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


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WORST_CASE(Omega(n^1), ?)
proof of /export/starexec/sandbox2/benchmark/theBenchmark.koat
# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty


The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(n^1, INF).

(0) CpxIntTrs
(1) Loat Proof [FINISHED, 975 ms]
(2) BOUNDS(n^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



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



   Accelerated rule 1 with backward acceleration, yielding the new rule 7.

   Accelerated rule 1 with backward acceleration, yielding the new rule 8.

   Accelerated rule 2 with backward acceleration, yielding the new rule 9.

   Accelerated rule 2 with backward acceleration, yielding the new rule 10.

   Found no metering function for rule 3.

   Removing the simple loops: 1 2.

   Also removing duplicate rules:.



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

   Start location: f2

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

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

      8: f300 -> f300 : A'=0, B'=0, C'=free, [ A>=1 && B>=1 && free>=1 ], cost: B

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



Chained accelerated rules (with incoming rules):

   Start location: f2

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

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

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

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



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

   Start location: f2

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

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



### Computing asymptotic complexity ###



Fully simplified ITS problem

   Start location: f2

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

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



Computing asymptotic complexity for rule 12

   Solved the limit problem by the following transformations:

      Created initial limit problem:

      free (+/+!), 1+B (+), A (+/+!), B (+/+!) [not solved]



      removing all constraints (solved by SMT)

      resulting limit problem: [solved]



      applying transformation rule (C) using substitution {free==n,A==n,B==n}

      resulting limit problem:

      [solved]



   Solution:

      free / n

      A / n

      B / n

   Resulting cost 1+n has complexity: Poly(n^1)



Found new complexity Poly(n^1).



Obtained the following overall complexity (w.r.t. the length of the input n):

   Complexity:  Poly(n^1)

   Cpx degree:  1

   Solved cost: 1+n

   Rule cost:   1+B

   Rule guard:  [ A>=1 && B>=1 && free>=1 ]



WORST_CASE(Omega(n^1),?)


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(2)
BOUNDS(n^1, INF)