ITP001 Axioms: ITP140^5.ax
%------------------------------------------------------------------------------
% File : ITP140^5 : TPTP v8.2.0. Bugfixed v7.5.0.
% Domain : Interactive Theorem Proving
% Axioms : HOL4 set theory export, chainy mode
% Version : [BG+19] axioms.
% English :
% Refs : [BG+19] Brown et al. (2019), GRUNGE: A Grand Unified ATP Chall
% : [Gau20] Gauthier (2020), Email to Geoff Sutcliffe
% Source : [BG+19]
% Names : lift_ieee^2.ax [Gau20]
% : HL4140^5.ax [TPAP]
% Status : Satisfiable
% Syntax : Number of formulae : 52 ( 3 unt; 2 typ; 0 def)
% Number of atoms : 2399 ( 31 equ; 0 cnn)
% Maximal formula atoms : 88 ( 46 avg)
% Number of connectives : 3760 ( 13 ~; 0 |; 89 &;3533 @)
% ( 7 <=>; 118 =>; 0 <=; 0 <~>)
% Maximal formula depth : 28 ( 18 avg;3533 nst)
% Number of types : 1 ( 0 usr)
% Number of type conns : 4 ( 4 >; 0 *; 0 +; 0 <<)
% Number of symbols : 81 ( 80 usr; 78 con; 0-2 aty)
% Number of variables : 198 ( 3 ^ 185 !; 10 ?; 198 :)
% SPC : TH0_SAT_EQU_NAR
% Comments :
% Bugfixes : v7.5.0 - Fixes to the axioms.
%------------------------------------------------------------------------------
thf(tp_c_2Elift__ieee_2Eerror,type,
c_2Elift__ieee_2Eerror: del > del > $i ).
thf(mem_c_2Elift__ieee_2Eerror,axiom,
! [A_27t: del,A_27w: del] : ( mem @ ( c_2Elift__ieee_2Eerror @ A_27t @ A_27w ) @ ( arr @ ( ty_2Ebool_2Eitself @ ( ty_2Epair_2Eprod @ A_27t @ A_27w ) ) @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) ) ).
thf(tp_c_2Elift__ieee_2Enormalizes,type,
c_2Elift__ieee_2Enormalizes: del > del > $i ).
thf(mem_c_2Elift__ieee_2Enormalizes,axiom,
! [A_27t: del,A_27w: del] : ( mem @ ( c_2Elift__ieee_2Enormalizes @ A_27t @ A_27w ) @ ( arr @ ( ty_2Ebool_2Eitself @ ( ty_2Epair_2Eprod @ A_27t @ A_27w ) ) @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) ) ).
thf(ax_thm_2Elift__ieee_2Eerror__def,axiom,
! [A_27t: del,A_27w: del,V0x: tp__ty_2Erealax_2Ereal] :
( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ ( c_2Elift__ieee_2Eerror @ A_27t @ A_27w ) @ ( c_2Ebool_2Ethe__value @ ( ty_2Epair_2Eprod @ A_27t @ A_27w ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( ap @ ( c_2Ebinary__ieee_2Efloat__to__real @ A_27t @ A_27w ) @ ( ap @ ( ap @ ( c_2Ebinary__ieee_2Eround @ A_27t @ A_27w ) @ ( inj__ty_2Ebinary__ieee_2Erounding @ fo__c_2Ebinary__ieee_2EroundTiesToEven ) ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) ) ).
thf(ax_thm_2Elift__ieee_2Enormalizes__def,axiom,
! [A_27t: del,A_27w: del,V0x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Elift__ieee_2Enormalizes @ A_27t @ A_27w ) @ ( c_2Ebool_2Ethe__value @ ( ty_2Epair_2Eprod @ A_27t @ A_27w ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) )
<=> ( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) @ ( ap @ ( c_2Ewords_2EINT__MAX @ A_27w ) @ ( c_2Ebool_2Ethe__value @ A_27w ) ) ) )
& ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Erealax_2Einv @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ ( ap @ ( c_2Ewords_2EINT__MAX @ A_27w ) @ ( c_2Ebool_2Ethe__value @ A_27w ) ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) @ ( ap @ c_2Ereal_2Eabs @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) )
& ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Eabs @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) @ ( ap @ ( c_2Ebinary__ieee_2Ethreshold @ A_27t @ A_27w ) @ ( c_2Ebool_2Ethe__value @ ( ty_2Epair_2Eprod @ A_27t @ A_27w ) ) ) ) ) ) ) ).
thf(conj_thm_2Elift__ieee_2Efloat__lt,axiom,
! [A_27a: del,A_27b: del,V0x: $i] :
( ( mem @ V0x @ ( ty_2Ebinary__ieee_2Efloat @ A_27a @ A_27b ) )
=> ! [V1y: $i] :
( ( mem @ V1y @ ( ty_2Ebinary__ieee_2Efloat @ A_27a @ A_27b ) )
=> ( ( ( p @ ( ap @ ( c_2Ebinary__ieee_2Efloat__is__finite @ A_27a @ A_27b ) @ V0x ) )
& ( p @ ( ap @ ( c_2Ebinary__ieee_2Efloat__is__finite @ A_27a @ A_27b ) @ V1y ) ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebinary__ieee_2Efloat__less__than @ A_27a @ A_27b ) @ V0x ) @ V1y ) )
<=> ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ ( c_2Ebinary__ieee_2Efloat__to__real @ A_27a @ A_27b ) @ V0x ) ) @ ( ap @ ( c_2Ebinary__ieee_2Efloat__to__real @ A_27a @ A_27b ) @ V1y ) ) ) ) ) ) ) ).
thf(conj_thm_2Elift__ieee_2Efloat__le,axiom,
! [A_27a: del,A_27b: del,V0x: $i] :
( ( mem @ V0x @ ( ty_2Ebinary__ieee_2Efloat @ A_27a @ A_27b ) )
=> ! [V1y: $i] :
( ( mem @ V1y @ ( ty_2Ebinary__ieee_2Efloat @ A_27a @ A_27b ) )
=> ( ( ( p @ ( ap @ ( c_2Ebinary__ieee_2Efloat__is__finite @ A_27a @ A_27b ) @ V0x ) )
& ( p @ ( ap @ ( c_2Ebinary__ieee_2Efloat__is__finite @ A_27a @ A_27b ) @ V1y ) ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebinary__ieee_2Efloat__less__equal @ A_27a @ A_27b ) @ V0x ) @ V1y ) )
<=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ ( c_2Ebinary__ieee_2Efloat__to__real @ A_27a @ A_27b ) @ V0x ) ) @ ( ap @ ( c_2Ebinary__ieee_2Efloat__to__real @ A_27a @ A_27b ) @ V1y ) ) ) ) ) ) ) ).
thf(conj_thm_2Elift__ieee_2Efloat__gt,axiom,
! [A_27a: del,A_27b: del,V0x: $i] :
( ( mem @ V0x @ ( ty_2Ebinary__ieee_2Efloat @ A_27a @ A_27b ) )
=> ! [V1y: $i] :
( ( mem @ V1y @ ( ty_2Ebinary__ieee_2Efloat @ A_27a @ A_27b ) )
=> ( ( ( p @ ( ap @ ( c_2Ebinary__ieee_2Efloat__is__finite @ A_27a @ A_27b ) @ V0x ) )
& ( p @ ( ap @ ( c_2Ebinary__ieee_2Efloat__is__finite @ A_27a @ A_27b ) @ V1y ) ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebinary__ieee_2Efloat__greater__than @ A_27a @ A_27b ) @ V0x ) @ V1y ) )
<=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__gt @ ( ap @ ( c_2Ebinary__ieee_2Efloat__to__real @ A_27a @ A_27b ) @ V0x ) ) @ ( ap @ ( c_2Ebinary__ieee_2Efloat__to__real @ A_27a @ A_27b ) @ V1y ) ) ) ) ) ) ) ).
thf(conj_thm_2Elift__ieee_2Efloat__ge,axiom,
! [A_27a: del,A_27b: del,V0x: $i] :
( ( mem @ V0x @ ( ty_2Ebinary__ieee_2Efloat @ A_27a @ A_27b ) )
=> ! [V1y: $i] :
( ( mem @ V1y @ ( ty_2Ebinary__ieee_2Efloat @ A_27a @ A_27b ) )
=> ( ( ( p @ ( ap @ ( c_2Ebinary__ieee_2Efloat__is__finite @ A_27a @ A_27b ) @ V0x ) )
& ( p @ ( ap @ ( c_2Ebinary__ieee_2Efloat__is__finite @ A_27a @ A_27b ) @ V1y ) ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebinary__ieee_2Efloat__greater__equal @ A_27a @ A_27b ) @ V0x ) @ V1y ) )
<=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__ge @ ( ap @ ( c_2Ebinary__ieee_2Efloat__to__real @ A_27a @ A_27b ) @ V0x ) ) @ ( ap @ ( c_2Ebinary__ieee_2Efloat__to__real @ A_27a @ A_27b ) @ V1y ) ) ) ) ) ) ) ).
thf(conj_thm_2Elift__ieee_2Efloat__eq,axiom,
! [A_27a: del,A_27b: del,V0x: $i] :
( ( mem @ V0x @ ( ty_2Ebinary__ieee_2Efloat @ A_27a @ A_27b ) )
=> ! [V1y: $i] :
( ( mem @ V1y @ ( ty_2Ebinary__ieee_2Efloat @ A_27a @ A_27b ) )
=> ( ( ( p @ ( ap @ ( c_2Ebinary__ieee_2Efloat__is__finite @ A_27a @ A_27b ) @ V0x ) )
& ( p @ ( ap @ ( c_2Ebinary__ieee_2Efloat__is__finite @ A_27a @ A_27b ) @ V1y ) ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebinary__ieee_2Efloat__equal @ A_27a @ A_27b ) @ V0x ) @ V1y ) )
<=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( c_2Ebinary__ieee_2Efloat__to__real @ A_27a @ A_27b ) @ V0x ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( c_2Ebinary__ieee_2Efloat__to__real @ A_27a @ A_27b ) @ V1y ) ) ) ) ) ) ) ).
thf(conj_thm_2Elift__ieee_2Efloat__eq__refl,axiom,
! [A_27a: del,A_27b: del,V0x: $i] :
( ( mem @ V0x @ ( ty_2Ebinary__ieee_2Efloat @ A_27a @ A_27b ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebinary__ieee_2Efloat__equal @ A_27a @ A_27b ) @ V0x ) @ V0x ) )
<=> ~ ( p @ ( ap @ ( c_2Ebinary__ieee_2Efloat__is__nan @ A_27a @ A_27b ) @ V0x ) ) ) ) ).
thf(conj_thm_2Elift__ieee_2Eis__closest__exits,axiom,
! [A_27a: del,A_27b: del,V0x: tp__ty_2Erealax_2Ereal,V1s: $i] :
( ( mem @ V1s @ ( arr @ ( ty_2Ebinary__ieee_2Efloat @ A_27a @ A_27b ) @ bool ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ ( ty_2Ebinary__ieee_2Efloat @ A_27a @ A_27b ) ) @ V1s ) )
=> ( ( V1s
!= ( c_2Epred__set_2EEMPTY @ ( ty_2Ebinary__ieee_2Efloat @ A_27a @ A_27b ) ) )
=> ? [V2a: $i] :
( ( mem @ V2a @ ( ty_2Ebinary__ieee_2Efloat @ A_27a @ A_27b ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ebinary__ieee_2Eis__closest @ A_27a @ A_27b ) @ V1s ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ V2a ) ) ) ) ) ) ).
