LTS Termination Proof

by T2Cert

Input

Integer Transition System

Initial Location: 4

Transitions: (pre-variables and post-variables)

0	0	1:	0 ≤ 0 ∧ 0 ≤ 0 ∧ y_13_0 ≤ 0 ∧ rt_11_post − st_14_0 ≤ 0 ∧ − rt_11_post + st_14_0 ≤ 0 ∧ rt_11_0 − rt_11_post ≤ 0 ∧ − rt_11_0 + rt_11_post ≤ 0 ∧ − y_13_post + y_13_post ≤ 0 ∧ y_13_post − y_13_post ≤ 0 ∧ − y_13_0 + y_13_0 ≤ 0 ∧ y_13_0 − y_13_0 ≤ 0 ∧ − x_15_post + x_15_post ≤ 0 ∧ x_15_post − x_15_post ≤ 0 ∧ − x_15_0 + x_15_0 ≤ 0 ∧ x_15_0 − x_15_0 ≤ 0 ∧ − st_14_0 + st_14_0 ≤ 0 ∧ st_14_0 − st_14_0 ≤ 0
0	1	2:	1 − y_13_0 ≤ 0 ∧ − y_13_post + y_13_post ≤ 0 ∧ y_13_post − y_13_post ≤ 0 ∧ − y_13_0 + y_13_0 ≤ 0 ∧ y_13_0 − y_13_0 ≤ 0 ∧ − x_15_post + x_15_post ≤ 0 ∧ x_15_post − x_15_post ≤ 0 ∧ − x_15_0 + x_15_0 ≤ 0 ∧ x_15_0 − x_15_0 ≤ 0 ∧ − st_14_0 + st_14_0 ≤ 0 ∧ st_14_0 − st_14_0 ≤ 0 ∧ − rt_11_post + rt_11_post ≤ 0 ∧ rt_11_post − rt_11_post ≤ 0 ∧ − rt_11_0 + rt_11_0 ≤ 0 ∧ rt_11_0 − rt_11_0 ≤ 0
2	2	1:	0 ≤ 0 ∧ 0 ≤ 0 ∧ x_15_0 ≤ 0 ∧ rt_11_post − st_14_0 ≤ 0 ∧ − rt_11_post + st_14_0 ≤ 0 ∧ rt_11_0 − rt_11_post ≤ 0 ∧ − rt_11_0 + rt_11_post ≤ 0 ∧ − y_13_post + y_13_post ≤ 0 ∧ y_13_post − y_13_post ≤ 0 ∧ − y_13_0 + y_13_0 ≤ 0 ∧ y_13_0 − y_13_0 ≤ 0 ∧ − x_15_post + x_15_post ≤ 0 ∧ x_15_post − x_15_post ≤ 0 ∧ − x_15_0 + x_15_0 ≤ 0 ∧ x_15_0 − x_15_0 ≤ 0 ∧ − st_14_0 + st_14_0 ≤ 0 ∧ st_14_0 − st_14_0 ≤ 0
2	3	3:	0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 1 − x_15_0 ≤ 0 ∧ − x_15_0 + x_15_post + y_13_0 ≤ 0 ∧ x_15_0 − x_15_post − y_13_0 ≤ 0 ∧ −1 − y_13_0 + y_13_post ≤ 0 ∧ 1 + y_13_0 − y_13_post ≤ 0 ∧ x_15_0 − x_15_post ≤ 0 ∧ − x_15_0 + x_15_post ≤ 0 ∧ y_13_0 − y_13_post ≤ 0 ∧ − y_13_0 + y_13_post ≤ 0 ∧ − st_14_0 + st_14_0 ≤ 0 ∧ st_14_0 − st_14_0 ≤ 0 ∧ − rt_11_post + rt_11_post ≤ 0 ∧ rt_11_post − rt_11_post ≤ 0 ∧ − rt_11_0 + rt_11_0 ≤ 0 ∧ rt_11_0 − rt_11_0 ≤ 0
3	4	2:	− y_13_post + y_13_post ≤ 0 ∧ y_13_post − y_13_post ≤ 0 ∧ − y_13_0 + y_13_0 ≤ 0 ∧ y_13_0 − y_13_0 ≤ 0 ∧ − x_15_post + x_15_post ≤ 0 ∧ x_15_post − x_15_post ≤ 0 ∧ − x_15_0 + x_15_0 ≤ 0 ∧ x_15_0 − x_15_0 ≤ 0 ∧ − st_14_0 + st_14_0 ≤ 0 ∧ st_14_0 − st_14_0 ≤ 0 ∧ − rt_11_post + rt_11_post ≤ 0 ∧ rt_11_post − rt_11_post ≤ 0 ∧ − rt_11_0 + rt_11_0 ≤ 0 ∧ rt_11_0 − rt_11_0 ≤ 0
4	5	0:	− y_13_post + y_13_post ≤ 0 ∧ y_13_post − y_13_post ≤ 0 ∧ − y_13_0 + y_13_0 ≤ 0 ∧ y_13_0 − y_13_0 ≤ 0 ∧ − x_15_post + x_15_post ≤ 0 ∧ x_15_post − x_15_post ≤ 0 ∧ − x_15_0 + x_15_0 ≤ 0 ∧ x_15_0 − x_15_0 ≤ 0 ∧ − st_14_0 + st_14_0 ≤ 0 ∧ st_14_0 − st_14_0 ≤ 0 ∧ − rt_11_post + rt_11_post ≤ 0 ∧ rt_11_post − rt_11_post ≤ 0 ∧ − rt_11_0 + rt_11_0 ≤ 0 ∧ rt_11_0 − rt_11_0 ≤ 0

