LTS Termination Proof

by T2Cert

Input

Integer Transition System

Initial Location: 3

Transitions: (pre-variables and post-variables)

0	0	1:	0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ − x_0 ≤ 0 ∧ − x_0 + x_post + 3⋅y_0 ≤ 0 ∧ x_0 − x_post − 3⋅y_0 ≤ 0 ∧ − y_0 + y_post ≤ 0 ∧ y_0 − y_post ≤ 0 ∧ x_0 − x_post ≤ 0 ∧ − x_0 + x_post ≤ 0 ∧ y_0 − y_post ≤ 0 ∧ − y_0 + y_post ≤ 0
1	1	0:	− y_post + y_post ≤ 0 ∧ y_post − y_post ≤ 0 ∧ − y_0 + y_0 ≤ 0 ∧ y_0 − y_0 ≤ 0 ∧ − x_post + x_post ≤ 0 ∧ x_post − x_post ≤ 0 ∧ − x_0 + x_0 ≤ 0 ∧ x_0 − x_0 ≤ 0
2	2	0:	0 ≤ 0 ∧ 0 ≤ 0 ∧ −1 + y_post ≤ 0 ∧ 1 − y_post ≤ 0 ∧ y_0 − y_post ≤ 0 ∧ − y_0 + y_post ≤ 0 ∧ − x_post + x_post ≤ 0 ∧ x_post − x_post ≤ 0 ∧ − x_0 + x_0 ≤ 0 ∧ x_0 − x_0 ≤ 0
3	3	2:	− y_post + y_post ≤ 0 ∧ y_post − y_post ≤ 0 ∧ − y_0 + y_0 ≤ 0 ∧ y_0 − y_0 ≤ 0 ∧ − x_post + x_post ≤ 0 ∧ x_post − x_post ≤ 0 ∧ − x_0 + x_0 ≤ 0 ∧ x_0 − x_0 ≤ 0

Proof

1 Invariant Updates

The following invariants are asserted.

0:	−1 + y_post ≤ 0 ∧ −1 + y_0 ≤ 0
1:	−1 + y_post ≤ 0 ∧ −1 + y_0 ≤ 0
2:	TRUE
3:	TRUE

The invariants are proved as follows.

IMPACT Invariant Proof

nodes (location) invariant:

0	(0)	−1 + y_post ≤ 0 ∧ −1 + y_0 ≤ 0
1	(1)	−1 + y_post ≤ 0 ∧ −1 + y_0 ≤ 0
2	(2)	TRUE
3	(3)	TRUE

initial node: 3
cover edges:
transition edges:

0 0 1

1 1 0

2 2 0

3 3 2

2 Switch to Cooperation Termination Proof

We consider the following cutpoint-transitions:

− y_post + y_post ≤ 0 ∧ y_post − y_post ≤ 0 ∧ − y_0 + y_0 ≤ 0 ∧ y_0 − y_0 ≤ 0 ∧ − x_post + x_post ≤ 0 ∧ x_post − x_post ≤ 0 ∧ − x_0 + x_0 ≤ 0 ∧ x_0 − x_0 ≤ 0

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

3 Transition Removal

We remove transitions 2, 3 using the following ranking functions, which are bounded by −11.

3:	0
2:	0
0:	0
1:	0
3:	−4
2:	−5
0:	−6
1:	−6
0_var_snapshot:	−6
0*:	−6

Hints:

5	lexWeak[ [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] ]
1	lexWeak[ [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] ]
3	lexStrict[ [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.

0* 7 0: − y_post + y_post ≤ 0 ∧ y_post − y_post ≤ 0 ∧ − y_0 + y_0 ≤ 0 ∧ y_0 − y_0 ≤ 0 ∧ − x_post + x_post ≤ 0 ∧ x_post − x_post ≤ 0 ∧ − x_0 + x_0 ≤ 0 ∧ x_0 − x_0 ≤ 0

5 Location Addition

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

0 5 0_var_snapshot: − y_post + y_post ≤ 0 ∧ y_post − y_post ≤ 0 ∧ − y_0 + y_0 ≤ 0 ∧ y_0 − y_0 ≤ 0 ∧ − x_post + x_post ≤ 0 ∧ x_post − x_post ≤ 0 ∧ − x_0 + x_0 ≤ 0 ∧ x_0 − x_0 ≤ 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 { 0, 1, 0_var_snapshot, 0* }.

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 4.

