WORST_CASE(Omega(1),?)

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

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

Checking for constant complexity:
   The following rule is satisfiable with cost >= 1, yielding constant complexity:
      9: l4 -> l3 : x^0'=x^post_10, y^0'=y^post_10, [ x^0==x^post_10 && y^0==y^post_10 ], cost: 1

Simplified all rules, resulting in:
   Start location: l4
      0: l0 -> l1 : x^0'=1+x^0, [ -x^0<=y^0 && 1+x^0<=y^0 ], cost: 1
      1: l0 -> l1 : y^0'=1+y^0, [ x^0<=y^0 && 1+x^0<=-y^0 ], cost: 1
      2: l0 -> l1 : x^0'=-1+x^0, [ y^0<=1-x^0 && 1+y^0<=x^0 ], cost: 1
      3: l0 -> l1 : y^0'=-1+y^0, [ y^0<=x^0 && 2-y^0<=x^0 ], cost: 1
      4: l1 -> l2 : [ 1+x^0<=0 ], cost: 1
      5: l1 -> l2 : [ 1<=x^0 ], cost: 1
      6: l2 -> l0 : [ 1+y^0<=0 ], cost: 1
      7: l2 -> l0 : [ 1<=y^0 ], cost: 1
      8: l3 -> l1 : [], cost: 1
      9: l4 -> l3 : [], cost: 1

### Simplification by acceleration and chaining ###

Eliminated locations (on linear paths):
   Start location: l4
      0: l0 -> l1 : x^0'=1+x^0, [ -x^0<=y^0 && 1+x^0<=y^0 ], cost: 1
      1: l0 -> l1 : y^0'=1+y^0, [ x^0<=y^0 && 1+x^0<=-y^0 ], cost: 1
      2: l0 -> l1 : x^0'=-1+x^0, [ y^0<=1-x^0 && 1+y^0<=x^0 ], cost: 1
      3: l0 -> l1 : y^0'=-1+y^0, [ y^0<=x^0 && 2-y^0<=x^0 ], cost: 1
      4: l1 -> l2 : [ 1+x^0<=0 ], cost: 1
      5: l1 -> l2 : [ 1<=x^0 ], cost: 1
      6: l2 -> l0 : [ 1+y^0<=0 ], cost: 1
      7: l2 -> l0 : [ 1<=y^0 ], cost: 1
     10: l4 -> l1 : [], cost: 2

Eliminated locations (on tree-shaped paths):
   Start location: l4
      0: l0 -> l1 : x^0'=1+x^0, [ -x^0<=y^0 && 1+x^0<=y^0 ], cost: 1
      1: l0 -> l1 : y^0'=1+y^0, [ x^0<=y^0 && 1+x^0<=-y^0 ], cost: 1
      2: l0 -> l1 : x^0'=-1+x^0, [ y^0<=1-x^0 && 1+y^0<=x^0 ], cost: 1
      3: l0 -> l1 : y^0'=-1+y^0, [ y^0<=x^0 && 2-y^0<=x^0 ], cost: 1
     11: l1 -> l0 : [ 1+x^0<=0 && 1+y^0<=0 ], cost: 2
     12: l1 -> l0 : [ 1+x^0<=0 && 1<=y^0 ], cost: 2
     13: l1 -> l0 : [ 1<=x^0 && 1+y^0<=0 ], cost: 2
     14: l1 -> l0 : [ 1<=x^0 && 1<=y^0 ], cost: 2
     10: l4 -> l1 : [], cost: 2

