ITP001 Axioms: ITP086^7.ax
%------------------------------------------------------------------------------
% File : ITP086^7 : 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 : ordinal.ax [Gau19]
% : HL4086^7.ax [TPAP]
% Status : Satisfiable
% Syntax : Number of formulae : 310 ( 70 unt; 93 typ; 0 def)
% Number of atoms : 795 ( 206 equ; 76 cnn)
% Maximal formula atoms : 17 ( 2 avg)
% Number of connectives : 4511 ( 76 ~; 11 |; 126 &;4122 @)
% ( 56 <=>; 120 =>; 0 <=; 0 <~>)
% Maximal formula depth : 23 ( 8 avg;4122 nst)
% Number of types : 4 ( 3 usr)
% Number of type conns : 251 ( 251 >; 0 *; 0 +; 0 <<)
% Number of symbols : 92 ( 90 usr; 5 con; 0-5 aty)
% Number of variables : 783 ( 23 ^ 655 !; 25 ?; 783 :)
% ( 80 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_SAT_EQU_NAR
% Comments :
% Bugfixes : v7.5.0 - Fixes to the axioms.
%------------------------------------------------------------------------------
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_2Eone_2Eone,type,
tyop_2Eone_2Eone: $tType ).
thf(tyop_2Eoption_2Eoption,type,
tyop_2Eoption_2Eoption: $tType > $tType ).
thf(tyop_2Eordinal_2Eordinal,type,
tyop_2Eordinal_2Eordinal: $tType > $tType ).
thf(tyop_2Epair_2Eprod,type,
tyop_2Epair_2Eprod: $tType > $tType > $tType ).
thf(tyop_2Esum_2Esum,type,
tyop_2Esum_2Esum: $tType > $tType > $tType ).
thf(tyop_2Ewellorder_2Ewellorder,type,
tyop_2Ewellorder_2Ewellorder: $tType > $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_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_2Epred__set_2EBIGUNION,type,
c_2Epred__set_2EBIGUNION:
!>[A_27a: $tType] : ( ( ( A_27a > $o ) > $o ) > A_27a > $o ) ).
thf(c_2Epred__set_2EBIJ,type,
c_2Epred__set_2EBIJ:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > A_27b ) > ( A_27a > $o ) > ( A_27b > $o ) > $o ) ).
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_2Elist_2ECONS,type,
c_2Elist_2ECONS:
!>[A_27a: $tType] : ( A_27a > ( tyop_2Elist_2Elist @ A_27a ) > ( tyop_2Elist_2Elist @ A_27a ) ) ).
thf(c_2Epred__set_2EDELETE,type,
c_2Epred__set_2EDELETE:
!>[A_27a: $tType] : ( ( A_27a > $o ) > A_27a > A_27a > $o ) ).
thf(c_2Elist_2EEL,type,
c_2Elist_2EEL:
!>[A_27a: $tType] : ( tyop_2Enum_2Enum > ( tyop_2Elist_2Elist @ A_27a ) > A_27a ) ).
thf(c_2Epred__set_2EEMPTY,type,
c_2Epred__set_2EEMPTY:
!>[A_27a: $tType] : ( A_27a > $o ) ).
thf(c_2Earithmetic_2EEXP,type,
c_2Earithmetic_2EEXP: tyop_2Enum_2Enum > tyop_2Enum_2Enum > tyop_2Enum_2Enum ).
thf(c_2Ebool_2EF,type,
c_2Ebool_2EF: $o ).
thf(c_2Epred__set_2EGSPEC,type,
c_2Epred__set_2EGSPEC:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27b > ( tyop_2Epair_2Eprod @ A_27a @ $o ) ) > A_27a > $o ) ).
thf(c_2Elist_2EHD,type,
c_2Elist_2EHD:
!>[A_27a: $tType] : ( ( tyop_2Elist_2Elist @ A_27a ) > A_27a ) ).
thf(c_2Epred__set_2EIMAGE,type,
c_2Epred__set_2EIMAGE:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > A_27b ) > ( A_27a > $o ) > A_27b > $o ) ).
thf(c_2Ebool_2EIN,type,
c_2Ebool_2EIN:
!>[A_27a: $tType] : ( A_27a > ( A_27a > $o ) > $o ) ).
thf(c_2Epred__set_2EINSERT,type,
c_2Epred__set_2EINSERT:
!>[A_27a: $tType] : ( A_27a > ( A_27a > $o ) > A_27a > $o ) ).
thf(c_2Elist_2ELENGTH,type,
c_2Elist_2ELENGTH:
!>[A_27a: $tType] : ( ( tyop_2Elist_2Elist @ A_27a ) > tyop_2Enum_2Enum ) ).
thf(c_2Elist_2ELIST__TO__SET,type,
c_2Elist_2ELIST__TO__SET:
!>[A_27a: $tType] : ( ( tyop_2Elist_2Elist @ A_27a ) > A_27a > $o ) ).
thf(c_2Elist_2ENIL,type,
c_2Elist_2ENIL:
!>[A_27a: $tType] : ( tyop_2Elist_2Elist @ A_27a ) ).
thf(c_2Eoption_2ENONE,type,
c_2Eoption_2ENONE:
!>[A_27a: $tType] : ( tyop_2Eoption_2Eoption @ A_27a ) ).
thf(c_2Earithmetic_2ENUMERAL,type,
c_2Earithmetic_2ENUMERAL: tyop_2Enum_2Enum > tyop_2Enum_2Enum ).
thf(c_2Epred__set_2EPSUBSET,type,
c_2Epred__set_2EPSUBSET:
!>[A_27a: $tType] : ( ( A_27a > $o ) > ( A_27a > $o ) > $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_2Epair_2ESND,type,
c_2Epair_2ESND:
!>[A_27a: $tType,A_27b: $tType] : ( ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) > A_27b ) ).
thf(c_2Eoption_2ESOME,type,
c_2Eoption_2ESOME:
!>[A_27a: $tType] : ( A_27a > ( tyop_2Eoption_2Eoption @ A_27a ) ) ).
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_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_2Epred__set_2EUNIV,type,
c_2Epred__set_2EUNIV:
!>[A_27a: $tType] : ( A_27a > $o ) ).
thf(c_2Erelation_2EWF,type,
c_2Erelation_2EWF:
!>[A_27a: $tType] : ( ( A_27a > A_27a > $o ) > $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_2Eordinal_2EallOrds,type,
c_2Eordinal_2EallOrds:
!>[A_27a: $tType] : ( tyop_2Ewellorder_2Ewellorder @ ( tyop_2Eordinal_2Eordinal @ A_27a ) ) ).
thf(c_2Ecardinal_2Ecardeq,type,
c_2Ecardinal_2Ecardeq:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > $o ) > ( A_27b > $o ) > $o ) ).
thf(c_2Ecardinal_2Ecardleq,type,
c_2Ecardinal_2Ecardleq:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > $o ) > ( A_27b > $o ) > $o ) ).
thf(c_2Epred__set_2Ecountable,type,
c_2Epred__set_2Ecountable:
!>[A_27a: $tType] : ( ( A_27a > $o ) > $o ) ).
thf(c_2Eordinal_2Edclose,type,
c_2Eordinal_2Edclose:
!>[A_27a: $tType] : ( ( ( tyop_2Eordinal_2Eordinal @ A_27a ) > $o ) > ( tyop_2Eordinal_2Eordinal @ A_27a ) > $o ) ).
thf(c_2Eordinal_2Edownward__closed,type,
c_2Eordinal_2Edownward__closed:
!>[A_27a: $tType] : ( ( ( tyop_2Eordinal_2Eordinal @ A_27a ) > $o ) > $o ) ).
thf(c_2Ewellorder_2EelsOf,type,
c_2Ewellorder_2EelsOf:
!>[A_27a: $tType] : ( ( tyop_2Ewellorder_2Ewellorder @ A_27a ) > A_27a > $o ) ).
thf(c_2Eordinal_2Eepsilon0,type,
c_2Eordinal_2Eepsilon0:
!>[A_27a: $tType] : ( tyop_2Eordinal_2Eordinal @ A_27a ) ).
thf(c_2Eordinal_2Eeval__poly,type,
c_2Eordinal_2Eeval__poly:
!>[A_27a: $tType] : ( ( tyop_2Eordinal_2Eordinal @ A_27a ) > ( tyop_2Elist_2Elist @ ( tyop_2Epair_2Eprod @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ ( tyop_2Eordinal_2Eordinal @ A_27a ) ) ) > ( tyop_2Eordinal_2Eordinal @ A_27a ) ) ).
thf(c_2Eordinal_2EfromNat,type,
c_2Eordinal_2EfromNat:
!>[A_27a: $tType] : ( tyop_2Enum_2Enum > ( tyop_2Eordinal_2Eordinal @ A_27a ) ) ).
thf(c_2Eordinal_2Eis__polyform,type,
c_2Eordinal_2Eis__polyform:
!>[A_27a: $tType,A_27b: $tType] : ( ( tyop_2Eordinal_2Eordinal @ A_27a ) > ( tyop_2Elist_2Elist @ ( tyop_2Epair_2Eprod @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ ( tyop_2Eordinal_2Eordinal @ A_27b ) ) ) > $o ) ).
thf(c_2Eset__relation_2Emaximal__elements,type,
c_2Eset__relation_2Emaximal__elements:
!>[A_27a: $tType] : ( ( A_27a > $o ) > ( ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) > $o ) > A_27a > $o ) ).
thf(c_2Eordinal_2Eoleast,type,
c_2Eordinal_2Eoleast:
!>[A_27a: $tType] : ( ( ( tyop_2Eordinal_2Eordinal @ A_27a ) > $o ) > ( tyop_2Eordinal_2Eordinal @ A_27a ) ) ).
thf(c_2Eordinal_2Eomax,type,
c_2Eordinal_2Eomax:
!>[A_27a: $tType] : ( ( ( tyop_2Eordinal_2Eordinal @ A_27a ) > $o ) > ( tyop_2Eoption_2Eoption @ ( tyop_2Eordinal_2Eordinal @ A_27a ) ) ) ).
thf(c_2Eordinal_2Eomega,type,
c_2Eordinal_2Eomega:
!>[A_27a: $tType] : ( tyop_2Eordinal_2Eordinal @ A_27a ) ).
thf(c_2Eordinal_2EordADD,type,
c_2Eordinal_2EordADD:
!>[A_27a: $tType] : ( ( tyop_2Eordinal_2Eordinal @ A_27a ) > ( tyop_2Eordinal_2Eordinal @ A_27a ) > ( tyop_2Eordinal_2Eordinal @ A_27a ) ) ).
thf(c_2Eordinal_2EordDIV,type,
c_2Eordinal_2EordDIV:
!>[A_27a: $tType] : ( ( tyop_2Eordinal_2Eordinal @ A_27a ) > ( tyop_2Eordinal_2Eordinal @ A_27a ) > ( tyop_2Eordinal_2Eordinal @ A_27a ) ) ).
thf(c_2Eordinal_2EordEXP,type,
c_2Eordinal_2EordEXP:
!>[A_27a: $tType] : ( ( tyop_2Eordinal_2Eordinal @ A_27a ) > ( tyop_2Eordinal_2Eordinal @ A_27a ) > ( tyop_2Eordinal_2Eordinal @ A_27a ) ) ).
thf(c_2Eordinal_2EordMOD,type,
c_2Eordinal_2EordMOD:
!>[A_27a: $tType] : ( ( tyop_2Eordinal_2Eordinal @ A_27a ) > ( tyop_2Eordinal_2Eordinal @ A_27a ) > ( tyop_2Eordinal_2Eordinal @ A_27a ) ) ).
thf(c_2Eordinal_2EordMULT,type,
c_2Eordinal_2EordMULT:
!>[A_27a: $tType] : ( ( tyop_2Eordinal_2Eordinal @ A_27a ) > ( tyop_2Eordinal_2Eordinal @ A_27a ) > ( tyop_2Eordinal_2Eordinal @ A_27a ) ) ).
thf(c_2Eordinal_2EordSUC,type,
c_2Eordinal_2EordSUC:
!>[A_27a: $tType] : ( ( tyop_2Eordinal_2Eordinal @ A_27a ) > ( tyop_2Eordinal_2Eordinal @ A_27a ) ) ).
thf(c_2Ewellorder_2Eorderiso,type,
c_2Ewellorder_2Eorderiso:
!>[A_27a: $tType,A_27b: $tType] : ( ( tyop_2Ewellorder_2Ewellorder @ A_27a ) > ( tyop_2Ewellorder_2Ewellorder @ A_27b ) > $o ) ).
thf(c_2Ewellorder_2Eorderlt,type,
c_2Ewellorder_2Eorderlt:
!>[A_27a: $tType,A_27b: $tType] : ( ( tyop_2Ewellorder_2Ewellorder @ A_27a ) > ( tyop_2Ewellorder_2Ewellorder @ A_27b ) > $o ) ).
thf(c_2Eordinal_2Eordinal__ABS,type,
c_2Eordinal_2Eordinal__ABS:
!>[A_27a: $tType] : ( ( tyop_2Ewellorder_2Ewellorder @ ( tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ A_27a ) ) > ( tyop_2Eordinal_2Eordinal @ A_27a ) ) ).
thf(c_2Eordinal_2Eordinal__ABS__CLASS,type,
c_2Eordinal_2Eordinal__ABS__CLASS:
!>[A_27a: $tType] : ( ( ( tyop_2Ewellorder_2Ewellorder @ ( tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ A_27a ) ) > $o ) > ( tyop_2Eordinal_2Eordinal @ A_27a ) ) ).
thf(c_2Eordinal_2Eordinal__REP,type,
c_2Eordinal_2Eordinal__REP:
!>[A_27a: $tType] : ( ( tyop_2Eordinal_2Eordinal @ A_27a ) > ( tyop_2Ewellorder_2Ewellorder @ ( tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ A_27a ) ) ) ).
thf(c_2Eordinal_2Eordinal__REP__CLASS,type,
c_2Eordinal_2Eordinal__REP__CLASS:
!>[A_27a: $tType] : ( ( tyop_2Eordinal_2Eordinal @ A_27a ) > ( tyop_2Ewellorder_2Ewellorder @ ( tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ A_27a ) ) > $o ) ).
thf(c_2Eordinal_2Eordlt,type,
c_2Eordinal_2Eordlt:
!>[A_27a: $tType] : ( ( tyop_2Eordinal_2Eordinal @ A_27a ) > ( tyop_2Eordinal_2Eordinal @ A_27a ) > $o ) ).
thf(c_2Eordinal_2Epolyform,type,
c_2Eordinal_2Epolyform:
!>[A_27a: $tType] : ( ( tyop_2Eordinal_2Eordinal @ A_27a ) > ( tyop_2Eordinal_2Eordinal @ A_27a ) > ( tyop_2Elist_2Elist @ ( tyop_2Epair_2Eprod @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ ( tyop_2Eordinal_2Eordinal @ A_27a ) ) ) ) ).
thf(c_2Eordinal_2Epreds,type,
c_2Eordinal_2Epreds:
!>[A_27a: $tType] : ( ( tyop_2Eordinal_2Eordinal @ A_27a ) > ( tyop_2Eordinal_2Eordinal @ A_27a ) > $o ) ).
thf(c_2Eoption_2Esome,type,
c_2Eoption_2Esome:
!>[A_27a: $tType] : ( ( A_27a > $o ) > ( tyop_2Eoption_2Eoption @ A_27a ) ) ).
thf(c_2Eset__relation_2Estrict,type,
c_2Eset__relation_2Estrict:
!>[A_27a: $tType] : ( ( ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) > $o ) > ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) > $o ) ).
thf(c_2Eordinal_2Esup,type,
c_2Eordinal_2Esup:
!>[A_27a: $tType] : ( ( ( tyop_2Eordinal_2Eordinal @ A_27a ) > $o ) > ( tyop_2Eordinal_2Eordinal @ A_27a ) ) ).
thf(c_2Ewellorder_2Ewellorder,type,
c_2Ewellorder_2Ewellorder:
!>[A_27a: $tType] : ( ( ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) > $o ) > $o ) ).
thf(c_2Ewellorder_2Ewellorder__ABS,type,
c_2Ewellorder_2Ewellorder__ABS:
!>[A_27a: $tType] : ( ( ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) > $o ) > ( tyop_2Ewellorder_2Ewellorder @ A_27a ) ) ).
thf(c_2Ewellorder_2Ewellorder__REP,type,
c_2Ewellorder_2Ewellorder__REP:
!>[A_27a: $tType] : ( ( tyop_2Ewellorder_2Ewellorder @ A_27a ) > ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) > $o ) ).
thf(c_2Ewellorder_2Ewobound,type,
c_2Ewellorder_2Ewobound:
!>[A_27a: $tType] : ( A_27a > ( tyop_2Ewellorder_2Ewellorder @ A_27a ) > ( tyop_2Ewellorder_2Ewellorder @ A_27a ) ) ).
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_2Eordinal_2EordADD__def,axiom,
! [A_27a: $tType,V0b: tyop_2Eordinal_2Eordinal @ A_27a] :
( ( ( c_2Eordinal_2EordADD @ A_27a @ V0b @ ( c_2Eordinal_2EfromNat @ A_27a @ c_2Enum_2E0 ) )
= V0b )
& ! [V1a: tyop_2Eordinal_2Eordinal @ A_27a] :
( ( c_2Eordinal_2EordADD @ A_27a @ V0b @ ( c_2Eordinal_2EordSUC @ A_27a @ V1a ) )
= ( c_2Eordinal_2EordSUC @ A_27a @ ( c_2Eordinal_2EordADD @ A_27a @ V0b @ V1a ) ) )
& ! [V2a: tyop_2Eordinal_2Eordinal @ A_27a] :
( ( ( c_2Eordinal_2Eordlt @ A_27a @ ( c_2Eordinal_2EfromNat @ A_27a @ c_2Enum_2E0 ) @ V2a )
& ( ( c_2Eordinal_2Eomax @ A_27a @ ( c_2Eordinal_2Epreds @ A_27a @ V2a ) )
= ( c_2Eoption_2ENONE @ ( tyop_2Eordinal_2Eordinal @ A_27a ) ) ) )
=> ( ( c_2Eordinal_2EordADD @ A_27a @ V0b @ V2a )
= ( c_2Eordinal_2Esup @ A_27a @ ( c_2Epred__set_2EIMAGE @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ ( c_2Eordinal_2EordADD @ A_27a @ V0b ) @ ( c_2Eordinal_2Epreds @ A_27a @ V2a ) ) ) ) ) ) ).
thf(thm_2Eordinal_2Eomega__def,axiom,
! [A_27a: $tType] :
( ( c_2Eordinal_2Eomega @ A_27a )
= ( c_2Eordinal_2Esup @ A_27a
@ ( c_2Epred__set_2EGSPEC @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ tyop_2Enum_2Enum
@ ^ [V0i: tyop_2Enum_2Enum] : ( c_2Epair_2E_2C @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ $o @ ( c_2Eordinal_2EfromNat @ A_27a @ V0i ) @ c_2Ebool_2ET ) ) ) ) ).
thf(thm_2Eordinal_2Edclose__def,axiom,
! [A_27a: $tType,V0s: ( tyop_2Eordinal_2Eordinal @ A_27a ) > $o] :
( ( c_2Eordinal_2Edclose @ A_27a @ V0s )
= ( c_2Epred__set_2EGSPEC @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ ( tyop_2Eordinal_2Eordinal @ A_27a )
@ ^ [V1x: tyop_2Eordinal_2Eordinal @ A_27a] :
( c_2Epair_2E_2C @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ $o @ V1x
@ ( c_2Ebool_2E_3F @ ( tyop_2Eordinal_2Eordinal @ A_27a )
@ ^ [V2y: tyop_2Eordinal_2Eordinal @ A_27a] : ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ V2y @ V0s ) @ ( c_2Eordinal_2Eordlt @ A_27a @ V1x @ V2y ) ) ) ) ) ) ).
thf(thm_2Eordinal_2Eordinal__TY__DEF,axiom,
! [A_27a: $tType] :
? [V0rep: ( tyop_2Eordinal_2Eordinal @ A_27a ) > ( tyop_2Ewellorder_2Ewellorder @ ( tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ A_27a ) ) > $o] :
( c_2Ebool_2ETYPE__DEFINITION @ ( ( tyop_2Ewellorder_2Ewellorder @ ( tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ A_27a ) ) > $o ) @ ( tyop_2Eordinal_2Eordinal @ A_27a )
@ ^ [V1c: ( tyop_2Ewellorder_2Ewellorder @ ( tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ A_27a ) ) > $o] :
( c_2Ebool_2E_3F @ ( tyop_2Ewellorder_2Ewellorder @ ( tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ A_27a ) )
@ ^ [V2r: tyop_2Ewellorder_2Ewellorder @ ( tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ A_27a )] : ( c_2Ebool_2E_2F_5C @ ( c_2Ewellorder_2Eorderiso @ ( tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ A_27a ) @ ( tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ A_27a ) @ V2r @ V2r ) @ ( c_2Emin_2E_3D @ ( ( tyop_2Ewellorder_2Ewellorder @ ( tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ A_27a ) ) > $o ) @ V1c @ ( c_2Ewellorder_2Eorderiso @ ( tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ A_27a ) @ ( tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ A_27a ) @ V2r ) ) ) )
@ V0rep ) ).
thf(thm_2Eordinal_2Eordinal__bijections,axiom,
! [A_27a: $tType] :
( ! [V0a: tyop_2Eordinal_2Eordinal @ A_27a] :
( ( c_2Eordinal_2Eordinal__ABS__CLASS @ A_27a @ ( c_2Eordinal_2Eordinal__REP__CLASS @ A_27a @ V0a ) )
= V0a )
& ! [V1r: ( tyop_2Ewellorder_2Ewellorder @ ( tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ A_27a ) ) > $o] :
( ( ^ [V2c: ( tyop_2Ewellorder_2Ewellorder @ ( tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ A_27a ) ) > $o] :
( c_2Ebool_2E_3F @ ( tyop_2Ewellorder_2Ewellorder @ ( tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ A_27a ) )
@ ^ [V3r: tyop_2Ewellorder_2Ewellorder @ ( tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ A_27a )] : ( c_2Ebool_2E_2F_5C @ ( c_2Ewellorder_2Eorderiso @ ( tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ A_27a ) @ ( tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ A_27a ) @ V3r @ V3r ) @ ( c_2Emin_2E_3D @ ( ( tyop_2Ewellorder_2Ewellorder @ ( tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ A_27a ) ) > $o ) @ V2c @ ( c_2Ewellorder_2Eorderiso @ ( tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ A_27a ) @ ( tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ A_27a ) @ V3r ) ) ) )
@ V1r )
<=> ( ( c_2Eordinal_2Eordinal__REP__CLASS @ A_27a @ ( c_2Eordinal_2Eordinal__ABS__CLASS @ A_27a @ V1r ) )
= V1r ) ) ) ).
