LTS Termination Proof

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Input

Integer Transition System

Proof

1 Switch to Cooperation Termination Proof

We consider the following cutpoint-transitions:
1 13 1: n_post + n_post ≤ 0n_postn_post ≤ 0n_0 + n_0 ≤ 0n_0n_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 ≤ 0guard_post + guard_post ≤ 0guard_postguard_post ≤ 0guard_0 + guard_0 ≤ 0guard_0guard_0 ≤ 0
5 20 5: n_post + n_post ≤ 0n_postn_post ≤ 0n_0 + n_0 ≤ 0n_0n_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 ≤ 0guard_post + guard_post ≤ 0guard_postguard_post ≤ 0guard_0 + guard_0 ≤ 0guard_0guard_0 ≤ 0
and for every transition t, a duplicate t is considered.

2 Transition Removal

We remove transitions 2, 11, 12 using the following ranking functions, which are bounded by −15.

9: 0
8: 0
0: 0
1: 0
2: 0
3: 0
5: 0
6: 0
7: 0
4: 0
9: −5
8: −6
0: −7
1: −7
2: −7
3: −7
5: −7
6: −7
7: −7
1_var_snapshot: −7
1*: −7
5_var_snapshot: −7
5*: −7
4: −10

3 Location Addition

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

1* 16 1: n_post + n_post ≤ 0n_postn_post ≤ 0n_0 + n_0 ≤ 0n_0n_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 ≤ 0guard_post + guard_post ≤ 0guard_postguard_post ≤ 0guard_0 + guard_0 ≤ 0guard_0guard_0 ≤ 0

4 Location Addition

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

1 14 1_var_snapshot: n_post + n_post ≤ 0n_postn_post ≤ 0n_0 + n_0 ≤ 0n_0n_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 ≤ 0guard_post + guard_post ≤ 0guard_postguard_post ≤ 0guard_0 + guard_0 ≤ 0guard_0guard_0 ≤ 0

5 Location Addition

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

5* 23 5: n_post + n_post ≤ 0n_postn_post ≤ 0n_0 + n_0 ≤ 0n_0n_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 ≤ 0guard_post + guard_post ≤ 0guard_postguard_post ≤ 0guard_0 + guard_0 ≤ 0guard_0guard_0 ≤ 0

6 Location Addition

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

5 21 5_var_snapshot: n_post + n_post ≤ 0n_postn_post ≤ 0n_0 + n_0 ≤ 0n_0n_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 ≤ 0guard_post + guard_post ≤ 0guard_postguard_post ≤ 0guard_0 + guard_0 ≤ 0guard_0guard_0 ≤ 0

7 SCC Decomposition

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

7.1 SCC Subproblem 1/1

Here we consider the SCC { 0, 1, 2, 3, 5, 6, 7, 1_var_snapshot, 1*, 5_var_snapshot, 5* }.

7.1.1 Transition Removal

We remove transition 3 using the following ranking functions, which are bounded by 5.

0: −5⋅i_0 + 5⋅n_0
1: 3 − 5⋅i_0 + 5⋅n_0
2: −5⋅i_0 + 5⋅n_0
3: 1 − 5⋅i_0 + 5⋅n_0
5: −5⋅i_0 + 5⋅n_0
6: −5⋅i_0 + 5⋅n_0
7: −5 − 5⋅i_0 + 5⋅n_0
1_var_snapshot: 2 − 5⋅i_0 + 5⋅n_0
1*: 4 − 5⋅i_0 + 5⋅n_0
5_var_snapshot: −5⋅i_0 + 5⋅n_0
5*: −5⋅i_0 + 5⋅n_0

7.1.2 Transition Removal

We remove transition 1 using the following ranking functions, which are bounded by 6.

0: 1 − 6⋅j_0 + 6⋅n_0
1: −1 − 6⋅j_0 + 6⋅n_0
2: −6⋅j_0 + 6⋅n_0
3: −2 − 6⋅j_0 + 6⋅n_0
5: 2 − 6⋅j_0 + 6⋅n_0
6: −2 − 6⋅j_0 + 6⋅n_0
7: −1 − 6⋅j_0 + 6⋅n_0
1_var_snapshot: −1 − 6⋅j_0 + 6⋅n_0
1*: −6⋅j_0 + 6⋅n_0
5_var_snapshot: 2 − 6⋅j_0 + 6⋅n_0
5*: 3 − 6⋅j_0 + 6⋅n_0

7.1.3 Transition Removal

We remove transitions 14, 16, 21, 23, 0, 4, 5, 6, 7, 8, 9, 10 using the following ranking functions, which are bounded by −10.

0: −6
1: −8
2: 0
3: −10
5: −4
6: −2
7: −1
1_var_snapshot: −9
1*: −7
5_var_snapshot: −5
5*: −3

7.1.4 Splitting Cut-Point Transitions

We consider 2 subproblems corresponding to sets of cut-point transitions as follows.

7.1.4.1 Cut-Point Subproblem 1/2

Here we consider cut-point transition 13.

7.1.4.1.1 Splitting Cut-Point Transitions

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

7.1.4.2 Cut-Point Subproblem 2/2

Here we consider cut-point transition 20.

7.1.4.2.1 Splitting Cut-Point Transitions

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

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