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

Initial linear ITS problem
   Start location: f0
      0: f0 -> facsum : [], cost: 0
      1: facsum -> fac : facsum : x'=-1+x, [ x>0 ], cost: 1
      2: facsum -> sink : [ x<=0 ], cost: 1
      3: fac -> fac : x'=-1+x, [ x>1 ], cost: 1
      4: fac -> sink : [ x<=1 ], cost: 1

Checking for constant complexity:
   Could not prove constant complexity.

Removed unreachable and leaf rules:
   Start location: f0
      0: f0 -> facsum : [], cost: 0
      1: facsum -> fac : facsum : x'=-1+x, [ x>0 ], cost: 1
      3: fac -> fac : x'=-1+x, [ x>1 ], cost: 1

### Simplification by acceleration and chaining ###

Accelerating simple loops of location 2.
   Accelerating the following rules:
      3: fac -> fac : x'=-1+x, [ x>1 ], cost: 1

   Accelerated rule 3 with metering function -1+x, yielding the new rule 5.
   Removing the simple loops: 3.

Accelerated all simple loops using metering functions (where possible):
   Start location: f0
      0: f0 -> facsum : [], cost: 0
      1: facsum -> fac : facsum : x'=-1+x, [ x>0 ], cost: 1
      5: fac -> fac : x'=1, [ x>1 ], cost: -1+x

Chained accelerated rules (with incoming rules):
   Start location: f0
      0: f0 -> facsum : [], cost: 0
      1: facsum -> fac : facsum : x'=-1+x, [ x>0 ], cost: 1
      6: facsum -> fac : x'=1, facsum : x'=-1+x, [ x>1 ], cost: x

Removed locations with no outgoing rules from right-hand sides
   Start location: f0
      0: f0 -> facsum : [], cost: 0
      7: facsum -> facsum : x'=-1+x, [ x>0 ], cost: 1
      8: facsum -> facsum : x'=-1+x, [ x>1 ], cost: x

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

   Accelerated rule 7 with metering function x, yielding the new rule 9.
   Accelerated rule 8 with metering function -1+x, yielding the new rule 10.
   Removing the simple loops: 7 8.

Accelerated all simple loops using metering functions (where possible):
   Start location: f0
      0: f0 -> facsum : [], cost: 0
      9: facsum -> facsum : x'=0, [ x>0 ], cost: x
     10: facsum -> facsum : x'=1, [ x>1 ], cost: -1/2-1/2*(-1+x)^2+(-1+x)*x+1/2*x

Chained accelerated rules (with incoming rules):
   Start location: f0
      0: f0 -> facsum : [], cost: 0
     11: f0 -> facsum : x'=0, [ x>0 ], cost: x
     12: f0 -> facsum : x'=1, [ x>1 ], cost: -1/2-1/2*(-1+x)^2+(-1+x)*x+1/2*x

Removed unreachable locations (and leaf rules with constant cost):
   Start location: f0
     11: f0 -> facsum : x'=0, [ x>0 ], cost: x
     12: f0 -> facsum : x'=1, [ x>1 ], cost: -1/2-1/2*(-1+x)^2+(-1+x)*x+1/2*x

### Obtained a tail recursive problem, continuing simplification ###

### Computing asymptotic complexity ###

Fully simplified ITS problem
   Start location: f0
     11: f0 -> facsum : x'=0, [ x>0 ], cost: x
     12: f0 -> facsum : x'=1, [ x>1 ], cost: -1/2-1/2*(-1+x)^2+(-1+x)*x+1/2*x

Computing asymptotic complexity for rule 11
   Solved the limit problem by the following transformations:
      Created initial limit problem:
      x (+) [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 n has complexity: Poly(n^1)

Found new complexity Poly(n^1).

Computing asymptotic complexity for rule 12
   Solved the limit problem by the following transformations:
      Created initial limit problem:
      -1+x (+/+!), -1+1/2*x+1/2*x^2 (+) [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 -1+1/2*n+1/2*n^2 has complexity: Poly(n^2)

Found new complexity Poly(n^2).

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

WORST_CASE(Omega(n^2),?)