/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(Omega(n^1), 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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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(n^1, max(4, 4 + 2 * Arg_6) + nat(Arg_6) + nat(2 * Arg_6) + max(4 + 54 * Arg_1, 58 + 54 * Arg_3, 58, 58 + 54 * Arg_5) + nat(2 * Arg_6 * max(28 + 27 * Arg_3, 28 + 27 * Arg_5, 1 + 27 * Arg_1, 28))).

(0) CpxIntTrs
(1) Koat2 Proof [FINISHED, 968 ms]
(2) BOUNDS(1, max(4, 4 + 2 * Arg_6) + nat(Arg_6) + nat(2 * Arg_6) + max(4 + 54 * Arg_1, 58 + 54 * Arg_3, 58, 58 + 54 * Arg_5) + nat(2 * Arg_6 * max(28 + 27 * Arg_3, 28 + 27 * Arg_5, 1 + 27 * Arg_1, 28)))
(3) Loat Proof [FINISHED, 588 ms]
(4) BOUNDS(n^1, INF)


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

(0)
Obligation:
Complexity Int TRS consisting of the following rules:
l0(A, B, C, F, X, Y, Z) -> Com_1(l1(X, Y, Z, F, X, Y, Z)) :|: TRUE
l1(A, B, C, F, X, Y, Z) -> Com_1(l1(A + B, B + C, C - 1, F, X, Y, Z)) :|: A >= 1
l1(A, B, C, F, X, Y, Z) -> Com_1(l2(A, B, C, F - 1, X, Y, Z)) :|: 0 >= A
l2(A, B, C, F, X, Y, Z) -> Com_1(l1(X, Y, Z, F, X, Y, Z)) :|: F >= 1
l2(A, B, C, F, X, Y, Z) -> Com_1(l3(A, B, C, F, X, Y, Z)) :|: 0 >= F

The start-symbols are:[l0_7]


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

(1) Koat2 Proof (FINISHED)
YES( ?, 3+1+27*max([1, max([Arg_1, max([1+Arg_5, 1+Arg_3])])])+max([0, Arg_6*(1+27*max([1, max([Arg_1, max([1+Arg_5, 1+Arg_3])])]))])+max([4, 4+2*Arg_6])+max([0, Arg_6*(1+27*max([1, max([Arg_1, max([1+Arg_5, 1+Arg_3])])]))])+max([0, Arg_6])+max([0, 2*Arg_6])+27*max([1, max([Arg_1, max([1+Arg_5, 1+Arg_3])])]) {O(n^2)})



Initial Complexity Problem:

  Start:  l0  

  Program_Vars:  Arg_0, Arg_1, Arg_2, Arg_3, Arg_4, Arg_5, Arg_6  

  Temp_Vars:    

  Locations:  l0, l1, l2, l3  

  Transitions:

  0: l0->l1

  1: l1->l1

  2: l1->l2

  3: l2->l1

  4: l2->l3  



Timebounds: 

  Overall timebound: 3+1+27*max([1, max([Arg_1, max([1+Arg_5, 1+Arg_3])])])+max([0, Arg_6*(1+27*max([1, max([Arg_1, max([1+Arg_5, 1+Arg_3])])]))])+max([4, 4+2*Arg_6])+max([0, Arg_6*(1+27*max([1, max([Arg_1, max([1+Arg_5, 1+Arg_3])])]))])+max([0, Arg_6])+max([0, 2*Arg_6])+27*max([1, max([Arg_1, max([1+Arg_5, 1+Arg_3])])]) {O(n^2)}

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

  1: l1->l1: 1+1+27*max([1, max([Arg_1, max([1+Arg_5, 1+Arg_3])])])+max([0, Arg_6*(1+27*max([1, max([Arg_1, max([1+Arg_5, 1+Arg_3])])]))])+27*max([1, max([Arg_1, max([1+Arg_5, 1+Arg_3])])])+max([0, Arg_6*(1+27*max([1, max([Arg_1, max([1+Arg_5, 1+Arg_3])])]))]) {O(n^2)}

  2: l1->l2: max([4, 4+2*Arg_6])+max([0, 2*Arg_6]) {O(n)}

  3: l2->l1: max([0, Arg_6]) {O(n)}

  4: l2->l3: 1 {O(1)}



Costbounds:

