ITP001 Axioms: ITP126^7.ax
%------------------------------------------------------------------------------
% File : ITP126^7 : TPTP v8.2.0. Bugfixed v7.5.0.
% Domain : Interactive Theorem Proving
% Axioms : HOL4 syntactic export, chainy mode
% Version : [BG+19] axioms.
% English :
% Refs : [BG+19] Brown et al. (2019), GRUNGE: A Grand Unified ATP Chall
% : [Gau19] Gauthier (2019), Email to Geoff Sutcliffe
% Source : [BG+19]
% Names : rat.ax [Gau19]
% : HL4126^7.ax [TPAP]
% Status : Satisfiable
% Syntax : Number of formulae : 347 ( 129 unt; 88 typ; 0 def)
% Number of atoms : 670 ( 350 equ; 47 cnn)
% Maximal formula atoms : 12 ( 1 avg)
% Number of connectives : 2392 ( 47 ~; 15 |; 76 &;2114 @)
% ( 68 <=>; 72 =>; 0 <=; 0 <~>)
% Maximal formula depth : 13 ( 4 avg;2114 nst)
% Number of types : 6 ( 5 usr)
% Number of type conns : 156 ( 156 >; 0 *; 0 +; 0 <<)
% Number of symbols : 85 ( 83 usr; 9 con; 0-5 aty)
% Number of variables : 525 ( 13 ^ 488 !; 6 ?; 525 :)
% ( 18 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_SAT_EQU_NAR
% Comments :
% Bugfixes : v7.5.0 - Fixes to the axioms.
%------------------------------------------------------------------------------
thf(tyop_2Efrac_2Efrac,type,
tyop_2Efrac_2Efrac: $tType ).
thf(tyop_2Einteger_2Eint,type,
tyop_2Einteger_2Eint: $tType ).
thf(tyop_2Emin_2Ebool,type,
tyop_2Emin_2Ebool: $tType ).
thf(tyop_2Emin_2Efun,type,
tyop_2Emin_2Efun: $tType > $tType > $tType ).
thf(tyop_2Enum_2Enum,type,
tyop_2Enum_2Enum: $tType ).
thf(tyop_2Epair_2Eprod,type,
tyop_2Epair_2Eprod: $tType > $tType > $tType ).
thf(tyop_2Erat_2Erat,type,
tyop_2Erat_2Erat: $tType ).
thf(c_2Ebool_2E_21,type,
c_2Ebool_2E_21:
!>[A_27a: $tType] : ( ( A_27a > $o ) > $o ) ).
thf(c_2Earithmetic_2E_2A,type,
c_2Earithmetic_2E_2A: tyop_2Enum_2Enum > tyop_2Enum_2Enum > tyop_2Enum_2Enum ).
thf(c_2Earithmetic_2E_2B,type,
c_2Earithmetic_2E_2B: tyop_2Enum_2Enum > tyop_2Enum_2Enum > tyop_2Enum_2Enum ).
thf(c_2Epair_2E_2C,type,
c_2Epair_2E_2C:
!>[A_27a: $tType,A_27b: $tType] : ( A_27a > A_27b > ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) ) ).
thf(c_2Earithmetic_2E_2D,type,
c_2Earithmetic_2E_2D: tyop_2Enum_2Enum > tyop_2Enum_2Enum > tyop_2Enum_2Enum ).
thf(c_2Ebool_2E_2F_5C,type,
c_2Ebool_2E_2F_5C: $o > $o > $o ).
thf(c_2Enum_2E0,type,
c_2Enum_2E0: tyop_2Enum_2Enum ).
thf(c_2Eprim__rec_2E_3C,type,
c_2Eprim__rec_2E_3C: tyop_2Enum_2Enum > tyop_2Enum_2Enum > $o ).
thf(c_2Earithmetic_2E_3C_3D,type,
c_2Earithmetic_2E_3C_3D: tyop_2Enum_2Enum > tyop_2Enum_2Enum > $o ).
thf(c_2Emin_2E_3D,type,
c_2Emin_2E_3D:
!>[A_27a: $tType] : ( A_27a > A_27a > $o ) ).
thf(c_2Emin_2E_3D_3D_3E,type,
c_2Emin_2E_3D_3D_3E: $o > $o > $o ).
thf(c_2Ebool_2E_3F,type,
c_2Ebool_2E_3F:
!>[A_27a: $tType] : ( ( A_27a > $o ) > $o ) ).
thf(c_2Emin_2E_40,type,
c_2Emin_2E_40:
!>[A_27a: $tType] : ( ( A_27a > $o ) > A_27a ) ).
thf(c_2Einteger_2EABS,type,
c_2Einteger_2EABS: tyop_2Einteger_2Eint > tyop_2Einteger_2Eint ).
thf(c_2Earithmetic_2EBIT1,type,
c_2Earithmetic_2EBIT1: tyop_2Enum_2Enum > tyop_2Enum_2Enum ).
thf(c_2Earithmetic_2EBIT2,type,
c_2Earithmetic_2EBIT2: tyop_2Enum_2Enum > tyop_2Enum_2Enum ).
thf(c_2Ebool_2ECOND,type,
c_2Ebool_2ECOND:
!>[A_27a: $tType] : ( $o > A_27a > A_27a > A_27a ) ).
thf(c_2Ebool_2EF,type,
c_2Ebool_2EF: $o ).
thf(c_2Ecombin_2EI,type,
c_2Ecombin_2EI:
!>[A_27a: $tType] : ( A_27a > A_27a ) ).
thf(c_2Earithmetic_2ENUMERAL,type,
c_2Earithmetic_2ENUMERAL: tyop_2Enum_2Enum > tyop_2Enum_2Enum ).
thf(c_2Einteger_2ENum,type,
c_2Einteger_2ENum: tyop_2Einteger_2Eint > tyop_2Enum_2Enum ).
thf(c_2Ebool_2EONE__ONE,type,
c_2Ebool_2EONE__ONE:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > A_27b ) > $o ) ).
thf(c_2Equotient_2EQUOTIENT,type,
c_2Equotient_2EQUOTIENT:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > A_27a > $o ) > ( A_27a > A_27b ) > ( A_27b > A_27a ) > $o ) ).
thf(c_2Erat_2ERATD,type,
c_2Erat_2ERATD: tyop_2Erat_2Erat > tyop_2Enum_2Enum ).
thf(c_2Erat_2ERATN,type,
c_2Erat_2ERATN: tyop_2Erat_2Erat > tyop_2Einteger_2Eint ).
thf(c_2EintExtension_2ESGN,type,
c_2EintExtension_2ESGN: tyop_2Einteger_2Eint > tyop_2Einteger_2Eint ).
thf(c_2Enum_2ESUC,type,
c_2Enum_2ESUC: tyop_2Enum_2Enum > tyop_2Enum_2Enum ).
thf(c_2Ebool_2ET,type,
c_2Ebool_2ET: $o ).
thf(c_2Ebool_2ETYPE__DEFINITION,type,
c_2Ebool_2ETYPE__DEFINITION:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > $o ) > ( A_27b > A_27a ) > $o ) ).
thf(c_2Erelation_2EWF,type,
c_2Erelation_2EWF:
!>[A_27a: $tType] : ( ( A_27a > A_27a > $o ) > $o ) ).
thf(c_2Erelation_2EWFREC,type,
c_2Erelation_2EWFREC:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > A_27a > $o ) > ( ( A_27a > A_27b ) > A_27a > A_27b ) > A_27a > A_27b ) ).
thf(c_2Earithmetic_2EZERO,type,
c_2Earithmetic_2EZERO: tyop_2Enum_2Enum ).
thf(c_2Ebool_2E_5C_2F,type,
c_2Ebool_2E_5C_2F: $o > $o > $o ).
thf(c_2Efrac_2Eabs__frac,type,
c_2Efrac_2Eabs__frac: ( tyop_2Epair_2Eprod @ tyop_2Einteger_2Eint @ tyop_2Einteger_2Eint ) > tyop_2Efrac_2Efrac ).
thf(c_2Erat_2Eabs__rat,type,
c_2Erat_2Eabs__rat: tyop_2Efrac_2Efrac > tyop_2Erat_2Erat ).
thf(c_2Erat_2Eabs__rat__CLASS,type,
c_2Erat_2Eabs__rat__CLASS: ( tyop_2Efrac_2Efrac > $o ) > tyop_2Erat_2Erat ).
thf(c_2Efrac_2Efrac__0,type,
c_2Efrac_2Efrac__0: tyop_2Efrac_2Efrac ).
thf(c_2Efrac_2Efrac__1,type,
c_2Efrac_2Efrac__1: tyop_2Efrac_2Efrac ).
thf(c_2Efrac_2Efrac__add,type,
c_2Efrac_2Efrac__add: tyop_2Efrac_2Efrac > tyop_2Efrac_2Efrac > tyop_2Efrac_2Efrac ).
thf(c_2Efrac_2Efrac__ainv,type,
c_2Efrac_2Efrac__ainv: tyop_2Efrac_2Efrac > tyop_2Efrac_2Efrac ).
thf(c_2Efrac_2Efrac__div,type,
c_2Efrac_2Efrac__div: tyop_2Efrac_2Efrac > tyop_2Efrac_2Efrac > tyop_2Efrac_2Efrac ).
thf(c_2Efrac_2Efrac__dnm,type,
c_2Efrac_2Efrac__dnm: tyop_2Efrac_2Efrac > tyop_2Einteger_2Eint ).
thf(c_2Efrac_2Efrac__minv,type,
c_2Efrac_2Efrac__minv: tyop_2Efrac_2Efrac > tyop_2Efrac_2Efrac ).
thf(c_2Efrac_2Efrac__mul,type,
c_2Efrac_2Efrac__mul: tyop_2Efrac_2Efrac > tyop_2Efrac_2Efrac > tyop_2Efrac_2Efrac ).
thf(c_2Efrac_2Efrac__nmr,type,
c_2Efrac_2Efrac__nmr: tyop_2Efrac_2Efrac > tyop_2Einteger_2Eint ).
thf(c_2Efrac_2Efrac__save,type,
c_2Efrac_2Efrac__save: tyop_2Einteger_2Eint > tyop_2Enum_2Enum > tyop_2Efrac_2Efrac ).
thf(c_2Efrac_2Efrac__sgn,type,
c_2Efrac_2Efrac__sgn: tyop_2Efrac_2Efrac > tyop_2Einteger_2Eint ).
thf(c_2Efrac_2Efrac__sub,type,
c_2Efrac_2Efrac__sub: tyop_2Efrac_2Efrac > tyop_2Efrac_2Efrac > tyop_2Efrac_2Efrac ).
thf(c_2Einteger_2Eint__add,type,
c_2Einteger_2Eint__add: tyop_2Einteger_2Eint > tyop_2Einteger_2Eint > tyop_2Einteger_2Eint ).
thf(c_2Einteger_2Eint__gt,type,
c_2Einteger_2Eint__gt: tyop_2Einteger_2Eint > tyop_2Einteger_2Eint > $o ).
thf(c_2Einteger_2Eint__le,type,
c_2Einteger_2Eint__le: tyop_2Einteger_2Eint > tyop_2Einteger_2Eint > $o ).
thf(c_2Einteger_2Eint__lt,type,
c_2Einteger_2Eint__lt: tyop_2Einteger_2Eint > tyop_2Einteger_2Eint > $o ).
thf(c_2Einteger_2Eint__mul,type,
c_2Einteger_2Eint__mul: tyop_2Einteger_2Eint > tyop_2Einteger_2Eint > tyop_2Einteger_2Eint ).
thf(c_2Einteger_2Eint__neg,type,
c_2Einteger_2Eint__neg: tyop_2Einteger_2Eint > tyop_2Einteger_2Eint ).
thf(c_2Einteger_2Eint__of__num,type,
c_2Einteger_2Eint__of__num: tyop_2Enum_2Enum > tyop_2Einteger_2Eint ).
thf(c_2Einteger_2Eint__sub,type,
c_2Einteger_2Eint__sub: tyop_2Einteger_2Eint > tyop_2Einteger_2Eint > tyop_2Einteger_2Eint ).
thf(c_2Earithmetic_2Enum__CASE,type,
c_2Earithmetic_2Enum__CASE:
!>[A_27a: $tType] : ( tyop_2Enum_2Enum > A_27a > ( tyop_2Enum_2Enum > A_27a ) > A_27a ) ).
thf(c_2Erat_2Erat__0,type,
c_2Erat_2Erat__0: tyop_2Erat_2Erat ).
thf(c_2Erat_2Erat__1,type,
c_2Erat_2Erat__1: tyop_2Erat_2Erat ).
thf(c_2Erat_2Erat__add,type,
c_2Erat_2Erat__add: tyop_2Erat_2Erat > tyop_2Erat_2Erat > tyop_2Erat_2Erat ).
thf(c_2Erat_2Erat__ainv,type,
c_2Erat_2Erat__ainv: tyop_2Erat_2Erat > tyop_2Erat_2Erat ).
thf(c_2Erat_2Erat__cons,type,
c_2Erat_2Erat__cons: tyop_2Einteger_2Eint > tyop_2Einteger_2Eint > tyop_2Erat_2Erat ).
thf(c_2Erat_2Erat__div,type,
c_2Erat_2Erat__div: tyop_2Erat_2Erat > tyop_2Erat_2Erat > tyop_2Erat_2Erat ).
thf(c_2Erat_2Erat__dnm,type,
c_2Erat_2Erat__dnm: tyop_2Erat_2Erat > tyop_2Einteger_2Eint ).
thf(c_2Erat_2Erat__equiv,type,
c_2Erat_2Erat__equiv: tyop_2Efrac_2Efrac > tyop_2Efrac_2Efrac > $o ).
thf(c_2Erat_2Erat__geq,type,
c_2Erat_2Erat__geq: tyop_2Erat_2Erat > tyop_2Erat_2Erat > $o ).
thf(c_2Erat_2Erat__gre,type,
c_2Erat_2Erat__gre: tyop_2Erat_2Erat > tyop_2Erat_2Erat > $o ).
thf(c_2Erat_2Erat__leq,type,
c_2Erat_2Erat__leq: tyop_2Erat_2Erat > tyop_2Erat_2Erat > $o ).
thf(c_2Erat_2Erat__les,type,
c_2Erat_2Erat__les: tyop_2Erat_2Erat > tyop_2Erat_2Erat > $o ).
thf(c_2Erat_2Erat__max,type,
c_2Erat_2Erat__max: tyop_2Erat_2Erat > tyop_2Erat_2Erat > tyop_2Erat_2Erat ).
thf(c_2Erat_2Erat__min,type,
c_2Erat_2Erat__min: tyop_2Erat_2Erat > tyop_2Erat_2Erat > tyop_2Erat_2Erat ).
thf(c_2Erat_2Erat__minv,type,
c_2Erat_2Erat__minv: tyop_2Erat_2Erat > tyop_2Erat_2Erat ).
thf(c_2Erat_2Erat__mul,type,
c_2Erat_2Erat__mul: tyop_2Erat_2Erat > tyop_2Erat_2Erat > tyop_2Erat_2Erat ).
thf(c_2Erat_2Erat__nmr,type,
c_2Erat_2Erat__nmr: tyop_2Erat_2Erat > tyop_2Einteger_2Eint ).
thf(c_2Erat_2Erat__of__int,type,
c_2Erat_2Erat__of__int: tyop_2Einteger_2Eint > tyop_2Erat_2Erat ).
thf(c_2Erat_2Erat__of__num,type,
c_2Erat_2Erat__of__num: tyop_2Enum_2Enum > tyop_2Erat_2Erat ).
thf(c_2Erat_2Erat__sgn,type,
c_2Erat_2Erat__sgn: tyop_2Erat_2Erat > tyop_2Einteger_2Eint ).
thf(c_2Erat_2Erat__sub,type,
c_2Erat_2Erat__sub: tyop_2Erat_2Erat > tyop_2Erat_2Erat > tyop_2Erat_2Erat ).
thf(c_2Erat_2Erep__rat,type,
c_2Erat_2Erep__rat: tyop_2Erat_2Erat > tyop_2Efrac_2Efrac ).
thf(c_2Erat_2Erep__rat__CLASS,type,
c_2Erat_2Erep__rat__CLASS: tyop_2Erat_2Erat > tyop_2Efrac_2Efrac > $o ).
thf(c_2Ebool_2E_7E,type,
c_2Ebool_2E_7E: $o > $o ).
thf(logicdef_2E_2F_5C,axiom,
! [V0: $o,V1: $o] :
( ( c_2Ebool_2E_2F_5C @ V0 @ V1 )
<=> ( V0
& V1 ) ) ).
thf(logicdef_2E_5C_2F,axiom,
! [V0: $o,V1: $o] :
( ( c_2Ebool_2E_5C_2F @ V0 @ V1 )
<=> ( V0
| V1 ) ) ).
thf(logicdef_2E_7E,axiom,
! [V0: $o] :
( ( c_2Ebool_2E_7E @ V0 )
<=> ( (~) @ V0 ) ) ).
thf(logicdef_2E_3D_3D_3E,axiom,
! [V0: $o,V1: $o] :
( ( c_2Emin_2E_3D_3D_3E @ V0 @ V1 )
<=> ( V0
=> V1 ) ) ).
thf(logicdef_2E_3D,axiom,
! [A_27a: $tType,V0: A_27a,V1: A_27a] :
( ( c_2Emin_2E_3D @ A_27a @ V0 @ V1 )
<=> ( V0 = V1 ) ) ).
thf(quantdef_2E_21,axiom,
! [A_27a: $tType,V0f: A_27a > $o] :
( ( c_2Ebool_2E_21 @ A_27a @ V0f )
<=> ! [V1x: A_27a] : ( V0f @ V1x ) ) ).
thf(quantdef_2E_3F,axiom,
! [A_27a: $tType,V0f: A_27a > $o] :
( ( c_2Ebool_2E_3F @ A_27a @ V0f )
<=> ? [V1x: A_27a] : ( V0f @ V1x ) ) ).
thf(thm_2Erat_2Erat__equiv__def,axiom,
! [V0f1: tyop_2Efrac_2Efrac,V1f2: tyop_2Efrac_2Efrac] :
( ( c_2Erat_2Erat__equiv @ V0f1 @ V1f2 )
<=> ( ( c_2Einteger_2Eint__mul @ ( c_2Efrac_2Efrac__nmr @ V0f1 ) @ ( c_2Efrac_2Efrac__dnm @ V1f2 ) )
= ( c_2Einteger_2Eint__mul @ ( c_2Efrac_2Efrac__nmr @ V1f2 ) @ ( c_2Efrac_2Efrac__dnm @ V0f1 ) ) ) ) ).
thf(thm_2Erat_2Erat__TY__DEF,axiom,
? [V0rep: tyop_2Erat_2Erat > tyop_2Efrac_2Efrac > $o] :
( c_2Ebool_2ETYPE__DEFINITION @ ( tyop_2Efrac_2Efrac > $o ) @ tyop_2Erat_2Erat
@ ^ [V1c: tyop_2Efrac_2Efrac > $o] :
( c_2Ebool_2E_3F @ tyop_2Efrac_2Efrac
@ ^ [V2r: tyop_2Efrac_2Efrac] : ( c_2Ebool_2E_2F_5C @ ( c_2Erat_2Erat__equiv @ V2r @ V2r ) @ ( c_2Emin_2E_3D @ ( tyop_2Efrac_2Efrac > $o ) @ V1c @ ( c_2Erat_2Erat__equiv @ V2r ) ) ) )
@ V0rep ) ).
thf(thm_2Erat_2Erat__bijections,axiom,
( ! [V0a: tyop_2Erat_2Erat] :
( ( c_2Erat_2Eabs__rat__CLASS @ ( c_2Erat_2Erep__rat__CLASS @ V0a ) )
= V0a )
& ! [V1r: tyop_2Efrac_2Efrac > $o] :
( ( ^ [V2c: tyop_2Efrac_2Efrac > $o] :
( c_2Ebool_2E_3F @ tyop_2Efrac_2Efrac
@ ^ [V3r: tyop_2Efrac_2Efrac] : ( c_2Ebool_2E_2F_5C @ ( c_2Erat_2Erat__equiv @ V3r @ V3r ) @ ( c_2Emin_2E_3D @ ( tyop_2Efrac_2Efrac > $o ) @ V2c @ ( c_2Erat_2Erat__equiv @ V3r ) ) ) )
@ V1r )
<=> ( ( c_2Erat_2Erep__rat__CLASS @ ( c_2Erat_2Eabs__rat__CLASS @ V1r ) )
= V1r ) ) ) ).
