## TPTP Problem File: SEU753^2.p

View Solutions - Solve Problem

```%------------------------------------------------------------------------------
% File     : SEU753^2 : TPTP v8.1.2. Released v3.7.0.
% Domain   : Set Theory
% Problem  : Typed Set Theory - Laws for Typed Sets - DeMorgan Laws
% Version  : Especial > Reduced > Especial.
% English  : (! A:i.! X:i.in X (powerset A) -> (! Y:i.in Y (powerset A) ->
%            setminus A (binintersect X Y) = binunion (setminus A X) (setminus A Y)))

% Refs     : [Bro08] Brown (2008), Email to G. Sutcliffe
% Source   : [Bro08]
% Names    : ZFC255l [Bro08]

% Status   : Theorem
% Rating   : 0.38 v8.1.0, 0.36 v7.5.0, 0.43 v7.4.0, 0.33 v7.3.0, 0.44 v7.2.0, 0.38 v7.1.0, 0.50 v7.0.0, 0.43 v6.4.0, 0.50 v6.3.0, 0.60 v6.2.0, 0.43 v6.1.0, 0.57 v5.5.0, 0.50 v5.4.0, 0.60 v4.1.0, 0.67 v4.0.1, 1.00 v4.0.0, 0.67 v3.7.0
% Syntax   : Number of formulae    :   18 (   6 unt;  11 typ;   6 def)
%            Number of atoms       :   48 (   8 equ;   0 cnn)
%            Maximal formula atoms :    9 (   6 avg)
%            Number of connectives :  137 (   0   ~;   0   |;   0   &; 108   @)
%                                         (   0 <=>;  29  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   14 (   3 avg)
%            Number of types       :    2 (   0 usr)
%            Number of type conns  :    9 (   9   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :   12 (  11 usr;   6 con; 0-2 aty)
%            Number of variables   :   24 (   0   ^  24   !;   0   ?;  24   :)
% SPC      : TH0_THM_EQU_NAR

%------------------------------------------------------------------------------
thf(in_type,type,
in: \$i > \$i > \$o ).

thf(powerset_type,type,
powerset: \$i > \$i ).

thf(binunion_type,type,
binunion: \$i > \$i > \$i ).

thf(binintersect_type,type,
binintersect: \$i > \$i > \$i ).

thf(setminus_type,type,
setminus: \$i > \$i > \$i ).

thf(binintersectT_lem_type,type,
binintersectT_lem: \$o ).

thf(binintersectT_lem,definition,
( binintersectT_lem
= ( ! [A: \$i,X: \$i] :
( ( in @ X @ ( powerset @ A ) )
=> ! [Y: \$i] :
( ( in @ Y @ ( powerset @ A ) )
=> ( in @ ( binintersect @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).

thf(binunionT_lem_type,type,
binunionT_lem: \$o ).

thf(binunionT_lem,definition,
( binunionT_lem
= ( ! [A: \$i,X: \$i] :
( ( in @ X @ ( powerset @ A ) )
=> ! [Y: \$i] :
( ( in @ Y @ ( powerset @ A ) )
=> ( in @ ( binunion @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).

thf(complementT_lem_type,type,
complementT_lem: \$o ).

thf(complementT_lem,definition,
( complementT_lem
= ( ! [A: \$i,X: \$i] :
( ( in @ X @ ( powerset @ A ) )
=> ( in @ ( setminus @ A @ X ) @ ( powerset @ A ) ) ) ) ) ).

thf(setextT_type,type,
setextT: \$o ).

thf(setextT,definition,
( setextT
= ( ! [A: \$i,X: \$i] :
( ( in @ X @ ( powerset @ A ) )
=> ! [Y: \$i] :
( ( in @ Y @ ( powerset @ A ) )
=> ( ! [Xx: \$i] :
( ( in @ Xx @ A )
=> ( ( in @ Xx @ X )
=> ( in @ Xx @ Y ) ) )
=> ( ! [Xx: \$i] :
( ( in @ Xx @ A )
=> ( ( in @ Xx @ Y )
=> ( in @ Xx @ X ) ) )
=> ( X = Y ) ) ) ) ) ) ) ).

thf(demorgan1a_type,type,
demorgan1a: \$o ).

thf(demorgan1a,definition,
( demorgan1a
= ( ! [A: \$i,X: \$i] :
( ( in @ X @ ( powerset @ A ) )
=> ! [Y: \$i] :
( ( in @ Y @ ( powerset @ A ) )
=> ! [Xx: \$i] :
( ( in @ Xx @ A )
=> ( ( in @ Xx @ ( setminus @ A @ ( binintersect @ X @ Y ) ) )
=> ( in @ Xx @ ( binunion @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) ) ) ) ) ) ) ) ).

thf(demorgan1b_type,type,
demorgan1b: \$o ).

thf(demorgan1b,definition,
( demorgan1b
= ( ! [A: \$i,X: \$i] :
( ( in @ X @ ( powerset @ A ) )
=> ! [Y: \$i] :
( ( in @ Y @ ( powerset @ A ) )
=> ! [Xx: \$i] :
( ( in @ Xx @ A )
=> ( ( in @ Xx @ ( binunion @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) )
=> ( in @ Xx @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ) ).

thf(demorgan1,conjecture,
( binintersectT_lem
=> ( binunionT_lem
=> ( complementT_lem
=> ( setextT
=> ( demorgan1a
=> ( demorgan1b
=> ! [A: \$i,X: \$i] :
( ( in @ X @ ( powerset @ A ) )
=> ! [Y: \$i] :
( ( in @ Y @ ( powerset @ A ) )
=> ( ( setminus @ A @ ( binintersect @ X @ Y ) )
= ( binunion @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) ) ) ) ) ) ) ) ) ) ).

%------------------------------------------------------------------------------
```