ITP001 Axioms: ITP140^7.ax
%------------------------------------------------------------------------------
% File : ITP140^7 : 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 : lift_ieee.ax [Gau19]
% : HL4140^7.ax [TPAP]
% Status : Satisfiable
% Syntax : Number of formulae : 134 ( 7 unt; 79 typ; 0 def)
% Number of atoms : 223 ( 36 equ; 14 cnn)
% Maximal formula atoms : 8 ( 1 avg)
% Number of connectives : 2400 ( 14 ~; 1 |; 88 &;2241 @)
% ( 10 <=>; 46 =>; 0 <=; 0 <~>)
% Maximal formula depth : 20 ( 13 avg;2241 nst)
% Number of types : 7 ( 6 usr)
% Number of type conns : 140 ( 140 >; 0 *; 0 +; 0 <<)
% Number of symbols : 75 ( 73 usr; 4 con; 0-6 aty)
% Number of variables : 281 ( 3 ^ 196 !; 11 ?; 281 :)
% ( 71 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_SAT_EQU_NAR
% Comments :
% Bugfixes : v7.5.0 - Fixes to the axioms.
%------------------------------------------------------------------------------
thf(tyop_2Ebinary__ieee_2Eflags,type,
tyop_2Ebinary__ieee_2Eflags: $tType ).
thf(tyop_2Ebinary__ieee_2Efloat,type,
tyop_2Ebinary__ieee_2Efloat: $tType > $tType > $tType ).
thf(tyop_2Ebinary__ieee_2Erounding,type,
tyop_2Ebinary__ieee_2Erounding: $tType ).
thf(tyop_2Ebool_2Eitself,type,
tyop_2Ebool_2Eitself: $tType > $tType ).
thf(tyop_2Efcp_2Ecart,type,
tyop_2Efcp_2Ecart: $tType > $tType > $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_2Eone_2Eone,type,
tyop_2Eone_2Eone: $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_2Earithmetic_2E_2B,type,
c_2Earithmetic_2E_2B: tyop_2Enum_2Enum > tyop_2Enum_2Enum > tyop_2Enum_2Enum ).
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_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_2Epred__set_2EEMPTY,type,
c_2Epred__set_2EEMPTY:
!>[A_27a: $tType] : ( A_27a > $o ) ).
thf(c_2Epred__set_2EFINITE,type,
c_2Epred__set_2EFINITE:
!>[A_27a: $tType] : ( ( A_27a > $o ) > $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_2Ewords_2EINT__MAX,type,
c_2Ewords_2EINT__MAX:
!>[A_27a: $tType] : ( ( tyop_2Ebool_2Eitself @ A_27a ) > tyop_2Enum_2Enum ) ).
thf(c_2Earithmetic_2ENUMERAL,type,
c_2Earithmetic_2ENUMERAL: tyop_2Enum_2Enum > tyop_2Enum_2Enum ).
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_2Epred__set_2EUNIV,type,
c_2Epred__set_2EUNIV:
!>[A_27a: $tType] : ( A_27a > $o ) ).
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_2Ebinary__ieee_2Eclosest__such,type,
c_2Ebinary__ieee_2Eclosest__such:
!>[A_27a: $tType,A_27b: $tType] : ( ( ( tyop_2Ebinary__ieee_2Efloat @ A_27a @ A_27b ) > $o ) > ( ( tyop_2Ebinary__ieee_2Efloat @ A_27a @ A_27b ) > $o ) > tyop_2Erealax_2Ereal > ( tyop_2Ebinary__ieee_2Efloat @ A_27a @ A_27b ) ) ).
thf(c_2Efcp_2Edimindex,type,
c_2Efcp_2Edimindex:
!>[A_27a: $tType] : ( ( tyop_2Ebool_2Eitself @ A_27a ) > tyop_2Enum_2Enum ) ).
thf(c_2Elift__ieee_2Eerror,type,
c_2Elift__ieee_2Eerror:
!>[A_27t: $tType,A_27w: $tType] : ( ( tyop_2Ebool_2Eitself @ ( tyop_2Epair_2Eprod @ A_27t @ A_27w ) ) > tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal ) ).
thf(c_2Ebinary__ieee_2Efloat__Sign,type,
c_2Ebinary__ieee_2Efloat__Sign:
!>[A_27t: $tType,A_27w: $tType] : ( ( tyop_2Ebinary__ieee_2Efloat @ A_27t @ A_27w ) > ( tyop_2Efcp_2Ecart @ $o @ tyop_2Eone_2Eone ) ) ).
thf(c_2Ebinary__ieee_2Efloat__add,type,
c_2Ebinary__ieee_2Efloat__add:
!>[A_27t: $tType,A_27w: $tType] : ( tyop_2Ebinary__ieee_2Erounding > ( tyop_2Ebinary__ieee_2Efloat @ A_27t @ A_27w ) > ( tyop_2Ebinary__ieee_2Efloat @ A_27t @ A_27w ) > ( tyop_2Epair_2Eprod @ tyop_2Ebinary__ieee_2Eflags @ ( tyop_2Ebinary__ieee_2Efloat @ A_27t @ A_27w ) ) ) ).
thf(c_2Ebinary__ieee_2Efloat__div,type,
c_2Ebinary__ieee_2Efloat__div:
!>[A_27t: $tType,A_27w: $tType] : ( tyop_2Ebinary__ieee_2Erounding > ( tyop_2Ebinary__ieee_2Efloat @ A_27t @ A_27w ) > ( tyop_2Ebinary__ieee_2Efloat @ A_27t @ A_27w ) > ( tyop_2Epair_2Eprod @ tyop_2Ebinary__ieee_2Eflags @ ( tyop_2Ebinary__ieee_2Efloat @ A_27t @ A_27w ) ) ) ).
thf(c_2Ebinary__ieee_2Efloat__equal,type,
c_2Ebinary__ieee_2Efloat__equal:
!>[A_27t: $tType,A_27w: $tType] : ( ( tyop_2Ebinary__ieee_2Efloat @ A_27t @ A_27w ) > ( tyop_2Ebinary__ieee_2Efloat @ A_27t @ A_27w ) > $o ) ).
thf(c_2Ebinary__ieee_2Efloat__greater__equal,type,
c_2Ebinary__ieee_2Efloat__greater__equal:
!>[A_27t: $tType,A_27w: $tType] : ( ( tyop_2Ebinary__ieee_2Efloat @ A_27t @ A_27w ) > ( tyop_2Ebinary__ieee_2Efloat @ A_27t @ A_27w ) > $o ) ).
thf(c_2Ebinary__ieee_2Efloat__greater__than,type,
c_2Ebinary__ieee_2Efloat__greater__than:
!>[A_27t: $tType,A_27w: $tType] : ( ( tyop_2Ebinary__ieee_2Efloat @ A_27t @ A_27w ) > ( tyop_2Ebinary__ieee_2Efloat @ A_27t @ A_27w ) > $o ) ).
thf(c_2Ebinary__ieee_2Efloat__is__finite,type,
c_2Ebinary__ieee_2Efloat__is__finite:
!>[A_27t: $tType,A_27w: $tType] : ( ( tyop_2Ebinary__ieee_2Efloat @ A_27t @ A_27w ) > $o ) ).
thf(c_2Ebinary__ieee_2Efloat__is__nan,type,
c_2Ebinary__ieee_2Efloat__is__nan:
!>[A_27t: $tType,A_27w: $tType] : ( ( tyop_2Ebinary__ieee_2Efloat @ A_27t @ A_27w ) > $o ) ).
thf(c_2Ebinary__ieee_2Efloat__is__zero,type,
c_2Ebinary__ieee_2Efloat__is__zero:
!>[A_27t: $tType,A_27w: $tType] : ( ( tyop_2Ebinary__ieee_2Efloat @ A_27t @ A_27w ) > $o ) ).
thf(c_2Ebinary__ieee_2Efloat__less__equal,type,
c_2Ebinary__ieee_2Efloat__less__equal:
!>[A_27t: $tType,A_27w: $tType] : ( ( tyop_2Ebinary__ieee_2Efloat @ A_27t @ A_27w ) > ( tyop_2Ebinary__ieee_2Efloat @ A_27t @ A_27w ) > $o ) ).
thf(c_2Ebinary__ieee_2Efloat__less__than,type,
c_2Ebinary__ieee_2Efloat__less__than:
!>[A_27t: $tType,A_27w: $tType] : ( ( tyop_2Ebinary__ieee_2Efloat @ A_27t @ A_27w ) > ( tyop_2Ebinary__ieee_2Efloat @ A_27t @ A_27w ) > $o ) ).
thf(c_2Ebinary__ieee_2Efloat__mul,type,
c_2Ebinary__ieee_2Efloat__mul:
!>[A_27t: $tType,A_27w: $tType] : ( tyop_2Ebinary__ieee_2Erounding > ( tyop_2Ebinary__ieee_2Efloat @ A_27t @ A_27w ) > ( tyop_2Ebinary__ieee_2Efloat @ A_27t @ A_27w ) > ( tyop_2Epair_2Eprod @ tyop_2Ebinary__ieee_2Eflags @ ( tyop_2Ebinary__ieee_2Efloat @ A_27t @ A_27w ) ) ) ).
thf(c_2Ebinary__ieee_2Efloat__mul__add,type,
c_2Ebinary__ieee_2Efloat__mul__add:
!>[A_27t: $tType,A_27w: $tType] : ( tyop_2Ebinary__ieee_2Erounding > ( tyop_2Ebinary__ieee_2Efloat @ A_27t @ A_27w ) > ( tyop_2Ebinary__ieee_2Efloat @ A_27t @ A_27w ) > ( tyop_2Ebinary__ieee_2Efloat @ A_27t @ A_27w ) > ( tyop_2Epair_2Eprod @ tyop_2Ebinary__ieee_2Eflags @ ( tyop_2Ebinary__ieee_2Efloat @ A_27t @ A_27w ) ) ) ).
thf(c_2Ebinary__ieee_2Efloat__mul__sub,type,
c_2Ebinary__ieee_2Efloat__mul__sub:
!>[A_27t: $tType,A_27w: $tType] : ( tyop_2Ebinary__ieee_2Erounding > ( tyop_2Ebinary__ieee_2Efloat @ A_27t @ A_27w ) > ( tyop_2Ebinary__ieee_2Efloat @ A_27t @ A_27w ) > ( tyop_2Ebinary__ieee_2Efloat @ A_27t @ A_27w ) > ( tyop_2Epair_2Eprod @ tyop_2Ebinary__ieee_2Eflags @ ( tyop_2Ebinary__ieee_2Efloat @ A_27t @ A_27w ) ) ) ).
thf(c_2Ebinary__ieee_2Efloat__round,type,
c_2Ebinary__ieee_2Efloat__round:
!>[A_27t: $tType,A_27w: $tType] : ( tyop_2Ebinary__ieee_2Erounding > $o > tyop_2Erealax_2Ereal > ( tyop_2Ebinary__ieee_2Efloat @ A_27t @ A_27w ) ) ).
thf(c_2Ebinary__ieee_2Efloat__sqrt,type,
c_2Ebinary__ieee_2Efloat__sqrt:
!>[A_27t: $tType,A_27w: $tType] : ( tyop_2Ebinary__ieee_2Erounding > ( tyop_2Ebinary__ieee_2Efloat @ A_27t @ A_27w ) > ( tyop_2Epair_2Eprod @ tyop_2Ebinary__ieee_2Eflags @ ( tyop_2Ebinary__ieee_2Efloat @ A_27t @ A_27w ) ) ) ).
