/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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, 1212 ms]
(2) BOUNDS(INF, INF)


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

(0)
Obligation:
Complexity Int TRS consisting of the following rules:
f8(A, B, C, D, E, F, G, H, I, J, K, L, M) -> Com_1(f32(A, B, N, D, E, F, G, H, I, J, K, L, M)) :|: A >= B
f15(A, B, C, D, E, F, G, H, I, J, K, L, M) -> Com_1(f8(A, B, C, N, E, F, G, H, I, J, K, L, M)) :|: TRUE
f300(A, B, C, D, E, F, G, H, I, J, K, L, M) -> Com_1(f8(1 + A, B, C, D, E, F, G, H, I, J, K, L, M)) :|: B >= E
f8(A, B, C, D, E, F, G, H, I, J, K, L, M) -> Com_1(f8(1 + A, B, C, D, E, F, G, H, I, J, K, L, M)) :|: B >= 1 + A && B >= E
f13(A, B, C, D, E, F, G, H, I, J, K, L, M) -> Com_1(f1(A, B, N, D, E, O, P, 0, 2, 0, 0, L, M)) :|: TRUE
f12(A, B, C, D, E, F, G, H, I, J, K, L, M) -> Com_1(f1(A, B, N, D, E, O, P, 0, I, 0, 0, 1 + I, M)) :|: 4 >= I
f10(A, B, C, D, E, F, G, H, I, J, K, L, M) -> Com_1(f1(A, B, N, D, E, O, P, 0, I, 0, 0, L, 2)) :|: TRUE
f8(A, B, C, D, E, F, G, H, I, J, K, L, M) -> Com_1(f1(A, B, N, D, E, O, P, 0, I, 0, 0, L, M)) :|: B >= 1 + A && E >= 1 + B
f1(A, B, C, D, E, F, G, H, I, J, K, L, M) -> Com_1(f1(A, B, N, D, E, O, P, 0, I, 0, 0, L, M)) :|: E >= 1 + B && 4 >= M
f300(A, B, C, D, E, F, G, H, I, J, K, L, M) -> Com_1(f1(A, B, N, D, E, O, P, 0, I, 0, 0, L, M)) :|: E >= 1 + B
f12(A, B, C, D, E, F, G, H, I, J, K, L, M) -> Com_1(f300(A, 1 + B, N, D, E, O, G, 1, I, 1, 1, L, M)) :|: I >= 5
f1(A, B, C, D, E, F, G, H, I, J, K, L, M) -> Com_1(f300(A, 1 + B, N, D, E, O, G, 1, I, 1, 1, L, M)) :|: E >= 1 + B && M >= 5

The start-symbols are:[f15_13]


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

(1) Loat Proof (FINISHED)


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



Initial linear ITS problem

   Start location: f15

      0: f8 -> f32 : C'=free, [ A>=B ], cost: 1

      3: f8 -> f8 : A'=1+A, [ B>=1+A && B>=E ], cost: 1

      7: f8 -> f1 : C'=free_12, F'=free_13, G'=free_11, H'=0, J'=0, K'=0, [ B>=1+A && E>=1+B ], cost: 1

      1: f15 -> f8 : D'=free_1, [], cost: 1

      2: f300 -> f8 : A'=1+A, [ B>=E ], cost: 1

      9: f300 -> f1 : C'=free_18, F'=free_19, G'=free_17, H'=0, J'=0, K'=0, [ E>=1+B ], cost: 1

      4: f13 -> f1 : C'=free_3, F'=free_4, G'=free_2, H'=0, Q'=2, J'=0, K'=0, [], cost: 1

      5: f12 -> f1 : C'=free_6, F'=free_7, G'=free_5, H'=0, J'=0, K'=0, L'=1+Q, [ 4>=Q ], cost: 1

     10: f12 -> f300 : B'=1+B, C'=free_20, F'=free_21, H'=1, J'=1, K'=1, [ Q>=5 ], cost: 1

      6: f10 -> f1 : C'=free_9, F'=free_10, G'=free_8, H'=0, J'=0, K'=0, M'=2, [], cost: 1

