LTS Termination Proof

by T2Cert

Input

Integer Transition System

Proof

1 Invariant Updates

The following invariants are asserted.

0: TRUE
1: TRUE
2: 10 − i57_0 ≤ 0
3: 10 − i57_0 ≤ 010 − i911_0 ≤ 0
4: 10 − i57_0 ≤ 0
5: TRUE
6: TRUE

The invariants are proved as follows.

IMPACT Invariant Proof

2 Switch to Cooperation Termination Proof

We consider the following cutpoint-transitions:
0 8 0: x79_post + x79_post ≤ 0x79_postx79_post ≤ 0x79_0 + x79_0 ≤ 0x79_0x79_0 ≤ 0x35_post + x35_post ≤ 0x35_postx35_post ≤ 0x35_0 + x35_0 ≤ 0x35_0x35_0 ≤ 0tmp_post + tmp_post ≤ 0tmp_posttmp_post ≤ 0tmp_0 + tmp_0 ≤ 0tmp_0tmp_0 ≤ 0i911_post + i911_post ≤ 0i911_posti911_post ≤ 0i911_0 + i911_0 ≤ 0i911_0i911_0 ≤ 0i57_post + i57_post ≤ 0i57_posti57_post ≤ 0i57_0 + i57_0 ≤ 0i57_0i57_0 ≤ 0
4 15 4: x79_post + x79_post ≤ 0x79_postx79_post ≤ 0x79_0 + x79_0 ≤ 0x79_0x79_0 ≤ 0x35_post + x35_post ≤ 0x35_postx35_post ≤ 0x35_0 + x35_0 ≤ 0x35_0x35_0 ≤ 0tmp_post + tmp_post ≤ 0tmp_posttmp_post ≤ 0tmp_0 + tmp_0 ≤ 0tmp_0tmp_0 ≤ 0i911_post + i911_post ≤ 0i911_posti911_post ≤ 0i911_0 + i911_0 ≤ 0i911_0i911_0 ≤ 0i57_post + i57_post ≤ 0i57_posti57_post ≤ 0i57_0 + i57_0 ≤ 0i57_0i57_0 ≤ 0
and for every transition t, a duplicate t is considered.

3 Transition Removal

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

6: 0
5: 0
0: 0
1: 0
2: 0
4: 0
3: 0
6: −6
5: −7
0: −8
1: −8
0_var_snapshot: −8
0*: −8
2: −9
4: −9
4_var_snapshot: −9
4*: −9
3: −10
Hints:
9 lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]
16 lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]
0 lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]
2 lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]
3 lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]
5 lexWeak[ [0, 0, 0, 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, 0, 0, 0, 0] , [0, 0, 0, 0, 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, 0, 0, 0, 0, 0] , [0, 0, 0, 0, 0, 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, 0, 0, 0, 0, 0, 0] , [0, 0, 0, 0, 0, 0, 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, 0, 0, 0, 0] , [0, 0, 0, 0, 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.

0* 11 0: x79_post + x79_post ≤ 0x79_postx79_post ≤ 0x79_0 + x79_0 ≤ 0x79_0x79_0 ≤ 0x35_post + x35_post ≤ 0x35_postx35_post ≤ 0x35_0 + x35_0 ≤ 0x35_0x35_0 ≤ 0tmp_post + tmp_post ≤ 0tmp_posttmp_post ≤ 0tmp_0 + tmp_0 ≤ 0tmp_0tmp_0 ≤ 0i911_post + i911_post ≤ 0i911_posti911_post ≤ 0i911_0 + i911_0 ≤ 0i911_0i911_0 ≤ 0i57_post + i57_post ≤ 0i57_posti57_post ≤ 0i57_0 + i57_0 ≤ 0i57_0i57_0 ≤ 0

5 Location Addition

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

0 9 0_var_snapshot: x79_post + x79_post ≤ 0x79_postx79_post ≤ 0x79_0 + x79_0 ≤ 0x79_0x79_0 ≤ 0x35_post + x35_post ≤ 0x35_postx35_post ≤ 0x35_0 + x35_0 ≤ 0x35_0x35_0 ≤ 0tmp_post + tmp_post ≤ 0tmp_posttmp_post ≤ 0tmp_0 + tmp_0 ≤ 0tmp_0tmp_0 ≤ 0i911_post + i911_post ≤ 0i911_posti911_post ≤ 0i911_0 + i911_0 ≤ 0i911_0i911_0 ≤ 0i57_post + i57_post ≤ 0i57_posti57_post ≤ 0i57_0 + i57_0 ≤ 0i57_0i57_0 ≤ 0

