LTS Termination Proof

by T2Cert

Input

Integer Transition System

Initial Location: 7

Transitions: (pre-variables and post-variables)

0	0	1:	1 − copied_0 ≤ 0 ∧ − e_0 + olde_0 ≤ 0 ∧ n_0 − oldn_0 ≤ 0 ∧ − oldn_post + oldn_post ≤ 0 ∧ oldn_post − oldn_post ≤ 0 ∧ − oldn_0 + oldn_0 ≤ 0 ∧ oldn_0 − oldn_0 ≤ 0 ∧ − olde_post + olde_post ≤ 0 ∧ olde_post − olde_post ≤ 0 ∧ − olde_0 + olde_0 ≤ 0 ∧ olde_0 − olde_0 ≤ 0 ∧ − n_post + n_post ≤ 0 ∧ n_post − n_post ≤ 0 ∧ − n_0 + n_0 ≤ 0 ∧ n_0 − n_0 ≤ 0 ∧ − e_post + e_post ≤ 0 ∧ e_post − e_post ≤ 0 ∧ − e_0 + e_0 ≤ 0 ∧ e_0 − e_0 ≤ 0 ∧ − copied_post + copied_post ≤ 0 ∧ copied_post − copied_post ≤ 0 ∧ − copied_0 + copied_0 ≤ 0 ∧ copied_0 − copied_0 ≤ 0
0	1	2:	0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ copied_0 ≤ 0 ∧ −1 + copied_post ≤ 0 ∧ 1 − copied_post ≤ 0 ∧ − n_0 + oldn_post ≤ 0 ∧ n_0 − oldn_post ≤ 0 ∧ − e_0 + olde_post ≤ 0 ∧ e_0 − olde_post ≤ 0 ∧ 1 − e_0 ≤ 0 ∧ −100 + n_0 ≤ 0 ∧ −11 − n_0 + n_post ≤ 0 ∧ 11 + n_0 − n_post ≤ 0 ∧ −1 − e_0 + e_post ≤ 0 ∧ 1 + e_0 − e_post ≤ 0 ∧ copied_0 − copied_post ≤ 0 ∧ − copied_0 + copied_post ≤ 0 ∧ e_0 − e_post ≤ 0 ∧ − e_0 + e_post ≤ 0 ∧ n_0 − n_post ≤ 0 ∧ − n_0 + n_post ≤ 0 ∧ olde_0 − olde_post ≤ 0 ∧ − olde_0 + olde_post ≤ 0 ∧ oldn_0 − oldn_post ≤ 0 ∧ − oldn_0 + oldn_post ≤ 0
2	2	0:	− oldn_post + oldn_post ≤ 0 ∧ oldn_post − oldn_post ≤ 0 ∧ − oldn_0 + oldn_0 ≤ 0 ∧ oldn_0 − oldn_0 ≤ 0 ∧ − olde_post + olde_post ≤ 0 ∧ olde_post − olde_post ≤ 0 ∧ − olde_0 + olde_0 ≤ 0 ∧ olde_0 − olde_0 ≤ 0 ∧ − n_post + n_post ≤ 0 ∧ n_post − n_post ≤ 0 ∧ − n_0 + n_0 ≤ 0 ∧ n_0 − n_0 ≤ 0 ∧ − e_post + e_post ≤ 0 ∧ e_post − e_post ≤ 0 ∧ − e_0 + e_0 ≤ 0 ∧ e_0 − e_0 ≤ 0 ∧ − copied_post + copied_post ≤ 0 ∧ copied_post − copied_post ≤ 0 ∧ − copied_0 + copied_0 ≤ 0 ∧ copied_0 − copied_0 ≤ 0