      8: f1 -> f1 : C'=free_15, F'=free_16, G'=free_14, H'=0, J'=0, K'=0, [ E>=1+B && 4>=M ], cost: 1

     11: f1 -> f300 : B'=1+B, C'=free_22, F'=free_23, H'=1, J'=1, K'=1, [ E>=1+B && M>=5 ], cost: 1



Checking for constant complexity:

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

      1: f15 -> f8 : D'=free_1, [], cost: 1



Removed unreachable and leaf rules:

   Start location: f15

      3: f8 -> f8 : A'=1+A, [ B>=1+A && B>=E ], cost: 1

      7: f8 -> f1 : C'=free_12, F'=free_13, G'=free_11, H'=0, J'=0, K'=0, [ B>=1+A && E>=1+B ], cost: 1

      1: f15 -> f8 : D'=free_1, [], cost: 1

      2: f300 -> f8 : A'=1+A, [ B>=E ], cost: 1

      9: f300 -> f1 : C'=free_18, F'=free_19, G'=free_17, H'=0, J'=0, K'=0, [ E>=1+B ], cost: 1

      8: f1 -> f1 : C'=free_15, F'=free_16, G'=free_14, H'=0, J'=0, K'=0, [ E>=1+B && 4>=M ], cost: 1

     11: f1 -> f300 : B'=1+B, C'=free_22, F'=free_23, H'=1, J'=1, K'=1, [ E>=1+B && M>=5 ], cost: 1



### Simplification by acceleration and chaining ###



Accelerating simple loops of location 0.

   Accelerating the following rules:

      3: f8 -> f8 : A'=1+A, [ B>=1+A && B>=E ], cost: 1



   Accelerated rule 3 with metering function -A+B, yielding the new rule 12.

   Removing the simple loops: 3.



Accelerating simple loops of location 6.

   Accelerating the following rules:

      8: f1 -> f1 : C'=free_15, F'=free_16, G'=free_14, H'=0, J'=0, K'=0, [ E>=1+B && 4>=M ], cost: 1



   Accelerated rule 8 with NONTERM, yielding the new rule 13.

   Removing the simple loops: 8.



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

   Start location: f15

      7: f8 -> f1 : C'=free_12, F'=free_13, G'=free_11, H'=0, J'=0, K'=0, [ B>=1+A && E>=1+B ], cost: 1

     12: f8 -> f8 : A'=B, [ B>=1+A && B>=E ], cost: -A+B

      1: f15 -> f8 : D'=free_1, [], cost: 1

      2: f300 -> f8 : A'=1+A, [ B>=E ], cost: 1

      9: f300 -> f1 : C'=free_18, F'=free_19, G'=free_17, H'=0, J'=0, K'=0, [ E>=1+B ], cost: 1

     11: f1 -> f300 : B'=1+B, C'=free_22, F'=free_23, H'=1, J'=1, K'=1, [ E>=1+B && M>=5 ], cost: 1

     13: f1 -> [9] : [ E>=1+B && 4>=M ], cost: NONTERM



Chained accelerated rules (with incoming rules):

   Start location: f15

      7: f8 -> f1 : C'=free_12, F'=free_13, G'=free_11, H'=0, J'=0, K'=0, [ B>=1+A && E>=1+B ], cost: 1

     16: f8 -> [9] : C'=free_12, F'=free_13, G'=free_11, H'=0, J'=0, K'=0, [ B>=1+A && E>=1+B && 4>=M ], cost: NONTERM

      1: f15 -> f8 : D'=free_1, [], cost: 1

     14: f15 -> f8 : A'=B, D'=free_1, [ B>=1+A && B>=E ], cost: 1-A+B

      2: f300 -> f8 : A'=1+A, [ B>=E ], cost: 1

      9: f300 -> f1 : C'=free_18, F'=free_19, G'=free_17, H'=0, J'=0, K'=0, [ E>=1+B ], cost: 1

