NO

Initial ITS

Start location: [1]
Program variables: i j
   0: [1] -> [2] : j'=0, T, cost: 1
   1: [2] -> [3] : i'=0, T, cost: 1
   2: [3] -> [4] : i-j <= 0, cost: 1
   9: [3] -> [9] : i-j > 0, cost: 1
   3: [4] -> [6] : -1-i+j < 0, cost: 1
   6: [4] -> [5] : -1-i+j >= 0, cost: 1
   7: [5] -> [8] : i'=1+i, T, cost: 1
   4: [6] -> [7] : j'=2+j, T, cost: 1
   5: [7] -> [5] : T, cost: 1
   8: [8] -> [3] : T, cost: 1

Chained Linear Paths

Start location: [1]
Program variables: i j
  10: [1] -> [3] : i'=0, j'=0, T, cost: 1
   2: [3] -> [4] : i-j <= 0, cost: 1
   9: [3] -> [9] : i-j > 0, cost: 1
   6: [4] -> [5] : -1-i+j >= 0, cost: 1
  12: [4] -> [5] : j'=2+j, -1-i+j < 0, cost: 1
  13: [5] -> [3] : i'=1+i, T, cost: 1

	Eliminating location [2] by chaining:

	Applied chaining

	First rule:
	[1] -> [2] : j'=0, T, cost: 1
	Second rule:
	[2] -> [3] : i'=0, T, cost: 1
	New rule:
	[1] -> [3] : i'=0, j'=0, T, cost: 1
	
	Applied deletion

	Removed the following rules: 0 1
	
	Eliminating location [6] by chaining:

	Applied chaining

	First rule:
	[4] -> [6] : -1-i+j < 0, cost: 1
	Second rule:
	[6] -> [7] : j'=2+j, T, cost: 1
	New rule:
	[4] -> [7] : j'=2+j, -1-i+j < 0, cost: 1
	
	Applied deletion

	Removed the following rules: 3 4
	
	Eliminating location [7] by chaining:

	Applied chaining

	First rule:
	[4] -> [7] : j'=2+j, -1-i+j < 0, cost: 1
	Second rule:
	[7] -> [5] : T, cost: 1
	New rule:
	[4] -> [5] : j'=2+j, -1-i+j < 0, cost: 1
	
	Applied deletion

	Removed the following rules: 5 11
	
	Eliminating location [8] by chaining:

	Applied chaining

	First rule:
	[5] -> [8] : i'=1+i, T, cost: 1
	Second rule:
	[8] -> [3] : T, cost: 1
	New rule:
	[5] -> [3] : i'=1+i, T, cost: 1
	
	Applied deletion

	Removed the following rules: 7 8

Step with 10

	Trace

	10[T]
	
	Blocked

	[{}, {}]

Step with 2

	Trace

	10[T], 2[(i-j <= 0)]
	
	Blocked

	[{}, {9[T]}, {}]

Step with 12

	Trace

	10[T], 2[(i-j <= 0)], 12[(-1-i+j < 0)]
	
	Blocked

	[{}, {9[T]}, {6[T]}, {}]

Step with 13

	Trace

	10[T], 2[(i-j <= 0)], 12[(-1-i+j < 0)], 13[T]
	
	Blocked

	[{}, {9[T]}, {6[T]}, {}, {}]

Acceleration Failed

marked recursive suffix as redundant

Step with 2

	Trace

	10[T], 2[(i-j <= 0)], 12[(-1-i+j < 0)], 13[T], 2[(i-j <= 0)]
	
	Blocked

	[{}, {9[T]}, {6[T]}, {}, {}, {}]

Accelerate

Start location: [1]
Program variables: i j
  10: [1] -> [3] : i'=0, j'=0, T, cost: 1
   2: [3] -> [4] : i-j <= 0, cost: 1
   9: [3] -> [9] : i-j > 0, cost: 1
   6: [4] -> [5] : -1-i+j >= 0, cost: 1
  12: [4] -> [5] : j'=2+j, -1-i+j < 0, cost: 1
  14: [4] -> [4] : i'=i+n, j'=2*n+j, (1-i+j >= 0 /\ 2+i-n-j > 0 /\ -1+n >= 0), cost: 1
  13: [5] -> [3] : i'=1+i, T, cost: 1

	Loop Acceleration

	Original rule:
	[4] -> [4] : i'=1+i, j'=2+j, (-1-i+j < 0 /\ -1+i-j <= 0), cost: 1
	New rule:
	[4] -> [4] : i'=i+n, j'=2*n+j, (1-i+j >= 0 /\ 2+i-n-j > 0 /\ -1+n >= 0), cost: 1