0	3	3:	0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ copied_0 ≤ 0 ∧ −1 + copied_post ≤ 0 ∧ 1 − copied_post ≤ 0 ∧ − n_0 + oldn_post ≤ 0 ∧ n_0 − oldn_post ≤ 0 ∧ − e_0 + olde_post ≤ 0 ∧ e_0 − olde_post ≤ 0 ∧ 1 − e_0 ≤ 0 ∧ 101 − n_0 ≤ 0 ∧ 10 − n_0 + n_post ≤ 0 ∧ −10 + n_0 − n_post ≤ 0 ∧ 1 − e_0 + e_post ≤ 0 ∧ −1 + e_0 − e_post ≤ 0 ∧ copied_0 − copied_post ≤ 0 ∧ − copied_0 + copied_post ≤ 0 ∧ e_0 − e_post ≤ 0 ∧ − e_0 + e_post ≤ 0 ∧ n_0 − n_post ≤ 0 ∧ − n_0 + n_post ≤ 0 ∧ olde_0 − olde_post ≤ 0 ∧ − olde_0 + olde_post ≤ 0 ∧ oldn_0 − oldn_post ≤ 0 ∧ − oldn_0 + oldn_post ≤ 0
3	4	0:	− oldn_post + oldn_post ≤ 0 ∧ oldn_post − oldn_post ≤ 0 ∧ − oldn_0 + oldn_0 ≤ 0 ∧ oldn_0 − oldn_0 ≤ 0 ∧ − olde_post + olde_post ≤ 0 ∧ olde_post − olde_post ≤ 0 ∧ − olde_0 + olde_0 ≤ 0 ∧ olde_0 − olde_0 ≤ 0 ∧ − n_post + n_post ≤ 0 ∧ n_post − n_post ≤ 0 ∧ − n_0 + n_0 ≤ 0 ∧ n_0 − n_0 ≤ 0 ∧ − e_post + e_post ≤ 0 ∧ e_post − e_post ≤ 0 ∧ − e_0 + e_0 ≤ 0 ∧ e_0 − e_0 ≤ 0 ∧ − copied_post + copied_post ≤ 0 ∧ copied_post − copied_post ≤ 0 ∧ − copied_0 + copied_0 ≤ 0 ∧ copied_0 − copied_0 ≤ 0
0	5	4:	0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 1 − e_0 ≤ 0 ∧ −100 + n_0 ≤ 0 ∧ −11 − n_0 + n_post ≤ 0 ∧ 11 + n_0 − n_post ≤ 0 ∧ −1 − e_0 + e_post ≤ 0 ∧ 1 + e_0 − e_post ≤ 0 ∧ e_0 − e_post ≤ 0 ∧ − e_0 + e_post ≤ 0 ∧ n_0 − n_post ≤ 0 ∧ − n_0 + n_post ≤ 0 ∧ − oldn_post + oldn_post ≤ 0 ∧ oldn_post − oldn_post ≤ 0 ∧ − oldn_0 + oldn_0 ≤ 0 ∧ oldn_0 − oldn_0 ≤ 0 ∧ − olde_post + olde_post ≤ 0 ∧ olde_post − olde_post ≤ 0 ∧ − olde_0 + olde_0 ≤ 0 ∧ olde_0 − olde_0 ≤ 0 ∧ − copied_post + copied_post ≤ 0 ∧ copied_post − copied_post ≤ 0 ∧ − copied_0 + copied_0 ≤ 0 ∧ copied_0 − copied_0 ≤ 0
4	6	0:	− oldn_post + oldn_post ≤ 0 ∧ oldn_post − oldn_post ≤ 0 ∧ − oldn_0 + oldn_0 ≤ 0 ∧ oldn_0 − oldn_0 ≤ 0 ∧ − olde_post + olde_post ≤ 0 ∧ olde_post − olde_post ≤ 0 ∧ − olde_0 + olde_0 ≤ 0 ∧ olde_0 − olde_0 ≤ 0 ∧ − n_post + n_post ≤ 0 ∧ n_post − n_post ≤ 0 ∧ − n_0 + n_0 ≤ 0 ∧ n_0 − n_0 ≤ 0 ∧ − e_post + e_post ≤ 0 ∧ e_post − e_post ≤ 0 ∧ − e_0 + e_0 ≤ 0 ∧ e_0 − e_0 ≤ 0 ∧ − copied_post + copied_post ≤ 0 ∧ copied_post − copied_post ≤ 0 ∧ − copied_0 + copied_0 ≤ 0 ∧ copied_0 − copied_0 ≤ 0
