LTS Termination Proof

by T2Cert

Input

Integer Transition System

Initial Location: 12

Transitions: (pre-variables and post-variables)

0	0	1:	− y8_post + y8_post ≤ 0 ∧ y8_post − y8_post ≤ 0 ∧ − y8_0 + y8_0 ≤ 0 ∧ y8_0 − y8_0 ≤ 0 ∧ − y6_post + y6_post ≤ 0 ∧ y6_post − y6_post ≤ 0 ∧ − y6_0 + y6_0 ≤ 0 ∧ y6_0 − y6_0 ≤ 0 ∧ − y4_post + y4_post ≤ 0 ∧ y4_post − y4_post ≤ 0 ∧ − y4_0 + y4_0 ≤ 0 ∧ y4_0 − y4_0 ≤ 0 ∧ − j_post + j_post ≤ 0 ∧ j_post − j_post ≤ 0 ∧ − j_0 + j_0 ≤ 0 ∧ j_0 − j_0 ≤ 0 ∧ − i_post + i_post ≤ 0 ∧ i_post − i_post ≤ 0 ∧ − i_0 + i_0 ≤ 0 ∧ i_0 − i_0 ≤ 0
2	1	3:	0 ≤ 0 ∧ 0 ≤ 0 ∧ 100 − i_0 ≤ 0 ∧ −100 + j_post ≤ 0 ∧ 100 − j_post ≤ 0 ∧ j_0 − j_post ≤ 0 ∧ − j_0 + j_post ≤ 0 ∧ − y8_post + y8_post ≤ 0 ∧ y8_post − y8_post ≤ 0 ∧ − y8_0 + y8_0 ≤ 0 ∧ y8_0 − y8_0 ≤ 0 ∧ − y6_post + y6_post ≤ 0 ∧ y6_post − y6_post ≤ 0 ∧ − y6_0 + y6_0 ≤ 0 ∧ y6_0 − y6_0 ≤ 0 ∧ − y4_post + y4_post ≤ 0 ∧ y4_post − y4_post ≤ 0 ∧ − y4_0 + y4_0 ≤ 0 ∧ y4_0 − y4_0 ≤ 0 ∧ − i_post + i_post ≤ 0 ∧ i_post − i_post ≤ 0 ∧ − i_0 + i_0 ≤ 0 ∧ i_0 − i_0 ≤ 0
2	2	0:	0 ≤ 0 ∧ 0 ≤ 0 ∧ −99 + i_0 ≤ 0 ∧ − i_0 + y4_post ≤ 0 ∧ i_0 − y4_post ≤ 0 ∧ y4_0 − y4_post ≤ 0 ∧ − y4_0 + y4_post ≤ 0 ∧ − y8_post + y8_post ≤ 0 ∧ y8_post − y8_post ≤ 0 ∧ − y8_0 + y8_0 ≤ 0 ∧ y8_0 − y8_0 ≤ 0 ∧ − y6_post + y6_post ≤ 0 ∧ y6_post − y6_post ≤ 0 ∧ − y6_0 + y6_0 ≤ 0 ∧ y6_0 − y6_0 ≤ 0 ∧ − j_post + j_post ≤ 0 ∧ j_post − j_post ≤ 0 ∧ − j_0 + j_0 ≤ 0 ∧ j_0 − j_0 ≤ 0 ∧ − i_post + i_post ≤ 0 ∧ i_post − i_post ≤ 0 ∧ − i_0 + i_0 ≤ 0 ∧ i_0 − i_0 ≤ 0