thf(thm_2Erat_2Erep__rat__def,axiom,
! [V0a: tyop_2Erat_2Erat] :
( ( c_2Erat_2Erep__rat @ V0a )
= ( c_2Emin_2E_40 @ tyop_2Efrac_2Efrac @ ( c_2Erat_2Erep__rat__CLASS @ V0a ) ) ) ).
thf(thm_2Erat_2Eabs__rat__def,axiom,
! [V0r: tyop_2Efrac_2Efrac] :
( ( c_2Erat_2Eabs__rat @ V0r )
= ( c_2Erat_2Eabs__rat__CLASS @ ( c_2Erat_2Erat__equiv @ V0r ) ) ) ).
thf(thm_2Erat_2Erat__nmr__def,axiom,
! [V0r: tyop_2Erat_2Erat] :
( ( c_2Erat_2Erat__nmr @ V0r )
= ( c_2Efrac_2Efrac__nmr @ ( c_2Erat_2Erep__rat @ V0r ) ) ) ).
thf(thm_2Erat_2Erat__dnm__def,axiom,
! [V0r: tyop_2Erat_2Erat] :
( ( c_2Erat_2Erat__dnm @ V0r )
= ( c_2Efrac_2Efrac__dnm @ ( c_2Erat_2Erep__rat @ V0r ) ) ) ).
thf(thm_2Erat_2Erat__sgn__def,axiom,
! [V0r: tyop_2Erat_2Erat] :
( ( c_2Erat_2Erat__sgn @ V0r )
= ( c_2Efrac_2Efrac__sgn @ ( c_2Erat_2Erep__rat @ V0r ) ) ) ).
thf(thm_2Erat_2Erat__0__def,axiom,
( c_2Erat_2Erat__0
= ( c_2Erat_2Eabs__rat @ c_2Efrac_2Efrac__0 ) ) ).
thf(thm_2Erat_2Erat__1__def,axiom,
( c_2Erat_2Erat__1
= ( c_2Erat_2Eabs__rat @ c_2Efrac_2Efrac__1 ) ) ).
thf(thm_2Erat_2Erat__ainv__def,axiom,
! [V0r1: tyop_2Erat_2Erat] :
( ( c_2Erat_2Erat__ainv @ V0r1 )
= ( c_2Erat_2Eabs__rat @ ( c_2Efrac_2Efrac__ainv @ ( c_2Erat_2Erep__rat @ V0r1 ) ) ) ) ).
thf(thm_2Erat_2Erat__minv__def,axiom,
! [V0r1: tyop_2Erat_2Erat] :
( ( c_2Erat_2Erat__minv @ V0r1 )
= ( c_2Erat_2Eabs__rat @ ( c_2Efrac_2Efrac__minv @ ( c_2Erat_2Erep__rat @ V0r1 ) ) ) ) ).
thf(thm_2Erat_2Erat__add__def,axiom,
! [V0r1: tyop_2Erat_2Erat,V1r2: tyop_2Erat_2Erat] :
( ( c_2Erat_2Erat__add @ V0r1 @ V1r2 )
= ( c_2Erat_2Eabs__rat @ ( c_2Efrac_2Efrac__add @ ( c_2Erat_2Erep__rat @ V0r1 ) @ ( c_2Erat_2Erep__rat @ V1r2 ) ) ) ) ).
thf(thm_2Erat_2Erat__sub__def,axiom,
! [V0r1: tyop_2Erat_2Erat,V1r2: tyop_2Erat_2Erat] :
( ( c_2Erat_2Erat__sub @ V0r1 @ V1r2 )
= ( c_2Erat_2Eabs__rat @ ( c_2Efrac_2Efrac__sub @ ( c_2Erat_2Erep__rat @ V0r1 ) @ ( c_2Erat_2Erep__rat @ V1r2 ) ) ) ) ).
thf(thm_2Erat_2Erat__mul__def,axiom,
! [V0r1: tyop_2Erat_2Erat,V1r2: tyop_2Erat_2Erat] :
( ( c_2Erat_2Erat__mul @ V0r1 @ V1r2 )
= ( c_2Erat_2Eabs__rat @ ( c_2Efrac_2Efrac__mul @ ( c_2Erat_2Erep__rat @ V0r1 ) @ ( c_2Erat_2Erep__rat @ V1r2 ) ) ) ) ).
thf(thm_2Erat_2Erat__div__def,axiom,
! [V0r1: tyop_2Erat_2Erat,V1r2: tyop_2Erat_2Erat] :
( ( c_2Erat_2Erat__div @ V0r1 @ V1r2 )
= ( c_2Erat_2Eabs__rat @ ( c_2Efrac_2Efrac__div @ ( c_2Erat_2Erep__rat @ V0r1 ) @ ( c_2Erat_2Erep__rat @ V1r2 ) ) ) ) ).
thf(thm_2Erat_2Erat__les__def,axiom,
! [V0r1: tyop_2Erat_2Erat,V1r2: tyop_2Erat_2Erat] :
( ( c_2Erat_2Erat__les @ V0r1 @ V1r2 )
<=> ( ( c_2Erat_2Erat__sgn @ ( c_2Erat_2Erat__sub @ V1r2 @ V0r1 ) )
= ( c_2Einteger_2Eint__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ) ).
thf(thm_2Erat_2Erat__gre__def,axiom,
! [V0r1: tyop_2Erat_2Erat,V1r2: tyop_2Erat_2Erat] :
( ( c_2Erat_2Erat__gre @ V0r1 @ V1r2 )
= ( c_2Erat_2Erat__les @ V1r2 @ V0r1 ) ) ).
thf(thm_2Erat_2Erat__leq__def,axiom,
! [V0r1: tyop_2Erat_2Erat,V1r2: tyop_2Erat_2Erat] :
( ( c_2Erat_2Erat__leq @ V0r1 @ V1r2 )
<=> ( ( c_2Erat_2Erat__les @ V0r1 @ V1r2 )
| ( V0r1 = V1r2 ) ) ) ).
thf(thm_2Erat_2Erat__geq__def,axiom,
! [V0r1: tyop_2Erat_2Erat,V1r2: tyop_2Erat_2Erat] :
( ( c_2Erat_2Erat__geq @ V0r1 @ V1r2 )
= ( c_2Erat_2Erat__leq @ V1r2 @ V0r1 ) ) ).
thf(thm_2Erat_2Erat__cons__def,axiom,
! [V0nmr: tyop_2Einteger_2Eint,V1dnm: tyop_2Einteger_2Eint] :
( ( c_2Erat_2Erat__cons @ V0nmr @ V1dnm )
= ( c_2Erat_2Eabs__rat @ ( c_2Efrac_2Eabs__frac @ ( c_2Epair_2E_2C @ tyop_2Einteger_2Eint @ tyop_2Einteger_2Eint @ ( c_2Einteger_2Eint__mul @ ( c_2Einteger_2Eint__mul @ ( c_2EintExtension_2ESGN @ V0nmr ) @ ( c_2EintExtension_2ESGN @ V1dnm ) ) @ ( c_2Einteger_2EABS @ V0nmr ) ) @ ( c_2Einteger_2EABS @ V1dnm ) ) ) ) ) ).
thf(thm_2Erat_2Erat__of__num__primitive__def,axiom,
( c_2Erat_2Erat__of__num
= ( c_2Erelation_2EWFREC @ tyop_2Enum_2Enum @ tyop_2Erat_2Erat
@ ( c_2Emin_2E_40 @ ( tyop_2Enum_2Enum > tyop_2Enum_2Enum > $o )
@ ^ [V0R: tyop_2Enum_2Enum > tyop_2Enum_2Enum > $o] :
( c_2Ebool_2E_2F_5C @ ( c_2Erelation_2EWF @ tyop_2Enum_2Enum @ V0R )
@ ( c_2Ebool_2E_21 @ tyop_2Enum_2Enum
@ ^ [V1n: tyop_2Enum_2Enum] : ( V0R @ ( c_2Enum_2ESUC @ V1n ) @ ( c_2Enum_2ESUC @ ( c_2Enum_2ESUC @ V1n ) ) ) ) ) )
@ ^ [V2rat__of__num: tyop_2Enum_2Enum > tyop_2Erat_2Erat,V3a: tyop_2Enum_2Enum] :
( c_2Earithmetic_2Enum__CASE @ tyop_2Erat_2Erat @ V3a @ ( c_2Ecombin_2EI @ tyop_2Erat_2Erat @ c_2Erat_2Erat__0 )
@ ^ [V4v: tyop_2Enum_2Enum] :
( c_2Earithmetic_2Enum__CASE @ tyop_2Erat_2Erat @ V4v @ ( c_2Ecombin_2EI @ tyop_2Erat_2Erat @ c_2Erat_2Erat__1 )
@ ^ [V5n: tyop_2Enum_2Enum] : ( c_2Ecombin_2EI @ tyop_2Erat_2Erat @ ( c_2Erat_2Erat__add @ ( V2rat__of__num @ ( c_2Enum_2ESUC @ V5n ) ) @ c_2Erat_2Erat__1 ) ) ) ) ) ) ).
thf(thm_2Erat_2Erat__of__int__def,axiom,
! [V0i: tyop_2Einteger_2Eint] :
( ( c_2Erat_2Erat__of__int @ V0i )
= ( c_2Ebool_2ECOND @ tyop_2Erat_2Erat @ ( c_2Einteger_2Eint__lt @ V0i @ ( c_2Einteger_2Eint__of__num @ c_2Enum_2E0 ) ) @ ( c_2Erat_2Erat__ainv @ ( c_2Erat_2Erat__of__num @ ( c_2Einteger_2ENum @ ( c_2Einteger_2Eint__neg @ V0i ) ) ) ) @ ( c_2Erat_2Erat__of__num @ ( c_2Einteger_2ENum @ V0i ) ) ) ) ).
thf(thm_2Erat_2ERATND__THM,axiom,
! [V0r: tyop_2Erat_2Erat] :
( ( V0r
= ( c_2Erat_2Erat__div @ ( c_2Erat_2Erat__of__int @ ( c_2Erat_2ERATN @ V0r ) ) @ ( c_2Erat_2Erat__of__num @ ( c_2Erat_2ERATD @ V0r ) ) ) )
& ( c_2Eprim__rec_2E_3C @ c_2Enum_2E0 @ ( c_2Erat_2ERATD @ V0r ) )
& ( ( ( c_2Erat_2ERATN @ V0r )
= ( c_2Einteger_2Eint__of__num @ c_2Enum_2E0 ) )
=> ( ( c_2Erat_2ERATD @ V0r )
= ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) )
& ! [V1n_27: tyop_2Einteger_2Eint,V2d_27: tyop_2Enum_2Enum] :
( ( ( V0r
= ( c_2Erat_2Erat__div @ ( c_2Erat_2Erat__of__int @ V1n_27 ) @ ( c_2Erat_2Erat__of__num @ V2d_27 ) ) )
& ( c_2Eprim__rec_2E_3C @ c_2Enum_2E0 @ V2d_27 ) )
=> ( c_2Einteger_2Eint__le @ ( c_2Einteger_2EABS @ ( c_2Erat_2ERATN @ V0r ) ) @ ( c_2Einteger_2EABS @ V1n_27 ) ) ) ) ).
thf(thm_2Erat_2Erat__min__def,axiom,
! [V0r1: tyop_2Erat_2Erat,V1r2: tyop_2Erat_2Erat] :
( ( c_2Erat_2Erat__min @ V0r1 @ V1r2 )
= ( c_2Ebool_2ECOND @ tyop_2Erat_2Erat @ ( c_2Erat_2Erat__les @ V0r1 @ V1r2 ) @ V0r1 @ V1r2 ) ) ).
thf(thm_2Erat_2Erat__max__def,axiom,
! [V0r1: tyop_2Erat_2Erat,V1r2: tyop_2Erat_2Erat] :
( ( c_2Erat_2Erat__max @ V0r1 @ V1r2 )
= ( c_2Ebool_2ECOND @ tyop_2Erat_2Erat @ ( c_2Erat_2Erat__gre @ V0r1 @ V1r2 ) @ V0r1 @ V1r2 ) ) ).
thf(thm_2Erat_2ERAT__EQUIV__REF,axiom,
! [V0a: tyop_2Efrac_2Efrac] : ( c_2Erat_2Erat__equiv @ V0a @ V0a ) ).
thf(thm_2Erat_2ERAT__EQUIV__SYM,axiom,
! [V0a: tyop_2Efrac_2Efrac,V1b: tyop_2Efrac_2Efrac] :
( ( c_2Erat_2Erat__equiv @ V0a @ V1b )
= ( c_2Erat_2Erat__equiv @ V1b @ V0a ) ) ).
thf(thm_2Erat_2ERAT__EQUIV__NMR__Z__IFF,axiom,
! [V0a: tyop_2Efrac_2Efrac,V1b: tyop_2Efrac_2Efrac] :
( ( c_2Erat_2Erat__equiv @ V0a @ V1b )
=> ( ( ( c_2Efrac_2Efrac__nmr @ V0a )
= ( c_2Einteger_2Eint__of__num @ c_2Enum_2E0 ) )
<=> ( ( c_2Efrac_2Efrac__nmr @ V1b )
= ( c_2Einteger_2Eint__of__num @ c_2Enum_2E0 ) ) ) ) ).
thf(thm_2Erat_2ERAT__EQUIV__NMR__GTZ__IFF,axiom,
! [V0a: tyop_2Efrac_2Efrac,V1b: tyop_2Efrac_2Efrac] :
( ( c_2Erat_2Erat__equiv @ V0a @ V1b )
=> ( ( c_2Einteger_2Eint__gt @ ( c_2Efrac_2Efrac__nmr @ V0a ) @ ( c_2Einteger_2Eint__of__num @ c_2Enum_2E0 ) )
= ( c_2Einteger_2Eint__gt @ ( c_2Efrac_2Efrac__nmr @ V1b ) @ ( c_2Einteger_2Eint__of__num @ c_2Enum_2E0 ) ) ) ) ).
thf(thm_2Erat_2ERAT__EQUIV__NMR__LTZ__IFF,axiom,
! [V0a: tyop_2Efrac_2Efrac,V1b: tyop_2Efrac_2Efrac] :
( ( c_2Erat_2Erat__equiv @ V0a @ V1b )
=> ( ( c_2Einteger_2Eint__lt @ ( c_2Efrac_2Efrac__nmr @ V0a ) @ ( c_2Einteger_2Eint__of__num @ c_2Enum_2E0 ) )
= ( c_2Einteger_2Eint__lt @ ( c_2Efrac_2Efrac__nmr @ V1b ) @ ( c_2Einteger_2Eint__of__num @ c_2Enum_2E0 ) ) ) ) ).
thf(thm_2Erat_2ERAT__NMR__Z__IFF__EQUIV,axiom,
! [V0a: tyop_2Efrac_2Efrac,V1b: tyop_2Efrac_2Efrac] :
( ( ( c_2Efrac_2Efrac__nmr @ V0a )
= ( c_2Einteger_2Eint__of__num @ c_2Enum_2E0 ) )
=> ( ( c_2Erat_2Erat__equiv @ V0a @ V1b )
<=> ( ( c_2Efrac_2Efrac__nmr @ V1b )
= ( c_2Einteger_2Eint__of__num @ c_2Enum_2E0 ) ) ) ) ).
thf(thm_2Erat_2ERAT__EQUIV__TRANS,axiom,
! [V0a: tyop_2Efrac_2Efrac,V1b: tyop_2Efrac_2Efrac,V2c: tyop_2Efrac_2Efrac] :
( ( ( c_2Erat_2Erat__equiv @ V0a @ V1b )
& ( c_2Erat_2Erat__equiv @ V1b @ V2c ) )
=> ( c_2Erat_2Erat__equiv @ V0a @ V2c ) ) ).
thf(thm_2Erat_2ERAT__EQUIV,axiom,
! [V0f1: tyop_2Efrac_2Efrac,V1f2: tyop_2Efrac_2Efrac] :
( ( c_2Erat_2Erat__equiv @ V0f1 @ V1f2 )
<=> ( ( c_2Erat_2Erat__equiv @ V0f1 )
= ( c_2Erat_2Erat__equiv @ V1f2 ) ) ) ).
thf(thm_2Erat_2ERAT__EQUIV__ALT,axiom,
! [V0a: tyop_2Efrac_2Efrac] :
( ( c_2Erat_2Erat__equiv @ V0a )
= ( ^ [V1x: tyop_2Efrac_2Efrac] :
( c_2Ebool_2E_3F @ tyop_2Einteger_2Eint
@ ^ [V2b: tyop_2Einteger_2Eint] :
( c_2Ebool_2E_3F @ tyop_2Einteger_2Eint
@ ^ [V3c: tyop_2Einteger_2Eint] : ( c_2Ebool_2E_2F_5C @ ( c_2Einteger_2Eint__lt @ ( c_2Einteger_2Eint__of__num @ c_2Enum_2E0 ) @ V2b ) @ ( c_2Ebool_2E_2F_5C @ ( c_2Einteger_2Eint__lt @ ( c_2Einteger_2Eint__of__num @ c_2Enum_2E0 ) @ V3c ) @ ( c_2Emin_2E_3D @ tyop_2Efrac_2Efrac @ ( c_2Efrac_2Efrac__mul @ V0a @ ( c_2Efrac_2Eabs__frac @ ( c_2Epair_2E_2C @ tyop_2Einteger_2Eint @ tyop_2Einteger_2Eint @ V2b @ V2b ) ) ) @ ( c_2Efrac_2Efrac__mul @ V1x @ ( c_2Efrac_2Eabs__frac @ ( c_2Epair_2E_2C @ tyop_2Einteger_2Eint @ tyop_2Einteger_2Eint @ V3c @ V3c ) ) ) ) ) ) ) ) ) ) ).
thf(thm_2Erat_2Erat__ABS__REP__CLASS,axiom,
( ! [V0a: tyop_2Erat_2Erat] :
( ( c_2Erat_2Eabs__rat__CLASS @ ( c_2Erat_2Erep__rat__CLASS @ V0a ) )
= V0a )
& ! [V1c: tyop_2Efrac_2Efrac > $o] :
( ? [V2r: tyop_2Efrac_2Efrac] :
( ( c_2Erat_2Erat__equiv @ V2r @ V2r )
& ( V1c
= ( c_2Erat_2Erat__equiv @ V2r ) ) )
<=> ( ( c_2Erat_2Erep__rat__CLASS @ ( c_2Erat_2Eabs__rat__CLASS @ V1c ) )
= V1c ) ) ) ).
thf(thm_2Erat_2Erat__QUOTIENT,axiom,
c_2Equotient_2EQUOTIENT @ tyop_2Efrac_2Efrac @ tyop_2Erat_2Erat @ c_2Erat_2Erat__equiv @ c_2Erat_2Eabs__rat @ c_2Erat_2Erep__rat ).
thf(thm_2Erat_2Erat__def,axiom,
c_2Equotient_2EQUOTIENT @ tyop_2Efrac_2Efrac @ tyop_2Erat_2Erat @ c_2Erat_2Erat__equiv @ c_2Erat_2Eabs__rat @ c_2Erat_2Erep__rat ).
thf(thm_2Erat_2Erat__type__thm,axiom,
( ! [V0a: tyop_2Erat_2Erat] :
( ( c_2Erat_2Eabs__rat @ ( c_2Erat_2Erep__rat @ V0a ) )
= V0a )
& ! [V1r: tyop_2Efrac_2Efrac,V2s: tyop_2Efrac_2Efrac] :
( ( c_2Erat_2Erat__equiv @ V1r @ V2s )
<=> ( ( c_2Erat_2Eabs__rat @ V1r )
= ( c_2Erat_2Eabs__rat @ V2s ) ) ) ) ).
