LTS Termination Proof

by T2Cert

Input

• Initial Location: 10
• Transitions: (pre-variables and post-variables)

  0	0	1:		1 + i34_0 ≤ 0 ∧ − i8_post + i8_post ≤ 0 ∧ i8_post − i8_post ≤ 0 ∧ − i8_0 + i8_0 ≤ 0 ∧ i8_0 − i8_0 ≤ 0 ∧ − i6_post + i6_post ≤ 0 ∧ i6_post − i6_post ≤ 0 ∧ − i6_0 + i6_0 ≤ 0 ∧ i6_0 − i6_0 ≤ 0 ∧ − i34_post + i34_post ≤ 0 ∧ i34_post − i34_post ≤ 0 ∧ − i34_0 + i34_0 ≤ 0 ∧ i34_0 − i34_0 ≤ 0 ∧ − i2_post + i2_post ≤ 0 ∧ i2_post − i2_post ≤ 0 ∧ − i2_0 + i2_0 ≤ 0 ∧ i2_0 − i2_0 ≤ 0
  0	1	2:		0 ≤ 0 ∧ 0 ≤ 0 ∧ − i34_0 ≤ 0 ∧ 1 − i34_0 + i34_post ≤ 0 ∧ −1 + i34_0 − i34_post ≤ 0 ∧ i34_0 − i34_post ≤ 0 ∧ − i34_0 + i34_post ≤ 0 ∧ − i8_post + i8_post ≤ 0 ∧ i8_post − i8_post ≤ 0 ∧ − i8_0 + i8_0 ≤ 0 ∧ i8_0 − i8_0 ≤ 0 ∧ − i6_post + i6_post ≤ 0 ∧ i6_post − i6_post ≤ 0 ∧ − i6_0 + i6_0 ≤ 0 ∧ i6_0 − i6_0 ≤ 0 ∧ − i2_post + i2_post ≤ 0 ∧ i2_post − i2_post ≤ 0 ∧ − i2_0 + i2_0 ≤ 0 ∧ i2_0 − i2_0 ≤ 0
  3	2	1:		1 − i2_0 ≤ 0 ∧ − i8_post + i8_post ≤ 0 ∧ i8_post − i8_post ≤ 0 ∧ − i8_0 + i8_0 ≤ 0 ∧ i8_0 − i8_0 ≤ 0 ∧ − i6_post + i6_post ≤ 0 ∧ i6_post − i6_post ≤ 0 ∧ − i6_0 + i6_0 ≤ 0 ∧ i6_0 − i6_0 ≤ 0 ∧ − i34_post + i34_post ≤ 0 ∧ i34_post − i34_post ≤ 0 ∧ − i34_0 + i34_0 ≤ 0 ∧ i34_0 − i34_0 ≤ 0 ∧ − i2_post + i2_post ≤ 0 ∧ i2_post − i2_post ≤ 0 ∧ − i2_0 + i2_0 ≤ 0 ∧ i2_0 − i2_0 ≤ 0
  3	3	1:		1 + i2_0 ≤ 0 ∧ − i8_post + i8_post ≤ 0 ∧ i8_post − i8_post ≤ 0 ∧ − i8_0 + i8_0 ≤ 0 ∧ i8_0 − i8_0 ≤ 0 ∧ − i6_post + i6_post ≤ 0 ∧ i6_post − i6_post ≤ 0 ∧ − i6_0 + i6_0 ≤ 0 ∧ i6_0 − i6_0 ≤ 0 ∧ − i34_post + i34_post ≤ 0 ∧ i34_post − i34_post ≤ 0 ∧ − i34_0 + i34_0 ≤ 0 ∧ i34_0 − i34_0 ≤ 0 ∧ − i2_post + i2_post ≤ 0 ∧ i2_post − i2_post ≤ 0 ∧ − i2_0 + i2_0 ≤ 0 ∧ i2_0 − i2_0 ≤ 0
  3	4	2:		0 ≤ 0 ∧ 0 ≤ 0 ∧ i2_0 ≤ 0 ∧ − i2_0 ≤ 0 ∧ −999 + i34_post ≤ 0 ∧ 999 − i34_post ≤ 0 ∧ i34_0 − i34_post ≤ 0 ∧ − i34_0 + i34_post ≤ 0 ∧ − i8_post + i8_post ≤ 0 ∧ i8_post − i8_post ≤ 0 ∧ − i8_0 + i8_0 ≤ 0 ∧ i8_0 − i8_0 ≤ 0 ∧ − i6_post + i6_post ≤ 0 ∧ i6_post − i6_post ≤ 0 ∧ − i6_0 + i6_0 ≤ 0 ∧ i6_0 − i6_0 ≤ 0 ∧ − i2_post + i2_post ≤ 0 ∧ i2_post − i2_post ≤ 0 ∧ − i2_0 + i2_0 ≤ 0 ∧ i2_0 − i2_0 ≤ 0
