ITP001 Axioms: ITP065^7.ax
%------------------------------------------------------------------------------
% File : ITP065^7 : TPTP v9.0.0. Bugfixed v7.5.0.
% Domain : Interactive Theorem Proving
% Axioms : HOL4 syntactic export, chainy mode
% Version : [BG+19] axioms.
% English :
% Refs : [BG+19] Brown et al. (2019), GRUNGE: A Grand Unified ATP Chall
% : [Gau19] Gauthier (2019), Email to Geoff Sutcliffe
% Source : [BG+19]
% Names : finite_map.ax [Gau19]
% : HL4065^7.ax [TPAP]
% Status : Satisfiable
% Syntax : Number of formulae : 393 ( 112 unt; 112 typ; 0 def)
% Number of atoms : 1037 ( 313 equ; 31 cnn)
% Maximal formula atoms : 24 ( 2 avg)
% Number of connectives : 7703 ( 31 ~; 10 |; 115 &;7331 @)
% ( 62 <=>; 154 =>; 0 <=; 0 <~>)
% Maximal formula depth : 22 ( 10 avg;7331 nst)
% Number of types : 4 ( 3 usr)
% Number of type conns : 493 ( 493 >; 0 *; 0 +; 0 <<)
% Number of symbols : 111 ( 109 usr; 4 con; 0-7 aty)
% Number of variables : 1678 ( 68 ^1427 !; 24 ?;1678 :)
% ( 159 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_SAT_EQU_NAR
% Comments :
% Bugfixes : v7.5.0 - Fixes to the axioms.
%------------------------------------------------------------------------------
thf(tyop_2Efinite__map_2Efmap,type,
tyop_2Efinite__map_2Efmap: $tType > $tType > $tType ).
thf(tyop_2Elist_2Elist,type,
tyop_2Elist_2Elist: $tType > $tType ).
thf(tyop_2Emin_2Ebool,type,
tyop_2Emin_2Ebool: $tType ).
thf(tyop_2Emin_2Efun,type,
tyop_2Emin_2Efun: $tType > $tType > $tType ).
thf(tyop_2Enum_2Enum,type,
tyop_2Enum_2Enum: $tType ).
thf(tyop_2Eone_2Eone,type,
tyop_2Eone_2Eone: $tType ).
thf(tyop_2Eoption_2Eoption,type,
tyop_2Eoption_2Eoption: $tType > $tType ).
thf(tyop_2Epair_2Eprod,type,
tyop_2Epair_2Eprod: $tType > $tType > $tType ).
thf(tyop_2Esum_2Esum,type,
tyop_2Esum_2Esum: $tType > $tType > $tType ).
thf(c_2Ebool_2E_21,type,
c_2Ebool_2E_21:
!>[A_27a: $tType] : ( ( A_27a > $o ) > $o ) ).
thf(c_2Earithmetic_2E_2B,type,
c_2Earithmetic_2E_2B: tyop_2Enum_2Enum > tyop_2Enum_2Enum > tyop_2Enum_2Enum ).
thf(c_2Epair_2E_2C,type,
c_2Epair_2E_2C:
!>[A_27a: $tType,A_27b: $tType] : ( A_27a > A_27b > ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) ) ).
thf(c_2Ebool_2E_2F_5C,type,
c_2Ebool_2E_2F_5C: $o > $o > $o ).
thf(c_2Enum_2E0,type,
c_2Enum_2E0: tyop_2Enum_2Enum ).
thf(c_2Eprim__rec_2E_3C,type,
c_2Eprim__rec_2E_3C: tyop_2Enum_2Enum > tyop_2Enum_2Enum > $o ).
thf(c_2Emin_2E_3D,type,
c_2Emin_2E_3D:
!>[A_27a: $tType] : ( A_27a > A_27a > $o ) ).
thf(c_2Emin_2E_3D_3D_3E,type,
c_2Emin_2E_3D_3D_3E: $o > $o > $o ).
thf(c_2Ebool_2E_3F,type,
c_2Ebool_2E_3F:
!>[A_27a: $tType] : ( ( A_27a > $o ) > $o ) ).
thf(c_2Elist_2EALL__DISTINCT,type,
c_2Elist_2EALL__DISTINCT:
!>[A_27a: $tType] : ( ( tyop_2Elist_2Elist @ A_27a ) > $o ) ).
thf(c_2Elist_2EAPPEND,type,
c_2Elist_2EAPPEND:
!>[A_27a: $tType] : ( ( tyop_2Elist_2Elist @ A_27a ) > ( tyop_2Elist_2Elist @ A_27a ) > ( tyop_2Elist_2Elist @ A_27a ) ) ).
thf(c_2Ebool_2EARB,type,
c_2Ebool_2EARB:
!>[A_27a: $tType] : A_27a ).
thf(c_2Ecombin_2EASSOC,type,
c_2Ecombin_2EASSOC:
!>[A_27a: $tType] : ( ( A_27a > A_27a > A_27a ) > $o ) ).
thf(c_2Epred__set_2EBIJ,type,
c_2Epred__set_2EBIJ:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > A_27b ) > ( A_27a > $o ) > ( A_27b > $o ) > $o ) ).
thf(c_2Earithmetic_2EBIT1,type,
c_2Earithmetic_2EBIT1: tyop_2Enum_2Enum > tyop_2Enum_2Enum ).
thf(c_2Ecombin_2EC,type,
c_2Ecombin_2EC:
!>[A_27a: $tType,A_27b: $tType,A_27c: $tType] : ( ( A_27a > A_27b > A_27c ) > A_27b > A_27a > A_27c ) ).
thf(c_2Epred__set_2ECARD,type,
c_2Epred__set_2ECARD:
!>[A_27a: $tType] : ( ( A_27a > $o ) > tyop_2Enum_2Enum ) ).
thf(c_2Ecombin_2ECOMM,type,
c_2Ecombin_2ECOMM:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > A_27a > A_27b ) > $o ) ).
thf(c_2Epred__set_2ECOMPL,type,
c_2Epred__set_2ECOMPL:
!>[A_27a: $tType] : ( ( A_27a > $o ) > A_27a > $o ) ).
thf(c_2Ebool_2ECOND,type,
c_2Ebool_2ECOND:
!>[A_27a: $tType] : ( $o > A_27a > A_27a > A_27a ) ).
thf(c_2Elist_2ECONS,type,
c_2Elist_2ECONS:
!>[A_27a: $tType] : ( A_27a > ( tyop_2Elist_2Elist @ A_27a ) > ( tyop_2Elist_2Elist @ A_27a ) ) ).
thf(c_2Epred__set_2EDELETE,type,
c_2Epred__set_2EDELETE:
!>[A_27a: $tType] : ( ( A_27a > $o ) > A_27a > A_27a > $o ) ).
thf(c_2Epred__set_2EDIFF,type,
c_2Epred__set_2EDIFF:
!>[A_27a: $tType] : ( ( A_27a > $o ) > ( A_27a > $o ) > A_27a > $o ) ).
thf(c_2Epred__set_2EDISJOINT,type,
c_2Epred__set_2EDISJOINT:
!>[A_27a: $tType] : ( ( A_27a > $o ) > ( A_27a > $o ) > $o ) ).
thf(c_2Efinite__map_2EDRESTRICT,type,
c_2Efinite__map_2EDRESTRICT:
!>[A_27a: $tType,A_27b: $tType] : ( ( tyop_2Efinite__map_2Efmap @ A_27a @ A_27b ) > ( A_27a > $o ) > ( tyop_2Efinite__map_2Efmap @ A_27a @ A_27b ) ) ).
thf(c_2Elist_2EEL,type,
c_2Elist_2EEL:
!>[A_27a: $tType] : ( tyop_2Enum_2Enum > ( tyop_2Elist_2Elist @ A_27a ) > A_27a ) ).
thf(c_2Epred__set_2EEMPTY,type,
c_2Epred__set_2EEMPTY:
!>[A_27a: $tType] : ( A_27a > $o ) ).
thf(c_2Elist_2EEVERY,type,
c_2Elist_2EEVERY:
!>[A_27a: $tType] : ( ( A_27a > $o ) > ( tyop_2Elist_2Elist @ A_27a ) > $o ) ).
thf(c_2Efinite__map_2EFAPPLY,type,
c_2Efinite__map_2EFAPPLY:
!>[A_27a: $tType,A_27b: $tType] : ( ( tyop_2Efinite__map_2Efmap @ A_27a @ A_27b ) > A_27a > A_27b ) ).
thf(c_2Efinite__map_2EFCARD,type,
c_2Efinite__map_2EFCARD:
!>[A_27a: $tType,A_27b: $tType] : ( ( tyop_2Efinite__map_2Efmap @ A_27a @ A_27b ) > tyop_2Enum_2Enum ) ).
thf(c_2Efinite__map_2EFDOM,type,
c_2Efinite__map_2EFDOM:
!>[A_27a: $tType,A_27b: $tType] : ( ( tyop_2Efinite__map_2Efmap @ A_27a @ A_27b ) > A_27a > $o ) ).
thf(c_2Efinite__map_2EFEMPTY,type,
c_2Efinite__map_2EFEMPTY:
!>[A_27a: $tType,A_27b: $tType] : ( tyop_2Efinite__map_2Efmap @ A_27a @ A_27b ) ).
thf(c_2Efinite__map_2EFEVERY,type,
c_2Efinite__map_2EFEVERY:
!>[A_27a: $tType,A_27b: $tType] : ( ( ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) > $o ) > ( tyop_2Efinite__map_2Efmap @ A_27a @ A_27b ) > $o ) ).
thf(c_2Epred__set_2EFINITE,type,
c_2Epred__set_2EFINITE:
!>[A_27a: $tType] : ( ( A_27a > $o ) > $o ) ).
thf(c_2Efinite__map_2EFLOOKUP,type,
c_2Efinite__map_2EFLOOKUP:
!>[A_27a: $tType,A_27b: $tType] : ( ( tyop_2Efinite__map_2Efmap @ A_27a @ A_27b ) > A_27a > ( tyop_2Eoption_2Eoption @ A_27b ) ) ).
thf(c_2Efinite__map_2EFMAP__MAP2,type,
c_2Efinite__map_2EFMAP__MAP2:
!>[A_27a: $tType,A_27b: $tType,A_27c: $tType] : ( ( ( tyop_2Epair_2Eprod @ A_27a @ A_27c ) > A_27b ) > ( tyop_2Efinite__map_2Efmap @ A_27a @ A_27c ) > ( tyop_2Efinite__map_2Efmap @ A_27a @ A_27b ) ) ).
thf(c_2Efinite__map_2EFMERGE,type,
c_2Efinite__map_2EFMERGE:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > A_27a > A_27a ) > ( tyop_2Efinite__map_2Efmap @ A_27b @ A_27a ) > ( tyop_2Efinite__map_2Efmap @ A_27b @ A_27a ) > ( tyop_2Efinite__map_2Efmap @ A_27b @ A_27a ) ) ).
thf(c_2Elist_2EFOLDL,type,
c_2Elist_2EFOLDL:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27b > A_27a > A_27b ) > A_27b > ( tyop_2Elist_2Elist @ A_27a ) > A_27b ) ).
thf(c_2Elist_2EFOLDL2,type,
c_2Elist_2EFOLDL2:
!>[A_27a: $tType,A_27b: $tType,A_27c: $tType] : ( ( A_27a > A_27b > A_27c > A_27a ) > A_27a > ( tyop_2Elist_2Elist @ A_27b ) > ( tyop_2Elist_2Elist @ A_27c ) > A_27a ) ).
thf(c_2Elist_2EFOLDR,type,
c_2Elist_2EFOLDR:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > A_27b > A_27b ) > A_27b > ( tyop_2Elist_2Elist @ A_27a ) > A_27b ) ).
thf(c_2Efinite__map_2EFRANGE,type,
c_2Efinite__map_2EFRANGE:
!>[A_27a: $tType,A_27b: $tType] : ( ( tyop_2Efinite__map_2Efmap @ A_27a @ A_27b ) > A_27b > $o ) ).
thf(c_2Epair_2EFST,type,
c_2Epair_2EFST:
!>[A_27a: $tType,A_27b: $tType] : ( ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) > A_27a ) ).
thf(c_2Efinite__map_2EFUNION,type,
c_2Efinite__map_2EFUNION:
!>[A_27a: $tType,A_27b: $tType] : ( ( tyop_2Efinite__map_2Efmap @ A_27a @ A_27b ) > ( tyop_2Efinite__map_2Efmap @ A_27a @ A_27b ) > ( tyop_2Efinite__map_2Efmap @ A_27a @ A_27b ) ) ).
thf(c_2Efinite__map_2EFUN__FMAP,type,
c_2Efinite__map_2EFUN__FMAP:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > A_27b ) > ( A_27a > $o ) > ( tyop_2Efinite__map_2Efmap @ A_27a @ A_27b ) ) ).
thf(c_2Efinite__map_2EFUPDATE,type,
c_2Efinite__map_2EFUPDATE:
!>[A_27a: $tType,A_27b: $tType] : ( ( tyop_2Efinite__map_2Efmap @ A_27a @ A_27b ) > ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) > ( tyop_2Efinite__map_2Efmap @ A_27a @ A_27b ) ) ).
thf(c_2Efinite__map_2EFUPDATE__LIST,type,
c_2Efinite__map_2EFUPDATE__LIST:
!>[A_27a: $tType,A_27b: $tType] : ( ( tyop_2Efinite__map_2Efmap @ A_27a @ A_27b ) > ( tyop_2Elist_2Elist @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) ) > ( tyop_2Efinite__map_2Efmap @ A_27a @ A_27b ) ) ).
thf(c_2Epred__set_2EGSPEC,type,
c_2Epred__set_2EGSPEC:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27b > ( tyop_2Epair_2Eprod @ A_27a @ $o ) ) > A_27a > $o ) ).
thf(c_2Epred__set_2EIMAGE,type,
c_2Epred__set_2EIMAGE:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > A_27b ) > ( A_27a > $o ) > A_27b > $o ) ).
thf(c_2Ebool_2EIN,type,
c_2Ebool_2EIN:
!>[A_27a: $tType] : ( A_27a > ( A_27a > $o ) > $o ) ).
thf(c_2Epred__set_2EINJ,type,
c_2Epred__set_2EINJ:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > A_27b ) > ( A_27a > $o ) > ( A_27b > $o ) > $o ) ).
thf(c_2Esum_2EINL,type,
c_2Esum_2EINL:
!>[A_27a: $tType,A_27b: $tType] : ( A_27a > ( tyop_2Esum_2Esum @ A_27a @ A_27b ) ) ).
thf(c_2Esum_2EINR,type,
c_2Esum_2EINR:
!>[A_27a: $tType,A_27b: $tType] : ( A_27b > ( tyop_2Esum_2Esum @ A_27a @ A_27b ) ) ).
thf(c_2Epred__set_2EINSERT,type,
c_2Epred__set_2EINSERT:
!>[A_27a: $tType] : ( A_27a > ( A_27a > $o ) > A_27a > $o ) ).
thf(c_2Epred__set_2EINTER,type,
c_2Epred__set_2EINTER:
!>[A_27a: $tType] : ( ( A_27a > $o ) > ( A_27a > $o ) > A_27a > $o ) ).
thf(c_2Esum_2EISL,type,
c_2Esum_2EISL:
!>[A_27a: $tType,A_27b: $tType] : ( ( tyop_2Esum_2Esum @ A_27a @ A_27b ) > $o ) ).
thf(c_2Ecombin_2EK,type,
c_2Ecombin_2EK:
!>[A_27a: $tType,A_27b: $tType] : ( A_27a > A_27b > A_27a ) ).
thf(c_2Elist_2ELENGTH,type,
c_2Elist_2ELENGTH:
!>[A_27a: $tType] : ( ( tyop_2Elist_2Elist @ A_27a ) > tyop_2Enum_2Enum ) ).
thf(c_2Ebool_2ELET,type,
c_2Ebool_2ELET:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > A_27b ) > A_27a > A_27b ) ).
thf(c_2Epred__set_2ELINV,type,
c_2Epred__set_2ELINV:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > A_27b ) > ( A_27a > $o ) > A_27b > A_27a ) ).
thf(c_2Elist_2ELIST__REL,type,
c_2Elist_2ELIST__REL:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > A_27b > $o ) > ( tyop_2Elist_2Elist @ A_27a ) > ( tyop_2Elist_2Elist @ A_27b ) > $o ) ).
thf(c_2Elist_2ELIST__TO__SET,type,
c_2Elist_2ELIST__TO__SET:
!>[A_27a: $tType] : ( ( tyop_2Elist_2Elist @ A_27a ) > A_27a > $o ) ).
thf(c_2Elist_2EMAP,type,
c_2Elist_2EMAP:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > A_27b ) > ( tyop_2Elist_2Elist @ A_27a ) > ( tyop_2Elist_2Elist @ A_27b ) ) ).
thf(c_2Elist_2EMAP2,type,
c_2Elist_2EMAP2:
!>[A_27a: $tType,A_27b: $tType,A_27c: $tType] : ( ( A_27b > A_27c > A_27a ) > ( tyop_2Elist_2Elist @ A_27b ) > ( tyop_2Elist_2Elist @ A_27c ) > ( tyop_2Elist_2Elist @ A_27a ) ) ).
thf(c_2Efinite__map_2EMAP__KEYS,type,
c_2Efinite__map_2EMAP__KEYS:
!>[A_27a: $tType,A_27b: $tType,A_27c: $tType] : ( ( A_27a > A_27b ) > ( tyop_2Efinite__map_2Efmap @ A_27a @ A_27c ) > ( tyop_2Efinite__map_2Efmap @ A_27b @ A_27c ) ) ).
thf(c_2Elist_2ENIL,type,
c_2Elist_2ENIL:
!>[A_27a: $tType] : ( tyop_2Elist_2Elist @ A_27a ) ).
thf(c_2Eoption_2ENONE,type,
c_2Eoption_2ENONE:
!>[A_27a: $tType] : ( tyop_2Eoption_2Eoption @ A_27a ) ).
thf(c_2Earithmetic_2ENUMERAL,type,
c_2Earithmetic_2ENUMERAL: tyop_2Enum_2Enum > tyop_2Enum_2Enum ).
thf(c_2Eoption_2EOPTION__BIND,type,
c_2Eoption_2EOPTION__BIND:
!>[A_27a: $tType,A_27b: $tType] : ( ( tyop_2Eoption_2Eoption @ A_27b ) > ( A_27b > ( tyop_2Eoption_2Eoption @ A_27a ) ) > ( tyop_2Eoption_2Eoption @ A_27a ) ) ).
thf(c_2Eoption_2EOPTREL,type,
c_2Eoption_2EOPTREL:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > A_27b > $o ) > ( tyop_2Eoption_2Eoption @ A_27a ) > ( tyop_2Eoption_2Eoption @ A_27b ) > $o ) ).
thf(c_2Esum_2EOUTL,type,
c_2Esum_2EOUTL:
!>[A_27a: $tType,A_27b: $tType] : ( ( tyop_2Esum_2Esum @ A_27a @ A_27b ) > A_27a ) ).
thf(c_2Esorting_2EPERM,type,
c_2Esorting_2EPERM:
!>[A_27a: $tType] : ( ( tyop_2Elist_2Elist @ A_27a ) > ( tyop_2Elist_2Elist @ A_27a ) > $o ) ).
thf(c_2Elist_2EREVERSE,type,
c_2Elist_2EREVERSE:
!>[A_27a: $tType] : ( ( tyop_2Elist_2Elist @ A_27a ) > ( tyop_2Elist_2Elist @ A_27a ) ) ).
thf(c_2Efinite__map_2ERRESTRICT,type,
c_2Efinite__map_2ERRESTRICT:
!>[A_27a: $tType,A_27b: $tType] : ( ( tyop_2Efinite__map_2Efmap @ A_27a @ A_27b ) > ( A_27b > $o ) > ( tyop_2Efinite__map_2Efmap @ A_27a @ A_27b ) ) ).
thf(c_2Epair_2ESND,type,
c_2Epair_2ESND:
!>[A_27a: $tType,A_27b: $tType] : ( ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) > A_27b ) ).
thf(c_2Elist_2ESNOC,type,
c_2Elist_2ESNOC:
!>[A_27a: $tType] : ( A_27a > ( tyop_2Elist_2Elist @ A_27a ) > ( tyop_2Elist_2Elist @ A_27a ) ) ).
thf(c_2Eoption_2ESOME,type,
c_2Eoption_2ESOME:
!>[A_27a: $tType] : ( A_27a > ( tyop_2Eoption_2Eoption @ A_27a ) ) ).
thf(c_2Efinite__map_2ESUBMAP,type,
c_2Efinite__map_2ESUBMAP:
!>[A_27a: $tType,A_27b: $tType] : ( ( tyop_2Efinite__map_2Efmap @ A_27a @ A_27b ) > ( tyop_2Efinite__map_2Efmap @ A_27a @ A_27b ) > $o ) ).
thf(c_2Epred__set_2ESUBSET,type,
c_2Epred__set_2ESUBSET:
!>[A_27a: $tType] : ( ( A_27a > $o ) > ( A_27a > $o ) > $o ) ).
thf(c_2Enum_2ESUC,type,
c_2Enum_2ESUC: tyop_2Enum_2Enum > tyop_2Enum_2Enum ).
thf(c_2Epred__set_2ESUM__IMAGE,type,
c_2Epred__set_2ESUM__IMAGE:
!>[A_27a: $tType] : ( ( A_27a > tyop_2Enum_2Enum ) > ( A_27a > $o ) > tyop_2Enum_2Enum ) ).
thf(c_2Ebool_2ETYPE__DEFINITION,type,
c_2Ebool_2ETYPE__DEFINITION:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > $o ) > ( A_27b > A_27a ) > $o ) ).
thf(c_2Epair_2EUNCURRY,type,
c_2Epair_2EUNCURRY:
!>[A_27a: $tType,A_27b: $tType,A_27c: $tType] : ( ( A_27a > A_27b > A_27c ) > ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) > A_27c ) ).
thf(c_2Epred__set_2EUNION,type,
c_2Epred__set_2EUNION:
!>[A_27a: $tType] : ( ( A_27a > $o ) > ( A_27a > $o ) > A_27a > $o ) ).
thf(c_2Epred__set_2EUNIV,type,
c_2Epred__set_2EUNIV:
!>[A_27a: $tType] : ( A_27a > $o ) ).
thf(c_2Earithmetic_2EZERO,type,
c_2Earithmetic_2EZERO: tyop_2Enum_2Enum ).
thf(c_2Elist_2EZIP,type,
c_2Elist_2EZIP:
!>[A_27a: $tType,A_27b: $tType] : ( ( tyop_2Epair_2Eprod @ ( tyop_2Elist_2Elist @ A_27a ) @ ( tyop_2Elist_2Elist @ A_27b ) ) > ( tyop_2Elist_2Elist @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) ) ) ).
thf(c_2Ebool_2E_5C_2F,type,
c_2Ebool_2E_5C_2F: $o > $o > $o ).
thf(c_2Efinite__map_2Ef__o,type,
c_2Efinite__map_2Ef__o:
!>[A_27a: $tType,A_27b: $tType,A_27c: $tType] : ( ( tyop_2Efinite__map_2Efmap @ A_27b @ A_27c ) > ( A_27a > A_27b ) > ( tyop_2Efinite__map_2Efmap @ A_27a @ A_27c ) ) ).
thf(c_2Efinite__map_2Ef__o__f,type,
c_2Efinite__map_2Ef__o__f:
!>[A_27a: $tType,A_27b: $tType,A_27c: $tType] : ( ( tyop_2Efinite__map_2Efmap @ A_27b @ A_27c ) > ( tyop_2Efinite__map_2Efmap @ A_27a @ A_27b ) > ( tyop_2Efinite__map_2Efmap @ A_27a @ A_27c ) ) ).
thf(c_2Efinite__map_2Efdomsub,type,
c_2Efinite__map_2Efdomsub:
!>[A_27a: $tType,A_27b: $tType] : ( ( tyop_2Efinite__map_2Efmap @ A_27a @ A_27b ) > A_27a > ( tyop_2Efinite__map_2Efmap @ A_27a @ A_27b ) ) ).
thf(c_2Efinite__map_2Efmap__ABS,type,
c_2Efinite__map_2Efmap__ABS:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > ( tyop_2Esum_2Esum @ A_27b @ tyop_2Eone_2Eone ) ) > ( tyop_2Efinite__map_2Efmap @ A_27a @ A_27b ) ) ).
thf(c_2Efinite__map_2Efmap__EQ__UPTO,type,
c_2Efinite__map_2Efmap__EQ__UPTO:
!>[A_27a: $tType,A_27b: $tType] : ( ( tyop_2Efinite__map_2Efmap @ A_27a @ A_27b ) > ( tyop_2Efinite__map_2Efmap @ A_27a @ A_27b ) > ( A_27a > $o ) > $o ) ).
thf(c_2Efinite__map_2Efmap__REP,type,
c_2Efinite__map_2Efmap__REP:
!>[A_27a: $tType,A_27b: $tType] : ( ( tyop_2Efinite__map_2Efmap @ A_27a @ A_27b ) > A_27a > ( tyop_2Esum_2Esum @ A_27b @ tyop_2Eone_2Eone ) ) ).
thf(c_2Efinite__map_2Efmap__inverse,type,
c_2Efinite__map_2Efmap__inverse:
!>[A_27a: $tType,A_27b: $tType] : ( ( tyop_2Efinite__map_2Efmap @ A_27a @ A_27b ) > ( tyop_2Efinite__map_2Efmap @ A_27b @ A_27a ) > $o ) ).
thf(c_2Efinite__map_2Efmap__rel,type,
c_2Efinite__map_2Efmap__rel:
!>[A_27a: $tType,A_27b: $tType,A_27c: $tType] : ( ( A_27a > A_27b > $o ) > ( tyop_2Efinite__map_2Efmap @ A_27c @ A_27a ) > ( tyop_2Efinite__map_2Efmap @ A_27c @ A_27b ) > $o ) ).
thf(c_2Efinite__map_2Efmap__size,type,
c_2Efinite__map_2Efmap__size:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > tyop_2Enum_2Enum ) > ( A_27b > tyop_2Enum_2Enum ) > ( tyop_2Efinite__map_2Efmap @ A_27a @ A_27b ) > tyop_2Enum_2Enum ) ).
thf(c_2Efinite__map_2Eis__fmap,type,
c_2Efinite__map_2Eis__fmap:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > ( tyop_2Esum_2Esum @ A_27b @ tyop_2Eone_2Eone ) ) > $o ) ).
thf(c_2Ecombin_2Eo,type,
c_2Ecombin_2Eo:
!>[A_27a: $tType,A_27b: $tType,A_27c: $tType] : ( ( A_27c > A_27b ) > ( A_27a > A_27c ) > A_27a > A_27b ) ).
thf(c_2Efinite__map_2Eo__f,type,
c_2Efinite__map_2Eo__f:
!>[A_27a: $tType,A_27b: $tType,A_27c: $tType] : ( ( A_27b > A_27c ) > ( tyop_2Efinite__map_2Efmap @ A_27a @ A_27b ) > ( tyop_2Efinite__map_2Efmap @ A_27a @ A_27c ) ) ).
thf(c_2Eone_2Eone,type,
c_2Eone_2Eone: tyop_2Eone_2Eone ).
thf(c_2Eoption_2Eoption__CASE,type,
c_2Eoption_2Eoption__CASE:
!>[A_27a: $tType,A_27b: $tType] : ( ( tyop_2Eoption_2Eoption @ A_27a ) > A_27b > ( A_27a > A_27b ) > A_27b ) ).
thf(c_2Eoption_2Esome,type,
c_2Eoption_2Esome:
!>[A_27a: $tType] : ( ( A_27a > $o ) > ( tyop_2Eoption_2Eoption @ A_27a ) ) ).
thf(c_2Ebool_2E_7E,type,
c_2Ebool_2E_7E: $o > $o ).
thf(logicdef_2E_2F_5C,axiom,
! [V0: $o,V1: $o] :
( ( c_2Ebool_2E_2F_5C @ V0 @ V1 )
<=> ( V0
& V1 ) ) ).
thf(logicdef_2E_5C_2F,axiom,
! [V0: $o,V1: $o] :
( ( c_2Ebool_2E_5C_2F @ V0 @ V1 )
<=> ( V0
| V1 ) ) ).
thf(logicdef_2E_7E,axiom,
! [V0: $o] :
( ( c_2Ebool_2E_7E @ V0 )
<=> ( (~) @ V0 ) ) ).
thf(logicdef_2E_3D_3D_3E,axiom,
! [V0: $o,V1: $o] :
( ( c_2Emin_2E_3D_3D_3E @ V0 @ V1 )
<=> ( V0
=> V1 ) ) ).
thf(logicdef_2E_3D,axiom,
! [A_27a: $tType,V0: A_27a,V1: A_27a] :
( ( c_2Emin_2E_3D @ A_27a @ V0 @ V1 )
<=> ( V0 = V1 ) ) ).
thf(quantdef_2E_21,axiom,
! [A_27a: $tType,V0f: A_27a > $o] :
( ( c_2Ebool_2E_21 @ A_27a @ V0f )
<=> ! [V1x: A_27a] : ( V0f @ V1x ) ) ).
thf(quantdef_2E_3F,axiom,
! [A_27a: $tType,V0f: A_27a > $o] :
( ( c_2Ebool_2E_3F @ A_27a @ V0f )
<=> ? [V1x: A_27a] : ( V0f @ V1x ) ) ).
thf(thm_2Efinite__map_2Efmap__inverse__def,axiom,
! [A_27a: $tType,A_27b: $tType,V0m1: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b,V1m2: tyop_2Efinite__map_2Efmap @ A_27b @ A_27a] :
( ( c_2Efinite__map_2Efmap__inverse @ A_27a @ A_27b @ V0m1 @ V1m2 )
<=> ! [V2k: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V2k @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b @ V0m1 ) )
=> ? [V3v: A_27b] :
( ( ( c_2Efinite__map_2EFLOOKUP @ A_27a @ A_27b @ V0m1 @ V2k )
= ( c_2Eoption_2ESOME @ A_27b @ V3v ) )
& ( ( c_2Efinite__map_2EFLOOKUP @ A_27b @ A_27a @ V1m2 @ V3v )
= ( c_2Eoption_2ESOME @ A_27a @ V2k ) ) ) ) ) ).
thf(thm_2Efinite__map_2EFMAP__MAP2__def,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0f: ( tyop_2Epair_2Eprod @ A_27a @ A_27c ) > A_27b,V1m: tyop_2Efinite__map_2Efmap @ A_27a @ A_27c] :
( ( c_2Efinite__map_2EFMAP__MAP2 @ A_27a @ A_27b @ A_27c @ V0f @ V1m )
= ( c_2Efinite__map_2EFUN__FMAP @ A_27a @ A_27b
@ ^ [V2x: A_27a] : ( V0f @ ( c_2Epair_2E_2C @ A_27a @ A_27c @ V2x @ ( c_2Efinite__map_2EFAPPLY @ A_27a @ A_27c @ V1m @ V2x ) ) )
@ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27c @ V1m ) ) ) ).
thf(thm_2Efinite__map_2EFUPDATE__LIST,axiom,
! [A_27a: $tType,A_27b: $tType] :
( ( c_2Efinite__map_2EFUPDATE__LIST @ A_27a @ A_27b )
= ( c_2Elist_2EFOLDL @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ ( tyop_2Efinite__map_2Efmap @ A_27a @ A_27b ) @ ( c_2Efinite__map_2EFUPDATE @ A_27a @ A_27b ) ) ) ).
