/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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, 547 ms]
(2) BOUNDS(INF, INF)


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(0)
Obligation:
Complexity Int TRS consisting of the following rules:
f2(A, B) -> Com_1(f2(-(1) + D, B)) :|: 0 >= C && 0 >= A + 1
f2(A, B) -> Com_1(f2(1 + D, B)) :|: C >= 2 && 0 >= A + 1
f2(A, B) -> Com_1(f2(-(1) + D, B)) :|: 0 >= 2 + C && A >= 1
f2(A, B) -> Com_1(f2(1 + D, B)) :|: C >= 0 && A >= 1
f3(A, B) -> Com_1(f2(A, B)) :|: TRUE
f2(A, B) -> Com_1(f300(0, D)) :|: 0 >= A + 1
f2(A, B) -> Com_1(f300(0, D)) :|: A >= 1
f2(A, B) -> Com_1(f300(A, D)) :|: A >= 0 && A <= 0

The start-symbols are:[f3_2]


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(1) Loat Proof (FINISHED)


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



Initial linear ITS problem

   Start location: f3

      0: f2 -> f2 : A'=-1+free_1, [ 0>=free && 0>=1+A ], cost: 1

      1: f2 -> f2 : A'=1+free_3, [ free_2>=2 && 0>=1+A ], cost: 1

      2: f2 -> f2 : A'=-1+free_5, [ 0>=2+free_4 && A>=1 ], cost: 1

      3: f2 -> f2 : A'=1+free_7, [ free_6>=0 && A>=1 ], cost: 1

      5: f2 -> f300 : A'=0, B'=free_8, [ 0>=1+A ], cost: 1

      6: f2 -> f300 : A'=0, B'=free_9, [ A>=1 ], cost: 1

      7: f2 -> f300 : B'=free_10, [ A==0 ], cost: 1

      4: f3 -> f2 : [], cost: 1



Checking for constant complexity:

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

      4: f3 -> f2 : [], cost: 1



Removed unreachable and leaf rules:

   Start location: f3

      0: f2 -> f2 : A'=-1+free_1, [ 0>=free && 0>=1+A ], cost: 1

      1: f2 -> f2 : A'=1+free_3, [ free_2>=2 && 0>=1+A ], cost: 1

      2: f2 -> f2 : A'=-1+free_5, [ 0>=2+free_4 && A>=1 ], cost: 1

      3: f2 -> f2 : A'=1+free_7, [ free_6>=0 && A>=1 ], cost: 1

      4: f3 -> f2 : [], cost: 1



Simplified all rules, resulting in:

   Start location: f3

      0: f2 -> f2 : A'=-1+free_1, [ 0>=1+A ], cost: 1

      1: f2 -> f2 : A'=1+free_3, [ 0>=1+A ], cost: 1

      2: f2 -> f2 : A'=-1+free_5, [ A>=1 ], cost: 1

      3: f2 -> f2 : A'=1+free_7, [ A>=1 ], cost: 1

      4: f3 -> f2 : [], cost: 1



### Simplification by acceleration and chaining ###



Accelerating simple loops of location 0.

   Accelerating the following rules:

      0: f2 -> f2 : A'=-1+free_1, [ 0>=1+A ], cost: 1

      1: f2 -> f2 : A'=1+free_3, [ 0>=1+A ], cost: 1

      2: f2 -> f2 : A'=-1+free_5, [ A>=1 ], cost: 1

      3: f2 -> f2 : A'=1+free_7, [ A>=1 ], cost: 1



   Accelerated rule 0 with NONTERM (after strengthening guard), yielding the new rule 8.

   Accelerated rule 1 with NONTERM (after strengthening guard), yielding the new rule 9.

   Accelerated rule 2 with NONTERM (after strengthening guard), yielding the new rule 10.

   Accelerated rule 3 with NONTERM (after strengthening guard), yielding the new rule 11.

   Removing the simple loops:.



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

   Start location: f3

      0: f2 -> f2 : A'=-1+free_1, [ 0>=1+A ], cost: 1

      1: f2 -> f2 : A'=1+free_3, [ 0>=1+A ], cost: 1

      2: f2 -> f2 : A'=-1+free_5, [ A>=1 ], cost: 1

      3: f2 -> f2 : A'=1+free_7, [ A>=1 ], cost: 1

      8: f2 -> [3] : [ 0>=1+A && 0>=free_1 ], cost: NONTERM

      9: f2 -> [3] : [ 0>=1+A && 0>=2+free_3 ], cost: NONTERM

     10: f2 -> [3] : [ A>=1 && -1+free_5>=1 ], cost: NONTERM

     11: f2 -> [3] : [ A>=1 && 1+free_7>=1 ], cost: NONTERM

      4: f3 -> f2 : [], cost: 1



Chained accelerated rules (with incoming rules):

   Start location: f3

      4: f3 -> f2 : [], cost: 1

     12: f3 -> f2 : A'=-1+free_1, [ 0>=1+A ], cost: 2

     13: f3 -> f2 : A'=1+free_3, [ 0>=1+A ], cost: 2

     14: f3 -> f2 : A'=-1+free_5, [ A>=1 ], cost: 2

     15: f3 -> f2 : A'=1+free_7, [ A>=1 ], cost: 2

     16: f3 -> [3] : [ 0>=1+A ], cost: NONTERM

     17: f3 -> [3] : [ 0>=1+A ], cost: NONTERM

     18: f3 -> [3] : [ A>=1 ], cost: NONTERM

     19: f3 -> [3] : [ A>=1 ], cost: NONTERM



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

   Start location: f3

     16: f3 -> [3] : [ 0>=1+A ], cost: NONTERM

     17: f3 -> [3] : [ 0>=1+A ], cost: NONTERM

     18: f3 -> [3] : [ A>=1 ], cost: NONTERM

     19: f3 -> [3] : [ A>=1 ], cost: NONTERM



### Computing asymptotic complexity ###



Fully simplified ITS problem

   Start location: f3

     17: f3 -> [3] : [ 0>=1+A ], cost: NONTERM

     19: f3 -> [3] : [ A>=1 ], cost: NONTERM



Computing asymptotic complexity for rule 17

   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+A ]



NO


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(2)
BOUNDS(INF, INF)