thf(thm_2Eordinal_2Eordinal__REP__def,axiom,
! [A_27a: $tType,V0a: tyop_2Eordinal_2Eordinal @ A_27a] :
( ( c_2Eordinal_2Eordinal__REP @ A_27a @ V0a )
= ( c_2Emin_2E_40 @ ( tyop_2Ewellorder_2Ewellorder @ ( tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ A_27a ) ) @ ( c_2Eordinal_2Eordinal__REP__CLASS @ A_27a @ V0a ) ) ) ).
thf(thm_2Eordinal_2Eordinal__ABS__def,axiom,
! [A_27a: $tType,V0r: tyop_2Ewellorder_2Ewellorder @ ( tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ A_27a )] :
( ( c_2Eordinal_2Eordinal__ABS @ A_27a @ V0r )
= ( c_2Eordinal_2Eordinal__ABS__CLASS @ A_27a @ ( c_2Ewellorder_2Eorderiso @ ( tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ A_27a ) @ ( tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ A_27a ) @ V0r ) ) ) ).
thf(thm_2Eordinal_2Eordlt__def,axiom,
! [A_27a: $tType,V0T1: tyop_2Eordinal_2Eordinal @ A_27a,V1T2: tyop_2Eordinal_2Eordinal @ A_27a] :
( ( c_2Eordinal_2Eordlt @ A_27a @ V0T1 @ V1T2 )
= ( c_2Ewellorder_2Eorderlt @ ( tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ A_27a ) @ ( tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ A_27a ) @ ( c_2Eordinal_2Eordinal__REP @ A_27a @ V0T1 ) @ ( c_2Eordinal_2Eordinal__REP @ A_27a @ V1T2 ) ) ) ).
thf(thm_2Eordinal_2EallOrds__def,axiom,
! [A_27a: $tType] :
( ( c_2Eordinal_2EallOrds @ A_27a )
= ( c_2Ewellorder_2Ewellorder__ABS @ ( tyop_2Eordinal_2Eordinal @ A_27a )
@ ( c_2Epred__set_2EGSPEC @ ( tyop_2Epair_2Eprod @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ ( tyop_2Eordinal_2Eordinal @ A_27a ) ) @ ( tyop_2Epair_2Eprod @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ ( tyop_2Eordinal_2Eordinal @ A_27a ) )
@ ( c_2Epair_2EUNCURRY @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ ( tyop_2Epair_2Eprod @ ( tyop_2Epair_2Eprod @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ ( tyop_2Eordinal_2Eordinal @ A_27a ) ) @ $o )
@ ^ [V0x: tyop_2Eordinal_2Eordinal @ A_27a,V1y: tyop_2Eordinal_2Eordinal @ A_27a] : ( c_2Epair_2E_2C @ ( tyop_2Epair_2Eprod @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ ( tyop_2Eordinal_2Eordinal @ A_27a ) ) @ $o @ ( c_2Epair_2E_2C @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ V0x @ V1y ) @ ( c_2Ebool_2E_5C_2F @ ( c_2Emin_2E_3D @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ V0x @ V1y ) @ ( c_2Eordinal_2Eordlt @ A_27a @ V0x @ V1y ) ) ) ) ) ) ) ).
thf(thm_2Eordinal_2Epreds__def,axiom,
! [A_27a: $tType,V0w: tyop_2Eordinal_2Eordinal @ A_27a] :
( ( c_2Eordinal_2Epreds @ A_27a @ V0w )
= ( c_2Epred__set_2EGSPEC @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ ( tyop_2Eordinal_2Eordinal @ A_27a )
@ ^ [V1w0: tyop_2Eordinal_2Eordinal @ A_27a] : ( c_2Epair_2E_2C @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ $o @ V1w0 @ ( c_2Eordinal_2Eordlt @ A_27a @ V1w0 @ V0w ) ) ) ) ).
thf(thm_2Eordinal_2Edownward__closed__def,axiom,
! [A_27a: $tType,V0s: ( tyop_2Eordinal_2Eordinal @ A_27a ) > $o] :
( ( c_2Eordinal_2Edownward__closed @ A_27a @ V0s )
<=> ! [V1a: tyop_2Eordinal_2Eordinal @ A_27a,V2b: tyop_2Eordinal_2Eordinal @ A_27a] :
( ( ( c_2Ebool_2EIN @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ V1a @ V0s )
& ( c_2Eordinal_2Eordlt @ A_27a @ V2b @ V1a ) )
=> ( c_2Ebool_2EIN @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ V2b @ V0s ) ) ) ).
thf(thm_2Eordinal_2Eoleast__def,axiom,
! [A_27a: $tType,V0P: ( tyop_2Eordinal_2Eordinal @ A_27a ) > $o] :
( ( c_2Eordinal_2Eoleast @ A_27a @ V0P )
= ( c_2Emin_2E_40 @ ( tyop_2Eordinal_2Eordinal @ A_27a )
@ ^ [V1x: tyop_2Eordinal_2Eordinal @ A_27a] :
( c_2Ebool_2E_2F_5C @ ( V0P @ V1x )
@ ( c_2Ebool_2E_21 @ ( tyop_2Eordinal_2Eordinal @ A_27a )
@ ^ [V2y: tyop_2Eordinal_2Eordinal @ A_27a] : ( c_2Emin_2E_3D_3D_3E @ ( c_2Eordinal_2Eordlt @ A_27a @ V2y @ V1x ) @ ( c_2Ebool_2E_7E @ ( V0P @ V2y ) ) ) ) ) ) ) ).
thf(thm_2Eordinal_2EordSUC__def,axiom,
! [A_27a: $tType,V0a: tyop_2Eordinal_2Eordinal @ A_27a] :
( ( c_2Eordinal_2EordSUC @ A_27a @ V0a )
= ( c_2Eordinal_2Eoleast @ A_27a
@ ^ [V1b: tyop_2Eordinal_2Eordinal @ A_27a] : ( c_2Eordinal_2Eordlt @ A_27a @ V0a @ V1b ) ) ) ).
thf(thm_2Eordinal_2EfromNat__def,axiom,
! [A_27a: $tType] :
( ( ( c_2Eordinal_2EfromNat @ A_27a @ c_2Enum_2E0 )
= ( c_2Eordinal_2Eoleast @ A_27a
@ ^ [V0a: tyop_2Eordinal_2Eordinal @ A_27a] : c_2Ebool_2ET ) )
& ! [V1n: tyop_2Enum_2Enum] :
( ( c_2Eordinal_2EfromNat @ A_27a @ ( c_2Enum_2ESUC @ V1n ) )
= ( c_2Eordinal_2EordSUC @ A_27a @ ( c_2Eordinal_2EfromNat @ A_27a @ V1n ) ) ) ) ).
thf(thm_2Eordinal_2Esup__def,axiom,
! [A_27a: $tType,V0ordset: ( tyop_2Eordinal_2Eordinal @ A_27a ) > $o] :
( ( c_2Eordinal_2Esup @ A_27a @ V0ordset )
= ( c_2Eordinal_2Eoleast @ A_27a
@ ^ [V1a: tyop_2Eordinal_2Eordinal @ A_27a] : ( c_2Ebool_2E_7E @ ( c_2Ebool_2EIN @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ V1a @ ( c_2Epred__set_2EBIGUNION @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ ( c_2Epred__set_2EIMAGE @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ ( ( tyop_2Eordinal_2Eordinal @ A_27a ) > $o ) @ ( c_2Eordinal_2Epreds @ A_27a ) @ V0ordset ) ) ) ) ) ) ).
thf(thm_2Eordinal_2Eomax__def,axiom,
! [A_27a: $tType,V0s: ( tyop_2Eordinal_2Eordinal @ A_27a ) > $o] :
( ( c_2Eordinal_2Eomax @ A_27a @ V0s )
= ( c_2Eoption_2Esome @ ( tyop_2Eordinal_2Eordinal @ A_27a )
@ ^ [V1a: tyop_2Eordinal_2Eordinal @ A_27a] :
( c_2Emin_2E_3D @ ( ( tyop_2Eordinal_2Eordinal @ A_27a ) > $o )
@ ( c_2Eset__relation_2Emaximal__elements @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ V0s
@ ( c_2Epred__set_2EGSPEC @ ( tyop_2Epair_2Eprod @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ ( tyop_2Eordinal_2Eordinal @ A_27a ) ) @ ( tyop_2Epair_2Eprod @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ ( tyop_2Eordinal_2Eordinal @ A_27a ) )
@ ( c_2Epair_2EUNCURRY @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ ( tyop_2Epair_2Eprod @ ( tyop_2Epair_2Eprod @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ ( tyop_2Eordinal_2Eordinal @ A_27a ) ) @ $o )
@ ^ [V2x: tyop_2Eordinal_2Eordinal @ A_27a,V3y: tyop_2Eordinal_2Eordinal @ A_27a] : ( c_2Epair_2E_2C @ ( tyop_2Epair_2Eprod @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ ( tyop_2Eordinal_2Eordinal @ A_27a ) ) @ $o @ ( c_2Epair_2E_2C @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ V2x @ V3y ) @ ( c_2Ebool_2E_7E @ ( c_2Eordinal_2Eordlt @ A_27a @ V3y @ V2x ) ) ) ) ) )
@ ( c_2Epred__set_2EINSERT @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ V1a @ ( c_2Epred__set_2EEMPTY @ ( tyop_2Eordinal_2Eordinal @ A_27a ) ) ) ) ) ) ).
thf(thm_2Eordinal_2EordMULT__def,axiom,
! [A_27a: $tType,V0b: tyop_2Eordinal_2Eordinal @ A_27a] :
( ( ( c_2Eordinal_2EordMULT @ A_27a @ V0b @ ( c_2Eordinal_2EfromNat @ A_27a @ c_2Enum_2E0 ) )
= ( c_2Eordinal_2EfromNat @ A_27a @ c_2Enum_2E0 ) )
& ! [V1a: tyop_2Eordinal_2Eordinal @ A_27a] :
( ( c_2Eordinal_2EordMULT @ A_27a @ V0b @ ( c_2Eordinal_2EordSUC @ A_27a @ V1a ) )
= ( c_2Eordinal_2EordADD @ A_27a @ ( c_2Eordinal_2EordMULT @ A_27a @ V0b @ V1a ) @ V0b ) )
& ! [V2a: tyop_2Eordinal_2Eordinal @ A_27a] :
( ( ( c_2Eordinal_2Eordlt @ A_27a @ ( c_2Eordinal_2EfromNat @ A_27a @ c_2Enum_2E0 ) @ V2a )
& ( ( c_2Eordinal_2Eomax @ A_27a @ ( c_2Eordinal_2Epreds @ A_27a @ V2a ) )
= ( c_2Eoption_2ENONE @ ( tyop_2Eordinal_2Eordinal @ A_27a ) ) ) )
=> ( ( c_2Eordinal_2EordMULT @ A_27a @ V0b @ V2a )
= ( c_2Eordinal_2Esup @ A_27a @ ( c_2Epred__set_2EIMAGE @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ ( c_2Eordinal_2EordMULT @ A_27a @ V0b ) @ ( c_2Eordinal_2Epreds @ A_27a @ V2a ) ) ) ) ) ) ).
thf(thm_2Eordinal_2EordDIVISION,axiom,
! [A_27a: $tType,V0a: tyop_2Eordinal_2Eordinal @ A_27a,V1b: tyop_2Eordinal_2Eordinal @ A_27a] :
( ( c_2Eordinal_2Eordlt @ A_27a @ ( c_2Eordinal_2EfromNat @ A_27a @ c_2Enum_2E0 ) @ V1b )
=> ( ( V0a
= ( c_2Eordinal_2EordADD @ A_27a @ ( c_2Eordinal_2EordMULT @ A_27a @ V1b @ ( c_2Eordinal_2EordDIV @ A_27a @ V0a @ V1b ) ) @ ( c_2Eordinal_2EordMOD @ A_27a @ V0a @ V1b ) ) )
& ( c_2Eordinal_2Eordlt @ A_27a @ ( c_2Eordinal_2EordMOD @ A_27a @ V0a @ V1b ) @ V1b ) ) ) ).
thf(thm_2Eordinal_2EordEXP__def,axiom,
! [A_27a: $tType] :
( ! [V0a: tyop_2Eordinal_2Eordinal @ A_27a] :
( ( c_2Eordinal_2EordEXP @ A_27a @ V0a @ ( c_2Eordinal_2EfromNat @ A_27a @ c_2Enum_2E0 ) )
= ( c_2Eordinal_2EfromNat @ A_27a @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) )
& ! [V1a: tyop_2Eordinal_2Eordinal @ A_27a,V2a_27: tyop_2Eordinal_2Eordinal @ A_27a] :
( ( c_2Eordinal_2EordEXP @ A_27a @ V1a @ ( c_2Eordinal_2EordSUC @ A_27a @ V2a_27 ) )
= ( c_2Eordinal_2EordMULT @ A_27a @ ( c_2Eordinal_2EordEXP @ A_27a @ V1a @ V2a_27 ) @ V1a ) )
& ! [V3a: tyop_2Eordinal_2Eordinal @ A_27a,V4a_27: tyop_2Eordinal_2Eordinal @ A_27a] :
( ( ( c_2Eordinal_2Eordlt @ A_27a @ ( c_2Eordinal_2EfromNat @ A_27a @ c_2Enum_2E0 ) @ V4a_27 )
& ( ( c_2Eordinal_2Eomax @ A_27a @ ( c_2Eordinal_2Epreds @ A_27a @ V4a_27 ) )
= ( c_2Eoption_2ENONE @ ( tyop_2Eordinal_2Eordinal @ A_27a ) ) ) )
=> ( ( c_2Eordinal_2EordEXP @ A_27a @ V3a @ V4a_27 )
= ( c_2Eordinal_2Esup @ A_27a @ ( c_2Epred__set_2EIMAGE @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ ( c_2Eordinal_2EordEXP @ A_27a @ V3a ) @ ( c_2Eordinal_2Epreds @ A_27a @ V4a_27 ) ) ) ) ) ) ).
thf(thm_2Eordinal_2Eepsilon0__def,axiom,
! [A_27a: $tType] :
( ( c_2Eordinal_2Eepsilon0 @ A_27a )
= ( c_2Eordinal_2Eoleast @ A_27a
@ ^ [V0x: tyop_2Eordinal_2Eordinal @ A_27a] : ( c_2Emin_2E_3D @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ ( c_2Eordinal_2EordEXP @ A_27a @ ( c_2Eordinal_2Eomega @ A_27a ) @ V0x ) @ V0x ) ) ) ).
thf(thm_2Eordinal_2Epolyform__def,axiom,
! [A_27a: $tType,V0a: tyop_2Eordinal_2Eordinal @ A_27a,V1b: tyop_2Eordinal_2Eordinal @ A_27a] :
( ( c_2Eordinal_2Eordlt @ A_27a @ ( c_2Eordinal_2EfromNat @ A_27a @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) @ V0a )
=> ( ( c_2Eordinal_2Eis__polyform @ A_27a @ A_27a @ V0a @ ( c_2Eordinal_2Epolyform @ A_27a @ V0a @ V1b ) )
& ( V1b
= ( c_2Eordinal_2Eeval__poly @ A_27a @ V0a @ ( c_2Eordinal_2Epolyform @ A_27a @ V0a @ V1b ) ) ) ) ) ).
thf(thm_2Eordinal_2Elt__suppreds,axiom,
! [A_27a: $tType,V0a: tyop_2Eordinal_2Eordinal @ A_27a,V1b: tyop_2Eordinal_2Eordinal @ A_27a] :
( ( c_2Eordinal_2Eordlt @ A_27a @ V1b @ ( c_2Eordinal_2Esup @ A_27a @ ( c_2Eordinal_2Epreds @ A_27a @ V0a ) ) )
<=> ? [V2d: tyop_2Eordinal_2Eordinal @ A_27a] :
( ( c_2Eordinal_2Eordlt @ A_27a @ V2d @ V0a )
& ( c_2Eordinal_2Eordlt @ A_27a @ V1b @ V2d ) ) ) ).
thf(thm_2Eordinal_2EordADD__fromNat__omega,axiom,
! [A_27a: $tType,V0n: tyop_2Enum_2Enum] :
( ( c_2Eordinal_2EordADD @ A_27a @ ( c_2Eordinal_2EfromNat @ A_27a @ V0n ) @ ( c_2Eordinal_2Eomega @ A_27a ) )
= ( c_2Eordinal_2Eomega @ A_27a ) ) ).
thf(thm_2Eordinal_2Eomega__islimit,axiom,
! [A_27a: $tType] :
( ( c_2Eordinal_2Eomax @ A_27a @ ( c_2Eordinal_2Epreds @ A_27a @ ( c_2Eordinal_2Eomega @ A_27a ) ) )
= ( c_2Eoption_2ENONE @ ( tyop_2Eordinal_2Eordinal @ A_27a ) ) ) ).
thf(thm_2Eordinal_2Eomax__preds__omega,axiom,
! [A_27a: $tType] :
( ( c_2Eordinal_2Eomax @ A_27a @ ( c_2Eordinal_2Epreds @ A_27a @ ( c_2Eordinal_2Eomega @ A_27a ) ) )
= ( c_2Eoption_2ENONE @ ( tyop_2Eordinal_2Eordinal @ A_27a ) ) ) ).
thf(thm_2Eordinal_2EordADD__fromNat,axiom,
! [A_27a: $tType,V0n: tyop_2Enum_2Enum,V1m: tyop_2Enum_2Enum] :
( ( c_2Eordinal_2EordADD @ A_27a @ ( c_2Eordinal_2EfromNat @ A_27a @ V0n ) @ ( c_2Eordinal_2EfromNat @ A_27a @ V1m ) )
= ( c_2Eordinal_2EfromNat @ A_27a @ ( c_2Earithmetic_2E_2B @ V0n @ V1m ) ) ) ).
thf(thm_2Eordinal_2Eubsup__thm,axiom,
! [A_27a: $tType,V0s: ( tyop_2Eordinal_2Eordinal @ A_27a ) > $o,V1b: tyop_2Eordinal_2Eordinal @ A_27a] :
( ! [V2a: tyop_2Eordinal_2Eordinal @ A_27a] :
( ( c_2Ebool_2EIN @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ V2a @ V0s )
=> ( c_2Eordinal_2Eordlt @ A_27a @ V2a @ V1b ) )
=> ! [V3c: tyop_2Eordinal_2Eordinal @ A_27a] :
( ( c_2Eordinal_2Eordlt @ A_27a @ V3c @ ( c_2Eordinal_2Esup @ A_27a @ V0s ) )
<=> ? [V4d: tyop_2Eordinal_2Eordinal @ A_27a] :
( ( c_2Ebool_2EIN @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ V4d @ V0s )
& ( c_2Eordinal_2Eordlt @ A_27a @ V3c @ V4d ) ) ) ) ).
thf(thm_2Eordinal_2EordADD__0L,axiom,
! [A_27a: $tType,V0a: tyop_2Eordinal_2Eordinal @ A_27a] :
( ( c_2Eordinal_2EordADD @ A_27a @ ( c_2Eordinal_2EfromNat @ A_27a @ c_2Enum_2E0 ) @ V0a )
= V0a ) ).
thf(thm_2Eordinal_2Eord__RECURSION,axiom,
! [A_27a: $tType,A_27b: $tType,V0z: A_27b,V1sf: ( tyop_2Eordinal_2Eordinal @ A_27a ) > A_27b > A_27b,V2lf: ( tyop_2Eordinal_2Eordinal @ A_27a ) > ( A_27b > $o ) > A_27b] :
? [V3h: ( tyop_2Eordinal_2Eordinal @ A_27a ) > A_27b] :
( ( ( V3h @ ( c_2Eordinal_2EfromNat @ A_27a @ c_2Enum_2E0 ) )
= V0z )
& ! [V4a: tyop_2Eordinal_2Eordinal @ A_27a] :
( ( V3h @ ( c_2Eordinal_2EordSUC @ A_27a @ V4a ) )
= ( V1sf @ V4a @ ( V3h @ V4a ) ) )
& ! [V5a: tyop_2Eordinal_2Eordinal @ A_27a] :
( ( ( c_2Eordinal_2Eordlt @ A_27a @ ( c_2Eordinal_2EfromNat @ A_27a @ c_2Enum_2E0 ) @ V5a )
& ( ( c_2Eordinal_2Eomax @ A_27a @ ( c_2Eordinal_2Epreds @ A_27a @ V5a ) )
= ( c_2Eoption_2ENONE @ ( tyop_2Eordinal_2Eordinal @ A_27a ) ) ) )
=> ( ( V3h @ V5a )
= ( V2lf @ V5a @ ( c_2Epred__set_2EIMAGE @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ A_27b @ V3h @ ( c_2Eordinal_2Epreds @ A_27a @ V5a ) ) ) ) ) ) ).
thf(thm_2Eordinal_2EfromNat__eq__omega,axiom,
! [A_27a: $tType,V0n: tyop_2Enum_2Enum] :
( (~)
@ ( ( c_2Eordinal_2EfromNat @ A_27a @ V0n )
= ( c_2Eordinal_2Eomega @ A_27a ) ) ) ).
thf(thm_2Eordinal_2EfromNat__lt__omega,axiom,
! [A_27a: $tType,V0n: tyop_2Enum_2Enum] : ( c_2Eordinal_2Eordlt @ A_27a @ ( c_2Eordinal_2EfromNat @ A_27a @ V0n ) @ ( c_2Eordinal_2Eomega @ A_27a ) ) ).
thf(thm_2Eordinal_2Elt__omega,axiom,
! [A_27a: $tType,V0a: tyop_2Eordinal_2Eordinal @ A_27a] :
( ( c_2Eordinal_2Eordlt @ A_27a @ V0a @ ( c_2Eordinal_2Eomega @ A_27a ) )
<=> ? [V1m: tyop_2Enum_2Enum] :
( V0a
= ( c_2Eordinal_2EfromNat @ A_27a @ V1m ) ) ) ).
