LTS Termination Proof

by T2Cert

Input

Integer Transition System

Initial Location: 6

Transitions: (pre-variables and post-variables)

0	0	1:	− x79_post + x79_post ≤ 0 ∧ x79_post − x79_post ≤ 0 ∧ − x79_0 + x79_0 ≤ 0 ∧ x79_0 − x79_0 ≤ 0 ∧ − x35_post + x35_post ≤ 0 ∧ x35_post − x35_post ≤ 0 ∧ − x35_0 + x35_0 ≤ 0 ∧ x35_0 − x35_0 ≤ 0 ∧ − tmp_post + tmp_post ≤ 0 ∧ tmp_post − tmp_post ≤ 0 ∧ − tmp_0 + tmp_0 ≤ 0 ∧ tmp_0 − tmp_0 ≤ 0 ∧ − i911_post + i911_post ≤ 0 ∧ i911_post − i911_post ≤ 0 ∧ − i911_0 + i911_0 ≤ 0 ∧ i911_0 − i911_0 ≤ 0 ∧ − i57_post + i57_post ≤ 0 ∧ i57_post − i57_post ≤ 0 ∧ − i57_0 + i57_0 ≤ 0 ∧ i57_0 − i57_0 ≤ 0
2	1	3:	10 − i911_0 ≤ 0 ∧ − x79_post + x79_post ≤ 0 ∧ x79_post − x79_post ≤ 0 ∧ − x79_0 + x79_0 ≤ 0 ∧ x79_0 − x79_0 ≤ 0 ∧ − x35_post + x35_post ≤ 0 ∧ x35_post − x35_post ≤ 0 ∧ − x35_0 + x35_0 ≤ 0 ∧ x35_0 − x35_0 ≤ 0 ∧ − tmp_post + tmp_post ≤ 0 ∧ tmp_post − tmp_post ≤ 0 ∧ − tmp_0 + tmp_0 ≤ 0 ∧ tmp_0 − tmp_0 ≤ 0 ∧ − i911_post + i911_post ≤ 0 ∧ i911_post − i911_post ≤ 0 ∧ − i911_0 + i911_0 ≤ 0 ∧ i911_0 − i911_0 ≤ 0 ∧ − i57_post + i57_post ≤ 0 ∧ i57_post − i57_post ≤ 0 ∧ − i57_0 + i57_0 ≤ 0 ∧ i57_0 − i57_0 ≤ 0
2	2	4:	0 ≤ 0 ∧ 0 ≤ 0 ∧ −9 + i911_0 ≤ 0 ∧ −1 − i911_0 + i911_post ≤ 0 ∧ 1 + i911_0 − i911_post ≤ 0 ∧ i911_0 − i911_post ≤ 0 ∧ − i911_0 + i911_post ≤ 0 ∧ − x79_post + x79_post ≤ 0 ∧ x79_post − x79_post ≤ 0 ∧ − x79_0 + x79_0 ≤ 0 ∧ x79_0 − x79_0 ≤ 0 ∧ − x35_post + x35_post ≤ 0 ∧ x35_post − x35_post ≤ 0 ∧ − x35_0 + x35_0 ≤ 0 ∧ x35_0 − x35_0 ≤ 0 ∧ − tmp_post + tmp_post ≤ 0 ∧ tmp_post − tmp_post ≤ 0 ∧ − tmp_0 + tmp_0 ≤ 0 ∧ tmp_0 − tmp_0 ≤ 0 ∧ − i57_post + i57_post ≤ 0 ∧ i57_post − i57_post ≤ 0 ∧ − i57_0 + i57_0 ≤ 0 ∧ i57_0 − i57_0 ≤ 0
4	3	2:	− x79_post + x79_post ≤ 0 ∧ x79_post − x79_post ≤ 0 ∧ − x79_0 + x79_0 ≤ 0 ∧ x79_0 − x79_0 ≤ 0 ∧ − x35_post + x35_post ≤ 0 ∧ x35_post − x35_post ≤ 0 ∧ − x35_0 + x35_0 ≤ 0 ∧ x35_0 − x35_0 ≤ 0 ∧ − tmp_post + tmp_post ≤ 0 ∧ tmp_post − tmp_post ≤ 0 ∧ − tmp_0 + tmp_0 ≤ 0 ∧ tmp_0 − tmp_0 ≤ 0 ∧ − i911_post + i911_post ≤ 0 ∧ i911_post − i911_post ≤ 0 ∧ − i911_0 + i911_0 ≤ 0 ∧ i911_0 − i911_0 ≤ 0 ∧ − i57_post + i57_post ≤ 0 ∧ i57_post − i57_post ≤ 0 ∧ − i57_0 + i57_0 ≤ 0 ∧ i57_0 − i57_0 ≤ 0