thf(thm_2Efinite__map_2ESUBMAP__DEF,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b,V1g: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b] :
( ( c_2Efinite__map_2ESUBMAP @ A_27a @ A_27b @ V0f @ V1g )
<=> ! [V2x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V2x @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b @ V0f ) )
=> ( ( c_2Ebool_2EIN @ A_27a @ V2x @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b @ V1g ) )
& ( ( c_2Efinite__map_2EFAPPLY @ A_27a @ A_27b @ V0f @ V2x )
= ( c_2Efinite__map_2EFAPPLY @ A_27a @ A_27b @ V1g @ V2x ) ) ) ) ) ).
thf(thm_2Efinite__map_2EFCARD__DEF,axiom,
! [A_27a: $tType,A_27b: $tType,V0fm: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b] :
( ( c_2Efinite__map_2EFCARD @ A_27a @ A_27b @ V0fm )
= ( c_2Epred__set_2ECARD @ A_27a @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b @ V0fm ) ) ) ).
thf(thm_2Efinite__map_2EFDOM__DEF,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b,V1x: A_27a] :
( ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b @ V0f @ V1x )
= ( c_2Esum_2EISL @ A_27b @ tyop_2Eone_2Eone @ ( c_2Efinite__map_2Efmap__REP @ A_27a @ A_27b @ V0f @ V1x ) ) ) ).
thf(thm_2Efinite__map_2EFAPPLY__DEF,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b,V1x: A_27a] :
( ( c_2Efinite__map_2EFAPPLY @ A_27a @ A_27b @ V0f @ V1x )
= ( c_2Esum_2EOUTL @ A_27b @ tyop_2Eone_2Eone @ ( c_2Efinite__map_2Efmap__REP @ A_27a @ A_27b @ V0f @ V1x ) ) ) ).
thf(thm_2Efinite__map_2EFEMPTY__DEF,axiom,
! [A_27a: $tType,A_27b: $tType] :
( ( c_2Efinite__map_2EFEMPTY @ A_27a @ A_27b )
= ( c_2Efinite__map_2Efmap__ABS @ A_27a @ A_27b
@ ^ [V0a: A_27a] : ( c_2Esum_2EINR @ A_27b @ tyop_2Eone_2Eone @ c_2Eone_2Eone ) ) ) ).
thf(thm_2Efinite__map_2EFUPDATE__DEF,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b,V1x: A_27a,V2y: A_27b] :
( ( c_2Efinite__map_2EFUPDATE @ A_27a @ A_27b @ V0f @ ( c_2Epair_2E_2C @ A_27a @ A_27b @ V1x @ V2y ) )
= ( c_2Efinite__map_2Efmap__ABS @ A_27a @ A_27b
@ ^ [V3a: A_27a] : ( c_2Ebool_2ECOND @ ( tyop_2Esum_2Esum @ A_27b @ tyop_2Eone_2Eone ) @ ( c_2Emin_2E_3D @ A_27a @ V3a @ V1x ) @ ( c_2Esum_2EINL @ A_27b @ tyop_2Eone_2Eone @ V2y ) @ ( c_2Efinite__map_2Efmap__REP @ A_27a @ A_27b @ V0f @ V3a ) ) ) ) ).
thf(thm_2Efinite__map_2Efmap__ISO__DEF,axiom,
! [A_27a: $tType,A_27b: $tType] :
( ! [V0a: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b] :
( ( c_2Efinite__map_2Efmap__ABS @ A_27a @ A_27b @ ( c_2Efinite__map_2Efmap__REP @ A_27a @ A_27b @ V0a ) )
= V0a )
& ! [V1r: A_27a > ( tyop_2Esum_2Esum @ A_27b @ tyop_2Eone_2Eone )] :
( ( c_2Efinite__map_2Eis__fmap @ A_27a @ A_27b @ V1r )
<=> ( ( c_2Efinite__map_2Efmap__REP @ A_27a @ A_27b @ ( c_2Efinite__map_2Efmap__ABS @ A_27a @ A_27b @ V1r ) )
= V1r ) ) ) ).
thf(thm_2Efinite__map_2Efmap__TY__DEF,axiom,
! [A_27a: $tType,A_27b: $tType] :
? [V0rep: ( tyop_2Efinite__map_2Efmap @ A_27a @ A_27b ) > A_27a > ( tyop_2Esum_2Esum @ A_27b @ tyop_2Eone_2Eone )] : ( c_2Ebool_2ETYPE__DEFINITION @ ( A_27a > ( tyop_2Esum_2Esum @ A_27b @ tyop_2Eone_2Eone ) ) @ ( tyop_2Efinite__map_2Efmap @ A_27a @ A_27b ) @ ( c_2Efinite__map_2Eis__fmap @ A_27a @ A_27b ) @ V0rep ) ).
thf(thm_2Efinite__map_2Eis__fmap__def,axiom,
! [A_27a: $tType,A_27b: $tType] :
( ( c_2Efinite__map_2Eis__fmap @ A_27a @ A_27b )
= ( ^ [V0a0: A_27a > ( tyop_2Esum_2Esum @ A_27b @ tyop_2Eone_2Eone )] :
( c_2Ebool_2E_21 @ ( ( A_27a > ( tyop_2Esum_2Esum @ A_27b @ tyop_2Eone_2Eone ) ) > $o )
@ ^ [V1is__fmap_27: ( A_27a > ( tyop_2Esum_2Esum @ A_27b @ tyop_2Eone_2Eone ) ) > $o] :
( c_2Emin_2E_3D_3D_3E
@ ( c_2Ebool_2E_21 @ ( A_27a > ( tyop_2Esum_2Esum @ A_27b @ tyop_2Eone_2Eone ) )
@ ^ [V2a0: A_27a > ( tyop_2Esum_2Esum @ A_27b @ tyop_2Eone_2Eone )] :
( c_2Emin_2E_3D_3D_3E
@ ( c_2Ebool_2E_5C_2F
@ ( c_2Emin_2E_3D @ ( A_27a > ( tyop_2Esum_2Esum @ A_27b @ tyop_2Eone_2Eone ) ) @ V2a0
@ ^ [V3a: A_27a] : ( c_2Esum_2EINR @ A_27b @ tyop_2Eone_2Eone @ c_2Eone_2Eone ) )
@ ( c_2Ebool_2E_3F @ ( A_27a > ( tyop_2Esum_2Esum @ A_27b @ tyop_2Eone_2Eone ) )
@ ^ [V4f: A_27a > ( tyop_2Esum_2Esum @ A_27b @ tyop_2Eone_2Eone )] :
( c_2Ebool_2E_3F @ A_27a
@ ^ [V5a: A_27a] :
( c_2Ebool_2E_3F @ A_27b
@ ^ [V6b: A_27b] :
( c_2Ebool_2E_2F_5C
@ ( c_2Emin_2E_3D @ ( A_27a > ( tyop_2Esum_2Esum @ A_27b @ tyop_2Eone_2Eone ) ) @ V2a0
@ ^ [V7x: A_27a] : ( c_2Ebool_2ECOND @ ( tyop_2Esum_2Esum @ A_27b @ tyop_2Eone_2Eone ) @ ( c_2Emin_2E_3D @ A_27a @ V7x @ V5a ) @ ( c_2Esum_2EINL @ A_27b @ tyop_2Eone_2Eone @ V6b ) @ ( V4f @ V7x ) ) )
@ ( V1is__fmap_27 @ V4f ) ) ) ) ) )
@ ( V1is__fmap_27 @ V2a0 ) ) )
@ ( V1is__fmap_27 @ V0a0 ) ) ) ) ) ).
thf(thm_2Efinite__map_2EDRESTRICT__DEF,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b,V1r: A_27a > $o] :
( ( ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b @ ( c_2Efinite__map_2EDRESTRICT @ A_27a @ A_27b @ V0f @ V1r ) )
= ( c_2Epred__set_2EINTER @ A_27a @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b @ V0f ) @ V1r ) )
& ! [V2x: A_27a] :
( ( c_2Efinite__map_2EFAPPLY @ A_27a @ A_27b @ ( c_2Efinite__map_2EDRESTRICT @ A_27a @ A_27b @ V0f @ V1r ) @ V2x )
= ( c_2Ebool_2ECOND @ A_27b @ ( c_2Ebool_2EIN @ A_27a @ V2x @ ( c_2Epred__set_2EINTER @ A_27a @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b @ V0f ) @ V1r ) ) @ ( c_2Efinite__map_2EFAPPLY @ A_27a @ A_27b @ V0f @ V2x ) @ ( c_2Efinite__map_2EFAPPLY @ A_27a @ A_27b @ ( c_2Efinite__map_2EFEMPTY @ A_27a @ A_27b ) @ V2x ) ) ) ) ).
thf(thm_2Efinite__map_2EFUNION__DEF,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b,V1g: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b] :
( ( ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b @ ( c_2Efinite__map_2EFUNION @ A_27a @ A_27b @ V0f @ V1g ) )
= ( c_2Epred__set_2EUNION @ A_27a @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b @ V0f ) @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b @ V1g ) ) )
& ! [V2x: A_27a] :
( ( c_2Efinite__map_2EFAPPLY @ A_27a @ A_27b @ ( c_2Efinite__map_2EFUNION @ A_27a @ A_27b @ V0f @ V1g ) @ V2x )
= ( c_2Ebool_2ECOND @ A_27b @ ( c_2Ebool_2EIN @ A_27a @ V2x @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b @ V0f ) ) @ ( c_2Efinite__map_2EFAPPLY @ A_27a @ A_27b @ V0f @ V2x ) @ ( c_2Efinite__map_2EFAPPLY @ A_27a @ A_27b @ V1g @ V2x ) ) ) ) ).
thf(thm_2Efinite__map_2Efmap__domsub,axiom,
! [A_27a: $tType,A_27b: $tType,V0fm: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b,V1k: A_27a] :
( ( c_2Efinite__map_2Efdomsub @ A_27a @ A_27b @ V0fm @ V1k )
= ( c_2Efinite__map_2EDRESTRICT @ A_27a @ A_27b @ V0fm @ ( c_2Epred__set_2ECOMPL @ A_27a @ ( c_2Epred__set_2EINSERT @ A_27a @ V1k @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) ) ) ) ).
thf(thm_2Efinite__map_2Ef__o__DEF,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0f: tyop_2Efinite__map_2Efmap @ A_27b @ A_27c,V1g: A_27a > A_27b] :
( ( c_2Efinite__map_2Ef__o @ A_27a @ A_27b @ A_27c @ V0f @ V1g )
= ( c_2Efinite__map_2Ef__o__f @ A_27a @ A_27b @ A_27c @ V0f
@ ( c_2Efinite__map_2EFUN__FMAP @ A_27a @ A_27b @ V1g
@ ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a
@ ^ [V2x: A_27a] : ( c_2Epair_2E_2C @ A_27a @ $o @ V2x @ ( c_2Ebool_2EIN @ A_27b @ ( V1g @ V2x ) @ ( c_2Efinite__map_2EFDOM @ A_27b @ A_27c @ V0f ) ) ) ) ) ) ) ).
thf(thm_2Efinite__map_2EFUN__FMAP__DEF,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: A_27a > A_27b,V1P: A_27a > $o] :
( ( c_2Epred__set_2EFINITE @ A_27a @ V1P )
=> ( ( ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b @ ( c_2Efinite__map_2EFUN__FMAP @ A_27a @ A_27b @ V0f @ V1P ) )
= V1P )
& ! [V2x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V2x @ V1P )
=> ( ( c_2Efinite__map_2EFAPPLY @ A_27a @ A_27b @ ( c_2Efinite__map_2EFUN__FMAP @ A_27a @ A_27b @ V0f @ V1P ) @ V2x )
= ( V0f @ V2x ) ) ) ) ) ).
thf(thm_2Efinite__map_2ERRESTRICT__DEF,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b,V1r: A_27b > $o] :
( ( ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b @ ( c_2Efinite__map_2ERRESTRICT @ A_27a @ A_27b @ V0f @ V1r ) )
= ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a
@ ^ [V2x: A_27a] : ( c_2Epair_2E_2C @ A_27a @ $o @ V2x @ ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ A_27a @ V2x @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b @ V0f ) ) @ ( c_2Ebool_2EIN @ A_27b @ ( c_2Efinite__map_2EFAPPLY @ A_27a @ A_27b @ V0f @ V2x ) @ V1r ) ) ) ) )
& ! [V3x: A_27a] :
( ( c_2Efinite__map_2EFAPPLY @ A_27a @ A_27b @ ( c_2Efinite__map_2ERRESTRICT @ A_27a @ A_27b @ V0f @ V1r ) @ V3x )
= ( c_2Ebool_2ECOND @ A_27b @ ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ A_27a @ V3x @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b @ V0f ) ) @ ( c_2Ebool_2EIN @ A_27b @ ( c_2Efinite__map_2EFAPPLY @ A_27a @ A_27b @ V0f @ V3x ) @ V1r ) ) @ ( c_2Efinite__map_2EFAPPLY @ A_27a @ A_27b @ V0f @ V3x ) @ ( c_2Efinite__map_2EFAPPLY @ A_27a @ A_27b @ ( c_2Efinite__map_2EFEMPTY @ A_27a @ A_27b ) @ V3x ) ) ) ) ).
thf(thm_2Efinite__map_2EFRANGE__DEF,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b] :
( ( c_2Efinite__map_2EFRANGE @ A_27a @ A_27b @ V0f )
= ( c_2Epred__set_2EGSPEC @ A_27b @ A_27b
@ ^ [V1y: A_27b] :
( c_2Epair_2E_2C @ A_27b @ $o @ V1y
@ ( c_2Ebool_2E_3F @ A_27a
@ ^ [V2x: A_27a] : ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ A_27a @ V2x @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b @ V0f ) ) @ ( c_2Emin_2E_3D @ A_27b @ ( c_2Efinite__map_2EFAPPLY @ A_27a @ A_27b @ V0f @ V2x ) @ V1y ) ) ) ) ) ) ).
thf(thm_2Efinite__map_2Eo__f__DEF,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0f: A_27b > A_27c,V1g: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b] :
( ( ( c_2Efinite__map_2EFDOM @ A_27a @ A_27c @ ( c_2Efinite__map_2Eo__f @ A_27a @ A_27b @ A_27c @ V0f @ V1g ) )
= ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b @ V1g ) )
& ! [V2x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V2x @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27c @ ( c_2Efinite__map_2Eo__f @ A_27a @ A_27b @ A_27c @ V0f @ V1g ) ) )
=> ( ( c_2Efinite__map_2EFAPPLY @ A_27a @ A_27c @ ( c_2Efinite__map_2Eo__f @ A_27a @ A_27b @ A_27c @ V0f @ V1g ) @ V2x )
= ( V0f @ ( c_2Efinite__map_2EFAPPLY @ A_27a @ A_27b @ V1g @ V2x ) ) ) ) ) ).
thf(thm_2Efinite__map_2Ef__o__f__DEF,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0f: tyop_2Efinite__map_2Efmap @ A_27b @ A_27c,V1g: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b] :
( ( ( c_2Efinite__map_2EFDOM @ A_27a @ A_27c @ ( c_2Efinite__map_2Ef__o__f @ A_27a @ A_27b @ A_27c @ V0f @ V1g ) )
= ( c_2Epred__set_2EINTER @ A_27a @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b @ V1g )
@ ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a
@ ^ [V2x: A_27a] : ( c_2Epair_2E_2C @ A_27a @ $o @ V2x @ ( c_2Ebool_2EIN @ A_27b @ ( c_2Efinite__map_2EFAPPLY @ A_27a @ A_27b @ V1g @ V2x ) @ ( c_2Efinite__map_2EFDOM @ A_27b @ A_27c @ V0f ) ) ) ) ) )
& ! [V3x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V3x @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27c @ ( c_2Efinite__map_2Ef__o__f @ A_27a @ A_27b @ A_27c @ V0f @ V1g ) ) )
=> ( ( c_2Efinite__map_2EFAPPLY @ A_27a @ A_27c @ ( c_2Efinite__map_2Ef__o__f @ A_27a @ A_27b @ A_27c @ V0f @ V1g ) @ V3x )
= ( c_2Efinite__map_2EFAPPLY @ A_27b @ A_27c @ V0f @ ( c_2Efinite__map_2EFAPPLY @ A_27a @ A_27b @ V1g @ V3x ) ) ) ) ) ).
thf(thm_2Efinite__map_2EFEVERY__DEF,axiom,
! [A_27a: $tType,A_27b: $tType,V0P: ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) > $o,V1f: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b] :
( ( c_2Efinite__map_2EFEVERY @ A_27a @ A_27b @ V0P @ V1f )
<=> ! [V2x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V2x @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b @ V1f ) )
=> ( V0P @ ( c_2Epair_2E_2C @ A_27a @ A_27b @ V2x @ ( c_2Efinite__map_2EFAPPLY @ A_27a @ A_27b @ V1f @ V2x ) ) ) ) ) ).
thf(thm_2Efinite__map_2EFLOOKUP__DEF,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b,V1x: A_27a] :
( ( c_2Efinite__map_2EFLOOKUP @ A_27a @ A_27b @ V0f @ V1x )
= ( c_2Ebool_2ECOND @ ( tyop_2Eoption_2Eoption @ A_27b ) @ ( c_2Ebool_2EIN @ A_27a @ V1x @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b @ V0f ) ) @ ( c_2Eoption_2ESOME @ A_27b @ ( c_2Efinite__map_2EFAPPLY @ A_27a @ A_27b @ V0f @ V1x ) ) @ ( c_2Eoption_2ENONE @ A_27b ) ) ) ).
thf(thm_2Efinite__map_2EFMERGE__DEF,axiom,
! [A_27a: $tType,A_27b: $tType,V0m: A_27a > A_27a > A_27a,V1f: tyop_2Efinite__map_2Efmap @ A_27b @ A_27a,V2g: tyop_2Efinite__map_2Efmap @ A_27b @ A_27a] :
( ( ( c_2Efinite__map_2EFDOM @ A_27b @ A_27a @ ( c_2Efinite__map_2EFMERGE @ A_27a @ A_27b @ V0m @ V1f @ V2g ) )
= ( c_2Epred__set_2EUNION @ A_27b @ ( c_2Efinite__map_2EFDOM @ A_27b @ A_27a @ V1f ) @ ( c_2Efinite__map_2EFDOM @ A_27b @ A_27a @ V2g ) ) )
& ! [V3x: A_27b] :
( ( c_2Efinite__map_2EFAPPLY @ A_27b @ A_27a @ ( c_2Efinite__map_2EFMERGE @ A_27a @ A_27b @ V0m @ V1f @ V2g ) @ V3x )
= ( c_2Ebool_2ECOND @ A_27a @ ( c_2Ebool_2E_7E @ ( c_2Ebool_2EIN @ A_27b @ V3x @ ( c_2Efinite__map_2EFDOM @ A_27b @ A_27a @ V1f ) ) ) @ ( c_2Efinite__map_2EFAPPLY @ A_27b @ A_27a @ V2g @ V3x ) @ ( c_2Ebool_2ECOND @ A_27a @ ( c_2Ebool_2E_7E @ ( c_2Ebool_2EIN @ A_27b @ V3x @ ( c_2Efinite__map_2EFDOM @ A_27b @ A_27a @ V2g ) ) ) @ ( c_2Efinite__map_2EFAPPLY @ A_27b @ A_27a @ V1f @ V3x ) @ ( V0m @ ( c_2Efinite__map_2EFAPPLY @ A_27b @ A_27a @ V1f @ V3x ) @ ( c_2Efinite__map_2EFAPPLY @ A_27b @ A_27a @ V2g @ V3x ) ) ) ) ) ) ).
thf(thm_2Efinite__map_2EMAP__KEYS__def,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0f: A_27a > A_27b,V1fm: tyop_2Efinite__map_2Efmap @ A_27a @ A_27c] :
( ( ( c_2Efinite__map_2EFDOM @ A_27b @ A_27c @ ( c_2Efinite__map_2EMAP__KEYS @ A_27a @ A_27b @ A_27c @ V0f @ V1fm ) )
= ( c_2Epred__set_2EIMAGE @ A_27a @ A_27b @ V0f @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27c @ V1fm ) ) )
& ( ( c_2Epred__set_2EINJ @ A_27a @ A_27b @ V0f @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27c @ V1fm ) @ ( c_2Epred__set_2EUNIV @ A_27b ) )
=> ! [V2x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V2x @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27c @ V1fm ) )
=> ( ( c_2Efinite__map_2EFAPPLY @ A_27b @ A_27c @ ( c_2Efinite__map_2EMAP__KEYS @ A_27a @ A_27b @ A_27c @ V0f @ V1fm ) @ ( V0f @ V2x ) )
= ( c_2Efinite__map_2EFAPPLY @ A_27a @ A_27c @ V1fm @ V2x ) ) ) ) ) ).
thf(thm_2Efinite__map_2Efmap__rel__def,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0R: A_27a > A_27b > $o,V1f1: tyop_2Efinite__map_2Efmap @ A_27c @ A_27a,V2f2: tyop_2Efinite__map_2Efmap @ A_27c @ A_27b] :
( ( c_2Efinite__map_2Efmap__rel @ A_27a @ A_27b @ A_27c @ V0R @ V1f1 @ V2f2 )
<=> ( ( ( c_2Efinite__map_2EFDOM @ A_27c @ A_27b @ V2f2 )
= ( c_2Efinite__map_2EFDOM @ A_27c @ A_27a @ V1f1 ) )
& ! [V3x: A_27c] :
( ( c_2Ebool_2EIN @ A_27c @ V3x @ ( c_2Efinite__map_2EFDOM @ A_27c @ A_27a @ V1f1 ) )
=> ( V0R @ ( c_2Efinite__map_2EFAPPLY @ A_27c @ A_27a @ V1f1 @ V3x ) @ ( c_2Efinite__map_2EFAPPLY @ A_27c @ A_27b @ V2f2 @ V3x ) ) ) ) ) ).
thf(thm_2Efinite__map_2Efmap__EQ__UPTO__def,axiom,
! [A_27a: $tType,A_27b: $tType,V0f1: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b,V1f2: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b,V2vs: A_27a > $o] :
( ( c_2Efinite__map_2Efmap__EQ__UPTO @ A_27a @ A_27b @ V0f1 @ V1f2 @ V2vs )
<=> ( ( ( c_2Epred__set_2EINTER @ A_27a @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b @ V0f1 ) @ ( c_2Epred__set_2ECOMPL @ A_27a @ V2vs ) )
= ( c_2Epred__set_2EINTER @ A_27a @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b @ V1f2 ) @ ( c_2Epred__set_2ECOMPL @ A_27a @ V2vs ) ) )
& ! [V3x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V3x @ ( c_2Epred__set_2EINTER @ A_27a @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b @ V0f1 ) @ ( c_2Epred__set_2ECOMPL @ A_27a @ V2vs ) ) )
=> ( ( c_2Efinite__map_2EFAPPLY @ A_27a @ A_27b @ V0f1 @ V3x )
= ( c_2Efinite__map_2EFAPPLY @ A_27a @ A_27b @ V1f2 @ V3x ) ) ) ) ) ).
thf(thm_2Efinite__map_2Efmap__size__def,axiom,
! [A_27a: $tType,A_27b: $tType,V0kz: A_27a > tyop_2Enum_2Enum,V1vz: A_27b > tyop_2Enum_2Enum,V2fm: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b] :
( ( c_2Efinite__map_2Efmap__size @ A_27a @ A_27b @ V0kz @ V1vz @ V2fm )
= ( c_2Epred__set_2ESUM__IMAGE @ A_27a
@ ^ [V3k: A_27a] : ( c_2Earithmetic_2E_2B @ ( V0kz @ V3k ) @ ( V1vz @ ( c_2Efinite__map_2EFAPPLY @ A_27a @ A_27b @ V2fm @ V3k ) ) )
@ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b @ V2fm ) ) ) ).
thf(thm_2Efinite__map_2Efevery__funion,axiom,
! [A_27a: $tType,A_27b: $tType,V0P: ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) > $o,V1m1: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b,V2m2: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b] :
( ( ( c_2Efinite__map_2EFEVERY @ A_27a @ A_27b @ V0P @ V1m1 )
& ( c_2Efinite__map_2EFEVERY @ A_27a @ A_27b @ V0P @ V2m2 ) )
=> ( c_2Efinite__map_2EFEVERY @ A_27a @ A_27b @ V0P @ ( c_2Efinite__map_2EFUNION @ A_27a @ A_27b @ V1m1 @ V2m2 ) ) ) ).
thf(thm_2Efinite__map_2Edrestrict__iter__list,axiom,
! [A_27a: $tType,A_27b: $tType,V0m: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b,V1l: tyop_2Elist_2Elist @ A_27a] :
( ( c_2Elist_2EFOLDR @ A_27a @ ( tyop_2Efinite__map_2Efmap @ A_27a @ A_27b )
@ ^ [V2k: A_27a,V3m: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b] : ( c_2Efinite__map_2Efdomsub @ A_27a @ A_27b @ V3m @ V2k )
@ V0m
@ V1l )
= ( c_2Efinite__map_2EDRESTRICT @ A_27a @ A_27b @ V0m @ ( c_2Epred__set_2ECOMPL @ A_27a @ ( c_2Elist_2ELIST__TO__SET @ A_27a @ V1l ) ) ) ) ).
thf(thm_2Efinite__map_2Edisjoint__drestrict,axiom,
! [A_27a: $tType,A_27b: $tType,V0s: A_27a > $o,V1m: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b] :
( ( c_2Epred__set_2EDISJOINT @ A_27a @ V0s @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b @ V1m ) )
=> ( ( c_2Efinite__map_2EDRESTRICT @ A_27a @ A_27b @ V1m @ ( c_2Epred__set_2ECOMPL @ A_27a @ V0s ) )
= V1m ) ) ).
thf(thm_2Efinite__map_2Efmap__to__list,axiom,
! [A_27a: $tType,A_27b: $tType,V0m: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b] :
? [V1l: tyop_2Elist_2Elist @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b )] :
( ( c_2Elist_2EALL__DISTINCT @ A_27a @ ( c_2Elist_2EMAP @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ A_27a @ ( c_2Epair_2EFST @ A_27a @ A_27b ) @ V1l ) )
& ( V0m
= ( c_2Efinite__map_2EFUPDATE__LIST @ A_27a @ A_27b @ ( c_2Efinite__map_2EFEMPTY @ A_27a @ A_27b ) @ V1l ) ) ) ).
thf(thm_2Efinite__map_2Efupdate__list__foldl,axiom,
! [A_27a: $tType,A_27b: $tType,V0m: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b,V1l: tyop_2Elist_2Elist @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b )] :
( ( c_2Elist_2EFOLDL @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ ( tyop_2Efinite__map_2Efmap @ A_27a @ A_27b )
@ ^ [V2env: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b] :
( c_2Epair_2EUNCURRY @ A_27a @ A_27b @ ( tyop_2Efinite__map_2Efmap @ A_27a @ A_27b )
@ ^ [V3k: A_27a,V4v: A_27b] : ( c_2Efinite__map_2EFUPDATE @ A_27a @ A_27b @ V2env @ ( c_2Epair_2E_2C @ A_27a @ A_27b @ V3k @ V4v ) ) )
@ V0m
@ V1l )
= ( c_2Efinite__map_2EFUPDATE__LIST @ A_27a @ A_27b @ V0m @ V1l ) ) ).
thf(thm_2Efinite__map_2Efupdate__list__foldr,axiom,
! [A_27a: $tType,A_27b: $tType,V0m: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b,V1l: tyop_2Elist_2Elist @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b )] :
( ( c_2Elist_2EFOLDR @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ ( tyop_2Efinite__map_2Efmap @ A_27a @ A_27b )
@ ( c_2Epair_2EUNCURRY @ A_27a @ A_27b @ ( ( tyop_2Efinite__map_2Efmap @ A_27a @ A_27b ) > ( tyop_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
@ ^ [V2k: A_27a,V3v: A_27b,V4env: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b] : ( c_2Efinite__map_2EFUPDATE @ A_27a @ A_27b @ V4env @ ( c_2Epair_2E_2C @ A_27a @ A_27b @ V2k @ V3v ) ) )
@ V0m
@ V1l )
= ( c_2Efinite__map_2EFUPDATE__LIST @ A_27a @ A_27b @ V0m @ ( c_2Elist_2EREVERSE @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ V1l ) ) ) ).
thf(thm_2Efinite__map_2EFUPDATE__EQ__FUPDATE__LIST,axiom,
! [A_27a: $tType,A_27b: $tType,V0fm: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b,V1kv: tyop_2Epair_2Eprod @ A_27a @ A_27b] :
( ( c_2Efinite__map_2EFUPDATE @ A_27a @ A_27b @ V0fm @ V1kv )
= ( c_2Efinite__map_2EFUPDATE__LIST @ A_27a @ A_27b @ V0fm @ ( c_2Elist_2ECONS @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ V1kv @ ( c_2Elist_2ENIL @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) ) ) ) ) ).
thf(thm_2Efinite__map_2Eflookup__thm,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b,V1x: A_27a,V2v: A_27b] :
( ( ( ( c_2Efinite__map_2EFLOOKUP @ A_27a @ A_27b @ V0f @ V1x )
= ( c_2Eoption_2ENONE @ A_27b ) )
<=> ( (~) @ ( c_2Ebool_2EIN @ A_27a @ V1x @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b @ V0f ) ) ) )
& ( ( ( c_2Efinite__map_2EFLOOKUP @ A_27a @ A_27b @ V0f @ V1x )
= ( c_2Eoption_2ESOME @ A_27b @ V2v ) )
<=> ( ( c_2Ebool_2EIN @ A_27a @ V1x @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b @ V0f ) )
& ( ( c_2Efinite__map_2EFAPPLY @ A_27a @ A_27b @ V0f @ V1x )
= V2v ) ) ) ) ).
thf(thm_2Efinite__map_2Efdom__fupdate__list2,axiom,
! [A_27a: $tType,A_27b: $tType,V0kvl: tyop_2Elist_2Elist @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ),V1fm: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b] :
( ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b @ ( c_2Efinite__map_2EFUPDATE__LIST @ A_27a @ A_27b @ V1fm @ V0kvl ) )
= ( c_2Epred__set_2EUNION @ A_27a @ ( c_2Epred__set_2EDIFF @ A_27a @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b @ V1fm ) @ ( c_2Elist_2ELIST__TO__SET @ A_27a @ ( c_2Elist_2EMAP @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ A_27a @ ( c_2Epair_2EFST @ A_27a @ A_27b ) @ V0kvl ) ) ) @ ( c_2Elist_2ELIST__TO__SET @ A_27a @ ( c_2Elist_2EMAP @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ A_27a @ ( c_2Epair_2EFST @ A_27a @ A_27b ) @ V0kvl ) ) ) ) ).
thf(thm_2Efinite__map_2Efupdate__list__map,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,A_27d: $tType,V0l: tyop_2Elist_2Elist @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ),V1f: A_27b > A_27c,V2x: A_27a,V3y: A_27d] :
( ( c_2Ebool_2EIN @ A_27a @ V2x @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b @ ( c_2Efinite__map_2EFUPDATE__LIST @ A_27a @ A_27b @ ( c_2Efinite__map_2EFEMPTY @ A_27a @ A_27b ) @ V0l ) ) )
=> ( ( c_2Efinite__map_2EFAPPLY @ A_27a @ A_27c
@ ( c_2Efinite__map_2EFUPDATE__LIST @ A_27a @ A_27c @ ( c_2Efinite__map_2EFEMPTY @ A_27a @ A_27c )
@ ( c_2Elist_2EMAP @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ ( tyop_2Epair_2Eprod @ A_27a @ A_27c )
@ ( c_2Epair_2EUNCURRY @ A_27a @ A_27b @ ( tyop_2Epair_2Eprod @ A_27a @ A_27c )
@ ^ [V4a: A_27a,V5b: A_27b] : ( c_2Epair_2E_2C @ A_27a @ A_27c @ V4a @ ( V1f @ V5b ) ) )
@ V0l ) )
@ V2x )
= ( V1f @ ( c_2Efinite__map_2EFAPPLY @ A_27a @ A_27b @ ( c_2Efinite__map_2EFUPDATE__LIST @ A_27a @ A_27b @ ( c_2Efinite__map_2EFEMPTY @ A_27a @ A_27b ) @ V0l ) @ V2x ) ) ) ) ).
