TPTP Problem File: ANA090^1.p
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% File : ANA090^1 : TPTP v8.2.0. Released v7.0.0.
% Domain : Analysis
% Problem : REAL_SUP_EQ_INF
% Version : Especial.
% English :
% Refs : [Kal16] Kalisyk (2016), Email to Geoff Sutcliffe
% Source : [Kal16]
% Names : REAL_SUP_EQ_INF_.p [Kal16]
% Status : Theorem
% Rating : 0.33 v8.1.0, 0.75 v7.5.0, 1.00 v7.1.0
% Syntax : Number of formulae : 25 ( 5 unt; 13 typ; 0 def)
% Number of atoms : 61 ( 19 equ; 0 cnn)
% Maximal formula atoms : 8 ( 5 avg)
% Number of connectives : 145 ( 4 ~; 1 |; 7 &; 119 @)
% ( 0 <=>; 14 =>; 0 <=; 0 <~>)
% Maximal formula depth : 11 ( 6 avg)
% Number of types : 2 ( 1 usr)
% Number of type conns : 34 ( 34 >; 0 *; 0 +; 0 <<)
% Number of symbols : 13 ( 12 usr; 0 con; 1-4 aty)
% Number of variables : 43 ( 0 ^; 34 !; 4 ?; 43 :)
% ( 5 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : Exported from core HOL Light.
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thf('thf_type_type/realax/real',type,
'type/realax/real': $tType ).
thf('thf_const_const/sets/sup',type,
'const/sets/sup': ( 'type/realax/real' > $o ) > 'type/realax/real' ).
thf('thf_const_const/sets/INSERT',type,
'const/sets/INSERT':
!>[A: $tType] : ( A > ( A > $o ) > A > $o ) ).
thf('thf_const_const/sets/inf',type,
'const/sets/inf': ( 'type/realax/real' > $o ) > 'type/realax/real' ).
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('thf_const_const/realax/real_neg',type,
'const/realax/real_neg': 'type/realax/real' > 'type/realax/real' ).
thf('thf_const_const/realax/real_min',type,
'const/realax/real_min': 'type/realax/real' > 'type/realax/real' > 'type/realax/real' ).
thf('thf_const_const/realax/real_max',type,
'const/realax/real_max': 'type/realax/real' > 'type/realax/real' > 'type/realax/real' ).
thf('thf_const_const/realax/real_le',type,
'const/realax/real_le': 'type/realax/real' > 'type/realax/real' > $o ).
thf('thf_const_const/realax/real_abs',type,
'const/realax/real_abs': 'type/realax/real' > 'type/realax/real' ).
thf('thf_const_const/class/COND',type,
'const/class/COND':
!>[A: $tType] : ( $o > A > A > A ) ).
thf('thm/sets/INF_INSERT_FINITE_',axiom,
! [A: 'type/realax/real',A0: 'type/realax/real' > $o] :
( ( 'const/sets/FINITE' @ 'type/realax/real' @ A0 )
=> ( ( 'const/sets/inf' @ ( 'const/sets/INSERT' @ 'type/realax/real' @ A @ A0 ) )
= ( 'const/class/COND' @ 'type/realax/real'
@ ( A0
= ( 'const/sets/EMPTY' @ 'type/realax/real' ) )
@ A
@ ( 'const/realax/real_min' @ A @ ( 'const/sets/inf' @ A0 ) ) ) ) ) ).
thf('thm/sets/FINITE_EMPTY_',axiom,
! [A: $tType] : ( 'const/sets/FINITE' @ A @ ( 'const/sets/EMPTY' @ A ) ) ).
thf('thm/sets/SUP_INSERT_FINITE_',axiom,
! [A: 'type/realax/real',A0: 'type/realax/real' > $o] :
( ( 'const/sets/FINITE' @ 'type/realax/real' @ A0 )
=> ( ( 'const/sets/sup' @ ( 'const/sets/INSERT' @ 'type/realax/real' @ A @ A0 ) )
= ( 'const/class/COND' @ 'type/realax/real'
@ ( A0
= ( 'const/sets/EMPTY' @ 'type/realax/real' ) )
@ A
@ ( 'const/realax/real_max' @ A @ ( 'const/sets/sup' @ A0 ) ) ) ) ) ).
thf('thm/sets/NOT_IN_EMPTY_',axiom,
! [A: $tType,A0: A] :
~ ( 'const/sets/IN' @ A @ A0 @ ( '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/IN_',axiom,
! [A: $tType,P: A > $o,A0: A] :
( ( 'const/sets/IN' @ A @ A0 @ P )
= ( P @ A0 ) ) ).
