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

Initial linear ITS problem
   Start location: f0
      0: f0 -> f1 : y'=0, [], cost: 1
      1: f1 -> f1 : x'=-1+x, y'=x+y, [ x>0 ], cost: 1
      2: f1 -> f2 : z'=y, [ x<=0 ], cost: 1
      3: f2 -> f3 : u'=-1+z, [ z>0 ], cost: 1
      4: f3 -> f3 : u'=-free+u, [ u>0 && free>0 ], cost: 1
      5: f3 -> f2 : z'=-1+z, [ u<=0 ], cost: 1

Checking for constant complexity:
   The following rule is satisfiable with cost >= 1, yielding constant complexity:
      0: f0 -> f1 : y'=0, [], cost: 1

### Simplification by acceleration and chaining ###

Accelerating simple loops of location 1.
   Accelerating the following rules:
      1: f1 -> f1 : x'=-1+x, y'=x+y, [ x>0 ], cost: 1

   Accelerated rule 1 with metering function x, yielding the new rule 6.
   Removing the simple loops: 1.

Accelerating simple loops of location 3.
   Accelerating the following rules:
      4: f3 -> f3 : u'=-free+u, [ u>0 && free>0 ], cost: 1

      During metering: Instantiating temporary variables by {free==1}
   Accelerated rule 4 with metering function u, yielding the new rule 7.
   Removing the simple loops: 4.

Accelerated all simple loops using metering functions (where possible):
   Start location: f0
      0: f0 -> f1 : y'=0, [], cost: 1
      2: f1 -> f2 : z'=y, [ x<=0 ], cost: 1
      6: f1 -> f1 : x'=0, y'=1/2*x+1/2*x^2+y, [ x>0 ], cost: x
      3: f2 -> f3 : u'=-1+z, [ z>0 ], cost: 1
      5: f3 -> f2 : z'=-1+z, [ u<=0 ], cost: 1
      7: f3 -> f3 : u'=0, [ u>0 ], cost: u

Chained accelerated rules (with incoming rules):
   Start location: f0
      0: f0 -> f1 : y'=0, [], cost: 1
      8: f0 -> f1 : x'=0, y'=1/2*x+1/2*x^2, [ x>0 ], cost: 1+x
      2: f1 -> f2 : z'=y, [ x<=0 ], cost: 1
      3: f2 -> f3 : u'=-1+z, [ z>0 ], cost: 1
      9: f2 -> f3 : u'=0, [ -1+z>0 ], cost: z
      5: f3 -> f2 : z'=-1+z, [ u<=0 ], cost: 1

Eliminated locations (on tree-shaped paths):
   Start location: f0
     10: f0 -> f2 : y'=0, z'=0, [ x<=0 ], cost: 2
     11: f0 -> f2 : x'=0, y'=1/2*x+1/2*x^2, z'=1/2*x+1/2*x^2, [ x>0 ], cost: 2+x
     12: f2 -> f2 : u'=-1+z, z'=-1+z, [ z>0 && -1+z<=0 ], cost: 2
     13: f2 -> f2 : u'=0, z'=-1+z, [ -1+z>0 ], cost: 1+z

Accelerating simple loops of location 2.
   Simplified some of the simple loops (and removed duplicate rules).
   Accelerating the following rules:
     12: f2 -> f2 : u'=-1+z, z'=-1+z, [ 1-z==0 ], cost: 2
     13: f2 -> f2 : u'=0, z'=-1+z, [ -1+z>0 ], cost: 1+z

   Accelerated rule 12 with metering function z, yielding the new rule 14.
   Accelerated rule 13 with metering function -1+z, yielding the new rule 15.
   Removing the simple loops: 12 13.