  Overall costbound: 3+1+27*max([1, max([Arg_1, max([1+Arg_5, 1+Arg_3])])])+max([0, Arg_6*(1+27*max([1, max([Arg_1, max([1+Arg_5, 1+Arg_3])])]))])+max([4, 4+2*Arg_6])+max([0, Arg_6*(1+27*max([1, max([Arg_1, max([1+Arg_5, 1+Arg_3])])]))])+max([0, Arg_6])+max([0, 2*Arg_6])+27*max([1, max([Arg_1, max([1+Arg_5, 1+Arg_3])])]) {O(n^2)}

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

  1: l1->l1: 1+1+27*max([1, max([Arg_1, max([1+Arg_5, 1+Arg_3])])])+max([0, Arg_6*(1+27*max([1, max([Arg_1, max([1+Arg_5, 1+Arg_3])])]))])+27*max([1, max([Arg_1, max([1+Arg_5, 1+Arg_3])])])+max([0, Arg_6*(1+27*max([1, max([Arg_1, max([1+Arg_5, 1+Arg_3])])]))]) {O(n^2)}

  2: l1->l2: max([4, 4+2*Arg_6])+max([0, 2*Arg_6]) {O(n)}

  3: l2->l1: max([0, Arg_6]) {O(n)}

  4: l2->l3: 1 {O(1)}



Sizebounds:

`Lower:

  0: l0->l1, Arg_0: Arg_1 {O(n)}

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

  0: l0->l1, Arg_2: Arg_3 {O(n)}

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

  0: l0->l1, Arg_4: Arg_5 {O(n)}

  0: l0->l1, Arg_5: Arg_5 {O(n)}

  0: l0->l1, Arg_6: Arg_6 {O(n)}

  1: l1->l1, Arg_0: min([-(-1-Arg_3), min([-(-1-Arg_3), -(-1+(1+2*27*max([1, max([Arg_1, max([1+Arg_5, 1+Arg_3])])])+max([0, Arg_6*(1+27*max([1, max([Arg_1, max([1+Arg_5, 1+Arg_3])])]))])+max([1, 1+Arg_6*(1+27*max([1, max([Arg_1, max([1+Arg_5, 1+Arg_3])])]))]))*max([0, max([-(Arg_5), 1+-(Arg_5)+2*27*max([1, max([Arg_1, max([1+Arg_5, 1+Arg_3])])])+max([0, Arg_6*(1+27*max([1, max([Arg_1, max([1+Arg_5, 1+Arg_3])])]))])+max([1, 1+Arg_6*(1+27*max([1, max([Arg_1, max([1+Arg_5, 1+Arg_3])])]))])])])+max([-(Arg_5), max([-(Arg_3), max([-(Arg_5), 1+-(Arg_5)+2*27*max([1, max([Arg_1, max([1+Arg_5, 1+Arg_3])])])+max([0, Arg_6*(1+27*max([1, max([Arg_1, max([1+Arg_5, 1+Arg_3])])]))])+max([1, 1+Arg_6*(1+27*max([1, max([Arg_1, max([1+Arg_5, 1+Arg_3])])]))])])])]))])]) {O(n^4)}

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

  1: l1->l1, Arg_2: min([Arg_5, min([Arg_3, min([Arg_5, -(1+-(Arg_5)+2*27*max([1, max([Arg_1, max([1+Arg_5, 1+Arg_3])])])+max([0, Arg_6*(1+27*max([1, max([Arg_1, max([1+Arg_5, 1+Arg_3])])]))])+max([1, 1+Arg_6*(1+27*max([1, max([Arg_1, max([1+Arg_5, 1+Arg_3])])]))]))])])])+(-1+min([0, -(Arg_6*(1+27*max([1, max([Arg_1, max([1+Arg_5, 1+Arg_3])])])))])+-1-max([0, Arg_6*(1+27*max([1, max([Arg_1, max([1+Arg_5, 1+Arg_3])])]))])+-27*max([1, max([Arg_1, max([1+Arg_5, 1+Arg_3])])])+-27*max([1, max([Arg_1, max([1+Arg_5, 1+Arg_3])])]))*max([0, max([-(Arg_5), 1+-(Arg_5)+2*27*max([1, max([Arg_1, max([1+Arg_5, 1+Arg_3])])])+max([0, Arg_6*(1+27*max([1, max([Arg_1, max([1+Arg_5, 1+Arg_3])])]))])+max([1, 1+Arg_6*(1+27*max([1, max([Arg_1, max([1+Arg_5, 1+Arg_3])])]))])])]) {O(n^4)}

