LTS Termination Proof

by T2Cert

Input

• Initial Location: 12
• Transitions: (pre-variables and post-variables)

  0	0	1:		− k6_post + k6_post ≤ 0 ∧ k6_post − k6_post ≤ 0 ∧ − k6_0 + k6_0 ≤ 0 ∧ k6_0 − k6_0 ≤ 0 ∧ − j5_post + j5_post ≤ 0 ∧ j5_post − j5_post ≤ 0 ∧ − j5_0 + j5_0 ≤ 0 ∧ j5_0 − j5_0 ≤ 0 ∧ − i4_post + i4_post ≤ 0 ∧ i4_post − i4_post ≤ 0 ∧ − i4_0 + i4_0 ≤ 0 ∧ i4_0 − i4_0 ≤ 0
  2	1	3:		− k6_post + k6_post ≤ 0 ∧ k6_post − k6_post ≤ 0 ∧ − k6_0 + k6_0 ≤ 0 ∧ k6_0 − k6_0 ≤ 0 ∧ − j5_post + j5_post ≤ 0 ∧ j5_post − j5_post ≤ 0 ∧ − j5_0 + j5_0 ≤ 0 ∧ j5_0 − j5_0 ≤ 0 ∧ − i4_post + i4_post ≤ 0 ∧ i4_post − i4_post ≤ 0 ∧ − i4_0 + i4_0 ≤ 0 ∧ i4_0 − i4_0 ≤ 0
  4	2	5:		− k6_post + k6_post ≤ 0 ∧ k6_post − k6_post ≤ 0 ∧ − k6_0 + k6_0 ≤ 0 ∧ k6_0 − k6_0 ≤ 0 ∧ − j5_post + j5_post ≤ 0 ∧ j5_post − j5_post ≤ 0 ∧ − j5_0 + j5_0 ≤ 0 ∧ j5_0 − j5_0 ≤ 0 ∧ − i4_post + i4_post ≤ 0 ∧ i4_post − i4_post ≤ 0 ∧ − i4_0 + i4_0 ≤ 0 ∧ i4_0 − i4_0 ≤ 0
  6	3	7:		− k6_post + k6_post ≤ 0 ∧ k6_post − k6_post ≤ 0 ∧ − k6_0 + k6_0 ≤ 0 ∧ k6_0 − k6_0 ≤ 0 ∧ − j5_post + j5_post ≤ 0 ∧ j5_post − j5_post ≤ 0 ∧ − j5_0 + j5_0 ≤ 0 ∧ j5_0 − j5_0 ≤ 0 ∧ − i4_post + i4_post ≤ 0 ∧ i4_post − i4_post ≤ 0 ∧ − i4_0 + i4_0 ≤ 0 ∧ i4_0 − i4_0 ≤ 0
  8	4	6:		0 ≤ 0 ∧ 0 ≤ 0 ∧ 6 − k6_0 ≤ 0 ∧ −1 − j5_0 + j5_post ≤ 0 ∧ 1 + j5_0 − j5_post ≤ 0 ∧ j5_0 − j5_post ≤ 0 ∧ − j5_0 + j5_post ≤ 0 ∧ − k6_post + k6_post ≤ 0 ∧ k6_post − k6_post ≤ 0 ∧ − k6_0 + k6_0 ≤ 0 ∧ k6_0 − k6_0 ≤ 0 ∧ − i4_post + i4_post ≤ 0 ∧ i4_post − i4_post ≤ 0 ∧ − i4_0 + i4_0 ≤ 0 ∧ i4_0 − i4_0 ≤ 0
  8	5	9:		0 ≤ 0 ∧ 0 ≤ 0 ∧ −5 + k6_0 ≤ 0 ∧ −1 − k6_0 + k6_post ≤ 0 ∧ 1 + k6_0 − k6_post ≤ 0 ∧ k6_0 − k6_post ≤ 0 ∧ − k6_0 + k6_post ≤ 0 ∧ − j5_post + j5_post ≤ 0 ∧ j5_post − j5_post ≤ 0 ∧ − j5_0 + j5_0 ≤ 0 ∧ j5_0 − j5_0 ≤ 0 ∧ − i4_post + i4_post ≤ 0 ∧ i4_post − i4_post ≤ 0 ∧ − i4_0 + i4_0 ≤ 0 ∧ i4_0 − i4_0 ≤ 0
