## TPTP Problem File: SEV482^1.p

View Solutions - Solve Problem

```%------------------------------------------------------------------------------
% File     : SEV482^1 : TPTP v7.5.0. Released v7.0.0.
% Domain   : Analysis
% Problem  : FORALL_IN_CROSS
% Version  : Especial.
% English  :

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

% Status   : Theorem
% Rating   : 0.25 v7.5.0, 0.67 v7.2.0, 0.75 v7.1.0
% Syntax   : Number of formulae    :    7 (   0 unit;   4 type;   0 defn)
%            Number of atoms       :   59 (   3 equality;  43 variable)
%            Maximal formula depth :   15 (  10 average)
%            Number of connectives :   62 (   0   ~;   0   |;   2   &;  58   @)
%                                         (   0 <=>;   2  =>;   0  <=;   0 <~>)
%                                         (   0  ~|;   0  ~&)
%            Number of type conns  :   18 (  18   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :    6 (   4   :;   0   =;   0  @=)
%                                         (   0  !!;   0  ??;   0 @@+;   0 @@-)
%            Number of variables   :   25 (   0 sgn;  20   !;   0   ?;   0   ^)
%                                         (  25   :;   5  !>;   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/sets/IN',type,(
'const/sets/IN':
!>[A: \$tType] :
( A > ( A > \$o ) > \$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/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/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/FORALL_IN_CROSS_',conjecture,(
! [A: \$tType,A0: \$tType,P: ( 'type/pair/prod' @ A @ A0 ) > \$o,A1: A > \$o,A2: A0 > \$o] :
( ( ! [A3: 'type/pair/prod' @ A @ A0] :
( ( 'const/sets/IN' @ ( 'type/pair/prod' @ A @ A0 ) @ A3 @ ( 'const/sets/CROSS' @ A @ A0 @ A1 @ A2 ) )
=> ( P @ A3 ) ) )
= ( ! [A3: A,A4: A0] :
( ( ( 'const/sets/IN' @ A @ A3 @ A1 )
& ( 'const/sets/IN' @ A0 @ A4 @ A2 ) )
=> ( P @ ( 'const/pair/,' @ A @ A0 @ A3 @ A4 ) ) ) ) ) )).

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