Eliminated locations (on tree-shaped paths):
   Start location: l4
     15: l1 -> l1 : y^0'=1+y^0, [ 1+x^0<=0 && 1+y^0<=0 && x^0<=y^0 && 1+x^0<=-y^0 ], cost: 3
     16: l1 -> l1 : x^0'=-1+x^0, [ 1+x^0<=0 && 1+y^0<=0 && y^0<=1-x^0 && 1+y^0<=x^0 ], cost: 3
     17: l1 -> l1 : x^0'=1+x^0, [ 1+x^0<=0 && 1<=y^0 && -x^0<=y^0 && 1+x^0<=y^0 ], cost: 3
     18: l1 -> l1 : y^0'=1+y^0, [ 1+x^0<=0 && 1<=y^0 && x^0<=y^0 && 1+x^0<=-y^0 ], cost: 3
     19: l1 -> l1 : x^0'=-1+x^0, [ 1<=x^0 && 1+y^0<=0 && y^0<=1-x^0 && 1+y^0<=x^0 ], cost: 3
     20: l1 -> l1 : y^0'=-1+y^0, [ 1<=x^0 && 1+y^0<=0 && y^0<=x^0 && 2-y^0<=x^0 ], cost: 3
     21: l1 -> l1 : x^0'=1+x^0, [ 1<=x^0 && 1<=y^0 && -x^0<=y^0 && 1+x^0<=y^0 ], cost: 3
     22: l1 -> l1 : y^0'=-1+y^0, [ 1<=x^0 && 1<=y^0 && y^0<=x^0 && 2-y^0<=x^0 ], cost: 3
     10: l4 -> l1 : [], cost: 2

Accelerating simple loops of location 1.
   Accelerating the following rules:
     15: l1 -> l1 : y^0'=1+y^0, [ 1+x^0<=0 && 1+y^0<=0 && x^0<=y^0 && 1+x^0<=-y^0 ], cost: 3
     16: l1 -> l1 : x^0'=-1+x^0, [ 1+x^0<=0 && 1+y^0<=0 && y^0<=1-x^0 && 1+y^0<=x^0 ], cost: 3
     17: l1 -> l1 : x^0'=1+x^0, [ 1+x^0<=0 && 1<=y^0 && -x^0<=y^0 && 1+x^0<=y^0 ], cost: 3
     18: l1 -> l1 : y^0'=1+y^0, [ 1+x^0<=0 && 1<=y^0 && x^0<=y^0 && 1+x^0<=-y^0 ], cost: 3
     19: l1 -> l1 : x^0'=-1+x^0, [ 1<=x^0 && 1+y^0<=0 && y^0<=1-x^0 && 1+y^0<=x^0 ], cost: 3
     20: l1 -> l1 : y^0'=-1+y^0, [ 1<=x^0 && 1+y^0<=0 && y^0<=x^0 && 2-y^0<=x^0 ], cost: 3
     21: l1 -> l1 : x^0'=1+x^0, [ 1<=x^0 && 1<=y^0 && -x^0<=y^0 && 1+x^0<=y^0 ], cost: 3
     22: l1 -> l1 : y^0'=-1+y^0, [ 1<=x^0 && 1<=y^0 && y^0<=x^0 && 2-y^0<=x^0 ], cost: 3

   Accelerated rule 15 with backward acceleration, yielding the new rule 23.
   Accelerated rule 16 with backward acceleration, yielding the new rule 24.
   Accelerated rule 17 with backward acceleration, yielding the new rule 25.
   Accelerated rule 18 with backward acceleration, yielding the new rule 26.
   Accelerated rule 19 with backward acceleration, yielding the new rule 27.
   Accelerated rule 20 with backward acceleration, yielding the new rule 28.
   Accelerated rule 21 with backward acceleration, yielding the new rule 29.
   Accelerated rule 22 with backward acceleration, yielding the new rule 30.
   Accelerated rule 22 with backward acceleration, yielding the new rule 31.
[accelerate] Nesting with 9 inner and 8 outer candidates
   Removing the simple loops: 15 16 17 18 19 20 21 22.

