/export/starexec/sandbox2/solver/bin/starexec_run_complexity /export/starexec/sandbox2/benchmark/theBenchmark.koat /export/starexec/sandbox2/output/output_files


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


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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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, 2144 ms]
(2) BOUNDS(INF, INF)


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

(0)
Obligation:
Complexity Int TRS consisting of the following rules:
f0(A, B, C, D, E, F, G, H) -> Com_1(f15(10, 35, 285, I, I, 0, G, H)) :|: TRUE
f15(A, B, C, D, E, F, G, H) -> Com_1(f25(A, B, C, D, E, F, I, 1)) :|: A >= F + 1
f25(A, B, C, D, E, F, G, H) -> Com_1(f25(A, B, C, D, E, F, I, H + 1)) :|: E >= H + 1
f41(A, B, C, D, E, F, G, H) -> Com_1(f15(A, B, C, D, E, F + 1, G, H)) :|: E >= B
f41(A, B, C, D, E, F, G, H) -> Com_1(f15(A, B, C, D, E + 1, F + 1, G, H)) :|: B >= E + 1
f25(A, B, C, D, E, F, G, H) -> Com_1(f41(A, B, C, D, E, F, G, H)) :|: H >= E && I >= J + 1
f25(A, B, C, D, E, F, G, H) -> Com_1(f41(A, B, C, D, E, F, G, H)) :|: H >= E
f25(A, B, C, D, E, F, G, H) -> Com_1(f15(A, B, C, D, E - 1, F + 1, G, H)) :|: H >= E
f15(A, B, C, D, E, F, G, H) -> Com_1(f48(A, B, C, D, E, F, G, H)) :|: F >= A

The start-symbols are:[f0_8]


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

(1) Loat Proof (FINISHED)


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



Initial linear ITS problem

   Start location: f0

      0: f0 -> f15 : A'=10, B'=35, C'=285, D'=free, E'=free, F'=0, [], cost: 1

      1: f15 -> f25 : G'=free_1, H'=1, [ A>=1+F ], cost: 1

      8: f15 -> f48 : [ F>=A ], cost: 1

      2: f25 -> f25 : G'=free_2, H'=1+H, [ E>=1+H ], cost: 1

      5: f25 -> f41 : [ H>=E && free_4>=1+free_3 ], cost: 1

      6: f25 -> f41 : [ H>=E ], cost: 1

      7: f25 -> f15 : E'=-1+E, F'=1+F, [ H>=E ], cost: 1

      3: f41 -> f15 : F'=1+F, [ E>=B ], cost: 1

      4: f41 -> f15 : E'=1+E, F'=1+F, [ B>=1+E ], cost: 1



Checking for constant complexity:

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

      0: f0 -> f15 : A'=10, B'=35, C'=285, D'=free, E'=free, F'=0, [], cost: 1



Removed unreachable and leaf rules:

   Start location: f0

      0: f0 -> f15 : A'=10, B'=35, C'=285, D'=free, E'=free, F'=0, [], cost: 1

      1: f15 -> f25 : G'=free_1, H'=1, [ A>=1+F ], cost: 1

      2: f25 -> f25 : G'=free_2, H'=1+H, [ E>=1+H ], cost: 1

      5: f25 -> f41 : [ H>=E && free_4>=1+free_3 ], cost: 1

      6: f25 -> f41 : [ H>=E ], cost: 1

      7: f25 -> f15 : E'=-1+E, F'=1+F, [ H>=E ], cost: 1

      3: f41 -> f15 : F'=1+F, [ E>=B ], cost: 1

      4: f41 -> f15 : E'=1+E, F'=1+F, [ B>=1+E ], cost: 1



Simplified all rules, resulting in:

   Start location: f0

      0: f0 -> f15 : A'=10, B'=35, C'=285, D'=free, E'=free, F'=0, [], cost: 1

      1: f15 -> f25 : G'=free_1, H'=1, [ A>=1+F ], cost: 1

      2: f25 -> f25 : G'=free_2, H'=1+H, [ E>=1+H ], cost: 1

      6: f25 -> f41 : [ H>=E ], cost: 1

      7: f25 -> f15 : E'=-1+E, F'=1+F, [ H>=E ], cost: 1

      3: f41 -> f15 : F'=1+F, [ E>=B ], cost: 1

      4: f41 -> f15 : E'=1+E, F'=1+F, [ B>=1+E ], cost: 1



### Simplification by acceleration and chaining ###



Accelerating simple loops of location 2.

