ITP001 Axioms: ITP140^5.ax


%------------------------------------------------------------------------------
% File     : ITP140^5 : TPTP v9.0.0. 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 ) ) ) ) ) ) ).

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