LTS Termination Proof

by T2Cert

Input

Integer Transition System

Initial Location: 2

Transitions: (pre-variables and post-variables)

0	0	1:		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 ∧ i_1 ≤ 0 ∧ − i_1 ≤ 0 ∧ − i_1 + tmp10_post ≤ 0 ∧ i_1 − tmp10_post ≤ 0 ∧ −1 − i_1 + i_2 ≤ 0 ∧ 1 + i_1 − i_2 ≤ 0 ∧ − i_2 + tmp___011_post ≤ 0 ∧ i_2 − tmp___011_post ≤ 0 ∧ −1 − i_2 + i_3 ≤ 0 ∧ 1 + i_2 − i_3 ≤ 0 ∧ − i_3 + tmp13_post ≤ 0 ∧ i_3 − tmp13_post ≤ 0 ∧ −1 − i_3 + i_4 ≤ 0 ∧ 1 + i_3 − i_4 ≤ 0 ∧ − i_4 + tmp___014_post ≤ 0 ∧ i_4 − tmp___014_post ≤ 0 ∧ −1 − i_4 + i_post ≤ 0 ∧ 1 + i_4 − i_post ≤ 0 ∧ i_0 − i_post ≤ 0 ∧ − i_0 + i_post ≤ 0 ∧ tmp10_0 − tmp10_post ≤ 0 ∧ − tmp10_0 + tmp10_post ≤ 0 ∧ tmp13_0 − tmp13_post ≤ 0 ∧ − tmp13_0 + tmp13_post ≤ 0 ∧ tmp___011_0 − tmp___011_post ≤ 0 ∧ − tmp___011_0 + tmp___011_post ≤ 0 ∧ tmp___014_0 − tmp___014_post ≤ 0 ∧ − tmp___014_0 + tmp___014_post ≤ 0
2	1	0:		− tmp___014_post + tmp___014_post ≤ 0 ∧ tmp___014_post − tmp___014_post ≤ 0 ∧ − tmp___014_0 + tmp___014_0 ≤ 0 ∧ tmp___014_0 − tmp___014_0 ≤ 0 ∧ − tmp___011_post + tmp___011_post ≤ 0 ∧ tmp___011_post − tmp___011_post ≤ 0 ∧ − tmp___011_0 + tmp___011_0 ≤ 0 ∧ tmp___011_0 − tmp___011_0 ≤ 0 ∧ − tmp13_post + tmp13_post ≤ 0 ∧ tmp13_post − tmp13_post ≤ 0 ∧ − tmp13_0 + tmp13_0 ≤ 0 ∧ tmp13_0 − tmp13_0 ≤ 0 ∧ − tmp10_post + tmp10_post ≤ 0 ∧ tmp10_post − tmp10_post ≤ 0 ∧ − tmp10_0 + tmp10_0 ≤ 0 ∧ tmp10_0 − tmp10_0 ≤ 0 ∧ − i_post + i_post ≤ 0 ∧ i_post − i_post ≤ 0 ∧ − i_4 + i_4 ≤ 0 ∧ i_4 − i_4 ≤ 0 ∧ − i_3 + i_3 ≤ 0 ∧ i_3 − i_3 ≤ 0 ∧ − i_2 + i_2 ≤ 0 ∧ i_2 − i_2 ≤ 0 ∧ − i_1 + i_1 ≤ 0 ∧ i_1 − i_1 ≤ 0 ∧ − i_0 + i_0 ≤ 0 ∧ i_0 − i_0 ≤ 0

Proof

1 Invariant Updates

The following invariants are asserted.

0:	TRUE
1:	tmp10_post ≤ 0 ∧ i_1 ≤ 0 ∧ − i_1 ≤ 0 ∧ tmp10_0 ≤ 0
2:	TRUE

The invariants are proved as follows.

IMPACT Invariant Proof

nodes (location) invariant:

0	(0)	TRUE
1	(1)	tmp10_post ≤ 0 ∧ i_1 ≤ 0 ∧ − i_1 ≤ 0 ∧ tmp10_0 ≤ 0
2	(2)	TRUE

initial node: 2
cover edges:
transition edges:

0 0 1

2 1 0

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 using the following ranking functions, which are bounded by −8.

2:	0
0:	0
1:	0
2:	−4
0:	−5
1:	−6

4 SCC Decomposition

There exist no SCC in the program graph.

Tool configuration

T2Cert

version: 1.0