/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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, 394 ms]
(2) BOUNDS(INF, INF)


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

(0)
Obligation:
Complexity Int TRS consisting of the following rules:
f300(A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, R, S, T, U) -> Com_1(f1(V, W, X, Y, Z, F, G, H, I, J, K, L, M, N, O, P, Q, R, S, T, U)) :|: TRUE
f1(A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, R, S, T, U) -> Com_1(f1(A, B, C, D, E, F, G, 256, V, W, X, Y, Z, B1, C1, D1, Q, R, S, T, U)) :|: A1 >= 1 && F + 1 >= G && F + 1 <= G && H >= 256 && H <= 256
f1(A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, R, S, T, U) -> Com_1(f1(A, B, C, D, E, F, G, H, V, W, X, Y, Z, B1, C1, P, Q, R, S, T, U)) :|: 0 >= H && F + 1 >= G && F + 1 <= G
f1(A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, R, S, T, U) -> Com_1(f1(A, B, C, D, E, F, G, H, V, W, X, Y, Z, B1, C1, D1, 256, R, S, T, U)) :|: A1 >= 1 && G >= 1 + F && G >= 2 + F && Q >= 256 && Q <= 256
f1(A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, R, S, T, U) -> Com_1(f1(A, B, C, D, E, F, G, H, V, W, X, Y, Z, B1, C1, P, Q, R, S, T, U)) :|: 0 >= Q && G >= 1 + F && G >= 2 + F
f1(A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, R, S, T, U) -> Com_1(f2(A, B, C, D, E, F, G, H, V, W, K, L, M, N, O, P, Q, 0, 0, 0, U)) :|: F >= G
f1(A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, R, S, T, U) -> Com_1(f3(A, B, C, D, E, F, G, H, V, W, X, Y, Z, B1, C1, D1, Q, H, H, H, A1)) :|: H >= 1 && H >= 257 && F + 1 >= G && F + 1 <= G
f1(A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, R, S, T, U) -> Com_1(f3(A, B, C, D, E, F, G, H, V, W, X, Y, Z, B1, C1, D1, Q, H, H, H, A1)) :|: H >= 1 && 255 >= H && F + 1 >= G && F + 1 <= G
f1(A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, R, S, T, U) -> Com_1(f3(A, B, C, D, E, F, G, H, V, W, X, Y, Z, B1, C1, D1, Q, Q, Q, Q, A1)) :|: Q >= 1 && Q >= 257 && G >= 1 + F && G >= 2 + F
f1(A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, R, S, T, U) -> Com_1(f3(A, B, C, D, E, F, G, H, V, W, X, Y, Z, B1, C1, D1, Q, Q, Q, Q, A1)) :|: Q >= 1 && 255 >= Q && G >= 1 + F && G >= 2 + F

The start-symbols are:[f300_21]


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

(1) Loat Proof (FINISHED)


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



Initial linear ITS problem

   Start location: f300

      0: f300 -> f1 : A'=free_1, B'=free_3, C'=free_4, D'=free, E'=free_2, [], cost: 1

      1: f1 -> f1 : H'=256, Q'=free_7, J'=free_10, K'=free_13, L'=free_6, M'=free_9, N'=free_8, O'=free_5, P'=free_11, [ free_12>=1 && 1+F==G && H==256 ], cost: 1

      2: f1 -> f1 : Q'=free_16, J'=free_19, K'=free_20, L'=free_15, M'=free_18, N'=free_17, O'=free_14, [ 0>=H && 1+F==G ], cost: 1

      3: f1 -> f1 : Q'=free_23, J'=free_26, K'=free_29, L'=free_22, M'=free_25, N'=free_24, O'=free_21, P'=free_27, Q_1'=256, [ free_28>=1 && G>=1+F && G>=2+F && Q_1==256 ], cost: 1

      4: f1 -> f1 : Q'=free_32, J'=free_35, K'=free_36, L'=free_31, M'=free_34, N'=free_33, O'=free_30, [ 0>=Q_1 && G>=1+F && G>=2+F ], cost: 1

      5: f1 -> f2 : A1'=B, B'=C, B1'=D, C'=E, C1'=F, D'=G, D1'=H, E'=free_37, F'=free_38, G'=K, H'=L, Q'=M, J'=N, K'=O, L'=P, M'=Q_1, N'=0, O'=0, P'=0, Q_1'=U, [ F>=G ], cost: 1

