WORST_CASE(Omega(n^1),?)

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

Initial linear ITS problem
   Start location: l5
      0: l0 -> l1 : op1^0'=op1^post_1, op2^0'=op2^post_1, [ op2^0<=0 && 0<=op2^0 && 0<=op1^0 && op1^0==op1^post_1 && op2^0==op2^post_1 ], cost: 1
      6: l1 -> l4 : op1^0'=op1^post_7, op2^0'=op2^post_7, [ 1<=op1^0 && op1^post_7==-1+op1^0 && op2^post_7==1+op2^0 ], cost: 1
      1: l2 -> l3 : op1^0'=op1^post_2, op2^0'=op2^post_2, [ 1<=op2^0 && op1^post_2==1+op1^0 && op2^post_2==-1+op2^0 ], cost: 1
      3: l2 -> l4 : op1^0'=op1^post_4, op2^0'=op2^post_4, [ 1<=op1^0 && op1^post_4==-1+op1^0 && op2^post_4==1+op2^0 ], cost: 1
      2: l3 -> l2 : op1^0'=op1^post_3, op2^0'=op2^post_3, [ op1^0==op1^post_3 && op2^0==op2^post_3 ], cost: 1
      4: l4 -> l2 : op1^0'=op1^post_5, op2^0'=op2^post_5, [ 1<=op2^0 && op2^post_5==-1+op2^0 && op1^0==op1^post_5 ], cost: 1
      5: l4 -> l1 : op1^0'=op1^post_6, op2^0'=op2^post_6, [ op1^0==op1^post_6 && op2^0==op2^post_6 ], cost: 1
      7: l5 -> l0 : op1^0'=op1^post_8, op2^0'=op2^post_8, [ op1^0==op1^post_8 && op2^0==op2^post_8 ], cost: 1

Checking for constant complexity:
   The following rule is satisfiable with cost >= 1, yielding constant complexity:
      7: l5 -> l0 : op1^0'=op1^post_8, op2^0'=op2^post_8, [ op1^0==op1^post_8 && op2^0==op2^post_8 ], cost: 1

Simplified all rules, resulting in:
   Start location: l5
      0: l0 -> l1 : [ op2^0==0 && 0<=op1^0 ], cost: 1
      6: l1 -> l4 : op1^0'=-1+op1^0, op2^0'=1+op2^0, [ 1<=op1^0 ], cost: 1
      1: l2 -> l3 : op1^0'=1+op1^0, op2^0'=-1+op2^0, [ 1<=op2^0 ], cost: 1
      3: l2 -> l4 : op1^0'=-1+op1^0, op2^0'=1+op2^0, [ 1<=op1^0 ], cost: 1
      2: l3 -> l2 : [], cost: 1
      4: l4 -> l2 : op2^0'=-1+op2^0, [ 1<=op2^0 ], cost: 1
      5: l4 -> l1 : [], cost: 1
      7: l5 -> l0 : [], cost: 1

### Simplification by acceleration and chaining ###

Eliminated locations (on linear paths):
   Start location: l5
      6: l1 -> l4 : op1^0'=-1+op1^0, op2^0'=1+op2^0, [ 1<=op1^0 ], cost: 1
      3: l2 -> l4 : op1^0'=-1+op1^0, op2^0'=1+op2^0, [ 1<=op1^0 ], cost: 1
      9: l2 -> l2 : op1^0'=1+op1^0, op2^0'=-1+op2^0, [ 1<=op2^0 ], cost: 2
      4: l4 -> l2 : op2^0'=-1+op2^0, [ 1<=op2^0 ], cost: 1
      5: l4 -> l1 : [], cost: 1
      8: l5 -> l1 : [ op2^0==0 && 0<=op1^0 ], cost: 2

Accelerating simple loops of location 2.
   Accelerating the following rules:
      9: l2 -> l2 : op1^0'=1+op1^0, op2^0'=-1+op2^0, [ 1<=op2^0 ], cost: 2

   Accelerated rule 9 with metering function op2^0, yielding the new rule 10.
   Removing the simple loops: 9.