4	3	3:	0 ≤ 0 ∧ 0 ≤ 0 ∧ −1 − j_0 + j_post ≤ 0 ∧ 1 + j_0 − j_post ≤ 0 ∧ j_0 − j_post ≤ 0 ∧ − j_0 + j_post ≤ 0 ∧ − y8_post + y8_post ≤ 0 ∧ y8_post − y8_post ≤ 0 ∧ − y8_0 + y8_0 ≤ 0 ∧ y8_0 − y8_0 ≤ 0 ∧ − y6_post + y6_post ≤ 0 ∧ y6_post − y6_post ≤ 0 ∧ − y6_0 + y6_0 ≤ 0 ∧ y6_0 − y6_0 ≤ 0 ∧ − y4_post + y4_post ≤ 0 ∧ y4_post − y4_post ≤ 0 ∧ − y4_0 + y4_0 ≤ 0 ∧ y4_0 − y4_0 ≤ 0 ∧ − i_post + i_post ≤ 0 ∧ i_post − i_post ≤ 0 ∧ − i_0 + i_0 ≤ 0 ∧ i_0 − i_0 ≤ 0
5	4	2:	− y8_post + y8_post ≤ 0 ∧ y8_post − y8_post ≤ 0 ∧ − y8_0 + y8_0 ≤ 0 ∧ y8_0 − y8_0 ≤ 0 ∧ − y6_post + y6_post ≤ 0 ∧ y6_post − y6_post ≤ 0 ∧ − y6_0 + y6_0 ≤ 0 ∧ y6_0 − y6_0 ≤ 0 ∧ − y4_post + y4_post ≤ 0 ∧ y4_post − y4_post ≤ 0 ∧ − y4_0 + y4_0 ≤ 0 ∧ y4_0 − y4_0 ≤ 0 ∧ − j_post + j_post ≤ 0 ∧ j_post − j_post ≤ 0 ∧ − j_0 + j_0 ≤ 0 ∧ j_0 − j_0 ≤ 0 ∧ − i_post + i_post ≤ 0 ∧ i_post − i_post ≤ 0 ∧ − i_0 + i_0 ≤ 0 ∧ i_0 − i_0 ≤ 0
6	5	4:	− y8_post + y8_post ≤ 0 ∧ y8_post − y8_post ≤ 0 ∧ − y8_0 + y8_0 ≤ 0 ∧ y8_0 − y8_0 ≤ 0 ∧ − y6_post + y6_post ≤ 0 ∧ y6_post − y6_post ≤ 0 ∧ − y6_0 + y6_0 ≤ 0 ∧ y6_0 − y6_0 ≤ 0 ∧ − y4_post + y4_post ≤ 0 ∧ y4_post − y4_post ≤ 0 ∧ − y4_0 + y4_0 ≤ 0 ∧ y4_0 − y4_0 ≤ 0 ∧ − j_post + j_post ≤ 0 ∧ j_post − j_post ≤ 0 ∧ − j_0 + j_0 ≤ 0 ∧ j_0 − j_0 ≤ 0 ∧ − i_post + i_post ≤ 0 ∧ i_post − i_post ≤ 0 ∧ − i_0 + i_0 ≤ 0 ∧ i_0 − i_0 ≤ 0
7	6	8:	200 − j_0 ≤ 0 ∧ − y8_post + y8_post ≤ 0 ∧ y8_post − y8_post ≤ 0 ∧ − y8_0 + y8_0 ≤ 0 ∧ y8_0 − y8_0 ≤ 0 ∧ − y6_post + y6_post ≤ 0 ∧ y6_post − y6_post ≤ 0 ∧ − y6_0 + y6_0 ≤ 0 ∧ y6_0 − y6_0 ≤ 0 ∧ − y4_post + y4_post ≤ 0 ∧ y4_post − y4_post ≤ 0 ∧ − y4_0 + y4_0 ≤ 0 ∧ y4_0 − y4_0 ≤ 0 ∧ − j_post + j_post ≤ 0 ∧ j_post − j_post ≤ 0 ∧ − j_0 + j_0 ≤ 0 ∧ j_0 − j_0 ≤ 0 ∧ − i_post + i_post ≤ 0 ∧ i_post − i_post ≤ 0 ∧ − i_0 + i_0 ≤ 0 ∧ i_0 − i_0 ≤ 0
