LTS Termination Proof

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Input

Integer Transition System

Proof

1 Invariant Updates

The following invariants are asserted.

0: 1 − arg1P ≤ 06 − arg2P ≤ 05 − arg3P ≤ 01 − arg4P ≤ 03 − arg5P ≤ 01 − arg1 ≤ 06 − arg2 ≤ 05 − arg3 ≤ 01 − arg4 ≤ 03 − arg5 ≤ 0x137 ≤ 0x138 ≤ 0x139 ≤ 01 − x217 ≤ 01 − x218 ≤ 0
1: 1 − arg1P ≤ 06 − arg2P ≤ 05 − arg3P ≤ 01 − arg4P ≤ 03 − arg5P ≤ 01 − arg1 ≤ 06 − arg2 ≤ 05 − arg3 ≤ 01 − arg4 ≤ 03 − arg5 ≤ 0x137 ≤ 0x138 ≤ 0x139 ≤ 01 − x217 ≤ 01 − x218 ≤ 0
2: 1 − arg1P ≤ 08 − arg2P ≤ 05 − arg3P ≤ 01 − arg4P ≤ 03 − arg5P ≤ 0arg7P ≤ 01 − arg1 ≤ 08 − arg2 ≤ 05 − arg3 ≤ 01 − arg4 ≤ 03 − arg5 ≤ 0arg7 ≤ 0x137 ≤ 0x138 ≤ 0x139 ≤ 0
3: 1 − arg1P ≤ 08 − arg2P ≤ 05 − arg3P ≤ 01 − arg4P ≤ 03 − arg5P ≤ 0arg7P ≤ 01 − arg1 ≤ 08 − arg2 ≤ 05 − arg3 ≤ 01 − arg4 ≤ 03 − arg5 ≤ 0arg7 ≤ 0x137 ≤ 0x138 ≤ 0x139 ≤ 0
4: 1 − arg1P ≤ 010 − arg2P ≤ 05 − arg3P ≤ 01 − arg4P ≤ 05 − arg5P ≤ 01 − arg1 ≤ 010 − arg2 ≤ 05 − arg3 ≤ 01 − arg4 ≤ 05 − arg5 ≤ 0x137 ≤ 0x138 ≤ 0x139 ≤ 0
5: 1 − arg1P ≤ 010 − arg2P ≤ 05 − arg3P ≤ 01 − arg4P ≤ 05 − arg5P ≤ 01 − arg1 ≤ 010 − arg2 ≤ 05 − arg3 ≤ 01 − arg4 ≤ 05 − arg5 ≤ 0x137 ≤ 0x138 ≤ 0x139 ≤ 0
6: 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
7: 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
8: TRUE
9: 8 − arg1P ≤ 03 − arg2P ≤ 08 − arg1 ≤ 03 − arg2 ≤ 0x137 ≤ 0x138 ≤ 0x139 ≤ 0
11: 8 − arg1P ≤ 05 − arg2P ≤ 0arg3P ≤ 08 − arg1 ≤ 05 − arg2 ≤ 0arg3 ≤ 0x137 ≤ 0x138 ≤ 0x139 ≤ 0
12: 5 − arg1P ≤ 0arg2P ≤ 03 − arg3P ≤ 05 − arg1 ≤ 0arg2 ≤ 03 − arg3 ≤ 0x137 ≤ 0x138 ≤ 0x139 ≤ 01 − x217 ≤ 01 − x218 ≤ 0
13: 1 − arg1P ≤ 0arg12P ≤ 0arg12P ≤ 01 − arg1 ≤ 0arg12 ≤ 0arg12 ≤ 0
14: TRUE

The invariants are proved as follows.

