/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(Omega(n^1), 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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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(n^1, nat(1 + Arg_0) + max(3, -1 + 2 * Arg_0) + max(7, 1 + 3 * Arg_0) + max(3, 3 + Arg_0 * max(7, 1 + 3 * Arg_0)) + max(4, 2 * Arg_0) + nat(Arg_0)).

(0) CpxIntTrs
(1) Koat2 Proof [FINISHED, 2933 ms]
(2) BOUNDS(1, nat(1 + Arg_0) + max(3, -1 + 2 * Arg_0) + max(7, 1 + 3 * Arg_0) + max(3, 3 + Arg_0 * max(7, 1 + 3 * Arg_0)) + max(4, 2 * Arg_0) + nat(Arg_0))
(3) Loat Proof [FINISHED, 1000 ms]
(4) BOUNDS(n^1, INF)


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

(0)
Obligation:
Complexity Int TRS consisting of the following rules:
start(A, B, C, D, E, F, G, H, I, J) -> Com_1(stop(A, B, C, D, E, F, G, H, 1, J)) :|: 1 >= A && B >= C && B <= C && D >= E && D <= E && F >= A && F <= A && G >= H && G <= H && I >= J && I <= J
start(A, B, C, D, E, F, G, H, I, J) -> Com_1(lbl31(A, K, C, D, E, F, G, H, 1, J)) :|: A >= 2 && B >= C && B <= C && D >= E && D <= E && F >= A && F <= A && G >= H && G <= H && I >= J && I <= J
lbl43(A, B, C, D, E, F, G, H, I, J) -> Com_1(lbl43(A, B, C, D, E, F, G - 1, H, I, J)) :|: G >= 0 && I >= G + 2 && G + 1 >= 0 && A >= I + 1 && F >= A && F <= A
lbl43(A, B, C, D, E, F, G, H, I, J) -> Com_1(lbl13(A, B, C, I, E, F, G, H, 1 + I, J)) :|: I >= G + 2 && G + 1 >= 0 && A >= I + 1 && F >= A && F <= A
lbl31(A, B, C, D, E, F, G, H, I, J) -> Com_1(lbl43(A, B, C, D, E, F, I - 2, H, I, J)) :|: I >= 1 && A >= I + 1 && F >= A && F <= A
lbl31(A, B, C, D, E, F, G, H, I, J) -> Com_1(lbl13(A, B, C, I, E, F, I - 1, H, 1 + I, J)) :|: I >= 1 && A >= I + 1 && F >= A && F <= A
lbl13(A, B, C, D, E, F, G, H, I, J) -> Com_1(stop(A, B, C, D, E, F, G, H, I, J)) :|: G + A >= 2 && G + 1 >= 0 && A >= 2 + G && F >= A && F <= A && I >= A && I <= A && D + 1 >= A && D + 1 <= A
lbl13(A, B, C, D, E, F, G, H, I, J) -> Com_1(lbl31(A, K, C, D, E, F, G, H, I, J)) :|: A >= D + 2 && G + D >= 1 && G + 1 >= 0 && A >= D + 1 && D >= G + 1 && F >= A && F <= A && I >= D + 1 && I <= D + 1
start0(A, B, C, D, E, F, G, H, I, J) -> Com_1(start(A, C, C, E, E, A, H, H, J, J)) :|: TRUE

The start-symbols are:[start0_10]


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

(1) Koat2 Proof (FINISHED)
YES( ?, 3+max([0, Arg_0*max([7, 1+3*Arg_0])])+max([4, 2*Arg_0])+max([0, Arg_0])+max([0, 1+Arg_0])+max([3, -1+2*Arg_0])+max([7, 1+3*Arg_0]) {O(n^2)})



Initial Complexity Problem:

  Start:  start0  

  Program_Vars:  Arg_0, Arg_1, Arg_2, Arg_3, Arg_4, Arg_5, Arg_6, Arg_7, Arg_8, Arg_9  

  Temp_Vars:  K  

  Locations:  lbl13, lbl31, lbl43, start, start0, stop  

  Transitions:

  7: lbl13->lbl31

  6: lbl13->stop

  5: lbl31->lbl13

  4: lbl31->lbl43

  3: lbl43->lbl13

  2: lbl43->lbl43

  1: start->lbl31

  0: start->stop

  8: start0->start  



Timebounds: 

