proof simple : !x:t.?y:t.P(x) => P(y) =
begin
[x:t;
 [P(x);
  P(x)];
 P(x) => P(x);
 ?y:t.P(x) => P(y)];
!x:t.?y:t.P(x) => P(y)
end;

term  simple : !x:t.?y:t.P(x) => P(y) =
fn x => (x , fn u => u);
 
proof compose : (!x:t.A(x) => B(x)) => (!x:t.B(x) => C(x)) => (!x:t.A(x) => C(x)) =
begin
[(!x:t.A(x) => B(x));
 [(!x:t.B(x) => C(x));
  [a : t;
   [A(a);
    A(a) => B(a);
    B(a);
    B(a) => C(a);
    C(a)];
   A(a) => C(a)];
  (!x:t.A(x) => C(x))];
  (!x:t.B(x) => C(x)) => (!x:t.A(x) => C(x))];
(!x:t.A(x) => B(x)) => (!x:t.B(x) => C(x)) => (!x:t.A(x) => C(x));
end;

term  compose : (!x:t.A(x) => B(x)) => (!x:t.B(x) => C(x)) => (!x:t.A(x) => C(x)) =
fn f => fn g => fn a => fn x => (g a) ((f a) x);

proof dea1 : (?x:t.A(x) & B(x)) => (?x:t.A(x)) & (?x:t.B(x)) =
begin
[(?x:t.A(x) & B(x));
 [x:t, A(x) & B(x);
  A(x);
  ?x:t.A(x)];
 [x:t, A(x) & B(x);
  B(x);
  ?x:t.B(x)];
 (?x:t.A(x));
 (?x:t.B(x));
 (?x:t.A(x)) & (?x:t.B(x))];
(?x:t.A(x) & B(x)) => (?x:t.A(x)) & (?x:t.B(x));
end;

term dea1 : (?x:t.A(x) & B(x)) => (?x:t.A(x)) & (?x:t.B(x)) =
fn e => let (x , u) = e in ( (x , fst u) , (x , snd u));

% proof dea2 : ((?x:t.A(x)) & (?x:t.B(x))) => (?x:t.A(x) & B(x)) =
% begin
% [((?x:t.A(x)) & (?x:t.B(x)));
%  (?x:t.A(x));
%  (?x:t.B(x));
%  [x : t, A(x);
%   A(x) & B (x);   % UNJUSTIFIED
%   ?x:t.A(x) & B(x)]; 
%  (?x:t.A(x) & B(x))];
% ((?x:t.A(x)) & (?x:t.B(x))) => (?x:t.A(x) & B(x));
% end;

% proof dao1 : (!x:t.A(x) | B(x)) => (!x:t.A(x)) | (!x:t.B(x)) = 
% begin
% [(!x:t.A(x) | B(x));
%  [x:t;
%   A(x) | B(x);
%   [A(x);
%    A(x)];
%   [B(x);
%    A(x)];  %% unjustified
%   A(x)];
%  (!x:t.A(x));
%  (!x:t.A(x)) | (!x:t.B(x))];
% (!x:t.A(x) | B(x)) => (!x:t.A(x)) | (!x:t.B(x));
% end;

proof dao2 : ((!x:t.A(x)) | (!x:t.B(x))) => (!x:t.A(x) | B(x)) =
begin
[((!x:t.A(x)) | (!x:t.B(x)));
 [x:t;
  [(!x:t.A(x));
   A(x);
   A(x) | B(x)];
  [(!x:t.B(x));
   B(x);
   A(x) | B(x)];
  A(x) | B(x)];
 (!x:t.A(x) | B(x))];
((!x:t.A(x)) | (!x:t.B(x))) => (!x:t.A(x) | B(x));
end;

term dao2 :  ((!x:t.A(x)) | (!x:t.B(x))) => (!x:t.A(x) | B(x)) =
fn u => fn x => 
  case u of 
    inl u1 => inl (u1 x)
  | inr u1 => inr (u1 x)
  end; 
 
