LTS Termination Proof

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Input

Integer Transition System

Proof

1 Invariant Updates

The following invariants are asserted.

0: −1 + a_post ≤ 01 − a_post ≤ 01 − a_0 ≤ 0a_0 ≤ 0
1: −1 + a_post ≤ 01 − a_post ≤ 01 − a_0 ≤ 0a_0 ≤ 0
2: −1 + a_post ≤ 01 − a_post ≤ 0−1 + a_0 ≤ 01 − a_0 ≤ 0−1 + b_post ≤ 0−1 + b_0 ≤ 0
3: −1 + a_post ≤ 01 − a_post ≤ 0−1 + a_0 ≤ 01 − a_0 ≤ 0−1 + b_post ≤ 0b_0 ≤ 0
4: −1 + a_post ≤ 01 − a_post ≤ 0−1 + a_0 ≤ 01 − a_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:
0 8 0: b_post + b_post ≤ 0b_postb_post ≤ 0b_0 + b_0 ≤ 0b_0b_0 ≤ 0a_post + a_post ≤ 0a_posta_post ≤ 0a_0 + a_0 ≤ 0a_0a_0 ≤ 0
and for every transition t, a duplicate t is considered.

3 Transition Removal

We remove transitions 2, 3, 4, 5, 6, 7 using the following ranking functions, which are bounded by −17.

6: 0
5: 0
4: 0
0: 0
1: 0
2: 0
3: 0
6: −7
5: −8
4: −9
0: −10
1: −10
0_var_snapshot: −10
0*: −10
2: −14
3: −15

4 Location Addition

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

0* 11 0: b_post + b_post ≤ 0b_postb_post ≤ 0b_0 + b_0 ≤ 0b_0b_0 ≤ 0a_post + a_post ≤ 0a_posta_post ≤ 0a_0 + a_0 ≤ 0a_0a_0 ≤ 0

5 Location Addition

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

0 9 0_var_snapshot: b_post + b_post ≤ 0b_postb_post ≤ 0b_0 + b_0 ≤ 0b_0b_0 ≤ 0a_post + a_post ≤ 0a_posta_post ≤ 0a_0 + a_0 ≤ 0a_0a_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, 1, 0_var_snapshot, 0* }.

6.1.1 Transition Removal

We remove transitions 11, 0, 1 using the following ranking functions, which are bounded by −3.

0: −2
1: 0
0_var_snapshot: 0
0*: −1

6.1.2 Transition Removal

We remove transition 9 using the following ranking functions, which are bounded by −1.

0: 0
1: 0
0_var_snapshot: 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 8.

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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