ITP001 Axioms: ITP065^5.ax
%------------------------------------------------------------------------------
% File : ITP065^5 : TPTP v8.2.0. Bugfixed v7.5.0.
% Domain : Interactive Theorem Proving
% Axioms : HOL4 set theory export, chainy mode
% Version : [BG+19] axioms.
% English :
% Refs : [BG+19] Brown et al. (2019), GRUNGE: A Grand Unified ATP Chall
% : [Gau20] Gauthier (2020), Email to Geoff Sutcliffe
% Source : [BG+19]
% Names : finite_map^2.ax [Gau20]
% : HL4065^5.ax [TPAP]
% Status : Satisfiable
% Syntax : Number of formulae : 331 ( 8 unt; 29 typ; 0 def)
% Number of atoms : 5873 ( 305 equ; 0 cnn)
% Maximal formula atoms : 97 ( 17 avg)
% Number of connectives : 14132 ( 30 ~; 9 |; 136 &;12944 @)
% ( 62 <=>; 951 =>; 0 <=; 0 <~>)
% Maximal formula depth : 32 ( 15 avg;12944 nst)
% Number of types : 1 ( 0 usr)
% Number of type conns : 64 ( 64 >; 0 *; 0 +; 0 <<)
% Number of symbols : 122 ( 121 usr; 92 con; 0-3 aty)
% Number of variables : 1565 ( 68 ^1474 !; 23 ?;1565 :)
% SPC : TH0_SAT_EQU_NAR
% Comments :
% Bugfixes : v7.5.0 - Fixes to the axioms.
%------------------------------------------------------------------------------
thf(tp_ty_2Efinite__map_2Efmap,type,
ty_2Efinite__map_2Efmap: del > del > del ).
thf(tp_c_2Efinite__map_2EDRESTRICT,type,
c_2Efinite__map_2EDRESTRICT: del > del > $i ).
thf(mem_c_2Efinite__map_2EDRESTRICT,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2Efinite__map_2EDRESTRICT @ A_27a @ A_27b ) @ ( arr @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) @ ( arr @ ( arr @ A_27a @ bool ) @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) ) ) ) ).
thf(tp_c_2Efinite__map_2EFAPPLY,type,
c_2Efinite__map_2EFAPPLY: del > del > $i ).
thf(mem_c_2Efinite__map_2EFAPPLY,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2Efinite__map_2EFAPPLY @ A_27a @ A_27b ) @ ( arr @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) @ ( arr @ A_27a @ A_27b ) ) ) ).
thf(tp_c_2Efinite__map_2EFCARD,type,
c_2Efinite__map_2EFCARD: del > del > $i ).
thf(mem_c_2Efinite__map_2EFCARD,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2Efinite__map_2EFCARD @ A_27a @ A_27b ) @ ( arr @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) @ ty_2Enum_2Enum ) ) ).
thf(tp_c_2Efinite__map_2EFDOM,type,
c_2Efinite__map_2EFDOM: del > del > $i ).
thf(mem_c_2Efinite__map_2EFDOM,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b ) @ ( arr @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) @ ( arr @ A_27a @ bool ) ) ) ).
thf(tp_c_2Efinite__map_2EFEMPTY,type,
c_2Efinite__map_2EFEMPTY: del > del > $i ).
thf(mem_c_2Efinite__map_2EFEMPTY,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2Efinite__map_2EFEMPTY @ A_27a @ A_27b ) @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) ) ).
thf(tp_c_2Efinite__map_2EFEVERY,type,
c_2Efinite__map_2EFEVERY: del > del > $i ).
thf(mem_c_2Efinite__map_2EFEVERY,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2Efinite__map_2EFEVERY @ A_27a @ A_27b ) @ ( arr @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ bool ) @ ( arr @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) @ bool ) ) ) ).
thf(tp_c_2Efinite__map_2EFLOOKUP,type,
c_2Efinite__map_2EFLOOKUP: del > del > $i ).
thf(mem_c_2Efinite__map_2EFLOOKUP,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2Efinite__map_2EFLOOKUP @ A_27a @ A_27b ) @ ( arr @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) @ ( arr @ A_27a @ ( ty_2Eoption_2Eoption @ A_27b ) ) ) ) ).
thf(tp_c_2Efinite__map_2EFMAP__MAP2,type,
c_2Efinite__map_2EFMAP__MAP2: del > del > del > $i ).
thf(mem_c_2Efinite__map_2EFMAP__MAP2,axiom,
! [A_27a: del,A_27b: del,A_27c: del] : ( mem @ ( c_2Efinite__map_2EFMAP__MAP2 @ A_27a @ A_27b @ A_27c ) @ ( arr @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27c ) @ A_27b ) @ ( arr @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27c ) @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) ) ) ) ).
thf(tp_c_2Efinite__map_2EFMERGE,type,
c_2Efinite__map_2EFMERGE: del > del > $i ).
thf(mem_c_2Efinite__map_2EFMERGE,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2Efinite__map_2EFMERGE @ A_27a @ A_27b ) @ ( arr @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) @ ( arr @ ( ty_2Efinite__map_2Efmap @ A_27b @ A_27a ) @ ( arr @ ( ty_2Efinite__map_2Efmap @ A_27b @ A_27a ) @ ( ty_2Efinite__map_2Efmap @ A_27b @ A_27a ) ) ) ) ) ).
thf(tp_c_2Efinite__map_2EFRANGE,type,
c_2Efinite__map_2EFRANGE: del > del > $i ).
thf(mem_c_2Efinite__map_2EFRANGE,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2Efinite__map_2EFRANGE @ A_27a @ A_27b ) @ ( arr @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) @ ( arr @ A_27b @ bool ) ) ) ).
thf(tp_c_2Efinite__map_2EFUNION,type,
c_2Efinite__map_2EFUNION: del > del > $i ).
thf(mem_c_2Efinite__map_2EFUNION,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2Efinite__map_2EFUNION @ A_27a @ A_27b ) @ ( arr @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) @ ( arr @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) ) ) ) ).
thf(tp_c_2Efinite__map_2EFUN__FMAP,type,
c_2Efinite__map_2EFUN__FMAP: del > del > $i ).
thf(mem_c_2Efinite__map_2EFUN__FMAP,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2Efinite__map_2EFUN__FMAP @ A_27a @ A_27b ) @ ( arr @ ( arr @ A_27a @ A_27b ) @ ( arr @ ( arr @ A_27a @ bool ) @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) ) ) ) ).
thf(tp_c_2Efinite__map_2EFUPDATE,type,
c_2Efinite__map_2EFUPDATE: del > del > $i ).
thf(mem_c_2Efinite__map_2EFUPDATE,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2Efinite__map_2EFUPDATE @ A_27a @ A_27b ) @ ( arr @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) ) ) ) ).
thf(tp_c_2Efinite__map_2EFUPDATE__LIST,type,
c_2Efinite__map_2EFUPDATE__LIST: del > del > $i ).
thf(mem_c_2Efinite__map_2EFUPDATE__LIST,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2Efinite__map_2EFUPDATE__LIST @ A_27a @ A_27b ) @ ( arr @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) @ ( arr @ ( ty_2Elist_2Elist @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) ) @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) ) ) ) ).
thf(tp_c_2Efinite__map_2EMAP__KEYS,type,
c_2Efinite__map_2EMAP__KEYS: del > del > del > $i ).
thf(mem_c_2Efinite__map_2EMAP__KEYS,axiom,
! [A_27a: del,A_27b: del,A_27c: del] : ( mem @ ( c_2Efinite__map_2EMAP__KEYS @ A_27a @ A_27b @ A_27c ) @ ( arr @ ( arr @ A_27a @ A_27b ) @ ( arr @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27c ) @ ( ty_2Efinite__map_2Efmap @ A_27b @ A_27c ) ) ) ) ).
thf(tp_c_2Efinite__map_2ERRESTRICT,type,
c_2Efinite__map_2ERRESTRICT: del > del > $i ).
thf(mem_c_2Efinite__map_2ERRESTRICT,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2Efinite__map_2ERRESTRICT @ A_27a @ A_27b ) @ ( arr @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) @ ( arr @ ( arr @ A_27b @ bool ) @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) ) ) ) ).
thf(tp_c_2Efinite__map_2ESUBMAP,type,
c_2Efinite__map_2ESUBMAP: del > del > $i ).
thf(mem_c_2Efinite__map_2ESUBMAP,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2Efinite__map_2ESUBMAP @ A_27a @ A_27b ) @ ( arr @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) @ ( arr @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) @ bool ) ) ) ).
thf(tp_c_2Efinite__map_2Ef__o,type,
c_2Efinite__map_2Ef__o: del > del > del > $i ).
thf(mem_c_2Efinite__map_2Ef__o,axiom,
! [A_27a: del,A_27b: del,A_27c: del] : ( mem @ ( c_2Efinite__map_2Ef__o @ A_27a @ A_27b @ A_27c ) @ ( arr @ ( ty_2Efinite__map_2Efmap @ A_27b @ A_27c ) @ ( arr @ ( arr @ A_27a @ A_27b ) @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27c ) ) ) ) ).
thf(tp_c_2Efinite__map_2Ef__o__f,type,
c_2Efinite__map_2Ef__o__f: del > del > del > $i ).
thf(mem_c_2Efinite__map_2Ef__o__f,axiom,
! [A_27a: del,A_27b: del,A_27c: del] : ( mem @ ( c_2Efinite__map_2Ef__o__f @ A_27a @ A_27b @ A_27c ) @ ( arr @ ( ty_2Efinite__map_2Efmap @ A_27b @ A_27c ) @ ( arr @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27c ) ) ) ) ).
thf(tp_c_2Efinite__map_2Efdomsub,type,
c_2Efinite__map_2Efdomsub: del > del > $i ).
thf(mem_c_2Efinite__map_2Efdomsub,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2Efinite__map_2Efdomsub @ A_27a @ A_27b ) @ ( arr @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) @ ( arr @ A_27a @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) ) ) ) ).
thf(tp_c_2Efinite__map_2Efmap__ABS,type,
c_2Efinite__map_2Efmap__ABS: del > del > $i ).
thf(mem_c_2Efinite__map_2Efmap__ABS,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2Efinite__map_2Efmap__ABS @ A_27a @ A_27b ) @ ( arr @ ( arr @ A_27a @ ( ty_2Esum_2Esum @ A_27b @ ty_2Eone_2Eone ) ) @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) ) ) ).
thf(tp_c_2Efinite__map_2Efmap__EQ__UPTO,type,
c_2Efinite__map_2Efmap__EQ__UPTO: del > del > $i ).
thf(mem_c_2Efinite__map_2Efmap__EQ__UPTO,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2Efinite__map_2Efmap__EQ__UPTO @ A_27a @ A_27b ) @ ( arr @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) @ ( arr @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) ) ) ).
thf(tp_c_2Efinite__map_2Efmap__REP,type,
c_2Efinite__map_2Efmap__REP: del > del > $i ).
thf(mem_c_2Efinite__map_2Efmap__REP,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2Efinite__map_2Efmap__REP @ A_27a @ A_27b ) @ ( arr @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) @ ( arr @ A_27a @ ( ty_2Esum_2Esum @ A_27b @ ty_2Eone_2Eone ) ) ) ) ).
thf(tp_c_2Efinite__map_2Efmap__inverse,type,
c_2Efinite__map_2Efmap__inverse: del > del > $i ).
thf(mem_c_2Efinite__map_2Efmap__inverse,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2Efinite__map_2Efmap__inverse @ A_27a @ A_27b ) @ ( arr @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) @ ( arr @ ( ty_2Efinite__map_2Efmap @ A_27b @ A_27a ) @ bool ) ) ) ).
thf(tp_c_2Efinite__map_2Efmap__rel,type,
c_2Efinite__map_2Efmap__rel: del > del > del > $i ).
thf(mem_c_2Efinite__map_2Efmap__rel,axiom,
! [A_27a: del,A_27b: del,A_27c: del] : ( mem @ ( c_2Efinite__map_2Efmap__rel @ A_27a @ A_27b @ A_27c ) @ ( arr @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) @ ( arr @ ( ty_2Efinite__map_2Efmap @ A_27c @ A_27a ) @ ( arr @ ( ty_2Efinite__map_2Efmap @ A_27c @ A_27b ) @ bool ) ) ) ) ).
thf(tp_c_2Efinite__map_2Efmap__size,type,
c_2Efinite__map_2Efmap__size: del > del > $i ).
thf(mem_c_2Efinite__map_2Efmap__size,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2Efinite__map_2Efmap__size @ A_27a @ A_27b ) @ ( arr @ ( arr @ A_27a @ ty_2Enum_2Enum ) @ ( arr @ ( arr @ A_27b @ ty_2Enum_2Enum ) @ ( arr @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) @ ty_2Enum_2Enum ) ) ) ) ).
thf(tp_c_2Efinite__map_2Eis__fmap,type,
c_2Efinite__map_2Eis__fmap: del > del > $i ).
thf(mem_c_2Efinite__map_2Eis__fmap,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2Efinite__map_2Eis__fmap @ A_27a @ A_27b ) @ ( arr @ ( arr @ A_27a @ ( ty_2Esum_2Esum @ A_27b @ ty_2Eone_2Eone ) ) @ bool ) ) ).
thf(tp_c_2Efinite__map_2Eo__f,type,
c_2Efinite__map_2Eo__f: del > del > del > $i ).
thf(mem_c_2Efinite__map_2Eo__f,axiom,
! [A_27a: del,A_27b: del,A_27c: del] : ( mem @ ( c_2Efinite__map_2Eo__f @ A_27a @ A_27b @ A_27c ) @ ( arr @ ( arr @ A_27b @ A_27c ) @ ( arr @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27c ) ) ) ) ).
thf(ax_thm_2Efinite__map_2Eis__fmap__def,axiom,
! [A_27a: del,A_27b: del] :
( ( c_2Efinite__map_2Eis__fmap @ A_27a @ A_27b )
= ( lam @ ( arr @ A_27a @ ( ty_2Esum_2Esum @ A_27b @ ty_2Eone_2Eone ) )
@ ^ [V0a0: $i] :
( ap @ ( c_2Ebool_2E_21 @ ( arr @ ( arr @ A_27a @ ( ty_2Esum_2Esum @ A_27b @ ty_2Eone_2Eone ) ) @ bool ) )
@ ( lam @ ( arr @ ( arr @ A_27a @ ( ty_2Esum_2Esum @ A_27b @ ty_2Eone_2Eone ) ) @ bool )
@ ^ [V1is__fmap_27: $i] :
( ap
@ ( ap @ c_2Emin_2E_3D_3D_3E
@ ( ap @ ( c_2Ebool_2E_21 @ ( arr @ A_27a @ ( ty_2Esum_2Esum @ A_27b @ ty_2Eone_2Eone ) ) )
@ ( lam @ ( arr @ A_27a @ ( ty_2Esum_2Esum @ A_27b @ ty_2Eone_2Eone ) )
@ ^ [V2a0: $i] :
( ap
@ ( ap @ c_2Emin_2E_3D_3D_3E
@ ( ap
@ ( ap @ c_2Ebool_2E_5C_2F
@ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ( arr @ A_27a @ ( ty_2Esum_2Esum @ A_27b @ ty_2Eone_2Eone ) ) ) @ V2a0 )
@ ( lam @ A_27a
@ ^ [V3a: $i] : ( ap @ ( c_2Esum_2EINR @ A_27b @ ty_2Eone_2Eone ) @ ( inj__ty_2Eone_2Eone @ fo__c_2Eone_2Eone ) ) ) ) )
@ ( ap @ ( c_2Ebool_2E_3F @ ( arr @ A_27a @ ( ty_2Esum_2Esum @ A_27b @ ty_2Eone_2Eone ) ) )
@ ( lam @ ( arr @ A_27a @ ( ty_2Esum_2Esum @ A_27b @ ty_2Eone_2Eone ) )
@ ^ [V4f: $i] :
( ap @ ( c_2Ebool_2E_3F @ A_27a )
@ ( lam @ A_27a
@ ^ [V5a: $i] :
( ap @ ( c_2Ebool_2E_3F @ A_27b )
@ ( lam @ A_27b
@ ^ [V6b: $i] :
( ap
@ ( ap @ c_2Ebool_2E_2F_5C
@ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ( arr @ A_27a @ ( ty_2Esum_2Esum @ A_27b @ ty_2Eone_2Eone ) ) ) @ V2a0 )
@ ( lam @ A_27a
@ ^ [V7x: $i] : ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ( ty_2Esum_2Esum @ A_27b @ ty_2Eone_2Eone ) ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ A_27a ) @ V7x ) @ V5a ) ) @ ( ap @ ( c_2Esum_2EINL @ A_27b @ ty_2Eone_2Eone ) @ V6b ) ) @ ( ap @ V4f @ V7x ) ) ) ) )
@ ( ap @ V1is__fmap_27 @ V4f ) ) ) ) ) ) ) ) ) )
@ ( ap @ V1is__fmap_27 @ V2a0 ) ) ) ) )
@ ( ap @ V1is__fmap_27 @ V0a0 ) ) ) ) ) ) ).
thf(conj_thm_2Efinite__map_2Eis__fmap__rules,axiom,
! [A_27a: del,A_27b: del] :
( ( p
@ ( ap @ ( c_2Efinite__map_2Eis__fmap @ A_27a @ A_27b )
@ ( lam @ A_27a
@ ^ [V0a: $i] : ( ap @ ( c_2Esum_2EINR @ A_27b @ ty_2Eone_2Eone ) @ ( inj__ty_2Eone_2Eone @ fo__c_2Eone_2Eone ) ) ) ) )
& ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ ( ty_2Esum_2Esum @ A_27b @ ty_2Eone_2Eone ) ) )
=> ! [V2a: $i] :
( ( mem @ V2a @ A_27a )
=> ! [V3b: $i] :
( ( mem @ V3b @ A_27b )
=> ( ( p @ ( ap @ ( c_2Efinite__map_2Eis__fmap @ A_27a @ A_27b ) @ V1f ) )
=> ( p
@ ( ap @ ( c_2Efinite__map_2Eis__fmap @ A_27a @ A_27b )
@ ( lam @ A_27a
@ ^ [V4x: $i] : ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ( ty_2Esum_2Esum @ A_27b @ ty_2Eone_2Eone ) ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ A_27a ) @ V4x ) @ V2a ) ) @ ( ap @ ( c_2Esum_2EINL @ A_27b @ ty_2Eone_2Eone ) @ V3b ) ) @ ( ap @ V1f @ V4x ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Efinite__map_2Eis__fmap__ind,axiom,
! [A_27a: del,A_27b: del,V0is__fmap_27: $i] :
( ( mem @ V0is__fmap_27 @ ( arr @ ( arr @ A_27a @ ( ty_2Esum_2Esum @ A_27b @ ty_2Eone_2Eone ) ) @ bool ) )
=> ( ( ( p
@ ( ap @ V0is__fmap_27
@ ( lam @ A_27a
@ ^ [V1a: $i] : ( ap @ ( c_2Esum_2EINR @ A_27b @ ty_2Eone_2Eone ) @ ( inj__ty_2Eone_2Eone @ fo__c_2Eone_2Eone ) ) ) ) )
& ! [V2f: $i] :
( ( mem @ V2f @ ( arr @ A_27a @ ( ty_2Esum_2Esum @ A_27b @ ty_2Eone_2Eone ) ) )
=> ! [V3a: $i] :
( ( mem @ V3a @ A_27a )
=> ! [V4b: $i] :
( ( mem @ V4b @ A_27b )
=> ( ( p @ ( ap @ V0is__fmap_27 @ V2f ) )
=> ( p
@ ( ap @ V0is__fmap_27
@ ( lam @ A_27a
@ ^ [V5x: $i] : ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ( ty_2Esum_2Esum @ A_27b @ ty_2Eone_2Eone ) ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ A_27a ) @ V5x ) @ V3a ) ) @ ( ap @ ( c_2Esum_2EINL @ A_27b @ ty_2Eone_2Eone ) @ V4b ) ) @ ( ap @ V2f @ V5x ) ) ) ) ) ) ) ) ) )
=> ! [V6a0: $i] :
( ( mem @ V6a0 @ ( arr @ A_27a @ ( ty_2Esum_2Esum @ A_27b @ ty_2Eone_2Eone ) ) )
=> ( ( p @ ( ap @ ( c_2Efinite__map_2Eis__fmap @ A_27a @ A_27b ) @ V6a0 ) )
=> ( p @ ( ap @ V0is__fmap_27 @ V6a0 ) ) ) ) ) ) ).