thf(thm_2Eordinal_2EfromNat__ordlt,axiom,
! [A_27a: $tType,V0n: tyop_2Enum_2Enum,V1m: tyop_2Enum_2Enum] :
( ( c_2Eordinal_2Eordlt @ A_27a @ ( c_2Eordinal_2EfromNat @ A_27a @ V0n ) @ ( c_2Eordinal_2EfromNat @ A_27a @ V1m ) )
= ( c_2Eprim__rec_2E_3C @ V0n @ V1m ) ) ).
thf(thm_2Eordinal_2Eordlt__fromNat,axiom,
! [A_27a: $tType,V0n: tyop_2Enum_2Enum,V1x: tyop_2Eordinal_2Eordinal @ A_27a] :
( ( c_2Eordinal_2Eordlt @ A_27a @ V1x @ ( c_2Eordinal_2EfromNat @ A_27a @ V0n ) )
<=> ? [V2m: tyop_2Enum_2Enum] :
( ( V1x
= ( c_2Eordinal_2EfromNat @ A_27a @ V2m ) )
& ( c_2Eprim__rec_2E_3C @ V2m @ V0n ) ) ) ).
thf(thm_2Eordinal_2EfromNat__11,axiom,
! [A_27a: $tType,V0x: tyop_2Enum_2Enum,V1y: tyop_2Enum_2Enum] :
( ( ( c_2Eordinal_2EfromNat @ A_27a @ V0x )
= ( c_2Eordinal_2EfromNat @ A_27a @ V1y ) )
<=> ( V0x = V1y ) ) ).
thf(thm_2Eordinal_2Epreds__suple,axiom,
! [A_27a: $tType,V0s: ( tyop_2Eordinal_2Eordinal @ A_27a ) > $o,V1b: tyop_2Eordinal_2Eordinal @ A_27a] :
( ( ( c_2Eordinal_2Edownward__closed @ A_27a @ V0s )
& ( (~)
@ ( V0s
= ( c_2Epred__set_2EUNIV @ ( tyop_2Eordinal_2Eordinal @ A_27a ) ) ) ) )
=> ( ( (~) @ ( c_2Eordinal_2Eordlt @ A_27a @ V1b @ ( c_2Eordinal_2Esup @ A_27a @ V0s ) ) )
<=> ! [V2d: tyop_2Eordinal_2Eordinal @ A_27a] :
( ( c_2Ebool_2EIN @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ V2d @ V0s )
=> ( (~) @ ( c_2Eordinal_2Eordlt @ A_27a @ V1b @ V2d ) ) ) ) ) ).
thf(thm_2Eordinal_2Epreds__lesup,axiom,
! [A_27a: $tType,V0s: ( tyop_2Eordinal_2Eordinal @ A_27a ) > $o] :
( ( ( c_2Eordinal_2Edownward__closed @ A_27a @ V0s )
& ( (~)
@ ( V0s
= ( c_2Epred__set_2EUNIV @ ( tyop_2Eordinal_2Eordinal @ A_27a ) ) ) ) )
=> ! [V1d: tyop_2Eordinal_2Eordinal @ A_27a] :
( ( c_2Ebool_2EIN @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ V1d @ V0s )
=> ( (~) @ ( c_2Eordinal_2Eordlt @ A_27a @ ( c_2Eordinal_2Esup @ A_27a @ V0s ) @ V1d ) ) ) ) ).
thf(thm_2Eordinal_2Epreds__sup__thm,axiom,
! [A_27a: $tType,V0s: ( tyop_2Eordinal_2Eordinal @ A_27a ) > $o] :
( ( ( c_2Eordinal_2Edownward__closed @ A_27a @ V0s )
& ( (~)
@ ( V0s
= ( c_2Epred__set_2EUNIV @ ( tyop_2Eordinal_2Eordinal @ A_27a ) ) ) ) )
=> ! [V1b: tyop_2Eordinal_2Eordinal @ A_27a] :
( ( c_2Eordinal_2Eordlt @ A_27a @ V1b @ ( c_2Eordinal_2Esup @ A_27a @ V0s ) )
<=> ? [V2d: tyop_2Eordinal_2Eordinal @ A_27a] :
( ( c_2Ebool_2EIN @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ V2d @ V0s )
& ( c_2Eordinal_2Eordlt @ A_27a @ V1b @ V2d ) ) ) ) ).
thf(thm_2Eordinal_2Ecsup__suple,axiom,
! [A_27a: $tType,V0s: ( tyop_2Eordinal_2Eordinal @ A_27a ) > $o,V1b: tyop_2Eordinal_2Eordinal @ A_27a] :
( ( c_2Epred__set_2Ecountable @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ V0s )
=> ( ( (~) @ ( c_2Eordinal_2Eordlt @ A_27a @ V1b @ ( c_2Eordinal_2Esup @ A_27a @ V0s ) ) )
<=> ! [V2d: tyop_2Eordinal_2Eordinal @ A_27a] :
( ( c_2Ebool_2EIN @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ V2d @ V0s )
=> ( (~) @ ( c_2Eordinal_2Eordlt @ A_27a @ V1b @ V2d ) ) ) ) ) ).
thf(thm_2Eordinal_2Ecsup__lesup,axiom,
! [A_27a: $tType,V0s: ( tyop_2Eordinal_2Eordinal @ A_27a ) > $o] :
( ( c_2Epred__set_2Ecountable @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ V0s )
=> ! [V1d: tyop_2Eordinal_2Eordinal @ A_27a] :
( ( c_2Ebool_2EIN @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ V1d @ V0s )
=> ( (~) @ ( c_2Eordinal_2Eordlt @ A_27a @ ( c_2Eordinal_2Esup @ A_27a @ V0s ) @ V1d ) ) ) ) ).
thf(thm_2Eordinal_2Epreds__sup,axiom,
! [A_27a: $tType,V0s: ( tyop_2Eordinal_2Eordinal @ A_27a ) > $o] :
( ( c_2Ecardinal_2Ecardleq @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ ( tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ A_27a ) @ V0s @ ( c_2Epred__set_2EUNIV @ ( tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ A_27a ) ) )
=> ( ( c_2Eordinal_2Epreds @ A_27a @ ( c_2Eordinal_2Esup @ A_27a @ V0s ) )
= ( c_2Eordinal_2Edclose @ A_27a @ V0s ) ) ) ).
thf(thm_2Eordinal_2Esup__preds__omax__NONE,axiom,
! [A_27a: $tType,V0a: tyop_2Eordinal_2Eordinal @ A_27a] :
( ( ( c_2Eordinal_2Eomax @ A_27a @ ( c_2Eordinal_2Epreds @ A_27a @ V0a ) )
= ( c_2Eoption_2ENONE @ ( tyop_2Eordinal_2Eordinal @ A_27a ) ) )
<=> ( ( c_2Eordinal_2Esup @ A_27a @ ( c_2Eordinal_2Epreds @ A_27a @ V0a ) )
= V0a ) ) ).
thf(thm_2Eordinal_2Esimple__ord__induction,axiom,
! [A_27a: $tType,V0P: ( tyop_2Eordinal_2Eordinal @ A_27a ) > $o] :
( ( ( V0P @ ( c_2Eordinal_2EfromNat @ A_27a @ c_2Enum_2E0 ) )
& ! [V1a: tyop_2Eordinal_2Eordinal @ A_27a] :
( ( V0P @ V1a )
=> ( V0P @ ( c_2Eordinal_2EordSUC @ A_27a @ V1a ) ) )
& ! [V2a: tyop_2Eordinal_2Eordinal @ A_27a] :
( ( ( ( c_2Eordinal_2Eomax @ A_27a @ ( c_2Eordinal_2Epreds @ A_27a @ V2a ) )
= ( c_2Eoption_2ENONE @ ( tyop_2Eordinal_2Eordinal @ A_27a ) ) )
& ( c_2Eordinal_2Eordlt @ A_27a @ ( c_2Eordinal_2EfromNat @ A_27a @ c_2Enum_2E0 ) @ V2a )
& ! [V3b: tyop_2Eordinal_2Eordinal @ A_27a] :
( ( c_2Eordinal_2Eordlt @ A_27a @ V3b @ V2a )
=> ( V0P @ V3b ) ) )
=> ( V0P @ V2a ) ) )
=> ! [V4a: tyop_2Eordinal_2Eordinal @ A_27a] : ( V0P @ V4a ) ) ).
thf(thm_2Eordinal_2Eordinal__ABS__REP__CLASS,axiom,
! [A_27a: $tType] :
( ! [V0a: tyop_2Eordinal_2Eordinal @ A_27a] :
( ( c_2Eordinal_2Eordinal__ABS__CLASS @ A_27a @ ( c_2Eordinal_2Eordinal__REP__CLASS @ A_27a @ V0a ) )
= V0a )
& ! [V1c: ( tyop_2Ewellorder_2Ewellorder @ ( tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ A_27a ) ) > $o] :
( ? [V2r: tyop_2Ewellorder_2Ewellorder @ ( tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ A_27a )] :
( ( c_2Ewellorder_2Eorderiso @ ( tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ A_27a ) @ ( tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ A_27a ) @ V2r @ V2r )
& ( V1c
= ( c_2Ewellorder_2Eorderiso @ ( tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ A_27a ) @ ( tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ A_27a ) @ V2r ) ) )
<=> ( ( c_2Eordinal_2Eordinal__REP__CLASS @ A_27a @ ( c_2Eordinal_2Eordinal__ABS__CLASS @ A_27a @ V1c ) )
= V1c ) ) ) ).
thf(thm_2Eordinal_2Eordinal__QUOTIENT,axiom,
! [A_27a: $tType] : ( c_2Equotient_2EQUOTIENT @ ( tyop_2Ewellorder_2Ewellorder @ ( tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ A_27a ) ) @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ ( c_2Ewellorder_2Eorderiso @ ( tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ A_27a ) @ ( tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ A_27a ) ) @ ( c_2Eordinal_2Eordinal__ABS @ A_27a ) @ ( c_2Eordinal_2Eordinal__REP @ A_27a ) ) ).
thf(thm_2Eordinal_2Eordlt__REFL,axiom,
! [A_27a: $tType,V0w: tyop_2Eordinal_2Eordinal @ A_27a] : ( (~) @ ( c_2Eordinal_2Eordlt @ A_27a @ V0w @ V0w ) ) ).
thf(thm_2Eordinal_2Eordlt__TRANS,axiom,
! [A_27a: $tType,V0w1: tyop_2Eordinal_2Eordinal @ A_27a,V1w2: tyop_2Eordinal_2Eordinal @ A_27a,V2w3: tyop_2Eordinal_2Eordinal @ A_27a] :
( ( ( c_2Eordinal_2Eordlt @ A_27a @ V0w1 @ V1w2 )
& ( c_2Eordinal_2Eordlt @ A_27a @ V1w2 @ V2w3 ) )
=> ( c_2Eordinal_2Eordlt @ A_27a @ V0w1 @ V2w3 ) ) ).
thf(thm_2Eordinal_2Eordlt__WF,axiom,
! [A_27a: $tType] : ( c_2Erelation_2EWF @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ ( c_2Eordinal_2Eordlt @ A_27a ) ) ).
thf(thm_2Eordinal_2Eordlt__trichotomy,axiom,
! [A_27a: $tType,V0w2: tyop_2Eordinal_2Eordinal @ A_27a,V1w1: tyop_2Eordinal_2Eordinal @ A_27a] :
( ( c_2Eordinal_2Eordlt @ A_27a @ V1w1 @ V0w2 )
| ( V1w1 = V0w2 )
| ( c_2Eordinal_2Eordlt @ A_27a @ V0w2 @ V1w1 ) ) ).
thf(thm_2Eordinal_2Ewellorder__allOrds,axiom,
! [A_27a: $tType] :
( c_2Ewellorder_2Ewellorder @ ( tyop_2Eordinal_2Eordinal @ A_27a )
@ ( c_2Epred__set_2EGSPEC @ ( tyop_2Epair_2Eprod @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ ( tyop_2Eordinal_2Eordinal @ A_27a ) ) @ ( tyop_2Epair_2Eprod @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ ( tyop_2Eordinal_2Eordinal @ A_27a ) )
@ ( c_2Epair_2EUNCURRY @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ ( tyop_2Epair_2Eprod @ ( tyop_2Epair_2Eprod @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ ( tyop_2Eordinal_2Eordinal @ A_27a ) ) @ $o )
@ ^ [V0x: tyop_2Eordinal_2Eordinal @ A_27a,V1y: tyop_2Eordinal_2Eordinal @ A_27a] : ( c_2Epair_2E_2C @ ( tyop_2Epair_2Eprod @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ ( tyop_2Eordinal_2Eordinal @ A_27a ) ) @ $o @ ( c_2Epair_2E_2C @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ V0x @ V1y ) @ ( c_2Ebool_2E_5C_2F @ ( c_2Emin_2E_3D @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ V0x @ V1y ) @ ( c_2Eordinal_2Eordlt @ A_27a @ V0x @ V1y ) ) ) ) ) ) ).
thf(thm_2Eordinal_2EWIN__allOrds,axiom,
! [A_27a: $tType,V0y: tyop_2Eordinal_2Eordinal @ A_27a,V1x: tyop_2Eordinal_2Eordinal @ A_27a] :
( ( c_2Ebool_2EIN @ ( tyop_2Epair_2Eprod @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ ( tyop_2Eordinal_2Eordinal @ A_27a ) ) @ ( c_2Epair_2E_2C @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ V1x @ V0y ) @ ( c_2Eset__relation_2Estrict @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ ( c_2Ewellorder_2Ewellorder__REP @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ ( c_2Eordinal_2EallOrds @ A_27a ) ) ) )
= ( c_2Eordinal_2Eordlt @ A_27a @ V1x @ V0y ) ) ).
thf(thm_2Eordinal_2EelsOf__allOrds,axiom,
! [A_27a: $tType] :
( ( c_2Ewellorder_2EelsOf @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ ( c_2Eordinal_2EallOrds @ A_27a ) )
= ( c_2Epred__set_2EUNIV @ ( tyop_2Eordinal_2Eordinal @ A_27a ) ) ) ).
thf(thm_2Eordinal_2Eordlt__mkOrdinal,axiom,
! [A_27a: $tType,V0o2: tyop_2Eordinal_2Eordinal @ A_27a,V1o1: tyop_2Eordinal_2Eordinal @ A_27a] :
( ( c_2Eordinal_2Eordlt @ A_27a @ V1o1 @ V0o2 )
<=> ! [V2w1: tyop_2Ewellorder_2Ewellorder @ ( tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ A_27a ),V3w2: tyop_2Ewellorder_2Ewellorder @ ( tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ A_27a )] :
( ( ( ( c_2Eordinal_2Eordinal__ABS @ A_27a @ V2w1 )
= V1o1 )
& ( ( c_2Eordinal_2Eordinal__ABS @ A_27a @ V3w2 )
= V0o2 ) )
=> ( c_2Ewellorder_2Eorderlt @ ( tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ A_27a ) @ ( tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ A_27a ) @ V2w1 @ V3w2 ) ) ) ).
thf(thm_2Eordinal_2Eorderlt__iso__REFL,axiom,
! [A_27a: $tType,A_27b: $tType,V0w2: tyop_2Ewellorder_2Ewellorder @ A_27b,V1w1: tyop_2Ewellorder_2Ewellorder @ A_27a] :
( ( c_2Ewellorder_2Eorderiso @ A_27a @ A_27b @ V1w1 @ V0w2 )
=> ( (~) @ ( c_2Ewellorder_2Eorderlt @ A_27a @ A_27b @ V1w1 @ V0w2 ) ) ) ).
thf(thm_2Eordinal_2Eorderiso__wobound2,axiom,
! [A_27a: $tType,V0y: A_27a,V1x: A_27a,V2w: tyop_2Ewellorder_2Ewellorder @ A_27a] :
( ( c_2Ewellorder_2Eorderiso @ A_27a @ A_27a @ ( c_2Ewellorder_2Ewobound @ A_27a @ V1x @ V2w ) @ ( c_2Ewellorder_2Ewobound @ A_27a @ V0y @ V2w ) )
=> ( (~) @ ( c_2Ebool_2EIN @ ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) @ ( c_2Epair_2E_2C @ A_27a @ A_27a @ V1x @ V0y ) @ ( c_2Eset__relation_2Estrict @ A_27a @ ( c_2Ewellorder_2Ewellorder__REP @ A_27a @ V2w ) ) ) ) ) ).
thf(thm_2Eordinal_2Ewellorder__ordinal__isomorphism,axiom,
! [A_27a: $tType,V0w: tyop_2Ewellorder_2Ewellorder @ ( tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ A_27a )] : ( c_2Ewellorder_2Eorderiso @ ( tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ A_27a ) @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ V0w @ ( c_2Ewellorder_2Ewobound @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ ( c_2Eordinal_2Eordinal__ABS @ A_27a @ V0w ) @ ( c_2Eordinal_2EallOrds @ A_27a ) ) ) ).
thf(thm_2Eordinal_2EIN__preds,axiom,
! [A_27a: $tType,V0x: tyop_2Eordinal_2Eordinal @ A_27a,V1w: tyop_2Eordinal_2Eordinal @ A_27a] :
( ( c_2Ebool_2EIN @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ V0x @ ( c_2Eordinal_2Epreds @ A_27a @ V1w ) )
= ( c_2Eordinal_2Eordlt @ A_27a @ V0x @ V1w ) ) ).
thf(thm_2Eordinal_2Epreds__11,axiom,
! [A_27a: $tType,V0w2: tyop_2Eordinal_2Eordinal @ A_27a,V1w1: tyop_2Eordinal_2Eordinal @ A_27a] :
( ( ( c_2Eordinal_2Epreds @ A_27a @ V1w1 )
= ( c_2Eordinal_2Epreds @ A_27a @ V0w2 ) )
<=> ( V1w1 = V0w2 ) ) ).
thf(thm_2Eordinal_2Epreds__downward__closed,axiom,
! [A_27a: $tType,V0w: tyop_2Eordinal_2Eordinal @ A_27a] : ( c_2Eordinal_2Edownward__closed @ A_27a @ ( c_2Eordinal_2Epreds @ A_27a @ V0w ) ) ).
thf(thm_2Eordinal_2Epreds__bij,axiom,
! [A_27a: $tType] : ( c_2Epred__set_2EBIJ @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ ( ( tyop_2Eordinal_2Eordinal @ A_27a ) > $o ) @ ( c_2Eordinal_2Epreds @ A_27a ) @ ( c_2Epred__set_2EUNIV @ ( tyop_2Eordinal_2Eordinal @ A_27a ) ) @ ( c_2Epred__set_2EDELETE @ ( ( tyop_2Eordinal_2Eordinal @ A_27a ) > $o ) @ ( c_2Eordinal_2Edownward__closed @ A_27a ) @ ( c_2Epred__set_2EUNIV @ ( tyop_2Eordinal_2Eordinal @ A_27a ) ) ) ) ).
thf(thm_2Eordinal_2Epreds__lt__PSUBSET,axiom,
! [A_27a: $tType,V0w2: tyop_2Eordinal_2Eordinal @ A_27a,V1w1: tyop_2Eordinal_2Eordinal @ A_27a] :
( ( c_2Eordinal_2Eordlt @ A_27a @ V1w1 @ V0w2 )
= ( c_2Epred__set_2EPSUBSET @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ ( c_2Eordinal_2Epreds @ A_27a @ V1w1 ) @ ( c_2Eordinal_2Epreds @ A_27a @ V0w2 ) ) ) ).
thf(thm_2Eordinal_2Epreds__wobound,axiom,
! [A_27a: $tType,V0ord: tyop_2Eordinal_2Eordinal @ A_27a] :
( ( c_2Eordinal_2Epreds @ A_27a @ V0ord )
= ( c_2Ewellorder_2EelsOf @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ ( c_2Ewellorder_2Ewobound @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ V0ord @ ( c_2Eordinal_2EallOrds @ A_27a ) ) ) ) ).
thf(thm_2Eordinal_2Epreds__inj__univ,axiom,
! [A_27a: $tType,V0ord: tyop_2Eordinal_2Eordinal @ A_27a] : ( c_2Ecardinal_2Ecardleq @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ ( tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ A_27a ) @ ( c_2Eordinal_2Epreds @ A_27a @ V0ord ) @ ( c_2Epred__set_2EUNIV @ ( tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ A_27a ) ) ) ).
thf(thm_2Eordinal_2Eunitinf__univnum,axiom,
c_2Ecardinal_2Ecardeq @ ( tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ tyop_2Eone_2Eone ) @ tyop_2Enum_2Enum @ ( c_2Epred__set_2EUNIV @ ( tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ tyop_2Eone_2Eone ) ) @ ( c_2Epred__set_2EUNIV @ tyop_2Enum_2Enum ) ).
thf(thm_2Eordinal_2Ecord__countable__preds,axiom,
! [V0ord: tyop_2Eordinal_2Eordinal @ tyop_2Eone_2Eone] : ( c_2Epred__set_2Ecountable @ ( tyop_2Eordinal_2Eordinal @ tyop_2Eone_2Eone ) @ ( c_2Eordinal_2Epreds @ tyop_2Eone_2Eone @ V0ord ) ) ).
thf(thm_2Eordinal_2Euniv__ord__greater__cardinal,axiom,
! [A_27a: $tType] : ( (~) @ ( c_2Ecardinal_2Ecardleq @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ ( tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ A_27a ) @ ( c_2Epred__set_2EUNIV @ ( tyop_2Eordinal_2Eordinal @ A_27a ) ) @ ( c_2Epred__set_2EUNIV @ ( tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ A_27a ) ) ) ) ).
thf(thm_2Eordinal_2Euniv__cord__uncountable,axiom,
(~) @ ( c_2Epred__set_2Ecountable @ ( tyop_2Eordinal_2Eordinal @ tyop_2Eone_2Eone ) @ ( c_2Epred__set_2EUNIV @ ( tyop_2Eordinal_2Eordinal @ tyop_2Eone_2Eone ) ) ) ).
thf(thm_2Eordinal_2Eordle__lteq,axiom,
! [A_27a: $tType,V0b: tyop_2Eordinal_2Eordinal @ A_27a,V1a: tyop_2Eordinal_2Eordinal @ A_27a] :
( ( (~) @ ( c_2Eordinal_2Eordlt @ A_27a @ V0b @ V1a ) )
<=> ( ( c_2Eordinal_2Eordlt @ A_27a @ V1a @ V0b )
| ( V1a = V0b ) ) ) ).