thf(c_2Ebinary__ieee_2Efloat__sub,type,
c_2Ebinary__ieee_2Efloat__sub:
!>[A_27t: $tType,A_27w: $tType] : ( tyop_2Ebinary__ieee_2Erounding > ( tyop_2Ebinary__ieee_2Efloat @ A_27t @ A_27w ) > ( tyop_2Ebinary__ieee_2Efloat @ A_27t @ A_27w ) > ( tyop_2Epair_2Eprod @ tyop_2Ebinary__ieee_2Eflags @ ( tyop_2Ebinary__ieee_2Efloat @ A_27t @ A_27w ) ) ) ).
thf(c_2Ebinary__ieee_2Efloat__to__real,type,
c_2Ebinary__ieee_2Efloat__to__real:
!>[A_27t: $tType,A_27w: $tType] : ( ( tyop_2Ebinary__ieee_2Efloat @ A_27t @ A_27w ) > tyop_2Erealax_2Ereal ) ).
thf(c_2Erealax_2Einv,type,
c_2Erealax_2Einv: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal ).
thf(c_2Ebinary__ieee_2Eis__closest,type,
c_2Ebinary__ieee_2Eis__closest:
!>[A_27a: $tType,A_27b: $tType] : ( ( ( tyop_2Ebinary__ieee_2Efloat @ A_27a @ A_27b ) > $o ) > tyop_2Erealax_2Ereal > ( tyop_2Ebinary__ieee_2Efloat @ A_27a @ A_27b ) > $o ) ).
thf(c_2Ebinary__ieee_2Elargest,type,
c_2Ebinary__ieee_2Elargest:
!>[A_27t: $tType,A_27w: $tType] : ( ( tyop_2Ebool_2Eitself @ ( tyop_2Epair_2Eprod @ A_27t @ A_27w ) ) > tyop_2Erealax_2Ereal ) ).
thf(c_2Ewords_2En2w,type,
c_2Ewords_2En2w:
!>[A_27a: $tType] : ( tyop_2Enum_2Enum > ( tyop_2Efcp_2Ecart @ $o @ A_27a ) ) ).
thf(c_2Elift__ieee_2Enormalizes,type,
c_2Elift__ieee_2Enormalizes:
!>[A_27t: $tType,A_27w: $tType] : ( ( tyop_2Ebool_2Eitself @ ( tyop_2Epair_2Eprod @ A_27t @ A_27w ) ) > tyop_2Erealax_2Ereal > $o ) ).
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_2Ebinary__ieee_2Ereal__to__float,type,
c_2Ebinary__ieee_2Ereal__to__float:
!>[A_27a: $tType,A_27b: $tType] : ( tyop_2Ebinary__ieee_2Erounding > tyop_2Erealax_2Ereal > ( tyop_2Ebinary__ieee_2Efloat @ A_27a @ A_27b ) ) ).
thf(c_2Ebinary__ieee_2Eround,type,
c_2Ebinary__ieee_2Eround:
!>[A_27t: $tType,A_27w: $tType] : ( tyop_2Ebinary__ieee_2Erounding > tyop_2Erealax_2Ereal > ( tyop_2Ebinary__ieee_2Efloat @ A_27t @ A_27w ) ) ).
thf(c_2Ebinary__ieee_2EroundTiesToEven,type,
c_2Ebinary__ieee_2EroundTiesToEven: tyop_2Ebinary__ieee_2Erounding ).
thf(c_2Etransc_2Esqrt,type,
c_2Etransc_2Esqrt: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal ).
thf(c_2Ebool_2Ethe__value,type,
c_2Ebool_2Ethe__value:
!>[A_27a: $tType] : ( tyop_2Ebool_2Eitself @ A_27a ) ).
thf(c_2Ebinary__ieee_2Ethreshold,type,
c_2Ebinary__ieee_2Ethreshold:
!>[A_27t: $tType,A_27w: $tType] : ( ( tyop_2Ebool_2Eitself @ ( tyop_2Epair_2Eprod @ A_27t @ A_27w ) ) > tyop_2Erealax_2Ereal ) ).
thf(c_2Ebinary__ieee_2Eulp,type,
c_2Ebinary__ieee_2Eulp:
!>[A_27t: $tType,A_27w: $tType] : ( ( tyop_2Ebool_2Eitself @ ( tyop_2Epair_2Eprod @ A_27t @ A_27w ) ) > tyop_2Erealax_2Ereal ) ).
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_2Elift__ieee_2Eerror__def,axiom,
! [A_27t: $tType,A_27w: $tType,V0x: tyop_2Erealax_2Ereal] :
( ( c_2Elift__ieee_2Eerror @ A_27t @ A_27w @ ( c_2Ebool_2Ethe__value @ ( tyop_2Epair_2Eprod @ A_27t @ A_27w ) ) @ V0x )
= ( c_2Ereal_2Ereal__sub @ ( c_2Ebinary__ieee_2Efloat__to__real @ A_27t @ A_27w @ ( c_2Ebinary__ieee_2Eround @ A_27t @ A_27w @ c_2Ebinary__ieee_2EroundTiesToEven @ V0x ) ) @ V0x ) ) ).
thf(thm_2Elift__ieee_2Enormalizes__def,axiom,
! [A_27t: $tType,A_27w: $tType,V0x: tyop_2Erealax_2Ereal] :
( ( c_2Elift__ieee_2Enormalizes @ A_27t @ A_27w @ ( c_2Ebool_2Ethe__value @ ( tyop_2Epair_2Eprod @ A_27t @ A_27w ) ) @ V0x )
<=> ( ( c_2Eprim__rec_2E_3C @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) @ ( c_2Ewords_2EINT__MAX @ A_27w @ ( c_2Ebool_2Ethe__value @ A_27w ) ) )
& ( 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_2Ewords_2EINT__MAX @ A_27w @ ( c_2Ebool_2Ethe__value @ A_27w ) ) @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ) @ ( c_2Ereal_2Eabs @ V0x ) )
& ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Eabs @ V0x ) @ ( c_2Ebinary__ieee_2Ethreshold @ A_27t @ A_27w @ ( c_2Ebool_2Ethe__value @ ( tyop_2Epair_2Eprod @ A_27t @ A_27w ) ) ) ) ) ) ).
thf(thm_2Elift__ieee_2Efloat__lt,axiom,
! [A_27a: $tType,A_27b: $tType,V0x: tyop_2Ebinary__ieee_2Efloat @ A_27a @ A_27b,V1y: tyop_2Ebinary__ieee_2Efloat @ A_27a @ A_27b] :
( ( ( c_2Ebinary__ieee_2Efloat__is__finite @ A_27a @ A_27b @ V0x )
& ( c_2Ebinary__ieee_2Efloat__is__finite @ A_27a @ A_27b @ V1y ) )
=> ( ( c_2Ebinary__ieee_2Efloat__less__than @ A_27a @ A_27b @ V0x @ V1y )
= ( c_2Erealax_2Ereal__lt @ ( c_2Ebinary__ieee_2Efloat__to__real @ A_27a @ A_27b @ V0x ) @ ( c_2Ebinary__ieee_2Efloat__to__real @ A_27a @ A_27b @ V1y ) ) ) ) ).
thf(thm_2Elift__ieee_2Efloat__le,axiom,
! [A_27a: $tType,A_27b: $tType,V0x: tyop_2Ebinary__ieee_2Efloat @ A_27a @ A_27b,V1y: tyop_2Ebinary__ieee_2Efloat @ A_27a @ A_27b] :
( ( ( c_2Ebinary__ieee_2Efloat__is__finite @ A_27a @ A_27b @ V0x )
& ( c_2Ebinary__ieee_2Efloat__is__finite @ A_27a @ A_27b @ V1y ) )
=> ( ( c_2Ebinary__ieee_2Efloat__less__equal @ A_27a @ A_27b @ V0x @ V1y )
= ( c_2Ereal_2Ereal__lte @ ( c_2Ebinary__ieee_2Efloat__to__real @ A_27a @ A_27b @ V0x ) @ ( c_2Ebinary__ieee_2Efloat__to__real @ A_27a @ A_27b @ V1y ) ) ) ) ).
thf(thm_2Elift__ieee_2Efloat__gt,axiom,
! [A_27a: $tType,A_27b: $tType,V0x: tyop_2Ebinary__ieee_2Efloat @ A_27a @ A_27b,V1y: tyop_2Ebinary__ieee_2Efloat @ A_27a @ A_27b] :
( ( ( c_2Ebinary__ieee_2Efloat__is__finite @ A_27a @ A_27b @ V0x )
& ( c_2Ebinary__ieee_2Efloat__is__finite @ A_27a @ A_27b @ V1y ) )
=> ( ( c_2Ebinary__ieee_2Efloat__greater__than @ A_27a @ A_27b @ V0x @ V1y )
= ( c_2Ereal_2Ereal__gt @ ( c_2Ebinary__ieee_2Efloat__to__real @ A_27a @ A_27b @ V0x ) @ ( c_2Ebinary__ieee_2Efloat__to__real @ A_27a @ A_27b @ V1y ) ) ) ) ).
thf(thm_2Elift__ieee_2Efloat__ge,axiom,
! [A_27a: $tType,A_27b: $tType,V0x: tyop_2Ebinary__ieee_2Efloat @ A_27a @ A_27b,V1y: tyop_2Ebinary__ieee_2Efloat @ A_27a @ A_27b] :
( ( ( c_2Ebinary__ieee_2Efloat__is__finite @ A_27a @ A_27b @ V0x )
& ( c_2Ebinary__ieee_2Efloat__is__finite @ A_27a @ A_27b @ V1y ) )
=> ( ( c_2Ebinary__ieee_2Efloat__greater__equal @ A_27a @ A_27b @ V0x @ V1y )
= ( c_2Ereal_2Ereal__ge @ ( c_2Ebinary__ieee_2Efloat__to__real @ A_27a @ A_27b @ V0x ) @ ( c_2Ebinary__ieee_2Efloat__to__real @ A_27a @ A_27b @ V1y ) ) ) ) ).
thf(thm_2Elift__ieee_2Efloat__eq,axiom,
! [A_27a: $tType,A_27b: $tType,V0x: tyop_2Ebinary__ieee_2Efloat @ A_27a @ A_27b,V1y: tyop_2Ebinary__ieee_2Efloat @ A_27a @ A_27b] :
( ( ( c_2Ebinary__ieee_2Efloat__is__finite @ A_27a @ A_27b @ V0x )
& ( c_2Ebinary__ieee_2Efloat__is__finite @ A_27a @ A_27b @ V1y ) )
=> ( ( c_2Ebinary__ieee_2Efloat__equal @ A_27a @ A_27b @ V0x @ V1y )
<=> ( ( c_2Ebinary__ieee_2Efloat__to__real @ A_27a @ A_27b @ V0x )
= ( c_2Ebinary__ieee_2Efloat__to__real @ A_27a @ A_27b @ V1y ) ) ) ) ).
thf(thm_2Elift__ieee_2Efloat__eq__refl,axiom,
! [A_27a: $tType,A_27b: $tType,V0x: tyop_2Ebinary__ieee_2Efloat @ A_27a @ A_27b] :
( ( c_2Ebinary__ieee_2Efloat__equal @ A_27a @ A_27b @ V0x @ V0x )
<=> ( (~) @ ( c_2Ebinary__ieee_2Efloat__is__nan @ A_27a @ A_27b @ V0x ) ) ) ).
