LTS Termination Proof

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Input

Integer Transition System

Proof

1 Invariant Updates

The following invariants are asserted.

0: −400 + x1_post ≤ 0 ∧ 400 − x1_post ≤ 0 ∧ −400 + x1_0 ≤ 0 ∧ 400 − x1_0 ≤ 0
1: −400 + x1_post ≤ 0 ∧ 400 − x1_post ≤ 0 ∧ −400 + x1_0 ≤ 0 ∧ 400 − x1_0 ≤ 0
2: −400 + x1_post ≤ 0 ∧ 400 − x1_post ≤ 0 ∧ −400 + x1_0 ≤ 0 ∧ 400 − x1_0 ≤ 0
3: −400 + x1_post ≤ 0 ∧ 400 − x1_post ≤ 0 ∧ −400 + x1_0 ≤ 0 ∧ 400 − x1_0 ≤ 0
4: −400 + x1_post ≤ 0 ∧ 400 − x1_post ≤ 0 ∧ −400 + x1_0 ≤ 0 ∧ 400 − x1_0 ≤ 0
5: −400 + x1_post ≤ 0 ∧ 400 − x1_post ≤ 0 ∧ −400 + x1_0 ≤ 0 ∧ 400 − x1_0 ≤ 0
6: −400 + x1_post ≤ 0 ∧ 400 − x1_post ≤ 0 ∧ −400 + x1_0 ≤ 0 ∧ 400 − x1_0 ≤ 0
7: −400 + x1_post ≤ 0 ∧ 400 − x1_post ≤ 0 ∧ −400 + x1_0 ≤ 0 ∧ 400 − x1_0 ≤ 0
8: −400 + x1_post ≤ 0 ∧ 400 − x1_post ≤ 0 ∧ −400 + x1_0 ≤ 0 ∧ 400 − x1_0 ≤ 0
9: −400 + x1_post ≤ 0 ∧ 400 − x1_post ≤ 0 ∧ −400 + x1_0 ≤ 0 ∧ 400 − x1_0 ≤ 0
10: −400 + x1_post ≤ 0 ∧ 400 − x1_post ≤ 0 ∧ −400 + x1_0 ≤ 0 ∧ 400 − x1_0 ≤ 0
11: −400 + x1_post ≤ 0 ∧ 400 − x1_post ≤ 0 ∧ −400 + x1_0 ≤ 0 ∧ 400 − x1_0 ≤ 0
12: TRUE
13: TRUE

The invariants are proved as follows.