thf(thm_2Erat_2Erat__equiv__reps,axiom,
! [V0r2: tyop_2Erat_2Erat,V1r1: tyop_2Erat_2Erat] :
( ( c_2Erat_2Erat__equiv @ ( c_2Erat_2Erep__rat @ V1r1 ) @ ( c_2Erat_2Erep__rat @ V0r2 ) )
<=> ( V1r1 = V0r2 ) ) ).
thf(thm_2Erat_2Erat__equiv__rep__abs,axiom,
! [V0f: tyop_2Efrac_2Efrac] : ( c_2Erat_2Erat__equiv @ ( c_2Erat_2Erep__rat @ ( c_2Erat_2Eabs__rat @ V0f ) ) @ V0f ) ).
thf(thm_2Erat_2Erat__of__num__ind,axiom,
! [V0P: tyop_2Enum_2Enum > $o] :
( ( ( V0P @ c_2Enum_2E0 )
& ( V0P @ ( c_2Enum_2ESUC @ c_2Enum_2E0 ) )
& ! [V1n: tyop_2Enum_2Enum] :
( ( V0P @ ( c_2Enum_2ESUC @ V1n ) )
=> ( V0P @ ( c_2Enum_2ESUC @ ( c_2Enum_2ESUC @ V1n ) ) ) ) )
=> ! [V2v: tyop_2Enum_2Enum] : ( V0P @ V2v ) ) ).
thf(thm_2Erat_2Erat__of__num__def,axiom,
( ( ( c_2Erat_2Erat__of__num @ c_2Enum_2E0 )
= c_2Erat_2Erat__0 )
& ( ( c_2Erat_2Erat__of__num @ ( c_2Enum_2ESUC @ c_2Enum_2E0 ) )
= c_2Erat_2Erat__1 )
& ! [V0n: tyop_2Enum_2Enum] :
( ( c_2Erat_2Erat__of__num @ ( c_2Enum_2ESUC @ ( c_2Enum_2ESUC @ V0n ) ) )
= ( c_2Erat_2Erat__add @ ( c_2Erat_2Erat__of__num @ ( c_2Enum_2ESUC @ V0n ) ) @ c_2Erat_2Erat__1 ) ) ) ).
thf(thm_2Erat_2Erat__of__num__def__compute,axiom,
( ( ( c_2Erat_2Erat__of__num @ c_2Enum_2E0 )
= c_2Erat_2Erat__0 )
& ( ( c_2Erat_2Erat__of__num @ ( c_2Enum_2ESUC @ c_2Enum_2E0 ) )
= c_2Erat_2Erat__1 )
& ! [V0n: tyop_2Enum_2Enum] :
( ( c_2Erat_2Erat__of__num @ ( c_2Enum_2ESUC @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ V0n ) ) ) )
= ( c_2Erat_2Erat__add @ ( c_2Erat_2Erat__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ V0n ) ) ) @ c_2Erat_2Erat__1 ) )
& ! [V1n: tyop_2Enum_2Enum] :
( ( c_2Erat_2Erat__of__num @ ( c_2Enum_2ESUC @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ V1n ) ) ) )
= ( c_2Erat_2Erat__add @ ( c_2Erat_2Erat__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ V1n ) ) ) @ c_2Erat_2Erat__1 ) ) ) ).
thf(thm_2Erat_2Erat__0,axiom,
( ( c_2Erat_2Erat__of__num @ c_2Enum_2E0 )
= ( c_2Erat_2Eabs__rat @ c_2Efrac_2Efrac__0 ) ) ).
thf(thm_2Erat_2Erat__1,axiom,
( ( c_2Erat_2Erat__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) )
= ( c_2Erat_2Eabs__rat @ c_2Efrac_2Efrac__1 ) ) ).
thf(thm_2Erat_2ERAT,axiom,
! [V0r: tyop_2Erat_2Erat] :
( ( c_2Erat_2Eabs__rat @ ( c_2Erat_2Erep__rat @ V0r ) )
= V0r ) ).
thf(thm_2Erat_2ERAT__ABS__EQUIV,axiom,
! [V0f1: tyop_2Efrac_2Efrac,V1f2: tyop_2Efrac_2Efrac] :
( ( ( c_2Erat_2Eabs__rat @ V0f1 )
= ( c_2Erat_2Eabs__rat @ V1f2 ) )
<=> ( c_2Erat_2Erat__equiv @ V0f1 @ V1f2 ) ) ).
thf(thm_2Erat_2ERAT__EQ,axiom,
! [V0f1: tyop_2Efrac_2Efrac,V1f2: tyop_2Efrac_2Efrac] :
( ( ( c_2Erat_2Eabs__rat @ V0f1 )
= ( c_2Erat_2Eabs__rat @ V1f2 ) )
<=> ( ( c_2Einteger_2Eint__mul @ ( c_2Efrac_2Efrac__nmr @ V0f1 ) @ ( c_2Efrac_2Efrac__dnm @ V1f2 ) )
= ( c_2Einteger_2Eint__mul @ ( c_2Efrac_2Efrac__nmr @ V1f2 ) @ ( c_2Efrac_2Efrac__dnm @ V0f1 ) ) ) ) ).
thf(thm_2Erat_2ERAT__EQ__ALT,axiom,
! [V0r1: tyop_2Erat_2Erat,V1r2: tyop_2Erat_2Erat] :
( ( V0r1 = V1r2 )
<=> ( ( c_2Einteger_2Eint__mul @ ( c_2Erat_2Erat__nmr @ V0r1 ) @ ( c_2Erat_2Erat__dnm @ V1r2 ) )
= ( c_2Einteger_2Eint__mul @ ( c_2Erat_2Erat__nmr @ V1r2 ) @ ( c_2Erat_2Erat__dnm @ V0r1 ) ) ) ) ).
thf(thm_2Erat_2ERAT__NMREQ0__CONG,axiom,
! [V0f1: tyop_2Efrac_2Efrac] :
( ( ( c_2Efrac_2Efrac__nmr @ ( c_2Erat_2Erep__rat @ ( c_2Erat_2Eabs__rat @ V0f1 ) ) )
= ( c_2Einteger_2Eint__of__num @ c_2Enum_2E0 ) )
<=> ( ( c_2Efrac_2Efrac__nmr @ V0f1 )
= ( c_2Einteger_2Eint__of__num @ c_2Enum_2E0 ) ) ) ).
thf(thm_2Erat_2ERAT__NMRLT0__CONG,axiom,
! [V0f1: tyop_2Efrac_2Efrac] :
( ( c_2Einteger_2Eint__lt @ ( c_2Efrac_2Efrac__nmr @ ( c_2Erat_2Erep__rat @ ( c_2Erat_2Eabs__rat @ V0f1 ) ) ) @ ( c_2Einteger_2Eint__of__num @ c_2Enum_2E0 ) )
= ( c_2Einteger_2Eint__lt @ ( c_2Efrac_2Efrac__nmr @ V0f1 ) @ ( c_2Einteger_2Eint__of__num @ c_2Enum_2E0 ) ) ) ).
thf(thm_2Erat_2ERAT__NMRGT0__CONG,axiom,
! [V0f1: tyop_2Efrac_2Efrac] :
( ( c_2Einteger_2Eint__gt @ ( c_2Efrac_2Efrac__nmr @ ( c_2Erat_2Erep__rat @ ( c_2Erat_2Eabs__rat @ V0f1 ) ) ) @ ( c_2Einteger_2Eint__of__num @ c_2Enum_2E0 ) )
= ( c_2Einteger_2Eint__gt @ ( c_2Efrac_2Efrac__nmr @ V0f1 ) @ ( c_2Einteger_2Eint__of__num @ c_2Enum_2E0 ) ) ) ).
thf(thm_2Erat_2ERAT__SGN__CONG,axiom,
! [V0f1: tyop_2Efrac_2Efrac] :
( ( c_2Efrac_2Efrac__sgn @ ( c_2Erat_2Erep__rat @ ( c_2Erat_2Eabs__rat @ V0f1 ) ) )
= ( c_2Efrac_2Efrac__sgn @ V0f1 ) ) ).
thf(thm_2Erat_2ERAT__AINV__CONG,axiom,
! [V0x: tyop_2Efrac_2Efrac] :
( ( c_2Erat_2Eabs__rat @ ( c_2Efrac_2Efrac__ainv @ ( c_2Erat_2Erep__rat @ ( c_2Erat_2Eabs__rat @ V0x ) ) ) )
= ( c_2Erat_2Eabs__rat @ ( c_2Efrac_2Efrac__ainv @ V0x ) ) ) ).
thf(thm_2Erat_2EFRAC__MINV__EQUIV,axiom,
! [V0y: tyop_2Efrac_2Efrac,V1x: tyop_2Efrac_2Efrac] :
( ( (~)
@ ( ( c_2Efrac_2Efrac__nmr @ V0y )
= ( c_2Einteger_2Eint__of__num @ c_2Enum_2E0 ) ) )
=> ( ( c_2Erat_2Erat__equiv @ V1x @ V0y )
=> ( c_2Erat_2Erat__equiv @ ( c_2Efrac_2Efrac__minv @ V1x ) @ ( c_2Efrac_2Efrac__minv @ V0y ) ) ) ) ).
thf(thm_2Erat_2ERAT__MINV__CONG,axiom,
! [V0x: tyop_2Efrac_2Efrac] :
( ( (~)
@ ( ( c_2Efrac_2Efrac__nmr @ V0x )
= ( c_2Einteger_2Eint__of__num @ c_2Enum_2E0 ) ) )
=> ( ( c_2Erat_2Eabs__rat @ ( c_2Efrac_2Efrac__minv @ ( c_2Erat_2Erep__rat @ ( c_2Erat_2Eabs__rat @ V0x ) ) ) )
= ( c_2Erat_2Eabs__rat @ ( c_2Efrac_2Efrac__minv @ V0x ) ) ) ) ).
thf(thm_2Erat_2EFRAC__ADD__EQUIV1,axiom,
! [V0y: tyop_2Efrac_2Efrac,V1x_27: tyop_2Efrac_2Efrac,V2x: tyop_2Efrac_2Efrac] :
( ( c_2Erat_2Erat__equiv @ V2x @ V1x_27 )
=> ( c_2Erat_2Erat__equiv @ ( c_2Efrac_2Efrac__add @ V2x @ V0y ) @ ( c_2Efrac_2Efrac__add @ V1x_27 @ V0y ) ) ) ).
thf(thm_2Erat_2ERAT__ADD__CONG1,axiom,
! [V0x: tyop_2Efrac_2Efrac,V1y: tyop_2Efrac_2Efrac] :
( ( c_2Erat_2Eabs__rat @ ( c_2Efrac_2Efrac__add @ ( c_2Erat_2Erep__rat @ ( c_2Erat_2Eabs__rat @ V0x ) ) @ V1y ) )
= ( c_2Erat_2Eabs__rat @ ( c_2Efrac_2Efrac__add @ V0x @ V1y ) ) ) ).
thf(thm_2Erat_2ERAT__ADD__CONG2,axiom,
! [V0x: tyop_2Efrac_2Efrac,V1y: tyop_2Efrac_2Efrac] :
( ( c_2Erat_2Eabs__rat @ ( c_2Efrac_2Efrac__add @ V0x @ ( c_2Erat_2Erep__rat @ ( c_2Erat_2Eabs__rat @ V1y ) ) ) )
= ( c_2Erat_2Eabs__rat @ ( c_2Efrac_2Efrac__add @ V0x @ V1y ) ) ) ).
thf(thm_2Erat_2ERAT__ADD__CONG,axiom,
( ! [V0x: tyop_2Efrac_2Efrac,V1y: tyop_2Efrac_2Efrac] :
( ( c_2Erat_2Eabs__rat @ ( c_2Efrac_2Efrac__add @ ( c_2Erat_2Erep__rat @ ( c_2Erat_2Eabs__rat @ V0x ) ) @ V1y ) )
= ( c_2Erat_2Eabs__rat @ ( c_2Efrac_2Efrac__add @ V0x @ V1y ) ) )
& ! [V2x: tyop_2Efrac_2Efrac,V3y: tyop_2Efrac_2Efrac] :
( ( c_2Erat_2Eabs__rat @ ( c_2Efrac_2Efrac__add @ V2x @ ( c_2Erat_2Erep__rat @ ( c_2Erat_2Eabs__rat @ V3y ) ) ) )
= ( c_2Erat_2Eabs__rat @ ( c_2Efrac_2Efrac__add @ V2x @ V3y ) ) ) ) ).
thf(thm_2Erat_2EFRAC__MUL__EQUIV1,axiom,
! [V0y: tyop_2Efrac_2Efrac,V1x_27: tyop_2Efrac_2Efrac,V2x: tyop_2Efrac_2Efrac] :
( ( c_2Erat_2Erat__equiv @ V2x @ V1x_27 )
=> ( c_2Erat_2Erat__equiv @ ( c_2Efrac_2Efrac__mul @ V2x @ V0y ) @ ( c_2Efrac_2Efrac__mul @ V1x_27 @ V0y ) ) ) ).
thf(thm_2Erat_2EFRAC__MUL__EQUIV2,axiom,
! [V0y: tyop_2Efrac_2Efrac,V1x_27: tyop_2Efrac_2Efrac,V2x: tyop_2Efrac_2Efrac] :
( ( c_2Erat_2Erat__equiv @ V2x @ V1x_27 )
=> ( c_2Erat_2Erat__equiv @ ( c_2Efrac_2Efrac__mul @ V0y @ V2x ) @ ( c_2Efrac_2Efrac__mul @ V0y @ V1x_27 ) ) ) ).
thf(thm_2Erat_2ERAT__MUL__CONG1,axiom,
! [V0x: tyop_2Efrac_2Efrac,V1y: tyop_2Efrac_2Efrac] :
( ( c_2Erat_2Eabs__rat @ ( c_2Efrac_2Efrac__mul @ ( c_2Erat_2Erep__rat @ ( c_2Erat_2Eabs__rat @ V0x ) ) @ V1y ) )
= ( c_2Erat_2Eabs__rat @ ( c_2Efrac_2Efrac__mul @ V0x @ V1y ) ) ) ).
thf(thm_2Erat_2ERAT__MUL__CONG2,axiom,
! [V0x: tyop_2Efrac_2Efrac,V1y: tyop_2Efrac_2Efrac] :
( ( c_2Erat_2Eabs__rat @ ( c_2Efrac_2Efrac__mul @ V0x @ ( c_2Erat_2Erep__rat @ ( c_2Erat_2Eabs__rat @ V1y ) ) ) )
= ( c_2Erat_2Eabs__rat @ ( c_2Efrac_2Efrac__mul @ V0x @ V1y ) ) ) ).
thf(thm_2Erat_2ERAT__MUL__CONG,axiom,
( ! [V0x: tyop_2Efrac_2Efrac,V1y: tyop_2Efrac_2Efrac] :
( ( c_2Erat_2Eabs__rat @ ( c_2Efrac_2Efrac__mul @ ( c_2Erat_2Erep__rat @ ( c_2Erat_2Eabs__rat @ V0x ) ) @ V1y ) )
= ( c_2Erat_2Eabs__rat @ ( c_2Efrac_2Efrac__mul @ V0x @ V1y ) ) )
& ! [V2x: tyop_2Efrac_2Efrac,V3y: tyop_2Efrac_2Efrac] :
( ( c_2Erat_2Eabs__rat @ ( c_2Efrac_2Efrac__mul @ V2x @ ( c_2Erat_2Erep__rat @ ( c_2Erat_2Eabs__rat @ V3y ) ) ) )
= ( c_2Erat_2Eabs__rat @ ( c_2Efrac_2Efrac__mul @ V2x @ V3y ) ) ) ) ).
thf(thm_2Erat_2ERAT__SUB__CONG1,axiom,
! [V0x: tyop_2Efrac_2Efrac,V1y: tyop_2Efrac_2Efrac] :
( ( c_2Erat_2Eabs__rat @ ( c_2Efrac_2Efrac__sub @ ( c_2Erat_2Erep__rat @ ( c_2Erat_2Eabs__rat @ V0x ) ) @ V1y ) )
= ( c_2Erat_2Eabs__rat @ ( c_2Efrac_2Efrac__sub @ V0x @ V1y ) ) ) ).
thf(thm_2Erat_2ERAT__SUB__CONG2,axiom,
! [V0x: tyop_2Efrac_2Efrac,V1y: tyop_2Efrac_2Efrac] :
( ( c_2Erat_2Eabs__rat @ ( c_2Efrac_2Efrac__sub @ V0x @ ( c_2Erat_2Erep__rat @ ( c_2Erat_2Eabs__rat @ V1y ) ) ) )
= ( c_2Erat_2Eabs__rat @ ( c_2Efrac_2Efrac__sub @ V0x @ V1y ) ) ) ).
thf(thm_2Erat_2ERAT__SUB__CONG,axiom,
( ! [V0x: tyop_2Efrac_2Efrac,V1y: tyop_2Efrac_2Efrac] :
( ( c_2Erat_2Eabs__rat @ ( c_2Efrac_2Efrac__sub @ ( c_2Erat_2Erep__rat @ ( c_2Erat_2Eabs__rat @ V0x ) ) @ V1y ) )
= ( c_2Erat_2Eabs__rat @ ( c_2Efrac_2Efrac__sub @ V0x @ V1y ) ) )
& ! [V2x: tyop_2Efrac_2Efrac,V3y: tyop_2Efrac_2Efrac] :
( ( c_2Erat_2Eabs__rat @ ( c_2Efrac_2Efrac__sub @ V2x @ ( c_2Erat_2Erep__rat @ ( c_2Erat_2Eabs__rat @ V3y ) ) ) )
= ( c_2Erat_2Eabs__rat @ ( c_2Efrac_2Efrac__sub @ V2x @ V3y ) ) ) ) ).
thf(thm_2Erat_2ERAT__DIV__CONG1,axiom,
! [V0x: tyop_2Efrac_2Efrac,V1y: tyop_2Efrac_2Efrac] :
( ( (~)
@ ( ( c_2Efrac_2Efrac__nmr @ V1y )
= ( c_2Einteger_2Eint__of__num @ c_2Enum_2E0 ) ) )
=> ( ( c_2Erat_2Eabs__rat @ ( c_2Efrac_2Efrac__div @ ( c_2Erat_2Erep__rat @ ( c_2Erat_2Eabs__rat @ V0x ) ) @ V1y ) )
= ( c_2Erat_2Eabs__rat @ ( c_2Efrac_2Efrac__div @ V0x @ V1y ) ) ) ) ).
thf(thm_2Erat_2ERAT__DIV__CONG2,axiom,
! [V0x: tyop_2Efrac_2Efrac,V1y: tyop_2Efrac_2Efrac] :
( ( (~)
@ ( ( c_2Efrac_2Efrac__nmr @ V1y )
= ( c_2Einteger_2Eint__of__num @ c_2Enum_2E0 ) ) )
=> ( ( c_2Erat_2Eabs__rat @ ( c_2Efrac_2Efrac__div @ V0x @ ( c_2Erat_2Erep__rat @ ( c_2Erat_2Eabs__rat @ V1y ) ) ) )
= ( c_2Erat_2Eabs__rat @ ( c_2Efrac_2Efrac__div @ V0x @ V1y ) ) ) ) ).
thf(thm_2Erat_2ERAT__DIV__CONG,axiom,
( ! [V0x: tyop_2Efrac_2Efrac,V1y: tyop_2Efrac_2Efrac] :
( ( (~)
@ ( ( c_2Efrac_2Efrac__nmr @ V1y )
= ( c_2Einteger_2Eint__of__num @ c_2Enum_2E0 ) ) )
=> ( ( c_2Erat_2Eabs__rat @ ( c_2Efrac_2Efrac__div @ ( c_2Erat_2Erep__rat @ ( c_2Erat_2Eabs__rat @ V0x ) ) @ V1y ) )
= ( c_2Erat_2Eabs__rat @ ( c_2Efrac_2Efrac__div @ V0x @ V1y ) ) ) )
& ! [V2x: tyop_2Efrac_2Efrac,V3y: tyop_2Efrac_2Efrac] :
( ( (~)
@ ( ( c_2Efrac_2Efrac__nmr @ V3y )
= ( c_2Einteger_2Eint__of__num @ c_2Enum_2E0 ) ) )
=> ( ( c_2Erat_2Eabs__rat @ ( c_2Efrac_2Efrac__div @ V2x @ ( c_2Erat_2Erep__rat @ ( c_2Erat_2Eabs__rat @ V3y ) ) ) )
= ( c_2Erat_2Eabs__rat @ ( c_2Efrac_2Efrac__div @ V2x @ V3y ) ) ) ) ) ).
