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I've started reading the HoTT book and am trying to understand its formal treatment of Martin-Löf type theory. Something small in the "first presentation" of the appendix has tripped me up; see the screenshot attached below for reference. In particular, the last two "alternative style" rules are not entirely clear to me. For one, surely the second should read "if $x:A\vdash b:B(x)$ then $\lambda x.b:\Pi_{(x:A)}B(x)$"? This seems to me a rather important point as $B$ and $B(x)$ are distinct terms. For the other, surely the third holds only if $x$ does not occur free in $B$? This may be implicit but it seems unclear to me.

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Atticus Stonestrom
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  • @DanielSchepler I was considering this as well, but surely that interpretation would then make the first alternative style rule incorrect? In that one aren't we considering "$B(x)$" as the term obtained by an application instance applied to $B$ and to $x$? – Atticus Stonestrom Mar 10 '20 at 01:03
  • Oh, I missed that you were looking at the "alternative style" rules at the end, not the first set of rules. – Daniel Schepler Mar 10 '20 at 01:05
  • @DanielSchepler Oh no worries, my bad. Do you have any insight on my question? I feel like I might be missing something obvious but am not sure – Atticus Stonestrom Mar 10 '20 at 01:15
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    I think you're right on both counts - in fact I'd put the restriction that $x$ is not free in $B$ on all three rules. (Though to be honest, for foundations I usually prefer using de Bruijn indices to avoid these types of shadowing issues altogether, and let the versions with named symbols be treated as purely syntactic sugar.) – Daniel Schepler Mar 10 '20 at 01:15
  • @DanielSchepler Fantastic, thank you so much! Definitely agree re: de Bruijn indices – Atticus Stonestrom Mar 10 '20 at 01:19
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    This typo is known. It's probably worth just opening a pull request to fix it. – SCappella Mar 10 '20 at 03:34
  • @SCappella right on, thank you! – Atticus Stonestrom Mar 10 '20 at 11:55

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