LTS Termination Proof

by T2Cert

Input

Integer Transition System

Initial Location: 4

Transitions: (pre-variables and post-variables)

0	0	1:	− tmp_post + tmp_post ≤ 0 ∧ tmp_post − tmp_post ≤ 0 ∧ − tmp_0 + tmp_0 ≤ 0 ∧ tmp_0 − tmp_0 ≤ 0 ∧ − j_post + j_post ≤ 0 ∧ j_post − j_post ≤ 0 ∧ − j_0 + j_0 ≤ 0 ∧ j_0 − j_0 ≤ 0 ∧ − i_post + i_post ≤ 0 ∧ i_post − i_post ≤ 0 ∧ − i_0 + i_0 ≤ 0 ∧ i_0 − i_0 ≤ 0
0	1	1:	− tmp_post + tmp_post ≤ 0 ∧ tmp_post − tmp_post ≤ 0 ∧ − tmp_0 + tmp_0 ≤ 0 ∧ tmp_0 − tmp_0 ≤ 0 ∧ − j_post + j_post ≤ 0 ∧ j_post − j_post ≤ 0 ∧ − j_0 + j_0 ≤ 0 ∧ j_0 − j_0 ≤ 0 ∧ − i_post + i_post ≤ 0 ∧ i_post − i_post ≤ 0 ∧ − i_0 + i_0 ≤ 0 ∧ i_0 − i_0 ≤ 0
1	2	2:	− tmp_post + tmp_post ≤ 0 ∧ tmp_post − tmp_post ≤ 0 ∧ − tmp_0 + tmp_0 ≤ 0 ∧ tmp_0 − tmp_0 ≤ 0 ∧ − j_post + j_post ≤ 0 ∧ j_post − j_post ≤ 0 ∧ − j_0 + j_0 ≤ 0 ∧ j_0 − j_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 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ i_post ≤ 0 ∧ − i_post ≤ 0 ∧ −1 + j_post ≤ 0 ∧ 1 − j_post ≤ 0 ∧ i_0 − i_post ≤ 0 ∧ − i_0 + i_post ≤ 0 ∧ j_0 − j_post ≤ 0 ∧ − j_0 + j_post ≤ 0 ∧ tmp_0 − tmp_post ≤ 0 ∧ − tmp_0 + tmp_post ≤ 0
4	4	3:	− tmp_post + tmp_post ≤ 0 ∧ tmp_post − tmp_post ≤ 0 ∧ − tmp_0 + tmp_0 ≤ 0 ∧ tmp_0 − tmp_0 ≤ 0 ∧ − j_post + j_post ≤ 0 ∧ j_post − j_post ≤ 0 ∧ − j_0 + j_0 ≤ 0 ∧ j_0 − j_0 ≤ 0 ∧ − i_post + i_post ≤ 0 ∧ i_post − i_post ≤ 0 ∧ − i_0 + i_0 ≤ 0 ∧ i_0 − i_0 ≤ 0

Proof

1 Invariant Updates

The following invariants are asserted.

0:	i_post ≤ 0 ∧ − i_post ≤ 0 ∧ −1 + j_post ≤ 0 ∧ 1 − j_post ≤ 0 ∧ i_0 ≤ 0 ∧ − i_0 ≤ 0 ∧ −1 + j_0 ≤ 0 ∧ 1 − j_0 ≤ 0
1:	i_post ≤ 0 ∧ − i_post ≤ 0 ∧ −1 + j_post ≤ 0 ∧ 1 − j_post ≤ 0 ∧ i_0 ≤ 0 ∧ − i_0 ≤ 0 ∧ −1 + j_0 ≤ 0 ∧ 1 − j_0 ≤ 0
2:	i_post ≤ 0 ∧ − i_post ≤ 0 ∧ −1 + j_post ≤ 0 ∧ 1 − j_post ≤ 0 ∧ i_0 ≤ 0 ∧ − i_0 ≤ 0 ∧ −1 + j_0 ≤ 0 ∧ 1 − j_0 ≤ 0
3:	TRUE
4:	TRUE

The invariants are proved as follows.

IMPACT Invariant Proof

nodes (location) invariant:

0	(0)	i_post ≤ 0 ∧ − i_post ≤ 0 ∧ −1 + j_post ≤ 0 ∧ 1 − j_post ≤ 0 ∧ i_0 ≤ 0 ∧ − i_0 ≤ 0 ∧ −1 + j_0 ≤ 0 ∧ 1 − j_0 ≤ 0
1	(1)	i_post ≤ 0 ∧ − i_post ≤ 0 ∧ −1 + j_post ≤ 0 ∧ 1 − j_post ≤ 0 ∧ i_0 ≤ 0 ∧ − i_0 ≤ 0 ∧ −1 + j_0 ≤ 0 ∧ 1 − j_0 ≤ 0
2	(2)	i_post ≤ 0 ∧ − i_post ≤ 0 ∧ −1 + j_post ≤ 0 ∧ 1 − j_post ≤ 0 ∧ i_0 ≤ 0 ∧ − i_0 ≤ 0 ∧ −1 + j_0 ≤ 0 ∧ 1 − j_0 ≤ 0
3	(3)	TRUE
4	(4)	TRUE

initial node: 4
cover edges:
transition edges:

0 0 1

0 1 1

1 2 2

3 3 0

4 4 3

2 Switch to Cooperation Termination Proof

We consider the following cutpoint-transitions:

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

3 Transition Removal

We remove transitions 0, 1, 2, 3, 4 using the following ranking functions, which are bounded by −12.

4:	0
3:	0
0:	0
1:	0
2:	0
4:	−6
3:	−7
0:	−8
1:	−9
2:	−10

Hints:

0	lexStrict[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] , [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]
1	lexStrict[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] , [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]
2	lexStrict[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] , [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]
3	lexStrict[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] , [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]
4	lexStrict[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] , [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]

4 SCC Decomposition

There exist no SCC in the program graph.

Tool configuration

T2Cert

version: 1.0