ITP001 Axioms: ITP139^7.ax
%------------------------------------------------------------------------------
% File : ITP139^7 : TPTP v8.2.0. 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 ) ) ) ) ) ) ).
%------------------------------------------------------------------------------