IMPACT Invariant Proof

2 Switch to Cooperation Termination Proof

We consider the following cutpoint-transitions:
13 18 13: x436 + x436 ≤ 0x436x436 ≤ 0x401 + x401 ≤ 0x401x401 ≤ 0x400 + x400 ≤ 0x400x400 ≤ 0x359 + x359 ≤ 0x359x359 ≤ 0x219 + x219 ≤ 0x219x219 ≤ 0x218 + x218 ≤ 0x218x218 ≤ 0x217 + x217 ≤ 0x217x217 ≤ 0x201 + x201 ≤ 0x201x201 ≤ 0x200 + x200 ≤ 0x200x200 ≤ 0x199 + x199 ≤ 0x199x199 ≤ 0x185 + x185 ≤ 0x185x185 ≤ 0x184 + x184 ≤ 0x184x184 ≤ 0x183 + x183 ≤ 0x183x183 ≤ 0x182 + x182 ≤ 0x182x182 ≤ 0x168 + x168 ≤ 0x168x168 ≤ 0x167 + x167 ≤ 0x167x167 ≤ 0x166 + x166 ≤ 0x166x166 ≤ 0x165 + x165 ≤ 0x165x165 ≤ 0x139 + x139 ≤ 0x139x139 ≤ 0x138 + x138 ≤ 0x138x138 ≤ 0x137 + x137 ≤ 0x137x137 ≤ 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 ≤ 0arg32P + arg32P ≤ 0arg32Parg32P ≤ 0arg32 + arg32 ≤ 0arg32arg32 ≤ 0arg31P + arg31P ≤ 0arg31Parg31P ≤ 0arg31 + arg31 ≤ 0arg31arg31 ≤ 0arg30P + arg30P ≤ 0arg30Parg30P ≤ 0arg30 + arg30 ≤ 0arg30arg30 ≤ 0arg3 + arg3 ≤ 0arg3arg3 ≤ 0arg2P + arg2P ≤ 0arg2Parg2P ≤ 0arg29P + arg29P ≤ 0arg29Parg29P ≤ 0arg29 + arg29 ≤ 0arg29arg29 ≤ 0arg28P + arg28P ≤ 0arg28Parg28P ≤ 0arg28 + arg28 ≤ 0arg28arg28 ≤ 0arg27P + arg27P ≤ 0arg27Parg27P ≤ 0arg27 + arg27 ≤ 0arg27arg27 ≤ 0arg26P + arg26P ≤ 0arg26Parg26P ≤ 0arg26 + arg26 ≤ 0arg26arg26 ≤ 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, 2, 3, 4, 6, 7, 8, 9, 10, 11, 12, 13, 14, 17 using the following ranking functions, which are bounded by −33.

14: 0
8: 0
9: 0
11: 0
2: 0
3: 0
4: 0
5: 0
12: 0
0: 0
1: 0
6: 0
7: 0
13: 0
14: −15
8: −16
9: −17
11: −18
2: −19
3: −20
4: −21
5: −22
12: −23
0: −24
1: −25
6: −26
7: −27
13: −28
13_var_snapshot: −28
13*: −28