thf(conj_thm_2Efinite__map_2Eis__fmap__strongind,axiom,
! [A_27a: del,A_27b: del,V0is__fmap_27: $i] :
( ( mem @ V0is__fmap_27 @ ( arr @ ( arr @ A_27a @ ( ty_2Esum_2Esum @ A_27b @ ty_2Eone_2Eone ) ) @ bool ) )
=> ( ( ( p
@ ( ap @ V0is__fmap_27
@ ( lam @ A_27a
@ ^ [V1a: $i] : ( ap @ ( c_2Esum_2EINR @ A_27b @ ty_2Eone_2Eone ) @ ( inj__ty_2Eone_2Eone @ fo__c_2Eone_2Eone ) ) ) ) )
& ! [V2f: $i] :
( ( mem @ V2f @ ( arr @ A_27a @ ( ty_2Esum_2Esum @ A_27b @ ty_2Eone_2Eone ) ) )
=> ! [V3a: $i] :
( ( mem @ V3a @ A_27a )
=> ! [V4b: $i] :
( ( mem @ V4b @ A_27b )
=> ( ( ( p @ ( ap @ ( c_2Efinite__map_2Eis__fmap @ A_27a @ A_27b ) @ V2f ) )
& ( p @ ( ap @ V0is__fmap_27 @ V2f ) ) )
=> ( p
@ ( ap @ V0is__fmap_27
@ ( lam @ A_27a
@ ^ [V5x: $i] : ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ( ty_2Esum_2Esum @ A_27b @ ty_2Eone_2Eone ) ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ A_27a ) @ V5x ) @ V3a ) ) @ ( ap @ ( c_2Esum_2EINL @ A_27b @ ty_2Eone_2Eone ) @ V4b ) ) @ ( ap @ V2f @ V5x ) ) ) ) ) ) ) ) ) )
=> ! [V6a0: $i] :
( ( mem @ V6a0 @ ( arr @ A_27a @ ( ty_2Esum_2Esum @ A_27b @ ty_2Eone_2Eone ) ) )
=> ( ( p @ ( ap @ ( c_2Efinite__map_2Eis__fmap @ A_27a @ A_27b ) @ V6a0 ) )
=> ( p @ ( ap @ V0is__fmap_27 @ V6a0 ) ) ) ) ) ) ).
thf(conj_thm_2Efinite__map_2Eis__fmap__cases,axiom,
! [A_27a: del,A_27b: del,V0a0: $i] :
( ( mem @ V0a0 @ ( arr @ A_27a @ ( ty_2Esum_2Esum @ A_27b @ ty_2Eone_2Eone ) ) )
=> ( ( p @ ( ap @ ( c_2Efinite__map_2Eis__fmap @ A_27a @ A_27b ) @ V0a0 ) )
<=> ( ( V0a0
= ( lam @ A_27a
@ ^ [V1a: $i] : ( ap @ ( c_2Esum_2EINR @ A_27b @ ty_2Eone_2Eone ) @ ( inj__ty_2Eone_2Eone @ fo__c_2Eone_2Eone ) ) ) )
| ? [V2f: $i] :
( ( mem @ V2f @ ( arr @ A_27a @ ( ty_2Esum_2Esum @ A_27b @ ty_2Eone_2Eone ) ) )
& ? [V3a: $i] :
( ( mem @ V3a @ A_27a )
& ? [V4b: $i] :
( ( mem @ V4b @ A_27b )
& ( V0a0
= ( lam @ A_27a
@ ^ [V5x: $i] : ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ( ty_2Esum_2Esum @ A_27b @ ty_2Eone_2Eone ) ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ A_27a ) @ V5x ) @ V3a ) ) @ ( ap @ ( c_2Esum_2EINL @ A_27b @ ty_2Eone_2Eone ) @ V4b ) ) @ ( ap @ V2f @ V5x ) ) ) )
& ( p @ ( ap @ ( c_2Efinite__map_2Eis__fmap @ A_27a @ A_27b ) @ V2f ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Efinite__map_2Efmap__TY__DEF,axiom,
! [A_27a: del,A_27b: del] :
? [V0rep: $i] :
( ( mem @ V0rep @ ( arr @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) @ ( arr @ A_27a @ ( ty_2Esum_2Esum @ A_27b @ ty_2Eone_2Eone ) ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2ETYPE__DEFINITION @ ( arr @ A_27a @ ( ty_2Esum_2Esum @ A_27b @ ty_2Eone_2Eone ) ) @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) ) @ ( c_2Efinite__map_2Eis__fmap @ A_27a @ A_27b ) ) @ V0rep ) ) ) ).
thf(ax_thm_2Efinite__map_2Efmap__ISO__DEF,axiom,
! [A_27a: del,A_27b: del] :
( ! [V0a: $i] :
( ( mem @ V0a @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ( ( ap @ ( c_2Efinite__map_2Efmap__ABS @ A_27a @ A_27b ) @ ( ap @ ( c_2Efinite__map_2Efmap__REP @ A_27a @ A_27b ) @ V0a ) )
= V0a ) )
& ! [V1r: $i] :
( ( mem @ V1r @ ( arr @ A_27a @ ( ty_2Esum_2Esum @ A_27b @ ty_2Eone_2Eone ) ) )
=> ( ( p @ ( ap @ ( c_2Efinite__map_2Eis__fmap @ A_27a @ A_27b ) @ V1r ) )
<=> ( ( ap @ ( c_2Efinite__map_2Efmap__REP @ A_27a @ A_27b ) @ ( ap @ ( c_2Efinite__map_2Efmap__ABS @ A_27a @ A_27b ) @ V1r ) )
= V1r ) ) ) ) ).
thf(ax_thm_2Efinite__map_2EFUPDATE__DEF,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ! [V2y: $i] :
( ( mem @ V2y @ A_27b )
=> ( ( ap @ ( ap @ ( c_2Efinite__map_2EFUPDATE @ A_27a @ A_27b ) @ V0f ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27b ) @ V1x ) @ V2y ) )
= ( ap @ ( c_2Efinite__map_2Efmap__ABS @ A_27a @ A_27b )
@ ( lam @ A_27a
@ ^ [V3a: $i] : ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ( ty_2Esum_2Esum @ A_27b @ ty_2Eone_2Eone ) ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ A_27a ) @ V3a ) @ V1x ) ) @ ( ap @ ( c_2Esum_2EINL @ A_27b @ ty_2Eone_2Eone ) @ V2y ) ) @ ( ap @ ( ap @ ( c_2Efinite__map_2Efmap__REP @ A_27a @ A_27b ) @ V0f ) @ V3a ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Efinite__map_2EFEMPTY__DEF,axiom,
! [A_27a: del,A_27b: del] :
( ( c_2Efinite__map_2EFEMPTY @ A_27a @ A_27b )
= ( ap @ ( c_2Efinite__map_2Efmap__ABS @ A_27a @ A_27b )
@ ( lam @ A_27a
@ ^ [V0a: $i] : ( ap @ ( c_2Esum_2EINR @ A_27b @ ty_2Eone_2Eone ) @ ( inj__ty_2Eone_2Eone @ fo__c_2Eone_2Eone ) ) ) ) ) ).
thf(ax_thm_2Efinite__map_2EFAPPLY__DEF,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ( ( ap @ ( ap @ ( c_2Efinite__map_2EFAPPLY @ A_27a @ A_27b ) @ V0f ) @ V1x )
= ( ap @ ( c_2Esum_2EOUTL @ A_27b @ ty_2Eone_2Eone ) @ ( ap @ ( ap @ ( c_2Efinite__map_2Efmap__REP @ A_27a @ A_27b ) @ V0f ) @ V1x ) ) ) ) ) ).
thf(ax_thm_2Efinite__map_2EFDOM__DEF,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b ) @ V0f ) @ V1x ) )
<=> ( p @ ( ap @ ( c_2Esum_2EISL @ A_27b @ ty_2Eone_2Eone ) @ ( ap @ ( ap @ ( c_2Efinite__map_2Efmap__REP @ A_27a @ A_27b ) @ V0f ) @ V1x ) ) ) ) ) ) ).
thf(conj_thm_2Efinite__map_2EFAPPLY__FUPDATE,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ! [V2y: $i] :
( ( mem @ V2y @ A_27b )
=> ( ( ap @ ( ap @ ( c_2Efinite__map_2EFAPPLY @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EFUPDATE @ A_27a @ A_27b ) @ V0f ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27b ) @ V1x ) @ V2y ) ) ) @ V1x )
= V2y ) ) ) ) ).
thf(conj_thm_2Efinite__map_2ENOT__EQ__FAPPLY,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ! [V1a: $i] :
( ( mem @ V1a @ A_27a )
=> ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ! [V3y: $i] :
( ( mem @ V3y @ A_27b )
=> ( ( V1a != V2x )
=> ( ( ap @ ( ap @ ( c_2Efinite__map_2EFAPPLY @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EFUPDATE @ A_27a @ A_27b ) @ V0f ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27b ) @ V2x ) @ V3y ) ) ) @ V1a )
= ( ap @ ( ap @ ( c_2Efinite__map_2EFAPPLY @ A_27a @ A_27b ) @ V0f ) @ V1a ) ) ) ) ) ) ) ).
thf(conj_thm_2Efinite__map_2EFUPDATE__COMMUTES,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ! [V1a: $i] :
( ( mem @ V1a @ A_27a )
=> ! [V2b: $i] :
( ( mem @ V2b @ A_27b )
=> ! [V3c: $i] :
( ( mem @ V3c @ A_27a )
=> ! [V4d: $i] :
( ( mem @ V4d @ A_27b )
=> ( ( V1a != V3c )
=> ( ( ap @ ( ap @ ( c_2Efinite__map_2EFUPDATE @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EFUPDATE @ A_27a @ A_27b ) @ V0f ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27b ) @ V1a ) @ V2b ) ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27b ) @ V3c ) @ V4d ) )
= ( ap @ ( ap @ ( c_2Efinite__map_2EFUPDATE @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EFUPDATE @ A_27a @ A_27b ) @ V0f ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27b ) @ V3c ) @ V4d ) ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27b ) @ V1a ) @ V2b ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Efinite__map_2EFUPDATE__EQ,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ! [V1a: $i] :
( ( mem @ V1a @ A_27a )
=> ! [V2b: $i] :
( ( mem @ V2b @ A_27b )
=> ! [V3c: $i] :
( ( mem @ V3c @ A_27b )
=> ( ( ap @ ( ap @ ( c_2Efinite__map_2EFUPDATE @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EFUPDATE @ A_27a @ A_27b ) @ V0f ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27b ) @ V1a ) @ V2b ) ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27b ) @ V1a ) @ V3c ) )
= ( ap @ ( ap @ ( c_2Efinite__map_2EFUPDATE @ A_27a @ A_27b ) @ V0f ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27b ) @ V1a ) @ V3c ) ) ) ) ) ) ) ).
thf(conj_thm_2Efinite__map_2EFDOM__FEMPTY,axiom,
! [A_27a: del,A_27b: del] :
( ( ap @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b ) @ ( c_2Efinite__map_2EFEMPTY @ A_27a @ A_27b ) )
= ( c_2Epred__set_2EEMPTY @ A_27a ) ) ).
thf(conj_thm_2Efinite__map_2EFDOM__FUPDATE,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ! [V1a: $i] :
( ( mem @ V1a @ A_27a )
=> ! [V2b: $i] :
( ( mem @ V2b @ A_27b )
=> ( ( ap @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EFUPDATE @ A_27a @ A_27b ) @ V0f ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27b ) @ V1a ) @ V2b ) ) )
= ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ A_27a ) @ V1a ) @ ( ap @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b ) @ V0f ) ) ) ) ) ) ).
thf(conj_thm_2Efinite__map_2EFAPPLY__FUPDATE__THM,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ! [V1a: $i] :
( ( mem @ V1a @ A_27a )
=> ! [V2b: $i] :
( ( mem @ V2b @ A_27b )
=> ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ( ( ap @ ( ap @ ( c_2Efinite__map_2EFAPPLY @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EFUPDATE @ A_27a @ A_27b ) @ V0f ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27b ) @ V1a ) @ V2b ) ) ) @ V3x )
= ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ A_27b ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ A_27a ) @ V3x ) @ V1a ) ) @ V2b ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EFAPPLY @ A_27a @ A_27b ) @ V0f ) @ V3x ) ) ) ) ) ) ) ).
thf(conj_thm_2Efinite__map_2ENOT__EQ__FEMPTY__FUPDATE,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ! [V1a: $i] :
( ( mem @ V1a @ A_27a )
=> ! [V2b: $i] :
( ( mem @ V2b @ A_27b )
=> ( ( c_2Efinite__map_2EFEMPTY @ A_27a @ A_27b )
!= ( ap @ ( ap @ ( c_2Efinite__map_2EFUPDATE @ A_27a @ A_27b ) @ V0f ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27b ) @ V1a ) @ V2b ) ) ) ) ) ) ).
thf(conj_thm_2Efinite__map_2EFDOM__EQ__FDOM__FUPDATE,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V1x ) @ ( ap @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b ) @ V0f ) ) )
=> ! [V2y: $i] :
( ( mem @ V2y @ A_27b )
=> ( ( ap @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EFUPDATE @ A_27a @ A_27b ) @ V0f ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27b ) @ V1x ) @ V2y ) ) )
= ( ap @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b ) @ V0f ) ) ) ) ) ) ).
thf(conj_thm_2Efinite__map_2Efmap__SIMPLE__INDUCT,axiom,
! [A_27a: del,A_27b: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) @ bool ) )
=> ( ( ( p @ ( ap @ V0P @ ( c_2Efinite__map_2EFEMPTY @ A_27a @ A_27b ) ) )
& ! [V1f: $i] :
( ( mem @ V1f @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ( ( p @ ( ap @ V0P @ V1f ) )
=> ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ! [V3y: $i] :
( ( mem @ V3y @ A_27b )
=> ( p @ ( ap @ V0P @ ( ap @ ( ap @ ( c_2Efinite__map_2EFUPDATE @ A_27a @ A_27b ) @ V1f ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27b ) @ V2x ) @ V3y ) ) ) ) ) ) ) ) )
=> ! [V4f: $i] :
( ( mem @ V4f @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ( p @ ( ap @ V0P @ V4f ) ) ) ) ) ).
thf(conj_thm_2Efinite__map_2EFDOM__EQ__EMPTY,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ( ( ( ap @ ( 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(conj_thm_2Efinite__map_2EFDOM__EQ__EMPTY__SYM,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ( ( ( c_2Epred__set_2EEMPTY @ A_27a )
= ( ap @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b ) @ V0f ) )
<=> ( V0f
= ( c_2Efinite__map_2EFEMPTY @ A_27a @ A_27b ) ) ) ) ).
thf(conj_thm_2Efinite__map_2EFDOM__F__FEMPTY1,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ( ! [V1a: $i] :
( ( mem @ V1a @ A_27a )
=> ~ ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V1a ) @ ( ap @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b ) @ V0f ) ) ) )
<=> ( V0f
= ( c_2Efinite__map_2EFEMPTY @ A_27a @ A_27b ) ) ) ) ).
thf(conj_thm_2Efinite__map_2EFDOM__FINITE,axiom,
! [A_27a: del,A_27b: del,V0fm: $i] :
( ( mem @ V0fm @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ ( ap @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b ) @ V0fm ) ) ) ) ).
thf(ax_thm_2Efinite__map_2EFCARD__DEF,axiom,
! [A_27a: del,A_27b: del,V0fm: $i] :
( ( mem @ V0fm @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Efinite__map_2EFCARD @ A_27a @ A_27b ) @ V0fm ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Epred__set_2ECARD @ A_27a ) @ ( ap @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b ) @ V0fm ) ) ) ) ) ).
thf(conj_thm_2Efinite__map_2EFCARD__FEMPTY,axiom,
! [A_27a: del,A_27b: del] :
( ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Efinite__map_2EFCARD @ A_27a @ A_27b ) @ ( c_2Efinite__map_2EFEMPTY @ A_27a @ A_27b ) ) )
= fo__c_2Enum_2E0 ) ).
thf(conj_thm_2Efinite__map_2EFCARD__FUPDATE,axiom,
! [A_27a: del,A_27b: del,V0fm: $i] :
( ( mem @ V0fm @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ! [V1a: $i] :
( ( mem @ V1a @ A_27a )
=> ! [V2b: $i] :
( ( mem @ V2b @ A_27b )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Efinite__map_2EFCARD @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EFUPDATE @ A_27a @ A_27b ) @ V0fm ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27b ) @ V1a ) @ V2b ) ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V1a ) @ ( ap @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b ) @ V0fm ) ) ) @ ( ap @ ( c_2Efinite__map_2EFCARD @ A_27a @ A_27b ) @ V0fm ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) @ ( ap @ ( c_2Efinite__map_2EFCARD @ A_27a @ A_27b ) @ V0fm ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Efinite__map_2EFCARD__0__FEMPTY,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ( ( ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Efinite__map_2EFCARD @ A_27a @ A_27b ) @ V0f ) )
= fo__c_2Enum_2E0 )
<=> ( V0f
= ( c_2Efinite__map_2EFEMPTY @ A_27a @ A_27b ) ) ) ) ).
thf(conj_thm_2Efinite__map_2EFCARD__SUC,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ! [V1n: tp__ty_2Enum_2Enum] :
( ( ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Efinite__map_2EFCARD @ A_27a @ A_27b ) @ V0f ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) )
<=> ? [V2f_27: $i] :
( ( mem @ V2f_27 @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
& ? [V3x: $i] :
( ( mem @ V3x @ A_27a )
& ? [V4y: $i] :
( ( mem @ V4y @ A_27b )
& ~ ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3x ) @ ( ap @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b ) @ V2f_27 ) ) )
& ( ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Efinite__map_2EFCARD @ A_27a @ A_27b ) @ V2f_27 ) )
= V1n )
& ( V0f
= ( ap @ ( ap @ ( c_2Efinite__map_2EFUPDATE @ A_27a @ A_27b ) @ V2f_27 ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27b ) @ V3x ) @ V4y ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Efinite__map_2Efmap__INDUCT,axiom,
! [A_27a: del,A_27b: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) @ bool ) )
=> ( ( ( p @ ( ap @ V0P @ ( c_2Efinite__map_2EFEMPTY @ A_27a @ A_27b ) ) )
& ! [V1f: $i] :
( ( mem @ V1f @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ( ( p @ ( ap @ V0P @ V1f ) )
=> ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ! [V3y: $i] :
( ( mem @ V3y @ A_27b )
=> ( ~ ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x ) @ ( ap @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b ) @ V1f ) ) )
=> ( p @ ( ap @ V0P @ ( ap @ ( ap @ ( c_2Efinite__map_2EFUPDATE @ A_27a @ A_27b ) @ V1f ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27b ) @ V2x ) @ V3y ) ) ) ) ) ) ) ) ) )
=> ! [V4f: $i] :
( ( mem @ V4f @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ( p @ ( ap @ V0P @ V4f ) ) ) ) ) ).
