WORST_CASE(Omega(n^1),?)

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

Initial linear ITS problem
   Start location: l4
      0: l0 -> l1 : __disjvr_0^0'=__disjvr_0^post_1, __disjvr_1^0'=__disjvr_1^post_1, 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 && __disjvr_0^0==__disjvr_0^post_1 && __disjvr_1^0==__disjvr_1^post_1 && y^0==y^post_1 ], cost: 1
      1: l0 -> l1 : __disjvr_0^0'=__disjvr_0^post_2, __disjvr_1^0'=__disjvr_1^post_2, 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 && __disjvr_0^0==__disjvr_0^post_2 && __disjvr_1^0==__disjvr_1^post_2 && x^0==x^post_2 ], cost: 1
      2: l0 -> l1 : __disjvr_0^0'=__disjvr_0^post_3, __disjvr_1^0'=__disjvr_1^post_3, 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 && __disjvr_0^0==__disjvr_0^post_3 && __disjvr_1^0==__disjvr_1^post_3 && y^0==y^post_3 ], cost: 1
      3: l0 -> l1 : __disjvr_0^0'=__disjvr_0^post_4, __disjvr_1^0'=__disjvr_1^post_4, 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 && __disjvr_0^0==__disjvr_0^post_4 && __disjvr_1^0==__disjvr_1^post_4 && x^0==x^post_4 ], cost: 1
      4: l1 -> l2 : __disjvr_0^0'=__disjvr_0^post_5, __disjvr_1^0'=__disjvr_1^post_5, x^0'=x^post_5, y^0'=y^post_5, [ __disjvr_0^post_5==__disjvr_0^0 && __disjvr_0^0==__disjvr_0^post_5 && __disjvr_1^0==__disjvr_1^post_5 && x^0==x^post_5 && y^0==y^post_5 ], cost: 1
      5: l2 -> l0 : __disjvr_0^0'=__disjvr_0^post_6, __disjvr_1^0'=__disjvr_1^post_6, x^0'=x^post_6, y^0'=y^post_6, [ __disjvr_1^post_6==__disjvr_1^0 && __disjvr_0^0==__disjvr_0^post_6 && __disjvr_1^0==__disjvr_1^post_6 && x^0==x^post_6 && y^0==y^post_6 ], cost: 1
      6: l3 -> l1 : __disjvr_0^0'=__disjvr_0^post_7, __disjvr_1^0'=__disjvr_1^post_7, x^0'=x^post_7, y^0'=y^post_7, [ __disjvr_0^0==__disjvr_0^post_7 && __disjvr_1^0==__disjvr_1^post_7 && x^0==x^post_7 && y^0==y^post_7 ], cost: 1
      7: l4 -> l3 : __disjvr_0^0'=__disjvr_0^post_8, __disjvr_1^0'=__disjvr_1^post_8, x^0'=x^post_8, y^0'=y^post_8, [ __disjvr_0^0==__disjvr_0^post_8 && __disjvr_1^0==__disjvr_1^post_8 && x^0==x^post_8 && y^0==y^post_8 ], cost: 1

Checking for constant complexity:
   The following rule is satisfiable with cost >= 1, yielding constant complexity:
      7: l4 -> l3 : __disjvr_0^0'=__disjvr_0^post_8, __disjvr_1^0'=__disjvr_1^post_8, x^0'=x^post_8, y^0'=y^post_8, [ __disjvr_0^0==__disjvr_0^post_8 && __disjvr_1^0==__disjvr_1^post_8 && x^0==x^post_8 && y^0==y^post_8 ], 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 : [], cost: 1
      5: l2 -> l0 : [], cost: 1
      6: l3 -> l1 : [], cost: 1
      7: 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
      9: l1 -> l0 : [], cost: 2
      8: l4 -> l1 : [], cost: 2

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

Accelerating simple loops of location 1.
   Accelerating the following rules:
     10: l1 -> l1 : x^0'=1+x^0, [ -x^0<=y^0 && 1+x^0<=y^0 ], cost: 3
     11: l1 -> l1 : y^0'=1+y^0, [ x^0<=y^0 && 1+x^0<=-y^0 ], cost: 3
     12: l1 -> l1 : x^0'=-1+x^0, [ y^0<=1-x^0 && 1+y^0<=x^0 ], cost: 3
     13: l1 -> l1 : y^0'=-1+y^0, [ y^0<=x^0 && 2-y^0<=x^0 ], cost: 3

