LTS Termination Proof

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Input

Integer Transition System

Proof

1 Invariant Updates

The following invariants are asserted.

0: TRUE
1: 1 − arg1P ≤ 0arg2P ≤ 01 − arg1 ≤ 0arg2 ≤ 0
2: arg1P ≤ 0arg1 ≤ 0
3: 1 − arg1P ≤ 01 − arg2P ≤ 01 − arg1 ≤ 01 − arg2 ≤ 0
5: arg1P ≤ 0arg1 ≤ 01 − arg2 ≤ 0
6: arg1P ≤ 0arg2P ≤ 0arg1 ≤ 0arg2 ≤ 0
8: 1 − arg2P ≤ 01 − arg3P ≤ 0arg4P ≤ 01 − arg2 ≤ 01 − arg3 ≤ 0arg4 ≤ 0
9: 1 − arg2P ≤ 02 − arg3P ≤ 01 − arg1 ≤ 01 − arg2 ≤ 02 − arg3 ≤ 0
10: 1 − arg2P ≤ 01 − arg3P ≤ 0arg4P ≤ 01 − arg2 ≤ 01 − arg3 ≤ 0arg4 ≤ 0
11: 1 − arg2P ≤ 02 − arg3P ≤ 01 − arg1 ≤ 01 − arg2 ≤ 02 − arg3 ≤ 0
12: 1 − arg1P ≤ 0arg2P ≤ 0arg4P ≤ 01 − arg1 ≤ 0arg2 ≤ 0arg4 ≤ 0
13: TRUE
14: TRUE

The invariants are proved as follows.

IMPACT Invariant Proof

2 Switch to Cooperation Termination Proof

We consider the following cutpoint-transitions:
2 36 2: arg9P + arg9P ≤ 0arg9Parg9P ≤ 0arg9 + arg9 ≤ 0arg9arg9 ≤ 0arg8P + arg8P ≤ 0arg8Parg8P ≤ 0arg8 + arg8 ≤ 0arg8arg8 ≤ 0arg7P + arg7P ≤ 0arg7Parg7P ≤ 0arg7 + arg7 ≤ 0arg7arg7 ≤ 0arg6P + arg6P ≤ 0arg6Parg6P ≤ 0arg6 + arg6 ≤ 0arg6arg6 ≤ 0arg5P + arg5P ≤ 0arg5Parg5P ≤ 0arg5 + arg5 ≤ 0arg5arg5 ≤ 0arg4P + 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 43 6: arg9P + arg9P ≤ 0arg9Parg9P ≤ 0arg9 + arg9 ≤ 0arg9arg9 ≤ 0arg8P + arg8P ≤ 0arg8Parg8P ≤ 0arg8 + arg8 ≤ 0arg8arg8 ≤ 0arg7P + arg7P ≤ 0arg7Parg7P ≤ 0arg7 + arg7 ≤ 0arg7arg7 ≤ 0arg6P + arg6P ≤ 0arg6Parg6P ≤ 0arg6 + arg6 ≤ 0arg6arg6 ≤ 0arg5P + arg5P ≤ 0arg5Parg5P ≤ 0arg5 + arg5 ≤ 0arg5arg5 ≤ 0arg4P + 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.

3 Transition Removal

We remove transitions 0, 1, 2, 3, 5, 6, 7, 8, 9, 32, 33, 34, 35 using the following ranking functions, which are bounded by −23.

14: 0
0: 0
1: 0
3: 0
2: 0
12: 0
5: 0
6: 0
8: 0
9: 0
10: 0
11: 0
13: 0
14: −9
0: −10
1: −11
3: −12
2: −13
12: −13
2_var_snapshot: −13
2*: −13
5: −16
6: −17
8: −17
9: −17
10: −17
11: −17
6_var_snapshot: −17
6*: −17
13: −21

