LTS Termination Proof

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Input

Integer Transition System

Proof

1 Invariant Updates

The following invariants are asserted.

0: TRUE
1: TRUE
2: TRUE
3: 100 − i_0 ≤ 0
4: 100 − i_0 ≤ 0
5: TRUE
6: 100 − i_0 ≤ 0
7: 100 − i_0 ≤ 0
8: 100 − i_0 ≤ 0200 − j_0 ≤ 0
9: TRUE
10: TRUE
11: TRUE
12: TRUE

The invariants are proved as follows.

IMPACT Invariant Proof

2 Switch to Cooperation Termination Proof

We consider the following cutpoint-transitions:
3 14 3: y8_post + y8_post ≤ 0y8_posty8_post ≤ 0y8_0 + y8_0 ≤ 0y8_0y8_0 ≤ 0y6_post + y6_post ≤ 0y6_posty6_post ≤ 0y6_0 + y6_0 ≤ 0y6_0y6_0 ≤ 0y4_post + y4_post ≤ 0y4_posty4_post ≤ 0y4_0 + y4_0 ≤ 0y4_0y4_0 ≤ 0j_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 21 5: y8_post + y8_post ≤ 0y8_posty8_post ≤ 0y8_0 + y8_0 ≤ 0y8_0y8_0 ≤ 0y6_post + y6_post ≤ 0y6_posty6_post ≤ 0y6_0 + y6_0 ≤ 0y6_0y6_0 ≤ 0y4_post + y4_post ≤ 0y4_posty4_post ≤ 0y4_0 + y4_0 ≤ 0y4_0y4_0 ≤ 0j_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 1, 6, 12, 13 using the following ranking functions, which are bounded by −17.

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

3* 17 3: y8_post + y8_post ≤ 0y8_posty8_post ≤ 0y8_0 + y8_0 ≤ 0y8_0y8_0 ≤ 0y6_post + y6_post ≤ 0y6_posty6_post ≤ 0y6_0 + y6_0 ≤ 0y6_0y6_0 ≤ 0y4_post + y4_post ≤ 0y4_posty4_post ≤ 0y4_0 + y4_0 ≤ 0y4_0y4_0 ≤ 0j_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.

3 15 3_var_snapshot: y8_post + y8_post ≤ 0y8_posty8_post ≤ 0y8_0 + y8_0 ≤ 0y8_0y8_0 ≤ 0y6_post + y6_post ≤ 0y6_posty6_post ≤ 0y6_0 + y6_0 ≤ 0y6_0y6_0 ≤ 0y4_post + y4_post ≤ 0y4_posty4_post ≤ 0y4_0 + y4_0 ≤ 0y4_0y4_0 ≤ 0j_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.

5* 24 5: y8_post + y8_post ≤ 0y8_posty8_post ≤ 0y8_0 + y8_0 ≤ 0y8_0y8_0 ≤ 0y6_post + y6_post ≤ 0y6_posty6_post ≤ 0y6_0 + y6_0 ≤ 0y6_0y6_0 ≤ 0y4_post + y4_post ≤ 0y4_posty4_post ≤ 0y4_0 + y4_0 ≤ 0y4_0y4_0 ≤ 0j_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.

5 22 5_var_snapshot: y8_post + y8_post ≤ 0y8_posty8_post ≤ 0y8_0 + y8_0 ≤ 0y8_0y8_0 ≤ 0y6_post + y6_post ≤ 0y6_posty6_post ≤ 0y6_0 + y6_0 ≤ 0y6_0y6_0 ≤ 0y4_post + y4_post ≤ 0y4_posty4_post ≤ 0y4_0 + y4_0 ≤ 0y4_0y4_0 ≤ 0j_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 { 3, 4, 6, 7, 3_var_snapshot, 3* }.

8.1.1 Transition Removal

We remove transition 7 using the following ranking functions, which are bounded by −1196.

3: 1 − 6⋅j_0
4: −3 − 6⋅j_0
6: −2 − 6⋅j_0
7: −1 − 6⋅j_0
3_var_snapshot: −6⋅j_0
3*: 2 − 6⋅j_0
Hints:
15 lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0] ]
17 lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0] ]
3 lexWeak[ [0, 0, 0, 0, 6, 0, 6, 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, 6, 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, 6, 0, 0, 0, 0] , [0, 0, 0, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]
8 lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0] ]

8.1.2 Transition Removal

We remove transitions 15, 17, 3, 5, 8 using the following ranking functions, which are bounded by −2.

3: 0
4: 100 + i_0
6: 3⋅i_0
7: −2
3_var_snapshot: −1
3*: i_0
Hints:
15 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] ]
17 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] ]
3 lexStrict[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 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, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]
5 lexStrict[ [2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0] , [3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]
8 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] ]

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 14.

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, 1, 2, 5, 9, 10, 5_var_snapshot, 5* }.

8.2.1 Transition Removal

We remove transition 2 using the following ranking functions, which are bounded by −791.

0: 1 − 8⋅i_0
1: −8⋅i_0
2: 2 − 8⋅i_0
5: 4 − 8⋅i_0
9: −2 − 8⋅i_0
10: −1 − 8⋅i_0
5_var_snapshot: 3 − 8⋅i_0
5*: 5 − 8⋅i_0
Hints:
22 lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 8] ]
24 lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 8] ]
0 lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 8] ]
2 lexStrict[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 8] , [0, 0, 8, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]
4 lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 8] ]
9 lexWeak[ [0, 0, 0, 8, 0, 8, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]
10 lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 8] ]
11 lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 8] ]

8.2.2 Transition Removal

We remove transitions 22, 24, 0, 4, 9, 10, 11 using the following ranking functions, which are bounded by −5.

0: 2
1: 1
2: −5
5: −3
9: −1
10: 0
5_var_snapshot: −4
5*: −2
Hints:
22 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] ]
24 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 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 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] ]
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, 0, 0, 0, 0] ]
10 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, 0, 0, 0] ]

8.2.3 Splitting Cut-Point Transitions

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

8.2.3.1 Cut-Point Subproblem 1/1

Here we consider cut-point transition 21.

8.2.3.1.1 Splitting Cut-Point Transitions

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

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