/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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, 857 ms]
(2) BOUNDS(1, INF)


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(0)
Obligation:
Complexity Int TRS consisting of the following rules:
f15(A, B, C, D, E) -> Com_1(f16(A, B, C, D, E)) :|: A >= B + 1
f15(A, B, C, D, E) -> Com_1(f16(A, B, C, D, E)) :|: B >= 1 + A
f0(A, B, C, D, E) -> Com_1(f7(F, B, F, D, E)) :|: 0 >= F
f0(A, B, C, D, E) -> Com_1(f7(F, B, F, D, E)) :|: F >= 101
f0(A, B, C, D, E) -> Com_1(f15(F, 0, F, G, E)) :|: 100 >= F && F >= 1
f16(A, B, C, D, E) -> Com_1(f15(A, B + 1, C, D, E)) :|: TRUE
f23(A, B, C, D, E) -> Com_1(f23(A, B, C, D, E + 1)) :|: B >= E + 1
f23(A, B, C, D, E) -> Com_1(f7(A, B, C, D, E)) :|: E >= B
f15(A, B, C, D, E) -> Com_1(f23(A, A - 1, C, D, 0)) :|: A >= B && A <= B
f16(A, B, C, D, E) -> Com_1(f23(A, B - 1, C, D, 0)) :|: TRUE

The start-symbols are:[f0_5]


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(1) Loat Proof (FINISHED)


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



Initial linear ITS problem

   Start location: f0

      0: f15 -> f16 : [ A>=1+B ], cost: 1

      1: f15 -> f16 : [ B>=1+A ], cost: 1

      8: f15 -> f23 : B'=-1+A, E'=0, [ A==B ], cost: 1

      2: f0 -> f7 : A'=free, C'=free, [ 0>=free ], cost: 1

      3: f0 -> f7 : A'=free_1, C'=free_1, [ free_1>=101 ], cost: 1

      4: f0 -> f15 : A'=free_2, B'=0, C'=free_2, D'=free_3, [ 100>=free_2 && free_2>=1 ], cost: 1

      5: f16 -> f15 : B'=1+B, [], cost: 1

      9: f16 -> f23 : B'=-1+B, E'=0, [], cost: 1

      6: f23 -> f23 : E'=1+E, [ B>=1+E ], cost: 1

      7: f23 -> f7 : [ E>=B ], cost: 1



Checking for constant complexity:

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

      2: f0 -> f7 : A'=free, C'=free, [ 0>=free ], cost: 1



Removed unreachable and leaf rules:

   Start location: f0

      0: f15 -> f16 : [ A>=1+B ], cost: 1

      1: f15 -> f16 : [ B>=1+A ], cost: 1

      8: f15 -> f23 : B'=-1+A, E'=0, [ A==B ], cost: 1

      4: f0 -> f15 : A'=free_2, B'=0, C'=free_2, D'=free_3, [ 100>=free_2 && free_2>=1 ], cost: 1

      5: f16 -> f15 : B'=1+B, [], cost: 1

      9: f16 -> f23 : B'=-1+B, E'=0, [], cost: 1

      6: f23 -> f23 : E'=1+E, [ B>=1+E ], cost: 1



### Simplification by acceleration and chaining ###



Accelerating simple loops of location 3.

   Accelerating the following rules:

      6: f23 -> f23 : E'=1+E, [ B>=1+E ], cost: 1



   Accelerated rule 6 with metering function -E+B, yielding the new rule 10.

   Removing the simple loops: 6.



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

   Start location: f0

      0: f15 -> f16 : [ A>=1+B ], cost: 1

      1: f15 -> f16 : [ B>=1+A ], cost: 1

      8: f15 -> f23 : B'=-1+A, E'=0, [ A==B ], cost: 1

      4: f0 -> f15 : A'=free_2, B'=0, C'=free_2, D'=free_3, [ 100>=free_2 && free_2>=1 ], cost: 1

      5: f16 -> f15 : B'=1+B, [], cost: 1

      9: f16 -> f23 : B'=-1+B, E'=0, [], cost: 1

     10: f23 -> f23 : E'=B, [ B>=1+E ], cost: -E+B



Chained accelerated rules (with incoming rules):

   Start location: f0

      0: f15 -> f16 : [ A>=1+B ], cost: 1

      1: f15 -> f16 : [ B>=1+A ], cost: 1

      8: f15 -> f23 : B'=-1+A, E'=0, [ A==B ], cost: 1

     11: f15 -> f23 : B'=-1+A, E'=-1+A, [ A==B && -1+A>=1 ], cost: A

      4: f0 -> f15 : A'=free_2, B'=0, C'=free_2, D'=free_3, [ 100>=free_2 && free_2>=1 ], cost: 1

      5: f16 -> f15 : B'=1+B, [], cost: 1

      9: f16 -> f23 : B'=-1+B, E'=0, [], cost: 1

     12: f16 -> f23 : B'=-1+B, E'=-1+B, [ -1+B>=1 ], cost: B



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

   Start location: f0

      0: f15 -> f16 : [ A>=1+B ], cost: 1

      1: f15 -> f16 : [ B>=1+A ], cost: 1

     11: f15 -> f23 : B'=-1+A, E'=-1+A, [ A==B && -1+A>=1 ], cost: A

      4: f0 -> f15 : A'=free_2, B'=0, C'=free_2, D'=free_3, [ 100>=free_2 && free_2>=1 ], cost: 1

