LTS Termination Proof

by T2Cert

Input

Integer Transition System

Initial Location: 3

Transitions: (pre-variables and post-variables)

0	0	1:	0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 1 − arg1P ≤ 0 ∧ 27 − arg2P ≤ 0 ∧ −27 + arg2P ≤ 0 ∧ 28 − arg3P ≤ 0 ∧ −28 + arg3P ≤ 0 ∧ − arg1P + arg1 ≤ 0 ∧ arg1P − arg1 ≤ 0 ∧ − arg2P + arg2 ≤ 0 ∧ arg2P − arg2 ≤ 0 ∧ − arg3P + arg3 ≤ 0 ∧ arg3P − arg3 ≤ 0
1	1	1:	0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ −3 + arg1P − arg1 ≤ 0 ∧ 1 − arg3 ≤ 0 ∧ 1 − arg1 ≤ 0 ∧ 4 − arg1P ≤ 0 ∧ −1 − arg2P + arg2 ≤ 0 ∧ 1 + arg2P − arg2 ≤ 0 ∧ arg2 − arg3P ≤ 0 ∧ − arg2 + arg3P ≤ 0 ∧ − arg1P + arg1 ≤ 0 ∧ arg1P − arg1 ≤ 0 ∧ − arg2P + arg2 ≤ 0 ∧ arg2P − arg2 ≤ 0 ∧ − arg3P + arg3 ≤ 0 ∧ arg3P − arg3 ≤ 0
1	2	2:	0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 1 + arg1P − arg1 ≤ 0 ∧ arg3 ≤ 0 ∧ 1 − arg1 ≤ 0 ∧ − arg1P ≤ 0 ∧ − arg1P + arg1 ≤ 0 ∧ arg1P − arg1 ≤ 0 ∧ − arg2P + arg2 ≤ 0 ∧ arg2P − arg2 ≤ 0 ∧ − arg3P + arg3 ≤ 0 ∧ arg3P − arg3 ≤ 0
2	3	2:	0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 1 + arg1P − arg1 ≤ 0 ∧ 1 − arg1 ≤ 0 ∧ − arg1P ≤ 0 ∧ − arg1P + arg1 ≤ 0 ∧ arg1P − arg1 ≤ 0 ∧ − arg2P + arg2 ≤ 0 ∧ arg2P − arg2 ≤ 0 ∧ − arg3P + arg3 ≤ 0 ∧ arg3P − arg3 ≤ 0
3	4	0:	0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ − arg1P + arg1 ≤ 0 ∧ arg1P − arg1 ≤ 0 ∧ − arg2P + arg2 ≤ 0 ∧ arg2P − arg2 ≤ 0 ∧ − arg3P + arg3 ≤ 0 ∧ arg3P − arg3 ≤ 0

Proof

1 Invariant Updates

The following invariants are asserted.

0:	TRUE
1:	1 − arg1P ≤ 0 ∧ 1 − arg1 ≤ 0
2:	− arg1P ≤ 0 ∧ − arg1 ≤ 0
3:	TRUE

The invariants are proved as follows.

IMPACT Invariant Proof

nodes (location) invariant:

0	(0)	TRUE
1	(1)	1 − arg1P ≤ 0 ∧ 1 − arg1 ≤ 0
2	(2)	− arg1P ≤ 0 ∧ − arg1 ≤ 0
3	(3)	TRUE

initial node: 3
cover edges:
transition edges:

0 0 1

1 1 1

1 2 2

2 3 2

3 4 0

2 Switch to Cooperation Termination Proof

We consider the following cutpoint-transitions:

1	5	1:	− arg3P + arg3P ≤ 0 ∧ arg3P − arg3P ≤ 0 ∧ − arg3 + arg3 ≤ 0 ∧ arg3 − arg3 ≤ 0 ∧ − arg2P + arg2P ≤ 0 ∧ arg2P − arg2P ≤ 0 ∧ − arg2 + arg2 ≤ 0 ∧ arg2 − arg2 ≤ 0 ∧ − arg1P + arg1P ≤ 0 ∧ arg1P − arg1P ≤ 0 ∧ − arg1 + arg1 ≤ 0 ∧ arg1 − arg1 ≤ 0
2	12	2:	− arg3P + arg3P ≤ 0 ∧ arg3P − arg3P ≤ 0 ∧ − arg3 + arg3 ≤ 0 ∧ arg3 − arg3 ≤ 0 ∧ − arg2P + arg2P ≤ 0 ∧ arg2P − arg2P ≤ 0 ∧ − arg2 + arg2 ≤ 0 ∧ arg2 − arg2 ≤ 0 ∧ − arg1P + arg1P ≤ 0 ∧ arg1P − arg1P ≤ 0 ∧ − arg1 + arg1 ≤ 0 ∧ arg1 − arg1 ≤ 0

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

3 Transition Removal

We remove transitions 0, 2, 4 using the following ranking functions, which are bounded by −15.

