LTS Termination Proof

by T2Cert

Input

Integer Transition System

Initial Location: 6

Transitions: (pre-variables and post-variables)

0	0	1:	0 ≤ 0 ∧ 0 ≤ 0 ∧ ___const_100_0 − i_0 ≤ 0 ∧ − ___const_100_0 + j_post ≤ 0 ∧ ___const_100_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 ∧ − ___const_200_0 + ___const_200_0 ≤ 0 ∧ ___const_200_0 − ___const_200_0 ≤ 0 ∧ − ___const_100_0 + ___const_100_0 ≤ 0 ∧ ___const_100_0 − ___const_100_0 ≤ 0
0	1	2:	0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 1 − ___const_100_0 + i_0 ≤ 0 ∧ − i_0 + y4_post ≤ 0 ∧ i_0 − y4_post ≤ 0 ∧ − i_0 + y6_post ≤ 0 ∧ i_0 − y6_post ≤ 0 ∧ −1 − i_0 + i_post ≤ 0 ∧ 1 + i_0 − i_post ≤ 0 ∧ i_0 − i_post ≤ 0 ∧ − i_0 + i_post ≤ 0 ∧ y4_0 − y4_post ≤ 0 ∧ − y4_0 + y4_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 ∧ − j_post + j_post ≤ 0 ∧ j_post − j_post ≤ 0 ∧ − j_0 + j_0 ≤ 0 ∧ j_0 − j_0 ≤ 0 ∧ − ___const_200_0 + ___const_200_0 ≤ 0 ∧ ___const_200_0 − ___const_200_0 ≤ 0 ∧ − ___const_100_0 + ___const_100_0 ≤ 0 ∧ ___const_100_0 − ___const_100_0 ≤ 0
2	2	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 ∧ − ___const_200_0 + ___const_200_0 ≤ 0 ∧ ___const_200_0 − ___const_200_0 ≤ 0 ∧ − ___const_100_0 + ___const_100_0 ≤ 0 ∧ ___const_100_0 − ___const_100_0 ≤ 0
3	3	4:	___const_200_0 − 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 ∧ − ___const_200_0 + ___const_200_0 ≤ 0 ∧ ___const_200_0 − ___const_200_0 ≤ 0 ∧ − ___const_100_0 + ___const_100_0 ≤ 0 ∧ ___const_100_0 − ___const_100_0 ≤ 0
3	4	1:	0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 1 − ___const_200_0 + j_0 ≤ 0 ∧ − j_0 + y8_post ≤ 0 ∧ j_0 − y8_post ≤ 0 ∧ −1 − j_0 + j_post ≤ 0 ∧ 1 + j_0 − j_post ≤ 0 ∧ j_0 − j_post ≤ 0 ∧ − j_0 + j_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 ∧ − i_post + i_post ≤ 0 ∧ i_post − i_post ≤ 0 ∧ − i_0 + i_0 ≤ 0 ∧ i_0 − i_0 ≤ 0 ∧ − ___const_200_0 + ___const_200_0 ≤ 0 ∧ ___const_200_0 − ___const_200_0 ≤ 0 ∧ − ___const_100_0 + ___const_100_0 ≤ 0 ∧ ___const_100_0 − ___const_100_0 ≤ 0
1	5	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 ∧ − ___const_200_0 + ___const_200_0 ≤ 0 ∧ ___const_200_0 − ___const_200_0 ≤ 0 ∧ − ___const_100_0 + ___const_100_0 ≤ 0 ∧ ___const_100_0 − ___const_100_0 ≤ 0
5	6	2:	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 ∧ − ___const_200_0 + ___const_200_0 ≤ 0 ∧ ___const_200_0 − ___const_200_0 ≤ 0 ∧ − ___const_100_0 + ___const_100_0 ≤ 0 ∧ ___const_100_0 − ___const_100_0 ≤ 0
6	7	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 ∧ − ___const_200_0 + ___const_200_0 ≤ 0 ∧ ___const_200_0 − ___const_200_0 ≤ 0 ∧ − ___const_100_0 + ___const_100_0 ≤ 0 ∧ ___const_100_0 − ___const_100_0 ≤ 0