7	7	6:	0 ≤ 0 ∧ 0 ≤ 0 ∧ −199 + j_0 ≤ 0 ∧ − j_0 + y8_post ≤ 0 ∧ j_0 − y8_post ≤ 0 ∧ y8_0 − y8_post ≤ 0 ∧ − y8_0 + y8_post ≤ 0 ∧ − y6_post + y6_post ≤ 0 ∧ y6_post − y6_post ≤ 0 ∧ − y6_0 + y6_0 ≤ 0 ∧ y6_0 − y6_0 ≤ 0 ∧ − y4_post + y4_post ≤ 0 ∧ y4_post − y4_post ≤ 0 ∧ − y4_0 + y4_0 ≤ 0 ∧ y4_0 − y4_0 ≤ 0 ∧ − j_post + j_post ≤ 0 ∧ j_post − j_post ≤ 0 ∧ − j_0 + j_0 ≤ 0 ∧ j_0 − j_0 ≤ 0 ∧ − i_post + i_post ≤ 0 ∧ i_post − i_post ≤ 0 ∧ − i_0 + i_0 ≤ 0 ∧ i_0 − i_0 ≤ 0
3	8	7:	− y8_post + y8_post ≤ 0 ∧ y8_post − y8_post ≤ 0 ∧ − y8_0 + y8_0 ≤ 0 ∧ y8_0 − y8_0 ≤ 0 ∧ − y6_post + y6_post ≤ 0 ∧ y6_post − y6_post ≤ 0 ∧ − y6_0 + y6_0 ≤ 0 ∧ y6_0 − y6_0 ≤ 0 ∧ − y4_post + y4_post ≤ 0 ∧ y4_post − y4_post ≤ 0 ∧ − y4_0 + y4_0 ≤ 0 ∧ y4_0 − y4_0 ≤ 0 ∧ − j_post + j_post ≤ 0 ∧ j_post − j_post ≤ 0 ∧ − j_0 + j_0 ≤ 0 ∧ j_0 − j_0 ≤ 0 ∧ − i_post + i_post ≤ 0 ∧ i_post − i_post ≤ 0 ∧ − i_0 + i_0 ≤ 0 ∧ i_0 − i_0 ≤ 0
9	9	5:	0 ≤ 0 ∧ 0 ≤ 0 ∧ −1 − i_0 + i_post ≤ 0 ∧ 1 + i_0 − i_post ≤ 0 ∧ i_0 − i_post ≤ 0 ∧ − i_0 + i_post ≤ 0 ∧ − y8_post + y8_post ≤ 0 ∧ y8_post − y8_post ≤ 0 ∧ − y8_0 + y8_0 ≤ 0 ∧ y8_0 − y8_0 ≤ 0 ∧ − y6_post + y6_post ≤ 0 ∧ y6_post − y6_post ≤ 0 ∧ − y6_0 + y6_0 ≤ 0 ∧ y6_0 − y6_0 ≤ 0 ∧ − y4_post + y4_post ≤ 0 ∧ y4_post − y4_post ≤ 0 ∧ − y4_0 + y4_0 ≤ 0 ∧ y4_0 − y4_0 ≤ 0 ∧ − j_post + j_post ≤ 0 ∧ j_post − j_post ≤ 0 ∧ − j_0 + j_0 ≤ 0 ∧ j_0 − j_0 ≤ 0