thf(conj_thm_2Efinite__map_2EFM__PULL__APART,axiom,
! [A_27a: del,A_27b: del,V0fm: $i] :
( ( mem @ V0fm @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ! [V1k: $i] :
( ( mem @ V1k @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V1k ) @ ( ap @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b ) @ V0fm ) ) )
=> ? [V2fm0: $i] :
( ( mem @ V2fm0 @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
& ? [V3v: $i] :
( ( mem @ V3v @ A_27b )
& ( V0fm
= ( ap @ ( ap @ ( c_2Efinite__map_2EFUPDATE @ A_27a @ A_27b ) @ V2fm0 ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27b ) @ V1k ) @ V3v ) ) )
& ~ ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V1k ) @ ( ap @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b ) @ V2fm0 ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Efinite__map_2ENOT__FDOM__FAPPLY__FEMPTY,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ( ~ ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V1x ) @ ( ap @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b ) @ V0f ) ) )
=> ( ( ap @ ( ap @ ( c_2Efinite__map_2EFAPPLY @ A_27a @ A_27b ) @ V0f ) @ V1x )
= ( ap @ ( ap @ ( c_2Efinite__map_2EFAPPLY @ A_27a @ A_27b ) @ ( c_2Efinite__map_2EFEMPTY @ A_27a @ A_27b ) ) @ V1x ) ) ) ) ) ).
thf(conj_thm_2Efinite__map_2Efmap__EQ,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ( ( ( ( ap @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b ) @ V0f )
= ( ap @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b ) @ V1g ) )
& ( ( ap @ ( c_2Efinite__map_2EFAPPLY @ A_27a @ A_27b ) @ V0f )
= ( ap @ ( c_2Efinite__map_2EFAPPLY @ A_27a @ A_27b ) @ V1g ) ) )
<=> ( V0f = V1g ) ) ) ) ).
thf(conj_thm_2Efinite__map_2Efmap__EQ__THM,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ( ( ( ( ap @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b ) @ V0f )
= ( ap @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b ) @ V1g ) )
& ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x ) @ ( ap @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b ) @ V0f ) ) )
=> ( ( ap @ ( ap @ ( c_2Efinite__map_2EFAPPLY @ A_27a @ A_27b ) @ V0f ) @ V2x )
= ( ap @ ( ap @ ( c_2Efinite__map_2EFAPPLY @ A_27a @ A_27b ) @ V1g ) @ V2x ) ) ) ) )
<=> ( V0f = V1g ) ) ) ) ).
thf(conj_thm_2Efinite__map_2Efmap__EXT,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ( ( V0f = V1g )
<=> ( ( ( ap @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b ) @ V0f )
= ( ap @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b ) @ V1g ) )
& ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x ) @ ( ap @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b ) @ V0f ) ) )
=> ( ( ap @ ( ap @ ( c_2Efinite__map_2EFAPPLY @ A_27a @ A_27b ) @ V0f ) @ V2x )
= ( ap @ ( ap @ ( c_2Efinite__map_2EFAPPLY @ A_27a @ A_27b ) @ V1g ) @ V2x ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Efinite__map_2ESUBMAP__DEF,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Efinite__map_2ESUBMAP @ A_27a @ A_27b ) @ V0f ) @ V1g ) )
<=> ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x ) @ ( ap @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b ) @ V0f ) ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x ) @ ( ap @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b ) @ V1g ) ) )
& ( ( ap @ ( ap @ ( c_2Efinite__map_2EFAPPLY @ A_27a @ A_27b ) @ V0f ) @ V2x )
= ( ap @ ( ap @ ( c_2Efinite__map_2EFAPPLY @ A_27a @ A_27b ) @ V1g ) @ V2x ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Efinite__map_2ESUBMAP__FEMPTY,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ( p @ ( ap @ ( ap @ ( c_2Efinite__map_2ESUBMAP @ A_27a @ A_27b ) @ ( c_2Efinite__map_2EFEMPTY @ A_27a @ A_27b ) ) @ V0f ) ) ) ).
thf(conj_thm_2Efinite__map_2ESUBMAP__REFL,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ( p @ ( ap @ ( ap @ ( c_2Efinite__map_2ESUBMAP @ A_27a @ A_27b ) @ V0f ) @ V0f ) ) ) ).
thf(conj_thm_2Efinite__map_2ESUBMAP__ANTISYM,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Efinite__map_2ESUBMAP @ A_27a @ A_27b ) @ V0f ) @ V1g ) )
& ( p @ ( ap @ ( ap @ ( c_2Efinite__map_2ESUBMAP @ A_27a @ A_27b ) @ V1g ) @ V0f ) ) )
<=> ( V0f = V1g ) ) ) ) ).
thf(conj_thm_2Efinite__map_2ESUBMAP__TRANS,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ! [V2h: $i] :
( ( mem @ V2h @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Efinite__map_2ESUBMAP @ A_27a @ A_27b ) @ V0f ) @ V1g ) )
& ( p @ ( ap @ ( ap @ ( c_2Efinite__map_2ESUBMAP @ A_27a @ A_27b ) @ V1g ) @ V2h ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Efinite__map_2ESUBMAP @ A_27a @ A_27b ) @ V0f ) @ V2h ) ) ) ) ) ) ).
thf(conj_thm_2Efinite__map_2EEQ__FDOM__SUBMAP,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ( ( V0f = V1g )
<=> ( ( p @ ( ap @ ( ap @ ( c_2Efinite__map_2ESUBMAP @ A_27a @ A_27b ) @ V0f ) @ V1g ) )
& ( ( ap @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b ) @ V0f )
= ( ap @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b ) @ V1g ) ) ) ) ) ) ).
thf(conj_thm_2Efinite__map_2ESUBMAP__FUPDATE__EQN,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ! [V2y: $i] :
( ( mem @ V2y @ A_27b )
=> ( ( p @ ( ap @ ( ap @ ( c_2Efinite__map_2ESUBMAP @ A_27a @ A_27b ) @ V0f ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EFUPDATE @ A_27a @ A_27b ) @ V0f ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27b ) @ V1x ) @ V2y ) ) ) )
<=> ( ~ ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V1x ) @ ( ap @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b ) @ V0f ) ) )
| ( ( ( ap @ ( ap @ ( c_2Efinite__map_2EFAPPLY @ A_27a @ A_27b ) @ V0f ) @ V1x )
= V2y )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V1x ) @ ( ap @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b ) @ V0f ) ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Efinite__map_2EDRESTRICT__DEF,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ! [V1r: $i] :
( ( mem @ V1r @ ( arr @ A_27a @ bool ) )
=> ( ( ( ap @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EDRESTRICT @ A_27a @ A_27b ) @ V0f ) @ V1r ) )
= ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ A_27a ) @ ( ap @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b ) @ V0f ) ) @ V1r ) )
& ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ( ( ap @ ( ap @ ( c_2Efinite__map_2EFAPPLY @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EDRESTRICT @ A_27a @ A_27b ) @ V0f ) @ V1r ) ) @ V2x )
= ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ A_27b ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ A_27a ) @ ( ap @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b ) @ V0f ) ) @ V1r ) ) ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EFAPPLY @ A_27a @ A_27b ) @ V0f ) @ V2x ) ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EFAPPLY @ A_27a @ A_27b ) @ ( c_2Efinite__map_2EFEMPTY @ A_27a @ A_27b ) ) @ V2x ) ) ) ) ) ) ) ).
thf(conj_thm_2Efinite__map_2EDRESTRICT__FEMPTY,axiom,
! [A_27a: del,A_27b: del,V0r: $i] :
( ( mem @ V0r @ ( arr @ A_27a @ bool ) )
=> ( ( ap @ ( ap @ ( 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(conj_thm_2Efinite__map_2EDRESTRICT__FUPDATE,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ! [V1r: $i] :
( ( mem @ V1r @ ( arr @ A_27a @ bool ) )
=> ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ! [V3y: $i] :
( ( mem @ V3y @ A_27b )
=> ( ( ap @ ( ap @ ( c_2Efinite__map_2EDRESTRICT @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EFUPDATE @ A_27a @ A_27b ) @ V0f ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27b ) @ V2x ) @ V3y ) ) ) @ V1r )
= ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x ) @ V1r ) ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EFUPDATE @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EDRESTRICT @ A_27a @ A_27b ) @ V0f ) @ V1r ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27b ) @ V2x ) @ V3y ) ) ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EDRESTRICT @ A_27a @ A_27b ) @ V0f ) @ V1r ) ) ) ) ) ) ) ).
thf(conj_thm_2Efinite__map_2ESTRONG__DRESTRICT__FUPDATE,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ! [V1r: $i] :
( ( mem @ V1r @ ( arr @ A_27a @ bool ) )
=> ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ! [V3y: $i] :
( ( mem @ V3y @ A_27b )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x ) @ V1r ) )
=> ( ( ap @ ( ap @ ( c_2Efinite__map_2EDRESTRICT @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EFUPDATE @ A_27a @ A_27b ) @ V0f ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27b ) @ V2x ) @ V3y ) ) ) @ V1r )
= ( ap @ ( ap @ ( c_2Efinite__map_2EFUPDATE @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EDRESTRICT @ A_27a @ A_27b ) @ V0f ) @ ( ap @ ( ap @ ( c_2Epred__set_2EDELETE @ A_27a ) @ V1r ) @ V2x ) ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27b ) @ V2x ) @ V3y ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Efinite__map_2EFDOM__DRESTRICT,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0f: $i] :
( ( mem @ V0f @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ! [V1r: $i] :
( ( mem @ V1r @ ( arr @ A_27a @ bool ) )
=> ! [V2x: $i] :
( ( mem @ V2x @ A_27c )
=> ( ( ap @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EDRESTRICT @ A_27a @ A_27b ) @ V0f ) @ V1r ) )
= ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ A_27a ) @ ( ap @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b ) @ V0f ) ) @ V1r ) ) ) ) ) ).
thf(conj_thm_2Efinite__map_2ENOT__FDOM__DRESTRICT,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ( ~ ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V1x ) @ ( ap @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b ) @ V0f ) ) )
=> ( ( ap @ ( ap @ ( c_2Efinite__map_2EDRESTRICT @ A_27a @ A_27b ) @ V0f ) @ ( ap @ ( c_2Epred__set_2ECOMPL @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ A_27a ) @ V1x ) @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) ) )
= V0f ) ) ) ) ).
thf(conj_thm_2Efinite__map_2EDRESTRICT__SUBMAP,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ! [V1r: $i] :
( ( mem @ V1r @ ( arr @ A_27a @ bool ) )
=> ( p @ ( ap @ ( ap @ ( c_2Efinite__map_2ESUBMAP @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EDRESTRICT @ A_27a @ A_27b ) @ V0f ) @ V1r ) ) @ V0f ) ) ) ) ).
thf(conj_thm_2Efinite__map_2EDRESTRICT__DRESTRICT,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ! [V1P: $i] :
( ( mem @ V1P @ ( arr @ A_27a @ bool ) )
=> ! [V2Q: $i] :
( ( mem @ V2Q @ ( arr @ A_27a @ bool ) )
=> ( ( ap @ ( ap @ ( c_2Efinite__map_2EDRESTRICT @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EDRESTRICT @ A_27a @ A_27b ) @ V0f ) @ V1P ) ) @ V2Q )
= ( ap @ ( ap @ ( c_2Efinite__map_2EDRESTRICT @ A_27a @ A_27b ) @ V0f ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ A_27a ) @ V1P ) @ V2Q ) ) ) ) ) ) ).
thf(conj_thm_2Efinite__map_2EDRESTRICT__IS__FEMPTY,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ( ( ap @ ( ap @ ( c_2Efinite__map_2EDRESTRICT @ A_27a @ A_27b ) @ V0f ) @ ( c_2Epred__set_2EEMPTY @ A_27a ) )
= ( c_2Efinite__map_2EFEMPTY @ A_27a @ A_27b ) ) ) ).
thf(conj_thm_2Efinite__map_2EFUPDATE__DRESTRICT,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ! [V2y: $i] :
( ( mem @ V2y @ A_27b )
=> ( ( ap @ ( ap @ ( c_2Efinite__map_2EFUPDATE @ A_27a @ A_27b ) @ V0f ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27b ) @ V1x ) @ V2y ) )
= ( ap @ ( ap @ ( c_2Efinite__map_2EFUPDATE @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EDRESTRICT @ A_27a @ A_27b ) @ V0f ) @ ( ap @ ( c_2Epred__set_2ECOMPL @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ A_27a ) @ V1x ) @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) ) ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27b ) @ V1x ) @ V2y ) ) ) ) ) ) ).
thf(conj_thm_2Efinite__map_2ESTRONG__DRESTRICT__FUPDATE__THM,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ! [V1r: $i] :
( ( mem @ V1r @ ( arr @ A_27a @ bool ) )
=> ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ! [V3y: $i] :
( ( mem @ V3y @ A_27b )
=> ( ( ap @ ( ap @ ( c_2Efinite__map_2EDRESTRICT @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EFUPDATE @ A_27a @ A_27b ) @ V0f ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27b ) @ V2x ) @ V3y ) ) ) @ V1r )
= ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x ) @ V1r ) ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EFUPDATE @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EDRESTRICT @ A_27a @ A_27b ) @ V0f ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ A_27a ) @ ( ap @ ( c_2Epred__set_2ECOMPL @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ A_27a ) @ V2x ) @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) ) ) @ V1r ) ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27b ) @ V2x ) @ V3y ) ) ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EDRESTRICT @ A_27a @ A_27b ) @ V0f ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ A_27a ) @ ( ap @ ( c_2Epred__set_2ECOMPL @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ A_27a ) @ V2x ) @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) ) ) @ V1r ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Efinite__map_2EDRESTRICT__UNIV,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ( ( ap @ ( ap @ ( c_2Efinite__map_2EDRESTRICT @ A_27a @ A_27b ) @ V0f ) @ ( c_2Epred__set_2EUNIV @ A_27a ) )
= V0f ) ) ).
thf(conj_thm_2Efinite__map_2ESUBMAP__DRESTRICT,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ! [V1P: $i] :
( ( mem @ V1P @ ( arr @ A_27a @ bool ) )
=> ( p @ ( ap @ ( ap @ ( c_2Efinite__map_2ESUBMAP @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EDRESTRICT @ A_27a @ A_27b ) @ V0f ) @ V1P ) ) @ V0f ) ) ) ) ).
thf(conj_thm_2Efinite__map_2EDRESTRICT__EQ__DRESTRICT,axiom,
! [A_27a: del,A_27b: del,V0f1: $i] :
( ( mem @ V0f1 @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ! [V1f2: $i] :
( ( mem @ V1f2 @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ! [V2s1: $i] :
( ( mem @ V2s1 @ ( arr @ A_27a @ bool ) )
=> ! [V3s2: $i] :
( ( mem @ V3s2 @ ( arr @ A_27a @ bool ) )
=> ( ( ( ap @ ( ap @ ( c_2Efinite__map_2EDRESTRICT @ A_27a @ A_27b ) @ V0f1 ) @ V2s1 )
= ( ap @ ( ap @ ( c_2Efinite__map_2EDRESTRICT @ A_27a @ A_27b ) @ V1f2 ) @ V3s2 ) )
<=> ( ( p @ ( ap @ ( ap @ ( c_2Efinite__map_2ESUBMAP @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EDRESTRICT @ A_27a @ A_27b ) @ V0f1 ) @ V2s1 ) ) @ V1f2 ) )
& ( p @ ( ap @ ( ap @ ( c_2Efinite__map_2ESUBMAP @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EDRESTRICT @ A_27a @ A_27b ) @ V1f2 ) @ V3s2 ) ) @ V0f1 ) )
& ( ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ A_27a ) @ V2s1 ) @ ( ap @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b ) @ V0f1 ) )
= ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ A_27a ) @ V3s2 ) @ ( ap @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b ) @ V1f2 ) ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Efinite__map_2EFUNION__DEF,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ( ( ( ap @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EFUNION @ A_27a @ A_27b ) @ V0f ) @ V1g ) )
= ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ A_27a ) @ ( ap @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b ) @ V0f ) ) @ ( ap @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b ) @ V1g ) ) )
& ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ( ( ap @ ( ap @ ( c_2Efinite__map_2EFAPPLY @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EFUNION @ A_27a @ A_27b ) @ V0f ) @ V1g ) ) @ V2x )
= ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ A_27b ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x ) @ ( ap @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b ) @ V0f ) ) ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EFAPPLY @ A_27a @ A_27b ) @ V0f ) @ V2x ) ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EFAPPLY @ A_27a @ A_27b ) @ V1g ) @ V2x ) ) ) ) ) ) ) ).
thf(conj_thm_2Efinite__map_2EFUNION__FEMPTY__1,axiom,
! [A_27a: del,A_27b: del,V0g: $i] :
( ( mem @ V0g @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ( ( ap @ ( ap @ ( c_2Efinite__map_2EFUNION @ A_27a @ A_27b ) @ ( c_2Efinite__map_2EFEMPTY @ A_27a @ A_27b ) ) @ V0g )
= V0g ) ) ).
thf(conj_thm_2Efinite__map_2EFUNION__FEMPTY__2,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ( ( ap @ ( ap @ ( c_2Efinite__map_2EFUNION @ A_27a @ A_27b ) @ V0f ) @ ( c_2Efinite__map_2EFEMPTY @ A_27a @ A_27b ) )
= V0f ) ) ).
thf(conj_thm_2Efinite__map_2EFUNION__FUPDATE__1,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ! [V3y: $i] :
( ( mem @ V3y @ A_27b )
=> ( ( ap @ ( ap @ ( c_2Efinite__map_2EFUNION @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EFUPDATE @ A_27a @ A_27b ) @ V0f ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27b ) @ V2x ) @ V3y ) ) ) @ V1g )
= ( ap @ ( ap @ ( c_2Efinite__map_2EFUPDATE @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EFUNION @ A_27a @ A_27b ) @ V0f ) @ V1g ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27b ) @ V2x ) @ V3y ) ) ) ) ) ) ) ).
thf(conj_thm_2Efinite__map_2EFUNION__FUPDATE__2,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ! [V3y: $i] :
( ( mem @ V3y @ A_27b )
=> ( ( ap @ ( ap @ ( c_2Efinite__map_2EFUNION @ A_27a @ A_27b ) @ V0f ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EFUPDATE @ A_27a @ A_27b ) @ V1g ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27b ) @ V2x ) @ V3y ) ) )
= ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x ) @ ( ap @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b ) @ V0f ) ) ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EFUNION @ A_27a @ A_27b ) @ V0f ) @ V1g ) ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EFUPDATE @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EFUNION @ A_27a @ A_27b ) @ V0f ) @ V1g ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27b ) @ V2x ) @ V3y ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Efinite__map_2EFDOM__FUNION,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0f: $i] :
( ( mem @ V0f @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ! [V2x: $i] :
( ( mem @ V2x @ A_27c )
=> ( ( ap @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EFUNION @ A_27a @ A_27b ) @ V0f ) @ V1g ) )
= ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ A_27a ) @ ( ap @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b ) @ V0f ) ) @ ( ap @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b ) @ V1g ) ) ) ) ) ) ).
thf(conj_thm_2Efinite__map_2EFUNION__IDEMPOT,axiom,
! [A_27a: del,A_27b: del,V0fm: $i] :
( ( mem @ V0fm @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ( ( ap @ ( ap @ ( c_2Efinite__map_2EFUNION @ A_27a @ A_27b ) @ V0fm ) @ V0fm )
= V0fm ) ) ).
