ITP001 Axioms: ITP081^7.ax
%------------------------------------------------------------------------------
% File : ITP081^7 : TPTP v8.2.0. Bugfixed v7.5.0.
% Domain : Interactive Theorem Proving
% Axioms : HOL4 syntactic export, chainy mode
% Version : [BG+19] axioms.
% English :
% Refs : [BG+19] Brown et al. (2019), GRUNGE: A Grand Unified ATP Chall
% : [Gau19] Gauthier (2019), Email to Geoff Sutcliffe
% Source : [BG+19]
% Names : alist.ax [Gau19]
% : HL4081^7.ax [TPAP]
% Status : Satisfiable
% Syntax : Number of formulae : 132 ( 28 unt; 67 typ; 0 def)
% Number of atoms : 161 ( 72 equ; 4 cnn)
% Maximal formula atoms : 6 ( 1 avg)
% Number of connectives : 1820 ( 4 ~; 1 |; 17 &;1753 @)
% ( 13 <=>; 32 =>; 0 <=; 0 <~>)
% Maximal formula depth : 21 ( 9 avg;1753 nst)
% Number of types : 3 ( 2 usr)
% Number of type conns : 171 ( 171 >; 0 *; 0 +; 0 <<)
% Number of symbols : 67 ( 65 usr; 1 con; 0-6 aty)
% Number of variables : 401 ( 25 ^ 289 !; 1 ?; 401 :)
% ( 86 !>; 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_2Eoption_2Eoption,type,
tyop_2Eoption_2Eoption: $tType > $tType ).
thf(tyop_2Epair_2Eprod,type,
tyop_2Epair_2Eprod: $tType > $tType > $tType ).
thf(c_2Ebool_2E_21,type,
c_2Ebool_2E_21:
!>[A_27a: $tType] : ( ( A_27a > $o ) > $o ) ).
thf(c_2Epair_2E_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_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_2Ealist_2EALOOKUP,type,
c_2Ealist_2EALOOKUP:
!>[A_27a: $tType,A_27b: $tType] : ( ( tyop_2Elist_2Elist @ ( tyop_2Epair_2Eprod @ A_27b @ A_27a ) ) > A_27b > ( tyop_2Eoption_2Eoption @ A_27a ) ) ).
thf(c_2Elist_2EAPPEND,type,
c_2Elist_2EAPPEND:
!>[A_27a: $tType] : ( ( tyop_2Elist_2Elist @ A_27a ) > ( tyop_2Elist_2Elist @ A_27a ) > ( tyop_2Elist_2Elist @ A_27a ) ) ).
thf(c_2Epred__set_2ECARD,type,
c_2Epred__set_2ECARD:
!>[A_27a: $tType] : ( ( A_27a > $o ) > tyop_2Enum_2Enum ) ).
thf(c_2Ebool_2ECOND,type,
c_2Ebool_2ECOND:
!>[A_27a: $tType] : ( $o > A_27a > A_27a > A_27a ) ).
thf(c_2Elist_2ECONS,type,
c_2Elist_2ECONS:
!>[A_27a: $tType] : ( A_27a > ( tyop_2Elist_2Elist @ A_27a ) > ( tyop_2Elist_2Elist @ A_27a ) ) ).
thf(c_2Elist_2EEL,type,
c_2Elist_2EEL:
!>[A_27a: $tType] : ( tyop_2Enum_2Enum > ( tyop_2Elist_2Elist @ A_27a ) > A_27a ) ).
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_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_2Elist_2EFILTER,type,
c_2Elist_2EFILTER:
!>[A_27a: $tType] : ( ( A_27a > $o ) > ( tyop_2Elist_2Elist @ A_27a ) > ( tyop_2Elist_2Elist @ A_27a ) ) ).
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_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_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_2Ecombin_2EI,type,
c_2Ecombin_2EI:
!>[A_27a: $tType] : ( A_27a > A_27a ) ).
thf(c_2Epred__set_2EIMAGE,type,
c_2Epred__set_2EIMAGE:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > A_27b ) > ( A_27a > $o ) > A_27b > $o ) ).
thf(c_2Ebool_2EIN,type,
c_2Ebool_2EIN:
!>[A_27a: $tType] : ( A_27a > ( A_27a > $o ) > $o ) ).
thf(c_2Epred__set_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_2Ewhile_2ELEAST,type,
c_2Ewhile_2ELEAST: ( tyop_2Enum_2Enum > $o ) > tyop_2Enum_2Enum ).
thf(c_2Elist_2ELENGTH,type,
c_2Elist_2ELENGTH:
!>[A_27a: $tType] : ( ( tyop_2Elist_2Elist @ A_27a ) > tyop_2Enum_2Enum ) ).
thf(c_2Elist_2ELIST__TO__SET,type,
c_2Elist_2ELIST__TO__SET:
!>[A_27a: $tType] : ( ( tyop_2Elist_2Elist @ A_27a ) > A_27a > $o ) ).
thf(c_2Elist_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_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_2Eoption_2EOPTION__MAP,type,
c_2Eoption_2EOPTION__MAP:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > A_27b ) > ( tyop_2Eoption_2Eoption @ A_27a ) > ( tyop_2Eoption_2Eoption @ A_27b ) ) ).
thf(c_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_2Elist_2ESET__TO__LIST,type,
c_2Elist_2ESET__TO__LIST:
!>[A_27a: $tType] : ( ( A_27a > $o ) > ( tyop_2Elist_2Elist @ A_27a ) ) ).
thf(c_2Epair_2ESND,type,
c_2Epair_2ESND:
!>[A_27a: $tType,A_27b: $tType] : ( ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) > A_27b ) ).
thf(c_2Eoption_2ESOME,type,
c_2Eoption_2ESOME:
!>[A_27a: $tType] : ( A_27a > ( tyop_2Eoption_2Eoption @ A_27a ) ) ).
thf(c_2Esorting_2ESTABLE,type,
c_2Esorting_2ESTABLE:
!>[A_27a: $tType] : ( ( ( A_27a > A_27a > $o ) > ( tyop_2Elist_2Elist @ A_27a ) > ( tyop_2Elist_2Elist @ A_27a ) ) > ( A_27a > A_27a > $o ) > $o ) ).
thf(c_2Epred__set_2ESUBSET,type,
c_2Epred__set_2ESUBSET:
!>[A_27a: $tType] : ( ( A_27a > $o ) > ( A_27a > $o ) > $o ) ).
thf(c_2Eoption_2ETHE,type,
c_2Eoption_2ETHE:
!>[A_27a: $tType] : ( ( tyop_2Eoption_2Eoption @ A_27a ) > A_27a ) ).
thf(c_2Epair_2EUNCURRY,type,
c_2Epair_2EUNCURRY:
!>[A_27a: $tType,A_27b: $tType,A_27c: $tType] : ( ( A_27a > A_27b > A_27c ) > ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) > A_27c ) ).
thf(c_2Epred__set_2EUNIV,type,
c_2Epred__set_2EUNIV:
!>[A_27a: $tType] : ( A_27a > $o ) ).