IMPACT Invariant Proof

2 Switch to Cooperation Termination Proof

We consider the following cutpoint-transitions:
2 19 2: − x1_post + x1_post ≤ 0 ∧ x1_post − x1_post ≤ 0 ∧ − x1_0 + x1_0 ≤ 0 ∧ x1_0 − x1_0 ≤ 0 ∧ − l5_post + l5_post ≤ 0 ∧ l5_post − l5_post ≤ 0 ∧ − l5_0 + l5_0 ≤ 0 ∧ l5_0 − l5_0 ≤ 0 ∧ − k4_post + k4_post ≤ 0 ∧ k4_post − k4_post ≤ 0 ∧ − k4_0 + k4_0 ≤ 0 ∧ k4_0 − k4_0 ≤ 0 ∧ − j3_post + j3_post ≤ 0 ∧ j3_post − j3_post ≤ 0 ∧ − j3_0 + j3_0 ≤ 0 ∧ j3_0 − j3_0 ≤ 0 ∧ − i2_post + i2_post ≤ 0 ∧ i2_post − i2_post ≤ 0 ∧ − i2_0 + i2_0 ≤ 0 ∧ i2_0 − i2_0 ≤ 0
3 26 3: − x1_post + x1_post ≤ 0 ∧ x1_post − x1_post ≤ 0 ∧ − x1_0 + x1_0 ≤ 0 ∧ x1_0 − x1_0 ≤ 0 ∧ − l5_post + l5_post ≤ 0 ∧ l5_post − l5_post ≤ 0 ∧ − l5_0 + l5_0 ≤ 0 ∧ l5_0 − l5_0 ≤ 0 ∧ − k4_post + k4_post ≤ 0 ∧ k4_post − k4_post ≤ 0 ∧ − k4_0 + k4_0 ≤ 0 ∧ k4_0 − k4_0 ≤ 0 ∧ − j3_post + j3_post ≤ 0 ∧ j3_post − j3_post ≤ 0 ∧ − j3_0 + j3_0 ≤ 0 ∧ j3_0 − j3_0 ≤ 0 ∧ − i2_post + i2_post ≤ 0 ∧ i2_post − i2_post ≤ 0 ∧ − i2_0 + i2_0 ≤ 0 ∧ i2_0 − i2_0 ≤ 0
6 33 6: − x1_post + x1_post ≤ 0 ∧ x1_post − x1_post ≤ 0 ∧ − x1_0 + x1_0 ≤ 0 ∧ x1_0 − x1_0 ≤ 0 ∧ − l5_post + l5_post ≤ 0 ∧ l5_post − l5_post ≤ 0 ∧ − l5_0 + l5_0 ≤ 0 ∧ l5_0 − l5_0 ≤ 0 ∧ − k4_post + k4_post ≤ 0 ∧ k4_post − k4_post ≤ 0 ∧ − k4_0 + k4_0 ≤ 0 ∧ k4_0 − k4_0 ≤ 0 ∧ − j3_post + j3_post ≤ 0 ∧ j3_post − j3_post ≤ 0 ∧ − j3_0 + j3_0 ≤ 0 ∧ j3_0 − j3_0 ≤ 0 ∧ − i2_post + i2_post ≤ 0 ∧ i2_post − i2_post ≤ 0 ∧ − i2_0 + i2_0 ≤ 0 ∧ i2_0 − i2_0 ≤ 0
9 40 9: − x1_post + x1_post ≤ 0 ∧ x1_post − x1_post ≤ 0 ∧ − x1_0 + x1_0 ≤ 0 ∧ x1_0 − x1_0 ≤ 0 ∧ − l5_post + l5_post ≤ 0 ∧ l5_post − l5_post ≤ 0 ∧ − l5_0 + l5_0 ≤ 0 ∧ l5_0 − l5_0 ≤ 0 ∧ − k4_post + k4_post ≤ 0 ∧ k4_post − k4_post ≤ 0 ∧ − k4_0 + k4_0 ≤ 0 ∧ k4_0 − k4_0 ≤ 0 ∧ − j3_post + j3_post ≤ 0 ∧ j3_post − j3_post ≤ 0 ∧ − j3_0 + j3_0 ≤ 0 ∧ j3_0 − j3_0 ≤ 0 ∧ − i2_post + i2_post ≤ 0 ∧ i2_post − i2_post ≤ 0 ∧ − i2_0 + i2_0 ≤ 0 ∧ i2_0 − i2_0 ≤ 0
and for every transition t, a duplicate t is considered.

3 Transition Removal

We remove transitions 0, 4, 9, 17, 18 using the following ranking functions, which are bounded by −21.

13: 0
12: 0
0: 0
2: 0
3: 0
4: 0
6: 0
7: 0
8: 0
9: 0
10: 0
11: 0
1: 0
5: 0
13: −6
12: −7
2: −8
4: −8
6: −8
7: −8
8: −8
9: −8
10: −8
11: −8
0: −8
3: −8
2_var_snapshot: −8
2*: −8
3_var_snapshot: −8
3*: −8
6_var_snapshot: −8
6*: −8
9_var_snapshot: −8
9*: −8
1: −15
5: −16

4 Location Addition

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

2* 22 2: − x1_post + x1_post ≤ 0 ∧ x1_post − x1_post ≤ 0 ∧ − x1_0 + x1_0 ≤ 0 ∧ x1_0 − x1_0 ≤ 0 ∧ − l5_post + l5_post ≤ 0 ∧ l5_post − l5_post ≤ 0 ∧ − l5_0 + l5_0 ≤ 0 ∧ l5_0 − l5_0 ≤ 0 ∧ − k4_post + k4_post ≤ 0 ∧ k4_post − k4_post ≤ 0 ∧ − k4_0 + k4_0 ≤ 0 ∧ k4_0 − k4_0 ≤ 0 ∧ − j3_post + j3_post ≤ 0 ∧ j3_post − j3_post ≤ 0 ∧ − j3_0 + j3_0 ≤ 0 ∧ j3_0 − j3_0 ≤ 0 ∧ − i2_post + i2_post ≤ 0 ∧ i2_post − i2_post ≤ 0 ∧ − i2_0 + i2_0 ≤ 0 ∧ i2_0 − i2_0 ≤ 0