Proof

1 Switch to Cooperation Termination Proof

We consider the following cutpoint-transitions:

− y_13_post + y_13_post ≤ 0 ∧ y_13_post − y_13_post ≤ 0 ∧ − y_13_0 + y_13_0 ≤ 0 ∧ y_13_0 − y_13_0 ≤ 0 ∧ − x_15_post + x_15_post ≤ 0 ∧ x_15_post − x_15_post ≤ 0 ∧ − x_15_0 + x_15_0 ≤ 0 ∧ x_15_0 − x_15_0 ≤ 0 ∧ − st_14_0 + st_14_0 ≤ 0 ∧ st_14_0 − st_14_0 ≤ 0 ∧ − rt_11_post + rt_11_post ≤ 0 ∧ rt_11_post − rt_11_post ≤ 0 ∧ − rt_11_0 + rt_11_0 ≤ 0 ∧ rt_11_0 − rt_11_0 ≤ 0

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

2 Transition Removal

We remove transitions 0, 1, 2, 5 using the following ranking functions, which are bounded by −13.

4:	0
0:	0
2:	0
3:	0
1:	0
4:	−5
0:	−6
2:	−7
3:	−7
2_var_snapshot:	−7
2*:	−7
1:	−11

3 Location Addition

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

2* 9 2: − y_13_post + y_13_post ≤ 0 ∧ y_13_post − y_13_post ≤ 0 ∧ − y_13_0 + y_13_0 ≤ 0 ∧ y_13_0 − y_13_0 ≤ 0 ∧ − x_15_post + x_15_post ≤ 0 ∧ x_15_post − x_15_post ≤ 0 ∧ − x_15_0 + x_15_0 ≤ 0 ∧ x_15_0 − x_15_0 ≤ 0 ∧ − st_14_0 + st_14_0 ≤ 0 ∧ st_14_0 − st_14_0 ≤ 0 ∧ − rt_11_post + rt_11_post ≤ 0 ∧ rt_11_post − rt_11_post ≤ 0 ∧ − rt_11_0 + rt_11_0 ≤ 0 ∧ rt_11_0 − rt_11_0 ≤ 0

4 Location Addition

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

2 7 2_var_snapshot: − y_13_post + y_13_post ≤ 0 ∧ y_13_post − y_13_post ≤ 0 ∧ − y_13_0 + y_13_0 ≤ 0 ∧ y_13_0 − y_13_0 ≤ 0 ∧ − x_15_post + x_15_post ≤ 0 ∧ x_15_post − x_15_post ≤ 0 ∧ − x_15_0 + x_15_0 ≤ 0 ∧ x_15_0 − x_15_0 ≤ 0 ∧ − st_14_0 + st_14_0 ≤ 0 ∧ st_14_0 − st_14_0 ≤ 0 ∧ − rt_11_post + rt_11_post ≤ 0 ∧ rt_11_post − rt_11_post ≤ 0 ∧ − rt_11_0 + rt_11_0 ≤ 0 ∧ rt_11_0 − rt_11_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 { 2, 3, 2_var_snapshot, 2* }.

5.1.1 Splitting Cut-Point Transitions

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

5.1.1.1 Cut-Point Subproblem 1/1

Here we consider cut-point transition 6.

