LTS Termination Proof

by T2Cert

Input

Integer Transition System

Initial Location: 2

Transitions: (pre-variables and post-variables)

0	0	1:	0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ − arg2P ≤ 0 ∧ − arg2 ≤ 0 ∧ − arg1P ≤ 0 ∧ 1 − arg1 ≤ 0 ∧ 1 + arg1P − arg3P ≤ 0 ∧ −1 − arg1P + arg3P ≤ 0 ∧ − arg1P + arg1 ≤ 0 ∧ arg1P − arg1 ≤ 0 ∧ − arg2P + arg2 ≤ 0 ∧ arg2P − arg2 ≤ 0 ∧ − arg3P + arg3 ≤ 0 ∧ arg3P − arg3 ≤ 0
1	1	1:	0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ − arg2 + arg3 ≤ 0 ∧ −1 − arg1 ≤ 0 ∧ 1 − arg1P + arg1 ≤ 0 ∧ −1 + arg1P − arg1 ≤ 0 ∧ 2 + arg1 − arg3P ≤ 0 ∧ −2 − arg1 + arg3P ≤ 0 ∧ − arg1P + arg1 ≤ 0 ∧ arg1P − arg1 ≤ 0 ∧ − arg3P + arg3 ≤ 0 ∧ arg3P − arg3 ≤ 0 ∧ − arg2P + arg2P ≤ 0 ∧ arg2P − arg2P ≤ 0 ∧ − arg2 + arg2 ≤ 0 ∧ arg2 − arg2 ≤ 0
2	2	0:	0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ − arg1P + arg1 ≤ 0 ∧ arg1P − arg1 ≤ 0 ∧ − arg2P + arg2 ≤ 0 ∧ arg2P − arg2 ≤ 0 ∧ − arg3P + arg3 ≤ 0 ∧ arg3P − arg3 ≤ 0

Proof

1 Invariant Updates

The following invariants are asserted.

0:	TRUE
1:	− arg2P ≤ 0 ∧ − arg2 ≤ 0
2:	TRUE

The invariants are proved as follows.

IMPACT Invariant Proof

nodes (location) invariant:

0	(0)	TRUE
1	(1)	− arg2P ≤ 0 ∧ − arg2 ≤ 0
2	(2)	TRUE

initial node: 2
cover edges:
transition edges:

0 0 1

1 1 1

2 2 0

2 Switch to Cooperation Termination Proof

We consider the following cutpoint-transitions:

− arg3P + arg3P ≤ 0 ∧ arg3P − arg3P ≤ 0 ∧ − arg3 + arg3 ≤ 0 ∧ arg3 − arg3 ≤ 0 ∧ − arg2P + arg2P ≤ 0 ∧ arg2P − arg2P ≤ 0 ∧ − arg2 + arg2 ≤ 0 ∧ arg2 − arg2 ≤ 0 ∧ − arg1P + arg1P ≤ 0 ∧ arg1P − arg1P ≤ 0 ∧ − arg1 + arg1 ≤ 0 ∧ arg1 − arg1 ≤ 0

and for every transition t, a duplicate t is considered.

3 Transition Removal

We remove transitions 0, 2 using the following ranking functions, which are bounded by −11.

2:	0
0:	0
1:	0
2:	−4
0:	−5
1:	−6
1_var_snapshot:	−6
1*:	−6

Hints:

4	lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]
1	lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]
0	lexStrict[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] , [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]
2	lexStrict[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] , [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]

4 Location Addition

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

1* 6 1: − arg3P + arg3P ≤ 0 ∧ arg3P − arg3P ≤ 0 ∧ − arg3 + arg3 ≤ 0 ∧ arg3 − arg3 ≤ 0 ∧ − arg2P + arg2P ≤ 0 ∧ arg2P − arg2P ≤ 0 ∧ − arg2 + arg2 ≤ 0 ∧ arg2 − arg2 ≤ 0 ∧ − arg1P + arg1P ≤ 0 ∧ arg1P − arg1P ≤ 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.

1 4 1_var_snapshot: − arg3P + arg3P ≤ 0 ∧ arg3P − arg3P ≤ 0 ∧ − arg3 + arg3 ≤ 0 ∧ arg3 − arg3 ≤ 0 ∧ − arg2P + arg2P ≤ 0 ∧ arg2P − arg2P ≤ 0 ∧ − arg2 + arg2 ≤ 0 ∧ arg2 − arg2 ≤ 0 ∧ − arg1P + arg1P ≤ 0 ∧ arg1P − arg1P ≤ 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 { 1, 1_var_snapshot, 1* }.

