/export/starexec/sandbox2/solver/bin/starexec_run_complexity /export/starexec/sandbox2/benchmark/theBenchmark.koat /export/starexec/sandbox2/output/output_files


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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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

(0) CpxIntTrs
(1) Loat Proof [FINISHED, 634 ms]
(2) BOUNDS(INF, INF)


----------------------------------------

(0)
Obligation:
Complexity Int TRS consisting of the following rules:
f21(A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q) -> Com_1(f29(A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q)) :|: 0 >= A
f41(A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q) -> Com_1(f41(A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q)) :|: TRUE
f43(A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q) -> Com_1(f46(A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q)) :|: TRUE
f29(A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q) -> Com_1(f41(A, R, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q)) :|: A >= 1
f29(A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q) -> Com_1(f41(A, S, C, 0, R, R, R, H, I, J, K, L, M, N, O, P, Q)) :|: 0 >= A && 999 + C >= R
f29(A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q) -> Com_1(f41(1, S, C, 0, R, R, R, H, I, J, K, L, M, N, O, P, Q)) :|: 0 >= A && R >= C + 1000
f21(A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q) -> Com_1(f29(0, B, R, D, E, F, G, 0, R, R, K, L, M, N, O, P, Q)) :|: A >= 1
f0(A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q) -> Com_1(f21(1, B, C, D, E, F, G, H, I, J, K, R, K, N, O, P, Q)) :|: 0 >= K
f0(A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q) -> Com_1(f21(1, B, C, D, E, F, G, H, I, J, S, R, 0, 1, S, S, S)) :|: S >= 1 && K >= 1
f0(A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q) -> Com_1(f41(1, B, C, D, E, F, G, H, I, J, S, R, 0, 1, S, S, S)) :|: 0 >= S && K >= 1

The start-symbols are:[f0_17]


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(1) Loat Proof (FINISHED)


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



Initial linear ITS problem

   Start location: f0

      0: f21 -> f29 : [ 0>=A ], cost: 1

      6: f21 -> f29 : A'=0, C'=free_5, H'=0, Q'=free_5, J'=free_5, [ A>=1 ], cost: 1

      1: f41 -> f41 : [], cost: 1

      2: f43 -> f46 : [], cost: 1

      3: f29 -> f41 : B'=free, [ A>=1 ], cost: 1

      4: f29 -> f41 : B'=free_2, D'=0, E'=free_1, F'=free_1, G'=free_1, [ 0>=A && 999+C>=free_1 ], cost: 1

      5: f29 -> f41 : A'=1, B'=free_4, D'=0, E'=free_3, F'=free_3, G'=free_3, [ 0>=A && free_3>=1000+C ], cost: 1

      7: f0 -> f21 : A'=1, L'=free_6, M'=K, [ 0>=K ], cost: 1

      8: f0 -> f21 : A'=1, K'=free_8, L'=free_7, M'=0, N'=1, O'=free_8, P'=free_8, Q_1'=free_8, [ free_8>=1 && K>=1 ], cost: 1

      9: f0 -> f41 : A'=1, K'=free_10, L'=free_9, M'=0, N'=1, O'=free_10, P'=free_10, Q_1'=free_10, [ 0>=free_10 && K>=1 ], cost: 1



Checking for constant complexity:

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

      7: f0 -> f21 : A'=1, L'=free_6, M'=K, [ 0>=K ], cost: 1



Removed unreachable and leaf rules:

   Start location: f0

      0: f21 -> f29 : [ 0>=A ], cost: 1

      6: f21 -> f29 : A'=0, C'=free_5, H'=0, Q'=free_5, J'=free_5, [ A>=1 ], cost: 1

      1: f41 -> f41 : [], cost: 1

      3: f29 -> f41 : B'=free, [ A>=1 ], cost: 1

      4: f29 -> f41 : B'=free_2, D'=0, E'=free_1, F'=free_1, G'=free_1, [ 0>=A && 999+C>=free_1 ], cost: 1

      5: f29 -> f41 : A'=1, B'=free_4, D'=0, E'=free_3, F'=free_3, G'=free_3, [ 0>=A && free_3>=1000+C ], cost: 1

      7: f0 -> f21 : A'=1, L'=free_6, M'=K, [ 0>=K ], cost: 1

      8: f0 -> f21 : A'=1, K'=free_8, L'=free_7, M'=0, N'=1, O'=free_8, P'=free_8, Q_1'=free_8, [ free_8>=1 && K>=1 ], cost: 1

      9: f0 -> f41 : A'=1, K'=free_10, L'=free_9, M'=0, N'=1, O'=free_10, P'=free_10, Q_1'=free_10, [ 0>=free_10 && K>=1 ], cost: 1



### Simplification by acceleration and chaining ###



Accelerating simple loops of location 1.

