LTS Termination Proof

by T2Cert

Input

Integer Transition System

Initial Location: 6

Transitions: (pre-variables and post-variables)

0	0	1:	0 ≤ 0 ∧ 0 ≤ 0 ∧ 50 − i5_0 ≤ 0 ∧ i_post ≤ 0 ∧ − i_post ≤ 0 ∧ i_0 − i_post ≤ 0 ∧ − i_0 + i_post ≤ 0 ∧ − tmp_post + tmp_post ≤ 0 ∧ tmp_post − tmp_post ≤ 0 ∧ − tmp_0 + tmp_0 ≤ 0 ∧ tmp_0 − tmp_0 ≤ 0 ∧ − i5_post + i5_post ≤ 0 ∧ i5_post − i5_post ≤ 0 ∧ − i5_1 + i5_1 ≤ 0 ∧ i5_1 − i5_1 ≤ 0 ∧ − i5_0 + i5_0 ≤ 0 ∧ i5_0 − i5_0 ≤ 0
0	1	2:	0 ≤ 0 ∧ 0 ≤ 0 ∧ −49 + i5_0 ≤ 0 ∧ −1 − i5_0 + i5_post ≤ 0 ∧ 1 + i5_0 − i5_post ≤ 0 ∧ i5_0 − i5_post ≤ 0 ∧ − i5_0 + i5_post ≤ 0 ∧ − tmp_post + tmp_post ≤ 0 ∧ tmp_post − tmp_post ≤ 0 ∧ − tmp_0 + tmp_0 ≤ 0 ∧ tmp_0 − tmp_0 ≤ 0 ∧ − i_post + i_post ≤ 0 ∧ i_post − i_post ≤ 0 ∧ − i_0 + i_0 ≤ 0 ∧ i_0 − i_0 ≤ 0 ∧ − i5_1 + i5_1 ≤ 0 ∧ i5_1 − i5_1 ≤ 0
2	2	0:	− tmp_post + tmp_post ≤ 0 ∧ tmp_post − tmp_post ≤ 0 ∧ − tmp_0 + tmp_0 ≤ 0 ∧ tmp_0 − tmp_0 ≤ 0 ∧ − i_post + i_post ≤ 0 ∧ i_post − i_post ≤ 0 ∧ − i_0 + i_0 ≤ 0 ∧ i_0 − i_0 ≤ 0 ∧ − i5_post + i5_post ≤ 0 ∧ i5_post − i5_post ≤ 0 ∧ − i5_1 + i5_1 ≤ 0 ∧ i5_1 − i5_1 ≤ 0 ∧ − i5_0 + i5_0 ≤ 0 ∧ i5_0 − i5_0 ≤ 0
1	3	3:	− tmp_post + tmp_post ≤ 0 ∧ tmp_post − tmp_post ≤ 0 ∧ − tmp_0 + tmp_0 ≤ 0 ∧ tmp_0 − tmp_0 ≤ 0 ∧ − i_post + i_post ≤ 0 ∧ i_post − i_post ≤ 0 ∧ − i_0 + i_0 ≤ 0 ∧ i_0 − i_0 ≤ 0 ∧ − i5_post + i5_post ≤ 0 ∧ i5_post − i5_post ≤ 0 ∧ − i5_1 + i5_1 ≤ 0 ∧ i5_1 − i5_1 ≤ 0 ∧ − i5_0 + i5_0 ≤ 0 ∧ i5_0 − i5_0 ≤ 0
3	4	4:	50 − i_0 ≤ 0 ∧ − tmp_post + tmp_post ≤ 0 ∧ tmp_post − tmp_post ≤ 0 ∧ − tmp_0 + tmp_0 ≤ 0 ∧ tmp_0 − tmp_0 ≤ 0 ∧ − i_post + i_post ≤ 0 ∧ i_post − i_post ≤ 0 ∧ − i_0 + i_0 ≤ 0 ∧ i_0 − i_0 ≤ 0 ∧ − i5_post + i5_post ≤ 0 ∧ i5_post − i5_post ≤ 0 ∧ − i5_1 + i5_1 ≤ 0 ∧ i5_1 − i5_1 ≤ 0 ∧ − i5_0 + i5_0 ≤ 0 ∧ i5_0 − i5_0 ≤ 0
3	5	1:	0 ≤ 0 ∧ 0 ≤ 0 ∧ −49 + i_0 ≤ 0 ∧ −1 − i_0 + i_post ≤ 0 ∧ 1 + i_0 − i_post ≤ 0 ∧ i_0 − i_post ≤ 0 ∧ − i_0 + i_post ≤ 0 ∧ − tmp_post + tmp_post ≤ 0 ∧ tmp_post − tmp_post ≤ 0 ∧ − tmp_0 + tmp_0 ≤ 0 ∧ tmp_0 − tmp_0 ≤ 0 ∧ − i5_post + i5_post ≤ 0 ∧ i5_post − i5_post ≤ 0 ∧ − i5_1 + i5_1 ≤ 0 ∧ i5_1 − i5_1 ≤ 0 ∧ − i5_0 + i5_0 ≤ 0 ∧ i5_0 − i5_0 ≤ 0
5	6	2:	0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ i_post ≤ 0 ∧ − i_post ≤ 0 ∧ i5_1 ≤ 0 ∧ − i5_1 ≤ 0 ∧ i5_post ≤ 0 ∧ − i5_post ≤ 0 ∧ i5_0 − i5_post ≤ 0 ∧ − i5_0 + i5_post ≤ 0 ∧ i_0 − i_post ≤ 0 ∧ − i_0 + i_post ≤ 0 ∧ tmp_0 − tmp_post ≤ 0 ∧ − tmp_0 + tmp_post ≤ 0
6	7	5:	− tmp_post + tmp_post ≤ 0 ∧ tmp_post − tmp_post ≤ 0 ∧ − tmp_0 + tmp_0 ≤ 0 ∧ tmp_0 − tmp_0 ≤ 0 ∧ − i_post + i_post ≤ 0 ∧ i_post − i_post ≤ 0 ∧ − i_0 + i_0 ≤ 0 ∧ i_0 − i_0 ≤ 0 ∧ − i5_post + i5_post ≤ 0 ∧ i5_post − i5_post ≤ 0 ∧ − i5_1 + i5_1 ≤ 0 ∧ i5_1 − i5_1 ≤ 0 ∧ − i5_0 + i5_0 ≤ 0 ∧ i5_0 − i5_0 ≤ 0