thf(thm_2Elift__ieee_2Eis__closest__exits,axiom,
! [A_27a: $tType,A_27b: $tType,V0x: tyop_2Erealax_2Ereal,V1s: ( tyop_2Ebinary__ieee_2Efloat @ A_27a @ A_27b ) > $o] :
( ( c_2Epred__set_2EFINITE @ ( tyop_2Ebinary__ieee_2Efloat @ A_27a @ A_27b ) @ V1s )
=> ( ( (~)
@ ( V1s
= ( c_2Epred__set_2EEMPTY @ ( tyop_2Ebinary__ieee_2Efloat @ A_27a @ A_27b ) ) ) )
=> ? [V2a: tyop_2Ebinary__ieee_2Efloat @ A_27a @ A_27b] : ( c_2Ebinary__ieee_2Eis__closest @ A_27a @ A_27b @ V1s @ V0x @ V2a ) ) ) ).
thf(thm_2Elift__ieee_2Eclosest__is__everything,axiom,
! [A_27a: $tType,A_27b: $tType,V0p: ( tyop_2Ebinary__ieee_2Efloat @ A_27a @ A_27b ) > $o,V1s: ( tyop_2Ebinary__ieee_2Efloat @ A_27a @ A_27b ) > $o,V2x: tyop_2Erealax_2Ereal] :
( ( c_2Epred__set_2EFINITE @ ( tyop_2Ebinary__ieee_2Efloat @ A_27a @ A_27b ) @ V1s )
=> ( ( (~)
@ ( V1s
= ( c_2Epred__set_2EEMPTY @ ( tyop_2Ebinary__ieee_2Efloat @ A_27a @ A_27b ) ) ) )
=> ( ( c_2Ebinary__ieee_2Eis__closest @ A_27a @ A_27b @ V1s @ V2x @ ( c_2Ebinary__ieee_2Eclosest__such @ A_27a @ A_27b @ V0p @ V1s @ V2x ) )
& ( ? [V3b: tyop_2Ebinary__ieee_2Efloat @ A_27a @ A_27b] :
( ( c_2Ebinary__ieee_2Eis__closest @ A_27a @ A_27b @ V1s @ V2x @ V3b )
& ( V0p @ V3b ) )
=> ( V0p @ ( c_2Ebinary__ieee_2Eclosest__such @ A_27a @ A_27b @ V0p @ V1s @ V2x ) ) ) ) ) ) ).
thf(thm_2Elift__ieee_2Eclosest__in__set,axiom,
! [A_27a: $tType,A_27b: $tType,V0p: ( tyop_2Ebinary__ieee_2Efloat @ A_27a @ A_27b ) > $o,V1s: ( tyop_2Ebinary__ieee_2Efloat @ A_27a @ A_27b ) > $o,V2x: tyop_2Erealax_2Ereal] :
( ( c_2Epred__set_2EFINITE @ ( tyop_2Ebinary__ieee_2Efloat @ A_27a @ A_27b ) @ V1s )
=> ( ( (~)
@ ( V1s
= ( c_2Epred__set_2EEMPTY @ ( tyop_2Ebinary__ieee_2Efloat @ A_27a @ A_27b ) ) ) )
=> ( c_2Ebool_2EIN @ ( tyop_2Ebinary__ieee_2Efloat @ A_27a @ A_27b ) @ ( c_2Ebinary__ieee_2Eclosest__such @ A_27a @ A_27b @ V0p @ V1s @ V2x ) @ V1s ) ) ) ).
thf(thm_2Elift__ieee_2Eclosest__is__closest,axiom,
! [A_27a: $tType,A_27b: $tType,V0p: ( tyop_2Ebinary__ieee_2Efloat @ A_27a @ A_27b ) > $o,V1s: ( tyop_2Ebinary__ieee_2Efloat @ A_27a @ A_27b ) > $o,V2x: tyop_2Erealax_2Ereal] :
( ( c_2Epred__set_2EFINITE @ ( tyop_2Ebinary__ieee_2Efloat @ A_27a @ A_27b ) @ V1s )
=> ( ( (~)
@ ( V1s
= ( c_2Epred__set_2EEMPTY @ ( tyop_2Ebinary__ieee_2Efloat @ A_27a @ A_27b ) ) ) )
=> ( c_2Ebinary__ieee_2Eis__closest @ A_27a @ A_27b @ V1s @ V2x @ ( c_2Ebinary__ieee_2Eclosest__such @ A_27a @ A_27b @ V0p @ V1s @ V2x ) ) ) ) ).
thf(thm_2Elift__ieee_2Efloat__finite,axiom,
! [A_27t: $tType,A_27w: $tType] : ( c_2Epred__set_2EFINITE @ ( tyop_2Ebinary__ieee_2Efloat @ A_27t @ A_27w ) @ ( c_2Epred__set_2EUNIV @ ( tyop_2Ebinary__ieee_2Efloat @ A_27t @ A_27w ) ) ) ).
thf(thm_2Elift__ieee_2Eis__finite__finite,axiom,
! [A_27a: $tType,A_27b: $tType] :
( c_2Epred__set_2EFINITE @ ( tyop_2Ebinary__ieee_2Efloat @ A_27a @ A_27b )
@ ( c_2Epred__set_2EGSPEC @ ( tyop_2Ebinary__ieee_2Efloat @ A_27a @ A_27b ) @ ( tyop_2Ebinary__ieee_2Efloat @ A_27a @ A_27b )
@ ^ [V0a: tyop_2Ebinary__ieee_2Efloat @ A_27a @ A_27b] : ( c_2Epair_2E_2C @ ( tyop_2Ebinary__ieee_2Efloat @ A_27a @ A_27b ) @ $o @ V0a @ ( c_2Ebinary__ieee_2Efloat__is__finite @ A_27a @ A_27b @ V0a ) ) ) ) ).
thf(thm_2Elift__ieee_2Eis__finite__nonempty,axiom,
! [A_27a: $tType,A_27b: $tType] :
( (~)
@ ( ( c_2Epred__set_2EGSPEC @ ( tyop_2Ebinary__ieee_2Efloat @ A_27a @ A_27b ) @ ( tyop_2Ebinary__ieee_2Efloat @ A_27a @ A_27b )
@ ^ [V0a: tyop_2Ebinary__ieee_2Efloat @ A_27a @ A_27b] : ( c_2Epair_2E_2C @ ( tyop_2Ebinary__ieee_2Efloat @ A_27a @ A_27b ) @ $o @ V0a @ ( c_2Ebinary__ieee_2Efloat__is__finite @ A_27a @ A_27b @ V0a ) ) )
= ( c_2Epred__set_2EEMPTY @ ( tyop_2Ebinary__ieee_2Efloat @ A_27a @ A_27b ) ) ) ) ).
thf(thm_2Elift__ieee_2Eis__finite__closest,axiom,
! [A_27a: $tType,A_27b: $tType,V0p: ( tyop_2Ebinary__ieee_2Efloat @ A_27a @ A_27b ) > $o,V1x: tyop_2Erealax_2Ereal] :
( c_2Ebinary__ieee_2Efloat__is__finite @ A_27a @ A_27b
@ ( c_2Ebinary__ieee_2Eclosest__such @ A_27a @ A_27b @ V0p
@ ( c_2Epred__set_2EGSPEC @ ( tyop_2Ebinary__ieee_2Efloat @ A_27a @ A_27b ) @ ( tyop_2Ebinary__ieee_2Efloat @ A_27a @ A_27b )
@ ^ [V2a: tyop_2Ebinary__ieee_2Efloat @ A_27a @ A_27b] : ( c_2Epair_2E_2C @ ( tyop_2Ebinary__ieee_2Efloat @ A_27a @ A_27b ) @ $o @ V2a @ ( c_2Ebinary__ieee_2Efloat__is__finite @ A_27a @ A_27b @ V2a ) ) )
@ V1x ) ) ).
thf(thm_2Elift__ieee_2Efloat__to__real__finite,axiom,
! [A_27t: $tType,A_27w: $tType,V0x: tyop_2Ebinary__ieee_2Efloat @ A_27t @ A_27w] :
( ( c_2Ebinary__ieee_2Efloat__is__finite @ A_27t @ A_27w @ V0x )
=> ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Eabs @ ( c_2Ebinary__ieee_2Efloat__to__real @ A_27t @ A_27w @ V0x ) ) @ ( c_2Ebinary__ieee_2Elargest @ A_27t @ A_27w @ ( c_2Ebool_2Ethe__value @ ( tyop_2Epair_2Eprod @ A_27t @ A_27w ) ) ) ) ) ).
thf(thm_2Elift__ieee_2Efloat__to__real__threshold,axiom,
! [A_27t: $tType,A_27w: $tType,V0x: tyop_2Ebinary__ieee_2Efloat @ A_27t @ A_27w] :
( ( c_2Ebinary__ieee_2Efloat__is__finite @ A_27t @ A_27w @ V0x )
=> ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Eabs @ ( c_2Ebinary__ieee_2Efloat__to__real @ A_27t @ A_27w @ V0x ) ) @ ( c_2Ebinary__ieee_2Ethreshold @ A_27t @ A_27w @ ( c_2Ebool_2Ethe__value @ ( tyop_2Epair_2Eprod @ A_27t @ A_27w ) ) ) ) ) ).
thf(thm_2Elift__ieee_2Eerror__at__worst__lemma,axiom,
! [A_27t: $tType,A_27w: $tType,V0a: tyop_2Ebinary__ieee_2Efloat @ A_27t @ A_27w,V1x: tyop_2Erealax_2Ereal] :
( ( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Eabs @ V1x ) @ ( c_2Ebinary__ieee_2Ethreshold @ A_27t @ A_27w @ ( c_2Ebool_2Ethe__value @ ( tyop_2Epair_2Eprod @ A_27t @ A_27w ) ) ) )
& ( c_2Ebinary__ieee_2Efloat__is__finite @ A_27t @ A_27w @ V0a ) )
=> ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Eabs @ ( c_2Elift__ieee_2Eerror @ A_27t @ A_27w @ ( c_2Ebool_2Ethe__value @ ( tyop_2Epair_2Eprod @ A_27t @ A_27w ) ) @ V1x ) ) @ ( c_2Ereal_2Eabs @ ( c_2Ereal_2Ereal__sub @ ( c_2Ebinary__ieee_2Efloat__to__real @ A_27t @ A_27w @ V0a ) @ V1x ) ) ) ) ).
thf(thm_2Elift__ieee_2Eerror__is__zero,axiom,
! [A_27t: $tType,A_27w: $tType,V0a: tyop_2Ebinary__ieee_2Efloat @ A_27t @ A_27w,V1x: tyop_2Erealax_2Ereal] :
( ( ( c_2Ebinary__ieee_2Efloat__is__finite @ A_27t @ A_27w @ V0a )
& ( ( c_2Ebinary__ieee_2Efloat__to__real @ A_27t @ A_27w @ V0a )
= V1x ) )
=> ( ( c_2Elift__ieee_2Eerror @ A_27t @ A_27w @ ( c_2Ebool_2Ethe__value @ ( tyop_2Epair_2Eprod @ A_27t @ A_27w ) ) @ V1x )
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ) ).
