proof k : A => B => A =
begin
[A;
 [B;
  A];
 B => A];
A => B => A;
end;

proof andcomm : (A & B) => B & A =
begin
[A & B;
 A;
 B;
 B & A];
(A & B) => B & A;
end;

proof curry : ((A & B) => C) => A => B => C = 
begin
[A & B => C;
 [A;
  [B;
   A & B;
   C];
  B => C];
 A => B => C];
((A & B) => C) => A => B => C; 
end;

proof distribandor1 : ((A & B) | C) => (A | C) & (B | C) =
begin
[A & B | C;
 [A & B;
  A;
  B;
  A | C;
  B | C;
  (A | C)  &  (B | C)];
 [C;
  A | C;
  B | C;
  (A | C)  &  (B | C)];
 (A | C)  &  (B | C)];
((A & B) | C) => (A | C) & (B | C);
end;

proof clue : (G => (C & B) | (R & H)) => (~B => S) => (R | H) => ~(G & S) => ~(R & C) => ~(B & H) => (~G)
= 
begin
[(G => (C & B) | (R & H));
 [(~B => S);
  [R | H;
   [~(G & S);
    [~(R & C);
     [~(B & H);
      [G;
       (C & B) | (R & H);
       [(C & B);
	C;
	B;
	[R;
	 R & C;
	 F];
	[H;
	 B & H;
	 F];
        F];
       [(R & H);
        H;
	[B;
	 B & H;
	 F];
	~B;
	S;
	(G & S);
        F];       
       F];
      ~G];
     ~(B & H) => (~G)];
    ~(R & C) => ~(B & H) => (~G)];
   ~(G & S) => ~(R & C) => ~(B & H) => (~G)];
  (R | H) => ~(G & S) => ~(R & C) => ~(B & H) => (~G)];
 (~B => S) => (R | H) => ~(G & S) => ~(R & C) => ~(B & H) => (~G)];
(G => (C & B) | (R & H)) => (~B => S) => (R | H) => ~(G & S) => ~(R & C) => ~(B & H) => ~G;
end;