LTS Termination Proof

by T2Cert

Input

Integer Transition System

Initial Location: 4

Transitions: (pre-variables and post-variables)

0	0	1:	− tmp3_post + tmp3_post ≤ 0 ∧ tmp3_post − tmp3_post ≤ 0 ∧ − tmp3_0 + tmp3_0 ≤ 0 ∧ tmp3_0 − tmp3_0 ≤ 0 ∧ − tmp2_post + tmp2_post ≤ 0 ∧ tmp2_post − tmp2_post ≤ 0 ∧ − tmp2_0 + tmp2_0 ≤ 0 ∧ tmp2_0 − tmp2_0 ≤ 0 ∧ − tmp1_post + tmp1_post ≤ 0 ∧ tmp1_post − tmp1_post ≤ 0 ∧ − tmp1_0 + tmp1_0 ≤ 0 ∧ tmp1_0 − tmp1_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
2	1	0:	− tmp3_post + tmp3_post ≤ 0 ∧ tmp3_post − tmp3_post ≤ 0 ∧ − tmp3_0 + tmp3_0 ≤ 0 ∧ tmp3_0 − tmp3_0 ≤ 0 ∧ − tmp2_post + tmp2_post ≤ 0 ∧ tmp2_post − tmp2_post ≤ 0 ∧ − tmp2_0 + tmp2_0 ≤ 0 ∧ tmp2_0 − tmp2_0 ≤ 0 ∧ − tmp1_post + tmp1_post ≤ 0 ∧ tmp1_post − tmp1_post ≤ 0 ∧ − tmp1_0 + tmp1_0 ≤ 0 ∧ tmp1_0 − tmp1_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
2	2	0:	− tmp3_post + tmp3_post ≤ 0 ∧ tmp3_post − tmp3_post ≤ 0 ∧ − tmp3_0 + tmp3_0 ≤ 0 ∧ tmp3_0 − tmp3_0 ≤ 0 ∧ − tmp2_post + tmp2_post ≤ 0 ∧ tmp2_post − tmp2_post ≤ 0 ∧ − tmp2_0 + tmp2_0 ≤ 0 ∧ tmp2_0 − tmp2_0 ≤ 0 ∧ − tmp1_post + tmp1_post ≤ 0 ∧ tmp1_post − tmp1_post ≤ 0 ∧ − tmp1_0 + tmp1_0 ≤ 0 ∧ tmp1_0 − tmp1_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	2:	0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 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 ∧ tmp1_0 − tmp1_post ≤ 0 ∧ − tmp1_0 + tmp1_post ≤ 0 ∧ tmp2_0 − tmp2_post ≤ 0 ∧ − tmp2_0 + tmp2_post ≤ 0 ∧ tmp3_0 − tmp3_post ≤ 0 ∧ − tmp3_0 + tmp3_post ≤ 0
4	4	3:	− tmp3_post + tmp3_post ≤ 0 ∧ tmp3_post − tmp3_post ≤ 0 ∧ − tmp3_0 + tmp3_0 ≤ 0 ∧ tmp3_0 − tmp3_0 ≤ 0 ∧ − tmp2_post + tmp2_post ≤ 0 ∧ tmp2_post − tmp2_post ≤ 0 ∧ − tmp2_0 + tmp2_0 ≤ 0 ∧ tmp2_0 − tmp2_0 ≤ 0 ∧ − tmp1_post + tmp1_post ≤ 0 ∧ tmp1_post − tmp1_post ≤ 0 ∧ − tmp1_0 + tmp1_0 ≤ 0 ∧ tmp1_0 − tmp1_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

2 1 0

2 2 0

3 3 2

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

4 SCC Decomposition

There exist no SCC in the program graph.

Tool configuration

T2Cert

version: 1.0