ITP001 Axioms: ITP049^7.ax
%------------------------------------------------------------------------------
% File : ITP049^7 : 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 : llist.ax [Gau19]
% : HL4049^7.ax [TPAP]
% Status : Satisfiable
% Syntax : Number of formulae : 329 ( 91 unt; 104 typ; 0 def)
% Number of atoms : 637 ( 333 equ; 15 cnn)
% Maximal formula atoms : 15 ( 1 avg)
% Number of connectives : 5116 ( 15 ~; 19 |; 142 &;4775 @)
% ( 60 <=>; 105 =>; 0 <=; 0 <~>)
% Maximal formula depth : 26 ( 7 avg;4775 nst)
% Number of types : 3 ( 2 usr)
% Number of type conns : 358 ( 358 >; 0 *; 0 +; 0 <<)
% Number of symbols : 104 ( 102 usr; 5 con; 0-7 aty)
% Number of variables : 1101 ( 116 ^ 815 !; 65 ?;1101 :)
% ( 105 !>; 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_2Ellist_2Ellist,type,
tyop_2Ellist_2Ellist: $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_2Eoption_2Eoption,type,
tyop_2Eoption_2Eoption: $tType > $tType ).
thf(tyop_2Epair_2Eprod,type,
tyop_2Epair_2Eprod: $tType > $tType > $tType ).
thf(c_2Ebool_2E_21,type,
c_2Ebool_2E_21:
!>[A_27a: $tType] : ( ( A_27a > $o ) > $o ) ).
thf(c_2Epair_2E_23_23,type,
c_2Epair_2E_23_23:
!>[A_27a: $tType,A_27b: $tType,A_27c: $tType,A_27d: $tType] : ( ( A_27a > A_27c ) > ( A_27b > A_27d ) > ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) > ( tyop_2Epair_2Eprod @ A_27c @ A_27d ) ) ).
thf(c_2Earithmetic_2E_2B,type,
c_2Earithmetic_2E_2B: tyop_2Enum_2Enum > tyop_2Enum_2Enum > tyop_2Enum_2Enum ).
thf(c_2Epair_2E_2C,type,
c_2Epair_2E_2C:
!>[A_27a: $tType,A_27b: $tType] : ( A_27a > A_27b > ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) ) ).
thf(c_2Earithmetic_2E_2D,type,
c_2Earithmetic_2E_2D: tyop_2Enum_2Enum > tyop_2Enum_2Enum > tyop_2Enum_2Enum ).
thf(c_2Ebool_2E_2F_5C,type,
c_2Ebool_2E_2F_5C: $o > $o > $o ).
thf(c_2Enum_2E0,type,
c_2Enum_2E0: tyop_2Enum_2Enum ).
thf(c_2Eprim__rec_2E_3C,type,
c_2Eprim__rec_2E_3C: tyop_2Enum_2Enum > tyop_2Enum_2Enum > $o ).
thf(c_2Earithmetic_2E_3C_3D,type,
c_2Earithmetic_2E_3C_3D: tyop_2Enum_2Enum > tyop_2Enum_2Enum > $o ).
thf(c_2Emin_2E_3D,type,
c_2Emin_2E_3D:
!>[A_27a: $tType] : ( A_27a > A_27a > $o ) ).
thf(c_2Emin_2E_3D_3D_3E,type,
c_2Emin_2E_3D_3D_3E: $o > $o > $o ).
thf(c_2Ebool_2E_3F,type,
c_2Ebool_2E_3F:
!>[A_27a: $tType] : ( ( A_27a > $o ) > $o ) ).
thf(c_2Ebool_2E_3F_21,type,
c_2Ebool_2E_3F_21:
!>[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_2Elist_2EAPPEND,type,
c_2Elist_2EAPPEND:
!>[A_27a: $tType] : ( ( tyop_2Elist_2Elist @ A_27a ) > ( tyop_2Elist_2Elist @ A_27a ) > ( tyop_2Elist_2Elist @ A_27a ) ) ).
thf(c_2Earithmetic_2EBIT1,type,
c_2Earithmetic_2EBIT1: tyop_2Enum_2Enum > tyop_2Enum_2Enum ).
thf(c_2Earithmetic_2EBIT2,type,
c_2Earithmetic_2EBIT2: tyop_2Enum_2Enum > tyop_2Enum_2Enum ).
thf(c_2Epred__set_2ECHOICE,type,
c_2Epred__set_2ECHOICE:
!>[A_27a: $tType] : ( ( A_27a > $o ) > A_27a ) ).
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_2EDIFF,type,
c_2Epred__set_2EDIFF:
!>[A_27a: $tType] : ( ( A_27a > $o ) > ( A_27a > $o ) > A_27a > $o ) ).
thf(c_2Elist_2EDROP,type,
c_2Elist_2EDROP:
!>[A_27a: $tType] : ( tyop_2Enum_2Enum > ( tyop_2Elist_2Elist @ A_27a ) > ( tyop_2Elist_2Elist @ A_27a ) ) ).
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_2Ebool_2EF,type,
c_2Ebool_2EF: $o ).
thf(c_2Epair_2EFST,type,
c_2Epair_2EFST:
!>[A_27a: $tType,A_27b: $tType] : ( ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) > A_27a ) ).
thf(c_2Earithmetic_2EFUNPOW,type,
c_2Earithmetic_2EFUNPOW:
!>[A_27a: $tType] : ( ( A_27a > A_27a ) > tyop_2Enum_2Enum > A_27a > A_27a ) ).
thf(c_2Elist_2EGENLIST,type,
c_2Elist_2EGENLIST:
!>[A_27a: $tType] : ( ( tyop_2Enum_2Enum > A_27a ) > tyop_2Enum_2Enum > ( tyop_2Elist_2Elist @ A_27a ) ) ).
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_2Ecombin_2EI,type,
c_2Ecombin_2EI:
!>[A_27a: $tType] : ( A_27a > A_27a ) ).
thf(c_2Eoption_2EIS__SOME,type,
c_2Eoption_2EIS__SOME:
!>[A_27a: $tType] : ( ( tyop_2Eoption_2Eoption @ A_27a ) > $o ) ).
thf(c_2Ecombin_2EK,type,
c_2Ecombin_2EK:
!>[A_27a: $tType,A_27b: $tType] : ( A_27a > A_27b > A_27a ) ).
thf(c_2Ellist_2ELAPPEND,type,
c_2Ellist_2ELAPPEND:
!>[A_27a: $tType] : ( ( tyop_2Ellist_2Ellist @ A_27a ) > ( tyop_2Ellist_2Ellist @ A_27a ) > ( tyop_2Ellist_2Ellist @ A_27a ) ) ).
thf(c_2Ellist_2ELCONS,type,
c_2Ellist_2ELCONS:
!>[A_27a: $tType] : ( A_27a > ( tyop_2Ellist_2Ellist @ A_27a ) > ( tyop_2Ellist_2Ellist @ A_27a ) ) ).
thf(c_2Ellist_2ELDROP,type,
c_2Ellist_2ELDROP:
!>[A_27a: $tType] : ( tyop_2Enum_2Enum > ( tyop_2Ellist_2Ellist @ A_27a ) > ( tyop_2Eoption_2Eoption @ ( tyop_2Ellist_2Ellist @ A_27a ) ) ) ).
thf(c_2Elist_2ELENGTH,type,
c_2Elist_2ELENGTH:
!>[A_27a: $tType] : ( ( tyop_2Elist_2Elist @ A_27a ) > tyop_2Enum_2Enum ) ).
thf(c_2Ebool_2ELET,type,
c_2Ebool_2ELET:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > A_27b ) > A_27a > A_27b ) ).
thf(c_2Ellist_2ELFILTER,type,
c_2Ellist_2ELFILTER:
!>[A_27a: $tType] : ( ( A_27a > $o ) > ( tyop_2Ellist_2Ellist @ A_27a ) > ( tyop_2Ellist_2Ellist @ A_27a ) ) ).
thf(c_2Ellist_2ELFINITE,type,
c_2Ellist_2ELFINITE:
!>[A_27a: $tType] : ( ( tyop_2Ellist_2Ellist @ A_27a ) > $o ) ).
thf(c_2Ellist_2ELFLATTEN,type,
c_2Ellist_2ELFLATTEN:
!>[A_27a: $tType] : ( ( tyop_2Ellist_2Ellist @ ( tyop_2Ellist_2Ellist @ A_27a ) ) > ( tyop_2Ellist_2Ellist @ A_27a ) ) ).
thf(c_2Ellist_2ELGENLIST,type,
c_2Ellist_2ELGENLIST:
!>[A_27a: $tType] : ( ( tyop_2Enum_2Enum > A_27a ) > ( tyop_2Eoption_2Eoption @ tyop_2Enum_2Enum ) > ( tyop_2Ellist_2Ellist @ A_27a ) ) ).
thf(c_2Ellist_2ELHD,type,
c_2Ellist_2ELHD:
!>[A_27a: $tType] : ( ( tyop_2Ellist_2Ellist @ A_27a ) > ( tyop_2Eoption_2Eoption @ A_27a ) ) ).
thf(c_2Ellist_2ELLENGTH,type,
c_2Ellist_2ELLENGTH:
!>[A_27a: $tType] : ( ( tyop_2Ellist_2Ellist @ A_27a ) > ( tyop_2Eoption_2Eoption @ tyop_2Enum_2Enum ) ) ).
thf(c_2Ellist_2ELMAP,type,
c_2Ellist_2ELMAP:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > A_27b ) > ( tyop_2Ellist_2Ellist @ A_27a ) > ( tyop_2Ellist_2Ellist @ A_27b ) ) ).
thf(c_2Ellist_2ELNIL,type,
c_2Ellist_2ELNIL:
!>[A_27a: $tType] : ( tyop_2Ellist_2Ellist @ A_27a ) ).
thf(c_2Ellist_2ELNTH,type,
c_2Ellist_2ELNTH:
!>[A_27a: $tType] : ( tyop_2Enum_2Enum > ( tyop_2Ellist_2Ellist @ A_27a ) > ( tyop_2Eoption_2Eoption @ A_27a ) ) ).
thf(c_2Ellist_2ELPREFIX,type,
c_2Ellist_2ELPREFIX:
!>[A_27a: $tType] : ( ( tyop_2Ellist_2Ellist @ A_27a ) > ( tyop_2Ellist_2Ellist @ A_27a ) > $o ) ).
thf(c_2Ellist_2ELREPEAT,type,
c_2Ellist_2ELREPEAT:
!>[A_27a: $tType] : ( ( tyop_2Elist_2Elist @ A_27a ) > ( tyop_2Ellist_2Ellist @ A_27a ) ) ).
thf(c_2Ellist_2ELTAKE,type,
c_2Ellist_2ELTAKE:
!>[A_27a: $tType] : ( tyop_2Enum_2Enum > ( tyop_2Ellist_2Ellist @ A_27a ) > ( tyop_2Eoption_2Eoption @ ( tyop_2Elist_2Elist @ A_27a ) ) ) ).
thf(c_2Ellist_2ELTL,type,
c_2Ellist_2ELTL:
!>[A_27a: $tType] : ( ( tyop_2Ellist_2Ellist @ A_27a ) > ( tyop_2Eoption_2Eoption @ ( tyop_2Ellist_2Ellist @ A_27a ) ) ) ).
thf(c_2Ellist_2ELTL__HD,type,
c_2Ellist_2ELTL__HD:
!>[A_27a: $tType] : ( ( tyop_2Ellist_2Ellist @ A_27a ) > ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ ( tyop_2Ellist_2Ellist @ A_27a ) @ A_27a ) ) ) ).
thf(c_2Ellist_2ELUNFOLD,type,
c_2Ellist_2ELUNFOLD:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27b > ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ A_27b @ A_27a ) ) ) > A_27b > ( tyop_2Ellist_2Ellist @ A_27a ) ) ).
thf(c_2Ellist_2ELUNZIP,type,
c_2Ellist_2ELUNZIP:
!>[A_27a: $tType,A_27b: $tType] : ( ( tyop_2Ellist_2Ellist @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) ) > ( tyop_2Epair_2Eprod @ ( tyop_2Ellist_2Ellist @ A_27a ) @ ( tyop_2Ellist_2Ellist @ A_27b ) ) ) ).
thf(c_2Ellist_2ELZIP,type,
c_2Ellist_2ELZIP:
!>[A_27a: $tType,A_27b: $tType] : ( ( tyop_2Epair_2Eprod @ ( tyop_2Ellist_2Ellist @ A_27a ) @ ( tyop_2Ellist_2Ellist @ A_27b ) ) > ( tyop_2Ellist_2Ellist @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) ) ) ).
thf(c_2Earithmetic_2EMOD,type,
c_2Earithmetic_2EMOD: tyop_2Enum_2Enum > tyop_2Enum_2Enum > tyop_2Enum_2Enum ).
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_2Elist_2ENULL,type,
c_2Elist_2ENULL:
!>[A_27a: $tType] : ( ( tyop_2Elist_2Elist @ A_27a ) > $o ) ).
thf(c_2Earithmetic_2ENUMERAL,type,
c_2Earithmetic_2ENUMERAL: tyop_2Enum_2Enum > tyop_2Enum_2Enum ).
thf(c_2Eoption_2EOPTION__BIND,type,
c_2Eoption_2EOPTION__BIND:
!>[A_27a: $tType,A_27b: $tType] : ( ( tyop_2Eoption_2Eoption @ A_27b ) > ( A_27b > ( tyop_2Eoption_2Eoption @ A_27a ) ) > ( tyop_2Eoption_2Eoption @ A_27a ) ) ).
thf(c_2Eoption_2EOPTION__JOIN,type,
c_2Eoption_2EOPTION__JOIN:
!>[A_27a: $tType] : ( ( tyop_2Eoption_2Eoption @ ( tyop_2Eoption_2Eoption @ A_27a ) ) > ( tyop_2Eoption_2Eoption @ A_27a ) ) ).
thf(c_2Eoption_2EOPTION__MAP,type,
c_2Eoption_2EOPTION__MAP:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > A_27b ) > ( tyop_2Eoption_2Eoption @ A_27a ) > ( tyop_2Eoption_2Eoption @ A_27b ) ) ).
thf(c_2Eprim__rec_2EPRE,type,
c_2Eprim__rec_2EPRE: tyop_2Enum_2Enum > tyop_2Enum_2Enum ).
thf(c_2Epair_2ESND,type,
c_2Epair_2ESND:
!>[A_27a: $tType,A_27b: $tType] : ( ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) > A_27b ) ).
thf(c_2Eoption_2ESOME,type,
c_2Eoption_2ESOME:
!>[A_27a: $tType] : ( A_27a > ( tyop_2Eoption_2Eoption @ A_27a ) ) ).
thf(c_2Epred__set_2ESUBSET,type,
c_2Epred__set_2ESUBSET:
!>[A_27a: $tType] : ( ( A_27a > $o ) > ( A_27a > $o ) > $o ) ).
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_2Elist_2ETAKE,type,
c_2Elist_2ETAKE:
!>[A_27a: $tType] : ( tyop_2Enum_2Enum > ( tyop_2Elist_2Elist @ A_27a ) > ( tyop_2Elist_2Elist @ A_27a ) ) ).
thf(c_2Eoption_2ETHE,type,
c_2Eoption_2ETHE:
!>[A_27a: $tType] : ( ( tyop_2Eoption_2Eoption @ A_27a ) > A_27a ) ).
thf(c_2Elist_2ETL,type,
c_2Elist_2ETL:
!>[A_27a: $tType] : ( ( tyop_2Elist_2Elist @ A_27a ) > ( tyop_2Elist_2Elist @ A_27a ) ) ).
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_2EUNION,type,
c_2Epred__set_2EUNION:
!>[A_27a: $tType] : ( ( A_27a > $o ) > ( A_27a > $o ) > A_27a > $o ) ).
thf(c_2Earithmetic_2EZERO,type,
c_2Earithmetic_2EZERO: tyop_2Enum_2Enum ).
thf(c_2Ebool_2E_5C_2F,type,
c_2Ebool_2E_5C_2F: $o > $o > $o ).
thf(c_2Eset__relation_2Edomain,type,
c_2Eset__relation_2Edomain:
!>[A_27a: $tType,A_27b: $tType] : ( ( ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) > $o ) > A_27a > $o ) ).
thf(c_2Ellist_2Eevery,type,
c_2Ellist_2Eevery:
!>[A_27a: $tType] : ( ( A_27a > $o ) > ( tyop_2Ellist_2Ellist @ A_27a ) > $o ) ).
thf(c_2Ellist_2Eexists,type,
c_2Ellist_2Eexists:
!>[A_27a: $tType] : ( ( A_27a > $o ) > ( tyop_2Ellist_2Ellist @ A_27a ) > $o ) ).
thf(c_2Eset__relation_2Efinite__prefixes,type,
c_2Eset__relation_2Efinite__prefixes:
!>[A_27a: $tType,A_27b: $tType] : ( ( ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) > $o ) > ( A_27b > $o ) > $o ) ).
thf(c_2Ellist_2EfromList,type,
c_2Ellist_2EfromList:
!>[A_27a: $tType] : ( ( tyop_2Elist_2Elist @ A_27a ) > ( tyop_2Ellist_2Ellist @ A_27a ) ) ).
thf(c_2Elist_2EisPREFIX,type,
c_2Elist_2EisPREFIX:
!>[A_27a: $tType] : ( ( tyop_2Elist_2Elist @ A_27a ) > ( tyop_2Elist_2Elist @ A_27a ) > $o ) ).
thf(c_2Eset__relation_2Elinear__order,type,
c_2Eset__relation_2Elinear__order:
!>[A_27a: $tType] : ( ( ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) > $o ) > ( A_27a > $o ) > $o ) ).
thf(c_2Ellist_2Elinear__order__to__list__f,type,
c_2Ellist_2Elinear__order__to__list__f:
!>[A_27a: $tType] : ( ( ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) > $o ) > ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ ( ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) > $o ) @ A_27a ) ) ) ).
thf(c_2Ellist_2Ellength__rel,type,
c_2Ellist_2Ellength__rel:
!>[A_27a: $tType] : ( ( tyop_2Ellist_2Ellist @ A_27a ) > tyop_2Enum_2Enum > $o ) ).
thf(c_2Ellist_2Ellist__abs,type,
c_2Ellist_2Ellist__abs:
!>[A_27a: $tType] : ( ( tyop_2Enum_2Enum > ( tyop_2Eoption_2Eoption @ A_27a ) ) > ( tyop_2Ellist_2Ellist @ A_27a ) ) ).
thf(c_2Ellist_2Ellist__rep,type,
c_2Ellist_2Ellist__rep:
!>[A_27a: $tType] : ( ( tyop_2Ellist_2Ellist @ A_27a ) > tyop_2Enum_2Enum > ( tyop_2Eoption_2Eoption @ A_27a ) ) ).
thf(c_2Ellist_2Elrep__ok,type,
c_2Ellist_2Elrep__ok:
!>[A_27a: $tType] : ( ( tyop_2Enum_2Enum > ( tyop_2Eoption_2Eoption @ A_27a ) ) > $o ) ).
thf(c_2Eset__relation_2Eminimal__elements,type,
c_2Eset__relation_2Eminimal__elements:
!>[A_27a: $tType] : ( ( A_27a > $o ) > ( ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) > $o ) > A_27a > $o ) ).
thf(c_2Ecombin_2Eo,type,
c_2Ecombin_2Eo:
!>[A_27a: $tType,A_27b: $tType,A_27c: $tType] : ( ( A_27c > A_27b ) > ( A_27a > A_27c ) > A_27a > A_27b ) ).
thf(c_2Eoption_2Eoption__CASE,type,
c_2Eoption_2Eoption__CASE:
!>[A_27a: $tType,A_27b: $tType] : ( ( tyop_2Eoption_2Eoption @ A_27a ) > A_27b > ( A_27a > A_27b ) > A_27b ) ).
thf(c_2Epair_2Epair__CASE,type,
c_2Epair_2Epair__CASE:
!>[A_27a: $tType,A_27b: $tType,A_27c: $tType] : ( ( tyop_2Epair_2Eprod @ A_27b @ A_27c ) > ( A_27b > A_27c > A_27a ) > A_27a ) ).
thf(c_2Eset__relation_2Erange,type,
c_2Eset__relation_2Erange:
!>[A_27a: $tType,A_27b: $tType] : ( ( ( tyop_2Epair_2Eprod @ A_27b @ A_27a ) > $o ) > A_27a > $o ) ).
thf(c_2Eset__relation_2Errestrict,type,
c_2Eset__relation_2Errestrict:
!>[A_27a: $tType] : ( ( ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) > $o ) > ( A_27a > $o ) > ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) > $o ) ).
thf(c_2Ellist_2EtoList,type,
c_2Ellist_2EtoList:
!>[A_27a: $tType] : ( ( tyop_2Ellist_2Ellist @ A_27a ) > ( tyop_2Eoption_2Eoption @ ( tyop_2Elist_2Elist @ 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_2Ellist_2Elrep__ok__def,axiom,
! [A_27a: $tType] :
( ( c_2Ellist_2Elrep__ok @ A_27a )
= ( ^ [V0a0: tyop_2Enum_2Enum > ( tyop_2Eoption_2Eoption @ A_27a )] :
( c_2Ebool_2E_3F @ ( ( tyop_2Enum_2Enum > ( tyop_2Eoption_2Eoption @ A_27a ) ) > $o )
@ ^ [V1lrep__ok_27: ( tyop_2Enum_2Enum > ( tyop_2Eoption_2Eoption @ A_27a ) ) > $o] :
( c_2Ebool_2E_2F_5C @ ( V1lrep__ok_27 @ V0a0 )
@ ( c_2Ebool_2E_21 @ ( tyop_2Enum_2Enum > ( tyop_2Eoption_2Eoption @ A_27a ) )
@ ^ [V2a0: tyop_2Enum_2Enum > ( tyop_2Eoption_2Eoption @ A_27a )] :
( c_2Emin_2E_3D_3D_3E @ ( V1lrep__ok_27 @ V2a0 )
@ ( c_2Ebool_2E_5C_2F
@ ( c_2Emin_2E_3D @ ( tyop_2Enum_2Enum > ( tyop_2Eoption_2Eoption @ A_27a ) ) @ V2a0
@ ^ [V3n: tyop_2Enum_2Enum] : ( c_2Eoption_2ENONE @ A_27a ) )
@ ( c_2Ebool_2E_3F @ A_27a
@ ^ [V4h: A_27a] :
( c_2Ebool_2E_3F @ ( tyop_2Enum_2Enum > ( tyop_2Eoption_2Eoption @ A_27a ) )
@ ^ [V5t: tyop_2Enum_2Enum > ( tyop_2Eoption_2Eoption @ A_27a )] :
( c_2Ebool_2E_2F_5C
@ ( c_2Emin_2E_3D @ ( tyop_2Enum_2Enum > ( tyop_2Eoption_2Eoption @ A_27a ) ) @ V2a0
@ ^ [V6n: tyop_2Enum_2Enum] : ( c_2Ebool_2ECOND @ ( tyop_2Eoption_2Eoption @ A_27a ) @ ( c_2Emin_2E_3D @ tyop_2Enum_2Enum @ V6n @ c_2Enum_2E0 ) @ ( c_2Eoption_2ESOME @ A_27a @ V4h ) @ ( V5t @ ( c_2Earithmetic_2E_2D @ V6n @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ) )
@ ( V1lrep__ok_27 @ V5t ) ) ) ) ) ) ) ) ) ) ) ).
thf(thm_2Ellist_2Ellist__TY__DEF,axiom,
! [A_27a: $tType] :
? [V0rep: ( tyop_2Ellist_2Ellist @ A_27a ) > tyop_2Enum_2Enum > ( tyop_2Eoption_2Eoption @ A_27a )] : ( c_2Ebool_2ETYPE__DEFINITION @ ( tyop_2Enum_2Enum > ( tyop_2Eoption_2Eoption @ A_27a ) ) @ ( tyop_2Ellist_2Ellist @ A_27a ) @ ( c_2Ellist_2Elrep__ok @ A_27a ) @ V0rep ) ).
thf(thm_2Ellist_2Ellist__absrep,axiom,
! [A_27a: $tType] :
( ! [V0a: tyop_2Ellist_2Ellist @ A_27a] :
( ( c_2Ellist_2Ellist__abs @ A_27a @ ( c_2Ellist_2Ellist__rep @ A_27a @ V0a ) )
= V0a )
& ! [V1r: tyop_2Enum_2Enum > ( tyop_2Eoption_2Eoption @ A_27a )] :
( ( c_2Ellist_2Elrep__ok @ A_27a @ V1r )
<=> ( ( c_2Ellist_2Ellist__rep @ A_27a @ ( c_2Ellist_2Ellist__abs @ A_27a @ V1r ) )
= V1r ) ) ) ).
thf(thm_2Ellist_2ELNIL,axiom,
! [A_27a: $tType] :
( ( c_2Ellist_2ELNIL @ A_27a )
= ( c_2Ellist_2Ellist__abs @ A_27a
@ ^ [V0n: tyop_2Enum_2Enum] : ( c_2Eoption_2ENONE @ A_27a ) ) ) ).
thf(thm_2Ellist_2ELCONS,axiom,
! [A_27a: $tType,V0h: A_27a,V1t: tyop_2Ellist_2Ellist @ A_27a] :
( ( c_2Ellist_2ELCONS @ A_27a @ V0h @ V1t )
= ( c_2Ellist_2Ellist__abs @ A_27a
@ ^ [V2n: tyop_2Enum_2Enum] : ( c_2Ebool_2ECOND @ ( tyop_2Eoption_2Eoption @ A_27a ) @ ( c_2Emin_2E_3D @ tyop_2Enum_2Enum @ V2n @ c_2Enum_2E0 ) @ ( c_2Eoption_2ESOME @ A_27a @ V0h ) @ ( c_2Ellist_2Ellist__rep @ A_27a @ V1t @ ( c_2Earithmetic_2E_2D @ V2n @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ) ) ) ).
thf(thm_2Ellist_2ELTL__HD__def,axiom,
! [A_27a: $tType,V0ll: tyop_2Ellist_2Ellist @ A_27a] :
( ( c_2Ellist_2ELTL__HD @ A_27a @ V0ll )
= ( c_2Eoption_2Eoption__CASE @ A_27a @ ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ ( tyop_2Ellist_2Ellist @ A_27a ) @ A_27a ) ) @ ( c_2Ellist_2Ellist__rep @ A_27a @ V0ll @ c_2Enum_2E0 ) @ ( c_2Eoption_2ENONE @ ( tyop_2Epair_2Eprod @ ( tyop_2Ellist_2Ellist @ A_27a ) @ A_27a ) )
@ ^ [V1h: A_27a] : ( c_2Eoption_2ESOME @ ( tyop_2Epair_2Eprod @ ( tyop_2Ellist_2Ellist @ A_27a ) @ A_27a ) @ ( c_2Epair_2E_2C @ ( tyop_2Ellist_2Ellist @ A_27a ) @ A_27a @ ( c_2Ellist_2Ellist__abs @ A_27a @ ( c_2Ecombin_2Eo @ tyop_2Enum_2Enum @ ( tyop_2Eoption_2Eoption @ A_27a ) @ tyop_2Enum_2Enum @ ( c_2Ellist_2Ellist__rep @ A_27a @ V0ll ) @ c_2Enum_2ESUC ) ) @ V1h ) ) ) ) ).
thf(thm_2Ellist_2ELHD,axiom,
! [A_27a: $tType,V0ll: tyop_2Ellist_2Ellist @ A_27a] :
( ( c_2Ellist_2ELHD @ A_27a @ V0ll )
= ( c_2Ellist_2Ellist__rep @ A_27a @ V0ll @ c_2Enum_2E0 ) ) ).