  7	6	4:		0 ≤ 0 ∧ 0 ≤ 0 ∧ 6 − j5_0 ≤ 0 ∧ −1 − i4_0 + i4_post ≤ 0 ∧ 1 + i4_0 − i4_post ≤ 0 ∧ i4_0 − i4_post ≤ 0 ∧ − i4_0 + i4_post ≤ 0 ∧ − k6_post + k6_post ≤ 0 ∧ k6_post − k6_post ≤ 0 ∧ − k6_0 + k6_0 ≤ 0 ∧ k6_0 − k6_0 ≤ 0 ∧ − j5_post + j5_post ≤ 0 ∧ j5_post − j5_post ≤ 0 ∧ − j5_0 + j5_0 ≤ 0 ∧ j5_0 − j5_0 ≤ 0
  7	7	9:		0 ≤ 0 ∧ 0 ≤ 0 ∧ −5 + j5_0 ≤ 0 ∧ −1 + k6_post ≤ 0 ∧ 1 − k6_post ≤ 0 ∧ k6_0 − k6_post ≤ 0 ∧ − k6_0 + k6_post ≤ 0 ∧ − j5_post + j5_post ≤ 0 ∧ j5_post − j5_post ≤ 0 ∧ − j5_0 + j5_0 ≤ 0 ∧ j5_0 − j5_0 ≤ 0 ∧ − i4_post + i4_post ≤ 0 ∧ i4_post − i4_post ≤ 0 ∧ − i4_0 + i4_0 ≤ 0 ∧ i4_0 − i4_0 ≤ 0
  5	8	10:		6 − i4_0 ≤ 0 ∧ − k6_post + k6_post ≤ 0 ∧ k6_post − k6_post ≤ 0 ∧ − k6_0 + k6_0 ≤ 0 ∧ k6_0 − k6_0 ≤ 0 ∧ − j5_post + j5_post ≤ 0 ∧ j5_post − j5_post ≤ 0 ∧ − j5_0 + j5_0 ≤ 0 ∧ j5_0 − j5_0 ≤ 0 ∧ − i4_post + i4_post ≤ 0 ∧ i4_post − i4_post ≤ 0 ∧ − i4_0 + i4_0 ≤ 0 ∧ i4_0 − i4_0 ≤ 0
  5	9	6:		0 ≤ 0 ∧ 0 ≤ 0 ∧ −5 + i4_0 ≤ 0 ∧ −1 + j5_post ≤ 0 ∧ 1 − j5_post ≤ 0 ∧ j5_0 − j5_post ≤ 0 ∧ − j5_0 + j5_post ≤ 0 ∧ − k6_post + k6_post ≤ 0 ∧ k6_post − k6_post ≤ 0 ∧ − k6_0 + k6_0 ≤ 0 ∧ k6_0 − k6_0 ≤ 0 ∧ − i4_post + i4_post ≤ 0 ∧ i4_post − i4_post ≤ 0 ∧ − i4_0 + i4_0 ≤ 0 ∧ i4_0 − i4_0 ≤ 0
  9	10	8:		− k6_post + k6_post ≤ 0 ∧ k6_post − k6_post ≤ 0 ∧ − k6_0 + k6_0 ≤ 0 ∧ k6_0 − k6_0 ≤ 0 ∧ − j5_post + j5_post ≤ 0 ∧ j5_post − j5_post ≤ 0 ∧ − j5_0 + j5_0 ≤ 0 ∧ j5_0 − j5_0 ≤ 0 ∧ − i4_post + i4_post ≤ 0 ∧ i4_post − i4_post ≤ 0 ∧ − i4_0 + i4_0 ≤ 0 ∧ i4_0 − i4_0 ≤ 0
  3	11	0:		0 ≤ 0 ∧ 0 ≤ 0 ∧ 6 − j5_0 ≤ 0 ∧ −1 − i4_0 + i4_post ≤ 0 ∧ 1 + i4_0 − i4_post ≤ 0 ∧ i4_0 − i4_post ≤ 0 ∧ − i4_0 + i4_post ≤ 0 ∧ − k6_post + k6_post ≤ 0 ∧ k6_post − k6_post ≤ 0 ∧ − k6_0 + k6_0 ≤ 0 ∧ k6_0 − k6_0 ≤ 0 ∧ − j5_post + j5_post ≤ 0 ∧ j5_post − j5_post ≤ 0 ∧ − j5_0 + j5_0 ≤ 0 ∧ j5_0 − j5_0 ≤ 0
