/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(1, max(29 + 27 * Arg_1, 29 + 27 * Arg_2, 2 + 27 * Arg_0, 29) + nat(Arg_3) + nat(Arg_3 * max(28 + 27 * Arg_3, 28, 1 + 27 * Arg_3)) + max(2, 2 + 2 * Arg_3)).

(0) CpxIntTrs
(1) Koat2 Proof [FINISHED, 523 ms]
(2) BOUNDS(1, max(29 + 27 * Arg_1, 29 + 27 * Arg_2, 2 + 27 * Arg_0, 29) + nat(Arg_3) + nat(Arg_3 * max(28 + 27 * Arg_3, 28, 1 + 27 * Arg_3)) + max(2, 2 + 2 * Arg_3))


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(0)
Obligation:
Complexity Int TRS consisting of the following rules:
l0(A, B, C, F) -> Com_1(l1(A, B, C, F)) :|: TRUE
l1(A, B, C, F) -> Com_1(l1(A + B, B + C, C - 1, F)) :|: A >= 1
l1(A, B, C, F) -> Com_1(l2(A, B, C, F - 1)) :|: 0 >= A
l2(A, B, C, F) -> Com_1(l1(F, F, F, F)) :|: F >= 1

The start-symbols are:[l0_4]


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(1) Koat2 Proof (FINISHED)
YES( ?, 2+27*max([1, max([Arg_0, max([1+Arg_2, 1+Arg_1])])])+max([0, Arg_3])+max([0, Arg_3*(1+27*max([1, max([Arg_3, 1+Arg_3])]))])+max([2, 2+2*Arg_3]) {O(n^2)})



Initial Complexity Problem:

  Start:  l0  

  Program_Vars:  Arg_0, Arg_1, Arg_2, Arg_3  

  Temp_Vars:    

  Locations:  l0, l1, l2  

  Transitions:

  0: l0->l1

  1: l1->l1

  2: l1->l2

  3: l2->l1  



Timebounds: 

  Overall timebound: 2+27*max([1, max([Arg_0, max([1+Arg_2, 1+Arg_1])])])+max([0, Arg_3])+max([0, Arg_3*(1+27*max([1, max([Arg_3, 1+Arg_3])]))])+max([2, 2+2*Arg_3]) {O(n^2)}

  0: l0->l1: 1 {O(1)}

  1: l1->l1: 1+27*max([1, max([Arg_0, max([1+Arg_2, 1+Arg_1])])])+max([0, Arg_3*(1+27*max([1, max([Arg_3, 1+Arg_3])]))]) {O(n^2)}

  2: l1->l2: max([2, 2+2*Arg_3]) {O(n)}

  3: l2->l1: max([0, Arg_3]) {O(n)}



Costbounds:

  Overall costbound: 2+27*max([1, max([Arg_0, max([1+Arg_2, 1+Arg_1])])])+max([0, Arg_3])+max([0, Arg_3*(1+27*max([1, max([Arg_3, 1+Arg_3])]))])+max([2, 2+2*Arg_3]) {O(n^2)}

  0: l0->l1: 1 {O(1)}

  1: l1->l1: 1+27*max([1, max([Arg_0, max([1+Arg_2, 1+Arg_1])])])+max([0, Arg_3*(1+27*max([1, max([Arg_3, 1+Arg_3])]))]) {O(n^2)}

  2: l1->l2: max([2, 2+2*Arg_3]) {O(n)}

  3: l2->l1: max([0, Arg_3]) {O(n)}



Sizebounds:

`Lower:

  0: l0->l1, Arg_0: Arg_0 {O(n)}

  0: l0->l1, Arg_1: Arg_1 {O(n)}

  0: l0->l1, Arg_2: Arg_2 {O(n)}

  0: l0->l1, Arg_3: Arg_3 {O(n)}

  1: l1->l1, Arg_0: min([2, min([-(-1-Arg_1), -(-1+(1+27*max([1, max([Arg_0, max([1+Arg_2, 1+Arg_1])])])+max([0, Arg_3*(1+27*max([1, max([Arg_3, 1+Arg_3])]))]))*max([0, max([-1, max([-(Arg_2), 1+27*max([1, max([Arg_0, max([1+Arg_2, 1+Arg_1])])])+max([-1, -(Arg_2)])+max([0, Arg_3*(1+27*max([1, max([Arg_3, 1+Arg_3])]))])])])])+max([-1, max([-(Arg_2), max([-(Arg_1), 1+27*max([1, max([Arg_0, max([1+Arg_2, 1+Arg_1])])])+max([-1, -(Arg_2)])+max([0, Arg_3*(1+27*max([1, max([Arg_3, 1+Arg_3])]))])])])]))])]) {O(n^4)}

  1: l1->l1, Arg_1: min([1, min([Arg_2, min([Arg_1, -(1+27*max([1, max([Arg_0, max([1+Arg_2, 1+Arg_1])])])+max([-1, -(Arg_2)])+max([0, Arg_3*(1+27*max([1, max([Arg_3, 1+Arg_3])]))]))])])])+(-1+min([0, -(Arg_3*(1+27*max([1, max([Arg_3, 1+Arg_3])])))])+-27*max([1, max([Arg_0, max([1+Arg_2, 1+Arg_1])])]))*max([0, max([-1, max([-(Arg_2), 1+27*max([1, max([Arg_0, max([1+Arg_2, 1+Arg_1])])])+max([-1, -(Arg_2)])+max([0, Arg_3*(1+27*max([1, max([Arg_3, 1+Arg_3])]))])])])]) {O(n^4)}

  1: l1->l1, Arg_2: -1+min([1, Arg_2])-max([0, Arg_3*(1+27*max([1, max([Arg_3, 1+Arg_3])]))])+-27*max([1, max([Arg_0, max([1+Arg_2, 1+Arg_1])])]) {O(n^2)}

  1: l1->l1, Arg_3: min([1, Arg_3]) {O(n)}

  2: l1->l2, Arg_0: min([2, min([Arg_0, min([-(-1-Arg_1), -(-1+(1+27*max([1, max([Arg_0, max([1+Arg_2, 1+Arg_1])])])+max([0, Arg_3*(1+27*max([1, max([Arg_3, 1+Arg_3])]))]))*max([0, max([-1, max([-(Arg_2), 1+27*max([1, max([Arg_0, max([1+Arg_2, 1+Arg_1])])])+max([-1, -(Arg_2)])+max([0, Arg_3*(1+27*max([1, max([Arg_3, 1+Arg_3])]))])])])])+max([-1, max([-(Arg_2), max([-(Arg_1), 1+27*max([1, max([Arg_0, max([1+Arg_2, 1+Arg_1])])])+max([-1, -(Arg_2)])+max([0, Arg_3*(1+27*max([1, max([Arg_3, 1+Arg_3])]))])])])]))])])]) {O(n^4)}

  2: l1->l2, Arg_1: min([Arg_1, -((1+27*max([1, max([Arg_0, max([1+Arg_2, 1+Arg_1])])])+max([0, Arg_3*(1+27*max([1, max([Arg_3, 1+Arg_3])]))]))*max([0, max([-1, max([-(Arg_2), 1+27*max([1, max([Arg_0, max([1+Arg_2, 1+Arg_1])])])+max([-1, -(Arg_2)])+max([0, Arg_3*(1+27*max([1, max([Arg_3, 1+Arg_3])]))])])])])+max([-1, max([-(Arg_2), max([-(Arg_1), 1+27*max([1, max([Arg_0, max([1+Arg_2, 1+Arg_1])])])+max([-1, -(Arg_2)])+max([0, Arg_3*(1+27*max([1, max([Arg_3, 1+Arg_3])]))])])])]))]) {O(n^4)}

