WORST_CASE(Omega(n^1),?)

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

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

Checking for constant complexity:
   The following rule is satisfiable with cost >= 1, yielding constant complexity:
     10: l7 -> l6 : c^0'=c^post_11, ox^0'=ox^post_11, x^0'=x^post_11, y^0'=y^post_11, [ c^0==c^post_11 && ox^0==ox^post_11 && x^0==x^post_11 && y^0==y^post_11 ], cost: 1

Removed unreachable and leaf rules:
   Start location: l7
      0: l0 -> l1 : c^0'=c^post_1, ox^0'=ox^post_1, x^0'=x^post_1, y^0'=y^post_1, [ 1+x^0<=ox^0 && c^0==c^post_1 && ox^0==ox^post_1 && x^0==x^post_1 && y^0==y^post_1 ], cost: 1
      8: l1 -> l5 : c^0'=c^post_9, ox^0'=ox^post_9, x^0'=x^post_9, y^0'=y^post_9, [ 1<=x^0 && 1<=y^0 && c^0==c^post_9 && ox^0==ox^post_9 && x^0==x^post_9 && y^0==y^post_9 ], cost: 1
      2: l3 -> l1 : c^0'=c^post_3, ox^0'=ox^post_3, x^0'=x^post_3, y^0'=y^post_3, [ c^0<=0 && c^0==c^post_3 && ox^0==ox^post_3 && x^0==x^post_3 && y^0==y^post_3 ], cost: 1
      3: l3 -> l1 : c^0'=c^post_4, ox^0'=ox^post_4, x^0'=x^post_4, y^0'=y^post_4, [ ox^post_4==x^0 && c^post_4==1 && x^0==x^post_4 && y^0==y^post_4 ], cost: 1
      4: l4 -> l3 : c^0'=c^post_5, ox^0'=ox^post_5, x^0'=x^post_5, y^0'=y^post_5, [ c^0<=0 && c^0==c^post_5 && ox^0==ox^post_5 && x^0==x^post_5 && y^0==y^post_5 ], cost: 1
      5: l4 -> l0 : c^0'=c^post_6, ox^0'=ox^post_6, x^0'=x^post_6, y^0'=y^post_6, [ 1<=c^0 && c^0==c^post_6 && ox^0==ox^post_6 && x^0==x^post_6 && y^0==y^post_6 ], cost: 1
      6: l5 -> l4 : c^0'=c^post_7, ox^0'=ox^post_7, x^0'=x^post_7, y^0'=y^post_7, [ y^post_7==-1+y^0 && c^0==c^post_7 && ox^0==ox^post_7 && x^0==x^post_7 ], cost: 1
      7: l5 -> l4 : c^0'=c^post_8, ox^0'=ox^post_8, x^0'=x^post_8, y^0'=y^post_8, [ x^post_8==-1+x^0 && c^0==c^post_8 && ox^0==ox^post_8 && y^0==y^post_8 ], cost: 1
      9: l6 -> l1 : c^0'=c^post_10, ox^0'=ox^post_10, x^0'=x^post_10, y^0'=y^post_10, [ c^0<=0 && c^0==c^post_10 && ox^0==ox^post_10 && x^0==x^post_10 && y^0==y^post_10 ], cost: 1
     10: l7 -> l6 : c^0'=c^post_11, ox^0'=ox^post_11, x^0'=x^post_11, y^0'=y^post_11, [ c^0==c^post_11 && ox^0==ox^post_11 && x^0==x^post_11 && y^0==y^post_11 ], cost: 1

Simplified all rules, resulting in:
   Start location: l7
      0: l0 -> l1 : [ 1+x^0<=ox^0 ], cost: 1
      8: l1 -> l5 : [ 1<=x^0 && 1<=y^0 ], cost: 1
      2: l3 -> l1 : [ c^0<=0 ], cost: 1
      3: l3 -> l1 : c^0'=1, ox^0'=x^0, [], cost: 1
      4: l4 -> l3 : [ c^0<=0 ], cost: 1
      5: l4 -> l0 : [ 1<=c^0 ], cost: 1
      6: l5 -> l4 : y^0'=-1+y^0, [], cost: 1
      7: l5 -> l4 : x^0'=-1+x^0, [], cost: 1
      9: l6 -> l1 : [ c^0<=0 ], cost: 1
     10: l7 -> l6 : [], cost: 1

### Simplification by acceleration and chaining ###

Eliminated locations (on linear paths):
   Start location: l7
      8: l1 -> l5 : [ 1<=x^0 && 1<=y^0 ], cost: 1
      2: l3 -> l1 : [ c^0<=0 ], cost: 1
      3: l3 -> l1 : c^0'=1, ox^0'=x^0, [], cost: 1
      4: l4 -> l3 : [ c^0<=0 ], cost: 1
     12: l4 -> l1 : [ 1<=c^0 && 1+x^0<=ox^0 ], cost: 2
      6: l5 -> l4 : y^0'=-1+y^0, [], cost: 1
      7: l5 -> l4 : x^0'=-1+x^0, [], cost: 1
     11: l7 -> l1 : [ c^0<=0 ], cost: 2

Eliminated locations (on tree-shaped paths):
   Start location: l7
     13: l1 -> l4 : y^0'=-1+y^0, [ 1<=x^0 && 1<=y^0 ], cost: 2
     14: l1 -> l4 : x^0'=-1+x^0, [ 1<=x^0 && 1<=y^0 ], cost: 2
     12: l4 -> l1 : [ 1<=c^0 && 1+x^0<=ox^0 ], cost: 2
     15: l4 -> l1 : [ c^0<=0 ], cost: 2
     16: l4 -> l1 : c^0'=1, ox^0'=x^0, [ c^0<=0 ], cost: 2
     11: l7 -> l1 : [ c^0<=0 ], cost: 2

