Characterization of Groebner Bases in terms of uniqueness of remainders Let $I$ be an ideal of a polynomial ring $R=k[x_1,\ldots,x_n]$ over a field $k$. A Groebner basis of $I$ is a finite generating set $\{g_1,\ldots,g_m\}$ such that every leading monomial (according to a fixed monomial order, e.g. lexicographical) of a polynomial in $I$ is divided by the leading monomial of some $g_i$. 
There are many characterizations of being a Groebner basis. For example:

Theorem
Let $\{g_1,...,g_m\}$ be a subset of an ideal $I$ of $R=k[x_1,\ldots,x_n]$.
  Then $G$ is a Groebner basis if and only if $G$ satisfy the following
  property: for all $f\in I$, whenever $f=g+r$ with $g\in I$ and $r\in R$ 
  satisfy that no monomial of $r$ is divisible by any
  $\operatorname{LM}(g_i)$, then $r=0$. ($\operatorname{LM}$ denotes
  the leading monomial)

Now, let's recall the multivariate division algorithm. Every polynomial $f$ can be divided by an (ordered) set $(g_1,\ldots,g_m)$ of polynomials, obtaining polynomial quotients $a_i$ and a polynomial remainder $r$ such that $$f = a_1g_1 + \cdots + a_mg_m + r$$ and there is not any monomial of $r$ that is divisible by some $\operatorname{LM}(g_i)$.
The result of the division algorithm depends on the order of the tuple, the quotients and the remainder are not unique in general. However, if $\{g_1,\ldots,g_m\}$ is a Groebner basis, the characterization above easily shows that the remainder is unique, no matter the order in which the $g_i$'s are listed.
My question is the following:

Does this property characterizes a Groebner basis?

More precisely, I'm asking if the following conjecture is true:

Conjecture
Let $\{g_1,\ldots,g_m\}$ be a set of polynomials such that for each
  polynomial $f$, the division of $f$ by
  $(g_{\sigma(1)},\ldots,g_{\sigma(m)})$ yields the same remainder for all permutations $\sigma$. Then
  $\{g_1,\ldots,g_m\}$ is a Groebner basis (of $\langle g_1,\ldots,g_m\rangle$).

I have not been able to give a counter-example of this (verifying the hypothesis is hard), and I have the impression that this is true, but it seems hard to prove.
Can anyone help me with this?
 A: The answer to your question depends on your definition of multivariate division algorithm.
A good algorithm in any case might be the following: Let $(g_1,\ldots,g_n)\in R^n$ be a fixed tuple of polynomials. Consider the following replacement rules:
$\operatorname{LM}(g_i) \mapsto (\operatorname{LM}(g_i)-g_i)$
The algorithm now does the following: Start with some $f \in R$. Whenever
a monomial $m$ of $f$ can be written as a multiple $m = a \operatorname{LM}(g_i)$ of a leading monomial of a $g_i$ with $a$ a monomial, replace that monomial $m$ according to the above rule by $a(\operatorname{LM}(g_i)-g_i)$. There is some choice here, as there might be multiple $g_i$ that can be used.
In this case the following statement is true:

If for all $f \in R$ the outcome of the above algorithm is independent of
  the choices in the algorithm, $(g_1, \ldots, g_n)$ is a Grobner basis.

This statement can be found for example in Kreuzer, Robbiano: Computational Commutative Algebra 1 (2000), Theorem 2.4.1, Condition $C_3$.
