% The typing rules are no longer operational,
% so we have to give derivations for type-checking.

% Extrinsic

_ : of (lam [x] x) (all [a] a => a)
  = tp_alli [a:tp] tp_lam [x:exp] [u:of x a] u.

_ : of (lam [x] app x x) ((all [a] a) => (all [a] a))
  = tp_lam [x:exp] [u:of x (all [a] a)]
     tp_app u (tp_alle ((all [a] a) => (all [a] a)) u).

one : tp = all [a] a => a.
_ : of (lam [x] app x x) (one => one)
  = tp_lam [x] [u:of x one] tp_app u (tp_alle one u).

% Intrinsic

%query 1 *
D : of* (tlam* [a] lam* a [x:term*] x) T.

%query 0 *
D : of* (lam* T1 [x] app* x x) T.

one = all [a] a => a.
%query 1 *
D : of* (lam* one [x:term*] app* (tapp* x one) x) T.

% Indexed

_ = tlam' [a] lam' [x:term a] x.
_ = tlam' [a] lam' [x] x.
% _ = lam [x] app x x. % fails
% constraints remain in next example
% _ = lam [x] app (tapp x _) x.

one = all [a] a => a.
id : term (one) = tlam' [a] lam' [x:term a] x.
selfApp : term (one => one)
	= lam' [x:term one] app' (tapp' x one) x.