thf(thm_2Elift__ieee_2Erelative__error,axiom,
! [A_27t: $tType,A_27w: $tType,V0x: tyop_2Erealax_2Ereal] :
( ( c_2Elift__ieee_2Enormalizes @ A_27t @ A_27w @ ( c_2Ebool_2Ethe__value @ ( tyop_2Epair_2Eprod @ A_27t @ A_27w ) ) @ 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_2E_2B @ ( c_2Efcp_2Edimindex @ A_27t @ ( c_2Ebool_2Ethe__value @ A_27t ) ) @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ) )
& ( ( c_2Ebinary__ieee_2Efloat__to__real @ A_27t @ A_27w @ ( c_2Ebinary__ieee_2Eround @ A_27t @ A_27w @ c_2Ebinary__ieee_2EroundTiesToEven @ 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_2Elift__ieee_2Efloat__round__finite,axiom,
! [A_27t: $tType,A_27w: $tType,V0b: $o,V1x: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Eabs @ V1x ) @ ( c_2Ebinary__ieee_2Ethreshold @ A_27t @ A_27w @ ( c_2Ebool_2Ethe__value @ ( tyop_2Epair_2Eprod @ A_27t @ A_27w ) ) ) )
=> ( c_2Ebinary__ieee_2Efloat__is__finite @ A_27t @ A_27w @ ( c_2Ebinary__ieee_2Efloat__round @ A_27t @ A_27w @ c_2Ebinary__ieee_2EroundTiesToEven @ V0b @ V1x ) ) ) ).
thf(thm_2Elift__ieee_2Efloat__add,axiom,
! [A_27t: $tType,A_27w: $tType,V0a: tyop_2Ebinary__ieee_2Efloat @ A_27t @ A_27w,V1b: tyop_2Ebinary__ieee_2Efloat @ A_27t @ A_27w] :
( ( ( c_2Ebinary__ieee_2Efloat__is__finite @ A_27t @ A_27w @ V0a )
& ( c_2Ebinary__ieee_2Efloat__is__finite @ A_27t @ A_27w @ V1b )
& ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Eabs @ ( c_2Erealax_2Ereal__add @ ( c_2Ebinary__ieee_2Efloat__to__real @ A_27t @ A_27w @ V0a ) @ ( c_2Ebinary__ieee_2Efloat__to__real @ A_27t @ A_27w @ V1b ) ) ) @ ( c_2Ebinary__ieee_2Ethreshold @ A_27t @ A_27w @ ( c_2Ebool_2Ethe__value @ ( tyop_2Epair_2Eprod @ A_27t @ A_27w ) ) ) ) )
=> ( ( c_2Ebinary__ieee_2Efloat__is__finite @ A_27t @ A_27w @ ( c_2Epair_2ESND @ tyop_2Ebinary__ieee_2Eflags @ ( tyop_2Ebinary__ieee_2Efloat @ A_27t @ A_27w ) @ ( c_2Ebinary__ieee_2Efloat__add @ A_27t @ A_27w @ c_2Ebinary__ieee_2EroundTiesToEven @ V0a @ V1b ) ) )
& ( ( c_2Ebinary__ieee_2Efloat__to__real @ A_27t @ A_27w @ ( c_2Epair_2ESND @ tyop_2Ebinary__ieee_2Eflags @ ( tyop_2Ebinary__ieee_2Efloat @ A_27t @ A_27w ) @ ( c_2Ebinary__ieee_2Efloat__add @ A_27t @ A_27w @ c_2Ebinary__ieee_2EroundTiesToEven @ V0a @ V1b ) ) )
= ( c_2Erealax_2Ereal__add @ ( c_2Erealax_2Ereal__add @ ( c_2Ebinary__ieee_2Efloat__to__real @ A_27t @ A_27w @ V0a ) @ ( c_2Ebinary__ieee_2Efloat__to__real @ A_27t @ A_27w @ V1b ) ) @ ( c_2Elift__ieee_2Eerror @ A_27t @ A_27w @ ( c_2Ebool_2Ethe__value @ ( tyop_2Epair_2Eprod @ A_27t @ A_27w ) ) @ ( c_2Erealax_2Ereal__add @ ( c_2Ebinary__ieee_2Efloat__to__real @ A_27t @ A_27w @ V0a ) @ ( c_2Ebinary__ieee_2Efloat__to__real @ A_27t @ A_27w @ V1b ) ) ) ) ) ) ) ).
thf(thm_2Elift__ieee_2Efloat__sub,axiom,
! [A_27t: $tType,A_27w: $tType,V0a: tyop_2Ebinary__ieee_2Efloat @ A_27t @ A_27w,V1b: tyop_2Ebinary__ieee_2Efloat @ A_27t @ A_27w] :
( ( ( c_2Ebinary__ieee_2Efloat__is__finite @ A_27t @ A_27w @ V0a )
& ( c_2Ebinary__ieee_2Efloat__is__finite @ A_27t @ A_27w @ V1b )
& ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Eabs @ ( c_2Ereal_2Ereal__sub @ ( c_2Ebinary__ieee_2Efloat__to__real @ A_27t @ A_27w @ V0a ) @ ( c_2Ebinary__ieee_2Efloat__to__real @ A_27t @ A_27w @ V1b ) ) ) @ ( c_2Ebinary__ieee_2Ethreshold @ A_27t @ A_27w @ ( c_2Ebool_2Ethe__value @ ( tyop_2Epair_2Eprod @ A_27t @ A_27w ) ) ) ) )
=> ( ( c_2Ebinary__ieee_2Efloat__is__finite @ A_27t @ A_27w @ ( c_2Epair_2ESND @ tyop_2Ebinary__ieee_2Eflags @ ( tyop_2Ebinary__ieee_2Efloat @ A_27t @ A_27w ) @ ( c_2Ebinary__ieee_2Efloat__sub @ A_27t @ A_27w @ c_2Ebinary__ieee_2EroundTiesToEven @ V0a @ V1b ) ) )
& ( ( c_2Ebinary__ieee_2Efloat__to__real @ A_27t @ A_27w @ ( c_2Epair_2ESND @ tyop_2Ebinary__ieee_2Eflags @ ( tyop_2Ebinary__ieee_2Efloat @ A_27t @ A_27w ) @ ( c_2Ebinary__ieee_2Efloat__sub @ A_27t @ A_27w @ c_2Ebinary__ieee_2EroundTiesToEven @ V0a @ V1b ) ) )
= ( c_2Erealax_2Ereal__add @ ( c_2Ereal_2Ereal__sub @ ( c_2Ebinary__ieee_2Efloat__to__real @ A_27t @ A_27w @ V0a ) @ ( c_2Ebinary__ieee_2Efloat__to__real @ A_27t @ A_27w @ V1b ) ) @ ( c_2Elift__ieee_2Eerror @ A_27t @ A_27w @ ( c_2Ebool_2Ethe__value @ ( tyop_2Epair_2Eprod @ A_27t @ A_27w ) ) @ ( c_2Ereal_2Ereal__sub @ ( c_2Ebinary__ieee_2Efloat__to__real @ A_27t @ A_27w @ V0a ) @ ( c_2Ebinary__ieee_2Efloat__to__real @ A_27t @ A_27w @ V1b ) ) ) ) ) ) ) ).
thf(thm_2Elift__ieee_2Efloat__mul,axiom,
! [A_27t: $tType,A_27w: $tType,V0a: tyop_2Ebinary__ieee_2Efloat @ A_27t @ A_27w,V1b: tyop_2Ebinary__ieee_2Efloat @ A_27t @ A_27w] :
( ( ( c_2Ebinary__ieee_2Efloat__is__finite @ A_27t @ A_27w @ V0a )
& ( c_2Ebinary__ieee_2Efloat__is__finite @ A_27t @ A_27w @ V1b )
& ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Eabs @ ( c_2Erealax_2Ereal__mul @ ( c_2Ebinary__ieee_2Efloat__to__real @ A_27t @ A_27w @ V0a ) @ ( c_2Ebinary__ieee_2Efloat__to__real @ A_27t @ A_27w @ V1b ) ) ) @ ( c_2Ebinary__ieee_2Ethreshold @ A_27t @ A_27w @ ( c_2Ebool_2Ethe__value @ ( tyop_2Epair_2Eprod @ A_27t @ A_27w ) ) ) ) )
=> ( ( c_2Ebinary__ieee_2Efloat__is__finite @ A_27t @ A_27w @ ( c_2Epair_2ESND @ tyop_2Ebinary__ieee_2Eflags @ ( tyop_2Ebinary__ieee_2Efloat @ A_27t @ A_27w ) @ ( c_2Ebinary__ieee_2Efloat__mul @ A_27t @ A_27w @ c_2Ebinary__ieee_2EroundTiesToEven @ V0a @ V1b ) ) )
& ( ( c_2Ebinary__ieee_2Efloat__to__real @ A_27t @ A_27w @ ( c_2Epair_2ESND @ tyop_2Ebinary__ieee_2Eflags @ ( tyop_2Ebinary__ieee_2Efloat @ A_27t @ A_27w ) @ ( c_2Ebinary__ieee_2Efloat__mul @ A_27t @ A_27w @ c_2Ebinary__ieee_2EroundTiesToEven @ V0a @ V1b ) ) )
= ( c_2Erealax_2Ereal__add @ ( c_2Erealax_2Ereal__mul @ ( c_2Ebinary__ieee_2Efloat__to__real @ A_27t @ A_27w @ V0a ) @ ( c_2Ebinary__ieee_2Efloat__to__real @ A_27t @ A_27w @ V1b ) ) @ ( c_2Elift__ieee_2Eerror @ A_27t @ A_27w @ ( c_2Ebool_2Ethe__value @ ( tyop_2Epair_2Eprod @ A_27t @ A_27w ) ) @ ( c_2Erealax_2Ereal__mul @ ( c_2Ebinary__ieee_2Efloat__to__real @ A_27t @ A_27w @ V0a ) @ ( c_2Ebinary__ieee_2Efloat__to__real @ A_27t @ A_27w @ V1b ) ) ) ) ) ) ) ).
thf(thm_2Elift__ieee_2Efloat__div,axiom,
! [A_27t: $tType,A_27w: $tType,V0a: tyop_2Ebinary__ieee_2Efloat @ A_27t @ A_27w,V1b: tyop_2Ebinary__ieee_2Efloat @ A_27t @ A_27w] :
( ( ( c_2Ebinary__ieee_2Efloat__is__finite @ A_27t @ A_27w @ V0a )
& ( c_2Ebinary__ieee_2Efloat__is__finite @ A_27t @ A_27w @ V1b )
& ( (~) @ ( c_2Ebinary__ieee_2Efloat__is__zero @ A_27t @ A_27w @ V1b ) )
& ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Eabs @ ( c_2Ereal_2E_2F @ ( c_2Ebinary__ieee_2Efloat__to__real @ A_27t @ A_27w @ V0a ) @ ( c_2Ebinary__ieee_2Efloat__to__real @ A_27t @ A_27w @ V1b ) ) ) @ ( c_2Ebinary__ieee_2Ethreshold @ A_27t @ A_27w @ ( c_2Ebool_2Ethe__value @ ( tyop_2Epair_2Eprod @ A_27t @ A_27w ) ) ) ) )
=> ( ( c_2Ebinary__ieee_2Efloat__is__finite @ A_27t @ A_27w @ ( c_2Epair_2ESND @ tyop_2Ebinary__ieee_2Eflags @ ( tyop_2Ebinary__ieee_2Efloat @ A_27t @ A_27w ) @ ( c_2Ebinary__ieee_2Efloat__div @ A_27t @ A_27w @ c_2Ebinary__ieee_2EroundTiesToEven @ V0a @ V1b ) ) )
& ( ( c_2Ebinary__ieee_2Efloat__to__real @ A_27t @ A_27w @ ( c_2Epair_2ESND @ tyop_2Ebinary__ieee_2Eflags @ ( tyop_2Ebinary__ieee_2Efloat @ A_27t @ A_27w ) @ ( c_2Ebinary__ieee_2Efloat__div @ A_27t @ A_27w @ c_2Ebinary__ieee_2EroundTiesToEven @ V0a @ V1b ) ) )
= ( c_2Erealax_2Ereal__add @ ( c_2Ereal_2E_2F @ ( c_2Ebinary__ieee_2Efloat__to__real @ A_27t @ A_27w @ V0a ) @ ( c_2Ebinary__ieee_2Efloat__to__real @ A_27t @ A_27w @ V1b ) ) @ ( c_2Elift__ieee_2Eerror @ A_27t @ A_27w @ ( c_2Ebool_2Ethe__value @ ( tyop_2Epair_2Eprod @ A_27t @ A_27w ) ) @ ( c_2Ereal_2E_2F @ ( c_2Ebinary__ieee_2Efloat__to__real @ A_27t @ A_27w @ V0a ) @ ( c_2Ebinary__ieee_2Efloat__to__real @ A_27t @ A_27w @ V1b ) ) ) ) ) ) ) ).
