ITP001 Axioms: ITP139^7.ax


%------------------------------------------------------------------------------
% File     : ITP139^7 : TPTP v9.0.0. Bugfixed v7.5.0.
% Domain   : Interactive Theorem Proving
% Axioms   : HOL4 syntactic export, chainy 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    : float.ax [Gau19]
%          : HL4139^7.ax [TPAP]

% Status   : Satisfiable
% Syntax   : Number of formulae    :  163 (  18 unt;  93 typ;   0 def)
%            Number of atoms       :  260 (  36 equ;  28 cnn)
%            Maximal formula atoms :   30 (   1 avg)
%            Number of connectives : 1893 (  28   ~;   8   |;  89   &;1721   @)
%                                         (  13 <=>;  34  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   30 (   9 avg;1721 nst)
%            Number of types       :    6 (   5 usr)
%            Number of type conns  :  153 ( 153   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :   90 (  88 usr;   9 con; 0-5 aty)
%            Number of variables   :  145 (   6   ^ 112   !;  10   ?; 145   :)
%                                         (  17  !>;   0  ?*;   0  @-;   0  @+)
% SPC      : TH1_SAT_EQU_NAR

% Comments :
% Bugfixes : v7.5.0 - Fixes to the axioms.
%------------------------------------------------------------------------------
thf(tyop_2Eieee_2Efloat,type,
    tyop_2Eieee_2Efloat: $tType ).

thf(tyop_2Eieee_2Eroundmode,type,
    tyop_2Eieee_2Eroundmode: $tType ).

thf(tyop_2Emin_2Ebool,type,
    tyop_2Emin_2Ebool: $tType ).

thf(tyop_2Emin_2Efun,type,
    tyop_2Emin_2Efun: $tType > $tType > $tType ).

thf(tyop_2Enum_2Enum,type,
    tyop_2Enum_2Enum: $tType ).

thf(tyop_2Epair_2Eprod,type,
    tyop_2Epair_2Eprod: $tType > $tType > $tType ).

thf(tyop_2Erealax_2Ereal,type,
    tyop_2Erealax_2Ereal: $tType ).

thf(c_2Ebool_2E_21,type,
    c_2Ebool_2E_21: 
      !>[A_27a: $tType] : ( ( A_27a > $o ) > $o ) ).

thf(c_2Epair_2E_2C,type,
    c_2Epair_2E_2C: 
      !>[A_27a: $tType,A_27b: $tType] : ( A_27a > A_27b > ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) ) ).

thf(c_2Earithmetic_2E_2D,type,
    c_2Earithmetic_2E_2D: tyop_2Enum_2Enum > tyop_2Enum_2Enum > tyop_2Enum_2Enum ).

thf(c_2Ereal_2E_2F,type,
    c_2Ereal_2E_2F: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal ).

thf(c_2Ebool_2E_2F_5C,type,
    c_2Ebool_2E_2F_5C: $o > $o > $o ).

thf(c_2Enum_2E0,type,
    c_2Enum_2E0: tyop_2Enum_2Enum ).

thf(c_2Eprim__rec_2E_3C,type,
    c_2Eprim__rec_2E_3C: tyop_2Enum_2Enum > tyop_2Enum_2Enum > $o ).

thf(c_2Earithmetic_2E_3C_3D,type,
    c_2Earithmetic_2E_3C_3D: tyop_2Enum_2Enum > tyop_2Enum_2Enum > $o ).

thf(c_2Emin_2E_3D,type,
    c_2Emin_2E_3D: 
      !>[A_27a: $tType] : ( A_27a > A_27a > $o ) ).

thf(c_2Emin_2E_3D_3D_3E,type,
    c_2Emin_2E_3D_3D_3E: $o > $o > $o ).

thf(c_2Ebool_2E_3F,type,
    c_2Ebool_2E_3F: 
      !>[A_27a: $tType] : ( ( A_27a > $o ) > $o ) ).

thf(c_2Earithmetic_2EBIT1,type,
    c_2Earithmetic_2EBIT1: tyop_2Enum_2Enum > tyop_2Enum_2Enum ).

thf(c_2Earithmetic_2EBIT2,type,
    c_2Earithmetic_2EBIT2: tyop_2Enum_2Enum > tyop_2Enum_2Enum ).

thf(c_2Ebool_2ECOND,type,
    c_2Ebool_2ECOND: 
      !>[A_27a: $tType] : ( $o > A_27a > A_27a > A_27a ) ).

thf(c_2Epred__set_2EEMPTY,type,
    c_2Epred__set_2EEMPTY: 
      !>[A_27a: $tType] : ( A_27a > $o ) ).

thf(c_2Earithmetic_2EEXP,type,
    c_2Earithmetic_2EEXP: tyop_2Enum_2Enum > tyop_2Enum_2Enum > tyop_2Enum_2Enum ).

thf(c_2Epred__set_2EFINITE,type,
    c_2Epred__set_2EFINITE: 
      !>[A_27a: $tType] : ( ( A_27a > $o ) > $o ) ).

thf(c_2Epair_2EFST,type,
    c_2Epair_2EFST: 
      !>[A_27a: $tType,A_27b: $tType] : ( ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) > A_27a ) ).

thf(c_2Eieee_2EFinite,type,
    c_2Eieee_2EFinite: tyop_2Eieee_2Efloat > $o ).

thf(c_2Epred__set_2EGSPEC,type,
    c_2Epred__set_2EGSPEC: 
      !>[A_27a: $tType,A_27b: $tType] : ( ( A_27b > ( tyop_2Epair_2Eprod @ A_27a @ $o ) ) > A_27a > $o ) ).

thf(c_2Ebool_2EIN,type,
    c_2Ebool_2EIN: 
      !>[A_27a: $tType] : ( A_27a > ( A_27a > $o ) > $o ) ).

thf(c_2Eieee_2EInfinity,type,
    c_2Eieee_2EInfinity: tyop_2Eieee_2Efloat > $o ).

thf(c_2Eieee_2EIsdenormal,type,
    c_2Eieee_2EIsdenormal: tyop_2Eieee_2Efloat > $o ).

thf(c_2Eieee_2EIsnan,type,
    c_2Eieee_2EIsnan: tyop_2Eieee_2Efloat > $o ).

thf(c_2Eieee_2EIsnormal,type,
    c_2Eieee_2EIsnormal: tyop_2Eieee_2Efloat > $o ).

thf(c_2Eieee_2EIszero,type,
    c_2Eieee_2EIszero: tyop_2Eieee_2Efloat > $o ).

thf(c_2Eieee_2EMinus__infinity,type,
    c_2Eieee_2EMinus__infinity: tyop_2Eieee_2Efloat ).

thf(c_2Eieee_2EMinus__zero,type,
    c_2Eieee_2EMinus__zero: tyop_2Eieee_2Efloat ).

thf(c_2Earithmetic_2ENUMERAL,type,
    c_2Earithmetic_2ENUMERAL: tyop_2Enum_2Enum > tyop_2Enum_2Enum ).

thf(c_2Eieee_2EPlus__infinity,type,
    c_2Eieee_2EPlus__infinity: tyop_2Eieee_2Efloat ).

thf(c_2Eieee_2EPlus__zero,type,
    c_2Eieee_2EPlus__zero: tyop_2Eieee_2Efloat ).

thf(c_2Epair_2ESND,type,
    c_2Epair_2ESND: 
      !>[A_27a: $tType,A_27b: $tType] : ( ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) > A_27b ) ).

thf(c_2Enum_2ESUC,type,
    c_2Enum_2ESUC: tyop_2Enum_2Enum > tyop_2Enum_2Enum ).

thf(c_2Eieee_2ETo__nearest,type,
    c_2Eieee_2ETo__nearest: tyop_2Eieee_2Eroundmode ).

thf(c_2Eieee_2EVal,type,
    c_2Eieee_2EVal: tyop_2Eieee_2Efloat > tyop_2Erealax_2Ereal ).

thf(c_2Earithmetic_2EZERO,type,
    c_2Earithmetic_2EZERO: tyop_2Enum_2Enum ).

thf(c_2Ebool_2E_5C_2F,type,
    c_2Ebool_2E_5C_2F: $o > $o > $o ).