5 Location Addition

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

2 20 2_var_snapshot: − x1_post + x1_post ≤ 0 ∧ x1_post − x1_post ≤ 0 ∧ − x1_0 + x1_0 ≤ 0 ∧ x1_0 − x1_0 ≤ 0 ∧ − l5_post + l5_post ≤ 0 ∧ l5_post − l5_post ≤ 0 ∧ − l5_0 + l5_0 ≤ 0 ∧ l5_0 − l5_0 ≤ 0 ∧ − k4_post + k4_post ≤ 0 ∧ k4_post − k4_post ≤ 0 ∧ − k4_0 + k4_0 ≤ 0 ∧ k4_0 − k4_0 ≤ 0 ∧ − j3_post + j3_post ≤ 0 ∧ j3_post − j3_post ≤ 0 ∧ − j3_0 + j3_0 ≤ 0 ∧ j3_0 − j3_0 ≤ 0 ∧ − i2_post + i2_post ≤ 0 ∧ i2_post − i2_post ≤ 0 ∧ − i2_0 + i2_0 ≤ 0 ∧ i2_0 − i2_0 ≤ 0

6 Location Addition

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

3* 29 3: − x1_post + x1_post ≤ 0 ∧ x1_post − x1_post ≤ 0 ∧ − x1_0 + x1_0 ≤ 0 ∧ x1_0 − x1_0 ≤ 0 ∧ − l5_post + l5_post ≤ 0 ∧ l5_post − l5_post ≤ 0 ∧ − l5_0 + l5_0 ≤ 0 ∧ l5_0 − l5_0 ≤ 0 ∧ − k4_post + k4_post ≤ 0 ∧ k4_post − k4_post ≤ 0 ∧ − k4_0 + k4_0 ≤ 0 ∧ k4_0 − k4_0 ≤ 0 ∧ − j3_post + j3_post ≤ 0 ∧ j3_post − j3_post ≤ 0 ∧ − j3_0 + j3_0 ≤ 0 ∧ j3_0 − j3_0 ≤ 0 ∧ − i2_post + i2_post ≤ 0 ∧ i2_post − i2_post ≤ 0 ∧ − i2_0 + i2_0 ≤ 0 ∧ i2_0 − i2_0 ≤ 0

7 Location Addition

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

3 27 3_var_snapshot: − x1_post + x1_post ≤ 0 ∧ x1_post − x1_post ≤ 0 ∧ − x1_0 + x1_0 ≤ 0 ∧ x1_0 − x1_0 ≤ 0 ∧ − l5_post + l5_post ≤ 0 ∧ l5_post − l5_post ≤ 0 ∧ − l5_0 + l5_0 ≤ 0 ∧ l5_0 − l5_0 ≤ 0 ∧ − k4_post + k4_post ≤ 0 ∧ k4_post − k4_post ≤ 0 ∧ − k4_0 + k4_0 ≤ 0 ∧ k4_0 − k4_0 ≤ 0 ∧ − j3_post + j3_post ≤ 0 ∧ j3_post − j3_post ≤ 0 ∧ − j3_0 + j3_0 ≤ 0 ∧ j3_0 − j3_0 ≤ 0 ∧ − i2_post + i2_post ≤ 0 ∧ i2_post − i2_post ≤ 0 ∧ − i2_0 + i2_0 ≤ 0 ∧ i2_0 − i2_0 ≤ 0

8 Location Addition

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

6* 36 6: − x1_post + x1_post ≤ 0 ∧ x1_post − x1_post ≤ 0 ∧ − x1_0 + x1_0 ≤ 0 ∧ x1_0 − x1_0 ≤ 0 ∧ − l5_post + l5_post ≤ 0 ∧ l5_post − l5_post ≤ 0 ∧ − l5_0 + l5_0 ≤ 0 ∧ l5_0 − l5_0 ≤ 0 ∧ − k4_post + k4_post ≤ 0 ∧ k4_post − k4_post ≤ 0 ∧ − k4_0 + k4_0 ≤ 0 ∧ k4_0 − k4_0 ≤ 0 ∧ − j3_post + j3_post ≤ 0 ∧ j3_post − j3_post ≤ 0 ∧ − j3_0 + j3_0 ≤ 0 ∧ j3_0 − j3_0 ≤ 0 ∧ − i2_post + i2_post ≤ 0 ∧ i2_post − i2_post ≤ 0 ∧ − i2_0 + i2_0 ≤ 0 ∧ i2_0 − i2_0 ≤ 0

