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 ≤ 01 − arg1 ≤ 0
2: 1 − arg1P ≤ 01 − arg1 ≤ 0
3: TRUE
5: 1 − arg1 ≤ 01 − arg2 ≤ 0
7: TRUE

The invariants are proved as follows.

IMPACT Invariant Proof

2 Switch to Cooperation Termination Proof

We consider the following cutpoint-transitions:
1 16 1: x50 + x50 ≤ 0x50x50 ≤ 0x47 + x47 ≤ 0x47x47 ≤ 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
2 23 2: x50 + x50 ≤ 0x50x50 ≤ 0x47 + x47 ≤ 0x47x47 ≤ 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
3 30 3: x50 + x50 ≤ 0x50x50 ≤ 0x47 + x47 ≤ 0x47x47 ≤ 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, 3, 15 using the following ranking functions, which are bounded by −17.

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

4 Location Addition

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

1* 19 1: x50 + x50 ≤ 0x50x50 ≤ 0x47 + x47 ≤ 0x47x47 ≤ 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.

1 17 1_var_snapshot: x50 + x50 ≤ 0x50x50 ≤ 0x47 + x47 ≤ 0x47x47 ≤ 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.

2* 26 2: x50 + x50 ≤ 0x50x50 ≤ 0x47 + x47 ≤ 0x47x47 ≤ 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.

2 24 2_var_snapshot: x50 + x50 ≤ 0x50x50 ≤ 0x47 + x47 ≤ 0x47x47 ≤ 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 Location Addition

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

3* 33 3: x50 + x50 ≤ 0x50x50 ≤ 0x47 + x47 ≤ 0x47x47 ≤ 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

9 Location Addition

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

3 31 3_var_snapshot: x50 + x50 ≤ 0x50x50 ≤ 0x47 + x47 ≤ 0x47x47 ≤ 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

10 SCC Decomposition

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

10.1 SCC Subproblem 1/2

Here we consider the SCC { 3, 5, 3_var_snapshot, 3* }.

10.1.1 Transition Removal

We remove transitions 8, 9, 10, 12, 13, 14 using the following ranking functions, which are bounded by 5.

3: 1 + 4⋅arg1 + 3⋅arg2
5: −1 + 4⋅arg1 + 3⋅arg2
3_var_snapshot: 4⋅arg1 + 3⋅arg2
3*: 2 + 4⋅arg1 + 3⋅arg2

10.1.2 Transition Removal

We remove transitions 31, 33 using the following ranking functions, which are bounded by −1.

3: 0
5: 0
3_var_snapshot: −1
3*: 1

10.1.3 Splitting Cut-Point Transitions

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

10.1.3.1 Cut-Point Subproblem 1/1

Here we consider cut-point transition 30.

10.1.3.1.1 Splitting Cut-Point Transitions

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

10.2 SCC Subproblem 2/2

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

10.2.1 Transition Removal

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

1: 2 − 4⋅arg2 + 4⋅arg3
2: −4⋅arg2 + 4⋅arg4
1_var_snapshot: 1 − 4⋅arg2 + 4⋅arg3
1*: 3 − 4⋅arg2 + 4⋅arg3
2_var_snapshot: −4⋅arg2 + 4⋅arg4
2*: −4⋅arg2 + 4⋅arg4

10.2.2 Transition Removal

We remove transitions 17, 19, 2 using the following ranking functions, which are bounded by −3.

1: −2
2: 0
1_var_snapshot: −3⋅arg1P
1*: −1
2_var_snapshot: 0
2*: 0

10.2.3 Transition Removal

We remove transitions 4, 5, 7 using the following ranking functions, which are bounded by −1.

1: 0
2: 1 − 3⋅arg3 + 3⋅arg4
1_var_snapshot: 0
1*: 0
2_var_snapshot: −3⋅arg3 + 3⋅arg4
2*: 2 − 3⋅arg3 + 3⋅arg4

10.2.4 Transition Removal

We remove transitions 24, 26 using the following ranking functions, which are bounded by −1.

1: 0
2: 0
1_var_snapshot: 0
1*: 0
2_var_snapshot: arg1
2*: arg1P

10.2.5 Splitting Cut-Point Transitions

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

10.2.5.1 Cut-Point Subproblem 1/2

Here we consider cut-point transition 16.

10.2.5.1.1 Splitting Cut-Point Transitions

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

10.2.5.2 Cut-Point Subproblem 2/2

Here we consider cut-point transition 23.

10.2.5.2.1 Splitting Cut-Point Transitions

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

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