LTS Termination Proof

by T2Cert

Input

• Initial Location: 7
• Transitions: (pre-variables and post-variables)

  0	0	1:		− y_17_post + y_17_post ≤ 0 ∧ y_17_post − y_17_post ≤ 0 ∧ − y_17_0 + y_17_0 ≤ 0 ∧ y_17_0 − y_17_0 ≤ 0 ∧ − x_13_post + x_13_post ≤ 0 ∧ x_13_post − x_13_post ≤ 0 ∧ − x_13_0 + x_13_0 ≤ 0 ∧ x_13_0 − x_13_0 ≤ 0 ∧ − st_16_post + st_16_post ≤ 0 ∧ st_16_post − st_16_post ≤ 0 ∧ − st_16_0 + st_16_0 ≤ 0 ∧ st_16_0 − st_16_0 ≤ 0 ∧ − st_14_0 + st_14_0 ≤ 0 ∧ st_14_0 − st_14_0 ≤ 0 ∧ − rv_15_post + rv_15_post ≤ 0 ∧ rv_15_post − rv_15_post ≤ 0 ∧ − rv_15_0 + rv_15_0 ≤ 0 ∧ rv_15_0 − rv_15_0 ≤ 0 ∧ − rt_11_post + rt_11_post ≤ 0 ∧ rt_11_post − rt_11_post ≤ 0 ∧ − rt_11_0 + rt_11_0 ≤ 0 ∧ rt_11_0 − rt_11_0 ≤ 0 ∧ − nd_12_post + nd_12_post ≤ 0 ∧ nd_12_post − nd_12_post ≤ 0 ∧ − nd_12_1 + nd_12_1 ≤ 0 ∧ nd_12_1 − nd_12_1 ≤ 0 ∧ − nd_12_0 + nd_12_0 ≤ 0 ∧ nd_12_0 − nd_12_0 ≤ 0
  1	1	2:		0 ≤ 0 ∧ 0 ≤ 0 ∧ x_13_0 ≤ 0 ∧ rt_11_post − st_14_0 ≤ 0 ∧ − rt_11_post + st_14_0 ≤ 0 ∧ rt_11_0 − rt_11_post ≤ 0 ∧ − rt_11_0 + rt_11_post ≤ 0 ∧ − y_17_post + y_17_post ≤ 0 ∧ y_17_post − y_17_post ≤ 0 ∧ − y_17_0 + y_17_0 ≤ 0 ∧ y_17_0 − y_17_0 ≤ 0 ∧ − x_13_post + x_13_post ≤ 0 ∧ x_13_post − x_13_post ≤ 0 ∧ − x_13_0 + x_13_0 ≤ 0 ∧ x_13_0 − x_13_0 ≤ 0 ∧ − st_16_post + st_16_post ≤ 0 ∧ st_16_post − st_16_post ≤ 0 ∧ − st_16_0 + st_16_0 ≤ 0 ∧ st_16_0 − st_16_0 ≤ 0 ∧ − st_14_0 + st_14_0 ≤ 0 ∧ st_14_0 − st_14_0 ≤ 0 ∧ − rv_15_post + rv_15_post ≤ 0 ∧ rv_15_post − rv_15_post ≤ 0 ∧ − rv_15_0 + rv_15_0 ≤ 0 ∧ rv_15_0 − rv_15_0 ≤ 0 ∧ − nd_12_post + nd_12_post ≤ 0 ∧ nd_12_post − nd_12_post ≤ 0 ∧ − nd_12_1 + nd_12_1 ≤ 0 ∧ nd_12_1 − nd_12_1 ≤ 0 ∧ − nd_12_0 + nd_12_0 ≤ 0 ∧ nd_12_0 − nd_12_0 ≤ 0
  1	2	3:		0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 1 − x_13_0 ≤ 0 ∧ − nd_12_1 + rv_15_post ≤ 0 ∧ nd_12_1 − rv_15_post ≤ 0 ∧ − rv_15_post ≤ 0 ∧ rv_15_post ≤ 0 ∧ 1 − y_17_0 + y_17_post ≤ 0 ∧ −1 + y_17_0 − y_17_post ≤ 0 ∧ 2 − y_17_post ≤ 0 ∧ nd_12_0 − nd_12_post ≤ 0 ∧ − nd_12_0 + nd_12_post ≤ 0 ∧ rv_15_0 − rv_15_post ≤ 0 ∧ − rv_15_0 + rv_15_post ≤ 0 ∧ st_16_0 − st_16_post ≤ 0 ∧ − st_16_0 + st_16_post ≤ 0 ∧ y_17_0 − y_17_post ≤ 0 ∧ − y_17_0 + y_17_post ≤ 0 ∧ − x_13_post + x_13_post ≤ 0 ∧ x_13_post − x_13_post ≤ 0 ∧ − x_13_0 + x_13_0 ≤ 0 ∧ x_13_0 − x_13_0 ≤ 0 ∧ − st_14_0 + st_14_0 ≤ 0 ∧ st_14_0 − st_14_0 ≤ 0 ∧ − rt_11_post + rt_11_post ≤ 0 ∧ rt_11_post − rt_11_post ≤ 0 ∧ − rt_11_0 + rt_11_0 ≤ 0 ∧ rt_11_0 − rt_11_0 ≤ 0
  3	3	1:		− y_17_post + y_17_post ≤ 0 ∧ y_17_post − y_17_post ≤ 0 ∧ − y_17_0 + y_17_0 ≤ 0 ∧ y_17_0 − y_17_0 ≤ 0 ∧ − x_13_post + x_13_post ≤ 0 ∧ x_13_post − x_13_post ≤ 0 ∧ − x_13_0 + x_13_0 ≤ 0 ∧ x_13_0 − x_13_0 ≤ 0 ∧ − st_16_post + st_16_post ≤ 0 ∧ st_16_post − st_16_post ≤ 0 ∧ − st_16_0 + st_16_0 ≤ 0 ∧ st_16_0 − st_16_0 ≤ 0 ∧ − st_14_0 + st_14_0 ≤ 0 ∧ st_14_0 − st_14_0 ≤ 0 ∧ − rv_15_post + rv_15_post ≤ 0 ∧ rv_15_post − rv_15_post ≤ 0 ∧ − rv_15_0 + rv_15_0 ≤ 0 ∧ rv_15_0 − rv_15_0 ≤ 0 ∧ − rt_11_post + rt_11_post ≤ 0 ∧ rt_11_post − rt_11_post ≤ 0 ∧ − rt_11_0 + rt_11_0 ≤ 0 ∧ rt_11_0 − rt_11_0 ≤ 0 ∧ − nd_12_post + nd_12_post ≤ 0 ∧ nd_12_post − nd_12_post ≤ 0 ∧ − nd_12_1 + nd_12_1 ≤ 0 ∧ nd_12_1 − nd_12_1 ≤ 0 ∧ − nd_12_0 + nd_12_0 ≤ 0 ∧ nd_12_0 − nd_12_0 ≤ 0
  1	4	4:		0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 1 − x_13_0 ≤ 0 ∧ − nd_12_1 + rv_15_post ≤ 0 ∧ nd_12_1 − rv_15_post ≤ 0 ∧ nd_12_0 − nd_12_post ≤ 0 ∧ − nd_12_0 + nd_12_post ≤ 0 ∧ rv_15_0 − rv_15_post ≤ 0 ∧ − rv_15_0 + rv_15_post ≤ 0 ∧ − y_17_post + y_17_post ≤ 0 ∧ y_17_post − y_17_post ≤ 0 ∧ − y_17_0 + y_17_0 ≤ 0 ∧ y_17_0 − y_17_0 ≤ 0 ∧ − x_13_post + x_13_post ≤ 0 ∧ x_13_post − x_13_post ≤ 0 ∧ − x_13_0 + x_13_0 ≤ 0 ∧ x_13_0 − x_13_0 ≤ 0 ∧ − st_16_post + st_16_post ≤ 0 ∧ st_16_post − st_16_post ≤ 0 ∧ − st_16_0 + st_16_0 ≤ 0 ∧ st_16_0 − st_16_0 ≤ 0 ∧ − st_14_0 + st_14_0 ≤ 0 ∧ st_14_0 − st_14_0 ≤ 0 ∧ − rt_11_post + rt_11_post ≤ 0 ∧ rt_11_post − rt_11_post ≤ 0 ∧ − rt_11_0 + rt_11_0 ≤ 0 ∧ rt_11_0 − rt_11_0 ≤ 0
