pisigma-0.1.0.1: examples/Universe.pi

:l Maybe.pi

data : Type;
El : data -> Type;

data = ( l : {empty maybe sigma box} ) * 
       case l of {
         empty -> Unit
       | maybe -> [data]
       | sigma -> [(a : data) * (El a -> data)]
       | box -> [^ data] };  

El = \ a -> split a with (la,a') ->
            case la of {
	      empty -> {}
	    | maybe -> Maybe [El a']
	    | sigma -> split a' with (b,c) ->
	      	         [(x:El b)*(El (c x))]
            | box -> [El (! a')] };

unit : data;
unit = ('maybe,('empty,'unit));

un : El unit;
un = ('nothing,'unit);

bool : data;
bool = ('maybe, unit);

tt : El bool;
tt = ('nothing,'unit);

ff : El bool;
ff = ('just,('nothing,'unit));

nat : data;
nat = ('sigma,(bool,\ b -> split b with (lb,b') ->
      		      	     case lb of {
			       nothing -> unit
			     | just -> ('box, [nat]) }));
      		      	     
zero : El nat;
zero = (tt,un);

succ : El nat -> El nat;
succ = \ n -> (ff,n);