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

Proof

1 Switch to Cooperation Termination Proof

We consider the following cutpoint-transitions:

− 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.

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

3 Location Addition

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

1* 6 1: − arg2P + arg2P ≤ 0 ∧ arg2P − arg2P ≤ 0 ∧ − arg2 + arg2 ≤ 0 ∧ arg2 − arg2 ≤ 0 ∧ − arg1P + arg1P ≤ 0 ∧ arg1P − arg1P ≤ 0 ∧ − arg1 + arg1 ≤ 0 ∧ arg1 − arg1 ≤ 0

4 Location Addition

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

1 4 1_var_snapshot: − 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 SCC Decomposition

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

5.1 SCC Subproblem 1/1

Here we consider the SCC { 1, 1_var_snapshot, 1* }.

5.1.1 Splitting Cut-Point Transitions

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

5.1.1.1 Cut-Point Subproblem 1/1

Here we consider cut-point transition 3.

5.1.1.1.1 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

5.1.1.1.2 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

5.1.1.1.3 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

5.1.1.1.4 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

5.1.1.1.5 Invariant Updates

The following invariants are asserted.

0:	TRUE
1:	arg1 − arg1P + arg2 − arg2P ≤ 0 ∧ − arg1P − arg2P ≤ 0 ∧ − arg2 ≤ 0 ∨ arg1 − arg1P + arg2 − arg2P ≤ 0 ∧ −1 − arg1P + arg2 − arg2P ≤ 0
2:	TRUE
1:	arg1 − arg1P + arg2 − arg2P ≤ 0 ∧ − arg1P − arg2P ≤ 0 ∧ − arg2 ≤ 0 ∨ arg1 − arg1P + arg2 − arg2P ≤ 0 ∧ −1 − arg1P + arg2 − arg2P ≤ 0 ∧ 1 − __snapshot_1_arg1P − __snapshot_1_arg2P + arg1P + arg2P ≤ 0 ∧ − __snapshot_1_arg1P − __snapshot_1_arg2P ≤ 0 ∨ arg1 − arg1P + arg2 − arg2P ≤ 0 ∧ −1 − arg1P + arg2 − arg2P ≤ 0
1_var_snapshot:	− __snapshot_1_arg1P − __snapshot_1_arg2P + arg1 + arg2 ≤ 0 ∧ − __snapshot_1_arg1P − __snapshot_1_arg2P ≤ 0 ∧ − arg2 ≤ 0 ∨ − __snapshot_1_arg1P − __snapshot_1_arg2P + arg1 + arg2 ≤ 0 ∧ −1 − __snapshot_1_arg1P − __snapshot_1_arg2P + arg2 ≤ 0
1*:	arg1 − arg1P + arg2 − arg2P ≤ 0 ∧ −1 − arg1P + arg2 − arg2P ≤ 0 ∧ 1 − __snapshot_1_arg1P − __snapshot_1_arg2P + arg1P + arg2P ≤ 0 ∧ − __snapshot_1_arg1P − __snapshot_1_arg2P ≤ 0

The invariants are proved as follows.

IMPACT Invariant Proof

nodes (location) invariant:

0	(2)	TRUE
1	(0)	TRUE
2	(1)	arg1 − arg1P + arg2 − arg2P ≤ 0 ∧ − arg1P − arg2P ≤ 0 ∧ − arg2 ≤ 0
3	(1)	arg1 − arg1P + arg2 − arg2P ≤ 0 ∧ −1 − arg1P + arg2 − arg2P ≤ 0
4	(1)	arg1 − arg1P + arg2 − arg2P ≤ 0 ∧ − arg1P − arg2P ≤ 0 ∧ − arg2 ≤ 0
5	(1_var_snapshot)	− __snapshot_1_arg1P − __snapshot_1_arg2P + arg1 + arg2 ≤ 0 ∧ − __snapshot_1_arg1P − __snapshot_1_arg2P ≤ 0 ∧ − arg2 ≤ 0
12	(1*)	arg1 − arg1P + arg2 − arg2P ≤ 0 ∧ −1 − arg1P + arg2 − arg2P ≤ 0 ∧ 1 − __snapshot_1_arg1P − __snapshot_1_arg2P + arg1P + arg2P ≤ 0 ∧ − __snapshot_1_arg1P − __snapshot_1_arg2P ≤ 0
13	(1)	arg1 − arg1P + arg2 − arg2P ≤ 0 ∧ −1 − arg1P + arg2 − arg2P ≤ 0 ∧ 1 − __snapshot_1_arg1P − __snapshot_1_arg2P + arg1P + arg2P ≤ 0 ∧ − __snapshot_1_arg1P − __snapshot_1_arg2P ≤ 0
14	(1_var_snapshot)	− __snapshot_1_arg1P − __snapshot_1_arg2P + arg1 + arg2 ≤ 0 ∧ −1 − __snapshot_1_arg1P − __snapshot_1_arg2P + arg2 ≤ 0
15	(1_var_snapshot)	− __snapshot_1_arg1P − __snapshot_1_arg2P + arg1 + arg2 ≤ 0 ∧ −1 − __snapshot_1_arg1P − __snapshot_1_arg2P + arg2 ≤ 0
23	(1)	arg1 − arg1P + arg2 − arg2P ≤ 0 ∧ −1 − arg1P + arg2 − arg2P ≤ 0
24	(1)	arg1 − arg1P + arg2 − arg2P ≤ 0 ∧ −1 − arg1P + arg2 − arg2P ≤ 0
28	(1*)	arg1 − arg1P + arg2 − arg2P ≤ 0 ∧ −1 − arg1P + arg2 − arg2P ≤ 0 ∧ 1 − __snapshot_1_arg1P − __snapshot_1_arg2P + arg1P + arg2P ≤ 0 ∧ − __snapshot_1_arg1P − __snapshot_1_arg2P ≤ 0
37	(1_var_snapshot)	− __snapshot_1_arg1P − __snapshot_1_arg2P + arg1 + arg2 ≤ 0 ∧ −1 − __snapshot_1_arg1P − __snapshot_1_arg2P + arg2 ≤ 0

initial node: 0

cover edges:

→ 15

Hint: distribute conclusion

	[1, 0]
	[0, 1]

→ 3

Hint: distribute conclusion

	[1, 0]
	[0, 1]

→ 12

Hint: distribute conclusion

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

→ 15

Hint: distribute conclusion

	[1, 0]
	[0, 1]

transition edges:

2 1

Hint: auto

0 2

Hint: distribute conclusion

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

1 3

Hint: distribute conclusion

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

3 4

Hint: distribute conclusion

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

1 23

Hint: distribute conclusion

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

3 24

Hint: distribute conclusion

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

4 5

Hint: distribute conclusion

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

1 12

Hint: distribute conclusion

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

6 13

Hint: distribute conclusion

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

4 14

Hint: distribute conclusion

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

4 15

Hint: distribute conclusion

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

1 28

Hint: distribute conclusion

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

4 37

Hint: distribute conclusion

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

5.1.1.1.6 Transition Removal

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

1:	arg1P + arg2P
1_var_snapshot:	__snapshot_1_arg1P + __snapshot_1_arg2P
1*:	__snapshot_1_arg1P + __snapshot_1_arg2P

Hints:

distribute assertion

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

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

distribute assertion

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

5.1.1.1.7 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] ]
	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] ]
	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] ]

distribute assertion

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

5.1.1.1.8 Splitting Cut-Point Transitions

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

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