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

4 Location Addition

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

1* 8 1: − arg3P + arg3P ≤ 0 ∧ arg3P − arg3P ≤ 0 ∧ − arg3 + arg3 ≤ 0 ∧ arg3 − arg3 ≤ 0 ∧ − arg2P + arg2P ≤ 0 ∧ arg2P − arg2P ≤ 0 ∧ − arg2 + arg2 ≤ 0 ∧ arg2 − arg2 ≤ 0 ∧ − arg1P + arg1P ≤ 0 ∧ arg1P − arg1P ≤ 0 ∧ − arg1 + arg1 ≤ 0 ∧ arg1 − arg1 ≤ 0

5 Location Addition

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

1 6 1_var_snapshot: − arg3P + arg3P ≤ 0 ∧ arg3P − arg3P ≤ 0 ∧ − arg3 + arg3 ≤ 0 ∧ arg3 − arg3 ≤ 0 ∧ − arg2P + arg2P ≤ 0 ∧ arg2P − arg2P ≤ 0 ∧ − arg2 + arg2 ≤ 0 ∧ arg2 − arg2 ≤ 0 ∧ − arg1P + arg1P ≤ 0 ∧ arg1P − arg1P ≤ 0 ∧ − arg1 + arg1 ≤ 0 ∧ arg1 − arg1 ≤ 0

6 Location Addition

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

2* 15 2: − arg3P + arg3P ≤ 0 ∧ arg3P − arg3P ≤ 0 ∧ − arg3 + arg3 ≤ 0 ∧ arg3 − arg3 ≤ 0 ∧ − arg2P + arg2P ≤ 0 ∧ arg2P − arg2P ≤ 0 ∧ − arg2 + arg2 ≤ 0 ∧ arg2 − arg2 ≤ 0 ∧ − arg1P + arg1P ≤ 0 ∧ arg1P − arg1P ≤ 0 ∧ − arg1 + arg1 ≤ 0 ∧ arg1 − arg1 ≤ 0

7 Location Addition

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

2 13 2_var_snapshot: − arg3P + arg3P ≤ 0 ∧ arg3P − arg3P ≤ 0 ∧ − arg3 + arg3 ≤ 0 ∧ arg3 − arg3 ≤ 0 ∧ − arg2P + arg2P ≤ 0 ∧ arg2P − arg2P ≤ 0 ∧ − arg2 + arg2 ≤ 0 ∧ arg2 − arg2 ≤ 0 ∧ − arg1P + arg1P ≤ 0 ∧ arg1P − arg1P ≤ 0 ∧ − arg1 + arg1 ≤ 0 ∧ arg1 − arg1 ≤ 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, 2_var_snapshot, 2* }.

8.1.1 Transition Removal

We remove transitions 13, 15, 3 using the following ranking functions, which are bounded by −1.

2:	1 + 3⋅arg1
2_var_snapshot:	3⋅arg1
2*:	2 + 3⋅arg1

8.1.2 Splitting Cut-Point Transitions

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

8.1.2.1 Cut-Point Subproblem 1/1

Here we consider cut-point transition 12.

8.1.2.1.1 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 { 1, 1_var_snapshot, 1* }.

8.2.1 Splitting Cut-Point Transitions

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

8.2.1.1 Cut-Point Subproblem 1/1

Here we consider cut-point transition 5.

8.2.1.1.1 Fresh Variable Addition

The new variable __snapshot_1_arg3P is introduced. The transition formulas are extended as follows:

6:	__snapshot_1_arg3P ≤ arg3P ∧ arg3P ≤ __snapshot_1_arg3P
8:	__snapshot_1_arg3P ≤ __snapshot_1_arg3P ∧ __snapshot_1_arg3P ≤ __snapshot_1_arg3P
1:	__snapshot_1_arg3P ≤ __snapshot_1_arg3P ∧ __snapshot_1_arg3P ≤ __snapshot_1_arg3P

8.2.1.1.2 Fresh Variable Addition

The new variable __snapshot_1_arg3 is introduced. The transition formulas are extended as follows:

6:	__snapshot_1_arg3 ≤ arg3 ∧ arg3 ≤ __snapshot_1_arg3
8:	__snapshot_1_arg3 ≤ __snapshot_1_arg3 ∧ __snapshot_1_arg3 ≤ __snapshot_1_arg3
1:	__snapshot_1_arg3 ≤ __snapshot_1_arg3 ∧ __snapshot_1_arg3 ≤ __snapshot_1_arg3

