LTS Termination Proof

by T2Cert

Input

Integer Transition System

Initial Location: 10

Transitions: (pre-variables and post-variables)

0	0	1:	− y4_post + y4_post ≤ 0 ∧ y4_post − y4_post ≤ 0 ∧ − y4_0 + y4_0 ≤ 0 ∧ y4_0 − y4_0 ≤ 0 ∧ − x3_post + x3_post ≤ 0 ∧ x3_post − x3_post ≤ 0 ∧ − x3_0 + x3_0 ≤ 0 ∧ x3_0 − x3_0 ≤ 0 ∧ − tmp5_post + tmp5_post ≤ 0 ∧ tmp5_post − tmp5_post ≤ 0 ∧ − tmp5_0 + tmp5_0 ≤ 0 ∧ tmp5_0 − tmp5_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:	− y4_post + y4_post ≤ 0 ∧ y4_post − y4_post ≤ 0 ∧ − y4_0 + y4_0 ≤ 0 ∧ y4_0 − y4_0 ≤ 0 ∧ − x3_post + x3_post ≤ 0 ∧ x3_post − x3_post ≤ 0 ∧ − x3_0 + x3_0 ≤ 0 ∧ x3_0 − x3_0 ≤ 0 ∧ − tmp5_post + tmp5_post ≤ 0 ∧ tmp5_post − tmp5_post ≤ 0 ∧ − tmp5_0 + tmp5_0 ≤ 0 ∧ tmp5_0 − tmp5_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	2	2:	− y4_post + y4_post ≤ 0 ∧ y4_post − y4_post ≤ 0 ∧ − y4_0 + y4_0 ≤ 0 ∧ y4_0 − y4_0 ≤ 0 ∧ − x3_post + x3_post ≤ 0 ∧ x3_post − x3_post ≤ 0 ∧ − x3_0 + x3_0 ≤ 0 ∧ x3_0 − x3_0 ≤ 0 ∧ − tmp5_post + tmp5_post ≤ 0 ∧ tmp5_post − tmp5_post ≤ 0 ∧ − tmp5_0 + tmp5_0 ≤ 0 ∧ tmp5_0 − tmp5_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	2:	− y4_post + y4_post ≤ 0 ∧ y4_post − y4_post ≤ 0 ∧ − y4_0 + y4_0 ≤ 0 ∧ y4_0 − y4_0 ≤ 0 ∧ − x3_post + x3_post ≤ 0 ∧ x3_post − x3_post ≤ 0 ∧ − x3_0 + x3_0 ≤ 0 ∧ x3_0 − x3_0 ≤ 0 ∧ − tmp5_post + tmp5_post ≤ 0 ∧ tmp5_post − tmp5_post ≤ 0 ∧ − tmp5_0 + tmp5_0 ≤ 0 ∧ tmp5_0 − tmp5_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	4	6:	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 ∧ − y4_post + y4_post ≤ 0 ∧ y4_post − y4_post ≤ 0 ∧ − y4_0 + y4_0 ≤ 0 ∧ y4_0 − y4_0 ≤ 0 ∧ − x3_post + x3_post ≤ 0 ∧ x3_post − x3_post ≤ 0 ∧ − x3_0 + x3_0 ≤ 0 ∧ x3_0 − x3_0 ≤ 0 ∧ − tmp5_post + tmp5_post ≤ 0 ∧ tmp5_post − tmp5_post ≤ 0 ∧ − tmp5_0 + tmp5_0 ≤ 0 ∧ tmp5_0 − tmp5_0 ≤ 0 ∧ − i_post + i_post ≤ 0 ∧ i_post − i_post ≤ 0 ∧ − i_0 + i_0 ≤ 0 ∧ i_0 − i_0 ≤ 0