thf(thm_2Eordinal_2Eordle__ANTISYM,axiom,
! [A_27a: $tType,V0b: tyop_2Eordinal_2Eordinal @ A_27a,V1a: tyop_2Eordinal_2Eordinal @ A_27a] :
( ( ( (~) @ ( c_2Eordinal_2Eordlt @ A_27a @ V0b @ V1a ) )
& ( (~) @ ( c_2Eordinal_2Eordlt @ A_27a @ V1a @ V0b ) ) )
=> ( V1a = V0b ) ) ).
thf(thm_2Eordinal_2Eordle__TRANS,axiom,
! [A_27a: $tType,V0x: tyop_2Eordinal_2Eordinal @ A_27a,V1y: tyop_2Eordinal_2Eordinal @ A_27a,V2z: tyop_2Eordinal_2Eordinal @ A_27a] :
( ( ( (~) @ ( c_2Eordinal_2Eordlt @ A_27a @ V1y @ V0x ) )
& ( (~) @ ( c_2Eordinal_2Eordlt @ A_27a @ V2z @ V1y ) ) )
=> ( (~) @ ( c_2Eordinal_2Eordlt @ A_27a @ V2z @ V0x ) ) ) ).
thf(thm_2Eordinal_2Eordlet__TRANS,axiom,
! [A_27a: $tType,V0x: tyop_2Eordinal_2Eordinal @ A_27a,V1y: tyop_2Eordinal_2Eordinal @ A_27a,V2z: tyop_2Eordinal_2Eordinal @ A_27a] :
( ( ( (~) @ ( c_2Eordinal_2Eordlt @ A_27a @ V1y @ V0x ) )
& ( c_2Eordinal_2Eordlt @ A_27a @ V1y @ V2z ) )
=> ( c_2Eordinal_2Eordlt @ A_27a @ V0x @ V2z ) ) ).
thf(thm_2Eordinal_2Eordlte__TRANS,axiom,
! [A_27a: $tType,V0x: tyop_2Eordinal_2Eordinal @ A_27a,V1y: tyop_2Eordinal_2Eordinal @ A_27a,V2z: tyop_2Eordinal_2Eordinal @ A_27a] :
( ( ( c_2Eordinal_2Eordlt @ A_27a @ V0x @ V1y )
& ( (~) @ ( c_2Eordinal_2Eordlt @ A_27a @ V2z @ V1y ) ) )
=> ( c_2Eordinal_2Eordlt @ A_27a @ V0x @ V2z ) ) ).
thf(thm_2Eordinal_2Eoleast__intro,axiom,
! [A_27a: $tType,V0Q: ( tyop_2Eordinal_2Eordinal @ A_27a ) > $o,V1P: ( tyop_2Eordinal_2Eordinal @ A_27a ) > $o] :
( ( ? [V2a: tyop_2Eordinal_2Eordinal @ A_27a] : ( V1P @ V2a )
& ! [V3a: tyop_2Eordinal_2Eordinal @ A_27a] :
( ( ! [V4b: tyop_2Eordinal_2Eordinal @ A_27a] :
( ( c_2Eordinal_2Eordlt @ A_27a @ V4b @ V3a )
=> ( (~) @ ( V1P @ V4b ) ) )
& ( V1P @ V3a ) )
=> ( V0Q @ V3a ) ) )
=> ( V0Q @ ( c_2Eordinal_2Eoleast @ A_27a @ V1P ) ) ) ).
thf(thm_2Eordinal_2EfromNat__def__compute,axiom,
! [A_27a: $tType] :
( ( ( c_2Eordinal_2EfromNat @ A_27a @ c_2Enum_2E0 )
= ( c_2Eordinal_2Eoleast @ A_27a
@ ^ [V0a: tyop_2Eordinal_2Eordinal @ A_27a] : c_2Ebool_2ET ) )
& ! [V1n: tyop_2Enum_2Enum] :
( ( c_2Eordinal_2EfromNat @ A_27a @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ V1n ) ) )
= ( c_2Eordinal_2EordSUC @ A_27a @ ( c_2Eordinal_2EfromNat @ A_27a @ ( c_2Earithmetic_2E_2D @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ V1n ) ) @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ) )
& ! [V2n: tyop_2Enum_2Enum] :
( ( c_2Eordinal_2EfromNat @ A_27a @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ V2n ) ) )
= ( c_2Eordinal_2EordSUC @ A_27a @ ( c_2Eordinal_2EfromNat @ A_27a @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ V2n ) ) ) ) ) ) ).
thf(thm_2Eordinal_2EfromNat__SUC,axiom,
! [A_27a: $tType,V0n: tyop_2Enum_2Enum] :
( ( c_2Eordinal_2EfromNat @ A_27a @ ( c_2Enum_2ESUC @ V0n ) )
= ( c_2Eordinal_2EordSUC @ A_27a @ ( c_2Eordinal_2EfromNat @ A_27a @ V0n ) ) ) ).
thf(thm_2Eordinal_2Eordlt__ZERO,axiom,
! [A_27a: $tType,V0a: tyop_2Eordinal_2Eordinal @ A_27a] : ( (~) @ ( c_2Eordinal_2Eordlt @ A_27a @ V0a @ ( c_2Eordinal_2EfromNat @ A_27a @ c_2Enum_2E0 ) ) ) ).
thf(thm_2Eordinal_2Epreds__surj,axiom,
! [A_27a: $tType,V0x: ( tyop_2Eordinal_2Eordinal @ A_27a ) > $o] :
( ( ( c_2Eordinal_2Edownward__closed @ A_27a @ V0x )
& ( (~)
@ ( V0x
= ( c_2Epred__set_2EUNIV @ ( tyop_2Eordinal_2Eordinal @ A_27a ) ) ) ) )
=> ? [V1y: tyop_2Eordinal_2Eordinal @ A_27a] :
( ( c_2Eordinal_2Epreds @ A_27a @ V1y )
= V0x ) ) ).
thf(thm_2Eordinal_2Eno__maximal__ordinal,axiom,
! [A_27a: $tType,V0a: tyop_2Eordinal_2Eordinal @ A_27a] :
? [V1b: tyop_2Eordinal_2Eordinal @ A_27a] : ( c_2Eordinal_2Eordlt @ A_27a @ V0a @ V1b ) ).
thf(thm_2Eordinal_2Eordlt__SUC,axiom,
! [A_27a: $tType,V0a: tyop_2Eordinal_2Eordinal @ A_27a] : ( c_2Eordinal_2Eordlt @ A_27a @ V0a @ ( c_2Eordinal_2EordSUC @ A_27a @ V0a ) ) ).
thf(thm_2Eordinal_2EordSUC__ZERO,axiom,
! [A_27a: $tType,V0a: tyop_2Eordinal_2Eordinal @ A_27a] :
( (~)
@ ( ( c_2Eordinal_2EordSUC @ A_27a @ V0a )
= ( c_2Eordinal_2EfromNat @ A_27a @ c_2Enum_2E0 ) ) ) ).
thf(thm_2Eordinal_2Eordlt__DISCRETE1,axiom,
! [A_27a: $tType,V0b: tyop_2Eordinal_2Eordinal @ A_27a,V1a: tyop_2Eordinal_2Eordinal @ A_27a] :
( (~)
@ ( ( c_2Eordinal_2Eordlt @ A_27a @ V1a @ V0b )
& ( c_2Eordinal_2Eordlt @ A_27a @ V0b @ ( c_2Eordinal_2EordSUC @ A_27a @ V1a ) ) ) ) ).
thf(thm_2Eordinal_2Eordlt__SUC__DISCRETE,axiom,
! [A_27a: $tType,V0b: tyop_2Eordinal_2Eordinal @ A_27a,V1a: tyop_2Eordinal_2Eordinal @ A_27a] :
( ( c_2Eordinal_2Eordlt @ A_27a @ V1a @ ( c_2Eordinal_2EordSUC @ A_27a @ V0b ) )
<=> ( ( c_2Eordinal_2Eordlt @ A_27a @ V1a @ V0b )
| ( V1a = V0b ) ) ) ).
thf(thm_2Eordinal_2EordSUC__MONO,axiom,
! [A_27a: $tType,V0b: tyop_2Eordinal_2Eordinal @ A_27a,V1a: tyop_2Eordinal_2Eordinal @ A_27a] :
( ( c_2Eordinal_2Eordlt @ A_27a @ ( c_2Eordinal_2EordSUC @ A_27a @ V1a ) @ ( c_2Eordinal_2EordSUC @ A_27a @ V0b ) )
= ( c_2Eordinal_2Eordlt @ A_27a @ V1a @ V0b ) ) ).
thf(thm_2Eordinal_2EordSUC__11,axiom,
! [A_27a: $tType,V0b: tyop_2Eordinal_2Eordinal @ A_27a,V1a: tyop_2Eordinal_2Eordinal @ A_27a] :
( ( ( c_2Eordinal_2EordSUC @ A_27a @ V1a )
= ( c_2Eordinal_2EordSUC @ A_27a @ V0b ) )
<=> ( V1a = V0b ) ) ).
thf(thm_2Eordinal_2Eord__induction,axiom,
! [A_27a: $tType,V0P: ( tyop_2Eordinal_2Eordinal @ A_27a ) > $o] :
( ! [V1min: tyop_2Eordinal_2Eordinal @ A_27a] :
( ! [V2b: tyop_2Eordinal_2Eordinal @ A_27a] :
( ( c_2Eordinal_2Eordlt @ A_27a @ V2b @ V1min )
=> ( V0P @ V2b ) )
=> ( V0P @ V1min ) )
=> ! [V3a: tyop_2Eordinal_2Eordinal @ A_27a] : ( V0P @ V3a ) ) ).
thf(thm_2Eordinal_2Esup__thm,axiom,
! [A_27a: $tType,V0s: ( tyop_2Eordinal_2Eordinal @ A_27a ) > $o] :
( ( c_2Ecardinal_2Ecardleq @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ ( tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ A_27a ) @ V0s @ ( c_2Epred__set_2EUNIV @ ( tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ A_27a ) ) )
=> ! [V1a: tyop_2Eordinal_2Eordinal @ A_27a] :
( ( c_2Eordinal_2Eordlt @ A_27a @ V1a @ ( c_2Eordinal_2Esup @ A_27a @ V0s ) )
<=> ? [V2b: tyop_2Eordinal_2Eordinal @ A_27a] :
( ( c_2Ebool_2EIN @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ V2b @ V0s )
& ( c_2Eordinal_2Eordlt @ A_27a @ V1a @ V2b ) ) ) ) ).
thf(thm_2Eordinal_2Esuple__thm,axiom,
! [A_27a: $tType,V0b: tyop_2Eordinal_2Eordinal @ A_27a,V1s: ( tyop_2Eordinal_2Eordinal @ A_27a ) > $o] :
( ( ( c_2Ecardinal_2Ecardleq @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ ( tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ A_27a ) @ V1s @ ( c_2Epred__set_2EUNIV @ ( tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ A_27a ) ) )
& ( c_2Ebool_2EIN @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ V0b @ V1s ) )
=> ( (~) @ ( c_2Eordinal_2Eordlt @ A_27a @ ( c_2Eordinal_2Esup @ A_27a @ V1s ) @ V0b ) ) ) ).
thf(thm_2Eordinal_2Esup__eq__sup,axiom,
! [A_27a: $tType,V0s2: ( tyop_2Eordinal_2Eordinal @ A_27a ) > $o,V1s1: ( tyop_2Eordinal_2Eordinal @ A_27a ) > $o] :
( ( ( c_2Ecardinal_2Ecardleq @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ ( tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ A_27a ) @ V1s1 @ ( c_2Epred__set_2EUNIV @ ( tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ A_27a ) ) )
& ( c_2Ecardinal_2Ecardleq @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ ( tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ A_27a ) @ V0s2 @ ( c_2Epred__set_2EUNIV @ ( tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ A_27a ) ) )
& ! [V2a: tyop_2Eordinal_2Eordinal @ A_27a] :
( ( c_2Ebool_2EIN @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ V2a @ V1s1 )
=> ? [V3b: tyop_2Eordinal_2Eordinal @ A_27a] :
( ( c_2Ebool_2EIN @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ V3b @ V0s2 )
& ( (~) @ ( c_2Eordinal_2Eordlt @ A_27a @ V3b @ V2a ) ) ) )
& ! [V4b: tyop_2Eordinal_2Eordinal @ A_27a] :
( ( c_2Ebool_2EIN @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ V4b @ V0s2 )
=> ? [V5a: tyop_2Eordinal_2Eordinal @ A_27a] :
( ( c_2Ebool_2EIN @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ V5a @ V1s1 )
& ( (~) @ ( c_2Eordinal_2Eordlt @ A_27a @ V5a @ V4b ) ) ) ) )
=> ( ( c_2Eordinal_2Esup @ A_27a @ V1s1 )
= ( c_2Eordinal_2Esup @ A_27a @ V0s2 ) ) ) ).
thf(thm_2Eordinal_2EUnum__cle__Uinf,axiom,
! [A_27a: $tType] : ( c_2Ecardinal_2Ecardleq @ tyop_2Enum_2Enum @ ( tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ A_27a ) @ ( c_2Epred__set_2EUNIV @ tyop_2Enum_2Enum ) @ ( c_2Epred__set_2EUNIV @ ( tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ A_27a ) ) ) ).
thf(thm_2Eordinal_2Ecsup__thm,axiom,
! [A_27a: $tType,V0s: ( tyop_2Eordinal_2Eordinal @ A_27a ) > $o] :
( ( c_2Epred__set_2Ecountable @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ V0s )
=> ! [V1b: tyop_2Eordinal_2Eordinal @ A_27a] :
( ( c_2Eordinal_2Eordlt @ A_27a @ V1b @ ( c_2Eordinal_2Esup @ A_27a @ V0s ) )
<=> ? [V2d: tyop_2Eordinal_2Eordinal @ A_27a] :
( ( c_2Ebool_2EIN @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ V2d @ V0s )
& ( c_2Eordinal_2Eordlt @ A_27a @ V1b @ V2d ) ) ) ) ).
thf(thm_2Eordinal_2Epredimage__sup__thm,axiom,
! [A_27a: $tType,V0f: ( tyop_2Eordinal_2Eordinal @ A_27a ) > ( tyop_2Eordinal_2Eordinal @ A_27a ),V1a: tyop_2Eordinal_2Eordinal @ A_27a,V2b: tyop_2Eordinal_2Eordinal @ A_27a] :
( ( c_2Eordinal_2Eordlt @ A_27a @ V2b @ ( c_2Eordinal_2Esup @ A_27a @ ( c_2Epred__set_2EIMAGE @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ V0f @ ( c_2Eordinal_2Epreds @ A_27a @ V1a ) ) ) )
<=> ? [V3d: tyop_2Eordinal_2Eordinal @ A_27a] :
( ( c_2Eordinal_2Eordlt @ A_27a @ V3d @ V1a )
& ( c_2Eordinal_2Eordlt @ A_27a @ V2b @ ( V0f @ V3d ) ) ) ) ).
thf(thm_2Eordinal_2Epredimage__suplt__ELIM,axiom,
! [A_27a: $tType,V0f: ( tyop_2Eordinal_2Eordinal @ A_27a ) > ( tyop_2Eordinal_2Eordinal @ A_27a ),V1b: tyop_2Eordinal_2Eordinal @ A_27a,V2a: tyop_2Eordinal_2Eordinal @ A_27a] :
( ( c_2Eordinal_2Eordlt @ A_27a @ ( c_2Eordinal_2Esup @ A_27a @ ( c_2Epred__set_2EIMAGE @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ V0f @ ( c_2Eordinal_2Epreds @ A_27a @ V2a ) ) ) @ V1b )
=> ! [V3d: tyop_2Eordinal_2Eordinal @ A_27a] :
( ( c_2Eordinal_2Eordlt @ A_27a @ V3d @ V2a )
=> ( (~) @ ( c_2Eordinal_2Eordlt @ A_27a @ V1b @ ( V0f @ V3d ) ) ) ) ) ).
thf(thm_2Eordinal_2Esuppred__suplt__ELIM,axiom,
! [A_27a: $tType,V0b: tyop_2Eordinal_2Eordinal @ A_27a,V1a: tyop_2Eordinal_2Eordinal @ A_27a] :
( ( c_2Eordinal_2Eordlt @ A_27a @ ( c_2Eordinal_2Esup @ A_27a @ ( c_2Eordinal_2Epreds @ A_27a @ V1a ) ) @ V0b )
=> ! [V2d: tyop_2Eordinal_2Eordinal @ A_27a] :
( ( c_2Eordinal_2Eordlt @ A_27a @ V2d @ V1a )
=> ( (~) @ ( c_2Eordinal_2Eordlt @ A_27a @ V0b @ V2d ) ) ) ) ).
thf(thm_2Eordinal_2Esup__EMPTY,axiom,
! [A_27a: $tType] :
( ( c_2Eordinal_2Esup @ A_27a @ ( c_2Epred__set_2EEMPTY @ ( tyop_2Eordinal_2Eordinal @ A_27a ) ) )
= ( c_2Eordinal_2EfromNat @ A_27a @ c_2Enum_2E0 ) ) ).
thf(thm_2Eordinal_2Esup__SING,axiom,
! [A_27a: $tType,V0a: tyop_2Eordinal_2Eordinal @ A_27a] :
( ( c_2Eordinal_2Esup @ A_27a @ ( c_2Epred__set_2EINSERT @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ V0a @ ( c_2Epred__set_2EEMPTY @ ( tyop_2Eordinal_2Eordinal @ A_27a ) ) ) )
= V0a ) ).
thf(thm_2Eordinal_2Esup__preds__SUC,axiom,
! [A_27a: $tType,V0a: tyop_2Eordinal_2Eordinal @ A_27a] :
( ( c_2Eordinal_2Esup @ A_27a @ ( c_2Eordinal_2Epreds @ A_27a @ ( c_2Eordinal_2EordSUC @ A_27a @ V0a ) ) )
= V0a ) ).
thf(thm_2Eordinal_2Epreds__ordSUC,axiom,
! [A_27a: $tType,V0a: tyop_2Eordinal_2Eordinal @ A_27a] :
( ( c_2Eordinal_2Epreds @ A_27a @ ( c_2Eordinal_2EordSUC @ A_27a @ V0a ) )
= ( c_2Epred__set_2EINSERT @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ V0a @ ( c_2Eordinal_2Epreds @ A_27a @ V0a ) ) ) ).
thf(thm_2Eordinal_2EcountableOrds__dclosed,axiom,
! [A_27a: $tType,V0b: tyop_2Eordinal_2Eordinal @ A_27a,V1a: tyop_2Eordinal_2Eordinal @ A_27a] :
( ( ( c_2Eordinal_2Eordlt @ A_27a @ V1a @ V0b )
& ( c_2Epred__set_2Ecountable @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ ( c_2Eordinal_2Epreds @ A_27a @ V0b ) ) )
=> ( c_2Epred__set_2Ecountable @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ ( c_2Eordinal_2Epreds @ A_27a @ V1a ) ) ) ).
thf(thm_2Eordinal_2Eomax__SOME,axiom,
! [A_27a: $tType,V0s: ( tyop_2Eordinal_2Eordinal @ A_27a ) > $o,V1a: tyop_2Eordinal_2Eordinal @ A_27a] :
( ( ( c_2Eordinal_2Eomax @ A_27a @ V0s )
= ( c_2Eoption_2ESOME @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ V1a ) )
<=> ( ( c_2Ebool_2EIN @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ V1a @ V0s )
& ! [V2b: tyop_2Eordinal_2Eordinal @ A_27a] :
( ( c_2Ebool_2EIN @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ V2b @ V0s )
=> ( (~) @ ( c_2Eordinal_2Eordlt @ A_27a @ V1a @ V2b ) ) ) ) ) ).
thf(thm_2Eordinal_2Eomax__NONE,axiom,
! [A_27a: $tType,V0s: ( tyop_2Eordinal_2Eordinal @ A_27a ) > $o] :
( ( ( c_2Eordinal_2Eomax @ A_27a @ V0s )
= ( c_2Eoption_2ENONE @ ( tyop_2Eordinal_2Eordinal @ A_27a ) ) )
<=> ! [V1a: tyop_2Eordinal_2Eordinal @ A_27a] :
( ( c_2Ebool_2EIN @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ V1a @ V0s )
=> ? [V2b: tyop_2Eordinal_2Eordinal @ A_27a] :
( ( c_2Ebool_2EIN @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ V2b @ V0s )
& ( c_2Eordinal_2Eordlt @ A_27a @ V1a @ V2b ) ) ) ) ).
thf(thm_2Eordinal_2Eomax__EMPTY,axiom,
! [A_27a: $tType] :
( ( c_2Eordinal_2Eomax @ A_27a @ ( c_2Epred__set_2EEMPTY @ ( tyop_2Eordinal_2Eordinal @ A_27a ) ) )
= ( c_2Eoption_2ENONE @ ( tyop_2Eordinal_2Eordinal @ A_27a ) ) ) ).
thf(thm_2Eordinal_2Epreds__0,axiom,
! [A_27a: $tType] :
( ( c_2Eordinal_2Epreds @ A_27a @ ( c_2Eordinal_2EfromNat @ A_27a @ c_2Enum_2E0 ) )
= ( c_2Epred__set_2EEMPTY @ ( tyop_2Eordinal_2Eordinal @ A_27a ) ) ) ).
thf(thm_2Eordinal_2Eordleq0,axiom,
! [A_27a: $tType,V0x: tyop_2Eordinal_2Eordinal @ A_27a] :
( ( (~) @ ( c_2Eordinal_2Eordlt @ A_27a @ ( c_2Eordinal_2EfromNat @ A_27a @ c_2Enum_2E0 ) @ V0x ) )
<=> ( V0x
= ( c_2Eordinal_2EfromNat @ A_27a @ c_2Enum_2E0 ) ) ) ).
thf(thm_2Eordinal_2Epreds__EQ__EMPTY,axiom,
! [A_27a: $tType,V0x: tyop_2Eordinal_2Eordinal @ A_27a] :
( ( ( c_2Eordinal_2Epreds @ A_27a @ V0x )
= ( c_2Epred__set_2EEMPTY @ ( tyop_2Eordinal_2Eordinal @ A_27a ) ) )
<=> ( V0x
= ( c_2Eordinal_2EfromNat @ A_27a @ c_2Enum_2E0 ) ) ) ).
thf(thm_2Eordinal_2Eomax__sup,axiom,
! [A_27a: $tType,V0s: ( tyop_2Eordinal_2Eordinal @ A_27a ) > $o,V1a: tyop_2Eordinal_2Eordinal @ A_27a] :
( ( ( c_2Eordinal_2Eomax @ A_27a @ V0s )
= ( c_2Eoption_2ESOME @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ V1a ) )
=> ( ( c_2Eordinal_2Esup @ A_27a @ V0s )
= V1a ) ) ).