thf(c_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_2Ealist_2Ealist__range,type,
c_2Ealist_2Ealist__range:
!>[A_27a: $tType,A_27b: $tType] : ( ( tyop_2Elist_2Elist @ ( tyop_2Epair_2Eprod @ A_27b @ A_27a ) ) > A_27a > $o ) ).
thf(c_2Ealist_2Ealist__to__fmap,type,
c_2Ealist_2Ealist__to__fmap:
!>[A_27a: $tType,A_27b: $tType] : ( ( tyop_2Elist_2Elist @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) ) > ( tyop_2Efinite__map_2Efmap @ A_27a @ A_27b ) ) ).
thf(c_2Ealist_2Efmap__to__alist,type,
c_2Ealist_2Efmap__to__alist:
!>[A_27a: $tType,A_27b: $tType] : ( ( tyop_2Efinite__map_2Efmap @ A_27a @ A_27b ) > ( tyop_2Elist_2Elist @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) ) ) ).
thf(c_2Erelation_2Einv__image,type,
c_2Erelation_2Einv__image:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27b > A_27b > $o ) > ( A_27a > A_27b ) > A_27a > A_27a > $o ) ).
thf(c_2Ecombin_2Eo,type,
c_2Ecombin_2Eo:
!>[A_27a: $tType,A_27b: $tType,A_27c: $tType] : ( ( A_27c > A_27b ) > ( A_27a > A_27c ) > A_27a > A_27b ) ).
thf(c_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_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_2Erelation_2Etotal,type,
c_2Erelation_2Etotal:
!>[A_27a: $tType] : ( ( A_27a > A_27a > $o ) > $o ) ).
thf(c_2Erelation_2Etransitive,type,
c_2Erelation_2Etransitive:
!>[A_27a: $tType] : ( ( A_27a > A_27a > $o ) > $o ) ).
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_2Ealist_2Efmap__to__alist__def,axiom,
! [A_27a: $tType,A_27b: $tType,V0s: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b] :
( ( c_2Ealist_2Efmap__to__alist @ A_27a @ A_27b @ V0s )
= ( c_2Elist_2EMAP @ A_27a @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b )
@ ^ [V1k: A_27a] : ( c_2Epair_2E_2C @ A_27a @ A_27b @ V1k @ ( c_2Efinite__map_2EFAPPLY @ A_27a @ A_27b @ V0s @ V1k ) )
@ ( c_2Elist_2ESET__TO__LIST @ A_27a @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b @ V0s ) ) ) ) ).
thf(thm_2Ealist_2Ealist__to__fmap__def,axiom,
! [A_27a: $tType,A_27b: $tType,V0s: tyop_2Elist_2Elist @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b )] :
( ( c_2Ealist_2Ealist__to__fmap @ A_27a @ A_27b @ V0s )
= ( 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 ) )
@ ^ [V1k: A_27a,V2v: A_27b,V3f: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b] : ( c_2Efinite__map_2EFUPDATE @ A_27a @ A_27b @ V3f @ ( c_2Epair_2E_2C @ A_27a @ A_27b @ V1k @ V2v ) ) )
@ ( c_2Efinite__map_2EFEMPTY @ A_27a @ A_27b )
@ V0s ) ) ).
thf(thm_2Ealist_2Ealist__range__def,axiom,
! [A_27a: $tType,A_27b: $tType,V0m: tyop_2Elist_2Elist @ ( tyop_2Epair_2Eprod @ A_27b @ A_27a )] :
( ( c_2Ealist_2Ealist__range @ A_27a @ A_27b @ V0m )
= ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a
@ ^ [V1v: A_27a] :
( c_2Epair_2E_2C @ A_27a @ $o @ V1v
@ ( c_2Ebool_2E_3F @ A_27b
@ ^ [V2k: A_27b] : ( c_2Emin_2E_3D @ ( tyop_2Eoption_2Eoption @ A_27a ) @ ( c_2Ealist_2EALOOKUP @ A_27a @ A_27b @ V0m @ V2k ) @ ( c_2Eoption_2ESOME @ A_27a @ V1v ) ) ) ) ) ) ).
thf(thm_2Ealist_2Efmap__to__alist__FEMPTY,axiom,
! [A_27a: $tType,A_27b: $tType] :
( ( c_2Ealist_2Efmap__to__alist @ A_27a @ A_27b @ ( c_2Efinite__map_2EFEMPTY @ A_27a @ A_27b ) )
= ( c_2Elist_2ENIL @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) ) ) ).
thf(thm_2Ealist_2Ealist__to__fmap__thm,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,A_27d: $tType,V0v: A_27d,V1t: tyop_2Elist_2Elist @ ( tyop_2Epair_2Eprod @ A_27c @ A_27d ),V2k: A_27c] :
( ( ( c_2Ealist_2Ealist__to__fmap @ A_27a @ A_27b @ ( c_2Elist_2ENIL @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) ) )
= ( c_2Efinite__map_2EFEMPTY @ A_27a @ A_27b ) )
& ( ( c_2Ealist_2Ealist__to__fmap @ A_27c @ A_27d @ ( c_2Elist_2ECONS @ ( tyop_2Epair_2Eprod @ A_27c @ A_27d ) @ ( c_2Epair_2E_2C @ A_27c @ A_27d @ V2k @ V0v ) @ V1t ) )
= ( c_2Efinite__map_2EFUPDATE @ A_27c @ A_27d @ ( c_2Ealist_2Ealist__to__fmap @ A_27c @ A_27d @ V1t ) @ ( c_2Epair_2E_2C @ A_27c @ A_27d @ V2k @ V0v ) ) ) ) ).
thf(thm_2Ealist_2EALOOKUP__ind,axiom,
! [A_27a: $tType,A_27b: $tType,V0P: ( tyop_2Elist_2Elist @ ( tyop_2Epair_2Eprod @ A_27b @ A_27a ) ) > A_27b > $o] :
( ( ! [V1q: A_27b] : ( V0P @ ( c_2Elist_2ENIL @ ( tyop_2Epair_2Eprod @ A_27b @ A_27a ) ) @ V1q )
& ! [V2x: A_27b,V3y: A_27a,V4t: tyop_2Elist_2Elist @ ( tyop_2Epair_2Eprod @ A_27b @ A_27a ),V5q: A_27b] :
( ( ( (~) @ ( V2x = V5q ) )
=> ( V0P @ V4t @ V5q ) )
=> ( V0P @ ( c_2Elist_2ECONS @ ( tyop_2Epair_2Eprod @ A_27b @ A_27a ) @ ( c_2Epair_2E_2C @ A_27b @ A_27a @ V2x @ V3y ) @ V4t ) @ V5q ) ) )
=> ! [V6v: tyop_2Elist_2Elist @ ( tyop_2Epair_2Eprod @ A_27b @ A_27a ),V7v1: A_27b] : ( V0P @ V6v @ V7v1 ) ) ).
