TPTP Problem File: ITP023^2.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : ITP023^2 : TPTP v8.2.0. Bugfixed v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : HOL4 set theory export of thm_2Ereal__topology_2EBOUNDED__BALL.p, bushy mode
% Version : [BG+19] axioms.
% English :
% Refs : [BG+19] Brown et al. (2019), GRUNGE: A Grand Unified ATP Chall
% : [Gau19] Gauthier (2019), Email to Geoff Sutcliffe
% Source : [BG+19]
% Names : thm_2Ereal__topology_2EBOUNDED__BALL.p [Gau19]
% : HL411001^2.p [TPAP]
% Status : Theorem
% Rating : 0.40 v8.2.0, 0.46 v8.1.0, 0.36 v7.5.0
% Syntax : Number of formulae : 93 ( 8 unt; 31 typ; 0 def)
% Number of atoms : 410 ( 12 equ; 0 cnn)
% Maximal formula atoms : 21 ( 6 avg)
% Number of connectives : 757 ( 47 ~; 34 |; 27 &; 511 @)
% ( 44 <=>; 94 =>; 0 <=; 0 <~>)
% Maximal formula depth : 15 ( 8 avg)
% Number of types : 5 ( 3 usr)
% Number of type conns : 25 ( 25 >; 0 *; 0 +; 0 <<)
% Number of symbols : 36 ( 33 usr; 19 con; 0-2 aty)
% Number of variables : 97 ( 0 ^; 95 !; 2 ?; 97 :)
% SPC : TH0_THM_EQU_NAR
% Comments :
% Bugfixes : v7.5.0 - Bugfixes in axioms and export.
%------------------------------------------------------------------------------
include('Axioms/ITP001/ITP001^2.ax').
%------------------------------------------------------------------------------
thf(tp_c_2Ebool_2ET,type,
c_2Ebool_2ET: $i ).
thf(mem_c_2Ebool_2ET,axiom,
mem @ c_2Ebool_2ET @ bool ).
thf(ax_true_p,axiom,
p @ c_2Ebool_2ET ).
thf(tp_c_2Ebool_2E_3F,type,
c_2Ebool_2E_3F: del > $i ).
thf(mem_c_2Ebool_2E_3F,axiom,
! [A_27a: del] : ( mem @ ( c_2Ebool_2E_3F @ A_27a ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) ).
thf(ax_ex_p,axiom,
! [A: del,Q: $i] :
( ( mem @ Q @ ( arr @ A @ bool ) )
=> ( ( p @ ( ap @ ( c_2Ebool_2E_3F @ A ) @ Q ) )
<=> ? [X: $i] :
( ( mem @ X @ A )
& ( p @ ( ap @ Q @ X ) ) ) ) ) ).
thf(tp_ty_2Erealax_2Ereal,type,
ty_2Erealax_2Ereal: del ).
thf(stp_ty_2Erealax_2Ereal,type,
tp__ty_2Erealax_2Ereal: $tType ).
thf(stp_inj_ty_2Erealax_2Ereal,type,
inj__ty_2Erealax_2Ereal: tp__ty_2Erealax_2Ereal > $i ).
thf(stp_surj_ty_2Erealax_2Ereal,type,
surj__ty_2Erealax_2Ereal: $i > tp__ty_2Erealax_2Ereal ).
thf(stp_inj_surj_ty_2Erealax_2Ereal,axiom,
! [X: tp__ty_2Erealax_2Ereal] :
( ( surj__ty_2Erealax_2Ereal @ ( inj__ty_2Erealax_2Ereal @ X ) )
= X ) ).
thf(stp_inj_mem_ty_2Erealax_2Ereal,axiom,
! [X: tp__ty_2Erealax_2Ereal] : ( mem @ ( inj__ty_2Erealax_2Ereal @ X ) @ ty_2Erealax_2Ereal ) ).
thf(stp_iso_mem_ty_2Erealax_2Ereal,axiom,
! [X: $i] :
( ( mem @ X @ ty_2Erealax_2Ereal )
=> ( X
= ( inj__ty_2Erealax_2Ereal @ ( surj__ty_2Erealax_2Ereal @ X ) ) ) ) ).
thf(tp_ty_2Epair_2Eprod,type,
ty_2Epair_2Eprod: del > del > del ).
thf(stp_c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal,type,
tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal: $tType ).
