WORST_CASE(Omega(n^1),?)

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

Initial linear ITS problem
   Start location: l8
      0: l0 -> l1 : c^0'=c^post_1, ox^0'=ox^post_1, oy^0'=oy^post_1, x^0'=x^post_1, y^0'=y^post_1, [ 1+y^0<=0 && c^0==c^post_1 && ox^0==ox^post_1 && oy^0==oy^post_1 && x^0==x^post_1 && y^0==y^post_1 ], cost: 1
      1: l0 -> l1 : c^0'=c^post_2, ox^0'=ox^post_2, oy^0'=oy^post_2, x^0'=x^post_2, y^0'=y^post_2, [ oy^0<=y^0 && c^0==c^post_2 && ox^0==ox^post_2 && oy^0==oy^post_2 && x^0==x^post_2 && y^0==y^post_2 ], cost: 1
      2: l2 -> l0 : c^0'=c^post_3, ox^0'=ox^post_3, oy^0'=oy^post_3, x^0'=x^post_3, y^0'=y^post_3, [ 1+x^0<=0 && c^0==c^post_3 && ox^0==ox^post_3 && oy^0==oy^post_3 && x^0==x^post_3 && y^0==y^post_3 ], cost: 1
      3: l2 -> l0 : c^0'=c^post_4, ox^0'=ox^post_4, oy^0'=oy^post_4, x^0'=x^post_4, y^0'=y^post_4, [ ox^0<=x^0 && c^0==c^post_4 && ox^0==ox^post_4 && oy^0==oy^post_4 && x^0==x^post_4 && y^0==y^post_4 ], cost: 1
      4: l2 -> l3 : c^0'=c^post_5, ox^0'=ox^post_5, oy^0'=oy^post_5, x^0'=x^post_5, y^0'=y^post_5, [ x^post_5==x^0 && c^0==c^post_5 && ox^0==ox^post_5 && oy^0==oy^post_5 && y^0==y^post_5 ], cost: 1
     11: l3 -> l6 : c^0'=c^post_12, ox^0'=ox^post_12, oy^0'=oy^post_12, x^0'=x^post_12, y^0'=y^post_12, [ 1<=x^0 && 1<=y^0 && c^0==c^post_12 && ox^0==ox^post_12 && oy^0==oy^post_12 && x^0==x^post_12 && y^0==y^post_12 ], cost: 1
      5: l4 -> l3 : c^0'=c^post_6, ox^0'=ox^post_6, oy^0'=oy^post_6, x^0'=x^post_6, y^0'=y^post_6, [ c^0<=0 && c^0==c^post_6 && ox^0==ox^post_6 && oy^0==oy^post_6 && x^0==x^post_6 && y^0==y^post_6 ], cost: 1
      6: l4 -> l3 : c^0'=c^post_7, ox^0'=ox^post_7, oy^0'=oy^post_7, x^0'=x^post_7, y^0'=y^post_7, [ ox^post_7==x^0 && oy^post_7==y^0 && c^post_7==1 && x^0==x^post_7 && y^0==y^post_7 ], cost: 1
      7: l5 -> l4 : c^0'=c^post_8, ox^0'=ox^post_8, oy^0'=oy^post_8, x^0'=x^post_8, y^0'=y^post_8, [ c^0<=0 && c^0==c^post_8 && ox^0==ox^post_8 && oy^0==oy^post_8 && x^0==x^post_8 && y^0==y^post_8 ], cost: 1
      8: l5 -> l2 : c^0'=c^post_9, ox^0'=ox^post_9, oy^0'=oy^post_9, x^0'=x^post_9, y^0'=y^post_9, [ 1<=c^0 && c^0==c^post_9 && ox^0==ox^post_9 && oy^0==oy^post_9 && x^0==x^post_9 && y^0==y^post_9 ], cost: 1
      9: l6 -> l5 : c^0'=c^post_10, ox^0'=ox^post_10, oy^0'=oy^post_10, x^0'=x^post_10, y^0'=y^post_10, [ y^post_10==-1+y^0 && c^0==c^post_10 && ox^0==ox^post_10 && oy^0==oy^post_10 && x^0==x^post_10 ], cost: 1
     10: l6 -> l5 : c^0'=c^post_11, ox^0'=ox^post_11, oy^0'=oy^post_11, x^0'=x^post_11, y^0'=y^post_11, [ x^post_11==-1+x^0 && c^0==c^post_11 && ox^0==ox^post_11 && oy^0==oy^post_11 && y^0==y^post_11 ], cost: 1
     12: l7 -> l3 : c^0'=c^post_13, ox^0'=ox^post_13, oy^0'=oy^post_13, x^0'=x^post_13, y^0'=y^post_13, [ c^0<=0 && c^0==c^post_13 && ox^0==ox^post_13 && oy^0==oy^post_13 && x^0==x^post_13 && y^0==y^post_13 ], cost: 1
     13: l8 -> l7 : c^0'=c^post_14, ox^0'=ox^post_14, oy^0'=oy^post_14, x^0'=x^post_14, y^0'=y^post_14, [ c^0==c^post_14 && ox^0==ox^post_14 && oy^0==oy^post_14 && x^0==x^post_14 && y^0==y^post_14 ], cost: 1