1	4	4:	0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 10 − i57_0 ≤ 0 ∧ i911_post ≤ 0 ∧ − i911_post ≤ 0 ∧ i911_0 − i911_post ≤ 0 ∧ − i911_0 + i911_post ≤ 0 ∧ x79_0 − x79_post ≤ 0 ∧ − x79_0 + x79_post ≤ 0 ∧ − x35_post + x35_post ≤ 0 ∧ x35_post − x35_post ≤ 0 ∧ − x35_0 + x35_0 ≤ 0 ∧ x35_0 − x35_0 ≤ 0 ∧ − tmp_post + tmp_post ≤ 0 ∧ tmp_post − tmp_post ≤ 0 ∧ − tmp_0 + tmp_0 ≤ 0 ∧ tmp_0 − tmp_0 ≤ 0 ∧ − i57_post + i57_post ≤ 0 ∧ i57_post − i57_post ≤ 0 ∧ − i57_0 + i57_0 ≤ 0 ∧ i57_0 − i57_0 ≤ 0
1	5	0:	0 ≤ 0 ∧ 0 ≤ 0 ∧ −9 + i57_0 ≤ 0 ∧ −1 − i57_0 + i57_post ≤ 0 ∧ 1 + i57_0 − i57_post ≤ 0 ∧ i57_0 − i57_post ≤ 0 ∧ − i57_0 + i57_post ≤ 0 ∧ − x79_post + x79_post ≤ 0 ∧ x79_post − x79_post ≤ 0 ∧ − x79_0 + x79_0 ≤ 0 ∧ x79_0 − x79_0 ≤ 0 ∧ − x35_post + x35_post ≤ 0 ∧ x35_post − x35_post ≤ 0 ∧ − x35_0 + x35_0 ≤ 0 ∧ x35_0 − x35_0 ≤ 0 ∧ − tmp_post + tmp_post ≤ 0 ∧ tmp_post − tmp_post ≤ 0 ∧ − tmp_0 + tmp_0 ≤ 0 ∧ tmp_0 − tmp_0 ≤ 0 ∧ − i911_post + i911_post ≤ 0 ∧ i911_post − i911_post ≤ 0 ∧ − i911_0 + i911_0 ≤ 0 ∧ i911_0 − i911_0 ≤ 0
5	6	0:	0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ i57_post ≤ 0 ∧ − i57_post ≤ 0 ∧ i57_0 − i57_post ≤ 0 ∧ − i57_0 + i57_post ≤ 0 ∧ tmp_0 − tmp_post ≤ 0 ∧ − tmp_0 + tmp_post ≤ 0 ∧ x35_0 − x35_post ≤ 0 ∧ − x35_0 + x35_post ≤ 0 ∧ − x79_post + x79_post ≤ 0 ∧ x79_post − x79_post ≤ 0 ∧ − x79_0 + x79_0 ≤ 0 ∧ x79_0 − x79_0 ≤ 0 ∧ − i911_post + i911_post ≤ 0 ∧ i911_post − i911_post ≤ 0 ∧ − i911_0 + i911_0 ≤ 0 ∧ i911_0 − i911_0 ≤ 0
6	7	5:	− x79_post + x79_post ≤ 0 ∧ x79_post − x79_post ≤ 0 ∧ − x79_0 + x79_0 ≤ 0 ∧ x79_0 − x79_0 ≤ 0 ∧ − x35_post + x35_post ≤ 0 ∧ x35_post − x35_post ≤ 0 ∧ − x35_0 + x35_0 ≤ 0 ∧ x35_0 − x35_0 ≤ 0 ∧ − tmp_post + tmp_post ≤ 0 ∧ tmp_post − tmp_post ≤ 0 ∧ − tmp_0 + tmp_0 ≤ 0 ∧ tmp_0 − tmp_0 ≤ 0 ∧ − i911_post + i911_post ≤ 0 ∧ i911_post − i911_post ≤ 0 ∧ − i911_0 + i911_0 ≤ 0 ∧ i911_0 − i911_0 ≤ 0 ∧ − i57_post + i57_post ≤ 0 ∧ i57_post − i57_post ≤ 0 ∧ − i57_0 + i57_0 ≤ 0 ∧ i57_0 − i57_0 ≤ 0