6.1.1.1.1 Fresh Variable Addition

The new variable __snapshot_0_y_post is introduced. The transition formulas are extended as follows:

5:	__snapshot_0_y_post ≤ y_post ∧ y_post ≤ __snapshot_0_y_post
7:	__snapshot_0_y_post ≤ __snapshot_0_y_post ∧ __snapshot_0_y_post ≤ __snapshot_0_y_post
0:	__snapshot_0_y_post ≤ __snapshot_0_y_post ∧ __snapshot_0_y_post ≤ __snapshot_0_y_post
1:	__snapshot_0_y_post ≤ __snapshot_0_y_post ∧ __snapshot_0_y_post ≤ __snapshot_0_y_post

6.1.1.1.2 Fresh Variable Addition

The new variable __snapshot_0_y_0 is introduced. The transition formulas are extended as follows:

5:	__snapshot_0_y_0 ≤ y_0 ∧ y_0 ≤ __snapshot_0_y_0
7:	__snapshot_0_y_0 ≤ __snapshot_0_y_0 ∧ __snapshot_0_y_0 ≤ __snapshot_0_y_0
0:	__snapshot_0_y_0 ≤ __snapshot_0_y_0 ∧ __snapshot_0_y_0 ≤ __snapshot_0_y_0
1:	__snapshot_0_y_0 ≤ __snapshot_0_y_0 ∧ __snapshot_0_y_0 ≤ __snapshot_0_y_0

6.1.1.1.3 Fresh Variable Addition

The new variable __snapshot_0_x_post is introduced. The transition formulas are extended as follows:

5:	__snapshot_0_x_post ≤ x_post ∧ x_post ≤ __snapshot_0_x_post
7:	__snapshot_0_x_post ≤ __snapshot_0_x_post ∧ __snapshot_0_x_post ≤ __snapshot_0_x_post
0:	__snapshot_0_x_post ≤ __snapshot_0_x_post ∧ __snapshot_0_x_post ≤ __snapshot_0_x_post
1:	__snapshot_0_x_post ≤ __snapshot_0_x_post ∧ __snapshot_0_x_post ≤ __snapshot_0_x_post

6.1.1.1.4 Fresh Variable Addition

The new variable __snapshot_0_x_0 is introduced. The transition formulas are extended as follows:

5:	__snapshot_0_x_0 ≤ x_0 ∧ x_0 ≤ __snapshot_0_x_0
7:	__snapshot_0_x_0 ≤ __snapshot_0_x_0 ∧ __snapshot_0_x_0 ≤ __snapshot_0_x_0
0:	__snapshot_0_x_0 ≤ __snapshot_0_x_0 ∧ __snapshot_0_x_0 ≤ __snapshot_0_x_0
1:	__snapshot_0_x_0 ≤ __snapshot_0_x_0 ∧ __snapshot_0_x_0 ≤ __snapshot_0_x_0

6.1.1.1.5 Invariant Updates

The following invariants are asserted.

0:	1 − 3⋅y_0 ≤ 0 ∧ −1 + y_post ≤ 0 ∧ −1 + y_0 ≤ 0
1:	1 − 3⋅y_0 ≤ 0 ∧ −1 + y_post ≤ 0 ∧ −1 + y_0 ≤ 0
2:	TRUE
3:	TRUE
0:	1 − 3⋅y_0 ≤ 0 ∧ −1 + y_post ≤ 0 ∧ −1 + y_0 ≤ 0 ∨ 1 − __snapshot_0_x_0 + x_0 ≤ 0 ∧ 1 − 3⋅y_0 ≤ 0 ∧ − __snapshot_0_x_0 ≤ 0 ∧ −1 + y_post ≤ 0 ∧ −1 + y_0 ≤ 0
1:	1 − __snapshot_0_x_0 + x_0 ≤ 0 ∧ 1 − 3⋅y_0 ≤ 0 ∧ − __snapshot_0_x_0 ≤ 0 ∧ −1 + y_post ≤ 0 ∧ −1 + y_0 ≤ 0
0_var_snapshot:	− __snapshot_0_x_0 + x_0 ≤ 0 ∧ 1 − __snapshot_0_x_0 + x_0 − 3⋅y_0 ≤ 0 ∧ 1 − 3⋅y_0 ≤ 0 ∧ −1 + y_post ≤ 0 ∧ −1 + y_0 ≤ 0
0*:	1 − __snapshot_0_x_0 + x_0 ≤ 0 ∧ 1 − 3⋅y_0 ≤ 0 ∧ − __snapshot_0_x_0 ≤ 0 ∧ −1 + y_post ≤ 0 ∧ −1 + y_0 ≤ 0

The invariants are proved as follows.