Checking for constant complexity:
   The following rule is satisfiable with cost >= 1, yielding constant complexity:
     13: l8 -> l7 : c^0'=c^post_14, ox^0'=ox^post_14, oy^0'=oy^post_14, x^0'=x^post_14, y^0'=y^post_14, [ c^0==c^post_14 && ox^0==ox^post_14 && oy^0==oy^post_14 && x^0==x^post_14 && y^0==y^post_14 ], cost: 1

Removed unreachable and leaf rules:
   Start location: l8
      4: l2 -> l3 : c^0'=c^post_5, ox^0'=ox^post_5, oy^0'=oy^post_5, x^0'=x^post_5, y^0'=y^post_5, [ x^post_5==x^0 && c^0==c^post_5 && ox^0==ox^post_5 && oy^0==oy^post_5 && y^0==y^post_5 ], cost: 1
     11: l3 -> l6 : c^0'=c^post_12, ox^0'=ox^post_12, oy^0'=oy^post_12, x^0'=x^post_12, y^0'=y^post_12, [ 1<=x^0 && 1<=y^0 && c^0==c^post_12 && ox^0==ox^post_12 && oy^0==oy^post_12 && x^0==x^post_12 && y^0==y^post_12 ], cost: 1
      5: l4 -> l3 : c^0'=c^post_6, ox^0'=ox^post_6, oy^0'=oy^post_6, x^0'=x^post_6, y^0'=y^post_6, [ c^0<=0 && c^0==c^post_6 && ox^0==ox^post_6 && oy^0==oy^post_6 && x^0==x^post_6 && y^0==y^post_6 ], cost: 1
      6: l4 -> l3 : c^0'=c^post_7, ox^0'=ox^post_7, oy^0'=oy^post_7, x^0'=x^post_7, y^0'=y^post_7, [ ox^post_7==x^0 && oy^post_7==y^0 && c^post_7==1 && x^0==x^post_7 && y^0==y^post_7 ], cost: 1
      7: l5 -> l4 : c^0'=c^post_8, ox^0'=ox^post_8, oy^0'=oy^post_8, x^0'=x^post_8, y^0'=y^post_8, [ c^0<=0 && c^0==c^post_8 && ox^0==ox^post_8 && oy^0==oy^post_8 && x^0==x^post_8 && y^0==y^post_8 ], cost: 1
      8: l5 -> l2 : c^0'=c^post_9, ox^0'=ox^post_9, oy^0'=oy^post_9, x^0'=x^post_9, y^0'=y^post_9, [ 1<=c^0 && c^0==c^post_9 && ox^0==ox^post_9 && oy^0==oy^post_9 && x^0==x^post_9 && y^0==y^post_9 ], cost: 1
      9: l6 -> l5 : c^0'=c^post_10, ox^0'=ox^post_10, oy^0'=oy^post_10, x^0'=x^post_10, y^0'=y^post_10, [ y^post_10==-1+y^0 && c^0==c^post_10 && ox^0==ox^post_10 && oy^0==oy^post_10 && x^0==x^post_10 ], cost: 1
     10: l6 -> l5 : c^0'=c^post_11, ox^0'=ox^post_11, oy^0'=oy^post_11, x^0'=x^post_11, y^0'=y^post_11, [ x^post_11==-1+x^0 && c^0==c^post_11 && ox^0==ox^post_11 && oy^0==oy^post_11 && y^0==y^post_11 ], cost: 1
     12: l7 -> l3 : c^0'=c^post_13, ox^0'=ox^post_13, oy^0'=oy^post_13, x^0'=x^post_13, y^0'=y^post_13, [ c^0<=0 && c^0==c^post_13 && ox^0==ox^post_13 && oy^0==oy^post_13 && x^0==x^post_13 && y^0==y^post_13 ], cost: 1
     13: l8 -> l7 : c^0'=c^post_14, ox^0'=ox^post_14, oy^0'=oy^post_14, x^0'=x^post_14, y^0'=y^post_14, [ c^0==c^post_14 && ox^0==ox^post_14 && oy^0==oy^post_14 && x^0==x^post_14 && y^0==y^post_14 ], cost: 1

