/export/starexec/sandbox/solver/bin/starexec_run_complexity /export/starexec/sandbox/benchmark/theBenchmark.koat /export/starexec/sandbox/output/output_files


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


KILLED
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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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(1, INF).

(0) CpxIntTrs
(1) Loat Proof [FINISHED, 1498 ms]
(2) BOUNDS(1, INF)


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

(0)
Obligation:
Complexity Int TRS consisting of the following rules:
f0(A, B) -> Com_1(f2(C, D)) :|: C >= 1 && C >= 5 && C >= 2 && C >= 3
f0(A, B) -> Com_1(f2(1, C)) :|: 0 >= 4 && 0 >= 1
f0(A, B) -> Com_1(f2(C, D)) :|: C >= 1 && C >= 5 && 0 >= C && C >= 3
f0(A, B) -> Com_1(f2(3, C)) :|: TRUE
f0(A, B) -> Com_1(f2(1, C)) :|: 0 >= 1
f0(A, B) -> Com_1(f2(3, C)) :|: 0 >= 3
f2(A, B) -> Com_1(f2(B, C)) :|: B >= 5 && B >= 2 && B >= 3 && A >= 2 * B && A <= 2 * B
f2(A, B) -> Com_1(f2(B, C)) :|: B >= 5 && B >= 2 && 1 >= B && A >= 2 * B && A <= 2 * B
f2(A, B) -> Com_1(f2(B, C)) :|: B >= 5 && 0 >= B && B >= 3 && A >= 2 * B && A <= 2 * B
f2(A, B) -> Com_1(f2(B, C)) :|: B >= 5 && 0 >= B && 1 >= B && A >= 2 * B && A <= 2 * B
f2(A, B) -> Com_1(f2(B, C)) :|: B >= 3 && B <= 3 && A >= 6 && A <= 6
f2(A, B) -> Com_1(f2(B, C)) :|: 3 >= B && B >= 2 && 1 >= B && A >= 2 * B && A <= 2 * B
f2(A, B) -> Com_1(f2(B, C)) :|: 0 >= 3 && B >= 3 && B <= 3 && A >= 6 && A <= 6
f2(A, B) -> Com_1(f2(B, C)) :|: 3 >= B && 0 >= B && 1 >= B && A >= 2 * B && A <= 2 * B
f2(A, B) -> Com_1(f2(6 * B + 4, C)) :|: 6 * B >= 1 && 6 * B + 2 >= 0 && 6 * B + 1 >= 0 && A >= 2 * B + 1 && A <= 2 * B + 1
f2(A, B) -> Com_1(f2(6 * B + 4, C)) :|: 6 * B >= 1 && 6 * B + 2 >= 0 && 0 >= 3 + 6 * B && A >= 2 * B + 1 && A <= 2 * B + 1
f2(A, B) -> Com_1(f2(6 * B + 4, C)) :|: 6 * B >= 1 && 0 >= 4 + 6 * B && 6 * B + 1 >= 0 && A >= 2 * B + 1 && A <= 2 * B + 1
f2(A, B) -> Com_1(f2(6 * B + 4, C)) :|: 6 * B >= 1 && 0 >= 4 + 6 * B && 0 >= 3 + 6 * B && A >= 2 * B + 1 && A <= 2 * B + 1
f2(A, B) -> Com_1(f2(6 * B + 4, C)) :|: 0 >= 6 * B + 1 && 6 * B + 2 >= 0 && 6 * B + 1 >= 0 && A >= 2 * B + 1 && A <= 2 * B + 1
f2(A, B) -> Com_1(f2(6 * B + 4, C)) :|: 0 >= 6 * B + 1 && 6 * B + 2 >= 0 && 0 >= 3 + 6 * B && A >= 2 * B + 1 && A <= 2 * B + 1
f2(A, B) -> Com_1(f2(6 * B + 4, C)) :|: 0 >= 6 * B + 1 && 0 >= 4 + 6 * B && 6 * B + 1 >= 0 && A >= 2 * B + 1 && A <= 2 * B + 1
f2(A, B) -> Com_1(f2(6 * B + 4, C)) :|: 0 >= 6 * B + 1 && 0 >= 4 + 6 * B && 0 >= 3 + 6 * B && A >= 2 * B + 1 && A <= 2 * B + 1

The start-symbols are:[f0_2]


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

(1) Loat Proof (FINISHED)


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



Initial linear ITS problem

   Start location: f0

      0: f0 -> f2 : A'=free, B'=free_1, [ free>=1 && free>=5 && free>=2 && free>=3 ], cost: 1