proof daa1 : (!x:t.A(x) & B(x)) => (!x:t.A(x)) & (!x:t.B(x)) =
begin
[(!x:t.A(x) & B(x));
 [x:t;
  A(x) & B(x);
  A(x)];
 [x:t;
  A(x) & B(x);
  B(x)];
 (!x:t.A(x));
 (!x:t.B(x));
 (!x:t.A(x)) & (!x:t.B(x))];
(!x:t.A(x) & B(x)) => (!x:t.A(x)) & (!x:t.B(x));
end;

term  daa1 : (!x:t.A(x) & B(x)) => (!x:t.A(x)) & (!x:t.B(x)) = 
fn f => (fn x => fst (f x) , fn x => snd (f x));

proof daa2 : ((!x:t.A(x)) & (!x:t.B(x))) => (!x:t.A(x) & B(x)) =
begin
[((!x:t.A(x)) & (!x:t.B(x)));
 (!x:t.A(x));
 (!x:t.B(x));
 [x:t;
  A(x);
  B(x);
  A(x) & B(x)];
 (!x:t.A(x) & B(x))];
((!x:t.A(x)) & (!x:t.B(x))) => (!x:t.A(x) & B(x));
end;

term daa2 : ((!x:t.A(x)) & (!x:t.B(x))) => (!x:t.A(x) & B(x)) =
fn p => fn x => ((fst p) x, (snd p) x);

proof deo1 : (?x:t.A(x) | B(x)) => (?x:t.A(x)) | (?x:t.B(x)) =
begin
[(?x:t.A(x) | B(x));
 [x : t, A(x) | B(x);
  [A(x);
   ?x:t.A(x);
   (?x:t.A(x)) | (?x:t.B(x))];
  [B(x);
   ?x:t.B(x);
   (?x:t.A(x)) | (?x:t.B(x))];
  (?x:t.A(x)) | (?x:t.B(x))];
 (?x:t.A(x)) | (?x:t.B(x))];
(?x:t.A(x) | B(x)) => (?x:t.A(x)) | (?x:t.B(x));
end;

term  deo1 : (?x:t.A(x) | B(x)) => (?x:t.A(x)) | (?x:t.B(x)) = 
fn p => let (x, u) = p in case u of inl u1 => inl (x , u1) | inr u2 => inr (x , u2) end;

proof deo2 : ((?x:t.A(x)) | (?x:t.B(x))) => (?x:t.A(x) | B(x)) =
begin
[((?x:t.A(x)) | (?x:t.B(x)));
 [(?x:t.A(x));
  [x:t , A(x);
   A(x) | B(x);
   (?x:t.A(x) | B(x))];
  (?x:t.A(x) | B(x))];
 [(?x:t.B(x));
  [x:t , B(x);
   A(x) | B(x);
   (?x:t.A(x) | B(x))];
  (?x:t.A(x) | B(x))];
 (?x:t.A(x) | B(x))];
((?x:t.A(x)) | (?x:t.B(x))) => (?x:t.A(x) | B(x));
end;

term  deo2 : ((?x:t.A(x)) | (?x:t.B(x))) => (?x:t.A(x) | B(x)) =
fn v => case v of 
          inl p1 => let (x , u) = p1 in (x , inl u)
	| inr p2 => let (x , u) = p2 in (x , inr u)
	end;
 
proof dm1 : ~(?x:t.A(x)) => !x:t.~A(x) =
begin
[~(?x:t.A(x));
 [x:t;
  [A(x);
   ?x:t.A(x);
   F];
  ~A(x)];
 !x:t.~A(x)];
~(?x:t.A(x)) => !x:t.~A(x);
end;

term dm1 : ~(?x:t.A(x)) => !x:t.~A(x) =
fn k => fn x => fn u => k (x,u);

proof dm2 : (!x:t.~A(x)) => ~(?x:t.A(x)) =
begin
[(!x:t.~A(x));
 [(?x:t.A(x));
  [x:t , A(x);
   ~A(x);
   F];
  F];
 ~(?x:t.A(x))];
(!x:t.~A(x)) => ~(?x:t.A(x));
end;

term dm2 : (!x:t.~A(x)) => ~(?x:t.A(x)) =
fn u => fn p => let (x , v) = p in u x v;