thf(thm_2Elift__ieee_2Efloat__sqrt,axiom,
! [A_27t: $tType,A_27w: $tType,V0a: tyop_2Ebinary__ieee_2Efloat @ A_27t @ A_27w] :
( ( ( c_2Ebinary__ieee_2Efloat__is__finite @ A_27t @ A_27w @ V0a )
& ( ( c_2Ebinary__ieee_2Efloat__Sign @ A_27t @ A_27w @ V0a )
= ( c_2Ewords_2En2w @ tyop_2Eone_2Eone @ c_2Enum_2E0 ) )
& ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Eabs @ ( c_2Etransc_2Esqrt @ ( c_2Ebinary__ieee_2Efloat__to__real @ A_27t @ A_27w @ V0a ) ) ) @ ( c_2Ebinary__ieee_2Ethreshold @ A_27t @ A_27w @ ( c_2Ebool_2Ethe__value @ ( tyop_2Epair_2Eprod @ A_27t @ A_27w ) ) ) ) )
=> ( ( c_2Ebinary__ieee_2Efloat__is__finite @ A_27t @ A_27w @ ( c_2Epair_2ESND @ tyop_2Ebinary__ieee_2Eflags @ ( tyop_2Ebinary__ieee_2Efloat @ A_27t @ A_27w ) @ ( c_2Ebinary__ieee_2Efloat__sqrt @ A_27t @ A_27w @ c_2Ebinary__ieee_2EroundTiesToEven @ V0a ) ) )
& ( ( c_2Ebinary__ieee_2Efloat__to__real @ A_27t @ A_27w @ ( c_2Epair_2ESND @ tyop_2Ebinary__ieee_2Eflags @ ( tyop_2Ebinary__ieee_2Efloat @ A_27t @ A_27w ) @ ( c_2Ebinary__ieee_2Efloat__sqrt @ A_27t @ A_27w @ c_2Ebinary__ieee_2EroundTiesToEven @ V0a ) ) )
= ( c_2Erealax_2Ereal__add @ ( c_2Etransc_2Esqrt @ ( c_2Ebinary__ieee_2Efloat__to__real @ A_27t @ A_27w @ V0a ) ) @ ( c_2Elift__ieee_2Eerror @ A_27t @ A_27w @ ( c_2Ebool_2Ethe__value @ ( tyop_2Epair_2Eprod @ A_27t @ A_27w ) ) @ ( c_2Etransc_2Esqrt @ ( c_2Ebinary__ieee_2Efloat__to__real @ A_27t @ A_27w @ V0a ) ) ) ) ) ) ) ).
thf(thm_2Elift__ieee_2Efloat__mul__add,axiom,
! [A_27t: $tType,A_27w: $tType,V0a: tyop_2Ebinary__ieee_2Efloat @ A_27t @ A_27w,V1b: tyop_2Ebinary__ieee_2Efloat @ A_27t @ A_27w,V2c: tyop_2Ebinary__ieee_2Efloat @ A_27t @ A_27w] :
( ( ( c_2Ebinary__ieee_2Efloat__is__finite @ A_27t @ A_27w @ V0a )
& ( c_2Ebinary__ieee_2Efloat__is__finite @ A_27t @ A_27w @ V1b )
& ( c_2Ebinary__ieee_2Efloat__is__finite @ A_27t @ A_27w @ V2c )
& ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Eabs @ ( c_2Erealax_2Ereal__add @ ( c_2Erealax_2Ereal__mul @ ( c_2Ebinary__ieee_2Efloat__to__real @ A_27t @ A_27w @ V0a ) @ ( c_2Ebinary__ieee_2Efloat__to__real @ A_27t @ A_27w @ V1b ) ) @ ( c_2Ebinary__ieee_2Efloat__to__real @ A_27t @ A_27w @ V2c ) ) ) @ ( c_2Ebinary__ieee_2Ethreshold @ A_27t @ A_27w @ ( c_2Ebool_2Ethe__value @ ( tyop_2Epair_2Eprod @ A_27t @ A_27w ) ) ) ) )
=> ( ( c_2Ebinary__ieee_2Efloat__is__finite @ A_27t @ A_27w @ ( c_2Epair_2ESND @ tyop_2Ebinary__ieee_2Eflags @ ( tyop_2Ebinary__ieee_2Efloat @ A_27t @ A_27w ) @ ( c_2Ebinary__ieee_2Efloat__mul__add @ A_27t @ A_27w @ c_2Ebinary__ieee_2EroundTiesToEven @ V0a @ V1b @ V2c ) ) )
& ( ( c_2Ebinary__ieee_2Efloat__to__real @ A_27t @ A_27w @ ( c_2Epair_2ESND @ tyop_2Ebinary__ieee_2Eflags @ ( tyop_2Ebinary__ieee_2Efloat @ A_27t @ A_27w ) @ ( c_2Ebinary__ieee_2Efloat__mul__add @ A_27t @ A_27w @ c_2Ebinary__ieee_2EroundTiesToEven @ V0a @ V1b @ V2c ) ) )
= ( c_2Erealax_2Ereal__add @ ( c_2Erealax_2Ereal__add @ ( c_2Erealax_2Ereal__mul @ ( c_2Ebinary__ieee_2Efloat__to__real @ A_27t @ A_27w @ V0a ) @ ( c_2Ebinary__ieee_2Efloat__to__real @ A_27t @ A_27w @ V1b ) ) @ ( c_2Ebinary__ieee_2Efloat__to__real @ A_27t @ A_27w @ V2c ) ) @ ( c_2Elift__ieee_2Eerror @ A_27t @ A_27w @ ( c_2Ebool_2Ethe__value @ ( tyop_2Epair_2Eprod @ A_27t @ A_27w ) ) @ ( c_2Erealax_2Ereal__add @ ( c_2Erealax_2Ereal__mul @ ( c_2Ebinary__ieee_2Efloat__to__real @ A_27t @ A_27w @ V0a ) @ ( c_2Ebinary__ieee_2Efloat__to__real @ A_27t @ A_27w @ V1b ) ) @ ( c_2Ebinary__ieee_2Efloat__to__real @ A_27t @ A_27w @ V2c ) ) ) ) ) ) ) ).
thf(thm_2Elift__ieee_2Efloat__mul__sub,axiom,
! [A_27t: $tType,A_27w: $tType,V0a: tyop_2Ebinary__ieee_2Efloat @ A_27t @ A_27w,V1b: tyop_2Ebinary__ieee_2Efloat @ A_27t @ A_27w,V2c: tyop_2Ebinary__ieee_2Efloat @ A_27t @ A_27w] :
( ( ( c_2Ebinary__ieee_2Efloat__is__finite @ A_27t @ A_27w @ V0a )
& ( c_2Ebinary__ieee_2Efloat__is__finite @ A_27t @ A_27w @ V1b )
& ( c_2Ebinary__ieee_2Efloat__is__finite @ A_27t @ A_27w @ V2c )
& ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Eabs @ ( c_2Ereal_2Ereal__sub @ ( c_2Erealax_2Ereal__mul @ ( c_2Ebinary__ieee_2Efloat__to__real @ A_27t @ A_27w @ V0a ) @ ( c_2Ebinary__ieee_2Efloat__to__real @ A_27t @ A_27w @ V1b ) ) @ ( c_2Ebinary__ieee_2Efloat__to__real @ A_27t @ A_27w @ V2c ) ) ) @ ( c_2Ebinary__ieee_2Ethreshold @ A_27t @ A_27w @ ( c_2Ebool_2Ethe__value @ ( tyop_2Epair_2Eprod @ A_27t @ A_27w ) ) ) ) )
=> ( ( c_2Ebinary__ieee_2Efloat__is__finite @ A_27t @ A_27w @ ( c_2Epair_2ESND @ tyop_2Ebinary__ieee_2Eflags @ ( tyop_2Ebinary__ieee_2Efloat @ A_27t @ A_27w ) @ ( c_2Ebinary__ieee_2Efloat__mul__sub @ A_27t @ A_27w @ c_2Ebinary__ieee_2EroundTiesToEven @ V0a @ V1b @ V2c ) ) )
& ( ( c_2Ebinary__ieee_2Efloat__to__real @ A_27t @ A_27w @ ( c_2Epair_2ESND @ tyop_2Ebinary__ieee_2Eflags @ ( tyop_2Ebinary__ieee_2Efloat @ A_27t @ A_27w ) @ ( c_2Ebinary__ieee_2Efloat__mul__sub @ A_27t @ A_27w @ c_2Ebinary__ieee_2EroundTiesToEven @ V0a @ V1b @ V2c ) ) )
= ( c_2Erealax_2Ereal__add @ ( c_2Ereal_2Ereal__sub @ ( c_2Erealax_2Ereal__mul @ ( c_2Ebinary__ieee_2Efloat__to__real @ A_27t @ A_27w @ V0a ) @ ( c_2Ebinary__ieee_2Efloat__to__real @ A_27t @ A_27w @ V1b ) ) @ ( c_2Ebinary__ieee_2Efloat__to__real @ A_27t @ A_27w @ V2c ) ) @ ( c_2Elift__ieee_2Eerror @ A_27t @ A_27w @ ( c_2Ebool_2Ethe__value @ ( tyop_2Epair_2Eprod @ A_27t @ A_27w ) ) @ ( c_2Ereal_2Ereal__sub @ ( c_2Erealax_2Ereal__mul @ ( c_2Ebinary__ieee_2Efloat__to__real @ A_27t @ A_27w @ V0a ) @ ( c_2Ebinary__ieee_2Efloat__to__real @ A_27t @ A_27w @ V1b ) ) @ ( c_2Ebinary__ieee_2Efloat__to__real @ A_27t @ A_27w @ V2c ) ) ) ) ) ) ) ).
