LTS Termination Proof

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Input

Integer Transition System

Initial Location: 6

Transitions: (pre-variables and post-variables)

0	0	1:	0 ≤ 0 ∧ 0 ≤ 0 ∧ 10 − i4_0 ≤ 0 ∧ i7_post ≤ 0 ∧ − i7_post ≤ 0 ∧ i7_0 − i7_post ≤ 0 ∧ − i7_0 + i7_post ≤ 0 ∧ − tmp_post + tmp_post ≤ 0 ∧ tmp_post − tmp_post ≤ 0 ∧ − tmp_0 + tmp_0 ≤ 0 ∧ tmp_0 − tmp_0 ≤ 0 ∧ − i4_post + i4_post ≤ 0 ∧ i4_post − i4_post ≤ 0 ∧ − i4_0 + i4_0 ≤ 0 ∧ i4_0 − i4_0 ≤ 0
0	1	2:	0 ≤ 0 ∧ 0 ≤ 0 ∧ −9 + i4_0 ≤ 0 ∧ −1 − i4_0 + i4_post ≤ 0 ∧ 1 + i4_0 − i4_post ≤ 0 ∧ i4_0 − i4_post ≤ 0 ∧ − i4_0 + i4_post ≤ 0 ∧ − tmp_post + tmp_post ≤ 0 ∧ tmp_post − tmp_post ≤ 0 ∧ − tmp_0 + tmp_0 ≤ 0 ∧ tmp_0 − tmp_0 ≤ 0 ∧ − i7_post + i7_post ≤ 0 ∧ i7_post − i7_post ≤ 0 ∧ − i7_0 + i7_0 ≤ 0 ∧ i7_0 − i7_0 ≤ 0
2	2	0:	− tmp_post + tmp_post ≤ 0 ∧ tmp_post − tmp_post ≤ 0 ∧ − tmp_0 + tmp_0 ≤ 0 ∧ tmp_0 − tmp_0 ≤ 0 ∧ − i7_post + i7_post ≤ 0 ∧ i7_post − i7_post ≤ 0 ∧ − i7_0 + i7_0 ≤ 0 ∧ i7_0 − i7_0 ≤ 0 ∧ − i4_post + i4_post ≤ 0 ∧ i4_post − i4_post ≤ 0 ∧ − i4_0 + i4_0 ≤ 0 ∧ i4_0 − i4_0 ≤ 0
1	3	3:	− tmp_post + tmp_post ≤ 0 ∧ tmp_post − tmp_post ≤ 0 ∧ − tmp_0 + tmp_0 ≤ 0 ∧ tmp_0 − tmp_0 ≤ 0 ∧ − i7_post + i7_post ≤ 0 ∧ i7_post − i7_post ≤ 0 ∧ − i7_0 + i7_0 ≤ 0 ∧ i7_0 − i7_0 ≤ 0 ∧ − i4_post + i4_post ≤ 0 ∧ i4_post − i4_post ≤ 0 ∧ − i4_0 + i4_0 ≤ 0 ∧ i4_0 − i4_0 ≤ 0
3	4	4:	10 − i7_0 ≤ 0 ∧ − tmp_post + tmp_post ≤ 0 ∧ tmp_post − tmp_post ≤ 0 ∧ − tmp_0 + tmp_0 ≤ 0 ∧ tmp_0 − tmp_0 ≤ 0 ∧ − i7_post + i7_post ≤ 0 ∧ i7_post − i7_post ≤ 0 ∧ − i7_0 + i7_0 ≤ 0 ∧ i7_0 − i7_0 ≤ 0 ∧ − i4_post + i4_post ≤ 0 ∧ i4_post − i4_post ≤ 0 ∧ − i4_0 + i4_0 ≤ 0 ∧ i4_0 − i4_0 ≤ 0
3	5	1:	0 ≤ 0 ∧ 0 ≤ 0 ∧ −9 + i7_0 ≤ 0 ∧ −1 − i7_0 + i7_post ≤ 0 ∧ 1 + i7_0 − i7_post ≤ 0 ∧ i7_0 − i7_post ≤ 0 ∧ − i7_0 + i7_post ≤ 0 ∧ − tmp_post + tmp_post ≤ 0 ∧ tmp_post − tmp_post ≤ 0 ∧ − tmp_0 + tmp_0 ≤ 0 ∧ tmp_0 − tmp_0 ≤ 0 ∧ − i4_post + i4_post ≤ 0 ∧ i4_post − i4_post ≤ 0 ∧ − i4_0 + i4_0 ≤ 0 ∧ i4_0 − i4_0 ≤ 0
5	6	2:	0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ i4_post ≤ 0 ∧ − i4_post ≤ 0 ∧ i4_0 − i4_post ≤ 0 ∧ − i4_0 + i4_post ≤ 0 ∧ tmp_0 − tmp_post ≤ 0 ∧ − tmp_0 + tmp_post ≤ 0 ∧ − i7_post + i7_post ≤ 0 ∧ i7_post − i7_post ≤ 0 ∧ − i7_0 + i7_0 ≤ 0 ∧ i7_0 − i7_0 ≤ 0
6	7	5:	− tmp_post + tmp_post ≤ 0 ∧ tmp_post − tmp_post ≤ 0 ∧ − tmp_0 + tmp_0 ≤ 0 ∧ tmp_0 − tmp_0 ≤ 0 ∧ − i7_post + i7_post ≤ 0 ∧ i7_post − i7_post ≤ 0 ∧ − i7_0 + i7_0 ≤ 0 ∧ i7_0 − i7_0 ≤ 0 ∧ − i4_post + i4_post ≤ 0 ∧ i4_post − i4_post ≤ 0 ∧ − i4_0 + i4_0 ≤ 0 ∧ i4_0 − i4_0 ≤ 0

