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


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


WORST_CASE(Omega(n^1), ?)
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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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(n^1, INF).

(0) CpxIntTrs
(1) Loat Proof [FINISHED, 3861 ms]
(2) BOUNDS(n^1, INF)


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

(0)
Obligation:
Complexity Int TRS consisting of the following rules:
start(A, B, C, D, E, F, G, H) -> Com_1(stop(A, B, C, D, E, F, G, H)) :|: 0 >= A + 1 && B >= C && B <= C && D >= A && D <= A && E >= F && E <= F && G >= H && G <= H
start(A, B, C, D, E, F, G, H) -> Com_1(lbl42(A, B - 1, C, D, E, F, G, H)) :|: A >= 0 && C >= 0 && B >= C && B <= C && D >= A && D <= A && E >= F && E <= F && G >= H && G <= H
start(A, B, C, D, E, F, G, H) -> Com_1(cut(A, B, C, D - 1, E, F, G, H)) :|: A >= 0 && B >= C && B <= C && D >= A && D <= A && E >= F && E <= F && G >= H && G <= H
start(A, B, C, D, E, F, G, H) -> Com_1(lbl72(A, 1 + B, C, D - 1, B, F, G, H)) :|: H >= C && A >= 0 && B >= C && B <= C && D >= A && D <= A && E >= F && E <= F && G >= H && G <= H
lbl72(A, B, C, D, E, F, G, H) -> Com_1(cut(A, B, C, D, E, F, G, H)) :|: A >= D + 1 && D + 1 >= 0 && H + 1 >= B && E + 1 >= B && E + 1 <= B && G >= H && G <= H
lbl72(A, B, C, D, E, F, G, H) -> Com_1(lbl72(A, 1 + B, C, D, B, F, G, H)) :|: H >= B && A >= D + 1 && D + 1 >= 0 && H + 1 >= B && E + 1 >= B && E + 1 <= B && G >= H && G <= H
lbl42(A, B, C, D, E, F, G, H) -> Com_1(lbl42(A, B - 1, C, D, E, F, G, H)) :|: B >= 0 && B + 1 >= 0 && D >= 0 && A >= D && G >= H && G <= H
lbl42(A, B, C, D, E, F, G, H) -> Com_1(cut(A, B, C, D - 1, E, F, G, H)) :|: B + 1 >= 0 && D >= 0 && A >= D && G >= H && G <= H
lbl42(A, B, C, D, E, F, G, H) -> Com_1(lbl72(A, 1 + B, C, D - 1, B, F, G, H)) :|: H >= B && B + 1 >= 0 && D >= 0 && A >= D && G >= H && G <= H
cut(A, B, C, D, E, F, G, H) -> Com_1(stop(A, B, C, D, E, F, G, H)) :|: A >= 0 && D + 1 >= 0 && D + 1 <= 0 && G >= H && G <= H
cut(A, B, C, D, E, F, G, H) -> Com_1(lbl42(A, B - 1, C, D, E, F, G, H)) :|: D >= 0 && B >= 0 && D + 1 >= 0 && A >= D + 1 && G >= H && G <= H
cut(A, B, C, D, E, F, G, H) -> Com_1(cut(A, B, C, D - 1, E, F, G, H)) :|: D >= 0 && D + 1 >= 0 && A >= D + 1 && G >= H && G <= H
cut(A, B, C, D, E, F, G, H) -> Com_1(lbl72(A, 1 + B, C, D - 1, B, F, G, H)) :|: H >= B && D >= 0 && D + 1 >= 0 && A >= D + 1 && G >= H && G <= H
start0(A, B, C, D, E, F, G, H) -> Com_1(start(A, C, C, A, F, F, H, H)) :|: TRUE

The start-symbols are:[start0_8]


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

(1) Loat Proof (FINISHED)


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



Initial linear ITS problem

   Start location: start0

      0: start -> stop : [ 0>=1+A && B==C && D==A && E==F && G==H ], cost: 1

      1: start -> lbl42 : B'=-1+B, [ A>=0 && C>=0 && B==C && D==A && E==F && G==H ], cost: 1

      2: start -> cut : D'=-1+D, [ A>=0 && B==C && D==A && E==F && G==H ], cost: 1

      3: start -> lbl72 : B'=1+B, D'=-1+D, E'=B, [ H>=C && A>=0 && B==C && D==A && E==F && G==H ], cost: 1

      4: lbl72 -> cut : [ A>=1+D && 1+D>=0 && 1+H>=B && 1+E==B && G==H ], cost: 1

      5: lbl72 -> lbl72 : B'=1+B, E'=B, [ H>=B && A>=1+D && 1+D>=0 && 1+H>=B && 1+E==B && G==H ], cost: 1

      6: lbl42 -> lbl42 : B'=-1+B, [ B>=0 && 1+B>=0 && D>=0 && A>=D && G==H ], cost: 1

      7: lbl42 -> cut : D'=-1+D, [ 1+B>=0 && D>=0 && A>=D && G==H ], cost: 1

      8: lbl42 -> lbl72 : B'=1+B, D'=-1+D, E'=B, [ H>=B && 1+B>=0 && D>=0 && A>=D && G==H ], cost: 1

      9: cut -> stop : [ A>=0 && 1+D==0 && G==H ], cost: 1

     10: cut -> lbl42 : B'=-1+B, [ D>=0 && B>=0 && 1+D>=0 && A>=1+D && G==H ], cost: 1

     11: cut -> cut : D'=-1+D, [ D>=0 && 1+D>=0 && A>=1+D && G==H ], cost: 1

     12: cut -> lbl72 : B'=1+B, D'=-1+D, E'=B, [ H>=B && D>=0 && 1+D>=0 && A>=1+D && G==H ], cost: 1

     13: start0 -> start : B'=C, D'=A, E'=F, G'=H, [], cost: 1



Checking for constant complexity:

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

     13: start0 -> start : B'=C, D'=A, E'=F, G'=H, [], cost: 1



Removed unreachable and leaf rules:

   Start location: start0

      1: start -> lbl42 : B'=-1+B, [ A>=0 && C>=0 && B==C && D==A && E==F && G==H ], cost: 1

      2: start -> cut : D'=-1+D, [ A>=0 && B==C && D==A && E==F && G==H ], cost: 1

      3: start -> lbl72 : B'=1+B, D'=-1+D, E'=B, [ H>=C && A>=0 && B==C && D==A && E==F && G==H ], cost: 1

      4: lbl72 -> cut : [ A>=1+D && 1+D>=0 && 1+H>=B && 1+E==B && G==H ], cost: 1

      5: lbl72 -> lbl72 : B'=1+B, E'=B, [ H>=B && A>=1+D && 1+D>=0 && 1+H>=B && 1+E==B && G==H ], cost: 1

      6: lbl42 -> lbl42 : B'=-1+B, [ B>=0 && 1+B>=0 && D>=0 && A>=D && G==H ], cost: 1

      7: lbl42 -> cut : D'=-1+D, [ 1+B>=0 && D>=0 && A>=D && G==H ], cost: 1

      8: lbl42 -> lbl72 : B'=1+B, D'=-1+D, E'=B, [ H>=B && 1+B>=0 && D>=0 && A>=D && G==H ], cost: 1

     10: cut -> lbl42 : B'=-1+B, [ D>=0 && B>=0 && 1+D>=0 && A>=1+D && G==H ], cost: 1

     11: cut -> cut : D'=-1+D, [ D>=0 && 1+D>=0 && A>=1+D && G==H ], cost: 1

     12: cut -> lbl72 : B'=1+B, D'=-1+D, E'=B, [ H>=B && D>=0 && 1+D>=0 && A>=1+D && G==H ], cost: 1

     13: start0 -> start : B'=C, D'=A, E'=F, G'=H, [], cost: 1



Simplified all rules, resulting in:

   Start location: start0

      1: start -> lbl42 : B'=-1+B, [ A>=0 && C>=0 && B==C && D==A && E==F && G==H ], cost: 1

      2: start -> cut : D'=-1+D, [ A>=0 && B==C && D==A && E==F && G==H ], cost: 1

      3: start -> lbl72 : B'=1+B, D'=-1+D, E'=B, [ H>=C && A>=0 && B==C && D==A && E==F && G==H ], cost: 1

      4: lbl72 -> cut : [ A>=1+D && 1+D>=0 && 1+H>=B && 1+E==B && G==H ], cost: 1

      5: lbl72 -> lbl72 : B'=1+B, E'=B, [ H>=B && A>=1+D && 1+D>=0 && 1+E==B && G==H ], cost: 1

      6: lbl42 -> lbl42 : B'=-1+B, [ B>=0 && D>=0 && A>=D && G==H ], cost: 1

      7: lbl42 -> cut : D'=-1+D, [ 1+B>=0 && D>=0 && A>=D && G==H ], cost: 1

      8: lbl42 -> lbl72 : B'=1+B, D'=-1+D, E'=B, [ H>=B && 1+B>=0 && D>=0 && A>=D && G==H ], cost: 1

     10: cut -> lbl42 : B'=-1+B, [ D>=0 && B>=0 && A>=1+D && G==H ], cost: 1

     11: cut -> cut : D'=-1+D, [ D>=0 && A>=1+D && G==H ], cost: 1

     12: cut -> lbl72 : B'=1+B, D'=-1+D, E'=B, [ H>=B && D>=0 && A>=1+D && G==H ], cost: 1

     13: start0 -> start : B'=C, D'=A, E'=F, G'=H, [], cost: 1



### Simplification by acceleration and chaining ###



Accelerating simple loops of location 1.

   Accelerating the following rules:

      5: lbl72 -> lbl72 : B'=1+B, E'=B, [ H>=B && A>=1+D && 1+D>=0 && 1+E==B && G==H ], cost: 1



   Accelerated rule 5 with metering function 1+H-B, yielding the new rule 14.

   Removing the simple loops: 5.



Accelerating simple loops of location 2.

   Accelerating the following rules:

      6: lbl42 -> lbl42 : B'=-1+B, [ B>=0 && D>=0 && A>=D && G==H ], cost: 1



   Accelerated rule 6 with metering function 1+B, yielding the new rule 15.

   Removing the simple loops: 6.



Accelerating simple loops of location 3.

   Accelerating the following rules:

     11: cut -> cut : D'=-1+D, [ D>=0 && A>=1+D && G==H ], cost: 1



   Accelerated rule 11 with metering function 1+D, yielding the new rule 16.

   Removing the simple loops: 11.



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

   Start location: start0

      1: start -> lbl42 : B'=-1+B, [ A>=0 && C>=0 && B==C && D==A && E==F && G==H ], cost: 1

      2: start -> cut : D'=-1+D, [ A>=0 && B==C && D==A && E==F && G==H ], cost: 1

      3: start -> lbl72 : B'=1+B, D'=-1+D, E'=B, [ H>=C && A>=0 && B==C && D==A && E==F && G==H ], cost: 1

      4: lbl72 -> cut : [ A>=1+D && 1+D>=0 && 1+H>=B && 1+E==B && G==H ], cost: 1

     14: lbl72 -> lbl72 : B'=1+H, E'=H, [ H>=B && A>=1+D && 1+D>=0 && 1+E==B && G==H ], cost: 1+H-B

      7: lbl42 -> cut : D'=-1+D, [ 1+B>=0 && D>=0 && A>=D && G==H ], cost: 1

      8: lbl42 -> lbl72 : B'=1+B, D'=-1+D, E'=B, [ H>=B && 1+B>=0 && D>=0 && A>=D && G==H ], cost: 1

     15: lbl42 -> lbl42 : B'=-1, [ B>=0 && D>=0 && A>=D && G==H ], cost: 1+B

     10: cut -> lbl42 : B'=-1+B, [ D>=0 && B>=0 && A>=1+D && G==H ], cost: 1

     12: cut -> lbl72 : B'=1+B, D'=-1+D, E'=B, [ H>=B && D>=0 && A>=1+D && G==H ], cost: 1