thf(thm_2Ellist_2ELTL,axiom,
! [A_27a: $tType,V0ll: tyop_2Ellist_2Ellist @ A_27a] :
( ( c_2Ellist_2ELTL @ A_27a @ V0ll )
= ( c_2Eoption_2Eoption__CASE @ A_27a @ ( tyop_2Eoption_2Eoption @ ( tyop_2Ellist_2Ellist @ A_27a ) ) @ ( c_2Ellist_2ELHD @ A_27a @ V0ll ) @ ( c_2Eoption_2ENONE @ ( tyop_2Ellist_2Ellist @ A_27a ) )
@ ^ [V1v: A_27a] :
( c_2Eoption_2ESOME @ ( tyop_2Ellist_2Ellist @ A_27a )
@ ( c_2Ellist_2Ellist__abs @ A_27a
@ ^ [V2n: tyop_2Enum_2Enum] : ( c_2Ellist_2Ellist__rep @ A_27a @ V0ll @ ( c_2Earithmetic_2E_2B @ V2n @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ) ) ) ) ).
thf(thm_2Ellist_2ELNTH,axiom,
! [A_27a: $tType] :
( ! [V0ll: tyop_2Ellist_2Ellist @ A_27a] :
( ( c_2Ellist_2ELNTH @ A_27a @ c_2Enum_2E0 @ V0ll )
= ( c_2Ellist_2ELHD @ A_27a @ V0ll ) )
& ! [V1n: tyop_2Enum_2Enum,V2ll: tyop_2Ellist_2Ellist @ A_27a] :
( ( c_2Ellist_2ELNTH @ A_27a @ ( c_2Enum_2ESUC @ V1n ) @ V2ll )
= ( c_2Eoption_2EOPTION__JOIN @ A_27a @ ( c_2Eoption_2EOPTION__MAP @ ( tyop_2Ellist_2Ellist @ A_27a ) @ ( tyop_2Eoption_2Eoption @ A_27a ) @ ( c_2Ellist_2ELNTH @ A_27a @ V1n ) @ ( c_2Ellist_2ELTL @ A_27a @ V2ll ) ) ) ) ) ).
thf(thm_2Ellist_2ELUNFOLD__def,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: A_27b > ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ A_27b @ A_27a ) ),V1z: A_27b] :
( ( c_2Ellist_2ELUNFOLD @ A_27a @ A_27b @ V0f @ V1z )
= ( c_2Ellist_2Ellist__abs @ A_27a
@ ^ [V2n: tyop_2Enum_2Enum] :
( c_2Eoption_2EOPTION__MAP @ ( tyop_2Epair_2Eprod @ A_27b @ A_27a ) @ A_27a @ ( c_2Epair_2ESND @ A_27b @ A_27a )
@ ( c_2Earithmetic_2EFUNPOW @ ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ A_27b @ A_27a ) )
@ ^ [V3m: tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ A_27b @ A_27a )] : ( c_2Eoption_2EOPTION__BIND @ ( tyop_2Epair_2Eprod @ A_27b @ A_27a ) @ ( tyop_2Epair_2Eprod @ A_27b @ A_27a ) @ V3m @ ( c_2Epair_2EUNCURRY @ A_27b @ A_27a @ ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ A_27b @ A_27a ) ) @ ( c_2Ecombin_2Eo @ A_27b @ ( A_27a > ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ A_27b @ A_27a ) ) ) @ ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ A_27b @ A_27a ) ) @ ( c_2Ecombin_2EK @ ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ A_27b @ A_27a ) ) @ A_27a ) @ V0f ) ) )
@ V2n
@ ( V0f @ V1z ) ) ) ) ) ).
thf(thm_2Ellist_2ELTAKE,axiom,
! [A_27a: $tType] :
( ! [V0ll: tyop_2Ellist_2Ellist @ A_27a] :
( ( c_2Ellist_2ELTAKE @ A_27a @ c_2Enum_2E0 @ V0ll )
= ( c_2Eoption_2ESOME @ ( tyop_2Elist_2Elist @ A_27a ) @ ( c_2Elist_2ENIL @ A_27a ) ) )
& ! [V1n: tyop_2Enum_2Enum,V2ll: tyop_2Ellist_2Ellist @ A_27a] :
( ( c_2Ellist_2ELTAKE @ A_27a @ ( c_2Enum_2ESUC @ V1n ) @ V2ll )
= ( c_2Eoption_2Eoption__CASE @ A_27a @ ( tyop_2Eoption_2Eoption @ ( tyop_2Elist_2Elist @ A_27a ) ) @ ( c_2Ellist_2ELHD @ A_27a @ V2ll ) @ ( c_2Eoption_2ENONE @ ( tyop_2Elist_2Elist @ A_27a ) )
@ ^ [V3hd: A_27a] :
( c_2Eoption_2Eoption__CASE @ ( tyop_2Elist_2Elist @ A_27a ) @ ( tyop_2Eoption_2Eoption @ ( tyop_2Elist_2Elist @ A_27a ) ) @ ( c_2Ellist_2ELTAKE @ A_27a @ V1n @ ( c_2Eoption_2ETHE @ ( tyop_2Ellist_2Ellist @ A_27a ) @ ( c_2Ellist_2ELTL @ A_27a @ V2ll ) ) ) @ ( c_2Eoption_2ENONE @ ( tyop_2Elist_2Elist @ A_27a ) )
@ ^ [V4tl: tyop_2Elist_2Elist @ A_27a] : ( c_2Eoption_2ESOME @ ( tyop_2Elist_2Elist @ A_27a ) @ ( c_2Elist_2ECONS @ A_27a @ V3hd @ V4tl ) ) ) ) ) ) ).
thf(thm_2Ellist_2ELMAP,axiom,
! [A_27a: $tType,A_27b: $tType] :
( ! [V0f: A_27a > A_27b] :
( ( c_2Ellist_2ELMAP @ A_27a @ A_27b @ V0f @ ( c_2Ellist_2ELNIL @ A_27a ) )
= ( c_2Ellist_2ELNIL @ A_27b ) )
& ! [V1f: A_27a > A_27b,V2h: A_27a,V3t: tyop_2Ellist_2Ellist @ A_27a] :
( ( c_2Ellist_2ELMAP @ A_27a @ A_27b @ V1f @ ( c_2Ellist_2ELCONS @ A_27a @ V2h @ V3t ) )
= ( c_2Ellist_2ELCONS @ A_27b @ ( V1f @ V2h ) @ ( c_2Ellist_2ELMAP @ A_27a @ A_27b @ V1f @ V3t ) ) ) ) ).
thf(thm_2Ellist_2ELAPPEND,axiom,
! [A_27a: $tType] :
( ! [V0x: tyop_2Ellist_2Ellist @ A_27a] :
( ( c_2Ellist_2ELAPPEND @ A_27a @ ( c_2Ellist_2ELNIL @ A_27a ) @ V0x )
= V0x )
& ! [V1h: A_27a,V2t: tyop_2Ellist_2Ellist @ A_27a,V3x: tyop_2Ellist_2Ellist @ A_27a] :
( ( c_2Ellist_2ELAPPEND @ A_27a @ ( c_2Ellist_2ELCONS @ A_27a @ V1h @ V2t ) @ V3x )
= ( c_2Ellist_2ELCONS @ A_27a @ V1h @ ( c_2Ellist_2ELAPPEND @ A_27a @ V2t @ V3x ) ) ) ) ).
thf(thm_2Ellist_2ELFINITE__def,axiom,
! [A_27a: $tType] :
( ( c_2Ellist_2ELFINITE @ A_27a )
= ( ^ [V0a0: tyop_2Ellist_2Ellist @ A_27a] :
( c_2Ebool_2E_21 @ ( ( tyop_2Ellist_2Ellist @ A_27a ) > $o )
@ ^ [V1LFINITE_27: ( tyop_2Ellist_2Ellist @ A_27a ) > $o] :
( c_2Emin_2E_3D_3D_3E
@ ( c_2Ebool_2E_21 @ ( tyop_2Ellist_2Ellist @ A_27a )
@ ^ [V2a0: tyop_2Ellist_2Ellist @ A_27a] :
( c_2Emin_2E_3D_3D_3E
@ ( c_2Ebool_2E_5C_2F @ ( c_2Emin_2E_3D @ ( tyop_2Ellist_2Ellist @ A_27a ) @ V2a0 @ ( c_2Ellist_2ELNIL @ A_27a ) )
@ ( c_2Ebool_2E_3F @ A_27a
@ ^ [V3h: A_27a] :
( c_2Ebool_2E_3F @ ( tyop_2Ellist_2Ellist @ A_27a )
@ ^ [V4t: tyop_2Ellist_2Ellist @ A_27a] : ( c_2Ebool_2E_2F_5C @ ( c_2Emin_2E_3D @ ( tyop_2Ellist_2Ellist @ A_27a ) @ V2a0 @ ( c_2Ellist_2ELCONS @ A_27a @ V3h @ V4t ) ) @ ( V1LFINITE_27 @ V4t ) ) ) ) )
@ ( V1LFINITE_27 @ V2a0 ) ) )
@ ( V1LFINITE_27 @ V0a0 ) ) ) ) ) ).
thf(thm_2Ellist_2Ellength__rel__def,axiom,
! [A_27a: $tType] :
( ( c_2Ellist_2Ellength__rel @ A_27a )
= ( ^ [V0a0: tyop_2Ellist_2Ellist @ A_27a,V1a1: tyop_2Enum_2Enum] :
( c_2Ebool_2E_21 @ ( ( tyop_2Ellist_2Ellist @ A_27a ) > tyop_2Enum_2Enum > $o )
@ ^ [V2llength__rel_27: ( tyop_2Ellist_2Ellist @ A_27a ) > tyop_2Enum_2Enum > $o] :
( c_2Emin_2E_3D_3D_3E
@ ( c_2Ebool_2E_21 @ ( tyop_2Ellist_2Ellist @ A_27a )
@ ^ [V3a0: tyop_2Ellist_2Ellist @ A_27a] :
( c_2Ebool_2E_21 @ tyop_2Enum_2Enum
@ ^ [V4a1: tyop_2Enum_2Enum] :
( c_2Emin_2E_3D_3D_3E
@ ( c_2Ebool_2E_5C_2F @ ( c_2Ebool_2E_2F_5C @ ( c_2Emin_2E_3D @ ( tyop_2Ellist_2Ellist @ A_27a ) @ V3a0 @ ( c_2Ellist_2ELNIL @ A_27a ) ) @ ( c_2Emin_2E_3D @ tyop_2Enum_2Enum @ V4a1 @ c_2Enum_2E0 ) )
@ ( c_2Ebool_2E_3F @ A_27a
@ ^ [V5h: A_27a] :
( c_2Ebool_2E_3F @ tyop_2Enum_2Enum
@ ^ [V6n: tyop_2Enum_2Enum] :
( c_2Ebool_2E_3F @ ( tyop_2Ellist_2Ellist @ A_27a )
@ ^ [V7t: tyop_2Ellist_2Ellist @ A_27a] : ( c_2Ebool_2E_2F_5C @ ( c_2Emin_2E_3D @ ( tyop_2Ellist_2Ellist @ A_27a ) @ V3a0 @ ( c_2Ellist_2ELCONS @ A_27a @ V5h @ V7t ) ) @ ( c_2Ebool_2E_2F_5C @ ( c_2Emin_2E_3D @ tyop_2Enum_2Enum @ V4a1 @ ( c_2Enum_2ESUC @ V6n ) ) @ ( V2llength__rel_27 @ V7t @ V6n ) ) ) ) ) ) )
@ ( V2llength__rel_27 @ V3a0 @ V4a1 ) ) ) )
@ ( V2llength__rel_27 @ V0a0 @ V1a1 ) ) ) ) ) ).
thf(thm_2Ellist_2ELLENGTH,axiom,
! [A_27a: $tType,V0ll: tyop_2Ellist_2Ellist @ A_27a] :
( ( c_2Ellist_2ELLENGTH @ A_27a @ V0ll )
= ( c_2Ebool_2ECOND @ ( tyop_2Eoption_2Eoption @ tyop_2Enum_2Enum ) @ ( c_2Ellist_2ELFINITE @ A_27a @ V0ll )
@ ( c_2Eoption_2ESOME @ tyop_2Enum_2Enum
@ ( c_2Emin_2E_40 @ tyop_2Enum_2Enum
@ ^ [V1n: tyop_2Enum_2Enum] : ( c_2Ellist_2Ellength__rel @ A_27a @ V0ll @ V1n ) ) )
@ ( c_2Eoption_2ENONE @ tyop_2Enum_2Enum ) ) ) ).
thf(thm_2Ellist_2EtoList,axiom,
! [A_27a: $tType,V0ll: tyop_2Ellist_2Ellist @ A_27a] :
( ( c_2Ellist_2EtoList @ A_27a @ V0ll )
= ( c_2Ebool_2ECOND @ ( tyop_2Eoption_2Eoption @ ( tyop_2Elist_2Elist @ A_27a ) ) @ ( c_2Ellist_2ELFINITE @ A_27a @ V0ll ) @ ( c_2Ellist_2ELTAKE @ A_27a @ ( c_2Eoption_2ETHE @ tyop_2Enum_2Enum @ ( c_2Ellist_2ELLENGTH @ A_27a @ V0ll ) ) @ V0ll ) @ ( c_2Eoption_2ENONE @ ( tyop_2Elist_2Elist @ A_27a ) ) ) ) ).
thf(thm_2Ellist_2EfromList__def,axiom,
! [A_27a: $tType] :
( ( ( c_2Ellist_2EfromList @ A_27a @ ( c_2Elist_2ENIL @ A_27a ) )
= ( c_2Ellist_2ELNIL @ A_27a ) )
& ! [V0h: A_27a,V1t: tyop_2Elist_2Elist @ A_27a] :
( ( c_2Ellist_2EfromList @ A_27a @ ( c_2Elist_2ECONS @ A_27a @ V0h @ V1t ) )
= ( c_2Ellist_2ELCONS @ A_27a @ V0h @ ( c_2Ellist_2EfromList @ A_27a @ V1t ) ) ) ) ).
thf(thm_2Ellist_2ELDROP,axiom,
! [A_27a: $tType] :
( ! [V0ll: tyop_2Ellist_2Ellist @ A_27a] :
( ( c_2Ellist_2ELDROP @ A_27a @ c_2Enum_2E0 @ V0ll )
= ( c_2Eoption_2ESOME @ ( tyop_2Ellist_2Ellist @ A_27a ) @ V0ll ) )
& ! [V1n: tyop_2Enum_2Enum,V2ll: tyop_2Ellist_2Ellist @ A_27a] :
( ( c_2Ellist_2ELDROP @ A_27a @ ( c_2Enum_2ESUC @ V1n ) @ V2ll )
= ( c_2Eoption_2EOPTION__JOIN @ ( tyop_2Ellist_2Ellist @ A_27a ) @ ( c_2Eoption_2EOPTION__MAP @ ( tyop_2Ellist_2Ellist @ A_27a ) @ ( tyop_2Eoption_2Eoption @ ( tyop_2Ellist_2Ellist @ A_27a ) ) @ ( c_2Ellist_2ELDROP @ A_27a @ V1n ) @ ( c_2Ellist_2ELTL @ A_27a @ V2ll ) ) ) ) ) ).
thf(thm_2Ellist_2Eexists__def,axiom,
! [A_27a: $tType] :
( ( c_2Ellist_2Eexists @ A_27a )
= ( ^ [V0P: A_27a > $o,V1a0: tyop_2Ellist_2Ellist @ A_27a] :
( c_2Ebool_2E_21 @ ( ( tyop_2Ellist_2Ellist @ A_27a ) > $o )
@ ^ [V2exists_27: ( tyop_2Ellist_2Ellist @ A_27a ) > $o] :
( c_2Emin_2E_3D_3D_3E
@ ( c_2Ebool_2E_21 @ ( tyop_2Ellist_2Ellist @ A_27a )
@ ^ [V3a0: tyop_2Ellist_2Ellist @ A_27a] :
( c_2Emin_2E_3D_3D_3E
@ ( c_2Ebool_2E_5C_2F
@ ( c_2Ebool_2E_3F @ A_27a
@ ^ [V4h: A_27a] :
( c_2Ebool_2E_3F @ ( tyop_2Ellist_2Ellist @ A_27a )
@ ^ [V5t: tyop_2Ellist_2Ellist @ A_27a] : ( c_2Ebool_2E_2F_5C @ ( c_2Emin_2E_3D @ ( tyop_2Ellist_2Ellist @ A_27a ) @ V3a0 @ ( c_2Ellist_2ELCONS @ A_27a @ V4h @ V5t ) ) @ ( V0P @ V4h ) ) ) )
@ ( c_2Ebool_2E_3F @ A_27a
@ ^ [V6h: A_27a] :
( c_2Ebool_2E_3F @ ( tyop_2Ellist_2Ellist @ A_27a )
@ ^ [V7t: tyop_2Ellist_2Ellist @ A_27a] : ( c_2Ebool_2E_2F_5C @ ( c_2Emin_2E_3D @ ( tyop_2Ellist_2Ellist @ A_27a ) @ V3a0 @ ( c_2Ellist_2ELCONS @ A_27a @ V6h @ V7t ) ) @ ( V2exists_27 @ V7t ) ) ) ) )
@ ( V2exists_27 @ V3a0 ) ) )
@ ( V2exists_27 @ V1a0 ) ) ) ) ) ).
thf(thm_2Ellist_2Eevery__def,axiom,
! [A_27a: $tType,V0P: A_27a > $o,V1ll: tyop_2Ellist_2Ellist @ A_27a] :
( ( c_2Ellist_2Eevery @ A_27a @ V0P @ V1ll )
<=> ( (~) @ ( c_2Ellist_2Eexists @ A_27a @ ( c_2Ecombin_2Eo @ A_27a @ $o @ $o @ c_2Ebool_2E_7E @ V0P ) @ V1ll ) ) ) ).
thf(thm_2Ellist_2ELFILTER,axiom,
! [A_27a: $tType,V0P: A_27a > $o,V1ll: tyop_2Ellist_2Ellist @ A_27a] :
( ( c_2Ellist_2ELFILTER @ A_27a @ V0P @ V1ll )
= ( c_2Ebool_2ECOND @ ( tyop_2Ellist_2Ellist @ A_27a ) @ ( c_2Ebool_2E_7E @ ( c_2Ellist_2Eexists @ A_27a @ V0P @ V1ll ) ) @ ( c_2Ellist_2ELNIL @ A_27a ) @ ( c_2Ebool_2ECOND @ ( tyop_2Ellist_2Ellist @ A_27a ) @ ( V0P @ ( c_2Eoption_2ETHE @ A_27a @ ( c_2Ellist_2ELHD @ A_27a @ V1ll ) ) ) @ ( c_2Ellist_2ELCONS @ A_27a @ ( c_2Eoption_2ETHE @ A_27a @ ( c_2Ellist_2ELHD @ A_27a @ V1ll ) ) @ ( c_2Ellist_2ELFILTER @ A_27a @ V0P @ ( c_2Eoption_2ETHE @ ( tyop_2Ellist_2Ellist @ A_27a ) @ ( c_2Ellist_2ELTL @ A_27a @ V1ll ) ) ) ) @ ( c_2Ellist_2ELFILTER @ A_27a @ V0P @ ( c_2Eoption_2ETHE @ ( tyop_2Ellist_2Ellist @ A_27a ) @ ( c_2Ellist_2ELTL @ A_27a @ V1ll ) ) ) ) ) ) ).
thf(thm_2Ellist_2ELFLATTEN,axiom,
! [A_27a: $tType,V0ll: tyop_2Ellist_2Ellist @ ( tyop_2Ellist_2Ellist @ A_27a )] :
( ( c_2Ellist_2ELFLATTEN @ A_27a @ V0ll )
= ( c_2Ebool_2ECOND @ ( tyop_2Ellist_2Ellist @ A_27a ) @ ( c_2Ellist_2Eevery @ ( tyop_2Ellist_2Ellist @ A_27a ) @ ( c_2Emin_2E_3D @ ( tyop_2Ellist_2Ellist @ A_27a ) @ ( c_2Ellist_2ELNIL @ A_27a ) ) @ V0ll ) @ ( c_2Ellist_2ELNIL @ A_27a ) @ ( c_2Ebool_2ECOND @ ( tyop_2Ellist_2Ellist @ A_27a ) @ ( c_2Emin_2E_3D @ ( tyop_2Ellist_2Ellist @ A_27a ) @ ( c_2Eoption_2ETHE @ ( tyop_2Ellist_2Ellist @ A_27a ) @ ( c_2Ellist_2ELHD @ ( tyop_2Ellist_2Ellist @ A_27a ) @ V0ll ) ) @ ( c_2Ellist_2ELNIL @ A_27a ) ) @ ( c_2Ellist_2ELFLATTEN @ A_27a @ ( c_2Eoption_2ETHE @ ( tyop_2Ellist_2Ellist @ ( tyop_2Ellist_2Ellist @ A_27a ) ) @ ( c_2Ellist_2ELTL @ ( tyop_2Ellist_2Ellist @ A_27a ) @ V0ll ) ) ) @ ( c_2Ellist_2ELCONS @ A_27a @ ( c_2Eoption_2ETHE @ A_27a @ ( c_2Ellist_2ELHD @ A_27a @ ( c_2Eoption_2ETHE @ ( tyop_2Ellist_2Ellist @ A_27a ) @ ( c_2Ellist_2ELHD @ ( tyop_2Ellist_2Ellist @ A_27a ) @ V0ll ) ) ) ) @ ( c_2Ellist_2ELFLATTEN @ A_27a @ ( c_2Ellist_2ELCONS @ ( tyop_2Ellist_2Ellist @ A_27a ) @ ( c_2Eoption_2ETHE @ ( tyop_2Ellist_2Ellist @ A_27a ) @ ( c_2Ellist_2ELTL @ A_27a @ ( c_2Eoption_2ETHE @ ( tyop_2Ellist_2Ellist @ A_27a ) @ ( c_2Ellist_2ELHD @ ( tyop_2Ellist_2Ellist @ A_27a ) @ V0ll ) ) ) ) @ ( c_2Eoption_2ETHE @ ( tyop_2Ellist_2Ellist @ ( tyop_2Ellist_2Ellist @ A_27a ) ) @ ( c_2Ellist_2ELTL @ ( tyop_2Ellist_2Ellist @ A_27a ) @ V0ll ) ) ) ) ) ) ) ) ).
thf(thm_2Ellist_2ELZIP__THM,axiom,
! [A_27a: $tType,A_27b: $tType] :
( ! [V0l1: tyop_2Ellist_2Ellist @ A_27a] :
( ( c_2Ellist_2ELZIP @ A_27a @ A_27b @ ( c_2Epair_2E_2C @ ( tyop_2Ellist_2Ellist @ A_27a ) @ ( tyop_2Ellist_2Ellist @ A_27b ) @ V0l1 @ ( c_2Ellist_2ELNIL @ A_27b ) ) )
= ( c_2Ellist_2ELNIL @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) ) )
& ! [V1l2: tyop_2Ellist_2Ellist @ A_27b] :
( ( c_2Ellist_2ELZIP @ A_27a @ A_27b @ ( c_2Epair_2E_2C @ ( tyop_2Ellist_2Ellist @ A_27a ) @ ( tyop_2Ellist_2Ellist @ A_27b ) @ ( c_2Ellist_2ELNIL @ A_27a ) @ V1l2 ) )
= ( c_2Ellist_2ELNIL @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) ) )
& ! [V2h1: A_27a,V3h2: A_27b,V4t1: tyop_2Ellist_2Ellist @ A_27a,V5t2: tyop_2Ellist_2Ellist @ A_27b] :
( ( c_2Ellist_2ELZIP @ A_27a @ A_27b @ ( c_2Epair_2E_2C @ ( tyop_2Ellist_2Ellist @ A_27a ) @ ( tyop_2Ellist_2Ellist @ A_27b ) @ ( c_2Ellist_2ELCONS @ A_27a @ V2h1 @ V4t1 ) @ ( c_2Ellist_2ELCONS @ A_27b @ V3h2 @ V5t2 ) ) )
= ( c_2Ellist_2ELCONS @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ ( c_2Epair_2E_2C @ A_27a @ A_27b @ V2h1 @ V3h2 ) @ ( c_2Ellist_2ELZIP @ A_27a @ A_27b @ ( c_2Epair_2E_2C @ ( tyop_2Ellist_2Ellist @ A_27a ) @ ( tyop_2Ellist_2Ellist @ A_27b ) @ V4t1 @ V5t2 ) ) ) ) ) ).
thf(thm_2Ellist_2ELUNZIP__THM,axiom,
! [A_27a: $tType,A_27b: $tType] :
( ( ( c_2Ellist_2ELUNZIP @ A_27a @ A_27b @ ( c_2Ellist_2ELNIL @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) ) )
= ( c_2Epair_2E_2C @ ( tyop_2Ellist_2Ellist @ A_27a ) @ ( tyop_2Ellist_2Ellist @ A_27b ) @ ( c_2Ellist_2ELNIL @ A_27a ) @ ( c_2Ellist_2ELNIL @ A_27b ) ) )
& ! [V0x: A_27a,V1y: A_27b,V2t: tyop_2Ellist_2Ellist @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b )] :
( ( c_2Ellist_2ELUNZIP @ A_27a @ A_27b @ ( c_2Ellist_2ELCONS @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ ( c_2Epair_2E_2C @ A_27a @ A_27b @ V0x @ V1y ) @ V2t ) )
= ( c_2Ebool_2ELET @ ( tyop_2Epair_2Eprod @ ( tyop_2Ellist_2Ellist @ A_27a ) @ ( tyop_2Ellist_2Ellist @ A_27b ) ) @ ( tyop_2Epair_2Eprod @ ( tyop_2Ellist_2Ellist @ A_27a ) @ ( tyop_2Ellist_2Ellist @ A_27b ) )
@ ( c_2Epair_2EUNCURRY @ ( tyop_2Ellist_2Ellist @ A_27a ) @ ( tyop_2Ellist_2Ellist @ A_27b ) @ ( tyop_2Epair_2Eprod @ ( tyop_2Ellist_2Ellist @ A_27a ) @ ( tyop_2Ellist_2Ellist @ A_27b ) )
@ ^ [V3ll1: tyop_2Ellist_2Ellist @ A_27a,V4ll2: tyop_2Ellist_2Ellist @ A_27b] : ( c_2Epair_2E_2C @ ( tyop_2Ellist_2Ellist @ A_27a ) @ ( tyop_2Ellist_2Ellist @ A_27b ) @ ( c_2Ellist_2ELCONS @ A_27a @ V0x @ V3ll1 ) @ ( c_2Ellist_2ELCONS @ A_27b @ V1y @ V4ll2 ) ) )
@ ( c_2Ellist_2ELUNZIP @ A_27a @ A_27b @ V2t ) ) ) ) ).
thf(thm_2Ellist_2Elinear__order__to__list__f__def,axiom,
! [A_27a: $tType,V0lo: ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) > $o] :
( ( c_2Ellist_2Elinear__order__to__list__f @ A_27a @ V0lo )
= ( c_2Ebool_2ELET @ ( A_27a > $o ) @ ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ ( ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) > $o ) @ A_27a ) )
@ ^ [V1min: A_27a > $o] : ( c_2Ebool_2ECOND @ ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ ( ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) > $o ) @ A_27a ) ) @ ( c_2Emin_2E_3D @ ( A_27a > $o ) @ V1min @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) @ ( c_2Eoption_2ENONE @ ( tyop_2Epair_2Eprod @ ( ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) > $o ) @ A_27a ) ) @ ( c_2Eoption_2ESOME @ ( tyop_2Epair_2Eprod @ ( ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) > $o ) @ A_27a ) @ ( c_2Epair_2E_2C @ ( ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) > $o ) @ A_27a @ ( c_2Eset__relation_2Errestrict @ A_27a @ V0lo @ ( c_2Epred__set_2EDIFF @ A_27a @ ( c_2Epred__set_2EUNION @ A_27a @ ( c_2Eset__relation_2Edomain @ A_27a @ A_27a @ V0lo ) @ ( c_2Eset__relation_2Erange @ A_27a @ A_27a @ V0lo ) ) @ V1min ) ) @ ( c_2Epred__set_2ECHOICE @ A_27a @ V1min ) ) ) )
@ ( c_2Eset__relation_2Eminimal__elements @ A_27a @ ( c_2Epred__set_2EUNION @ A_27a @ ( c_2Eset__relation_2Edomain @ A_27a @ A_27a @ V0lo ) @ ( c_2Eset__relation_2Erange @ A_27a @ A_27a @ V0lo ) ) @ V0lo ) ) ) ).