  Overall timebound: 3+max([0, Arg_0*max([7, 1+3*Arg_0])])+max([4, 2*Arg_0])+max([0, Arg_0])+max([0, 1+Arg_0])+max([3, -1+2*Arg_0])+max([7, 1+3*Arg_0]) {O(n^2)}

  6: lbl13->stop: 1 {O(1)}

  7: lbl13->lbl31: max([3, -1+2*Arg_0]) {O(n)}

  4: lbl31->lbl43: max([0, 1+Arg_0]) {O(n)}

  5: lbl31->lbl13: max([3, -1+2*Arg_0]) {O(n)}

  2: lbl43->lbl43: max([0, Arg_0*max([7, 1+3*Arg_0])])+max([7, 1+3*Arg_0]) {O(n^2)}

  3: lbl43->lbl13: max([0, Arg_0]) {O(n)}

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

  1: start->lbl31: 1 {O(1)}

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



Costbounds:

  Overall costbound: 3+max([0, Arg_0*max([7, 1+3*Arg_0])])+max([4, 2*Arg_0])+max([0, Arg_0])+max([0, 1+Arg_0])+max([3, -1+2*Arg_0])+max([7, 1+3*Arg_0]) {O(n^2)}

  6: lbl13->stop: 1 {O(1)}

  7: lbl13->lbl31: max([3, -1+2*Arg_0]) {O(n)}

  4: lbl31->lbl43: max([0, 1+Arg_0]) {O(n)}

  5: lbl31->lbl13: max([3, -1+2*Arg_0]) {O(n)}

  2: lbl43->lbl43: max([0, Arg_0*max([7, 1+3*Arg_0])])+max([7, 1+3*Arg_0]) {O(n^2)}

  3: lbl43->lbl13: max([0, Arg_0]) {O(n)}

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

  1: start->lbl31: 1 {O(1)}

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



Sizebounds:

`Lower:

  6: lbl13->stop, Arg_0: 2 {O(1)}

  6: lbl13->stop, Arg_2: Arg_2 {O(n)}

  6: lbl13->stop, Arg_3: 1 {O(1)}

  6: lbl13->stop, Arg_4: Arg_4 {O(n)}

  6: lbl13->stop, Arg_5: 2 {O(1)}

  6: lbl13->stop, Arg_6: -1 {O(1)}

  6: lbl13->stop, Arg_7: Arg_7 {O(n)}

  6: lbl13->stop, Arg_8: 2 {O(1)}

  6: lbl13->stop, Arg_9: Arg_9 {O(n)}

  7: lbl13->lbl31, Arg_0: 3 {O(1)}

  7: lbl13->lbl31, Arg_2: Arg_2 {O(n)}

  7: lbl13->lbl31, Arg_3: 1 {O(1)}

  7: lbl13->lbl31, Arg_4: Arg_4 {O(n)}

  7: lbl13->lbl31, Arg_5: 3 {O(1)}

  7: lbl13->lbl31, Arg_6: -1 {O(1)}

  7: lbl13->lbl31, Arg_7: Arg_7 {O(n)}

  7: lbl13->lbl31, Arg_8: 2 {O(1)}

  7: lbl13->lbl31, Arg_9: Arg_9 {O(n)}

  4: lbl31->lbl43, Arg_0: 2 {O(1)}

  4: lbl31->lbl43, Arg_2: Arg_2 {O(n)}

  4: lbl31->lbl43, Arg_3: min([1, Arg_4]) {O(n)}

  4: lbl31->lbl43, Arg_4: Arg_4 {O(n)}

  4: lbl31->lbl43, Arg_5: 2 {O(1)}

  4: lbl31->lbl43, Arg_6: -1 {O(1)}

  4: lbl31->lbl43, Arg_7: Arg_7 {O(n)}

  4: lbl31->lbl43, Arg_8: 1 {O(1)}

  4: lbl31->lbl43, Arg_9: Arg_9 {O(n)}

  5: lbl31->lbl13, Arg_0: 2 {O(1)}

  5: lbl31->lbl13, Arg_2: Arg_2 {O(n)}

  5: lbl31->lbl13, Arg_3: 1 {O(1)}

  5: lbl31->lbl13, Arg_4: Arg_4 {O(n)}

  5: lbl31->lbl13, Arg_5: 2 {O(1)}

  5: lbl31->lbl13, Arg_6: 0 {O(1)}

  5: lbl31->lbl13, Arg_7: Arg_7 {O(n)}

  5: lbl31->lbl13, Arg_8: 2 {O(1)}

  5: lbl31->lbl13, Arg_9: Arg_9 {O(n)}

  2: lbl43->lbl43, Arg_0: 3 {O(1)}

  2: lbl43->lbl43, Arg_2: Arg_2 {O(n)}

  2: lbl43->lbl43, Arg_3: min([1, Arg_4]) {O(n)}

  2: lbl43->lbl43, Arg_4: Arg_4 {O(n)}

  2: lbl43->lbl43, Arg_5: 3 {O(1)}

  2: lbl43->lbl43, Arg_6: -1 {O(1)}

  2: lbl43->lbl43, Arg_7: Arg_7 {O(n)}

  2: lbl43->lbl43, Arg_8: 2 {O(1)}

  2: lbl43->lbl43, Arg_9: Arg_9 {O(n)}

  3: lbl43->lbl13, Arg_0: 2 {O(1)}

  3: lbl43->lbl13, Arg_2: Arg_2 {O(n)}

  3: lbl43->lbl13, Arg_3: 1 {O(1)}

  3: lbl43->lbl13, Arg_4: Arg_4 {O(n)}

  3: lbl43->lbl13, Arg_5: 2 {O(1)}

  3: lbl43->lbl13, Arg_6: -1 {O(1)}

  3: lbl43->lbl13, Arg_7: Arg_7 {O(n)}

  3: lbl43->lbl13, Arg_8: 2 {O(1)}

  3: lbl43->lbl13, Arg_9: Arg_9 {O(n)}

  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_4 {O(n)}

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

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

  0: start->stop, Arg_6: Arg_7 {O(n)}

  0: start->stop, Arg_7: Arg_7 {O(n)}

  0: start->stop, Arg_8: 1 {O(1)}

  0: start->stop, Arg_9: Arg_9 {O(n)}

  1: start->lbl31, Arg_0: 2 {O(1)}

  1: start->lbl31, Arg_2: Arg_2 {O(n)}

  1: start->lbl31, Arg_3: Arg_4 {O(n)}

  1: start->lbl31, Arg_4: Arg_4 {O(n)}

  1: start->lbl31, Arg_5: 2 {O(1)}

  1: start->lbl31, Arg_6: Arg_7 {O(n)}

  1: start->lbl31, Arg_7: Arg_7 {O(n)}

  1: start->lbl31, Arg_8: 1 {O(1)}

  1: start->lbl31, Arg_9: Arg_9 {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)}

  8: start0->start, Arg_6: Arg_7 {O(n)}

  8: start0->start, Arg_7: Arg_7 {O(n)}

  8: start0->start, Arg_8: Arg_9 {O(n)}

  8: start0->start, Arg_9: Arg_9 {O(n)}

`Upper:

  6: lbl13->stop, Arg_0: Arg_0 {O(n)}

  6: lbl13->stop, Arg_2: Arg_2 {O(n)}

  6: lbl13->stop, Arg_3: max([1, 1+max([0, Arg_0])+max([3, -1+2*Arg_0])]) {O(n)}

  6: lbl13->stop, Arg_4: Arg_4 {O(n)}

  6: lbl13->stop, Arg_5: Arg_0 {O(n)}

  6: lbl13->stop, Arg_6: max([0, max([-1, max([max([0, Arg_0])+max([3, -1+2*Arg_0]), -1+max([0, Arg_0])+max([3, -1+2*Arg_0])])])]) {O(n)}

  6: lbl13->stop, Arg_7: Arg_7 {O(n)}

  6: lbl13->stop, Arg_8: 1+max([0, Arg_0])+max([3, -1+2*Arg_0]) {O(n)}

  6: lbl13->stop, Arg_9: Arg_9 {O(n)}

  7: lbl13->lbl31, Arg_0: Arg_0 {O(n)}

  7: lbl13->lbl31, Arg_2: Arg_2 {O(n)}

  7: lbl13->lbl31, Arg_3: max([1, 1+max([0, Arg_0])+max([3, -1+2*Arg_0])]) {O(n)}

  7: lbl13->lbl31, Arg_4: Arg_4 {O(n)}

  7: lbl13->lbl31, Arg_5: Arg_0 {O(n)}

  7: lbl13->lbl31, Arg_6: max([0, max([-1, max([max([0, Arg_0])+max([3, -1+2*Arg_0]), -1+max([0, Arg_0])+max([3, -1+2*Arg_0])])])]) {O(n)}