     16: cut -> cut : D'=-1, [ D>=0 && A>=1+D && G==H ], cost: 1+D

     13: start0 -> start : B'=C, D'=A, E'=F, G'=H, [], cost: 1



Chained accelerated rules (with incoming rules):

   Start location: start0

      1: start -> lbl42 : B'=-1+B, [ A>=0 && C>=0 && B==C && D==A && E==F && G==H ], cost: 1

      2: start -> cut : D'=-1+D, [ A>=0 && B==C && D==A && E==F && G==H ], cost: 1

      3: start -> lbl72 : B'=1+B, D'=-1+D, E'=B, [ H>=C && A>=0 && B==C && D==A && E==F && G==H ], cost: 1

     17: start -> lbl72 : B'=1+H, D'=-1+D, E'=H, [ H>=C && A>=0 && B==C && D==A && E==F && G==H && H>=1+B && D>=0 ], cost: 1+H-B

     20: start -> lbl42 : B'=-1, [ A>=0 && C>=0 && B==C && D==A && E==F && G==H && -1+B>=0 && D>=0 ], cost: 1+B

     22: start -> cut : D'=-1, [ A>=0 && B==C && D==A && E==F && G==H && -1+D>=0 ], cost: 1+D

      4: lbl72 -> cut : [ A>=1+D && 1+D>=0 && 1+H>=B && 1+E==B && G==H ], cost: 1

     23: lbl72 -> cut : D'=-1, [ A>=1+D && 1+H>=B && 1+E==B && G==H && D>=0 ], cost: 2+D

      7: lbl42 -> cut : D'=-1+D, [ 1+B>=0 && D>=0 && A>=D && G==H ], cost: 1

      8: lbl42 -> lbl72 : B'=1+B, D'=-1+D, E'=B, [ H>=B && 1+B>=0 && D>=0 && A>=D && G==H ], cost: 1

     18: lbl42 -> lbl72 : B'=1+H, D'=-1+D, E'=H, [ 1+B>=0 && D>=0 && A>=D && G==H && H>=1+B ], cost: 1+H-B

     24: lbl42 -> cut : D'=-1, [ 1+B>=0 && A>=D && G==H && -1+D>=0 ], cost: 1+D

     10: cut -> lbl42 : B'=-1+B, [ D>=0 && B>=0 && A>=1+D && G==H ], cost: 1

     12: cut -> lbl72 : B'=1+B, D'=-1+D, E'=B, [ H>=B && D>=0 && A>=1+D && G==H ], cost: 1

     19: cut -> lbl72 : B'=1+H, D'=-1+D, E'=H, [ D>=0 && A>=1+D && G==H && H>=1+B ], cost: 1+H-B

     21: cut -> lbl42 : B'=-1, [ D>=0 && A>=1+D && G==H && -1+B>=0 ], cost: 1+B

     13: start0 -> start : B'=C, D'=A, E'=F, G'=H, [], cost: 1



Eliminated locations (on tree-shaped paths):

   Start location: start0

      4: lbl72 -> cut : [ A>=1+D && 1+D>=0 && 1+H>=B && 1+E==B && G==H ], cost: 1

     23: lbl72 -> cut : D'=-1, [ A>=1+D && 1+H>=B && 1+E==B && G==H && D>=0 ], cost: 2+D

      7: lbl42 -> cut : D'=-1+D, [ 1+B>=0 && D>=0 && A>=D && G==H ], cost: 1

      8: lbl42 -> lbl72 : B'=1+B, D'=-1+D, E'=B, [ H>=B && 1+B>=0 && D>=0 && A>=D && G==H ], cost: 1

     18: lbl42 -> lbl72 : B'=1+H, D'=-1+D, E'=H, [ 1+B>=0 && D>=0 && A>=D && G==H && H>=1+B ], cost: 1+H-B

     24: lbl42 -> cut : D'=-1, [ 1+B>=0 && A>=D && G==H && -1+D>=0 ], cost: 1+D

     10: cut -> lbl42 : B'=-1+B, [ D>=0 && B>=0 && A>=1+D && G==H ], cost: 1

     12: cut -> lbl72 : B'=1+B, D'=-1+D, E'=B, [ H>=B && D>=0 && A>=1+D && G==H ], cost: 1

     19: cut -> lbl72 : B'=1+H, D'=-1+D, E'=H, [ D>=0 && A>=1+D && G==H && H>=1+B ], cost: 1+H-B

     21: cut -> lbl42 : B'=-1, [ D>=0 && A>=1+D && G==H && -1+B>=0 ], cost: 1+B

     25: start0 -> lbl42 : B'=-1+C, D'=A, E'=F, G'=H, [ A>=0 && C>=0 ], cost: 2

     26: start0 -> cut : B'=C, D'=-1+A, E'=F, G'=H, [ A>=0 ], cost: 2

     27: start0 -> lbl72 : B'=1+C, D'=-1+A, E'=C, G'=H, [ H>=C && A>=0 ], cost: 2

     28: start0 -> lbl72 : B'=1+H, D'=-1+A, E'=H, G'=H, [ A>=0 && H>=1+C ], cost: 2-C+H

     29: start0 -> lbl42 : B'=-1, D'=A, E'=F, G'=H, [ A>=0 && -1+C>=0 ], cost: 2+C

     30: start0 -> cut : B'=C, D'=-1, E'=F, G'=H, [ -1+A>=0 ], cost: 2+A



Eliminated location lbl72 (as a last resort):

   Start location: start0

      7: lbl42 -> cut : D'=-1+D, [ 1+B>=0 && D>=0 && A>=D && G==H ], cost: 1

     24: lbl42 -> cut : D'=-1, [ 1+B>=0 && A>=D && G==H && -1+D>=0 ], cost: 1+D

     31: lbl42 -> cut : B'=1+B, D'=-1+D, E'=B, [ H>=B && 1+B>=0 && D>=0 && A>=D && G==H ], cost: 2