thf(stp_inj_c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal,type,
inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal > $i ).
thf(stp_surj_c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal,type,
surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal: $i > tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal ).
thf(stp_inj_surj_c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal,axiom,
! [X: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal] :
( ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ X ) )
= X ) ).
thf(stp_inj_mem_c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal,axiom,
! [X: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal] : ( mem @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ X ) @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) ).
thf(stp_iso_mem_c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal,axiom,
! [X: $i] :
( ( mem @ X @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ( X
= ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ X ) ) ) ) ).
thf(tp_c_2Ereal__topology_2Eball,type,
c_2Ereal__topology_2Eball: $i ).
thf(mem_c_2Ereal__topology_2Eball,axiom,
mem @ c_2Ereal__topology_2Eball @ ( arr @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) ).
thf(tp_c_2Epred__set_2ESUBSET,type,
c_2Epred__set_2ESUBSET: del > $i ).
thf(mem_c_2Epred__set_2ESUBSET,axiom,
! [A_27a: del] : ( mem @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ ( arr @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) ) ).
thf(tp_c_2Epair_2E_2C,type,
c_2Epair_2E_2C: del > del > $i ).
thf(mem_c_2Epair_2E_2C,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2Epair_2E_2C @ A_27a @ A_27b ) @ ( arr @ A_27a @ ( arr @ A_27b @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) ) ) ) ).
thf(tp_c_2Ereal__topology_2Ecball,type,
c_2Ereal__topology_2Ecball: $i ).
thf(mem_c_2Ereal__topology_2Ecball,axiom,
mem @ c_2Ereal__topology_2Ecball @ ( arr @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) ).
thf(tp_c_2Ereal__topology_2Ebounded__def,type,
c_2Ereal__topology_2Ebounded__def: $i ).
thf(mem_c_2Ereal__topology_2Ebounded__def,axiom,
mem @ c_2Ereal__topology_2Ebounded__def @ ( arr @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ bool ) ).
thf(tp_c_2Ebool_2EF,type,
c_2Ebool_2EF: $i ).
thf(mem_c_2Ebool_2EF,axiom,
mem @ c_2Ebool_2EF @ bool ).
thf(ax_false_p,axiom,
~ ( p @ c_2Ebool_2EF ) ).
thf(tp_c_2Emin_2E_3D_3D_3E,type,
c_2Emin_2E_3D_3D_3E: $i ).
thf(mem_c_2Emin_2E_3D_3D_3E,axiom,
mem @ c_2Emin_2E_3D_3D_3E @ ( arr @ bool @ ( arr @ bool @ bool ) ) ).
thf(ax_imp_p,axiom,
! [Q: $i] :
( ( mem @ Q @ bool )
=> ! [R: $i] :
( ( mem @ R @ bool )
=> ( ( p @ ( ap @ ( ap @ c_2Emin_2E_3D_3D_3E @ Q ) @ R ) )
<=> ( ( p @ Q )
=> ( p @ R ) ) ) ) ) ).
thf(tp_c_2Ebool_2E_5C_2F,type,
c_2Ebool_2E_5C_2F: $i ).
thf(mem_c_2Ebool_2E_5C_2F,axiom,
mem @ c_2Ebool_2E_5C_2F @ ( arr @ bool @ ( arr @ bool @ bool ) ) ).
thf(ax_or_p,axiom,
! [Q: $i] :
( ( mem @ Q @ bool )
=> ! [R: $i] :
( ( mem @ R @ bool )
=> ( ( p @ ( ap @ ( ap @ c_2Ebool_2E_5C_2F @ Q ) @ R ) )
<=> ( ( p @ Q )
| ( p @ R ) ) ) ) ) ).
thf(tp_c_2Ebool_2E_2F_5C,type,
c_2Ebool_2E_2F_5C: $i ).
thf(mem_c_2Ebool_2E_2F_5C,axiom,
mem @ c_2Ebool_2E_2F_5C @ ( arr @ bool @ ( arr @ bool @ bool ) ) ).
thf(ax_and_p,axiom,
! [Q: $i] :
( ( mem @ Q @ bool )
=> ! [R: $i] :
( ( mem @ R @ bool )
=> ( ( p @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ Q ) @ R ) )
<=> ( ( p @ Q )
& ( p @ R ) ) ) ) ) ).