thf(thm_2Erat_2ERAT__NMRDNM__EQ,axiom,
! [V0f1: tyop_2Efrac_2Efrac] :
( ( ( c_2Erat_2Eabs__rat @ ( c_2Efrac_2Eabs__frac @ ( c_2Epair_2E_2C @ tyop_2Einteger_2Eint @ tyop_2Einteger_2Eint @ ( c_2Efrac_2Efrac__nmr @ V0f1 ) @ ( c_2Efrac_2Efrac__dnm @ V0f1 ) ) ) )
= ( c_2Erat_2Erat__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) )
<=> ( ( c_2Efrac_2Efrac__nmr @ V0f1 )
= ( c_2Efrac_2Efrac__dnm @ V0f1 ) ) ) ).
thf(thm_2Erat_2ERAT__AINV__CALCULATE,axiom,
! [V0f1: tyop_2Efrac_2Efrac] :
( ( c_2Erat_2Erat__ainv @ ( c_2Erat_2Eabs__rat @ V0f1 ) )
= ( c_2Erat_2Eabs__rat @ ( c_2Efrac_2Efrac__ainv @ V0f1 ) ) ) ).
thf(thm_2Erat_2ERAT__MINV__CALCULATE,axiom,
! [V0f1: tyop_2Efrac_2Efrac] :
( ( (~)
@ ( ( c_2Einteger_2Eint__of__num @ c_2Enum_2E0 )
= ( c_2Efrac_2Efrac__nmr @ V0f1 ) ) )
=> ( ( c_2Erat_2Erat__minv @ ( c_2Erat_2Eabs__rat @ V0f1 ) )
= ( c_2Erat_2Eabs__rat @ ( c_2Efrac_2Efrac__minv @ V0f1 ) ) ) ) ).
thf(thm_2Erat_2ERAT__ADD__CALCULATE,axiom,
! [V0f1: tyop_2Efrac_2Efrac,V1f2: tyop_2Efrac_2Efrac] :
( ( c_2Erat_2Erat__add @ ( c_2Erat_2Eabs__rat @ V0f1 ) @ ( c_2Erat_2Eabs__rat @ V1f2 ) )
= ( c_2Erat_2Eabs__rat @ ( c_2Efrac_2Efrac__add @ V0f1 @ V1f2 ) ) ) ).
thf(thm_2Erat_2ERAT__SUB__CALCULATE,axiom,
! [V0f1: tyop_2Efrac_2Efrac,V1f2: tyop_2Efrac_2Efrac] :
( ( c_2Erat_2Erat__sub @ ( c_2Erat_2Eabs__rat @ V0f1 ) @ ( c_2Erat_2Eabs__rat @ V1f2 ) )
= ( c_2Erat_2Eabs__rat @ ( c_2Efrac_2Efrac__sub @ V0f1 @ V1f2 ) ) ) ).
thf(thm_2Erat_2ERAT__MUL__CALCULATE,axiom,
! [V0f1: tyop_2Efrac_2Efrac,V1f2: tyop_2Efrac_2Efrac] :
( ( c_2Erat_2Erat__mul @ ( c_2Erat_2Eabs__rat @ V0f1 ) @ ( c_2Erat_2Eabs__rat @ V1f2 ) )
= ( c_2Erat_2Eabs__rat @ ( c_2Efrac_2Efrac__mul @ V0f1 @ V1f2 ) ) ) ).
thf(thm_2Erat_2ERAT__DIV__CALCULATE,axiom,
! [V0f1: tyop_2Efrac_2Efrac,V1f2: tyop_2Efrac_2Efrac] :
( ( (~)
@ ( ( c_2Efrac_2Efrac__nmr @ V1f2 )
= ( c_2Einteger_2Eint__of__num @ c_2Enum_2E0 ) ) )
=> ( ( c_2Erat_2Erat__div @ ( c_2Erat_2Eabs__rat @ V0f1 ) @ ( c_2Erat_2Eabs__rat @ V1f2 ) )
= ( c_2Erat_2Eabs__rat @ ( c_2Efrac_2Efrac__div @ V0f1 @ V1f2 ) ) ) ) ).
thf(thm_2Erat_2ERAT__EQ__CALCULATE,axiom,
! [V0f1: tyop_2Efrac_2Efrac,V1f2: tyop_2Efrac_2Efrac] :
( ( ( c_2Erat_2Eabs__rat @ V0f1 )
= ( c_2Erat_2Eabs__rat @ V1f2 ) )
<=> ( ( c_2Einteger_2Eint__mul @ ( c_2Efrac_2Efrac__nmr @ V0f1 ) @ ( c_2Efrac_2Efrac__dnm @ V1f2 ) )
= ( c_2Einteger_2Eint__mul @ ( c_2Efrac_2Efrac__nmr @ V1f2 ) @ ( c_2Efrac_2Efrac__dnm @ V0f1 ) ) ) ) ).
thf(thm_2Erat_2ERAT__LES__CALCULATE,axiom,
! [V0f1: tyop_2Efrac_2Efrac,V1f2: tyop_2Efrac_2Efrac] :
( ( c_2Erat_2Erat__les @ ( c_2Erat_2Eabs__rat @ V0f1 ) @ ( c_2Erat_2Eabs__rat @ V1f2 ) )
= ( c_2Einteger_2Eint__lt @ ( c_2Einteger_2Eint__mul @ ( c_2Efrac_2Efrac__nmr @ V0f1 ) @ ( c_2Efrac_2Efrac__dnm @ V1f2 ) ) @ ( c_2Einteger_2Eint__mul @ ( c_2Efrac_2Efrac__nmr @ V1f2 ) @ ( c_2Efrac_2Efrac__dnm @ V0f1 ) ) ) ) ).
thf(thm_2Erat_2ERAT__LEQ__CALCULATE,axiom,
! [V0f1: tyop_2Efrac_2Efrac,V1f2: tyop_2Efrac_2Efrac] :
( ( c_2Erat_2Erat__leq @ ( c_2Erat_2Eabs__rat @ V0f1 ) @ ( c_2Erat_2Eabs__rat @ V1f2 ) )
= ( c_2Einteger_2Eint__le @ ( c_2Einteger_2Eint__mul @ ( c_2Efrac_2Efrac__nmr @ V0f1 ) @ ( c_2Efrac_2Efrac__dnm @ V1f2 ) ) @ ( c_2Einteger_2Eint__mul @ ( c_2Efrac_2Efrac__nmr @ V1f2 ) @ ( c_2Efrac_2Efrac__dnm @ V0f1 ) ) ) ) ).
thf(thm_2Erat_2ERAT__OF__NUM__CALCULATE,axiom,
! [V0n1: tyop_2Enum_2Enum] :
( ( c_2Erat_2Erat__of__num @ V0n1 )
= ( c_2Erat_2Eabs__rat @ ( c_2Efrac_2Eabs__frac @ ( c_2Epair_2E_2C @ tyop_2Einteger_2Eint @ tyop_2Einteger_2Eint @ ( c_2Einteger_2Eint__of__num @ V0n1 ) @ ( c_2Einteger_2Eint__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ) ) ) ).
thf(thm_2Erat_2ERAT__OF__NUM__LEQ,axiom,
! [V0b: tyop_2Enum_2Enum,V1a: tyop_2Enum_2Enum] :
( ( c_2Erat_2Erat__leq @ ( c_2Erat_2Erat__of__num @ V1a ) @ ( c_2Erat_2Erat__of__num @ V0b ) )
= ( c_2Earithmetic_2E_3C_3D @ V1a @ V0b ) ) ).
thf(thm_2Erat_2ERAT__OF__NUM__LES,axiom,
! [V0b: tyop_2Enum_2Enum,V1a: tyop_2Enum_2Enum] :
( ( c_2Erat_2Erat__les @ ( c_2Erat_2Erat__of__num @ V1a ) @ ( c_2Erat_2Erat__of__num @ V0b ) )
= ( c_2Eprim__rec_2E_3C @ V1a @ V0b ) ) ).
thf(thm_2Erat_2ERAT__EQ0__NMR,axiom,
! [V0r1: tyop_2Erat_2Erat] :
( ( V0r1
= ( c_2Erat_2Erat__of__num @ c_2Enum_2E0 ) )
<=> ( ( c_2Erat_2Erat__nmr @ V0r1 )
= ( c_2Einteger_2Eint__of__num @ c_2Enum_2E0 ) ) ) ).
thf(thm_2Erat_2ERAT__0LES__NMR,axiom,
! [V0r1: tyop_2Erat_2Erat] :
( ( c_2Erat_2Erat__les @ ( c_2Erat_2Erat__of__num @ c_2Enum_2E0 ) @ V0r1 )
= ( c_2Einteger_2Eint__lt @ ( c_2Einteger_2Eint__of__num @ c_2Enum_2E0 ) @ ( c_2Erat_2Erat__nmr @ V0r1 ) ) ) ).
thf(thm_2Erat_2ERAT__LES0__NMR,axiom,
! [V0r1: tyop_2Erat_2Erat] :
( ( c_2Erat_2Erat__les @ V0r1 @ ( c_2Erat_2Erat__of__num @ c_2Enum_2E0 ) )
= ( c_2Einteger_2Eint__lt @ ( c_2Erat_2Erat__nmr @ V0r1 ) @ ( c_2Einteger_2Eint__of__num @ c_2Enum_2E0 ) ) ) ).
thf(thm_2Erat_2ERAT__0LEQ__NMR,axiom,
! [V0r1: tyop_2Erat_2Erat] :
( ( c_2Erat_2Erat__leq @ ( c_2Erat_2Erat__of__num @ c_2Enum_2E0 ) @ V0r1 )
= ( c_2Einteger_2Eint__le @ ( c_2Einteger_2Eint__of__num @ c_2Enum_2E0 ) @ ( c_2Erat_2Erat__nmr @ V0r1 ) ) ) ).
thf(thm_2Erat_2ERAT__LEQ0__NMR,axiom,
! [V0r1: tyop_2Erat_2Erat] :
( ( c_2Erat_2Erat__leq @ V0r1 @ ( c_2Erat_2Erat__of__num @ c_2Enum_2E0 ) )
= ( c_2Einteger_2Eint__le @ ( c_2Erat_2Erat__nmr @ V0r1 ) @ ( c_2Einteger_2Eint__of__num @ c_2Enum_2E0 ) ) ) ).
thf(thm_2Erat_2ERAT__ADD__ASSOC,axiom,
! [V0a: tyop_2Erat_2Erat,V1b: tyop_2Erat_2Erat,V2c: tyop_2Erat_2Erat] :
( ( c_2Erat_2Erat__add @ V0a @ ( c_2Erat_2Erat__add @ V1b @ V2c ) )
= ( c_2Erat_2Erat__add @ ( c_2Erat_2Erat__add @ V0a @ V1b ) @ V2c ) ) ).
thf(thm_2Erat_2ERAT__MUL__ASSOC,axiom,
! [V0a: tyop_2Erat_2Erat,V1b: tyop_2Erat_2Erat,V2c: tyop_2Erat_2Erat] :
( ( c_2Erat_2Erat__mul @ V0a @ ( c_2Erat_2Erat__mul @ V1b @ V2c ) )
= ( c_2Erat_2Erat__mul @ ( c_2Erat_2Erat__mul @ V0a @ V1b ) @ V2c ) ) ).
thf(thm_2Erat_2ERAT__ADD__COMM,axiom,
! [V0a: tyop_2Erat_2Erat,V1b: tyop_2Erat_2Erat] :
( ( c_2Erat_2Erat__add @ V0a @ V1b )
= ( c_2Erat_2Erat__add @ V1b @ V0a ) ) ).
thf(thm_2Erat_2ERAT__MUL__COMM,axiom,
! [V0a: tyop_2Erat_2Erat,V1b: tyop_2Erat_2Erat] :
( ( c_2Erat_2Erat__mul @ V0a @ V1b )
= ( c_2Erat_2Erat__mul @ V1b @ V0a ) ) ).
thf(thm_2Erat_2ERAT__ADD__RID,axiom,
! [V0a: tyop_2Erat_2Erat] :
( ( c_2Erat_2Erat__add @ V0a @ ( c_2Erat_2Erat__of__num @ c_2Enum_2E0 ) )
= V0a ) ).
thf(thm_2Erat_2ERAT__ADD__LID,axiom,
! [V0a: tyop_2Erat_2Erat] :
( ( c_2Erat_2Erat__add @ ( c_2Erat_2Erat__of__num @ c_2Enum_2E0 ) @ V0a )
= V0a ) ).
thf(thm_2Erat_2ERAT__MUL__RID,axiom,
! [V0a: tyop_2Erat_2Erat] :
( ( c_2Erat_2Erat__mul @ V0a @ ( c_2Erat_2Erat__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) )
= V0a ) ).
thf(thm_2Erat_2ERAT__MUL__LID,axiom,
! [V0a: tyop_2Erat_2Erat] :
( ( c_2Erat_2Erat__mul @ ( c_2Erat_2Erat__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) @ V0a )
= V0a ) ).
thf(thm_2Erat_2ERAT__ADD__RINV,axiom,
! [V0a: tyop_2Erat_2Erat] :
( ( c_2Erat_2Erat__add @ V0a @ ( c_2Erat_2Erat__ainv @ V0a ) )
= ( c_2Erat_2Erat__of__num @ c_2Enum_2E0 ) ) ).
thf(thm_2Erat_2ERAT__ADD__LINV,axiom,
! [V0a: tyop_2Erat_2Erat] :
( ( c_2Erat_2Erat__add @ ( c_2Erat_2Erat__ainv @ V0a ) @ V0a )
= ( c_2Erat_2Erat__of__num @ c_2Enum_2E0 ) ) ).
thf(thm_2Erat_2ERAT__MUL__RINV,axiom,
! [V0a: tyop_2Erat_2Erat] :
( ( (~)
@ ( V0a
= ( c_2Erat_2Erat__of__num @ c_2Enum_2E0 ) ) )
=> ( ( c_2Erat_2Erat__mul @ V0a @ ( c_2Erat_2Erat__minv @ V0a ) )
= ( c_2Erat_2Erat__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ) ).
thf(thm_2Erat_2ERAT__MUL__LINV,axiom,
! [V0a: tyop_2Erat_2Erat] :
( ( (~)
@ ( V0a
= ( c_2Erat_2Erat__of__num @ c_2Enum_2E0 ) ) )
=> ( ( c_2Erat_2Erat__mul @ ( c_2Erat_2Erat__minv @ V0a ) @ V0a )
= ( c_2Erat_2Erat__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ) ).
thf(thm_2Erat_2ERAT__RDISTRIB,axiom,
! [V0a: tyop_2Erat_2Erat,V1b: tyop_2Erat_2Erat,V2c: tyop_2Erat_2Erat] :
( ( c_2Erat_2Erat__mul @ ( c_2Erat_2Erat__add @ V0a @ V1b ) @ V2c )
= ( c_2Erat_2Erat__add @ ( c_2Erat_2Erat__mul @ V0a @ V2c ) @ ( c_2Erat_2Erat__mul @ V1b @ V2c ) ) ) ).
thf(thm_2Erat_2ERAT__LDISTRIB,axiom,
! [V0a: tyop_2Erat_2Erat,V1b: tyop_2Erat_2Erat,V2c: tyop_2Erat_2Erat] :
( ( c_2Erat_2Erat__mul @ V2c @ ( c_2Erat_2Erat__add @ V0a @ V1b ) )
= ( c_2Erat_2Erat__add @ ( c_2Erat_2Erat__mul @ V2c @ V0a ) @ ( c_2Erat_2Erat__mul @ V2c @ V1b ) ) ) ).
thf(thm_2Erat_2ERAT__1__NOT__0,axiom,
( (~)
@ ( ( c_2Erat_2Erat__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) )
= ( c_2Erat_2Erat__of__num @ c_2Enum_2E0 ) ) ) ).
thf(thm_2Erat_2ERAT__MUL__LZERO,axiom,
! [V0r1: tyop_2Erat_2Erat] :
( ( c_2Erat_2Erat__mul @ ( c_2Erat_2Erat__of__num @ c_2Enum_2E0 ) @ V0r1 )
= ( c_2Erat_2Erat__of__num @ c_2Enum_2E0 ) ) ).
thf(thm_2Erat_2ERAT__MUL__RZERO,axiom,
! [V0r1: tyop_2Erat_2Erat] :
( ( c_2Erat_2Erat__mul @ V0r1 @ ( c_2Erat_2Erat__of__num @ c_2Enum_2E0 ) )
= ( c_2Erat_2Erat__of__num @ c_2Enum_2E0 ) ) ).
thf(thm_2Erat_2ERAT__SUB__ADDAINV,axiom,
! [V0r1: tyop_2Erat_2Erat,V1r2: tyop_2Erat_2Erat] :
( ( c_2Erat_2Erat__sub @ V0r1 @ V1r2 )
= ( c_2Erat_2Erat__add @ V0r1 @ ( c_2Erat_2Erat__ainv @ V1r2 ) ) ) ).
thf(thm_2Erat_2ERAT__DIV__MULMINV,axiom,
! [V0r1: tyop_2Erat_2Erat,V1r2: tyop_2Erat_2Erat] :
( ( c_2Erat_2Erat__div @ V0r1 @ V1r2 )
= ( c_2Erat_2Erat__mul @ V0r1 @ ( c_2Erat_2Erat__minv @ V1r2 ) ) ) ).
thf(thm_2Erat_2ERAT__DIV__0,axiom,
! [V0x: tyop_2Erat_2Erat] :
( ( c_2Erat_2Erat__div @ ( c_2Erat_2Erat__of__num @ c_2Enum_2E0 ) @ V0x )
= ( c_2Erat_2Erat__of__num @ c_2Enum_2E0 ) ) ).
thf(thm_2Erat_2ERAT__AINV__0,axiom,
( ( c_2Erat_2Erat__ainv @ ( c_2Erat_2Erat__of__num @ c_2Enum_2E0 ) )
= ( c_2Erat_2Erat__of__num @ c_2Enum_2E0 ) ) ).
thf(thm_2Erat_2ERAT__AINV__AINV,axiom,
! [V0r1: tyop_2Erat_2Erat] :
( ( c_2Erat_2Erat__ainv @ ( c_2Erat_2Erat__ainv @ V0r1 ) )
= V0r1 ) ).
thf(thm_2Erat_2ERAT__AINV__ADD,axiom,
! [V0r1: tyop_2Erat_2Erat,V1r2: tyop_2Erat_2Erat] :
( ( c_2Erat_2Erat__ainv @ ( c_2Erat_2Erat__add @ V0r1 @ V1r2 ) )
= ( c_2Erat_2Erat__add @ ( c_2Erat_2Erat__ainv @ V0r1 ) @ ( c_2Erat_2Erat__ainv @ V1r2 ) ) ) ).
thf(thm_2Erat_2ERAT__AINV__SUB,axiom,
! [V0r1: tyop_2Erat_2Erat,V1r2: tyop_2Erat_2Erat] :
( ( c_2Erat_2Erat__ainv @ ( c_2Erat_2Erat__sub @ V0r1 @ V1r2 ) )
= ( c_2Erat_2Erat__sub @ V1r2 @ V0r1 ) ) ).
thf(thm_2Erat_2ERAT__AINV__RMUL,axiom,
! [V0r1: tyop_2Erat_2Erat,V1r2: tyop_2Erat_2Erat] :
( ( c_2Erat_2Erat__ainv @ ( c_2Erat_2Erat__mul @ V0r1 @ V1r2 ) )
= ( c_2Erat_2Erat__mul @ V0r1 @ ( c_2Erat_2Erat__ainv @ V1r2 ) ) ) ).