     32: lbl42 -> cut : B'=1+B, D'=-1, E'=B, [ H>=B && 1+B>=0 && A>=D && G==H && -1+D>=0 ], cost: 2+D

     35: lbl42 -> cut : B'=1+H, D'=-1+D, E'=H, [ 1+B>=0 && D>=0 && A>=D && G==H && H>=1+B ], cost: 2+H-B

     36: lbl42 -> cut : B'=1+H, D'=-1, E'=H, [ 1+B>=0 && A>=D && G==H && H>=1+B && -1+D>=0 ], cost: 2+D+H-B

     10: cut -> lbl42 : B'=-1+B, [ D>=0 && B>=0 && A>=1+D && G==H ], cost: 1

     21: cut -> lbl42 : B'=-1, [ D>=0 && A>=1+D && G==H && -1+B>=0 ], cost: 1+B

     33: cut -> cut : B'=1+B, D'=-1+D, E'=B, [ H>=B && D>=0 && A>=1+D && G==H ], cost: 2

     34: cut -> cut : B'=1+B, D'=-1, E'=B, [ H>=B && A>=1+D && G==H && -1+D>=0 ], cost: 2+D

     37: cut -> cut : B'=1+H, D'=-1+D, E'=H, [ D>=0 && A>=1+D && G==H && H>=1+B ], cost: 2+H-B

     38: cut -> cut : B'=1+H, D'=-1, E'=H, [ A>=1+D && G==H && H>=1+B && -1+D>=0 ], cost: 2+D+H-B

     25: start0 -> lbl42 : B'=-1+C, D'=A, E'=F, G'=H, [ A>=0 && C>=0 ], cost: 2

     26: start0 -> cut : B'=C, D'=-1+A, E'=F, G'=H, [ A>=0 ], cost: 2

     29: start0 -> lbl42 : B'=-1, D'=A, E'=F, G'=H, [ A>=0 && -1+C>=0 ], cost: 2+C

     30: start0 -> cut : B'=C, D'=-1, E'=F, G'=H, [ -1+A>=0 ], cost: 2+A

     39: start0 -> cut : B'=1+C, D'=-1+A, E'=C, G'=H, [ H>=C && A>=0 ], cost: 3

     40: start0 -> cut : B'=1+C, D'=-1, E'=C, G'=H, [ H>=C && -1+A>=0 ], cost: 3+A

     41: start0 -> cut : B'=1+H, D'=-1+A, E'=H, G'=H, [ A>=0 && H>=1+C ], cost: 3-C+H

     42: start0 -> cut : B'=1+H, D'=-1, E'=H, G'=H, [ H>=1+C && -1+A>=0 ], cost: 3-C+A+H



Applied pruning (of leafs and parallel rules):

   Start location: start0

      7: lbl42 -> cut : D'=-1+D, [ 1+B>=0 && D>=0 && A>=D && G==H ], cost: 1

     24: lbl42 -> cut : D'=-1, [ 1+B>=0 && A>=D && G==H && -1+D>=0 ], cost: 1+D

     32: lbl42 -> cut : B'=1+B, D'=-1, E'=B, [ H>=B && 1+B>=0 && A>=D && G==H && -1+D>=0 ], cost: 2+D

     35: lbl42 -> cut : B'=1+H, D'=-1+D, E'=H, [ 1+B>=0 && D>=0 && A>=D && G==H && H>=1+B ], cost: 2+H-B

     36: lbl42 -> cut : B'=1+H, D'=-1, E'=H, [ 1+B>=0 && A>=D && G==H && H>=1+B && -1+D>=0 ], cost: 2+D+H-B

     10: cut -> lbl42 : B'=-1+B, [ D>=0 && B>=0 && A>=1+D && G==H ], cost: 1

     21: cut -> lbl42 : B'=-1, [ D>=0 && A>=1+D && G==H && -1+B>=0 ], cost: 1+B

     33: cut -> cut : B'=1+B, D'=-1+D, E'=B, [ H>=B && D>=0 && A>=1+D && G==H ], cost: 2

     34: cut -> cut : B'=1+B, D'=-1, E'=B, [ H>=B && A>=1+D && G==H && -1+D>=0 ], cost: 2+D

     37: cut -> cut : B'=1+H, D'=-1+D, E'=H, [ D>=0 && A>=1+D && G==H && H>=1+B ], cost: 2+H-B

     38: cut -> cut : B'=1+H, D'=-1, E'=H, [ A>=1+D && G==H && H>=1+B && -1+D>=0 ], cost: 2+D+H-B

     25: start0 -> lbl42 : B'=-1+C, D'=A, E'=F, G'=H, [ A>=0 && C>=0 ], cost: 2

     26: start0 -> cut : B'=C, D'=-1+A, E'=F, G'=H, [ A>=0 ], cost: 2

     29: start0 -> lbl42 : B'=-1, D'=A, E'=F, G'=H, [ A>=0 && -1+C>=0 ], cost: 2+C

     30: start0 -> cut : B'=C, D'=-1, E'=F, G'=H, [ -1+A>=0 ], cost: 2+A

     40: start0 -> cut : B'=1+C, D'=-1, E'=C, G'=H, [ H>=C && -1+A>=0 ], cost: 3+A

     41: start0 -> cut : B'=1+H, D'=-1+A, E'=H, G'=H, [ A>=0 && H>=1+C ], cost: 3-C+H

     42: start0 -> cut : B'=1+H, D'=-1, E'=H, G'=H, [ H>=1+C && -1+A>=0 ], cost: 3-C+A+H



Accelerating simple loops of location 3.

   Accelerating the following rules:

     33: cut -> cut : B'=1+B, D'=-1+D, E'=B, [ H>=B && D>=0 && A>=1+D && G==H ], cost: 2

     34: cut -> cut : B'=1+B, D'=-1, E'=B, [ H>=B && A>=1+D && G==H && -1+D>=0 ], cost: 2+D

     37: cut -> cut : B'=1+H, D'=-1+D, E'=H, [ D>=0 && A>=1+D && G==H && H>=1+B ], cost: 2+H-B

     38: cut -> cut : B'=1+H, D'=-1, E'=H, [ A>=1+D && G==H && H>=1+B && -1+D>=0 ], cost: 2+D+H-B



   Found no metering function for rule 33.

