I think I finally understand why I keep asking stupid questions and
find statements in class so strange.
"You keep using that word.
I do not think it means what you think it means"
-- inigo Montoya 1987 (in The Princess Bride)
It is like an artist (programmer) taking an art theory class.
We're using the same words but we don't mean the same thing.
You are using types "ABOUT programming" and I'm using types
"about PROGRAMMING".
You use types as a meta-language to talk ABOUT programs
and use type-erasure to eliminate them at runtime. When you say
a program is "correct", you mean "type correct". So a 'plus' function
(\x:nat \y:nat 7) which always returns 7 is still "type correct".
I use types as first-class objects that I can create, modify, and
pass around at run time. When I say a program is "correct" I mean
that the computed result is "correct" in that it agrees with what a
user would expect.
The insight is that we are using the same words but different meanings.
This may be why computer algebra and proof theory seem to be disjoint
disciplines in computational mathematics.
The upside is that I'll finally stop asking stupid questions. Well,
maybe not, as I seem to have a genius for generating them :-)
Tim