ITP001 Axioms: ITP030^5.ax
%------------------------------------------------------------------------------
% File : ITP030^5 : TPTP v8.2.0. Bugfixed v7.5.0.
% Domain : Interactive Theorem Proving
% Axioms : HOL4 set theory export, chainy mode
% Version : [BG+19] axioms.
% English :
% Refs : [BG+19] Brown et al. (2019), GRUNGE: A Grand Unified ATP Chall
% : [Gau20] Gauthier (2020), Email to Geoff Sutcliffe
% Source : [BG+19]
% Names : list^2.ax [Gau20]
% : HL4030^5.ax [TPAP]
% Status : Satisfiable
% Syntax : Number of formulae : 611 ( 15 unt; 69 typ; 0 def)
% Number of atoms : 10116 ( 657 equ; 0 cnn)
% Maximal formula atoms : 85 ( 16 avg)
% Number of connectives : 21655 ( 52 ~; 25 |; 404 &;19449 @)
% ( 187 <=>;1538 =>; 0 <=; 0 <~>)
% Maximal formula depth : 48 ( 13 avg;19449 nst)
% Number of types : 3 ( 2 usr)
% Number of type conns : 83 ( 83 >; 0 *; 0 +; 0 <<)
% Number of symbols : 161 ( 158 usr; 96 con; 0-3 aty)
% Number of variables : 2417 ( 99 ^2224 !; 94 ?;2417 :)
% SPC : TH0_SAT_EQU_NAR
% Comments :
% Bugfixes : v7.5.0 - Fixes to the axioms.
%------------------------------------------------------------------------------
thf(tp_ty_2Elist_2Elist,type,
ty_2Elist_2Elist: del > del ).
thf(tp_c_2Elist_2EALL__DISTINCT,type,
c_2Elist_2EALL__DISTINCT: del > $i ).
thf(mem_c_2Elist_2EALL__DISTINCT,axiom,
! [A_27a: del] : ( mem @ ( c_2Elist_2EALL__DISTINCT @ A_27a ) @ ( arr @ ( ty_2Elist_2Elist @ A_27a ) @ bool ) ) ).
thf(tp_c_2Elist_2EAPPEND,type,
c_2Elist_2EAPPEND: del > $i ).
thf(mem_c_2Elist_2EAPPEND,axiom,
! [A_27a: del] : ( mem @ ( c_2Elist_2EAPPEND @ A_27a ) @ ( arr @ ( ty_2Elist_2Elist @ A_27a ) @ ( arr @ ( ty_2Elist_2Elist @ A_27a ) @ ( ty_2Elist_2Elist @ A_27a ) ) ) ) ).
thf(tp_c_2Elist_2ECONS,type,
c_2Elist_2ECONS: del > $i ).
thf(mem_c_2Elist_2ECONS,axiom,
! [A_27a: del] : ( mem @ ( c_2Elist_2ECONS @ A_27a ) @ ( arr @ A_27a @ ( arr @ ( ty_2Elist_2Elist @ A_27a ) @ ( ty_2Elist_2Elist @ A_27a ) ) ) ) ).
thf(tp_c_2Elist_2EDROP,type,
c_2Elist_2EDROP: del > $i ).
thf(mem_c_2Elist_2EDROP,axiom,
! [A_27a: del] : ( mem @ ( c_2Elist_2EDROP @ A_27a ) @ ( arr @ ty_2Enum_2Enum @ ( arr @ ( ty_2Elist_2Elist @ A_27a ) @ ( ty_2Elist_2Elist @ A_27a ) ) ) ) ).
thf(tp_c_2Elist_2EEL,type,
c_2Elist_2EEL: del > $i ).
thf(mem_c_2Elist_2EEL,axiom,
! [A_27a: del] : ( mem @ ( c_2Elist_2EEL @ A_27a ) @ ( arr @ ty_2Enum_2Enum @ ( arr @ ( ty_2Elist_2Elist @ A_27a ) @ A_27a ) ) ) ).
thf(tp_c_2Elist_2EEVERY,type,
c_2Elist_2EEVERY: del > $i ).
thf(mem_c_2Elist_2EEVERY,axiom,
! [A_27a: del] : ( mem @ ( c_2Elist_2EEVERY @ A_27a ) @ ( arr @ ( arr @ A_27a @ bool ) @ ( arr @ ( ty_2Elist_2Elist @ A_27a ) @ bool ) ) ) ).
thf(tp_c_2Elist_2EEVERYi,type,
c_2Elist_2EEVERYi: del > $i ).
thf(mem_c_2Elist_2EEVERYi,axiom,
! [A_27a: del] : ( mem @ ( c_2Elist_2EEVERYi @ A_27a ) @ ( arr @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) @ ( arr @ ( ty_2Elist_2Elist @ A_27a ) @ bool ) ) ) ).
thf(tp_c_2Elist_2EEXISTS,type,
c_2Elist_2EEXISTS: del > $i ).
thf(mem_c_2Elist_2EEXISTS,axiom,
! [A_27a: del] : ( mem @ ( c_2Elist_2EEXISTS @ A_27a ) @ ( arr @ ( arr @ A_27a @ bool ) @ ( arr @ ( ty_2Elist_2Elist @ A_27a ) @ bool ) ) ) ).
thf(tp_c_2Elist_2EFILTER,type,
c_2Elist_2EFILTER: del > $i ).
thf(mem_c_2Elist_2EFILTER,axiom,
! [A_27a: del] : ( mem @ ( c_2Elist_2EFILTER @ A_27a ) @ ( arr @ ( arr @ A_27a @ bool ) @ ( arr @ ( ty_2Elist_2Elist @ A_27a ) @ ( ty_2Elist_2Elist @ A_27a ) ) ) ) ).
thf(tp_c_2Elist_2EFIND,type,
c_2Elist_2EFIND: del > $i ).
thf(mem_c_2Elist_2EFIND,axiom,
! [A_27a: del] : ( mem @ ( c_2Elist_2EFIND @ A_27a ) @ ( arr @ ( arr @ A_27a @ bool ) @ ( arr @ ( ty_2Elist_2Elist @ A_27a ) @ ( ty_2Eoption_2Eoption @ A_27a ) ) ) ) ).
thf(tp_c_2Elist_2EFLAT,type,
c_2Elist_2EFLAT: del > $i ).
thf(mem_c_2Elist_2EFLAT,axiom,
! [A_27a: del] : ( mem @ ( c_2Elist_2EFLAT @ A_27a ) @ ( arr @ ( ty_2Elist_2Elist @ ( ty_2Elist_2Elist @ A_27a ) ) @ ( ty_2Elist_2Elist @ A_27a ) ) ) ).
thf(tp_c_2Elist_2EFOLDL,type,
c_2Elist_2EFOLDL: del > del > $i ).
thf(mem_c_2Elist_2EFOLDL,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2Elist_2EFOLDL @ A_27a @ A_27b ) @ ( arr @ ( arr @ A_27b @ ( arr @ A_27a @ A_27b ) ) @ ( arr @ A_27b @ ( arr @ ( ty_2Elist_2Elist @ A_27a ) @ A_27b ) ) ) ) ).
thf(tp_c_2Elist_2EFOLDL2,type,
c_2Elist_2EFOLDL2: del > del > del > $i ).
thf(mem_c_2Elist_2EFOLDL2,axiom,
! [A_27a: del,A_27b: del,A_27c: del] : ( mem @ ( c_2Elist_2EFOLDL2 @ A_27a @ A_27b @ A_27c ) @ ( arr @ ( arr @ A_27a @ ( arr @ A_27b @ ( arr @ A_27c @ A_27a ) ) ) @ ( arr @ A_27a @ ( arr @ ( ty_2Elist_2Elist @ A_27b ) @ ( arr @ ( ty_2Elist_2Elist @ A_27c ) @ A_27a ) ) ) ) ) ).
thf(tp_c_2Elist_2EFOLDR,type,
c_2Elist_2EFOLDR: del > del > $i ).
thf(mem_c_2Elist_2EFOLDR,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2Elist_2EFOLDR @ A_27a @ A_27b ) @ ( arr @ ( arr @ A_27a @ ( arr @ A_27b @ A_27b ) ) @ ( arr @ A_27b @ ( arr @ ( ty_2Elist_2Elist @ A_27a ) @ A_27b ) ) ) ) ).
thf(tp_c_2Elist_2EFRONT,type,
c_2Elist_2EFRONT: del > $i ).
thf(mem_c_2Elist_2EFRONT,axiom,
! [A_27a: del] : ( mem @ ( c_2Elist_2EFRONT @ A_27a ) @ ( arr @ ( ty_2Elist_2Elist @ A_27a ) @ ( ty_2Elist_2Elist @ A_27a ) ) ) ).
thf(tp_c_2Elist_2EGENLIST,type,
c_2Elist_2EGENLIST: del > $i ).
thf(mem_c_2Elist_2EGENLIST,axiom,
! [A_27a: del] : ( mem @ ( c_2Elist_2EGENLIST @ A_27a ) @ ( arr @ ( arr @ ty_2Enum_2Enum @ A_27a ) @ ( arr @ ty_2Enum_2Enum @ ( ty_2Elist_2Elist @ A_27a ) ) ) ) ).
thf(tp_c_2Elist_2EGENLIST__AUX,type,
c_2Elist_2EGENLIST__AUX: del > $i ).
thf(mem_c_2Elist_2EGENLIST__AUX,axiom,
! [A_27a: del] : ( mem @ ( c_2Elist_2EGENLIST__AUX @ A_27a ) @ ( arr @ ( arr @ ty_2Enum_2Enum @ A_27a ) @ ( arr @ ty_2Enum_2Enum @ ( arr @ ( ty_2Elist_2Elist @ A_27a ) @ ( ty_2Elist_2Elist @ A_27a ) ) ) ) ) ).
thf(tp_c_2Elist_2EHD,type,
c_2Elist_2EHD: del > $i ).
thf(mem_c_2Elist_2EHD,axiom,
! [A_27a: del] : ( mem @ ( c_2Elist_2EHD @ A_27a ) @ ( arr @ ( ty_2Elist_2Elist @ A_27a ) @ A_27a ) ) ).
thf(tp_c_2Elist_2EINDEX__FIND,type,
c_2Elist_2EINDEX__FIND: del > $i ).
thf(mem_c_2Elist_2EINDEX__FIND,axiom,
! [A_27a: del] : ( mem @ ( c_2Elist_2EINDEX__FIND @ A_27a ) @ ( arr @ ty_2Enum_2Enum @ ( arr @ ( arr @ A_27a @ bool ) @ ( arr @ ( ty_2Elist_2Elist @ A_27a ) @ ( ty_2Eoption_2Eoption @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ A_27a ) ) ) ) ) ) ).
thf(tp_c_2Elist_2EINDEX__OF,type,
c_2Elist_2EINDEX__OF: del > $i ).
thf(mem_c_2Elist_2EINDEX__OF,axiom,
! [A_27a: del] : ( mem @ ( c_2Elist_2EINDEX__OF @ A_27a ) @ ( arr @ A_27a @ ( arr @ ( ty_2Elist_2Elist @ A_27a ) @ ( ty_2Eoption_2Eoption @ ty_2Enum_2Enum ) ) ) ) ).
thf(tp_c_2Elist_2ELAST,type,
c_2Elist_2ELAST: del > $i ).
thf(mem_c_2Elist_2ELAST,axiom,
! [A_27a: del] : ( mem @ ( c_2Elist_2ELAST @ A_27a ) @ ( arr @ ( ty_2Elist_2Elist @ A_27a ) @ A_27a ) ) ).
thf(tp_c_2Elist_2ELEN,type,
c_2Elist_2ELEN: del > $i ).
thf(mem_c_2Elist_2ELEN,axiom,
! [A_27a: del] : ( mem @ ( c_2Elist_2ELEN @ A_27a ) @ ( arr @ ( ty_2Elist_2Elist @ A_27a ) @ ( arr @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) ) ) ).
thf(tp_c_2Elist_2ELENGTH,type,
c_2Elist_2ELENGTH: del > $i ).
thf(mem_c_2Elist_2ELENGTH,axiom,
! [A_27a: del] : ( mem @ ( c_2Elist_2ELENGTH @ A_27a ) @ ( arr @ ( ty_2Elist_2Elist @ A_27a ) @ ty_2Enum_2Enum ) ) ).
thf(tp_c_2Elist_2ELIST__APPLY,type,
c_2Elist_2ELIST__APPLY: del > del > $i ).
thf(mem_c_2Elist_2ELIST__APPLY,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2Elist_2ELIST__APPLY @ A_27a @ A_27b ) @ ( arr @ ( ty_2Elist_2Elist @ ( arr @ A_27b @ A_27a ) ) @ ( arr @ ( ty_2Elist_2Elist @ A_27b ) @ ( ty_2Elist_2Elist @ A_27a ) ) ) ) ).
thf(tp_c_2Elist_2ELIST__BIND,type,
c_2Elist_2ELIST__BIND: del > del > $i ).
thf(mem_c_2Elist_2ELIST__BIND,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2Elist_2ELIST__BIND @ A_27a @ A_27b ) @ ( arr @ ( ty_2Elist_2Elist @ A_27b ) @ ( arr @ ( arr @ A_27b @ ( ty_2Elist_2Elist @ A_27a ) ) @ ( ty_2Elist_2Elist @ A_27a ) ) ) ) ).
thf(stp_c_ty_2Elist_2Elist_ty_2Eone_2Eone,type,
tp__c_ty_2Elist_2Elist_ty_2Eone_2Eone: $tType ).
thf(stp_inj_c_ty_2Elist_2Elist_ty_2Eone_2Eone,type,
inj__c_ty_2Elist_2Elist_ty_2Eone_2Eone: tp__c_ty_2Elist_2Elist_ty_2Eone_2Eone > $i ).
thf(stp_surj_c_ty_2Elist_2Elist_ty_2Eone_2Eone,type,
surj__c_ty_2Elist_2Elist_ty_2Eone_2Eone: $i > tp__c_ty_2Elist_2Elist_ty_2Eone_2Eone ).
thf(stp_inj_surj_c_ty_2Elist_2Elist_ty_2Eone_2Eone,axiom,
! [X: tp__c_ty_2Elist_2Elist_ty_2Eone_2Eone] :
( ( surj__c_ty_2Elist_2Elist_ty_2Eone_2Eone @ ( inj__c_ty_2Elist_2Elist_ty_2Eone_2Eone @ X ) )
= X ) ).
thf(stp_inj_mem_c_ty_2Elist_2Elist_ty_2Eone_2Eone,axiom,
! [X: tp__c_ty_2Elist_2Elist_ty_2Eone_2Eone] : ( mem @ ( inj__c_ty_2Elist_2Elist_ty_2Eone_2Eone @ X ) @ ( ty_2Elist_2Elist @ ty_2Eone_2Eone ) ) ).
thf(stp_iso_mem_c_ty_2Elist_2Elist_ty_2Eone_2Eone,axiom,
! [X: $i] :
( ( mem @ X @ ( ty_2Elist_2Elist @ ty_2Eone_2Eone ) )
=> ( X
= ( inj__c_ty_2Elist_2Elist_ty_2Eone_2Eone @ ( surj__c_ty_2Elist_2Elist_ty_2Eone_2Eone @ X ) ) ) ) ).
thf(tp_c_2Elist_2ELIST__GUARD,type,
c_2Elist_2ELIST__GUARD: $i ).
thf(mem_c_2Elist_2ELIST__GUARD,axiom,
mem @ c_2Elist_2ELIST__GUARD @ ( arr @ bool @ ( ty_2Elist_2Elist @ ty_2Eone_2Eone ) ) ).
thf(tp_c_2Elist_2ELIST__IGNORE__BIND,type,
c_2Elist_2ELIST__IGNORE__BIND: del > del > $i ).
thf(mem_c_2Elist_2ELIST__IGNORE__BIND,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2Elist_2ELIST__IGNORE__BIND @ A_27a @ A_27b ) @ ( arr @ ( ty_2Elist_2Elist @ A_27b ) @ ( arr @ ( ty_2Elist_2Elist @ A_27a ) @ ( ty_2Elist_2Elist @ A_27a ) ) ) ) ).
thf(tp_c_2Elist_2ELIST__LIFT2,type,
c_2Elist_2ELIST__LIFT2: del > del > del > $i ).
thf(mem_c_2Elist_2ELIST__LIFT2,axiom,
! [A_27a: del,A_27b: del,A_27c: del] : ( mem @ ( c_2Elist_2ELIST__LIFT2 @ A_27a @ A_27b @ A_27c ) @ ( arr @ ( arr @ A_27b @ ( arr @ A_27c @ A_27a ) ) @ ( arr @ ( ty_2Elist_2Elist @ A_27b ) @ ( arr @ ( ty_2Elist_2Elist @ A_27c ) @ ( ty_2Elist_2Elist @ A_27a ) ) ) ) ) ).
thf(tp_c_2Elist_2ELIST__REL,type,
c_2Elist_2ELIST__REL: del > del > $i ).
thf(mem_c_2Elist_2ELIST__REL,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2Elist_2ELIST__REL @ A_27a @ A_27b ) @ ( arr @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) @ ( arr @ ( ty_2Elist_2Elist @ A_27a ) @ ( arr @ ( ty_2Elist_2Elist @ A_27b ) @ bool ) ) ) ) ).
thf(tp_c_2Elist_2ELIST__TO__SET,type,
c_2Elist_2ELIST__TO__SET: del > $i ).
thf(mem_c_2Elist_2ELIST__TO__SET,axiom,
! [A_27a: del] : ( mem @ ( c_2Elist_2ELIST__TO__SET @ A_27a ) @ ( arr @ ( ty_2Elist_2Elist @ A_27a ) @ ( arr @ A_27a @ bool ) ) ) ).
thf(tp_c_2Elist_2ELLEX,type,
c_2Elist_2ELLEX: del > $i ).
thf(mem_c_2Elist_2ELLEX,axiom,
! [A_27a: del] : ( mem @ ( c_2Elist_2ELLEX @ A_27a ) @ ( arr @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) @ ( arr @ ( ty_2Elist_2Elist @ A_27a ) @ ( arr @ ( ty_2Elist_2Elist @ A_27a ) @ bool ) ) ) ) ).
thf(tp_c_2Elist_2ELRC,type,
c_2Elist_2ELRC: del > $i ).
thf(mem_c_2Elist_2ELRC,axiom,
! [A_27a: del] : ( mem @ ( c_2Elist_2ELRC @ A_27a ) @ ( arr @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) @ ( arr @ ( ty_2Elist_2Elist @ A_27a ) @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) ) ) ) ).
thf(tp_c_2Elist_2ELUPDATE,type,
c_2Elist_2ELUPDATE: del > $i ).
thf(mem_c_2Elist_2ELUPDATE,axiom,
! [A_27a: del] : ( mem @ ( c_2Elist_2ELUPDATE @ A_27a ) @ ( arr @ A_27a @ ( arr @ ty_2Enum_2Enum @ ( arr @ ( ty_2Elist_2Elist @ A_27a ) @ ( ty_2Elist_2Elist @ A_27a ) ) ) ) ) ).
thf(tp_c_2Elist_2EMAP,type,
c_2Elist_2EMAP: del > del > $i ).
thf(mem_c_2Elist_2EMAP,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2Elist_2EMAP @ A_27a @ A_27b ) @ ( arr @ ( arr @ A_27a @ A_27b ) @ ( arr @ ( ty_2Elist_2Elist @ A_27a ) @ ( ty_2Elist_2Elist @ A_27b ) ) ) ) ).
thf(tp_c_2Elist_2EMAP2,type,
c_2Elist_2EMAP2: del > del > del > $i ).
thf(mem_c_2Elist_2EMAP2,axiom,
! [A_27a: del,A_27b: del,A_27c: del] : ( mem @ ( c_2Elist_2EMAP2 @ A_27a @ A_27b @ A_27c ) @ ( arr @ ( arr @ A_27b @ ( arr @ A_27c @ A_27a ) ) @ ( arr @ ( ty_2Elist_2Elist @ A_27b ) @ ( arr @ ( ty_2Elist_2Elist @ A_27c ) @ ( ty_2Elist_2Elist @ A_27a ) ) ) ) ) ).
thf(tp_c_2Elist_2ENIL,type,
c_2Elist_2ENIL: del > $i ).
thf(mem_c_2Elist_2ENIL,axiom,
! [A_27a: del] : ( mem @ ( c_2Elist_2ENIL @ A_27a ) @ ( ty_2Elist_2Elist @ A_27a ) ) ).
thf(tp_c_2Elist_2ENULL,type,
c_2Elist_2ENULL: del > $i ).
thf(mem_c_2Elist_2ENULL,axiom,
! [A_27a: del] : ( mem @ ( c_2Elist_2ENULL @ A_27a ) @ ( arr @ ( ty_2Elist_2Elist @ A_27a ) @ bool ) ) ).
thf(tp_c_2Elist_2EOPT__MMAP,type,
c_2Elist_2EOPT__MMAP: del > del > $i ).
thf(mem_c_2Elist_2EOPT__MMAP,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2Elist_2EOPT__MMAP @ A_27a @ A_27b ) @ ( arr @ ( arr @ A_27b @ ( ty_2Eoption_2Eoption @ A_27a ) ) @ ( arr @ ( ty_2Elist_2Elist @ A_27b ) @ ( ty_2Eoption_2Eoption @ ( ty_2Elist_2Elist @ A_27a ) ) ) ) ) ).
thf(tp_c_2Elist_2EPAD__LEFT,type,
c_2Elist_2EPAD__LEFT: del > $i ).
thf(mem_c_2Elist_2EPAD__LEFT,axiom,
! [A_27a: del] : ( mem @ ( c_2Elist_2EPAD__LEFT @ A_27a ) @ ( arr @ A_27a @ ( arr @ ty_2Enum_2Enum @ ( arr @ ( ty_2Elist_2Elist @ A_27a ) @ ( ty_2Elist_2Elist @ A_27a ) ) ) ) ) ).
thf(tp_c_2Elist_2EPAD__RIGHT,type,
c_2Elist_2EPAD__RIGHT: del > $i ).
thf(mem_c_2Elist_2EPAD__RIGHT,axiom,
! [A_27a: del] : ( mem @ ( c_2Elist_2EPAD__RIGHT @ A_27a ) @ ( arr @ A_27a @ ( arr @ ty_2Enum_2Enum @ ( arr @ ( ty_2Elist_2Elist @ A_27a ) @ ( ty_2Elist_2Elist @ A_27a ) ) ) ) ) ).
thf(tp_c_2Elist_2EREV,type,
c_2Elist_2EREV: del > $i ).
thf(mem_c_2Elist_2EREV,axiom,
! [A_27a: del] : ( mem @ ( c_2Elist_2EREV @ A_27a ) @ ( arr @ ( ty_2Elist_2Elist @ A_27a ) @ ( arr @ ( ty_2Elist_2Elist @ A_27a ) @ ( ty_2Elist_2Elist @ A_27a ) ) ) ) ).
thf(tp_c_2Elist_2EREVERSE,type,
c_2Elist_2EREVERSE: del > $i ).
thf(mem_c_2Elist_2EREVERSE,axiom,
! [A_27a: del] : ( mem @ ( c_2Elist_2EREVERSE @ A_27a ) @ ( arr @ ( ty_2Elist_2Elist @ A_27a ) @ ( ty_2Elist_2Elist @ A_27a ) ) ) ).
thf(tp_c_2Elist_2ESET__TO__LIST,type,
c_2Elist_2ESET__TO__LIST: del > $i ).
thf(mem_c_2Elist_2ESET__TO__LIST,axiom,
! [A_27a: del] : ( mem @ ( c_2Elist_2ESET__TO__LIST @ A_27a ) @ ( arr @ ( arr @ A_27a @ bool ) @ ( ty_2Elist_2Elist @ A_27a ) ) ) ).
thf(tp_c_2Elist_2ESHORTLEX,type,
c_2Elist_2ESHORTLEX: del > $i ).
thf(mem_c_2Elist_2ESHORTLEX,axiom,
! [A_27a: del] : ( mem @ ( c_2Elist_2ESHORTLEX @ A_27a ) @ ( arr @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) @ ( arr @ ( ty_2Elist_2Elist @ A_27a ) @ ( arr @ ( ty_2Elist_2Elist @ A_27a ) @ bool ) ) ) ) ).
thf(tp_c_2Elist_2ESNOC,type,
c_2Elist_2ESNOC: del > $i ).
thf(mem_c_2Elist_2ESNOC,axiom,
! [A_27a: del] : ( mem @ ( c_2Elist_2ESNOC @ A_27a ) @ ( arr @ A_27a @ ( arr @ ( ty_2Elist_2Elist @ A_27a ) @ ( ty_2Elist_2Elist @ A_27a ) ) ) ) ).
thf(stp_c_ty_2Elist_2Elist_ty_2Enum_2Enum,type,
tp__c_ty_2Elist_2Elist_ty_2Enum_2Enum: $tType ).
thf(stp_inj_c_ty_2Elist_2Elist_ty_2Enum_2Enum,type,
inj__c_ty_2Elist_2Elist_ty_2Enum_2Enum: tp__c_ty_2Elist_2Elist_ty_2Enum_2Enum > $i ).
thf(stp_surj_c_ty_2Elist_2Elist_ty_2Enum_2Enum,type,
surj__c_ty_2Elist_2Elist_ty_2Enum_2Enum: $i > tp__c_ty_2Elist_2Elist_ty_2Enum_2Enum ).
thf(stp_inj_surj_c_ty_2Elist_2Elist_ty_2Enum_2Enum,axiom,
! [X: tp__c_ty_2Elist_2Elist_ty_2Enum_2Enum] :
( ( surj__c_ty_2Elist_2Elist_ty_2Enum_2Enum @ ( inj__c_ty_2Elist_2Elist_ty_2Enum_2Enum @ X ) )
= X ) ).
thf(stp_inj_mem_c_ty_2Elist_2Elist_ty_2Enum_2Enum,axiom,
! [X: tp__c_ty_2Elist_2Elist_ty_2Enum_2Enum] : ( mem @ ( inj__c_ty_2Elist_2Elist_ty_2Enum_2Enum @ X ) @ ( ty_2Elist_2Elist @ ty_2Enum_2Enum ) ) ).
thf(stp_iso_mem_c_ty_2Elist_2Elist_ty_2Enum_2Enum,axiom,
! [X: $i] :
( ( mem @ X @ ( ty_2Elist_2Elist @ ty_2Enum_2Enum ) )
=> ( X
= ( inj__c_ty_2Elist_2Elist_ty_2Enum_2Enum @ ( surj__c_ty_2Elist_2Elist_ty_2Enum_2Enum @ X ) ) ) ) ).
thf(tp_c_2Elist_2ESUM,type,
c_2Elist_2ESUM: $i ).
thf(mem_c_2Elist_2ESUM,axiom,
mem @ c_2Elist_2ESUM @ ( arr @ ( ty_2Elist_2Elist @ ty_2Enum_2Enum ) @ ty_2Enum_2Enum ) ).
thf(stp_fo_c_2Elist_2ESUM,type,
fo__c_2Elist_2ESUM: tp__c_ty_2Elist_2Elist_ty_2Enum_2Enum > tp__ty_2Enum_2Enum ).
thf(stp_eq_fo_c_2Elist_2ESUM,axiom,
! [X0: tp__c_ty_2Elist_2Elist_ty_2Enum_2Enum] :
( ( inj__ty_2Enum_2Enum @ ( fo__c_2Elist_2ESUM @ X0 ) )
= ( ap @ c_2Elist_2ESUM @ ( inj__c_ty_2Elist_2Elist_ty_2Enum_2Enum @ X0 ) ) ) ).
thf(tp_c_2Elist_2ESUM__ACC,type,
c_2Elist_2ESUM__ACC: $i ).
thf(mem_c_2Elist_2ESUM__ACC,axiom,
mem @ c_2Elist_2ESUM__ACC @ ( arr @ ( ty_2Elist_2Elist @ ty_2Enum_2Enum ) @ ( arr @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) ) ).
thf(stp_fo_c_2Elist_2ESUM__ACC,type,
fo__c_2Elist_2ESUM__ACC: tp__c_ty_2Elist_2Elist_ty_2Enum_2Enum > tp__ty_2Enum_2Enum > tp__ty_2Enum_2Enum ).
thf(stp_eq_fo_c_2Elist_2ESUM__ACC,axiom,
! [X0: tp__c_ty_2Elist_2Elist_ty_2Enum_2Enum,X1: tp__ty_2Enum_2Enum] :
( ( inj__ty_2Enum_2Enum @ ( fo__c_2Elist_2ESUM__ACC @ X0 @ X1 ) )
= ( ap @ ( ap @ c_2Elist_2ESUM__ACC @ ( inj__c_ty_2Elist_2Elist_ty_2Enum_2Enum @ X0 ) ) @ ( inj__ty_2Enum_2Enum @ X1 ) ) ) ).
thf(tp_c_2Elist_2ETAKE,type,
c_2Elist_2ETAKE: del > $i ).
thf(mem_c_2Elist_2ETAKE,axiom,
! [A_27a: del] : ( mem @ ( c_2Elist_2ETAKE @ A_27a ) @ ( arr @ ty_2Enum_2Enum @ ( arr @ ( ty_2Elist_2Elist @ A_27a ) @ ( ty_2Elist_2Elist @ A_27a ) ) ) ) ).
thf(tp_c_2Elist_2ETL,type,
c_2Elist_2ETL: del > $i ).
thf(mem_c_2Elist_2ETL,axiom,
! [A_27a: del] : ( mem @ ( c_2Elist_2ETL @ A_27a ) @ ( arr @ ( ty_2Elist_2Elist @ A_27a ) @ ( ty_2Elist_2Elist @ A_27a ) ) ) ).
thf(tp_c_2Elist_2EUNIQUE,type,
c_2Elist_2EUNIQUE: del > $i ).
thf(mem_c_2Elist_2EUNIQUE,axiom,
! [A_27a: del] : ( mem @ ( c_2Elist_2EUNIQUE @ A_27a ) @ ( arr @ A_27a @ ( arr @ ( ty_2Elist_2Elist @ A_27a ) @ bool ) ) ) ).
thf(tp_c_2Elist_2EUNZIP,type,
c_2Elist_2EUNZIP: del > del > $i ).
thf(mem_c_2Elist_2EUNZIP,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2Elist_2EUNZIP @ A_27a @ A_27b ) @ ( arr @ ( ty_2Elist_2Elist @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) ) @ ( ty_2Epair_2Eprod @ ( ty_2Elist_2Elist @ A_27a ) @ ( ty_2Elist_2Elist @ A_27b ) ) ) ) ).
thf(tp_c_2Elist_2EZIP,type,
c_2Elist_2EZIP: del > del > $i ).
thf(mem_c_2Elist_2EZIP,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2Elist_2EZIP @ A_27a @ A_27b ) @ ( arr @ ( ty_2Epair_2Eprod @ ( ty_2Elist_2Elist @ A_27a ) @ ( ty_2Elist_2Elist @ A_27b ) ) @ ( ty_2Elist_2Elist @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) ) ) ) ).
thf(tp_c_2Elist_2EdropWhile,type,
c_2Elist_2EdropWhile: del > $i ).
thf(mem_c_2Elist_2EdropWhile,axiom,
! [A_27a: del] : ( mem @ ( c_2Elist_2EdropWhile @ A_27a ) @ ( arr @ ( arr @ A_27a @ bool ) @ ( arr @ ( ty_2Elist_2Elist @ A_27a ) @ ( ty_2Elist_2Elist @ A_27a ) ) ) ) ).
thf(tp_c_2Elist_2EisPREFIX,type,
c_2Elist_2EisPREFIX: del > $i ).
thf(mem_c_2Elist_2EisPREFIX,axiom,
! [A_27a: del] : ( mem @ ( c_2Elist_2EisPREFIX @ A_27a ) @ ( arr @ ( ty_2Elist_2Elist @ A_27a ) @ ( arr @ ( ty_2Elist_2Elist @ A_27a ) @ bool ) ) ) ).
thf(tp_c_2Elist_2Elist__CASE,type,
c_2Elist_2Elist__CASE: del > del > $i ).
thf(mem_c_2Elist_2Elist__CASE,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2Elist_2Elist__CASE @ A_27a @ A_27b ) @ ( arr @ ( ty_2Elist_2Elist @ A_27a ) @ ( arr @ A_27b @ ( arr @ ( arr @ A_27a @ ( arr @ ( ty_2Elist_2Elist @ A_27a ) @ A_27b ) ) @ A_27b ) ) ) ) ).
thf(tp_c_2Elist_2Elist__size,type,
c_2Elist_2Elist__size: del > $i ).
thf(mem_c_2Elist_2Elist__size,axiom,
! [A_27a: del] : ( mem @ ( c_2Elist_2Elist__size @ A_27a ) @ ( arr @ ( arr @ A_27a @ ty_2Enum_2Enum ) @ ( arr @ ( ty_2Elist_2Elist @ A_27a ) @ ty_2Enum_2Enum ) ) ) ).
thf(tp_c_2Elist_2Enub,type,
c_2Elist_2Enub: del > $i ).
thf(mem_c_2Elist_2Enub,axiom,
! [A_27a: del] : ( mem @ ( c_2Elist_2Enub @ A_27a ) @ ( arr @ ( ty_2Elist_2Elist @ A_27a ) @ ( ty_2Elist_2Elist @ A_27a ) ) ) ).
thf(tp_c_2Elist_2EoEL,type,
c_2Elist_2EoEL: del > $i ).
thf(mem_c_2Elist_2EoEL,axiom,
! [A_27a: del] : ( mem @ ( c_2Elist_2EoEL @ A_27a ) @ ( arr @ ty_2Enum_2Enum @ ( arr @ ( ty_2Elist_2Elist @ A_27a ) @ ( ty_2Eoption_2Eoption @ A_27a ) ) ) ) ).
thf(tp_c_2Elist_2EoHD,type,
c_2Elist_2EoHD: del > $i ).
thf(mem_c_2Elist_2EoHD,axiom,
! [A_27a: del] : ( mem @ ( c_2Elist_2EoHD @ A_27a ) @ ( arr @ ( ty_2Elist_2Elist @ A_27a ) @ ( ty_2Eoption_2Eoption @ A_27a ) ) ) ).
thf(tp_c_2Elist_2EsplitAtPki,type,
c_2Elist_2EsplitAtPki: del > del > $i ).
thf(mem_c_2Elist_2EsplitAtPki,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2Elist_2EsplitAtPki @ A_27a @ A_27b ) @ ( arr @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27b @ bool ) ) @ ( arr @ ( arr @ ( ty_2Elist_2Elist @ A_27b ) @ ( arr @ ( ty_2Elist_2Elist @ A_27b ) @ A_27a ) ) @ ( arr @ ( ty_2Elist_2Elist @ A_27b ) @ A_27a ) ) ) ) ).
thf(ax_thm_2Elist_2Elist__TY__DEF,axiom,
! [A_27a: del] :
? [V0rep: $i] :
( ( mem @ V0rep @ ( arr @ ( ty_2Elist_2Elist @ A_27a ) @ ( ty_2Eind__type_2Erecspace @ A_27a ) ) )
& ( p
@ ( ap
@ ( ap @ ( c_2Ebool_2ETYPE__DEFINITION @ ( ty_2Eind__type_2Erecspace @ A_27a ) @ ( ty_2Elist_2Elist @ A_27a ) )
@ ( lam @ ( ty_2Eind__type_2Erecspace @ A_27a )
@ ^ [V1a0_27: $i] :
( ap @ ( c_2Ebool_2E_21 @ ( arr @ ( ty_2Eind__type_2Erecspace @ A_27a ) @ bool ) )
@ ( lam @ ( arr @ ( ty_2Eind__type_2Erecspace @ A_27a ) @ bool )
@ ^ [V2_27list_27: $i] :
( ap
@ ( ap @ c_2Emin_2E_3D_3D_3E
@ ( ap @ ( c_2Ebool_2E_21 @ ( ty_2Eind__type_2Erecspace @ A_27a ) )
@ ( lam @ ( ty_2Eind__type_2Erecspace @ A_27a )
@ ^ [V3a0_27: $i] :
( ap
@ ( ap @ c_2Emin_2E_3D_3D_3E
@ ( ap
@ ( ap @ c_2Ebool_2E_5C_2F
@ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ( ty_2Eind__type_2Erecspace @ A_27a ) ) @ V3a0_27 )
@ ( ap @ ( ap @ ( ap @ ( c_2Eind__type_2ECONSTR @ A_27a ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( c_2Ebool_2EARB @ A_27a ) )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V4n: $i] : ( c_2Eind__type_2EBOTTOM @ A_27a ) ) ) ) )
@ ( ap @ ( c_2Ebool_2E_3F @ A_27a )
@ ( lam @ A_27a
@ ^ [V5a0: $i] :
( ap @ ( c_2Ebool_2E_3F @ ( ty_2Eind__type_2Erecspace @ A_27a ) )
@ ( lam @ ( ty_2Eind__type_2Erecspace @ A_27a )
@ ^ [V6a1: $i] :
( ap
@ ( ap @ c_2Ebool_2E_2F_5C
@ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ( ty_2Eind__type_2Erecspace @ A_27a ) ) @ V3a0_27 )
@ ( ap
@ ( ap
@ ( lam @ A_27a
@ ^ [V7a0: $i] :
( lam @ ( ty_2Eind__type_2Erecspace @ A_27a )
@ ^ [V8a1: $i] :
( ap @ ( ap @ ( ap @ ( c_2Eind__type_2ECONSTR @ A_27a ) @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ V7a0 )
@ ( ap @ ( ap @ ( c_2Eind__type_2EFCONS @ ( ty_2Eind__type_2Erecspace @ A_27a ) ) @ V8a1 )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V9n: $i] : ( c_2Eind__type_2EBOTTOM @ A_27a ) ) ) ) ) )
@ V5a0 )
@ V6a1 ) ) )
@ ( ap @ V2_27list_27 @ V6a1 ) ) ) ) ) ) ) )
@ ( ap @ V2_27list_27 @ V3a0_27 ) ) ) ) )
@ ( ap @ V2_27list_27 @ V1a0_27 ) ) ) ) ) )
@ V0rep ) ) ) ).
thf(ax_thm_2Elist_2Elist__case__def,axiom,
! [A_27a: del,A_27b: del] :
( ! [V0v: $i] :
( ( mem @ V0v @ A_27b )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ ( arr @ ( ty_2Elist_2Elist @ A_27a ) @ A_27b ) ) )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Elist_2Elist__CASE @ A_27a @ A_27b ) @ ( c_2Elist_2ENIL @ A_27a ) ) @ V0v ) @ V1f )
= V0v ) ) )
& ! [V2a0: $i] :
( ( mem @ V2a0 @ A_27a )
=> ! [V3a1: $i] :
( ( mem @ V3a1 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V4v: $i] :
( ( mem @ V4v @ A_27b )
=> ! [V5f: $i] :
( ( mem @ V5f @ ( arr @ A_27a @ ( arr @ ( ty_2Elist_2Elist @ A_27a ) @ A_27b ) ) )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Elist_2Elist__CASE @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V2a0 ) @ V3a1 ) ) @ V4v ) @ V5f )
= ( ap @ ( ap @ V5f @ V2a0 ) @ V3a1 ) ) ) ) ) ) ) ).
thf(ax_thm_2Elist_2Elist__size__def,axiom,
! [A_27a: del] :
( ! [V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ ( c_2Elist_2Elist__size @ A_27a ) @ V0f ) @ ( c_2Elist_2ENIL @ A_27a ) ) )
= fo__c_2Enum_2E0 ) )
& ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V2a0: $i] :
( ( mem @ V2a0 @ A_27a )
=> ! [V3a1: $i] :
( ( mem @ V3a1 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ ( c_2Elist_2Elist__size @ A_27a ) @ V1f ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V2a0 ) @ V3a1 ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( ap @ V1f @ V2a0 ) ) @ ( ap @ ( ap @ ( c_2Elist_2Elist__size @ A_27a ) @ V1f ) @ V3a1 ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2Edatatype__list,axiom,
! [A_27a: del,V0list: $i] :
( ( mem @ V0list @ ( arr @ ( ty_2Elist_2Elist @ A_27a ) @ ( arr @ ( arr @ A_27a @ ( arr @ ( ty_2Elist_2Elist @ A_27a ) @ ( ty_2Elist_2Elist @ A_27a ) ) ) @ bool ) ) )
=> ( p @ ( ap @ ( c_2Ebool_2EDATATYPE @ bool ) @ ( ap @ ( ap @ V0list @ ( c_2Elist_2ENIL @ A_27a ) ) @ ( c_2Elist_2ECONS @ A_27a ) ) ) ) ) ).
thf(conj_thm_2Elist_2Elist__11,axiom,
! [A_27a: del,V0a0: $i] :
( ( mem @ V0a0 @ A_27a )
=> ! [V1a1: $i] :
( ( mem @ V1a1 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V2a0_27: $i] :
( ( mem @ V2a0_27 @ A_27a )
=> ! [V3a1_27: $i] :
( ( mem @ V3a1_27 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V0a0 ) @ V1a1 )
= ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V2a0_27 ) @ V3a1_27 ) )
<=> ( ( V0a0 = V2a0_27 )
& ( V1a1 = V3a1_27 ) ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2Elist__distinct,axiom,
! [A_27a: del,V0a1: $i] :
( ( mem @ V0a1 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V1a0: $i] :
( ( mem @ V1a0 @ A_27a )
=> ( ( c_2Elist_2ENIL @ A_27a )
!= ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V1a0 ) @ V0a1 ) ) ) ) ).
thf(conj_thm_2Elist_2Elist__Axiom,axiom,
! [A_27a: del,A_27b: del,V0f0: $i] :
( ( mem @ V0f0 @ A_27b )
=> ! [V1f1: $i] :
( ( mem @ V1f1 @ ( arr @ A_27a @ ( arr @ ( ty_2Elist_2Elist @ A_27a ) @ ( arr @ A_27b @ A_27b ) ) ) )
=> ? [V2fn: $i] :
( ( mem @ V2fn @ ( arr @ ( ty_2Elist_2Elist @ A_27a ) @ A_27b ) )
& ( ( ap @ V2fn @ ( c_2Elist_2ENIL @ A_27a ) )
= V0f0 )
& ! [V3a0: $i] :
( ( mem @ V3a0 @ A_27a )
=> ! [V4a1: $i] :
( ( mem @ V4a1 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( ap @ V2fn @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V3a0 ) @ V4a1 ) )
= ( ap @ ( ap @ ( ap @ V1f1 @ V3a0 ) @ V4a1 ) @ ( ap @ V2fn @ V4a1 ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2Elist__case__cong,axiom,
! [A_27a: del,A_27b: del,V0v_27: $i] :
( ( mem @ V0v_27 @ A_27b )
=> ! [V1f_27: $i] :
( ( mem @ V1f_27 @ ( arr @ A_27a @ ( arr @ ( ty_2Elist_2Elist @ A_27a ) @ A_27b ) ) )
=> ! [V2M: $i] :
( ( mem @ V2M @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V3M_27: $i] :
( ( mem @ V3M_27 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V4v: $i] :
( ( mem @ V4v @ A_27b )
=> ! [V5f: $i] :
( ( mem @ V5f @ ( arr @ A_27a @ ( arr @ ( ty_2Elist_2Elist @ A_27a ) @ A_27b ) ) )
=> ( ( ( V2M = V3M_27 )
& ( ( V3M_27
= ( c_2Elist_2ENIL @ A_27a ) )
=> ( V4v = V0v_27 ) )
& ! [V6a0: $i] :
( ( mem @ V6a0 @ A_27a )
=> ! [V7a1: $i] :
( ( mem @ V7a1 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( V3M_27
= ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V6a0 ) @ V7a1 ) )
=> ( ( ap @ ( ap @ V5f @ V6a0 ) @ V7a1 )
= ( ap @ ( ap @ V1f_27 @ V6a0 ) @ V7a1 ) ) ) ) ) )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Elist_2Elist__CASE @ A_27a @ A_27b ) @ V2M ) @ V4v ) @ V5f )
= ( ap @ ( ap @ ( ap @ ( c_2Elist_2Elist__CASE @ A_27a @ A_27b ) @ V3M_27 ) @ V0v_27 ) @ V1f_27 ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2Elist__case__eq,axiom,
! [A_27a: del,A_27b: del,V0x: $i] :
( ( mem @ V0x @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V1v: $i] :
( ( mem @ V1v @ A_27b )
=> ! [V2f: $i] :
( ( mem @ V2f @ ( arr @ A_27a @ ( arr @ ( ty_2Elist_2Elist @ A_27a ) @ A_27b ) ) )
=> ! [V3v_27: $i] :
( ( mem @ V3v_27 @ A_27b )
=> ( ( ( ap @ ( ap @ ( ap @ ( c_2Elist_2Elist__CASE @ A_27a @ A_27b ) @ V0x ) @ V1v ) @ V2f )
= V3v_27 )
<=> ( ( ( V0x
= ( c_2Elist_2ENIL @ A_27a ) )
& ( V1v = V3v_27 ) )
| ? [V4a: $i] :
( ( mem @ V4a @ A_27a )
& ? [V5l: $i] :
( ( mem @ V5l @ ( ty_2Elist_2Elist @ A_27a ) )
& ( V0x
= ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V4a ) @ V5l ) )
& ( ( ap @ ( ap @ V2f @ V4a ) @ V5l )
= V3v_27 ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2Elist__Axiom__old,axiom,
! [A_27a: del,A_27b: del,V0x: $i] :
( ( mem @ V0x @ A_27b )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27b @ ( arr @ A_27a @ ( arr @ ( ty_2Elist_2Elist @ A_27a ) @ A_27b ) ) ) )
=> ( p
@ ( ap @ ( c_2Ebool_2E_3F_21 @ ( arr @ ( ty_2Elist_2Elist @ A_27a ) @ A_27b ) )
@ ( lam @ ( arr @ ( ty_2Elist_2Elist @ A_27a ) @ A_27b )
@ ^ [V2fn1: $i] :
( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ A_27b ) @ ( ap @ V2fn1 @ ( c_2Elist_2ENIL @ A_27a ) ) ) @ V0x ) )
@ ( ap @ ( c_2Ebool_2E_21 @ A_27a )
@ ( lam @ A_27a
@ ^ [V3h: $i] :
( ap @ ( c_2Ebool_2E_21 @ ( ty_2Elist_2Elist @ A_27a ) )
@ ( lam @ ( ty_2Elist_2Elist @ A_27a )
@ ^ [V4t: $i] : ( ap @ ( ap @ ( c_2Emin_2E_3D @ A_27b ) @ ( ap @ V2fn1 @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V3h ) @ V4t ) ) ) @ ( ap @ ( ap @ ( ap @ V1f @ ( ap @ V2fn1 @ V4t ) ) @ V3h ) @ V4t ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Elist_2ENULL__DEF,axiom,
! [A_27a: del] :
( ( ( p @ ( ap @ ( c_2Elist_2ENULL @ A_27a ) @ ( c_2Elist_2ENIL @ A_27a ) ) )
<=> $true )
& ! [V0h: $i] :
( ( mem @ V0h @ A_27a )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( p @ ( ap @ ( c_2Elist_2ENULL @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V0h ) @ V1t ) ) )
<=> $false ) ) ) ) ).
thf(ax_thm_2Elist_2EHD,axiom,
! [A_27a: del,V0h: $i] :
( ( mem @ V0h @ A_27a )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( ap @ ( c_2Elist_2EHD @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V0h ) @ V1t ) )
= V0h ) ) ) ).
thf(ax_thm_2Elist_2ETL__DEF,axiom,
! [A_27a: del] :
( ( ( ap @ ( c_2Elist_2ETL @ A_27a ) @ ( c_2Elist_2ENIL @ A_27a ) )
= ( c_2Elist_2ENIL @ A_27a ) )
& ! [V0h: $i] :
( ( mem @ V0h @ A_27a )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( ap @ ( c_2Elist_2ETL @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V0h ) @ V1t ) )
= V1t ) ) ) ) ).
thf(conj_thm_2Elist_2ETL,axiom,
! [A_27a: del,V0h: $i] :
( ( mem @ V0h @ A_27a )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( ap @ ( c_2Elist_2ETL @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V0h ) @ V1t ) )
= V1t ) ) ) ).
thf(ax_thm_2Elist_2ESUM,axiom,
( ( ( surj__ty_2Enum_2Enum @ ( ap @ c_2Elist_2ESUM @ ( c_2Elist_2ENIL @ ty_2Enum_2Enum ) ) )
= fo__c_2Enum_2E0 )
& ! [V0h: tp__ty_2Enum_2Enum,V1t: tp__c_ty_2Elist_2Elist_ty_2Enum_2Enum] :
( ( surj__ty_2Enum_2Enum @ ( ap @ c_2Elist_2ESUM @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V0h ) ) @ ( inj__c_ty_2Elist_2Elist_ty_2Enum_2Enum @ V1t ) ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( inj__ty_2Enum_2Enum @ V0h ) ) @ ( ap @ c_2Elist_2ESUM @ ( inj__c_ty_2Elist_2Elist_ty_2Enum_2Enum @ V1t ) ) ) ) ) ) ).
thf(ax_thm_2Elist_2EAPPEND,axiom,
! [A_27a: del] :
( ! [V0l: $i] :
( ( mem @ V0l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ ( c_2Elist_2ENIL @ A_27a ) ) @ V0l )
= V0l ) )
& ! [V1l1: $i] :
( ( mem @ V1l1 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V2l2: $i] :
( ( mem @ V2l2 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V3h: $i] :
( ( mem @ V3h @ A_27a )
=> ( ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V3h ) @ V1l1 ) ) @ V2l2 )
= ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V3h ) @ ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ V1l1 ) @ V2l2 ) ) ) ) ) ) ) ).
thf(ax_thm_2Elist_2EFLAT,axiom,
! [A_27a: del] :
( ( ( ap @ ( c_2Elist_2EFLAT @ A_27a ) @ ( c_2Elist_2ENIL @ ( ty_2Elist_2Elist @ A_27a ) ) )
= ( c_2Elist_2ENIL @ A_27a ) )
& ! [V0h: $i] :
( ( mem @ V0h @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( ty_2Elist_2Elist @ ( ty_2Elist_2Elist @ A_27a ) ) )
=> ( ( ap @ ( c_2Elist_2EFLAT @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ( ty_2Elist_2Elist @ A_27a ) ) @ V0h ) @ V1t ) )
= ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ V0h ) @ ( ap @ ( c_2Elist_2EFLAT @ A_27a ) @ V1t ) ) ) ) ) ) ).
thf(ax_thm_2Elist_2ELENGTH,axiom,
! [A_27a: del] :
( ( ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ ( c_2Elist_2ENIL @ A_27a ) ) )
= fo__c_2Enum_2E0 )
& ! [V0h: $i] :
( ( mem @ V0h @ A_27a )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V0h ) @ V1t ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ c_2Enum_2ESUC @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ V1t ) ) ) ) ) ) ) ).
thf(ax_thm_2Elist_2EMAP,axiom,
! [A_27a: del,A_27b: del] :
( ! [V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27b ) )
=> ( ( ap @ ( ap @ ( c_2Elist_2EMAP @ A_27a @ A_27b ) @ V0f ) @ ( c_2Elist_2ENIL @ A_27a ) )
= ( c_2Elist_2ENIL @ A_27b ) ) )
& ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ A_27b ) )
=> ! [V2h: $i] :
( ( mem @ V2h @ A_27a )
=> ! [V3t: $i] :
( ( mem @ V3t @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( ap @ ( ap @ ( c_2Elist_2EMAP @ A_27a @ A_27b ) @ V1f ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V2h ) @ V3t ) )
= ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27b ) @ ( ap @ V1f @ V2h ) ) @ ( ap @ ( ap @ ( c_2Elist_2EMAP @ A_27a @ A_27b ) @ V1f ) @ V3t ) ) ) ) ) ) ) ).
thf(ax_thm_2Elist_2ELIST__TO__SET__DEF,axiom,
! [A_27a: del] :
( ! [V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Elist_2ELIST__TO__SET @ A_27a ) @ ( c_2Elist_2ENIL @ A_27a ) ) @ V0x ) )
<=> $false ) )
& ! [V1h: $i] :
( ( mem @ V1h @ A_27a )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Elist_2ELIST__TO__SET @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V1h ) @ V2t ) ) @ V3x ) )
<=> ( ( V3x = V1h )
| ( p @ ( ap @ ( ap @ ( c_2Elist_2ELIST__TO__SET @ A_27a ) @ V2t ) @ V3x ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2ELIST__TO__SET,axiom,
! [A_27a: del,A_27b: del,V0h: $i] :
( ( mem @ V0h @ A_27b )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( ty_2Elist_2Elist @ A_27b ) )
=> ( ( ( ap @ ( c_2Elist_2ELIST__TO__SET @ A_27a ) @ ( c_2Elist_2ENIL @ A_27a ) )
= ( c_2Epred__set_2EEMPTY @ A_27a ) )
& ( ( ap @ ( c_2Elist_2ELIST__TO__SET @ A_27b ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27b ) @ V0h ) @ V1t ) )
= ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ A_27b ) @ V0h ) @ ( ap @ ( c_2Elist_2ELIST__TO__SET @ A_27b ) @ V1t ) ) ) ) ) ) ).
