LTS Termination Proof

by T2Cert

Input

Integer Transition System

Initial Location: 5

Transitions: (pre-variables and post-variables)

0	0	1:	0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ x_13_0 − x_13_post ≤ 0 ∧ − x_13_0 + x_13_post ≤ 0 ∧ y_16_0 − y_16_post ≤ 0 ∧ − y_16_0 + y_16_post ≤ 0 ∧ − y_33_post + y_33_post ≤ 0 ∧ y_33_post − y_33_post ≤ 0 ∧ − y_33_0 + y_33_0 ≤ 0 ∧ y_33_0 − y_33_0 ≤ 0 ∧ − y_28_post + y_28_post ≤ 0 ∧ y_28_post − y_28_post ≤ 0 ∧ − y_28_0 + y_28_0 ≤ 0 ∧ y_28_0 − y_28_0 ≤ 0 ∧ − x_32_post + x_32_post ≤ 0 ∧ x_32_post − x_32_post ≤ 0 ∧ − x_32_0 + x_32_0 ≤ 0 ∧ x_32_0 − x_32_0 ≤ 0 ∧ − x_27_post + x_27_post ≤ 0 ∧ x_27_post − x_27_post ≤ 0 ∧ − x_27_0 + x_27_0 ≤ 0 ∧ x_27_0 − x_27_0 ≤ 0
1	1	2:	0 ≤ 0 ∧ 0 ≤ 0 ∧ 1 − x_13_0 ≤ 0 ∧ −5000 + y_16_post ≤ 0 ∧ 5000 − y_16_post ≤ 0 ∧ 1 − y_16_post ≤ 0 ∧ 1 − x_13_0 ≤ 0 ∧ y_16_0 − y_16_post ≤ 0 ∧ − y_16_0 + y_16_post ≤ 0 ∧ − y_33_post + y_33_post ≤ 0 ∧ y_33_post − y_33_post ≤ 0 ∧ − y_33_0 + y_33_0 ≤ 0 ∧ y_33_0 − y_33_0 ≤ 0 ∧ − y_28_post + y_28_post ≤ 0 ∧ y_28_post − y_28_post ≤ 0 ∧ − y_28_0 + y_28_0 ≤ 0 ∧ y_28_0 − y_28_0 ≤ 0 ∧ − x_32_post + x_32_post ≤ 0 ∧ x_32_post − x_32_post ≤ 0 ∧ − x_32_0 + x_32_0 ≤ 0 ∧ x_32_0 − x_32_0 ≤ 0 ∧ − x_27_post + x_27_post ≤ 0 ∧ x_27_post − x_27_post ≤ 0 ∧ − x_27_0 + x_27_0 ≤ 0 ∧ x_27_0 − x_27_0 ≤ 0 ∧ − x_13_post + x_13_post ≤ 0 ∧ x_13_post − x_13_post ≤ 0 ∧ − x_13_0 + x_13_0 ≤ 0 ∧ x_13_0 − x_13_0 ≤ 0
2	3	1:	y_16_0 ≤ 0 ∧ y_16_0 ≤ 0 ∧ − y_33_post + y_33_post ≤ 0 ∧ y_33_post − y_33_post ≤ 0 ∧ − y_33_0 + y_33_0 ≤ 0 ∧ y_33_0 − y_33_0 ≤ 0 ∧ − y_28_post + y_28_post ≤ 0 ∧ y_28_post − y_28_post ≤ 0 ∧ − y_28_0 + y_28_0 ≤ 0 ∧ y_28_0 − y_28_0 ≤ 0 ∧ − y_16_post + y_16_post ≤ 0 ∧ y_16_post − y_16_post ≤ 0 ∧ − y_16_0 + y_16_0 ≤ 0 ∧ y_16_0 − y_16_0 ≤ 0 ∧ − x_32_post + x_32_post ≤ 0 ∧ x_32_post − x_32_post ≤ 0 ∧ − x_32_0 + x_32_0 ≤ 0 ∧ x_32_0 − x_32_0 ≤ 0 ∧ − x_27_post + x_27_post ≤ 0 ∧ x_27_post − x_27_post ≤ 0 ∧ − x_27_0 + x_27_0 ≤ 0 ∧ x_27_0 − x_27_0 ≤ 0 ∧ − x_13_post + x_13_post ≤ 0 ∧ x_13_post − x_13_post ≤ 0 ∧ − x_13_0 + x_13_0 ≤ 0 ∧ x_13_0 − x_13_0 ≤ 0
