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, z^0'=z^post_1, [ x^0==x^post_1 && y^0==y^post_1 && z^0==z^post_1 ], cost: 1
      4: l1 -> l2 : x^0'=x^post_5, y^0'=y^post_5, z^0'=z^post_5, [ 1+x^0<=y^0 && x^0==x^post_5 && y^0==y^post_5 && z^0==z^post_5 ], cost: 1
      1: l2 -> l1 : x^0'=x^post_2, y^0'=y^post_2, z^0'=z^post_2, [ z^0<=y^0 && x^post_2==1+x^0 && y^0==y^post_2 && z^0==z^post_2 ], cost: 1
      2: l2 -> l3 : x^0'=x^post_3, y^0'=y^post_3, z^0'=z^post_3, [ 1+y^0<=z^0 && y^post_3==1+y^0 && x^0==x^post_3 && z^0==z^post_3 ], cost: 1
      3: l3 -> l2 : x^0'=x^post_4, y^0'=y^post_4, z^0'=z^post_4, [ x^0==x^post_4 && y^0==y^post_4 && z^0==z^post_4 ], cost: 1
      5: l4 -> l0 : x^0'=x^post_6, y^0'=y^post_6, z^0'=z^post_6, [ x^0==x^post_6 && y^0==y^post_6 && z^0==z^post_6 ], cost: 1

Checking for constant complexity:
   The following rule is satisfiable with cost >= 1, yielding constant complexity:
      5: l4 -> l0 : x^0'=x^post_6, y^0'=y^post_6, z^0'=z^post_6, [ x^0==x^post_6 && y^0==y^post_6 && z^0==z^post_6 ], cost: 1

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

### Simplification by acceleration and chaining ###

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

Accelerating simple loops of location 2.
   Accelerating the following rules:
      7: l2 -> l2 : y^0'=1+y^0, [ 1+y^0<=z^0 ], cost: 2

   Accelerated rule 7 with metering function z^0-y^0, yielding the new rule 8.
   Removing the simple loops: 7.

Accelerated all simple loops using metering functions (where possible):
   Start location: l4
      4: l1 -> l2 : [ 1+x^0<=y^0 ], cost: 1
      1: l2 -> l1 : x^0'=1+x^0, [ z^0<=y^0 ], cost: 1
      8: l2 -> l2 : y^0'=z^0, [ 1+y^0<=z^0 ], cost: 2*z^0-2*y^0
      6: l4 -> l1 : [], cost: 2

Chained accelerated rules (with incoming rules):
   Start location: l4
      4: l1 -> l2 : [ 1+x^0<=y^0 ], cost: 1
      9: l1 -> l2 : y^0'=z^0, [ 1+x^0<=y^0 && 1+y^0<=z^0 ], cost: 1+2*z^0-2*y^0
      1: l2 -> l1 : x^0'=1+x^0, [ z^0<=y^0 ], cost: 1
      6: l4 -> l1 : [], cost: 2

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

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

   Accelerated rule 10 with metering function -x^0+y^0, yielding the new rule 12.
   Found no metering function for rule 11.
   Removing the simple loops: 10.

Accelerated all simple loops using metering functions (where possible):
   Start location: l4
     11: l1 -> l1 : x^0'=1+x^0, y^0'=z^0, [ 1+x^0<=y^0 && 1+y^0<=z^0 ], cost: 2+2*z^0-2*y^0
     12: l1 -> l1 : x^0'=y^0, [ 1+x^0<=y^0 && z^0<=y^0 ], cost: -2*x^0+2*y^0
      6: l4 -> l1 : [], cost: 2

Chained accelerated rules (with incoming rules):
   Start location: l4
      6: l4 -> l1 : [], cost: 2
     13: l4 -> l1 : x^0'=1+x^0, y^0'=z^0, [ 1+x^0<=y^0 && 1+y^0<=z^0 ], cost: 4+2*z^0-2*y^0
     14: l4 -> l1 : x^0'=y^0, [ 1+x^0<=y^0 && z^0<=y^0 ], cost: 2-2*x^0+2*y^0

Removed unreachable locations (and leaf rules with constant cost):
   Start location: l4
     13: l4 -> l1 : x^0'=1+x^0, y^0'=z^0, [ 1+x^0<=y^0 && 1+y^0<=z^0 ], cost: 4+2*z^0-2*y^0
     14: l4 -> l1 : x^0'=y^0, [ 1+x^0<=y^0 && z^0<=y^0 ], cost: 2-2*x^0+2*y^0

### Computing asymptotic complexity ###

Fully simplified ITS problem
   Start location: l4
     13: l4 -> l1 : x^0'=1+x^0, y^0'=z^0, [ 1+x^0<=y^0 && 1+y^0<=z^0 ], cost: 4+2*z^0-2*y^0
     14: l4 -> l1 : x^0'=y^0, [ 1+x^0<=y^0 && z^0<=y^0 ], cost: 2-2*x^0+2*y^0

Computing asymptotic complexity for rule 13
   Solved the limit problem by the following transformations:
      Created initial limit problem:
      z^0-y^0 (+/+!), -x^0+y^0 (+/+!), 4+2*z^0-2*y^0 (+) [not solved]

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

      applying transformation rule (C) using substitution {z^0==n,x^0==0,y^0==1}
      resulting limit problem:
      [solved]

   Solution:
      z^0 / n
      x^0 / 0
      y^0 / 1
   Resulting cost 2+2*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+2*n
   Rule cost:   4+2*z^0-2*y^0
   Rule guard:  [ 1+x^0<=y^0 && 1+y^0<=z^0 ]

WORST_CASE(Omega(n^1),?)