% this proof is incomplete!  
proof example : A & B =
begin
A;   % unjustified
B;   % unjustified
A & B;
end;

% this proof is incomplete!  
proof example2 : (A & B) & (A & B) =
begin
A;    % unjustified
B;    % unjustified
A & B;
(A & B) & (A & B)
end;

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

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

proof distribimpand1 : (A => (B & C)) => (A => B) & (A => C) =
begin
[A => (B & C);
 [A;
  B & C;
  B];
 A => B;
 [A;
  B & C;
  C];
 A => C;
 (A => B) & (A => C)];
(A => (B & C)) => (A => B) & (A => C);
end;

proof distriborimp1 : ((A | B) => C) => (A => C) & (B => C) =
begin
[(A | B) => C;
 [A;
  A | B;
  C];
 A => C;
 [B;
  A | B;
  C];
 B => C;
 (A => C) & (B => C)];
((A | B) => C) => (A => C) & (B => C);
end;