In the classic the calculus of constructions paper there is a rule that states(page 7 of the pdf, page 101 of the original document)This rule would mean that any context is reducible to a member of that context.  This seems like it shouldn't be correct, as it would entail1  Nat3  Nat1  3if Nat is a context.I think the best interpretation is that the lower delta was meant to be an M.  Especially considering the rules given on the next page.So is this simply a typo, or some subtle logical rule that I don't understand?
Typo in the calculus of constructions paper?
lo.logic;calculus of constructions;sequent calculus
You are correct, there is an error in that paper, and the rule should indeed read:$$\frac{\Gamma\vdash M:\Delta}{\Gamma\vdash M\cong M} $$the use of jugements of this style for equality (sometimes called typed equality) originates already in Martin-Lf, I think (see here for example). It's often replaced with an untyped operational definition in modern treatments, where there is no jugement of the form $\Gamma\vdash N\cong M$, and conversion is defined on raw terms.Somewhat counter-intuitively, proving that the system with typed conversion is equivalent to the one without types is very difficult, and was settled in 2010 by Siles and Herbelin.