  2	5	0:		− i8_post + i8_post ≤ 0 ∧ i8_post − i8_post ≤ 0 ∧ − i8_0 + i8_0 ≤ 0 ∧ i8_0 − i8_0 ≤ 0 ∧ − i6_post + i6_post ≤ 0 ∧ i6_post − i6_post ≤ 0 ∧ − i6_0 + i6_0 ≤ 0 ∧ i6_0 − i6_0 ≤ 0 ∧ − i34_post + i34_post ≤ 0 ∧ i34_post − i34_post ≤ 0 ∧ − i34_0 + i34_0 ≤ 0 ∧ i34_0 − i34_0 ≤ 0 ∧ − i2_post + i2_post ≤ 0 ∧ i2_post − i2_post ≤ 0 ∧ − i2_0 + i2_0 ≤ 0 ∧ i2_0 − i2_0 ≤ 0
  4	6	5:		− i8_post + i8_post ≤ 0 ∧ i8_post − i8_post ≤ 0 ∧ − i8_0 + i8_0 ≤ 0 ∧ i8_0 − i8_0 ≤ 0 ∧ − i6_post + i6_post ≤ 0 ∧ i6_post − i6_post ≤ 0 ∧ − i6_0 + i6_0 ≤ 0 ∧ i6_0 − i6_0 ≤ 0 ∧ − i34_post + i34_post ≤ 0 ∧ i34_post − i34_post ≤ 0 ∧ − i34_0 + i34_0 ≤ 0 ∧ i34_0 − i34_0 ≤ 0 ∧ − i2_post + i2_post ≤ 0 ∧ i2_post − i2_post ≤ 0 ∧ − i2_0 + i2_0 ≤ 0 ∧ i2_0 − i2_0 ≤ 0
  6	7	7:		1 + i8_0 ≤ 0 ∧ − i8_post + i8_post ≤ 0 ∧ i8_post − i8_post ≤ 0 ∧ − i8_0 + i8_0 ≤ 0 ∧ i8_0 − i8_0 ≤ 0 ∧ − i6_post + i6_post ≤ 0 ∧ i6_post − i6_post ≤ 0 ∧ − i6_0 + i6_0 ≤ 0 ∧ i6_0 − i6_0 ≤ 0 ∧ − i34_post + i34_post ≤ 0 ∧ i34_post − i34_post ≤ 0 ∧ − i34_0 + i34_0 ≤ 0 ∧ i34_0 − i34_0 ≤ 0 ∧ − i2_post + i2_post ≤ 0 ∧ i2_post − i2_post ≤ 0 ∧ − i2_0 + i2_0 ≤ 0 ∧ i2_0 − i2_0 ≤ 0
  6	8	8:		0 ≤ 0 ∧ 0 ≤ 0 ∧ − i8_0 ≤ 0 ∧ 1 − i8_0 + i8_post ≤ 0 ∧ −1 + i8_0 − i8_post ≤ 0 ∧ i8_0 − i8_post ≤ 0 ∧ − i8_0 + i8_post ≤ 0 ∧ − i6_post + i6_post ≤ 0 ∧ i6_post − i6_post ≤ 0 ∧ − i6_0 + i6_0 ≤ 0 ∧ i6_0 − i6_0 ≤ 0 ∧ − i34_post + i34_post ≤ 0 ∧ i34_post − i34_post ≤ 0 ∧ − i34_0 + i34_0 ≤ 0 ∧ i34_0 − i34_0 ≤ 0 ∧ − i2_post + i2_post ≤ 0 ∧ i2_post − i2_post ≤ 0 ∧ − i2_0 + i2_0 ≤ 0 ∧ i2_0 − i2_0 ≤ 0
  8	9	6:		− i8_post + i8_post ≤ 0 ∧ i8_post − i8_post ≤ 0 ∧ − i8_0 + i8_0 ≤ 0 ∧ i8_0 − i8_0 ≤ 0 ∧ − i6_post + i6_post ≤ 0 ∧ i6_post − i6_post ≤ 0 ∧ − i6_0 + i6_0 ≤ 0 ∧ i6_0 − i6_0 ≤ 0 ∧ − i34_post + i34_post ≤ 0 ∧ i34_post − i34_post ≤ 0 ∧ − i34_0 + i34_0 ≤ 0 ∧ i34_0 − i34_0 ≤ 0 ∧ − i2_post + i2_post ≤ 0 ∧ i2_post − i2_post ≤ 0 ∧ − i2_0 + i2_0 ≤ 0 ∧ i2_0 − i2_0 ≤ 0