thf(ax_thm_2Elist_2EFILTER,axiom,
! [A_27a: del] :
( ! [V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ( ( ap @ ( ap @ ( c_2Elist_2EFILTER @ A_27a ) @ V0P ) @ ( c_2Elist_2ENIL @ A_27a ) )
= ( c_2Elist_2ENIL @ A_27a ) ) )
& ! [V1P: $i] :
( ( mem @ V1P @ ( arr @ A_27a @ bool ) )
=> ! [V2h: $i] :
( ( mem @ V2h @ A_27a )
=> ! [V3t: $i] :
( ( mem @ V3t @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( ap @ ( ap @ ( c_2Elist_2EFILTER @ A_27a ) @ V1P ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V2h ) @ V3t ) )
= ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ( ty_2Elist_2Elist @ A_27a ) ) @ ( ap @ V1P @ V2h ) ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V2h ) @ ( ap @ ( ap @ ( c_2Elist_2EFILTER @ A_27a ) @ V1P ) @ V3t ) ) ) @ ( ap @ ( ap @ ( c_2Elist_2EFILTER @ A_27a ) @ V1P ) @ V3t ) ) ) ) ) ) ) ).
thf(ax_thm_2Elist_2EFOLDR,axiom,
! [A_27a: del,A_27b: del] :
( ! [V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ( arr @ A_27b @ A_27b ) ) )
=> ! [V1e: $i] :
( ( mem @ V1e @ A_27b )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Elist_2EFOLDR @ A_27a @ A_27b ) @ V0f ) @ V1e ) @ ( c_2Elist_2ENIL @ A_27a ) )
= V1e ) ) )
& ! [V2f: $i] :
( ( mem @ V2f @ ( arr @ A_27a @ ( arr @ A_27b @ A_27b ) ) )
=> ! [V3e: $i] :
( ( mem @ V3e @ A_27b )
=> ! [V4x: $i] :
( ( mem @ V4x @ A_27a )
=> ! [V5l: $i] :
( ( mem @ V5l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Elist_2EFOLDR @ A_27a @ A_27b ) @ V2f ) @ V3e ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V4x ) @ V5l ) )
= ( ap @ ( ap @ V2f @ V4x ) @ ( ap @ ( ap @ ( ap @ ( c_2Elist_2EFOLDR @ A_27a @ A_27b ) @ V2f ) @ V3e ) @ V5l ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Elist_2EFOLDL,axiom,
! [A_27a: del,A_27b: del] :
( ! [V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27b @ ( arr @ A_27a @ A_27b ) ) )
=> ! [V1e: $i] :
( ( mem @ V1e @ A_27b )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Elist_2EFOLDL @ A_27a @ A_27b ) @ V0f ) @ V1e ) @ ( c_2Elist_2ENIL @ A_27a ) )
= V1e ) ) )
& ! [V2f: $i] :
( ( mem @ V2f @ ( arr @ A_27b @ ( arr @ A_27a @ A_27b ) ) )
=> ! [V3e: $i] :
( ( mem @ V3e @ A_27b )
=> ! [V4x: $i] :
( ( mem @ V4x @ A_27a )
=> ! [V5l: $i] :
( ( mem @ V5l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Elist_2EFOLDL @ A_27a @ A_27b ) @ V2f ) @ V3e ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V4x ) @ V5l ) )
= ( ap @ ( ap @ ( ap @ ( c_2Elist_2EFOLDL @ A_27a @ A_27b ) @ V2f ) @ ( ap @ ( ap @ V2f @ V3e ) @ V4x ) ) @ V5l ) ) ) ) ) ) ) ).
thf(ax_thm_2Elist_2EEVERY__DEF,axiom,
! [A_27a: del] :
( ! [V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Elist_2EEVERY @ A_27a ) @ V0P ) @ ( c_2Elist_2ENIL @ A_27a ) ) )
<=> $true ) )
& ! [V1P: $i] :
( ( mem @ V1P @ ( arr @ A_27a @ bool ) )
=> ! [V2h: $i] :
( ( mem @ V2h @ A_27a )
=> ! [V3t: $i] :
( ( mem @ V3t @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Elist_2EEVERY @ A_27a ) @ V1P ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V2h ) @ V3t ) ) )
<=> ( ( p @ ( ap @ V1P @ V2h ) )
& ( p @ ( ap @ ( ap @ ( c_2Elist_2EEVERY @ A_27a ) @ V1P ) @ V3t ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Elist_2EEXISTS__DEF,axiom,
! [A_27a: del] :
( ! [V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Elist_2EEXISTS @ A_27a ) @ V0P ) @ ( c_2Elist_2ENIL @ A_27a ) ) )
<=> $false ) )
& ! [V1P: $i] :
( ( mem @ V1P @ ( arr @ A_27a @ bool ) )
=> ! [V2h: $i] :
( ( mem @ V2h @ A_27a )
=> ! [V3t: $i] :
( ( mem @ V3t @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Elist_2EEXISTS @ A_27a ) @ V1P ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V2h ) @ V3t ) ) )
<=> ( ( p @ ( ap @ V1P @ V2h ) )
| ( p @ ( ap @ ( ap @ ( c_2Elist_2EEXISTS @ A_27a ) @ V1P ) @ V3t ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Elist_2EEL,axiom,
! [A_27a: del] :
( ! [V0l: $i] :
( ( mem @ V0l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( ap @ ( ap @ ( c_2Elist_2EEL @ A_27a ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ V0l )
= ( ap @ ( c_2Elist_2EHD @ A_27a ) @ V0l ) ) )
& ! [V1l: $i] :
( ( mem @ V1l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V2n: tp__ty_2Enum_2Enum] :
( ( ap @ ( ap @ ( c_2Elist_2EEL @ A_27a ) @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) @ V1l )
= ( ap @ ( ap @ ( c_2Elist_2EEL @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) @ ( ap @ ( c_2Elist_2ETL @ A_27a ) @ V1l ) ) ) ) ) ).
thf(conj_thm_2Elist_2EMAP2__ind,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ ( arr @ A_27b @ ( arr @ A_27c @ A_27a ) ) @ ( arr @ ( ty_2Elist_2Elist @ A_27b ) @ ( arr @ ( ty_2Elist_2Elist @ A_27c ) @ bool ) ) ) )
=> ( ( ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27b @ ( arr @ A_27c @ A_27a ) ) )
=> ! [V2h1: $i] :
( ( mem @ V2h1 @ A_27b )
=> ! [V3t1: $i] :
( ( mem @ V3t1 @ ( ty_2Elist_2Elist @ A_27b ) )
=> ! [V4h2: $i] :
( ( mem @ V4h2 @ A_27c )
=> ! [V5t2: $i] :
( ( mem @ V5t2 @ ( ty_2Elist_2Elist @ A_27c ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ V0P @ V1f ) @ V3t1 ) @ V5t2 ) )
=> ( p @ ( ap @ ( ap @ ( ap @ V0P @ V1f ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27b ) @ V2h1 ) @ V3t1 ) ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27c ) @ V4h2 ) @ V5t2 ) ) ) ) ) ) ) ) )
& ! [V6f: $i] :
( ( mem @ V6f @ ( arr @ A_27b @ ( arr @ A_27c @ A_27a ) ) )
=> ! [V7y: $i] :
( ( mem @ V7y @ ( ty_2Elist_2Elist @ A_27c ) )
=> ( p @ ( ap @ ( ap @ ( ap @ V0P @ V6f ) @ ( c_2Elist_2ENIL @ A_27b ) ) @ V7y ) ) ) )
& ! [V8f: $i] :
( ( mem @ V8f @ ( arr @ A_27b @ ( arr @ A_27c @ A_27a ) ) )
=> ! [V9v4: $i] :
( ( mem @ V9v4 @ A_27b )
=> ! [V10v5: $i] :
( ( mem @ V10v5 @ ( ty_2Elist_2Elist @ A_27b ) )
=> ( p @ ( ap @ ( ap @ ( ap @ V0P @ V8f ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27b ) @ V9v4 ) @ V10v5 ) ) @ ( c_2Elist_2ENIL @ A_27c ) ) ) ) ) ) )
=> ! [V11v: $i] :
( ( mem @ V11v @ ( arr @ A_27b @ ( arr @ A_27c @ A_27a ) ) )
=> ! [V12v1: $i] :
( ( mem @ V12v1 @ ( ty_2Elist_2Elist @ A_27b ) )
=> ! [V13v2: $i] :
( ( mem @ V13v2 @ ( ty_2Elist_2Elist @ A_27c ) )
=> ( p @ ( ap @ ( ap @ ( ap @ V0P @ V11v ) @ V12v1 ) @ V13v2 ) ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2EMAP2__DEF,axiom,
! [A_27a: del,A_27b: del,A_27c: del] :
( ! [V0t2: $i] :
( ( mem @ V0t2 @ ( ty_2Elist_2Elist @ A_27c ) )
=> ! [V1t1: $i] :
( ( mem @ V1t1 @ ( ty_2Elist_2Elist @ A_27b ) )
=> ! [V2h2: $i] :
( ( mem @ V2h2 @ A_27c )
=> ! [V3h1: $i] :
( ( mem @ V3h1 @ A_27b )
=> ! [V4f: $i] :
( ( mem @ V4f @ ( arr @ A_27b @ ( arr @ A_27c @ A_27a ) ) )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Elist_2EMAP2 @ A_27a @ A_27b @ A_27c ) @ V4f ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27b ) @ V3h1 ) @ V1t1 ) ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27c ) @ V2h2 ) @ V0t2 ) )
= ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ ( ap @ ( ap @ V4f @ V3h1 ) @ V2h2 ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Elist_2EMAP2 @ A_27a @ A_27b @ A_27c ) @ V4f ) @ V1t1 ) @ V0t2 ) ) ) ) ) ) ) )
& ! [V5y: $i] :
( ( mem @ V5y @ ( ty_2Elist_2Elist @ A_27c ) )
=> ! [V6f: $i] :
( ( mem @ V6f @ ( arr @ A_27b @ ( arr @ A_27c @ A_27a ) ) )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Elist_2EMAP2 @ A_27a @ A_27b @ A_27c ) @ V6f ) @ ( c_2Elist_2ENIL @ A_27b ) ) @ V5y )
= ( c_2Elist_2ENIL @ A_27a ) ) ) )
& ! [V7v5: $i] :
( ( mem @ V7v5 @ ( ty_2Elist_2Elist @ A_27b ) )
=> ! [V8v4: $i] :
( ( mem @ V8v4 @ A_27b )
=> ! [V9f: $i] :
( ( mem @ V9f @ ( arr @ A_27b @ ( arr @ A_27c @ A_27a ) ) )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Elist_2EMAP2 @ A_27a @ A_27b @ A_27c ) @ V9f ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27b ) @ V8v4 ) @ V7v5 ) ) @ ( c_2Elist_2ENIL @ A_27c ) )
= ( c_2Elist_2ENIL @ A_27a ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2EMAP2,axiom,
! [A_27a: del,A_27b: del,A_27c: del,A_27d: del,A_27e: del,A_27f: del] :
( ! [V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ( arr @ A_27b @ A_27c ) ) )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Elist_2EMAP2 @ A_27c @ A_27a @ A_27b ) @ V0f ) @ ( c_2Elist_2ENIL @ A_27a ) ) @ ( c_2Elist_2ENIL @ A_27b ) )
= ( c_2Elist_2ENIL @ A_27c ) ) )
& ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27d @ ( arr @ A_27e @ A_27f ) ) )
=> ! [V2h1: $i] :
( ( mem @ V2h1 @ A_27d )
=> ! [V3t1: $i] :
( ( mem @ V3t1 @ ( ty_2Elist_2Elist @ A_27d ) )
=> ! [V4h2: $i] :
( ( mem @ V4h2 @ A_27e )
=> ! [V5t2: $i] :
( ( mem @ V5t2 @ ( ty_2Elist_2Elist @ A_27e ) )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Elist_2EMAP2 @ A_27f @ A_27d @ A_27e ) @ V1f ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27d ) @ V2h1 ) @ V3t1 ) ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27e ) @ V4h2 ) @ V5t2 ) )
= ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27f ) @ ( ap @ ( ap @ V1f @ V2h1 ) @ V4h2 ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Elist_2EMAP2 @ A_27f @ A_27d @ A_27e ) @ V1f ) @ V3t1 ) @ V5t2 ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2EMAP2__NIL,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27b @ ( arr @ A_27c @ A_27a ) ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ ( ty_2Elist_2Elist @ A_27b ) )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Elist_2EMAP2 @ A_27a @ A_27b @ A_27c ) @ V0f ) @ V1x ) @ ( c_2Elist_2ENIL @ A_27c ) )
= ( c_2Elist_2ENIL @ A_27a ) ) ) ) ).
thf(conj_thm_2Elist_2ELENGTH__MAP2,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ( arr @ A_27b @ A_27c ) ) )
=> ! [V1xs: $i] :
( ( mem @ V1xs @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V2ys: $i] :
( ( mem @ V2ys @ ( ty_2Elist_2Elist @ A_27b ) )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Elist_2ELENGTH @ A_27c ) @ ( ap @ ( ap @ ( ap @ ( c_2Elist_2EMAP2 @ A_27c @ A_27a @ A_27b ) @ V0f ) @ V1xs ) @ V2ys ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Earithmetic_2EMIN @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ V1xs ) ) @ ( ap @ ( c_2Elist_2ELENGTH @ A_27b ) @ V2ys ) ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2EEL__MAP2,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ( arr @ A_27b @ A_27c ) ) )
=> ! [V1ts: $i] :
( ( mem @ V1ts @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V2tt: $i] :
( ( mem @ V2tt @ ( ty_2Elist_2Elist @ A_27b ) )
=> ! [V3n: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V3n ) ) @ ( ap @ ( ap @ c_2Earithmetic_2EMIN @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ V1ts ) ) @ ( ap @ ( c_2Elist_2ELENGTH @ A_27b ) @ V2tt ) ) ) )
=> ( ( ap @ ( ap @ ( c_2Elist_2EEL @ A_27c ) @ ( inj__ty_2Enum_2Enum @ V3n ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Elist_2EMAP2 @ A_27c @ A_27a @ A_27b ) @ V0f ) @ V1ts ) @ V2tt ) )
= ( ap @ ( ap @ V0f @ ( ap @ ( ap @ ( c_2Elist_2EEL @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V3n ) ) @ V1ts ) ) @ ( ap @ ( ap @ ( c_2Elist_2EEL @ A_27b ) @ ( inj__ty_2Enum_2Enum @ V3n ) ) @ V2tt ) ) ) ) ) ) ) ).
thf(ax_thm_2Elist_2EINDEX__FIND__def,axiom,
! [A_27a: del] :
( ! [V0i: tp__ty_2Enum_2Enum,V1P: $i] :
( ( mem @ V1P @ ( arr @ A_27a @ bool ) )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Elist_2EINDEX__FIND @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V0i ) ) @ V1P ) @ ( c_2Elist_2ENIL @ A_27a ) )
= ( c_2Eoption_2ENONE @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ A_27a ) ) ) )
& ! [V2i: tp__ty_2Enum_2Enum,V3P: $i] :
( ( mem @ V3P @ ( arr @ A_27a @ bool ) )
=> ! [V4h: $i] :
( ( mem @ V4h @ A_27a )
=> ! [V5t: $i] :
( ( mem @ V5t @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Elist_2EINDEX__FIND @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V2i ) ) @ V3P ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V4h ) @ V5t ) )
= ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ( ty_2Eoption_2Eoption @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ A_27a ) ) ) @ ( ap @ V3P @ V4h ) ) @ ( ap @ ( c_2Eoption_2ESOME @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V2i ) ) @ V4h ) ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Elist_2EINDEX__FIND @ A_27a ) @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V2i ) ) ) @ V3P ) @ V5t ) ) ) ) ) ) ) ).
thf(ax_thm_2Elist_2EFIND__def,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ( ( ap @ ( c_2Elist_2EFIND @ A_27a ) @ V0P )
= ( ap @ ( ap @ ( c_2Ecombin_2Eo @ ( ty_2Elist_2Elist @ A_27a ) @ ( ty_2Eoption_2Eoption @ A_27a ) @ ( ty_2Eoption_2Eoption @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ A_27a ) ) ) @ ( ap @ ( c_2Eoption_2EOPTION__MAP @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ A_27a ) @ A_27a ) @ ( c_2Epair_2ESND @ ty_2Enum_2Enum @ A_27a ) ) ) @ ( ap @ ( ap @ ( c_2Elist_2EINDEX__FIND @ A_27a ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ V0P ) ) ) ) ).
thf(ax_thm_2Elist_2EINDEX__OF__def,axiom,
! [A_27a: del,V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ( ( ap @ ( c_2Elist_2EINDEX__OF @ A_27a ) @ V0x )
= ( ap @ ( ap @ ( c_2Ecombin_2Eo @ ( ty_2Elist_2Elist @ A_27a ) @ ( ty_2Eoption_2Eoption @ ty_2Enum_2Enum ) @ ( ty_2Eoption_2Eoption @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ A_27a ) ) ) @ ( ap @ ( c_2Eoption_2EOPTION__MAP @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ A_27a ) @ ty_2Enum_2Enum ) @ ( c_2Epair_2EFST @ ty_2Enum_2Enum @ A_27a ) ) ) @ ( ap @ ( ap @ ( c_2Elist_2EINDEX__FIND @ A_27a ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( ap @ ( c_2Emin_2E_3D @ A_27a ) @ V0x ) ) ) ) ) ).
thf(conj_thm_2Elist_2ENULL,axiom,
! [A_27a: del] :
( ( p @ ( ap @ ( c_2Elist_2ENULL @ A_27a ) @ ( c_2Elist_2ENIL @ A_27a ) ) )
& ! [V0h: $i] :
( ( mem @ V0h @ A_27a )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( ty_2Elist_2Elist @ A_27a ) )
=> ~ ( p @ ( ap @ ( c_2Elist_2ENULL @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V0h ) @ V1t ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2Elist__INDUCT0,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ ( ty_2Elist_2Elist @ A_27a ) @ bool ) )
=> ( ( ( p @ ( ap @ V0P @ ( c_2Elist_2ENIL @ A_27a ) ) )
& ! [V1l: $i] :
( ( mem @ V1l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( p @ ( ap @ V0P @ V1l ) )
=> ! [V2a: $i] :
( ( mem @ V2a @ A_27a )
=> ( p @ ( ap @ V0P @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V2a ) @ V1l ) ) ) ) ) ) )
=> ! [V3l: $i] :
( ( mem @ V3l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( p @ ( ap @ V0P @ V3l ) ) ) ) ) ).
thf(conj_thm_2Elist_2Elist__INDUCT,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ ( ty_2Elist_2Elist @ A_27a ) @ bool ) )
=> ( ( ( p @ ( ap @ V0P @ ( c_2Elist_2ENIL @ A_27a ) ) )
& ! [V1t: $i] :
( ( mem @ V1t @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( p @ ( ap @ V0P @ V1t ) )
=> ! [V2h: $i] :
( ( mem @ V2h @ A_27a )
=> ( p @ ( ap @ V0P @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V2h ) @ V1t ) ) ) ) ) ) )
=> ! [V3l: $i] :
( ( mem @ V3l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( p @ ( ap @ V0P @ V3l ) ) ) ) ) ).
thf(conj_thm_2Elist_2Elist__induction,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ ( ty_2Elist_2Elist @ A_27a ) @ bool ) )
=> ( ( ( p @ ( ap @ V0P @ ( c_2Elist_2ENIL @ A_27a ) ) )
& ! [V1t: $i] :
( ( mem @ V1t @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( p @ ( ap @ V0P @ V1t ) )
=> ! [V2h: $i] :
( ( mem @ V2h @ A_27a )
=> ( p @ ( ap @ V0P @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V2h ) @ V1t ) ) ) ) ) ) )
=> ! [V3l: $i] :
( ( mem @ V3l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( p @ ( ap @ V0P @ V3l ) ) ) ) ) ).
thf(conj_thm_2Elist_2Elist__CASES,axiom,
! [A_27a: del,V0l: $i] :
( ( mem @ V0l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( V0l
= ( c_2Elist_2ENIL @ A_27a ) )
| ? [V1h: $i] :
( ( mem @ V1h @ A_27a )
& ? [V2t: $i] :
( ( mem @ V2t @ ( ty_2Elist_2Elist @ A_27a ) )
& ( V0l
= ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V1h ) @ V2t ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2Elist__nchotomy,axiom,
! [A_27a: del,V0l: $i] :
( ( mem @ V0l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( V0l
= ( c_2Elist_2ENIL @ A_27a ) )
| ? [V1h: $i] :
( ( mem @ V1h @ A_27a )
& ? [V2t: $i] :
( ( mem @ V2t @ ( ty_2Elist_2Elist @ A_27a ) )
& ( V0l
= ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V1h ) @ V2t ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2Elist__case__compute,axiom,
! [A_27a: del,A_27b: del,V0b: $i] :
( ( mem @ V0b @ A_27b )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ ( arr @ ( ty_2Elist_2Elist @ A_27a ) @ A_27b ) ) )
=> ! [V2l: $i] :
( ( mem @ V2l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Elist_2Elist__CASE @ A_27a @ A_27b ) @ V2l ) @ V0b ) @ V1f )
= ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ A_27b ) @ ( ap @ ( c_2Elist_2ENULL @ A_27a ) @ V2l ) ) @ V0b ) @ ( ap @ ( ap @ V1f @ ( ap @ ( c_2Elist_2EHD @ A_27a ) @ V2l ) ) @ ( ap @ ( c_2Elist_2ETL @ A_27a ) @ V2l ) ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2ECONS__11,axiom,
! [A_27a: del,V0a0: $i] :
( ( mem @ V0a0 @ A_27a )
=> ! [V1a1: $i] :
( ( mem @ V1a1 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V2a0_27: $i] :
( ( mem @ V2a0_27 @ A_27a )
=> ! [V3a1_27: $i] :
( ( mem @ V3a1_27 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V0a0 ) @ V1a1 )
= ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V2a0_27 ) @ V3a1_27 ) )
<=> ( ( V0a0 = V2a0_27 )
& ( V1a1 = V3a1_27 ) ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2ENOT__NIL__CONS,axiom,
! [A_27a: del,V0a1: $i] :
( ( mem @ V0a1 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V1a0: $i] :
( ( mem @ V1a0 @ A_27a )
=> ( ( c_2Elist_2ENIL @ A_27a )
!= ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V1a0 ) @ V0a1 ) ) ) ) ).
thf(conj_thm_2Elist_2ENOT__CONS__NIL,axiom,
! [A_27a: del,V0a1: $i] :
( ( mem @ V0a1 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V1a0: $i] :
( ( mem @ V1a0 @ A_27a )
=> ( ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V1a0 ) @ V0a1 )
!= ( c_2Elist_2ENIL @ A_27a ) ) ) ) ).
thf(conj_thm_2Elist_2ELIST__NOT__EQ,axiom,
! [A_27a: del,V0l1: $i] :
( ( mem @ V0l1 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V1l2: $i] :
( ( mem @ V1l2 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( V0l1 != V1l2 )
=> ! [V2h1: $i] :
( ( mem @ V2h1 @ A_27a )
=> ! [V3h2: $i] :
( ( mem @ V3h2 @ A_27a )
=> ( ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V2h1 ) @ V0l1 )
!= ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V3h2 ) @ V1l2 ) ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2ENOT__EQ__LIST,axiom,
! [A_27a: del,V0h1: $i] :
( ( mem @ V0h1 @ A_27a )
=> ! [V1h2: $i] :
( ( mem @ V1h2 @ A_27a )
=> ( ( V0h1 != V1h2 )
=> ! [V2l1: $i] :
( ( mem @ V2l1 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V3l2: $i] :
( ( mem @ V3l2 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V0h1 ) @ V2l1 )
!= ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V1h2 ) @ V3l2 ) ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2EEQ__LIST,axiom,
! [A_27a: del,V0h1: $i] :
( ( mem @ V0h1 @ A_27a )
=> ! [V1h2: $i] :
( ( mem @ V1h2 @ A_27a )
=> ( ( V0h1 = V1h2 )
=> ! [V2l1: $i] :
( ( mem @ V2l1 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V3l2: $i] :
( ( mem @ V3l2 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( V2l1 = V3l2 )
=> ( ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V0h1 ) @ V2l1 )
= ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V1h2 ) @ V3l2 ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2ECONS,axiom,
! [A_27a: del,V0l: $i] :
( ( mem @ V0l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ~ ( p @ ( ap @ ( c_2Elist_2ENULL @ A_27a ) @ V0l ) )
=> ( ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ ( ap @ ( c_2Elist_2EHD @ A_27a ) @ V0l ) ) @ ( ap @ ( c_2Elist_2ETL @ A_27a ) @ V0l ) )
= V0l ) ) ) ).
thf(conj_thm_2Elist_2EAPPEND__NIL,axiom,
! [A_27a: del,V0l: $i] :
( ( mem @ V0l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ V0l ) @ ( c_2Elist_2ENIL @ A_27a ) )
= V0l ) ) ).
thf(conj_thm_2Elist_2EAPPEND__ASSOC,axiom,
! [A_27a: del,V0l1: $i] :
( ( mem @ V0l1 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V1l2: $i] :
( ( mem @ V1l2 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V2l3: $i] :
( ( mem @ V2l3 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ V0l1 ) @ ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ V1l2 ) @ V2l3 ) )
= ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ V0l1 ) @ V1l2 ) ) @ V2l3 ) ) ) ) ) ).
thf(conj_thm_2Elist_2ELENGTH__APPEND,axiom,
! [A_27a: del,V0l1: $i] :
( ( mem @ V0l1 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V1l2: $i] :
( ( mem @ V1l2 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ V0l1 ) @ V1l2 ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ V0l1 ) ) @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ V1l2 ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2EMAP__APPEND,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27b ) )
=> ! [V1l1: $i] :
( ( mem @ V1l1 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V2l2: $i] :
( ( mem @ V2l2 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( ap @ ( ap @ ( c_2Elist_2EMAP @ A_27a @ A_27b ) @ V0f ) @ ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ V1l1 ) @ V2l2 ) )
= ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27b ) @ ( ap @ ( ap @ ( c_2Elist_2EMAP @ A_27a @ A_27b ) @ V0f ) @ V1l1 ) ) @ ( ap @ ( ap @ ( c_2Elist_2EMAP @ A_27a @ A_27b ) @ V0f ) @ V2l2 ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2EMAP__ID,axiom,
! [A_27a: del,V0l: $i] :
( ( mem @ V0l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( ( ap
@ ( ap @ ( c_2Elist_2EMAP @ A_27a @ A_27a )
@ ( lam @ A_27a
@ ^ [V1x: $i] : V1x ) )
@ V0l )
= V0l )
& ( ( ap @ ( ap @ ( c_2Elist_2EMAP @ A_27a @ A_27a ) @ ( c_2Ecombin_2EI @ A_27a ) ) @ V0l )
= V0l ) ) ) ).
thf(conj_thm_2Elist_2ELENGTH__MAP,axiom,
! [A_27a: del,A_27b: del,V0l: $i] :
( ( mem @ V0l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ A_27b ) )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Elist_2ELENGTH @ A_27b ) @ ( ap @ ( ap @ ( c_2Elist_2EMAP @ A_27a @ A_27b ) @ V1f ) @ V0l ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ V0l ) ) ) ) ) ).
thf(conj_thm_2Elist_2EMAP__EQ__NIL,axiom,
! [A_27a: del,A_27b: del,V0l: $i] :
( ( mem @ V0l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ A_27b ) )
=> ( ( ( ( ap @ ( ap @ ( c_2Elist_2EMAP @ A_27a @ A_27b ) @ V1f ) @ V0l )
= ( c_2Elist_2ENIL @ A_27b ) )
<=> ( V0l
= ( c_2Elist_2ENIL @ A_27a ) ) )
& ( ( ( c_2Elist_2ENIL @ A_27b )
= ( ap @ ( ap @ ( c_2Elist_2EMAP @ A_27a @ A_27b ) @ V1f ) @ V0l ) )
<=> ( V0l
= ( c_2Elist_2ENIL @ A_27a ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2EMAP__EQ__CONS,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27b ) )
=> ! [V1l: $i] :
( ( mem @ V1l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V2h: $i] :
( ( mem @ V2h @ A_27b )
=> ! [V3t: $i] :
( ( mem @ V3t @ ( ty_2Elist_2Elist @ A_27b ) )
=> ( ( ( ap @ ( ap @ ( c_2Elist_2EMAP @ A_27a @ A_27b ) @ V0f ) @ V1l )
= ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27b ) @ V2h ) @ V3t ) )
<=> ? [V4x0: $i] :
( ( mem @ V4x0 @ A_27a )
& ? [V5t0: $i] :
( ( mem @ V5t0 @ ( ty_2Elist_2Elist @ A_27a ) )
& ( V1l
= ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V4x0 ) @ V5t0 ) )
& ( V2h
= ( ap @ V0f @ V4x0 ) )
& ( V3t
= ( ap @ ( ap @ ( c_2Elist_2EMAP @ A_27a @ A_27b ) @ V0f ) @ V5t0 ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2EMAP__EQ__SING,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27b ) )
=> ! [V1l: $i] :
( ( mem @ V1l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V2x: $i] :
( ( mem @ V2x @ A_27b )
=> ( ( ( ap @ ( ap @ ( c_2Elist_2EMAP @ A_27a @ A_27b ) @ V0f ) @ V1l )
= ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27b ) @ V2x ) @ ( c_2Elist_2ENIL @ A_27b ) ) )
<=> ? [V3x0: $i] :
( ( mem @ V3x0 @ A_27a )
& ( V1l
= ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V3x0 ) @ ( c_2Elist_2ENIL @ A_27a ) ) )
& ( V2x
= ( ap @ V0f @ V3x0 ) ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2EMAP__EQ__f,axiom,
! [A_27a: del,A_27b: del,V0f1: $i] :
( ( mem @ V0f1 @ ( arr @ A_27a @ A_27b ) )
=> ! [V1f2: $i] :
( ( mem @ V1f2 @ ( arr @ A_27a @ A_27b ) )
=> ! [V2l: $i] :
( ( mem @ V2l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( ( ap @ ( ap @ ( c_2Elist_2EMAP @ A_27a @ A_27b ) @ V0f1 ) @ V2l )
= ( ap @ ( ap @ ( c_2Elist_2EMAP @ A_27a @ A_27b ) @ V1f2 ) @ V2l ) )
<=> ! [V3e: $i] :
( ( mem @ V3e @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3e ) @ ( ap @ ( c_2Elist_2ELIST__TO__SET @ A_27a ) @ V2l ) ) )
=> ( ( ap @ V0f1 @ V3e )
= ( ap @ V1f2 @ V3e ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2EMAP__o,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27b @ A_27c ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ A_27a @ A_27b ) )
=> ( ( ap @ ( c_2Elist_2EMAP @ A_27a @ A_27c ) @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ A_27a @ A_27c @ A_27b ) @ V0f ) @ V1g ) )
= ( ap @ ( ap @ ( c_2Ecombin_2Eo @ ( ty_2Elist_2Elist @ A_27a ) @ ( ty_2Elist_2Elist @ A_27c ) @ ( ty_2Elist_2Elist @ A_27b ) ) @ ( ap @ ( c_2Elist_2EMAP @ A_27b @ A_27c ) @ V0f ) ) @ ( ap @ ( c_2Elist_2EMAP @ A_27a @ A_27b ) @ V1g ) ) ) ) ) ).
thf(conj_thm_2Elist_2EMAP__MAP__o,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27b @ A_27c ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ A_27a @ A_27b ) )
=> ! [V2l: $i] :
( ( mem @ V2l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( ap @ ( ap @ ( c_2Elist_2EMAP @ A_27b @ A_27c ) @ V0f ) @ ( ap @ ( ap @ ( c_2Elist_2EMAP @ A_27a @ A_27b ) @ V1g ) @ V2l ) )
= ( ap @ ( ap @ ( c_2Elist_2EMAP @ A_27a @ A_27c ) @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ A_27a @ A_27c @ A_27b ) @ V0f ) @ V1g ) ) @ V2l ) ) ) ) ) ).
thf(conj_thm_2Elist_2EEL__MAP,axiom,
! [A_27a: del,A_27b: del,V0n: tp__ty_2Enum_2Enum,V1l: $i] :
( ( mem @ V1l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V0n ) ) @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ V1l ) ) )
=> ! [V2f: $i] :
( ( mem @ V2f @ ( arr @ A_27a @ A_27b ) )
=> ( ( ap @ ( ap @ ( c_2Elist_2EEL @ A_27b ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) @ ( ap @ ( ap @ ( c_2Elist_2EMAP @ A_27a @ A_27b ) @ V2f ) @ V1l ) )
= ( ap @ V2f @ ( ap @ ( ap @ ( c_2Elist_2EEL @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) @ V1l ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2EEL__APPEND__EQN,axiom,
! [A_27a: del,V0l1: $i] :
( ( mem @ V0l1 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V1l2: $i] :
( ( mem @ V1l2 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V2n: tp__ty_2Enum_2Enum] :
( ( ap @ ( ap @ ( c_2Elist_2EEL @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) @ ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ V0l1 ) @ V1l2 ) )
= ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ A_27a ) @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V2n ) ) @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ V0l1 ) ) ) @ ( ap @ ( ap @ ( c_2Elist_2EEL @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) @ V0l1 ) ) @ ( ap @ ( ap @ ( c_2Elist_2EEL @ A_27a ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ ( inj__ty_2Enum_2Enum @ V2n ) ) @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ V0l1 ) ) ) @ V1l2 ) ) ) ) ) ).
thf(conj_thm_2Elist_2EMAP__TL,axiom,
! [A_27a: del,A_27b: del,V0l: $i] :
( ( mem @ V0l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ A_27b ) )
=> ( ~ ( p @ ( ap @ ( c_2Elist_2ENULL @ A_27a ) @ V0l ) )
=> ( ( ap @ ( ap @ ( c_2Elist_2EMAP @ A_27a @ A_27b ) @ V1f ) @ ( ap @ ( c_2Elist_2ETL @ A_27a ) @ V0l ) )
= ( ap @ ( c_2Elist_2ETL @ A_27b ) @ ( ap @ ( ap @ ( c_2Elist_2EMAP @ A_27a @ A_27b ) @ V1f ) @ V0l ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2EEVERY__EL,axiom,
! [A_27a: del,V0l: $i] :
( ( mem @ V0l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V1P: $i] :
( ( mem @ V1P @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Elist_2EEVERY @ A_27a ) @ V1P ) @ V0l ) )
<=> ! [V2n: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V2n ) ) @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ V0l ) ) )
=> ( p @ ( ap @ V1P @ ( ap @ ( ap @ ( c_2Elist_2EEL @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) @ V0l ) ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2EEVERY__CONJ,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ! [V1Q: $i] :
( ( mem @ V1Q @ ( arr @ A_27a @ bool ) )
=> ! [V2l: $i] :
( ( mem @ V2l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( p
@ ( ap
@ ( ap @ ( c_2Elist_2EEVERY @ A_27a )
@ ( lam @ A_27a
@ ^ [V3x: $i] : ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ V0P @ V3x ) ) @ ( ap @ V1Q @ V3x ) ) ) )
@ V2l ) )
<=> ( ( p @ ( ap @ ( ap @ ( c_2Elist_2EEVERY @ A_27a ) @ V0P ) @ V2l ) )
& ( p @ ( ap @ ( ap @ ( c_2Elist_2EEVERY @ A_27a ) @ V1Q ) @ V2l ) ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2EEVERY__MEM,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ! [V1l: $i] :
( ( mem @ V1l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Elist_2EEVERY @ A_27a ) @ V0P ) @ V1l ) )
<=> ! [V2e: $i] :
( ( mem @ V2e @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2e ) @ ( ap @ ( c_2Elist_2ELIST__TO__SET @ A_27a ) @ V1l ) ) )
=> ( p @ ( ap @ V0P @ V2e ) ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2EEVERY__MAP,axiom,
! [A_27a: del,A_27b: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27b @ bool ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ A_27b ) )
=> ! [V2l: $i] :
( ( mem @ V2l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Elist_2EEVERY @ A_27b ) @ V0P ) @ ( ap @ ( ap @ ( c_2Elist_2EMAP @ A_27a @ A_27b ) @ V1f ) @ V2l ) ) )
<=> ( p
@ ( ap
@ ( ap @ ( c_2Elist_2EEVERY @ A_27a )
@ ( lam @ A_27a
@ ^ [V3x: $i] : ( ap @ V0P @ ( ap @ V1f @ V3x ) ) ) )
@ V2l ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2EEVERY__SIMP,axiom,
! [A_27a: del,V0c: $i] :
( ( mem @ V0c @ bool )
=> ! [V1l: $i] :
( ( mem @ V1l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( p
@ ( ap
@ ( ap @ ( c_2Elist_2EEVERY @ A_27a )
@ ( lam @ A_27a
@ ^ [V2x: $i] : V0c ) )
@ V1l ) )
<=> ( ( V1l
= ( c_2Elist_2ENIL @ A_27a ) )
| ( p @ V0c ) ) ) ) ) ).
thf(conj_thm_2Elist_2EMONO__EVERY,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ! [V1Q: $i] :
( ( mem @ V1Q @ ( arr @ A_27a @ bool ) )
=> ! [V2l: $i] :
( ( mem @ V2l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ( ( p @ ( ap @ V0P @ V3x ) )
=> ( p @ ( ap @ V1Q @ V3x ) ) ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Elist_2EEVERY @ A_27a ) @ V0P ) @ V2l ) )
=> ( p @ ( ap @ ( ap @ ( c_2Elist_2EEVERY @ A_27a ) @ V1Q ) @ V2l ) ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2EEXISTS__MEM,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ! [V1l: $i] :
( ( mem @ V1l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Elist_2EEXISTS @ A_27a ) @ V0P ) @ V1l ) )
<=> ? [V2e: $i] :
( ( mem @ V2e @ A_27a )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2e ) @ ( ap @ ( c_2Elist_2ELIST__TO__SET @ A_27a ) @ V1l ) ) )
& ( p @ ( ap @ V0P @ V2e ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2EEXISTS__MAP,axiom,
! [A_27a: del,A_27b: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27b @ bool ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ A_27b ) )
=> ! [V2l: $i] :
( ( mem @ V2l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Elist_2EEXISTS @ A_27b ) @ V0P ) @ ( ap @ ( ap @ ( c_2Elist_2EMAP @ A_27a @ A_27b ) @ V1f ) @ V2l ) ) )
<=> ( p
@ ( ap
@ ( ap @ ( c_2Elist_2EEXISTS @ A_27a )
@ ( lam @ A_27a
@ ^ [V3x: $i] : ( ap @ V0P @ ( ap @ V1f @ V3x ) ) ) )
@ V2l ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2EEXISTS__SIMP,axiom,
! [A_27a: del,V0c: $i] :
( ( mem @ V0c @ bool )
=> ! [V1l: $i] :
( ( mem @ V1l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( p
@ ( ap
@ ( ap @ ( c_2Elist_2EEXISTS @ A_27a )
@ ( lam @ A_27a
@ ^ [V2x: $i] : V0c ) )
@ V1l ) )
<=> ( ( V1l
!= ( c_2Elist_2ENIL @ A_27a ) )
& ( p @ V0c ) ) ) ) ) ).
thf(conj_thm_2Elist_2EMONO__EXISTS,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ! [V1Q: $i] :
( ( mem @ V1Q @ ( arr @ A_27a @ bool ) )
=> ! [V2l: $i] :
( ( mem @ V2l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ( ( p @ ( ap @ V0P @ V3x ) )
=> ( p @ ( ap @ V1Q @ V3x ) ) ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Elist_2EEXISTS @ A_27a ) @ V0P ) @ V2l ) )
=> ( p @ ( ap @ ( ap @ ( c_2Elist_2EEXISTS @ A_27a ) @ V1Q ) @ V2l ) ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2EEVERY__NOT__EXISTS,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ! [V1l: $i] :
( ( mem @ V1l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Elist_2EEVERY @ A_27a ) @ V0P ) @ V1l ) )
<=> ~ ( p
@ ( ap
@ ( ap @ ( c_2Elist_2EEXISTS @ A_27a )
@ ( lam @ A_27a
@ ^ [V2x: $i] : ( ap @ c_2Ebool_2E_7E @ ( ap @ V0P @ V2x ) ) ) )
@ V1l ) ) ) ) ) ).
thf(conj_thm_2Elist_2EEXISTS__NOT__EVERY,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ! [V1l: $i] :
( ( mem @ V1l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Elist_2EEXISTS @ A_27a ) @ V0P ) @ V1l ) )
<=> ~ ( p
@ ( ap
@ ( ap @ ( c_2Elist_2EEVERY @ A_27a )
@ ( lam @ A_27a
@ ^ [V2x: $i] : ( ap @ c_2Ebool_2E_7E @ ( ap @ V0P @ V2x ) ) ) )
@ V1l ) ) ) ) ) ).
thf(conj_thm_2Elist_2EMEM__APPEND,axiom,
! [A_27a: del,V0e: $i] :
( ( mem @ V0e @ A_27a )
=> ! [V1l1: $i] :
( ( mem @ V1l1 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V2l2: $i] :
( ( mem @ V2l2 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V0e ) @ ( ap @ ( c_2Elist_2ELIST__TO__SET @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ V1l1 ) @ V2l2 ) ) ) )
<=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V0e ) @ ( ap @ ( c_2Elist_2ELIST__TO__SET @ A_27a ) @ V1l1 ) ) )
| ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V0e ) @ ( ap @ ( c_2Elist_2ELIST__TO__SET @ A_27a ) @ V2l2 ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2EMEM__FILTER,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ! [V1L: $i] :
( ( mem @ V1L @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x ) @ ( ap @ ( c_2Elist_2ELIST__TO__SET @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2EFILTER @ A_27a ) @ V0P ) @ V1L ) ) ) )
<=> ( ( p @ ( ap @ V0P @ V2x ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x ) @ ( ap @ ( c_2Elist_2ELIST__TO__SET @ A_27a ) @ V1L ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2EMEM__FLAT,axiom,
! [A_27a: del,V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ! [V1L: $i] :
( ( mem @ V1L @ ( ty_2Elist_2Elist @ ( ty_2Elist_2Elist @ A_27a ) ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V0x ) @ ( ap @ ( c_2Elist_2ELIST__TO__SET @ A_27a ) @ ( ap @ ( c_2Elist_2EFLAT @ A_27a ) @ V1L ) ) ) )
<=> ? [V2l: $i] :
( ( mem @ V2l @ ( ty_2Elist_2Elist @ A_27a ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Elist_2Elist @ A_27a ) ) @ V2l ) @ ( ap @ ( c_2Elist_2ELIST__TO__SET @ ( ty_2Elist_2Elist @ A_27a ) ) @ V1L ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V0x ) @ ( ap @ ( c_2Elist_2ELIST__TO__SET @ A_27a ) @ V2l ) ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2EFLAT__APPEND,axiom,
! [A_27a: del,V0l1: $i] :
( ( mem @ V0l1 @ ( ty_2Elist_2Elist @ ( ty_2Elist_2Elist @ A_27a ) ) )
=> ! [V1l2: $i] :
( ( mem @ V1l2 @ ( ty_2Elist_2Elist @ ( ty_2Elist_2Elist @ A_27a ) ) )
=> ( ( ap @ ( c_2Elist_2EFLAT @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ ( ty_2Elist_2Elist @ A_27a ) ) @ V0l1 ) @ V1l2 ) )
= ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ ( ap @ ( c_2Elist_2EFLAT @ A_27a ) @ V0l1 ) ) @ ( ap @ ( c_2Elist_2EFLAT @ A_27a ) @ V1l2 ) ) ) ) ) ).
thf(conj_thm_2Elist_2EFLAT__compute,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0t: $i] :
( ( mem @ V0t @ ( ty_2Elist_2Elist @ ( ty_2Elist_2Elist @ A_27b ) ) )
=> ! [V1h: $i] :
( ( mem @ V1h @ A_27c )
=> ! [V2t1: $i] :
( ( mem @ V2t1 @ ( ty_2Elist_2Elist @ A_27c ) )
=> ! [V3t2: $i] :
( ( mem @ V3t2 @ ( ty_2Elist_2Elist @ ( ty_2Elist_2Elist @ A_27c ) ) )
=> ( ( ( ap @ ( c_2Elist_2EFLAT @ A_27a ) @ ( c_2Elist_2ENIL @ ( ty_2Elist_2Elist @ A_27a ) ) )
= ( c_2Elist_2ENIL @ A_27a ) )
& ( ( ap @ ( c_2Elist_2EFLAT @ A_27b ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ( ty_2Elist_2Elist @ A_27b ) ) @ ( c_2Elist_2ENIL @ A_27b ) ) @ V0t ) )
= ( ap @ ( c_2Elist_2EFLAT @ A_27b ) @ V0t ) )
& ( ( ap @ ( c_2Elist_2EFLAT @ A_27c ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ( ty_2Elist_2Elist @ A_27c ) ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27c ) @ V1h ) @ V2t1 ) ) @ V3t2 ) )
= ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27c ) @ V1h ) @ ( ap @ ( c_2Elist_2EFLAT @ A_27c ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ( ty_2Elist_2Elist @ A_27c ) ) @ V2t1 ) @ V3t2 ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2EEVERY__APPEND,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ! [V1l1: $i] :
( ( mem @ V1l1 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V2l2: $i] :
( ( mem @ V2l2 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Elist_2EEVERY @ A_27a ) @ V0P ) @ ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ V1l1 ) @ V2l2 ) ) )
<=> ( ( p @ ( ap @ ( ap @ ( c_2Elist_2EEVERY @ A_27a ) @ V0P ) @ V1l1 ) )
& ( p @ ( ap @ ( ap @ ( c_2Elist_2EEVERY @ A_27a ) @ V0P ) @ V2l2 ) ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2EEXISTS__APPEND,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ! [V1l1: $i] :
( ( mem @ V1l1 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V2l2: $i] :
( ( mem @ V2l2 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Elist_2EEXISTS @ A_27a ) @ V0P ) @ ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ V1l1 ) @ V2l2 ) ) )
<=> ( ( p @ ( ap @ ( ap @ ( c_2Elist_2EEXISTS @ A_27a ) @ V0P ) @ V1l1 ) )
| ( p @ ( ap @ ( ap @ ( c_2Elist_2EEXISTS @ A_27a ) @ V0P ) @ V2l2 ) ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2ENOT__EVERY,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ! [V1l: $i] :
( ( mem @ V1l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ~ ( p @ ( ap @ ( ap @ ( c_2Elist_2EEVERY @ A_27a ) @ V0P ) @ V1l ) )
<=> ( p @ ( ap @ ( ap @ ( c_2Elist_2EEXISTS @ A_27a ) @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ A_27a @ bool @ bool ) @ c_2Ebool_2E_7E ) @ V0P ) ) @ V1l ) ) ) ) ) ).
thf(conj_thm_2Elist_2ENOT__EXISTS,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ! [V1l: $i] :
( ( mem @ V1l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ~ ( p @ ( ap @ ( ap @ ( c_2Elist_2EEXISTS @ A_27a ) @ V0P ) @ V1l ) )
<=> ( p @ ( ap @ ( ap @ ( c_2Elist_2EEVERY @ A_27a ) @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ A_27a @ bool @ bool ) @ c_2Ebool_2E_7E ) @ V0P ) ) @ V1l ) ) ) ) ) ).
thf(conj_thm_2Elist_2EMEM__MAP,axiom,
! [A_27a: del,A_27b: del,V0l: $i] :
( ( mem @ V0l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ A_27b ) )
=> ! [V2x: $i] :
( ( mem @ V2x @ A_27b )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27b ) @ V2x ) @ ( ap @ ( c_2Elist_2ELIST__TO__SET @ A_27b ) @ ( ap @ ( ap @ ( c_2Elist_2EMAP @ A_27a @ A_27b ) @ V1f ) @ V0l ) ) ) )
<=> ? [V3y: $i] :
( ( mem @ V3y @ A_27a )
& ( V2x
= ( ap @ V1f @ V3y ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3y ) @ ( ap @ ( c_2Elist_2ELIST__TO__SET @ A_27a ) @ V0l ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2ELENGTH__NIL,axiom,
! [A_27a: del,V0l: $i] :
( ( mem @ V0l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ V0l ) )
= fo__c_2Enum_2E0 )
<=> ( V0l
= ( c_2Elist_2ENIL @ A_27a ) ) ) ) ).
thf(conj_thm_2Elist_2ELENGTH__NIL__SYM,axiom,
! [A_27a: del,V0l: $i] :
( ( mem @ V0l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( fo__c_2Enum_2E0
= ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ V0l ) ) )
<=> ( V0l
= ( c_2Elist_2ENIL @ A_27a ) ) ) ) ).
thf(conj_thm_2Elist_2ENULL__EQ,axiom,
! [A_27a: del,V0l: $i] :
( ( mem @ V0l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( p @ ( ap @ ( c_2Elist_2ENULL @ A_27a ) @ V0l ) )
<=> ( V0l
= ( c_2Elist_2ENIL @ A_27a ) ) ) ) ).
thf(conj_thm_2Elist_2ENULL__LENGTH,axiom,
! [A_27a: del,V0l: $i] :
( ( mem @ V0l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( p @ ( ap @ ( c_2Elist_2ENULL @ A_27a ) @ V0l ) )
<=> ( ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ V0l ) )
= fo__c_2Enum_2E0 ) ) ) ).