thf(thm_2Ealist_2EALOOKUP__def,axiom,
! [A_27a: $tType,A_27b: $tType] :
( ! [V0q: A_27b] :
( ( c_2Ealist_2EALOOKUP @ A_27a @ A_27b @ ( c_2Elist_2ENIL @ ( tyop_2Epair_2Eprod @ A_27b @ A_27a ) ) @ V0q )
= ( c_2Eoption_2ENONE @ A_27a ) )
& ! [V1y: A_27a,V2x: A_27b,V3t: tyop_2Elist_2Elist @ ( tyop_2Epair_2Eprod @ A_27b @ A_27a ),V4q: A_27b] :
( ( c_2Ealist_2EALOOKUP @ A_27a @ A_27b @ ( c_2Elist_2ECONS @ ( tyop_2Epair_2Eprod @ A_27b @ A_27a ) @ ( c_2Epair_2E_2C @ A_27b @ A_27a @ V2x @ V1y ) @ V3t ) @ V4q )
= ( c_2Ebool_2ECOND @ ( tyop_2Eoption_2Eoption @ A_27a ) @ ( c_2Emin_2E_3D @ A_27b @ V2x @ V4q ) @ ( c_2Eoption_2ESOME @ A_27a @ V1y ) @ ( c_2Ealist_2EALOOKUP @ A_27a @ A_27b @ V3t @ V4q ) ) ) ) ).
thf(thm_2Ealist_2EALOOKUP__FAILS,axiom,
! [A_27a: $tType,A_27b: $tType,V0x: A_27b,V1l: tyop_2Elist_2Elist @ ( tyop_2Epair_2Eprod @ A_27b @ A_27a )] :
( ( ( c_2Ealist_2EALOOKUP @ A_27a @ A_27b @ V1l @ V0x )
= ( c_2Eoption_2ENONE @ A_27a ) )
<=> ! [V2k: A_27b,V3v: A_27a] :
( ( 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 ) @ V1l ) )
=> ( (~) @ ( V2k = V0x ) ) ) ) ).
thf(thm_2Ealist_2EALOOKUP__NONE,axiom,
! [A_27a: $tType,A_27b: $tType,V0l: tyop_2Elist_2Elist @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ),V1x: A_27a] :
( ( ( c_2Ealist_2EALOOKUP @ A_27b @ A_27a @ V0l @ V1x )
= ( c_2Eoption_2ENONE @ A_27b ) )
<=> ( (~) @ ( c_2Ebool_2EIN @ A_27a @ V1x @ ( 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 ) @ V0l ) ) ) ) ) ).
thf(thm_2Ealist_2EALOOKUP__TABULATE,axiom,
! [A_27a: $tType,A_27b: $tType,V0x: A_27a,V1l: tyop_2Elist_2Elist @ A_27a,V2f: A_27a > A_27b] :
( ( c_2Ebool_2EIN @ A_27a @ V0x @ ( c_2Elist_2ELIST__TO__SET @ A_27a @ V1l ) )
=> ( ( c_2Ealist_2EALOOKUP @ A_27b @ A_27a
@ ( c_2Elist_2EMAP @ A_27a @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b )
@ ^ [V3k: A_27a] : ( c_2Epair_2E_2C @ A_27a @ A_27b @ V3k @ ( V2f @ V3k ) )
@ V1l )
@ V0x )
= ( c_2Eoption_2ESOME @ A_27b @ ( V2f @ V0x ) ) ) ) ).
thf(thm_2Ealist_2EALOOKUP__EQ__FLOOKUP,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,A_27d: $tType,V0fm: tyop_2Efinite__map_2Efmap @ A_27c @ A_27d,V1al: tyop_2Elist_2Elist @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b )] :
( ( ( c_2Efinite__map_2EFLOOKUP @ A_27a @ A_27b @ ( c_2Ealist_2Ealist__to__fmap @ A_27a @ A_27b @ V1al ) )
= ( c_2Ealist_2EALOOKUP @ A_27b @ A_27a @ V1al ) )
& ( ( c_2Ealist_2EALOOKUP @ A_27d @ A_27c @ ( c_2Ealist_2Efmap__to__alist @ A_27c @ A_27d @ V0fm ) )
= ( c_2Efinite__map_2EFLOOKUP @ A_27c @ A_27d @ V0fm ) ) ) ).
thf(thm_2Ealist_2EMEM__fmap__to__alist,axiom,
! [A_27a: $tType,A_27b: $tType,V0y: A_27b,V1x: A_27a,V2fm: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b] :
( ( c_2Ebool_2EIN @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ ( c_2Epair_2E_2C @ A_27a @ A_27b @ V1x @ V0y ) @ ( c_2Elist_2ELIST__TO__SET @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ ( c_2Ealist_2Efmap__to__alist @ A_27a @ A_27b @ V2fm ) ) )
<=> ( ( c_2Ebool_2EIN @ A_27a @ V1x @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b @ V2fm ) )
& ( ( c_2Efinite__map_2EFAPPLY @ A_27a @ A_27b @ V2fm @ V1x )
= V0y ) ) ) ).
thf(thm_2Ealist_2EMEM__fmap__to__alist__FLOOKUP,axiom,
! [A_27a: $tType,A_27b: $tType,V0p: tyop_2Epair_2Eprod @ A_27a @ A_27b,V1fm: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b] :
( ( c_2Ebool_2EIN @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ V0p @ ( c_2Elist_2ELIST__TO__SET @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ ( c_2Ealist_2Efmap__to__alist @ A_27a @ A_27b @ V1fm ) ) )
<=> ( ( c_2Efinite__map_2EFLOOKUP @ A_27a @ A_27b @ V1fm @ ( c_2Epair_2EFST @ A_27a @ A_27b @ V0p ) )
= ( c_2Eoption_2ESOME @ A_27b @ ( c_2Epair_2ESND @ A_27a @ A_27b @ V0p ) ) ) ) ).
thf(thm_2Ealist_2EMEM__pair__fmap__to__alist__FLOOKUP,axiom,
! [A_27a: $tType,A_27b: $tType,V0x: A_27a,V1y: A_27b,V2fm: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b] :
( ( c_2Ebool_2EIN @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ ( c_2Epair_2E_2C @ A_27a @ A_27b @ V0x @ V1y ) @ ( c_2Elist_2ELIST__TO__SET @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ ( c_2Ealist_2Efmap__to__alist @ A_27a @ A_27b @ V2fm ) ) )
<=> ( ( c_2Efinite__map_2EFLOOKUP @ A_27a @ A_27b @ V2fm @ V0x )
= ( c_2Eoption_2ESOME @ A_27b @ V1y ) ) ) ).
thf(thm_2Ealist_2ELENGTH__fmap__to__alist,axiom,
! [A_27a: $tType,A_27b: $tType,V0fm: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b] :
( ( c_2Elist_2ELENGTH @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ ( c_2Ealist_2Efmap__to__alist @ A_27a @ A_27b @ V0fm ) )
= ( c_2Epred__set_2ECARD @ A_27a @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b @ V0fm ) ) ) ).