   Accelerated rule 10 with metering function -x^0+y^0, yielding the new rule 14.
   Accelerated rule 11 with metering function -x^0-y^0, yielding the new rule 15.
   Accelerated rule 12 with metering function x^0-y^0, yielding the new rule 16.
   Accelerated rule 13 with metering function -1+x^0+y^0, yielding the new rule 17.
   Removing the simple loops: 10 11 12 13.

Accelerated all simple loops using metering functions (where possible):
   Start location: l4
     14: l1 -> l1 : x^0'=y^0, [ -x^0<=y^0 && 1+x^0<=y^0 ], cost: -3*x^0+3*y^0
     15: l1 -> l1 : y^0'=-x^0, [ x^0<=y^0 && 1+x^0<=-y^0 ], cost: -3*x^0-3*y^0
     16: l1 -> l1 : x^0'=y^0, [ y^0<=1-x^0 && 1+y^0<=x^0 ], cost: 3*x^0-3*y^0
     17: l1 -> l1 : y^0'=1-x^0, [ y^0<=x^0 && 2-y^0<=x^0 ], cost: -3+3*x^0+3*y^0
      8: l4 -> l1 : [], cost: 2

Chained accelerated rules (with incoming rules):
   Start location: l4
      8: l4 -> l1 : [], cost: 2
     18: l4 -> l1 : x^0'=y^0, [ -x^0<=y^0 && 1+x^0<=y^0 ], cost: 2-3*x^0+3*y^0
     19: l4 -> l1 : y^0'=-x^0, [ x^0<=y^0 && 1+x^0<=-y^0 ], cost: 2-3*x^0-3*y^0
     20: l4 -> l1 : x^0'=y^0, [ y^0<=1-x^0 && 1+y^0<=x^0 ], cost: 2+3*x^0-3*y^0
     21: l4 -> l1 : y^0'=1-x^0, [ y^0<=x^0 && 2-y^0<=x^0 ], cost: -1+3*x^0+3*y^0

Removed unreachable locations (and leaf rules with constant cost):
   Start location: l4
     18: l4 -> l1 : x^0'=y^0, [ -x^0<=y^0 && 1+x^0<=y^0 ], cost: 2-3*x^0+3*y^0
     19: l4 -> l1 : y^0'=-x^0, [ x^0<=y^0 && 1+x^0<=-y^0 ], cost: 2-3*x^0-3*y^0
     20: l4 -> l1 : x^0'=y^0, [ y^0<=1-x^0 && 1+y^0<=x^0 ], cost: 2+3*x^0-3*y^0
     21: l4 -> l1 : y^0'=1-x^0, [ y^0<=x^0 && 2-y^0<=x^0 ], cost: -1+3*x^0+3*y^0

### Computing asymptotic complexity ###

Fully simplified ITS problem
   Start location: l4
     18: l4 -> l1 : x^0'=y^0, [ -x^0<=y^0 && 1+x^0<=y^0 ], cost: 2-3*x^0+3*y^0
     19: l4 -> l1 : y^0'=-x^0, [ x^0<=y^0 && 1+x^0<=-y^0 ], cost: 2-3*x^0-3*y^0
     20: l4 -> l1 : x^0'=y^0, [ y^0<=1-x^0 && 1+y^0<=x^0 ], cost: 2+3*x^0-3*y^0
     21: l4 -> l1 : y^0'=1-x^0, [ y^0<=x^0 && 2-y^0<=x^0 ], cost: -1+3*x^0+3*y^0

Computing asymptotic complexity for rule 18
   Solved the limit problem by the following transformations:
      Created initial limit problem:
      -x^0+y^0 (+/+!), 1+x^0+y^0 (+/+!), 2-3*x^0+3*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+6*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+6*n
   Rule cost:   2-3*x^0+3*y^0
   Rule guard:  [ -x^0<=y^0 && 1+x^0<=y^0 ]

WORST_CASE(Omega(n^1),?)