/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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, 609 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) -> Com_1(f29(A, B, C, D, E, F, G, H, I, J, K, L, M, N, O)) :|: 0 >= A
f41(A, B, C, D, E, F, G, H, I, J, K, L, M, N, O) -> Com_1(f41(A, B, C, D, E, F, G, H, I, J, K, L, M, N, O)) :|: TRUE
f43(A, B, C, D, E, F, G, H, I, J, K, L, M, N, O) -> Com_1(f46(A, B, C, D, E, F, G, H, I, J, K, L, M, N, O)) :|: TRUE
f29(A, B, C, D, E, F, G, H, I, J, K, L, M, N, O) -> Com_1(f41(A, B, C, D, E, F, G, H, I, J, K, L, M, N, O)) :|: A >= 1
f29(A, B, C, D, E, F, G, H, I, J, K, L, M, N, O) -> Com_1(f41(A, B, P, 0, P, P, G, H, I, J, K, L, M, N, O)) :|: 0 >= A && 999 + B >= P
f29(A, B, C, D, E, F, G, H, I, J, K, L, M, N, O) -> Com_1(f41(1, B, P, 0, P, P, G, H, I, J, K, L, M, N, O)) :|: 0 >= A && P >= B + 1000
f21(A, B, C, D, E, F, G, H, I, J, K, L, M, N, O) -> Com_1(f29(0, P, P, D, E, F, 0, P, I, J, K, L, M, N, O)) :|: A >= 1
f0(A, B, C, D, E, F, G, H, I, J, K, L, M, N, O) -> Com_1(f21(1, B, C, D, E, F, G, H, P, P, K, L, M, N, O)) :|: 0 >= P
f0(A, B, C, D, E, F, G, H, I, J, K, L, M, N, O) -> Com_1(f21(1, B, C, D, E, F, G, H, P, 0, 1, Q, Q, Q, Q)) :|: P >= 1 && Q >= 1
f0(A, B, C, D, E, F, G, H, I, J, K, L, M, N, O) -> Com_1(f41(1, B, C, D, E, F, G, H, P, 0, 1, Q, Q, Q, Q)) :|: P >= 1 && 0 >= Q

The start-symbols are:[f0_15]


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

(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, B'=free_2, C'=free_2, G'=0, H'=free_2, [ A>=1 ], cost: 1

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

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

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

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

      5: f29 -> f41 : A'=1, C'=free_1, D'=0, E'=free_1, F'=free_1, [ 0>=A && free_1>=1000+B ], cost: 1

      7: f0 -> f21 : A'=1, Q'=free_3, J'=free_3, [ 0>=free_3 ], cost: 1

      8: f0 -> f21 : A'=1, Q'=free_4, J'=0, K'=1, L'=free_5, M'=free_5, N'=free_5, O'=free_5, [ free_4>=1 && free_5>=1 ], cost: 1

      9: f0 -> f41 : A'=1, Q'=free_6, J'=0, K'=1, L'=free_7, M'=free_7, N'=free_7, O'=free_7, [ free_6>=1 && 0>=free_7 ], cost: 1



Checking for constant complexity:

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

      7: f0 -> f21 : A'=1, Q'=free_3, J'=free_3, [ 0>=free_3 ], cost: 1



Removed unreachable and leaf rules:

   Start location: f0

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

      6: f21 -> f29 : A'=0, B'=free_2, C'=free_2, G'=0, H'=free_2, [ A>=1 ], cost: 1

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

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

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

      5: f29 -> f41 : A'=1, C'=free_1, D'=0, E'=free_1, F'=free_1, [ 0>=A && free_1>=1000+B ], cost: 1

      7: f0 -> f21 : A'=1, Q'=free_3, J'=free_3, [ 0>=free_3 ], cost: 1

      8: f0 -> f21 : A'=1, Q'=free_4, J'=0, K'=1, L'=free_5, M'=free_5, N'=free_5, O'=free_5, [ free_4>=1 && free_5>=1 ], cost: 1

      9: f0 -> f41 : A'=1, Q'=free_6, J'=0, K'=1, L'=free_7, M'=free_7, N'=free_7, O'=free_7, [ free_6>=1 && 0>=free_7 ], 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, B'=free_2, C'=free_2, G'=0, H'=free_2, [ A>=1 ], cost: 1

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

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

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

      5: f29 -> f41 : A'=1, C'=free_1, D'=0, E'=free_1, F'=free_1, [ 0>=A && free_1>=1000+B ], cost: 1

      7: f0 -> f21 : A'=1, Q'=free_3, J'=free_3, [ 0>=free_3 ], cost: 1

      8: f0 -> f21 : A'=1, Q'=free_4, J'=0, K'=1, L'=free_5, M'=free_5, N'=free_5, O'=free_5, [ free_4>=1 && free_5>=1 ], cost: 1

      9: f0 -> f41 : A'=1, Q'=free_6, J'=0, K'=1, L'=free_7, M'=free_7, N'=free_7, O'=free_7, [ free_6>=1 && 0>=free_7 ], cost: 1



Chained accelerated rules (with incoming rules):

   Start location: f0

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

      6: f21 -> f29 : A'=0, B'=free_2, C'=free_2, G'=0, H'=free_2, [ A>=1 ], cost: 1

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

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

      5: f29 -> f41 : A'=1, C'=free_1, D'=0, E'=free_1, F'=free_1, [ 0>=A && free_1>=1000+B ], cost: 1

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

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

     13: f29 -> [6] : A'=1, C'=free_1, D'=0, E'=free_1, F'=free_1, [ 0>=A && free_1>=1000+B ], cost: NONTERM

