/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^2))
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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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, nat(2 * Arg_0 * max(-4 + 4 * Arg_0, 8) + min(8 + -8 * Arg_0, -16)) + nat(-1 + Arg_0) + max(-8 + 8 * Arg_0, 8) + max(4, 2 * Arg_0) + max(6, 6 + Arg_0)).

(0) CpxIntTrs
(1) Koat2 Proof [FINISHED, 3395 ms]
(2) BOUNDS(1, nat(2 * Arg_0 * max(-4 + 4 * Arg_0, 8) + min(8 + -8 * Arg_0, -16)) + nat(-1 + Arg_0) + max(-8 + 8 * Arg_0, 8) + max(4, 2 * Arg_0) + max(6, 6 + 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+max([1, 1+Arg_0])+max([0, (-1+Arg_0)*max([8, -4+4*Arg_0])])+max([0, -1+Arg_0])+max([4, -4+4*Arg_0])+max([4, 2*Arg_0])+max([0, (-1+Arg_0)*max([8, -4+4*Arg_0])])+max([4, -4+4*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:

  lbl101(Arg_0,Arg_1,Arg_2,Arg_3,Arg_4,Arg_5) -> lbl101(Arg_0,(2)*Arg_1,Arg_2,Arg_3,Arg_4,Arg_5):|:Arg_5 <= Arg_0 && 2 <= Arg_5 && 4 <= Arg_3+Arg_5 && Arg_3 <= Arg_5 && 4 <= Arg_1+Arg_5 && 4 <= Arg_0+Arg_5 && Arg_0 <= Arg_5 && Arg_3 <= Arg_0 && 2 <= Arg_3 && 4 <= Arg_1+Arg_3 && 4 <= Arg_0+Arg_3 && 2 <= Arg_1 && 4 <= Arg_0+Arg_1 && 2 <= Arg_0 && Arg_1+1 <= Arg_3 && 2 <= Arg_1 && Arg_1+2 <= (2)*Arg_3 && Arg_3 <= Arg_0 && Arg_5 <= Arg_0 && Arg_0 <= Arg_5

  lbl101(Arg_0,Arg_1,Arg_2,Arg_3,Arg_4,Arg_5) -> lbl121(Arg_0,Arg_1,Arg_2,Arg_3-1,Arg_4,Arg_5):|:Arg_5 <= Arg_0 && 2 <= Arg_5 && 4 <= Arg_3+Arg_5 && Arg_3 <= Arg_5 && 4 <= Arg_1+Arg_5 && 4 <= Arg_0+Arg_5 && Arg_0 <= Arg_5 && Arg_3 <= Arg_0 && 2 <= Arg_3 && 4 <= Arg_1+Arg_3 && 4 <= Arg_0+Arg_3 && 2 <= Arg_1 && 4 <= Arg_0+Arg_1 && 2 <= Arg_0 && Arg_3 <= Arg_1 && 2 <= Arg_1 && Arg_1+2 <= (2)*Arg_3 && Arg_3 <= Arg_0 && Arg_5 <= Arg_0 && Arg_0 <= Arg_5

  lbl121(Arg_0,Arg_1,Arg_2,Arg_3,Arg_4,Arg_5) -> lbl101(Arg_0,2,Arg_2,Arg_3,Arg_4,Arg_5):|:Arg_5 <= Arg_0 && 0 <= Arg_5 && 0 <= 1+Arg_3+Arg_5 && 1+Arg_3 <= Arg_5 && 1 <= Arg_1+Arg_5 && 0 <= Arg_0+Arg_5 && Arg_0 <= Arg_5 && 1+Arg_3 <= Arg_1 && 1+Arg_3 <= Arg_0 && 0 <= 1+Arg_3 && 0 <= Arg_1+Arg_3 && 0 <= 1+Arg_0+Arg_3 && 1 <= Arg_1 && 1 <= Arg_0+Arg_1 && 0 <= Arg_0 && 2 <= Arg_3 && Arg_3+1 <= Arg_0 && Arg_3+1 <= Arg_1 && 1 <= Arg_1 && 0 <= 1+Arg_3 && Arg_5 <= Arg_0 && Arg_0 <= Arg_5

  lbl121(Arg_0,Arg_1,Arg_2,Arg_3,Arg_4,Arg_5) -> lbl121(Arg_0,1,Arg_2,Arg_3-1,Arg_4,Arg_5):|:Arg_5 <= Arg_0 && 0 <= Arg_5 && 0 <= 1+Arg_3+Arg_5 && 1+Arg_3 <= Arg_5 && 1 <= Arg_1+Arg_5 && 0 <= Arg_0+Arg_5 && Arg_0 <= Arg_5 && 1+Arg_3 <= Arg_1 && 1+Arg_3 <= Arg_0 && 0 <= 1+Arg_3 && 0 <= Arg_1+Arg_3 && 0 <= 1+Arg_0+Arg_3 && 1 <= Arg_1 && 1 <= Arg_0+Arg_1 && 0 <= Arg_0 && 0 <= Arg_3 && Arg_3 <= 1 && Arg_3+1 <= Arg_0 && Arg_3+1 <= Arg_1 && 1 <= Arg_1 && 0 <= 1+Arg_3 && Arg_5 <= Arg_0 && Arg_0 <= Arg_5