   Found no metering function for rule 34.

   Found no metering function for rule 37.

   Found no metering function for rule 38.

   Removing the simple loops:.



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

   Start location: start0

      7: lbl42 -> cut : D'=-1+D, [ 1+B>=0 && D>=0 && A>=D && G==H ], cost: 1

     24: lbl42 -> cut : D'=-1, [ 1+B>=0 && A>=D && G==H && -1+D>=0 ], cost: 1+D

     32: lbl42 -> cut : B'=1+B, D'=-1, E'=B, [ H>=B && 1+B>=0 && A>=D && G==H && -1+D>=0 ], cost: 2+D

     35: lbl42 -> cut : B'=1+H, D'=-1+D, E'=H, [ 1+B>=0 && D>=0 && A>=D && G==H && H>=1+B ], cost: 2+H-B

     36: lbl42 -> cut : B'=1+H, D'=-1, E'=H, [ 1+B>=0 && A>=D && G==H && H>=1+B && -1+D>=0 ], cost: 2+D+H-B

     10: cut -> lbl42 : B'=-1+B, [ D>=0 && B>=0 && A>=1+D && G==H ], cost: 1

     21: cut -> lbl42 : B'=-1, [ D>=0 && A>=1+D && G==H && -1+B>=0 ], cost: 1+B

     33: cut -> cut : B'=1+B, D'=-1+D, E'=B, [ H>=B && D>=0 && A>=1+D && G==H ], cost: 2

     34: cut -> cut : B'=1+B, D'=-1, E'=B, [ H>=B && A>=1+D && G==H && -1+D>=0 ], cost: 2+D

     37: cut -> cut : B'=1+H, D'=-1+D, E'=H, [ D>=0 && A>=1+D && G==H && H>=1+B ], cost: 2+H-B

     38: cut -> cut : B'=1+H, D'=-1, E'=H, [ A>=1+D && G==H && H>=1+B && -1+D>=0 ], cost: 2+D+H-B

     25: start0 -> lbl42 : B'=-1+C, D'=A, E'=F, G'=H, [ A>=0 && C>=0 ], cost: 2

     26: start0 -> cut : B'=C, D'=-1+A, E'=F, G'=H, [ A>=0 ], cost: 2

     29: start0 -> lbl42 : B'=-1, D'=A, E'=F, G'=H, [ A>=0 && -1+C>=0 ], cost: 2+C

     30: start0 -> cut : B'=C, D'=-1, E'=F, G'=H, [ -1+A>=0 ], cost: 2+A

     40: start0 -> cut : B'=1+C, D'=-1, E'=C, G'=H, [ H>=C && -1+A>=0 ], cost: 3+A

     41: start0 -> cut : B'=1+H, D'=-1+A, E'=H, G'=H, [ A>=0 && H>=1+C ], cost: 3-C+H

     42: start0 -> cut : B'=1+H, D'=-1, E'=H, G'=H, [ H>=1+C && -1+A>=0 ], cost: 3-C+A+H



Chained accelerated rules (with incoming rules):

   Start location: start0

      7: lbl42 -> cut : D'=-1+D, [ 1+B>=0 && D>=0 && A>=D && G==H ], cost: 1

     24: lbl42 -> cut : D'=-1, [ 1+B>=0 && A>=D && G==H && -1+D>=0 ], cost: 1+D

     32: lbl42 -> cut : B'=1+B, D'=-1, E'=B, [ H>=B && 1+B>=0 && A>=D && G==H && -1+D>=0 ], cost: 2+D

     35: lbl42 -> cut : B'=1+H, D'=-1+D, E'=H, [ 1+B>=0 && D>=0 && A>=D && G==H && H>=1+B ], cost: 2+H-B

     36: lbl42 -> cut : B'=1+H, D'=-1, E'=H, [ 1+B>=0 && A>=D && G==H && H>=1+B && -1+D>=0 ], cost: 2+D+H-B

     43: lbl42 -> cut : B'=1+B, D'=-2+D, E'=B, [ 1+B>=0 && A>=D && G==H && H>=B && -1+D>=0 ], cost: 3

     45: lbl42 -> cut : B'=1+B, D'=-1, E'=B, [ 1+B>=0 && A>=D && G==H && H>=B && -2+D>=0 ], cost: 2+D

     47: lbl42 -> cut : B'=1+H, D'=-2+D, E'=H, [ 1+B>=0 && A>=D && G==H && -1+D>=0 && H>=1+B ], cost: 3+H-B

     49: lbl42 -> cut : B'=1+H, D'=-1, E'=H, [ 1+B>=0 && A>=D && G==H && H>=1+B && -2+D>=0 ], cost: 2+D+H-B

     10: cut -> lbl42 : B'=-1+B, [ D>=0 && B>=0 && A>=1+D && G==H ], cost: 1

     21: cut -> lbl42 : B'=-1, [ D>=0 && A>=1+D && G==H && -1+B>=0 ], cost: 1+B

     25: start0 -> lbl42 : B'=-1+C, D'=A, E'=F, G'=H, [ A>=0 && C>=0 ], cost: 2

     26: start0 -> cut : B'=C, D'=-1+A, E'=F, G'=H, [ A>=0 ], cost: 2

     29: start0 -> lbl42 : B'=-1, D'=A, E'=F, G'=H, [ A>=0 && -1+C>=0 ], cost: 2+C

     30: start0 -> cut : B'=C, D'=-1, E'=F, G'=H, [ -1+A>=0 ], cost: 2+A

     40: start0 -> cut : B'=1+C, D'=-1, E'=C, G'=H, [ H>=C && -1+A>=0 ], cost: 3+A

     41: start0 -> cut : B'=1+H, D'=-1+A, E'=H, G'=H, [ A>=0 && H>=1+C ], cost: 3-C+H

     42: start0 -> cut : B'=1+H, D'=-1, E'=H, G'=H, [ H>=1+C && -1+A>=0 ], cost: 3-C+A+H

