LTS Termination Proof

by T2Cert

Input

Integer Transition System

Proof

1 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 ∧ − ___const_5_0 + ___const_5_0 ≤ 0 ∧ ___const_5_0 − ___const_5_0 ≤ 0 ∧ − ___const_400_0 + ___const_400_0 ≤ 0 ∧ ___const_400_0 − ___const_400_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 ∧ − ___const_5_0 + ___const_5_0 ≤ 0 ∧ ___const_5_0 − ___const_5_0 ≤ 0 ∧ − ___const_400_0 + ___const_400_0 ≤ 0 ∧ ___const_400_0 − ___const_400_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 ∧ − ___const_5_0 + ___const_5_0 ≤ 0 ∧ ___const_5_0 − ___const_5_0 ≤ 0 ∧ − ___const_400_0 + ___const_400_0 ≤ 0 ∧ ___const_400_0 − ___const_400_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 ∧ − ___const_5_0 + ___const_5_0 ≤ 0 ∧ ___const_5_0 − ___const_5_0 ≤ 0 ∧ − ___const_400_0 + ___const_400_0 ≤ 0 ∧ ___const_400_0 − ___const_400_0 ≤ 0
and for every transition t, a duplicate t is considered.

2 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

3 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 ∧ − ___const_5_0 + ___const_5_0 ≤ 0 ∧ ___const_5_0 − ___const_5_0 ≤ 0 ∧ − ___const_400_0 + ___const_400_0 ≤ 0 ∧ ___const_400_0 − ___const_400_0 ≤ 0

4 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 ∧ − ___const_5_0 + ___const_5_0 ≤ 0 ∧ ___const_5_0 − ___const_5_0 ≤ 0 ∧ − ___const_400_0 + ___const_400_0 ≤ 0 ∧ ___const_400_0 − ___const_400_0 ≤ 0

5 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 ∧ − ___const_5_0 + ___const_5_0 ≤ 0 ∧ ___const_5_0 − ___const_5_0 ≤ 0 ∧ − ___const_400_0 + ___const_400_0 ≤ 0 ∧ ___const_400_0 − ___const_400_0 ≤ 0

6 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 ∧ − ___const_5_0 + ___const_5_0 ≤ 0 ∧ ___const_5_0 − ___const_5_0 ≤ 0 ∧ − ___const_400_0 + ___const_400_0 ≤ 0 ∧ ___const_400_0 − ___const_400_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: − 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 ∧ − ___const_5_0 + ___const_5_0 ≤ 0 ∧ ___const_5_0 − ___const_5_0 ≤ 0 ∧ − ___const_400_0 + ___const_400_0 ≤ 0 ∧ ___const_400_0 − ___const_400_0 ≤ 0

8 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 ∧ − ___const_5_0 + ___const_5_0 ≤ 0 ∧ ___const_5_0 − ___const_5_0 ≤ 0 ∧ − ___const_400_0 + ___const_400_0 ≤ 0 ∧ ___const_400_0 − ___const_400_0 ≤ 0

9 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 ∧ − ___const_5_0 + ___const_5_0 ≤ 0 ∧ ___const_5_0 − ___const_5_0 ≤ 0 ∧ − ___const_400_0 + ___const_400_0 ≤ 0 ∧ ___const_400_0 − ___const_400_0 ≤ 0

10 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 ∧ − ___const_5_0 + ___const_5_0 ≤ 0 ∧ ___const_5_0 − ___const_5_0 ≤ 0 ∧ − ___const_400_0 + ___const_400_0 ≤ 0 ∧ ___const_400_0 − ___const_400_0 ≤ 0

11 SCC Decomposition

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

11.1 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* }.

11.1.1 Transition Removal

We remove transition 1 using the following ranking functions, which are bounded by 5.

2: 5⋅___const_5_0 − 5⋅i2_0
4: 5⋅___const_5_0 − 5⋅i2_0
6: 5⋅___const_5_0 − 5⋅i2_0
7: 5⋅___const_5_0 − 5⋅i2_0
8: 5⋅___const_5_0 − 5⋅i2_0
9: 5⋅___const_5_0 − 5⋅i2_0
10: 5⋅___const_5_0 − 5⋅i2_0
11: 5⋅___const_5_0 − 5⋅i2_0
0: 1 + 5⋅___const_5_0 − 5⋅i2_0
3: 3 + 5⋅___const_5_0 − 5⋅i2_0
2_var_snapshot: 5⋅___const_5_0 − 5⋅i2_0
2*: 5⋅___const_5_0 − 5⋅i2_0
3_var_snapshot: 2 + 5⋅___const_5_0 − 5⋅i2_0
3*: 4 + 5⋅___const_5_0 − 5⋅i2_0
6_var_snapshot: 5⋅___const_5_0 − 5⋅i2_0
6*: 5⋅___const_5_0 − 5⋅i2_0
9_var_snapshot: 5⋅___const_5_0 − 5⋅i2_0
9*: 5⋅___const_5_0 − 5⋅i2_0

11.1.2 Transition Removal

We remove transition 16 using the following ranking functions, which are bounded by 12.

