LTS Termination Proof

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Input

Integer Transition System

Proof

1 Invariant Updates

The following invariants are asserted.

0: 2 − j_0 ≤ 0
1: 2 − j_0 ≤ 0
2: TRUE
3: TRUE
4: 2 − j_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:
3 8 3: j_post + j_post ≤ 0j_postj_post ≤ 0j_0 + j_0 ≤ 0j_0j_0 ≤ 0
and for every transition t, a duplicate t is considered.

3 Transition Removal

We remove transitions 0, 1, 2, 5, 6, 7 using the following ranking functions, which are bounded by −17.

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

4 Location Addition

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

3* 11 3: j_post + j_post ≤ 0j_postj_post ≤ 0j_0 + j_0 ≤ 0j_0j_0 ≤ 0

5 Location Addition

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

3 9 3_var_snapshot: j_post + j_post ≤ 0j_postj_post ≤ 0j_0 + j_0 ≤ 0j_0j_0 ≤ 0

6 SCC Decomposition

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

6.1 SCC Subproblem 1/1

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

6.1.1 Transition Removal

We remove transition 3 using the following ranking functions, which are bounded by −6.

2: −1 − 4⋅j_0
3: 1 − 4⋅j_0
3_var_snapshot: −4⋅j_0
3*: 2 − 4⋅j_0
Hints:
9 lexWeak[ [0, 0, 0, 4] ]
11 lexWeak[ [0, 0, 0, 4] ]
3 lexStrict[ [0, 0, 0, 0, 4, 0, 4] , [0, 0, 4, 0, 0, 0, 0] ]
4 lexWeak[ [0, 0, 0, 0, 0, 4] ]

6.1.2 Transition Removal

We remove transitions 11, 4 using the following ranking functions, which are bounded by −1.

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

6.1.3 Transition Removal

We remove transition 9 using the following ranking functions, which are bounded by 0.

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

6.1.4 Splitting Cut-Point Transitions

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

6.1.4.1 Cut-Point Subproblem 1/1

Here we consider cut-point transition 8.

6.1.4.1.1 Splitting Cut-Point Transitions

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

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