ITP001 Axioms: ITP089^7.ax
%------------------------------------------------------------------------------
% File : ITP089^7 : 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 : toto.ax [Gau19]
% : HL4089^7.ax [TPAP]
% Status : Satisfiable
% Syntax : Number of formulae : 225 ( 50 unt; 94 typ; 0 def)
% Number of atoms : 421 ( 239 equ; 19 cnn)
% Maximal formula atoms : 18 ( 1 avg)
% Number of connectives : 2002 ( 19 ~; 8 |; 110 &;1755 @)
% ( 34 <=>; 76 =>; 0 <=; 0 <~>)
% Maximal formula depth : 28 ( 7 avg;1755 nst)
% Number of types : 6 ( 5 usr)
% Number of type conns : 367 ( 367 >; 0 *; 0 +; 0 <<)
% Number of symbols : 91 ( 89 usr; 13 con; 0-6 aty)
% Number of variables : 568 ( 19 ^ 476 !; 12 ?; 568 :)
% ( 61 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_SAT_EQU_NAR
% Comments :
% Bugfixes : v7.5.0 - Fixes to the axioms.
%------------------------------------------------------------------------------
thf(tyop_2Eind__type_2Erecspace,type,
tyop_2Eind__type_2Erecspace: $tType > $tType ).
thf(tyop_2Elist_2Elist,type,
tyop_2Elist_2Elist: $tType > $tType ).
thf(tyop_2Emin_2Ebool,type,
tyop_2Emin_2Ebool: $tType ).
thf(tyop_2Emin_2Efun,type,
tyop_2Emin_2Efun: $tType > $tType > $tType ).
thf(tyop_2Enum_2Enum,type,
tyop_2Enum_2Enum: $tType ).
thf(tyop_2Epair_2Eprod,type,
tyop_2Epair_2Eprod: $tType > $tType > $tType ).
thf(tyop_2Estring_2Echar,type,
tyop_2Estring_2Echar: $tType ).
thf(tyop_2EternaryComparisons_2Eordering,type,
tyop_2EternaryComparisons_2Eordering: $tType ).
thf(tyop_2Etoto_2Enum__dt,type,
tyop_2Etoto_2Enum__dt: $tType ).
thf(tyop_2Etoto_2Etoto,type,
tyop_2Etoto_2Etoto: $tType > $tType ).
thf(c_2Ebool_2E_21,type,
c_2Ebool_2E_21:
!>[A_27a: $tType] : ( ( A_27a > $o ) > $o ) ).
thf(c_2Earithmetic_2E_2B,type,
c_2Earithmetic_2E_2B: tyop_2Enum_2Enum > tyop_2Enum_2Enum > tyop_2Enum_2Enum ).
thf(c_2Epair_2E_2C,type,
c_2Epair_2E_2C:
!>[A_27a: $tType,A_27b: $tType] : ( A_27a > A_27b > ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) ) ).
thf(c_2Earithmetic_2E_2D,type,
c_2Earithmetic_2E_2D: tyop_2Enum_2Enum > tyop_2Enum_2Enum > tyop_2Enum_2Enum ).
thf(c_2Ebool_2E_2F_5C,type,
c_2Ebool_2E_2F_5C: $o > $o > $o ).
thf(c_2Enum_2E0,type,
c_2Enum_2E0: tyop_2Enum_2Enum ).
thf(c_2Eprim__rec_2E_3C,type,
c_2Eprim__rec_2E_3C: tyop_2Enum_2Enum > tyop_2Enum_2Enum > $o ).
thf(c_2Emin_2E_3D,type,
c_2Emin_2E_3D:
!>[A_27a: $tType] : ( A_27a > A_27a > $o ) ).
thf(c_2Emin_2E_3D_3D_3E,type,
c_2Emin_2E_3D_3D_3E: $o > $o > $o ).
thf(c_2Ebool_2E_3F,type,
c_2Ebool_2E_3F:
!>[A_27a: $tType] : ( ( A_27a > $o ) > $o ) ).
thf(c_2Emin_2E_40,type,
c_2Emin_2E_40:
!>[A_27a: $tType] : ( ( A_27a > $o ) > A_27a ) ).
thf(c_2Ebool_2EARB,type,
c_2Ebool_2EARB:
!>[A_27a: $tType] : A_27a ).
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_2Eind__type_2EBOTTOM,type,
c_2Eind__type_2EBOTTOM:
!>[A_27a: $tType] : ( tyop_2Eind__type_2Erecspace @ A_27a ) ).
thf(c_2Estring_2ECHR,type,
c_2Estring_2ECHR: tyop_2Enum_2Enum > tyop_2Estring_2Echar ).
thf(c_2Ebool_2ECOND,type,
c_2Ebool_2ECOND:
!>[A_27a: $tType] : ( $o > A_27a > A_27a > A_27a ) ).
thf(c_2Elist_2ECONS,type,
c_2Elist_2ECONS:
!>[A_27a: $tType] : ( A_27a > ( tyop_2Elist_2Elist @ A_27a ) > ( tyop_2Elist_2Elist @ A_27a ) ) ).
thf(c_2Eind__type_2ECONSTR,type,
c_2Eind__type_2ECONSTR:
!>[A_27a: $tType] : ( tyop_2Enum_2Enum > A_27a > ( tyop_2Enum_2Enum > ( tyop_2Eind__type_2Erecspace @ A_27a ) ) > ( tyop_2Eind__type_2Erecspace @ A_27a ) ) ).
thf(c_2Ebool_2EDATATYPE,type,
c_2Ebool_2EDATATYPE:
!>[A_27a: $tType] : ( A_27a > $o ) ).
thf(c_2Earithmetic_2EDIV2,type,
c_2Earithmetic_2EDIV2: tyop_2Enum_2Enum > tyop_2Enum_2Enum ).
thf(c_2EternaryComparisons_2EEQUAL,type,
c_2EternaryComparisons_2EEQUAL: tyop_2EternaryComparisons_2Eordering ).
thf(c_2Ebool_2EF,type,
c_2Ebool_2EF: $o ).
thf(c_2Eind__type_2EFCONS,type,
c_2Eind__type_2EFCONS:
!>[A_27a: $tType] : ( A_27a > ( tyop_2Enum_2Enum > A_27a ) > tyop_2Enum_2Enum > A_27a ) ).
thf(c_2Epair_2EFST,type,
c_2Epair_2EFST:
!>[A_27a: $tType,A_27b: $tType] : ( ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) > A_27a ) ).
thf(c_2EternaryComparisons_2EGREATER,type,
c_2EternaryComparisons_2EGREATER: tyop_2EternaryComparisons_2Eordering ).
thf(c_2Ecombin_2EI,type,
c_2Ecombin_2EI:
!>[A_27a: $tType] : ( A_27a > A_27a ) ).
thf(c_2EternaryComparisons_2ELESS,type,
c_2EternaryComparisons_2ELESS: tyop_2EternaryComparisons_2Eordering ).
thf(c_2Epair_2ELEX,type,
c_2Epair_2ELEX:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > A_27a > $o ) > ( A_27b > A_27b > $o ) > ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) > ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) > $o ) ).
thf(c_2Erelation_2ELinearOrder,type,
c_2Erelation_2ELinearOrder:
!>[A_27a: $tType] : ( ( A_27a > A_27a > $o ) > $o ) ).
thf(c_2Etoto_2EListOrd,type,
c_2Etoto_2EListOrd:
!>[A_27a: $tType] : ( ( tyop_2Etoto_2Etoto @ A_27a ) > ( tyop_2Elist_2Elist @ A_27a ) > ( tyop_2Elist_2Elist @ A_27a ) > tyop_2EternaryComparisons_2Eordering ) ).
thf(c_2Elist_2ENIL,type,
c_2Elist_2ENIL:
!>[A_27a: $tType] : ( tyop_2Elist_2Elist @ A_27a ) ).
thf(c_2Earithmetic_2ENUMERAL,type,
c_2Earithmetic_2ENUMERAL: tyop_2Enum_2Enum > tyop_2Enum_2Enum ).
thf(c_2Earithmetic_2EODD,type,
c_2Earithmetic_2EODD: tyop_2Enum_2Enum > $o ).
thf(c_2Ebool_2EONE__ONE,type,
c_2Ebool_2EONE__ONE:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > A_27b ) > $o ) ).
thf(c_2Estring_2EORD,type,
c_2Estring_2EORD: tyop_2Estring_2Echar > tyop_2Enum_2Enum ).
thf(c_2Epair_2ESND,type,
c_2Epair_2ESND:
!>[A_27a: $tType,A_27b: $tType] : ( ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) > A_27b ) ).
thf(c_2Erelation_2ESTRORD,type,
c_2Erelation_2ESTRORD:
!>[A_27a: $tType] : ( ( A_27a > A_27a > $o ) > A_27a > A_27a > $o ) ).
thf(c_2Enum_2ESUC,type,
c_2Enum_2ESUC: tyop_2Enum_2Enum > tyop_2Enum_2Enum ).
thf(c_2Erelation_2EStrongLinearOrder,type,
c_2Erelation_2EStrongLinearOrder:
!>[A_27a: $tType] : ( ( A_27a > A_27a > $o ) > $o ) ).
thf(c_2Etoto_2EStrongLinearOrder__of__TO,type,
c_2Etoto_2EStrongLinearOrder__of__TO:
!>[A_27a: $tType] : ( ( A_27a > A_27a > tyop_2EternaryComparisons_2Eordering ) > A_27a > A_27a > $o ) ).
thf(c_2Erelation_2EStrongOrder,type,
c_2Erelation_2EStrongOrder:
!>[A_27g: $tType] : ( ( A_27g > A_27g > $o ) > $o ) ).
thf(c_2Ebool_2ET,type,
c_2Ebool_2ET: $o ).
thf(c_2Etoto_2ETO,type,
c_2Etoto_2ETO:
!>[A_27a: $tType] : ( ( A_27a > A_27a > tyop_2EternaryComparisons_2Eordering ) > ( tyop_2Etoto_2Etoto @ A_27a ) ) ).
thf(c_2Etoto_2ETO__inv,type,
c_2Etoto_2ETO__inv:
!>[A_27a: $tType] : ( ( A_27a > A_27a > tyop_2EternaryComparisons_2Eordering ) > A_27a > A_27a > tyop_2EternaryComparisons_2Eordering ) ).
thf(c_2Etoto_2ETO__of__LinearOrder,type,
c_2Etoto_2ETO__of__LinearOrder:
!>[A_27a: $tType] : ( ( A_27a > A_27a > $o ) > A_27a > A_27a > tyop_2EternaryComparisons_2Eordering ) ).
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_2Etoto_2ETotOrd,type,
c_2Etoto_2ETotOrd:
!>[A_27a: $tType] : ( ( A_27a > A_27a > tyop_2EternaryComparisons_2Eordering ) > $o ) ).
thf(c_2Epair_2EUNCURRY,type,
c_2Epair_2EUNCURRY:
!>[A_27a: $tType,A_27b: $tType,A_27c: $tType] : ( ( A_27a > A_27b > A_27c ) > ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) > A_27c ) ).
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_2Erelation_2EWeakLinearOrder,type,
c_2Erelation_2EWeakLinearOrder:
!>[A_27a: $tType] : ( ( A_27a > A_27a > $o ) > $o ) ).
thf(c_2Etoto_2EWeakLinearOrder__of__TO,type,
c_2Etoto_2EWeakLinearOrder__of__TO:
!>[A_27a: $tType] : ( ( A_27a > A_27a > tyop_2EternaryComparisons_2Eordering ) > A_27a > A_27a > $o ) ).
thf(c_2Earithmetic_2EZERO,type,
c_2Earithmetic_2EZERO: tyop_2Enum_2Enum ).
thf(c_2Ebool_2E_5C_2F,type,
c_2Ebool_2E_5C_2F: $o > $o > $o ).
thf(c_2Etoto_2Eapto,type,
c_2Etoto_2Eapto:
!>[A_27a: $tType] : ( ( tyop_2Etoto_2Etoto @ A_27a ) > A_27a > A_27a > tyop_2EternaryComparisons_2Eordering ) ).
thf(c_2Etoto_2Ebit1,type,
c_2Etoto_2Ebit1: tyop_2Etoto_2Enum__dt > tyop_2Etoto_2Enum__dt ).
thf(c_2Etoto_2Ebit2,type,
c_2Etoto_2Ebit2: tyop_2Etoto_2Enum__dt > tyop_2Etoto_2Enum__dt ).
thf(c_2Etoto_2EcharOrd,type,
c_2Etoto_2EcharOrd: tyop_2Estring_2Echar > tyop_2Estring_2Echar > tyop_2EternaryComparisons_2Eordering ).
thf(c_2Estring_2Echar__lt,type,
c_2Estring_2Echar__lt: tyop_2Estring_2Echar > tyop_2Estring_2Echar > $o ).
thf(c_2Etoto_2Echarto,type,
c_2Etoto_2Echarto: tyop_2Etoto_2Etoto @ tyop_2Estring_2Echar ).
thf(c_2Etoto_2EimageOrd,type,
c_2Etoto_2EimageOrd:
!>[A_27a: $tType,A_27c: $tType] : ( ( A_27a > A_27c ) > ( A_27c > A_27c > tyop_2EternaryComparisons_2Eordering ) > A_27a > A_27a > tyop_2EternaryComparisons_2Eordering ) ).
thf(c_2Erelation_2Einv,type,
c_2Erelation_2Einv:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > A_27b > $o ) > A_27b > A_27a > $o ) ).
thf(c_2Erelation_2Eirreflexive,type,
c_2Erelation_2Eirreflexive:
!>[A_27a: $tType] : ( ( A_27a > A_27a > $o ) > $o ) ).
thf(c_2Etoto_2ElexTO,type,
c_2Etoto_2ElexTO:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > A_27a > tyop_2EternaryComparisons_2Eordering ) > ( A_27b > A_27b > tyop_2EternaryComparisons_2Eordering ) > ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) > ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) > tyop_2EternaryComparisons_2Eordering ) ).