10	10	9:	− y8_post + y8_post ≤ 0 ∧ y8_post − y8_post ≤ 0 ∧ − y8_0 + y8_0 ≤ 0 ∧ y8_0 − y8_0 ≤ 0 ∧ − y6_post + y6_post ≤ 0 ∧ y6_post − y6_post ≤ 0 ∧ − y6_0 + y6_0 ≤ 0 ∧ y6_0 − y6_0 ≤ 0 ∧ − y4_post + y4_post ≤ 0 ∧ y4_post − y4_post ≤ 0 ∧ − y4_0 + y4_0 ≤ 0 ∧ y4_0 − y4_0 ≤ 0 ∧ − j_post + j_post ≤ 0 ∧ j_post − j_post ≤ 0 ∧ − j_0 + j_0 ≤ 0 ∧ j_0 − j_0 ≤ 0 ∧ − i_post + i_post ≤ 0 ∧ i_post − i_post ≤ 0 ∧ − i_0 + i_0 ≤ 0 ∧ i_0 − i_0 ≤ 0
1	11	10:	0 ≤ 0 ∧ 0 ≤ 0 ∧ − i_0 + y6_post ≤ 0 ∧ i_0 − y6_post ≤ 0 ∧ y6_0 − y6_post ≤ 0 ∧ − y6_0 + y6_post ≤ 0 ∧ − y8_post + y8_post ≤ 0 ∧ y8_post − y8_post ≤ 0 ∧ − y8_0 + y8_0 ≤ 0 ∧ y8_0 − y8_0 ≤ 0 ∧ − y4_post + y4_post ≤ 0 ∧ y4_post − y4_post ≤ 0 ∧ − y4_0 + y4_0 ≤ 0 ∧ y4_0 − y4_0 ≤ 0 ∧ − j_post + j_post ≤ 0 ∧ j_post − j_post ≤ 0 ∧ − j_0 + j_0 ≤ 0 ∧ j_0 − j_0 ≤ 0 ∧ − i_post + i_post ≤ 0 ∧ i_post − i_post ≤ 0 ∧ − i_0 + i_0 ≤ 0 ∧ i_0 − i_0 ≤ 0
11	12	5:	0 ≤ 0 ∧ 0 ≤ 0 ∧ i_post ≤ 0 ∧ − i_post ≤ 0 ∧ i_0 − i_post ≤ 0 ∧ − i_0 + i_post ≤ 0 ∧ − y8_post + y8_post ≤ 0 ∧ y8_post − y8_post ≤ 0 ∧ − y8_0 + y8_0 ≤ 0 ∧ y8_0 − y8_0 ≤ 0 ∧ − y6_post + y6_post ≤ 0 ∧ y6_post − y6_post ≤ 0 ∧ − y6_0 + y6_0 ≤ 0 ∧ y6_0 − y6_0 ≤ 0 ∧ − y4_post + y4_post ≤ 0 ∧ y4_post − y4_post ≤ 0 ∧ − y4_0 + y4_0 ≤ 0 ∧ y4_0 − y4_0 ≤ 0 ∧ − j_post + j_post ≤ 0 ∧ j_post − j_post ≤ 0 ∧ − j_0 + j_0 ≤ 0 ∧ j_0 − j_0 ≤ 0
12	13	11:	− y8_post + y8_post ≤ 0 ∧ y8_post − y8_post ≤ 0 ∧ − y8_0 + y8_0 ≤ 0 ∧ y8_0 − y8_0 ≤ 0 ∧ − y6_post + y6_post ≤ 0 ∧ y6_post − y6_post ≤ 0 ∧ − y6_0 + y6_0 ≤ 0 ∧ y6_0 − y6_0 ≤ 0 ∧ − y4_post + y4_post ≤ 0 ∧ y4_post − y4_post ≤ 0 ∧ − y4_0 + y4_0 ≤ 0 ∧ y4_0 − y4_0 ≤ 0 ∧ − j_post + j_post ≤ 0 ∧ j_post − j_post ≤ 0 ∧ − j_0 + j_0 ≤ 0 ∧ j_0 − j_0 ≤ 0 ∧ − i_post + i_post ≤ 0 ∧ i_post − i_post ≤ 0 ∧ − i_0 + i_0 ≤ 0 ∧ i_0 − i_0 ≤ 0