     15: f300 -> f8 : A'=B, [ B>=E && B>=2+A ], cost: -A+B

     17: f300 -> [9] : C'=free_18, F'=free_19, G'=free_17, H'=0, J'=0, K'=0, [ E>=1+B && 4>=M ], cost: NONTERM

     11: f1 -> f300 : B'=1+B, C'=free_22, F'=free_23, H'=1, J'=1, K'=1, [ E>=1+B && M>=5 ], cost: 1



Eliminated locations (on tree-shaped paths):

   Start location: f15

      7: f8 -> f1 : C'=free_12, F'=free_13, G'=free_11, H'=0, J'=0, K'=0, [ B>=1+A && E>=1+B ], cost: 1

     16: f8 -> [9] : C'=free_12, F'=free_13, G'=free_11, H'=0, J'=0, K'=0, [ B>=1+A && E>=1+B && 4>=M ], cost: NONTERM

      1: f15 -> f8 : D'=free_1, [], cost: 1

     14: f15 -> f8 : A'=B, D'=free_1, [ B>=1+A && B>=E ], cost: 1-A+B

     18: f1 -> f8 : A'=1+A, B'=1+B, C'=free_22, F'=free_23, H'=1, J'=1, K'=1, [ E>=1+B && M>=5 && 1+B>=E ], cost: 2

     19: f1 -> f1 : B'=1+B, C'=free_18, F'=free_19, G'=free_17, H'=0, J'=0, K'=0, [ M>=5 && E>=2+B ], cost: 2

     20: f1 -> f8 : A'=1+B, B'=1+B, C'=free_22, F'=free_23, H'=1, J'=1, K'=1, [ E>=1+B && M>=5 && 1+B>=E && 1+B>=2+A ], cost: 2-A+B



Accelerating simple loops of location 6.

   Accelerating the following rules:

     19: f1 -> f1 : B'=1+B, C'=free_18, F'=free_19, G'=free_17, H'=0, J'=0, K'=0, [ M>=5 && E>=2+B ], cost: 2



   Accelerated rule 19 with metering function -1+E-B, yielding the new rule 21.

   Removing the simple loops: 19.



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

   Start location: f15

      7: f8 -> f1 : C'=free_12, F'=free_13, G'=free_11, H'=0, J'=0, K'=0, [ B>=1+A && E>=1+B ], cost: 1

     16: f8 -> [9] : C'=free_12, F'=free_13, G'=free_11, H'=0, J'=0, K'=0, [ B>=1+A && E>=1+B && 4>=M ], cost: NONTERM

      1: f15 -> f8 : D'=free_1, [], cost: 1

     14: f15 -> f8 : A'=B, D'=free_1, [ B>=1+A && B>=E ], cost: 1-A+B

     18: f1 -> f8 : A'=1+A, B'=1+B, C'=free_22, F'=free_23, H'=1, J'=1, K'=1, [ E>=1+B && M>=5 && 1+B>=E ], cost: 2

     20: f1 -> f8 : A'=1+B, B'=1+B, C'=free_22, F'=free_23, H'=1, J'=1, K'=1, [ E>=1+B && M>=5 && 1+B>=E && 1+B>=2+A ], cost: 2-A+B

     21: f1 -> f1 : B'=-1+E, C'=free_18, F'=free_19, G'=free_17, H'=0, J'=0, K'=0, [ M>=5 && E>=2+B ], cost: -2+2*E-2*B



Chained accelerated rules (with incoming rules):

   Start location: f15

      7: f8 -> f1 : C'=free_12, F'=free_13, G'=free_11, H'=0, J'=0, K'=0, [ B>=1+A && E>=1+B ], cost: 1

     16: f8 -> [9] : C'=free_12, F'=free_13, G'=free_11, H'=0, J'=0, K'=0, [ B>=1+A && E>=1+B && 4>=M ], cost: NONTERM

     22: f8 -> f1 : B'=-1+E, C'=free_18, F'=free_19, G'=free_17, H'=0, J'=0, K'=0, [ B>=1+A && M>=5 && E>=2+B ], cost: -1+2*E-2*B