2	4	4:	0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 1 − y_16_0 ≤ 0 ∧ 1 − x_13_0 + x_13_post ≤ 0 ∧ −1 + x_13_0 − x_13_post ≤ 0 ∧ 1 − y_16_0 + y_16_post ≤ 0 ∧ −1 + y_16_0 − y_16_post ≤ 0 ∧ 1 + x_13_post − x_32_post ≤ 0 ∧ −1 − x_13_post + x_32_post ≤ 0 ∧ 1 + y_16_post − y_33_post ≤ 0 ∧ −1 − y_16_post + y_33_post ≤ 0 ∧ 1 − y_33_post ≤ 0 ∧ x_13_0 − x_13_post ≤ 0 ∧ − x_13_0 + x_13_post ≤ 0 ∧ x_32_0 − x_32_post ≤ 0 ∧ − x_32_0 + x_32_post ≤ 0 ∧ y_16_0 − y_16_post ≤ 0 ∧ − y_16_0 + y_16_post ≤ 0 ∧ y_33_0 − y_33_post ≤ 0 ∧ − y_33_0 + y_33_post ≤ 0 ∧ − y_28_post + y_28_post ≤ 0 ∧ y_28_post − y_28_post ≤ 0 ∧ − y_28_0 + y_28_0 ≤ 0 ∧ y_28_0 − y_28_0 ≤ 0 ∧ − x_27_post + x_27_post ≤ 0 ∧ x_27_post − x_27_post ≤ 0 ∧ − x_27_0 + x_27_0 ≤ 0 ∧ x_27_0 − x_27_0 ≤ 0
4	5	2:	− y_33_post + y_33_post ≤ 0 ∧ y_33_post − y_33_post ≤ 0 ∧ − y_33_0 + y_33_0 ≤ 0 ∧ y_33_0 − y_33_0 ≤ 0 ∧ − y_28_post + y_28_post ≤ 0 ∧ y_28_post − y_28_post ≤ 0 ∧ − y_28_0 + y_28_0 ≤ 0 ∧ y_28_0 − y_28_0 ≤ 0 ∧ − y_16_post + y_16_post ≤ 0 ∧ y_16_post − y_16_post ≤ 0 ∧ − y_16_0 + y_16_0 ≤ 0 ∧ y_16_0 − y_16_0 ≤ 0 ∧ − x_32_post + x_32_post ≤ 0 ∧ x_32_post − x_32_post ≤ 0 ∧ − x_32_0 + x_32_0 ≤ 0 ∧ x_32_0 − x_32_0 ≤ 0 ∧ − x_27_post + x_27_post ≤ 0 ∧ x_27_post − x_27_post ≤ 0 ∧ − x_27_0 + x_27_0 ≤ 0 ∧ x_27_0 − x_27_0 ≤ 0 ∧ − x_13_post + x_13_post ≤ 0 ∧ x_13_post − x_13_post ≤ 0 ∧ − x_13_0 + x_13_0 ≤ 0 ∧ x_13_0 − x_13_0 ≤ 0
5	6	0:	− y_33_post + y_33_post ≤ 0 ∧ y_33_post − y_33_post ≤ 0 ∧ − y_33_0 + y_33_0 ≤ 0 ∧ y_33_0 − y_33_0 ≤ 0 ∧ − y_28_post + y_28_post ≤ 0 ∧ y_28_post − y_28_post ≤ 0 ∧ − y_28_0 + y_28_0 ≤ 0 ∧ y_28_0 − y_28_0 ≤ 0 ∧ − y_16_post + y_16_post ≤ 0 ∧ y_16_post − y_16_post ≤ 0 ∧ − y_16_0 + y_16_0 ≤ 0 ∧ y_16_0 − y_16_0 ≤ 0 ∧ − x_32_post + x_32_post ≤ 0 ∧ x_32_post − x_32_post ≤ 0 ∧ − x_32_0 + x_32_0 ≤ 0 ∧ x_32_0 − x_32_0 ≤ 0 ∧ − x_27_post + x_27_post ≤ 0 ∧ x_27_post − x_27_post ≤ 0 ∧ − x_27_0 + x_27_0 ≤ 0 ∧ x_27_0 − x_27_0 ≤ 0 ∧ − x_13_post + x_13_post ≤ 0 ∧ x_13_post − x_13_post ≤ 0 ∧ − x_13_0 + x_13_0 ≤ 0 ∧ x_13_0 − x_13_0 ≤ 0

