%%% Lambda-Calculus Fragment of Mini-ML.
%%% Author: Frank Pfenning

% Simple types
tp : type.				%name tp T.

arrow : tp -> tp -> tp.			% T1 => T2

% Expressions
exp : type.				%name exp E.

lam : (exp -> exp) -> exp.		% lam x. E
app : exp -> exp -> exp.		% (E1 E2)

% Type inference 
% |- E : T  (expression E has type T)

of : exp -> tp -> type.			%name of P.
%mode of +E *T.
% %mode of +E +T.  % incorrect at tp_app
% %mode of +E -T.  % incorrect at tp_lam

tp_lam : of (lam E) (arrow T1 T2)	% |- lam x. E : T1 => T2
	  <- ({x:exp}			% if  x:T1 |- E : T2.
		of x T1 -> of (E x) T2).

tp_app : of (app E1 E2) T1		% |- E1 E2 : T1
	  <- of E1 (arrow T2 T1)	% if  |- E1 : T2 => T1
	  <- of E2 T2.			% and |- E2 : T2.

% Evaluation (call-by-value) 
% E ==> V  (expression E evaluates to value V)

eval : exp -> exp -> type.		%name eval D.
%mode eval +E -V.

ev_lam  : eval (lam E) (lam E).		% lam x.E ==> lam x.E.

ev_app  : eval (app E1 E2) V		% E1 E2 ==> V
	    <- eval E1 (lam E1')	% if  E1 ==> lam x. E1'
	    <- eval E2 V2		% and E2 ==> V2
	    <- eval (E1' V2) V.		% and [V2/x]E1' ==> V.

% Type inference terminates
%terminates E (of E T).

% %terminates E (eval E V).   % fails for ev_app

% Type preservation
%theorem
tps : forall* {E:exp} {V:exp} {T:tp}
       forall {D:eval E V} {P:of E T}
       exists {Q:of V T}
       true.

%prove 5 D (tps D P Q).

% Applying type preservation
e0 = (app (lam [x] x) (lam [y] y)).
%solve p0 : of e0 T.
%solve d0 : eval e0 V.

% the following line is commented out because tps does not 
% have a proof term any more.
% %solve tps0 : tps d0 p0 Q.      

% Example of regular contexts

exp' : type.

app' : exp' -> exp' -> exp'.
lam' : (exp' -> exp') -> exp'.

cp : exp -> exp' -> type.
cp_app : cp (app D1 D2) (app' E1 E2) 
         <- cp D1 E1
         <- cp D2 E2.
cp_lam : cp (lam ([x:exp] D x)) (lam' ([y:exp'] E y))
         <- ({x:exp} {y:exp'} cp x y -> cp (D x) (E y)). 


%theorem cpt :  forallG (pi {x:exp}{y:exp'}{u:cp x y})
		forall {E:exp}
		exists {E':exp'}{D: cp E E'} 
		true.
%prove 5 {E} (cpt E _ _).