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
3: arg1P ≤ 01 − arg1 ≤ 0
4: TRUE

The invariants are proved as follows.

IMPACT Invariant Proof

2 Switch to Cooperation Termination Proof

We consider the following cutpoint-transitions:
1 8 1: arg2P + arg2P ≤ 0arg2Parg2P ≤ 0arg2 + arg2 ≤ 0arg2arg2 ≤ 0arg1P + arg1P ≤ 0arg1Parg1P ≤ 0arg1 + arg1 ≤ 0arg1arg1 ≤ 0
2 15 2: arg2P + arg2P ≤ 0arg2Parg2P ≤ 0arg2 + arg2 ≤ 0arg2arg2 ≤ 0arg1P + arg1P ≤ 0arg1Parg1P ≤ 0arg1 + arg1 ≤ 0arg1arg1 ≤ 0
3 22 3: arg2P + 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, 7 using the following ranking functions, which are bounded by −15.

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

4 Location Addition

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

1* 11 1: arg2P + 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 9 1_var_snapshot: arg2P + 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.

2* 18 2: arg2P + 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.

2 16 2_var_snapshot: arg2P + arg2P ≤ 0arg2Parg2P ≤ 0arg2 + arg2 ≤ 0arg2arg2 ≤ 0arg1P + arg1P ≤ 0arg1Parg1P ≤ 0arg1 + arg1 ≤ 0arg1arg1 ≤ 0

8 Location Addition

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

3* 25 3: arg2P + arg2P ≤ 0arg2Parg2P ≤ 0arg2 + arg2 ≤ 0arg2arg2 ≤ 0arg1P + arg1P ≤ 0arg1Parg1P ≤ 0arg1 + arg1 ≤ 0arg1arg1 ≤ 0

9 Location Addition

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

3 23 3_var_snapshot: arg2P + arg2P ≤ 0arg2Parg2P ≤ 0arg2 + arg2 ≤ 0arg2arg2 ≤ 0arg1P + arg1P ≤ 0arg1Parg1P ≤ 0arg1 + arg1 ≤ 0arg1arg1 ≤ 0

10 SCC Decomposition

We consider subproblems for each of the 1 SCC(s) of the program graph.

10.1 SCC Subproblem 1/1

Here we consider the SCC { 1, 2, 3, 1_var_snapshot, 1*, 2_var_snapshot, 2*, 3_var_snapshot, 3* }.

10.1.1 Transition Removal

We remove transitions 1, 2, 5, 6 using the following ranking functions, which are bounded by 5.

1: −2 + 7⋅arg1 + 9⋅arg2
2: 9⋅arg1 + 2⋅arg2
3: 14⋅arg1 + 2⋅arg2
1_var_snapshot: −3 + 7⋅arg1 + 9⋅arg2
1*: −1 + 7⋅arg1 + 9⋅arg2
2_var_snapshot: 9⋅arg1 + 2⋅arg2
2*: 1 + 9⋅arg1 + 2⋅arg2
3_var_snapshot: 14⋅arg1 + 2⋅arg2
3*: 1 + 14⋅arg1 + 2⋅arg2

10.1.2 Transition Removal

We remove transitions 9, 11, 23, 25, 3, 4 using the following ranking functions, which are bounded by −3.

1: −2
2: 1 + 3⋅arg2
3: 0
1_var_snapshot: −3
1*: −1
2_var_snapshot: 3⋅arg2
2*: 2 + 3⋅arg2
3_var_snapshot: −1
3*: 1

10.1.3 Transition Removal

We remove transitions 16, 18 using the following ranking functions, which are bounded by −1.

1: 0
2: 0
3: 0
1_var_snapshot: 0
1*: 0
2_var_snapshot: −1
2*: 1
3_var_snapshot: 0
3*: 0

10.1.4 Splitting Cut-Point Transitions

We consider 3 subproblems corresponding to sets of cut-point transitions as follows.

10.1.4.1 Cut-Point Subproblem 1/3

Here we consider cut-point transition 8.

10.1.4.1.1 Splitting Cut-Point Transitions

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

10.1.4.2 Cut-Point Subproblem 2/3

Here we consider cut-point transition 15.

10.1.4.2.1 Splitting Cut-Point Transitions

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

10.1.4.3 Cut-Point Subproblem 3/3

Here we consider cut-point transition 22.

10.1.4.3.1 Splitting Cut-Point Transitions

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

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