->m1: max([0, 1+Arg_4]) {O(n)}

  3: m1->m1: max([0, 1+max([2, 2*Arg_1])+max([-2, -2*Arg_2])]) {O(n)}

  4: m1->m1: max([0, 1+Arg_4]) {O(n)}

  0: start->m1: 1 {O(1)}



Costbounds:

  Overall costbound: 1+max([0, 1+Arg_4])+max([0, 1+max([2, 2*Arg_1])+max([-2, -2*Arg_2])])+max([0, 1+Arg_4])+max([0, 1+Arg_1]) {O(n)}

  1: m1->m1: max([0, 1+Arg_1]) {O(n)}

  2: m1->m1: max([0, 1+Arg_4]) {O(n)}

  3: m1->m1: max([0, 1+max([2, 2*Arg_1])+max([-2, -2*Arg_2])]) {O(n)}

  4: m1->m1: max([0, 1+Arg_4]) {O(n)}

  0: start->m1: 1 {O(1)}



Sizebounds:

`Lower:

  1: m1->m1, Arg_0: 0 {O(1)}

  1: m1->m1, Arg_1: 1 {O(1)}

  1: m1->m1, Arg_2: 1 {O(1)}

  1: m1->m1, Arg_3: 0 {O(1)}

  1: m1->m1, Arg_4: 0 {O(1)}

  1: m1->m1, Arg_5: 0 {O(1)}

  2: m1->m1, Arg_0: 0 {O(1)}

  2: m1->m1, Arg_1: 1 {O(1)}

  2: m1->m1, Arg_2: 2 {O(1)}

  2: m1->m1, Arg_3: 0 {O(1)}

  2: m1->m1, Arg_4: 0 {O(1)}

  2: m1->m1, Arg_5: 0 {O(1)}

  3: m1->m1, Arg_0: 0 {O(1)}

  3: m1->m1, Arg_1: 1 {O(1)}

  3: m1->m1, Arg_2: 1 {O(1)}

  3: m1->m1, Arg_3: 0 {O(1)}

  3: m1->m1, Arg_4: 0 {O(1)}

  3: m1->m1, Arg_5: 0 {O(1)}

  4: m1->m1, Arg_0: 0 {O(1)}

  4: m1->m1, Arg_1: 1 {O(1)}

  4: m1->m1, Arg_2: 1 {O(1)}

  4: m1->m1, Arg_3: 0 {O(1)}

  4: m1->m1, Arg_4: 0 {O(1)}

  4: m1->m1, Arg_5: 0 {O(1)}

  0: start->m1, Arg_0: 0 {O(1)}

  0: start->m1, Arg_1: 1 {O(1)}

  0: start->m1, Arg_2: 1 {O(1)}

  0: start->m1, Arg_3: 0 {O(1)}

  0: start->m1, Arg_4: 0 {O(1)}

  0: start->m1, Arg_5: 0 {O(1)}

`Upper:

  1: m1->m1, Arg_0: max([Arg_0+max([0, 1+Arg_4])+max([0, 1+Arg_4]), Arg_0+max([0, 1+Arg_4])]) {O(n)}

  1: m1->m1, Arg_1: Arg_1 {O(n)}

  1: m1->m1, Arg_2: Arg_2+max([0, 1+max([2, 2*Arg_1])+max([-2, -2*Arg_2])])+max([0, 1+Arg_1]) {O(n)}

  1: m1->m1, Arg_3: Arg_3 {O(n)}

  1: m1->m1, Arg_4: Arg_4 {O(n)}

  1: m1->m1, Arg_5: max([Arg_5+max([0, 1+Arg_4])+max([0, 1+max([2, 2*Arg_1])+max([-2, -2*Arg_2])])+max([0, 1+Arg_4]), Arg_5+max([0, 1+Arg_4])+max([0, 1+max([2, 2*Arg_1])+max([-2, -2*Arg_2])])])+max([0, 1+Arg_1]) {O(n)}

  2: m1->m1, Arg_0: Arg_0+max([0, 1+Arg_4])+max([0, 1+Arg_4]) {O(n)}

  2: m1->m1, Arg_1: Arg_1 {O(n)}

  2: m1->m1, Arg_2: Arg_2+max([0, 1+max([2, 2*Arg_1])+max([-2, -2*Arg_2])]) {O(n)}

  2: m1->m1, Arg_3: Arg_3 {O(n)}

  2: m1->m1, Arg_4: Arg_4 {O(n)}

  2: m1->m1, Arg_5: Arg_5+max([0, 1+Arg_4])+max([0, 1+max([2, 2*Arg_1])+max([-2, -2*Arg_2])])+max([0, 1+Arg_4]) {O(n)}

  3: m1->m1, Arg_0: Arg_0+max([0, 1+Arg_4]) {O(n)}

  3: m1->m1, Arg_1: Arg_1 {O(n)}

  3: m1->m1, Arg_2: Arg_2+max([0, 1+max([2, 2*Arg_1])+max([-2, -2*Arg_2])]) {O(n)}

  3: m1->m1, Arg_3: Arg_3 {O(n)}

  3: m1->m1, Arg_4: Arg_4 {O(n)}

  3: m1->m1, Arg_5: Arg_5+max([0, 1+Arg_4])+max([0, 1+max([2, 2*Arg_1])+max([-2, -2*Arg_2])]) {O(n)}

  4: m1->m1, Arg_0: Arg_0+max([0, 1+Arg_4]) {O(n)}

  4: m1->m1, Arg_1: Arg_1 {O(n)}

  4: m1->m1, Arg_2: Arg_2+max([0, 1+max([2, 2*Arg_1])+max([-2, -2*Arg_2])]) {O(n)}

  4: m1->m1, Arg_3: Arg_3 {O(n)}

  4: m1->m1, Arg_4: Arg_4 {O(n)}

  4: m1->m1, Arg_5: Arg_5+max([0, 1+Arg_4])+max([0, 1+max([2, 2*Arg_1])+max([-2, -2*Arg_2])]) {O(n)}

  0: start->m1, Arg_0: Arg_0 {O(n)}

  0: start->m1, Arg_1: Arg_1 {O(n)}

  0: start->m1, Arg_2: Arg_2 {O(n)}

  0: start->m1, Arg_3: Arg_3 {O(n)}

  0: start->m1, Arg_4: Arg_4 {O(n)}

  0: start->m1, Arg_5: Arg_5 {O(n)}


----------------------------------------

(2)
BOUNDS(1, max(1, 2 + Arg_4) + nat(max(3, 1 + 2 * Arg_1) + max(-2 * Arg_2, -2)) + nat(1 + Arg_4) + nat(1 + Arg_1))