thf(thm_2Efinite__map_2Efmap__rel__sym,axiom,
! [A_27a: $tType,A_27b: $tType,V0R: A_27a > A_27a > $o] :
( ! [V1x: A_27a,V2y: A_27a] :
( ( V0R @ V1x @ V2y )
=> ( V0R @ V2y @ V1x ) )
=> ! [V3x: tyop_2Efinite__map_2Efmap @ A_27b @ A_27a,V4y: tyop_2Efinite__map_2Efmap @ A_27b @ A_27a] :
( ( c_2Efinite__map_2Efmap__rel @ A_27a @ A_27a @ A_27b @ V0R @ V3x @ V4y )
=> ( c_2Efinite__map_2Efmap__rel @ A_27a @ A_27a @ A_27b @ V0R @ V4y @ V3x ) ) ) ).
thf(thm_2Efinite__map_2Efmap__rel__trans,axiom,
! [A_27a: $tType,A_27b: $tType,V0R: A_27a > A_27a > $o] :
( ! [V1x: A_27a,V2y: A_27a,V3z: A_27a] :
( ( ( V0R @ V1x @ V2y )
& ( V0R @ V2y @ V3z ) )
=> ( V0R @ V1x @ V3z ) )
=> ! [V4x: tyop_2Efinite__map_2Efmap @ A_27b @ A_27a,V5y: tyop_2Efinite__map_2Efmap @ A_27b @ A_27a,V6z: tyop_2Efinite__map_2Efmap @ A_27b @ A_27a] :
( ( ( c_2Efinite__map_2Efmap__rel @ A_27a @ A_27a @ A_27b @ V0R @ V4x @ V5y )
& ( c_2Efinite__map_2Efmap__rel @ A_27a @ A_27a @ A_27b @ V0R @ V5y @ V6z ) )
=> ( c_2Efinite__map_2Efmap__rel @ A_27a @ A_27a @ A_27b @ V0R @ V4x @ V6z ) ) ) ).
thf(thm_2Efinite__map_2Ef__o__f__FUPDATE__compose,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0f1: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b,V1f2: tyop_2Efinite__map_2Efmap @ A_27c @ A_27a,V2k: A_27c,V3x: A_27a,V4v: A_27b] :
( ( ( (~) @ ( c_2Ebool_2EIN @ A_27a @ V3x @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b @ V0f1 ) ) )
& ( (~) @ ( c_2Ebool_2EIN @ A_27a @ V3x @ ( c_2Efinite__map_2EFRANGE @ A_27c @ A_27a @ V1f2 ) ) ) )
=> ( ( c_2Efinite__map_2Ef__o__f @ A_27c @ A_27a @ A_27b @ ( c_2Efinite__map_2EFUPDATE @ A_27a @ A_27b @ V0f1 @ ( c_2Epair_2E_2C @ A_27a @ A_27b @ V3x @ V4v ) ) @ ( c_2Efinite__map_2EFUPDATE @ A_27c @ A_27a @ V1f2 @ ( c_2Epair_2E_2C @ A_27c @ A_27a @ V2k @ V3x ) ) )
= ( c_2Efinite__map_2EFUPDATE @ A_27c @ A_27b @ ( c_2Efinite__map_2Ef__o__f @ A_27c @ A_27a @ A_27b @ V0f1 @ V1f2 ) @ ( c_2Epair_2E_2C @ A_27c @ A_27b @ V2k @ V4v ) ) ) ) ).
thf(thm_2Efinite__map_2EDRESTRICT__SUBSET,axiom,
! [A_27a: $tType,A_27b: $tType,V0f1: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b,V1f2: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b,V2s: A_27a > $o,V3t: A_27a > $o] :
( ( ( ( c_2Efinite__map_2EDRESTRICT @ A_27a @ A_27b @ V0f1 @ V2s )
= ( c_2Efinite__map_2EDRESTRICT @ A_27a @ A_27b @ V1f2 @ V2s ) )
& ( c_2Epred__set_2ESUBSET @ A_27a @ V3t @ V2s ) )
=> ( ( c_2Efinite__map_2EDRESTRICT @ A_27a @ A_27b @ V0f1 @ V3t )
= ( c_2Efinite__map_2EDRESTRICT @ A_27a @ A_27b @ V1f2 @ V3t ) ) ) ).
thf(thm_2Efinite__map_2EDRESTRICT__FDOM,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b] :
( ( c_2Efinite__map_2EDRESTRICT @ A_27a @ A_27b @ V0f @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b @ V0f ) )
= V0f ) ).
thf(thm_2Efinite__map_2EFRANGE__DRESTRICT__SUBSET,axiom,
! [A_27a: $tType,A_27b: $tType,V0s: A_27b > $o,V1fm: tyop_2Efinite__map_2Efmap @ A_27b @ A_27a] : ( c_2Epred__set_2ESUBSET @ A_27a @ ( c_2Efinite__map_2EFRANGE @ A_27b @ A_27a @ ( c_2Efinite__map_2EDRESTRICT @ A_27b @ A_27a @ V1fm @ V0s ) ) @ ( c_2Efinite__map_2EFRANGE @ A_27b @ A_27a @ V1fm ) ) ).
thf(thm_2Efinite__map_2EIN__FRANGE__DOMSUB__suff,axiom,
! [A_27a: $tType,A_27b: $tType,V0k: A_27b,V1fm: tyop_2Efinite__map_2Efmap @ A_27b @ A_27a,V2P: A_27a > $o] :
( ! [V3v: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V3v @ ( c_2Efinite__map_2EFRANGE @ A_27b @ A_27a @ V1fm ) )
=> ( V2P @ V3v ) )
=> ! [V4v: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V4v @ ( c_2Efinite__map_2EFRANGE @ A_27b @ A_27a @ ( c_2Efinite__map_2Efdomsub @ A_27b @ A_27a @ V1fm @ V0k ) ) )
=> ( V2P @ V4v ) ) ) ).
thf(thm_2Efinite__map_2EFRANGE__DOMSUB__SUBSET,axiom,
! [A_27a: $tType,A_27b: $tType,V0k: A_27b,V1fm: tyop_2Efinite__map_2Efmap @ A_27b @ A_27a] : ( c_2Epred__set_2ESUBSET @ A_27a @ ( c_2Efinite__map_2EFRANGE @ A_27b @ A_27a @ ( c_2Efinite__map_2Efdomsub @ A_27b @ A_27a @ V1fm @ V0k ) ) @ ( c_2Efinite__map_2EFRANGE @ A_27b @ A_27a @ V1fm ) ) ).
thf(thm_2Efinite__map_2EIN__FRANGE__FUNION__suff,axiom,
! [A_27a: $tType,A_27b: $tType,V0f2: tyop_2Efinite__map_2Efmap @ A_27b @ A_27a,V1f1: tyop_2Efinite__map_2Efmap @ A_27b @ A_27a,V2P: A_27a > $o] :
( ( ! [V3v: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V3v @ ( c_2Efinite__map_2EFRANGE @ A_27b @ A_27a @ V1f1 ) )
=> ( V2P @ V3v ) )
& ! [V4v: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V4v @ ( c_2Efinite__map_2EFRANGE @ A_27b @ A_27a @ V0f2 ) )
=> ( V2P @ V4v ) ) )
=> ! [V5v: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V5v @ ( c_2Efinite__map_2EFRANGE @ A_27b @ A_27a @ ( c_2Efinite__map_2EFUNION @ A_27b @ A_27a @ V1f1 @ V0f2 ) ) )
=> ( V2P @ V5v ) ) ) ).
thf(thm_2Efinite__map_2EFRANGE__FUNION__SUBSET,axiom,
! [A_27a: $tType,A_27b: $tType,V0f2: tyop_2Efinite__map_2Efmap @ A_27b @ A_27a,V1f1: tyop_2Efinite__map_2Efmap @ A_27b @ A_27a] : ( c_2Epred__set_2ESUBSET @ A_27a @ ( c_2Efinite__map_2EFRANGE @ A_27b @ A_27a @ ( c_2Efinite__map_2EFUNION @ A_27b @ A_27a @ V1f1 @ V0f2 ) ) @ ( c_2Epred__set_2EUNION @ A_27a @ ( c_2Efinite__map_2EFRANGE @ A_27b @ A_27a @ V1f1 ) @ ( c_2Efinite__map_2EFRANGE @ A_27b @ A_27a @ V0f2 ) ) ) ).
thf(thm_2Efinite__map_2EIN__FRANGE__FUPDATE__LIST__suff,axiom,
! [A_27a: $tType,A_27b: $tType,V0ls: tyop_2Elist_2Elist @ ( tyop_2Epair_2Eprod @ A_27b @ A_27a ),V1fm: tyop_2Efinite__map_2Efmap @ A_27b @ A_27a,V2P: A_27a > $o] :
( ( ! [V3v: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V3v @ ( c_2Efinite__map_2EFRANGE @ A_27b @ A_27a @ V1fm ) )
=> ( V2P @ V3v ) )
& ! [V4v: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V4v @ ( c_2Elist_2ELIST__TO__SET @ A_27a @ ( c_2Elist_2EMAP @ ( tyop_2Epair_2Eprod @ A_27b @ A_27a ) @ A_27a @ ( c_2Epair_2ESND @ A_27b @ A_27a ) @ V0ls ) ) )
=> ( V2P @ V4v ) ) )
=> ! [V5v: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V5v @ ( c_2Efinite__map_2EFRANGE @ A_27b @ A_27a @ ( c_2Efinite__map_2EFUPDATE__LIST @ A_27b @ A_27a @ V1fm @ V0ls ) ) )
=> ( V2P @ V5v ) ) ) ).
thf(thm_2Efinite__map_2EFRANGE__FUPDATE__LIST__SUBSET,axiom,
! [A_27a: $tType,A_27b: $tType,V0ls: tyop_2Elist_2Elist @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ),V1fm: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b] : ( c_2Epred__set_2ESUBSET @ A_27b @ ( c_2Efinite__map_2EFRANGE @ A_27a @ A_27b @ ( c_2Efinite__map_2EFUPDATE__LIST @ A_27a @ A_27b @ V1fm @ V0ls ) ) @ ( c_2Epred__set_2EUNION @ A_27b @ ( c_2Efinite__map_2EFRANGE @ A_27a @ A_27b @ V1fm ) @ ( c_2Elist_2ELIST__TO__SET @ A_27b @ ( c_2Elist_2EMAP @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ A_27b @ ( c_2Epair_2ESND @ A_27a @ A_27b ) @ V0ls ) ) ) ) ).
thf(thm_2Efinite__map_2EIN__FRANGE__FLOOKUP,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b,V1v: A_27b] :
( ( c_2Ebool_2EIN @ A_27b @ V1v @ ( c_2Efinite__map_2EFRANGE @ A_27a @ A_27b @ V0f ) )
<=> ? [V2k: A_27a] :
( ( c_2Efinite__map_2EFLOOKUP @ A_27a @ A_27b @ V0f @ V2k )
= ( c_2Eoption_2ESOME @ A_27b @ V1v ) ) ) ).
thf(thm_2Efinite__map_2EIN__FRANGE,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b,V1v: A_27b] :
( ( c_2Ebool_2EIN @ A_27b @ V1v @ ( c_2Efinite__map_2EFRANGE @ A_27a @ A_27b @ V0f ) )
<=> ? [V2k: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V2k @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b @ V0f ) )
& ( ( c_2Efinite__map_2EFAPPLY @ A_27a @ A_27b @ V0f @ V2k )
= V1v ) ) ) ).
thf(thm_2Efinite__map_2EFUPDATE__LIST__ALL__DISTINCT__REVERSE,axiom,
! [A_27a: $tType,A_27b: $tType,V0ls: tyop_2Elist_2Elist @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b )] :
( ( c_2Elist_2EALL__DISTINCT @ A_27a @ ( c_2Elist_2EMAP @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ A_27a @ ( c_2Epair_2EFST @ A_27a @ A_27b ) @ V0ls ) )
=> ! [V1fm: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b] :
( ( c_2Efinite__map_2EFUPDATE__LIST @ A_27a @ A_27b @ V1fm @ ( c_2Elist_2EREVERSE @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ V0ls ) )
= ( c_2Efinite__map_2EFUPDATE__LIST @ A_27a @ A_27b @ V1fm @ V0ls ) ) ) ).
thf(thm_2Efinite__map_2EFUPDATE__LIST__ALL__DISTINCT__APPLY__MEM,axiom,
! [A_27a: $tType,A_27b: $tType,V0P: A_27a > $o,V1ls: tyop_2Elist_2Elist @ ( tyop_2Epair_2Eprod @ A_27b @ A_27a ),V2k: A_27b,V3v: A_27a,V4fm: tyop_2Efinite__map_2Efmap @ A_27b @ A_27a] :
( ( ( c_2Elist_2EALL__DISTINCT @ A_27b @ ( c_2Elist_2EMAP @ ( tyop_2Epair_2Eprod @ A_27b @ A_27a ) @ A_27b @ ( c_2Epair_2EFST @ A_27b @ A_27a ) @ V1ls ) )
& ( c_2Ebool_2EIN @ ( tyop_2Epair_2Eprod @ A_27b @ A_27a ) @ ( c_2Epair_2E_2C @ A_27b @ A_27a @ V2k @ V3v ) @ ( c_2Elist_2ELIST__TO__SET @ ( tyop_2Epair_2Eprod @ A_27b @ A_27a ) @ V1ls ) )
& ( V0P @ V3v ) )
=> ( V0P @ ( c_2Efinite__map_2EFAPPLY @ A_27b @ A_27a @ ( c_2Efinite__map_2EFUPDATE__LIST @ A_27b @ A_27a @ V4fm @ V1ls ) @ V2k ) ) ) ).
thf(thm_2Efinite__map_2EFUPDATE__SAME__LIST__APPLY,axiom,
! [A_27a: $tType,A_27b: $tType,V0kvl: tyop_2Elist_2Elist @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ),V1fm1: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b,V2fm2: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b,V3x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V3x @ ( c_2Elist_2ELIST__TO__SET @ A_27a @ ( c_2Elist_2EMAP @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ A_27a @ ( c_2Epair_2EFST @ A_27a @ A_27b ) @ V0kvl ) ) )
=> ( ( c_2Efinite__map_2EFAPPLY @ A_27a @ A_27b @ ( c_2Efinite__map_2EFUPDATE__LIST @ A_27a @ A_27b @ V1fm1 @ V0kvl ) @ V3x )
= ( c_2Efinite__map_2EFAPPLY @ A_27a @ A_27b @ ( c_2Efinite__map_2EFUPDATE__LIST @ A_27a @ A_27b @ V2fm2 @ V0kvl ) @ V3x ) ) ) ).
thf(thm_2Efinite__map_2EFUPDATE__SAME__APPLY,axiom,
! [A_27a: $tType,A_27b: $tType,V0x: A_27a,V1kv: tyop_2Epair_2Eprod @ A_27a @ A_27b,V2fm2: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b,V3fm1: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b] :
( ( ( V0x
= ( c_2Epair_2EFST @ A_27a @ A_27b @ V1kv ) )
| ( ( c_2Efinite__map_2EFAPPLY @ A_27a @ A_27b @ V3fm1 @ V0x )
= ( c_2Efinite__map_2EFAPPLY @ A_27a @ A_27b @ V2fm2 @ V0x ) ) )
=> ( ( c_2Efinite__map_2EFAPPLY @ A_27a @ A_27b @ ( c_2Efinite__map_2EFUPDATE @ A_27a @ A_27b @ V3fm1 @ V1kv ) @ V0x )
= ( c_2Efinite__map_2EFAPPLY @ A_27a @ A_27b @ ( c_2Efinite__map_2EFUPDATE @ A_27a @ A_27b @ V2fm2 @ V1kv ) @ V0x ) ) ) ).
thf(thm_2Efinite__map_2EFUPDATE__LIST__APPLY__HO__THM,axiom,
! [A_27a: $tType,A_27b: $tType,V0P: A_27a > $o,V1f: tyop_2Efinite__map_2Efmap @ A_27b @ A_27a,V2kvl: tyop_2Elist_2Elist @ ( tyop_2Epair_2Eprod @ A_27b @ A_27a ),V3k: A_27b] :
( ( ? [V4n: tyop_2Enum_2Enum] :
( ( c_2Eprim__rec_2E_3C @ V4n @ ( c_2Elist_2ELENGTH @ ( tyop_2Epair_2Eprod @ A_27b @ A_27a ) @ V2kvl ) )
& ( V3k
= ( c_2Elist_2EEL @ A_27b @ V4n @ ( c_2Elist_2EMAP @ ( tyop_2Epair_2Eprod @ A_27b @ A_27a ) @ A_27b @ ( c_2Epair_2EFST @ A_27b @ A_27a ) @ V2kvl ) ) )
& ( V0P @ ( c_2Elist_2EEL @ A_27a @ V4n @ ( c_2Elist_2EMAP @ ( tyop_2Epair_2Eprod @ A_27b @ A_27a ) @ A_27a @ ( c_2Epair_2ESND @ A_27b @ A_27a ) @ V2kvl ) ) )
& ! [V5m: tyop_2Enum_2Enum] :
( ( ( c_2Eprim__rec_2E_3C @ V4n @ V5m )
& ( c_2Eprim__rec_2E_3C @ V5m @ ( c_2Elist_2ELENGTH @ ( tyop_2Epair_2Eprod @ A_27b @ A_27a ) @ V2kvl ) ) )
=> ( (~)
@ ( ( c_2Elist_2EEL @ A_27b @ V5m @ ( c_2Elist_2EMAP @ ( tyop_2Epair_2Eprod @ A_27b @ A_27a ) @ A_27b @ ( c_2Epair_2EFST @ A_27b @ A_27a ) @ V2kvl ) )
= V3k ) ) ) )
| ( ( (~) @ ( c_2Ebool_2EIN @ A_27b @ V3k @ ( c_2Elist_2ELIST__TO__SET @ A_27b @ ( c_2Elist_2EMAP @ ( tyop_2Epair_2Eprod @ A_27b @ A_27a ) @ A_27b @ ( c_2Epair_2EFST @ A_27b @ A_27a ) @ V2kvl ) ) ) )
& ( V0P @ ( c_2Efinite__map_2EFAPPLY @ A_27b @ A_27a @ V1f @ V3k ) ) ) )
=> ( V0P @ ( c_2Efinite__map_2EFAPPLY @ A_27b @ A_27a @ ( c_2Efinite__map_2EFUPDATE__LIST @ A_27b @ A_27a @ V1f @ V2kvl ) @ V3k ) ) ) ).
thf(thm_2Efinite__map_2EFUPDATE__LIST__APPLY__NOT__MEM__matchable,axiom,
! [A_27a: $tType,A_27b: $tType,V0kvl: tyop_2Elist_2Elist @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ),V1f: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b,V2k: A_27a,V3v: A_27b] :
( ( ( (~) @ ( c_2Ebool_2EIN @ A_27a @ V2k @ ( c_2Elist_2ELIST__TO__SET @ A_27a @ ( c_2Elist_2EMAP @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ A_27a @ ( c_2Epair_2EFST @ A_27a @ A_27b ) @ V0kvl ) ) ) )
& ( V3v
= ( c_2Efinite__map_2EFAPPLY @ A_27a @ A_27b @ V1f @ V2k ) ) )
=> ( ( c_2Efinite__map_2EFAPPLY @ A_27a @ A_27b @ ( c_2Efinite__map_2EFUPDATE__LIST @ A_27a @ A_27b @ V1f @ V0kvl ) @ V2k )
= V3v ) ) ).
thf(thm_2Efinite__map_2EDRESTRICT__FUNION__SUBSET,axiom,
! [A_27a: $tType,A_27b: $tType,V0s2: A_27a > $o,V1s1: A_27a > $o,V2g: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b,V3f: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b] :
( ( c_2Epred__set_2ESUBSET @ A_27a @ V0s2 @ V1s1 )
=> ? [V4h: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b] :
( ( c_2Efinite__map_2EFUNION @ A_27a @ A_27b @ ( c_2Efinite__map_2EDRESTRICT @ A_27a @ A_27b @ V3f @ V1s1 ) @ V2g )
= ( c_2Efinite__map_2EFUNION @ A_27a @ A_27b @ ( c_2Efinite__map_2EDRESTRICT @ A_27a @ A_27b @ V3f @ V0s2 ) @ V4h ) ) ) ).
thf(thm_2Efinite__map_2EFOLDL2__FUPDATE__LIST__paired,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,A_27d: $tType,A_27e: $tType,V0f1: A_27a > A_27b > A_27c > A_27d,V1f2: A_27a > A_27b > A_27c > A_27e,V2bs: tyop_2Elist_2Elist @ A_27a,V3cs: tyop_2Elist_2Elist @ ( tyop_2Epair_2Eprod @ A_27b @ A_27c ),V4a: tyop_2Efinite__map_2Efmap @ A_27d @ A_27e] :
( ( ( c_2Elist_2ELENGTH @ A_27a @ V2bs )
= ( c_2Elist_2ELENGTH @ ( tyop_2Epair_2Eprod @ A_27b @ A_27c ) @ V3cs ) )
=> ( ( c_2Elist_2EFOLDL2 @ ( tyop_2Efinite__map_2Efmap @ A_27d @ A_27e ) @ A_27a @ ( tyop_2Epair_2Eprod @ A_27b @ A_27c )
@ ^ [V5fm: tyop_2Efinite__map_2Efmap @ A_27d @ A_27e,V6b: A_27a] :
( c_2Epair_2EUNCURRY @ A_27b @ A_27c @ ( tyop_2Efinite__map_2Efmap @ A_27d @ A_27e )
@ ^ [V7c: A_27b,V8d: A_27c] : ( c_2Efinite__map_2EFUPDATE @ A_27d @ A_27e @ V5fm @ ( c_2Epair_2E_2C @ A_27d @ A_27e @ ( V0f1 @ V6b @ V7c @ V8d ) @ ( V1f2 @ V6b @ V7c @ V8d ) ) ) )
@ V4a
@ V2bs
@ V3cs )
= ( c_2Efinite__map_2EFUPDATE__LIST @ A_27d @ A_27e @ V4a
@ ( c_2Elist_2EZIP @ A_27d @ A_27e
@ ( c_2Epair_2E_2C @ ( tyop_2Elist_2Elist @ A_27d ) @ ( tyop_2Elist_2Elist @ A_27e )
@ ( c_2Elist_2EMAP2 @ A_27d @ A_27a @ ( tyop_2Epair_2Eprod @ A_27b @ A_27c )
@ ^ [V9b: A_27a] : ( c_2Epair_2EUNCURRY @ A_27b @ A_27c @ A_27d @ ( V0f1 @ V9b ) )
@ V2bs
@ V3cs )
@ ( c_2Elist_2EMAP2 @ A_27e @ A_27a @ ( tyop_2Epair_2Eprod @ A_27b @ A_27c )
@ ^ [V10b: A_27a] : ( c_2Epair_2EUNCURRY @ A_27b @ A_27c @ A_27e @ ( V1f2 @ V10b ) )
@ V2bs
@ V3cs ) ) ) ) ) ) ).
thf(thm_2Efinite__map_2EFOLDL2__FUPDATE__LIST,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,A_27d: $tType,V0f1: A_27a > A_27b > A_27c,V1f2: A_27a > A_27b > A_27d,V2bs: tyop_2Elist_2Elist @ A_27a,V3cs: tyop_2Elist_2Elist @ A_27b,V4a: tyop_2Efinite__map_2Efmap @ A_27c @ A_27d] :
( ( ( c_2Elist_2ELENGTH @ A_27a @ V2bs )
= ( c_2Elist_2ELENGTH @ A_27b @ V3cs ) )
=> ( ( c_2Elist_2EFOLDL2 @ ( tyop_2Efinite__map_2Efmap @ A_27c @ A_27d ) @ A_27a @ A_27b
@ ^ [V5fm: tyop_2Efinite__map_2Efmap @ A_27c @ A_27d,V6b: A_27a,V7c: A_27b] : ( c_2Efinite__map_2EFUPDATE @ A_27c @ A_27d @ V5fm @ ( c_2Epair_2E_2C @ A_27c @ A_27d @ ( V0f1 @ V6b @ V7c ) @ ( V1f2 @ V6b @ V7c ) ) )
@ V4a
@ V2bs
@ V3cs )
= ( c_2Efinite__map_2EFUPDATE__LIST @ A_27c @ A_27d @ V4a @ ( c_2Elist_2EZIP @ A_27c @ A_27d @ ( c_2Epair_2E_2C @ ( tyop_2Elist_2Elist @ A_27c ) @ ( tyop_2Elist_2Elist @ A_27d ) @ ( c_2Elist_2EMAP2 @ A_27c @ A_27a @ A_27b @ V0f1 @ V2bs @ V3cs ) @ ( c_2Elist_2EMAP2 @ A_27d @ A_27a @ A_27b @ V1f2 @ V2bs @ V3cs ) ) ) ) ) ) ).
thf(thm_2Efinite__map_2EDRESTRICT__EQ__DRESTRICT__SAME,axiom,
! [A_27a: $tType,A_27b: $tType,V0s: A_27a > $o,V1f2: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b,V2f1: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b] :
( ( ( c_2Efinite__map_2EDRESTRICT @ A_27a @ A_27b @ V2f1 @ V0s )
= ( c_2Efinite__map_2EDRESTRICT @ A_27a @ A_27b @ V1f2 @ V0s ) )
<=> ( ( ( c_2Epred__set_2EINTER @ A_27a @ V0s @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b @ V2f1 ) )
= ( c_2Epred__set_2EINTER @ A_27a @ V0s @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b @ V1f2 ) ) )
& ! [V3x: A_27a] :
( ( ( c_2Ebool_2EIN @ A_27a @ V3x @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b @ V2f1 ) )
& ( c_2Ebool_2EIN @ A_27a @ V3x @ V0s ) )
=> ( ( c_2Efinite__map_2EFAPPLY @ A_27a @ A_27b @ V2f1 @ V3x )
= ( c_2Efinite__map_2EFAPPLY @ A_27a @ A_27b @ V1f2 @ V3x ) ) ) ) ) ).
thf(thm_2Efinite__map_2EDRESTRICT__FUNION__SAME,axiom,
! [A_27a: $tType,A_27b: $tType,V0fm: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b,V1s: A_27a > $o] :
( ( c_2Efinite__map_2EFUNION @ A_27a @ A_27b @ ( c_2Efinite__map_2EDRESTRICT @ A_27a @ A_27b @ V0fm @ V1s ) @ V0fm )
= V0fm ) ).
thf(thm_2Efinite__map_2EDRESTRICT__SUBSET__SUBMAP__gen,axiom,
! [A_27a: $tType,A_27b: $tType,V0f1: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b,V1f2: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b,V2s: A_27a > $o,V3t: A_27a > $o] :
( ( ( c_2Efinite__map_2ESUBMAP @ A_27a @ A_27b @ ( c_2Efinite__map_2EDRESTRICT @ A_27a @ A_27b @ V0f1 @ V2s ) @ ( c_2Efinite__map_2EDRESTRICT @ A_27a @ A_27b @ V1f2 @ V2s ) )
& ( c_2Epred__set_2ESUBSET @ A_27a @ V3t @ V2s ) )
=> ( c_2Efinite__map_2ESUBMAP @ A_27a @ A_27b @ ( c_2Efinite__map_2EDRESTRICT @ A_27a @ A_27b @ V0f1 @ V3t ) @ ( c_2Efinite__map_2EDRESTRICT @ A_27a @ A_27b @ V1f2 @ V3t ) ) ) ).
thf(thm_2Efinite__map_2EDRESTRICT__DOMSUB,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b,V1s: A_27a > $o,V2k: A_27a] :
( ( c_2Efinite__map_2Efdomsub @ A_27a @ A_27b @ ( c_2Efinite__map_2EDRESTRICT @ A_27a @ A_27b @ V0f @ V1s ) @ V2k )
= ( c_2Efinite__map_2EDRESTRICT @ A_27a @ A_27b @ V0f @ ( c_2Epred__set_2EDELETE @ A_27a @ V1s @ V2k ) ) ) ).
thf(thm_2Efinite__map_2EDOMSUB__SUBMAP,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b,V1g: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b,V2x: A_27a] :
( ( ( c_2Efinite__map_2ESUBMAP @ A_27a @ A_27b @ V0f @ V1g )
& ( (~) @ ( c_2Ebool_2EIN @ A_27a @ V2x @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b @ V0f ) ) ) )
=> ( c_2Efinite__map_2ESUBMAP @ A_27a @ A_27b @ V0f @ ( c_2Efinite__map_2Efdomsub @ A_27a @ A_27b @ V1g @ V2x ) ) ) ).
thf(thm_2Efinite__map_2ESUBMAP__DOMSUB__gen,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b,V1g: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b,V2k: A_27a] :
( ( c_2Efinite__map_2ESUBMAP @ A_27a @ A_27b @ ( c_2Efinite__map_2Efdomsub @ A_27a @ A_27b @ V0f @ V2k ) @ V1g )
= ( c_2Efinite__map_2ESUBMAP @ A_27a @ A_27b @ ( c_2Efinite__map_2Efdomsub @ A_27a @ A_27b @ V0f @ V2k ) @ ( c_2Efinite__map_2Efdomsub @ A_27a @ A_27b @ V1g @ V2k ) ) ) ).
thf(thm_2Efinite__map_2ESUBMAP__mono__FUPDATE,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b,V1g: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b,V2x: A_27a,V3y: A_27b] :
( ( c_2Efinite__map_2ESUBMAP @ A_27a @ A_27b @ ( c_2Efinite__map_2Efdomsub @ A_27a @ A_27b @ V0f @ V2x ) @ ( c_2Efinite__map_2Efdomsub @ A_27a @ A_27b @ V1g @ V2x ) )
=> ( c_2Efinite__map_2ESUBMAP @ A_27a @ A_27b @ ( c_2Efinite__map_2EFUPDATE @ A_27a @ A_27b @ V0f @ ( c_2Epair_2E_2C @ A_27a @ A_27b @ V2x @ V3y ) ) @ ( c_2Efinite__map_2EFUPDATE @ A_27a @ A_27b @ V1g @ ( c_2Epair_2E_2C @ A_27a @ A_27b @ V2x @ V3y ) ) ) ) ).
thf(thm_2Efinite__map_2EIMAGE__FRANGE,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0f: A_27a > A_27b,V1fm: tyop_2Efinite__map_2Efmap @ A_27c @ A_27a] :
( ( c_2Epred__set_2EIMAGE @ A_27a @ A_27b @ V0f @ ( c_2Efinite__map_2EFRANGE @ A_27c @ A_27a @ V1fm ) )
= ( c_2Efinite__map_2EFRANGE @ A_27c @ A_27b @ ( c_2Efinite__map_2Eo__f @ A_27c @ A_27a @ A_27b @ V0f @ V1fm ) ) ) ).
thf(thm_2Efinite__map_2EDRESTRICTED__FUNION,axiom,
! [A_27a: $tType,A_27b: $tType,V0f1: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b,V1f2: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b,V2s: A_27a > $o] :
( ( c_2Efinite__map_2EDRESTRICT @ A_27a @ A_27b @ ( c_2Efinite__map_2EFUNION @ A_27a @ A_27b @ V0f1 @ V1f2 ) @ V2s )
= ( c_2Efinite__map_2EFUNION @ A_27a @ A_27b @ ( c_2Efinite__map_2EDRESTRICT @ A_27a @ A_27b @ V0f1 @ V2s ) @ ( c_2Efinite__map_2EDRESTRICT @ A_27a @ A_27b @ V1f2 @ ( c_2Epred__set_2EDIFF @ A_27a @ V2s @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b @ V0f1 ) ) ) ) ) ).