Proof

1 Invariant Updates

The following invariants are asserted.

0:	TRUE
1:	TRUE
2:	TRUE
4:	1 − y_33_post ≤ 0 ∧ 1 − y_33_0 ≤ 0
5:	TRUE

The invariants are proved as follows.

IMPACT Invariant Proof

nodes (location) invariant:

0	(0)	TRUE
1	(1)	TRUE
2	(2)	TRUE
4	(4)	1 − y_33_post ≤ 0 ∧ 1 − y_33_0 ≤ 0
5	(5)	TRUE

initial node: 5
cover edges:
transition edges:

0 0 1

1 1 2

2 3 1

2 4 4

4 5 2

5 6 0

2 Switch to Cooperation Termination Proof

We consider the following cutpoint-transitions:

1	7	1:	− y_33_post + y_33_post ≤ 0 ∧ y_33_post − y_33_post ≤ 0 ∧ − y_33_0 + y_33_0 ≤ 0 ∧ y_33_0 − y_33_0 ≤ 0 ∧ − y_28_post + y_28_post ≤ 0 ∧ y_28_post − y_28_post ≤ 0 ∧ − y_28_0 + y_28_0 ≤ 0 ∧ y_28_0 − y_28_0 ≤ 0 ∧ − y_16_post + y_16_post ≤ 0 ∧ y_16_post − y_16_post ≤ 0 ∧ − y_16_0 + y_16_0 ≤ 0 ∧ y_16_0 − y_16_0 ≤ 0 ∧ − x_32_post + x_32_post ≤ 0 ∧ x_32_post − x_32_post ≤ 0 ∧ − x_32_0 + x_32_0 ≤ 0 ∧ x_32_0 − x_32_0 ≤ 0 ∧ − x_27_post + x_27_post ≤ 0 ∧ x_27_post − x_27_post ≤ 0 ∧ − x_27_0 + x_27_0 ≤ 0 ∧ x_27_0 − x_27_0 ≤ 0 ∧ − x_13_post + x_13_post ≤ 0 ∧ x_13_post − x_13_post ≤ 0 ∧ − x_13_0 + x_13_0 ≤ 0 ∧ x_13_0 − x_13_0 ≤ 0
2	14	2:	− y_33_post + y_33_post ≤ 0 ∧ y_33_post − y_33_post ≤ 0 ∧ − y_33_0 + y_33_0 ≤ 0 ∧ y_33_0 − y_33_0 ≤ 0 ∧ − y_28_post + y_28_post ≤ 0 ∧ y_28_post − y_28_post ≤ 0 ∧ − y_28_0 + y_28_0 ≤ 0 ∧ y_28_0 − y_28_0 ≤ 0 ∧ − y_16_post + y_16_post ≤ 0 ∧ y_16_post − y_16_post ≤ 0 ∧ − y_16_0 + y_16_0 ≤ 0 ∧ y_16_0 − y_16_0 ≤ 0 ∧ − x_32_post + x_32_post ≤ 0 ∧ x_32_post − x_32_post ≤ 0 ∧ − x_32_0 + x_32_0 ≤ 0 ∧ x_32_0 − x_32_0 ≤ 0 ∧ − x_27_post + x_27_post ≤ 0 ∧ x_27_post − x_27_post ≤ 0 ∧ − x_27_0 + x_27_0 ≤ 0 ∧ x_27_0 − x_27_0 ≤ 0 ∧ − x_13_post + x_13_post ≤ 0 ∧ x_13_post − x_13_post ≤ 0 ∧ − x_13_0 + x_13_0 ≤ 0 ∧ x_13_0 − x_13_0 ≤ 0

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

3 Transition Removal

We remove transitions 0, 6 using the following ranking functions, which are bounded by −13.

