/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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, 207 ms]
(2) BOUNDS(1, INF)


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

(0)
Obligation:
Complexity Int TRS consisting of the following rules:
f1(A, B) -> Com_1(f2(A, B)) :|: 0 >= A + B + 1 && A >= 1
f2(A, B) -> Com_1(f2(A - B, B)) :|: A >= 0
f2(A, B) -> Com_1(f3(A, B)) :|: 0 >= A + 1

The start-symbols are:[f1_2]


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

(1) Loat Proof (FINISHED)


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



Initial linear ITS problem

   Start location: f1

      0: f1 -> f2 : [ 0>=1+A+B && A>=1 ], cost: 1

      1: f2 -> f2 : A'=A-B, [ A>=0 ], cost: 1

      2: f2 -> f3 : [ 0>=1+A ], cost: 1



Checking for constant complexity:

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

      0: f1 -> f2 : [ 0>=1+A+B && A>=1 ], cost: 1



Removed unreachable and leaf rules:

   Start location: f1

      0: f1 -> f2 : [ 0>=1+A+B && A>=1 ], cost: 1

      1: f2 -> f2 : A'=A-B, [ A>=0 ], cost: 1



### Simplification by acceleration and chaining ###



Accelerating simple loops of location 1.

   Accelerating the following rules:

      1: f2 -> f2 : A'=A-B, [ A>=0 ], cost: 1



   Found no metering function for rule 1.

   Removing the simple loops:.



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

   Start location: f1

      0: f1 -> f2 : [ 0>=1+A+B && A>=1 ], cost: 1

      1: f2 -> f2 : A'=A-B, [ A>=0 ], cost: 1



Chained accelerated rules (with incoming rules):

   Start location: f1

      0: f1 -> f2 : [ 0>=1+A+B && A>=1 ], cost: 1

      3: f1 -> f2 : A'=A-B, [ 0>=1+A+B && A>=1 ], 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:  [ 0>=1+A+B && A>=1 ]



WORST_CASE(Omega(1),?)


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

(2)
BOUNDS(1, INF)