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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(1, max(4 + 8 * Arg_3, 12 + 8 * Arg_4, 12) + max(27 + 27 * Arg_2, 27 + 27 * Arg_1, 27, 27 * Arg_0)).

(0) CpxIntTrs
(1) Koat2 Proof [FINISHED, 597 ms]
(2) BOUNDS(1, max(4 + 8 * Arg_3, 12 + 8 * Arg_4, 12) + max(27 + 27 * Arg_2, 27 + 27 * Arg_1, 27, 27 * Arg_0))


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

The start-symbols are:[l0_5]


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(1) Koat2 Proof (FINISHED)
YES( ?, 3+1+8*max([1, max([Arg_3, 1+Arg_4])])+27*max([1, max([Arg_0, max([1+Arg_2, 1+Arg_1])])]) {O(n)})



Initial Complexity Problem:

  Start:  l0  

  Program_Vars:  Arg_0, Arg_1, Arg_2, Arg_3, Arg_4  

  Temp_Vars:    

  Locations:  l0, l1, l2  

  Transitions:

  0: l0->l1

  1: l1->l1

  2: l1->l2

  3: l2->l2  



Timebounds: 

  Overall timebound: 3+1+8*max([1, max([Arg_3, 1+Arg_4])])+27*max([1, max([Arg_0, max([1+Arg_2, 1+Arg_1])])]) {O(n)}

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

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

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

  3: l2->l2: 1+8*max([1, max([Arg_3, 1+Arg_4])]) {O(n)}



Costbounds:

  Overall costbound: 3+1+8*max([1, max([Arg_3, 1+Arg_4])])+27*max([1, max([Arg_0, max([1+Arg_2, 1+Arg_1])])]) {O(n)}

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

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

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

  3: l2->l2: 1+8*max([1, max([Arg_3, 1+Arg_4])]) {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)}

  0: l0->l1, Arg_4: Arg_4 {O(n)}

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

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

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

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

  1: l1->l1, Arg_4: Arg_4 {O(n)}

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

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

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

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

  2: l1->l2, Arg_4: Arg_4 {O(n)}

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

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

  3: l2->l2, Arg_2: min([Arg_2, -(1+-(Arg_2)+27*max([1, max([Arg_0, max([1+Arg_2, 1+Arg_1])])]))]) {O(n)}

  3: l2->l2, Arg_3: min([-(-(Arg_4)+8*max([1, max([Arg_3, 1+Arg_4])])), -(-1-Arg_4)]) {O(n)}

  3: l2->l2, Arg_4: -1+Arg_4+-8*max([1, max([Arg_3, 1+Arg_4])]) {O(n)}

`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)}

  0: l0->l1, Arg_4: Arg_4 {O(n)}

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

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

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

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

  1: l1->l1, Arg_4: Arg_4 {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_2])+max([Arg_2, max([Arg_1, Arg_2])])]) {O(n^2)}

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

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

  2: l1->l2, Arg_4: Arg_4 {O(n)}

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

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

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

  3: l2->l2, Arg_3: (1+8*max([1, max([Arg_3, 1+Arg_4])]))*max([0, Arg_4])+max([Arg_4, max([Arg_3, Arg_4])]) {O(n^2)}

  3: l2->l2, Arg_4: Arg_4 {O(n)}


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(2)
BOUNDS(1, max(4 + 8 * Arg_3, 12 + 8 * Arg_4, 12) + max(27 + 27 * Arg_2, 27 + 27 * Arg_1, 27, 27 * Arg_0))