LTS Termination Proof

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Input

Integer Transition System

Initial Location: 7

Transitions: (pre-variables and post-variables)

0	0	1:	0 ≤ 0 ∧ 0 ≤ 0 ∧ −1 − i1_0 + i1_post ≤ 0 ∧ 1 + i1_0 − i1_post ≤ 0 ∧ i1_0 − i1_post ≤ 0 ∧ − i1_0 + i1_post ≤ 0
2	1	0:	0 ≤ 0 ∧ 0 ≤ 0 ∧ − i1_post + i1_post ≤ 0 ∧ i1_post − i1_post ≤ 0 ∧ − i1_0 + i1_0 ≤ 0 ∧ i1_0 − i1_0 ≤ 0
3	2	2:	0 ≤ 0 ∧ 0 ≤ 0 ∧ − i1_post + i1_post ≤ 0 ∧ i1_post − i1_post ≤ 0 ∧ − i1_0 + i1_0 ≤ 0 ∧ i1_0 − i1_0 ≤ 0
3	3	0:	0 ≤ 0 ∧ 0 ≤ 0 ∧ − i1_post + i1_post ≤ 0 ∧ i1_post − i1_post ≤ 0 ∧ − i1_0 + i1_0 ≤ 0 ∧ i1_0 − i1_0 ≤ 0
3	4	2:	0 ≤ 0 ∧ 0 ≤ 0 ∧ − i1_post + i1_post ≤ 0 ∧ i1_post − i1_post ≤ 0 ∧ − i1_0 + i1_0 ≤ 0 ∧ i1_0 − i1_0 ≤ 0
4	5	5:	42 − i1_0 ≤ 0 ∧ − i1_post + i1_post ≤ 0 ∧ i1_post − i1_post ≤ 0 ∧ − i1_0 + i1_0 ≤ 0 ∧ i1_0 − i1_0 ≤ 0
4	6	3:	−41 + i1_0 ≤ 0 ∧ − i1_post + i1_post ≤ 0 ∧ i1_post − i1_post ≤ 0 ∧ − i1_0 + i1_0 ≤ 0 ∧ i1_0 − i1_0 ≤ 0
1	7	4:	0 ≤ 0 ∧ 0 ≤ 0 ∧ − i1_post + i1_post ≤ 0 ∧ i1_post − i1_post ≤ 0 ∧ − i1_0 + i1_0 ≤ 0 ∧ i1_0 − i1_0 ≤ 0
6	8	1:	0 ≤ 0 ∧ 0 ≤ 0 ∧ i1_post ≤ 0 ∧ − i1_post ≤ 0 ∧ i1_0 − i1_post ≤ 0 ∧ − i1_0 + i1_post ≤ 0
7	9	6:	0 ≤ 0 ∧ 0 ≤ 0 ∧ − i1_post + i1_post ≤ 0 ∧ i1_post − i1_post ≤ 0 ∧ − i1_0 + i1_0 ≤ 0 ∧ i1_0 − i1_0 ≤ 0

Proof

1 Invariant Updates

The following invariants are asserted.

0:	TRUE
1:	TRUE
2:	TRUE
3:	TRUE
4:	TRUE
5:	42 − i1_0 ≤ 0
6:	TRUE
7:	TRUE

The invariants are proved as follows.

IMPACT Invariant Proof

nodes (location) invariant:

0	(0)	TRUE
1	(1)	TRUE
2	(2)	TRUE
3	(3)	TRUE
4	(4)	TRUE
5	(5)	42 − i1_0 ≤ 0
6	(6)	TRUE
7	(7)	TRUE

initial node: 7
cover edges:
transition edges:

0 0 1

1 7 4

2 1 0

3 2 2

3 3 0

3 4 2

4 5 5

4 6 3

6 8 1

7 9 6

2 Switch to Cooperation Termination Proof

We consider the following cutpoint-transitions:

− i1_post + i1_post ≤ 0 ∧ i1_post − i1_post ≤ 0 ∧ − i1_0 + i1_0 ≤ 0 ∧ i1_0 − i1_0 ≤ 0

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

3 Transition Removal

We remove transitions 5, 8, 9 using the following ranking functions, which are bounded by −13.