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

  1: l1->l1, Arg_4: -1+Arg_5+min([0, -(Arg_6*(1+27*max([1, max([Arg_1, max([1+Arg_5, 1+Arg_3])])])))])+-1-max([0, Arg_6*(1+27*max([1, max([Arg_1, max([1+Arg_5, 1+Arg_3])])]))])+-27*max([1, max([Arg_1, max([1+Arg_5, 1+Arg_3])])])+-27*max([1, max([Arg_1, max([1+Arg_5, 1+Arg_3])])]) {O(n^2)}

  1: l1->l1, Arg_5: Arg_5 {O(n)}

  1: l1->l1, Arg_6: min([1, Arg_6]) {O(n)}

  2: l1->l2, Arg_0: min([Arg_1, min([-(-1-Arg_3), min([-(-1-Arg_3), -(-1+(1+2*27*max([1, max([Arg_1, max([1+Arg_5, 1+Arg_3])])])+max([0, Arg_6*(1+27*max([1, max([Arg_1, max([1+Arg_5, 1+Arg_3])])]))])+max([1, 1+Arg_6*(1+27*max([1, max([Arg_1, max([1+Arg_5, 1+Arg_3])])]))]))*max([0, max([-(Arg_5), 1+-(Arg_5)+2*27*max([1, max([Arg_1, max([1+Arg_5, 1+Arg_3])])])+max([0, Arg_6*(1+27*max([1, max([Arg_1, max([1+Arg_5, 1+Arg_3])])]))])+max([1, 1+Arg_6*(1+27*max([1, max([Arg_1, max([1+Arg_5, 1+Arg_3])])]))])])])+max([-(Arg_5), max([-(Arg_3), max([-(Arg_5), 1+-(Arg_5)+2*27*max([1, max([Arg_1, max([1+Arg_5, 1+Arg_3])])])+max([0, Arg_6*(1+27*max([1, max([Arg_1, max([1+Arg_5, 1+Arg_3])])]))])+max([1, 1+Arg_6*(1+27*max([1, max([Arg_1, max([1+Arg_5, 1+Arg_3])])]))])])])]))])])]) {O(n^4)}

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

  2: l1->l2, Arg_2: min([Arg_3, -((1+2*27*max([1, max([Arg_1, max([1+Arg_5, 1+Arg_3])])])+max([0, Arg_6*(1+27*max([1, max([Arg_1, max([1+Arg_5, 1+Arg_3])])]))])+max([1, 1+Arg_6*(1+27*max([1, max([Arg_1, max([1+Arg_5, 1+Arg_3])])]))]))*max([0, max([-(Arg_5), 1+-(Arg_5)+2*27*max([1, max([Arg_1, max([1+Arg_5, 1+Arg_3])])])+max([0, Arg_6*(1+27*max([1, max([Arg_1, max([1+Arg_5, 1+Arg_3])])]))])+max([1, 1+Arg_6*(1+27*max([1, max([Arg_1, max([1+Arg_5, 1+Arg_3])])]))])])])+max([-(Arg_5), max([-(Arg_3), max([-(Arg_5), 1+-(Arg_5)+2*27*max([1, max([Arg_1, max([1+Arg_5, 1+Arg_3])])])+max([0, Arg_6*(1+27*max([1, max([Arg_1, max([1+Arg_5, 1+Arg_3])])]))])+max([1, 1+Arg_6*(1+27*max([1, max([Arg_1, max([1+Arg_5, 1+Arg_3])])]))])])])]))]) {O(n^4)}

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

  2: l1->l2, Arg_4: min([Arg_5, -(1+-(Arg_5)+2*27*max([1, max([Arg_1, max([1+Arg_5, 1+Arg_3])])])+max([0, Arg_6*(1+27*max([1, max([Arg_1, max([1+Arg_5, 1+Arg_3])])]))])+max([1, 1+Arg_6*(1+27*max([1, max([Arg_1, max([1+Arg_5, 1+Arg_3])])]))]))]) {O(n^2)}