thf(thm_2Ealist_2Efmap__to__alist__to__fmap,axiom,
! [A_27a: $tType,A_27b: $tType,V0fm: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b] :
( ( c_2Ealist_2Ealist__to__fmap @ A_27a @ A_27b @ ( c_2Ealist_2Efmap__to__alist @ A_27a @ A_27b @ V0fm ) )
= V0fm ) ).
thf(thm_2Ealist_2EALOOKUP__MEM,axiom,
! [A_27a: $tType,A_27b: $tType,V0al: tyop_2Elist_2Elist @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ),V1k: A_27a,V2v: A_27b] :
( ( ( c_2Ealist_2EALOOKUP @ A_27b @ A_27a @ V0al @ V1k )
= ( c_2Eoption_2ESOME @ A_27b @ V2v ) )
=> ( c_2Ebool_2EIN @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ ( c_2Epair_2E_2C @ A_27a @ A_27b @ V1k @ V2v ) @ ( c_2Elist_2ELIST__TO__SET @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ V0al ) ) ) ).
thf(thm_2Ealist_2EALOOKUP__SOME__FAPPLY__alist__to__fmap,axiom,
! [A_27a: $tType,A_27b: $tType,V0al: tyop_2Elist_2Elist @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ),V1k: A_27a,V2v: A_27b] :
( ( ( c_2Ealist_2EALOOKUP @ A_27b @ A_27a @ V0al @ V1k )
= ( c_2Eoption_2ESOME @ A_27b @ V2v ) )
=> ( ( c_2Efinite__map_2EFAPPLY @ A_27a @ A_27b @ ( c_2Ealist_2Ealist__to__fmap @ A_27a @ A_27b @ V0al ) @ V1k )
= V2v ) ) ).
thf(thm_2Ealist_2Ealist__to__fmap__FAPPLY__MEM,axiom,
! [A_27a: $tType,A_27b: $tType,V0al: tyop_2Elist_2Elist @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ),V1z: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V1z @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b @ ( c_2Ealist_2Ealist__to__fmap @ A_27a @ A_27b @ V0al ) ) )
=> ( c_2Ebool_2EIN @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ ( c_2Epair_2E_2C @ A_27a @ A_27b @ V1z @ ( c_2Efinite__map_2EFAPPLY @ A_27a @ A_27b @ ( c_2Ealist_2Ealist__to__fmap @ A_27a @ A_27b @ V0al ) @ V1z ) ) @ ( c_2Elist_2ELIST__TO__SET @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ V0al ) ) ) ).
thf(thm_2Ealist_2EALOOKUP__MAP,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0f: A_27a > A_27b,V1al: tyop_2Elist_2Elist @ ( tyop_2Epair_2Eprod @ A_27c @ A_27a )] :
( ( c_2Ealist_2EALOOKUP @ A_27b @ A_27c
@ ( c_2Elist_2EMAP @ ( tyop_2Epair_2Eprod @ A_27c @ A_27a ) @ ( tyop_2Epair_2Eprod @ A_27c @ A_27b )
@ ( c_2Epair_2EUNCURRY @ A_27c @ A_27a @ ( tyop_2Epair_2Eprod @ A_27c @ A_27b )
@ ^ [V2x: A_27c,V3y: A_27a] : ( c_2Epair_2E_2C @ A_27c @ A_27b @ V2x @ ( V0f @ V3y ) ) )
@ V1al ) )
= ( c_2Ecombin_2Eo @ A_27c @ ( tyop_2Eoption_2Eoption @ A_27b ) @ ( tyop_2Eoption_2Eoption @ A_27a ) @ ( c_2Eoption_2EOPTION__MAP @ A_27a @ A_27b @ V0f ) @ ( c_2Ealist_2EALOOKUP @ A_27a @ A_27c @ V1al ) ) ) ).
thf(thm_2Ealist_2EFDOM__alist__to__fmap,axiom,
! [A_27a: $tType,A_27b: $tType,V0al: tyop_2Elist_2Elist @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b )] :
( ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b @ ( c_2Ealist_2Ealist__to__fmap @ A_27a @ A_27b @ V0al ) )
= ( 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 ) @ V0al ) ) ) ).
thf(thm_2Ealist_2Ealist__to__fmap__prefix,axiom,
! [A_27a: $tType,A_27b: $tType,V0ls: tyop_2Elist_2Elist @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ),V1l1: tyop_2Elist_2Elist @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ),V2l2: tyop_2Elist_2Elist @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b )] :
( ( ( c_2Ealist_2Ealist__to__fmap @ A_27a @ A_27b @ V1l1 )
= ( c_2Ealist_2Ealist__to__fmap @ A_27a @ A_27b @ V2l2 ) )
=> ( ( c_2Ealist_2Ealist__to__fmap @ A_27a @ A_27b @ ( c_2Elist_2EAPPEND @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ V0ls @ V1l1 ) )
= ( c_2Ealist_2Ealist__to__fmap @ A_27a @ A_27b @ ( c_2Elist_2EAPPEND @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ V0ls @ V2l2 ) ) ) ) ).
thf(thm_2Ealist_2Ealist__to__fmap__APPEND,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 )] :
( ( c_2Ealist_2Ealist__to__fmap @ A_27a @ A_27b @ ( c_2Elist_2EAPPEND @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ V0l1 @ V1l2 ) )
= ( c_2Efinite__map_2EFUNION @ A_27a @ A_27b @ ( c_2Ealist_2Ealist__to__fmap @ A_27a @ A_27b @ V0l1 ) @ ( c_2Ealist_2Ealist__to__fmap @ A_27a @ A_27b @ V1l2 ) ) ) ).
thf(thm_2Ealist_2EALOOKUP__prefix,axiom,
! [A_27a: $tType,A_27b: $tType,V0v: A_27b,V1ls: tyop_2Elist_2Elist @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ),V2k: A_27a,V3ls2: tyop_2Elist_2Elist @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b )] :
( ( ( ( c_2Ealist_2EALOOKUP @ A_27b @ A_27a @ V1ls @ V2k )
= ( c_2Eoption_2ESOME @ A_27b @ V0v ) )
=> ( ( c_2Ealist_2EALOOKUP @ A_27b @ A_27a @ ( c_2Elist_2EAPPEND @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ V1ls @ V3ls2 ) @ V2k )
= ( c_2Eoption_2ESOME @ A_27b @ V0v ) ) )
& ( ( ( c_2Ealist_2EALOOKUP @ A_27b @ A_27a @ V1ls @ V2k )
= ( c_2Eoption_2ENONE @ A_27b ) )
=> ( ( c_2Ealist_2EALOOKUP @ A_27b @ A_27a @ ( c_2Elist_2EAPPEND @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ V1ls @ V3ls2 ) @ V2k )
= ( c_2Ealist_2EALOOKUP @ A_27b @ A_27a @ V3ls2 @ V2k ) ) ) ) ).