thf(conj_thm_2Elift__ieee_2Eclosest__is__everything,axiom,
! [A_27a: del,A_27b: del,V0p: $i] :
( ( mem @ V0p @ ( arr @ ( ty_2Ebinary__ieee_2Efloat @ A_27a @ A_27b ) @ bool ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ( ty_2Ebinary__ieee_2Efloat @ A_27a @ A_27b ) @ bool ) )
=> ! [V2x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ ( ty_2Ebinary__ieee_2Efloat @ A_27a @ A_27b ) ) @ V1s ) )
=> ( ( V1s
!= ( c_2Epred__set_2EEMPTY @ ( ty_2Ebinary__ieee_2Efloat @ A_27a @ A_27b ) ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ebinary__ieee_2Eis__closest @ A_27a @ A_27b ) @ V1s ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Ebinary__ieee_2Eclosest__such @ A_27a @ A_27b ) @ V0p ) @ V1s ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) ) )
& ( ? [V3b: $i] :
( ( mem @ V3b @ ( ty_2Ebinary__ieee_2Efloat @ A_27a @ A_27b ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ebinary__ieee_2Eis__closest @ A_27a @ A_27b ) @ V1s ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) @ V3b ) )
& ( p @ ( ap @ V0p @ V3b ) ) )
=> ( p @ ( ap @ V0p @ ( ap @ ( ap @ ( ap @ ( c_2Ebinary__ieee_2Eclosest__such @ A_27a @ A_27b ) @ V0p ) @ V1s ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elift__ieee_2Eclosest__in__set,axiom,
! [A_27a: del,A_27b: del,V0p: $i] :
( ( mem @ V0p @ ( arr @ ( ty_2Ebinary__ieee_2Efloat @ A_27a @ A_27b ) @ bool ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ( ty_2Ebinary__ieee_2Efloat @ A_27a @ A_27b ) @ bool ) )
=> ! [V2x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ ( ty_2Ebinary__ieee_2Efloat @ A_27a @ A_27b ) ) @ V1s ) )
=> ( ( V1s
!= ( c_2Epred__set_2EEMPTY @ ( ty_2Ebinary__ieee_2Efloat @ A_27a @ A_27b ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Ebinary__ieee_2Efloat @ A_27a @ A_27b ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Ebinary__ieee_2Eclosest__such @ A_27a @ A_27b ) @ V0p ) @ V1s ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) ) @ V1s ) ) ) ) ) ) ).
thf(conj_thm_2Elift__ieee_2Eclosest__is__closest,axiom,
! [A_27a: del,A_27b: del,V0p: $i] :
( ( mem @ V0p @ ( arr @ ( ty_2Ebinary__ieee_2Efloat @ A_27a @ A_27b ) @ bool ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ( ty_2Ebinary__ieee_2Efloat @ A_27a @ A_27b ) @ bool ) )
=> ! [V2x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ ( ty_2Ebinary__ieee_2Efloat @ A_27a @ A_27b ) ) @ V1s ) )
=> ( ( V1s
!= ( c_2Epred__set_2EEMPTY @ ( ty_2Ebinary__ieee_2Efloat @ A_27a @ A_27b ) ) )
=> ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ebinary__ieee_2Eis__closest @ A_27a @ A_27b ) @ V1s ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Ebinary__ieee_2Eclosest__such @ A_27a @ A_27b ) @ V0p ) @ V1s ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elift__ieee_2Efloat__finite,axiom,
! [A_27t: del,A_27w: del] : ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ ( ty_2Ebinary__ieee_2Efloat @ A_27t @ A_27w ) ) @ ( c_2Epred__set_2EUNIV @ ( ty_2Ebinary__ieee_2Efloat @ A_27t @ A_27w ) ) ) ) ).
thf(conj_thm_2Elift__ieee_2Eis__finite__finite,axiom,
! [A_27a: del,A_27b: del] :
( p
@ ( ap @ ( c_2Epred__set_2EFINITE @ ( ty_2Ebinary__ieee_2Efloat @ A_27a @ A_27b ) )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ( ty_2Ebinary__ieee_2Efloat @ A_27a @ A_27b ) @ ( ty_2Ebinary__ieee_2Efloat @ A_27a @ A_27b ) )
@ ( lam @ ( ty_2Ebinary__ieee_2Efloat @ A_27a @ A_27b )
@ ^ [V0a: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Ebinary__ieee_2Efloat @ A_27a @ A_27b ) @ bool ) @ V0a ) @ ( ap @ ( c_2Ebinary__ieee_2Efloat__is__finite @ A_27a @ A_27b ) @ V0a ) ) ) ) ) ) ).
thf(conj_thm_2Elift__ieee_2Eis__finite__nonempty,axiom,
! [A_27a: del,A_27b: del] :
( ( ap @ ( c_2Epred__set_2EGSPEC @ ( ty_2Ebinary__ieee_2Efloat @ A_27a @ A_27b ) @ ( ty_2Ebinary__ieee_2Efloat @ A_27a @ A_27b ) )
@ ( lam @ ( ty_2Ebinary__ieee_2Efloat @ A_27a @ A_27b )
@ ^ [V0a: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Ebinary__ieee_2Efloat @ A_27a @ A_27b ) @ bool ) @ V0a ) @ ( ap @ ( c_2Ebinary__ieee_2Efloat__is__finite @ A_27a @ A_27b ) @ V0a ) ) ) )
!= ( c_2Epred__set_2EEMPTY @ ( ty_2Ebinary__ieee_2Efloat @ A_27a @ A_27b ) ) ) ).
thf(conj_thm_2Elift__ieee_2Eis__finite__closest,axiom,
! [A_27a: del,A_27b: del,V0p: $i] :
( ( mem @ V0p @ ( arr @ ( ty_2Ebinary__ieee_2Efloat @ A_27a @ A_27b ) @ bool ) )
=> ! [V1x: tp__ty_2Erealax_2Ereal] :
( p
@ ( ap @ ( c_2Ebinary__ieee_2Efloat__is__finite @ A_27a @ A_27b )
@ ( ap
@ ( ap @ ( ap @ ( c_2Ebinary__ieee_2Eclosest__such @ A_27a @ A_27b ) @ V0p )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ( ty_2Ebinary__ieee_2Efloat @ A_27a @ A_27b ) @ ( ty_2Ebinary__ieee_2Efloat @ A_27a @ A_27b ) )
@ ( lam @ ( ty_2Ebinary__ieee_2Efloat @ A_27a @ A_27b )
@ ^ [V2a: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Ebinary__ieee_2Efloat @ A_27a @ A_27b ) @ bool ) @ V2a ) @ ( ap @ ( c_2Ebinary__ieee_2Efloat__is__finite @ A_27a @ A_27b ) @ V2a ) ) ) ) )
@ ( inj__ty_2Erealax_2Ereal @ V1x ) ) ) ) ) ).
thf(conj_thm_2Elift__ieee_2Efloat__to__real__finite,axiom,
! [A_27t: del,A_27w: del,V0x: $i] :
( ( mem @ V0x @ ( ty_2Ebinary__ieee_2Efloat @ A_27t @ A_27w ) )
=> ( ( p @ ( ap @ ( c_2Ebinary__ieee_2Efloat__is__finite @ A_27t @ A_27w ) @ V0x ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Eabs @ ( ap @ ( c_2Ebinary__ieee_2Efloat__to__real @ A_27t @ A_27w ) @ V0x ) ) ) @ ( ap @ ( c_2Ebinary__ieee_2Elargest @ A_27t @ A_27w ) @ ( c_2Ebool_2Ethe__value @ ( ty_2Epair_2Eprod @ A_27t @ A_27w ) ) ) ) ) ) ) ).
thf(conj_thm_2Elift__ieee_2Efloat__to__real__threshold,axiom,
! [A_27t: del,A_27w: del,V0x: $i] :
( ( mem @ V0x @ ( ty_2Ebinary__ieee_2Efloat @ A_27t @ A_27w ) )
=> ( ( p @ ( ap @ ( c_2Ebinary__ieee_2Efloat__is__finite @ A_27t @ A_27w ) @ V0x ) )
=> ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Eabs @ ( ap @ ( c_2Ebinary__ieee_2Efloat__to__real @ A_27t @ A_27w ) @ V0x ) ) ) @ ( ap @ ( c_2Ebinary__ieee_2Ethreshold @ A_27t @ A_27w ) @ ( c_2Ebool_2Ethe__value @ ( ty_2Epair_2Eprod @ A_27t @ A_27w ) ) ) ) ) ) ) ).
thf(conj_thm_2Elift__ieee_2Eerror__at__worst__lemma,axiom,
! [A_27t: del,A_27w: del,V0a: $i] :
( ( mem @ V0a @ ( ty_2Ebinary__ieee_2Efloat @ A_27t @ A_27w ) )
=> ! [V1x: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Eabs @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) ) @ ( ap @ ( c_2Ebinary__ieee_2Ethreshold @ A_27t @ A_27w ) @ ( c_2Ebool_2Ethe__value @ ( ty_2Epair_2Eprod @ A_27t @ A_27w ) ) ) ) )
& ( p @ ( ap @ ( c_2Ebinary__ieee_2Efloat__is__finite @ A_27t @ A_27w ) @ V0a ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Eabs @ ( ap @ ( ap @ ( c_2Elift__ieee_2Eerror @ A_27t @ A_27w ) @ ( c_2Ebool_2Ethe__value @ ( ty_2Epair_2Eprod @ A_27t @ A_27w ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) ) ) @ ( ap @ c_2Ereal_2Eabs @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( ap @ ( c_2Ebinary__ieee_2Efloat__to__real @ A_27t @ A_27w ) @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) ) ) ) ) ) ).
thf(conj_thm_2Elift__ieee_2Eerror__is__zero,axiom,
! [A_27t: del,A_27w: del,V0a: $i] :
( ( mem @ V0a @ ( ty_2Ebinary__ieee_2Efloat @ A_27t @ A_27w ) )
=> ! [V1x: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( c_2Ebinary__ieee_2Efloat__is__finite @ A_27t @ A_27w ) @ V0a ) )
& ( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( c_2Ebinary__ieee_2Efloat__to__real @ A_27t @ A_27w ) @ V0a ) )
= V1x ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ ( c_2Elift__ieee_2Eerror @ A_27t @ A_27w ) @ ( c_2Ebool_2Ethe__value @ ( ty_2Epair_2Eprod @ A_27t @ A_27w ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) ) ).