thf(conj_thm_2Elist_2ELENGTH__CONS,axiom,
! [A_27a: del,V0l: $i] :
( ( mem @ V0l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V1n: tp__ty_2Enum_2Enum] :
( ( ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ V0l ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) )
<=> ? [V2h: $i] :
( ( mem @ V2h @ A_27a )
& ? [V3l_27: $i] :
( ( mem @ V3l_27 @ ( ty_2Elist_2Elist @ A_27a ) )
& ( ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ V3l_27 ) )
= V1n )
& ( V0l
= ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V2h ) @ V3l_27 ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2ELENGTH__EQ__CONS,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ ( ty_2Elist_2Elist @ A_27a ) @ bool ) )
=> ! [V1n: tp__ty_2Enum_2Enum] :
( ! [V2l: $i] :
( ( mem @ V2l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ V2l ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) )
=> ( p @ ( ap @ V0P @ V2l ) ) ) )
<=> ! [V3l: $i] :
( ( mem @ V3l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ V3l ) )
= V1n )
=> ( p
@ ( ap
@ ( lam @ ( ty_2Elist_2Elist @ A_27a )
@ ^ [V4l: $i] :
( ap @ ( c_2Ebool_2E_21 @ A_27a )
@ ( lam @ A_27a
@ ^ [V5x: $i] : ( ap @ V0P @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V5x ) @ V4l ) ) ) ) )
@ V3l ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2ELENGTH__EQ__SUM,axiom,
! [A_27a: del,V0l: $i] :
( ( mem @ V0l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V1n1: tp__ty_2Enum_2Enum,V2n2: tp__ty_2Enum_2Enum] :
( ( ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ V0l ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( inj__ty_2Enum_2Enum @ V1n1 ) ) @ ( inj__ty_2Enum_2Enum @ V2n2 ) ) ) )
<=> ? [V3l1: $i] :
( ( mem @ V3l1 @ ( ty_2Elist_2Elist @ A_27a ) )
& ? [V4l2: $i] :
( ( mem @ V4l2 @ ( ty_2Elist_2Elist @ A_27a ) )
& ( ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ V3l1 ) )
= V1n1 )
& ( ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ V4l2 ) )
= V2n2 )
& ( V0l
= ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ V3l1 ) @ V4l2 ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2ELENGTH__EQ__NUM,axiom,
! [A_27a: del] :
( ! [V0l: $i] :
( ( mem @ V0l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ V0l ) )
= fo__c_2Enum_2E0 )
<=> ( V0l
= ( c_2Elist_2ENIL @ A_27a ) ) ) )
& ! [V1l: $i] :
( ( mem @ V1l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V2n: tp__ty_2Enum_2Enum] :
( ( ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ V1l ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) )
<=> ? [V3h: $i] :
( ( mem @ V3h @ A_27a )
& ? [V4l_27: $i] :
( ( mem @ V4l_27 @ ( ty_2Elist_2Elist @ A_27a ) )
& ( ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ V4l_27 ) )
= V2n )
& ( V1l
= ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V3h ) @ V4l_27 ) ) ) ) ) )
& ! [V5l: $i] :
( ( mem @ V5l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V6n1: tp__ty_2Enum_2Enum,V7n2: tp__ty_2Enum_2Enum] :
( ( ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ V5l ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( inj__ty_2Enum_2Enum @ V6n1 ) ) @ ( inj__ty_2Enum_2Enum @ V7n2 ) ) ) )
<=> ? [V8l1: $i] :
( ( mem @ V8l1 @ ( ty_2Elist_2Elist @ A_27a ) )
& ? [V9l2: $i] :
( ( mem @ V9l2 @ ( ty_2Elist_2Elist @ A_27a ) )
& ( ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ V8l1 ) )
= V6n1 )
& ( ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ V9l2 ) )
= V7n2 )
& ( V5l
= ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ V8l1 ) @ V9l2 ) ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2ELENGTH__EQ__NUM__compute,axiom,
! [A_27a: del] :
( ! [V0l: $i] :
( ( mem @ V0l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ V0l ) )
= fo__c_2Enum_2E0 )
<=> ( V0l
= ( c_2Elist_2ENIL @ A_27a ) ) ) )
& ! [V1l: $i] :
( ( mem @ V1l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V2n: tp__ty_2Enum_2Enum] :
( ( ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ V1l ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) ) )
<=> ? [V3h: $i] :
( ( mem @ V3h @ A_27a )
& ? [V4l_27: $i] :
( ( mem @ V4l_27 @ ( ty_2Elist_2Elist @ A_27a ) )
& ( ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ V4l_27 ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) )
& ( V1l
= ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V3h ) @ V4l_27 ) ) ) ) ) )
& ! [V5l: $i] :
( ( mem @ V5l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V6n: tp__ty_2Enum_2Enum] :
( ( ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ V5l ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ V6n ) ) ) ) )
<=> ? [V7h: $i] :
( ( mem @ V7h @ A_27a )
& ? [V8l_27: $i] :
( ( mem @ V8l_27 @ ( ty_2Elist_2Elist @ A_27a ) )
& ( ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ V8l_27 ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ V6n ) ) ) ) )
& ( V5l
= ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V7h ) @ V8l_27 ) ) ) ) ) )
& ! [V9l: $i] :
( ( mem @ V9l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V10n1: tp__ty_2Enum_2Enum,V11n2: tp__ty_2Enum_2Enum] :
( ( ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ V9l ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( inj__ty_2Enum_2Enum @ V10n1 ) ) @ ( inj__ty_2Enum_2Enum @ V11n2 ) ) ) )
<=> ? [V12l1: $i] :
( ( mem @ V12l1 @ ( ty_2Elist_2Elist @ A_27a ) )
& ? [V13l2: $i] :
( ( mem @ V13l2 @ ( ty_2Elist_2Elist @ A_27a ) )
& ( ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ V12l1 ) )
= V10n1 )
& ( ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ V13l2 ) )
= V11n2 )
& ( V9l
= ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ V12l1 ) @ V13l2 ) ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2ELENGTH__EQ__NIL,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ ( ty_2Elist_2Elist @ A_27a ) @ bool ) )
=> ( ! [V1l: $i] :
( ( mem @ V1l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ V1l ) )
= fo__c_2Enum_2E0 )
=> ( p @ ( ap @ V0P @ V1l ) ) ) )
<=> ( p @ ( ap @ V0P @ ( c_2Elist_2ENIL @ A_27a ) ) ) ) ) ).
thf(conj_thm_2Elist_2ECONS__ACYCLIC,axiom,
! [A_27a: del,V0l: $i] :
( ( mem @ V0l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ( ( V0l
!= ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V1x ) @ V0l ) )
& ( ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V1x ) @ V0l )
!= V0l ) ) ) ) ).
thf(conj_thm_2Elist_2EAPPEND__eq__NIL,axiom,
! [A_27a: del] :
( ! [V0l1: $i] :
( ( mem @ V0l1 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V1l2: $i] :
( ( mem @ V1l2 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( ( c_2Elist_2ENIL @ A_27a )
= ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ V0l1 ) @ V1l2 ) )
<=> ( ( V0l1
= ( c_2Elist_2ENIL @ A_27a ) )
& ( V1l2
= ( c_2Elist_2ENIL @ A_27a ) ) ) ) ) )
& ! [V2l1: $i] :
( ( mem @ V2l1 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V3l2: $i] :
( ( mem @ V3l2 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ V2l1 ) @ V3l2 )
= ( c_2Elist_2ENIL @ A_27a ) )
<=> ( ( V2l1
= ( c_2Elist_2ENIL @ A_27a ) )
& ( V3l2
= ( c_2Elist_2ENIL @ A_27a ) ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2EMAP__EQ__APPEND,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27b ) )
=> ! [V1l: $i] :
( ( mem @ V1l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V2l1: $i] :
( ( mem @ V2l1 @ ( ty_2Elist_2Elist @ A_27b ) )
=> ! [V3l2: $i] :
( ( mem @ V3l2 @ ( ty_2Elist_2Elist @ A_27b ) )
=> ( ( ( ap @ ( ap @ ( c_2Elist_2EMAP @ A_27a @ A_27b ) @ V0f ) @ V1l )
= ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27b ) @ V2l1 ) @ V3l2 ) )
<=> ? [V4l10: $i] :
( ( mem @ V4l10 @ ( ty_2Elist_2Elist @ A_27a ) )
& ? [V5l20: $i] :
( ( mem @ V5l20 @ ( ty_2Elist_2Elist @ A_27a ) )
& ( V1l
= ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ V4l10 ) @ V5l20 ) )
& ( V2l1
= ( ap @ ( ap @ ( c_2Elist_2EMAP @ A_27a @ A_27b ) @ V0f ) @ V4l10 ) )
& ( V3l2
= ( ap @ ( ap @ ( c_2Elist_2EMAP @ A_27a @ A_27b ) @ V0f ) @ V5l20 ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2EAPPEND__EQ__SING,axiom,
! [A_27a: del,V0l1: $i] :
( ( mem @ V0l1 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V1l2: $i] :
( ( mem @ V1l2 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V2e: $i] :
( ( mem @ V2e @ A_27a )
=> ( ( ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ V0l1 ) @ V1l2 )
= ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V2e ) @ ( c_2Elist_2ENIL @ A_27a ) ) )
<=> ( ( ( V0l1
= ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V2e ) @ ( c_2Elist_2ENIL @ A_27a ) ) )
& ( V1l2
= ( c_2Elist_2ENIL @ A_27a ) ) )
| ( ( V0l1
= ( c_2Elist_2ENIL @ A_27a ) )
& ( V1l2
= ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V2e ) @ ( c_2Elist_2ENIL @ A_27a ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2EAPPEND__11,axiom,
! [A_27a: del] :
( ! [V0l1: $i] :
( ( mem @ V0l1 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V1l2: $i] :
( ( mem @ V1l2 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V2l3: $i] :
( ( mem @ V2l3 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ V0l1 ) @ V1l2 )
= ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ V0l1 ) @ V2l3 ) )
<=> ( V1l2 = V2l3 ) ) ) ) )
& ! [V3l1: $i] :
( ( mem @ V3l1 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V4l2: $i] :
( ( mem @ V4l2 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V5l3: $i] :
( ( mem @ V5l3 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ V4l2 ) @ V3l1 )
= ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ V5l3 ) @ V3l1 ) )
<=> ( V4l2 = V5l3 ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2EAPPEND__LENGTH__EQ,axiom,
! [A_27a: del,V0l1: $i] :
( ( mem @ V0l1 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V1l1_27: $i] :
( ( mem @ V1l1_27 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ V0l1 ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ V1l1_27 ) ) )
=> ! [V2l2: $i] :
( ( mem @ V2l2 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V3l2_27: $i] :
( ( mem @ V3l2_27 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ V2l2 ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ V3l2_27 ) ) )
=> ( ( ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ V0l1 ) @ V2l2 )
= ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ V1l1_27 ) @ V3l2_27 ) )
<=> ( ( V0l1 = V1l1_27 )
& ( V2l2 = V3l2_27 ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2EAPPEND__11__LENGTH,axiom,
! [A_27a: del] :
( ! [V0l1: $i] :
( ( mem @ V0l1 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V1l2: $i] :
( ( mem @ V1l2 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V2l1_27: $i] :
( ( mem @ V2l1_27 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V3l2_27: $i] :
( ( mem @ V3l2_27 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ V0l1 ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ V2l1_27 ) ) )
=> ( ( ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ V0l1 ) @ V1l2 )
= ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ V2l1_27 ) @ V3l2_27 ) )
<=> ( ( V0l1 = V2l1_27 )
& ( V1l2 = V3l2_27 ) ) ) ) ) ) ) )
& ! [V4l1: $i] :
( ( mem @ V4l1 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V5l2: $i] :
( ( mem @ V5l2 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V6l1_27: $i] :
( ( mem @ V6l1_27 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V7l2_27: $i] :
( ( mem @ V7l2_27 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ V5l2 ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ V7l2_27 ) ) )
=> ( ( ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ V4l1 ) @ V5l2 )
= ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ V6l1_27 ) @ V7l2_27 ) )
<=> ( ( V4l1 = V6l1_27 )
& ( V5l2 = V7l2_27 ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2EAPPEND__EQ__SELF,axiom,
! [A_27a: del] :
( ! [V0l1: $i] :
( ( mem @ V0l1 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V1l2: $i] :
( ( mem @ V1l2 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ V0l1 ) @ V1l2 )
= V0l1 )
<=> ( V1l2
= ( c_2Elist_2ENIL @ A_27a ) ) ) ) )
& ! [V2l1: $i] :
( ( mem @ V2l1 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V3l2: $i] :
( ( mem @ V3l2 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ V2l1 ) @ V3l2 )
= V3l2 )
<=> ( V2l1
= ( c_2Elist_2ENIL @ A_27a ) ) ) ) )
& ! [V4l1: $i] :
( ( mem @ V4l1 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V5l2: $i] :
( ( mem @ V5l2 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( V4l1
= ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ V4l1 ) @ V5l2 ) )
<=> ( V5l2
= ( c_2Elist_2ENIL @ A_27a ) ) ) ) )
& ! [V6l1: $i] :
( ( mem @ V6l1 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V7l2: $i] :
( ( mem @ V7l2 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( V7l2
= ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ V6l1 ) @ V7l2 ) )
<=> ( V6l1
= ( c_2Elist_2ENIL @ A_27a ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2EMEM__SPLIT,axiom,
! [A_27a: del,V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ! [V1l: $i] :
( ( mem @ V1l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V0x ) @ ( ap @ ( c_2Elist_2ELIST__TO__SET @ A_27a ) @ V1l ) ) )
<=> ? [V2l1: $i] :
( ( mem @ V2l1 @ ( ty_2Elist_2Elist @ A_27a ) )
& ? [V3l2: $i] :
( ( mem @ V3l2 @ ( ty_2Elist_2Elist @ A_27a ) )
& ( V1l
= ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ V2l1 ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V0x ) @ V3l2 ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2ELIST__EQ__REWRITE,axiom,
! [A_27a: del,V0l1: $i] :
( ( mem @ V0l1 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V1l2: $i] :
( ( mem @ V1l2 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( V0l1 = V1l2 )
<=> ( ( ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ V0l1 ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ V1l2 ) ) )
& ! [V2x: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V2x ) ) @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ V0l1 ) ) )
=> ( ( ap @ ( ap @ ( c_2Elist_2EEL @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V2x ) ) @ V0l1 )
= ( ap @ ( ap @ ( c_2Elist_2EEL @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V2x ) ) @ V1l2 ) ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2ELIST__EQ,axiom,
! [A_27a: del,V0l1: $i] :
( ( mem @ V0l1 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V1l2: $i] :
( ( mem @ V1l2 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( ( ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ V0l1 ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ V1l2 ) ) )
& ! [V2x: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V2x ) ) @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ V0l1 ) ) )
=> ( ( ap @ ( ap @ ( c_2Elist_2EEL @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V2x ) ) @ V0l1 )
= ( ap @ ( ap @ ( c_2Elist_2EEL @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V2x ) ) @ V1l2 ) ) ) )
=> ( V0l1 = V1l2 ) ) ) ) ).
thf(conj_thm_2Elist_2EFOLDL__EQ__FOLDR,axiom,
! [A_27a: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) )
=> ! [V1l: $i] :
( ( mem @ V1l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V2e: $i] :
( ( mem @ V2e @ A_27a )
=> ( ( ( p @ ( ap @ ( c_2Ecombin_2EASSOC @ A_27a ) @ V0f ) )
& ( p @ ( ap @ ( c_2Ecombin_2ECOMM @ A_27a @ A_27a ) @ V0f ) ) )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Elist_2EFOLDL @ A_27a @ A_27a ) @ V0f ) @ V2e ) @ V1l )
= ( ap @ ( ap @ ( ap @ ( c_2Elist_2EFOLDR @ A_27a @ A_27a ) @ V0f ) @ V2e ) @ V1l ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2EFOLDR__CONS,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27b ) )
=> ! [V1ls: $i] :
( ( mem @ V1ls @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V2a: $i] :
( ( mem @ V2a @ ( ty_2Elist_2Elist @ A_27b ) )
=> ( ( ap
@ ( ap
@ ( ap @ ( c_2Elist_2EFOLDR @ A_27a @ ( ty_2Elist_2Elist @ A_27b ) )
@ ( lam @ A_27a
@ ^ [V3x: $i] :
( lam @ ( ty_2Elist_2Elist @ A_27b )
@ ^ [V4y: $i] : ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27b ) @ ( ap @ V0f @ V3x ) ) @ V4y ) ) ) )
@ V2a )
@ V1ls )
= ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27b ) @ ( ap @ ( ap @ ( c_2Elist_2EMAP @ A_27a @ A_27b ) @ V0f ) @ V1ls ) ) @ V2a ) ) ) ) ) ).
thf(conj_thm_2Elist_2ELENGTH__TL,axiom,
! [A_27a: del,V0l: $i] :
( ( mem @ V0l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ V0l ) ) )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ ( ap @ ( c_2Elist_2ETL @ A_27a ) @ V0l ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ V0l ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2EFILTER__EQ__NIL,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ! [V1l: $i] :
( ( mem @ V1l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( ( ap @ ( ap @ ( c_2Elist_2EFILTER @ A_27a ) @ V0P ) @ V1l )
= ( c_2Elist_2ENIL @ A_27a ) )
<=> ( p
@ ( ap
@ ( ap @ ( c_2Elist_2EEVERY @ A_27a )
@ ( lam @ A_27a
@ ^ [V2x: $i] : ( ap @ c_2Ebool_2E_7E @ ( ap @ V0P @ V2x ) ) ) )
@ V1l ) ) ) ) ) ).
thf(conj_thm_2Elist_2EFILTER__NEQ__NIL,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ! [V1l: $i] :
( ( mem @ V1l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( ( ap @ ( ap @ ( c_2Elist_2EFILTER @ A_27a ) @ V0P ) @ V1l )
!= ( c_2Elist_2ENIL @ A_27a ) )
<=> ? [V2x: $i] :
( ( mem @ V2x @ A_27a )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x ) @ ( ap @ ( c_2Elist_2ELIST__TO__SET @ A_27a ) @ V1l ) ) )
& ( p @ ( ap @ V0P @ V2x ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2EFILTER__EQ__ID,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ! [V1l: $i] :
( ( mem @ V1l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( ( ap @ ( ap @ ( c_2Elist_2EFILTER @ A_27a ) @ V0P ) @ V1l )
= V1l )
<=> ( p @ ( ap @ ( ap @ ( c_2Elist_2EEVERY @ A_27a ) @ V0P ) @ V1l ) ) ) ) ) ).
thf(conj_thm_2Elist_2EFILTER__NEQ__ID,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ! [V1l: $i] :
( ( mem @ V1l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( ( ap @ ( ap @ ( c_2Elist_2EFILTER @ A_27a ) @ V0P ) @ V1l )
!= V1l )
<=> ? [V2x: $i] :
( ( mem @ V2x @ A_27a )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x ) @ ( ap @ ( c_2Elist_2ELIST__TO__SET @ A_27a ) @ V1l ) ) )
& ~ ( p @ ( ap @ V0P @ V2x ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2EFILTER__EQ__CONS,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ! [V1l: $i] :
( ( mem @ V1l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V2h: $i] :
( ( mem @ V2h @ A_27a )
=> ! [V3lr: $i] :
( ( mem @ V3lr @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( ( ap @ ( ap @ ( c_2Elist_2EFILTER @ A_27a ) @ V0P ) @ V1l )
= ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V2h ) @ V3lr ) )
<=> ? [V4l1: $i] :
( ( mem @ V4l1 @ ( ty_2Elist_2Elist @ A_27a ) )
& ? [V5l2: $i] :
( ( mem @ V5l2 @ ( ty_2Elist_2Elist @ A_27a ) )
& ( V1l
= ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ V4l1 ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V2h ) @ ( c_2Elist_2ENIL @ A_27a ) ) ) ) @ V5l2 ) )
& ( ( ap @ ( ap @ ( c_2Elist_2EFILTER @ A_27a ) @ V0P ) @ V4l1 )
= ( c_2Elist_2ENIL @ A_27a ) )
& ( ( ap @ ( ap @ ( c_2Elist_2EFILTER @ A_27a ) @ V0P ) @ V5l2 )
= V3lr )
& ( p @ ( ap @ V0P @ V2h ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2EFILTER__APPEND__DISTRIB,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ! [V1L: $i] :
( ( mem @ V1L @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V2M: $i] :
( ( mem @ V2M @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( ap @ ( ap @ ( c_2Elist_2EFILTER @ A_27a ) @ V0P ) @ ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ V1L ) @ V2M ) )
= ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2EFILTER @ A_27a ) @ V0P ) @ V1L ) ) @ ( ap @ ( ap @ ( c_2Elist_2EFILTER @ A_27a ) @ V0P ) @ V2M ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2EMEM,axiom,
! [A_27a: del] :
( ! [V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V0x ) @ ( ap @ ( c_2Elist_2ELIST__TO__SET @ A_27a ) @ ( c_2Elist_2ENIL @ A_27a ) ) ) )
<=> $false ) )
& ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ! [V2h: $i] :
( ( mem @ V2h @ A_27a )
=> ! [V3t: $i] :
( ( mem @ V3t @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V1x ) @ ( ap @ ( c_2Elist_2ELIST__TO__SET @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V2h ) @ V3t ) ) ) )
<=> ( ( V1x = V2h )
| ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V1x ) @ ( ap @ ( c_2Elist_2ELIST__TO__SET @ A_27a ) @ V3t ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2EFILTER__EQ__APPEND,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ! [V1l: $i] :
( ( mem @ V1l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V2l1: $i] :
( ( mem @ V2l1 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V3l2: $i] :
( ( mem @ V3l2 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( ( ap @ ( ap @ ( c_2Elist_2EFILTER @ A_27a ) @ V0P ) @ V1l )
= ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ V2l1 ) @ V3l2 ) )
<=> ? [V4l3: $i] :
( ( mem @ V4l3 @ ( ty_2Elist_2Elist @ A_27a ) )
& ? [V5l4: $i] :
( ( mem @ V5l4 @ ( ty_2Elist_2Elist @ A_27a ) )
& ( V1l
= ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ V4l3 ) @ V5l4 ) )
& ( ( ap @ ( ap @ ( c_2Elist_2EFILTER @ A_27a ) @ V0P ) @ V4l3 )
= V2l1 )
& ( ( ap @ ( ap @ ( c_2Elist_2EFILTER @ A_27a ) @ V0P ) @ V5l4 )
= V3l2 ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2EEVERY__FILTER,axiom,
! [A_27a: del,V0P1: $i] :
( ( mem @ V0P1 @ ( arr @ A_27a @ bool ) )
=> ! [V1P2: $i] :
( ( mem @ V1P2 @ ( arr @ A_27a @ bool ) )
=> ! [V2l: $i] :
( ( mem @ V2l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Elist_2EEVERY @ A_27a ) @ V0P1 ) @ ( ap @ ( ap @ ( c_2Elist_2EFILTER @ A_27a ) @ V1P2 ) @ V2l ) ) )
<=> ( p
@ ( ap
@ ( ap @ ( c_2Elist_2EEVERY @ A_27a )
@ ( lam @ A_27a
@ ^ [V3x: $i] : ( ap @ ( ap @ c_2Emin_2E_3D_3D_3E @ ( ap @ V1P2 @ V3x ) ) @ ( ap @ V0P1 @ V3x ) ) ) )
@ V2l ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2EEVERY__FILTER__IMP,axiom,
! [A_27a: del,V0P1: $i] :
( ( mem @ V0P1 @ ( arr @ A_27a @ bool ) )
=> ! [V1P2: $i] :
( ( mem @ V1P2 @ ( arr @ A_27a @ bool ) )
=> ! [V2l: $i] :
( ( mem @ V2l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Elist_2EEVERY @ A_27a ) @ V0P1 ) @ V2l ) )
=> ( p @ ( ap @ ( ap @ ( c_2Elist_2EEVERY @ A_27a ) @ V0P1 ) @ ( ap @ ( ap @ ( c_2Elist_2EFILTER @ A_27a ) @ V1P2 ) @ V2l ) ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2EFILTER__COND__REWRITE,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ! [V1l: $i] :
( ( mem @ V1l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( ( ap @ ( ap @ ( c_2Elist_2EFILTER @ A_27a ) @ V0P ) @ ( c_2Elist_2ENIL @ A_27a ) )
= ( c_2Elist_2ENIL @ A_27a ) )
& ! [V2h: $i] :
( ( mem @ V2h @ A_27a )
=> ( ( p @ ( ap @ V0P @ V2h ) )
=> ( ( ap @ ( ap @ ( c_2Elist_2EFILTER @ A_27a ) @ V0P ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V2h ) @ V1l ) )
= ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V2h ) @ ( ap @ ( ap @ ( c_2Elist_2EFILTER @ A_27a ) @ V0P ) @ V1l ) ) ) ) )
& ! [V3h: $i] :
( ( mem @ V3h @ A_27a )
=> ( ~ ( p @ ( ap @ V0P @ V3h ) )
=> ( ( ap @ ( ap @ ( c_2Elist_2EFILTER @ A_27a ) @ V0P ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V3h ) @ V1l ) )
= ( ap @ ( ap @ ( c_2Elist_2EFILTER @ A_27a ) @ V0P ) @ V1l ) ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2ENOT__NULL__MEM,axiom,
! [A_27a: del,V0l: $i] :
( ( mem @ V0l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ~ ( p @ ( ap @ ( c_2Elist_2ENULL @ A_27a ) @ V0l ) )
<=> ? [V1e: $i] :
( ( mem @ V1e @ A_27a )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V1e ) @ ( ap @ ( c_2Elist_2ELIST__TO__SET @ A_27a ) @ V0l ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2EEL__compute,axiom,
! [A_27a: del,V0l: $i] :
( ( mem @ V0l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V1n: tp__ty_2Enum_2Enum] :
( ( ap @ ( ap @ ( c_2Elist_2EEL @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) @ V0l )
= ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ A_27a ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ ( c_2Elist_2EHD @ A_27a ) @ V0l ) ) @ ( ap @ ( ap @ ( c_2Elist_2EEL @ A_27a ) @ ( ap @ c_2Eprim__rec_2EPRE @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) @ ( ap @ ( c_2Elist_2ETL @ A_27a ) @ V0l ) ) ) ) ) ).
thf(conj_thm_2Elist_2EEL__simp,axiom,
! [A_27a: del,V0n: tp__ty_2Enum_2Enum,V1l: $i] :
( ( mem @ V1l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( ( ap @ ( ap @ ( c_2Elist_2EEL @ A_27a ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) ) @ V1l )
= ( ap @ ( ap @ ( c_2Elist_2EEL @ A_27a ) @ ( ap @ c_2Eprim__rec_2EPRE @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) ) ) @ ( ap @ ( c_2Elist_2ETL @ A_27a ) @ V1l ) ) )
& ( ( ap @ ( ap @ ( c_2Elist_2EEL @ A_27a ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) ) @ V1l )
= ( ap @ ( ap @ ( c_2Elist_2EEL @ A_27a ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) ) @ ( ap @ ( c_2Elist_2ETL @ A_27a ) @ V1l ) ) ) ) ) ).
thf(conj_thm_2Elist_2EEL__restricted,axiom,
! [A_27a: del,A_27b: del,V0n: tp__ty_2Enum_2Enum,V1l: $i] :
( ( mem @ V1l @ A_27b )
=> ! [V2ls: $i] :
( ( mem @ V2ls @ ( ty_2Elist_2Elist @ A_27b ) )
=> ( ( ( ap @ ( c_2Elist_2EEL @ A_27a ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) )
= ( c_2Elist_2EHD @ A_27a ) )
& ( ( ap @ ( ap @ ( c_2Elist_2EEL @ A_27b ) @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27b ) @ V1l ) @ V2ls ) )
= ( ap @ ( ap @ ( c_2Elist_2EEL @ A_27b ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) @ V2ls ) ) ) ) ) ).
thf(conj_thm_2Elist_2EEL__simp__restricted,axiom,
! [A_27a: del,V0n: tp__ty_2Enum_2Enum,V1l: $i] :
( ( mem @ V1l @ A_27a )
=> ! [V2ls: $i] :
( ( mem @ V2ls @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( ( ap @ ( ap @ ( c_2Elist_2EEL @ A_27a ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V1l ) @ V2ls ) )
= ( ap @ ( ap @ ( c_2Elist_2EEL @ A_27a ) @ ( ap @ c_2Eprim__rec_2EPRE @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) ) ) @ V2ls ) )
& ( ( ap @ ( ap @ ( c_2Elist_2EEL @ A_27a ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V1l ) @ V2ls ) )
= ( ap @ ( ap @ ( c_2Elist_2EEL @ A_27a ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) ) @ V2ls ) ) ) ) ) ).
thf(conj_thm_2Elist_2ESUM__eq__0,axiom,
! [V0ls: tp__c_ty_2Elist_2Elist_ty_2Enum_2Enum] :
( ( ( surj__ty_2Enum_2Enum @ ( ap @ c_2Elist_2ESUM @ ( inj__c_ty_2Elist_2Elist_ty_2Enum_2Enum @ V0ls ) ) )
= fo__c_2Enum_2E0 )
<=> ! [V1x: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V1x ) ) @ ( ap @ ( c_2Elist_2ELIST__TO__SET @ ty_2Enum_2Enum ) @ ( inj__c_ty_2Elist_2Elist_ty_2Enum_2Enum @ V0ls ) ) ) )
=> ( V1x = fo__c_2Enum_2E0 ) ) ) ).
thf(conj_thm_2Elist_2ENULL__FILTER,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ! [V1ls: $i] :
( ( mem @ V1ls @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( p @ ( ap @ ( c_2Elist_2ENULL @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2EFILTER @ A_27a ) @ V0P ) @ V1ls ) ) )
<=> ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x ) @ ( ap @ ( c_2Elist_2ELIST__TO__SET @ A_27a ) @ V1ls ) ) )
=> ~ ( p @ ( ap @ V0P @ V2x ) ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2EWF__LIST__PRED,axiom,
! [A_27a: del] :
( p
@ ( ap @ ( c_2Erelation_2EWF @ ( ty_2Elist_2Elist @ A_27a ) )
@ ( lam @ ( ty_2Elist_2Elist @ A_27a )
@ ^ [V0L1: $i] :
( lam @ ( ty_2Elist_2Elist @ A_27a )
@ ^ [V1L2: $i] :
( ap @ ( c_2Ebool_2E_3F @ A_27a )
@ ( lam @ A_27a
@ ^ [V2h: $i] : ( ap @ ( ap @ ( c_2Emin_2E_3D @ ( ty_2Elist_2Elist @ A_27a ) ) @ V1L2 ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V2h ) @ V0L1 ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2ELIST__REL__rules,axiom,
! [A_27a: del,A_27b: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Elist_2ELIST__REL @ A_27a @ A_27b ) @ V0R ) @ ( c_2Elist_2ENIL @ A_27a ) ) @ ( c_2Elist_2ENIL @ A_27b ) ) )
& ! [V1h1: $i] :
( ( mem @ V1h1 @ A_27a )
=> ! [V2h2: $i] :
( ( mem @ V2h2 @ A_27b )
=> ! [V3t1: $i] :
( ( mem @ V3t1 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V4t2: $i] :
( ( mem @ V4t2 @ ( ty_2Elist_2Elist @ A_27b ) )
=> ( ( ( p @ ( ap @ ( ap @ V0R @ V1h1 ) @ V2h2 ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( c_2Elist_2ELIST__REL @ A_27a @ A_27b ) @ V0R ) @ V3t1 ) @ V4t2 ) ) )
=> ( p @ ( ap @ ( ap @ ( ap @ ( c_2Elist_2ELIST__REL @ A_27a @ A_27b ) @ V0R ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V1h1 ) @ V3t1 ) ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27b ) @ V2h2 ) @ V4t2 ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2ELIST__REL__ind,axiom,
! [A_27a: del,A_27b: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) )
=> ! [V1LIST__REL_27: $i] :
( ( mem @ V1LIST__REL_27 @ ( arr @ ( ty_2Elist_2Elist @ A_27a ) @ ( arr @ ( ty_2Elist_2Elist @ A_27b ) @ bool ) ) )
=> ( ( ( p @ ( ap @ ( ap @ V1LIST__REL_27 @ ( c_2Elist_2ENIL @ A_27a ) ) @ ( c_2Elist_2ENIL @ A_27b ) ) )
& ! [V2h1: $i] :
( ( mem @ V2h1 @ A_27a )
=> ! [V3h2: $i] :
( ( mem @ V3h2 @ A_27b )
=> ! [V4t1: $i] :
( ( mem @ V4t1 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V5t2: $i] :
( ( mem @ V5t2 @ ( ty_2Elist_2Elist @ A_27b ) )
=> ( ( ( p @ ( ap @ ( ap @ V0R @ V2h1 ) @ V3h2 ) )
& ( p @ ( ap @ ( ap @ V1LIST__REL_27 @ V4t1 ) @ V5t2 ) ) )
=> ( p @ ( ap @ ( ap @ V1LIST__REL_27 @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V2h1 ) @ V4t1 ) ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27b ) @ V3h2 ) @ V5t2 ) ) ) ) ) ) ) ) )
=> ! [V6a0: $i] :
( ( mem @ V6a0 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V7a1: $i] :
( ( mem @ V7a1 @ ( ty_2Elist_2Elist @ A_27b ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Elist_2ELIST__REL @ A_27a @ A_27b ) @ V0R ) @ V6a0 ) @ V7a1 ) )
=> ( p @ ( ap @ ( ap @ V1LIST__REL_27 @ V6a0 ) @ V7a1 ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2ELIST__REL__strongind,axiom,
! [A_27a: del,A_27b: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) )
=> ! [V1LIST__REL_27: $i] :
( ( mem @ V1LIST__REL_27 @ ( arr @ ( ty_2Elist_2Elist @ A_27a ) @ ( arr @ ( ty_2Elist_2Elist @ A_27b ) @ bool ) ) )
=> ( ( ( p @ ( ap @ ( ap @ V1LIST__REL_27 @ ( c_2Elist_2ENIL @ A_27a ) ) @ ( c_2Elist_2ENIL @ A_27b ) ) )
& ! [V2h1: $i] :
( ( mem @ V2h1 @ A_27a )
=> ! [V3h2: $i] :
( ( mem @ V3h2 @ A_27b )
=> ! [V4t1: $i] :
( ( mem @ V4t1 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V5t2: $i] :
( ( mem @ V5t2 @ ( ty_2Elist_2Elist @ A_27b ) )
=> ( ( ( p @ ( ap @ ( ap @ V0R @ V2h1 ) @ V3h2 ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( c_2Elist_2ELIST__REL @ A_27a @ A_27b ) @ V0R ) @ V4t1 ) @ V5t2 ) )
& ( p @ ( ap @ ( ap @ V1LIST__REL_27 @ V4t1 ) @ V5t2 ) ) )
=> ( p @ ( ap @ ( ap @ V1LIST__REL_27 @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V2h1 ) @ V4t1 ) ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27b ) @ V3h2 ) @ V5t2 ) ) ) ) ) ) ) ) )
=> ! [V6a0: $i] :
( ( mem @ V6a0 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V7a1: $i] :
( ( mem @ V7a1 @ ( ty_2Elist_2Elist @ A_27b ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Elist_2ELIST__REL @ A_27a @ A_27b ) @ V0R ) @ V6a0 ) @ V7a1 ) )
=> ( p @ ( ap @ ( ap @ V1LIST__REL_27 @ V6a0 ) @ V7a1 ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2ELIST__REL__cases,axiom,
! [A_27a: del,A_27b: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) )
=> ! [V1a0: $i] :
( ( mem @ V1a0 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V2a1: $i] :
( ( mem @ V2a1 @ ( ty_2Elist_2Elist @ A_27b ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Elist_2ELIST__REL @ A_27a @ A_27b ) @ V0R ) @ V1a0 ) @ V2a1 ) )
<=> ( ( ( V1a0
= ( c_2Elist_2ENIL @ A_27a ) )
& ( V2a1
= ( c_2Elist_2ENIL @ A_27b ) ) )
| ? [V3h1: $i] :
( ( mem @ V3h1 @ A_27a )
& ? [V4h2: $i] :
( ( mem @ V4h2 @ A_27b )
& ? [V5t1: $i] :
( ( mem @ V5t1 @ ( ty_2Elist_2Elist @ A_27a ) )
& ? [V6t2: $i] :
( ( mem @ V6t2 @ ( ty_2Elist_2Elist @ A_27b ) )
& ( V1a0
= ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V3h1 ) @ V5t1 ) )
& ( V2a1
= ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27b ) @ V4h2 ) @ V6t2 ) )
& ( p @ ( ap @ ( ap @ V0R @ V3h1 ) @ V4h2 ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( c_2Elist_2ELIST__REL @ A_27a @ A_27b ) @ V0R ) @ V5t1 ) @ V6t2 ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2ELIST__REL__EL__EQN,axiom,
! [A_27a: del,A_27b: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) )
=> ! [V1l1: $i] :
( ( mem @ V1l1 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V2l2: $i] :
( ( mem @ V2l2 @ ( ty_2Elist_2Elist @ A_27b ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Elist_2ELIST__REL @ A_27a @ A_27b ) @ V0R ) @ V1l1 ) @ V2l2 ) )
<=> ( ( ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ V1l1 ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Elist_2ELENGTH @ A_27b ) @ V2l2 ) ) )
& ! [V3n: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V3n ) ) @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ V1l1 ) ) )
=> ( p @ ( ap @ ( ap @ V0R @ ( ap @ ( ap @ ( c_2Elist_2EEL @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V3n ) ) @ V1l1 ) ) @ ( ap @ ( ap @ ( c_2Elist_2EEL @ A_27b ) @ ( inj__ty_2Enum_2Enum @ V3n ) ) @ V2l2 ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2ELIST__REL__def,axiom,
! [A_27a: del,A_27b: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) )
=> ! [V1a: $i] :
( ( mem @ V1a @ A_27a )
=> ! [V2as: $i] :
( ( mem @ V2as @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V3b: $i] :
( ( mem @ V3b @ A_27b )
=> ! [V4bs: $i] :
( ( mem @ V4bs @ ( ty_2Elist_2Elist @ A_27b ) )
=> ( ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Elist_2ELIST__REL @ A_27a @ A_27b ) @ V0R ) @ ( c_2Elist_2ENIL @ A_27a ) ) @ ( c_2Elist_2ENIL @ A_27b ) ) )
<=> $true )
& ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Elist_2ELIST__REL @ A_27a @ A_27b ) @ V0R ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V1a ) @ V2as ) ) @ ( c_2Elist_2ENIL @ A_27b ) ) )
<=> $false )
& ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Elist_2ELIST__REL @ A_27a @ A_27b ) @ V0R ) @ ( c_2Elist_2ENIL @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27b ) @ V3b ) @ V4bs ) ) )
<=> $false )
& ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Elist_2ELIST__REL @ A_27a @ A_27b ) @ V0R ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V1a ) @ V2as ) ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27b ) @ V3b ) @ V4bs ) ) )
<=> ( ( p @ ( ap @ ( ap @ V0R @ V1a ) @ V3b ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( c_2Elist_2ELIST__REL @ A_27a @ A_27b ) @ V0R ) @ V2as ) @ V4bs ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2ELIST__REL__mono,axiom,
! [A_27a: del,A_27b: del,V0R1: $i] :
( ( mem @ V0R1 @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) )
=> ! [V1R2: $i] :
( ( mem @ V1R2 @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) )
=> ! [V2l1: $i] :
( ( mem @ V2l1 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V3l2: $i] :
( ( mem @ V3l2 @ ( ty_2Elist_2Elist @ A_27b ) )
=> ( ! [V4x: $i] :
( ( mem @ V4x @ A_27a )
=> ! [V5y: $i] :
( ( mem @ V5y @ A_27b )
=> ( ( p @ ( ap @ ( ap @ V0R1 @ V4x ) @ V5y ) )
=> ( p @ ( ap @ ( ap @ V1R2 @ V4x ) @ V5y ) ) ) ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Elist_2ELIST__REL @ A_27a @ A_27b ) @ V0R1 ) @ V2l1 ) @ V3l2 ) )
=> ( p @ ( ap @ ( ap @ ( ap @ ( c_2Elist_2ELIST__REL @ A_27a @ A_27b ) @ V1R2 ) @ V2l1 ) @ V3l2 ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2ELIST__REL__NIL,axiom,
! [A_27a: del,A_27b: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) )
=> ! [V1y: $i] :
( ( mem @ V1y @ ( ty_2Elist_2Elist @ A_27b ) )
=> ! [V2x: $i] :
( ( mem @ V2x @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Elist_2ELIST__REL @ A_27a @ A_27b ) @ V0R ) @ ( c_2Elist_2ENIL @ A_27a ) ) @ V1y ) )
<=> ( V1y
= ( c_2Elist_2ENIL @ A_27b ) ) )
& ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Elist_2ELIST__REL @ A_27a @ A_27b ) @ V0R ) @ V2x ) @ ( c_2Elist_2ENIL @ A_27b ) ) )
<=> ( V2x
= ( c_2Elist_2ENIL @ A_27a ) ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2ELIST__REL__CONS1,axiom,
! [A_27a: del,A_27b: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) )
=> ! [V1h: $i] :
( ( mem @ V1h @ A_27a )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V3xs: $i] :
( ( mem @ V3xs @ ( ty_2Elist_2Elist @ A_27b ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Elist_2ELIST__REL @ A_27a @ A_27b ) @ V0R ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V1h ) @ V2t ) ) @ V3xs ) )
<=> ? [V4h_27: $i] :
( ( mem @ V4h_27 @ A_27b )
& ? [V5t_27: $i] :
( ( mem @ V5t_27 @ ( ty_2Elist_2Elist @ A_27b ) )
& ( V3xs
= ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27b ) @ V4h_27 ) @ V5t_27 ) )
& ( p @ ( ap @ ( ap @ V0R @ V1h ) @ V4h_27 ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( c_2Elist_2ELIST__REL @ A_27a @ A_27b ) @ V0R ) @ V2t ) @ V5t_27 ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2ELIST__REL__CONS2,axiom,
! [A_27a: del,A_27b: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) )
=> ! [V1xs: $i] :
( ( mem @ V1xs @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V2h: $i] :
( ( mem @ V2h @ A_27b )
=> ! [V3t: $i] :
( ( mem @ V3t @ ( ty_2Elist_2Elist @ A_27b ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Elist_2ELIST__REL @ A_27a @ A_27b ) @ V0R ) @ V1xs ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27b ) @ V2h ) @ V3t ) ) )
<=> ? [V4h_27: $i] :
( ( mem @ V4h_27 @ A_27a )
& ? [V5t_27: $i] :
( ( mem @ V5t_27 @ ( ty_2Elist_2Elist @ A_27a ) )
& ( V1xs
= ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V4h_27 ) @ V5t_27 ) )
& ( p @ ( ap @ ( ap @ V0R @ V4h_27 ) @ V2h ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( c_2Elist_2ELIST__REL @ A_27a @ A_27b ) @ V0R ) @ V5t_27 ) @ V3t ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2ELIST__REL__CONJ,axiom,
! [A_27a: del,A_27b: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) )
=> ! [V1Q: $i] :
( ( mem @ V1Q @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) )
=> ! [V2l1: $i] :
( ( mem @ V2l1 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V3l2: $i] :
( ( mem @ V3l2 @ ( ty_2Elist_2Elist @ A_27b ) )
=> ( ( p
@ ( ap
@ ( ap
@ ( ap @ ( c_2Elist_2ELIST__REL @ A_27a @ A_27b )
@ ( lam @ A_27a
@ ^ [V4a: $i] :
( lam @ A_27b
@ ^ [V5b: $i] : ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ V0P @ V4a ) @ V5b ) ) @ ( ap @ ( ap @ V1Q @ V4a ) @ V5b ) ) ) ) )
@ V2l1 )
@ V3l2 ) )
<=> ( ( p
@ ( ap
@ ( ap
@ ( ap @ ( c_2Elist_2ELIST__REL @ A_27a @ A_27b )
@ ( lam @ A_27a
@ ^ [V6a: $i] :
( lam @ A_27b
@ ^ [V7b: $i] : ( ap @ ( ap @ V0P @ V6a ) @ V7b ) ) ) )
@ V2l1 )
@ V3l2 ) )
& ( p
@ ( ap
@ ( ap
@ ( ap @ ( c_2Elist_2ELIST__REL @ A_27a @ A_27b )
@ ( lam @ A_27a
@ ^ [V8a: $i] :
( lam @ A_27b
@ ^ [V9b: $i] : ( ap @ ( ap @ V1Q @ V8a ) @ V9b ) ) ) )
@ V2l1 )
@ V3l2 ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2ELIST__REL__MAP1,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27c @ A_27a ) )
=> ! [V2l1: $i] :
( ( mem @ V2l1 @ ( ty_2Elist_2Elist @ A_27c ) )
=> ! [V3l2: $i] :
( ( mem @ V3l2 @ ( ty_2Elist_2Elist @ A_27b ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Elist_2ELIST__REL @ A_27a @ A_27b ) @ V0R ) @ ( ap @ ( ap @ ( c_2Elist_2EMAP @ A_27c @ A_27a ) @ V1f ) @ V2l1 ) ) @ V3l2 ) )
<=> ( p @ ( ap @ ( ap @ ( ap @ ( c_2Elist_2ELIST__REL @ A_27c @ A_27b ) @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ A_27c @ ( arr @ A_27b @ bool ) @ A_27a ) @ V0R ) @ V1f ) ) @ V2l1 ) @ V3l2 ) ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2ELIST__REL__MAP2,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) )
=> ! [V1l1: $i] :
( ( mem @ V1l1 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V2f: $i] :
( ( mem @ V2f @ ( arr @ A_27c @ A_27b ) )
=> ! [V3l2: $i] :
( ( mem @ V3l2 @ ( ty_2Elist_2Elist @ A_27c ) )
=> ( ( p
@ ( ap
@ ( ap
@ ( ap @ ( c_2Elist_2ELIST__REL @ A_27a @ A_27b )
@ ( lam @ A_27a
@ ^ [V4a: $i] :
( lam @ A_27b
@ ^ [V5b: $i] : ( ap @ ( ap @ V0R @ V4a ) @ V5b ) ) ) )
@ V1l1 )
@ ( ap @ ( ap @ ( c_2Elist_2EMAP @ A_27c @ A_27b ) @ V2f ) @ V3l2 ) ) )
<=> ( p
@ ( ap
@ ( ap
@ ( ap @ ( c_2Elist_2ELIST__REL @ A_27a @ A_27c )
@ ( lam @ A_27a
@ ^ [V6a: $i] :
( lam @ A_27c
@ ^ [V7b: $i] : ( ap @ ( ap @ V0R @ V6a ) @ ( ap @ V2f @ V7b ) ) ) ) )
@ V1l1 )
@ V3l2 ) ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2ELIST__REL__LENGTH,axiom,
! [A_27a: del,A_27b: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V2y: $i] :
( ( mem @ V2y @ ( ty_2Elist_2Elist @ A_27b ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Elist_2ELIST__REL @ A_27a @ A_27b ) @ V0R ) @ V1x ) @ V2y ) )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ V1x ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Elist_2ELENGTH @ A_27b ) @ V2y ) ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2ELIST__REL__SPLIT1,axiom,
! [A_27a: del,A_27b: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) )
=> ! [V1xs2: $i] :
( ( mem @ V1xs2 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V2xs1: $i] :
( ( mem @ V2xs1 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V3zs: $i] :
( ( mem @ V3zs @ ( ty_2Elist_2Elist @ A_27b ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Elist_2ELIST__REL @ A_27a @ A_27b ) @ V0P ) @ ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ V2xs1 ) @ V1xs2 ) ) @ V3zs ) )
<=> ? [V4ys1: $i] :
( ( mem @ V4ys1 @ ( ty_2Elist_2Elist @ A_27b ) )
& ? [V5ys2: $i] :
( ( mem @ V5ys2 @ ( ty_2Elist_2Elist @ A_27b ) )
& ( V3zs
= ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27b ) @ V4ys1 ) @ V5ys2 ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( c_2Elist_2ELIST__REL @ A_27a @ A_27b ) @ V0P ) @ V2xs1 ) @ V4ys1 ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( c_2Elist_2ELIST__REL @ A_27a @ A_27b ) @ V0P ) @ V1xs2 ) @ V5ys2 ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2ELIST__REL__SPLIT2,axiom,
! [A_27a: del,A_27b: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27b @ ( arr @ A_27a @ bool ) ) )
=> ! [V1xs2: $i] :
( ( mem @ V1xs2 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V2xs1: $i] :
( ( mem @ V2xs1 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V3zs: $i] :
( ( mem @ V3zs @ ( ty_2Elist_2Elist @ A_27b ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Elist_2ELIST__REL @ A_27b @ A_27a ) @ V0P ) @ V3zs ) @ ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ V2xs1 ) @ V1xs2 ) ) )
<=> ? [V4ys1: $i] :
( ( mem @ V4ys1 @ ( ty_2Elist_2Elist @ A_27b ) )
& ? [V5ys2: $i] :
( ( mem @ V5ys2 @ ( ty_2Elist_2Elist @ A_27b ) )
& ( V3zs
= ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27b ) @ V4ys1 ) @ V5ys2 ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( c_2Elist_2ELIST__REL @ A_27b @ A_27a ) @ V0P ) @ V4ys1 ) @ V2xs1 ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( c_2Elist_2ELIST__REL @ A_27b @ A_27a ) @ V0P ) @ V5ys2 ) @ V1xs2 ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2Elist__size__cong,axiom,
! [A_27a: del,V0M: $i] :
( ( mem @ V0M @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V1N: $i] :
( ( mem @ V1N @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V2f: $i] :
( ( mem @ V2f @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V3f_27: $i] :
( ( mem @ V3f_27 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( ( ( V0M = V1N )
& ! [V4x: $i] :
( ( mem @ V4x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V4x ) @ ( ap @ ( c_2Elist_2ELIST__TO__SET @ A_27a ) @ V1N ) ) )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ V2f @ V4x ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ V3f_27 @ V4x ) ) ) ) ) )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ ( c_2Elist_2Elist__size @ A_27a ) @ V2f ) @ V0M ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ ( c_2Elist_2Elist__size @ A_27a ) @ V3f_27 ) @ V1N ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2EFOLDR__CONG,axiom,
! [A_27a: del,A_27b: del,V0l: $i] :
( ( mem @ V0l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V1l_27: $i] :
( ( mem @ V1l_27 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V2b: $i] :
( ( mem @ V2b @ A_27b )
=> ! [V3b_27: $i] :
( ( mem @ V3b_27 @ A_27b )
=> ! [V4f: $i] :
( ( mem @ V4f @ ( arr @ A_27a @ ( arr @ A_27b @ A_27b ) ) )
=> ! [V5f_27: $i] :
( ( mem @ V5f_27 @ ( arr @ A_27a @ ( arr @ A_27b @ A_27b ) ) )
=> ( ( ( V0l = V1l_27 )
& ( V2b = V3b_27 )
& ! [V6x: $i] :
( ( mem @ V6x @ A_27a )
=> ! [V7a: $i] :
( ( mem @ V7a @ A_27b )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V6x ) @ ( ap @ ( c_2Elist_2ELIST__TO__SET @ A_27a ) @ V1l_27 ) ) )
=> ( ( ap @ ( ap @ V4f @ V6x ) @ V7a )
= ( ap @ ( ap @ V5f_27 @ V6x ) @ V7a ) ) ) ) ) )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Elist_2EFOLDR @ A_27a @ A_27b ) @ V4f ) @ V2b ) @ V0l )
= ( ap @ ( ap @ ( ap @ ( c_2Elist_2EFOLDR @ A_27a @ A_27b ) @ V5f_27 ) @ V3b_27 ) @ V1l_27 ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2EFOLDL__CONG,axiom,
! [A_27a: del,A_27b: del,V0l: $i] :
( ( mem @ V0l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V1l_27: $i] :
( ( mem @ V1l_27 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V2b: $i] :
( ( mem @ V2b @ A_27b )
=> ! [V3b_27: $i] :
( ( mem @ V3b_27 @ A_27b )
=> ! [V4f: $i] :
( ( mem @ V4f @ ( arr @ A_27b @ ( arr @ A_27a @ A_27b ) ) )
=> ! [V5f_27: $i] :
( ( mem @ V5f_27 @ ( arr @ A_27b @ ( arr @ A_27a @ A_27b ) ) )
=> ( ( ( V0l = V1l_27 )
& ( V2b = V3b_27 )
& ! [V6x: $i] :
( ( mem @ V6x @ A_27a )
=> ! [V7a: $i] :
( ( mem @ V7a @ A_27b )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V6x ) @ ( ap @ ( c_2Elist_2ELIST__TO__SET @ A_27a ) @ V1l_27 ) ) )
=> ( ( ap @ ( ap @ V4f @ V7a ) @ V6x )
= ( ap @ ( ap @ V5f_27 @ V7a ) @ V6x ) ) ) ) ) )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Elist_2EFOLDL @ A_27a @ A_27b ) @ V4f ) @ V2b ) @ V0l )
= ( ap @ ( ap @ ( ap @ ( c_2Elist_2EFOLDL @ A_27a @ A_27b ) @ V5f_27 ) @ V3b_27 ) @ V1l_27 ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2EMAP__CONG,axiom,
! [A_27a: del,A_27b: del,V0l1: $i] :
( ( mem @ V0l1 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V1l2: $i] :
( ( mem @ V1l2 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V2f: $i] :
( ( mem @ V2f @ ( arr @ A_27a @ A_27b ) )
=> ! [V3f_27: $i] :
( ( mem @ V3f_27 @ ( arr @ A_27a @ A_27b ) )
=> ( ( ( V0l1 = V1l2 )
& ! [V4x: $i] :
( ( mem @ V4x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V4x ) @ ( ap @ ( c_2Elist_2ELIST__TO__SET @ A_27a ) @ V1l2 ) ) )
=> ( ( ap @ V2f @ V4x )
= ( ap @ V3f_27 @ V4x ) ) ) ) )
=> ( ( ap @ ( ap @ ( c_2Elist_2EMAP @ A_27a @ A_27b ) @ V2f ) @ V0l1 )
= ( ap @ ( ap @ ( c_2Elist_2EMAP @ A_27a @ A_27b ) @ V3f_27 ) @ V1l2 ) ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2EMAP2__CONG,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0l1: $i] :
( ( mem @ V0l1 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V1l1_27: $i] :
( ( mem @ V1l1_27 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V2l2: $i] :
( ( mem @ V2l2 @ ( ty_2Elist_2Elist @ A_27b ) )
=> ! [V3l2_27: $i] :
( ( mem @ V3l2_27 @ ( ty_2Elist_2Elist @ A_27b ) )
=> ! [V4f: $i] :
( ( mem @ V4f @ ( arr @ A_27a @ ( arr @ A_27b @ A_27c ) ) )
=> ! [V5f_27: $i] :
( ( mem @ V5f_27 @ ( arr @ A_27a @ ( arr @ A_27b @ A_27c ) ) )
=> ( ( ( V0l1 = V1l1_27 )
& ( V2l2 = V3l2_27 )
& ! [V6x: $i] :
( ( mem @ V6x @ A_27a )
=> ! [V7y: $i] :
( ( mem @ V7y @ A_27b )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V6x ) @ ( ap @ ( c_2Elist_2ELIST__TO__SET @ A_27a ) @ V1l1_27 ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27b ) @ V7y ) @ ( ap @ ( c_2Elist_2ELIST__TO__SET @ A_27b ) @ V3l2_27 ) ) ) )
=> ( ( ap @ ( ap @ V4f @ V6x ) @ V7y )
= ( ap @ ( ap @ V5f_27 @ V6x ) @ V7y ) ) ) ) ) )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Elist_2EMAP2 @ A_27c @ A_27a @ A_27b ) @ V4f ) @ V0l1 ) @ V2l2 )
= ( ap @ ( ap @ ( ap @ ( c_2Elist_2EMAP2 @ A_27c @ A_27a @ A_27b ) @ V5f_27 ) @ V1l1_27 ) @ V3l2_27 ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2EEXISTS__CONG,axiom,
! [A_27a: del,V0l1: $i] :
( ( mem @ V0l1 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V1l2: $i] :
( ( mem @ V1l2 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V2P: $i] :
( ( mem @ V2P @ ( arr @ A_27a @ bool ) )
=> ! [V3P_27: $i] :
( ( mem @ V3P_27 @ ( arr @ A_27a @ bool ) )
=> ( ( ( V0l1 = V1l2 )
& ! [V4x: $i] :
( ( mem @ V4x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V4x ) @ ( ap @ ( c_2Elist_2ELIST__TO__SET @ A_27a ) @ V1l2 ) ) )
=> ( ( p @ ( ap @ V2P @ V4x ) )
<=> ( p @ ( ap @ V3P_27 @ V4x ) ) ) ) ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Elist_2EEXISTS @ A_27a ) @ V2P ) @ V0l1 ) )
<=> ( p @ ( ap @ ( ap @ ( c_2Elist_2EEXISTS @ A_27a ) @ V3P_27 ) @ V1l2 ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2EEVERY__CONG,axiom,
! [A_27a: del,V0l1: $i] :
( ( mem @ V0l1 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V1l2: $i] :
( ( mem @ V1l2 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V2P: $i] :
( ( mem @ V2P @ ( arr @ A_27a @ bool ) )
=> ! [V3P_27: $i] :
( ( mem @ V3P_27 @ ( arr @ A_27a @ bool ) )
=> ( ( ( V0l1 = V1l2 )
& ! [V4x: $i] :
( ( mem @ V4x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V4x ) @ ( ap @ ( c_2Elist_2ELIST__TO__SET @ A_27a ) @ V1l2 ) ) )
=> ( ( p @ ( ap @ V2P @ V4x ) )
<=> ( p @ ( ap @ V3P_27 @ V4x ) ) ) ) ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Elist_2EEVERY @ A_27a ) @ V2P ) @ V0l1 ) )
<=> ( p @ ( ap @ ( ap @ ( c_2Elist_2EEVERY @ A_27a ) @ V3P_27 ) @ V1l2 ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2EEVERY__MONOTONIC,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ! [V1Q: $i] :
( ( mem @ V1Q @ ( arr @ A_27a @ bool ) )
=> ( ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ( ( p @ ( ap @ V0P @ V2x ) )
=> ( p @ ( ap @ V1Q @ V2x ) ) ) )
=> ! [V3l: $i] :
( ( mem @ V3l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Elist_2EEVERY @ A_27a ) @ V0P ) @ V3l ) )
=> ( p @ ( ap @ ( ap @ ( c_2Elist_2EEVERY @ A_27a ) @ V1Q ) @ V3l ) ) ) ) ) ) ) ).