thf(thm_2Ealist_2EALOOKUP__APPEND,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 ),V2k: A_27a] :
( ( c_2Ealist_2EALOOKUP @ A_27b @ A_27a @ ( c_2Elist_2EAPPEND @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ V0l1 @ V1l2 ) @ V2k )
= ( c_2Eoption_2Eoption__CASE @ A_27b @ ( tyop_2Eoption_2Eoption @ A_27b ) @ ( c_2Ealist_2EALOOKUP @ A_27b @ A_27a @ V0l1 @ V2k ) @ ( c_2Ealist_2EALOOKUP @ A_27b @ A_27a @ V1l2 @ V2k )
@ ^ [V3v: A_27b] : ( c_2Eoption_2ESOME @ A_27b @ V3v ) ) ) ).
thf(thm_2Ealist_2EFUPDATE__LIST__EQ__APPEND__REVERSE,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_2Efinite__map_2EFUPDATE__LIST @ A_27a @ A_27b @ V1fm @ V0ls )
= ( c_2Ealist_2Ealist__to__fmap @ A_27a @ A_27b @ ( c_2Elist_2EAPPEND @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ ( c_2Elist_2EREVERSE @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ V0ls ) @ ( c_2Ealist_2Efmap__to__alist @ A_27a @ A_27b @ V1fm ) ) ) ) ).
thf(thm_2Ealist_2EFLOOKUP__FUPDATE__LIST__ALOOKUP__SOME,axiom,
! [A_27a: $tType,A_27b: $tType,V0v: A_27a,V1ls: tyop_2Elist_2Elist @ ( tyop_2Epair_2Eprod @ A_27b @ A_27a ),V2k: A_27b,V3fm: tyop_2Efinite__map_2Efmap @ A_27b @ A_27a] :
( ( ( c_2Ealist_2EALOOKUP @ A_27a @ A_27b @ V1ls @ V2k )
= ( c_2Eoption_2ESOME @ A_27a @ V0v ) )
=> ( ( c_2Efinite__map_2EFLOOKUP @ A_27b @ A_27a @ ( c_2Efinite__map_2EFUPDATE__LIST @ A_27b @ A_27a @ V3fm @ ( c_2Elist_2EREVERSE @ ( tyop_2Epair_2Eprod @ A_27b @ A_27a ) @ V1ls ) ) @ V2k )
= ( c_2Eoption_2ESOME @ A_27a @ V0v ) ) ) ).
thf(thm_2Ealist_2EFLOOKUP__FUPDATE__LIST__ALOOKUP__NONE,axiom,
! [A_27a: $tType,A_27b: $tType,V0ls: tyop_2Elist_2Elist @ ( tyop_2Epair_2Eprod @ A_27b @ A_27a ),V1k: A_27b,V2fm: tyop_2Efinite__map_2Efmap @ A_27b @ A_27a] :
( ( ( c_2Ealist_2EALOOKUP @ A_27a @ A_27b @ V0ls @ V1k )
= ( c_2Eoption_2ENONE @ A_27a ) )
=> ( ( c_2Efinite__map_2EFLOOKUP @ A_27b @ A_27a @ ( c_2Efinite__map_2EFUPDATE__LIST @ A_27b @ A_27a @ V2fm @ ( c_2Elist_2EREVERSE @ ( tyop_2Epair_2Eprod @ A_27b @ A_27a ) @ V0ls ) ) @ V1k )
= ( c_2Efinite__map_2EFLOOKUP @ A_27b @ A_27a @ V2fm @ V1k ) ) ) ).
thf(thm_2Ealist_2EFUNION__alist__to__fmap,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_2Efinite__map_2EFUNION @ A_27a @ A_27b @ ( c_2Ealist_2Ealist__to__fmap @ A_27a @ A_27b @ V0ls ) @ V1fm )
= ( c_2Efinite__map_2EFUPDATE__LIST @ A_27a @ A_27b @ V1fm @ ( c_2Elist_2EREVERSE @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ V0ls ) ) ) ).
thf(thm_2Ealist_2Ealist__to__fmap__MAP,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,A_27d: $tType,V0f1: A_27a > A_27b,V1f2: A_27c > A_27d,V2al: tyop_2Elist_2Elist @ ( tyop_2Epair_2Eprod @ A_27a @ A_27c )] :
( ( c_2Epred__set_2EINJ @ A_27a @ A_27b @ V0f1 @ ( c_2Elist_2ELIST__TO__SET @ A_27a @ ( c_2Elist_2EMAP @ ( tyop_2Epair_2Eprod @ A_27a @ A_27c ) @ A_27a @ ( c_2Epair_2EFST @ A_27a @ A_27c ) @ V2al ) ) @ ( c_2Epred__set_2EUNIV @ A_27b ) )
=> ( ( c_2Ealist_2Ealist__to__fmap @ A_27b @ A_27d
@ ( c_2Elist_2EMAP @ ( tyop_2Epair_2Eprod @ A_27a @ A_27c ) @ ( tyop_2Epair_2Eprod @ A_27b @ A_27d )
@ ( c_2Epair_2EUNCURRY @ A_27a @ A_27c @ ( tyop_2Epair_2Eprod @ A_27b @ A_27d )
@ ^ [V3x: A_27a,V4y: A_27c] : ( c_2Epair_2E_2C @ A_27b @ A_27d @ ( V0f1 @ V3x ) @ ( V1f2 @ V4y ) ) )
@ V2al ) )
= ( c_2Efinite__map_2EMAP__KEYS @ A_27a @ A_27b @ A_27d @ V0f1 @ ( c_2Efinite__map_2Eo__f @ A_27a @ A_27c @ A_27d @ V1f2 @ ( c_2Ealist_2Ealist__to__fmap @ A_27a @ A_27c @ V2al ) ) ) ) ) ).
thf(thm_2Ealist_2Ealist__to__fmap__to__alist,axiom,
! [A_27a: $tType,A_27b: $tType,V0al: tyop_2Elist_2Elist @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b )] :
( ( c_2Ealist_2Efmap__to__alist @ A_27a @ A_27b @ ( c_2Ealist_2Ealist__to__fmap @ A_27a @ A_27b @ V0al ) )
= ( c_2Elist_2EMAP @ A_27a @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b )
@ ^ [V1k: A_27a] : ( c_2Epair_2E_2C @ A_27a @ A_27b @ V1k @ ( c_2Eoption_2ETHE @ A_27b @ ( c_2Ealist_2EALOOKUP @ A_27b @ A_27a @ V0al @ V1k ) ) )
@ ( c_2Elist_2ESET__TO__LIST @ 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 ) @ V0al ) ) ) ) ) ).
thf(thm_2Ealist_2Ealist__to__fmap__to__alist__PERM,axiom,
! [A_27a: $tType,A_27b: $tType,V0al: 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 ) @ V0al ) )
=> ( c_2Esorting_2EPERM @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ ( c_2Ealist_2Efmap__to__alist @ A_27a @ A_27b @ ( c_2Ealist_2Ealist__to__fmap @ A_27a @ A_27b @ V0al ) ) @ V0al ) ) ).