Simplified all rules, resulting in:
   Start location: l8
      4: l2 -> l3 : [], cost: 1
     11: l3 -> l6 : [ 1<=x^0 && 1<=y^0 ], cost: 1
      5: l4 -> l3 : [ c^0<=0 ], cost: 1
      6: l4 -> l3 : c^0'=1, ox^0'=x^0, oy^0'=y^0, [], cost: 1
      7: l5 -> l4 : [ c^0<=0 ], cost: 1
      8: l5 -> l2 : [ 1<=c^0 ], cost: 1
      9: l6 -> l5 : y^0'=-1+y^0, [], cost: 1
     10: l6 -> l5 : x^0'=-1+x^0, [], cost: 1
     12: l7 -> l3 : [ c^0<=0 ], cost: 1
     13: l8 -> l7 : [], cost: 1

### Simplification by acceleration and chaining ###

Eliminated locations (on linear paths):
   Start location: l8
     11: l3 -> l6 : [ 1<=x^0 && 1<=y^0 ], cost: 1
      5: l4 -> l3 : [ c^0<=0 ], cost: 1
      6: l4 -> l3 : c^0'=1, ox^0'=x^0, oy^0'=y^0, [], cost: 1
      7: l5 -> l4 : [ c^0<=0 ], cost: 1
     15: l5 -> l3 : [ 1<=c^0 ], cost: 2
      9: l6 -> l5 : y^0'=-1+y^0, [], cost: 1
     10: l6 -> l5 : x^0'=-1+x^0, [], cost: 1
     14: l8 -> l3 : [ c^0<=0 ], cost: 2

Eliminated locations (on tree-shaped paths):
   Start location: l8
     16: l3 -> l5 : y^0'=-1+y^0, [ 1<=x^0 && 1<=y^0 ], cost: 2
     17: l3 -> l5 : x^0'=-1+x^0, [ 1<=x^0 && 1<=y^0 ], cost: 2
     15: l5 -> l3 : [ 1<=c^0 ], cost: 2
     18: l5 -> l3 : [ c^0<=0 ], cost: 2
     19: l5 -> l3 : c^0'=1, ox^0'=x^0, oy^0'=y^0, [ c^0<=0 ], cost: 2
     14: l8 -> l3 : [ c^0<=0 ], cost: 2

Eliminated locations (on tree-shaped paths):
   Start location: l8
     20: l3 -> l3 : y^0'=-1+y^0, [ 1<=x^0 && 1<=y^0 && 1<=c^0 ], cost: 4
     21: l3 -> l3 : y^0'=-1+y^0, [ 1<=x^0 && 1<=y^0 && c^0<=0 ], cost: 4
     22: l3 -> l3 : c^0'=1, ox^0'=x^0, oy^0'=-1+y^0, y^0'=-1+y^0, [ 1<=x^0 && 1<=y^0 && c^0<=0 ], cost: 4
     23: l3 -> l3 : x^0'=-1+x^0, [ 1<=x^0 && 1<=y^0 && 1<=c^0 ], cost: 4
     24: l3 -> l3 : x^0'=-1+x^0, [ 1<=x^0 && 1<=y^0 && c^0<=0 ], cost: 4
     25: l3 -> l3 : c^0'=1, ox^0'=-1+x^0, oy^0'=y^0, x^0'=-1+x^0, [ 1<=x^0 && 1<=y^0 && c^0<=0 ], cost: 4
     14: l8 -> l3 : [ c^0<=0 ], cost: 2

Accelerating simple loops of location 3.
   Accelerating the following rules:
     20: l3 -> l3 : y^0'=-1+y^0, [ 1<=x^0 && 1<=y^0 && 1<=c^0 ], cost: 4
     21: l3 -> l3 : y^0'=-1+y^0, [ 1<=x^0 && 1<=y^0 && c^0<=0 ], cost: 4
     22: l3 -> l3 : c^0'=1, ox^0'=x^0, oy^0'=-1+y^0, y^0'=-1+y^0, [ 1<=x^0 && 1<=y^0 && c^0<=0 ], cost: 4
     23: l3 -> l3 : x^0'=-1+x^0, [ 1<=x^0 && 1<=y^0 && 1<=c^0 ], cost: 4
     24: l3 -> l3 : x^0'=-1+x^0, [ 1<=x^0 && 1<=y^0 && c^0<=0 ], cost: 4
     25: l3 -> l3 : c^0'=1, ox^0'=-1+x^0, oy^0'=y^0, x^0'=-1+x^0, [ 1<=x^0 && 1<=y^0 && c^0<=0 ], cost: 4

