LTS Termination Proof

by T2Cert

Input

Integer Transition System

Initial Location: 10

Transitions: (pre-variables and post-variables)

0	0	1:	0 ≤ 0 ∧ 0 ≤ 0 ∧ −2 − x_0 + x_post ≤ 0 ∧ 2 + x_0 − x_post ≤ 0 ∧ x_0 − x_post ≤ 0 ∧ − x_0 + x_post ≤ 0 ∧ − tmp_post + tmp_post ≤ 0 ∧ tmp_post − tmp_post ≤ 0 ∧ − tmp_0 + tmp_0 ≤ 0 ∧ tmp_0 − tmp_0 ≤ 0 ∧ − i_post + i_post ≤ 0 ∧ i_post − i_post ≤ 0 ∧ − i_0 + i_0 ≤ 0 ∧ i_0 − i_0 ≤ 0 ∧ − N_0 + N_0 ≤ 0 ∧ N_0 − N_0 ≤ 0
0	1	1:	− x_post + x_post ≤ 0 ∧ x_post − x_post ≤ 0 ∧ − x_0 + x_0 ≤ 0 ∧ x_0 − x_0 ≤ 0 ∧ − tmp_post + tmp_post ≤ 0 ∧ tmp_post − tmp_post ≤ 0 ∧ − tmp_0 + tmp_0 ≤ 0 ∧ tmp_0 − tmp_0 ≤ 0 ∧ − i_post + i_post ≤ 0 ∧ i_post − i_post ≤ 0 ∧ − i_0 + i_0 ≤ 0 ∧ i_0 − i_0 ≤ 0 ∧ − N_0 + N_0 ≤ 0 ∧ N_0 − N_0 ≤ 0
2	2	3:	N_0 − i_0 ≤ 0 ∧ − x_post + x_post ≤ 0 ∧ x_post − x_post ≤ 0 ∧ − x_0 + x_0 ≤ 0 ∧ x_0 − x_0 ≤ 0 ∧ − tmp_post + tmp_post ≤ 0 ∧ tmp_post − tmp_post ≤ 0 ∧ − tmp_0 + tmp_0 ≤ 0 ∧ tmp_0 − tmp_0 ≤ 0 ∧ − i_post + i_post ≤ 0 ∧ i_post − i_post ≤ 0 ∧ − i_0 + i_0 ≤ 0 ∧ i_0 − i_0 ≤ 0 ∧ − N_0 + N_0 ≤ 0 ∧ N_0 − N_0 ≤ 0
2	3	0:	1 − N_0 + i_0 ≤ 0 ∧ − x_post + x_post ≤ 0 ∧ x_post − x_post ≤ 0 ∧ − x_0 + x_0 ≤ 0 ∧ x_0 − x_0 ≤ 0 ∧ − tmp_post + tmp_post ≤ 0 ∧ tmp_post − tmp_post ≤ 0 ∧ − tmp_0 + tmp_0 ≤ 0 ∧ tmp_0 − tmp_0 ≤ 0 ∧ − i_post + i_post ≤ 0 ∧ i_post − i_post ≤ 0 ∧ − i_0 + i_0 ≤ 0 ∧ i_0 − i_0 ≤ 0 ∧ − N_0 + N_0 ≤ 0 ∧ N_0 − N_0 ≤ 0
4	4	2:	− x_post + x_post ≤ 0 ∧ x_post − x_post ≤ 0 ∧ − x_0 + x_0 ≤ 0 ∧ x_0 − x_0 ≤ 0 ∧ − tmp_post + tmp_post ≤ 0 ∧ tmp_post − tmp_post ≤ 0 ∧ − tmp_0 + tmp_0 ≤ 0 ∧ tmp_0 − tmp_0 ≤ 0 ∧ − i_post + i_post ≤ 0 ∧ i_post − i_post ≤ 0 ∧ − i_0 + i_0 ≤ 0 ∧ i_0 − i_0 ≤ 0 ∧ − N_0 + N_0 ≤ 0 ∧ N_0 − N_0 ≤ 0
