## TPTP Problem File: SEV484^1.p

View Solutions - Solve Problem

```%------------------------------------------------------------------------------
% File     : SEV484^1 : TPTP v7.5.0. Released v7.0.0.
% Domain   : Analysis
% Problem  : CROSS_UNIONS_UNIONS
% Version  : Especial.
% English  :

% Refs     : [Kal16] Kalisyk (2016), Email to Geoff Sutcliffe
% Source   : [Kal16]
% Names    : CROSS_UNIONS_UNIONS_.p [Kal16]

% Status   : Theorem
% Rating   : 1.00 v7.2.0, 0.75 v7.1.0
% Syntax   : Number of formulae    :   18 (   0 unit;   8 type;   0 defn)
%            Number of atoms       :  185 (  12 equality; 122 variable)
%            Maximal formula depth :   18 (  10 average)
%            Number of connectives :  152 (   0   ~;   0   |;   5   &; 147   @)
%                                         (   0 <=>;   0  =>;   0  <=;   0 <~>)
%                                         (   0  ~|;   0  ~&)
%            Number of type conns  :   59 (  59   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :   10 (   8   :;   0   =;   0  @=)
%                                         (   0  !!;   0  ??;   0 @@+;   0 @@-)
%            Number of variables   :   67 (   0 sgn;  40   !;  12   ?;   6   ^)
%                                         (  67   :;   9  !>;   0  ?*)
%                                         (   0  @-;   0  @+)
% SPC      : TH1_THM_EQU_NAR

% Comments : Exported from core HOL Light.
%------------------------------------------------------------------------------
thf('thf_type_type/pair/prod',type,(
'type/pair/prod': \$tType > \$tType > \$tType )).

thf('thf_const_const/trivia/I',type,(
'const/trivia/I':
!>[A: \$tType] :
( A > A ) )).

thf('thf_const_const/sets/UNIONS',type,(
'const/sets/UNIONS':
!>[A: \$tType] :
( ( ( A > \$o ) > \$o ) > A > \$o ) )).

thf('thf_const_const/sets/SETSPEC',type,(
'const/sets/SETSPEC':
!>[A: \$tType] :
( A > \$o > A > \$o ) )).

thf('thf_const_const/sets/IN',type,(
'const/sets/IN':
!>[A: \$tType] :
( A > ( A > \$o ) > \$o ) )).

thf('thf_const_const/sets/GSPEC',type,(
'const/sets/GSPEC':
!>[A: \$tType] :
( ( A > \$o ) > A > \$o ) )).

thf('thf_const_const/sets/CROSS',type,(
'const/sets/CROSS':
!>[A: \$tType,A0: \$tType] :
( ( A > \$o ) > ( A0 > \$o ) > ( 'type/pair/prod' @ A @ A0 ) > \$o ) )).

thf('thf_const_const/pair/,',type,(
'const/pair/,':
!>[A: \$tType,B: \$tType] :
( A > B > ( 'type/pair/prod' @ A @ B ) ) )).

thf('thm/trivia/I_THM_',axiom,(
! [A: \$tType,A0: A] :
( ( 'const/trivia/I' @ A @ A0 )
= A0 ) )).

thf('thm/sets/IN_',axiom,(
! [A: \$tType,P: A > \$o,A0: A] :
( ( 'const/sets/IN' @ A @ A0 @ P )
= ( P @ A0 ) ) )).

thf('thm/sets/IN_UNIONS_',axiom,(
! [A: \$tType,A0: ( A > \$o ) > \$o,A1: A] :
( ( 'const/sets/IN' @ A @ A1 @ ( 'const/sets/UNIONS' @ A @ A0 ) )
= ( ? [A2: A > \$o] :
( ( 'const/sets/IN' @ ( A > \$o ) @ A2 @ A0 )
& ( 'const/sets/IN' @ A @ A1 @ A2 ) ) ) ) )).

thf('thm/sets/IN_CROSS_',axiom,(
! [A: \$tType,A0: \$tType,A1: A,A2: A0,A3: A > \$o,A4: A0 > \$o] :
( ( 'const/sets/IN' @ ( 'type/pair/prod' @ A @ A0 ) @ ( 'const/pair/,' @ A @ A0 @ A1 @ A2 ) @ ( 'const/sets/CROSS' @ A @ A0 @ A3 @ A4 ) )
= ( ( 'const/sets/IN' @ A @ A1 @ A3 )
& ( 'const/sets/IN' @ A0 @ A2 @ A4 ) ) ) )).