Proof

1 Invariant Updates

The following invariants are asserted.

0:	TRUE
1:	TRUE
2:	10 − i57_0 ≤ 0
3:	10 − i57_0 ≤ 0 ∧ 10 − i911_0 ≤ 0
4:	10 − i57_0 ≤ 0
5:	TRUE
6:	TRUE

The invariants are proved as follows.

IMPACT Invariant Proof

nodes (location) invariant:

0	(0)	TRUE
1	(1)	TRUE
2	(2)	10 − i57_0 ≤ 0
3	(3)	10 − i57_0 ≤ 0 ∧ 10 − i911_0 ≤ 0
4	(4)	10 − i57_0 ≤ 0
5	(5)	TRUE
6	(6)	TRUE

initial node: 6
cover edges:
transition edges:

0 0 1

1 4 4

1 5 0

2 1 3

2 2 4

4 3 2

5 6 0

6 7 5

2 Switch to Cooperation Termination Proof

We consider the following cutpoint-transitions:

0	8	0:	− x79_post + x79_post ≤ 0 ∧ x79_post − x79_post ≤ 0 ∧ − x79_0 + x79_0 ≤ 0 ∧ x79_0 − x79_0 ≤ 0 ∧ − x35_post + x35_post ≤ 0 ∧ x35_post − x35_post ≤ 0 ∧ − x35_0 + x35_0 ≤ 0 ∧ x35_0 − x35_0 ≤ 0 ∧ − tmp_post + tmp_post ≤ 0 ∧ tmp_post − tmp_post ≤ 0 ∧ − tmp_0 + tmp_0 ≤ 0 ∧ tmp_0 − tmp_0 ≤ 0 ∧ − i911_post + i911_post ≤ 0 ∧ i911_post − i911_post ≤ 0 ∧ − i911_0 + i911_0 ≤ 0 ∧ i911_0 − i911_0 ≤ 0 ∧ − i57_post + i57_post ≤ 0 ∧ i57_post − i57_post ≤ 0 ∧ − i57_0 + i57_0 ≤ 0 ∧ i57_0 − i57_0 ≤ 0
4	15	4:	− x79_post + x79_post ≤ 0 ∧ x79_post − x79_post ≤ 0 ∧ − x79_0 + x79_0 ≤ 0 ∧ x79_0 − x79_0 ≤ 0 ∧ − x35_post + x35_post ≤ 0 ∧ x35_post − x35_post ≤ 0 ∧ − x35_0 + x35_0 ≤ 0 ∧ x35_0 − x35_0 ≤ 0 ∧ − tmp_post + tmp_post ≤ 0 ∧ tmp_post − tmp_post ≤ 0 ∧ − tmp_0 + tmp_0 ≤ 0 ∧ tmp_0 − tmp_0 ≤ 0 ∧ − i911_post + i911_post ≤ 0 ∧ i911_post − i911_post ≤ 0 ∧ − i911_0 + i911_0 ≤ 0 ∧ i911_0 − i911_0 ≤ 0 ∧ − i57_post + i57_post ≤ 0 ∧ i57_post − i57_post ≤ 0 ∧ − i57_0 + i57_0 ≤ 0 ∧ i57_0 − i57_0 ≤ 0

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

3 Transition Removal

We remove transitions 1, 4, 6, 7 using the following ranking functions, which are bounded by −17.

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

4 Location Addition

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

0* 11 0: − x79_post + x79_post ≤ 0 ∧ x79_post − x79_post ≤ 0 ∧ − x79_0 + x79_0 ≤ 0 ∧ x79_0 − x79_0 ≤ 0 ∧ − x35_post + x35_post ≤ 0 ∧ x35_post − x35_post ≤ 0 ∧ − x35_0 + x35_0 ≤ 0 ∧ x35_0 − x35_0 ≤ 0 ∧ − tmp_post + tmp_post ≤ 0 ∧ tmp_post − tmp_post ≤ 0 ∧ − tmp_0 + tmp_0 ≤ 0 ∧ tmp_0 − tmp_0 ≤ 0 ∧ − i911_post + i911_post ≤ 0 ∧ i911_post − i911_post ≤ 0 ∧ − i911_0 + i911_0 ≤ 0 ∧ i911_0 − i911_0 ≤ 0 ∧ − i57_post + i57_post ≤ 0 ∧ i57_post − i57_post ≤ 0 ∧ − i57_0 + i57_0 ≤ 0 ∧ i57_0 − i57_0 ≤ 0

