WORST_CASE(Omega(1),?)

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

Initial linear ITS problem
   Start location: l3
      0: l0 -> l1 : i^0'=i^post_1, x^0'=x^post_1, y^0'=y^post_1, z^0'=z^post_1, [ 1<=x^0 && z^post_1==y^0+i^0 && x^post_1==x^0-z^post_1 && i^post_1==1+i^0 && y^0==y^post_1 ], cost: 1
      1: l1 -> l0 : i^0'=i^post_2, x^0'=x^post_2, y^0'=y^post_2, z^0'=z^post_2, [ i^0==i^post_2 && x^0==x^post_2 && y^0==y^post_2 && z^0==z^post_2 ], cost: 1
      2: l2 -> l0 : i^0'=i^post_3, x^0'=x^post_3, y^0'=y^post_3, z^0'=z^post_3, [ i^post_3==0 && y^post_3==y^post_3 && x^0==x^post_3 && z^0==z^post_3 ], cost: 1
      3: l3 -> l2 : i^0'=i^post_4, x^0'=x^post_4, y^0'=y^post_4, z^0'=z^post_4, [ i^0==i^post_4 && x^0==x^post_4 && y^0==y^post_4 && z^0==z^post_4 ], cost: 1

Checking for constant complexity:
   The following rule is satisfiable with cost >= 1, yielding constant complexity:
      3: l3 -> l2 : i^0'=i^post_4, x^0'=x^post_4, y^0'=y^post_4, z^0'=z^post_4, [ i^0==i^post_4 && x^0==x^post_4 && y^0==y^post_4 && z^0==z^post_4 ], cost: 1

Simplified all rules, resulting in:
   Start location: l3
      0: l0 -> l1 : i^0'=1+i^0, x^0'=x^0-y^0-i^0, z^0'=y^0+i^0, [ 1<=x^0 ], cost: 1
      1: l1 -> l0 : [], cost: 1
      2: l2 -> l0 : i^0'=0, y^0'=y^post_3, [], cost: 1
      3: l3 -> l2 : [], cost: 1

### Simplification by acceleration and chaining ###

Eliminated locations (on linear paths):
   Start location: l3
      5: l0 -> l0 : i^0'=1+i^0, x^0'=x^0-y^0-i^0, z^0'=y^0+i^0, [ 1<=x^0 ], cost: 2
      4: l3 -> l0 : i^0'=0, y^0'=y^post_3, [], cost: 2

Accelerating simple loops of location 0.
   Accelerating the following rules:
      5: l0 -> l0 : i^0'=1+i^0, x^0'=x^0-y^0-i^0, z^0'=y^0+i^0, [ 1<=x^0 ], cost: 2

[test] deduced pseudo-invariant 1-y^0-i^0<=0, also trying -1+y^0+i^0<=-1
   Accelerated rule 5 with backward acceleration, yielding the new rule 6.
   Accelerated rule 5 with backward acceleration, yielding the new rule 7.
[accelerate] Nesting with 2 inner and 1 outer candidates

Accelerated all simple loops using metering functions (where possible):
   Start location: l3
      5: l0 -> l0 : i^0'=1+i^0, x^0'=x^0-y^0-i^0, z^0'=y^0+i^0, [ 1<=x^0 ], cost: 2
      6: l0 -> l0 : i^0'=k+i^0, x^0'=-1/2*k^2+1/2*k-k*i^0+x^0-k*y^0, z^0'=-1+k+y^0+i^0, [ 1-y^0-i^0<=0 && k>=1 && 1<=-1/2+1/2*k+x^0-(-1+k)*y^0-(-1+k)*i^0-1/2*(-1+k)^2 ], cost: 2*k
      7: l0 -> l0 : i^0'=1-y^0, x^0'=1/2+x^0+(-1+y^0+i^0)*i^0-1/2*y^0-1/2*(-1+y^0+i^0)^2-1/2*i^0+(-1+y^0+i^0)*y^0, z^0'=0, [ 1<=x^0 && 1-y^0-i^0>=1 ], cost: 2-2*y^0-2*i^0
      4: l3 -> l0 : i^0'=0, y^0'=y^post_3, [], cost: 2

Chained accelerated rules (with incoming rules):
   Start location: l3
      4: l3 -> l0 : i^0'=0, y^0'=y^post_3, [], cost: 2
      8: l3 -> l0 : i^0'=1, x^0'=x^0-y^post_3, y^0'=y^post_3, z^0'=y^post_3, [ 1<=x^0 ], cost: 4
      9: l3 -> l0 : i^0'=k, x^0'=-1/2*k^2+1/2*k+x^0-k*y^post_3, y^0'=y^post_3, z^0'=-1+k+y^post_3, [ 1-y^post_3<=0 && k>=1 && 1<=-1/2+1/2*k+x^0-1/2*(-1+k)^2-(-1+k)*y^post_3 ], cost: 2+2*k
     10: l3 -> l0 : i^0'=1-y^post_3, x^0'=1/2+y^post_3*(-1+y^post_3)+x^0-1/2*y^post_3-1/2*(-1+y^post_3)^2, y^0'=y^post_3, z^0'=0, [ 1<=x^0 && 1-y^post_3>=1 ], cost: 4-2*y^post_3

Removed unreachable locations (and leaf rules with constant cost):
   Start location: l3
      9: l3 -> l0 : i^0'=k, x^0'=-1/2*k^2+1/2*k+x^0-k*y^post_3, y^0'=y^post_3, z^0'=-1+k+y^post_3, [ 1-y^post_3<=0 && k>=1 && 1<=-1/2+1/2*k+x^0-1/2*(-1+k)^2-(-1+k)*y^post_3 ], cost: 2+2*k
     10: l3 -> l0 : i^0'=1-y^post_3, x^0'=1/2+y^post_3*(-1+y^post_3)+x^0-1/2*y^post_3-1/2*(-1+y^post_3)^2, y^0'=y^post_3, z^0'=0, [ 1<=x^0 && 1-y^post_3>=1 ], cost: 4-2*y^post_3

### Computing asymptotic complexity ###

Fully simplified ITS problem
   Start location: l3
      9: l3 -> l0 : i^0'=k, x^0'=-1/2*k^2+1/2*k+x^0-k*y^post_3, y^0'=y^post_3, z^0'=-1+k+y^post_3, [ 1-y^post_3<=0 && k>=1 && 1<=-1/2+1/2*k+x^0-1/2*(-1+k)^2-(-1+k)*y^post_3 ], cost: 2+2*k
     10: l3 -> l0 : i^0'=1-y^post_3, x^0'=1/2+y^post_3*(-1+y^post_3)+x^0-1/2*y^post_3-1/2*(-1+y^post_3)^2, y^0'=y^post_3, z^0'=0, [ 1<=x^0 && 1-y^post_3>=1 ], cost: 4-2*y^post_3

Computing asymptotic complexity for rule 10
   Resulting cost 0 has complexity: Unknown

Computing asymptotic complexity for rule 9
   Resulting cost 0 has complexity: Unknown

Obtained the following overall complexity (w.r.t. the length of the input n):
   Complexity:  Constant
   Cpx degree:  0
   Solved cost: 1
   Rule cost:   1
   Rule guard:  [ i^0==i^post_4 && x^0==x^post_4 && y^0==y^post_4 && z^0==z^post_4 ]

WORST_CASE(Omega(1),?)