  3	12	2:		0 ≤ 0 ∧ 0 ≤ 0 ∧ −5 + j5_0 ≤ 0 ∧ −1 − j5_0 + j5_post ≤ 0 ∧ 1 + j5_0 − j5_post ≤ 0 ∧ j5_0 − j5_post ≤ 0 ∧ − j5_0 + j5_post ≤ 0 ∧ − k6_post + k6_post ≤ 0 ∧ k6_post − k6_post ≤ 0 ∧ − k6_0 + k6_0 ≤ 0 ∧ k6_0 − k6_0 ≤ 0 ∧ − i4_post + i4_post ≤ 0 ∧ i4_post − i4_post ≤ 0 ∧ − i4_0 + i4_0 ≤ 0 ∧ i4_0 − i4_0 ≤ 0
  1	13	4:		0 ≤ 0 ∧ 0 ≤ 0 ∧ 6 − i4_0 ≤ 0 ∧ −1 + i4_post ≤ 0 ∧ 1 − i4_post ≤ 0 ∧ i4_0 − i4_post ≤ 0 ∧ − i4_0 + i4_post ≤ 0 ∧ − k6_post + k6_post ≤ 0 ∧ k6_post − k6_post ≤ 0 ∧ − k6_0 + k6_0 ≤ 0 ∧ k6_0 − k6_0 ≤ 0 ∧ − j5_post + j5_post ≤ 0 ∧ j5_post − j5_post ≤ 0 ∧ − j5_0 + j5_0 ≤ 0 ∧ j5_0 − j5_0 ≤ 0
  1	14	2:		0 ≤ 0 ∧ 0 ≤ 0 ∧ −5 + i4_0 ≤ 0 ∧ −1 + j5_post ≤ 0 ∧ 1 − j5_post ≤ 0 ∧ j5_0 − j5_post ≤ 0 ∧ − j5_0 + j5_post ≤ 0 ∧ − k6_post + k6_post ≤ 0 ∧ k6_post − k6_post ≤ 0 ∧ − k6_0 + k6_0 ≤ 0 ∧ k6_0 − k6_0 ≤ 0 ∧ − i4_post + i4_post ≤ 0 ∧ i4_post − i4_post ≤ 0 ∧ − i4_0 + i4_0 ≤ 0 ∧ i4_0 − i4_0 ≤ 0
  11	15	0:		0 ≤ 0 ∧ 0 ≤ 0 ∧ −1 + i4_post ≤ 0 ∧ 1 − i4_post ≤ 0 ∧ i4_0 − i4_post ≤ 0 ∧ − i4_0 + i4_post ≤ 0 ∧ − k6_post + k6_post ≤ 0 ∧ k6_post − k6_post ≤ 0 ∧ − k6_0 + k6_0 ≤ 0 ∧ k6_0 − k6_0 ≤ 0 ∧ − j5_post + j5_post ≤ 0 ∧ j5_post − j5_post ≤ 0 ∧ − j5_0 + j5_0 ≤ 0 ∧ j5_0 − j5_0 ≤ 0
  12	16	11:		− k6_post + k6_post ≤ 0 ∧ k6_post − k6_post ≤ 0 ∧ − k6_0 + k6_0 ≤ 0 ∧ k6_0 − k6_0 ≤ 0 ∧ − j5_post + j5_post ≤ 0 ∧ j5_post − j5_post ≤ 0 ∧ − j5_0 + j5_0 ≤ 0 ∧ j5_0 − j5_0 ≤ 0 ∧ − i4_post + i4_post ≤ 0 ∧ i4_post − i4_post ≤ 0 ∧ − i4_0 + i4_0 ≤ 0 ∧ i4_0 − i4_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)		TRUE
  1		(1)		TRUE
  2		(2)		TRUE
  3		(3)		TRUE
  4		(4)		TRUE
  5		(5)		TRUE
  6		(6)		TRUE
  7		(7)		TRUE
  8		(8)		TRUE
  9		(9)		TRUE
  10		(10)		6 − i4_0 ≤ 0
  11		(11)		TRUE
  12		(12)		TRUE