   Accelerating the following rules:

      2: f25 -> f25 : G'=free_2, H'=1+H, [ E>=1+H ], cost: 1



   Accelerated rule 2 with metering function -H+E, yielding the new rule 9.

   Removing the simple loops: 2.



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

   Start location: f0

      0: f0 -> f15 : A'=10, B'=35, C'=285, D'=free, E'=free, F'=0, [], cost: 1

      1: f15 -> f25 : G'=free_1, H'=1, [ A>=1+F ], cost: 1

      6: f25 -> f41 : [ H>=E ], cost: 1

      7: f25 -> f15 : E'=-1+E, F'=1+F, [ H>=E ], cost: 1

      9: f25 -> f25 : G'=free_2, H'=E, [ E>=1+H ], cost: -H+E

      3: f41 -> f15 : F'=1+F, [ E>=B ], cost: 1

      4: f41 -> f15 : E'=1+E, F'=1+F, [ B>=1+E ], cost: 1



Chained accelerated rules (with incoming rules):

   Start location: f0

      0: f0 -> f15 : A'=10, B'=35, C'=285, D'=free, E'=free, F'=0, [], cost: 1

      1: f15 -> f25 : G'=free_1, H'=1, [ A>=1+F ], cost: 1

     10: f15 -> f25 : G'=free_2, H'=E, [ A>=1+F && E>=2 ], cost: E

      6: f25 -> f41 : [ H>=E ], cost: 1

      7: f25 -> f15 : E'=-1+E, F'=1+F, [ H>=E ], cost: 1

      3: f41 -> f15 : F'=1+F, [ E>=B ], cost: 1

      4: f41 -> f15 : E'=1+E, F'=1+F, [ B>=1+E ], cost: 1



Eliminated locations (on tree-shaped paths):

   Start location: f0

      0: f0 -> f15 : A'=10, B'=35, C'=285, D'=free, E'=free, F'=0, [], cost: 1

     11: f15 -> f41 : G'=free_1, H'=1, [ A>=1+F && 1>=E ], cost: 2

     12: f15 -> f15 : E'=-1+E, F'=1+F, G'=free_1, H'=1, [ A>=1+F && 1>=E ], cost: 2

     13: f15 -> f41 : G'=free_2, H'=E, [ A>=1+F && E>=2 ], cost: 1+E

     14: f15 -> f15 : E'=-1+E, F'=1+F, G'=free_2, H'=E, [ A>=1+F && E>=2 ], cost: 1+E

      3: f41 -> f15 : F'=1+F, [ E>=B ], cost: 1

      4: f41 -> f15 : E'=1+E, F'=1+F, [ B>=1+E ], cost: 1



Accelerating simple loops of location 1.

   Accelerating the following rules:

     12: f15 -> f15 : E'=-1+E, F'=1+F, G'=free_1, H'=1, [ A>=1+F && 1>=E ], cost: 2

     14: f15 -> f15 : E'=-1+E, F'=1+F, G'=free_2, H'=E, [ A>=1+F && E>=2 ], cost: 1+E



   Accelerated rule 12 with metering function -F+A, yielding the new rule 15.

   Found no metering function for rule 14.

   Removing the simple loops: 12.



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

   Start location: f0

      0: f0 -> f15 : A'=10, B'=35, C'=285, D'=free, E'=free, F'=0, [], cost: 1

     11: f15 -> f41 : G'=free_1, H'=1, [ A>=1+F && 1>=E ], cost: 2

     13: f15 -> f41 : G'=free_2, H'=E, [ A>=1+F && E>=2 ], cost: 1+E

     14: f15 -> f15 : E'=-1+E, F'=1+F, G'=free_2, H'=E, [ A>=1+F && E>=2 ], cost: 1+E

     15: f15 -> f15 : E'=F-A+E, F'=A, G'=free_1, H'=1, [ A>=1+F && 1>=E ], cost: -2*F+2*A

      3: f41 -> f15 : F'=1+F, [ E>=B ], cost: 1

      4: f41 -> f15 : E'=1+E, F'=1+F, [ B>=1+E ], cost: 1



Chained accelerated rules (with incoming rules):