Proof

1 Invariant Updates

The following invariants are asserted.

0:	TRUE
1:	TRUE
2:	TRUE
3:	100 − i_0 ≤ 0
4:	100 − i_0 ≤ 0
5:	TRUE
6:	100 − i_0 ≤ 0
7:	100 − i_0 ≤ 0
8:	100 − i_0 ≤ 0 ∧ 200 − j_0 ≤ 0
9:	TRUE
10:	TRUE
11:	TRUE
12:	TRUE

The invariants are proved as follows.

IMPACT Invariant Proof

nodes (location) invariant:

0	(0)	TRUE
1	(1)	TRUE
2	(2)	TRUE
3	(3)	100 − i_0 ≤ 0
4	(4)	100 − i_0 ≤ 0
5	(5)	TRUE
6	(6)	100 − i_0 ≤ 0
7	(7)	100 − i_0 ≤ 0
8	(8)	100 − i_0 ≤ 0 ∧ 200 − j_0 ≤ 0
9	(9)	TRUE
10	(10)	TRUE
11	(11)	TRUE
12	(12)	TRUE

initial node: 12
cover edges:
transition edges:

0 0 1

1 11 10

2 1 3

2 2 0

3 8 7

4 3 3

5 4 2

6 5 4

7 6 8

7 7 6

9 9 5

10 10 9

11 12 5

12 13 11

2 Switch to Cooperation Termination Proof

We consider the following cutpoint-transitions:

3	14	3:	− y8_post + y8_post ≤ 0 ∧ y8_post − y8_post ≤ 0 ∧ − y8_0 + y8_0 ≤ 0 ∧ y8_0 − y8_0 ≤ 0 ∧ − y6_post + y6_post ≤ 0 ∧ y6_post − y6_post ≤ 0 ∧ − y6_0 + y6_0 ≤ 0 ∧ y6_0 − y6_0 ≤ 0 ∧ − y4_post + y4_post ≤ 0 ∧ y4_post − y4_post ≤ 0 ∧ − y4_0 + y4_0 ≤ 0 ∧ y4_0 − y4_0 ≤ 0 ∧ − j_post + j_post ≤ 0 ∧ j_post − j_post ≤ 0 ∧ − j_0 + j_0 ≤ 0 ∧ j_0 − j_0 ≤ 0 ∧ − i_post + i_post ≤ 0 ∧ i_post − i_post ≤ 0 ∧ − i_0 + i_0 ≤ 0 ∧ i_0 − i_0 ≤ 0
5	21	5:	− y8_post + y8_post ≤ 0 ∧ y8_post − y8_post ≤ 0 ∧ − y8_0 + y8_0 ≤ 0 ∧ y8_0 − y8_0 ≤ 0 ∧ − y6_post + y6_post ≤ 0 ∧ y6_post − y6_post ≤ 0 ∧ − y6_0 + y6_0 ≤ 0 ∧ y6_0 − y6_0 ≤ 0 ∧ − y4_post + y4_post ≤ 0 ∧ y4_post − y4_post ≤ 0 ∧ − y4_0 + y4_0 ≤ 0 ∧ y4_0 − y4_0 ≤ 0 ∧ − j_post + j_post ≤ 0 ∧ j_post − j_post ≤ 0 ∧ − j_0 + j_0 ≤ 0 ∧ j_0 − j_0 ≤ 0 ∧ − i_post + i_post ≤ 0 ∧ i_post − i_post ≤ 0 ∧ − i_0 + i_0 ≤ 0 ∧ i_0 − i_0 ≤ 0

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

3 Transition Removal

We remove transitions 1, 6, 12, 13 using the following ranking functions, which are bounded by −17.