thf(thm_2Eordinal_2Epreds__omax__SOME__SUC,axiom,
! [A_27a: $tType,V0b: tyop_2Eordinal_2Eordinal @ A_27a,V1a: tyop_2Eordinal_2Eordinal @ A_27a] :
( ( ( c_2Eordinal_2Eomax @ A_27a @ ( c_2Eordinal_2Epreds @ A_27a @ V1a ) )
= ( c_2Eoption_2ESOME @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ V0b ) )
<=> ( V1a
= ( c_2Eordinal_2EordSUC @ A_27a @ V0b ) ) ) ).
thf(thm_2Eordinal_2Eomax__preds__SUC,axiom,
! [A_27a: $tType,V0x: tyop_2Eordinal_2Eordinal @ A_27a] :
( ( c_2Eordinal_2Eomax @ A_27a @ ( c_2Eordinal_2Epreds @ A_27a @ ( c_2Eordinal_2EordSUC @ A_27a @ V0x ) ) )
= ( c_2Eoption_2ESOME @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ V0x ) ) ).
thf(thm_2Eordinal_2EORD__ONE,axiom,
! [A_27a: $tType] :
( ( c_2Eordinal_2EordSUC @ A_27a @ ( c_2Eordinal_2EfromNat @ A_27a @ c_2Enum_2E0 ) )
= ( c_2Eordinal_2EfromNat @ A_27a @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ).
thf(thm_2Eordinal_2EordSUC__NUMERAL,axiom,
! [A_27a: $tType,V0n: tyop_2Enum_2Enum] :
( ( c_2Eordinal_2EordSUC @ A_27a @ ( c_2Eordinal_2EfromNat @ A_27a @ ( c_2Earithmetic_2ENUMERAL @ V0n ) ) )
= ( c_2Eordinal_2EfromNat @ A_27a @ ( c_2Earithmetic_2E_2B @ ( c_2Earithmetic_2ENUMERAL @ V0n ) @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ) ).
thf(thm_2Eordinal_2EordZERO__ltSUC,axiom,
! [A_27a: $tType,V0x: tyop_2Eordinal_2Eordinal @ A_27a] : ( c_2Eordinal_2Eordlt @ A_27a @ ( c_2Eordinal_2EfromNat @ A_27a @ c_2Enum_2E0 ) @ ( c_2Eordinal_2EordSUC @ A_27a @ V0x ) ) ).
thf(thm_2Eordinal_2Eordlt__CANCEL__ADDR,axiom,
! [A_27a: $tType,V0b: tyop_2Eordinal_2Eordinal @ A_27a,V1a: tyop_2Eordinal_2Eordinal @ A_27a] :
( ( c_2Eordinal_2Eordlt @ A_27a @ V1a @ ( c_2Eordinal_2EordADD @ A_27a @ V1a @ V0b ) )
= ( c_2Eordinal_2Eordlt @ A_27a @ ( c_2Eordinal_2EfromNat @ A_27a @ c_2Enum_2E0 ) @ V0b ) ) ).
thf(thm_2Eordinal_2Eordlt__CANCEL__ADDL,axiom,
! [A_27a: $tType,V0b: tyop_2Eordinal_2Eordinal @ A_27a,V1a: tyop_2Eordinal_2Eordinal @ A_27a] :
( ( c_2Eordinal_2Eordlt @ A_27a @ ( c_2Eordinal_2EordADD @ A_27a @ V1a @ V0b ) @ V1a )
= c_2Ebool_2EF ) ).
thf(thm_2Eordinal_2EordADD__CANCEL1,axiom,
! [A_27a: $tType] :
( ! [V0c: tyop_2Eordinal_2Eordinal @ A_27a,V1a: tyop_2Eordinal_2Eordinal @ A_27a] :
( ( V1a
= ( c_2Eordinal_2EordADD @ A_27a @ V1a @ V0c ) )
<=> ( V0c
= ( c_2Eordinal_2EfromNat @ A_27a @ c_2Enum_2E0 ) ) )
& ! [V2c: tyop_2Eordinal_2Eordinal @ A_27a,V3a: tyop_2Eordinal_2Eordinal @ A_27a] :
( ( ( c_2Eordinal_2EordADD @ A_27a @ V3a @ V2c )
= V3a )
<=> ( V2c
= ( c_2Eordinal_2EfromNat @ A_27a @ c_2Enum_2E0 ) ) ) ) ).
thf(thm_2Eordinal_2EordADD__MONO,axiom,
! [A_27a: $tType,V0b: tyop_2Eordinal_2Eordinal @ A_27a,V1a: tyop_2Eordinal_2Eordinal @ A_27a,V2c: tyop_2Eordinal_2Eordinal @ A_27a] :
( ( c_2Eordinal_2Eordlt @ A_27a @ V1a @ V0b )
=> ( c_2Eordinal_2Eordlt @ A_27a @ ( c_2Eordinal_2EordADD @ A_27a @ V2c @ V1a ) @ ( c_2Eordinal_2EordADD @ A_27a @ V2c @ V0b ) ) ) ).
thf(thm_2Eordinal_2Eordlt__CANCEL,axiom,
! [A_27a: $tType,V0b: tyop_2Eordinal_2Eordinal @ A_27a,V1a: tyop_2Eordinal_2Eordinal @ A_27a,V2c: tyop_2Eordinal_2Eordinal @ A_27a] :
( ( c_2Eordinal_2Eordlt @ A_27a @ ( c_2Eordinal_2EordADD @ A_27a @ V2c @ V1a ) @ ( c_2Eordinal_2EordADD @ A_27a @ V2c @ V0b ) )
= ( c_2Eordinal_2Eordlt @ A_27a @ V1a @ V0b ) ) ).
thf(thm_2Eordinal_2EordADD__RIGHT__CANCEL,axiom,
! [A_27a: $tType,V0b: tyop_2Eordinal_2Eordinal @ A_27a,V1a: tyop_2Eordinal_2Eordinal @ A_27a,V2c: tyop_2Eordinal_2Eordinal @ A_27a] :
( ( ( c_2Eordinal_2EordADD @ A_27a @ V1a @ V0b )
= ( c_2Eordinal_2EordADD @ A_27a @ V1a @ V2c ) )
<=> ( V0b = V2c ) ) ).
thf(thm_2Eordinal_2EleqLEFT__CANCEL,axiom,
! [A_27a: $tType,V0x: tyop_2Eordinal_2Eordinal @ A_27a,V1a: tyop_2Eordinal_2Eordinal @ A_27a] : ( (~) @ ( c_2Eordinal_2Eordlt @ A_27a @ ( c_2Eordinal_2EordADD @ A_27a @ V1a @ V0x ) @ V0x ) ) ).
thf(thm_2Eordinal_2Eordlt__EXISTS__ADD,axiom,
! [A_27a: $tType,V0a: tyop_2Eordinal_2Eordinal @ A_27a,V1b: tyop_2Eordinal_2Eordinal @ A_27a] :
( ( c_2Eordinal_2Eordlt @ A_27a @ V0a @ V1b )
<=> ? [V2c: tyop_2Eordinal_2Eordinal @ A_27a] :
( ( (~)
@ ( V2c
= ( c_2Eordinal_2EfromNat @ A_27a @ c_2Enum_2E0 ) ) )
& ( V1b
= ( c_2Eordinal_2EordADD @ A_27a @ V0a @ V2c ) ) ) ) ).
thf(thm_2Eordinal_2Eordle__EXISTS__ADD,axiom,
! [A_27a: $tType,V0a: tyop_2Eordinal_2Eordinal @ A_27a,V1b: tyop_2Eordinal_2Eordinal @ A_27a] :
( ( (~) @ ( c_2Eordinal_2Eordlt @ A_27a @ V1b @ V0a ) )
<=> ? [V2c: tyop_2Eordinal_2Eordinal @ A_27a] :
( V1b
= ( c_2Eordinal_2EordADD @ A_27a @ V0a @ V2c ) ) ) ).
thf(thm_2Eordinal_2Eordle__CANCEL__ADDR,axiom,
! [A_27a: $tType,V0x: tyop_2Eordinal_2Eordinal @ A_27a,V1a: tyop_2Eordinal_2Eordinal @ A_27a] : ( (~) @ ( c_2Eordinal_2Eordlt @ A_27a @ ( c_2Eordinal_2EordADD @ A_27a @ V0x @ V1a ) @ V0x ) ) ).
thf(thm_2Eordinal_2Edclose__BIGUNION,axiom,
! [A_27a: $tType,V0s: ( tyop_2Eordinal_2Eordinal @ A_27a ) > $o] :
( ( c_2Eordinal_2Edclose @ A_27a @ V0s )
= ( c_2Epred__set_2EBIGUNION @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ ( c_2Epred__set_2EIMAGE @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ ( ( tyop_2Eordinal_2Eordinal @ A_27a ) > $o ) @ ( c_2Eordinal_2Epreds @ A_27a ) @ V0s ) ) ) ).
thf(thm_2Eordinal_2EcountableOrds__uncountable,axiom,
! [A_27a: $tType] :
( (~)
@ ( c_2Epred__set_2Ecountable @ ( tyop_2Eordinal_2Eordinal @ A_27a )
@ ( c_2Epred__set_2EGSPEC @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ ( tyop_2Eordinal_2Eordinal @ A_27a )
@ ^ [V0a: tyop_2Eordinal_2Eordinal @ A_27a] : ( c_2Epair_2E_2C @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ $o @ V0a @ ( c_2Epred__set_2Ecountable @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ ( c_2Eordinal_2Epreds @ A_27a @ V0a ) ) ) ) ) ) ).
thf(thm_2Eordinal_2Edclose__cardleq__univinf,axiom,
! [A_27a: $tType,V0s: ( tyop_2Eordinal_2Eordinal @ A_27a ) > $o] :
( ( c_2Ecardinal_2Ecardleq @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ ( tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ A_27a ) @ V0s @ ( c_2Epred__set_2EUNIV @ ( tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ A_27a ) ) )
=> ( c_2Ecardinal_2Ecardleq @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ ( tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ A_27a ) @ ( c_2Eordinal_2Edclose @ A_27a @ V0s ) @ ( c_2Epred__set_2EUNIV @ ( tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ A_27a ) ) ) ) ).
thf(thm_2Eordinal_2Esup__lt__implies,axiom,
! [A_27a: $tType,V0s: ( tyop_2Eordinal_2Eordinal @ A_27a ) > $o,V1b: tyop_2Eordinal_2Eordinal @ A_27a,V2a: tyop_2Eordinal_2Eordinal @ A_27a] :
( ( ( c_2Ecardinal_2Ecardleq @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ ( tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ A_27a ) @ V0s @ ( c_2Epred__set_2EUNIV @ ( tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ A_27a ) ) )
& ( c_2Eordinal_2Eordlt @ A_27a @ ( c_2Eordinal_2Esup @ A_27a @ V0s ) @ V2a )
& ( c_2Ebool_2EIN @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ V1b @ V0s ) )
=> ( c_2Eordinal_2Eordlt @ A_27a @ V1b @ V2a ) ) ).
thf(thm_2Eordinal_2Esup__eq__max,axiom,
! [A_27a: $tType,V0s: ( tyop_2Eordinal_2Eordinal @ A_27a ) > $o,V1a: tyop_2Eordinal_2Eordinal @ A_27a] :
( ( ! [V2b: tyop_2Eordinal_2Eordinal @ A_27a] :
( ( c_2Ebool_2EIN @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ V2b @ V0s )
=> ( (~) @ ( c_2Eordinal_2Eordlt @ A_27a @ V1a @ V2b ) ) )
& ( c_2Ebool_2EIN @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ V1a @ V0s ) )
=> ( ( c_2Eordinal_2Esup @ A_27a @ V0s )
= V1a ) ) ).
thf(thm_2Eordinal_2Esup__eq__SUC,axiom,
! [A_27a: $tType,V0s: ( tyop_2Eordinal_2Eordinal @ A_27a ) > $o,V1a: tyop_2Eordinal_2Eordinal @ A_27a] :
( ( ( c_2Ecardinal_2Ecardleq @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ ( tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ A_27a ) @ V0s @ ( c_2Epred__set_2EUNIV @ ( tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ A_27a ) ) )
& ( ( c_2Eordinal_2Esup @ A_27a @ V0s )
= ( c_2Eordinal_2EordSUC @ A_27a @ V1a ) ) )
=> ( c_2Ebool_2EIN @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ ( c_2Eordinal_2EordSUC @ A_27a @ V1a ) @ V0s ) ) ).
thf(thm_2Eordinal_2Egeneric__continuity,axiom,
! [A_27a: $tType,V0f: ( tyop_2Eordinal_2Eordinal @ A_27a ) > ( tyop_2Eordinal_2Eordinal @ A_27a )] :
( ( ! [V1a: tyop_2Eordinal_2Eordinal @ A_27a] :
( ( ( c_2Eordinal_2Eordlt @ A_27a @ ( c_2Eordinal_2EfromNat @ A_27a @ c_2Enum_2E0 ) @ V1a )
& ( ( c_2Eordinal_2Eomax @ A_27a @ ( c_2Eordinal_2Epreds @ A_27a @ V1a ) )
= ( c_2Eoption_2ENONE @ ( tyop_2Eordinal_2Eordinal @ A_27a ) ) ) )
=> ( ( V0f @ V1a )
= ( c_2Eordinal_2Esup @ A_27a @ ( c_2Epred__set_2EIMAGE @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ V0f @ ( c_2Eordinal_2Epreds @ A_27a @ V1a ) ) ) ) )
& ! [V2x: tyop_2Eordinal_2Eordinal @ A_27a,V3y: tyop_2Eordinal_2Eordinal @ A_27a] :
( ( (~) @ ( c_2Eordinal_2Eordlt @ A_27a @ V3y @ V2x ) )
=> ( (~) @ ( c_2Eordinal_2Eordlt @ A_27a @ ( V0f @ V3y ) @ ( V0f @ V2x ) ) ) ) )
=> ! [V4s: ( tyop_2Eordinal_2Eordinal @ A_27a ) > $o] :
( ( ( c_2Ecardinal_2Ecardleq @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ ( tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ A_27a ) @ V4s @ ( c_2Epred__set_2EUNIV @ ( tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ A_27a ) ) )
& ( (~)
@ ( V4s
= ( c_2Epred__set_2EEMPTY @ ( tyop_2Eordinal_2Eordinal @ A_27a ) ) ) ) )
=> ( ( V0f @ ( c_2Eordinal_2Esup @ A_27a @ V4s ) )
= ( c_2Eordinal_2Esup @ A_27a @ ( c_2Epred__set_2EIMAGE @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ V0f @ V4s ) ) ) ) ) ).
thf(thm_2Eordinal_2Eord__CASES,axiom,
! [A_27a: $tType,V0a: tyop_2Eordinal_2Eordinal @ A_27a] :
( ( V0a
= ( c_2Eordinal_2EfromNat @ A_27a @ c_2Enum_2E0 ) )
| ? [V1a0: tyop_2Eordinal_2Eordinal @ A_27a] :
( V0a
= ( c_2Eordinal_2EordSUC @ A_27a @ V1a0 ) )
| ( ( c_2Eordinal_2Eordlt @ A_27a @ ( c_2Eordinal_2EfromNat @ A_27a @ c_2Enum_2E0 ) @ V0a )
& ( ( c_2Eordinal_2Eomax @ A_27a @ ( c_2Eordinal_2Epreds @ A_27a @ V0a ) )
= ( c_2Eoption_2ENONE @ ( tyop_2Eordinal_2Eordinal @ A_27a ) ) ) ) ) ).
thf(thm_2Eordinal_2Eislimit__0,axiom,
! [A_27a: $tType] :
( ( c_2Eordinal_2Eomax @ A_27a @ ( c_2Eordinal_2Epreds @ A_27a @ ( c_2Eordinal_2EfromNat @ A_27a @ c_2Enum_2E0 ) ) )
= ( c_2Eoption_2ENONE @ ( tyop_2Eordinal_2Eordinal @ A_27a ) ) ) ).
thf(thm_2Eordinal_2Eordinal__IVT,axiom,
! [A_27a: $tType,V0f: ( tyop_2Eordinal_2Eordinal @ A_27a ) > ( tyop_2Eordinal_2Eordinal @ A_27a ),V1c: tyop_2Eordinal_2Eordinal @ A_27a,V2a2: tyop_2Eordinal_2Eordinal @ A_27a,V3a1: tyop_2Eordinal_2Eordinal @ A_27a] :
( ( ! [V4a: tyop_2Eordinal_2Eordinal @ A_27a] :
( ( ( c_2Eordinal_2Eordlt @ A_27a @ ( c_2Eordinal_2EfromNat @ A_27a @ c_2Enum_2E0 ) @ V4a )
& ( ( c_2Eordinal_2Eomax @ A_27a @ ( c_2Eordinal_2Epreds @ A_27a @ V4a ) )
= ( c_2Eoption_2ENONE @ ( tyop_2Eordinal_2Eordinal @ A_27a ) ) ) )
=> ( ( V0f @ V4a )
= ( c_2Eordinal_2Esup @ A_27a @ ( c_2Epred__set_2EIMAGE @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ V0f @ ( c_2Eordinal_2Epreds @ A_27a @ V4a ) ) ) ) )
& ! [V5x: tyop_2Eordinal_2Eordinal @ A_27a,V6y: tyop_2Eordinal_2Eordinal @ A_27a] :
( ( (~) @ ( c_2Eordinal_2Eordlt @ A_27a @ V6y @ V5x ) )
=> ( (~) @ ( c_2Eordinal_2Eordlt @ A_27a @ ( V0f @ V6y ) @ ( V0f @ V5x ) ) ) )
& ( c_2Eordinal_2Eordlt @ A_27a @ V3a1 @ V2a2 )
& ( (~) @ ( c_2Eordinal_2Eordlt @ A_27a @ V1c @ ( V0f @ V3a1 ) ) )
& ( c_2Eordinal_2Eordlt @ A_27a @ V1c @ ( V0f @ V2a2 ) ) )
=> ? [V7b: tyop_2Eordinal_2Eordinal @ A_27a] :
( ( (~) @ ( c_2Eordinal_2Eordlt @ A_27a @ V7b @ V3a1 ) )
& ( c_2Eordinal_2Eordlt @ A_27a @ V7b @ V2a2 )
& ( (~) @ ( c_2Eordinal_2Eordlt @ A_27a @ V1c @ ( V0f @ V7b ) ) )
& ( c_2Eordinal_2Eordlt @ A_27a @ V1c @ ( V0f @ ( c_2Eordinal_2EordSUC @ A_27a @ V7b ) ) ) ) ) ).
thf(thm_2Eordinal_2EordADD__continuous,axiom,
! [A_27a: $tType,V0a: tyop_2Eordinal_2Eordinal @ A_27a,V1s: ( tyop_2Eordinal_2Eordinal @ A_27a ) > $o] :
( ( ( c_2Ecardinal_2Ecardleq @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ ( tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ A_27a ) @ V1s @ ( c_2Epred__set_2EUNIV @ ( tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ A_27a ) ) )
& ( (~)
@ ( V1s
= ( c_2Epred__set_2EEMPTY @ ( tyop_2Eordinal_2Eordinal @ A_27a ) ) ) ) )
=> ( ( c_2Eordinal_2EordADD @ A_27a @ V0a @ ( c_2Eordinal_2Esup @ A_27a @ V1s ) )
= ( c_2Eordinal_2Esup @ A_27a @ ( c_2Epred__set_2EIMAGE @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ ( c_2Eordinal_2EordADD @ A_27a @ V0a ) @ V1s ) ) ) ) ).
thf(thm_2Eordinal_2EordADD__ASSOC,axiom,
! [A_27a: $tType,V0a: tyop_2Eordinal_2Eordinal @ A_27a,V1b: tyop_2Eordinal_2Eordinal @ A_27a,V2c: tyop_2Eordinal_2Eordinal @ A_27a] :
( ( c_2Eordinal_2EordADD @ A_27a @ V0a @ ( c_2Eordinal_2EordADD @ A_27a @ V1b @ V2c ) )
= ( c_2Eordinal_2EordADD @ A_27a @ ( c_2Eordinal_2EordADD @ A_27a @ V0a @ V1b ) @ V2c ) ) ).
thf(thm_2Eordinal_2EADD1R,axiom,
! [A_27a: $tType,V0a: tyop_2Eordinal_2Eordinal @ A_27a] :
( ( c_2Eordinal_2EordADD @ A_27a @ V0a @ ( c_2Eordinal_2EfromNat @ A_27a @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) )
= ( c_2Eordinal_2EordSUC @ A_27a @ V0a ) ) ).
thf(thm_2Eordinal_2EordADD__weak__MONO,axiom,
! [A_27a: $tType,V0c: tyop_2Eordinal_2Eordinal @ A_27a,V1a: tyop_2Eordinal_2Eordinal @ A_27a,V2b: tyop_2Eordinal_2Eordinal @ A_27a] :
( ( c_2Eordinal_2Eordlt @ A_27a @ V1a @ V2b )
=> ( (~) @ ( c_2Eordinal_2Eordlt @ A_27a @ ( c_2Eordinal_2EordADD @ A_27a @ V2b @ V0c ) @ ( c_2Eordinal_2EordADD @ A_27a @ V1a @ V0c ) ) ) ) ).
thf(thm_2Eordinal_2EordMULT__0L,axiom,
! [A_27a: $tType,V0a: tyop_2Eordinal_2Eordinal @ A_27a] :
( ( c_2Eordinal_2EordMULT @ A_27a @ ( c_2Eordinal_2EfromNat @ A_27a @ c_2Enum_2E0 ) @ V0a )
= ( c_2Eordinal_2EfromNat @ A_27a @ c_2Enum_2E0 ) ) ).
thf(thm_2Eordinal_2EordMULT__0R,axiom,
! [A_27a: $tType,V0a: tyop_2Eordinal_2Eordinal @ A_27a] :
( ( c_2Eordinal_2EordMULT @ A_27a @ V0a @ ( c_2Eordinal_2EfromNat @ A_27a @ c_2Enum_2E0 ) )
= ( c_2Eordinal_2EfromNat @ A_27a @ c_2Enum_2E0 ) ) ).
thf(thm_2Eordinal_2EordMULT__1L,axiom,
! [A_27a: $tType,V0a: tyop_2Eordinal_2Eordinal @ A_27a] :
( ( c_2Eordinal_2EordMULT @ A_27a @ ( c_2Eordinal_2EfromNat @ A_27a @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) @ V0a )
= V0a ) ).
