WORST_CASE(Omega(n^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 metering function -y^0, yielding the new rule 23.
   Accelerated rule 16 with metering function x^0-y^0, yielding the new rule 24.
   Accelerated rule 17 with metering function -x^0, yielding the new rule 25.
   Accelerated rule 18 with metering function -x^0-y^0, yielding the new rule 26.
   Accelerated rule 19 with metering function x^0, yielding the new rule 27.
   Accelerated rule 20 with metering function -1+x^0+y^0, yielding the new rule 28.
   Accelerated rule 21 with metering function -x^0+y^0, yielding the new rule 29.
   Accelerated rule 22 with metering function y^0, yielding the new rule 30.
   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 && 1+y^0<=0 && x^0<=y^0 && 1+x^0<=-y^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 && 1+y^0<=x^0 ], cost: 3*x^0-3*y^0
     25: l1 -> l1 : x^0'=0, [ 1+x^0<=0 && 1<=y^0 && -x^0<=y^0 && 1+x^0<=y^0 ], cost: -3*x^0
     26: l1 -> l1 : y^0'=-x^0, [ 1+x^0<=0 && 1<=y^0 && x^0<=y^0 && 1+x^0<=-y^0 ], cost: -3*x^0-3*y^0
     27: l1 -> l1 : x^0'=0, [ 1<=x^0 && 1+y^0<=0 && y^0<=1-x^0 && 1+y^0<=x^0 ], cost: 3*x^0
     28: l1 -> l1 : y^0'=1-x^0, [ 1<=x^0 && 1+y^0<=0 && y^0<=x^0 && 2-y^0<=x^0 ], cost: -3+3*x^0+3*y^0
     29: l1 -> l1 : x^0'=y^0, [ 1<=x^0 && 1<=y^0 && -x^0<=y^0 && 1+x^0<=y^0 ], cost: -3*x^0+3*y^0
     30: l1 -> l1 : y^0'=0, [ 1<=x^0 && 1<=y^0 && y^0<=x^0 && 2-y^0<=x^0 ], cost: 3*y^0
     10: l4 -> l1 : [], cost: 2

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

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

### Computing asymptotic complexity ###

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

Computing asymptotic complexity for rule 31
   Simplified the guard:
     31: l4 -> l1 : y^0'=0, [ 1+y^0<=0 && x^0<=y^0 ], cost: 2-3*y^0
   Solved the limit problem by the following transformations:
      Created initial limit problem:
      1-x^0+y^0 (+/+!), 2-3*y^0 (+), -y^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}
      resulting limit problem:
      [solved]

   Solution:
      x^0 / -n
      y^0 / -n
   Resulting cost 2+3*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+3*n
   Rule cost:   2-3*y^0
   Rule guard:  [ 1+y^0<=0 && x^0<=y^0 ]

WORST_CASE(Omega(n^1),?)