LTS Termination Proof

by T2Cert

Input

Integer Transition System

Initial Location: 8

Transitions: (pre-variables and post-variables)

0	0	1:	3 − i_0 ≤ 0 ∧ − i_post + i_post ≤ 0 ∧ i_post − i_post ≤ 0 ∧ − i_0 + i_0 ≤ 0 ∧ i_0 − i_0 ≤ 0
0	1	2:	0 ≤ 0 ∧ 0 ≤ 0 ∧ −2 + i_0 ≤ 0 ∧ −1 − i_0 + i_post ≤ 0 ∧ 1 + i_0 − i_post ≤ 0 ∧ i_0 − i_post ≤ 0 ∧ − i_0 + i_post ≤ 0
3	2	0:	0 ≤ 0 ∧ 0 ≤ 0 ∧ − i_post + i_post ≤ 0 ∧ i_post − i_post ≤ 0 ∧ − i_0 + i_0 ≤ 0 ∧ i_0 − i_0 ≤ 0
3	3	0:	0 ≤ 0 ∧ 0 ≤ 0 ∧ − i_post + i_post ≤ 0 ∧ i_post − i_post ≤ 0 ∧ − i_0 + i_0 ≤ 0 ∧ i_0 − i_0 ≤ 0
3	4	1:	0 ≤ 0 ∧ 0 ≤ 0 ∧ − i_post + i_post ≤ 0 ∧ i_post − i_post ≤ 0 ∧ − i_0 + i_0 ≤ 0 ∧ i_0 − i_0 ≤ 0
2	5	3:	0 ≤ 0 ∧ 0 ≤ 0 ∧ − i_post + i_post ≤ 0 ∧ i_post − i_post ≤ 0 ∧ − i_0 + i_0 ≤ 0 ∧ i_0 − i_0 ≤ 0
4	6	5:	0 ≤ 0 ∧ 0 ≤ 0 ∧ − i_post + i_post ≤ 0 ∧ i_post − i_post ≤ 0 ∧ − i_0 + i_0 ≤ 0 ∧ i_0 − i_0 ≤ 0
1	7	6:	0 ≤ 0 ∧ 0 ≤ 0 ∧ − i_post + i_post ≤ 0 ∧ i_post − i_post ≤ 0 ∧ − i_0 + i_0 ≤ 0 ∧ i_0 − i_0 ≤ 0
6	8	4:	−1 + i_0 ≤ 0 ∧ − i_post + i_post ≤ 0 ∧ i_post − i_post ≤ 0 ∧ − i_0 + i_0 ≤ 0 ∧ i_0 − i_0 ≤ 0
6	9	4:	2 − i_0 ≤ 0 ∧ − i_post + i_post ≤ 0 ∧ i_post − i_post ≤ 0 ∧ − i_0 + i_0 ≤ 0 ∧ i_0 − i_0 ≤ 0
7	10	2:	0 ≤ 0 ∧ 0 ≤ 0 ∧ i_post ≤ 0 ∧ − i_post ≤ 0 ∧ i_0 − i_post ≤ 0 ∧ − i_0 + i_post ≤ 0
8	11	7:	0 ≤ 0 ∧ 0 ≤ 0 ∧ − i_post + i_post ≤ 0 ∧ i_post − i_post ≤ 0 ∧ − i_0 + i_0 ≤ 0 ∧ i_0 − i_0 ≤ 0

Proof

1 Switch to Cooperation Termination Proof

We consider the following cutpoint-transitions:

− i_post + i_post ≤ 0 ∧ i_post − i_post ≤ 0 ∧ − i_0 + i_0 ≤ 0 ∧ i_0 − i_0 ≤ 0

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

2 Transition Removal

We remove transitions 0, 4, 6, 7, 8, 9, 10, 11 using the following ranking functions, which are bounded by −19.

8:	0
7:	0
0:	0
2:	0
3:	0
1:	0
6:	0
4:	0
5:	0
8:	−8
7:	−9
0:	−10
2:	−10
3:	−10
2_var_snapshot:	−10
2*:	−10
1:	−11
6:	−12
4:	−13
5:	−14

Hints:

13	lexWeak[ [0, 0, 0, 0] ]
1	lexWeak[ [0, 0, 0, 0, 0, 0, 0] ]
2	lexWeak[ [0, 0, 0, 0, 0, 0] ]
3	lexWeak[ [0, 0, 0, 0, 0, 0] ]
5	lexWeak[ [0, 0, 0, 0, 0, 0] ]
0	lexStrict[ [0, 0, 0, 0, 0] , [0, 0, 0, 0, 0] ]
4	lexStrict[ [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] ]
7	lexStrict[ [0, 0, 0, 0, 0, 0] , [0, 0, 0, 0, 0, 0] ]
8	lexStrict[ [0, 0, 0, 0, 0] , [0, 0, 0, 0, 0] ]
9	lexStrict[ [0, 0, 0, 0, 0] , [0, 0, 0, 0, 0] ]
10	lexStrict[ [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, 0] ]

3 Location Addition

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

2* 15 2: − i_post + i_post ≤ 0 ∧ i_post − i_post ≤ 0 ∧ − i_0 + i_0 ≤ 0 ∧ i_0 − i_0 ≤ 0

4 Location Addition

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

2 13 2_var_snapshot: − i_post + i_post ≤ 0 ∧ i_post − i_post ≤ 0 ∧ − i_0 + i_0 ≤ 0 ∧ i_0 − i_0 ≤ 0

5 SCC Decomposition

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

5.1 SCC Subproblem 1/1

Here we consider the SCC { 0, 2, 3, 2_var_snapshot, 2* }.

5.1.1 Transition Removal

We remove transition 1 using the following ranking functions, which are bounded by −13.

0:	−2 − 5⋅i_0
2:	1 − 5⋅i_0
3:	−1 − 5⋅i_0
2_var_snapshot:	−5⋅i_0
2*:	2 − 5⋅i_0

Hints:

13	lexWeak[ [0, 0, 0, 5] ]
15	lexWeak[ [0, 0, 0, 5] ]
1	lexStrict[ [0, 0, 0, 0, 5, 0, 5] , [0, 0, 5, 0, 0, 0, 0] ]
2	lexWeak[ [0, 0, 0, 0, 0, 5] ]
3	lexWeak[ [0, 0, 0, 0, 0, 5] ]
5	lexWeak[ [0, 0, 0, 0, 0, 5] ]

5.1.2 Transition Removal

We remove transitions 13, 15, 3, 5 using the following ranking functions, which are bounded by −2.

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

Hints:

13	lexStrict[ [0, 0, 0, 0] , [0, 0, 0, 0] ]
15	lexStrict[ [0, 0, 0, 0] , [0, 0, 0, 0] ]
2	lexWeak[ [0, 0, 0, 0, 0, 0] ]
3	lexStrict[ [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] ]

5.1.3 Transition Removal

We remove transition 2 using the following ranking functions, which are bounded by 0.

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

Hints:

2	lexStrict[ [0, 0, 0, 0, 0, 0] , [0, 0, 0, 0, 0, 0] ]

5.1.4 Splitting Cut-Point Transitions

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

5.1.4.1 Cut-Point Subproblem 1/1

Here we consider cut-point transition 12.

5.1.4.1.1 Splitting Cut-Point Transitions

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

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