thf(c_2Etoto_2Elextoto,type,
c_2Etoto_2Elextoto:
!>[A_27a: $tType,A_27b: $tType] : ( ( tyop_2Etoto_2Etoto @ A_27a ) > ( tyop_2Etoto_2Etoto @ A_27b ) > ( tyop_2Etoto_2Etoto @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) ) ) ).
thf(c_2Etoto_2Elistorder,type,
c_2Etoto_2Elistorder:
!>[A_27a: $tType] : ( ( A_27a > A_27a > $o ) > ( tyop_2Elist_2Elist @ A_27a ) > ( tyop_2Elist_2Elist @ A_27a ) > $o ) ).
thf(c_2Etoto_2Elistoto,type,
c_2Etoto_2Elistoto:
!>[A_27a: $tType] : ( ( tyop_2Etoto_2Etoto @ A_27a ) > ( tyop_2Etoto_2Etoto @ ( tyop_2Elist_2Elist @ A_27a ) ) ) ).
thf(c_2Etoto_2EnumOrd,type,
c_2Etoto_2EnumOrd: tyop_2Enum_2Enum > tyop_2Enum_2Enum > tyop_2EternaryComparisons_2Eordering ).
thf(c_2Etoto_2Enum__dtOrd,type,
c_2Etoto_2Enum__dtOrd: tyop_2Etoto_2Enum__dt > tyop_2Etoto_2Enum__dt > tyop_2EternaryComparisons_2Eordering ).
thf(c_2Etoto_2Enum__dt__CASE,type,
c_2Etoto_2Enum__dt__CASE:
!>[A_27a: $tType] : ( tyop_2Etoto_2Enum__dt > A_27a > ( tyop_2Etoto_2Enum__dt > A_27a ) > ( tyop_2Etoto_2Enum__dt > A_27a ) > A_27a ) ).
thf(c_2Etoto_2Enum__dt__size,type,
c_2Etoto_2Enum__dt__size: tyop_2Etoto_2Enum__dt > tyop_2Enum_2Enum ).
thf(c_2Etoto_2Enum__to__dt,type,
c_2Etoto_2Enum__to__dt: tyop_2Enum_2Enum > tyop_2Etoto_2Enum__dt ).
thf(c_2Etoto_2Enumto,type,
c_2Etoto_2Enumto: tyop_2Etoto_2Etoto @ tyop_2Enum_2Enum ).
thf(c_2EternaryComparisons_2Eordering__CASE,type,
c_2EternaryComparisons_2Eordering__CASE:
!>[A_27a: $tType] : ( tyop_2EternaryComparisons_2Eordering > A_27a > A_27a > A_27a > A_27a ) ).
thf(c_2Etoto_2Eqk__numOrd,type,
c_2Etoto_2Eqk__numOrd: tyop_2Enum_2Enum > tyop_2Enum_2Enum > tyop_2EternaryComparisons_2Eordering ).
thf(c_2Etoto_2Eqk__numto,type,
c_2Etoto_2Eqk__numto: tyop_2Etoto_2Etoto @ tyop_2Enum_2Enum ).
thf(c_2Etoto_2Estringto,type,
c_2Etoto_2Estringto: tyop_2Etoto_2Etoto @ ( tyop_2Elist_2Elist @ tyop_2Estring_2Echar ) ).
thf(c_2Etoto_2Etoto__inv,type,
c_2Etoto_2Etoto__inv:
!>[A_27a: $tType] : ( ( tyop_2Etoto_2Etoto @ A_27a ) > ( tyop_2Etoto_2Etoto @ A_27a ) ) ).
thf(c_2Etoto_2Etoto__of__LinearOrder,type,
c_2Etoto_2Etoto__of__LinearOrder:
!>[A_27a: $tType] : ( ( A_27a > A_27a > $o ) > ( tyop_2Etoto_2Etoto @ A_27a ) ) ).
thf(c_2Erelation_2Etransitive,type,
c_2Erelation_2Etransitive:
!>[A_27a: $tType] : ( ( A_27a > A_27a > $o ) > $o ) ).
thf(c_2Erelation_2Etrichotomous,type,
c_2Erelation_2Etrichotomous:
!>[A_27a: $tType] : ( ( A_27a > A_27a > $o ) > $o ) ).
thf(c_2Etoto_2Ezer,type,
c_2Etoto_2Ezer: tyop_2Etoto_2Enum__dt ).
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_2Etoto_2ETotOrd,axiom,
! [A_27a: $tType,V0c: A_27a > A_27a > tyop_2EternaryComparisons_2Eordering] :
( ( c_2Etoto_2ETotOrd @ A_27a @ V0c )
<=> ( ! [V1x: A_27a,V2y: A_27a] :
( ( ( V0c @ V1x @ V2y )
= c_2EternaryComparisons_2EEQUAL )
<=> ( V1x = V2y ) )
& ! [V3x: A_27a,V4y: A_27a] :
( ( ( V0c @ V3x @ V4y )
= c_2EternaryComparisons_2EGREATER )
<=> ( ( V0c @ V4y @ V3x )
= c_2EternaryComparisons_2ELESS ) )
& ! [V5x: A_27a,V6y: A_27a,V7z: A_27a] :
( ( ( ( V0c @ V5x @ V6y )
= c_2EternaryComparisons_2ELESS )
& ( ( V0c @ V6y @ V7z )
= c_2EternaryComparisons_2ELESS ) )
=> ( ( V0c @ V5x @ V7z )
= c_2EternaryComparisons_2ELESS ) ) ) ) ).
thf(thm_2Etoto_2ETO__of__LinearOrder,axiom,
! [A_27a: $tType,V0r: A_27a > A_27a > $o,V1x: A_27a,V2y: A_27a] :
( ( c_2Etoto_2ETO__of__LinearOrder @ A_27a @ V0r @ V1x @ V2y )
= ( c_2Ebool_2ECOND @ tyop_2EternaryComparisons_2Eordering @ ( c_2Emin_2E_3D @ A_27a @ V1x @ V2y ) @ c_2EternaryComparisons_2EEQUAL @ ( c_2Ebool_2ECOND @ tyop_2EternaryComparisons_2Eordering @ ( V0r @ V1x @ V2y ) @ c_2EternaryComparisons_2ELESS @ c_2EternaryComparisons_2EGREATER ) ) ) ).
thf(thm_2Etoto_2Etoto__TY__DEF,axiom,
! [A_27a: $tType] :
? [V0rep: ( tyop_2Etoto_2Etoto @ A_27a ) > A_27a > A_27a > tyop_2EternaryComparisons_2Eordering] : ( c_2Ebool_2ETYPE__DEFINITION @ ( A_27a > A_27a > tyop_2EternaryComparisons_2Eordering ) @ ( tyop_2Etoto_2Etoto @ A_27a ) @ ( c_2Etoto_2ETotOrd @ A_27a ) @ V0rep ) ).
thf(thm_2Etoto_2Eto__bij,axiom,
! [A_27a: $tType] :
( ! [V0a: tyop_2Etoto_2Etoto @ A_27a] :
( ( c_2Etoto_2ETO @ A_27a @ ( c_2Etoto_2Eapto @ A_27a @ V0a ) )
= V0a )
& ! [V1r: A_27a > A_27a > tyop_2EternaryComparisons_2Eordering] :
( ( c_2Etoto_2ETotOrd @ A_27a @ V1r )
<=> ( ( c_2Etoto_2Eapto @ A_27a @ ( c_2Etoto_2ETO @ A_27a @ V1r ) )
= V1r ) ) ) ).
thf(thm_2Etoto_2EWeakLinearOrder__of__TO,axiom,
! [A_27a: $tType,V0c: A_27a > A_27a > tyop_2EternaryComparisons_2Eordering,V1x: A_27a,V2y: A_27a] :
( ( c_2Etoto_2EWeakLinearOrder__of__TO @ A_27a @ V0c @ V1x @ V2y )
= ( c_2EternaryComparisons_2Eordering__CASE @ $o @ ( V0c @ V1x @ V2y ) @ c_2Ebool_2ET @ c_2Ebool_2ET @ c_2Ebool_2EF ) ) ).
thf(thm_2Etoto_2EStrongLinearOrder__of__TO,axiom,
! [A_27a: $tType,V0c: A_27a > A_27a > tyop_2EternaryComparisons_2Eordering,V1x: A_27a,V2y: A_27a] :
( ( c_2Etoto_2EStrongLinearOrder__of__TO @ A_27a @ V0c @ V1x @ V2y )
= ( c_2EternaryComparisons_2Eordering__CASE @ $o @ ( V0c @ V1x @ V2y ) @ c_2Ebool_2ET @ c_2Ebool_2EF @ c_2Ebool_2EF ) ) ).
thf(thm_2Etoto_2Etoto__of__LinearOrder,axiom,
! [A_27a: $tType,V0r: A_27a > A_27a > $o] :
( ( c_2Etoto_2Etoto__of__LinearOrder @ A_27a @ V0r )
= ( c_2Etoto_2ETO @ A_27a @ ( c_2Etoto_2ETO__of__LinearOrder @ A_27a @ V0r ) ) ) ).
thf(thm_2Etoto_2ETO__inv,axiom,
! [A_27a: $tType,V0c: A_27a > A_27a > tyop_2EternaryComparisons_2Eordering,V1x: A_27a,V2y: A_27a] :
( ( c_2Etoto_2ETO__inv @ A_27a @ V0c @ V1x @ V2y )
= ( V0c @ V2y @ V1x ) ) ).
thf(thm_2Etoto_2Etoto__inv,axiom,
! [A_27a: $tType,V0c: tyop_2Etoto_2Etoto @ A_27a] :
( ( c_2Etoto_2Etoto__inv @ A_27a @ V0c )
= ( c_2Etoto_2ETO @ A_27a @ ( c_2Etoto_2ETO__inv @ A_27a @ ( c_2Etoto_2Eapto @ A_27a @ V0c ) ) ) ) ).
thf(thm_2Etoto_2ElexTO,axiom,
! [A_27a: $tType,A_27b: $tType,V0R: A_27a > A_27a > tyop_2EternaryComparisons_2Eordering,V1V: A_27b > A_27b > tyop_2EternaryComparisons_2Eordering] :
( ( c_2Etoto_2ElexTO @ A_27a @ A_27b @ V0R @ V1V )
= ( c_2Etoto_2ETO__of__LinearOrder @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ ( c_2Epair_2ELEX @ A_27a @ A_27b @ ( c_2Etoto_2EStrongLinearOrder__of__TO @ A_27a @ V0R ) @ ( c_2Etoto_2EStrongLinearOrder__of__TO @ A_27b @ V1V ) ) ) ) ).
thf(thm_2Etoto_2Elextoto,axiom,
! [A_27a: $tType,A_27b: $tType,V0c: tyop_2Etoto_2Etoto @ A_27a,V1v: tyop_2Etoto_2Etoto @ A_27b] :
( ( c_2Etoto_2Elextoto @ A_27a @ A_27b @ V0c @ V1v )
= ( c_2Etoto_2ETO @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ ( c_2Etoto_2ElexTO @ A_27a @ A_27b @ ( c_2Etoto_2Eapto @ A_27a @ V0c ) @ ( c_2Etoto_2Eapto @ A_27b @ V1v ) ) ) ) ).
thf(thm_2Etoto_2EnumOrd,axiom,
( c_2Etoto_2EnumOrd
= ( c_2Etoto_2ETO__of__LinearOrder @ tyop_2Enum_2Enum @ c_2Eprim__rec_2E_3C ) ) ).
thf(thm_2Etoto_2Enumto,axiom,
( c_2Etoto_2Enumto
= ( c_2Etoto_2ETO @ tyop_2Enum_2Enum @ c_2Etoto_2EnumOrd ) ) ).
thf(thm_2Etoto_2Enum__dt__TY__DEF,axiom,
? [V0rep: tyop_2Etoto_2Enum__dt > ( tyop_2Eind__type_2Erecspace @ $o )] :
( c_2Ebool_2ETYPE__DEFINITION @ ( tyop_2Eind__type_2Erecspace @ $o ) @ tyop_2Etoto_2Enum__dt
@ ^ [V1a0: tyop_2Eind__type_2Erecspace @ $o] :
( c_2Ebool_2E_21 @ ( ( tyop_2Eind__type_2Erecspace @ $o ) > $o )
@ ^ [V2_27num__dt_27: ( tyop_2Eind__type_2Erecspace @ $o ) > $o] :
( c_2Emin_2E_3D_3D_3E
@ ( c_2Ebool_2E_21 @ ( tyop_2Eind__type_2Erecspace @ $o )
@ ^ [V3a0: tyop_2Eind__type_2Erecspace @ $o] :
( c_2Emin_2E_3D_3D_3E
@ ( c_2Ebool_2E_5C_2F
@ ( c_2Emin_2E_3D @ ( tyop_2Eind__type_2Erecspace @ $o ) @ V3a0
@ ( c_2Eind__type_2ECONSTR @ $o @ c_2Enum_2E0 @ ( c_2Ebool_2EARB @ $o )
@ ^ [V4n: tyop_2Enum_2Enum] : ( c_2Eind__type_2EBOTTOM @ $o ) ) )
@ ( c_2Ebool_2E_5C_2F
@ ( c_2Ebool_2E_3F @ ( tyop_2Eind__type_2Erecspace @ $o )
@ ^ [V5a: tyop_2Eind__type_2Erecspace @ $o] :
( c_2Ebool_2E_2F_5C
@ ( c_2Emin_2E_3D @ ( tyop_2Eind__type_2Erecspace @ $o ) @ V3a0
@ ( ^ [V6a: tyop_2Eind__type_2Erecspace @ $o] :
( c_2Eind__type_2ECONSTR @ $o @ ( c_2Enum_2ESUC @ c_2Enum_2E0 ) @ ( c_2Ebool_2EARB @ $o )
@ ( c_2Eind__type_2EFCONS @ ( tyop_2Eind__type_2Erecspace @ $o ) @ V6a
@ ^ [V7n: tyop_2Enum_2Enum] : ( c_2Eind__type_2EBOTTOM @ $o ) ) )
@ V5a ) )
@ ( V2_27num__dt_27 @ V5a ) ) )
@ ( c_2Ebool_2E_3F @ ( tyop_2Eind__type_2Erecspace @ $o )
@ ^ [V8a: tyop_2Eind__type_2Erecspace @ $o] :
( c_2Ebool_2E_2F_5C
@ ( c_2Emin_2E_3D @ ( tyop_2Eind__type_2Erecspace @ $o ) @ V3a0
@ ( ^ [V9a: tyop_2Eind__type_2Erecspace @ $o] :
( c_2Eind__type_2ECONSTR @ $o @ ( c_2Enum_2ESUC @ ( c_2Enum_2ESUC @ c_2Enum_2E0 ) ) @ ( c_2Ebool_2EARB @ $o )
@ ( c_2Eind__type_2EFCONS @ ( tyop_2Eind__type_2Erecspace @ $o ) @ V9a
@ ^ [V10n: tyop_2Enum_2Enum] : ( c_2Eind__type_2EBOTTOM @ $o ) ) )
@ V8a ) )
@ ( V2_27num__dt_27 @ V8a ) ) ) ) )
@ ( V2_27num__dt_27 @ V3a0 ) ) )
@ ( V2_27num__dt_27 @ V1a0 ) ) )
@ V0rep ) ).