6	5	7:	− y4_post + y4_post ≤ 0 ∧ y4_post − y4_post ≤ 0 ∧ − y4_0 + y4_0 ≤ 0 ∧ y4_0 − y4_0 ≤ 0 ∧ − x3_post + x3_post ≤ 0 ∧ x3_post − x3_post ≤ 0 ∧ − x3_0 + x3_0 ≤ 0 ∧ x3_0 − x3_0 ≤ 0 ∧ − tmp5_post + tmp5_post ≤ 0 ∧ tmp5_post − tmp5_post ≤ 0 ∧ − tmp5_0 + tmp5_0 ≤ 0 ∧ tmp5_0 − tmp5_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	6	5:	0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ − j_0 + x3_post ≤ 0 ∧ j_0 − x3_post ≤ 0 ∧ −1 − j_0 + y4_post ≤ 0 ∧ 1 + j_0 − y4_post ≤ 0 ∧ tmp5_0 − tmp5_post ≤ 0 ∧ − tmp5_0 + tmp5_post ≤ 0 ∧ x3_0 − x3_post ≤ 0 ∧ − x3_0 + x3_post ≤ 0 ∧ y4_0 − y4_post ≤ 0 ∧ − y4_0 + y4_post ≤ 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	7	5:	− y4_post + y4_post ≤ 0 ∧ y4_post − y4_post ≤ 0 ∧ − y4_0 + y4_0 ≤ 0 ∧ y4_0 − y4_0 ≤ 0 ∧ − x3_post + x3_post ≤ 0 ∧ x3_post − x3_post ≤ 0 ∧ − x3_0 + x3_0 ≤ 0 ∧ x3_0 − x3_0 ≤ 0 ∧ − tmp5_post + tmp5_post ≤ 0 ∧ tmp5_post − tmp5_post ≤ 0 ∧ − tmp5_0 + tmp5_0 ≤ 0 ∧ tmp5_0 − tmp5_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	8	0:	0 ≤ 0 ∧ 0 ≤ 0 ∧ i_0 − j_0 ≤ 0 ∧ 1 − i_0 + i_post ≤ 0 ∧ −1 + i_0 − i_post ≤ 0 ∧ i_0 − i_post ≤ 0 ∧ − i_0 + i_post ≤ 0 ∧ − y4_post + y4_post ≤ 0 ∧ y4_post − y4_post ≤ 0 ∧ − y4_0 + y4_0 ≤ 0 ∧ y4_0 − y4_0 ≤ 0 ∧ − x3_post + x3_post ≤ 0 ∧ x3_post − x3_post ≤ 0 ∧ − x3_0 + x3_0 ≤ 0 ∧ x3_0 − x3_0 ≤ 0 ∧ − tmp5_post + tmp5_post ≤ 0 ∧ tmp5_post − tmp5_post ≤ 0 ∧ − tmp5_0 + tmp5_0 ≤ 0 ∧ tmp5_0 − tmp5_0 ≤ 0 ∧ − j_post + j_post ≤ 0 ∧ j_post − j_post ≤ 0 ∧ − j_0 + j_0 ≤ 0 ∧ j_0 − j_0 ≤ 0