  lbl121(Arg_0,Arg_1,Arg_2,Arg_3,Arg_4,Arg_5) -> stop(Arg_0,Arg_1,Arg_2,Arg_3,Arg_4,Arg_5):|:Arg_5 <= Arg_0 && 0 <= Arg_5 && 0 <= 1+Arg_3+Arg_5 && 1+Arg_3 <= Arg_5 && 1 <= Arg_1+Arg_5 && 0 <= Arg_0+Arg_5 && Arg_0 <= Arg_5 && 1+Arg_3 <= Arg_1 && 1+Arg_3 <= Arg_0 && 0 <= 1+Arg_3 && 0 <= Arg_1+Arg_3 && 0 <= 1+Arg_0+Arg_3 && 1 <= Arg_1 && 1 <= Arg_0+Arg_1 && 0 <= Arg_0 && 0 <= Arg_0 && 0 <= Arg_1 && 1 <= Arg_1 && Arg_3+1 <= 0 && 0 <= 1+Arg_3 && Arg_5 <= Arg_0 && Arg_0 <= Arg_5

  start(Arg_0,Arg_1,Arg_2,Arg_3,Arg_4,Arg_5) -> lbl101(Arg_0,2,Arg_2,Arg_5,Arg_4,Arg_5):|:Arg_5 <= Arg_0 && Arg_0 <= Arg_5 && Arg_4 <= Arg_3 && Arg_3 <= Arg_4 && Arg_2 <= Arg_1 && Arg_1 <= Arg_2 && 2 <= Arg_0 && Arg_1 <= Arg_2 && Arg_2 <= Arg_1 && Arg_3 <= Arg_4 && Arg_4 <= Arg_3 && Arg_5 <= Arg_0 && Arg_0 <= Arg_5

  start(Arg_0,Arg_1,Arg_2,Arg_3,Arg_4,Arg_5) -> lbl121(Arg_0,1,Arg_2,Arg_5-1,Arg_4,Arg_5):|:Arg_5 <= Arg_0 && Arg_0 <= Arg_5 && Arg_4 <= Arg_3 && Arg_3 <= Arg_4 && Arg_2 <= Arg_1 && Arg_1 <= Arg_2 && 0 <= Arg_0 && Arg_0 <= 1 && Arg_1 <= Arg_2 && Arg_2 <= Arg_1 && Arg_3 <= Arg_4 && Arg_4 <= Arg_3 && Arg_5 <= Arg_0 && Arg_0 <= Arg_5

  start(Arg_0,Arg_1,Arg_2,Arg_3,Arg_4,Arg_5) -> stop(Arg_0,Arg_1,Arg_2,Arg_5,Arg_4,Arg_5):|:Arg_5 <= Arg_0 && Arg_0 <= Arg_5 && Arg_4 <= Arg_3 && Arg_3 <= Arg_4 && Arg_2 <= Arg_1 && Arg_1 <= Arg_2 && Arg_0+1 <= 0 && Arg_1 <= Arg_2 && Arg_2 <= Arg_1 && Arg_3 <= Arg_4 && Arg_4 <= Arg_3 && Arg_5 <= Arg_0 && Arg_0 <= Arg_5

  start0(Arg_0,Arg_1,Arg_2,Arg_3,Arg_4,Arg_5) -> start(Arg_0,Arg_2,Arg_2,Arg_4,Arg_4,Arg_0):|:  



Timebounds: 

  Overall timebound: 5+max([1, 1+Arg_0])+max([0, (-1+Arg_0)*max([8, -4+4*Arg_0])])+max([0, -1+Arg_0])+max([4, -4+4*Arg_0])+max([4, 2*Arg_0])+max([0, (-1+Arg_0)*max([8, -4+4*Arg_0])])+max([4, -4+4*Arg_0]) {O(n^2)}

  6: lbl101->lbl101: max([4, -4+4*Arg_0])+max([0, (-1+Arg_0)*max([8, -4+4*Arg_0])])+max([4, -4+4*Arg_0])+max([0, (-1+Arg_0)*max([8, -4+4*Arg_0])]) {O(n^2)}

  7: lbl101->lbl121: max([4, 2*Arg_0]) {O(n)}

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

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

  5: lbl121->lbl101: max([0, -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+max([1, 1+Arg_0])+max([0, (-1+Arg_0)*max([8, -4+4*Arg_0])])+max([0, -1+Arg_0])+max([4, -4+4*Arg_0])+max([4, 2*Arg_0])+max([0, (-1+Arg_0)*max([8, -4+4*Arg_0])])+max([4, -4+4*Arg_0]) {O(n^2)}

  6: lbl101->lbl101: max([4, -4+4*Arg_0])+max([0, (-1+Arg_0)*max([8, -4+4*Arg_0])])+max([4, -4+4*Arg_0])+max([0, (-1+Arg_0)*max([8, -4+4*Arg_0])]) {O(n^2)}

  7: lbl101->lbl121: max([4, 2*Arg_0]) {O(n)}

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

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

  5: lbl121->lbl101: max([0, -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^(max([4, -4+4*Arg_0])+max([0, (-1+Arg_0)*max([8, -4+4*Arg_0])])+max([4, -4+4*Arg_0])+max([0, (-1+Arg_0)*max([8, -4+4*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^(max([4, -4+4*Arg_0])+max([0, (-1+Arg_0)*max([8, -4+4*Arg_0])])+max([4, -4+4*Arg_0])+max([0, (-1+Arg_0)*max([8, -4+4*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, nat(2 * Arg_0 * max(-4 + 4 * Arg_0, 8) + min(8 + -8 * Arg_0, -16)) + nat(-1 + Arg_0) + max(-8 + 8 * Arg_0, 8) + max(4, 2 * Arg_0) + max(6, 6 + Arg_0))