thf(ax_thm_2Efinite__map_2EFMERGE__DEF,axiom,
! [A_27a: del,A_27b: del,V0m: $i] :
( ( mem @ V0m @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( ty_2Efinite__map_2Efmap @ A_27b @ A_27a ) )
=> ! [V2g: $i] :
( ( mem @ V2g @ ( ty_2Efinite__map_2Efmap @ A_27b @ A_27a ) )
=> ( ( ( ap @ ( c_2Efinite__map_2EFDOM @ A_27b @ A_27a ) @ ( ap @ ( ap @ ( ap @ ( c_2Efinite__map_2EFMERGE @ A_27a @ A_27b ) @ V0m ) @ V1f ) @ V2g ) )
= ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ A_27b ) @ ( ap @ ( c_2Efinite__map_2EFDOM @ A_27b @ A_27a ) @ V1f ) ) @ ( ap @ ( c_2Efinite__map_2EFDOM @ A_27b @ A_27a ) @ V2g ) ) )
& ! [V3x: $i] :
( ( mem @ V3x @ A_27b )
=> ( ( ap @ ( ap @ ( c_2Efinite__map_2EFAPPLY @ A_27b @ A_27a ) @ ( ap @ ( ap @ ( ap @ ( c_2Efinite__map_2EFMERGE @ A_27a @ A_27b ) @ V0m ) @ V1f ) @ V2g ) ) @ V3x )
= ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ A_27a ) @ ( ap @ c_2Ebool_2E_7E @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27b ) @ V3x ) @ ( ap @ ( c_2Efinite__map_2EFDOM @ A_27b @ A_27a ) @ V1f ) ) ) ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EFAPPLY @ A_27b @ A_27a ) @ V2g ) @ V3x ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ A_27a ) @ ( ap @ c_2Ebool_2E_7E @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27b ) @ V3x ) @ ( ap @ ( c_2Efinite__map_2EFDOM @ A_27b @ A_27a ) @ V2g ) ) ) ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EFAPPLY @ A_27b @ A_27a ) @ V1f ) @ V3x ) ) @ ( ap @ ( ap @ V0m @ ( ap @ ( ap @ ( c_2Efinite__map_2EFAPPLY @ A_27b @ A_27a ) @ V1f ) @ V3x ) ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EFAPPLY @ A_27b @ A_27a ) @ V2g ) @ V3x ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Efinite__map_2EFMERGE__FEMPTY,axiom,
! [A_27a: del,A_27b: del,V0m: $i] :
( ( mem @ V0m @ ( arr @ A_27b @ ( arr @ A_27b @ A_27b ) ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ( ( ( ap @ ( ap @ ( ap @ ( c_2Efinite__map_2EFMERGE @ A_27b @ A_27a ) @ V0m ) @ V1f ) @ ( c_2Efinite__map_2EFEMPTY @ A_27a @ A_27b ) )
= V1f )
& ( ( ap @ ( ap @ ( ap @ ( c_2Efinite__map_2EFMERGE @ A_27b @ A_27a ) @ V0m ) @ ( c_2Efinite__map_2EFEMPTY @ A_27a @ A_27b ) ) @ V1f )
= V1f ) ) ) ) ).
thf(conj_thm_2Efinite__map_2EFDOM__FMERGE,axiom,
! [A_27a: del,A_27b: del,V0m: $i] :
( ( mem @ V0m @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( ty_2Efinite__map_2Efmap @ A_27b @ A_27a ) )
=> ! [V2g: $i] :
( ( mem @ V2g @ ( ty_2Efinite__map_2Efmap @ A_27b @ A_27a ) )
=> ( ( ap @ ( c_2Efinite__map_2EFDOM @ A_27b @ A_27a ) @ ( ap @ ( ap @ ( ap @ ( c_2Efinite__map_2EFMERGE @ A_27a @ A_27b ) @ V0m ) @ V1f ) @ V2g ) )
= ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ A_27b ) @ ( ap @ ( c_2Efinite__map_2EFDOM @ A_27b @ A_27a ) @ V1f ) ) @ ( ap @ ( c_2Efinite__map_2EFDOM @ A_27b @ A_27a ) @ V2g ) ) ) ) ) ) ).
thf(conj_thm_2Efinite__map_2EFMERGE__FUNION,axiom,
! [A_27a: del,A_27b: del] :
( ( c_2Efinite__map_2EFUNION @ A_27a @ A_27b )
= ( ap @ ( c_2Efinite__map_2EFMERGE @ A_27b @ A_27a )
@ ( lam @ A_27b
@ ^ [V0x: $i] :
( lam @ A_27b
@ ^ [V1y: $i] : V0x ) ) ) ) ).
thf(conj_thm_2Efinite__map_2EFUNION__FMERGE,axiom,
! [A_27a: del,A_27b: del,V0f1: $i] :
( ( mem @ V0f1 @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ! [V1f2: $i] :
( ( mem @ V1f2 @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ! [V2m: $i] :
( ( mem @ V2m @ ( arr @ A_27b @ ( arr @ A_27b @ A_27b ) ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2EDISJOINT @ A_27a ) @ ( ap @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b ) @ V0f1 ) ) @ ( ap @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b ) @ V1f2 ) ) )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Efinite__map_2EFMERGE @ A_27b @ A_27a ) @ V2m ) @ V0f1 ) @ V1f2 )
= ( ap @ ( ap @ ( c_2Efinite__map_2EFUNION @ A_27a @ A_27b ) @ V0f1 ) @ V1f2 ) ) ) ) ) ) ).
thf(conj_thm_2Efinite__map_2EFMERGE__NO__CHANGE,axiom,
! [A_27a: del,A_27b: del,V0m: $i] :
( ( mem @ V0m @ ( arr @ A_27b @ ( arr @ A_27b @ A_27b ) ) )
=> ! [V1f1: $i] :
( ( mem @ V1f1 @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ! [V2f2: $i] :
( ( mem @ V2f2 @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ( ( ( ( ap @ ( ap @ ( ap @ ( c_2Efinite__map_2EFMERGE @ A_27b @ A_27a ) @ V0m ) @ V1f1 ) @ V2f2 )
= V1f1 )
<=> ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3x ) @ ( ap @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b ) @ V2f2 ) ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3x ) @ ( ap @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b ) @ V1f1 ) ) )
& ( ( ap @ ( ap @ V0m @ ( ap @ ( ap @ ( c_2Efinite__map_2EFAPPLY @ A_27a @ A_27b ) @ V1f1 ) @ V3x ) ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EFAPPLY @ A_27a @ A_27b ) @ V2f2 ) @ V3x ) )
= ( ap @ ( ap @ ( c_2Efinite__map_2EFAPPLY @ A_27a @ A_27b ) @ V1f1 ) @ V3x ) ) ) ) ) )
& ( ( ( ap @ ( ap @ ( ap @ ( c_2Efinite__map_2EFMERGE @ A_27b @ A_27a ) @ V0m ) @ V1f1 ) @ V2f2 )
= V2f2 )
<=> ! [V4x: $i] :
( ( mem @ V4x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V4x ) @ ( ap @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b ) @ V1f1 ) ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V4x ) @ ( ap @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b ) @ V2f2 ) ) )
& ( ( ap @ ( ap @ V0m @ ( ap @ ( ap @ ( c_2Efinite__map_2EFAPPLY @ A_27a @ A_27b ) @ V1f1 ) @ V4x ) ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EFAPPLY @ A_27a @ A_27b ) @ V2f2 ) @ V4x ) )
= ( ap @ ( ap @ ( c_2Efinite__map_2EFAPPLY @ A_27a @ A_27b ) @ V2f2 ) @ V4x ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Efinite__map_2EFMERGE__COMM,axiom,
! [A_27a: del,A_27b: del,V0m: $i] :
( ( mem @ V0m @ ( arr @ A_27b @ ( arr @ A_27b @ A_27b ) ) )
=> ( ( p @ ( ap @ ( c_2Ecombin_2ECOMM @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) ) @ ( ap @ ( c_2Efinite__map_2EFMERGE @ A_27b @ A_27a ) @ V0m ) ) )
<=> ( p @ ( ap @ ( c_2Ecombin_2ECOMM @ A_27b @ A_27b ) @ V0m ) ) ) ) ).
thf(conj_thm_2Efinite__map_2EFMERGE__ASSOC,axiom,
! [A_27a: del,A_27b: del,V0m: $i] :
( ( mem @ V0m @ ( arr @ A_27b @ ( arr @ A_27b @ A_27b ) ) )
=> ( ( p @ ( ap @ ( c_2Ecombin_2EASSOC @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) ) @ ( ap @ ( c_2Efinite__map_2EFMERGE @ A_27b @ A_27a ) @ V0m ) ) )
<=> ( p @ ( ap @ ( c_2Ecombin_2EASSOC @ A_27b ) @ V0m ) ) ) ) ).
thf(conj_thm_2Efinite__map_2EFMERGE__DRESTRICT,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27b @ ( arr @ A_27b @ A_27b ) ) )
=> ! [V1st1: $i] :
( ( mem @ V1st1 @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ! [V2st2: $i] :
( ( mem @ V2st2 @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ! [V3vs: $i] :
( ( mem @ V3vs @ ( arr @ A_27a @ bool ) )
=> ( ( ap @ ( ap @ ( c_2Efinite__map_2EDRESTRICT @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( ap @ ( c_2Efinite__map_2EFMERGE @ A_27b @ A_27a ) @ V0f ) @ V1st1 ) @ V2st2 ) ) @ V3vs )
= ( ap @ ( ap @ ( ap @ ( c_2Efinite__map_2EFMERGE @ A_27b @ A_27a ) @ V0f ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EDRESTRICT @ A_27a @ A_27b ) @ V1st1 ) @ V3vs ) ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EDRESTRICT @ A_27a @ A_27b ) @ V2st2 ) @ V3vs ) ) ) ) ) ) ) ).
thf(conj_thm_2Efinite__map_2EFMERGE__EQ__FEMPTY,axiom,
! [A_27a: del,A_27b: del,V0m: $i] :
( ( mem @ V0m @ ( arr @ A_27b @ ( arr @ A_27b @ A_27b ) ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ! [V2g: $i] :
( ( mem @ V2g @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ( ( ( ap @ ( ap @ ( ap @ ( c_2Efinite__map_2EFMERGE @ A_27b @ A_27a ) @ V0m ) @ V1f ) @ V2g )
= ( c_2Efinite__map_2EFEMPTY @ A_27a @ A_27b ) )
<=> ( ( V1f
= ( c_2Efinite__map_2EFEMPTY @ A_27a @ A_27b ) )
& ( V2g
= ( c_2Efinite__map_2EFEMPTY @ A_27a @ A_27b ) ) ) ) ) ) ) ).
thf(ax_thm_2Efinite__map_2EFLOOKUP__DEF,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ( ( ap @ ( ap @ ( c_2Efinite__map_2EFLOOKUP @ A_27a @ A_27b ) @ V0f ) @ V1x )
= ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ( ty_2Eoption_2Eoption @ A_27b ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V1x ) @ ( ap @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b ) @ V0f ) ) ) @ ( ap @ ( c_2Eoption_2ESOME @ A_27b ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EFAPPLY @ A_27a @ A_27b ) @ V0f ) @ V1x ) ) ) @ ( c_2Eoption_2ENONE @ A_27b ) ) ) ) ) ).
thf(conj_thm_2Efinite__map_2EFLOOKUP__EMPTY,axiom,
! [A_27a: del,A_27b: del,V0k: $i] :
( ( mem @ V0k @ A_27b )
=> ( ( ap @ ( ap @ ( c_2Efinite__map_2EFLOOKUP @ A_27b @ A_27a ) @ ( c_2Efinite__map_2EFEMPTY @ A_27b @ A_27a ) ) @ V0k )
= ( c_2Eoption_2ENONE @ A_27a ) ) ) ).
thf(conj_thm_2Efinite__map_2EFLOOKUP__UPDATE,axiom,
! [A_27a: del,A_27b: del,V0fm: $i] :
( ( mem @ V0fm @ ( ty_2Efinite__map_2Efmap @ A_27b @ A_27a ) )
=> ! [V1k1: $i] :
( ( mem @ V1k1 @ A_27b )
=> ! [V2v: $i] :
( ( mem @ V2v @ A_27a )
=> ! [V3k2: $i] :
( ( mem @ V3k2 @ A_27b )
=> ( ( ap @ ( ap @ ( c_2Efinite__map_2EFLOOKUP @ A_27b @ A_27a ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EFUPDATE @ A_27b @ A_27a ) @ V0fm ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27b @ A_27a ) @ V1k1 ) @ V2v ) ) ) @ V3k2 )
= ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ( ty_2Eoption_2Eoption @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ A_27b ) @ V1k1 ) @ V3k2 ) ) @ ( ap @ ( c_2Eoption_2ESOME @ A_27a ) @ V2v ) ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EFLOOKUP @ A_27b @ A_27a ) @ V0fm ) @ V3k2 ) ) ) ) ) ) ) ).
thf(conj_thm_2Efinite__map_2EFLOOKUP__SUBMAP,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ! [V2k: $i] :
( ( mem @ V2k @ A_27a )
=> ! [V3v: $i] :
( ( mem @ V3v @ A_27b )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Efinite__map_2ESUBMAP @ A_27a @ A_27b ) @ V0f ) @ V1g ) )
& ( ( ap @ ( ap @ ( c_2Efinite__map_2EFLOOKUP @ A_27a @ A_27b ) @ V0f ) @ V2k )
= ( ap @ ( c_2Eoption_2ESOME @ A_27b ) @ V3v ) ) )
=> ( ( ap @ ( ap @ ( c_2Efinite__map_2EFLOOKUP @ A_27a @ A_27b ) @ V1g ) @ V2k )
= ( ap @ ( c_2Eoption_2ESOME @ A_27b ) @ V3v ) ) ) ) ) ) ) ).
thf(conj_thm_2Efinite__map_2ESUBMAP__FUPDATE__FLOOKUP,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ! [V2y: $i] :
( ( mem @ V2y @ A_27b )
=> ( ( p @ ( ap @ ( ap @ ( c_2Efinite__map_2ESUBMAP @ A_27a @ A_27b ) @ V0f ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EFUPDATE @ A_27a @ A_27b ) @ V0f ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27b ) @ V1x ) @ V2y ) ) ) )
<=> ( ( ( ap @ ( ap @ ( c_2Efinite__map_2EFLOOKUP @ A_27a @ A_27b ) @ V0f ) @ V1x )
= ( c_2Eoption_2ENONE @ A_27b ) )
| ( ( ap @ ( ap @ ( c_2Efinite__map_2EFLOOKUP @ A_27a @ A_27b ) @ V0f ) @ V1x )
= ( ap @ ( c_2Eoption_2ESOME @ A_27b ) @ V2y ) ) ) ) ) ) ) ).
thf(conj_thm_2Efinite__map_2EFLOOKUP__FUNION,axiom,
! [A_27a: del,A_27b: del,V0f1: $i] :
( ( mem @ V0f1 @ ( ty_2Efinite__map_2Efmap @ A_27b @ A_27a ) )
=> ! [V1f2: $i] :
( ( mem @ V1f2 @ ( ty_2Efinite__map_2Efmap @ A_27b @ A_27a ) )
=> ! [V2k: $i] :
( ( mem @ V2k @ A_27b )
=> ( ( ap @ ( ap @ ( c_2Efinite__map_2EFLOOKUP @ A_27b @ A_27a ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EFUNION @ A_27b @ A_27a ) @ V0f1 ) @ V1f2 ) ) @ V2k )
= ( ap @ ( ap @ ( ap @ ( c_2Eoption_2Eoption__CASE @ A_27a @ ( ty_2Eoption_2Eoption @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EFLOOKUP @ A_27b @ A_27a ) @ V0f1 ) @ V2k ) ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EFLOOKUP @ A_27b @ A_27a ) @ V1f2 ) @ V2k ) )
@ ( lam @ A_27a
@ ^ [V3v: $i] : ( ap @ ( c_2Eoption_2ESOME @ A_27a ) @ V3v ) ) ) ) ) ) ) ).
thf(conj_thm_2Efinite__map_2EFLOOKUP__EXT,axiom,
! [A_27a: del,A_27b: del,V0f1: $i] :
( ( mem @ V0f1 @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ! [V1f2: $i] :
( ( mem @ V1f2 @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ( ( V0f1 = V1f2 )
<=> ( ( ap @ ( c_2Efinite__map_2EFLOOKUP @ A_27a @ A_27b ) @ V0f1 )
= ( ap @ ( c_2Efinite__map_2EFLOOKUP @ A_27a @ A_27b ) @ V1f2 ) ) ) ) ) ).
thf(conj_thm_2Efinite__map_2Efmap__eq__flookup,axiom,
! [A_27a: del,A_27b: del,V0f1: $i] :
( ( mem @ V0f1 @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ! [V1f2: $i] :
( ( mem @ V1f2 @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ( ( V0f1 = V1f2 )
<=> ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ( ( ap @ ( ap @ ( c_2Efinite__map_2EFLOOKUP @ A_27a @ A_27b ) @ V0f1 ) @ V2x )
= ( ap @ ( ap @ ( c_2Efinite__map_2EFLOOKUP @ A_27a @ A_27b ) @ V1f2 ) @ V2x ) ) ) ) ) ) ).
thf(conj_thm_2Efinite__map_2EFLOOKUP__DRESTRICT,axiom,
! [A_27a: del,A_27b: del,V0fm: $i] :
( ( mem @ V0fm @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ! [V2k: $i] :
( ( mem @ V2k @ A_27a )
=> ( ( ap @ ( ap @ ( c_2Efinite__map_2EFLOOKUP @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EDRESTRICT @ A_27a @ A_27b ) @ V0fm ) @ V1s ) ) @ V2k )
= ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ( ty_2Eoption_2Eoption @ A_27b ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2k ) @ V1s ) ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EFLOOKUP @ A_27a @ A_27b ) @ V0fm ) @ V2k ) ) @ ( c_2Eoption_2ENONE @ A_27b ) ) ) ) ) ) ).
thf(ax_thm_2Efinite__map_2EFEVERY__DEF,axiom,
! [A_27a: del,A_27b: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ bool ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Efinite__map_2EFEVERY @ A_27a @ A_27b ) @ V0P ) @ V1f ) )
<=> ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x ) @ ( ap @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b ) @ V1f ) ) )
=> ( p @ ( ap @ V0P @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27b ) @ V2x ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EFAPPLY @ A_27a @ A_27b ) @ V1f ) @ V2x ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Efinite__map_2EFEVERY__FEMPTY,axiom,
! [A_27a: del,A_27b: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ bool ) )
=> ( p @ ( ap @ ( ap @ ( c_2Efinite__map_2EFEVERY @ A_27a @ A_27b ) @ V0P ) @ ( c_2Efinite__map_2EFEMPTY @ A_27a @ A_27b ) ) ) ) ).
