/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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, 846 ms]
(2) BOUNDS(INF, INF)


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

(0)
Obligation:
Complexity Int TRS consisting of the following rules:
f51(A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, R, S, T, U, V) -> Com_1(f51(A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, R, S, T, U, V)) :|: TRUE
f53(A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, R, S, T, U, V) -> Com_1(f56(A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, R, S, T, U, V)) :|: TRUE
f11(A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, R, S, T, U, V) -> Com_1(f51(A, B, 0, 0, E, F, G, H, I, J, K, L, M, N, O, P, Q, R, S, T, U, V)) :|: A >= B
f37(A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, R, S, T, U, V) -> Com_1(f51(A, B, 0, 0, E, F, G, H, I, J, K, L, M, N, O, P, Q, R, S, T, U, V)) :|: E >= 3
f37(A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, R, S, T, U, V) -> Com_1(f51(A, B, 0, 0, E, F, G, H, I, J, K, L, M, N, O, P, Q, R, S, T, U, V)) :|: 1 >= E
f37(A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, R, S, T, U, V) -> Com_1(f51(A, B, 0, 0, 2, G, G, H, I, J, K, L, M, N, O, P, Q, R, S, T, U, V)) :|: E >= 2 && E <= 2
f11(A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, R, S, T, U, V) -> Com_1(f51(A, B, 0, 0, E, F, W, X, Y, Z, D, W, W, W, O, P, Q, R, S, T, U, V)) :|: W >= 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, U, V) -> Com_1(f37(A, B, C, D, A1, F, W, X, Y, Z, D, W, W, W, W, Q, 0, A1, A1, A1, 0, V)) :|: B >= A + 1 && 0 >= W && A1 >= 2
f11(A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, R, S, T, U, V) -> Com_1(f37(A, B, C, D, A1, F, W, X, Y, Z, D, W, W, W, W, Q, 0, A1, A1, A1, 0, V)) :|: B >= A + 1 && 0 >= W && 0 >= A1
f11(A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, R, S, T, U, V) -> Com_1(f11(A + 1, B, C, D, 1, F, W, X, Y, Z, D, W, W, W, W, Q, Q, 1, 1, 1, 0, V)) :|: 0 >= W && B >= A + 1
f0(A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, R, S, T, U, V) -> Com_1(f11(A, B, 0, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, R, S, T, U, 0)) :|: TRUE

The start-symbols are:[f0_22]


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

(1) Loat Proof (FINISHED)


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



Initial linear ITS problem

   Start location: f0

      0: f51 -> f51 : [], cost: 1

      1: f53 -> f56 : A1'=B, B'=C, C'=D, D'=E, E'=F, F'=G, G'=H, H'=Q, Q'=J, J'=K, K'=L, L'=M, M'=N, N'=O, O'=P, P'=Q_1, Q_1'=R, R'=S, S'=T, T'=U, U'=V, [], cost: 1

      2: f11 -> f51 : C'=0, D'=0, [ A>=B ], cost: 1

      6: f11 -> f51 : C'=0, D'=0, G'=free_2, H'=free_3, Q'=free, J'=free_1, K'=D, L'=free_2, M'=free_2, N'=free_2, [ free_2>=1 && B>=1+A ], cost: 1

      7: f11 -> f37 : E'=free_6, G'=free_8, H'=free_4, Q'=free_5, J'=free_7, K'=D, L'=free_8, M'=free_8, N'=free_8, O'=free_8, P'=Q_1, Q_1'=0, R'=free_6, S'=free_6, T'=free_6, U'=0, [ B>=1+A && 0>=free_8 && free_6>=2 ], cost: 1

      8: f11 -> f37 : E'=free_11, G'=free_13, H'=free_9, Q'=free_10, J'=free_12, K'=D, L'=free_13, M'=free_13, N'=free_13, O'=free_13, P'=Q_1, Q_1'=0, R'=free_11, S'=free_11, T'=free_11, U'=0, [ B>=1+A && 0>=free_13 && 0>=free_11 ], cost: 1

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

      3: f37 -> f51 : C'=0, D'=0, [ E>=3 ], cost: 1

      4: f37 -> f51 : C'=0, D'=0, [ 1>=E ], cost: 1

      5: f37 -> f51 : C'=0, D'=0, E'=2, F'=G, [ E==2 ], cost: 1

     10: f0 -> f11 : C'=0, V'=0, [], cost: 1



Checking for constant complexity:

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

     10: f0 -> f11 : C'=0, V'=0, [], cost: 1



Removed unreachable and leaf rules:

   Start location: f0

      0: f51 -> f51 : [], cost: 1

      2: f11 -> f51 : C'=0, D'=0, [ A>=B ], cost: 1

      6: f11 -> f51 : C'=0, D'=0, G'=free_2, H'=free_3, Q'=free, J'=free_1, K'=D, L'=free_2, M'=free_2, N'=free_2, [ free_2>=1 && B>=1+A ], cost: 1