8.2.1.1.3 Fresh Variable Addition

The new variable __snapshot_1_arg2P is introduced. The transition formulas are extended as follows:

6:	__snapshot_1_arg2P ≤ arg2P ∧ arg2P ≤ __snapshot_1_arg2P
8:	__snapshot_1_arg2P ≤ __snapshot_1_arg2P ∧ __snapshot_1_arg2P ≤ __snapshot_1_arg2P
1:	__snapshot_1_arg2P ≤ __snapshot_1_arg2P ∧ __snapshot_1_arg2P ≤ __snapshot_1_arg2P

8.2.1.1.4 Fresh Variable Addition

The new variable __snapshot_1_arg2 is introduced. The transition formulas are extended as follows:

6:	__snapshot_1_arg2 ≤ arg2 ∧ arg2 ≤ __snapshot_1_arg2
8:	__snapshot_1_arg2 ≤ __snapshot_1_arg2 ∧ __snapshot_1_arg2 ≤ __snapshot_1_arg2
1:	__snapshot_1_arg2 ≤ __snapshot_1_arg2 ∧ __snapshot_1_arg2 ≤ __snapshot_1_arg2

8.2.1.1.5 Fresh Variable Addition

The new variable __snapshot_1_arg1P is introduced. The transition formulas are extended as follows:

6:	__snapshot_1_arg1P ≤ arg1P ∧ arg1P ≤ __snapshot_1_arg1P
8:	__snapshot_1_arg1P ≤ __snapshot_1_arg1P ∧ __snapshot_1_arg1P ≤ __snapshot_1_arg1P
1:	__snapshot_1_arg1P ≤ __snapshot_1_arg1P ∧ __snapshot_1_arg1P ≤ __snapshot_1_arg1P

8.2.1.1.6 Fresh Variable Addition

The new variable __snapshot_1_arg1 is introduced. The transition formulas are extended as follows:

6:	__snapshot_1_arg1 ≤ arg1 ∧ arg1 ≤ __snapshot_1_arg1
8:	__snapshot_1_arg1 ≤ __snapshot_1_arg1 ∧ __snapshot_1_arg1 ≤ __snapshot_1_arg1
1:	__snapshot_1_arg1 ≤ __snapshot_1_arg1 ∧ __snapshot_1_arg1 ≤ __snapshot_1_arg1

8.2.1.1.7 Invariant Updates

The following invariants are asserted.

0:	TRUE
1:	1 + arg2 − arg3P ≤ 0 ∧ − arg3P ≤ 0 ∧ 1 − arg1P ≤ 0 ∧ 1 − arg1 ≤ 0 ∨ −1 + arg3 − arg3P ≤ 0 ∧ 1 + arg2 − arg3P ≤ 0 ∧ 1 − arg1P ≤ 0 ∧ 1 − arg1 ≤ 0
2:	− arg1P ≤ 0 ∧ − arg1 ≤ 0
3:	TRUE
1:	1 + arg2 − arg3P ≤ 0 ∧ − arg3P ≤ 0 ∧ 1 − arg1P ≤ 0 ∧ 1 − arg1 ≤ 0 ∨ −1 + arg3 − arg3P ≤ 0 ∧ 1 − __snapshot_1_arg3P + arg3P ≤ 0 ∧ 1 + arg2 − arg3P ≤ 0 ∧ − __snapshot_1_arg3P ≤ 0 ∧ 1 − arg1P ≤ 0 ∧ 1 − arg1 ≤ 0 ∨ −1 + arg3 − arg3P ≤ 0 ∧ 1 + arg2 − arg3P ≤ 0 ∧ 1 − arg1P ≤ 0 ∧ 1 − arg1 ≤ 0
1_var_snapshot:	1 − __snapshot_1_arg3P + arg2 ≤ 0 ∧ − __snapshot_1_arg3P ≤ 0 ∧ 1 − arg1P ≤ 0 ∧ 1 − arg1 ≤ 0 ∨ 1 − __snapshot_1_arg3P + arg2 ≤ 0 ∧ −1 − __snapshot_1_arg3P + arg3 ≤ 0 ∧ 1 − arg1P ≤ 0 ∧ 1 − arg1 ≤ 0
1*:	−1 + arg3 − arg3P ≤ 0 ∧ 1 − __snapshot_1_arg3P + arg3P ≤ 0 ∧ 1 + arg2 − arg3P ≤ 0 ∧ − __snapshot_1_arg3P ≤ 0 ∧ 1 − arg1P ≤ 0 ∧ 1 − arg1 ≤ 0

The invariants are proved as follows.