thf(conj_thm_2Efinite__map_2EFEVERY__FUPDATE,axiom,
! [A_27a: del,A_27b: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ bool ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ! [V3y: $i] :
( ( mem @ V3y @ A_27b )
=> ( ( p @ ( ap @ ( ap @ ( c_2Efinite__map_2EFEVERY @ A_27a @ A_27b ) @ V0P ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EFUPDATE @ A_27a @ A_27b ) @ V1f ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27b ) @ V2x ) @ V3y ) ) ) )
<=> ( ( p @ ( ap @ V0P @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27b ) @ V2x ) @ V3y ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Efinite__map_2EFEVERY @ A_27a @ A_27b ) @ V0P ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EDRESTRICT @ A_27a @ A_27b ) @ V1f ) @ ( ap @ ( c_2Epred__set_2ECOMPL @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ A_27a ) @ V2x ) @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Efinite__map_2EFEVERY__FLOOKUP,axiom,
! [A_27a: del,A_27b: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ bool ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ! [V2k: $i] :
( ( mem @ V2k @ A_27a )
=> ! [V3v: $i] :
( ( mem @ V3v @ A_27b )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Efinite__map_2EFEVERY @ A_27a @ A_27b ) @ V0P ) @ V1f ) )
& ( ( ap @ ( ap @ ( c_2Efinite__map_2EFLOOKUP @ A_27a @ A_27b ) @ V1f ) @ V2k )
= ( ap @ ( c_2Eoption_2ESOME @ A_27b ) @ V3v ) ) )
=> ( p @ ( ap @ V0P @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27b ) @ V2k ) @ V3v ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Efinite__map_2Ef__o__f__DEF,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0f: $i] :
( ( mem @ V0f @ ( ty_2Efinite__map_2Efmap @ A_27b @ A_27c ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ( ( ( ap @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27c ) @ ( ap @ ( ap @ ( c_2Efinite__map_2Ef__o__f @ A_27a @ A_27b @ A_27c ) @ V0f ) @ V1g ) )
= ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ A_27a ) @ ( ap @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b ) @ V1g ) )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a )
@ ( lam @ A_27a
@ ^ [V2x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ bool ) @ V2x ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27b ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EFAPPLY @ A_27a @ A_27b ) @ V1g ) @ V2x ) ) @ ( ap @ ( c_2Efinite__map_2EFDOM @ A_27b @ A_27c ) @ V0f ) ) ) ) ) ) )
& ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3x ) @ ( ap @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27c ) @ ( ap @ ( ap @ ( c_2Efinite__map_2Ef__o__f @ A_27a @ A_27b @ A_27c ) @ V0f ) @ V1g ) ) ) )
=> ( ( ap @ ( ap @ ( c_2Efinite__map_2EFAPPLY @ A_27a @ A_27c ) @ ( ap @ ( ap @ ( c_2Efinite__map_2Ef__o__f @ A_27a @ A_27b @ A_27c ) @ V0f ) @ V1g ) ) @ V3x )
= ( ap @ ( ap @ ( c_2Efinite__map_2EFAPPLY @ A_27b @ A_27c ) @ V0f ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EFAPPLY @ A_27a @ A_27b ) @ V1g ) @ V3x ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Efinite__map_2Ef__o__f__FEMPTY__1,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0f: $i] :
( ( mem @ V0f @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ( ( ap @ ( ap @ ( 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(conj_thm_2Efinite__map_2Ef__o__f__FEMPTY__2,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0f: $i] :
( ( mem @ V0f @ ( ty_2Efinite__map_2Efmap @ A_27b @ A_27c ) )
=> ( ( ap @ ( ap @ ( 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(ax_thm_2Efinite__map_2Eo__f__DEF,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27b @ A_27c ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ( ( ( ap @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27c ) @ ( ap @ ( ap @ ( c_2Efinite__map_2Eo__f @ A_27a @ A_27b @ A_27c ) @ V0f ) @ V1g ) )
= ( ap @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b ) @ V1g ) )
& ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x ) @ ( ap @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27c ) @ ( ap @ ( ap @ ( c_2Efinite__map_2Eo__f @ A_27a @ A_27b @ A_27c ) @ V0f ) @ V1g ) ) ) )
=> ( ( ap @ ( ap @ ( c_2Efinite__map_2EFAPPLY @ A_27a @ A_27c ) @ ( ap @ ( ap @ ( c_2Efinite__map_2Eo__f @ A_27a @ A_27b @ A_27c ) @ V0f ) @ V1g ) ) @ V2x )
= ( ap @ V0f @ ( ap @ ( ap @ ( c_2Efinite__map_2EFAPPLY @ A_27a @ A_27b ) @ V1g ) @ V2x ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Efinite__map_2Eo__f__FDOM,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27b @ A_27c ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ( ( ap @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b ) @ V1g )
= ( ap @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27c ) @ ( ap @ ( ap @ ( c_2Efinite__map_2Eo__f @ A_27a @ A_27b @ A_27c ) @ V0f ) @ V1g ) ) ) ) ) ).
thf(conj_thm_2Efinite__map_2EFDOM__o__f,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27b @ A_27c ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ( ( ap @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27c ) @ ( ap @ ( ap @ ( c_2Efinite__map_2Eo__f @ A_27a @ A_27b @ A_27c ) @ V0f ) @ V1g ) )
= ( ap @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b ) @ V1g ) ) ) ) ).
thf(conj_thm_2Efinite__map_2Eo__f__FAPPLY,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27b @ A_27c ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x ) @ ( ap @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b ) @ V1g ) ) )
=> ( ( ap @ ( ap @ ( c_2Efinite__map_2EFAPPLY @ A_27a @ A_27c ) @ ( ap @ ( ap @ ( c_2Efinite__map_2Eo__f @ A_27a @ A_27b @ A_27c ) @ V0f ) @ V1g ) ) @ V2x )
= ( ap @ V0f @ ( ap @ ( ap @ ( c_2Efinite__map_2EFAPPLY @ A_27a @ A_27b ) @ V1g ) @ V2x ) ) ) ) ) ) ) ).
thf(conj_thm_2Efinite__map_2Eo__f__FEMPTY,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27c @ A_27b ) )
=> ( ( ap @ ( ap @ ( 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(conj_thm_2Efinite__map_2EFEVERY__o__f,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0m: $i] :
( ( mem @ V0m @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ! [V1P: $i] :
( ( mem @ V1P @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27c ) @ bool ) )
=> ! [V2f: $i] :
( ( mem @ V2f @ ( arr @ A_27b @ A_27c ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Efinite__map_2EFEVERY @ A_27a @ A_27c ) @ V1P ) @ ( ap @ ( ap @ ( c_2Efinite__map_2Eo__f @ A_27a @ A_27b @ A_27c ) @ V2f ) @ V0m ) ) )
<=> ( p
@ ( ap
@ ( ap @ ( c_2Efinite__map_2EFEVERY @ A_27a @ A_27b )
@ ( lam @ ( ty_2Epair_2Eprod @ A_27a @ A_27b )
@ ^ [V3x: $i] : ( ap @ V1P @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27c ) @ ( ap @ ( c_2Epair_2EFST @ A_27a @ A_27b ) @ V3x ) ) @ ( ap @ V2f @ ( ap @ ( c_2Epair_2ESND @ A_27a @ A_27b ) @ V3x ) ) ) ) ) )
@ V0m ) ) ) ) ) ) ).
thf(conj_thm_2Efinite__map_2Eo__f__o__f,axiom,
! [A_27a: del,A_27b: del,A_27c: del,A_27d: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27c @ A_27b ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ A_27d @ A_27c ) )
=> ! [V2h: $i] :
( ( mem @ V2h @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27d ) )
=> ( ( ap @ ( ap @ ( c_2Efinite__map_2Eo__f @ A_27a @ A_27c @ A_27b ) @ V0f ) @ ( ap @ ( ap @ ( c_2Efinite__map_2Eo__f @ A_27a @ A_27d @ A_27c ) @ V1g ) @ V2h ) )
= ( ap @ ( ap @ ( c_2Efinite__map_2Eo__f @ A_27a @ A_27d @ A_27b ) @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ A_27d @ A_27b @ A_27c ) @ V0f ) @ V1g ) ) @ V2h ) ) ) ) ) ).
thf(conj_thm_2Efinite__map_2EFLOOKUP__o__f,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27c @ A_27a ) )
=> ! [V1fm: $i] :
( ( mem @ V1fm @ ( ty_2Efinite__map_2Efmap @ A_27b @ A_27c ) )
=> ! [V2k: $i] :
( ( mem @ V2k @ A_27b )
=> ( ( ap @ ( ap @ ( c_2Efinite__map_2EFLOOKUP @ A_27b @ A_27a ) @ ( ap @ ( ap @ ( c_2Efinite__map_2Eo__f @ A_27b @ A_27c @ A_27a ) @ V0f ) @ V1fm ) ) @ V2k )
= ( ap @ ( ap @ ( ap @ ( c_2Eoption_2Eoption__CASE @ A_27c @ ( ty_2Eoption_2Eoption @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EFLOOKUP @ A_27b @ A_27c ) @ V1fm ) @ V2k ) ) @ ( c_2Eoption_2ENONE @ A_27a ) )
@ ( lam @ A_27c
@ ^ [V3v: $i] : ( ap @ ( c_2Eoption_2ESOME @ A_27a ) @ ( ap @ V0f @ V3v ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Efinite__map_2EFRANGE__DEF,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ( ( ap @ ( c_2Efinite__map_2EFRANGE @ A_27a @ A_27b ) @ V0f )
= ( ap @ ( c_2Epred__set_2EGSPEC @ A_27b @ A_27b )
@ ( lam @ A_27b
@ ^ [V1y: $i] :
( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27b @ bool ) @ V1y )
@ ( ap @ ( c_2Ebool_2E_3F @ A_27a )
@ ( lam @ A_27a
@ ^ [V2x: $i] : ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x ) @ ( ap @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b ) @ V0f ) ) ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ A_27b ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EFAPPLY @ A_27a @ A_27b ) @ V0f ) @ V2x ) ) @ V1y ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Efinite__map_2EFRANGE__FEMPTY,axiom,
! [A_27a: del,A_27b: del] :
( ( ap @ ( c_2Efinite__map_2EFRANGE @ A_27b @ A_27a ) @ ( c_2Efinite__map_2EFEMPTY @ A_27b @ A_27a ) )
= ( c_2Epred__set_2EEMPTY @ A_27a ) ) ).
thf(conj_thm_2Efinite__map_2EFRANGE__FUPDATE,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ! [V2y: $i] :
( ( mem @ V2y @ A_27b )
=> ( ( ap @ ( c_2Efinite__map_2EFRANGE @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EFUPDATE @ A_27a @ A_27b ) @ V0f ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27b ) @ V1x ) @ V2y ) ) )
= ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ A_27b ) @ V2y ) @ ( ap @ ( c_2Efinite__map_2EFRANGE @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EDRESTRICT @ A_27a @ A_27b ) @ V0f ) @ ( ap @ ( c_2Epred__set_2ECOMPL @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ A_27a ) @ V1x ) @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Efinite__map_2ESUBMAP__FRANGE,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Efinite__map_2ESUBMAP @ A_27a @ A_27b ) @ V0f ) @ V1g ) )
=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27b ) @ ( ap @ ( c_2Efinite__map_2EFRANGE @ A_27a @ A_27b ) @ V0f ) ) @ ( ap @ ( c_2Efinite__map_2EFRANGE @ A_27a @ A_27b ) @ V1g ) ) ) ) ) ) ).
thf(conj_thm_2Efinite__map_2EFINITE__FRANGE,axiom,
! [A_27a: del,A_27b: del,V0fm: $i] :
( ( mem @ V0fm @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27b ) @ ( ap @ ( c_2Efinite__map_2EFRANGE @ A_27a @ A_27b ) @ V0fm ) ) ) ) ).
thf(conj_thm_2Efinite__map_2Eo__f__FRANGE,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( ty_2Efinite__map_2Efmap @ A_27b @ A_27a ) )
=> ! [V2f: $i] :
( ( mem @ V2f @ ( arr @ A_27a @ A_27c ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V0x ) @ ( ap @ ( c_2Efinite__map_2EFRANGE @ A_27b @ A_27a ) @ V1g ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27c ) @ ( ap @ V2f @ V0x ) ) @ ( ap @ ( c_2Efinite__map_2EFRANGE @ A_27b @ A_27c ) @ ( ap @ ( ap @ ( c_2Efinite__map_2Eo__f @ A_27b @ A_27a @ A_27c ) @ V2f ) @ V1g ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Efinite__map_2EFRANGE__FLOOKUP,axiom,
! [A_27a: del,A_27b: del,V0v: $i] :
( ( mem @ V0v @ A_27a )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( ty_2Efinite__map_2Efmap @ A_27b @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V0v ) @ ( ap @ ( c_2Efinite__map_2EFRANGE @ A_27b @ A_27a ) @ V1f ) ) )
<=> ? [V2k: $i] :
( ( mem @ V2k @ A_27b )
& ( ( ap @ ( ap @ ( c_2Efinite__map_2EFLOOKUP @ A_27b @ A_27a ) @ V1f ) @ V2k )
= ( ap @ ( c_2Eoption_2ESOME @ A_27a ) @ V0v ) ) ) ) ) ) ).
thf(conj_thm_2Efinite__map_2EFRANGE__FUNION,axiom,
! [A_27a: del,A_27b: del,V0fm1: $i] :
( ( mem @ V0fm1 @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ! [V1fm2: $i] :
( ( mem @ V1fm2 @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2EDISJOINT @ A_27a ) @ ( ap @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b ) @ V0fm1 ) ) @ ( ap @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b ) @ V1fm2 ) ) )
=> ( ( ap @ ( c_2Efinite__map_2EFRANGE @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EFUNION @ A_27a @ A_27b ) @ V0fm1 ) @ V1fm2 ) )
= ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ A_27b ) @ ( ap @ ( c_2Efinite__map_2EFRANGE @ A_27a @ A_27b ) @ V0fm1 ) ) @ ( ap @ ( c_2Efinite__map_2EFRANGE @ A_27a @ A_27b ) @ V1fm2 ) ) ) ) ) ) ).
thf(ax_thm_2Efinite__map_2ERRESTRICT__DEF,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ! [V1r: $i] :
( ( mem @ V1r @ ( arr @ A_27b @ bool ) )
=> ( ( ( ap @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Efinite__map_2ERRESTRICT @ A_27a @ A_27b ) @ V0f ) @ V1r ) )
= ( ap @ ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a )
@ ( lam @ A_27a
@ ^ [V2x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ bool ) @ V2x ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x ) @ ( ap @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b ) @ V0f ) ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27b ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EFAPPLY @ A_27a @ A_27b ) @ V0f ) @ V2x ) ) @ V1r ) ) ) ) ) )
& ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ( ( ap @ ( ap @ ( c_2Efinite__map_2EFAPPLY @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Efinite__map_2ERRESTRICT @ A_27a @ A_27b ) @ V0f ) @ V1r ) ) @ V3x )
= ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ A_27b ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3x ) @ ( ap @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b ) @ V0f ) ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27b ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EFAPPLY @ A_27a @ A_27b ) @ V0f ) @ V3x ) ) @ V1r ) ) ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EFAPPLY @ A_27a @ A_27b ) @ V0f ) @ V3x ) ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EFAPPLY @ A_27a @ A_27b ) @ ( c_2Efinite__map_2EFEMPTY @ A_27a @ A_27b ) ) @ V3x ) ) ) ) ) ) ) ).
thf(conj_thm_2Efinite__map_2ERRESTRICT__FEMPTY,axiom,
! [A_27a: del,A_27b: del,V0r: $i] :
( ( mem @ V0r @ ( arr @ A_27a @ bool ) )
=> ( ( ap @ ( ap @ ( 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(conj_thm_2Efinite__map_2ERRESTRICT__FUPDATE,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ! [V1r: $i] :
( ( mem @ V1r @ ( arr @ A_27b @ bool ) )
=> ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ! [V3y: $i] :
( ( mem @ V3y @ A_27b )
=> ( ( ap @ ( ap @ ( c_2Efinite__map_2ERRESTRICT @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EFUPDATE @ A_27a @ A_27b ) @ V0f ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27b ) @ V2x ) @ V3y ) ) ) @ V1r )
= ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27b ) @ V3y ) @ V1r ) ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EFUPDATE @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Efinite__map_2ERRESTRICT @ A_27a @ A_27b ) @ V0f ) @ V1r ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27b ) @ V2x ) @ V3y ) ) ) @ ( ap @ ( ap @ ( c_2Efinite__map_2ERRESTRICT @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EDRESTRICT @ A_27a @ A_27b ) @ V0f ) @ ( ap @ ( c_2Epred__set_2ECOMPL @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ A_27a ) @ V2x ) @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) ) ) ) @ V1r ) ) ) ) ) ) ) ).
thf(ax_thm_2Efinite__map_2EFUN__FMAP__DEF,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27b ) )
=> ! [V1P: $i] :
( ( mem @ V1P @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V1P ) )
=> ( ( ( ap @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EFUN__FMAP @ A_27a @ A_27b ) @ V0f ) @ V1P ) )
= V1P )
& ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x ) @ V1P ) )
=> ( ( ap @ ( ap @ ( c_2Efinite__map_2EFAPPLY @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EFUN__FMAP @ A_27a @ A_27b ) @ V0f ) @ V1P ) ) @ V2x )
= ( ap @ V0f @ V2x ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Efinite__map_2EFUN__FMAP__EMPTY,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27b ) )
=> ( ( ap @ ( ap @ ( 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(conj_thm_2Efinite__map_2EFRANGE__FMAP,axiom,
! [A_27a: del,A_27b: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ A_27b ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V0P ) )
=> ( ( ap @ ( c_2Efinite__map_2EFRANGE @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EFUN__FMAP @ A_27a @ A_27b ) @ V1f ) @ V0P ) )
= ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27b ) @ V1f ) @ V0P ) ) ) ) ) ).
thf(conj_thm_2Efinite__map_2EFDOM__FMAP,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27b ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V1s ) )
=> ( ( ap @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EFUN__FMAP @ A_27a @ A_27b ) @ V0f ) @ V1s ) )
= V1s ) ) ) ) ).
thf(conj_thm_2Efinite__map_2EFLOOKUP__FUN__FMAP,axiom,
! [A_27a: del,A_27b: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ A_27b ) )
=> ! [V2k: $i] :
( ( mem @ V2k @ A_27a )
=> ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V0P ) )
=> ( ( ap @ ( ap @ ( c_2Efinite__map_2EFLOOKUP @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EFUN__FMAP @ A_27a @ A_27b ) @ V1f ) @ V0P ) ) @ V2k )
= ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ( ty_2Eoption_2Eoption @ A_27b ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2k ) @ V0P ) ) @ ( ap @ ( c_2Eoption_2ESOME @ A_27b ) @ ( ap @ V1f @ V2k ) ) ) @ ( c_2Eoption_2ENONE @ A_27b ) ) ) ) ) ) ) ).