thf(thm_2Efinite__map_2EDRESTRICT__SUBSET__SUBMAP,axiom,
! [A_27a: $tType,A_27b: $tType,V0s2: A_27a > $o,V1s1: A_27a > $o,V2f: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b] :
( ( c_2Epred__set_2ESUBSET @ A_27a @ V1s1 @ V0s2 )
=> ( c_2Efinite__map_2ESUBMAP @ A_27a @ A_27b @ ( c_2Efinite__map_2EDRESTRICT @ A_27a @ A_27b @ V2f @ V1s1 ) @ ( c_2Efinite__map_2EDRESTRICT @ A_27a @ A_27b @ V2f @ V0s2 ) ) ) ).
thf(thm_2Efinite__map_2EDRESTRICT__SUBMAP__gen,axiom,
! [A_27a: $tType,A_27b: $tType,V0g: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b,V1f: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b,V2P: A_27a > $o] :
( ( c_2Efinite__map_2ESUBMAP @ A_27a @ A_27b @ V1f @ V0g )
=> ( c_2Efinite__map_2ESUBMAP @ A_27a @ A_27b @ ( c_2Efinite__map_2EDRESTRICT @ A_27a @ A_27b @ V1f @ V2P ) @ V0g ) ) ).
thf(thm_2Efinite__map_2EIN__FRANGE__o__f__suff,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0fm: tyop_2Efinite__map_2Efmap @ A_27b @ A_27a,V1f: A_27a > A_27c,V2P: A_27c > $o] :
( ! [V3v: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V3v @ ( c_2Efinite__map_2EFRANGE @ A_27b @ A_27a @ V0fm ) )
=> ( V2P @ ( V1f @ V3v ) ) )
=> ! [V4v: A_27c] :
( ( c_2Ebool_2EIN @ A_27c @ V4v @ ( c_2Efinite__map_2EFRANGE @ A_27b @ A_27c @ ( c_2Efinite__map_2Eo__f @ A_27b @ A_27a @ A_27c @ V1f @ V0fm ) ) )
=> ( V2P @ V4v ) ) ) ).
thf(thm_2Efinite__map_2EIN__FRANGE__FUPDATE__suff,axiom,
! [A_27a: $tType,A_27b: $tType,V0kv: tyop_2Epair_2Eprod @ A_27b @ A_27a,V1fm: tyop_2Efinite__map_2Efmap @ A_27b @ A_27a,V2P: A_27a > $o] :
( ( ! [V3v: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V3v @ ( c_2Efinite__map_2EFRANGE @ A_27b @ A_27a @ V1fm ) )
=> ( V2P @ V3v ) )
& ( V2P @ ( c_2Epair_2ESND @ A_27b @ A_27a @ V0kv ) ) )
=> ! [V4v: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V4v @ ( c_2Efinite__map_2EFRANGE @ A_27b @ A_27a @ ( c_2Efinite__map_2EFUPDATE @ A_27b @ A_27a @ V1fm @ V0kv ) ) )
=> ( V2P @ V4v ) ) ) ).
thf(thm_2Efinite__map_2EFRANGE__FUPDATE__SUBSET,axiom,
! [A_27a: $tType,A_27b: $tType,V0kv: tyop_2Epair_2Eprod @ A_27b @ A_27a,V1fm: tyop_2Efinite__map_2Efmap @ A_27b @ A_27a] : ( c_2Epred__set_2ESUBSET @ A_27a @ ( c_2Efinite__map_2EFRANGE @ A_27b @ A_27a @ ( c_2Efinite__map_2EFUPDATE @ A_27b @ A_27a @ V1fm @ V0kv ) ) @ ( c_2Epred__set_2EUNION @ A_27a @ ( c_2Efinite__map_2EFRANGE @ A_27b @ A_27a @ V1fm ) @ ( c_2Epred__set_2EINSERT @ A_27a @ ( c_2Epair_2ESND @ A_27b @ A_27a @ V0kv ) @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) ) ) ).
thf(thm_2Efinite__map_2EIN__FRANGE__DRESTRICT__suff,axiom,
! [A_27a: $tType,A_27b: $tType,V0s: A_27b > $o,V1fm: tyop_2Efinite__map_2Efmap @ A_27b @ A_27a,V2P: A_27a > $o] :
( ! [V3v: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V3v @ ( c_2Efinite__map_2EFRANGE @ A_27b @ A_27a @ V1fm ) )
=> ( V2P @ V3v ) )
=> ! [V4v: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V4v @ ( c_2Efinite__map_2EFRANGE @ A_27b @ A_27a @ ( c_2Efinite__map_2EDRESTRICT @ A_27b @ A_27a @ V1fm @ V0s ) ) )
=> ( V2P @ V4v ) ) ) ).
thf(thm_2Efinite__map_2EFUNION__ASSOC,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b,V1g: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b,V2h: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b] :
( ( c_2Efinite__map_2EFUNION @ A_27a @ A_27b @ V0f @ ( c_2Efinite__map_2EFUNION @ A_27a @ A_27b @ V1g @ V2h ) )
= ( c_2Efinite__map_2EFUNION @ A_27a @ A_27b @ ( c_2Efinite__map_2EFUNION @ A_27a @ A_27b @ V0f @ V1g ) @ V2h ) ) ).
thf(thm_2Efinite__map_2EFUNION__COMM,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b,V1g: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b] :
( ( c_2Epred__set_2EDISJOINT @ A_27a @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b @ V0f ) @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b @ V1g ) )
=> ( ( c_2Efinite__map_2EFUNION @ A_27a @ A_27b @ V0f @ V1g )
= ( c_2Efinite__map_2EFUNION @ A_27a @ A_27b @ V1g @ V0f ) ) ) ).
thf(thm_2Efinite__map_2EDOMSUB__FUNION,axiom,
! [A_27a: $tType,A_27b: $tType,V0k: A_27a,V1g: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b,V2f: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b] :
( ( c_2Efinite__map_2Efdomsub @ A_27a @ A_27b @ ( c_2Efinite__map_2EFUNION @ A_27a @ A_27b @ V2f @ V1g ) @ V0k )
= ( c_2Efinite__map_2EFUNION @ A_27a @ A_27b @ ( c_2Efinite__map_2Efdomsub @ A_27a @ A_27b @ V2f @ V0k ) @ ( c_2Efinite__map_2Efdomsub @ A_27a @ A_27b @ V1g @ V0k ) ) ) ).
thf(thm_2Efinite__map_2EFUNION__EQ__IMPL,axiom,
! [A_27a: $tType,A_27b: $tType,V0f1: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b,V1f2: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b,V2f3: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b] :
( ( ( c_2Epred__set_2EDISJOINT @ A_27a @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b @ V0f1 ) @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b @ V1f2 ) )
& ( c_2Epred__set_2EDISJOINT @ A_27a @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b @ V0f1 ) @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b @ V2f3 ) )
& ( V1f2 = V2f3 ) )
=> ( ( c_2Efinite__map_2EFUNION @ A_27a @ A_27b @ V0f1 @ V1f2 )
= ( c_2Efinite__map_2EFUNION @ A_27a @ A_27b @ V0f1 @ V2f3 ) ) ) ).
thf(thm_2Efinite__map_2EFUNION__EQ,axiom,
! [A_27a: $tType,A_27b: $tType,V0f1: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b,V1f2: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b,V2f3: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b] :
( ( ( c_2Epred__set_2EDISJOINT @ A_27a @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b @ V0f1 ) @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b @ V1f2 ) )
& ( c_2Epred__set_2EDISJOINT @ A_27a @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b @ V0f1 ) @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b @ V2f3 ) ) )
=> ( ( ( c_2Efinite__map_2EFUNION @ A_27a @ A_27b @ V0f1 @ V1f2 )
= ( c_2Efinite__map_2EFUNION @ A_27a @ A_27b @ V0f1 @ V2f3 ) )
<=> ( V1f2 = V2f3 ) ) ) ).
thf(thm_2Efinite__map_2EFEMPTY__SUBMAP,axiom,
! [A_27a: $tType,A_27b: $tType,V0h: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b] :
( ( c_2Efinite__map_2ESUBMAP @ A_27a @ A_27b @ V0h @ ( c_2Efinite__map_2EFEMPTY @ A_27a @ A_27b ) )
<=> ( V0h
= ( c_2Efinite__map_2EFEMPTY @ A_27a @ A_27b ) ) ) ).
thf(thm_2Efinite__map_2ESUBMAP__FUNION__ID,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,A_27d: $tType] :
( ! [V0f1: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b,V1f2: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b] : ( c_2Efinite__map_2ESUBMAP @ A_27a @ A_27b @ V0f1 @ ( c_2Efinite__map_2EFUNION @ A_27a @ A_27b @ V0f1 @ V1f2 ) )
& ! [V2f1: tyop_2Efinite__map_2Efmap @ A_27c @ A_27d,V3f2: tyop_2Efinite__map_2Efmap @ A_27c @ A_27d] :
( ( c_2Epred__set_2EDISJOINT @ A_27c @ ( c_2Efinite__map_2EFDOM @ A_27c @ A_27d @ V2f1 ) @ ( c_2Efinite__map_2EFDOM @ A_27c @ A_27d @ V3f2 ) )
=> ( c_2Efinite__map_2ESUBMAP @ A_27c @ A_27d @ V3f2 @ ( c_2Efinite__map_2EFUNION @ A_27c @ A_27d @ V2f1 @ V3f2 ) ) ) ) ).
thf(thm_2Efinite__map_2ESUBMAP__FUNION,axiom,
! [A_27a: $tType,A_27b: $tType,V0f1: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b,V1f2: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b,V2f3: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b] :
( ( ( c_2Efinite__map_2ESUBMAP @ A_27a @ A_27b @ V0f1 @ V1f2 )
| ( ( c_2Epred__set_2EDISJOINT @ A_27a @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b @ V0f1 ) @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b @ V1f2 ) )
& ( c_2Efinite__map_2ESUBMAP @ A_27a @ A_27b @ V0f1 @ V2f3 ) ) )
=> ( c_2Efinite__map_2ESUBMAP @ A_27a @ A_27b @ V0f1 @ ( c_2Efinite__map_2EFUNION @ A_27a @ A_27b @ V1f2 @ V2f3 ) ) ) ).
thf(thm_2Efinite__map_2ESUBMAP__FUNION__EQ,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,A_27d: $tType] :
( ! [V0f1: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b,V1f2: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b,V2f3: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b] :
( ( c_2Epred__set_2EDISJOINT @ A_27a @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b @ V0f1 ) @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b @ V1f2 ) )
=> ( ( c_2Efinite__map_2ESUBMAP @ A_27a @ A_27b @ V0f1 @ ( c_2Efinite__map_2EFUNION @ A_27a @ A_27b @ V1f2 @ V2f3 ) )
= ( c_2Efinite__map_2ESUBMAP @ A_27a @ A_27b @ V0f1 @ V2f3 ) ) )
& ! [V3f1: tyop_2Efinite__map_2Efmap @ A_27c @ A_27d,V4f2: tyop_2Efinite__map_2Efmap @ A_27c @ A_27d,V5f3: tyop_2Efinite__map_2Efmap @ A_27c @ A_27d] :
( ( c_2Epred__set_2EDISJOINT @ A_27c @ ( c_2Efinite__map_2EFDOM @ A_27c @ A_27d @ V3f1 ) @ ( c_2Epred__set_2EDIFF @ A_27c @ ( c_2Efinite__map_2EFDOM @ A_27c @ A_27d @ V5f3 ) @ ( c_2Efinite__map_2EFDOM @ A_27c @ A_27d @ V4f2 ) ) )
=> ( ( c_2Efinite__map_2ESUBMAP @ A_27c @ A_27d @ V3f1 @ ( c_2Efinite__map_2EFUNION @ A_27c @ A_27d @ V4f2 @ V5f3 ) )
= ( c_2Efinite__map_2ESUBMAP @ A_27c @ A_27d @ V3f1 @ V4f2 ) ) ) ) ).
thf(thm_2Efinite__map_2EFUNION__EQ__FEMPTY,axiom,
! [A_27a: $tType,A_27b: $tType,V0h1: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b,V1h2: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b] :
( ( ( c_2Efinite__map_2EFUNION @ A_27a @ A_27b @ V0h1 @ V1h2 )
= ( c_2Efinite__map_2EFEMPTY @ A_27a @ A_27b ) )
<=> ( ( V0h1
= ( c_2Efinite__map_2EFEMPTY @ A_27a @ A_27b ) )
& ( V1h2
= ( c_2Efinite__map_2EFEMPTY @ A_27a @ A_27b ) ) ) ) ).
thf(thm_2Efinite__map_2EFEVERY__DRESTRICT__COMPL,axiom,
! [A_27a: $tType,A_27b: $tType,V0v: A_27b,V1s: A_27a > $o,V2k: A_27a,V3f: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b,V4P: ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) > $o] :
( ( c_2Efinite__map_2EFEVERY @ A_27a @ A_27b @ V4P @ ( c_2Efinite__map_2EDRESTRICT @ A_27a @ A_27b @ ( c_2Efinite__map_2EFUPDATE @ A_27a @ A_27b @ V3f @ ( c_2Epair_2E_2C @ A_27a @ A_27b @ V2k @ V0v ) ) @ ( c_2Epred__set_2ECOMPL @ A_27a @ V1s ) ) )
<=> ( ( ( (~) @ ( c_2Ebool_2EIN @ A_27a @ V2k @ V1s ) )
=> ( V4P @ ( c_2Epair_2E_2C @ A_27a @ A_27b @ V2k @ V0v ) ) )
& ( c_2Efinite__map_2EFEVERY @ A_27a @ A_27b @ V4P @ ( c_2Efinite__map_2EDRESTRICT @ A_27a @ A_27b @ V3f @ ( c_2Epred__set_2ECOMPL @ A_27a @ ( c_2Epred__set_2EINSERT @ A_27a @ V2k @ V1s ) ) ) ) ) ) ).
thf(thm_2Efinite__map_2EFUPDATE__ELIM,axiom,
! [A_27a: $tType,A_27b: $tType,V0k: A_27a,V1v: A_27b,V2f: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b] :
( ( ( c_2Ebool_2EIN @ A_27a @ V0k @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b @ V2f ) )
& ( ( c_2Efinite__map_2EFAPPLY @ A_27a @ A_27b @ V2f @ V0k )
= V1v ) )
=> ( ( c_2Efinite__map_2EFUPDATE @ A_27a @ A_27b @ V2f @ ( c_2Epair_2E_2C @ A_27a @ A_27b @ V0k @ V1v ) )
= V2f ) ) ).
thf(thm_2Efinite__map_2EFEVERY__STRENGTHEN__THM,axiom,
! [A_27a: $tType,A_27b: $tType,V0y: A_27b,V1x: A_27a,V2f: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b,V3P: ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) > $o] :
( ( c_2Efinite__map_2EFEVERY @ A_27a @ A_27b @ V3P @ ( c_2Efinite__map_2EFEMPTY @ A_27a @ A_27b ) )
& ( ( ( c_2Efinite__map_2EFEVERY @ A_27a @ A_27b @ V3P @ V2f )
& ( V3P @ ( c_2Epair_2E_2C @ A_27a @ A_27b @ V1x @ V0y ) ) )
=> ( c_2Efinite__map_2EFEVERY @ A_27a @ A_27b @ V3P @ ( c_2Efinite__map_2EFUPDATE @ A_27a @ A_27b @ V2f @ ( c_2Epair_2E_2C @ A_27a @ A_27b @ V1x @ V0y ) ) ) ) ) ).
thf(thm_2Efinite__map_2EFMAP__MAP2__FUPDATE,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0x: A_27a,V1v: A_27c,V2m: tyop_2Efinite__map_2Efmap @ A_27a @ A_27c,V3f: ( tyop_2Epair_2Eprod @ A_27a @ A_27c ) > A_27b] :
( ( c_2Efinite__map_2EFMAP__MAP2 @ A_27a @ A_27b @ A_27c @ V3f @ ( c_2Efinite__map_2EFUPDATE @ A_27a @ A_27c @ V2m @ ( c_2Epair_2E_2C @ A_27a @ A_27c @ V0x @ V1v ) ) )
= ( c_2Efinite__map_2EFUPDATE @ A_27a @ A_27b @ ( c_2Efinite__map_2EFMAP__MAP2 @ A_27a @ A_27b @ A_27c @ V3f @ V2m ) @ ( c_2Epair_2E_2C @ A_27a @ A_27b @ V0x @ ( V3f @ ( c_2Epair_2E_2C @ A_27a @ A_27c @ V0x @ V1v ) ) ) ) ) ).
thf(thm_2Efinite__map_2EFMAP__MAP2__FEMPTY,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0f: ( tyop_2Epair_2Eprod @ A_27a @ A_27c ) > A_27b] :
( ( c_2Efinite__map_2EFMAP__MAP2 @ A_27a @ A_27b @ A_27c @ V0f @ ( c_2Efinite__map_2EFEMPTY @ A_27a @ A_27c ) )
= ( c_2Efinite__map_2EFEMPTY @ A_27a @ A_27b ) ) ).
thf(thm_2Efinite__map_2EFMAP__MAP2__THM,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0m: tyop_2Efinite__map_2Efmap @ A_27a @ A_27c,V1f: ( tyop_2Epair_2Eprod @ A_27a @ A_27c ) > A_27b] :
( ( ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b @ ( c_2Efinite__map_2EFMAP__MAP2 @ A_27a @ A_27b @ A_27c @ V1f @ V0m ) )
= ( c_2Efinite__map_2EFDOM @ A_27a @ A_27c @ V0m ) )
& ! [V2x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V2x @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27c @ V0m ) )
=> ( ( c_2Efinite__map_2EFAPPLY @ A_27a @ A_27b @ ( c_2Efinite__map_2EFMAP__MAP2 @ A_27a @ A_27b @ A_27c @ V1f @ V0m ) @ V2x )
= ( V1f @ ( c_2Epair_2E_2C @ A_27a @ A_27c @ V2x @ ( c_2Efinite__map_2EFAPPLY @ A_27a @ A_27c @ V0m @ V2x ) ) ) ) ) ) ).
thf(thm_2Efinite__map_2EFUPDATE__PURGE,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b,V1x: A_27a,V2y: A_27b] :
( ( c_2Efinite__map_2EFUPDATE @ A_27a @ A_27b @ V0f @ ( c_2Epair_2E_2C @ A_27a @ A_27b @ V1x @ V2y ) )
= ( c_2Efinite__map_2EFUPDATE @ A_27a @ A_27b @ ( c_2Efinite__map_2Efdomsub @ A_27a @ A_27b @ V0f @ V1x ) @ ( c_2Epair_2E_2C @ A_27a @ A_27b @ V1x @ V2y ) ) ) ).
thf(thm_2Efinite__map_2EFMEQ__SINGLE__SIMPLE__DISJ__ELIM,axiom,
! [A_27a: $tType,A_27b: $tType,V0fm: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b,V1k: A_27a,V2v: A_27b,V3ck: A_27a,V4cv: A_27b] :
( ( ( c_2Efinite__map_2EFUPDATE @ A_27a @ A_27b @ V0fm @ ( c_2Epair_2E_2C @ A_27a @ A_27b @ V1k @ V2v ) )
= ( c_2Efinite__map_2EFUPDATE @ A_27a @ A_27b @ ( c_2Efinite__map_2EFEMPTY @ A_27a @ A_27b ) @ ( c_2Epair_2E_2C @ A_27a @ A_27b @ V3ck @ V4cv ) ) )
<=> ( ( V1k = V3ck )
& ( V2v = V4cv )
& ( ( V0fm
= ( c_2Efinite__map_2EFEMPTY @ A_27a @ A_27b ) )
| ? [V5v_27: A_27b] :
( V0fm
= ( c_2Efinite__map_2EFUPDATE @ A_27a @ A_27b @ ( c_2Efinite__map_2EFEMPTY @ A_27a @ A_27b ) @ ( c_2Epair_2E_2C @ A_27a @ A_27b @ V1k @ V5v_27 ) ) ) ) ) ) ).
thf(thm_2Efinite__map_2EFMEQ__SINGLE__SIMPLE__ELIM,axiom,
! [A_27a: $tType,A_27b: $tType,V0P: ( tyop_2Efinite__map_2Efmap @ A_27a @ A_27b ) > $o,V1k: A_27a,V2v: A_27b,V3ck: A_27a,V4cv: A_27b,V5nv: A_27b] :
( ? [V6fm: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b] :
( ( ( c_2Efinite__map_2EFUPDATE @ A_27a @ A_27b @ V6fm @ ( c_2Epair_2E_2C @ A_27a @ A_27b @ V1k @ V2v ) )
= ( c_2Efinite__map_2EFUPDATE @ A_27a @ A_27b @ ( c_2Efinite__map_2EFEMPTY @ A_27a @ A_27b ) @ ( c_2Epair_2E_2C @ A_27a @ A_27b @ V3ck @ V4cv ) ) )
& ( V0P @ ( c_2Efinite__map_2EFUPDATE @ A_27a @ A_27b @ V6fm @ ( c_2Epair_2E_2C @ A_27a @ A_27b @ V1k @ V5nv ) ) ) )
<=> ( ( V1k = V3ck )
& ( V2v = V4cv )
& ( V0P @ ( c_2Efinite__map_2EFUPDATE @ A_27a @ A_27b @ ( c_2Efinite__map_2EFEMPTY @ A_27a @ A_27b ) @ ( c_2Epair_2E_2C @ A_27a @ A_27b @ V3ck @ V5nv ) ) ) ) ) ).
thf(thm_2Efinite__map_2EFMEQ__ENUMERATE__CASES,axiom,
! [A_27a: $tType,A_27b: $tType,V0f1: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b,V1kvl: tyop_2Elist_2Elist @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ),V2p: tyop_2Epair_2Eprod @ A_27a @ A_27b] :
( ( ( c_2Efinite__map_2EFUPDATE @ A_27a @ A_27b @ V0f1 @ V2p )
= ( c_2Efinite__map_2EFUPDATE__LIST @ A_27a @ A_27b @ ( c_2Efinite__map_2EFEMPTY @ A_27a @ A_27b ) @ V1kvl ) )
=> ( c_2Ebool_2EIN @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ V2p @ ( c_2Elist_2ELIST__TO__SET @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ V1kvl ) ) ) ).
thf(thm_2Efinite__map_2EFUPDATE__LIST__SAME__KEYS__UNWIND,axiom,
! [A_27a: $tType,A_27b: $tType,V0f1: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b,V1f2: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b,V2kvl1: tyop_2Elist_2Elist @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ),V3kvl2: tyop_2Elist_2Elist @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b )] :
( ( ( ( c_2Efinite__map_2EFUPDATE__LIST @ A_27a @ A_27b @ V0f1 @ V2kvl1 )
= ( c_2Efinite__map_2EFUPDATE__LIST @ A_27a @ A_27b @ V1f2 @ V3kvl2 ) )
& ( ( c_2Elist_2EMAP @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ A_27a @ ( c_2Epair_2EFST @ A_27a @ A_27b ) @ V2kvl1 )
= ( c_2Elist_2EMAP @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ A_27a @ ( c_2Epair_2EFST @ A_27a @ A_27b ) @ V3kvl2 ) )
& ( c_2Elist_2EALL__DISTINCT @ A_27a @ ( c_2Elist_2EMAP @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ A_27a @ ( c_2Epair_2EFST @ A_27a @ A_27b ) @ V2kvl1 ) ) )
=> ( ( V2kvl1 = V3kvl2 )
& ! [V4kvl: tyop_2Elist_2Elist @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b )] :
( ( ( c_2Elist_2EMAP @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ A_27a @ ( c_2Epair_2EFST @ A_27a @ A_27b ) @ V4kvl )
= ( c_2Elist_2EMAP @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ A_27a @ ( c_2Epair_2EFST @ A_27a @ A_27b ) @ V2kvl1 ) )
=> ( ( c_2Efinite__map_2EFUPDATE__LIST @ A_27a @ A_27b @ V0f1 @ V4kvl )
= ( c_2Efinite__map_2EFUPDATE__LIST @ A_27a @ A_27b @ V1f2 @ V4kvl ) ) ) ) ) ).
thf(thm_2Efinite__map_2EFUPDATE__LIST__SAME__UPDATE,axiom,
! [A_27a: $tType,A_27b: $tType,V0kvl: tyop_2Elist_2Elist @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ),V1f1: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b,V2f2: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b] :
( ( ( c_2Efinite__map_2EFUPDATE__LIST @ A_27a @ A_27b @ V1f1 @ V0kvl )
= ( c_2Efinite__map_2EFUPDATE__LIST @ A_27a @ A_27b @ V2f2 @ V0kvl ) )
<=> ( ( c_2Efinite__map_2EDRESTRICT @ A_27a @ A_27b @ V1f1 @ ( c_2Epred__set_2ECOMPL @ A_27a @ ( c_2Elist_2ELIST__TO__SET @ A_27a @ ( c_2Elist_2EMAP @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ A_27a @ ( c_2Epair_2EFST @ A_27a @ A_27b ) @ V0kvl ) ) ) )
= ( c_2Efinite__map_2EDRESTRICT @ A_27a @ A_27b @ V2f2 @ ( c_2Epred__set_2ECOMPL @ A_27a @ ( c_2Elist_2ELIST__TO__SET @ A_27a @ ( c_2Elist_2EMAP @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ A_27a @ ( c_2Epair_2EFST @ A_27a @ A_27b ) @ V0kvl ) ) ) ) ) ) ).
thf(thm_2Efinite__map_2EFDOM__FUPDATE__LIST,axiom,
! [A_27a: $tType,A_27b: $tType,V0kvl: tyop_2Elist_2Elist @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ),V1fm: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b] :
( ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b @ ( c_2Efinite__map_2EFUPDATE__LIST @ A_27a @ A_27b @ V1fm @ V0kvl ) )
= ( c_2Epred__set_2EUNION @ A_27a @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b @ V1fm ) @ ( c_2Elist_2ELIST__TO__SET @ A_27a @ ( c_2Elist_2EMAP @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ A_27a @ ( c_2Epair_2EFST @ A_27a @ A_27b ) @ V0kvl ) ) ) ) ).
thf(thm_2Efinite__map_2EFUPD11__SAME__UPDATE,axiom,
! [A_27a: $tType,A_27b: $tType,V0f1: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b,V1f2: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b,V2k: A_27a,V3v: A_27b] :
( ( ( c_2Efinite__map_2EFUPDATE @ A_27a @ A_27b @ V0f1 @ ( c_2Epair_2E_2C @ A_27a @ A_27b @ V2k @ V3v ) )
= ( c_2Efinite__map_2EFUPDATE @ A_27a @ A_27b @ V1f2 @ ( c_2Epair_2E_2C @ A_27a @ A_27b @ V2k @ V3v ) ) )
<=> ( ( c_2Efinite__map_2EDRESTRICT @ A_27a @ A_27b @ V0f1 @ ( c_2Epred__set_2ECOMPL @ A_27a @ ( c_2Epred__set_2EINSERT @ A_27a @ V2k @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) ) )
= ( c_2Efinite__map_2EDRESTRICT @ A_27a @ A_27b @ V1f2 @ ( c_2Epred__set_2ECOMPL @ A_27a @ ( c_2Epred__set_2EINSERT @ A_27a @ V2k @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) ) ) ) ) ).
thf(thm_2Efinite__map_2EFUPD__SAME__KEY__UNWIND,axiom,
! [A_27a: $tType,A_27b: $tType,V0f1: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b,V1f2: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b,V2k: A_27a,V3v1: A_27b,V4v2: A_27b] :
( ( ( c_2Efinite__map_2EFUPDATE @ A_27a @ A_27b @ V0f1 @ ( c_2Epair_2E_2C @ A_27a @ A_27b @ V2k @ V3v1 ) )
= ( c_2Efinite__map_2EFUPDATE @ A_27a @ A_27b @ V1f2 @ ( c_2Epair_2E_2C @ A_27a @ A_27b @ V2k @ V4v2 ) ) )
=> ( ( V3v1 = V4v2 )
& ! [V5v: A_27b] :
( ( c_2Efinite__map_2EFUPDATE @ A_27a @ A_27b @ V0f1 @ ( c_2Epair_2E_2C @ A_27a @ A_27b @ V2k @ V5v ) )
= ( c_2Efinite__map_2EFUPDATE @ A_27a @ A_27b @ V1f2 @ ( c_2Epair_2E_2C @ A_27a @ A_27b @ V2k @ V5v ) ) ) ) ) ).
thf(thm_2Efinite__map_2EFUPD11__SAME__BASE,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b,V1k1: A_27a,V2v1: A_27b,V3k2: A_27a,V4v2: A_27b] :
( ( ( c_2Efinite__map_2EFUPDATE @ A_27a @ A_27b @ V0f @ ( c_2Epair_2E_2C @ A_27a @ A_27b @ V1k1 @ V2v1 ) )
= ( c_2Efinite__map_2EFUPDATE @ A_27a @ A_27b @ V0f @ ( c_2Epair_2E_2C @ A_27a @ A_27b @ V3k2 @ V4v2 ) ) )
<=> ( ( ( V1k1 = V3k2 )
& ( V2v1 = V4v2 ) )
| ( ( (~) @ ( V1k1 = V3k2 ) )
& ( c_2Ebool_2EIN @ A_27a @ V1k1 @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b @ V0f ) )
& ( c_2Ebool_2EIN @ A_27a @ V3k2 @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b @ V0f ) )
& ( ( c_2Efinite__map_2EFUPDATE @ A_27a @ A_27b @ V0f @ ( c_2Epair_2E_2C @ A_27a @ A_27b @ V1k1 @ V2v1 ) )
= V0f )
& ( ( c_2Efinite__map_2EFUPDATE @ A_27a @ A_27b @ V0f @ ( c_2Epair_2E_2C @ A_27a @ A_27b @ V3k2 @ V4v2 ) )
= V0f ) ) ) ) ).
thf(thm_2Efinite__map_2ESAME__KEY__UPDATES__DIFFER,axiom,
! [A_27a: $tType,A_27b: $tType,V0f1: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b,V1f2: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b,V2k: A_27a,V3v1: A_27b,V4v2: A_27b] :
( ( (~) @ ( V3v1 = V4v2 ) )
=> ( (~)
@ ( ( c_2Efinite__map_2EFUPDATE @ A_27a @ A_27b @ V0f1 @ ( c_2Epair_2E_2C @ A_27a @ A_27b @ V2k @ V3v1 ) )
= ( c_2Efinite__map_2EFUPDATE @ A_27a @ A_27b @ V1f2 @ ( c_2Epair_2E_2C @ A_27a @ A_27b @ V2k @ V4v2 ) ) ) ) ) ).
thf(thm_2Efinite__map_2EFUPD11__SAME__NEW__KEY,axiom,
! [A_27a: $tType,A_27b: $tType,V0f1: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b,V1f2: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b,V2k: A_27a,V3v1: A_27b,V4v2: A_27b] :
( ( ( (~) @ ( c_2Ebool_2EIN @ A_27a @ V2k @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b @ V0f1 ) ) )
& ( (~) @ ( c_2Ebool_2EIN @ A_27a @ V2k @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b @ V1f2 ) ) ) )
=> ( ( ( c_2Efinite__map_2EFUPDATE @ A_27a @ A_27b @ V0f1 @ ( c_2Epair_2E_2C @ A_27a @ A_27b @ V2k @ V3v1 ) )
= ( c_2Efinite__map_2EFUPDATE @ A_27a @ A_27b @ V1f2 @ ( c_2Epair_2E_2C @ A_27a @ A_27b @ V2k @ V4v2 ) ) )
<=> ( ( V0f1 = V1f2 )
& ( V3v1 = V4v2 ) ) ) ) ).