thf(thm_2Eordinal_2EordMULT__1R,axiom,
! [A_27a: $tType,V0a: tyop_2Eordinal_2Eordinal @ A_27a] :
( ( c_2Eordinal_2EordMULT @ A_27a @ V0a @ ( c_2Eordinal_2EfromNat @ A_27a @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) )
= V0a ) ).
thf(thm_2Eordinal_2EordMULT__2R,axiom,
! [A_27a: $tType,V0a: tyop_2Eordinal_2Eordinal @ A_27a] :
( ( c_2Eordinal_2EordMULT @ A_27a @ V0a @ ( c_2Eordinal_2EfromNat @ A_27a @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) )
= ( c_2Eordinal_2EordADD @ A_27a @ V0a @ V0a ) ) ).
thf(thm_2Eordinal_2Eislimit__SUC__lt,axiom,
! [A_27a: $tType,V0b: tyop_2Eordinal_2Eordinal @ A_27a,V1a: tyop_2Eordinal_2Eordinal @ A_27a] :
( ( ( ( c_2Eordinal_2Eomax @ A_27a @ ( c_2Eordinal_2Epreds @ A_27a @ V0b ) )
= ( c_2Eoption_2ENONE @ ( tyop_2Eordinal_2Eordinal @ A_27a ) ) )
& ( c_2Eordinal_2Eordlt @ A_27a @ V1a @ V0b ) )
=> ( c_2Eordinal_2Eordlt @ A_27a @ ( c_2Eordinal_2EordSUC @ A_27a @ V1a ) @ V0b ) ) ).
thf(thm_2Eordinal_2EordMULT__lt__MONO__R,axiom,
! [A_27a: $tType,V0a: tyop_2Eordinal_2Eordinal @ A_27a,V1b: tyop_2Eordinal_2Eordinal @ A_27a,V2c: tyop_2Eordinal_2Eordinal @ A_27a] :
( ( ( c_2Eordinal_2Eordlt @ A_27a @ V0a @ V1b )
& ( c_2Eordinal_2Eordlt @ A_27a @ ( c_2Eordinal_2EfromNat @ A_27a @ c_2Enum_2E0 ) @ V2c ) )
=> ( c_2Eordinal_2Eordlt @ A_27a @ ( c_2Eordinal_2EordMULT @ A_27a @ V2c @ V0a ) @ ( c_2Eordinal_2EordMULT @ A_27a @ V2c @ V1b ) ) ) ).
thf(thm_2Eordinal_2EordMULT__le__MONO__R,axiom,
! [A_27a: $tType,V0a: tyop_2Eordinal_2Eordinal @ A_27a,V1b: tyop_2Eordinal_2Eordinal @ A_27a,V2c: tyop_2Eordinal_2Eordinal @ A_27a] :
( ( (~) @ ( c_2Eordinal_2Eordlt @ A_27a @ V1b @ V0a ) )
=> ( (~) @ ( c_2Eordinal_2Eordlt @ A_27a @ ( c_2Eordinal_2EordMULT @ A_27a @ V2c @ V1b ) @ ( c_2Eordinal_2EordMULT @ A_27a @ V2c @ V0a ) ) ) ) ).
thf(thm_2Eordinal_2EordMULT__lt__MONO__R__EQN,axiom,
! [A_27a: $tType,V0c: tyop_2Eordinal_2Eordinal @ A_27a,V1b: tyop_2Eordinal_2Eordinal @ A_27a,V2a: tyop_2Eordinal_2Eordinal @ A_27a] :
( ( c_2Eordinal_2Eordlt @ A_27a @ ( c_2Eordinal_2EordMULT @ A_27a @ V0c @ V2a ) @ ( c_2Eordinal_2EordMULT @ A_27a @ V0c @ V1b ) )
<=> ( ( c_2Eordinal_2Eordlt @ A_27a @ V2a @ V1b )
& ( c_2Eordinal_2Eordlt @ A_27a @ ( c_2Eordinal_2EfromNat @ A_27a @ c_2Enum_2E0 ) @ V0c ) ) ) ).
thf(thm_2Eordinal_2EordADD__le__MONO__L,axiom,
! [A_27a: $tType,V0z: tyop_2Eordinal_2Eordinal @ A_27a,V1y: tyop_2Eordinal_2Eordinal @ A_27a,V2x: tyop_2Eordinal_2Eordinal @ A_27a] :
( ( (~) @ ( c_2Eordinal_2Eordlt @ A_27a @ V1y @ V2x ) )
=> ( (~) @ ( c_2Eordinal_2Eordlt @ A_27a @ ( c_2Eordinal_2EordADD @ A_27a @ V1y @ V0z ) @ ( c_2Eordinal_2EordADD @ A_27a @ V2x @ V0z ) ) ) ) ).
thf(thm_2Eordinal_2EordMULT__le__MONO__L,axiom,
! [A_27a: $tType,V0a: tyop_2Eordinal_2Eordinal @ A_27a,V1b: tyop_2Eordinal_2Eordinal @ A_27a,V2c: tyop_2Eordinal_2Eordinal @ A_27a] :
( ( (~) @ ( c_2Eordinal_2Eordlt @ A_27a @ V1b @ V0a ) )
=> ( (~) @ ( c_2Eordinal_2Eordlt @ A_27a @ ( c_2Eordinal_2EordMULT @ A_27a @ V1b @ V2c ) @ ( c_2Eordinal_2EordMULT @ A_27a @ V0a @ V2c ) ) ) ) ).
thf(thm_2Eordinal_2EordMULT__CANCEL__R,axiom,
! [A_27a: $tType,V0z: tyop_2Eordinal_2Eordinal @ A_27a,V1y: tyop_2Eordinal_2Eordinal @ A_27a,V2x: tyop_2Eordinal_2Eordinal @ A_27a] :
( ( ( c_2Eordinal_2EordMULT @ A_27a @ V0z @ V2x )
= ( c_2Eordinal_2EordMULT @ A_27a @ V0z @ V1y ) )
<=> ( ( V0z
= ( c_2Eordinal_2EfromNat @ A_27a @ c_2Enum_2E0 ) )
| ( V2x = V1y ) ) ) ).
thf(thm_2Eordinal_2EordMULT__continuous,axiom,
! [A_27a: $tType,V0a: tyop_2Eordinal_2Eordinal @ A_27a,V1s: ( tyop_2Eordinal_2Eordinal @ A_27a ) > $o] :
( ( c_2Ecardinal_2Ecardleq @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ ( tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ A_27a ) @ V1s @ ( c_2Epred__set_2EUNIV @ ( tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ A_27a ) ) )
=> ( ( c_2Eordinal_2EordMULT @ A_27a @ V0a @ ( c_2Eordinal_2Esup @ A_27a @ V1s ) )
= ( c_2Eordinal_2Esup @ A_27a @ ( c_2Epred__set_2EIMAGE @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ ( c_2Eordinal_2EordMULT @ A_27a @ V0a ) @ V1s ) ) ) ) ).
thf(thm_2Eordinal_2EordMULT__fromNat,axiom,
! [A_27a: $tType,V0n: tyop_2Enum_2Enum,V1m: tyop_2Enum_2Enum] :
( ( c_2Eordinal_2EordMULT @ A_27a @ ( c_2Eordinal_2EfromNat @ A_27a @ V0n ) @ ( c_2Eordinal_2EfromNat @ A_27a @ V1m ) )
= ( c_2Eordinal_2EfromNat @ A_27a @ ( c_2Earithmetic_2E_2A @ V0n @ V1m ) ) ) ).
thf(thm_2Eordinal_2Eomega__MUL__fromNat,axiom,
! [A_27a: $tType,V0n: tyop_2Enum_2Enum] :
( ( c_2Eprim__rec_2E_3C @ c_2Enum_2E0 @ V0n )
=> ( ( c_2Eordinal_2EordMULT @ A_27a @ ( c_2Eordinal_2EfromNat @ A_27a @ V0n ) @ ( c_2Eordinal_2Eomega @ A_27a ) )
= ( c_2Eordinal_2Eomega @ A_27a ) ) ) ).
thf(thm_2Eordinal_2EordMULT__LDISTRIB,axiom,
! [A_27a: $tType,V0a: tyop_2Eordinal_2Eordinal @ A_27a,V1b: tyop_2Eordinal_2Eordinal @ A_27a,V2c: tyop_2Eordinal_2Eordinal @ A_27a] :
( ( c_2Eordinal_2EordMULT @ A_27a @ V2c @ ( c_2Eordinal_2EordADD @ A_27a @ V0a @ V1b ) )
= ( c_2Eordinal_2EordADD @ A_27a @ ( c_2Eordinal_2EordMULT @ A_27a @ V2c @ V0a ) @ ( c_2Eordinal_2EordMULT @ A_27a @ V2c @ V1b ) ) ) ).
thf(thm_2Eordinal_2EordMULT__ASSOC,axiom,
! [A_27a: $tType,V0a: tyop_2Eordinal_2Eordinal @ A_27a,V1b: tyop_2Eordinal_2Eordinal @ A_27a,V2c: tyop_2Eordinal_2Eordinal @ A_27a] :
( ( c_2Eordinal_2EordMULT @ A_27a @ V0a @ ( c_2Eordinal_2EordMULT @ A_27a @ V1b @ V2c ) )
= ( c_2Eordinal_2EordMULT @ A_27a @ ( c_2Eordinal_2EordMULT @ A_27a @ V0a @ V1b ) @ V2c ) ) ).
thf(thm_2Eordinal_2EordDIV__UNIQUE,axiom,
! [A_27a: $tType,V0a: tyop_2Eordinal_2Eordinal @ A_27a,V1b: tyop_2Eordinal_2Eordinal @ A_27a,V2q: tyop_2Eordinal_2Eordinal @ A_27a,V3r: tyop_2Eordinal_2Eordinal @ A_27a] :
( ( ( c_2Eordinal_2Eordlt @ A_27a @ ( c_2Eordinal_2EfromNat @ A_27a @ c_2Enum_2E0 ) @ V1b )
& ( V0a
= ( c_2Eordinal_2EordADD @ A_27a @ ( c_2Eordinal_2EordMULT @ A_27a @ V1b @ V2q ) @ V3r ) )
& ( c_2Eordinal_2Eordlt @ A_27a @ V3r @ V1b ) )
=> ( ( c_2Eordinal_2EordDIV @ A_27a @ V0a @ V1b )
= V2q ) ) ).
thf(thm_2Eordinal_2EordMOD__UNIQUE,axiom,
! [A_27a: $tType,V0a: tyop_2Eordinal_2Eordinal @ A_27a,V1b: tyop_2Eordinal_2Eordinal @ A_27a,V2q: tyop_2Eordinal_2Eordinal @ A_27a,V3r: tyop_2Eordinal_2Eordinal @ A_27a] :
( ( ( c_2Eordinal_2Eordlt @ A_27a @ ( c_2Eordinal_2EfromNat @ A_27a @ c_2Enum_2E0 ) @ V1b )
& ( V0a
= ( c_2Eordinal_2EordADD @ A_27a @ ( c_2Eordinal_2EordMULT @ A_27a @ V1b @ V2q ) @ V3r ) )
& ( c_2Eordinal_2Eordlt @ A_27a @ V3r @ V1b ) )
=> ( ( c_2Eordinal_2EordMOD @ A_27a @ V0a @ V1b )
= V3r ) ) ).
thf(thm_2Eordinal_2EordEXP__1R,axiom,
! [A_27a: $tType,V0a: tyop_2Eordinal_2Eordinal @ A_27a] :
( ( c_2Eordinal_2EordEXP @ A_27a @ V0a @ ( c_2Eordinal_2EfromNat @ A_27a @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) )
= V0a ) ).
thf(thm_2Eordinal_2EordEXP__1L,axiom,
! [A_27a: $tType,V0a: tyop_2Eordinal_2Eordinal @ A_27a] :
( ( c_2Eordinal_2EordEXP @ A_27a @ ( c_2Eordinal_2EfromNat @ A_27a @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) @ V0a )
= ( c_2Eordinal_2EfromNat @ A_27a @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ).
thf(thm_2Eordinal_2EordEXP__2R,axiom,
! [A_27a: $tType,V0a: tyop_2Eordinal_2Eordinal @ A_27a] :
( ( c_2Eordinal_2EordEXP @ A_27a @ V0a @ ( c_2Eordinal_2EfromNat @ A_27a @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) )
= ( c_2Eordinal_2EordMULT @ A_27a @ V0a @ V0a ) ) ).
thf(thm_2Eordinal_2EordEXP__fromNat,axiom,
! [A_27a: $tType,V0x: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum] :
( ( c_2Eordinal_2EordEXP @ A_27a @ ( c_2Eordinal_2EfromNat @ A_27a @ V0x ) @ ( c_2Eordinal_2EfromNat @ A_27a @ V1n ) )
= ( c_2Eordinal_2EfromNat @ A_27a @ ( c_2Earithmetic_2EEXP @ V0x @ V1n ) ) ) ).
thf(thm_2Eordinal_2EordEXP__le__MONO__L,axiom,
! [A_27a: $tType,V0x: tyop_2Eordinal_2Eordinal @ A_27a,V1a: tyop_2Eordinal_2Eordinal @ A_27a,V2b: tyop_2Eordinal_2Eordinal @ A_27a] :
( ( (~) @ ( c_2Eordinal_2Eordlt @ A_27a @ V2b @ V1a ) )
=> ( (~) @ ( c_2Eordinal_2Eordlt @ A_27a @ ( c_2Eordinal_2EordEXP @ A_27a @ V2b @ V0x ) @ ( c_2Eordinal_2EordEXP @ A_27a @ V1a @ V0x ) ) ) ) ).
thf(thm_2Eordinal_2EIFF__ZERO__lt,axiom,
! [A_27a: $tType,V0x: tyop_2Eordinal_2Eordinal @ A_27a] :
( ( ( (~)
@ ( V0x
= ( c_2Eordinal_2EfromNat @ A_27a @ c_2Enum_2E0 ) ) )
<=> ( c_2Eordinal_2Eordlt @ A_27a @ ( c_2Eordinal_2EfromNat @ A_27a @ c_2Enum_2E0 ) @ V0x ) )
& ( ( (~) @ ( c_2Eordinal_2Eordlt @ A_27a @ V0x @ ( c_2Eordinal_2EfromNat @ A_27a @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) )
<=> ( c_2Eordinal_2Eordlt @ A_27a @ ( c_2Eordinal_2EfromNat @ A_27a @ c_2Enum_2E0 ) @ V0x ) ) ) ).
thf(thm_2Eordinal_2Eislimit__SUC,axiom,
! [A_27a: $tType,V0x: tyop_2Eordinal_2Eordinal @ A_27a] :
( ( ( c_2Eordinal_2Eomax @ A_27a @ ( c_2Eordinal_2Epreds @ A_27a @ ( c_2Eordinal_2EordSUC @ A_27a @ V0x ) ) )
= ( c_2Eoption_2ENONE @ ( tyop_2Eordinal_2Eordinal @ A_27a ) ) )
<=> c_2Ebool_2EF ) ).
thf(thm_2Eordinal_2Eislimit__fromNat,axiom,
! [A_27a: $tType,V0x: tyop_2Enum_2Enum] :
( ( ( c_2Eordinal_2Eomax @ A_27a @ ( c_2Eordinal_2Epreds @ A_27a @ ( c_2Eordinal_2EfromNat @ A_27a @ V0x ) ) )
= ( c_2Eoption_2ENONE @ ( tyop_2Eordinal_2Eordinal @ A_27a ) ) )
<=> ( V0x = c_2Enum_2E0 ) ) ).
thf(thm_2Eordinal_2EordEXP__ZERO__limit,axiom,
! [A_27a: $tType,V0x: tyop_2Eordinal_2Eordinal @ A_27a] :
( ( ( c_2Eordinal_2Eomax @ A_27a @ ( c_2Eordinal_2Epreds @ A_27a @ V0x ) )
= ( c_2Eoption_2ENONE @ ( tyop_2Eordinal_2Eordinal @ A_27a ) ) )
=> ( ( c_2Eordinal_2EordEXP @ A_27a @ ( c_2Eordinal_2EfromNat @ A_27a @ c_2Enum_2E0 ) @ V0x )
= ( c_2Eordinal_2EfromNat @ A_27a @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ) ).
thf(thm_2Eordinal_2EordEXP__ZERO__nonlimit,axiom,
! [A_27a: $tType,V0x: tyop_2Eordinal_2Eordinal @ A_27a] :
( ( (~)
@ ( ( c_2Eordinal_2Eomax @ A_27a @ ( c_2Eordinal_2Epreds @ A_27a @ V0x ) )
= ( c_2Eoption_2ENONE @ ( tyop_2Eordinal_2Eordinal @ A_27a ) ) ) )
=> ( ( c_2Eordinal_2EordEXP @ A_27a @ ( c_2Eordinal_2EfromNat @ A_27a @ c_2Enum_2E0 ) @ V0x )
= ( c_2Eordinal_2EfromNat @ A_27a @ c_2Enum_2E0 ) ) ) ).
thf(thm_2Eordinal_2Esup__EQ__0,axiom,
! [A_27a: $tType,V0s: ( tyop_2Eordinal_2Eordinal @ A_27a ) > $o] :
( ( c_2Ecardinal_2Ecardleq @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ ( tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ A_27a ) @ V0s @ ( c_2Epred__set_2EUNIV @ ( tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ A_27a ) ) )
=> ( ( ( c_2Eordinal_2Esup @ A_27a @ V0s )
= ( c_2Eordinal_2EfromNat @ A_27a @ c_2Enum_2E0 ) )
<=> ( ( V0s
= ( c_2Epred__set_2EEMPTY @ ( tyop_2Eordinal_2Eordinal @ A_27a ) ) )
| ( V0s
= ( c_2Epred__set_2EINSERT @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ ( c_2Eordinal_2EfromNat @ A_27a @ c_2Enum_2E0 ) @ ( c_2Epred__set_2EEMPTY @ ( tyop_2Eordinal_2Eordinal @ A_27a ) ) ) ) ) ) ) ).
thf(thm_2Eordinal_2EordADD__EQ__0,axiom,
! [A_27a: $tType,V0y: tyop_2Eordinal_2Eordinal @ A_27a,V1x: tyop_2Eordinal_2Eordinal @ A_27a] :
( ( ( c_2Eordinal_2EordADD @ A_27a @ V1x @ V0y )
= ( c_2Eordinal_2EfromNat @ A_27a @ c_2Enum_2E0 ) )
<=> ( ( V1x
= ( c_2Eordinal_2EfromNat @ A_27a @ c_2Enum_2E0 ) )
& ( V0y
= ( c_2Eordinal_2EfromNat @ A_27a @ c_2Enum_2E0 ) ) ) ) ).
thf(thm_2Eordinal_2EIMAGE__EQ__SING,axiom,
! [A_27a: $tType,A_27b: $tType,V0x: A_27a,V1s: A_27b > $o,V2f: A_27b > A_27a] :
( ( ( c_2Epred__set_2EIMAGE @ A_27b @ A_27a @ V2f @ V1s )
= ( c_2Epred__set_2EINSERT @ A_27a @ V0x @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) )
<=> ( ? [V3y: A_27b] : ( c_2Ebool_2EIN @ A_27b @ V3y @ V1s )
& ! [V4y: A_27b] :
( ( c_2Ebool_2EIN @ A_27b @ V4y @ V1s )
=> ( ( V2f @ V4y )
= V0x ) ) ) ) ).
thf(thm_2Eordinal_2EordMULT__EQ__0,axiom,
! [A_27a: $tType,V0x: tyop_2Eordinal_2Eordinal @ A_27a,V1y: tyop_2Eordinal_2Eordinal @ A_27a] :
( ( ( c_2Eordinal_2EordMULT @ A_27a @ V0x @ V1y )
= ( c_2Eordinal_2EfromNat @ A_27a @ c_2Enum_2E0 ) )
<=> ( ( V0x
= ( c_2Eordinal_2EfromNat @ A_27a @ c_2Enum_2E0 ) )
| ( V1y
= ( c_2Eordinal_2EfromNat @ A_27a @ c_2Enum_2E0 ) ) ) ) ).
thf(thm_2Eordinal_2EordEXP__EQ__0,axiom,
! [A_27a: $tType,V0y: tyop_2Eordinal_2Eordinal @ A_27a,V1x: tyop_2Eordinal_2Eordinal @ A_27a] :
( ( ( c_2Eordinal_2EordEXP @ A_27a @ V1x @ V0y )
= ( c_2Eordinal_2EfromNat @ A_27a @ c_2Enum_2E0 ) )
<=> ( ( V1x
= ( c_2Eordinal_2EfromNat @ A_27a @ c_2Enum_2E0 ) )
& ( (~)
@ ( ( c_2Eordinal_2Eomax @ A_27a @ ( c_2Eordinal_2Epreds @ A_27a @ V0y ) )
= ( c_2Eoption_2ENONE @ ( tyop_2Eordinal_2Eordinal @ A_27a ) ) ) ) ) ) ).
thf(thm_2Eordinal_2EZERO__lt__ordEXP__I,axiom,
! [A_27a: $tType,V0a: tyop_2Eordinal_2Eordinal @ A_27a,V1x: tyop_2Eordinal_2Eordinal @ A_27a] :
( ( c_2Eordinal_2Eordlt @ A_27a @ ( c_2Eordinal_2EfromNat @ A_27a @ c_2Enum_2E0 ) @ V0a )
=> ( c_2Eordinal_2Eordlt @ A_27a @ ( c_2Eordinal_2EfromNat @ A_27a @ c_2Enum_2E0 ) @ ( c_2Eordinal_2EordEXP @ A_27a @ V0a @ V1x ) ) ) ).
thf(thm_2Eordinal_2EZERO__lt__ordEXP,axiom,
! [A_27a: $tType,V0x: tyop_2Eordinal_2Eordinal @ A_27a,V1a: tyop_2Eordinal_2Eordinal @ A_27a] :
( ( c_2Eordinal_2Eordlt @ A_27a @ ( c_2Eordinal_2EfromNat @ A_27a @ c_2Enum_2E0 ) @ ( c_2Eordinal_2EordEXP @ A_27a @ V1a @ V0x ) )
<=> ( ( c_2Eordinal_2Eordlt @ A_27a @ ( c_2Eordinal_2EfromNat @ A_27a @ c_2Enum_2E0 ) @ V1a )
| ( ( c_2Eordinal_2Eomax @ A_27a @ ( c_2Eordinal_2Epreds @ A_27a @ V0x ) )
= ( c_2Eoption_2ENONE @ ( tyop_2Eordinal_2Eordinal @ A_27a ) ) ) ) ) ).