thf(thm_2Erat_2ERAT__AINV__LMUL,axiom,
! [V0r1: tyop_2Erat_2Erat,V1r2: tyop_2Erat_2Erat] :
( ( c_2Erat_2Erat__ainv @ ( c_2Erat_2Erat__mul @ V0r1 @ V1r2 ) )
= ( c_2Erat_2Erat__mul @ ( c_2Erat_2Erat__ainv @ V0r1 ) @ V1r2 ) ) ).
thf(thm_2Erat_2ERAT__AINV__EQ,axiom,
! [V0r1: tyop_2Erat_2Erat,V1r2: tyop_2Erat_2Erat] :
( ( ( c_2Erat_2Erat__ainv @ V0r1 )
= V1r2 )
<=> ( V0r1
= ( c_2Erat_2Erat__ainv @ V1r2 ) ) ) ).
thf(thm_2Erat_2ERAT__EQ__AINV,axiom,
! [V0r1: tyop_2Erat_2Erat,V1r2: tyop_2Erat_2Erat] :
( ( ( c_2Erat_2Erat__ainv @ V0r1 )
= ( c_2Erat_2Erat__ainv @ V1r2 ) )
<=> ( V0r1 = V1r2 ) ) ).
thf(thm_2Erat_2ERAT__AINV__MINV,axiom,
! [V0r1: tyop_2Erat_2Erat] :
( ( (~)
@ ( V0r1
= ( c_2Erat_2Erat__of__num @ c_2Enum_2E0 ) ) )
=> ( ( c_2Erat_2Erat__ainv @ ( c_2Erat_2Erat__minv @ V0r1 ) )
= ( c_2Erat_2Erat__minv @ ( c_2Erat_2Erat__ainv @ V0r1 ) ) ) ) ).
thf(thm_2Erat_2ERAT__SUB__RDISTRIB,axiom,
! [V0a: tyop_2Erat_2Erat,V1b: tyop_2Erat_2Erat,V2c: tyop_2Erat_2Erat] :
( ( c_2Erat_2Erat__mul @ ( c_2Erat_2Erat__sub @ V0a @ V1b ) @ V2c )
= ( c_2Erat_2Erat__sub @ ( c_2Erat_2Erat__mul @ V0a @ V2c ) @ ( c_2Erat_2Erat__mul @ V1b @ V2c ) ) ) ).
thf(thm_2Erat_2ERAT__SUB__LDISTRIB,axiom,
! [V0a: tyop_2Erat_2Erat,V1b: tyop_2Erat_2Erat,V2c: tyop_2Erat_2Erat] :
( ( c_2Erat_2Erat__mul @ V2c @ ( c_2Erat_2Erat__sub @ V0a @ V1b ) )
= ( c_2Erat_2Erat__sub @ ( c_2Erat_2Erat__mul @ V2c @ V0a ) @ ( c_2Erat_2Erat__mul @ V2c @ V1b ) ) ) ).
thf(thm_2Erat_2ERAT__SUB__LID,axiom,
! [V0r1: tyop_2Erat_2Erat] :
( ( c_2Erat_2Erat__sub @ ( c_2Erat_2Erat__of__num @ c_2Enum_2E0 ) @ V0r1 )
= ( c_2Erat_2Erat__ainv @ V0r1 ) ) ).
thf(thm_2Erat_2ERAT__SUB__RID,axiom,
! [V0r1: tyop_2Erat_2Erat] :
( ( c_2Erat_2Erat__sub @ V0r1 @ ( c_2Erat_2Erat__of__num @ c_2Enum_2E0 ) )
= V0r1 ) ).
thf(thm_2Erat_2ERAT__SUB__ID,axiom,
! [V0r: tyop_2Erat_2Erat] :
( ( c_2Erat_2Erat__sub @ V0r @ V0r )
= ( c_2Erat_2Erat__of__num @ c_2Enum_2E0 ) ) ).
thf(thm_2Erat_2ERAT__EQ__SUB0,axiom,
! [V0r1: tyop_2Erat_2Erat,V1r2: tyop_2Erat_2Erat] :
( ( ( c_2Erat_2Erat__sub @ V0r1 @ V1r2 )
= ( c_2Erat_2Erat__of__num @ c_2Enum_2E0 ) )
<=> ( V0r1 = V1r2 ) ) ).
thf(thm_2Erat_2ERAT__EQ__0SUB,axiom,
! [V0r1: tyop_2Erat_2Erat,V1r2: tyop_2Erat_2Erat] :
( ( ( c_2Erat_2Erat__of__num @ c_2Enum_2E0 )
= ( c_2Erat_2Erat__sub @ V0r1 @ V1r2 ) )
<=> ( V0r1 = V1r2 ) ) ).
thf(thm_2Erat_2ERAT__SGN__CALCULATE,axiom,
! [V0f1: tyop_2Efrac_2Efrac] :
( ( c_2Erat_2Erat__sgn @ ( c_2Erat_2Eabs__rat @ V0f1 ) )
= ( c_2Efrac_2Efrac__sgn @ V0f1 ) ) ).
thf(thm_2Erat_2ERAT__SGN__CLAUSES,axiom,
! [V0r1: tyop_2Erat_2Erat] :
( ( ( ( c_2Erat_2Erat__sgn @ V0r1 )
= ( c_2Einteger_2Eint__neg @ ( c_2Einteger_2Eint__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) )
<=> ( c_2Erat_2Erat__les @ V0r1 @ ( c_2Erat_2Erat__of__num @ c_2Enum_2E0 ) ) )
& ( ( ( c_2Erat_2Erat__sgn @ V0r1 )
= ( c_2Einteger_2Eint__of__num @ c_2Enum_2E0 ) )
<=> ( V0r1
= ( c_2Erat_2Erat__of__num @ c_2Enum_2E0 ) ) )
& ( ( ( c_2Erat_2Erat__sgn @ V0r1 )
= ( c_2Einteger_2Eint__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) )
<=> ( c_2Erat_2Erat__gre @ V0r1 @ ( c_2Erat_2Erat__of__num @ c_2Enum_2E0 ) ) ) ) ).
thf(thm_2Erat_2ERAT__SGN__0,axiom,
( ( c_2Erat_2Erat__sgn @ ( c_2Erat_2Erat__of__num @ c_2Enum_2E0 ) )
= ( c_2Einteger_2Eint__of__num @ c_2Enum_2E0 ) ) ).
thf(thm_2Erat_2ERAT__SGN__AINV,axiom,
! [V0r1: tyop_2Erat_2Erat] :
( ( c_2Einteger_2Eint__neg @ ( c_2Erat_2Erat__sgn @ ( c_2Erat_2Erat__ainv @ V0r1 ) ) )
= ( c_2Erat_2Erat__sgn @ V0r1 ) ) ).
thf(thm_2Erat_2ERAT__SGN__MUL,axiom,
! [V0r1: tyop_2Erat_2Erat,V1r2: tyop_2Erat_2Erat] :
( ( c_2Erat_2Erat__sgn @ ( c_2Erat_2Erat__mul @ V0r1 @ V1r2 ) )
= ( c_2Einteger_2Eint__mul @ ( c_2Erat_2Erat__sgn @ V0r1 ) @ ( c_2Erat_2Erat__sgn @ V1r2 ) ) ) ).
thf(thm_2Erat_2ERAT__SGN__MINV,axiom,
! [V0r1: tyop_2Erat_2Erat] :
( ( (~)
@ ( V0r1
= ( c_2Erat_2Erat__of__num @ c_2Enum_2E0 ) ) )
=> ( ( c_2Erat_2Erat__sgn @ ( c_2Erat_2Erat__minv @ V0r1 ) )
= ( c_2Erat_2Erat__sgn @ V0r1 ) ) ) ).
thf(thm_2Erat_2ERAT__SGN__TOTAL,axiom,
! [V0r1: tyop_2Erat_2Erat] :
( ( ( c_2Erat_2Erat__sgn @ V0r1 )
= ( c_2Einteger_2Eint__neg @ ( c_2Einteger_2Eint__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) )
| ( ( c_2Erat_2Erat__sgn @ V0r1 )
= ( c_2Einteger_2Eint__of__num @ c_2Enum_2E0 ) )
| ( ( c_2Erat_2Erat__sgn @ V0r1 )
= ( c_2Einteger_2Eint__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ) ).
thf(thm_2Erat_2ERAT__SGN__COMPLEMENT,axiom,
! [V0r1: tyop_2Erat_2Erat] :
( ( ( (~)
@ ( ( c_2Erat_2Erat__sgn @ V0r1 )
= ( c_2Einteger_2Eint__neg @ ( c_2Einteger_2Eint__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ) )
<=> ( ( ( c_2Erat_2Erat__sgn @ V0r1 )
= ( c_2Einteger_2Eint__of__num @ c_2Enum_2E0 ) )
| ( ( c_2Erat_2Erat__sgn @ V0r1 )
= ( c_2Einteger_2Eint__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ) )
& ( ( (~)
@ ( ( c_2Erat_2Erat__sgn @ V0r1 )
= ( c_2Einteger_2Eint__of__num @ c_2Enum_2E0 ) ) )
<=> ( ( ( c_2Erat_2Erat__sgn @ V0r1 )
= ( c_2Einteger_2Eint__neg @ ( c_2Einteger_2Eint__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) )
| ( ( c_2Erat_2Erat__sgn @ V0r1 )
= ( c_2Einteger_2Eint__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ) )
& ( ( (~)
@ ( ( c_2Erat_2Erat__sgn @ V0r1 )
= ( c_2Einteger_2Eint__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) )
<=> ( ( ( c_2Erat_2Erat__sgn @ V0r1 )
= ( c_2Einteger_2Eint__neg @ ( c_2Einteger_2Eint__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) )
| ( ( c_2Erat_2Erat__sgn @ V0r1 )
= ( c_2Einteger_2Eint__of__num @ c_2Enum_2E0 ) ) ) ) ) ).
thf(thm_2Erat_2ERAT__LES__REF,axiom,
! [V0r1: tyop_2Erat_2Erat] : ( (~) @ ( c_2Erat_2Erat__les @ V0r1 @ V0r1 ) ) ).
thf(thm_2Erat_2ERAT__LES__ANTISYM,axiom,
! [V0r1: tyop_2Erat_2Erat,V1r2: tyop_2Erat_2Erat] :
( ( c_2Erat_2Erat__les @ V0r1 @ V1r2 )
=> ( (~) @ ( c_2Erat_2Erat__les @ V1r2 @ V0r1 ) ) ) ).
thf(thm_2Erat_2ERAT__LES__TRANS,axiom,
! [V0r1: tyop_2Erat_2Erat,V1r2: tyop_2Erat_2Erat,V2r3: tyop_2Erat_2Erat] :
( ( ( c_2Erat_2Erat__les @ V0r1 @ V1r2 )
& ( c_2Erat_2Erat__les @ V1r2 @ V2r3 ) )
=> ( c_2Erat_2Erat__les @ V0r1 @ V2r3 ) ) ).
thf(thm_2Erat_2ERAT__LES__TOTAL,axiom,
! [V0r1: tyop_2Erat_2Erat,V1r2: tyop_2Erat_2Erat] :
( ( c_2Erat_2Erat__les @ V0r1 @ V1r2 )
| ( V0r1 = V1r2 )
| ( c_2Erat_2Erat__les @ V1r2 @ V0r1 ) ) ).
thf(thm_2Erat_2ERAT__LEQ__REF,axiom,
! [V0r1: tyop_2Erat_2Erat] : ( c_2Erat_2Erat__leq @ V0r1 @ V0r1 ) ).
thf(thm_2Erat_2ERAT__LEQ__ANTISYM,axiom,
! [V0r1: tyop_2Erat_2Erat,V1r2: tyop_2Erat_2Erat] :
( ( ( c_2Erat_2Erat__leq @ V0r1 @ V1r2 )
& ( c_2Erat_2Erat__leq @ V1r2 @ V0r1 ) )
=> ( V0r1 = V1r2 ) ) ).
thf(thm_2Erat_2ERAT__LEQ__TRANS,axiom,
! [V0r1: tyop_2Erat_2Erat,V1r2: tyop_2Erat_2Erat,V2r3: tyop_2Erat_2Erat] :
( ( ( c_2Erat_2Erat__leq @ V0r1 @ V1r2 )
& ( c_2Erat_2Erat__leq @ V1r2 @ V2r3 ) )
=> ( c_2Erat_2Erat__leq @ V0r1 @ V2r3 ) ) ).
thf(thm_2Erat_2ERAT__LES__01,axiom,
c_2Erat_2Erat__les @ ( c_2Erat_2Erat__of__num @ c_2Enum_2E0 ) @ ( c_2Erat_2Erat__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ).
thf(thm_2Erat_2ERAT__LES__IMP__LEQ,axiom,
! [V0r1: tyop_2Erat_2Erat,V1r2: tyop_2Erat_2Erat] :
( ( c_2Erat_2Erat__les @ V0r1 @ V1r2 )
=> ( c_2Erat_2Erat__leq @ V0r1 @ V1r2 ) ) ).
thf(thm_2Erat_2ERAT__LES__IMP__NEQ,axiom,
! [V0r1: tyop_2Erat_2Erat,V1r2: tyop_2Erat_2Erat] :
( ( c_2Erat_2Erat__les @ V0r1 @ V1r2 )
=> ( (~) @ ( V0r1 = V1r2 ) ) ) ).
thf(thm_2Erat_2ERAT__LEQ__LES,axiom,
! [V0r1: tyop_2Erat_2Erat,V1r2: tyop_2Erat_2Erat] :
( ( (~) @ ( c_2Erat_2Erat__les @ V1r2 @ V0r1 ) )
<=> ( c_2Erat_2Erat__leq @ V0r1 @ V1r2 ) ) ).
thf(thm_2Erat_2ERAT__LES__LEQ,axiom,
! [V0r1: tyop_2Erat_2Erat,V1r2: tyop_2Erat_2Erat] :
( ( (~) @ ( c_2Erat_2Erat__leq @ V1r2 @ V0r1 ) )
<=> ( c_2Erat_2Erat__les @ V0r1 @ V1r2 ) ) ).
thf(thm_2Erat_2ERAT__LES__LEQ2,axiom,
! [V0r1: tyop_2Erat_2Erat,V1r2: tyop_2Erat_2Erat] :
( ( c_2Erat_2Erat__les @ V0r1 @ V1r2 )
<=> ( ( c_2Erat_2Erat__leq @ V0r1 @ V1r2 )
& ( (~) @ ( c_2Erat_2Erat__leq @ V1r2 @ V0r1 ) ) ) ) ).
thf(thm_2Erat_2ERAT__LES__LEQ__TRANS,axiom,
! [V0a: tyop_2Erat_2Erat,V1b: tyop_2Erat_2Erat,V2c: tyop_2Erat_2Erat] :
( ( ( c_2Erat_2Erat__les @ V0a @ V1b )
& ( c_2Erat_2Erat__leq @ V1b @ V2c ) )
=> ( c_2Erat_2Erat__les @ V0a @ V2c ) ) ).
thf(thm_2Erat_2ERAT__LEQ__LES__TRANS,axiom,
! [V0a: tyop_2Erat_2Erat,V1b: tyop_2Erat_2Erat,V2c: tyop_2Erat_2Erat] :
( ( ( c_2Erat_2Erat__leq @ V0a @ V1b )
& ( c_2Erat_2Erat__les @ V1b @ V2c ) )
=> ( c_2Erat_2Erat__les @ V0a @ V2c ) ) ).
thf(thm_2Erat_2ERAT__0LES__0LES__ADD,axiom,
! [V0r1: tyop_2Erat_2Erat,V1r2: tyop_2Erat_2Erat] :
( ( c_2Erat_2Erat__les @ ( c_2Erat_2Erat__of__num @ c_2Enum_2E0 ) @ V0r1 )
=> ( ( c_2Erat_2Erat__les @ ( c_2Erat_2Erat__of__num @ c_2Enum_2E0 ) @ V1r2 )
=> ( c_2Erat_2Erat__les @ ( c_2Erat_2Erat__of__num @ c_2Enum_2E0 ) @ ( c_2Erat_2Erat__add @ V0r1 @ V1r2 ) ) ) ) ).
thf(thm_2Erat_2ERAT__LES0__LES0__ADD,axiom,
! [V0r1: tyop_2Erat_2Erat,V1r2: tyop_2Erat_2Erat] :
( ( c_2Erat_2Erat__les @ V0r1 @ ( c_2Erat_2Erat__of__num @ c_2Enum_2E0 ) )
=> ( ( c_2Erat_2Erat__les @ V1r2 @ ( c_2Erat_2Erat__of__num @ c_2Enum_2E0 ) )
=> ( c_2Erat_2Erat__les @ ( c_2Erat_2Erat__add @ V0r1 @ V1r2 ) @ ( c_2Erat_2Erat__of__num @ c_2Enum_2E0 ) ) ) ) ).
thf(thm_2Erat_2ERAT__0LES__0LEQ__ADD,axiom,
! [V0r1: tyop_2Erat_2Erat,V1r2: tyop_2Erat_2Erat] :
( ( c_2Erat_2Erat__les @ ( c_2Erat_2Erat__of__num @ c_2Enum_2E0 ) @ V0r1 )
=> ( ( c_2Erat_2Erat__leq @ ( c_2Erat_2Erat__of__num @ c_2Enum_2E0 ) @ V1r2 )
=> ( c_2Erat_2Erat__les @ ( c_2Erat_2Erat__of__num @ c_2Enum_2E0 ) @ ( c_2Erat_2Erat__add @ V0r1 @ V1r2 ) ) ) ) ).
thf(thm_2Erat_2ERAT__LES0__LEQ0__ADD,axiom,
! [V0r1: tyop_2Erat_2Erat,V1r2: tyop_2Erat_2Erat] :
( ( c_2Erat_2Erat__les @ V0r1 @ ( c_2Erat_2Erat__of__num @ c_2Enum_2E0 ) )
=> ( ( c_2Erat_2Erat__leq @ V1r2 @ ( c_2Erat_2Erat__of__num @ c_2Enum_2E0 ) )
=> ( c_2Erat_2Erat__les @ ( c_2Erat_2Erat__add @ V0r1 @ V1r2 ) @ ( c_2Erat_2Erat__of__num @ c_2Enum_2E0 ) ) ) ) ).
thf(thm_2Erat_2ERAT__LSUB__EQ,axiom,
! [V0r1: tyop_2Erat_2Erat,V1r2: tyop_2Erat_2Erat,V2r3: tyop_2Erat_2Erat] :
( ( ( c_2Erat_2Erat__sub @ V0r1 @ V1r2 )
= V2r3 )
<=> ( V0r1
= ( c_2Erat_2Erat__add @ V1r2 @ V2r3 ) ) ) ).
thf(thm_2Erat_2ERAT__RSUB__EQ,axiom,
! [V0r1: tyop_2Erat_2Erat,V1r2: tyop_2Erat_2Erat,V2r3: tyop_2Erat_2Erat] :
( ( V0r1
= ( c_2Erat_2Erat__sub @ V1r2 @ V2r3 ) )
<=> ( ( c_2Erat_2Erat__add @ V0r1 @ V2r3 )
= V1r2 ) ) ).
thf(thm_2Erat_2ERAT__LDIV__EQ,axiom,
! [V0r1: tyop_2Erat_2Erat,V1r2: tyop_2Erat_2Erat,V2r3: tyop_2Erat_2Erat] :
( ( (~)
@ ( V1r2
= ( c_2Erat_2Erat__of__num @ c_2Enum_2E0 ) ) )
=> ( ( ( c_2Erat_2Erat__div @ V0r1 @ V1r2 )
= V2r3 )
<=> ( V0r1
= ( c_2Erat_2Erat__mul @ V1r2 @ V2r3 ) ) ) ) ).