5 Location Addition

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

0 9 0_var_snapshot: − x79_post + x79_post ≤ 0 ∧ x79_post − x79_post ≤ 0 ∧ − x79_0 + x79_0 ≤ 0 ∧ x79_0 − x79_0 ≤ 0 ∧ − x35_post + x35_post ≤ 0 ∧ x35_post − x35_post ≤ 0 ∧ − x35_0 + x35_0 ≤ 0 ∧ x35_0 − x35_0 ≤ 0 ∧ − tmp_post + tmp_post ≤ 0 ∧ tmp_post − tmp_post ≤ 0 ∧ − tmp_0 + tmp_0 ≤ 0 ∧ tmp_0 − tmp_0 ≤ 0 ∧ − i911_post + i911_post ≤ 0 ∧ i911_post − i911_post ≤ 0 ∧ − i911_0 + i911_0 ≤ 0 ∧ i911_0 − i911_0 ≤ 0 ∧ − i57_post + i57_post ≤ 0 ∧ i57_post − i57_post ≤ 0 ∧ − i57_0 + i57_0 ≤ 0 ∧ i57_0 − i57_0 ≤ 0

6 Location Addition

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

4* 18 4: − x79_post + x79_post ≤ 0 ∧ x79_post − x79_post ≤ 0 ∧ − x79_0 + x79_0 ≤ 0 ∧ x79_0 − x79_0 ≤ 0 ∧ − x35_post + x35_post ≤ 0 ∧ x35_post − x35_post ≤ 0 ∧ − x35_0 + x35_0 ≤ 0 ∧ x35_0 − x35_0 ≤ 0 ∧ − tmp_post + tmp_post ≤ 0 ∧ tmp_post − tmp_post ≤ 0 ∧ − tmp_0 + tmp_0 ≤ 0 ∧ tmp_0 − tmp_0 ≤ 0 ∧ − i911_post + i911_post ≤ 0 ∧ i911_post − i911_post ≤ 0 ∧ − i911_0 + i911_0 ≤ 0 ∧ i911_0 − i911_0 ≤ 0 ∧ − i57_post + i57_post ≤ 0 ∧ i57_post − i57_post ≤ 0 ∧ − i57_0 + i57_0 ≤ 0 ∧ i57_0 − i57_0 ≤ 0

7 Location Addition

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

4 16 4_var_snapshot: − x79_post + x79_post ≤ 0 ∧ x79_post − x79_post ≤ 0 ∧ − x79_0 + x79_0 ≤ 0 ∧ x79_0 − x79_0 ≤ 0 ∧ − x35_post + x35_post ≤ 0 ∧ x35_post − x35_post ≤ 0 ∧ − x35_0 + x35_0 ≤ 0 ∧ x35_0 − x35_0 ≤ 0 ∧ − tmp_post + tmp_post ≤ 0 ∧ tmp_post − tmp_post ≤ 0 ∧ − tmp_0 + tmp_0 ≤ 0 ∧ tmp_0 − tmp_0 ≤ 0 ∧ − i911_post + i911_post ≤ 0 ∧ i911_post − i911_post ≤ 0 ∧ − i911_0 + i911_0 ≤ 0 ∧ i911_0 − i911_0 ≤ 0 ∧ − i57_post + i57_post ≤ 0 ∧ i57_post − i57_post ≤ 0 ∧ − i57_0 + i57_0 ≤ 0 ∧ i57_0 − i57_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 { 2, 4, 4_var_snapshot, 4* }.

8.1.1 Transition Removal

We remove transition 2 using the following ranking functions, which are bounded by −38.

2:	−1 − 4⋅i911_0
4:	1 − 4⋅i911_0
4_var_snapshot:	−4⋅i911_0
4*:	2 − 4⋅i911_0

8.1.2 Transition Removal

We remove transitions 16, 18 using the following ranking functions, which are bounded by −11.

2:	−1 − 2⋅i57_0
4:	−10
4_var_snapshot:	−2⋅i57_0
4*:	0

8.1.3 Transition Removal

We remove transition 3 using the following ranking functions, which are bounded by 9.

2:	0
4:	0
4_var_snapshot:	i57_0
4*:	0

8.1.4 Splitting Cut-Point Transitions

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

8.1.4.1 Cut-Point Subproblem 1/1

Here we consider cut-point transition 15.

8.1.4.1.1 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, 1, 0_var_snapshot, 0* }.

8.2.1 Transition Removal

We remove transition 5 using the following ranking functions, which are bounded by −28.

0:	2 − 3⋅i57_0
1:	−3⋅i57_0
0_var_snapshot:	1 − 3⋅i57_0
0*:	2 − 3⋅i57_0

8.2.2 Transition Removal

We remove transitions 9, 11 using the following ranking functions, which are bounded by −1.

0:	0
1:	−2
0_var_snapshot:	−1
0*:	1

8.2.3 Transition Removal

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

0:	0
1:	0
0_var_snapshot:	1
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 8.

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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