thf(thm_2Efinite__map_2EFUPD11__SAME__KEY__AND__BASE,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b,V1k: A_27a,V2v1: A_27b,V3v2: A_27b] :
( ( ( c_2Efinite__map_2EFUPDATE @ A_27a @ A_27b @ V0f @ ( c_2Epair_2E_2C @ A_27a @ A_27b @ V1k @ V2v1 ) )
= ( c_2Efinite__map_2EFUPDATE @ A_27a @ A_27b @ V0f @ ( c_2Epair_2E_2C @ A_27a @ A_27b @ V1k @ V3v2 ) ) )
<=> ( V2v1 = V3v2 ) ) ).
thf(thm_2Efinite__map_2EFUPDATE__LIST__SNOC,axiom,
! [A_27a: $tType,A_27b: $tType,V0xs: tyop_2Elist_2Elist @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ),V1x: tyop_2Epair_2Eprod @ A_27a @ A_27b,V2fm: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b] :
( ( c_2Efinite__map_2EFUPDATE__LIST @ A_27a @ A_27b @ V2fm @ ( c_2Elist_2ESNOC @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ V1x @ V0xs ) )
= ( c_2Efinite__map_2EFUPDATE @ A_27a @ A_27b @ ( c_2Efinite__map_2EFUPDATE__LIST @ A_27a @ A_27b @ V2fm @ V0xs ) @ V1x ) ) ).
thf(thm_2Efinite__map_2EFOLDL__FUPDATE__LIST,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0f1: A_27a > A_27b,V1f2: A_27a > A_27c,V2ls: tyop_2Elist_2Elist @ A_27a,V3a: tyop_2Efinite__map_2Efmap @ A_27b @ A_27c] :
( ( c_2Elist_2EFOLDL @ A_27a @ ( tyop_2Efinite__map_2Efmap @ A_27b @ A_27c )
@ ^ [V4fm: tyop_2Efinite__map_2Efmap @ A_27b @ A_27c,V5k: A_27a] : ( c_2Efinite__map_2EFUPDATE @ A_27b @ A_27c @ V4fm @ ( c_2Epair_2E_2C @ A_27b @ A_27c @ ( V0f1 @ V5k ) @ ( V1f2 @ V5k ) ) )
@ V3a
@ V2ls )
= ( c_2Efinite__map_2EFUPDATE__LIST @ A_27b @ A_27c @ V3a
@ ( c_2Elist_2EMAP @ A_27a @ ( tyop_2Epair_2Eprod @ A_27b @ A_27c )
@ ^ [V6k: A_27a] : ( c_2Epair_2E_2C @ A_27b @ A_27c @ ( V0f1 @ V6k ) @ ( V1f2 @ V6k ) )
@ V2ls ) ) ) ).
thf(thm_2Efinite__map_2EFUPDATE__LIST__APPLY__MEM,axiom,
! [A_27a: $tType,A_27b: $tType,V0kvl: tyop_2Elist_2Elist @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ),V1f: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b,V2k: A_27a,V3v: A_27b,V4n: tyop_2Enum_2Enum] :
( ( ( c_2Eprim__rec_2E_3C @ V4n @ ( c_2Elist_2ELENGTH @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ V0kvl ) )
& ( V2k
= ( c_2Elist_2EEL @ A_27a @ V4n @ ( c_2Elist_2EMAP @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ A_27a @ ( c_2Epair_2EFST @ A_27a @ A_27b ) @ V0kvl ) ) )
& ( V3v
= ( c_2Elist_2EEL @ A_27b @ V4n @ ( c_2Elist_2EMAP @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ A_27b @ ( c_2Epair_2ESND @ A_27a @ A_27b ) @ V0kvl ) ) )
& ! [V5m: tyop_2Enum_2Enum] :
( ( ( c_2Eprim__rec_2E_3C @ V4n @ V5m )
& ( c_2Eprim__rec_2E_3C @ V5m @ ( c_2Elist_2ELENGTH @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ V0kvl ) ) )
=> ( (~)
@ ( ( c_2Elist_2EEL @ A_27a @ V5m @ ( c_2Elist_2EMAP @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ A_27a @ ( c_2Epair_2EFST @ A_27a @ A_27b ) @ V0kvl ) )
= V2k ) ) ) )
=> ( ( c_2Efinite__map_2EFAPPLY @ A_27a @ A_27b @ ( c_2Efinite__map_2EFUPDATE__LIST @ A_27a @ A_27b @ V1f @ V0kvl ) @ V2k )
= V3v ) ) ).
thf(thm_2Efinite__map_2EFEVERY__FUPDATE__LIST,axiom,
! [A_27a: $tType,A_27b: $tType,V0kvl: tyop_2Elist_2Elist @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ),V1fm: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b,V2P: ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) > $o] :
( ( c_2Elist_2EALL__DISTINCT @ A_27a @ ( c_2Elist_2EMAP @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ A_27a @ ( c_2Epair_2EFST @ A_27a @ A_27b ) @ V0kvl ) )
=> ( ( c_2Efinite__map_2EFEVERY @ A_27a @ A_27b @ V2P @ ( c_2Efinite__map_2EFUPDATE__LIST @ A_27a @ A_27b @ V1fm @ V0kvl ) )
<=> ( ( c_2Elist_2EEVERY @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ V2P @ V0kvl )
& ( c_2Efinite__map_2EFEVERY @ A_27a @ A_27b @ V2P @ ( c_2Efinite__map_2EDRESTRICT @ A_27a @ A_27b @ V1fm @ ( c_2Epred__set_2ECOMPL @ A_27a @ ( c_2Elist_2ELIST__TO__SET @ A_27a @ ( c_2Elist_2EMAP @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ A_27a @ ( c_2Epair_2EFST @ A_27a @ A_27b ) @ V0kvl ) ) ) ) ) ) ) ) ).
thf(thm_2Efinite__map_2EFUPDATE__FUPDATE__LIST__MEM,axiom,
! [A_27a: $tType,A_27b: $tType,V0v: A_27b,V1kvl: tyop_2Elist_2Elist @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ),V2k: A_27a,V3fm: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b] :
( ( c_2Ebool_2EIN @ A_27a @ V2k @ ( c_2Elist_2ELIST__TO__SET @ A_27a @ ( c_2Elist_2EMAP @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ A_27a @ ( c_2Epair_2EFST @ A_27a @ A_27b ) @ V1kvl ) ) )
=> ( ( c_2Efinite__map_2EFUPDATE__LIST @ A_27a @ A_27b @ ( c_2Efinite__map_2EFUPDATE @ A_27a @ A_27b @ V3fm @ ( c_2Epair_2E_2C @ A_27a @ A_27b @ V2k @ V0v ) ) @ V1kvl )
= ( c_2Efinite__map_2EFUPDATE__LIST @ A_27a @ A_27b @ V3fm @ V1kvl ) ) ) ).
thf(thm_2Efinite__map_2EFUPDATE__FUPDATE__LIST__COMMUTES,axiom,
! [A_27a: $tType,A_27b: $tType,V0v: A_27b,V1kvl: tyop_2Elist_2Elist @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ),V2k: A_27a,V3fm: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b] :
( ( (~) @ ( c_2Ebool_2EIN @ A_27a @ V2k @ ( c_2Elist_2ELIST__TO__SET @ A_27a @ ( c_2Elist_2EMAP @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ A_27a @ ( c_2Epair_2EFST @ A_27a @ A_27b ) @ V1kvl ) ) ) )
=> ( ( c_2Efinite__map_2EFUPDATE__LIST @ A_27a @ A_27b @ ( c_2Efinite__map_2EFUPDATE @ A_27a @ A_27b @ V3fm @ ( c_2Epair_2E_2C @ A_27a @ A_27b @ V2k @ V0v ) ) @ V1kvl )
= ( c_2Efinite__map_2EFUPDATE @ A_27a @ A_27b @ ( c_2Efinite__map_2EFUPDATE__LIST @ A_27a @ A_27b @ V3fm @ V1kvl ) @ ( c_2Epair_2E_2C @ A_27a @ A_27b @ V2k @ V0v ) ) ) ) ).
thf(thm_2Efinite__map_2EFUPDATE__LIST__APPEND,axiom,
! [A_27a: $tType,A_27b: $tType,V0kvl2: tyop_2Elist_2Elist @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ),V1kvl1: tyop_2Elist_2Elist @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ),V2fm: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b] :
( ( c_2Efinite__map_2EFUPDATE__LIST @ A_27a @ A_27b @ V2fm @ ( c_2Elist_2EAPPEND @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ V1kvl1 @ V0kvl2 ) )
= ( c_2Efinite__map_2EFUPDATE__LIST @ A_27a @ A_27b @ ( c_2Efinite__map_2EFUPDATE__LIST @ A_27a @ A_27b @ V2fm @ V1kvl1 ) @ V0kvl2 ) ) ).
thf(thm_2Efinite__map_2EFUPDATE__LIST__APPLY__NOT__MEM,axiom,
! [A_27a: $tType,A_27b: $tType,V0kvl: tyop_2Elist_2Elist @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ),V1f: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b,V2k: A_27a] :
( ( (~) @ ( c_2Ebool_2EIN @ A_27a @ V2k @ ( c_2Elist_2ELIST__TO__SET @ A_27a @ ( c_2Elist_2EMAP @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ A_27a @ ( c_2Epair_2EFST @ A_27a @ A_27b ) @ V0kvl ) ) ) )
=> ( ( c_2Efinite__map_2EFAPPLY @ A_27a @ A_27b @ ( c_2Efinite__map_2EFUPDATE__LIST @ A_27a @ A_27b @ V1f @ V0kvl ) @ V2k )
= ( c_2Efinite__map_2EFAPPLY @ A_27a @ A_27b @ V1f @ V2k ) ) ) ).
thf(thm_2Efinite__map_2EFUPDATE__LIST__THM,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b] :
( ( ( c_2Efinite__map_2EFUPDATE__LIST @ A_27a @ A_27b @ V0f @ ( c_2Elist_2ENIL @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) ) )
= V0f )
& ! [V1h: tyop_2Epair_2Eprod @ A_27a @ A_27b,V2t: tyop_2Elist_2Elist @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b )] :
( ( c_2Efinite__map_2EFUPDATE__LIST @ A_27a @ A_27b @ V0f @ ( c_2Elist_2ECONS @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ V1h @ V2t ) )
= ( c_2Efinite__map_2EFUPDATE__LIST @ A_27a @ A_27b @ ( c_2Efinite__map_2EFUPDATE @ A_27a @ A_27b @ V0f @ V1h ) @ V2t ) ) ) ).
thf(thm_2Efinite__map_2ESUBMAP__FUPDATE,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b,V1g: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b,V2x: A_27a,V3y: A_27b] :
( ( c_2Efinite__map_2ESUBMAP @ A_27a @ A_27b @ ( c_2Efinite__map_2EFUPDATE @ A_27a @ A_27b @ V0f @ ( c_2Epair_2E_2C @ A_27a @ A_27b @ V2x @ V3y ) ) @ V1g )
<=> ( ( c_2Ebool_2EIN @ A_27a @ V2x @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b @ V1g ) )
& ( ( c_2Efinite__map_2EFAPPLY @ A_27a @ A_27b @ V1g @ V2x )
= V3y )
& ( c_2Efinite__map_2ESUBMAP @ A_27a @ A_27b @ ( c_2Efinite__map_2Efdomsub @ A_27a @ A_27b @ V0f @ V2x ) @ ( c_2Efinite__map_2Efdomsub @ A_27a @ A_27b @ V1g @ V2x ) ) ) ) ).
thf(thm_2Efinite__map_2ESUBMAP__TRANS,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b,V1g: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b,V2h: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b] :
( ( ( c_2Efinite__map_2ESUBMAP @ A_27a @ A_27b @ V0f @ V1g )
& ( c_2Efinite__map_2ESUBMAP @ A_27a @ A_27b @ V1g @ V2h ) )
=> ( c_2Efinite__map_2ESUBMAP @ A_27a @ A_27b @ V0f @ V2h ) ) ).
thf(thm_2Efinite__map_2ESUBMAP__ANTISYM,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b,V1g: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b] :
( ( ( c_2Efinite__map_2ESUBMAP @ A_27a @ A_27b @ V0f @ V1g )
& ( c_2Efinite__map_2ESUBMAP @ A_27a @ A_27b @ V1g @ V0f ) )
<=> ( V0f = V1g ) ) ).
thf(thm_2Efinite__map_2ESUBMAP__REFL,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b] : ( c_2Efinite__map_2ESUBMAP @ A_27a @ A_27b @ V0f @ V0f ) ).
thf(thm_2Efinite__map_2ESUBMAP__FEMPTY,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b] : ( c_2Efinite__map_2ESUBMAP @ A_27a @ A_27b @ ( c_2Efinite__map_2EFEMPTY @ A_27a @ A_27b ) @ V0f ) ).
thf(thm_2Efinite__map_2Efmap__EXT,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b,V1g: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b] :
( ( V0f = V1g )
<=> ( ( ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b @ V0f )
= ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b @ V1g ) )
& ! [V2x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V2x @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b @ V0f ) )
=> ( ( c_2Efinite__map_2EFAPPLY @ A_27a @ A_27b @ V0f @ V2x )
= ( c_2Efinite__map_2EFAPPLY @ A_27a @ A_27b @ V1g @ V2x ) ) ) ) ) ).
thf(thm_2Efinite__map_2Efmap__EQ__THM,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b,V1g: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b] :
( ( ( ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b @ V0f )
= ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b @ V1g ) )
& ! [V2x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V2x @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b @ V0f ) )
=> ( ( c_2Efinite__map_2EFAPPLY @ A_27a @ A_27b @ V0f @ V2x )
= ( c_2Efinite__map_2EFAPPLY @ A_27a @ A_27b @ V1g @ V2x ) ) ) )
<=> ( V0f = V1g ) ) ).
thf(thm_2Efinite__map_2Efmap__EQ,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b,V1g: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b] :
( ( ( ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b @ V0f )
= ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b @ V1g ) )
& ( ( c_2Efinite__map_2EFAPPLY @ A_27a @ A_27b @ V0f )
= ( c_2Efinite__map_2EFAPPLY @ A_27a @ A_27b @ V1g ) ) )
<=> ( V0f = V1g ) ) ).
thf(thm_2Efinite__map_2ENOT__FDOM__FAPPLY__FEMPTY,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b,V1x: A_27a] :
( ( (~) @ ( c_2Ebool_2EIN @ A_27a @ V1x @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b @ V0f ) ) )
=> ( ( c_2Efinite__map_2EFAPPLY @ A_27a @ A_27b @ V0f @ V1x )
= ( c_2Efinite__map_2EFAPPLY @ A_27a @ A_27b @ ( c_2Efinite__map_2EFEMPTY @ A_27a @ A_27b ) @ V1x ) ) ) ).
thf(thm_2Efinite__map_2EFM__PULL__APART,axiom,
! [A_27a: $tType,A_27b: $tType,V0fm: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b,V1k: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V1k @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b @ V0fm ) )
=> ? [V2fm0: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b,V3v: A_27b] :
( ( V0fm
= ( c_2Efinite__map_2EFUPDATE @ A_27a @ A_27b @ V2fm0 @ ( c_2Epair_2E_2C @ A_27a @ A_27b @ V1k @ V3v ) ) )
& ( (~) @ ( c_2Ebool_2EIN @ A_27a @ V1k @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b @ V2fm0 ) ) ) ) ) ).
thf(thm_2Efinite__map_2Efmap__INDUCT,axiom,
! [A_27a: $tType,A_27b: $tType,V0P: ( tyop_2Efinite__map_2Efmap @ A_27a @ A_27b ) > $o] :
( ( ( V0P @ ( c_2Efinite__map_2EFEMPTY @ A_27a @ A_27b ) )
& ! [V1f: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b] :
( ( V0P @ V1f )
=> ! [V2x: A_27a,V3y: A_27b] :
( ( (~) @ ( c_2Ebool_2EIN @ A_27a @ V2x @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b @ V1f ) ) )
=> ( V0P @ ( c_2Efinite__map_2EFUPDATE @ A_27a @ A_27b @ V1f @ ( c_2Epair_2E_2C @ A_27a @ A_27b @ V2x @ V3y ) ) ) ) ) )
=> ! [V4f: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b] : ( V0P @ V4f ) ) ).
thf(thm_2Efinite__map_2EFCARD__SUC,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b,V1n: tyop_2Enum_2Enum] :
( ( ( c_2Efinite__map_2EFCARD @ A_27a @ A_27b @ V0f )
= ( c_2Enum_2ESUC @ V1n ) )
<=> ? [V2f_27: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b,V3x: A_27a,V4y: A_27b] :
( ( (~) @ ( c_2Ebool_2EIN @ A_27a @ V3x @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b @ V2f_27 ) ) )
& ( ( c_2Efinite__map_2EFCARD @ A_27a @ A_27b @ V2f_27 )
= V1n )
& ( V0f
= ( c_2Efinite__map_2EFUPDATE @ A_27a @ A_27b @ V2f_27 @ ( c_2Epair_2E_2C @ A_27a @ A_27b @ V3x @ V4y ) ) ) ) ) ).
thf(thm_2Efinite__map_2EFCARD__0__FEMPTY,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b] :
( ( ( c_2Efinite__map_2EFCARD @ A_27a @ A_27b @ V0f )
= c_2Enum_2E0 )
<=> ( V0f
= ( c_2Efinite__map_2EFEMPTY @ A_27a @ A_27b ) ) ) ).
thf(thm_2Efinite__map_2EFCARD__FUPDATE,axiom,
! [A_27a: $tType,A_27b: $tType,V0fm: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b,V1a: A_27a,V2b: A_27b] :
( ( c_2Efinite__map_2EFCARD @ A_27a @ A_27b @ ( c_2Efinite__map_2EFUPDATE @ A_27a @ A_27b @ V0fm @ ( c_2Epair_2E_2C @ A_27a @ A_27b @ V1a @ V2b ) ) )
= ( c_2Ebool_2ECOND @ tyop_2Enum_2Enum @ ( c_2Ebool_2EIN @ A_27a @ V1a @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b @ V0fm ) ) @ ( c_2Efinite__map_2EFCARD @ A_27a @ A_27b @ V0fm ) @ ( c_2Earithmetic_2E_2B @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) @ ( c_2Efinite__map_2EFCARD @ A_27a @ A_27b @ V0fm ) ) ) ) ).
thf(thm_2Efinite__map_2EFCARD__FEMPTY,axiom,
! [A_27a: $tType,A_27b: $tType] :
( ( c_2Efinite__map_2EFCARD @ A_27a @ A_27b @ ( c_2Efinite__map_2EFEMPTY @ A_27a @ A_27b ) )
= c_2Enum_2E0 ) ).
thf(thm_2Efinite__map_2EFDOM__FINITE,axiom,
! [A_27a: $tType,A_27b: $tType,V0fm: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b] : ( c_2Epred__set_2EFINITE @ A_27a @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b @ V0fm ) ) ).
thf(thm_2Efinite__map_2EFDOM__F__FEMPTY1,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b] :
( ! [V1a: A_27a] : ( (~) @ ( c_2Ebool_2EIN @ A_27a @ V1a @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b @ V0f ) ) )
<=> ( V0f
= ( c_2Efinite__map_2EFEMPTY @ A_27a @ A_27b ) ) ) ).
thf(thm_2Efinite__map_2EFDOM__EQ__EMPTY__SYM,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b] :
( ( ( c_2Epred__set_2EEMPTY @ A_27a )
= ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b @ V0f ) )
<=> ( V0f
= ( c_2Efinite__map_2EFEMPTY @ A_27a @ A_27b ) ) ) ).
thf(thm_2Efinite__map_2EFDOM__EQ__EMPTY,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b] :
( ( ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b @ V0f )
= ( c_2Epred__set_2EEMPTY @ A_27a ) )
<=> ( V0f
= ( c_2Efinite__map_2EFEMPTY @ A_27a @ A_27b ) ) ) ).
thf(thm_2Efinite__map_2Efmap__SIMPLE__INDUCT,axiom,
! [A_27a: $tType,A_27b: $tType,V0P: ( tyop_2Efinite__map_2Efmap @ A_27a @ A_27b ) > $o] :
( ( ( V0P @ ( c_2Efinite__map_2EFEMPTY @ A_27a @ A_27b ) )
& ! [V1f: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b] :
( ( V0P @ V1f )
=> ! [V2x: A_27a,V3y: A_27b] : ( V0P @ ( c_2Efinite__map_2EFUPDATE @ A_27a @ A_27b @ V1f @ ( c_2Epair_2E_2C @ A_27a @ A_27b @ V2x @ V3y ) ) ) ) )
=> ! [V4f: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b] : ( V0P @ V4f ) ) ).
thf(thm_2Efinite__map_2EFDOM__EQ__FDOM__FUPDATE,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b,V1x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V1x @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b @ V0f ) )
=> ! [V2y: A_27b] :
( ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b @ ( c_2Efinite__map_2EFUPDATE @ A_27a @ A_27b @ V0f @ ( c_2Epair_2E_2C @ A_27a @ A_27b @ V1x @ V2y ) ) )
= ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b @ V0f ) ) ) ).
thf(thm_2Efinite__map_2ENOT__EQ__FEMPTY__FUPDATE,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b,V1a: A_27a,V2b: A_27b] :
( (~)
@ ( ( c_2Efinite__map_2EFEMPTY @ A_27a @ A_27b )
= ( c_2Efinite__map_2EFUPDATE @ A_27a @ A_27b @ V0f @ ( c_2Epair_2E_2C @ A_27a @ A_27b @ V1a @ V2b ) ) ) ) ).
thf(thm_2Efinite__map_2EFAPPLY__FUPDATE__THM,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b,V1a: A_27a,V2b: A_27b,V3x: A_27a] :
( ( c_2Efinite__map_2EFAPPLY @ A_27a @ A_27b @ ( c_2Efinite__map_2EFUPDATE @ A_27a @ A_27b @ V0f @ ( c_2Epair_2E_2C @ A_27a @ A_27b @ V1a @ V2b ) ) @ V3x )
= ( c_2Ebool_2ECOND @ A_27b @ ( c_2Emin_2E_3D @ A_27a @ V3x @ V1a ) @ V2b @ ( c_2Efinite__map_2EFAPPLY @ A_27a @ A_27b @ V0f @ V3x ) ) ) ).
thf(thm_2Efinite__map_2EFDOM__FUPDATE,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b,V1a: A_27a,V2b: A_27b] :
( ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b @ ( c_2Efinite__map_2EFUPDATE @ A_27a @ A_27b @ V0f @ ( c_2Epair_2E_2C @ A_27a @ A_27b @ V1a @ V2b ) ) )
= ( c_2Epred__set_2EINSERT @ A_27a @ V1a @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b @ V0f ) ) ) ).
thf(thm_2Efinite__map_2EFDOM__FEMPTY,axiom,
! [A_27a: $tType,A_27b: $tType] :
( ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b @ ( c_2Efinite__map_2EFEMPTY @ A_27a @ A_27b ) )
= ( c_2Epred__set_2EEMPTY @ A_27a ) ) ).
thf(thm_2Efinite__map_2EFUPDATE__EQ,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b,V1a: A_27a,V2b: A_27b,V3c: A_27b] :
( ( c_2Efinite__map_2EFUPDATE @ A_27a @ A_27b @ ( c_2Efinite__map_2EFUPDATE @ A_27a @ A_27b @ V0f @ ( c_2Epair_2E_2C @ A_27a @ A_27b @ V1a @ V2b ) ) @ ( c_2Epair_2E_2C @ A_27a @ A_27b @ V1a @ V3c ) )
= ( c_2Efinite__map_2EFUPDATE @ A_27a @ A_27b @ V0f @ ( c_2Epair_2E_2C @ A_27a @ A_27b @ V1a @ V3c ) ) ) ).
thf(thm_2Efinite__map_2EFUPDATE__COMMUTES,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b,V1a: A_27a,V2b: A_27b,V3c: A_27a,V4d: A_27b] :
( ( (~) @ ( V1a = V3c ) )
=> ( ( c_2Efinite__map_2EFUPDATE @ A_27a @ A_27b @ ( c_2Efinite__map_2EFUPDATE @ A_27a @ A_27b @ V0f @ ( c_2Epair_2E_2C @ A_27a @ A_27b @ V1a @ V2b ) ) @ ( c_2Epair_2E_2C @ A_27a @ A_27b @ V3c @ V4d ) )
= ( c_2Efinite__map_2EFUPDATE @ A_27a @ A_27b @ ( c_2Efinite__map_2EFUPDATE @ A_27a @ A_27b @ V0f @ ( c_2Epair_2E_2C @ A_27a @ A_27b @ V3c @ V4d ) ) @ ( c_2Epair_2E_2C @ A_27a @ A_27b @ V1a @ V2b ) ) ) ) ).
thf(thm_2Efinite__map_2ENOT__EQ__FAPPLY,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b,V1a: A_27a,V2x: A_27a,V3y: A_27b] :
( ( (~) @ ( V1a = V2x ) )
=> ( ( c_2Efinite__map_2EFAPPLY @ A_27a @ A_27b @ ( c_2Efinite__map_2EFUPDATE @ A_27a @ A_27b @ V0f @ ( c_2Epair_2E_2C @ A_27a @ A_27b @ V2x @ V3y ) ) @ V1a )
= ( c_2Efinite__map_2EFAPPLY @ A_27a @ A_27b @ V0f @ V1a ) ) ) ).
thf(thm_2Efinite__map_2EFAPPLY__FUPDATE,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b,V1x: A_27a,V2y: A_27b] :
( ( c_2Efinite__map_2EFAPPLY @ A_27a @ A_27b @ ( c_2Efinite__map_2EFUPDATE @ A_27a @ A_27b @ V0f @ ( c_2Epair_2E_2C @ A_27a @ A_27b @ V1x @ V2y ) ) @ V1x )
= V2y ) ).
thf(thm_2Efinite__map_2Eis__fmap__cases,axiom,
! [A_27a: $tType,A_27b: $tType,V0a0: A_27a > ( tyop_2Esum_2Esum @ A_27b @ tyop_2Eone_2Eone )] :
( ( c_2Efinite__map_2Eis__fmap @ A_27a @ A_27b @ V0a0 )
<=> ( ( V0a0
= ( ^ [V1a: A_27a] : ( c_2Esum_2EINR @ A_27b @ tyop_2Eone_2Eone @ c_2Eone_2Eone ) ) )
| ? [V2f: A_27a > ( tyop_2Esum_2Esum @ A_27b @ tyop_2Eone_2Eone ),V3a: A_27a,V4b: A_27b] :
( ( V0a0
= ( ^ [V5x: A_27a] : ( c_2Ebool_2ECOND @ ( tyop_2Esum_2Esum @ A_27b @ tyop_2Eone_2Eone ) @ ( c_2Emin_2E_3D @ A_27a @ V5x @ V3a ) @ ( c_2Esum_2EINL @ A_27b @ tyop_2Eone_2Eone @ V4b ) @ ( V2f @ V5x ) ) ) )
& ( c_2Efinite__map_2Eis__fmap @ A_27a @ A_27b @ V2f ) ) ) ) ).
thf(thm_2Efinite__map_2Eis__fmap__strongind,axiom,
! [A_27a: $tType,A_27b: $tType,V0is__fmap_27: ( A_27a > ( tyop_2Esum_2Esum @ A_27b @ tyop_2Eone_2Eone ) ) > $o] :
( ( ( V0is__fmap_27
@ ^ [V1a: A_27a] : ( c_2Esum_2EINR @ A_27b @ tyop_2Eone_2Eone @ c_2Eone_2Eone ) )
& ! [V2f: A_27a > ( tyop_2Esum_2Esum @ A_27b @ tyop_2Eone_2Eone ),V3a: A_27a,V4b: A_27b] :
( ( ( c_2Efinite__map_2Eis__fmap @ A_27a @ A_27b @ V2f )
& ( V0is__fmap_27 @ V2f ) )
=> ( V0is__fmap_27
@ ^ [V5x: A_27a] : ( c_2Ebool_2ECOND @ ( tyop_2Esum_2Esum @ A_27b @ tyop_2Eone_2Eone ) @ ( c_2Emin_2E_3D @ A_27a @ V5x @ V3a ) @ ( c_2Esum_2EINL @ A_27b @ tyop_2Eone_2Eone @ V4b ) @ ( V2f @ V5x ) ) ) ) )
=> ! [V6a0: A_27a > ( tyop_2Esum_2Esum @ A_27b @ tyop_2Eone_2Eone )] :
( ( c_2Efinite__map_2Eis__fmap @ A_27a @ A_27b @ V6a0 )
=> ( V0is__fmap_27 @ V6a0 ) ) ) ).
thf(thm_2Efinite__map_2Eis__fmap__ind,axiom,
! [A_27a: $tType,A_27b: $tType,V0is__fmap_27: ( A_27a > ( tyop_2Esum_2Esum @ A_27b @ tyop_2Eone_2Eone ) ) > $o] :
( ( ( V0is__fmap_27
@ ^ [V1a: A_27a] : ( c_2Esum_2EINR @ A_27b @ tyop_2Eone_2Eone @ c_2Eone_2Eone ) )
& ! [V2f: A_27a > ( tyop_2Esum_2Esum @ A_27b @ tyop_2Eone_2Eone ),V3a: A_27a,V4b: A_27b] :
( ( V0is__fmap_27 @ V2f )
=> ( V0is__fmap_27
@ ^ [V5x: A_27a] : ( c_2Ebool_2ECOND @ ( tyop_2Esum_2Esum @ A_27b @ tyop_2Eone_2Eone ) @ ( c_2Emin_2E_3D @ A_27a @ V5x @ V3a ) @ ( c_2Esum_2EINL @ A_27b @ tyop_2Eone_2Eone @ V4b ) @ ( V2f @ V5x ) ) ) ) )
=> ! [V6a0: A_27a > ( tyop_2Esum_2Esum @ A_27b @ tyop_2Eone_2Eone )] :
( ( c_2Efinite__map_2Eis__fmap @ A_27a @ A_27b @ V6a0 )
=> ( V0is__fmap_27 @ V6a0 ) ) ) ).
thf(thm_2Efinite__map_2Eis__fmap__rules,axiom,
! [A_27a: $tType,A_27b: $tType] :
( ( c_2Efinite__map_2Eis__fmap @ A_27a @ A_27b
@ ^ [V0a: A_27a] : ( c_2Esum_2EINR @ A_27b @ tyop_2Eone_2Eone @ c_2Eone_2Eone ) )
& ! [V1f: A_27a > ( tyop_2Esum_2Esum @ A_27b @ tyop_2Eone_2Eone ),V2a: A_27a,V3b: A_27b] :
( ( c_2Efinite__map_2Eis__fmap @ A_27a @ A_27b @ V1f )
=> ( c_2Efinite__map_2Eis__fmap @ A_27a @ A_27b
@ ^ [V4x: A_27a] : ( c_2Ebool_2ECOND @ ( tyop_2Esum_2Esum @ A_27b @ tyop_2Eone_2Eone ) @ ( c_2Emin_2E_3D @ A_27a @ V4x @ V2a ) @ ( c_2Esum_2EINL @ A_27b @ tyop_2Eone_2Eone @ V3b ) @ ( V1f @ V4x ) ) ) ) ) ).
thf(thm_2Efinite__map_2EFUNION__FEMPTY__1,axiom,
! [A_27a: $tType,A_27b: $tType,V0g: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b] :
( ( c_2Efinite__map_2EFUNION @ A_27a @ A_27b @ ( c_2Efinite__map_2EFEMPTY @ A_27a @ A_27b ) @ V0g )
= V0g ) ).
thf(thm_2Efinite__map_2EFUNION__FEMPTY__2,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b] :
( ( c_2Efinite__map_2EFUNION @ A_27a @ A_27b @ V0f @ ( c_2Efinite__map_2EFEMPTY @ A_27a @ A_27b ) )
= V0f ) ).