thf(thm_2Erat_2ERAT__RDIV__EQ,axiom,
! [V0r1: tyop_2Erat_2Erat,V1r2: tyop_2Erat_2Erat,V2r3: tyop_2Erat_2Erat] :
( ( (~)
@ ( V2r3
= ( c_2Erat_2Erat__of__num @ c_2Enum_2E0 ) ) )
=> ( ( V0r1
= ( c_2Erat_2Erat__div @ V1r2 @ V2r3 ) )
<=> ( ( c_2Erat_2Erat__mul @ V0r1 @ V2r3 )
= V1r2 ) ) ) ).
thf(thm_2Erat_2ERAT__AINV__ONE__ONE,axiom,
c_2Ebool_2EONE__ONE @ tyop_2Erat_2Erat @ tyop_2Erat_2Erat @ c_2Erat_2Erat__ainv ).
thf(thm_2Erat_2ERAT__ADD__ONE__ONE,axiom,
! [V0r1: tyop_2Erat_2Erat] : ( c_2Ebool_2EONE__ONE @ tyop_2Erat_2Erat @ tyop_2Erat_2Erat @ ( c_2Erat_2Erat__add @ V0r1 ) ) ).
thf(thm_2Erat_2ERAT__MUL__ONE__ONE,axiom,
! [V0r1: tyop_2Erat_2Erat] :
( ( (~)
@ ( V0r1
= ( c_2Erat_2Erat__of__num @ c_2Enum_2E0 ) ) )
<=> ( c_2Ebool_2EONE__ONE @ tyop_2Erat_2Erat @ tyop_2Erat_2Erat @ ( c_2Erat_2Erat__mul @ V0r1 ) ) ) ).
thf(thm_2Erat_2ERAT__EQ__LADD,axiom,
! [V0r1: tyop_2Erat_2Erat,V1r2: tyop_2Erat_2Erat,V2r3: tyop_2Erat_2Erat] :
( ( ( c_2Erat_2Erat__add @ V2r3 @ V0r1 )
= ( c_2Erat_2Erat__add @ V2r3 @ V1r2 ) )
<=> ( V0r1 = V1r2 ) ) ).
thf(thm_2Erat_2ERAT__EQ__RADD,axiom,
! [V0r1: tyop_2Erat_2Erat,V1r2: tyop_2Erat_2Erat,V2r3: tyop_2Erat_2Erat] :
( ( ( c_2Erat_2Erat__add @ V0r1 @ V2r3 )
= ( c_2Erat_2Erat__add @ V1r2 @ V2r3 ) )
<=> ( V0r1 = V1r2 ) ) ).
thf(thm_2Erat_2ERAT__EQ__RMUL,axiom,
! [V0r1: tyop_2Erat_2Erat,V1r2: tyop_2Erat_2Erat,V2r3: tyop_2Erat_2Erat] :
( ( (~)
@ ( V2r3
= ( c_2Erat_2Erat__of__num @ c_2Enum_2E0 ) ) )
=> ( ( ( c_2Erat_2Erat__mul @ V0r1 @ V2r3 )
= ( c_2Erat_2Erat__mul @ V1r2 @ V2r3 ) )
<=> ( V0r1 = V1r2 ) ) ) ).
thf(thm_2Erat_2ERAT__EQ__LMUL,axiom,
! [V0r1: tyop_2Erat_2Erat,V1r2: tyop_2Erat_2Erat,V2r3: tyop_2Erat_2Erat] :
( ( (~)
@ ( V2r3
= ( c_2Erat_2Erat__of__num @ c_2Enum_2E0 ) ) )
=> ( ( ( c_2Erat_2Erat__mul @ V2r3 @ V0r1 )
= ( c_2Erat_2Erat__mul @ V2r3 @ V1r2 ) )
<=> ( V0r1 = V1r2 ) ) ) ).
thf(thm_2Erat_2ERAT__LES__RADD,axiom,
! [V0r1: tyop_2Erat_2Erat,V1r2: tyop_2Erat_2Erat,V2r3: tyop_2Erat_2Erat] :
( ( c_2Erat_2Erat__les @ ( c_2Erat_2Erat__add @ V0r1 @ V2r3 ) @ ( c_2Erat_2Erat__add @ V1r2 @ V2r3 ) )
= ( c_2Erat_2Erat__les @ V0r1 @ V1r2 ) ) ).
thf(thm_2Erat_2ERAT__LES__LADD,axiom,
! [V0r1: tyop_2Erat_2Erat,V1r2: tyop_2Erat_2Erat,V2r3: tyop_2Erat_2Erat] :
( ( c_2Erat_2Erat__les @ ( c_2Erat_2Erat__add @ V2r3 @ V0r1 ) @ ( c_2Erat_2Erat__add @ V2r3 @ V1r2 ) )
= ( c_2Erat_2Erat__les @ V0r1 @ V1r2 ) ) ).
thf(thm_2Erat_2ERAT__LEQ__RADD,axiom,
! [V0r1: tyop_2Erat_2Erat,V1r2: tyop_2Erat_2Erat,V2r3: tyop_2Erat_2Erat] :
( ( c_2Erat_2Erat__leq @ ( c_2Erat_2Erat__add @ V0r1 @ V2r3 ) @ ( c_2Erat_2Erat__add @ V1r2 @ V2r3 ) )
= ( c_2Erat_2Erat__leq @ V0r1 @ V1r2 ) ) ).
thf(thm_2Erat_2ERAT__LEQ__LADD,axiom,
! [V0r1: tyop_2Erat_2Erat,V1r2: tyop_2Erat_2Erat,V2r3: tyop_2Erat_2Erat] :
( ( c_2Erat_2Erat__leq @ ( c_2Erat_2Erat__add @ V2r3 @ V0r1 ) @ ( c_2Erat_2Erat__add @ V2r3 @ V1r2 ) )
= ( c_2Erat_2Erat__leq @ V0r1 @ V1r2 ) ) ).
thf(thm_2Erat_2ERAT__ADD__MONO,axiom,
! [V0a: tyop_2Erat_2Erat,V1b: tyop_2Erat_2Erat,V2c: tyop_2Erat_2Erat,V3d: tyop_2Erat_2Erat] :
( ( ( c_2Erat_2Erat__leq @ V0a @ V1b )
& ( c_2Erat_2Erat__leq @ V2c @ V3d ) )
=> ( c_2Erat_2Erat__leq @ ( c_2Erat_2Erat__add @ V0a @ V2c ) @ ( c_2Erat_2Erat__add @ V1b @ V3d ) ) ) ).
thf(thm_2Erat_2ERAT__LES__AINV,axiom,
! [V0r1: tyop_2Erat_2Erat,V1r2: tyop_2Erat_2Erat] :
( ( c_2Erat_2Erat__les @ ( c_2Erat_2Erat__ainv @ V0r1 ) @ ( c_2Erat_2Erat__ainv @ V1r2 ) )
= ( c_2Erat_2Erat__les @ V1r2 @ V0r1 ) ) ).
thf(thm_2Erat_2ERAT__LSUB__LES,axiom,
! [V0r1: tyop_2Erat_2Erat,V1r2: tyop_2Erat_2Erat,V2r3: tyop_2Erat_2Erat] :
( ( c_2Erat_2Erat__les @ ( c_2Erat_2Erat__sub @ V0r1 @ V1r2 ) @ V2r3 )
= ( c_2Erat_2Erat__les @ V0r1 @ ( c_2Erat_2Erat__add @ V1r2 @ V2r3 ) ) ) ).
thf(thm_2Erat_2ERAT__RSUB__LES,axiom,
! [V0r1: tyop_2Erat_2Erat,V1r2: tyop_2Erat_2Erat,V2r3: tyop_2Erat_2Erat] :
( ( c_2Erat_2Erat__les @ V0r1 @ ( c_2Erat_2Erat__sub @ V1r2 @ V2r3 ) )
= ( c_2Erat_2Erat__les @ ( c_2Erat_2Erat__add @ V0r1 @ V2r3 ) @ V1r2 ) ) ).
thf(thm_2Erat_2ERAT__LSUB__LEQ,axiom,
! [V0r1: tyop_2Erat_2Erat,V1r2: tyop_2Erat_2Erat,V2r3: tyop_2Erat_2Erat] :
( ( c_2Erat_2Erat__leq @ ( c_2Erat_2Erat__sub @ V0r1 @ V1r2 ) @ V2r3 )
= ( c_2Erat_2Erat__leq @ V0r1 @ ( c_2Erat_2Erat__add @ V1r2 @ V2r3 ) ) ) ).
thf(thm_2Erat_2ERAT__RSUB__LEQ,axiom,
! [V0r1: tyop_2Erat_2Erat,V1r2: tyop_2Erat_2Erat,V2r3: tyop_2Erat_2Erat] :
( ( c_2Erat_2Erat__leq @ V0r1 @ ( c_2Erat_2Erat__sub @ V1r2 @ V2r3 ) )
= ( c_2Erat_2Erat__leq @ ( c_2Erat_2Erat__add @ V0r1 @ V2r3 ) @ V1r2 ) ) ).
thf(thm_2Erat_2ERAT__LES__RMUL__POS,axiom,
! [V0r1: tyop_2Erat_2Erat,V1r2: tyop_2Erat_2Erat,V2r3: tyop_2Erat_2Erat] :
( ( c_2Erat_2Erat__les @ ( c_2Erat_2Erat__of__num @ c_2Enum_2E0 ) @ V2r3 )
=> ( ( c_2Erat_2Erat__les @ ( c_2Erat_2Erat__mul @ V0r1 @ V2r3 ) @ ( c_2Erat_2Erat__mul @ V1r2 @ V2r3 ) )
= ( c_2Erat_2Erat__les @ V0r1 @ V1r2 ) ) ) ).
thf(thm_2Erat_2ERAT__LES__LMUL__POS,axiom,
! [V0r1: tyop_2Erat_2Erat,V1r2: tyop_2Erat_2Erat,V2r3: tyop_2Erat_2Erat] :
( ( c_2Erat_2Erat__les @ ( c_2Erat_2Erat__of__num @ c_2Enum_2E0 ) @ V2r3 )
=> ( ( c_2Erat_2Erat__les @ ( c_2Erat_2Erat__mul @ V2r3 @ V0r1 ) @ ( c_2Erat_2Erat__mul @ V2r3 @ V1r2 ) )
= ( c_2Erat_2Erat__les @ V0r1 @ V1r2 ) ) ) ).
thf(thm_2Erat_2ERAT__LES__RMUL__NEG,axiom,
! [V0r1: tyop_2Erat_2Erat,V1r2: tyop_2Erat_2Erat,V2r3: tyop_2Erat_2Erat] :
( ( c_2Erat_2Erat__les @ V2r3 @ ( c_2Erat_2Erat__of__num @ c_2Enum_2E0 ) )
=> ( ( c_2Erat_2Erat__les @ ( c_2Erat_2Erat__mul @ V1r2 @ V2r3 ) @ ( c_2Erat_2Erat__mul @ V0r1 @ V2r3 ) )
= ( c_2Erat_2Erat__les @ V0r1 @ V1r2 ) ) ) ).
thf(thm_2Erat_2ERAT__LES__LMUL__NEG,axiom,
! [V0r1: tyop_2Erat_2Erat,V1r2: tyop_2Erat_2Erat,V2r3: tyop_2Erat_2Erat] :
( ( c_2Erat_2Erat__les @ V2r3 @ ( c_2Erat_2Erat__of__num @ c_2Enum_2E0 ) )
=> ( ( c_2Erat_2Erat__les @ ( c_2Erat_2Erat__mul @ V2r3 @ V1r2 ) @ ( c_2Erat_2Erat__mul @ V2r3 @ V0r1 ) )
= ( c_2Erat_2Erat__les @ V0r1 @ V1r2 ) ) ) ).
thf(thm_2Erat_2ERAT__AINV__LES,axiom,
! [V0r1: tyop_2Erat_2Erat,V1r2: tyop_2Erat_2Erat] :
( ( c_2Erat_2Erat__les @ ( c_2Erat_2Erat__ainv @ V0r1 ) @ V1r2 )
= ( c_2Erat_2Erat__les @ ( c_2Erat_2Erat__ainv @ V1r2 ) @ V0r1 ) ) ).
thf(thm_2Erat_2ERAT__LDIV__LES__POS,axiom,
! [V0r1: tyop_2Erat_2Erat,V1r2: tyop_2Erat_2Erat,V2r3: tyop_2Erat_2Erat] :
( ( c_2Erat_2Erat__les @ ( c_2Erat_2Erat__of__num @ c_2Enum_2E0 ) @ V1r2 )
=> ( ( c_2Erat_2Erat__les @ ( c_2Erat_2Erat__div @ V0r1 @ V1r2 ) @ V2r3 )
= ( c_2Erat_2Erat__les @ V0r1 @ ( c_2Erat_2Erat__mul @ V1r2 @ V2r3 ) ) ) ) ).
thf(thm_2Erat_2ERAT__LDIV__LES__NEG,axiom,
! [V0r1: tyop_2Erat_2Erat,V1r2: tyop_2Erat_2Erat,V2r3: tyop_2Erat_2Erat] :
( ( c_2Erat_2Erat__les @ V1r2 @ ( c_2Erat_2Erat__of__num @ c_2Enum_2E0 ) )
=> ( ( c_2Erat_2Erat__les @ ( c_2Erat_2Erat__div @ V0r1 @ V1r2 ) @ V2r3 )
= ( c_2Erat_2Erat__les @ ( c_2Erat_2Erat__mul @ V1r2 @ V2r3 ) @ V0r1 ) ) ) ).
thf(thm_2Erat_2ERAT__RDIV__LES__POS,axiom,
! [V0r1: tyop_2Erat_2Erat,V1r2: tyop_2Erat_2Erat,V2r3: tyop_2Erat_2Erat] :
( ( c_2Erat_2Erat__les @ ( c_2Erat_2Erat__of__num @ c_2Enum_2E0 ) @ V2r3 )
=> ( ( c_2Erat_2Erat__les @ V0r1 @ ( c_2Erat_2Erat__div @ V1r2 @ V2r3 ) )
= ( c_2Erat_2Erat__les @ ( c_2Erat_2Erat__mul @ V0r1 @ V2r3 ) @ V1r2 ) ) ) ).
thf(thm_2Erat_2ERAT__RDIV__LES__NEG,axiom,
! [V0r1: tyop_2Erat_2Erat,V1r2: tyop_2Erat_2Erat,V2r3: tyop_2Erat_2Erat] :
( ( c_2Erat_2Erat__les @ V2r3 @ ( c_2Erat_2Erat__of__num @ c_2Enum_2E0 ) )
=> ( ( c_2Erat_2Erat__les @ V0r1 @ ( c_2Erat_2Erat__div @ V1r2 @ V2r3 ) )
= ( c_2Erat_2Erat__les @ V1r2 @ ( c_2Erat_2Erat__mul @ V0r1 @ V2r3 ) ) ) ) ).
thf(thm_2Erat_2ERAT__LDIV__LEQ__POS,axiom,
! [V0r1: tyop_2Erat_2Erat,V1r2: tyop_2Erat_2Erat,V2r3: tyop_2Erat_2Erat] :
( ( c_2Erat_2Erat__les @ ( c_2Erat_2Erat__of__num @ c_2Enum_2E0 ) @ V1r2 )
=> ( ( c_2Erat_2Erat__leq @ ( c_2Erat_2Erat__div @ V0r1 @ V1r2 ) @ V2r3 )
= ( c_2Erat_2Erat__leq @ V0r1 @ ( c_2Erat_2Erat__mul @ V1r2 @ V2r3 ) ) ) ) ).
thf(thm_2Erat_2ERAT__LDIV__LEQ__NEG,axiom,
! [V0r1: tyop_2Erat_2Erat,V1r2: tyop_2Erat_2Erat,V2r3: tyop_2Erat_2Erat] :
( ( c_2Erat_2Erat__les @ V1r2 @ ( c_2Erat_2Erat__of__num @ c_2Enum_2E0 ) )
=> ( ( c_2Erat_2Erat__leq @ ( c_2Erat_2Erat__div @ V0r1 @ V1r2 ) @ V2r3 )
= ( c_2Erat_2Erat__leq @ ( c_2Erat_2Erat__mul @ V1r2 @ V2r3 ) @ V0r1 ) ) ) ).
thf(thm_2Erat_2ERAT__RDIV__LEQ__POS,axiom,
! [V0r1: tyop_2Erat_2Erat,V1r2: tyop_2Erat_2Erat,V2r3: tyop_2Erat_2Erat] :
( ( c_2Erat_2Erat__les @ ( c_2Erat_2Erat__of__num @ c_2Enum_2E0 ) @ V2r3 )
=> ( ( c_2Erat_2Erat__leq @ V0r1 @ ( c_2Erat_2Erat__div @ V1r2 @ V2r3 ) )
= ( c_2Erat_2Erat__leq @ ( c_2Erat_2Erat__mul @ V0r1 @ V2r3 ) @ V1r2 ) ) ) ).
thf(thm_2Erat_2ERAT__RDIV__LEQ__NEG,axiom,
! [V0r1: tyop_2Erat_2Erat,V1r2: tyop_2Erat_2Erat,V2r3: tyop_2Erat_2Erat] :
( ( c_2Erat_2Erat__les @ V2r3 @ ( c_2Erat_2Erat__of__num @ c_2Enum_2E0 ) )
=> ( ( c_2Erat_2Erat__leq @ V0r1 @ ( c_2Erat_2Erat__div @ V1r2 @ V2r3 ) )
= ( c_2Erat_2Erat__leq @ V1r2 @ ( c_2Erat_2Erat__mul @ V0r1 @ V2r3 ) ) ) ) ).
thf(thm_2Erat_2ERAT__LES__SUB0,axiom,
! [V0r1: tyop_2Erat_2Erat,V1r2: tyop_2Erat_2Erat] :
( ( c_2Erat_2Erat__les @ ( c_2Erat_2Erat__sub @ V0r1 @ V1r2 ) @ ( c_2Erat_2Erat__of__num @ c_2Enum_2E0 ) )
= ( c_2Erat_2Erat__les @ V0r1 @ V1r2 ) ) ).
thf(thm_2Erat_2ERAT__LES__0SUB,axiom,
! [V0r1: tyop_2Erat_2Erat,V1r2: tyop_2Erat_2Erat] :
( ( c_2Erat_2Erat__les @ ( c_2Erat_2Erat__of__num @ c_2Enum_2E0 ) @ ( c_2Erat_2Erat__sub @ V0r1 @ V1r2 ) )
= ( c_2Erat_2Erat__les @ V1r2 @ V0r1 ) ) ).
thf(thm_2Erat_2ERAT__MINV__LES,axiom,
! [V0r1: tyop_2Erat_2Erat] :
( ( c_2Erat_2Erat__les @ ( c_2Erat_2Erat__of__num @ c_2Enum_2E0 ) @ V0r1 )
=> ( ( ( c_2Erat_2Erat__les @ ( c_2Erat_2Erat__minv @ V0r1 ) @ ( c_2Erat_2Erat__of__num @ c_2Enum_2E0 ) )
= ( c_2Erat_2Erat__les @ V0r1 @ ( c_2Erat_2Erat__of__num @ c_2Enum_2E0 ) ) )
& ( ( c_2Erat_2Erat__les @ ( c_2Erat_2Erat__of__num @ c_2Enum_2E0 ) @ ( c_2Erat_2Erat__minv @ V0r1 ) )
= ( c_2Erat_2Erat__les @ ( c_2Erat_2Erat__of__num @ c_2Enum_2E0 ) @ V0r1 ) ) ) ) ).
