LTS Termination Proof

by T2Cert

Input

Integer Transition System

Proof

1 Invariant Updates

The following invariants are asserted.

0: − arg1P ≤ 0 ∧ 8 − arg2P ≤ 0 ∧ −1 + arg5P ≤ 0 ∧ 1 − arg5P ≤ 0 ∧ arg6P ≤ 0 ∧ − arg6P ≤ 0 ∧ arg7P ≤ 0 ∧ − arg7P ≤ 0 ∧ − arg1 ≤ 0 ∧ 8 − arg2 ≤ 0 ∧ −1 + arg5 ≤ 0 ∧ 1 − arg5 ≤ 0 ∧ arg6 ≤ 0 ∧ − arg6 ≤ 0 ∧ arg7 ≤ 0 ∧ − arg7 ≤ 0
1: − arg1P ≤ 0 ∧ 10 − arg2P ≤ 0 ∧ arg3P ≤ 0 ∧ − arg3P ≤ 0 ∧ arg4P ≤ 0 ∧ − arg4P ≤ 0 ∧ arg8P ≤ 0 ∧ − arg8P ≤ 0 ∧ arg9P ≤ 0 ∧ − arg9P ≤ 0 ∧ arg10P ≤ 0 ∧ − arg10P ≤ 0 ∧ −1 + arg19P ≤ 0 ∧ arg20P ≤ 0 ∧ arg23P ≤ 0 ∧ − arg1 ≤ 0 ∧ 10 − arg2 ≤ 0 ∧ arg3 ≤ 0 ∧ − arg3 ≤ 0 ∧ arg4 ≤ 0 ∧ − arg4 ≤ 0 ∧ arg8 ≤ 0 ∧ − arg8 ≤ 0 ∧ arg9 ≤ 0 ∧ − arg9 ≤ 0 ∧ arg10 ≤ 0 ∧ − arg10 ≤ 0 ∧ −1 + arg19 ≤ 0 ∧ arg20 ≤ 0 ∧ arg23 ≤ 0
2: TRUE
3: 7 − arg1P ≤ 0 ∧ 7 − arg1 ≤ 0 ∧ − x25 ≤ 0
5: 10 − arg1P ≤ 0 ∧ 3 − arg2P ≤ 0 ∧ 1 − arg3P ≤ 0 ∧ 10 − arg1 ≤ 0 ∧ 3 − arg2 ≤ 0 ∧ 1 − arg3 ≤ 0 ∧ 1 − arg4 ≤ 0 ∧ − x25 ≤ 0 ∧ 1 − x49 ≤ 0 ∧ 1 − x50 ≤ 0
6: 10 − arg1P ≤ 0 ∧ 3 − arg2P ≤ 0 ∧ 1 − arg3P ≤ 0 ∧ 10 − arg1 ≤ 0 ∧ 3 − arg2 ≤ 0 ∧ 1 − arg3 ≤ 0 ∧ 1 − arg4 ≤ 0 ∧ − x25 ≤ 0 ∧ 1 − x65 ≤ 0
7: 1 − arg1P ≤ 0 ∧ 6 − arg2P ≤ 0 ∧ 1 − arg3P ≤ 0 ∧ 3 − arg4P ≤ 0 ∧ 1 − arg1 ≤ 0 ∧ 6 − arg2 ≤ 0 ∧ 1 − arg3 ≤ 0 ∧ 3 − arg4 ≤ 0 ∧ − x25 ≤ 0
8: 1 − arg1P ≤ 0 ∧ 6 − arg2P ≤ 0 ∧ 5 − arg3P ≤ 0 ∧ 1 − arg4P ≤ 0 ∧ 3 − arg5P ≤ 0 ∧ 1 − arg1 ≤ 0 ∧ 6 − arg2 ≤ 0 ∧ 5 − arg3 ≤ 0 ∧ 1 − arg4 ≤ 0 ∧ 3 − arg5 ≤ 0 ∧ − x25 ≤ 0
9: 1 − arg1P ≤ 0 ∧ arg12P ≤ 0 ∧ − arg12P ≤ 0 ∧ 1 − arg1 ≤ 0 ∧ arg12 ≤ 0 ∧ − arg12 ≤ 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 ≤ 0 ∧ x67 − x67 ≤ 0 ∧ − x66 + x66 ≤ 0 ∧ x66 − x66 ≤ 0 ∧ − x65 + x65 ≤ 0 ∧ x65 − x65 ≤ 0 ∧ − x51 + x51 ≤ 0 ∧ x51 − x51 ≤ 0 ∧ − x50 + x50 ≤ 0 ∧ x50 − x50 ≤ 0 ∧ − x49 + x49 ≤ 0 ∧ x49 − x49 ≤ 0 ∧ − x25 + x25 ≤ 0 ∧ x25 − x25 ≤ 0 ∧ − x247 + x247 ≤ 0 ∧ x247 − x247 ≤ 0 ∧ − x212 + x212 ≤ 0 ∧ x212 − x212 ≤ 0 ∧ − x211 + x211 ≤ 0 ∧ x211 − x211 ≤ 0 ∧ − x170 + x170 ≤ 0 ∧ x170 − x170 ≤ 0 ∧ − arg9P + arg9P ≤ 0 ∧ arg9P − arg9P ≤ 0 ∧ − arg9 + arg9 ≤ 0 ∧ arg9 − arg9 ≤ 0 ∧ − arg8P + arg8P ≤ 0 ∧ arg8P − arg8P ≤ 0 ∧ − arg8 + arg8 ≤ 0 ∧ arg8 − arg8 ≤ 0 ∧ − arg7P + arg7P ≤ 0 ∧ arg7P − arg7P ≤ 0 ∧ − arg7 + arg7 ≤ 0 ∧ arg7 − arg7 ≤ 0 ∧ − arg6P + arg6P ≤ 0 ∧ arg6P − arg6P ≤ 0 ∧ − arg6 + arg6 ≤ 0 ∧ arg6 − arg6 ≤ 0 ∧ − arg5P + arg5P ≤ 0 ∧ arg5P − arg5P ≤ 0 ∧ − arg5 + arg5 ≤ 0 ∧ arg5 − arg5 ≤ 0 ∧ − arg4P + arg4P ≤ 0 ∧ arg4P − arg4P ≤ 0 ∧ − arg4 + arg4 ≤ 0 ∧ arg4 − arg4 ≤ 0 ∧ − arg3P + arg3P ≤ 0 ∧ arg3P − arg3P ≤ 0 ∧ − arg3 + arg3 ≤ 0 ∧ arg3 − arg3 ≤ 0 ∧ − arg2P + arg2P ≤ 0 ∧ arg2P − arg2P ≤ 0 ∧ − arg25P + arg25P ≤ 0 ∧ arg25P − arg25P ≤ 0 ∧ − arg25 + arg25 ≤ 0 ∧ arg25 − arg25 ≤ 0 ∧ − arg24P + arg24P ≤ 0 ∧ arg24P − arg24P ≤ 0 ∧ − arg24 + arg24 ≤ 0 ∧ arg24 − arg24 ≤ 0 ∧ − arg23P + arg23P ≤ 0 ∧ arg23P − arg23P ≤ 0 ∧ − arg23 + arg23 ≤ 0 ∧ arg23 − arg23 ≤ 0 ∧ − arg22P + arg22P ≤ 0 ∧ arg22P − arg22P ≤ 0 ∧ − arg22 + arg22 ≤ 0 ∧ arg22 − arg22 ≤ 0 ∧ − arg21P + arg21P ≤ 0 ∧ arg21P − arg21P ≤ 0 ∧ − arg21 + arg21 ≤ 0 ∧ arg21 − arg21 ≤ 0 ∧ − arg20P + arg20P ≤ 0 ∧ arg20P − arg20P ≤ 0 ∧ − arg20 + arg20 ≤ 0 ∧ arg20 − arg20 ≤ 0 ∧ − arg2 + arg2 ≤ 0 ∧ arg2 − arg2 ≤ 0 ∧ − arg1P + arg1P ≤ 0 ∧ arg1P − arg1P ≤ 0 ∧ − arg19P + arg19P ≤ 0 ∧ arg19P − arg19P ≤ 0 ∧ − arg19 + arg19 ≤ 0 ∧ arg19 − arg19 ≤ 0 ∧ − arg18P + arg18P ≤ 0 ∧ arg18P − arg18P ≤ 0 ∧ − arg18 + arg18 ≤ 0 ∧ arg18 − arg18 ≤ 0 ∧ − arg17P + arg17P ≤ 0 ∧ arg17P − arg17P ≤ 0 ∧ − arg17 + arg17 ≤ 0 ∧ arg17 − arg17 ≤ 0 ∧ − arg16P + arg16P ≤ 0 ∧ arg16P − arg16P ≤ 0 ∧ − arg16 + arg16 ≤ 0 ∧ arg16 − arg16 ≤ 0 ∧ − arg15P + arg15P ≤ 0 ∧ arg15P − arg15P ≤ 0 ∧ − arg15 + arg15 ≤ 0 ∧ arg15 − arg15 ≤ 0 ∧ − arg14P + arg14P ≤ 0 ∧ arg14P − arg14P ≤ 0 ∧ − arg14 + arg14 ≤ 0 ∧ arg14 − arg14 ≤ 0 ∧ − arg13P + arg13P ≤ 0 ∧ arg13P − arg13P ≤ 0 ∧ − arg13 + arg13 ≤ 0 ∧ arg13 − arg13 ≤ 0 ∧ − arg12P + arg12P ≤ 0 ∧ arg12P − arg12P ≤ 0 ∧ − arg12 + arg12 ≤ 0 ∧ arg12 − arg12 ≤ 0 ∧ − arg11P + arg11P ≤ 0 ∧ arg11P − arg11P ≤ 0 ∧ − arg11 + arg11 ≤ 0 ∧ arg11 − arg11 ≤ 0 ∧ − arg10P + arg10P ≤ 0 ∧ arg10P − arg10P ≤ 0 ∧ − arg10 + arg10 ≤ 0 ∧ arg10 − arg10 ≤ 0 ∧ − arg1 + arg1 ≤ 0 ∧ arg1 − arg1 ≤ 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 ≤ 0 ∧ x67 − x67 ≤ 0 ∧ − x66 + x66 ≤ 0 ∧ x66 − x66 ≤ 0 ∧ − x65 + x65 ≤ 0 ∧ x65 − x65 ≤ 0 ∧ − x51 + x51 ≤ 0 ∧ x51 − x51 ≤ 0 ∧ − x50 + x50 ≤ 0 ∧ x50 − x50 ≤ 0 ∧ − x49 + x49 ≤ 0 ∧ x49 − x49 ≤ 0 ∧ − x25 + x25 ≤ 0 ∧ x25 − x25 ≤ 0 ∧ − x247 + x247 ≤ 0 ∧ x247 − x247 ≤ 0 ∧ − x212 + x212 ≤ 0 ∧ x212 − x212 ≤ 0 ∧ − x211 + x211 ≤ 0 ∧ x211 − x211 ≤ 0 ∧ − x170 + x170 ≤ 0 ∧ x170 − x170 ≤ 0 ∧ − arg9P + arg9P ≤ 0 ∧ arg9P − arg9P ≤ 0 ∧ − arg9 + arg9 ≤ 0 ∧ arg9 − arg9 ≤ 0 ∧ − arg8P + arg8P ≤ 0 ∧ arg8P − arg8P ≤ 0 ∧ − arg8 + arg8 ≤ 0 ∧ arg8 − arg8 ≤ 0 ∧ − arg7P + arg7P ≤ 0 ∧ arg7P − arg7P ≤ 0 ∧ − arg7 + arg7 ≤ 0 ∧ arg7 − arg7 ≤ 0 ∧ − arg6P + arg6P ≤ 0 ∧ arg6P − arg6P ≤ 0 ∧ − arg6 + arg6 ≤ 0 ∧ arg6 − arg6 ≤ 0 ∧ − arg5P + arg5P ≤ 0 ∧ arg5P − arg5P ≤ 0 ∧ − arg5 + arg5 ≤ 0 ∧ arg5 − arg5 ≤ 0 ∧ − arg4P + arg4P ≤ 0 ∧ arg4P − arg4P ≤ 0 ∧ − arg4 + arg4 ≤ 0 ∧ arg4 − arg4 ≤ 0 ∧ − arg3P + arg3P ≤ 0 ∧ arg3P − arg3P ≤ 0 ∧ − arg3 + arg3 ≤ 0 ∧ arg3 − arg3 ≤ 0 ∧ − arg2P + arg2P ≤ 0 ∧ arg2P − arg2P ≤ 0 ∧ − arg25P + arg25P ≤ 0 ∧ arg25P − arg25P ≤ 0 ∧ − arg25 + arg25 ≤ 0 ∧ arg25 − arg25 ≤ 0 ∧ − arg24P + arg24P ≤ 0 ∧ arg24P − arg24P ≤ 0 ∧ − arg24 + arg24 ≤ 0 ∧ arg24 − arg24 ≤ 0 ∧ − arg23P + arg23P ≤ 0 ∧ arg23P − arg23P ≤ 0 ∧ − arg23 + arg23 ≤ 0 ∧ arg23 − arg23 ≤ 0 ∧ − arg22P + arg22P ≤ 0 ∧ arg22P − arg22P ≤ 0 ∧ − arg22 + arg22 ≤ 0 ∧ arg22 − arg22 ≤ 0 ∧ − arg21P + arg21P ≤ 0 ∧ arg21P − arg21P ≤ 0 ∧ − arg21 + arg21 ≤ 0 ∧ arg21 − arg21 ≤ 0 ∧ − arg20P + arg20P ≤ 0 ∧ arg20P − arg20P ≤ 0 ∧ − arg20 + arg20 ≤ 0 ∧ arg20 − arg20 ≤ 0 ∧ − arg2 + arg2 ≤ 0 ∧ arg2 − arg2 ≤ 0 ∧ − arg1P + arg1P ≤ 0 ∧ arg1P − arg1P ≤ 0 ∧ − arg19P + arg19P ≤ 0 ∧ arg19P − arg19P ≤ 0 ∧ − arg19 + arg19 ≤ 0 ∧ arg19 − arg19 ≤ 0 ∧ − arg18P + arg18P ≤ 0 ∧ arg18P − arg18P ≤ 0 ∧ − arg18 + arg18 ≤ 0 ∧ arg18 − arg18 ≤ 0 ∧ − arg17P + arg17P ≤ 0 ∧ arg17P − arg17P ≤ 0 ∧ − arg17 + arg17 ≤ 0 ∧ arg17 − arg17 ≤ 0 ∧ − arg16P + arg16P ≤ 0 ∧ arg16P − arg16P ≤ 0 ∧ − arg16 + arg16 ≤ 0 ∧ arg16 − arg16 ≤ 0 ∧ − arg15P + arg15P ≤ 0 ∧ arg15P − arg15P ≤ 0 ∧ − arg15 + arg15 ≤ 0 ∧ arg15 − arg15 ≤ 0 ∧ − arg14P + arg14P ≤ 0 ∧ arg14P − arg14P ≤ 0 ∧ − arg14 + arg14 ≤ 0 ∧ arg14 − arg14 ≤ 0 ∧ − arg13P + arg13P ≤ 0 ∧ arg13P − arg13P ≤ 0 ∧ − arg13 + arg13 ≤ 0 ∧ arg13 − arg13 ≤ 0 ∧ − arg12P + arg12P ≤ 0 ∧ arg12P − arg12P ≤ 0 ∧ − arg12 + arg12 ≤ 0 ∧ arg12 − arg12 ≤ 0 ∧ − arg11P + arg11P ≤ 0 ∧ arg11P − arg11P ≤ 0 ∧ − arg11 + arg11 ≤ 0 ∧ arg11 − arg11 ≤ 0 ∧ − arg10P + arg10P ≤ 0 ∧ arg10P − arg10P ≤ 0 ∧ − arg10 + arg10 ≤ 0 ∧ arg10 − arg10 ≤ 0 ∧ − arg1 + arg1 ≤ 0 ∧ arg1 − arg1 ≤ 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 ≤ 0 ∧ x67 − x67 ≤ 0 ∧ − x66 + x66 ≤ 0 ∧ x66 − x66 ≤ 0 ∧ − x65 + x65 ≤ 0 ∧ x65 − x65 ≤ 0 ∧ − x51 + x51 ≤ 0 ∧ x51 − x51 ≤ 0 ∧ − x50 + x50 ≤ 0 ∧ x50 − x50 ≤ 0 ∧ − x49 + x49 ≤ 0 ∧ x49 − x49 ≤ 0 ∧ − x25 + x25 ≤ 0 ∧ x25 − x25 ≤ 0 ∧ − x247 + x247 ≤ 0 ∧ x247 − x247 ≤ 0 ∧ − x212 + x212 ≤ 0 ∧ x212 − x212 ≤ 0 ∧ − x211 + x211 ≤ 0 ∧ x211 − x211 ≤ 0 ∧ − x170 + x170 ≤ 0 ∧ x170 − x170 ≤ 0 ∧ − arg9P + arg9P ≤ 0 ∧ arg9P − arg9P ≤ 0 ∧ − arg9 + arg9 ≤ 0 ∧ arg9 − arg9 ≤ 0 ∧ − arg8P + arg8P ≤ 0 ∧ arg8P − arg8P ≤ 0 ∧ − arg8 + arg8 ≤ 0 ∧ arg8 − arg8 ≤ 0 ∧ − arg7P + arg7P ≤ 0 ∧ arg7P − arg7P ≤ 0 ∧ − arg7 + arg7 ≤ 0 ∧ arg7 − arg7 ≤ 0 ∧ − arg6P + arg6P ≤ 0 ∧ arg6P − arg6P ≤ 0 ∧ − arg6 + arg6 ≤ 0 ∧ arg6 − arg6 ≤ 0 ∧ − arg5P + arg5P ≤ 0 ∧ arg5P − arg5P ≤ 0 ∧ − arg5 + arg5 ≤ 0 ∧ arg5 − arg5 ≤ 0 ∧ − arg4P + arg4P ≤ 0 ∧ arg4P − arg4P ≤ 0 ∧ − arg4 + arg4 ≤ 0 ∧ arg4 − arg4 ≤ 0 ∧ − arg3P + arg3P ≤ 0 ∧ arg3P − arg3P ≤ 0 ∧ − arg3 + arg3 ≤ 0 ∧ arg3 − arg3 ≤ 0 ∧ − arg2P + arg2P ≤ 0 ∧ arg2P − arg2P ≤ 0 ∧ − arg25P + arg25P ≤ 0 ∧ arg25P − arg25P ≤ 0 ∧ − arg25 + arg25 ≤ 0 ∧ arg25 − arg25 ≤ 0 ∧ − arg24P + arg24P ≤ 0 ∧ arg24P − arg24P ≤ 0 ∧ − arg24 + arg24 ≤ 0 ∧ arg24 − arg24 ≤ 0 ∧ − arg23P + arg23P ≤ 0 ∧ arg23P − arg23P ≤ 0 ∧ − arg23 + arg23 ≤ 0 ∧ arg23 − arg23 ≤ 0 ∧ − arg22P + arg22P ≤ 0 ∧ arg22P − arg22P ≤ 0 ∧ − arg22 + arg22 ≤ 0 ∧ arg22 − arg22 ≤ 0 ∧ − arg21P + arg21P ≤ 0 ∧ arg21P − arg21P ≤ 0 ∧ − arg21 + arg21 ≤ 0 ∧ arg21 − arg21 ≤ 0 ∧ − arg20P + arg20P ≤ 0 ∧ arg20P − arg20P ≤ 0 ∧ − arg20 + arg20 ≤ 0 ∧ arg20 − arg20 ≤ 0 ∧ − arg2 + arg2 ≤ 0 ∧ arg2 − arg2 ≤ 0 ∧ − arg1P + arg1P ≤ 0 ∧ arg1P − arg1P ≤ 0 ∧ − arg19P + arg19P ≤ 0 ∧ arg19P − arg19P ≤ 0 ∧ − arg19 + arg19 ≤ 0 ∧ arg19 − arg19 ≤ 0 ∧ − arg18P + arg18P ≤ 0 ∧ arg18P − arg18P ≤ 0 ∧ − arg18 + arg18 ≤ 0 ∧ arg18 − arg18 ≤ 0 ∧ − arg17P + arg17P ≤ 0 ∧ arg17P − arg17P ≤ 0 ∧ − arg17 + arg17 ≤ 0 ∧ arg17 − arg17 ≤ 0 ∧ − arg16P + arg16P ≤ 0 ∧ arg16P − arg16P ≤ 0 ∧ − arg16 + arg16 ≤ 0 ∧ arg16 − arg16 ≤ 0 ∧ − arg15P + arg15P ≤ 0 ∧ arg15P − arg15P ≤ 0 ∧ − arg15 + arg15 ≤ 0 ∧ arg15 − arg15 ≤ 0 ∧ − arg14P + arg14P ≤ 0 ∧ arg14P − arg14P ≤ 0 ∧ − arg14 + arg14 ≤ 0 ∧ arg14 − arg14 ≤ 0 ∧ − arg13P + arg13P ≤ 0 ∧ arg13P − arg13P ≤ 0 ∧ − arg13 + arg13 ≤ 0 ∧ arg13 − arg13 ≤ 0 ∧ − arg12P + arg12P ≤ 0 ∧ arg12P − arg12P ≤ 0 ∧ − arg12 + arg12 ≤ 0 ∧ arg12 − arg12 ≤ 0 ∧ − arg11P + arg11P ≤ 0 ∧ arg11P − arg11P ≤ 0 ∧ − arg11 + arg11 ≤ 0 ∧ arg11 − arg11 ≤ 0 ∧ − arg10P + arg10P ≤ 0 ∧ arg10P − arg10P ≤ 0 ∧ − arg10 + arg10 ≤ 0 ∧ arg10 − arg10 ≤ 0 ∧ − arg1 + arg1 ≤ 0 ∧ arg1 − arg1 ≤ 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 17, 11, 12 using the following ranking functions, which are bounded by −3.

9: −1
9_var_snapshot: −2
9*: 0

6.1.2 Transition Removal

We remove transition 15 using the following ranking functions, which are bounded by 0.

9: arg1P
9_var_snapshot: 0
9*: 0

6.1.3 Splitting Cut-Point Transitions

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

6.1.3.1 Cut-Point Subproblem 1/1

Here we consider cut-point transition 14.

6.1.3.1.1 Splitting Cut-Point Transitions

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

Tool configuration

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