Heh, I find it amusing. But I have had prior experience from figuring out a similar proof. I used Pierce's Law (aka call/cc), but it too unfolds into the same story when evaluated.
On 2024-11-02 17:02, Brent Yorgey wrote:
You may find this helpful:
https://www.cs.cmu.edu/~cmartens/if/dem.html <https://www.cs.cmu.edu/ ~cmartens/if/dem.html>
Actually, you probably won't find it helpful, but at least it is amusing. And after you figure out your proof, you can go back and figure out what this story has to do with it.
-Brent
On Fri, Nov 1, 2024 at 8:45 AM Dmitriy Matrosov <sgf.dma@gmail.com <mailto:sgf.dma@gmail.com>> wrote:
Hi.
I've reading "Type theory and formal proof, an introduction"
book and in chapter 7.4 authors say, that in constructive
logic plus either excluded third law (ET) or double negation
law (DN) we can derive the other. And then authors derive DN
from ET in calculus of constructions. But they didn't say
anything (yet?) about the vice versa: deriving ET from DN in
calculus of constructions.
I've tried to do this, but the best i can come up with is
just case analysis:
- assume, that 'a' is true, then 'a or not a' is also true
(by 'or-intro' rule).
- or if after assuming that 'a' is true we can derive
bottom, then 'not a' is true (by 'not-intro' rule).
Then using the 'or-intro' rule we again end up with 'a or (not
a)' being true.
- assume 'not a' and if we can derive bottom again, then
'not (not a)' is true. Then by using DN we again end up
with 'a' being true. etc.
I.e. I may reduce any (not.. (n times) .. not a) into either
'a' or 'not a' by using DN and function composition. Thus, i
probable can derive ET from DN using induction, but i can't
code neither induction, nor above case analysis in calculus
of constructions.
So, is it possible to derive ET from DN in calculus of
constructions? If it is, i'd appreciate not a direct answer,
but a hint on how to do this. And if it is not,
well, probably, authors will explain this later in the book
and I should just continue reading.
Thanks.
_______________________________________________
Haskell-Cafe mailing list
To (un)subscribe, modify options or view archives go to:
http://mail.haskell.org/cgi-bin/mailman/listinfo/haskell-cafe
<http://mail.haskell.org/cgi-bin/mailman/listinfo/haskell-cafe>
Only members subscribed via the mailman list are allowed to post.
_______________________________________________
Haskell-Cafe mailing list
To (un)subscribe, modify options or view archives go to:
http://mail.haskell.org/cgi-bin/mailman/listinfo/haskell-cafe
Only members subscribed via the mailman list are allowed to post.
_______________________________________________
Haskell-Cafe mailing list
To (un)subscribe, modify options or view archives go to:
http://mail.haskell.org/cgi-bin/mailman/listinfo/haskell-cafe
Only members subscribed via the mailman list are allowed to post.