thf(thm_2Efinite__map_2EFUNION__FUPDATE__1,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b,V1g: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b,V2x: A_27a,V3y: A_27b] :
( ( c_2Efinite__map_2EFUNION @ A_27a @ A_27b @ ( c_2Efinite__map_2EFUPDATE @ A_27a @ A_27b @ V0f @ ( c_2Epair_2E_2C @ A_27a @ A_27b @ V2x @ V3y ) ) @ V1g )
= ( c_2Efinite__map_2EFUPDATE @ A_27a @ A_27b @ ( c_2Efinite__map_2EFUNION @ A_27a @ A_27b @ V0f @ V1g ) @ ( c_2Epair_2E_2C @ A_27a @ A_27b @ V2x @ V3y ) ) ) ).
thf(thm_2Efinite__map_2EFUNION__FUPDATE__2,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b,V1g: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b,V2x: A_27a,V3y: A_27b] :
( ( c_2Efinite__map_2EFUNION @ A_27a @ A_27b @ V0f @ ( c_2Efinite__map_2EFUPDATE @ A_27a @ A_27b @ V1g @ ( c_2Epair_2E_2C @ A_27a @ A_27b @ V2x @ V3y ) ) )
= ( c_2Ebool_2ECOND @ ( tyop_2Efinite__map_2Efmap @ A_27a @ A_27b ) @ ( c_2Ebool_2EIN @ A_27a @ V2x @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b @ V0f ) ) @ ( c_2Efinite__map_2EFUNION @ A_27a @ A_27b @ V0f @ V1g ) @ ( c_2Efinite__map_2EFUPDATE @ A_27a @ A_27b @ ( c_2Efinite__map_2EFUNION @ A_27a @ A_27b @ V0f @ V1g ) @ ( c_2Epair_2E_2C @ A_27a @ A_27b @ V2x @ V3y ) ) ) ) ).
thf(thm_2Efinite__map_2EEQ__FDOM__SUBMAP,axiom,
! [A_27a: $tType,A_27b: $tType,V0g: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b,V1f: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b] :
( ( V1f = V0g )
<=> ( ( c_2Efinite__map_2ESUBMAP @ A_27a @ A_27b @ V1f @ V0g )
& ( ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b @ V1f )
= ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b @ V0g ) ) ) ) ).
thf(thm_2Efinite__map_2ESUBMAP__FUPDATE__EQN,axiom,
! [A_27a: $tType,A_27b: $tType,V0y: A_27b,V1x: A_27a,V2f: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b] :
( ( c_2Efinite__map_2ESUBMAP @ A_27a @ A_27b @ V2f @ ( c_2Efinite__map_2EFUPDATE @ A_27a @ A_27b @ V2f @ ( c_2Epair_2E_2C @ A_27a @ A_27b @ V1x @ V0y ) ) )
<=> ( ( (~) @ ( c_2Ebool_2EIN @ A_27a @ V1x @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b @ V2f ) ) )
| ( ( ( c_2Efinite__map_2EFAPPLY @ A_27a @ A_27b @ V2f @ V1x )
= V0y )
& ( c_2Ebool_2EIN @ A_27a @ V1x @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b @ V2f ) ) ) ) ) ).
thf(thm_2Efinite__map_2EDRESTRICT__FEMPTY,axiom,
! [A_27a: $tType,A_27b: $tType,V0r: A_27a > $o] :
( ( c_2Efinite__map_2EDRESTRICT @ A_27a @ A_27b @ ( c_2Efinite__map_2EFEMPTY @ A_27a @ A_27b ) @ V0r )
= ( c_2Efinite__map_2EFEMPTY @ A_27a @ A_27b ) ) ).
thf(thm_2Efinite__map_2EDRESTRICT__FUPDATE,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b,V1r: A_27a > $o,V2x: A_27a,V3y: A_27b] :
( ( c_2Efinite__map_2EDRESTRICT @ A_27a @ A_27b @ ( c_2Efinite__map_2EFUPDATE @ A_27a @ A_27b @ V0f @ ( c_2Epair_2E_2C @ A_27a @ A_27b @ V2x @ V3y ) ) @ V1r )
= ( c_2Ebool_2ECOND @ ( tyop_2Efinite__map_2Efmap @ A_27a @ A_27b ) @ ( c_2Ebool_2EIN @ A_27a @ V2x @ V1r ) @ ( c_2Efinite__map_2EFUPDATE @ A_27a @ A_27b @ ( c_2Efinite__map_2EDRESTRICT @ A_27a @ A_27b @ V0f @ V1r ) @ ( c_2Epair_2E_2C @ A_27a @ A_27b @ V2x @ V3y ) ) @ ( c_2Efinite__map_2EDRESTRICT @ A_27a @ A_27b @ V0f @ V1r ) ) ) ).
thf(thm_2Efinite__map_2ESTRONG__DRESTRICT__FUPDATE,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b,V1r: A_27a > $o,V2x: A_27a,V3y: A_27b] :
( ( c_2Ebool_2EIN @ A_27a @ V2x @ V1r )
=> ( ( c_2Efinite__map_2EDRESTRICT @ A_27a @ A_27b @ ( c_2Efinite__map_2EFUPDATE @ A_27a @ A_27b @ V0f @ ( c_2Epair_2E_2C @ A_27a @ A_27b @ V2x @ V3y ) ) @ V1r )
= ( c_2Efinite__map_2EFUPDATE @ A_27a @ A_27b @ ( c_2Efinite__map_2EDRESTRICT @ A_27a @ A_27b @ V0f @ ( c_2Epred__set_2EDELETE @ A_27a @ V1r @ V2x ) ) @ ( c_2Epair_2E_2C @ A_27a @ A_27b @ V2x @ V3y ) ) ) ) ).
thf(thm_2Efinite__map_2EFDOM__DRESTRICT,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0f: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b,V1r: A_27a > $o,V2x: A_27c] :
( ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b @ ( c_2Efinite__map_2EDRESTRICT @ A_27a @ A_27b @ V0f @ V1r ) )
= ( c_2Epred__set_2EINTER @ A_27a @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b @ V0f ) @ V1r ) ) ).
thf(thm_2Efinite__map_2ENOT__FDOM__DRESTRICT,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b,V1x: A_27a] :
( ( (~) @ ( c_2Ebool_2EIN @ A_27a @ V1x @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b @ V0f ) ) )
=> ( ( c_2Efinite__map_2EDRESTRICT @ A_27a @ A_27b @ V0f @ ( c_2Epred__set_2ECOMPL @ A_27a @ ( c_2Epred__set_2EINSERT @ A_27a @ V1x @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) ) )
= V0f ) ) ).
thf(thm_2Efinite__map_2EDRESTRICT__SUBMAP,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b,V1r: A_27a > $o] : ( c_2Efinite__map_2ESUBMAP @ A_27a @ A_27b @ ( c_2Efinite__map_2EDRESTRICT @ A_27a @ A_27b @ V0f @ V1r ) @ V0f ) ).
thf(thm_2Efinite__map_2EDRESTRICT__DRESTRICT,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b,V1P: A_27a > $o,V2Q: A_27a > $o] :
( ( c_2Efinite__map_2EDRESTRICT @ A_27a @ A_27b @ ( c_2Efinite__map_2EDRESTRICT @ A_27a @ A_27b @ V0f @ V1P ) @ V2Q )
= ( c_2Efinite__map_2EDRESTRICT @ A_27a @ A_27b @ V0f @ ( c_2Epred__set_2EINTER @ A_27a @ V1P @ V2Q ) ) ) ).
thf(thm_2Efinite__map_2EDRESTRICT__IS__FEMPTY,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b] :
( ( c_2Efinite__map_2EDRESTRICT @ A_27a @ A_27b @ V0f @ ( c_2Epred__set_2EEMPTY @ A_27a ) )
= ( c_2Efinite__map_2EFEMPTY @ A_27a @ A_27b ) ) ).
thf(thm_2Efinite__map_2EFUPDATE__DRESTRICT,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b,V1x: A_27a,V2y: A_27b] :
( ( c_2Efinite__map_2EFUPDATE @ A_27a @ A_27b @ V0f @ ( c_2Epair_2E_2C @ A_27a @ A_27b @ V1x @ V2y ) )
= ( c_2Efinite__map_2EFUPDATE @ A_27a @ A_27b @ ( c_2Efinite__map_2EDRESTRICT @ A_27a @ A_27b @ V0f @ ( c_2Epred__set_2ECOMPL @ A_27a @ ( c_2Epred__set_2EINSERT @ A_27a @ V1x @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) ) ) @ ( c_2Epair_2E_2C @ A_27a @ A_27b @ V1x @ V2y ) ) ) ).
thf(thm_2Efinite__map_2ESTRONG__DRESTRICT__FUPDATE__THM,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b,V1r: A_27a > $o,V2x: A_27a,V3y: A_27b] :
( ( c_2Efinite__map_2EDRESTRICT @ A_27a @ A_27b @ ( c_2Efinite__map_2EFUPDATE @ A_27a @ A_27b @ V0f @ ( c_2Epair_2E_2C @ A_27a @ A_27b @ V2x @ V3y ) ) @ V1r )
= ( c_2Ebool_2ECOND @ ( tyop_2Efinite__map_2Efmap @ A_27a @ A_27b ) @ ( c_2Ebool_2EIN @ A_27a @ V2x @ V1r ) @ ( c_2Efinite__map_2EFUPDATE @ A_27a @ A_27b @ ( c_2Efinite__map_2EDRESTRICT @ A_27a @ A_27b @ V0f @ ( c_2Epred__set_2EINTER @ A_27a @ ( c_2Epred__set_2ECOMPL @ A_27a @ ( c_2Epred__set_2EINSERT @ A_27a @ V2x @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) ) @ V1r ) ) @ ( c_2Epair_2E_2C @ A_27a @ A_27b @ V2x @ V3y ) ) @ ( c_2Efinite__map_2EDRESTRICT @ A_27a @ A_27b @ V0f @ ( c_2Epred__set_2EINTER @ A_27a @ ( c_2Epred__set_2ECOMPL @ A_27a @ ( c_2Epred__set_2EINSERT @ A_27a @ V2x @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) ) @ V1r ) ) ) ) ).
thf(thm_2Efinite__map_2EDRESTRICT__UNIV,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b] :
( ( c_2Efinite__map_2EDRESTRICT @ A_27a @ A_27b @ V0f @ ( c_2Epred__set_2EUNIV @ A_27a ) )
= V0f ) ).
thf(thm_2Efinite__map_2ESUBMAP__DRESTRICT,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b,V1P: A_27a > $o] : ( c_2Efinite__map_2ESUBMAP @ A_27a @ A_27b @ ( c_2Efinite__map_2EDRESTRICT @ A_27a @ A_27b @ V0f @ V1P ) @ V0f ) ).
thf(thm_2Efinite__map_2EDRESTRICT__EQ__DRESTRICT,axiom,
! [A_27a: $tType,A_27b: $tType,V0f1: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b,V1f2: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b,V2s1: A_27a > $o,V3s2: A_27a > $o] :
( ( ( c_2Efinite__map_2EDRESTRICT @ A_27a @ A_27b @ V0f1 @ V2s1 )
= ( c_2Efinite__map_2EDRESTRICT @ A_27a @ A_27b @ V1f2 @ V3s2 ) )
<=> ( ( c_2Efinite__map_2ESUBMAP @ A_27a @ A_27b @ ( c_2Efinite__map_2EDRESTRICT @ A_27a @ A_27b @ V0f1 @ V2s1 ) @ V1f2 )
& ( c_2Efinite__map_2ESUBMAP @ A_27a @ A_27b @ ( c_2Efinite__map_2EDRESTRICT @ A_27a @ A_27b @ V1f2 @ V3s2 ) @ V0f1 )
& ( ( c_2Epred__set_2EINTER @ A_27a @ V2s1 @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b @ V0f1 ) )
= ( c_2Epred__set_2EINTER @ A_27a @ V3s2 @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b @ V1f2 ) ) ) ) ) ).
thf(thm_2Efinite__map_2EFDOM__FUNION,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0f: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b,V1g: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b,V2x: A_27c] :
( ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b @ ( c_2Efinite__map_2EFUNION @ A_27a @ A_27b @ V0f @ V1g ) )
= ( c_2Epred__set_2EUNION @ A_27a @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b @ V0f ) @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b @ V1g ) ) ) ).
thf(thm_2Efinite__map_2EFMERGE__DOMSUB,axiom,
! [A_27a: $tType,A_27b: $tType,V0m: A_27a > A_27a > A_27a,V1m1: tyop_2Efinite__map_2Efmap @ A_27b @ A_27a,V2m2: tyop_2Efinite__map_2Efmap @ A_27b @ A_27a,V3k: A_27b] :
( ( c_2Efinite__map_2Efdomsub @ A_27b @ A_27a @ ( c_2Efinite__map_2EFMERGE @ A_27a @ A_27b @ V0m @ V1m1 @ V2m2 ) @ V3k )
= ( c_2Efinite__map_2EFMERGE @ A_27a @ A_27b @ V0m @ ( c_2Efinite__map_2Efdomsub @ A_27b @ A_27a @ V1m1 @ V3k ) @ ( c_2Efinite__map_2Efdomsub @ A_27b @ A_27a @ V2m2 @ V3k ) ) ) ).
thf(thm_2Efinite__map_2ESUBMAP__DOMSUB,axiom,
! [A_27a: $tType,A_27b: $tType,V0k: A_27a,V1f: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b] : ( c_2Efinite__map_2ESUBMAP @ A_27a @ A_27b @ ( c_2Efinite__map_2Efdomsub @ A_27a @ A_27b @ V1f @ V0k ) @ V1f ) ).
thf(thm_2Efinite__map_2Efmap__CASES,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b] :
( ( V0f
= ( c_2Efinite__map_2EFEMPTY @ A_27a @ A_27b ) )
| ? [V1g: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b,V2x: A_27a,V3y: A_27b] :
( V0f
= ( c_2Efinite__map_2EFUPDATE @ A_27a @ A_27b @ V1g @ ( c_2Epair_2E_2C @ A_27a @ A_27b @ V2x @ V3y ) ) ) ) ).
thf(thm_2Efinite__map_2EDOMSUB__NOT__IN__DOM,axiom,
! [A_27a: $tType,A_27b: $tType,V0k: A_27a,V1fm: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b] :
( ( (~) @ ( c_2Ebool_2EIN @ A_27a @ V0k @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b @ V1fm ) ) )
=> ( ( c_2Efinite__map_2Efdomsub @ A_27a @ A_27b @ V1fm @ V0k )
= V1fm ) ) ).
thf(thm_2Efinite__map_2Eo__f__FUPDATE,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0v: A_27c,V1k: A_27a,V2fm: tyop_2Efinite__map_2Efmap @ A_27a @ A_27c,V3f: A_27c > A_27b] :
( ( c_2Efinite__map_2Eo__f @ A_27a @ A_27c @ A_27b @ V3f @ ( c_2Efinite__map_2EFUPDATE @ A_27a @ A_27c @ V2fm @ ( c_2Epair_2E_2C @ A_27a @ A_27c @ V1k @ V0v ) ) )
= ( c_2Efinite__map_2EFUPDATE @ A_27a @ A_27b @ ( c_2Efinite__map_2Eo__f @ A_27a @ A_27c @ A_27b @ V3f @ V2fm ) @ ( c_2Epair_2E_2C @ A_27a @ A_27b @ V1k @ ( V3f @ V0v ) ) ) ) ).
thf(thm_2Efinite__map_2EDOMSUB__COMMUTES,axiom,
! [A_27a: $tType,A_27b: $tType,V0k2: A_27a,V1k1: A_27a,V2fm: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b] :
( ( c_2Efinite__map_2Efdomsub @ A_27a @ A_27b @ ( c_2Efinite__map_2Efdomsub @ A_27a @ A_27b @ V2fm @ V1k1 ) @ V0k2 )
= ( c_2Efinite__map_2Efdomsub @ A_27a @ A_27b @ ( c_2Efinite__map_2Efdomsub @ A_27a @ A_27b @ V2fm @ V0k2 ) @ V1k1 ) ) ).
thf(thm_2Efinite__map_2EDOMSUB__IDEM,axiom,
! [A_27a: $tType,A_27b: $tType,V0k: A_27a,V1fm: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b] :
( ( c_2Efinite__map_2Efdomsub @ A_27a @ A_27b @ ( c_2Efinite__map_2Efdomsub @ A_27a @ A_27b @ V1fm @ V0k ) @ V0k )
= ( c_2Efinite__map_2Efdomsub @ A_27a @ A_27b @ V1fm @ V0k ) ) ).
thf(thm_2Efinite__map_2Eo__f__DOMSUB,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0k: A_27a,V1g: A_27c > A_27b,V2fm: tyop_2Efinite__map_2Efmap @ A_27a @ A_27c] :
( ( c_2Efinite__map_2Efdomsub @ A_27a @ A_27b @ ( c_2Efinite__map_2Eo__f @ A_27a @ A_27c @ A_27b @ V1g @ V2fm ) @ V0k )
= ( c_2Efinite__map_2Eo__f @ A_27a @ A_27c @ A_27b @ V1g @ ( c_2Efinite__map_2Efdomsub @ A_27a @ A_27c @ V2fm @ V0k ) ) ) ).
thf(thm_2Efinite__map_2EFRANGE__FUPDATE__DOMSUB,axiom,
! [A_27a: $tType,A_27b: $tType,V0fm: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b,V1k: A_27a,V2v: A_27b] :
( ( c_2Efinite__map_2EFRANGE @ A_27a @ A_27b @ ( c_2Efinite__map_2EFUPDATE @ A_27a @ A_27b @ V0fm @ ( c_2Epair_2E_2C @ A_27a @ A_27b @ V1k @ V2v ) ) )
= ( c_2Epred__set_2EINSERT @ A_27b @ V2v @ ( c_2Efinite__map_2EFRANGE @ A_27a @ A_27b @ ( c_2Efinite__map_2Efdomsub @ A_27a @ A_27b @ V0fm @ V1k ) ) ) ) ).
thf(thm_2Efinite__map_2EDOMSUB__FLOOKUP__THM,axiom,
! [A_27a: $tType,A_27b: $tType,V0fm: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b,V1k1: A_27a,V2k2: A_27a] :
( ( c_2Efinite__map_2EFLOOKUP @ A_27a @ A_27b @ ( c_2Efinite__map_2Efdomsub @ A_27a @ A_27b @ V0fm @ V1k1 ) @ V2k2 )
= ( c_2Ebool_2ECOND @ ( tyop_2Eoption_2Eoption @ A_27b ) @ ( c_2Emin_2E_3D @ A_27a @ V1k1 @ V2k2 ) @ ( c_2Eoption_2ENONE @ A_27b ) @ ( c_2Efinite__map_2EFLOOKUP @ A_27a @ A_27b @ V0fm @ V2k2 ) ) ) ).
thf(thm_2Efinite__map_2EDOMSUB__FLOOKUP__NEQ,axiom,
! [A_27a: $tType,A_27b: $tType,V0fm: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b,V1k1: A_27a,V2k2: A_27a] :
( ( (~) @ ( V1k1 = V2k2 ) )
=> ( ( c_2Efinite__map_2EFLOOKUP @ A_27a @ A_27b @ ( c_2Efinite__map_2Efdomsub @ A_27a @ A_27b @ V0fm @ V1k1 ) @ V2k2 )
= ( c_2Efinite__map_2EFLOOKUP @ A_27a @ A_27b @ V0fm @ V2k2 ) ) ) ).
thf(thm_2Efinite__map_2EDOMSUB__FLOOKUP,axiom,
! [A_27a: $tType,A_27b: $tType,V0fm: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b,V1k: A_27a] :
( ( c_2Efinite__map_2EFLOOKUP @ A_27a @ A_27b @ ( c_2Efinite__map_2Efdomsub @ A_27a @ A_27b @ V0fm @ V1k ) @ V1k )
= ( c_2Eoption_2ENONE @ A_27b ) ) ).
thf(thm_2Efinite__map_2EDOMSUB__FAPPLY__THM,axiom,
! [A_27a: $tType,A_27b: $tType,V0fm: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b,V1k1: A_27a,V2k2: A_27a] :
( ( c_2Efinite__map_2EFAPPLY @ A_27a @ A_27b @ ( c_2Efinite__map_2Efdomsub @ A_27a @ A_27b @ V0fm @ V1k1 ) @ V2k2 )
= ( c_2Ebool_2ECOND @ A_27b @ ( c_2Emin_2E_3D @ A_27a @ V1k1 @ V2k2 ) @ ( c_2Efinite__map_2EFAPPLY @ A_27a @ A_27b @ ( c_2Efinite__map_2EFEMPTY @ A_27a @ A_27b ) @ V2k2 ) @ ( c_2Efinite__map_2EFAPPLY @ A_27a @ A_27b @ V0fm @ V2k2 ) ) ) ).
thf(thm_2Efinite__map_2EDOMSUB__FAPPLY__NEQ,axiom,
! [A_27a: $tType,A_27b: $tType,V0fm: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b,V1k1: A_27a,V2k2: A_27a] :
( ( (~) @ ( V1k1 = V2k2 ) )
=> ( ( c_2Efinite__map_2EFAPPLY @ A_27a @ A_27b @ ( c_2Efinite__map_2Efdomsub @ A_27a @ A_27b @ V0fm @ V1k1 ) @ V2k2 )
= ( c_2Efinite__map_2EFAPPLY @ A_27a @ A_27b @ V0fm @ V2k2 ) ) ) ).
thf(thm_2Efinite__map_2EDOMSUB__FAPPLY,axiom,
! [A_27a: $tType,A_27b: $tType,V0fm: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b,V1k: A_27a] :
( ( c_2Efinite__map_2EFAPPLY @ A_27a @ A_27b @ ( c_2Efinite__map_2Efdomsub @ A_27a @ A_27b @ V0fm @ V1k ) @ V1k )
= ( c_2Efinite__map_2EFAPPLY @ A_27a @ A_27b @ ( c_2Efinite__map_2EFEMPTY @ A_27a @ A_27b ) @ V1k ) ) ).
thf(thm_2Efinite__map_2EFDOM__DOMSUB,axiom,
! [A_27a: $tType,A_27b: $tType,V0fm: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b,V1k: A_27a] :
( ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b @ ( c_2Efinite__map_2Efdomsub @ A_27a @ A_27b @ V0fm @ V1k ) )
= ( c_2Epred__set_2EDELETE @ A_27a @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b @ V0fm ) @ V1k ) ) ).
thf(thm_2Efinite__map_2EDOMSUB__FUPDATE__THM,axiom,
! [A_27a: $tType,A_27b: $tType,V0fm: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b,V1k1: A_27a,V2k2: A_27a,V3v: A_27b] :
( ( c_2Efinite__map_2Efdomsub @ A_27a @ A_27b @ ( c_2Efinite__map_2EFUPDATE @ A_27a @ A_27b @ V0fm @ ( c_2Epair_2E_2C @ A_27a @ A_27b @ V1k1 @ V3v ) ) @ V2k2 )
= ( c_2Ebool_2ECOND @ ( tyop_2Efinite__map_2Efmap @ A_27a @ A_27b ) @ ( c_2Emin_2E_3D @ A_27a @ V1k1 @ V2k2 ) @ ( c_2Efinite__map_2Efdomsub @ A_27a @ A_27b @ V0fm @ V2k2 ) @ ( c_2Efinite__map_2EFUPDATE @ A_27a @ A_27b @ ( c_2Efinite__map_2Efdomsub @ A_27a @ A_27b @ V0fm @ V2k2 ) @ ( c_2Epair_2E_2C @ A_27a @ A_27b @ V1k1 @ V3v ) ) ) ) ).
thf(thm_2Efinite__map_2EDOMSUB__FUPDATE__NEQ,axiom,
! [A_27a: $tType,A_27b: $tType,V0fm: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b,V1k1: A_27a,V2k2: A_27a,V3v: A_27b] :
( ( (~) @ ( V1k1 = V2k2 ) )
=> ( ( c_2Efinite__map_2Efdomsub @ A_27a @ A_27b @ ( c_2Efinite__map_2EFUPDATE @ A_27a @ A_27b @ V0fm @ ( c_2Epair_2E_2C @ A_27a @ A_27b @ V1k1 @ V3v ) ) @ V2k2 )
= ( c_2Efinite__map_2EFUPDATE @ A_27a @ A_27b @ ( c_2Efinite__map_2Efdomsub @ A_27a @ A_27b @ V0fm @ V2k2 ) @ ( c_2Epair_2E_2C @ A_27a @ A_27b @ V1k1 @ V3v ) ) ) ) ).
thf(thm_2Efinite__map_2EDOMSUB__FUPDATE,axiom,
! [A_27a: $tType,A_27b: $tType,V0fm: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b,V1k: A_27a,V2v: A_27b] :
( ( c_2Efinite__map_2Efdomsub @ A_27a @ A_27b @ ( c_2Efinite__map_2EFUPDATE @ A_27a @ A_27b @ V0fm @ ( c_2Epair_2E_2C @ A_27a @ A_27b @ V1k @ V2v ) ) @ V1k )
= ( c_2Efinite__map_2Efdomsub @ A_27a @ A_27b @ V0fm @ V1k ) ) ).
thf(thm_2Efinite__map_2EDOMSUB__FEMPTY,axiom,
! [A_27a: $tType,A_27b: $tType,V0k: A_27a] :
( ( c_2Efinite__map_2Efdomsub @ A_27a @ A_27b @ ( c_2Efinite__map_2EFEMPTY @ A_27a @ A_27b ) @ V0k )
= ( c_2Efinite__map_2EFEMPTY @ A_27a @ A_27b ) ) ).
thf(thm_2Efinite__map_2Ef__o__ASSOC,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,A_27d: $tType,V0h: A_27c > A_27a,V1g: A_27a > A_27b,V2f: tyop_2Efinite__map_2Efmap @ A_27b @ A_27d] :
( ( ! [V3x: A_27a,V4y: A_27a] :
( ( ( V1g @ V3x )
= ( V1g @ V4y ) )
<=> ( V3x = V4y ) )
& ! [V5x: A_27c,V6y: A_27c] :
( ( ( V0h @ V5x )
= ( V0h @ V6y ) )
<=> ( V5x = V6y ) ) )
=> ( ( c_2Efinite__map_2Ef__o @ A_27c @ A_27a @ A_27d @ ( c_2Efinite__map_2Ef__o @ A_27a @ A_27b @ A_27d @ V2f @ V1g ) @ V0h )
= ( c_2Efinite__map_2Ef__o @ A_27c @ A_27b @ A_27d @ V2f @ ( c_2Ecombin_2Eo @ A_27c @ A_27b @ A_27a @ V1g @ V0h ) ) ) ) ).
thf(thm_2Efinite__map_2EFINITE__PRED__11,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0g: A_27a > A_27b] :
( ! [V1x: A_27a,V2y: A_27a] :
( ( ( V0g @ V1x )
= ( V0g @ V2y ) )
<=> ( V1x = V2y ) )
=> ! [V3f: tyop_2Efinite__map_2Efmap @ A_27b @ A_27c] :
( c_2Epred__set_2EFINITE @ A_27a
@ ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a
@ ^ [V4x: A_27a] : ( c_2Epair_2E_2C @ A_27a @ $o @ V4x @ ( c_2Ebool_2EIN @ A_27b @ ( V0g @ V4x ) @ ( c_2Efinite__map_2EFDOM @ A_27b @ A_27c @ V3f ) ) ) ) ) ) ).
thf(thm_2Efinite__map_2EFAPPLY__f__o,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0f: tyop_2Efinite__map_2Efmap @ A_27b @ A_27c,V1g: A_27a > A_27b] :
( ( c_2Epred__set_2EFINITE @ A_27a
@ ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a
@ ^ [V2x: A_27a] : ( c_2Epair_2E_2C @ A_27a @ $o @ V2x @ ( c_2Ebool_2EIN @ A_27b @ ( V1g @ V2x ) @ ( c_2Efinite__map_2EFDOM @ A_27b @ A_27c @ V0f ) ) ) ) )
=> ! [V3x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V3x @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27c @ ( c_2Efinite__map_2Ef__o @ A_27a @ A_27b @ A_27c @ V0f @ V1g ) ) )
=> ( ( c_2Efinite__map_2EFAPPLY @ A_27a @ A_27c @ ( c_2Efinite__map_2Ef__o @ A_27a @ A_27b @ A_27c @ V0f @ V1g ) @ V3x )
= ( c_2Efinite__map_2EFAPPLY @ A_27b @ A_27c @ V0f @ ( V1g @ V3x ) ) ) ) ) ).
thf(thm_2Efinite__map_2Ef__o__FUPDATE,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0fm: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b,V1k: A_27a,V2v: A_27b,V3g: A_27c > A_27a] :
( ( ( c_2Epred__set_2EFINITE @ A_27c
@ ( c_2Epred__set_2EGSPEC @ A_27c @ A_27c
@ ^ [V4x: A_27c] : ( c_2Epair_2E_2C @ A_27c @ $o @ V4x @ ( c_2Ebool_2EIN @ A_27a @ ( V3g @ V4x ) @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b @ V0fm ) ) ) ) )
& ( c_2Epred__set_2EFINITE @ A_27c
@ ( c_2Epred__set_2EGSPEC @ A_27c @ A_27c
@ ^ [V5x: A_27c] : ( c_2Epair_2E_2C @ A_27c @ $o @ V5x @ ( c_2Emin_2E_3D @ A_27a @ ( V3g @ V5x ) @ V1k ) ) ) ) )
=> ( ( c_2Efinite__map_2Ef__o @ A_27c @ A_27a @ A_27b @ ( c_2Efinite__map_2EFUPDATE @ A_27a @ A_27b @ V0fm @ ( c_2Epair_2E_2C @ A_27a @ A_27b @ V1k @ V2v ) ) @ V3g )
= ( c_2Efinite__map_2EFMERGE @ A_27b @ A_27c @ ( c_2Ecombin_2EC @ A_27b @ A_27b @ A_27b @ ( c_2Ecombin_2EK @ A_27b @ A_27b ) ) @ ( c_2Efinite__map_2Ef__o @ A_27c @ A_27a @ A_27b @ V0fm @ V3g )
@ ( c_2Efinite__map_2EFUN__FMAP @ A_27c @ A_27b @ ( c_2Ecombin_2EK @ A_27b @ A_27c @ V2v )
@ ( c_2Epred__set_2EGSPEC @ A_27c @ A_27c
@ ^ [V6x: A_27c] : ( c_2Epair_2E_2C @ A_27c @ $o @ V6x @ ( c_2Emin_2E_3D @ A_27a @ ( V3g @ V6x ) @ V1k ) ) ) ) ) ) ) ).
thf(thm_2Efinite__map_2Ef__o__FEMPTY,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0g: A_27a > A_27b] :
( ( c_2Efinite__map_2Ef__o @ A_27a @ A_27b @ A_27c @ ( c_2Efinite__map_2EFEMPTY @ A_27b @ A_27c ) @ V0g )
= ( c_2Efinite__map_2EFEMPTY @ A_27a @ A_27c ) ) ).
thf(thm_2Efinite__map_2EFDOM__f__o,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0f: tyop_2Efinite__map_2Efmap @ A_27b @ A_27c,V1g: A_27a > A_27b] :
( ( c_2Epred__set_2EFINITE @ A_27a
@ ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a
@ ^ [V2x: A_27a] : ( c_2Epair_2E_2C @ A_27a @ $o @ V2x @ ( c_2Ebool_2EIN @ A_27b @ ( V1g @ V2x ) @ ( c_2Efinite__map_2EFDOM @ A_27b @ A_27c @ V0f ) ) ) ) )
=> ( ( c_2Efinite__map_2EFDOM @ A_27a @ A_27c @ ( c_2Efinite__map_2Ef__o @ A_27a @ A_27b @ A_27c @ V0f @ V1g ) )
= ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a
@ ^ [V3x: A_27a] : ( c_2Epair_2E_2C @ A_27a @ $o @ V3x @ ( c_2Ebool_2EIN @ A_27b @ ( V1g @ V3x ) @ ( c_2Efinite__map_2EFDOM @ A_27b @ A_27c @ V0f ) ) ) ) ) ) ).