thf(ax_thm_2Elist_2EZIP__def,axiom,
! [A_27a: del,A_27b: del] :
( ! [V0l2: $i] :
( ( mem @ V0l2 @ ( ty_2Elist_2Elist @ A_27b ) )
=> ( ( ap @ ( c_2Elist_2EZIP @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Elist_2Elist @ A_27a ) @ ( ty_2Elist_2Elist @ A_27b ) ) @ ( c_2Elist_2ENIL @ A_27a ) ) @ V0l2 ) )
= ( c_2Elist_2ENIL @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) ) ) )
& ! [V1l1: $i] :
( ( mem @ V1l1 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( ap @ ( c_2Elist_2EZIP @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Elist_2Elist @ A_27a ) @ ( ty_2Elist_2Elist @ A_27b ) ) @ V1l1 ) @ ( c_2Elist_2ENIL @ A_27b ) ) )
= ( c_2Elist_2ENIL @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) ) ) )
& ! [V2x1: $i] :
( ( mem @ V2x1 @ A_27a )
=> ! [V3l1: $i] :
( ( mem @ V3l1 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V4x2: $i] :
( ( mem @ V4x2 @ A_27b )
=> ! [V5l2: $i] :
( ( mem @ V5l2 @ ( ty_2Elist_2Elist @ A_27b ) )
=> ( ( ap @ ( c_2Elist_2EZIP @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Elist_2Elist @ A_27a ) @ ( ty_2Elist_2Elist @ A_27b ) ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V2x1 ) @ V3l1 ) ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27b ) @ V4x2 ) @ V5l2 ) ) )
= ( ap @ ( ap @ ( c_2Elist_2ECONS @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27b ) @ V2x1 ) @ V4x2 ) ) @ ( ap @ ( c_2Elist_2EZIP @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Elist_2Elist @ A_27a ) @ ( ty_2Elist_2Elist @ A_27b ) ) @ V3l1 ) @ V5l2 ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2EZIP,axiom,
! [A_27a: del,A_27b: del,A_27c: del,A_27d: del] :
( ( ( ap @ ( c_2Elist_2EZIP @ A_27c @ A_27d ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Elist_2Elist @ A_27c ) @ ( ty_2Elist_2Elist @ A_27d ) ) @ ( c_2Elist_2ENIL @ A_27c ) ) @ ( c_2Elist_2ENIL @ A_27d ) ) )
= ( c_2Elist_2ENIL @ ( ty_2Epair_2Eprod @ A_27c @ A_27d ) ) )
& ! [V0x1: $i] :
( ( mem @ V0x1 @ A_27a )
=> ! [V1l1: $i] :
( ( mem @ V1l1 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V2x2: $i] :
( ( mem @ V2x2 @ A_27b )
=> ! [V3l2: $i] :
( ( mem @ V3l2 @ ( ty_2Elist_2Elist @ A_27b ) )
=> ( ( ap @ ( c_2Elist_2EZIP @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Elist_2Elist @ A_27a ) @ ( ty_2Elist_2Elist @ A_27b ) ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V0x1 ) @ V1l1 ) ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27b ) @ V2x2 ) @ V3l2 ) ) )
= ( ap @ ( ap @ ( c_2Elist_2ECONS @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27b ) @ V0x1 ) @ V2x2 ) ) @ ( ap @ ( c_2Elist_2EZIP @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Elist_2Elist @ A_27a ) @ ( ty_2Elist_2Elist @ A_27b ) ) @ V1l1 ) @ V3l2 ) ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Elist_2EUNZIP,axiom,
! [A_27a: del,A_27b: del] :
( ( ( ap @ ( c_2Elist_2EUNZIP @ A_27a @ A_27b ) @ ( c_2Elist_2ENIL @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) ) )
= ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Elist_2Elist @ A_27a ) @ ( ty_2Elist_2Elist @ A_27b ) ) @ ( c_2Elist_2ENIL @ A_27a ) ) @ ( c_2Elist_2ENIL @ A_27b ) ) )
& ! [V0x: $i] :
( ( mem @ V0x @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) )
=> ! [V1l: $i] :
( ( mem @ V1l @ ( ty_2Elist_2Elist @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) ) )
=> ( ( ap @ ( c_2Elist_2EUNZIP @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) ) @ V0x ) @ V1l ) )
= ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Elist_2Elist @ A_27a ) @ ( ty_2Elist_2Elist @ A_27b ) ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ ( ap @ ( c_2Epair_2EFST @ A_27a @ A_27b ) @ V0x ) ) @ ( ap @ ( c_2Epair_2EFST @ ( ty_2Elist_2Elist @ A_27a ) @ ( ty_2Elist_2Elist @ A_27b ) ) @ ( ap @ ( c_2Elist_2EUNZIP @ A_27a @ A_27b ) @ V1l ) ) ) ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27b ) @ ( ap @ ( c_2Epair_2ESND @ A_27a @ A_27b ) @ V0x ) ) @ ( ap @ ( c_2Epair_2ESND @ ( ty_2Elist_2Elist @ A_27a ) @ ( ty_2Elist_2Elist @ A_27b ) ) @ ( ap @ ( c_2Elist_2EUNZIP @ A_27a @ A_27b ) @ V1l ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2EUNZIP__THM,axiom,
! [A_27a: del,A_27b: del,V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ! [V1y: $i] :
( ( mem @ V1y @ A_27b )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( ty_2Elist_2Elist @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) ) )
=> ( ( ( ap @ ( c_2Elist_2EUNZIP @ A_27a @ A_27b ) @ ( c_2Elist_2ENIL @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) ) )
= ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Elist_2Elist @ A_27a ) @ ( ty_2Elist_2Elist @ A_27b ) ) @ ( c_2Elist_2ENIL @ A_27a ) ) @ ( c_2Elist_2ENIL @ A_27b ) ) )
& ( ( ap @ ( c_2Elist_2EUNZIP @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27b ) @ V0x ) @ V1y ) ) @ V2t ) )
= ( ap
@ ( ap @ ( c_2Ebool_2ELET @ ( ty_2Epair_2Eprod @ ( ty_2Elist_2Elist @ A_27a ) @ ( ty_2Elist_2Elist @ A_27b ) ) @ ( ty_2Epair_2Eprod @ ( ty_2Elist_2Elist @ A_27a ) @ ( ty_2Elist_2Elist @ A_27b ) ) )
@ ( ap @ ( c_2Epair_2EUNCURRY @ ( ty_2Elist_2Elist @ A_27a ) @ ( ty_2Elist_2Elist @ A_27b ) @ ( ty_2Epair_2Eprod @ ( ty_2Elist_2Elist @ A_27a ) @ ( ty_2Elist_2Elist @ A_27b ) ) )
@ ( lam @ ( ty_2Elist_2Elist @ A_27a )
@ ^ [V3L1: $i] :
( lam @ ( ty_2Elist_2Elist @ A_27b )
@ ^ [V4L2: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Elist_2Elist @ A_27a ) @ ( ty_2Elist_2Elist @ A_27b ) ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V0x ) @ V3L1 ) ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27b ) @ V1y ) @ V4L2 ) ) ) ) ) )
@ ( ap @ ( c_2Elist_2EUNZIP @ A_27a @ A_27b ) @ V2t ) ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2EUNZIP__MAP,axiom,
! [A_27a: del,A_27b: del,V0L: $i] :
( ( mem @ V0L @ ( ty_2Elist_2Elist @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) ) )
=> ( ( ap @ ( c_2Elist_2EUNZIP @ A_27a @ A_27b ) @ V0L )
= ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Elist_2Elist @ A_27a ) @ ( ty_2Elist_2Elist @ A_27b ) ) @ ( ap @ ( ap @ ( c_2Elist_2EMAP @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ A_27a ) @ ( c_2Epair_2EFST @ A_27a @ A_27b ) ) @ V0L ) ) @ ( ap @ ( ap @ ( c_2Elist_2EMAP @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ A_27b ) @ ( c_2Epair_2ESND @ A_27a @ A_27b ) ) @ V0L ) ) ) ) ).
thf(conj_thm_2Elist_2ELENGTH__ZIP,axiom,
! [A_27a: del,A_27b: del,V0l1: $i] :
( ( mem @ V0l1 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V1l2: $i] :
( ( mem @ V1l2 @ ( ty_2Elist_2Elist @ A_27b ) )
=> ( ( ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ V0l1 ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Elist_2ELENGTH @ A_27b ) @ V1l2 ) ) )
=> ( ( ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Elist_2ELENGTH @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) ) @ ( ap @ ( c_2Elist_2EZIP @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Elist_2Elist @ A_27a ) @ ( ty_2Elist_2Elist @ A_27b ) ) @ V0l1 ) @ V1l2 ) ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ V0l1 ) ) )
& ( ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Elist_2ELENGTH @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) ) @ ( ap @ ( c_2Elist_2EZIP @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Elist_2Elist @ A_27a ) @ ( ty_2Elist_2Elist @ A_27b ) ) @ V0l1 ) @ V1l2 ) ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Elist_2ELENGTH @ A_27b ) @ V1l2 ) ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2ELENGTH__UNZIP,axiom,
! [A_27a: del,A_27b: del,V0pl: $i] :
( ( mem @ V0pl @ ( ty_2Elist_2Elist @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) ) )
=> ( ( ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ ( ap @ ( c_2Epair_2EFST @ ( ty_2Elist_2Elist @ A_27a ) @ ( ty_2Elist_2Elist @ A_27b ) ) @ ( ap @ ( c_2Elist_2EUNZIP @ A_27a @ A_27b ) @ V0pl ) ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Elist_2ELENGTH @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) ) @ V0pl ) ) )
& ( ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Elist_2ELENGTH @ A_27b ) @ ( ap @ ( c_2Epair_2ESND @ ( ty_2Elist_2Elist @ A_27a ) @ ( ty_2Elist_2Elist @ A_27b ) ) @ ( ap @ ( c_2Elist_2EUNZIP @ A_27a @ A_27b ) @ V0pl ) ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Elist_2ELENGTH @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) ) @ V0pl ) ) ) ) ) ).
thf(conj_thm_2Elist_2EZIP__UNZIP,axiom,
! [A_27a: del,A_27b: del,V0l: $i] :
( ( mem @ V0l @ ( ty_2Elist_2Elist @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) ) )
=> ( ( ap @ ( c_2Elist_2EZIP @ A_27a @ A_27b ) @ ( ap @ ( c_2Elist_2EUNZIP @ A_27a @ A_27b ) @ V0l ) )
= V0l ) ) ).
thf(conj_thm_2Elist_2EUNZIP__ZIP,axiom,
! [A_27a: del,A_27b: del,V0l1: $i] :
( ( mem @ V0l1 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V1l2: $i] :
( ( mem @ V1l2 @ ( ty_2Elist_2Elist @ A_27b ) )
=> ( ( ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ V0l1 ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Elist_2ELENGTH @ A_27b ) @ V1l2 ) ) )
=> ( ( ap @ ( c_2Elist_2EUNZIP @ A_27a @ A_27b ) @ ( ap @ ( c_2Elist_2EZIP @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Elist_2Elist @ A_27a ) @ ( ty_2Elist_2Elist @ A_27b ) ) @ V0l1 ) @ V1l2 ) ) )
= ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Elist_2Elist @ A_27a ) @ ( ty_2Elist_2Elist @ A_27b ) ) @ V0l1 ) @ V1l2 ) ) ) ) ) ).
thf(conj_thm_2Elist_2EZIP__MAP,axiom,
! [A_27a: del,A_27b: del,A_27c: del,A_27d: del,V0l1: $i] :
( ( mem @ V0l1 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V1l2: $i] :
( ( mem @ V1l2 @ ( ty_2Elist_2Elist @ A_27b ) )
=> ! [V2f1: $i] :
( ( mem @ V2f1 @ ( arr @ A_27a @ A_27c ) )
=> ! [V3f2: $i] :
( ( mem @ V3f2 @ ( arr @ A_27b @ A_27d ) )
=> ( ( ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ V0l1 ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Elist_2ELENGTH @ A_27b ) @ V1l2 ) ) )
=> ( ( ( ap @ ( c_2Elist_2EZIP @ A_27c @ A_27b ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Elist_2Elist @ A_27c ) @ ( ty_2Elist_2Elist @ A_27b ) ) @ ( ap @ ( ap @ ( c_2Elist_2EMAP @ A_27a @ A_27c ) @ V2f1 ) @ V0l1 ) ) @ V1l2 ) )
= ( ap
@ ( ap @ ( c_2Elist_2EMAP @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ ( ty_2Epair_2Eprod @ A_27c @ A_27b ) )
@ ( lam @ ( ty_2Epair_2Eprod @ A_27a @ A_27b )
@ ^ [V4p: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27c @ A_27b ) @ ( ap @ V2f1 @ ( ap @ ( c_2Epair_2EFST @ A_27a @ A_27b ) @ V4p ) ) ) @ ( ap @ ( c_2Epair_2ESND @ A_27a @ A_27b ) @ V4p ) ) ) )
@ ( ap @ ( c_2Elist_2EZIP @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Elist_2Elist @ A_27a ) @ ( ty_2Elist_2Elist @ A_27b ) ) @ V0l1 ) @ V1l2 ) ) ) )
& ( ( ap @ ( c_2Elist_2EZIP @ A_27a @ A_27d ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Elist_2Elist @ A_27a ) @ ( ty_2Elist_2Elist @ A_27d ) ) @ V0l1 ) @ ( ap @ ( ap @ ( c_2Elist_2EMAP @ A_27b @ A_27d ) @ V3f2 ) @ V1l2 ) ) )
= ( ap
@ ( ap @ ( c_2Elist_2EMAP @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ ( ty_2Epair_2Eprod @ A_27a @ A_27d ) )
@ ( lam @ ( ty_2Epair_2Eprod @ A_27a @ A_27b )
@ ^ [V5p: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27d ) @ ( ap @ ( c_2Epair_2EFST @ A_27a @ A_27b ) @ V5p ) ) @ ( ap @ V3f2 @ ( ap @ ( c_2Epair_2ESND @ A_27a @ A_27b ) @ V5p ) ) ) ) )
@ ( ap @ ( c_2Elist_2EZIP @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Elist_2Elist @ A_27a ) @ ( ty_2Elist_2Elist @ A_27b ) ) @ V0l1 ) @ V1l2 ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2EMEM__ZIP,axiom,
! [A_27a: del,A_27b: del,V0l1: $i] :
( ( mem @ V0l1 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V1l2: $i] :
( ( mem @ V1l2 @ ( ty_2Elist_2Elist @ A_27b ) )
=> ! [V2p: $i] :
( ( mem @ V2p @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) )
=> ( ( ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ V0l1 ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Elist_2ELENGTH @ A_27b ) @ V1l2 ) ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) ) @ V2p ) @ ( ap @ ( c_2Elist_2ELIST__TO__SET @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) ) @ ( ap @ ( c_2Elist_2EZIP @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Elist_2Elist @ A_27a ) @ ( ty_2Elist_2Elist @ A_27b ) ) @ V0l1 ) @ V1l2 ) ) ) ) )
<=> ? [V3n: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V3n ) ) @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ V0l1 ) ) )
& ( V2p
= ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Elist_2EEL @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V3n ) ) @ V0l1 ) ) @ ( ap @ ( ap @ ( c_2Elist_2EEL @ A_27b ) @ ( inj__ty_2Enum_2Enum @ V3n ) ) @ V1l2 ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2EEL__ZIP,axiom,
! [A_27a: del,A_27b: del,V0l1: $i] :
( ( mem @ V0l1 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V1l2: $i] :
( ( mem @ V1l2 @ ( ty_2Elist_2Elist @ A_27b ) )
=> ! [V2n: tp__ty_2Enum_2Enum] :
( ( ( ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ V0l1 ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Elist_2ELENGTH @ A_27b ) @ V1l2 ) ) )
& ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V2n ) ) @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ V0l1 ) ) ) )
=> ( ( ap @ ( ap @ ( c_2Elist_2EEL @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) @ ( ap @ ( c_2Elist_2EZIP @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Elist_2Elist @ A_27a ) @ ( ty_2Elist_2Elist @ A_27b ) ) @ V0l1 ) @ V1l2 ) ) )
= ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Elist_2EEL @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) @ V0l1 ) ) @ ( ap @ ( ap @ ( c_2Elist_2EEL @ A_27b ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) @ V1l2 ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2EMAP2__ZIP,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0l1: $i] :
( ( mem @ V0l1 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V1l2: $i] :
( ( mem @ V1l2 @ ( ty_2Elist_2Elist @ A_27b ) )
=> ( ( ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ V0l1 ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Elist_2ELENGTH @ A_27b ) @ V1l2 ) ) )
=> ! [V2f: $i] :
( ( mem @ V2f @ ( arr @ A_27a @ ( arr @ A_27b @ A_27c ) ) )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Elist_2EMAP2 @ A_27c @ A_27a @ A_27b ) @ V2f ) @ V0l1 ) @ V1l2 )
= ( ap @ ( ap @ ( c_2Elist_2EMAP @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ A_27c ) @ ( ap @ ( c_2Epair_2EUNCURRY @ A_27a @ A_27b @ A_27c ) @ V2f ) ) @ ( ap @ ( c_2Elist_2EZIP @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Elist_2Elist @ A_27a ) @ ( ty_2Elist_2Elist @ A_27b ) ) @ V0l1 ) @ V1l2 ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2EMAP2__MAP,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0l1: $i] :
( ( mem @ V0l1 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V1l2: $i] :
( ( mem @ V1l2 @ ( ty_2Elist_2Elist @ A_27b ) )
=> ( ( ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ V0l1 ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Elist_2ELENGTH @ A_27b ) @ V1l2 ) ) )
=> ! [V2f: $i] :
( ( mem @ V2f @ ( arr @ A_27a @ ( arr @ A_27b @ A_27c ) ) )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Elist_2EMAP2 @ A_27c @ A_27a @ A_27b ) @ V2f ) @ V0l1 ) @ V1l2 )
= ( ap @ ( ap @ ( c_2Elist_2EMAP @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ A_27c ) @ ( ap @ ( c_2Epair_2EUNCURRY @ A_27a @ A_27b @ A_27c ) @ V2f ) ) @ ( ap @ ( c_2Elist_2EZIP @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Elist_2Elist @ A_27a ) @ ( ty_2Elist_2Elist @ A_27b ) ) @ V0l1 ) @ V1l2 ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2EMAP__ZIP,axiom,
! [A_27a: del,A_27b: del,A_27c: del,A_27d: del,V0l1: $i] :
( ( mem @ V0l1 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V1l2: $i] :
( ( mem @ V1l2 @ ( ty_2Elist_2Elist @ A_27b ) )
=> ! [V2f: $i] :
( ( mem @ V2f @ ( arr @ A_27a @ A_27c ) )
=> ! [V3g: $i] :
( ( mem @ V3g @ ( arr @ A_27b @ A_27d ) )
=> ( ( ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ V0l1 ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Elist_2ELENGTH @ A_27b ) @ V1l2 ) ) )
=> ( ( ( ap @ ( ap @ ( c_2Elist_2EMAP @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ A_27a ) @ ( c_2Epair_2EFST @ A_27a @ A_27b ) ) @ ( ap @ ( c_2Elist_2EZIP @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Elist_2Elist @ A_27a ) @ ( ty_2Elist_2Elist @ A_27b ) ) @ V0l1 ) @ V1l2 ) ) )
= V0l1 )
& ( ( ap @ ( ap @ ( c_2Elist_2EMAP @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ A_27b ) @ ( c_2Epair_2ESND @ A_27a @ A_27b ) ) @ ( ap @ ( c_2Elist_2EZIP @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Elist_2Elist @ A_27a ) @ ( ty_2Elist_2Elist @ A_27b ) ) @ V0l1 ) @ V1l2 ) ) )
= V1l2 )
& ( ( ap @ ( ap @ ( c_2Elist_2EMAP @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ A_27c ) @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ A_27c @ A_27a ) @ V2f ) @ ( c_2Epair_2EFST @ A_27a @ A_27b ) ) ) @ ( ap @ ( c_2Elist_2EZIP @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Elist_2Elist @ A_27a ) @ ( ty_2Elist_2Elist @ A_27b ) ) @ V0l1 ) @ V1l2 ) ) )
= ( ap @ ( ap @ ( c_2Elist_2EMAP @ A_27a @ A_27c ) @ V2f ) @ V0l1 ) )
& ( ( ap @ ( ap @ ( c_2Elist_2EMAP @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ A_27d ) @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ A_27d @ A_27b ) @ V3g ) @ ( c_2Epair_2ESND @ A_27a @ A_27b ) ) ) @ ( ap @ ( c_2Elist_2EZIP @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Elist_2Elist @ A_27a ) @ ( ty_2Elist_2Elist @ A_27b ) ) @ V0l1 ) @ V1l2 ) ) )
= ( ap @ ( ap @ ( c_2Elist_2EMAP @ A_27b @ A_27d ) @ V3g ) @ V1l2 ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2EMEM__EL,axiom,
! [A_27a: del,V0l: $i] :
( ( mem @ V0l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V1x ) @ ( ap @ ( c_2Elist_2ELIST__TO__SET @ A_27a ) @ V0l ) ) )
<=> ? [V2n: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V2n ) ) @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ V0l ) ) )
& ( V1x
= ( ap @ ( ap @ ( c_2Elist_2EEL @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) @ V0l ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2ESUM__MAP__PLUS__ZIP,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ ty_2Enum_2Enum ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V2h: $i] :
( ( mem @ V2h @ ( arr @ A_27b @ ty_2Enum_2Enum ) )
=> ! [V3ls1: $i] :
( ( mem @ V3ls1 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V4ls2: $i] :
( ( mem @ V4ls2 @ ( ty_2Elist_2Elist @ A_27b ) )
=> ( ( ( ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ V3ls1 ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Elist_2ELENGTH @ A_27b ) @ V4ls2 ) ) )
& ! [V5x: $i] :
( ( mem @ V5x @ A_27a )
=> ! [V6y: $i] :
( ( mem @ V6y @ A_27b )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ V0f @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27b ) @ V5x ) @ V6y ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( ap @ V1g @ V5x ) ) @ ( ap @ V2h @ V6y ) ) ) ) ) ) )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ c_2Elist_2ESUM @ ( ap @ ( ap @ ( c_2Elist_2EMAP @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ ty_2Enum_2Enum ) @ V0f ) @ ( ap @ ( c_2Elist_2EZIP @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Elist_2Elist @ A_27a ) @ ( ty_2Elist_2Elist @ A_27b ) ) @ V3ls1 ) @ V4ls2 ) ) ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( ap @ c_2Elist_2ESUM @ ( ap @ ( ap @ ( c_2Elist_2EMAP @ A_27a @ ty_2Enum_2Enum ) @ V1g ) @ V3ls1 ) ) ) @ ( ap @ c_2Elist_2ESUM @ ( ap @ ( ap @ ( c_2Elist_2EMAP @ A_27b @ ty_2Enum_2Enum ) @ V2h ) @ V4ls2 ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2ELIST__REL__EVERY__ZIP,axiom,
! [A_27a: del,A_27b: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) )
=> ! [V1l1: $i] :
( ( mem @ V1l1 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V2l2: $i] :
( ( mem @ V2l2 @ ( ty_2Elist_2Elist @ A_27b ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Elist_2ELIST__REL @ A_27a @ A_27b ) @ V0R ) @ V1l1 ) @ V2l2 ) )
<=> ( ( ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ V1l1 ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Elist_2ELENGTH @ A_27b ) @ V2l2 ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Elist_2EEVERY @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) ) @ ( ap @ ( c_2Epair_2EUNCURRY @ A_27a @ A_27b @ bool ) @ V0R ) ) @ ( ap @ ( c_2Elist_2EZIP @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Elist_2Elist @ A_27a ) @ ( ty_2Elist_2Elist @ A_27b ) ) @ V1l1 ) @ V2l2 ) ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Elist_2EREVERSE__DEF,axiom,
! [A_27a: del] :
( ( ( ap @ ( c_2Elist_2EREVERSE @ A_27a ) @ ( c_2Elist_2ENIL @ A_27a ) )
= ( c_2Elist_2ENIL @ A_27a ) )
& ! [V0h: $i] :
( ( mem @ V0h @ A_27a )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( ap @ ( c_2Elist_2EREVERSE @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V0h ) @ V1t ) )
= ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ ( ap @ ( c_2Elist_2EREVERSE @ A_27a ) @ V1t ) ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V0h ) @ ( c_2Elist_2ENIL @ A_27a ) ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2EREVERSE__APPEND,axiom,
! [A_27a: del,V0l1: $i] :
( ( mem @ V0l1 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V1l2: $i] :
( ( mem @ V1l2 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( ap @ ( c_2Elist_2EREVERSE @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ V0l1 ) @ V1l2 ) )
= ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ ( ap @ ( c_2Elist_2EREVERSE @ A_27a ) @ V1l2 ) ) @ ( ap @ ( c_2Elist_2EREVERSE @ A_27a ) @ V0l1 ) ) ) ) ) ).
thf(conj_thm_2Elist_2EREVERSE__REVERSE,axiom,
! [A_27a: del,V0l: $i] :
( ( mem @ V0l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( ap @ ( c_2Elist_2EREVERSE @ A_27a ) @ ( ap @ ( c_2Elist_2EREVERSE @ A_27a ) @ V0l ) )
= V0l ) ) ).
thf(conj_thm_2Elist_2EREVERSE__11,axiom,
! [A_27a: del,V0l1: $i] :
( ( mem @ V0l1 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V1l2: $i] :
( ( mem @ V1l2 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( ( ap @ ( c_2Elist_2EREVERSE @ A_27a ) @ V0l1 )
= ( ap @ ( c_2Elist_2EREVERSE @ A_27a ) @ V1l2 ) )
<=> ( V0l1 = V1l2 ) ) ) ) ).
thf(conj_thm_2Elist_2EMEM__REVERSE,axiom,
! [A_27a: del,V0l: $i] :
( ( mem @ V0l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V1x ) @ ( ap @ ( c_2Elist_2ELIST__TO__SET @ A_27a ) @ ( ap @ ( c_2Elist_2EREVERSE @ A_27a ) @ V0l ) ) ) )
<=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V1x ) @ ( ap @ ( c_2Elist_2ELIST__TO__SET @ A_27a ) @ V0l ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2ELENGTH__REVERSE,axiom,
! [A_27a: del,V0l: $i] :
( ( mem @ V0l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ ( ap @ ( c_2Elist_2EREVERSE @ A_27a ) @ V0l ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ V0l ) ) ) ) ).
thf(conj_thm_2Elist_2EREVERSE__EQ__NIL,axiom,
! [A_27a: del,V0l: $i] :
( ( mem @ V0l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( ( ap @ ( c_2Elist_2EREVERSE @ A_27a ) @ V0l )
= ( c_2Elist_2ENIL @ A_27a ) )
<=> ( V0l
= ( c_2Elist_2ENIL @ A_27a ) ) ) ) ).
thf(conj_thm_2Elist_2EREVERSE__EQ__SING,axiom,
! [A_27a: del,V0l: $i] :
( ( mem @ V0l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V1e: $i] :
( ( mem @ V1e @ A_27a )
=> ( ( ( ap @ ( c_2Elist_2EREVERSE @ A_27a ) @ V0l )
= ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V1e ) @ ( c_2Elist_2ENIL @ A_27a ) ) )
<=> ( V0l
= ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V1e ) @ ( c_2Elist_2ENIL @ A_27a ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2EFILTER__REVERSE,axiom,
! [A_27a: del,V0l: $i] :
( ( mem @ V0l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V1P: $i] :
( ( mem @ V1P @ ( arr @ A_27a @ bool ) )
=> ( ( ap @ ( ap @ ( c_2Elist_2EFILTER @ A_27a ) @ V1P ) @ ( ap @ ( c_2Elist_2EREVERSE @ A_27a ) @ V0l ) )
= ( ap @ ( c_2Elist_2EREVERSE @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2EFILTER @ A_27a ) @ V1P ) @ V0l ) ) ) ) ) ).
thf(ax_thm_2Elist_2ELAST__DEF,axiom,
! [A_27a: del,V0h: $i] :
( ( mem @ V0h @ A_27a )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( ap @ ( c_2Elist_2ELAST @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V0h ) @ V1t ) )
= ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ A_27a ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ( ty_2Elist_2Elist @ A_27a ) ) @ V1t ) @ ( c_2Elist_2ENIL @ A_27a ) ) ) @ V0h ) @ ( ap @ ( c_2Elist_2ELAST @ A_27a ) @ V1t ) ) ) ) ) ).
thf(ax_thm_2Elist_2EFRONT__DEF,axiom,
! [A_27a: del,V0h: $i] :
( ( mem @ V0h @ A_27a )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( ap @ ( c_2Elist_2EFRONT @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V0h ) @ V1t ) )
= ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ( ty_2Elist_2Elist @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ( ty_2Elist_2Elist @ A_27a ) ) @ V1t ) @ ( c_2Elist_2ENIL @ A_27a ) ) ) @ ( c_2Elist_2ENIL @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V0h ) @ ( ap @ ( c_2Elist_2EFRONT @ A_27a ) @ V1t ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2ELAST__CONS,axiom,
! [A_27a: del] :
( ! [V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ( ( ap @ ( c_2Elist_2ELAST @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V0x ) @ ( c_2Elist_2ENIL @ A_27a ) ) )
= V0x ) )
& ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ! [V2y: $i] :
( ( mem @ V2y @ A_27a )
=> ! [V3z: $i] :
( ( mem @ V3z @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( ap @ ( c_2Elist_2ELAST @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V1x ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V2y ) @ V3z ) ) )
= ( ap @ ( c_2Elist_2ELAST @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V2y ) @ V3z ) ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2ELAST__EL,axiom,
! [A_27a: del,V0ls: $i] :
( ( mem @ V0ls @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( V0ls
!= ( c_2Elist_2ENIL @ A_27a ) )
=> ( ( ap @ ( c_2Elist_2ELAST @ A_27a ) @ V0ls )
= ( ap @ ( ap @ ( c_2Elist_2EEL @ A_27a ) @ ( ap @ c_2Eprim__rec_2EPRE @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ V0ls ) ) ) @ V0ls ) ) ) ) ).
thf(conj_thm_2Elist_2ELAST__MAP,axiom,
! [A_27a: del,A_27b: del,V0l: $i] :
( ( mem @ V0l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ A_27b ) )
=> ( ( V0l
!= ( c_2Elist_2ENIL @ A_27a ) )
=> ( ( ap @ ( c_2Elist_2ELAST @ A_27b ) @ ( ap @ ( ap @ ( c_2Elist_2EMAP @ A_27a @ A_27b ) @ V1f ) @ V0l ) )
= ( ap @ V1f @ ( ap @ ( c_2Elist_2ELAST @ A_27a ) @ V0l ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2EFRONT__CONS,axiom,
! [A_27a: del] :
( ! [V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ( ( ap @ ( c_2Elist_2EFRONT @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V0x ) @ ( c_2Elist_2ENIL @ A_27a ) ) )
= ( c_2Elist_2ENIL @ A_27a ) ) )
& ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ! [V2y: $i] :
( ( mem @ V2y @ A_27a )
=> ! [V3z: $i] :
( ( mem @ V3z @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( ap @ ( c_2Elist_2EFRONT @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V1x ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V2y ) @ V3z ) ) )
= ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V1x ) @ ( ap @ ( c_2Elist_2EFRONT @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V2y ) @ V3z ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2ELENGTH__FRONT__CONS,axiom,
! [A_27a: del,V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ! [V1xs: $i] :
( ( mem @ V1xs @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ ( ap @ ( c_2Elist_2EFRONT @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V0x ) @ V1xs ) ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ V1xs ) ) ) ) ) ).
thf(conj_thm_2Elist_2EFRONT__CONS__EQ__NIL,axiom,
! [A_27a: del] :
( ! [V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ! [V1xs: $i] :
( ( mem @ V1xs @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( ( ap @ ( c_2Elist_2EFRONT @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V0x ) @ V1xs ) )
= ( c_2Elist_2ENIL @ A_27a ) )
<=> ( V1xs
= ( c_2Elist_2ENIL @ A_27a ) ) ) ) )
& ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ! [V3xs: $i] :
( ( mem @ V3xs @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( ( c_2Elist_2ENIL @ A_27a )
= ( ap @ ( c_2Elist_2EFRONT @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V2x ) @ V3xs ) ) )
<=> ( V3xs
= ( c_2Elist_2ENIL @ A_27a ) ) ) ) )
& ! [V4x: $i] :
( ( mem @ V4x @ A_27a )
=> ! [V5xs: $i] :
( ( mem @ V5xs @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( p @ ( ap @ ( c_2Elist_2ENULL @ A_27a ) @ ( ap @ ( c_2Elist_2EFRONT @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V4x ) @ V5xs ) ) ) )
<=> ( p @ ( ap @ ( c_2Elist_2ENULL @ A_27a ) @ V5xs ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2EAPPEND__FRONT__LAST,axiom,
! [A_27a: del,V0l: $i] :
( ( mem @ V0l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( V0l
!= ( c_2Elist_2ENIL @ A_27a ) )
=> ( ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ ( ap @ ( c_2Elist_2EFRONT @ A_27a ) @ V0l ) ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ ( ap @ ( c_2Elist_2ELAST @ A_27a ) @ V0l ) ) @ ( c_2Elist_2ENIL @ A_27a ) ) )
= V0l ) ) ) ).
thf(conj_thm_2Elist_2ELAST__CONS__cond,axiom,
! [A_27a: del,V0h: $i] :
( ( mem @ V0h @ A_27a )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( ap @ ( c_2Elist_2ELAST @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V0h ) @ V1t ) )
= ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ A_27a ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ( ty_2Elist_2Elist @ A_27a ) ) @ V1t ) @ ( c_2Elist_2ENIL @ A_27a ) ) ) @ V0h ) @ ( ap @ ( c_2Elist_2ELAST @ A_27a ) @ V1t ) ) ) ) ) ).
thf(conj_thm_2Elist_2ELAST__APPEND__CONS,axiom,
! [A_27a: del,V0h: $i] :
( ( mem @ V0h @ A_27a )
=> ! [V1l1: $i] :
( ( mem @ V1l1 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V2l2: $i] :
( ( mem @ V2l2 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( ap @ ( c_2Elist_2ELAST @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ V1l1 ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V0h ) @ V2l2 ) ) )
= ( ap @ ( c_2Elist_2ELAST @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V0h ) @ V2l2 ) ) ) ) ) ) ).
thf(ax_thm_2Elist_2ETAKE__def,axiom,
! [A_27a: del] :
( ! [V0n: tp__ty_2Enum_2Enum] :
( ( ap @ ( ap @ ( c_2Elist_2ETAKE @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) @ ( c_2Elist_2ENIL @ A_27a ) )
= ( c_2Elist_2ENIL @ A_27a ) )
& ! [V1n: tp__ty_2Enum_2Enum,V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ! [V3xs: $i] :
( ( mem @ V3xs @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( ap @ ( ap @ ( c_2Elist_2ETAKE @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V2x ) @ V3xs ) )
= ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ( ty_2Elist_2Elist @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( c_2Elist_2ENIL @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V2x ) @ ( ap @ ( ap @ ( c_2Elist_2ETAKE @ A_27a ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ ( inj__ty_2Enum_2Enum @ V1n ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ V3xs ) ) ) ) ) ) ) ).
thf(ax_thm_2Elist_2EDROP__def,axiom,
! [A_27a: del] :
( ! [V0n: tp__ty_2Enum_2Enum] :
( ( ap @ ( ap @ ( c_2Elist_2EDROP @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) @ ( c_2Elist_2ENIL @ A_27a ) )
= ( c_2Elist_2ENIL @ A_27a ) )
& ! [V1n: tp__ty_2Enum_2Enum,V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ! [V3xs: $i] :
( ( mem @ V3xs @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( ap @ ( ap @ ( c_2Elist_2EDROP @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V2x ) @ V3xs ) )
= ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ( ty_2Elist_2Elist @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V2x ) @ V3xs ) ) @ ( ap @ ( ap @ ( c_2Elist_2EDROP @ A_27a ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ ( inj__ty_2Enum_2Enum @ V1n ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ V3xs ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2ETAKE__nil,axiom,
! [A_27a: del,V0n: tp__ty_2Enum_2Enum] :
( ( ap @ ( ap @ ( c_2Elist_2ETAKE @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) @ ( c_2Elist_2ENIL @ A_27a ) )
= ( c_2Elist_2ENIL @ A_27a ) ) ).
thf(conj_thm_2Elist_2ETAKE__cons,axiom,
! [A_27a: del,V0n: tp__ty_2Enum_2Enum,V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ! [V2xs: $i] :
( ( mem @ V2xs @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) )
=> ( ( ap @ ( ap @ ( c_2Elist_2ETAKE @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V1x ) @ V2xs ) )
= ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V1x ) @ ( ap @ ( ap @ ( c_2Elist_2ETAKE @ A_27a ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ ( inj__ty_2Enum_2Enum @ V0n ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ V2xs ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2EDROP__nil,axiom,
! [A_27a: del,V0n: tp__ty_2Enum_2Enum] :
( ( ap @ ( ap @ ( c_2Elist_2EDROP @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) @ ( c_2Elist_2ENIL @ A_27a ) )
= ( c_2Elist_2ENIL @ A_27a ) ) ).
thf(conj_thm_2Elist_2EDROP__cons,axiom,
! [A_27a: del,V0n: tp__ty_2Enum_2Enum,V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ! [V2xs: $i] :
( ( mem @ V2xs @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) )
=> ( ( ap @ ( ap @ ( c_2Elist_2EDROP @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V1x ) @ V2xs ) )
= ( ap @ ( ap @ ( c_2Elist_2EDROP @ A_27a ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ ( inj__ty_2Enum_2Enum @ V0n ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ V2xs ) ) ) ) ) ).
thf(conj_thm_2Elist_2ETAKE__0,axiom,
! [A_27a: del,V0l: $i] :
( ( mem @ V0l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( ap @ ( ap @ ( c_2Elist_2ETAKE @ A_27a ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ V0l )
= ( c_2Elist_2ENIL @ A_27a ) ) ) ).
thf(conj_thm_2Elist_2ETAKE__LENGTH__ID,axiom,
! [A_27a: del,V0l: $i] :
( ( mem @ V0l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( ap @ ( ap @ ( c_2Elist_2ETAKE @ A_27a ) @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ V0l ) ) @ V0l )
= V0l ) ) ).
thf(conj_thm_2Elist_2ELENGTH__TAKE,axiom,
! [A_27a: del,V0n: tp__ty_2Enum_2Enum,V1l: $i] :
( ( mem @ V1l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ V0n ) ) @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ V1l ) ) )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2ETAKE @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) @ V1l ) ) )
= V0n ) ) ) ).
thf(conj_thm_2Elist_2EMAP__TAKE,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27b ) )
=> ! [V1n: tp__ty_2Enum_2Enum,V2l: $i] :
( ( mem @ V2l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( ap @ ( ap @ ( c_2Elist_2EMAP @ A_27a @ A_27b ) @ V0f ) @ ( ap @ ( ap @ ( c_2Elist_2ETAKE @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) @ V2l ) )
= ( ap @ ( ap @ ( c_2Elist_2ETAKE @ A_27b ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) @ ( ap @ ( ap @ ( c_2Elist_2EMAP @ A_27a @ A_27b ) @ V0f ) @ V2l ) ) ) ) ) ).
thf(conj_thm_2Elist_2ETAKE__APPEND1,axiom,
! [A_27a: del,V0l1: $i] :
( ( mem @ V0l1 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V1l2: $i] :
( ( mem @ V1l2 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V2n: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ V2n ) ) @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ V0l1 ) ) )
=> ( ( ap @ ( ap @ ( c_2Elist_2ETAKE @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) @ ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ V0l1 ) @ V1l2 ) )
= ( ap @ ( ap @ ( c_2Elist_2ETAKE @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) @ V0l1 ) ) ) ) ) ).
thf(conj_thm_2Elist_2ETAKE__APPEND2,axiom,
! [A_27a: del,V0l1: $i] :
( ( mem @ V0l1 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V1l2: $i] :
( ( mem @ V1l2 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V2n: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ V0l1 ) ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) )
=> ( ( ap @ ( ap @ ( c_2Elist_2ETAKE @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) @ ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ V0l1 ) @ V1l2 ) )
= ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ V0l1 ) @ ( ap @ ( ap @ ( c_2Elist_2ETAKE @ A_27a ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ ( inj__ty_2Enum_2Enum @ V2n ) ) @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ V0l1 ) ) ) @ V1l2 ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2EDROP__0,axiom,
! [A_27a: del,V0l: $i] :
( ( mem @ V0l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( ap @ ( ap @ ( c_2Elist_2EDROP @ A_27a ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ V0l )
= V0l ) ) ).
thf(conj_thm_2Elist_2ETAKE__DROP,axiom,
! [A_27a: del,V0n: tp__ty_2Enum_2Enum,V1l: $i] :
( ( mem @ V1l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2ETAKE @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) @ V1l ) ) @ ( ap @ ( ap @ ( c_2Elist_2EDROP @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) @ V1l ) )
= V1l ) ) ).
thf(conj_thm_2Elist_2ELENGTH__DROP,axiom,
! [A_27a: del,V0n: tp__ty_2Enum_2Enum,V1l: $i] :
( ( mem @ V1l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2EDROP @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) @ V1l ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ V1l ) ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) ) ) ).
thf(conj_thm_2Elist_2EMEM__DROP,axiom,
! [A_27a: del,V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ! [V1ls: $i] :
( ( mem @ V1ls @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V2n: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V0x ) @ ( ap @ ( c_2Elist_2ELIST__TO__SET @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2EDROP @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) @ V1ls ) ) ) )
<=> ( ( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V2n ) ) @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ V1ls ) ) )
& ( V0x
= ( ap @ ( ap @ ( c_2Elist_2EEL @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) @ V1ls ) ) )
| ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V0x ) @ ( ap @ ( c_2Elist_2ELIST__TO__SET @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2EDROP @ A_27a ) @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) @ V1ls ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2EDROP__NIL,axiom,
! [A_27a: del,V0ls: $i] :
( ( mem @ V0ls @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V1n: tp__ty_2Enum_2Enum] :
( ( ( ap @ ( ap @ ( c_2Elist_2EDROP @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) @ V0ls )
= ( c_2Elist_2ENIL @ A_27a ) )
<=> ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3E_3D @ ( inj__ty_2Enum_2Enum @ V1n ) ) @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ V0ls ) ) ) ) ) ).