thf(ax_thm_2Efinite__map_2Ef__o__DEF,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0f: $i] :
( ( mem @ V0f @ ( ty_2Efinite__map_2Efmap @ A_27b @ A_27c ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ A_27a @ A_27b ) )
=> ( ( ap @ ( ap @ ( c_2Efinite__map_2Ef__o @ A_27a @ A_27b @ A_27c ) @ V0f ) @ V1g )
= ( ap @ ( ap @ ( c_2Efinite__map_2Ef__o__f @ A_27a @ A_27b @ A_27c ) @ V0f )
@ ( ap @ ( ap @ ( c_2Efinite__map_2EFUN__FMAP @ A_27a @ A_27b ) @ V1g )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a )
@ ( lam @ A_27a
@ ^ [V2x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ bool ) @ V2x ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27b ) @ ( ap @ V1g @ V2x ) ) @ ( ap @ ( c_2Efinite__map_2EFDOM @ A_27b @ A_27c ) @ V0f ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Efinite__map_2EFDOM__f__o,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0f: $i] :
( ( mem @ V0f @ ( ty_2Efinite__map_2Efmap @ A_27b @ A_27c ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ A_27a @ A_27b ) )
=> ( ( p
@ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a )
@ ( lam @ A_27a
@ ^ [V2x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ bool ) @ V2x ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27b ) @ ( ap @ V1g @ V2x ) ) @ ( ap @ ( c_2Efinite__map_2EFDOM @ A_27b @ A_27c ) @ V0f ) ) ) ) ) ) )
=> ( ( ap @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27c ) @ ( ap @ ( ap @ ( c_2Efinite__map_2Ef__o @ A_27a @ A_27b @ A_27c ) @ V0f ) @ V1g ) )
= ( ap @ ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a )
@ ( lam @ A_27a
@ ^ [V3x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ bool ) @ V3x ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27b ) @ ( ap @ V1g @ V3x ) ) @ ( ap @ ( c_2Efinite__map_2EFDOM @ A_27b @ A_27c ) @ V0f ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Efinite__map_2Ef__o__FEMPTY,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0g: $i] :
( ( mem @ V0g @ ( arr @ A_27a @ A_27b ) )
=> ( ( ap @ ( ap @ ( 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(conj_thm_2Efinite__map_2Ef__o__FUPDATE,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0fm: $i] :
( ( mem @ V0fm @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ! [V1k: $i] :
( ( mem @ V1k @ A_27a )
=> ! [V2v: $i] :
( ( mem @ V2v @ A_27b )
=> ! [V3g: $i] :
( ( mem @ V3g @ ( arr @ A_27c @ A_27a ) )
=> ( ( ( p
@ ( ap @ ( c_2Epred__set_2EFINITE @ A_27c )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ A_27c @ A_27c )
@ ( lam @ A_27c
@ ^ [V4x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27c @ bool ) @ V4x ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ ( ap @ V3g @ V4x ) ) @ ( ap @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b ) @ V0fm ) ) ) ) ) ) )
& ( p
@ ( ap @ ( c_2Epred__set_2EFINITE @ A_27c )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ A_27c @ A_27c )
@ ( lam @ A_27c
@ ^ [V5x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27c @ bool ) @ V5x ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ A_27a ) @ ( ap @ V3g @ V5x ) ) @ V1k ) ) ) ) ) ) )
=> ( ( ap @ ( ap @ ( c_2Efinite__map_2Ef__o @ A_27c @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EFUPDATE @ A_27a @ A_27b ) @ V0fm ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27b ) @ V1k ) @ V2v ) ) ) @ V3g )
= ( ap @ ( ap @ ( ap @ ( c_2Efinite__map_2EFMERGE @ A_27b @ A_27c ) @ ( ap @ ( c_2Ecombin_2EC @ A_27b @ A_27b @ A_27b ) @ ( c_2Ecombin_2EK @ A_27b @ A_27b ) ) ) @ ( ap @ ( ap @ ( c_2Efinite__map_2Ef__o @ A_27c @ A_27a @ A_27b ) @ V0fm ) @ V3g ) )
@ ( ap @ ( ap @ ( c_2Efinite__map_2EFUN__FMAP @ A_27c @ A_27b ) @ ( ap @ ( c_2Ecombin_2EK @ A_27b @ A_27c ) @ V2v ) )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ A_27c @ A_27c )
@ ( lam @ A_27c
@ ^ [V6x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27c @ bool ) @ V6x ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ A_27a ) @ ( ap @ V3g @ V6x ) ) @ V1k ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Efinite__map_2EFAPPLY__f__o,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0f: $i] :
( ( mem @ V0f @ ( ty_2Efinite__map_2Efmap @ A_27b @ A_27c ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ A_27a @ A_27b ) )
=> ( ( p
@ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a )
@ ( lam @ A_27a
@ ^ [V2x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ bool ) @ V2x ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27b ) @ ( ap @ V1g @ V2x ) ) @ ( ap @ ( c_2Efinite__map_2EFDOM @ A_27b @ A_27c ) @ V0f ) ) ) ) ) ) )
=> ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3x ) @ ( ap @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27c ) @ ( ap @ ( ap @ ( c_2Efinite__map_2Ef__o @ A_27a @ A_27b @ A_27c ) @ V0f ) @ V1g ) ) ) )
=> ( ( ap @ ( ap @ ( c_2Efinite__map_2EFAPPLY @ A_27a @ A_27c ) @ ( ap @ ( ap @ ( c_2Efinite__map_2Ef__o @ A_27a @ A_27b @ A_27c ) @ V0f ) @ V1g ) ) @ V3x )
= ( ap @ ( ap @ ( c_2Efinite__map_2EFAPPLY @ A_27b @ A_27c ) @ V0f ) @ ( ap @ V1g @ V3x ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Efinite__map_2EFINITE__PRED__11,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0g: $i] :
( ( mem @ V0g @ ( arr @ A_27a @ A_27b ) )
=> ( ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ! [V2y: $i] :
( ( mem @ V2y @ A_27a )
=> ( ( ( ap @ V0g @ V1x )
= ( ap @ V0g @ V2y ) )
<=> ( V1x = V2y ) ) ) )
=> ! [V3f: $i] :
( ( mem @ V3f @ ( ty_2Efinite__map_2Efmap @ A_27b @ A_27c ) )
=> ( p
@ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a )
@ ( lam @ A_27a
@ ^ [V4x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ bool ) @ V4x ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27b ) @ ( ap @ V0g @ V4x ) ) @ ( ap @ ( c_2Efinite__map_2EFDOM @ A_27b @ A_27c ) @ V3f ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Efinite__map_2Ef__o__ASSOC,axiom,
! [A_27a: del,A_27b: del,A_27c: del,A_27d: del,V0g: $i] :
( ( mem @ V0g @ ( arr @ A_27a @ A_27b ) )
=> ! [V1h: $i] :
( ( mem @ V1h @ ( arr @ A_27c @ A_27a ) )
=> ! [V2f: $i] :
( ( mem @ V2f @ ( ty_2Efinite__map_2Efmap @ A_27b @ A_27d ) )
=> ( ( ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ! [V4y: $i] :
( ( mem @ V4y @ A_27a )
=> ( ( ( ap @ V0g @ V3x )
= ( ap @ V0g @ V4y ) )
<=> ( V3x = V4y ) ) ) )
& ! [V5x: $i] :
( ( mem @ V5x @ A_27c )
=> ! [V6y: $i] :
( ( mem @ V6y @ A_27c )
=> ( ( ( ap @ V1h @ V5x )
= ( ap @ V1h @ V6y ) )
<=> ( V5x = V6y ) ) ) ) )
=> ( ( ap @ ( ap @ ( c_2Efinite__map_2Ef__o @ A_27c @ A_27a @ A_27d ) @ ( ap @ ( ap @ ( c_2Efinite__map_2Ef__o @ A_27a @ A_27b @ A_27d ) @ V2f ) @ V0g ) ) @ V1h )
= ( ap @ ( ap @ ( c_2Efinite__map_2Ef__o @ A_27c @ A_27b @ A_27d ) @ V2f ) @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ A_27c @ A_27b @ A_27a ) @ V0g ) @ V1h ) ) ) ) ) ) ) ).
thf(ax_thm_2Efinite__map_2Efmap__domsub,axiom,
! [A_27a: del,A_27b: del,V0fm: $i] :
( ( mem @ V0fm @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ! [V1k: $i] :
( ( mem @ V1k @ A_27a )
=> ( ( ap @ ( ap @ ( c_2Efinite__map_2Efdomsub @ A_27a @ A_27b ) @ V0fm ) @ V1k )
= ( ap @ ( ap @ ( c_2Efinite__map_2EDRESTRICT @ A_27a @ A_27b ) @ V0fm ) @ ( ap @ ( c_2Epred__set_2ECOMPL @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ A_27a ) @ V1k ) @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) ) ) ) ) ) ).
thf(conj_thm_2Efinite__map_2EDOMSUB__FEMPTY,axiom,
! [A_27a: del,A_27b: del,V0k: $i] :
( ( mem @ V0k @ A_27a )
=> ( ( ap @ ( ap @ ( 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(conj_thm_2Efinite__map_2EDOMSUB__FUPDATE,axiom,
! [A_27a: del,A_27b: del,V0fm: $i] :
( ( mem @ V0fm @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ! [V1k: $i] :
( ( mem @ V1k @ A_27a )
=> ! [V2v: $i] :
( ( mem @ V2v @ A_27b )
=> ( ( ap @ ( ap @ ( c_2Efinite__map_2Efdomsub @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EFUPDATE @ A_27a @ A_27b ) @ V0fm ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27b ) @ V1k ) @ V2v ) ) ) @ V1k )
= ( ap @ ( ap @ ( c_2Efinite__map_2Efdomsub @ A_27a @ A_27b ) @ V0fm ) @ V1k ) ) ) ) ) ).
thf(conj_thm_2Efinite__map_2EDOMSUB__FUPDATE__NEQ,axiom,
! [A_27a: del,A_27b: del,V0fm: $i] :
( ( mem @ V0fm @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ! [V1k1: $i] :
( ( mem @ V1k1 @ A_27a )
=> ! [V2k2: $i] :
( ( mem @ V2k2 @ A_27a )
=> ! [V3v: $i] :
( ( mem @ V3v @ A_27b )
=> ( ( V1k1 != V2k2 )
=> ( ( ap @ ( ap @ ( c_2Efinite__map_2Efdomsub @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EFUPDATE @ A_27a @ A_27b ) @ V0fm ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27b ) @ V1k1 ) @ V3v ) ) ) @ V2k2 )
= ( ap @ ( ap @ ( c_2Efinite__map_2EFUPDATE @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Efinite__map_2Efdomsub @ A_27a @ A_27b ) @ V0fm ) @ V2k2 ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27b ) @ V1k1 ) @ V3v ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Efinite__map_2EDOMSUB__FUPDATE__THM,axiom,
! [A_27a: del,A_27b: del,V0fm: $i] :
( ( mem @ V0fm @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ! [V1k1: $i] :
( ( mem @ V1k1 @ A_27a )
=> ! [V2k2: $i] :
( ( mem @ V2k2 @ A_27a )
=> ! [V3v: $i] :
( ( mem @ V3v @ A_27b )
=> ( ( ap @ ( ap @ ( c_2Efinite__map_2Efdomsub @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EFUPDATE @ A_27a @ A_27b ) @ V0fm ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27b ) @ V1k1 ) @ V3v ) ) ) @ V2k2 )
= ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ A_27a ) @ V1k1 ) @ V2k2 ) ) @ ( ap @ ( ap @ ( c_2Efinite__map_2Efdomsub @ A_27a @ A_27b ) @ V0fm ) @ V2k2 ) ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EFUPDATE @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Efinite__map_2Efdomsub @ A_27a @ A_27b ) @ V0fm ) @ V2k2 ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27b ) @ V1k1 ) @ V3v ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Efinite__map_2EFDOM__DOMSUB,axiom,
! [A_27a: del,A_27b: del,V0fm: $i] :
( ( mem @ V0fm @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ! [V1k: $i] :
( ( mem @ V1k @ A_27a )
=> ( ( ap @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Efinite__map_2Efdomsub @ A_27a @ A_27b ) @ V0fm ) @ V1k ) )
= ( ap @ ( ap @ ( c_2Epred__set_2EDELETE @ A_27a ) @ ( ap @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b ) @ V0fm ) ) @ V1k ) ) ) ) ).
thf(conj_thm_2Efinite__map_2EDOMSUB__FAPPLY,axiom,
! [A_27a: del,A_27b: del,V0fm: $i] :
( ( mem @ V0fm @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ! [V1k: $i] :
( ( mem @ V1k @ A_27a )
=> ( ( ap @ ( ap @ ( c_2Efinite__map_2EFAPPLY @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Efinite__map_2Efdomsub @ A_27a @ A_27b ) @ V0fm ) @ V1k ) ) @ V1k )
= ( ap @ ( ap @ ( c_2Efinite__map_2EFAPPLY @ A_27a @ A_27b ) @ ( c_2Efinite__map_2EFEMPTY @ A_27a @ A_27b ) ) @ V1k ) ) ) ) ).
thf(conj_thm_2Efinite__map_2EDOMSUB__FAPPLY__NEQ,axiom,
! [A_27a: del,A_27b: del,V0fm: $i] :
( ( mem @ V0fm @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ! [V1k1: $i] :
( ( mem @ V1k1 @ A_27a )
=> ! [V2k2: $i] :
( ( mem @ V2k2 @ A_27a )
=> ( ( V1k1 != V2k2 )
=> ( ( ap @ ( ap @ ( c_2Efinite__map_2EFAPPLY @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Efinite__map_2Efdomsub @ A_27a @ A_27b ) @ V0fm ) @ V1k1 ) ) @ V2k2 )
= ( ap @ ( ap @ ( c_2Efinite__map_2EFAPPLY @ A_27a @ A_27b ) @ V0fm ) @ V2k2 ) ) ) ) ) ) ).
thf(conj_thm_2Efinite__map_2EDOMSUB__FAPPLY__THM,axiom,
! [A_27a: del,A_27b: del,V0fm: $i] :
( ( mem @ V0fm @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ! [V1k1: $i] :
( ( mem @ V1k1 @ A_27a )
=> ! [V2k2: $i] :
( ( mem @ V2k2 @ A_27a )
=> ( ( ap @ ( ap @ ( c_2Efinite__map_2EFAPPLY @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Efinite__map_2Efdomsub @ A_27a @ A_27b ) @ V0fm ) @ V1k1 ) ) @ V2k2 )
= ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ A_27b ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ A_27a ) @ V1k1 ) @ V2k2 ) ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EFAPPLY @ A_27a @ A_27b ) @ ( c_2Efinite__map_2EFEMPTY @ A_27a @ A_27b ) ) @ V2k2 ) ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EFAPPLY @ A_27a @ A_27b ) @ V0fm ) @ V2k2 ) ) ) ) ) ) ).
thf(conj_thm_2Efinite__map_2EDOMSUB__FLOOKUP,axiom,
! [A_27a: del,A_27b: del,V0fm: $i] :
( ( mem @ V0fm @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ! [V1k: $i] :
( ( mem @ V1k @ A_27a )
=> ( ( ap @ ( ap @ ( c_2Efinite__map_2EFLOOKUP @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Efinite__map_2Efdomsub @ A_27a @ A_27b ) @ V0fm ) @ V1k ) ) @ V1k )
= ( c_2Eoption_2ENONE @ A_27b ) ) ) ) ).
thf(conj_thm_2Efinite__map_2EDOMSUB__FLOOKUP__NEQ,axiom,
! [A_27a: del,A_27b: del,V0fm: $i] :
( ( mem @ V0fm @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ! [V1k1: $i] :
( ( mem @ V1k1 @ A_27a )
=> ! [V2k2: $i] :
( ( mem @ V2k2 @ A_27a )
=> ( ( V1k1 != V2k2 )
=> ( ( ap @ ( ap @ ( c_2Efinite__map_2EFLOOKUP @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Efinite__map_2Efdomsub @ A_27a @ A_27b ) @ V0fm ) @ V1k1 ) ) @ V2k2 )
= ( ap @ ( ap @ ( c_2Efinite__map_2EFLOOKUP @ A_27a @ A_27b ) @ V0fm ) @ V2k2 ) ) ) ) ) ) ).
thf(conj_thm_2Efinite__map_2EDOMSUB__FLOOKUP__THM,axiom,
! [A_27a: del,A_27b: del,V0fm: $i] :
( ( mem @ V0fm @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ! [V1k1: $i] :
( ( mem @ V1k1 @ A_27a )
=> ! [V2k2: $i] :
( ( mem @ V2k2 @ A_27a )
=> ( ( ap @ ( ap @ ( c_2Efinite__map_2EFLOOKUP @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Efinite__map_2Efdomsub @ A_27a @ A_27b ) @ V0fm ) @ V1k1 ) ) @ V2k2 )
= ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ( ty_2Eoption_2Eoption @ A_27b ) ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ A_27a ) @ V1k1 ) @ V2k2 ) ) @ ( c_2Eoption_2ENONE @ A_27b ) ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EFLOOKUP @ A_27a @ A_27b ) @ V0fm ) @ V2k2 ) ) ) ) ) ) ).
thf(conj_thm_2Efinite__map_2EFRANGE__FUPDATE__DOMSUB,axiom,
! [A_27a: del,A_27b: del,V0fm: $i] :
( ( mem @ V0fm @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ! [V1k: $i] :
( ( mem @ V1k @ A_27a )
=> ! [V2v: $i] :
( ( mem @ V2v @ A_27b )
=> ( ( ap @ ( c_2Efinite__map_2EFRANGE @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EFUPDATE @ A_27a @ A_27b ) @ V0fm ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27b ) @ V1k ) @ V2v ) ) )
= ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ A_27b ) @ V2v ) @ ( ap @ ( c_2Efinite__map_2EFRANGE @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Efinite__map_2Efdomsub @ A_27a @ A_27b ) @ V0fm ) @ V1k ) ) ) ) ) ) ) ).
thf(conj_thm_2Efinite__map_2Eo__f__DOMSUB,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0g: $i] :
( ( mem @ V0g @ ( arr @ A_27c @ A_27b ) )
=> ! [V1fm: $i] :
( ( mem @ V1fm @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27c ) )
=> ! [V2k: $i] :
( ( mem @ V2k @ A_27a )
=> ( ( ap @ ( ap @ ( c_2Efinite__map_2Efdomsub @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Efinite__map_2Eo__f @ A_27a @ A_27c @ A_27b ) @ V0g ) @ V1fm ) ) @ V2k )
= ( ap @ ( ap @ ( c_2Efinite__map_2Eo__f @ A_27a @ A_27c @ A_27b ) @ V0g ) @ ( ap @ ( ap @ ( c_2Efinite__map_2Efdomsub @ A_27a @ A_27c ) @ V1fm ) @ V2k ) ) ) ) ) ) ).
thf(conj_thm_2Efinite__map_2EDOMSUB__IDEM,axiom,
! [A_27a: del,A_27b: del,V0fm: $i] :
( ( mem @ V0fm @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ! [V1k: $i] :
( ( mem @ V1k @ A_27a )
=> ( ( ap @ ( ap @ ( c_2Efinite__map_2Efdomsub @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Efinite__map_2Efdomsub @ A_27a @ A_27b ) @ V0fm ) @ V1k ) ) @ V1k )
= ( ap @ ( ap @ ( c_2Efinite__map_2Efdomsub @ A_27a @ A_27b ) @ V0fm ) @ V1k ) ) ) ) ).
thf(conj_thm_2Efinite__map_2EDOMSUB__COMMUTES,axiom,
! [A_27a: del,A_27b: del,V0fm: $i] :
( ( mem @ V0fm @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ! [V1k1: $i] :
( ( mem @ V1k1 @ A_27a )
=> ! [V2k2: $i] :
( ( mem @ V2k2 @ A_27a )
=> ( ( ap @ ( ap @ ( c_2Efinite__map_2Efdomsub @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Efinite__map_2Efdomsub @ A_27a @ A_27b ) @ V0fm ) @ V1k1 ) ) @ V2k2 )
= ( ap @ ( ap @ ( c_2Efinite__map_2Efdomsub @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Efinite__map_2Efdomsub @ A_27a @ A_27b ) @ V0fm ) @ V2k2 ) ) @ V1k1 ) ) ) ) ) ).
thf(conj_thm_2Efinite__map_2Eo__f__FUPDATE,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27c @ A_27b ) )
=> ! [V1fm: $i] :
( ( mem @ V1fm @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27c ) )
=> ! [V2k: $i] :
( ( mem @ V2k @ A_27a )
=> ! [V3v: $i] :
( ( mem @ V3v @ A_27c )
=> ( ( ap @ ( ap @ ( c_2Efinite__map_2Eo__f @ A_27a @ A_27c @ A_27b ) @ V0f ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EFUPDATE @ A_27a @ A_27c ) @ V1fm ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27c ) @ V2k ) @ V3v ) ) )
= ( ap @ ( ap @ ( c_2Efinite__map_2EFUPDATE @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Efinite__map_2Eo__f @ A_27a @ A_27c @ A_27b ) @ V0f ) @ V1fm ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27b ) @ V2k ) @ ( ap @ V0f @ V3v ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Efinite__map_2EDOMSUB__NOT__IN__DOM,axiom,
! [A_27a: del,A_27b: del,V0k: $i] :
( ( mem @ V0k @ A_27a )
=> ! [V1fm: $i] :
( ( mem @ V1fm @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ( ~ ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V0k ) @ ( ap @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b ) @ V1fm ) ) )
=> ( ( ap @ ( ap @ ( c_2Efinite__map_2Efdomsub @ A_27a @ A_27b ) @ V1fm ) @ V0k )
= V1fm ) ) ) ) ).
thf(conj_thm_2Efinite__map_2Efmap__CASES,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ( ( V0f
= ( c_2Efinite__map_2EFEMPTY @ A_27a @ A_27b ) )
| ? [V1g: $i] :
( ( mem @ V1g @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
& ? [V2x: $i] :
( ( mem @ V2x @ A_27a )
& ? [V3y: $i] :
( ( mem @ V3y @ A_27b )
& ( V0f
= ( ap @ ( ap @ ( c_2Efinite__map_2EFUPDATE @ A_27a @ A_27b ) @ V1g ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27b ) @ V2x ) @ V3y ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Efinite__map_2ESUBMAP__DOMSUB,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ! [V1k: $i] :
( ( mem @ V1k @ A_27a )
=> ( p @ ( ap @ ( ap @ ( c_2Efinite__map_2ESUBMAP @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Efinite__map_2Efdomsub @ A_27a @ A_27b ) @ V0f ) @ V1k ) ) @ V0f ) ) ) ) ).
