/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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, 792 ms]
(2) BOUNDS(INF, INF)


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(0)
Obligation:
Complexity Int TRS consisting of the following rules:
f2(A, B) -> Com_1(f1(C, D)) :|: 0 >= C + 1 && A >= 14 && A <= 14
f2(A, B) -> Com_1(f2(C, B)) :|: C >= 0 && A >= 14 && A <= 14
f2(A, B) -> Com_1(f1(-(1) + A, C)) :|: A >= 15 && 0 >= A
f2(A, B) -> Com_1(f1(-(1) + A, C)) :|: 13 >= A && 0 >= A
f2(A, B) -> Com_1(f2(-(1) + A, B)) :|: A >= 15 && A >= 1
f2(A, B) -> Com_1(f2(-(1) + A, B)) :|: 13 >= A && A >= 1
f300(A, B) -> Com_1(f2(A, B)) :|: TRUE

The start-symbols are:[f300_2]


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(1) Loat Proof (FINISHED)


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



Initial linear ITS problem

   Start location: f300

      0: f2 -> f1 : A'=free, B'=free_1, [ 0>=1+free && A==14 ], cost: 1

      1: f2 -> f2 : A'=free_2, [ free_2>=0 && A==14 ], cost: 1

      2: f2 -> f1 : A'=-1+A, B'=free_3, [ A>=15 && 0>=A ], cost: 1

      3: f2 -> f1 : A'=-1+A, B'=free_4, [ 13>=A && 0>=A ], cost: 1

      4: f2 -> f2 : A'=-1+A, [ A>=15 && A>=1 ], cost: 1

      5: f2 -> f2 : A'=-1+A, [ 13>=A && A>=1 ], cost: 1

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



Checking for constant complexity:

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

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



Removed unreachable and leaf rules:

   Start location: f300

      1: f2 -> f2 : A'=free_2, [ free_2>=0 && A==14 ], cost: 1

      4: f2 -> f2 : A'=-1+A, [ A>=15 && A>=1 ], cost: 1

      5: f2 -> f2 : A'=-1+A, [ 13>=A && A>=1 ], cost: 1

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



Simplified all rules, resulting in:

   Start location: f300

      1: f2 -> f2 : A'=free_2, [ free_2>=0 && A==14 ], cost: 1

      4: f2 -> f2 : A'=-1+A, [ A>=15 ], cost: 1

      5: f2 -> f2 : A'=-1+A, [ 13>=A && A>=1 ], cost: 1

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



### Simplification by acceleration and chaining ###



Accelerating simple loops of location 0.

   Accelerating the following rules:

      1: f2 -> f2 : A'=free_2, [ free_2>=0 && A==14 ], cost: 1

      4: f2 -> f2 : A'=-1+A, [ A>=15 ], cost: 1

      5: f2 -> f2 : A'=-1+A, [ 13>=A && A>=1 ], cost: 1



      During metering: Instantiating temporary variables by {free_2==0}

   Accelerated rule 1 with metering function meter (where 14*meter==-13+A), yielding the new rule 7.

   Accelerated rule 4 with metering function -14+A, yielding the new rule 8.

   Accelerated rule 5 with metering function A, yielding the new rule 9.

   Nested simple loops 1 (outer loop) and 8 (inner loop) with NONTERM, resulting in the new rules: 10, 11.

   Nested simple loops 1 (outer loop) and 9 (inner loop) with metering function meter_1 (where 14*meter_1==-13+A), resulting in the new rules: 12.

   Removing the simple loops: 1 4 5.



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

   Start location: f300

      7: f2 -> f2 : A'=0, [ A==14 && 14*meter==-13+A && meter>=1 ], cost: meter

      8: f2 -> f2 : A'=14, [ A>=15 ], cost: -14+A

      9: f2 -> f2 : A'=0, [ 13>=A && A>=1 ], cost: A

     10: f2 -> [3] : [ A==14 && free_2>=15 ], cost: NONTERM

     11: f2 -> [3] : A'=14, [ A>=15 && free_2>=15 ], cost: NONTERM

     12: f2 -> f2 : A'=0, [ A==14 && 13>=free_2 && free_2>=1 && 14*meter_1==-13+A && meter_1>=1 ], cost: free_2*meter_1+meter_1

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



Chained accelerated rules (with incoming rules):

   Start location: f300

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

     13: f300 -> f2 : A'=14, [ A>=15 ], cost: -13+A

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

     15: f300 -> [3] : [ A==14 ], cost: NONTERM

     16: f300 -> [3] : A'=14, [ A>=15 ], cost: NONTERM



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

   Start location: f300

     13: f300 -> f2 : A'=14, [ A>=15 ], cost: -13+A

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

     15: f300 -> [3] : [ A==14 ], cost: NONTERM

     16: f300 -> [3] : A'=14, [ A>=15 ], cost: NONTERM



### Computing asymptotic complexity ###



Fully simplified ITS problem

   Start location: f300

     13: f300 -> f2 : A'=14, [ A>=15 ], cost: -13+A

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

     15: f300 -> [3] : [ A==14 ], cost: NONTERM

     16: f300 -> [3] : A'=14, [ A>=15 ], cost: NONTERM



Computing asymptotic complexity for rule 13

   Solved the limit problem by the following transformations:

      Created initial limit problem:

      -14+A (+/+!), -13+A (+) [not solved]



      removing all constraints (solved by SMT)

      resulting limit problem: [solved]



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

      resulting limit problem:

      [solved]



   Solution:

      A / n

   Resulting cost -13+n has complexity: Poly(n^1)



Found new complexity Poly(n^1).



Computing asymptotic complexity for rule 15

   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:  [ A==14 ]



NO


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(2)
BOUNDS(INF, INF)