thf(thm_2Ellist_2ELPREFIX__def,axiom,
! [A_27a: $tType,V0l1: tyop_2Ellist_2Ellist @ A_27a,V1l2: tyop_2Ellist_2Ellist @ A_27a] :
( ( c_2Ellist_2ELPREFIX @ A_27a @ V0l1 @ V1l2 )
= ( c_2Eoption_2Eoption__CASE @ ( tyop_2Elist_2Elist @ A_27a ) @ $o @ ( c_2Ellist_2EtoList @ A_27a @ V0l1 ) @ ( c_2Emin_2E_3D @ ( tyop_2Ellist_2Ellist @ A_27a ) @ V0l1 @ V1l2 )
@ ^ [V2xs: tyop_2Elist_2Elist @ A_27a] :
( c_2Eoption_2Eoption__CASE @ ( tyop_2Elist_2Elist @ A_27a ) @ $o @ ( c_2Ellist_2EtoList @ A_27a @ V1l2 ) @ ( c_2Emin_2E_3D @ ( tyop_2Eoption_2Eoption @ ( tyop_2Elist_2Elist @ A_27a ) ) @ ( c_2Ellist_2ELTAKE @ A_27a @ ( c_2Elist_2ELENGTH @ A_27a @ V2xs ) @ V1l2 ) @ ( c_2Eoption_2ESOME @ ( tyop_2Elist_2Elist @ A_27a ) @ V2xs ) )
@ ^ [V3ys: tyop_2Elist_2Elist @ A_27a] : ( c_2Elist_2EisPREFIX @ A_27a @ V2xs @ V3ys ) ) ) ) ).
thf(thm_2Ellist_2ELGENLIST__def,axiom,
! [A_27a: $tType] :
( ! [V0f: tyop_2Enum_2Enum > A_27a] :
( ( c_2Ellist_2ELGENLIST @ A_27a @ V0f @ ( c_2Eoption_2ENONE @ tyop_2Enum_2Enum ) )
= ( c_2Ellist_2ELUNFOLD @ A_27a @ tyop_2Enum_2Enum
@ ^ [V1n: tyop_2Enum_2Enum] : ( c_2Eoption_2ESOME @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ A_27a ) @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ A_27a @ ( c_2Earithmetic_2E_2B @ V1n @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) @ ( V0f @ V1n ) ) )
@ c_2Enum_2E0 ) )
& ! [V2f: tyop_2Enum_2Enum > A_27a,V3lim: tyop_2Enum_2Enum] :
( ( c_2Ellist_2ELGENLIST @ A_27a @ V2f @ ( c_2Eoption_2ESOME @ tyop_2Enum_2Enum @ V3lim ) )
= ( c_2Ellist_2ELUNFOLD @ A_27a @ tyop_2Enum_2Enum
@ ^ [V4n: tyop_2Enum_2Enum] : ( c_2Ebool_2ECOND @ ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ A_27a ) ) @ ( c_2Eprim__rec_2E_3C @ V4n @ V3lim ) @ ( c_2Eoption_2ESOME @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ A_27a ) @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ A_27a @ ( c_2Earithmetic_2E_2B @ V4n @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) @ ( V2f @ V4n ) ) ) @ ( c_2Eoption_2ENONE @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ A_27a ) ) )
@ c_2Enum_2E0 ) ) ) ).
thf(thm_2Ellist_2ELREPEAT__def,axiom,
! [A_27a: $tType,V0l: tyop_2Elist_2Elist @ A_27a] :
( ( c_2Ellist_2ELREPEAT @ A_27a @ V0l )
= ( c_2Ebool_2ECOND @ ( tyop_2Ellist_2Ellist @ A_27a ) @ ( c_2Elist_2ENULL @ A_27a @ V0l ) @ ( c_2Ellist_2ELNIL @ A_27a )
@ ( c_2Ellist_2ELGENLIST @ A_27a
@ ^ [V1n: tyop_2Enum_2Enum] : ( c_2Elist_2EEL @ A_27a @ ( c_2Earithmetic_2EMOD @ V1n @ ( c_2Elist_2ELENGTH @ A_27a @ V0l ) ) @ V0l )
@ ( c_2Eoption_2ENONE @ tyop_2Enum_2Enum ) ) ) ) ).
thf(thm_2Ellist_2Elrep__ok__rules,axiom,
! [A_27a: $tType] :
( ( c_2Ellist_2Elrep__ok @ A_27a
@ ^ [V0n: tyop_2Enum_2Enum] : ( c_2Eoption_2ENONE @ A_27a ) )
& ! [V1h: A_27a,V2t: tyop_2Enum_2Enum > ( tyop_2Eoption_2Eoption @ A_27a )] :
( ( c_2Ellist_2Elrep__ok @ A_27a @ V2t )
=> ( c_2Ellist_2Elrep__ok @ A_27a
@ ^ [V3n: tyop_2Enum_2Enum] : ( c_2Ebool_2ECOND @ ( tyop_2Eoption_2Eoption @ A_27a ) @ ( c_2Emin_2E_3D @ tyop_2Enum_2Enum @ V3n @ c_2Enum_2E0 ) @ ( c_2Eoption_2ESOME @ A_27a @ V1h ) @ ( V2t @ ( c_2Earithmetic_2E_2D @ V3n @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ) ) ) ) ).
thf(thm_2Ellist_2Elrep__ok__coind,axiom,
! [A_27a: $tType,V0lrep__ok_27: ( tyop_2Enum_2Enum > ( tyop_2Eoption_2Eoption @ A_27a ) ) > $o] :
( ! [V1a0: tyop_2Enum_2Enum > ( tyop_2Eoption_2Eoption @ A_27a )] :
( ( V0lrep__ok_27 @ V1a0 )
=> ( ( V1a0
= ( ^ [V2n: tyop_2Enum_2Enum] : ( c_2Eoption_2ENONE @ A_27a ) ) )
| ? [V3h: A_27a,V4t: tyop_2Enum_2Enum > ( tyop_2Eoption_2Eoption @ A_27a )] :
( ( V1a0
= ( ^ [V5n: tyop_2Enum_2Enum] : ( c_2Ebool_2ECOND @ ( tyop_2Eoption_2Eoption @ A_27a ) @ ( c_2Emin_2E_3D @ tyop_2Enum_2Enum @ V5n @ c_2Enum_2E0 ) @ ( c_2Eoption_2ESOME @ A_27a @ V3h ) @ ( V4t @ ( c_2Earithmetic_2E_2D @ V5n @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ) ) )
& ( V0lrep__ok_27 @ V4t ) ) ) )
=> ! [V6a0: tyop_2Enum_2Enum > ( tyop_2Eoption_2Eoption @ A_27a )] :
( ( V0lrep__ok_27 @ V6a0 )
=> ( c_2Ellist_2Elrep__ok @ A_27a @ V6a0 ) ) ) ).
thf(thm_2Ellist_2Elrep__ok__cases,axiom,
! [A_27a: $tType,V0a0: tyop_2Enum_2Enum > ( tyop_2Eoption_2Eoption @ A_27a )] :
( ( c_2Ellist_2Elrep__ok @ A_27a @ V0a0 )
<=> ( ( V0a0
= ( ^ [V1n: tyop_2Enum_2Enum] : ( c_2Eoption_2ENONE @ A_27a ) ) )
| ? [V2h: A_27a,V3t: tyop_2Enum_2Enum > ( tyop_2Eoption_2Eoption @ A_27a )] :
( ( V0a0
= ( ^ [V4n: tyop_2Enum_2Enum] : ( c_2Ebool_2ECOND @ ( tyop_2Eoption_2Eoption @ A_27a ) @ ( c_2Emin_2E_3D @ tyop_2Enum_2Enum @ V4n @ c_2Enum_2E0 ) @ ( c_2Eoption_2ESOME @ A_27a @ V2h ) @ ( V3t @ ( c_2Earithmetic_2E_2D @ V4n @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ) ) )
& ( c_2Ellist_2Elrep__ok @ A_27a @ V3t ) ) ) ) ).
thf(thm_2Ellist_2Elrep__ok__alt,axiom,
! [A_27a: $tType,V0f: tyop_2Enum_2Enum > ( tyop_2Eoption_2Eoption @ A_27a )] :
( ( c_2Ellist_2Elrep__ok @ A_27a @ V0f )
<=> ! [V1n: tyop_2Enum_2Enum] :
( ( c_2Eoption_2EIS__SOME @ A_27a @ ( V0f @ ( c_2Enum_2ESUC @ V1n ) ) )
=> ( c_2Eoption_2EIS__SOME @ A_27a @ ( V0f @ V1n ) ) ) ) ).
thf(thm_2Ellist_2Elrep__ok__MAP,axiom,
! [A_27a: $tType,A_27b: $tType,V0g: tyop_2Enum_2Enum > ( tyop_2Eoption_2Eoption @ A_27b ),V1f: A_27b > A_27a] :
( ( c_2Ellist_2Elrep__ok @ A_27a
@ ^ [V2n: tyop_2Enum_2Enum] : ( c_2Eoption_2EOPTION__MAP @ A_27b @ A_27a @ V1f @ ( V0g @ V2n ) ) )
= ( c_2Ellist_2Elrep__ok @ A_27b @ V0g ) ) ).
thf(thm_2Ellist_2Elrep__ok__FUNPOW__BIND,axiom,
! [A_27a: $tType,V0g: A_27a > ( tyop_2Eoption_2Eoption @ A_27a ),V1fz: tyop_2Eoption_2Eoption @ A_27a] :
( c_2Ellist_2Elrep__ok @ A_27a
@ ^ [V2n: tyop_2Enum_2Enum] :
( c_2Earithmetic_2EFUNPOW @ ( tyop_2Eoption_2Eoption @ A_27a )
@ ^ [V3m: tyop_2Eoption_2Eoption @ A_27a] : ( c_2Eoption_2EOPTION__BIND @ A_27a @ A_27a @ V3m @ V0g )
@ V2n
@ V1fz ) ) ).
thf(thm_2Ellist_2Ellist__rep__LCONS,axiom,
! [A_27a: $tType,V0t: tyop_2Ellist_2Ellist @ A_27a,V1h: A_27a] :
( ( c_2Ellist_2Ellist__rep @ A_27a @ ( c_2Ellist_2ELCONS @ A_27a @ V1h @ V0t ) )
= ( ^ [V2n: tyop_2Enum_2Enum] : ( c_2Ebool_2ECOND @ ( tyop_2Eoption_2Eoption @ A_27a ) @ ( c_2Emin_2E_3D @ tyop_2Enum_2Enum @ V2n @ c_2Enum_2E0 ) @ ( c_2Eoption_2ESOME @ A_27a @ V1h ) @ ( c_2Ellist_2Ellist__rep @ A_27a @ V0t @ ( c_2Earithmetic_2E_2D @ V2n @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ) ) ) ).
thf(thm_2Ellist_2Ellist__rep__LNIL,axiom,
! [A_27a: $tType] :
( ( c_2Ellist_2Ellist__rep @ A_27a @ ( c_2Ellist_2ELNIL @ A_27a ) )
= ( ^ [V0n: tyop_2Enum_2Enum] : ( c_2Eoption_2ENONE @ A_27a ) ) ) ).
thf(thm_2Ellist_2ELTL__HD__LNIL,axiom,
! [A_27a: $tType] :
( ( c_2Ellist_2ELTL__HD @ A_27a @ ( c_2Ellist_2ELNIL @ A_27a ) )
= ( c_2Eoption_2ENONE @ ( tyop_2Epair_2Eprod @ ( tyop_2Ellist_2Ellist @ A_27a ) @ A_27a ) ) ) ).
thf(thm_2Ellist_2ELTL__HD__LCONS,axiom,
! [A_27a: $tType,V0t: tyop_2Ellist_2Ellist @ A_27a,V1h: A_27a] :
( ( c_2Ellist_2ELTL__HD @ A_27a @ ( c_2Ellist_2ELCONS @ A_27a @ V1h @ V0t ) )
= ( c_2Eoption_2ESOME @ ( tyop_2Epair_2Eprod @ ( tyop_2Ellist_2Ellist @ A_27a ) @ A_27a ) @ ( c_2Epair_2E_2C @ ( tyop_2Ellist_2Ellist @ A_27a ) @ A_27a @ V0t @ V1h ) ) ) ).
thf(thm_2Ellist_2ELTL__HD__HD,axiom,
! [A_27a: $tType,V0ll: tyop_2Ellist_2Ellist @ A_27a] :
( ( c_2Ellist_2ELHD @ A_27a @ V0ll )
= ( c_2Eoption_2EOPTION__MAP @ ( tyop_2Epair_2Eprod @ ( tyop_2Ellist_2Ellist @ A_27a ) @ A_27a ) @ A_27a @ ( c_2Epair_2ESND @ ( tyop_2Ellist_2Ellist @ A_27a ) @ A_27a ) @ ( c_2Ellist_2ELTL__HD @ A_27a @ V0ll ) ) ) ).
thf(thm_2Ellist_2ELTL__HD__TL,axiom,
! [A_27a: $tType,V0ll: tyop_2Ellist_2Ellist @ A_27a] :
( ( c_2Ellist_2ELTL @ A_27a @ V0ll )
= ( c_2Eoption_2EOPTION__MAP @ ( tyop_2Epair_2Eprod @ ( tyop_2Ellist_2Ellist @ A_27a ) @ A_27a ) @ ( tyop_2Ellist_2Ellist @ A_27a ) @ ( c_2Epair_2EFST @ ( tyop_2Ellist_2Ellist @ A_27a ) @ A_27a ) @ ( c_2Ellist_2ELTL__HD @ A_27a @ V0ll ) ) ) ).
thf(thm_2Ellist_2ELHD__LCONS,axiom,
! [A_27a: $tType,V0t: tyop_2Ellist_2Ellist @ A_27a,V1h: A_27a] :
( ( c_2Ellist_2ELHD @ A_27a @ ( c_2Ellist_2ELCONS @ A_27a @ V1h @ V0t ) )
= ( c_2Eoption_2ESOME @ A_27a @ V1h ) ) ).
thf(thm_2Ellist_2ELTL__LCONS,axiom,
! [A_27a: $tType,V0t: tyop_2Ellist_2Ellist @ A_27a,V1h: A_27a] :
( ( c_2Ellist_2ELTL @ A_27a @ ( c_2Ellist_2ELCONS @ A_27a @ V1h @ V0t ) )
= ( c_2Eoption_2ESOME @ ( tyop_2Ellist_2Ellist @ A_27a ) @ V0t ) ) ).
thf(thm_2Ellist_2ELHDTL__CONS__THM,axiom,
! [A_27a: $tType,V0h: A_27a,V1t: tyop_2Ellist_2Ellist @ A_27a] :
( ( ( c_2Ellist_2ELHD @ A_27a @ ( c_2Ellist_2ELCONS @ A_27a @ V0h @ V1t ) )
= ( c_2Eoption_2ESOME @ A_27a @ V0h ) )
& ( ( c_2Ellist_2ELTL @ A_27a @ ( c_2Ellist_2ELCONS @ A_27a @ V0h @ V1t ) )
= ( c_2Eoption_2ESOME @ ( tyop_2Ellist_2Ellist @ A_27a ) @ V1t ) ) ) ).
thf(thm_2Ellist_2Ellist__CASES,axiom,
! [A_27a: $tType,V0l: tyop_2Ellist_2Ellist @ A_27a] :
( ( V0l
= ( c_2Ellist_2ELNIL @ A_27a ) )
| ? [V1h: A_27a,V2t: tyop_2Ellist_2Ellist @ A_27a] :
( V0l
= ( c_2Ellist_2ELCONS @ A_27a @ V1h @ V2t ) ) ) ).
thf(thm_2Ellist_2ELTL__HD__11,axiom,
! [A_27a: $tType,V0ll2: tyop_2Ellist_2Ellist @ A_27a,V1ll1: tyop_2Ellist_2Ellist @ A_27a] :
( ( ( c_2Ellist_2ELTL__HD @ A_27a @ V1ll1 )
= ( c_2Ellist_2ELTL__HD @ A_27a @ V0ll2 ) )
=> ( V1ll1 = V0ll2 ) ) ).
thf(thm_2Ellist_2ELHD__THM,axiom,
! [A_27a: $tType,A_27b: $tType] :
( ( ( c_2Ellist_2ELHD @ A_27a @ ( c_2Ellist_2ELNIL @ A_27a ) )
= ( c_2Eoption_2ENONE @ A_27a ) )
& ! [V0h: A_27b,V1t: tyop_2Ellist_2Ellist @ A_27b] :
( ( c_2Ellist_2ELHD @ A_27b @ ( c_2Ellist_2ELCONS @ A_27b @ V0h @ V1t ) )
= ( c_2Eoption_2ESOME @ A_27b @ V0h ) ) ) ).
thf(thm_2Ellist_2ELTL__THM,axiom,
! [A_27a: $tType,A_27b: $tType] :
( ( ( c_2Ellist_2ELTL @ A_27a @ ( c_2Ellist_2ELNIL @ A_27a ) )
= ( c_2Eoption_2ENONE @ ( tyop_2Ellist_2Ellist @ A_27a ) ) )
& ! [V0h: A_27b,V1t: tyop_2Ellist_2Ellist @ A_27b] :
( ( c_2Ellist_2ELTL @ A_27b @ ( c_2Ellist_2ELCONS @ A_27b @ V0h @ V1t ) )
= ( c_2Eoption_2ESOME @ ( tyop_2Ellist_2Ellist @ A_27b ) @ V1t ) ) ) ).
thf(thm_2Ellist_2ELCONS__NOT__NIL,axiom,
! [A_27a: $tType,V0h: A_27a,V1t: tyop_2Ellist_2Ellist @ A_27a] :
( ( (~)
@ ( ( c_2Ellist_2ELCONS @ A_27a @ V0h @ V1t )
= ( c_2Ellist_2ELNIL @ A_27a ) ) )
& ( (~)
@ ( ( c_2Ellist_2ELNIL @ A_27a )
= ( c_2Ellist_2ELCONS @ A_27a @ V0h @ V1t ) ) ) ) ).
thf(thm_2Ellist_2ELCONS__11,axiom,
! [A_27a: $tType,V0h1: A_27a,V1t1: tyop_2Ellist_2Ellist @ A_27a,V2h2: A_27a,V3t2: tyop_2Ellist_2Ellist @ A_27a] :
( ( ( c_2Ellist_2ELCONS @ A_27a @ V0h1 @ V1t1 )
= ( c_2Ellist_2ELCONS @ A_27a @ V2h2 @ V3t2 ) )
<=> ( ( V0h1 = V2h2 )
& ( V1t1 = V3t2 ) ) ) ).
thf(thm_2Ellist_2ELTL__HD__iff,axiom,
! [A_27a: $tType,V0x: tyop_2Ellist_2Ellist @ A_27a,V1t: tyop_2Ellist_2Ellist @ A_27a,V2h: A_27a] :
( ( ( ( c_2Ellist_2ELTL__HD @ A_27a @ V0x )
= ( c_2Eoption_2ESOME @ ( tyop_2Epair_2Eprod @ ( tyop_2Ellist_2Ellist @ A_27a ) @ A_27a ) @ ( c_2Epair_2E_2C @ ( tyop_2Ellist_2Ellist @ A_27a ) @ A_27a @ V1t @ V2h ) ) )
<=> ( V0x
= ( c_2Ellist_2ELCONS @ A_27a @ V2h @ V1t ) ) )
& ( ( ( c_2Ellist_2ELTL__HD @ A_27a @ V0x )
= ( c_2Eoption_2ENONE @ ( tyop_2Epair_2Eprod @ ( tyop_2Ellist_2Ellist @ A_27a ) @ A_27a ) ) )
<=> ( V0x
= ( c_2Ellist_2ELNIL @ A_27a ) ) ) ) ).
thf(thm_2Ellist_2ELHD__EQ__NONE,axiom,
! [A_27a: $tType,V0ll: tyop_2Ellist_2Ellist @ A_27a] :
( ( ( ( c_2Ellist_2ELHD @ A_27a @ V0ll )
= ( c_2Eoption_2ENONE @ A_27a ) )
<=> ( V0ll
= ( c_2Ellist_2ELNIL @ A_27a ) ) )
& ( ( ( c_2Eoption_2ENONE @ A_27a )
= ( c_2Ellist_2ELHD @ A_27a @ V0ll ) )
<=> ( V0ll
= ( c_2Ellist_2ELNIL @ A_27a ) ) ) ) ).
thf(thm_2Ellist_2ELTL__EQ__NONE,axiom,
! [A_27a: $tType,V0ll: tyop_2Ellist_2Ellist @ A_27a] :
( ( ( ( c_2Ellist_2ELTL @ A_27a @ V0ll )
= ( c_2Eoption_2ENONE @ ( tyop_2Ellist_2Ellist @ A_27a ) ) )
<=> ( V0ll
= ( c_2Ellist_2ELNIL @ A_27a ) ) )
& ( ( ( c_2Eoption_2ENONE @ ( tyop_2Ellist_2Ellist @ A_27a ) )
= ( c_2Ellist_2ELTL @ A_27a @ V0ll ) )
<=> ( V0ll
= ( c_2Ellist_2ELNIL @ A_27a ) ) ) ) ).
thf(thm_2Ellist_2ELHDTL__EQ__SOME,axiom,
! [A_27a: $tType,V0h: A_27a,V1t: tyop_2Ellist_2Ellist @ A_27a,V2ll: tyop_2Ellist_2Ellist @ A_27a] :
( ( V2ll
= ( c_2Ellist_2ELCONS @ A_27a @ V0h @ V1t ) )
<=> ( ( ( c_2Ellist_2ELHD @ A_27a @ V2ll )
= ( c_2Eoption_2ESOME @ A_27a @ V0h ) )
& ( ( c_2Ellist_2ELTL @ A_27a @ V2ll )
= ( c_2Eoption_2ESOME @ ( tyop_2Ellist_2Ellist @ A_27a ) @ V1t ) ) ) ) ).
thf(thm_2Ellist_2ELNTH__THM,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType] :
( ! [V0n: tyop_2Enum_2Enum] :
( ( c_2Ellist_2ELNTH @ A_27a @ V0n @ ( c_2Ellist_2ELNIL @ A_27a ) )
= ( c_2Eoption_2ENONE @ A_27a ) )
& ! [V1h: A_27b,V2t: tyop_2Ellist_2Ellist @ A_27b] :
( ( c_2Ellist_2ELNTH @ A_27b @ c_2Enum_2E0 @ ( c_2Ellist_2ELCONS @ A_27b @ V1h @ V2t ) )
= ( c_2Eoption_2ESOME @ A_27b @ V1h ) )
& ! [V3n: tyop_2Enum_2Enum,V4h: A_27c,V5t: tyop_2Ellist_2Ellist @ A_27c] :
( ( c_2Ellist_2ELNTH @ A_27c @ ( c_2Enum_2ESUC @ V3n ) @ ( c_2Ellist_2ELCONS @ A_27c @ V4h @ V5t ) )
= ( c_2Ellist_2ELNTH @ A_27c @ V3n @ V5t ) ) ) ).
thf(thm_2Ellist_2ELNTH__rep,axiom,
! [A_27a: $tType,V0i: tyop_2Enum_2Enum,V1ll: tyop_2Ellist_2Ellist @ A_27a] :
( ( c_2Ellist_2ELNTH @ A_27a @ V0i @ V1ll )
= ( c_2Ellist_2Ellist__rep @ A_27a @ V1ll @ V0i ) ) ).
thf(thm_2Ellist_2ELNTH__EQ,axiom,
! [A_27a: $tType,V0ll1: tyop_2Ellist_2Ellist @ A_27a,V1ll2: tyop_2Ellist_2Ellist @ A_27a] :
( ( V0ll1 = V1ll2 )
<=> ! [V2n: tyop_2Enum_2Enum] :
( ( c_2Ellist_2ELNTH @ A_27a @ V2n @ V0ll1 )
= ( c_2Ellist_2ELNTH @ A_27a @ V2n @ V1ll2 ) ) ) ).
thf(thm_2Ellist_2ELUNFOLD,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: A_27a > ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) ),V1x: A_27a] :
( ( c_2Ellist_2ELUNFOLD @ A_27b @ A_27a @ V0f @ V1x )
= ( c_2Eoption_2Eoption__CASE @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ ( tyop_2Ellist_2Ellist @ A_27b ) @ ( V0f @ V1x ) @ ( c_2Ellist_2ELNIL @ A_27b )
@ ^ [V2v: tyop_2Epair_2Eprod @ A_27a @ A_27b] :
( c_2Epair_2Epair__CASE @ ( tyop_2Ellist_2Ellist @ A_27b ) @ A_27a @ A_27b @ V2v
@ ^ [V3v1: A_27a,V4v2: A_27b] : ( c_2Ellist_2ELCONS @ A_27b @ V4v2 @ ( c_2Ellist_2ELUNFOLD @ A_27b @ A_27a @ V0f @ V3v1 ) ) ) ) ) ).
thf(thm_2Ellist_2ELUNFOLD__UNIQUE,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: A_27a > ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) ),V1g: A_27a > ( tyop_2Ellist_2Ellist @ A_27b )] :
( ! [V2x: A_27a] :
( ( V1g @ V2x )
= ( c_2Eoption_2Eoption__CASE @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ ( tyop_2Ellist_2Ellist @ A_27b ) @ ( V0f @ V2x ) @ ( c_2Ellist_2ELNIL @ A_27b )
@ ^ [V3v: tyop_2Epair_2Eprod @ A_27a @ A_27b] :
( c_2Epair_2Epair__CASE @ ( tyop_2Ellist_2Ellist @ A_27b ) @ A_27a @ A_27b @ V3v
@ ^ [V4v1: A_27a,V5v2: A_27b] : ( c_2Ellist_2ELCONS @ A_27b @ V5v2 @ ( V1g @ V4v1 ) ) ) ) )
=> ! [V6y: A_27a] :
( ( V1g @ V6y )
= ( c_2Ellist_2ELUNFOLD @ A_27b @ A_27a @ V0f @ V6y ) ) ) ).
thf(thm_2Ellist_2ELUNFOLD__LTL__HD,axiom,
! [A_27a: $tType,V0ll: tyop_2Ellist_2Ellist @ A_27a] :
( ( c_2Ellist_2ELUNFOLD @ A_27a @ ( tyop_2Ellist_2Ellist @ A_27a ) @ ( c_2Ellist_2ELTL__HD @ A_27a ) @ V0ll )
= V0ll ) ).
thf(thm_2Ellist_2ELTL__HD__LUNFOLD,axiom,
! [A_27a: $tType,A_27b: $tType,V0x: A_27b,V1f: A_27b > ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ A_27b @ A_27a ) )] :
( ( c_2Ellist_2ELTL__HD @ A_27a @ ( c_2Ellist_2ELUNFOLD @ A_27a @ A_27b @ V1f @ V0x ) )
= ( c_2Eoption_2EOPTION__MAP @ ( tyop_2Epair_2Eprod @ A_27b @ A_27a ) @ ( tyop_2Epair_2Eprod @ ( tyop_2Ellist_2Ellist @ A_27a ) @ A_27a ) @ ( c_2Epair_2E_23_23 @ A_27b @ A_27a @ ( tyop_2Ellist_2Ellist @ A_27a ) @ A_27a @ ( c_2Ellist_2ELUNFOLD @ A_27a @ A_27b @ V1f ) @ ( c_2Ecombin_2EI @ A_27a ) ) @ ( V1f @ V0x ) ) ) ).