thf(thm_2Etoto_2Enum__dt__case__def,axiom,
! [A_27a: $tType] :
( ! [V0v: A_27a,V1f: tyop_2Etoto_2Enum__dt > A_27a,V2f1: tyop_2Etoto_2Enum__dt > A_27a] :
( ( c_2Etoto_2Enum__dt__CASE @ A_27a @ c_2Etoto_2Ezer @ V0v @ V1f @ V2f1 )
= V0v )
& ! [V3a: tyop_2Etoto_2Enum__dt,V4v: A_27a,V5f: tyop_2Etoto_2Enum__dt > A_27a,V6f1: tyop_2Etoto_2Enum__dt > A_27a] :
( ( c_2Etoto_2Enum__dt__CASE @ A_27a @ ( c_2Etoto_2Ebit1 @ V3a ) @ V4v @ V5f @ V6f1 )
= ( V5f @ V3a ) )
& ! [V7a: tyop_2Etoto_2Enum__dt,V8v: A_27a,V9f: tyop_2Etoto_2Enum__dt > A_27a,V10f1: tyop_2Etoto_2Enum__dt > A_27a] :
( ( c_2Etoto_2Enum__dt__CASE @ A_27a @ ( c_2Etoto_2Ebit2 @ V7a ) @ V8v @ V9f @ V10f1 )
= ( V10f1 @ V7a ) ) ) ).
thf(thm_2Etoto_2Enum__dt__size__def,axiom,
( ( ( c_2Etoto_2Enum__dt__size @ c_2Etoto_2Ezer )
= c_2Enum_2E0 )
& ! [V0a: tyop_2Etoto_2Enum__dt] :
( ( c_2Etoto_2Enum__dt__size @ ( c_2Etoto_2Ebit1 @ V0a ) )
= ( c_2Earithmetic_2E_2B @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) @ ( c_2Etoto_2Enum__dt__size @ V0a ) ) )
& ! [V1a: tyop_2Etoto_2Enum__dt] :
( ( c_2Etoto_2Enum__dt__size @ ( c_2Etoto_2Ebit2 @ V1a ) )
= ( c_2Earithmetic_2E_2B @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) @ ( c_2Etoto_2Enum__dt__size @ V1a ) ) ) ) ).
thf(thm_2Etoto_2Enum__to__dt__primitive,axiom,
( c_2Etoto_2Enum__to__dt
= ( c_2Erelation_2EWFREC @ tyop_2Enum_2Enum @ tyop_2Etoto_2Enum__dt
@ ( 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_2F_5C
@ ( c_2Ebool_2E_21 @ tyop_2Enum_2Enum
@ ^ [V1n: tyop_2Enum_2Enum] : ( c_2Emin_2E_3D_3D_3E @ ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2E_7E @ ( c_2Emin_2E_3D @ tyop_2Enum_2Enum @ V1n @ c_2Enum_2E0 ) ) @ ( c_2Earithmetic_2EODD @ V1n ) ) @ ( V0R @ ( c_2Earithmetic_2EDIV2 @ ( c_2Earithmetic_2E_2D @ V1n @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) @ V1n ) ) )
@ ( c_2Ebool_2E_21 @ tyop_2Enum_2Enum
@ ^ [V2n: tyop_2Enum_2Enum] : ( c_2Emin_2E_3D_3D_3E @ ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2E_7E @ ( c_2Emin_2E_3D @ tyop_2Enum_2Enum @ V2n @ c_2Enum_2E0 ) ) @ ( c_2Ebool_2E_7E @ ( c_2Earithmetic_2EODD @ V2n ) ) ) @ ( V0R @ ( c_2Earithmetic_2EDIV2 @ ( c_2Earithmetic_2E_2D @ V2n @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) ) @ V2n ) ) ) ) ) )
@ ^ [V3num__to__dt: tyop_2Enum_2Enum > tyop_2Etoto_2Enum__dt,V4a: tyop_2Enum_2Enum] : ( c_2Ecombin_2EI @ tyop_2Etoto_2Enum__dt @ ( c_2Ebool_2ECOND @ tyop_2Etoto_2Enum__dt @ ( c_2Emin_2E_3D @ tyop_2Enum_2Enum @ V4a @ c_2Enum_2E0 ) @ c_2Etoto_2Ezer @ ( c_2Ebool_2ECOND @ tyop_2Etoto_2Enum__dt @ ( c_2Earithmetic_2EODD @ V4a ) @ ( c_2Etoto_2Ebit1 @ ( V3num__to__dt @ ( c_2Earithmetic_2EDIV2 @ ( c_2Earithmetic_2E_2D @ V4a @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ) ) @ ( c_2Etoto_2Ebit2 @ ( V3num__to__dt @ ( c_2Earithmetic_2EDIV2 @ ( c_2Earithmetic_2E_2D @ V4a @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) ) ) ) ) ) ) ) ) ).
thf(thm_2Etoto_2Eqk__numOrd__def,axiom,
! [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum] :
( ( c_2Etoto_2Eqk__numOrd @ V0m @ V1n )
= ( c_2Etoto_2Enum__dtOrd @ ( c_2Etoto_2Enum__to__dt @ V0m ) @ ( c_2Etoto_2Enum__to__dt @ V1n ) ) ) ).
thf(thm_2Etoto_2Eqk__numto,axiom,
( c_2Etoto_2Eqk__numto
= ( c_2Etoto_2ETO @ tyop_2Enum_2Enum @ c_2Etoto_2Eqk__numOrd ) ) ).
thf(thm_2Etoto_2EcharOrd,axiom,
! [V0a: tyop_2Estring_2Echar,V1b: tyop_2Estring_2Echar] :
( ( c_2Etoto_2EcharOrd @ V0a @ V1b )
= ( c_2Etoto_2EnumOrd @ ( c_2Estring_2EORD @ V0a ) @ ( c_2Estring_2EORD @ V1b ) ) ) ).
thf(thm_2Etoto_2Echarto,axiom,
( c_2Etoto_2Echarto
= ( c_2Etoto_2ETO @ tyop_2Estring_2Echar @ c_2Etoto_2EcharOrd ) ) ).
thf(thm_2Etoto_2EListOrd,axiom,
! [A_27a: $tType,V0c: tyop_2Etoto_2Etoto @ A_27a] :
( ( c_2Etoto_2EListOrd @ A_27a @ V0c )
= ( c_2Etoto_2ETO__of__LinearOrder @ ( tyop_2Elist_2Elist @ A_27a ) @ ( c_2Etoto_2Elistorder @ A_27a @ ( c_2Etoto_2EStrongLinearOrder__of__TO @ A_27a @ ( c_2Etoto_2Eapto @ A_27a @ V0c ) ) ) ) ) ).
thf(thm_2Etoto_2Elistoto,axiom,
! [A_27a: $tType,V0c: tyop_2Etoto_2Etoto @ A_27a] :
( ( c_2Etoto_2Elistoto @ A_27a @ V0c )
= ( c_2Etoto_2ETO @ ( tyop_2Elist_2Elist @ A_27a ) @ ( c_2Etoto_2EListOrd @ A_27a @ V0c ) ) ) ).
thf(thm_2Etoto_2Estringto,axiom,
( c_2Etoto_2Estringto
= ( c_2Etoto_2Elistoto @ tyop_2Estring_2Echar @ c_2Etoto_2Echarto ) ) ).
thf(thm_2Etoto_2EimageOrd,axiom,
! [A_27a: $tType,A_27c: $tType,V0f: A_27a > A_27c,V1cp: A_27c > A_27c > tyop_2EternaryComparisons_2Eordering,V2a: A_27a,V3b: A_27a] :
( ( c_2Etoto_2EimageOrd @ A_27a @ A_27c @ V0f @ V1cp @ V2a @ V3b )
= ( V1cp @ ( V0f @ V2a ) @ ( V0f @ V3b ) ) ) ).
thf(thm_2Etoto_2EStrongLinearOrderExists,axiom,
! [A_27a: $tType] :
? [V0R: A_27a > A_27a > $o] : ( c_2Erelation_2EStrongLinearOrder @ A_27a @ V0R ) ).
thf(thm_2Etoto_2Etrichotomous__ALT,axiom,
! [A_27a: $tType,V0R: A_27a > A_27a > $o] :
( ( c_2Erelation_2Etrichotomous @ A_27a @ V0R )
<=> ! [V1x: A_27a,V2y: A_27a] :
( ( ( (~) @ ( V0R @ V1x @ V2y ) )
& ( (~) @ ( V0R @ V2y @ V1x ) ) )
=> ( V1x = V2y ) ) ) ).
thf(thm_2Etoto_2ETotOrd__TO__of__LO,axiom,
! [A_27a: $tType,V0r: A_27a > A_27a > $o] :
( ( c_2Erelation_2ELinearOrder @ A_27a @ V0r )
=> ( c_2Etoto_2ETotOrd @ A_27a @ ( c_2Etoto_2ETO__of__LinearOrder @ A_27a @ V0r ) ) ) ).
thf(thm_2Etoto_2ESPLIT__PAIRS,axiom,
! [A_27a: $tType,A_27b: $tType,V0x: tyop_2Epair_2Eprod @ A_27a @ A_27b,V1y: tyop_2Epair_2Eprod @ A_27a @ A_27b] :
( ( V0x = V1y )
<=> ( ( ( c_2Epair_2EFST @ A_27a @ A_27b @ V0x )
= ( c_2Epair_2EFST @ A_27a @ A_27b @ V1y ) )
& ( ( c_2Epair_2ESND @ A_27a @ A_27b @ V0x )
= ( c_2Epair_2ESND @ A_27a @ A_27b @ V1y ) ) ) ) ).
thf(thm_2Etoto_2Eall__cpn__distinct,axiom,
( ( (~) @ ( c_2EternaryComparisons_2ELESS = c_2EternaryComparisons_2EEQUAL ) )
& ( (~) @ ( c_2EternaryComparisons_2ELESS = c_2EternaryComparisons_2EGREATER ) )
& ( (~) @ ( c_2EternaryComparisons_2EEQUAL = c_2EternaryComparisons_2EGREATER ) )
& ( (~) @ ( c_2EternaryComparisons_2EEQUAL = c_2EternaryComparisons_2ELESS ) )
& ( (~) @ ( c_2EternaryComparisons_2EGREATER = c_2EternaryComparisons_2ELESS ) )
& ( (~) @ ( c_2EternaryComparisons_2EGREATER = c_2EternaryComparisons_2EEQUAL ) ) ) ).
thf(thm_2Etoto_2ETO__exists,axiom,
! [A_27a: $tType] :
? [V0x: A_27a > A_27a > tyop_2EternaryComparisons_2Eordering] : ( c_2Etoto_2ETotOrd @ A_27a @ V0x ) ).
thf(thm_2Etoto_2ETO__apto__ID,axiom,
! [A_27a: $tType,V0a: tyop_2Etoto_2Etoto @ A_27a] :
( ( c_2Etoto_2ETO @ A_27a @ ( c_2Etoto_2Eapto @ A_27a @ V0a ) )
= V0a ) ).
thf(thm_2Etoto_2ETO__apto__TO__ID,axiom,
! [A_27a: $tType,V0r: A_27a > A_27a > tyop_2EternaryComparisons_2Eordering] :
( ( c_2Etoto_2ETotOrd @ A_27a @ V0r )
<=> ( ( c_2Etoto_2Eapto @ A_27a @ ( c_2Etoto_2ETO @ A_27a @ V0r ) )
= V0r ) ) ).
thf(thm_2Etoto_2ETO__11,axiom,
! [A_27a: $tType,V0r: A_27a > A_27a > tyop_2EternaryComparisons_2Eordering,V1r_27: A_27a > A_27a > tyop_2EternaryComparisons_2Eordering] :
( ( c_2Etoto_2ETotOrd @ A_27a @ V0r )
=> ( ( c_2Etoto_2ETotOrd @ A_27a @ V1r_27 )
=> ( ( ( c_2Etoto_2ETO @ A_27a @ V0r )
= ( c_2Etoto_2ETO @ A_27a @ V1r_27 ) )
<=> ( V0r = V1r_27 ) ) ) ) ).
thf(thm_2Etoto_2Eonto__apto,axiom,
! [A_27a: $tType,V0r: A_27a > A_27a > tyop_2EternaryComparisons_2Eordering] :
( ( c_2Etoto_2ETotOrd @ A_27a @ V0r )
<=> ? [V1a: tyop_2Etoto_2Etoto @ A_27a] :
( V0r
= ( c_2Etoto_2Eapto @ A_27a @ V1a ) ) ) ).
thf(thm_2Etoto_2ETO__onto,axiom,
! [A_27a: $tType,V0a: tyop_2Etoto_2Etoto @ A_27a] :
? [V1r: A_27a > A_27a > tyop_2EternaryComparisons_2Eordering] :
( ( V0a
= ( c_2Etoto_2ETO @ A_27a @ V1r ) )
& ( c_2Etoto_2ETotOrd @ A_27a @ V1r ) ) ).