9 Location Addition

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

6 34 6_var_snapshot: − x1_post + x1_post ≤ 0 ∧ x1_post − x1_post ≤ 0 ∧ − x1_0 + x1_0 ≤ 0 ∧ x1_0 − x1_0 ≤ 0 ∧ − l5_post + l5_post ≤ 0 ∧ l5_post − l5_post ≤ 0 ∧ − l5_0 + l5_0 ≤ 0 ∧ l5_0 − l5_0 ≤ 0 ∧ − k4_post + k4_post ≤ 0 ∧ k4_post − k4_post ≤ 0 ∧ − k4_0 + k4_0 ≤ 0 ∧ k4_0 − k4_0 ≤ 0 ∧ − j3_post + j3_post ≤ 0 ∧ j3_post − j3_post ≤ 0 ∧ − j3_0 + j3_0 ≤ 0 ∧ j3_0 − j3_0 ≤ 0 ∧ − i2_post + i2_post ≤ 0 ∧ i2_post − i2_post ≤ 0 ∧ − i2_0 + i2_0 ≤ 0 ∧ i2_0 − i2_0 ≤ 0

10 Location Addition

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

9* 43 9: − x1_post + x1_post ≤ 0 ∧ x1_post − x1_post ≤ 0 ∧ − x1_0 + x1_0 ≤ 0 ∧ x1_0 − x1_0 ≤ 0 ∧ − l5_post + l5_post ≤ 0 ∧ l5_post − l5_post ≤ 0 ∧ − l5_0 + l5_0 ≤ 0 ∧ l5_0 − l5_0 ≤ 0 ∧ − k4_post + k4_post ≤ 0 ∧ k4_post − k4_post ≤ 0 ∧ − k4_0 + k4_0 ≤ 0 ∧ k4_0 − k4_0 ≤ 0 ∧ − j3_post + j3_post ≤ 0 ∧ j3_post − j3_post ≤ 0 ∧ − j3_0 + j3_0 ≤ 0 ∧ j3_0 − j3_0 ≤ 0 ∧ − i2_post + i2_post ≤ 0 ∧ i2_post − i2_post ≤ 0 ∧ − i2_0 + i2_0 ≤ 0 ∧ i2_0 − i2_0 ≤ 0

11 Location Addition

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

9 41 9_var_snapshot: − x1_post + x1_post ≤ 0 ∧ x1_post − x1_post ≤ 0 ∧ − x1_0 + x1_0 ≤ 0 ∧ x1_0 − x1_0 ≤ 0 ∧ − l5_post + l5_post ≤ 0 ∧ l5_post − l5_post ≤ 0 ∧ − l5_0 + l5_0 ≤ 0 ∧ l5_0 − l5_0 ≤ 0 ∧ − k4_post + k4_post ≤ 0 ∧ k4_post − k4_post ≤ 0 ∧ − k4_0 + k4_0 ≤ 0 ∧ k4_0 − k4_0 ≤ 0 ∧ − j3_post + j3_post ≤ 0 ∧ j3_post − j3_post ≤ 0 ∧ − j3_0 + j3_0 ≤ 0 ∧ j3_0 − j3_0 ≤ 0 ∧ − i2_post + i2_post ≤ 0 ∧ i2_post − i2_post ≤ 0 ∧ − i2_0 + i2_0 ≤ 0 ∧ i2_0 − i2_0 ≤ 0

12 SCC Decomposition

We consider subproblems for each of the 1 SCC(s) of the program graph.

12.1 SCC Subproblem 1/1

Here we consider the SCC { 2, 4, 6, 7, 8, 9, 10, 11, 0, 3, 2_var_snapshot, 2*, 3_var_snapshot, 3*, 6_var_snapshot, 6*, 9_var_snapshot, 9* }.

12.1.1 Transition Removal

We remove transition 1 using the following ranking functions, which are bounded by −6005.

2: −1601⋅i2_0
4: −1601⋅i2_0
6: −1601⋅i2_0
7: −1601⋅i2_0
8: −1601⋅i2_0
9: −1601⋅i2_0
10: −1601⋅i2_0
11: −1601⋅i2_0
0: −1601⋅i2_0 + x1_0
3: 1200 − 1601⋅i2_0
2_var_snapshot: −1601⋅i2_0
2*: −1601⋅i2_0
3_var_snapshot: 800 − 1601⋅i2_0
3*: −1601⋅i2_0 + 4⋅x1_0
6_var_snapshot: −1601⋅i2_0
6*: −1601⋅i2_0
9_var_snapshot: −1601⋅i2_0
9*: −1601⋅i2_0

12.1.2 Transition Removal

We remove transition 16 using the following ranking functions, which are bounded by −5605.