   Accelerated rule 20 with metering function y^0, yielding the new rule 26.
   Accelerated rule 21 with metering function y^0, yielding the new rule 27.
   Found no metering function for rule 22.
   Accelerated rule 23 with metering function x^0, yielding the new rule 28.
   Accelerated rule 24 with metering function x^0, yielding the new rule 29.
   Found no metering function for rule 25.
   Removing the simple loops: 20 21 23 24.

Accelerated all simple loops using metering functions (where possible):
   Start location: l8
     22: l3 -> l3 : c^0'=1, ox^0'=x^0, oy^0'=-1+y^0, y^0'=-1+y^0, [ 1<=x^0 && 1<=y^0 && c^0<=0 ], cost: 4
     25: l3 -> l3 : c^0'=1, ox^0'=-1+x^0, oy^0'=y^0, x^0'=-1+x^0, [ 1<=x^0 && 1<=y^0 && c^0<=0 ], cost: 4
     26: l3 -> l3 : y^0'=0, [ 1<=x^0 && 1<=y^0 && 1<=c^0 ], cost: 4*y^0
     27: l3 -> l3 : y^0'=0, [ 1<=x^0 && 1<=y^0 && c^0<=0 ], cost: 4*y^0
     28: l3 -> l3 : x^0'=0, [ 1<=x^0 && 1<=y^0 && 1<=c^0 ], cost: 4*x^0
     29: l3 -> l3 : x^0'=0, [ 1<=x^0 && 1<=y^0 && c^0<=0 ], cost: 4*x^0
     14: l8 -> l3 : [ c^0<=0 ], cost: 2

Chained accelerated rules (with incoming rules):
   Start location: l8
     14: l8 -> l3 : [ c^0<=0 ], cost: 2
     30: l8 -> l3 : c^0'=1, ox^0'=x^0, oy^0'=-1+y^0, y^0'=-1+y^0, [ c^0<=0 && 1<=x^0 && 1<=y^0 ], cost: 6
     31: l8 -> l3 : c^0'=1, ox^0'=-1+x^0, oy^0'=y^0, x^0'=-1+x^0, [ c^0<=0 && 1<=x^0 && 1<=y^0 ], cost: 6
     32: l8 -> l3 : y^0'=0, [ c^0<=0 && 1<=x^0 && 1<=y^0 ], cost: 2+4*y^0
     33: l8 -> l3 : x^0'=0, [ c^0<=0 && 1<=x^0 && 1<=y^0 ], cost: 2+4*x^0

Removed unreachable locations (and leaf rules with constant cost):
   Start location: l8
     32: l8 -> l3 : y^0'=0, [ c^0<=0 && 1<=x^0 && 1<=y^0 ], cost: 2+4*y^0
     33: l8 -> l3 : x^0'=0, [ c^0<=0 && 1<=x^0 && 1<=y^0 ], cost: 2+4*x^0

### Computing asymptotic complexity ###

Fully simplified ITS problem
   Start location: l8
     32: l8 -> l3 : y^0'=0, [ c^0<=0 && 1<=x^0 && 1<=y^0 ], cost: 2+4*y^0
     33: l8 -> l3 : x^0'=0, [ c^0<=0 && 1<=x^0 && 1<=y^0 ], cost: 2+4*x^0

Computing asymptotic complexity for rule 32
   Solved the limit problem by the following transformations:
      Created initial limit problem:
      x^0 (+/+!), 1-c^0 (+/+!), y^0 (+/+!), 2+4*y^0 (+) [not solved]

      removing all constraints (solved by SMT)
      resulting limit problem: [solved]

      applying transformation rule (C) using substitution {x^0==n,c^0==-n,y^0==n}
      resulting limit problem:
      [solved]

   Solution:
      x^0 / n
      c^0 / -n
      y^0 / n
   Resulting cost 2+4*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: 2+4*n
   Rule cost:   2+4*y^0
   Rule guard:  [ c^0<=0 && 1<=x^0 && 1<=y^0 ]

WORST_CASE(Omega(n^1),?)