LTS Termination Proof

by T2Cert

Input

Integer Transition System

Proof

1 Switch to Cooperation Termination Proof

We consider the following cutpoint-transitions:
2 12 2: i_post + i_post ≤ 0i_posti_post ≤ 0i_0 + i_0 ≤ 0i_0i_0 ≤ 0
and for every transition t, a duplicate t is considered.

2 Transition Removal

We remove transitions 0, 4, 6, 7, 8, 9, 10, 11 using the following ranking functions, which are bounded by −19.

8: 0
7: 0
0: 0
2: 0
3: 0
1: 0
6: 0
4: 0
5: 0
8: −8
7: −9
0: −10
2: −10
3: −10
2_var_snapshot: −10
2*: −10
1: −11
6: −12
4: −13
5: −14
Hints:
13 lexWeak[ [0, 0, 0, 0] ]
1 lexWeak[ [0, 0, 0, 0, 0, 0, 0] ]
2 lexWeak[ [0, 0, 0, 0, 0, 0] ]
3 lexWeak[ [0, 0, 0, 0, 0, 0] ]
5 lexWeak[ [0, 0, 0, 0, 0, 0] ]
0 lexStrict[ [0, 0, 0, 0, 0] , [0, 0, 0, 0, 0] ]
4 lexStrict[ [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] ]
7 lexStrict[ [0, 0, 0, 0, 0, 0] , [0, 0, 0, 0, 0, 0] ]
8 lexStrict[ [0, 0, 0, 0, 0] , [0, 0, 0, 0, 0] ]
9 lexStrict[ [0, 0, 0, 0, 0] , [0, 0, 0, 0, 0] ]
10 lexStrict[ [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] ]

3 Location Addition

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

2* 15 2: i_post + i_post ≤ 0i_posti_post ≤ 0i_0 + i_0 ≤ 0i_0i_0 ≤ 0

4 Location Addition

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

2 13 2_var_snapshot: i_post + i_post ≤ 0i_posti_post ≤ 0i_0 + i_0 ≤ 0i_0i_0 ≤ 0

5 SCC Decomposition

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

5.1 SCC Subproblem 1/1

Here we consider the SCC { 0, 2, 3, 2_var_snapshot, 2* }.

5.1.1 Transition Removal

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

0: −2 − 5⋅i_0
2: 1 − 5⋅i_0
3: −1 − 5⋅i_0
2_var_snapshot: −5⋅i_0
2*: 2 − 5⋅i_0
Hints:
13 lexWeak[ [0, 0, 0, 5] ]
15 lexWeak[ [0, 0, 0, 5] ]
1 lexStrict[ [0, 0, 0, 0, 5, 0, 5] , [0, 0, 5, 0, 0, 0, 0] ]
2 lexWeak[ [0, 0, 0, 0, 0, 5] ]
3 lexWeak[ [0, 0, 0, 0, 0, 5] ]
5 lexWeak[ [0, 0, 0, 0, 0, 5] ]

5.1.2 Transition Removal

We remove transitions 13, 15, 3, 5 using the following ranking functions, which are bounded by −2.

0: −2
2: 1
3: −1
2_var_snapshot: 0
2*: 2
Hints:
13 lexStrict[ [0, 0, 0, 0] , [0, 0, 0, 0] ]
15 lexStrict[ [0, 0, 0, 0] , [0, 0, 0, 0] ]
2 lexWeak[ [0, 0, 0, 0, 0, 0] ]
3 lexStrict[ [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] ]

5.1.3 Transition Removal

We remove transition 2 using the following ranking functions, which are bounded by 0.

0: 0
2: 0
3: 1
2_var_snapshot: 0
2*: 0
Hints:
2 lexStrict[ [0, 0, 0, 0, 0, 0] , [0, 0, 0, 0, 0, 0] ]

5.1.4 Splitting Cut-Point Transitions

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

5.1.4.1 Cut-Point Subproblem 1/1

Here we consider cut-point transition 12.

5.1.4.1.1 Splitting Cut-Point Transitions

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

Tool configuration

T2Cert