thf(conj_thm_2Elift__ieee_2Erelative__error,axiom,
! [A_27t: del,A_27w: del,V0x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Elift__ieee_2Enormalizes @ A_27t @ A_27w ) @ ( c_2Ebool_2Ethe__value @ ( ty_2Epair_2Eprod @ A_27t @ A_27w ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) )
=> ? [V1e: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Eabs @ ( inj__ty_2Erealax_2Ereal @ V1e ) ) ) @ ( ap @ ( ap @ c_2Ereal_2E_2F @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( ap @ ( c_2Efcp_2Edimindex @ A_27t ) @ ( c_2Ebool_2Ethe__value @ A_27t ) ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) )
& ( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( c_2Ebinary__ieee_2Efloat__to__real @ A_27t @ A_27w ) @ ( ap @ ( ap @ ( c_2Ebinary__ieee_2Eround @ A_27t @ A_27w ) @ ( inj__ty_2Ebinary__ieee_2Erounding @ fo__c_2Ebinary__ieee_2EroundTiesToEven ) ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V1e ) ) ) ) ) ) ) ).
thf(conj_thm_2Elift__ieee_2Efloat__round__finite,axiom,
! [A_27t: del,A_27w: del,V0b: $i] :
( ( mem @ V0b @ bool )
=> ! [V1x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Eabs @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) ) @ ( ap @ ( c_2Ebinary__ieee_2Ethreshold @ A_27t @ A_27w ) @ ( c_2Ebool_2Ethe__value @ ( ty_2Epair_2Eprod @ A_27t @ A_27w ) ) ) ) )
=> ( p @ ( ap @ ( c_2Ebinary__ieee_2Efloat__is__finite @ A_27t @ A_27w ) @ ( ap @ ( ap @ ( ap @ ( c_2Ebinary__ieee_2Efloat__round @ A_27t @ A_27w ) @ ( inj__ty_2Ebinary__ieee_2Erounding @ fo__c_2Ebinary__ieee_2EroundTiesToEven ) ) @ V0b ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) ) ) ) ) ).
thf(conj_thm_2Elift__ieee_2Efloat__add,axiom,
! [A_27t: del,A_27w: del,V0a: $i] :
( ( mem @ V0a @ ( ty_2Ebinary__ieee_2Efloat @ A_27t @ A_27w ) )
=> ! [V1b: $i] :
( ( mem @ V1b @ ( ty_2Ebinary__ieee_2Efloat @ A_27t @ A_27w ) )
=> ( ( ( p @ ( ap @ ( c_2Ebinary__ieee_2Efloat__is__finite @ A_27t @ A_27w ) @ V0a ) )
& ( p @ ( ap @ ( c_2Ebinary__ieee_2Efloat__is__finite @ A_27t @ A_27w ) @ V1b ) )
& ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Eabs @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( ap @ ( c_2Ebinary__ieee_2Efloat__to__real @ A_27t @ A_27w ) @ V0a ) ) @ ( ap @ ( c_2Ebinary__ieee_2Efloat__to__real @ A_27t @ A_27w ) @ V1b ) ) ) ) @ ( ap @ ( c_2Ebinary__ieee_2Ethreshold @ A_27t @ A_27w ) @ ( c_2Ebool_2Ethe__value @ ( ty_2Epair_2Eprod @ A_27t @ A_27w ) ) ) ) ) )
=> ( ( p @ ( ap @ ( c_2Ebinary__ieee_2Efloat__is__finite @ A_27t @ A_27w ) @ ( ap @ ( c_2Epair_2ESND @ ty_2Ebinary__ieee_2Eflags @ ( ty_2Ebinary__ieee_2Efloat @ A_27t @ A_27w ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Ebinary__ieee_2Efloat__add @ A_27t @ A_27w ) @ ( inj__ty_2Ebinary__ieee_2Erounding @ fo__c_2Ebinary__ieee_2EroundTiesToEven ) ) @ V0a ) @ V1b ) ) ) )
& ( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( c_2Ebinary__ieee_2Efloat__to__real @ A_27t @ A_27w ) @ ( ap @ ( c_2Epair_2ESND @ ty_2Ebinary__ieee_2Eflags @ ( ty_2Ebinary__ieee_2Efloat @ A_27t @ A_27w ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Ebinary__ieee_2Efloat__add @ A_27t @ A_27w ) @ ( inj__ty_2Ebinary__ieee_2Erounding @ fo__c_2Ebinary__ieee_2EroundTiesToEven ) ) @ V0a ) @ V1b ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( ap @ ( c_2Ebinary__ieee_2Efloat__to__real @ A_27t @ A_27w ) @ V0a ) ) @ ( ap @ ( c_2Ebinary__ieee_2Efloat__to__real @ A_27t @ A_27w ) @ V1b ) ) ) @ ( ap @ ( ap @ ( c_2Elift__ieee_2Eerror @ A_27t @ A_27w ) @ ( c_2Ebool_2Ethe__value @ ( ty_2Epair_2Eprod @ A_27t @ A_27w ) ) ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( ap @ ( c_2Ebinary__ieee_2Efloat__to__real @ A_27t @ A_27w ) @ V0a ) ) @ ( ap @ ( c_2Ebinary__ieee_2Efloat__to__real @ A_27t @ A_27w ) @ V1b ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elift__ieee_2Efloat__sub,axiom,
! [A_27t: del,A_27w: del,V0a: $i] :
( ( mem @ V0a @ ( ty_2Ebinary__ieee_2Efloat @ A_27t @ A_27w ) )
=> ! [V1b: $i] :
( ( mem @ V1b @ ( ty_2Ebinary__ieee_2Efloat @ A_27t @ A_27w ) )
=> ( ( ( p @ ( ap @ ( c_2Ebinary__ieee_2Efloat__is__finite @ A_27t @ A_27w ) @ V0a ) )
& ( p @ ( ap @ ( c_2Ebinary__ieee_2Efloat__is__finite @ A_27t @ A_27w ) @ V1b ) )
& ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Eabs @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( ap @ ( c_2Ebinary__ieee_2Efloat__to__real @ A_27t @ A_27w ) @ V0a ) ) @ ( ap @ ( c_2Ebinary__ieee_2Efloat__to__real @ A_27t @ A_27w ) @ V1b ) ) ) ) @ ( ap @ ( c_2Ebinary__ieee_2Ethreshold @ A_27t @ A_27w ) @ ( c_2Ebool_2Ethe__value @ ( ty_2Epair_2Eprod @ A_27t @ A_27w ) ) ) ) ) )
=> ( ( p @ ( ap @ ( c_2Ebinary__ieee_2Efloat__is__finite @ A_27t @ A_27w ) @ ( ap @ ( c_2Epair_2ESND @ ty_2Ebinary__ieee_2Eflags @ ( ty_2Ebinary__ieee_2Efloat @ A_27t @ A_27w ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Ebinary__ieee_2Efloat__sub @ A_27t @ A_27w ) @ ( inj__ty_2Ebinary__ieee_2Erounding @ fo__c_2Ebinary__ieee_2EroundTiesToEven ) ) @ V0a ) @ V1b ) ) ) )
& ( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( c_2Ebinary__ieee_2Efloat__to__real @ A_27t @ A_27w ) @ ( ap @ ( c_2Epair_2ESND @ ty_2Ebinary__ieee_2Eflags @ ( ty_2Ebinary__ieee_2Efloat @ A_27t @ A_27w ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Ebinary__ieee_2Efloat__sub @ A_27t @ A_27w ) @ ( inj__ty_2Ebinary__ieee_2Erounding @ fo__c_2Ebinary__ieee_2EroundTiesToEven ) ) @ V0a ) @ V1b ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( ap @ ( c_2Ebinary__ieee_2Efloat__to__real @ A_27t @ A_27w ) @ V0a ) ) @ ( ap @ ( c_2Ebinary__ieee_2Efloat__to__real @ A_27t @ A_27w ) @ V1b ) ) ) @ ( ap @ ( ap @ ( c_2Elift__ieee_2Eerror @ A_27t @ A_27w ) @ ( c_2Ebool_2Ethe__value @ ( ty_2Epair_2Eprod @ A_27t @ A_27w ) ) ) @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( ap @ ( c_2Ebinary__ieee_2Efloat__to__real @ A_27t @ A_27w ) @ V0a ) ) @ ( ap @ ( c_2Ebinary__ieee_2Efloat__to__real @ A_27t @ A_27w ) @ V1b ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elift__ieee_2Efloat__mul,axiom,
! [A_27t: del,A_27w: del,V0a: $i] :
( ( mem @ V0a @ ( ty_2Ebinary__ieee_2Efloat @ A_27t @ A_27w ) )
=> ! [V1b: $i] :
( ( mem @ V1b @ ( ty_2Ebinary__ieee_2Efloat @ A_27t @ A_27w ) )
=> ( ( ( p @ ( ap @ ( c_2Ebinary__ieee_2Efloat__is__finite @ A_27t @ A_27w ) @ V0a ) )
& ( p @ ( ap @ ( c_2Ebinary__ieee_2Efloat__is__finite @ A_27t @ A_27w ) @ V1b ) )
& ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Eabs @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ ( c_2Ebinary__ieee_2Efloat__to__real @ A_27t @ A_27w ) @ V0a ) ) @ ( ap @ ( c_2Ebinary__ieee_2Efloat__to__real @ A_27t @ A_27w ) @ V1b ) ) ) ) @ ( ap @ ( c_2Ebinary__ieee_2Ethreshold @ A_27t @ A_27w ) @ ( c_2Ebool_2Ethe__value @ ( ty_2Epair_2Eprod @ A_27t @ A_27w ) ) ) ) ) )
=> ( ( p @ ( ap @ ( c_2Ebinary__ieee_2Efloat__is__finite @ A_27t @ A_27w ) @ ( ap @ ( c_2Epair_2ESND @ ty_2Ebinary__ieee_2Eflags @ ( ty_2Ebinary__ieee_2Efloat @ A_27t @ A_27w ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Ebinary__ieee_2Efloat__mul @ A_27t @ A_27w ) @ ( inj__ty_2Ebinary__ieee_2Erounding @ fo__c_2Ebinary__ieee_2EroundTiesToEven ) ) @ V0a ) @ V1b ) ) ) )
& ( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( c_2Ebinary__ieee_2Efloat__to__real @ A_27t @ A_27w ) @ ( ap @ ( c_2Epair_2ESND @ ty_2Ebinary__ieee_2Eflags @ ( ty_2Ebinary__ieee_2Efloat @ A_27t @ A_27w ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Ebinary__ieee_2Efloat__mul @ A_27t @ A_27w ) @ ( inj__ty_2Ebinary__ieee_2Erounding @ fo__c_2Ebinary__ieee_2EroundTiesToEven ) ) @ V0a ) @ V1b ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ ( c_2Ebinary__ieee_2Efloat__to__real @ A_27t @ A_27w ) @ V0a ) ) @ ( ap @ ( c_2Ebinary__ieee_2Efloat__to__real @ A_27t @ A_27w ) @ V1b ) ) ) @ ( ap @ ( ap @ ( c_2Elift__ieee_2Eerror @ A_27t @ A_27w ) @ ( c_2Ebool_2Ethe__value @ ( ty_2Epair_2Eprod @ A_27t @ A_27w ) ) ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ ( c_2Ebinary__ieee_2Efloat__to__real @ A_27t @ A_27w ) @ V0a ) ) @ ( ap @ ( c_2Ebinary__ieee_2Efloat__to__real @ A_27t @ A_27w ) @ V1b ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elift__ieee_2Efloat__div,axiom,
! [A_27t: del,A_27w: del,V0a: $i] :
( ( mem @ V0a @ ( ty_2Ebinary__ieee_2Efloat @ A_27t @ A_27w ) )
=> ! [V1b: $i] :
( ( mem @ V1b @ ( ty_2Ebinary__ieee_2Efloat @ A_27t @ A_27w ) )
=> ( ( ( p @ ( ap @ ( c_2Ebinary__ieee_2Efloat__is__finite @ A_27t @ A_27w ) @ V0a ) )
& ( p @ ( ap @ ( c_2Ebinary__ieee_2Efloat__is__finite @ A_27t @ A_27w ) @ V1b ) )
& ~ ( p @ ( ap @ ( c_2Ebinary__ieee_2Efloat__is__zero @ A_27t @ A_27w ) @ V1b ) )
& ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Eabs @ ( ap @ ( ap @ c_2Ereal_2E_2F @ ( ap @ ( c_2Ebinary__ieee_2Efloat__to__real @ A_27t @ A_27w ) @ V0a ) ) @ ( ap @ ( c_2Ebinary__ieee_2Efloat__to__real @ A_27t @ A_27w ) @ V1b ) ) ) ) @ ( ap @ ( c_2Ebinary__ieee_2Ethreshold @ A_27t @ A_27w ) @ ( c_2Ebool_2Ethe__value @ ( ty_2Epair_2Eprod @ A_27t @ A_27w ) ) ) ) ) )
=> ( ( p @ ( ap @ ( c_2Ebinary__ieee_2Efloat__is__finite @ A_27t @ A_27w ) @ ( ap @ ( c_2Epair_2ESND @ ty_2Ebinary__ieee_2Eflags @ ( ty_2Ebinary__ieee_2Efloat @ A_27t @ A_27w ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Ebinary__ieee_2Efloat__div @ A_27t @ A_27w ) @ ( inj__ty_2Ebinary__ieee_2Erounding @ fo__c_2Ebinary__ieee_2EroundTiesToEven ) ) @ V0a ) @ V1b ) ) ) )
& ( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( c_2Ebinary__ieee_2Efloat__to__real @ A_27t @ A_27w ) @ ( ap @ ( c_2Epair_2ESND @ ty_2Ebinary__ieee_2Eflags @ ( ty_2Ebinary__ieee_2Efloat @ A_27t @ A_27w ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Ebinary__ieee_2Efloat__div @ A_27t @ A_27w ) @ ( inj__ty_2Ebinary__ieee_2Erounding @ fo__c_2Ebinary__ieee_2EroundTiesToEven ) ) @ V0a ) @ V1b ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( ap @ ( ap @ c_2Ereal_2E_2F @ ( ap @ ( c_2Ebinary__ieee_2Efloat__to__real @ A_27t @ A_27w ) @ V0a ) ) @ ( ap @ ( c_2Ebinary__ieee_2Efloat__to__real @ A_27t @ A_27w ) @ V1b ) ) ) @ ( ap @ ( ap @ ( c_2Elift__ieee_2Eerror @ A_27t @ A_27w ) @ ( c_2Ebool_2Ethe__value @ ( ty_2Epair_2Eprod @ A_27t @ A_27w ) ) ) @ ( ap @ ( ap @ c_2Ereal_2E_2F @ ( ap @ ( c_2Ebinary__ieee_2Efloat__to__real @ A_27t @ A_27w ) @ V0a ) ) @ ( ap @ ( c_2Ebinary__ieee_2Efloat__to__real @ A_27t @ A_27w ) @ V1b ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elift__ieee_2Efloat__sqrt,axiom,
! [A_27t: del,A_27w: del,V0a: $i] :
( ( mem @ V0a @ ( ty_2Ebinary__ieee_2Efloat @ A_27t @ A_27w ) )
=> ( ( ( p @ ( ap @ ( c_2Ebinary__ieee_2Efloat__is__finite @ A_27t @ A_27w ) @ V0a ) )
& ( ( surj__c_ty_2Efcp_2Ecart_o_ty_2Eone_2Eone @ ( ap @ ( c_2Ebinary__ieee_2Efloat__Sign @ A_27t @ A_27w ) @ V0a ) )
= ( surj__c_ty_2Efcp_2Ecart_o_ty_2Eone_2Eone @ ( ap @ ( c_2Ewords_2En2w @ ty_2Eone_2Eone ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
& ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Eabs @ ( ap @ c_2Etransc_2Esqrt @ ( ap @ ( c_2Ebinary__ieee_2Efloat__to__real @ A_27t @ A_27w ) @ V0a ) ) ) ) @ ( ap @ ( c_2Ebinary__ieee_2Ethreshold @ A_27t @ A_27w ) @ ( c_2Ebool_2Ethe__value @ ( ty_2Epair_2Eprod @ A_27t @ A_27w ) ) ) ) ) )
=> ( ( p @ ( ap @ ( c_2Ebinary__ieee_2Efloat__is__finite @ A_27t @ A_27w ) @ ( ap @ ( c_2Epair_2ESND @ ty_2Ebinary__ieee_2Eflags @ ( ty_2Ebinary__ieee_2Efloat @ A_27t @ A_27w ) ) @ ( ap @ ( ap @ ( c_2Ebinary__ieee_2Efloat__sqrt @ A_27t @ A_27w ) @ ( inj__ty_2Ebinary__ieee_2Erounding @ fo__c_2Ebinary__ieee_2EroundTiesToEven ) ) @ V0a ) ) ) )
& ( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( c_2Ebinary__ieee_2Efloat__to__real @ A_27t @ A_27w ) @ ( ap @ ( c_2Epair_2ESND @ ty_2Ebinary__ieee_2Eflags @ ( ty_2Ebinary__ieee_2Efloat @ A_27t @ A_27w ) ) @ ( ap @ ( ap @ ( c_2Ebinary__ieee_2Efloat__sqrt @ A_27t @ A_27w ) @ ( inj__ty_2Ebinary__ieee_2Erounding @ fo__c_2Ebinary__ieee_2EroundTiesToEven ) ) @ V0a ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( ap @ c_2Etransc_2Esqrt @ ( ap @ ( c_2Ebinary__ieee_2Efloat__to__real @ A_27t @ A_27w ) @ V0a ) ) ) @ ( ap @ ( ap @ ( c_2Elift__ieee_2Eerror @ A_27t @ A_27w ) @ ( c_2Ebool_2Ethe__value @ ( ty_2Epair_2Eprod @ A_27t @ A_27w ) ) ) @ ( ap @ c_2Etransc_2Esqrt @ ( ap @ ( c_2Ebinary__ieee_2Efloat__to__real @ A_27t @ A_27w ) @ V0a ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elift__ieee_2Efloat__mul__add,axiom,
! [A_27t: del,A_27w: del,V0a: $i] :
( ( mem @ V0a @ ( ty_2Ebinary__ieee_2Efloat @ A_27t @ A_27w ) )
=> ! [V1b: $i] :
( ( mem @ V1b @ ( ty_2Ebinary__ieee_2Efloat @ A_27t @ A_27w ) )
=> ! [V2c: $i] :
( ( mem @ V2c @ ( ty_2Ebinary__ieee_2Efloat @ A_27t @ A_27w ) )
=> ( ( ( p @ ( ap @ ( c_2Ebinary__ieee_2Efloat__is__finite @ A_27t @ A_27w ) @ V0a ) )
& ( p @ ( ap @ ( c_2Ebinary__ieee_2Efloat__is__finite @ A_27t @ A_27w ) @ V1b ) )
& ( p @ ( ap @ ( c_2Ebinary__ieee_2Efloat__is__finite @ A_27t @ A_27w ) @ V2c ) )
& ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Eabs @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ ( c_2Ebinary__ieee_2Efloat__to__real @ A_27t @ A_27w ) @ V0a ) ) @ ( ap @ ( c_2Ebinary__ieee_2Efloat__to__real @ A_27t @ A_27w ) @ V1b ) ) ) @ ( ap @ ( c_2Ebinary__ieee_2Efloat__to__real @ A_27t @ A_27w ) @ V2c ) ) ) ) @ ( ap @ ( c_2Ebinary__ieee_2Ethreshold @ A_27t @ A_27w ) @ ( c_2Ebool_2Ethe__value @ ( ty_2Epair_2Eprod @ A_27t @ A_27w ) ) ) ) ) )
=> ( ( p @ ( ap @ ( c_2Ebinary__ieee_2Efloat__is__finite @ A_27t @ A_27w ) @ ( ap @ ( c_2Epair_2ESND @ ty_2Ebinary__ieee_2Eflags @ ( ty_2Ebinary__ieee_2Efloat @ A_27t @ A_27w ) ) @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Ebinary__ieee_2Efloat__mul__add @ A_27t @ A_27w ) @ ( inj__ty_2Ebinary__ieee_2Erounding @ fo__c_2Ebinary__ieee_2EroundTiesToEven ) ) @ V0a ) @ V1b ) @ V2c ) ) ) )
& ( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( c_2Ebinary__ieee_2Efloat__to__real @ A_27t @ A_27w ) @ ( ap @ ( c_2Epair_2ESND @ ty_2Ebinary__ieee_2Eflags @ ( ty_2Ebinary__ieee_2Efloat @ A_27t @ A_27w ) ) @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Ebinary__ieee_2Efloat__mul__add @ A_27t @ A_27w ) @ ( inj__ty_2Ebinary__ieee_2Erounding @ fo__c_2Ebinary__ieee_2EroundTiesToEven ) ) @ V0a ) @ V1b ) @ V2c ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ ( c_2Ebinary__ieee_2Efloat__to__real @ A_27t @ A_27w ) @ V0a ) ) @ ( ap @ ( c_2Ebinary__ieee_2Efloat__to__real @ A_27t @ A_27w ) @ V1b ) ) ) @ ( ap @ ( c_2Ebinary__ieee_2Efloat__to__real @ A_27t @ A_27w ) @ V2c ) ) ) @ ( ap @ ( ap @ ( c_2Elift__ieee_2Eerror @ A_27t @ A_27w ) @ ( c_2Ebool_2Ethe__value @ ( ty_2Epair_2Eprod @ A_27t @ A_27w ) ) ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ ( c_2Ebinary__ieee_2Efloat__to__real @ A_27t @ A_27w ) @ V0a ) ) @ ( ap @ ( c_2Ebinary__ieee_2Efloat__to__real @ A_27t @ A_27w ) @ V1b ) ) ) @ ( ap @ ( c_2Ebinary__ieee_2Efloat__to__real @ A_27t @ A_27w ) @ V2c ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elift__ieee_2Efloat__mul__sub,axiom,