thf(thm_2Etoto_2ETotOrd__apto,axiom,
! [A_27a: $tType,V0c: tyop_2Etoto_2Etoto @ A_27a] : ( c_2Etoto_2ETotOrd @ A_27a @ ( c_2Etoto_2Eapto @ A_27a @ V0c ) ) ).
thf(thm_2Etoto_2ETO__apto__TO__IMP,axiom,
! [A_27a: $tType,V0r: A_27a > A_27a > tyop_2EternaryComparisons_2Eordering] :
( ( c_2Etoto_2ETotOrd @ A_27a @ V0r )
=> ( ( c_2Etoto_2Eapto @ A_27a @ ( c_2Etoto_2ETO @ A_27a @ V0r ) )
= V0r ) ) ).
thf(thm_2Etoto_2Etoto__thm,axiom,
! [A_27a: $tType,V0c: tyop_2Etoto_2Etoto @ A_27a] :
( ! [V1x: A_27a,V2y: A_27a] :
( ( ( c_2Etoto_2Eapto @ A_27a @ V0c @ V1x @ V2y )
= c_2EternaryComparisons_2EEQUAL )
<=> ( V1x = V2y ) )
& ! [V3x: A_27a,V4y: A_27a] :
( ( ( c_2Etoto_2Eapto @ A_27a @ V0c @ V3x @ V4y )
= c_2EternaryComparisons_2EGREATER )
<=> ( ( c_2Etoto_2Eapto @ A_27a @ V0c @ V4y @ V3x )
= c_2EternaryComparisons_2ELESS ) )
& ! [V5x: A_27a,V6y: A_27a,V7z: A_27a] :
( ( ( ( c_2Etoto_2Eapto @ A_27a @ V0c @ V5x @ V6y )
= c_2EternaryComparisons_2ELESS )
& ( ( c_2Etoto_2Eapto @ A_27a @ V0c @ V6y @ V7z )
= c_2EternaryComparisons_2ELESS ) )
=> ( ( c_2Etoto_2Eapto @ A_27a @ V0c @ V5x @ V7z )
= c_2EternaryComparisons_2ELESS ) ) ) ).
thf(thm_2Etoto_2ETO__equal__eq,axiom,
! [A_27a: $tType,V0c: A_27a > A_27a > tyop_2EternaryComparisons_2Eordering] :
( ( c_2Etoto_2ETotOrd @ A_27a @ V0c )
=> ! [V1x: A_27a,V2y: A_27a] :
( ( ( V0c @ V1x @ V2y )
= c_2EternaryComparisons_2EEQUAL )
<=> ( V1x = V2y ) ) ) ).
thf(thm_2Etoto_2Etoto__equal__eq,axiom,
! [A_27a: $tType,V0c: tyop_2Etoto_2Etoto @ A_27a,V1x: A_27a,V2y: A_27a] :
( ( ( c_2Etoto_2Eapto @ A_27a @ V0c @ V1x @ V2y )
= c_2EternaryComparisons_2EEQUAL )
<=> ( V1x = V2y ) ) ).
thf(thm_2Etoto_2Etoto__equal__imp__eq,axiom,
! [A_27a: $tType,V0c: tyop_2Etoto_2Etoto @ A_27a,V1x: A_27a,V2y: A_27a] :
( ( ( c_2Etoto_2Eapto @ A_27a @ V0c @ V1x @ V2y )
= c_2EternaryComparisons_2EEQUAL )
=> ( V1x = V2y ) ) ).
thf(thm_2Etoto_2ETO__refl,axiom,
! [A_27a: $tType,V0c: A_27a > A_27a > tyop_2EternaryComparisons_2Eordering] :
( ( c_2Etoto_2ETotOrd @ A_27a @ V0c )
=> ! [V1x: A_27a] :
( ( V0c @ V1x @ V1x )
= c_2EternaryComparisons_2EEQUAL ) ) ).
thf(thm_2Etoto_2Etoto__refl,axiom,
! [A_27a: $tType,V0c: tyop_2Etoto_2Etoto @ A_27a,V1x: A_27a] :
( ( c_2Etoto_2Eapto @ A_27a @ V0c @ V1x @ V1x )
= c_2EternaryComparisons_2EEQUAL ) ).
thf(thm_2Etoto_2Etoto__equal__sym,axiom,
! [A_27a: $tType,V0c: tyop_2Etoto_2Etoto @ A_27a,V1x: A_27a,V2y: A_27a] :
( ( ( c_2Etoto_2Eapto @ A_27a @ V0c @ V1x @ V2y )
= c_2EternaryComparisons_2EEQUAL )
<=> ( ( c_2Etoto_2Eapto @ A_27a @ V0c @ V2y @ V1x )
= c_2EternaryComparisons_2EEQUAL ) ) ).
thf(thm_2Etoto_2ETO__antisym,axiom,
! [A_27a: $tType,V0c: A_27a > A_27a > tyop_2EternaryComparisons_2Eordering] :
( ( c_2Etoto_2ETotOrd @ A_27a @ V0c )
=> ! [V1x: A_27a,V2y: A_27a] :
( ( ( V0c @ V1x @ V2y )
= c_2EternaryComparisons_2EGREATER )
<=> ( ( V0c @ V2y @ V1x )
= c_2EternaryComparisons_2ELESS ) ) ) ).
thf(thm_2Etoto_2Etoto__antisym,axiom,
! [A_27a: $tType,V0c: tyop_2Etoto_2Etoto @ A_27a,V1x: A_27a,V2y: A_27a] :
( ( ( c_2Etoto_2Eapto @ A_27a @ V0c @ V1x @ V2y )
= c_2EternaryComparisons_2EGREATER )
<=> ( ( c_2Etoto_2Eapto @ A_27a @ V0c @ V2y @ V1x )
= c_2EternaryComparisons_2ELESS ) ) ).
thf(thm_2Etoto_2Etoto__not__less__refl,axiom,
! [A_27a: $tType,V0cmp: tyop_2Etoto_2Etoto @ A_27a,V1h: A_27a] :
( ( ( c_2Etoto_2Eapto @ A_27a @ V0cmp @ V1h @ V1h )
= c_2EternaryComparisons_2ELESS )
<=> c_2Ebool_2EF ) ).
thf(thm_2Etoto_2Etoto__swap__cases,axiom,
! [A_27a: $tType,V0c: tyop_2Etoto_2Etoto @ A_27a,V1x: A_27a,V2y: A_27a] :
( ( c_2Etoto_2Eapto @ A_27a @ V0c @ V2y @ V1x )
= ( c_2EternaryComparisons_2Eordering__CASE @ tyop_2EternaryComparisons_2Eordering @ ( c_2Etoto_2Eapto @ A_27a @ V0c @ V1x @ V2y ) @ c_2EternaryComparisons_2EGREATER @ c_2EternaryComparisons_2EEQUAL @ c_2EternaryComparisons_2ELESS ) ) ).
thf(thm_2Etoto_2Etoto__glneq,axiom,
! [A_27a: $tType] :
( ! [V0c: tyop_2Etoto_2Etoto @ A_27a,V1x: A_27a,V2y: A_27a] :
( ( ( c_2Etoto_2Eapto @ A_27a @ V0c @ V1x @ V2y )
= c_2EternaryComparisons_2ELESS )
=> ( (~) @ ( V1x = V2y ) ) )
& ! [V3c: tyop_2Etoto_2Etoto @ A_27a,V4x: A_27a,V5y: A_27a] :
( ( ( c_2Etoto_2Eapto @ A_27a @ V3c @ V4x @ V5y )
= c_2EternaryComparisons_2EGREATER )
=> ( (~) @ ( V4x = V5y ) ) ) ) ).
thf(thm_2Etoto_2Etoto__cpn__eqn,axiom,
! [A_27a: $tType] :
( ! [V0c: tyop_2Etoto_2Etoto @ A_27a,V1x: A_27a,V2y: A_27a] :
( ( ( c_2Etoto_2Eapto @ A_27a @ V0c @ V1x @ V2y )
= c_2EternaryComparisons_2EEQUAL )
=> ( V1x = V2y ) )
& ! [V3c: tyop_2Etoto_2Etoto @ A_27a,V4x: A_27a,V5y: A_27a] :
( ( ( c_2Etoto_2Eapto @ A_27a @ V3c @ V4x @ V5y )
= c_2EternaryComparisons_2ELESS )
=> ( (~) @ ( V4x = V5y ) ) )
& ! [V6c: tyop_2Etoto_2Etoto @ A_27a,V7x: A_27a,V8y: A_27a] :
( ( ( c_2Etoto_2Eapto @ A_27a @ V6c @ V7x @ V8y )
= c_2EternaryComparisons_2EGREATER )
=> ( (~) @ ( V7x = V8y ) ) ) ) ).
thf(thm_2Etoto_2ETO__cpn__eqn,axiom,
! [A_27a: $tType,V0c: A_27a > A_27a > tyop_2EternaryComparisons_2Eordering] :
( ( c_2Etoto_2ETotOrd @ A_27a @ V0c )
=> ( ! [V1x: A_27a,V2y: A_27a] :
( ( ( V0c @ V1x @ V2y )
= c_2EternaryComparisons_2ELESS )
=> ( (~) @ ( V1x = V2y ) ) )
& ! [V3x: A_27a,V4y: A_27a] :
( ( ( V0c @ V3x @ V4y )
= c_2EternaryComparisons_2EGREATER )
=> ( (~) @ ( V3x = V4y ) ) )
& ! [V5x: A_27a,V6y: A_27a] :
( ( ( V0c @ V5x @ V6y )
= c_2EternaryComparisons_2EEQUAL )
=> ( V5x = V6y ) ) ) ) ).
thf(thm_2Etoto_2ENOT__EQ__LESS__IMP,axiom,
! [A_27a: $tType,V0cmp: tyop_2Etoto_2Etoto @ A_27a,V1x: A_27a,V2y: A_27a] :
( ( (~)
@ ( ( c_2Etoto_2Eapto @ A_27a @ V0cmp @ V1x @ V2y )
= c_2EternaryComparisons_2ELESS ) )
=> ( ( V1x = V2y )
| ( ( c_2Etoto_2Eapto @ A_27a @ V0cmp @ V2y @ V1x )
= c_2EternaryComparisons_2ELESS ) ) ) ).
thf(thm_2Etoto_2EtotoEEtrans,axiom,
! [A_27a: $tType,V0c: tyop_2Etoto_2Etoto @ A_27a,V1x: A_27a,V2y: A_27a,V3z: A_27a] :
( ( ( ( ( c_2Etoto_2Eapto @ A_27a @ V0c @ V1x @ V2y )
= c_2EternaryComparisons_2EEQUAL )
& ( ( c_2Etoto_2Eapto @ A_27a @ V0c @ V2y @ V3z )
= c_2EternaryComparisons_2EEQUAL ) )
=> ( ( c_2Etoto_2Eapto @ A_27a @ V0c @ V1x @ V3z )
= c_2EternaryComparisons_2EEQUAL ) )
& ( ( ( ( c_2Etoto_2Eapto @ A_27a @ V0c @ V1x @ V2y )
= c_2EternaryComparisons_2EEQUAL )
& ( ( c_2Etoto_2Eapto @ A_27a @ V0c @ V3z @ V2y )
= c_2EternaryComparisons_2EEQUAL ) )
=> ( ( c_2Etoto_2Eapto @ A_27a @ V0c @ V1x @ V3z )
= c_2EternaryComparisons_2EEQUAL ) ) ) ).
thf(thm_2Etoto_2EtotoLLtrans,axiom,
! [A_27a: $tType,V0c: tyop_2Etoto_2Etoto @ A_27a,V1x: A_27a,V2y: A_27a,V3z: A_27a] :
( ( ( ( c_2Etoto_2Eapto @ A_27a @ V0c @ V1x @ V2y )
= c_2EternaryComparisons_2ELESS )
& ( ( c_2Etoto_2Eapto @ A_27a @ V0c @ V2y @ V3z )
= c_2EternaryComparisons_2ELESS ) )
=> ( ( c_2Etoto_2Eapto @ A_27a @ V0c @ V1x @ V3z )
= c_2EternaryComparisons_2ELESS ) ) ).
thf(thm_2Etoto_2EtotoLGtrans,axiom,
! [A_27a: $tType,V0c: tyop_2Etoto_2Etoto @ A_27a,V1x: A_27a,V2y: A_27a,V3z: A_27a] :
( ( ( ( c_2Etoto_2Eapto @ A_27a @ V0c @ V1x @ V2y )
= c_2EternaryComparisons_2ELESS )
& ( ( c_2Etoto_2Eapto @ A_27a @ V0c @ V3z @ V2y )
= c_2EternaryComparisons_2EGREATER ) )
=> ( ( c_2Etoto_2Eapto @ A_27a @ V0c @ V1x @ V3z )
= c_2EternaryComparisons_2ELESS ) ) ).
thf(thm_2Etoto_2EtotoGGtrans,axiom,
! [A_27a: $tType,V0c: tyop_2Etoto_2Etoto @ A_27a,V1x: A_27a,V2y: A_27a,V3z: A_27a] :
( ( ( ( c_2Etoto_2Eapto @ A_27a @ V0c @ V2y @ V1x )
= c_2EternaryComparisons_2EGREATER )
& ( ( c_2Etoto_2Eapto @ A_27a @ V0c @ V3z @ V2y )
= c_2EternaryComparisons_2EGREATER ) )
=> ( ( c_2Etoto_2Eapto @ A_27a @ V0c @ V1x @ V3z )
= c_2EternaryComparisons_2ELESS ) ) ).
thf(thm_2Etoto_2EtotoGLtrans,axiom,
! [A_27a: $tType,V0c: tyop_2Etoto_2Etoto @ A_27a,V1x: A_27a,V2y: A_27a,V3z: A_27a] :
( ( ( ( c_2Etoto_2Eapto @ A_27a @ V0c @ V2y @ V1x )
= c_2EternaryComparisons_2EGREATER )
& ( ( c_2Etoto_2Eapto @ A_27a @ V0c @ V2y @ V3z )
= c_2EternaryComparisons_2ELESS ) )
=> ( ( c_2Etoto_2Eapto @ A_27a @ V0c @ V1x @ V3z )
= c_2EternaryComparisons_2ELESS ) ) ).