thf(c_2Ereal_2Eabs,type,
    c_2Ereal_2Eabs: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal ).

thf(c_2Eieee_2Ebias,type,
    c_2Eieee_2Ebias: ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum ) > tyop_2Enum_2Enum ).

thf(c_2Eieee_2Ebottomfloat,type,
    c_2Eieee_2Ebottomfloat: ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum ) > ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum ) ) ).

thf(c_2Eieee_2Eclosest,type,
    c_2Eieee_2Eclosest: 
      !>[A_27a: $tType] : ( ( A_27a > tyop_2Erealax_2Ereal ) > ( A_27a > $o ) > ( A_27a > $o ) > tyop_2Erealax_2Ereal > A_27a ) ).

thf(c_2Eieee_2Edefloat,type,
    c_2Eieee_2Edefloat: tyop_2Eieee_2Efloat > ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum ) ) ).

thf(c_2Eieee_2Eemax,type,
    c_2Eieee_2Eemax: ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum ) > tyop_2Enum_2Enum ).

thf(c_2Efloat_2Eerror,type,
    c_2Efloat_2Eerror: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal ).

thf(c_2Eieee_2Eexponent,type,
    c_2Eieee_2Eexponent: ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum ) ) > tyop_2Enum_2Enum ).

thf(c_2Eieee_2Eexpwidth,type,
    c_2Eieee_2Eexpwidth: ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum ) > tyop_2Enum_2Enum ).

thf(c_2Eieee_2Efloat,type,
    c_2Eieee_2Efloat: ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum ) ) > tyop_2Eieee_2Efloat ).

thf(c_2Eieee_2Efloat__add,type,
    c_2Eieee_2Efloat__add: tyop_2Eieee_2Efloat > tyop_2Eieee_2Efloat > tyop_2Eieee_2Efloat ).

thf(c_2Eieee_2Efloat__div,type,
    c_2Eieee_2Efloat__div: tyop_2Eieee_2Efloat > tyop_2Eieee_2Efloat > tyop_2Eieee_2Efloat ).

thf(c_2Eieee_2Efloat__eq,type,
    c_2Eieee_2Efloat__eq: tyop_2Eieee_2Efloat > tyop_2Eieee_2Efloat > $o ).

thf(c_2Eieee_2Efloat__format,type,
    c_2Eieee_2Efloat__format: tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum ).

thf(c_2Eieee_2Efloat__ge,type,
    c_2Eieee_2Efloat__ge: tyop_2Eieee_2Efloat > tyop_2Eieee_2Efloat > $o ).

thf(c_2Eieee_2Efloat__gt,type,
    c_2Eieee_2Efloat__gt: tyop_2Eieee_2Efloat > tyop_2Eieee_2Efloat > $o ).

thf(c_2Eieee_2Efloat__le,type,
    c_2Eieee_2Efloat__le: tyop_2Eieee_2Efloat > tyop_2Eieee_2Efloat > $o ).

thf(c_2Eieee_2Efloat__lt,type,
    c_2Eieee_2Efloat__lt: tyop_2Eieee_2Efloat > tyop_2Eieee_2Efloat > $o ).

thf(c_2Eieee_2Efloat__mul,type,
    c_2Eieee_2Efloat__mul: tyop_2Eieee_2Efloat > tyop_2Eieee_2Efloat > tyop_2Eieee_2Efloat ).

thf(c_2Eieee_2Efloat__sub,type,
    c_2Eieee_2Efloat__sub: tyop_2Eieee_2Efloat > tyop_2Eieee_2Efloat > tyop_2Eieee_2Efloat ).

thf(c_2Eieee_2Efraction,type,
    c_2Eieee_2Efraction: ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum ) ) > tyop_2Enum_2Enum ).

thf(c_2Eieee_2Efracwidth,type,
    c_2Eieee_2Efracwidth: ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum ) > tyop_2Enum_2Enum ).

thf(c_2Erealax_2Einv,type,
    c_2Erealax_2Einv: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal ).

thf(c_2Eieee_2Eis__closest,type,
    c_2Eieee_2Eis__closest: 
      !>[A_27a: $tType] : ( ( A_27a > tyop_2Erealax_2Ereal ) > ( A_27a > $o ) > tyop_2Erealax_2Ereal > A_27a > $o ) ).

thf(c_2Eieee_2Eis__finite,type,
    c_2Eieee_2Eis__finite: ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum ) > ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum ) ) > $o ).

thf(c_2Eieee_2Eis__valid,type,
    c_2Eieee_2Eis__valid: ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum ) > ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum ) ) > $o ).

thf(c_2Eieee_2Elargest,type,
    c_2Eieee_2Elargest: ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum ) > tyop_2Erealax_2Ereal ).

thf(c_2Eieee_2Eminus__infinity,type,
    c_2Eieee_2Eminus__infinity: ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum ) > ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum ) ) ).

thf(c_2Eieee_2Eminus__zero,type,
    c_2Eieee_2Eminus__zero: ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum ) > ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum ) ) ).

thf(c_2Efloat_2Enormalizes,type,
    c_2Efloat_2Enormalizes: tyop_2Erealax_2Ereal > $o ).

thf(c_2Eieee_2Eplus__infinity,type,
    c_2Eieee_2Eplus__infinity: ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum ) > ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum ) ) ).

thf(c_2Eieee_2Eplus__zero,type,
    c_2Eieee_2Eplus__zero: ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum ) > ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum ) ) ).

thf(c_2Ereal_2Epow,type,
    c_2Ereal_2Epow: tyop_2Erealax_2Ereal > tyop_2Enum_2Enum > tyop_2Erealax_2Ereal ).

thf(c_2Erealax_2Ereal__add,type,
    c_2Erealax_2Ereal__add: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal ).

thf(c_2Ereal_2Ereal__ge,type,
    c_2Ereal_2Ereal__ge: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal > $o ).

thf(c_2Ereal_2Ereal__gt,type,
    c_2Ereal_2Ereal__gt: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal > $o ).

thf(c_2Erealax_2Ereal__lt,type,
    c_2Erealax_2Ereal__lt: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal > $o ).

thf(c_2Ereal_2Ereal__lte,type,
    c_2Ereal_2Ereal__lte: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal > $o ).

thf(c_2Erealax_2Ereal__mul,type,
    c_2Erealax_2Ereal__mul: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal ).

thf(c_2Erealax_2Ereal__neg,type,
    c_2Erealax_2Ereal__neg: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal ).

thf(c_2Ereal_2Ereal__of__num,type,
    c_2Ereal_2Ereal__of__num: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal ).

thf(c_2Ereal_2Ereal__sub,type,
    c_2Ereal_2Ereal__sub: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal ).

thf(c_2Eieee_2Eround,type,
    c_2Eieee_2Eround: ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum ) > tyop_2Eieee_2Eroundmode > tyop_2Erealax_2Ereal > ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum ) ) ).

thf(c_2Eieee_2Esign,type,
    c_2Eieee_2Esign: ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum ) ) > tyop_2Enum_2Enum ).

thf(c_2Eieee_2Ethreshold,type,
    c_2Eieee_2Ethreshold: ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum ) > tyop_2Erealax_2Ereal ).

thf(c_2Eieee_2Etopfloat,type,
    c_2Eieee_2Etopfloat: ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum ) > ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum ) ) ).

thf(c_2Eieee_2Evalof,type,
    c_2Eieee_2Evalof: ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum ) > ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum ) ) > tyop_2Erealax_2Ereal ).

thf(c_2Eieee_2Ezerosign,type,
    c_2Eieee_2Ezerosign: ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum ) > tyop_2Enum_2Enum > ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum ) ) > ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum ) ) ).

thf(c_2Ebool_2E_7E,type,
    c_2Ebool_2E_7E: $o > $o ).

thf(logicdef_2E_2F_5C,axiom,
    ! [V0: $o,V1: $o] :
      ( ( c_2Ebool_2E_2F_5C @ V0 @ V1 )
    <=> ( V0
        & V1 ) ) ).

thf(logicdef_2E_5C_2F,axiom,
    ! [V0: $o,V1: $o] :
      ( ( c_2Ebool_2E_5C_2F @ V0 @ V1 )
    <=> ( V0
        | V1 ) ) ).

