/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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, 938 ms]
(2) BOUNDS(INF, INF)


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(0)
Obligation:
Complexity Int TRS consisting of the following rules:
f0(A, B, C, D, E) -> Com_1(f3(0, 0, C, D, E)) :|: TRUE
f3(A, B, C, D, E) -> Com_1(f3(A, B, C - 1, F, E)) :|: C >= 1 && F >= 1
f3(A, B, C, D, E) -> Com_1(f3(A, B, C - 2, F, E)) :|: C >= 1 && 0 >= F
f3(A, B, C, D, E) -> Com_1(f6(A, B, C, D, F)) :|: 0 >= C
f6(A, B, C, D, E) -> Com_1(f6(1, B, C, D, F)) :|: E >= 1
f6(A, B, C, D, E) -> Com_1(f6(0, B, C, D, F)) :|: 0 >= E

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: f0 -> f3 : A'=0, B'=0, [], cost: 1

      1: f3 -> f3 : C'=-1+C, D'=free, [ C>=1 && free>=1 ], cost: 1

      2: f3 -> f3 : C'=-2+C, D'=free_1, [ C>=1 && 0>=free_1 ], cost: 1

      3: f3 -> f6 : E'=free_2, [ 0>=C ], cost: 1

      4: f6 -> f6 : A'=1, E'=free_3, [ E>=1 ], cost: 1

      5: f6 -> f6 : A'=0, E'=free_4, [ 0>=E ], cost: 1



Checking for constant complexity:

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

      0: f0 -> f3 : A'=0, B'=0, [], cost: 1



### Simplification by acceleration and chaining ###



Accelerating simple loops of location 1.

   Accelerating the following rules:

      1: f3 -> f3 : C'=-1+C, D'=free, [ C>=1 && free>=1 ], cost: 1

      2: f3 -> f3 : C'=-2+C, D'=free_1, [ C>=1 && 0>=free_1 ], cost: 1



   Accelerated rule 1 with metering function C, yielding the new rule 6.

   Accelerated rule 2 with metering function meter (where 2*meter==C), yielding the new rule 7.

      During metering: Instantiating temporary variables by {meter==1}

   Removing the simple loops: 1 2.



Accelerating simple loops of location 2.

   Accelerating the following rules:

      4: f6 -> f6 : A'=1, E'=free_3, [ E>=1 ], cost: 1

      5: f6 -> f6 : A'=0, E'=free_4, [ 0>=E ], cost: 1



   Accelerated rule 4 with NONTERM (after strengthening guard), yielding the new rule 8.

   Accelerated rule 5 with NONTERM (after strengthening guard), yielding the new rule 9.

   Removing the simple loops:.



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

   Start location: f0

      0: f0 -> f3 : A'=0, B'=0, [], cost: 1

      3: f3 -> f6 : E'=free_2, [ 0>=C ], cost: 1

      6: f3 -> f3 : C'=0, D'=free, [ C>=1 && free>=1 ], cost: C

      7: f3 -> f3 : C'=C-2*meter, D'=free_1, [ C>=1 && 0>=free_1 && 2*meter==C && meter>=1 ], cost: meter

      4: f6 -> f6 : A'=1, E'=free_3, [ E>=1 ], cost: 1

      5: f6 -> f6 : A'=0, E'=free_4, [ 0>=E ], cost: 1

      8: f6 -> [4] : [ E>=1 && free_3>=1 ], cost: NONTERM

      9: f6 -> [4] : [ 0>=E && 0>=free_4 ], cost: NONTERM



Chained accelerated rules (with incoming rules):

   Start location: f0

      0: f0 -> f3 : A'=0, B'=0, [], cost: 1

     10: f0 -> f3 : A'=0, B'=0, C'=0, D'=free, [ C>=1 && free>=1 ], cost: 1+C

     11: f0 -> f3 : A'=0, B'=0, C'=C-2*meter, D'=free_1, [ C>=1 && 0>=free_1 && 2*meter==C && meter>=1 ], cost: 1+meter