thf(thm_2Ealist_2EALOOKUP__LEAST__EL,axiom,
! [A_27a: $tType,A_27b: $tType,V0ls: tyop_2Elist_2Elist @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ),V1k: A_27a] :
( ( c_2Ealist_2EALOOKUP @ A_27b @ A_27a @ V0ls @ V1k )
= ( c_2Ebool_2ECOND @ ( tyop_2Eoption_2Eoption @ A_27b ) @ ( c_2Ebool_2EIN @ A_27a @ V1k @ ( 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 ) @ V0ls ) ) )
@ ( c_2Eoption_2ESOME @ A_27b
@ ( c_2Elist_2EEL @ A_27b
@ ( c_2Ewhile_2ELEAST
@ ^ [V2n: tyop_2Enum_2Enum] : ( c_2Emin_2E_3D @ A_27a @ ( c_2Elist_2EEL @ A_27a @ V2n @ ( c_2Elist_2EMAP @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ A_27a @ ( c_2Epair_2EFST @ A_27a @ A_27b ) @ V0ls ) ) @ V1k ) )
@ ( c_2Elist_2EMAP @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ A_27b @ ( c_2Epair_2ESND @ A_27a @ A_27b ) @ V0ls ) ) )
@ ( c_2Eoption_2ENONE @ A_27b ) ) ) ).
thf(thm_2Ealist_2EALOOKUP__ALL__DISTINCT__MEM,axiom,
! [A_27a: $tType,A_27b: $tType,V0v: A_27b,V1k: A_27a,V2al: 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 ) @ V2al ) )
& ( c_2Ebool_2EIN @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ ( c_2Epair_2E_2C @ A_27a @ A_27b @ V1k @ V0v ) @ ( c_2Elist_2ELIST__TO__SET @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ V2al ) ) )
=> ( ( c_2Ealist_2EALOOKUP @ A_27b @ A_27a @ V2al @ V1k )
= ( c_2Eoption_2ESOME @ A_27b @ V0v ) ) ) ).
thf(thm_2Ealist_2EALL__DISTINCT__fmap__to__alist__keys,axiom,
! [A_27a: $tType,A_27b: $tType,V0fm: 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 ) @ ( c_2Ealist_2Efmap__to__alist @ A_27a @ A_27b @ V0fm ) ) ) ).
thf(thm_2Ealist_2Efmap__to__alist__inj,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_2Ealist_2Efmap__to__alist @ A_27a @ A_27b @ V0f1 )
= ( c_2Ealist_2Efmap__to__alist @ A_27a @ A_27b @ V1f2 ) )
=> ( V0f1 = V1f2 ) ) ).
thf(thm_2Ealist_2Efmap__to__alist__preserves__FDOM,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0fm1: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b,V1fm2: tyop_2Efinite__map_2Efmap @ A_27a @ A_27c] :
( ( ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b @ V0fm1 )
= ( c_2Efinite__map_2EFDOM @ A_27a @ A_27c @ V1fm2 ) )
=> ( ( c_2Elist_2EMAP @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ A_27a @ ( c_2Epair_2EFST @ A_27a @ A_27b ) @ ( c_2Ealist_2Efmap__to__alist @ A_27a @ A_27b @ V0fm1 ) )
= ( c_2Elist_2EMAP @ ( tyop_2Epair_2Eprod @ A_27a @ A_27c ) @ A_27a @ ( c_2Epair_2EFST @ A_27a @ A_27c ) @ ( c_2Ealist_2Efmap__to__alist @ A_27a @ A_27c @ V1fm2 ) ) ) ) ).
thf(thm_2Ealist_2EPERM__fmap__to__alist,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_2Esorting_2EPERM @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ ( c_2Ealist_2Efmap__to__alist @ A_27a @ A_27b @ V1fm1 ) @ ( c_2Ealist_2Efmap__to__alist @ A_27a @ A_27b @ V0fm2 ) )
<=> ( V1fm1 = V0fm2 ) ) ).
thf(thm_2Ealist_2Ealist__to__fmap__PERM,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 )] :
( ( ( c_2Esorting_2EPERM @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ V0l1 @ V1l2 )
& ( 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 ) @ V0l1 ) ) )
=> ( ( c_2Ealist_2Ealist__to__fmap @ A_27a @ A_27b @ V0l1 )
= ( c_2Ealist_2Ealist__to__fmap @ A_27a @ A_27b @ V1l2 ) ) ) ).
thf(thm_2Ealist_2EALOOKUP__ALL__DISTINCT__EL,axiom,
! [A_27a: $tType,A_27b: $tType,V0ls: tyop_2Elist_2Elist @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ),V1n: tyop_2Enum_2Enum] :
( ( ( c_2Eprim__rec_2E_3C @ V1n @ ( c_2Elist_2ELENGTH @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ V0ls ) )
& ( 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_2Ealist_2EALOOKUP @ A_27b @ A_27a @ V0ls @ ( c_2Epair_2EFST @ A_27a @ A_27b @ ( c_2Elist_2EEL @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ V1n @ V0ls ) ) )
= ( c_2Eoption_2ESOME @ A_27b @ ( c_2Epair_2ESND @ A_27a @ A_27b @ ( c_2Elist_2EEL @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ V1n @ V0ls ) ) ) ) ) ).
thf(thm_2Ealist_2EALOOKUP__ZIP__MAP__SND,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,A_27d: $tType,V0l1: tyop_2Elist_2Elist @ A_27a,V1l2: tyop_2Elist_2Elist @ A_27b,V2k: A_27c,V3f: A_27b > A_27d] :
( ( ( c_2Elist_2ELENGTH @ A_27a @ V0l1 )
= ( c_2Elist_2ELENGTH @ A_27b @ V1l2 ) )
=> ( ( c_2Ealist_2EALOOKUP @ A_27d @ A_27a @ ( c_2Elist_2EZIP @ A_27a @ A_27d @ ( c_2Epair_2E_2C @ ( tyop_2Elist_2Elist @ A_27a ) @ ( tyop_2Elist_2Elist @ A_27d ) @ V0l1 @ ( c_2Elist_2EMAP @ A_27b @ A_27d @ V3f @ V1l2 ) ) ) )
= ( c_2Ecombin_2Eo @ A_27a @ ( tyop_2Eoption_2Eoption @ A_27d ) @ ( tyop_2Eoption_2Eoption @ A_27b ) @ ( c_2Eoption_2EOPTION__MAP @ A_27b @ A_27d @ V3f ) @ ( c_2Ealist_2EALOOKUP @ A_27b @ A_27a @ ( c_2Elist_2EZIP @ A_27a @ A_27b @ ( c_2Epair_2E_2C @ ( tyop_2Elist_2Elist @ A_27a ) @ ( tyop_2Elist_2Elist @ A_27b ) @ V0l1 @ V1l2 ) ) ) ) ) ) ).
thf(thm_2Ealist_2EALOOKUP__FILTER,axiom,
! [A_27a: $tType,A_27b: $tType,V0P: A_27a > $o,V1ls: tyop_2Elist_2Elist @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ),V2x: A_27a] :
( ( c_2Ealist_2EALOOKUP @ A_27b @ A_27a
@ ( c_2Elist_2EFILTER @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b )
@ ( c_2Epair_2EUNCURRY @ A_27a @ A_27b @ $o
@ ^ [V3k: A_27a,V4v: A_27b] : ( V0P @ V3k ) )
@ V1ls )
@ V2x )
= ( c_2Ebool_2ECOND @ ( tyop_2Eoption_2Eoption @ A_27b ) @ ( V0P @ V2x ) @ ( c_2Ealist_2EALOOKUP @ A_27b @ A_27a @ V1ls @ V2x ) @ ( c_2Eoption_2ENONE @ A_27b ) ) ) ).