Proof

1 Invariant Updates

The following invariants are asserted.

0:	i_post ≤ 0 ∧ − i_post ≤ 0 ∧ i5_1 ≤ 0 ∧ − i5_1 ≤ 0 ∧ i_0 ≤ 0 ∧ − i_0 ≤ 0
1:	i5_1 ≤ 0 ∧ − i5_1 ≤ 0 ∧ 50 − i5_0 ≤ 0
2:	i_post ≤ 0 ∧ − i_post ≤ 0 ∧ i5_1 ≤ 0 ∧ − i5_1 ≤ 0 ∧ i_0 ≤ 0 ∧ − i_0 ≤ 0
3:	i5_1 ≤ 0 ∧ − i5_1 ≤ 0 ∧ 50 − i5_0 ≤ 0
4:	i5_1 ≤ 0 ∧ − i5_1 ≤ 0 ∧ 50 − i5_0 ≤ 0 ∧ 50 − i_0 ≤ 0
5:	TRUE
6:	TRUE

The invariants are proved as follows.

IMPACT Invariant Proof

nodes (location) invariant:

0	(0)	i_post ≤ 0 ∧ − i_post ≤ 0 ∧ i5_1 ≤ 0 ∧ − i5_1 ≤ 0 ∧ i_0 ≤ 0 ∧ − i_0 ≤ 0
1	(1)	i5_1 ≤ 0 ∧ − i5_1 ≤ 0 ∧ 50 − i5_0 ≤ 0
2	(2)	i_post ≤ 0 ∧ − i_post ≤ 0 ∧ i5_1 ≤ 0 ∧ − i5_1 ≤ 0 ∧ i_0 ≤ 0 ∧ − i_0 ≤ 0
3	(3)	i5_1 ≤ 0 ∧ − i5_1 ≤ 0 ∧ 50 − i5_0 ≤ 0
4	(4)	i5_1 ≤ 0 ∧ − i5_1 ≤ 0 ∧ 50 − i5_0 ≤ 0 ∧ 50 − i_0 ≤ 0
5	(5)	TRUE
6	(6)	TRUE

initial node: 6
cover edges:
transition edges:

0 0 1

0 1 2

1 3 3

2 2 0

3 4 4

3 5 1

5 6 2

6 7 5

2 Switch to Cooperation Termination Proof

We consider the following cutpoint-transitions:

1	8	1:	− tmp_post + tmp_post ≤ 0 ∧ tmp_post − tmp_post ≤ 0 ∧ − tmp_0 + tmp_0 ≤ 0 ∧ tmp_0 − tmp_0 ≤ 0 ∧ − i_post + i_post ≤ 0 ∧ i_post − i_post ≤ 0 ∧ − i_0 + i_0 ≤ 0 ∧ i_0 − i_0 ≤ 0 ∧ − i5_post + i5_post ≤ 0 ∧ i5_post − i5_post ≤ 0 ∧ − i5_1 + i5_1 ≤ 0 ∧ i5_1 − i5_1 ≤ 0 ∧ − i5_0 + i5_0 ≤ 0 ∧ i5_0 − i5_0 ≤ 0
2	15	2:	− tmp_post + tmp_post ≤ 0 ∧ tmp_post − tmp_post ≤ 0 ∧ − tmp_0 + tmp_0 ≤ 0 ∧ tmp_0 − tmp_0 ≤ 0 ∧ − i_post + i_post ≤ 0 ∧ i_post − i_post ≤ 0 ∧ − i_0 + i_0 ≤ 0 ∧ i_0 − i_0 ≤ 0 ∧ − i5_post + i5_post ≤ 0 ∧ i5_post − i5_post ≤ 0 ∧ − i5_1 + i5_1 ≤ 0 ∧ i5_1 − i5_1 ≤ 0 ∧ − i5_0 + i5_0 ≤ 0 ∧ i5_0 − i5_0 ≤ 0

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

3 Transition Removal

We remove transitions 0, 4, 6, 7 using the following ranking functions, which are bounded by −17.