      3: f3 -> f6 : E'=free_2, [ 0>=C ], cost: 1

     12: f3 -> f6 : A'=1, E'=free_3, [ 0>=C ], cost: 2

     13: f3 -> f6 : A'=0, E'=free_4, [ 0>=C ], cost: 2

     14: f3 -> [4] : E'=free_2, [ 0>=C && free_2>=1 ], cost: NONTERM

     15: f3 -> [4] : E'=free_2, [ 0>=C && 0>=free_2 ], cost: NONTERM



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

   Start location: f0

      0: f0 -> f3 : A'=0, B'=0, [], cost: 1

     10: f0 -> f3 : A'=0, B'=0, C'=0, D'=free, [ C>=1 && free>=1 ], cost: 1+C

     11: f0 -> f3 : A'=0, B'=0, C'=C-2*meter, D'=free_1, [ C>=1 && 0>=free_1 && 2*meter==C && meter>=1 ], cost: 1+meter

     14: f3 -> [4] : E'=free_2, [ 0>=C && free_2>=1 ], cost: NONTERM

     15: f3 -> [4] : E'=free_2, [ 0>=C && 0>=free_2 ], cost: NONTERM



Eliminated locations (on tree-shaped paths):

   Start location: f0

     16: f0 -> [4] : A'=0, B'=0, E'=free_2, [ 0>=C && free_2>=1 ], cost: NONTERM

     17: f0 -> [4] : A'=0, B'=0, E'=free_2, [ 0>=C && 0>=free_2 ], cost: NONTERM

     18: f0 -> [4] : A'=0, B'=0, C'=0, D'=free, E'=free_2, [ C>=1 && free>=1 && free_2>=1 ], cost: NONTERM

     19: f0 -> [4] : A'=0, B'=0, C'=0, D'=free, E'=free_2, [ C>=1 && free>=1 && 0>=free_2 ], cost: NONTERM

     20: f0 -> [4] : A'=0, B'=0, C'=C-2*meter, D'=free_1, E'=free_2, [ C>=1 && 0>=free_1 && 2*meter==C && meter>=1 && free_2>=1 ], cost: NONTERM

     21: f0 -> [4] : A'=0, B'=0, C'=C-2*meter, D'=free_1, E'=free_2, [ C>=1 && 0>=free_1 && 2*meter==C && meter>=1 && 0>=free_2 ], cost: NONTERM



Applied pruning (of leafs and parallel rules):

   Start location: f0

     16: f0 -> [4] : A'=0, B'=0, E'=free_2, [ 0>=C && free_2>=1 ], cost: NONTERM

     17: f0 -> [4] : A'=0, B'=0, E'=free_2, [ 0>=C && 0>=free_2 ], cost: NONTERM

     18: f0 -> [4] : A'=0, B'=0, C'=0, D'=free, E'=free_2, [ C>=1 && free>=1 && free_2>=1 ], cost: NONTERM

     19: f0 -> [4] : A'=0, B'=0, C'=0, D'=free, E'=free_2, [ C>=1 && free>=1 && 0>=free_2 ], cost: NONTERM

     21: f0 -> [4] : A'=0, B'=0, C'=C-2*meter, D'=free_1, E'=free_2, [ C>=1 && 0>=free_1 && 2*meter==C && meter>=1 && 0>=free_2 ], cost: NONTERM



### Computing asymptotic complexity ###



Fully simplified ITS problem

   Start location: f0

     16: f0 -> [4] : A'=0, B'=0, E'=free_2, [ 0>=C && free_2>=1 ], cost: NONTERM

     17: f0 -> [4] : A'=0, B'=0, E'=free_2, [ 0>=C && 0>=free_2 ], cost: NONTERM

     18: f0 -> [4] : A'=0, B'=0, C'=0, D'=free, E'=free_2, [ C>=1 && free>=1 && free_2>=1 ], cost: NONTERM

     19: f0 -> [4] : A'=0, B'=0, C'=0, D'=free, E'=free_2, [ C>=1 && free>=1 && 0>=free_2 ], cost: NONTERM

     21: f0 -> [4] : A'=0, B'=0, C'=C-2*meter, D'=free_1, E'=free_2, [ C>=1 && 0>=free_1 && 2*meter==C && meter>=1 && 0>=free_2 ], cost: NONTERM



Computing asymptotic complexity for rule 16

   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>=C && free_2>=1 ]



NO


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(2)
BOUNDS(INF, INF)