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

4 Location Addition

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

1* 10 1: − y_33_post + y_33_post ≤ 0 ∧ y_33_post − y_33_post ≤ 0 ∧ − y_33_0 + y_33_0 ≤ 0 ∧ y_33_0 − y_33_0 ≤ 0 ∧ − y_28_post + y_28_post ≤ 0 ∧ y_28_post − y_28_post ≤ 0 ∧ − y_28_0 + y_28_0 ≤ 0 ∧ y_28_0 − y_28_0 ≤ 0 ∧ − y_16_post + y_16_post ≤ 0 ∧ y_16_post − y_16_post ≤ 0 ∧ − y_16_0 + y_16_0 ≤ 0 ∧ y_16_0 − y_16_0 ≤ 0 ∧ − x_32_post + x_32_post ≤ 0 ∧ x_32_post − x_32_post ≤ 0 ∧ − x_32_0 + x_32_0 ≤ 0 ∧ x_32_0 − x_32_0 ≤ 0 ∧ − x_27_post + x_27_post ≤ 0 ∧ x_27_post − x_27_post ≤ 0 ∧ − x_27_0 + x_27_0 ≤ 0 ∧ x_27_0 − x_27_0 ≤ 0 ∧ − x_13_post + x_13_post ≤ 0 ∧ x_13_post − x_13_post ≤ 0 ∧ − x_13_0 + x_13_0 ≤ 0 ∧ x_13_0 − x_13_0 ≤ 0

5 Location Addition

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

1 8 1_var_snapshot: − y_33_post + y_33_post ≤ 0 ∧ y_33_post − y_33_post ≤ 0 ∧ − y_33_0 + y_33_0 ≤ 0 ∧ y_33_0 − y_33_0 ≤ 0 ∧ − y_28_post + y_28_post ≤ 0 ∧ y_28_post − y_28_post ≤ 0 ∧ − y_28_0 + y_28_0 ≤ 0 ∧ y_28_0 − y_28_0 ≤ 0 ∧ − y_16_post + y_16_post ≤ 0 ∧ y_16_post − y_16_post ≤ 0 ∧ − y_16_0 + y_16_0 ≤ 0 ∧ y_16_0 − y_16_0 ≤ 0 ∧ − x_32_post + x_32_post ≤ 0 ∧ x_32_post − x_32_post ≤ 0 ∧ − x_32_0 + x_32_0 ≤ 0 ∧ x_32_0 − x_32_0 ≤ 0 ∧ − x_27_post + x_27_post ≤ 0 ∧ x_27_post − x_27_post ≤ 0 ∧ − x_27_0 + x_27_0 ≤ 0 ∧ x_27_0 − x_27_0 ≤ 0 ∧ − x_13_post + x_13_post ≤ 0 ∧ x_13_post − x_13_post ≤ 0 ∧ − x_13_0 + x_13_0 ≤ 0 ∧ x_13_0 − x_13_0 ≤ 0