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

4 Location Addition

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

1* 11 1: − tmp_post + tmp_post ≤ 0 ∧ tmp_post − tmp_post ≤ 0 ∧ − tmp_0 + tmp_0 ≤ 0 ∧ tmp_0 − tmp_0 ≤ 0 ∧ − i_post + i_post ≤ 0 ∧ i_post − i_post ≤ 0 ∧ − i_0 + i_0 ≤ 0 ∧ i_0 − i_0 ≤ 0 ∧ − i5_post + i5_post ≤ 0 ∧ i5_post − i5_post ≤ 0 ∧ − i5_1 + i5_1 ≤ 0 ∧ i5_1 − i5_1 ≤ 0 ∧ − i5_0 + i5_0 ≤ 0 ∧ i5_0 − i5_0 ≤ 0

5 Location Addition

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

1 9 1_var_snapshot: − tmp_post + tmp_post ≤ 0 ∧ tmp_post − tmp_post ≤ 0 ∧ − tmp_0 + tmp_0 ≤ 0 ∧ tmp_0 − tmp_0 ≤ 0 ∧ − i_post + i_post ≤ 0 ∧ i_post − i_post ≤ 0 ∧ − i_0 + i_0 ≤ 0 ∧ i_0 − i_0 ≤ 0 ∧ − i5_post + i5_post ≤ 0 ∧ i5_post − i5_post ≤ 0 ∧ − i5_1 + i5_1 ≤ 0 ∧ i5_1 − i5_1 ≤ 0 ∧ − i5_0 + i5_0 ≤ 0 ∧ i5_0 − i5_0 ≤ 0

6 Location Addition

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

2* 18 2: − tmp_post + tmp_post ≤ 0 ∧ tmp_post − tmp_post ≤ 0 ∧ − tmp_0 + tmp_0 ≤ 0 ∧ tmp_0 − tmp_0 ≤ 0 ∧ − i_post + i_post ≤ 0 ∧ i_post − i_post ≤ 0 ∧ − i_0 + i_0 ≤ 0 ∧ i_0 − i_0 ≤ 0 ∧ − i5_post + i5_post ≤ 0 ∧ i5_post − i5_post ≤ 0 ∧ − i5_1 + i5_1 ≤ 0 ∧ i5_1 − i5_1 ≤ 0 ∧ − i5_0 + i5_0 ≤ 0 ∧ i5_0 − i5_0 ≤ 0

7 Location Addition

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

2 16 2_var_snapshot: − tmp_post + tmp_post ≤ 0 ∧ tmp_post − tmp_post ≤ 0 ∧ − tmp_0 + tmp_0 ≤ 0 ∧ tmp_0 − tmp_0 ≤ 0 ∧ − i_post + i_post ≤ 0 ∧ i_post − i_post ≤ 0 ∧ − i_0 + i_0 ≤ 0 ∧ i_0 − i_0 ≤ 0 ∧ − i5_post + i5_post ≤ 0 ∧ i5_post − i5_post ≤ 0 ∧ − i5_1 + i5_1 ≤ 0 ∧ i5_1 − i5_1 ≤ 0 ∧ − i5_0 + i5_0 ≤ 0 ∧ i5_0 − i5_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, 1_var_snapshot, 1* }.

8.1.1 Transition Removal

We remove transition 5 using the following ranking functions, which are bounded by −197.

1:	2 − 4⋅i_0
3:	−4⋅i_0
1_var_snapshot:	1 − 4⋅i_0
1*:	3 − 4⋅i_0

8.1.2 Transition Removal

We remove transitions 9, 3 using the following ranking functions, which are bounded by −1.

1:	i5_0
3:	−1
1_var_snapshot:	0
1*:	2⋅i5_0

8.1.3 Transition Removal

We remove transition 11 using the following ranking functions, which are bounded by 49.

1:	0
3:	0
1_var_snapshot:	0
1*:	i5_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 8.

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, 2, 2_var_snapshot, 2* }.

8.2.1 Transition Removal

We remove transition 1 using the following ranking functions, which are bounded by −198.

0:	−1 − 4⋅i5_0
2:	1 − 4⋅i5_0
2_var_snapshot:	−4⋅i5_0
2*:	2 − 4⋅i5_0

8.2.2 Transition Removal

We remove transitions 16, 18, 2 using the following ranking functions, which are bounded by −1.

0:	−1
2:	1
2_var_snapshot:	0
2*:	2

8.2.3 Splitting Cut-Point Transitions

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

8.2.3.1 Cut-Point Subproblem 1/1

Here we consider cut-point transition 15.

8.2.3.1.1 Splitting Cut-Point Transitions

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

Tool configuration

T2Cert

version: 1.0