thf(thm_2Ealist_2EALOOKUP__APPEND__same,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 ),V2l: tyop_2Elist_2Elist @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b )] :
( ( ( c_2Ealist_2EALOOKUP @ A_27b @ A_27a @ V0l1 )
= ( c_2Ealist_2EALOOKUP @ A_27b @ A_27a @ V1l2 ) )
=> ( ( c_2Ealist_2EALOOKUP @ A_27b @ A_27a @ ( c_2Elist_2EAPPEND @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ V0l1 @ V2l ) )
= ( c_2Ealist_2EALOOKUP @ A_27b @ A_27a @ ( c_2Elist_2EAPPEND @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ V1l2 @ V2l ) ) ) ) ).
thf(thm_2Ealist_2EALOOKUP__IN__FRANGE,axiom,
! [A_27a: $tType,A_27b: $tType,V0ls: tyop_2Elist_2Elist @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ),V1k: A_27a,V2v: A_27b] :
( ( ( c_2Ealist_2EALOOKUP @ A_27b @ A_27a @ V0ls @ V1k )
= ( c_2Eoption_2ESOME @ A_27b @ V2v ) )
=> ( c_2Ebool_2EIN @ A_27b @ V2v @ ( c_2Efinite__map_2EFRANGE @ A_27a @ A_27b @ ( c_2Ealist_2Ealist__to__fmap @ A_27a @ A_27b @ V0ls ) ) ) ) ).
thf(thm_2Ealist_2EFRANGE__alist__to__fmap__SUBSET,axiom,
! [A_27a: $tType,A_27b: $tType,V0ls: tyop_2Elist_2Elist @ ( tyop_2Epair_2Eprod @ A_27b @ A_27a )] : ( c_2Epred__set_2ESUBSET @ A_27a @ ( c_2Efinite__map_2EFRANGE @ A_27b @ A_27a @ ( c_2Ealist_2Ealist__to__fmap @ A_27b @ A_27a @ V0ls ) ) @ ( c_2Epred__set_2EIMAGE @ ( tyop_2Epair_2Eprod @ A_27b @ A_27a ) @ A_27a @ ( c_2Epair_2ESND @ A_27b @ A_27a ) @ ( c_2Elist_2ELIST__TO__SET @ ( tyop_2Epair_2Eprod @ A_27b @ A_27a ) @ V0ls ) ) ) ).
thf(thm_2Ealist_2EIN__FRANGE__alist__to__fmap__suff,axiom,
! [A_27a: $tType,A_27b: $tType,V0ls: tyop_2Elist_2Elist @ ( tyop_2Epair_2Eprod @ A_27b @ A_27a ),V1P: A_27a > $o] :
( ! [V2v: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V2v @ ( 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 ) ) )
=> ( V1P @ V2v ) )
=> ! [V3v: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V3v @ ( c_2Efinite__map_2EFRANGE @ A_27b @ A_27a @ ( c_2Ealist_2Ealist__to__fmap @ A_27b @ A_27a @ V0ls ) ) )
=> ( V1P @ V3v ) ) ) ).
thf(thm_2Ealist_2Ealist__to__fmap__MAP__matchable,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,A_27d: $tType,V0f1: A_27a > A_27b,V1f2: A_27c > A_27d,V2al: tyop_2Elist_2Elist @ ( tyop_2Epair_2Eprod @ A_27a @ A_27c ),V3mal: tyop_2Elist_2Elist @ ( tyop_2Epair_2Eprod @ A_27b @ A_27d ),V4v: tyop_2Efinite__map_2Efmap @ A_27b @ A_27d] :
( ( ( c_2Epred__set_2EINJ @ A_27a @ A_27b @ V0f1 @ ( c_2Elist_2ELIST__TO__SET @ A_27a @ ( c_2Elist_2EMAP @ ( tyop_2Epair_2Eprod @ A_27a @ A_27c ) @ A_27a @ ( c_2Epair_2EFST @ A_27a @ A_27c ) @ V2al ) ) @ ( c_2Epred__set_2EUNIV @ A_27b ) )
& ( V3mal
= ( c_2Elist_2EMAP @ ( tyop_2Epair_2Eprod @ A_27a @ A_27c ) @ ( tyop_2Epair_2Eprod @ A_27b @ A_27d )
@ ( c_2Epair_2EUNCURRY @ A_27a @ A_27c @ ( tyop_2Epair_2Eprod @ A_27b @ A_27d )
@ ^ [V5x: A_27a,V6y: A_27c] : ( c_2Epair_2E_2C @ A_27b @ A_27d @ ( V0f1 @ V5x ) @ ( V1f2 @ V6y ) ) )
@ V2al ) )
& ( V4v
= ( c_2Efinite__map_2EMAP__KEYS @ A_27a @ A_27b @ A_27d @ V0f1 @ ( c_2Efinite__map_2Eo__f @ A_27a @ A_27c @ A_27d @ V1f2 @ ( c_2Ealist_2Ealist__to__fmap @ A_27a @ A_27c @ V2al ) ) ) ) )
=> ( ( c_2Ealist_2Ealist__to__fmap @ A_27b @ A_27d @ V3mal )
= V4v ) ) ).
thf(thm_2Ealist_2EMAP__values__fmap__to__alist,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_2Elist_2EMAP @ ( tyop_2Epair_2Eprod @ A_27c @ A_27a ) @ ( tyop_2Epair_2Eprod @ A_27c @ A_27b )
@ ( c_2Epair_2EUNCURRY @ A_27c @ A_27a @ ( tyop_2Epair_2Eprod @ A_27c @ A_27b )
@ ^ [V2k: A_27c,V3v: A_27a] : ( c_2Epair_2E_2C @ A_27c @ A_27b @ V2k @ ( V0f @ V3v ) ) )
@ ( c_2Ealist_2Efmap__to__alist @ A_27c @ A_27a @ V1fm ) )
= ( c_2Ealist_2Efmap__to__alist @ A_27c @ A_27b @ ( c_2Efinite__map_2Eo__f @ A_27c @ A_27a @ A_27b @ V0f @ V1fm ) ) ) ).
thf(thm_2Ealist_2EMAP__KEYS__I,axiom,
! [A_27a: $tType,A_27b: $tType,V0fm: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b] :
( ( c_2Efinite__map_2EMAP__KEYS @ A_27a @ A_27a @ A_27b @ ( c_2Ecombin_2EI @ A_27a ) @ V0fm )
= V0fm ) ).
