LTS Termination Proof

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Input

Integer Transition System

Proof

1 Invariant Updates

The following invariants are asserted.

0: TRUE
1: arg1P ≤ 0arg2P ≤ 02 − arg3P ≤ 0arg1 ≤ 0arg2 ≤ 02 − arg3 ≤ 0x5 ≤ 0
2: 1 − arg1P ≤ 01 − arg1 ≤ 0x5 ≤ 0
3: 2 − arg1P ≤ 02 − arg1 ≤ 0x5 ≤ 0
4: 2 − arg1P ≤ 01 − arg2P ≤ 02 − arg1 ≤ 01 − arg2 ≤ 0x5 ≤ 0
5: arg1P ≤ 01 − arg2P ≤ 01 − arg3P ≤ 0arg4P ≤ 0arg1 ≤ 01 − arg2 ≤ 01 − arg3 ≤ 0arg4 ≤ 0x5 ≤ 0
6: 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: x87 + x87 ≤ 0x87x87 ≤ 0x86 + x86 ≤ 0x86x86 ≤ 0x79 + x79 ≤ 0x79x79 ≤ 0x78 + x78 ≤ 0x78x78 ≤ 0x5 + x5 ≤ 0x5x5 ≤ 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: x87 + x87 ≤ 0x87x87 ≤ 0x86 + x86 ≤ 0x86x86 ≤ 0x79 + x79 ≤ 0x79x79 ≤ 0x78 + x78 ≤ 0x78x78 ≤ 0x5 + x5 ≤ 0x5x5 ≤ 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: x87 + x87 ≤ 0x87x87 ≤ 0x86 + x86 ≤ 0x86x86 ≤ 0x79 + x79 ≤ 0x79x79 ≤ 0x78 + x78 ≤ 0x78x78 ≤ 0x5 + x5 ≤ 0x5x5 ≤ 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
4 37 4: x87 + x87 ≤ 0x87x87 ≤ 0x86 + x86 ≤ 0x86x86 ≤ 0x79 + x79 ≤ 0x79x79 ≤ 0x78 + x78 ≤ 0x78x78 ≤ 0x5 + x5 ≤ 0x5x5 ≤ 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, 4, 5, 6, 7, 8, 13, 14, 15 using the following ranking functions, which are bounded by −23.

6: 0
0: 0
1: 0
2: 0
4: 0
3: 0
5: 0
6: −7
0: −8
1: −9
1_var_snapshot: −9
1*: −9
2: −12
2_var_snapshot: −12
2*: −12
4: −15
4_var_snapshot: −15
4*: −15
3: −18
5: −18
3_var_snapshot: −18
3*: −18

4 Location Addition

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

1* 19 1: x87 + x87 ≤ 0x87x87 ≤ 0x86 + x86 ≤ 0x86x86 ≤ 0x79 + x79 ≤ 0x79x79 ≤ 0x78 + x78 ≤ 0x78x78 ≤ 0x5 + x5 ≤ 0x5x5 ≤ 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: x87 + x87 ≤ 0x87x87 ≤ 0x86 + x86 ≤ 0x86x86 ≤ 0x79 + x79 ≤ 0x79x79 ≤ 0x78 + x78 ≤ 0x78x78 ≤ 0x5 + x5 ≤ 0x5x5 ≤ 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: x87 + x87 ≤ 0x87x87 ≤ 0x86 + x86 ≤ 0x86x86 ≤ 0x79 + x79 ≤ 0x79x79 ≤ 0x78 + x78 ≤ 0x78x78 ≤ 0x5 + x5 ≤ 0x5x5 ≤ 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: x87 + x87 ≤ 0x87x87 ≤ 0x86 + x86 ≤ 0x86x86 ≤ 0x79 + x79 ≤ 0x79x79 ≤ 0x78 + x78 ≤ 0x78x78 ≤ 0x5 + x5 ≤ 0x5x5 ≤ 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: x87 + x87 ≤ 0x87x87 ≤ 0x86 + x86 ≤ 0x86x86 ≤ 0x79 + x79 ≤ 0x79x79 ≤ 0x78 + x78 ≤ 0x78x78 ≤ 0x5 + x5 ≤ 0x5x5 ≤ 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: x87 + x87 ≤ 0x87x87 ≤ 0x86 + x86 ≤ 0x86x86 ≤ 0x79 + x79 ≤ 0x79x79 ≤ 0x78 + x78 ≤ 0x78x78 ≤ 0x5 + x5 ≤ 0x5x5 ≤ 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 Location Addition

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

4* 40 4: x87 + x87 ≤ 0x87x87 ≤ 0x86 + x86 ≤ 0x86x86 ≤ 0x79 + x79 ≤ 0x79x79 ≤ 0x78 + x78 ≤ 0x78x78 ≤ 0x5 + x5 ≤ 0x5x5 ≤ 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

11 Location Addition

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

4 38 4_var_snapshot: x87 + x87 ≤ 0x87x87 ≤ 0x86 + x86 ≤ 0x86x86 ≤ 0x79 + x79 ≤ 0x79x79 ≤ 0x78 + x78 ≤ 0x78x78 ≤ 0x5 + x5 ≤ 0x5x5 ≤ 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

12 SCC Decomposition

We consider subproblems for each of the 4 SCC(s) of the program graph.

12.1 SCC Subproblem 1/4

Here we consider the SCC { 4, 4_var_snapshot, 4* }.

12.1.1 Transition Removal

We remove transition 12 using the following ranking functions, which are bounded by 2.

4: 1 + 3⋅arg4
4_var_snapshot: 3⋅arg4
4*: 2 + 3⋅arg4

12.1.2 Transition Removal

We remove transitions 38, 40 using the following ranking functions, which are bounded by −1.

4: 0
4_var_snapshot: arg1
4*: arg1P

12.1.3 Splitting Cut-Point Transitions

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

12.1.3.1 Cut-Point Subproblem 1/1

Here we consider cut-point transition 37.

12.1.3.1.1 Splitting Cut-Point Transitions

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

12.2 SCC Subproblem 2/4

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

12.2.1 Transition Removal

We remove transition 3 using the following ranking functions, which are bounded by 2.

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

12.2.2 Transition Removal

We remove transitions 24, 26 using the following ranking functions, which are bounded by −2.

2: −1
2_var_snapshot: −2
2*: 0

12.2.3 Splitting Cut-Point Transitions

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

12.2.3.1 Cut-Point Subproblem 1/1

Here we consider cut-point transition 23.

12.2.3.1.1 Splitting Cut-Point Transitions

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

12.3 SCC Subproblem 3/4

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

12.3.1 Transition Removal

We remove transitions 9, 10, 11 using the following ranking functions, which are bounded by −1.

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

12.3.2 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*: arg1P

12.3.3 Splitting Cut-Point Transitions

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

12.3.3.1 Cut-Point Subproblem 1/1

Here we consider cut-point transition 30.

12.3.3.1.1 Splitting Cut-Point Transitions

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

12.4 SCC Subproblem 4/4

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

12.4.1 Transition Removal

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

1: 1 + 3⋅arg4
1_var_snapshot: 3⋅arg4
1*: 2 + 3⋅arg4

12.4.2 Transition Removal

We remove transitions 17, 19 using the following ranking functions, which are bounded by −1.

1: 0
1_var_snapshot: arg3
1*: 1

12.4.3 Splitting Cut-Point Transitions

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

12.4.3.1 Cut-Point Subproblem 1/1

Here we consider cut-point transition 16.

12.4.3.1.1 Splitting Cut-Point Transitions

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

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