LTS Termination Proof

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Input

Integer Transition System

Proof

1 Invariant Updates

The following invariants are asserted.

0: TRUE
1: 2 − i_0 ≤ 0
2: TRUE
3: 2 − i_0 ≤ 0
4: 2 − i_0 ≤ 02 − j_0 ≤ 0
5: 2 − i_0 ≤ 02 − j_0 ≤ 0
6: 2 − i_0 ≤ 02 − j_0 ≤ 0
7: TRUE
8: TRUE

The invariants are proved as follows.

IMPACT Invariant Proof

2 Switch to Cooperation Termination Proof

We consider the following cutpoint-transitions:
1 11 1: j_post + j_post ≤ 0j_postj_post ≤ 0j_0 + j_0 ≤ 0j_0j_0 ≤ 0i_post + i_post ≤ 0i_posti_post ≤ 0i_0 + i_0 ≤ 0i_0i_0 ≤ 0
2 18 2: j_post + j_post ≤ 0j_postj_post ≤ 0j_0 + j_0 ≤ 0j_0j_0 ≤ 0i_post + i_post ≤ 0i_posti_post ≤ 0i_0 + i_0 ≤ 0i_0i_0 ≤ 0
and for every transition t, a duplicate t is considered.

3 Transition Removal

We remove transitions 0, 4, 5, 6, 7, 9, 10 using the following ranking functions, which are bounded by −21.

8: 0
7: 0
0: 0
2: 0
1: 0
3: 0
6: 0
4: 0
5: 0
8: −8
7: −9
0: −10
2: −10
2_var_snapshot: −10
2*: −10
1: −11
3: −11
1_var_snapshot: −11
1*: −11
6: −12
4: −13
5: −14
Hints:
12 lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 0] ]
19 lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0] ]
1 lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]
2 lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0] ]
3 lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 0] ]
8 lexWeak[ [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] ]
4 lexStrict[ [0, 0, 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, 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] ]
7 lexStrict[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0] , [0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]
9 lexStrict[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0] , [0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]
10 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* 14 1: j_post + j_post ≤ 0j_postj_post ≤ 0j_0 + j_0 ≤ 0j_0j_0 ≤ 0i_post + i_post ≤ 0i_posti_post ≤ 0i_0 + i_0 ≤ 0i_0i_0 ≤ 0

5 Location Addition

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

1 12 1_var_snapshot: j_post + j_post ≤ 0j_postj_post ≤ 0j_0 + j_0 ≤ 0j_0j_0 ≤ 0i_post + i_post ≤ 0i_posti_post ≤ 0i_0 + i_0 ≤ 0i_0i_0 ≤ 0

6 Location Addition

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

2* 21 2: j_post + j_post ≤ 0j_postj_post ≤ 0j_0 + j_0 ≤ 0j_0j_0 ≤ 0i_post + i_post ≤ 0i_posti_post ≤ 0i_0 + i_0 ≤ 0i_0i_0 ≤ 0

7 Location Addition

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

2 19 2_var_snapshot: j_post + j_post ≤ 0j_postj_post ≤ 0j_0 + j_0 ≤ 0j_0j_0 ≤ 0i_post + i_post ≤ 0i_posti_post ≤ 0i_0 + i_0 ≤ 0i_0i_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 8 using the following ranking functions, which are bounded by −4.

1: 2 − 3⋅j_0
3: −3⋅j_0
1_var_snapshot: 1 − 3⋅j_0
1*: 2 − 3⋅j_0
Hints:
12 lexWeak[ [0, 0, 0, 0, 3, 0, 0, 0, 0] ]
14 lexWeak[ [0, 0, 0, 0, 3, 0, 0, 0, 0] ]
3 lexWeak[ [0, 0, 0, 0, 3, 0, 0, 0, 0] ]
8 lexStrict[ [0, 0, 0, 0, 0, 3, 0, 3, 0, 0, 0, 0] , [0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0] ]

8.1.2 Transition Removal

We remove transitions 12, 14, 3 using the following ranking functions, which are bounded by −1.

1: −1 + i_0
3: −1
1_var_snapshot: 0
1*: i_0
Hints:
12 lexStrict[ [1, 0, 0, 0, 0, 0, 0, 0, 0] , [1, 0, 0, 0, 0, 0, 0, 0, 0] ]
14 lexStrict[ [0, 0, 0, 0, 0, 0, 0, 1, 0] , [1, 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] ]

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 11.

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 −6.

0: −1 − 4⋅i_0
2: 1 − 4⋅i_0
2_var_snapshot: −4⋅i_0
2*: 2 − 4⋅i_0
Hints:
19 lexWeak[ [0, 0, 0, 0, 0, 0, 0, 4] ]
21 lexWeak[ [0, 0, 0, 0, 0, 0, 0, 4] ]
1 lexStrict[ [0, 0, 0, 0, 4, 0, 4, 0, 0, 0, 0] , [0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0] ]
2 lexWeak[ [0, 0, 0, 0, 0, 0, 0, 4] ]

8.2.2 Transition Removal

We remove transitions 19, 2 using the following ranking functions, which are bounded by −2.

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

8.2.3 Transition Removal

We remove transition 21 using the following ranking functions, which are bounded by 0.

0: 0
2: 0
2_var_snapshot: 0
2*: 1
Hints:
21 lexStrict[ [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 18.

8.2.4.1.1 Splitting Cut-Point Transitions

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

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