5	5	6:	0 ≤ 0 ∧ 0 ≤ 0 ∧ tmp_post ≤ 0 ∧ − tmp_post ≤ 0 ∧ tmp_0 − tmp_post ≤ 0 ∧ − tmp_0 + tmp_post ≤ 0 ∧ − x_post + x_post ≤ 0 ∧ x_post − x_post ≤ 0 ∧ − x_0 + x_0 ≤ 0 ∧ x_0 − x_0 ≤ 0 ∧ − i_post + i_post ≤ 0 ∧ i_post − i_post ≤ 0 ∧ − i_0 + i_0 ≤ 0 ∧ i_0 − i_0 ≤ 0 ∧ − N_0 + N_0 ≤ 0 ∧ N_0 − N_0 ≤ 0
7	6	5:	2 + N_0 − x_0 ≤ 0 ∧ − x_post + x_post ≤ 0 ∧ x_post − x_post ≤ 0 ∧ − x_0 + x_0 ≤ 0 ∧ x_0 − x_0 ≤ 0 ∧ − tmp_post + tmp_post ≤ 0 ∧ tmp_post − tmp_post ≤ 0 ∧ − tmp_0 + tmp_0 ≤ 0 ∧ tmp_0 − tmp_0 ≤ 0 ∧ − i_post + i_post ≤ 0 ∧ i_post − i_post ≤ 0 ∧ − i_0 + i_0 ≤ 0 ∧ i_0 − i_0 ≤ 0 ∧ − N_0 + N_0 ≤ 0 ∧ N_0 − N_0 ≤ 0
7	7	5:	− N_0 + x_0 ≤ 0 ∧ − x_post + x_post ≤ 0 ∧ x_post − x_post ≤ 0 ∧ − x_0 + x_0 ≤ 0 ∧ x_0 − x_0 ≤ 0 ∧ − tmp_post + tmp_post ≤ 0 ∧ tmp_post − tmp_post ≤ 0 ∧ − tmp_0 + tmp_0 ≤ 0 ∧ tmp_0 − tmp_0 ≤ 0 ∧ − i_post + i_post ≤ 0 ∧ i_post − i_post ≤ 0 ∧ − i_0 + i_0 ≤ 0 ∧ i_0 − i_0 ≤ 0 ∧ − N_0 + N_0 ≤ 0 ∧ N_0 − N_0 ≤ 0
7	8	6:	0 ≤ 0 ∧ 0 ≤ 0 ∧ −1 − N_0 + x_0 ≤ 0 ∧ 1 + N_0 − x_0 ≤ 0 ∧ −1 + tmp_post ≤ 0 ∧ 1 − tmp_post ≤ 0 ∧ tmp_0 − tmp_post ≤ 0 ∧ − tmp_0 + tmp_post ≤ 0 ∧ − x_post + x_post ≤ 0 ∧ x_post − x_post ≤ 0 ∧ − x_0 + x_0 ≤ 0 ∧ x_0 − x_0 ≤ 0 ∧ − i_post + i_post ≤ 0 ∧ i_post − i_post ≤ 0 ∧ − i_0 + i_0 ≤ 0 ∧ i_0 − i_0 ≤ 0 ∧ − N_0 + N_0 ≤ 0 ∧ N_0 − N_0 ≤ 0
6	9	8:	− x_post + x_post ≤ 0 ∧ x_post − x_post ≤ 0 ∧ − x_0 + x_0 ≤ 0 ∧ x_0 − x_0 ≤ 0 ∧ − tmp_post + tmp_post ≤ 0 ∧ tmp_post − tmp_post ≤ 0 ∧ − tmp_0 + tmp_0 ≤ 0 ∧ tmp_0 − tmp_0 ≤ 0 ∧ − i_post + i_post ≤ 0 ∧ i_post − i_post ≤ 0 ∧ − i_0 + i_0 ≤ 0 ∧ i_0 − i_0 ≤ 0 ∧ − N_0 + N_0 ≤ 0 ∧ N_0 − N_0 ≤ 0
3	10	7:	1 + N_0 − x_0 ≤ 0 ∧ − x_post + x_post ≤ 0 ∧ x_post − x_post ≤ 0 ∧ − x_0 + x_0 ≤ 0 ∧ x_0 − x_0 ≤ 0 ∧ − tmp_post + tmp_post ≤ 0 ∧ tmp_post − tmp_post ≤ 0 ∧ − tmp_0 + tmp_0 ≤ 0 ∧ tmp_0 − tmp_0 ≤ 0 ∧ − i_post + i_post ≤ 0 ∧ i_post − i_post ≤ 0 ∧ − i_0 + i_0 ≤ 0 ∧ i_0 − i_0 ≤ 0 ∧ − N_0 + N_0 ≤ 0 ∧ N_0 − N_0 ≤ 0