thf(logicdef_2E_7E,axiom,
    ! [V0: $o] :
      ( ( c_2Ebool_2E_7E @ V0 )
    <=> ( (~) @ V0 ) ) ).

thf(logicdef_2E_3D_3D_3E,axiom,
    ! [V0: $o,V1: $o] :
      ( ( c_2Emin_2E_3D_3D_3E @ V0 @ V1 )
    <=> ( V0
       => V1 ) ) ).

thf(logicdef_2E_3D,axiom,
    ! [A_27a: $tType,V0: A_27a,V1: A_27a] :
      ( ( c_2Emin_2E_3D @ A_27a @ V0 @ V1 )
    <=> ( V0 = V1 ) ) ).

thf(quantdef_2E_21,axiom,
    ! [A_27a: $tType,V0f: A_27a > $o] :
      ( ( c_2Ebool_2E_21 @ A_27a @ V0f )
    <=> ! [V1x: A_27a] : ( V0f @ V1x ) ) ).

thf(quantdef_2E_3F,axiom,
    ! [A_27a: $tType,V0f: A_27a > $o] :
      ( ( c_2Ebool_2E_3F @ A_27a @ V0f )
    <=> ? [V1x: A_27a] : ( V0f @ V1x ) ) ).

thf(thm_2Efloat_2Eerror__def,axiom,
    ! [V0x: tyop_2Erealax_2Ereal] :
      ( ( c_2Efloat_2Eerror @ V0x )
      = ( c_2Ereal_2Ereal__sub @ ( c_2Eieee_2EVal @ ( c_2Eieee_2Efloat @ ( c_2Eieee_2Eround @ c_2Eieee_2Efloat__format @ c_2Eieee_2ETo__nearest @ V0x ) ) ) @ V0x ) ) ).

thf(thm_2Efloat_2Enormalizes__def,axiom,
    ! [V0x: tyop_2Erealax_2Ereal] :
      ( ( c_2Efloat_2Enormalizes @ V0x )
    <=> ( ( c_2Ereal_2Ereal__lte @ ( c_2Erealax_2Einv @ ( c_2Ereal_2Epow @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) @ ( c_2Earithmetic_2E_2D @ ( c_2Eieee_2Ebias @ c_2Eieee_2Efloat__format ) @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ) @ ( c_2Ereal_2Eabs @ V0x ) )
        & ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Eabs @ V0x ) @ ( c_2Eieee_2Ethreshold @ c_2Eieee_2Efloat__format ) ) ) ) ).

thf(thm_2Efloat_2ESIGN,axiom,
    ! [V0a: tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum )] :
      ( ( c_2Eieee_2Esign @ V0a )
      = ( c_2Epair_2EFST @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum ) @ V0a ) ) ).

thf(thm_2Efloat_2EEXPONENT,axiom,
    ! [V0a: tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum )] :
      ( ( c_2Eieee_2Eexponent @ V0a )
      = ( c_2Epair_2EFST @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum @ ( c_2Epair_2ESND @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum ) @ V0a ) ) ) ).

thf(thm_2Efloat_2EFRACTION,axiom,
    ! [V0a: tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum )] :
      ( ( c_2Eieee_2Efraction @ V0a )
      = ( c_2Epair_2ESND @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum @ ( c_2Epair_2ESND @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum ) @ V0a ) ) ) ).

thf(thm_2Efloat_2EIS__VALID,axiom,
    ! [V0X: tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum,V1a: tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum )] :
      ( ( c_2Eieee_2Eis__valid @ V0X @ V1a )
    <=> ( ( c_2Eprim__rec_2E_3C @ ( c_2Eieee_2Esign @ V1a ) @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) )
        & ( c_2Eprim__rec_2E_3C @ ( c_2Eieee_2Eexponent @ V1a ) @ ( c_2Earithmetic_2EEXP @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) @ ( c_2Eieee_2Eexpwidth @ V0X ) ) )
        & ( c_2Eprim__rec_2E_3C @ ( c_2Eieee_2Efraction @ V1a ) @ ( c_2Earithmetic_2EEXP @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) @ ( c_2Eieee_2Efracwidth @ V0X ) ) ) ) ) ).

thf(thm_2Efloat_2EVALOF,axiom,
    ! [V0X: tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum,V1a: tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum )] :
      ( ( c_2Eieee_2Evalof @ V0X @ V1a )
      = ( c_2Ebool_2ECOND @ tyop_2Erealax_2Ereal @ ( c_2Emin_2E_3D @ tyop_2Enum_2Enum @ ( c_2Eieee_2Eexponent @ V1a ) @ c_2Enum_2E0 ) @ ( c_2Erealax_2Ereal__mul @ ( c_2Erealax_2Ereal__mul @ ( c_2Ereal_2Epow @ ( c_2Erealax_2Ereal__neg @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) @ ( c_2Eieee_2Esign @ V1a ) ) @ ( c_2Ereal_2E_2F @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) @ ( c_2Ereal_2Epow @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) @ ( c_2Eieee_2Ebias @ V0X ) ) ) ) @ ( c_2Ereal_2E_2F @ ( c_2Ereal_2Ereal__of__num @ ( c_2Eieee_2Efraction @ V1a ) ) @ ( c_2Ereal_2Epow @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) @ ( c_2Eieee_2Efracwidth @ V0X ) ) ) ) @ ( c_2Erealax_2Ereal__mul @ ( c_2Erealax_2Ereal__mul @ ( c_2Ereal_2Epow @ ( c_2Erealax_2Ereal__neg @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) @ ( c_2Eieee_2Esign @ V1a ) ) @ ( c_2Ereal_2E_2F @ ( c_2Ereal_2Epow @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) @ ( c_2Eieee_2Eexponent @ V1a ) ) @ ( c_2Ereal_2Epow @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) @ ( c_2Eieee_2Ebias @ V0X ) ) ) ) @ ( c_2Erealax_2Ereal__add @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) @ ( c_2Ereal_2E_2F @ ( c_2Ereal_2Ereal__of__num @ ( c_2Eieee_2Efraction @ V1a ) ) @ ( c_2Ereal_2Epow @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) @ ( c_2Eieee_2Efracwidth @ V0X ) ) ) ) ) ) ) ).

thf(thm_2Efloat_2EIS__VALID__DEFLOAT,axiom,
    ! [V0a: tyop_2Eieee_2Efloat] : ( c_2Eieee_2Eis__valid @ c_2Eieee_2Efloat__format @ ( c_2Eieee_2Edefloat @ V0a ) ) ).

thf(thm_2Efloat_2EIS__FINITE__EXPLICIT,axiom,
    ! [V0a: tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum )] :
      ( ( c_2Eieee_2Eis__finite @ c_2Eieee_2Efloat__format @ V0a )
    <=> ( ( c_2Eprim__rec_2E_3C @ ( c_2Eieee_2Esign @ V0a ) @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) )
        & ( c_2Eprim__rec_2E_3C @ ( c_2Eieee_2Eexponent @ V0a ) @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ) ) ) ) ) )
        & ( c_2Eprim__rec_2E_3C @ ( c_2Eieee_2Efraction @ V0a ) @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

thf(thm_2Efloat_2EFLOAT__CASES,axiom,
    ! [V0a: tyop_2Eieee_2Efloat] :
      ( ( c_2Eieee_2EIsnan @ V0a )
      | ( c_2Eieee_2EInfinity @ V0a )
      | ( c_2Eieee_2EIsnormal @ V0a )
      | ( c_2Eieee_2EIsdenormal @ V0a )
      | ( c_2Eieee_2EIszero @ V0a ) ) ).

thf(thm_2Efloat_2EFLOAT__CASES__FINITE,axiom,
    ! [V0a: tyop_2Eieee_2Efloat] :
      ( ( c_2Eieee_2EIsnan @ V0a )
      | ( c_2Eieee_2EInfinity @ V0a )
      | ( c_2Eieee_2EFinite @ V0a ) ) ).