Proof

1 Invariant Updates

The following invariants are asserted.

0:	TRUE
1:	10 − i4_0 ≤ 0
2:	TRUE
3:	10 − i4_0 ≤ 0
4:	10 − i4_0 ≤ 0 ∧ 10 − i7_0 ≤ 0
5:	TRUE
6:	TRUE

The invariants are proved as follows.

IMPACT Invariant Proof

nodes (location) invariant:

0	(0)	TRUE
1	(1)	10 − i4_0 ≤ 0
2	(2)	TRUE
3	(3)	10 − i4_0 ≤ 0
4	(4)	10 − i4_0 ≤ 0 ∧ 10 − i7_0 ≤ 0
5	(5)	TRUE
6	(6)	TRUE

initial node: 6
cover edges:
transition edges:

0 0 1

0 1 2

1 3 3

2 2 0

3 4 4

3 5 1

5 6 2

6 7 5

2 Switch to Cooperation Termination Proof

We consider the following cutpoint-transitions:

1	8	1:	− tmp_post + tmp_post ≤ 0 ∧ tmp_post − tmp_post ≤ 0 ∧ − tmp_0 + tmp_0 ≤ 0 ∧ tmp_0 − tmp_0 ≤ 0 ∧ − i7_post + i7_post ≤ 0 ∧ i7_post − i7_post ≤ 0 ∧ − i7_0 + i7_0 ≤ 0 ∧ i7_0 − i7_0 ≤ 0 ∧ − i4_post + i4_post ≤ 0 ∧ i4_post − i4_post ≤ 0 ∧ − i4_0 + i4_0 ≤ 0 ∧ i4_0 − i4_0 ≤ 0
2	15	2:	− tmp_post + tmp_post ≤ 0 ∧ tmp_post − tmp_post ≤ 0 ∧ − tmp_0 + tmp_0 ≤ 0 ∧ tmp_0 − tmp_0 ≤ 0 ∧ − i7_post + i7_post ≤ 0 ∧ i7_post − i7_post ≤ 0 ∧ − i7_0 + i7_0 ≤ 0 ∧ i7_0 − i7_0 ≤ 0 ∧ − i4_post + i4_post ≤ 0 ∧ i4_post − i4_post ≤ 0 ∧ − i4_0 + i4_0 ≤ 0 ∧ i4_0 − i4_0 ≤ 0

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

3 Transition Removal

We remove transitions 0, 4, 6, 7 using the following ranking functions, which are bounded by −17.

6:	0
5:	0
0:	0
2:	0
1:	0
3:	0
4:	0
6:	−6
5:	−7
0:	−8
2:	−8
2_var_snapshot:	−8
2*:	−8
1:	−9
3:	−9
1_var_snapshot:	−9
1*:	−9
4:	−10

Hints:

9	lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]
16	lexWeak[ [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] ]
2	lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]
3	lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]
5	lexWeak[ [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] ]
4	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] ]
6	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] ]
7	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* 11 1: − tmp_post + tmp_post ≤ 0 ∧ tmp_post − tmp_post ≤ 0 ∧ − tmp_0 + tmp_0 ≤ 0 ∧ tmp_0 − tmp_0 ≤ 0 ∧ − i7_post + i7_post ≤ 0 ∧ i7_post − i7_post ≤ 0 ∧ − i7_0 + i7_0 ≤ 0 ∧ i7_0 − i7_0 ≤ 0 ∧ − i4_post + i4_post ≤ 0 ∧ i4_post − i4_post ≤ 0 ∧ − i4_0 + i4_0 ≤ 0 ∧ i4_0 − i4_0 ≤ 0

