LTS Termination Proof

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Input

Integer Transition System

Proof

1 Invariant Updates

The following invariants are asserted.

0: p_post ≤ 0p_post ≤ 0p_0 ≤ 0p_0 ≤ 0
1: −1 + p_post ≤ 01 − p_post ≤ 0−1 + p_0 ≤ 01 − p_0 ≤ 0y_0 ≤ 0
2: p_post ≤ 0p_post ≤ 0p_0 ≤ 0p_0 ≤ 0
3: TRUE
4: TRUE

The invariants are proved as follows.

IMPACT Invariant Proof

2 Switch to Cooperation Termination Proof

We consider the following cutpoint-transitions:
2 5 2: y_post + y_post ≤ 0y_posty_post ≤ 0y_0 + y_0 ≤ 0y_0y_0 ≤ 0p_post + p_post ≤ 0p_postp_post ≤ 0p_0 + p_0 ≤ 0p_0p_0 ≤ 0
and for every transition t, a duplicate t is considered.

3 Transition Removal

We remove transitions 0, 3, 4 using the following ranking functions, which are bounded by −13.

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

4 Location Addition

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

2* 8 2: y_post + y_post ≤ 0y_posty_post ≤ 0y_0 + y_0 ≤ 0y_0y_0 ≤ 0p_post + p_post ≤ 0p_postp_post ≤ 0p_0 + p_0 ≤ 0p_0p_0 ≤ 0

5 Location Addition

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

2 6 2_var_snapshot: y_post + y_post ≤ 0y_posty_post ≤ 0y_0 + y_0 ≤ 0y_0y_0 ≤ 0p_post + p_post ≤ 0p_postp_post ≤ 0p_0 + p_0 ≤ 0p_0p_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 { 0, 2, 2_var_snapshot, 2* }.

6.1.1 Transition Removal

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

0: −1 + 4⋅y_0
2: 1 + 4⋅y_0
2_var_snapshot: 4⋅y_0
2*: 2 + 4⋅y_0
Hints:
6 lexWeak[ [0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0] ]
8 lexWeak[ [0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0] ]
1 lexStrict[ [0, 0, 0, 0, 0, 0, 0, 4, 0, 4, 0, 0, 0, 0, 0] , [0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0] ]
2 lexWeak[ [0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0] ]

6.1.2 Transition Removal

We remove transitions 6, 8, 2 using the following ranking functions, which are bounded by −1.

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

6.1.3 Splitting Cut-Point Transitions

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

6.1.3.1 Cut-Point Subproblem 1/1

Here we consider cut-point transition 5.

6.1.3.1.1 Splitting Cut-Point Transitions

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

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