Accelerated all simple loops using metering functions (where possible):
   Start location: l5
      6: l1 -> l4 : op1^0'=-1+op1^0, op2^0'=1+op2^0, [ 1<=op1^0 ], cost: 1
      3: l2 -> l4 : op1^0'=-1+op1^0, op2^0'=1+op2^0, [ 1<=op1^0 ], cost: 1
     10: l2 -> l2 : op1^0'=op1^0+op2^0, op2^0'=0, [ 1<=op2^0 ], cost: 2*op2^0
      4: l4 -> l2 : op2^0'=-1+op2^0, [ 1<=op2^0 ], cost: 1
      5: l4 -> l1 : [], cost: 1
      8: l5 -> l1 : [ op2^0==0 && 0<=op1^0 ], cost: 2

Chained accelerated rules (with incoming rules):
   Start location: l5
      6: l1 -> l4 : op1^0'=-1+op1^0, op2^0'=1+op2^0, [ 1<=op1^0 ], cost: 1
      3: l2 -> l4 : op1^0'=-1+op1^0, op2^0'=1+op2^0, [ 1<=op1^0 ], cost: 1
      4: l4 -> l2 : op2^0'=-1+op2^0, [ 1<=op2^0 ], cost: 1
      5: l4 -> l1 : [], cost: 1
     11: l4 -> l2 : op1^0'=-1+op1^0+op2^0, op2^0'=0, [ 1<=-1+op2^0 ], cost: -1+2*op2^0
      8: l5 -> l1 : [ op2^0==0 && 0<=op1^0 ], cost: 2

Eliminated locations (on tree-shaped paths):
   Start location: l5
      6: l1 -> l4 : op1^0'=-1+op1^0, op2^0'=1+op2^0, [ 1<=op1^0 ], cost: 1
      5: l4 -> l1 : [], cost: 1
     12: l4 -> l4 : op1^0'=-1+op1^0, op2^0'=op2^0, [ 1<=op2^0 && 1<=op1^0 ], cost: 2
     13: l4 -> l4 : op1^0'=-2+op1^0+op2^0, op2^0'=1, [ 1<=-1+op2^0 && 1<=-1+op1^0+op2^0 ], cost: 2*op2^0
      8: l5 -> l1 : [ op2^0==0 && 0<=op1^0 ], cost: 2

Accelerating simple loops of location 4.
   Simplified some of the simple loops (and removed duplicate rules).
   Accelerating the following rules:
     12: l4 -> l4 : op1^0'=-1+op1^0, [ 1<=op2^0 && 1<=op1^0 ], cost: 2
     13: l4 -> l4 : op1^0'=-2+op1^0+op2^0, op2^0'=1, [ 1<=-1+op2^0 && 1<=-1+op1^0+op2^0 ], cost: 2*op2^0

   Accelerated rule 12 with metering function op1^0, yielding the new rule 14.
   Found no metering function for rule 13.
   Removing the simple loops: 12.

Accelerated all simple loops using metering functions (where possible):
   Start location: l5
      6: l1 -> l4 : op1^0'=-1+op1^0, op2^0'=1+op2^0, [ 1<=op1^0 ], cost: 1
      5: l4 -> l1 : [], cost: 1
     13: l4 -> l4 : op1^0'=-2+op1^0+op2^0, op2^0'=1, [ 1<=-1+op2^0 && 1<=-1+op1^0+op2^0 ], cost: 2*op2^0
     14: l4 -> l4 : op1^0'=0, [ 1<=op2^0 && 1<=op1^0 ], cost: 2*op1^0
      8: l5 -> l1 : [ op2^0==0 && 0<=op1^0 ], cost: 2

Chained accelerated rules (with incoming rules):
   Start location: l5
      6: l1 -> l4 : op1^0'=-1+op1^0, op2^0'=1+op2^0, [ 1<=op1^0 ], cost: 1
     15: l1 -> l4 : op1^0'=-2+op1^0+op2^0, op2^0'=1, [ 1<=op1^0 && 1<=op2^0 && 1<=-1+op1^0+op2^0 ], cost: 3+2*op2^0
     16: l1 -> l4 : op1^0'=0, op2^0'=1+op2^0, [ 1<=1+op2^0 && 1<=-1+op1^0 ], cost: -1+2*op1^0
      5: l4 -> l1 : [], cost: 1
      8: l5 -> l1 : [ op2^0==0 && 0<=op1^0 ], cost: 2