thf(thm_2Efloat_2EFLOAT__DISTINCT,axiom,
    ! [V0a: tyop_2Eieee_2Efloat] :
      ( ( (~)
        @ ( ( c_2Eieee_2EIsnan @ V0a )
          & ( c_2Eieee_2EInfinity @ V0a ) ) )
      & ( (~)
        @ ( ( c_2Eieee_2EIsnan @ V0a )
          & ( c_2Eieee_2EIsnormal @ V0a ) ) )
      & ( (~)
        @ ( ( c_2Eieee_2EIsnan @ V0a )
          & ( c_2Eieee_2EIsdenormal @ V0a ) ) )
      & ( (~)
        @ ( ( c_2Eieee_2EIsnan @ V0a )
          & ( c_2Eieee_2EIszero @ V0a ) ) )
      & ( (~)
        @ ( ( c_2Eieee_2EInfinity @ V0a )
          & ( c_2Eieee_2EIsnormal @ V0a ) ) )
      & ( (~)
        @ ( ( c_2Eieee_2EInfinity @ V0a )
          & ( c_2Eieee_2EIsdenormal @ V0a ) ) )
      & ( (~)
        @ ( ( c_2Eieee_2EInfinity @ V0a )
          & ( c_2Eieee_2EIszero @ V0a ) ) )
      & ( (~)
        @ ( ( c_2Eieee_2EIsnormal @ V0a )
          & ( c_2Eieee_2EIsdenormal @ V0a ) ) )
      & ( (~)
        @ ( ( c_2Eieee_2EIsnormal @ V0a )
          & ( c_2Eieee_2EIszero @ V0a ) ) )
      & ( (~)
        @ ( ( c_2Eieee_2EIsdenormal @ V0a )
          & ( c_2Eieee_2EIszero @ V0a ) ) ) ) ).

thf(thm_2Efloat_2EFLOAT__DISTINCT__FINITE,axiom,
    ! [V0a: tyop_2Eieee_2Efloat] :
      ( ( (~)
        @ ( ( c_2Eieee_2EIsnan @ V0a )
          & ( c_2Eieee_2EInfinity @ V0a ) ) )
      & ( (~)
        @ ( ( c_2Eieee_2EIsnan @ V0a )
          & ( c_2Eieee_2EFinite @ V0a ) ) )
      & ( (~)
        @ ( ( c_2Eieee_2EInfinity @ V0a )
          & ( c_2Eieee_2EFinite @ V0a ) ) ) ) ).

thf(thm_2Efloat_2EFLOAT__INFINITIES__SIGNED,axiom,
    ( ( ( c_2Eieee_2Esign @ ( c_2Eieee_2Edefloat @ c_2Eieee_2EPlus__infinity ) )
      = c_2Enum_2E0 )
    & ( ( c_2Eieee_2Esign @ ( c_2Eieee_2Edefloat @ c_2Eieee_2EMinus__infinity ) )
      = ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ).

thf(thm_2Efloat_2EINFINITY__IS__INFINITY,axiom,
    ( ( c_2Eieee_2EInfinity @ c_2Eieee_2EPlus__infinity )
    & ( c_2Eieee_2EInfinity @ c_2Eieee_2EMinus__infinity ) ) ).

thf(thm_2Efloat_2EZERO__IS__ZERO,axiom,
    ( ( c_2Eieee_2EIszero @ c_2Eieee_2EPlus__zero )
    & ( c_2Eieee_2EIszero @ c_2Eieee_2EMinus__zero ) ) ).

thf(thm_2Efloat_2EINFINITY__NOT__NAN,axiom,
    ( ( (~) @ ( c_2Eieee_2EIsnan @ c_2Eieee_2EPlus__infinity ) )
    & ( (~) @ ( c_2Eieee_2EIsnan @ c_2Eieee_2EMinus__infinity ) ) ) ).

thf(thm_2Efloat_2EZERO__NOT__NAN,axiom,
    ( ( (~) @ ( c_2Eieee_2EIsnan @ c_2Eieee_2EPlus__zero ) )
    & ( (~) @ ( c_2Eieee_2EIsnan @ c_2Eieee_2EMinus__zero ) ) ) ).

thf(thm_2Efloat_2EFLOAT__INFINITIES,axiom,
    ! [V0a: tyop_2Eieee_2Efloat] :
      ( ( c_2Eieee_2EInfinity @ V0a )
    <=> ( ( c_2Eieee_2Efloat__eq @ V0a @ c_2Eieee_2EPlus__infinity )
        | ( c_2Eieee_2Efloat__eq @ V0a @ c_2Eieee_2EMinus__infinity ) ) ) ).

thf(thm_2Efloat_2EFLOAT__INFINITES__DISTINCT,axiom,
    ! [V0a: tyop_2Eieee_2Efloat] :
      ( (~)
      @ ( ( c_2Eieee_2Efloat__eq @ V0a @ c_2Eieee_2EPlus__infinity )
        & ( c_2Eieee_2Efloat__eq @ V0a @ c_2Eieee_2EMinus__infinity ) ) ) ).

thf(thm_2Efloat_2EFLOAT__LT,axiom,
    ! [V0a: tyop_2Eieee_2Efloat,V1b: tyop_2Eieee_2Efloat] :
      ( ( ( c_2Eieee_2EFinite @ V0a )
        & ( c_2Eieee_2EFinite @ V1b ) )
     => ( ( c_2Eieee_2Efloat__lt @ V0a @ V1b )
        = ( c_2Erealax_2Ereal__lt @ ( c_2Eieee_2EVal @ V0a ) @ ( c_2Eieee_2EVal @ V1b ) ) ) ) ).

thf(thm_2Efloat_2EFLOAT__GT,axiom,
    ! [V0a: tyop_2Eieee_2Efloat,V1b: tyop_2Eieee_2Efloat] :
      ( ( ( c_2Eieee_2EFinite @ V0a )
        & ( c_2Eieee_2EFinite @ V1b ) )
     => ( ( c_2Eieee_2Efloat__gt @ V0a @ V1b )
        = ( c_2Ereal_2Ereal__gt @ ( c_2Eieee_2EVal @ V0a ) @ ( c_2Eieee_2EVal @ V1b ) ) ) ) ).

thf(thm_2Efloat_2EFLOAT__LE,axiom,
    ! [V0a: tyop_2Eieee_2Efloat,V1b: tyop_2Eieee_2Efloat] :
      ( ( ( c_2Eieee_2EFinite @ V0a )
        & ( c_2Eieee_2EFinite @ V1b ) )
     => ( ( c_2Eieee_2Efloat__le @ V0a @ V1b )
        = ( c_2Ereal_2Ereal__lte @ ( c_2Eieee_2EVal @ V0a ) @ ( c_2Eieee_2EVal @ V1b ) ) ) ) ).

thf(thm_2Efloat_2EFLOAT__GE,axiom,
    ! [V0a: tyop_2Eieee_2Efloat,V1b: tyop_2Eieee_2Efloat] :
      ( ( ( c_2Eieee_2EFinite @ V0a )
        & ( c_2Eieee_2EFinite @ V1b ) )
     => ( ( c_2Eieee_2Efloat__ge @ V0a @ V1b )
        = ( c_2Ereal_2Ereal__ge @ ( c_2Eieee_2EVal @ V0a ) @ ( c_2Eieee_2EVal @ V1b ) ) ) ) ).

thf(thm_2Efloat_2EFLOAT__EQ,axiom,
    ! [V0a: tyop_2Eieee_2Efloat,V1b: tyop_2Eieee_2Efloat] :
      ( ( ( c_2Eieee_2EFinite @ V0a )
        & ( c_2Eieee_2EFinite @ V1b ) )
     => ( ( c_2Eieee_2Efloat__eq @ V0a @ V1b )
      <=> ( ( c_2Eieee_2EVal @ V0a )
          = ( c_2Eieee_2EVal @ V1b ) ) ) ) ).

thf(thm_2Efloat_2EFLOAT__EQ__REFL,axiom,
    ! [V0a: tyop_2Eieee_2Efloat] :
      ( ( c_2Eieee_2Efloat__eq @ V0a @ V0a )
    <=> ( (~) @ ( c_2Eieee_2EIsnan @ V0a ) ) ) ).

