Removing `circular rules'

Say that I have a huge list (>300.000 elements) of polynomial equations and I want to simplify these by

1. looking for some simple equations of the form x[i] == _ or _ == x[i],
2. converting these equations to a list of rules of the form x[i] -> pol[i] where pol[i] is a polynomial that can depend on x[1],...,x[n] (but typically only depends on a few variables), and
3. substituting these rules repeatedly via ReplaceRepeated in order to reduce the number of variables in the system.

The problem that typically arises is that some of these rules might be circular. For example from the system

{ x[1] == x[2] * x[3] * x[5], x[4] == x[5] * x[6], x[5] == x[1], ... } 

I would get a list of rules that looks like

 { x[1] -> x[2] * x[3] * x[5], x[4] -> x[5] * x[6], x[5] -> x[1], ... } 

Using ReplaceRepeated will run forever.

The question I have is: what would be an efficient way to select the biggest subset of a list of rules of the form x[i] -> pol[i] such that this subset is non-circular, i.e. ReplaceRepeated would not run forever? It may be assumed that any 'self-circular' rule (in which a variable appears both on the LHS and RHS) has already been removed.

I used to have an implementation that looks like

clashingRulesQ[ a_ -> b_, c_ -> d_ ] := With[ { vb = Variables[b], vd = Variables[d] }, Or[ a == c, MemberQ[a] @ vd, MemberQ[c] @ vb ] ] removeCircularRules[ ruleList_ ] := DeleteDuplicates[ ruleList, clashingRulesQ ] 

but this one removes more rules than necessary since it would, e.g. reduce

{ x[1] -> x[2], x[2] -> x[3] } 

to

{ x[1] -> x[2] }