LTS Termination Proof

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Input

Integer Transition System

Proof

1 Switch to Cooperation Termination Proof

We consider the following cutpoint-transitions:
1 6 1: arg3P + arg3P ≤ 0arg3Parg3P ≤ 0arg3 + arg3 ≤ 0arg3arg3 ≤ 0arg2P + arg2P ≤ 0arg2Parg2P ≤ 0arg2 + arg2 ≤ 0arg2arg2 ≤ 0arg1P + arg1P ≤ 0arg1Parg1P ≤ 0arg1 + arg1 ≤ 0arg1arg1 ≤ 0
2 13 2: 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.

2 Transition Removal

We remove transitions 0, 3, 5 using the following ranking functions, which are bounded by −15.

3: 0
0: 0
1: 0
2: 0
3: −5
0: −6
1: −7
1_var_snapshot: −7
1*: −7
2: −10
2_var_snapshot: −10
2*: −10

3 Location Addition

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

1* 9 1: arg3P + arg3P ≤ 0arg3Parg3P ≤ 0arg3 + arg3 ≤ 0arg3arg3 ≤ 0arg2P + arg2P ≤ 0arg2Parg2P ≤ 0arg2 + arg2 ≤ 0arg2arg2 ≤ 0arg1P + arg1P ≤ 0arg1Parg1P ≤ 0arg1 + arg1 ≤ 0arg1arg1 ≤ 0

4 Location Addition

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

1 7 1_var_snapshot: 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.

2* 16 2: 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.

2 14 2_var_snapshot: arg3P + arg3P ≤ 0arg3Parg3P ≤ 0arg3 + arg3 ≤ 0arg3arg3 ≤ 0arg2P + arg2P ≤ 0arg2Parg2P ≤ 0arg2 + arg2 ≤ 0arg2arg2 ≤ 0arg1P + arg1P ≤ 0arg1Parg1P ≤ 0arg1 + arg1 ≤ 0arg1arg1 ≤ 0

7 SCC Decomposition

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

7.1 SCC Subproblem 1/2

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

7.1.1 Transition Removal

We remove transition 4 using the following ranking functions, which are bounded by 2.

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

7.1.2 Transition Removal

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

2: 0
2_var_snapshot: −1
2*: 1

7.1.3 Transition Removal

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

2: −1
2_var_snapshot: 0
2*: 0

7.1.4 Splitting Cut-Point Transitions

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

7.1.4.1 Cut-Point Subproblem 1/1

Here we consider cut-point transition 13.

7.1.4.1.1 Splitting Cut-Point Transitions

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

7.2 SCC Subproblem 2/2

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

7.2.1 Splitting Cut-Point Transitions

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

7.2.1.1 Cut-Point Subproblem 1/1

Here we consider cut-point transition 6.

7.2.1.1.1 Fresh Variable Addition

The new variable __snapshot_1_arg3P is introduced. The transition formulas are extended as follows:

7: __snapshot_1_arg3Parg3Parg3P__snapshot_1_arg3P
9: __snapshot_1_arg3P__snapshot_1_arg3P__snapshot_1_arg3P__snapshot_1_arg3P
1: __snapshot_1_arg3P__snapshot_1_arg3P__snapshot_1_arg3P__snapshot_1_arg3P
2: __snapshot_1_arg3P__snapshot_1_arg3P__snapshot_1_arg3P__snapshot_1_arg3P

7.2.1.1.2 Fresh Variable Addition

The new variable __snapshot_1_arg3 is introduced. The transition formulas are extended as follows:

7: __snapshot_1_arg3arg3arg3__snapshot_1_arg3
9: __snapshot_1_arg3__snapshot_1_arg3__snapshot_1_arg3__snapshot_1_arg3
1: __snapshot_1_arg3__snapshot_1_arg3__snapshot_1_arg3__snapshot_1_arg3
2: __snapshot_1_arg3__snapshot_1_arg3__snapshot_1_arg3__snapshot_1_arg3