  7: lbl13->lbl31, Arg_7: Arg_7 {O(n)}

  7: lbl13->lbl31, Arg_8: 1+max([0, Arg_0])+max([3, -1+2*Arg_0]) {O(n)}

  7: lbl13->lbl31, Arg_9: Arg_9 {O(n)}

  4: lbl31->lbl43, Arg_0: Arg_0 {O(n)}

  4: lbl31->lbl43, Arg_2: Arg_2 {O(n)}

  4: lbl31->lbl43, Arg_3: max([1, max([Arg_4, 1+max([0, Arg_0])+max([3, -1+2*Arg_0])])]) {O(n)}

  4: lbl31->lbl43, Arg_4: Arg_4 {O(n)}

  4: lbl31->lbl43, Arg_5: Arg_0 {O(n)}

  4: lbl31->lbl43, Arg_6: max([-1, -1+max([0, Arg_0])+max([3, -1+2*Arg_0])]) {O(n)}

  4: lbl31->lbl43, Arg_7: Arg_7 {O(n)}

  4: lbl31->lbl43, Arg_8: 1+max([0, Arg_0])+max([3, -1+2*Arg_0]) {O(n)}

  4: lbl31->lbl43, Arg_9: Arg_9 {O(n)}

  5: lbl31->lbl13, Arg_0: Arg_0 {O(n)}

  5: lbl31->lbl13, Arg_2: Arg_2 {O(n)}

  5: lbl31->lbl13, Arg_3: max([1, 1+max([0, Arg_0])+max([3, -1+2*Arg_0])]) {O(n)}

  5: lbl31->lbl13, Arg_4: Arg_4 {O(n)}

  5: lbl31->lbl13, Arg_5: Arg_0 {O(n)}

  5: lbl31->lbl13, Arg_6: max([0, max([0, Arg_0])+max([3, -1+2*Arg_0])]) {O(n)}

  5: lbl31->lbl13, Arg_7: Arg_7 {O(n)}

  5: lbl31->lbl13, Arg_8: 1+max([0, Arg_0])+max([3, -1+2*Arg_0]) {O(n)}

  5: lbl31->lbl13, Arg_9: Arg_9 {O(n)}

  2: lbl43->lbl43, Arg_0: Arg_0 {O(n)}

  2: lbl43->lbl43, Arg_2: Arg_2 {O(n)}

  2: lbl43->lbl43, Arg_3: max([1, max([Arg_4, 1+max([0, Arg_0])+max([3, -1+2*Arg_0])])]) {O(n)}

  2: lbl43->lbl43, Arg_4: Arg_4 {O(n)}

  2: lbl43->lbl43, Arg_5: Arg_0 {O(n)}

  2: lbl43->lbl43, Arg_6: max([-1, -1+max([0, Arg_0])+max([3, -1+2*Arg_0])]) {O(n)}

  2: lbl43->lbl43, Arg_7: Arg_7 {O(n)}

  2: lbl43->lbl43, Arg_8: 1+max([0, Arg_0])+max([3, -1+2*Arg_0]) {O(n)}

  2: lbl43->lbl43, Arg_9: Arg_9 {O(n)}

  3: lbl43->lbl13, Arg_0: Arg_0 {O(n)}

  3: lbl43->lbl13, Arg_2: Arg_2 {O(n)}

  3: lbl43->lbl13, Arg_3: 1+max([0, Arg_0])+max([3, -1+2*Arg_0]) {O(n)}

  3: lbl43->lbl13, Arg_4: Arg_4 {O(n)}

  3: lbl43->lbl13, Arg_5: Arg_0 {O(n)}

  3: lbl43->lbl13, Arg_6: max([-1, -1+max([0, Arg_0])+max([3, -1+2*Arg_0])]) {O(n)}

  3: lbl43->lbl13, Arg_7: Arg_7 {O(n)}

  3: lbl43->lbl13, Arg_8: 1+max([0, Arg_0])+max([3, -1+2*Arg_0]) {O(n)}

  3: lbl43->lbl13, Arg_9: Arg_9 {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: Arg_4 {O(n)}