IMPACT Invariant Proof

nodes (location) invariant:

0	(3)	TRUE
1	(2)	TRUE
2	(0)	1 − 3⋅y_0 ≤ 0 ∧ −1 + y_post ≤ 0 ∧ −1 + y_0 ≤ 0
3	(1)	1 − 3⋅y_0 ≤ 0 ∧ −1 + y_post ≤ 0 ∧ −1 + y_0 ≤ 0
4	(0)	1 − 3⋅y_0 ≤ 0 ∧ −1 + y_post ≤ 0 ∧ −1 + y_0 ≤ 0
5	(0_var_snapshot)	− __snapshot_0_x_0 + x_0 ≤ 0 ∧ 1 − __snapshot_0_x_0 + x_0 − 3⋅y_0 ≤ 0 ∧ 1 − 3⋅y_0 ≤ 0 ∧ −1 + y_post ≤ 0 ∧ −1 + y_0 ≤ 0
10	(0)	1 − 3⋅y_0 ≤ 0 ∧ −1 + y_post ≤ 0 ∧ −1 + y_0 ≤ 0
14	(1)	1 − __snapshot_0_x_0 + x_0 ≤ 0 ∧ 1 − 3⋅y_0 ≤ 0 ∧ − __snapshot_0_x_0 ≤ 0 ∧ −1 + y_post ≤ 0 ∧ −1 + y_0 ≤ 0
15	(0*)	1 − __snapshot_0_x_0 + x_0 ≤ 0 ∧ 1 − 3⋅y_0 ≤ 0 ∧ − __snapshot_0_x_0 ≤ 0 ∧ −1 + y_post ≤ 0 ∧ −1 + y_0 ≤ 0
16	(0)	1 − __snapshot_0_x_0 + x_0 ≤ 0 ∧ 1 − 3⋅y_0 ≤ 0 ∧ − __snapshot_0_x_0 ≤ 0 ∧ −1 + y_post ≤ 0 ∧ −1 + y_0 ≤ 0
17	(0_var_snapshot)	− __snapshot_0_x_0 + x_0 ≤ 0 ∧ 1 − __snapshot_0_x_0 + x_0 − 3⋅y_0 ≤ 0 ∧ 1 − 3⋅y_0 ≤ 0 ∧ −1 + y_post ≤ 0 ∧ −1 + y_0 ≤ 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, 1, 0, 0]
	[0, 0, 1, 0, 0]
	[0, 0, 0, 1, 0]
	[0, 0, 0, 0, 1]

transition edges:

3 1

Hint: auto

2 2

Hint: distribute conclusion

	[0, 0, 0, 3, 0, 3, 0, 0, 0, 0]
	[0, 0, 1, 0, 0, 0, 0, 0, 0, 0]
	[0, 0, 1, 0, 1, 0, 0, 0, 0, 0]

0 3

Hint: distribute conclusion

	[1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 3]
	[0, 0, 1, 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, 1, 0, 0, 0, 1, 0]

4 4

Hint: distribute conclusion

	[1, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0]
	[0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0]
	[0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0]

1 10

Hint: distribute conclusion

	[1, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0]
	[0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0]
	[0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0]

5 5

Hint: distribute conclusion

	[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1]
	[1, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1]
	[1, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 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, 0, 0, 0, 0]
	[0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]

0 14

Hint: distribute conclusion

	[1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1]
	[0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0]
	[1, 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, 1]
	[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, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0]

1 15

Hint: distribute conclusion

	[1, 0, 0, 0, 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, 0, 0, 0, 3, 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, 1, 0, 1, 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, 0, 0, 0, 0, 0, 0, 0, 0]

7 16

Hint: distribute conclusion

	[1, 0, 0, 0, 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, 0, 0, 0, 3, 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, 1, 0, 1, 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, 0, 0, 0, 0, 0, 0, 0, 0]

5 17

Hint: distribute conclusion

	[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1]
	[0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1]
	[0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
	[0, 0, 0, 1, 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, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]

6.1.1.1.6 Transition Removal

We remove transition 7 using the following ranking functions, which are bounded by −2.

0:	x_0
1:	__snapshot_0_x_0
0_var_snapshot:	__snapshot_0_x_0
0*:	__snapshot_0_x_0

Hints:

distribute assertion

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

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

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, 1, 0] ]

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

6.1.1.1.7 Transition Removal

We remove transition 5 using the following ranking functions, which are bounded by −6.

0:	−1
0_var_snapshot:	−2
1:	−3
0*:	−4

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, 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] ]

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

6.1.1.1.8 Splitting Cut-Point Transitions

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

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