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


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WORST_CASE(NON_POLY, ?)
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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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(INF, INF).

(0) CpxIntTrs
(1) Loat Proof [FINISHED, 432 ms]
(2) BOUNDS(INF, INF)


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(0)
Obligation:
Complexity Int TRS consisting of the following rules:
f7(A, B) -> Com_1(f7(C, B)) :|: 0 >= A + 1
f7(A, B) -> Com_1(f7(C, B)) :|: A >= 1
f13(A, B) -> Com_1(f13(A, B)) :|: TRUE
f15(A, B) -> Com_1(f17(A, B)) :|: TRUE
f7(A, B) -> Com_1(f13(0, 1)) :|: A >= 0 && A <= 0
f0(A, B) -> Com_1(f7(C, 1)) :|: TRUE

The start-symbols are:[f0_2]


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(1) Loat Proof (FINISHED)


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



Initial linear ITS problem

   Start location: f0

      0: f7 -> f7 : A'=free, [ 0>=1+A ], cost: 1

      1: f7 -> f7 : A'=free_1, [ A>=1 ], cost: 1

      4: f7 -> f13 : A'=0, B'=1, [ A==0 ], cost: 1

      2: f13 -> f13 : [], cost: 1

      3: f15 -> f17 : [], cost: 1

      5: f0 -> f7 : A'=free_2, B'=1, [], cost: 1



Checking for constant complexity:

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

      5: f0 -> f7 : A'=free_2, B'=1, [], cost: 1



Removed unreachable and leaf rules:

   Start location: f0

      0: f7 -> f7 : A'=free, [ 0>=1+A ], cost: 1

      1: f7 -> f7 : A'=free_1, [ A>=1 ], cost: 1

      4: f7 -> f13 : A'=0, B'=1, [ A==0 ], cost: 1

      2: f13 -> f13 : [], cost: 1

      5: f0 -> f7 : A'=free_2, B'=1, [], cost: 1



### Simplification by acceleration and chaining ###



Accelerating simple loops of location 0.

   Accelerating the following rules:

      0: f7 -> f7 : A'=free, [ 0>=1+A ], cost: 1

      1: f7 -> f7 : A'=free_1, [ A>=1 ], cost: 1



   Accelerated rule 0 with NONTERM (after strengthening guard), yielding the new rule 6.

   Accelerated rule 1 with NONTERM (after strengthening guard), yielding the new rule 7.

   Removing the simple loops:.



Accelerating simple loops of location 1.

   Accelerating the following rules:

      2: f13 -> f13 : [], cost: 1



   Accelerated rule 2 with NONTERM, yielding the new rule 8.

   Removing the simple loops: 2.



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

   Start location: f0

      0: f7 -> f7 : A'=free, [ 0>=1+A ], cost: 1

      1: f7 -> f7 : A'=free_1, [ A>=1 ], cost: 1

      4: f7 -> f13 : A'=0, B'=1, [ A==0 ], cost: 1

      6: f7 -> [5] : [ 0>=1+A && 0>=1+free ], cost: NONTERM

      7: f7 -> [5] : [ A>=1 && free_1>=1 ], cost: NONTERM

      8: f13 -> [6] : [], cost: NONTERM

      5: f0 -> f7 : A'=free_2, B'=1, [], cost: 1



Chained accelerated rules (with incoming rules):

   Start location: f0

      4: f7 -> f13 : A'=0, B'=1, [ A==0 ], cost: 1

     13: f7 -> [6] : A'=0, B'=1, [ A==0 ], cost: NONTERM

      5: f0 -> f7 : A'=free_2, B'=1, [], cost: 1

      9: f0 -> f7 : A'=free, B'=1, [], cost: 2

     10: f0 -> f7 : A'=free_1, B'=1, [], cost: 2

     11: f0 -> [5] : A'=free_2, B'=1, [ 0>=1+free_2 ], cost: NONTERM

     12: f0 -> [5] : A'=free_2, B'=1, [ free_2>=1 ], cost: NONTERM



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

   Start location: f0

     13: f7 -> [6] : A'=0, B'=1, [ A==0 ], cost: NONTERM

      5: f0 -> f7 : A'=free_2, B'=1, [], cost: 1

      9: f0 -> f7 : A'=free, B'=1, [], cost: 2

     10: f0 -> f7 : A'=free_1, B'=1, [], cost: 2

     11: f0 -> [5] : A'=free_2, B'=1, [ 0>=1+free_2 ], cost: NONTERM

     12: f0 -> [5] : A'=free_2, B'=1, [ free_2>=1 ], cost: NONTERM



Eliminated locations (on tree-shaped paths):

   Start location: f0

     11: f0 -> [5] : A'=free_2, B'=1, [ 0>=1+free_2 ], cost: NONTERM

     12: f0 -> [5] : A'=free_2, B'=1, [ free_2>=1 ], cost: NONTERM

     14: f0 -> [6] : A'=0, B'=1, [ free_2==0 ], cost: NONTERM

     15: f0 -> [6] : A'=0, B'=1, [ free==0 ], cost: NONTERM

     16: f0 -> [6] : A'=0, B'=1, [ free_1==0 ], cost: NONTERM



### Computing asymptotic complexity ###



Fully simplified ITS problem

   Start location: f0

     11: f0 -> [5] : A'=free_2, B'=1, [ 0>=1+free_2 ], cost: NONTERM

     12: f0 -> [5] : A'=free_2, B'=1, [ free_2>=1 ], cost: NONTERM

     14: f0 -> [6] : A'=0, B'=1, [ free_2==0 ], cost: NONTERM

     15: f0 -> [6] : A'=0, B'=1, [ free==0 ], cost: NONTERM

     16: f0 -> [6] : A'=0, B'=1, [ free_1==0 ], cost: NONTERM



Computing asymptotic complexity for rule 11

   Guard is satisfiable, yielding nontermination

   Resulting cost NONTERM has complexity: Nonterm



Found new complexity Nonterm.



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

   Complexity:  Nonterm

   Cpx degree:  Nonterm

   Solved cost: NONTERM

   Rule cost:   NONTERM

   Rule guard:  [ 0>=1+free_2 ]



NO


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(2)
BOUNDS(INF, INF)