! [A_27t: del,A_27w: del,V0a: $i] :
( ( mem @ V0a @ ( ty_2Ebinary__ieee_2Efloat @ A_27t @ A_27w ) )
=> ! [V1b: $i] :
( ( mem @ V1b @ ( ty_2Ebinary__ieee_2Efloat @ A_27t @ A_27w ) )
=> ! [V2c: $i] :
( ( mem @ V2c @ ( ty_2Ebinary__ieee_2Efloat @ A_27t @ A_27w ) )
=> ( ( ( p @ ( ap @ ( c_2Ebinary__ieee_2Efloat__is__finite @ A_27t @ A_27w ) @ V0a ) )
& ( p @ ( ap @ ( c_2Ebinary__ieee_2Efloat__is__finite @ A_27t @ A_27w ) @ V1b ) )
& ( p @ ( ap @ ( c_2Ebinary__ieee_2Efloat__is__finite @ A_27t @ A_27w ) @ V2c ) )
& ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Eabs @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ ( c_2Ebinary__ieee_2Efloat__to__real @ A_27t @ A_27w ) @ V0a ) ) @ ( ap @ ( c_2Ebinary__ieee_2Efloat__to__real @ A_27t @ A_27w ) @ V1b ) ) ) @ ( ap @ ( c_2Ebinary__ieee_2Efloat__to__real @ A_27t @ A_27w ) @ V2c ) ) ) ) @ ( ap @ ( c_2Ebinary__ieee_2Ethreshold @ A_27t @ A_27w ) @ ( c_2Ebool_2Ethe__value @ ( ty_2Epair_2Eprod @ A_27t @ A_27w ) ) ) ) ) )
=> ( ( p @ ( ap @ ( c_2Ebinary__ieee_2Efloat__is__finite @ A_27t @ A_27w ) @ ( ap @ ( c_2Epair_2ESND @ ty_2Ebinary__ieee_2Eflags @ ( ty_2Ebinary__ieee_2Efloat @ A_27t @ A_27w ) ) @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Ebinary__ieee_2Efloat__mul__sub @ A_27t @ A_27w ) @ ( inj__ty_2Ebinary__ieee_2Erounding @ fo__c_2Ebinary__ieee_2EroundTiesToEven ) ) @ V0a ) @ V1b ) @ V2c ) ) ) )
& ( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( c_2Ebinary__ieee_2Efloat__to__real @ A_27t @ A_27w ) @ ( ap @ ( c_2Epair_2ESND @ ty_2Ebinary__ieee_2Eflags @ ( ty_2Ebinary__ieee_2Efloat @ A_27t @ A_27w ) ) @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Ebinary__ieee_2Efloat__mul__sub @ A_27t @ A_27w ) @ ( inj__ty_2Ebinary__ieee_2Erounding @ fo__c_2Ebinary__ieee_2EroundTiesToEven ) ) @ V0a ) @ V1b ) @ V2c ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ ( c_2Ebinary__ieee_2Efloat__to__real @ A_27t @ A_27w ) @ V0a ) ) @ ( ap @ ( c_2Ebinary__ieee_2Efloat__to__real @ A_27t @ A_27w ) @ V1b ) ) ) @ ( ap @ ( c_2Ebinary__ieee_2Efloat__to__real @ A_27t @ A_27w ) @ V2c ) ) ) @ ( ap @ ( ap @ ( c_2Elift__ieee_2Eerror @ A_27t @ A_27w ) @ ( c_2Ebool_2Ethe__value @ ( ty_2Epair_2Eprod @ A_27t @ A_27w ) ) ) @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ ( c_2Ebinary__ieee_2Efloat__to__real @ A_27t @ A_27w ) @ V0a ) ) @ ( ap @ ( c_2Ebinary__ieee_2Efloat__to__real @ A_27t @ A_27w ) @ V1b ) ) ) @ ( ap @ ( c_2Ebinary__ieee_2Efloat__to__real @ A_27t @ A_27w ) @ V2c ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elift__ieee_2Efloat__add__finite,axiom,
! [A_27t: del,A_27w: del,V0a: $i] :
( ( mem @ V0a @ ( ty_2Ebinary__ieee_2Efloat @ A_27t @ A_27w ) )
=> ! [V1b: $i] :
( ( mem @ V1b @ ( ty_2Ebinary__ieee_2Efloat @ A_27t @ A_27w ) )
=> ( ( ( p @ ( ap @ ( c_2Ebinary__ieee_2Efloat__is__finite @ A_27t @ A_27w ) @ V0a ) )
& ( p @ ( ap @ ( c_2Ebinary__ieee_2Efloat__is__finite @ A_27t @ A_27w ) @ V1b ) )
& ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Eabs @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( ap @ ( c_2Ebinary__ieee_2Efloat__to__real @ A_27t @ A_27w ) @ V0a ) ) @ ( ap @ ( c_2Ebinary__ieee_2Efloat__to__real @ A_27t @ A_27w ) @ V1b ) ) ) ) @ ( ap @ ( c_2Ebinary__ieee_2Ethreshold @ A_27t @ A_27w ) @ ( c_2Ebool_2Ethe__value @ ( ty_2Epair_2Eprod @ A_27t @ A_27w ) ) ) ) ) )
=> ( p @ ( ap @ ( c_2Ebinary__ieee_2Efloat__is__finite @ A_27t @ A_27w ) @ ( ap @ ( c_2Epair_2ESND @ ty_2Ebinary__ieee_2Eflags @ ( ty_2Ebinary__ieee_2Efloat @ A_27t @ A_27w ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Ebinary__ieee_2Efloat__add @ A_27t @ A_27w ) @ ( inj__ty_2Ebinary__ieee_2Erounding @ fo__c_2Ebinary__ieee_2EroundTiesToEven ) ) @ V0a ) @ V1b ) ) ) ) ) ) ) ).
thf(conj_thm_2Elift__ieee_2Efloat__sub__finite,axiom,
! [A_27t: del,A_27w: del,V0a: $i] :
( ( mem @ V0a @ ( ty_2Ebinary__ieee_2Efloat @ A_27t @ A_27w ) )
=> ! [V1b: $i] :
( ( mem @ V1b @ ( ty_2Ebinary__ieee_2Efloat @ A_27t @ A_27w ) )
=> ( ( ( p @ ( ap @ ( c_2Ebinary__ieee_2Efloat__is__finite @ A_27t @ A_27w ) @ V0a ) )
& ( p @ ( ap @ ( c_2Ebinary__ieee_2Efloat__is__finite @ A_27t @ A_27w ) @ V1b ) )
& ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Eabs @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( ap @ ( c_2Ebinary__ieee_2Efloat__to__real @ A_27t @ A_27w ) @ V0a ) ) @ ( ap @ ( c_2Ebinary__ieee_2Efloat__to__real @ A_27t @ A_27w ) @ V1b ) ) ) ) @ ( ap @ ( c_2Ebinary__ieee_2Ethreshold @ A_27t @ A_27w ) @ ( c_2Ebool_2Ethe__value @ ( ty_2Epair_2Eprod @ A_27t @ A_27w ) ) ) ) ) )
=> ( p @ ( ap @ ( c_2Ebinary__ieee_2Efloat__is__finite @ A_27t @ A_27w ) @ ( ap @ ( c_2Epair_2ESND @ ty_2Ebinary__ieee_2Eflags @ ( ty_2Ebinary__ieee_2Efloat @ A_27t @ A_27w ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Ebinary__ieee_2Efloat__sub @ A_27t @ A_27w ) @ ( inj__ty_2Ebinary__ieee_2Erounding @ fo__c_2Ebinary__ieee_2EroundTiesToEven ) ) @ V0a ) @ V1b ) ) ) ) ) ) ) ).
thf(conj_thm_2Elift__ieee_2Efloat__mul__finite,axiom,
! [A_27t: del,A_27w: del,V0a: $i] :
( ( mem @ V0a @ ( ty_2Ebinary__ieee_2Efloat @ A_27t @ A_27w ) )
=> ! [V1b: $i] :
( ( mem @ V1b @ ( ty_2Ebinary__ieee_2Efloat @ A_27t @ A_27w ) )
=> ( ( ( p @ ( ap @ ( c_2Ebinary__ieee_2Efloat__is__finite @ A_27t @ A_27w ) @ V0a ) )
& ( p @ ( ap @ ( c_2Ebinary__ieee_2Efloat__is__finite @ A_27t @ A_27w ) @ V1b ) )
& ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Eabs @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ ( c_2Ebinary__ieee_2Efloat__to__real @ A_27t @ A_27w ) @ V0a ) ) @ ( ap @ ( c_2Ebinary__ieee_2Efloat__to__real @ A_27t @ A_27w ) @ V1b ) ) ) ) @ ( ap @ ( c_2Ebinary__ieee_2Ethreshold @ A_27t @ A_27w ) @ ( c_2Ebool_2Ethe__value @ ( ty_2Epair_2Eprod @ A_27t @ A_27w ) ) ) ) ) )
=> ( p @ ( ap @ ( c_2Ebinary__ieee_2Efloat__is__finite @ A_27t @ A_27w ) @ ( ap @ ( c_2Epair_2ESND @ ty_2Ebinary__ieee_2Eflags @ ( ty_2Ebinary__ieee_2Efloat @ A_27t @ A_27w ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Ebinary__ieee_2Efloat__mul @ A_27t @ A_27w ) @ ( inj__ty_2Ebinary__ieee_2Erounding @ fo__c_2Ebinary__ieee_2EroundTiesToEven ) ) @ V0a ) @ V1b ) ) ) ) ) ) ) ).
thf(conj_thm_2Elift__ieee_2Efloat__div__finite,axiom,
! [A_27t: del,A_27w: del,V0a: $i] :
( ( mem @ V0a @ ( ty_2Ebinary__ieee_2Efloat @ A_27t @ A_27w ) )
=> ! [V1b: $i] :
( ( mem @ V1b @ ( ty_2Ebinary__ieee_2Efloat @ A_27t @ A_27w ) )
=> ( ( ( p @ ( ap @ ( c_2Ebinary__ieee_2Efloat__is__finite @ A_27t @ A_27w ) @ V0a ) )
& ( p @ ( ap @ ( c_2Ebinary__ieee_2Efloat__is__finite @ A_27t @ A_27w ) @ V1b ) )
& ~ ( p @ ( ap @ ( c_2Ebinary__ieee_2Efloat__is__zero @ A_27t @ A_27w ) @ V1b ) )
& ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Eabs @ ( ap @ ( ap @ c_2Ereal_2E_2F @ ( ap @ ( c_2Ebinary__ieee_2Efloat__to__real @ A_27t @ A_27w ) @ V0a ) ) @ ( ap @ ( c_2Ebinary__ieee_2Efloat__to__real @ A_27t @ A_27w ) @ V1b ) ) ) ) @ ( ap @ ( c_2Ebinary__ieee_2Ethreshold @ A_27t @ A_27w ) @ ( c_2Ebool_2Ethe__value @ ( ty_2Epair_2Eprod @ A_27t @ A_27w ) ) ) ) ) )
=> ( p @ ( ap @ ( c_2Ebinary__ieee_2Efloat__is__finite @ A_27t @ A_27w ) @ ( ap @ ( c_2Epair_2ESND @ ty_2Ebinary__ieee_2Eflags @ ( ty_2Ebinary__ieee_2Efloat @ A_27t @ A_27w ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Ebinary__ieee_2Efloat__div @ A_27t @ A_27w ) @ ( inj__ty_2Ebinary__ieee_2Erounding @ fo__c_2Ebinary__ieee_2EroundTiesToEven ) ) @ V0a ) @ V1b ) ) ) ) ) ) ) ).