4 Location Addition

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

13* 21 13: x436 + x436 ≤ 0x436x436 ≤ 0x401 + x401 ≤ 0x401x401 ≤ 0x400 + x400 ≤ 0x400x400 ≤ 0x359 + x359 ≤ 0x359x359 ≤ 0x219 + x219 ≤ 0x219x219 ≤ 0x218 + x218 ≤ 0x218x218 ≤ 0x217 + x217 ≤ 0x217x217 ≤ 0x201 + x201 ≤ 0x201x201 ≤ 0x200 + x200 ≤ 0x200x200 ≤ 0x199 + x199 ≤ 0x199x199 ≤ 0x185 + x185 ≤ 0x185x185 ≤ 0x184 + x184 ≤ 0x184x184 ≤ 0x183 + x183 ≤ 0x183x183 ≤ 0x182 + x182 ≤ 0x182x182 ≤ 0x168 + x168 ≤ 0x168x168 ≤ 0x167 + x167 ≤ 0x167x167 ≤ 0x166 + x166 ≤ 0x166x166 ≤ 0x165 + x165 ≤ 0x165x165 ≤ 0x139 + x139 ≤ 0x139x139 ≤ 0x138 + x138 ≤ 0x138x138 ≤ 0x137 + x137 ≤ 0x137x137 ≤ 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 ≤ 0arg32P + arg32P ≤ 0arg32Parg32P ≤ 0arg32 + arg32 ≤ 0arg32arg32 ≤ 0arg31P + arg31P ≤ 0arg31Parg31P ≤ 0arg31 + arg31 ≤ 0arg31arg31 ≤ 0arg30P + arg30P ≤ 0arg30Parg30P ≤ 0arg30 + arg30 ≤ 0arg30arg30 ≤ 0arg3 + arg3 ≤ 0arg3arg3 ≤ 0arg2P + arg2P ≤ 0arg2Parg2P ≤ 0arg29P + arg29P ≤ 0arg29Parg29P ≤ 0arg29 + arg29 ≤ 0arg29arg29 ≤ 0arg28P + arg28P ≤ 0arg28Parg28P ≤ 0arg28 + arg28 ≤ 0arg28arg28 ≤ 0arg27P + arg27P ≤ 0arg27Parg27P ≤ 0arg27 + arg27 ≤ 0arg27arg27 ≤ 0arg26P + arg26P ≤ 0arg26Parg26P ≤ 0arg26 + arg26 ≤ 0arg26arg26 ≤ 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.

13 19 13_var_snapshot: x436 + x436 ≤ 0x436x436 ≤ 0x401 + x401 ≤ 0x401x401 ≤ 0x400 + x400 ≤ 0x400x400 ≤ 0x359 + x359 ≤ 0x359x359 ≤ 0x219 + x219 ≤ 0x219x219 ≤ 0x218 + x218 ≤ 0x218x218 ≤ 0x217 + x217 ≤ 0x217x217 ≤ 0x201 + x201 ≤ 0x201x201 ≤ 0x200 + x200 ≤ 0x200x200 ≤ 0x199 + x199 ≤ 0x199x199 ≤ 0x185 + x185 ≤ 0x185x185 ≤ 0x184 + x184 ≤ 0x184x184 ≤ 0x183 + x183 ≤ 0x183x183 ≤ 0x182 + x182 ≤ 0x182x182 ≤ 0x168 + x168 ≤ 0x168x168 ≤ 0x167 + x167 ≤ 0x167x167 ≤ 0x166 + x166 ≤ 0x166x166 ≤ 0x165 + x165 ≤ 0x165x165 ≤ 0x139 + x139 ≤ 0x139x139 ≤ 0x138 + x138 ≤ 0x138x138 ≤ 0x137 + x137 ≤ 0x137x137 ≤ 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 ≤ 0arg32P + arg32P ≤ 0arg32Parg32P ≤ 0arg32 + arg32 ≤ 0arg32arg32 ≤ 0arg31P + arg31P ≤ 0arg31Parg31P ≤ 0arg31 + arg31 ≤ 0arg31arg31 ≤ 0arg30P + arg30P ≤ 0arg30Parg30P ≤ 0arg30 + arg30 ≤ 0arg30arg30 ≤ 0arg3 + arg3 ≤ 0arg3arg3 ≤ 0arg2P + arg2P ≤ 0arg2Parg2P ≤ 0arg29P + arg29P ≤ 0arg29Parg29P ≤ 0arg29 + arg29 ≤ 0arg29arg29 ≤ 0arg28P + arg28P ≤ 0arg28Parg28P ≤ 0arg28 + arg28 ≤ 0arg28arg28 ≤ 0arg27P + arg27P ≤ 0arg27Parg27P ≤ 0arg27 + arg27 ≤ 0arg27arg27 ≤ 0arg26P + arg26P ≤ 0arg26Parg26P ≤ 0arg26 + arg26 ≤ 0arg26arg26 ≤ 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 { 13, 13_var_snapshot, 13* }.

6.1.1 Transition Removal

We remove transitions 19, 21, 15, 16 using the following ranking functions, which are bounded by −1.

13: 1
13_var_snapshot: 0
13*: 2

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 18.

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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