/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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, 824 ms]
(2) BOUNDS(INF, INF)


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

(0)
Obligation:
Complexity Int TRS consisting of the following rules:
f47(A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, R, S, T) -> Com_1(f47(A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, R, S, T)) :|: TRUE
f49(A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, R, S, T) -> Com_1(f52(A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, R, S, T)) :|: TRUE
f11(A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, R, S, T) -> Com_1(f47(A, B, 0, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, R, S, T)) :|: A >= B
f35(A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, R, S, T) -> Com_1(f47(A, B, 0, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, R, S, T)) :|: D >= 3
f35(A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, R, S, T) -> Com_1(f47(A, B, 0, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, R, S, T)) :|: 1 >= D
f35(A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, R, S, T) -> Com_1(f47(A, B, 0, 2, F, F, G, H, I, J, K, L, M, N, O, P, Q, R, S, T)) :|: D >= 2 && D <= 2
f11(A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, R, S, T) -> Com_1(f47(A, B, 0, D, E, U, V, W, X, C, U, U, M, N, O, P, Q, R, S, T)) :|: U >= 1 && B >= A + 1
f11(A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, R, S, T) -> Com_1(f35(A, B, C, Y, E, U, V, W, X, C, U, U, U, O, 0, Y, Y, 0, S, T)) :|: B >= A + 1 && 0 >= U && Y >= 2
f11(A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, R, S, T) -> Com_1(f35(A, B, C, Y, E, U, V, W, X, C, U, U, U, O, 0, Y, Y, 0, S, T)) :|: B >= A + 1 && 0 >= U && 0 >= Y
f11(A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, R, S, T) -> Com_1(f11(A + 1, B, C, 1, E, U, V, W, X, C, U, U, U, O, O, 1, 1, 0, S, T)) :|: 0 >= U && B >= A + 1
f0(A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, R, S, T) -> Com_1(f11(A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, R, 0, 0)) :|: TRUE

The start-symbols are:[f0_20]


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

(1) Loat Proof (FINISHED)


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



Initial linear ITS problem

   Start location: f0

      0: f47 -> f47 : [], cost: 1

      1: f49 -> f52 : [], cost: 1

      2: f11 -> f47 : C'=0, [ A>=B ], cost: 1

      6: f11 -> f47 : C'=0, F'=free_2, G'=free_3, H'=free, Q'=free_1, J'=C, K'=free_2, L'=free_2, [ free_2>=1 && B>=1+A ], cost: 1

      7: f11 -> f35 : D'=free_6, F'=free_8, G'=free_4, H'=free_5, Q'=free_7, J'=C, K'=free_8, L'=free_8, M'=free_8, N'=O, O'=0, P'=free_6, Q_1'=free_6, R'=0, [ B>=1+A && 0>=free_8 && free_6>=2 ], cost: 1

      8: f11 -> f35 : D'=free_11, F'=free_13, G'=free_9, H'=free_10, Q'=free_12, J'=C, K'=free_13, L'=free_13, M'=free_13, N'=O, O'=0, P'=free_11, Q_1'=free_11, R'=0, [ B>=1+A && 0>=free_13 && 0>=free_11 ], cost: 1

      9: f11 -> f11 : A'=1+A, D'=1, F'=free_16, G'=free_17, H'=free_14, Q'=free_15, J'=C, K'=free_16, L'=free_16, M'=free_16, N'=O, P'=1, Q_1'=1, R'=0, [ 0>=free_16 && B>=1+A ], cost: 1

      3: f35 -> f47 : C'=0, [ D>=3 ], cost: 1

      4: f35 -> f47 : C'=0, [ 1>=D ], cost: 1

      5: f35 -> f47 : C'=0, D'=2, E'=F, [ D==2 ], cost: 1

     10: f0 -> f11 : S'=0, T'=0, [], cost: 1



Checking for constant complexity:

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

     10: f0 -> f11 : S'=0, T'=0, [], cost: 1



Removed unreachable and leaf rules:

