WORST_CASE(Omega(0),?)

Initial ITS
Start location: l4
   0: l0 -> l1 : i^0'=i^post0, n^0'=n^post0, j^0'=j^post0, (n^0-n^post0 == 0 /\ -j^post0+j^0 == 0 /\ -1-i^0+i^post0 == 0 /\ i^0-j^0 <= 0), cost: 1
   1: l0 -> l2 : i^0'=i^post1, n^0'=n^post1, j^0'=j^post1, (n^0-n^post1 == 0 /\ i^0-i^post1 == 0 /\ 1-i^0+j^0 <= 0 /\ -1-j^0+j^post1 == 0), cost: 1
   3: l1 -> l0 : i^0'=i^post3, n^0'=n^post3, j^0'=j^post3, (1+i^0-n^0 <= 0 /\ j^post3 == 0 /\ i^0-i^post3 == 0 /\ n^0-n^post3 == 0), cost: 1
   2: l2 -> l0 : i^0'=i^post2, n^0'=n^post2, j^0'=j^post2, (n^0-n^post2 == 0 /\ j^0-j^post2 == 0 /\ i^0-i^post2 == 0), cost: 1
   4: l3 -> l1 : i^0'=i^post4, n^0'=n^post4, j^0'=j^post4, (i^post4 == 0 /\ n^0-n^post4 == 0 /\ j^0-j^post4 == 0), cost: 1
   5: l4 -> l3 : i^0'=i^post5, n^0'=n^post5, j^0'=j^post5, (i^0-i^post5 == 0 /\ n^0-n^post5 == 0 /\ j^0-j^post5 == 0), cost: 1


Applied preprocessing
Original rule:
l0 -> l1 : i^0'=i^post0, n^0'=n^post0, j^0'=j^post0, (n^0-n^post0 == 0 /\ -j^post0+j^0 == 0 /\ -1-i^0+i^post0 == 0 /\ i^0-j^0 <= 0), cost: 1
New rule:
l0 -> l1 : i^0'=1+i^0, i^0-j^0 <= 0, cost: 1

Applied preprocessing
Original rule:
l0 -> l2 : i^0'=i^post1, n^0'=n^post1, j^0'=j^post1, (n^0-n^post1 == 0 /\ i^0-i^post1 == 0 /\ 1-i^0+j^0 <= 0 /\ -1-j^0+j^post1 == 0), cost: 1
New rule:
l0 -> l2 : j^0'=1+j^0, 1-i^0+j^0 <= 0, cost: 1

Applied preprocessing
Original rule:
l2 -> l0 : i^0'=i^post2, n^0'=n^post2, j^0'=j^post2, (n^0-n^post2 == 0 /\ j^0-j^post2 == 0 /\ i^0-i^post2 == 0), cost: 1
New rule:
l2 -> l0 : TRUE, cost: 1

Applied preprocessing
Original rule:
l1 -> l0 : i^0'=i^post3, n^0'=n^post3, j^0'=j^post3, (1+i^0-n^0 <= 0 /\ j^post3 == 0 /\ i^0-i^post3 == 0 /\ n^0-n^post3 == 0), cost: 1
New rule:
l1 -> l0 : j^0'=0, 1+i^0-n^0 <= 0, cost: 1

Applied preprocessing
Original rule:
l3 -> l1 : i^0'=i^post4, n^0'=n^post4, j^0'=j^post4, (i^post4 == 0 /\ n^0-n^post4 == 0 /\ j^0-j^post4 == 0), cost: 1
New rule:
l3 -> l1 : i^0'=0, TRUE, cost: 1

Applied preprocessing
Original rule:
l4 -> l3 : i^0'=i^post5, n^0'=n^post5, j^0'=j^post5, (i^0-i^post5 == 0 /\ n^0-n^post5 == 0 /\ j^0-j^post5 == 0), cost: 1
New rule:
l4 -> l3 : TRUE, cost: 1

