LTS Termination Proof

by T2Cert

Input

Integer Transition System

Proof

1 Invariant Updates

The following invariants are asserted.

0: arg1P ≤ 08 − arg2P ≤ 0−1 + arg5P ≤ 01 − arg5P ≤ 0arg6P ≤ 0arg6P ≤ 0arg7P ≤ 0arg7P ≤ 0arg1 ≤ 08 − arg2 ≤ 0−1 + arg5 ≤ 01 − arg5 ≤ 0arg6 ≤ 0arg6 ≤ 0arg7 ≤ 0arg7 ≤ 0
1: arg1P ≤ 010 − arg2P ≤ 0arg3P ≤ 0arg3P ≤ 0arg4P ≤ 0arg4P ≤ 0arg8P ≤ 0arg8P ≤ 0arg9P ≤ 0arg9P ≤ 0arg10P ≤ 0arg10P ≤ 0−1 + arg19P ≤ 0arg22P ≤ 0arg23P ≤ 0arg1 ≤ 010 − arg2 ≤ 0arg3 ≤ 0arg3 ≤ 0arg4 ≤ 0arg4 ≤ 0arg8 ≤ 0arg8 ≤ 0arg9 ≤ 0arg9 ≤ 0arg10 ≤ 0arg10 ≤ 0−1 + arg19 ≤ 0arg22 ≤ 0arg23 ≤ 0
2: TRUE
3: 8 − arg1P ≤ 03 − arg2P ≤ 08 − arg1 ≤ 03 − arg2 ≤ 0x25 ≤ 0x26 ≤ 0x27 ≤ 0
5: 10 − arg1P ≤ 03 − arg2P ≤ 01 − arg3P ≤ 010 − arg1 ≤ 03 − arg2 ≤ 01 − arg3 ≤ 0x25 ≤ 0x26 ≤ 0x27 ≤ 0
6: 10 − arg1P ≤ 03 − arg2P ≤ 01 − arg3P ≤ 010 − arg1 ≤ 03 − arg2 ≤ 01 − arg3 ≤ 0x25 ≤ 0x26 ≤ 0x27 ≤ 0
7: 1 − arg1P ≤ 06 − arg2P ≤ 01 − arg3P ≤ 03 − arg4P ≤ 01 − arg1 ≤ 06 − arg2 ≤ 01 − arg3 ≤ 03 − arg4 ≤ 0x25 ≤ 0x26 ≤ 0x27 ≤ 0
8: 1 − arg1P ≤ 06 − arg2P ≤ 05 − arg3P ≤ 01 − arg4P ≤ 03 − arg5P ≤ 01 − arg1 ≤ 06 − arg2 ≤ 05 − arg3 ≤ 01 − arg4 ≤ 03 − arg5 ≤ 0x25 ≤ 0x26 ≤ 0x27 ≤ 0
9: 1 − arg1P ≤ 0arg12P ≤ 0arg12P ≤ 01 − arg1 ≤ 0arg12 ≤ 0arg12 ≤ 0
10: TRUE

The invariants are proved as follows.