thf(thm_2Efloat_2EIS__VALID__SPECIAL,axiom,
    ! [V0X: tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum] :
      ( ( c_2Eieee_2Eis__valid @ V0X @ ( c_2Eieee_2Eminus__infinity @ V0X ) )
      & ( c_2Eieee_2Eis__valid @ V0X @ ( c_2Eieee_2Eplus__infinity @ V0X ) )
      & ( c_2Eieee_2Eis__valid @ V0X @ ( c_2Eieee_2Etopfloat @ V0X ) )
      & ( c_2Eieee_2Eis__valid @ V0X @ ( c_2Eieee_2Ebottomfloat @ V0X ) )
      & ( c_2Eieee_2Eis__valid @ V0X @ ( c_2Eieee_2Eplus__zero @ V0X ) )
      & ( c_2Eieee_2Eis__valid @ V0X @ ( c_2Eieee_2Eminus__zero @ V0X ) ) ) ).

thf(thm_2Efloat_2EIS__CLOSEST__EXISTS,axiom,
    ! [V0v: ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum ) ) > tyop_2Erealax_2Ereal,V1x: tyop_2Erealax_2Ereal,V2s: ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum ) ) > $o] :
      ( ( c_2Epred__set_2EFINITE @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum ) ) @ V2s )
     => ( ( (~)
          @ ( V2s
            = ( c_2Epred__set_2EEMPTY @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum ) ) ) ) )
       => ? [V3a: tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum )] : ( c_2Eieee_2Eis__closest @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum ) ) @ V0v @ V2s @ V1x @ V3a ) ) ) ).

thf(thm_2Efloat_2ECLOSEST__IS__EVERYTHING,axiom,
    ! [V0v: ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum ) ) > tyop_2Erealax_2Ereal,V1p: ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum ) ) > $o,V2s: ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum ) ) > $o,V3x: tyop_2Erealax_2Ereal] :
      ( ( c_2Epred__set_2EFINITE @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum ) ) @ V2s )
     => ( ( (~)
          @ ( V2s
            = ( c_2Epred__set_2EEMPTY @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum ) ) ) ) )
       => ( ( c_2Eieee_2Eis__closest @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum ) ) @ V0v @ V2s @ V3x @ ( c_2Eieee_2Eclosest @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum ) ) @ V0v @ V1p @ V2s @ V3x ) )
          & ( ? [V4b: tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum )] :
                ( ( c_2Eieee_2Eis__closest @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum ) ) @ V0v @ V2s @ V3x @ V4b )
                & ( V1p @ V4b ) )
           => ( V1p @ ( c_2Eieee_2Eclosest @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum ) ) @ V0v @ V1p @ V2s @ V3x ) ) ) ) ) ) ).

thf(thm_2Efloat_2ECLOSEST__IN__SET,axiom,
    ! [V0v: ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum ) ) > tyop_2Erealax_2Ereal,V1p: ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum ) ) > $o,V2x: tyop_2Erealax_2Ereal,V3s: ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum ) ) > $o] :
      ( ( c_2Epred__set_2EFINITE @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum ) ) @ V3s )
     => ( ( (~)
          @ ( V3s
            = ( c_2Epred__set_2EEMPTY @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum ) ) ) ) )
       => ( c_2Ebool_2EIN @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum ) ) @ ( c_2Eieee_2Eclosest @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum ) ) @ V0v @ V1p @ V3s @ V2x ) @ V3s ) ) ) ).

thf(thm_2Efloat_2ECLOSEST__IS__CLOSEST,axiom,
    ! [V0v: ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum ) ) > tyop_2Erealax_2Ereal,V1p: ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum ) ) > $o,V2x: tyop_2Erealax_2Ereal,V3s: ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum ) ) > $o] :
      ( ( c_2Epred__set_2EFINITE @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum ) ) @ V3s )
     => ( ( (~)
          @ ( V3s
            = ( c_2Epred__set_2EEMPTY @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum ) ) ) ) )
       => ( c_2Eieee_2Eis__closest @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum ) ) @ V0v @ V3s @ V2x @ ( c_2Eieee_2Eclosest @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum ) ) @ V0v @ V1p @ V3s @ V2x ) ) ) ) ).

thf(thm_2Efloat_2EIS__VALID__FINITE,axiom,
    ! [V0X: tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum] :
      ( c_2Epred__set_2EFINITE @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum ) )
      @ ( c_2Epred__set_2EGSPEC @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum ) ) @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum ) )
        @ ^ [V1a: tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum )] : ( c_2Epair_2E_2C @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum ) ) @ $o @ V1a @ ( c_2Eieee_2Eis__valid @ V0X @ V1a ) ) ) ) ).

thf(thm_2Efloat_2EIS__FINITE__FINITE,axiom,
    ! [V0X: tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum] :
      ( c_2Epred__set_2EFINITE @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum ) )
      @ ( c_2Epred__set_2EGSPEC @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum ) ) @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum ) )
        @ ^ [V1a: tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum )] : ( c_2Epair_2E_2C @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum ) ) @ $o @ V1a @ ( c_2Eieee_2Eis__finite @ V0X @ V1a ) ) ) ) ).

thf(thm_2Efloat_2EIS__VALID__NONEMPTY,axiom,
    ! [V0X: tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum] :
      ( (~)
      @ ( ( c_2Epred__set_2EGSPEC @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum ) ) @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum ) )
          @ ^ [V1a: tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum )] : ( c_2Epair_2E_2C @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum ) ) @ $o @ V1a @ ( c_2Eieee_2Eis__valid @ V0X @ V1a ) ) )
        = ( c_2Epred__set_2EEMPTY @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum ) ) ) ) ) ).

thf(thm_2Efloat_2EIS__FINITE__NONEMPTY,axiom,
    ! [V0X: tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum] :
      ( (~)
      @ ( ( c_2Epred__set_2EGSPEC @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum ) ) @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum ) )
          @ ^ [V1a: tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum )] : ( c_2Epair_2E_2C @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum ) ) @ $o @ V1a @ ( c_2Eieee_2Eis__finite @ V0X @ V1a ) ) )
        = ( c_2Epred__set_2EEMPTY @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum ) ) ) ) ) ).

thf(thm_2Efloat_2EIS__FINITE__CLOSEST,axiom,
    ! [V0X: tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum,V1v: ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum ) ) > tyop_2Erealax_2Ereal,V2p: ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum ) ) > $o,V3x: tyop_2Erealax_2Ereal] :
      ( c_2Eieee_2Eis__finite @ V0X
      @ ( c_2Eieee_2Eclosest @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum ) ) @ V1v @ V2p
        @ ( c_2Epred__set_2EGSPEC @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum ) ) @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum ) )
          @ ^ [V4a: tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum )] : ( c_2Epair_2E_2C @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum ) ) @ $o @ V4a @ ( c_2Eieee_2Eis__finite @ V0X @ V4a ) ) )
        @ V3x ) ) ).

thf(thm_2Efloat_2EIS__VALID__CLOSEST,axiom,
    ! [V0X: tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum,V1v: ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum ) ) > tyop_2Erealax_2Ereal,V2p: ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum ) ) > $o,V3x: tyop_2Erealax_2Ereal] :
      ( c_2Eieee_2Eis__valid @ V0X
      @ ( c_2Eieee_2Eclosest @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum ) ) @ V1v @ V2p
        @ ( c_2Epred__set_2EGSPEC @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum ) ) @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum ) )
          @ ^ [V4a: tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum )] : ( c_2Epair_2E_2C @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum ) ) @ $o @ V4a @ ( c_2Eieee_2Eis__finite @ V0X @ V4a ) ) )
        @ V3x ) ) ).

thf(thm_2Efloat_2EIS__VALID__ROUND,axiom,
    ! [V0X: tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum,V1x: tyop_2Erealax_2Ereal] : ( c_2Eieee_2Eis__valid @ V0X @ ( c_2Eieee_2Eround @ V0X @ c_2Eieee_2ETo__nearest @ V1x ) ) ).

thf(thm_2Efloat_2EDEFLOAT__FLOAT__ROUND,axiom,
    ! [V0x: tyop_2Erealax_2Ereal] :
      ( ( c_2Eieee_2Edefloat @ ( c_2Eieee_2Efloat @ ( c_2Eieee_2Eround @ c_2Eieee_2Efloat__format @ c_2Eieee_2ETo__nearest @ V0x ) ) )
      = ( c_2Eieee_2Eround @ c_2Eieee_2Efloat__format @ c_2Eieee_2ETo__nearest @ V0x ) ) ).

