/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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, 212 ms]
(2) BOUNDS(INF, INF)


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(0)
Obligation:
Complexity Int TRS consisting of the following rules:
f1(A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P) -> Com_1(f1(A, 1 + B, D, Q, D, R, B, H, I, J, K, L, M, N, O, P)) :|: A >= B + 1 && B >= 0
f1(A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P) -> Com_1(f4(R, S, U, X, W, F, G, Q, T, V, Y, Z, C, N, O, P)) :|: B >= A && B >= 0 && S >= Q && Q >= 2
f3(A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P) -> Com_1(f4(T, V, S, Y, X, F, G, R, U, W, Z, B1, O, Q, O, P)) :|: 0 >= R && 0 >= A1
f3(A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P) -> Com_1(f1(R, 2, U, S, U, F, G, R, T, U, K, L, M, Q, T, V)) :|: R >= 2
f3(A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P) -> Com_1(f4(R, S, U, Y, X, F, G, 1, T, W, Z, B1, D, Q, V, P)) :|: TRUE

The start-symbols are:[f3_16]


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(1) Loat Proof (FINISHED)


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



Initial linear ITS problem

   Start location: f3

      0: f1 -> f1 : B'=1+B, C'=D, D'=free, E'=D, F'=free_1, G'=B, [ A>=1+B && B>=0 ], cost: 1

      1: f1 -> f4 : A'=free_2, A1'=free_5, B'=free_11, B1'=free_4, C'=free_7, D'=F, E'=G, F'=free_8, G'=free_10, H'=free_3, Q'=free_6, J'=free_9, K'=C, L'=N, M'=O, N'=P, [ B>=A && B>=0 && free_5>=free_8 && free_8>=2 ], cost: 1

      2: f3 -> f4 : A'=free_12, A1'=free_15, B'=free_23, B1'=free_14, C'=free_18, D'=F, E'=G, F'=free_19, G'=free_22, H'=free_13, Q'=free_17, J'=free_21, K'=N, L'=N, M'=free_16, N'=P, [ 0>=free_19 && 0>=free_20 ], cost: 1

      3: f3 -> f1 : A'=free_24, B'=2, C'=free_26, D'=free_29, E'=free_26, H'=free_24, Q'=free_25, J'=free_26, N'=free_25, O'=free_27, P'=free_28, [ free_24>=2 ], cost: 1

      4: f3 -> f4 : A'=free_30, A1'=free_33, B'=free_40, B1'=free_32, C'=free_36, D'=F, E'=G, F'=1, G'=free_37, H'=free_39, Q'=free_31, J'=free_35, K'=D, L'=free_34, M'=free_38, N'=P, [], cost: 1



Checking for constant complexity:

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

      2: f3 -> f4 : A'=free_12, A1'=free_15, B'=free_23, B1'=free_14, C'=free_18, D'=F, E'=G, F'=free_19, G'=free_22, H'=free_13, Q'=free_17, J'=free_21, K'=N, L'=N, M'=free_16, N'=P, [ 0>=free_19 && 0>=free_20 ], cost: 1



Removed unreachable and leaf rules:

   Start location: f3

      0: f1 -> f1 : B'=1+B, C'=D, D'=free, E'=D, F'=free_1, G'=B, [ A>=1+B && B>=0 ], cost: 1

      3: f3 -> f1 : A'=free_24, B'=2, C'=free_26, D'=free_29, E'=free_26, H'=free_24, Q'=free_25, J'=free_26, N'=free_25, O'=free_27, P'=free_28, [ free_24>=2 ], cost: 1



### Simplification by acceleration and chaining ###



Accelerating simple loops of location 0.

   Accelerating the following rules:

      0: f1 -> f1 : B'=1+B, C'=D, D'=free, E'=D, F'=free_1, G'=B, [ A>=1+B && B>=0 ], cost: 1



   Accelerated rule 0 with metering function -B+A, yielding the new rule 5.

   Removing the simple loops: 0.



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

   Start location: f3

      5: f1 -> f1 : B'=A, C'=free, D'=free, E'=free, F'=free_1, G'=-1+A, [ A>=1+B && B>=0 ], cost: -B+A

      3: f3 -> f1 : A'=free_24, B'=2, C'=free_26, D'=free_29, E'=free_26, H'=free_24, Q'=free_25, J'=free_26, N'=free_25, O'=free_27, P'=free_28, [ free_24>=2 ], cost: 1



Chained accelerated rules (with incoming rules):

   Start location: f3

      3: f3 -> f1 : A'=free_24, B'=2, C'=free_26, D'=free_29, E'=free_26, H'=free_24, Q'=free_25, J'=free_26, N'=free_25, O'=free_27, P'=free_28, [ free_24>=2 ], cost: 1

      6: f3 -> f1 : A'=free_24, B'=free_24, C'=free, D'=free, E'=free, F'=free_1, G'=-1+free_24, H'=free_24, Q'=free_25, J'=free_26, N'=free_25, O'=free_27, P'=free_28, [ free_24>=3 ], cost: -1+free_24



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

   Start location: f3

      6: f3 -> f1 : A'=free_24, B'=free_24, C'=free, D'=free, E'=free, F'=free_1, G'=-1+free_24, H'=free_24, Q'=free_25, J'=free_26, N'=free_25, O'=free_27, P'=free_28, [ free_24>=3 ], cost: -1+free_24



### Computing asymptotic complexity ###



Fully simplified ITS problem

   Start location: f3

      6: f3 -> f1 : A'=free_24, B'=free_24, C'=free, D'=free, E'=free, F'=free_1, G'=-1+free_24, H'=free_24, Q'=free_25, J'=free_26, N'=free_25, O'=free_27, P'=free_28, [ free_24>=3 ], cost: -1+free_24



Computing asymptotic complexity for rule 6

   Solved the limit problem by the following transformations:

      Created initial limit problem:

      -2+free_24 (+/+!), -1+free_24 (+) [not solved]



      removing all constraints (solved by SMT)

      resulting limit problem: [solved]



      applying transformation rule (C) using substitution {free_24==n}

      resulting limit problem:

      [solved]



   Solution:

      free_24 / n

   Resulting cost -1+n has complexity: Unbounded



Found new complexity Unbounded.



Obtained the following overall complexity (w.r.t. the length of the input n):

   Complexity:  Unbounded

   Cpx degree:  Unbounded

   Solved cost: -1+n

   Rule cost:   -1+free_24

   Rule guard:  [ free_24>=3 ]



WORST_CASE(INF,?)


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(2)
BOUNDS(INF, INF)