thf(thm_2Etoto_2EtotoLEtrans,axiom,
! [A_27a: $tType,V0c: tyop_2Etoto_2Etoto @ A_27a,V1x: A_27a,V2y: A_27a,V3z: A_27a] :
( ( ( ( c_2Etoto_2Eapto @ A_27a @ V0c @ V1x @ V2y )
= c_2EternaryComparisons_2ELESS )
& ( ( c_2Etoto_2Eapto @ A_27a @ V0c @ V2y @ V3z )
= c_2EternaryComparisons_2EEQUAL ) )
=> ( ( c_2Etoto_2Eapto @ A_27a @ V0c @ V1x @ V3z )
= c_2EternaryComparisons_2ELESS ) ) ).
thf(thm_2Etoto_2EtotoELtrans,axiom,
! [A_27a: $tType,V0c: tyop_2Etoto_2Etoto @ A_27a,V1x: A_27a,V2y: A_27a,V3z: A_27a] :
( ( ( ( c_2Etoto_2Eapto @ A_27a @ V0c @ V1x @ V2y )
= c_2EternaryComparisons_2EEQUAL )
& ( ( c_2Etoto_2Eapto @ A_27a @ V0c @ V2y @ V3z )
= c_2EternaryComparisons_2ELESS ) )
=> ( ( c_2Etoto_2Eapto @ A_27a @ V0c @ V1x @ V3z )
= c_2EternaryComparisons_2ELESS ) ) ).
thf(thm_2Etoto_2Etoto__trans__less,axiom,
! [A_27a: $tType] :
( ! [V0c: tyop_2Etoto_2Etoto @ A_27a,V1x: A_27a,V2y: A_27a,V3z: A_27a] :
( ( ( ( c_2Etoto_2Eapto @ A_27a @ V0c @ V1x @ V2y )
= c_2EternaryComparisons_2ELESS )
& ( ( c_2Etoto_2Eapto @ A_27a @ V0c @ V2y @ V3z )
= c_2EternaryComparisons_2ELESS ) )
=> ( ( c_2Etoto_2Eapto @ A_27a @ V0c @ V1x @ V3z )
= c_2EternaryComparisons_2ELESS ) )
& ! [V4c: tyop_2Etoto_2Etoto @ A_27a,V5x: A_27a,V6y: A_27a,V7z: A_27a] :
( ( ( ( c_2Etoto_2Eapto @ A_27a @ V4c @ V5x @ V6y )
= c_2EternaryComparisons_2ELESS )
& ( ( c_2Etoto_2Eapto @ A_27a @ V4c @ V7z @ V6y )
= c_2EternaryComparisons_2EGREATER ) )
=> ( ( c_2Etoto_2Eapto @ A_27a @ V4c @ V5x @ V7z )
= c_2EternaryComparisons_2ELESS ) )
& ! [V8c: tyop_2Etoto_2Etoto @ A_27a,V9x: A_27a,V10y: A_27a,V11z: A_27a] :
( ( ( ( c_2Etoto_2Eapto @ A_27a @ V8c @ V10y @ V9x )
= c_2EternaryComparisons_2EGREATER )
& ( ( c_2Etoto_2Eapto @ A_27a @ V8c @ V11z @ V10y )
= c_2EternaryComparisons_2EGREATER ) )
=> ( ( c_2Etoto_2Eapto @ A_27a @ V8c @ V9x @ V11z )
= c_2EternaryComparisons_2ELESS ) )
& ! [V12c: tyop_2Etoto_2Etoto @ A_27a,V13x: A_27a,V14y: A_27a,V15z: A_27a] :
( ( ( ( c_2Etoto_2Eapto @ A_27a @ V12c @ V14y @ V13x )
= c_2EternaryComparisons_2EGREATER )
& ( ( c_2Etoto_2Eapto @ A_27a @ V12c @ V14y @ V15z )
= c_2EternaryComparisons_2ELESS ) )
=> ( ( c_2Etoto_2Eapto @ A_27a @ V12c @ V13x @ V15z )
= c_2EternaryComparisons_2ELESS ) )
& ! [V16c: tyop_2Etoto_2Etoto @ A_27a,V17x: A_27a,V18y: A_27a,V19z: A_27a] :
( ( ( ( c_2Etoto_2Eapto @ A_27a @ V16c @ V17x @ V18y )
= c_2EternaryComparisons_2ELESS )
& ( ( c_2Etoto_2Eapto @ A_27a @ V16c @ V18y @ V19z )
= c_2EternaryComparisons_2EEQUAL ) )
=> ( ( c_2Etoto_2Eapto @ A_27a @ V16c @ V17x @ V19z )
= c_2EternaryComparisons_2ELESS ) )
& ! [V20c: tyop_2Etoto_2Etoto @ A_27a,V21x: A_27a,V22y: A_27a,V23z: A_27a] :
( ( ( ( c_2Etoto_2Eapto @ A_27a @ V20c @ V21x @ V22y )
= c_2EternaryComparisons_2EEQUAL )
& ( ( c_2Etoto_2Eapto @ A_27a @ V20c @ V22y @ V23z )
= c_2EternaryComparisons_2ELESS ) )
=> ( ( c_2Etoto_2Eapto @ A_27a @ V20c @ V21x @ V23z )
= c_2EternaryComparisons_2ELESS ) ) ) ).
thf(thm_2Etoto_2EWeak__Weak__of,axiom,
! [A_27a: $tType,V0c: tyop_2Etoto_2Etoto @ A_27a] : ( c_2Erelation_2EWeakLinearOrder @ A_27a @ ( c_2Etoto_2EWeakLinearOrder__of__TO @ A_27a @ ( c_2Etoto_2Eapto @ A_27a @ V0c ) ) ) ).
thf(thm_2Etoto_2ESTRORD__SLO,axiom,
! [A_27a: $tType,V0R: A_27a > A_27a > $o] :
( ( c_2Erelation_2EWeakLinearOrder @ A_27a @ V0R )
=> ( c_2Erelation_2EStrongLinearOrder @ A_27a @ ( c_2Erelation_2ESTRORD @ A_27a @ V0R ) ) ) ).
thf(thm_2Etoto_2EStrongof__toto__STRORD,axiom,
! [A_27a: $tType,V0c: tyop_2Etoto_2Etoto @ A_27a] :
( ( c_2Etoto_2EStrongLinearOrder__of__TO @ A_27a @ ( c_2Etoto_2Eapto @ A_27a @ V0c ) )
= ( c_2Erelation_2ESTRORD @ A_27a @ ( c_2Etoto_2EWeakLinearOrder__of__TO @ A_27a @ ( c_2Etoto_2Eapto @ A_27a @ V0c ) ) ) ) ).
thf(thm_2Etoto_2EStrong__Strong__of,axiom,
! [A_27a: $tType,V0c: tyop_2Etoto_2Etoto @ A_27a] : ( c_2Erelation_2EStrongLinearOrder @ A_27a @ ( c_2Etoto_2EStrongLinearOrder__of__TO @ A_27a @ ( c_2Etoto_2Eapto @ A_27a @ V0c ) ) ) ).
thf(thm_2Etoto_2EStrong__Strong__of__TO,axiom,
! [A_27a: $tType,V0c: A_27a > A_27a > tyop_2EternaryComparisons_2Eordering] :
( ( c_2Etoto_2ETotOrd @ A_27a @ V0c )
=> ( c_2Erelation_2EStrongLinearOrder @ A_27a @ ( c_2Etoto_2EStrongLinearOrder__of__TO @ A_27a @ V0c ) ) ) ).
thf(thm_2Etoto_2ETotOrd__TO__of__Weak,axiom,
! [A_27a: $tType,V0r: A_27a > A_27a > $o] :
( ( c_2Erelation_2EWeakLinearOrder @ A_27a @ V0r )
=> ( c_2Etoto_2ETotOrd @ A_27a @ ( c_2Etoto_2ETO__of__LinearOrder @ A_27a @ V0r ) ) ) ).
thf(thm_2Etoto_2ETotOrd__TO__of__Strong,axiom,
! [A_27a: $tType,V0r: A_27a > A_27a > $o] :
( ( c_2Erelation_2EStrongLinearOrder @ A_27a @ V0r )
=> ( c_2Etoto_2ETotOrd @ A_27a @ ( c_2Etoto_2ETO__of__LinearOrder @ A_27a @ V0r ) ) ) ).
thf(thm_2Etoto_2Etoto__Weak__thm,axiom,
! [A_27a: $tType,V0c: tyop_2Etoto_2Etoto @ A_27a] :
( ( c_2Etoto_2Etoto__of__LinearOrder @ A_27a @ ( c_2Etoto_2EWeakLinearOrder__of__TO @ A_27a @ ( c_2Etoto_2Eapto @ A_27a @ V0c ) ) )
= V0c ) ).
thf(thm_2Etoto_2Etoto__Strong__thm,axiom,
! [A_27a: $tType,V0c: tyop_2Etoto_2Etoto @ A_27a] :
( ( c_2Etoto_2Etoto__of__LinearOrder @ A_27a @ ( c_2Etoto_2EStrongLinearOrder__of__TO @ A_27a @ ( c_2Etoto_2Eapto @ A_27a @ V0c ) ) )
= V0c ) ).
thf(thm_2Etoto_2EWeak__toto__thm,axiom,
! [A_27a: $tType,V0r: A_27a > A_27a > $o] :
( ( c_2Erelation_2EWeakLinearOrder @ A_27a @ V0r )
=> ( ( c_2Etoto_2EWeakLinearOrder__of__TO @ A_27a @ ( c_2Etoto_2Eapto @ A_27a @ ( c_2Etoto_2Etoto__of__LinearOrder @ A_27a @ V0r ) ) )
= V0r ) ) ).
thf(thm_2Etoto_2EStrong__toto__thm,axiom,
! [A_27a: $tType,V0r: A_27a > A_27a > $o] :
( ( c_2Erelation_2EStrongLinearOrder @ A_27a @ V0r )
=> ( ( c_2Etoto_2EStrongLinearOrder__of__TO @ A_27a @ ( c_2Etoto_2Eapto @ A_27a @ ( c_2Etoto_2Etoto__of__LinearOrder @ A_27a @ V0r ) ) )
= V0r ) ) ).
thf(thm_2Etoto_2ETotOrd__inv,axiom,
! [A_27a: $tType,V0c: A_27a > A_27a > tyop_2EternaryComparisons_2Eordering] :
( ( c_2Etoto_2ETotOrd @ A_27a @ V0c )
=> ( c_2Etoto_2ETotOrd @ A_27a @ ( c_2Etoto_2ETO__inv @ A_27a @ V0c ) ) ) ).
thf(thm_2Etoto_2Einv__TO,axiom,
! [A_27a: $tType,V0r: A_27a > A_27a > tyop_2EternaryComparisons_2Eordering] :
( ( c_2Etoto_2ETotOrd @ A_27a @ V0r )
=> ( ( c_2Etoto_2Etoto__inv @ A_27a @ ( c_2Etoto_2ETO @ A_27a @ V0r ) )
= ( c_2Etoto_2ETO @ A_27a @ ( c_2Etoto_2ETO__inv @ A_27a @ V0r ) ) ) ) ).
thf(thm_2Etoto_2Eapto__inv,axiom,
! [A_27a: $tType,V0c: tyop_2Etoto_2Etoto @ A_27a] :
( ( c_2Etoto_2Eapto @ A_27a @ ( c_2Etoto_2Etoto__inv @ A_27a @ V0c ) )
= ( c_2Etoto_2ETO__inv @ A_27a @ ( c_2Etoto_2Eapto @ A_27a @ V0c ) ) ) ).
thf(thm_2Etoto_2EWeak__toto__inv,axiom,
! [A_27a: $tType,V0c: tyop_2Etoto_2Etoto @ A_27a] :
( ( c_2Etoto_2EWeakLinearOrder__of__TO @ A_27a @ ( c_2Etoto_2Eapto @ A_27a @ ( c_2Etoto_2Etoto__inv @ A_27a @ V0c ) ) )
= ( c_2Erelation_2Einv @ A_27a @ A_27a @ ( c_2Etoto_2EWeakLinearOrder__of__TO @ A_27a @ ( c_2Etoto_2Eapto @ A_27a @ V0c ) ) ) ) ).
thf(thm_2Etoto_2EStrong__toto__inv,axiom,
! [A_27a: $tType,V0c: tyop_2Etoto_2Etoto @ A_27a] :
( ( c_2Etoto_2EStrongLinearOrder__of__TO @ A_27a @ ( c_2Etoto_2Eapto @ A_27a @ ( c_2Etoto_2Etoto__inv @ A_27a @ V0c ) ) )
= ( c_2Erelation_2Einv @ A_27a @ A_27a @ ( c_2Etoto_2EStrongLinearOrder__of__TO @ A_27a @ ( c_2Etoto_2Eapto @ A_27a @ V0c ) ) ) ) ).
thf(thm_2Etoto_2ETO__inv__TO__inv,axiom,
! [A_27a: $tType,V0c: A_27a > A_27a > tyop_2EternaryComparisons_2Eordering] :
( ( c_2Etoto_2ETO__inv @ A_27a @ ( c_2Etoto_2ETO__inv @ A_27a @ V0c ) )
= V0c ) ).
thf(thm_2Etoto_2Etoto__inv__toto__inv,axiom,
! [A_27a: $tType,V0c: tyop_2Etoto_2Etoto @ A_27a] :
( ( c_2Etoto_2Etoto__inv @ A_27a @ ( c_2Etoto_2Etoto__inv @ A_27a @ V0c ) )
= V0c ) ).
thf(thm_2Etoto_2ETO__inv__Ord,axiom,
! [A_27a: $tType,V0r: A_27a > A_27a > $o] :
( ( c_2Etoto_2ETO__of__LinearOrder @ A_27a @ ( c_2Erelation_2Einv @ A_27a @ A_27a @ V0r ) )
= ( c_2Etoto_2ETO__inv @ A_27a @ ( c_2Etoto_2ETO__of__LinearOrder @ A_27a @ V0r ) ) ) ).
