LTS Termination Proof

by T2Cert

Input

Integer Transition System

Proof

1 Invariant Updates

The following invariants are asserted.

0: i_1 ≤ 0i_1 ≤ 0
1: i_1 ≤ 0i_1 ≤ 0
2: i_1 ≤ 0i_1 ≤ 0i_0 ≤ 0i2_post ≤ 0i2_0 ≤ 0
3: i_1 ≤ 0i_1 ≤ 0i_0 ≤ 0
4: i_1 ≤ 0i_1 ≤ 0i_0 ≤ 0
5: i_1 ≤ 0i_1 ≤ 0i_0 ≤ 0
6: i_1 ≤ 0i_1 ≤ 0
7: i_1 ≤ 0i_1 ≤ 0i_0 ≤ 0
8: i_1 ≤ 0i_1 ≤ 0i_0 ≤ 0
9: i_1 ≤ 0i_1 ≤ 0i_0 ≤ 0
10: i_1 ≤ 0i_1 ≤ 0
11: i_1 ≤ 0i_1 ≤ 01 − i_0 ≤ 0
12: i_1 ≤ 0i_1 ≤ 0i_0 ≤ 0i2_post ≤ 0i2_0 ≤ 0
13: i_1 ≤ 0i_1 ≤ 0i_0 ≤ 0i2_post ≤ 0i2_0 ≤ 0
14: i_1 ≤ 0i_1 ≤ 0i_0 ≤ 0i2_post ≤ 0i2_0 ≤ 0
15: i_1 ≤ 0i_1 ≤ 0i_0 ≤ 0i2_post ≤ 0i2_0 ≤ 0
16: i_1 ≤ 0i_1 ≤ 0i_0 ≤ 0i2_post ≤ 0i2_0 ≤ 0
17: TRUE
18: TRUE

The invariants are proved as follows.

IMPACT Invariant Proof

2 Switch to Cooperation Termination Proof

We consider the following cutpoint-transitions:
1 28 1: r_post + r_post ≤ 0r_postr_post ≤ 0r_0 + r_0 ≤ 0r_0r_0 ≤ 0i_post + i_post ≤ 0i_posti_post ≤ 0i_1 + i_1 ≤ 0i_1i_1 ≤ 0i_0 + i_0 ≤ 0i_0i_0 ≤ 0i2_post + i2_post ≤ 0i2_posti2_post ≤ 0i2_0 + i2_0 ≤ 0i2_0i2_0 ≤ 0
6 35 6: r_post + r_post ≤ 0r_postr_post ≤ 0r_0 + r_0 ≤ 0r_0r_0 ≤ 0i_post + i_post ≤ 0i_posti_post ≤ 0i_1 + i_1 ≤ 0i_1i_1 ≤ 0i_0 + i_0 ≤ 0i_0i_0 ≤ 0i2_post + i2_post ≤ 0i2_posti2_post ≤ 0i2_0 + i2_0 ≤ 0i2_0i2_0 ≤ 0
and for every transition t, a duplicate t is considered.

3 Transition Removal

We remove transitions 0, 13, 26, 27 using the following ranking functions, which are bounded by −17.

18: 0
17: 0
0: 0
2: 0
6: 0
12: 0
13: 0
14: 0
15: 0
16: 0
1: 0
3: 0
4: 0
5: 0
7: 0
8: 0
9: 0
10: 0
11: 0
18: −6
17: −7
0: −8
2: −8
6: −8
12: −8
13: −8
14: −8
15: −8
16: −8
6_var_snapshot: −8
6*: −8
1: −9
3: −9
4: −9
5: −9
7: −9
8: −9
9: −9
10: −9
1_var_snapshot: −9
1*: −9
11: −10
Hints:
29 lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]
36 lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]
1 lexWeak[ [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] ]
3 lexWeak[ [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] ]
5 lexWeak[ [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] ]
7 lexWeak[ [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] ]
9 lexWeak[ [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] ]
11 lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]
12 lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]
14 lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]
15 lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]
16 lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]
17 lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]
18 lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]
19 lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]
20 lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]
21 lexWeak[ [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] ]
23 lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]
24 lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]
25 lexWeak[ [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] ]
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] ]
26 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] ]
27 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] ]

4 Location Addition

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

1* 31 1: r_post + r_post ≤ 0r_postr_post ≤ 0r_0 + r_0 ≤ 0r_0r_0 ≤ 0i_post + i_post ≤ 0i_posti_post ≤ 0i_1 + i_1 ≤ 0i_1i_1 ≤ 0i_0 + i_0 ≤ 0i_0i_0 ≤ 0i2_post + i2_post ≤ 0i2_posti2_post ≤ 0i2_0 + i2_0 ≤ 0i2_0i2_0 ≤ 0

5 Location Addition

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

1 29 1_var_snapshot: r_post + r_post ≤ 0r_postr_post ≤ 0r_0 + r_0 ≤ 0r_0r_0 ≤ 0i_post + i_post ≤ 0i_posti_post ≤ 0i_1 + i_1 ≤ 0i_1i_1 ≤ 0i_0 + i_0 ≤ 0i_0i_0 ≤ 0i2_post + i2_post ≤ 0i2_posti2_post ≤ 0i2_0 + i2_0 ≤ 0i2_0i2_0 ≤ 0