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/INF_',axiom,
! [A: 'type/realax/real' > $o] :
( ( ( A
!= ( 'const/sets/EMPTY' @ 'type/realax/real' ) )
& ? [A0: 'type/realax/real'] :
! [A1: 'type/realax/real'] :
( ( 'const/sets/IN' @ 'type/realax/real' @ A1 @ A )
=> ( 'const/realax/real_le' @ A0 @ A1 ) ) )
=> ( ! [A0: 'type/realax/real'] :
( ( 'const/sets/IN' @ 'type/realax/real' @ A0 @ A )
=> ( 'const/realax/real_le' @ ( 'const/sets/inf' @ A ) @ A0 ) )
& ! [A0: 'type/realax/real'] :
( ! [A1: 'type/realax/real'] :
( ( 'const/sets/IN' @ 'type/realax/real' @ A1 @ A )
=> ( 'const/realax/real_le' @ A0 @ A1 ) )
=> ( 'const/realax/real_le' @ A0 @ ( 'const/sets/inf' @ A ) ) ) ) ) ).
thf('thm/real/REAL_ABS_BOUNDS_',axiom,
! [A: 'type/realax/real',A0: 'type/realax/real'] :
( ( 'const/realax/real_le' @ ( 'const/realax/real_abs' @ A ) @ A0 )
= ( ( 'const/realax/real_le' @ ( 'const/realax/real_neg' @ A0 ) @ A )
& ( 'const/realax/real_le' @ A @ A0 ) ) ) ).
thf('thm/sets/SUP_',axiom,
! [A: 'type/realax/real' > $o] :
( ( ( A
!= ( 'const/sets/EMPTY' @ 'type/realax/real' ) )
& ? [A0: 'type/realax/real'] :
! [A1: 'type/realax/real'] :
( ( 'const/sets/IN' @ 'type/realax/real' @ A1 @ A )
=> ( 'const/realax/real_le' @ A1 @ A0 ) ) )
=> ( ! [A0: 'type/realax/real'] :
( ( 'const/sets/IN' @ 'type/realax/real' @ A0 @ A )
=> ( 'const/realax/real_le' @ A0 @ ( 'const/sets/sup' @ A ) ) )
& ! [A0: 'type/realax/real'] :
( ! [A1: 'type/realax/real'] :
( ( 'const/sets/IN' @ 'type/realax/real' @ A1 @ A )
=> ( 'const/realax/real_le' @ A1 @ A0 ) )
=> ( 'const/realax/real_le' @ ( 'const/sets/sup' @ A ) @ A0 ) ) ) ) ).
thf('thm/realax/REAL_LE_ANTISYM_',axiom,
! [A: 'type/realax/real',A0: 'type/realax/real'] :
( ( ( 'const/realax/real_le' @ A @ A0 )
& ( 'const/realax/real_le' @ A0 @ A ) )
= ( A = A0 ) ) ).
thf('thm/sets/REAL_SUP_EQ_INF_',conjecture,
! [A: 'type/realax/real' > $o] :
( ( ( A
!= ( 'const/sets/EMPTY' @ 'type/realax/real' ) )
& ? [B: 'type/realax/real'] :
! [A0: 'type/realax/real'] :
( ( 'const/sets/IN' @ 'type/realax/real' @ A0 @ A )
=> ( 'const/realax/real_le' @ ( 'const/realax/real_abs' @ A0 ) @ B ) ) )
=> ( ( ( 'const/sets/sup' @ A )
= ( 'const/sets/inf' @ A ) )
= ( ? [A0: 'type/realax/real'] :
( A
= ( 'const/sets/INSERT' @ 'type/realax/real' @ A0 @ ( 'const/sets/EMPTY' @ 'type/realax/real' ) ) ) ) ) ) ).
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