thf(thm_2Eordinal_2EordEXP__lt__MONO__R,axiom,
! [A_27a: $tType,V0y: tyop_2Eordinal_2Eordinal @ A_27a,V1x: tyop_2Eordinal_2Eordinal @ A_27a,V2a: tyop_2Eordinal_2Eordinal @ A_27a] :
( ( ( c_2Eordinal_2Eordlt @ A_27a @ ( c_2Eordinal_2EfromNat @ A_27a @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) @ V2a )
& ( c_2Eordinal_2Eordlt @ A_27a @ V1x @ V0y ) )
=> ( c_2Eordinal_2Eordlt @ A_27a @ ( c_2Eordinal_2EordEXP @ A_27a @ V2a @ V1x ) @ ( c_2Eordinal_2EordEXP @ A_27a @ V2a @ V0y ) ) ) ).
thf(thm_2Eordinal_2EordEXP__lt__IFF,axiom,
! [A_27a: $tType,V0x: tyop_2Eordinal_2Eordinal @ A_27a,V1y: tyop_2Eordinal_2Eordinal @ A_27a,V2a: tyop_2Eordinal_2Eordinal @ A_27a] :
( ( c_2Eordinal_2Eordlt @ A_27a @ ( c_2Eordinal_2EfromNat @ A_27a @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) @ V2a )
=> ( ( c_2Eordinal_2Eordlt @ A_27a @ ( c_2Eordinal_2EordEXP @ A_27a @ V2a @ V0x ) @ ( c_2Eordinal_2EordEXP @ A_27a @ V2a @ V1y ) )
= ( c_2Eordinal_2Eordlt @ A_27a @ V0x @ V1y ) ) ) ).
thf(thm_2Eordinal_2EordEXP__le__MONO__R,axiom,
! [A_27a: $tType,V0x: tyop_2Eordinal_2Eordinal @ A_27a,V1y: tyop_2Eordinal_2Eordinal @ A_27a,V2a: tyop_2Eordinal_2Eordinal @ A_27a] :
( ( ( c_2Eordinal_2Eordlt @ A_27a @ ( c_2Eordinal_2EfromNat @ A_27a @ c_2Enum_2E0 ) @ V2a )
& ( (~) @ ( c_2Eordinal_2Eordlt @ A_27a @ V1y @ V0x ) ) )
=> ( (~) @ ( c_2Eordinal_2Eordlt @ A_27a @ ( c_2Eordinal_2EordEXP @ A_27a @ V2a @ V1y ) @ ( c_2Eordinal_2EordEXP @ A_27a @ V2a @ V0x ) ) ) ) ).
thf(thm_2Eordinal_2EordEXP__continuous,axiom,
! [A_27a: $tType,V0a: tyop_2Eordinal_2Eordinal @ A_27a,V1s: ( tyop_2Eordinal_2Eordinal @ A_27a ) > $o] :
( ( ( c_2Eordinal_2Eordlt @ A_27a @ ( c_2Eordinal_2EfromNat @ A_27a @ c_2Enum_2E0 ) @ V0a )
& ( c_2Ecardinal_2Ecardleq @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ ( tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ A_27a ) @ V1s @ ( c_2Epred__set_2EUNIV @ ( tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ A_27a ) ) )
& ( (~)
@ ( V1s
= ( c_2Epred__set_2EEMPTY @ ( tyop_2Eordinal_2Eordinal @ A_27a ) ) ) ) )
=> ( ( c_2Eordinal_2EordEXP @ A_27a @ V0a @ ( c_2Eordinal_2Esup @ A_27a @ V1s ) )
= ( c_2Eordinal_2Esup @ A_27a @ ( c_2Epred__set_2EIMAGE @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ ( c_2Eordinal_2EordEXP @ A_27a @ V0a ) @ V1s ) ) ) ) ).
thf(thm_2Eordinal_2EordEXP__ADD,axiom,
! [A_27a: $tType,V0z: tyop_2Eordinal_2Eordinal @ A_27a,V1y: tyop_2Eordinal_2Eordinal @ A_27a,V2x: tyop_2Eordinal_2Eordinal @ A_27a] :
( ( c_2Eordinal_2Eordlt @ A_27a @ ( c_2Eordinal_2EfromNat @ A_27a @ c_2Enum_2E0 ) @ V2x )
=> ( ( c_2Eordinal_2EordEXP @ A_27a @ V2x @ ( c_2Eordinal_2EordADD @ A_27a @ V1y @ V0z ) )
= ( c_2Eordinal_2EordMULT @ A_27a @ ( c_2Eordinal_2EordEXP @ A_27a @ V2x @ V1y ) @ ( c_2Eordinal_2EordEXP @ A_27a @ V2x @ V0z ) ) ) ) ).
thf(thm_2Eordinal_2EordEXP__MUL,axiom,
! [A_27a: $tType,V0z: tyop_2Eordinal_2Eordinal @ A_27a,V1y: tyop_2Eordinal_2Eordinal @ A_27a,V2x: tyop_2Eordinal_2Eordinal @ A_27a] :
( ( c_2Eordinal_2Eordlt @ A_27a @ ( c_2Eordinal_2EfromNat @ A_27a @ c_2Enum_2E0 ) @ V2x )
=> ( ( c_2Eordinal_2EordEXP @ A_27a @ V2x @ ( c_2Eordinal_2EordMULT @ A_27a @ V1y @ V0z ) )
= ( c_2Eordinal_2EordEXP @ A_27a @ ( c_2Eordinal_2EordEXP @ A_27a @ V2x @ V1y ) @ V0z ) ) ) ).
thf(thm_2Eordinal_2Efixpoints__exist,axiom,
! [A_27a: $tType,V0f: ( tyop_2Eordinal_2Eordinal @ A_27a ) > ( tyop_2Eordinal_2Eordinal @ A_27a )] :
( ( ! [V1s: ( tyop_2Eordinal_2Eordinal @ A_27a ) > $o] :
( ( ( (~)
@ ( V1s
= ( c_2Epred__set_2EEMPTY @ ( tyop_2Eordinal_2Eordinal @ A_27a ) ) ) )
& ( c_2Ecardinal_2Ecardleq @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ ( tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ A_27a ) @ V1s @ ( c_2Epred__set_2EUNIV @ ( tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ A_27a ) ) ) )
=> ( ( V0f @ ( c_2Eordinal_2Esup @ A_27a @ V1s ) )
= ( c_2Eordinal_2Esup @ A_27a @ ( c_2Epred__set_2EIMAGE @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ V0f @ V1s ) ) ) )
& ! [V2x: tyop_2Eordinal_2Eordinal @ A_27a] : ( (~) @ ( c_2Eordinal_2Eordlt @ A_27a @ ( V0f @ V2x ) @ V2x ) ) )
=> ! [V3a: tyop_2Eordinal_2Eordinal @ A_27a] :
? [V4b: tyop_2Eordinal_2Eordinal @ A_27a] :
( ( (~) @ ( c_2Eordinal_2Eordlt @ A_27a @ V4b @ V3a ) )
& ( ( V0f @ V4b )
= V4b ) ) ) ).
thf(thm_2Eordinal_2Ex__le__ordEXP__x,axiom,
! [A_27a: $tType,V0a: tyop_2Eordinal_2Eordinal @ A_27a,V1x: tyop_2Eordinal_2Eordinal @ A_27a] :
( ( c_2Eordinal_2Eordlt @ A_27a @ ( c_2Eordinal_2EfromNat @ A_27a @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) @ V0a )
=> ( (~) @ ( c_2Eordinal_2Eordlt @ A_27a @ ( c_2Eordinal_2EordEXP @ A_27a @ V0a @ V1x ) @ V1x ) ) ) ).
thf(thm_2Eordinal_2Eepsilon0__fixpoint,axiom,
! [A_27a: $tType] :
( ( c_2Eordinal_2EordEXP @ A_27a @ ( c_2Eordinal_2Eomega @ A_27a ) @ ( c_2Eordinal_2Eepsilon0 @ A_27a ) )
= ( c_2Eordinal_2Eepsilon0 @ A_27a ) ) ).
thf(thm_2Eordinal_2Eepsilon0__least__fixpoint,axiom,
! [A_27a: $tType,V0a: tyop_2Eordinal_2Eordinal @ A_27a] :
( ( c_2Eordinal_2Eordlt @ A_27a @ V0a @ ( c_2Eordinal_2Eepsilon0 @ A_27a ) )
=> ( ( c_2Eordinal_2Eordlt @ A_27a @ V0a @ ( c_2Eordinal_2EordEXP @ A_27a @ ( c_2Eordinal_2Eomega @ A_27a ) @ V0a ) )
& ( c_2Eordinal_2Eordlt @ A_27a @ ( c_2Eordinal_2EordEXP @ A_27a @ ( c_2Eordinal_2Eomega @ A_27a ) @ V0a ) @ ( c_2Eordinal_2Eepsilon0 @ A_27a ) ) ) ) ).
thf(thm_2Eordinal_2Eomega__lt__epsilon0,axiom,
! [A_27a: $tType] : ( c_2Eordinal_2Eordlt @ A_27a @ ( c_2Eordinal_2Eomega @ A_27a ) @ ( c_2Eordinal_2Eepsilon0 @ A_27a ) ) ).
thf(thm_2Eordinal_2EfromNat__lt__epsilon0,axiom,
! [A_27a: $tType,V0n: tyop_2Enum_2Enum] : ( c_2Eordinal_2Eordlt @ A_27a @ ( c_2Eordinal_2EfromNat @ A_27a @ V0n ) @ ( c_2Eordinal_2Eepsilon0 @ A_27a ) ) ).
thf(thm_2Eordinal_2Eadd__nat__islimit,axiom,
! [A_27a: $tType,V0n: tyop_2Enum_2Enum,V1a: tyop_2Eordinal_2Eordinal @ A_27a] :
( ( c_2Eprim__rec_2E_3C @ c_2Enum_2E0 @ V0n )
=> ( ( ( c_2Eordinal_2Eomax @ A_27a @ ( c_2Eordinal_2Epreds @ A_27a @ ( c_2Eordinal_2EordADD @ A_27a @ V1a @ ( c_2Eordinal_2EfromNat @ A_27a @ V0n ) ) ) )
= ( c_2Eoption_2ENONE @ ( tyop_2Eordinal_2Eordinal @ A_27a ) ) )
<=> c_2Ebool_2EF ) ) ).
thf(thm_2Eordinal_2Estrict__continuity__preserves__islimit,axiom,
! [A_27a: $tType,V0f: ( tyop_2Eordinal_2Eordinal @ A_27a ) > ( tyop_2Eordinal_2Eordinal @ A_27a ),V1a: tyop_2Eordinal_2Eordinal @ A_27a] :
( ( ! [V2s: ( tyop_2Eordinal_2Eordinal @ A_27a ) > $o] :
( ( ( c_2Ecardinal_2Ecardleq @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ ( tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ A_27a ) @ V2s @ ( c_2Epred__set_2EUNIV @ ( tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ A_27a ) ) )
& ( (~)
@ ( V2s
= ( c_2Epred__set_2EEMPTY @ ( tyop_2Eordinal_2Eordinal @ A_27a ) ) ) ) )
=> ( ( V0f @ ( c_2Eordinal_2Esup @ A_27a @ V2s ) )
= ( c_2Eordinal_2Esup @ A_27a @ ( c_2Epred__set_2EIMAGE @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ V0f @ V2s ) ) ) )
& ! [V3x: tyop_2Eordinal_2Eordinal @ A_27a,V4y: tyop_2Eordinal_2Eordinal @ A_27a] :
( ( c_2Eordinal_2Eordlt @ A_27a @ V3x @ V4y )
=> ( c_2Eordinal_2Eordlt @ A_27a @ ( V0f @ V3x ) @ ( V0f @ V4y ) ) )
& ( ( c_2Eordinal_2Eomax @ A_27a @ ( c_2Eordinal_2Epreds @ A_27a @ V1a ) )
= ( c_2Eoption_2ENONE @ ( tyop_2Eordinal_2Eordinal @ A_27a ) ) )
& ( (~)
@ ( V1a
= ( c_2Eordinal_2EfromNat @ A_27a @ c_2Enum_2E0 ) ) ) )
=> ( ( c_2Eordinal_2Eomax @ A_27a @ ( c_2Eordinal_2Epreds @ A_27a @ ( V0f @ V1a ) ) )
= ( c_2Eoption_2ENONE @ ( tyop_2Eordinal_2Eordinal @ A_27a ) ) ) ) ).
thf(thm_2Eordinal_2Eadd__omega__islimit,axiom,
! [A_27a: $tType,V0a: tyop_2Eordinal_2Eordinal @ A_27a] :
( ( c_2Eordinal_2Eomax @ A_27a @ ( c_2Eordinal_2Epreds @ A_27a @ ( c_2Eordinal_2EordADD @ A_27a @ V0a @ ( c_2Eordinal_2Eomega @ A_27a ) ) ) )
= ( c_2Eoption_2ENONE @ ( tyop_2Eordinal_2Eordinal @ A_27a ) ) ) ).
thf(thm_2Eordinal_2Eislimit__mul__R,axiom,
! [A_27a: $tType,V0b: tyop_2Eordinal_2Eordinal @ A_27a,V1a: tyop_2Eordinal_2Eordinal @ A_27a] :
( ( ( c_2Eordinal_2Eomax @ A_27a @ ( c_2Eordinal_2Epreds @ A_27a @ V1a ) )
= ( c_2Eoption_2ENONE @ ( tyop_2Eordinal_2Eordinal @ A_27a ) ) )
=> ( ( c_2Eordinal_2Eomax @ A_27a @ ( c_2Eordinal_2Epreds @ A_27a @ ( c_2Eordinal_2EordMULT @ A_27a @ V0b @ V1a ) ) )
= ( c_2Eoption_2ENONE @ ( tyop_2Eordinal_2Eordinal @ A_27a ) ) ) ) ).
thf(thm_2Eordinal_2Emul__omega__islimit,axiom,
! [A_27a: $tType,V0a: tyop_2Eordinal_2Eordinal @ A_27a] :
( ( c_2Eordinal_2Eomax @ A_27a @ ( c_2Eordinal_2Epreds @ A_27a @ ( c_2Eordinal_2EordMULT @ A_27a @ ( c_2Eordinal_2Eomega @ A_27a ) @ V0a ) ) )
= ( c_2Eoption_2ENONE @ ( tyop_2Eordinal_2Eordinal @ A_27a ) ) ) ).
thf(thm_2Eordinal_2Eomega__exp__islimit,axiom,
! [A_27a: $tType,V0a: tyop_2Eordinal_2Eordinal @ A_27a] :
( ( c_2Eordinal_2Eordlt @ A_27a @ ( c_2Eordinal_2EfromNat @ A_27a @ c_2Enum_2E0 ) @ V0a )
=> ( ( c_2Eordinal_2Eomax @ A_27a @ ( c_2Eordinal_2Epreds @ A_27a @ ( c_2Eordinal_2EordEXP @ A_27a @ ( c_2Eordinal_2Eomega @ A_27a ) @ V0a ) ) )
= ( c_2Eoption_2ENONE @ ( tyop_2Eordinal_2Eordinal @ A_27a ) ) ) ) ).
thf(thm_2Eordinal_2Eexpbound__add,axiom,
! [A_27a: $tType,V0a: tyop_2Eordinal_2Eordinal @ A_27a,V1x: tyop_2Eordinal_2Eordinal @ A_27a,V2y: tyop_2Eordinal_2Eordinal @ A_27a] :
( ( ( c_2Eordinal_2Eordlt @ A_27a @ V1x @ ( c_2Eordinal_2EordEXP @ A_27a @ ( c_2Eordinal_2Eomega @ A_27a ) @ V0a ) )
& ( c_2Eordinal_2Eordlt @ A_27a @ V2y @ ( c_2Eordinal_2EordEXP @ A_27a @ ( c_2Eordinal_2Eomega @ A_27a ) @ V0a ) ) )
=> ( c_2Eordinal_2Eordlt @ A_27a @ ( c_2Eordinal_2EordADD @ A_27a @ V1x @ V2y ) @ ( c_2Eordinal_2EordEXP @ A_27a @ ( c_2Eordinal_2Eomega @ A_27a ) @ V0a ) ) ) ).
thf(thm_2Eordinal_2EaddL__fixpoint__iff,axiom,
! [A_27a: $tType,V0b: tyop_2Eordinal_2Eordinal @ A_27a,V1a: tyop_2Eordinal_2Eordinal @ A_27a] :
( ( ( c_2Eordinal_2EordADD @ A_27a @ V1a @ V0b )
= V0b )
<=> ( (~) @ ( c_2Eordinal_2Eordlt @ A_27a @ V0b @ ( c_2Eordinal_2EordMULT @ A_27a @ V1a @ ( c_2Eordinal_2Eomega @ A_27a ) ) ) ) ) ).
thf(thm_2Eordinal_2EordADD__under__epsilon0,axiom,
! [A_27a: $tType,V0y: tyop_2Eordinal_2Eordinal @ A_27a,V1x: tyop_2Eordinal_2Eordinal @ A_27a] :
( ( ( c_2Eordinal_2Eordlt @ A_27a @ V1x @ ( c_2Eordinal_2Eepsilon0 @ A_27a ) )
& ( c_2Eordinal_2Eordlt @ A_27a @ V0y @ ( c_2Eordinal_2Eepsilon0 @ A_27a ) ) )
=> ( c_2Eordinal_2Eordlt @ A_27a @ ( c_2Eordinal_2EordADD @ A_27a @ V1x @ V0y ) @ ( c_2Eordinal_2Eepsilon0 @ A_27a ) ) ) ).
thf(thm_2Eordinal_2EordMUL__under__epsilon0,axiom,
! [A_27a: $tType,V0y: tyop_2Eordinal_2Eordinal @ A_27a,V1x: tyop_2Eordinal_2Eordinal @ A_27a] :
( ( ( c_2Eordinal_2Eordlt @ A_27a @ V1x @ ( c_2Eordinal_2Eepsilon0 @ A_27a ) )
& ( c_2Eordinal_2Eordlt @ A_27a @ V0y @ ( c_2Eordinal_2Eepsilon0 @ A_27a ) ) )
=> ( c_2Eordinal_2Eordlt @ A_27a @ ( c_2Eordinal_2EordMULT @ A_27a @ V1x @ V0y ) @ ( c_2Eordinal_2Eepsilon0 @ A_27a ) ) ) ).
thf(thm_2Eordinal_2EordEXP__under__epsilon0,axiom,
! [A_27a: $tType,V0b: tyop_2Eordinal_2Eordinal @ A_27a,V1a: tyop_2Eordinal_2Eordinal @ A_27a] :
( ( ( c_2Eordinal_2Eordlt @ A_27a @ V1a @ ( c_2Eordinal_2Eepsilon0 @ A_27a ) )
& ( c_2Eordinal_2Eordlt @ A_27a @ V0b @ ( c_2Eordinal_2Eepsilon0 @ A_27a ) ) )
=> ( c_2Eordinal_2Eordlt @ A_27a @ ( c_2Eordinal_2EordEXP @ A_27a @ V1a @ V0b ) @ ( c_2Eordinal_2Eepsilon0 @ A_27a ) ) ) ).
thf(thm_2Eordinal_2Eeval__poly__ind,axiom,
! [A_27a: $tType,V0P: ( tyop_2Eordinal_2Eordinal @ A_27a ) > ( tyop_2Elist_2Elist @ ( tyop_2Epair_2Eprod @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ ( tyop_2Eordinal_2Eordinal @ A_27a ) ) ) > $o] :
( ( ! [V1a: tyop_2Eordinal_2Eordinal @ A_27a] : ( V0P @ V1a @ ( c_2Elist_2ENIL @ ( tyop_2Epair_2Eprod @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ ( tyop_2Eordinal_2Eordinal @ A_27a ) ) ) )
& ! [V2a: tyop_2Eordinal_2Eordinal @ A_27a,V3c: tyop_2Eordinal_2Eordinal @ A_27a,V4e: tyop_2Eordinal_2Eordinal @ A_27a,V5t: tyop_2Elist_2Elist @ ( tyop_2Epair_2Eprod @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ ( tyop_2Eordinal_2Eordinal @ A_27a ) )] :
( ( V0P @ V2a @ V5t )
=> ( V0P @ V2a @ ( c_2Elist_2ECONS @ ( tyop_2Epair_2Eprod @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ ( tyop_2Eordinal_2Eordinal @ A_27a ) ) @ ( c_2Epair_2E_2C @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ V3c @ V4e ) @ V5t ) ) ) )
=> ! [V6v: tyop_2Eordinal_2Eordinal @ A_27a,V7v1: tyop_2Elist_2Elist @ ( tyop_2Epair_2Eprod @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ ( tyop_2Eordinal_2Eordinal @ A_27a ) )] : ( V0P @ V6v @ V7v1 ) ) ).
thf(thm_2Eordinal_2Eeval__poly__def,axiom,
! [A_27a: $tType] :
( ! [V0a: tyop_2Eordinal_2Eordinal @ A_27a] :
( ( c_2Eordinal_2Eeval__poly @ A_27a @ V0a @ ( c_2Elist_2ENIL @ ( tyop_2Epair_2Eprod @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ ( tyop_2Eordinal_2Eordinal @ A_27a ) ) ) )
= ( c_2Eordinal_2EfromNat @ A_27a @ c_2Enum_2E0 ) )
& ! [V1t: tyop_2Elist_2Elist @ ( tyop_2Epair_2Eprod @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ ( tyop_2Eordinal_2Eordinal @ A_27a ) ),V2e: tyop_2Eordinal_2Eordinal @ A_27a,V3c: tyop_2Eordinal_2Eordinal @ A_27a,V4a: tyop_2Eordinal_2Eordinal @ A_27a] :
( ( c_2Eordinal_2Eeval__poly @ A_27a @ V4a @ ( c_2Elist_2ECONS @ ( tyop_2Epair_2Eprod @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ ( tyop_2Eordinal_2Eordinal @ A_27a ) ) @ ( c_2Epair_2E_2C @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ V3c @ V2e ) @ V1t ) )
= ( c_2Eordinal_2EordADD @ A_27a @ ( c_2Eordinal_2EordMULT @ A_27a @ ( c_2Eordinal_2EordEXP @ A_27a @ V4a @ V2e ) @ V3c ) @ ( c_2Eordinal_2Eeval__poly @ A_27a @ V4a @ V1t ) ) ) ) ).