IMPACT Invariant Proof

2 Switch to Cooperation Termination Proof

We consider the following cutpoint-transitions:
9 14 9: x67 + x67 ≤ 0x67x67 ≤ 0x53 + x53 ≤ 0x53x53 ≤ 0x27 + x27 ≤ 0x27x27 ≤ 0x26 + x26 ≤ 0x26x26 ≤ 0x25 + x25 ≤ 0x25x25 ≤ 0x247 + x247 ≤ 0x247x247 ≤ 0x212 + x212 ≤ 0x212x212 ≤ 0x211 + x211 ≤ 0x211x211 ≤ 0x170 + x170 ≤ 0x170x170 ≤ 0arg9P + 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 ≤ 0arg25P + arg25P ≤ 0arg25Parg25P ≤ 0arg25 + arg25 ≤ 0arg25arg25 ≤ 0arg24P + arg24P ≤ 0arg24Parg24P ≤ 0arg24 + arg24 ≤ 0arg24arg24 ≤ 0arg23P + arg23P ≤ 0arg23Parg23P ≤ 0arg23 + arg23 ≤ 0arg23arg23 ≤ 0arg22P + arg22P ≤ 0arg22Parg22P ≤ 0arg22 + arg22 ≤ 0arg22arg22 ≤ 0arg21P + arg21P ≤ 0arg21Parg21P ≤ 0arg21 + arg21 ≤ 0arg21arg21 ≤ 0arg20P + arg20P ≤ 0arg20Parg20P ≤ 0arg20 + arg20 ≤ 0arg20arg20 ≤ 0arg2 + arg2 ≤ 0arg2arg2 ≤ 0arg1P + arg1P ≤ 0arg1Parg1P ≤ 0arg19P + arg19P ≤ 0arg19Parg19P ≤ 0arg19 + arg19 ≤ 0arg19arg19 ≤ 0arg18P + arg18P ≤ 0arg18Parg18P ≤ 0arg18 + arg18 ≤ 0arg18arg18 ≤ 0arg17P + arg17P ≤ 0arg17Parg17P ≤ 0arg17 + arg17 ≤ 0arg17arg17 ≤ 0arg16P + arg16P ≤ 0arg16Parg16P ≤ 0arg16 + arg16 ≤ 0arg16arg16 ≤ 0arg15P + arg15P ≤ 0arg15Parg15P ≤ 0arg15 + arg15 ≤ 0arg15arg15 ≤ 0arg14P + arg14P ≤ 0arg14Parg14P ≤ 0arg14 + arg14 ≤ 0arg14arg14 ≤ 0arg13P + arg13P ≤ 0arg13Parg13P ≤ 0arg13 + arg13 ≤ 0arg13arg13 ≤ 0arg12P + arg12P ≤ 0arg12Parg12P ≤ 0arg12 + arg12 ≤ 0arg12arg12 ≤ 0arg11P + arg11P ≤ 0arg11Parg11P ≤ 0arg11 + arg11 ≤ 0arg11arg11 ≤ 0arg10P + arg10P ≤ 0arg10Parg10P ≤ 0arg10 + arg10 ≤ 0arg10arg10 ≤ 0arg1 + arg1 ≤ 0arg1arg1 ≤ 0
and for every transition t, a duplicate t is considered.

3 Transition Removal

We remove transitions 0, 1, 3, 4, 5, 6, 7, 8, 9, 10, 13 using the following ranking functions, which are bounded by −25.

10: 0
2: 0
3: 0
5: 0
6: 0
7: 0
8: 0
0: 0
1: 0
9: 0
10: −11
2: −12
3: −13
5: −14
6: −15
7: −16
8: −17
0: −18
1: −19
9: −20
9_var_snapshot: −20
9*: −20

4 Location Addition

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

9* 17 9: x67 + x67 ≤ 0x67x67 ≤ 0x53 + x53 ≤ 0x53x53 ≤ 0x27 + x27 ≤ 0x27x27 ≤ 0x26 + x26 ≤ 0x26x26 ≤ 0x25 + x25 ≤ 0x25x25 ≤ 0x247 + x247 ≤ 0x247x247 ≤ 0x212 + x212 ≤ 0x212x212 ≤ 0x211 + x211 ≤ 0x211x211 ≤ 0x170 + x170 ≤ 0x170x170 ≤ 0arg9P + 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 ≤ 0arg25P + arg25P ≤ 0arg25Parg25P ≤ 0arg25 + arg25 ≤ 0arg25arg25 ≤ 0arg24P + arg24P ≤ 0arg24Parg24P ≤ 0arg24 + arg24 ≤ 0arg24arg24 ≤ 0arg23P + arg23P ≤ 0arg23Parg23P ≤ 0arg23 + arg23 ≤ 0arg23arg23 ≤ 0arg22P + arg22P ≤ 0arg22Parg22P ≤ 0arg22 + arg22 ≤ 0arg22arg22 ≤ 0arg21P + arg21P ≤ 0arg21Parg21P ≤ 0arg21 + arg21 ≤ 0arg21arg21 ≤ 0arg20P + arg20P ≤ 0arg20Parg20P ≤ 0arg20 + arg20 ≤ 0arg20arg20 ≤ 0arg2 + arg2 ≤ 0arg2arg2 ≤ 0arg1P + arg1P ≤ 0arg1Parg1P ≤ 0arg19P + arg19P ≤ 0arg19Parg19P ≤ 0arg19 + arg19 ≤ 0arg19arg19 ≤ 0arg18P + arg18P ≤ 0arg18Parg18P ≤ 0arg18 + arg18 ≤ 0arg18arg18 ≤ 0arg17P + arg17P ≤ 0arg17Parg17P ≤ 0arg17 + arg17 ≤ 0arg17arg17 ≤ 0arg16P + arg16P ≤ 0arg16Parg16P ≤ 0arg16 + arg16 ≤ 0arg16arg16 ≤ 0arg15P + arg15P ≤ 0arg15Parg15P ≤ 0arg15 + arg15 ≤ 0arg15arg15 ≤ 0arg14P + arg14P ≤ 0arg14Parg14P ≤ 0arg14 + arg14 ≤ 0arg14arg14 ≤ 0arg13P + arg13P ≤ 0arg13Parg13P ≤ 0arg13 + arg13 ≤ 0arg13arg13 ≤ 0arg12P + arg12P ≤ 0arg12Parg12P ≤ 0arg12 + arg12 ≤ 0arg12arg12 ≤ 0arg11P + arg11P ≤ 0arg11Parg11P ≤ 0arg11 + arg11 ≤ 0arg11arg11 ≤ 0arg10P + arg10P ≤ 0arg10Parg10P ≤ 0arg10 + arg10 ≤ 0arg10arg10 ≤ 0arg1 + arg1 ≤ 0arg1arg1 ≤ 0

