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


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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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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, 312 ms]
(2) BOUNDS(1, INF)


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(0)
Obligation:
Complexity Int TRS consisting of the following rules:
f0(A, B, C, D, E, F) -> Com_1(f0(-(1) + A, C, -(1) + C, A, E, F)) :|: A >= 1
f1(A, B, C, D, E, F) -> Com_1(f0(-(1) + A, B, -(1) + C, D, C, A)) :|: A >= 1 && C >= 1
f0(A, B, C, D, E, F) -> Com_1(f0(5000, B, C, D, E, F)) :|: 0 >= A && C >= 1
f3(A, B, C, D, E, F) -> Com_1(f0(5000, B, G, D, E, F)) :|: G >= 1

The start-symbols are:[f3_6]


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(1) Loat Proof (FINISHED)


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



Initial linear ITS problem

   Start location: f3

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

      2: f0 -> f0 : A'=5000, [ 0>=A && C>=1 ], cost: 1

      1: f1 -> f0 : A'=-1+A, C'=-1+C, E'=C, F'=A, [ A>=1 && C>=1 ], cost: 1

      3: f3 -> f0 : A'=5000, C'=free, [ free>=1 ], cost: 1



Checking for constant complexity:

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

      3: f3 -> f0 : A'=5000, C'=free, [ free>=1 ], cost: 1



Removed unreachable and leaf rules:

   Start location: f3

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

      2: f0 -> f0 : A'=5000, [ 0>=A && C>=1 ], cost: 1

      3: f3 -> f0 : A'=5000, C'=free, [ free>=1 ], cost: 1



### Simplification by acceleration and chaining ###



Accelerating simple loops of location 0.

   Accelerating the following rules:

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

      2: f0 -> f0 : A'=5000, [ 0>=A && C>=1 ], cost: 1



   Accelerated rule 0 with metering function A, yielding the new rule 4.

   Found no metering function for rule 2.

   Removing the simple loops: 0.



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

   Start location: f3

      2: f0 -> f0 : A'=5000, [ 0>=A && C>=1 ], cost: 1

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

      3: f3 -> f0 : A'=5000, C'=free, [ free>=1 ], cost: 1



Chained accelerated rules (with incoming rules):

   Start location: f3

      3: f3 -> f0 : A'=5000, C'=free, [ free>=1 ], cost: 1

      5: f3 -> f0 : A'=0, B'=-4999+free, C'=-5000+free, D'=1, [ free>=1 ], cost: 5001



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

   Start location: f3

     <empty>



### Computing asymptotic complexity ###



Fully simplified ITS problem

   Start location: f3

     <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>=1 ]



WORST_CASE(Omega(1),?)


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(2)
BOUNDS(1, INF)