3	11	7:	1 − N_0 + x_0 ≤ 0 ∧ − x_post + x_post ≤ 0 ∧ x_post − x_post ≤ 0 ∧ − x_0 + x_0 ≤ 0 ∧ x_0 − x_0 ≤ 0 ∧ − tmp_post + tmp_post ≤ 0 ∧ tmp_post − tmp_post ≤ 0 ∧ − tmp_0 + tmp_0 ≤ 0 ∧ tmp_0 − tmp_0 ≤ 0 ∧ − i_post + i_post ≤ 0 ∧ i_post − i_post ≤ 0 ∧ − i_0 + i_0 ≤ 0 ∧ i_0 − i_0 ≤ 0 ∧ − N_0 + N_0 ≤ 0 ∧ N_0 − N_0 ≤ 0
3	12	6:	0 ≤ 0 ∧ 0 ≤ 0 ∧ − N_0 + x_0 ≤ 0 ∧ N_0 − x_0 ≤ 0 ∧ −1 + tmp_post ≤ 0 ∧ 1 − tmp_post ≤ 0 ∧ tmp_0 − tmp_post ≤ 0 ∧ − tmp_0 + tmp_post ≤ 0 ∧ − x_post + x_post ≤ 0 ∧ x_post − x_post ≤ 0 ∧ − x_0 + x_0 ≤ 0 ∧ x_0 − x_0 ≤ 0 ∧ − i_post + i_post ≤ 0 ∧ i_post − i_post ≤ 0 ∧ − i_0 + i_0 ≤ 0 ∧ i_0 − i_0 ≤ 0 ∧ − N_0 + N_0 ≤ 0 ∧ N_0 − N_0 ≤ 0
1	13	4:	0 ≤ 0 ∧ 0 ≤ 0 ∧ −1 − i_0 + i_post ≤ 0 ∧ 1 + i_0 − i_post ≤ 0 ∧ i_0 − i_post ≤ 0 ∧ − i_0 + i_post ≤ 0 ∧ − x_post + x_post ≤ 0 ∧ x_post − x_post ≤ 0 ∧ − x_0 + x_0 ≤ 0 ∧ x_0 − x_0 ≤ 0 ∧ − tmp_post + tmp_post ≤ 0 ∧ tmp_post − tmp_post ≤ 0 ∧ − tmp_0 + tmp_0 ≤ 0 ∧ tmp_0 − tmp_0 ≤ 0 ∧ − N_0 + N_0 ≤ 0 ∧ N_0 − N_0 ≤ 0
9	14	4:	0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ x_post ≤ 0 ∧ − x_post ≤ 0 ∧ i_post ≤ 0 ∧ − i_post ≤ 0 ∧ i_0 − i_post ≤ 0 ∧ − i_0 + i_post ≤ 0 ∧ x_0 − x_post ≤ 0 ∧ − x_0 + x_post ≤ 0 ∧ − tmp_post + tmp_post ≤ 0 ∧ tmp_post − tmp_post ≤ 0 ∧ − tmp_0 + tmp_0 ≤ 0 ∧ tmp_0 − tmp_0 ≤ 0 ∧ − N_0 + N_0 ≤ 0 ∧ N_0 − N_0 ≤ 0
10	15	9:	− x_post + x_post ≤ 0 ∧ x_post − x_post ≤ 0 ∧ − x_0 + x_0 ≤ 0 ∧ x_0 − x_0 ≤ 0 ∧ − tmp_post + tmp_post ≤ 0 ∧ tmp_post − tmp_post ≤ 0 ∧ − tmp_0 + tmp_0 ≤ 0 ∧ tmp_0 − tmp_0 ≤ 0 ∧ − i_post + i_post ≤ 0 ∧ i_post − i_post ≤ 0 ∧ − i_0 + i_0 ≤ 0 ∧ i_0 − i_0 ≤ 0 ∧ − N_0 + N_0 ≤ 0 ∧ N_0 − N_0 ≤ 0