2: 1 + 13⋅___const_5_0 − 13⋅j3_0
4: 13⋅___const_5_0 − 13⋅j3_0
6: −11 + 13⋅___const_5_0 − 13⋅j3_0
7: −11 + 13⋅___const_5_0 − 13⋅j3_0
8: −11 + 13⋅___const_5_0 − 13⋅j3_0
9: −11 + 13⋅___const_5_0 − 13⋅j3_0
10: −11 + 13⋅___const_5_0 − 13⋅j3_0
11: −11 + 13⋅___const_5_0 − 13⋅j3_0
0: −4 + 13⋅___const_5_0 − 13⋅j3_0
3: −2 + 13⋅___const_5_0 − 13⋅j3_0
2_var_snapshot: 1 + 13⋅___const_5_0 − 13⋅j3_0
2*: 2 + 13⋅___const_5_0 − 13⋅j3_0
3_var_snapshot: −3 + 13⋅___const_5_0 − 13⋅j3_0
3*: −1 + 13⋅___const_5_0 − 13⋅j3_0
6_var_snapshot: −11 + 13⋅___const_5_0 − 13⋅j3_0
6*: −11 + 13⋅___const_5_0 − 13⋅j3_0
9_var_snapshot: −11 + 13⋅___const_5_0 − 13⋅j3_0
9*: −11 + 13⋅___const_5_0 − 13⋅j3_0

11.1.3 Transition Removal

We remove transitions 20, 22, 27, 29, 2, 3, 13, 15 using the following ranking functions, which are bounded by −8.

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

11.1.4 Transition Removal

We remove transition 14 using the following ranking functions, which are bounded by 10.

2: 0
4: 0
6: 1 + 11⋅___const_5_0 − 11⋅k4_0
7: 11⋅___const_5_0 − 11⋅k4_0
8: −9 + 11⋅___const_5_0 − 11⋅k4_0
9: −9 + 11⋅___const_5_0 − 11⋅k4_0
10: −9 + 11⋅___const_5_0 − 11⋅k4_0
11: −9 + 11⋅___const_5_0 − 11⋅k4_0
0: 0
3: 0
2_var_snapshot: 0
2*: 0
3_var_snapshot: 0
3*: 0
6_var_snapshot: 1 + 11⋅___const_5_0 − 11⋅k4_0
6*: 2 + 11⋅___const_5_0 − 11⋅k4_0
9_var_snapshot: −9 + 11⋅___const_5_0 − 11⋅k4_0
9*: −9 + 11⋅___const_5_0 − 11⋅k4_0

11.1.5 Transition Removal

We remove transition 12 using the following ranking functions, which are bounded by 4.

2: 0
4: 0
6: −2 + 5⋅___const_5_0 − 5⋅l5_0
7: −4 + 5⋅___const_5_0 − 5⋅l5_0
8: −1 + 5⋅___const_5_0 − 5⋅l5_0
9: 2 + 5⋅___const_5_0 − 5⋅l5_0
10: 5⋅___const_5_0 − 5⋅l5_0
11: −1 + 5⋅___const_5_0 − 5⋅l5_0
0: 0
3: 0
2_var_snapshot: 0
2*: 0
3_var_snapshot: 0
3*: 0
6_var_snapshot: −3 + 5⋅___const_5_0 − 5⋅l5_0
6*: −1 + 5⋅___const_5_0 − 5⋅l5_0
9_var_snapshot: 1 + 5⋅___const_5_0 − 5⋅l5_0
9*: 3 + 5⋅___const_5_0 − 5⋅l5_0

11.1.6 Transition Removal

We remove transitions 34, 36, 41, 43, 5, 6, 7, 8, 10, 11 using the following ranking functions, which are bounded by −4.

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

11.1.7 Splitting Cut-Point Transitions

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

11.1.7.1 Cut-Point Subproblem 1/4

Here we consider cut-point transition 19.

11.1.7.1.1 Splitting Cut-Point Transitions

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

11.1.7.2 Cut-Point Subproblem 2/4

Here we consider cut-point transition 26.

11.1.7.2.1 Splitting Cut-Point Transitions

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

11.1.7.3 Cut-Point Subproblem 3/4

Here we consider cut-point transition 33.

11.1.7.3.1 Splitting Cut-Point Transitions

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

11.1.7.4 Cut-Point Subproblem 4/4

Here we consider cut-point transition 40.

11.1.7.4.1 Splitting Cut-Point Transitions

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

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