  5	10	8:		0 ≤ 0 ∧ 0 ≤ 0 ∧ 1 + i6_0 ≤ 0 ∧ −999 + i8_post ≤ 0 ∧ 999 − i8_post ≤ 0 ∧ i8_0 − i8_post ≤ 0 ∧ − i8_0 + i8_post ≤ 0 ∧ − i6_post + i6_post ≤ 0 ∧ i6_post − i6_post ≤ 0 ∧ − i6_0 + i6_0 ≤ 0 ∧ i6_0 − i6_0 ≤ 0 ∧ − i34_post + i34_post ≤ 0 ∧ i34_post − i34_post ≤ 0 ∧ − i34_0 + i34_0 ≤ 0 ∧ i34_0 − i34_0 ≤ 0 ∧ − i2_post + i2_post ≤ 0 ∧ i2_post − i2_post ≤ 0 ∧ − i2_0 + i2_0 ≤ 0 ∧ i2_0 − i2_0 ≤ 0
  5	11	4:		0 ≤ 0 ∧ 0 ≤ 0 ∧ − i6_0 ≤ 0 ∧ 1 − i6_0 + i6_post ≤ 0 ∧ −1 + i6_0 − i6_post ≤ 0 ∧ i6_0 − i6_post ≤ 0 ∧ − i6_0 + i6_post ≤ 0 ∧ − i8_post + i8_post ≤ 0 ∧ i8_post − i8_post ≤ 0 ∧ − i8_0 + i8_0 ≤ 0 ∧ i8_0 − i8_0 ≤ 0 ∧ − i34_post + i34_post ≤ 0 ∧ i34_post − i34_post ≤ 0 ∧ − i34_0 + i34_0 ≤ 0 ∧ i34_0 − i34_0 ≤ 0 ∧ − i2_post + i2_post ≤ 0 ∧ i2_post − i2_post ≤ 0 ∧ − i2_0 + i2_0 ≤ 0 ∧ i2_0 − i2_0 ≤ 0
  1	12	4:		0 ≤ 0 ∧ 0 ≤ 0 ∧ −999 + i6_post ≤ 0 ∧ 999 − i6_post ≤ 0 ∧ i6_0 − i6_post ≤ 0 ∧ − i6_0 + i6_post ≤ 0 ∧ − i8_post + i8_post ≤ 0 ∧ i8_post − i8_post ≤ 0 ∧ − i8_0 + i8_0 ≤ 0 ∧ i8_0 − i8_0 ≤ 0 ∧ − i34_post + i34_post ≤ 0 ∧ i34_post − i34_post ≤ 0 ∧ − i34_0 + i34_0 ≤ 0 ∧ i34_0 − i34_0 ≤ 0 ∧ − i2_post + i2_post ≤ 0 ∧ i2_post − i2_post ≤ 0 ∧ − i2_0 + i2_0 ≤ 0 ∧ i2_0 − i2_0 ≤ 0
  9	13	3:		0 ≤ 0 ∧ 0 ≤ 0 ∧ −1 + i2_post ≤ 0 ∧ 1 − i2_post ≤ 0 ∧ i2_0 − i2_post ≤ 0 ∧ − i2_0 + i2_post ≤ 0 ∧ − i8_post + i8_post ≤ 0 ∧ i8_post − i8_post ≤ 0 ∧ − i8_0 + i8_0 ≤ 0 ∧ i8_0 − i8_0 ≤ 0 ∧ − i6_post + i6_post ≤ 0 ∧ i6_post − i6_post ≤ 0 ∧ − i6_0 + i6_0 ≤ 0 ∧ i6_0 − i6_0 ≤ 0 ∧ − i34_post + i34_post ≤ 0 ∧ i34_post − i34_post ≤ 0 ∧ − i34_0 + i34_0 ≤ 0 ∧ i34_0 − i34_0 ≤ 0
  10	14	9:		− i8_post + i8_post ≤ 0 ∧ i8_post − i8_post ≤ 0 ∧ − i8_0 + i8_0 ≤ 0 ∧ i8_0 − i8_0 ≤ 0 ∧ − i6_post + i6_post ≤ 0 ∧ i6_post − i6_post ≤ 0 ∧ − i6_0 + i6_0 ≤ 0 ∧ i6_0 − i6_0 ≤ 0 ∧ − i34_post + i34_post ≤ 0 ∧ i34_post − i34_post ≤ 0 ∧ − i34_0 + i34_0 ≤ 0 ∧ i34_0 − i34_0 ≤ 0 ∧ − i2_post + i2_post ≤ 0 ∧ i2_post − i2_post ≤ 0 ∧ − i2_0 + i2_0 ≤ 0 ∧ i2_0 − i2_0 ≤ 0