   Start location: f0

      0: f0 -> f15 : A'=10, B'=35, C'=285, D'=free, E'=free, F'=0, [], cost: 1

     16: f0 -> f15 : A'=10, B'=35, C'=285, D'=free, E'=-1+free, F'=1, G'=free_2, H'=free, [ free>=2 ], cost: 2+free

     19: f0 -> f15 : A'=10, B'=35, C'=285, D'=free, E'=-10+free, F'=10, G'=free_1, H'=1, [ 1>=free ], cost: 21

     11: f15 -> f41 : G'=free_1, H'=1, [ A>=1+F && 1>=E ], cost: 2

     13: f15 -> f41 : G'=free_2, H'=E, [ A>=1+F && E>=2 ], cost: 1+E

      3: f41 -> f15 : F'=1+F, [ E>=B ], cost: 1

      4: f41 -> f15 : E'=1+E, F'=1+F, [ B>=1+E ], cost: 1

     17: f41 -> f15 : E'=-1+E, F'=2+F, G'=free_2, H'=E, [ E>=B && A>=2+F && E>=2 ], cost: 2+E

     18: f41 -> f15 : F'=2+F, G'=free_2, H'=1+E, [ B>=1+E && A>=2+F && 1+E>=2 ], cost: 3+E

     20: f41 -> f15 : E'=1+F-A+E, F'=A, G'=free_1, H'=1, [ E>=B && A>=2+F && 1>=E ], cost: -1-2*F+2*A

     21: f41 -> f15 : E'=2+F-A+E, F'=A, G'=free_1, H'=1, [ B>=1+E && A>=2+F && 1>=1+E ], cost: -1-2*F+2*A



Eliminated locations (on tree-shaped paths):

   Start location: f0

      0: f0 -> f15 : A'=10, B'=35, C'=285, D'=free, E'=free, F'=0, [], cost: 1

     16: f0 -> f15 : A'=10, B'=35, C'=285, D'=free, E'=-1+free, F'=1, G'=free_2, H'=free, [ free>=2 ], cost: 2+free

     19: f0 -> f15 : A'=10, B'=35, C'=285, D'=free, E'=-10+free, F'=10, G'=free_1, H'=1, [ 1>=free ], cost: 21

     22: f15 -> f15 : F'=1+F, G'=free_1, H'=1, [ A>=1+F && 1>=E && E>=B ], cost: 3

     23: f15 -> f15 : E'=1+E, F'=1+F, G'=free_1, H'=1, [ A>=1+F && 1>=E && B>=1+E ], cost: 3

     24: f15 -> f15 : F'=2+F, G'=free_2, H'=1+E, [ 1>=E && B>=1+E && A>=2+F && 1+E>=2 ], cost: 5+E

     25: f15 -> f15 : E'=1+F-A+E, F'=A, G'=free_1, H'=1, [ 1>=E && E>=B && A>=2+F ], cost: 1-2*F+2*A

     26: f15 -> f15 : E'=2+F-A+E, F'=A, G'=free_1, H'=1, [ B>=1+E && A>=2+F && 1>=1+E ], cost: 1-2*F+2*A

     27: f15 -> f15 : F'=1+F, G'=free_2, H'=E, [ A>=1+F && E>=2 && E>=B ], cost: 2+E

     28: f15 -> f15 : E'=1+E, F'=1+F, G'=free_2, H'=E, [ A>=1+F && E>=2 && B>=1+E ], cost: 2+E

     29: f15 -> f15 : E'=-1+E, F'=2+F, G'=free_2, H'=E, [ E>=2 && E>=B && A>=2+F ], cost: 3+2*E

     30: f15 -> f15 : F'=2+F, G'=free_2, H'=1+E, [ E>=2 && B>=1+E && A>=2+F ], cost: 4+2*E

     31: f15 -> [7] : [ A>=1+F && E>=2 ], cost: 1+E



Applied pruning (of leafs and parallel rules):

   Start location: f0

      0: f0 -> f15 : A'=10, B'=35, C'=285, D'=free, E'=free, F'=0, [], cost: 1

     16: f0 -> f15 : A'=10, B'=35, C'=285, D'=free, E'=-1+free, F'=1, G'=free_2, H'=free, [ free>=2 ], cost: 2+free

     19: f0 -> f15 : A'=10, B'=35, C'=285, D'=free, E'=-10+free, F'=10, G'=free_1, H'=1, [ 1>=free ], cost: 21

