LTS Termination Proof

by T2Cert

Input

Integer Transition System

Proof

1 Invariant Updates

The following invariants are asserted.

0: TRUE
1: 10 − i4_0 ≤ 0
2: TRUE
3: 10 − i4_0 ≤ 0
4: 10 − i4_0 ≤ 010 − i7_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:
1 8 1: tmp_post + tmp_post ≤ 0tmp_posttmp_post ≤ 0tmp_0 + tmp_0 ≤ 0tmp_0tmp_0 ≤ 0i7_post + i7_post ≤ 0i7_posti7_post ≤ 0i7_0 + i7_0 ≤ 0i7_0i7_0 ≤ 0i4_post + i4_post ≤ 0i4_posti4_post ≤ 0i4_0 + i4_0 ≤ 0i4_0i4_0 ≤ 0
2 15 2: tmp_post + tmp_post ≤ 0tmp_posttmp_post ≤ 0tmp_0 + tmp_0 ≤ 0tmp_0tmp_0 ≤ 0i7_post + i7_post ≤ 0i7_posti7_post ≤ 0i7_0 + i7_0 ≤ 0i7_0i7_0 ≤ 0i4_post + i4_post ≤ 0i4_posti4_post ≤ 0i4_0 + i4_0 ≤ 0i4_0i4_0 ≤ 0
and for every transition t, a duplicate t is considered.

3 Transition Removal

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

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

4 Location Addition

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

1* 11 1: tmp_post + tmp_post ≤ 0tmp_posttmp_post ≤ 0tmp_0 + tmp_0 ≤ 0tmp_0tmp_0 ≤ 0i7_post + i7_post ≤ 0i7_posti7_post ≤ 0i7_0 + i7_0 ≤ 0i7_0i7_0 ≤ 0i4_post + i4_post ≤ 0i4_posti4_post ≤ 0i4_0 + i4_0 ≤ 0i4_0i4_0 ≤ 0

5 Location Addition

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

1 9 1_var_snapshot: tmp_post + tmp_post ≤ 0tmp_posttmp_post ≤ 0tmp_0 + tmp_0 ≤ 0tmp_0tmp_0 ≤ 0i7_post + i7_post ≤ 0i7_posti7_post ≤ 0i7_0 + i7_0 ≤ 0i7_0i7_0 ≤ 0i4_post + i4_post ≤ 0i4_posti4_post ≤ 0i4_0 + i4_0 ≤ 0i4_0i4_0 ≤ 0

6 Location Addition

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

2* 18 2: tmp_post + tmp_post ≤ 0tmp_posttmp_post ≤ 0tmp_0 + tmp_0 ≤ 0tmp_0tmp_0 ≤ 0i7_post + i7_post ≤ 0i7_posti7_post ≤ 0i7_0 + i7_0 ≤ 0i7_0i7_0 ≤ 0i4_post + i4_post ≤ 0i4_posti4_post ≤ 0i4_0 + i4_0 ≤ 0i4_0i4_0 ≤ 0

7 Location Addition

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

2 16 2_var_snapshot: tmp_post + tmp_post ≤ 0tmp_posttmp_post ≤ 0tmp_0 + tmp_0 ≤ 0tmp_0tmp_0 ≤ 0i7_post + i7_post ≤ 0i7_posti7_post ≤ 0i7_0 + i7_0 ≤ 0i7_0i7_0 ≤ 0i4_post + i4_post ≤ 0i4_posti4_post ≤ 0i4_0 + i4_0 ≤ 0i4_0i4_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 { 1, 3, 1_var_snapshot, 1* }.

8.1.1 Transition Removal

We remove transition 5 using the following ranking functions, which are bounded by −30.

1: −3⋅i7_0
3: −2 − 3⋅i7_0
1_var_snapshot: −1 − 3⋅i7_0
1*: −3⋅i7_0
Hints:
9 lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0] ]
11 lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0] ]
3 lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0] ]
5 lexStrict[ [0, 0, 0, 0, 0, 3, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0] , [0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]

8.1.2 Transition Removal

We remove transitions 9, 11, 3 using the following ranking functions, which are bounded by −1.

1: i4_0
3: i4_0
1_var_snapshot: 0
1*: 1 + i4_0
Hints:
9 lexStrict[ [1, 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] ]
11 lexStrict[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0] , [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]
3 lexStrict[ [1, 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] ]

8.1.3 Splitting Cut-Point Transitions

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

8.1.3.1 Cut-Point Subproblem 1/1

Here we consider cut-point transition 8.

8.1.3.1.1 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, 2, 2_var_snapshot, 2* }.

8.2.1 Transition Removal

We remove transition 1 using the following ranking functions, which are bounded by −38.

0: −1 − 4⋅i4_0
2: 1 − 4⋅i4_0
2_var_snapshot: −4⋅i4_0
2*: 2 − 4⋅i4_0
Hints:
16 lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4] ]
18 lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4] ]
1 lexStrict[ [0, 0, 0, 0, 4, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0] , [0, 0, 4, 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, 4] ]

8.2.2 Transition Removal

We remove transitions 16, 18 using the following ranking functions, which are bounded by −1.

0: −2
2: 0
2_var_snapshot: −1
2*: 1
Hints:
16 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] ]
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] ]
2 lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]

8.2.3 Transition Removal

We remove transition 2 using the following ranking functions, which are bounded by −1.

0: −1
2: 0
2_var_snapshot: 0
2*: 0
Hints:
2 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] ]

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 15.

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