thf(thm_2Efloat_2EDEFLOAT__FLOAT__ZEROSIGN__ROUND,axiom,
    ! [V0x: tyop_2Erealax_2Ereal,V1b: tyop_2Enum_2Enum] :
      ( ( c_2Eieee_2Edefloat @ ( c_2Eieee_2Efloat @ ( c_2Eieee_2Ezerosign @ c_2Eieee_2Efloat__format @ V1b @ ( c_2Eieee_2Eround @ c_2Eieee_2Efloat__format @ c_2Eieee_2ETo__nearest @ V0x ) ) ) )
      = ( c_2Eieee_2Ezerosign @ c_2Eieee_2Efloat__format @ V1b @ ( c_2Eieee_2Eround @ c_2Eieee_2Efloat__format @ c_2Eieee_2ETo__nearest @ V0x ) ) ) ).

thf(thm_2Efloat_2EVALOF__DEFLOAT__FLOAT__ZEROSIGN__ROUND,axiom,
    ! [V0x: tyop_2Erealax_2Ereal,V1b: tyop_2Enum_2Enum] :
      ( ( c_2Eieee_2Evalof @ c_2Eieee_2Efloat__format @ ( c_2Eieee_2Edefloat @ ( c_2Eieee_2Efloat @ ( c_2Eieee_2Ezerosign @ c_2Eieee_2Efloat__format @ V1b @ ( c_2Eieee_2Eround @ c_2Eieee_2Efloat__format @ c_2Eieee_2ETo__nearest @ V0x ) ) ) ) )
      = ( c_2Eieee_2Evalof @ c_2Eieee_2Efloat__format @ ( c_2Eieee_2Eround @ c_2Eieee_2Efloat__format @ c_2Eieee_2ETo__nearest @ V0x ) ) ) ).

thf(thm_2Efloat_2EISFINITE,axiom,
    ! [V0a: tyop_2Eieee_2Efloat] :
      ( ( c_2Eieee_2EFinite @ V0a )
      = ( c_2Eieee_2Eis__finite @ c_2Eieee_2Efloat__format @ ( c_2Eieee_2Edefloat @ V0a ) ) ) ).

thf(thm_2Efloat_2EVAL__FINITE,axiom,
    ! [V0a: tyop_2Eieee_2Efloat] :
      ( ( c_2Eieee_2EFinite @ V0a )
     => ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Eabs @ ( c_2Eieee_2EVal @ V0a ) ) @ ( c_2Eieee_2Elargest @ c_2Eieee_2Efloat__format ) ) ) ).

thf(thm_2Efloat_2EFLOAT__THRESHOLD__EXPLICIT,axiom,
    ( ( c_2Eieee_2Ethreshold @ c_2Eieee_2Efloat__format )
    = ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT2 @ ( c_2Earithmetic_2EBIT2 @ ( c_2Earithmetic_2EBIT2 @ ( c_2Earithmetic_2EBIT2 @ ( c_2Earithmetic_2EBIT2 @ ( c_2Earithmetic_2EBIT2 @ ( c_2Earithmetic_2EBIT2 @ ( c_2Earithmetic_2EBIT2 @ ( c_2Earithmetic_2EBIT2 @ ( c_2Earithmetic_2EBIT2 @ ( c_2Earithmetic_2EBIT2 @ ( c_2Earithmetic_2EBIT2 @ ( c_2Earithmetic_2EBIT2 @ ( c_2Earithmetic_2EBIT2 @ ( c_2Earithmetic_2EBIT2 @ ( c_2Earithmetic_2EBIT2 @ ( c_2Earithmetic_2EBIT2 @ ( c_2Earithmetic_2EBIT2 @ ( c_2Earithmetic_2EBIT2 @ ( c_2Earithmetic_2EBIT2 @ ( c_2Earithmetic_2EBIT2 @ ( c_2Earithmetic_2EBIT2 @ ( c_2Earithmetic_2EBIT2 @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

thf(thm_2Efloat_2EFLOAT__LARGEST__EXPLICIT,axiom,
    ( ( c_2Eieee_2Elargest @ c_2Eieee_2Efloat__format )
    = ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT2 @ ( c_2Earithmetic_2EBIT2 @ ( c_2Earithmetic_2EBIT2 @ ( c_2Earithmetic_2EBIT2 @ ( c_2Earithmetic_2EBIT2 @ ( c_2Earithmetic_2EBIT2 @ ( c_2Earithmetic_2EBIT2 @ ( c_2Earithmetic_2EBIT2 @ ( c_2Earithmetic_2EBIT2 @ ( c_2Earithmetic_2EBIT2 @ ( c_2Earithmetic_2EBIT2 @ ( c_2Earithmetic_2EBIT2 @ ( c_2Earithmetic_2EBIT2 @ ( c_2Earithmetic_2EBIT2 @ ( c_2Earithmetic_2EBIT2 @ ( c_2Earithmetic_2EBIT2 @ ( c_2Earithmetic_2EBIT2 @ ( c_2Earithmetic_2EBIT2 @ ( c_2Earithmetic_2EBIT2 @ ( c_2Earithmetic_2EBIT2 @ ( c_2Earithmetic_2EBIT2 @ ( c_2Earithmetic_2EBIT2 @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

thf(thm_2Efloat_2EVAL__THRESHOLD,axiom,
    ! [V0a: tyop_2Eieee_2Efloat] :
      ( ( c_2Eieee_2EFinite @ V0a )
     => ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Eabs @ ( c_2Eieee_2EVal @ V0a ) ) @ ( c_2Eieee_2Ethreshold @ c_2Eieee_2Efloat__format ) ) ) ).

thf(thm_2Efloat_2EERROR__IS__ZERO,axiom,
    ! [V0a: tyop_2Eieee_2Efloat,V1x: tyop_2Erealax_2Ereal] :
      ( ( ( c_2Eieee_2EFinite @ V0a )
        & ( ( c_2Eieee_2EVal @ V0a )
          = V1x ) )
     => ( ( c_2Efloat_2Eerror @ V1x )
        = ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ) ).

thf(thm_2Efloat_2EERROR__BOUND__NORM__STRONG,axiom,
    ! [V0x: tyop_2Erealax_2Ereal,V1j: tyop_2Enum_2Enum] :
      ( ( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Eabs @ V0x ) @ ( c_2Eieee_2Ethreshold @ c_2Eieee_2Efloat__format ) )
        & ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Eabs @ V0x ) @ ( c_2Ereal_2E_2F @ ( c_2Ereal_2Epow @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) @ ( c_2Enum_2ESUC @ V1j ) ) @ ( c_2Ereal_2Epow @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ ( c_2Earithmetic_2EBIT2 @ ( c_2Earithmetic_2EBIT2 @ ( c_2Earithmetic_2EBIT2 @ ( c_2Earithmetic_2EBIT2 @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) ) ) ) ) ) ) ) )
     => ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Eabs @ ( c_2Efloat_2Eerror @ V0x ) ) @ ( c_2Ereal_2E_2F @ ( c_2Ereal_2Epow @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) @ V1j ) @ ( c_2Ereal_2Epow @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ ( c_2Earithmetic_2EBIT2 @ ( c_2Earithmetic_2EBIT2 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT2 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ) ) ) ) ) ) ) ) ).