thf(tp_c_2Ebool_2E_7E,type,
c_2Ebool_2E_7E: $i ).
thf(mem_c_2Ebool_2E_7E,axiom,
mem @ c_2Ebool_2E_7E @ ( arr @ bool @ bool ) ).
thf(ax_neg_p,axiom,
! [Q: $i] :
( ( mem @ Q @ bool )
=> ( ( p @ ( ap @ c_2Ebool_2E_7E @ Q ) )
<=> ~ ( p @ Q ) ) ) ).
thf(tp_c_2Emin_2E_3D,type,
c_2Emin_2E_3D: del > $i ).
thf(mem_c_2Emin_2E_3D,axiom,
! [A_27a: del] : ( mem @ ( c_2Emin_2E_3D @ A_27a ) @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) ) ).
thf(ax_eq_p,axiom,
! [A: del,X: $i] :
( ( mem @ X @ A )
=> ! [Y: $i] :
( ( mem @ Y @ A )
=> ( ( p @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ A ) @ X ) @ Y ) )
<=> ( X = Y ) ) ) ) ).
thf(tp_c_2Ebool_2E_21,type,
c_2Ebool_2E_21: del > $i ).
thf(mem_c_2Ebool_2E_21,axiom,
! [A_27a: del] : ( mem @ ( c_2Ebool_2E_21 @ A_27a ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) ).
thf(ax_all_p,axiom,
! [A: del,Q: $i] :
( ( mem @ Q @ ( arr @ A @ bool ) )
=> ( ( p @ ( ap @ ( c_2Ebool_2E_21 @ A ) @ Q ) )
<=> ! [X: $i] :
( ( mem @ X @ A )
=> ( p @ ( ap @ Q @ X ) ) ) ) ) ).
thf(conj_thm_2Ebool_2ETRUTH,axiom,
$true ).
thf(conj_thm_2Ebool_2EIMP__ANTISYM__AX,axiom,
! [V0t1: $i] :
( ( mem @ V0t1 @ bool )
=> ! [V1t2: $i] :
( ( mem @ V1t2 @ bool )
=> ( ( ( p @ V0t1 )
=> ( p @ V1t2 ) )
=> ( ( ( p @ V1t2 )
=> ( p @ V0t1 ) )
=> ( ( p @ V0t1 )
<=> ( p @ V1t2 ) ) ) ) ) ) ).
thf(conj_thm_2Ebool_2EIMP__F,axiom,
! [V0t: $i] :
( ( mem @ V0t @ bool )
=> ( ( ( p @ V0t )
=> $false )
=> ~ ( p @ V0t ) ) ) ).
thf(conj_thm_2Ebool_2EF__IMP,axiom,
! [V0t: $i] :
( ( mem @ V0t @ bool )
=> ( ~ ( p @ V0t )
=> ( ( p @ V0t )
=> $false ) ) ) ).
thf(conj_thm_2Ebool_2EIMP__CLAUSES,axiom,
! [V0t: $i] :
( ( mem @ V0t @ bool )
=> ( ( ( $true
=> ( p @ V0t ) )
<=> ( p @ V0t ) )
& ( ( ( p @ V0t )
=> $true )
<=> $true )
& ( ( $false
=> ( p @ V0t ) )
<=> $true )
& ( ( ( p @ V0t )
=> ( p @ V0t ) )
<=> $true )
& ( ( ( p @ V0t )
=> $false )
<=> ~ ( p @ V0t ) ) ) ) ).
thf(conj_thm_2Ebool_2ENOT__CLAUSES,axiom,
( ! [V0t: $i] :
( ( mem @ V0t @ bool )
=> ( ~ ~ ( p @ V0t )
<=> ( p @ V0t ) ) )
& ( ~ $true
<=> $false )
& ( ~ $false
<=> $true ) ) ).
thf(conj_thm_2Ebool_2EEQ__SYM__EQ,axiom,
! [A_27a: del,V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ! [V1y: $i] :
( ( mem @ V1y @ A_27a )
=> ( ( V0x = V1y )
<=> ( V1y = V0x ) ) ) ) ).