thf(thm_2Elift__ieee_2Efloat__add__finite,axiom,
! [A_27t: $tType,A_27w: $tType,V0a: tyop_2Ebinary__ieee_2Efloat @ A_27t @ A_27w,V1b: tyop_2Ebinary__ieee_2Efloat @ A_27t @ A_27w] :
( ( ( c_2Ebinary__ieee_2Efloat__is__finite @ A_27t @ A_27w @ V0a )
& ( c_2Ebinary__ieee_2Efloat__is__finite @ A_27t @ A_27w @ V1b )
& ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Eabs @ ( c_2Erealax_2Ereal__add @ ( c_2Ebinary__ieee_2Efloat__to__real @ A_27t @ A_27w @ V0a ) @ ( c_2Ebinary__ieee_2Efloat__to__real @ A_27t @ A_27w @ V1b ) ) ) @ ( c_2Ebinary__ieee_2Ethreshold @ A_27t @ A_27w @ ( c_2Ebool_2Ethe__value @ ( tyop_2Epair_2Eprod @ A_27t @ A_27w ) ) ) ) )
=> ( c_2Ebinary__ieee_2Efloat__is__finite @ A_27t @ A_27w @ ( c_2Epair_2ESND @ tyop_2Ebinary__ieee_2Eflags @ ( tyop_2Ebinary__ieee_2Efloat @ A_27t @ A_27w ) @ ( c_2Ebinary__ieee_2Efloat__add @ A_27t @ A_27w @ c_2Ebinary__ieee_2EroundTiesToEven @ V0a @ V1b ) ) ) ) ).
thf(thm_2Elift__ieee_2Efloat__sub__finite,axiom,
! [A_27t: $tType,A_27w: $tType,V0a: tyop_2Ebinary__ieee_2Efloat @ A_27t @ A_27w,V1b: tyop_2Ebinary__ieee_2Efloat @ A_27t @ A_27w] :
( ( ( c_2Ebinary__ieee_2Efloat__is__finite @ A_27t @ A_27w @ V0a )
& ( c_2Ebinary__ieee_2Efloat__is__finite @ A_27t @ A_27w @ V1b )
& ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Eabs @ ( c_2Ereal_2Ereal__sub @ ( c_2Ebinary__ieee_2Efloat__to__real @ A_27t @ A_27w @ V0a ) @ ( c_2Ebinary__ieee_2Efloat__to__real @ A_27t @ A_27w @ V1b ) ) ) @ ( c_2Ebinary__ieee_2Ethreshold @ A_27t @ A_27w @ ( c_2Ebool_2Ethe__value @ ( tyop_2Epair_2Eprod @ A_27t @ A_27w ) ) ) ) )
=> ( c_2Ebinary__ieee_2Efloat__is__finite @ A_27t @ A_27w @ ( c_2Epair_2ESND @ tyop_2Ebinary__ieee_2Eflags @ ( tyop_2Ebinary__ieee_2Efloat @ A_27t @ A_27w ) @ ( c_2Ebinary__ieee_2Efloat__sub @ A_27t @ A_27w @ c_2Ebinary__ieee_2EroundTiesToEven @ V0a @ V1b ) ) ) ) ).
thf(thm_2Elift__ieee_2Efloat__mul__finite,axiom,
! [A_27t: $tType,A_27w: $tType,V0a: tyop_2Ebinary__ieee_2Efloat @ A_27t @ A_27w,V1b: tyop_2Ebinary__ieee_2Efloat @ A_27t @ A_27w] :
( ( ( c_2Ebinary__ieee_2Efloat__is__finite @ A_27t @ A_27w @ V0a )
& ( c_2Ebinary__ieee_2Efloat__is__finite @ A_27t @ A_27w @ V1b )
& ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Eabs @ ( c_2Erealax_2Ereal__mul @ ( c_2Ebinary__ieee_2Efloat__to__real @ A_27t @ A_27w @ V0a ) @ ( c_2Ebinary__ieee_2Efloat__to__real @ A_27t @ A_27w @ V1b ) ) ) @ ( c_2Ebinary__ieee_2Ethreshold @ A_27t @ A_27w @ ( c_2Ebool_2Ethe__value @ ( tyop_2Epair_2Eprod @ A_27t @ A_27w ) ) ) ) )
=> ( c_2Ebinary__ieee_2Efloat__is__finite @ A_27t @ A_27w @ ( c_2Epair_2ESND @ tyop_2Ebinary__ieee_2Eflags @ ( tyop_2Ebinary__ieee_2Efloat @ A_27t @ A_27w ) @ ( c_2Ebinary__ieee_2Efloat__mul @ A_27t @ A_27w @ c_2Ebinary__ieee_2EroundTiesToEven @ V0a @ V1b ) ) ) ) ).
thf(thm_2Elift__ieee_2Efloat__div__finite,axiom,
! [A_27t: $tType,A_27w: $tType,V0a: tyop_2Ebinary__ieee_2Efloat @ A_27t @ A_27w,V1b: tyop_2Ebinary__ieee_2Efloat @ A_27t @ A_27w] :
( ( ( c_2Ebinary__ieee_2Efloat__is__finite @ A_27t @ A_27w @ V0a )
& ( c_2Ebinary__ieee_2Efloat__is__finite @ A_27t @ A_27w @ V1b )
& ( (~) @ ( c_2Ebinary__ieee_2Efloat__is__zero @ A_27t @ A_27w @ V1b ) )
& ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Eabs @ ( c_2Ereal_2E_2F @ ( c_2Ebinary__ieee_2Efloat__to__real @ A_27t @ A_27w @ V0a ) @ ( c_2Ebinary__ieee_2Efloat__to__real @ A_27t @ A_27w @ V1b ) ) ) @ ( c_2Ebinary__ieee_2Ethreshold @ A_27t @ A_27w @ ( c_2Ebool_2Ethe__value @ ( tyop_2Epair_2Eprod @ A_27t @ A_27w ) ) ) ) )
=> ( c_2Ebinary__ieee_2Efloat__is__finite @ A_27t @ A_27w @ ( c_2Epair_2ESND @ tyop_2Ebinary__ieee_2Eflags @ ( tyop_2Ebinary__ieee_2Efloat @ A_27t @ A_27w ) @ ( c_2Ebinary__ieee_2Efloat__div @ A_27t @ A_27w @ c_2Ebinary__ieee_2EroundTiesToEven @ V0a @ V1b ) ) ) ) ).
thf(thm_2Elift__ieee_2Efloat__sqrt__finite,axiom,
! [A_27t: $tType,A_27w: $tType,V0a: tyop_2Ebinary__ieee_2Efloat @ A_27t @ A_27w] :
( ( ( c_2Ebinary__ieee_2Efloat__is__finite @ A_27t @ A_27w @ V0a )
& ( ( c_2Ebinary__ieee_2Efloat__Sign @ A_27t @ A_27w @ V0a )
= ( c_2Ewords_2En2w @ tyop_2Eone_2Eone @ c_2Enum_2E0 ) )
& ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Eabs @ ( c_2Etransc_2Esqrt @ ( c_2Ebinary__ieee_2Efloat__to__real @ A_27t @ A_27w @ V0a ) ) ) @ ( c_2Ebinary__ieee_2Ethreshold @ A_27t @ A_27w @ ( c_2Ebool_2Ethe__value @ ( tyop_2Epair_2Eprod @ A_27t @ A_27w ) ) ) ) )
=> ( c_2Ebinary__ieee_2Efloat__is__finite @ A_27t @ A_27w @ ( c_2Epair_2ESND @ tyop_2Ebinary__ieee_2Eflags @ ( tyop_2Ebinary__ieee_2Efloat @ A_27t @ A_27w ) @ ( c_2Ebinary__ieee_2Efloat__sqrt @ A_27t @ A_27w @ c_2Ebinary__ieee_2EroundTiesToEven @ V0a ) ) ) ) ).
thf(thm_2Elift__ieee_2Efloat__mul__add__finite,axiom,
! [A_27t: $tType,A_27w: $tType,V0a: tyop_2Ebinary__ieee_2Efloat @ A_27t @ A_27w,V1b: tyop_2Ebinary__ieee_2Efloat @ A_27t @ A_27w,V2c: tyop_2Ebinary__ieee_2Efloat @ A_27t @ A_27w] :
( ( ( c_2Ebinary__ieee_2Efloat__is__finite @ A_27t @ A_27w @ V0a )
& ( c_2Ebinary__ieee_2Efloat__is__finite @ A_27t @ A_27w @ V1b )
& ( c_2Ebinary__ieee_2Efloat__is__finite @ A_27t @ A_27w @ V2c )
& ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Eabs @ ( c_2Erealax_2Ereal__add @ ( c_2Erealax_2Ereal__mul @ ( c_2Ebinary__ieee_2Efloat__to__real @ A_27t @ A_27w @ V0a ) @ ( c_2Ebinary__ieee_2Efloat__to__real @ A_27t @ A_27w @ V1b ) ) @ ( c_2Ebinary__ieee_2Efloat__to__real @ A_27t @ A_27w @ V2c ) ) ) @ ( c_2Ebinary__ieee_2Ethreshold @ A_27t @ A_27w @ ( c_2Ebool_2Ethe__value @ ( tyop_2Epair_2Eprod @ A_27t @ A_27w ) ) ) ) )
=> ( c_2Ebinary__ieee_2Efloat__is__finite @ A_27t @ A_27w @ ( c_2Epair_2ESND @ tyop_2Ebinary__ieee_2Eflags @ ( tyop_2Ebinary__ieee_2Efloat @ A_27t @ A_27w ) @ ( c_2Ebinary__ieee_2Efloat__mul__add @ A_27t @ A_27w @ c_2Ebinary__ieee_2EroundTiesToEven @ V0a @ V1b @ V2c ) ) ) ) ).
thf(thm_2Elift__ieee_2Efloat__mul__sub__finite,axiom,
! [A_27t: $tType,A_27w: $tType,V0a: tyop_2Ebinary__ieee_2Efloat @ A_27t @ A_27w,V1b: tyop_2Ebinary__ieee_2Efloat @ A_27t @ A_27w,V2c: tyop_2Ebinary__ieee_2Efloat @ A_27t @ A_27w] :
( ( ( c_2Ebinary__ieee_2Efloat__is__finite @ A_27t @ A_27w @ V0a )
& ( c_2Ebinary__ieee_2Efloat__is__finite @ A_27t @ A_27w @ V1b )
& ( c_2Ebinary__ieee_2Efloat__is__finite @ A_27t @ A_27w @ V2c )
& ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Eabs @ ( c_2Ereal_2Ereal__sub @ ( c_2Erealax_2Ereal__mul @ ( c_2Ebinary__ieee_2Efloat__to__real @ A_27t @ A_27w @ V0a ) @ ( c_2Ebinary__ieee_2Efloat__to__real @ A_27t @ A_27w @ V1b ) ) @ ( c_2Ebinary__ieee_2Efloat__to__real @ A_27t @ A_27w @ V2c ) ) ) @ ( c_2Ebinary__ieee_2Ethreshold @ A_27t @ A_27w @ ( c_2Ebool_2Ethe__value @ ( tyop_2Epair_2Eprod @ A_27t @ A_27w ) ) ) ) )
=> ( c_2Ebinary__ieee_2Efloat__is__finite @ A_27t @ A_27w @ ( c_2Epair_2ESND @ tyop_2Ebinary__ieee_2Eflags @ ( tyop_2Ebinary__ieee_2Efloat @ A_27t @ A_27w ) @ ( c_2Ebinary__ieee_2Efloat__mul__sub @ A_27t @ A_27w @ c_2Ebinary__ieee_2EroundTiesToEven @ V0a @ V1b @ V2c ) ) ) ) ).