7	9	8:	1 − i_0 + j_0 ≤ 0 ∧ − y4_post + y4_post ≤ 0 ∧ y4_post − y4_post ≤ 0 ∧ − y4_0 + y4_0 ≤ 0 ∧ y4_0 − y4_0 ≤ 0 ∧ − x3_post + x3_post ≤ 0 ∧ x3_post − x3_post ≤ 0 ∧ − x3_0 + x3_0 ≤ 0 ∧ x3_0 − x3_0 ≤ 0 ∧ − tmp5_post + tmp5_post ≤ 0 ∧ tmp5_post − tmp5_post ≤ 0 ∧ − tmp5_0 + tmp5_0 ≤ 0 ∧ tmp5_0 − tmp5_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	10	4:	1 + i_0 ≤ 0 ∧ − y4_post + y4_post ≤ 0 ∧ y4_post − y4_post ≤ 0 ∧ − y4_0 + y4_0 ≤ 0 ∧ y4_0 − y4_0 ≤ 0 ∧ − x3_post + x3_post ≤ 0 ∧ x3_post − x3_post ≤ 0 ∧ − x3_0 + x3_0 ≤ 0 ∧ x3_0 − x3_0 ≤ 0 ∧ − tmp5_post + tmp5_post ≤ 0 ∧ tmp5_post − tmp5_post ≤ 0 ∧ − tmp5_0 + tmp5_0 ≤ 0 ∧ tmp5_0 − tmp5_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	6:	− i_0 ≤ 0 ∧ − y4_post + y4_post ≤ 0 ∧ y4_post − y4_post ≤ 0 ∧ − y4_0 + y4_0 ≤ 0 ∧ y4_0 − y4_0 ≤ 0 ∧ − x3_post + x3_post ≤ 0 ∧ x3_post − x3_post ≤ 0 ∧ − x3_0 + x3_0 ≤ 0 ∧ x3_0 − x3_0 ≤ 0 ∧ − tmp5_post + tmp5_post ≤ 0 ∧ tmp5_post − tmp5_post ≤ 0 ∧ − tmp5_0 + tmp5_0 ≤ 0 ∧ tmp5_0 − tmp5_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	12	0:	0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ j_post ≤ 0 ∧ − j_post ≤ 0 ∧ −4 + i_post ≤ 0 ∧ 4 − i_post ≤ 0 ∧ i_0 − i_post ≤ 0 ∧ − i_0 + i_post ≤ 0 ∧ j_0 − j_post ≤ 0 ∧ − j_0 + j_post ≤ 0 ∧ − y4_post + y4_post ≤ 0 ∧ y4_post − y4_post ≤ 0 ∧ − y4_0 + y4_0 ≤ 0 ∧ y4_0 − y4_0 ≤ 0 ∧ − x3_post + x3_post ≤ 0 ∧ x3_post − x3_post ≤ 0 ∧ − x3_0 + x3_0 ≤ 0 ∧ x3_0 − x3_0 ≤ 0 ∧ − tmp5_post + tmp5_post ≤ 0 ∧ tmp5_post − tmp5_post ≤ 0 ∧ − tmp5_0 + tmp5_0 ≤ 0 ∧ tmp5_0 − tmp5_0 ≤ 0
10	13	9:	− y4_post + y4_post ≤ 0 ∧ y4_post − y4_post ≤ 0 ∧ − y4_0 + y4_0 ≤ 0 ∧ y4_0 − y4_0 ≤ 0 ∧ − x3_post + x3_post ≤ 0 ∧ x3_post − x3_post ≤ 0 ∧ − x3_0 + x3_0 ≤ 0 ∧ x3_0 − x3_0 ≤ 0 ∧ − tmp5_post + tmp5_post ≤ 0 ∧ tmp5_post − tmp5_post ≤ 0 ∧ − tmp5_0 + tmp5_0 ≤ 0 ∧ tmp5_0 − tmp5_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:	1 + i_0 ≤ 0
3:	1 + i_0 ≤ 0
4:	1 + i_0 ≤ 0
5:	TRUE
6:	TRUE
7:	TRUE
8:	TRUE
9:	TRUE
10:	TRUE