  4	5	5:		1 − rv_15_0 ≤ 0 ∧ − y_17_post + y_17_post ≤ 0 ∧ y_17_post − y_17_post ≤ 0 ∧ − y_17_0 + y_17_0 ≤ 0 ∧ y_17_0 − y_17_0 ≤ 0 ∧ − x_13_post + x_13_post ≤ 0 ∧ x_13_post − x_13_post ≤ 0 ∧ − x_13_0 + x_13_0 ≤ 0 ∧ x_13_0 − x_13_0 ≤ 0 ∧ − st_16_post + st_16_post ≤ 0 ∧ st_16_post − st_16_post ≤ 0 ∧ − st_16_0 + st_16_0 ≤ 0 ∧ st_16_0 − st_16_0 ≤ 0 ∧ − st_14_0 + st_14_0 ≤ 0 ∧ st_14_0 − st_14_0 ≤ 0 ∧ − rv_15_post + rv_15_post ≤ 0 ∧ rv_15_post − rv_15_post ≤ 0 ∧ − rv_15_0 + rv_15_0 ≤ 0 ∧ rv_15_0 − rv_15_0 ≤ 0 ∧ − rt_11_post + rt_11_post ≤ 0 ∧ rt_11_post − rt_11_post ≤ 0 ∧ − rt_11_0 + rt_11_0 ≤ 0 ∧ rt_11_0 − rt_11_0 ≤ 0 ∧ − nd_12_post + nd_12_post ≤ 0 ∧ nd_12_post − nd_12_post ≤ 0 ∧ − nd_12_1 + nd_12_1 ≤ 0 ∧ nd_12_1 − nd_12_1 ≤ 0 ∧ − nd_12_0 + nd_12_0 ≤ 0 ∧ nd_12_0 − nd_12_0 ≤ 0
  4	6	5:		1 + rv_15_0 ≤ 0 ∧ − y_17_post + y_17_post ≤ 0 ∧ y_17_post − y_17_post ≤ 0 ∧ − y_17_0 + y_17_0 ≤ 0 ∧ y_17_0 − y_17_0 ≤ 0 ∧ − x_13_post + x_13_post ≤ 0 ∧ x_13_post − x_13_post ≤ 0 ∧ − x_13_0 + x_13_0 ≤ 0 ∧ x_13_0 − x_13_0 ≤ 0 ∧ − st_16_post + st_16_post ≤ 0 ∧ st_16_post − st_16_post ≤ 0 ∧ − st_16_0 + st_16_0 ≤ 0 ∧ st_16_0 − st_16_0 ≤ 0 ∧ − st_14_0 + st_14_0 ≤ 0 ∧ st_14_0 − st_14_0 ≤ 0 ∧ − rv_15_post + rv_15_post ≤ 0 ∧ rv_15_post − rv_15_post ≤ 0 ∧ − rv_15_0 + rv_15_0 ≤ 0 ∧ rv_15_0 − rv_15_0 ≤ 0 ∧ − rt_11_post + rt_11_post ≤ 0 ∧ rt_11_post − rt_11_post ≤ 0 ∧ − rt_11_0 + rt_11_0 ≤ 0 ∧ rt_11_0 − rt_11_0 ≤ 0 ∧ − nd_12_post + nd_12_post ≤ 0 ∧ nd_12_post − nd_12_post ≤ 0 ∧ − nd_12_1 + nd_12_1 ≤ 0 ∧ nd_12_1 − nd_12_1 ≤ 0 ∧ − nd_12_0 + nd_12_0 ≤ 0 ∧ nd_12_0 − nd_12_0 ≤ 0
  5	7	6:		0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 1 − x_13_0 + x_13_post ≤ 0 ∧ −1 + x_13_0 − x_13_post ≤ 0 ∧ − nd_12_1 + y_17_post ≤ 0 ∧ nd_12_1 − y_17_post ≤ 0 ∧ nd_12_0 − nd_12_post ≤ 0 ∧ − nd_12_0 + nd_12_post ≤ 0 ∧ x_13_0 − x_13_post ≤ 0 ∧ − x_13_0 + x_13_post ≤ 0 ∧ y_17_0 − y_17_post ≤ 0 ∧ − y_17_0 + y_17_post ≤ 0 ∧ − st_16_post + st_16_post ≤ 0 ∧ st_16_post − st_16_post ≤ 0 ∧ − st_16_0 + st_16_0 ≤ 0 ∧ st_16_0 − st_16_0 ≤ 0 ∧ − st_14_0 + st_14_0 ≤ 0 ∧ st_14_0 − st_14_0 ≤ 0 ∧ − rv_15_post + rv_15_post ≤ 0 ∧ rv_15_post − rv_15_post ≤ 0 ∧ − rv_15_0 + rv_15_0 ≤ 0 ∧ rv_15_0 − rv_15_0 ≤ 0 ∧ − rt_11_post + rt_11_post ≤ 0 ∧ rt_11_post − rt_11_post ≤ 0 ∧ − rt_11_0 + rt_11_0 ≤ 0 ∧ rt_11_0 − rt_11_0 ≤ 0
