/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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, 330 ms]
(2) BOUNDS(1, INF)


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(0)
Obligation:
Complexity Int TRS consisting of the following rules:
f2(A, B, C, D) -> Com_1(f300(A, B, C, D)) :|: TRUE
f300(A, B, C, D) -> Com_1(f300(-(1) + B, -(1) + B, E, D)) :|: A >= 1 && B >= 1 && E >= 1 && B + A >= 1
f300(A, B, C, D) -> Com_1(f300(-(1) + B, -(1) + B, E, D)) :|: A >= 1 && B >= 1 && 0 >= E + 1 && B + A >= 1
f300(A, B, C, D) -> Com_1(f300(-(1) + A, -(2) + A, 0, D)) :|: A >= 1 && B + A >= 1 && B >= 1
f300(A, B, C, D) -> Com_1(f1(A, B, C, E)) :|: A >= 1 && 0 >= B + A && B >= 1
f300(A, B, C, D) -> Com_1(f1(A, B, C, E)) :|: B >= 1 && 0 >= A
f300(A, B, C, D) -> Com_1(f1(A, B, C, E)) :|: 0 >= B

The start-symbols are:[f2_4]


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(1) Loat Proof (FINISHED)


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



Initial linear ITS problem

   Start location: f2

      0: f2 -> f300 : [], cost: 1

      1: f300 -> f300 : A'=-1+B, B'=-1+B, C'=free, [ A>=1 && B>=1 && free>=1 && A+B>=1 ], cost: 1

      2: f300 -> f300 : A'=-1+B, B'=-1+B, C'=free_1, [ A>=1 && B>=1 && 0>=1+free_1 && A+B>=1 ], cost: 1

      3: f300 -> f300 : A'=-1+A, B'=-2+A, C'=0, [ A>=1 && A+B>=1 && B>=1 ], cost: 1

      4: f300 -> f1 : D'=free_2, [ A>=1 && 0>=A+B && B>=1 ], cost: 1

      5: f300 -> f1 : D'=free_3, [ B>=1 && 0>=A ], cost: 1

      6: f300 -> f1 : D'=free_4, [ 0>=B ], cost: 1



Checking for constant complexity:

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

      0: f2 -> f300 : [], cost: 1



Removed unreachable and leaf rules:

   Start location: f2

      0: f2 -> f300 : [], cost: 1

      1: f300 -> f300 : A'=-1+B, B'=-1+B, C'=free, [ A>=1 && B>=1 && free>=1 && A+B>=1 ], cost: 1

      2: f300 -> f300 : A'=-1+B, B'=-1+B, C'=free_1, [ A>=1 && B>=1 && 0>=1+free_1 && A+B>=1 ], cost: 1

      3: f300 -> f300 : A'=-1+A, B'=-2+A, C'=0, [ A>=1 && A+B>=1 && B>=1 ], cost: 1



### Simplification by acceleration and chaining ###



Accelerating simple loops of location 1.

   Accelerating the following rules:

      1: f300 -> f300 : A'=-1+B, B'=-1+B, C'=free, [ A>=1 && B>=1 && free>=1 ], cost: 1

      2: f300 -> f300 : A'=-1+B, B'=-1+B, C'=free_1, [ A>=1 && B>=1 && 0>=1+free_1 ], cost: 1

      3: f300 -> f300 : A'=-1+A, B'=-2+A, C'=0, [ A>=1 && A+B>=1 && B>=1 ], cost: 1



   Found no metering function for rule 1.

   Found no metering function for rule 2.

   Found no metering function for rule 3.

   Removing the simple loops:.



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

   Start location: f2

      0: f2 -> f300 : [], cost: 1

      1: f300 -> f300 : A'=-1+B, B'=-1+B, C'=free, [ A>=1 && B>=1 && free>=1 ], cost: 1

      2: f300 -> f300 : A'=-1+B, B'=-1+B, C'=free_1, [ A>=1 && B>=1 && 0>=1+free_1 ], cost: 1

      3: f300 -> f300 : A'=-1+A, B'=-2+A, C'=0, [ A>=1 && A+B>=1 && B>=1 ], cost: 1



Chained accelerated rules (with incoming rules):

   Start location: f2

      0: f2 -> f300 : [], cost: 1

      7: f2 -> f300 : A'=-1+B, B'=-1+B, C'=free, [ A>=1 && B>=1 && free>=1 ], cost: 2

      8: f2 -> f300 : A'=-1+B, B'=-1+B, C'=free_1, [ A>=1 && B>=1 && 0>=1+free_1 ], cost: 2

      9: f2 -> f300 : A'=-1+A, B'=-2+A, C'=0, [ A>=1 && A+B>=1 && B>=1 ], cost: 2



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

   Start location: f2

     <empty>



### Computing asymptotic complexity ###



Fully simplified ITS problem

   Start location: f2

     <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)