• initial node: 12
• cover edges:
• transition edges:

  0	0 1
  1	13 4
  1	14 2
  2	1 3
  3	11 0
  3	12 2
  4	2 5
  5	8 10
  5	9 6
  6	3 7
  7	6 4
  7	7 9
  8	4 6
  8	5 9
  9	10 8
  11	15 0
  12	16 11

2 Switch to Cooperation Termination Proof

3 Transition Removal

We remove transitions 8, 13, 15, 16 using the following ranking functions, which are bounded by −23.

4 Location Addition

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

0* 20 0: − k6_post + k6_post ≤ 0 ∧ k6_post − k6_post ≤ 0 ∧ − k6_0 + k6_0 ≤ 0 ∧ k6_0 − k6_0 ≤ 0 ∧ − j5_post + j5_post ≤ 0 ∧ j5_post − j5_post ≤ 0 ∧ − j5_0 + j5_0 ≤ 0 ∧ j5_0 − j5_0 ≤ 0 ∧ − i4_post + i4_post ≤ 0 ∧ i4_post − i4_post ≤ 0 ∧ − i4_0 + i4_0 ≤ 0 ∧ i4_0 − i4_0 ≤ 0

5 Location Addition

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

0 18 0_var_snapshot: − k6_post + k6_post ≤ 0 ∧ k6_post − k6_post ≤ 0 ∧ − k6_0 + k6_0 ≤ 0 ∧ k6_0 − k6_0 ≤ 0 ∧ − j5_post + j5_post ≤ 0 ∧ j5_post − j5_post ≤ 0 ∧ − j5_0 + j5_0 ≤ 0 ∧ j5_0 − j5_0 ≤ 0 ∧ − i4_post + i4_post ≤ 0 ∧ i4_post − i4_post ≤ 0 ∧ − i4_0 + i4_0 ≤ 0 ∧ i4_0 − i4_0 ≤ 0

6 Location Addition

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

2* 27 2: − k6_post + k6_post ≤ 0 ∧ k6_post − k6_post ≤ 0 ∧ − k6_0 + k6_0 ≤ 0 ∧ k6_0 − k6_0 ≤ 0 ∧ − j5_post + j5_post ≤ 0 ∧ j5_post − j5_post ≤ 0 ∧ − j5_0 + j5_0 ≤ 0 ∧ j5_0 − j5_0 ≤ 0 ∧ − i4_post + i4_post ≤ 0 ∧ i4_post − i4_post ≤ 0 ∧ − i4_0 + i4_0 ≤ 0 ∧ i4_0 − i4_0 ≤ 0

7 Location Addition

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

2 25 2_var_snapshot: − k6_post + k6_post ≤ 0 ∧ k6_post − k6_post ≤ 0 ∧ − k6_0 + k6_0 ≤ 0 ∧ k6_0 − k6_0 ≤ 0 ∧ − j5_post + j5_post ≤ 0 ∧ j5_post − j5_post ≤ 0 ∧ − j5_0 + j5_0 ≤ 0 ∧ j5_0 − j5_0 ≤ 0 ∧ − i4_post + i4_post ≤ 0 ∧ i4_post − i4_post ≤ 0 ∧ − i4_0 + i4_0 ≤ 0 ∧ i4_0 − i4_0 ≤ 0

8 Location Addition

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

4* 34 4: − k6_post + k6_post ≤ 0 ∧ k6_post − k6_post ≤ 0 ∧ − k6_0 + k6_0 ≤ 0 ∧ k6_0 − k6_0 ≤ 0 ∧ − j5_post + j5_post ≤ 0 ∧ j5_post − j5_post ≤ 0 ∧ − j5_0 + j5_0 ≤ 0 ∧ j5_0 − j5_0 ≤ 0 ∧ − i4_post + i4_post ≤ 0 ∧ i4_post − i4_post ≤ 0 ∧ − i4_0 + i4_0 ≤ 0 ∧ i4_0 − i4_0 ≤ 0

9 Location Addition

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

4 32 4_var_snapshot: − k6_post + k6_post ≤ 0 ∧ k6_post − k6_post ≤ 0 ∧ − k6_0 + k6_0 ≤ 0 ∧ k6_0 − k6_0 ≤ 0 ∧ − j5_post + j5_post ≤ 0 ∧ j5_post − j5_post ≤ 0 ∧ − j5_0 + j5_0 ≤ 0 ∧ j5_0 − j5_0 ≤ 0 ∧ − i4_post + i4_post ≤ 0 ∧ i4_post − i4_post ≤ 0 ∧ − i4_0 + i4_0 ≤ 0 ∧ i4_0 − i4_0 ≤ 0

10 Location Addition

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

6* 41 6: − k6_post + k6_post ≤ 0 ∧ k6_post − k6_post ≤ 0 ∧ − k6_0 + k6_0 ≤ 0 ∧ k6_0 − k6_0 ≤ 0 ∧ − j5_post + j5_post ≤ 0 ∧ j5_post − j5_post ≤ 0 ∧ − j5_0 + j5_0 ≤ 0 ∧ j5_0 − j5_0 ≤ 0 ∧ − i4_post + i4_post ≤ 0 ∧ i4_post − i4_post ≤ 0 ∧ − i4_0 + i4_0 ≤ 0 ∧ i4_0 − i4_0 ≤ 0