  6	8	1:		− y_17_post + y_17_post ≤ 0 ∧ y_17_post − y_17_post ≤ 0 ∧ − y_17_0 + y_17_0 ≤ 0 ∧ y_17_0 − y_17_0 ≤ 0 ∧ − x_13_post + x_13_post ≤ 0 ∧ x_13_post − x_13_post ≤ 0 ∧ − x_13_0 + x_13_0 ≤ 0 ∧ x_13_0 − x_13_0 ≤ 0 ∧ − st_16_post + st_16_post ≤ 0 ∧ st_16_post − st_16_post ≤ 0 ∧ − st_16_0 + st_16_0 ≤ 0 ∧ st_16_0 − st_16_0 ≤ 0 ∧ − st_14_0 + st_14_0 ≤ 0 ∧ st_14_0 − st_14_0 ≤ 0 ∧ − rv_15_post + rv_15_post ≤ 0 ∧ rv_15_post − rv_15_post ≤ 0 ∧ − rv_15_0 + rv_15_0 ≤ 0 ∧ rv_15_0 − rv_15_0 ≤ 0 ∧ − rt_11_post + rt_11_post ≤ 0 ∧ rt_11_post − rt_11_post ≤ 0 ∧ − rt_11_0 + rt_11_0 ≤ 0 ∧ rt_11_0 − rt_11_0 ≤ 0 ∧ − nd_12_post + nd_12_post ≤ 0 ∧ nd_12_post − nd_12_post ≤ 0 ∧ − nd_12_1 + nd_12_1 ≤ 0 ∧ nd_12_1 − nd_12_1 ≤ 0 ∧ − nd_12_0 + nd_12_0 ≤ 0 ∧ nd_12_0 − nd_12_0 ≤ 0
  7	9	0:		− y_17_post + y_17_post ≤ 0 ∧ y_17_post − y_17_post ≤ 0 ∧ − y_17_0 + y_17_0 ≤ 0 ∧ y_17_0 − y_17_0 ≤ 0 ∧ − x_13_post + x_13_post ≤ 0 ∧ x_13_post − x_13_post ≤ 0 ∧ − x_13_0 + x_13_0 ≤ 0 ∧ x_13_0 − x_13_0 ≤ 0 ∧ − st_16_post + st_16_post ≤ 0 ∧ st_16_post − st_16_post ≤ 0 ∧ − st_16_0 + st_16_0 ≤ 0 ∧ st_16_0 − st_16_0 ≤ 0 ∧ − st_14_0 + st_14_0 ≤ 0 ∧ st_14_0 − st_14_0 ≤ 0 ∧ − rv_15_post + rv_15_post ≤ 0 ∧ rv_15_post − rv_15_post ≤ 0 ∧ − rv_15_0 + rv_15_0 ≤ 0 ∧ rv_15_0 − rv_15_0 ≤ 0 ∧ − rt_11_post + rt_11_post ≤ 0 ∧ rt_11_post − rt_11_post ≤ 0 ∧ − rt_11_0 + rt_11_0 ≤ 0 ∧ rt_11_0 − rt_11_0 ≤ 0 ∧ − nd_12_post + nd_12_post ≤ 0 ∧ nd_12_post − nd_12_post ≤ 0 ∧ − nd_12_1 + nd_12_1 ≤ 0 ∧ nd_12_1 − nd_12_1 ≤ 0 ∧ − nd_12_0 + nd_12_0 ≤ 0 ∧ nd_12_0 − nd_12_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)		x_13_0 ≤ 0
  3		(3)		rv_15_post ≤ 0 ∧ − rv_15_post ≤ 0 ∧ 2 − y_17_post ≤ 0 ∧ 1 − x_13_0 ≤ 0 ∧ 2 − y_17_0 ≤ 0 ∧ rv_15_0 ≤ 0 ∧ − rv_15_0 ≤ 0
  4		(4)		1 − x_13_0 ≤ 0
  5		(5)		1 − x_13_0 ≤ 0
  6		(6)		TRUE
  7		(7)		TRUE