thf(thm_2Ealist_2Ealist__to__fmap__MAP__values,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0f: A_27a > A_27b,V1al: tyop_2Elist_2Elist @ ( tyop_2Epair_2Eprod @ A_27c @ A_27a )] :
( ( c_2Ealist_2Ealist__to__fmap @ A_27c @ A_27b
@ ( c_2Elist_2EMAP @ ( tyop_2Epair_2Eprod @ A_27c @ A_27a ) @ ( tyop_2Epair_2Eprod @ A_27c @ A_27b )
@ ( c_2Epair_2EUNCURRY @ A_27c @ A_27a @ ( tyop_2Epair_2Eprod @ A_27c @ A_27b )
@ ^ [V2k: A_27c,V3v: A_27a] : ( c_2Epair_2E_2C @ A_27c @ A_27b @ V2k @ ( V0f @ V3v ) ) )
@ V1al ) )
= ( c_2Efinite__map_2Eo__f @ A_27c @ A_27a @ A_27b @ V0f @ ( c_2Ealist_2Ealist__to__fmap @ A_27c @ A_27a @ V1al ) ) ) ).
thf(thm_2Ealist_2Eset__MAP__FST__fmap__to__alist,axiom,
! [A_27a: $tType,A_27b: $tType,V0fm: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b] :
( ( 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 ) @ ( c_2Ealist_2Efmap__to__alist @ A_27a @ A_27b @ V0fm ) ) )
= ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b @ V0fm ) ) ).
thf(thm_2Ealist_2Ealookup__distinct__reverse,axiom,
! [A_27a: $tType,A_27b: $tType,V0l: tyop_2Elist_2Elist @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ),V1k: A_27a] :
( ( 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 ) @ V0l ) )
=> ( ( c_2Ealist_2EALOOKUP @ A_27b @ A_27a @ ( c_2Elist_2EREVERSE @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ V0l ) @ V1k )
= ( c_2Ealist_2EALOOKUP @ A_27b @ A_27a @ V0l @ V1k ) ) ) ).
thf(thm_2Ealist_2Eflookup__fupdate__list,axiom,
! [A_27a: $tType,A_27b: $tType,V0l: tyop_2Elist_2Elist @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ),V1k: A_27a,V2m: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b] :
( ( c_2Efinite__map_2EFLOOKUP @ A_27a @ A_27b @ ( c_2Efinite__map_2EFUPDATE__LIST @ A_27a @ A_27b @ V2m @ V0l ) @ V1k )
= ( c_2Eoption_2Eoption__CASE @ A_27b @ ( tyop_2Eoption_2Eoption @ A_27b ) @ ( c_2Ealist_2EALOOKUP @ A_27b @ A_27a @ ( c_2Elist_2EREVERSE @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ V0l ) @ V1k ) @ ( c_2Efinite__map_2EFLOOKUP @ A_27a @ A_27b @ V2m @ V1k )
@ ^ [V3v: A_27b] : ( c_2Eoption_2ESOME @ A_27b @ V3v ) ) ) ).
thf(thm_2Ealist_2Efupdate__list__funion,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_2Efinite__map_2EFUPDATE__LIST @ A_27a @ A_27b @ V0m @ V1l )
= ( c_2Efinite__map_2EFUNION @ A_27a @ A_27b @ ( c_2Efinite__map_2EFUPDATE__LIST @ A_27a @ A_27b @ ( c_2Efinite__map_2EFEMPTY @ A_27a @ A_27b ) @ V1l ) @ V0m ) ) ).
thf(thm_2Ealist_2Emem__to__flookup,axiom,
! [A_27a: $tType,A_27b: $tType,V0x: A_27a,V1y: A_27b,V2l: 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 ) @ V2l ) )
& ( c_2Ebool_2EIN @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ ( c_2Epair_2E_2C @ A_27a @ A_27b @ V0x @ V1y ) @ ( c_2Elist_2ELIST__TO__SET @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ V2l ) ) )
=> ( ( c_2Efinite__map_2EFLOOKUP @ A_27a @ A_27b @ ( c_2Efinite__map_2EFUPDATE__LIST @ A_27a @ A_27b @ ( c_2Efinite__map_2EFEMPTY @ A_27a @ A_27b ) @ V2l ) @ V0x )
= ( c_2Eoption_2ESOME @ A_27b @ V1y ) ) ) ).
thf(thm_2Ealist_2Ealookup__filter,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0f: A_27a,V1l: tyop_2Elist_2Elist @ ( tyop_2Epair_2Eprod @ A_27b @ A_27c ),V2x: A_27b] :
( ( c_2Ealist_2EALOOKUP @ A_27c @ A_27b @ V1l @ V2x )
= ( c_2Ealist_2EALOOKUP @ A_27c @ A_27b
@ ( c_2Elist_2EFILTER @ ( tyop_2Epair_2Eprod @ A_27b @ A_27c )
@ ( c_2Epair_2EUNCURRY @ A_27b @ A_27c @ $o
@ ^ [V3x_27: A_27b,V4y: A_27c] : ( c_2Emin_2E_3D @ A_27b @ V2x @ V3x_27 ) )
@ V1l )
@ V2x ) ) ).
thf(thm_2Ealist_2Ealookup__stable__sorted,axiom,
! [A_27a: $tType,A_27b: $tType,V0R: A_27a > A_27a > $o,V1sort: ( ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) > ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) > $o ) > ( tyop_2Elist_2Elist @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) ) > ( tyop_2Elist_2Elist @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) ),V2x: A_27a,V3l: tyop_2Elist_2Elist @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b )] :
( ( ( c_2Erelation_2Etransitive @ A_27a @ V0R )
& ( c_2Erelation_2Etotal @ A_27a @ V0R )
& ( c_2Esorting_2ESTABLE @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ V1sort @ ( c_2Erelation_2Einv__image @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ A_27a @ V0R @ ( c_2Epair_2EFST @ A_27a @ A_27b ) ) ) )
=> ( ( c_2Ealist_2EALOOKUP @ A_27b @ A_27a @ ( V1sort @ ( c_2Erelation_2Einv__image @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ A_27a @ V0R @ ( c_2Epair_2EFST @ A_27a @ A_27b ) ) @ V3l ) @ V2x )
= ( c_2Ealist_2EALOOKUP @ A_27b @ A_27a @ V3l @ V2x ) ) ) ).
thf(thm_2Ealist_2EALOOKUP__ALL__DISTINCT__PERM__same,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 )] :
( ( ( 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 ) @ V0l1 ) )
& ( c_2Esorting_2EPERM @ A_27a @ ( c_2Elist_2EMAP @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ A_27a @ ( c_2Epair_2EFST @ A_27a @ A_27b ) @ V0l1 ) @ ( c_2Elist_2EMAP @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ A_27a @ ( c_2Epair_2EFST @ A_27a @ A_27b ) @ V1l2 ) )
& ( ( c_2Elist_2ELIST__TO__SET @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ V0l1 )
= ( c_2Elist_2ELIST__TO__SET @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ V1l2 ) ) )
=> ( ( c_2Ealist_2EALOOKUP @ A_27b @ A_27a @ V0l1 )
= ( c_2Ealist_2EALOOKUP @ A_27b @ A_27a @ V1l2 ) ) ) ).
%------------------------------------------------------------------------------