thf(thm_2Erat_2ERAT__MUL__SIGN__CASES,axiom,
! [V0p: tyop_2Erat_2Erat,V1q: tyop_2Erat_2Erat] :
( ( ( c_2Erat_2Erat__les @ ( c_2Erat_2Erat__of__num @ c_2Enum_2E0 ) @ ( c_2Erat_2Erat__mul @ V0p @ V1q ) )
<=> ( ( ( c_2Erat_2Erat__les @ ( c_2Erat_2Erat__of__num @ c_2Enum_2E0 ) @ V0p )
& ( c_2Erat_2Erat__les @ ( c_2Erat_2Erat__of__num @ c_2Enum_2E0 ) @ V1q ) )
| ( ( c_2Erat_2Erat__les @ V0p @ ( c_2Erat_2Erat__of__num @ c_2Enum_2E0 ) )
& ( c_2Erat_2Erat__les @ V1q @ ( c_2Erat_2Erat__of__num @ c_2Enum_2E0 ) ) ) ) )
& ( ( c_2Erat_2Erat__les @ ( c_2Erat_2Erat__mul @ V0p @ V1q ) @ ( c_2Erat_2Erat__of__num @ c_2Enum_2E0 ) )
<=> ( ( ( c_2Erat_2Erat__les @ ( c_2Erat_2Erat__of__num @ c_2Enum_2E0 ) @ V0p )
& ( c_2Erat_2Erat__les @ V1q @ ( c_2Erat_2Erat__of__num @ c_2Enum_2E0 ) ) )
| ( ( c_2Erat_2Erat__les @ V0p @ ( c_2Erat_2Erat__of__num @ c_2Enum_2E0 ) )
& ( c_2Erat_2Erat__les @ ( c_2Erat_2Erat__of__num @ c_2Enum_2E0 ) @ V1q ) ) ) ) ) ).
thf(thm_2Erat_2ERAT__NO__ZERODIV,axiom,
! [V0r1: tyop_2Erat_2Erat,V1r2: tyop_2Erat_2Erat] :
( ( ( V0r1
= ( c_2Erat_2Erat__of__num @ c_2Enum_2E0 ) )
| ( V1r2
= ( c_2Erat_2Erat__of__num @ c_2Enum_2E0 ) ) )
<=> ( ( c_2Erat_2Erat__mul @ V0r1 @ V1r2 )
= ( c_2Erat_2Erat__of__num @ c_2Enum_2E0 ) ) ) ).
thf(thm_2Erat_2ERAT__NO__ZERODIV__THM,axiom,
! [V0r1: tyop_2Erat_2Erat,V1r2: tyop_2Erat_2Erat] :
( ( ( c_2Erat_2Erat__mul @ V0r1 @ V1r2 )
= ( c_2Erat_2Erat__of__num @ c_2Enum_2E0 ) )
<=> ( ( V0r1
= ( c_2Erat_2Erat__of__num @ c_2Enum_2E0 ) )
| ( V1r2
= ( c_2Erat_2Erat__of__num @ c_2Enum_2E0 ) ) ) ) ).
thf(thm_2Erat_2ERAT__NO__ZERODIV__NEG,axiom,
! [V0r1: tyop_2Erat_2Erat,V1r2: tyop_2Erat_2Erat] :
( ( (~)
@ ( ( c_2Erat_2Erat__mul @ V0r1 @ V1r2 )
= ( c_2Erat_2Erat__of__num @ c_2Enum_2E0 ) ) )
<=> ( ( (~)
@ ( V0r1
= ( c_2Erat_2Erat__of__num @ c_2Enum_2E0 ) ) )
& ( (~)
@ ( V1r2
= ( c_2Erat_2Erat__of__num @ c_2Enum_2E0 ) ) ) ) ) ).
thf(thm_2Erat_2ERAT__NO__IDDIV,axiom,
! [V0r1: tyop_2Erat_2Erat,V1r2: tyop_2Erat_2Erat] :
( ( ( c_2Erat_2Erat__mul @ V0r1 @ V1r2 )
= V1r2 )
<=> ( ( V0r1
= ( c_2Erat_2Erat__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) )
| ( V1r2
= ( c_2Erat_2Erat__of__num @ c_2Enum_2E0 ) ) ) ) ).
thf(thm_2Erat_2ERDIV__MUL__OUT,axiom,
! [V0r3: tyop_2Erat_2Erat,V1r2: tyop_2Erat_2Erat,V2r1: tyop_2Erat_2Erat] :
( ( c_2Erat_2Erat__mul @ V2r1 @ ( c_2Erat_2Erat__div @ V1r2 @ V0r3 ) )
= ( c_2Erat_2Erat__div @ ( c_2Erat_2Erat__mul @ V2r1 @ V1r2 ) @ V0r3 ) ) ).
thf(thm_2Erat_2ELDIV__MUL__OUT,axiom,
! [V0r3: tyop_2Erat_2Erat,V1r2: tyop_2Erat_2Erat,V2r1: tyop_2Erat_2Erat] :
( ( c_2Erat_2Erat__mul @ ( c_2Erat_2Erat__div @ V2r1 @ V1r2 ) @ V0r3 )
= ( c_2Erat_2Erat__div @ ( c_2Erat_2Erat__mul @ V2r1 @ V0r3 ) @ V1r2 ) ) ).
thf(thm_2Erat_2ERAT__DENSE__THM,axiom,
! [V0r1: tyop_2Erat_2Erat,V1r3: tyop_2Erat_2Erat] :
( ( c_2Erat_2Erat__les @ V0r1 @ V1r3 )
=> ? [V2r2: tyop_2Erat_2Erat] :
( ( c_2Erat_2Erat__les @ V0r1 @ V2r2 )
& ( c_2Erat_2Erat__les @ V2r2 @ V1r3 ) ) ) ).
thf(thm_2Erat_2ERAT__SAVE,axiom,
! [V0r1: tyop_2Erat_2Erat] :
? [V1a1: tyop_2Einteger_2Eint,V2b1: tyop_2Enum_2Enum] :
( V0r1
= ( c_2Erat_2Eabs__rat @ ( c_2Efrac_2Efrac__save @ V1a1 @ V2b1 ) ) ) ).
thf(thm_2Erat_2ERAT__SAVE__MINV,axiom,
! [V0a1: tyop_2Einteger_2Eint,V1b1: tyop_2Enum_2Enum] :
( ( (~)
@ ( ( c_2Erat_2Eabs__rat @ ( c_2Efrac_2Efrac__save @ V0a1 @ V1b1 ) )
= ( c_2Erat_2Erat__of__num @ c_2Enum_2E0 ) ) )
=> ( ( c_2Erat_2Erat__minv @ ( c_2Erat_2Eabs__rat @ ( c_2Efrac_2Efrac__save @ V0a1 @ V1b1 ) ) )
= ( c_2Erat_2Eabs__rat @ ( c_2Efrac_2Efrac__save @ ( c_2Einteger_2Eint__mul @ ( c_2EintExtension_2ESGN @ V0a1 ) @ ( c_2Einteger_2Eint__add @ ( c_2Einteger_2Eint__of__num @ V1b1 ) @ ( c_2Einteger_2Eint__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ) @ ( c_2Einteger_2ENum @ ( c_2Einteger_2Eint__sub @ ( c_2Einteger_2EABS @ V0a1 ) @ ( c_2Einteger_2Eint__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ) ) ) ) ) ).
thf(thm_2Erat_2ERAT__SAVE__TO__CONS,axiom,
! [V0a1: tyop_2Einteger_2Eint,V1b1: tyop_2Enum_2Enum] :
( ( c_2Erat_2Eabs__rat @ ( c_2Efrac_2Efrac__save @ V0a1 @ V1b1 ) )
= ( c_2Erat_2Erat__cons @ V0a1 @ ( c_2Einteger_2Eint__add @ ( c_2Einteger_2Eint__of__num @ V1b1 ) @ ( c_2Einteger_2Eint__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ) ) ).
thf(thm_2Erat_2ERAT__OF__NUM,axiom,
! [A_27a: $tType,V0n: A_27a] :
( ( ( c_2Erat_2Erat__of__num @ c_2Enum_2E0 )
= c_2Erat_2Erat__0 )
& ! [V1n: tyop_2Enum_2Enum] :
( ( c_2Erat_2Erat__of__num @ ( c_2Enum_2ESUC @ V1n ) )
= ( c_2Erat_2Erat__add @ ( c_2Erat_2Erat__of__num @ V1n ) @ c_2Erat_2Erat__1 ) ) ) ).
thf(thm_2Erat_2ERAT__SAVE__NUM,axiom,
! [V0n: tyop_2Enum_2Enum] :
( ( c_2Erat_2Erat__of__num @ V0n )
= ( c_2Erat_2Eabs__rat @ ( c_2Efrac_2Efrac__save @ ( c_2Einteger_2Eint__of__num @ V0n ) @ c_2Enum_2E0 ) ) ) ).
thf(thm_2Erat_2ERAT__CONS__TO__NUM,axiom,
! [V0n: tyop_2Enum_2Enum] :
( ( ( c_2Erat_2Erat__cons @ ( c_2Einteger_2Eint__of__num @ V0n ) @ ( c_2Einteger_2Eint__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) )
= ( c_2Erat_2Erat__of__num @ V0n ) )
& ( ( c_2Erat_2Erat__cons @ ( c_2Einteger_2Eint__neg @ ( c_2Einteger_2Eint__of__num @ V0n ) ) @ ( c_2Einteger_2Eint__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) )
= ( c_2Erat_2Erat__ainv @ ( c_2Erat_2Erat__of__num @ V0n ) ) ) ) ).
thf(thm_2Erat_2ERAT__0,axiom,
( c_2Erat_2Erat__0
= ( c_2Erat_2Erat__of__num @ c_2Enum_2E0 ) ) ).
thf(thm_2Erat_2ERAT__1,axiom,
( c_2Erat_2Erat__1
= ( c_2Erat_2Erat__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ).
thf(thm_2Erat_2ERAT__MINV__1,axiom,
( ( c_2Erat_2Erat__minv @ ( c_2Erat_2Erat__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) )
= ( c_2Erat_2Erat__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ).
thf(thm_2Erat_2ERAT__DIV__1,axiom,
! [V0r: tyop_2Erat_2Erat] :
( ( c_2Erat_2Erat__div @ V0r @ ( c_2Erat_2Erat__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) )
= V0r ) ).
thf(thm_2Erat_2ERAT__DIV__NEG1,axiom,
! [V0r: tyop_2Erat_2Erat] :
( ( c_2Erat_2Erat__div @ V0r @ ( c_2Erat_2Erat__ainv @ ( c_2Erat_2Erat__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) )
= ( c_2Erat_2Erat__ainv @ V0r ) ) ).
thf(thm_2Erat_2ERAT__DIV__INV,axiom,
! [V0r: tyop_2Erat_2Erat] :
( ( (~)
@ ( V0r
= ( c_2Erat_2Erat__of__num @ c_2Enum_2E0 ) ) )
=> ( ( c_2Erat_2Erat__div @ V0r @ V0r )
= ( c_2Erat_2Erat__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ) ).
thf(thm_2Erat_2ERAT__MINV__MUL,axiom,
! [V0b: tyop_2Erat_2Erat,V1a: tyop_2Erat_2Erat] :
( ( ( (~)
@ ( V1a
= ( c_2Erat_2Erat__of__num @ c_2Enum_2E0 ) ) )
& ( (~)
@ ( V0b
= ( c_2Erat_2Erat__of__num @ c_2Enum_2E0 ) ) ) )
=> ( ( c_2Erat_2Erat__minv @ ( c_2Erat_2Erat__mul @ V1a @ V0b ) )
= ( c_2Erat_2Erat__mul @ ( c_2Erat_2Erat__minv @ V1a ) @ ( c_2Erat_2Erat__minv @ V0b ) ) ) ) ).
thf(thm_2Erat_2ERAT__DIVDIV__MUL,axiom,
! [V0d: tyop_2Erat_2Erat,V1c: tyop_2Erat_2Erat,V2b: tyop_2Erat_2Erat,V3a: tyop_2Erat_2Erat] :
( ( ( (~)
@ ( V2b
= ( c_2Erat_2Erat__of__num @ c_2Enum_2E0 ) ) )
& ( (~)
@ ( V0d
= ( c_2Erat_2Erat__of__num @ c_2Enum_2E0 ) ) ) )
=> ( ( c_2Erat_2Erat__mul @ ( c_2Erat_2Erat__div @ V3a @ V2b ) @ ( c_2Erat_2Erat__div @ V1c @ V0d ) )
= ( c_2Erat_2Erat__div @ ( c_2Erat_2Erat__mul @ V3a @ V1c ) @ ( c_2Erat_2Erat__mul @ V2b @ V0d ) ) ) ) ).
thf(thm_2Erat_2ERAT__DIVDIV__ADD,axiom,
! [V0y: tyop_2Erat_2Erat,V1x: tyop_2Erat_2Erat,V2b: tyop_2Erat_2Erat,V3a: tyop_2Erat_2Erat] :
( ( ( (~)
@ ( V0y
= ( c_2Erat_2Erat__of__num @ c_2Enum_2E0 ) ) )
& ( (~)
@ ( V2b
= ( c_2Erat_2Erat__of__num @ c_2Enum_2E0 ) ) ) )
=> ( ( c_2Erat_2Erat__add @ ( c_2Erat_2Erat__div @ V1x @ V0y ) @ ( c_2Erat_2Erat__div @ V3a @ V2b ) )
= ( c_2Erat_2Erat__div @ ( c_2Erat_2Erat__add @ ( c_2Erat_2Erat__mul @ V1x @ V2b ) @ ( c_2Erat_2Erat__mul @ V3a @ V0y ) ) @ ( c_2Erat_2Erat__mul @ V0y @ V2b ) ) ) ) ).
thf(thm_2Erat_2ERAT__DIV__AINV,axiom,
! [V0y: tyop_2Erat_2Erat,V1x: tyop_2Erat_2Erat] :
( ( c_2Erat_2Erat__ainv @ ( c_2Erat_2Erat__div @ V1x @ V0y ) )
= ( c_2Erat_2Erat__div @ ( c_2Erat_2Erat__ainv @ V1x ) @ V0y ) ) ).
thf(thm_2Erat_2ERAT__MINV__EQ__0,axiom,
! [V0r: tyop_2Erat_2Erat] :
( ( (~)
@ ( V0r
= ( c_2Erat_2Erat__of__num @ c_2Enum_2E0 ) ) )
=> ( (~)
@ ( ( c_2Erat_2Erat__minv @ V0r )
= ( c_2Erat_2Erat__of__num @ c_2Enum_2E0 ) ) ) ) ).
thf(thm_2Erat_2ERAT__DIV__MINV,axiom,
! [V0y: tyop_2Erat_2Erat,V1x: tyop_2Erat_2Erat] :
( ( ( (~)
@ ( V1x
= ( c_2Erat_2Erat__of__num @ c_2Enum_2E0 ) ) )
& ( (~)
@ ( V0y
= ( c_2Erat_2Erat__of__num @ c_2Enum_2E0 ) ) ) )
=> ( ( c_2Erat_2Erat__minv @ ( c_2Erat_2Erat__div @ V1x @ V0y ) )
= ( c_2Erat_2Erat__div @ V0y @ V1x ) ) ) ).
thf(thm_2Erat_2ERAT__DIV__EQ0,axiom,
! [V0n: tyop_2Erat_2Erat,V1d: tyop_2Erat_2Erat] :
( ( (~)
@ ( V1d
= ( c_2Erat_2Erat__of__num @ c_2Enum_2E0 ) ) )
=> ( ( ( ( c_2Erat_2Erat__div @ V0n @ V1d )
= ( c_2Erat_2Erat__of__num @ c_2Enum_2E0 ) )
<=> ( V0n
= ( c_2Erat_2Erat__of__num @ c_2Enum_2E0 ) ) )
& ( ( ( c_2Erat_2Erat__of__num @ c_2Enum_2E0 )
= ( c_2Erat_2Erat__div @ V0n @ V1d ) )
<=> ( V0n
= ( c_2Erat_2Erat__of__num @ c_2Enum_2E0 ) ) ) ) ) ).
thf(thm_2Erat_2ERAT__ADD__NUM__CALCULATE,axiom,
( ! [V0n: tyop_2Enum_2Enum,V1m: tyop_2Enum_2Enum] :
( ( c_2Erat_2Erat__add @ ( c_2Erat_2Erat__of__num @ V0n ) @ ( c_2Erat_2Erat__of__num @ V1m ) )
= ( c_2Erat_2Erat__of__num @ ( c_2Earithmetic_2E_2B @ V0n @ V1m ) ) )
& ! [V2n: tyop_2Enum_2Enum,V3m: tyop_2Enum_2Enum] :
( ( c_2Erat_2Erat__add @ ( c_2Erat_2Erat__ainv @ ( c_2Erat_2Erat__of__num @ V2n ) ) @ ( c_2Erat_2Erat__of__num @ V3m ) )
= ( c_2Ebool_2ECOND @ tyop_2Erat_2Erat @ ( c_2Earithmetic_2E_3C_3D @ V2n @ V3m ) @ ( c_2Erat_2Erat__of__num @ ( c_2Earithmetic_2E_2D @ V3m @ V2n ) ) @ ( c_2Erat_2Erat__ainv @ ( c_2Erat_2Erat__of__num @ ( c_2Earithmetic_2E_2D @ V2n @ V3m ) ) ) ) )
& ! [V4n: tyop_2Enum_2Enum,V5m: tyop_2Enum_2Enum] :
( ( c_2Erat_2Erat__add @ ( c_2Erat_2Erat__of__num @ V4n ) @ ( c_2Erat_2Erat__ainv @ ( c_2Erat_2Erat__of__num @ V5m ) ) )
= ( c_2Ebool_2ECOND @ tyop_2Erat_2Erat @ ( c_2Earithmetic_2E_3C_3D @ V5m @ V4n ) @ ( c_2Erat_2Erat__of__num @ ( c_2Earithmetic_2E_2D @ V4n @ V5m ) ) @ ( c_2Erat_2Erat__ainv @ ( c_2Erat_2Erat__of__num @ ( c_2Earithmetic_2E_2D @ V5m @ V4n ) ) ) ) )
& ! [V6n: tyop_2Enum_2Enum,V7m: tyop_2Enum_2Enum] :
( ( c_2Erat_2Erat__add @ ( c_2Erat_2Erat__ainv @ ( c_2Erat_2Erat__of__num @ V6n ) ) @ ( c_2Erat_2Erat__ainv @ ( c_2Erat_2Erat__of__num @ V7m ) ) )
= ( c_2Erat_2Erat__ainv @ ( c_2Erat_2Erat__of__num @ ( c_2Earithmetic_2E_2B @ V6n @ V7m ) ) ) ) ) ).
thf(thm_2Erat_2ERAT__MUL__NUM__CALCULATE,axiom,
( ! [V0n: tyop_2Enum_2Enum,V1m: tyop_2Enum_2Enum] :
( ( c_2Erat_2Erat__mul @ ( c_2Erat_2Erat__of__num @ V0n ) @ ( c_2Erat_2Erat__of__num @ V1m ) )
= ( c_2Erat_2Erat__of__num @ ( c_2Earithmetic_2E_2A @ V0n @ V1m ) ) )
& ! [V2n: tyop_2Enum_2Enum,V3m: tyop_2Enum_2Enum] :
( ( c_2Erat_2Erat__mul @ ( c_2Erat_2Erat__ainv @ ( c_2Erat_2Erat__of__num @ V2n ) ) @ ( c_2Erat_2Erat__of__num @ V3m ) )
= ( c_2Erat_2Erat__ainv @ ( c_2Erat_2Erat__of__num @ ( c_2Earithmetic_2E_2A @ V2n @ V3m ) ) ) )
& ! [V4n: tyop_2Enum_2Enum,V5m: tyop_2Enum_2Enum] :
( ( c_2Erat_2Erat__mul @ ( c_2Erat_2Erat__of__num @ V4n ) @ ( c_2Erat_2Erat__ainv @ ( c_2Erat_2Erat__of__num @ V5m ) ) )
= ( c_2Erat_2Erat__ainv @ ( c_2Erat_2Erat__of__num @ ( c_2Earithmetic_2E_2A @ V4n @ V5m ) ) ) )
& ! [V6n: tyop_2Enum_2Enum,V7m: tyop_2Enum_2Enum] :
( ( c_2Erat_2Erat__mul @ ( c_2Erat_2Erat__ainv @ ( c_2Erat_2Erat__of__num @ V6n ) ) @ ( c_2Erat_2Erat__ainv @ ( c_2Erat_2Erat__of__num @ V7m ) ) )
= ( c_2Erat_2Erat__of__num @ ( c_2Earithmetic_2E_2A @ V6n @ V7m ) ) ) ) ).
