/export/starexec/sandbox2/solver/bin/starexec_run_complexity /export/starexec/sandbox2/benchmark/theBenchmark.koat /export/starexec/sandbox2/output/output_files


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MAYBE
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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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(1, INF).

(0) CpxIntTrs
(1) Loat Proof [FINISHED, 115 ms]
(2) BOUNDS(1, INF)


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(0)
Obligation:
Complexity Int TRS consisting of the following rules:
f0(A, B, C, D) -> Com_1(f6(1, C, C, D)) :|: TRUE
f6(A, B, C, D) -> Com_1(f6(A, E, C, E)) :|: B >= A + 2 && E * E >= C + 1
f6(A, B, C, D) -> Com_1(f6(E, B, C, E)) :|: B >= A + 2 && C >= E * E
f6(A, B, C, D) -> Com_1(f16(A, B, C, D)) :|: A + 1 >= B

The start-symbols are:[f0_4]


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(1) Loat Proof (FINISHED)


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



Initial linear ITS problem

   Start location: f0

      0: f0 -> f6 : A'=1, B'=C, [], cost: 1

      1: f6 -> f6 : B'=free, D'=free, [ B>=2+A && free^2>=1+C ], cost: 1

      2: f6 -> f6 : A'=free_1, D'=free_1, [ B>=2+A && C>=free_1^2 ], cost: 1

      3: f6 -> f16 : [ 1+A>=B ], cost: 1



Checking for constant complexity:

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

      0: f0 -> f6 : A'=1, B'=C, [], cost: 1



Removed unreachable and leaf rules:

   Start location: f0

      0: f0 -> f6 : A'=1, B'=C, [], cost: 1

      1: f6 -> f6 : B'=free, D'=free, [ B>=2+A && free^2>=1+C ], cost: 1

      2: f6 -> f6 : A'=free_1, D'=free_1, [ B>=2+A && C>=free_1^2 ], cost: 1



### Simplification by acceleration and chaining ###



Accelerating simple loops of location 1.

   Accelerating the following rules:

      1: f6 -> f6 : B'=free, D'=free, [ B>=2+A && free^2>=1+C ], cost: 1

      2: f6 -> f6 : A'=free_1, D'=free_1, [ B>=2+A && C>=free_1^2 ], cost: 1



   Found no metering function for rule 1 (rule is too complicated).

   Found no metering function for rule 2 (rule is too complicated).

   Removing the simple loops:.



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

   Start location: f0

      0: f0 -> f6 : A'=1, B'=C, [], cost: 1

      1: f6 -> f6 : B'=free, D'=free, [ B>=2+A && free^2>=1+C ], cost: 1

      2: f6 -> f6 : A'=free_1, D'=free_1, [ B>=2+A && C>=free_1^2 ], cost: 1



Chained accelerated rules (with incoming rules):

   Start location: f0

      0: f0 -> f6 : A'=1, B'=C, [], cost: 1

      4: f0 -> f6 : A'=1, B'=free, D'=free, [ C>=3 && free^2>=1+C ], cost: 2

      5: f0 -> f6 : A'=free_1, B'=C, D'=free_1, [ C>=3 && C>=free_1^2 ], 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)