      6: f1 -> f3 : A1'=B, B'=C, B1'=D, C'=E, C1'=F, D'=G, D1'=H, E'=free_41, F'=free_44, G'=free_47, H'=free_40, Q'=free_43, J'=free_42, K'=free_39, L'=free_45, M'=Q_1, N'=H, O'=H, P'=H, Q_1'=free_46, [ H>=1 && H>=257 && 1+F==G ], cost: 1

      7: f1 -> f3 : A1'=B, B'=C, B1'=D, C'=E, C1'=F, D'=G, D1'=H, E'=free_50, F'=free_53, G'=free_56, H'=free_49, Q'=free_52, J'=free_51, K'=free_48, L'=free_54, M'=Q_1, N'=H, O'=H, P'=H, Q_1'=free_55, [ H>=1 && 255>=H && 1+F==G ], cost: 1

      8: f1 -> f3 : A1'=B, B'=C, B1'=D, C'=E, C1'=F, D'=G, D1'=H, E'=free_59, F'=free_62, G'=free_65, H'=free_58, Q'=free_61, J'=free_60, K'=free_57, L'=free_63, M'=Q_1, N'=Q_1, O'=Q_1, P'=Q_1, Q_1'=free_64, [ Q_1>=1 && Q_1>=257 && G>=1+F && G>=2+F ], cost: 1

      9: f1 -> f3 : A1'=B, B'=C, B1'=D, C'=E, C1'=F, D'=G, D1'=H, E'=free_68, F'=free_71, G'=free_74, H'=free_67, Q'=free_70, J'=free_69, K'=free_66, L'=free_72, M'=Q_1, N'=Q_1, O'=Q_1, P'=Q_1, Q_1'=free_73, [ Q_1>=1 && 255>=Q_1 && G>=1+F && G>=2+F ], cost: 1



Checking for constant complexity:

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

      0: f300 -> f1 : A'=free_1, B'=free_3, C'=free_4, D'=free, E'=free_2, [], cost: 1



Removed unreachable and leaf rules:

   Start location: f300

      0: f300 -> f1 : A'=free_1, B'=free_3, C'=free_4, D'=free, E'=free_2, [], cost: 1

      1: f1 -> f1 : H'=256, Q'=free_7, J'=free_10, K'=free_13, L'=free_6, M'=free_9, N'=free_8, O'=free_5, P'=free_11, [ free_12>=1 && 1+F==G && H==256 ], cost: 1

      2: f1 -> f1 : Q'=free_16, J'=free_19, K'=free_20, L'=free_15, M'=free_18, N'=free_17, O'=free_14, [ 0>=H && 1+F==G ], cost: 1

      3: f1 -> f1 : Q'=free_23, J'=free_26, K'=free_29, L'=free_22, M'=free_25, N'=free_24, O'=free_21, P'=free_27, Q_1'=256, [ free_28>=1 && G>=1+F && G>=2+F && Q_1==256 ], cost: 1

      4: f1 -> f1 : Q'=free_32, J'=free_35, K'=free_36, L'=free_31, M'=free_34, N'=free_33, O'=free_30, [ 0>=Q_1 && G>=1+F && G>=2+F ], cost: 1



Simplified all rules, resulting in:

   Start location: f300

      0: f300 -> f1 : A'=free_1, B'=free_3, C'=free_4, D'=free, E'=free_2, [], cost: 1

      1: f1 -> f1 : H'=256, Q'=free_7, J'=free_10, K'=free_13, L'=free_6, M'=free_9, N'=free_8, O'=free_5, P'=free_11, [ 1+F==G && H==256 ], cost: 1

      2: f1 -> f1 : Q'=free_16, J'=free_19, K'=free_20, L'=free_15, M'=free_18, N'=free_17, O'=free_14, [ 0>=H && 1+F==G ], cost: 1

      3: f1 -> f1 : Q'=free_23, J'=free_26, K'=free_29, L'=free_22, M'=free_25, N'=free_24, O'=free_21, P'=free_27, Q_1'=256, [ G>=2+F && Q_1==256 ], cost: 1

      4: f1 -> f1 : Q'=free_32, J'=free_35, K'=free_36, L'=free_31, M'=free_34, N'=free_33, O'=free_30, [ 0>=Q_1 && G>=2+F ], cost: 1



### Simplification by acceleration and chaining ###



Accelerating simple loops of location 1.