Simplified rules
Start location: l4
   6: l0 -> l1 : i^0'=1+i^0, i^0-j^0 <= 0, cost: 1
   7: l0 -> l2 : j^0'=1+j^0, 1-i^0+j^0 <= 0, cost: 1
   9: l1 -> l0 : j^0'=0, 1+i^0-n^0 <= 0, cost: 1
   8: l2 -> l0 : TRUE, cost: 1
  10: l3 -> l1 : i^0'=0, TRUE, cost: 1
  11: l4 -> l3 : TRUE, cost: 1


Eliminating location l3 by chaining:

Applied chaining
First rule:
l4 -> l3 : TRUE, cost: 1
Second rule:
l3 -> l1 : i^0'=0, TRUE, cost: 1
New rule:
l4 -> l1 : i^0'=0, TRUE, cost: 2

Applied deletion
Removed the following rules: 10 11

Eliminating location l2 by chaining:

Applied chaining
First rule:
l0 -> l2 : j^0'=1+j^0, 1-i^0+j^0 <= 0, cost: 1
Second rule:
l2 -> l0 : TRUE, cost: 1
New rule:
l0 -> l0 : j^0'=1+j^0, 1-i^0+j^0 <= 0, cost: 2

Applied deletion
Removed the following rules: 7 8

Eliminated locations on linear paths
Start location: l4
   6: l0 -> l1 : i^0'=1+i^0, i^0-j^0 <= 0, cost: 1
  13: l0 -> l0 : j^0'=1+j^0, 1-i^0+j^0 <= 0, cost: 2
   9: l1 -> l0 : j^0'=0, 1+i^0-n^0 <= 0, cost: 1
  12: l4 -> l1 : i^0'=0, TRUE, cost: 2


Applied acceleration
Original rule:
l0 -> l0 : j^0'=1+j^0, 1-i^0+j^0 <= 0, cost: 2
New rule:
l0 -> l0 : j^0'=n0+j^0, (n0 >= 0 /\ i^0-n0-j^0 >= 0), cost: 2*n0

Applied instantiation
Original rule:
l0 -> l0 : j^0'=n0+j^0, (n0 >= 0 /\ i^0-n0-j^0 >= 0), cost: 2*n0
New rule:
l0 -> l0 : j^0'=i^0, (0 >= 0 /\ i^0-j^0 >= 0), cost: 2*i^0-2*j^0

Applied simplification
Original rule:
l0 -> l0 : j^0'=i^0, (0 >= 0 /\ i^0-j^0 >= 0), cost: 2*i^0-2*j^0
New rule:
l0 -> l0 : j^0'=i^0, i^0-j^0 >= 0, cost: 2*i^0-2*j^0

Applied deletion
Removed the following rules: 13

Accelerated simple loops
Start location: l4
   6: l0 -> l1 : i^0'=1+i^0, i^0-j^0 <= 0, cost: 1
  15: l0 -> l0 : j^0'=i^0, i^0-j^0 >= 0, cost: 2*i^0-2*j^0
   9: l1 -> l0 : j^0'=0, 1+i^0-n^0 <= 0, cost: 1
  12: l4 -> l1 : i^0'=0, TRUE, cost: 2


Applied chaining
First rule:
l1 -> l0 : j^0'=0, 1+i^0-n^0 <= 0, cost: 1
Second rule:
l0 -> l0 : j^0'=i^0, i^0-j^0 >= 0, cost: 2*i^0-2*j^0
New rule:
l1 -> l0 : j^0'=i^0, (i^0 >= 0 /\ 1+i^0-n^0 <= 0), cost: 1+2*i^0

Applied deletion
Removed the following rules: 15

Chained accelerated rules with incoming rules
Start location: l4
   6: l0 -> l1 : i^0'=1+i^0, i^0-j^0 <= 0, cost: 1
   9: l1 -> l0 : j^0'=0, 1+i^0-n^0 <= 0, cost: 1
  16: l1 -> l0 : j^0'=i^0, (i^0 >= 0 /\ 1+i^0-n^0 <= 0), cost: 1+2*i^0
  12: l4 -> l1 : i^0'=0, TRUE, cost: 2


Eliminating location l0 by chaining:

Applied chaining
First rule:
l1 -> l0 : j^0'=0, 1+i^0-n^0 <= 0, cost: 1
Second rule:
l0 -> l1 : i^0'=1+i^0, i^0-j^0 <= 0, cost: 1
New rule:
l1 -> l1 : i^0'=1+i^0, j^0'=0, (i^0 <= 0 /\ 1+i^0-n^0 <= 0), cost: 2