      5: f16 -> f15 : B'=1+B, [], cost: 1

     12: f16 -> f23 : B'=-1+B, E'=-1+B, [ -1+B>=1 ], cost: B



Eliminated locations (on tree-shaped paths):

   Start location: f0

     11: f15 -> f23 : B'=-1+A, E'=-1+A, [ A==B && -1+A>=1 ], cost: A

     13: f15 -> f15 : B'=1+B, [ A>=1+B ], cost: 2

     14: f15 -> f23 : B'=-1+B, E'=-1+B, [ A>=1+B && -1+B>=1 ], cost: 1+B

     15: f15 -> f15 : B'=1+B, [ B>=1+A ], cost: 2

     16: f15 -> f23 : B'=-1+B, E'=-1+B, [ B>=1+A && -1+B>=1 ], cost: 1+B

      4: f0 -> f15 : A'=free_2, B'=0, C'=free_2, D'=free_3, [ 100>=free_2 && free_2>=1 ], cost: 1



Accelerating simple loops of location 0.

   Accelerating the following rules:

     13: f15 -> f15 : B'=1+B, [ A>=1+B ], cost: 2

     15: f15 -> f15 : B'=1+B, [ B>=1+A ], cost: 2



   Accelerated rule 13 with metering function A-B, yielding the new rule 17.

   Accelerated rule 15 with NONTERM, yielding the new rule 18.

   Removing the simple loops: 13 15.



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

   Start location: f0

     11: f15 -> f23 : B'=-1+A, E'=-1+A, [ A==B && -1+A>=1 ], cost: A

     14: f15 -> f23 : B'=-1+B, E'=-1+B, [ A>=1+B && -1+B>=1 ], cost: 1+B

     16: f15 -> f23 : B'=-1+B, E'=-1+B, [ B>=1+A && -1+B>=1 ], cost: 1+B

     17: f15 -> f15 : B'=A, [ A>=1+B ], cost: 2*A-2*B

     18: f15 -> [6] : [ B>=1+A ], cost: NONTERM

      4: f0 -> f15 : A'=free_2, B'=0, C'=free_2, D'=free_3, [ 100>=free_2 && free_2>=1 ], cost: 1



Chained accelerated rules (with incoming rules):

   Start location: f0

     11: f15 -> f23 : B'=-1+A, E'=-1+A, [ A==B && -1+A>=1 ], cost: A

     14: f15 -> f23 : B'=-1+B, E'=-1+B, [ A>=1+B && -1+B>=1 ], cost: 1+B

     16: f15 -> f23 : B'=-1+B, E'=-1+B, [ B>=1+A && -1+B>=1 ], cost: 1+B

      4: f0 -> f15 : A'=free_2, B'=0, C'=free_2, D'=free_3, [ 100>=free_2 && free_2>=1 ], cost: 1

     19: f0 -> f15 : A'=free_2, B'=free_2, C'=free_2, D'=free_3, [ 100>=free_2 && free_2>=1 ], cost: 1+2*free_2



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

   Start location: f0

     11: f15 -> f23 : B'=-1+A, E'=-1+A, [ A==B && -1+A>=1 ], cost: A

     14: f15 -> f23 : B'=-1+B, E'=-1+B, [ A>=1+B && -1+B>=1 ], cost: 1+B

     16: f15 -> f23 : B'=-1+B, E'=-1+B, [ B>=1+A && -1+B>=1 ], cost: 1+B

      4: f0 -> f15 : A'=free_2, B'=0, C'=free_2, D'=free_3, [ 100>=free_2 && free_2>=1 ], cost: 1

     19: f0 -> f15 : A'=free_2, B'=free_2, C'=free_2, D'=free_3, [ 100>=free_2 && free_2>=1 ], cost: 1+2*free_2



Eliminated locations (on tree-shaped paths):

   Start location: f0

     20: f0 -> f23 : A'=free_2, B'=-1+free_2, C'=free_2, D'=free_3, E'=-1+free_2, [ 100>=free_2 && -1+free_2>=1 ], cost: 1+3*free_2

     21: f0 -> [7] : [ 100>=free_2 && free_2>=1 ], cost: 1+2*free_2



### Computing asymptotic complexity ###



Fully simplified ITS problem

   Start location: f0

     20: f0 -> f23 : A'=free_2, B'=-1+free_2, C'=free_2, D'=free_3, E'=-1+free_2, [ 100>=free_2 && -1+free_2>=1 ], cost: 1+3*free_2

     21: f0 -> [7] : [ 100>=free_2 && free_2>=1 ], cost: 1+2*free_2



Computing asymptotic complexity for rule 20

   Could not solve the limit problem.

   Resulting cost 0 has complexity: Unknown



Computing asymptotic complexity for rule 21

   Could not solve the limit problem.

   Resulting cost 0 has complexity: Unknown



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>=free ]



WORST_CASE(Omega(1),?)


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(2)
BOUNDS(1, INF)