   Start location: f0

      0: f47 -> f47 : [], cost: 1

      2: f11 -> f47 : C'=0, [ A>=B ], cost: 1

      6: f11 -> f47 : C'=0, F'=free_2, G'=free_3, H'=free, Q'=free_1, J'=C, K'=free_2, L'=free_2, [ free_2>=1 && B>=1+A ], cost: 1

      7: f11 -> f35 : D'=free_6, F'=free_8, G'=free_4, H'=free_5, Q'=free_7, J'=C, K'=free_8, L'=free_8, M'=free_8, N'=O, O'=0, P'=free_6, Q_1'=free_6, R'=0, [ B>=1+A && 0>=free_8 && free_6>=2 ], cost: 1

      8: f11 -> f35 : D'=free_11, F'=free_13, G'=free_9, H'=free_10, Q'=free_12, J'=C, K'=free_13, L'=free_13, M'=free_13, N'=O, O'=0, P'=free_11, Q_1'=free_11, R'=0, [ B>=1+A && 0>=free_13 && 0>=free_11 ], cost: 1

      9: f11 -> f11 : A'=1+A, D'=1, F'=free_16, G'=free_17, H'=free_14, Q'=free_15, J'=C, K'=free_16, L'=free_16, M'=free_16, N'=O, P'=1, Q_1'=1, R'=0, [ 0>=free_16 && B>=1+A ], cost: 1

      3: f35 -> f47 : C'=0, [ D>=3 ], cost: 1

      4: f35 -> f47 : C'=0, [ 1>=D ], cost: 1

      5: f35 -> f47 : C'=0, D'=2, E'=F, [ D==2 ], cost: 1

     10: f0 -> f11 : S'=0, T'=0, [], cost: 1



### Simplification by acceleration and chaining ###



Accelerating simple loops of location 0.

   Accelerating the following rules:

      0: f47 -> f47 : [], cost: 1



   Accelerated rule 0 with NONTERM, yielding the new rule 11.

   Removing the simple loops: 0.



Accelerating simple loops of location 2.

   Accelerating the following rules:

      9: f11 -> f11 : A'=1+A, D'=1, F'=free_16, G'=free_17, H'=free_14, Q'=free_15, J'=C, K'=free_16, L'=free_16, M'=free_16, N'=O, P'=1, Q_1'=1, R'=0, [ 0>=free_16 && B>=1+A ], cost: 1



   Accelerated rule 9 with metering function -A+B, yielding the new rule 12.

   Removing the simple loops: 9.



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

   Start location: f0

     11: f47 -> [6] : [], cost: NONTERM

      2: f11 -> f47 : C'=0, [ A>=B ], cost: 1

      6: f11 -> f47 : C'=0, F'=free_2, G'=free_3, H'=free, Q'=free_1, J'=C, K'=free_2, L'=free_2, [ free_2>=1 && B>=1+A ], cost: 1

      7: f11 -> f35 : D'=free_6, F'=free_8, G'=free_4, H'=free_5, Q'=free_7, J'=C, K'=free_8, L'=free_8, M'=free_8, N'=O, O'=0, P'=free_6, Q_1'=free_6, R'=0, [ B>=1+A && 0>=free_8 && free_6>=2 ], cost: 1

      8: f11 -> f35 : D'=free_11, F'=free_13, G'=free_9, H'=free_10, Q'=free_12, J'=C, K'=free_13, L'=free_13, M'=free_13, N'=O, O'=0, P'=free_11, Q_1'=free_11, R'=0, [ B>=1+A && 0>=free_13 && 0>=free_11 ], cost: 1

     12: f11 -> f11 : A'=B, D'=1, F'=free_16, G'=free_17, H'=free_14, Q'=free_15, J'=C, K'=free_16, L'=free_16, M'=free_16, N'=O, P'=1, Q_1'=1, R'=0, [ 0>=free_16 && B>=1+A ], cost: -A+B