   Accelerating the following rules:

      1: f41 -> f41 : [], cost: 1



   Accelerated rule 1 with NONTERM, yielding the new rule 10.

   Removing the simple loops: 1.



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

   Start location: f0

      0: f21 -> f29 : [ 0>=A ], cost: 1

      6: f21 -> f29 : A'=0, C'=free_5, H'=0, Q'=free_5, J'=free_5, [ A>=1 ], cost: 1

     10: f41 -> [6] : [], cost: NONTERM

      3: f29 -> f41 : B'=free, [ A>=1 ], cost: 1

      4: f29 -> f41 : B'=free_2, D'=0, E'=free_1, F'=free_1, G'=free_1, [ 0>=A && 999+C>=free_1 ], cost: 1

      5: f29 -> f41 : A'=1, B'=free_4, D'=0, E'=free_3, F'=free_3, G'=free_3, [ 0>=A && free_3>=1000+C ], cost: 1

      7: f0 -> f21 : A'=1, L'=free_6, M'=K, [ 0>=K ], cost: 1

      8: f0 -> f21 : A'=1, K'=free_8, L'=free_7, M'=0, N'=1, O'=free_8, P'=free_8, Q_1'=free_8, [ free_8>=1 && K>=1 ], cost: 1

      9: f0 -> f41 : A'=1, K'=free_10, L'=free_9, M'=0, N'=1, O'=free_10, P'=free_10, Q_1'=free_10, [ 0>=free_10 && K>=1 ], cost: 1



Chained accelerated rules (with incoming rules):

   Start location: f0

      0: f21 -> f29 : [ 0>=A ], cost: 1

      6: f21 -> f29 : A'=0, C'=free_5, H'=0, Q'=free_5, J'=free_5, [ A>=1 ], cost: 1

      3: f29 -> f41 : B'=free, [ A>=1 ], cost: 1

      4: f29 -> f41 : B'=free_2, D'=0, E'=free_1, F'=free_1, G'=free_1, [ 0>=A && 999+C>=free_1 ], cost: 1

      5: f29 -> f41 : A'=1, B'=free_4, D'=0, E'=free_3, F'=free_3, G'=free_3, [ 0>=A && free_3>=1000+C ], cost: 1

     11: f29 -> [6] : B'=free, [ A>=1 ], cost: NONTERM

     12: f29 -> [6] : B'=free_2, D'=0, E'=free_1, F'=free_1, G'=free_1, [ 0>=A && 999+C>=free_1 ], cost: NONTERM

     13: f29 -> [6] : A'=1, B'=free_4, D'=0, E'=free_3, F'=free_3, G'=free_3, [ 0>=A && free_3>=1000+C ], cost: NONTERM

      7: f0 -> f21 : A'=1, L'=free_6, M'=K, [ 0>=K ], cost: 1

      8: f0 -> f21 : A'=1, K'=free_8, L'=free_7, M'=0, N'=1, O'=free_8, P'=free_8, Q_1'=free_8, [ free_8>=1 && K>=1 ], cost: 1

      9: f0 -> f41 : A'=1, K'=free_10, L'=free_9, M'=0, N'=1, O'=free_10, P'=free_10, Q_1'=free_10, [ 0>=free_10 && K>=1 ], cost: 1

     14: f0 -> [6] : A'=1, K'=free_10, L'=free_9, M'=0, N'=1, O'=free_10, P'=free_10, Q_1'=free_10, [ 0>=free_10 && K>=1 ], cost: NONTERM



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

   Start location: f0

      0: f21 -> f29 : [ 0>=A ], cost: 1

      6: f21 -> f29 : A'=0, C'=free_5, H'=0, Q'=free_5, J'=free_5, [ A>=1 ], cost: 1

     11: f29 -> [6] : B'=free, [ A>=1 ], cost: NONTERM

     12: f29 -> [6] : B'=free_2, D'=0, E'=free_1, F'=free_1, G'=free_1, [ 0>=A && 999+C>=free_1 ], cost: NONTERM

     13: f29 -> [6] : A'=1, B'=free_4, D'=0, E'=free_3, F'=free_3, G'=free_3, [ 0>=A && free_3>=1000+C ], cost: NONTERM

      7: f0 -> f21 : A'=1, L'=free_6, M'=K, [ 0>=K ], cost: 1

      8: f0 -> f21 : A'=1, K'=free_8, L'=free_7, M'=0, N'=1, O'=free_8, P'=free_8, Q_1'=free_8, [ free_8>=1 && K>=1 ], cost: 1

     14: f0 -> [6] : A'=1, K'=free_10, L'=free_9, M'=0, N'=1, O'=free_10, P'=free_10, Q_1'=free_10, [ 0>=free_10 && K>=1 ], cost: NONTERM



Eliminated locations (on tree-shaped paths):

