## TPTP Problem File: SEV474^1.p

View Solutions - Solve Problem

```%------------------------------------------------------------------------------
% File     : SEV474^1 : TPTP v7.5.0. Released v7.0.0.
% Domain   : Analysis
% Problem  : INFINITE_IMAGE
% Version  : Especial.
% English  :

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

% Status   : Theorem
% Rating   : 1.00 v7.2.0, 0.75 v7.1.0
% Syntax   : Number of formulae    :   13 (   0 unit;   5 type;   0 defn)
%            Number of atoms       :  111 (  14 equality;  77 variable)
%            Maximal formula depth :   15 (   8 average)
%            Number of connectives :   76 (   1   ~;   0   |;   6   &;  64   @)
%                                         (   0 <=>;   5  =>;   0  <=;   0 <~>)
%                                         (   0  ~|;   0  ~&)
%            Number of type conns  :   26 (  26   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :    7 (   5   :;   0   =;   0  @=)
%                                         (   0  !!;   0  ??;   0 @@+;   0 @@-)
%            Number of variables   :   41 (   0 sgn;  33   !;   2   ?;   0   ^)
%                                         (  41   :;   6  !>;   0  ?*)
%                                         (   0  @-;   0  @+)
% SPC      : TH1_THM_EQU_NAR

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

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

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

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

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

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_IMAGE_',axiom,(
! [B: \$tType,A: \$tType,A0: B,A1: A > \$o,A2: A > B] :
( ( 'const/sets/IN' @ B @ A0 @ ( 'const/sets/IMAGE' @ A @ B @ A2 @ A1 ) )
= ( ? [A3: A] :
( ( A0
= ( A2 @ A3 ) )
& ( 'const/sets/IN' @ A @ A3 @ A1 ) ) ) ) )).

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/FINITE_IMAGE_',axiom,(
! [B: \$tType,A: \$tType,A0: A > B,A1: A > \$o] :
( ( 'const/sets/FINITE' @ A @ A1 )
=> ( 'const/sets/FINITE' @ B @ ( 'const/sets/IMAGE' @ A @ B @ A0 @ A1 ) ) ) )).

thf('thm/sets/INFINITE_',axiom,(
! [A: \$tType,A0: A > \$o] :
( ( 'const/sets/INFINITE' @ A @ A0 )
= ( ~ ( 'const/sets/FINITE' @ A @ A0 ) ) ) )).

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

thf('thm/sets/INFINITE_IMAGE_',conjecture,(
! [B: \$tType,A: \$tType,A0: A > B,A1: A > \$o] :
( ( ( 'const/sets/INFINITE' @ A @ A1 )
& ! [A2: A,A3: A] :
( ( ( 'const/sets/IN' @ A @ A2 @ A1 )
& ( 'const/sets/IN' @ A @ A3 @ A1 )
& ( ( A0 @ A2 )
= ( A0 @ A3 ) ) )
=> ( A2 = A3 ) ) )
=> ( 'const/sets/INFINITE' @ B @ ( 'const/sets/IMAGE' @ A @ B @ A0 @ A1 ) ) ) )).

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