      3: f35 -> f47 : C'=0, [ D>=3 ], cost: 1

      4: f35 -> f47 : C'=0, [ 1>=D ], cost: 1

      5: f35 -> f47 : C'=0, D'=2, E'=F, [ D==2 ], cost: 1

     10: f0 -> f11 : S'=0, T'=0, [], cost: 1



Chained accelerated rules (with incoming rules):

   Start location: f0

      2: f11 -> f47 : C'=0, [ A>=B ], cost: 1

      6: f11 -> f47 : C'=0, F'=free_2, G'=free_3, H'=free, Q'=free_1, J'=C, K'=free_2, L'=free_2, [ free_2>=1 && B>=1+A ], cost: 1

      7: f11 -> f35 : D'=free_6, F'=free_8, G'=free_4, H'=free_5, Q'=free_7, J'=C, K'=free_8, L'=free_8, M'=free_8, N'=O, O'=0, P'=free_6, Q_1'=free_6, R'=0, [ B>=1+A && 0>=free_8 && free_6>=2 ], cost: 1

      8: f11 -> f35 : D'=free_11, F'=free_13, G'=free_9, H'=free_10, Q'=free_12, J'=C, K'=free_13, L'=free_13, M'=free_13, N'=O, O'=0, P'=free_11, Q_1'=free_11, R'=0, [ B>=1+A && 0>=free_13 && 0>=free_11 ], cost: 1

     13: f11 -> [6] : C'=0, [ A>=B ], cost: NONTERM

     17: f11 -> [6] : C'=0, F'=free_2, G'=free_3, H'=free, Q'=free_1, J'=C, K'=free_2, L'=free_2, [ free_2>=1 && B>=1+A ], cost: NONTERM

      3: f35 -> f47 : C'=0, [ D>=3 ], cost: 1

      4: f35 -> f47 : C'=0, [ 1>=D ], cost: 1

      5: f35 -> f47 : C'=0, D'=2, E'=F, [ D==2 ], cost: 1

     14: f35 -> [6] : C'=0, [ D>=3 ], cost: NONTERM

     15: f35 -> [6] : C'=0, [ 1>=D ], cost: NONTERM

     16: f35 -> [6] : C'=0, D'=2, E'=F, [ D==2 ], cost: NONTERM

     10: f0 -> f11 : S'=0, T'=0, [], cost: 1

     18: f0 -> f11 : A'=B, D'=1, F'=free_16, G'=free_17, H'=free_14, Q'=free_15, J'=C, K'=free_16, L'=free_16, M'=free_16, N'=O, P'=1, Q_1'=1, R'=0, S'=0, T'=0, [ 0>=free_16 && B>=1+A ], cost: 1-A+B



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

   Start location: f0

      7: f11 -> f35 : D'=free_6, F'=free_8, G'=free_4, H'=free_5, Q'=free_7, J'=C, K'=free_8, L'=free_8, M'=free_8, N'=O, O'=0, P'=free_6, Q_1'=free_6, R'=0, [ B>=1+A && 0>=free_8 && free_6>=2 ], cost: 1

      8: f11 -> f35 : D'=free_11, F'=free_13, G'=free_9, H'=free_10, Q'=free_12, J'=C, K'=free_13, L'=free_13, M'=free_13, N'=O, O'=0, P'=free_11, Q_1'=free_11, R'=0, [ B>=1+A && 0>=free_13 && 0>=free_11 ], cost: 1

     13: f11 -> [6] : C'=0, [ A>=B ], cost: NONTERM

     17: f11 -> [6] : C'=0, F'=free_2, G'=free_3, H'=free, Q'=free_1, J'=C, K'=free_2, L'=free_2, [ free_2>=1 && B>=1+A ], cost: NONTERM