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

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

  0: start->stop, Arg_6: Arg_7 {O(n)}

  0: start->stop, Arg_7: Arg_7 {O(n)}

  0: start->stop, Arg_8: 1 {O(1)}

  0: start->stop, Arg_9: Arg_9 {O(n)}

  1: start->lbl31, Arg_0: Arg_0 {O(n)}

  1: start->lbl31, Arg_2: Arg_2 {O(n)}

  1: start->lbl31, Arg_3: Arg_4 {O(n)}

  1: start->lbl31, Arg_4: Arg_4 {O(n)}

  1: start->lbl31, Arg_5: Arg_0 {O(n)}

  1: start->lbl31, Arg_6: Arg_7 {O(n)}

  1: start->lbl31, Arg_7: Arg_7 {O(n)}

  1: start->lbl31, Arg_8: 1 {O(1)}

  1: start->lbl31, Arg_9: Arg_9 {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)}

  8: start0->start, Arg_6: Arg_7 {O(n)}

  8: start0->start, Arg_7: Arg_7 {O(n)}

  8: start0->start, Arg_8: Arg_9 {O(n)}

  8: start0->start, Arg_9: Arg_9 {O(n)}


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

(2)
BOUNDS(1, nat(1 + Arg_0) + max(3, -1 + 2 * Arg_0) + max(7, 1 + 3 * Arg_0) + max(3, 3 + Arg_0 * max(7, 1 + 3 * Arg_0)) + max(4, 2 * Arg_0) + nat(Arg_0))

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

(3) Loat Proof (FINISHED)


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



Initial linear ITS problem

   Start location: start0

      0: start -> stop : Q'=1, [ 1>=A && B==C && D==E && F==A && G==H && Q==J ], cost: 1

      1: start -> lbl31 : B'=free, Q'=1, [ A>=2 && B==C && D==E && F==A && G==H && Q==J ], cost: 1

      2: lbl43 -> lbl43 : G'=-1+G, [ G>=0 && Q>=2+G && 1+G>=0 && A>=1+Q && F==A ], cost: 1

      3: lbl43 -> lbl13 : D'=Q, Q'=1+Q, [ Q>=2+G && 1+G>=0 && A>=1+Q && F==A ], cost: 1

      4: lbl31 -> lbl43 : G'=-2+Q, [ Q>=1 && A>=1+Q && F==A ], cost: 1

      5: lbl31 -> lbl13 : D'=Q, G'=-1+Q, Q'=1+Q, [ Q>=1 && A>=1+Q && F==A ], cost: 1

      6: lbl13 -> stop : [ G+A>=2 && 1+G>=0 && A>=2+G && F==A && Q==A && 1+D==A ], cost: 1

      7: lbl13 -> lbl31 : B'=free_1, [ A>=2+D && G+D>=1 && 1+G>=0 && A>=1+D && D>=1+G && F==A && Q==1+D ], cost: 1

      8: start0 -> start : B'=C, D'=E, F'=A, G'=H, Q'=J, [], cost: 1



Checking for constant complexity:

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

      8: start0 -> start : B'=C, D'=E, F'=A, G'=H, Q'=J, [], cost: 1



Removed unreachable and leaf rules:

   Start location: start0

      1: start -> lbl31 : B'=free, Q'=1, [ A>=2 && B==C && D==E && F==A && G==H && Q==J ], cost: 1

      2: lbl43 -> lbl43 : G'=-1+G, [ G>=0 && Q>=2+G && 1+G>=0 && A>=1+Q && F==A ], cost: 1

      3: lbl43 -> lbl13 : D'=Q, Q'=1+Q, [ Q>=2+G && 1+G>=0 && A>=1+Q && F==A ], cost: 1

      4: lbl31 -> lbl43 : G'=-2+Q, [ Q>=1 && A>=1+Q && F==A ], cost: 1

      5: lbl31 -> lbl13 : D'=Q, G'=-1+Q, Q'=1+Q, [ Q>=1 && A>=1+Q && F==A ], cost: 1

      7: lbl13 -> lbl31 : B'=free_1, [ A>=2+D && G+D>=1 && 1+G>=0 && A>=1+D && D>=1+G && F==A && Q==1+D ], cost: 1

      8: start0 -> start : B'=C, D'=E, F'=A, G'=H, Q'=J, [], cost: 1



Simplified all rules, resulting in:

   Start location: start0

      1: start -> lbl31 : B'=free, Q'=1, [ A>=2 && B==C && D==E && F==A && G==H && Q==J ], cost: 1

