/export/starexec/sandbox/solver/bin/starexec_run_complexity /export/starexec/sandbox/benchmark/theBenchmark.koat /export/starexec/sandbox/output/output_files


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


MAYBE
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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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, INF).

(0) CpxIntTrs
(1) Loat Proof [FINISHED, 429 ms]
(2) BOUNDS(1, INF)


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

(0)
Obligation:
Complexity Int TRS consisting of the following rules:
f0(A, B, C, D, E, F) -> Com_1(f0(-(A), B + A, C + A, D, E, F)) :|: 0 >= A + 1
f0(A, B, C, D, E, F) -> Com_1(f0(A + B, -(B), C, D + B, E, F)) :|: 0 >= B + 1
f0(A, B, C, D, E, F) -> Com_1(f0(A, B + D, C, -(D), E + D, F)) :|: 0 >= D + 1
f0(A, B, C, D, E, F) -> Com_1(f0(A, B, C + E, D + E, -(E), F)) :|: 0 >= E + 1
f0(A, B, C, D, E, F) -> Com_1(f0(A + C, B, -(C), D, E + C, F)) :|: 0 >= C + 1
f1(A, B, C, D, E, F) -> Com_1(f0(G, H, K, I, J, G + H + I + J + K)) :|: G + H + I + J + K >= 1

The start-symbols are:[f1_6]


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

(1) Loat Proof (FINISHED)


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



Initial linear ITS problem

   Start location: f1

      0: f0 -> f0 : A'=-A, B'=A+B, C'=C+A, [ 0>=1+A ], cost: 1

      1: f0 -> f0 : A'=A+B, B'=-B, D'=D+B, [ 0>=1+B ], cost: 1

      2: f0 -> f0 : B'=D+B, D'=-D, E'=D+E, [ 0>=1+D ], cost: 1

      3: f0 -> f0 : C'=C+E, D'=D+E, E'=-E, [ 0>=1+E ], cost: 1

      4: f0 -> f0 : A'=C+A, C'=-C, E'=C+E, [ 0>=1+C ], cost: 1

      5: f1 -> f0 : A'=free_1, B'=free_3, C'=free_2, D'=free_4, E'=free, F'=free_4+free_1+free_2+free_3+free, [ free_4+free_1+free_2+free_3+free>=1 ], cost: 1



Checking for constant complexity:

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

      5: f1 -> f0 : A'=free_1, B'=free_3, C'=free_2, D'=free_4, E'=free, F'=free_4+free_1+free_2+free_3+free, [ free_4+free_1+free_2+free_3+free>=1 ], cost: 1



### Simplification by acceleration and chaining ###



Accelerating simple loops of location 0.

   Accelerating the following rules:

      0: f0 -> f0 : A'=-A, B'=A+B, C'=C+A, [ 0>=1+A ], cost: 1

      1: f0 -> f0 : A'=A+B, B'=-B, D'=D+B, [ 0>=1+B ], cost: 1

      2: f0 -> f0 : B'=D+B, D'=-D, E'=D+E, [ 0>=1+D ], cost: 1

      3: f0 -> f0 : C'=C+E, D'=D+E, E'=-E, [ 0>=1+E ], cost: 1

      4: f0 -> f0 : A'=C+A, C'=-C, E'=C+E, [ 0>=1+C ], cost: 1



   Found no metering function for rule 0.

   Found no metering function for rule 1.

   Found no metering function for rule 2.

   Found no metering function for rule 3.

   Found no metering function for rule 4.

   Removing the simple loops:.



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

   Start location: f1

      0: f0 -> f0 : A'=-A, B'=A+B, C'=C+A, [ 0>=1+A ], cost: 1

      1: f0 -> f0 : A'=A+B, B'=-B, D'=D+B, [ 0>=1+B ], cost: 1

      2: f0 -> f0 : B'=D+B, D'=-D, E'=D+E, [ 0>=1+D ], cost: 1

      3: f0 -> f0 : C'=C+E, D'=D+E, E'=-E, [ 0>=1+E ], cost: 1

      4: f0 -> f0 : A'=C+A, C'=-C, E'=C+E, [ 0>=1+C ], cost: 1

      5: f1 -> f0 : A'=free_1, B'=free_3, C'=free_2, D'=free_4, E'=free, F'=free_4+free_1+free_2+free_3+free, [ free_4+free_1+free_2+free_3+free>=1 ], cost: 1



Chained accelerated rules (with incoming rules):

   Start location: f1

      5: f1 -> f0 : A'=free_1, B'=free_3, C'=free_2, D'=free_4, E'=free, F'=free_4+free_1+free_2+free_3+free, [ free_4+free_1+free_2+free_3+free>=1 ], cost: 1

      6: f1 -> f0 : A'=-free_1, B'=free_1+free_3, C'=free_1+free_2, D'=free_4, E'=free, F'=free_4+free_1+free_2+free_3+free, [ free_4+free_1+free_2+free_3+free>=1 && 0>=1+free_1 ], cost: 2

      7: f1 -> f0 : A'=free_1+free_3, B'=-free_3, C'=free_2, D'=free_4+free_3, E'=free, F'=free_4+free_1+free_2+free_3+free, [ free_4+free_1+free_2+free_3+free>=1 && 0>=1+free_3 ], cost: 2

      8: f1 -> f0 : A'=free_1, B'=free_4+free_3, C'=free_2, D'=-free_4, E'=free_4+free, F'=free_4+free_1+free_2+free_3+free, [ free_4+free_1+free_2+free_3+free>=1 && 0>=1+free_4 ], cost: 2

      9: f1 -> f0 : A'=free_1, B'=free_3, C'=free_2+free, D'=free_4+free, E'=-free, F'=free_4+free_1+free_2+free_3+free, [ free_4+free_1+free_2+free_3+free>=1 && 0>=1+free ], cost: 2

     10: f1 -> f0 : A'=free_1+free_2, B'=free_3, C'=-free_2, D'=free_4, E'=free_2+free, F'=free_4+free_1+free_2+free_3+free, [ free_4+free_1+free_2+free_3+free>=1 && 0>=1+free_2 ], cost: 2



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

   Start location: f1

     <empty>



### Computing asymptotic complexity ###



Fully simplified ITS problem

   Start location: f1

     <empty>



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

   Complexity:  Constant

   Cpx degree:  0

   Solved cost: 1

   Rule cost:   1

   Rule guard:  [ free_4+free_1+free_2+free_3+free>=1 ]



WORST_CASE(Omega(1),?)


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

(2)
BOUNDS(1, INF)