6.1.1 Splitting Cut-Point Transitions

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

6.1.1.1 Cut-Point Subproblem 1/1

Here we consider cut-point transition 3.

6.1.1.1.1 Fresh Variable Addition

The new variable __snapshot_1_arg3P is introduced. The transition formulas are extended as follows:

4:	__snapshot_1_arg3P ≤ arg3P ∧ arg3P ≤ __snapshot_1_arg3P
6:	__snapshot_1_arg3P ≤ __snapshot_1_arg3P ∧ __snapshot_1_arg3P ≤ __snapshot_1_arg3P
1:	__snapshot_1_arg3P ≤ __snapshot_1_arg3P ∧ __snapshot_1_arg3P ≤ __snapshot_1_arg3P

6.1.1.1.2 Fresh Variable Addition

The new variable __snapshot_1_arg3 is introduced. The transition formulas are extended as follows:

4:	__snapshot_1_arg3 ≤ arg3 ∧ arg3 ≤ __snapshot_1_arg3
6:	__snapshot_1_arg3 ≤ __snapshot_1_arg3 ∧ __snapshot_1_arg3 ≤ __snapshot_1_arg3
1:	__snapshot_1_arg3 ≤ __snapshot_1_arg3 ∧ __snapshot_1_arg3 ≤ __snapshot_1_arg3

6.1.1.1.3 Fresh Variable Addition

The new variable __snapshot_1_arg2P is introduced. The transition formulas are extended as follows:

4:	__snapshot_1_arg2P ≤ arg2P ∧ arg2P ≤ __snapshot_1_arg2P
6:	__snapshot_1_arg2P ≤ __snapshot_1_arg2P ∧ __snapshot_1_arg2P ≤ __snapshot_1_arg2P
1:	__snapshot_1_arg2P ≤ __snapshot_1_arg2P ∧ __snapshot_1_arg2P ≤ __snapshot_1_arg2P

6.1.1.1.4 Fresh Variable Addition

The new variable __snapshot_1_arg2 is introduced. The transition formulas are extended as follows:

4:	__snapshot_1_arg2 ≤ arg2 ∧ arg2 ≤ __snapshot_1_arg2
6:	__snapshot_1_arg2 ≤ __snapshot_1_arg2 ∧ __snapshot_1_arg2 ≤ __snapshot_1_arg2
1:	__snapshot_1_arg2 ≤ __snapshot_1_arg2 ∧ __snapshot_1_arg2 ≤ __snapshot_1_arg2

6.1.1.1.5 Fresh Variable Addition

The new variable __snapshot_1_arg1P is introduced. The transition formulas are extended as follows:

4:	__snapshot_1_arg1P ≤ arg1P ∧ arg1P ≤ __snapshot_1_arg1P
6:	__snapshot_1_arg1P ≤ __snapshot_1_arg1P ∧ __snapshot_1_arg1P ≤ __snapshot_1_arg1P
1:	__snapshot_1_arg1P ≤ __snapshot_1_arg1P ∧ __snapshot_1_arg1P ≤ __snapshot_1_arg1P

6.1.1.1.6 Fresh Variable Addition

The new variable __snapshot_1_arg1 is introduced. The transition formulas are extended as follows:

4:	__snapshot_1_arg1 ≤ arg1 ∧ arg1 ≤ __snapshot_1_arg1
6:	__snapshot_1_arg1 ≤ __snapshot_1_arg1 ∧ __snapshot_1_arg1 ≤ __snapshot_1_arg1
1:	__snapshot_1_arg1 ≤ __snapshot_1_arg1 ∧ __snapshot_1_arg1 ≤ __snapshot_1_arg1

6.1.1.1.7 Invariant Updates

The following invariants are asserted.

0:	TRUE
1:	−1 − arg1 + arg3 ≤ 0 ∧ − arg2P ≤ 0 ∧ − arg2 ≤ 0
2:	TRUE
1:	−1 − arg1 + arg3 ≤ 0 ∧ − arg2P ≤ 0 ∧ − arg2 ≤ 0 ∨ − __snapshot_1_arg2 + __snapshot_1_arg3 ≤ 0 ∧ −1 − arg1 + arg3 ≤ 0 ∧ 1 − __snapshot_1_arg2 + __snapshot_1_arg3 + arg2 − arg3 ≤ 0 ∧ − arg2P ≤ 0 ∧ − arg2 ≤ 0
1_var_snapshot:	−1 − __snapshot_1_arg2 + __snapshot_1_arg3 − arg1 + arg2 ≤ 0 ∧ − __snapshot_1_arg2 + __snapshot_1_arg3 + arg2 − arg3 ≤ 0 ∧ − arg2P ≤ 0 ∧ − arg2 ≤ 0
1*:	− __snapshot_1_arg2 + __snapshot_1_arg3 ≤ 0 ∧ −1 − arg1 + arg3 ≤ 0 ∧ 1 − __snapshot_1_arg2 + __snapshot_1_arg3 + arg2 − arg3 ≤ 0 ∧ − arg2P ≤ 0 ∧ − arg2 ≤ 0