      1: f15 -> f8 : D'=free_1, [], cost: 1

     14: f15 -> f8 : A'=B, D'=free_1, [ B>=1+A && B>=E ], cost: 1-A+B

     18: f1 -> f8 : A'=1+A, B'=1+B, C'=free_22, F'=free_23, H'=1, J'=1, K'=1, [ E>=1+B && M>=5 && 1+B>=E ], cost: 2

     20: f1 -> f8 : A'=1+B, B'=1+B, C'=free_22, F'=free_23, H'=1, J'=1, K'=1, [ E>=1+B && M>=5 && 1+B>=E && 1+B>=2+A ], cost: 2-A+B



Eliminated locations (on tree-shaped paths):

   Start location: f15

     16: f8 -> [9] : C'=free_12, F'=free_13, G'=free_11, H'=0, J'=0, K'=0, [ B>=1+A && E>=1+B && 4>=M ], cost: NONTERM

     23: f8 -> f8 : A'=1+A, B'=1+B, C'=free_22, F'=free_23, G'=free_11, H'=1, J'=1, K'=1, [ B>=1+A && E>=1+B && M>=5 && 1+B>=E ], cost: 3

     24: f8 -> f8 : A'=1+B, B'=1+B, C'=free_22, F'=free_23, G'=free_11, H'=1, J'=1, K'=1, [ B>=1+A && E>=1+B && M>=5 && 1+B>=E ], cost: 3-A+B

     25: f8 -> f8 : A'=1+A, B'=E, C'=free_22, F'=free_23, G'=free_17, H'=1, J'=1, K'=1, [ B>=1+A && M>=5 && E>=2+B ], cost: 1+2*E-2*B

     26: f8 -> f8 : A'=E, B'=E, C'=free_22, F'=free_23, G'=free_17, H'=1, J'=1, K'=1, [ B>=1+A && M>=5 && E>=2+B && E>=2+A ], cost: -A+3*E-2*B

      1: f15 -> f8 : D'=free_1, [], cost: 1

     14: f15 -> f8 : A'=B, D'=free_1, [ B>=1+A && B>=E ], cost: 1-A+B



Accelerating simple loops of location 0.

   Simplified some of the simple loops (and removed duplicate rules).

   Accelerating the following rules:

     23: f8 -> f8 : A'=1+A, B'=1+B, C'=free_22, F'=free_23, G'=free_11, H'=1, J'=1, K'=1, [ B>=1+A && 1-E+B==0 && M>=5 ], cost: 3

     24: f8 -> f8 : A'=1+B, B'=1+B, C'=free_22, F'=free_23, G'=free_11, H'=1, J'=1, K'=1, [ B>=1+A && 1-E+B==0 && M>=5 ], cost: 3-A+B

     25: f8 -> f8 : A'=1+A, B'=E, C'=free_22, F'=free_23, G'=free_17, H'=1, J'=1, K'=1, [ B>=1+A && M>=5 && E>=2+B ], cost: 1+2*E-2*B

     26: f8 -> f8 : A'=E, B'=E, C'=free_22, F'=free_23, G'=free_17, H'=1, J'=1, K'=1, [ B>=1+A && M>=5 && E>=2+B && E>=2+A ], cost: -A+3*E-2*B



   Accelerated rule 23 with metering function E-B, yielding the new rule 27.

   Accelerated rule 24 with metering function -1+E-B, yielding the new rule 28.

   Found no metering function for rule 25.

   Found no metering function for rule 26.

   Removing the simple loops: 23 24.