     14: f35 -> [6] : C'=0, [ D>=3 ], cost: NONTERM

     15: f35 -> [6] : C'=0, [ 1>=D ], cost: NONTERM

     16: f35 -> [6] : C'=0, D'=2, E'=F, [ D==2 ], cost: NONTERM

     10: f0 -> f11 : S'=0, T'=0, [], cost: 1

     18: f0 -> f11 : A'=B, D'=1, F'=free_16, G'=free_17, H'=free_14, Q'=free_15, J'=C, K'=free_16, L'=free_16, M'=free_16, N'=O, P'=1, Q_1'=1, R'=0, S'=0, T'=0, [ 0>=free_16 && B>=1+A ], cost: 1-A+B



Eliminated locations (on tree-shaped paths):

   Start location: f0

     14: f35 -> [6] : C'=0, [ D>=3 ], cost: NONTERM

     15: f35 -> [6] : C'=0, [ 1>=D ], cost: NONTERM

     16: f35 -> [6] : C'=0, D'=2, E'=F, [ D==2 ], cost: NONTERM

     19: f0 -> f35 : D'=free_6, F'=free_8, G'=free_4, H'=free_5, Q'=free_7, J'=C, K'=free_8, L'=free_8, M'=free_8, N'=O, O'=0, P'=free_6, Q_1'=free_6, R'=0, S'=0, T'=0, [ B>=1+A && 0>=free_8 && free_6>=2 ], cost: 2

     20: f0 -> f35 : D'=free_11, F'=free_13, G'=free_9, H'=free_10, Q'=free_12, J'=C, K'=free_13, L'=free_13, M'=free_13, N'=O, O'=0, P'=free_11, Q_1'=free_11, R'=0, S'=0, T'=0, [ B>=1+A && 0>=free_13 && 0>=free_11 ], cost: 2

     21: f0 -> [6] : C'=0, S'=0, T'=0, [ A>=B ], cost: NONTERM

     22: f0 -> [6] : C'=0, F'=free_2, G'=free_3, H'=free, Q'=free_1, J'=C, K'=free_2, L'=free_2, S'=0, T'=0, [ free_2>=1 && B>=1+A ], cost: NONTERM

     23: f0 -> [6] : A'=B, C'=0, D'=1, F'=free_16, G'=free_17, H'=free_14, Q'=free_15, J'=C, K'=free_16, L'=free_16, M'=free_16, N'=O, P'=1, Q_1'=1, R'=0, S'=0, T'=0, [ 0>=free_16 && B>=1+A ], cost: NONTERM

     24: f0 -> [8] : [ 0>=free_16 && B>=1+A ], cost: 1-A+B



Eliminated locations (on tree-shaped paths):

   Start location: f0

     21: f0 -> [6] : C'=0, S'=0, T'=0, [ A>=B ], cost: NONTERM

     22: f0 -> [6] : C'=0, F'=free_2, G'=free_3, H'=free, Q'=free_1, J'=C, K'=free_2, L'=free_2, S'=0, T'=0, [ free_2>=1 && B>=1+A ], cost: NONTERM

     23: f0 -> [6] : A'=B, C'=0, D'=1, F'=free_16, G'=free_17, H'=free_14, Q'=free_15, J'=C, K'=free_16, L'=free_16, M'=free_16, N'=O, P'=1, Q_1'=1, R'=0, S'=0, T'=0, [ 0>=free_16 && B>=1+A ], cost: NONTERM

     24: f0 -> [8] : [ 0>=free_16 && B>=1+A ], cost: 1-A+B

     25: f0 -> [6] : C'=0, D'=free_6, F'=free_8, G'=free_4, H'=free_5, Q'=free_7, J'=C, K'=free_8, L'=free_8, M'=free_8, N'=O, O'=0, P'=free_6, Q_1'=free_6, R'=0, S'=0, T'=0, [ B>=1+A && 0>=free_8 && free_6>=3 ], cost: NONTERM

     26: f0 -> [6] : C'=0, D'=2, E'=free_8, F'=free_8, G'=free_4, H'=free_5, Q'=free_7, J'=C, K'=free_8, L'=free_8, M'=free_8, N'=O, O'=0, P'=free_6, Q_1'=free_6, R'=0, S'=0, T'=0, [ B>=1+A && 0>=free_8 && free_6==2 ], cost: NONTERM