Proof

1 Invariant Updates

The following invariants are asserted.

The invariants are proved as follows.

IMPACT Invariant Proof

• nodes (location) invariant:

  0		(0)		−1 + i2_post ≤ 0 ∧ 1 − i2_post ≤ 0 ∧ 1 − i2_0 ≤ 0 ∧ i2_0 ≤ 0
  1		(1)		−1 + i2_post ≤ 0 ∧ 1 − i2_post ≤ 0 ∧ −1 + i2_0 ≤ 0 ∧ 1 − i2_0 ≤ 0
  2		(2)		−1 + i2_post ≤ 0 ∧ 1 − i2_post ≤ 0 ∧ 1 − i2_0 ≤ 0 ∧ i2_0 ≤ 0
  3		(3)		−1 + i2_post ≤ 0 ∧ 1 − i2_post ≤ 0 ∧ −1 + i2_0 ≤ 0 ∧ 1 − i2_0 ≤ 0
  4		(4)		−1 + i2_post ≤ 0 ∧ 1 − i2_post ≤ 0 ∧ −1 + i2_0 ≤ 0 ∧ 1 − i2_0 ≤ 0
  5		(5)		−1 + i2_post ≤ 0 ∧ 1 − i2_post ≤ 0 ∧ −1 + i2_0 ≤ 0 ∧ 1 − i2_0 ≤ 0
  6		(6)		−1 + i2_post ≤ 0 ∧ 1 − i2_post ≤ 0 ∧ −1 + i2_0 ≤ 0 ∧ 1 − i2_0 ≤ 0 ∧ 1 + i6_0 ≤ 0
  7		(7)		−1 + i2_post ≤ 0 ∧ 1 − i2_post ≤ 0 ∧ −1 + i2_0 ≤ 0 ∧ 1 − i2_0 ≤ 0 ∧ 1 + i6_0 ≤ 0 ∧ 1 + i8_0 ≤ 0
  8		(8)		−1 + i2_post ≤ 0 ∧ 1 − i2_post ≤ 0 ∧ −1 + i2_0 ≤ 0 ∧ 1 − i2_0 ≤ 0 ∧ 1 + i6_0 ≤ 0
  9		(9)		TRUE
  10		(10)		TRUE

• initial node: 10
• cover edges:
• transition edges:

  0	0 1
  0	1 2
  1	12 4
  2	5 0
  3	2 1
  3	3 1
  3	4 2
  4	6 5
  5	10 8
  5	11 4
  6	7 7
  6	8 8
  8	9 6
  9	13 3
  10	14 9

2 Switch to Cooperation Termination Proof

3 Transition Removal

We remove transitions 0, 2, 3, 4, 7, 10, 12, 13, 14 using the following ranking functions, which are bounded by −25.