  2: l1->l2, Arg_5: Arg_5 {O(n)}

  2: l1->l2, Arg_6: min([0, -(1-Arg_6)]) {O(n)}

  3: l2->l1, Arg_0: Arg_1 {O(n)}

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

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

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

  3: l2->l1, Arg_4: Arg_5 {O(n)}

  3: l2->l1, Arg_5: Arg_5 {O(n)}

  3: l2->l1, Arg_6: 1 {O(1)}

  4: l2->l3, Arg_0: min([Arg_1, min([-(-1-Arg_3), min([-(-1-Arg_3), -(-1+(1+2*27*max([1, max([Arg_1, max([1+Arg_5, 1+Arg_3])])])+max([0, Arg_6*(1+27*max([1, max([Arg_1, max([1+Arg_5, 1+Arg_3])])]))])+max([1, 1+Arg_6*(1+27*max([1, max([Arg_1, max([1+Arg_5, 1+Arg_3])])]))]))*max([0, max([-(Arg_5), 1+-(Arg_5)+2*27*max([1, max([Arg_1, max([1+Arg_5, 1+Arg_3])])])+max([0, Arg_6*(1+27*max([1, max([Arg_1, max([1+Arg_5, 1+Arg_3])])]))])+max([1, 1+Arg_6*(1+27*max([1, max([Arg_1, max([1+Arg_5, 1+Arg_3])])]))])])])+max([-(Arg_5), max([-(Arg_3), max([-(Arg_5), 1+-(Arg_5)+2*27*max([1, max([Arg_1, max([1+Arg_5, 1+Arg_3])])])+max([0, Arg_6*(1+27*max([1, max([Arg_1, max([1+Arg_5, 1+Arg_3])])]))])+max([1, 1+Arg_6*(1+27*max([1, max([Arg_1, max([1+Arg_5, 1+Arg_3])])]))])])])]))])])]) {O(n^4)}

  4: l2->l3, Arg_1: Arg_1 {O(n)}

  4: l2->l3, Arg_2: min([Arg_3, -((1+2*27*max([1, max([Arg_1, max([1+Arg_5, 1+Arg_3])])])+max([0, Arg_6*(1+27*max([1, max([Arg_1, max([1+Arg_5, 1+Arg_3])])]))])+max([1, 1+Arg_6*(1+27*max([1, max([Arg_1, max([1+Arg_5, 1+Arg_3])])]))]))*max([0, max([-(Arg_5), 1+-(Arg_5)+2*27*max([1, max([Arg_1, max([1+Arg_5, 1+Arg_3])])])+max([0, Arg_6*(1+27*max([1, max([Arg_1, max([1+Arg_5, 1+Arg_3])])]))])+max([1, 1+Arg_6*(1+27*max([1, max([Arg_1, max([1+Arg_5, 1+Arg_3])])]))])])])+max([-(Arg_5), max([-(Arg_3), max([-(Arg_5), 1+-(Arg_5)+2*27*max([1, max([Arg_1, max([1+Arg_5, 1+Arg_3])])])+max([0, Arg_6*(1+27*max([1, max([Arg_1, max([1+Arg_5, 1+Arg_3])])]))])+max([1, 1+Arg_6*(1+27*max([1, max([Arg_1, max([1+Arg_5, 1+Arg_3])])]))])])])]))]) {O(n^4)}

  4: l2->l3, Arg_3: Arg_3 {O(n)}

  4: l2->l3, Arg_4: min([Arg_5, -(1+-(Arg_5)+2*27*max([1, max([Arg_1, max([1+Arg_5, 1+Arg_3])])])+max([0, Arg_6*(1+27*max([1, max([Arg_1, max([1+Arg_5, 1+Arg_3])])]))])+max([1, 1+Arg_6*(1+27*max([1, max([Arg_1, max([1+Arg_5, 1+Arg_3])])]))]))]) {O(n^2)}

  4: l2->l3, Arg_5: Arg_5 {O(n)}

  4: l2->l3, Arg_6: min([0, -(1-Arg_6)]) {O(n)}

`Upper:

  0: l0->l1, Arg_0: Arg_1 {O(n)}

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

  0: l0->l1, Arg_2: Arg_3 {O(n)}

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

  0: l0->l1, Arg_4: Arg_5 {O(n)}

  0: l0->l1, Arg_5: Arg_5 {O(n)}

  0: l0->l1, Arg_6: Arg_6 {O(n)}

  1: l1->l1, Arg_0: (1+1+27*max([1, max([Arg_1, max([1+Arg_5, 1+Arg_3])])])+max([0, Arg_6*(1+27*max([1, max([Arg_1, max([1+Arg_5, 1+Arg_3])])]))])+27*max([1, max([Arg_1, max([1+Arg_5, 1+Arg_3])])])+max([0, Arg_6*(1+27*max([1, max([Arg_1, max([1+Arg_5, 1+Arg_3])])]))]))*max([0, max([Arg_3, (1+1+27*max([1, max([Arg_1, max([1+Arg_5, 1+Arg_3])])])+max([0, Arg_6*(1+27*max([1, max([Arg_1, max([1+Arg_5, 1+Arg_3])])]))])+27*max([1, max([Arg_1, max([1+Arg_5, 1+Arg_3])])])+max([0, Arg_6*(1+27*max([1, max([Arg_1, max([1+Arg_5, 1+Arg_3])])]))]))*max([0, Arg_5])+max([Arg_5, max([Arg_5, max([Arg_3, max([Arg_3, Arg_5])])])])])])+max([Arg_3, max([Arg_1, max([Arg_3, (1+1+27*max([1, max([Arg_1, max([1+Arg_5, 1+Arg_3])])])+max([0, Arg_6*(1+27*max([1, max([Arg_1, max([1+Arg_5, 1+Arg_3])])]))])+27*max([1, max([Arg_1, max([1+Arg_5, 1+Arg_3])])])+max([0, Arg_6*(1+27*max([1, max([Arg_1, max([1+Arg_5, 1+Arg_3])])]))]))*max([0, Arg_5])+max([Arg_5, max([Arg_5, max([Arg_3, max([Arg_3, Arg_5])])])])])])]) {O(n^5)}

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

  1: l1->l1, Arg_2: (1+1+27*max([1, max([Arg_1, max([1+Arg_5, 1+Arg_3])])])+max([0, Arg_6*(1+27*max([1, max([Arg_1, max([1+Arg_5, 1+Arg_3])])]))])+27*max([1, max([Arg_1, max([1+Arg_5, 1+Arg_3])])])+max([0, Arg_6*(1+27*max([1, max([Arg_1, max([1+Arg_5, 1+Arg_3])])]))]))*max([0, Arg_5])+max([Arg_5, max([Arg_5, max([Arg_3, max([Arg_3, Arg_5])])])]) {O(n^3)}

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

  1: l1->l1, Arg_4: Arg_5 {O(n)}

  1: l1->l1, Arg_5: Arg_5 {O(n)}

  1: l1->l1, Arg_6: Arg_6 {O(n)}

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

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

  2: l1->l2, Arg_2: max([Arg_3, (1+1+27*max([1, max([Arg_1, max([1+Arg_5, 1+Arg_3])])])+max([0, Arg_6*(1+27*max([1, max([Arg_1, max([1+Arg_5, 1+Arg_3])])]))])+27*max([1, max([Arg_1, max([1+Arg_5, 1+Arg_3])])])+max([0, Arg_6*(1+27*max([1, max([Arg_1, max([1+Arg_5, 1+Arg_3])])]))]))*max([0, Arg_5])+max([Arg_5, max([Arg_5, max([Arg_3, max([Arg_3, Arg_5])])])])]) {O(n^3)}

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

  2: l1->l2, Arg_4: Arg_5 {O(n)}

  2: l1->l2, Arg_5: Arg_5 {O(n)}

  2: l1->l2, Arg_6: Arg_6 {O(n)}

  3: l2->l1, Arg_0: Arg_1 {O(n)}

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

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

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

  3: l2->l1, Arg_4: Arg_5 {O(n)}

  3: l2->l1, Arg_5: Arg_5 {O(n)}

  3: l2->l1, Arg_6: Arg_6 {O(n)}

  4: l2->l3, Arg_0: 0 {O(1)}

  4: l2->l3, Arg_1: Arg_1 {O(n)}

  4: l2->l3, Arg_2: max([Arg_3, (1+1+27*max([1, max([Arg_1, max([1+Arg_5, 1+Arg_3])])])+max([0, Arg_6*(1+27*max([1, max([Arg_1, max([1+Arg_5, 1+Arg_3])])]))])+27*max([1, max([Arg_1, max([1+Arg_5, 1+Arg_3])])])+max([0, Arg_6*(1+27*max([1, max([Arg_1, max([1+Arg_5, 1+Arg_3])])]))]))*max([0, Arg_5])+max([Arg_5, max([Arg_5, max([Arg_3, max([Arg_3, Arg_5])])])])]) {O(n^3)}