0	7	5:	0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 1 − e_0 ≤ 0 ∧ 101 − n_0 ≤ 0 ∧ 10 − n_0 + n_post ≤ 0 ∧ −10 + n_0 − n_post ≤ 0 ∧ 1 − e_0 + e_post ≤ 0 ∧ −1 + e_0 − e_post ≤ 0 ∧ e_0 − e_post ≤ 0 ∧ − e_0 + e_post ≤ 0 ∧ n_0 − n_post ≤ 0 ∧ − n_0 + n_post ≤ 0 ∧ − oldn_post + oldn_post ≤ 0 ∧ oldn_post − oldn_post ≤ 0 ∧ − oldn_0 + oldn_0 ≤ 0 ∧ oldn_0 − oldn_0 ≤ 0 ∧ − olde_post + olde_post ≤ 0 ∧ olde_post − olde_post ≤ 0 ∧ − olde_0 + olde_0 ≤ 0 ∧ olde_0 − olde_0 ≤ 0 ∧ − copied_post + copied_post ≤ 0 ∧ copied_post − copied_post ≤ 0 ∧ − copied_0 + copied_0 ≤ 0 ∧ copied_0 − copied_0 ≤ 0
5	8	0:	− oldn_post + oldn_post ≤ 0 ∧ oldn_post − oldn_post ≤ 0 ∧ − oldn_0 + oldn_0 ≤ 0 ∧ oldn_0 − oldn_0 ≤ 0 ∧ − olde_post + olde_post ≤ 0 ∧ olde_post − olde_post ≤ 0 ∧ − olde_0 + olde_0 ≤ 0 ∧ olde_0 − olde_0 ≤ 0 ∧ − n_post + n_post ≤ 0 ∧ n_post − n_post ≤ 0 ∧ − n_0 + n_0 ≤ 0 ∧ n_0 − n_0 ≤ 0 ∧ − e_post + e_post ≤ 0 ∧ e_post − e_post ≤ 0 ∧ − e_0 + e_0 ≤ 0 ∧ e_0 − e_0 ≤ 0 ∧ − copied_post + copied_post ≤ 0 ∧ copied_post − copied_post ≤ 0 ∧ − copied_0 + copied_0 ≤ 0 ∧ copied_0 − copied_0 ≤ 0
6	9	0:	0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ −1 + e_post ≤ 0 ∧ 1 − e_post ≤ 0 ∧ copied_post ≤ 0 ∧ − copied_post ≤ 0 ∧ copied_0 − copied_post ≤ 0 ∧ − copied_0 + copied_post ≤ 0 ∧ e_0 − e_post ≤ 0 ∧ − e_0 + e_post ≤ 0 ∧ n_0 − n_post ≤ 0 ∧ − n_0 + n_post ≤ 0 ∧ − oldn_post + oldn_post ≤ 0 ∧ oldn_post − oldn_post ≤ 0 ∧ − oldn_0 + oldn_0 ≤ 0 ∧ oldn_0 − oldn_0 ≤ 0 ∧ − olde_post + olde_post ≤ 0 ∧ olde_post − olde_post ≤ 0 ∧ − olde_0 + olde_0 ≤ 0 ∧ olde_0 − olde_0 ≤ 0
7	10	6:	− oldn_post + oldn_post ≤ 0 ∧ oldn_post − oldn_post ≤ 0 ∧ − oldn_0 + oldn_0 ≤ 0 ∧ oldn_0 − oldn_0 ≤ 0 ∧ − olde_post + olde_post ≤ 0 ∧ olde_post − olde_post ≤ 0 ∧ − olde_0 + olde_0 ≤ 0 ∧ olde_0 − olde_0 ≤ 0 ∧ − n_post + n_post ≤ 0 ∧ n_post − n_post ≤ 0 ∧ − n_0 + n_0 ≤ 0 ∧ n_0 − n_0 ≤ 0 ∧ − e_post + e_post ≤ 0 ∧ e_post − e_post ≤ 0 ∧ − e_0 + e_0 ≤ 0 ∧ e_0 − e_0 ≤ 0 ∧ − copied_post + copied_post ≤ 0 ∧ copied_post − copied_post ≤ 0 ∧ − copied_0 + copied_0 ≤ 0 ∧ copied_0 − copied_0 ≤ 0

