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 ∧ − arg2P ≤ 0 ∧ − arg2 ≤ 0 ∧ − arg3P ≤ 0 ∧ − arg1P ≤ 0 ∧ 1 − arg1 ≤ 0 ∧ − arg1P + arg1 ≤ 0 ∧ arg1P − arg1 ≤ 0 ∧ − arg2P + arg2 ≤ 0 ∧ arg2P − arg2 ≤ 0 ∧ − arg3P + arg3 ≤ 0 ∧ arg3P − arg3 ≤ 0
1	1	2:	1 − arg2 + arg3 ≤ 0 ∧ − 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
1	2	2:	arg2 − arg3 ≤ 0 ∧ 1 − arg1 + arg3 ≤ 0 ∧ − 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	3	1:	0 ≤ 0 ∧ 0 ≤ 0 ∧ 1 − arg2 + arg3 ≤ 0 ∧ −1 − arg2P + arg2 ≤ 0 ∧ 1 + arg2P − arg2 ≤ 0 ∧ − arg2P + arg2 ≤ 0 ∧ arg2P − arg2 ≤ 0 ∧ − arg3P + arg3P ≤ 0 ∧ arg3P − arg3P ≤ 0 ∧ − arg3 + arg3 ≤ 0 ∧ arg3 − arg3 ≤ 0 ∧ − arg1P + arg1P ≤ 0 ∧ arg1P − arg1P ≤ 0 ∧ − arg1 + arg1 ≤ 0 ∧ arg1 − arg1 ≤ 0
2	4	1:	arg2 − arg3 ≤ 0 ∧ arg1 − arg3 ≤ 0 ∧ − 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	5	1:	0 ≤ 0 ∧ 0 ≤ 0 ∧ arg2 − arg3 ≤ 0 ∧ 1 − arg1 + arg3 ≤ 0 ∧ −1 − arg1P + arg1 ≤ 0 ∧ 1 + arg1P − arg1 ≤ 0 ∧ − arg1P + arg1 ≤ 0 ∧ arg1P − arg1 ≤ 0 ∧ − arg3P + arg3P ≤ 0 ∧ arg3P − arg3P ≤ 0 ∧ − arg3 + arg3 ≤ 0 ∧ arg3 − arg3 ≤ 0 ∧ − arg2P + arg2P ≤ 0 ∧ arg2P − arg2P ≤ 0 ∧ − arg2 + arg2 ≤ 0 ∧ arg2 − arg2 ≤ 0
3	6	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:	− arg3P ≤ 0 ∧ − arg3 ≤ 0
2:	− arg3P ≤ 0 ∧ − arg3 ≤ 0
3:	TRUE

The invariants are proved as follows.

IMPACT Invariant Proof

nodes (location) invariant:

0	(0)	TRUE
1	(1)	− arg3P ≤ 0 ∧ − arg3 ≤ 0
2	(2)	− arg3P ≤ 0 ∧ − arg3 ≤ 0
3	(3)	TRUE

initial node: 3
cover edges:
transition edges:

0 0 1

1 1 2

1 2 2

2 3 1

2 4 1

2 5 1

3 6 0

2 Switch to Cooperation Termination Proof

We consider the following cutpoint-transitions:

− 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, 6 using the following ranking functions, which are bounded by −11.

3:	0
0:	0
1:	0
2:	0
3:	−4
0:	−5
1:	−6
2:	−6
1_var_snapshot:	−6
1*:	−6

4 Location Addition

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

1* 10 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 8 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 SCC Decomposition

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

6.1 SCC Subproblem 1/1

Here we consider the SCC { 1, 2, 1_var_snapshot, 1* }.

6.1.1 Transition Removal

We remove transition 5 using the following ranking functions, which are bounded by 0.

1:	arg1
2:	arg1
1_var_snapshot:	arg1
1*:	arg1

6.1.2 Transition Removal

We remove transition 3 using the following ranking functions, which are bounded by 0.

1:	arg2
2:	arg2
1_var_snapshot:	arg2
1*:	arg2

6.1.3 Splitting Cut-Point Transitions

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

6.1.3.1 Cut-Point Subproblem 1/1

Here we consider cut-point transition 7.

6.1.3.1.1 Invariant Updates

The following invariants are asserted.

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

The invariants are proved as follows.

IMPACT Invariant Proof

nodes (location) invariant:

0	(3)	TRUE
1	(0)	TRUE
2	(1)	− arg3P ≤ 0 ∧ − arg3 ≤ 0
3	(2)	− arg3P ≤ 0 ∧ − arg3 ≤ 0
4	(2)	− arg3P ≤ 0 ∧ − arg3 ≤ 0
5	(1)	− arg3P ≤ 0 ∧ − arg3 ≤ 0
6	(1_var_snapshot)	− arg3P ≤ 0 ∧ − arg3 ≤ 0
11	(1)	− arg3P ≤ 0 ∧ − arg3 ≤ 0
12	(1)	− arg3P ≤ 0 ∧ − arg3 ≤ 0
13	(1)	− arg3P ≤ 0 ∧ − arg3 ≤ 0
18	(2)	1 − arg2 + arg3 ≤ 0 ∧ − arg3P ≤ 0 ∧ − arg3 ≤ 0
19	(2)	1 − arg1 + arg3 ≤ 0 ∧ − arg3P ≤ 0 ∧ − arg3 ≤ 0
20	(1*)	1 ≤ 0 ∧ − arg3P ≤ 0 ∧ − arg3 ≤ 0
21	(1*)	1 ≤ 0 ∧ − arg3P ≤ 0 ∧ − arg3 ≤ 0

initial node: 0
cover edges:

3 → 4

11 → 2

12 → 2

13 → 2
transition edges:

0 6 1

1 0 2

2 1 3

2 2 4

2 7 5

4 3 11

4 4 12

4 5 13

5 8 6

6 1 18

6 2 19

18 4 21

19 4 20

6.1.3.1.2 Transition Removal

We remove transition 8 using the following ranking functions, which are bounded by −6.

1:	−1
1_var_snapshot:	−2
2:	−3
1*:	−4

6.1.3.1.3 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	6 1
1	0 2
2	1 3
2	2 4
2	7 5
4	3 11
4	4 12
4	5 13
5	8 6
6	1 18
6	2 19
18	4 21
19	4 20

3	→ 4
11	→ 2
12	→ 2
13	→ 2