thf(conj_thm_2Efinite__map_2EFMERGE__DOMSUB,axiom,
! [A_27a: del,A_27b: del,V0m: $i] :
( ( mem @ V0m @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) )
=> ! [V1m1: $i] :
( ( mem @ V1m1 @ ( ty_2Efinite__map_2Efmap @ A_27b @ A_27a ) )
=> ! [V2m2: $i] :
( ( mem @ V2m2 @ ( ty_2Efinite__map_2Efmap @ A_27b @ A_27a ) )
=> ! [V3k: $i] :
( ( mem @ V3k @ A_27b )
=> ( ( ap @ ( ap @ ( c_2Efinite__map_2Efdomsub @ A_27b @ A_27a ) @ ( ap @ ( ap @ ( ap @ ( c_2Efinite__map_2EFMERGE @ A_27a @ A_27b ) @ V0m ) @ V1m1 ) @ V2m2 ) ) @ V3k )
= ( ap @ ( ap @ ( ap @ ( c_2Efinite__map_2EFMERGE @ A_27a @ A_27b ) @ V0m ) @ ( ap @ ( ap @ ( c_2Efinite__map_2Efdomsub @ A_27b @ A_27a ) @ V1m1 ) @ V3k ) ) @ ( ap @ ( ap @ ( c_2Efinite__map_2Efdomsub @ A_27b @ A_27a ) @ V2m2 ) @ V3k ) ) ) ) ) ) ) ).
thf(conj_thm_2Efinite__map_2ESUBMAP__FUPDATE,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ! [V3y: $i] :
( ( mem @ V3y @ A_27b )
=> ( ( p @ ( ap @ ( ap @ ( c_2Efinite__map_2ESUBMAP @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EFUPDATE @ A_27a @ A_27b ) @ V0f ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27b ) @ V2x ) @ V3y ) ) ) @ V1g ) )
<=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x ) @ ( ap @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b ) @ V1g ) ) )
& ( ( ap @ ( ap @ ( c_2Efinite__map_2EFAPPLY @ A_27a @ A_27b ) @ V1g ) @ V2x )
= V3y )
& ( p @ ( ap @ ( ap @ ( c_2Efinite__map_2ESUBMAP @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Efinite__map_2Efdomsub @ A_27a @ A_27b ) @ V0f ) @ V2x ) ) @ ( ap @ ( ap @ ( c_2Efinite__map_2Efdomsub @ A_27a @ A_27b ) @ V1g ) @ V2x ) ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Efinite__map_2EFUPDATE__LIST,axiom,
! [A_27a: del,A_27b: del] :
( ( c_2Efinite__map_2EFUPDATE__LIST @ A_27a @ A_27b )
= ( ap @ ( c_2Elist_2EFOLDL @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) ) @ ( c_2Efinite__map_2EFUPDATE @ A_27a @ A_27b ) ) ) ).
thf(conj_thm_2Efinite__map_2EFUPDATE__LIST__THM,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ( ( ( ap @ ( ap @ ( c_2Efinite__map_2EFUPDATE__LIST @ A_27a @ A_27b ) @ V0f ) @ ( c_2Elist_2ENIL @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) ) )
= V0f )
& ! [V1h: $i] :
( ( mem @ V1h @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( ty_2Elist_2Elist @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) ) )
=> ( ( ap @ ( ap @ ( c_2Efinite__map_2EFUPDATE__LIST @ A_27a @ A_27b ) @ V0f ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) ) @ V1h ) @ V2t ) )
= ( ap @ ( ap @ ( c_2Efinite__map_2EFUPDATE__LIST @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EFUPDATE @ A_27a @ A_27b ) @ V0f ) @ V1h ) ) @ V2t ) ) ) ) ) ) ).
thf(conj_thm_2Efinite__map_2EFUPDATE__LIST__APPLY__NOT__MEM,axiom,
! [A_27a: del,A_27b: del,V0kvl: $i] :
( ( mem @ V0kvl @ ( ty_2Elist_2Elist @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ! [V2k: $i] :
( ( mem @ V2k @ A_27a )
=> ( ~ ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2k ) @ ( ap @ ( c_2Elist_2ELIST__TO__SET @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2EMAP @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ A_27a ) @ ( c_2Epair_2EFST @ A_27a @ A_27b ) ) @ V0kvl ) ) ) )
=> ( ( ap @ ( ap @ ( c_2Efinite__map_2EFAPPLY @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EFUPDATE__LIST @ A_27a @ A_27b ) @ V1f ) @ V0kvl ) ) @ V2k )
= ( ap @ ( ap @ ( c_2Efinite__map_2EFAPPLY @ A_27a @ A_27b ) @ V1f ) @ V2k ) ) ) ) ) ) ).
thf(conj_thm_2Efinite__map_2EFUPDATE__LIST__APPEND,axiom,
! [A_27a: del,A_27b: del,V0fm: $i] :
( ( mem @ V0fm @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ! [V1kvl1: $i] :
( ( mem @ V1kvl1 @ ( ty_2Elist_2Elist @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) ) )
=> ! [V2kvl2: $i] :
( ( mem @ V2kvl2 @ ( ty_2Elist_2Elist @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) ) )
=> ( ( ap @ ( ap @ ( c_2Efinite__map_2EFUPDATE__LIST @ A_27a @ A_27b ) @ V0fm ) @ ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) ) @ V1kvl1 ) @ V2kvl2 ) )
= ( ap @ ( ap @ ( c_2Efinite__map_2EFUPDATE__LIST @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EFUPDATE__LIST @ A_27a @ A_27b ) @ V0fm ) @ V1kvl1 ) ) @ V2kvl2 ) ) ) ) ) ).
thf(conj_thm_2Efinite__map_2EFUPDATE__FUPDATE__LIST__COMMUTES,axiom,
! [A_27a: del,A_27b: del,V0k: $i] :
( ( mem @ V0k @ A_27a )
=> ! [V1kvl: $i] :
( ( mem @ V1kvl @ ( ty_2Elist_2Elist @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) ) )
=> ! [V2fm: $i] :
( ( mem @ V2fm @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ! [V3v: $i] :
( ( mem @ V3v @ A_27b )
=> ( ~ ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V0k ) @ ( ap @ ( c_2Elist_2ELIST__TO__SET @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2EMAP @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ A_27a ) @ ( c_2Epair_2EFST @ A_27a @ A_27b ) ) @ V1kvl ) ) ) )
=> ( ( ap @ ( ap @ ( c_2Efinite__map_2EFUPDATE__LIST @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EFUPDATE @ A_27a @ A_27b ) @ V2fm ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27b ) @ V0k ) @ V3v ) ) ) @ V1kvl )
= ( ap @ ( ap @ ( c_2Efinite__map_2EFUPDATE @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EFUPDATE__LIST @ A_27a @ A_27b ) @ V2fm ) @ V1kvl ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27b ) @ V0k ) @ V3v ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Efinite__map_2EFUPDATE__FUPDATE__LIST__MEM,axiom,
! [A_27a: del,A_27b: del,V0k: $i] :
( ( mem @ V0k @ A_27a )
=> ! [V1kvl: $i] :
( ( mem @ V1kvl @ ( ty_2Elist_2Elist @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) ) )
=> ! [V2fm: $i] :
( ( mem @ V2fm @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ! [V3v: $i] :
( ( mem @ V3v @ A_27b )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V0k ) @ ( ap @ ( c_2Elist_2ELIST__TO__SET @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2EMAP @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ A_27a ) @ ( c_2Epair_2EFST @ A_27a @ A_27b ) ) @ V1kvl ) ) ) )
=> ( ( ap @ ( ap @ ( c_2Efinite__map_2EFUPDATE__LIST @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EFUPDATE @ A_27a @ A_27b ) @ V2fm ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27b ) @ V0k ) @ V3v ) ) ) @ V1kvl )
= ( ap @ ( ap @ ( c_2Efinite__map_2EFUPDATE__LIST @ A_27a @ A_27b ) @ V2fm ) @ V1kvl ) ) ) ) ) ) ) ).
thf(conj_thm_2Efinite__map_2EFEVERY__FUPDATE__LIST,axiom,
! [A_27a: del,A_27b: del,V0kvl: $i] :
( ( mem @ V0kvl @ ( ty_2Elist_2Elist @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) ) )
=> ! [V1P: $i] :
( ( mem @ V1P @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ bool ) )
=> ! [V2fm: $i] :
( ( mem @ V2fm @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ( ( p @ ( ap @ ( c_2Elist_2EALL__DISTINCT @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2EMAP @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ A_27a ) @ ( c_2Epair_2EFST @ A_27a @ A_27b ) ) @ V0kvl ) ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Efinite__map_2EFEVERY @ A_27a @ A_27b ) @ V1P ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EFUPDATE__LIST @ A_27a @ A_27b ) @ V2fm ) @ V0kvl ) ) )
<=> ( ( p @ ( ap @ ( ap @ ( c_2Elist_2EEVERY @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) ) @ V1P ) @ V0kvl ) )
& ( p @ ( ap @ ( ap @ ( c_2Efinite__map_2EFEVERY @ A_27a @ A_27b ) @ V1P ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EDRESTRICT @ A_27a @ A_27b ) @ V2fm ) @ ( ap @ ( c_2Epred__set_2ECOMPL @ A_27a ) @ ( ap @ ( c_2Elist_2ELIST__TO__SET @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2EMAP @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ A_27a ) @ ( c_2Epair_2EFST @ A_27a @ A_27b ) ) @ V0kvl ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Efinite__map_2EFUPDATE__LIST__APPLY__MEM,axiom,
! [A_27a: del,A_27b: del,V0kvl: $i] :
( ( mem @ V0kvl @ ( ty_2Elist_2Elist @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ! [V2k: $i] :
( ( mem @ V2k @ A_27a )
=> ! [V3v: $i] :
( ( mem @ V3v @ A_27b )
=> ! [V4n: tp__ty_2Enum_2Enum] :
( ( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V4n ) ) @ ( ap @ ( c_2Elist_2ELENGTH @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) ) @ V0kvl ) ) )
& ( V2k
= ( ap @ ( ap @ ( c_2Elist_2EEL @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V4n ) ) @ ( ap @ ( ap @ ( c_2Elist_2EMAP @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ A_27a ) @ ( c_2Epair_2EFST @ A_27a @ A_27b ) ) @ V0kvl ) ) )
& ( V3v
= ( ap @ ( ap @ ( c_2Elist_2EEL @ A_27b ) @ ( inj__ty_2Enum_2Enum @ V4n ) ) @ ( ap @ ( ap @ ( c_2Elist_2EMAP @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ A_27b ) @ ( c_2Epair_2ESND @ A_27a @ A_27b ) ) @ V0kvl ) ) )
& ! [V5m: tp__ty_2Enum_2Enum] :
( ( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V4n ) ) @ ( inj__ty_2Enum_2Enum @ V5m ) ) )
& ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V5m ) ) @ ( ap @ ( c_2Elist_2ELENGTH @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) ) @ V0kvl ) ) ) )
=> ( ( ap @ ( ap @ ( c_2Elist_2EEL @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V5m ) ) @ ( ap @ ( ap @ ( c_2Elist_2EMAP @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ A_27a ) @ ( c_2Epair_2EFST @ A_27a @ A_27b ) ) @ V0kvl ) )
!= V2k ) ) )
=> ( ( ap @ ( ap @ ( c_2Efinite__map_2EFAPPLY @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EFUPDATE__LIST @ A_27a @ A_27b ) @ V1f ) @ V0kvl ) ) @ V2k )
= V3v ) ) ) ) ) ) ).
thf(conj_thm_2Efinite__map_2EFOLDL__FUPDATE__LIST,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0f1: $i] :
( ( mem @ V0f1 @ ( arr @ A_27a @ A_27b ) )
=> ! [V1f2: $i] :
( ( mem @ V1f2 @ ( arr @ A_27a @ A_27c ) )
=> ! [V2ls: $i] :
( ( mem @ V2ls @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V3a: $i] :
( ( mem @ V3a @ ( ty_2Efinite__map_2Efmap @ A_27b @ A_27c ) )
=> ( ( ap
@ ( ap
@ ( ap @ ( c_2Elist_2EFOLDL @ A_27a @ ( ty_2Efinite__map_2Efmap @ A_27b @ A_27c ) )
@ ( lam @ ( ty_2Efinite__map_2Efmap @ A_27b @ A_27c )
@ ^ [V4fm: $i] :
( lam @ A_27a
@ ^ [V5k: $i] : ( ap @ ( ap @ ( c_2Efinite__map_2EFUPDATE @ A_27b @ A_27c ) @ V4fm ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27b @ A_27c ) @ ( ap @ V0f1 @ V5k ) ) @ ( ap @ V1f2 @ V5k ) ) ) ) ) )
@ V3a )
@ V2ls )
= ( ap @ ( ap @ ( c_2Efinite__map_2EFUPDATE__LIST @ A_27b @ A_27c ) @ V3a )
@ ( ap
@ ( ap @ ( c_2Elist_2EMAP @ A_27a @ ( ty_2Epair_2Eprod @ A_27b @ A_27c ) )
@ ( lam @ A_27a
@ ^ [V6k: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27b @ A_27c ) @ ( ap @ V0f1 @ V6k ) ) @ ( ap @ V1f2 @ V6k ) ) ) )
@ V2ls ) ) ) ) ) ) ) ).
thf(conj_thm_2Efinite__map_2EFUPDATE__LIST__SNOC,axiom,
! [A_27a: del,A_27b: del,V0xs: $i] :
( ( mem @ V0xs @ ( ty_2Elist_2Elist @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) )
=> ! [V2fm: $i] :
( ( mem @ V2fm @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ( ( ap @ ( ap @ ( c_2Efinite__map_2EFUPDATE__LIST @ A_27a @ A_27b ) @ V2fm ) @ ( ap @ ( ap @ ( c_2Elist_2ESNOC @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) ) @ V1x ) @ V0xs ) )
= ( ap @ ( ap @ ( c_2Efinite__map_2EFUPDATE @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EFUPDATE__LIST @ A_27a @ A_27b ) @ V2fm ) @ V0xs ) ) @ V1x ) ) ) ) ) ).
thf(conj_thm_2Efinite__map_2EFUPD11__SAME__KEY__AND__BASE,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ! [V1k: $i] :
( ( mem @ V1k @ A_27a )
=> ! [V2v1: $i] :
( ( mem @ V2v1 @ A_27b )
=> ! [V3v2: $i] :
( ( mem @ V3v2 @ A_27b )
=> ( ( ( ap @ ( ap @ ( c_2Efinite__map_2EFUPDATE @ A_27a @ A_27b ) @ V0f ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27b ) @ V1k ) @ V2v1 ) )
= ( ap @ ( ap @ ( c_2Efinite__map_2EFUPDATE @ A_27a @ A_27b ) @ V0f ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27b ) @ V1k ) @ V3v2 ) ) )
<=> ( V2v1 = V3v2 ) ) ) ) ) ) ).
thf(conj_thm_2Efinite__map_2EFUPD11__SAME__NEW__KEY,axiom,
! [A_27a: del,A_27b: del,V0f1: $i] :
( ( mem @ V0f1 @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ! [V1f2: $i] :
( ( mem @ V1f2 @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ! [V2k: $i] :
( ( mem @ V2k @ A_27a )
=> ! [V3v1: $i] :
( ( mem @ V3v1 @ A_27b )
=> ! [V4v2: $i] :
( ( mem @ V4v2 @ A_27b )
=> ( ( ~ ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2k ) @ ( ap @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b ) @ V0f1 ) ) )
& ~ ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2k ) @ ( ap @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b ) @ V1f2 ) ) ) )
=> ( ( ( ap @ ( ap @ ( c_2Efinite__map_2EFUPDATE @ A_27a @ A_27b ) @ V0f1 ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27b ) @ V2k ) @ V3v1 ) )
= ( ap @ ( ap @ ( c_2Efinite__map_2EFUPDATE @ A_27a @ A_27b ) @ V1f2 ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27b ) @ V2k ) @ V4v2 ) ) )
<=> ( ( V0f1 = V1f2 )
& ( V3v1 = V4v2 ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Efinite__map_2ESAME__KEY__UPDATES__DIFFER,axiom,
! [A_27a: del,A_27b: del,V0f1: $i] :
( ( mem @ V0f1 @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ! [V1f2: $i] :
( ( mem @ V1f2 @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ! [V2k: $i] :
( ( mem @ V2k @ A_27a )
=> ! [V3v1: $i] :
( ( mem @ V3v1 @ A_27b )
=> ! [V4v2: $i] :
( ( mem @ V4v2 @ A_27b )
=> ( ( V3v1 != V4v2 )
=> ( ( ap @ ( ap @ ( c_2Efinite__map_2EFUPDATE @ A_27a @ A_27b ) @ V0f1 ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27b ) @ V2k ) @ V3v1 ) )
!= ( ap @ ( ap @ ( c_2Efinite__map_2EFUPDATE @ A_27a @ A_27b ) @ V1f2 ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27b ) @ V2k ) @ V4v2 ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Efinite__map_2EFUPD11__SAME__BASE,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ! [V1k1: $i] :
( ( mem @ V1k1 @ A_27a )
=> ! [V2v1: $i] :
( ( mem @ V2v1 @ A_27b )
=> ! [V3k2: $i] :
( ( mem @ V3k2 @ A_27a )
=> ! [V4v2: $i] :
( ( mem @ V4v2 @ A_27b )
=> ( ( ( ap @ ( ap @ ( c_2Efinite__map_2EFUPDATE @ A_27a @ A_27b ) @ V0f ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27b ) @ V1k1 ) @ V2v1 ) )
= ( ap @ ( ap @ ( c_2Efinite__map_2EFUPDATE @ A_27a @ A_27b ) @ V0f ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27b ) @ V3k2 ) @ V4v2 ) ) )
<=> ( ( ( V1k1 = V3k2 )
& ( V2v1 = V4v2 ) )
| ( ( V1k1 != V3k2 )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V1k1 ) @ ( ap @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b ) @ V0f ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3k2 ) @ ( ap @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b ) @ V0f ) ) )
& ( ( ap @ ( ap @ ( c_2Efinite__map_2EFUPDATE @ A_27a @ A_27b ) @ V0f ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27b ) @ V1k1 ) @ V2v1 ) )
= V0f )
& ( ( ap @ ( ap @ ( c_2Efinite__map_2EFUPDATE @ A_27a @ A_27b ) @ V0f ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27b ) @ V3k2 ) @ V4v2 ) )
= V0f ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Efinite__map_2EFUPD__SAME__KEY__UNWIND,axiom,
! [A_27a: del,A_27b: del,V0f1: $i] :
( ( mem @ V0f1 @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ! [V1f2: $i] :
( ( mem @ V1f2 @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ! [V2k: $i] :
( ( mem @ V2k @ A_27a )
=> ! [V3v1: $i] :
( ( mem @ V3v1 @ A_27b )
=> ! [V4v2: $i] :
( ( mem @ V4v2 @ A_27b )
=> ( ( ( ap @ ( ap @ ( c_2Efinite__map_2EFUPDATE @ A_27a @ A_27b ) @ V0f1 ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27b ) @ V2k ) @ V3v1 ) )
= ( ap @ ( ap @ ( c_2Efinite__map_2EFUPDATE @ A_27a @ A_27b ) @ V1f2 ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27b ) @ V2k ) @ V4v2 ) ) )
=> ( ( V3v1 = V4v2 )
& ! [V5v: $i] :
( ( mem @ V5v @ A_27b )
=> ( ( ap @ ( ap @ ( c_2Efinite__map_2EFUPDATE @ A_27a @ A_27b ) @ V0f1 ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27b ) @ V2k ) @ V5v ) )
= ( ap @ ( ap @ ( c_2Efinite__map_2EFUPDATE @ A_27a @ A_27b ) @ V1f2 ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27b ) @ V2k ) @ V5v ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Efinite__map_2EFUPD11__SAME__UPDATE,axiom,
! [A_27a: del,A_27b: del,V0f1: $i] :
( ( mem @ V0f1 @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ! [V1f2: $i] :
( ( mem @ V1f2 @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ! [V2k: $i] :
( ( mem @ V2k @ A_27a )
=> ! [V3v: $i] :
( ( mem @ V3v @ A_27b )
=> ( ( ( ap @ ( ap @ ( c_2Efinite__map_2EFUPDATE @ A_27a @ A_27b ) @ V0f1 ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27b ) @ V2k ) @ V3v ) )
= ( ap @ ( ap @ ( c_2Efinite__map_2EFUPDATE @ A_27a @ A_27b ) @ V1f2 ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27b ) @ V2k ) @ V3v ) ) )
<=> ( ( ap @ ( ap @ ( c_2Efinite__map_2EDRESTRICT @ A_27a @ A_27b ) @ V0f1 ) @ ( ap @ ( c_2Epred__set_2ECOMPL @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ A_27a ) @ V2k ) @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) ) )
= ( ap @ ( ap @ ( c_2Efinite__map_2EDRESTRICT @ A_27a @ A_27b ) @ V1f2 ) @ ( ap @ ( c_2Epred__set_2ECOMPL @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ A_27a ) @ V2k ) @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Efinite__map_2EFDOM__FUPDATE__LIST,axiom,
! [A_27a: del,A_27b: del,V0kvl: $i] :
( ( mem @ V0kvl @ ( ty_2Elist_2Elist @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) ) )
=> ! [V1fm: $i] :
( ( mem @ V1fm @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ( ( ap @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EFUPDATE__LIST @ A_27a @ A_27b ) @ V1fm ) @ V0kvl ) )
= ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ A_27a ) @ ( ap @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b ) @ V1fm ) ) @ ( ap @ ( c_2Elist_2ELIST__TO__SET @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2EMAP @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ A_27a ) @ ( c_2Epair_2EFST @ A_27a @ A_27b ) ) @ V0kvl ) ) ) ) ) ) ).