5 Location Addition

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

9 15 9_var_snapshot: x67 + x67 ≤ 0x67x67 ≤ 0x53 + x53 ≤ 0x53x53 ≤ 0x27 + x27 ≤ 0x27x27 ≤ 0x26 + x26 ≤ 0x26x26 ≤ 0x25 + x25 ≤ 0x25x25 ≤ 0x247 + x247 ≤ 0x247x247 ≤ 0x212 + x212 ≤ 0x212x212 ≤ 0x211 + x211 ≤ 0x211x211 ≤ 0x170 + x170 ≤ 0x170x170 ≤ 0arg9P + 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 ≤ 0arg25P + arg25P ≤ 0arg25Parg25P ≤ 0arg25 + arg25 ≤ 0arg25arg25 ≤ 0arg24P + arg24P ≤ 0arg24Parg24P ≤ 0arg24 + arg24 ≤ 0arg24arg24 ≤ 0arg23P + arg23P ≤ 0arg23Parg23P ≤ 0arg23 + arg23 ≤ 0arg23arg23 ≤ 0arg22P + arg22P ≤ 0arg22Parg22P ≤ 0arg22 + arg22 ≤ 0arg22arg22 ≤ 0arg21P + arg21P ≤ 0arg21Parg21P ≤ 0arg21 + arg21 ≤ 0arg21arg21 ≤ 0arg20P + arg20P ≤ 0arg20Parg20P ≤ 0arg20 + arg20 ≤ 0arg20arg20 ≤ 0arg2 + arg2 ≤ 0arg2arg2 ≤ 0arg1P + arg1P ≤ 0arg1Parg1P ≤ 0arg19P + arg19P ≤ 0arg19Parg19P ≤ 0arg19 + arg19 ≤ 0arg19arg19 ≤ 0arg18P + arg18P ≤ 0arg18Parg18P ≤ 0arg18 + arg18 ≤ 0arg18arg18 ≤ 0arg17P + arg17P ≤ 0arg17Parg17P ≤ 0arg17 + arg17 ≤ 0arg17arg17 ≤ 0arg16P + arg16P ≤ 0arg16Parg16P ≤ 0arg16 + arg16 ≤ 0arg16arg16 ≤ 0arg15P + arg15P ≤ 0arg15Parg15P ≤ 0arg15 + arg15 ≤ 0arg15arg15 ≤ 0arg14P + arg14P ≤ 0arg14Parg14P ≤ 0arg14 + arg14 ≤ 0arg14arg14 ≤ 0arg13P + arg13P ≤ 0arg13Parg13P ≤ 0arg13 + arg13 ≤ 0arg13arg13 ≤ 0arg12P + arg12P ≤ 0arg12Parg12P ≤ 0arg12 + arg12 ≤ 0arg12arg12 ≤ 0arg11P + arg11P ≤ 0arg11Parg11P ≤ 0arg11 + arg11 ≤ 0arg11arg11 ≤ 0arg10P + arg10P ≤ 0arg10Parg10P ≤ 0arg10 + arg10 ≤ 0arg10arg10 ≤ 0arg1 + arg1 ≤ 0arg1arg1 ≤ 0

6 SCC Decomposition

We consider subproblems for each of the 1 SCC(s) of the program graph.

6.1 SCC Subproblem 1/1

Here we consider the SCC { 9, 9_var_snapshot, 9* }.

6.1.1 Transition Removal

We remove transitions 15, 17, 11, 12 using the following ranking functions, which are bounded by −3.

9: −1
9_var_snapshot: −2
9*: 0

6.1.2 Splitting Cut-Point Transitions

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

6.1.2.1 Cut-Point Subproblem 1/1

Here we consider cut-point transition 14.

6.1.2.1.1 Splitting Cut-Point Transitions

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

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