      2: lbl43 -> lbl43 : G'=-1+G, [ G>=0 && Q>=2+G && A>=1+Q && F==A ], cost: 1

      3: lbl43 -> lbl13 : D'=Q, Q'=1+Q, [ Q>=2+G && 1+G>=0 && A>=1+Q && F==A ], cost: 1

      4: lbl31 -> lbl43 : G'=-2+Q, [ Q>=1 && A>=1+Q && F==A ], cost: 1

      5: lbl31 -> lbl13 : D'=Q, G'=-1+Q, Q'=1+Q, [ Q>=1 && A>=1+Q && F==A ], cost: 1

      7: lbl13 -> lbl31 : B'=free_1, [ A>=2+D && G+D>=1 && 1+G>=0 && D>=1+G && F==A && Q==1+D ], cost: 1

      8: start0 -> start : B'=C, D'=E, F'=A, G'=H, Q'=J, [], cost: 1



### Simplification by acceleration and chaining ###



Accelerating simple loops of location 1.

   Accelerating the following rules:

      2: lbl43 -> lbl43 : G'=-1+G, [ G>=0 && Q>=2+G && A>=1+Q && F==A ], cost: 1



   Accelerated rule 2 with metering function 1+G, yielding the new rule 9.

   Removing the simple loops: 2.



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

   Start location: start0

      1: start -> lbl31 : B'=free, Q'=1, [ A>=2 && B==C && D==E && F==A && G==H && Q==J ], cost: 1

      3: lbl43 -> lbl13 : D'=Q, Q'=1+Q, [ Q>=2+G && 1+G>=0 && A>=1+Q && F==A ], cost: 1

      9: lbl43 -> lbl43 : G'=-1, [ G>=0 && Q>=2+G && A>=1+Q && F==A ], cost: 1+G

      4: lbl31 -> lbl43 : G'=-2+Q, [ Q>=1 && A>=1+Q && F==A ], cost: 1

      5: lbl31 -> lbl13 : D'=Q, G'=-1+Q, Q'=1+Q, [ Q>=1 && A>=1+Q && F==A ], cost: 1

      7: lbl13 -> lbl31 : B'=free_1, [ A>=2+D && G+D>=1 && 1+G>=0 && D>=1+G && F==A && Q==1+D ], cost: 1

      8: start0 -> start : B'=C, D'=E, F'=A, G'=H, Q'=J, [], cost: 1



Chained accelerated rules (with incoming rules):

   Start location: start0

      1: start -> lbl31 : B'=free, Q'=1, [ A>=2 && B==C && D==E && F==A && G==H && Q==J ], cost: 1

      3: lbl43 -> lbl13 : D'=Q, Q'=1+Q, [ Q>=2+G && 1+G>=0 && A>=1+Q && F==A ], cost: 1

      4: lbl31 -> lbl43 : G'=-2+Q, [ Q>=1 && A>=1+Q && F==A ], cost: 1

      5: lbl31 -> lbl13 : D'=Q, G'=-1+Q, Q'=1+Q, [ Q>=1 && A>=1+Q && F==A ], cost: 1

     10: lbl31 -> lbl43 : G'=-1, [ A>=1+Q && F==A && -2+Q>=0 ], cost: Q

      7: lbl13 -> lbl31 : B'=free_1, [ A>=2+D && G+D>=1 && 1+G>=0 && D>=1+G && F==A && Q==1+D ], cost: 1

      8: start0 -> start : B'=C, D'=E, F'=A, G'=H, Q'=J, [], cost: 1



Eliminated locations (on linear paths):

   Start location: start0

      3: lbl43 -> lbl13 : D'=Q, Q'=1+Q, [ Q>=2+G && 1+G>=0 && A>=1+Q && F==A ], cost: 1

      4: lbl31 -> lbl43 : G'=-2+Q, [ Q>=1 && A>=1+Q && F==A ], cost: 1

      5: lbl31 -> lbl13 : D'=Q, G'=-1+Q, Q'=1+Q, [ Q>=1 && A>=1+Q && F==A ], cost: 1

     10: lbl31 -> lbl43 : G'=-1, [ A>=1+Q && F==A && -2+Q>=0 ], cost: Q

      7: lbl13 -> lbl31 : B'=free_1, [ A>=2+D && G+D>=1 && 1+G>=0 && D>=1+G && F==A && Q==1+D ], cost: 1