6 Location Addition

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

2* 17 2: − y_33_post + y_33_post ≤ 0 ∧ y_33_post − y_33_post ≤ 0 ∧ − y_33_0 + y_33_0 ≤ 0 ∧ y_33_0 − y_33_0 ≤ 0 ∧ − y_28_post + y_28_post ≤ 0 ∧ y_28_post − y_28_post ≤ 0 ∧ − y_28_0 + y_28_0 ≤ 0 ∧ y_28_0 − y_28_0 ≤ 0 ∧ − y_16_post + y_16_post ≤ 0 ∧ y_16_post − y_16_post ≤ 0 ∧ − y_16_0 + y_16_0 ≤ 0 ∧ y_16_0 − y_16_0 ≤ 0 ∧ − x_32_post + x_32_post ≤ 0 ∧ x_32_post − x_32_post ≤ 0 ∧ − x_32_0 + x_32_0 ≤ 0 ∧ x_32_0 − x_32_0 ≤ 0 ∧ − x_27_post + x_27_post ≤ 0 ∧ x_27_post − x_27_post ≤ 0 ∧ − x_27_0 + x_27_0 ≤ 0 ∧ x_27_0 − x_27_0 ≤ 0 ∧ − x_13_post + x_13_post ≤ 0 ∧ x_13_post − x_13_post ≤ 0 ∧ − x_13_0 + x_13_0 ≤ 0 ∧ x_13_0 − x_13_0 ≤ 0

7 Location Addition

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

2 15 2_var_snapshot: − y_33_post + y_33_post ≤ 0 ∧ y_33_post − y_33_post ≤ 0 ∧ − y_33_0 + y_33_0 ≤ 0 ∧ y_33_0 − y_33_0 ≤ 0 ∧ − y_28_post + y_28_post ≤ 0 ∧ y_28_post − y_28_post ≤ 0 ∧ − y_28_0 + y_28_0 ≤ 0 ∧ y_28_0 − y_28_0 ≤ 0 ∧ − y_16_post + y_16_post ≤ 0 ∧ y_16_post − y_16_post ≤ 0 ∧ − y_16_0 + y_16_0 ≤ 0 ∧ y_16_0 − y_16_0 ≤ 0 ∧ − x_32_post + x_32_post ≤ 0 ∧ x_32_post − x_32_post ≤ 0 ∧ − x_32_0 + x_32_0 ≤ 0 ∧ x_32_0 − x_32_0 ≤ 0 ∧ − x_27_post + x_27_post ≤ 0 ∧ x_27_post − x_27_post ≤ 0 ∧ − x_27_0 + x_27_0 ≤ 0 ∧ x_27_0 − x_27_0 ≤ 0 ∧ − x_13_post + x_13_post ≤ 0 ∧ x_13_post − x_13_post ≤ 0 ∧ − x_13_0 + x_13_0 ≤ 0 ∧ x_13_0 − x_13_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 { 1, 2, 4, 1_var_snapshot, 1*, 2_var_snapshot, 2* }.

8.1.1 Transition Removal

We remove transition 1 using the following ranking functions, which are bounded by −9995.

1:	−20000 + 10006⋅x_13_0
2:	−5000 + 10006⋅x_13_0 − 5⋅y_16_0
4:	1 + 10006⋅x_13_0 − 5⋅y_16_0
1_var_snapshot:	−20000 + 10006⋅x_13_0
1*:	−15000 + 10006⋅x_13_0
2_var_snapshot:	−10000 + 10006⋅x_13_0 − 5⋅y_16_0
2*:	10006⋅x_13_0 − 5⋅y_16_0

8.1.2 Transition Removal

We remove transition 4 using the following ranking functions, which are bounded by 1.

1:	−4 + 4⋅y_16_0
2:	−1 + 4⋅y_16_0
4:	1 + 4⋅y_16_0
1_var_snapshot:	−5 + 4⋅y_16_0
1*:	−3 + 4⋅y_16_0
2_var_snapshot:	−2 + 4⋅y_16_0
2*:	4⋅y_16_0

8.1.3 Transition Removal

We remove transitions 8, 10, 15, 17, 3, 5 using the following ranking functions, which are bounded by −3.

1:	−2
2:	1
4:	3⋅y_33_0
1_var_snapshot:	−3
1*:	−1
2_var_snapshot:	0
2*:	2

8.1.4 Splitting Cut-Point Transitions

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

8.1.4.1 Cut-Point Subproblem 1/2

Here we consider cut-point transition 7.

8.1.4.1.1 Splitting Cut-Point Transitions

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

8.1.4.2 Cut-Point Subproblem 2/2

Here we consider cut-point transition 14.

8.1.4.2.1 Splitting Cut-Point Transitions

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

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