12:	0
11:	0
0:	0
1:	0
2:	0
5:	0
9:	0
10:	0
3:	0
4:	0
6:	0
7:	0
8:	0
12:	−6
11:	−7
0:	−8
1:	−8
2:	−8
5:	−8
9:	−8
10:	−8
5_var_snapshot:	−8
5*:	−8
3:	−9
4:	−9
6:	−9
7:	−9
3_var_snapshot:	−9
3*:	−9
8:	−10

Hints:

15	lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]
22	lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]
0	lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]
2	lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]
3	lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]
4	lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]
5	lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]
7	lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]
8	lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]
9	lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]
10	lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]
11	lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]
1	lexStrict[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] , [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]
6	lexStrict[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] , [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]
12	lexStrict[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] , [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]
13	lexStrict[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] , [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]

4 Location Addition

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

3* 17 3: − y8_post + y8_post ≤ 0 ∧ y8_post − y8_post ≤ 0 ∧ − y8_0 + y8_0 ≤ 0 ∧ y8_0 − y8_0 ≤ 0 ∧ − y6_post + y6_post ≤ 0 ∧ y6_post − y6_post ≤ 0 ∧ − y6_0 + y6_0 ≤ 0 ∧ y6_0 − y6_0 ≤ 0 ∧ − y4_post + y4_post ≤ 0 ∧ y4_post − y4_post ≤ 0 ∧ − y4_0 + y4_0 ≤ 0 ∧ y4_0 − y4_0 ≤ 0 ∧ − j_post + j_post ≤ 0 ∧ j_post − j_post ≤ 0 ∧ − j_0 + j_0 ≤ 0 ∧ j_0 − j_0 ≤ 0 ∧ − i_post + i_post ≤ 0 ∧ i_post − i_post ≤ 0 ∧ − i_0 + i_0 ≤ 0 ∧ i_0 − i_0 ≤ 0

5 Location Addition

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

3 15 3_var_snapshot: − y8_post + y8_post ≤ 0 ∧ y8_post − y8_post ≤ 0 ∧ − y8_0 + y8_0 ≤ 0 ∧ y8_0 − y8_0 ≤ 0 ∧ − y6_post + y6_post ≤ 0 ∧ y6_post − y6_post ≤ 0 ∧ − y6_0 + y6_0 ≤ 0 ∧ y6_0 − y6_0 ≤ 0 ∧ − y4_post + y4_post ≤ 0 ∧ y4_post − y4_post ≤ 0 ∧ − y4_0 + y4_0 ≤ 0 ∧ y4_0 − y4_0 ≤ 0 ∧ − j_post + j_post ≤ 0 ∧ j_post − j_post ≤ 0 ∧ − j_0 + j_0 ≤ 0 ∧ j_0 − j_0 ≤ 0 ∧ − i_post + i_post ≤ 0 ∧ i_post − i_post ≤ 0 ∧ − i_0 + i_0 ≤ 0 ∧ i_0 − i_0 ≤ 0

6 Location Addition

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

5* 24 5: − y8_post + y8_post ≤ 0 ∧ y8_post − y8_post ≤ 0 ∧ − y8_0 + y8_0 ≤ 0 ∧ y8_0 − y8_0 ≤ 0 ∧ − y6_post + y6_post ≤ 0 ∧ y6_post − y6_post ≤ 0 ∧ − y6_0 + y6_0 ≤ 0 ∧ y6_0 − y6_0 ≤ 0 ∧ − y4_post + y4_post ≤ 0 ∧ y4_post − y4_post ≤ 0 ∧ − y4_0 + y4_0 ≤ 0 ∧ y4_0 − y4_0 ≤ 0 ∧ − j_post + j_post ≤ 0 ∧ j_post − j_post ≤ 0 ∧ − j_0 + j_0 ≤ 0 ∧ j_0 − j_0 ≤ 0 ∧ − i_post + i_post ≤ 0 ∧ i_post − i_post ≤ 0 ∧ − i_0 + i_0 ≤ 0 ∧ i_0 − i_0 ≤ 0

