LTS Termination Proof

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Input

Integer Transition System

Proof

1 Invariant Updates

The following invariants are asserted.

0: TRUE
1: arg1P ≤ 0arg1 ≤ 0
2: TRUE
4: TRUE
5: TRUE

The invariants are proved as follows.

IMPACT Invariant Proof

2 Switch to Cooperation Termination Proof

We consider the following cutpoint-transitions:
1 11 1: arg3P + arg3P ≤ 0arg3Parg3P ≤ 0arg3 + arg3 ≤ 0arg3arg3 ≤ 0arg2P + arg2P ≤ 0arg2Parg2P ≤ 0arg2 + arg2 ≤ 0arg2arg2 ≤ 0arg1P + arg1P ≤ 0arg1Parg1P ≤ 0arg1 + arg1 ≤ 0arg1arg1 ≤ 0
4 18 4: arg3P + arg3P ≤ 0arg3Parg3P ≤ 0arg3 + arg3 ≤ 0arg3arg3 ≤ 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, 2, 3, 4, 5, 7, 10 using the following ranking functions, which are bounded by −17.

5: 0
0: 0
1: 0
2: 0
4: 0
5: −6
0: −7
1: −8
1_var_snapshot: −8
1*: −8
2: −11
4: −12
4_var_snapshot: −12
4*: −12

4 Location Addition

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

1* 14 1: arg3P + arg3P ≤ 0arg3Parg3P ≤ 0arg3 + arg3 ≤ 0arg3arg3 ≤ 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.

1 12 1_var_snapshot: arg3P + arg3P ≤ 0arg3Parg3P ≤ 0arg3 + arg3 ≤ 0arg3arg3 ≤ 0arg2P + arg2P ≤ 0arg2Parg2P ≤ 0arg2 + arg2 ≤ 0arg2arg2 ≤ 0arg1P + arg1P ≤ 0arg1Parg1P ≤ 0arg1 + arg1 ≤ 0arg1arg1 ≤ 0

6 Location Addition

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

4* 21 4: arg3P + arg3P ≤ 0arg3Parg3P ≤ 0arg3 + arg3 ≤ 0arg3arg3 ≤ 0arg2P + arg2P ≤ 0arg2Parg2P ≤ 0arg2 + arg2 ≤ 0arg2arg2 ≤ 0arg1P + arg1P ≤ 0arg1Parg1P ≤ 0arg1 + arg1 ≤ 0arg1arg1 ≤ 0

7 Location Addition

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

4 19 4_var_snapshot: arg3P + arg3P ≤ 0arg3Parg3P ≤ 0arg3 + arg3 ≤ 0arg3arg3 ≤ 0arg2P + arg2P ≤ 0arg2Parg2P ≤ 0arg2 + arg2 ≤ 0arg2arg2 ≤ 0arg1P + arg1P ≤ 0arg1Parg1P ≤ 0arg1 + arg1 ≤ 0arg1arg1 ≤ 0

8 SCC Decomposition

We consider subproblems for each of the 2 SCC(s) of the program graph.

8.1 SCC Subproblem 1/2

Here we consider the SCC { 4, 4_var_snapshot, 4* }.

8.1.1 Transition Removal

We remove transition 9 using the following ranking functions, which are bounded by 0.

4: arg1
4_var_snapshot: arg1
4*: arg1

8.1.2 Transition Removal

We remove transition 8 using the following ranking functions, which are bounded by 1.

4: 2⋅arg2
4_var_snapshot: 2⋅arg2
4*: 1 + 2⋅arg2

8.1.3 Transition Removal

We remove transition 19 using the following ranking functions, which are bounded by −1.

4: 0
4_var_snapshot: −1
4*: 1

8.1.4 Transition Removal

We remove transition 21 using the following ranking functions, which are bounded by −1.

4: −1
4_var_snapshot: 0
4*: 0

8.1.5 Splitting Cut-Point Transitions

We consider 1 subproblems corresponding to sets of cut-point transitions as follows.

8.1.5.1 Cut-Point Subproblem 1/1

Here we consider cut-point transition 18.

8.1.5.1.1 Splitting Cut-Point Transitions

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

8.2 SCC Subproblem 2/2

Here we consider the SCC { 1, 1_var_snapshot, 1* }.

8.2.1 Transition Removal

We remove transition 1 using the following ranking functions, which are bounded by 2.

1: 1 + 3⋅arg2
1_var_snapshot: 3⋅arg2
1*: 2 + 3⋅arg2

8.2.2 Transition Removal

We remove transitions 12, 14 using the following ranking functions, which are bounded by −1.

1: 0
1_var_snapshot: −1
1*: 1

8.2.3 Splitting Cut-Point Transitions

We consider 1 subproblems corresponding to sets of cut-point transitions as follows.

8.2.3.1 Cut-Point Subproblem 1/1

Here we consider cut-point transition 11.

8.2.3.1.1 Splitting Cut-Point Transitions

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

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