thf(thm_2Efloat_2EREAL__IN__BINADE,axiom,
    ! [V0x: tyop_2Erealax_2Ereal] :
      ( ( c_2Efloat_2Enormalizes @ V0x )
     => ? [V1j: tyop_2Enum_2Enum] :
          ( ( c_2Earithmetic_2E_3C_3D @ V1j @ ( c_2Earithmetic_2E_2D @ ( c_2Eieee_2Eemax @ c_2Eieee_2Efloat__format ) @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) )
          & ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2E_2F @ ( c_2Ereal_2Epow @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) @ V1j ) @ ( c_2Ereal_2Epow @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ ( c_2Earithmetic_2EBIT2 @ ( c_2Earithmetic_2EBIT2 @ ( c_2Earithmetic_2EBIT2 @ ( c_2Earithmetic_2EBIT2 @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) ) ) ) ) ) ) @ ( c_2Ereal_2Eabs @ V0x ) )
          & ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Eabs @ V0x ) @ ( c_2Ereal_2E_2F @ ( c_2Ereal_2Epow @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) @ ( c_2Enum_2ESUC @ V1j ) ) @ ( c_2Ereal_2Epow @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ ( c_2Earithmetic_2EBIT2 @ ( c_2Earithmetic_2EBIT2 @ ( c_2Earithmetic_2EBIT2 @ ( c_2Earithmetic_2EBIT2 @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) ) ) ) ) ) ) ) ) ) ).

thf(thm_2Efloat_2EERROR__BOUND__NORM__STRONG__NORMALIZE,axiom,
    ! [V0x: tyop_2Erealax_2Ereal] :
      ( ( c_2Efloat_2Enormalizes @ V0x )
     => ? [V1j: tyop_2Enum_2Enum] : ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Eabs @ ( c_2Efloat_2Eerror @ V0x ) ) @ ( c_2Ereal_2E_2F @ ( c_2Ereal_2Epow @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) @ V1j ) @ ( c_2Ereal_2Epow @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ ( c_2Earithmetic_2EBIT2 @ ( c_2Earithmetic_2EBIT2 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT2 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ) ) ) ) ) ) ) ) ).

thf(thm_2Efloat_2ERELATIVE__ERROR,axiom,
    ! [V0x: tyop_2Erealax_2Ereal] :
      ( ( c_2Efloat_2Enormalizes @ V0x )
     => ? [V1e: tyop_2Erealax_2Ereal] :
          ( ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Eabs @ V1e ) @ ( c_2Ereal_2E_2F @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) @ ( c_2Ereal_2Epow @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) ) ) ) ) )
          & ( ( c_2Eieee_2EVal @ ( c_2Eieee_2Efloat @ ( c_2Eieee_2Eround @ c_2Eieee_2Efloat__format @ c_2Eieee_2ETo__nearest @ V0x ) ) )
            = ( c_2Erealax_2Ereal__mul @ V0x @ ( c_2Erealax_2Ereal__add @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) @ V1e ) ) ) ) ) ).

thf(thm_2Efloat_2EDEFLOAT__FLOAT__ZEROSIGN__ROUND__FINITE,axiom,
    ! [V0b: tyop_2Enum_2Enum,V1x: tyop_2Erealax_2Ereal] :
      ( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Eabs @ V1x ) @ ( c_2Eieee_2Ethreshold @ c_2Eieee_2Efloat__format ) )
     => ( c_2Eieee_2Eis__finite @ c_2Eieee_2Efloat__format @ ( c_2Eieee_2Edefloat @ ( c_2Eieee_2Efloat @ ( c_2Eieee_2Ezerosign @ c_2Eieee_2Efloat__format @ V0b @ ( c_2Eieee_2Eround @ c_2Eieee_2Efloat__format @ c_2Eieee_2ETo__nearest @ V1x ) ) ) ) ) ) ).

thf(thm_2Efloat_2EFLOAT__ADD,axiom,
    ! [V0a: tyop_2Eieee_2Efloat,V1b: tyop_2Eieee_2Efloat] :
      ( ( ( c_2Eieee_2EFinite @ V0a )
        & ( c_2Eieee_2EFinite @ V1b )
        & ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Eabs @ ( c_2Erealax_2Ereal__add @ ( c_2Eieee_2EVal @ V0a ) @ ( c_2Eieee_2EVal @ V1b ) ) ) @ ( c_2Eieee_2Ethreshold @ c_2Eieee_2Efloat__format ) ) )
     => ( ( c_2Eieee_2EFinite @ ( c_2Eieee_2Efloat__add @ V0a @ V1b ) )
        & ( ( c_2Eieee_2EVal @ ( c_2Eieee_2Efloat__add @ V0a @ V1b ) )
          = ( c_2Erealax_2Ereal__add @ ( c_2Erealax_2Ereal__add @ ( c_2Eieee_2EVal @ V0a ) @ ( c_2Eieee_2EVal @ V1b ) ) @ ( c_2Efloat_2Eerror @ ( c_2Erealax_2Ereal__add @ ( c_2Eieee_2EVal @ V0a ) @ ( c_2Eieee_2EVal @ V1b ) ) ) ) ) ) ) ).

thf(thm_2Efloat_2EFLOAT__SUB,axiom,
    ! [V0a: tyop_2Eieee_2Efloat,V1b: tyop_2Eieee_2Efloat] :
      ( ( ( c_2Eieee_2EFinite @ V0a )
        & ( c_2Eieee_2EFinite @ V1b )
        & ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Eabs @ ( c_2Ereal_2Ereal__sub @ ( c_2Eieee_2EVal @ V0a ) @ ( c_2Eieee_2EVal @ V1b ) ) ) @ ( c_2Eieee_2Ethreshold @ c_2Eieee_2Efloat__format ) ) )
     => ( ( c_2Eieee_2EFinite @ ( c_2Eieee_2Efloat__sub @ V0a @ V1b ) )
        & ( ( c_2Eieee_2EVal @ ( c_2Eieee_2Efloat__sub @ V0a @ V1b ) )
          = ( c_2Erealax_2Ereal__add @ ( c_2Ereal_2Ereal__sub @ ( c_2Eieee_2EVal @ V0a ) @ ( c_2Eieee_2EVal @ V1b ) ) @ ( c_2Efloat_2Eerror @ ( c_2Ereal_2Ereal__sub @ ( c_2Eieee_2EVal @ V0a ) @ ( c_2Eieee_2EVal @ V1b ) ) ) ) ) ) ) ).

thf(thm_2Efloat_2EFLOAT__MUL,axiom,
    ! [V0a: tyop_2Eieee_2Efloat,V1b: tyop_2Eieee_2Efloat] :
      ( ( ( c_2Eieee_2EFinite @ V0a )
        & ( c_2Eieee_2EFinite @ V1b )
        & ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Eabs @ ( c_2Erealax_2Ereal__mul @ ( c_2Eieee_2EVal @ V0a ) @ ( c_2Eieee_2EVal @ V1b ) ) ) @ ( c_2Eieee_2Ethreshold @ c_2Eieee_2Efloat__format ) ) )
     => ( ( c_2Eieee_2EFinite @ ( c_2Eieee_2Efloat__mul @ V0a @ V1b ) )
        & ( ( c_2Eieee_2EVal @ ( c_2Eieee_2Efloat__mul @ V0a @ V1b ) )
          = ( c_2Erealax_2Ereal__add @ ( c_2Erealax_2Ereal__mul @ ( c_2Eieee_2EVal @ V0a ) @ ( c_2Eieee_2EVal @ V1b ) ) @ ( c_2Efloat_2Eerror @ ( c_2Erealax_2Ereal__mul @ ( c_2Eieee_2EVal @ V0a ) @ ( c_2Eieee_2EVal @ V1b ) ) ) ) ) ) ) ).

thf(thm_2Efloat_2EFLOAT__DIV,axiom,
    ! [V0a: tyop_2Eieee_2Efloat,V1b: tyop_2Eieee_2Efloat] :
      ( ( ( c_2Eieee_2EFinite @ V0a )
        & ( c_2Eieee_2EFinite @ V1b )
        & ( (~) @ ( c_2Eieee_2EIszero @ V1b ) )
        & ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Eabs @ ( c_2Ereal_2E_2F @ ( c_2Eieee_2EVal @ V0a ) @ ( c_2Eieee_2EVal @ V1b ) ) ) @ ( c_2Eieee_2Ethreshold @ c_2Eieee_2Efloat__format ) ) )
     => ( ( c_2Eieee_2EFinite @ ( c_2Eieee_2Efloat__div @ V0a @ V1b ) )
        & ( ( c_2Eieee_2EVal @ ( c_2Eieee_2Efloat__div @ V0a @ V1b ) )
          = ( c_2Erealax_2Ereal__add @ ( c_2Ereal_2E_2F @ ( c_2Eieee_2EVal @ V0a ) @ ( c_2Eieee_2EVal @ V1b ) ) @ ( c_2Efloat_2Eerror @ ( c_2Ereal_2E_2F @ ( c_2Eieee_2EVal @ V0a ) @ ( c_2Eieee_2EVal @ V1b ) ) ) ) ) ) ) ).