thf(thm_2Etoto_2ETO__of__less__rel,axiom,
! [A_27a: $tType,V0r: A_27a > A_27a > $o] :
( ( c_2Erelation_2EStrongLinearOrder @ A_27a @ V0r )
=> ! [V1x: A_27a,V2y: A_27a] :
( ( ( c_2Etoto_2ETO__of__LinearOrder @ A_27a @ V0r @ V1x @ V2y )
= c_2EternaryComparisons_2ELESS )
<=> ( V0r @ V1x @ V2y ) ) ) ).
thf(thm_2Etoto_2ETO__of__greater__ler,axiom,
! [A_27a: $tType,V0r: A_27a > A_27a > $o] :
( ( c_2Erelation_2EStrongLinearOrder @ A_27a @ V0r )
=> ! [V1x: A_27a,V2y: A_27a] :
( ( ( c_2Etoto_2ETO__of__LinearOrder @ A_27a @ V0r @ V1x @ V2y )
= c_2EternaryComparisons_2EGREATER )
<=> ( V0r @ V2y @ V1x ) ) ) ).
thf(thm_2Etoto_2Etoto__equal__imp,axiom,
! [A_27a: $tType,V0cmp: tyop_2Etoto_2Etoto @ A_27a,V1phi: A_27a > A_27a > $o] :
( ( ( c_2Erelation_2ELinearOrder @ A_27a @ V1phi )
& ( V0cmp
= ( c_2Etoto_2Etoto__of__LinearOrder @ A_27a @ V1phi ) ) )
=> ! [V2x: A_27a,V3y: A_27a] :
( ( ( V2x = V3y )
<=> c_2Ebool_2ET )
=> ( ( c_2Etoto_2Eapto @ A_27a @ V0cmp @ V2x @ V3y )
= c_2EternaryComparisons_2EEQUAL ) ) ) ).
thf(thm_2Etoto_2Etoto__unequal__imp,axiom,
! [A_27a: $tType,V0cmp: tyop_2Etoto_2Etoto @ A_27a,V1phi: A_27a > A_27a > $o] :
( ( ( c_2Erelation_2ELinearOrder @ A_27a @ V1phi )
& ( V0cmp
= ( c_2Etoto_2Etoto__of__LinearOrder @ A_27a @ V1phi ) ) )
=> ! [V2x: A_27a,V3y: A_27a] :
( ( ( V2x = V3y )
<=> c_2Ebool_2EF )
=> ( c_2Ebool_2ECOND @ $o @ ( V1phi @ V2x @ V3y ) @ ( c_2Emin_2E_3D @ tyop_2EternaryComparisons_2Eordering @ ( c_2Etoto_2Eapto @ A_27a @ V0cmp @ V2x @ V3y ) @ c_2EternaryComparisons_2ELESS ) @ ( c_2Emin_2E_3D @ tyop_2EternaryComparisons_2Eordering @ ( c_2Etoto_2Eapto @ A_27a @ V0cmp @ V2x @ V3y ) @ c_2EternaryComparisons_2EGREATER ) ) ) ) ).
thf(thm_2Etoto_2EStrongOrder__ALT,axiom,
! [A_27a: $tType,V0Z: A_27a > A_27a > $o] :
( ( c_2Erelation_2EStrongOrder @ A_27a @ V0Z )
<=> ( ( c_2Erelation_2Eirreflexive @ A_27a @ V0Z )
& ( c_2Erelation_2Etransitive @ A_27a @ V0Z ) ) ) ).
thf(thm_2Etoto_2ELEX__ALT,axiom,
! [A_27a: $tType,A_27b: $tType,V0R: A_27a > A_27a > $o,V1U: A_27b > A_27b > $o,V2c: tyop_2Epair_2Eprod @ A_27a @ A_27b,V3d: tyop_2Epair_2Eprod @ A_27a @ A_27b] :
( ( c_2Epair_2ELEX @ A_27a @ A_27b @ V0R @ V1U @ V2c @ V3d )
<=> ( ( V0R @ ( c_2Epair_2EFST @ A_27a @ A_27b @ V2c ) @ ( c_2Epair_2EFST @ A_27a @ A_27b @ V3d ) )
| ( ( ( c_2Epair_2EFST @ A_27a @ A_27b @ V2c )
= ( c_2Epair_2EFST @ A_27a @ A_27b @ V3d ) )
& ( V1U @ ( c_2Epair_2ESND @ A_27a @ A_27b @ V2c ) @ ( c_2Epair_2ESND @ A_27a @ A_27b @ V3d ) ) ) ) ) ).
thf(thm_2Etoto_2ESLO__LEX,axiom,
! [A_27a: $tType,A_27b: $tType,V0R: A_27a > A_27a > $o,V1V: A_27b > A_27b > $o] :
( ( ( c_2Erelation_2EStrongLinearOrder @ A_27a @ V0R )
& ( c_2Erelation_2EStrongLinearOrder @ A_27b @ V1V ) )
=> ( c_2Erelation_2EStrongLinearOrder @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ ( c_2Epair_2ELEX @ A_27a @ A_27b @ V0R @ V1V ) ) ) ).
thf(thm_2Etoto_2ElexTO__thm,axiom,
! [A_27a: $tType,A_27b: $tType,V0R: A_27a > A_27a > tyop_2EternaryComparisons_2Eordering,V1V: A_27b > A_27b > tyop_2EternaryComparisons_2Eordering] :
( ( ( c_2Etoto_2ETotOrd @ A_27a @ V0R )
& ( c_2Etoto_2ETotOrd @ A_27b @ V1V ) )
=> ! [V2x: tyop_2Epair_2Eprod @ A_27a @ A_27b,V3y: tyop_2Epair_2Eprod @ A_27a @ A_27b] :
( ( c_2Etoto_2ElexTO @ A_27a @ A_27b @ V0R @ V1V @ V2x @ V3y )
= ( c_2EternaryComparisons_2Eordering__CASE @ tyop_2EternaryComparisons_2Eordering @ ( V0R @ ( c_2Epair_2EFST @ A_27a @ A_27b @ V2x ) @ ( c_2Epair_2EFST @ A_27a @ A_27b @ V3y ) ) @ c_2EternaryComparisons_2ELESS @ ( V1V @ ( c_2Epair_2ESND @ A_27a @ A_27b @ V2x ) @ ( c_2Epair_2ESND @ A_27a @ A_27b @ V3y ) ) @ c_2EternaryComparisons_2EGREATER ) ) ) ).
thf(thm_2Etoto_2ElexTO__ALT,axiom,
! [A_27a: $tType,A_27b: $tType,V0R: A_27a > A_27a > tyop_2EternaryComparisons_2Eordering,V1V: A_27b > A_27b > tyop_2EternaryComparisons_2Eordering] :
( ( ( c_2Etoto_2ETotOrd @ A_27a @ V0R )
& ( c_2Etoto_2ETotOrd @ A_27b @ V1V ) )
=> ( c_2Ebool_2E_21 @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b )
@ ( c_2Epair_2EUNCURRY @ A_27a @ A_27b @ $o
@ ^ [V2r: A_27a,V3u: A_27b] :
( c_2Ebool_2E_21 @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b )
@ ( c_2Epair_2EUNCURRY @ A_27a @ A_27b @ $o
@ ^ [V4r_27: A_27a,V5u_27: A_27b] : ( c_2Emin_2E_3D @ tyop_2EternaryComparisons_2Eordering @ ( c_2Etoto_2ElexTO @ A_27a @ A_27b @ V0R @ V1V @ ( c_2Epair_2E_2C @ A_27a @ A_27b @ V2r @ V3u ) @ ( c_2Epair_2E_2C @ A_27a @ A_27b @ V4r_27 @ V5u_27 ) ) @ ( c_2EternaryComparisons_2Eordering__CASE @ tyop_2EternaryComparisons_2Eordering @ ( V0R @ V2r @ V4r_27 ) @ c_2EternaryComparisons_2ELESS @ ( V1V @ V3u @ V5u_27 ) @ c_2EternaryComparisons_2EGREATER ) ) ) ) ) ) ) ).
thf(thm_2Etoto_2ETO__lexTO,axiom,
! [A_27a: $tType,A_27b: $tType,V0R: A_27a > A_27a > tyop_2EternaryComparisons_2Eordering,V1V: A_27b > A_27b > tyop_2EternaryComparisons_2Eordering] :
( ( ( c_2Etoto_2ETotOrd @ A_27a @ V0R )
& ( c_2Etoto_2ETotOrd @ A_27b @ V1V ) )
=> ( c_2Etoto_2ETotOrd @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ ( c_2Etoto_2ElexTO @ A_27a @ A_27b @ V0R @ V1V ) ) ) ).
thf(thm_2Etoto_2Epre__aplextoto,axiom,
! [A_27a: $tType,A_27b: $tType,V0c: tyop_2Etoto_2Etoto @ A_27a,V1v: tyop_2Etoto_2Etoto @ A_27b,V2x: tyop_2Epair_2Eprod @ A_27a @ A_27b,V3y: tyop_2Epair_2Eprod @ A_27a @ A_27b] :
( ( c_2Etoto_2Eapto @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ ( c_2Etoto_2Elextoto @ A_27a @ A_27b @ V0c @ V1v ) @ V2x @ V3y )
= ( c_2EternaryComparisons_2Eordering__CASE @ tyop_2EternaryComparisons_2Eordering @ ( c_2Etoto_2Eapto @ A_27a @ V0c @ ( c_2Epair_2EFST @ A_27a @ A_27b @ V2x ) @ ( c_2Epair_2EFST @ A_27a @ A_27b @ V3y ) ) @ c_2EternaryComparisons_2ELESS @ ( c_2Etoto_2Eapto @ A_27b @ V1v @ ( c_2Epair_2ESND @ A_27a @ A_27b @ V2x ) @ ( c_2Epair_2ESND @ A_27a @ A_27b @ V3y ) ) @ c_2EternaryComparisons_2EGREATER ) ) ).
thf(thm_2Etoto_2Eaplextoto,axiom,
! [A_27a: $tType,A_27b: $tType,V0c: tyop_2Etoto_2Etoto @ A_27a,V1v: tyop_2Etoto_2Etoto @ A_27b,V2x1: A_27a,V3x2: A_27b,V4y1: A_27a,V5y2: A_27b] :
( ( c_2Etoto_2Eapto @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ ( c_2Etoto_2Elextoto @ A_27a @ A_27b @ V0c @ V1v ) @ ( c_2Epair_2E_2C @ A_27a @ A_27b @ V2x1 @ V3x2 ) @ ( c_2Epair_2E_2C @ A_27a @ A_27b @ V4y1 @ V5y2 ) )
= ( c_2EternaryComparisons_2Eordering__CASE @ tyop_2EternaryComparisons_2Eordering @ ( c_2Etoto_2Eapto @ A_27a @ V0c @ V2x1 @ V4y1 ) @ c_2EternaryComparisons_2ELESS @ ( c_2Etoto_2Eapto @ A_27b @ V1v @ V3x2 @ V5y2 ) @ c_2EternaryComparisons_2EGREATER ) ) ).
thf(thm_2Etoto_2EStrongLinearOrder__LESS,axiom,
c_2Erelation_2EStrongLinearOrder @ tyop_2Enum_2Enum @ c_2Eprim__rec_2E_3C ).
thf(thm_2Etoto_2ETO__numOrd,axiom,
c_2Etoto_2ETotOrd @ tyop_2Enum_2Enum @ c_2Etoto_2EnumOrd ).
thf(thm_2Etoto_2Eapnumto__thm,axiom,
( ( c_2Etoto_2Eapto @ tyop_2Enum_2Enum @ c_2Etoto_2Enumto )
= c_2Etoto_2EnumOrd ) ).
thf(thm_2Etoto_2EnumeralOrd,axiom,
! [V0x: tyop_2Enum_2Enum,V1y: tyop_2Enum_2Enum] :
( ( ( c_2Etoto_2EnumOrd @ c_2Earithmetic_2EZERO @ c_2Earithmetic_2EZERO )
= c_2EternaryComparisons_2EEQUAL )
& ( ( c_2Etoto_2EnumOrd @ c_2Earithmetic_2EZERO @ ( c_2Earithmetic_2EBIT1 @ V1y ) )
= c_2EternaryComparisons_2ELESS )
& ( ( c_2Etoto_2EnumOrd @ c_2Earithmetic_2EZERO @ ( c_2Earithmetic_2EBIT2 @ V1y ) )
= c_2EternaryComparisons_2ELESS )
& ( ( c_2Etoto_2EnumOrd @ ( c_2Earithmetic_2EBIT1 @ V0x ) @ c_2Earithmetic_2EZERO )
= c_2EternaryComparisons_2EGREATER )
& ( ( c_2Etoto_2EnumOrd @ ( c_2Earithmetic_2EBIT2 @ V0x ) @ c_2Earithmetic_2EZERO )
= c_2EternaryComparisons_2EGREATER )
& ( ( c_2Etoto_2EnumOrd @ ( c_2Earithmetic_2EBIT1 @ V0x ) @ ( c_2Earithmetic_2EBIT1 @ V1y ) )
= ( c_2Etoto_2EnumOrd @ V0x @ V1y ) )
& ( ( c_2Etoto_2EnumOrd @ ( c_2Earithmetic_2EBIT2 @ V0x ) @ ( c_2Earithmetic_2EBIT2 @ V1y ) )
= ( c_2Etoto_2EnumOrd @ V0x @ V1y ) )
& ( ( c_2Etoto_2EnumOrd @ ( c_2Earithmetic_2EBIT1 @ V0x ) @ ( c_2Earithmetic_2EBIT2 @ V1y ) )
= ( c_2EternaryComparisons_2Eordering__CASE @ tyop_2EternaryComparisons_2Eordering @ ( c_2Etoto_2EnumOrd @ V0x @ V1y ) @ c_2EternaryComparisons_2ELESS @ c_2EternaryComparisons_2ELESS @ c_2EternaryComparisons_2EGREATER ) )
& ( ( c_2Etoto_2EnumOrd @ ( c_2Earithmetic_2EBIT2 @ V0x ) @ ( c_2Earithmetic_2EBIT1 @ V1y ) )
= ( c_2EternaryComparisons_2Eordering__CASE @ tyop_2EternaryComparisons_2Eordering @ ( c_2Etoto_2EnumOrd @ V0x @ V1y ) @ c_2EternaryComparisons_2ELESS @ c_2EternaryComparisons_2EGREATER @ c_2EternaryComparisons_2EGREATER ) ) ) ).