Eliminated locations (on tree-shaped paths):
   Start location: l7
     17: l1 -> l1 : y^0'=-1+y^0, [ 1<=x^0 && 1<=y^0 && 1<=c^0 && 1+x^0<=ox^0 ], cost: 4
     18: l1 -> l1 : y^0'=-1+y^0, [ 1<=x^0 && 1<=y^0 && c^0<=0 ], cost: 4
     19: l1 -> l1 : c^0'=1, ox^0'=x^0, y^0'=-1+y^0, [ 1<=x^0 && 1<=y^0 && c^0<=0 ], cost: 4
     20: l1 -> l1 : x^0'=-1+x^0, [ 1<=x^0 && 1<=y^0 && 1<=c^0 && x^0<=ox^0 ], cost: 4
     21: l1 -> l1 : x^0'=-1+x^0, [ 1<=x^0 && 1<=y^0 && c^0<=0 ], cost: 4
     22: l1 -> l1 : c^0'=1, ox^0'=-1+x^0, x^0'=-1+x^0, [ 1<=x^0 && 1<=y^0 && c^0<=0 ], cost: 4
     11: l7 -> l1 : [ c^0<=0 ], cost: 2

Accelerating simple loops of location 1.
   Accelerating the following rules:
     17: l1 -> l1 : y^0'=-1+y^0, [ 1<=x^0 && 1<=y^0 && 1<=c^0 && 1+x^0<=ox^0 ], cost: 4
     18: l1 -> l1 : y^0'=-1+y^0, [ 1<=x^0 && 1<=y^0 && c^0<=0 ], cost: 4
     19: l1 -> l1 : c^0'=1, ox^0'=x^0, y^0'=-1+y^0, [ 1<=x^0 && 1<=y^0 && c^0<=0 ], cost: 4
     20: l1 -> l1 : x^0'=-1+x^0, [ 1<=x^0 && 1<=y^0 && 1<=c^0 && x^0<=ox^0 ], cost: 4
     21: l1 -> l1 : x^0'=-1+x^0, [ 1<=x^0 && 1<=y^0 && c^0<=0 ], cost: 4
     22: l1 -> l1 : c^0'=1, ox^0'=-1+x^0, x^0'=-1+x^0, [ 1<=x^0 && 1<=y^0 && c^0<=0 ], cost: 4

   Accelerated rule 17 with metering function y^0, yielding the new rule 23.
   Accelerated rule 18 with metering function y^0, yielding the new rule 24.
   Found no metering function for rule 19.
   Accelerated rule 20 with metering function x^0, yielding the new rule 25.
   Accelerated rule 21 with metering function x^0, yielding the new rule 26.
   Found no metering function for rule 22.
   Removing the simple loops: 17 18 20 21.

Accelerated all simple loops using metering functions (where possible):
   Start location: l7
     19: l1 -> l1 : c^0'=1, ox^0'=x^0, y^0'=-1+y^0, [ 1<=x^0 && 1<=y^0 && c^0<=0 ], cost: 4
     22: l1 -> l1 : c^0'=1, ox^0'=-1+x^0, x^0'=-1+x^0, [ 1<=x^0 && 1<=y^0 && c^0<=0 ], cost: 4
     23: l1 -> l1 : y^0'=0, [ 1<=x^0 && 1<=y^0 && 1<=c^0 && 1+x^0<=ox^0 ], cost: 4*y^0
     24: l1 -> l1 : y^0'=0, [ 1<=x^0 && 1<=y^0 && c^0<=0 ], cost: 4*y^0
     25: l1 -> l1 : x^0'=0, [ 1<=x^0 && 1<=y^0 && 1<=c^0 && x^0<=ox^0 ], cost: 4*x^0
     26: l1 -> l1 : x^0'=0, [ 1<=x^0 && 1<=y^0 && c^0<=0 ], cost: 4*x^0
     11: l7 -> l1 : [ c^0<=0 ], cost: 2

Chained accelerated rules (with incoming rules):
   Start location: l7
     11: l7 -> l1 : [ c^0<=0 ], cost: 2
     27: l7 -> l1 : c^0'=1, ox^0'=x^0, y^0'=-1+y^0, [ c^0<=0 && 1<=x^0 && 1<=y^0 ], cost: 6
     28: l7 -> l1 : c^0'=1, ox^0'=-1+x^0, x^0'=-1+x^0, [ c^0<=0 && 1<=x^0 && 1<=y^0 ], cost: 6
     29: l7 -> l1 : y^0'=0, [ c^0<=0 && 1<=x^0 && 1<=y^0 ], cost: 2+4*y^0
     30: l7 -> l1 : 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: l7
     29: l7 -> l1 : y^0'=0, [ c^0<=0 && 1<=x^0 && 1<=y^0 ], cost: 2+4*y^0
     30: l7 -> l1 : 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: l7
     29: l7 -> l1 : y^0'=0, [ c^0<=0 && 1<=x^0 && 1<=y^0 ], cost: 2+4*y^0
     30: l7 -> l1 : x^0'=0, [ c^0<=0 && 1<=x^0 && 1<=y^0 ], cost: 2+4*x^0

Computing asymptotic complexity for rule 29
   Solved the limit problem by the following transformations:
      Created initial limit problem:
      x^0 (+/+!), 2+4*y^0 (+), y^0 (+/+!), 1-c^0 (+/+!) [not solved]

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

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

   Solution:
      x^0 / n
      y^0 / n
      c^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),?)