   Start location: f0

     11: f29 -> [6] : B'=free, [ A>=1 ], cost: NONTERM

     12: f29 -> [6] : B'=free_2, D'=0, E'=free_1, F'=free_1, G'=free_1, [ 0>=A && 999+C>=free_1 ], cost: NONTERM

     13: f29 -> [6] : A'=1, B'=free_4, D'=0, E'=free_3, F'=free_3, G'=free_3, [ 0>=A && free_3>=1000+C ], cost: NONTERM

     14: f0 -> [6] : A'=1, K'=free_10, L'=free_9, M'=0, N'=1, O'=free_10, P'=free_10, Q_1'=free_10, [ 0>=free_10 && K>=1 ], cost: NONTERM

     15: f0 -> f29 : A'=0, C'=free_5, H'=0, Q'=free_5, J'=free_5, L'=free_6, M'=K, [ 0>=K ], cost: 2

     16: f0 -> f29 : A'=0, C'=free_5, H'=0, Q'=free_5, J'=free_5, K'=free_8, L'=free_7, M'=0, N'=1, O'=free_8, P'=free_8, Q_1'=free_8, [ free_8>=1 && K>=1 ], cost: 2



Eliminated locations (on tree-shaped paths):

   Start location: f0

     14: f0 -> [6] : A'=1, K'=free_10, L'=free_9, M'=0, N'=1, O'=free_10, P'=free_10, Q_1'=free_10, [ 0>=free_10 && K>=1 ], cost: NONTERM

     17: f0 -> [6] : A'=0, B'=free_2, C'=free_5, D'=0, E'=free_1, F'=free_1, G'=free_1, H'=0, Q'=free_5, J'=free_5, L'=free_6, M'=K, [ 0>=K && 999+free_5>=free_1 ], cost: NONTERM

     18: f0 -> [6] : A'=1, B'=free_4, C'=free_5, D'=0, E'=free_3, F'=free_3, G'=free_3, H'=0, Q'=free_5, J'=free_5, L'=free_6, M'=K, [ 0>=K && free_3>=1000+free_5 ], cost: NONTERM

     19: f0 -> [6] : A'=0, B'=free_2, C'=free_5, D'=0, E'=free_1, F'=free_1, G'=free_1, H'=0, Q'=free_5, J'=free_5, K'=free_8, L'=free_7, M'=0, N'=1, O'=free_8, P'=free_8, Q_1'=free_8, [ free_8>=1 && K>=1 && 999+free_5>=free_1 ], cost: NONTERM

     20: f0 -> [6] : A'=1, B'=free_4, C'=free_5, D'=0, E'=free_3, F'=free_3, G'=free_3, H'=0, Q'=free_5, J'=free_5, K'=free_8, L'=free_7, M'=0, N'=1, O'=free_8, P'=free_8, Q_1'=free_8, [ free_8>=1 && K>=1 && free_3>=1000+free_5 ], cost: NONTERM



### Computing asymptotic complexity ###



Fully simplified ITS problem

   Start location: f0

     14: f0 -> [6] : A'=1, K'=free_10, L'=free_9, M'=0, N'=1, O'=free_10, P'=free_10, Q_1'=free_10, [ 0>=free_10 && K>=1 ], cost: NONTERM

     17: f0 -> [6] : A'=0, B'=free_2, C'=free_5, D'=0, E'=free_1, F'=free_1, G'=free_1, H'=0, Q'=free_5, J'=free_5, L'=free_6, M'=K, [ 0>=K && 999+free_5>=free_1 ], cost: NONTERM

     18: f0 -> [6] : A'=1, B'=free_4, C'=free_5, D'=0, E'=free_3, F'=free_3, G'=free_3, H'=0, Q'=free_5, J'=free_5, L'=free_6, M'=K, [ 0>=K && free_3>=1000+free_5 ], cost: NONTERM

     19: f0 -> [6] : A'=0, B'=free_2, C'=free_5, D'=0, E'=free_1, F'=free_1, G'=free_1, H'=0, Q'=free_5, J'=free_5, K'=free_8, L'=free_7, M'=0, N'=1, O'=free_8, P'=free_8, Q_1'=free_8, [ free_8>=1 && K>=1 && 999+free_5>=free_1 ], cost: NONTERM

     20: f0 -> [6] : A'=1, B'=free_4, C'=free_5, D'=0, E'=free_3, F'=free_3, G'=free_3, H'=0, Q'=free_5, J'=free_5, K'=free_8, L'=free_7, M'=0, N'=1, O'=free_8, P'=free_8, Q_1'=free_8, [ free_8>=1 && K>=1 && free_3>=1000+free_5 ], cost: NONTERM



Computing asymptotic complexity for rule 14

   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>=free_10 && K>=1 ]



NO


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(2)
BOUNDS(INF, INF)