thf(conj_thm_2Ebool_2EEQ__CLAUSES,axiom,
! [V0t: $i] :
( ( mem @ V0t @ bool )
=> ( ( ( $true
<=> ( p @ V0t ) )
<=> ( p @ V0t ) )
& ( ( ( p @ V0t )
<=> $true )
<=> ( p @ V0t ) )
& ( ( $false
<=> ( p @ V0t ) )
<=> ~ ( p @ V0t ) )
& ( ( ( p @ V0t )
<=> $false )
<=> ~ ( p @ V0t ) ) ) ) ).
thf(conj_thm_2Ebool_2ENOT__FORALL__THM,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ( ~ ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ( p @ ( ap @ V0P @ V1x ) ) )
<=> ? [V2x: $i] :
( ( mem @ V2x @ A_27a )
& ~ ( p @ ( ap @ V0P @ V2x ) ) ) ) ) ).
thf(conj_thm_2Ebool_2EDISJ__ASSOC,axiom,
! [V0A: $i] :
( ( mem @ V0A @ bool )
=> ! [V1B: $i] :
( ( mem @ V1B @ bool )
=> ! [V2C: $i] :
( ( mem @ V2C @ bool )
=> ( ( ( p @ V0A )
| ( p @ V1B )
| ( p @ V2C ) )
<=> ( ( p @ V0A )
| ( p @ V1B )
| ( p @ V2C ) ) ) ) ) ) ).
thf(conj_thm_2Ebool_2EDISJ__SYM,axiom,
! [V0A: $i] :
( ( mem @ V0A @ bool )
=> ! [V1B: $i] :
( ( mem @ V1B @ bool )
=> ( ( ( p @ V0A )
| ( p @ V1B ) )
<=> ( ( p @ V1B )
| ( p @ V0A ) ) ) ) ) ).
thf(conj_thm_2Ebool_2EDE__MORGAN__THM,axiom,
! [V0A: $i] :
( ( mem @ V0A @ bool )
=> ! [V1B: $i] :
( ( mem @ V1B @ bool )
=> ( ( ~ ( ( p @ V0A )
& ( p @ V1B ) )
<=> ( ~ ( p @ V0A )
| ~ ( p @ V1B ) ) )
& ( ~ ( ( p @ V0A )
| ( p @ V1B ) )
<=> ( ~ ( p @ V0A )
& ~ ( p @ V1B ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EBALL__SUBSET__CBALL,axiom,
! [V0x: tp__ty_2Erealax_2Ereal,V1e: tp__ty_2Erealax_2Ereal] : ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal__topology_2Eball @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( inj__ty_2Erealax_2Ereal @ V1e ) ) ) ) @ ( ap @ c_2Ereal__topology_2Ecball @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( inj__ty_2Erealax_2Ereal @ V1e ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EBOUNDED__SUBSET,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ c_2Ereal__topology_2Ebounded__def @ V1t ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V0s ) @ V1t ) ) )
=> ( p @ ( ap @ c_2Ereal__topology_2Ebounded__def @ V0s ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EBOUNDED__CBALL,axiom,
! [V0x: tp__ty_2Erealax_2Ereal,V1e: tp__ty_2Erealax_2Ereal] : ( p @ ( ap @ c_2Ereal__topology_2Ebounded__def @ ( ap @ c_2Ereal__topology_2Ecball @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( inj__ty_2Erealax_2Ereal @ V1e ) ) ) ) ) ).
thf(conj_thm_2Esat_2ENOT__NOT,axiom,
! [V0t: $i] :
( ( mem @ V0t @ bool )
=> ( ~ ~ ( p @ V0t )
<=> ( p @ V0t ) ) ) ).
thf(conj_thm_2Esat_2EAND__INV__IMP,axiom,
! [V0A: $i] :
( ( mem @ V0A @ bool )
=> ( ( p @ V0A )
=> ( ~ ( p @ V0A )
=> $false ) ) ) ).
thf(conj_thm_2Esat_2EOR__DUAL2,axiom,
! [V0A: $i] :
( ( mem @ V0A @ bool )
=> ! [V1B: $i] :
( ( mem @ V1B @ bool )
=> ( ( ~ ( ( p @ V0A )
| ( p @ V1B ) )
=> $false )
<=> ( ( ( p @ V0A )
=> $false )
=> ( ~ ( p @ V1B )
=> $false ) ) ) ) ) ).