thf(thm_2Efloat_2EFLOAT__ADD__FINITE,axiom,
    ! [V0b: tyop_2Eieee_2Efloat,V1a: tyop_2Eieee_2Efloat] :
      ( ( ( c_2Eieee_2EFinite @ V1a )
        & ( c_2Eieee_2EFinite @ V0b )
        & ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Eabs @ ( c_2Erealax_2Ereal__add @ ( c_2Eieee_2EVal @ V1a ) @ ( c_2Eieee_2EVal @ V0b ) ) ) @ ( c_2Eieee_2Ethreshold @ c_2Eieee_2Efloat__format ) ) )
     => ( c_2Eieee_2EFinite @ ( c_2Eieee_2Efloat__add @ V1a @ V0b ) ) ) ).

thf(thm_2Efloat_2EFLOAT__SUB__FINITE,axiom,
    ! [V0b: tyop_2Eieee_2Efloat,V1a: tyop_2Eieee_2Efloat] :
      ( ( ( c_2Eieee_2EFinite @ V1a )
        & ( c_2Eieee_2EFinite @ V0b )
        & ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Eabs @ ( c_2Ereal_2Ereal__sub @ ( c_2Eieee_2EVal @ V1a ) @ ( c_2Eieee_2EVal @ V0b ) ) ) @ ( c_2Eieee_2Ethreshold @ c_2Eieee_2Efloat__format ) ) )
     => ( c_2Eieee_2EFinite @ ( c_2Eieee_2Efloat__sub @ V1a @ V0b ) ) ) ).

thf(thm_2Efloat_2EFLOAT__MUL__FINITE,axiom,
    ! [V0b: tyop_2Eieee_2Efloat,V1a: tyop_2Eieee_2Efloat] :
      ( ( ( c_2Eieee_2EFinite @ V1a )
        & ( c_2Eieee_2EFinite @ V0b )
        & ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Eabs @ ( c_2Erealax_2Ereal__mul @ ( c_2Eieee_2EVal @ V1a ) @ ( c_2Eieee_2EVal @ V0b ) ) ) @ ( c_2Eieee_2Ethreshold @ c_2Eieee_2Efloat__format ) ) )
     => ( c_2Eieee_2EFinite @ ( c_2Eieee_2Efloat__mul @ V1a @ V0b ) ) ) ).

thf(thm_2Efloat_2EFLOAT__ADD__RELATIVE,axiom,
    ! [V0a: tyop_2Eieee_2Efloat,V1b: tyop_2Eieee_2Efloat] :
      ( ( ( c_2Eieee_2EFinite @ V0a )
        & ( c_2Eieee_2EFinite @ V1b )
        & ( c_2Efloat_2Enormalizes @ ( c_2Erealax_2Ereal__add @ ( c_2Eieee_2EVal @ V0a ) @ ( c_2Eieee_2EVal @ V1b ) ) ) )
     => ( ( c_2Eieee_2EFinite @ ( c_2Eieee_2Efloat__add @ V0a @ V1b ) )
        & ? [V2e: tyop_2Erealax_2Ereal] :
            ( ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Eabs @ V2e ) @ ( c_2Ereal_2E_2F @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) @ ( c_2Ereal_2Epow @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) ) ) ) ) )
            & ( ( c_2Eieee_2EVal @ ( c_2Eieee_2Efloat__add @ V0a @ V1b ) )
              = ( c_2Erealax_2Ereal__mul @ ( c_2Erealax_2Ereal__add @ ( c_2Eieee_2EVal @ V0a ) @ ( c_2Eieee_2EVal @ V1b ) ) @ ( c_2Erealax_2Ereal__add @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) @ V2e ) ) ) ) ) ) ).

thf(thm_2Efloat_2EFLOAT__SUB__RELATIVE,axiom,
    ! [V0a: tyop_2Eieee_2Efloat,V1b: tyop_2Eieee_2Efloat] :
      ( ( ( c_2Eieee_2EFinite @ V0a )
        & ( c_2Eieee_2EFinite @ V1b )
        & ( c_2Efloat_2Enormalizes @ ( c_2Ereal_2Ereal__sub @ ( c_2Eieee_2EVal @ V0a ) @ ( c_2Eieee_2EVal @ V1b ) ) ) )
     => ( ( c_2Eieee_2EFinite @ ( c_2Eieee_2Efloat__sub @ V0a @ V1b ) )
        & ? [V2e: tyop_2Erealax_2Ereal] :
            ( ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Eabs @ V2e ) @ ( c_2Ereal_2E_2F @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) @ ( c_2Ereal_2Epow @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) ) ) ) ) )
            & ( ( c_2Eieee_2EVal @ ( c_2Eieee_2Efloat__sub @ V0a @ V1b ) )
              = ( c_2Erealax_2Ereal__mul @ ( c_2Ereal_2Ereal__sub @ ( c_2Eieee_2EVal @ V0a ) @ ( c_2Eieee_2EVal @ V1b ) ) @ ( c_2Erealax_2Ereal__add @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) @ V2e ) ) ) ) ) ) ).

thf(thm_2Efloat_2EFLOAT__MUL__RELATIVE,axiom,
    ! [V0a: tyop_2Eieee_2Efloat,V1b: tyop_2Eieee_2Efloat] :
      ( ( ( c_2Eieee_2EFinite @ V0a )
        & ( c_2Eieee_2EFinite @ V1b )
        & ( c_2Efloat_2Enormalizes @ ( c_2Erealax_2Ereal__mul @ ( c_2Eieee_2EVal @ V0a ) @ ( c_2Eieee_2EVal @ V1b ) ) ) )
     => ( ( c_2Eieee_2EFinite @ ( c_2Eieee_2Efloat__mul @ V0a @ V1b ) )
        & ? [V2e: tyop_2Erealax_2Ereal] :
            ( ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Eabs @ V2e ) @ ( c_2Ereal_2E_2F @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) @ ( c_2Ereal_2Epow @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) ) ) ) ) )
            & ( ( c_2Eieee_2EVal @ ( c_2Eieee_2Efloat__mul @ V0a @ V1b ) )
              = ( c_2Erealax_2Ereal__mul @ ( c_2Erealax_2Ereal__mul @ ( c_2Eieee_2EVal @ V0a ) @ ( c_2Eieee_2EVal @ V1b ) ) @ ( c_2Erealax_2Ereal__add @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) @ V2e ) ) ) ) ) ) ).

thf(thm_2Efloat_2EFLOAT__DIV__RELATIVE,axiom,
    ! [V0a: tyop_2Eieee_2Efloat,V1b: tyop_2Eieee_2Efloat] :
      ( ( ( c_2Eieee_2EFinite @ V0a )
        & ( c_2Eieee_2EFinite @ V1b )
        & ( (~) @ ( c_2Eieee_2EIszero @ V1b ) )
        & ( c_2Efloat_2Enormalizes @ ( c_2Ereal_2E_2F @ ( c_2Eieee_2EVal @ V0a ) @ ( c_2Eieee_2EVal @ V1b ) ) ) )
     => ( ( c_2Eieee_2EFinite @ ( c_2Eieee_2Efloat__div @ V0a @ V1b ) )
        & ? [V2e: tyop_2Erealax_2Ereal] :
            ( ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Eabs @ V2e ) @ ( c_2Ereal_2E_2F @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) @ ( c_2Ereal_2Epow @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) ) ) ) ) )
            & ( ( c_2Eieee_2EVal @ ( c_2Eieee_2Efloat__div @ V0a @ V1b ) )
              = ( c_2Erealax_2Ereal__mul @ ( c_2Ereal_2E_2F @ ( c_2Eieee_2EVal @ V0a ) @ ( c_2Eieee_2EVal @ V1b ) ) @ ( c_2Erealax_2Ereal__add @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) @ V2e ) ) ) ) ) ) ).

%------------------------------------------------------------------------------