     25: f15 -> f15 : E'=1+F-A+E, F'=A, G'=free_1, H'=1, [ 1>=E && E>=B && A>=2+F ], cost: 1-2*F+2*A

     26: f15 -> f15 : E'=2+F-A+E, F'=A, G'=free_1, H'=1, [ B>=1+E && A>=2+F && 1>=1+E ], cost: 1-2*F+2*A

     27: f15 -> f15 : F'=1+F, G'=free_2, H'=E, [ A>=1+F && E>=2 && E>=B ], cost: 2+E

     28: f15 -> f15 : E'=1+E, F'=1+F, G'=free_2, H'=E, [ A>=1+F && E>=2 && B>=1+E ], cost: 2+E

     30: f15 -> f15 : F'=2+F, G'=free_2, H'=1+E, [ E>=2 && B>=1+E && A>=2+F ], cost: 4+2*E

     31: f15 -> [7] : [ A>=1+F && E>=2 ], cost: 1+E



Accelerating simple loops of location 1.

   Accelerating the following rules:

     25: f15 -> f15 : E'=1+F-A+E, F'=A, G'=free_1, H'=1, [ 1>=E && E>=B && A>=2+F ], cost: 1-2*F+2*A

     26: f15 -> f15 : E'=2+F-A+E, F'=A, G'=free_1, H'=1, [ B>=1+E && A>=2+F && 1>=1+E ], cost: 1-2*F+2*A

     27: f15 -> f15 : F'=1+F, G'=free_2, H'=E, [ A>=1+F && E>=2 && E>=B ], cost: 2+E

     28: f15 -> f15 : E'=1+E, F'=1+F, G'=free_2, H'=E, [ A>=1+F && E>=2 && B>=1+E ], cost: 2+E

     30: f15 -> f15 : F'=2+F, G'=free_2, H'=1+E, [ E>=2 && B>=1+E && A>=2+F ], cost: 4+2*E



   Found no metering function for rule 25.

   Found no metering function for rule 26.

   Accelerated rule 27 with metering function -F+A, yielding the new rule 32.

   Found no metering function for rule 28.

   Accelerated rule 30 with metering function meter (where 2*meter==-1-F+A), yielding the new rule 33.

      During metering: Instantiating temporary variables by {meter==1}

      During metering: Instantiating temporary variables by {meter==1}

   Removing the simple loops: 27 30.



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

   Start location: f0

      0: f0 -> f15 : A'=10, B'=35, C'=285, D'=free, E'=free, F'=0, [], cost: 1

     16: f0 -> f15 : A'=10, B'=35, C'=285, D'=free, E'=-1+free, F'=1, G'=free_2, H'=free, [ free>=2 ], cost: 2+free

     19: f0 -> f15 : A'=10, B'=35, C'=285, D'=free, E'=-10+free, F'=10, G'=free_1, H'=1, [ 1>=free ], cost: 21

     25: f15 -> f15 : E'=1+F-A+E, F'=A, G'=free_1, H'=1, [ 1>=E && E>=B && A>=2+F ], cost: 1-2*F+2*A

     26: f15 -> f15 : E'=2+F-A+E, F'=A, G'=free_1, H'=1, [ B>=1+E && A>=2+F && 1>=1+E ], cost: 1-2*F+2*A

     28: f15 -> f15 : E'=1+E, F'=1+F, G'=free_2, H'=E, [ A>=1+F && E>=2 && B>=1+E ], cost: 2+E

     31: f15 -> [7] : [ A>=1+F && E>=2 ], cost: 1+E

     32: f15 -> f15 : F'=A, G'=free_2, H'=E, [ A>=1+F && E>=2 && E>=B ], cost: -2*F+2*A-(F-A)*E

     33: f15 -> f15 : F'=F+2*meter, G'=free_2, H'=1+E, [ E>=2 && B>=1+E && A>=2+F && 2*meter==-1-F+A && meter>=1 ], cost: 2*meter*E+4*meter



Chained accelerated rules (with incoming rules):

   Start location: f0

      0: f0 -> f15 : A'=10, B'=35, C'=285, D'=free, E'=free, F'=0, [], cost: 1

     16: f0 -> f15 : A'=10, B'=35, C'=285, D'=free, E'=-1+free, F'=1, G'=free_2, H'=free, [ free>=2 ], cost: 2+free

