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 ∧ − j_post + j_post ≤ 0 ∧ j_post − j_post ≤ 0 ∧ − j_0 + j_0 ≤ 0 ∧ j_0 − j_0 ≤ 0
0	1	1:	0 ≤ 0 ∧ 0 ≤ 0 ∧ − j_post + j_post ≤ 0 ∧ j_post − j_post ≤ 0 ∧ − j_0 + j_0 ≤ 0 ∧ j_0 − j_0 ≤ 0
2	2	0:	2 − j_0 ≤ 0 ∧ − j_post + j_post ≤ 0 ∧ j_post − j_post ≤ 0 ∧ − j_0 + j_0 ≤ 0 ∧ j_0 − j_0 ≤ 0
2	3	3:	0 ≤ 0 ∧ 0 ≤ 0 ∧ −1 + j_0 ≤ 0 ∧ −1 − j_0 + j_post ≤ 0 ∧ 1 + j_0 − j_post ≤ 0 ∧ j_0 − j_post ≤ 0 ∧ − j_0 + j_post ≤ 0
3	4	2:	0 ≤ 0 ∧ 0 ≤ 0 ∧ − j_post + j_post ≤ 0 ∧ j_post − j_post ≤ 0 ∧ − j_0 + j_0 ≤ 0 ∧ j_0 − j_0 ≤ 0
1	5	4:	0 ≤ 0 ∧ 0 ≤ 0 ∧ − j_post + j_post ≤ 0 ∧ j_post − j_post ≤ 0 ∧ − j_0 + j_0 ≤ 0 ∧ j_0 − j_0 ≤ 0
5	6	3:	0 ≤ 0 ∧ 0 ≤ 0 ∧ j_post ≤ 0 ∧ − j_post ≤ 0 ∧ j_0 − j_post ≤ 0 ∧ − j_0 + j_post ≤ 0
6	7	5:	0 ≤ 0 ∧ 0 ≤ 0 ∧ − j_post + j_post ≤ 0 ∧ j_post − j_post ≤ 0 ∧ − j_0 + j_0 ≤ 0 ∧ j_0 − j_0 ≤ 0

Proof

1 Invariant Updates

The following invariants are asserted.

0:	2 − j_0 ≤ 0
1:	2 − j_0 ≤ 0
2:	TRUE
3:	TRUE
4:	2 − j_0 ≤ 0
5:	TRUE
6:	TRUE

The invariants are proved as follows.

IMPACT Invariant Proof

nodes (location) invariant:

0	(0)	2 − j_0 ≤ 0
1	(1)	2 − j_0 ≤ 0
2	(2)	TRUE
3	(3)	TRUE
4	(4)	2 − j_0 ≤ 0
5	(5)	TRUE
6	(6)	TRUE

initial node: 6
cover edges:
transition edges:

0 0 1

0 1 1

1 5 4

2 2 0

2 3 3

3 4 2

5 6 3

6 7 5

2 Switch to Cooperation Termination Proof

We consider the following cutpoint-transitions:

− j_post + j_post ≤ 0 ∧ j_post − j_post ≤ 0 ∧ − j_0 + j_0 ≤ 0 ∧ j_0 − j_0 ≤ 0

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

3 Transition Removal

We remove transitions 0, 1, 2, 5, 6, 7 using the following ranking functions, which are bounded by −17.

6:	0
5:	0
2:	0
3:	0
0:	0
1:	0
4:	0
6:	−7
5:	−8
2:	−9
3:	−9
3_var_snapshot:	−9
3*:	−9
0:	−10
1:	−11
4:	−12

4 Location Addition

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

3* 11 3: − j_post + j_post ≤ 0 ∧ j_post − j_post ≤ 0 ∧ − j_0 + j_0 ≤ 0 ∧ j_0 − j_0 ≤ 0

5 Location Addition

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

3 9 3_var_snapshot: − j_post + j_post ≤ 0 ∧ j_post − j_post ≤ 0 ∧ − j_0 + j_0 ≤ 0 ∧ j_0 − j_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 { 2, 3, 3_var_snapshot, 3* }.

6.1.1 Transition Removal

We remove transition 3 using the following ranking functions, which are bounded by −6.

2:	−1 − 4⋅j_0
3:	1 − 4⋅j_0
3_var_snapshot:	−4⋅j_0
3*:	2 − 4⋅j_0

6.1.2 Transition Removal

We remove transitions 11, 4 using the following ranking functions, which are bounded by −1.

2:	−1
3:	1
3_var_snapshot:	0
3*:	2

6.1.3 Transition Removal

We remove transition 9 using the following ranking functions, which are bounded by 0.

2:	0
3:	1
3_var_snapshot:	0
3*:	0

6.1.4 Splitting Cut-Point Transitions

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

6.1.4.1 Cut-Point Subproblem 1/1

Here we consider cut-point transition 8.

6.1.4.1.1 Splitting Cut-Point Transitions

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

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