4 Location Addition

The following skip-transition is inserted and corresponding redirections w.r.t. the old location are performed.

2* 18 2: − i8_post + i8_post ≤ 0 ∧ i8_post − i8_post ≤ 0 ∧ − i8_0 + i8_0 ≤ 0 ∧ i8_0 − i8_0 ≤ 0 ∧ − i6_post + i6_post ≤ 0 ∧ i6_post − i6_post ≤ 0 ∧ − i6_0 + i6_0 ≤ 0 ∧ i6_0 − i6_0 ≤ 0 ∧ − i34_post + i34_post ≤ 0 ∧ i34_post − i34_post ≤ 0 ∧ − i34_0 + i34_0 ≤ 0 ∧ i34_0 − i34_0 ≤ 0 ∧ − i2_post + i2_post ≤ 0 ∧ i2_post − i2_post ≤ 0 ∧ − i2_0 + i2_0 ≤ 0 ∧ i2_0 − i2_0 ≤ 0

5 Location Addition

The following skip-transition is inserted and corresponding redirections w.r.t. the old location are performed.

2 16 2_var_snapshot: − i8_post + i8_post ≤ 0 ∧ i8_post − i8_post ≤ 0 ∧ − i8_0 + i8_0 ≤ 0 ∧ i8_0 − i8_0 ≤ 0 ∧ − i6_post + i6_post ≤ 0 ∧ i6_post − i6_post ≤ 0 ∧ − i6_0 + i6_0 ≤ 0 ∧ i6_0 − i6_0 ≤ 0 ∧ − i34_post + i34_post ≤ 0 ∧ i34_post − i34_post ≤ 0 ∧ − i34_0 + i34_0 ≤ 0 ∧ i34_0 − i34_0 ≤ 0 ∧ − i2_post + i2_post ≤ 0 ∧ i2_post − i2_post ≤ 0 ∧ − i2_0 + i2_0 ≤ 0 ∧ i2_0 − i2_0 ≤ 0

6 Location Addition

The following skip-transition is inserted and corresponding redirections w.r.t. the old location are performed.

4* 25 4: − i8_post + i8_post ≤ 0 ∧ i8_post − i8_post ≤ 0 ∧ − i8_0 + i8_0 ≤ 0 ∧ i8_0 − i8_0 ≤ 0 ∧ − i6_post + i6_post ≤ 0 ∧ i6_post − i6_post ≤ 0 ∧ − i6_0 + i6_0 ≤ 0 ∧ i6_0 − i6_0 ≤ 0 ∧ − i34_post + i34_post ≤ 0 ∧ i34_post − i34_post ≤ 0 ∧ − i34_0 + i34_0 ≤ 0 ∧ i34_0 − i34_0 ≤ 0 ∧ − i2_post + i2_post ≤ 0 ∧ i2_post − i2_post ≤ 0 ∧ − i2_0 + i2_0 ≤ 0 ∧ i2_0 − i2_0 ≤ 0

7 Location Addition

The following skip-transition is inserted and corresponding redirections w.r.t. the old location are performed.

4 23 4_var_snapshot: − i8_post + i8_post ≤ 0 ∧ i8_post − i8_post ≤ 0 ∧ − i8_0 + i8_0 ≤ 0 ∧ i8_0 − i8_0 ≤ 0 ∧ − i6_post + i6_post ≤ 0 ∧ i6_post − i6_post ≤ 0 ∧ − i6_0 + i6_0 ≤ 0 ∧ i6_0 − i6_0 ≤ 0 ∧ − i34_post + i34_post ≤ 0 ∧ i34_post − i34_post ≤ 0 ∧ − i34_0 + i34_0 ≤ 0 ∧ i34_0 − i34_0 ≤ 0 ∧ − i2_post + i2_post ≤ 0 ∧ i2_post − i2_post ≤ 0 ∧ − i2_0 + i2_0 ≤ 0 ∧ i2_0 − i2_0 ≤ 0

8 Location Addition

The following skip-transition is inserted and corresponding redirections w.r.t. the old location are performed.

