atom : type.
prop : type.

%% Atoms, including some sample ones for examples
atom : type.  %name atom P.
a : atom.
b : atom.
c : atom.
d : atom.
e : atom.
f : atom.

%% Propositions, omitting true (tt) and false (ff)
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 ?.


left : prop -> type.
right : prop -> type.

init : left (? P) -> right (? P).

/\L1 : (left A -> right C)
    -> (left (A /\ B) -> right C).

/\L2 : (left B -> right C)
    -> (left (A /\ B) -> right C).

/\R : right A
   -> right B
   -> right (A /\ B).


=>R : (left A -> right B)
   -> right (A => B).

=>L : right A
   -> (left B -> right C)
   -> (left (A => B) -> right C).