The invariants are proved as follows.

IMPACT Invariant Proof

nodes (location) invariant:

0	(0)	TRUE
1	(1)	TRUE
2	(2)	1 + i_0 ≤ 0
3	(3)	1 + i_0 ≤ 0
4	(4)	1 + i_0 ≤ 0
5	(5)	TRUE
6	(6)	TRUE
7	(7)	TRUE
8	(8)	TRUE
9	(9)	TRUE
10	(10)	TRUE

initial node: 10
cover edges:
transition edges:

0 0 1

1 10 4

1 11 6

2 1 3

4 2 2

4 3 2

5 4 6

6 5 7

7 8 0

7 9 8

8 6 5

8 7 5

9 12 0

10 13 9

2 Switch to Cooperation Termination Proof

We consider the following cutpoint-transitions:

0	14	0:	− y4_post + y4_post ≤ 0 ∧ y4_post − y4_post ≤ 0 ∧ − y4_0 + y4_0 ≤ 0 ∧ y4_0 − y4_0 ≤ 0 ∧ − x3_post + x3_post ≤ 0 ∧ x3_post − x3_post ≤ 0 ∧ − x3_0 + x3_0 ≤ 0 ∧ x3_0 − x3_0 ≤ 0 ∧ − tmp5_post + tmp5_post ≤ 0 ∧ tmp5_post − tmp5_post ≤ 0 ∧ − tmp5_0 + tmp5_0 ≤ 0 ∧ tmp5_0 − tmp5_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	21	6:	− y4_post + y4_post ≤ 0 ∧ y4_post − y4_post ≤ 0 ∧ − y4_0 + y4_0 ≤ 0 ∧ y4_0 − y4_0 ≤ 0 ∧ − x3_post + x3_post ≤ 0 ∧ x3_post − x3_post ≤ 0 ∧ − x3_0 + x3_0 ≤ 0 ∧ x3_0 − x3_0 ≤ 0 ∧ − tmp5_post + tmp5_post ≤ 0 ∧ tmp5_post − tmp5_post ≤ 0 ∧ − tmp5_0 + tmp5_0 ≤ 0 ∧ tmp5_0 − tmp5_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, 2, 3, 10, 12, 13 using the following ranking functions, which are bounded by −19.

10:	0
9:	0
0:	0
1:	0
5:	0
6:	0
7:	0
8:	0
4:	0
2:	0
3:	0
10:	−7
9:	−8
0:	−9
1:	−9
5:	−9
6:	−9
7:	−9
8:	−9
0_var_snapshot:	−9
0*:	−9
6_var_snapshot:	−9
6*:	−9
4:	−12
2:	−13
3:	−14

Hints:

15	lexWeak[ [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] ]
4	lexWeak[ [0, 0, 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] ]
6	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] ]
7	lexWeak[ [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, 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] ]
11	lexWeak[ [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] ]
2	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] ]
3	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] ]
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, 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, 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.

0* 17 0: − y4_post + y4_post ≤ 0 ∧ y4_post − y4_post ≤ 0 ∧ − y4_0 + y4_0 ≤ 0 ∧ y4_0 − y4_0 ≤ 0 ∧ − x3_post + x3_post ≤ 0 ∧ x3_post − x3_post ≤ 0 ∧ − x3_0 + x3_0 ≤ 0 ∧ x3_0 − x3_0 ≤ 0 ∧ − tmp5_post + tmp5_post ≤ 0 ∧ tmp5_post − tmp5_post ≤ 0 ∧ − tmp5_0 + tmp5_0 ≤ 0 ∧ tmp5_0 − tmp5_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.

0 15 0_var_snapshot: − y4_post + y4_post ≤ 0 ∧ y4_post − y4_post ≤ 0 ∧ − y4_0 + y4_0 ≤ 0 ∧ y4_0 − y4_0 ≤ 0 ∧ − x3_post + x3_post ≤ 0 ∧ x3_post − x3_post ≤ 0 ∧ − x3_0 + x3_0 ≤ 0 ∧ x3_0 − x3_0 ≤ 0 ∧ − tmp5_post + tmp5_post ≤ 0 ∧ tmp5_post − tmp5_post ≤ 0 ∧ − tmp5_0 + tmp5_0 ≤ 0 ∧ tmp5_0 − tmp5_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.

6* 24 6: − y4_post + y4_post ≤ 0 ∧ y4_post − y4_post ≤ 0 ∧ − y4_0 + y4_0 ≤ 0 ∧ y4_0 − y4_0 ≤ 0 ∧ − x3_post + x3_post ≤ 0 ∧ x3_post − x3_post ≤ 0 ∧ − x3_0 + x3_0 ≤ 0 ∧ x3_0 − x3_0 ≤ 0 ∧ − tmp5_post + tmp5_post ≤ 0 ∧ tmp5_post − tmp5_post ≤ 0 ∧ − tmp5_0 + tmp5_0 ≤ 0 ∧ tmp5_0 − tmp5_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.