      7: f11 -> f37 : E'=free_6, G'=free_8, H'=free_4, Q'=free_5, J'=free_7, K'=D, L'=free_8, M'=free_8, N'=free_8, O'=free_8, P'=Q_1, Q_1'=0, R'=free_6, S'=free_6, T'=free_6, U'=0, [ B>=1+A && 0>=free_8 && free_6>=2 ], cost: 1

      8: f11 -> f37 : E'=free_11, G'=free_13, H'=free_9, Q'=free_10, J'=free_12, K'=D, L'=free_13, M'=free_13, N'=free_13, O'=free_13, P'=Q_1, Q_1'=0, R'=free_11, S'=free_11, T'=free_11, U'=0, [ B>=1+A && 0>=free_13 && 0>=free_11 ], cost: 1

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

      3: f37 -> f51 : C'=0, D'=0, [ E>=3 ], cost: 1

      4: f37 -> f51 : C'=0, D'=0, [ 1>=E ], cost: 1

      5: f37 -> f51 : C'=0, D'=0, E'=2, F'=G, [ E==2 ], cost: 1

     10: f0 -> f11 : C'=0, V'=0, [], cost: 1



### Simplification by acceleration and chaining ###



Accelerating simple loops of location 0.

   Accelerating the following rules:

      0: f51 -> f51 : [], 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, E'=1, G'=free_16, H'=free_17, Q'=free_14, J'=free_15, K'=D, L'=free_16, M'=free_16, N'=free_16, O'=free_16, P'=Q_1, R'=1, S'=1, T'=1, U'=0, [ 0>=free_16 && B>=1+A ], cost: 1



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

   Removing the simple loops: 9.



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

   Start location: f0

     11: f51 -> [6] : [], cost: NONTERM

      2: f11 -> f51 : C'=0, D'=0, [ A>=B ], cost: 1

      6: f11 -> f51 : C'=0, D'=0, G'=free_2, H'=free_3, Q'=free, J'=free_1, K'=D, L'=free_2, M'=free_2, N'=free_2, [ free_2>=1 && B>=1+A ], cost: 1

      7: f11 -> f37 : E'=free_6, G'=free_8, H'=free_4, Q'=free_5, J'=free_7, K'=D, L'=free_8, M'=free_8, N'=free_8, O'=free_8, P'=Q_1, Q_1'=0, R'=free_6, S'=free_6, T'=free_6, U'=0, [ B>=1+A && 0>=free_8 && free_6>=2 ], cost: 1

      8: f11 -> f37 : E'=free_11, G'=free_13, H'=free_9, Q'=free_10, J'=free_12, K'=D, L'=free_13, M'=free_13, N'=free_13, O'=free_13, P'=Q_1, Q_1'=0, R'=free_11, S'=free_11, T'=free_11, U'=0, [ B>=1+A && 0>=free_13 && 0>=free_11 ], cost: 1

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

      3: f37 -> f51 : C'=0, D'=0, [ E>=3 ], cost: 1

      4: f37 -> f51 : C'=0, D'=0, [ 1>=E ], cost: 1

      5: f37 -> f51 : C'=0, D'=0, E'=2, F'=G, [ E==2 ], cost: 1

     10: f0 -> f11 : C'=0, V'=0, [], cost: 1



Chained accelerated rules (with incoming rules):

   Start location: f0

      2: f11 -> f51 : C'=0, D'=0, [ A>=B ], cost: 1

      6: f11 -> f51 : C'=0, D'=0, G'=free_2, H'=free_3, Q'=free, J'=free_1, K'=D, L'=free_2, M'=free_2, N'=free_2, [ free_2>=1 && B>=1+A ], cost: 1

      7: f11 -> f37 : E'=free_6, G'=free_8, H'=free_4, Q'=free_5, J'=free_7, K'=D, L'=free_8, M'=free_8, N'=free_8, O'=free_8, P'=Q_1, Q_1'=0, R'=free_6, S'=free_6, T'=free_6, U'=0, [ B>=1+A && 0>=free_8 && free_6>=2 ], cost: 1

      8: f11 -> f37 : E'=free_11, G'=free_13, H'=free_9, Q'=free_10, J'=free_12, K'=D, L'=free_13, M'=free_13, N'=free_13, O'=free_13, P'=Q_1, Q_1'=0, R'=free_11, S'=free_11, T'=free_11, U'=0, [ B>=1+A && 0>=free_13 && 0>=free_11 ], cost: 1