     27: f0 -> [6] : C'=0, D'=free_11, F'=free_13, G'=free_9, H'=free_10, Q'=free_12, J'=C, K'=free_13, L'=free_13, M'=free_13, N'=O, O'=0, P'=free_11, Q_1'=free_11, R'=0, S'=0, T'=0, [ B>=1+A && 0>=free_13 && 0>=free_11 ], cost: NONTERM



Applied pruning (of leafs and parallel rules):

   Start location: f0

     21: f0 -> [6] : C'=0, S'=0, T'=0, [ A>=B ], cost: NONTERM

     22: f0 -> [6] : C'=0, F'=free_2, G'=free_3, H'=free, Q'=free_1, J'=C, K'=free_2, L'=free_2, S'=0, T'=0, [ free_2>=1 && B>=1+A ], cost: NONTERM

     23: f0 -> [6] : A'=B, C'=0, D'=1, F'=free_16, G'=free_17, H'=free_14, Q'=free_15, J'=C, K'=free_16, L'=free_16, M'=free_16, N'=O, P'=1, Q_1'=1, R'=0, S'=0, T'=0, [ 0>=free_16 && B>=1+A ], cost: NONTERM

     24: f0 -> [8] : [ 0>=free_16 && B>=1+A ], cost: 1-A+B

     25: f0 -> [6] : C'=0, D'=free_6, F'=free_8, G'=free_4, H'=free_5, Q'=free_7, J'=C, K'=free_8, L'=free_8, M'=free_8, N'=O, O'=0, P'=free_6, Q_1'=free_6, R'=0, S'=0, T'=0, [ B>=1+A && 0>=free_8 && free_6>=3 ], cost: NONTERM

     27: f0 -> [6] : C'=0, D'=free_11, F'=free_13, G'=free_9, H'=free_10, Q'=free_12, J'=C, K'=free_13, L'=free_13, M'=free_13, N'=O, O'=0, P'=free_11, Q_1'=free_11, R'=0, S'=0, T'=0, [ B>=1+A && 0>=free_13 && 0>=free_11 ], cost: NONTERM



### Computing asymptotic complexity ###



Fully simplified ITS problem

   Start location: f0

     21: f0 -> [6] : C'=0, S'=0, T'=0, [ A>=B ], cost: NONTERM

     22: f0 -> [6] : C'=0, F'=free_2, G'=free_3, H'=free, Q'=free_1, J'=C, K'=free_2, L'=free_2, S'=0, T'=0, [ free_2>=1 && B>=1+A ], cost: NONTERM

     23: f0 -> [6] : A'=B, C'=0, D'=1, F'=free_16, G'=free_17, H'=free_14, Q'=free_15, J'=C, K'=free_16, L'=free_16, M'=free_16, N'=O, P'=1, Q_1'=1, R'=0, S'=0, T'=0, [ 0>=free_16 && B>=1+A ], cost: NONTERM

     24: f0 -> [8] : [ 0>=free_16 && B>=1+A ], cost: 1-A+B

     25: f0 -> [6] : C'=0, D'=free_6, F'=free_8, G'=free_4, H'=free_5, Q'=free_7, J'=C, K'=free_8, L'=free_8, M'=free_8, N'=O, O'=0, P'=free_6, Q_1'=free_6, R'=0, S'=0, T'=0, [ B>=1+A && 0>=free_8 && free_6>=3 ], cost: NONTERM

     27: f0 -> [6] : C'=0, D'=free_11, F'=free_13, G'=free_9, H'=free_10, Q'=free_12, J'=C, K'=free_13, L'=free_13, M'=free_13, N'=O, O'=0, P'=free_11, Q_1'=free_11, R'=0, S'=0, T'=0, [ B>=1+A && 0>=free_13 && 0>=free_11 ], cost: NONTERM



Computing asymptotic complexity for rule 21

   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>=B ]



NO


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

(2)
BOUNDS(INF, INF)