LTS Termination Proof

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Input

Integer Transition System

Proof

1 Invariant Updates

The following invariants are asserted.

0: TRUE
1: 20 − j_0 ≤ 0
2: TRUE
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: 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, 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

4 Location Addition

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

2* 8 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

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: 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 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 −54.

0: 4 − 3⋅j_0
2: 2 − 3⋅j_0
2_var_snapshot: −3⋅j_0
2*: 2 − 3⋅j_0

6.1.2 Transition Removal

We remove transitions 6, 8, 2 using the following ranking functions, which are bounded by −3.

0: −3
2: −1
2_var_snapshot: −2
2*: 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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