     44: start0 -> cut : B'=1+C, D'=-2+A, E'=C, G'=H, [ H>=C && -1+A>=0 ], cost: 4

     46: start0 -> cut : B'=1+C, D'=-1, E'=C, G'=H, [ H>=C && -2+A>=0 ], cost: 3+A

     48: start0 -> cut : B'=1+H, D'=-2+A, E'=H, G'=H, [ -1+A>=0 && H>=1+C ], cost: 4-C+H

     50: start0 -> cut : B'=1+H, D'=-1, E'=H, G'=H, [ H>=1+C && -2+A>=0 ], cost: 3-C+A+H



Eliminated location lbl42 (as a last resort):

   Start location: start0

     51: cut -> cut : B'=-1+B, D'=-1+D, [ D>=0 && B>=0 && A>=1+D && G==H ], cost: 2

     52: cut -> cut : B'=-1+B, D'=-1, [ B>=0 && A>=1+D && G==H && -1+D>=0 ], cost: 2+D

     53: cut -> cut : B'=B, D'=-1, E'=-1+B, [ B>=0 && A>=1+D && G==H && H>=-1+B && -1+D>=0 ], cost: 3+D

     54: cut -> cut : B'=1+H, D'=-1+D, E'=H, [ D>=0 && B>=0 && A>=1+D && G==H && H>=B ], cost: 4+H-B

     55: cut -> cut : B'=1+H, D'=-1, E'=H, [ B>=0 && A>=1+D && G==H && H>=B && -1+D>=0 ], cost: 4+D+H-B

     56: cut -> cut : B'=B, D'=-2+D, E'=-1+B, [ B>=0 && A>=1+D && G==H && H>=-1+B && -1+D>=0 ], cost: 4

     57: cut -> cut : B'=B, D'=-1, E'=-1+B, [ B>=0 && A>=1+D && G==H && H>=-1+B && -2+D>=0 ], cost: 3+D

     58: cut -> cut : B'=1+H, D'=-2+D, E'=H, [ B>=0 && A>=1+D && G==H && -1+D>=0 && H>=B ], cost: 5+H-B

     59: cut -> cut : B'=1+H, D'=-1, E'=H, [ B>=0 && A>=1+D && G==H && H>=B && -2+D>=0 ], cost: 4+D+H-B

     60: cut -> cut : B'=-1, D'=-1+D, [ D>=0 && A>=1+D && G==H && -1+B>=0 ], cost: 2+B

     61: cut -> cut : B'=-1, D'=-1, [ A>=1+D && G==H && -1+B>=0 && -1+D>=0 ], cost: 2+D+B

     62: cut -> cut : B'=0, D'=-1, E'=-1, [ A>=1+D && G==H && -1+B>=0 && H>=-1 && -1+D>=0 ], cost: 3+D+B

     63: cut -> cut : B'=1+H, D'=-1+D, E'=H, [ D>=0 && A>=1+D && G==H && -1+B>=0 && H>=0 ], cost: 4+H+B

     64: cut -> cut : B'=1+H, D'=-1, E'=H, [ A>=1+D && G==H && -1+B>=0 && H>=0 && -1+D>=0 ], cost: 4+D+H+B

     65: cut -> cut : B'=0, D'=-2+D, E'=-1, [ A>=1+D && G==H && -1+B>=0 && H>=-1 && -1+D>=0 ], cost: 4+B

     66: cut -> cut : B'=0, D'=-1, E'=-1, [ A>=1+D && G==H && -1+B>=0 && H>=-1 && -2+D>=0 ], cost: 3+D+B

     67: cut -> cut : B'=1+H, D'=-2+D, E'=H, [ A>=1+D && G==H && -1+B>=0 && -1+D>=0 && H>=0 ], cost: 5+H+B

     68: cut -> cut : B'=1+H, D'=-1, E'=H, [ A>=1+D && G==H && -1+B>=0 && H>=0 && -2+D>=0 ], cost: 4+D+H+B

     26: start0 -> cut : B'=C, D'=-1+A, E'=F, G'=H, [ A>=0 ], cost: 2

     30: start0 -> cut : B'=C, D'=-1, E'=F, G'=H, [ -1+A>=0 ], cost: 2+A

     40: start0 -> cut : B'=1+C, D'=-1, E'=C, G'=H, [ H>=C && -1+A>=0 ], cost: 3+A

     41: start0 -> cut : B'=1+H, D'=-1+A, E'=H, G'=H, [ A>=0 && H>=1+C ], cost: 3-C+H

     42: start0 -> cut : B'=1+H, D'=-1, E'=H, G'=H, [ H>=1+C && -1+A>=0 ], cost: 3-C+A+H

     44: start0 -> cut : B'=1+C, D'=-2+A, E'=C, G'=H, [ H>=C && -1+A>=0 ], cost: 4

     46: start0 -> cut : B'=1+C, D'=-1, E'=C, G'=H, [ H>=C && -2+A>=0 ], cost: 3+A

     48: start0 -> cut : B'=1+H, D'=-2+A, E'=H, G'=H, [ -1+A>=0 && H>=1+C ], cost: 4-C+H

     50: start0 -> cut : B'=1+H, D'=-1, E'=H, G'=H, [ H>=1+C && -2+A>=0 ], cost: 3-C+A+H

     69: start0 -> cut : B'=-1+C, D'=-1+A, E'=F, G'=H, [ A>=0 && C>=0 ], cost: 3

     70: start0 -> cut : B'=-1+C, D'=-1, E'=F, G'=H, [ C>=0 && -1+A>=0 ], cost: 3+A

     71: start0 -> cut : B'=C, D'=-1, E'=-1+C, G'=H, [ C>=0 && H>=-1+C && -1+A>=0 ], cost: 4+A

     72: start0 -> cut : B'=1+H, D'=-1+A, E'=H, G'=H, [ A>=0 && C>=0 && H>=C ], cost: 5-C+H