thf(thm_2Eordinal_2Eis__polyform__ind,axiom,
! [A_27a: $tType,A_27b: $tType,V0P: ( tyop_2Eordinal_2Eordinal @ A_27a ) > ( tyop_2Elist_2Elist @ ( tyop_2Epair_2Eprod @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ ( tyop_2Eordinal_2Eordinal @ A_27b ) ) ) > $o] :
( ( ! [V1a: tyop_2Eordinal_2Eordinal @ A_27a] : ( V0P @ V1a @ ( c_2Elist_2ENIL @ ( tyop_2Epair_2Eprod @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ ( tyop_2Eordinal_2Eordinal @ A_27b ) ) ) )
& ! [V2a: tyop_2Eordinal_2Eordinal @ A_27a,V3c: tyop_2Eordinal_2Eordinal @ A_27a,V4e: tyop_2Eordinal_2Eordinal @ A_27b] : ( V0P @ V2a @ ( c_2Elist_2ECONS @ ( tyop_2Epair_2Eprod @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ ( tyop_2Eordinal_2Eordinal @ A_27b ) ) @ ( c_2Epair_2E_2C @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ ( tyop_2Eordinal_2Eordinal @ A_27b ) @ V3c @ V4e ) @ ( c_2Elist_2ENIL @ ( tyop_2Epair_2Eprod @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ ( tyop_2Eordinal_2Eordinal @ A_27b ) ) ) ) )
& ! [V5a: tyop_2Eordinal_2Eordinal @ A_27a,V6c1: tyop_2Eordinal_2Eordinal @ A_27a,V7e1: tyop_2Eordinal_2Eordinal @ A_27b,V8c2: tyop_2Eordinal_2Eordinal @ A_27a,V9e2: tyop_2Eordinal_2Eordinal @ A_27b,V10t: tyop_2Elist_2Elist @ ( tyop_2Epair_2Eprod @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ ( tyop_2Eordinal_2Eordinal @ A_27b ) )] :
( ( V0P @ V5a @ ( c_2Elist_2ECONS @ ( tyop_2Epair_2Eprod @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ ( tyop_2Eordinal_2Eordinal @ A_27b ) ) @ ( c_2Epair_2E_2C @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ ( tyop_2Eordinal_2Eordinal @ A_27b ) @ V8c2 @ V9e2 ) @ V10t ) )
=> ( V0P @ V5a @ ( c_2Elist_2ECONS @ ( tyop_2Epair_2Eprod @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ ( tyop_2Eordinal_2Eordinal @ A_27b ) ) @ ( c_2Epair_2E_2C @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ ( tyop_2Eordinal_2Eordinal @ A_27b ) @ V6c1 @ V7e1 ) @ ( c_2Elist_2ECONS @ ( tyop_2Epair_2Eprod @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ ( tyop_2Eordinal_2Eordinal @ A_27b ) ) @ ( c_2Epair_2E_2C @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ ( tyop_2Eordinal_2Eordinal @ A_27b ) @ V8c2 @ V9e2 ) @ V10t ) ) ) ) )
=> ! [V11v: tyop_2Eordinal_2Eordinal @ A_27a,V12v1: tyop_2Elist_2Elist @ ( tyop_2Epair_2Eprod @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ ( tyop_2Eordinal_2Eordinal @ A_27b ) )] : ( V0P @ V11v @ V12v1 ) ) ).
thf(thm_2Eordinal_2Eis__polyform__def,axiom,
! [A_27a: $tType,A_27b: $tType] :
( ! [V0a: tyop_2Eordinal_2Eordinal @ A_27a] :
( ( c_2Eordinal_2Eis__polyform @ A_27a @ A_27b @ V0a @ ( c_2Elist_2ENIL @ ( tyop_2Epair_2Eprod @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ ( tyop_2Eordinal_2Eordinal @ A_27b ) ) ) )
= c_2Ebool_2ET )
& ! [V1e: tyop_2Eordinal_2Eordinal @ A_27b,V2c: tyop_2Eordinal_2Eordinal @ A_27a,V3a: tyop_2Eordinal_2Eordinal @ A_27a] :
( ( c_2Eordinal_2Eis__polyform @ A_27a @ A_27b @ V3a @ ( c_2Elist_2ECONS @ ( tyop_2Epair_2Eprod @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ ( tyop_2Eordinal_2Eordinal @ A_27b ) ) @ ( c_2Epair_2E_2C @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ ( tyop_2Eordinal_2Eordinal @ A_27b ) @ V2c @ V1e ) @ ( c_2Elist_2ENIL @ ( tyop_2Epair_2Eprod @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ ( tyop_2Eordinal_2Eordinal @ A_27b ) ) ) ) )
<=> ( ( c_2Eordinal_2Eordlt @ A_27a @ ( c_2Eordinal_2EfromNat @ A_27a @ c_2Enum_2E0 ) @ V2c )
& ( c_2Eordinal_2Eordlt @ A_27a @ V2c @ V3a ) ) )
& ! [V4t: tyop_2Elist_2Elist @ ( tyop_2Epair_2Eprod @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ ( tyop_2Eordinal_2Eordinal @ A_27b ) ),V5e2: tyop_2Eordinal_2Eordinal @ A_27b,V6e1: tyop_2Eordinal_2Eordinal @ A_27b,V7c2: tyop_2Eordinal_2Eordinal @ A_27a,V8c1: tyop_2Eordinal_2Eordinal @ A_27a,V9a: tyop_2Eordinal_2Eordinal @ A_27a] :
( ( c_2Eordinal_2Eis__polyform @ A_27a @ A_27b @ V9a @ ( c_2Elist_2ECONS @ ( tyop_2Epair_2Eprod @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ ( tyop_2Eordinal_2Eordinal @ A_27b ) ) @ ( c_2Epair_2E_2C @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ ( tyop_2Eordinal_2Eordinal @ A_27b ) @ V8c1 @ V6e1 ) @ ( c_2Elist_2ECONS @ ( tyop_2Epair_2Eprod @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ ( tyop_2Eordinal_2Eordinal @ A_27b ) ) @ ( c_2Epair_2E_2C @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ ( tyop_2Eordinal_2Eordinal @ A_27b ) @ V7c2 @ V5e2 ) @ V4t ) ) )
<=> ( ( c_2Eordinal_2Eordlt @ A_27a @ ( c_2Eordinal_2EfromNat @ A_27a @ c_2Enum_2E0 ) @ V8c1 )
& ( c_2Eordinal_2Eordlt @ A_27a @ V8c1 @ V9a )
& ( c_2Eordinal_2Eordlt @ A_27b @ V5e2 @ V6e1 )
& ( c_2Eordinal_2Eis__polyform @ A_27a @ A_27b @ V9a @ ( c_2Elist_2ECONS @ ( tyop_2Epair_2Eprod @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ ( tyop_2Eordinal_2Eordinal @ A_27b ) ) @ ( c_2Epair_2E_2C @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ ( tyop_2Eordinal_2Eordinal @ A_27b ) @ V7c2 @ V5e2 ) @ V4t ) ) ) ) ) ).
thf(thm_2Eordinal_2Eis__polyform__ELthm,axiom,
! [A_27a: $tType,A_27b: $tType,V0ces: tyop_2Elist_2Elist @ ( tyop_2Epair_2Eprod @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ ( tyop_2Eordinal_2Eordinal @ A_27b ) ),V1a: tyop_2Eordinal_2Eordinal @ A_27a] :
( ( c_2Eordinal_2Eis__polyform @ A_27a @ A_27b @ V1a @ V0ces )
<=> ( ! [V2i: tyop_2Enum_2Enum,V3j: tyop_2Enum_2Enum] :
( ( ( c_2Eprim__rec_2E_3C @ V2i @ V3j )
& ( c_2Eprim__rec_2E_3C @ V3j @ ( c_2Elist_2ELENGTH @ ( tyop_2Epair_2Eprod @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ ( tyop_2Eordinal_2Eordinal @ A_27b ) ) @ V0ces ) ) )
=> ( c_2Eordinal_2Eordlt @ A_27b @ ( c_2Epair_2ESND @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ ( tyop_2Eordinal_2Eordinal @ A_27b ) @ ( c_2Elist_2EEL @ ( tyop_2Epair_2Eprod @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ ( tyop_2Eordinal_2Eordinal @ A_27b ) ) @ V3j @ V0ces ) ) @ ( c_2Epair_2ESND @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ ( tyop_2Eordinal_2Eordinal @ A_27b ) @ ( c_2Elist_2EEL @ ( tyop_2Epair_2Eprod @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ ( tyop_2Eordinal_2Eordinal @ A_27b ) ) @ V2i @ V0ces ) ) ) )
& ! [V4c: tyop_2Eordinal_2Eordinal @ A_27a,V5e: tyop_2Eordinal_2Eordinal @ A_27b] :
( ( c_2Ebool_2EIN @ ( tyop_2Epair_2Eprod @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ ( tyop_2Eordinal_2Eordinal @ A_27b ) ) @ ( c_2Epair_2E_2C @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ ( tyop_2Eordinal_2Eordinal @ A_27b ) @ V4c @ V5e ) @ ( c_2Elist_2ELIST__TO__SET @ ( tyop_2Epair_2Eprod @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ ( tyop_2Eordinal_2Eordinal @ A_27b ) ) @ V0ces ) )
=> ( ( c_2Eordinal_2Eordlt @ A_27a @ ( c_2Eordinal_2EfromNat @ A_27a @ c_2Enum_2E0 ) @ V4c )
& ( c_2Eordinal_2Eordlt @ A_27a @ V4c @ V1a ) ) ) ) ) ).
thf(thm_2Eordinal_2Epolyform__exists,axiom,
! [A_27a: $tType,V0a: tyop_2Eordinal_2Eordinal @ A_27a,V1b: tyop_2Eordinal_2Eordinal @ A_27a] :
( ( c_2Eordinal_2Eordlt @ A_27a @ ( c_2Eordinal_2EfromNat @ A_27a @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) @ V0a )
=> ? [V2ces: tyop_2Elist_2Elist @ ( tyop_2Epair_2Eprod @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ ( tyop_2Eordinal_2Eordinal @ A_27a ) )] :
( ( c_2Eordinal_2Eis__polyform @ A_27a @ A_27a @ V0a @ V2ces )
& ( V1b
= ( c_2Eordinal_2Eeval__poly @ A_27a @ V0a @ V2ces ) ) ) ) ).
thf(thm_2Eordinal_2ECNF__thm,axiom,
! [A_27a: $tType,V0b: tyop_2Eordinal_2Eordinal @ A_27a] :
( ( c_2Eordinal_2Eis__polyform @ A_27a @ A_27a @ ( c_2Eordinal_2Eomega @ A_27a ) @ ( c_2Eordinal_2Epolyform @ A_27a @ ( c_2Eordinal_2Eomega @ A_27a ) @ V0b ) )
& ( V0b
= ( c_2Eordinal_2Eeval__poly @ A_27a @ ( c_2Eordinal_2Eomega @ A_27a ) @ ( c_2Eordinal_2Epolyform @ A_27a @ ( c_2Eordinal_2Eomega @ A_27a ) @ V0b ) ) ) ) ).
thf(thm_2Eordinal_2Epolyform__0,axiom,
! [A_27a: $tType,V0a: tyop_2Eordinal_2Eordinal @ A_27a] :
( ( c_2Eordinal_2Eordlt @ A_27a @ ( c_2Eordinal_2EfromNat @ A_27a @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) @ V0a )
=> ( ( c_2Eordinal_2Epolyform @ A_27a @ V0a @ ( c_2Eordinal_2EfromNat @ A_27a @ c_2Enum_2E0 ) )
= ( c_2Elist_2ENIL @ ( tyop_2Epair_2Eprod @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ ( tyop_2Eordinal_2Eordinal @ A_27a ) ) ) ) ) ).
thf(thm_2Eordinal_2Epolyform__EQ__NIL,axiom,
! [A_27a: $tType,V0x: tyop_2Eordinal_2Eordinal @ A_27a,V1a: tyop_2Eordinal_2Eordinal @ A_27a] :
( ( c_2Eordinal_2Eordlt @ A_27a @ ( c_2Eordinal_2EfromNat @ A_27a @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) @ V1a )
=> ( ( ( c_2Eordinal_2Epolyform @ A_27a @ V1a @ V0x )
= ( c_2Elist_2ENIL @ ( tyop_2Epair_2Eprod @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ ( tyop_2Eordinal_2Eordinal @ A_27a ) ) ) )
<=> ( V0x
= ( c_2Eordinal_2EfromNat @ A_27a @ c_2Enum_2E0 ) ) ) ) ).
thf(thm_2Eordinal_2Eis__polyform__CONS__E,axiom,
! [A_27a: $tType,A_27b: $tType,V0t: tyop_2Elist_2Elist @ ( tyop_2Epair_2Eprod @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ ( tyop_2Eordinal_2Eordinal @ A_27b ) ),V1e: tyop_2Eordinal_2Eordinal @ A_27b,V2c: tyop_2Eordinal_2Eordinal @ A_27a,V3a: tyop_2Eordinal_2Eordinal @ A_27a] :
( ( c_2Eordinal_2Eis__polyform @ A_27a @ A_27b @ V3a @ ( c_2Elist_2ECONS @ ( tyop_2Epair_2Eprod @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ ( tyop_2Eordinal_2Eordinal @ A_27b ) ) @ ( c_2Epair_2E_2C @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ ( tyop_2Eordinal_2Eordinal @ A_27b ) @ V2c @ V1e ) @ V0t ) )
=> ( ( c_2Eordinal_2Eordlt @ A_27a @ ( c_2Eordinal_2EfromNat @ A_27a @ c_2Enum_2E0 ) @ V2c )
& ( c_2Eordinal_2Eordlt @ A_27a @ V2c @ V3a )
& ( c_2Eordinal_2Eis__polyform @ A_27a @ A_27b @ V3a @ V0t ) ) ) ).
thf(thm_2Eordinal_2Eis__polyform__head__dominates__tail,axiom,
! [A_27a: $tType,V0t: tyop_2Elist_2Elist @ ( tyop_2Epair_2Eprod @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ ( tyop_2Eordinal_2Eordinal @ A_27a ) ),V1e: tyop_2Eordinal_2Eordinal @ A_27a,V2c: tyop_2Eordinal_2Eordinal @ A_27a,V3a: tyop_2Eordinal_2Eordinal @ A_27a] :
( ( ( c_2Eordinal_2Eordlt @ A_27a @ ( c_2Eordinal_2EfromNat @ A_27a @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) @ V3a )
& ( c_2Eordinal_2Eis__polyform @ A_27a @ A_27a @ V3a @ ( c_2Elist_2ECONS @ ( tyop_2Epair_2Eprod @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ ( tyop_2Eordinal_2Eordinal @ A_27a ) ) @ ( c_2Epair_2E_2C @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ V2c @ V1e ) @ V0t ) ) )
=> ( c_2Eordinal_2Eordlt @ A_27a @ ( c_2Eordinal_2Eeval__poly @ A_27a @ V3a @ V0t ) @ ( c_2Eordinal_2EordEXP @ A_27a @ V3a @ V1e ) ) ) ).
thf(thm_2Eordinal_2Ecx__lt__x,axiom,
! [A_27a: $tType,V0x: tyop_2Eordinal_2Eordinal @ A_27a,V1c: tyop_2Eordinal_2Eordinal @ A_27a] :
( ( c_2Eordinal_2Eordlt @ A_27a @ ( c_2Eordinal_2EordMULT @ A_27a @ V0x @ V1c ) @ V0x )
<=> ( ( c_2Eordinal_2Eordlt @ A_27a @ ( c_2Eordinal_2EfromNat @ A_27a @ c_2Enum_2E0 ) @ V0x )
& ( V1c
= ( c_2Eordinal_2EfromNat @ A_27a @ c_2Enum_2E0 ) ) ) ) ).
thf(thm_2Eordinal_2Epolyform__UNIQUE,axiom,
! [A_27a: $tType,V0a: tyop_2Eordinal_2Eordinal @ A_27a,V1b: tyop_2Eordinal_2Eordinal @ A_27a,V2ces: tyop_2Elist_2Elist @ ( tyop_2Epair_2Eprod @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ ( tyop_2Eordinal_2Eordinal @ A_27a ) )] :
( ( ( c_2Eordinal_2Eordlt @ A_27a @ ( c_2Eordinal_2EfromNat @ A_27a @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) @ V0a )
& ( c_2Eordinal_2Eis__polyform @ A_27a @ A_27a @ V0a @ V2ces )
& ( V1b
= ( c_2Eordinal_2Eeval__poly @ A_27a @ V0a @ V2ces ) ) )
=> ( ( c_2Eordinal_2Epolyform @ A_27a @ V0a @ V1b )
= V2ces ) ) ).
thf(thm_2Eordinal_2Epolyform__eval__poly,axiom,
! [A_27a: $tType,V0b: tyop_2Elist_2Elist @ ( tyop_2Epair_2Eprod @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ ( tyop_2Eordinal_2Eordinal @ A_27a ) ),V1a: tyop_2Eordinal_2Eordinal @ A_27a] :
( ( ( c_2Eordinal_2Eordlt @ A_27a @ ( c_2Eordinal_2EfromNat @ A_27a @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) @ V1a )
& ( c_2Eordinal_2Eis__polyform @ A_27a @ A_27a @ V1a @ V0b ) )
=> ( ( c_2Eordinal_2Epolyform @ A_27a @ V1a @ ( c_2Eordinal_2Eeval__poly @ A_27a @ V1a @ V0b ) )
= V0b ) ) ).
thf(thm_2Eordinal_2ECNF__nat,axiom,
! [A_27a: $tType,V0n: tyop_2Enum_2Enum] :
( ( c_2Eordinal_2Epolyform @ A_27a @ ( c_2Eordinal_2Eomega @ A_27a ) @ ( c_2Eordinal_2EfromNat @ A_27a @ V0n ) )
= ( c_2Ebool_2ECOND @ ( tyop_2Elist_2Elist @ ( tyop_2Epair_2Eprod @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ ( tyop_2Eordinal_2Eordinal @ A_27a ) ) ) @ ( c_2Emin_2E_3D @ tyop_2Enum_2Enum @ V0n @ c_2Enum_2E0 ) @ ( c_2Elist_2ENIL @ ( tyop_2Epair_2Eprod @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ ( tyop_2Eordinal_2Eordinal @ A_27a ) ) ) @ ( c_2Elist_2ECONS @ ( tyop_2Epair_2Eprod @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ ( tyop_2Eordinal_2Eordinal @ A_27a ) ) @ ( c_2Epair_2E_2C @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ ( c_2Eordinal_2EfromNat @ A_27a @ V0n ) @ ( c_2Eordinal_2EfromNat @ A_27a @ c_2Enum_2E0 ) ) @ ( c_2Elist_2ENIL @ ( tyop_2Epair_2Eprod @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ ( tyop_2Eordinal_2Eordinal @ A_27a ) ) ) ) ) ) ).
thf(thm_2Eordinal_2EordLOG__correct,axiom,
! [A_27a: $tType,V0x: tyop_2Eordinal_2Eordinal @ A_27a,V1b: tyop_2Eordinal_2Eordinal @ A_27a] :
( ( ( c_2Eordinal_2Eordlt @ A_27a @ ( c_2Eordinal_2EfromNat @ A_27a @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) @ V1b )
& ( c_2Eordinal_2Eordlt @ A_27a @ ( c_2Eordinal_2EfromNat @ A_27a @ c_2Enum_2E0 ) @ V0x ) )
=> ( ( (~) @ ( c_2Eordinal_2Eordlt @ A_27a @ V0x @ ( c_2Eordinal_2EordEXP @ A_27a @ V1b @ ( c_2Epair_2ESND @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ ( c_2Elist_2EHD @ ( tyop_2Epair_2Eprod @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ ( tyop_2Eordinal_2Eordinal @ A_27a ) ) @ ( c_2Eordinal_2Epolyform @ A_27a @ V1b @ V0x ) ) ) ) ) )
& ! [V2a: tyop_2Eordinal_2Eordinal @ A_27a] :
( ( c_2Eordinal_2Eordlt @ A_27a @ ( c_2Epair_2ESND @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ ( c_2Elist_2EHD @ ( tyop_2Epair_2Eprod @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ ( tyop_2Eordinal_2Eordinal @ A_27a ) ) @ ( c_2Eordinal_2Epolyform @ A_27a @ V1b @ V0x ) ) ) @ V2a )
=> ( c_2Eordinal_2Eordlt @ A_27a @ V0x @ ( c_2Eordinal_2EordEXP @ A_27a @ V1b @ V2a ) ) ) ) ) ).
thf(thm_2Eordinal_2Eolog__correct,axiom,
! [A_27a: $tType,V0x: tyop_2Eordinal_2Eordinal @ A_27a] :
( ( c_2Eordinal_2Eordlt @ A_27a @ ( c_2Eordinal_2EfromNat @ A_27a @ c_2Enum_2E0 ) @ V0x )
=> ( ( (~) @ ( c_2Eordinal_2Eordlt @ A_27a @ V0x @ ( c_2Eordinal_2EordEXP @ A_27a @ ( c_2Eordinal_2Eomega @ A_27a ) @ ( c_2Epair_2ESND @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ ( c_2Elist_2EHD @ ( tyop_2Epair_2Eprod @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ ( tyop_2Eordinal_2Eordinal @ A_27a ) ) @ ( c_2Eordinal_2Epolyform @ A_27a @ ( c_2Eordinal_2Eomega @ A_27a ) @ V0x ) ) ) ) ) )
& ! [V1a: tyop_2Eordinal_2Eordinal @ A_27a] :
( ( c_2Eordinal_2Eordlt @ A_27a @ ( c_2Epair_2ESND @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ ( c_2Elist_2EHD @ ( tyop_2Epair_2Eprod @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ ( tyop_2Eordinal_2Eordinal @ A_27a ) ) @ ( c_2Eordinal_2Epolyform @ A_27a @ ( c_2Eordinal_2Eomega @ A_27a ) @ V0x ) ) ) @ V1a )
=> ( c_2Eordinal_2Eordlt @ A_27a @ V0x @ ( c_2Eordinal_2EordEXP @ A_27a @ ( c_2Eordinal_2Eomega @ A_27a ) @ V1a ) ) ) ) ) ).
%------------------------------------------------------------------------------