thf(thm_2Ellist_2ELNTH__LUNFOLD,axiom,
! [A_27a: $tType,A_27b: $tType,V0x: A_27b,V1n: tyop_2Enum_2Enum,V2f: A_27b > ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ A_27b @ A_27a ) )] :
( ( ( c_2Ellist_2ELNTH @ A_27a @ c_2Enum_2E0 @ ( c_2Ellist_2ELUNFOLD @ A_27a @ A_27b @ V2f @ V0x ) )
= ( c_2Eoption_2EOPTION__MAP @ ( tyop_2Epair_2Eprod @ A_27b @ A_27a ) @ A_27a @ ( c_2Epair_2ESND @ A_27b @ A_27a ) @ ( V2f @ V0x ) ) )
& ( ( c_2Ellist_2ELNTH @ A_27a @ ( c_2Enum_2ESUC @ V1n ) @ ( c_2Ellist_2ELUNFOLD @ A_27a @ A_27b @ V2f @ V0x ) )
= ( c_2Eoption_2Eoption__CASE @ ( tyop_2Epair_2Eprod @ A_27b @ A_27a ) @ ( tyop_2Eoption_2Eoption @ A_27a ) @ ( V2f @ V0x ) @ ( c_2Eoption_2ENONE @ A_27a )
@ ^ [V3v: tyop_2Epair_2Eprod @ A_27b @ A_27a] :
( c_2Epair_2Epair__CASE @ ( tyop_2Eoption_2Eoption @ A_27a ) @ A_27b @ A_27a @ V3v
@ ^ [V4tx: A_27b,V5hx: A_27a] : ( c_2Ellist_2ELNTH @ A_27a @ V1n @ ( c_2Ellist_2ELUNFOLD @ A_27a @ A_27b @ V2f @ V4tx ) ) ) ) ) ) ).
thf(thm_2Ellist_2ELNTH__LUNFOLD__compute,axiom,
! [A_27a: $tType,A_27b: $tType,V0x: A_27b,V1f: A_27b > ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ A_27b @ A_27a ) )] :
( ( ( c_2Ellist_2ELNTH @ A_27a @ c_2Enum_2E0 @ ( c_2Ellist_2ELUNFOLD @ A_27a @ A_27b @ V1f @ V0x ) )
= ( c_2Eoption_2EOPTION__MAP @ ( tyop_2Epair_2Eprod @ A_27b @ A_27a ) @ A_27a @ ( c_2Epair_2ESND @ A_27b @ A_27a ) @ ( V1f @ V0x ) ) )
& ! [V2n: tyop_2Enum_2Enum] :
( ( c_2Ellist_2ELNTH @ A_27a @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ V2n ) ) @ ( c_2Ellist_2ELUNFOLD @ A_27a @ A_27b @ V1f @ V0x ) )
= ( c_2Eoption_2Eoption__CASE @ ( tyop_2Epair_2Eprod @ A_27b @ A_27a ) @ ( tyop_2Eoption_2Eoption @ A_27a ) @ ( V1f @ V0x ) @ ( c_2Eoption_2ENONE @ A_27a )
@ ^ [V3v: tyop_2Epair_2Eprod @ A_27b @ A_27a] :
( c_2Epair_2Epair__CASE @ ( tyop_2Eoption_2Eoption @ A_27a ) @ A_27b @ A_27a @ V3v
@ ^ [V4tx: A_27b,V5hx: A_27a] : ( c_2Ellist_2ELNTH @ A_27a @ ( c_2Earithmetic_2E_2D @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ V2n ) ) @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) @ ( c_2Ellist_2ELUNFOLD @ A_27a @ A_27b @ V1f @ V4tx ) ) ) ) )
& ! [V6n: tyop_2Enum_2Enum] :
( ( c_2Ellist_2ELNTH @ A_27a @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ V6n ) ) @ ( c_2Ellist_2ELUNFOLD @ A_27a @ A_27b @ V1f @ V0x ) )
= ( c_2Eoption_2Eoption__CASE @ ( tyop_2Epair_2Eprod @ A_27b @ A_27a ) @ ( tyop_2Eoption_2Eoption @ A_27a ) @ ( V1f @ V0x ) @ ( c_2Eoption_2ENONE @ A_27a )
@ ^ [V7v: tyop_2Epair_2Eprod @ A_27b @ A_27a] :
( c_2Epair_2Epair__CASE @ ( tyop_2Eoption_2Eoption @ A_27a ) @ A_27b @ A_27a @ V7v
@ ^ [V8tx: A_27b,V9hx: A_27a] : ( c_2Ellist_2ELNTH @ A_27a @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ V6n ) ) @ ( c_2Ellist_2ELUNFOLD @ A_27a @ A_27b @ V1f @ V8tx ) ) ) ) ) ) ).
thf(thm_2Ellist_2ELHD__LUNFOLD,axiom,
! [A_27a: $tType,A_27b: $tType,V0x: A_27b,V1f: A_27b > ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ A_27b @ A_27a ) )] :
( ( c_2Ellist_2ELHD @ A_27a @ ( c_2Ellist_2ELUNFOLD @ A_27a @ A_27b @ V1f @ V0x ) )
= ( c_2Eoption_2EOPTION__MAP @ ( tyop_2Epair_2Eprod @ A_27b @ A_27a ) @ A_27a @ ( c_2Epair_2ESND @ A_27b @ A_27a ) @ ( V1f @ V0x ) ) ) ).
thf(thm_2Ellist_2ELTL__LUNFOLD,axiom,
! [A_27a: $tType,A_27b: $tType,V0x: A_27b,V1f: A_27b > ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ A_27b @ A_27a ) )] :
( ( c_2Ellist_2ELTL @ A_27a @ ( c_2Ellist_2ELUNFOLD @ A_27a @ A_27b @ V1f @ V0x ) )
= ( c_2Eoption_2EOPTION__MAP @ ( tyop_2Epair_2Eprod @ A_27b @ A_27a ) @ ( tyop_2Ellist_2Ellist @ A_27a ) @ ( c_2Ecombin_2Eo @ ( tyop_2Epair_2Eprod @ A_27b @ A_27a ) @ ( tyop_2Ellist_2Ellist @ A_27a ) @ A_27b @ ( c_2Ellist_2ELUNFOLD @ A_27a @ A_27b @ V1f ) @ ( c_2Epair_2EFST @ A_27b @ A_27a ) ) @ ( V1f @ V0x ) ) ) ).
thf(thm_2Ellist_2Ellist__Axiom__1,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: A_27a > ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) )] :
? [V1g: A_27a > ( tyop_2Ellist_2Ellist @ A_27b )] :
! [V2x: A_27a] :
( ( V1g @ V2x )
= ( c_2Eoption_2Eoption__CASE @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ ( tyop_2Ellist_2Ellist @ A_27b ) @ ( V0f @ V2x ) @ ( c_2Ellist_2ELNIL @ A_27b )
@ ^ [V3v: tyop_2Epair_2Eprod @ A_27a @ A_27b] :
( c_2Epair_2Epair__CASE @ ( tyop_2Ellist_2Ellist @ A_27b ) @ A_27a @ A_27b @ V3v
@ ^ [V4a: A_27a,V5b: A_27b] : ( c_2Ellist_2ELCONS @ A_27b @ V5b @ ( V1g @ V4a ) ) ) ) ) ).
thf(thm_2Ellist_2Ellist__Axiom__1ue,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: A_27a > ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) )] :
( c_2Ebool_2E_3F_21 @ ( A_27a > ( tyop_2Ellist_2Ellist @ A_27b ) )
@ ^ [V1g: A_27a > ( tyop_2Ellist_2Ellist @ A_27b )] :
( c_2Ebool_2E_21 @ A_27a
@ ^ [V2x: A_27a] :
( c_2Emin_2E_3D @ ( tyop_2Ellist_2Ellist @ A_27b ) @ ( V1g @ V2x )
@ ( c_2Eoption_2Eoption__CASE @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ ( tyop_2Ellist_2Ellist @ A_27b ) @ ( V0f @ V2x ) @ ( c_2Ellist_2ELNIL @ A_27b )
@ ^ [V3v: tyop_2Epair_2Eprod @ A_27a @ A_27b] :
( c_2Epair_2Epair__CASE @ ( tyop_2Ellist_2Ellist @ A_27b ) @ A_27a @ A_27b @ V3v
@ ^ [V4a: A_27a,V5b: A_27b] : ( c_2Ellist_2ELCONS @ A_27b @ V5b @ ( V1g @ V4a ) ) ) ) ) ) ) ).
thf(thm_2Ellist_2Ellist__ue__Axiom,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: A_27a > ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) )] :
( c_2Ebool_2E_3F_21 @ ( A_27a > ( tyop_2Ellist_2Ellist @ A_27b ) )
@ ^ [V1g: A_27a > ( tyop_2Ellist_2Ellist @ A_27b )] :
( c_2Ebool_2E_2F_5C
@ ( c_2Ebool_2E_21 @ A_27a
@ ^ [V2x: A_27a] : ( c_2Emin_2E_3D @ ( tyop_2Eoption_2Eoption @ A_27b ) @ ( c_2Ellist_2ELHD @ A_27b @ ( V1g @ V2x ) ) @ ( c_2Eoption_2EOPTION__MAP @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ A_27b @ ( c_2Epair_2ESND @ A_27a @ A_27b ) @ ( V0f @ V2x ) ) ) )
@ ( c_2Ebool_2E_21 @ A_27a
@ ^ [V3x: A_27a] : ( c_2Emin_2E_3D @ ( tyop_2Eoption_2Eoption @ ( tyop_2Ellist_2Ellist @ A_27b ) ) @ ( c_2Ellist_2ELTL @ A_27b @ ( V1g @ V3x ) ) @ ( c_2Eoption_2EOPTION__MAP @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ ( tyop_2Ellist_2Ellist @ A_27b ) @ ( c_2Ecombin_2Eo @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ ( tyop_2Ellist_2Ellist @ A_27b ) @ A_27a @ V1g @ ( c_2Epair_2EFST @ A_27a @ A_27b ) ) @ ( V0f @ V3x ) ) ) ) ) ) ).
thf(thm_2Ellist_2Ellist__Axiom,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: A_27a > ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) )] :
? [V1g: A_27a > ( tyop_2Ellist_2Ellist @ A_27b )] :
( ! [V2x: A_27a] :
( ( c_2Ellist_2ELHD @ A_27b @ ( V1g @ V2x ) )
= ( c_2Eoption_2EOPTION__MAP @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ A_27b @ ( c_2Epair_2ESND @ A_27a @ A_27b ) @ ( V0f @ V2x ) ) )
& ! [V3x: A_27a] :
( ( c_2Ellist_2ELTL @ A_27b @ ( V1g @ V3x ) )
= ( c_2Eoption_2EOPTION__MAP @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ ( tyop_2Ellist_2Ellist @ A_27b ) @ ( c_2Ecombin_2Eo @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ ( tyop_2Ellist_2Ellist @ A_27b ) @ A_27a @ V1g @ ( c_2Epair_2EFST @ A_27a @ A_27b ) ) @ ( V0f @ V3x ) ) ) ) ).
thf(thm_2Ellist_2ELUNFOLD__BISIMULATION,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0f1: A_27a > ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) ),V1f2: A_27c > ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ A_27c @ A_27b ) ),V2x1: A_27a,V3x2: A_27c] :
( ( ( c_2Ellist_2ELUNFOLD @ A_27b @ A_27a @ V0f1 @ V2x1 )
= ( c_2Ellist_2ELUNFOLD @ A_27b @ A_27c @ V1f2 @ V3x2 ) )
<=> ? [V4R: A_27a > A_27c > $o] :
( ( V4R @ V2x1 @ V3x2 )
& ! [V5y1: A_27a,V6y2: A_27c] :
( ( V4R @ V5y1 @ V6y2 )
=> ( ( ( ( V0f1 @ V5y1 )
= ( c_2Eoption_2ENONE @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) ) )
& ( ( V1f2 @ V6y2 )
= ( c_2Eoption_2ENONE @ ( tyop_2Epair_2Eprod @ A_27c @ A_27b ) ) ) )
| ? [V7h: A_27b,V8t1: A_27a,V9t2: A_27c] :
( ( ( V0f1 @ V5y1 )
= ( c_2Eoption_2ESOME @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ ( c_2Epair_2E_2C @ A_27a @ A_27b @ V8t1 @ V7h ) ) )
& ( ( V1f2 @ V6y2 )
= ( c_2Eoption_2ESOME @ ( tyop_2Epair_2Eprod @ A_27c @ A_27b ) @ ( c_2Epair_2E_2C @ A_27c @ A_27b @ V9t2 @ V7h ) ) )
& ( V4R @ V8t1 @ V9t2 ) ) ) ) ) ) ).
thf(thm_2Ellist_2ELLIST__BISIMULATION0,axiom,
! [A_27a: $tType,V0ll1: tyop_2Ellist_2Ellist @ A_27a,V1ll2: tyop_2Ellist_2Ellist @ A_27a] :
( ( V0ll1 = V1ll2 )
<=> ? [V2R: ( tyop_2Ellist_2Ellist @ A_27a ) > ( tyop_2Ellist_2Ellist @ A_27a ) > $o] :
( ( V2R @ V0ll1 @ V1ll2 )
& ! [V3ll3: tyop_2Ellist_2Ellist @ A_27a,V4ll4: tyop_2Ellist_2Ellist @ A_27a] :
( ( V2R @ V3ll3 @ V4ll4 )
=> ( ( ( V3ll3
= ( c_2Ellist_2ELNIL @ A_27a ) )
& ( V4ll4
= ( c_2Ellist_2ELNIL @ A_27a ) ) )
| ? [V5h: A_27a,V6t1: tyop_2Ellist_2Ellist @ A_27a,V7t2: tyop_2Ellist_2Ellist @ A_27a] :
( ( V3ll3
= ( c_2Ellist_2ELCONS @ A_27a @ V5h @ V6t1 ) )
& ( V4ll4
= ( c_2Ellist_2ELCONS @ A_27a @ V5h @ V7t2 ) )
& ( V2R @ V6t1 @ V7t2 ) ) ) ) ) ) ).
thf(thm_2Ellist_2ELLIST__BISIMULATION,axiom,
! [A_27a: $tType,V0ll1: tyop_2Ellist_2Ellist @ A_27a,V1ll2: tyop_2Ellist_2Ellist @ A_27a] :
( ( V0ll1 = V1ll2 )
<=> ? [V2R: ( tyop_2Ellist_2Ellist @ A_27a ) > ( tyop_2Ellist_2Ellist @ A_27a ) > $o] :
( ( V2R @ V0ll1 @ V1ll2 )
& ! [V3ll3: tyop_2Ellist_2Ellist @ A_27a,V4ll4: tyop_2Ellist_2Ellist @ A_27a] :
( ( V2R @ V3ll3 @ V4ll4 )
=> ( ( ( V3ll3
= ( c_2Ellist_2ELNIL @ A_27a ) )
& ( V4ll4
= ( c_2Ellist_2ELNIL @ A_27a ) ) )
| ( ( ( c_2Ellist_2ELHD @ A_27a @ V3ll3 )
= ( c_2Ellist_2ELHD @ A_27a @ V4ll4 ) )
& ( V2R @ ( c_2Eoption_2ETHE @ ( tyop_2Ellist_2Ellist @ A_27a ) @ ( c_2Ellist_2ELTL @ A_27a @ V3ll3 ) ) @ ( c_2Eoption_2ETHE @ ( tyop_2Ellist_2Ellist @ A_27a ) @ ( c_2Ellist_2ELTL @ A_27a @ V4ll4 ) ) ) ) ) ) ) ) ).
thf(thm_2Ellist_2ELLIST__STRONG__BISIMULATION,axiom,
! [A_27a: $tType,V0ll1: tyop_2Ellist_2Ellist @ A_27a,V1ll2: tyop_2Ellist_2Ellist @ A_27a] :
( ( V0ll1 = V1ll2 )
<=> ? [V2R: ( tyop_2Ellist_2Ellist @ A_27a ) > ( tyop_2Ellist_2Ellist @ A_27a ) > $o] :
( ( V2R @ V0ll1 @ V1ll2 )
& ! [V3ll3: tyop_2Ellist_2Ellist @ A_27a,V4ll4: tyop_2Ellist_2Ellist @ A_27a] :
( ( V2R @ V3ll3 @ V4ll4 )
=> ( ( V3ll3 = V4ll4 )
| ? [V5h: A_27a,V6t1: tyop_2Ellist_2Ellist @ A_27a,V7t2: tyop_2Ellist_2Ellist @ A_27a] :
( ( V3ll3
= ( c_2Ellist_2ELCONS @ A_27a @ V5h @ V6t1 ) )
& ( V4ll4
= ( c_2Ellist_2ELCONS @ A_27a @ V5h @ V7t2 ) )
& ( V2R @ V6t1 @ V7t2 ) ) ) ) ) ) ).
thf(thm_2Ellist_2ELTAKE__LUNFOLD,axiom,
! [A_27a: $tType,A_27b: $tType,V0x: A_27b,V1n: tyop_2Enum_2Enum,V2f: A_27b > ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ A_27b @ A_27a ) )] :
( ( ( c_2Ellist_2ELTAKE @ A_27a @ c_2Enum_2E0 @ ( c_2Ellist_2ELUNFOLD @ A_27a @ A_27b @ V2f @ V0x ) )
= ( c_2Eoption_2ESOME @ ( tyop_2Elist_2Elist @ A_27a ) @ ( c_2Elist_2ENIL @ A_27a ) ) )
& ( ( c_2Ellist_2ELTAKE @ A_27a @ ( c_2Enum_2ESUC @ V1n ) @ ( c_2Ellist_2ELUNFOLD @ A_27a @ A_27b @ V2f @ V0x ) )
= ( c_2Eoption_2Eoption__CASE @ ( tyop_2Epair_2Eprod @ A_27b @ A_27a ) @ ( tyop_2Eoption_2Eoption @ ( tyop_2Elist_2Elist @ A_27a ) ) @ ( V2f @ V0x ) @ ( c_2Eoption_2ENONE @ ( tyop_2Elist_2Elist @ A_27a ) )
@ ^ [V3v: tyop_2Epair_2Eprod @ A_27b @ A_27a] :
( c_2Epair_2Epair__CASE @ ( tyop_2Eoption_2Eoption @ ( tyop_2Elist_2Elist @ A_27a ) ) @ A_27b @ A_27a @ V3v
@ ^ [V4tx: A_27b,V5hx: A_27a] : ( c_2Eoption_2EOPTION__MAP @ ( tyop_2Elist_2Elist @ A_27a ) @ ( tyop_2Elist_2Elist @ A_27a ) @ ( c_2Elist_2ECONS @ A_27a @ V5hx ) @ ( c_2Ellist_2ELTAKE @ A_27a @ V1n @ ( c_2Ellist_2ELUNFOLD @ A_27a @ A_27b @ V2f @ V4tx ) ) ) ) ) ) ) ).
thf(thm_2Ellist_2ELTAKE__THM,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType] :
( ! [V0l: tyop_2Ellist_2Ellist @ A_27a] :
( ( c_2Ellist_2ELTAKE @ A_27a @ c_2Enum_2E0 @ V0l )
= ( c_2Eoption_2ESOME @ ( tyop_2Elist_2Elist @ A_27a ) @ ( c_2Elist_2ENIL @ A_27a ) ) )
& ! [V1n: tyop_2Enum_2Enum] :
( ( c_2Ellist_2ELTAKE @ A_27b @ ( c_2Enum_2ESUC @ V1n ) @ ( c_2Ellist_2ELNIL @ A_27b ) )
= ( c_2Eoption_2ENONE @ ( tyop_2Elist_2Elist @ A_27b ) ) )
& ! [V2n: tyop_2Enum_2Enum,V3h: A_27c,V4t: tyop_2Ellist_2Ellist @ A_27c] :
( ( c_2Ellist_2ELTAKE @ A_27c @ ( c_2Enum_2ESUC @ V2n ) @ ( c_2Ellist_2ELCONS @ A_27c @ V3h @ V4t ) )
= ( c_2Eoption_2EOPTION__MAP @ ( tyop_2Elist_2Elist @ A_27c ) @ ( tyop_2Elist_2Elist @ A_27c ) @ ( c_2Elist_2ECONS @ A_27c @ V3h ) @ ( c_2Ellist_2ELTAKE @ A_27c @ V2n @ V4t ) ) ) ) ).
thf(thm_2Ellist_2ELTAKE__SNOC__LNTH,axiom,
! [A_27a: $tType,V0n: tyop_2Enum_2Enum,V1ll: tyop_2Ellist_2Ellist @ A_27a] :
( ( c_2Ellist_2ELTAKE @ A_27a @ ( c_2Enum_2ESUC @ V0n ) @ V1ll )
= ( c_2Eoption_2Eoption__CASE @ ( tyop_2Elist_2Elist @ A_27a ) @ ( tyop_2Eoption_2Eoption @ ( tyop_2Elist_2Elist @ A_27a ) ) @ ( c_2Ellist_2ELTAKE @ A_27a @ V0n @ V1ll ) @ ( c_2Eoption_2ENONE @ ( tyop_2Elist_2Elist @ A_27a ) )
@ ^ [V2l: tyop_2Elist_2Elist @ A_27a] :
( c_2Eoption_2Eoption__CASE @ A_27a @ ( tyop_2Eoption_2Eoption @ ( tyop_2Elist_2Elist @ A_27a ) ) @ ( c_2Ellist_2ELNTH @ A_27a @ V0n @ V1ll ) @ ( c_2Eoption_2ENONE @ ( tyop_2Elist_2Elist @ A_27a ) )
@ ^ [V3e: A_27a] : ( c_2Eoption_2ESOME @ ( tyop_2Elist_2Elist @ A_27a ) @ ( c_2Elist_2EAPPEND @ A_27a @ V2l @ ( c_2Elist_2ECONS @ A_27a @ V3e @ ( c_2Elist_2ENIL @ A_27a ) ) ) ) ) ) ) ).
thf(thm_2Ellist_2ELTAKE__EQ__NONE__LNTH,axiom,
! [A_27a: $tType,V0n: tyop_2Enum_2Enum,V1ll: tyop_2Ellist_2Ellist @ A_27a] :
( ( ( c_2Ellist_2ELTAKE @ A_27a @ V0n @ V1ll )
= ( c_2Eoption_2ENONE @ ( tyop_2Elist_2Elist @ A_27a ) ) )
=> ( ( c_2Ellist_2ELNTH @ A_27a @ V0n @ V1ll )
= ( c_2Eoption_2ENONE @ A_27a ) ) ) ).
thf(thm_2Ellist_2ELTAKE__NIL__EQ__SOME,axiom,
! [A_27a: $tType,V0l: tyop_2Elist_2Elist @ A_27a,V1m: tyop_2Enum_2Enum] :
( ( ( c_2Ellist_2ELTAKE @ A_27a @ V1m @ ( c_2Ellist_2ELNIL @ A_27a ) )
= ( c_2Eoption_2ESOME @ ( tyop_2Elist_2Elist @ A_27a ) @ V0l ) )
<=> ( ( V1m = c_2Enum_2E0 )
& ( V0l
= ( c_2Elist_2ENIL @ A_27a ) ) ) ) ).
thf(thm_2Ellist_2ELTAKE__NIL__EQ__NONE,axiom,
! [A_27a: $tType,V0m: tyop_2Enum_2Enum] :
( ( ( c_2Ellist_2ELTAKE @ A_27a @ V0m @ ( c_2Ellist_2ELNIL @ A_27a ) )
= ( c_2Eoption_2ENONE @ ( tyop_2Elist_2Elist @ A_27a ) ) )
<=> ( c_2Eprim__rec_2E_3C @ c_2Enum_2E0 @ V0m ) ) ).
thf(thm_2Ellist_2ELTAKE__EQ,axiom,
! [A_27a: $tType,V0ll1: tyop_2Ellist_2Ellist @ A_27a,V1ll2: tyop_2Ellist_2Ellist @ A_27a] :
( ( V0ll1 = V1ll2 )
<=> ! [V2n: tyop_2Enum_2Enum] :
( ( c_2Ellist_2ELTAKE @ A_27a @ V2n @ V0ll1 )
= ( c_2Ellist_2ELTAKE @ A_27a @ V2n @ V1ll2 ) ) ) ).
thf(thm_2Ellist_2ELTAKE__CONS__EQ__NONE,axiom,
! [A_27a: $tType,V0m: tyop_2Enum_2Enum,V1h: A_27a,V2t: tyop_2Ellist_2Ellist @ A_27a] :
( ( ( c_2Ellist_2ELTAKE @ A_27a @ V0m @ ( c_2Ellist_2ELCONS @ A_27a @ V1h @ V2t ) )
= ( c_2Eoption_2ENONE @ ( tyop_2Elist_2Elist @ A_27a ) ) )
<=> ? [V3n: tyop_2Enum_2Enum] :
( ( V0m
= ( c_2Enum_2ESUC @ V3n ) )
& ( ( c_2Ellist_2ELTAKE @ A_27a @ V3n @ V2t )
= ( c_2Eoption_2ENONE @ ( tyop_2Elist_2Elist @ A_27a ) ) ) ) ) ).
thf(thm_2Ellist_2ELTAKE__CONS__EQ__SOME,axiom,
! [A_27a: $tType,V0m: tyop_2Enum_2Enum,V1h: A_27a,V2t: tyop_2Ellist_2Ellist @ A_27a,V3l: tyop_2Elist_2Elist @ A_27a] :
( ( ( c_2Ellist_2ELTAKE @ A_27a @ V0m @ ( c_2Ellist_2ELCONS @ A_27a @ V1h @ V2t ) )
= ( c_2Eoption_2ESOME @ ( tyop_2Elist_2Elist @ A_27a ) @ V3l ) )
<=> ( ( ( V0m = c_2Enum_2E0 )
& ( V3l
= ( c_2Elist_2ENIL @ A_27a ) ) )
| ? [V4n: tyop_2Enum_2Enum,V5l_27: tyop_2Elist_2Elist @ A_27a] :
( ( V0m
= ( c_2Enum_2ESUC @ V4n ) )
& ( ( c_2Ellist_2ELTAKE @ A_27a @ V4n @ V2t )
= ( c_2Eoption_2ESOME @ ( tyop_2Elist_2Elist @ A_27a ) @ V5l_27 ) )
& ( V3l
= ( c_2Elist_2ECONS @ A_27a @ V1h @ V5l_27 ) ) ) ) ) ).
thf(thm_2Ellist_2ELTAKE__EQ__SOME__CONS,axiom,
! [A_27a: $tType,V0n: tyop_2Enum_2Enum,V1l: tyop_2Ellist_2Ellist @ A_27a,V2x: tyop_2Elist_2Elist @ A_27a] :
( ( ( c_2Ellist_2ELTAKE @ A_27a @ V0n @ V1l )
= ( c_2Eoption_2ESOME @ ( tyop_2Elist_2Elist @ A_27a ) @ V2x ) )
=> ! [V3h: A_27a] :
? [V4y: tyop_2Elist_2Elist @ A_27a] :
( ( c_2Ellist_2ELTAKE @ A_27a @ V0n @ ( c_2Ellist_2ELCONS @ A_27a @ V3h @ V1l ) )
= ( c_2Eoption_2ESOME @ ( tyop_2Elist_2Elist @ A_27a ) @ V4y ) ) ) ).
thf(thm_2Ellist_2ELMAP__APPEND,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: A_27a > A_27b,V1ll1: tyop_2Ellist_2Ellist @ A_27a,V2ll2: tyop_2Ellist_2Ellist @ A_27a] :
( ( c_2Ellist_2ELMAP @ A_27a @ A_27b @ V0f @ ( c_2Ellist_2ELAPPEND @ A_27a @ V1ll1 @ V2ll2 ) )
= ( c_2Ellist_2ELAPPEND @ A_27b @ ( c_2Ellist_2ELMAP @ A_27a @ A_27b @ V0f @ V1ll1 ) @ ( c_2Ellist_2ELMAP @ A_27a @ A_27b @ V0f @ V2ll2 ) ) ) ).