Proof

1 Invariant Updates

The following invariants are asserted.

0:	− copied_post ≤ 0 ∧ − copied_0 ≤ 0
1:	− copied_post ≤ 0 ∧ 1 − copied_0 ≤ 0
2:	−1 + copied_post ≤ 0 ∧ 1 − copied_post ≤ 0 ∧ −1 + copied_0 ≤ 0 ∧ 1 − copied_0 ≤ 0
3:	−1 + copied_post ≤ 0 ∧ 1 − copied_post ≤ 0 ∧ −1 + copied_0 ≤ 0 ∧ 1 − copied_0 ≤ 0
4:	− copied_post ≤ 0 ∧ − copied_0 ≤ 0
5:	− copied_post ≤ 0 ∧ − copied_0 ≤ 0
6:	TRUE
7:	TRUE

The invariants are proved as follows.

IMPACT Invariant Proof

nodes (location) invariant:

0	(0)	− copied_post ≤ 0 ∧ − copied_0 ≤ 0
1	(1)	− copied_post ≤ 0 ∧ 1 − copied_0 ≤ 0
2	(2)	−1 + copied_post ≤ 0 ∧ 1 − copied_post ≤ 0 ∧ −1 + copied_0 ≤ 0 ∧ 1 − copied_0 ≤ 0
3	(3)	−1 + copied_post ≤ 0 ∧ 1 − copied_post ≤ 0 ∧ −1 + copied_0 ≤ 0 ∧ 1 − copied_0 ≤ 0
4	(4)	− copied_post ≤ 0 ∧ − copied_0 ≤ 0
5	(5)	− copied_post ≤ 0 ∧ − copied_0 ≤ 0
6	(6)	TRUE
7	(7)	TRUE

initial node: 7
cover edges:
transition edges:

0 0 1

0 1 2

0 3 3

0 5 4

0 7 5

2 2 0

3 4 0

4 6 0

5 8 0

6 9 0

7 10 6

2 Switch to Cooperation Termination Proof

We consider the following cutpoint-transitions:

− oldn_post + oldn_post ≤ 0 ∧ oldn_post − oldn_post ≤ 0 ∧ − oldn_0 + oldn_0 ≤ 0 ∧ oldn_0 − oldn_0 ≤ 0 ∧ − olde_post + olde_post ≤ 0 ∧ olde_post − olde_post ≤ 0 ∧ − olde_0 + olde_0 ≤ 0 ∧ olde_0 − olde_0 ≤ 0 ∧ − n_post + n_post ≤ 0 ∧ n_post − n_post ≤ 0 ∧ − n_0 + n_0 ≤ 0 ∧ n_0 − n_0 ≤ 0 ∧ − e_post + e_post ≤ 0 ∧ e_post − e_post ≤ 0 ∧ − e_0 + e_0 ≤ 0 ∧ e_0 − e_0 ≤ 0 ∧ − copied_post + copied_post ≤ 0 ∧ copied_post − copied_post ≤ 0 ∧ − copied_0 + copied_0 ≤ 0 ∧ copied_0 − copied_0 ≤ 0

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

3 Transition Removal

We remove transitions 0, 9, 10 using the following ranking functions, which are bounded by −13.