   Accelerating the following rules:

      1: f1 -> f1 : H'=256, Q'=free_7, J'=free_10, K'=free_13, L'=free_6, M'=free_9, N'=free_8, O'=free_5, P'=free_11, [ 1+F==G && H==256 ], cost: 1

      2: f1 -> f1 : Q'=free_16, J'=free_19, K'=free_20, L'=free_15, M'=free_18, N'=free_17, O'=free_14, [ 0>=H && 1+F==G ], cost: 1

      3: f1 -> f1 : Q'=free_23, J'=free_26, K'=free_29, L'=free_22, M'=free_25, N'=free_24, O'=free_21, P'=free_27, Q_1'=256, [ G>=2+F && Q_1==256 ], cost: 1

      4: f1 -> f1 : Q'=free_32, J'=free_35, K'=free_36, L'=free_31, M'=free_34, N'=free_33, O'=free_30, [ 0>=Q_1 && G>=2+F ], cost: 1



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

   Accelerated rule 2 with NONTERM, yielding the new rule 11.

   Accelerated rule 3 with NONTERM, yielding the new rule 12.

   Accelerated rule 4 with NONTERM, yielding the new rule 13.

   Removing the simple loops: 1 2 3 4.



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

   Start location: f300

      0: f300 -> f1 : A'=free_1, B'=free_3, C'=free_4, D'=free, E'=free_2, [], cost: 1

     10: f1 -> [4] : [ 1+F==G && H==256 ], cost: NONTERM

     11: f1 -> [4] : [ 0>=H && 1+F==G ], cost: NONTERM

     12: f1 -> [4] : [ G>=2+F && Q_1==256 ], cost: NONTERM

     13: f1 -> [4] : [ 0>=Q_1 && G>=2+F ], cost: NONTERM



Chained accelerated rules (with incoming rules):

   Start location: f300

      0: f300 -> f1 : A'=free_1, B'=free_3, C'=free_4, D'=free, E'=free_2, [], cost: 1

     14: f300 -> [4] : A'=free_1, B'=free_3, C'=free_4, D'=free, E'=free_2, [ 1+F==G && H==256 ], cost: NONTERM

     15: f300 -> [4] : A'=free_1, B'=free_3, C'=free_4, D'=free, E'=free_2, [ 0>=H && 1+F==G ], cost: NONTERM

     16: f300 -> [4] : A'=free_1, B'=free_3, C'=free_4, D'=free, E'=free_2, [ G>=2+F && Q_1==256 ], cost: NONTERM

     17: f300 -> [4] : A'=free_1, B'=free_3, C'=free_4, D'=free, E'=free_2, [ 0>=Q_1 && G>=2+F ], cost: NONTERM



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

   Start location: f300

     14: f300 -> [4] : A'=free_1, B'=free_3, C'=free_4, D'=free, E'=free_2, [ 1+F==G && H==256 ], cost: NONTERM

     15: f300 -> [4] : A'=free_1, B'=free_3, C'=free_4, D'=free, E'=free_2, [ 0>=H && 1+F==G ], cost: NONTERM

     16: f300 -> [4] : A'=free_1, B'=free_3, C'=free_4, D'=free, E'=free_2, [ G>=2+F && Q_1==256 ], cost: NONTERM

     17: f300 -> [4] : A'=free_1, B'=free_3, C'=free_4, D'=free, E'=free_2, [ 0>=Q_1 && G>=2+F ], cost: NONTERM



### Computing asymptotic complexity ###



Fully simplified ITS problem

   Start location: f300

     14: f300 -> [4] : A'=free_1, B'=free_3, C'=free_4, D'=free, E'=free_2, [ 1+F==G && H==256 ], cost: NONTERM

     15: f300 -> [4] : A'=free_1, B'=free_3, C'=free_4, D'=free, E'=free_2, [ 0>=H && 1+F==G ], cost: NONTERM

     16: f300 -> [4] : A'=free_1, B'=free_3, C'=free_4, D'=free, E'=free_2, [ G>=2+F && Q_1==256 ], cost: NONTERM

     17: f300 -> [4] : A'=free_1, B'=free_3, C'=free_4, D'=free, E'=free_2, [ 0>=Q_1 && G>=2+F ], 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:  [ 1+F==G && H==256 ]



NO


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(2)
BOUNDS(INF, INF)