LTS Termination Proof

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Input

Integer Transition System

Proof

1 Invariant Updates

The following invariants are asserted.

0: −10 + x_post ≤ 010 − x_post ≤ 0−10 + x_0 ≤ 010 − x_0 ≤ 0
1: −10 + x_post ≤ 010 − x_post ≤ 010 − x_0 ≤ 0
2: −10 + x_post ≤ 010 − x_post ≤ 0−10 + x_0 ≤ 010 − x_0 ≤ 0
3: j_post ≤ 0j_post ≤ 0−10 + x_post ≤ 010 − x_post ≤ 0j_0 ≤ 0j_0 ≤ 0−10 + x_0 ≤ 010 − x_0 ≤ 0
4: −10 + x_post ≤ 010 − x_post ≤ 010 − x_0 ≤ 0
5: −10 + x_post ≤ 010 − x_post ≤ 010 − x_0 ≤ 0
6: TRUE
7: TRUE

The invariants are proved as follows.

IMPACT Invariant Proof

2 Switch to Cooperation Termination Proof

We consider the following cutpoint-transitions:
2 10 2: x_post + x_post ≤ 0x_postx_post ≤ 0x_0 + x_0 ≤ 0x_0x_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 ≤ 0
and for every transition t, a duplicate t is considered.

3 Transition Removal

We remove transitions 0, 2, 3, 5, 6, 7, 8, 9 using the following ranking functions, which are bounded by −19.

7: 0
6: 0
3: 0
0: 0
2: 0
1: 0
5: 0
4: 0
7: −8
6: −9
3: −10
0: −11
2: −11
2_var_snapshot: −11
2*: −11
1: −12
5: −13
4: −17

4 Location Addition

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

2* 13 2: x_post + x_post ≤ 0x_postx_post ≤ 0x_0 + x_0 ≤ 0x_0x_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 ≤ 0

5 Location Addition

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

2 11 2_var_snapshot: x_post + x_post ≤ 0x_postx_post ≤ 0x_0 + x_0 ≤ 0x_0x_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 ≤ 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 −128.

0: −13⋅i_0x_0
2: 1 − 13⋅i_0
2_var_snapshot: −13⋅i_0
2*: 2 − 13⋅i_0

6.1.2 Transition Removal

We remove transitions 11, 13, 4 using the following ranking functions, which are bounded by −11.

0: −1 − x_post
2: 0
2_var_snapshot: x_post
2*: x_post

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 10.

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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