/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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, 225 ms]
(2) BOUNDS(1, INF)


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(0)
Obligation:
Complexity Int TRS consisting of the following rules:
f0(A, B, C) -> Com_1(f1(A, B, C)) :|: TRUE
f1(A, B, C) -> Com_1(f1(A + 1, B + A + 1, C)) :|: A >= B
f1(A, B, C) -> Com_1(f1(A - C, B - 2 * C, C - 1)) :|: A >= B

The start-symbols are:[f0_3]


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(1) Loat Proof (FINISHED)


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



Initial linear ITS problem

   Start location: f0

      0: f0 -> f1 : [], cost: 1

      1: f1 -> f1 : A'=1+A, B'=1+A+B, [ A>=B ], cost: 1

      2: f1 -> f1 : A'=-C+A, B'=-2*C+B, C'=-1+C, [ A>=B ], cost: 1



Checking for constant complexity:

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

      0: f0 -> f1 : [], cost: 1



### Simplification by acceleration and chaining ###



Accelerating simple loops of location 1.

   Accelerating the following rules:

      1: f1 -> f1 : A'=1+A, B'=1+A+B, [ A>=B ], cost: 1

      2: f1 -> f1 : A'=-C+A, B'=-2*C+B, C'=-1+C, [ A>=B ], cost: 1



   Found no metering function for rule 1.

   Found no metering function for rule 2.

   Removing the simple loops:.



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

   Start location: f0

      0: f0 -> f1 : [], cost: 1

      1: f1 -> f1 : A'=1+A, B'=1+A+B, [ A>=B ], cost: 1

      2: f1 -> f1 : A'=-C+A, B'=-2*C+B, C'=-1+C, [ A>=B ], cost: 1



Chained accelerated rules (with incoming rules):

   Start location: f0

      0: f0 -> f1 : [], cost: 1

      3: f0 -> f1 : A'=1+A, B'=1+A+B, [ A>=B ], cost: 2

      4: f0 -> f1 : A'=-C+A, B'=-2*C+B, C'=-1+C, [ A>=B ], cost: 2



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

   Start location: f0

     <empty>



### Computing asymptotic complexity ###



Fully simplified ITS problem

   Start location: f0

     <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:  []



WORST_CASE(Omega(1),?)


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(2)
BOUNDS(1, INF)