• initial node: 7
• cover edges:
• transition edges:

  0	0 1
  1	1 2
  1	2 3
  1	4 4
  3	3 1
  4	5 5
  4	6 5
  5	7 6
  6	8 1
  7	9 0

2 Switch to Cooperation Termination Proof

3 Transition Removal

We remove transitions 0, 1, 9 using the following ranking functions, which are bounded by −13.

4 Location Addition

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

1* 13 1: − y_17_post + y_17_post ≤ 0 ∧ y_17_post − y_17_post ≤ 0 ∧ − y_17_0 + y_17_0 ≤ 0 ∧ y_17_0 − y_17_0 ≤ 0 ∧ − x_13_post + x_13_post ≤ 0 ∧ x_13_post − x_13_post ≤ 0 ∧ − x_13_0 + x_13_0 ≤ 0 ∧ x_13_0 − x_13_0 ≤ 0 ∧ − st_16_post + st_16_post ≤ 0 ∧ st_16_post − st_16_post ≤ 0 ∧ − st_16_0 + st_16_0 ≤ 0 ∧ st_16_0 − st_16_0 ≤ 0 ∧ − st_14_0 + st_14_0 ≤ 0 ∧ st_14_0 − st_14_0 ≤ 0 ∧ − rv_15_post + rv_15_post ≤ 0 ∧ rv_15_post − rv_15_post ≤ 0 ∧ − rv_15_0 + rv_15_0 ≤ 0 ∧ rv_15_0 − rv_15_0 ≤ 0 ∧ − rt_11_post + rt_11_post ≤ 0 ∧ rt_11_post − rt_11_post ≤ 0 ∧ − rt_11_0 + rt_11_0 ≤ 0 ∧ rt_11_0 − rt_11_0 ≤ 0 ∧ − nd_12_post + nd_12_post ≤ 0 ∧ nd_12_post − nd_12_post ≤ 0 ∧ − nd_12_1 + nd_12_1 ≤ 0 ∧ nd_12_1 − nd_12_1 ≤ 0 ∧ − nd_12_0 + nd_12_0 ≤ 0 ∧ nd_12_0 − nd_12_0 ≤ 0

5 Location Addition

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

1 11 1_var_snapshot: − y_17_post + y_17_post ≤ 0 ∧ y_17_post − y_17_post ≤ 0 ∧ − y_17_0 + y_17_0 ≤ 0 ∧ y_17_0 − y_17_0 ≤ 0 ∧ − x_13_post + x_13_post ≤ 0 ∧ x_13_post − x_13_post ≤ 0 ∧ − x_13_0 + x_13_0 ≤ 0 ∧ x_13_0 − x_13_0 ≤ 0 ∧ − st_16_post + st_16_post ≤ 0 ∧ st_16_post − st_16_post ≤ 0 ∧ − st_16_0 + st_16_0 ≤ 0 ∧ st_16_0 − st_16_0 ≤ 0 ∧ − st_14_0 + st_14_0 ≤ 0 ∧ st_14_0 − st_14_0 ≤ 0 ∧ − rv_15_post + rv_15_post ≤ 0 ∧ rv_15_post − rv_15_post ≤ 0 ∧ − rv_15_0 + rv_15_0 ≤ 0 ∧ rv_15_0 − rv_15_0 ≤ 0 ∧ − rt_11_post + rt_11_post ≤ 0 ∧ rt_11_post − rt_11_post ≤ 0 ∧ − rt_11_0 + rt_11_0 ≤ 0 ∧ rt_11_0 − rt_11_0 ≤ 0 ∧ − nd_12_post + nd_12_post ≤ 0 ∧ nd_12_post − nd_12_post ≤ 0 ∧ − nd_12_1 + nd_12_1 ≤ 0 ∧ nd_12_1 − nd_12_1 ≤ 0 ∧ − nd_12_0 + nd_12_0 ≤ 0 ∧ nd_12_0 − nd_12_0 ≤ 0

6 SCC Decomposition

We consider subproblems for each of the 1 SCC(s) of the program graph.

6.1 SCC Subproblem 1/1

Here we consider the SCC { 1, 3, 4, 5, 6, 1_var_snapshot, 1* }.

6.1.1 Transition Removal

We remove transitions 5, 6 using the following ranking functions, which are bounded by 2.

6.1.2 Transition Removal

We remove transitions 4, 7 using the following ranking functions, which are bounded by −1.

6.1.3 Transition Removal

We remove transition 8 using the following ranking functions, which are bounded by 0.

6.1.4 Transition Removal

We remove transitions 2, 3 using the following ranking functions, which are bounded by 7.

6.1.5 Transition Removal

We remove transitions 11, 13 using the following ranking functions, which are bounded by −2.

6.1.6 Splitting Cut-Point Transitions

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

6.1.6.1 Cut-Point Subproblem 1/1

6.1.6.1.1 Splitting Cut-Point Transitions

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

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