%% Sample tutch proofs in first-order logic
%% from lecture notes Section 4.2

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

proof AllEx : !y:t. (!x:t. A(x)) => ?x:t. A(x) =
begin
    [ c : t;
      [ !x:t. A(x);
        A(c);
        ?x:t. A(x) ];
      (!x:t. A(x)) => ?x:t. A(x)];
    !y:t. (!x:t. A(x)) => ?x:t. A(x)
end;