6 Location Addition

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

4* 18 4: x79_post + x79_post ≤ 0x79_postx79_post ≤ 0x79_0 + x79_0 ≤ 0x79_0x79_0 ≤ 0x35_post + x35_post ≤ 0x35_postx35_post ≤ 0x35_0 + x35_0 ≤ 0x35_0x35_0 ≤ 0tmp_post + tmp_post ≤ 0tmp_posttmp_post ≤ 0tmp_0 + tmp_0 ≤ 0tmp_0tmp_0 ≤ 0i911_post + i911_post ≤ 0i911_posti911_post ≤ 0i911_0 + i911_0 ≤ 0i911_0i911_0 ≤ 0i57_post + i57_post ≤ 0i57_posti57_post ≤ 0i57_0 + i57_0 ≤ 0i57_0i57_0 ≤ 0

7 Location Addition

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

4 16 4_var_snapshot: x79_post + x79_post ≤ 0x79_postx79_post ≤ 0x79_0 + x79_0 ≤ 0x79_0x79_0 ≤ 0x35_post + x35_post ≤ 0x35_postx35_post ≤ 0x35_0 + x35_0 ≤ 0x35_0x35_0 ≤ 0tmp_post + tmp_post ≤ 0tmp_posttmp_post ≤ 0tmp_0 + tmp_0 ≤ 0tmp_0tmp_0 ≤ 0i911_post + i911_post ≤ 0i911_posti911_post ≤ 0i911_0 + i911_0 ≤ 0i911_0i911_0 ≤ 0i57_post + i57_post ≤ 0i57_posti57_post ≤ 0i57_0 + i57_0 ≤ 0i57_0i57_0 ≤ 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, 4, 4_var_snapshot, 4* }.

8.1.1 Transition Removal

We remove transition 2 using the following ranking functions, which are bounded by −38.

2: −1 − 4⋅i911_0
4: 1 − 4⋅i911_0
4_var_snapshot: −4⋅i911_0
4*: 2 − 4⋅i911_0
Hints:
16 lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0] ]
18 lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0] ]
2 lexStrict[ [0, 0, 0, 0, 0, 4, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] , [0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]
3 lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0] ]

8.1.2 Transition Removal

We remove transitions 16, 18 using the following ranking functions, which are bounded by −11.

2: −1 − 2⋅i57_0
4: −10
4_var_snapshot: −2⋅i57_0
4*: 0
Hints:
16 lexStrict[ [2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2] , [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]
18 lexStrict[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] , [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]
3 lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2] ]

8.1.3 Transition Removal

We remove transition 3 using the following ranking functions, which are bounded by 9.

2: 0
4: 0
4_var_snapshot: i57_0
4*: 0
Hints:
3 lexStrict[ [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] , [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]

8.1.4 Splitting Cut-Point Transitions

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

8.1.4.1 Cut-Point Subproblem 1/1

Here we consider cut-point transition 15.

8.1.4.1.1 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 { 0, 1, 0_var_snapshot, 0* }.

8.2.1 Transition Removal

We remove transition 5 using the following ranking functions, which are bounded by −28.

0: 2 − 3⋅i57_0
1: −3⋅i57_0
0_var_snapshot: 1 − 3⋅i57_0
0*: 2 − 3⋅i57_0
Hints:
9 lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3] ]
11 lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3] ]
0 lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3] ]
5 lexStrict[ [0, 0, 0, 0, 3, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] , [0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]

8.2.2 Transition Removal

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

0: 0
1: −2
0_var_snapshot: −1
0*: 1
Hints:
9 lexStrict[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] , [0, 0, 0, 0, 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, 0, 0, 0, 0] , [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]
0 lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]

8.2.3 Transition Removal

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

0: 0
1: −1
0_var_snapshot: 0
0*: 0
Hints:
0 lexStrict[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] , [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]

8.2.4 Splitting Cut-Point Transitions

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

8.2.4.1 Cut-Point Subproblem 1/1

Here we consider cut-point transition 8.

8.2.4.1.1 Splitting Cut-Point Transitions

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

Tool configuration

T2Cert