      1: f0 -> f2 : A'=1, B'=free_2, [ 0>=4 && 0>=1 ], cost: 1

      2: f0 -> f2 : A'=free_3, B'=free_4, [ free_3>=1 && free_3>=5 && 0>=free_3 && free_3>=3 ], cost: 1

      3: f0 -> f2 : A'=3, B'=free_5, [], cost: 1

      4: f0 -> f2 : A'=1, B'=free_6, [ 0>=1 ], cost: 1

      5: f0 -> f2 : A'=3, B'=free_7, [ 0>=3 ], cost: 1

      6: f2 -> f2 : A'=B, B'=free_8, [ B>=5 && B>=2 && B>=3 && A==2*B ], cost: 1

      7: f2 -> f2 : A'=B, B'=free_9, [ B>=5 && B>=2 && 1>=B && A==2*B ], cost: 1

      8: f2 -> f2 : A'=B, B'=free_10, [ B>=5 && 0>=B && B>=3 && A==2*B ], cost: 1

      9: f2 -> f2 : A'=B, B'=free_11, [ B>=5 && 0>=B && 1>=B && A==2*B ], cost: 1

     10: f2 -> f2 : A'=B, B'=free_12, [ B==3 && A==6 ], cost: 1

     11: f2 -> f2 : A'=B, B'=free_13, [ 3>=B && B>=2 && 1>=B && A==2*B ], cost: 1

     12: f2 -> f2 : A'=B, B'=free_14, [ 0>=3 && B==3 && A==6 ], cost: 1

     13: f2 -> f2 : A'=B, B'=free_15, [ 3>=B && 0>=B && 1>=B && A==2*B ], cost: 1

     14: f2 -> f2 : A'=4+6*B, B'=free_16, [ 6*B>=1 && 2+6*B>=0 && 1+6*B>=0 && A==1+2*B ], cost: 1

     15: f2 -> f2 : A'=4+6*B, B'=free_17, [ 6*B>=1 && 2+6*B>=0 && 0>=3+6*B && A==1+2*B ], cost: 1

     16: f2 -> f2 : A'=4+6*B, B'=free_18, [ 6*B>=1 && 0>=4+6*B && 1+6*B>=0 && A==1+2*B ], cost: 1

     17: f2 -> f2 : A'=4+6*B, B'=free_19, [ 6*B>=1 && 0>=4+6*B && 0>=3+6*B && A==1+2*B ], cost: 1

     18: f2 -> f2 : A'=4+6*B, B'=free_20, [ 0>=1+6*B && 2+6*B>=0 && 1+6*B>=0 && A==1+2*B ], cost: 1

     19: f2 -> f2 : A'=4+6*B, B'=free_21, [ 0>=1+6*B && 2+6*B>=0 && 0>=3+6*B && A==1+2*B ], cost: 1

     20: f2 -> f2 : A'=4+6*B, B'=free_22, [ 0>=1+6*B && 0>=4+6*B && 1+6*B>=0 && A==1+2*B ], cost: 1

     21: f2 -> f2 : A'=4+6*B, B'=free_23, [ 0>=1+6*B && 0>=4+6*B && 0>=3+6*B && A==1+2*B ], cost: 1



Checking for constant complexity:

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

      0: f0 -> f2 : A'=free, B'=free_1, [ free>=1 && free>=5 && free>=2 && free>=3 ], cost: 1



Removed rules with unsatisfiable guard:

   Start location: f0

      0: f0 -> f2 : A'=free, B'=free_1, [ free>=1 && free>=5 && free>=2 && free>=3 ], cost: 1

      3: f0 -> f2 : A'=3, B'=free_5, [], cost: 1

      6: f2 -> f2 : A'=B, B'=free_8, [ B>=5 && B>=2 && B>=3 && A==2*B ], cost: 1

     10: f2 -> f2 : A'=B, B'=free_12, [ B==3 && A==6 ], cost: 1

     13: f2 -> f2 : A'=B, B'=free_15, [ 3>=B && 0>=B && 1>=B && A==2*B ], cost: 1

     14: f2 -> f2 : A'=4+6*B, B'=free_16, [ 6*B>=1 && 2+6*B>=0 && 1+6*B>=0 && A==1+2*B ], cost: 1

     21: f2 -> f2 : A'=4+6*B, B'=free_23, [ 0>=1+6*B && 0>=4+6*B && 0>=3+6*B && A==1+2*B ], cost: 1



Simplified all rules, resulting in:

   Start location: f0

      0: f0 -> f2 : A'=free, B'=free_1, [ free>=5 ], cost: 1

      3: f0 -> f2 : A'=3, B'=free_5, [], cost: 1

      6: f2 -> f2 : A'=B, B'=free_8, [ B>=5 && A==2*B ], cost: 1

     10: f2 -> f2 : A'=B, B'=free_12, [ B==3 && A==6 ], cost: 1

     13: f2 -> f2 : A'=B, B'=free_15, [ 0>=B && A==2*B ], cost: 1

     14: f2 -> f2 : A'=4+6*B, B'=free_16, [ 6*B>=1 && A==1+2*B ], cost: 1

     21: f2 -> f2 : A'=4+6*B, B'=free_23, [ 0>=4+6*B && A==1+2*B ], cost: 1



### Simplification by acceleration and chaining ###



Accelerating simple loops of location 1.

   Accelerating the following rules:

      6: f2 -> f2 : A'=B, B'=free_8, [ B>=5 && A==2*B ], cost: 1

     10: f2 -> f2 : A'=B, B'=free_12, [ B==3 && A==6 ], cost: 1

     13: f2 -> f2 : A'=B, B'=free_15, [ 0>=B && A==2*B ], cost: 1

     14: f2 -> f2 : A'=4+6*B, B'=free_16, [ 6*B>=1 && A==1+2*B ], cost: 1

     21: f2 -> f2 : A'=4+6*B, B'=free_23, [ 0>=4+6*B && A==1+2*B ], cost: 1



      During metering: Instantiating temporary variables by {free_8==5}

   Accelerated rule 6 with metering function meter (where 5*meter==-9+B) (after strengthening guard), yielding the new rule 22.

   Accelerated rule 10 with metering function meter_1 (where 3*meter_1==-6+A), yielding the new rule 23.

   Found no metering function for rule 13.

   Accelerated rule 14 with NONTERM (after strengthening guard), yielding the new rule 24.

   Accelerated rule 21 with NONTERM (after strengthening guard), yielding the new rule 25.

   Removing the simple loops: 10.



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

   Start location: f0

      0: f0 -> f2 : A'=free, B'=free_1, [ free>=5 ], cost: 1

      3: f0 -> f2 : A'=3, B'=free_5, [], cost: 1

      6: f2 -> f2 : A'=B, B'=free_8, [ B>=5 && A==2*B ], cost: 1

     13: f2 -> f2 : A'=B, B'=free_15, [ 0>=B && A==2*B ], cost: 1

     14: f2 -> f2 : A'=4+6*B, B'=free_16, [ 6*B>=1 && A==1+2*B ], cost: 1

     21: f2 -> f2 : A'=4+6*B, B'=free_23, [ 0>=4+6*B && A==1+2*B ], cost: 1

     22: f2 -> f2 : A'=5, B'=5, [ A==2*B && B==10 && 5*meter==-9+B && meter>=1 ], cost: meter

     23: f2 -> f2 : A'=free_12, B'=free_12, [ B==3 && A==6 && 3*meter_1==-6+A && meter_1>=1 ], cost: meter_1

     24: f2 -> [2] : [ 6*B>=1 && A==1+2*B && 6*free_16>=1 && 4+6*B==1+2*free_16 ], cost: NONTERM

     25: f2 -> [2] : [ 0>=4+6*B && A==1+2*B && 0>=4+6*free_23 && 4+6*B==1+2*free_23 ], cost: NONTERM



Chained accelerated rules (with incoming rules):

   Start location: f0

      0: f0 -> f2 : A'=free, B'=free_1, [ free>=5 ], cost: 1

      3: f0 -> f2 : A'=3, B'=free_5, [], cost: 1

     26: f0 -> f2 : A'=free_1, B'=free_8, [ 2*free_1>=5 && free_1>=5 ], cost: 2

     27: f0 -> f2 : A'=4+6*free_1, B'=free_16, [ 1+2*free_1>=5 && 6*free_1>=1 ], cost: 2

     28: f0 -> f2 : A'=10, B'=free_16, [], cost: 2



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

   Start location: f0

     <empty>



### Computing asymptotic complexity ###



Fully simplified ITS problem

   Start location: f0

     <empty>



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

   Complexity:  Constant

   Cpx degree:  0

   Solved cost: 1

   Rule cost:   1

   Rule guard:  [ free>=1 && free>=5 && free>=2 && free>=3 ]



WORST_CASE(Omega(1),?)


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

(2)
BOUNDS(1, INF)