Accelerated all simple loops using metering functions (where possible):
   Start location: l4
     23: l1 -> l1 : y^0'=0, [ 1+x^0<=0 && x^0<=y^0 && 1+x^0<=-y^0 && -y^0>=0 ], cost: -3*y^0
     24: l1 -> l1 : x^0'=y^0, [ 1+x^0<=0 && 1+y^0<=0 && y^0<=1-x^0 && -y^0+x^0>=0 ], cost: -3*y^0+3*x^0
     25: l1 -> l1 : x^0'=0, [ 1<=y^0 && -x^0<=y^0 && 1+x^0<=y^0 && -x^0>=0 ], cost: -3*x^0
     26: l1 -> l1 : y^0'=-x^0, [ 1+x^0<=0 && 1<=y^0 && x^0<=y^0 && -y^0-x^0>=0 ], cost: -3*y^0-3*x^0
     27: l1 -> l1 : x^0'=0, [ 1+y^0<=0 && y^0<=1-x^0 && 1+y^0<=x^0 && x^0>=0 ], cost: 3*x^0
     28: l1 -> l1 : y^0'=1-x^0, [ 1<=x^0 && 1+y^0<=0 && y^0<=x^0 && -1+y^0+x^0>=0 ], cost: -3+3*y^0+3*x^0
     29: l1 -> l1 : x^0'=y^0, [ 1<=x^0 && 1<=y^0 && -x^0<=y^0 && y^0-x^0>=0 ], cost: 3*y^0-3*x^0
     30: l1 -> l1 : y^0'=0, [ 1<=x^0 && y^0<=x^0 && y^0>=0 ], cost: 3*y^0
     31: l1 -> l1 : y^0'=1-x^0, [ 1<=x^0 && y^0<=x^0 && -1+y^0+x^0>=0 && 1<=2-x^0 ], cost: -3+3*y^0+3*x^0
     10: l4 -> l1 : [], cost: 2

Chained accelerated rules (with incoming rules):
   Start location: l4
     10: l4 -> l1 : [], cost: 2
     32: l4 -> l1 : y^0'=0, [ 1+x^0<=0 && x^0<=y^0 && 1+x^0<=-y^0 && -y^0>=0 ], cost: 2-3*y^0
     33: l4 -> l1 : x^0'=y^0, [ 1+x^0<=0 && 1+y^0<=0 && y^0<=1-x^0 && -y^0+x^0>=0 ], cost: 2-3*y^0+3*x^0
     34: l4 -> l1 : x^0'=0, [ 1<=y^0 && -x^0<=y^0 && 1+x^0<=y^0 && -x^0>=0 ], cost: 2-3*x^0
     35: l4 -> l1 : y^0'=-x^0, [ 1+x^0<=0 && 1<=y^0 && x^0<=y^0 && -y^0-x^0>=0 ], cost: 2-3*y^0-3*x^0
     36: l4 -> l1 : x^0'=0, [ 1+y^0<=0 && y^0<=1-x^0 && 1+y^0<=x^0 && x^0>=0 ], cost: 2+3*x^0
     37: l4 -> l1 : y^0'=1-x^0, [ 1<=x^0 && 1+y^0<=0 && y^0<=x^0 && -1+y^0+x^0>=0 ], cost: -1+3*y^0+3*x^0
     38: l4 -> l1 : x^0'=y^0, [ 1<=x^0 && 1<=y^0 && -x^0<=y^0 && y^0-x^0>=0 ], cost: 2+3*y^0-3*x^0
     39: l4 -> l1 : y^0'=0, [ 1<=x^0 && y^0<=x^0 && y^0>=0 ], cost: 2+3*y^0
     40: l4 -> l1 : y^0'=1-x^0, [ 1-x^0==0 && y^0<=x^0 && -1+y^0+x^0>=0 ], cost: -1+3*y^0+3*x^0

Removed unreachable locations (and leaf rules with constant cost):
   Start location: l4
     32: l4 -> l1 : y^0'=0, [ 1+x^0<=0 && x^0<=y^0 && 1+x^0<=-y^0 && -y^0>=0 ], cost: 2-3*y^0
     33: l4 -> l1 : x^0'=y^0, [ 1+x^0<=0 && 1+y^0<=0 && y^0<=1-x^0 && -y^0+x^0>=0 ], cost: 2-3*y^0+3*x^0
     34: l4 -> l1 : x^0'=0, [ 1<=y^0 && -x^0<=y^0 && 1+x^0<=y^0 && -x^0>=0 ], cost: 2-3*x^0
     35: l4 -> l1 : y^0'=-x^0, [ 1+x^0<=0 && 1<=y^0 && x^0<=y^0 && -y^0-x^0>=0 ], cost: 2-3*y^0-3*x^0
     36: l4 -> l1 : x^0'=0, [ 1+y^0<=0 && y^0<=1-x^0 && 1+y^0<=x^0 && x^0>=0 ], cost: 2+3*x^0
     37: l4 -> l1 : y^0'=1-x^0, [ 1<=x^0 && 1+y^0<=0 && y^0<=x^0 && -1+y^0+x^0>=0 ], cost: -1+3*y^0+3*x^0
     38: l4 -> l1 : x^0'=y^0, [ 1<=x^0 && 1<=y^0 && -x^0<=y^0 && y^0-x^0>=0 ], cost: 2+3*y^0-3*x^0
     39: l4 -> l1 : y^0'=0, [ 1<=x^0 && y^0<=x^0 && y^0>=0 ], cost: 2+3*y^0
     40: l4 -> l1 : y^0'=1-x^0, [ 1-x^0==0 && y^0<=x^0 && -1+y^0+x^0>=0 ], cost: -1+3*y^0+3*x^0

