## TPTP Problem File: SEV473^1.p

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```%------------------------------------------------------------------------------
% File     : SEV473^1 : TPTP v7.5.0. Released v7.0.0.
% Domain   : Analysis
% Problem  : FINITE_FINITE_UNIONS
% Version  : Especial.
% English  :

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

% Status   : Theorem
% Rating   : 1.00 v7.1.0
% Syntax   : Number of formulae    :   14 (   0 unit;   6 type;   0 defn)
%            Number of atoms       :   92 (   6 equality;  56 variable)
%            Maximal formula depth :   12 (   7 average)
%            Number of connectives :   69 (   1   ~;   1   |;   2   &;  60   @)
%                                         (   0 <=>;   5  =>;   0  <=;   0 <~>)
%                                         (   0  ~|;   0  ~&)
%            Number of type conns  :   36 (  36   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :    8 (   6   :;   0   =;   0  @=)
%                                         (   0  !!;   0  ??;   0 @@+;   0 @@-)
%            Number of variables   :   28 (   0 sgn;  22   !;   0   ?;   0   ^)
%                                         (  28   :;   6  !>;   0  ?*)
%                                         (   0  @-;   0  @+)
% SPC      : TH1_THM_EQU_NAR

% Comments : Exported from core HOL Light.
%------------------------------------------------------------------------------
thf('thf_const_const/sets/UNIONS',type,(
'const/sets/UNIONS':
!>[A: \$tType] :
( ( ( A > \$o ) > \$o ) > A > \$o ) )).

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

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

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

thf('thm/sets/FINITE_INDUCT_',axiom,(
! [A: \$tType,A0: ( A > \$o ) > \$o] :
( ( ( A0 @ ( 'const/sets/EMPTY' @ A ) )
& ! [A1: A,A2: A > \$o] :
( ( A0 @ A2 )
=> ( A0 @ ( 'const/sets/INSERT' @ A @ A1 @ A2 ) ) ) )
=> ! [A1: A > \$o] :
( ( 'const/sets/FINITE' @ A @ A1 )
=> ( A0 @ A1 ) ) ) )).

thf('thm/sets/FINITE_UNION_',axiom,(
! [A: \$tType,A0: A > \$o,A1: A > \$o] :
( ( 'const/sets/FINITE' @ A @ ( 'const/sets/UNION' @ A @ A0 @ A1 ) )
= ( ( 'const/sets/FINITE' @ A @ A0 )
& ( 'const/sets/FINITE' @ A @ A1 ) ) ) )).

thf('thm/sets/FINITE_RULES_0',axiom,(
! [A: \$tType] :
( 'const/sets/FINITE' @ A @ ( 'const/sets/EMPTY' @ A ) ) )).

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

thf('thm/sets/UNIONS_INSERT_',axiom,(
! [A: \$tType,A0: A > \$o,A1: ( A > \$o ) > \$o] :
( ( 'const/sets/UNIONS' @ A @ ( 'const/sets/INSERT' @ ( A > \$o ) @ A0 @ A1 ) )
= ( 'const/sets/UNION' @ A @ A0 @ ( 'const/sets/UNIONS' @ A @ A1 ) ) ) )).

thf('thm/sets/NOT_IN_EMPTY_',axiom,(
! [A: \$tType,A0: A] :
~ ( 'const/sets/IN' @ A @ A0 @ ( 'const/sets/EMPTY' @ A ) ) )).

thf('thm/sets/UNIONS_0_',axiom,(
! [A: \$tType] :
( ( 'const/sets/UNIONS' @ A @ ( 'const/sets/EMPTY' @ ( A > \$o ) ) )
= ( 'const/sets/EMPTY' @ A ) ) )).

thf('thm/sets/FINITE_FINITE_UNIONS_',conjecture,(
! [A: \$tType,A0: ( A > \$o ) > \$o] :
( ( 'const/sets/FINITE' @ ( A > \$o ) @ A0 )
=> ( ( 'const/sets/FINITE' @ A @ ( 'const/sets/UNIONS' @ A @ A0 ) )
= ( ! [A1: A > \$o] :
( ( 'const/sets/IN' @ ( A > \$o ) @ A1 @ A0 )
=> ( 'const/sets/FINITE' @ A @ A1 ) ) ) ) ) )).

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