thf(thm_2Etoto_2Edatatype__num__dt,axiom,
! [V0num__dt: tyop_2Etoto_2Enum__dt > ( tyop_2Etoto_2Enum__dt > tyop_2Etoto_2Enum__dt ) > ( tyop_2Etoto_2Enum__dt > tyop_2Etoto_2Enum__dt ) > $o] : ( c_2Ebool_2EDATATYPE @ $o @ ( V0num__dt @ c_2Etoto_2Ezer @ c_2Etoto_2Ebit1 @ c_2Etoto_2Ebit2 ) ) ).
thf(thm_2Etoto_2Enum__dt__11,axiom,
( ! [V0a: tyop_2Etoto_2Enum__dt,V1a_27: tyop_2Etoto_2Enum__dt] :
( ( ( c_2Etoto_2Ebit1 @ V0a )
= ( c_2Etoto_2Ebit1 @ V1a_27 ) )
<=> ( V0a = V1a_27 ) )
& ! [V2a: tyop_2Etoto_2Enum__dt,V3a_27: tyop_2Etoto_2Enum__dt] :
( ( ( c_2Etoto_2Ebit2 @ V2a )
= ( c_2Etoto_2Ebit2 @ V3a_27 ) )
<=> ( V2a = V3a_27 ) ) ) ).
thf(thm_2Etoto_2Enum__dt__distinct,axiom,
( ! [V0a: tyop_2Etoto_2Enum__dt] :
( (~)
@ ( c_2Etoto_2Ezer
= ( c_2Etoto_2Ebit1 @ V0a ) ) )
& ! [V1a: tyop_2Etoto_2Enum__dt] :
( (~)
@ ( c_2Etoto_2Ezer
= ( c_2Etoto_2Ebit2 @ V1a ) ) )
& ! [V2a_27: tyop_2Etoto_2Enum__dt,V3a: tyop_2Etoto_2Enum__dt] :
( (~)
@ ( ( c_2Etoto_2Ebit1 @ V3a )
= ( c_2Etoto_2Ebit2 @ V2a_27 ) ) ) ) ).
thf(thm_2Etoto_2Enum__dt__nchotomy,axiom,
! [V0nn: tyop_2Etoto_2Enum__dt] :
( ( V0nn = c_2Etoto_2Ezer )
| ? [V1n: tyop_2Etoto_2Enum__dt] :
( V0nn
= ( c_2Etoto_2Ebit1 @ V1n ) )
| ? [V2n: tyop_2Etoto_2Enum__dt] :
( V0nn
= ( c_2Etoto_2Ebit2 @ V2n ) ) ) ).
thf(thm_2Etoto_2Enum__dt__Axiom,axiom,
! [A_27a: $tType,V0f0: A_27a,V1f1: tyop_2Etoto_2Enum__dt > A_27a > A_27a,V2f2: tyop_2Etoto_2Enum__dt > A_27a > A_27a] :
? [V3fn: tyop_2Etoto_2Enum__dt > A_27a] :
( ( ( V3fn @ c_2Etoto_2Ezer )
= V0f0 )
& ! [V4a: tyop_2Etoto_2Enum__dt] :
( ( V3fn @ ( c_2Etoto_2Ebit1 @ V4a ) )
= ( V1f1 @ V4a @ ( V3fn @ V4a ) ) )
& ! [V5a: tyop_2Etoto_2Enum__dt] :
( ( V3fn @ ( c_2Etoto_2Ebit2 @ V5a ) )
= ( V2f2 @ V5a @ ( V3fn @ V5a ) ) ) ) ).
thf(thm_2Etoto_2Enum__dt__induction,axiom,
! [V0P: tyop_2Etoto_2Enum__dt > $o] :
( ( ( V0P @ c_2Etoto_2Ezer )
& ! [V1n: tyop_2Etoto_2Enum__dt] :
( ( V0P @ V1n )
=> ( V0P @ ( c_2Etoto_2Ebit1 @ V1n ) ) )
& ! [V2n: tyop_2Etoto_2Enum__dt] :
( ( V0P @ V2n )
=> ( V0P @ ( c_2Etoto_2Ebit2 @ V2n ) ) ) )
=> ! [V3n: tyop_2Etoto_2Enum__dt] : ( V0P @ V3n ) ) ).
thf(thm_2Etoto_2Enum__dt__case__cong,axiom,
! [A_27a: $tType,V0v_27: A_27a,V1f1_27: tyop_2Etoto_2Enum__dt > A_27a,V2f_27: tyop_2Etoto_2Enum__dt > A_27a,V3M: tyop_2Etoto_2Enum__dt,V4M_27: tyop_2Etoto_2Enum__dt,V5v: A_27a,V6f: tyop_2Etoto_2Enum__dt > A_27a,V7f1: tyop_2Etoto_2Enum__dt > A_27a] :
( ( ( V3M = V4M_27 )
& ( ( V4M_27 = c_2Etoto_2Ezer )
=> ( V5v = V0v_27 ) )
& ! [V8a: tyop_2Etoto_2Enum__dt] :
( ( V4M_27
= ( c_2Etoto_2Ebit1 @ V8a ) )
=> ( ( V6f @ V8a )
= ( V2f_27 @ V8a ) ) )
& ! [V9a: tyop_2Etoto_2Enum__dt] :
( ( V4M_27
= ( c_2Etoto_2Ebit2 @ V9a ) )
=> ( ( V7f1 @ V9a )
= ( V1f1_27 @ V9a ) ) ) )
=> ( ( c_2Etoto_2Enum__dt__CASE @ A_27a @ V3M @ V5v @ V6f @ V7f1 )
= ( c_2Etoto_2Enum__dt__CASE @ A_27a @ V4M_27 @ V0v_27 @ V2f_27 @ V1f1_27 ) ) ) ).
thf(thm_2Etoto_2Enum__dt__case__eq,axiom,
! [A_27a: $tType,V0x: tyop_2Etoto_2Enum__dt,V1v_27: A_27a,V2v: A_27a,V3f1: tyop_2Etoto_2Enum__dt > A_27a,V4f: tyop_2Etoto_2Enum__dt > A_27a] :
( ( ( c_2Etoto_2Enum__dt__CASE @ A_27a @ V0x @ V2v @ V4f @ V3f1 )
= V1v_27 )
<=> ( ( ( V0x = c_2Etoto_2Ezer )
& ( V2v = V1v_27 ) )
| ? [V5n: tyop_2Etoto_2Enum__dt] :
( ( V0x
= ( c_2Etoto_2Ebit1 @ V5n ) )
& ( ( V4f @ V5n )
= V1v_27 ) )
| ? [V6n: tyop_2Etoto_2Enum__dt] :
( ( V0x
= ( c_2Etoto_2Ebit2 @ V6n ) )
& ( ( V3f1 @ V6n )
= V1v_27 ) ) ) ) ).
thf(thm_2Etoto_2Enum__dtOrd__ind,axiom,
! [V0P: tyop_2Etoto_2Enum__dt > tyop_2Etoto_2Enum__dt > $o] :
( ( ( V0P @ c_2Etoto_2Ezer @ c_2Etoto_2Ezer )
& ! [V1x: tyop_2Etoto_2Enum__dt] : ( V0P @ c_2Etoto_2Ezer @ ( c_2Etoto_2Ebit1 @ V1x ) )
& ! [V2x: tyop_2Etoto_2Enum__dt] : ( V0P @ c_2Etoto_2Ezer @ ( c_2Etoto_2Ebit2 @ V2x ) )
& ! [V3x: tyop_2Etoto_2Enum__dt] : ( V0P @ ( c_2Etoto_2Ebit1 @ V3x ) @ c_2Etoto_2Ezer )
& ! [V4x: tyop_2Etoto_2Enum__dt] : ( V0P @ ( c_2Etoto_2Ebit2 @ V4x ) @ c_2Etoto_2Ezer )
& ! [V5x: tyop_2Etoto_2Enum__dt,V6y: tyop_2Etoto_2Enum__dt] : ( V0P @ ( c_2Etoto_2Ebit1 @ V5x ) @ ( c_2Etoto_2Ebit2 @ V6y ) )
& ! [V7x: tyop_2Etoto_2Enum__dt,V8y: tyop_2Etoto_2Enum__dt] : ( V0P @ ( c_2Etoto_2Ebit2 @ V7x ) @ ( c_2Etoto_2Ebit1 @ V8y ) )
& ! [V9x: tyop_2Etoto_2Enum__dt,V10y: tyop_2Etoto_2Enum__dt] :
( ( V0P @ V9x @ V10y )
=> ( V0P @ ( c_2Etoto_2Ebit1 @ V9x ) @ ( c_2Etoto_2Ebit1 @ V10y ) ) )
& ! [V11x: tyop_2Etoto_2Enum__dt,V12y: tyop_2Etoto_2Enum__dt] :
( ( V0P @ V11x @ V12y )
=> ( V0P @ ( c_2Etoto_2Ebit2 @ V11x ) @ ( c_2Etoto_2Ebit2 @ V12y ) ) ) )
=> ! [V13v: tyop_2Etoto_2Enum__dt,V14v1: tyop_2Etoto_2Enum__dt] : ( V0P @ V13v @ V14v1 ) ) ).
thf(thm_2Etoto_2Enum__dtOrd,axiom,
( ( ( c_2Etoto_2Enum__dtOrd @ c_2Etoto_2Ezer @ c_2Etoto_2Ezer )
= c_2EternaryComparisons_2EEQUAL )
& ! [V0x: tyop_2Etoto_2Enum__dt] :
( ( c_2Etoto_2Enum__dtOrd @ c_2Etoto_2Ezer @ ( c_2Etoto_2Ebit1 @ V0x ) )
= c_2EternaryComparisons_2ELESS )
& ! [V1x: tyop_2Etoto_2Enum__dt] :
( ( c_2Etoto_2Enum__dtOrd @ c_2Etoto_2Ezer @ ( c_2Etoto_2Ebit2 @ V1x ) )
= c_2EternaryComparisons_2ELESS )
& ! [V2x: tyop_2Etoto_2Enum__dt] :
( ( c_2Etoto_2Enum__dtOrd @ ( c_2Etoto_2Ebit1 @ V2x ) @ c_2Etoto_2Ezer )
= c_2EternaryComparisons_2EGREATER )
& ! [V3x: tyop_2Etoto_2Enum__dt] :
( ( c_2Etoto_2Enum__dtOrd @ ( c_2Etoto_2Ebit2 @ V3x ) @ c_2Etoto_2Ezer )
= c_2EternaryComparisons_2EGREATER )
& ! [V4y: tyop_2Etoto_2Enum__dt,V5x: tyop_2Etoto_2Enum__dt] :
( ( c_2Etoto_2Enum__dtOrd @ ( c_2Etoto_2Ebit1 @ V5x ) @ ( c_2Etoto_2Ebit2 @ V4y ) )
= c_2EternaryComparisons_2ELESS )
& ! [V6y: tyop_2Etoto_2Enum__dt,V7x: tyop_2Etoto_2Enum__dt] :
( ( c_2Etoto_2Enum__dtOrd @ ( c_2Etoto_2Ebit2 @ V7x ) @ ( c_2Etoto_2Ebit1 @ V6y ) )
= c_2EternaryComparisons_2EGREATER )
& ! [V8y: tyop_2Etoto_2Enum__dt,V9x: tyop_2Etoto_2Enum__dt] :
( ( c_2Etoto_2Enum__dtOrd @ ( c_2Etoto_2Ebit1 @ V9x ) @ ( c_2Etoto_2Ebit1 @ V8y ) )
= ( c_2Etoto_2Enum__dtOrd @ V9x @ V8y ) )
& ! [V10y: tyop_2Etoto_2Enum__dt,V11x: tyop_2Etoto_2Enum__dt] :
( ( c_2Etoto_2Enum__dtOrd @ ( c_2Etoto_2Ebit2 @ V11x ) @ ( c_2Etoto_2Ebit2 @ V10y ) )
= ( c_2Etoto_2Enum__dtOrd @ V11x @ V10y ) ) ) ).
thf(thm_2Etoto_2ETO__qk__numOrd,axiom,
c_2Etoto_2ETotOrd @ tyop_2Enum_2Enum @ c_2Etoto_2Eqk__numOrd ).
thf(thm_2Etoto_2Eqk__numeralOrd,axiom,
! [V0x: tyop_2Enum_2Enum,V1y: tyop_2Enum_2Enum] :
( ( ( c_2Etoto_2Eqk__numOrd @ c_2Earithmetic_2EZERO @ c_2Earithmetic_2EZERO )
= c_2EternaryComparisons_2EEQUAL )
& ( ( c_2Etoto_2Eqk__numOrd @ c_2Earithmetic_2EZERO @ ( c_2Earithmetic_2EBIT1 @ V1y ) )
= c_2EternaryComparisons_2ELESS )
& ( ( c_2Etoto_2Eqk__numOrd @ c_2Earithmetic_2EZERO @ ( c_2Earithmetic_2EBIT2 @ V1y ) )
= c_2EternaryComparisons_2ELESS )
& ( ( c_2Etoto_2Eqk__numOrd @ ( c_2Earithmetic_2EBIT1 @ V0x ) @ c_2Earithmetic_2EZERO )
= c_2EternaryComparisons_2EGREATER )
& ( ( c_2Etoto_2Eqk__numOrd @ ( c_2Earithmetic_2EBIT2 @ V0x ) @ c_2Earithmetic_2EZERO )
= c_2EternaryComparisons_2EGREATER )
& ( ( c_2Etoto_2Eqk__numOrd @ ( c_2Earithmetic_2EBIT1 @ V0x ) @ ( c_2Earithmetic_2EBIT1 @ V1y ) )
= ( c_2Etoto_2Eqk__numOrd @ V0x @ V1y ) )
& ( ( c_2Etoto_2Eqk__numOrd @ ( c_2Earithmetic_2EBIT2 @ V0x ) @ ( c_2Earithmetic_2EBIT2 @ V1y ) )
= ( c_2Etoto_2Eqk__numOrd @ V0x @ V1y ) )
& ( ( c_2Etoto_2Eqk__numOrd @ ( c_2Earithmetic_2EBIT1 @ V0x ) @ ( c_2Earithmetic_2EBIT2 @ V1y ) )
= c_2EternaryComparisons_2ELESS )
& ( ( c_2Etoto_2Eqk__numOrd @ ( c_2Earithmetic_2EBIT2 @ V0x ) @ ( c_2Earithmetic_2EBIT1 @ V1y ) )
= c_2EternaryComparisons_2EGREATER ) ) ).