  2: l1->l2, Arg_2: min([Arg_2, -(1+27*max([1, max([Arg_0, max([1+Arg_2, 1+Arg_1])])])+max([-1, -(Arg_2)])+max([0, Arg_3*(1+27*max([1, max([Arg_3, 1+Arg_3])]))]))]) {O(n^2)}

  2: l1->l2, Arg_3: min([0, -(1-Arg_3)]) {O(n)}

  3: l2->l1, Arg_0: 1 {O(1)}

  3: l2->l1, Arg_1: 1 {O(1)}

  3: l2->l1, Arg_2: 1 {O(1)}

  3: l2->l1, Arg_3: 1 {O(1)}

`Upper:

  0: l0->l1, Arg_0: Arg_0 {O(n)}

  0: l0->l1, Arg_1: Arg_1 {O(n)}

  0: l0->l1, Arg_2: Arg_2 {O(n)}

  0: l0->l1, Arg_3: Arg_3 {O(n)}

  1: l1->l1, Arg_0: (1+27*max([1, max([Arg_0, max([1+Arg_2, 1+Arg_1])])])+max([0, Arg_3*(1+27*max([1, max([Arg_3, 1+Arg_3])]))]))*max([0, max([Arg_1, max([Arg_3, (1+27*max([1, max([Arg_0, max([1+Arg_2, 1+Arg_1])])])+max([0, Arg_3*(1+27*max([1, max([Arg_3, 1+Arg_3])]))]))*max([0, max([Arg_2, Arg_3])])+max([Arg_3, max([Arg_1, max([Arg_2, max([Arg_2, Arg_3])])])])])])])+max([Arg_0, max([Arg_3, max([Arg_1, max([Arg_3, (1+27*max([1, max([Arg_0, max([1+Arg_2, 1+Arg_1])])])+max([0, Arg_3*(1+27*max([1, max([Arg_3, 1+Arg_3])]))]))*max([0, max([Arg_2, Arg_3])])+max([Arg_3, max([Arg_1, max([Arg_2, max([Arg_2, Arg_3])])])])])])])]) {O(n^5)}

  1: l1->l1, Arg_1: (1+27*max([1, max([Arg_0, max([1+Arg_2, 1+Arg_1])])])+max([0, Arg_3*(1+27*max([1, max([Arg_3, 1+Arg_3])]))]))*max([0, max([Arg_2, Arg_3])])+max([Arg_3, max([Arg_1, max([Arg_2, max([Arg_2, Arg_3])])])]) {O(n^3)}

  1: l1->l1, Arg_2: max([Arg_2, Arg_3]) {O(n)}

  1: l1->l1, Arg_3: Arg_3 {O(n)}

  2: l1->l2, Arg_0: 0 {O(1)}

  2: l1->l2, Arg_1: max([Arg_1, (1+27*max([1, max([Arg_0, max([1+Arg_2, 1+Arg_1])])])+max([0, Arg_3*(1+27*max([1, max([Arg_3, 1+Arg_3])]))]))*max([0, max([Arg_2, Arg_3])])+max([Arg_3, max([Arg_1, max([Arg_2, max([Arg_2, Arg_3])])])])]) {O(n^3)}

  2: l1->l2, Arg_2: max([Arg_2, max([Arg_2, Arg_3])]) {O(n)}

  2: l1->l2, Arg_3: Arg_3 {O(n)}

  3: l2->l1, Arg_0: Arg_3 {O(n)}

  3: l2->l1, Arg_1: Arg_3 {O(n)}

  3: l2->l1, Arg_2: Arg_3 {O(n)}

  3: l2->l1, Arg_3: Arg_3 {O(n)}


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(2)
BOUNDS(1, max(29 + 27 * Arg_1, 29 + 27 * Arg_2, 2 + 27 * Arg_0, 29) + nat(Arg_3) + nat(Arg_3 * max(28 + 27 * Arg_3, 28, 1 + 27 * Arg_3)) + max(2, 2 + 2 * Arg_3))