## TPTP Problem File: SEV260^5.p

View Solutions - Solve Problem

```%------------------------------------------------------------------------------
% File     : SEV260^5 : TPTP v7.5.0. Released v4.0.0.
% Domain   : Set Theory (Sets of sets)
% Problem  : TPS problem CLOSED-THM1
% Version  : Especial.
% English  : The inverse image of a closed set under a continuous function is
%            closed.

% Refs     : [Bro09] Brown (2009), Email to Geoff Sutcliffe
% Source   : [Bro09]
% Names    : tps_0474 [Bro09]
%          : CLOSED-THM1 [TPS]

% Status   : Theorem
% Rating   : 0.45 v7.5.0, 0.29 v7.4.0, 0.33 v7.2.0, 0.25 v7.1.0, 0.38 v7.0.0, 0.43 v6.4.0, 0.50 v6.3.0, 0.40 v6.2.0, 0.43 v5.5.0, 0.50 v5.4.0, 0.60 v4.1.0, 0.33 v4.0.1, 0.67 v4.0.0
% Syntax   : Number of formulae    :    3 (   0 unit;   2 type;   0 defn)
%            Number of atoms       :   94 (  12 equality;  78 variable)
%            Maximal formula depth :   23 (   9 average)
%            Number of connectives :   73 (   4   ~;   0   |;  18   &;  34   @)
%                                         (   0 <=>;  17  =>;   0  <=;   0 <~>)
%                                         (   0  ~|;   0  ~&)
%            Number of type conns  :   31 (  31   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :    5 (   2   :;   0   =)
%            Number of variables   :   39 (   4 sgn;  25   !;   2   ?;  12   ^)
%                                         (  39   :;   0  !>;   0  ?*)
%                                         (   0  @-;   0  @+)
% SPC      : TH0_THM_EQU_NAR

% Comments : This problem is from the TPS library. Copyright (c) 2009 The TPS
%            project in the Department of Mathematical Sciences at Carnegie
%          : Polymorphic definitions expanded.
%------------------------------------------------------------------------------
thf(a_type,type,(
a: \$tType )).

thf(b_type,type,(
b: \$tType )).

thf(cCLOSED_THM1_pme,conjecture,(
! [T: ( a > \$o ) > \$o,S: ( b > \$o ) > \$o,Xf: a > b] :
( ( ! [R: a > \$o] :
( ( R
= ( ^ [Xx: a] : \$false ) )
=> ( T @ R ) )
& ! [R: a > \$o] :
( ( R
= ( ^ [Xx: a] : ~ ( \$false ) ) )
=> ( T @ R ) )
& ! [K: ( a > \$o ) > \$o,R: a > \$o] :
( ( ! [Xx: a > \$o] :
( ( K @ Xx )
=> ( T @ Xx ) )
& ( R
= ( ^ [Xx: a] :
? [S0: a > \$o] :
( ( K @ S0 )
& ( S0 @ Xx ) ) ) ) )
=> ( T @ R ) )
& ! [Y: a > \$o,Z: a > \$o,S0: a > \$o] :
( ( ( T @ Y )
& ( T @ Z )
& ( S0
= ( ^ [Xx: a] :
( ( Y @ Xx )
& ( Z @ Xx ) ) ) ) )
=> ( T @ S0 ) )
& ! [R: b > \$o] :
( ( R
= ( ^ [Xx: b] : \$false ) )
=> ( S @ R ) )
& ! [R: b > \$o] :
( ( R
= ( ^ [Xx: b] : ~ ( \$false ) ) )
=> ( S @ R ) )
& ! [K: ( b > \$o ) > \$o,R: b > \$o] :
( ( ! [Xx: b > \$o] :
( ( K @ Xx )
=> ( S @ Xx ) )
& ( R
= ( ^ [Xx: b] :
? [S0: b > \$o] :
( ( K @ S0 )
& ( S0 @ Xx ) ) ) ) )
=> ( S @ R ) )
& ! [Y: b > \$o,Z: b > \$o,S0: b > \$o] :
( ( ( S @ Y )
& ( S @ Z )
& ( S0
= ( ^ [Xx: b] :
( ( Y @ Xx )
& ( Z @ Xx ) ) ) ) )
=> ( S @ S0 ) )
& ! [X: b > \$o] :
( ( S @ X )
=> ! [Y: a > \$o] :
( ( Y
= ( ^ [Xb: a] :
( X @ ( Xf @ Xb ) ) ) )
=> ( T @ Y ) ) ) )
=> ! [X: b > \$o] :
( ! [R: b > \$o] :
( ( R
= ( ^ [Xx: b] :
~ ( X @ Xx ) ) )
=> ( S @ R ) )
=> ! [Y: a > \$o] :
( ( Y
= ( ^ [Xb: a] :
( X @ ( Xf @ Xb ) ) ) )
=> ! [R: a > \$o] :
( ( R
= ( ^ [Xx: a] :
~ ( Y @ Xx ) ) )
=> ( T @ R ) ) ) ) ) )).

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