8* 32 8: − i8_post + i8_post ≤ 0 ∧ i8_post − i8_post ≤ 0 ∧ − i8_0 + i8_0 ≤ 0 ∧ i8_0 − i8_0 ≤ 0 ∧ − i6_post + i6_post ≤ 0 ∧ i6_post − i6_post ≤ 0 ∧ − i6_0 + i6_0 ≤ 0 ∧ i6_0 − i6_0 ≤ 0 ∧ − i34_post + i34_post ≤ 0 ∧ i34_post − i34_post ≤ 0 ∧ − i34_0 + i34_0 ≤ 0 ∧ i34_0 − i34_0 ≤ 0 ∧ − i2_post + i2_post ≤ 0 ∧ i2_post − i2_post ≤ 0 ∧ − i2_0 + i2_0 ≤ 0 ∧ i2_0 − i2_0 ≤ 0

9 Location Addition

The following skip-transition is inserted and corresponding redirections w.r.t. the old location are performed.

8 30 8_var_snapshot: − i8_post + i8_post ≤ 0 ∧ i8_post − i8_post ≤ 0 ∧ − i8_0 + i8_0 ≤ 0 ∧ i8_0 − i8_0 ≤ 0 ∧ − i6_post + i6_post ≤ 0 ∧ i6_post − i6_post ≤ 0 ∧ − i6_0 + i6_0 ≤ 0 ∧ i6_0 − i6_0 ≤ 0 ∧ − i34_post + i34_post ≤ 0 ∧ i34_post − i34_post ≤ 0 ∧ − i34_0 + i34_0 ≤ 0 ∧ i34_0 − i34_0 ≤ 0 ∧ − i2_post + i2_post ≤ 0 ∧ i2_post − i2_post ≤ 0 ∧ − i2_0 + i2_0 ≤ 0 ∧ i2_0 − i2_0 ≤ 0

10 SCC Decomposition

We consider subproblems for each of the 3 SCC(s) of the program graph.

10.1 SCC Subproblem 1/3

Here we consider the SCC { 6, 8, 8_var_snapshot, 8* }.

10.1.1 Transition Removal

We remove transition 8 using the following ranking functions, which are bounded by −4.

10.1.2 Transition Removal

We remove transitions 30, 32, 9 using the following ranking functions, which are bounded by −2.

10.1.3 Splitting Cut-Point Transitions

We consider 1 subproblems corresponding to sets of cut-point transitions as follows.

10.1.3.1 Cut-Point Subproblem 1/1

10.1.3.1.1 Splitting Cut-Point Transitions

There remain no cut-point transition to consider. Hence the cooperation termination is trivial.

10.2 SCC Subproblem 2/3

Here we consider the SCC { 4, 5, 4_var_snapshot, 4* }.

10.2.1 Transition Removal

We remove transition 11 using the following ranking functions, which are bounded by −1.

10.2.2 Transition Removal

We remove transitions 23, 25, 6 using the following ranking functions, which are bounded by −2.

10.2.3 Splitting Cut-Point Transitions

We consider 1 subproblems corresponding to sets of cut-point transitions as follows.

10.2.3.1 Cut-Point Subproblem 1/1

10.2.3.1.1 Splitting Cut-Point Transitions

There remain no cut-point transition to consider. Hence the cooperation termination is trivial.

10.3 SCC Subproblem 3/3

Here we consider the SCC { 0, 2, 2_var_snapshot, 2* }.

10.3.1 Transition Removal

We remove transitions 16, 18, 1, 5 using the following ranking functions, which are bounded by −2.

10.3.2 Splitting Cut-Point Transitions

We consider 1 subproblems corresponding to sets of cut-point transitions as follows.

10.3.2.1 Cut-Point Subproblem 1/1

10.3.2.1.1 Splitting Cut-Point Transitions

There remain no cut-point transition to consider. Hence the cooperation termination is trivial.

Tool configuration

T2Cert

• version: 1.0