Applied chaining
First rule:
l1 -> l0 : j^0'=i^0, (i^0 >= 0 /\ 1+i^0-n^0 <= 0), cost: 1+2*i^0
Second rule:
l0 -> l1 : i^0'=1+i^0, i^0-j^0 <= 0, cost: 1
New rule:
l1 -> l1 : i^0'=1+i^0, j^0'=i^0, (0 <= 0 /\ i^0 >= 0 /\ 1+i^0-n^0 <= 0), cost: 2+2*i^0

Applied simplification
Original rule:
l1 -> l1 : i^0'=1+i^0, j^0'=i^0, (0 <= 0 /\ i^0 >= 0 /\ 1+i^0-n^0 <= 0), cost: 2+2*i^0
New rule:
l1 -> l1 : i^0'=1+i^0, j^0'=i^0, (i^0 >= 0 /\ 1+i^0-n^0 <= 0), cost: 2+2*i^0

Applied deletion
Removed the following rules: 6 9 16

Eliminated locations on tree-shaped paths
Start location: l4
  17: l1 -> l1 : i^0'=1+i^0, j^0'=0, (i^0 <= 0 /\ 1+i^0-n^0 <= 0), cost: 2
  18: l1 -> l1 : i^0'=1+i^0, j^0'=i^0, (i^0 >= 0 /\ 1+i^0-n^0 <= 0), cost: 2+2*i^0
  12: l4 -> l1 : i^0'=0, TRUE, cost: 2


Applied acceleration
Original rule:
l1 -> l1 : i^0'=1+i^0, j^0'=0, (i^0 <= 0 /\ 1+i^0-n^0 <= 0), cost: 2
New rule:
l1 -> l1 : i^0'=i^0+n4, j^0'=0, (-1+n4 >= 0 /\ -i^0+n^0-n4 >= 0 /\ 1-i^0-n4 >= 0), cost: 2*n4

Applied instantiation
Original rule:
l1 -> l1 : i^0'=i^0+n4, j^0'=0, (-1+n4 >= 0 /\ -i^0+n^0-n4 >= 0 /\ 1-i^0-n4 >= 0), cost: 2*n4
New rule:
l1 -> l1 : i^0'=1, j^0'=0, (0 >= 0 /\ -1+n^0 >= 0 /\ -i^0 >= 0), cost: 2-2*i^0

Applied instantiation
Original rule:
l1 -> l1 : i^0'=i^0+n4, j^0'=0, (-1+n4 >= 0 /\ -i^0+n^0-n4 >= 0 /\ 1-i^0-n4 >= 0), cost: 2*n4
New rule:
l1 -> l1 : i^0'=n^0, j^0'=0, (0 >= 0 /\ -1-i^0+n^0 >= 0 /\ 1-n^0 >= 0), cost: -2*i^0+2*n^0

Applied acceleration
Original rule:
l1 -> l1 : i^0'=1+i^0, j^0'=i^0, (i^0 >= 0 /\ 1+i^0-n^0 <= 0), cost: 2+2*i^0
New rule:
l1 -> l1 : i^0'=i^0+n7, j^0'=-1+i^0+n7, (i^0 >= 0 /\ -i^0+n^0-n7 >= 0 /\ -1+n7 >= 0), cost: n7+n7^2+2*i^0*n7

Applied instantiation
Original rule:
l1 -> l1 : i^0'=i^0+n7, j^0'=-1+i^0+n7, (i^0 >= 0 /\ -i^0+n^0-n7 >= 0 /\ -1+n7 >= 0), cost: n7+n7^2+2*i^0*n7
New rule:
l1 -> l1 : i^0'=n^0, j^0'=-1+n^0, (0 >= 0 /\ i^0 >= 0 /\ -1-i^0+n^0 >= 0), cost: -i^0-2*i^0*(i^0-n^0)+n^0+(i^0-n^0)^2