thf(thm_2Efinite__map_2EFLOOKUP__FUN__FMAP,axiom,
! [A_27a: $tType,A_27b: $tType,V0k: A_27a,V1f: A_27a > A_27b,V2P: A_27a > $o] :
( ( c_2Epred__set_2EFINITE @ A_27a @ V2P )
=> ( ( c_2Efinite__map_2EFLOOKUP @ A_27a @ A_27b @ ( c_2Efinite__map_2EFUN__FMAP @ A_27a @ A_27b @ V1f @ V2P ) @ V0k )
= ( c_2Ebool_2ECOND @ ( tyop_2Eoption_2Eoption @ A_27b ) @ ( c_2Ebool_2EIN @ A_27a @ V0k @ V2P ) @ ( c_2Eoption_2ESOME @ A_27b @ ( V1f @ V0k ) ) @ ( c_2Eoption_2ENONE @ A_27b ) ) ) ) ).
thf(thm_2Efinite__map_2EFDOM__FMAP,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: A_27a > A_27b,V1s: A_27a > $o] :
( ( c_2Epred__set_2EFINITE @ A_27a @ V1s )
=> ( ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b @ ( c_2Efinite__map_2EFUN__FMAP @ A_27a @ A_27b @ V0f @ V1s ) )
= V1s ) ) ).
thf(thm_2Efinite__map_2EFRANGE__FMAP,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: A_27a > A_27b,V1P: A_27a > $o] :
( ( c_2Epred__set_2EFINITE @ A_27a @ V1P )
=> ( ( c_2Efinite__map_2EFRANGE @ A_27a @ A_27b @ ( c_2Efinite__map_2EFUN__FMAP @ A_27a @ A_27b @ V0f @ V1P ) )
= ( c_2Epred__set_2EIMAGE @ A_27a @ A_27b @ V0f @ V1P ) ) ) ).
thf(thm_2Efinite__map_2EFUN__FMAP__EMPTY,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: A_27a > A_27b] :
( ( c_2Efinite__map_2EFUN__FMAP @ A_27a @ A_27b @ V0f @ ( c_2Epred__set_2EEMPTY @ A_27a ) )
= ( c_2Efinite__map_2EFEMPTY @ A_27a @ A_27b ) ) ).
thf(thm_2Efinite__map_2ERRESTRICT__FUPDATE,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b,V1r: A_27b > $o,V2x: A_27a,V3y: A_27b] :
( ( c_2Efinite__map_2ERRESTRICT @ A_27a @ A_27b @ ( c_2Efinite__map_2EFUPDATE @ A_27a @ A_27b @ V0f @ ( c_2Epair_2E_2C @ A_27a @ A_27b @ V2x @ V3y ) ) @ V1r )
= ( c_2Ebool_2ECOND @ ( tyop_2Efinite__map_2Efmap @ A_27a @ A_27b ) @ ( c_2Ebool_2EIN @ A_27b @ V3y @ V1r ) @ ( c_2Efinite__map_2EFUPDATE @ A_27a @ A_27b @ ( c_2Efinite__map_2ERRESTRICT @ A_27a @ A_27b @ V0f @ V1r ) @ ( c_2Epair_2E_2C @ A_27a @ A_27b @ V2x @ V3y ) ) @ ( c_2Efinite__map_2ERRESTRICT @ A_27a @ A_27b @ ( c_2Efinite__map_2EDRESTRICT @ A_27a @ A_27b @ V0f @ ( c_2Epred__set_2ECOMPL @ A_27a @ ( c_2Epred__set_2EINSERT @ A_27a @ V2x @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) ) ) @ V1r ) ) ) ).
thf(thm_2Efinite__map_2ERRESTRICT__FEMPTY,axiom,
! [A_27a: $tType,A_27b: $tType,V0r: A_27a > $o] :
( ( c_2Efinite__map_2ERRESTRICT @ A_27b @ A_27a @ ( c_2Efinite__map_2EFEMPTY @ A_27b @ A_27a ) @ V0r )
= ( c_2Efinite__map_2EFEMPTY @ A_27b @ A_27a ) ) ).
thf(thm_2Efinite__map_2EFRANGE__FUNION,axiom,
! [A_27a: $tType,A_27b: $tType,V0fm2: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b,V1fm1: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b] :
( ( c_2Epred__set_2EDISJOINT @ A_27a @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b @ V1fm1 ) @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b @ V0fm2 ) )
=> ( ( c_2Efinite__map_2EFRANGE @ A_27a @ A_27b @ ( c_2Efinite__map_2EFUNION @ A_27a @ A_27b @ V1fm1 @ V0fm2 ) )
= ( c_2Epred__set_2EUNION @ A_27b @ ( c_2Efinite__map_2EFRANGE @ A_27a @ A_27b @ V1fm1 ) @ ( c_2Efinite__map_2EFRANGE @ A_27a @ A_27b @ V0fm2 ) ) ) ) ).
thf(thm_2Efinite__map_2EFRANGE__FLOOKUP,axiom,
! [A_27a: $tType,A_27b: $tType,V0v: A_27a,V1f: tyop_2Efinite__map_2Efmap @ A_27b @ A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V0v @ ( c_2Efinite__map_2EFRANGE @ A_27b @ A_27a @ V1f ) )
<=> ? [V2k: A_27b] :
( ( c_2Efinite__map_2EFLOOKUP @ A_27b @ A_27a @ V1f @ V2k )
= ( c_2Eoption_2ESOME @ A_27a @ V0v ) ) ) ).
thf(thm_2Efinite__map_2Eo__f__FRANGE,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0x: A_27a,V1g: tyop_2Efinite__map_2Efmap @ A_27b @ A_27a,V2f: A_27a > A_27c] :
( ( c_2Ebool_2EIN @ A_27a @ V0x @ ( c_2Efinite__map_2EFRANGE @ A_27b @ A_27a @ V1g ) )
=> ( c_2Ebool_2EIN @ A_27c @ ( V2f @ V0x ) @ ( c_2Efinite__map_2EFRANGE @ A_27b @ A_27c @ ( c_2Efinite__map_2Eo__f @ A_27b @ A_27a @ A_27c @ V2f @ V1g ) ) ) ) ).
thf(thm_2Efinite__map_2EFINITE__FRANGE,axiom,
! [A_27a: $tType,A_27b: $tType,V0fm: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b] : ( c_2Epred__set_2EFINITE @ A_27b @ ( c_2Efinite__map_2EFRANGE @ A_27a @ A_27b @ V0fm ) ) ).
thf(thm_2Efinite__map_2ESUBMAP__FRANGE,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b,V1g: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b] :
( ( c_2Efinite__map_2ESUBMAP @ A_27a @ A_27b @ V0f @ V1g )
=> ( c_2Epred__set_2ESUBSET @ A_27b @ ( c_2Efinite__map_2EFRANGE @ A_27a @ A_27b @ V0f ) @ ( c_2Efinite__map_2EFRANGE @ A_27a @ A_27b @ V1g ) ) ) ).
thf(thm_2Efinite__map_2EFRANGE__FUPDATE,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b,V1x: A_27a,V2y: A_27b] :
( ( c_2Efinite__map_2EFRANGE @ A_27a @ A_27b @ ( c_2Efinite__map_2EFUPDATE @ A_27a @ A_27b @ V0f @ ( c_2Epair_2E_2C @ A_27a @ A_27b @ V1x @ V2y ) ) )
= ( c_2Epred__set_2EINSERT @ A_27b @ V2y @ ( c_2Efinite__map_2EFRANGE @ A_27a @ A_27b @ ( c_2Efinite__map_2EDRESTRICT @ A_27a @ A_27b @ V0f @ ( c_2Epred__set_2ECOMPL @ A_27a @ ( c_2Epred__set_2EINSERT @ A_27a @ V1x @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) ) ) ) ) ) ).
thf(thm_2Efinite__map_2EFRANGE__FEMPTY,axiom,
! [A_27a: $tType,A_27b: $tType] :
( ( c_2Efinite__map_2EFRANGE @ A_27b @ A_27a @ ( c_2Efinite__map_2EFEMPTY @ A_27b @ A_27a ) )
= ( c_2Epred__set_2EEMPTY @ A_27a ) ) ).
thf(thm_2Efinite__map_2EFLOOKUP__o__f,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0k: A_27b,V1fm: tyop_2Efinite__map_2Efmap @ A_27b @ A_27c,V2f: A_27c > A_27a] :
( ( c_2Efinite__map_2EFLOOKUP @ A_27b @ A_27a @ ( c_2Efinite__map_2Eo__f @ A_27b @ A_27c @ A_27a @ V2f @ V1fm ) @ V0k )
= ( c_2Eoption_2Eoption__CASE @ A_27c @ ( tyop_2Eoption_2Eoption @ A_27a ) @ ( c_2Efinite__map_2EFLOOKUP @ A_27b @ A_27c @ V1fm @ V0k ) @ ( c_2Eoption_2ENONE @ A_27a )
@ ^ [V3v: A_27c] : ( c_2Eoption_2ESOME @ A_27a @ ( V2f @ V3v ) ) ) ) ).
thf(thm_2Efinite__map_2Eo__f__o__f,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,A_27d: $tType,V0h: tyop_2Efinite__map_2Efmap @ A_27a @ A_27d,V1g: A_27d > A_27c,V2f: A_27c > A_27b] :
( ( c_2Efinite__map_2Eo__f @ A_27a @ A_27c @ A_27b @ V2f @ ( c_2Efinite__map_2Eo__f @ A_27a @ A_27d @ A_27c @ V1g @ V0h ) )
= ( c_2Efinite__map_2Eo__f @ A_27a @ A_27d @ A_27b @ ( c_2Ecombin_2Eo @ A_27d @ A_27b @ A_27c @ V2f @ V1g ) @ V0h ) ) ).
thf(thm_2Efinite__map_2EFEVERY__o__f,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0m: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b,V1P: ( tyop_2Epair_2Eprod @ A_27a @ A_27c ) > $o,V2f: A_27b > A_27c] :
( ( c_2Efinite__map_2EFEVERY @ A_27a @ A_27c @ V1P @ ( c_2Efinite__map_2Eo__f @ A_27a @ A_27b @ A_27c @ V2f @ V0m ) )
= ( c_2Efinite__map_2EFEVERY @ A_27a @ A_27b
@ ^ [V3x: tyop_2Epair_2Eprod @ A_27a @ A_27b] : ( V1P @ ( c_2Epair_2E_2C @ A_27a @ A_27c @ ( c_2Epair_2EFST @ A_27a @ A_27b @ V3x ) @ ( V2f @ ( c_2Epair_2ESND @ A_27a @ A_27b @ V3x ) ) ) )
@ V0m ) ) ).
thf(thm_2Efinite__map_2Eo__f__FEMPTY,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0f: A_27c > A_27b] :
( ( c_2Efinite__map_2Eo__f @ A_27a @ A_27c @ A_27b @ V0f @ ( c_2Efinite__map_2EFEMPTY @ A_27a @ A_27c ) )
= ( c_2Efinite__map_2EFEMPTY @ A_27a @ A_27b ) ) ).
thf(thm_2Efinite__map_2Eo__f__FAPPLY,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0f: A_27b > A_27c,V1g: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b,V2x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V2x @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b @ V1g ) )
=> ( ( c_2Efinite__map_2EFAPPLY @ A_27a @ A_27c @ ( c_2Efinite__map_2Eo__f @ A_27a @ A_27b @ A_27c @ V0f @ V1g ) @ V2x )
= ( V0f @ ( c_2Efinite__map_2EFAPPLY @ A_27a @ A_27b @ V1g @ V2x ) ) ) ) ).
thf(thm_2Efinite__map_2EFDOM__o__f,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0f: A_27b > A_27c,V1g: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b] :
( ( c_2Efinite__map_2EFDOM @ A_27a @ A_27c @ ( c_2Efinite__map_2Eo__f @ A_27a @ A_27b @ A_27c @ V0f @ V1g ) )
= ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b @ V1g ) ) ).
thf(thm_2Efinite__map_2Eo__f__FDOM,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0f: A_27b > A_27c,V1g: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b] :
( ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b @ V1g )
= ( c_2Efinite__map_2EFDOM @ A_27a @ A_27c @ ( c_2Efinite__map_2Eo__f @ A_27a @ A_27b @ A_27c @ V0f @ V1g ) ) ) ).
thf(thm_2Efinite__map_2Ef__o__f__FEMPTY__2,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0f: tyop_2Efinite__map_2Efmap @ A_27b @ A_27c] :
( ( c_2Efinite__map_2Ef__o__f @ A_27a @ A_27b @ A_27c @ V0f @ ( c_2Efinite__map_2EFEMPTY @ A_27a @ A_27b ) )
= ( c_2Efinite__map_2EFEMPTY @ A_27a @ A_27c ) ) ).
thf(thm_2Efinite__map_2Ef__o__f__FEMPTY__1,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0f: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b] :
( ( c_2Efinite__map_2Ef__o__f @ A_27a @ A_27b @ A_27c @ ( c_2Efinite__map_2EFEMPTY @ A_27b @ A_27c ) @ V0f )
= ( c_2Efinite__map_2EFEMPTY @ A_27a @ A_27c ) ) ).
thf(thm_2Efinite__map_2EFEVERY__FLOOKUP,axiom,
! [A_27a: $tType,A_27b: $tType,V0v: A_27b,V1k: A_27a,V2f: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b,V3P: ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) > $o] :
( ( ( c_2Efinite__map_2EFEVERY @ A_27a @ A_27b @ V3P @ V2f )
& ( ( c_2Efinite__map_2EFLOOKUP @ A_27a @ A_27b @ V2f @ V1k )
= ( c_2Eoption_2ESOME @ A_27b @ V0v ) ) )
=> ( V3P @ ( c_2Epair_2E_2C @ A_27a @ A_27b @ V1k @ V0v ) ) ) ).
thf(thm_2Efinite__map_2EFEVERY__FUPDATE,axiom,
! [A_27a: $tType,A_27b: $tType,V0P: ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) > $o,V1f: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b,V2x: A_27a,V3y: A_27b] :
( ( c_2Efinite__map_2EFEVERY @ A_27a @ A_27b @ V0P @ ( c_2Efinite__map_2EFUPDATE @ A_27a @ A_27b @ V1f @ ( c_2Epair_2E_2C @ A_27a @ A_27b @ V2x @ V3y ) ) )
<=> ( ( V0P @ ( c_2Epair_2E_2C @ A_27a @ A_27b @ V2x @ V3y ) )
& ( c_2Efinite__map_2EFEVERY @ A_27a @ A_27b @ V0P @ ( c_2Efinite__map_2EDRESTRICT @ A_27a @ A_27b @ V1f @ ( c_2Epred__set_2ECOMPL @ A_27a @ ( c_2Epred__set_2EINSERT @ A_27a @ V2x @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) ) ) ) ) ) ).
thf(thm_2Efinite__map_2EFEVERY__FEMPTY,axiom,
! [A_27a: $tType,A_27b: $tType,V0P: ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) > $o] : ( c_2Efinite__map_2EFEVERY @ A_27a @ A_27b @ V0P @ ( c_2Efinite__map_2EFEMPTY @ A_27a @ A_27b ) ) ).
thf(thm_2Efinite__map_2EFLOOKUP__DRESTRICT,axiom,
! [A_27a: $tType,A_27b: $tType,V0fm: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b,V1s: A_27a > $o,V2k: A_27a] :
( ( c_2Efinite__map_2EFLOOKUP @ A_27a @ A_27b @ ( c_2Efinite__map_2EDRESTRICT @ A_27a @ A_27b @ V0fm @ V1s ) @ V2k )
= ( c_2Ebool_2ECOND @ ( tyop_2Eoption_2Eoption @ A_27b ) @ ( c_2Ebool_2EIN @ A_27a @ V2k @ V1s ) @ ( c_2Efinite__map_2EFLOOKUP @ A_27a @ A_27b @ V0fm @ V2k ) @ ( c_2Eoption_2ENONE @ A_27b ) ) ) ).
thf(thm_2Efinite__map_2Efmap__eq__flookup,axiom,
! [A_27a: $tType,A_27b: $tType,V0f2: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b,V1f1: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b] :
( ( V1f1 = V0f2 )
<=> ! [V2x: A_27a] :
( ( c_2Efinite__map_2EFLOOKUP @ A_27a @ A_27b @ V1f1 @ V2x )
= ( c_2Efinite__map_2EFLOOKUP @ A_27a @ A_27b @ V0f2 @ V2x ) ) ) ).
thf(thm_2Efinite__map_2EFLOOKUP__EXT,axiom,
! [A_27a: $tType,A_27b: $tType,V0f2: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b,V1f1: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b] :
( ( V1f1 = V0f2 )
<=> ( ( c_2Efinite__map_2EFLOOKUP @ A_27a @ A_27b @ V1f1 )
= ( c_2Efinite__map_2EFLOOKUP @ A_27a @ A_27b @ V0f2 ) ) ) ).
thf(thm_2Efinite__map_2EFLOOKUP__FUNION,axiom,
! [A_27a: $tType,A_27b: $tType,V0k: A_27b,V1f2: tyop_2Efinite__map_2Efmap @ A_27b @ A_27a,V2f1: tyop_2Efinite__map_2Efmap @ A_27b @ A_27a] :
( ( c_2Efinite__map_2EFLOOKUP @ A_27b @ A_27a @ ( c_2Efinite__map_2EFUNION @ A_27b @ A_27a @ V2f1 @ V1f2 ) @ V0k )
= ( c_2Eoption_2Eoption__CASE @ A_27a @ ( tyop_2Eoption_2Eoption @ A_27a ) @ ( c_2Efinite__map_2EFLOOKUP @ A_27b @ A_27a @ V2f1 @ V0k ) @ ( c_2Efinite__map_2EFLOOKUP @ A_27b @ A_27a @ V1f2 @ V0k )
@ ^ [V3v: A_27a] : ( c_2Eoption_2ESOME @ A_27a @ V3v ) ) ) ).
thf(thm_2Efinite__map_2ESUBMAP__FUPDATE__FLOOKUP,axiom,
! [A_27a: $tType,A_27b: $tType,V0y: A_27b,V1x: A_27a,V2f: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b] :
( ( c_2Efinite__map_2ESUBMAP @ A_27a @ A_27b @ V2f @ ( c_2Efinite__map_2EFUPDATE @ A_27a @ A_27b @ V2f @ ( c_2Epair_2E_2C @ A_27a @ A_27b @ V1x @ V0y ) ) )
<=> ( ( ( c_2Efinite__map_2EFLOOKUP @ A_27a @ A_27b @ V2f @ V1x )
= ( c_2Eoption_2ENONE @ A_27b ) )
| ( ( c_2Efinite__map_2EFLOOKUP @ A_27a @ A_27b @ V2f @ V1x )
= ( c_2Eoption_2ESOME @ A_27b @ V0y ) ) ) ) ).
thf(thm_2Efinite__map_2EFLOOKUP__SUBMAP,axiom,
! [A_27a: $tType,A_27b: $tType,V0v: A_27b,V1k: A_27a,V2g: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b,V3f: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b] :
( ( ( c_2Efinite__map_2ESUBMAP @ A_27a @ A_27b @ V3f @ V2g )
& ( ( c_2Efinite__map_2EFLOOKUP @ A_27a @ A_27b @ V3f @ V1k )
= ( c_2Eoption_2ESOME @ A_27b @ V0v ) ) )
=> ( ( c_2Efinite__map_2EFLOOKUP @ A_27a @ A_27b @ V2g @ V1k )
= ( c_2Eoption_2ESOME @ A_27b @ V0v ) ) ) ).
thf(thm_2Efinite__map_2EFLOOKUP__UPDATE,axiom,
! [A_27a: $tType,A_27b: $tType,V0v: A_27a,V1k2: A_27b,V2k1: A_27b,V3fm: tyop_2Efinite__map_2Efmap @ A_27b @ A_27a] :
( ( c_2Efinite__map_2EFLOOKUP @ A_27b @ A_27a @ ( c_2Efinite__map_2EFUPDATE @ A_27b @ A_27a @ V3fm @ ( c_2Epair_2E_2C @ A_27b @ A_27a @ V2k1 @ V0v ) ) @ V1k2 )
= ( c_2Ebool_2ECOND @ ( tyop_2Eoption_2Eoption @ A_27a ) @ ( c_2Emin_2E_3D @ A_27b @ V2k1 @ V1k2 ) @ ( c_2Eoption_2ESOME @ A_27a @ V0v ) @ ( c_2Efinite__map_2EFLOOKUP @ A_27b @ A_27a @ V3fm @ V1k2 ) ) ) ).
thf(thm_2Efinite__map_2EFLOOKUP__EMPTY,axiom,
! [A_27a: $tType,A_27b: $tType,V0k: A_27b] :
( ( c_2Efinite__map_2EFLOOKUP @ A_27b @ A_27a @ ( c_2Efinite__map_2EFEMPTY @ A_27b @ A_27a ) @ V0k )
= ( c_2Eoption_2ENONE @ A_27a ) ) ).
thf(thm_2Efinite__map_2EFMERGE__EQ__FEMPTY,axiom,
! [A_27a: $tType,A_27b: $tType,V0m: A_27b > A_27b > A_27b,V1g: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b,V2f: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b] :
( ( ( c_2Efinite__map_2EFMERGE @ A_27b @ A_27a @ V0m @ V2f @ V1g )
= ( c_2Efinite__map_2EFEMPTY @ A_27a @ A_27b ) )
<=> ( ( V2f
= ( c_2Efinite__map_2EFEMPTY @ A_27a @ A_27b ) )
& ( V1g
= ( c_2Efinite__map_2EFEMPTY @ A_27a @ A_27b ) ) ) ) ).
thf(thm_2Efinite__map_2EFMERGE__DRESTRICT,axiom,
! [A_27a: $tType,A_27b: $tType,V0vs: A_27a > $o,V1st2: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b,V2st1: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b,V3f: A_27b > A_27b > A_27b] :
( ( c_2Efinite__map_2EDRESTRICT @ A_27a @ A_27b @ ( c_2Efinite__map_2EFMERGE @ A_27b @ A_27a @ V3f @ V2st1 @ V1st2 ) @ V0vs )
= ( c_2Efinite__map_2EFMERGE @ A_27b @ A_27a @ V3f @ ( c_2Efinite__map_2EDRESTRICT @ A_27a @ A_27b @ V2st1 @ V0vs ) @ ( c_2Efinite__map_2EDRESTRICT @ A_27a @ A_27b @ V1st2 @ V0vs ) ) ) ).
thf(thm_2Efinite__map_2EFMERGE__ASSOC,axiom,
! [A_27a: $tType,A_27b: $tType,V0m: A_27b > A_27b > A_27b] :
( ( c_2Ecombin_2EASSOC @ ( tyop_2Efinite__map_2Efmap @ A_27a @ A_27b ) @ ( c_2Efinite__map_2EFMERGE @ A_27b @ A_27a @ V0m ) )
= ( c_2Ecombin_2EASSOC @ A_27b @ V0m ) ) ).
thf(thm_2Efinite__map_2EFMERGE__COMM,axiom,
! [A_27a: $tType,A_27b: $tType,V0m: A_27b > A_27b > A_27b] :
( ( c_2Ecombin_2ECOMM @ ( tyop_2Efinite__map_2Efmap @ A_27a @ A_27b ) @ ( tyop_2Efinite__map_2Efmap @ A_27a @ A_27b ) @ ( c_2Efinite__map_2EFMERGE @ A_27b @ A_27a @ V0m ) )
= ( c_2Ecombin_2ECOMM @ A_27b @ A_27b @ V0m ) ) ).
thf(thm_2Efinite__map_2EFMERGE__NO__CHANGE,axiom,
! [A_27a: $tType,A_27b: $tType,V0m: A_27b > A_27b > A_27b,V1f2: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b,V2f1: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b] :
( ( ( ( c_2Efinite__map_2EFMERGE @ A_27b @ A_27a @ V0m @ V2f1 @ V1f2 )
= V2f1 )
<=> ! [V3x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V3x @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b @ V1f2 ) )
=> ( ( c_2Ebool_2EIN @ A_27a @ V3x @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b @ V2f1 ) )
& ( ( V0m @ ( c_2Efinite__map_2EFAPPLY @ A_27a @ A_27b @ V2f1 @ V3x ) @ ( c_2Efinite__map_2EFAPPLY @ A_27a @ A_27b @ V1f2 @ V3x ) )
= ( c_2Efinite__map_2EFAPPLY @ A_27a @ A_27b @ V2f1 @ V3x ) ) ) ) )
& ( ( ( c_2Efinite__map_2EFMERGE @ A_27b @ A_27a @ V0m @ V2f1 @ V1f2 )
= V1f2 )
<=> ! [V4x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V4x @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b @ V2f1 ) )
=> ( ( c_2Ebool_2EIN @ A_27a @ V4x @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b @ V1f2 ) )
& ( ( V0m @ ( c_2Efinite__map_2EFAPPLY @ A_27a @ A_27b @ V2f1 @ V4x ) @ ( c_2Efinite__map_2EFAPPLY @ A_27a @ A_27b @ V1f2 @ V4x ) )
= ( c_2Efinite__map_2EFAPPLY @ A_27a @ A_27b @ V1f2 @ V4x ) ) ) ) ) ) ).
thf(thm_2Efinite__map_2EFUNION__FMERGE,axiom,
! [A_27a: $tType,A_27b: $tType,V0f1: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b,V1f2: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b,V2m: A_27b > A_27b > A_27b] :
( ( c_2Epred__set_2EDISJOINT @ A_27a @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b @ V0f1 ) @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b @ V1f2 ) )
=> ( ( c_2Efinite__map_2EFMERGE @ A_27b @ A_27a @ V2m @ V0f1 @ V1f2 )
= ( c_2Efinite__map_2EFUNION @ A_27a @ A_27b @ V0f1 @ V1f2 ) ) ) ).
thf(thm_2Efinite__map_2EFMERGE__FUNION,axiom,
! [A_27a: $tType,A_27b: $tType] :
( ( c_2Efinite__map_2EFUNION @ A_27a @ A_27b )
= ( c_2Efinite__map_2EFMERGE @ A_27b @ A_27a
@ ^ [V0x: A_27b,V1y: A_27b] : V0x ) ) ).
thf(thm_2Efinite__map_2EFDOM__FMERGE,axiom,
! [A_27a: $tType,A_27b: $tType,V0m: A_27a > A_27a > A_27a,V1f: tyop_2Efinite__map_2Efmap @ A_27b @ A_27a,V2g: tyop_2Efinite__map_2Efmap @ A_27b @ A_27a] :
( ( c_2Efinite__map_2EFDOM @ A_27b @ A_27a @ ( c_2Efinite__map_2EFMERGE @ A_27a @ A_27b @ V0m @ V1f @ V2g ) )
= ( c_2Epred__set_2EUNION @ A_27b @ ( c_2Efinite__map_2EFDOM @ A_27b @ A_27a @ V1f ) @ ( c_2Efinite__map_2EFDOM @ A_27b @ A_27a @ V2g ) ) ) ).
thf(thm_2Efinite__map_2EFMERGE__FEMPTY,axiom,
! [A_27a: $tType,A_27b: $tType,V0m: A_27b > A_27b > A_27b,V1f: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b] :
( ( ( c_2Efinite__map_2EFMERGE @ A_27b @ A_27a @ V0m @ V1f @ ( c_2Efinite__map_2EFEMPTY @ A_27a @ A_27b ) )
= V1f )
& ( ( c_2Efinite__map_2EFMERGE @ A_27b @ A_27a @ V0m @ ( c_2Efinite__map_2EFEMPTY @ A_27a @ A_27b ) @ V1f )
= V1f ) ) ).
thf(thm_2Efinite__map_2EFUNION__IDEMPOT,axiom,
! [A_27a: $tType,A_27b: $tType,V0fm: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b] :
( ( c_2Efinite__map_2EFUNION @ A_27a @ A_27b @ V0fm @ V0fm )
= V0fm ) ).
thf(thm_2Efinite__map_2EDRESTRICT__FUNION,axiom,
! [A_27a: $tType,A_27b: $tType,V0h: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b,V1s1: A_27a > $o,V2s2: A_27a > $o] :
( ( c_2Efinite__map_2EFUNION @ A_27a @ A_27b @ ( c_2Efinite__map_2EDRESTRICT @ A_27a @ A_27b @ V0h @ V1s1 ) @ ( c_2Efinite__map_2EDRESTRICT @ A_27a @ A_27b @ V0h @ V2s2 ) )
= ( c_2Efinite__map_2EDRESTRICT @ A_27a @ A_27b @ V0h @ ( c_2Epred__set_2EUNION @ A_27a @ V1s1 @ V2s2 ) ) ) ).
thf(thm_2Efinite__map_2EDRESTRICT__EQ__FUNION,axiom,
! [A_27a: $tType,A_27b: $tType,V0h: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b,V1h1: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b,V2h2: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b] :
( ( ( c_2Epred__set_2EDISJOINT @ A_27a @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b @ V1h1 ) @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b @ V2h2 ) )
& ( ( c_2Efinite__map_2EFUNION @ A_27a @ A_27b @ V1h1 @ V2h2 )
= V0h ) )
=> ( V2h2
= ( c_2Efinite__map_2EDRESTRICT @ A_27a @ A_27b @ V0h @ ( c_2Epred__set_2ECOMPL @ A_27a @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b @ V1h1 ) ) ) ) ) ).
thf(thm_2Efinite__map_2EIN__FDOM__FOLDR__UNION,axiom,
! [A_27a: $tType,A_27b: $tType,V0x: A_27a,V1hL: tyop_2Elist_2Elist @ ( tyop_2Efinite__map_2Efmap @ A_27a @ A_27b )] :
( ( c_2Ebool_2EIN @ A_27a @ V0x @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b @ ( c_2Elist_2EFOLDR @ ( tyop_2Efinite__map_2Efmap @ A_27a @ A_27b ) @ ( tyop_2Efinite__map_2Efmap @ A_27a @ A_27b ) @ ( c_2Efinite__map_2EFUNION @ A_27a @ A_27b ) @ ( c_2Efinite__map_2EFEMPTY @ A_27a @ A_27b ) @ V1hL ) ) )
<=> ? [V2h: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b] :
( ( c_2Ebool_2EIN @ ( tyop_2Efinite__map_2Efmap @ A_27a @ A_27b ) @ V2h @ ( c_2Elist_2ELIST__TO__SET @ ( tyop_2Efinite__map_2Efmap @ A_27a @ A_27b ) @ V1hL ) )
& ( c_2Ebool_2EIN @ A_27a @ V0x @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b @ V2h ) ) ) ) ).
thf(thm_2Efinite__map_2EDRESTRICT__FUNION__DRESTRICT__COMPL,axiom,
! [A_27a: $tType,A_27b: $tType,V0s: A_27a > $o,V1f: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b] :
( ( c_2Efinite__map_2EFUNION @ A_27a @ A_27b @ ( c_2Efinite__map_2EDRESTRICT @ A_27a @ A_27b @ V1f @ V0s ) @ ( c_2Efinite__map_2EDRESTRICT @ A_27a @ A_27b @ V1f @ ( c_2Epred__set_2ECOMPL @ A_27a @ V0s ) ) )
= V1f ) ).
thf(thm_2Efinite__map_2EDRESTRICT__IDEMPOT,axiom,
! [A_27a: $tType,A_27b: $tType,V0s: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b,V1vs: A_27a > $o] :
( ( c_2Efinite__map_2EDRESTRICT @ A_27a @ A_27b @ ( c_2Efinite__map_2EDRESTRICT @ A_27a @ A_27b @ V0s @ V1vs ) @ V1vs )
= ( c_2Efinite__map_2EDRESTRICT @ A_27a @ A_27b @ V0s @ V1vs ) ) ).