IMPACT Invariant Proof

nodes (location) invariant:

0	(3)	TRUE
1	(0)	TRUE
2	(1)	1 + arg2 − arg3P ≤ 0 ∧ − arg3P ≤ 0 ∧ 1 − arg1P ≤ 0 ∧ 1 − arg1 ≤ 0
3	(1)	−1 + arg3 − arg3P ≤ 0 ∧ 1 + arg2 − arg3P ≤ 0 ∧ 1 − arg1P ≤ 0 ∧ 1 − arg1 ≤ 0
4	(2)	− arg1P ≤ 0 ∧ − arg1 ≤ 0
5	(1)	1 + arg2 − arg3P ≤ 0 ∧ − arg3P ≤ 0 ∧ 1 − arg1P ≤ 0 ∧ 1 − arg1 ≤ 0
6	(1_var_snapshot)	1 − __snapshot_1_arg3P + arg2 ≤ 0 ∧ − __snapshot_1_arg3P ≤ 0 ∧ 1 − arg1P ≤ 0 ∧ 1 − arg1 ≤ 0
13	(1*)	−1 + arg3 − arg3P ≤ 0 ∧ 1 − __snapshot_1_arg3P + arg3P ≤ 0 ∧ 1 + arg2 − arg3P ≤ 0 ∧ − __snapshot_1_arg3P ≤ 0 ∧ 1 − arg1P ≤ 0 ∧ 1 − arg1 ≤ 0
14	(1)	−1 + arg3 − arg3P ≤ 0 ∧ 1 − __snapshot_1_arg3P + arg3P ≤ 0 ∧ 1 + arg2 − arg3P ≤ 0 ∧ − __snapshot_1_arg3P ≤ 0 ∧ 1 − arg1P ≤ 0 ∧ 1 − arg1 ≤ 0
15	(1_var_snapshot)	1 − __snapshot_1_arg3P + arg2 ≤ 0 ∧ −1 − __snapshot_1_arg3P + arg3 ≤ 0 ∧ 1 − arg1P ≤ 0 ∧ 1 − arg1 ≤ 0
16	(1_var_snapshot)	1 − __snapshot_1_arg3P + arg2 ≤ 0 ∧ −1 − __snapshot_1_arg3P + arg3 ≤ 0 ∧ 1 − arg1P ≤ 0 ∧ 1 − arg1 ≤ 0
20	(2)	− arg1P ≤ 0 ∧ − arg1 ≤ 0
27	(1)	−1 + arg3 − arg3P ≤ 0 ∧ 1 + arg2 − arg3P ≤ 0 ∧ 1 − arg1P ≤ 0 ∧ 1 − arg1 ≤ 0
28	(2)	− arg1P ≤ 0 ∧ − arg1 ≤ 0
29	(1)	−1 + arg3 − arg3P ≤ 0 ∧ 1 + arg2 − arg3P ≤ 0 ∧ 1 − arg1P ≤ 0 ∧ 1 − arg1 ≤ 0
33	(1*)	−1 + arg3 − arg3P ≤ 0 ∧ 1 − __snapshot_1_arg3P + arg3P ≤ 0 ∧ 1 + arg2 − arg3P ≤ 0 ∧ − __snapshot_1_arg3P ≤ 0 ∧ 1 − arg1P ≤ 0 ∧ 1 − arg1 ≤ 0
42	(1_var_snapshot)	1 − __snapshot_1_arg3P + arg2 ≤ 0 ∧ −1 − __snapshot_1_arg3P + arg3 ≤ 0 ∧ 1 − arg1P ≤ 0 ∧ 1 − arg1 ≤ 0

initial node: 0
cover edges:

15 → 16

20 → 4

27 → 3

28 → 4

33 → 13

42 → 16
transition edges:

0 4 1

1 0 2

2 1 3

2 2 4

2 5 5

3 1 27

3 2 28

3 5 29

4 3 20

5 6 6

6 1 13

13 8 14

14 6 15

14 6 16

16 1 33

29 6 42

8.2.1.1.8 Transition Removal

We remove transition 8 using the following ranking functions, which are bounded by −2.

1:	arg3P
1_var_snapshot:	__snapshot_1_arg3P
1*:	__snapshot_1_arg3P

8.2.1.1.9 Transition Removal

We remove transition 6 using the following ranking functions, which are bounded by −5.

1:	−1
1_var_snapshot:	−2
1*:	−3

8.2.1.1.10 Splitting Cut-Point Transitions

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

Tool configuration

T2Cert

version: 1.0

0	4 1
1	0 2
2	1 3
2	2 4
2	5 5
3	1 27
3	2 28
3	5 29
4	3 20
5	6 6
6	1 13
13	8 14
14	6 15
14	6 16
16	1 33
29	6 42

15	→ 16
20	→ 4
27	→ 3
28	→ 4
33	→ 13
42	→ 16