thf(thm_2Ellist_2ELAPPEND__EQ__LNIL,axiom,
! [A_27a: $tType,V0l2: tyop_2Ellist_2Ellist @ A_27a,V1l1: tyop_2Ellist_2Ellist @ A_27a] :
( ( ( c_2Ellist_2ELAPPEND @ A_27a @ V1l1 @ V0l2 )
= ( c_2Ellist_2ELNIL @ A_27a ) )
<=> ( ( V1l1
= ( c_2Ellist_2ELNIL @ A_27a ) )
& ( V0l2
= ( c_2Ellist_2ELNIL @ A_27a ) ) ) ) ).
thf(thm_2Ellist_2ELAPPEND__ASSOC,axiom,
! [A_27a: $tType,V0ll1: tyop_2Ellist_2Ellist @ A_27a,V1ll2: tyop_2Ellist_2Ellist @ A_27a,V2ll3: tyop_2Ellist_2Ellist @ A_27a] :
( ( c_2Ellist_2ELAPPEND @ A_27a @ ( c_2Ellist_2ELAPPEND @ A_27a @ V0ll1 @ V1ll2 ) @ V2ll3 )
= ( c_2Ellist_2ELAPPEND @ A_27a @ V0ll1 @ ( c_2Ellist_2ELAPPEND @ A_27a @ V1ll2 @ V2ll3 ) ) ) ).
thf(thm_2Ellist_2ELMAP__MAP,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0f: A_27a > A_27b,V1g: A_27c > A_27a,V2ll: tyop_2Ellist_2Ellist @ A_27c] :
( ( c_2Ellist_2ELMAP @ A_27a @ A_27b @ V0f @ ( c_2Ellist_2ELMAP @ A_27c @ A_27a @ V1g @ V2ll ) )
= ( c_2Ellist_2ELMAP @ A_27c @ A_27b @ ( c_2Ecombin_2Eo @ A_27c @ A_27b @ A_27a @ V0f @ V1g ) @ V2ll ) ) ).
thf(thm_2Ellist_2ELAPPEND__NIL__2ND,axiom,
! [A_27a: $tType,V0ll: tyop_2Ellist_2Ellist @ A_27a] :
( ( c_2Ellist_2ELAPPEND @ A_27a @ V0ll @ ( c_2Ellist_2ELNIL @ A_27a ) )
= V0ll ) ).
thf(thm_2Ellist_2ELHD__LAPPEND,axiom,
! [A_27a: $tType,V0l2: tyop_2Ellist_2Ellist @ A_27a,V1l1: tyop_2Ellist_2Ellist @ A_27a] :
( ( c_2Ellist_2ELHD @ A_27a @ ( c_2Ellist_2ELAPPEND @ A_27a @ V1l1 @ V0l2 ) )
= ( c_2Ebool_2ECOND @ ( tyop_2Eoption_2Eoption @ A_27a ) @ ( c_2Emin_2E_3D @ ( tyop_2Ellist_2Ellist @ A_27a ) @ V1l1 @ ( c_2Ellist_2ELNIL @ A_27a ) ) @ ( c_2Ellist_2ELHD @ A_27a @ V0l2 ) @ ( c_2Ellist_2ELHD @ A_27a @ V1l1 ) ) ) ).
thf(thm_2Ellist_2ELTL__LAPPEND,axiom,
! [A_27a: $tType,V0l2: tyop_2Ellist_2Ellist @ A_27a,V1l1: tyop_2Ellist_2Ellist @ A_27a] :
( ( c_2Ellist_2ELTL @ A_27a @ ( c_2Ellist_2ELAPPEND @ A_27a @ V1l1 @ V0l2 ) )
= ( c_2Ebool_2ECOND @ ( tyop_2Eoption_2Eoption @ ( tyop_2Ellist_2Ellist @ A_27a ) ) @ ( c_2Emin_2E_3D @ ( tyop_2Ellist_2Ellist @ A_27a ) @ V1l1 @ ( c_2Ellist_2ELNIL @ A_27a ) ) @ ( c_2Ellist_2ELTL @ A_27a @ V0l2 ) @ ( c_2Eoption_2ESOME @ ( tyop_2Ellist_2Ellist @ A_27a ) @ ( c_2Ellist_2ELAPPEND @ A_27a @ ( c_2Eoption_2ETHE @ ( tyop_2Ellist_2Ellist @ A_27a ) @ ( c_2Ellist_2ELTL @ A_27a @ V1l1 ) ) @ V0l2 ) ) ) ) ).
thf(thm_2Ellist_2ELTAKE__LAPPEND1,axiom,
! [A_27a: $tType,V0n: tyop_2Enum_2Enum,V1l1: tyop_2Ellist_2Ellist @ A_27a,V2l2: tyop_2Ellist_2Ellist @ A_27a] :
( ( c_2Eoption_2EIS__SOME @ ( tyop_2Elist_2Elist @ A_27a ) @ ( c_2Ellist_2ELTAKE @ A_27a @ V0n @ V1l1 ) )
=> ( ( c_2Ellist_2ELTAKE @ A_27a @ V0n @ ( c_2Ellist_2ELAPPEND @ A_27a @ V1l1 @ V2l2 ) )
= ( c_2Ellist_2ELTAKE @ A_27a @ V0n @ V1l1 ) ) ) ).
thf(thm_2Ellist_2ELFINITE__rules,axiom,
! [A_27a: $tType] :
( ( c_2Ellist_2ELFINITE @ A_27a @ ( c_2Ellist_2ELNIL @ A_27a ) )
& ! [V0h: A_27a,V1t: tyop_2Ellist_2Ellist @ A_27a] :
( ( c_2Ellist_2ELFINITE @ A_27a @ V1t )
=> ( c_2Ellist_2ELFINITE @ A_27a @ ( c_2Ellist_2ELCONS @ A_27a @ V0h @ V1t ) ) ) ) ).
thf(thm_2Ellist_2ELFINITE__ind,axiom,
! [A_27a: $tType,V0LFINITE_27: ( tyop_2Ellist_2Ellist @ A_27a ) > $o] :
( ( ( V0LFINITE_27 @ ( c_2Ellist_2ELNIL @ A_27a ) )
& ! [V1h: A_27a,V2t: tyop_2Ellist_2Ellist @ A_27a] :
( ( V0LFINITE_27 @ V2t )
=> ( V0LFINITE_27 @ ( c_2Ellist_2ELCONS @ A_27a @ V1h @ V2t ) ) ) )
=> ! [V3a0: tyop_2Ellist_2Ellist @ A_27a] :
( ( c_2Ellist_2ELFINITE @ A_27a @ V3a0 )
=> ( V0LFINITE_27 @ V3a0 ) ) ) ).
thf(thm_2Ellist_2ELFINITE__strongind,axiom,
! [A_27a: $tType,V0LFINITE_27: ( tyop_2Ellist_2Ellist @ A_27a ) > $o] :
( ( ( V0LFINITE_27 @ ( c_2Ellist_2ELNIL @ A_27a ) )
& ! [V1h: A_27a,V2t: tyop_2Ellist_2Ellist @ A_27a] :
( ( ( c_2Ellist_2ELFINITE @ A_27a @ V2t )
& ( V0LFINITE_27 @ V2t ) )
=> ( V0LFINITE_27 @ ( c_2Ellist_2ELCONS @ A_27a @ V1h @ V2t ) ) ) )
=> ! [V3a0: tyop_2Ellist_2Ellist @ A_27a] :
( ( c_2Ellist_2ELFINITE @ A_27a @ V3a0 )
=> ( V0LFINITE_27 @ V3a0 ) ) ) ).
thf(thm_2Ellist_2ELFINITE__cases,axiom,
! [A_27a: $tType,V0a0: tyop_2Ellist_2Ellist @ A_27a] :
( ( c_2Ellist_2ELFINITE @ A_27a @ V0a0 )
<=> ( ( V0a0
= ( c_2Ellist_2ELNIL @ A_27a ) )
| ? [V1h: A_27a,V2t: tyop_2Ellist_2Ellist @ A_27a] :
( ( V0a0
= ( c_2Ellist_2ELCONS @ A_27a @ V1h @ V2t ) )
& ( c_2Ellist_2ELFINITE @ A_27a @ V2t ) ) ) ) ).
thf(thm_2Ellist_2ELFINITE__THM,axiom,
! [A_27a: $tType,A_27b: $tType] :
( ( ( c_2Ellist_2ELFINITE @ A_27a @ ( c_2Ellist_2ELNIL @ A_27a ) )
= c_2Ebool_2ET )
& ! [V0h: A_27b,V1t: tyop_2Ellist_2Ellist @ A_27b] :
( ( c_2Ellist_2ELFINITE @ A_27b @ ( c_2Ellist_2ELCONS @ A_27b @ V0h @ V1t ) )
= ( c_2Ellist_2ELFINITE @ A_27b @ V1t ) ) ) ).
thf(thm_2Ellist_2ELFINITE,axiom,
! [A_27a: $tType,V0ll: tyop_2Ellist_2Ellist @ A_27a] :
( ( c_2Ellist_2ELFINITE @ A_27a @ V0ll )
<=> ? [V1n: tyop_2Enum_2Enum] :
( ( c_2Ellist_2ELTAKE @ A_27a @ V1n @ V0ll )
= ( c_2Eoption_2ENONE @ ( tyop_2Elist_2Elist @ A_27a ) ) ) ) ).
thf(thm_2Ellist_2Ellength__rel__rules,axiom,
! [A_27a: $tType] :
( ( c_2Ellist_2Ellength__rel @ A_27a @ ( c_2Ellist_2ELNIL @ A_27a ) @ c_2Enum_2E0 )
& ! [V0h: A_27a,V1n: tyop_2Enum_2Enum,V2t: tyop_2Ellist_2Ellist @ A_27a] :
( ( c_2Ellist_2Ellength__rel @ A_27a @ V2t @ V1n )
=> ( c_2Ellist_2Ellength__rel @ A_27a @ ( c_2Ellist_2ELCONS @ A_27a @ V0h @ V2t ) @ ( c_2Enum_2ESUC @ V1n ) ) ) ) ).
thf(thm_2Ellist_2Ellength__rel__ind,axiom,
! [A_27a: $tType,V0llength__rel_27: ( tyop_2Ellist_2Ellist @ A_27a ) > tyop_2Enum_2Enum > $o] :
( ( ( V0llength__rel_27 @ ( c_2Ellist_2ELNIL @ A_27a ) @ c_2Enum_2E0 )
& ! [V1h: A_27a,V2n: tyop_2Enum_2Enum,V3t: tyop_2Ellist_2Ellist @ A_27a] :
( ( V0llength__rel_27 @ V3t @ V2n )
=> ( V0llength__rel_27 @ ( c_2Ellist_2ELCONS @ A_27a @ V1h @ V3t ) @ ( c_2Enum_2ESUC @ V2n ) ) ) )
=> ! [V4a0: tyop_2Ellist_2Ellist @ A_27a,V5a1: tyop_2Enum_2Enum] :
( ( c_2Ellist_2Ellength__rel @ A_27a @ V4a0 @ V5a1 )
=> ( V0llength__rel_27 @ V4a0 @ V5a1 ) ) ) ).
thf(thm_2Ellist_2Ellength__rel__strongind,axiom,
! [A_27a: $tType,V0llength__rel_27: ( tyop_2Ellist_2Ellist @ A_27a ) > tyop_2Enum_2Enum > $o] :
( ( ( V0llength__rel_27 @ ( c_2Ellist_2ELNIL @ A_27a ) @ c_2Enum_2E0 )
& ! [V1h: A_27a,V2n: tyop_2Enum_2Enum,V3t: tyop_2Ellist_2Ellist @ A_27a] :
( ( ( c_2Ellist_2Ellength__rel @ A_27a @ V3t @ V2n )
& ( V0llength__rel_27 @ V3t @ V2n ) )
=> ( V0llength__rel_27 @ ( c_2Ellist_2ELCONS @ A_27a @ V1h @ V3t ) @ ( c_2Enum_2ESUC @ V2n ) ) ) )
=> ! [V4a0: tyop_2Ellist_2Ellist @ A_27a,V5a1: tyop_2Enum_2Enum] :
( ( c_2Ellist_2Ellength__rel @ A_27a @ V4a0 @ V5a1 )
=> ( V0llength__rel_27 @ V4a0 @ V5a1 ) ) ) ).
thf(thm_2Ellist_2Ellength__rel__cases,axiom,
! [A_27a: $tType,V0a0: tyop_2Ellist_2Ellist @ A_27a,V1a1: tyop_2Enum_2Enum] :
( ( c_2Ellist_2Ellength__rel @ A_27a @ V0a0 @ V1a1 )
<=> ( ( ( V0a0
= ( c_2Ellist_2ELNIL @ A_27a ) )
& ( V1a1 = c_2Enum_2E0 ) )
| ? [V2h: A_27a,V3n: tyop_2Enum_2Enum,V4t: tyop_2Ellist_2Ellist @ A_27a] :
( ( V0a0
= ( c_2Ellist_2ELCONS @ A_27a @ V2h @ V4t ) )
& ( V1a1
= ( c_2Enum_2ESUC @ V3n ) )
& ( c_2Ellist_2Ellength__rel @ A_27a @ V4t @ V3n ) ) ) ) ).
thf(thm_2Ellist_2ELLENGTH__THM,axiom,
! [A_27a: $tType,A_27b: $tType] :
( ( ( c_2Ellist_2ELLENGTH @ A_27a @ ( c_2Ellist_2ELNIL @ A_27a ) )
= ( c_2Eoption_2ESOME @ tyop_2Enum_2Enum @ c_2Enum_2E0 ) )
& ! [V0h: A_27b,V1t: tyop_2Ellist_2Ellist @ A_27b] :
( ( c_2Ellist_2ELLENGTH @ A_27b @ ( c_2Ellist_2ELCONS @ A_27b @ V0h @ V1t ) )
= ( c_2Eoption_2EOPTION__MAP @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum @ c_2Enum_2ESUC @ ( c_2Ellist_2ELLENGTH @ A_27b @ V1t ) ) ) ) ).
thf(thm_2Ellist_2ELLENGTH__0,axiom,
! [A_27a: $tType,V0x: tyop_2Ellist_2Ellist @ A_27a] :
( ( ( c_2Ellist_2ELLENGTH @ A_27a @ V0x )
= ( c_2Eoption_2ESOME @ tyop_2Enum_2Enum @ c_2Enum_2E0 ) )
<=> ( V0x
= ( c_2Ellist_2ELNIL @ A_27a ) ) ) ).
thf(thm_2Ellist_2ELFINITE__HAS__LENGTH,axiom,
! [A_27a: $tType,V0ll: tyop_2Ellist_2Ellist @ A_27a] :
( ( c_2Ellist_2ELFINITE @ A_27a @ V0ll )
=> ? [V1n: tyop_2Enum_2Enum] :
( ( c_2Ellist_2ELLENGTH @ A_27a @ V0ll )
= ( c_2Eoption_2ESOME @ tyop_2Enum_2Enum @ V1n ) ) ) ).
thf(thm_2Ellist_2ENOT__LFINITE__NO__LENGTH,axiom,
! [A_27a: $tType,V0ll: tyop_2Ellist_2Ellist @ A_27a] :
( ( (~) @ ( c_2Ellist_2ELFINITE @ A_27a @ V0ll ) )
=> ( ( c_2Ellist_2ELLENGTH @ A_27a @ V0ll )
= ( c_2Eoption_2ENONE @ tyop_2Enum_2Enum ) ) ) ).
thf(thm_2Ellist_2ELFINITE__LLENGTH,axiom,
! [A_27a: $tType,V0ll: tyop_2Ellist_2Ellist @ A_27a] :
( ( c_2Ellist_2ELFINITE @ A_27a @ V0ll )
<=> ? [V1n: tyop_2Enum_2Enum] :
( ( c_2Ellist_2ELLENGTH @ A_27a @ V0ll )
= ( c_2Eoption_2ESOME @ tyop_2Enum_2Enum @ V1n ) ) ) ).
thf(thm_2Ellist_2ELFINITE__INDUCTION,axiom,
! [A_27a: $tType,V0P: ( tyop_2Ellist_2Ellist @ A_27a ) > $o] :
( ( ( V0P @ ( c_2Ellist_2ELNIL @ A_27a ) )
& ! [V1h: A_27a,V2t: tyop_2Ellist_2Ellist @ A_27a] :
( ( V0P @ V2t )
=> ( V0P @ ( c_2Ellist_2ELCONS @ A_27a @ V1h @ V2t ) ) ) )
=> ! [V3a0: tyop_2Ellist_2Ellist @ A_27a] :
( ( c_2Ellist_2ELFINITE @ A_27a @ V3a0 )
=> ( V0P @ V3a0 ) ) ) ).
thf(thm_2Ellist_2ELFINITE__STRONG__INDUCTION,axiom,
! [A_27a: $tType,V0P: ( tyop_2Ellist_2Ellist @ A_27a ) > $o] :
( ( ( V0P @ ( c_2Ellist_2ELNIL @ A_27a ) )
& ! [V1h: A_27a,V2t: tyop_2Ellist_2Ellist @ A_27a] :
( ( ( c_2Ellist_2ELFINITE @ A_27a @ V2t )
& ( V0P @ V2t ) )
=> ( V0P @ ( c_2Ellist_2ELCONS @ A_27a @ V1h @ V2t ) ) ) )
=> ! [V3a0: tyop_2Ellist_2Ellist @ A_27a] :
( ( c_2Ellist_2ELFINITE @ A_27a @ V3a0 )
=> ( V0P @ V3a0 ) ) ) ).
thf(thm_2Ellist_2ELFINITE__MAP,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: A_27a > A_27b,V1ll: tyop_2Ellist_2Ellist @ A_27a] :
( ( c_2Ellist_2ELFINITE @ A_27b @ ( c_2Ellist_2ELMAP @ A_27a @ A_27b @ V0f @ V1ll ) )
= ( c_2Ellist_2ELFINITE @ A_27a @ V1ll ) ) ).
thf(thm_2Ellist_2ELFINITE__APPEND,axiom,
! [A_27a: $tType,V0ll1: tyop_2Ellist_2Ellist @ A_27a,V1ll2: tyop_2Ellist_2Ellist @ A_27a] :
( ( c_2Ellist_2ELFINITE @ A_27a @ ( c_2Ellist_2ELAPPEND @ A_27a @ V0ll1 @ V1ll2 ) )
<=> ( ( c_2Ellist_2ELFINITE @ A_27a @ V0ll1 )
& ( c_2Ellist_2ELFINITE @ A_27a @ V1ll2 ) ) ) ).
thf(thm_2Ellist_2ELTAKE__LNTH__EL,axiom,
! [A_27a: $tType,V0n: tyop_2Enum_2Enum,V1ll: tyop_2Ellist_2Ellist @ A_27a,V2m: tyop_2Enum_2Enum,V3l: tyop_2Elist_2Elist @ A_27a] :
( ( ( ( c_2Ellist_2ELTAKE @ A_27a @ V0n @ V1ll )
= ( c_2Eoption_2ESOME @ ( tyop_2Elist_2Elist @ A_27a ) @ V3l ) )
& ( c_2Eprim__rec_2E_3C @ V2m @ V0n ) )
=> ( ( c_2Ellist_2ELNTH @ A_27a @ V2m @ V1ll )
= ( c_2Eoption_2ESOME @ A_27a @ ( c_2Elist_2EEL @ A_27a @ V2m @ V3l ) ) ) ) ).
thf(thm_2Ellist_2ENOT__LFINITE__APPEND,axiom,
! [A_27a: $tType,V0ll1: tyop_2Ellist_2Ellist @ A_27a,V1ll2: tyop_2Ellist_2Ellist @ A_27a] :
( ( (~) @ ( c_2Ellist_2ELFINITE @ A_27a @ V0ll1 ) )
=> ( ( c_2Ellist_2ELAPPEND @ A_27a @ V0ll1 @ V1ll2 )
= V0ll1 ) ) ).
thf(thm_2Ellist_2ELFINITE__LAPPEND__IMP__NIL,axiom,
! [A_27a: $tType,V0ll: tyop_2Ellist_2Ellist @ A_27a] :
( ( c_2Ellist_2ELFINITE @ A_27a @ V0ll )
=> ! [V1l2: tyop_2Ellist_2Ellist @ A_27a] :
( ( ( c_2Ellist_2ELAPPEND @ A_27a @ V0ll @ V1l2 )
= V0ll )
=> ( V1l2
= ( c_2Ellist_2ELNIL @ A_27a ) ) ) ) ).
thf(thm_2Ellist_2ELLENGTH__MAP,axiom,
! [A_27a: $tType,A_27b: $tType,V0ll: tyop_2Ellist_2Ellist @ A_27a,V1f: A_27a > A_27b] :
( ( c_2Ellist_2ELLENGTH @ A_27b @ ( c_2Ellist_2ELMAP @ A_27a @ A_27b @ V1f @ V0ll ) )
= ( c_2Ellist_2ELLENGTH @ A_27a @ V0ll ) ) ).
thf(thm_2Ellist_2ELLENGTH__APPEND,axiom,
! [A_27a: $tType,V0ll1: tyop_2Ellist_2Ellist @ A_27a,V1ll2: tyop_2Ellist_2Ellist @ A_27a] :
( ( c_2Ellist_2ELLENGTH @ A_27a @ ( c_2Ellist_2ELAPPEND @ A_27a @ V0ll1 @ V1ll2 ) )
= ( c_2Ebool_2ECOND @ ( tyop_2Eoption_2Eoption @ tyop_2Enum_2Enum ) @ ( c_2Ebool_2E_2F_5C @ ( c_2Ellist_2ELFINITE @ A_27a @ V0ll1 ) @ ( c_2Ellist_2ELFINITE @ A_27a @ V1ll2 ) ) @ ( c_2Eoption_2ESOME @ tyop_2Enum_2Enum @ ( c_2Earithmetic_2E_2B @ ( c_2Eoption_2ETHE @ tyop_2Enum_2Enum @ ( c_2Ellist_2ELLENGTH @ A_27a @ V0ll1 ) ) @ ( c_2Eoption_2ETHE @ tyop_2Enum_2Enum @ ( c_2Ellist_2ELLENGTH @ A_27a @ V1ll2 ) ) ) ) @ ( c_2Eoption_2ENONE @ tyop_2Enum_2Enum ) ) ) ).
thf(thm_2Ellist_2EtoList__THM,axiom,
! [A_27a: $tType,A_27b: $tType] :
( ( ( c_2Ellist_2EtoList @ A_27a @ ( c_2Ellist_2ELNIL @ A_27a ) )
= ( c_2Eoption_2ESOME @ ( tyop_2Elist_2Elist @ A_27a ) @ ( c_2Elist_2ENIL @ A_27a ) ) )
& ! [V0h: A_27b,V1t: tyop_2Ellist_2Ellist @ A_27b] :
( ( c_2Ellist_2EtoList @ A_27b @ ( c_2Ellist_2ELCONS @ A_27b @ V0h @ V1t ) )
= ( c_2Eoption_2EOPTION__MAP @ ( tyop_2Elist_2Elist @ A_27b ) @ ( tyop_2Elist_2Elist @ A_27b ) @ ( c_2Elist_2ECONS @ A_27b @ V0h ) @ ( c_2Ellist_2EtoList @ A_27b @ V1t ) ) ) ) ).
thf(thm_2Ellist_2EfromList__EQ__LNIL,axiom,
! [A_27a: $tType,V0l: tyop_2Elist_2Elist @ A_27a] :
( ( ( c_2Ellist_2EfromList @ A_27a @ V0l )
= ( c_2Ellist_2ELNIL @ A_27a ) )
<=> ( V0l
= ( c_2Elist_2ENIL @ A_27a ) ) ) ).
thf(thm_2Ellist_2ELHD__fromList,axiom,
! [A_27a: $tType,V0l: tyop_2Elist_2Elist @ A_27a] :
( ( c_2Ellist_2ELHD @ A_27a @ ( c_2Ellist_2EfromList @ A_27a @ V0l ) )
= ( c_2Ebool_2ECOND @ ( tyop_2Eoption_2Eoption @ A_27a ) @ ( c_2Elist_2ENULL @ A_27a @ V0l ) @ ( c_2Eoption_2ENONE @ A_27a ) @ ( c_2Eoption_2ESOME @ A_27a @ ( c_2Elist_2EHD @ A_27a @ V0l ) ) ) ) ).
thf(thm_2Ellist_2ELTL__fromList,axiom,
! [A_27a: $tType,V0l: tyop_2Elist_2Elist @ A_27a] :
( ( c_2Ellist_2ELTL @ A_27a @ ( c_2Ellist_2EfromList @ A_27a @ V0l ) )
= ( c_2Ebool_2ECOND @ ( tyop_2Eoption_2Eoption @ ( tyop_2Ellist_2Ellist @ A_27a ) ) @ ( c_2Elist_2ENULL @ A_27a @ V0l ) @ ( c_2Eoption_2ENONE @ ( tyop_2Ellist_2Ellist @ A_27a ) ) @ ( c_2Eoption_2ESOME @ ( tyop_2Ellist_2Ellist @ A_27a ) @ ( c_2Ellist_2EfromList @ A_27a @ ( c_2Elist_2ETL @ A_27a @ V0l ) ) ) ) ) ).
thf(thm_2Ellist_2ELFINITE__fromList,axiom,
! [A_27a: $tType,V0l: tyop_2Elist_2Elist @ A_27a] : ( c_2Ellist_2ELFINITE @ A_27a @ ( c_2Ellist_2EfromList @ A_27a @ V0l ) ) ).
thf(thm_2Ellist_2ELLENGTH__fromList,axiom,
! [A_27a: $tType,V0l: tyop_2Elist_2Elist @ A_27a] :
( ( c_2Ellist_2ELLENGTH @ A_27a @ ( c_2Ellist_2EfromList @ A_27a @ V0l ) )
= ( c_2Eoption_2ESOME @ tyop_2Enum_2Enum @ ( c_2Elist_2ELENGTH @ A_27a @ V0l ) ) ) ).
thf(thm_2Ellist_2ELTAKE__fromList,axiom,
! [A_27a: $tType,V0l: tyop_2Elist_2Elist @ A_27a] :
( ( c_2Ellist_2ELTAKE @ A_27a @ ( c_2Elist_2ELENGTH @ A_27a @ V0l ) @ ( c_2Ellist_2EfromList @ A_27a @ V0l ) )
= ( c_2Eoption_2ESOME @ ( tyop_2Elist_2Elist @ A_27a ) @ V0l ) ) ).
thf(thm_2Ellist_2Efrom__toList,axiom,
! [A_27a: $tType,V0l: tyop_2Elist_2Elist @ A_27a] :
( ( c_2Ellist_2EtoList @ A_27a @ ( c_2Ellist_2EfromList @ A_27a @ V0l ) )
= ( c_2Eoption_2ESOME @ ( tyop_2Elist_2Elist @ A_27a ) @ V0l ) ) ).
thf(thm_2Ellist_2ELFINITE__toList,axiom,
! [A_27a: $tType,V0ll: tyop_2Ellist_2Ellist @ A_27a] :
( ( c_2Ellist_2ELFINITE @ A_27a @ V0ll )
=> ? [V1l: tyop_2Elist_2Elist @ A_27a] :
( ( c_2Ellist_2EtoList @ A_27a @ V0ll )
= ( c_2Eoption_2ESOME @ ( tyop_2Elist_2Elist @ A_27a ) @ V1l ) ) ) ).
thf(thm_2Ellist_2ELFINITE__toList__SOME,axiom,
! [A_27a: $tType,V0ll: tyop_2Ellist_2Ellist @ A_27a] :
( ( c_2Ellist_2ELFINITE @ A_27a @ V0ll )
= ( c_2Eoption_2EIS__SOME @ ( tyop_2Elist_2Elist @ A_27a ) @ ( c_2Ellist_2EtoList @ A_27a @ V0ll ) ) ) ).