thf(conj_thm_2Elift__ieee_2Efloat__sqrt__finite,axiom,
! [A_27t: del,A_27w: del,V0a: $i] :
( ( mem @ V0a @ ( ty_2Ebinary__ieee_2Efloat @ A_27t @ A_27w ) )
=> ( ( ( p @ ( ap @ ( c_2Ebinary__ieee_2Efloat__is__finite @ A_27t @ A_27w ) @ V0a ) )
& ( ( surj__c_ty_2Efcp_2Ecart_o_ty_2Eone_2Eone @ ( ap @ ( c_2Ebinary__ieee_2Efloat__Sign @ A_27t @ A_27w ) @ V0a ) )
= ( surj__c_ty_2Efcp_2Ecart_o_ty_2Eone_2Eone @ ( ap @ ( c_2Ewords_2En2w @ ty_2Eone_2Eone ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
& ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Eabs @ ( ap @ c_2Etransc_2Esqrt @ ( ap @ ( c_2Ebinary__ieee_2Efloat__to__real @ A_27t @ A_27w ) @ V0a ) ) ) ) @ ( ap @ ( c_2Ebinary__ieee_2Ethreshold @ A_27t @ A_27w ) @ ( c_2Ebool_2Ethe__value @ ( ty_2Epair_2Eprod @ A_27t @ A_27w ) ) ) ) ) )
=> ( p @ ( ap @ ( c_2Ebinary__ieee_2Efloat__is__finite @ A_27t @ A_27w ) @ ( ap @ ( c_2Epair_2ESND @ ty_2Ebinary__ieee_2Eflags @ ( ty_2Ebinary__ieee_2Efloat @ A_27t @ A_27w ) ) @ ( ap @ ( ap @ ( c_2Ebinary__ieee_2Efloat__sqrt @ A_27t @ A_27w ) @ ( inj__ty_2Ebinary__ieee_2Erounding @ fo__c_2Ebinary__ieee_2EroundTiesToEven ) ) @ V0a ) ) ) ) ) ) ).
thf(conj_thm_2Elift__ieee_2Efloat__mul__add__finite,axiom,
! [A_27t: del,A_27w: del,V0a: $i] :
( ( mem @ V0a @ ( ty_2Ebinary__ieee_2Efloat @ A_27t @ A_27w ) )
=> ! [V1b: $i] :
( ( mem @ V1b @ ( ty_2Ebinary__ieee_2Efloat @ A_27t @ A_27w ) )
=> ! [V2c: $i] :
( ( mem @ V2c @ ( ty_2Ebinary__ieee_2Efloat @ A_27t @ A_27w ) )
=> ( ( ( p @ ( ap @ ( c_2Ebinary__ieee_2Efloat__is__finite @ A_27t @ A_27w ) @ V0a ) )
& ( p @ ( ap @ ( c_2Ebinary__ieee_2Efloat__is__finite @ A_27t @ A_27w ) @ V1b ) )
& ( p @ ( ap @ ( c_2Ebinary__ieee_2Efloat__is__finite @ A_27t @ A_27w ) @ V2c ) )
& ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Eabs @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ ( c_2Ebinary__ieee_2Efloat__to__real @ A_27t @ A_27w ) @ V0a ) ) @ ( ap @ ( c_2Ebinary__ieee_2Efloat__to__real @ A_27t @ A_27w ) @ V1b ) ) ) @ ( ap @ ( c_2Ebinary__ieee_2Efloat__to__real @ A_27t @ A_27w ) @ V2c ) ) ) ) @ ( ap @ ( c_2Ebinary__ieee_2Ethreshold @ A_27t @ A_27w ) @ ( c_2Ebool_2Ethe__value @ ( ty_2Epair_2Eprod @ A_27t @ A_27w ) ) ) ) ) )
=> ( p @ ( ap @ ( c_2Ebinary__ieee_2Efloat__is__finite @ A_27t @ A_27w ) @ ( ap @ ( c_2Epair_2ESND @ ty_2Ebinary__ieee_2Eflags @ ( ty_2Ebinary__ieee_2Efloat @ A_27t @ A_27w ) ) @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Ebinary__ieee_2Efloat__mul__add @ A_27t @ A_27w ) @ ( inj__ty_2Ebinary__ieee_2Erounding @ fo__c_2Ebinary__ieee_2EroundTiesToEven ) ) @ V0a ) @ V1b ) @ V2c ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elift__ieee_2Efloat__mul__sub__finite,axiom,
! [A_27t: del,A_27w: del,V0a: $i] :
( ( mem @ V0a @ ( ty_2Ebinary__ieee_2Efloat @ A_27t @ A_27w ) )
=> ! [V1b: $i] :
( ( mem @ V1b @ ( ty_2Ebinary__ieee_2Efloat @ A_27t @ A_27w ) )
=> ! [V2c: $i] :
( ( mem @ V2c @ ( ty_2Ebinary__ieee_2Efloat @ A_27t @ A_27w ) )
=> ( ( ( p @ ( ap @ ( c_2Ebinary__ieee_2Efloat__is__finite @ A_27t @ A_27w ) @ V0a ) )
& ( p @ ( ap @ ( c_2Ebinary__ieee_2Efloat__is__finite @ A_27t @ A_27w ) @ V1b ) )
& ( p @ ( ap @ ( c_2Ebinary__ieee_2Efloat__is__finite @ A_27t @ A_27w ) @ V2c ) )
& ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Eabs @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ ( c_2Ebinary__ieee_2Efloat__to__real @ A_27t @ A_27w ) @ V0a ) ) @ ( ap @ ( c_2Ebinary__ieee_2Efloat__to__real @ A_27t @ A_27w ) @ V1b ) ) ) @ ( ap @ ( c_2Ebinary__ieee_2Efloat__to__real @ A_27t @ A_27w ) @ V2c ) ) ) ) @ ( ap @ ( c_2Ebinary__ieee_2Ethreshold @ A_27t @ A_27w ) @ ( c_2Ebool_2Ethe__value @ ( ty_2Epair_2Eprod @ A_27t @ A_27w ) ) ) ) ) )
=> ( p @ ( ap @ ( c_2Ebinary__ieee_2Efloat__is__finite @ A_27t @ A_27w ) @ ( ap @ ( c_2Epair_2ESND @ ty_2Ebinary__ieee_2Eflags @ ( ty_2Ebinary__ieee_2Efloat @ A_27t @ A_27w ) ) @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Ebinary__ieee_2Efloat__mul__sub @ A_27t @ A_27w ) @ ( inj__ty_2Ebinary__ieee_2Erounding @ fo__c_2Ebinary__ieee_2EroundTiesToEven ) ) @ V0a ) @ V1b ) @ V2c ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elift__ieee_2Efloat__add__relative,axiom,
! [A_27t: del,A_27w: del,V0a: $i] :
( ( mem @ V0a @ ( ty_2Ebinary__ieee_2Efloat @ A_27t @ A_27w ) )
=> ! [V1b: $i] :
( ( mem @ V1b @ ( ty_2Ebinary__ieee_2Efloat @ A_27t @ A_27w ) )
=> ( ( ( p @ ( ap @ ( c_2Ebinary__ieee_2Efloat__is__finite @ A_27t @ A_27w ) @ V0a ) )
& ( p @ ( ap @ ( c_2Ebinary__ieee_2Efloat__is__finite @ A_27t @ A_27w ) @ V1b ) )
& ( p @ ( ap @ ( ap @ ( c_2Elift__ieee_2Enormalizes @ A_27t @ A_27w ) @ ( c_2Ebool_2Ethe__value @ ( ty_2Epair_2Eprod @ A_27t @ A_27w ) ) ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( ap @ ( c_2Ebinary__ieee_2Efloat__to__real @ A_27t @ A_27w ) @ V0a ) ) @ ( ap @ ( c_2Ebinary__ieee_2Efloat__to__real @ A_27t @ A_27w ) @ V1b ) ) ) ) )
=> ( ( p @ ( ap @ ( c_2Ebinary__ieee_2Efloat__is__finite @ A_27t @ A_27w ) @ ( ap @ ( c_2Epair_2ESND @ ty_2Ebinary__ieee_2Eflags @ ( ty_2Ebinary__ieee_2Efloat @ A_27t @ A_27w ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Ebinary__ieee_2Efloat__add @ A_27t @ A_27w ) @ ( inj__ty_2Ebinary__ieee_2Erounding @ fo__c_2Ebinary__ieee_2EroundTiesToEven ) ) @ V0a ) @ V1b ) ) ) )
& ? [V2e: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Eabs @ ( inj__ty_2Erealax_2Ereal @ V2e ) ) ) @ ( ap @ ( ap @ c_2Ereal_2E_2F @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( ap @ ( c_2Efcp_2Edimindex @ A_27t ) @ ( c_2Ebool_2Ethe__value @ A_27t ) ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) )
& ( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( c_2Ebinary__ieee_2Efloat__to__real @ A_27t @ A_27w ) @ ( ap @ ( c_2Epair_2ESND @ ty_2Ebinary__ieee_2Eflags @ ( ty_2Ebinary__ieee_2Efloat @ A_27t @ A_27w ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Ebinary__ieee_2Efloat__add @ A_27t @ A_27w ) @ ( inj__ty_2Ebinary__ieee_2Erounding @ fo__c_2Ebinary__ieee_2EroundTiesToEven ) ) @ V0a ) @ V1b ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( ap @ ( c_2Ebinary__ieee_2Efloat__to__real @ A_27t @ A_27w ) @ V0a ) ) @ ( ap @ ( c_2Ebinary__ieee_2Efloat__to__real @ A_27t @ A_27w ) @ V1b ) ) ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V2e ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elift__ieee_2Efloat__sub__relative,axiom,
! [A_27t: del,A_27w: del,V0a: $i] :
( ( mem @ V0a @ ( ty_2Ebinary__ieee_2Efloat @ A_27t @ A_27w ) )
=> ! [V1b: $i] :
( ( mem @ V1b @ ( ty_2Ebinary__ieee_2Efloat @ A_27t @ A_27w ) )
=> ( ( ( p @ ( ap @ ( c_2Ebinary__ieee_2Efloat__is__finite @ A_27t @ A_27w ) @ V0a ) )
& ( p @ ( ap @ ( c_2Ebinary__ieee_2Efloat__is__finite @ A_27t @ A_27w ) @ V1b ) )
& ( p @ ( ap @ ( ap @ ( c_2Elift__ieee_2Enormalizes @ A_27t @ A_27w ) @ ( c_2Ebool_2Ethe__value @ ( ty_2Epair_2Eprod @ A_27t @ A_27w ) ) ) @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( ap @ ( c_2Ebinary__ieee_2Efloat__to__real @ A_27t @ A_27w ) @ V0a ) ) @ ( ap @ ( c_2Ebinary__ieee_2Efloat__to__real @ A_27t @ A_27w ) @ V1b ) ) ) ) )
=> ( ( p @ ( ap @ ( c_2Ebinary__ieee_2Efloat__is__finite @ A_27t @ A_27w ) @ ( ap @ ( c_2Epair_2ESND @ ty_2Ebinary__ieee_2Eflags @ ( ty_2Ebinary__ieee_2Efloat @ A_27t @ A_27w ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Ebinary__ieee_2Efloat__sub @ A_27t @ A_27w ) @ ( inj__ty_2Ebinary__ieee_2Erounding @ fo__c_2Ebinary__ieee_2EroundTiesToEven ) ) @ V0a ) @ V1b ) ) ) )
& ? [V2e: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Eabs @ ( inj__ty_2Erealax_2Ereal @ V2e ) ) ) @ ( ap @ ( ap @ c_2Ereal_2E_2F @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( ap @ ( c_2Efcp_2Edimindex @ A_27t ) @ ( c_2Ebool_2Ethe__value @ A_27t ) ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) )
& ( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( c_2Ebinary__ieee_2Efloat__to__real @ A_27t @ A_27w ) @ ( ap @ ( c_2Epair_2ESND @ ty_2Ebinary__ieee_2Eflags @ ( ty_2Ebinary__ieee_2Efloat @ A_27t @ A_27w ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Ebinary__ieee_2Efloat__sub @ A_27t @ A_27w ) @ ( inj__ty_2Ebinary__ieee_2Erounding @ fo__c_2Ebinary__ieee_2EroundTiesToEven ) ) @ V0a ) @ V1b ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( ap @ ( c_2Ebinary__ieee_2Efloat__to__real @ A_27t @ A_27w ) @ V0a ) ) @ ( ap @ ( c_2Ebinary__ieee_2Efloat__to__real @ A_27t @ A_27w ) @ V1b ) ) ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V2e ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elift__ieee_2Efloat__mul__relative,axiom,
! [A_27t: del,A_27w: del,V0a: $i] :
( ( mem @ V0a @ ( ty_2Ebinary__ieee_2Efloat @ A_27t @ A_27w ) )
=> ! [V1b: $i] :
( ( mem @ V1b @ ( ty_2Ebinary__ieee_2Efloat @ A_27t @ A_27w ) )
=> ( ( ( p @ ( ap @ ( c_2Ebinary__ieee_2Efloat__is__finite @ A_27t @ A_27w ) @ V0a ) )
& ( p @ ( ap @ ( c_2Ebinary__ieee_2Efloat__is__finite @ A_27t @ A_27w ) @ V1b ) )
& ( p @ ( ap @ ( ap @ ( c_2Elift__ieee_2Enormalizes @ A_27t @ A_27w ) @ ( c_2Ebool_2Ethe__value @ ( ty_2Epair_2Eprod @ A_27t @ A_27w ) ) ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ ( c_2Ebinary__ieee_2Efloat__to__real @ A_27t @ A_27w ) @ V0a ) ) @ ( ap @ ( c_2Ebinary__ieee_2Efloat__to__real @ A_27t @ A_27w ) @ V1b ) ) ) ) )
=> ( ( p @ ( ap @ ( c_2Ebinary__ieee_2Efloat__is__finite @ A_27t @ A_27w ) @ ( ap @ ( c_2Epair_2ESND @ ty_2Ebinary__ieee_2Eflags @ ( ty_2Ebinary__ieee_2Efloat @ A_27t @ A_27w ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Ebinary__ieee_2Efloat__mul @ A_27t @ A_27w ) @ ( inj__ty_2Ebinary__ieee_2Erounding @ fo__c_2Ebinary__ieee_2EroundTiesToEven ) ) @ V0a ) @ V1b ) ) ) )
& ? [V2e: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Eabs @ ( inj__ty_2Erealax_2Ereal @ V2e ) ) ) @ ( ap @ ( ap @ c_2Ereal_2E_2F @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( ap @ ( c_2Efcp_2Edimindex @ A_27t ) @ ( c_2Ebool_2Ethe__value @ A_27t ) ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) )
& ( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( c_2Ebinary__ieee_2Efloat__to__real @ A_27t @ A_27w ) @ ( ap @ ( c_2Epair_2ESND @ ty_2Ebinary__ieee_2Eflags @ ( ty_2Ebinary__ieee_2Efloat @ A_27t @ A_27w ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Ebinary__ieee_2Efloat__mul @ A_27t @ A_27w ) @ ( inj__ty_2Ebinary__ieee_2Erounding @ fo__c_2Ebinary__ieee_2EroundTiesToEven ) ) @ V0a ) @ V1b ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ ( c_2Ebinary__ieee_2Efloat__to__real @ A_27t @ A_27w ) @ V0a ) ) @ ( ap @ ( c_2Ebinary__ieee_2Efloat__to__real @ A_27t @ A_27w ) @ V1b ) ) ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V2e ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elift__ieee_2Efloat__div__relative,axiom,
! [A_27t: del,A_27w: del,V0a: $i] :
( ( mem @ V0a @ ( ty_2Ebinary__ieee_2Efloat @ A_27t @ A_27w ) )
=> ! [V1b: $i] :
( ( mem @ V1b @ ( ty_2Ebinary__ieee_2Efloat @ A_27t @ A_27w ) )
=> ( ( ( p @ ( ap @ ( c_2Ebinary__ieee_2Efloat__is__finite @ A_27t @ A_27w ) @ V0a ) )
& ( p @ ( ap @ ( c_2Ebinary__ieee_2Efloat__is__finite @ A_27t @ A_27w ) @ V1b ) )
& ~ ( p @ ( ap @ ( c_2Ebinary__ieee_2Efloat__is__zero @ A_27t @ A_27w ) @ V1b ) )
& ( p @ ( ap @ ( ap @ ( c_2Elift__ieee_2Enormalizes @ A_27t @ A_27w ) @ ( c_2Ebool_2Ethe__value @ ( ty_2Epair_2Eprod @ A_27t @ A_27w ) ) ) @ ( ap @ ( ap @ c_2Ereal_2E_2F @ ( ap @ ( c_2Ebinary__ieee_2Efloat__to__real @ A_27t @ A_27w ) @ V0a ) ) @ ( ap @ ( c_2Ebinary__ieee_2Efloat__to__real @ A_27t @ A_27w ) @ V1b ) ) ) ) )
=> ( ( p @ ( ap @ ( c_2Ebinary__ieee_2Efloat__is__finite @ A_27t @ A_27w ) @ ( ap @ ( c_2Epair_2ESND @ ty_2Ebinary__ieee_2Eflags @ ( ty_2Ebinary__ieee_2Efloat @ A_27t @ A_27w ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Ebinary__ieee_2Efloat__div @ A_27t @ A_27w ) @ ( inj__ty_2Ebinary__ieee_2Erounding @ fo__c_2Ebinary__ieee_2EroundTiesToEven ) ) @ V0a ) @ V1b ) ) ) )
& ? [V2e: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Eabs @ ( inj__ty_2Erealax_2Ereal @ V2e ) ) ) @ ( ap @ ( ap @ c_2Ereal_2E_2F @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( ap @ ( c_2Efcp_2Edimindex @ A_27t ) @ ( c_2Ebool_2Ethe__value @ A_27t ) ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) )
& ( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( c_2Ebinary__ieee_2Efloat__to__real @ A_27t @ A_27w ) @ ( ap @ ( c_2Epair_2ESND @ ty_2Ebinary__ieee_2Eflags @ ( ty_2Ebinary__ieee_2Efloat @ A_27t @ A_27w ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Ebinary__ieee_2Efloat__div @ A_27t @ A_27w ) @ ( inj__ty_2Ebinary__ieee_2Erounding @ fo__c_2Ebinary__ieee_2EroundTiesToEven ) ) @ V0a ) @ V1b ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ ( ap @ c_2Ereal_2E_2F @ ( ap @ ( c_2Ebinary__ieee_2Efloat__to__real @ A_27t @ A_27w ) @ V0a ) ) @ ( ap @ ( c_2Ebinary__ieee_2Efloat__to__real @ A_27t @ A_27w ) @ V1b ) ) ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V2e ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elift__ieee_2Efloat__sqrt__relative,axiom,
! [A_27t: del,A_27w: del,V0a: $i] :
( ( mem @ V0a @ ( ty_2Ebinary__ieee_2Efloat @ A_27t @ A_27w ) )
=> ( ( ( p @ ( ap @ ( c_2Ebinary__ieee_2Efloat__is__finite @ A_27t @ A_27w ) @ V0a ) )
& ( ( surj__c_ty_2Efcp_2Ecart_o_ty_2Eone_2Eone @ ( ap @ ( c_2Ebinary__ieee_2Efloat__Sign @ A_27t @ A_27w ) @ V0a ) )
= ( surj__c_ty_2Efcp_2Ecart_o_ty_2Eone_2Eone @ ( ap @ ( c_2Ewords_2En2w @ ty_2Eone_2Eone ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Elift__ieee_2Enormalizes @ A_27t @ A_27w ) @ ( c_2Ebool_2Ethe__value @ ( ty_2Epair_2Eprod @ A_27t @ A_27w ) ) ) @ ( ap @ c_2Etransc_2Esqrt @ ( ap @ ( c_2Ebinary__ieee_2Efloat__to__real @ A_27t @ A_27w ) @ V0a ) ) ) ) )
=> ( ( p @ ( ap @ ( c_2Ebinary__ieee_2Efloat__is__finite @ A_27t @ A_27w ) @ ( ap @ ( c_2Epair_2ESND @ ty_2Ebinary__ieee_2Eflags @ ( ty_2Ebinary__ieee_2Efloat @ A_27t @ A_27w ) ) @ ( ap @ ( ap @ ( c_2Ebinary__ieee_2Efloat__sqrt @ A_27t @ A_27w ) @ ( inj__ty_2Ebinary__ieee_2Erounding @ fo__c_2Ebinary__ieee_2EroundTiesToEven ) ) @ V0a ) ) ) )
& ? [V1e: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Eabs @ ( inj__ty_2Erealax_2Ereal @ V1e ) ) ) @ ( ap @ ( ap @ c_2Ereal_2E_2F @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( ap @ ( c_2Efcp_2Edimindex @ A_27t ) @ ( c_2Ebool_2Ethe__value @ A_27t ) ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) )
& ( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( c_2Ebinary__ieee_2Efloat__to__real @ A_27t @ A_27w ) @ ( ap @ ( c_2Epair_2ESND @ ty_2Ebinary__ieee_2Eflags @ ( ty_2Ebinary__ieee_2Efloat @ A_27t @ A_27w ) ) @ ( ap @ ( ap @ ( c_2Ebinary__ieee_2Efloat__sqrt @ A_27t @ A_27w ) @ ( inj__ty_2Ebinary__ieee_2Erounding @ fo__c_2Ebinary__ieee_2EroundTiesToEven ) ) @ V0a ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ c_2Etransc_2Esqrt @ ( ap @ ( c_2Ebinary__ieee_2Efloat__to__real @ A_27t @ A_27w ) @ V0a ) ) ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V1e ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elift__ieee_2Efloat__mul__add__relative,axiom,
! [A_27t: del,A_27w: del,V0a: $i] :
( ( mem @ V0a @ ( ty_2Ebinary__ieee_2Efloat @ A_27t @ A_27w ) )
=> ! [V1b: $i] :
( ( mem @ V1b @ ( ty_2Ebinary__ieee_2Efloat @ A_27t @ A_27w ) )
=> ! [V2c: $i] :
( ( mem @ V2c @ ( ty_2Ebinary__ieee_2Efloat @ A_27t @ A_27w ) )
=> ( ( ( p @ ( ap @ ( c_2Ebinary__ieee_2Efloat__is__finite @ A_27t @ A_27w ) @ V0a ) )
& ( p @ ( ap @ ( c_2Ebinary__ieee_2Efloat__is__finite @ A_27t @ A_27w ) @ V1b ) )
& ( p @ ( ap @ ( c_2Ebinary__ieee_2Efloat__is__finite @ A_27t @ A_27w ) @ V2c ) )