     11: start0 -> lbl31 : B'=free, D'=E, F'=A, G'=H, Q'=1, [ A>=2 ], cost: 2



Eliminated locations (on tree-shaped paths):

   Start location: start0

     14: lbl31 -> lbl31 : B'=free_1, D'=Q, G'=-1+Q, Q'=1+Q, [ Q>=1 && F==A && A>=2+Q && -1+2*Q>=1 ], cost: 2

     15: lbl31 -> lbl31 : B'=free_1, D'=Q, G'=-2+Q, Q'=1+Q, [ Q>=1 && F==A && A>=2+Q && -2+2*Q>=1 ], cost: 3

     16: lbl31 -> lbl31 : B'=free_1, D'=Q, G'=-1, Q'=1+Q, [ F==A && -2+Q>=0 && A>=2+Q ], cost: 2+Q

     11: start0 -> lbl31 : B'=free, D'=E, F'=A, G'=H, Q'=1, [ A>=2 ], cost: 2



Accelerating simple loops of location 2.

   Accelerating the following rules:

     14: lbl31 -> lbl31 : B'=free_1, D'=Q, G'=-1+Q, Q'=1+Q, [ Q>=1 && F==A && A>=2+Q && -1+2*Q>=1 ], cost: 2

     15: lbl31 -> lbl31 : B'=free_1, D'=Q, G'=-2+Q, Q'=1+Q, [ Q>=1 && F==A && A>=2+Q && -2+2*Q>=1 ], cost: 3

     16: lbl31 -> lbl31 : B'=free_1, D'=Q, G'=-1, Q'=1+Q, [ F==A && -2+Q>=0 && A>=2+Q ], cost: 2+Q



   Accelerated rule 14 with metering function -1+A-Q, yielding the new rule 17.

   Accelerated rule 15 with metering function -1+A-Q, yielding the new rule 18.

   Accelerated rule 16 with metering function -1+A-Q, yielding the new rule 19.

   Removing the simple loops: 14 15 16.



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

   Start location: start0

     17: lbl31 -> lbl31 : B'=free_1, D'=-2+A, G'=-3+A, Q'=-1+A, [ Q>=1 && F==A && A>=2+Q && -1+2*Q>=1 ], cost: -2+2*A-2*Q

     18: lbl31 -> lbl31 : B'=free_1, D'=-2+A, G'=-4+A, Q'=-1+A, [ Q>=1 && F==A && A>=2+Q && -2+2*Q>=1 ], cost: -3+3*A-3*Q

     19: lbl31 -> lbl31 : B'=free_1, D'=-2+A, G'=-1, Q'=-1+A, [ F==A && -2+Q>=0 && A>=2+Q ], cost: -3/2+(-1+A-Q)*Q+1/2*(-1+A-Q)^2+3/2*A-3/2*Q

     11: start0 -> lbl31 : B'=free, D'=E, F'=A, G'=H, Q'=1, [ A>=2 ], cost: 2



Chained accelerated rules (with incoming rules):

   Start location: start0

     11: start0 -> lbl31 : B'=free, D'=E, F'=A, G'=H, Q'=1, [ A>=2 ], cost: 2

     20: start0 -> lbl31 : B'=free_1, D'=-2+A, F'=A, G'=-3+A, Q'=-1+A, [ A>=3 ], cost: -2+2*A



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

   Start location: start0

     20: start0 -> lbl31 : B'=free_1, D'=-2+A, F'=A, G'=-3+A, Q'=-1+A, [ A>=3 ], cost: -2+2*A



### Computing asymptotic complexity ###



Fully simplified ITS problem

   Start location: start0

     20: start0 -> lbl31 : B'=free_1, D'=-2+A, F'=A, G'=-3+A, Q'=-1+A, [ A>=3 ], cost: -2+2*A



Computing asymptotic complexity for rule 20

   Solved the limit problem by the following transformations:

      Created initial limit problem:

      -2+2*A (+), -2+A (+/+!) [not solved]



      removing all constraints (solved by SMT)

      resulting limit problem: [solved]



      applying transformation rule (C) using substitution {A==n}

      resulting limit problem:

      [solved]



   Solution:

      A / n

   Resulting cost -2+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: -2+2*n

   Rule cost:   -2+2*A

   Rule guard:  [ A>=3 ]



WORST_CASE(Omega(n^1),?)


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(4)
BOUNDS(n^1, INF)