Accelerated all simple loops using metering functions (where possible):
   Start location: f0
     10: f0 -> f2 : y'=0, z'=0, [ x<=0 ], cost: 2
     11: f0 -> f2 : x'=0, y'=1/2*x+1/2*x^2, z'=1/2*x+1/2*x^2, [ x>0 ], cost: 2+x
     14: f2 -> f2 : u'=0, z'=0, [ 1-z==0 ], cost: 2*z
     15: f2 -> f2 : u'=0, z'=1, [ -1+z>0 ], cost: -3/2-1/2*(-1+z)^2+3/2*z+z*(-1+z)

Chained accelerated rules (with incoming rules):
   Start location: f0
     10: f0 -> f2 : y'=0, z'=0, [ x<=0 ], cost: 2
     11: f0 -> f2 : x'=0, y'=1/2*x+1/2*x^2, z'=1/2*x+1/2*x^2, [ x>0 ], cost: 2+x
     16: f0 -> f2 : u'=0, x'=0, y'=1/2*x+1/2*x^2, z'=0, [ x>0 && 1-1/2*x-1/2*x^2==0 ], cost: 2+2*x+x^2
     17: f0 -> f2 : u'=0, x'=0, y'=1/2*x+1/2*x^2, z'=1, [ x>0 && -1+1/2*x+1/2*x^2>0 ], cost: 1/2+7/4*x+3/4*x^2-1/8*(-2+x+x^2)^2+1/4*(x+x^2)*(-2+x+x^2)

Removed unreachable locations (and leaf rules with constant cost):
   Start location: f0
     11: f0 -> f2 : x'=0, y'=1/2*x+1/2*x^2, z'=1/2*x+1/2*x^2, [ x>0 ], cost: 2+x
     16: f0 -> f2 : u'=0, x'=0, y'=1/2*x+1/2*x^2, z'=0, [ x>0 && 1-1/2*x-1/2*x^2==0 ], cost: 2+2*x+x^2
     17: f0 -> f2 : u'=0, x'=0, y'=1/2*x+1/2*x^2, z'=1, [ x>0 && -1+1/2*x+1/2*x^2>0 ], cost: 1/2+7/4*x+3/4*x^2-1/8*(-2+x+x^2)^2+1/4*(x+x^2)*(-2+x+x^2)

### Computing asymptotic complexity ###

Fully simplified ITS problem
   Start location: f0
     11: f0 -> f2 : x'=0, y'=1/2*x+1/2*x^2, z'=1/2*x+1/2*x^2, [ x>0 ], cost: 2+x
     16: f0 -> f2 : u'=0, x'=0, y'=1/2*x+1/2*x^2, z'=0, [ x>0 && 1-1/2*x-1/2*x^2==0 ], cost: 2+2*x+x^2
     17: f0 -> f2 : u'=0, x'=0, y'=1/2*x+1/2*x^2, z'=1, [ x>0 && -1+1/2*x+1/2*x^2>0 ], cost: 1/2+7/4*x+3/4*x^2-1/8*(-2+x+x^2)^2+1/4*(x+x^2)*(-2+x+x^2)

Computing asymptotic complexity for rule 11
   Solved the limit problem by the following transformations:
      Created initial limit problem:
      x (+/+!), 2+x (+) [not solved]

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

      applying transformation rule (C) using substitution {x==n}
      resulting limit problem:
      [solved]

   Solution:
      x / n
   Resulting cost 2+n has complexity: Poly(n^1)

Found new complexity Poly(n^1).

Computing asymptotic complexity for rule 16
   Could not solve the limit problem.
   Resulting cost 0 has complexity: Unknown

Computing asymptotic complexity for rule 17
   Solved the limit problem by the following transformations:
      Created initial limit problem:
      x (+/+!), -1+1/2*x+1/2*x^2 (+/+!), 7/4*x+7/8*x^2+1/4*x^3+1/8*x^4 (+) [not solved]

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

      applying transformation rule (C) using substitution {x==n}
      resulting limit problem:
      [solved]

   Solution:
      x / n
   Resulting cost 7/8*n^2+1/4*n^3+7/4*n+1/8*n^4 has complexity: Poly(n^4)

Found new complexity Poly(n^4).

Obtained the following overall complexity (w.r.t. the length of the input n):
   Complexity:  Poly(n^4)
   Cpx degree:  4
   Solved cost: 7/8*n^2+1/4*n^3+7/4*n+1/8*n^4
   Rule cost:   1/2+7/4*x+3/4*x^2-1/8*(-2+x+x^2)^2+1/4*(x+x^2)*(-2+x+x^2)
   Rule guard:  [ x>0 && -1+1/2*x+1/2*x^2>0 ]

WORST_CASE(Omega(n^4),?)