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 : type.   %name proof M.

pair : proof -> proof -> proof.
fst : proof -> proof.
snd : proof -> proof.
fn : prop -> (proof -> proof) -> proof.
app : proof -> proof -> proof.

% above representation allows invalid proofs.
% proof checking judgement M $ A picks out the valid ones.

$ : proof -> prop -> type.  %name $ D.
%infix none 8 $.
%mode $ +M -A.

/\I : pair M N $ A /\ B
    <- M $ A
    <- N $ B.

/\EL : fst M $ A
     <- M $ A /\ B.

/\ER : snd M $ B
     <- M $ A /\ B.

=>I : (fn A [x] M(x)) $ A => B
    <- ({x} x $ A -> M(x) $ B).

=>E : app M N $ B
    <- M $ A => B
    <- N $ A.