     73: start0 -> cut : B'=1+H, D'=-1, E'=H, G'=H, [ C>=0 && H>=C && -1+A>=0 ], cost: 5-C+A+H

     74: start0 -> cut : B'=C, D'=-2+A, E'=-1+C, G'=H, [ C>=0 && H>=-1+C && -1+A>=0 ], cost: 5

     75: start0 -> cut : B'=C, D'=-1, E'=-1+C, G'=H, [ C>=0 && H>=-1+C && -2+A>=0 ], cost: 4+A

     76: start0 -> cut : B'=1+H, D'=-2+A, E'=H, G'=H, [ C>=0 && -1+A>=0 && H>=C ], cost: 6-C+H

     77: start0 -> cut : B'=1+H, D'=-1, E'=H, G'=H, [ C>=0 && H>=C && -2+A>=0 ], cost: 5-C+A+H

     78: start0 -> cut : B'=-1, D'=-1+A, E'=F, G'=H, [ A>=0 && -1+C>=0 ], cost: 3+C

     79: start0 -> cut : B'=-1, D'=-1, E'=F, G'=H, [ -1+C>=0 && -1+A>=0 ], cost: 3+C+A

     80: start0 -> cut : B'=0, D'=-1, E'=-1, G'=H, [ -1+C>=0 && H>=-1 && -1+A>=0 ], cost: 4+C+A

     81: start0 -> cut : B'=1+H, D'=-1+A, E'=H, G'=H, [ A>=0 && -1+C>=0 && H>=0 ], cost: 5+C+H

     82: start0 -> cut : B'=1+H, D'=-1, E'=H, G'=H, [ -1+C>=0 && H>=0 && -1+A>=0 ], cost: 5+C+A+H

     83: start0 -> cut : B'=0, D'=-2+A, E'=-1, G'=H, [ -1+C>=0 && H>=-1 && -1+A>=0 ], cost: 5+C

     84: start0 -> cut : B'=0, D'=-1, E'=-1, G'=H, [ -1+C>=0 && H>=-1 && -2+A>=0 ], cost: 4+C+A

     85: start0 -> cut : B'=1+H, D'=-2+A, E'=H, G'=H, [ -1+C>=0 && -1+A>=0 && H>=0 ], cost: 6+C+H

     86: start0 -> cut : B'=1+H, D'=-1, E'=H, G'=H, [ -1+C>=0 && H>=0 && -2+A>=0 ], cost: 5+C+A+H



Applied pruning (of leafs and parallel rules):

   Start location: start0

     53: cut -> cut : B'=B, D'=-1, E'=-1+B, [ B>=0 && A>=1+D && G==H && H>=-1+B && -1+D>=0 ], cost: 3+D

     54: cut -> cut : B'=1+H, D'=-1+D, E'=H, [ D>=0 && B>=0 && A>=1+D && G==H && H>=B ], cost: 4+H-B

     57: cut -> cut : B'=B, D'=-1, E'=-1+B, [ B>=0 && A>=1+D && G==H && H>=-1+B && -2+D>=0 ], cost: 3+D

     58: cut -> cut : B'=1+H, D'=-2+D, E'=H, [ B>=0 && A>=1+D && G==H && -1+D>=0 && H>=B ], cost: 5+H-B

     62: cut -> cut : B'=0, D'=-1, E'=-1, [ A>=1+D && G==H && -1+B>=0 && H>=-1 && -1+D>=0 ], cost: 3+D+B

     73: start0 -> cut : B'=1+H, D'=-1, E'=H, G'=H, [ C>=0 && H>=C && -1+A>=0 ], cost: 5-C+A+H

     81: start0 -> cut : B'=1+H, D'=-1+A, E'=H, G'=H, [ A>=0 && -1+C>=0 && H>=0 ], cost: 5+C+H

     82: start0 -> cut : B'=1+H, D'=-1, E'=H, G'=H, [ -1+C>=0 && H>=0 && -1+A>=0 ], cost: 5+C+A+H

     85: start0 -> cut : B'=1+H, D'=-2+A, E'=H, G'=H, [ -1+C>=0 && -1+A>=0 && H>=0 ], cost: 6+C+H

     86: start0 -> cut : B'=1+H, D'=-1, E'=H, G'=H, [ -1+C>=0 && H>=0 && -2+A>=0 ], cost: 5+C+A+H



Accelerating simple loops of location 3.

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

   Accelerating the following rules:

     53: cut -> cut : D'=-1, E'=-1+B, [ B>=0 && A>=1+D && G==H && H>=-1+B && -1+D>=0 ], cost: 3+D

     54: cut -> cut : B'=1+H, D'=-1+D, E'=H, [ D>=0 && B>=0 && A>=1+D && G==H && H>=B ], cost: 4+H-B

     57: cut -> cut : D'=-1, E'=-1+B, [ B>=0 && A>=1+D && G==H && H>=-1+B && -2+D>=0 ], cost: 3+D

     58: cut -> cut : B'=1+H, D'=-2+D, E'=H, [ B>=0 && A>=1+D && G==H && -1+D>=0 && H>=B ], cost: 5+H-B

     62: cut -> cut : B'=0, D'=-1, E'=-1, [ A>=1+D && G==H && -1+B>=0 && H>=-1 && -1+D>=0 ], cost: 3+D+B



   Found no metering function for rule 53.

   Found no metering function for rule 54.

   Found no metering function for rule 57.

   Found no metering function for rule 58.

   Found no metering function for rule 62.

   Removing the simple loops:.