  4: l2->l3, Arg_3: Arg_3 {O(n)}

  4: l2->l3, Arg_4: Arg_5 {O(n)}

  4: l2->l3, Arg_5: Arg_5 {O(n)}

  4: l2->l3, Arg_6: 0 {O(1)}


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

(2)
BOUNDS(1, max(4, 4 + 2 * Arg_6) + nat(Arg_6) + nat(2 * Arg_6) + max(4 + 54 * Arg_1, 58 + 54 * Arg_3, 58, 58 + 54 * Arg_5) + nat(2 * Arg_6 * max(28 + 27 * Arg_3, 28 + 27 * Arg_5, 1 + 27 * Arg_1, 28)))

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

(3) Loat Proof (FINISHED)


### Pre-processing the ITS problem ###



Initial linear ITS problem

   Start location: l0

      0: l0 -> l1 : A'=B, C'=D, E'=F, [], cost: 1

      1: l1 -> l1 : A'=C+A, C'=C+E, E'=-1+E, [ A>=1 ], cost: 1

      2: l1 -> l2 : G'=-1+G, [ 0>=A ], cost: 1

      3: l2 -> l1 : A'=B, C'=D, E'=F, [ G>=1 ], cost: 1

      4: l2 -> l3 : [ 0>=G ], cost: 1



Checking for constant complexity:

   The following rule is satisfiable with cost >= 1, yielding constant complexity:

      0: l0 -> l1 : A'=B, C'=D, E'=F, [], cost: 1



Removed unreachable and leaf rules:

   Start location: l0

      0: l0 -> l1 : A'=B, C'=D, E'=F, [], cost: 1

      1: l1 -> l1 : A'=C+A, C'=C+E, E'=-1+E, [ A>=1 ], cost: 1

      2: l1 -> l2 : G'=-1+G, [ 0>=A ], cost: 1

      3: l2 -> l1 : A'=B, C'=D, E'=F, [ G>=1 ], cost: 1



### Simplification by acceleration and chaining ###



Accelerating simple loops of location 1.

   Accelerating the following rules:

      1: l1 -> l1 : A'=C+A, C'=C+E, E'=-1+E, [ A>=1 ], cost: 1



   Found no metering function for rule 1.

   Removing the simple loops:.



Accelerated all simple loops using metering functions (where possible):

   Start location: l0

      0: l0 -> l1 : A'=B, C'=D, E'=F, [], cost: 1

      1: l1 -> l1 : A'=C+A, C'=C+E, E'=-1+E, [ A>=1 ], cost: 1

      2: l1 -> l2 : G'=-1+G, [ 0>=A ], cost: 1

      3: l2 -> l1 : A'=B, C'=D, E'=F, [ G>=1 ], cost: 1



Chained accelerated rules (with incoming rules):

   Start location: l0

      0: l0 -> l1 : A'=B, C'=D, E'=F, [], cost: 1

      5: l0 -> l1 : A'=D+B, C'=F+D, E'=-1+F, [ B>=1 ], cost: 2

      2: l1 -> l2 : G'=-1+G, [ 0>=A ], cost: 1

      3: l2 -> l1 : A'=B, C'=D, E'=F, [ G>=1 ], cost: 1

      6: l2 -> l1 : A'=D+B, C'=F+D, E'=-1+F, [ G>=1 && B>=1 ], cost: 2



Eliminated locations (on tree-shaped paths):

   Start location: l0

      0: l0 -> l1 : A'=B, C'=D, E'=F, [], cost: 1

      5: l0 -> l1 : A'=D+B, C'=F+D, E'=-1+F, [ B>=1 ], cost: 2

      7: l1 -> l1 : A'=B, C'=D, E'=F, G'=-1+G, [ 0>=A && -1+G>=1 ], cost: 2

      8: l1 -> l1 : A'=D+B, C'=F+D, E'=-1+F, G'=-1+G, [ 0>=A && -1+G>=1 && B>=1 ], cost: 3



Accelerating simple loops of location 1.

