LTS Termination Proof

by T2Cert

Input

Integer Transition System

Proof

1 Switch to Cooperation Termination Proof

We consider the following cutpoint-transitions:
1 7 1: arg4P + arg4P ≤ 0arg4Parg4P ≤ 0arg4 + arg4 ≤ 0arg4arg4 ≤ 0arg3P + arg3P ≤ 0arg3Parg3P ≤ 0arg3 + arg3 ≤ 0arg3arg3 ≤ 0arg2P + arg2P ≤ 0arg2Parg2P ≤ 0arg2 + arg2 ≤ 0arg2arg2 ≤ 0arg1P + arg1P ≤ 0arg1Parg1P ≤ 0arg1 + arg1 ≤ 0arg1arg1 ≤ 0
2 14 2: arg4P + arg4P ≤ 0arg4Parg4P ≤ 0arg4 + arg4 ≤ 0arg4arg4 ≤ 0arg3P + 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, 6 using the following ranking functions, which are bounded by −13.

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

3 Location Addition

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

1* 10 1: arg4P + arg4P ≤ 0arg4Parg4P ≤ 0arg4 + arg4 ≤ 0arg4arg4 ≤ 0arg3P + 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 8 1_var_snapshot: arg4P + arg4P ≤ 0arg4Parg4P ≤ 0arg4 + arg4 ≤ 0arg4arg4 ≤ 0arg3P + 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* 17 2: arg4P + arg4P ≤ 0arg4Parg4P ≤ 0arg4 + arg4 ≤ 0arg4arg4 ≤ 0arg3P + 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 15 2_var_snapshot: arg4P + arg4P ≤ 0arg4Parg4P ≤ 0arg4 + arg4 ≤ 0arg4arg4 ≤ 0arg3P + 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 1 SCC(s) of the program graph.

7.1 SCC Subproblem 1/1

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

7.1.1 Transition Removal

We remove transition 1 using the following ranking functions, which are bounded by 0.

1: −2 + 4⋅arg1
2: 4⋅arg2 − 4⋅arg3
1_var_snapshot: −3 + 4⋅arg1
1*: −1 + 4⋅arg1
2_var_snapshot: 4⋅arg2 − 4⋅arg3
2*: 4⋅arg2 − 4⋅arg3

7.1.2 Transition Removal

We remove transitions 8, 10, 2, 4 using the following ranking functions, which are bounded by −3.

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

7.1.3 Splitting Cut-Point Transitions

We consider 2 subproblems corresponding to sets of cut-point transitions as follows.

7.1.3.1 Cut-Point Subproblem 1/2

Here we consider cut-point transition 7.

7.1.3.1.1 Splitting Cut-Point Transitions

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

7.1.3.2 Cut-Point Subproblem 2/2

Here we consider cut-point transition 14.

7.1.3.2.1 Fresh Variable Addition

The new variable __snapshot_2_arg4P is introduced. The transition formulas are extended as follows:

8: __snapshot_2_arg4P__snapshot_2_arg4P__snapshot_2_arg4P__snapshot_2_arg4P
15: __snapshot_2_arg4Parg4Parg4P__snapshot_2_arg4P
17: __snapshot_2_arg4P__snapshot_2_arg4P__snapshot_2_arg4P__snapshot_2_arg4P
3: __snapshot_2_arg4P__snapshot_2_arg4P__snapshot_2_arg4P__snapshot_2_arg4P
5: __snapshot_2_arg4P__snapshot_2_arg4P__snapshot_2_arg4P__snapshot_2_arg4P

7.1.3.2.2 Fresh Variable Addition

The new variable __snapshot_2_arg4 is introduced. The transition formulas are extended as follows:

8: __snapshot_2_arg4__snapshot_2_arg4__snapshot_2_arg4__snapshot_2_arg4
15: __snapshot_2_arg4arg4arg4__snapshot_2_arg4
17: __snapshot_2_arg4__snapshot_2_arg4__snapshot_2_arg4__snapshot_2_arg4
3: __snapshot_2_arg4__snapshot_2_arg4__snapshot_2_arg4__snapshot_2_arg4
5: __snapshot_2_arg4__snapshot_2_arg4__snapshot_2_arg4__snapshot_2_arg4

7.1.3.2.3 Fresh Variable Addition

The new variable __snapshot_2_arg3P is introduced. The transition formulas are extended as follows:

8: __snapshot_2_arg3P__snapshot_2_arg3P__snapshot_2_arg3P__snapshot_2_arg3P
15: __snapshot_2_arg3Parg3Parg3P__snapshot_2_arg3P
17: __snapshot_2_arg3P__snapshot_2_arg3P__snapshot_2_arg3P__snapshot_2_arg3P
3: __snapshot_2_arg3P__snapshot_2_arg3P__snapshot_2_arg3P__snapshot_2_arg3P
5: __snapshot_2_arg3P__snapshot_2_arg3P__snapshot_2_arg3P__snapshot_2_arg3P

7.1.3.2.4 Fresh Variable Addition

The new variable __snapshot_2_arg3 is introduced. The transition formulas are extended as follows:

8: __snapshot_2_arg3__snapshot_2_arg3__snapshot_2_arg3__snapshot_2_arg3
15: __snapshot_2_arg3arg3arg3__snapshot_2_arg3
17: __snapshot_2_arg3__snapshot_2_arg3__snapshot_2_arg3__snapshot_2_arg3
3: __snapshot_2_arg3__snapshot_2_arg3__snapshot_2_arg3__snapshot_2_arg3
5: __snapshot_2_arg3__snapshot_2_arg3__snapshot_2_arg3__snapshot_2_arg3

7.1.3.2.5 Fresh Variable Addition

The new variable __snapshot_2_arg2P is introduced. The transition formulas are extended as follows:

8: __snapshot_2_arg2P__snapshot_2_arg2P__snapshot_2_arg2P__snapshot_2_arg2P
15: __snapshot_2_arg2Parg2Parg2P__snapshot_2_arg2P
17: __snapshot_2_arg2P__snapshot_2_arg2P__snapshot_2_arg2P__snapshot_2_arg2P
3: __snapshot_2_arg2P__snapshot_2_arg2P__snapshot_2_arg2P__snapshot_2_arg2P
5: __snapshot_2_arg2P__snapshot_2_arg2P__snapshot_2_arg2P__snapshot_2_arg2P

7.1.3.2.6 Fresh Variable Addition

The new variable __snapshot_2_arg2 is introduced. The transition formulas are extended as follows:

8: __snapshot_2_arg2__snapshot_2_arg2__snapshot_2_arg2__snapshot_2_arg2
15: __snapshot_2_arg2arg2arg2__snapshot_2_arg2
17: __snapshot_2_arg2__snapshot_2_arg2__snapshot_2_arg2__snapshot_2_arg2
3: __snapshot_2_arg2__snapshot_2_arg2__snapshot_2_arg2__snapshot_2_arg2
5: __snapshot_2_arg2__snapshot_2_arg2__snapshot_2_arg2__snapshot_2_arg2

7.1.3.2.7 Fresh Variable Addition

The new variable __snapshot_2_arg1P is introduced. The transition formulas are extended as follows:

8: __snapshot_2_arg1P__snapshot_2_arg1P__snapshot_2_arg1P__snapshot_2_arg1P
15: __snapshot_2_arg1Parg1Parg1P__snapshot_2_arg1P
17: __snapshot_2_arg1P__snapshot_2_arg1P__snapshot_2_arg1P__snapshot_2_arg1P
3: __snapshot_2_arg1P__snapshot_2_arg1P__snapshot_2_arg1P__snapshot_2_arg1P
5: __snapshot_2_arg1P__snapshot_2_arg1P__snapshot_2_arg1P__snapshot_2_arg1P

7.1.3.2.8 Fresh Variable Addition

The new variable __snapshot_2_arg1 is introduced. The transition formulas are extended as follows:

8: __snapshot_2_arg1__snapshot_2_arg1__snapshot_2_arg1__snapshot_2_arg1
15: __snapshot_2_arg1arg1arg1__snapshot_2_arg1
17: __snapshot_2_arg1__snapshot_2_arg1__snapshot_2_arg1__snapshot_2_arg1
3: __snapshot_2_arg1__snapshot_2_arg1__snapshot_2_arg1__snapshot_2_arg1
5: __snapshot_2_arg1__snapshot_2_arg1__snapshot_2_arg1__snapshot_2_arg1

7.1.3.2.9 Invariant Updates

The following invariants are asserted.

0: TRUE
1: TRUE
2: 1 − arg4 ≤ 01 ≤ 0arg4 ≤ 0
3: TRUE
1: FALSE
2: 1 − arg4 ≤ 01 − __snapshot_2_arg3 + arg3 ≤ 01 − __snapshot_2_arg3 ≤ 0arg4 ≤ 0arg4 ≤ 0
1_var_snapshot: 1 ≤ 0
1*: 1 ≤ 0
2_var_snapshot: __snapshot_2_arg3 + arg3 ≤ 01 − __snapshot_2_arg3 + arg3arg4 ≤ 01 − arg4 ≤ 0__snapshot_2_arg3 + arg3 ≤ 0arg4 ≤ 0
2*: 1 ≤ 01 − __snapshot_2_arg3 + arg3 ≤ 01 − __snapshot_2_arg3 ≤ 0arg4 ≤ 0

The invariants are proved as follows.

IMPACT Invariant Proof

7.1.3.2.10 Transition Removal

We remove transition 17 using the following ranking functions, which are bounded by −1.

1: __snapshot_2_arg3
2: arg3
1_var_snapshot: __snapshot_2_arg3
1*: __snapshot_2_arg3
2_var_snapshot: __snapshot_2_arg3
2*: __snapshot_2_arg3

7.1.3.2.11 Transition Removal

We remove transition 15 using the following ranking functions, which are bounded by −8.

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

7.1.3.2.12 Splitting Cut-Point Transitions

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

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