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 ∧ − i_post + i_post ≤ 0 ∧ i_post − i_post ≤ 0 ∧ − i_1 + i_1 ≤ 0 ∧ i_1 − i_1 ≤ 0 ∧ − i_0 + i_0 ≤ 0 ∧ i_0 − i_0 ≤ 0
2	1	0:	20 − i_0 ≤ 0 ∧ − i_post + i_post ≤ 0 ∧ i_post − i_post ≤ 0 ∧ − i_1 + i_1 ≤ 0 ∧ i_1 − i_1 ≤ 0 ∧ − i_0 + i_0 ≤ 0 ∧ i_0 − i_0 ≤ 0
2	2	3:	0 ≤ 0 ∧ 0 ≤ 0 ∧ −19 + 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 ∧ − i_1 + i_1 ≤ 0 ∧ i_1 − i_1 ≤ 0
4	3	0:	i_0 ≤ 0 ∧ − i_post + i_post ≤ 0 ∧ i_post − i_post ≤ 0 ∧ − i_1 + i_1 ≤ 0 ∧ i_1 − i_1 ≤ 0 ∧ − i_0 + i_0 ≤ 0 ∧ i_0 − i_0 ≤ 0
4	4	3:	1 − i_0 ≤ 0 ∧ − i_post + i_post ≤ 0 ∧ i_post − i_post ≤ 0 ∧ − i_1 + i_1 ≤ 0 ∧ i_1 − i_1 ≤ 0 ∧ − i_0 + i_0 ≤ 0 ∧ i_0 − i_0 ≤ 0
3	5	2:	0 ≤ 0 ∧ 0 ≤ 0 ∧ − i_post + i_post ≤ 0 ∧ i_post − i_post ≤ 0 ∧ − i_1 + i_1 ≤ 0 ∧ i_1 − i_1 ≤ 0 ∧ − i_0 + i_0 ≤ 0 ∧ i_0 − i_0 ≤ 0
5	6	4:	0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ i_1 ≤ 0 ∧ − i_1 ≤ 0 ∧ i_0 − i_post ≤ 0 ∧ − i_0 + i_post ≤ 0
6	7	5:	0 ≤ 0 ∧ 0 ≤ 0 ∧ − i_post + i_post ≤ 0 ∧ i_post − i_post ≤ 0 ∧ − i_1 + i_1 ≤ 0 ∧ i_1 − i_1 ≤ 0 ∧ − i_0 + i_0 ≤ 0 ∧ i_0 − i_0 ≤ 0

Proof

1 Invariant Updates

The following invariants are asserted.

0:	i_1 ≤ 0 ∧ − i_1 ≤ 0
1:	i_1 ≤ 0 ∧ − i_1 ≤ 0
2:	i_1 ≤ 0 ∧ − i_1 ≤ 0
3:	i_1 ≤ 0 ∧ − i_1 ≤ 0
4:	i_1 ≤ 0 ∧ − i_1 ≤ 0
5:	TRUE
6:	TRUE

The invariants are proved as follows.

IMPACT Invariant Proof

nodes (location) invariant:

0	(0)	i_1 ≤ 0 ∧ − i_1 ≤ 0
1	(1)	i_1 ≤ 0 ∧ − i_1 ≤ 0
2	(2)	i_1 ≤ 0 ∧ − i_1 ≤ 0
3	(3)	i_1 ≤ 0 ∧ − i_1 ≤ 0
4	(4)	i_1 ≤ 0 ∧ − i_1 ≤ 0
5	(5)	TRUE
6	(6)	TRUE

initial node: 6
cover edges:
transition edges:

0 0 1

2 1 0

2 2 3

3 5 2

4 3 0

4 4 3

5 6 4

6 7 5

2 Switch to Cooperation Termination Proof

We consider the following cutpoint-transitions:

− i_post + i_post ≤ 0 ∧ i_post − i_post ≤ 0 ∧ − i_1 + i_1 ≤ 0 ∧ i_1 − i_1 ≤ 0 ∧ − i_0 + i_0 ≤ 0 ∧ i_0 − i_0 ≤ 0

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

3 Transition Removal

We remove transitions 0, 1, 3, 4, 6, 7 using the following ranking functions, which are bounded by −17.

6:	0
5:	0
4:	0
2:	0
3:	0
0:	0
1:	0
6:	−7
5:	−8
4:	−9
2:	−10
3:	−10
3_var_snapshot:	−10
3*:	−10
0:	−11
1:	−12

Hints:

9	lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0] ]
2	lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]
5	lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]
0	lexStrict[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0] , [0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]
1	lexStrict[ [0, 0, 0, 0, 0, 0, 0, 0, 0] , [0, 0, 0, 0, 0, 0, 0, 0, 0] ]
3	lexStrict[ [0, 0, 0, 0, 0, 0, 0, 0, 0] , [0, 0, 0, 0, 0, 0, 0, 0, 0] ]
4	lexStrict[ [0, 0, 0, 0, 0, 0, 0, 0, 0] , [0, 0, 0, 0, 0, 0, 0, 0, 0] ]
6	lexStrict[ [0, 0, 0, 0, 0, 0, 0, 0] , [0, 0, 0, 0, 0, 0, 0, 0] ]
7	lexStrict[ [0, 0, 0, 0, 0, 0, 0, 0] , [0, 0, 0, 0, 0, 0, 0, 0] ]

4 Location Addition

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

3* 11 3: − i_post + i_post ≤ 0 ∧ i_post − i_post ≤ 0 ∧ − i_1 + i_1 ≤ 0 ∧ i_1 − i_1 ≤ 0 ∧ − i_0 + i_0 ≤ 0 ∧ i_0 − i_0 ≤ 0

5 Location Addition

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

3 9 3_var_snapshot: − i_post + i_post ≤ 0 ∧ i_post − i_post ≤ 0 ∧ − i_1 + i_1 ≤ 0 ∧ i_1 − i_1 ≤ 0 ∧ − i_0 + i_0 ≤ 0 ∧ i_0 − i_0 ≤ 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 { 2, 3, 3_var_snapshot, 3* }.

6.1.1 Transition Removal

We remove transition 2 using the following ranking functions, which are bounded by −78.

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

Hints:

9	lexWeak[ [0, 0, 0, 0, 0, 0, 0, 4] ]
11	lexWeak[ [0, 0, 0, 0, 0, 0, 0, 4] ]
2	lexStrict[ [0, 0, 0, 0, 0, 0, 4, 0, 4, 0, 0] , [0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0] ]
5	lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 4] ]

6.1.2 Transition Removal

We remove transitions 9, 11, 5 using the following ranking functions, which are bounded by −1.

2:	−1
3:	1
3_var_snapshot:	0
3*:	2

Hints:

9	lexStrict[ [0, 0, 0, 0, 0, 0, 0, 0] , [0, 0, 0, 0, 0, 0, 0, 0] ]
11	lexStrict[ [0, 0, 0, 0, 0, 0, 0, 0] , [0, 0, 0, 0, 0, 0, 0, 0] ]
5	lexStrict[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0] , [0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]

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 8.

6.1.3.1.1 Splitting Cut-Point Transitions

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

Tool configuration

T2Cert

version: 1.0