thf(thm_2Ellist_2Eto__fromList,axiom,
! [A_27a: $tType,V0ll: tyop_2Ellist_2Ellist @ A_27a] :
( ( c_2Ellist_2ELFINITE @ A_27a @ V0ll )
=> ( ( c_2Ellist_2EfromList @ A_27a @ ( c_2Eoption_2ETHE @ ( tyop_2Elist_2Elist @ A_27a ) @ ( c_2Ellist_2EtoList @ A_27a @ V0ll ) ) )
= V0ll ) ) ).
thf(thm_2Ellist_2ELTAKE__LAPPEND2,axiom,
! [A_27a: $tType,V0n: tyop_2Enum_2Enum,V1l1: tyop_2Ellist_2Ellist @ A_27a,V2l2: tyop_2Ellist_2Ellist @ A_27a] :
( ( ( c_2Ellist_2ELTAKE @ A_27a @ V0n @ V1l1 )
= ( c_2Eoption_2ENONE @ ( tyop_2Elist_2Elist @ A_27a ) ) )
=> ( ( c_2Ellist_2ELTAKE @ A_27a @ V0n @ ( c_2Ellist_2ELAPPEND @ A_27a @ V1l1 @ V2l2 ) )
= ( c_2Eoption_2EOPTION__MAP @ ( tyop_2Elist_2Elist @ A_27a ) @ ( tyop_2Elist_2Elist @ A_27a ) @ ( c_2Elist_2EAPPEND @ A_27a @ ( c_2Eoption_2ETHE @ ( tyop_2Elist_2Elist @ A_27a ) @ ( c_2Ellist_2EtoList @ A_27a @ V1l1 ) ) ) @ ( c_2Ellist_2ELTAKE @ A_27a @ ( c_2Earithmetic_2E_2D @ V0n @ ( c_2Eoption_2ETHE @ tyop_2Enum_2Enum @ ( c_2Ellist_2ELLENGTH @ A_27a @ V1l1 ) ) ) @ V2l2 ) ) ) ) ).
thf(thm_2Ellist_2ELNTH__fromList,axiom,
! [A_27a: $tType,V0n: tyop_2Enum_2Enum,V1l: tyop_2Elist_2Elist @ A_27a] :
( ( c_2Ellist_2ELNTH @ A_27a @ V0n @ ( c_2Ellist_2EfromList @ A_27a @ V1l ) )
= ( c_2Ebool_2ECOND @ ( tyop_2Eoption_2Eoption @ A_27a ) @ ( c_2Eprim__rec_2E_3C @ V0n @ ( c_2Elist_2ELENGTH @ A_27a @ V1l ) ) @ ( c_2Eoption_2ESOME @ A_27a @ ( c_2Elist_2EEL @ A_27a @ V0n @ V1l ) ) @ ( c_2Eoption_2ENONE @ A_27a ) ) ) ).
thf(thm_2Ellist_2Elnth__fromList__some,axiom,
! [A_27a: $tType,V0n: tyop_2Enum_2Enum,V1l: tyop_2Elist_2Elist @ A_27a] :
( ( c_2Eprim__rec_2E_3C @ V0n @ ( c_2Elist_2ELENGTH @ A_27a @ V1l ) )
<=> ( ( c_2Ellist_2ELNTH @ A_27a @ V0n @ ( c_2Ellist_2EfromList @ A_27a @ V1l ) )
= ( c_2Eoption_2ESOME @ A_27a @ ( c_2Elist_2EEL @ A_27a @ V0n @ V1l ) ) ) ) ).
thf(thm_2Ellist_2ELDROP__FUNPOW,axiom,
! [A_27a: $tType,V0n: tyop_2Enum_2Enum,V1ll: tyop_2Ellist_2Ellist @ A_27a] :
( ( c_2Ellist_2ELDROP @ A_27a @ V0n @ V1ll )
= ( c_2Earithmetic_2EFUNPOW @ ( tyop_2Eoption_2Eoption @ ( tyop_2Ellist_2Ellist @ A_27a ) )
@ ^ [V2m: tyop_2Eoption_2Eoption @ ( tyop_2Ellist_2Ellist @ A_27a )] : ( c_2Eoption_2EOPTION__BIND @ ( tyop_2Ellist_2Ellist @ A_27a ) @ ( tyop_2Ellist_2Ellist @ A_27a ) @ V2m @ ( c_2Ellist_2ELTL @ A_27a ) )
@ V0n
@ ( c_2Eoption_2ESOME @ ( tyop_2Ellist_2Ellist @ A_27a ) @ V1ll ) ) ) ).
thf(thm_2Ellist_2ELDROP__THM,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType] :
( ! [V0ll: tyop_2Ellist_2Ellist @ A_27a] :
( ( c_2Ellist_2ELDROP @ A_27a @ c_2Enum_2E0 @ V0ll )
= ( c_2Eoption_2ESOME @ ( tyop_2Ellist_2Ellist @ A_27a ) @ V0ll ) )
& ! [V1n: tyop_2Enum_2Enum] :
( ( c_2Ellist_2ELDROP @ A_27b @ ( c_2Enum_2ESUC @ V1n ) @ ( c_2Ellist_2ELNIL @ A_27b ) )
= ( c_2Eoption_2ENONE @ ( tyop_2Ellist_2Ellist @ A_27b ) ) )
& ! [V2n: tyop_2Enum_2Enum,V3h: A_27c,V4t: tyop_2Ellist_2Ellist @ A_27c] :
( ( c_2Ellist_2ELDROP @ A_27c @ ( c_2Enum_2ESUC @ V2n ) @ ( c_2Ellist_2ELCONS @ A_27c @ V3h @ V4t ) )
= ( c_2Ellist_2ELDROP @ A_27c @ V2n @ V4t ) ) ) ).
thf(thm_2Ellist_2ELDROP1__THM,axiom,
! [A_27a: $tType] :
( ( c_2Ellist_2ELDROP @ A_27a @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) )
= ( c_2Ellist_2ELTL @ A_27a ) ) ).
thf(thm_2Ellist_2ELNTH__HD__LDROP,axiom,
! [A_27a: $tType,V0n: tyop_2Enum_2Enum,V1ll: tyop_2Ellist_2Ellist @ A_27a] :
( ( c_2Ellist_2ELNTH @ A_27a @ V0n @ V1ll )
= ( c_2Eoption_2EOPTION__BIND @ A_27a @ ( tyop_2Ellist_2Ellist @ A_27a ) @ ( c_2Ellist_2ELDROP @ A_27a @ V0n @ V1ll ) @ ( c_2Ellist_2ELHD @ A_27a ) ) ) ).
thf(thm_2Ellist_2ENOT__LFINITE__TAKE,axiom,
! [A_27a: $tType,V0ll: tyop_2Ellist_2Ellist @ A_27a] :
( ( (~) @ ( c_2Ellist_2ELFINITE @ A_27a @ V0ll ) )
=> ! [V1n: tyop_2Enum_2Enum] :
? [V2y: tyop_2Elist_2Elist @ A_27a] :
( ( c_2Ellist_2ELTAKE @ A_27a @ V1n @ V0ll )
= ( c_2Eoption_2ESOME @ ( tyop_2Elist_2Elist @ A_27a ) @ V2y ) ) ) ).
thf(thm_2Ellist_2ELFINITE__TAKE,axiom,
! [A_27a: $tType,V0n: tyop_2Enum_2Enum,V1ll: tyop_2Ellist_2Ellist @ A_27a] :
( ( ( c_2Ellist_2ELFINITE @ A_27a @ V1ll )
& ( c_2Earithmetic_2E_3C_3D @ V0n @ ( c_2Eoption_2ETHE @ tyop_2Enum_2Enum @ ( c_2Ellist_2ELLENGTH @ A_27a @ V1ll ) ) ) )
=> ? [V2y: tyop_2Elist_2Elist @ A_27a] :
( ( c_2Ellist_2ELTAKE @ A_27a @ V0n @ V1ll )
= ( c_2Eoption_2ESOME @ ( tyop_2Elist_2Elist @ A_27a ) @ V2y ) ) ) ).
thf(thm_2Ellist_2ENOT__LFINITE__DROP,axiom,
! [A_27a: $tType,V0ll: tyop_2Ellist_2Ellist @ A_27a] :
( ( (~) @ ( c_2Ellist_2ELFINITE @ A_27a @ V0ll ) )
=> ! [V1n: tyop_2Enum_2Enum] :
? [V2y: tyop_2Ellist_2Ellist @ A_27a] :
( ( c_2Ellist_2ELDROP @ A_27a @ V1n @ V0ll )
= ( c_2Eoption_2ESOME @ ( tyop_2Ellist_2Ellist @ A_27a ) @ V2y ) ) ) ).
thf(thm_2Ellist_2ELFINITE__DROP,axiom,
! [A_27a: $tType,V0n: tyop_2Enum_2Enum,V1ll: tyop_2Ellist_2Ellist @ A_27a] :
( ( ( c_2Ellist_2ELFINITE @ A_27a @ V1ll )
& ( c_2Earithmetic_2E_3C_3D @ V0n @ ( c_2Eoption_2ETHE @ tyop_2Enum_2Enum @ ( c_2Ellist_2ELLENGTH @ A_27a @ V1ll ) ) ) )
=> ? [V2y: tyop_2Ellist_2Ellist @ A_27a] :
( ( c_2Ellist_2ELDROP @ A_27a @ V0n @ V1ll )
= ( c_2Eoption_2ESOME @ ( tyop_2Ellist_2Ellist @ A_27a ) @ V2y ) ) ) ).
thf(thm_2Ellist_2ELTAKE__DROP,axiom,
! [A_27a: $tType] :
( ! [V0n: tyop_2Enum_2Enum,V1ll: tyop_2Ellist_2Ellist @ A_27a] :
( ( (~) @ ( c_2Ellist_2ELFINITE @ A_27a @ V1ll ) )
=> ( ( c_2Ellist_2ELAPPEND @ A_27a @ ( c_2Ellist_2EfromList @ A_27a @ ( c_2Eoption_2ETHE @ ( tyop_2Elist_2Elist @ A_27a ) @ ( c_2Ellist_2ELTAKE @ A_27a @ V0n @ V1ll ) ) ) @ ( c_2Eoption_2ETHE @ ( tyop_2Ellist_2Ellist @ A_27a ) @ ( c_2Ellist_2ELDROP @ A_27a @ V0n @ V1ll ) ) )
= V1ll ) )
& ! [V2n: tyop_2Enum_2Enum,V3ll: tyop_2Ellist_2Ellist @ A_27a] :
( ( ( c_2Ellist_2ELFINITE @ A_27a @ V3ll )
& ( c_2Earithmetic_2E_3C_3D @ V2n @ ( c_2Eoption_2ETHE @ tyop_2Enum_2Enum @ ( c_2Ellist_2ELLENGTH @ A_27a @ V3ll ) ) ) )
=> ( ( c_2Ellist_2ELAPPEND @ A_27a @ ( c_2Ellist_2EfromList @ A_27a @ ( c_2Eoption_2ETHE @ ( tyop_2Elist_2Elist @ A_27a ) @ ( c_2Ellist_2ELTAKE @ A_27a @ V2n @ V3ll ) ) ) @ ( c_2Eoption_2ETHE @ ( tyop_2Ellist_2Ellist @ A_27a ) @ ( c_2Ellist_2ELDROP @ A_27a @ V2n @ V3ll ) ) )
= V3ll ) ) ) ).
thf(thm_2Ellist_2ELDROP__ADD,axiom,
! [A_27a: $tType,V0k1: tyop_2Enum_2Enum,V1k2: tyop_2Enum_2Enum,V2x: tyop_2Ellist_2Ellist @ A_27a] :
( ( c_2Ellist_2ELDROP @ A_27a @ ( c_2Earithmetic_2E_2B @ V0k1 @ V1k2 ) @ V2x )
= ( c_2Eoption_2Eoption__CASE @ ( tyop_2Ellist_2Ellist @ A_27a ) @ ( tyop_2Eoption_2Eoption @ ( tyop_2Ellist_2Ellist @ A_27a ) ) @ ( c_2Ellist_2ELDROP @ A_27a @ V0k1 @ V2x ) @ ( c_2Eoption_2ENONE @ ( tyop_2Ellist_2Ellist @ A_27a ) )
@ ^ [V3ll: tyop_2Ellist_2Ellist @ A_27a] : ( c_2Ellist_2ELDROP @ A_27a @ V1k2 @ V3ll ) ) ) ).
thf(thm_2Ellist_2ELDROP__SOME__LLENGTH,axiom,
! [A_27a: $tType,V0n: tyop_2Enum_2Enum,V1m: tyop_2Enum_2Enum,V2ll: tyop_2Ellist_2Ellist @ A_27a,V3l: tyop_2Ellist_2Ellist @ A_27a] :
( ( ( ( c_2Ellist_2ELDROP @ A_27a @ V0n @ V2ll )
= ( c_2Eoption_2ESOME @ ( tyop_2Ellist_2Ellist @ A_27a ) @ V3l ) )
& ( ( c_2Ellist_2ELLENGTH @ A_27a @ V2ll )
= ( c_2Eoption_2ESOME @ tyop_2Enum_2Enum @ V1m ) ) )
=> ( ( c_2Earithmetic_2E_3C_3D @ V0n @ V1m )
& ( ( c_2Ellist_2ELLENGTH @ A_27a @ V3l )
= ( c_2Eoption_2ESOME @ tyop_2Enum_2Enum @ ( c_2Earithmetic_2E_2D @ V1m @ V0n ) ) ) ) ) ).
thf(thm_2Ellist_2ELFINITE__LNTH__NONE,axiom,
! [A_27a: $tType,V0ll: tyop_2Ellist_2Ellist @ A_27a] :
( ( c_2Ellist_2ELFINITE @ A_27a @ V0ll )
<=> ? [V1n: tyop_2Enum_2Enum] :
( ( c_2Ellist_2ELNTH @ A_27a @ V1n @ V0ll )
= ( c_2Eoption_2ENONE @ A_27a ) ) ) ).
thf(thm_2Ellist_2Einfinite__lnth__some,axiom,
! [A_27a: $tType,V0ll: tyop_2Ellist_2Ellist @ A_27a] :
( ( (~) @ ( c_2Ellist_2ELFINITE @ A_27a @ V0ll ) )
<=> ! [V1n: tyop_2Enum_2Enum] :
? [V2x: A_27a] :
( ( c_2Ellist_2ELNTH @ A_27a @ V1n @ V0ll )
= ( c_2Eoption_2ESOME @ A_27a @ V2x ) ) ) ).
thf(thm_2Ellist_2ELNTH__LAPPEND,axiom,
! [A_27a: $tType,V0n: tyop_2Enum_2Enum,V1l2: tyop_2Ellist_2Ellist @ A_27a,V2l1: tyop_2Ellist_2Ellist @ A_27a] :
( ( c_2Ellist_2ELNTH @ A_27a @ V0n @ ( c_2Ellist_2ELAPPEND @ A_27a @ V2l1 @ V1l2 ) )
= ( c_2Eoption_2Eoption__CASE @ tyop_2Enum_2Enum @ ( tyop_2Eoption_2Eoption @ A_27a ) @ ( c_2Ellist_2ELLENGTH @ A_27a @ V2l1 ) @ ( c_2Ellist_2ELNTH @ A_27a @ V0n @ V2l1 )
@ ^ [V3m: tyop_2Enum_2Enum] : ( c_2Ebool_2ECOND @ ( tyop_2Eoption_2Eoption @ A_27a ) @ ( c_2Eprim__rec_2E_3C @ V0n @ V3m ) @ ( c_2Ellist_2ELNTH @ A_27a @ V0n @ V2l1 ) @ ( c_2Ellist_2ELNTH @ A_27a @ ( c_2Earithmetic_2E_2D @ V0n @ V3m ) @ V1l2 ) ) ) ) ).
thf(thm_2Ellist_2ELNTH__ADD,axiom,
! [A_27a: $tType,V0a: tyop_2Enum_2Enum,V1b: tyop_2Enum_2Enum,V2ll: tyop_2Ellist_2Ellist @ A_27a] :
( ( c_2Ellist_2ELNTH @ A_27a @ ( c_2Earithmetic_2E_2B @ V0a @ V1b ) @ V2ll )
= ( c_2Eoption_2EOPTION__BIND @ A_27a @ ( tyop_2Ellist_2Ellist @ A_27a ) @ ( c_2Ellist_2ELDROP @ A_27a @ V0a @ V2ll ) @ ( c_2Ellist_2ELNTH @ A_27a @ V1b ) ) ) ).
thf(thm_2Ellist_2Elnth__some__down__closed,axiom,
! [A_27a: $tType,V0ll: tyop_2Ellist_2Ellist @ A_27a,V1x: A_27a,V2n1: tyop_2Enum_2Enum,V3n2: tyop_2Enum_2Enum] :
( ( ( ( c_2Ellist_2ELNTH @ A_27a @ V2n1 @ V0ll )
= ( c_2Eoption_2ESOME @ A_27a @ V1x ) )
& ( c_2Earithmetic_2E_3C_3D @ V3n2 @ V2n1 ) )
=> ? [V4y: A_27a] :
( ( c_2Ellist_2ELNTH @ A_27a @ V3n2 @ V0ll )
= ( c_2Eoption_2ESOME @ A_27a @ V4y ) ) ) ).
thf(thm_2Ellist_2Eexists__rules,axiom,
! [A_27a: $tType,V0P: A_27a > $o] :
( ! [V1h: A_27a,V2t: tyop_2Ellist_2Ellist @ A_27a] :
( ( V0P @ V1h )
=> ( c_2Ellist_2Eexists @ A_27a @ V0P @ ( c_2Ellist_2ELCONS @ A_27a @ V1h @ V2t ) ) )
& ! [V3h: A_27a,V4t: tyop_2Ellist_2Ellist @ A_27a] :
( ( c_2Ellist_2Eexists @ A_27a @ V0P @ V4t )
=> ( c_2Ellist_2Eexists @ A_27a @ V0P @ ( c_2Ellist_2ELCONS @ A_27a @ V3h @ V4t ) ) ) ) ).
thf(thm_2Ellist_2Eexists__ind,axiom,
! [A_27a: $tType,V0P: A_27a > $o,V1exists_27: ( tyop_2Ellist_2Ellist @ A_27a ) > $o] :
( ( ! [V2h: A_27a,V3t: tyop_2Ellist_2Ellist @ A_27a] :
( ( V0P @ V2h )
=> ( V1exists_27 @ ( c_2Ellist_2ELCONS @ A_27a @ V2h @ V3t ) ) )
& ! [V4h: A_27a,V5t: tyop_2Ellist_2Ellist @ A_27a] :
( ( V1exists_27 @ V5t )
=> ( V1exists_27 @ ( c_2Ellist_2ELCONS @ A_27a @ V4h @ V5t ) ) ) )
=> ! [V6a0: tyop_2Ellist_2Ellist @ A_27a] :
( ( c_2Ellist_2Eexists @ A_27a @ V0P @ V6a0 )
=> ( V1exists_27 @ V6a0 ) ) ) ).
thf(thm_2Ellist_2Eexists__strongind,axiom,
! [A_27a: $tType,V0P: A_27a > $o,V1exists_27: ( tyop_2Ellist_2Ellist @ A_27a ) > $o] :
( ( ! [V2h: A_27a,V3t: tyop_2Ellist_2Ellist @ A_27a] :
( ( V0P @ V2h )
=> ( V1exists_27 @ ( c_2Ellist_2ELCONS @ A_27a @ V2h @ V3t ) ) )
& ! [V4h: A_27a,V5t: tyop_2Ellist_2Ellist @ A_27a] :
( ( ( c_2Ellist_2Eexists @ A_27a @ V0P @ V5t )
& ( V1exists_27 @ V5t ) )
=> ( V1exists_27 @ ( c_2Ellist_2ELCONS @ A_27a @ V4h @ V5t ) ) ) )
=> ! [V6a0: tyop_2Ellist_2Ellist @ A_27a] :
( ( c_2Ellist_2Eexists @ A_27a @ V0P @ V6a0 )
=> ( V1exists_27 @ V6a0 ) ) ) ).
thf(thm_2Ellist_2Eexists__cases,axiom,
! [A_27a: $tType,V0P: A_27a > $o,V1a0: tyop_2Ellist_2Ellist @ A_27a] :
( ( c_2Ellist_2Eexists @ A_27a @ V0P @ V1a0 )
<=> ( ? [V2h: A_27a,V3t: tyop_2Ellist_2Ellist @ A_27a] :
( ( V1a0
= ( c_2Ellist_2ELCONS @ A_27a @ V2h @ V3t ) )
& ( V0P @ V2h ) )
| ? [V4h: A_27a,V5t: tyop_2Ellist_2Ellist @ A_27a] :
( ( V1a0
= ( c_2Ellist_2ELCONS @ A_27a @ V4h @ V5t ) )
& ( c_2Ellist_2Eexists @ A_27a @ V0P @ V5t ) ) ) ) ).
thf(thm_2Ellist_2Eexists__thm,axiom,
! [A_27a: $tType,V0t: tyop_2Ellist_2Ellist @ A_27a,V1h: A_27a,V2P: A_27a > $o] :
( ( ( c_2Ellist_2Eexists @ A_27a @ V2P @ ( c_2Ellist_2ELNIL @ A_27a ) )
= c_2Ebool_2EF )
& ( ( c_2Ellist_2Eexists @ A_27a @ V2P @ ( c_2Ellist_2ELCONS @ A_27a @ V1h @ V0t ) )
<=> ( ( V2P @ V1h )
| ( c_2Ellist_2Eexists @ A_27a @ V2P @ V0t ) ) ) ) ).
thf(thm_2Ellist_2Eexists__LNTH,axiom,
! [A_27a: $tType,V0P: A_27a > $o,V1l: tyop_2Ellist_2Ellist @ A_27a] :
( ( c_2Ellist_2Eexists @ A_27a @ V0P @ V1l )
<=> ? [V2n: tyop_2Enum_2Enum,V3e: A_27a] :
( ( ( c_2Eoption_2ESOME @ A_27a @ V3e )
= ( c_2Ellist_2ELNTH @ A_27a @ V2n @ V1l ) )
& ( V0P @ V3e ) ) ) ).
thf(thm_2Ellist_2EMONO__exists,axiom,
! [A_27a: $tType,V0l: tyop_2Ellist_2Ellist @ A_27a,V1Q: A_27a > $o,V2P: A_27a > $o] :
( ! [V3x: A_27a] :
( ( V2P @ V3x )
=> ( V1Q @ V3x ) )
=> ( ( c_2Ellist_2Eexists @ A_27a @ V2P @ V0l )
=> ( c_2Ellist_2Eexists @ A_27a @ V1Q @ V0l ) ) ) ).
thf(thm_2Ellist_2Eexists__strong__ind,axiom,
! [A_27a: $tType,V0P: A_27a > $o,V1Q: ( tyop_2Ellist_2Ellist @ A_27a ) > $o] :
( ( ! [V2h: A_27a,V3t: tyop_2Ellist_2Ellist @ A_27a] :
( ( V0P @ V2h )
=> ( V1Q @ ( c_2Ellist_2ELCONS @ A_27a @ V2h @ V3t ) ) )
& ! [V4h: A_27a,V5t: tyop_2Ellist_2Ellist @ A_27a] :
( ( ( V1Q @ V5t )
& ( c_2Ellist_2Eexists @ A_27a @ V0P @ V5t ) )
=> ( V1Q @ ( c_2Ellist_2ELCONS @ A_27a @ V4h @ V5t ) ) ) )
=> ! [V6a0: tyop_2Ellist_2Ellist @ A_27a] :
( ( c_2Ellist_2Eexists @ A_27a @ V0P @ V6a0 )
=> ( V1Q @ V6a0 ) ) ) ).
thf(thm_2Ellist_2Eexists__LDROP,axiom,
! [A_27a: $tType,V0ll: tyop_2Ellist_2Ellist @ A_27a,V1P: A_27a > $o] :
( ( c_2Ellist_2Eexists @ A_27a @ V1P @ V0ll )
<=> ? [V2n: tyop_2Enum_2Enum,V3a: A_27a,V4t: tyop_2Ellist_2Ellist @ A_27a] :
( ( ( c_2Ellist_2ELDROP @ A_27a @ V2n @ V0ll )
= ( c_2Eoption_2ESOME @ ( tyop_2Ellist_2Ellist @ A_27a ) @ ( c_2Ellist_2ELCONS @ A_27a @ V3a @ V4t ) ) )
& ( V1P @ V3a ) ) ) ).
thf(thm_2Ellist_2Eevery__coind,axiom,
! [A_27a: $tType,V0P: A_27a > $o,V1Q: ( tyop_2Ellist_2Ellist @ A_27a ) > $o] :
( ! [V2h: A_27a,V3t: tyop_2Ellist_2Ellist @ A_27a] :
( ( V1Q @ ( c_2Ellist_2ELCONS @ A_27a @ V2h @ V3t ) )
=> ( ( V0P @ V2h )
& ( V1Q @ V3t ) ) )
=> ! [V4ll: tyop_2Ellist_2Ellist @ A_27a] :
( ( V1Q @ V4ll )
=> ( c_2Ellist_2Eevery @ A_27a @ V0P @ V4ll ) ) ) ).
thf(thm_2Ellist_2Eevery__thm,axiom,
! [A_27a: $tType,V0t: tyop_2Ellist_2Ellist @ A_27a,V1h: A_27a,V2P: A_27a > $o] :
( ( ( c_2Ellist_2Eevery @ A_27a @ V2P @ ( c_2Ellist_2ELNIL @ A_27a ) )
= c_2Ebool_2ET )
& ( ( c_2Ellist_2Eevery @ A_27a @ V2P @ ( c_2Ellist_2ELCONS @ A_27a @ V1h @ V0t ) )
<=> ( ( V2P @ V1h )
& ( c_2Ellist_2Eevery @ A_27a @ V2P @ V0t ) ) ) ) ).
thf(thm_2Ellist_2ELL__ALL__THM,axiom,
! [A_27a: $tType,V0t: tyop_2Ellist_2Ellist @ A_27a,V1h: A_27a,V2P: A_27a > $o] :
( ( ( c_2Ellist_2Eevery @ A_27a @ V2P @ ( c_2Ellist_2ELNIL @ A_27a ) )
= c_2Ebool_2ET )
& ( ( c_2Ellist_2Eevery @ A_27a @ V2P @ ( c_2Ellist_2ELCONS @ A_27a @ V1h @ V0t ) )
<=> ( ( V2P @ V1h )
& ( c_2Ellist_2Eevery @ A_27a @ V2P @ V0t ) ) ) ) ).
thf(thm_2Ellist_2EMONO__every,axiom,
! [A_27a: $tType,V0l: tyop_2Ellist_2Ellist @ A_27a,V1Q: A_27a > $o,V2P: A_27a > $o] :
( ! [V3x: A_27a] :
( ( V2P @ V3x )
=> ( V1Q @ V3x ) )
=> ( ( c_2Ellist_2Eevery @ A_27a @ V2P @ V0l )
=> ( c_2Ellist_2Eevery @ A_27a @ V1Q @ V0l ) ) ) ).
thf(thm_2Ellist_2Eevery__strong__coind,axiom,
! [A_27a: $tType,V0P: A_27a > $o,V1Q: ( tyop_2Ellist_2Ellist @ A_27a ) > $o] :
( ( ! [V2h: A_27a,V3t: tyop_2Ellist_2Ellist @ A_27a] :
( ( V1Q @ ( c_2Ellist_2ELCONS @ A_27a @ V2h @ V3t ) )
=> ( V0P @ V2h ) )
& ! [V4h: A_27a,V5t: tyop_2Ellist_2Ellist @ A_27a] :
( ( V1Q @ ( c_2Ellist_2ELCONS @ A_27a @ V4h @ V5t ) )
=> ( ( V1Q @ V5t )
| ( c_2Ellist_2Eevery @ A_27a @ V0P @ V5t ) ) ) )
=> ! [V6ll: tyop_2Ellist_2Ellist @ A_27a] :
( ( V1Q @ V6ll )
=> ( c_2Ellist_2Eevery @ A_27a @ V0P @ V6ll ) ) ) ).