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

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

      3: f37 -> f51 : C'=0, D'=0, [ E>=3 ], cost: 1

      4: f37 -> f51 : C'=0, D'=0, [ 1>=E ], cost: 1

      5: f37 -> f51 : C'=0, D'=0, E'=2, F'=G, [ E==2 ], cost: 1

     14: f37 -> [6] : C'=0, D'=0, [ E>=3 ], cost: NONTERM

     15: f37 -> [6] : C'=0, D'=0, [ 1>=E ], cost: NONTERM

     16: f37 -> [6] : C'=0, D'=0, E'=2, F'=G, [ E==2 ], cost: NONTERM

     10: f0 -> f11 : C'=0, V'=0, [], cost: 1

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



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

   Start location: f0

      7: f11 -> f37 : E'=free_6, G'=free_8, H'=free_4, Q'=free_5, J'=free_7, K'=D, L'=free_8, M'=free_8, N'=free_8, O'=free_8, P'=Q_1, Q_1'=0, R'=free_6, S'=free_6, T'=free_6, U'=0, [ B>=1+A && 0>=free_8 && free_6>=2 ], cost: 1

      8: f11 -> f37 : E'=free_11, G'=free_13, H'=free_9, Q'=free_10, J'=free_12, K'=D, L'=free_13, M'=free_13, N'=free_13, O'=free_13, P'=Q_1, Q_1'=0, R'=free_11, S'=free_11, T'=free_11, U'=0, [ B>=1+A && 0>=free_13 && 0>=free_11 ], cost: 1

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

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

     14: f37 -> [6] : C'=0, D'=0, [ E>=3 ], cost: NONTERM

     15: f37 -> [6] : C'=0, D'=0, [ 1>=E ], cost: NONTERM

     16: f37 -> [6] : C'=0, D'=0, E'=2, F'=G, [ E==2 ], cost: NONTERM

     10: f0 -> f11 : C'=0, V'=0, [], cost: 1

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



Eliminated locations (on tree-shaped paths):

   Start location: f0

     14: f37 -> [6] : C'=0, D'=0, [ E>=3 ], cost: NONTERM

     15: f37 -> [6] : C'=0, D'=0, [ 1>=E ], cost: NONTERM

     16: f37 -> [6] : C'=0, D'=0, E'=2, F'=G, [ E==2 ], cost: NONTERM

     19: f0 -> f37 : C'=0, E'=free_6, G'=free_8, H'=free_4, Q'=free_5, J'=free_7, K'=D, L'=free_8, M'=free_8, N'=free_8, O'=free_8, P'=Q_1, Q_1'=0, R'=free_6, S'=free_6, T'=free_6, U'=0, V'=0, [ B>=1+A && 0>=free_8 && free_6>=2 ], cost: 2

     20: f0 -> f37 : C'=0, E'=free_11, G'=free_13, H'=free_9, Q'=free_10, J'=free_12, K'=D, L'=free_13, M'=free_13, N'=free_13, O'=free_13, P'=Q_1, Q_1'=0, R'=free_11, S'=free_11, T'=free_11, U'=0, V'=0, [ B>=1+A && 0>=free_13 && 0>=free_11 ], cost: 2

     21: f0 -> [6] : C'=0, D'=0, V'=0, [ A>=B ], cost: NONTERM

     22: f0 -> [6] : C'=0, D'=0, G'=free_2, H'=free_3, Q'=free, J'=free_1, K'=D, L'=free_2, M'=free_2, N'=free_2, V'=0, [ free_2>=1 && B>=1+A ], cost: NONTERM

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

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



Eliminated locations (on tree-shaped paths):

   Start location: f0

     21: f0 -> [6] : C'=0, D'=0, V'=0, [ A>=B ], cost: NONTERM

     22: f0 -> [6] : C'=0, D'=0, G'=free_2, H'=free_3, Q'=free, J'=free_1, K'=D, L'=free_2, M'=free_2, N'=free_2, V'=0, [ free_2>=1 && B>=1+A ], cost: NONTERM

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

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

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

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

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

     22: f0 -> [6] : C'=0, D'=0, G'=free_2, H'=free_3, Q'=free, J'=free_1, K'=D, L'=free_2, M'=free_2, N'=free_2, V'=0, [ free_2>=1 && B>=1+A ], cost: NONTERM

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

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

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

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

     22: f0 -> [6] : C'=0, D'=0, G'=free_2, H'=free_3, Q'=free, J'=free_1, K'=D, L'=free_2, M'=free_2, N'=free_2, V'=0, [ free_2>=1 && B>=1+A ], cost: NONTERM

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

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

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

     27: f0 -> [6] : C'=0, D'=0, E'=free_11, G'=free_13, H'=free_9, Q'=free_10, J'=free_12, K'=D, L'=free_13, M'=free_13, N'=free_13, O'=free_13, P'=Q_1, Q_1'=0, R'=free_11, S'=free_11, T'=free_11, U'=0, V'=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)