7:	0
6:	0
0:	0
1:	0
2:	0
3:	0
4:	0
5:	0
7:	−5
6:	−6
0:	−7
1:	−7
2:	−7
3:	−7
4:	−7
1_var_snapshot:	−7
1*:	−7
5:	−8

Hints:

11	lexWeak[ [0, 0, 0, 0] ]
0	lexWeak[ [0, 0, 0, 0, 0, 0] ]
1	lexWeak[ [0, 0, 0, 0, 0, 0] ]
2	lexWeak[ [0, 0, 0, 0, 0, 0] ]
3	lexWeak[ [0, 0, 0, 0, 0, 0] ]
4	lexWeak[ [0, 0, 0, 0, 0, 0] ]
6	lexWeak[ [0, 0, 0, 0, 0] ]
7	lexWeak[ [0, 0, 0, 0, 0, 0] ]
5	lexStrict[ [0, 0, 0, 0, 0] , [0, 0, 0, 0, 0] ]
8	lexStrict[ [0, 0, 0, 0, 0, 0] , [0, 0, 0, 0, 0, 0] ]
9	lexStrict[ [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* 13 1: − i1_post + i1_post ≤ 0 ∧ i1_post − i1_post ≤ 0 ∧ − i1_0 + i1_0 ≤ 0 ∧ i1_0 − i1_0 ≤ 0

5 Location Addition

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

1 11 1_var_snapshot: − i1_post + i1_post ≤ 0 ∧ i1_post − i1_post ≤ 0 ∧ − i1_0 + i1_0 ≤ 0 ∧ i1_0 − i1_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, 2, 3, 4, 1_var_snapshot, 1* }.

6.1.1 Transition Removal

We remove transition 6 using the following ranking functions, which are bounded by −207.

0:	−2 − 5⋅i1_0
1:	1 − 5⋅i1_0
2:	−2 − 5⋅i1_0
3:	−2 − 5⋅i1_0
4:	−1 − 5⋅i1_0
1_var_snapshot:	−5⋅i1_0
1*:	2 − 5⋅i1_0

Hints:

11	lexWeak[ [0, 0, 0, 5] ]
13	lexWeak[ [0, 0, 0, 5] ]
0	lexWeak[ [0, 0, 0, 5, 0, 5] ]
1	lexWeak[ [0, 0, 0, 0, 0, 5] ]
2	lexWeak[ [0, 0, 0, 0, 0, 5] ]
3	lexWeak[ [0, 0, 0, 0, 0, 5] ]
4	lexWeak[ [0, 0, 0, 0, 0, 5] ]
6	lexStrict[ [0, 0, 0, 0, 5] , [5, 0, 0, 0, 0] ]
7	lexWeak[ [0, 0, 0, 0, 0, 5] ]

6.1.2 Transition Removal

We remove transitions 11, 13, 0, 1, 2, 3, 4, 7 using the following ranking functions, which are bounded by −6.

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

Hints:

11	lexStrict[ [0, 0, 0, 0] , [0, 0, 0, 0] ]
13	lexStrict[ [0, 0, 0, 0] , [0, 0, 0, 0] ]
0	lexStrict[ [0, 0, 0, 0, 0, 0] , [0, 0, 0, 0, 0, 0] ]
1	lexStrict[ [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] ]
3	lexStrict[ [0, 0, 0, 0, 0, 0] , [0, 0, 0, 0, 0, 0] ]
4	lexStrict[ [0, 0, 0, 0, 0, 0] , [0, 0, 0, 0, 0, 0] ]
7	lexStrict[ [0, 0, 0, 0, 0, 0] , [0, 0, 0, 0, 0, 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 10.

6.1.3.1.1 Splitting Cut-Point Transitions

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

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