thf(thm_2Etoto_2Eap__qk__numto__thm,axiom,
( ( c_2Etoto_2Eapto @ tyop_2Enum_2Enum @ c_2Etoto_2Eqk__numto )
= c_2Etoto_2Eqk__numOrd ) ).
thf(thm_2Etoto_2ETO__charOrd,axiom,
c_2Etoto_2ETotOrd @ tyop_2Estring_2Echar @ c_2Etoto_2EcharOrd ).
thf(thm_2Etoto_2Eapcharto__thm,axiom,
( ( c_2Etoto_2Eapto @ tyop_2Estring_2Echar @ c_2Etoto_2Echarto )
= c_2Etoto_2EcharOrd ) ).
thf(thm_2Etoto_2EcharOrd__lt__lem,axiom,
! [V0a: tyop_2Enum_2Enum,V1b: tyop_2Enum_2Enum] :
( ( ( c_2Etoto_2EnumOrd @ V0a @ V1b )
= c_2EternaryComparisons_2ELESS )
=> ( ( ( c_2Eprim__rec_2E_3C @ V1b @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ) ) ) ) ) )
= c_2Ebool_2ET )
=> ( ( c_2Etoto_2EcharOrd @ ( c_2Estring_2ECHR @ V0a ) @ ( c_2Estring_2ECHR @ V1b ) )
= c_2EternaryComparisons_2ELESS ) ) ) ).
thf(thm_2Etoto_2EcharOrd__gt__lem,axiom,
! [V0a: tyop_2Enum_2Enum,V1b: tyop_2Enum_2Enum] :
( ( ( c_2Etoto_2EnumOrd @ V0a @ V1b )
= c_2EternaryComparisons_2EGREATER )
=> ( ( ( c_2Eprim__rec_2E_3C @ V0a @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ) ) ) ) ) )
= c_2Ebool_2ET )
=> ( ( c_2Etoto_2EcharOrd @ ( c_2Estring_2ECHR @ V0a ) @ ( c_2Estring_2ECHR @ V1b ) )
= c_2EternaryComparisons_2EGREATER ) ) ) ).
thf(thm_2Etoto_2EcharOrd__eq__lem,axiom,
! [V0a: tyop_2Enum_2Enum,V1b: tyop_2Enum_2Enum] :
( ( ( c_2Etoto_2EnumOrd @ V0a @ V1b )
= c_2EternaryComparisons_2EEQUAL )
=> ( ( c_2Etoto_2EcharOrd @ ( c_2Estring_2ECHR @ V0a ) @ ( c_2Estring_2ECHR @ V1b ) )
= c_2EternaryComparisons_2EEQUAL ) ) ).
thf(thm_2Etoto_2EcharOrd__thm,axiom,
( c_2Etoto_2EcharOrd
= ( c_2Etoto_2ETO__of__LinearOrder @ tyop_2Estring_2Echar @ c_2Estring_2Echar__lt ) ) ).
thf(thm_2Etoto_2Elistorder__ind,axiom,
! [A_27a: $tType,V0P: ( A_27a > A_27a > $o ) > ( tyop_2Elist_2Elist @ A_27a ) > ( tyop_2Elist_2Elist @ A_27a ) > $o] :
( ( ! [V1V: A_27a > A_27a > $o,V2l: tyop_2Elist_2Elist @ A_27a] : ( V0P @ V1V @ V2l @ ( c_2Elist_2ENIL @ A_27a ) )
& ! [V3V: A_27a > A_27a > $o,V4s: A_27a,V5m: tyop_2Elist_2Elist @ A_27a] : ( V0P @ V3V @ ( c_2Elist_2ENIL @ A_27a ) @ ( c_2Elist_2ECONS @ A_27a @ V4s @ V5m ) )
& ! [V6V: A_27a > A_27a > $o,V7r: A_27a,V8l: tyop_2Elist_2Elist @ A_27a,V9s: A_27a,V10m: tyop_2Elist_2Elist @ A_27a] :
( ( V0P @ V6V @ V8l @ V10m )
=> ( V0P @ V6V @ ( c_2Elist_2ECONS @ A_27a @ V7r @ V8l ) @ ( c_2Elist_2ECONS @ A_27a @ V9s @ V10m ) ) ) )
=> ! [V11v: A_27a > A_27a > $o,V12v1: tyop_2Elist_2Elist @ A_27a,V13v2: tyop_2Elist_2Elist @ A_27a] : ( V0P @ V11v @ V12v1 @ V13v2 ) ) ).
thf(thm_2Etoto_2Elistorder,axiom,
! [A_27a: $tType] :
( ! [V0l: tyop_2Elist_2Elist @ A_27a,V1V: A_27a > A_27a > $o] :
( ( c_2Etoto_2Elistorder @ A_27a @ V1V @ V0l @ ( c_2Elist_2ENIL @ A_27a ) )
= c_2Ebool_2EF )
& ! [V2s: A_27a,V3m: tyop_2Elist_2Elist @ A_27a,V4V: A_27a > A_27a > $o] :
( ( c_2Etoto_2Elistorder @ A_27a @ V4V @ ( c_2Elist_2ENIL @ A_27a ) @ ( c_2Elist_2ECONS @ A_27a @ V2s @ V3m ) )
= c_2Ebool_2ET )
& ! [V5s: A_27a,V6r: A_27a,V7m: tyop_2Elist_2Elist @ A_27a,V8l: tyop_2Elist_2Elist @ A_27a,V9V: A_27a > A_27a > $o] :
( ( c_2Etoto_2Elistorder @ A_27a @ V9V @ ( c_2Elist_2ECONS @ A_27a @ V6r @ V8l ) @ ( c_2Elist_2ECONS @ A_27a @ V5s @ V7m ) )
<=> ( ( V9V @ V6r @ V5s )
| ( ( V6r = V5s )
& ( c_2Etoto_2Elistorder @ A_27a @ V9V @ V8l @ V7m ) ) ) ) ) ).
thf(thm_2Etoto_2ESLO__listorder,axiom,
! [A_27a: $tType,V0V: A_27a > A_27a > $o] :
( ( c_2Erelation_2EStrongLinearOrder @ A_27a @ V0V )
=> ( c_2Erelation_2EStrongLinearOrder @ ( tyop_2Elist_2Elist @ A_27a ) @ ( c_2Etoto_2Elistorder @ A_27a @ V0V ) ) ) ).
thf(thm_2Etoto_2ETO__ListOrd,axiom,
! [A_27a: $tType,V0c: tyop_2Etoto_2Etoto @ A_27a] : ( c_2Etoto_2ETotOrd @ ( tyop_2Elist_2Elist @ A_27a ) @ ( c_2Etoto_2EListOrd @ A_27a @ V0c ) ) ).
thf(thm_2Etoto_2EListOrd__THM,axiom,
! [A_27a: $tType,V0c: tyop_2Etoto_2Etoto @ A_27a] :
( ( ( c_2Etoto_2EListOrd @ A_27a @ V0c @ ( c_2Elist_2ENIL @ A_27a ) @ ( c_2Elist_2ENIL @ A_27a ) )
= c_2EternaryComparisons_2EEQUAL )
& ! [V1b: A_27a,V2y: tyop_2Elist_2Elist @ A_27a] :
( ( c_2Etoto_2EListOrd @ A_27a @ V0c @ ( c_2Elist_2ENIL @ A_27a ) @ ( c_2Elist_2ECONS @ A_27a @ V1b @ V2y ) )
= c_2EternaryComparisons_2ELESS )
& ! [V3a: A_27a,V4x: tyop_2Elist_2Elist @ A_27a] :
( ( c_2Etoto_2EListOrd @ A_27a @ V0c @ ( c_2Elist_2ECONS @ A_27a @ V3a @ V4x ) @ ( c_2Elist_2ENIL @ A_27a ) )
= c_2EternaryComparisons_2EGREATER )
& ! [V5a: A_27a,V6x: tyop_2Elist_2Elist @ A_27a,V7b: A_27a,V8y: tyop_2Elist_2Elist @ A_27a] :
( ( c_2Etoto_2EListOrd @ A_27a @ V0c @ ( c_2Elist_2ECONS @ A_27a @ V5a @ V6x ) @ ( c_2Elist_2ECONS @ A_27a @ V7b @ V8y ) )
= ( c_2EternaryComparisons_2Eordering__CASE @ tyop_2EternaryComparisons_2Eordering @ ( c_2Etoto_2Eapto @ A_27a @ V0c @ V5a @ V7b ) @ c_2EternaryComparisons_2ELESS @ ( c_2Etoto_2EListOrd @ A_27a @ V0c @ V6x @ V8y ) @ c_2EternaryComparisons_2EGREATER ) ) ) ).
thf(thm_2Etoto_2Eaplistoto,axiom,
! [A_27a: $tType,V0c: tyop_2Etoto_2Etoto @ A_27a] :
( ( ( c_2Etoto_2Eapto @ ( tyop_2Elist_2Elist @ A_27a ) @ ( c_2Etoto_2Elistoto @ A_27a @ V0c ) @ ( c_2Elist_2ENIL @ A_27a ) @ ( c_2Elist_2ENIL @ A_27a ) )
= c_2EternaryComparisons_2EEQUAL )
& ! [V1b: A_27a,V2y: tyop_2Elist_2Elist @ A_27a] :
( ( c_2Etoto_2Eapto @ ( tyop_2Elist_2Elist @ A_27a ) @ ( c_2Etoto_2Elistoto @ A_27a @ V0c ) @ ( c_2Elist_2ENIL @ A_27a ) @ ( c_2Elist_2ECONS @ A_27a @ V1b @ V2y ) )
= c_2EternaryComparisons_2ELESS )
& ! [V3a: A_27a,V4x: tyop_2Elist_2Elist @ A_27a] :
( ( c_2Etoto_2Eapto @ ( tyop_2Elist_2Elist @ A_27a ) @ ( c_2Etoto_2Elistoto @ A_27a @ V0c ) @ ( c_2Elist_2ECONS @ A_27a @ V3a @ V4x ) @ ( c_2Elist_2ENIL @ A_27a ) )
= c_2EternaryComparisons_2EGREATER )
& ! [V5a: A_27a,V6x: tyop_2Elist_2Elist @ A_27a,V7b: A_27a,V8y: tyop_2Elist_2Elist @ A_27a] :
( ( c_2Etoto_2Eapto @ ( tyop_2Elist_2Elist @ A_27a ) @ ( c_2Etoto_2Elistoto @ A_27a @ V0c ) @ ( c_2Elist_2ECONS @ A_27a @ V5a @ V6x ) @ ( c_2Elist_2ECONS @ A_27a @ V7b @ V8y ) )
= ( c_2EternaryComparisons_2Eordering__CASE @ tyop_2EternaryComparisons_2Eordering @ ( c_2Etoto_2Eapto @ A_27a @ V0c @ V5a @ V7b ) @ c_2EternaryComparisons_2ELESS @ ( c_2Etoto_2Eapto @ ( tyop_2Elist_2Elist @ A_27a ) @ ( c_2Etoto_2Elistoto @ A_27a @ V0c ) @ V6x @ V8y ) @ c_2EternaryComparisons_2EGREATER ) ) ) ).
thf(thm_2Etoto_2ETO__injection,axiom,
! [A_27c: $tType,A_27d: $tType,V0cp: A_27c > A_27c > tyop_2EternaryComparisons_2Eordering] :
( ( c_2Etoto_2ETotOrd @ A_27c @ V0cp )
=> ! [V1f: A_27d > A_27c] :
( ( c_2Ebool_2EONE__ONE @ A_27d @ A_27c @ V1f )
=> ( c_2Etoto_2ETotOrd @ A_27d @ ( c_2Etoto_2EimageOrd @ A_27d @ A_27c @ V1f @ V0cp ) ) ) ) ).
thf(thm_2Etoto_2EStrongLinearOrder__of__TO__TO__of__LinearOrder,axiom,
! [A_27a: $tType,V0R: A_27a > A_27a > $o] :
( ( c_2Erelation_2Eirreflexive @ A_27a @ V0R )
=> ( ( c_2Etoto_2EStrongLinearOrder__of__TO @ A_27a @ ( c_2Etoto_2ETO__of__LinearOrder @ A_27a @ V0R ) )
= V0R ) ) ).
thf(thm_2Etoto_2ETO__of__LinearOrder__LEX,axiom,
! [A_27a: $tType,A_27b: $tType,V0R: A_27a > A_27a > $o,V1V: A_27b > A_27b > $o] :
( ( ( c_2Erelation_2Eirreflexive @ A_27a @ V0R )
& ( c_2Erelation_2Eirreflexive @ A_27b @ V1V ) )
=> ( ( c_2Etoto_2ETO__of__LinearOrder @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ ( c_2Epair_2ELEX @ A_27a @ A_27b @ V0R @ V1V ) )
= ( c_2Etoto_2ElexTO @ A_27a @ A_27b @ ( c_2Etoto_2ETO__of__LinearOrder @ A_27a @ V0R ) @ ( c_2Etoto_2ETO__of__LinearOrder @ A_27b @ V1V ) ) ) ) ).
%------------------------------------------------------------------------------