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

   Start location: f15

     16: f8 -> [9] : C'=free_12, F'=free_13, G'=free_11, H'=0, J'=0, K'=0, [ B>=1+A && E>=1+B && 4>=M ], cost: NONTERM

     25: f8 -> f8 : A'=1+A, B'=E, C'=free_22, F'=free_23, G'=free_17, H'=1, J'=1, K'=1, [ B>=1+A && M>=5 && E>=2+B ], cost: 1+2*E-2*B

     26: f8 -> f8 : A'=E, B'=E, C'=free_22, F'=free_23, G'=free_17, H'=1, J'=1, K'=1, [ B>=1+A && M>=5 && E>=2+B && E>=2+A ], cost: -A+3*E-2*B

     27: f8 -> f8 : A'=A+E-B, B'=E, C'=free_22, F'=free_23, G'=free_11, H'=1, J'=1, K'=1, [ B>=1+A && 1-E+B==0 && M>=5 ], cost: 3*E-3*B

     28: f8 -> f8 : A'=-1+E, B'=-1+E, C'=free_22, F'=free_23, G'=free_11, H'=1, J'=1, K'=1, [ B>=1+A && 1-E+B==0 && M>=5 && -1+E-B>=1 ], cost: -3+3*E-3*B

      1: f15 -> f8 : D'=free_1, [], cost: 1

     14: f15 -> f8 : A'=B, D'=free_1, [ B>=1+A && B>=E ], cost: 1-A+B



Chained accelerated rules (with incoming rules):

   Start location: f15

     16: f8 -> [9] : C'=free_12, F'=free_13, G'=free_11, H'=0, J'=0, K'=0, [ B>=1+A && E>=1+B && 4>=M ], cost: NONTERM

      1: f15 -> f8 : D'=free_1, [], cost: 1

     14: f15 -> f8 : A'=B, D'=free_1, [ B>=1+A && B>=E ], cost: 1-A+B

     29: f15 -> f8 : A'=1+A, B'=E, C'=free_22, D'=free_1, F'=free_23, G'=free_17, H'=1, J'=1, K'=1, [ B>=1+A && M>=5 && E>=2+B ], cost: 2+2*E-2*B

     30: f15 -> f8 : A'=E, B'=E, C'=free_22, D'=free_1, F'=free_23, G'=free_17, H'=1, J'=1, K'=1, [ B>=1+A && M>=5 && E>=2+B && E>=2+A ], cost: 1-A+3*E-2*B

     31: f15 -> f8 : A'=A+E-B, B'=E, C'=free_22, D'=free_1, F'=free_23, G'=free_11, H'=1, J'=1, K'=1, [ B>=1+A && 1-E+B==0 && M>=5 ], cost: 1+3*E-3*B



Eliminated locations (on tree-shaped paths):

   Start location: f15

     32: f15 -> [9] : C'=free_12, D'=free_1, F'=free_13, G'=free_11, H'=0, J'=0, K'=0, [ B>=1+A && E>=1+B && 4>=M ], cost: NONTERM

     33: f15 -> [12] : [ B>=1+A && B>=E ], cost: 1-A+B

     34: f15 -> [12] : [ B>=1+A && M>=5 && E>=2+B ], cost: 2+2*E-2*B

     35: f15 -> [12] : [ B>=1+A && M>=5 && E>=2+B && E>=2+A ], cost: 1-A+3*E-2*B

     36: f15 -> [12] : [ B>=1+A && 1-E+B==0 && M>=5 ], cost: 1+3*E-3*B



### Computing asymptotic complexity ###



Fully simplified ITS problem

   Start location: f15

     32: f15 -> [9] : C'=free_12, D'=free_1, F'=free_13, G'=free_11, H'=0, J'=0, K'=0, [ B>=1+A && E>=1+B && 4>=M ], cost: NONTERM

     33: f15 -> [12] : [ B>=1+A && B>=E ], cost: 1-A+B

     34: f15 -> [12] : [ B>=1+A && M>=5 && E>=2+B ], cost: 2+2*E-2*B

     35: f15 -> [12] : [ B>=1+A && M>=5 && E>=2+B && E>=2+A ], cost: 1-A+3*E-2*B

     36: f15 -> [12] : [ B>=1+A && 1-E+B==0 && M>=5 ], cost: 1+3*E-3*B



Computing asymptotic complexity for rule 32

   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:  [ B>=1+A && E>=1+B && 4>=M ]



NO


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

(2)
BOUNDS(INF, INF)