thf(thm_2Elift__ieee_2Efloat__add__relative,axiom,
! [A_27t: $tType,A_27w: $tType,V0a: tyop_2Ebinary__ieee_2Efloat @ A_27t @ A_27w,V1b: tyop_2Ebinary__ieee_2Efloat @ A_27t @ A_27w] :
( ( ( c_2Ebinary__ieee_2Efloat__is__finite @ A_27t @ A_27w @ V0a )
& ( c_2Ebinary__ieee_2Efloat__is__finite @ A_27t @ A_27w @ V1b )
& ( c_2Elift__ieee_2Enormalizes @ A_27t @ A_27w @ ( c_2Ebool_2Ethe__value @ ( tyop_2Epair_2Eprod @ A_27t @ A_27w ) ) @ ( c_2Erealax_2Ereal__add @ ( c_2Ebinary__ieee_2Efloat__to__real @ A_27t @ A_27w @ V0a ) @ ( c_2Ebinary__ieee_2Efloat__to__real @ A_27t @ A_27w @ V1b ) ) ) )
=> ( ( c_2Ebinary__ieee_2Efloat__is__finite @ A_27t @ A_27w @ ( c_2Epair_2ESND @ tyop_2Ebinary__ieee_2Eflags @ ( tyop_2Ebinary__ieee_2Efloat @ A_27t @ A_27w ) @ ( c_2Ebinary__ieee_2Efloat__add @ A_27t @ A_27w @ c_2Ebinary__ieee_2EroundTiesToEven @ 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_2E_2B @ ( c_2Efcp_2Edimindex @ A_27t @ ( c_2Ebool_2Ethe__value @ A_27t ) ) @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ) )
& ( ( c_2Ebinary__ieee_2Efloat__to__real @ A_27t @ A_27w @ ( c_2Epair_2ESND @ tyop_2Ebinary__ieee_2Eflags @ ( tyop_2Ebinary__ieee_2Efloat @ A_27t @ A_27w ) @ ( c_2Ebinary__ieee_2Efloat__add @ A_27t @ A_27w @ c_2Ebinary__ieee_2EroundTiesToEven @ V0a @ V1b ) ) )
= ( c_2Erealax_2Ereal__mul @ ( c_2Erealax_2Ereal__add @ ( c_2Ebinary__ieee_2Efloat__to__real @ A_27t @ A_27w @ V0a ) @ ( c_2Ebinary__ieee_2Efloat__to__real @ A_27t @ A_27w @ V1b ) ) @ ( c_2Erealax_2Ereal__add @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) @ V2e ) ) ) ) ) ) ).
thf(thm_2Elift__ieee_2Efloat__sub__relative,axiom,
! [A_27t: $tType,A_27w: $tType,V0a: tyop_2Ebinary__ieee_2Efloat @ A_27t @ A_27w,V1b: tyop_2Ebinary__ieee_2Efloat @ A_27t @ A_27w] :
( ( ( c_2Ebinary__ieee_2Efloat__is__finite @ A_27t @ A_27w @ V0a )
& ( c_2Ebinary__ieee_2Efloat__is__finite @ A_27t @ A_27w @ V1b )
& ( c_2Elift__ieee_2Enormalizes @ A_27t @ A_27w @ ( c_2Ebool_2Ethe__value @ ( tyop_2Epair_2Eprod @ A_27t @ A_27w ) ) @ ( c_2Ereal_2Ereal__sub @ ( c_2Ebinary__ieee_2Efloat__to__real @ A_27t @ A_27w @ V0a ) @ ( c_2Ebinary__ieee_2Efloat__to__real @ A_27t @ A_27w @ V1b ) ) ) )
=> ( ( c_2Ebinary__ieee_2Efloat__is__finite @ A_27t @ A_27w @ ( c_2Epair_2ESND @ tyop_2Ebinary__ieee_2Eflags @ ( tyop_2Ebinary__ieee_2Efloat @ A_27t @ A_27w ) @ ( c_2Ebinary__ieee_2Efloat__sub @ A_27t @ A_27w @ c_2Ebinary__ieee_2EroundTiesToEven @ 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_2E_2B @ ( c_2Efcp_2Edimindex @ A_27t @ ( c_2Ebool_2Ethe__value @ A_27t ) ) @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ) )
& ( ( c_2Ebinary__ieee_2Efloat__to__real @ A_27t @ A_27w @ ( c_2Epair_2ESND @ tyop_2Ebinary__ieee_2Eflags @ ( tyop_2Ebinary__ieee_2Efloat @ A_27t @ A_27w ) @ ( c_2Ebinary__ieee_2Efloat__sub @ A_27t @ A_27w @ c_2Ebinary__ieee_2EroundTiesToEven @ V0a @ V1b ) ) )
= ( c_2Erealax_2Ereal__mul @ ( c_2Ereal_2Ereal__sub @ ( c_2Ebinary__ieee_2Efloat__to__real @ A_27t @ A_27w @ V0a ) @ ( c_2Ebinary__ieee_2Efloat__to__real @ A_27t @ A_27w @ V1b ) ) @ ( c_2Erealax_2Ereal__add @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) @ V2e ) ) ) ) ) ) ).
thf(thm_2Elift__ieee_2Efloat__mul__relative,axiom,
! [A_27t: $tType,A_27w: $tType,V0a: tyop_2Ebinary__ieee_2Efloat @ A_27t @ A_27w,V1b: tyop_2Ebinary__ieee_2Efloat @ A_27t @ A_27w] :
( ( ( c_2Ebinary__ieee_2Efloat__is__finite @ A_27t @ A_27w @ V0a )
& ( c_2Ebinary__ieee_2Efloat__is__finite @ A_27t @ A_27w @ V1b )
& ( c_2Elift__ieee_2Enormalizes @ A_27t @ A_27w @ ( c_2Ebool_2Ethe__value @ ( tyop_2Epair_2Eprod @ A_27t @ A_27w ) ) @ ( c_2Erealax_2Ereal__mul @ ( c_2Ebinary__ieee_2Efloat__to__real @ A_27t @ A_27w @ V0a ) @ ( c_2Ebinary__ieee_2Efloat__to__real @ A_27t @ A_27w @ V1b ) ) ) )
=> ( ( c_2Ebinary__ieee_2Efloat__is__finite @ A_27t @ A_27w @ ( c_2Epair_2ESND @ tyop_2Ebinary__ieee_2Eflags @ ( tyop_2Ebinary__ieee_2Efloat @ A_27t @ A_27w ) @ ( c_2Ebinary__ieee_2Efloat__mul @ A_27t @ A_27w @ c_2Ebinary__ieee_2EroundTiesToEven @ 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_2E_2B @ ( c_2Efcp_2Edimindex @ A_27t @ ( c_2Ebool_2Ethe__value @ A_27t ) ) @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ) )
& ( ( c_2Ebinary__ieee_2Efloat__to__real @ A_27t @ A_27w @ ( c_2Epair_2ESND @ tyop_2Ebinary__ieee_2Eflags @ ( tyop_2Ebinary__ieee_2Efloat @ A_27t @ A_27w ) @ ( c_2Ebinary__ieee_2Efloat__mul @ A_27t @ A_27w @ c_2Ebinary__ieee_2EroundTiesToEven @ V0a @ V1b ) ) )
= ( c_2Erealax_2Ereal__mul @ ( c_2Erealax_2Ereal__mul @ ( c_2Ebinary__ieee_2Efloat__to__real @ A_27t @ A_27w @ V0a ) @ ( c_2Ebinary__ieee_2Efloat__to__real @ A_27t @ A_27w @ V1b ) ) @ ( c_2Erealax_2Ereal__add @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) @ V2e ) ) ) ) ) ) ).
thf(thm_2Elift__ieee_2Efloat__div__relative,axiom,
! [A_27t: $tType,A_27w: $tType,V0a: tyop_2Ebinary__ieee_2Efloat @ A_27t @ A_27w,V1b: tyop_2Ebinary__ieee_2Efloat @ A_27t @ A_27w] :
( ( ( c_2Ebinary__ieee_2Efloat__is__finite @ A_27t @ A_27w @ V0a )
& ( c_2Ebinary__ieee_2Efloat__is__finite @ A_27t @ A_27w @ V1b )
& ( (~) @ ( c_2Ebinary__ieee_2Efloat__is__zero @ A_27t @ A_27w @ V1b ) )
& ( c_2Elift__ieee_2Enormalizes @ A_27t @ A_27w @ ( c_2Ebool_2Ethe__value @ ( tyop_2Epair_2Eprod @ A_27t @ A_27w ) ) @ ( c_2Ereal_2E_2F @ ( c_2Ebinary__ieee_2Efloat__to__real @ A_27t @ A_27w @ V0a ) @ ( c_2Ebinary__ieee_2Efloat__to__real @ A_27t @ A_27w @ V1b ) ) ) )
=> ( ( c_2Ebinary__ieee_2Efloat__is__finite @ A_27t @ A_27w @ ( c_2Epair_2ESND @ tyop_2Ebinary__ieee_2Eflags @ ( tyop_2Ebinary__ieee_2Efloat @ A_27t @ A_27w ) @ ( c_2Ebinary__ieee_2Efloat__div @ A_27t @ A_27w @ c_2Ebinary__ieee_2EroundTiesToEven @ 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_2E_2B @ ( c_2Efcp_2Edimindex @ A_27t @ ( c_2Ebool_2Ethe__value @ A_27t ) ) @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ) )
& ( ( c_2Ebinary__ieee_2Efloat__to__real @ A_27t @ A_27w @ ( c_2Epair_2ESND @ tyop_2Ebinary__ieee_2Eflags @ ( tyop_2Ebinary__ieee_2Efloat @ A_27t @ A_27w ) @ ( c_2Ebinary__ieee_2Efloat__div @ A_27t @ A_27w @ c_2Ebinary__ieee_2EroundTiesToEven @ V0a @ V1b ) ) )
= ( c_2Erealax_2Ereal__mul @ ( c_2Ereal_2E_2F @ ( c_2Ebinary__ieee_2Efloat__to__real @ A_27t @ A_27w @ V0a ) @ ( c_2Ebinary__ieee_2Efloat__to__real @ A_27t @ A_27w @ V1b ) ) @ ( c_2Erealax_2Ereal__add @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) @ V2e ) ) ) ) ) ) ).
thf(thm_2Elift__ieee_2Efloat__sqrt__relative,axiom,
! [A_27t: $tType,A_27w: $tType,V0a: tyop_2Ebinary__ieee_2Efloat @ A_27t @ A_27w] :
( ( ( c_2Ebinary__ieee_2Efloat__is__finite @ A_27t @ A_27w @ V0a )
& ( ( c_2Ebinary__ieee_2Efloat__Sign @ A_27t @ A_27w @ V0a )
= ( c_2Ewords_2En2w @ tyop_2Eone_2Eone @ c_2Enum_2E0 ) )
& ( c_2Elift__ieee_2Enormalizes @ A_27t @ A_27w @ ( c_2Ebool_2Ethe__value @ ( tyop_2Epair_2Eprod @ A_27t @ A_27w ) ) @ ( c_2Etransc_2Esqrt @ ( c_2Ebinary__ieee_2Efloat__to__real @ A_27t @ A_27w @ V0a ) ) ) )
=> ( ( c_2Ebinary__ieee_2Efloat__is__finite @ A_27t @ A_27w @ ( c_2Epair_2ESND @ tyop_2Ebinary__ieee_2Eflags @ ( tyop_2Ebinary__ieee_2Efloat @ A_27t @ A_27w ) @ ( c_2Ebinary__ieee_2Efloat__sqrt @ A_27t @ A_27w @ c_2Ebinary__ieee_2EroundTiesToEven @ V0a ) ) )
& ? [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_2E_2B @ ( c_2Efcp_2Edimindex @ A_27t @ ( c_2Ebool_2Ethe__value @ A_27t ) ) @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ) )
& ( ( c_2Ebinary__ieee_2Efloat__to__real @ A_27t @ A_27w @ ( c_2Epair_2ESND @ tyop_2Ebinary__ieee_2Eflags @ ( tyop_2Ebinary__ieee_2Efloat @ A_27t @ A_27w ) @ ( c_2Ebinary__ieee_2Efloat__sqrt @ A_27t @ A_27w @ c_2Ebinary__ieee_2EroundTiesToEven @ V0a ) ) )
= ( c_2Erealax_2Ereal__mul @ ( c_2Etransc_2Esqrt @ ( c_2Ebinary__ieee_2Efloat__to__real @ A_27t @ A_27w @ V0a ) ) @ ( c_2Erealax_2Ereal__add @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) @ V1e ) ) ) ) ) ) ).
