atom : type.  %name atom P.
a : atom.
b : atom.
c : atom.
d : atom.
e : atom.

prop : type.  %name prop A.
=> : prop -> prop -> prop.  %infix right 10 =>.
\/ : prop -> prop -> prop.  %infix right 11 \/.
/\ : prop -> prop -> prop.  %infix right 12 /\.
?  : atom -> prop.          %prefix 14 ?.
ff : prop.
tt : prop.

proof : prop -> type.  %name proof M.

pair : proof A -> proof B -> proof (A /\ B).
fst : proof (A /\ B) -> proof A.
snd : proof (A /\ B) -> proof B.
fn : {A:prop} (proof A -> proof B) -> proof (A => B).
app : proof (A => B) -> proof A -> proof B.

% intrinsic representation *doesn't* allow invalid proofs!
% new type "proof A" combines old "proof" and "of" types.