thf(conj_thm_2Esat_2EOR__DUAL3,axiom,
! [V0A: $i] :
( ( mem @ V0A @ bool )
=> ! [V1B: $i] :
( ( mem @ V1B @ bool )
=> ( ( ~ ( ~ ( p @ V0A )
| ( p @ V1B ) )
=> $false )
<=> ( ( p @ V0A )
=> ( ~ ( p @ V1B )
=> $false ) ) ) ) ) ).
thf(conj_thm_2Esat_2EAND__INV2,axiom,
! [V0A: $i] :
( ( mem @ V0A @ bool )
=> ( ( ~ ( p @ V0A )
=> $false )
=> ( ( ( p @ V0A )
=> $false )
=> $false ) ) ) ).
thf(conj_thm_2Esat_2Edc__eq,axiom,
! [V0p: $i] :
( ( mem @ V0p @ bool )
=> ! [V1q: $i] :
( ( mem @ V1q @ bool )
=> ! [V2r: $i] :
( ( mem @ V2r @ bool )
=> ( ( ( p @ V0p )
<=> ( ( p @ V1q )
<=> ( p @ V2r ) ) )
<=> ( ( ( p @ V0p )
| ( p @ V1q )
| ( p @ V2r ) )
& ( ( p @ V0p )
| ~ ( p @ V2r )
| ~ ( p @ V1q ) )
& ( ( p @ V1q )
| ~ ( p @ V2r )
| ~ ( p @ V0p ) )
& ( ( p @ V2r )
| ~ ( p @ V1q )
| ~ ( p @ V0p ) ) ) ) ) ) ) ).
thf(conj_thm_2Esat_2Edc__conj,axiom,
! [V0p: $i] :
( ( mem @ V0p @ bool )
=> ! [V1q: $i] :
( ( mem @ V1q @ bool )
=> ! [V2r: $i] :
( ( mem @ V2r @ bool )
=> ( ( ( p @ V0p )
<=> ( ( p @ V1q )
& ( p @ V2r ) ) )
<=> ( ( ( p @ V0p )
| ~ ( p @ V1q )
| ~ ( p @ V2r ) )
& ( ( p @ V1q )
| ~ ( p @ V0p ) )
& ( ( p @ V2r )
| ~ ( p @ V0p ) ) ) ) ) ) ) ).
thf(conj_thm_2Esat_2Edc__disj,axiom,
! [V0p: $i] :
( ( mem @ V0p @ bool )
=> ! [V1q: $i] :
( ( mem @ V1q @ bool )
=> ! [V2r: $i] :
( ( mem @ V2r @ bool )
=> ( ( ( p @ V0p )
<=> ( ( p @ V1q )
| ( p @ V2r ) ) )
<=> ( ( ( p @ V0p )
| ~ ( p @ V1q ) )
& ( ( p @ V0p )
| ~ ( p @ V2r ) )
& ( ( p @ V1q )
| ( p @ V2r )
| ~ ( p @ V0p ) ) ) ) ) ) ) ).
thf(conj_thm_2Esat_2Edc__imp,axiom,
! [V0p: $i] :
( ( mem @ V0p @ bool )
=> ! [V1q: $i] :
( ( mem @ V1q @ bool )
=> ! [V2r: $i] :
( ( mem @ V2r @ bool )
=> ( ( ( p @ V0p )
<=> ( ( p @ V1q )
=> ( p @ V2r ) ) )
<=> ( ( ( p @ V0p )
| ( p @ V1q ) )
& ( ( p @ V0p )
| ~ ( p @ V2r ) )
& ( ~ ( p @ V1q )
| ( p @ V2r )
| ~ ( p @ V0p ) ) ) ) ) ) ) ).
thf(conj_thm_2Esat_2Edc__neg,axiom,
! [V0p: $i] :
( ( mem @ V0p @ bool )
=> ! [V1q: $i] :
( ( mem @ V1q @ bool )
=> ( ( ( p @ V0p )
<=> ~ ( p @ V1q ) )
<=> ( ( ( p @ V0p )
| ( p @ V1q ) )
& ( ~ ( p @ V1q )
| ~ ( p @ V0p ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EBOUNDED__BALL,conjecture,
! [V0x: tp__ty_2Erealax_2Ereal,V1e: tp__ty_2Erealax_2Ereal] : ( p @ ( ap @ c_2Ereal__topology_2Ebounded__def @ ( ap @ c_2Ereal__topology_2Eball @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( inj__ty_2Erealax_2Ereal @ V1e ) ) ) ) ) ).
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