thf(thm_2Elift__ieee_2Efloat__mul__add__relative,axiom,
! [A_27t: $tType,A_27w: $tType,V0a: tyop_2Ebinary__ieee_2Efloat @ A_27t @ A_27w,V1b: tyop_2Ebinary__ieee_2Efloat @ A_27t @ A_27w,V2c: tyop_2Ebinary__ieee_2Efloat @ A_27t @ A_27w] :
( ( ( c_2Ebinary__ieee_2Efloat__is__finite @ A_27t @ A_27w @ V0a )
& ( c_2Ebinary__ieee_2Efloat__is__finite @ A_27t @ A_27w @ V1b )
& ( c_2Ebinary__ieee_2Efloat__is__finite @ A_27t @ A_27w @ V2c )
& ( c_2Elift__ieee_2Enormalizes @ A_27t @ A_27w @ ( c_2Ebool_2Ethe__value @ ( tyop_2Epair_2Eprod @ A_27t @ A_27w ) ) @ ( c_2Erealax_2Ereal__add @ ( c_2Erealax_2Ereal__mul @ ( c_2Ebinary__ieee_2Efloat__to__real @ A_27t @ A_27w @ V0a ) @ ( c_2Ebinary__ieee_2Efloat__to__real @ A_27t @ A_27w @ V1b ) ) @ ( c_2Ebinary__ieee_2Efloat__to__real @ A_27t @ A_27w @ V2c ) ) ) )
=> ( ( c_2Ebinary__ieee_2Efloat__is__finite @ A_27t @ A_27w @ ( c_2Epair_2ESND @ tyop_2Ebinary__ieee_2Eflags @ ( tyop_2Ebinary__ieee_2Efloat @ A_27t @ A_27w ) @ ( c_2Ebinary__ieee_2Efloat__mul__add @ A_27t @ A_27w @ c_2Ebinary__ieee_2EroundTiesToEven @ V0a @ V1b @ V2c ) ) )
& ? [V3e: tyop_2Erealax_2Ereal] :
( ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Eabs @ V3e ) @ ( 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_2E_2B @ ( c_2Efcp_2Edimindex @ A_27t @ ( c_2Ebool_2Ethe__value @ A_27t ) ) @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ) )
& ( ( c_2Ebinary__ieee_2Efloat__to__real @ A_27t @ A_27w @ ( c_2Epair_2ESND @ tyop_2Ebinary__ieee_2Eflags @ ( tyop_2Ebinary__ieee_2Efloat @ A_27t @ A_27w ) @ ( c_2Ebinary__ieee_2Efloat__mul__add @ A_27t @ A_27w @ c_2Ebinary__ieee_2EroundTiesToEven @ V0a @ V1b @ V2c ) ) )
= ( c_2Erealax_2Ereal__mul @ ( c_2Erealax_2Ereal__add @ ( c_2Erealax_2Ereal__mul @ ( c_2Ebinary__ieee_2Efloat__to__real @ A_27t @ A_27w @ V0a ) @ ( c_2Ebinary__ieee_2Efloat__to__real @ A_27t @ A_27w @ V1b ) ) @ ( c_2Ebinary__ieee_2Efloat__to__real @ A_27t @ A_27w @ V2c ) ) @ ( c_2Erealax_2Ereal__add @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) @ V3e ) ) ) ) ) ) ).
thf(thm_2Elift__ieee_2Efloat__mul__sub__relative,axiom,
! [A_27t: $tType,A_27w: $tType,V0a: tyop_2Ebinary__ieee_2Efloat @ A_27t @ A_27w,V1b: tyop_2Ebinary__ieee_2Efloat @ A_27t @ A_27w,V2c: tyop_2Ebinary__ieee_2Efloat @ A_27t @ A_27w] :
( ( ( c_2Ebinary__ieee_2Efloat__is__finite @ A_27t @ A_27w @ V0a )
& ( c_2Ebinary__ieee_2Efloat__is__finite @ A_27t @ A_27w @ V1b )
& ( c_2Ebinary__ieee_2Efloat__is__finite @ A_27t @ A_27w @ V2c )
& ( c_2Elift__ieee_2Enormalizes @ A_27t @ A_27w @ ( c_2Ebool_2Ethe__value @ ( tyop_2Epair_2Eprod @ A_27t @ A_27w ) ) @ ( c_2Ereal_2Ereal__sub @ ( c_2Erealax_2Ereal__mul @ ( c_2Ebinary__ieee_2Efloat__to__real @ A_27t @ A_27w @ V0a ) @ ( c_2Ebinary__ieee_2Efloat__to__real @ A_27t @ A_27w @ V1b ) ) @ ( c_2Ebinary__ieee_2Efloat__to__real @ A_27t @ A_27w @ V2c ) ) ) )
=> ( ( c_2Ebinary__ieee_2Efloat__is__finite @ A_27t @ A_27w @ ( c_2Epair_2ESND @ tyop_2Ebinary__ieee_2Eflags @ ( tyop_2Ebinary__ieee_2Efloat @ A_27t @ A_27w ) @ ( c_2Ebinary__ieee_2Efloat__mul__sub @ A_27t @ A_27w @ c_2Ebinary__ieee_2EroundTiesToEven @ V0a @ V1b @ V2c ) ) )
& ? [V3e: tyop_2Erealax_2Ereal] :
( ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Eabs @ V3e ) @ ( 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_2E_2B @ ( c_2Efcp_2Edimindex @ A_27t @ ( c_2Ebool_2Ethe__value @ A_27t ) ) @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ) )
& ( ( c_2Ebinary__ieee_2Efloat__to__real @ A_27t @ A_27w @ ( c_2Epair_2ESND @ tyop_2Ebinary__ieee_2Eflags @ ( tyop_2Ebinary__ieee_2Efloat @ A_27t @ A_27w ) @ ( c_2Ebinary__ieee_2Efloat__mul__sub @ A_27t @ A_27w @ c_2Ebinary__ieee_2EroundTiesToEven @ V0a @ V1b @ V2c ) ) )
= ( c_2Erealax_2Ereal__mul @ ( c_2Ereal_2Ereal__sub @ ( c_2Erealax_2Ereal__mul @ ( c_2Ebinary__ieee_2Efloat__to__real @ A_27t @ A_27w @ V0a ) @ ( c_2Ebinary__ieee_2Efloat__to__real @ A_27t @ A_27w @ V1b ) ) @ ( c_2Ebinary__ieee_2Efloat__to__real @ A_27t @ A_27w @ V2c ) ) @ ( c_2Erealax_2Ereal__add @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) @ V3e ) ) ) ) ) ) ).
thf(thm_2Elift__ieee_2Efinite__float__within__threshold,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: tyop_2Ebinary__ieee_2Efloat @ A_27a @ A_27b] :
( ( c_2Ebinary__ieee_2Efloat__is__finite @ A_27a @ A_27b @ V0f )
=> ( ( (~) @ ( c_2Ereal_2Ereal__lte @ ( c_2Ebinary__ieee_2Efloat__to__real @ A_27a @ A_27b @ V0f ) @ ( c_2Erealax_2Ereal__neg @ ( c_2Ebinary__ieee_2Ethreshold @ A_27a @ A_27b @ ( c_2Ebool_2Ethe__value @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) ) ) ) ) )
& ( (~) @ ( c_2Ereal_2Ereal__ge @ ( c_2Ebinary__ieee_2Efloat__to__real @ A_27a @ A_27b @ V0f ) @ ( c_2Ebinary__ieee_2Ethreshold @ A_27a @ A_27b @ ( c_2Ebool_2Ethe__value @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) ) ) ) ) ) ) ).
thf(thm_2Elift__ieee_2Eround__finite__normal__float__id,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: tyop_2Ebinary__ieee_2Efloat @ A_27a @ A_27b] :
( ( ( c_2Ebinary__ieee_2Efloat__is__finite @ A_27a @ A_27b @ V0f )
& ( (~) @ ( c_2Ebinary__ieee_2Efloat__is__zero @ A_27a @ A_27b @ V0f ) ) )
=> ( ( c_2Ebinary__ieee_2Eround @ A_27a @ A_27b @ c_2Ebinary__ieee_2EroundTiesToEven @ ( c_2Ebinary__ieee_2Efloat__to__real @ A_27a @ A_27b @ V0f ) )
= V0f ) ) ).
thf(thm_2Elift__ieee_2Ereal__to__float__finite__normal__id,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: tyop_2Ebinary__ieee_2Efloat @ A_27a @ A_27b] :
( ( ( c_2Ebinary__ieee_2Efloat__is__finite @ A_27a @ A_27b @ V0f )
& ( (~) @ ( c_2Ebinary__ieee_2Efloat__is__zero @ A_27a @ A_27b @ V0f ) ) )
=> ( ( c_2Ebinary__ieee_2Ereal__to__float @ A_27a @ A_27b @ c_2Ebinary__ieee_2EroundTiesToEven @ ( c_2Ebinary__ieee_2Efloat__to__real @ A_27a @ A_27b @ V0f ) )
= V0f ) ) ).
thf(thm_2Elift__ieee_2Efloat__to__real__real__to__float__zero__id,axiom,
! [A_27a: $tType,A_27b: $tType] :
( ( c_2Ebinary__ieee_2Efloat__to__real @ A_27a @ A_27b @ ( c_2Ebinary__ieee_2Ereal__to__float @ A_27a @ A_27b @ c_2Ebinary__ieee_2EroundTiesToEven @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) )
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ).
thf(thm_2Elift__ieee_2Enon__representable__float__is__zero,axiom,
! [A_27a: $tType,A_27b: $tType,V0ff: tyop_2Erealax_2Ereal,V1P: $o] :
( ( c_2Ereal_2Ereal__lte @ ( c_2Erealax_2Ereal__mul @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) @ ( c_2Ereal_2Eabs @ V0ff ) ) @ ( c_2Ebinary__ieee_2Eulp @ A_27a @ A_27b @ ( c_2Ebool_2Ethe__value @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) ) ) )
=> ( ( c_2Ebinary__ieee_2Efloat__to__real @ A_27a @ A_27b @ ( c_2Ebinary__ieee_2Efloat__round @ A_27a @ A_27b @ c_2Ebinary__ieee_2EroundTiesToEven @ V1P @ V0ff ) )
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ) ).
%------------------------------------------------------------------------------