11 Location Addition

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

6 39 6_var_snapshot: − k6_post + k6_post ≤ 0 ∧ k6_post − k6_post ≤ 0 ∧ − k6_0 + k6_0 ≤ 0 ∧ k6_0 − k6_0 ≤ 0 ∧ − j5_post + j5_post ≤ 0 ∧ j5_post − j5_post ≤ 0 ∧ − j5_0 + j5_0 ≤ 0 ∧ j5_0 − j5_0 ≤ 0 ∧ − i4_post + i4_post ≤ 0 ∧ i4_post − i4_post ≤ 0 ∧ − i4_0 + i4_0 ≤ 0 ∧ i4_0 − i4_0 ≤ 0

12 Location Addition

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

9* 48 9: − k6_post + k6_post ≤ 0 ∧ k6_post − k6_post ≤ 0 ∧ − k6_0 + k6_0 ≤ 0 ∧ k6_0 − k6_0 ≤ 0 ∧ − j5_post + j5_post ≤ 0 ∧ j5_post − j5_post ≤ 0 ∧ − j5_0 + j5_0 ≤ 0 ∧ j5_0 − j5_0 ≤ 0 ∧ − i4_post + i4_post ≤ 0 ∧ i4_post − i4_post ≤ 0 ∧ − i4_0 + i4_0 ≤ 0 ∧ i4_0 − i4_0 ≤ 0

13 Location Addition

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

9 46 9_var_snapshot: − k6_post + k6_post ≤ 0 ∧ k6_post − k6_post ≤ 0 ∧ − k6_0 + k6_0 ≤ 0 ∧ k6_0 − k6_0 ≤ 0 ∧ − j5_post + j5_post ≤ 0 ∧ j5_post − j5_post ≤ 0 ∧ − j5_0 + j5_0 ≤ 0 ∧ j5_0 − j5_0 ≤ 0 ∧ − i4_post + i4_post ≤ 0 ∧ i4_post − i4_post ≤ 0 ∧ − i4_0 + i4_0 ≤ 0 ∧ i4_0 − i4_0 ≤ 0

14 SCC Decomposition

We consider subproblems for each of the 2 SCC(s) of the program graph.

14.1 SCC Subproblem 1/2

Here we consider the SCC { 4, 5, 6, 7, 8, 9, 4_var_snapshot, 4*, 6_var_snapshot, 6*, 9_var_snapshot, 9* }.

14.1.1 Transition Removal

We remove transition 9 using the following ranking functions, which are bounded by −60.

14.1.2 Transition Removal

We remove transition 7 using the following ranking functions, which are bounded by −20.

14.1.3 Transition Removal

We remove transition 5 using the following ranking functions, which are bounded by −17.

14.1.4 Transition Removal

We remove transitions 32, 34, 39, 41, 46, 48, 2, 3, 4, 6, 10 using the following ranking functions, which are bounded by −9.

14.1.5 Splitting Cut-Point Transitions

We consider 3 subproblems corresponding to sets of cut-point transitions as follows.

14.1.5.1 Cut-Point Subproblem 1/3

14.1.5.1.1 Splitting Cut-Point Transitions

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

14.1.5.2 Cut-Point Subproblem 2/3

14.1.5.2.1 Splitting Cut-Point Transitions

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

14.1.5.3 Cut-Point Subproblem 3/3

14.1.5.3.1 Splitting Cut-Point Transitions

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

14.2 SCC Subproblem 2/2

Here we consider the SCC { 0, 1, 2, 3, 0_var_snapshot, 0*, 2_var_snapshot, 2* }.

14.2.1 Transition Removal

We remove transition 14 using the following ranking functions, which are bounded by −42.

14.2.2 Transition Removal

We remove transition 12 using the following ranking functions, which are bounded by −16.

14.2.3 Transition Removal

We remove transitions 18, 25, 27, 0, 1, 11 using the following ranking functions, which are bounded by −4.

14.2.4 Transition Removal

We remove transition 20 using the following ranking functions, which are bounded by 0.

14.2.5 Splitting Cut-Point Transitions

We consider 2 subproblems corresponding to sets of cut-point transitions as follows.

14.2.5.1 Cut-Point Subproblem 1/2

14.2.5.1.1 Splitting Cut-Point Transitions

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

14.2.5.2 Cut-Point Subproblem 2/2

14.2.5.2.1 Splitting Cut-Point Transitions

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

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