> The divide between ALL and ESSENTIAL is that the latter only shows
> the steps with typecode '|-' (and the final step, whatever its
> typecode is). The other steps, those constructing the formulas and
> classes, are "syntactic", that is, their typecodes are wff, set,
> class (and indeed, their leaves are $f-statements).

Ohhh, yeah, that makes complete sense. I was expecting that metamath
somehow had an internal concept of what is merely syntactic and what is
essential, but I guess it's just by convention. In other words, if I
were to make a file with different typecodes, then ESSENTIAL mode would
not work so well? (that's easy to verify I guess)

Probably what I should do is give the user control over the typecodes
they want to consider essential.

> To see an extreme example, which is kind of abusing the system, see 
> http://us2.metamath.org/mpeuni/bj-0.html and 
> http://us2.metamath.org/mpeuni/bj-1.html (see also the head comment
> of that section, and compare the results of >show proof xxx/ESSENTIAL
> and /ALL on these).

Woah, that's real weird.

> A clear exposition of the notion of syntactic versus essential (which 
> extends beyond set.mm) is presented in Mario's article "Models for 
> Metamath" (the part about weakly grammatical databases, so you do not
> have to read through the (harder) part about models).

Oh, cool; I'll have to read that. Thanks for the tip.

- Philip

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