   Accelerating the following rules:

      7: l1 -> l1 : A'=B, C'=D, E'=F, G'=-1+G, [ 0>=A && -1+G>=1 ], cost: 2

      8: l1 -> l1 : A'=D+B, C'=F+D, E'=-1+F, G'=-1+G, [ 0>=A && -1+G>=1 && B>=1 ], cost: 3



   Accelerated rule 7 with metering function -1+G (after strengthening guard), yielding the new rule 9.

   Accelerated rule 8 with metering function -1+G (after strengthening guard), yielding the new rule 10.

   Removing the simple loops:.



Accelerated all simple loops using metering functions (where possible):

   Start location: l0

      0: l0 -> l1 : A'=B, C'=D, E'=F, [], cost: 1

      5: l0 -> l1 : A'=D+B, C'=F+D, E'=-1+F, [ B>=1 ], cost: 2

      7: l1 -> l1 : A'=B, C'=D, E'=F, G'=-1+G, [ 0>=A && -1+G>=1 ], cost: 2

      8: l1 -> l1 : A'=D+B, C'=F+D, E'=-1+F, G'=-1+G, [ 0>=A && -1+G>=1 && B>=1 ], cost: 3

      9: l1 -> l1 : A'=B, C'=D, E'=F, G'=1, [ 0>=A && -1+G>=1 && 0>=B ], cost: -2+2*G

     10: l1 -> l1 : A'=D+B, C'=F+D, E'=-1+F, G'=1, [ 0>=A && -1+G>=1 && B>=1 && 0>=D+B ], cost: -3+3*G



Chained accelerated rules (with incoming rules):

   Start location: l0

      0: l0 -> l1 : A'=B, C'=D, E'=F, [], cost: 1

      5: l0 -> l1 : A'=D+B, C'=F+D, E'=-1+F, [ B>=1 ], cost: 2

     11: l0 -> l1 : A'=B, C'=D, E'=F, G'=-1+G, [ 0>=B && -1+G>=1 ], cost: 3

     12: l0 -> l1 : A'=B, C'=D, E'=F, G'=-1+G, [ B>=1 && 0>=D+B && -1+G>=1 ], cost: 4

     13: l0 -> l1 : A'=D+B, C'=F+D, E'=-1+F, G'=-1+G, [ B>=1 && 0>=D+B && -1+G>=1 ], cost: 5

     14: l0 -> l1 : A'=B, C'=D, E'=F, G'=1, [ 0>=B && -1+G>=1 ], cost: -1+2*G

     15: l0 -> l1 : A'=D+B, C'=F+D, E'=-1+F, G'=1, [ B>=1 && 0>=D+B && -1+G>=1 ], cost: -1+3*G



Removed unreachable locations (and leaf rules with constant cost):

   Start location: l0

     14: l0 -> l1 : A'=B, C'=D, E'=F, G'=1, [ 0>=B && -1+G>=1 ], cost: -1+2*G

     15: l0 -> l1 : A'=D+B, C'=F+D, E'=-1+F, G'=1, [ B>=1 && 0>=D+B && -1+G>=1 ], cost: -1+3*G



### Computing asymptotic complexity ###



Fully simplified ITS problem

   Start location: l0

     14: l0 -> l1 : A'=B, C'=D, E'=F, G'=1, [ 0>=B && -1+G>=1 ], cost: -1+2*G

     15: l0 -> l1 : A'=D+B, C'=F+D, E'=-1+F, G'=1, [ B>=1 && 0>=D+B && -1+G>=1 ], cost: -1+3*G



Computing asymptotic complexity for rule 14

   Solved the limit problem by the following transformations:

      Created initial limit problem:

      -1+2*G (+), 1-B (+/+!), -1+G (+/+!) [not solved]



      removing all constraints (solved by SMT)

      resulting limit problem: [solved]



      applying transformation rule (C) using substitution {G==n,B==-n}

      resulting limit problem:

      [solved]



   Solution:

      G / n

      B / -n

   Resulting cost -1+2*n has complexity: Poly(n^1)



Found new complexity Poly(n^1).



Obtained the following overall complexity (w.r.t. the length of the input n):

   Complexity:  Poly(n^1)

   Cpx degree:  1

   Solved cost: -1+2*n

   Rule cost:   -1+2*G

   Rule guard:  [ 0>=B && -1+G>=1 ]



WORST_CASE(Omega(n^1),?)


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(4)
BOUNDS(n^1, INF)