5.1.1.1.1 Fresh Variable Addition

The new variable __snapshot_2_y_13_post is introduced. The transition formulas are extended as follows:

7:	__snapshot_2_y_13_post ≤ y_13_post ∧ y_13_post ≤ __snapshot_2_y_13_post
9:	__snapshot_2_y_13_post ≤ __snapshot_2_y_13_post ∧ __snapshot_2_y_13_post ≤ __snapshot_2_y_13_post
3:	__snapshot_2_y_13_post ≤ __snapshot_2_y_13_post ∧ __snapshot_2_y_13_post ≤ __snapshot_2_y_13_post
4:	__snapshot_2_y_13_post ≤ __snapshot_2_y_13_post ∧ __snapshot_2_y_13_post ≤ __snapshot_2_y_13_post

5.1.1.1.2 Fresh Variable Addition

The new variable __snapshot_2_y_13_0 is introduced. The transition formulas are extended as follows:

7:	__snapshot_2_y_13_0 ≤ y_13_0 ∧ y_13_0 ≤ __snapshot_2_y_13_0
9:	__snapshot_2_y_13_0 ≤ __snapshot_2_y_13_0 ∧ __snapshot_2_y_13_0 ≤ __snapshot_2_y_13_0
3:	__snapshot_2_y_13_0 ≤ __snapshot_2_y_13_0 ∧ __snapshot_2_y_13_0 ≤ __snapshot_2_y_13_0
4:	__snapshot_2_y_13_0 ≤ __snapshot_2_y_13_0 ∧ __snapshot_2_y_13_0 ≤ __snapshot_2_y_13_0

5.1.1.1.3 Fresh Variable Addition

The new variable __snapshot_2_x_15_post is introduced. The transition formulas are extended as follows:

7:	__snapshot_2_x_15_post ≤ x_15_post ∧ x_15_post ≤ __snapshot_2_x_15_post
9:	__snapshot_2_x_15_post ≤ __snapshot_2_x_15_post ∧ __snapshot_2_x_15_post ≤ __snapshot_2_x_15_post
3:	__snapshot_2_x_15_post ≤ __snapshot_2_x_15_post ∧ __snapshot_2_x_15_post ≤ __snapshot_2_x_15_post
4:	__snapshot_2_x_15_post ≤ __snapshot_2_x_15_post ∧ __snapshot_2_x_15_post ≤ __snapshot_2_x_15_post

5.1.1.1.4 Fresh Variable Addition

The new variable __snapshot_2_x_15_0 is introduced. The transition formulas are extended as follows:

7:	__snapshot_2_x_15_0 ≤ x_15_0 ∧ x_15_0 ≤ __snapshot_2_x_15_0
9:	__snapshot_2_x_15_0 ≤ __snapshot_2_x_15_0 ∧ __snapshot_2_x_15_0 ≤ __snapshot_2_x_15_0
3:	__snapshot_2_x_15_0 ≤ __snapshot_2_x_15_0 ∧ __snapshot_2_x_15_0 ≤ __snapshot_2_x_15_0
4:	__snapshot_2_x_15_0 ≤ __snapshot_2_x_15_0 ∧ __snapshot_2_x_15_0 ≤ __snapshot_2_x_15_0

5.1.1.1.5 Fresh Variable Addition

The new variable __snapshot_2_st_14_0 is introduced. The transition formulas are extended as follows:

7:	__snapshot_2_st_14_0 ≤ st_14_0 ∧ st_14_0 ≤ __snapshot_2_st_14_0
9:	__snapshot_2_st_14_0 ≤ __snapshot_2_st_14_0 ∧ __snapshot_2_st_14_0 ≤ __snapshot_2_st_14_0
3:	__snapshot_2_st_14_0 ≤ __snapshot_2_st_14_0 ∧ __snapshot_2_st_14_0 ≤ __snapshot_2_st_14_0
4:	__snapshot_2_st_14_0 ≤ __snapshot_2_st_14_0 ∧ __snapshot_2_st_14_0 ≤ __snapshot_2_st_14_0

5.1.1.1.6 Fresh Variable Addition

The new variable __snapshot_2_rt_11_post is introduced. The transition formulas are extended as follows:

7:	__snapshot_2_rt_11_post ≤ rt_11_post ∧ rt_11_post ≤ __snapshot_2_rt_11_post
9:	__snapshot_2_rt_11_post ≤ __snapshot_2_rt_11_post ∧ __snapshot_2_rt_11_post ≤ __snapshot_2_rt_11_post
3:	__snapshot_2_rt_11_post ≤ __snapshot_2_rt_11_post ∧ __snapshot_2_rt_11_post ≤ __snapshot_2_rt_11_post
4:	__snapshot_2_rt_11_post ≤ __snapshot_2_rt_11_post ∧ __snapshot_2_rt_11_post ≤ __snapshot_2_rt_11_post

