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: counter_post ≤ 0counter_post ≤ 0counter_0 ≤ 0counter_0 ≤ 0
4: TRUE
5: TRUE

The invariants are proved as follows.

IMPACT Invariant Proof

2 Switch to Cooperation Termination Proof

We consider the following cutpoint-transitions:
2 7 2: z_post + z_post ≤ 0z_postz_post ≤ 0z_0 + z_0 ≤ 0z_0z_0 ≤ 0y_0 + y_0 ≤ 0y_0y_0 ≤ 0counter_post + counter_post ≤ 0counter_postcounter_post ≤ 0counter_0 + counter_0 ≤ 0counter_0counter_0 ≤ 0___const_36_0 + ___const_36_0 ≤ 0___const_36_0___const_36_0 ≤ 0___const_127_0 + ___const_127_0 ≤ 0___const_127_0___const_127_0 ≤ 0
and for every transition t, a duplicate t is considered.

3 Transition Removal

We remove transitions 0, 2, 3, 5, 6 using the following ranking functions, which are bounded by −15.

5: 0
4: 0
3: 0
0: 0
2: 0
1: 0
5: −6
4: −7
3: −8
0: −9
2: −9
2_var_snapshot: −9
2*: −9
1: −13

4 Location Addition

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

2* 10 2: z_post + z_post ≤ 0z_postz_post ≤ 0z_0 + z_0 ≤ 0z_0z_0 ≤ 0y_0 + y_0 ≤ 0y_0y_0 ≤ 0counter_post + counter_post ≤ 0counter_postcounter_post ≤ 0counter_0 + counter_0 ≤ 0counter_0counter_0 ≤ 0___const_36_0 + ___const_36_0 ≤ 0___const_36_0___const_36_0 ≤ 0___const_127_0 + ___const_127_0 ≤ 0___const_127_0___const_127_0 ≤ 0

5 Location Addition

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

2 8 2_var_snapshot: z_post + z_post ≤ 0z_postz_post ≤ 0z_0 + z_0 ≤ 0z_0z_0 ≤ 0y_0 + y_0 ≤ 0y_0y_0 ≤ 0counter_post + counter_post ≤ 0counter_postcounter_post ≤ 0counter_0 + counter_0 ≤ 0counter_0counter_0 ≤ 0___const_36_0 + ___const_36_0 ≤ 0___const_36_0___const_36_0 ≤ 0___const_127_0 + ___const_127_0 ≤ 0___const_127_0___const_127_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 0.

0: −1 + 2⋅___const_36_0 − 2⋅counter_0
2: 2⋅___const_36_0 − 2⋅counter_0
2_var_snapshot: 2⋅___const_36_0 − 2⋅counter_0
2*: 2⋅___const_36_0 − 2⋅counter_0

6.1.2 Transition Removal

We remove transitions 8, 10, 4 using the following ranking functions, which are bounded by −1.

0: −1
2: 1
2_var_snapshot: 0
2*: 2

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 7.

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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