thf(conj_thm_2Elist_2EFOLDL2__ind,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ ( arr @ A_27a @ ( arr @ A_27b @ ( arr @ A_27c @ A_27a ) ) ) @ ( arr @ A_27a @ ( arr @ ( ty_2Elist_2Elist @ A_27b ) @ ( arr @ ( ty_2Elist_2Elist @ A_27c ) @ bool ) ) ) ) )
=> ( ( ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ ( arr @ A_27b @ ( arr @ A_27c @ A_27a ) ) ) )
=> ! [V2a: $i] :
( ( mem @ V2a @ A_27a )
=> ! [V3b: $i] :
( ( mem @ V3b @ A_27b )
=> ! [V4bs: $i] :
( ( mem @ V4bs @ ( ty_2Elist_2Elist @ A_27b ) )
=> ! [V5c: $i] :
( ( mem @ V5c @ A_27c )
=> ! [V6cs: $i] :
( ( mem @ V6cs @ ( ty_2Elist_2Elist @ A_27c ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( ap @ V0P @ V1f ) @ ( ap @ ( ap @ ( ap @ V1f @ V2a ) @ V3b ) @ V5c ) ) @ V4bs ) @ V6cs ) )
=> ( p @ ( ap @ ( ap @ ( ap @ ( ap @ V0P @ V1f ) @ V2a ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27b ) @ V3b ) @ V4bs ) ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27c ) @ V5c ) @ V6cs ) ) ) ) ) ) ) ) ) )
& ! [V7f: $i] :
( ( mem @ V7f @ ( arr @ A_27a @ ( arr @ A_27b @ ( arr @ A_27c @ A_27a ) ) ) )
=> ! [V8a: $i] :
( ( mem @ V8a @ A_27a )
=> ! [V9cs: $i] :
( ( mem @ V9cs @ ( ty_2Elist_2Elist @ A_27c ) )
=> ( p @ ( ap @ ( ap @ ( ap @ ( ap @ V0P @ V7f ) @ V8a ) @ ( c_2Elist_2ENIL @ A_27b ) ) @ V9cs ) ) ) ) )
& ! [V10f: $i] :
( ( mem @ V10f @ ( arr @ A_27a @ ( arr @ A_27b @ ( arr @ A_27c @ A_27a ) ) ) )
=> ! [V11a: $i] :
( ( mem @ V11a @ A_27a )
=> ! [V12v6: $i] :
( ( mem @ V12v6 @ A_27b )
=> ! [V13v7: $i] :
( ( mem @ V13v7 @ ( ty_2Elist_2Elist @ A_27b ) )
=> ( p @ ( ap @ ( ap @ ( ap @ ( ap @ V0P @ V10f ) @ V11a ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27b ) @ V12v6 ) @ V13v7 ) ) @ ( c_2Elist_2ENIL @ A_27c ) ) ) ) ) ) ) )
=> ! [V14v: $i] :
( ( mem @ V14v @ ( arr @ A_27a @ ( arr @ A_27b @ ( arr @ A_27c @ A_27a ) ) ) )
=> ! [V15v1: $i] :
( ( mem @ V15v1 @ A_27a )
=> ! [V16v2: $i] :
( ( mem @ V16v2 @ ( ty_2Elist_2Elist @ A_27b ) )
=> ! [V17v3: $i] :
( ( mem @ V17v3 @ ( ty_2Elist_2Elist @ A_27c ) )
=> ( p @ ( ap @ ( ap @ ( ap @ ( ap @ V0P @ V14v ) @ V15v1 ) @ V16v2 ) @ V17v3 ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2EFOLDL2__def,axiom,
! [A_27a: del,A_27b: del,A_27c: del] :
( ! [V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ( arr @ A_27b @ ( arr @ A_27c @ A_27a ) ) ) )
=> ! [V1cs: $i] :
( ( mem @ V1cs @ ( ty_2Elist_2Elist @ A_27c ) )
=> ! [V2c: $i] :
( ( mem @ V2c @ A_27c )
=> ! [V3bs: $i] :
( ( mem @ V3bs @ ( ty_2Elist_2Elist @ A_27b ) )
=> ! [V4b: $i] :
( ( mem @ V4b @ A_27b )
=> ! [V5a: $i] :
( ( mem @ V5a @ A_27a )
=> ( ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Elist_2EFOLDL2 @ A_27a @ A_27b @ A_27c ) @ V0f ) @ V5a ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27b ) @ V4b ) @ V3bs ) ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27c ) @ V2c ) @ V1cs ) )
= ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Elist_2EFOLDL2 @ A_27a @ A_27b @ A_27c ) @ V0f ) @ ( ap @ ( ap @ ( ap @ V0f @ V5a ) @ V4b ) @ V2c ) ) @ V3bs ) @ V1cs ) ) ) ) ) ) ) )
& ! [V6f: $i] :
( ( mem @ V6f @ ( arr @ A_27a @ ( arr @ A_27b @ ( arr @ A_27c @ A_27a ) ) ) )
=> ! [V7cs: $i] :
( ( mem @ V7cs @ ( ty_2Elist_2Elist @ A_27c ) )
=> ! [V8a: $i] :
( ( mem @ V8a @ A_27a )
=> ( ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Elist_2EFOLDL2 @ A_27a @ A_27b @ A_27c ) @ V6f ) @ V8a ) @ ( c_2Elist_2ENIL @ A_27b ) ) @ V7cs )
= V8a ) ) ) )
& ! [V9v7: $i] :
( ( mem @ V9v7 @ ( ty_2Elist_2Elist @ A_27b ) )
=> ! [V10v6: $i] :
( ( mem @ V10v6 @ A_27b )
=> ! [V11f: $i] :
( ( mem @ V11f @ ( arr @ A_27a @ ( arr @ A_27b @ ( arr @ A_27c @ A_27a ) ) ) )
=> ! [V12a: $i] :
( ( mem @ V12a @ A_27a )
=> ( ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Elist_2EFOLDL2 @ A_27a @ A_27b @ A_27c ) @ V11f ) @ V12a ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27b ) @ V10v6 ) @ V9v7 ) ) @ ( c_2Elist_2ENIL @ A_27c ) )
= V12a ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2EFOLDL2__cong,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0l1: $i] :
( ( mem @ V0l1 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V1l1_27: $i] :
( ( mem @ V1l1_27 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V2l2: $i] :
( ( mem @ V2l2 @ ( ty_2Elist_2Elist @ A_27b ) )
=> ! [V3l2_27: $i] :
( ( mem @ V3l2_27 @ ( ty_2Elist_2Elist @ A_27b ) )
=> ! [V4a: $i] :
( ( mem @ V4a @ A_27c )
=> ! [V5a_27: $i] :
( ( mem @ V5a_27 @ A_27c )
=> ! [V6f: $i] :
( ( mem @ V6f @ ( arr @ A_27c @ ( arr @ A_27a @ ( arr @ A_27b @ A_27c ) ) ) )
=> ! [V7f_27: $i] :
( ( mem @ V7f_27 @ ( arr @ A_27c @ ( arr @ A_27a @ ( arr @ A_27b @ A_27c ) ) ) )
=> ( ( ( V0l1 = V1l1_27 )
& ( V2l2 = V3l2_27 )
& ( V4a = V5a_27 )
& ! [V8z: $i] :
( ( mem @ V8z @ A_27c )
=> ! [V9b: $i] :
( ( mem @ V9b @ A_27a )
=> ! [V10c: $i] :
( ( mem @ V10c @ A_27b )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V9b ) @ ( ap @ ( c_2Elist_2ELIST__TO__SET @ A_27a ) @ V1l1_27 ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27b ) @ V10c ) @ ( ap @ ( c_2Elist_2ELIST__TO__SET @ A_27b ) @ V3l2_27 ) ) ) )
=> ( ( ap @ ( ap @ ( ap @ V6f @ V8z ) @ V9b ) @ V10c )
= ( ap @ ( ap @ ( ap @ V7f_27 @ V8z ) @ V9b ) @ V10c ) ) ) ) ) ) )
=> ( ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Elist_2EFOLDL2 @ A_27c @ A_27a @ A_27b ) @ V6f ) @ V4a ) @ V0l1 ) @ V2l2 )
= ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Elist_2EFOLDL2 @ A_27c @ A_27a @ A_27b ) @ V7f_27 ) @ V5a_27 ) @ V1l1_27 ) @ V3l2_27 ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2EFOLDL2__FOLDL,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0l1: $i] :
( ( mem @ V0l1 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V1l2: $i] :
( ( mem @ V1l2 @ ( ty_2Elist_2Elist @ A_27b ) )
=> ( ( ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ V0l1 ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Elist_2ELENGTH @ A_27b ) @ V1l2 ) ) )
=> ! [V2f: $i] :
( ( mem @ V2f @ ( arr @ A_27c @ ( arr @ A_27a @ ( arr @ A_27b @ A_27c ) ) ) )
=> ! [V3a: $i] :
( ( mem @ V3a @ A_27c )
=> ( ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Elist_2EFOLDL2 @ A_27c @ A_27a @ A_27b ) @ V2f ) @ V3a ) @ V0l1 ) @ V1l2 )
= ( ap
@ ( ap
@ ( ap @ ( c_2Elist_2EFOLDL @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ A_27c )
@ ( lam @ A_27c
@ ^ [V4a: $i] : ( ap @ ( c_2Epair_2EUNCURRY @ A_27a @ A_27b @ A_27c ) @ ( ap @ V2f @ V4a ) ) ) )
@ V3a )
@ ( ap @ ( c_2Elist_2EZIP @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Elist_2Elist @ A_27a ) @ ( ty_2Elist_2Elist @ A_27b ) ) @ V0l1 ) @ V1l2 ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2EEVERY2__cong,axiom,
! [A_27a: del,A_27b: del,V0l1: $i] :
( ( mem @ V0l1 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V1l1_27: $i] :
( ( mem @ V1l1_27 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V2l2: $i] :
( ( mem @ V2l2 @ ( ty_2Elist_2Elist @ A_27b ) )
=> ! [V3l2_27: $i] :
( ( mem @ V3l2_27 @ ( ty_2Elist_2Elist @ A_27b ) )
=> ! [V4P: $i] :
( ( mem @ V4P @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) )
=> ! [V5P_27: $i] :
( ( mem @ V5P_27 @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) )
=> ( ( ( V0l1 = V1l1_27 )
& ( V2l2 = V3l2_27 )
& ! [V6x: $i] :
( ( mem @ V6x @ A_27a )
=> ! [V7y: $i] :
( ( mem @ V7y @ A_27b )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V6x ) @ ( ap @ ( c_2Elist_2ELIST__TO__SET @ A_27a ) @ V1l1_27 ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27b ) @ V7y ) @ ( ap @ ( c_2Elist_2ELIST__TO__SET @ A_27b ) @ V3l2_27 ) ) ) )
=> ( ( p @ ( ap @ ( ap @ V4P @ V6x ) @ V7y ) )
<=> ( p @ ( ap @ ( ap @ V5P_27 @ V6x ) @ V7y ) ) ) ) ) ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Elist_2ELIST__REL @ A_27a @ A_27b ) @ V4P ) @ V0l1 ) @ V2l2 ) )
<=> ( p @ ( ap @ ( ap @ ( ap @ ( c_2Elist_2ELIST__REL @ A_27a @ A_27b ) @ V5P_27 ) @ V1l1_27 ) @ V3l2_27 ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2EMAP__EQ__EVERY2,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0f1: $i] :
( ( mem @ V0f1 @ ( arr @ A_27a @ A_27b ) )
=> ! [V1f2: $i] :
( ( mem @ V1f2 @ ( arr @ A_27c @ A_27b ) )
=> ! [V2l1: $i] :
( ( mem @ V2l1 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V3l2: $i] :
( ( mem @ V3l2 @ ( ty_2Elist_2Elist @ A_27c ) )
=> ( ( ( ap @ ( ap @ ( c_2Elist_2EMAP @ A_27a @ A_27b ) @ V0f1 ) @ V2l1 )
= ( ap @ ( ap @ ( c_2Elist_2EMAP @ A_27c @ A_27b ) @ V1f2 ) @ V3l2 ) )
<=> ( ( ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ V2l1 ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Elist_2ELENGTH @ A_27c ) @ V3l2 ) ) )
& ( p
@ ( ap
@ ( ap
@ ( ap @ ( c_2Elist_2ELIST__REL @ A_27a @ A_27c )
@ ( lam @ A_27a
@ ^ [V4x: $i] :
( lam @ A_27c
@ ^ [V5y: $i] : ( ap @ ( ap @ ( c_2Emin_2E_3D @ A_27b ) @ ( ap @ V0f1 @ V4x ) ) @ ( ap @ V1f2 @ V5y ) ) ) ) )
@ V2l1 )
@ V3l2 ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2EEVERY2__EVERY,axiom,
! [A_27a: del,A_27b: del,V0l1: $i] :
( ( mem @ V0l1 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V1l2: $i] :
( ( mem @ V1l2 @ ( ty_2Elist_2Elist @ A_27b ) )
=> ! [V2f: $i] :
( ( mem @ V2f @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Elist_2ELIST__REL @ A_27a @ A_27b ) @ V2f ) @ V0l1 ) @ V1l2 ) )
<=> ( ( ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ V0l1 ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Elist_2ELENGTH @ A_27b ) @ V1l2 ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Elist_2EEVERY @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) ) @ ( ap @ ( c_2Epair_2EUNCURRY @ A_27a @ A_27b @ bool ) @ V2f ) ) @ ( ap @ ( c_2Elist_2EZIP @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Elist_2Elist @ A_27a ) @ ( ty_2Elist_2Elist @ A_27b ) ) @ V0l1 ) @ V1l2 ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2EEVERY2__LENGTH,axiom,
! [A_27a: del,A_27b: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) )
=> ! [V1l1: $i] :
( ( mem @ V1l1 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V2l2: $i] :
( ( mem @ V2l2 @ ( ty_2Elist_2Elist @ A_27b ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Elist_2ELIST__REL @ A_27a @ A_27b ) @ V0P ) @ V1l1 ) @ V2l2 ) )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ V1l1 ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Elist_2ELENGTH @ A_27b ) @ V2l2 ) ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2EEVERY2__mono,axiom,
! [A_27a: del,A_27b: del,V0R1: $i] :
( ( mem @ V0R1 @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) )
=> ! [V1R2: $i] :
( ( mem @ V1R2 @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) )
=> ! [V2l1: $i] :
( ( mem @ V2l1 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V3l2: $i] :
( ( mem @ V3l2 @ ( ty_2Elist_2Elist @ A_27b ) )
=> ( ! [V4x: $i] :
( ( mem @ V4x @ A_27a )
=> ! [V5y: $i] :
( ( mem @ V5y @ A_27b )
=> ( ( p @ ( ap @ ( ap @ V0R1 @ V4x ) @ V5y ) )
=> ( p @ ( ap @ ( ap @ V1R2 @ V4x ) @ V5y ) ) ) ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Elist_2ELIST__REL @ A_27a @ A_27b ) @ V0R1 ) @ V2l1 ) @ V3l2 ) )
=> ( p @ ( ap @ ( ap @ ( ap @ ( c_2Elist_2ELIST__REL @ A_27a @ A_27b ) @ V1R2 ) @ V2l1 ) @ V3l2 ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Elist_2EALL__DISTINCT,axiom,
! [A_27a: del] :
( ( ( p @ ( ap @ ( c_2Elist_2EALL__DISTINCT @ A_27a ) @ ( c_2Elist_2ENIL @ A_27a ) ) )
<=> $true )
& ! [V0h: $i] :
( ( mem @ V0h @ A_27a )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( p @ ( ap @ ( c_2Elist_2EALL__DISTINCT @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V0h ) @ V1t ) ) )
<=> ( ~ ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V0h ) @ ( ap @ ( c_2Elist_2ELIST__TO__SET @ A_27a ) @ V1t ) ) )
& ( p @ ( ap @ ( c_2Elist_2EALL__DISTINCT @ A_27a ) @ V1t ) ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2EALL__DISTINCT__FILTER,axiom,
! [A_27a: del,V0l: $i] :
( ( mem @ V0l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( p @ ( ap @ ( c_2Elist_2EALL__DISTINCT @ A_27a ) @ V0l ) )
<=> ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V1x ) @ ( ap @ ( c_2Elist_2ELIST__TO__SET @ A_27a ) @ V0l ) ) )
=> ( ( ap @ ( ap @ ( c_2Elist_2EFILTER @ A_27a ) @ ( ap @ ( c_2Emin_2E_3D @ A_27a ) @ V1x ) ) @ V0l )
= ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V1x ) @ ( c_2Elist_2ENIL @ A_27a ) ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2EFILTER__ALL__DISTINCT,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ! [V1l: $i] :
( ( mem @ V1l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( p @ ( ap @ ( c_2Elist_2EALL__DISTINCT @ A_27a ) @ V1l ) )
=> ( p @ ( ap @ ( c_2Elist_2EALL__DISTINCT @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2EFILTER @ A_27a ) @ V0P ) @ V1l ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2EALL__DISTINCT__MAP,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27b ) )
=> ! [V1ls: $i] :
( ( mem @ V1ls @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( p @ ( ap @ ( c_2Elist_2EALL__DISTINCT @ A_27b ) @ ( ap @ ( ap @ ( c_2Elist_2EMAP @ A_27a @ A_27b ) @ V0f ) @ V1ls ) ) )
=> ( p @ ( ap @ ( c_2Elist_2EALL__DISTINCT @ A_27a ) @ V1ls ) ) ) ) ) ).
thf(conj_thm_2Elist_2EEL__ALL__DISTINCT__EL__EQ,axiom,
! [A_27a: del,V0l: $i] :
( ( mem @ V0l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( p @ ( ap @ ( c_2Elist_2EALL__DISTINCT @ A_27a ) @ V0l ) )
<=> ! [V1n1: tp__ty_2Enum_2Enum,V2n2: tp__ty_2Enum_2Enum] :
( ( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V1n1 ) ) @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ V0l ) ) )
& ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V2n2 ) ) @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ V0l ) ) ) )
=> ( ( ( ap @ ( ap @ ( c_2Elist_2EEL @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V1n1 ) ) @ V0l )
= ( ap @ ( ap @ ( c_2Elist_2EEL @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V2n2 ) ) @ V0l ) )
<=> ( V1n1 = V2n2 ) ) ) ) ) ).
thf(conj_thm_2Elist_2EALL__DISTINCT__EL__IMP,axiom,
! [A_27a: del,V0l: $i] :
( ( mem @ V0l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V1n1: tp__ty_2Enum_2Enum,V2n2: tp__ty_2Enum_2Enum] :
( ( ( p @ ( ap @ ( c_2Elist_2EALL__DISTINCT @ A_27a ) @ V0l ) )
& ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V1n1 ) ) @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ V0l ) ) )
& ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V2n2 ) ) @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ V0l ) ) ) )
=> ( ( ( ap @ ( ap @ ( c_2Elist_2EEL @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V1n1 ) ) @ V0l )
= ( ap @ ( ap @ ( c_2Elist_2EEL @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V2n2 ) ) @ V0l ) )
<=> ( V1n1 = V2n2 ) ) ) ) ).
thf(conj_thm_2Elist_2EALL__DISTINCT__APPEND,axiom,
! [A_27a: del,V0l1: $i] :
( ( mem @ V0l1 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V1l2: $i] :
( ( mem @ V1l2 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( p @ ( ap @ ( c_2Elist_2EALL__DISTINCT @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ V0l1 ) @ V1l2 ) ) )
<=> ( ( p @ ( ap @ ( c_2Elist_2EALL__DISTINCT @ A_27a ) @ V0l1 ) )
& ( p @ ( ap @ ( c_2Elist_2EALL__DISTINCT @ A_27a ) @ V1l2 ) )
& ! [V2e: $i] :
( ( mem @ V2e @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2e ) @ ( ap @ ( c_2Elist_2ELIST__TO__SET @ A_27a ) @ V0l1 ) ) )
=> ~ ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2e ) @ ( ap @ ( c_2Elist_2ELIST__TO__SET @ A_27a ) @ V1l2 ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2EALL__DISTINCT__SING,axiom,
! [A_27a: del,V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ( p @ ( ap @ ( c_2Elist_2EALL__DISTINCT @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V0x ) @ ( c_2Elist_2ENIL @ A_27a ) ) ) ) ) ).
thf(conj_thm_2Elist_2EALL__DISTINCT__ZIP,axiom,
! [A_27a: del,A_27b: del,V0l1: $i] :
( ( mem @ V0l1 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V1l2: $i] :
( ( mem @ V1l2 @ ( ty_2Elist_2Elist @ A_27b ) )
=> ( ( ( p @ ( ap @ ( c_2Elist_2EALL__DISTINCT @ A_27a ) @ V0l1 ) )
& ( ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ V0l1 ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Elist_2ELENGTH @ A_27b ) @ V1l2 ) ) ) )
=> ( p @ ( ap @ ( c_2Elist_2EALL__DISTINCT @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) ) @ ( ap @ ( c_2Elist_2EZIP @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Elist_2Elist @ A_27a ) @ ( ty_2Elist_2Elist @ A_27b ) ) @ V0l1 ) @ V1l2 ) ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2EALL__DISTINCT__ZIP__SWAP,axiom,
! [A_27a: del,A_27b: del,V0l1: $i] :
( ( mem @ V0l1 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V1l2: $i] :
( ( mem @ V1l2 @ ( ty_2Elist_2Elist @ A_27b ) )
=> ( ( ( p @ ( ap @ ( c_2Elist_2EALL__DISTINCT @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) ) @ ( ap @ ( c_2Elist_2EZIP @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Elist_2Elist @ A_27a ) @ ( ty_2Elist_2Elist @ A_27b ) ) @ V0l1 ) @ V1l2 ) ) ) )
& ( ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ V0l1 ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Elist_2ELENGTH @ A_27b ) @ V1l2 ) ) ) )
=> ( p @ ( ap @ ( c_2Elist_2EALL__DISTINCT @ ( ty_2Epair_2Eprod @ A_27b @ A_27a ) ) @ ( ap @ ( c_2Elist_2EZIP @ A_27b @ A_27a ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Elist_2Elist @ A_27b ) @ ( ty_2Elist_2Elist @ A_27a ) ) @ V1l2 ) @ V0l1 ) ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2EALL__DISTINCT__REVERSE,axiom,
! [A_27a: del,V0l: $i] :
( ( mem @ V0l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( p @ ( ap @ ( c_2Elist_2EALL__DISTINCT @ A_27a ) @ ( ap @ ( c_2Elist_2EREVERSE @ A_27a ) @ V0l ) ) )
<=> ( p @ ( ap @ ( c_2Elist_2EALL__DISTINCT @ A_27a ) @ V0l ) ) ) ) ).
thf(conj_thm_2Elist_2EALL__DISTINCT__FLAT__REVERSE,axiom,
! [A_27a: del,V0xs: $i] :
( ( mem @ V0xs @ ( ty_2Elist_2Elist @ ( ty_2Elist_2Elist @ A_27a ) ) )
=> ( ( p @ ( ap @ ( c_2Elist_2EALL__DISTINCT @ A_27a ) @ ( ap @ ( c_2Elist_2EFLAT @ A_27a ) @ ( ap @ ( c_2Elist_2EREVERSE @ ( ty_2Elist_2Elist @ A_27a ) ) @ V0xs ) ) ) )
<=> ( p @ ( ap @ ( c_2Elist_2EALL__DISTINCT @ A_27a ) @ ( ap @ ( c_2Elist_2EFLAT @ A_27a ) @ V0xs ) ) ) ) ) ).
thf(ax_thm_2Elist_2ELRC__def,axiom,
! [A_27a: del] :
( ! [V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ! [V2y: $i] :
( ( mem @ V2y @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Elist_2ELRC @ A_27a ) @ V0R ) @ ( c_2Elist_2ENIL @ A_27a ) ) @ V1x ) @ V2y ) )
<=> ( V1x = V2y ) ) ) ) )
& ! [V3R: $i] :
( ( mem @ V3R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V4h: $i] :
( ( mem @ V4h @ A_27a )
=> ! [V5t: $i] :
( ( mem @ V5t @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V6x: $i] :
( ( mem @ V6x @ A_27a )
=> ! [V7y: $i] :
( ( mem @ V7y @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Elist_2ELRC @ A_27a ) @ V3R ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V4h ) @ V5t ) ) @ V6x ) @ V7y ) )
<=> ( ( V6x = V4h )
& ? [V8z: $i] :
( ( mem @ V8z @ A_27a )
& ( p @ ( ap @ ( ap @ V3R @ V6x ) @ V8z ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Elist_2ELRC @ A_27a ) @ V3R ) @ V5t ) @ V8z ) @ V7y ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2ENRC__LRC,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1n: tp__ty_2Enum_2Enum,V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ! [V3y: $i] :
( ( mem @ V3y @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Earithmetic_2ENRC @ A_27a ) @ V0R ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) @ V2x ) @ V3y ) )
<=> ? [V4ls: $i] :
( ( mem @ V4ls @ ( ty_2Elist_2Elist @ A_27a ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Elist_2ELRC @ A_27a ) @ V0R ) @ V4ls ) @ V2x ) @ V3y ) )
& ( ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ V4ls ) )
= V1n ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2ELRC__MEM,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1ls: $i] :
( ( mem @ V1ls @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ! [V3y: $i] :
( ( mem @ V3y @ A_27a )
=> ! [V4e: $i] :
( ( mem @ V4e @ A_27a )
=> ( ( ( p @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Elist_2ELRC @ A_27a ) @ V0R ) @ V1ls ) @ V2x ) @ V3y ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V4e ) @ ( ap @ ( c_2Elist_2ELIST__TO__SET @ A_27a ) @ V1ls ) ) ) )
=> ? [V5z: $i] :
( ( mem @ V5z @ A_27a )
& ? [V6t: $i] :
( ( mem @ V6t @ ( ty_2Elist_2Elist @ A_27a ) )
& ( p @ ( ap @ ( ap @ V0R @ V4e ) @ V5z ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Elist_2ELRC @ A_27a ) @ V0R ) @ V6t ) @ V5z ) @ V3y ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2ELRC__MEM__right,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1h: $i] :
( ( mem @ V1h @ A_27a )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ! [V4y: $i] :
( ( mem @ V4y @ A_27a )
=> ! [V5e: $i] :
( ( mem @ V5e @ A_27a )
=> ( ( ( p @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Elist_2ELRC @ A_27a ) @ V0R ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V1h ) @ V2t ) ) @ V3x ) @ V4y ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V5e ) @ ( ap @ ( c_2Elist_2ELIST__TO__SET @ A_27a ) @ V2t ) ) ) )
=> ? [V6z: $i] :
( ( mem @ V6z @ A_27a )
& ? [V7p: $i] :
( ( mem @ V7p @ ( ty_2Elist_2Elist @ A_27a ) )
& ( p @ ( ap @ ( ap @ V0R @ V6z ) @ V5e ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Elist_2ELRC @ A_27a ) @ V0R ) @ V7p ) @ V3x ) @ V6z ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2ELIST__TO__SET__APPEND,axiom,
! [A_27a: del,V0l1: $i] :
( ( mem @ V0l1 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V1l2: $i] :
( ( mem @ V1l2 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( ap @ ( c_2Elist_2ELIST__TO__SET @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ V0l1 ) @ V1l2 ) )
= ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ A_27a ) @ ( ap @ ( c_2Elist_2ELIST__TO__SET @ A_27a ) @ V0l1 ) ) @ ( ap @ ( c_2Elist_2ELIST__TO__SET @ A_27a ) @ V1l2 ) ) ) ) ) ).
thf(conj_thm_2Elist_2EUNION__APPEND,axiom,
! [A_27a: del,V0l1: $i] :
( ( mem @ V0l1 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V1l2: $i] :
( ( mem @ V1l2 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ A_27a ) @ ( ap @ ( c_2Elist_2ELIST__TO__SET @ A_27a ) @ V0l1 ) ) @ ( ap @ ( c_2Elist_2ELIST__TO__SET @ A_27a ) @ V1l2 ) )
= ( ap @ ( c_2Elist_2ELIST__TO__SET @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ V0l1 ) @ V1l2 ) ) ) ) ) ).
thf(conj_thm_2Elist_2ELIST__TO__SET__EQ__EMPTY,axiom,
! [A_27a: del,V0l: $i] :
( ( mem @ V0l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( ( ( ap @ ( c_2Elist_2ELIST__TO__SET @ A_27a ) @ V0l )
= ( c_2Epred__set_2EEMPTY @ A_27a ) )
<=> ( V0l
= ( c_2Elist_2ENIL @ A_27a ) ) )
& ( ( ( c_2Epred__set_2EEMPTY @ A_27a )
= ( ap @ ( c_2Elist_2ELIST__TO__SET @ A_27a ) @ V0l ) )
<=> ( V0l
= ( c_2Elist_2ENIL @ A_27a ) ) ) ) ) ).
thf(conj_thm_2Elist_2EFINITE__LIST__TO__SET,axiom,
! [A_27a: del,V0l: $i] :
( ( mem @ V0l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ ( ap @ ( c_2Elist_2ELIST__TO__SET @ A_27a ) @ V0l ) ) ) ) ).
thf(conj_thm_2Elist_2ESUM__IMAGE__LIST__TO__SET__upper__bound,axiom,
! [A_27a: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V1ls: $i] :
( ( mem @ V1ls @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( ap @ ( ap @ ( c_2Epred__set_2ESUM__IMAGE @ A_27a ) @ V0f ) @ ( ap @ ( c_2Elist_2ELIST__TO__SET @ A_27a ) @ V1ls ) ) ) @ ( ap @ c_2Elist_2ESUM @ ( ap @ ( ap @ ( c_2Elist_2EMAP @ A_27a @ ty_2Enum_2Enum ) @ V0f ) @ V1ls ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2ESUM__MAP__MEM__bound,axiom,
! [A_27a: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ! [V2ls: $i] :
( ( mem @ V2ls @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V1x ) @ ( ap @ ( c_2Elist_2ELIST__TO__SET @ A_27a ) @ V2ls ) ) )
=> ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( ap @ V0f @ V1x ) ) @ ( ap @ c_2Elist_2ESUM @ ( ap @ ( ap @ ( c_2Elist_2EMAP @ A_27a @ ty_2Enum_2Enum ) @ V0f ) @ V2ls ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2EINJ__MAP__EQ,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27b ) )
=> ! [V1l1: $i] :
( ( mem @ V1l1 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V2l2: $i] :
( ( mem @ V2l2 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Epred__set_2EINJ @ A_27a @ A_27b ) @ V0f ) @ ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ A_27a ) @ ( ap @ ( c_2Elist_2ELIST__TO__SET @ A_27a ) @ V1l1 ) ) @ ( ap @ ( c_2Elist_2ELIST__TO__SET @ A_27a ) @ V2l2 ) ) ) @ ( c_2Epred__set_2EUNIV @ A_27b ) ) )
& ( ( ap @ ( ap @ ( c_2Elist_2EMAP @ A_27a @ A_27b ) @ V0f ) @ V1l1 )
= ( ap @ ( ap @ ( c_2Elist_2EMAP @ A_27a @ A_27b ) @ V0f ) @ V2l2 ) ) )
=> ( V1l1 = V2l2 ) ) ) ) ) ).
thf(conj_thm_2Elist_2EINJ__MAP__EQ__IFF,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27b ) )
=> ! [V1l1: $i] :
( ( mem @ V1l1 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V2l2: $i] :
( ( mem @ V2l2 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Epred__set_2EINJ @ A_27a @ A_27b ) @ V0f ) @ ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ A_27a ) @ ( ap @ ( c_2Elist_2ELIST__TO__SET @ A_27a ) @ V1l1 ) ) @ ( ap @ ( c_2Elist_2ELIST__TO__SET @ A_27a ) @ V2l2 ) ) ) @ ( c_2Epred__set_2EUNIV @ A_27b ) ) )
=> ( ( ( ap @ ( ap @ ( c_2Elist_2EMAP @ A_27a @ A_27b ) @ V0f ) @ V1l1 )
= ( ap @ ( ap @ ( c_2Elist_2EMAP @ A_27a @ A_27b ) @ V0f ) @ V2l2 ) )
<=> ( V1l1 = V2l2 ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2ECARD__LIST__TO__SET,axiom,
! [A_27a: del,V0ls: $i] :
( ( mem @ V0ls @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( ap @ ( c_2Epred__set_2ECARD @ A_27a ) @ ( ap @ ( c_2Elist_2ELIST__TO__SET @ A_27a ) @ V0ls ) ) ) @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ V0ls ) ) ) ) ).
thf(conj_thm_2Elist_2EALL__DISTINCT__CARD__LIST__TO__SET,axiom,
! [A_27a: del,V0ls: $i] :
( ( mem @ V0ls @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( p @ ( ap @ ( c_2Elist_2EALL__DISTINCT @ A_27a ) @ V0ls ) )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Epred__set_2ECARD @ A_27a ) @ ( ap @ ( c_2Elist_2ELIST__TO__SET @ A_27a ) @ V0ls ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ V0ls ) ) ) ) ) ).
thf(conj_thm_2Elist_2ECARD__LIST__TO__SET__ALL__DISTINCT,axiom,
! [A_27a: del,V0ls: $i] :
( ( mem @ V0ls @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Epred__set_2ECARD @ A_27a ) @ ( ap @ ( c_2Elist_2ELIST__TO__SET @ A_27a ) @ V0ls ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ V0ls ) ) )
=> ( p @ ( ap @ ( c_2Elist_2EALL__DISTINCT @ A_27a ) @ V0ls ) ) ) ) ).
thf(conj_thm_2Elist_2ELIST__TO__SET__REVERSE,axiom,
! [A_27a: del,V0ls: $i] :
( ( mem @ V0ls @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( ap @ ( c_2Elist_2ELIST__TO__SET @ A_27a ) @ ( ap @ ( c_2Elist_2EREVERSE @ A_27a ) @ V0ls ) )
= ( ap @ ( c_2Elist_2ELIST__TO__SET @ A_27a ) @ V0ls ) ) ) ).
thf(conj_thm_2Elist_2ELIST__TO__SET__THM,axiom,
! [A_27a: del,A_27b: del,V0h: $i] :
( ( mem @ V0h @ A_27b )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( ty_2Elist_2Elist @ A_27b ) )
=> ( ( ( ap @ ( c_2Elist_2ELIST__TO__SET @ A_27a ) @ ( c_2Elist_2ENIL @ A_27a ) )
= ( c_2Epred__set_2EEMPTY @ A_27a ) )
& ( ( ap @ ( c_2Elist_2ELIST__TO__SET @ A_27b ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27b ) @ V0h ) @ V1t ) )
= ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ A_27b ) @ V0h ) @ ( ap @ ( c_2Elist_2ELIST__TO__SET @ A_27b ) @ V1t ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2ELIST__TO__SET__MAP,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27b ) )
=> ! [V1l: $i] :
( ( mem @ V1l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( ap @ ( c_2Elist_2ELIST__TO__SET @ A_27b ) @ ( ap @ ( ap @ ( c_2Elist_2EMAP @ A_27a @ A_27b ) @ V0f ) @ V1l ) )
= ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27b ) @ V0f ) @ ( ap @ ( c_2Elist_2ELIST__TO__SET @ A_27a ) @ V1l ) ) ) ) ) ).
thf(conj_thm_2Elist_2ELIST__TO__SET__FILTER,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ! [V1l: $i] :
( ( mem @ V1l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( ap @ ( c_2Elist_2ELIST__TO__SET @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2EFILTER @ A_27a ) @ V0P ) @ V1l ) )
= ( ap
@ ( ap @ ( c_2Epred__set_2EINTER @ A_27a )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a )
@ ( lam @ A_27a
@ ^ [V2x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ bool ) @ V2x ) @ ( ap @ V0P @ V2x ) ) ) ) )
@ ( ap @ ( c_2Elist_2ELIST__TO__SET @ A_27a ) @ V1l ) ) ) ) ) ).
thf(ax_thm_2Elist_2ESET__TO__LIST__primitive__def,axiom,
! [A_27a: del] :
( ( c_2Elist_2ESET__TO__LIST @ A_27a )
= ( ap
@ ( ap @ ( c_2Erelation_2EWFREC @ ( arr @ A_27a @ bool ) @ ( ty_2Elist_2Elist @ A_27a ) )
@ ( ap @ ( c_2Emin_2E_40 @ ( arr @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) )
@ ( lam @ ( arr @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) )
@ ^ [V0R: $i] :
( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( c_2Erelation_2EWF @ ( arr @ A_27a @ bool ) ) @ V0R ) )
@ ( ap @ ( c_2Ebool_2E_21 @ ( arr @ A_27a @ bool ) )
@ ( lam @ ( arr @ A_27a @ bool )
@ ^ [V1s: $i] : ( ap @ ( ap @ c_2Emin_2E_3D_3D_3E @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V1s ) ) @ ( ap @ c_2Ebool_2E_7E @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ( arr @ A_27a @ bool ) ) @ V1s ) @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) ) ) ) @ ( ap @ ( ap @ V0R @ ( ap @ ( c_2Epred__set_2EREST @ A_27a ) @ V1s ) ) @ V1s ) ) ) ) ) ) ) )
@ ( lam @ ( arr @ ( arr @ A_27a @ bool ) @ ( ty_2Elist_2Elist @ A_27a ) )
@ ^ [V2SET__TO__LIST: $i] :
( lam @ ( arr @ A_27a @ bool )
@ ^ [V3a: $i] : ( ap @ ( c_2Ecombin_2EI @ ( ty_2Elist_2Elist @ A_27a ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ( ty_2Elist_2Elist @ A_27a ) ) @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V3a ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ( ty_2Elist_2Elist @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ( arr @ A_27a @ bool ) ) @ V3a ) @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) ) @ ( c_2Elist_2ENIL @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ ( ap @ ( c_2Epred__set_2ECHOICE @ A_27a ) @ V3a ) ) @ ( ap @ V2SET__TO__LIST @ ( ap @ ( c_2Epred__set_2EREST @ A_27a ) @ V3a ) ) ) ) ) @ ( c_2Ebool_2EARB @ ( ty_2Elist_2Elist @ A_27a ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2ESET__TO__LIST__THM,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V0s ) )
=> ( ( ap @ ( c_2Elist_2ESET__TO__LIST @ A_27a ) @ V0s )
= ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ( ty_2Elist_2Elist @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ( arr @ A_27a @ bool ) ) @ V0s ) @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) ) @ ( c_2Elist_2ENIL @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ ( ap @ ( c_2Epred__set_2ECHOICE @ A_27a ) @ V0s ) ) @ ( ap @ ( c_2Elist_2ESET__TO__LIST @ A_27a ) @ ( ap @ ( c_2Epred__set_2EREST @ A_27a ) @ V0s ) ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2ESET__TO__LIST__IND,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) )
=> ( ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ( ( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V1s ) )
& ( V1s
!= ( c_2Epred__set_2EEMPTY @ A_27a ) ) )
=> ( p @ ( ap @ V0P @ ( ap @ ( c_2Epred__set_2EREST @ A_27a ) @ V1s ) ) ) )
=> ( p @ ( ap @ V0P @ V1s ) ) ) )
=> ! [V2v: $i] :
( ( mem @ V2v @ ( arr @ A_27a @ bool ) )
=> ( p @ ( ap @ V0P @ V2v ) ) ) ) ) ).
thf(conj_thm_2Elist_2ESET__TO__LIST__EMPTY,axiom,
! [A_27a: del] :
( ( ap @ ( c_2Elist_2ESET__TO__LIST @ A_27a ) @ ( c_2Epred__set_2EEMPTY @ A_27a ) )
= ( c_2Elist_2ENIL @ A_27a ) ) ).
thf(conj_thm_2Elist_2ESET__TO__LIST__INV,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V0s ) )
=> ( ( ap @ ( c_2Elist_2ELIST__TO__SET @ A_27a ) @ ( ap @ ( c_2Elist_2ESET__TO__LIST @ A_27a ) @ V0s ) )
= V0s ) ) ) ).
thf(conj_thm_2Elist_2ESET__TO__LIST__CARD,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V0s ) )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ ( ap @ ( c_2Elist_2ESET__TO__LIST @ A_27a ) @ V0s ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Epred__set_2ECARD @ A_27a ) @ V0s ) ) ) ) ) ).
thf(conj_thm_2Elist_2ESET__TO__LIST__IN__MEM,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V0s ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V1x ) @ V0s ) )
<=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V1x ) @ ( ap @ ( c_2Elist_2ELIST__TO__SET @ A_27a ) @ ( ap @ ( c_2Elist_2ESET__TO__LIST @ A_27a ) @ V0s ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2EMEM__SET__TO__LIST,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V0s ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V1x ) @ ( ap @ ( c_2Elist_2ELIST__TO__SET @ A_27a ) @ ( ap @ ( c_2Elist_2ESET__TO__LIST @ A_27a ) @ V0s ) ) ) )
<=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V1x ) @ V0s ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2ESET__TO__LIST__SING,axiom,
! [A_27a: del,V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ( ( ap @ ( c_2Elist_2ESET__TO__LIST @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ A_27a ) @ V0x ) @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) )
= ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V0x ) @ ( c_2Elist_2ENIL @ A_27a ) ) ) ) ).
thf(conj_thm_2Elist_2EALL__DISTINCT__SET__TO__LIST,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V0s ) )
=> ( p @ ( ap @ ( c_2Elist_2EALL__DISTINCT @ A_27a ) @ ( ap @ ( c_2Elist_2ESET__TO__LIST @ A_27a ) @ V0s ) ) ) ) ) ).
thf(conj_thm_2Elist_2EITSET__eq__FOLDL__SET__TO__LIST,axiom,
! [A_27a: del,A_27b: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V0s ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ ( arr @ A_27b @ A_27b ) ) )
=> ! [V2a: $i] :
( ( mem @ V2a @ A_27b )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Epred__set_2EITSET @ A_27a @ A_27b ) @ V1f ) @ V0s ) @ V2a )
= ( ap @ ( ap @ ( ap @ ( c_2Elist_2EFOLDL @ A_27a @ A_27b ) @ ( ap @ ( c_2Ecombin_2EC @ A_27a @ A_27b @ A_27b ) @ V1f ) ) @ V2a ) @ ( ap @ ( c_2Elist_2ESET__TO__LIST @ A_27a ) @ V0s ) ) ) ) ) ) ) ).
thf(ax_thm_2Elist_2EisPREFIX,axiom,
! [A_27a: del] :
( ! [V0l: $i] :
( ( mem @ V0l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Elist_2EisPREFIX @ A_27a ) @ ( c_2Elist_2ENIL @ A_27a ) ) @ V0l ) )
<=> $true ) )
& ! [V1h: $i] :
( ( mem @ V1h @ A_27a )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V3l: $i] :
( ( mem @ V3l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Elist_2EisPREFIX @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V1h ) @ V2t ) ) @ V3l ) )
<=> ( p
@ ( ap @ ( ap @ ( ap @ ( c_2Elist_2Elist__CASE @ A_27a @ bool ) @ V3l ) @ c_2Ebool_2EF )
@ ( lam @ A_27a
@ ^ [V4h_27: $i] :
( lam @ ( ty_2Elist_2Elist @ A_27a )
@ ^ [V5t_27: $i] : ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ A_27a ) @ V1h ) @ V4h_27 ) ) @ ( ap @ ( ap @ ( c_2Elist_2EisPREFIX @ A_27a ) @ V2t ) @ V5t_27 ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2EisPREFIX__THM,axiom,
! [A_27a: del,V0l: $i] :
( ( mem @ V0l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V1h: $i] :
( ( mem @ V1h @ A_27a )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V3h1: $i] :
( ( mem @ V3h1 @ A_27a )
=> ! [V4t1: $i] :
( ( mem @ V4t1 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V5h2: $i] :
( ( mem @ V5h2 @ A_27a )
=> ! [V6t2: $i] :
( ( mem @ V6t2 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Elist_2EisPREFIX @ A_27a ) @ ( c_2Elist_2ENIL @ A_27a ) ) @ V0l ) )
<=> $true )
& ( ( p @ ( ap @ ( ap @ ( c_2Elist_2EisPREFIX @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V1h ) @ V2t ) ) @ ( c_2Elist_2ENIL @ A_27a ) ) )
<=> $false )
& ( ( p @ ( ap @ ( ap @ ( c_2Elist_2EisPREFIX @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V3h1 ) @ V4t1 ) ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V5h2 ) @ V6t2 ) ) )
<=> ( ( V3h1 = V5h2 )
& ( p @ ( ap @ ( ap @ ( c_2Elist_2EisPREFIX @ A_27a ) @ V4t1 ) @ V6t2 ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2EisPREFIX__NILR,axiom,
! [A_27a: del,V0x: $i] :
( ( mem @ V0x @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Elist_2EisPREFIX @ A_27a ) @ V0x ) @ ( c_2Elist_2ENIL @ A_27a ) ) )
<=> ( V0x
= ( c_2Elist_2ENIL @ A_27a ) ) ) ) ).
thf(conj_thm_2Elist_2EisPREFIX__CONSR,axiom,
! [A_27a: del,V0x: $i] :
( ( mem @ V0x @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V1y: $i] :
( ( mem @ V1y @ A_27a )
=> ! [V2ys: $i] :
( ( mem @ V2ys @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Elist_2EisPREFIX @ A_27a ) @ V0x ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V1y ) @ V2ys ) ) )
<=> ( ( V0x
= ( c_2Elist_2ENIL @ A_27a ) )
| ? [V3xs: $i] :
( ( mem @ V3xs @ ( ty_2Elist_2Elist @ A_27a ) )
& ( V0x
= ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V1y ) @ V3xs ) )
& ( p @ ( ap @ ( ap @ ( c_2Elist_2EisPREFIX @ A_27a ) @ V3xs ) @ V2ys ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Elist_2ESNOC,axiom,
! [A_27a: del] :
( ! [V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ( ( ap @ ( ap @ ( c_2Elist_2ESNOC @ A_27a ) @ V0x ) @ ( c_2Elist_2ENIL @ A_27a ) )
= ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V0x ) @ ( c_2Elist_2ENIL @ A_27a ) ) ) )
& ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ! [V2x_27: $i] :
( ( mem @ V2x_27 @ A_27a )
=> ! [V3l: $i] :
( ( mem @ V3l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( ap @ ( ap @ ( c_2Elist_2ESNOC @ A_27a ) @ V1x ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V2x_27 ) @ V3l ) )
= ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V2x_27 ) @ ( ap @ ( ap @ ( c_2Elist_2ESNOC @ A_27a ) @ V1x ) @ V3l ) ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2ELENGTH__SNOC,axiom,
! [A_27a: del,V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ! [V1l: $i] :
( ( mem @ V1l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2ESNOC @ A_27a ) @ V0x ) @ V1l ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ c_2Enum_2ESUC @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ V1l ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2ELAST__SNOC,axiom,
! [A_27a: del,V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ! [V1l: $i] :
( ( mem @ V1l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( ap @ ( c_2Elist_2ELAST @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2ESNOC @ A_27a ) @ V0x ) @ V1l ) )
= V0x ) ) ) ).
thf(conj_thm_2Elist_2EFRONT__SNOC,axiom,
! [A_27a: del,V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ! [V1l: $i] :
( ( mem @ V1l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( ap @ ( c_2Elist_2EFRONT @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2ESNOC @ A_27a ) @ V0x ) @ V1l ) )
= V1l ) ) ) ).
thf(conj_thm_2Elist_2ESNOC__APPEND,axiom,
! [A_27a: del,V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ! [V1l: $i] :
( ( mem @ V1l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( ap @ ( ap @ ( c_2Elist_2ESNOC @ A_27a ) @ V0x ) @ V1l )
= ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ V1l ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V0x ) @ ( c_2Elist_2ENIL @ A_27a ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2ELIST__TO__SET__SNOC,axiom,
! [A_27a: del,V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ! [V1ls: $i] :
( ( mem @ V1ls @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( ap @ ( c_2Elist_2ELIST__TO__SET @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2ESNOC @ A_27a ) @ V0x ) @ V1ls ) )
= ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ A_27a ) @ V0x ) @ ( ap @ ( c_2Elist_2ELIST__TO__SET @ A_27a ) @ V1ls ) ) ) ) ) ).
thf(conj_thm_2Elist_2EMAP__SNOC,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27b ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ! [V2l: $i] :
( ( mem @ V2l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( ap @ ( ap @ ( c_2Elist_2EMAP @ A_27a @ A_27b ) @ V0f ) @ ( ap @ ( ap @ ( c_2Elist_2ESNOC @ A_27a ) @ V1x ) @ V2l ) )
= ( ap @ ( ap @ ( c_2Elist_2ESNOC @ A_27b ) @ ( ap @ V0f @ V1x ) ) @ ( ap @ ( ap @ ( c_2Elist_2EMAP @ A_27a @ A_27b ) @ V0f ) @ V2l ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2EEL__SNOC,axiom,
! [A_27a: del,V0n: tp__ty_2Enum_2Enum,V1l: $i] :
( ( mem @ V1l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V0n ) ) @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ V1l ) ) )
=> ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ( ( ap @ ( ap @ ( c_2Elist_2EEL @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) @ ( ap @ ( ap @ ( c_2Elist_2ESNOC @ A_27a ) @ V2x ) @ V1l ) )
= ( ap @ ( ap @ ( c_2Elist_2EEL @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) @ V1l ) ) ) ) ) ).
thf(conj_thm_2Elist_2EEL__LENGTH__SNOC,axiom,
! [A_27a: del,V0l: $i] :
( ( mem @ V0l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ( ( ap @ ( ap @ ( c_2Elist_2EEL @ A_27a ) @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ V0l ) ) @ ( ap @ ( ap @ ( c_2Elist_2ESNOC @ A_27a ) @ V1x ) @ V0l ) )
= V1x ) ) ) ).
thf(conj_thm_2Elist_2EAPPEND__SNOC,axiom,
! [A_27a: del,V0l1: $i] :
( ( mem @ V0l1 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ! [V2l2: $i] :
( ( mem @ V2l2 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ V0l1 ) @ ( ap @ ( ap @ ( c_2Elist_2ESNOC @ A_27a ) @ V1x ) @ V2l2 ) )
= ( ap @ ( ap @ ( c_2Elist_2ESNOC @ A_27a ) @ V1x ) @ ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ V0l1 ) @ V2l2 ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2EEVERY__SNOC,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ! [V2l: $i] :
( ( mem @ V2l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Elist_2EEVERY @ A_27a ) @ V0P ) @ ( ap @ ( ap @ ( c_2Elist_2ESNOC @ A_27a ) @ V1x ) @ V2l ) ) )
<=> ( ( p @ ( ap @ ( ap @ ( c_2Elist_2EEVERY @ A_27a ) @ V0P ) @ V2l ) )
& ( p @ ( ap @ V0P @ V1x ) ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2EEXISTS__SNOC,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ! [V2l: $i] :
( ( mem @ V2l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Elist_2EEXISTS @ A_27a ) @ V0P ) @ ( ap @ ( ap @ ( c_2Elist_2ESNOC @ A_27a ) @ V1x ) @ V2l ) ) )
<=> ( ( p @ ( ap @ V0P @ V1x ) )
| ( p @ ( ap @ ( ap @ ( c_2Elist_2EEXISTS @ A_27a ) @ V0P ) @ V2l ) ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2EMEM__SNOC,axiom,
! [A_27a: del,V0y: $i] :
( ( mem @ V0y @ A_27a )
=> ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ! [V2l: $i] :
( ( mem @ V2l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V0y ) @ ( ap @ ( c_2Elist_2ELIST__TO__SET @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2ESNOC @ A_27a ) @ V1x ) @ V2l ) ) ) )
<=> ( ( V0y = V1x )
| ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V0y ) @ ( ap @ ( c_2Elist_2ELIST__TO__SET @ A_27a ) @ V2l ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2ESNOC__11,axiom,
! [A_27a: del,V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ! [V1y: $i] :
( ( mem @ V1y @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V2a: $i] :
( ( mem @ V2a @ A_27a )
=> ! [V3b: $i] :
( ( mem @ V3b @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( ( ap @ ( ap @ ( c_2Elist_2ESNOC @ A_27a ) @ V0x ) @ V1y )
= ( ap @ ( ap @ ( c_2Elist_2ESNOC @ A_27a ) @ V2a ) @ V3b ) )
<=> ( ( V0x = V2a )
& ( V1y = V3b ) ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2EREVERSE__SNOC__DEF,axiom,
! [A_27a: del,A_27b: del] :
( ( ( ap @ ( c_2Elist_2EREVERSE @ A_27b ) @ ( c_2Elist_2ENIL @ A_27b ) )
= ( c_2Elist_2ENIL @ A_27b ) )
& ! [V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ! [V1l: $i] :
( ( mem @ V1l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( ap @ ( c_2Elist_2EREVERSE @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V0x ) @ V1l ) )
= ( ap @ ( ap @ ( c_2Elist_2ESNOC @ A_27a ) @ V0x ) @ ( ap @ ( c_2Elist_2EREVERSE @ A_27a ) @ V1l ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2EREVERSE__SNOC,axiom,
! [A_27a: del,V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ! [V1l: $i] :
( ( mem @ V1l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( ap @ ( c_2Elist_2EREVERSE @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2ESNOC @ A_27a ) @ V0x ) @ V1l ) )
= ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V0x ) @ ( ap @ ( c_2Elist_2EREVERSE @ A_27a ) @ V1l ) ) ) ) ) ).
thf(conj_thm_2Elist_2ESNOC__Axiom,axiom,
! [A_27a: del,A_27b: del,V0e: $i] :
( ( mem @ V0e @ A_27b )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ ( arr @ ( ty_2Elist_2Elist @ A_27a ) @ ( arr @ A_27b @ A_27b ) ) ) )
=> ? [V2fn: $i] :
( ( mem @ V2fn @ ( arr @ ( ty_2Elist_2Elist @ A_27a ) @ A_27b ) )
& ( ( ap @ V2fn @ ( c_2Elist_2ENIL @ A_27a ) )
= V0e )
& ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ! [V4l: $i] :
( ( mem @ V4l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( ap @ V2fn @ ( ap @ ( ap @ ( c_2Elist_2ESNOC @ A_27a ) @ V3x ) @ V4l ) )
= ( ap @ ( ap @ ( ap @ V1f @ V3x ) @ V4l ) @ ( ap @ V2fn @ V4l ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2ESNOC__INDUCT,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ ( ty_2Elist_2Elist @ A_27a ) @ bool ) )
=> ( ( ( p @ ( ap @ V0P @ ( c_2Elist_2ENIL @ A_27a ) ) )
& ! [V1l: $i] :
( ( mem @ V1l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( p @ ( ap @ V0P @ V1l ) )
=> ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ( p @ ( ap @ V0P @ ( ap @ ( ap @ ( c_2Elist_2ESNOC @ A_27a ) @ V2x ) @ V1l ) ) ) ) ) ) )
=> ! [V3l: $i] :
( ( mem @ V3l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( p @ ( ap @ V0P @ V3l ) ) ) ) ) ).
thf(conj_thm_2Elist_2ESNOC__CASES,axiom,
! [A_27a: del,V0ll: $i] :
( ( mem @ V0ll @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( V0ll
= ( c_2Elist_2ENIL @ A_27a ) )
| ? [V1x: $i] :
( ( mem @ V1x @ A_27a )
& ? [V2l: $i] :
( ( mem @ V2l @ ( ty_2Elist_2Elist @ A_27a ) )
& ( V0ll
= ( ap @ ( ap @ ( c_2Elist_2ESNOC @ A_27a ) @ V1x ) @ V2l ) ) ) ) ) ) ).
thf(ax_thm_2Elist_2EGENLIST,axiom,
! [A_27a: del] :
( ! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ A_27a ) )
=> ( ( ap @ ( ap @ ( c_2Elist_2EGENLIST @ A_27a ) @ V0f ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) )
= ( c_2Elist_2ENIL @ A_27a ) ) )
& ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ ty_2Enum_2Enum @ A_27a ) )
=> ! [V2n: tp__ty_2Enum_2Enum] :
( ( ap @ ( ap @ ( c_2Elist_2EGENLIST @ A_27a ) @ V1f ) @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V2n ) ) )
= ( ap @ ( ap @ ( c_2Elist_2ESNOC @ A_27a ) @ ( ap @ V1f @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) @ ( ap @ ( ap @ ( c_2Elist_2EGENLIST @ A_27a ) @ V1f ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2ELENGTH__GENLIST,axiom,
! [A_27a: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ A_27a ) )
=> ! [V1n: tp__ty_2Enum_2Enum] :
( ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2EGENLIST @ A_27a ) @ V0f ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) )
= V1n ) ) ).
thf(ax_thm_2Elist_2EGENLIST__AUX,axiom,
! [A_27a: del] :
( ! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ A_27a ) )
=> ! [V1l: $i] :
( ( mem @ V1l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Elist_2EGENLIST__AUX @ A_27a ) @ V0f ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ V1l )
= V1l ) ) )
& ! [V2f: $i] :
( ( mem @ V2f @ ( arr @ ty_2Enum_2Enum @ A_27a ) )
=> ! [V3n: tp__ty_2Enum_2Enum,V4l: $i] :
( ( mem @ V4l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Elist_2EGENLIST__AUX @ A_27a ) @ V2f ) @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) @ V4l )
= ( ap @ ( ap @ ( ap @ ( c_2Elist_2EGENLIST__AUX @ A_27a ) @ V2f ) @ ( inj__ty_2Enum_2Enum @ V3n ) ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ ( ap @ V2f @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) @ V4l ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2EGENLIST__AUX__compute,axiom,
! [A_27a: del] :
( ! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ A_27a ) )
=> ! [V1l: $i] :
( ( mem @ V1l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Elist_2EGENLIST__AUX @ A_27a ) @ V0f ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ V1l )
= V1l ) ) )
& ! [V2f: $i] :
( ( mem @ V2f @ ( arr @ ty_2Enum_2Enum @ A_27a ) )
=> ! [V3n: tp__ty_2Enum_2Enum,V4l: $i] :
( ( mem @ V4l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Elist_2EGENLIST__AUX @ A_27a ) @ V2f ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) ) @ V4l )
= ( ap @ ( ap @ ( ap @ ( c_2Elist_2EGENLIST__AUX @ A_27a ) @ V2f ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ ( ap @ V2f @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) @ V4l ) ) ) ) )
& ! [V5f: $i] :
( ( mem @ V5f @ ( arr @ ty_2Enum_2Enum @ A_27a ) )
=> ! [V6n: tp__ty_2Enum_2Enum,V7l: $i] :
( ( mem @ V7l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Elist_2EGENLIST__AUX @ A_27a ) @ V5f ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ V6n ) ) ) ) @ V7l )
= ( ap @ ( ap @ ( ap @ ( c_2Elist_2EGENLIST__AUX @ A_27a ) @ V5f ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ V6n ) ) ) ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ ( ap @ V5f @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ V6n ) ) ) ) ) @ V7l ) ) ) ) ) ) ).
thf(ax_thm_2Elist_2EPAD__LEFT,axiom,
! [A_27a: del,V0c: $i] :
( ( mem @ V0c @ A_27a )
=> ! [V1n: tp__ty_2Enum_2Enum,V2s: $i] :
( ( mem @ V2s @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Elist_2EPAD__LEFT @ A_27a ) @ V0c ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) @ V2s )
= ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2EGENLIST @ A_27a ) @ ( ap @ ( c_2Ecombin_2EK @ A_27a @ ty_2Enum_2Enum ) @ V0c ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ ( inj__ty_2Enum_2Enum @ V1n ) ) @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ V2s ) ) ) ) @ V2s ) ) ) ) ).