6 22 6_var_snapshot: − y4_post + y4_post ≤ 0 ∧ y4_post − y4_post ≤ 0 ∧ − y4_0 + y4_0 ≤ 0 ∧ y4_0 − y4_0 ≤ 0 ∧ − x3_post + x3_post ≤ 0 ∧ x3_post − x3_post ≤ 0 ∧ − x3_0 + x3_0 ≤ 0 ∧ x3_0 − x3_0 ≤ 0 ∧ − tmp5_post + tmp5_post ≤ 0 ∧ tmp5_post − tmp5_post ≤ 0 ∧ − tmp5_0 + tmp5_0 ≤ 0 ∧ tmp5_0 − tmp5_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 1 SCC(s) of the program graph.

8.1 SCC Subproblem 1/1

Here we consider the SCC { 0, 1, 5, 6, 7, 8, 0_var_snapshot, 0*, 6_var_snapshot, 6* }.

8.1.1 Transition Removal

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

0:	2 + 5⋅i_0
1:	5⋅i_0
5:	−1 + 5⋅i_0
6:	−1 + 5⋅i_0
7:	−1 + 5⋅i_0
8:	−1 + 5⋅i_0
0_var_snapshot:	1 + 5⋅i_0
0*:	3 + 5⋅i_0
6_var_snapshot:	−1 + 5⋅i_0
6*:	−1 + 5⋅i_0

Hints:

15	lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 5, 0] ]
17	lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 5, 0] ]
22	lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 5, 0] ]
24	lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 5, 0] ]
0	lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 5, 0] ]
4	lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 5, 0] ]
5	lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 5, 0] ]
6	lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 5, 0] ]
7	lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 5, 0] ]
8	lexWeak[ [0, 0, 0, 5, 0, 5, 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, 5, 0] ]
11	lexStrict[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 5, 0] , [5, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]

8.1.2 Transition Removal

We remove transition 9 using the following ranking functions, which are bounded by 10.

0:	9 + i_0 + 10⋅i_post − 11⋅j_0
1:	8 + i_0 + 10⋅i_post − 11⋅j_0
5:	−8 + 11⋅i_0 − 11⋅j_0
6:	1 + 11⋅i_0 − 11⋅j_0
7:	11⋅i_0 − 11⋅j_0
8:	−1 + 11⋅i_0 − 11⋅j_0
0_var_snapshot:	8 + i_0 + 10⋅i_post − 11⋅j_0
0*:	10 + i_0 + 10⋅i_post − 11⋅j_0
6_var_snapshot:	11⋅i_0 − 11⋅j_0
6*:	2 + 11⋅i_0 − 11⋅j_0

Hints:

15	lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 11, 10, 0, 1, 0] ]
17	lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 11, 10, 0, 1, 0] ]
22	lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 11, 0, 0, 11, 0] ]
24	lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 11, 0, 0, 11, 0] ]
0	lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 11, 10, 0, 1, 0] ]
4	lexWeak[ [0, 0, 0, 11, 0, 11, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 11, 0] ]
5	lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 11, 0, 0, 11, 0] ]
6	lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 11, 0, 0, 11, 0] ]
7	lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 11, 0, 0, 11, 0] ]
8	lexWeak[ [0, 0, 0, 11, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 11] ]
9	lexStrict[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 11, 0, 0, 11, 0] , [11, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]

8.1.3 Transition Removal

We remove transitions 15, 17, 22, 24, 0, 4, 5, 6, 7, 8 using the following ranking functions, which are bounded by −4.

0:	−2
1:	−4
5:	4
6:	2
7:	0
8:	5
0_var_snapshot:	−3
0*:	−1
6_var_snapshot:	1
6*:	3

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] ]
17	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] ]
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, 0, 0, 0, 0] ]
5	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] ]
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, 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, 0] , [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, 0, 0, 0, 0] ]

8.1.4 Splitting Cut-Point Transitions

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

8.1.4.1 Cut-Point Subproblem 1/2

Here we consider cut-point transition 14.

8.1.4.1.1 Splitting Cut-Point Transitions

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

8.1.4.2 Cut-Point Subproblem 2/2

Here we consider cut-point transition 21.

8.1.4.2.1 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	10 4
1	11 6
2	1 3
4	2 2
4	3 2
5	4 6
6	5 7
7	8 0
7	9 8
8	6 5
8	7 5
9	12 0
10	13 9