5 Location Addition

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

1 9 1_var_snapshot: − tmp_post + tmp_post ≤ 0 ∧ tmp_post − tmp_post ≤ 0 ∧ − tmp_0 + tmp_0 ≤ 0 ∧ tmp_0 − tmp_0 ≤ 0 ∧ − i7_post + i7_post ≤ 0 ∧ i7_post − i7_post ≤ 0 ∧ − i7_0 + i7_0 ≤ 0 ∧ i7_0 − i7_0 ≤ 0 ∧ − i4_post + i4_post ≤ 0 ∧ i4_post − i4_post ≤ 0 ∧ − i4_0 + i4_0 ≤ 0 ∧ i4_0 − i4_0 ≤ 0

6 Location Addition

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

2* 18 2: − tmp_post + tmp_post ≤ 0 ∧ tmp_post − tmp_post ≤ 0 ∧ − tmp_0 + tmp_0 ≤ 0 ∧ tmp_0 − tmp_0 ≤ 0 ∧ − i7_post + i7_post ≤ 0 ∧ i7_post − i7_post ≤ 0 ∧ − i7_0 + i7_0 ≤ 0 ∧ i7_0 − i7_0 ≤ 0 ∧ − i4_post + i4_post ≤ 0 ∧ i4_post − i4_post ≤ 0 ∧ − i4_0 + i4_0 ≤ 0 ∧ i4_0 − i4_0 ≤ 0

7 Location Addition

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

2 16 2_var_snapshot: − tmp_post + tmp_post ≤ 0 ∧ tmp_post − tmp_post ≤ 0 ∧ − tmp_0 + tmp_0 ≤ 0 ∧ tmp_0 − tmp_0 ≤ 0 ∧ − i7_post + i7_post ≤ 0 ∧ i7_post − i7_post ≤ 0 ∧ − i7_0 + i7_0 ≤ 0 ∧ i7_0 − i7_0 ≤ 0 ∧ − i4_post + i4_post ≤ 0 ∧ i4_post − i4_post ≤ 0 ∧ − i4_0 + i4_0 ≤ 0 ∧ i4_0 − i4_0 ≤ 0

8 SCC Decomposition

We consider subproblems for each of the 2 SCC(s) of the program graph.

8.1 SCC Subproblem 1/2

Here we consider the SCC { 1, 3, 1_var_snapshot, 1* }.

8.1.1 Transition Removal

We remove transition 5 using the following ranking functions, which are bounded by −30.

1:	−3⋅i7_0
3:	−2 − 3⋅i7_0
1_var_snapshot:	−1 − 3⋅i7_0
1*:	−3⋅i7_0

Hints:

9	lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0] ]
11	lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0] ]
3	lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0] ]
5	lexStrict[ [0, 0, 0, 0, 0, 3, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0] , [0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]

8.1.2 Transition Removal

We remove transitions 9, 11, 3 using the following ranking functions, which are bounded by −1.

1:	i4_0
3:	− i4_0
1_var_snapshot:	0
1*:	1 + i4_0

Hints:

9	lexStrict[ [1, 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] ]
11	lexStrict[ [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] ]
3	lexStrict[ [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] ]

8.1.3 Splitting Cut-Point Transitions

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

8.1.3.1 Cut-Point Subproblem 1/1

Here we consider cut-point transition 8.

8.1.3.1.1 Splitting Cut-Point Transitions

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

8.2 SCC Subproblem 2/2

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

8.2.1 Transition Removal

We remove transition 1 using the following ranking functions, which are bounded by −38.

0:	−1 − 4⋅i4_0
2:	1 − 4⋅i4_0
2_var_snapshot:	−4⋅i4_0
2*:	2 − 4⋅i4_0

Hints:

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

8.2.2 Transition Removal

We remove transitions 16, 18 using the following ranking functions, which are bounded by −1.

0:	−2
2:	0
2_var_snapshot:	−1
2*:	1

Hints:

16	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] ]
18	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	lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]

8.2.3 Transition Removal

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

0:	−1
2:	0
2_var_snapshot:	0
2*:	0

Hints:

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] ]

8.2.4 Splitting Cut-Point Transitions

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

8.2.4.1 Cut-Point Subproblem 1/1

Here we consider cut-point transition 15.

8.2.4.1.1 Splitting Cut-Point Transitions

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

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