thf(thm_2Ellist_2ELFILTER__THM,axiom,
! [A_27a: $tType,A_27b: $tType] :
( ! [V0P: A_27a > $o] :
( ( c_2Ellist_2ELFILTER @ A_27a @ V0P @ ( c_2Ellist_2ELNIL @ A_27a ) )
= ( c_2Ellist_2ELNIL @ A_27a ) )
& ! [V1P: A_27b > $o,V2h: A_27b,V3t: tyop_2Ellist_2Ellist @ A_27b] :
( ( c_2Ellist_2ELFILTER @ A_27b @ V1P @ ( c_2Ellist_2ELCONS @ A_27b @ V2h @ V3t ) )
= ( c_2Ebool_2ECOND @ ( tyop_2Ellist_2Ellist @ A_27b ) @ ( V1P @ V2h ) @ ( c_2Ellist_2ELCONS @ A_27b @ V2h @ ( c_2Ellist_2ELFILTER @ A_27b @ V1P @ V3t ) ) @ ( c_2Ellist_2ELFILTER @ A_27b @ V1P @ V3t ) ) ) ) ).
thf(thm_2Ellist_2ELFILTER__NIL,axiom,
! [A_27a: $tType,V0P: A_27a > $o,V1ll: tyop_2Ellist_2Ellist @ A_27a] :
( ( c_2Ellist_2Eevery @ A_27a @ ( c_2Ecombin_2Eo @ A_27a @ $o @ $o @ c_2Ebool_2E_7E @ V0P ) @ V1ll )
=> ( ( c_2Ellist_2ELFILTER @ A_27a @ V0P @ V1ll )
= ( c_2Ellist_2ELNIL @ A_27a ) ) ) ).
thf(thm_2Ellist_2ELFILTER__EQ__NIL,axiom,
! [A_27a: $tType,V0P: A_27a > $o,V1ll: tyop_2Ellist_2Ellist @ A_27a] :
( ( ( c_2Ellist_2ELFILTER @ A_27a @ V0P @ V1ll )
= ( c_2Ellist_2ELNIL @ A_27a ) )
<=> ( c_2Ellist_2Eevery @ A_27a @ ( c_2Ecombin_2Eo @ A_27a @ $o @ $o @ c_2Ebool_2E_7E @ V0P ) @ V1ll ) ) ).
thf(thm_2Ellist_2ELFILTER__APPEND,axiom,
! [A_27a: $tType,V0P: A_27a > $o,V1ll1: tyop_2Ellist_2Ellist @ A_27a,V2ll2: tyop_2Ellist_2Ellist @ A_27a] :
( ( c_2Ellist_2ELFINITE @ A_27a @ V1ll1 )
=> ( ( c_2Ellist_2ELFILTER @ A_27a @ V0P @ ( c_2Ellist_2ELAPPEND @ A_27a @ V1ll1 @ V2ll2 ) )
= ( c_2Ellist_2ELAPPEND @ A_27a @ ( c_2Ellist_2ELFILTER @ A_27a @ V0P @ V1ll1 ) @ ( c_2Ellist_2ELFILTER @ A_27a @ V0P @ V2ll2 ) ) ) ) ).
thf(thm_2Ellist_2ELFLATTEN__THM,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,A_27d: $tType,V0t: tyop_2Ellist_2Ellist @ ( tyop_2Ellist_2Ellist @ A_27c )] :
( ( ( c_2Ellist_2ELFLATTEN @ A_27a @ ( c_2Ellist_2ELNIL @ ( tyop_2Ellist_2Ellist @ A_27a ) ) )
= ( c_2Ellist_2ELNIL @ A_27a ) )
& ! [V1tl: A_27b] :
( ( c_2Ellist_2ELFLATTEN @ A_27c @ ( c_2Ellist_2ELCONS @ ( tyop_2Ellist_2Ellist @ A_27c ) @ ( c_2Ellist_2ELNIL @ A_27c ) @ V0t ) )
= ( c_2Ellist_2ELFLATTEN @ A_27c @ V0t ) )
& ! [V2h: A_27d,V3t: tyop_2Ellist_2Ellist @ A_27d,V4tl: tyop_2Ellist_2Ellist @ ( tyop_2Ellist_2Ellist @ A_27d )] :
( ( c_2Ellist_2ELFLATTEN @ A_27d @ ( c_2Ellist_2ELCONS @ ( tyop_2Ellist_2Ellist @ A_27d ) @ ( c_2Ellist_2ELCONS @ A_27d @ V2h @ V3t ) @ V4tl ) )
= ( c_2Ellist_2ELCONS @ A_27d @ V2h @ ( c_2Ellist_2ELFLATTEN @ A_27d @ ( c_2Ellist_2ELCONS @ ( tyop_2Ellist_2Ellist @ A_27d ) @ V3t @ V4tl ) ) ) ) ) ).
thf(thm_2Ellist_2ELFLATTEN__APPEND,axiom,
! [A_27a: $tType,V0h: tyop_2Ellist_2Ellist @ A_27a,V1t: tyop_2Ellist_2Ellist @ ( tyop_2Ellist_2Ellist @ A_27a )] :
( ( c_2Ellist_2ELFLATTEN @ A_27a @ ( c_2Ellist_2ELCONS @ ( tyop_2Ellist_2Ellist @ A_27a ) @ V0h @ V1t ) )
= ( c_2Ellist_2ELAPPEND @ A_27a @ V0h @ ( c_2Ellist_2ELFLATTEN @ A_27a @ V1t ) ) ) ).
thf(thm_2Ellist_2ELFLATTEN__EQ__NIL,axiom,
! [A_27a: $tType,V0ll: tyop_2Ellist_2Ellist @ ( tyop_2Ellist_2Ellist @ A_27a )] :
( ( ( c_2Ellist_2ELFLATTEN @ A_27a @ V0ll )
= ( c_2Ellist_2ELNIL @ A_27a ) )
<=> ( c_2Ellist_2Eevery @ ( tyop_2Ellist_2Ellist @ A_27a ) @ ( c_2Emin_2E_3D @ ( tyop_2Ellist_2Ellist @ A_27a ) @ ( c_2Ellist_2ELNIL @ A_27a ) ) @ V0ll ) ) ).
thf(thm_2Ellist_2ELFLATTEN__SINGLETON,axiom,
! [A_27a: $tType,V0h: tyop_2Ellist_2Ellist @ A_27a] :
( ( c_2Ellist_2ELFLATTEN @ A_27a @ ( c_2Ellist_2ELCONS @ ( tyop_2Ellist_2Ellist @ A_27a ) @ V0h @ ( c_2Ellist_2ELNIL @ ( tyop_2Ellist_2Ellist @ A_27a ) ) ) )
= V0h ) ).
thf(thm_2Ellist_2ELZIP__LUNZIP,axiom,
! [A_27a: $tType,A_27b: $tType,V0ll: tyop_2Ellist_2Ellist @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b )] :
( ( c_2Ellist_2ELZIP @ A_27a @ A_27b @ ( c_2Ellist_2ELUNZIP @ A_27a @ A_27b @ V0ll ) )
= V0ll ) ).
thf(thm_2Ellist_2ELUNFOLD__THM,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: A_27a > ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) ),V1x: A_27a,V2v1: A_27a,V3v2: A_27b] :
( ( ( ( V0f @ V1x )
= ( c_2Eoption_2ENONE @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) ) )
=> ( ( c_2Ellist_2ELUNFOLD @ A_27b @ A_27a @ V0f @ V1x )
= ( c_2Ellist_2ELNIL @ A_27b ) ) )
& ( ( ( V0f @ V1x )
= ( c_2Eoption_2ESOME @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ ( c_2Epair_2E_2C @ A_27a @ A_27b @ V2v1 @ V3v2 ) ) )
=> ( ( c_2Ellist_2ELUNFOLD @ A_27b @ A_27a @ V0f @ V1x )
= ( c_2Ellist_2ELCONS @ A_27b @ V3v2 @ ( c_2Ellist_2ELUNFOLD @ A_27b @ A_27a @ V0f @ V2v1 ) ) ) ) ) ).
thf(thm_2Ellist_2ELLIST__EQ,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: A_27a > ( tyop_2Ellist_2Ellist @ A_27b ),V1g: A_27a > ( tyop_2Ellist_2Ellist @ A_27b )] :
( ! [V2x: A_27a] :
( ( ( ( V0f @ V2x )
= ( c_2Ellist_2ELNIL @ A_27b ) )
& ( ( V1g @ V2x )
= ( c_2Ellist_2ELNIL @ A_27b ) ) )
| ? [V3h: A_27b,V4y: A_27a] :
( ( ( V0f @ V2x )
= ( c_2Ellist_2ELCONS @ A_27b @ V3h @ ( V0f @ V4y ) ) )
& ( ( V1g @ V2x )
= ( c_2Ellist_2ELCONS @ A_27b @ V3h @ ( V1g @ V4y ) ) ) ) )
=> ! [V5x: A_27a] :
( ( V0f @ V5x )
= ( V1g @ V5x ) ) ) ).
thf(thm_2Ellist_2ELUNFOLD__EQ,axiom,
! [A_27a: $tType,A_27b: $tType,V0R: A_27a > ( tyop_2Ellist_2Ellist @ A_27b ) > $o,V1f: A_27a > ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) ),V2s: A_27a,V3ll: tyop_2Ellist_2Ellist @ A_27b] :
( ( ( V0R @ V2s @ V3ll )
& ! [V4s: A_27a,V5ll: tyop_2Ellist_2Ellist @ A_27b] :
( ( V0R @ V4s @ V5ll )
=> ( ( ( ( V1f @ V4s )
= ( c_2Eoption_2ENONE @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) ) )
& ( V5ll
= ( c_2Ellist_2ELNIL @ A_27b ) ) )
| ? [V6s_27: A_27a,V7x: A_27b,V8ll_27: tyop_2Ellist_2Ellist @ A_27b] :
( ( ( V1f @ V4s )
= ( c_2Eoption_2ESOME @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ ( c_2Epair_2E_2C @ A_27a @ A_27b @ V6s_27 @ V7x ) ) )
& ( ( c_2Ellist_2ELHD @ A_27b @ V5ll )
= ( c_2Eoption_2ESOME @ A_27b @ V7x ) )
& ( ( c_2Ellist_2ELTL @ A_27b @ V5ll )
= ( c_2Eoption_2ESOME @ ( tyop_2Ellist_2Ellist @ A_27b ) @ V8ll_27 ) )
& ( V0R @ V6s_27 @ V8ll_27 ) ) ) ) )
=> ( ( c_2Ellist_2ELUNFOLD @ A_27b @ A_27a @ V1f @ V2s )
= V3ll ) ) ).
thf(thm_2Ellist_2ELMAP__LUNFOLD,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0f: A_27a > A_27b,V1g: A_27c > ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ A_27c @ A_27a ) ),V2s: A_27c] :
( ( c_2Ellist_2ELMAP @ A_27a @ A_27b @ V0f @ ( c_2Ellist_2ELUNFOLD @ A_27a @ A_27c @ V1g @ V2s ) )
= ( c_2Ellist_2ELUNFOLD @ A_27b @ A_27c
@ ^ [V3s: A_27c] :
( c_2Eoption_2EOPTION__MAP @ ( tyop_2Epair_2Eprod @ A_27c @ A_27a ) @ ( tyop_2Epair_2Eprod @ A_27c @ A_27b )
@ ( c_2Epair_2EUNCURRY @ A_27c @ A_27a @ ( tyop_2Epair_2Eprod @ A_27c @ A_27b )
@ ^ [V4x: A_27c,V5y: A_27a] : ( c_2Epair_2E_2C @ A_27c @ A_27b @ V4x @ ( V0f @ V5y ) ) )
@ ( V1g @ V3s ) )
@ V2s ) ) ).
thf(thm_2Ellist_2ELNTH__LDROP,axiom,
! [A_27a: $tType,V0n: tyop_2Enum_2Enum,V1l: tyop_2Ellist_2Ellist @ A_27a,V2x: A_27a] :
( ( ( c_2Ellist_2ELNTH @ A_27a @ V0n @ V1l )
= ( c_2Eoption_2ESOME @ A_27a @ V2x ) )
=> ( ( c_2Ellist_2ELHD @ A_27a @ ( c_2Eoption_2ETHE @ ( tyop_2Ellist_2Ellist @ A_27a ) @ ( c_2Ellist_2ELDROP @ A_27a @ V0n @ V1l ) ) )
= ( c_2Eoption_2ESOME @ A_27a @ V2x ) ) ) ).
thf(thm_2Ellist_2ELAPPEND__fromList,axiom,
! [A_27a: $tType,V0l1: tyop_2Elist_2Elist @ A_27a,V1l2: tyop_2Elist_2Elist @ A_27a] :
( ( c_2Ellist_2ELAPPEND @ A_27a @ ( c_2Ellist_2EfromList @ A_27a @ V0l1 ) @ ( c_2Ellist_2EfromList @ A_27a @ V1l2 ) )
= ( c_2Ellist_2EfromList @ A_27a @ ( c_2Elist_2EAPPEND @ A_27a @ V0l1 @ V1l2 ) ) ) ).
thf(thm_2Ellist_2ELTAKE__LENGTH,axiom,
! [A_27a: $tType,V0n: tyop_2Enum_2Enum,V1ll: tyop_2Ellist_2Ellist @ A_27a,V2l: tyop_2Elist_2Elist @ A_27a] :
( ( ( c_2Ellist_2ELTAKE @ A_27a @ V0n @ V1ll )
= ( c_2Eoption_2ESOME @ ( tyop_2Elist_2Elist @ A_27a ) @ V2l ) )
=> ( V0n
= ( c_2Elist_2ELENGTH @ A_27a @ V2l ) ) ) ).
thf(thm_2Ellist_2ELTAKE__TAKE__LESS,axiom,
! [A_27a: $tType,V0n: tyop_2Enum_2Enum,V1m: tyop_2Enum_2Enum,V2ll: tyop_2Ellist_2Ellist @ A_27a,V3l: tyop_2Elist_2Elist @ A_27a] :
( ( ( ( c_2Ellist_2ELTAKE @ A_27a @ V0n @ V2ll )
= ( c_2Eoption_2ESOME @ ( tyop_2Elist_2Elist @ A_27a ) @ V3l ) )
& ( c_2Earithmetic_2E_3C_3D @ V1m @ V0n ) )
=> ( ( c_2Ellist_2ELTAKE @ A_27a @ V1m @ V2ll )
= ( c_2Eoption_2ESOME @ ( tyop_2Elist_2Elist @ A_27a ) @ ( c_2Elist_2ETAKE @ A_27a @ V1m @ V3l ) ) ) ) ).
thf(thm_2Ellist_2ELTAKE__LLENGTH__NONE,axiom,
! [A_27a: $tType,V0n: tyop_2Enum_2Enum,V1m: tyop_2Enum_2Enum,V2ll: tyop_2Ellist_2Ellist @ A_27a] :
( ( ( ( c_2Ellist_2ELLENGTH @ A_27a @ V2ll )
= ( c_2Eoption_2ESOME @ tyop_2Enum_2Enum @ V0n ) )
& ( c_2Eprim__rec_2E_3C @ V0n @ V1m ) )
=> ( ( c_2Ellist_2ELTAKE @ A_27a @ V1m @ V2ll )
= ( c_2Eoption_2ENONE @ ( tyop_2Elist_2Elist @ A_27a ) ) ) ) ).
thf(thm_2Ellist_2ELTAKE__LLENGTH__SOME,axiom,
! [A_27a: $tType,V0n: tyop_2Enum_2Enum,V1ll: tyop_2Ellist_2Ellist @ A_27a] :
( ( ( c_2Ellist_2ELLENGTH @ A_27a @ V1ll )
= ( c_2Eoption_2ESOME @ tyop_2Enum_2Enum @ V0n ) )
=> ? [V2l: tyop_2Elist_2Elist @ A_27a] :
( ( ( c_2Ellist_2ELTAKE @ A_27a @ V0n @ V1ll )
= ( c_2Eoption_2ESOME @ ( tyop_2Elist_2Elist @ A_27a ) @ V2l ) )
& ( ( c_2Ellist_2EtoList @ A_27a @ V1ll )
= ( c_2Eoption_2ESOME @ ( tyop_2Elist_2Elist @ A_27a ) @ V2l ) ) ) ) ).
thf(thm_2Ellist_2EtoList__LAPPEND__APPEND,axiom,
! [A_27a: $tType,V0x: tyop_2Elist_2Elist @ A_27a,V1l2: tyop_2Ellist_2Ellist @ A_27a,V2l1: tyop_2Ellist_2Ellist @ A_27a] :
( ( ( c_2Ellist_2EtoList @ A_27a @ ( c_2Ellist_2ELAPPEND @ A_27a @ V2l1 @ V1l2 ) )
= ( c_2Eoption_2ESOME @ ( tyop_2Elist_2Elist @ A_27a ) @ V0x ) )
=> ( V0x
= ( c_2Elist_2EAPPEND @ A_27a @ ( c_2Eoption_2ETHE @ ( tyop_2Elist_2Elist @ A_27a ) @ ( c_2Ellist_2EtoList @ A_27a @ V2l1 ) ) @ ( c_2Eoption_2ETHE @ ( tyop_2Elist_2Elist @ A_27a ) @ ( c_2Ellist_2EtoList @ A_27a @ V1l2 ) ) ) ) ) ).
thf(thm_2Ellist_2ELNTH__LLENGTH__NONE,axiom,
! [A_27a: $tType,V0x: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum,V2l: tyop_2Ellist_2Ellist @ A_27a] :
( ( ( ( c_2Ellist_2ELLENGTH @ A_27a @ V2l )
= ( c_2Eoption_2ESOME @ tyop_2Enum_2Enum @ V0x ) )
& ( c_2Earithmetic_2E_3C_3D @ V0x @ V1n ) )
=> ( ( c_2Ellist_2ELNTH @ A_27a @ V1n @ V2l )
= ( c_2Eoption_2ENONE @ A_27a ) ) ) ).
thf(thm_2Ellist_2ELNTH__NONE__MONO,axiom,
! [A_27a: $tType,V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum,V2l: tyop_2Ellist_2Ellist @ A_27a] :
( ( ( ( c_2Ellist_2ELNTH @ A_27a @ V0m @ V2l )
= ( c_2Eoption_2ENONE @ A_27a ) )
& ( c_2Earithmetic_2E_3C_3D @ V0m @ V1n ) )
=> ( ( c_2Ellist_2ELNTH @ A_27a @ V1n @ V2l )
= ( c_2Eoption_2ENONE @ A_27a ) ) ) ).
thf(thm_2Ellist_2Elinear__order__to__llist__eq,axiom,
! [A_27a: $tType,V0lo: ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) > $o,V1X: A_27a > $o] :
( ( ( c_2Eset__relation_2Elinear__order @ A_27a @ V0lo @ V1X )
& ( c_2Eset__relation_2Efinite__prefixes @ A_27a @ A_27a @ V0lo @ V1X ) )
=> ? [V2ll: tyop_2Ellist_2Ellist @ A_27a] :
( ( V1X
= ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a
@ ^ [V3x: A_27a] :
( c_2Epair_2E_2C @ A_27a @ $o @ V3x
@ ( c_2Ebool_2E_3F @ tyop_2Enum_2Enum
@ ^ [V4i: tyop_2Enum_2Enum] : ( c_2Emin_2E_3D @ ( tyop_2Eoption_2Eoption @ A_27a ) @ ( c_2Ellist_2ELNTH @ A_27a @ V4i @ V2ll ) @ ( c_2Eoption_2ESOME @ A_27a @ V3x ) ) ) ) ) )
& ( V0lo
= ( c_2Epred__set_2EGSPEC @ ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) @ ( tyop_2Epair_2Eprod @ A_27a @ A_27a )
@ ( c_2Epair_2EUNCURRY @ A_27a @ A_27a @ ( tyop_2Epair_2Eprod @ ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) @ $o )
@ ^ [V5x: A_27a,V6y: A_27a] :
( c_2Epair_2E_2C @ ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) @ $o @ ( c_2Epair_2E_2C @ A_27a @ A_27a @ V5x @ V6y )
@ ( c_2Ebool_2E_3F @ tyop_2Enum_2Enum
@ ^ [V7i: tyop_2Enum_2Enum] :
( c_2Ebool_2E_3F @ tyop_2Enum_2Enum
@ ^ [V8j: tyop_2Enum_2Enum] : ( c_2Ebool_2E_2F_5C @ ( c_2Earithmetic_2E_3C_3D @ V7i @ V8j ) @ ( c_2Ebool_2E_2F_5C @ ( c_2Emin_2E_3D @ ( tyop_2Eoption_2Eoption @ A_27a ) @ ( c_2Ellist_2ELNTH @ A_27a @ V7i @ V2ll ) @ ( c_2Eoption_2ESOME @ A_27a @ V5x ) ) @ ( c_2Emin_2E_3D @ ( tyop_2Eoption_2Eoption @ A_27a ) @ ( c_2Ellist_2ELNTH @ A_27a @ V8j @ V2ll ) @ ( c_2Eoption_2ESOME @ A_27a @ V6y ) ) ) ) ) ) ) ) ) )
& ! [V9i: tyop_2Enum_2Enum,V10j: tyop_2Enum_2Enum,V11x: A_27a] :
( ( ( ( c_2Ellist_2ELNTH @ A_27a @ V9i @ V2ll )
= ( c_2Eoption_2ESOME @ A_27a @ V11x ) )
& ( ( c_2Ellist_2ELNTH @ A_27a @ V10j @ V2ll )
= ( c_2Eoption_2ESOME @ A_27a @ V11x ) ) )
=> ( V9i = V10j ) ) ) ) ).
thf(thm_2Ellist_2Elinear__order__to__llist,axiom,
! [A_27a: $tType,V0lo: ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) > $o,V1X: A_27a > $o] :
( ( ( c_2Eset__relation_2Elinear__order @ A_27a @ V0lo @ V1X )
& ( c_2Eset__relation_2Efinite__prefixes @ A_27a @ A_27a @ V0lo @ V1X ) )
=> ? [V2ll: tyop_2Ellist_2Ellist @ A_27a] :
( ( V1X
= ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a
@ ^ [V3x: A_27a] :
( c_2Epair_2E_2C @ A_27a @ $o @ V3x
@ ( c_2Ebool_2E_3F @ tyop_2Enum_2Enum
@ ^ [V4i: tyop_2Enum_2Enum] : ( c_2Emin_2E_3D @ ( tyop_2Eoption_2Eoption @ A_27a ) @ ( c_2Ellist_2ELNTH @ A_27a @ V4i @ V2ll ) @ ( c_2Eoption_2ESOME @ A_27a @ V3x ) ) ) ) ) )
& ( c_2Epred__set_2ESUBSET @ ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) @ V0lo
@ ( c_2Epred__set_2EGSPEC @ ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) @ ( tyop_2Epair_2Eprod @ A_27a @ A_27a )
@ ( c_2Epair_2EUNCURRY @ A_27a @ A_27a @ ( tyop_2Epair_2Eprod @ ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) @ $o )
@ ^ [V5x: A_27a,V6y: A_27a] :
( c_2Epair_2E_2C @ ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) @ $o @ ( c_2Epair_2E_2C @ A_27a @ A_27a @ V5x @ V6y )
@ ( c_2Ebool_2E_3F @ tyop_2Enum_2Enum
@ ^ [V7i: tyop_2Enum_2Enum] :
( c_2Ebool_2E_3F @ tyop_2Enum_2Enum
@ ^ [V8j: tyop_2Enum_2Enum] : ( c_2Ebool_2E_2F_5C @ ( c_2Earithmetic_2E_3C_3D @ V7i @ V8j ) @ ( c_2Ebool_2E_2F_5C @ ( c_2Emin_2E_3D @ ( tyop_2Eoption_2Eoption @ A_27a ) @ ( c_2Ellist_2ELNTH @ A_27a @ V7i @ V2ll ) @ ( c_2Eoption_2ESOME @ A_27a @ V5x ) ) @ ( c_2Emin_2E_3D @ ( tyop_2Eoption_2Eoption @ A_27a ) @ ( c_2Ellist_2ELNTH @ A_27a @ V8j @ V2ll ) @ ( c_2Eoption_2ESOME @ A_27a @ V6y ) ) ) ) ) ) ) ) ) )
& ! [V9i: tyop_2Enum_2Enum,V10j: tyop_2Enum_2Enum,V11x: A_27a] :
( ( ( ( c_2Ellist_2ELNTH @ A_27a @ V9i @ V2ll )
= ( c_2Eoption_2ESOME @ A_27a @ V11x ) )
& ( ( c_2Ellist_2ELNTH @ A_27a @ V10j @ V2ll )
= ( c_2Eoption_2ESOME @ A_27a @ V11x ) ) )
=> ( V9i = V10j ) ) ) ) ).
thf(thm_2Ellist_2ELPREFIX__LNIL,axiom,
! [A_27a: $tType,V0ll: tyop_2Ellist_2Ellist @ A_27a] :
( ( c_2Ellist_2ELPREFIX @ A_27a @ ( c_2Ellist_2ELNIL @ A_27a ) @ V0ll )
& ( ( c_2Ellist_2ELPREFIX @ A_27a @ V0ll @ ( c_2Ellist_2ELNIL @ A_27a ) )
<=> ( V0ll
= ( c_2Ellist_2ELNIL @ A_27a ) ) ) ) ).
thf(thm_2Ellist_2ELPREFIX__LCONS,axiom,
! [A_27a: $tType,A_27b: $tType] :
( ! [V0ll: tyop_2Ellist_2Ellist @ A_27a,V1h: A_27a,V2t: tyop_2Ellist_2Ellist @ A_27a] :
( ( c_2Ellist_2ELPREFIX @ A_27a @ V0ll @ ( c_2Ellist_2ELCONS @ A_27a @ V1h @ V2t ) )
<=> ( ( V0ll
= ( c_2Ellist_2ELNIL @ A_27a ) )
| ? [V3l: tyop_2Ellist_2Ellist @ A_27a] :
( ( V0ll
= ( c_2Ellist_2ELCONS @ A_27a @ V1h @ V3l ) )
& ( c_2Ellist_2ELPREFIX @ A_27a @ V3l @ V2t ) ) ) )
& ! [V4h: A_27b,V5t: tyop_2Ellist_2Ellist @ A_27b,V6ll: tyop_2Ellist_2Ellist @ A_27b] :
( ( c_2Ellist_2ELPREFIX @ A_27b @ ( c_2Ellist_2ELCONS @ A_27b @ V4h @ V5t ) @ V6ll )
<=> ? [V7l: tyop_2Ellist_2Ellist @ A_27b] :
( ( V6ll
= ( c_2Ellist_2ELCONS @ A_27b @ V4h @ V7l ) )
& ( c_2Ellist_2ELPREFIX @ A_27b @ V5t @ V7l ) ) ) ) ).
thf(thm_2Ellist_2ELPREFIX__LUNFOLD,axiom,
! [A_27a: $tType,A_27b: $tType,V0n: A_27b,V1ll: tyop_2Ellist_2Ellist @ A_27a,V2f: A_27b > ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ A_27b @ A_27a ) )] :
( ( c_2Ellist_2ELPREFIX @ A_27a @ V1ll @ ( c_2Ellist_2ELUNFOLD @ A_27a @ A_27b @ V2f @ V0n ) )
= ( c_2Eoption_2Eoption__CASE @ ( tyop_2Epair_2Eprod @ A_27b @ A_27a ) @ $o @ ( V2f @ V0n ) @ ( c_2Emin_2E_3D @ ( tyop_2Ellist_2Ellist @ A_27a ) @ V1ll @ ( c_2Ellist_2ELNIL @ A_27a ) )
@ ^ [V3v: tyop_2Epair_2Eprod @ A_27b @ A_27a] :
( c_2Epair_2Epair__CASE @ $o @ A_27b @ A_27a @ V3v
@ ^ [V4n: A_27b,V5x: A_27a] :
( c_2Ebool_2E_21 @ A_27a
@ ^ [V6h: A_27a] :
( c_2Ebool_2E_21 @ ( tyop_2Ellist_2Ellist @ A_27a )
@ ^ [V7t: tyop_2Ellist_2Ellist @ A_27a] : ( c_2Emin_2E_3D_3D_3E @ ( c_2Emin_2E_3D @ ( tyop_2Ellist_2Ellist @ A_27a ) @ V1ll @ ( c_2Ellist_2ELCONS @ A_27a @ V6h @ V7t ) ) @ ( c_2Ebool_2E_2F_5C @ ( c_2Emin_2E_3D @ A_27a @ V6h @ V5x ) @ ( c_2Ellist_2ELPREFIX @ A_27a @ V7t @ ( c_2Ellist_2ELUNFOLD @ A_27a @ A_27b @ V2f @ V4n ) ) ) ) ) ) ) ) ) ).