thf(ax_thm_2Elist_2EPAD__RIGHT,axiom,
! [A_27a: del,V0c: $i] :
( ( mem @ V0c @ A_27a )
=> ! [V1n: tp__ty_2Enum_2Enum,V2s: $i] :
( ( mem @ V2s @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Elist_2EPAD__RIGHT @ A_27a ) @ V0c ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) @ V2s )
= ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ V2s ) @ ( ap @ ( ap @ ( c_2Elist_2EGENLIST @ A_27a ) @ ( ap @ ( c_2Ecombin_2EK @ A_27a @ ty_2Enum_2Enum ) @ V0c ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ ( inj__ty_2Enum_2Enum @ V1n ) ) @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ V2s ) ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2EMAP__GENLIST,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27b ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ ty_2Enum_2Enum @ A_27a ) )
=> ! [V2n: tp__ty_2Enum_2Enum] :
( ( ap @ ( ap @ ( c_2Elist_2EMAP @ A_27a @ A_27b ) @ V0f ) @ ( ap @ ( ap @ ( c_2Elist_2EGENLIST @ A_27a ) @ V1g ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) )
= ( ap @ ( ap @ ( c_2Elist_2EGENLIST @ A_27b ) @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ ty_2Enum_2Enum @ A_27b @ A_27a ) @ V0f ) @ V1g ) ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) ) ) ).
thf(conj_thm_2Elist_2EEL__GENLIST,axiom,
! [A_27a: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ A_27a ) )
=> ! [V1n: tp__ty_2Enum_2Enum,V2x: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V2x ) ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) )
=> ( ( ap @ ( ap @ ( c_2Elist_2EEL @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V2x ) ) @ ( ap @ ( ap @ ( c_2Elist_2EGENLIST @ A_27a ) @ V0f ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) )
= ( ap @ V0f @ ( inj__ty_2Enum_2Enum @ V2x ) ) ) ) ) ).
thf(conj_thm_2Elist_2EHD__GENLIST,axiom,
! [A_27a: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ A_27a ) )
=> ! [V1n: tp__ty_2Enum_2Enum] :
( ( ap @ ( c_2Elist_2EHD @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2EGENLIST @ A_27a ) @ V0f ) @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) )
= ( ap @ V0f @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ).
thf(conj_thm_2Elist_2EHD__GENLIST__COR,axiom,
! [A_27a: del,V0n: tp__ty_2Enum_2Enum,V1f: $i] :
( ( mem @ V1f @ ( arr @ ty_2Enum_2Enum @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) )
=> ( ( ap @ ( c_2Elist_2EHD @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2EGENLIST @ A_27a ) @ V1f ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) )
= ( ap @ V1f @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) ).
thf(conj_thm_2Elist_2EGENLIST__FUN__EQ,axiom,
! [A_27a: del,V0n: tp__ty_2Enum_2Enum,V1f: $i] :
( ( mem @ V1f @ ( arr @ ty_2Enum_2Enum @ A_27a ) )
=> ! [V2g: $i] :
( ( mem @ V2g @ ( arr @ ty_2Enum_2Enum @ A_27a ) )
=> ( ( ( ap @ ( ap @ ( c_2Elist_2EGENLIST @ A_27a ) @ V1f ) @ ( inj__ty_2Enum_2Enum @ V0n ) )
= ( ap @ ( ap @ ( c_2Elist_2EGENLIST @ A_27a ) @ V2g ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) )
<=> ! [V3x: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V3x ) ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) )
=> ( ( ap @ V1f @ ( inj__ty_2Enum_2Enum @ V3x ) )
= ( ap @ V2g @ ( inj__ty_2Enum_2Enum @ V3x ) ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2EGENLIST__APPEND,axiom,
! [A_27a: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ A_27a ) )
=> ! [V1a: tp__ty_2Enum_2Enum,V2b: tp__ty_2Enum_2Enum] :
( ( ap @ ( ap @ ( c_2Elist_2EGENLIST @ A_27a ) @ V0f ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( inj__ty_2Enum_2Enum @ V1a ) ) @ ( inj__ty_2Enum_2Enum @ V2b ) ) )
= ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2EGENLIST @ A_27a ) @ V0f ) @ ( inj__ty_2Enum_2Enum @ V2b ) ) )
@ ( ap
@ ( ap @ ( c_2Elist_2EGENLIST @ A_27a )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V3t: $i] : ( ap @ V0f @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ V3t ) @ ( inj__ty_2Enum_2Enum @ V2b ) ) ) ) )
@ ( inj__ty_2Enum_2Enum @ V1a ) ) ) ) ) ).
thf(conj_thm_2Elist_2EEVERY__GENLIST,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ ty_2Enum_2Enum @ A_27a ) )
=> ! [V2n: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ ( c_2Elist_2EEVERY @ A_27a ) @ V0P ) @ ( ap @ ( ap @ ( c_2Elist_2EGENLIST @ A_27a ) @ V1f ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) )
<=> ! [V3i: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V3i ) ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) )
=> ( p @ ( ap @ V0P @ ( ap @ V1f @ ( inj__ty_2Enum_2Enum @ V3i ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2EEXISTS__GENLIST,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ ty_2Enum_2Enum @ A_27a ) )
=> ! [V2n: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ ( c_2Elist_2EEXISTS @ A_27a ) @ V0P ) @ ( ap @ ( ap @ ( c_2Elist_2EGENLIST @ A_27a ) @ V1f ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) )
<=> ? [V3i: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V3i ) ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) )
& ( p @ ( ap @ V0P @ ( ap @ V1f @ ( inj__ty_2Enum_2Enum @ V3i ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2ETL__GENLIST,axiom,
! [A_27a: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ A_27a ) )
=> ! [V1n: tp__ty_2Enum_2Enum] :
( ( ap @ ( c_2Elist_2ETL @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2EGENLIST @ A_27a ) @ V0f ) @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) )
= ( ap @ ( ap @ ( c_2Elist_2EGENLIST @ A_27a ) @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ ty_2Enum_2Enum @ A_27a @ ty_2Enum_2Enum ) @ V0f ) @ c_2Enum_2ESUC ) ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) ) ).
thf(conj_thm_2Elist_2EZIP__GENLIST,axiom,
! [A_27a: del,A_27b: del,V0l: $i] :
( ( mem @ V0l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ ty_2Enum_2Enum @ A_27b ) )
=> ! [V2n: tp__ty_2Enum_2Enum] :
( ( ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ V0l ) )
= V2n )
=> ( ( ap @ ( c_2Elist_2EZIP @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Elist_2Elist @ A_27a ) @ ( ty_2Elist_2Elist @ A_27b ) ) @ V0l ) @ ( ap @ ( ap @ ( c_2Elist_2EGENLIST @ A_27b ) @ V1f ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) )
= ( ap
@ ( ap @ ( c_2Elist_2EGENLIST @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V3x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Elist_2EEL @ A_27a ) @ V3x ) @ V0l ) ) @ ( ap @ V1f @ V3x ) ) ) )
@ ( inj__ty_2Enum_2Enum @ V2n ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2EGENLIST__CONS,axiom,
! [A_27a: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ A_27a ) )
=> ! [V1n: tp__ty_2Enum_2Enum] :
( ( ap @ ( ap @ ( c_2Elist_2EGENLIST @ A_27a ) @ V0f ) @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V1n ) ) )
= ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ ( ap @ V0f @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ ( ap @ ( c_2Elist_2EGENLIST @ A_27a ) @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ ty_2Enum_2Enum @ A_27a @ ty_2Enum_2Enum ) @ V0f ) @ c_2Enum_2ESUC ) ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) ) ) ).
thf(conj_thm_2Elist_2ENULL__GENLIST,axiom,
! [A_27a: del,V0n: tp__ty_2Enum_2Enum,V1f: $i] :
( ( mem @ V1f @ ( arr @ ty_2Enum_2Enum @ A_27a ) )
=> ( ( p @ ( ap @ ( c_2Elist_2ENULL @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2EGENLIST @ A_27a ) @ V1f ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) )
<=> ( V0n = fo__c_2Enum_2E0 ) ) ) ).
thf(conj_thm_2Elist_2EGENLIST__GENLIST__AUX,axiom,
! [A_27a: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ A_27a ) )
=> ! [V1n: tp__ty_2Enum_2Enum] :
( ( ap @ ( ap @ ( c_2Elist_2EGENLIST @ A_27a ) @ V0f ) @ ( inj__ty_2Enum_2Enum @ V1n ) )
= ( ap @ ( ap @ ( ap @ ( c_2Elist_2EGENLIST__AUX @ A_27a ) @ V0f ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) @ ( c_2Elist_2ENIL @ A_27a ) ) ) ) ).
thf(conj_thm_2Elist_2EGENLIST__NUMERALS,axiom,
! [A_27a: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ A_27a ) )
=> ! [V1n: tp__ty_2Enum_2Enum] :
( ( ( ap @ ( ap @ ( c_2Elist_2EGENLIST @ A_27a ) @ V0f ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) )
= ( c_2Elist_2ENIL @ A_27a ) )
& ( ( ap @ ( ap @ ( c_2Elist_2EGENLIST @ A_27a ) @ V0f ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( inj__ty_2Enum_2Enum @ V1n ) ) )
= ( ap @ ( ap @ ( ap @ ( c_2Elist_2EGENLIST__AUX @ A_27a ) @ V0f ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) @ ( c_2Elist_2ENIL @ A_27a ) ) ) ) ) ).
thf(conj_thm_2Elist_2EMEM__GENLIST,axiom,
! [A_27a: del,V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ ty_2Enum_2Enum @ A_27a ) )
=> ! [V2n: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V0x ) @ ( ap @ ( c_2Elist_2ELIST__TO__SET @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2EGENLIST @ A_27a ) @ V1f ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) ) )
<=> ? [V3m: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V3m ) ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) )
& ( V0x
= ( ap @ V1f @ ( inj__ty_2Enum_2Enum @ V3m ) ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2EALL__DISTINCT__SNOC,axiom,
! [A_27a: del,V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ! [V1l: $i] :
( ( mem @ V1l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( p @ ( ap @ ( c_2Elist_2EALL__DISTINCT @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2ESNOC @ A_27a ) @ V0x ) @ V1l ) ) )
<=> ( ~ ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V0x ) @ ( ap @ ( c_2Elist_2ELIST__TO__SET @ A_27a ) @ V1l ) ) )
& ( p @ ( ap @ ( c_2Elist_2EALL__DISTINCT @ A_27a ) @ V1l ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2EALL__DISTINCT__GENLIST,axiom,
! [A_27a: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ A_27a ) )
=> ! [V1n: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( c_2Elist_2EALL__DISTINCT @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2EGENLIST @ A_27a ) @ V0f ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) )
<=> ! [V2m1: tp__ty_2Enum_2Enum,V3m2: tp__ty_2Enum_2Enum] :
( ( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V2m1 ) ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) )
& ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V3m2 ) ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) )
& ( ( ap @ V0f @ ( inj__ty_2Enum_2Enum @ V2m1 ) )
= ( ap @ V0f @ ( inj__ty_2Enum_2Enum @ V3m2 ) ) ) )
=> ( V2m1 = V3m2 ) ) ) ) ).
thf(conj_thm_2Elist_2EFOLDL__SNOC,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27b @ ( arr @ A_27a @ A_27b ) ) )
=> ! [V1e: $i] :
( ( mem @ V1e @ A_27b )
=> ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ! [V3l: $i] :
( ( mem @ V3l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Elist_2EFOLDL @ A_27a @ A_27b ) @ V0f ) @ V1e ) @ ( ap @ ( ap @ ( c_2Elist_2ESNOC @ A_27a ) @ V2x ) @ V3l ) )
= ( ap @ ( ap @ V0f @ ( ap @ ( ap @ ( ap @ ( c_2Elist_2EFOLDL @ A_27a @ A_27b ) @ V0f ) @ V1e ) @ V3l ) ) @ V2x ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2ESUM__SNOC,axiom,
! [V0x: tp__ty_2Enum_2Enum,V1l: tp__c_ty_2Elist_2Elist_ty_2Enum_2Enum] :
( ( surj__ty_2Enum_2Enum @ ( ap @ c_2Elist_2ESUM @ ( ap @ ( ap @ ( c_2Elist_2ESNOC @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V0x ) ) @ ( inj__c_ty_2Elist_2Elist_ty_2Enum_2Enum @ V1l ) ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( ap @ c_2Elist_2ESUM @ ( inj__c_ty_2Elist_2Elist_ty_2Enum_2Enum @ V1l ) ) ) @ ( inj__ty_2Enum_2Enum @ V0x ) ) ) ) ).
thf(conj_thm_2Elist_2ESUM__APPEND,axiom,
! [V0l1: tp__c_ty_2Elist_2Elist_ty_2Enum_2Enum,V1l2: tp__c_ty_2Elist_2Elist_ty_2Enum_2Enum] :
( ( surj__ty_2Enum_2Enum @ ( ap @ c_2Elist_2ESUM @ ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ ty_2Enum_2Enum ) @ ( inj__c_ty_2Elist_2Elist_ty_2Enum_2Enum @ V0l1 ) ) @ ( inj__c_ty_2Elist_2Elist_ty_2Enum_2Enum @ V1l2 ) ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( ap @ c_2Elist_2ESUM @ ( inj__c_ty_2Elist_2Elist_ty_2Enum_2Enum @ V0l1 ) ) ) @ ( ap @ c_2Elist_2ESUM @ ( inj__c_ty_2Elist_2Elist_ty_2Enum_2Enum @ V1l2 ) ) ) ) ) ).
thf(conj_thm_2Elist_2ESUM__MAP__FOLDL,axiom,
! [A_27a: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V1ls: $i] :
( ( mem @ V1ls @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ c_2Elist_2ESUM @ ( ap @ ( ap @ ( c_2Elist_2EMAP @ A_27a @ ty_2Enum_2Enum ) @ V0f ) @ V1ls ) ) )
= ( surj__ty_2Enum_2Enum
@ ( ap
@ ( ap
@ ( ap @ ( c_2Elist_2EFOLDL @ A_27a @ ty_2Enum_2Enum )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V2a: $i] :
( lam @ A_27a
@ ^ [V3e: $i] : ( ap @ ( ap @ c_2Earithmetic_2E_2B @ V2a ) @ ( ap @ V0f @ V3e ) ) ) ) )
@ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) )
@ V1ls ) ) ) ) ) ).
thf(conj_thm_2Elist_2ESUM__IMAGE__eq__SUM__MAP__SET__TO__LIST,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V0s ) )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ ( c_2Epred__set_2ESUM__IMAGE @ A_27a ) @ V1f ) @ V0s ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ c_2Elist_2ESUM @ ( ap @ ( ap @ ( c_2Elist_2EMAP @ A_27a @ ty_2Enum_2Enum ) @ V1f ) @ ( ap @ ( c_2Elist_2ESET__TO__LIST @ A_27a ) @ V0s ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2EEL__REVERSE,axiom,
! [A_27a: del,V0n: tp__ty_2Enum_2Enum,V1l: $i] :
( ( mem @ V1l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V0n ) ) @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ V1l ) ) )
=> ( ( ap @ ( ap @ ( c_2Elist_2EEL @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) @ ( ap @ ( c_2Elist_2EREVERSE @ A_27a ) @ V1l ) )
= ( ap @ ( ap @ ( c_2Elist_2EEL @ A_27a ) @ ( ap @ c_2Eprim__rec_2EPRE @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ V1l ) ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) ) @ V1l ) ) ) ) ).
thf(conj_thm_2Elist_2EREVERSE__GENLIST,axiom,
! [A_27a: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ A_27a ) )
=> ! [V1n: tp__ty_2Enum_2Enum] :
( ( ap @ ( c_2Elist_2EREVERSE @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2EGENLIST @ A_27a ) @ V0f ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) )
= ( ap
@ ( ap @ ( c_2Elist_2EGENLIST @ A_27a )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V2m: $i] : ( ap @ V0f @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ ( ap @ c_2Eprim__rec_2EPRE @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) @ V2m ) ) ) )
@ ( inj__ty_2Enum_2Enum @ V1n ) ) ) ) ).
thf(conj_thm_2Elist_2EFOLDL__UNION__BIGUNION,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) )
=> ! [V1ls: $i] :
( ( mem @ V1ls @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ A_27b @ bool ) )
=> ( ( ap
@ ( ap
@ ( ap @ ( c_2Elist_2EFOLDL @ A_27a @ ( arr @ A_27b @ bool ) )
@ ( lam @ ( arr @ A_27b @ bool )
@ ^ [V3s: $i] :
( lam @ A_27a
@ ^ [V4x: $i] : ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ A_27b ) @ V3s ) @ ( ap @ V0f @ V4x ) ) ) ) )
@ V2s )
@ V1ls )
= ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ A_27b ) @ V2s ) @ ( ap @ ( c_2Epred__set_2EBIGUNION @ A_27b ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ A_27a @ ( arr @ A_27b @ bool ) ) @ V0f ) @ ( ap @ ( c_2Elist_2ELIST__TO__SET @ A_27a ) @ V1ls ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2EFOLDL__UNION__BIGUNION__paired,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ( arr @ A_27b @ ( arr @ A_27c @ bool ) ) ) )
=> ! [V1ls: $i] :
( ( mem @ V1ls @ ( ty_2Elist_2Elist @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) ) )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ A_27c @ bool ) )
=> ( ( ap
@ ( ap
@ ( ap @ ( c_2Elist_2EFOLDL @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ ( arr @ A_27c @ bool ) )
@ ( lam @ ( arr @ A_27c @ bool )
@ ^ [V3s: $i] :
( ap @ ( c_2Epair_2EUNCURRY @ A_27a @ A_27b @ ( arr @ A_27c @ bool ) )
@ ( lam @ A_27a
@ ^ [V4x: $i] :
( lam @ A_27b
@ ^ [V5y: $i] : ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ A_27c ) @ V3s ) @ ( ap @ ( ap @ V0f @ V4x ) @ V5y ) ) ) ) ) ) )
@ V2s )
@ V1ls )
= ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ A_27c ) @ V2s ) @ ( ap @ ( c_2Epred__set_2EBIGUNION @ A_27c ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ ( arr @ A_27c @ bool ) ) @ ( ap @ ( c_2Epair_2EUNCURRY @ A_27a @ A_27b @ ( arr @ A_27c @ bool ) ) @ V0f ) ) @ ( ap @ ( c_2Elist_2ELIST__TO__SET @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) ) @ V1ls ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2EFOLDL__ZIP__SAME,axiom,
! [A_27a: del,A_27b: del,V0ls: $i] :
( ( mem @ V0ls @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27b @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ A_27b ) ) )
=> ! [V2e: $i] :
( ( mem @ V2e @ A_27b )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Elist_2EFOLDL @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ A_27b ) @ V1f ) @ V2e ) @ ( ap @ ( c_2Elist_2EZIP @ A_27a @ A_27a ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Elist_2Elist @ A_27a ) @ ( ty_2Elist_2Elist @ A_27a ) ) @ V0ls ) @ V0ls ) ) )
= ( ap
@ ( ap
@ ( ap @ ( c_2Elist_2EFOLDL @ A_27a @ A_27b )
@ ( lam @ A_27b
@ ^ [V3x: $i] :
( lam @ A_27a
@ ^ [V4y: $i] : ( ap @ ( ap @ V1f @ V3x ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V4y ) @ V4y ) ) ) ) )
@ V2e )
@ V0ls ) ) ) ) ) ).
thf(conj_thm_2Elist_2EMAP__ZIP__SAME,axiom,
! [A_27a: del,A_27b: del,V0ls: $i] :
( ( mem @ V0ls @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ A_27b ) )
=> ( ( ap @ ( ap @ ( c_2Elist_2EMAP @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ A_27b ) @ V1f ) @ ( ap @ ( c_2Elist_2EZIP @ A_27a @ A_27a ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Elist_2Elist @ A_27a ) @ ( ty_2Elist_2Elist @ A_27a ) ) @ V0ls ) @ V0ls ) ) )
= ( ap
@ ( ap @ ( c_2Elist_2EMAP @ A_27a @ A_27b )
@ ( lam @ A_27a
@ ^ [V2x: $i] : ( ap @ V1f @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V2x ) @ V2x ) ) ) )
@ V0ls ) ) ) ) ).
thf(conj_thm_2Elist_2EINFINITE__LIST__UNIV,axiom,
! [A_27a: del] :
~ ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ ( ty_2Elist_2Elist @ A_27a ) ) @ ( c_2Epred__set_2EUNIV @ ( ty_2Elist_2Elist @ A_27a ) ) ) ) ).
thf(ax_thm_2Elist_2ELEN__DEF,axiom,
! [A_27a: del] :
( ! [V0n: tp__ty_2Enum_2Enum] :
( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ ( c_2Elist_2ELEN @ A_27a ) @ ( c_2Elist_2ENIL @ A_27a ) ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) )
= V0n )
& ! [V1h: $i] :
( ( mem @ V1h @ A_27a )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V3n: tp__ty_2Enum_2Enum] :
( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ ( c_2Elist_2ELEN @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V1h ) @ V2t ) ) @ ( inj__ty_2Enum_2Enum @ V3n ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ ( c_2Elist_2ELEN @ A_27a ) @ V2t ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( inj__ty_2Enum_2Enum @ V3n ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Elist_2EREV__DEF,axiom,
! [A_27a: del] :
( ! [V0acc: $i] :
( ( mem @ V0acc @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( ap @ ( ap @ ( c_2Elist_2EREV @ A_27a ) @ ( c_2Elist_2ENIL @ A_27a ) ) @ V0acc )
= V0acc ) )
& ! [V1h: $i] :
( ( mem @ V1h @ A_27a )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V3acc: $i] :
( ( mem @ V3acc @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( ap @ ( ap @ ( c_2Elist_2EREV @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V1h ) @ V2t ) ) @ V3acc )
= ( ap @ ( ap @ ( c_2Elist_2EREV @ A_27a ) @ V2t ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V1h ) @ V3acc ) ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2ELEN__LENGTH__LEM,axiom,
! [A_27a: del,V0L: $i] :
( ( mem @ V0L @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V1n: tp__ty_2Enum_2Enum] :
( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ ( c_2Elist_2ELEN @ A_27a ) @ V0L ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ V0L ) ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) ) ) ).
thf(conj_thm_2Elist_2EREV__REVERSE__LEM,axiom,
! [A_27a: del,V0L1: $i] :
( ( mem @ V0L1 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V1L2: $i] :
( ( mem @ V1L2 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( ap @ ( ap @ ( c_2Elist_2EREV @ A_27a ) @ V0L1 ) @ V1L2 )
= ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ ( ap @ ( c_2Elist_2EREVERSE @ A_27a ) @ V0L1 ) ) @ V1L2 ) ) ) ) ).
thf(conj_thm_2Elist_2ELENGTH__LEN,axiom,
! [A_27a: del,V0L: $i] :
( ( mem @ V0L @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ V0L ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ ( c_2Elist_2ELEN @ A_27a ) @ V0L ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) ).
thf(conj_thm_2Elist_2EREVERSE__REV,axiom,
! [A_27a: del,V0L: $i] :
( ( mem @ V0L @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( ap @ ( c_2Elist_2EREVERSE @ A_27a ) @ V0L )
= ( ap @ ( ap @ ( c_2Elist_2EREV @ A_27a ) @ V0L ) @ ( c_2Elist_2ENIL @ A_27a ) ) ) ) ).
thf(ax_thm_2Elist_2ESUM__ACC__DEF,axiom,
( ! [V0acc: tp__ty_2Enum_2Enum] :
( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Elist_2ESUM__ACC @ ( c_2Elist_2ENIL @ ty_2Enum_2Enum ) ) @ ( inj__ty_2Enum_2Enum @ V0acc ) ) )
= V0acc )
& ! [V1h: tp__ty_2Enum_2Enum,V2t: tp__c_ty_2Elist_2Elist_ty_2Enum_2Enum,V3acc: tp__ty_2Enum_2Enum] :
( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Elist_2ESUM__ACC @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V1h ) ) @ ( inj__c_ty_2Elist_2Elist_ty_2Enum_2Enum @ V2t ) ) ) @ ( inj__ty_2Enum_2Enum @ V3acc ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Elist_2ESUM__ACC @ ( inj__c_ty_2Elist_2Elist_ty_2Enum_2Enum @ V2t ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( inj__ty_2Enum_2Enum @ V1h ) ) @ ( inj__ty_2Enum_2Enum @ V3acc ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2ESUM__ACC__SUM__LEM,axiom,
! [V0L: tp__c_ty_2Elist_2Elist_ty_2Enum_2Enum,V1n: tp__ty_2Enum_2Enum] :
( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Elist_2ESUM__ACC @ ( inj__c_ty_2Elist_2Elist_ty_2Enum_2Enum @ V0L ) ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( ap @ c_2Elist_2ESUM @ ( inj__c_ty_2Elist_2Elist_ty_2Enum_2Enum @ V0L ) ) ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) ) ).
thf(conj_thm_2Elist_2ESUM__SUM__ACC,axiom,
! [V0L: tp__c_ty_2Elist_2Elist_ty_2Enum_2Enum] :
( ( surj__ty_2Enum_2Enum @ ( ap @ c_2Elist_2ESUM @ ( inj__c_ty_2Elist_2Elist_ty_2Enum_2Enum @ V0L ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Elist_2ESUM__ACC @ ( inj__c_ty_2Elist_2Elist_ty_2Enum_2Enum @ V0L ) ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ).
thf(conj_thm_2Elist_2EEXISTS__LIST,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ ( ty_2Elist_2Elist @ A_27a ) @ bool ) )
=> ( ? [V1l: $i] :
( ( mem @ V1l @ ( ty_2Elist_2Elist @ A_27a ) )
& ( p @ ( ap @ V0P @ V1l ) ) )
<=> ( ( p @ ( ap @ V0P @ ( c_2Elist_2ENIL @ A_27a ) ) )
| ? [V2h: $i] :
( ( mem @ V2h @ A_27a )
& ? [V3t: $i] :
( ( mem @ V3t @ ( ty_2Elist_2Elist @ A_27a ) )
& ( p @ ( ap @ V0P @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V2h ) @ V3t ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2EFORALL__LIST,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ ( ty_2Elist_2Elist @ A_27a ) @ bool ) )
=> ( ! [V1l: $i] :
( ( mem @ V1l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( p @ ( ap @ V0P @ V1l ) ) )
<=> ( ( p @ ( ap @ V0P @ ( c_2Elist_2ENIL @ A_27a ) ) )
& ! [V2h: $i] :
( ( mem @ V2h @ A_27a )
=> ! [V3t: $i] :
( ( mem @ V3t @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( p @ ( ap @ V0P @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V2h ) @ V3t ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2EMEM__SPLIT__APPEND__first,axiom,
! [A_27a: del,V0e: $i] :
( ( mem @ V0e @ A_27a )
=> ! [V1l: $i] :
( ( mem @ V1l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V0e ) @ ( ap @ ( c_2Elist_2ELIST__TO__SET @ A_27a ) @ V1l ) ) )
<=> ? [V2pfx: $i] :
( ( mem @ V2pfx @ ( ty_2Elist_2Elist @ A_27a ) )
& ? [V3sfx: $i] :
( ( mem @ V3sfx @ ( ty_2Elist_2Elist @ A_27a ) )
& ( V1l
= ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ V2pfx ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V0e ) @ ( c_2Elist_2ENIL @ A_27a ) ) ) ) @ V3sfx ) )
& ~ ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V0e ) @ ( ap @ ( c_2Elist_2ELIST__TO__SET @ A_27a ) @ V2pfx ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2EMEM__SPLIT__APPEND__last,axiom,
! [A_27a: del,V0e: $i] :
( ( mem @ V0e @ A_27a )
=> ! [V1l: $i] :
( ( mem @ V1l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V0e ) @ ( ap @ ( c_2Elist_2ELIST__TO__SET @ A_27a ) @ V1l ) ) )
<=> ? [V2pfx: $i] :
( ( mem @ V2pfx @ ( ty_2Elist_2Elist @ A_27a ) )
& ? [V3sfx: $i] :
( ( mem @ V3sfx @ ( ty_2Elist_2Elist @ A_27a ) )
& ( V1l
= ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ V2pfx ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V0e ) @ ( c_2Elist_2ENIL @ A_27a ) ) ) ) @ V3sfx ) )
& ~ ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V0e ) @ ( ap @ ( c_2Elist_2ELIST__TO__SET @ A_27a ) @ V3sfx ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2EAPPEND__EQ__APPEND,axiom,
! [A_27a: del,V0l1: $i] :
( ( mem @ V0l1 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V1l2: $i] :
( ( mem @ V1l2 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V2m1: $i] :
( ( mem @ V2m1 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V3m2: $i] :
( ( mem @ V3m2 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ V0l1 ) @ V1l2 )
= ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ V2m1 ) @ V3m2 ) )
<=> ( ? [V4l: $i] :
( ( mem @ V4l @ ( ty_2Elist_2Elist @ A_27a ) )
& ( V0l1
= ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ V2m1 ) @ V4l ) )
& ( V3m2
= ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ V4l ) @ V1l2 ) ) )
| ? [V5l: $i] :
( ( mem @ V5l @ ( ty_2Elist_2Elist @ A_27a ) )
& ( V2m1
= ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ V0l1 ) @ V5l ) )
& ( V1l2
= ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ V5l ) @ V3m2 ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2EAPPEND__EQ__CONS,axiom,
! [A_27a: del,V0l1: $i] :
( ( mem @ V0l1 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V1l2: $i] :
( ( mem @ V1l2 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V2h: $i] :
( ( mem @ V2h @ A_27a )
=> ! [V3t: $i] :
( ( mem @ V3t @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ V0l1 ) @ V1l2 )
= ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V2h ) @ V3t ) )
<=> ( ( ( V0l1
= ( c_2Elist_2ENIL @ A_27a ) )
& ( V1l2
= ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V2h ) @ V3t ) ) )
| ? [V4lt: $i] :
( ( mem @ V4lt @ ( ty_2Elist_2Elist @ A_27a ) )
& ( V0l1
= ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V2h ) @ V4lt ) )
& ( V3t
= ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ V4lt ) @ V1l2 ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2EAPPEND__EQ__APPEND__MID,axiom,
! [A_27a: del,V0l1: $i] :
( ( mem @ V0l1 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V1e: $i] :
( ( mem @ V1e @ A_27a )
=> ! [V2l2: $i] :
( ( mem @ V2l2 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V3m1: $i] :
( ( mem @ V3m1 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V4m2: $i] :
( ( mem @ V4m2 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ V0l1 ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V1e ) @ ( c_2Elist_2ENIL @ A_27a ) ) ) ) @ V2l2 )
= ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ V3m1 ) @ V4m2 ) )
<=> ( ? [V5l: $i] :
( ( mem @ V5l @ ( ty_2Elist_2Elist @ A_27a ) )
& ( V3m1
= ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ V0l1 ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V1e ) @ ( c_2Elist_2ENIL @ A_27a ) ) ) ) @ V5l ) )
& ( V2l2
= ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ V5l ) @ V4m2 ) ) )
| ? [V6l: $i] :
( ( mem @ V6l @ ( ty_2Elist_2Elist @ A_27a ) )
& ( V0l1
= ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ V3m1 ) @ V6l ) )
& ( V4m2
= ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ V6l ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V1e ) @ ( c_2Elist_2ENIL @ A_27a ) ) ) ) @ V2l2 ) ) ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Elist_2ELUPDATE__def,axiom,
! [A_27a: del] :
( ! [V0e: $i] :
( ( mem @ V0e @ A_27a )
=> ! [V1n: tp__ty_2Enum_2Enum] :
( ( ap @ ( ap @ ( ap @ ( c_2Elist_2ELUPDATE @ A_27a ) @ V0e ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) @ ( c_2Elist_2ENIL @ A_27a ) )
= ( c_2Elist_2ENIL @ A_27a ) ) )
& ! [V2e: $i] :
( ( mem @ V2e @ A_27a )
=> ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ! [V4l: $i] :
( ( mem @ V4l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Elist_2ELUPDATE @ A_27a ) @ V2e ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V3x ) @ V4l ) )
= ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V2e ) @ V4l ) ) ) ) )
& ! [V5e: $i] :
( ( mem @ V5e @ A_27a )
=> ! [V6n: tp__ty_2Enum_2Enum,V7x: $i] :
( ( mem @ V7x @ A_27a )
=> ! [V8l: $i] :
( ( mem @ V8l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Elist_2ELUPDATE @ A_27a ) @ V5e ) @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V6n ) ) ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V7x ) @ V8l ) )
= ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V7x ) @ ( ap @ ( ap @ ( ap @ ( c_2Elist_2ELUPDATE @ A_27a ) @ V5e ) @ ( inj__ty_2Enum_2Enum @ V6n ) ) @ V8l ) ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2ELUPDATE__NIL,axiom,
! [A_27a: del,V0xs: $i] :
( ( mem @ V0xs @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V1n: tp__ty_2Enum_2Enum,V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ( ( ( ap @ ( ap @ ( ap @ ( c_2Elist_2ELUPDATE @ A_27a ) @ V2x ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) @ V0xs )
= ( c_2Elist_2ENIL @ A_27a ) )
<=> ( V0xs
= ( c_2Elist_2ENIL @ A_27a ) ) ) ) ) ).
thf(conj_thm_2Elist_2ELUPDATE__SEM,axiom,
! [A_27a: del] :
( ! [V0e: $i] :
( ( mem @ V0e @ A_27a )
=> ! [V1n: tp__ty_2Enum_2Enum,V2l: $i] :
( ( mem @ V2l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ ( ap @ ( ap @ ( ap @ ( c_2Elist_2ELUPDATE @ A_27a ) @ V0e ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) @ V2l ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ V2l ) ) ) ) )
& ! [V3e: $i] :
( ( mem @ V3e @ A_27a )
=> ! [V4n: tp__ty_2Enum_2Enum,V5l: $i] :
( ( mem @ V5l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V6p: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V6p ) ) @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ V5l ) ) )
=> ( ( ap @ ( ap @ ( c_2Elist_2EEL @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V6p ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Elist_2ELUPDATE @ A_27a ) @ V3e ) @ ( inj__ty_2Enum_2Enum @ V4n ) ) @ V5l ) )
= ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ A_27a ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V6p ) ) @ ( inj__ty_2Enum_2Enum @ V4n ) ) ) @ V3e ) @ ( ap @ ( ap @ ( c_2Elist_2EEL @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V6p ) ) @ V5l ) ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2EEL__LUPDATE,axiom,
! [A_27a: del,V0ys: $i] :
( ( mem @ V0ys @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ! [V2i: tp__ty_2Enum_2Enum,V3k: tp__ty_2Enum_2Enum] :
( ( ap @ ( ap @ ( c_2Elist_2EEL @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V2i ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Elist_2ELUPDATE @ A_27a ) @ V1x ) @ ( inj__ty_2Enum_2Enum @ V3k ) ) @ V0ys ) )
= ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ A_27a ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V2i ) ) @ ( inj__ty_2Enum_2Enum @ V3k ) ) ) @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V3k ) ) @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ V0ys ) ) ) ) @ V1x ) @ ( ap @ ( ap @ ( c_2Elist_2EEL @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V2i ) ) @ V0ys ) ) ) ) ) ).
thf(conj_thm_2Elist_2ELENGTH__LUPDATE,axiom,
! [A_27a: del,V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ! [V1n: tp__ty_2Enum_2Enum,V2ys: $i] :
( ( mem @ V2ys @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ ( ap @ ( ap @ ( ap @ ( c_2Elist_2ELUPDATE @ A_27a ) @ V0x ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) @ V2ys ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ V2ys ) ) ) ) ) ).
thf(conj_thm_2Elist_2ELUPDATE__LENGTH,axiom,
! [A_27a: del,V0xs: $i] :
( ( mem @ V0xs @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ! [V2y: $i] :
( ( mem @ V2y @ A_27a )
=> ! [V3ys: $i] :
( ( mem @ V3ys @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Elist_2ELUPDATE @ A_27a ) @ V1x ) @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ V0xs ) ) @ ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ V0xs ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V2y ) @ V3ys ) ) )
= ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ V0xs ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V1x ) @ V3ys ) ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2ELUPDATE__SNOC,axiom,
! [A_27a: del,V0ys: $i] :
( ( mem @ V0ys @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V1k: tp__ty_2Enum_2Enum,V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ! [V3y: $i] :
( ( mem @ V3y @ A_27a )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Elist_2ELUPDATE @ A_27a ) @ V2x ) @ ( inj__ty_2Enum_2Enum @ V1k ) ) @ ( ap @ ( ap @ ( c_2Elist_2ESNOC @ A_27a ) @ V3y ) @ V0ys ) )
= ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ( ty_2Elist_2Elist @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V1k ) ) @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ V0ys ) ) ) @ ( ap @ ( ap @ ( c_2Elist_2ESNOC @ A_27a ) @ V2x ) @ V0ys ) ) @ ( ap @ ( ap @ ( c_2Elist_2ESNOC @ A_27a ) @ V3y ) @ ( ap @ ( ap @ ( ap @ ( c_2Elist_2ELUPDATE @ A_27a ) @ V2x ) @ ( inj__ty_2Enum_2Enum @ V1k ) ) @ V0ys ) ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2EMEM__LUPDATE__E,axiom,
! [A_27a: del,V0l: $i] :
( ( mem @ V0l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ! [V2y: $i] :
( ( mem @ V2y @ A_27a )
=> ! [V3i: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V1x ) @ ( ap @ ( c_2Elist_2ELIST__TO__SET @ A_27a ) @ ( ap @ ( ap @ ( ap @ ( c_2Elist_2ELUPDATE @ A_27a ) @ V2y ) @ ( inj__ty_2Enum_2Enum @ V3i ) ) @ V0l ) ) ) )
=> ( ( V1x = V2y )
| ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V1x ) @ ( ap @ ( c_2Elist_2ELIST__TO__SET @ A_27a ) @ V0l ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2EMEM__LUPDATE,axiom,
! [A_27a: del,V0l: $i] :
( ( mem @ V0l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ! [V2y: $i] :
( ( mem @ V2y @ A_27a )
=> ! [V3i: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V1x ) @ ( ap @ ( c_2Elist_2ELIST__TO__SET @ A_27a ) @ ( ap @ ( ap @ ( ap @ ( c_2Elist_2ELUPDATE @ A_27a ) @ V2y ) @ ( inj__ty_2Enum_2Enum @ V3i ) ) @ V0l ) ) ) )
<=> ( ( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V3i ) ) @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ V0l ) ) )
& ( V1x = V2y ) )
| ? [V4j: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V4j ) ) @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ V0l ) ) )
& ( V3i != V4j )
& ( ( ap @ ( ap @ ( c_2Elist_2EEL @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V4j ) ) @ V0l )
= V1x ) ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2ELUPDATE__compute,axiom,
! [A_27a: del] :
( ! [V0e: $i] :
( ( mem @ V0e @ A_27a )
=> ! [V1n: tp__ty_2Enum_2Enum] :
( ( ap @ ( ap @ ( ap @ ( c_2Elist_2ELUPDATE @ A_27a ) @ V0e ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) @ ( c_2Elist_2ENIL @ A_27a ) )
= ( c_2Elist_2ENIL @ A_27a ) ) )
& ! [V2e: $i] :
( ( mem @ V2e @ A_27a )
=> ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ! [V4l: $i] :
( ( mem @ V4l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Elist_2ELUPDATE @ A_27a ) @ V2e ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V3x ) @ V4l ) )
= ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V2e ) @ V4l ) ) ) ) )
& ! [V5e: $i] :
( ( mem @ V5e @ A_27a )
=> ! [V6n: tp__ty_2Enum_2Enum,V7x: $i] :
( ( mem @ V7x @ A_27a )
=> ! [V8l: $i] :
( ( mem @ V8l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Elist_2ELUPDATE @ A_27a ) @ V5e ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ V6n ) ) ) ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V7x ) @ V8l ) )
= ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V7x ) @ ( ap @ ( ap @ ( ap @ ( c_2Elist_2ELUPDATE @ A_27a ) @ V5e ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ V6n ) ) ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ V8l ) ) ) ) ) )
& ! [V9e: $i] :
( ( mem @ V9e @ A_27a )
=> ! [V10n: tp__ty_2Enum_2Enum,V11x: $i] :
( ( mem @ V11x @ A_27a )
=> ! [V12l: $i] :
( ( mem @ V12l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Elist_2ELUPDATE @ A_27a ) @ V9e ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ V10n ) ) ) ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V11x ) @ V12l ) )
= ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V11x ) @ ( ap @ ( ap @ ( ap @ ( c_2Elist_2ELUPDATE @ A_27a ) @ V9e ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ V10n ) ) ) ) @ V12l ) ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2ELUPDATE__MAP,axiom,
! [A_27a: del,A_27b: del,V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ! [V1n: tp__ty_2Enum_2Enum,V2l: $i] :
( ( mem @ V2l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V3f: $i] :
( ( mem @ V3f @ ( arr @ A_27a @ A_27b ) )
=> ( ( ap @ ( ap @ ( c_2Elist_2EMAP @ A_27a @ A_27b ) @ V3f ) @ ( ap @ ( ap @ ( ap @ ( c_2Elist_2ELUPDATE @ A_27a ) @ V0x ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) @ V2l ) )
= ( ap @ ( ap @ ( ap @ ( c_2Elist_2ELUPDATE @ A_27b ) @ ( ap @ V3f @ V0x ) ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) @ ( ap @ ( ap @ ( c_2Elist_2EMAP @ A_27a @ A_27b ) @ V3f ) @ V2l ) ) ) ) ) ) ).
thf(ax_thm_2Elist_2EEVERYi__def,axiom,
! [A_27a: del] :
( ! [V0P: $i] :
( ( mem @ V0P @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Elist_2EEVERYi @ A_27a ) @ V0P ) @ ( c_2Elist_2ENIL @ A_27a ) ) )
<=> $true ) )
& ! [V1P: $i] :
( ( mem @ V1P @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) )
=> ! [V2h: $i] :
( ( mem @ V2h @ A_27a )
=> ! [V3t: $i] :
( ( mem @ V3t @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Elist_2EEVERYi @ A_27a ) @ V1P ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V2h ) @ V3t ) ) )
<=> ( ( p @ ( ap @ ( ap @ V1P @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ V2h ) )
& ( p @ ( ap @ ( ap @ ( c_2Elist_2EEVERYi @ A_27a ) @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) @ ty_2Enum_2Enum ) @ V1P ) @ c_2Enum_2ESUC ) ) @ V3t ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Elist_2EsplitAtPki__def,axiom,
! [A_27a: del,A_27b: del] :
( ! [V0P: $i] :
( ( mem @ V0P @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27b @ bool ) ) )
=> ! [V1k: $i] :
( ( mem @ V1k @ ( arr @ ( ty_2Elist_2Elist @ A_27b ) @ ( arr @ ( ty_2Elist_2Elist @ A_27b ) @ A_27a ) ) )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Elist_2EsplitAtPki @ A_27a @ A_27b ) @ V0P ) @ V1k ) @ ( c_2Elist_2ENIL @ A_27b ) )
= ( ap @ ( ap @ V1k @ ( c_2Elist_2ENIL @ A_27b ) ) @ ( c_2Elist_2ENIL @ A_27b ) ) ) ) )
& ! [V2P: $i] :
( ( mem @ V2P @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27b @ bool ) ) )
=> ! [V3k: $i] :
( ( mem @ V3k @ ( arr @ ( ty_2Elist_2Elist @ A_27b ) @ ( arr @ ( ty_2Elist_2Elist @ A_27b ) @ A_27a ) ) )
=> ! [V4h: $i] :
( ( mem @ V4h @ A_27b )
=> ! [V5t: $i] :
( ( mem @ V5t @ ( ty_2Elist_2Elist @ A_27b ) )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Elist_2EsplitAtPki @ A_27a @ A_27b ) @ V2P ) @ V3k ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27b ) @ V4h ) @ V5t ) )
= ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ A_27a ) @ ( ap @ ( ap @ V2P @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ V4h ) ) @ ( ap @ ( ap @ V3k @ ( c_2Elist_2ENIL @ A_27b ) ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27b ) @ V4h ) @ V5t ) ) )
@ ( ap
@ ( ap @ ( ap @ ( c_2Elist_2EsplitAtPki @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ ty_2Enum_2Enum @ ( arr @ A_27b @ bool ) @ ty_2Enum_2Enum ) @ V2P ) @ c_2Enum_2ESUC ) )
@ ( lam @ ( ty_2Elist_2Elist @ A_27b )
@ ^ [V6p: $i] :
( lam @ ( ty_2Elist_2Elist @ A_27b )
@ ^ [V7s: $i] : ( ap @ ( ap @ V3k @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27b ) @ V4h ) @ V6p ) ) @ V7s ) ) ) )
@ V5t ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2EsplitAtPki__APPEND,axiom,
! [A_27a: del,A_27b: del,V0l1: $i] :
( ( mem @ V0l1 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V1l2: $i] :
( ( mem @ V1l2 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V2P: $i] :
( ( mem @ V2P @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) )
=> ! [V3k: $i] :
( ( mem @ V3k @ ( arr @ ( ty_2Elist_2Elist @ A_27a ) @ ( arr @ ( ty_2Elist_2Elist @ A_27a ) @ A_27b ) ) )
=> ( ( ( p
@ ( ap
@ ( ap @ ( c_2Elist_2EEVERYi @ A_27a )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V4i: $i] : ( ap @ ( ap @ ( c_2Ecombin_2Eo @ A_27a @ bool @ bool ) @ c_2Ebool_2E_7E ) @ ( ap @ V2P @ V4i ) ) ) )
@ V0l1 ) )
& ( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ V1l2 ) ) )
=> ( p @ ( ap @ ( ap @ V2P @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ V0l1 ) ) @ ( ap @ ( c_2Elist_2EHD @ A_27a ) @ V1l2 ) ) ) ) )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Elist_2EsplitAtPki @ A_27b @ A_27a ) @ V2P ) @ V3k ) @ ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ V0l1 ) @ V1l2 ) )
= ( ap @ ( ap @ V3k @ V0l1 ) @ V1l2 ) ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2EsplitAtPki__EQN,axiom,
! [A_27a: del,A_27b: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27b @ bool ) ) )
=> ! [V1k: $i] :
( ( mem @ V1k @ ( arr @ ( ty_2Elist_2Elist @ A_27b ) @ ( arr @ ( ty_2Elist_2Elist @ A_27b ) @ A_27a ) ) )
=> ! [V2l: $i] :
( ( mem @ V2l @ ( ty_2Elist_2Elist @ A_27b ) )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Elist_2EsplitAtPki @ A_27a @ A_27b ) @ V0P ) @ V1k ) @ V2l )
= ( ap
@ ( ap
@ ( ap @ ( c_2Eoption_2Eoption__CASE @ ty_2Enum_2Enum @ A_27a )
@ ( ap @ c_2Ewhile_2EOLEAST
@ ( lam @ ty_2Enum_2Enum
@ ^ [V3i: $i] : ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ V3i ) @ ( ap @ ( c_2Elist_2ELENGTH @ A_27b ) @ V2l ) ) ) @ ( ap @ ( ap @ V0P @ V3i ) @ ( ap @ ( ap @ ( c_2Elist_2EEL @ A_27b ) @ V3i ) @ V2l ) ) ) ) ) )
@ ( ap @ ( ap @ V1k @ V2l ) @ ( c_2Elist_2ENIL @ A_27b ) ) )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V4i: $i] : ( ap @ ( ap @ V1k @ ( ap @ ( ap @ ( c_2Elist_2ETAKE @ A_27b ) @ V4i ) @ V2l ) ) @ ( ap @ ( ap @ ( c_2Elist_2EDROP @ A_27b ) @ V4i ) @ V2l ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2ETAKE__LENGTH__TOO__LONG,axiom,
! [A_27a: del,V0l: $i] :
( ( mem @ V0l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V1n: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ V0l ) ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) )
=> ( ( ap @ ( ap @ ( c_2Elist_2ETAKE @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) @ V0l )
= V0l ) ) ) ).
thf(conj_thm_2Elist_2EDROP__LENGTH__TOO__LONG,axiom,
! [A_27a: del,V0l: $i] :
( ( mem @ V0l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V1n: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ V0l ) ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) )
=> ( ( ap @ ( ap @ ( c_2Elist_2EDROP @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) @ V0l )
= ( c_2Elist_2ENIL @ A_27a ) ) ) ) ).
thf(conj_thm_2Elist_2ETAKE__splitAtPki,axiom,
! [A_27a: del,V0n: tp__ty_2Enum_2Enum,V1l: $i] :
( ( mem @ V1l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( ap @ ( ap @ ( c_2Elist_2ETAKE @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) @ V1l )
= ( ap @ ( ap @ ( ap @ ( c_2Elist_2EsplitAtPki @ ( ty_2Elist_2Elist @ A_27a ) @ A_27a ) @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) @ bool ) @ ( c_2Ecombin_2EK @ bool @ A_27a ) ) @ ( ap @ ( c_2Emin_2E_3D @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) ) @ ( c_2Ecombin_2EK @ ( ty_2Elist_2Elist @ A_27a ) @ ( ty_2Elist_2Elist @ A_27a ) ) ) @ V1l ) ) ) ).
thf(conj_thm_2Elist_2EDROP__splitAtPki,axiom,
! [A_27a: del,V0n: tp__ty_2Enum_2Enum,V1l: $i] :
( ( mem @ V1l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( ap @ ( ap @ ( c_2Elist_2EDROP @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) @ V1l )
= ( ap @ ( ap @ ( ap @ ( c_2Elist_2EsplitAtPki @ ( ty_2Elist_2Elist @ A_27a ) @ A_27a ) @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) @ bool ) @ ( c_2Ecombin_2EK @ bool @ A_27a ) ) @ ( ap @ ( c_2Emin_2E_3D @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) ) @ ( ap @ ( c_2Ecombin_2EK @ ( arr @ ( ty_2Elist_2Elist @ A_27a ) @ ( ty_2Elist_2Elist @ A_27a ) ) @ ( ty_2Elist_2Elist @ A_27a ) ) @ ( c_2Ecombin_2EI @ ( ty_2Elist_2Elist @ A_27a ) ) ) ) @ V1l ) ) ) ).
thf(ax_thm_2Elist_2ELIST__BIND__def,axiom,
! [A_27a: del,A_27b: del,V0l: $i] :
( ( mem @ V0l @ ( ty_2Elist_2Elist @ A_27b ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27b @ ( ty_2Elist_2Elist @ A_27a ) ) )
=> ( ( ap @ ( ap @ ( c_2Elist_2ELIST__BIND @ A_27a @ A_27b ) @ V0l ) @ V1f )
= ( ap @ ( c_2Elist_2EFLAT @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2EMAP @ A_27b @ ( ty_2Elist_2Elist @ A_27a ) ) @ V1f ) @ V0l ) ) ) ) ) ).
thf(conj_thm_2Elist_2ELIST__BIND__THM,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27b @ ( ty_2Elist_2Elist @ A_27a ) ) )
=> ! [V1h: $i] :
( ( mem @ V1h @ A_27b )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( ty_2Elist_2Elist @ A_27b ) )
=> ( ( ( ap @ ( ap @ ( c_2Elist_2ELIST__BIND @ A_27a @ A_27b ) @ ( c_2Elist_2ENIL @ A_27b ) ) @ V0f )
= ( c_2Elist_2ENIL @ A_27a ) )
& ( ( ap @ ( ap @ ( c_2Elist_2ELIST__BIND @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27b ) @ V1h ) @ V2t ) ) @ V0f )
= ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ ( ap @ V0f @ V1h ) ) @ ( ap @ ( ap @ ( c_2Elist_2ELIST__BIND @ A_27a @ A_27b ) @ V2t ) @ V0f ) ) ) ) ) ) ) ).
thf(ax_thm_2Elist_2ELIST__IGNORE__BIND__def,axiom,
! [A_27a: del,A_27b: del,V0m1: $i] :
( ( mem @ V0m1 @ ( ty_2Elist_2Elist @ A_27b ) )
=> ! [V1m2: $i] :
( ( mem @ V1m2 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( ap @ ( ap @ ( c_2Elist_2ELIST__IGNORE__BIND @ A_27a @ A_27b ) @ V0m1 ) @ V1m2 )
= ( ap @ ( ap @ ( c_2Elist_2ELIST__BIND @ A_27a @ A_27b ) @ V0m1 ) @ ( ap @ ( c_2Ecombin_2EK @ ( ty_2Elist_2Elist @ A_27a ) @ A_27b ) @ V1m2 ) ) ) ) ) ).
