LTS Termination Proof

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Input

Integer Transition System

Proof

1 Invariant Updates

The following invariants are asserted.

0: 1 − c_0 ≤ 0
1: TRUE
2: 1 − c_0 ≤ 0
3: c_0 ≤ 0
4: TRUE
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:
1 9 1: x_post + x_post ≤ 0x_postx_post ≤ 0x_0 + x_0 ≤ 0x_0x_0 ≤ 0ox_post + ox_post ≤ 0ox_postox_post ≤ 0ox_0 + ox_0 ≤ 0ox_0ox_0 ≤ 0c_post + c_post ≤ 0c_postc_post ≤ 0c_0 + c_0 ≤ 0c_0c_0 ≤ 0
and for every transition t, a duplicate t is considered.

3 Transition Removal

We remove transitions 1, 7, 8 using the following ranking functions, which are bounded by −13.

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

4 Location Addition

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

1* 12 1: x_post + x_post ≤ 0x_postx_post ≤ 0x_0 + x_0 ≤ 0x_0x_0 ≤ 0ox_post + ox_post ≤ 0ox_postox_post ≤ 0ox_0 + ox_0 ≤ 0ox_0ox_0 ≤ 0c_post + c_post ≤ 0c_postc_post ≤ 0c_0 + c_0 ≤ 0c_0c_0 ≤ 0

5 Location Addition

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

1 10 1_var_snapshot: x_post + x_post ≤ 0x_postx_post ≤ 0x_0 + x_0 ≤ 0x_0x_0 ≤ 0ox_post + ox_post ≤ 0ox_postox_post ≤ 0ox_0 + ox_0 ≤ 0ox_0ox_0 ≤ 0c_post + c_post ≤ 0c_postc_post ≤ 0c_0 + c_0 ≤ 0c_0c_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, 3, 4, 1_var_snapshot, 1* }.

6.1.1 Transition Removal

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

0: 3 + 7⋅x_0
1: 1 + 7⋅x_0
3: 3 + 7⋅x_0
4: 4 + 7⋅x_0
1_var_snapshot: 7⋅x_0
1*: 2 + 7⋅x_0

6.1.2 Transition Removal

We remove transitions 10, 12, 0, 2, 3, 4, 5 using the following ranking functions, which are bounded by −2.

0: 1
1: −1
3: 1
4: 2
1_var_snapshot: −2
1*: 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 9.

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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