7 Location Addition

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

5 22 5_var_snapshot: − y8_post + y8_post ≤ 0 ∧ y8_post − y8_post ≤ 0 ∧ − y8_0 + y8_0 ≤ 0 ∧ y8_0 − y8_0 ≤ 0 ∧ − y6_post + y6_post ≤ 0 ∧ y6_post − y6_post ≤ 0 ∧ − y6_0 + y6_0 ≤ 0 ∧ y6_0 − y6_0 ≤ 0 ∧ − y4_post + y4_post ≤ 0 ∧ y4_post − y4_post ≤ 0 ∧ − y4_0 + y4_0 ≤ 0 ∧ y4_0 − y4_0 ≤ 0 ∧ − j_post + j_post ≤ 0 ∧ j_post − j_post ≤ 0 ∧ − j_0 + j_0 ≤ 0 ∧ j_0 − j_0 ≤ 0 ∧ − i_post + i_post ≤ 0 ∧ i_post − i_post ≤ 0 ∧ − i_0 + i_0 ≤ 0 ∧ i_0 − i_0 ≤ 0

8 SCC Decomposition

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

8.1 SCC Subproblem 1/2

Here we consider the SCC { 3, 4, 6, 7, 3_var_snapshot, 3* }.

8.1.1 Transition Removal

We remove transition 7 using the following ranking functions, which are bounded by −1196.

3:	1 − 6⋅j_0
4:	−3 − 6⋅j_0
6:	−2 − 6⋅j_0
7:	−1 − 6⋅j_0
3_var_snapshot:	−6⋅j_0
3*:	2 − 6⋅j_0

Hints:

15	lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0] ]
17	lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0] ]
3	lexWeak[ [0, 0, 0, 0, 6, 0, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]
5	lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0] ]
7	lexStrict[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0] , [0, 0, 0, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]
8	lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0] ]

8.1.2 Transition Removal

We remove transitions 15, 17, 3, 5, 8 using the following ranking functions, which are bounded by −2.

3:	0
4:	100 + i_0
6:	3⋅i_0
7:	−2
3_var_snapshot:	−1
3*:	i_0

Hints:

15	lexStrict[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] , [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]
17	lexStrict[ [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] , [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]
3	lexStrict[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0] , [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]
5	lexStrict[ [2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0] , [3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]
8	lexStrict[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] , [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]

8.1.3 Splitting Cut-Point Transitions

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

8.1.3.1 Cut-Point Subproblem 1/1

Here we consider cut-point transition 14.

8.1.3.1.1 Splitting Cut-Point Transitions

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

8.2 SCC Subproblem 2/2

Here we consider the SCC { 0, 1, 2, 5, 9, 10, 5_var_snapshot, 5* }.

8.2.1 Transition Removal

We remove transition 2 using the following ranking functions, which are bounded by −791.

0:	1 − 8⋅i_0
1:	−8⋅i_0
2:	2 − 8⋅i_0
5:	4 − 8⋅i_0
9:	−2 − 8⋅i_0
10:	−1 − 8⋅i_0
5_var_snapshot:	3 − 8⋅i_0
5*:	5 − 8⋅i_0

Hints:

22	lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 8] ]
24	lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 8] ]
0	lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 8] ]
2	lexStrict[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 8] , [0, 0, 8, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]
4	lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 8] ]
9	lexWeak[ [0, 0, 0, 8, 0, 8, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]
10	lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 8] ]
11	lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 8] ]

8.2.2 Transition Removal

We remove transitions 22, 24, 0, 4, 9, 10, 11 using the following ranking functions, which are bounded by −5.

0:	2
1:	1
2:	−5
5:	−3
9:	−1
10:	0
5_var_snapshot:	−4
5*:	−2

Hints:

22	lexStrict[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] , [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]
24	lexStrict[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] , [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]
0	lexStrict[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] , [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]
4	lexStrict[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] , [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]
9	lexStrict[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] , [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]
10	lexStrict[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] , [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]
11	lexStrict[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] , [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]

8.2.3 Splitting Cut-Point Transitions

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

8.2.3.1 Cut-Point Subproblem 1/1

Here we consider cut-point transition 21.

8.2.3.1.1 Splitting Cut-Point Transitions

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

Tool configuration

T2Cert

version: 1.0

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