      7: f0 -> f21 : A'=1, Q'=free_3, J'=free_3, [ 0>=free_3 ], cost: 1

      8: f0 -> f21 : A'=1, Q'=free_4, J'=0, K'=1, L'=free_5, M'=free_5, N'=free_5, O'=free_5, [ free_4>=1 && free_5>=1 ], cost: 1

      9: f0 -> f41 : A'=1, Q'=free_6, J'=0, K'=1, L'=free_7, M'=free_7, N'=free_7, O'=free_7, [ free_6>=1 && 0>=free_7 ], cost: 1

     14: f0 -> [6] : A'=1, Q'=free_6, J'=0, K'=1, L'=free_7, M'=free_7, N'=free_7, O'=free_7, [ free_6>=1 && 0>=free_7 ], 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, B'=free_2, C'=free_2, G'=0, H'=free_2, [ A>=1 ], cost: 1

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

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

     13: f29 -> [6] : A'=1, C'=free_1, D'=0, E'=free_1, F'=free_1, [ 0>=A && free_1>=1000+B ], cost: NONTERM

      7: f0 -> f21 : A'=1, Q'=free_3, J'=free_3, [ 0>=free_3 ], cost: 1

      8: f0 -> f21 : A'=1, Q'=free_4, J'=0, K'=1, L'=free_5, M'=free_5, N'=free_5, O'=free_5, [ free_4>=1 && free_5>=1 ], cost: 1

     14: f0 -> [6] : A'=1, Q'=free_6, J'=0, K'=1, L'=free_7, M'=free_7, N'=free_7, O'=free_7, [ free_6>=1 && 0>=free_7 ], cost: NONTERM



Eliminated locations (on tree-shaped paths):

   Start location: f0

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

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

     13: f29 -> [6] : A'=1, C'=free_1, D'=0, E'=free_1, F'=free_1, [ 0>=A && free_1>=1000+B ], cost: NONTERM

     14: f0 -> [6] : A'=1, Q'=free_6, J'=0, K'=1, L'=free_7, M'=free_7, N'=free_7, O'=free_7, [ free_6>=1 && 0>=free_7 ], cost: NONTERM

     15: f0 -> f29 : A'=0, B'=free_2, C'=free_2, G'=0, H'=free_2, Q'=free_3, J'=free_3, [ 0>=free_3 ], cost: 2

     16: f0 -> f29 : A'=0, B'=free_2, C'=free_2, G'=0, H'=free_2, Q'=free_4, J'=0, K'=1, L'=free_5, M'=free_5, N'=free_5, O'=free_5, [ free_4>=1 && free_5>=1 ], cost: 2



Eliminated locations (on tree-shaped paths):

   Start location: f0

     14: f0 -> [6] : A'=1, Q'=free_6, J'=0, K'=1, L'=free_7, M'=free_7, N'=free_7, O'=free_7, [ free_6>=1 && 0>=free_7 ], cost: NONTERM

     17: f0 -> [6] : A'=0, B'=free_2, C'=free, D'=0, E'=free, F'=free, G'=0, H'=free_2, Q'=free_3, J'=free_3, [ 0>=free_3 && 999+free_2>=free ], cost: NONTERM

     18: f0 -> [6] : A'=1, B'=free_2, C'=free_1, D'=0, E'=free_1, F'=free_1, G'=0, H'=free_2, Q'=free_3, J'=free_3, [ 0>=free_3 && free_1>=1000+free_2 ], cost: NONTERM

     19: f0 -> [6] : A'=0, B'=free_2, C'=free, D'=0, E'=free, F'=free, G'=0, H'=free_2, Q'=free_4, J'=0, K'=1, L'=free_5, M'=free_5, N'=free_5, O'=free_5, [ free_4>=1 && free_5>=1 && 999+free_2>=free ], cost: NONTERM

     20: f0 -> [6] : A'=1, B'=free_2, C'=free_1, D'=0, E'=free_1, F'=free_1, G'=0, H'=free_2, Q'=free_4, J'=0, K'=1, L'=free_5, M'=free_5, N'=free_5, O'=free_5, [ free_4>=1 && free_5>=1 && free_1>=1000+free_2 ], cost: NONTERM



### Computing asymptotic complexity ###



Fully simplified ITS problem

   Start location: f0

     14: f0 -> [6] : A'=1, Q'=free_6, J'=0, K'=1, L'=free_7, M'=free_7, N'=free_7, O'=free_7, [ free_6>=1 && 0>=free_7 ], cost: NONTERM

     17: f0 -> [6] : A'=0, B'=free_2, C'=free, D'=0, E'=free, F'=free, G'=0, H'=free_2, Q'=free_3, J'=free_3, [ 0>=free_3 && 999+free_2>=free ], cost: NONTERM

     18: f0 -> [6] : A'=1, B'=free_2, C'=free_1, D'=0, E'=free_1, F'=free_1, G'=0, H'=free_2, Q'=free_3, J'=free_3, [ 0>=free_3 && free_1>=1000+free_2 ], cost: NONTERM

     19: f0 -> [6] : A'=0, B'=free_2, C'=free, D'=0, E'=free, F'=free, G'=0, H'=free_2, Q'=free_4, J'=0, K'=1, L'=free_5, M'=free_5, N'=free_5, O'=free_5, [ free_4>=1 && free_5>=1 && 999+free_2>=free ], cost: NONTERM

     20: f0 -> [6] : A'=1, B'=free_2, C'=free_1, D'=0, E'=free_1, F'=free_1, G'=0, H'=free_2, Q'=free_4, J'=0, K'=1, L'=free_5, M'=free_5, N'=free_5, O'=free_5, [ free_4>=1 && free_5>=1 && free_1>=1000+free_2 ], 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:  [ free_6>=1 && 0>=free_7 ]



NO


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(2)
BOUNDS(INF, INF)