		1+i-j > 0 [0]: montonic decrease yields 2+i-n-j > 0
		
		1+i-j > 0 [1]: eventual increase yields (1+i-j > 0 /\ 1 <= 0)
		
		1-i+j >= 0 [0]: monotonic increase yields 1-i+j >= 0
		
		Replacement map: {1+i-j > 0 -> 2+i-n-j > 0, 1-i+j >= 0 -> 1-i+j >= 0}

	Trace

	10[T], 2[(i-j <= 0)], 14[(1-i+j >= 0 /\ 2+i-n-j > 0 /\ -1+n >= 0)]
	
	Blocked

	[{}, {9[T]}, {6[T]}, {14[T]}]

Step with 6

	Trace

	10[T], 2[(i-j <= 0)], 14[(1-i+j >= 0 /\ 2+i-n-j > 0 /\ -1+n >= 0)], 6[(-1-i+j >= 0)]
	
	Blocked

	[{}, {9[T]}, {6[T]}, {14[T]}, {}]

Step with 13

	Trace

	10[T], 2[(i-j <= 0)], 14[(1-i+j >= 0 /\ 2+i-n-j > 0 /\ -1+n >= 0)], 6[(-1-i+j >= 0)], 13[T]
	
	Blocked

	[{}, {9[T]}, {6[T]}, {14[T]}, {}, {}]

Nonterm

Start location: [1]
Program variables: i j
  10: [1] -> [3] : i'=0, j'=0, T, cost: 1
   2: [3] -> [4] : i-j <= 0, cost: 1
   9: [3] -> [9] : i-j > 0, cost: 1
  15: [3] -> LoAT_sink : (0 >= 0 /\ 1 >= 0 /\ i-j >= 0 /\ 1-i+j >= 0 /\ -i+j >= 0), cost: NONTERM
   6: [4] -> [5] : -1-i+j >= 0, cost: 1
  12: [4] -> [5] : j'=2+j, -1-i+j < 0, cost: 1
  14: [4] -> [4] : i'=i+n, j'=2*n+j, (1-i+j >= 0 /\ 2+i-n-j > 0 /\ -1+n >= 0), cost: 1
  13: [5] -> [3] : i'=1+i, T, cost: 1

	unrolling

	Original rule:
	[3] -> [3] : i'=2+2*i-j, j'=2+2*i-j, (i-j <= 0 /\ i-j >= 0 /\ 1-i+j >= 0), cost: 1
	New rule:
	[3] -> [3] : i'=4+2*i-j, j'=4+2*i-j, (0 <= 0 /\ 0 >= 0 /\ 1 >= 0 /\ i-j <= 0 /\ i-j >= 0 /\ 1-i+j >= 0), cost: 1
	
	Certificate of Non-Termination

	Original rule:
	[3] -> [3] : i'=4+2*i-j, j'=4+2*i-j, (0 <= 0 /\ 0 >= 0 /\ 1 >= 0 /\ i-j <= 0 /\ i-j >= 0 /\ 1-i+j >= 0), cost: 1
	New rule:
	[3] -> LoAT_sink : (0 >= 0 /\ 1 >= 0 /\ i-j >= 0 /\ 1-i+j >= 0 /\ -i+j >= 0), cost: NONTERM

		0 >= 0 [0]: monotonic increase yields 0 >= 0
		
		1 >= 0 [0]: monotonic increase yields 1 >= 0
		
		i-j >= 0 [0]: monotonic increase yields i-j >= 0
		
		1-i+j >= 0 [0]: monotonic increase yields 1-i+j >= 0
		
		-i+j >= 0 [0]: monotonic increase yields -i+j >= 0
		
		Replacement map: {0 >= 0 -> 0 >= 0, 1 >= 0 -> 1 >= 0, i-j >= 0 -> i-j >= 0, 1-i+j >= 0 -> 1-i+j >= 0, -i+j >= 0 -> -i+j >= 0}

Step with 15

	Trace

	10[T], 15[(0 >= 0 /\ 1 >= 0 /\ i-j >= 0 /\ 1-i+j >= 0 /\ -i+j >= 0)]
	
	Blocked

	[{}, {9[T]}, {15[T]}]

Refute

	Counterexample

	[ i=0 j=0 ] 10 [ i=i j=j ] 15
	
NO

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