2: 800 − 1601⋅j3_0 + x1_0 + x1_post
4: −1601⋅j3_0 + x1_0 + x1_post
6: −1601⋅j3_0 + x1_0
7: −1601⋅j3_0 + x1_0
8: 400 − 1601⋅j3_0
9: −1601⋅j3_0 + x1_0
10: −1601⋅j3_0 + x1_post
11: −1601⋅j3_0 + x1_0
0: −1601⋅j3_0 + x1_post
3: 798 − 1601⋅j3_0
2_var_snapshot: 800 − 1601⋅j3_0 + x1_post
2*: −1601⋅j3_0 + x1_0 + 4⋅x1_post
3_var_snapshot: −3 − 1601⋅j3_0 + 2⋅x1_0
3*: 399 − 1601⋅j3_0 + x1_post
6_var_snapshot: −1601⋅j3_0 + x1_0
6*: −400 − 1601⋅j3_0 + x1_0 + x1_post
9_var_snapshot: 400 − 1601⋅j3_0
9*: −1601⋅j3_0 + x1_0

12.1.3 Transition Removal

We remove transition 14 using the following ranking functions, which are bounded by −23.

2: −401⋅k4_0 + x1_0
4: 399 − 401⋅k4_0 − x1_0
6: −396 − 6⋅k4_0 + x1_post
7: −398 − 6⋅k4_0 + x1_post
8: −6⋅k4_0
9: −6⋅k4_0
10: −6⋅k4_0
11: −6⋅k4_0
0: −401⋅k4_0 − 2⋅x1_post
3: 1 − 401⋅k4_0 − x1_0
2_var_snapshot: −401⋅k4_0 − x1_0 + x1_post
2*: 1977 − 401⋅k4_0
3_var_snapshot: −401⋅k4_0 − x1_0
3*: 398 − 401⋅k4_0 − 2⋅x1_0 + x1_post
6_var_snapshot: 3 − 6⋅k4_0
6*: 5 − 6⋅k4_0
9_var_snapshot: −6⋅k4_0
9*: −400 − 6⋅k4_0 + x1_post

12.1.4 Transition Removal

We remove transitions 20, 22, 27, 29, 34, 36, 2, 3, 5, 11, 13, 15 using the following ranking functions, which are bounded by −3601.

2: 1200 − 7⋅x1_0
4: 400 − 7⋅x1_0
6: 0
7: −6⋅x1_0 + 4⋅x1_post
8: 800
9: 800
10: 2⋅x1_post
11: 800
0: −1 − x1_0 − 8⋅x1_post
3: −8⋅x1_post
2_var_snapshot: 800 − 7⋅x1_0
2*: −7⋅x1_0 + 4⋅x1_post
3_var_snapshot: − x1_0 − 8⋅x1_post
3*: −7⋅x1_0
6_var_snapshot: 2000 − 6⋅x1_0
6*: − x1_0 + 2⋅x1_post
9_var_snapshot: 800
9*: 2⋅x1_0

12.1.5 Transition Removal

We remove transition 12 using the following ranking functions, which are bounded by 380.

2: 0
4: 0
6: 0
7: 0
8: −5⋅l5_0 + x1_0
9: 3 − 5⋅l5_0 + x1_0
10: 1 − 5⋅l5_0 + x1_0
11: −5⋅l5_0 + x1_0
0: 0
3: 0
2_var_snapshot: 0
2*: 0
3_var_snapshot: 0
3*: 0
6_var_snapshot: 0
6*: 0
9_var_snapshot: 402 − 5⋅l5_0
9*: 4 − 5⋅l5_0 + x1_post

12.1.6 Transition Removal

We remove transitions 41, 43, 6, 7, 8, 10 using the following ranking functions, which are bounded by −1201.

2: 0
4: 0
6: 0
7: 0
8: 0
9: −2⋅x1_post
10: −4⋅x1_0
11: 1
0: 0
3: 0
2_var_snapshot: 0
2*: 0
3_var_snapshot: 0
3*: 0
6_var_snapshot: 0
6*: 0
9_var_snapshot: −3⋅x1_0
9*: −400

12.1.7 Splitting Cut-Point Transitions

We consider 4 subproblems corresponding to sets of cut-point transitions as follows.

12.1.7.1 Cut-Point Subproblem 1/4

Here we consider cut-point transition 19.

12.1.7.1.1 Splitting Cut-Point Transitions

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

12.1.7.2 Cut-Point Subproblem 2/4

Here we consider cut-point transition 26.

12.1.7.2.1 Splitting Cut-Point Transitions

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

12.1.7.3 Cut-Point Subproblem 3/4

Here we consider cut-point transition 33.

12.1.7.3.1 Splitting Cut-Point Transitions

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

12.1.7.4 Cut-Point Subproblem 4/4

Here we consider cut-point transition 40.

12.1.7.4.1 Splitting Cut-Point Transitions

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

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