& ( p @ ( ap @ ( ap @ ( c_2Elift__ieee_2Enormalizes @ A_27t @ A_27w ) @ ( c_2Ebool_2Ethe__value @ ( ty_2Epair_2Eprod @ A_27t @ A_27w ) ) ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ ( c_2Ebinary__ieee_2Efloat__to__real @ A_27t @ A_27w ) @ V0a ) ) @ ( ap @ ( c_2Ebinary__ieee_2Efloat__to__real @ A_27t @ A_27w ) @ V1b ) ) ) @ ( ap @ ( c_2Ebinary__ieee_2Efloat__to__real @ A_27t @ A_27w ) @ V2c ) ) ) ) )
=> ( ( p @ ( ap @ ( c_2Ebinary__ieee_2Efloat__is__finite @ A_27t @ A_27w ) @ ( ap @ ( c_2Epair_2ESND @ ty_2Ebinary__ieee_2Eflags @ ( ty_2Ebinary__ieee_2Efloat @ A_27t @ A_27w ) ) @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Ebinary__ieee_2Efloat__mul__add @ A_27t @ A_27w ) @ ( inj__ty_2Ebinary__ieee_2Erounding @ fo__c_2Ebinary__ieee_2EroundTiesToEven ) ) @ V0a ) @ V1b ) @ V2c ) ) ) )
& ? [V3e: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Eabs @ ( inj__ty_2Erealax_2Ereal @ V3e ) ) ) @ ( ap @ ( ap @ c_2Ereal_2E_2F @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( ap @ ( c_2Efcp_2Edimindex @ A_27t ) @ ( c_2Ebool_2Ethe__value @ A_27t ) ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) )
& ( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( c_2Ebinary__ieee_2Efloat__to__real @ A_27t @ A_27w ) @ ( ap @ ( c_2Epair_2ESND @ ty_2Ebinary__ieee_2Eflags @ ( ty_2Ebinary__ieee_2Efloat @ A_27t @ A_27w ) ) @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Ebinary__ieee_2Efloat__mul__add @ A_27t @ A_27w ) @ ( inj__ty_2Ebinary__ieee_2Erounding @ fo__c_2Ebinary__ieee_2EroundTiesToEven ) ) @ V0a ) @ V1b ) @ V2c ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ ( c_2Ebinary__ieee_2Efloat__to__real @ A_27t @ A_27w ) @ V0a ) ) @ ( ap @ ( c_2Ebinary__ieee_2Efloat__to__real @ A_27t @ A_27w ) @ V1b ) ) ) @ ( ap @ ( c_2Ebinary__ieee_2Efloat__to__real @ A_27t @ A_27w ) @ V2c ) ) ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V3e ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elift__ieee_2Efloat__mul__sub__relative,axiom,
! [A_27t: del,A_27w: del,V0a: $i] :
( ( mem @ V0a @ ( ty_2Ebinary__ieee_2Efloat @ A_27t @ A_27w ) )
=> ! [V1b: $i] :
( ( mem @ V1b @ ( ty_2Ebinary__ieee_2Efloat @ A_27t @ A_27w ) )
=> ! [V2c: $i] :
( ( mem @ V2c @ ( ty_2Ebinary__ieee_2Efloat @ A_27t @ A_27w ) )
=> ( ( ( p @ ( ap @ ( c_2Ebinary__ieee_2Efloat__is__finite @ A_27t @ A_27w ) @ V0a ) )
& ( p @ ( ap @ ( c_2Ebinary__ieee_2Efloat__is__finite @ A_27t @ A_27w ) @ V1b ) )
& ( p @ ( ap @ ( c_2Ebinary__ieee_2Efloat__is__finite @ A_27t @ A_27w ) @ V2c ) )
& ( p @ ( ap @ ( ap @ ( c_2Elift__ieee_2Enormalizes @ A_27t @ A_27w ) @ ( c_2Ebool_2Ethe__value @ ( ty_2Epair_2Eprod @ A_27t @ A_27w ) ) ) @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ ( c_2Ebinary__ieee_2Efloat__to__real @ A_27t @ A_27w ) @ V0a ) ) @ ( ap @ ( c_2Ebinary__ieee_2Efloat__to__real @ A_27t @ A_27w ) @ V1b ) ) ) @ ( ap @ ( c_2Ebinary__ieee_2Efloat__to__real @ A_27t @ A_27w ) @ V2c ) ) ) ) )
=> ( ( p @ ( ap @ ( c_2Ebinary__ieee_2Efloat__is__finite @ A_27t @ A_27w ) @ ( ap @ ( c_2Epair_2ESND @ ty_2Ebinary__ieee_2Eflags @ ( ty_2Ebinary__ieee_2Efloat @ A_27t @ A_27w ) ) @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Ebinary__ieee_2Efloat__mul__sub @ A_27t @ A_27w ) @ ( inj__ty_2Ebinary__ieee_2Erounding @ fo__c_2Ebinary__ieee_2EroundTiesToEven ) ) @ V0a ) @ V1b ) @ V2c ) ) ) )
& ? [V3e: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Eabs @ ( inj__ty_2Erealax_2Ereal @ V3e ) ) ) @ ( ap @ ( ap @ c_2Ereal_2E_2F @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( ap @ ( c_2Efcp_2Edimindex @ A_27t ) @ ( c_2Ebool_2Ethe__value @ A_27t ) ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) )
& ( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( c_2Ebinary__ieee_2Efloat__to__real @ A_27t @ A_27w ) @ ( ap @ ( c_2Epair_2ESND @ ty_2Ebinary__ieee_2Eflags @ ( ty_2Ebinary__ieee_2Efloat @ A_27t @ A_27w ) ) @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Ebinary__ieee_2Efloat__mul__sub @ A_27t @ A_27w ) @ ( inj__ty_2Ebinary__ieee_2Erounding @ fo__c_2Ebinary__ieee_2EroundTiesToEven ) ) @ V0a ) @ V1b ) @ V2c ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ ( c_2Ebinary__ieee_2Efloat__to__real @ A_27t @ A_27w ) @ V0a ) ) @ ( ap @ ( c_2Ebinary__ieee_2Efloat__to__real @ A_27t @ A_27w ) @ V1b ) ) ) @ ( ap @ ( c_2Ebinary__ieee_2Efloat__to__real @ A_27t @ A_27w ) @ V2c ) ) ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V3e ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elift__ieee_2Efinite__float__within__threshold,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( ty_2Ebinary__ieee_2Efloat @ A_27a @ A_27b ) )
=> ( ( p @ ( ap @ ( c_2Ebinary__ieee_2Efloat__is__finite @ A_27a @ A_27b ) @ V0f ) )
=> ( ~ ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ ( c_2Ebinary__ieee_2Efloat__to__real @ A_27a @ A_27b ) @ V0f ) ) @ ( ap @ c_2Erealax_2Ereal__neg @ ( ap @ ( c_2Ebinary__ieee_2Ethreshold @ A_27a @ A_27b ) @ ( c_2Ebool_2Ethe__value @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) ) ) ) ) )
& ~ ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__ge @ ( ap @ ( c_2Ebinary__ieee_2Efloat__to__real @ A_27a @ A_27b ) @ V0f ) ) @ ( ap @ ( c_2Ebinary__ieee_2Ethreshold @ A_27a @ A_27b ) @ ( c_2Ebool_2Ethe__value @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elift__ieee_2Eround__finite__normal__float__id,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( ty_2Ebinary__ieee_2Efloat @ A_27a @ A_27b ) )
=> ( ( ( p @ ( ap @ ( c_2Ebinary__ieee_2Efloat__is__finite @ A_27a @ A_27b ) @ V0f ) )
& ~ ( p @ ( ap @ ( c_2Ebinary__ieee_2Efloat__is__zero @ A_27a @ A_27b ) @ V0f ) ) )
=> ( ( ap @ ( ap @ ( c_2Ebinary__ieee_2Eround @ A_27a @ A_27b ) @ ( inj__ty_2Ebinary__ieee_2Erounding @ fo__c_2Ebinary__ieee_2EroundTiesToEven ) ) @ ( ap @ ( c_2Ebinary__ieee_2Efloat__to__real @ A_27a @ A_27b ) @ V0f ) )
= V0f ) ) ) ).
thf(conj_thm_2Elift__ieee_2Ereal__to__float__finite__normal__id,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( ty_2Ebinary__ieee_2Efloat @ A_27a @ A_27b ) )
=> ( ( ( p @ ( ap @ ( c_2Ebinary__ieee_2Efloat__is__finite @ A_27a @ A_27b ) @ V0f ) )
& ~ ( p @ ( ap @ ( c_2Ebinary__ieee_2Efloat__is__zero @ A_27a @ A_27b ) @ V0f ) ) )
=> ( ( ap @ ( ap @ ( c_2Ebinary__ieee_2Ereal__to__float @ A_27a @ A_27b ) @ ( inj__ty_2Ebinary__ieee_2Erounding @ fo__c_2Ebinary__ieee_2EroundTiesToEven ) ) @ ( ap @ ( c_2Ebinary__ieee_2Efloat__to__real @ A_27a @ A_27b ) @ V0f ) )
= V0f ) ) ) ).
thf(conj_thm_2Elift__ieee_2Efloat__to__real__real__to__float__zero__id,axiom,
! [A_27a: del,A_27b: del] :
( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( c_2Ebinary__ieee_2Efloat__to__real @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Ebinary__ieee_2Ereal__to__float @ A_27a @ A_27b ) @ ( inj__ty_2Ebinary__ieee_2Erounding @ fo__c_2Ebinary__ieee_2EroundTiesToEven ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ).
thf(conj_thm_2Elift__ieee_2Enon__representable__float__is__zero,axiom,
! [A_27a: del,A_27b: del,V0ff: tp__ty_2Erealax_2Ereal,V1P: $i] :
( ( mem @ V1P @ bool )
=> ( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( ap @ c_2Ereal_2Eabs @ ( inj__ty_2Erealax_2Ereal @ V0ff ) ) ) ) @ ( ap @ ( c_2Ebinary__ieee_2Eulp @ A_27a @ A_27b ) @ ( c_2Ebool_2Ethe__value @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) ) ) ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( c_2Ebinary__ieee_2Efloat__to__real @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( ap @ ( c_2Ebinary__ieee_2Efloat__round @ A_27a @ A_27b ) @ ( inj__ty_2Ebinary__ieee_2Erounding @ fo__c_2Ebinary__ieee_2EroundTiesToEven ) ) @ V1P ) @ ( inj__ty_2Erealax_2Ereal @ V0ff ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) ) ).
%------------------------------------------------------------------------------