thf('thm/sets/IN_ELIM_THM_1',axiom,(
! [A: \$tType,A0: A > \$o,A1: A] :
( ( 'const/sets/IN' @ A @ A1
@ ( 'const/sets/GSPEC' @ A
@ ^ [A2: A] :
? [A3: A] :
( 'const/sets/SETSPEC' @ A @ A2 @ ( A0 @ A3 ) @ A3 ) ) )
= ( A0 @ A1 ) ) )).

thf('thm/sets/UNIONS_GSPEC_0',axiom,(
! [A: \$tType,A0: \$tType,P: A > \$o,A1: A > A0 > \$o] :
( ( 'const/sets/UNIONS' @ A0
@ ( 'const/sets/GSPEC' @ ( A0 > \$o )
@ ^ [A2: A0 > \$o] :
? [A3: A] :
( 'const/sets/SETSPEC' @ ( A0 > \$o ) @ A2 @ ( P @ A3 ) @ ( A1 @ A3 ) ) ) )
= ( 'const/sets/GSPEC' @ A0
@ ^ [A2: A0] :
? [A3: A0] :
( 'const/sets/SETSPEC' @ A0 @ A2
@ ? [A4: A] :
( ( P @ A4 )
& ( 'const/sets/IN' @ A0 @ A3 @ ( A1 @ A4 ) ) )
@ A3 ) ) ) )).

thf('thm/pair/FORALL_PAIR_THM_',axiom,(
! [A: \$tType,A0: \$tType,P: ( 'type/pair/prod' @ A @ A0 ) > \$o] :
( ( ! [A1: 'type/pair/prod' @ A @ A0] :
( P @ A1 ) )
= ( ! [A1: A,A2: A0] :
( P @ ( 'const/pair/,' @ A @ A0 @ A1 @ A2 ) ) ) ) )).

thf('thm/sets/EXTENSION_',axiom,(
! [A: \$tType,A0: A > \$o,A1: A > \$o] :
( ( A0 = A1 )
= ( ! [A2: A] :
( ( 'const/sets/IN' @ A @ A2 @ A0 )
= ( 'const/sets/IN' @ A @ A2 @ A1 ) ) ) ) )).

thf('thm/sets/UNIONS_GSPEC_1',axiom,(
! [A: \$tType,A0: \$tType,A1: \$tType,P: A > A0 > \$o,A2: A > A0 > A1 > \$o] :
( ( 'const/sets/UNIONS' @ A1
@ ( 'const/sets/GSPEC' @ ( A1 > \$o )
@ ^ [A3: A1 > \$o] :
? [A4: A,A5: A0] :
( 'const/sets/SETSPEC' @ ( A1 > \$o ) @ A3 @ ( P @ A4 @ A5 ) @ ( A2 @ A4 @ A5 ) ) ) )
= ( 'const/sets/GSPEC' @ A1
@ ^ [A3: A1] :
? [A4: A1] :
( 'const/sets/SETSPEC' @ A1 @ A3
@ ? [A5: A,A6: A0] :
( ( P @ A5 @ A6 )
& ( 'const/sets/IN' @ A1 @ A4 @ ( A2 @ A5 @ A6 ) ) )
@ A4 ) ) ) )).

thf('thm/sets/CROSS_UNIONS_UNIONS_',conjecture,(
! [A: \$tType,A0: \$tType,A1: ( A > \$o ) > \$o,A2: ( A0 > \$o ) > \$o] :
( ( 'const/sets/CROSS' @ A @ A0 @ ( 'const/sets/UNIONS' @ A @ A1 ) @ ( 'const/sets/UNIONS' @ A0 @ A2 ) )
= ( 'const/sets/UNIONS' @ ( 'type/pair/prod' @ A @ A0 )
@ ( 'const/sets/GSPEC' @ ( ( 'type/pair/prod' @ A @ A0 ) > \$o )
@ ^ [A3: ( 'type/pair/prod' @ A @ A0 ) > \$o] :
? [A4: A > \$o,A5: A0 > \$o] :
( 'const/sets/SETSPEC' @ ( ( 'type/pair/prod' @ A @ A0 ) > \$o ) @ A3
@ ( ( 'const/sets/IN' @ ( A > \$o ) @ A4 @ A1 )
& ( 'const/sets/IN' @ ( A0 > \$o ) @ A5 @ A2 ) )
@ ( 'const/sets/CROSS' @ A @ A0 @ A4 @ A5 ) ) ) ) ) )).

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