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 : 56 ( 9 unt; 4 typ; 0 def)
% Number of atoms : 2388 ( 33 equ)
% Maximal formula atoms : 10 ( 42 avg)
% Number of connectives : 225 ( 13 ~; 0 |; 89 &)
% ( 7 <=>; 116 =>; 0 <=; 0 <~>)
% Maximal formula depth : 15 ( 9 avg)
% Maximal term depth : 2 ( 1 avg)
% Number of FOOLs : 2124 (2124 fml; 0 var)
% Number of types : 1 ( 0 usr)
% Number of type conns : 8 ( 4 >; 4 *; 0 +; 0 <<)
% Number of predicates : 80 ( 79 usr; 30 prp; 0-2 aty)
% Number of functors : 4 ( 4 usr; 0 con; 2-2 aty)
% Number of variables : 201 ( 191 !; 10 ?; 201 :)
% SPC : TF0_SAT_EQU_NAR
% Comments :
% Bugfixes : v7.5.0 - Fixes to the axioms.
%------------------------------------------------------------------------------
tff(tp_c_2Elift__ieee_2Eerror,type,
c_2Elift__ieee_2Eerror: ( del * del ) > $i ).
tff(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))) ).
tff(tp_c_2Elift__ieee_2Enormalizes,type,
c_2Elift__ieee_2Enormalizes: ( del * del ) > $i ).
tff(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))) ).
tff(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))) ) ).
tff(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))))) ) ) ).
tff(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))) ) ) ) ) ).
tff(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))) ) ) ) ) ).
tff(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))) ) ) ) ) ).
tff(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))) ) ) ) ) ).
tff(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)) ) ) ) ) ) ).
tff(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)) ) ) ).
tff(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)) ) ) ) ) ).
tff(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)))) ) ) ) ) ) ) ).
tff(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)) ) ) ) ) ).
tff(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)))) ) ) ) ) ).
tff(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)))) ).
tff(lamtp_f3133,type,
f3133: ( del * del ) > $i ).
tff(lameq_f3133,axiom,
! [A_27a: del,A_27b: del,V0a: $i] : ( ap(f3133(A_27a,A_27b),V0a) = 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)) ) ).
tff(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)),f3133(A_27a,A_27b)))) ).
tff(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)),f3133(A_27a,A_27b)) != c_2Epred__set_2EEMPTY(ty_2Ebinary__ieee_2Efloat(A_27a,A_27b)) ) ).
tff(lamtp_f3134,type,
f3134: ( del * del ) > $i ).
tff(lameq_f3134,axiom,
! [A_27a: del,A_27b: del,V2a: $i] : ( ap(f3134(A_27a,A_27b),V2a) = 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)) ) ).
tff(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)),f3134(A_27a,A_27b))),inj__ty_2Erealax_2Ereal(V1x)))) ) ).
tff(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))))) ) ) ).
tff(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))))) ) ) ).
tff(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))))) ) ) ).
tff(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))) ) ) ) ).
tff(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)))) ) ) ) ).
tff(conj_thm_2Elift__ieee_2Efloat__round__finite,axiom,
! [A_27t: del,A_27w: del,V0b: tp__o,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)),inj__o(V0b)),inj__ty_2Erealax_2Ereal(V1x)))) ) ).
tff(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))))) ) ) ) ) ) ).
tff(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))))) ) ) ) ) ) ).
tff(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))))) ) ) ) ) ) ).
tff(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))))) ) ) ) ) ) ).
tff(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))))) ) ) ) ) ).
tff(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))))) ) ) ) ) ) ) ).
tff(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))))) ) ) ) ) ) ) ).
tff(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)))) ) ) ) ).
tff(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)))) ) ) ) ).
tff(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)))) ) ) ) ).
tff(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)))) ) ) ) ).
tff(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)))) ) ) ).
tff(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)))) ) ) ) ) ).
tff(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)))) ) ) ) ) ).
tff(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)))) ) ) ) ) ) ) ).
tff(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)))) ) ) ) ) ) ) ).
tff(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)))) ) ) ) ) ) ) ).
tff(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)))) ) ) ) ) ) ) ).
tff(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)))) ) ) ) ) ) ).
tff(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)))) ) ) ) ) ) ) ) ).
tff(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)))) ) ) ) ) ) ) ) ).
tff(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))))) ) ) ) ).
tff(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 ) ) ) ).
tff(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 ) ) ) ).
tff(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))) ) ).
tff(conj_thm_2Elift__ieee_2Enon__representable__float__is__zero,axiom,
! [A_27a: del,A_27b: del,V0ff: tp__ty_2Erealax_2Ereal,V1P: tp__o] :
( 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)),inj__o(V1P)),inj__ty_2Erealax_2Ereal(V0ff)))) = surj__ty_2Erealax_2Ereal(ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))) ) ) ).
%------------------------------------------------------------------------------