proof demorgan3 : ~(A | B) => ~A & ~B = 
begin
[~(A | B);
 [A;
  A | B;
  F];
 [B;
  A | B;
  F];
 ~A;
 ~B;
 ~A & ~B];
~(A | B) => ~A & ~B;
end;

% proof demorgan4 : ~(A & B) => ~A | ~B = 
% begin
% [~(A & B);
% %%% Options: (1) prove A & B, but you can't.
% %%%          (2) prove ~A or prove ~B
%  [A;
%   %%% stuck, and ~B is symmetric
%   F];
%  ~A;
%  ~A | ~B];
% ~(A & B) => ~A | ~B;
% end;

% proof imp1 : ((A => B) => C) => (A | C) & (B => C) =
% begin
% [(A => B) => C;
%  (A | C);  %% UNJUSTIFIED
%  [B;
%   [A; 
%    B];
%   A => B;
%   C];
%  (B => C);
%  (A | C) & (B => C)];
% ((A => B) => C) => (A | C) & (B => C)
% end;

proof imp2 : ((A | C) & (B => C)) => ((A => B) => C) =
begin
[((A | C) & (B => C));
 A | C;
 B => C;
 [A => B;
  [C;
   C];
  [A;
   B;
   C];
  C];  
 ((A => B) => C)];
((A | C) & (B => C)) => ((A => B) => C);
end;
 
proof orcomm : (A | B) => B | A =
begin
[A | B;
 [A; 
  B | A];
 [B;
  B | A];
 B | A];
(A | B) => B | A;
end;

annotated proof orcomm : (A | B) => B | A =
begin
[x : A | B;
 [a : A; 
  inr a : B | A];
 [b : B;
  inl b : B | A];
 case x of inl a => inr a | inr b => inl b end : B | A];
fn x => case x of inl a => inr a | inr b => inl b end : (A | B) => B | A;
end;

term orcomm : (A | B) => B | A =
fn x => case x of inl a => inr a | inr b => inl b end;

annotated proof curry : ((A & B) => C) => (A => (B => C)) = 
begin
[f : A & B => C;
 [a : A;
  [b : B;
   (a , b) : A & B;
   f (a , b) : C];
  fn b => f (a , b) : B => C];
 fn a => fn b => f (a , b) : A => B => C];
fn f => fn a => fn b => f (a , b) : ((A & B) => C) => A => B => C; 
end;

term curry : ((A & B) => C) => A => B => C = 
fn f => fn a => fn b => f (a , b) : ((A & B) => C) => A => B => C; 

term uncurry : (A => B => C) => ((A & B) => C) =
fn g => fn p => g (fst p) (snd p);