LTS Termination Proof

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Input

Integer Transition System

Proof

1 Invariant Updates

The following invariants are asserted.

0: TRUE
1: 1 − arg1 ≤ 01 − arg2 ≤ 0x2 ≤ 0
2: 1 − arg2 ≤ 0x2 ≤ 0x7 ≤ 0
3: TRUE

The invariants are proved as follows.

IMPACT Invariant Proof

2 Switch to Cooperation Termination Proof

We consider the following cutpoint-transitions:
2 4 2: x8 + x8 ≤ 0x8x8 ≤ 0x7 + x7 ≤ 0x7x7 ≤ 0x2 + x2 ≤ 0x2x2 ≤ 0arg2P + arg2P ≤ 0arg2Parg2P ≤ 0arg2 + arg2 ≤ 0arg2arg2 ≤ 0arg1P + arg1P ≤ 0arg1Parg1P ≤ 0arg1 + arg1 ≤ 0arg1arg1 ≤ 0
and for every transition t, a duplicate t is considered.

3 Transition Removal

We remove transitions 0, 1, 3 using the following ranking functions, which are bounded by −13.

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

4 Location Addition

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

2* 7 2: x8 + x8 ≤ 0x8x8 ≤ 0x7 + x7 ≤ 0x7x7 ≤ 0x2 + x2 ≤ 0x2x2 ≤ 0arg2P + arg2P ≤ 0arg2Parg2P ≤ 0arg2 + arg2 ≤ 0arg2arg2 ≤ 0arg1P + arg1P ≤ 0arg1Parg1P ≤ 0arg1 + arg1 ≤ 0arg1arg1 ≤ 0

5 Location Addition

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

2 5 2_var_snapshot: x8 + x8 ≤ 0x8x8 ≤ 0x7 + x7 ≤ 0x7x7 ≤ 0x2 + x2 ≤ 0x2x2 ≤ 0arg2P + arg2P ≤ 0arg2Parg2P ≤ 0arg2 + arg2 ≤ 0arg2arg2 ≤ 0arg1P + arg1P ≤ 0arg1Parg1P ≤ 0arg1 + arg1 ≤ 0arg1arg1 ≤ 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 { 2, 2_var_snapshot, 2* }.

6.1.1 Transition Removal

We remove transition 2 using the following ranking functions, which are bounded by 2.

2: 1 + 3⋅arg1
2_var_snapshot: 3⋅arg1
2*: 2 + 3⋅arg1

6.1.2 Transition Removal

We remove transitions 5, 7 using the following ranking functions, which are bounded by −1.

2: 0
2_var_snapshot: −1
2*: arg2

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 4.

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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