LTS Termination Proof

by T2Cert

Input

Integer Transition System

Proof

1 Invariant Updates

The following invariants are asserted.

0: TRUE
1: TRUE
2: 1 − arg1P ≤ 0arg2P ≤ 01 − arg1 ≤ 0arg2 ≤ 0x8 ≤ 0
4: arg1P ≤ 0arg1 ≤ 0x8 ≤ 01 − x14 ≤ 0
5: arg4P ≤ 01 − arg1 ≤ 0arg4 ≤ 0
6: 2 − arg1P ≤ 03 − arg2P ≤ 0arg3P ≤ 02 − arg1 ≤ 03 − arg2 ≤ 0arg3 ≤ 0x8 ≤ 01 − x14 ≤ 0
7: 1 − arg1P ≤ 04 − arg2P ≤ 01 − arg1 ≤ 04 − arg2 ≤ 0x8 ≤ 01 − x14 ≤ 0
8: TRUE

The invariants are proved as follows.

IMPACT Invariant Proof

2 Switch to Cooperation Termination Proof

We consider the following cutpoint-transitions:
1 17 1: x8 + x8 ≤ 0x8x8 ≤ 0x14 + x14 ≤ 0x14x14 ≤ 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
4 24 4: x8 + x8 ≤ 0x8x8 ≤ 0x14 + x14 ≤ 0x14x14 ≤ 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 31 5: x8 + x8 ≤ 0x8x8 ≤ 0x14 + x14 ≤ 0x14x14 ≤ 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, 3, 13, 14, 15, 16 using the following ranking functions, which are bounded by −21.

8: 0
0: 0
1: 0
5: 0
2: 0
4: 0
6: 0
7: 0
8: −7
0: −8
1: −9
5: −9
1_var_snapshot: −9
1*: −9
5_var_snapshot: −9
5*: −9
2: −14
4: −15
6: −15
4_var_snapshot: −15
4*: −15
7: −16

4 Location Addition

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

1* 20 1: x8 + x8 ≤ 0x8x8 ≤ 0x14 + x14 ≤ 0x14x14 ≤ 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.

1 18 1_var_snapshot: x8 + x8 ≤ 0x8x8 ≤ 0x14 + x14 ≤ 0x14x14 ≤ 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.

4* 27 4: x8 + x8 ≤ 0x8x8 ≤ 0x14 + x14 ≤ 0x14x14 ≤ 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.

4 25 4_var_snapshot: x8 + x8 ≤ 0x8x8 ≤ 0x14 + x14 ≤ 0x14x14 ≤ 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 Location Addition

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

5* 34 5: x8 + x8 ≤ 0x8x8 ≤ 0x14 + x14 ≤ 0x14x14 ≤ 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

9 Location Addition

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

5 32 5_var_snapshot: x8 + x8 ≤ 0x8x8 ≤ 0x14 + x14 ≤ 0x14x14 ≤ 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

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 { 1, 5, 1_var_snapshot, 1*, 5_var_snapshot, 5* }.

10.1.1 Transition Removal

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

1: 4 + 4⋅arg1
5: 4⋅arg1
1_var_snapshot: 2 + 4⋅arg1
1*: 6 + 4⋅arg1
5_var_snapshot: 4⋅arg1
5*: 4⋅arg1

10.1.2 Transition Removal

We remove transitions 18, 20, 6, 7 using the following ranking functions, which are bounded by −1.

1: 0
5: 1 − 3⋅arg3 + 3⋅arg4
1_var_snapshot: −1
1*: 1
5_var_snapshot: −3⋅arg3 + 3⋅arg4
5*: 2 − 3⋅arg3 + 3⋅arg4

10.1.3 Transition Removal

We remove transitions 32, 34 using the following ranking functions, which are bounded by −2.

1: 0
5: −1
1_var_snapshot: 0
1*: 0
5_var_snapshot: −2
5*: 0

10.1.4 Splitting Cut-Point Transitions

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

10.1.4.1 Cut-Point Subproblem 1/2

Here we consider cut-point transition 17.

10.1.4.1.1 Splitting Cut-Point Transitions

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

10.1.4.2 Cut-Point Subproblem 2/2

Here we consider cut-point transition 31.

10.1.4.2.1 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 { 4, 6, 4_var_snapshot, 4* }.

10.2.1 Transition Removal

We remove transitions 25, 27, 8, 9, 10, 11, 12 using the following ranking functions, which are bounded by −5.

4: −2 + 3⋅arg1
6: −5 + 3⋅arg1
4_var_snapshot: −4 + 3⋅arg1
4*: 3⋅arg1

10.2.2 Splitting Cut-Point Transitions

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

10.2.2.1 Cut-Point Subproblem 1/1

Here we consider cut-point transition 24.

10.2.2.1.1 Splitting Cut-Point Transitions

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

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