Proof

1 Invariant Updates

The following invariants are asserted.

0:	TRUE
1:	TRUE
2:	TRUE
3:	TRUE
4:	TRUE
5:	TRUE
6:	−1 + tmp_post ≤ 0 ∧ −1 + tmp_0 ≤ 0
7:	TRUE
8:	−1 + tmp_post ≤ 0 ∧ −1 + tmp_0 ≤ 0
9:	TRUE
10:	TRUE

The invariants are proved as follows.

IMPACT Invariant Proof

nodes (location) invariant:

0	(0)	TRUE
1	(1)	TRUE
2	(2)	TRUE
3	(3)	TRUE
4	(4)	TRUE
5	(5)	TRUE
6	(6)	−1 + tmp_post ≤ 0 ∧ −1 + tmp_0 ≤ 0
7	(7)	TRUE
8	(8)	−1 + tmp_post ≤ 0 ∧ −1 + tmp_0 ≤ 0
9	(9)	TRUE
10	(10)	TRUE

initial node: 10
cover edges:
transition edges:

0 0 1

0 1 1

1 13 4

2 2 3

2 3 0

3 10 7

3 11 7

3 12 6

4 4 2

5 5 6

6 9 8

7 6 5

7 7 5

7 8 6

9 14 4

10 15 9

2 Switch to Cooperation Termination Proof

We consider the following cutpoint-transitions:

− x_post + x_post ≤ 0 ∧ x_post − x_post ≤ 0 ∧ − x_0 + x_0 ≤ 0 ∧ x_0 − x_0 ≤ 0 ∧ − tmp_post + tmp_post ≤ 0 ∧ tmp_post − tmp_post ≤ 0 ∧ − tmp_0 + tmp_0 ≤ 0 ∧ tmp_0 − tmp_0 ≤ 0 ∧ − i_post + i_post ≤ 0 ∧ i_post − i_post ≤ 0 ∧ − i_0 + i_0 ≤ 0 ∧ i_0 − i_0 ≤ 0 ∧ − N_0 + N_0 ≤ 0 ∧ N_0 − N_0 ≤ 0

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

3 Transition Removal

We remove transitions 2, 5, 6, 7, 8, 9, 10, 11, 12, 14, 15 using the following ranking functions, which are bounded by −21.

10:	0
9:	0
0:	0
1:	0
2:	0
4:	0
3:	0
7:	0
5:	0
6:	0
8:	0
10:	−9
9:	−10
0:	−11
1:	−11
2:	−11
4:	−11
4_var_snapshot:	−11
4*:	−11
3:	−12
7:	−13
5:	−14
6:	−15
8:	−16

4 Location Addition

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

4* 19 4: − x_post + x_post ≤ 0 ∧ x_post − x_post ≤ 0 ∧ − x_0 + x_0 ≤ 0 ∧ x_0 − x_0 ≤ 0 ∧ − tmp_post + tmp_post ≤ 0 ∧ tmp_post − tmp_post ≤ 0 ∧ − tmp_0 + tmp_0 ≤ 0 ∧ tmp_0 − tmp_0 ≤ 0 ∧ − i_post + i_post ≤ 0 ∧ i_post − i_post ≤ 0 ∧ − i_0 + i_0 ≤ 0 ∧ i_0 − i_0 ≤ 0 ∧ − N_0 + N_0 ≤ 0 ∧ N_0 − N_0 ≤ 0

5 Location Addition

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

4 17 4_var_snapshot: − x_post + x_post ≤ 0 ∧ x_post − x_post ≤ 0 ∧ − x_0 + x_0 ≤ 0 ∧ x_0 − x_0 ≤ 0 ∧ − tmp_post + tmp_post ≤ 0 ∧ tmp_post − tmp_post ≤ 0 ∧ − tmp_0 + tmp_0 ≤ 0 ∧ tmp_0 − tmp_0 ≤ 0 ∧ − i_post + i_post ≤ 0 ∧ i_post − i_post ≤ 0 ∧ − i_0 + i_0 ≤ 0 ∧ i_0 − i_0 ≤ 0 ∧ − N_0 + N_0 ≤ 0 ∧ N_0 − N_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 { 0, 1, 2, 4, 4_var_snapshot, 4* }.

6.1.1 Transition Removal

We remove transition 3 using the following ranking functions, which are bounded by 7.

0:	1 + 6⋅N_0 − 6⋅i_0
1:	6⋅N_0 − 6⋅i_0
2:	2 + 6⋅N_0 − 6⋅i_0
4:	4 + 6⋅N_0 − 6⋅i_0
4_var_snapshot:	3 + 6⋅N_0 − 6⋅i_0
4*:	5 + 6⋅N_0 − 6⋅i_0

6.1.2 Transition Removal

We remove transitions 17, 19, 0, 1, 4, 13 using the following ranking functions, which are bounded by −4.

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

6.1.3 Splitting Cut-Point Transitions

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

6.1.3.1 Cut-Point Subproblem 1/1

Here we consider cut-point transition 16.

6.1.3.1.1 Splitting Cut-Point Transitions

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

Tool configuration

T2Cert

version: 1.0

0	0 1
0	1 1
1	13 4
2	2 3
2	3 0
3	10 7
3	11 7
3	12 6
4	4 2
5	5 6
6	9 8
7	6 5
7	7 5
7	8 6
9	14 4
10	15 9