5.1.1.1.7 Fresh Variable Addition

The new variable __snapshot_2_rt_11_0 is introduced. The transition formulas are extended as follows:

7:	__snapshot_2_rt_11_0 ≤ rt_11_0 ∧ rt_11_0 ≤ __snapshot_2_rt_11_0
9:	__snapshot_2_rt_11_0 ≤ __snapshot_2_rt_11_0 ∧ __snapshot_2_rt_11_0 ≤ __snapshot_2_rt_11_0
3:	__snapshot_2_rt_11_0 ≤ __snapshot_2_rt_11_0 ∧ __snapshot_2_rt_11_0 ≤ __snapshot_2_rt_11_0
4:	__snapshot_2_rt_11_0 ≤ __snapshot_2_rt_11_0 ∧ __snapshot_2_rt_11_0 ≤ __snapshot_2_rt_11_0

5.1.1.1.8 Invariant Updates

The following invariants are asserted.

0:	TRUE
1:	TRUE
2:	1 − y_13_0 ≤ 0
3:	1 − y_13_0 ≤ 0
4:	TRUE
2:	1 − y_13_0 ≤ 0 ∨ 1 − __snapshot_2_x_15_0 + x_15_0 ≤ 0 ∧ 1 − __snapshot_2_x_15_0 ≤ 0 ∧ 1 − y_13_0 ≤ 0
3:	1 − __snapshot_2_x_15_0 + x_15_0 ≤ 0 ∧ 1 − __snapshot_2_x_15_0 ≤ 0 ∧ 1 − y_13_0 ≤ 0
2_var_snapshot:	− __snapshot_2_x_15_0 + x_15_0 ≤ 0 ∧ 1 − __snapshot_2_x_15_0 + x_15_0 − y_13_0 ≤ 0 ∧ − y_13_0 ≤ 0
2*:	1 − __snapshot_2_x_15_0 + x_15_0 ≤ 0 ∧ 1 − __snapshot_2_x_15_0 ≤ 0 ∧ 1 − y_13_0 ≤ 0

The invariants are proved as follows.

IMPACT Invariant Proof

nodes (location) invariant:

0	(4)	TRUE
1	(0)	TRUE
2	(1)	TRUE
3	(2)	1 − y_13_0 ≤ 0
4	(1)	TRUE
5	(3)	1 − y_13_0 ≤ 0
6	(2)	1 − y_13_0 ≤ 0
7	(2_var_snapshot)	− __snapshot_2_x_15_0 + x_15_0 ≤ 0 ∧ 1 − __snapshot_2_x_15_0 + x_15_0 − y_13_0 ≤ 0 ∧ − y_13_0 ≤ 0
12	(2)	1 − y_13_0 ≤ 0
16	(3)	1 − __snapshot_2_x_15_0 + x_15_0 ≤ 0 ∧ 1 − __snapshot_2_x_15_0 ≤ 0 ∧ 1 − y_13_0 ≤ 0
17	(2*)	1 − __snapshot_2_x_15_0 + x_15_0 ≤ 0 ∧ 1 − __snapshot_2_x_15_0 ≤ 0 ∧ 1 − y_13_0 ≤ 0
18	(2)	1 − __snapshot_2_x_15_0 + x_15_0 ≤ 0 ∧ 1 − __snapshot_2_x_15_0 ≤ 0 ∧ 1 − y_13_0 ≤ 0
19	(2_var_snapshot)	− __snapshot_2_x_15_0 + x_15_0 ≤ 0 ∧ 1 − __snapshot_2_x_15_0 + x_15_0 − y_13_0 ≤ 0 ∧ − y_13_0 ≤ 0

initial node: 0
cover edges:

2 → 4

12 → 3

19 → 7
transition edges:

0 5 1

1 0 2

1 1 3

3 2 4

3 3 5

3 6 6

5 4 12

6 7 7

7 3 16

16 4 17

17 9 18

18 7 19

5.1.1.1.9 Transition Removal

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

2:	x_15_0
3:	__snapshot_2_x_15_0
2_var_snapshot:	__snapshot_2_x_15_0
2*:	__snapshot_2_x_15_0

5.1.1.1.10 Transition Removal

We remove transition 7 using the following ranking functions, which are bounded by −6.

2:	−1
2_var_snapshot:	−2
3:	−3
2*:	−4

5.1.1.1.11 Splitting Cut-Point Transitions

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

Tool configuration

T2Cert

version: 1.0

2	→ 4
12	→ 3
19	→ 7