Proof

1 Switch to Cooperation Termination Proof

We consider the following cutpoint-transitions:

1	8	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 ∧ − ___const_200_0 + ___const_200_0 ≤ 0 ∧ ___const_200_0 − ___const_200_0 ≤ 0 ∧ − ___const_100_0 + ___const_100_0 ≤ 0 ∧ ___const_100_0 − ___const_100_0 ≤ 0
2	15	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 ∧ − ___const_200_0 + ___const_200_0 ≤ 0 ∧ ___const_200_0 − ___const_200_0 ≤ 0 ∧ − ___const_100_0 + ___const_100_0 ≤ 0 ∧ ___const_100_0 − ___const_100_0 ≤ 0

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

2 Transition Removal

We remove transitions 0, 3, 6, 7 using the following ranking functions, which are bounded by −17.

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

3 Location Addition

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

1* 11 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 ∧ − ___const_200_0 + ___const_200_0 ≤ 0 ∧ ___const_200_0 − ___const_200_0 ≤ 0 ∧ − ___const_100_0 + ___const_100_0 ≤ 0 ∧ ___const_100_0 − ___const_100_0 ≤ 0

4 Location Addition

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

1 9 1_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 ∧ − ___const_200_0 + ___const_200_0 ≤ 0 ∧ ___const_200_0 − ___const_200_0 ≤ 0 ∧ − ___const_100_0 + ___const_100_0 ≤ 0 ∧ ___const_100_0 − ___const_100_0 ≤ 0

5 Location Addition

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

2* 18 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 ∧ − ___const_200_0 + ___const_200_0 ≤ 0 ∧ ___const_200_0 − ___const_200_0 ≤ 0 ∧ − ___const_100_0 + ___const_100_0 ≤ 0 ∧ ___const_100_0 − ___const_100_0 ≤ 0

6 Location Addition

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

2 16 2_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 ∧ − ___const_200_0 + ___const_200_0 ≤ 0 ∧ ___const_200_0 − ___const_200_0 ≤ 0 ∧ − ___const_100_0 + ___const_100_0 ≤ 0 ∧ ___const_100_0 − ___const_100_0 ≤ 0

7 SCC Decomposition

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

7.1 SCC Subproblem 1/2

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

7.1.1 Transition Removal

We remove transition 4 using the following ranking functions, which are bounded by 2.

1:	1 + 4⋅___const_200_0 − 4⋅j_0
3:	−1 + 4⋅___const_200_0 − 4⋅j_0
1_var_snapshot:	4⋅___const_200_0 − 4⋅j_0
1*:	2 + 4⋅___const_200_0 − 4⋅j_0

7.1.2 Transition Removal

We remove transitions 11, 5 using the following ranking functions, which are bounded by −2.

1:	0
3:	−2
1_var_snapshot:	−1
1*:	1

7.1.3 Transition Removal

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

1:	0
3:	0
1_var_snapshot:	−1
1*:	0

7.1.4 Splitting Cut-Point Transitions

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

7.1.4.1 Cut-Point Subproblem 1/1

Here we consider cut-point transition 8.

7.1.4.1.1 Splitting Cut-Point Transitions

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

7.2 SCC Subproblem 2/2

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

7.2.1 Transition Removal

We remove transition 1 using the following ranking functions, which are bounded by 2.

0:	−1 + 4⋅___const_100_0 − 4⋅i_0
2:	1 + 4⋅___const_100_0 − 4⋅i_0
2_var_snapshot:	4⋅___const_100_0 − 4⋅i_0
2*:	2 + 4⋅___const_100_0 − 4⋅i_0

7.2.2 Transition Removal

We remove transitions 18, 2 using the following ranking functions, which are bounded by −3.

0:	−3
2:	−1
2_var_snapshot:	−2
2*:	0

7.2.3 Transition Removal

We remove transition 16 using the following ranking functions, which are bounded by −1.

0:	0
2:	0
2_var_snapshot:	−1
2*:	0

7.2.4 Splitting Cut-Point Transitions

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

7.2.4.1 Cut-Point Subproblem 1/1

Here we consider cut-point transition 15.

7.2.4.1.1 Splitting Cut-Point Transitions

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

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