     19: f0 -> f15 : A'=10, B'=35, C'=285, D'=free, E'=-10+free, F'=10, G'=free_1, H'=1, [ 1>=free ], cost: 21

     34: f0 -> f15 : A'=10, B'=35, C'=285, D'=free, E'=-8+free, F'=10, G'=free_1, H'=1, [ 1>=1+free ], cost: 22

     35: f0 -> f15 : A'=10, B'=35, C'=285, D'=free, E'=1+free, F'=1, G'=free_2, H'=free, [ free>=2 && 35>=1+free ], cost: 3+free

     36: f0 -> f15 : A'=10, B'=35, C'=285, D'=free, E'=free, F'=2, G'=free_2, H'=-1+free, [ -1+free>=2 && 35>=free ], cost: 3+2*free

     37: f0 -> f15 : A'=10, B'=35, C'=285, D'=free, E'=free, F'=10, G'=free_2, H'=free, [ free>=35 ], cost: 21+10*free

     38: f0 -> f15 : A'=10, B'=35, C'=285, D'=free, E'=-1+free, F'=10, G'=free_2, H'=-1+free, [ -1+free>=35 ], cost: 11+10*free

     39: f0 -> f15 : A'=10, B'=35, C'=285, D'=free, E'=-1+free, F'=9, G'=free_2, H'=free, [ -1+free>=2 && 35>=free ], cost: 10+9*free

     31: f15 -> [7] : [ A>=1+F && E>=2 ], cost: 1+E



Eliminated locations (on tree-shaped paths):

   Start location: f0

     40: f0 -> [7] : A'=10, B'=35, C'=285, D'=free, E'=free, F'=0, [ free>=2 ], cost: 2+free

     41: f0 -> [7] : A'=10, B'=35, C'=285, D'=free, E'=-1+free, F'=1, G'=free_2, H'=free, [ -1+free>=2 ], cost: 2+2*free

     42: f0 -> [7] : A'=10, B'=35, C'=285, D'=free, E'=1+free, F'=1, G'=free_2, H'=free, [ free>=2 && 35>=1+free ], cost: 5+2*free

     43: f0 -> [7] : A'=10, B'=35, C'=285, D'=free, E'=free, F'=2, G'=free_2, H'=-1+free, [ -1+free>=2 && 35>=free ], cost: 4+3*free

     44: f0 -> [7] : A'=10, B'=35, C'=285, D'=free, E'=-1+free, F'=9, G'=free_2, H'=free, [ -1+free>=2 && 35>=free ], cost: 10+10*free

     45: f0 -> [9] : [ free>=35 ], cost: 21+10*free

     46: f0 -> [9] : [ -1+free>=35 ], cost: 11+10*free



### Computing asymptotic complexity ###



Fully simplified ITS problem

   Start location: f0

     40: f0 -> [7] : A'=10, B'=35, C'=285, D'=free, E'=free, F'=0, [ free>=2 ], cost: 2+free

     41: f0 -> [7] : A'=10, B'=35, C'=285, D'=free, E'=-1+free, F'=1, G'=free_2, H'=free, [ -1+free>=2 ], cost: 2+2*free

     42: f0 -> [7] : A'=10, B'=35, C'=285, D'=free, E'=1+free, F'=1, G'=free_2, H'=free, [ free>=2 && 35>=1+free ], cost: 5+2*free

     43: f0 -> [7] : A'=10, B'=35, C'=285, D'=free, E'=free, F'=2, G'=free_2, H'=-1+free, [ -1+free>=2 && 35>=free ], cost: 4+3*free

     44: f0 -> [7] : A'=10, B'=35, C'=285, D'=free, E'=-1+free, F'=9, G'=free_2, H'=free, [ -1+free>=2 && 35>=free ], cost: 10+10*free

     45: f0 -> [9] : [ free>=35 ], cost: 21+10*free

     46: f0 -> [9] : [ -1+free>=35 ], cost: 11+10*free



Computing asymptotic complexity for rule 40

   Solved the limit problem by the following transformations:

      Created initial limit problem:

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



      removing all constraints (solved by SMT)

      resulting limit problem: [solved]



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

      resulting limit problem:

      [solved]



   Solution:

      free / n

   Resulting cost 2+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: 2+n

   Rule cost:   2+free

   Rule guard:  [ free>=2 ]



WORST_CASE(INF,?)


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

(2)
BOUNDS(INF, INF)