/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^2))
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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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(10 + 2 * Arg_0, 8) + max(6, -2 + 4 * Arg_0) + max(1, 1 + Arg_0) + max(2, 2 + 2 * Arg_0) * max(5, -1 + 2 * Arg_0)).

(0) CpxIntTrs
(1) Koat2 Proof [FINISHED, 1772 ms]
(2) BOUNDS(1, max(10 + 2 * Arg_0, 8) + max(6, -2 + 4 * Arg_0) + max(1, 1 + Arg_0) + max(2, 2 + 2 * Arg_0) * max(5, -1 + 2 * Arg_0))


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(0)
Obligation:
Complexity Int TRS consisting of the following rules:
start(A, B, C, D, E, F) -> Com_1(stop(A, B, C, F, E, F)) :|: 0 >= A + 1 && B >= C && B <= C && D >= E && D <= E && F >= A && F <= A
start(A, B, C, D, E, F) -> Com_1(lbl121(A, 1, C, F - 1, E, F)) :|: A >= 0 && 1 >= A && B >= C && B <= C && D >= E && D <= E && F >= A && F <= A
start(A, B, C, D, E, F) -> Com_1(lbl101(A, 2, C, F, E, F)) :|: A >= 2 && B >= C && B <= C && D >= E && D <= E && F >= A && F <= A
lbl121(A, B, C, D, E, F) -> Com_1(stop(A, B, C, D, E, F)) :|: A >= 0 && B >= 0 && B >= 1 && D + 1 >= 0 && D + 1 <= 0 && F >= A && F <= A
lbl121(A, B, C, D, E, F) -> Com_1(lbl121(A, 1, C, D - 1, E, F)) :|: D >= 0 && 1 >= D && A >= D + 1 && B >= D + 1 && B >= 1 && D + 1 >= 0 && F >= A && F <= A
lbl121(A, B, C, D, E, F) -> Com_1(lbl101(A, 2, C, D, E, F)) :|: D >= 2 && A >= D + 1 && B >= D + 1 && B >= 1 && D + 1 >= 0 && F >= A && F <= A
lbl101(A, B, C, D, E, F) -> Com_1(lbl101(A, 2 * B, C, D, E, F)) :|: D >= B + 1 && B >= 2 && 2 * D >= B + 2 && A >= D && F >= A && F <= A
lbl101(A, B, C, D, E, F) -> Com_1(lbl121(A, B, C, D - 1, E, F)) :|: B >= D && B >= 2 && 2 * D >= B + 2 && A >= D && F >= A && F <= A
start0(A, B, C, D, E, F) -> Com_1(start(A, C, C, E, E, A)) :|: TRUE

The start-symbols are:[start0_6]


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(1) Koat2 Proof (FINISHED)
YES( ?, 5+1+max([1, 2+Arg_0])+max([3, -1+2*Arg_0])+max([1, 1+Arg_0])+max([3, -1+2*Arg_0])+2*max([1, 1+Arg_0])*max([5, -1+2*Arg_0])+max([1, 2+Arg_0]) {O(n^2)})



Initial Complexity Problem:

  Start:  start0  

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

  Temp_Vars:    

  Locations:  lbl101, lbl121, start, start0, stop  

  Transitions:

  6: lbl101->lbl101

  7: lbl101->lbl121

  5: lbl121->lbl101

  4: lbl121->lbl121

  3: lbl121->stop

  2: start->lbl101

  1: start->lbl121

  0: start->stop

  8: start0->start  



Timebounds: 

  Overall timebound: 5+1+max([1, 2+Arg_0])+max([3, -1+2*Arg_0])+max([1, 1+Arg_0])+max([3, -1+2*Arg_0])+2*max([1, 1+Arg_0])*max([5, -1+2*Arg_0])+max([1, 2+Arg_0]) {O(n^2)}

  6: lbl101->lbl101: 1+2*max([1, 1+Arg_0])*max([5, -1+2*Arg_0])+max([3, -1+2*Arg_0])+max([3, -1+2*Arg_0]) {O(n^2)}

  7: lbl101->lbl121: max([1, 2+Arg_0]) {O(n)}

  3: lbl121->stop: 1 {O(1)}

  4: lbl121->lbl121: max([1, 2+Arg_0]) {O(n)}

  5: lbl121->lbl101: max([1, 1+Arg_0]) {O(n)}

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

  1: start->lbl121: 1 {O(1)}

  2: start->lbl101: 1 {O(1)}

  8: start0->start: 1 {O(1)}



Costbounds:

  Overall costbound: 5+1+max([1, 2+Arg_0])+max([3, -1+2*Arg_0])+max([1, 1+Arg_0])+max([3, -1+2*Arg_0])+2*max([1, 1+Arg_0])*max([5, -1+2*Arg_0])+max([1, 2+Arg_0]) {O(n^2)}

  6: lbl101->lbl101: 1+2*max([1, 1+Arg_0])*max([5, -1+2*Arg_0])+max([3, -1+2*Arg_0])+max([3, -1+2*Arg_0]) {O(n^2)}

  7: lbl101->lbl121: max([1, 2+Arg_0]) {O(n)}

  3: lbl121->stop: 1 {O(1)}

  4: lbl121->lbl121: max([1, 2+Arg_0]) {O(n)}

  5: lbl121->lbl101: max([1, 1+Arg_0]) {O(n)}

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

  1: start->lbl121: 1 {O(1)}

  2: start->lbl101: 1 {O(1)}

  8: start0->start: 1 {O(1)}



Sizebounds:

`Lower:

  6: lbl101->lbl101, Arg_0: 3 {O(1)}

  6: lbl101->lbl101, Arg_1: 4 {O(1)}

  6: lbl101->lbl101, Arg_2: Arg_2 {O(n)}

  6: lbl101->lbl101, Arg_3: 3 {O(1)}

  6: lbl101->lbl101, Arg_4: Arg_4 {O(n)}

  6: lbl101->lbl101, Arg_5: 3 {O(1)}

  7: lbl101->lbl121, Arg_0: 2 {O(1)}

  7: lbl101->lbl121, Arg_1: 2 {O(1)}

  7: lbl101->lbl121, Arg_2: Arg_2 {O(n)}

  7: lbl101->lbl121, Arg_3: 1 {O(1)}

  7: lbl101->lbl121, Arg_4: Arg_4 {O(n)}

  7: lbl101->lbl121, Arg_5: 2 {O(1)}

  3: lbl121->stop, Arg_0: 0 {O(1)}

  3: lbl121->stop, Arg_1: 1 {O(1)}

  3: lbl121->stop, Arg_2: Arg_2 {O(n)}

  3: lbl121->stop, Arg_3: -1 {O(1)}

  3: lbl121->stop, Arg_4: Arg_4 {O(n)}

  3: lbl121->stop, Arg_5: 0 {O(1)}

  4: lbl121->lbl121, Arg_0: 1 {O(1)}

  4: lbl121->lbl121, Arg_1: 1 {O(1)}

  4: lbl121->lbl121, Arg_2: Arg_2 {O(n)}

  4: lbl121->lbl121, Arg_3: -1 {O(1)}

  4: lbl121->lbl121, Arg_4: Arg_4 {O(n)}

  4: lbl121->lbl121, Arg_5: 1 {O(1)}

  5: lbl121->lbl101, Arg_0: 3 {O(1)}

  5: lbl121->lbl101, Arg_1: 2 {O(1)}

  5: lbl121->lbl101, Arg_2: Arg_2 {O(n)}

  5: lbl121->lbl101, Arg_3: 2 {O(1)}

  5: lbl121->lbl101, Arg_4: Arg_4 {O(n)}

  5: lbl121->lbl101, Arg_5: 3 {O(1)}

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

  0: start->stop, Arg_1: Arg_2 {O(n)}

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

  0: start->stop, Arg_3: Arg_0 {O(n)}

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

  0: start->stop, Arg_5: Arg_0 {O(n)}

  1: start->lbl121, Arg_0: 0 {O(1)}

  1: start->lbl121, Arg_1: 1 {O(1)}

  1: start->lbl121, Arg_2: Arg_2 {O(n)}

  1: start->lbl121, Arg_3: -1 {O(1)}

  1: start->lbl121, Arg_4: Arg_4 {O(n)}

  1: start->lbl121, Arg_5: 0 {O(1)}

  2: start->lbl101, Arg_0: 2 {O(1)}

  2: start->lbl101, Arg_1: 2 {O(1)}

  2: start->lbl101, Arg_2: Arg_2 {O(n)}

  2: start->lbl101, Arg_3: 2 {O(1)}

  2: start->lbl101, Arg_4: Arg_4 {O(n)}

  2: start->lbl101, Arg_5: 2 {O(1)}

  8: start0->start, Arg_0: Arg_0 {O(n)}

  8: start0->start, Arg_1: Arg_2 {O(n)}

  8: start0->start, Arg_2: Arg_2 {O(n)}

  8: start0->start, Arg_3: Arg_4 {O(n)}

  8: start0->start, Arg_4: Arg_4 {O(n)}

  8: start0->start, Arg_5: Arg_0 {O(n)}

`Upper:

  6: lbl101->lbl101, Arg_0: Arg_0 {O(n)}

  6: lbl101->lbl101, Arg_1: 2*2^(1+2*max([1, 1+Arg_0])*max([5, -1+2*Arg_0])+max([3, -1+2*Arg_0])+max([3, -1+2*Arg_0])) {O(EXP)}

  6: lbl101->lbl101, Arg_2: Arg_2 {O(n)}

  6: lbl101->lbl101, Arg_3: Arg_0 {O(n)}

  6: lbl101->lbl101, Arg_4: Arg_4 {O(n)}

  6: lbl101->lbl101, Arg_5: Arg_0 {O(n)}

  7: lbl101->lbl121, Arg_0: Arg_0 {O(n)}

  7: lbl101->lbl121, Arg_1: max([2, 2*2^(1+2*max([1, 1+Arg_0])*max([5, -1+2*Arg_0])+max([3, -1+2*Arg_0])+max([3, -1+2*Arg_0]))]) {O(EXP)}

  7: lbl101->lbl121, Arg_2: Arg_2 {O(n)}

  7: lbl101->lbl121, Arg_3: Arg_0 {O(n)}

  7: lbl101->lbl121, Arg_4: Arg_4 {O(n)}

  7: lbl101->lbl121, Arg_5: Arg_0 {O(n)}

  3: lbl121->stop, Arg_0: max([1, max([1, Arg_0])]) {O(n)}

  3: lbl121->stop, Arg_1: 1 {O(1)}

  3: lbl121->stop, Arg_2: Arg_2 {O(n)}

  3: lbl121->stop, Arg_3: -1 {O(1)}

  3: lbl121->stop, Arg_4: Arg_4 {O(n)}

  3: lbl121->stop, Arg_5: max([1, max([1, Arg_0])]) {O(n)}

  4: lbl121->lbl121, Arg_0: max([1, Arg_0]) {O(n)}

  4: lbl121->lbl121, Arg_1: 1 {O(1)}

  4: lbl121->lbl121, Arg_2: Arg_2 {O(n)}

  4: lbl121->lbl121, Arg_3: 0 {O(1)}

  4: lbl121->lbl121, Arg_4: Arg_4 {O(n)}

  4: lbl121->lbl121, Arg_5: max([1, Arg_0]) {O(n)}

  5: lbl121->lbl101, Arg_0: Arg_0 {O(n)}

  5: lbl121->lbl101, Arg_1: 2 {O(1)}

  5: lbl121->lbl101, Arg_2: Arg_2 {O(n)}

  5: lbl121->lbl101, Arg_3: Arg_0 {O(n)}

  5: lbl121->lbl101, Arg_4: Arg_4 {O(n)}

  5: lbl121->lbl101, Arg_5: Arg_0 {O(n)}

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

  0: start->stop, Arg_1: Arg_2 {O(n)}

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

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

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

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

  1: start->lbl121, Arg_0: 1 {O(1)}

  1: start->lbl121, Arg_1: 1 {O(1)}

  1: start->lbl121, Arg_2: Arg_2 {O(n)}

  1: start->lbl121, Arg_3: 0 {O(1)}

  1: start->lbl121, Arg_4: Arg_4 {O(n)}

  1: start->lbl121, Arg_5: 1 {O(1)}

  2: start->lbl101, Arg_0: Arg_0 {O(n)}

  2: start->lbl101, Arg_1: 2 {O(1)}

  2: start->lbl101, Arg_2: Arg_2 {O(n)}

  2: start->lbl101, Arg_3: Arg_0 {O(n)}

  2: start->lbl101, Arg_4: Arg_4 {O(n)}

  2: start->lbl101, Arg_5: Arg_0 {O(n)}

  8: start0->start, Arg_0: Arg_0 {O(n)}

  8: start0->start, Arg_1: Arg_2 {O(n)}

  8: start0->start, Arg_2: Arg_2 {O(n)}

  8: start0->start, Arg_3: Arg_4 {O(n)}

  8: start0->start, Arg_4: Arg_4 {O(n)}

  8: start0->start, Arg_5: Arg_0 {O(n)}


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(2)
BOUNDS(1, max(10 + 2 * Arg_0, 8) + max(6, -2 + 4 * Arg_0) + max(1, 1 + Arg_0) + max(2, 2 + 2 * Arg_0) * max(5, -1 + 2 * Arg_0))