%%% The Mini-ML typing rules which can be used as a complete
%%% implementation of type inference.
%%% Author: Frank Pfenning, based on [Michaylov & Pfenning 92]

of : exp -> tp -> type.  %name of P

% Natural Numbers
tp_z     : of z nat.
tp_s     : of (s E) nat
	    <- of E nat.
tp_case  : of (case E1 E2 E3) T
	    <- of E1 nat
	    <- of E2 T
	    <- ({x:exp} of x nat -> of (E3 x) T).

% Pairs
tp_pair : of (pair E1 E2) (cross T1 T2)
	   <- of E1 T1
	   <- of E2 T2.
tp_fst  : of (fst E) T1
	   <- of E (cross T1 T2).
tp_snd  : of (snd E) T2
	   <- of E (cross T1 T2).

% Functions
tp_lam : of (lam E) (arrow T1 T2)
	  <- ({x:exp} of x T1 -> of (E x) T2).
tp_app : of (app E1 E2) T1
	  <- of E1 (arrow T2 T1)
	  <- of E2 T2.

% Definitions
tp_letv : of (letv E1 E2) T2
	   <- of E1 T1
	   <- ({x:exp} of x T1 -> of (E2 x) T2).
tp_letn : of (letn E1 E2) T2
	   <- of E1 T1
	   <- of (E2 E1) T2.

% Recursion
tp_fix : of (fix E) T
	  <- ({x:exp} of x T -> of (E x) T).