thf(thm_2Erat_2ERAT__EQ__NUM__CALCULATE,axiom,
( ! [V0n: tyop_2Enum_2Enum,V1m: tyop_2Enum_2Enum] :
( ( ( c_2Erat_2Erat__of__num @ V0n )
= ( c_2Erat_2Erat__of__num @ V1m ) )
<=> ( V0n = V1m ) )
& ! [V2n: tyop_2Enum_2Enum,V3m: tyop_2Enum_2Enum] :
( ( ( c_2Erat_2Erat__of__num @ V2n )
= ( c_2Erat_2Erat__ainv @ ( c_2Erat_2Erat__of__num @ V3m ) ) )
<=> ( ( V2n = c_2Enum_2E0 )
& ( V3m = c_2Enum_2E0 ) ) )
& ! [V4n: tyop_2Enum_2Enum,V5m: tyop_2Enum_2Enum] :
( ( ( c_2Erat_2Erat__ainv @ ( c_2Erat_2Erat__of__num @ V4n ) )
= ( c_2Erat_2Erat__of__num @ V5m ) )
<=> ( ( V4n = c_2Enum_2E0 )
& ( V5m = c_2Enum_2E0 ) ) )
& ! [V6n: tyop_2Enum_2Enum,V7m: tyop_2Enum_2Enum] :
( ( ( c_2Erat_2Erat__ainv @ ( c_2Erat_2Erat__of__num @ V6n ) )
= ( c_2Erat_2Erat__ainv @ ( c_2Erat_2Erat__of__num @ V7m ) ) )
<=> ( V6n = V7m ) ) ) ).
thf(thm_2Erat_2ERAT__LT__NUM__CALCULATE,axiom,
! [V0n: tyop_2Enum_2Enum,V1m: tyop_2Enum_2Enum,V2b: tyop_2Enum_2Enum,V3a: tyop_2Enum_2Enum] :
( ( ( c_2Erat_2Erat__les @ ( c_2Erat_2Erat__of__num @ V3a ) @ ( c_2Erat_2Erat__of__num @ V2b ) )
= ( c_2Eprim__rec_2E_3C @ V3a @ V2b ) )
& ( ( c_2Erat_2Erat__les @ ( c_2Erat_2Erat__ainv @ ( c_2Erat_2Erat__of__num @ V1m ) ) @ ( c_2Erat_2Erat__of__num @ V0n ) )
<=> ( ( c_2Eprim__rec_2E_3C @ c_2Enum_2E0 @ V1m )
| ( c_2Eprim__rec_2E_3C @ c_2Enum_2E0 @ V0n ) ) )
& ( ( c_2Erat_2Erat__les @ ( c_2Erat_2Erat__of__num @ V1m ) @ ( c_2Erat_2Erat__ainv @ ( c_2Erat_2Erat__of__num @ V0n ) ) )
= c_2Ebool_2EF )
& ( ( c_2Erat_2Erat__les @ ( c_2Erat_2Erat__ainv @ ( c_2Erat_2Erat__of__num @ V1m ) ) @ ( c_2Erat_2Erat__ainv @ ( c_2Erat_2Erat__of__num @ V0n ) ) )
= ( c_2Eprim__rec_2E_3C @ V0n @ V1m ) ) ) ).
thf(thm_2Erat_2ERAT__LE__NUM__CALCULATE,axiom,
! [V0n: tyop_2Enum_2Enum,V1m: tyop_2Enum_2Enum,V2b: tyop_2Enum_2Enum,V3a: tyop_2Enum_2Enum] :
( ( ( c_2Erat_2Erat__leq @ ( c_2Erat_2Erat__of__num @ V3a ) @ ( c_2Erat_2Erat__of__num @ V2b ) )
= ( c_2Earithmetic_2E_3C_3D @ V3a @ V2b ) )
& ( ( c_2Erat_2Erat__leq @ ( c_2Erat_2Erat__ainv @ ( c_2Erat_2Erat__of__num @ V1m ) ) @ ( c_2Erat_2Erat__of__num @ V0n ) )
= c_2Ebool_2ET )
& ( ( c_2Erat_2Erat__leq @ ( c_2Erat_2Erat__of__num @ V1m ) @ ( c_2Erat_2Erat__ainv @ ( c_2Erat_2Erat__of__num @ V0n ) ) )
<=> ( ( V1m = c_2Enum_2E0 )
& ( V0n = c_2Enum_2E0 ) ) )
& ( ( c_2Erat_2Erat__leq @ ( c_2Erat_2Erat__ainv @ ( c_2Erat_2Erat__of__num @ V1m ) ) @ ( c_2Erat_2Erat__ainv @ ( c_2Erat_2Erat__of__num @ V0n ) ) )
= ( c_2Earithmetic_2E_3C_3D @ V0n @ V1m ) ) ) ).
thf(thm_2Erat_2Erat__of__int__11,axiom,
! [V0i2: tyop_2Einteger_2Eint,V1i1: tyop_2Einteger_2Eint] :
( ( ( c_2Erat_2Erat__of__int @ V1i1 )
= ( c_2Erat_2Erat__of__int @ V0i2 ) )
<=> ( V1i1 = V0i2 ) ) ).
thf(thm_2Erat_2Erat__of__int__of__num,axiom,
! [V0x: tyop_2Enum_2Enum] :
( ( c_2Erat_2Erat__of__int @ ( c_2Einteger_2Eint__of__num @ V0x ) )
= ( c_2Erat_2Erat__of__num @ V0x ) ) ).
thf(thm_2Erat_2Erat__of__int__MUL,axiom,
! [V0y: tyop_2Einteger_2Eint,V1x: tyop_2Einteger_2Eint] :
( ( c_2Erat_2Erat__mul @ ( c_2Erat_2Erat__of__int @ V1x ) @ ( c_2Erat_2Erat__of__int @ V0y ) )
= ( c_2Erat_2Erat__of__int @ ( c_2Einteger_2Eint__mul @ V1x @ V0y ) ) ) ).
thf(thm_2Erat_2Erat__of__int__ADD,axiom,
! [V0y: tyop_2Einteger_2Eint,V1x: tyop_2Einteger_2Eint] :
( ( c_2Erat_2Erat__add @ ( c_2Erat_2Erat__of__int @ V1x ) @ ( c_2Erat_2Erat__of__int @ V0y ) )
= ( c_2Erat_2Erat__of__int @ ( c_2Einteger_2Eint__add @ V1x @ V0y ) ) ) ).
thf(thm_2Erat_2Erat__of__int__LE,axiom,
! [V0j: tyop_2Einteger_2Eint,V1i: tyop_2Einteger_2Eint] :
( ( c_2Erat_2Erat__leq @ ( c_2Erat_2Erat__of__int @ V1i ) @ ( c_2Erat_2Erat__of__int @ V0j ) )
= ( c_2Einteger_2Eint__le @ V1i @ V0j ) ) ).
thf(thm_2Erat_2Erat__of__int__LT,axiom,
! [V0j: tyop_2Einteger_2Eint,V1i: tyop_2Einteger_2Eint] :
( ( c_2Erat_2Erat__les @ ( c_2Erat_2Erat__of__int @ V1i ) @ ( c_2Erat_2Erat__of__int @ V0j ) )
= ( c_2Einteger_2Eint__lt @ V1i @ V0j ) ) ).
thf(thm_2Erat_2Erat__of__int__ainv,axiom,
! [V0i: tyop_2Einteger_2Eint] :
( ( c_2Erat_2Erat__of__int @ ( c_2Einteger_2Eint__neg @ V0i ) )
= ( c_2Erat_2Erat__ainv @ ( c_2Erat_2Erat__of__int @ V0i ) ) ) ).
thf(thm_2Erat_2ERAT__OF__INT__CALCULATE,axiom,
! [V0i: tyop_2Einteger_2Eint] :
( ( c_2Erat_2Erat__of__int @ V0i )
= ( c_2Erat_2Eabs__rat @ ( c_2Efrac_2Eabs__frac @ ( c_2Epair_2E_2C @ tyop_2Einteger_2Eint @ tyop_2Einteger_2Eint @ V0i @ ( c_2Einteger_2Eint__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ) ) ) ).
thf(thm_2Erat_2ERATD__NZERO,axiom,
! [V0r: tyop_2Erat_2Erat] :
( ( c_2Eprim__rec_2E_3C @ c_2Enum_2E0 @ ( c_2Erat_2ERATD @ V0r ) )
& ( (~)
@ ( ( c_2Erat_2ERATD @ V0r )
= c_2Enum_2E0 ) ) ) ).
thf(thm_2Erat_2ERATN__LEAST,axiom,
! [V0r: tyop_2Erat_2Erat,V1n_27: tyop_2Einteger_2Eint,V2d_27: tyop_2Enum_2Enum] :
( ( ( V0r
= ( c_2Erat_2Erat__div @ ( c_2Erat_2Erat__of__int @ V1n_27 ) @ ( c_2Erat_2Erat__of__num @ V2d_27 ) ) )
& ( c_2Eprim__rec_2E_3C @ c_2Enum_2E0 @ V2d_27 ) )
=> ( c_2Einteger_2Eint__le @ ( c_2Einteger_2EABS @ ( c_2Erat_2ERATN @ V0r ) ) @ ( c_2Einteger_2EABS @ V1n_27 ) ) ) ).
thf(thm_2Erat_2ERATN__RATD__EQ__THM,axiom,
! [V0r: tyop_2Erat_2Erat] :
( V0r
= ( c_2Erat_2Erat__div @ ( c_2Erat_2Erat__of__int @ ( c_2Erat_2ERATN @ V0r ) ) @ ( c_2Erat_2Erat__of__num @ ( c_2Erat_2ERATD @ V0r ) ) ) ) ).
thf(thm_2Erat_2ERATN__RATD__MULT,axiom,
! [V0r: tyop_2Erat_2Erat] :
( ( c_2Erat_2Erat__mul @ V0r @ ( c_2Erat_2Erat__of__num @ ( c_2Erat_2ERATD @ V0r ) ) )
= ( c_2Erat_2Erat__of__int @ ( c_2Erat_2ERATN @ V0r ) ) ) ).
thf(thm_2Erat_2ERATND__RAT__OF__NUM,axiom,
! [V0n: tyop_2Enum_2Enum] :
( ( ( c_2Erat_2ERATN @ ( c_2Erat_2Erat__of__num @ V0n ) )
= ( c_2Einteger_2Eint__of__num @ V0n ) )
& ( ( c_2Erat_2ERATD @ ( c_2Erat_2Erat__of__num @ V0n ) )
= ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ).
thf(thm_2Erat_2ERATN__EQ0,axiom,
! [V0r: tyop_2Erat_2Erat] :
( ( ( ( c_2Erat_2ERATN @ V0r )
= ( c_2Einteger_2Eint__of__num @ c_2Enum_2E0 ) )
<=> ( V0r
= ( c_2Erat_2Erat__of__num @ c_2Enum_2E0 ) ) )
& ( ( ( c_2Einteger_2Eint__of__num @ c_2Enum_2E0 )
= ( c_2Erat_2ERATN @ V0r ) )
<=> ( V0r
= ( c_2Erat_2Erat__of__num @ c_2Enum_2E0 ) ) ) ) ).
thf(thm_2Erat_2ERATN__SIGN,axiom,
! [V0x: tyop_2Erat_2Erat] :
( ( ( c_2Einteger_2Eint__lt @ ( c_2Einteger_2Eint__of__num @ c_2Enum_2E0 ) @ ( c_2Erat_2ERATN @ V0x ) )
= ( c_2Erat_2Erat__les @ ( c_2Erat_2Erat__of__num @ c_2Enum_2E0 ) @ V0x ) )
& ( ( c_2Einteger_2Eint__le @ ( c_2Einteger_2Eint__of__num @ c_2Enum_2E0 ) @ ( c_2Erat_2ERATN @ V0x ) )
= ( c_2Erat_2Erat__leq @ ( c_2Erat_2Erat__of__num @ c_2Enum_2E0 ) @ V0x ) )
& ( ( c_2Einteger_2Eint__lt @ ( c_2Erat_2ERATN @ V0x ) @ ( c_2Einteger_2Eint__of__num @ c_2Enum_2E0 ) )
= ( c_2Erat_2Erat__les @ V0x @ ( c_2Erat_2Erat__of__num @ c_2Enum_2E0 ) ) )
& ( ( c_2Einteger_2Eint__le @ ( c_2Erat_2ERATN @ V0x ) @ ( c_2Einteger_2Eint__of__num @ c_2Enum_2E0 ) )
= ( c_2Erat_2Erat__leq @ V0x @ ( c_2Erat_2Erat__of__num @ c_2Enum_2E0 ) ) ) ) ).
thf(thm_2Erat_2ERAT__AINV__SGN,axiom,
! [V0r: tyop_2Erat_2Erat] :
( ( ( c_2Erat_2Erat__les @ ( c_2Erat_2Erat__of__num @ c_2Enum_2E0 ) @ ( c_2Erat_2Erat__ainv @ V0r ) )
= ( c_2Erat_2Erat__les @ V0r @ ( c_2Erat_2Erat__of__num @ c_2Enum_2E0 ) ) )
& ( ( c_2Erat_2Erat__les @ ( c_2Erat_2Erat__ainv @ V0r ) @ ( c_2Erat_2Erat__of__num @ c_2Enum_2E0 ) )
= ( c_2Erat_2Erat__les @ ( c_2Erat_2Erat__of__num @ c_2Enum_2E0 ) @ V0r ) ) ) ).
thf(thm_2Erat_2ERATN__NEG,axiom,
! [V0r: tyop_2Erat_2Erat] :
( ( c_2Erat_2ERATN @ ( c_2Erat_2Erat__ainv @ V0r ) )
= ( c_2Einteger_2Eint__neg @ ( c_2Erat_2ERATN @ V0r ) ) ) ).
thf(thm_2Erat_2ERATD__NEG,axiom,
! [V0r: tyop_2Erat_2Erat] :
( ( c_2Erat_2ERATD @ ( c_2Erat_2Erat__ainv @ V0r ) )
= ( c_2Erat_2ERATD @ V0r ) ) ).
thf(thm_2Erat_2ERATN__RATD__RAT__OF__INT,axiom,
! [V0i: tyop_2Einteger_2Eint] :
( ( ( c_2Erat_2ERATN @ ( c_2Erat_2Erat__of__int @ V0i ) )
= V0i )
& ( ( c_2Erat_2ERATD @ ( c_2Erat_2Erat__of__int @ V0i ) )
= ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ).
thf(thm_2Erat_2ERATN__DIV__RATD,axiom,
! [V0r: tyop_2Erat_2Erat] :
( ( c_2Erat_2Erat__div @ ( c_2Erat_2Erat__of__int @ ( c_2Erat_2ERATN @ V0r ) ) @ ( c_2Erat_2Erat__of__num @ ( c_2Erat_2ERATD @ V0r ) ) )
= V0r ) ).
thf(thm_2Erat_2ERAT__AINV__EQ__NUM,axiom,
! [V0x: tyop_2Erat_2Erat,V1n: tyop_2Enum_2Enum] :
( ( ( c_2Erat_2Erat__ainv @ V0x )
= ( c_2Erat_2Erat__of__num @ V1n ) )
<=> ( V0x
= ( c_2Erat_2Erat__of__int @ ( c_2Einteger_2Eint__neg @ ( c_2Einteger_2Eint__of__num @ V1n ) ) ) ) ) ).
thf(thm_2Erat_2ERAT__SGN__NUM__COND,axiom,
! [V0n: tyop_2Enum_2Enum] :
( ( c_2Erat_2Erat__sgn @ ( c_2Erat_2Erat__of__num @ V0n ) )
= ( c_2Ebool_2ECOND @ tyop_2Einteger_2Eint @ ( c_2Emin_2E_3D @ tyop_2Enum_2Enum @ V0n @ c_2Enum_2E0 ) @ ( c_2Einteger_2Eint__of__num @ c_2Enum_2E0 ) @ ( c_2Einteger_2Eint__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ) ).
thf(thm_2Erat_2ERAT__SGN__AINV__RWT,axiom,
! [V0r: tyop_2Erat_2Erat] :
( ( c_2Erat_2Erat__sgn @ ( c_2Erat_2Erat__ainv @ V0r ) )
= ( c_2Einteger_2Eint__neg @ ( c_2Erat_2Erat__sgn @ V0r ) ) ) ).
thf(thm_2Erat_2ERAT__SGN__ALT,axiom,
! [V0r: tyop_2Erat_2Erat] :
( ( c_2Erat_2Erat__sgn @ V0r )
= ( c_2EintExtension_2ESGN @ ( c_2Erat_2ERATN @ V0r ) ) ) ).
thf(thm_2Erat_2ERAT__SGN__NUM__BITs,axiom,
! [V0n: tyop_2Enum_2Enum] :
( ( ( c_2Erat_2Erat__sgn @ ( c_2Erat_2Erat__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ V0n ) ) ) )
= ( c_2Einteger_2Eint__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) )
& ( ( c_2Erat_2Erat__sgn @ ( c_2Erat_2Erat__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ V0n ) ) ) )
= ( c_2Einteger_2Eint__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ) ).
thf(thm_2Erat_2ERAT__SGN__EQ0,axiom,
! [V0r: tyop_2Erat_2Erat] :
( ( ( ( c_2Erat_2Erat__sgn @ V0r )
= ( c_2Einteger_2Eint__of__num @ c_2Enum_2E0 ) )
<=> ( V0r
= ( c_2Erat_2Erat__of__num @ c_2Enum_2E0 ) ) )
& ( ( ( c_2Einteger_2Eint__of__num @ c_2Enum_2E0 )
= ( c_2Erat_2Erat__sgn @ V0r ) )
<=> ( V0r
= ( c_2Erat_2Erat__of__num @ c_2Enum_2E0 ) ) ) ) ).
thf(thm_2Erat_2ERAT__SGN__POS,axiom,
! [V0r: tyop_2Erat_2Erat] :
( ( ( c_2Erat_2Erat__sgn @ V0r )
= ( c_2Einteger_2Eint__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) )
<=> ( c_2Erat_2Erat__les @ ( c_2Erat_2Erat__of__num @ c_2Enum_2E0 ) @ V0r ) ) ).
thf(thm_2Erat_2ERAT__SGN__NEG,axiom,
! [V0r: tyop_2Erat_2Erat] :
( ( ( c_2Erat_2Erat__sgn @ V0r )
= ( c_2Einteger_2Eint__neg @ ( c_2Einteger_2Eint__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) )
<=> ( c_2Erat_2Erat__les @ V0r @ ( c_2Erat_2Erat__of__num @ c_2Enum_2E0 ) ) ) ).
thf(thm_2Erat_2ERAT__SGN__DIV,axiom,
! [V0n: tyop_2Erat_2Erat,V1d: tyop_2Erat_2Erat] :
( ( (~)
@ ( V1d
= ( c_2Erat_2Erat__of__num @ c_2Enum_2E0 ) ) )
=> ( ( c_2Erat_2Erat__sgn @ ( c_2Erat_2Erat__div @ V0n @ V1d ) )
= ( c_2Einteger_2Eint__mul @ ( c_2Erat_2Erat__sgn @ V0n ) @ ( c_2Erat_2Erat__sgn @ V1d ) ) ) ) ).
thf(thm_2Erat_2ERAT__MINV__RATND,axiom,
! [V0r: tyop_2Erat_2Erat] :
( ( (~)
@ ( V0r
= ( c_2Erat_2Erat__of__num @ c_2Enum_2E0 ) ) )
=> ( ( c_2Erat_2Erat__minv @ V0r )
= ( c_2Erat_2Erat__div @ ( c_2Erat_2Erat__mul @ ( c_2Erat_2Erat__of__int @ ( c_2Erat_2Erat__sgn @ V0r ) ) @ ( c_2Erat_2Erat__of__num @ ( c_2Erat_2ERATD @ V0r ) ) ) @ ( c_2Erat_2Erat__of__int @ ( c_2Einteger_2EABS @ ( c_2Erat_2ERATN @ V0r ) ) ) ) ) ) ).
%------------------------------------------------------------------------------