### Computing asymptotic complexity ###

Fully simplified ITS problem
   Start location: l4
     32: l4 -> l1 : y^0'=0, [ 1+x^0<=0 && x^0<=y^0 && 1+x^0<=-y^0 && -y^0>=0 ], cost: 2-3*y^0
     33: l4 -> l1 : x^0'=y^0, [ 1+x^0<=0 && 1+y^0<=0 && y^0<=1-x^0 && -y^0+x^0>=0 ], cost: 2-3*y^0+3*x^0
     34: l4 -> l1 : x^0'=0, [ 1<=y^0 && -x^0<=y^0 && 1+x^0<=y^0 && -x^0>=0 ], cost: 2-3*x^0
     35: l4 -> l1 : y^0'=-x^0, [ 1+x^0<=0 && 1<=y^0 && x^0<=y^0 && -y^0-x^0>=0 ], cost: 2-3*y^0-3*x^0
     36: l4 -> l1 : x^0'=0, [ 1+y^0<=0 && y^0<=1-x^0 && 1+y^0<=x^0 && x^0>=0 ], cost: 2+3*x^0
     37: l4 -> l1 : y^0'=1-x^0, [ 1<=x^0 && 1+y^0<=0 && y^0<=x^0 && -1+y^0+x^0>=0 ], cost: -1+3*y^0+3*x^0
     38: l4 -> l1 : x^0'=y^0, [ 1<=x^0 && 1<=y^0 && -x^0<=y^0 && y^0-x^0>=0 ], cost: 2+3*y^0-3*x^0
     39: l4 -> l1 : y^0'=0, [ 1<=x^0 && y^0<=x^0 && y^0>=0 ], cost: 2+3*y^0
     40: l4 -> l1 : y^0'=1-x^0, [ 1-x^0==0 && y^0<=x^0 && -1+y^0+x^0>=0 ], cost: -1+3*y^0+3*x^0

Computing asymptotic complexity for rule 39
   Resulting cost 0 has complexity: Unknown

Computing asymptotic complexity for rule 40
   Resulting cost 0 has complexity: Unknown

Computing asymptotic complexity for rule 32
   Simplified the guard:
     32: l4 -> l1 : y^0'=0, [ x^0<=y^0 && 1+x^0<=-y^0 && -y^0>=0 ], cost: 2-3*y^0
   Resulting cost 0 has complexity: Unknown

Computing asymptotic complexity for rule 33
   Resulting cost 0 has complexity: Unknown

Computing asymptotic complexity for rule 34
   Simplified the guard:
     34: l4 -> l1 : x^0'=0, [ -x^0<=y^0 && 1+x^0<=y^0 && -x^0>=0 ], cost: 2-3*x^0
   Resulting cost 0 has complexity: Unknown

Computing asymptotic complexity for rule 35
   Simplified the guard:
     35: l4 -> l1 : y^0'=-x^0, [ 1<=y^0 && -y^0-x^0>=0 ], cost: 2-3*y^0-3*x^0
   Resulting cost 0 has complexity: Unknown

Computing asymptotic complexity for rule 36
   Resulting cost 0 has complexity: Unknown

Computing asymptotic complexity for rule 37
   Simplified the guard:
     37: l4 -> l1 : y^0'=1-x^0, [ 1+y^0<=0 && -1+y^0+x^0>=0 ], cost: -1+3*y^0+3*x^0
   Resulting cost 0 has complexity: Unknown

Computing asymptotic complexity for rule 38
   Resulting cost 0 has complexity: Unknown

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:  [ x^0==x^post_10 && y^0==y^post_10 ]

WORST_CASE(Omega(1),?)