Eliminated locations (on tree-shaped paths):
   Start location: l5
     17: l1 -> l1 : op1^0'=-1+op1^0, op2^0'=1+op2^0, [ 1<=op1^0 ], cost: 2
     18: l1 -> l1 : op1^0'=-2+op1^0+op2^0, op2^0'=1, [ 1<=op1^0 && 1<=op2^0 && 1<=-1+op1^0+op2^0 ], cost: 4+2*op2^0
     19: l1 -> l1 : op1^0'=0, op2^0'=1+op2^0, [ 1<=1+op2^0 && 1<=-1+op1^0 ], cost: 2*op1^0
      8: l5 -> l1 : [ op2^0==0 && 0<=op1^0 ], cost: 2

Accelerating simple loops of location 1.
   Accelerating the following rules:
     17: l1 -> l1 : op1^0'=-1+op1^0, op2^0'=1+op2^0, [ 1<=op1^0 ], cost: 2
     18: l1 -> l1 : op1^0'=-2+op1^0+op2^0, op2^0'=1, [ 1<=op1^0 && 1<=op2^0 && 1<=-1+op1^0+op2^0 ], cost: 4+2*op2^0
     19: l1 -> l1 : op1^0'=0, op2^0'=1+op2^0, [ 1<=1+op2^0 && 1<=-1+op1^0 ], cost: 2*op1^0

   Accelerated rule 17 with metering function op1^0, yielding the new rule 20.
   Accelerated rule 18 with metering function op1^0, yielding the new rule 21.
   Found no metering function for rule 19.
   Removing the simple loops: 17 18.

Accelerated all simple loops using metering functions (where possible):
   Start location: l5
     19: l1 -> l1 : op1^0'=0, op2^0'=1+op2^0, [ 1<=1+op2^0 && 1<=-1+op1^0 ], cost: 2*op1^0
     20: l1 -> l1 : op1^0'=0, op2^0'=op1^0+op2^0, [ 1<=op1^0 ], cost: 2*op1^0
     21: l1 -> l1 : op1^0'=-1+op2^0, op2^0'=1, [ 1<=op1^0 && 1<=op2^0 && 1<=-1+op1^0+op2^0 ], cost: 6*op1^0
      8: l5 -> l1 : [ op2^0==0 && 0<=op1^0 ], cost: 2

Chained accelerated rules (with incoming rules):
   Start location: l5
      8: l5 -> l1 : [ op2^0==0 && 0<=op1^0 ], cost: 2
     22: l5 -> l1 : op1^0'=0, op2^0'=1+op2^0, [ op2^0==0 && 1<=-1+op1^0 ], cost: 2+2*op1^0
     23: l5 -> l1 : op1^0'=0, op2^0'=op1^0+op2^0, [ op2^0==0 && 1<=op1^0 ], cost: 2+2*op1^0

Removed unreachable locations (and leaf rules with constant cost):
   Start location: l5
     22: l5 -> l1 : op1^0'=0, op2^0'=1+op2^0, [ op2^0==0 && 1<=-1+op1^0 ], cost: 2+2*op1^0
     23: l5 -> l1 : op1^0'=0, op2^0'=op1^0+op2^0, [ op2^0==0 && 1<=op1^0 ], cost: 2+2*op1^0

### Computing asymptotic complexity ###

Fully simplified ITS problem
   Start location: l5
     22: l5 -> l1 : op1^0'=0, op2^0'=1+op2^0, [ op2^0==0 && 1<=-1+op1^0 ], cost: 2+2*op1^0
     23: l5 -> l1 : op1^0'=0, op2^0'=op1^0+op2^0, [ op2^0==0 && 1<=op1^0 ], cost: 2+2*op1^0

Computing asymptotic complexity for rule 22
   Solved the limit problem by the following transformations:
      Created initial limit problem:
      1-op2^0 (+/+!), 2+2*op1^0 (+), -1+op1^0 (+/+!), 1+op2^0 (+/+!) [not solved]

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

      applying transformation rule (C) using substitution {op1^0==n,op2^0==0}
      resulting limit problem:
      [solved]

   Solution:
      op1^0 / n
      op2^0 / 0
   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:   2+2*op1^0
   Rule guard:  [ op2^0==0 && 1<=-1+op1^0 ]

WORST_CASE(Omega(n^1),?)