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

4 Location Addition

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

0* 14 0: − oldn_post + oldn_post ≤ 0 ∧ oldn_post − oldn_post ≤ 0 ∧ − oldn_0 + oldn_0 ≤ 0 ∧ oldn_0 − oldn_0 ≤ 0 ∧ − olde_post + olde_post ≤ 0 ∧ olde_post − olde_post ≤ 0 ∧ − olde_0 + olde_0 ≤ 0 ∧ olde_0 − olde_0 ≤ 0 ∧ − n_post + n_post ≤ 0 ∧ n_post − n_post ≤ 0 ∧ − n_0 + n_0 ≤ 0 ∧ n_0 − n_0 ≤ 0 ∧ − e_post + e_post ≤ 0 ∧ e_post − e_post ≤ 0 ∧ − e_0 + e_0 ≤ 0 ∧ e_0 − e_0 ≤ 0 ∧ − copied_post + copied_post ≤ 0 ∧ copied_post − copied_post ≤ 0 ∧ − copied_0 + copied_0 ≤ 0 ∧ copied_0 − copied_0 ≤ 0

5 Location Addition

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

0 12 0_var_snapshot: − oldn_post + oldn_post ≤ 0 ∧ oldn_post − oldn_post ≤ 0 ∧ − oldn_0 + oldn_0 ≤ 0 ∧ oldn_0 − oldn_0 ≤ 0 ∧ − olde_post + olde_post ≤ 0 ∧ olde_post − olde_post ≤ 0 ∧ − olde_0 + olde_0 ≤ 0 ∧ olde_0 − olde_0 ≤ 0 ∧ − n_post + n_post ≤ 0 ∧ n_post − n_post ≤ 0 ∧ − n_0 + n_0 ≤ 0 ∧ n_0 − n_0 ≤ 0 ∧ − e_post + e_post ≤ 0 ∧ e_post − e_post ≤ 0 ∧ − e_0 + e_0 ≤ 0 ∧ e_0 − e_0 ≤ 0 ∧ − copied_post + copied_post ≤ 0 ∧ copied_post − copied_post ≤ 0 ∧ − copied_0 + copied_0 ≤ 0 ∧ copied_0 − copied_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 { 0, 2, 3, 4, 5, 0_var_snapshot, 0* }.

6.1.1 Transition Removal

We remove transitions 1, 5 using the following ranking functions, which are bounded by −719.

0:	−1 + 84⋅e_0 − 8⋅n_0
2:	copied_post + 84⋅e_0 − 8⋅n_0
3:	1 + 84⋅e_0 − 8⋅n_0
4:	1 + 84⋅e_0 − 8⋅n_0
5:	1 + 84⋅e_0 − 8⋅n_0
0_var_snapshot:	−2 + 84⋅e_0 − 8⋅n_0
0*:	84⋅e_0 − 8⋅n_0

6.1.2 Transition Removal

We remove transition 7 using the following ranking functions, which are bounded by 19.

0:	30⋅e_0
2:	1 + 30⋅e_0
3:	copied_post + 30⋅e_0
4:	1 + 30⋅e_0
5:	10 + 30⋅e_0
0_var_snapshot:	−10 + 30⋅e_0
0*:	30⋅e_0

6.1.3 Transition Removal

We remove transitions 2, 3, 4, 8 using the following ranking functions, which are bounded by −4.

0:	−1 − 4⋅copied_0
2:	1 − 4⋅copied_0
3:	−3⋅copied_0
4:	1 − 4⋅copied_0
5:	1
0_var_snapshot:	−2 − 4⋅copied_0
0*:	−4⋅copied_0

6.1.4 Transition Removal

We remove transitions 12, 6 using the following ranking functions, which are bounded by −3.

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

6.1.5 Transition Removal

We remove transition 14 using the following ranking functions, which are bounded by −1.

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

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

Here we consider cut-point transition 11.

6.1.6.1.1 Splitting Cut-Point Transitions

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

Tool configuration

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