6 Location Addition

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

6* 38 6: r_post + r_post ≤ 0r_postr_post ≤ 0r_0 + r_0 ≤ 0r_0r_0 ≤ 0i_post + i_post ≤ 0i_posti_post ≤ 0i_1 + i_1 ≤ 0i_1i_1 ≤ 0i_0 + i_0 ≤ 0i_0i_0 ≤ 0i2_post + i2_post ≤ 0i2_posti2_post ≤ 0i2_0 + i2_0 ≤ 0i2_0i2_0 ≤ 0

7 Location Addition

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

6 36 6_var_snapshot: r_post + r_post ≤ 0r_postr_post ≤ 0r_0 + r_0 ≤ 0r_0r_0 ≤ 0i_post + i_post ≤ 0i_posti_post ≤ 0i_1 + i_1 ≤ 0i_1i_1 ≤ 0i_0 + i_0 ≤ 0i_0i_0 ≤ 0i2_post + i2_post ≤ 0i2_posti2_post ≤ 0i2_0 + i2_0 ≤ 0i2_0i2_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 { 1, 3, 4, 5, 7, 8, 9, 10, 1_var_snapshot, 1* }.

8.1.1 Transition Removal

We remove transitions 2, 3, 4, 6, 7, 8, 9, 10, 12, 14 using the following ranking functions, which are bounded by −8.

1: 1 − 10⋅i_0
3: −7 − 10⋅i_0
4: −5 − 10⋅i_0
5: −6 − 10⋅i_0
7: −4 − 10⋅i_0
8: −3 − 10⋅i_0
9: −2 − 10⋅i_0
10: −1 − 10⋅i_0
1_var_snapshot: −10⋅i_0
1*: 2 − 10⋅i_0
Hints:
29 lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 10, 0, 0, 0, 0] ]
31 lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 10, 0, 0, 0, 0] ]
2 lexStrict[ [0, 0, 0, 0, 0, 0, 10, 0, 10, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] , [0, 0, 10, 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, 10, 0, 0, 0, 0] , [0, 0, 10, 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, 10, 0, 0, 0, 0] , [0, 0, 10, 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, 10, 0, 0, 0, 0] , [0, 0, 10, 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, 10, 0, 0, 0, 0] , [0, 0, 10, 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, 10, 0, 0, 0, 0] , [0, 0, 10, 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, 10, 0, 0, 0, 0] , [0, 0, 10, 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, 10, 0, 0, 0, 0] , [0, 0, 10, 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, 10, 0, 0, 0, 0] ]
12 lexStrict[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 10, 0, 0, 0, 0] , [0, 0, 10, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]
14 lexStrict[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 10, 0, 0, 0, 0] , [0, 0, 10, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]
16 lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 10, 0, 0, 0, 0] ]

8.1.2 Transition Removal

We remove transitions 29, 31, 11, 16 using the following ranking functions, which are bounded by −3.

1: −1
3: 0
4: 0
5: 0
7: 0
8: 0
9: 1
10: −3
1_var_snapshot: −2
1*: 0
Hints:
29 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] ]
31 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] ]
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] ]
16 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] ]

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 28.

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, 2, 6, 12, 13, 14, 15, 16, 6_var_snapshot, 6* }.

8.2.1 Transition Removal

We remove transitions 1, 15, 17, 18, 19, 20, 21, 22, 23, 24, 25 using the following ranking functions, which are bounded by −6.

0: 1 − 10⋅i_0
2: −10⋅i_0
6: 3 − 10⋅i_0
12: −5 − 10⋅i_0
13: −3 − 10⋅i_0
14: −4 − 10⋅i_0
15: −2 − 10⋅i_0
16: −1 − 10⋅i_0
6_var_snapshot: 2 − 10⋅i_0
6*: 4 − 10⋅i_0
Hints:
36 lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 10, 0, 0, 0, 0] ]
38 lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 10, 0, 0, 0, 0] ]
1 lexStrict[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 10] , [0, 0, 0, 0, 10, 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, 10, 0, 0, 0, 0] ]
15 lexStrict[ [0, 0, 0, 0, 0, 0, 0, 0, 10, 0, 10, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] , [0, 0, 10, 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, 10, 0, 0, 0, 0] , [0, 0, 10, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]
18 lexStrict[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 10, 0, 0, 0, 0] , [0, 0, 10, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]
19 lexStrict[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 10, 0, 0, 0, 0] , [0, 0, 10, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]
20 lexStrict[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 10, 0, 0, 0, 0] , [0, 0, 10, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]
21 lexStrict[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 10, 0, 0, 0, 0] , [0, 0, 10, 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, 10, 0, 0, 0, 0] , [0, 0, 10, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]
23 lexStrict[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 10, 0, 0, 0, 0] , [0, 0, 10, 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, 10, 0, 0, 0, 0] , [0, 0, 10, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]
25 lexStrict[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 10, 0, 0, 0, 0] , [0, 0, 10, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]

8.2.2 Transition Removal

We remove transitions 36, 38 using the following ranking functions, which are bounded by −1.

0: −2
2: 0
6: 0
12: 0
13: 0
14: 0
15: 0
16: 0
6_var_snapshot: −1
6*: 1
Hints:
36 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] ]
38 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] ]
5 lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]

8.2.3 Transition Removal

We remove transition 5 using the following ranking functions, which are bounded by 0.

0: 0
2: 0
6: 0
12: 0
13: 0
14: 0
15: 0
16: 0
6_var_snapshot: 1
6*: 0
Hints:
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] ]

8.2.4 Splitting Cut-Point Transitions

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

8.2.4.1 Cut-Point Subproblem 1/1

Here we consider cut-point transition 35.

8.2.4.1.1 Splitting Cut-Point Transitions

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

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