thf(thm_2Ellist_2ELPREFIX__REFL,axiom,
! [A_27a: $tType,V0ll: tyop_2Ellist_2Ellist @ A_27a] : ( c_2Ellist_2ELPREFIX @ A_27a @ V0ll @ V0ll ) ).
thf(thm_2Ellist_2ELPREFIX__ANTISYM,axiom,
! [A_27a: $tType,V0l2: tyop_2Ellist_2Ellist @ A_27a,V1l1: tyop_2Ellist_2Ellist @ A_27a] :
( ( ( c_2Ellist_2ELPREFIX @ A_27a @ V1l1 @ V0l2 )
& ( c_2Ellist_2ELPREFIX @ A_27a @ V0l2 @ V1l1 ) )
=> ( V1l1 = V0l2 ) ) ).
thf(thm_2Ellist_2ELPREFIX__TRANS,axiom,
! [A_27a: $tType,V0l3: tyop_2Ellist_2Ellist @ A_27a,V1l2: tyop_2Ellist_2Ellist @ A_27a,V2l1: tyop_2Ellist_2Ellist @ A_27a] :
( ( ( c_2Ellist_2ELPREFIX @ A_27a @ V2l1 @ V1l2 )
& ( c_2Ellist_2ELPREFIX @ A_27a @ V1l2 @ V0l3 ) )
=> ( c_2Ellist_2ELPREFIX @ A_27a @ V2l1 @ V0l3 ) ) ).
thf(thm_2Ellist_2ELPREFIX__fromList,axiom,
! [A_27a: $tType,V0l: tyop_2Elist_2Elist @ A_27a,V1ll: tyop_2Ellist_2Ellist @ A_27a] :
( ( c_2Ellist_2ELPREFIX @ A_27a @ ( c_2Ellist_2EfromList @ A_27a @ V0l ) @ V1ll )
= ( c_2Eoption_2Eoption__CASE @ ( tyop_2Elist_2Elist @ A_27a ) @ $o @ ( c_2Ellist_2EtoList @ A_27a @ V1ll ) @ ( c_2Emin_2E_3D @ ( tyop_2Eoption_2Eoption @ ( tyop_2Elist_2Elist @ A_27a ) ) @ ( c_2Ellist_2ELTAKE @ A_27a @ ( c_2Elist_2ELENGTH @ A_27a @ V0l ) @ V1ll ) @ ( c_2Eoption_2ESOME @ ( tyop_2Elist_2Elist @ A_27a ) @ V0l ) )
@ ^ [V2ys: tyop_2Elist_2Elist @ A_27a] : ( c_2Elist_2EisPREFIX @ A_27a @ V0l @ V2ys ) ) ) ).
thf(thm_2Ellist_2Eprefixes__lprefix__total,axiom,
! [A_27a: $tType,V0ll: tyop_2Ellist_2Ellist @ A_27a,V1l1: tyop_2Ellist_2Ellist @ A_27a,V2l2: tyop_2Ellist_2Ellist @ A_27a] :
( ( ( c_2Ellist_2ELPREFIX @ A_27a @ V1l1 @ V0ll )
& ( c_2Ellist_2ELPREFIX @ A_27a @ V2l2 @ V0ll ) )
=> ( ( c_2Ellist_2ELPREFIX @ A_27a @ V1l1 @ V2l2 )
| ( c_2Ellist_2ELPREFIX @ A_27a @ V2l2 @ V1l1 ) ) ) ).
thf(thm_2Ellist_2ELTAKE__IMP__LDROP,axiom,
! [A_27a: $tType,V0n: tyop_2Enum_2Enum,V1ll: tyop_2Ellist_2Ellist @ A_27a,V2l1: tyop_2Elist_2Elist @ A_27a] :
( ( ( c_2Ellist_2ELTAKE @ A_27a @ V0n @ V1ll )
= ( c_2Eoption_2ESOME @ ( tyop_2Elist_2Elist @ A_27a ) @ V2l1 ) )
=> ? [V3l2: tyop_2Ellist_2Ellist @ A_27a] :
( ( ( c_2Ellist_2ELDROP @ A_27a @ V0n @ V1ll )
= ( c_2Eoption_2ESOME @ ( tyop_2Ellist_2Ellist @ A_27a ) @ V3l2 ) )
& ( ( c_2Ellist_2ELAPPEND @ A_27a @ ( c_2Ellist_2EfromList @ A_27a @ V2l1 ) @ V3l2 )
= V1ll ) ) ) ).
thf(thm_2Ellist_2ELDROP__EQ__LNIL,axiom,
! [A_27a: $tType,V0n: tyop_2Enum_2Enum,V1ll: tyop_2Ellist_2Ellist @ A_27a] :
( ( ( c_2Ellist_2ELDROP @ A_27a @ V0n @ V1ll )
= ( c_2Eoption_2ESOME @ ( tyop_2Ellist_2Ellist @ A_27a ) @ ( c_2Ellist_2ELNIL @ A_27a ) ) )
<=> ( ( c_2Ellist_2ELLENGTH @ A_27a @ V1ll )
= ( c_2Eoption_2ESOME @ tyop_2Enum_2Enum @ V0n ) ) ) ).
thf(thm_2Ellist_2ELPREFIX__APPEND,axiom,
! [A_27a: $tType,V0l2: tyop_2Ellist_2Ellist @ A_27a,V1l1: tyop_2Ellist_2Ellist @ A_27a] :
( ( c_2Ellist_2ELPREFIX @ A_27a @ V1l1 @ V0l2 )
<=> ? [V2ll: tyop_2Ellist_2Ellist @ A_27a] :
( V0l2
= ( c_2Ellist_2ELAPPEND @ A_27a @ V1l1 @ V2ll ) ) ) ).
thf(thm_2Ellist_2ENOT__LFINITE__DROP__LFINITE,axiom,
! [A_27a: $tType,V0n: tyop_2Enum_2Enum,V1l: tyop_2Ellist_2Ellist @ A_27a,V2t: tyop_2Ellist_2Ellist @ A_27a] :
( ( ( (~) @ ( c_2Ellist_2ELFINITE @ A_27a @ V1l ) )
& ( ( c_2Ellist_2ELDROP @ A_27a @ V0n @ V1l )
= ( c_2Eoption_2ESOME @ ( tyop_2Ellist_2Ellist @ A_27a ) @ V2t ) ) )
=> ( (~) @ ( c_2Ellist_2ELFINITE @ A_27a @ V2t ) ) ) ).
thf(thm_2Ellist_2ELDROP__APPEND1,axiom,
! [A_27a: $tType,V0n: tyop_2Enum_2Enum,V1l2: tyop_2Ellist_2Ellist @ A_27a,V2l1: tyop_2Ellist_2Ellist @ A_27a,V3l: tyop_2Ellist_2Ellist @ A_27a] :
( ( ( c_2Ellist_2ELDROP @ A_27a @ V0n @ V2l1 )
= ( c_2Eoption_2ESOME @ ( tyop_2Ellist_2Ellist @ A_27a ) @ V3l ) )
=> ( ( c_2Ellist_2ELDROP @ A_27a @ V0n @ ( c_2Ellist_2ELAPPEND @ A_27a @ V2l1 @ V1l2 ) )
= ( c_2Eoption_2ESOME @ ( tyop_2Ellist_2Ellist @ A_27a ) @ ( c_2Ellist_2ELAPPEND @ A_27a @ V3l @ V1l2 ) ) ) ) ).
thf(thm_2Ellist_2ELDROP__fromList,axiom,
! [A_27a: $tType,V0ls: tyop_2Elist_2Elist @ A_27a,V1n: tyop_2Enum_2Enum] :
( ( c_2Ellist_2ELDROP @ A_27a @ V1n @ ( c_2Ellist_2EfromList @ A_27a @ V0ls ) )
= ( c_2Ebool_2ECOND @ ( tyop_2Eoption_2Eoption @ ( tyop_2Ellist_2Ellist @ A_27a ) ) @ ( c_2Earithmetic_2E_3C_3D @ V1n @ ( c_2Elist_2ELENGTH @ A_27a @ V0ls ) ) @ ( c_2Eoption_2ESOME @ ( tyop_2Ellist_2Ellist @ A_27a ) @ ( c_2Ellist_2EfromList @ A_27a @ ( c_2Elist_2EDROP @ A_27a @ V1n @ V0ls ) ) ) @ ( c_2Eoption_2ENONE @ ( tyop_2Ellist_2Ellist @ A_27a ) ) ) ) ).
thf(thm_2Ellist_2ELDROP__SUC,axiom,
! [A_27a: $tType,V0n: tyop_2Enum_2Enum,V1ls: tyop_2Ellist_2Ellist @ A_27a] :
( ( c_2Ellist_2ELDROP @ A_27a @ ( c_2Enum_2ESUC @ V0n ) @ V1ls )
= ( c_2Eoption_2EOPTION__BIND @ ( tyop_2Ellist_2Ellist @ A_27a ) @ ( tyop_2Ellist_2Ellist @ A_27a ) @ ( c_2Ellist_2ELDROP @ A_27a @ V0n @ V1ls ) @ ( c_2Ellist_2ELTL @ A_27a ) ) ) ).
thf(thm_2Ellist_2ELHD__LGENLIST,axiom,
! [A_27a: $tType,V0limopt: tyop_2Eoption_2Eoption @ tyop_2Enum_2Enum,V1f: tyop_2Enum_2Enum > A_27a] :
( ( c_2Ellist_2ELHD @ A_27a @ ( c_2Ellist_2ELGENLIST @ A_27a @ V1f @ V0limopt ) )
= ( c_2Ebool_2ECOND @ ( tyop_2Eoption_2Eoption @ A_27a ) @ ( c_2Emin_2E_3D @ ( tyop_2Eoption_2Eoption @ tyop_2Enum_2Enum ) @ V0limopt @ ( c_2Eoption_2ESOME @ tyop_2Enum_2Enum @ c_2Enum_2E0 ) ) @ ( c_2Eoption_2ENONE @ A_27a ) @ ( c_2Eoption_2ESOME @ A_27a @ ( V1f @ c_2Enum_2E0 ) ) ) ) ).
thf(thm_2Ellist_2ELTL__LGENLIST,axiom,
! [A_27a: $tType,V0limopt: tyop_2Eoption_2Eoption @ tyop_2Enum_2Enum,V1f: tyop_2Enum_2Enum > A_27a] :
( ( c_2Ellist_2ELTL @ A_27a @ ( c_2Ellist_2ELGENLIST @ A_27a @ V1f @ V0limopt ) )
= ( c_2Ebool_2ECOND @ ( tyop_2Eoption_2Eoption @ ( tyop_2Ellist_2Ellist @ A_27a ) ) @ ( c_2Emin_2E_3D @ ( tyop_2Eoption_2Eoption @ tyop_2Enum_2Enum ) @ V0limopt @ ( c_2Eoption_2ESOME @ tyop_2Enum_2Enum @ c_2Enum_2E0 ) ) @ ( c_2Eoption_2ENONE @ ( tyop_2Ellist_2Ellist @ A_27a ) ) @ ( c_2Eoption_2ESOME @ ( tyop_2Ellist_2Ellist @ A_27a ) @ ( c_2Ellist_2ELGENLIST @ A_27a @ ( c_2Ecombin_2Eo @ tyop_2Enum_2Enum @ A_27a @ tyop_2Enum_2Enum @ V1f @ c_2Enum_2ESUC ) @ ( c_2Eoption_2EOPTION__MAP @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum @ c_2Eprim__rec_2EPRE @ V0limopt ) ) ) ) ) ).
thf(thm_2Ellist_2Enumopt__BISIMULATION,axiom,
! [V0mopt: tyop_2Eoption_2Eoption @ tyop_2Enum_2Enum,V1nopt: tyop_2Eoption_2Eoption @ tyop_2Enum_2Enum] :
( ( V0mopt = V1nopt )
<=> ? [V2R: ( tyop_2Eoption_2Eoption @ tyop_2Enum_2Enum ) > ( tyop_2Eoption_2Eoption @ tyop_2Enum_2Enum ) > $o] :
( ( V2R @ V0mopt @ V1nopt )
& ! [V3m: tyop_2Eoption_2Eoption @ tyop_2Enum_2Enum,V4n: tyop_2Eoption_2Eoption @ tyop_2Enum_2Enum] :
( ( V2R @ V3m @ V4n )
=> ( ( ( V3m
= ( c_2Eoption_2ESOME @ tyop_2Enum_2Enum @ c_2Enum_2E0 ) )
& ( V4n
= ( c_2Eoption_2ESOME @ tyop_2Enum_2Enum @ c_2Enum_2E0 ) ) )
| ( ( (~)
@ ( V3m
= ( c_2Eoption_2ESOME @ tyop_2Enum_2Enum @ c_2Enum_2E0 ) ) )
& ( (~)
@ ( V4n
= ( c_2Eoption_2ESOME @ tyop_2Enum_2Enum @ c_2Enum_2E0 ) ) )
& ( V2R @ ( c_2Eoption_2EOPTION__MAP @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum @ c_2Eprim__rec_2EPRE @ V3m ) @ ( c_2Eoption_2EOPTION__MAP @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum @ c_2Eprim__rec_2EPRE @ V4n ) ) ) ) ) ) ) ).
thf(thm_2Ellist_2ELGENLIST__EQ__LNIL,axiom,
! [A_27a: $tType,V0n: tyop_2Eoption_2Eoption @ tyop_2Enum_2Enum,V1f: tyop_2Enum_2Enum > A_27a] :
( ( ( ( c_2Ellist_2ELGENLIST @ A_27a @ V1f @ V0n )
= ( c_2Ellist_2ELNIL @ A_27a ) )
<=> ( V0n
= ( c_2Eoption_2ESOME @ tyop_2Enum_2Enum @ c_2Enum_2E0 ) ) )
& ( ( ( c_2Ellist_2ELNIL @ A_27a )
= ( c_2Ellist_2ELGENLIST @ A_27a @ V1f @ V0n ) )
<=> ( V0n
= ( c_2Eoption_2ESOME @ tyop_2Enum_2Enum @ c_2Enum_2E0 ) ) ) ) ).
thf(thm_2Ellist_2ELFINITE__LGENLIST,axiom,
! [A_27a: $tType,V0n: tyop_2Eoption_2Eoption @ tyop_2Enum_2Enum,V1f: tyop_2Enum_2Enum > A_27a] :
( ( c_2Ellist_2ELFINITE @ A_27a @ ( c_2Ellist_2ELGENLIST @ A_27a @ V1f @ V0n ) )
<=> ( (~)
@ ( V0n
= ( c_2Eoption_2ENONE @ tyop_2Enum_2Enum ) ) ) ) ).
thf(thm_2Ellist_2ELTL__HD__LTL__LHD,axiom,
! [A_27a: $tType,V0l: tyop_2Ellist_2Ellist @ A_27a] :
( ( c_2Ellist_2ELTL__HD @ A_27a @ V0l )
= ( c_2Eoption_2EOPTION__BIND @ ( tyop_2Epair_2Eprod @ ( tyop_2Ellist_2Ellist @ A_27a ) @ A_27a ) @ A_27a @ ( c_2Ellist_2ELHD @ A_27a @ V0l )
@ ^ [V1h: A_27a] :
( c_2Eoption_2EOPTION__BIND @ ( tyop_2Epair_2Eprod @ ( tyop_2Ellist_2Ellist @ A_27a ) @ A_27a ) @ ( tyop_2Ellist_2Ellist @ A_27a ) @ ( c_2Ellist_2ELTL @ A_27a @ V0l )
@ ^ [V2t: tyop_2Ellist_2Ellist @ A_27a] : ( c_2Eoption_2ESOME @ ( tyop_2Epair_2Eprod @ ( tyop_2Ellist_2Ellist @ A_27a ) @ A_27a ) @ ( c_2Epair_2E_2C @ ( tyop_2Ellist_2Ellist @ A_27a ) @ A_27a @ V2t @ V1h ) ) ) ) ) ).
thf(thm_2Ellist_2ELGENLIST__SOME,axiom,
! [A_27a: $tType,V0f: tyop_2Enum_2Enum > A_27a] :
( ( ( c_2Ellist_2ELGENLIST @ A_27a @ V0f @ ( c_2Eoption_2ESOME @ tyop_2Enum_2Enum @ c_2Enum_2E0 ) )
= ( c_2Ellist_2ELNIL @ A_27a ) )
& ! [V1n: tyop_2Enum_2Enum] :
( ( c_2Ellist_2ELGENLIST @ A_27a @ V0f @ ( c_2Eoption_2ESOME @ tyop_2Enum_2Enum @ ( c_2Enum_2ESUC @ V1n ) ) )
= ( c_2Ellist_2ELCONS @ A_27a @ ( V0f @ c_2Enum_2E0 ) @ ( c_2Ellist_2ELGENLIST @ A_27a @ ( c_2Ecombin_2Eo @ tyop_2Enum_2Enum @ A_27a @ tyop_2Enum_2Enum @ V0f @ c_2Enum_2ESUC ) @ ( c_2Eoption_2ESOME @ tyop_2Enum_2Enum @ V1n ) ) ) ) ) ).
thf(thm_2Ellist_2ELGENLIST__SOME__compute,axiom,
! [A_27a: $tType,V0f: tyop_2Enum_2Enum > A_27a] :
( ( ( c_2Ellist_2ELGENLIST @ A_27a @ V0f @ ( c_2Eoption_2ESOME @ tyop_2Enum_2Enum @ c_2Enum_2E0 ) )
= ( c_2Ellist_2ELNIL @ A_27a ) )
& ! [V1n: tyop_2Enum_2Enum] :
( ( c_2Ellist_2ELGENLIST @ A_27a @ V0f @ ( c_2Eoption_2ESOME @ tyop_2Enum_2Enum @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ V1n ) ) ) )
= ( c_2Ellist_2ELCONS @ A_27a @ ( V0f @ c_2Enum_2E0 ) @ ( c_2Ellist_2ELGENLIST @ A_27a @ ( c_2Ecombin_2Eo @ tyop_2Enum_2Enum @ A_27a @ tyop_2Enum_2Enum @ V0f @ c_2Enum_2ESUC ) @ ( c_2Eoption_2ESOME @ tyop_2Enum_2Enum @ ( 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_2Ellist_2ELGENLIST @ A_27a @ V0f @ ( c_2Eoption_2ESOME @ tyop_2Enum_2Enum @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ V2n ) ) ) )
= ( c_2Ellist_2ELCONS @ A_27a @ ( V0f @ c_2Enum_2E0 ) @ ( c_2Ellist_2ELGENLIST @ A_27a @ ( c_2Ecombin_2Eo @ tyop_2Enum_2Enum @ A_27a @ tyop_2Enum_2Enum @ V0f @ c_2Enum_2ESUC ) @ ( c_2Eoption_2ESOME @ tyop_2Enum_2Enum @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ V2n ) ) ) ) ) ) ) ).
thf(thm_2Ellist_2ELNTH__LGENLIST,axiom,
! [A_27a: $tType,V0n: tyop_2Enum_2Enum,V1f: tyop_2Enum_2Enum > A_27a,V2lim: tyop_2Eoption_2Eoption @ tyop_2Enum_2Enum] :
( ( c_2Ellist_2ELNTH @ A_27a @ V0n @ ( c_2Ellist_2ELGENLIST @ A_27a @ V1f @ V2lim ) )
= ( c_2Eoption_2Eoption__CASE @ tyop_2Enum_2Enum @ ( tyop_2Eoption_2Eoption @ A_27a ) @ V2lim @ ( c_2Eoption_2ESOME @ A_27a @ ( V1f @ V0n ) )
@ ^ [V3lim0: tyop_2Enum_2Enum] : ( c_2Ebool_2ECOND @ ( tyop_2Eoption_2Eoption @ A_27a ) @ ( c_2Eprim__rec_2E_3C @ V0n @ V3lim0 ) @ ( c_2Eoption_2ESOME @ A_27a @ ( V1f @ V0n ) ) @ ( c_2Eoption_2ENONE @ A_27a ) ) ) ) ).
thf(thm_2Ellist_2ELNTH__LMAP,axiom,
! [A_27a: $tType,A_27b: $tType,V0n: tyop_2Enum_2Enum,V1f: A_27a > A_27b,V2l: tyop_2Ellist_2Ellist @ A_27a] :
( ( c_2Ellist_2ELNTH @ A_27b @ V0n @ ( c_2Ellist_2ELMAP @ A_27a @ A_27b @ V1f @ V2l ) )
= ( c_2Eoption_2EOPTION__MAP @ A_27a @ A_27b @ V1f @ ( c_2Ellist_2ELNTH @ A_27a @ V0n @ V2l ) ) ) ).
thf(thm_2Ellist_2ELLENGTH__LGENLIST,axiom,
! [A_27a: $tType,V0limopt: tyop_2Eoption_2Eoption @ tyop_2Enum_2Enum,V1f: tyop_2Enum_2Enum > A_27a] :
( ( c_2Ellist_2ELLENGTH @ A_27a @ ( c_2Ellist_2ELGENLIST @ A_27a @ V1f @ V0limopt ) )
= V0limopt ) ).
thf(thm_2Ellist_2ELMAP__LGENLIST,axiom,
! [A_27a: $tType,A_27b: $tType,V0limopt: tyop_2Eoption_2Eoption @ tyop_2Enum_2Enum,V1g: tyop_2Enum_2Enum > A_27b,V2f: A_27b > A_27a] :
( ( c_2Ellist_2ELMAP @ A_27b @ A_27a @ V2f @ ( c_2Ellist_2ELGENLIST @ A_27b @ V1g @ V0limopt ) )
= ( c_2Ellist_2ELGENLIST @ A_27a @ ( c_2Ecombin_2Eo @ tyop_2Enum_2Enum @ A_27a @ A_27b @ V2f @ V1g ) @ V0limopt ) ) ).
thf(thm_2Ellist_2ELGENLIST__EQ__CONS,axiom,
! [A_27a: $tType,V0t: tyop_2Ellist_2Ellist @ A_27a,V1h: A_27a,V2f: tyop_2Enum_2Enum > A_27a] :
( ( ( c_2Ellist_2ELGENLIST @ A_27a @ V2f @ ( c_2Eoption_2ENONE @ tyop_2Enum_2Enum ) )
= ( c_2Ellist_2ELCONS @ A_27a @ V1h @ V0t ) )
<=> ( ( V1h
= ( V2f @ c_2Enum_2E0 ) )
& ( V0t
= ( c_2Ellist_2ELGENLIST @ A_27a @ ( c_2Ecombin_2Eo @ tyop_2Enum_2Enum @ A_27a @ tyop_2Enum_2Enum @ V2f @ ( c_2Earithmetic_2E_2B @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) @ ( c_2Eoption_2ENONE @ tyop_2Enum_2Enum ) ) ) ) ) ).
thf(thm_2Ellist_2ELGENLIST__CHUNK__GENLIST,axiom,
! [A_27a: $tType,V0n: tyop_2Enum_2Enum,V1f: tyop_2Enum_2Enum > A_27a] :
( ( c_2Ellist_2ELGENLIST @ A_27a @ V1f @ ( c_2Eoption_2ENONE @ tyop_2Enum_2Enum ) )
= ( c_2Ellist_2ELAPPEND @ A_27a @ ( c_2Ellist_2EfromList @ A_27a @ ( c_2Elist_2EGENLIST @ A_27a @ V1f @ V0n ) ) @ ( c_2Ellist_2ELGENLIST @ A_27a @ ( c_2Ecombin_2Eo @ tyop_2Enum_2Enum @ A_27a @ tyop_2Enum_2Enum @ V1f @ ( c_2Earithmetic_2E_2B @ V0n ) ) @ ( c_2Eoption_2ENONE @ tyop_2Enum_2Enum ) ) ) ) ).
thf(thm_2Ellist_2ELREPEAT__thm,axiom,
! [A_27a: $tType,V0l: tyop_2Elist_2Elist @ A_27a] :
( ( c_2Ellist_2ELREPEAT @ A_27a @ V0l )
= ( c_2Ellist_2ELAPPEND @ A_27a @ ( c_2Ellist_2EfromList @ A_27a @ V0l ) @ ( c_2Ellist_2ELREPEAT @ A_27a @ V0l ) ) ) ).
thf(thm_2Ellist_2ELREPEAT__NIL,axiom,
! [A_27a: $tType] :
( ( c_2Ellist_2ELREPEAT @ A_27a @ ( c_2Elist_2ENIL @ A_27a ) )
= ( c_2Ellist_2ELNIL @ A_27a ) ) ).
thf(thm_2Ellist_2ELREPEAT__EQ__LNIL,axiom,
! [A_27a: $tType,V0l: tyop_2Elist_2Elist @ A_27a] :
( ( ( ( c_2Ellist_2ELREPEAT @ A_27a @ V0l )
= ( c_2Ellist_2ELNIL @ A_27a ) )
<=> ( V0l
= ( c_2Elist_2ENIL @ A_27a ) ) )
& ( ( ( c_2Ellist_2ELNIL @ A_27a )
= ( c_2Ellist_2ELREPEAT @ A_27a @ V0l ) )
<=> ( V0l
= ( c_2Elist_2ENIL @ A_27a ) ) ) ) ).
thf(thm_2Ellist_2ELHD__LREPEAT,axiom,
! [A_27a: $tType,V0l: tyop_2Elist_2Elist @ A_27a] :
( ( c_2Ellist_2ELHD @ A_27a @ ( c_2Ellist_2ELREPEAT @ A_27a @ V0l ) )
= ( c_2Ellist_2ELHD @ A_27a @ ( c_2Ellist_2EfromList @ A_27a @ V0l ) ) ) ).
thf(thm_2Ellist_2ELTL__LREPEAT,axiom,
! [A_27a: $tType,V0l: tyop_2Elist_2Elist @ A_27a] :
( ( c_2Ellist_2ELTL @ A_27a @ ( c_2Ellist_2ELREPEAT @ A_27a @ V0l ) )
= ( c_2Eoption_2EOPTION__MAP @ ( tyop_2Ellist_2Ellist @ A_27a ) @ ( tyop_2Ellist_2Ellist @ A_27a )
@ ^ [V1t: tyop_2Ellist_2Ellist @ A_27a] : ( c_2Ellist_2ELAPPEND @ A_27a @ V1t @ ( c_2Ellist_2ELREPEAT @ A_27a @ V0l ) )
@ ( c_2Ellist_2ELTL @ A_27a @ ( c_2Ellist_2EfromList @ A_27a @ V0l ) ) ) ) ).
thf(thm_2Ellist_2ELLENGTH__LREPEAT,axiom,
! [A_27a: $tType,V0l: tyop_2Elist_2Elist @ A_27a] :
( ( c_2Ellist_2ELLENGTH @ A_27a @ ( c_2Ellist_2ELREPEAT @ A_27a @ V0l ) )
= ( c_2Ebool_2ECOND @ ( tyop_2Eoption_2Eoption @ tyop_2Enum_2Enum ) @ ( c_2Elist_2ENULL @ A_27a @ V0l ) @ ( c_2Eoption_2ESOME @ tyop_2Enum_2Enum @ c_2Enum_2E0 ) @ ( c_2Eoption_2ENONE @ tyop_2Enum_2Enum ) ) ) ).
%------------------------------------------------------------------------------