The invariants are proved as follows.

IMPACT Invariant Proof

nodes (location) invariant:

0	(2)	TRUE
1	(0)	TRUE
2	(1)	−1 − arg1 + arg3 ≤ 0 ∧ − arg2P ≤ 0 ∧ − arg2 ≤ 0
3	(1)	−1 − arg1 + arg3 ≤ 0 ∧ − arg2P ≤ 0 ∧ − arg2 ≤ 0
4	(1)	−1 − arg1 + arg3 ≤ 0 ∧ − arg2P ≤ 0 ∧ − arg2 ≤ 0
5	(1_var_snapshot)	−1 − __snapshot_1_arg2 + __snapshot_1_arg3 − arg1 + arg2 ≤ 0 ∧ − __snapshot_1_arg2 + __snapshot_1_arg3 + arg2 − arg3 ≤ 0 ∧ − arg2P ≤ 0 ∧ − arg2 ≤ 0
12	(1*)	− __snapshot_1_arg2 + __snapshot_1_arg3 ≤ 0 ∧ −1 − arg1 + arg3 ≤ 0 ∧ 1 − __snapshot_1_arg2 + __snapshot_1_arg3 + arg2 − arg3 ≤ 0 ∧ − arg2P ≤ 0 ∧ − arg2 ≤ 0
13	(1)	− __snapshot_1_arg2 + __snapshot_1_arg3 ≤ 0 ∧ −1 − arg1 + arg3 ≤ 0 ∧ 1 − __snapshot_1_arg2 + __snapshot_1_arg3 + arg2 − arg3 ≤ 0 ∧ − arg2P ≤ 0 ∧ − arg2 ≤ 0
14	(1_var_snapshot)	−1 − __snapshot_1_arg2 + __snapshot_1_arg3 − arg1 + arg2 ≤ 0 ∧ − __snapshot_1_arg2 + __snapshot_1_arg3 + arg2 − arg3 ≤ 0 ∧ − arg2P ≤ 0 ∧ − arg2 ≤ 0

initial node: 0

cover edges:

→ 2

Hint: distribute conclusion

	[1, 0, 0]
	[0, 1, 0]
	[0, 0, 1]

→ 5

Hint: distribute conclusion

	[1, 0, 0, 0]
	[0, 1, 0, 0]
	[0, 0, 1, 0]
	[0, 0, 0, 1]

transition edges:

2 1

Hint: auto

0 2

Hint: distribute conclusion

	[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0]
	[0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
	[0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0]

1 3

Hint: distribute conclusion

	[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0]
	[0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0]
	[0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1]

3 4

Hint: distribute conclusion

	[1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1]
	[0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0]
	[0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0]

4 5

Hint: distribute conclusion

	[1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0]
	[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0]
	[0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
	[0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]

1 12

Hint: distribute conclusion

	[0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0]
	[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
	[1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0]
	[0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
	[0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]

6 13

Hint: distribute conclusion

	[1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0]
	[0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
	[0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0]
	[0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
	[0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]

4 14

Hint: distribute conclusion

	[0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0]
	[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0]
	[0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
	[0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]

6.1.1.1.8 Transition Removal

We remove transition 6 using the following ranking functions, which are bounded by −2.

1:	arg2 − arg3
1_var_snapshot:	__snapshot_1_arg2 − __snapshot_1_arg3
1*:	__snapshot_1_arg2 − __snapshot_1_arg3

Hints:

distribute assertion

	lexWeak[ [0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0] ]
	lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0] ]

lexStrict[ [0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] , [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]

lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0] ]

6.1.1.1.9 Transition Removal

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

1:	−1
1_var_snapshot:	−2
1*:	−3

Hints:

distribute assertion

	lexStrict[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] , [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]
	lexStrict[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] , [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]

lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]

6.1.1.1.10 Splitting Cut-Point Transitions

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

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