LTS Termination Proof

by T2Cert

Input

Integer Transition System

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:	− 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	6	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	7	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	8	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.

0:	TRUE
1:	TRUE
2:	x_13_0 ≤ 0
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:	1 − x_13_0 ≤ 0
5:	1 − x_13_0 ≤ 0
6:	TRUE
7:	TRUE

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

5 6 6

6 7 1

7 8 0

2 Switch to Cooperation Termination Proof

We consider the following cutpoint-transitions:

− 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

and for every transition t, a duplicate t is considered.

3 Transition Removal

We remove transitions 0, 1, 8 using the following ranking functions, which are bounded by −13.

7:	0
0:	0
1:	0
3:	0
4:	0
5:	0
6:	0
2:	0
7:	−5
0:	−6
1:	−7
3:	−7
4:	−7
5:	−7
6:	−7
1_var_snapshot:	−7
1*:	−7
2:	−11

4 Location Addition

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

1* 12 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 10 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 0.

1:	−1 + 3⋅x_13_0
3:	−1 + 3⋅x_13_0
4:	−1 + 3⋅x_13_0
5:	−2 + 3⋅x_13_0
6:	3⋅x_13_0
1_var_snapshot:	−1 + 3⋅x_13_0
1*:	−1 + 3⋅x_13_0

6.1.2 Transition Removal

We remove transitions 4, 7 using the following ranking functions, which are bounded by −2.

1:	−1
3:	−1
4:	−2
5:	0
6:	0
1_var_snapshot:	−1
1*:	−1

6.1.3 Transition Removal

We remove transitions 2, 3 using the following ranking functions, which are bounded by 8.

1:	−1 + 4⋅y_17_0
3:	1 + 4⋅y_17_0
4:	0
5:	0
6:	0
1_var_snapshot:	−2 + 4⋅y_17_0
1*:	4⋅y_17_0

6.1.4 Transition Removal

We remove transitions 10, 12 using the following ranking functions, which are bounded by −2.

1:	−1
3:	0
4:	0
5:	0
6:	0
1_var_snapshot:	−2
1*:	0

6.1.5 Splitting Cut-Point Transitions

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

6.1.5.1 Cut-Point Subproblem 1/1

Here we consider cut-point transition 9.

6.1.5.1.1 Splitting Cut-Point Transitions

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

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