thf(thm_2Efinite__map_2ESUBMAP__FUNION__ABSORPTION,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b,V1g: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b] :
( ( c_2Efinite__map_2ESUBMAP @ A_27a @ A_27b @ V0f @ V1g )
<=> ( ( c_2Efinite__map_2EFUNION @ A_27a @ A_27b @ V0f @ V1g )
= V1g ) ) ).
thf(thm_2Efinite__map_2EMAP__KEYS__witness,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType] :
( c_2Ebool_2ELET @ ( ( A_27a > A_27b ) > ( tyop_2Efinite__map_2Efmap @ A_27a @ A_27c ) > ( tyop_2Efinite__map_2Efmap @ A_27b @ A_27c ) ) @ $o
@ ^ [V0m: ( A_27a > A_27b ) > ( tyop_2Efinite__map_2Efmap @ A_27a @ A_27c ) > ( tyop_2Efinite__map_2Efmap @ A_27b @ A_27c )] :
( c_2Ebool_2E_21 @ ( A_27a > A_27b )
@ ^ [V1f: A_27a > A_27b] :
( c_2Ebool_2E_21 @ ( tyop_2Efinite__map_2Efmap @ A_27a @ A_27c )
@ ^ [V2fm: tyop_2Efinite__map_2Efmap @ A_27a @ A_27c] :
( c_2Ebool_2E_2F_5C @ ( c_2Emin_2E_3D @ ( A_27b > $o ) @ ( c_2Efinite__map_2EFDOM @ A_27b @ A_27c @ ( V0m @ V1f @ V2fm ) ) @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27b @ V1f @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27c @ V2fm ) ) )
@ ( c_2Emin_2E_3D_3D_3E @ ( c_2Epred__set_2EINJ @ A_27a @ A_27b @ V1f @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27c @ V2fm ) @ ( c_2Epred__set_2EUNIV @ A_27b ) )
@ ( c_2Ebool_2E_21 @ A_27a
@ ^ [V3x: A_27a] : ( c_2Emin_2E_3D_3D_3E @ ( c_2Ebool_2EIN @ A_27a @ V3x @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27c @ V2fm ) ) @ ( c_2Emin_2E_3D @ A_27c @ ( c_2Efinite__map_2EFAPPLY @ A_27b @ A_27c @ ( V0m @ V1f @ V2fm ) @ ( V1f @ V3x ) ) @ ( c_2Efinite__map_2EFAPPLY @ A_27a @ A_27c @ V2fm @ V3x ) ) ) ) ) ) ) )
@ ^ [V4f: A_27a > A_27b,V5fm: tyop_2Efinite__map_2Efmap @ A_27a @ A_27c] : ( c_2Ebool_2ECOND @ ( tyop_2Efinite__map_2Efmap @ A_27b @ A_27c ) @ ( c_2Epred__set_2EINJ @ A_27a @ A_27b @ V4f @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27c @ V5fm ) @ ( c_2Epred__set_2EUNIV @ A_27b ) ) @ ( c_2Efinite__map_2Ef__o__f @ A_27b @ A_27a @ A_27c @ V5fm @ ( c_2Efinite__map_2EFUN__FMAP @ A_27b @ A_27a @ ( c_2Epred__set_2ELINV @ A_27a @ A_27b @ V4f @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27c @ V5fm ) ) @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27b @ V4f @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27c @ V5fm ) ) ) ) @ ( c_2Efinite__map_2EFUN__FMAP @ A_27b @ A_27c @ ( c_2Ebool_2EARB @ ( A_27b > A_27c ) ) @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27b @ V4f @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27c @ V5fm ) ) ) ) ) ).
thf(thm_2Efinite__map_2EMAP__KEYS__FEMPTY,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0f: A_27a > A_27b] :
( ( c_2Efinite__map_2EMAP__KEYS @ A_27a @ A_27b @ A_27c @ V0f @ ( c_2Efinite__map_2EFEMPTY @ A_27a @ A_27c ) )
= ( c_2Efinite__map_2EFEMPTY @ A_27b @ A_27c ) ) ).
thf(thm_2Efinite__map_2EMAP__KEYS__FUPDATE,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0f: A_27a > A_27b,V1fm: tyop_2Efinite__map_2Efmap @ A_27a @ A_27c,V2k: A_27a,V3v: A_27c] :
( ( c_2Epred__set_2EINJ @ A_27a @ A_27b @ V0f @ ( c_2Epred__set_2EINSERT @ A_27a @ V2k @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27c @ V1fm ) ) @ ( c_2Epred__set_2EUNIV @ A_27b ) )
=> ( ( c_2Efinite__map_2EMAP__KEYS @ A_27a @ A_27b @ A_27c @ V0f @ ( c_2Efinite__map_2EFUPDATE @ A_27a @ A_27c @ V1fm @ ( c_2Epair_2E_2C @ A_27a @ A_27c @ V2k @ V3v ) ) )
= ( c_2Efinite__map_2EFUPDATE @ A_27b @ A_27c @ ( c_2Efinite__map_2EMAP__KEYS @ A_27a @ A_27b @ A_27c @ V0f @ V1fm ) @ ( c_2Epair_2E_2C @ A_27b @ A_27c @ ( V0f @ V2k ) @ V3v ) ) ) ) ).
thf(thm_2Efinite__map_2EMAP__KEYS__using__LINV,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0f: A_27a > A_27b,V1fm: tyop_2Efinite__map_2Efmap @ A_27a @ A_27c] :
( ( c_2Epred__set_2EINJ @ A_27a @ A_27b @ V0f @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27c @ V1fm ) @ ( c_2Epred__set_2EUNIV @ A_27b ) )
=> ( ( c_2Efinite__map_2EMAP__KEYS @ A_27a @ A_27b @ A_27c @ V0f @ V1fm )
= ( c_2Efinite__map_2Ef__o__f @ A_27b @ A_27a @ A_27c @ V1fm @ ( c_2Efinite__map_2EFUN__FMAP @ A_27b @ A_27a @ ( c_2Epred__set_2ELINV @ A_27a @ A_27b @ V0f @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27c @ V1fm ) ) @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27b @ V0f @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27c @ V1fm ) ) ) ) ) ) ).
thf(thm_2Efinite__map_2EMAP__KEYS__BIJ__LINV,axiom,
! [A_27a: $tType,V0t: tyop_2Efinite__map_2Efmap @ tyop_2Enum_2Enum @ A_27a,V1f: tyop_2Enum_2Enum > tyop_2Enum_2Enum] :
( ( c_2Epred__set_2EBIJ @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum @ V1f @ ( c_2Epred__set_2EUNIV @ tyop_2Enum_2Enum ) @ ( c_2Epred__set_2EUNIV @ tyop_2Enum_2Enum ) )
=> ( ( c_2Efinite__map_2EMAP__KEYS @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum @ A_27a @ V1f @ ( c_2Efinite__map_2EMAP__KEYS @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum @ A_27a @ ( c_2Epred__set_2ELINV @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum @ V1f @ ( c_2Epred__set_2EUNIV @ tyop_2Enum_2Enum ) ) @ V0t ) )
= V0t ) ) ).
thf(thm_2Efinite__map_2EFLOOKUP__MAP__KEYS,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0m: tyop_2Efinite__map_2Efmap @ A_27a @ A_27c,V1k: A_27b,V2f: A_27a > A_27b] :
( ( c_2Epred__set_2EINJ @ A_27a @ A_27b @ V2f @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27c @ V0m ) @ ( c_2Epred__set_2EUNIV @ A_27b ) )
=> ( ( c_2Efinite__map_2EFLOOKUP @ A_27b @ A_27c @ ( c_2Efinite__map_2EMAP__KEYS @ A_27a @ A_27b @ A_27c @ V2f @ V0m ) @ V1k )
= ( c_2Eoption_2EOPTION__BIND @ A_27c @ A_27a
@ ( c_2Eoption_2Esome @ A_27a
@ ^ [V3x: A_27a] : ( c_2Ebool_2E_2F_5C @ ( c_2Emin_2E_3D @ A_27b @ V1k @ ( V2f @ V3x ) ) @ ( c_2Ebool_2EIN @ A_27a @ V3x @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27c @ V0m ) ) ) )
@ ( c_2Efinite__map_2EFLOOKUP @ A_27a @ A_27c @ V0m ) ) ) ) ).
thf(thm_2Efinite__map_2EFLOOKUP__MAP__KEYS__MAPPED,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0m: tyop_2Efinite__map_2Efmap @ A_27a @ A_27c,V1k: A_27a,V2f: A_27a > A_27b] :
( ( c_2Epred__set_2EINJ @ A_27a @ A_27b @ V2f @ ( c_2Epred__set_2EUNIV @ A_27a ) @ ( c_2Epred__set_2EUNIV @ A_27b ) )
=> ( ( c_2Efinite__map_2EFLOOKUP @ A_27b @ A_27c @ ( c_2Efinite__map_2EMAP__KEYS @ A_27a @ A_27b @ A_27c @ V2f @ V0m ) @ ( V2f @ V1k ) )
= ( c_2Efinite__map_2EFLOOKUP @ A_27a @ A_27c @ V0m @ V1k ) ) ) ).
thf(thm_2Efinite__map_2EDRESTRICT__MAP__KEYS__IMAGE,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0s: A_27a > $o,V1fm: tyop_2Efinite__map_2Efmap @ A_27a @ A_27c,V2f: A_27a > A_27b] :
( ( c_2Epred__set_2EINJ @ A_27a @ A_27b @ V2f @ ( c_2Epred__set_2EUNIV @ A_27a ) @ ( c_2Epred__set_2EUNIV @ A_27b ) )
=> ( ( c_2Efinite__map_2EDRESTRICT @ A_27b @ A_27c @ ( c_2Efinite__map_2EMAP__KEYS @ A_27a @ A_27b @ A_27c @ V2f @ V1fm ) @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27b @ V2f @ V0s ) )
= ( c_2Efinite__map_2EMAP__KEYS @ A_27a @ A_27b @ A_27c @ V2f @ ( c_2Efinite__map_2EDRESTRICT @ A_27a @ A_27c @ V1fm @ V0s ) ) ) ) ).
thf(thm_2Efinite__map_2EDOMSUB__MAP__KEYS,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0s: A_27a,V1fm: tyop_2Efinite__map_2Efmap @ A_27a @ A_27c,V2f: A_27a > A_27b] :
( ( c_2Epred__set_2EBIJ @ A_27a @ A_27b @ V2f @ ( c_2Epred__set_2EUNIV @ A_27a ) @ ( c_2Epred__set_2EUNIV @ A_27b ) )
=> ( ( c_2Efinite__map_2Efdomsub @ A_27b @ A_27c @ ( c_2Efinite__map_2EMAP__KEYS @ A_27a @ A_27b @ A_27c @ V2f @ V1fm ) @ ( V2f @ V0s ) )
= ( c_2Efinite__map_2EMAP__KEYS @ A_27a @ A_27b @ A_27c @ V2f @ ( c_2Efinite__map_2Efdomsub @ A_27a @ A_27c @ V1fm @ V0s ) ) ) ) ).
thf(thm_2Efinite__map_2Efmap__rel__FUPDATE__same,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0v2: A_27b,V1v1: A_27a,V2k: A_27c,V3f2: tyop_2Efinite__map_2Efmap @ A_27c @ A_27b,V4f1: tyop_2Efinite__map_2Efmap @ A_27c @ A_27a,V5R: A_27a > A_27b > $o] :
( ( ( c_2Efinite__map_2Efmap__rel @ A_27a @ A_27b @ A_27c @ V5R @ V4f1 @ V3f2 )
& ( V5R @ V1v1 @ V0v2 ) )
=> ( c_2Efinite__map_2Efmap__rel @ A_27a @ A_27b @ A_27c @ V5R @ ( c_2Efinite__map_2EFUPDATE @ A_27c @ A_27a @ V4f1 @ ( c_2Epair_2E_2C @ A_27c @ A_27a @ V2k @ V1v1 ) ) @ ( c_2Efinite__map_2EFUPDATE @ A_27c @ A_27b @ V3f2 @ ( c_2Epair_2E_2C @ A_27c @ A_27b @ V2k @ V0v2 ) ) ) ) ).
thf(thm_2Efinite__map_2Efmap__rel__FUPDATE__LIST__same,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0R: A_27a > A_27b > $o,V1ls1: tyop_2Elist_2Elist @ ( tyop_2Epair_2Eprod @ A_27c @ A_27a ),V2ls2: tyop_2Elist_2Elist @ ( tyop_2Epair_2Eprod @ A_27c @ A_27b ),V3f1: tyop_2Efinite__map_2Efmap @ A_27c @ A_27a,V4f2: tyop_2Efinite__map_2Efmap @ A_27c @ A_27b] :
( ( ( c_2Efinite__map_2Efmap__rel @ A_27a @ A_27b @ A_27c @ V0R @ V3f1 @ V4f2 )
& ( ( c_2Elist_2EMAP @ ( tyop_2Epair_2Eprod @ A_27c @ A_27a ) @ A_27c @ ( c_2Epair_2EFST @ A_27c @ A_27a ) @ V1ls1 )
= ( c_2Elist_2EMAP @ ( tyop_2Epair_2Eprod @ A_27c @ A_27b ) @ A_27c @ ( c_2Epair_2EFST @ A_27c @ A_27b ) @ V2ls2 ) )
& ( c_2Elist_2ELIST__REL @ A_27a @ A_27b @ V0R @ ( c_2Elist_2EMAP @ ( tyop_2Epair_2Eprod @ A_27c @ A_27a ) @ A_27a @ ( c_2Epair_2ESND @ A_27c @ A_27a ) @ V1ls1 ) @ ( c_2Elist_2EMAP @ ( tyop_2Epair_2Eprod @ A_27c @ A_27b ) @ A_27b @ ( c_2Epair_2ESND @ A_27c @ A_27b ) @ V2ls2 ) ) )
=> ( c_2Efinite__map_2Efmap__rel @ A_27a @ A_27b @ A_27c @ V0R @ ( c_2Efinite__map_2EFUPDATE__LIST @ A_27c @ A_27a @ V3f1 @ V1ls1 ) @ ( c_2Efinite__map_2EFUPDATE__LIST @ A_27c @ A_27b @ V4f2 @ V2ls2 ) ) ) ).
thf(thm_2Efinite__map_2Efmap__rel__FEMPTY,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0R: A_27a > A_27b > $o] : ( c_2Efinite__map_2Efmap__rel @ A_27a @ A_27b @ A_27c @ V0R @ ( c_2Efinite__map_2EFEMPTY @ A_27c @ A_27a ) @ ( c_2Efinite__map_2EFEMPTY @ A_27c @ A_27b ) ) ).
thf(thm_2Efinite__map_2Efmap__rel__FEMPTY2,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: tyop_2Efinite__map_2Efmap @ A_27b @ A_27a,V1R: A_27a > A_27a > $o] :
( ( ( c_2Efinite__map_2Efmap__rel @ A_27a @ A_27a @ A_27b @ V1R @ ( c_2Efinite__map_2EFEMPTY @ A_27b @ A_27a ) @ V0f )
<=> ( V0f
= ( c_2Efinite__map_2EFEMPTY @ A_27b @ A_27a ) ) )
& ( ( c_2Efinite__map_2Efmap__rel @ A_27a @ A_27a @ A_27b @ V1R @ V0f @ ( c_2Efinite__map_2EFEMPTY @ A_27b @ A_27a ) )
<=> ( V0f
= ( c_2Efinite__map_2EFEMPTY @ A_27b @ A_27a ) ) ) ) ).
thf(thm_2Efinite__map_2Efmap__rel__refl,axiom,
! [A_27a: $tType,A_27b: $tType,V0x: tyop_2Efinite__map_2Efmap @ A_27b @ A_27a,V1R: A_27a > A_27a > $o] :
( ! [V2x: A_27a] : ( V1R @ V2x @ V2x )
=> ( c_2Efinite__map_2Efmap__rel @ A_27a @ A_27a @ A_27b @ V1R @ V0x @ V0x ) ) ).
thf(thm_2Efinite__map_2Efmap__rel__FUNION__rels,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0f4: tyop_2Efinite__map_2Efmap @ A_27c @ A_27b,V1f3: tyop_2Efinite__map_2Efmap @ A_27c @ A_27a,V2f2: tyop_2Efinite__map_2Efmap @ A_27c @ A_27b,V3f1: tyop_2Efinite__map_2Efmap @ A_27c @ A_27a,V4R: A_27a > A_27b > $o] :
( ( ( c_2Efinite__map_2Efmap__rel @ A_27a @ A_27b @ A_27c @ V4R @ V3f1 @ V2f2 )
& ( c_2Efinite__map_2Efmap__rel @ A_27a @ A_27b @ A_27c @ V4R @ V1f3 @ V0f4 ) )
=> ( c_2Efinite__map_2Efmap__rel @ A_27a @ A_27b @ A_27c @ V4R @ ( c_2Efinite__map_2EFUNION @ A_27c @ A_27a @ V3f1 @ V1f3 ) @ ( c_2Efinite__map_2EFUNION @ A_27c @ A_27b @ V2f2 @ V0f4 ) ) ) ).
thf(thm_2Efinite__map_2Efmap__rel__FUPDATE__I,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0v2: A_27b,V1v1: A_27a,V2k: A_27c,V3f2: tyop_2Efinite__map_2Efmap @ A_27c @ A_27b,V4f1: tyop_2Efinite__map_2Efmap @ A_27c @ A_27a,V5R: A_27a > A_27b > $o] :
( ( ( c_2Efinite__map_2Efmap__rel @ A_27a @ A_27b @ A_27c @ V5R @ ( c_2Efinite__map_2Efdomsub @ A_27c @ A_27a @ V4f1 @ V2k ) @ ( c_2Efinite__map_2Efdomsub @ A_27c @ A_27b @ V3f2 @ V2k ) )
& ( V5R @ V1v1 @ V0v2 ) )
=> ( c_2Efinite__map_2Efmap__rel @ A_27a @ A_27b @ A_27c @ V5R @ ( c_2Efinite__map_2EFUPDATE @ A_27c @ A_27a @ V4f1 @ ( c_2Epair_2E_2C @ A_27c @ A_27a @ V2k @ V1v1 ) ) @ ( c_2Efinite__map_2EFUPDATE @ A_27c @ A_27b @ V3f2 @ ( c_2Epair_2E_2C @ A_27c @ A_27b @ V2k @ V0v2 ) ) ) ) ).
thf(thm_2Efinite__map_2Efmap__rel__mono,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0f2: tyop_2Efinite__map_2Efmap @ A_27c @ A_27b,V1f1: tyop_2Efinite__map_2Efmap @ A_27c @ A_27a,V2R2: A_27a > A_27b > $o,V3R1: A_27a > A_27b > $o] :
( ! [V4x: A_27a,V5y: A_27b] :
( ( V3R1 @ V4x @ V5y )
=> ( V2R2 @ V4x @ V5y ) )
=> ( ( c_2Efinite__map_2Efmap__rel @ A_27a @ A_27b @ A_27c @ V3R1 @ V1f1 @ V0f2 )
=> ( c_2Efinite__map_2Efmap__rel @ A_27a @ A_27b @ A_27c @ V2R2 @ V1f1 @ V0f2 ) ) ) ).
thf(thm_2Efinite__map_2Efmap__rel__OPTREL__FLOOKUP,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0f2: tyop_2Efinite__map_2Efmap @ A_27c @ A_27b,V1f1: tyop_2Efinite__map_2Efmap @ A_27c @ A_27a,V2R: A_27a > A_27b > $o] :
( ( c_2Efinite__map_2Efmap__rel @ A_27a @ A_27b @ A_27c @ V2R @ V1f1 @ V0f2 )
<=> ! [V3k: A_27c] : ( c_2Eoption_2EOPTREL @ A_27a @ A_27b @ V2R @ ( c_2Efinite__map_2EFLOOKUP @ A_27c @ A_27a @ V1f1 @ V3k ) @ ( c_2Efinite__map_2EFLOOKUP @ A_27c @ A_27b @ V0f2 @ V3k ) ) ) ).
thf(thm_2Efinite__map_2Efmap__rel__FLOOKUP__imp,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0f2: tyop_2Efinite__map_2Efmap @ A_27c @ A_27b,V1f1: tyop_2Efinite__map_2Efmap @ A_27c @ A_27a,V2R: A_27a > A_27b > $o] :
( ( c_2Efinite__map_2Efmap__rel @ A_27a @ A_27b @ A_27c @ V2R @ V1f1 @ V0f2 )
=> ( ! [V3k: A_27c] :
( ( ( c_2Efinite__map_2EFLOOKUP @ A_27c @ A_27a @ V1f1 @ V3k )
= ( c_2Eoption_2ENONE @ A_27a ) )
=> ( ( c_2Efinite__map_2EFLOOKUP @ A_27c @ A_27b @ V0f2 @ V3k )
= ( c_2Eoption_2ENONE @ A_27b ) ) )
& ! [V4k: A_27c,V5v1: A_27a] :
( ( ( c_2Efinite__map_2EFLOOKUP @ A_27c @ A_27a @ V1f1 @ V4k )
= ( c_2Eoption_2ESOME @ A_27a @ V5v1 ) )
=> ? [V6v2: A_27b] :
( ( ( c_2Efinite__map_2EFLOOKUP @ A_27c @ A_27b @ V0f2 @ V4k )
= ( c_2Eoption_2ESOME @ A_27b @ V6v2 ) )
& ( V2R @ V5v1 @ V6v2 ) ) ) ) ) ).
thf(thm_2Efinite__map_2Efmap__EQ__UPTO______EMPTY,axiom,
! [A_27a: $tType,A_27b: $tType,V0f1: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b,V1f2: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b] :
( ( c_2Efinite__map_2Efmap__EQ__UPTO @ A_27a @ A_27b @ V0f1 @ V1f2 @ ( c_2Epred__set_2EEMPTY @ A_27a ) )
<=> ( V0f1 = V1f2 ) ) ).
thf(thm_2Efinite__map_2Efmap__EQ__UPTO______EQ,axiom,
! [A_27a: $tType,A_27b: $tType,V0vs: A_27a > $o,V1f: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b] : ( c_2Efinite__map_2Efmap__EQ__UPTO @ A_27a @ A_27b @ V1f @ V1f @ V0vs ) ).
thf(thm_2Efinite__map_2Efmap__EQ__UPTO______FUPDATE__BOTH,axiom,
! [A_27a: $tType,A_27b: $tType,V0f1: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b,V1f2: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b,V2ks: A_27a > $o,V3k: A_27a,V4v: A_27b] :
( ( c_2Efinite__map_2Efmap__EQ__UPTO @ A_27a @ A_27b @ V0f1 @ V1f2 @ V2ks )
=> ( c_2Efinite__map_2Efmap__EQ__UPTO @ A_27a @ A_27b @ ( c_2Efinite__map_2EFUPDATE @ A_27a @ A_27b @ V0f1 @ ( c_2Epair_2E_2C @ A_27a @ A_27b @ V3k @ V4v ) ) @ ( c_2Efinite__map_2EFUPDATE @ A_27a @ A_27b @ V1f2 @ ( c_2Epair_2E_2C @ A_27a @ A_27b @ V3k @ V4v ) ) @ ( c_2Epred__set_2EDELETE @ A_27a @ V2ks @ V3k ) ) ) ).
thf(thm_2Efinite__map_2Efmap__EQ__UPTO______FUPDATE__BOTH______NO__DELETE,axiom,
! [A_27a: $tType,A_27b: $tType,V0f1: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b,V1f2: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b,V2ks: A_27a > $o,V3k: A_27a,V4v: A_27b] :
( ( c_2Efinite__map_2Efmap__EQ__UPTO @ A_27a @ A_27b @ V0f1 @ V1f2 @ V2ks )
=> ( c_2Efinite__map_2Efmap__EQ__UPTO @ A_27a @ A_27b @ ( c_2Efinite__map_2EFUPDATE @ A_27a @ A_27b @ V0f1 @ ( c_2Epair_2E_2C @ A_27a @ A_27b @ V3k @ V4v ) ) @ ( c_2Efinite__map_2EFUPDATE @ A_27a @ A_27b @ V1f2 @ ( c_2Epair_2E_2C @ A_27a @ A_27b @ V3k @ V4v ) ) @ V2ks ) ) ).
thf(thm_2Efinite__map_2Efmap__EQ__UPTO______FUPDATE__SING,axiom,
! [A_27a: $tType,A_27b: $tType,V0f1: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b,V1f2: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b,V2ks: A_27a > $o,V3k: A_27a,V4v: A_27b] :
( ( c_2Efinite__map_2Efmap__EQ__UPTO @ A_27a @ A_27b @ V0f1 @ V1f2 @ V2ks )
=> ( c_2Efinite__map_2Efmap__EQ__UPTO @ A_27a @ A_27b @ ( c_2Efinite__map_2EFUPDATE @ A_27a @ A_27b @ V0f1 @ ( c_2Epair_2E_2C @ A_27a @ A_27b @ V3k @ V4v ) ) @ V1f2 @ ( c_2Epred__set_2EINSERT @ A_27a @ V3k @ V2ks ) ) ) ).
thf(thm_2Efinite__map_2Eo__f__FUNION,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0f2: tyop_2Efinite__map_2Efmap @ A_27a @ A_27c,V1f1: tyop_2Efinite__map_2Efmap @ A_27a @ A_27c,V2f: A_27c > A_27b] :
( ( c_2Efinite__map_2Eo__f @ A_27a @ A_27c @ A_27b @ V2f @ ( c_2Efinite__map_2EFUNION @ A_27a @ A_27c @ V1f1 @ V0f2 ) )
= ( c_2Efinite__map_2EFUNION @ A_27a @ A_27b @ ( c_2Efinite__map_2Eo__f @ A_27a @ A_27c @ A_27b @ V2f @ V1f1 ) @ ( c_2Efinite__map_2Eo__f @ A_27a @ A_27c @ A_27b @ V2f @ V0f2 ) ) ) ).
thf(thm_2Efinite__map_2EFDOM__FOLDR__DOMSUB,axiom,
! [A_27a: $tType,A_27b: $tType,V0ls: tyop_2Elist_2Elist @ A_27a,V1fm: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b] :
( ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b
@ ( c_2Elist_2EFOLDR @ A_27a @ ( tyop_2Efinite__map_2Efmap @ A_27a @ A_27b )
@ ^ [V2k: A_27a,V3m: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b] : ( c_2Efinite__map_2Efdomsub @ A_27a @ A_27b @ V3m @ V2k )
@ V1fm
@ V0ls ) )
= ( c_2Epred__set_2EDIFF @ A_27a @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b @ V1fm ) @ ( c_2Elist_2ELIST__TO__SET @ A_27a @ V0ls ) ) ) ).
thf(thm_2Efinite__map_2EFEVERY__SUBMAP,axiom,
! [A_27a: $tType,A_27b: $tType,V0fm0: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b,V1fm: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b,V2P: ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) > $o] :
( ( ( c_2Efinite__map_2EFEVERY @ A_27a @ A_27b @ V2P @ V1fm )
& ( c_2Efinite__map_2ESUBMAP @ A_27a @ A_27b @ V0fm0 @ V1fm ) )
=> ( c_2Efinite__map_2EFEVERY @ A_27a @ A_27b @ V2P @ V0fm0 ) ) ).
thf(thm_2Efinite__map_2EFEVERY__ALL__FLOOKUP,axiom,
! [A_27a: $tType,A_27b: $tType,V0P: ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) > $o,V1f: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b] :
( ( c_2Efinite__map_2EFEVERY @ A_27a @ A_27b @ V0P @ V1f )
<=> ! [V2k: A_27a,V3v: A_27b] :
( ( ( c_2Efinite__map_2EFLOOKUP @ A_27a @ A_27b @ V1f @ V2k )
= ( c_2Eoption_2ESOME @ A_27b @ V3v ) )
=> ( V0P @ ( c_2Epair_2E_2C @ A_27a @ A_27b @ V2k @ V3v ) ) ) ) ).
thf(thm_2Efinite__map_2EFUPDATE__LIST__CANCEL,axiom,
! [A_27a: $tType,A_27b: $tType,V0ls1: tyop_2Elist_2Elist @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ),V1fm: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b,V2ls2: tyop_2Elist_2Elist @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b )] :
( ! [V3k: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V3k @ ( c_2Elist_2ELIST__TO__SET @ A_27a @ ( c_2Elist_2EMAP @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ A_27a @ ( c_2Epair_2EFST @ A_27a @ A_27b ) @ V0ls1 ) ) )
=> ( c_2Ebool_2EIN @ A_27a @ V3k @ ( c_2Elist_2ELIST__TO__SET @ A_27a @ ( c_2Elist_2EMAP @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ A_27a @ ( c_2Epair_2EFST @ A_27a @ A_27b ) @ V2ls2 ) ) ) )
=> ( ( c_2Efinite__map_2EFUPDATE__LIST @ A_27a @ A_27b @ ( c_2Efinite__map_2EFUPDATE__LIST @ A_27a @ A_27b @ V1fm @ V0ls1 ) @ V2ls2 )
= ( c_2Efinite__map_2EFUPDATE__LIST @ A_27a @ A_27b @ V1fm @ V2ls2 ) ) ) ).
thf(thm_2Efinite__map_2EFUPDATE__EQ__FUNION,axiom,
! [A_27a: $tType,A_27b: $tType,V0fm: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b,V1kv: tyop_2Epair_2Eprod @ A_27a @ A_27b] :
( ( c_2Efinite__map_2EFUPDATE @ A_27a @ A_27b @ V0fm @ V1kv )
= ( c_2Efinite__map_2EFUNION @ A_27a @ A_27b @ ( c_2Efinite__map_2EFUPDATE @ A_27a @ A_27b @ ( c_2Efinite__map_2EFEMPTY @ A_27a @ A_27b ) @ V1kv ) @ V0fm ) ) ).
thf(thm_2Efinite__map_2EFUPDATE__LIST__APPEND__COMMUTES,axiom,
! [A_27a: $tType,A_27b: $tType,V0l1: tyop_2Elist_2Elist @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ),V1l2: tyop_2Elist_2Elist @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ),V2fm: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b] :
( ( c_2Epred__set_2EDISJOINT @ A_27a @ ( c_2Elist_2ELIST__TO__SET @ A_27a @ ( c_2Elist_2EMAP @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ A_27a @ ( c_2Epair_2EFST @ A_27a @ A_27b ) @ V0l1 ) ) @ ( c_2Elist_2ELIST__TO__SET @ A_27a @ ( c_2Elist_2EMAP @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ A_27a @ ( c_2Epair_2EFST @ A_27a @ A_27b ) @ V1l2 ) ) )
=> ( ( c_2Efinite__map_2EFUPDATE__LIST @ A_27a @ A_27b @ ( c_2Efinite__map_2EFUPDATE__LIST @ A_27a @ A_27b @ V2fm @ V0l1 ) @ V1l2 )
= ( c_2Efinite__map_2EFUPDATE__LIST @ A_27a @ A_27b @ ( c_2Efinite__map_2EFUPDATE__LIST @ A_27a @ A_27b @ V2fm @ V1l2 ) @ V0l1 ) ) ) ).
thf(thm_2Efinite__map_2EFUPDATE__LIST__ALL__DISTINCT__PERM,axiom,
! [A_27a: $tType,A_27b: $tType,V0ls: tyop_2Elist_2Elist @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ),V1ls_27: tyop_2Elist_2Elist @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ),V2fm: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b] :
( ( ( c_2Elist_2EALL__DISTINCT @ A_27a @ ( c_2Elist_2EMAP @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ A_27a @ ( c_2Epair_2EFST @ A_27a @ A_27b ) @ V0ls ) )
& ( c_2Esorting_2EPERM @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ V0ls @ V1ls_27 ) )
=> ( ( c_2Efinite__map_2EFUPDATE__LIST @ A_27a @ A_27b @ V2fm @ V0ls )
= ( c_2Efinite__map_2EFUPDATE__LIST @ A_27a @ A_27b @ V2fm @ V1ls_27 ) ) ) ).
%------------------------------------------------------------------------------