thf(conj_thm_2Elist_2ELIST__BIND__ID,axiom,
! [A_27a: del,V0l: $i] :
( ( mem @ V0l @ ( ty_2Elist_2Elist @ ( ty_2Elist_2Elist @ A_27a ) ) )
=> ( ( ( ap @ ( ap @ ( c_2Elist_2ELIST__BIND @ A_27a @ ( ty_2Elist_2Elist @ A_27a ) ) @ V0l )
@ ( lam @ ( ty_2Elist_2Elist @ A_27a )
@ ^ [V1x: $i] : V1x ) )
= ( ap @ ( c_2Elist_2EFLAT @ A_27a ) @ V0l ) )
& ( ( ap @ ( ap @ ( c_2Elist_2ELIST__BIND @ A_27a @ ( ty_2Elist_2Elist @ A_27a ) ) @ V0l ) @ ( c_2Ecombin_2EI @ ( ty_2Elist_2Elist @ A_27a ) ) )
= ( ap @ ( c_2Elist_2EFLAT @ A_27a ) @ V0l ) ) ) ) ).
thf(conj_thm_2Elist_2ELIST__BIND__APPEND,axiom,
! [A_27a: del,A_27b: del,V0l1: $i] :
( ( mem @ V0l1 @ ( ty_2Elist_2Elist @ A_27b ) )
=> ! [V1l2: $i] :
( ( mem @ V1l2 @ ( ty_2Elist_2Elist @ A_27b ) )
=> ! [V2f: $i] :
( ( mem @ V2f @ ( arr @ A_27b @ ( ty_2Elist_2Elist @ A_27a ) ) )
=> ( ( ap @ ( ap @ ( c_2Elist_2ELIST__BIND @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27b ) @ V0l1 ) @ V1l2 ) ) @ V2f )
= ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2ELIST__BIND @ A_27a @ A_27b ) @ V0l1 ) @ V2f ) ) @ ( ap @ ( ap @ ( c_2Elist_2ELIST__BIND @ A_27a @ A_27b ) @ V1l2 ) @ V2f ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2ELIST__BIND__MAP,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27c @ A_27b ) )
=> ! [V1l: $i] :
( ( mem @ V1l @ ( ty_2Elist_2Elist @ A_27c ) )
=> ! [V2g: $i] :
( ( mem @ V2g @ ( arr @ A_27b @ ( ty_2Elist_2Elist @ A_27a ) ) )
=> ( ( ap @ ( ap @ ( c_2Elist_2ELIST__BIND @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Elist_2EMAP @ A_27c @ A_27b ) @ V0f ) @ V1l ) ) @ V2g )
= ( ap @ ( ap @ ( c_2Elist_2ELIST__BIND @ A_27a @ A_27c ) @ V1l ) @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ A_27c @ ( ty_2Elist_2Elist @ A_27a ) @ A_27b ) @ V2g ) @ V0f ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2EMAP__LIST__BIND,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27b @ A_27a ) )
=> ! [V1l: $i] :
( ( mem @ V1l @ ( ty_2Elist_2Elist @ A_27c ) )
=> ! [V2g: $i] :
( ( mem @ V2g @ ( arr @ A_27c @ ( ty_2Elist_2Elist @ A_27b ) ) )
=> ( ( ap @ ( ap @ ( c_2Elist_2EMAP @ A_27b @ A_27a ) @ V0f ) @ ( ap @ ( ap @ ( c_2Elist_2ELIST__BIND @ A_27b @ A_27c ) @ V1l ) @ V2g ) )
= ( ap @ ( ap @ ( c_2Elist_2ELIST__BIND @ A_27a @ A_27c ) @ V1l ) @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ A_27c @ ( ty_2Elist_2Elist @ A_27a ) @ ( ty_2Elist_2Elist @ A_27b ) ) @ ( ap @ ( c_2Elist_2EMAP @ A_27b @ A_27a ) @ V0f ) ) @ V2g ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2ELIST__BIND__LIST__BIND,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0l: $i] :
( ( mem @ V0l @ ( ty_2Elist_2Elist @ A_27c ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ A_27c @ ( ty_2Elist_2Elist @ A_27b ) ) )
=> ! [V2f: $i] :
( ( mem @ V2f @ ( arr @ A_27b @ ( ty_2Elist_2Elist @ A_27a ) ) )
=> ( ( ap @ ( ap @ ( c_2Elist_2ELIST__BIND @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Elist_2ELIST__BIND @ A_27b @ A_27c ) @ V0l ) @ V1g ) ) @ V2f )
= ( ap @ ( ap @ ( c_2Elist_2ELIST__BIND @ A_27a @ A_27c ) @ V0l ) @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ A_27c @ ( ty_2Elist_2Elist @ A_27a ) @ ( ty_2Elist_2Elist @ A_27b ) ) @ ( ap @ ( ap @ ( c_2Ecombin_2EC @ ( ty_2Elist_2Elist @ A_27b ) @ ( arr @ A_27b @ ( ty_2Elist_2Elist @ A_27a ) ) @ ( ty_2Elist_2Elist @ A_27a ) ) @ ( c_2Elist_2ELIST__BIND @ A_27a @ A_27b ) ) @ V2f ) ) @ V1g ) ) ) ) ) ) ).
thf(ax_thm_2Elist_2ELIST__GUARD__def,axiom,
! [V0b: $i] :
( ( mem @ V0b @ bool )
=> ( ( surj__c_ty_2Elist_2Elist_ty_2Eone_2Eone @ ( ap @ c_2Elist_2ELIST__GUARD @ V0b ) )
= ( surj__c_ty_2Elist_2Elist_ty_2Eone_2Eone @ ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ( ty_2Elist_2Elist @ ty_2Eone_2Eone ) ) @ V0b ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ty_2Eone_2Eone ) @ ( inj__ty_2Eone_2Eone @ fo__c_2Eone_2Eone ) ) @ ( c_2Elist_2ENIL @ ty_2Eone_2Eone ) ) ) @ ( c_2Elist_2ENIL @ ty_2Eone_2Eone ) ) ) ) ) ).
thf(conj_thm_2Elist_2ESINGL__LIST__APPLY__L,axiom,
! [A_27a: del,A_27b: del,V0x: $i] :
( ( mem @ V0x @ A_27b )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27b @ ( ty_2Elist_2Elist @ A_27a ) ) )
=> ( ( ap @ ( ap @ ( c_2Elist_2ELIST__BIND @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27b ) @ V0x ) @ ( c_2Elist_2ENIL @ A_27b ) ) ) @ V1f )
= ( ap @ V1f @ V0x ) ) ) ) ).
thf(conj_thm_2Elist_2ESINGL__LIST__APPLY__R,axiom,
! [A_27a: del,V0l: $i] :
( ( mem @ V0l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( ap @ ( ap @ ( c_2Elist_2ELIST__BIND @ A_27a @ A_27a ) @ V0l )
@ ( lam @ A_27a
@ ^ [V1x: $i] : ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V1x ) @ ( c_2Elist_2ENIL @ A_27a ) ) ) )
= V0l ) ) ).
thf(ax_thm_2Elist_2ELIST__APPLY__def,axiom,
! [A_27a: del,A_27b: del,V0fs: $i] :
( ( mem @ V0fs @ ( ty_2Elist_2Elist @ ( arr @ A_27b @ A_27a ) ) )
=> ! [V1xs: $i] :
( ( mem @ V1xs @ ( ty_2Elist_2Elist @ A_27b ) )
=> ( ( ap @ ( ap @ ( c_2Elist_2ELIST__APPLY @ A_27a @ A_27b ) @ V0fs ) @ V1xs )
= ( ap @ ( ap @ ( c_2Elist_2ELIST__BIND @ A_27a @ ( arr @ A_27b @ A_27a ) ) @ V0fs ) @ ( ap @ ( ap @ ( c_2Ecombin_2EC @ ( arr @ A_27b @ A_27a ) @ ( ty_2Elist_2Elist @ A_27b ) @ ( ty_2Elist_2Elist @ A_27a ) ) @ ( c_2Elist_2EMAP @ A_27b @ A_27a ) ) @ V1xs ) ) ) ) ) ).
thf(ax_thm_2Elist_2ELIST__LIFT2__def,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27b @ ( arr @ A_27c @ A_27a ) ) )
=> ! [V1xs: $i] :
( ( mem @ V1xs @ ( ty_2Elist_2Elist @ A_27b ) )
=> ! [V2ys: $i] :
( ( mem @ V2ys @ ( ty_2Elist_2Elist @ A_27c ) )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Elist_2ELIST__LIFT2 @ A_27a @ A_27b @ A_27c ) @ V0f ) @ V1xs ) @ V2ys )
= ( ap @ ( ap @ ( c_2Elist_2ELIST__APPLY @ A_27a @ A_27c ) @ ( ap @ ( ap @ ( c_2Elist_2EMAP @ A_27b @ ( arr @ A_27c @ A_27a ) ) @ V0f ) @ V1xs ) ) @ V2ys ) ) ) ) ) ).
thf(conj_thm_2Elist_2ESINGL__APPLY__MAP,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27b @ A_27a ) )
=> ! [V1l: $i] :
( ( mem @ V1l @ ( ty_2Elist_2Elist @ A_27b ) )
=> ( ( ap @ ( ap @ ( c_2Elist_2ELIST__APPLY @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ( arr @ A_27b @ A_27a ) ) @ V0f ) @ ( c_2Elist_2ENIL @ ( arr @ A_27b @ A_27a ) ) ) ) @ V1l )
= ( ap @ ( ap @ ( c_2Elist_2EMAP @ A_27b @ A_27a ) @ V0f ) @ V1l ) ) ) ) ).
thf(conj_thm_2Elist_2ESINGL__SINGL__APPLY,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27b @ A_27a ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ A_27b )
=> ( ( ap @ ( ap @ ( c_2Elist_2ELIST__APPLY @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ( arr @ A_27b @ A_27a ) ) @ V0f ) @ ( c_2Elist_2ENIL @ ( arr @ A_27b @ A_27a ) ) ) ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27b ) @ V1x ) @ ( c_2Elist_2ENIL @ A_27b ) ) )
= ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ ( ap @ V0f @ V1x ) ) @ ( c_2Elist_2ENIL @ A_27a ) ) ) ) ) ).
thf(conj_thm_2Elist_2ESINGL__APPLY__PERMUTE,axiom,
! [A_27a: del,A_27b: del,V0fs: $i] :
( ( mem @ V0fs @ ( ty_2Elist_2Elist @ ( arr @ A_27b @ A_27a ) ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ A_27b )
=> ( ( ap @ ( ap @ ( c_2Elist_2ELIST__APPLY @ A_27a @ A_27b ) @ V0fs ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27b ) @ V1x ) @ ( c_2Elist_2ENIL @ A_27b ) ) )
= ( ap
@ ( ap @ ( c_2Elist_2ELIST__APPLY @ A_27a @ ( arr @ A_27b @ A_27a ) )
@ ( ap
@ ( ap @ ( c_2Elist_2ECONS @ ( arr @ ( arr @ A_27b @ A_27a ) @ A_27a ) )
@ ( lam @ ( arr @ A_27b @ A_27a )
@ ^ [V2f: $i] : ( ap @ V2f @ V1x ) ) )
@ ( c_2Elist_2ENIL @ ( arr @ ( arr @ A_27b @ A_27a ) @ A_27a ) ) ) )
@ V0fs ) ) ) ) ).
thf(conj_thm_2Elist_2EMAP__FLAT,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27b @ A_27a ) )
=> ! [V1l: $i] :
( ( mem @ V1l @ ( ty_2Elist_2Elist @ ( ty_2Elist_2Elist @ A_27b ) ) )
=> ( ( ap @ ( ap @ ( c_2Elist_2EMAP @ A_27b @ A_27a ) @ V0f ) @ ( ap @ ( c_2Elist_2EFLAT @ A_27b ) @ V1l ) )
= ( ap @ ( c_2Elist_2EFLAT @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2EMAP @ ( ty_2Elist_2Elist @ A_27b ) @ ( ty_2Elist_2Elist @ A_27a ) ) @ ( ap @ ( c_2Elist_2EMAP @ A_27b @ A_27a ) @ V0f ) ) @ V1l ) ) ) ) ) ).
thf(conj_thm_2Elist_2ELIST__APPLY__o,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0fs: $i] :
( ( mem @ V0fs @ ( ty_2Elist_2Elist @ ( arr @ A_27c @ A_27a ) ) )
=> ! [V1gs: $i] :
( ( mem @ V1gs @ ( ty_2Elist_2Elist @ ( arr @ A_27b @ A_27c ) ) )
=> ! [V2xs: $i] :
( ( mem @ V2xs @ ( ty_2Elist_2Elist @ A_27b ) )
=> ( ( ap @ ( ap @ ( c_2Elist_2ELIST__APPLY @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Elist_2ELIST__APPLY @ ( arr @ A_27b @ A_27a ) @ ( arr @ A_27b @ A_27c ) ) @ ( ap @ ( ap @ ( c_2Elist_2ELIST__APPLY @ ( arr @ ( arr @ A_27b @ A_27c ) @ ( arr @ A_27b @ A_27a ) ) @ ( arr @ A_27c @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ( arr @ ( arr @ A_27c @ A_27a ) @ ( arr @ ( arr @ A_27b @ A_27c ) @ ( arr @ A_27b @ A_27a ) ) ) ) @ ( c_2Ecombin_2Eo @ A_27b @ A_27a @ A_27c ) ) @ ( c_2Elist_2ENIL @ ( arr @ ( arr @ A_27c @ A_27a ) @ ( arr @ ( arr @ A_27b @ A_27c ) @ ( arr @ A_27b @ A_27a ) ) ) ) ) ) @ V0fs ) ) @ V1gs ) ) @ V2xs )
= ( ap @ ( ap @ ( c_2Elist_2ELIST__APPLY @ A_27a @ A_27c ) @ V0fs ) @ ( ap @ ( ap @ ( c_2Elist_2ELIST__APPLY @ A_27c @ A_27b ) @ V1gs ) @ V2xs ) ) ) ) ) ) ).
thf(ax_thm_2Elist_2ESHORTLEX__def,axiom,
! [A_27a: del] :
( ! [V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1l2: $i] :
( ( mem @ V1l2 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Elist_2ESHORTLEX @ A_27a ) @ V0R ) @ ( c_2Elist_2ENIL @ A_27a ) ) @ V1l2 ) )
<=> ( V1l2
!= ( c_2Elist_2ENIL @ A_27a ) ) ) ) )
& ! [V2R: $i] :
( ( mem @ V2R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V3h1: $i] :
( ( mem @ V3h1 @ A_27a )
=> ! [V4t1: $i] :
( ( mem @ V4t1 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V5l2: $i] :
( ( mem @ V5l2 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Elist_2ESHORTLEX @ A_27a ) @ V2R ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V3h1 ) @ V4t1 ) ) @ V5l2 ) )
<=> ( p
@ ( ap @ ( ap @ ( ap @ ( c_2Elist_2Elist__CASE @ A_27a @ bool ) @ V5l2 ) @ c_2Ebool_2EF )
@ ( lam @ A_27a
@ ^ [V6h2: $i] :
( lam @ ( ty_2Elist_2Elist @ A_27a )
@ ^ [V7t2: $i] : ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ bool ) @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ V4t1 ) ) @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ V7t2 ) ) ) @ c_2Ebool_2ET ) @ ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ bool ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ty_2Enum_2Enum ) @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ V4t1 ) ) @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ V7t2 ) ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ bool ) @ ( ap @ ( ap @ V2R @ V3h1 ) @ V6h2 ) ) @ c_2Ebool_2ET ) @ ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ bool ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ A_27a ) @ V3h1 ) @ V6h2 ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Elist_2ESHORTLEX @ A_27a ) @ V2R ) @ V4t1 ) @ V7t2 ) ) @ c_2Ebool_2EF ) ) ) @ c_2Ebool_2EF ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2ESHORTLEX__THM,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1h1: $i] :
( ( mem @ V1h1 @ A_27a )
=> ! [V2t1: $i] :
( ( mem @ V2t1 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V3h2: $i] :
( ( mem @ V3h2 @ A_27a )
=> ! [V4t2: $i] :
( ( mem @ V4t2 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ~ ( p @ ( ap @ ( ap @ ( ap @ ( c_2Elist_2ESHORTLEX @ A_27a ) @ V0R ) @ ( c_2Elist_2ENIL @ A_27a ) ) @ ( c_2Elist_2ENIL @ A_27a ) ) )
& ~ ( p @ ( ap @ ( ap @ ( ap @ ( c_2Elist_2ESHORTLEX @ A_27a ) @ V0R ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V1h1 ) @ V2t1 ) ) @ ( c_2Elist_2ENIL @ A_27a ) ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( c_2Elist_2ESHORTLEX @ A_27a ) @ V0R ) @ ( c_2Elist_2ENIL @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V3h2 ) @ V4t2 ) ) )
& ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Elist_2ESHORTLEX @ A_27a ) @ V0R ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V1h1 ) @ V2t1 ) ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V3h2 ) @ V4t2 ) ) )
<=> ( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ V2t1 ) ) @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ V4t2 ) ) )
| ( ( ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ V2t1 ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ V4t2 ) ) )
& ( ( p @ ( ap @ ( ap @ V0R @ V1h1 ) @ V3h2 ) )
| ( ( V1h1 = V3h2 )
& ( p @ ( ap @ ( ap @ ( ap @ ( c_2Elist_2ESHORTLEX @ A_27a ) @ V0R ) @ V2t1 ) @ V4t2 ) ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2ESHORTLEX__MONO,axiom,
! [A_27a: del,V0R1: $i] :
( ( mem @ V0R1 @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1R2: $i] :
( ( mem @ V1R2 @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V2x: $i] :
( ( mem @ V2x @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V3y: $i] :
( ( mem @ V3y @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ! [V4x: $i] :
( ( mem @ V4x @ A_27a )
=> ! [V5y: $i] :
( ( mem @ V5y @ A_27a )
=> ( ( p @ ( ap @ ( ap @ V0R1 @ V4x ) @ V5y ) )
=> ( p @ ( ap @ ( ap @ V1R2 @ V4x ) @ V5y ) ) ) ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Elist_2ESHORTLEX @ A_27a ) @ V0R1 ) @ V2x ) @ V3y ) )
=> ( p @ ( ap @ ( ap @ ( ap @ ( c_2Elist_2ESHORTLEX @ A_27a ) @ V1R2 ) @ V2x ) @ V3y ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2ESHORTLEX__NIL2,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1l: $i] :
( ( mem @ V1l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ~ ( p @ ( ap @ ( ap @ ( ap @ ( c_2Elist_2ESHORTLEX @ A_27a ) @ V0R ) @ V1l ) @ ( c_2Elist_2ENIL @ A_27a ) ) ) ) ) ).
thf(conj_thm_2Elist_2ESHORTLEX__transitive,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( p @ ( ap @ ( c_2Erelation_2Etransitive @ A_27a ) @ V0R ) )
=> ( p @ ( ap @ ( c_2Erelation_2Etransitive @ ( ty_2Elist_2Elist @ A_27a ) ) @ ( ap @ ( c_2Elist_2ESHORTLEX @ A_27a ) @ V0R ) ) ) ) ) ).
thf(conj_thm_2Elist_2ELENGTH__LT__SHORTLEX,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1l1: $i] :
( ( mem @ V1l1 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V2l2: $i] :
( ( mem @ V2l2 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ V1l1 ) ) @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ V2l2 ) ) )
=> ( p @ ( ap @ ( ap @ ( ap @ ( c_2Elist_2ESHORTLEX @ A_27a ) @ V0R ) @ V1l1 ) @ V2l2 ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2ESHORTLEX__LENGTH__LE,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1l1: $i] :
( ( mem @ V1l1 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V2l2: $i] :
( ( mem @ V2l2 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Elist_2ESHORTLEX @ A_27a ) @ V0R ) @ V1l1 ) @ V2l2 ) )
=> ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ V1l1 ) ) @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ V2l2 ) ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2ESHORTLEX__total,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( p @ ( ap @ ( c_2Erelation_2Etotal @ A_27a ) @ ( ap @ ( c_2Erelation_2ERC @ A_27a ) @ V0R ) ) )
=> ( p @ ( ap @ ( c_2Erelation_2Etotal @ ( ty_2Elist_2Elist @ A_27a ) ) @ ( ap @ ( c_2Erelation_2ERC @ ( ty_2Elist_2Elist @ A_27a ) ) @ ( ap @ ( c_2Elist_2ESHORTLEX @ A_27a ) @ V0R ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2EWF__SHORTLEX__same__lengths,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( p @ ( ap @ ( c_2Erelation_2EWF @ A_27a ) @ V0R ) )
=> ! [V1l: tp__ty_2Enum_2Enum,V2s: $i] :
( ( mem @ V2s @ ( arr @ ( ty_2Elist_2Elist @ A_27a ) @ bool ) )
=> ( ( ! [V3d: $i] :
( ( mem @ V3d @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Elist_2Elist @ A_27a ) ) @ V3d ) @ V2s ) )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ V3d ) )
= V1l ) ) )
& ? [V4a: $i] :
( ( mem @ V4a @ ( ty_2Elist_2Elist @ A_27a ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Elist_2Elist @ A_27a ) ) @ V4a ) @ V2s ) ) ) )
=> ? [V5b: $i] :
( ( mem @ V5b @ ( ty_2Elist_2Elist @ A_27a ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Elist_2Elist @ A_27a ) ) @ V5b ) @ V2s ) )
& ! [V6c: $i] :
( ( mem @ V6c @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Elist_2ESHORTLEX @ A_27a ) @ V0R ) @ V6c ) @ V5b ) )
=> ~ ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Elist_2Elist @ A_27a ) ) @ V6c ) @ V2s ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2EWF__SHORTLEX,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( p @ ( ap @ ( c_2Erelation_2EWF @ A_27a ) @ V0R ) )
=> ( p @ ( ap @ ( c_2Erelation_2EWF @ ( ty_2Elist_2Elist @ A_27a ) ) @ ( ap @ ( c_2Elist_2ESHORTLEX @ A_27a ) @ V0R ) ) ) ) ) ).
thf(ax_thm_2Elist_2ELLEX__def,axiom,
! [A_27a: del] :
( ! [V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1l2: $i] :
( ( mem @ V1l2 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Elist_2ELLEX @ A_27a ) @ V0R ) @ ( c_2Elist_2ENIL @ A_27a ) ) @ V1l2 ) )
<=> ( V1l2
!= ( c_2Elist_2ENIL @ A_27a ) ) ) ) )
& ! [V2R: $i] :
( ( mem @ V2R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V3h1: $i] :
( ( mem @ V3h1 @ A_27a )
=> ! [V4t1: $i] :
( ( mem @ V4t1 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V5l2: $i] :
( ( mem @ V5l2 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Elist_2ELLEX @ A_27a ) @ V2R ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V3h1 ) @ V4t1 ) ) @ V5l2 ) )
<=> ( p
@ ( ap @ ( ap @ ( ap @ ( c_2Elist_2Elist__CASE @ A_27a @ bool ) @ V5l2 ) @ c_2Ebool_2EF )
@ ( lam @ A_27a
@ ^ [V6h2: $i] :
( lam @ ( ty_2Elist_2Elist @ A_27a )
@ ^ [V7t2: $i] : ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ bool ) @ ( ap @ ( ap @ V2R @ V3h1 ) @ V6h2 ) ) @ c_2Ebool_2ET ) @ ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ bool ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ A_27a ) @ V3h1 ) @ V6h2 ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Elist_2ELLEX @ A_27a ) @ V2R ) @ V4t1 ) @ V7t2 ) ) @ c_2Ebool_2EF ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2ELLEX__THM,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1h1: $i] :
( ( mem @ V1h1 @ A_27a )
=> ! [V2t1: $i] :
( ( mem @ V2t1 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V3h2: $i] :
( ( mem @ V3h2 @ A_27a )
=> ! [V4t2: $i] :
( ( mem @ V4t2 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ~ ( p @ ( ap @ ( ap @ ( ap @ ( c_2Elist_2ELLEX @ A_27a ) @ V0R ) @ ( c_2Elist_2ENIL @ A_27a ) ) @ ( c_2Elist_2ENIL @ A_27a ) ) )
& ~ ( p @ ( ap @ ( ap @ ( ap @ ( c_2Elist_2ELLEX @ A_27a ) @ V0R ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V1h1 ) @ V2t1 ) ) @ ( c_2Elist_2ENIL @ A_27a ) ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( c_2Elist_2ELLEX @ A_27a ) @ V0R ) @ ( c_2Elist_2ENIL @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V3h2 ) @ V4t2 ) ) )
& ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Elist_2ELLEX @ A_27a ) @ V0R ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V1h1 ) @ V2t1 ) ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V3h2 ) @ V4t2 ) ) )
<=> ( ( p @ ( ap @ ( ap @ V0R @ V1h1 ) @ V3h2 ) )
| ( ( V1h1 = V3h2 )
& ( p @ ( ap @ ( ap @ ( ap @ ( c_2Elist_2ELLEX @ A_27a ) @ V0R ) @ V2t1 ) @ V4t2 ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2ELLEX__MONO,axiom,
! [A_27a: del,V0R1: $i] :
( ( mem @ V0R1 @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1R2: $i] :
( ( mem @ V1R2 @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V2x: $i] :
( ( mem @ V2x @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V3y: $i] :
( ( mem @ V3y @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ! [V4x: $i] :
( ( mem @ V4x @ A_27a )
=> ! [V5y: $i] :
( ( mem @ V5y @ A_27a )
=> ( ( p @ ( ap @ ( ap @ V0R1 @ V4x ) @ V5y ) )
=> ( p @ ( ap @ ( ap @ V1R2 @ V4x ) @ V5y ) ) ) ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Elist_2ELLEX @ A_27a ) @ V0R1 ) @ V2x ) @ V3y ) )
=> ( p @ ( ap @ ( ap @ ( ap @ ( c_2Elist_2ELLEX @ A_27a ) @ V1R2 ) @ V2x ) @ V3y ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2ELLEX__CONG,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1l1: $i] :
( ( mem @ V1l1 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V2l2: $i] :
( ( mem @ V2l2 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V3R_27: $i] :
( ( mem @ V3R_27 @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V4l1_27: $i] :
( ( mem @ V4l1_27 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V5l2_27: $i] :
( ( mem @ V5l2_27 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( ( V1l1 = V4l1_27 )
& ( V2l2 = V5l2_27 )
& ! [V6a: $i] :
( ( mem @ V6a @ A_27a )
=> ! [V7b: $i] :
( ( mem @ V7b @ A_27a )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V6a ) @ ( ap @ ( c_2Elist_2ELIST__TO__SET @ A_27a ) @ V4l1_27 ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V7b ) @ ( ap @ ( c_2Elist_2ELIST__TO__SET @ A_27a ) @ V5l2_27 ) ) ) )
=> ( ( p @ ( ap @ ( ap @ V0R @ V6a ) @ V7b ) )
<=> ( p @ ( ap @ ( ap @ V3R_27 @ V6a ) @ V7b ) ) ) ) ) ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Elist_2ELLEX @ A_27a ) @ V0R ) @ V1l1 ) @ V2l2 ) )
<=> ( p @ ( ap @ ( ap @ ( ap @ ( c_2Elist_2ELLEX @ A_27a ) @ V3R_27 ) @ V4l1_27 ) @ V5l2_27 ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2ELLEX__NIL2,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1l: $i] :
( ( mem @ V1l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ~ ( p @ ( ap @ ( ap @ ( ap @ ( c_2Elist_2ELLEX @ A_27a ) @ V0R ) @ V1l ) @ ( c_2Elist_2ENIL @ A_27a ) ) ) ) ) ).
thf(conj_thm_2Elist_2ELLEX__transitive,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( p @ ( ap @ ( c_2Erelation_2Etransitive @ A_27a ) @ V0R ) )
=> ( p @ ( ap @ ( c_2Erelation_2Etransitive @ ( ty_2Elist_2Elist @ A_27a ) ) @ ( ap @ ( c_2Elist_2ELLEX @ A_27a ) @ V0R ) ) ) ) ) ).
thf(conj_thm_2Elist_2ELLEX__total,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( p @ ( ap @ ( c_2Erelation_2Etotal @ A_27a ) @ ( ap @ ( c_2Erelation_2ERC @ A_27a ) @ V0R ) ) )
=> ( p @ ( ap @ ( c_2Erelation_2Etotal @ ( ty_2Elist_2Elist @ A_27a ) ) @ ( ap @ ( c_2Erelation_2ERC @ ( ty_2Elist_2Elist @ A_27a ) ) @ ( ap @ ( c_2Elist_2ELLEX @ A_27a ) @ V0R ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2ELLEX__not__WF,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ? [V1a: $i] :
( ( mem @ V1a @ A_27a )
& ? [V2b: $i] :
( ( mem @ V2b @ A_27a )
& ( p @ ( ap @ ( ap @ V0R @ V1a ) @ V2b ) ) ) )
=> ~ ( p @ ( ap @ ( c_2Erelation_2EWF @ ( ty_2Elist_2Elist @ A_27a ) ) @ ( ap @ ( c_2Elist_2ELLEX @ A_27a ) @ V0R ) ) ) ) ) ).
thf(conj_thm_2Elist_2ELLEX__EL__THM,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1l1: $i] :
( ( mem @ V1l1 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V2l2: $i] :
( ( mem @ V2l2 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Elist_2ELLEX @ A_27a ) @ V0R ) @ V1l1 ) @ V2l2 ) )
<=> ? [V3n: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ V3n ) ) @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ V1l1 ) ) )
& ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V3n ) ) @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ V2l2 ) ) )
& ( ( ap @ ( ap @ ( c_2Elist_2ETAKE @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V3n ) ) @ V1l1 )
= ( ap @ ( ap @ ( c_2Elist_2ETAKE @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V3n ) ) @ V2l2 ) )
& ( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V3n ) ) @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ V1l1 ) ) )
=> ( p @ ( ap @ ( ap @ V0R @ ( ap @ ( ap @ ( c_2Elist_2EEL @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V3n ) ) @ V1l1 ) ) @ ( ap @ ( ap @ ( c_2Elist_2EEL @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V3n ) ) @ V2l2 ) ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Elist_2Enub__def,axiom,
! [A_27a: del] :
( ( ( ap @ ( c_2Elist_2Enub @ A_27a ) @ ( c_2Elist_2ENIL @ A_27a ) )
= ( c_2Elist_2ENIL @ A_27a ) )
& ! [V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ! [V1l: $i] :
( ( mem @ V1l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( ap @ ( c_2Elist_2Enub @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V0x ) @ V1l ) )
= ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ( ty_2Elist_2Elist @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V0x ) @ ( ap @ ( c_2Elist_2ELIST__TO__SET @ A_27a ) @ V1l ) ) ) @ ( ap @ ( c_2Elist_2Enub @ A_27a ) @ V1l ) ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V0x ) @ ( ap @ ( c_2Elist_2Enub @ A_27a ) @ V1l ) ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2Enub__set,axiom,
! [A_27a: del,V0l: $i] :
( ( mem @ V0l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( ap @ ( c_2Elist_2ELIST__TO__SET @ A_27a ) @ ( ap @ ( c_2Elist_2Enub @ A_27a ) @ V0l ) )
= ( ap @ ( c_2Elist_2ELIST__TO__SET @ A_27a ) @ V0l ) ) ) ).
thf(conj_thm_2Elist_2Eall__distinct__nub,axiom,
! [A_27a: del,V0l: $i] :
( ( mem @ V0l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( p @ ( ap @ ( c_2Elist_2EALL__DISTINCT @ A_27a ) @ ( ap @ ( c_2Elist_2Enub @ A_27a ) @ V0l ) ) ) ) ).
thf(conj_thm_2Elist_2Enub__append,axiom,
! [A_27a: del,V0l1: $i] :
( ( mem @ V0l1 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V1l2: $i] :
( ( mem @ V1l2 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( ap @ ( c_2Elist_2Enub @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ V0l1 ) @ V1l2 ) )
= ( ap
@ ( ap @ ( c_2Elist_2EAPPEND @ A_27a )
@ ( ap @ ( c_2Elist_2Enub @ A_27a )
@ ( ap
@ ( ap @ ( c_2Elist_2EFILTER @ A_27a )
@ ( lam @ A_27a
@ ^ [V2x: $i] : ( ap @ c_2Ebool_2E_7E @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x ) @ ( ap @ ( c_2Elist_2ELIST__TO__SET @ A_27a ) @ V1l2 ) ) ) ) )
@ V0l1 ) ) )
@ ( ap @ ( c_2Elist_2Enub @ A_27a ) @ V1l2 ) ) ) ) ) ).
thf(conj_thm_2Elist_2Elist__to__set__diff,axiom,
! [A_27a: del,V0l1: $i] :
( ( mem @ V0l1 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V1l2: $i] :
( ( mem @ V1l2 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( ap @ ( ap @ ( c_2Epred__set_2EDIFF @ A_27a ) @ ( ap @ ( c_2Elist_2ELIST__TO__SET @ A_27a ) @ V1l2 ) ) @ ( ap @ ( c_2Elist_2ELIST__TO__SET @ A_27a ) @ V0l1 ) )
= ( ap @ ( c_2Elist_2ELIST__TO__SET @ A_27a )
@ ( ap
@ ( ap @ ( c_2Elist_2EFILTER @ A_27a )
@ ( lam @ A_27a
@ ^ [V2x: $i] : ( ap @ c_2Ebool_2E_7E @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x ) @ ( ap @ ( c_2Elist_2ELIST__TO__SET @ A_27a ) @ V0l1 ) ) ) ) )
@ V1l2 ) ) ) ) ) ).
thf(conj_thm_2Elist_2Elength__nub__append,axiom,
! [A_27a: del,V0l1: $i] :
( ( mem @ V0l1 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V1l2: $i] :
( ( mem @ V1l2 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ ( ap @ ( c_2Elist_2Enub @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ V0l1 ) @ V1l2 ) ) ) )
= ( surj__ty_2Enum_2Enum
@ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ ( ap @ ( c_2Elist_2Enub @ A_27a ) @ V0l1 ) ) )
@ ( ap @ ( c_2Elist_2ELENGTH @ A_27a )
@ ( ap @ ( c_2Elist_2Enub @ A_27a )
@ ( ap
@ ( ap @ ( c_2Elist_2EFILTER @ A_27a )
@ ( lam @ A_27a
@ ^ [V2x: $i] : ( ap @ c_2Ebool_2E_7E @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x ) @ ( ap @ ( c_2Elist_2ELIST__TO__SET @ A_27a ) @ V0l1 ) ) ) ) )
@ V1l2 ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2EALL__DISTINCT__DROP,axiom,
! [A_27a: del,V0ls: $i] :
( ( mem @ V0ls @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V1n: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( c_2Elist_2EALL__DISTINCT @ A_27a ) @ V0ls ) )
=> ( p @ ( ap @ ( c_2Elist_2EALL__DISTINCT @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2EDROP @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) @ V0ls ) ) ) ) ) ).
thf(conj_thm_2Elist_2EEXISTS__LIST__EQ__MAP,axiom,
! [A_27a: del,A_27b: del,V0ls: $i] :
( ( mem @ V0ls @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27b @ A_27a ) )
=> ( ( p
@ ( ap
@ ( ap @ ( c_2Elist_2EEVERY @ A_27a )
@ ( lam @ A_27a
@ ^ [V2x: $i] :
( ap @ ( c_2Ebool_2E_3F @ A_27b )
@ ( lam @ A_27b
@ ^ [V3y: $i] : ( ap @ ( ap @ ( c_2Emin_2E_3D @ A_27a ) @ V2x ) @ ( ap @ V1f @ V3y ) ) ) ) ) )
@ V0ls ) )
=> ? [V4l: $i] :
( ( mem @ V4l @ ( ty_2Elist_2Elist @ A_27b ) )
& ( V0ls
= ( ap @ ( ap @ ( c_2Elist_2EMAP @ A_27b @ A_27a ) @ V1f ) @ V4l ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2ELIST__TO__SET__FLAT,axiom,
! [A_27a: del,V0ls: $i] :
( ( mem @ V0ls @ ( ty_2Elist_2Elist @ ( ty_2Elist_2Elist @ A_27a ) ) )
=> ( ( ap @ ( c_2Elist_2ELIST__TO__SET @ A_27a ) @ ( ap @ ( c_2Elist_2EFLAT @ A_27a ) @ V0ls ) )
= ( ap @ ( c_2Epred__set_2EBIGUNION @ A_27a ) @ ( ap @ ( c_2Elist_2ELIST__TO__SET @ ( arr @ A_27a @ bool ) ) @ ( ap @ ( ap @ ( c_2Elist_2EMAP @ ( ty_2Elist_2Elist @ A_27a ) @ ( arr @ A_27a @ bool ) ) @ ( c_2Elist_2ELIST__TO__SET @ A_27a ) ) @ V0ls ) ) ) ) ) ).
thf(conj_thm_2Elist_2EMEM__APPEND__lemma,axiom,
! [A_27a: del,V0a: $i] :
( ( mem @ V0a @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V1b: $i] :
( ( mem @ V1b @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V2c: $i] :
( ( mem @ V2c @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V3d: $i] :
( ( mem @ V3d @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V4x: $i] :
( ( mem @ V4x @ A_27a )
=> ( ( ( ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ V0a ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V4x ) @ ( c_2Elist_2ENIL @ A_27a ) ) ) ) @ V1b )
= ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ V2c ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V4x ) @ ( c_2Elist_2ENIL @ A_27a ) ) ) ) @ V3d ) )
& ~ ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V4x ) @ ( ap @ ( c_2Elist_2ELIST__TO__SET @ A_27a ) @ V1b ) ) )
& ~ ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V4x ) @ ( ap @ ( c_2Elist_2ELIST__TO__SET @ A_27a ) @ V0a ) ) ) )
=> ( ( V0a = V2c )
& ( V1b = V3d ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2EEVERY2__REVERSE,axiom,
! [A_27a: del,A_27b: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) )
=> ! [V1l1: $i] :
( ( mem @ V1l1 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V2l2: $i] :
( ( mem @ V2l2 @ ( ty_2Elist_2Elist @ A_27b ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Elist_2ELIST__REL @ A_27a @ A_27b ) @ V0R ) @ V1l1 ) @ V2l2 ) )
=> ( p @ ( ap @ ( ap @ ( ap @ ( c_2Elist_2ELIST__REL @ A_27a @ A_27b ) @ V0R ) @ ( ap @ ( c_2Elist_2EREVERSE @ A_27a ) @ V1l1 ) ) @ ( ap @ ( c_2Elist_2EREVERSE @ A_27b ) @ V2l2 ) ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2ESUM__MAP__PLUS,axiom,
! [A_27a: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V2ls: $i] :
( ( mem @ V2ls @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( surj__ty_2Enum_2Enum
@ ( ap @ c_2Elist_2ESUM
@ ( ap
@ ( ap @ ( c_2Elist_2EMAP @ A_27a @ ty_2Enum_2Enum )
@ ( lam @ A_27a
@ ^ [V3x: $i] : ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( ap @ V0f @ V3x ) ) @ ( ap @ V1g @ V3x ) ) ) )
@ V2ls ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( ap @ c_2Elist_2ESUM @ ( ap @ ( ap @ ( c_2Elist_2EMAP @ A_27a @ ty_2Enum_2Enum ) @ V0f ) @ V2ls ) ) ) @ ( ap @ c_2Elist_2ESUM @ ( ap @ ( ap @ ( c_2Elist_2EMAP @ A_27a @ ty_2Enum_2Enum ) @ V1g ) @ V2ls ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2ETAKE__LENGTH__ID__rwt,axiom,
! [A_27a: del,V0l: $i] :
( ( mem @ V0l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V1m: tp__ty_2Enum_2Enum] :
( ( V1m
= ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ V0l ) ) )
=> ( ( ap @ ( ap @ ( c_2Elist_2ETAKE @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V1m ) ) @ V0l )
= V0l ) ) ) ).
thf(conj_thm_2Elist_2EZIP__DROP,axiom,
! [A_27a: del,A_27b: del,V0a: $i] :
( ( mem @ V0a @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V1b: $i] :
( ( mem @ V1b @ ( ty_2Elist_2Elist @ A_27b ) )
=> ! [V2n: tp__ty_2Enum_2Enum] :
( ( ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ V2n ) ) @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ V0a ) ) )
& ( ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ V0a ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Elist_2ELENGTH @ A_27b ) @ V1b ) ) ) )
=> ( ( ap @ ( c_2Elist_2EZIP @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Elist_2Elist @ A_27a ) @ ( ty_2Elist_2Elist @ A_27b ) ) @ ( ap @ ( ap @ ( c_2Elist_2EDROP @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) @ V0a ) ) @ ( ap @ ( ap @ ( c_2Elist_2EDROP @ A_27b ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) @ V1b ) ) )
= ( ap @ ( ap @ ( c_2Elist_2EDROP @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) @ ( ap @ ( c_2Elist_2EZIP @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Elist_2Elist @ A_27a ) @ ( ty_2Elist_2Elist @ A_27b ) ) @ V0a ) @ V1b ) ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2EGENLIST__EL,axiom,
! [A_27a: del,V0ls: $i] :
( ( mem @ V0ls @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ ty_2Enum_2Enum @ A_27a ) )
=> ! [V2n: tp__ty_2Enum_2Enum] :
( ( ( V2n
= ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ V0ls ) ) )
& ! [V3i: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V3i ) ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) )
=> ( ( ap @ V1f @ ( inj__ty_2Enum_2Enum @ V3i ) )
= ( ap @ ( ap @ ( c_2Elist_2EEL @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V3i ) ) @ V0ls ) ) ) )
=> ( ( ap @ ( ap @ ( c_2Elist_2EGENLIST @ A_27a ) @ V1f ) @ ( inj__ty_2Enum_2Enum @ V2n ) )
= V0ls ) ) ) ) ).
thf(conj_thm_2Elist_2EEVERY2__trans,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ! [V2y: $i] :
( ( mem @ V2y @ A_27a )
=> ! [V3z: $i] :
( ( mem @ V3z @ A_27a )
=> ( ( ( p @ ( ap @ ( ap @ V0R @ V1x ) @ V2y ) )
& ( p @ ( ap @ ( ap @ V0R @ V2y ) @ V3z ) ) )
=> ( p @ ( ap @ ( ap @ V0R @ V1x ) @ V3z ) ) ) ) ) )
=> ! [V4x: $i] :
( ( mem @ V4x @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V5y: $i] :
( ( mem @ V5y @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V6z: $i] :
( ( mem @ V6z @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Elist_2ELIST__REL @ A_27a @ A_27a ) @ V0R ) @ V4x ) @ V5y ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( c_2Elist_2ELIST__REL @ A_27a @ A_27a ) @ V0R ) @ V5y ) @ V6z ) ) )
=> ( p @ ( ap @ ( ap @ ( ap @ ( c_2Elist_2ELIST__REL @ A_27a @ A_27a ) @ V0R ) @ V4x ) @ V6z ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2EEVERY2__sym,axiom,
! [A_27a: del,A_27b: del,V0R1: $i] :
( ( mem @ V0R1 @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) )
=> ! [V1R2: $i] :
( ( mem @ V1R2 @ ( arr @ A_27b @ ( arr @ A_27a @ bool ) ) )
=> ( ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ! [V3y: $i] :
( ( mem @ V3y @ A_27b )
=> ( ( p @ ( ap @ ( ap @ V0R1 @ V2x ) @ V3y ) )
=> ( p @ ( ap @ ( ap @ V1R2 @ V3y ) @ V2x ) ) ) ) )
=> ! [V4x: $i] :
( ( mem @ V4x @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V5y: $i] :
( ( mem @ V5y @ ( ty_2Elist_2Elist @ A_27b ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Elist_2ELIST__REL @ A_27a @ A_27b ) @ V0R1 ) @ V4x ) @ V5y ) )
=> ( p @ ( ap @ ( ap @ ( ap @ ( c_2Elist_2ELIST__REL @ A_27b @ A_27a ) @ V1R2 ) @ V5y ) @ V4x ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2EEVERY2__LUPDATE__same,axiom,
! [A_27a: del,A_27b: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) )
=> ! [V1l1: $i] :
( ( mem @ V1l1 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V2l2: $i] :
( ( mem @ V2l2 @ ( ty_2Elist_2Elist @ A_27b ) )
=> ! [V3v1: $i] :
( ( mem @ V3v1 @ A_27a )
=> ! [V4v2: $i] :
( ( mem @ V4v2 @ A_27b )
=> ! [V5n: tp__ty_2Enum_2Enum] :
( ( ( p @ ( ap @ ( ap @ V0P @ V3v1 ) @ V4v2 ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( c_2Elist_2ELIST__REL @ A_27a @ A_27b ) @ V0P ) @ V1l1 ) @ V2l2 ) ) )
=> ( p @ ( ap @ ( ap @ ( ap @ ( c_2Elist_2ELIST__REL @ A_27a @ A_27b ) @ V0P ) @ ( ap @ ( ap @ ( ap @ ( c_2Elist_2ELUPDATE @ A_27a ) @ V3v1 ) @ ( inj__ty_2Enum_2Enum @ V5n ) ) @ V1l1 ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Elist_2ELUPDATE @ A_27b ) @ V4v2 ) @ ( inj__ty_2Enum_2Enum @ V5n ) ) @ V2l2 ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2EEVERY2__refl,axiom,
! [A_27a: del,V0ls: $i] :
( ( mem @ V0ls @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V1R: $i] :
( ( mem @ V1R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x ) @ ( ap @ ( c_2Elist_2ELIST__TO__SET @ A_27a ) @ V0ls ) ) )
=> ( p @ ( ap @ ( ap @ V1R @ V2x ) @ V2x ) ) ) )
=> ( p @ ( ap @ ( ap @ ( ap @ ( c_2Elist_2ELIST__REL @ A_27a @ A_27a ) @ V1R ) @ V0ls ) @ V0ls ) ) ) ) ) ).
thf(conj_thm_2Elist_2EEVERY2__THM,axiom,
! [A_27a: del,A_27b: del,A_27c: del,A_27d: del,A_27e: del,A_27f: del,A_27g: del,A_27h: del] :
( ! [V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) )
=> ! [V1ys: $i] :
( ( mem @ V1ys @ ( ty_2Elist_2Elist @ A_27b ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Elist_2ELIST__REL @ A_27a @ A_27b ) @ V0P ) @ ( c_2Elist_2ENIL @ A_27a ) ) @ V1ys ) )
<=> ( V1ys
= ( c_2Elist_2ENIL @ A_27b ) ) ) ) )
& ! [V2P: $i] :
( ( mem @ V2P @ ( arr @ A_27c @ ( arr @ A_27d @ bool ) ) )
=> ! [V3yys: $i] :
( ( mem @ V3yys @ ( ty_2Elist_2Elist @ A_27d ) )
=> ! [V4x: $i] :
( ( mem @ V4x @ A_27c )
=> ! [V5xs: $i] :
( ( mem @ V5xs @ ( ty_2Elist_2Elist @ A_27c ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Elist_2ELIST__REL @ A_27c @ A_27d ) @ V2P ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27c ) @ V4x ) @ V5xs ) ) @ V3yys ) )
<=> ? [V6y: $i] :
( ( mem @ V6y @ A_27d )
& ? [V7ys: $i] :
( ( mem @ V7ys @ ( ty_2Elist_2Elist @ A_27d ) )
& ( V3yys
= ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27d ) @ V6y ) @ V7ys ) )
& ( p @ ( ap @ ( ap @ V2P @ V4x ) @ V6y ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( c_2Elist_2ELIST__REL @ A_27c @ A_27d ) @ V2P ) @ V5xs ) @ V7ys ) ) ) ) ) ) ) ) )
& ! [V8P: $i] :
( ( mem @ V8P @ ( arr @ A_27e @ ( arr @ A_27f @ bool ) ) )
=> ! [V9xs: $i] :
( ( mem @ V9xs @ ( ty_2Elist_2Elist @ A_27e ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Elist_2ELIST__REL @ A_27e @ A_27f ) @ V8P ) @ V9xs ) @ ( c_2Elist_2ENIL @ A_27f ) ) )
<=> ( V9xs
= ( c_2Elist_2ENIL @ A_27e ) ) ) ) )
& ! [V10P: $i] :
( ( mem @ V10P @ ( arr @ A_27g @ ( arr @ A_27h @ bool ) ) )
=> ! [V11xxs: $i] :
( ( mem @ V11xxs @ ( ty_2Elist_2Elist @ A_27g ) )
=> ! [V12y: $i] :
( ( mem @ V12y @ A_27h )
=> ! [V13ys: $i] :
( ( mem @ V13ys @ ( ty_2Elist_2Elist @ A_27h ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Elist_2ELIST__REL @ A_27g @ A_27h ) @ V10P ) @ V11xxs ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27h ) @ V12y ) @ V13ys ) ) )
<=> ? [V14x: $i] :
( ( mem @ V14x @ A_27g )
& ? [V15xs: $i] :
( ( mem @ V15xs @ ( ty_2Elist_2Elist @ A_27g ) )
& ( V11xxs
= ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27g ) @ V14x ) @ V15xs ) )
& ( p @ ( ap @ ( ap @ V10P @ V14x ) @ V12y ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( c_2Elist_2ELIST__REL @ A_27g @ A_27h ) @ V10P ) @ V15xs ) @ V13ys ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2ELIST__REL__trans,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1l1: $i] :
( ( mem @ V1l1 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V2l2: $i] :
( ( mem @ V2l2 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V3l3: $i] :
( ( mem @ V3l3 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( ! [V4n: tp__ty_2Enum_2Enum] :
( ( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V4n ) ) @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ V1l1 ) ) )
& ( p @ ( ap @ ( ap @ V0R @ ( ap @ ( ap @ ( c_2Elist_2EEL @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V4n ) ) @ V1l1 ) ) @ ( ap @ ( ap @ ( c_2Elist_2EEL @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V4n ) ) @ V2l2 ) ) )
& ( p @ ( ap @ ( ap @ V0R @ ( ap @ ( ap @ ( c_2Elist_2EEL @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V4n ) ) @ V2l2 ) ) @ ( ap @ ( ap @ ( c_2Elist_2EEL @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V4n ) ) @ V3l3 ) ) ) )
=> ( p @ ( ap @ ( ap @ V0R @ ( ap @ ( ap @ ( c_2Elist_2EEL @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V4n ) ) @ V1l1 ) ) @ ( ap @ ( ap @ ( c_2Elist_2EEL @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V4n ) ) @ V3l3 ) ) ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( c_2Elist_2ELIST__REL @ A_27a @ A_27a ) @ V0R ) @ V1l1 ) @ V2l2 ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( c_2Elist_2ELIST__REL @ A_27a @ A_27a ) @ V0R ) @ V2l2 ) @ V3l3 ) ) )
=> ( p @ ( ap @ ( ap @ ( ap @ ( c_2Elist_2ELIST__REL @ A_27a @ A_27a ) @ V0R ) @ V1l1 ) @ V3l3 ) ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2ESWAP__REVERSE,axiom,
! [A_27a: del,V0l1: $i] :
( ( mem @ V0l1 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V1l2: $i] :
( ( mem @ V1l2 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( V0l1
= ( ap @ ( c_2Elist_2EREVERSE @ A_27a ) @ V1l2 ) )
<=> ( V1l2
= ( ap @ ( c_2Elist_2EREVERSE @ A_27a ) @ V0l1 ) ) ) ) ) ).
thf(conj_thm_2Elist_2ESWAP__REVERSE__SYM,axiom,
! [A_27a: del,V0l1: $i] :
( ( mem @ V0l1 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V1l2: $i] :
( ( mem @ V1l2 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( ( ap @ ( c_2Elist_2EREVERSE @ A_27a ) @ V0l1 )
= V1l2 )
<=> ( V0l1
= ( ap @ ( c_2Elist_2EREVERSE @ A_27a ) @ V1l2 ) ) ) ) ) ).
thf(conj_thm_2Elist_2EBIGUNION__IMAGE__set__SUBSET,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27b @ ( arr @ A_27a @ bool ) ) )
=> ! [V1ls: $i] :
( ( mem @ V1ls @ ( ty_2Elist_2Elist @ A_27b ) )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ ( ap @ ( c_2Epred__set_2EBIGUNION @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ A_27b @ ( arr @ A_27a @ bool ) ) @ V0f ) @ ( ap @ ( c_2Elist_2ELIST__TO__SET @ A_27b ) @ V1ls ) ) ) ) @ V2s ) )
<=> ! [V3x: $i] :
( ( mem @ V3x @ A_27b )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27b ) @ V3x ) @ ( ap @ ( c_2Elist_2ELIST__TO__SET @ A_27b ) @ V1ls ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ ( ap @ V0f @ V3x ) ) @ V2s ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2EIMAGE__EL__count__LENGTH,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27b ) )
=> ! [V1ls: $i] :
( ( mem @ V1ls @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( ap
@ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Enum_2Enum @ A_27b )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V2n: $i] : ( ap @ V0f @ ( ap @ ( ap @ ( c_2Elist_2EEL @ A_27a ) @ V2n ) @ V1ls ) ) ) )
@ ( ap @ c_2Epred__set_2Ecount @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ V1ls ) ) )
= ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27b ) @ V0f ) @ ( ap @ ( c_2Elist_2ELIST__TO__SET @ A_27a ) @ V1ls ) ) ) ) ) ).