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

   Start location: start0

     53: cut -> cut : D'=-1, E'=-1+B, [ B>=0 && A>=1+D && G==H && H>=-1+B && -1+D>=0 ], cost: 3+D

     54: cut -> cut : B'=1+H, D'=-1+D, E'=H, [ D>=0 && B>=0 && A>=1+D && G==H && H>=B ], cost: 4+H-B

     57: cut -> cut : D'=-1, E'=-1+B, [ B>=0 && A>=1+D && G==H && H>=-1+B && -2+D>=0 ], cost: 3+D

     58: cut -> cut : B'=1+H, D'=-2+D, E'=H, [ B>=0 && A>=1+D && G==H && -1+D>=0 && H>=B ], cost: 5+H-B

     62: cut -> cut : B'=0, D'=-1, E'=-1, [ A>=1+D && G==H && -1+B>=0 && H>=-1 && -1+D>=0 ], cost: 3+D+B

     73: start0 -> cut : B'=1+H, D'=-1, E'=H, G'=H, [ C>=0 && H>=C && -1+A>=0 ], cost: 5-C+A+H

     81: start0 -> cut : B'=1+H, D'=-1+A, E'=H, G'=H, [ A>=0 && -1+C>=0 && H>=0 ], cost: 5+C+H

     82: start0 -> cut : B'=1+H, D'=-1, E'=H, G'=H, [ -1+C>=0 && H>=0 && -1+A>=0 ], cost: 5+C+A+H

     85: start0 -> cut : B'=1+H, D'=-2+A, E'=H, G'=H, [ -1+C>=0 && -1+A>=0 && H>=0 ], cost: 6+C+H

     86: start0 -> cut : B'=1+H, D'=-1, E'=H, G'=H, [ -1+C>=0 && H>=0 && -2+A>=0 ], cost: 5+C+A+H



Chained accelerated rules (with incoming rules):

   Start location: start0

     73: start0 -> cut : B'=1+H, D'=-1, E'=H, G'=H, [ C>=0 && H>=C && -1+A>=0 ], cost: 5-C+A+H

     81: start0 -> cut : B'=1+H, D'=-1+A, E'=H, G'=H, [ A>=0 && -1+C>=0 && H>=0 ], cost: 5+C+H

     82: start0 -> cut : B'=1+H, D'=-1, E'=H, G'=H, [ -1+C>=0 && H>=0 && -1+A>=0 ], cost: 5+C+A+H

     85: start0 -> cut : B'=1+H, D'=-2+A, E'=H, G'=H, [ -1+C>=0 && -1+A>=0 && H>=0 ], cost: 6+C+H

     86: start0 -> cut : B'=1+H, D'=-1, E'=H, G'=H, [ -1+C>=0 && H>=0 && -2+A>=0 ], cost: 5+C+A+H

     87: start0 -> cut : B'=1+H, D'=-1, E'=H, G'=H, [ -1+C>=0 && H>=0 && -2+A>=0 ], cost: 7+C+A+H

     88: start0 -> cut : B'=1+H, D'=-1, E'=H, G'=H, [ -1+C>=0 && H>=0 && -3+A>=0 ], cost: 7+C+A+H

     89: start0 -> cut : B'=1+H, D'=-1, E'=H, G'=H, [ -1+C>=0 && H>=0 && -3+A>=0 ], cost: 7+C+A+H

     90: start0 -> cut : B'=1+H, D'=-1, E'=H, G'=H, [ -1+C>=0 && H>=0 && -4+A>=0 ], cost: 7+C+A+H

     91: start0 -> cut : B'=0, D'=-1, E'=-1, G'=H, [ -1+C>=0 && H>=0 && -2+A>=0 ], cost: 8+C+A+2*H

     92: start0 -> cut : B'=0, D'=-1, E'=-1, G'=H, [ -1+C>=0 && H>=0 && -3+A>=0 ], cost: 8+C+A+2*H



### Computing asymptotic complexity ###



Fully simplified ITS problem

   Start location: start0

     73: start0 -> cut : B'=1+H, D'=-1, E'=H, G'=H, [ C>=0 && H>=C && -1+A>=0 ], cost: 5-C+A+H

     81: start0 -> cut : B'=1+H, D'=-1+A, E'=H, G'=H, [ A>=0 && -1+C>=0 && H>=0 ], cost: 5+C+H

     82: start0 -> cut : B'=1+H, D'=-1, E'=H, G'=H, [ -1+C>=0 && H>=0 && -1+A>=0 ], cost: 5+C+A+H

     85: start0 -> cut : B'=1+H, D'=-2+A, E'=H, G'=H, [ -1+C>=0 && -1+A>=0 && H>=0 ], cost: 6+C+H

     87: start0 -> cut : B'=1+H, D'=-1, E'=H, G'=H, [ -1+C>=0 && H>=0 && -2+A>=0 ], cost: 7+C+A+H

     89: start0 -> cut : B'=1+H, D'=-1, E'=H, G'=H, [ -1+C>=0 && H>=0 && -3+A>=0 ], cost: 7+C+A+H

     90: start0 -> cut : B'=1+H, D'=-1, E'=H, G'=H, [ -1+C>=0 && H>=0 && -4+A>=0 ], cost: 7+C+A+H

     91: start0 -> cut : B'=0, D'=-1, E'=-1, G'=H, [ -1+C>=0 && H>=0 && -2+A>=0 ], cost: 8+C+A+2*H

     92: start0 -> cut : B'=0, D'=-1, E'=-1, G'=H, [ -1+C>=0 && H>=0 && -3+A>=0 ], cost: 8+C+A+2*H



Computing asymptotic complexity for rule 73

   Solved the limit problem by the following transformations:

      Created initial limit problem:

      1+C (+/+!), A (+/+!), 1-C+H (+/+!), 5-C+A+H (+) [not solved]



      removing all constraints (solved by SMT)

      resulting limit problem: [solved]



      applying transformation rule (C) using substitution {C==n,A==n,H==n}

      resulting limit problem:

      [solved]



   Solution:

      C / n

      A / n

      H / n

   Resulting cost 5+n has complexity: Poly(n^1)



Found new complexity Poly(n^1).



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

   Complexity:  Poly(n^1)

   Cpx degree:  1

   Solved cost: 5+n

   Rule cost:   5-C+A+H

   Rule guard:  [ C>=0 && H>=C && -1+A>=0 ]



WORST_CASE(Omega(n^1),?)


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

(2)
BOUNDS(n^1, INF)