LTS Termination Proof

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Input

Integer Transition System

Initial Location: 4

Transitions: (pre-variables and post-variables)

0	0	1:	0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ y_0 ≤ 0 ∧ 1 − x_0 + x_post ≤ 0 ∧ −1 + x_0 − x_post ≤ 0 ∧ −1 − y_0 + y_post ≤ 0 ∧ 1 + 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
0	1	2:	0 ≤ 0 ∧ 0 ≤ 0 ∧ 1 − y_0 ≤ 0 ∧ 1 − y_0 + y_post ≤ 0 ∧ −1 + y_0 − 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
2	2	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
3	3	1:	− 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
1	4	0:	0 ≤ 0 ∧ 0 ≤ 0 ∧ 1 − x_0 ≤ 0 ∧ − x_0 + y_post ≤ 0 ∧ x_0 − 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
4	5	3:	− 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 − x_0 ≤ 0
1:	TRUE
2:	1 − x_0 ≤ 0
3:	TRUE
4:	TRUE

The invariants are proved as follows.

IMPACT Invariant Proof

nodes (location) invariant:

0	(0)	1 − x_0 ≤ 0
1	(1)	TRUE
2	(2)	1 − x_0 ≤ 0
3	(3)	TRUE
4	(4)	TRUE

initial node: 4
cover edges:
transition edges:

0 0 1

0 1 2

1 4 0

2 2 0

3 3 1

4 5 3

2 Switch to Cooperation Termination Proof

We consider the following cutpoint-transitions:

0	6	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
1	13	1:	− 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 3, 5 using the following ranking functions, which are bounded by −13.

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

Hints:

7	lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 0] ]
14	lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0] ]
0	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] ]
2	lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 0] ]
4	lexWeak[ [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] ]
5	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* 9 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 7 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 Location Addition

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

1* 16 1: − 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

7 Location Addition

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

1 14 1_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

8 SCC Decomposition

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

8.1 SCC Subproblem 1/1

Here we consider the SCC { 0, 2, 1, 0_var_snapshot, 0*, 1_var_snapshot, 1* }.

8.1.1 Transition Removal

We remove transition 0 using the following ranking functions, which are bounded by 3.

0:	4⋅x_0
2:	4⋅x_0
1:	2 + 4⋅x_0
0_var_snapshot:	4⋅x_0
0*:	4⋅x_0
1_var_snapshot:	1 + 4⋅x_0
1*:	3 + 4⋅x_0

Hints:

7	lexWeak[ [0, 0, 0, 0, 0, 0, 0, 4, 0] ]
9	lexWeak[ [0, 0, 0, 0, 0, 0, 0, 4, 0] ]
14	lexWeak[ [0, 0, 0, 0, 0, 0, 4, 0] ]
16	lexWeak[ [0, 0, 0, 0, 0, 0, 4, 0] ]
0	lexStrict[ [0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 4, 0, 0, 0] , [4, 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, 4, 0] ]
2	lexWeak[ [0, 0, 0, 0, 0, 0, 0, 4, 0] ]
4	lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0] ]

8.1.2 Transition Removal

We remove transitions 14, 4 using the following ranking functions, which are bounded by 2.

0:	1 − 4⋅x_0 + 4⋅y_0
2:	3 − 4⋅x_0 + 4⋅y_0
1:	4
0_var_snapshot:	−4⋅x_0 + 4⋅y_0
0*:	2 − 4⋅x_0 + 4⋅y_0
1_var_snapshot:	3
1*:	4

Hints:

7	lexWeak[ [0, 0, 0, 4, 0, 0, 0, 0, 4] ]
9	lexWeak[ [0, 0, 0, 4, 0, 0, 0, 0, 4] ]
14	lexStrict[ [0, 0, 0, 0, 0, 0, 0, 0] , [0, 0, 0, 0, 0, 0, 0, 0] ]
16	lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0] ]
1	lexWeak[ [0, 0, 0, 0, 4, 0, 4, 0, 0, 0, 0, 4] ]
2	lexWeak[ [0, 0, 0, 4, 0, 0, 0, 0, 4] ]
4	lexStrict[ [0, 0, 0, 4, 0, 4, 0, 0, 0, 0, 4] , [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]

8.1.3 Transition Removal

We remove transitions 16, 1 using the following ranking functions, which are bounded by 0.

0:	−2 + 4⋅y_0
2:	4⋅y_0
1:	0
0_var_snapshot:	−3 + 4⋅y_0
0*:	−1 + 4⋅y_0
1_var_snapshot:	0
1*:	1

Hints:

7	lexWeak[ [0, 0, 0, 4, 0, 0, 0, 0, 0] ]
9	lexWeak[ [0, 0, 0, 4, 0, 0, 0, 0, 0] ]
16	lexStrict[ [0, 0, 0, 0, 0, 0, 0, 0] , [0, 0, 0, 0, 0, 0, 0, 0] ]
1	lexStrict[ [0, 0, 0, 0, 4, 0, 4, 0, 0, 0, 0, 0] , [0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0] ]
2	lexWeak[ [0, 0, 0, 4, 0, 0, 0, 0, 0] ]

8.1.4 Transition Removal

We remove transitions 7, 9, 2 using the following ranking functions, which are bounded by −1.

0:	0
2:	2⋅x_0
1:	0
0_var_snapshot:	−1
0*:	x_0
1_var_snapshot:	0
1*:	0

Hints:

7	lexStrict[ [0, 0, 0, 0, 0, 0, 0, 0, 0] , [0, 0, 0, 0, 0, 0, 0, 0, 0] ]
9	lexStrict[ [1, 0, 0, 0, 0, 0, 0, 0, 0] , [1, 0, 0, 0, 0, 0, 0, 0, 0] ]
2	lexStrict[ [1, 0, 0, 0, 0, 0, 0, 1, 0] , [2, 0, 0, 0, 0, 0, 0, 0, 0] ]

8.1.5 Splitting Cut-Point Transitions

We consider 2 subproblems corresponding to sets of cut-point transitions as follows.

8.1.5.1 Cut-Point Subproblem 1/2

Here we consider cut-point transition 6.

8.1.5.1.1 Splitting Cut-Point Transitions

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

8.1.5.2 Cut-Point Subproblem 2/2

Here we consider cut-point transition 13.

8.1.5.2.1 Splitting Cut-Point Transitions

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

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