4 Location Addition

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

2* 39 2: arg9P + arg9P ≤ 0arg9Parg9P ≤ 0arg9 + arg9 ≤ 0arg9arg9 ≤ 0arg8P + arg8P ≤ 0arg8Parg8P ≤ 0arg8 + arg8 ≤ 0arg8arg8 ≤ 0arg7P + arg7P ≤ 0arg7Parg7P ≤ 0arg7 + arg7 ≤ 0arg7arg7 ≤ 0arg6P + arg6P ≤ 0arg6Parg6P ≤ 0arg6 + arg6 ≤ 0arg6arg6 ≤ 0arg5P + arg5P ≤ 0arg5Parg5P ≤ 0arg5 + arg5 ≤ 0arg5arg5 ≤ 0arg4P + 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 37 2_var_snapshot: arg9P + arg9P ≤ 0arg9Parg9P ≤ 0arg9 + arg9 ≤ 0arg9arg9 ≤ 0arg8P + arg8P ≤ 0arg8Parg8P ≤ 0arg8 + arg8 ≤ 0arg8arg8 ≤ 0arg7P + arg7P ≤ 0arg7Parg7P ≤ 0arg7 + arg7 ≤ 0arg7arg7 ≤ 0arg6P + arg6P ≤ 0arg6Parg6P ≤ 0arg6 + arg6 ≤ 0arg6arg6 ≤ 0arg5P + arg5P ≤ 0arg5Parg5P ≤ 0arg5 + arg5 ≤ 0arg5arg5 ≤ 0arg4P + 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.

6* 46 6: arg9P + arg9P ≤ 0arg9Parg9P ≤ 0arg9 + arg9 ≤ 0arg9arg9 ≤ 0arg8P + arg8P ≤ 0arg8Parg8P ≤ 0arg8 + arg8 ≤ 0arg8arg8 ≤ 0arg7P + arg7P ≤ 0arg7Parg7P ≤ 0arg7 + arg7 ≤ 0arg7arg7 ≤ 0arg6P + arg6P ≤ 0arg6Parg6P ≤ 0arg6 + arg6 ≤ 0arg6arg6 ≤ 0arg5P + arg5P ≤ 0arg5Parg5P ≤ 0arg5 + arg5 ≤ 0arg5arg5 ≤ 0arg4P + 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 Location Addition

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

6 44 6_var_snapshot: arg9P + arg9P ≤ 0arg9Parg9P ≤ 0arg9 + arg9 ≤ 0arg9arg9 ≤ 0arg8P + arg8P ≤ 0arg8Parg8P ≤ 0arg8 + arg8 ≤ 0arg8arg8 ≤ 0arg7P + arg7P ≤ 0arg7Parg7P ≤ 0arg7 + arg7 ≤ 0arg7arg7 ≤ 0arg6P + arg6P ≤ 0arg6Parg6P ≤ 0arg6 + arg6 ≤ 0arg6arg6 ≤ 0arg5P + arg5P ≤ 0arg5Parg5P ≤ 0arg5 + arg5 ≤ 0arg5arg5 ≤ 0arg4P + arg4P ≤ 0arg4Parg4P ≤ 0arg4 + arg4 ≤ 0arg4arg4 ≤ 0arg3P + 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 { 2, 12, 2_var_snapshot, 2* }.

8.1.1 Transition Removal

We remove transitions 37, 39, 25, 26, 27, 28, 29, 30, 31 using the following ranking functions, which are bounded by −4.

2: −2 + 4⋅arg1
12: 4⋅arg2
2_var_snapshot: −3 + 4⋅arg1
2*: −1 + 4⋅arg1

8.1.2 Splitting Cut-Point Transitions

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

8.1.2.1 Cut-Point Subproblem 1/1

Here we consider cut-point transition 36.

8.1.2.1.1 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 { 6, 8, 9, 10, 11, 6_var_snapshot, 6* }.

8.2.1 Transition Removal

We remove transitions 11, 13, 14, 15, 16, 17, 18, 21, 22, 23, 24 using the following ranking functions, which are bounded by 0.

6: −1 + 6⋅arg3
8: −4 + 6⋅arg1
9: −5 + 6⋅arg1
10: −3 + 6⋅arg1
11: −4 + 6⋅arg1
6_var_snapshot: −2 + 6⋅arg3
6*: 6⋅arg3

8.2.2 Transition Removal

We remove transitions 44, 46, 12, 20 using the following ranking functions, which are bounded by −2.

6: −1
8: 1
9: 0
10: 2⋅arg2
11: 0
6_var_snapshot: −2
6*: 0

8.2.3 Transition Removal

We remove transition 19 using the following ranking functions, which are bounded by 0.

6: 0
8: 0
9: 0
10: arg2
11: 0
6_var_snapshot: 0
6*: 0

8.2.4 Splitting Cut-Point Transitions

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

8.2.4.1 Cut-Point Subproblem 1/1

Here we consider cut-point transition 43.

8.2.4.1.1 Splitting Cut-Point Transitions

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

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