Applied simplification
Original rule:
l1 -> l1 : i^0'=1, j^0'=0, (0 >= 0 /\ -1+n^0 >= 0 /\ -i^0 >= 0), cost: 2-2*i^0
New rule:
l1 -> l1 : i^0'=1, j^0'=0, (i^0 <= 0 /\ -1+n^0 >= 0), cost: 2-2*i^0

Applied simplification
Original rule:
l1 -> l1 : i^0'=n^0, j^0'=0, (0 >= 0 /\ -1-i^0+n^0 >= 0 /\ 1-n^0 >= 0), cost: -2*i^0+2*n^0
New rule:
l1 -> l1 : i^0'=n^0, j^0'=0, (-1-i^0+n^0 >= 0 /\ -1+n^0 <= 0), cost: -2*i^0+2*n^0

Applied simplification
Original rule:
l1 -> l1 : i^0'=n^0, j^0'=-1+n^0, (0 >= 0 /\ i^0 >= 0 /\ -1-i^0+n^0 >= 0), cost: -i^0-2*i^0*(i^0-n^0)+n^0+(i^0-n^0)^2
New rule:
l1 -> l1 : i^0'=n^0, j^0'=-1+n^0, (i^0 >= 0 /\ -1-i^0+n^0 >= 0), cost: -i^0-2*i^0*(i^0-n^0)+n^0+(i^0-n^0)^2

Applied deletion
Removed the following rules: 17 18

Accelerated simple loops
Start location: l4
  22: l1 -> l1 : i^0'=1, j^0'=0, (i^0 <= 0 /\ -1+n^0 >= 0), cost: 2-2*i^0
  23: l1 -> l1 : i^0'=n^0, j^0'=0, (-1-i^0+n^0 >= 0 /\ -1+n^0 <= 0), cost: -2*i^0+2*n^0
  24: l1 -> l1 : i^0'=n^0, j^0'=-1+n^0, (i^0 >= 0 /\ -1-i^0+n^0 >= 0), cost: -i^0-2*i^0*(i^0-n^0)+n^0+(i^0-n^0)^2
  12: l4 -> l1 : i^0'=0, TRUE, cost: 2


Applied chaining
First rule:
l4 -> l1 : i^0'=0, TRUE, cost: 2
Second rule:
l1 -> l1 : i^0'=1, j^0'=0, (i^0 <= 0 /\ -1+n^0 >= 0), cost: 2-2*i^0
New rule:
l4 -> l1 : i^0'=1, j^0'=0, -1+n^0 >= 0, cost: 4

Applied chaining
First rule:
l4 -> l1 : i^0'=0, TRUE, cost: 2
Second rule:
l1 -> l1 : i^0'=n^0, j^0'=0, (-1-i^0+n^0 >= 0 /\ -1+n^0 <= 0), cost: -2*i^0+2*n^0
New rule:
l4 -> l1 : i^0'=n^0, j^0'=0, -1+n^0 == 0, cost: 2+2*n^0

Applied chaining
First rule:
l4 -> l1 : i^0'=0, TRUE, cost: 2
Second rule:
l1 -> l1 : i^0'=n^0, j^0'=-1+n^0, (i^0 >= 0 /\ -1-i^0+n^0 >= 0), cost: -i^0-2*i^0*(i^0-n^0)+n^0+(i^0-n^0)^2
New rule:
l4 -> l1 : i^0'=n^0, j^0'=-1+n^0, -1+n^0 >= 0, cost: 2+n^0+n^0^2

Applied deletion
Removed the following rules: 22 23 24

Chained accelerated rules with incoming rules
Start location: l4
  12: l4 -> l1 : i^0'=0, TRUE, cost: 2
  25: l4 -> l1 : i^0'=1, j^0'=0, -1+n^0 >= 0, cost: 4
  26: l4 -> l1 : i^0'=n^0, j^0'=0, -1+n^0 == 0, cost: 2+2*n^0
  27: l4 -> l1 : i^0'=n^0, j^0'=-1+n^0, -1+n^0 >= 0, cost: 2+n^0+n^0^2


Removed unreachable locations and irrelevant leafs
Start location: l4
  <empty>


Computing asymptotic complexity

Proved the following lower bound
Complexity:  Unknown
Cpx degree: ?

Solved cost: 0
Rule cost:   0