7.2.1.1.3 Fresh Variable Addition

The new variable __snapshot_1_arg2P is introduced. The transition formulas are extended as follows:

7: __snapshot_1_arg2Parg2Parg2P__snapshot_1_arg2P
9: __snapshot_1_arg2P__snapshot_1_arg2P__snapshot_1_arg2P__snapshot_1_arg2P
1: __snapshot_1_arg2P__snapshot_1_arg2P__snapshot_1_arg2P__snapshot_1_arg2P
2: __snapshot_1_arg2P__snapshot_1_arg2P__snapshot_1_arg2P__snapshot_1_arg2P

7.2.1.1.4 Fresh Variable Addition

The new variable __snapshot_1_arg2 is introduced. The transition formulas are extended as follows:

7: __snapshot_1_arg2arg2arg2__snapshot_1_arg2
9: __snapshot_1_arg2__snapshot_1_arg2__snapshot_1_arg2__snapshot_1_arg2
1: __snapshot_1_arg2__snapshot_1_arg2__snapshot_1_arg2__snapshot_1_arg2
2: __snapshot_1_arg2__snapshot_1_arg2__snapshot_1_arg2__snapshot_1_arg2

7.2.1.1.5 Fresh Variable Addition

The new variable __snapshot_1_arg1P is introduced. The transition formulas are extended as follows:

7: __snapshot_1_arg1Parg1Parg1P__snapshot_1_arg1P
9: __snapshot_1_arg1P__snapshot_1_arg1P__snapshot_1_arg1P__snapshot_1_arg1P
1: __snapshot_1_arg1P__snapshot_1_arg1P__snapshot_1_arg1P__snapshot_1_arg1P
2: __snapshot_1_arg1P__snapshot_1_arg1P__snapshot_1_arg1P__snapshot_1_arg1P

7.2.1.1.6 Fresh Variable Addition

The new variable __snapshot_1_arg1 is introduced. The transition formulas are extended as follows:

7: __snapshot_1_arg1arg1arg1__snapshot_1_arg1
9: __snapshot_1_arg1__snapshot_1_arg1__snapshot_1_arg1__snapshot_1_arg1
1: __snapshot_1_arg1__snapshot_1_arg1__snapshot_1_arg1__snapshot_1_arg1
2: __snapshot_1_arg1__snapshot_1_arg1__snapshot_1_arg1__snapshot_1_arg1

7.2.1.1.7 Invariant Updates

The following invariants are asserted.

0: arg2arg2P ≤ 0
1: −1 − arg1 + arg2 ≤ 01 + arg1arg2P ≤ 0arg2P ≤ 0
2: TRUE
3: TRUE
1: −1 − arg1 + arg2 ≤ 01 + arg1arg2P ≤ 0arg2P ≤ 0−1 − arg1 + arg2 ≤ 01 − __snapshot_1_arg2P + arg2P ≤ 01 + arg1arg2P ≤ 0__snapshot_1_arg2P ≤ 0arg2P ≤ 0
1_var_snapshot: 1 − __snapshot_1_arg2P + arg1 ≤ 0−1 − arg1 + arg2 ≤ 0__snapshot_1_arg2P ≤ 0
1*: −1 − arg1 + arg2 ≤ 01 − __snapshot_1_arg2P + arg2P ≤ 01 + arg1arg2P ≤ 0__snapshot_1_arg2P ≤ 0arg2P ≤ 0

The invariants are proved as follows.

IMPACT Invariant Proof

7.2.1.1.8 Transition Removal

We remove transition 9 using the following ranking functions, which are bounded by −2.

1: arg2P
1_var_snapshot: __snapshot_1_arg2P
1*: __snapshot_1_arg2P

7.2.1.1.9 Transition Removal

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

1: −1
1_var_snapshot: −2
1*: −3

7.2.1.1.10 Splitting Cut-Point Transitions

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

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