thf(conj_thm_2Elist_2EGENLIST__EL__MAP,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27b ) )
=> ! [V1ls: $i] :
( ( mem @ V1ls @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( ap
@ ( ap @ ( c_2Elist_2EGENLIST @ A_27b )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V2n: $i] : ( ap @ V0f @ ( ap @ ( ap @ ( c_2Elist_2EEL @ A_27a ) @ V2n ) @ V1ls ) ) ) )
@ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ V1ls ) )
= ( ap @ ( ap @ ( c_2Elist_2EMAP @ A_27a @ A_27b ) @ V0f ) @ V1ls ) ) ) ) ).
thf(conj_thm_2Elist_2ELENGTH__FILTER__LEQ__MONO,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ! [V1Q: $i] :
( ( mem @ V1Q @ ( arr @ A_27a @ bool ) )
=> ( ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ( ( p @ ( ap @ V0P @ V2x ) )
=> ( p @ ( ap @ V1Q @ V2x ) ) ) )
=> ! [V3ls: $i] :
( ( mem @ V3ls @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2EFILTER @ A_27a ) @ V0P ) @ V3ls ) ) ) @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2EFILTER @ A_27a ) @ V1Q ) @ V3ls ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2ELIST__EQ__MAP__PAIR,axiom,
! [A_27a: del,A_27b: del,V0l1: $i] :
( ( mem @ V0l1 @ ( ty_2Elist_2Elist @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) ) )
=> ! [V1l2: $i] :
( ( mem @ V1l2 @ ( ty_2Elist_2Elist @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) ) )
=> ( ( ( ( ap @ ( ap @ ( c_2Elist_2EMAP @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ A_27a ) @ ( c_2Epair_2EFST @ A_27a @ A_27b ) ) @ V0l1 )
= ( ap @ ( ap @ ( c_2Elist_2EMAP @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ A_27a ) @ ( c_2Epair_2EFST @ A_27a @ A_27b ) ) @ V1l2 ) )
& ( ( ap @ ( ap @ ( c_2Elist_2EMAP @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ A_27b ) @ ( c_2Epair_2ESND @ A_27a @ A_27b ) ) @ V0l1 )
= ( ap @ ( ap @ ( c_2Elist_2EMAP @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ A_27b ) @ ( c_2Epair_2ESND @ A_27a @ A_27b ) ) @ V1l2 ) ) )
=> ( V0l1 = V1l2 ) ) ) ) ).
thf(conj_thm_2Elist_2ETAKE__SUM,axiom,
! [A_27a: del,V0n: tp__ty_2Enum_2Enum,V1m: tp__ty_2Enum_2Enum,V2l: $i] :
( ( mem @ V2l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( ap @ ( ap @ ( c_2Elist_2ETAKE @ A_27a ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( inj__ty_2Enum_2Enum @ V0n ) ) @ ( inj__ty_2Enum_2Enum @ V1m ) ) ) @ V2l )
= ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2ETAKE @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) @ V2l ) ) @ ( ap @ ( ap @ ( c_2Elist_2ETAKE @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V1m ) ) @ ( ap @ ( ap @ ( c_2Elist_2EDROP @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) @ V2l ) ) ) ) ) ).
thf(conj_thm_2Elist_2EALL__DISTINCT__FILTER__EL__IMP,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ! [V1l: $i] :
( ( mem @ V1l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V2n1: tp__ty_2Enum_2Enum,V3n2: tp__ty_2Enum_2Enum] :
( ( ( p @ ( ap @ ( c_2Elist_2EALL__DISTINCT @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2EFILTER @ A_27a ) @ V0P ) @ V1l ) ) )
& ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V2n1 ) ) @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ V1l ) ) )
& ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V3n2 ) ) @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ V1l ) ) )
& ( p @ ( ap @ V0P @ ( ap @ ( ap @ ( c_2Elist_2EEL @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V2n1 ) ) @ V1l ) ) )
& ( ( ap @ ( ap @ ( c_2Elist_2EEL @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V2n1 ) ) @ V1l )
= ( ap @ ( ap @ ( c_2Elist_2EEL @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V3n2 ) ) @ V1l ) ) )
=> ( V2n1 = V3n2 ) ) ) ) ).
thf(conj_thm_2Elist_2EFLAT__EQ__NIL,axiom,
! [A_27a: del,V0ls: $i] :
( ( mem @ V0ls @ ( ty_2Elist_2Elist @ ( ty_2Elist_2Elist @ A_27a ) ) )
=> ( ( ( ap @ ( c_2Elist_2EFLAT @ A_27a ) @ V0ls )
= ( c_2Elist_2ENIL @ A_27a ) )
<=> ( p @ ( ap @ ( ap @ ( c_2Elist_2EEVERY @ ( ty_2Elist_2Elist @ A_27a ) ) @ ( ap @ ( c_2Emin_2E_3D @ ( ty_2Elist_2Elist @ A_27a ) ) @ ( c_2Elist_2ENIL @ A_27a ) ) ) @ V0ls ) ) ) ) ).
thf(conj_thm_2Elist_2EALL__DISTINCT__MAP__INJ,axiom,
! [A_27a: del,A_27b: del,V0ls: $i] :
( ( mem @ V0ls @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ A_27b ) )
=> ( ( ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ! [V3y: $i] :
( ( mem @ V3y @ A_27a )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x ) @ ( ap @ ( c_2Elist_2ELIST__TO__SET @ A_27a ) @ V0ls ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3y ) @ ( ap @ ( c_2Elist_2ELIST__TO__SET @ A_27a ) @ V0ls ) ) )
& ( ( ap @ V1f @ V2x )
= ( ap @ V1f @ V3y ) ) )
=> ( V2x = V3y ) ) ) )
& ( p @ ( ap @ ( c_2Elist_2EALL__DISTINCT @ A_27a ) @ V0ls ) ) )
=> ( p @ ( ap @ ( c_2Elist_2EALL__DISTINCT @ A_27b ) @ ( ap @ ( ap @ ( c_2Elist_2EMAP @ A_27a @ A_27b ) @ V1f ) @ V0ls ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2ELENGTH__o__REVERSE,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27b @ ( ty_2Elist_2Elist @ A_27c ) ) )
=> ( ( ( ap @ ( ap @ ( c_2Ecombin_2Eo @ ( ty_2Elist_2Elist @ A_27a ) @ ty_2Enum_2Enum @ ( ty_2Elist_2Elist @ A_27a ) ) @ ( c_2Elist_2ELENGTH @ A_27a ) ) @ ( c_2Elist_2EREVERSE @ A_27a ) )
= ( c_2Elist_2ELENGTH @ A_27a ) )
& ( ( ap @ ( ap @ ( c_2Ecombin_2Eo @ A_27b @ ty_2Enum_2Enum @ ( ty_2Elist_2Elist @ A_27c ) ) @ ( c_2Elist_2ELENGTH @ A_27c ) ) @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ A_27b @ ( ty_2Elist_2Elist @ A_27c ) @ ( ty_2Elist_2Elist @ A_27c ) ) @ ( c_2Elist_2EREVERSE @ A_27c ) ) @ V0f ) )
= ( ap @ ( ap @ ( c_2Ecombin_2Eo @ A_27b @ ty_2Enum_2Enum @ ( ty_2Elist_2Elist @ A_27c ) ) @ ( c_2Elist_2ELENGTH @ A_27c ) ) @ V0f ) ) ) ) ).
thf(conj_thm_2Elist_2EREVERSE__o__REVERSE,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ( ty_2Elist_2Elist @ A_27b ) ) )
=> ( ( ap @ ( ap @ ( c_2Ecombin_2Eo @ A_27a @ ( ty_2Elist_2Elist @ A_27b ) @ ( ty_2Elist_2Elist @ A_27b ) ) @ ( c_2Elist_2EREVERSE @ A_27b ) ) @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ A_27a @ ( ty_2Elist_2Elist @ A_27b ) @ ( ty_2Elist_2Elist @ A_27b ) ) @ ( c_2Elist_2EREVERSE @ A_27b ) ) @ V0f ) )
= V0f ) ) ).
thf(conj_thm_2Elist_2EGENLIST__PLUS__APPEND,axiom,
! [V0a: tp__ty_2Enum_2Enum,V1n1: tp__ty_2Enum_2Enum,V2n2: tp__ty_2Enum_2Enum] :
( ( surj__c_ty_2Elist_2Elist_ty_2Enum_2Enum @ ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ ( c_2Elist_2EGENLIST @ ty_2Enum_2Enum ) @ ( ap @ c_2Earithmetic_2E_2B @ ( inj__ty_2Enum_2Enum @ V0a ) ) ) @ ( inj__ty_2Enum_2Enum @ V1n1 ) ) ) @ ( ap @ ( ap @ ( c_2Elist_2EGENLIST @ ty_2Enum_2Enum ) @ ( ap @ c_2Earithmetic_2E_2B @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( inj__ty_2Enum_2Enum @ V1n1 ) ) @ ( inj__ty_2Enum_2Enum @ V0a ) ) ) ) @ ( inj__ty_2Enum_2Enum @ V2n2 ) ) ) )
= ( surj__c_ty_2Elist_2Elist_ty_2Enum_2Enum @ ( ap @ ( ap @ ( c_2Elist_2EGENLIST @ ty_2Enum_2Enum ) @ ( ap @ c_2Earithmetic_2E_2B @ ( inj__ty_2Enum_2Enum @ V0a ) ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( inj__ty_2Enum_2Enum @ V1n1 ) ) @ ( inj__ty_2Enum_2Enum @ V2n2 ) ) ) ) ) ).
thf(conj_thm_2Elist_2ELIST__TO__SET__GENLIST,axiom,
! [A_27a: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ A_27a ) )
=> ! [V1n: tp__ty_2Enum_2Enum] :
( ( ap @ ( c_2Elist_2ELIST__TO__SET @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2EGENLIST @ A_27a ) @ V0f ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) )
= ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Enum_2Enum @ A_27a ) @ V0f ) @ ( ap @ c_2Epred__set_2Ecount @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) ) ) ).
thf(conj_thm_2Elist_2EMEM__ZIP__MEM__MAP,axiom,
! [A_27a: del,A_27b: del,V0ps: $i] :
( ( mem @ V0ps @ ( ty_2Epair_2Eprod @ ( ty_2Elist_2Elist @ A_27a ) @ ( ty_2Elist_2Elist @ A_27b ) ) )
=> ! [V1p: $i] :
( ( mem @ V1p @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) )
=> ( ( ( ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ ( ap @ ( c_2Epair_2EFST @ ( ty_2Elist_2Elist @ A_27a ) @ ( ty_2Elist_2Elist @ A_27b ) ) @ V0ps ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Elist_2ELENGTH @ A_27b ) @ ( ap @ ( c_2Epair_2ESND @ ( ty_2Elist_2Elist @ A_27a ) @ ( ty_2Elist_2Elist @ A_27b ) ) @ V0ps ) ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) ) @ V1p ) @ ( ap @ ( c_2Elist_2ELIST__TO__SET @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) ) @ ( ap @ ( c_2Elist_2EZIP @ A_27a @ A_27b ) @ V0ps ) ) ) ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ ( ap @ ( c_2Epair_2EFST @ A_27a @ A_27b ) @ V1p ) ) @ ( ap @ ( c_2Elist_2ELIST__TO__SET @ A_27a ) @ ( ap @ ( c_2Epair_2EFST @ ( ty_2Elist_2Elist @ A_27a ) @ ( ty_2Elist_2Elist @ A_27b ) ) @ V0ps ) ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27b ) @ ( ap @ ( c_2Epair_2ESND @ A_27a @ A_27b ) @ V1p ) ) @ ( ap @ ( c_2Elist_2ELIST__TO__SET @ A_27b ) @ ( ap @ ( c_2Epair_2ESND @ ( ty_2Elist_2Elist @ A_27a ) @ ( ty_2Elist_2Elist @ A_27b ) ) @ V0ps ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2EDISJOINT__GENLIST__PLUS,axiom,
! [V0x: $i] :
( ( mem @ V0x @ ( arr @ ty_2Enum_2Enum @ bool ) )
=> ! [V1n: tp__ty_2Enum_2Enum,V2a: tp__ty_2Enum_2Enum,V3b: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2EDISJOINT @ ty_2Enum_2Enum ) @ V0x ) @ ( ap @ ( c_2Elist_2ELIST__TO__SET @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ ( c_2Elist_2EGENLIST @ ty_2Enum_2Enum ) @ ( ap @ c_2Earithmetic_2E_2B @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( inj__ty_2Enum_2Enum @ V2a ) ) @ ( inj__ty_2Enum_2Enum @ V3b ) ) ) ) ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2EDISJOINT @ ty_2Enum_2Enum ) @ V0x ) @ ( ap @ ( c_2Elist_2ELIST__TO__SET @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ ( c_2Elist_2EGENLIST @ ty_2Enum_2Enum ) @ ( ap @ c_2Earithmetic_2E_2B @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) @ ( inj__ty_2Enum_2Enum @ V2a ) ) ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2EDISJOINT @ ty_2Enum_2Enum ) @ V0x ) @ ( ap @ ( c_2Elist_2ELIST__TO__SET @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ ( c_2Elist_2EGENLIST @ ty_2Enum_2Enum ) @ ( ap @ c_2Earithmetic_2E_2B @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( inj__ty_2Enum_2Enum @ V1n ) ) @ ( inj__ty_2Enum_2Enum @ V2a ) ) ) ) @ ( inj__ty_2Enum_2Enum @ V3b ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2EEVERY2__MAP,axiom,
! [A_27a: del,A_27b: del,A_27c: del,A_27d: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27c @ A_27a ) )
=> ! [V2l1: $i] :
( ( mem @ V2l1 @ ( ty_2Elist_2Elist @ A_27c ) )
=> ! [V3l2: $i] :
( ( mem @ V3l2 @ ( ty_2Elist_2Elist @ A_27b ) )
=> ! [V4Q: $i] :
( ( mem @ V4Q @ ( arr @ A_27c @ ( arr @ A_27d @ bool ) ) )
=> ! [V5g: $i] :
( ( mem @ V5g @ ( arr @ A_27b @ A_27d ) )
=> ( ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Elist_2ELIST__REL @ A_27a @ A_27b ) @ V0P ) @ ( ap @ ( ap @ ( c_2Elist_2EMAP @ A_27c @ A_27a ) @ V1f ) @ V2l1 ) ) @ V3l2 ) )
<=> ( p
@ ( ap
@ ( ap
@ ( ap @ ( c_2Elist_2ELIST__REL @ A_27c @ A_27b )
@ ( lam @ A_27c
@ ^ [V6x: $i] :
( lam @ A_27b
@ ^ [V7y: $i] : ( ap @ ( ap @ V0P @ ( ap @ V1f @ V6x ) ) @ V7y ) ) ) )
@ V2l1 )
@ V3l2 ) ) )
& ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Elist_2ELIST__REL @ A_27c @ A_27d ) @ V4Q ) @ V2l1 ) @ ( ap @ ( ap @ ( c_2Elist_2EMAP @ A_27b @ A_27d ) @ V5g ) @ V3l2 ) ) )
<=> ( p
@ ( ap
@ ( ap
@ ( ap @ ( c_2Elist_2ELIST__REL @ A_27c @ A_27b )
@ ( lam @ A_27c
@ ^ [V8x: $i] :
( lam @ A_27b
@ ^ [V9y: $i] : ( ap @ ( ap @ V4Q @ V8x ) @ ( ap @ V5g @ V9y ) ) ) ) )
@ V2l1 )
@ V3l2 ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2Eexists__list__GENLIST,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ ( ty_2Elist_2Elist @ A_27a ) @ bool ) )
=> ( ? [V1ls: $i] :
( ( mem @ V1ls @ ( ty_2Elist_2Elist @ A_27a ) )
& ( p @ ( ap @ V0P @ V1ls ) ) )
<=> ? [V2n: tp__ty_2Enum_2Enum,V3f: $i] :
( ( mem @ V3f @ ( arr @ ty_2Enum_2Enum @ A_27a ) )
& ( p @ ( ap @ V0P @ ( ap @ ( ap @ ( c_2Elist_2EGENLIST @ A_27a ) @ V3f ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2EEVERY__MEM__MONO,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ! [V1Q: $i] :
( ( mem @ V1Q @ ( arr @ A_27a @ bool ) )
=> ! [V2l: $i] :
( ( mem @ V2l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3x ) @ ( ap @ ( c_2Elist_2ELIST__TO__SET @ A_27a ) @ V2l ) ) )
& ( p @ ( ap @ V0P @ V3x ) ) )
=> ( p @ ( ap @ V1Q @ V3x ) ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Elist_2EEVERY @ A_27a ) @ V0P ) @ V2l ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Elist_2EEVERY @ A_27a ) @ V1Q ) @ V2l ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2EEVERY2__MEM__MONO,axiom,
! [A_27a: del,A_27b: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) )
=> ! [V1Q: $i] :
( ( mem @ V1Q @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) )
=> ! [V2l1: $i] :
( ( mem @ V2l1 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V3l2: $i] :
( ( mem @ V3l2 @ ( ty_2Elist_2Elist @ A_27b ) )
=> ( ( ! [V4x: $i] :
( ( mem @ V4x @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) ) @ V4x ) @ ( ap @ ( c_2Elist_2ELIST__TO__SET @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) ) @ ( ap @ ( c_2Elist_2EZIP @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Elist_2Elist @ A_27a ) @ ( ty_2Elist_2Elist @ A_27b ) ) @ V2l1 ) @ V3l2 ) ) ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Epair_2EUNCURRY @ A_27a @ A_27b @ bool ) @ V0P ) @ V4x ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Epair_2EUNCURRY @ A_27a @ A_27b @ bool ) @ V1Q ) @ V4x ) ) ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( c_2Elist_2ELIST__REL @ A_27a @ A_27b ) @ V0P ) @ V2l1 ) @ V3l2 ) ) )
=> ( p @ ( ap @ ( ap @ ( ap @ ( c_2Elist_2ELIST__REL @ A_27a @ A_27b ) @ V1Q ) @ V2l1 ) @ V3l2 ) ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2Emem__exists__set,axiom,
! [A_27a: del,A_27b: del,V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ! [V1y: $i] :
( ( mem @ V1y @ A_27b )
=> ! [V2l: $i] :
( ( mem @ V2l @ ( ty_2Elist_2Elist @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27b ) @ V0x ) @ V1y ) ) @ ( ap @ ( c_2Elist_2ELIST__TO__SET @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) ) @ V2l ) ) )
=> ? [V3z: $i] :
( ( mem @ V3z @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) )
& ( V0x
= ( ap @ ( c_2Epair_2EFST @ A_27a @ A_27b ) @ V3z ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) ) @ V3z ) @ ( ap @ ( c_2Elist_2ELIST__TO__SET @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) ) @ V2l ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2Eevery__zip__snd,axiom,
! [A_27a: del,A_27b: del,V0l1: $i] :
( ( mem @ V0l1 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V1l2: $i] :
( ( mem @ V1l2 @ ( ty_2Elist_2Elist @ A_27b ) )
=> ! [V2P: $i] :
( ( mem @ V2P @ ( arr @ A_27b @ bool ) )
=> ( ( ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ V0l1 ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Elist_2ELENGTH @ A_27b ) @ V1l2 ) ) )
=> ( ( p
@ ( ap
@ ( ap @ ( c_2Elist_2EEVERY @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) )
@ ( lam @ ( ty_2Epair_2Eprod @ A_27a @ A_27b )
@ ^ [V3x: $i] : ( ap @ V2P @ ( ap @ ( c_2Epair_2ESND @ A_27a @ A_27b ) @ V3x ) ) ) )
@ ( ap @ ( c_2Elist_2EZIP @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Elist_2Elist @ A_27a ) @ ( ty_2Elist_2Elist @ A_27b ) ) @ V0l1 ) @ V1l2 ) ) ) )
<=> ( p @ ( ap @ ( ap @ ( c_2Elist_2EEVERY @ A_27b ) @ V2P ) @ V1l2 ) ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2Eevery__zip__fst,axiom,
! [A_27a: del,A_27b: del,V0l1: $i] :
( ( mem @ V0l1 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V1l2: $i] :
( ( mem @ V1l2 @ ( ty_2Elist_2Elist @ A_27b ) )
=> ! [V2P: $i] :
( ( mem @ V2P @ ( arr @ A_27a @ bool ) )
=> ( ( ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ V0l1 ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Elist_2ELENGTH @ A_27b ) @ V1l2 ) ) )
=> ( ( p
@ ( ap
@ ( ap @ ( c_2Elist_2EEVERY @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) )
@ ( lam @ ( ty_2Epair_2Eprod @ A_27a @ A_27b )
@ ^ [V3x: $i] : ( ap @ V2P @ ( ap @ ( c_2Epair_2EFST @ A_27a @ A_27b ) @ V3x ) ) ) )
@ ( ap @ ( c_2Elist_2EZIP @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Elist_2Elist @ A_27a ) @ ( ty_2Elist_2Elist @ A_27b ) ) @ V0l1 ) @ V1l2 ) ) ) )
<=> ( p @ ( ap @ ( ap @ ( c_2Elist_2EEVERY @ A_27a ) @ V2P ) @ V0l1 ) ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2Eel__append3,axiom,
! [A_27a: del,V0l1: $i] :
( ( mem @ V0l1 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ! [V2l2: $i] :
( ( mem @ V2l2 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( ap @ ( ap @ ( c_2Elist_2EEL @ A_27a ) @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ V0l1 ) ) @ ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ V0l1 ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V1x ) @ ( c_2Elist_2ENIL @ A_27a ) ) ) ) @ V2l2 ) )
= V1x ) ) ) ) ).
thf(conj_thm_2Elist_2Elupdate__append,axiom,
! [A_27a: del,V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ! [V1n: tp__ty_2Enum_2Enum,V2l1: $i] :
( ( mem @ V2l1 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V3l2: $i] :
( ( mem @ V3l2 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V1n ) ) @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ V2l1 ) ) )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Elist_2ELUPDATE @ A_27a ) @ V0x ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) @ ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ V2l1 ) @ V3l2 ) )
= ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ ( ap @ ( ap @ ( ap @ ( c_2Elist_2ELUPDATE @ A_27a ) @ V0x ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) @ V2l1 ) ) @ V3l2 ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2Elupdate__append2,axiom,
! [A_27a: del,A_27b: del,V0v: $i] :
( ( mem @ V0v @ A_27a )
=> ! [V1l1: $i] :
( ( mem @ V1l1 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ! [V3l2: $i] :
( ( mem @ V3l2 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V4l3: $i] :
( ( mem @ V4l3 @ A_27b )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Elist_2ELUPDATE @ A_27a ) @ V0v ) @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ V1l1 ) ) @ ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ V1l1 ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V2x ) @ ( c_2Elist_2ENIL @ A_27a ) ) ) ) @ V3l2 ) )
= ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ V1l1 ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V0v ) @ ( c_2Elist_2ENIL @ A_27a ) ) ) ) @ V3l2 ) ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2EHD__REVERSE,axiom,
! [A_27a: del,V0x: $i] :
( ( mem @ V0x @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( V0x
!= ( c_2Elist_2ENIL @ A_27a ) )
=> ( ( ap @ ( c_2Elist_2EHD @ A_27a ) @ ( ap @ ( c_2Elist_2EREVERSE @ A_27a ) @ V0x ) )
= ( ap @ ( c_2Elist_2ELAST @ A_27a ) @ V0x ) ) ) ) ).
thf(conj_thm_2Elist_2ELAST__REVERSE,axiom,
! [A_27a: del,V0ls: $i] :
( ( mem @ V0ls @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( V0ls
!= ( c_2Elist_2ENIL @ A_27a ) )
=> ( ( ap @ ( c_2Elist_2ELAST @ A_27a ) @ ( ap @ ( c_2Elist_2EREVERSE @ A_27a ) @ V0ls ) )
= ( ap @ ( c_2Elist_2EHD @ A_27a ) @ V0ls ) ) ) ) ).
thf(conj_thm_2Elist_2ENOT__NIL__EQ__LENGTH__NOT__0,axiom,
! [A_27a: del,V0x: $i] :
( ( mem @ V0x @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( V0x
!= ( c_2Elist_2ENIL @ A_27a ) )
<=> ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ V0x ) ) ) ) ) ).
thf(conj_thm_2Elist_2Elast__drop,axiom,
! [A_27a: del,V0l: $i] :
( ( mem @ V0l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V1n: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V1n ) ) @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ V0l ) ) )
=> ( ( ap @ ( c_2Elist_2ELAST @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2EDROP @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) @ V0l ) )
= ( ap @ ( c_2Elist_2ELAST @ A_27a ) @ V0l ) ) ) ) ).
thf(ax_thm_2Elist_2EdropWhile__def,axiom,
! [A_27a: del] :
( ! [V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ( ( ap @ ( ap @ ( c_2Elist_2EdropWhile @ A_27a ) @ V0P ) @ ( c_2Elist_2ENIL @ A_27a ) )
= ( c_2Elist_2ENIL @ A_27a ) ) )
& ! [V1P: $i] :
( ( mem @ V1P @ ( arr @ A_27a @ bool ) )
=> ! [V2h: $i] :
( ( mem @ V2h @ A_27a )
=> ! [V3t: $i] :
( ( mem @ V3t @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( ap @ ( ap @ ( c_2Elist_2EdropWhile @ A_27a ) @ V1P ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V2h ) @ V3t ) )
= ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ( ty_2Elist_2Elist @ A_27a ) ) @ ( ap @ V1P @ V2h ) ) @ ( ap @ ( ap @ ( c_2Elist_2EdropWhile @ A_27a ) @ V1P ) @ V3t ) ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V2h ) @ V3t ) ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2EdropWhile__splitAtPki,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ( ( ap @ ( c_2Elist_2EdropWhile @ A_27a ) @ V0P )
= ( ap @ ( ap @ ( c_2Elist_2EsplitAtPki @ ( ty_2Elist_2Elist @ A_27a ) @ A_27a ) @ ( ap @ ( c_2Ecombin_2EC @ A_27a @ ty_2Enum_2Enum @ bool ) @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ A_27a @ ( arr @ ty_2Enum_2Enum @ bool ) @ bool ) @ ( c_2Ecombin_2EK @ bool @ ty_2Enum_2Enum ) ) @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ A_27a @ bool @ bool ) @ c_2Ebool_2E_7E ) @ V0P ) ) ) ) @ ( ap @ ( c_2Ecombin_2EK @ ( arr @ ( ty_2Elist_2Elist @ A_27a ) @ ( ty_2Elist_2Elist @ A_27a ) ) @ ( ty_2Elist_2Elist @ A_27a ) ) @ ( c_2Ecombin_2EI @ ( ty_2Elist_2Elist @ A_27a ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2EdropWhile__eq__nil,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ! [V1ls: $i] :
( ( mem @ V1ls @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( ( ap @ ( ap @ ( c_2Elist_2EdropWhile @ A_27a ) @ V0P ) @ V1ls )
= ( c_2Elist_2ENIL @ A_27a ) )
<=> ( p @ ( ap @ ( ap @ ( c_2Elist_2EEVERY @ A_27a ) @ V0P ) @ V1ls ) ) ) ) ) ).
thf(conj_thm_2Elist_2EMEM__dropWhile__IMP,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ! [V1ls: $i] :
( ( mem @ V1ls @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x ) @ ( ap @ ( c_2Elist_2ELIST__TO__SET @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2EdropWhile @ A_27a ) @ V0P ) @ V1ls ) ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x ) @ ( ap @ ( c_2Elist_2ELIST__TO__SET @ A_27a ) @ V1ls ) ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2EHD__dropWhile,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ! [V1ls: $i] :
( ( mem @ V1ls @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Elist_2EEXISTS @ A_27a ) @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ A_27a @ bool @ bool ) @ c_2Ebool_2E_7E ) @ V0P ) ) @ V1ls ) )
=> ~ ( p @ ( ap @ V0P @ ( ap @ ( c_2Elist_2EHD @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2EdropWhile @ A_27a ) @ V0P ) @ V1ls ) ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2ELENGTH__dropWhile__LESS__EQ,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ! [V1ls: $i] :
( ( mem @ V1ls @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2EdropWhile @ A_27a ) @ V0P ) @ V1ls ) ) ) @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ V1ls ) ) ) ) ) ).
thf(conj_thm_2Elist_2EdropWhile__APPEND__EVERY,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ! [V1l1: $i] :
( ( mem @ V1l1 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V2l2: $i] :
( ( mem @ V2l2 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Elist_2EEVERY @ A_27a ) @ V0P ) @ V1l1 ) )
=> ( ( ap @ ( ap @ ( c_2Elist_2EdropWhile @ A_27a ) @ V0P ) @ ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ V1l1 ) @ V2l2 ) )
= ( ap @ ( ap @ ( c_2Elist_2EdropWhile @ A_27a ) @ V0P ) @ V2l2 ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2EdropWhile__APPEND__EXISTS,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ! [V1l1: $i] :
( ( mem @ V1l1 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V2l2: $i] :
( ( mem @ V2l2 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Elist_2EEXISTS @ A_27a ) @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ A_27a @ bool @ bool ) @ c_2Ebool_2E_7E ) @ V0P ) ) @ V1l1 ) )
=> ( ( ap @ ( ap @ ( c_2Elist_2EdropWhile @ A_27a ) @ V0P ) @ ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ V1l1 ) @ V2l2 ) )
= ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2EdropWhile @ A_27a ) @ V0P ) @ V1l1 ) ) @ V2l2 ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2EEL__LENGTH__dropWhile__REVERSE,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ! [V1ls: $i] :
( ( mem @ V1ls @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V2k: tp__ty_2Enum_2Enum] :
( ( ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2EdropWhile @ A_27a ) @ V0P ) @ ( ap @ ( c_2Elist_2EREVERSE @ A_27a ) @ V1ls ) ) ) ) @ ( inj__ty_2Enum_2Enum @ V2k ) ) )
& ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V2k ) ) @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ V1ls ) ) ) )
=> ( p @ ( ap @ V0P @ ( ap @ ( ap @ ( c_2Elist_2EEL @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V2k ) ) @ V1ls ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2ELENGTH__TAKE__EQ,axiom,
! [A_27a: del,V0n: tp__ty_2Enum_2Enum,V1xs: $i] :
( ( mem @ V1xs @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2ETAKE @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) @ V1xs ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ V0n ) ) @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ V1xs ) ) ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ V1xs ) ) ) ) ) ).
thf(conj_thm_2Elist_2EIMP__EVERY__LUPDATE,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ! [V1xs: $i] :
( ( mem @ V1xs @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V2h: $i] :
( ( mem @ V2h @ A_27a )
=> ! [V3i: tp__ty_2Enum_2Enum] :
( ( ( p @ ( ap @ V0P @ V2h ) )
& ( p @ ( ap @ ( ap @ ( c_2Elist_2EEVERY @ A_27a ) @ V0P ) @ V1xs ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Elist_2EEVERY @ A_27a ) @ V0P ) @ ( ap @ ( ap @ ( ap @ ( c_2Elist_2ELUPDATE @ A_27a ) @ V2h ) @ ( inj__ty_2Enum_2Enum @ V3i ) ) @ V1xs ) ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2EMAP__APPEND__MAP__EQ,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0f1: $i] :
( ( mem @ V0f1 @ ( arr @ A_27a @ A_27c ) )
=> ! [V1g1: $i] :
( ( mem @ V1g1 @ ( arr @ A_27b @ A_27c ) )
=> ! [V2f2: $i] :
( ( mem @ V2f2 @ ( arr @ A_27a @ A_27c ) )
=> ! [V3g2: $i] :
( ( mem @ V3g2 @ ( arr @ A_27b @ A_27c ) )
=> ! [V4xs: $i] :
( ( mem @ V4xs @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V5ys: $i] :
( ( mem @ V5ys @ ( ty_2Elist_2Elist @ A_27b ) )
=> ( ( ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27c ) @ ( ap @ ( ap @ ( c_2Elist_2EMAP @ A_27a @ A_27c ) @ V0f1 ) @ V4xs ) ) @ ( ap @ ( ap @ ( c_2Elist_2EMAP @ A_27b @ A_27c ) @ V1g1 ) @ V5ys ) )
= ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27c ) @ ( ap @ ( ap @ ( c_2Elist_2EMAP @ A_27a @ A_27c ) @ V2f2 ) @ V4xs ) ) @ ( ap @ ( ap @ ( c_2Elist_2EMAP @ A_27b @ A_27c ) @ V3g2 ) @ V5ys ) ) )
<=> ( ( ( ap @ ( ap @ ( c_2Elist_2EMAP @ A_27a @ A_27c ) @ V0f1 ) @ V4xs )
= ( ap @ ( ap @ ( c_2Elist_2EMAP @ A_27a @ A_27c ) @ V2f2 ) @ V4xs ) )
& ( ( ap @ ( ap @ ( c_2Elist_2EMAP @ A_27b @ A_27c ) @ V1g1 ) @ V5ys )
= ( ap @ ( ap @ ( c_2Elist_2EMAP @ A_27b @ A_27c ) @ V3g2 ) @ V5ys ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2ELUPDATE__SOME__MAP,axiom,
! [A_27a: del,A_27b: del,V0xs: $i] :
( ( mem @ V0xs @ ( ty_2Elist_2Elist @ ( ty_2Eoption_2Eoption @ A_27a ) ) )
=> ! [V1n: tp__ty_2Enum_2Enum,V2f: $i] :
( ( mem @ V2f @ ( arr @ A_27a @ A_27b ) )
=> ! [V3h: $i] :
( ( mem @ V3h @ A_27a )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Elist_2ELUPDATE @ ( ty_2Eoption_2Eoption @ A_27b ) ) @ ( ap @ ( c_2Eoption_2ESOME @ A_27b ) @ ( ap @ V2f @ V3h ) ) ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) @ ( ap @ ( ap @ ( c_2Elist_2EMAP @ ( ty_2Eoption_2Eoption @ A_27a ) @ ( ty_2Eoption_2Eoption @ A_27b ) ) @ ( ap @ ( c_2Eoption_2EOPTION__MAP @ A_27a @ A_27b ) @ V2f ) ) @ V0xs ) )
= ( ap @ ( ap @ ( c_2Elist_2EMAP @ ( ty_2Eoption_2Eoption @ A_27a ) @ ( ty_2Eoption_2Eoption @ A_27b ) ) @ ( ap @ ( c_2Eoption_2EOPTION__MAP @ A_27a @ A_27b ) @ V2f ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Elist_2ELUPDATE @ ( ty_2Eoption_2Eoption @ A_27a ) ) @ ( ap @ ( c_2Eoption_2ESOME @ A_27a ) @ V3h ) ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) @ V0xs ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2EZIP__EQ__NIL,axiom,
! [A_27a: del,A_27b: del,V0l1: $i] :
( ( mem @ V0l1 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V1l2: $i] :
( ( mem @ V1l2 @ ( ty_2Elist_2Elist @ A_27b ) )
=> ( ( ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ V0l1 ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Elist_2ELENGTH @ A_27b ) @ V1l2 ) ) )
=> ( ( ( ap @ ( c_2Elist_2EZIP @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Elist_2Elist @ A_27a ) @ ( ty_2Elist_2Elist @ A_27b ) ) @ V0l1 ) @ V1l2 ) )
= ( c_2Elist_2ENIL @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) ) )
<=> ( ( V0l1
= ( c_2Elist_2ENIL @ A_27a ) )
& ( V1l2
= ( c_2Elist_2ENIL @ A_27b ) ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2ELUPDATE__SAME,axiom,
! [A_27a: del,V0n: tp__ty_2Enum_2Enum,V1ls: $i] :
( ( mem @ V1ls @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V0n ) ) @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ V1ls ) ) )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Elist_2ELUPDATE @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2EEL @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) @ V1ls ) ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) @ V1ls )
= V1ls ) ) ) ).
thf(ax_thm_2Elist_2EUNIQUE__DEF,axiom,
! [A_27a: del,V0e: $i] :
( ( mem @ V0e @ A_27a )
=> ! [V1L: $i] :
( ( mem @ V1L @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Elist_2EUNIQUE @ A_27a ) @ V0e ) @ V1L ) )
<=> ? [V2L1: $i] :
( ( mem @ V2L1 @ ( ty_2Elist_2Elist @ A_27a ) )
& ? [V3L2: $i] :
( ( mem @ V3L2 @ ( ty_2Elist_2Elist @ A_27a ) )
& ( ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ V2L1 ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V0e ) @ ( c_2Elist_2ENIL @ A_27a ) ) ) ) @ V3L2 )
= V1L )
& ~ ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V0e ) @ ( ap @ ( c_2Elist_2ELIST__TO__SET @ A_27a ) @ V2L1 ) ) )
& ~ ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V0e ) @ ( ap @ ( c_2Elist_2ELIST__TO__SET @ A_27a ) @ V3L2 ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2EUNIQUE__FILTER,axiom,
! [A_27a: del,V0e: $i] :
( ( mem @ V0e @ A_27a )
=> ! [V1L: $i] :
( ( mem @ V1L @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Elist_2EUNIQUE @ A_27a ) @ V0e ) @ V1L ) )
<=> ( ( ap @ ( ap @ ( c_2Elist_2EFILTER @ A_27a ) @ ( ap @ ( c_2Emin_2E_3D @ A_27a ) @ V0e ) ) @ V1L )
= ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V0e ) @ ( c_2Elist_2ENIL @ A_27a ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2EUNIQUE__LENGTH__FILTER,axiom,
! [A_27a: del,V0e: $i] :
( ( mem @ V0e @ A_27a )
=> ! [V1L: $i] :
( ( mem @ V1L @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Elist_2EUNIQUE @ A_27a ) @ V0e ) @ V1L ) )
<=> ( ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2EFILTER @ A_27a ) @ ( ap @ ( c_2Emin_2E_3D @ A_27a ) @ V0e ) ) @ V1L ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Elist_2EOPT__MMAP__def,axiom,
! [A_27a: del,A_27b: del] :
( ! [V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27b @ ( ty_2Eoption_2Eoption @ A_27a ) ) )
=> ( ( ap @ ( ap @ ( c_2Elist_2EOPT__MMAP @ A_27a @ A_27b ) @ V0f ) @ ( c_2Elist_2ENIL @ A_27b ) )
= ( ap @ ( c_2Eoption_2ESOME @ ( ty_2Elist_2Elist @ A_27a ) ) @ ( c_2Elist_2ENIL @ A_27a ) ) ) )
& ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27b @ ( ty_2Eoption_2Eoption @ A_27a ) ) )
=> ! [V2h0: $i] :
( ( mem @ V2h0 @ A_27b )
=> ! [V3t0: $i] :
( ( mem @ V3t0 @ ( ty_2Elist_2Elist @ A_27b ) )
=> ( ( ap @ ( ap @ ( c_2Elist_2EOPT__MMAP @ A_27a @ A_27b ) @ V1f ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27b ) @ V2h0 ) @ V3t0 ) )
= ( ap @ ( ap @ ( c_2Eoption_2EOPTION__BIND @ ( ty_2Elist_2Elist @ A_27a ) @ A_27a ) @ ( ap @ V1f @ V2h0 ) )
@ ( lam @ A_27a
@ ^ [V4h: $i] :
( ap @ ( ap @ ( c_2Eoption_2EOPTION__BIND @ ( ty_2Elist_2Elist @ A_27a ) @ ( ty_2Elist_2Elist @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Elist_2EOPT__MMAP @ A_27a @ A_27b ) @ V1f ) @ V3t0 ) )
@ ( lam @ ( ty_2Elist_2Elist @ A_27a )
@ ^ [V5t: $i] : ( ap @ ( c_2Eoption_2ESOME @ ( ty_2Elist_2Elist @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V4h ) @ V5t ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2EOPT__MMAP__cong,axiom,
! [A_27a: del,A_27b: del,V0f1: $i] :
( ( mem @ V0f1 @ ( arr @ A_27a @ ( ty_2Eoption_2Eoption @ A_27b ) ) )
=> ! [V1f2: $i] :
( ( mem @ V1f2 @ ( arr @ A_27a @ ( ty_2Eoption_2Eoption @ A_27b ) ) )
=> ! [V2x1: $i] :
( ( mem @ V2x1 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V3x2: $i] :
( ( mem @ V3x2 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( ( V2x1 = V3x2 )
& ! [V4a: $i] :
( ( mem @ V4a @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V4a ) @ ( ap @ ( c_2Elist_2ELIST__TO__SET @ A_27a ) @ V3x2 ) ) )
=> ( ( ap @ V0f1 @ V4a )
= ( ap @ V1f2 @ V4a ) ) ) ) )
=> ( ( ap @ ( ap @ ( c_2Elist_2EOPT__MMAP @ A_27b @ A_27a ) @ V0f1 ) @ V2x1 )
= ( ap @ ( ap @ ( c_2Elist_2EOPT__MMAP @ A_27b @ A_27a ) @ V1f2 ) @ V3x2 ) ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2ELAST__compute,axiom,
! [A_27a: del,A_27b: del] :
( ! [V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ( ( ap @ ( c_2Elist_2ELAST @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V0x ) @ ( c_2Elist_2ENIL @ A_27a ) ) )
= V0x ) )
& ! [V1h1: $i] :
( ( mem @ V1h1 @ A_27b )
=> ! [V2h2: $i] :
( ( mem @ V2h2 @ A_27b )
=> ! [V3t: $i] :
( ( mem @ V3t @ ( ty_2Elist_2Elist @ A_27b ) )
=> ( ( ap @ ( c_2Elist_2ELAST @ A_27b ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27b ) @ V1h1 ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27b ) @ V2h2 ) @ V3t ) ) )
= ( ap @ ( c_2Elist_2ELAST @ A_27b ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27b ) @ V2h2 ) @ V3t ) ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2ETAKE__compute,axiom,
! [A_27a: del,A_27b: del,A_27c: del] :
( ! [V0l: $i] :
( ( mem @ V0l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( ap @ ( ap @ ( c_2Elist_2ETAKE @ A_27a ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ V0l )
= ( c_2Elist_2ENIL @ A_27a ) ) )
& ! [V1n: tp__ty_2Enum_2Enum] :
( ( ap @ ( ap @ ( c_2Elist_2ETAKE @ A_27b ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) ) @ ( c_2Elist_2ENIL @ A_27b ) )
= ( c_2Elist_2ENIL @ A_27b ) )
& ! [V2n: tp__ty_2Enum_2Enum] :
( ( ap @ ( ap @ ( c_2Elist_2ETAKE @ A_27b ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) ) @ ( c_2Elist_2ENIL @ A_27b ) )
= ( c_2Elist_2ENIL @ A_27b ) )
& ! [V3n: tp__ty_2Enum_2Enum,V4h: $i] :
( ( mem @ V4h @ A_27c )
=> ! [V5t: $i] :
( ( mem @ V5t @ ( ty_2Elist_2Elist @ A_27c ) )
=> ( ( ap @ ( ap @ ( c_2Elist_2ETAKE @ A_27c ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27c ) @ V4h ) @ V5t ) )
= ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27c ) @ V4h ) @ ( ap @ ( ap @ ( c_2Elist_2ETAKE @ A_27c ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ V5t ) ) ) ) )
& ! [V6n: tp__ty_2Enum_2Enum,V7h: $i] :
( ( mem @ V7h @ A_27c )
=> ! [V8t: $i] :
( ( mem @ V8t @ ( ty_2Elist_2Elist @ A_27c ) )
=> ( ( ap @ ( ap @ ( c_2Elist_2ETAKE @ A_27c ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ V6n ) ) ) ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27c ) @ V7h ) @ V8t ) )
= ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27c ) @ V7h ) @ ( ap @ ( ap @ ( c_2Elist_2ETAKE @ A_27c ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ V6n ) ) ) ) @ V8t ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2EDROP__compute,axiom,
! [A_27a: del,A_27b: del,A_27c: del] :
( ! [V0l: $i] :
( ( mem @ V0l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( ap @ ( ap @ ( c_2Elist_2EDROP @ A_27a ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ V0l )
= V0l ) )
& ! [V1n: tp__ty_2Enum_2Enum] :
( ( ap @ ( ap @ ( c_2Elist_2EDROP @ A_27b ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) ) @ ( c_2Elist_2ENIL @ A_27b ) )
= ( c_2Elist_2ENIL @ A_27b ) )
& ! [V2n: tp__ty_2Enum_2Enum] :
( ( ap @ ( ap @ ( c_2Elist_2EDROP @ A_27b ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) ) @ ( c_2Elist_2ENIL @ A_27b ) )
= ( c_2Elist_2ENIL @ A_27b ) )
& ! [V3n: tp__ty_2Enum_2Enum,V4h: $i] :
( ( mem @ V4h @ A_27c )
=> ! [V5t: $i] :
( ( mem @ V5t @ ( ty_2Elist_2Elist @ A_27c ) )
=> ( ( ap @ ( ap @ ( c_2Elist_2EDROP @ A_27c ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27c ) @ V4h ) @ V5t ) )
= ( ap @ ( ap @ ( c_2Elist_2EDROP @ A_27c ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ V5t ) ) ) )
& ! [V6n: tp__ty_2Enum_2Enum,V7h: $i] :
( ( mem @ V7h @ A_27c )
=> ! [V8t: $i] :
( ( mem @ V8t @ ( ty_2Elist_2Elist @ A_27c ) )
=> ( ( ap @ ( ap @ ( c_2Elist_2EDROP @ A_27c ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ V6n ) ) ) ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27c ) @ V7h ) @ V8t ) )
= ( ap @ ( ap @ ( c_2Elist_2EDROP @ A_27c ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ V6n ) ) ) ) @ V8t ) ) ) ) ) ).
thf(ax_thm_2Elist_2EoHD__def,axiom,
! [A_27a: del,V0l: $i] :
( ( mem @ V0l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( ap @ ( c_2Elist_2EoHD @ A_27a ) @ V0l )
= ( ap @ ( ap @ ( ap @ ( c_2Elist_2Elist__CASE @ A_27a @ ( ty_2Eoption_2Eoption @ A_27a ) ) @ V0l ) @ ( c_2Eoption_2ENONE @ A_27a ) )
@ ( lam @ A_27a
@ ^ [V1h: $i] :
( lam @ ( ty_2Elist_2Elist @ A_27a )
@ ^ [V2v1: $i] : ( ap @ ( c_2Eoption_2ESOME @ A_27a ) @ V1h ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2EoHD__thm,axiom,
! [A_27a: del,A_27b: del,V0h: $i] :
( ( mem @ V0h @ A_27b )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( ty_2Elist_2Elist @ A_27b ) )
=> ( ( ( ap @ ( c_2Elist_2EoHD @ A_27a ) @ ( c_2Elist_2ENIL @ A_27a ) )
= ( c_2Eoption_2ENONE @ A_27a ) )
& ( ( ap @ ( c_2Elist_2EoHD @ A_27b ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27b ) @ V0h ) @ V1t ) )
= ( ap @ ( c_2Eoption_2ESOME @ A_27b ) @ V0h ) ) ) ) ) ).
thf(ax_thm_2Elist_2EoEL__def,axiom,
! [A_27a: del] :
( ! [V0n: tp__ty_2Enum_2Enum] :
( ( ap @ ( ap @ ( c_2Elist_2EoEL @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) @ ( c_2Elist_2ENIL @ A_27a ) )
= ( c_2Eoption_2ENONE @ A_27a ) )
& ! [V1n: tp__ty_2Enum_2Enum,V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ! [V3xs: $i] :
( ( mem @ V3xs @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( ap @ ( ap @ ( c_2Elist_2EoEL @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V2x ) @ V3xs ) )
= ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ( ty_2Eoption_2Eoption @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ ( c_2Eoption_2ESOME @ A_27a ) @ V2x ) ) @ ( ap @ ( ap @ ( c_2Elist_2EoEL @ A_27a ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ ( inj__ty_2Enum_2Enum @ V1n ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ V3xs ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2EoEL__THM,axiom,
! [A_27a: del,V0xs: $i] :
( ( mem @ V0xs @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V1n: tp__ty_2Enum_2Enum] :
( ( ap @ ( ap @ ( c_2Elist_2EoEL @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) @ V0xs )
= ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ( ty_2Eoption_2Eoption @ A_27a ) ) @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V1n ) ) @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ V0xs ) ) ) @ ( ap @ ( c_2Eoption_2ESOME @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2EEL @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) @ V0xs ) ) ) @ ( c_2Eoption_2ENONE @ A_27a ) ) ) ) ).
thf(conj_thm_2Elist_2EoEL__EQ__EL,axiom,
! [A_27a: del,V0xs: $i] :
( ( mem @ V0xs @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V1n: tp__ty_2Enum_2Enum,V2y: $i] :
( ( mem @ V2y @ A_27a )
=> ( ( ( ap @ ( ap @ ( c_2Elist_2EoEL @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) @ V0xs )
= ( ap @ ( c_2Eoption_2ESOME @ A_27a ) @ V2y ) )
<=> ( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V1n ) ) @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ V0xs ) ) )
& ( V2y
= ( ap @ ( ap @ ( c_2Elist_2EEL @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) @ V0xs ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2EoEL__DROP,axiom,
! [A_27a: del,V0n: tp__ty_2Enum_2Enum,V1m: tp__ty_2Enum_2Enum,V2xs: $i] :
( ( mem @ V2xs @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( ap @ ( ap @ ( c_2Elist_2EoEL @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) @ ( ap @ ( ap @ ( c_2Elist_2EDROP @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V1m ) ) @ V2xs ) )
= ( ap @ ( ap @ ( c_2Elist_2EoEL @ A_27a ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( inj__ty_2Enum_2Enum @ V1m ) ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) @ V2xs ) ) ) ).
thf(conj_thm_2Elist_2EoEL__TAKE__E,axiom,
! [A_27a: del,V0n: tp__ty_2Enum_2Enum,V1m: tp__ty_2Enum_2Enum,V2xs: $i] :
( ( mem @ V2xs @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ( ( ( ap @ ( ap @ ( c_2Elist_2EoEL @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) @ ( ap @ ( ap @ ( c_2Elist_2ETAKE @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V1m ) ) @ V2xs ) )
= ( ap @ ( c_2Eoption_2ESOME @ A_27a ) @ V3x ) )
=> ( ( ap @ ( ap @ ( c_2Elist_2EoEL @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) @ V2xs )
= ( ap @ ( c_2Eoption_2ESOME @ A_27a ) @ V3x ) ) ) ) ) ).
thf(conj_thm_2Elist_2EoEL__LUPDATE,axiom,
! [A_27a: del,V0xs: $i] :
( ( mem @ V0xs @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V1i: tp__ty_2Enum_2Enum,V2n: tp__ty_2Enum_2Enum,V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ( ( ap @ ( ap @ ( c_2Elist_2EoEL @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Elist_2ELUPDATE @ A_27a ) @ V3x ) @ ( inj__ty_2Enum_2Enum @ V1i ) ) @ V0xs ) )
= ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ( ty_2Eoption_2Eoption @ A_27a ) ) @ ( ap @ c_2Ebool_2E_7E @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V1i ) ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) ) @ ( ap @ ( ap @ ( c_2Elist_2EoEL @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) @ V0xs ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ( ty_2Eoption_2Eoption @ A_27a ) ) @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V1i ) ) @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ V0xs ) ) ) @ ( ap @ ( c_2Eoption_2ESOME @ A_27a ) @ V3x ) ) @ ( c_2Eoption_2ENONE @ A_27a ) ) ) ) ) ) ).
thf(conj_thm_2Elist_2Elazy__list__case__compute,axiom,
! [A_27a: del,A_27b: del] :
( ( c_2Elist_2Elist__CASE @ A_27a @ A_27b )
= ( lam @ ( ty_2Elist_2Elist @ A_27a )
@ ^ [V0l: $i] :
( lam @ A_27b
@ ^ [V1b: $i] :
( lam @ ( arr @ A_27a @ ( arr @ ( ty_2Elist_2Elist @ A_27a ) @ A_27b ) )
@ ^ [V2f: $i] : ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ A_27b ) @ ( ap @ ( c_2Elist_2ENULL @ A_27a ) @ V0l ) ) @ V1b ) @ ( ap @ ( ap @ V2f @ ( ap @ ( c_2Elist_2EHD @ A_27a ) @ V0l ) ) @ ( ap @ ( c_2Elist_2ETL @ A_27a ) @ V0l ) ) ) ) ) ) ) ).
%------------------------------------------------------------------------------