LTS Termination Proof

by T2Cert

Input

Integer Transition System

Initial Location: 5

Transitions: (pre-variables and post-variables)

0	0	1:	0 ≤ 0 ∧ 0 ≤ 0 ∧ x_0 ≤ 0 ∧ −1 + x_post ≤ 0 ∧ 1 − x_post ≤ 0 ∧ x_0 − x_post ≤ 0 ∧ − x_0 + x_post ≤ 0 ∧ − x_1 + x_1 ≤ 0 ∧ x_1 − x_1 ≤ 0
0	1	1:	0 ≤ 0 ∧ 0 ≤ 0 ∧ 1 − x_0 ≤ 0 ∧ −1 − x_0 + x_post ≤ 0 ∧ 1 + x_0 − x_post ≤ 0 ∧ x_0 − x_post ≤ 0 ∧ − x_0 + x_post ≤ 0 ∧ − x_1 + x_1 ≤ 0 ∧ x_1 − x_1 ≤ 0
2	2	3:	4 − x_0 ≤ 0 ∧ − x_post + x_post ≤ 0 ∧ x_post − x_post ≤ 0 ∧ − x_1 + x_1 ≤ 0 ∧ x_1 − x_1 ≤ 0 ∧ − x_0 + x_0 ≤ 0 ∧ x_0 − x_0 ≤ 0
2	3	0:	−3 + x_0 ≤ 0 ∧ − x_post + x_post ≤ 0 ∧ x_post − x_post ≤ 0 ∧ − x_1 + x_1 ≤ 0 ∧ x_1 − x_1 ≤ 0 ∧ − x_0 + x_0 ≤ 0 ∧ x_0 − x_0 ≤ 0
1	4	2:	0 ≤ 0 ∧ 0 ≤ 0 ∧ − x_post + x_post ≤ 0 ∧ x_post − x_post ≤ 0 ∧ − x_1 + x_1 ≤ 0 ∧ x_1 − x_1 ≤ 0 ∧ − x_0 + x_0 ≤ 0 ∧ x_0 − x_0 ≤ 0
4	5	1:	0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ −5 + x_1 ≤ 0 ∧ 5 − x_1 ≤ 0 ∧ x_0 − x_post ≤ 0 ∧ − x_0 + x_post ≤ 0
5	6	4:	0 ≤ 0 ∧ 0 ≤ 0 ∧ − x_post + x_post ≤ 0 ∧ x_post − x_post ≤ 0 ∧ − x_1 + x_1 ≤ 0 ∧ x_1 − x_1 ≤ 0 ∧ − x_0 + x_0 ≤ 0 ∧ x_0 − x_0 ≤ 0

Proof

1 Invariant Updates

The following invariants are asserted.

0:	−3 + x_0 ≤ 0 ∧ −5 + x_1 ≤ 0 ∧ 5 − x_1 ≤ 0
1:	−5 + x_1 ≤ 0 ∧ 5 − x_1 ≤ 0
2:	−5 + x_1 ≤ 0 ∧ 5 − x_1 ≤ 0
3:	4 − x_0 ≤ 0 ∧ −5 + x_1 ≤ 0 ∧ 5 − x_1 ≤ 0
4:	TRUE
5:	TRUE

The invariants are proved as follows.

IMPACT Invariant Proof

nodes (location) invariant:

0	(0)	−3 + x_0 ≤ 0 ∧ −5 + x_1 ≤ 0 ∧ 5 − x_1 ≤ 0
1	(1)	−5 + x_1 ≤ 0 ∧ 5 − x_1 ≤ 0
2	(2)	−5 + x_1 ≤ 0 ∧ 5 − x_1 ≤ 0
3	(3)	4 − x_0 ≤ 0 ∧ −5 + x_1 ≤ 0 ∧ 5 − x_1 ≤ 0
4	(4)	TRUE
5	(5)	TRUE

initial node: 5
cover edges:
transition edges:

0 0 1

0 1 1

1 4 2

2 2 3

2 3 0

4 5 1

5 6 4

2 Switch to Cooperation Termination Proof

We consider the following cutpoint-transitions:

− x_post + x_post ≤ 0 ∧ x_post − x_post ≤ 0 ∧ − x_1 + x_1 ≤ 0 ∧ x_1 − x_1 ≤ 0 ∧ − x_0 + x_0 ≤ 0 ∧ x_0 − x_0 ≤ 0

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

3 Transition Removal

We remove transitions 2, 5, 6 using the following ranking functions, which are bounded by −13.

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

Hints:

8	lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0] ]
0	lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]
1	lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]
3	lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 0] ]
4	lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]
2	lexStrict[ [0, 0, 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] ]
6	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.

1* 10 1: − x_post + x_post ≤ 0 ∧ x_post − x_post ≤ 0 ∧ − x_1 + x_1 ≤ 0 ∧ x_1 − x_1 ≤ 0 ∧ − x_0 + x_0 ≤ 0 ∧ x_0 − x_0 ≤ 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: − x_post + x_post ≤ 0 ∧ x_post − x_post ≤ 0 ∧ − x_1 + x_1 ≤ 0 ∧ x_1 − x_1 ≤ 0 ∧ − x_0 + x_0 ≤ 0 ∧ x_0 − x_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 { 0, 1, 2, 1_var_snapshot, 1* }.

6.1.1 Transition Removal

We remove transitions 0, 1 using the following ranking functions, which are bounded by −45.

0:	−13⋅x_0 − x_1
1:	−13⋅x_0 + x_1
2:	−13⋅x_0 − x_1
1_var_snapshot:	−13⋅x_0
1*:	6 − 13⋅x_0

Hints:

8	lexWeak[ [0, 1, 0, 0, 0, 0, 0, 13] ]
10	lexWeak[ [1, 0, 0, 0, 1, 0, 0, 13] ]
0	lexStrict[ [0, 1, 0, 0, 0, 13, 0, 13, 0, 13, 0, 0] , [0, 1, 0, 0, 0, 13, 0, 0, 0, 0, 0, 0] ]
1	lexStrict[ [0, 1, 0, 0, 0, 0, 0, 13, 0, 13, 0, 0] , [13, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]
3	lexWeak[ [0, 0, 0, 0, 0, 0, 1, 0, 13] ]
4	lexWeak[ [0, 1, 0, 0, 0, 0, 0, 1, 0, 13] ]

6.1.2 Transition Removal

We remove transitions 8, 10, 3, 4 using the following ranking functions, which are bounded by 4.

0:	0
1:	15
2:	−5 + 2⋅x_1
1_var_snapshot:	2⋅x_1
1*:	4⋅x_1

Hints:

8	lexStrict[ [2, 0, 0, 0, 2, 0, 0, 0] , [0, 0, 0, 0, 0, 0, 0, 0] ]
10	lexStrict[ [0, 4, 0, 0, 0, 0, 0, 0] , [0, 4, 0, 0, 0, 0, 0, 0] ]
3	lexStrict[ [0, 2, 0, 0, 0, 0, 0, 0, 0] , [0, 2, 0, 0, 0, 0, 0, 0, 0] ]
4	lexStrict[ [0, 0, 0, 0, 0, 0, 2, 0, 0, 0] , [0, 2, 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 7.

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