thf(conj_thm_2Efinite__map_2EFUPDATE__LIST__SAME__UPDATE,axiom,
! [A_27a: del,A_27b: del,V0kvl: $i] :
( ( mem @ V0kvl @ ( ty_2Elist_2Elist @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) ) )
=> ! [V1f1: $i] :
( ( mem @ V1f1 @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ! [V2f2: $i] :
( ( mem @ V2f2 @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ( ( ( ap @ ( ap @ ( c_2Efinite__map_2EFUPDATE__LIST @ A_27a @ A_27b ) @ V1f1 ) @ V0kvl )
= ( ap @ ( ap @ ( c_2Efinite__map_2EFUPDATE__LIST @ A_27a @ A_27b ) @ V2f2 ) @ V0kvl ) )
<=> ( ( ap @ ( ap @ ( c_2Efinite__map_2EDRESTRICT @ A_27a @ A_27b ) @ V1f1 ) @ ( ap @ ( c_2Epred__set_2ECOMPL @ A_27a ) @ ( ap @ ( c_2Elist_2ELIST__TO__SET @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2EMAP @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ A_27a ) @ ( c_2Epair_2EFST @ A_27a @ A_27b ) ) @ V0kvl ) ) ) )
= ( ap @ ( ap @ ( c_2Efinite__map_2EDRESTRICT @ A_27a @ A_27b ) @ V2f2 ) @ ( ap @ ( c_2Epred__set_2ECOMPL @ A_27a ) @ ( ap @ ( c_2Elist_2ELIST__TO__SET @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2EMAP @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ A_27a ) @ ( c_2Epair_2EFST @ A_27a @ A_27b ) ) @ V0kvl ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Efinite__map_2EFUPDATE__LIST__SAME__KEYS__UNWIND,axiom,
! [A_27a: del,A_27b: del,V0f1: $i] :
( ( mem @ V0f1 @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ! [V1f2: $i] :
( ( mem @ V1f2 @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ! [V2kvl1: $i] :
( ( mem @ V2kvl1 @ ( ty_2Elist_2Elist @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) ) )
=> ! [V3kvl2: $i] :
( ( mem @ V3kvl2 @ ( ty_2Elist_2Elist @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) ) )
=> ( ( ( ( ap @ ( ap @ ( c_2Efinite__map_2EFUPDATE__LIST @ A_27a @ A_27b ) @ V0f1 ) @ V2kvl1 )
= ( ap @ ( ap @ ( c_2Efinite__map_2EFUPDATE__LIST @ A_27a @ A_27b ) @ V1f2 ) @ V3kvl2 ) )
& ( ( ap @ ( ap @ ( c_2Elist_2EMAP @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ A_27a ) @ ( c_2Epair_2EFST @ A_27a @ A_27b ) ) @ V2kvl1 )
= ( ap @ ( ap @ ( c_2Elist_2EMAP @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ A_27a ) @ ( c_2Epair_2EFST @ A_27a @ A_27b ) ) @ V3kvl2 ) )
& ( p @ ( ap @ ( c_2Elist_2EALL__DISTINCT @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2EMAP @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ A_27a ) @ ( c_2Epair_2EFST @ A_27a @ A_27b ) ) @ V2kvl1 ) ) ) )
=> ( ( V2kvl1 = V3kvl2 )
& ! [V4kvl: $i] :
( ( mem @ V4kvl @ ( ty_2Elist_2Elist @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) ) )
=> ( ( ( ap @ ( ap @ ( c_2Elist_2EMAP @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ A_27a ) @ ( c_2Epair_2EFST @ A_27a @ A_27b ) ) @ V4kvl )
= ( ap @ ( ap @ ( c_2Elist_2EMAP @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ A_27a ) @ ( c_2Epair_2EFST @ A_27a @ A_27b ) ) @ V2kvl1 ) )
=> ( ( ap @ ( ap @ ( c_2Efinite__map_2EFUPDATE__LIST @ A_27a @ A_27b ) @ V0f1 ) @ V4kvl )
= ( ap @ ( ap @ ( c_2Efinite__map_2EFUPDATE__LIST @ A_27a @ A_27b ) @ V1f2 ) @ V4kvl ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Efinite__map_2EFMEQ__ENUMERATE__CASES,axiom,
! [A_27a: del,A_27b: del,V0f1: $i] :
( ( mem @ V0f1 @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ! [V1kvl: $i] :
( ( mem @ V1kvl @ ( ty_2Elist_2Elist @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) ) )
=> ! [V2p: $i] :
( ( mem @ V2p @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) )
=> ( ( ( ap @ ( ap @ ( c_2Efinite__map_2EFUPDATE @ A_27a @ A_27b ) @ V0f1 ) @ V2p )
= ( ap @ ( ap @ ( c_2Efinite__map_2EFUPDATE__LIST @ A_27a @ A_27b ) @ ( c_2Efinite__map_2EFEMPTY @ A_27a @ A_27b ) ) @ V1kvl ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) ) @ V2p ) @ ( ap @ ( c_2Elist_2ELIST__TO__SET @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) ) @ V1kvl ) ) ) ) ) ) ) ).
thf(conj_thm_2Efinite__map_2EFMEQ__SINGLE__SIMPLE__ELIM,axiom,
! [A_27a: del,A_27b: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) @ bool ) )
=> ! [V1k: $i] :
( ( mem @ V1k @ A_27a )
=> ! [V2v: $i] :
( ( mem @ V2v @ A_27b )
=> ! [V3ck: $i] :
( ( mem @ V3ck @ A_27a )
=> ! [V4cv: $i] :
( ( mem @ V4cv @ A_27b )
=> ! [V5nv: $i] :
( ( mem @ V5nv @ A_27b )
=> ( ? [V6fm: $i] :
( ( mem @ V6fm @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
& ( ( ap @ ( ap @ ( c_2Efinite__map_2EFUPDATE @ A_27a @ A_27b ) @ V6fm ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27b ) @ V1k ) @ V2v ) )
= ( ap @ ( ap @ ( c_2Efinite__map_2EFUPDATE @ A_27a @ A_27b ) @ ( c_2Efinite__map_2EFEMPTY @ A_27a @ A_27b ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27b ) @ V3ck ) @ V4cv ) ) )
& ( p @ ( ap @ V0P @ ( ap @ ( ap @ ( c_2Efinite__map_2EFUPDATE @ A_27a @ A_27b ) @ V6fm ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27b ) @ V1k ) @ V5nv ) ) ) ) )
<=> ( ( V1k = V3ck )
& ( V2v = V4cv )
& ( p @ ( ap @ V0P @ ( ap @ ( ap @ ( c_2Efinite__map_2EFUPDATE @ A_27a @ A_27b ) @ ( c_2Efinite__map_2EFEMPTY @ A_27a @ A_27b ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27b ) @ V3ck ) @ V5nv ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Efinite__map_2EFMEQ__SINGLE__SIMPLE__DISJ__ELIM,axiom,
! [A_27a: del,A_27b: del,V0fm: $i] :
( ( mem @ V0fm @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ! [V1k: $i] :
( ( mem @ V1k @ A_27a )
=> ! [V2v: $i] :
( ( mem @ V2v @ A_27b )
=> ! [V3ck: $i] :
( ( mem @ V3ck @ A_27a )
=> ! [V4cv: $i] :
( ( mem @ V4cv @ A_27b )
=> ( ( ( ap @ ( ap @ ( c_2Efinite__map_2EFUPDATE @ A_27a @ A_27b ) @ V0fm ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27b ) @ V1k ) @ V2v ) )
= ( ap @ ( ap @ ( c_2Efinite__map_2EFUPDATE @ A_27a @ A_27b ) @ ( c_2Efinite__map_2EFEMPTY @ A_27a @ A_27b ) ) @ ( ap @ ( ap @ ( 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: $i] :
( ( mem @ V5v_27 @ A_27b )
& ( V0fm
= ( ap @ ( ap @ ( c_2Efinite__map_2EFUPDATE @ A_27a @ A_27b ) @ ( c_2Efinite__map_2EFEMPTY @ A_27a @ A_27b ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27b ) @ V1k ) @ V5v_27 ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Efinite__map_2EFUPDATE__PURGE,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ! [V2y: $i] :
( ( mem @ V2y @ A_27b )
=> ( ( ap @ ( ap @ ( c_2Efinite__map_2EFUPDATE @ A_27a @ A_27b ) @ V0f ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27b ) @ V1x ) @ V2y ) )
= ( ap @ ( ap @ ( c_2Efinite__map_2EFUPDATE @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Efinite__map_2Efdomsub @ A_27a @ A_27b ) @ V0f ) @ V1x ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27b ) @ V1x ) @ V2y ) ) ) ) ) ) ).
thf(ax_thm_2Efinite__map_2EFMAP__MAP2__def,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27c ) @ A_27b ) )
=> ! [V1m: $i] :
( ( mem @ V1m @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27c ) )
=> ( ( ap @ ( ap @ ( c_2Efinite__map_2EFMAP__MAP2 @ A_27a @ A_27b @ A_27c ) @ V0f ) @ V1m )
= ( ap
@ ( ap @ ( c_2Efinite__map_2EFUN__FMAP @ A_27a @ A_27b )
@ ( lam @ A_27a
@ ^ [V2x: $i] : ( ap @ V0f @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27c ) @ V2x ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EFAPPLY @ A_27a @ A_27c ) @ V1m ) @ V2x ) ) ) ) )
@ ( ap @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27c ) @ V1m ) ) ) ) ) ).
thf(conj_thm_2Efinite__map_2EFMAP__MAP2__THM,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27c ) @ A_27b ) )
=> ! [V1m: $i] :
( ( mem @ V1m @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27c ) )
=> ( ( ( ap @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EFMAP__MAP2 @ A_27a @ A_27b @ A_27c ) @ V0f ) @ V1m ) )
= ( ap @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27c ) @ V1m ) )
& ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x ) @ ( ap @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27c ) @ V1m ) ) )
=> ( ( ap @ ( ap @ ( c_2Efinite__map_2EFAPPLY @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EFMAP__MAP2 @ A_27a @ A_27b @ A_27c ) @ V0f ) @ V1m ) ) @ V2x )
= ( ap @ V0f @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27c ) @ V2x ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EFAPPLY @ A_27a @ A_27c ) @ V1m ) @ V2x ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Efinite__map_2EFMAP__MAP2__FEMPTY,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27c ) @ A_27b ) )
=> ( ( ap @ ( ap @ ( 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(conj_thm_2Efinite__map_2EFMAP__MAP2__FUPDATE,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27c ) @ A_27b ) )
=> ! [V1m: $i] :
( ( mem @ V1m @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27c ) )
=> ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ! [V3v: $i] :
( ( mem @ V3v @ A_27c )
=> ( ( ap @ ( ap @ ( c_2Efinite__map_2EFMAP__MAP2 @ A_27a @ A_27b @ A_27c ) @ V0f ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EFUPDATE @ A_27a @ A_27c ) @ V1m ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27c ) @ V2x ) @ V3v ) ) )
= ( ap @ ( ap @ ( c_2Efinite__map_2EFUPDATE @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EFMAP__MAP2 @ A_27a @ A_27b @ A_27c ) @ V0f ) @ V1m ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27b ) @ V2x ) @ ( ap @ V0f @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27c ) @ V2x ) @ V3v ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Efinite__map_2EFEVERY__STRENGTHEN__THM,axiom,
! [A_27a: del,A_27b: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ bool ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ! [V3y: $i] :
( ( mem @ V3y @ A_27b )
=> ( ( p @ ( ap @ ( ap @ ( c_2Efinite__map_2EFEVERY @ A_27a @ A_27b ) @ V0P ) @ ( c_2Efinite__map_2EFEMPTY @ A_27a @ A_27b ) ) )
& ( ( ( p @ ( ap @ ( ap @ ( c_2Efinite__map_2EFEVERY @ A_27a @ A_27b ) @ V0P ) @ V1f ) )
& ( p @ ( ap @ V0P @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27b ) @ V2x ) @ V3y ) ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Efinite__map_2EFEVERY @ A_27a @ A_27b ) @ V0P ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EFUPDATE @ A_27a @ A_27b ) @ V1f ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27b ) @ V2x ) @ V3y ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Efinite__map_2EFUPDATE__ELIM,axiom,
! [A_27a: del,A_27b: del,V0k: $i] :
( ( mem @ V0k @ A_27a )
=> ! [V1v: $i] :
( ( mem @ V1v @ A_27b )
=> ! [V2f: $i] :
( ( mem @ V2f @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V0k ) @ ( ap @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b ) @ V2f ) ) )
& ( ( ap @ ( ap @ ( c_2Efinite__map_2EFAPPLY @ A_27a @ A_27b ) @ V2f ) @ V0k )
= V1v ) )
=> ( ( ap @ ( ap @ ( c_2Efinite__map_2EFUPDATE @ A_27a @ A_27b ) @ V2f ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27b ) @ V0k ) @ V1v ) )
= V2f ) ) ) ) ) ).
thf(conj_thm_2Efinite__map_2EFEVERY__DRESTRICT__COMPL,axiom,
! [A_27a: del,A_27b: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ bool ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ! [V2k: $i] :
( ( mem @ V2k @ A_27a )
=> ! [V3v: $i] :
( ( mem @ V3v @ A_27b )
=> ! [V4s: $i] :
( ( mem @ V4s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Efinite__map_2EFEVERY @ A_27a @ A_27b ) @ V0P ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EDRESTRICT @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EFUPDATE @ A_27a @ A_27b ) @ V1f ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27b ) @ V2k ) @ V3v ) ) ) @ ( ap @ ( c_2Epred__set_2ECOMPL @ A_27a ) @ V4s ) ) ) )
<=> ( ( ~ ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2k ) @ V4s ) )
=> ( p @ ( ap @ V0P @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27b ) @ V2k ) @ V3v ) ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Efinite__map_2EFEVERY @ A_27a @ A_27b ) @ V0P ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EDRESTRICT @ A_27a @ A_27b ) @ V1f ) @ ( ap @ ( c_2Epred__set_2ECOMPL @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ A_27a ) @ V2k ) @ V4s ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Efinite__map_2EFUNION__EQ__FEMPTY,axiom,
! [A_27a: del,A_27b: del,V0h1: $i] :
( ( mem @ V0h1 @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ! [V1h2: $i] :
( ( mem @ V1h2 @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ( ( ( ap @ ( ap @ ( 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(conj_thm_2Efinite__map_2ESUBMAP__FUNION__EQ,axiom,
! [A_27a: del,A_27b: del,A_27c: del,A_27d: del] :
( ! [V0f1: $i] :
( ( mem @ V0f1 @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ! [V1f2: $i] :
( ( mem @ V1f2 @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ! [V2f3: $i] :
( ( mem @ V2f3 @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2EDISJOINT @ A_27a ) @ ( ap @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b ) @ V0f1 ) ) @ ( ap @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b ) @ V1f2 ) ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Efinite__map_2ESUBMAP @ A_27a @ A_27b ) @ V0f1 ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EFUNION @ A_27a @ A_27b ) @ V1f2 ) @ V2f3 ) ) )
<=> ( p @ ( ap @ ( ap @ ( c_2Efinite__map_2ESUBMAP @ A_27a @ A_27b ) @ V0f1 ) @ V2f3 ) ) ) ) ) ) )
& ! [V3f1: $i] :
( ( mem @ V3f1 @ ( ty_2Efinite__map_2Efmap @ A_27c @ A_27d ) )
=> ! [V4f2: $i] :
( ( mem @ V4f2 @ ( ty_2Efinite__map_2Efmap @ A_27c @ A_27d ) )
=> ! [V5f3: $i] :
( ( mem @ V5f3 @ ( ty_2Efinite__map_2Efmap @ A_27c @ A_27d ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2EDISJOINT @ A_27c ) @ ( ap @ ( c_2Efinite__map_2EFDOM @ A_27c @ A_27d ) @ V3f1 ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EDIFF @ A_27c ) @ ( ap @ ( c_2Efinite__map_2EFDOM @ A_27c @ A_27d ) @ V5f3 ) ) @ ( ap @ ( c_2Efinite__map_2EFDOM @ A_27c @ A_27d ) @ V4f2 ) ) ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Efinite__map_2ESUBMAP @ A_27c @ A_27d ) @ V3f1 ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EFUNION @ A_27c @ A_27d ) @ V4f2 ) @ V5f3 ) ) )
<=> ( p @ ( ap @ ( ap @ ( c_2Efinite__map_2ESUBMAP @ A_27c @ A_27d ) @ V3f1 ) @ V4f2 ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Efinite__map_2ESUBMAP__FUNION,axiom,
! [A_27a: del,A_27b: del,V0f1: $i] :
( ( mem @ V0f1 @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ! [V1f2: $i] :
( ( mem @ V1f2 @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ! [V2f3: $i] :
( ( mem @ V2f3 @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Efinite__map_2ESUBMAP @ A_27a @ A_27b ) @ V0f1 ) @ V1f2 ) )
| ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2EDISJOINT @ A_27a ) @ ( ap @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b ) @ V0f1 ) ) @ ( ap @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b ) @ V1f2 ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Efinite__map_2ESUBMAP @ A_27a @ A_27b ) @ V0f1 ) @ V2f3 ) ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Efinite__map_2ESUBMAP @ A_27a @ A_27b ) @ V0f1 ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EFUNION @ A_27a @ A_27b ) @ V1f2 ) @ V2f3 ) ) ) ) ) ) ) ).
thf(conj_thm_2Efinite__map_2ESUBMAP__FUNION__ID,axiom,
! [A_27a: del,A_27b: del,A_27c: del,A_27d: del] :
( ! [V0f1: $i] :
( ( mem @ V0f1 @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ! [V1f2: $i] :
( ( mem @ V1f2 @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ( p @ ( ap @ ( ap @ ( c_2Efinite__map_2ESUBMAP @ A_27a @ A_27b ) @ V0f1 ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EFUNION @ A_27a @ A_27b ) @ V0f1 ) @ V1f2 ) ) ) ) )
& ! [V2f1: $i] :
( ( mem @ V2f1 @ ( ty_2Efinite__map_2Efmap @ A_27c @ A_27d ) )
=> ! [V3f2: $i] :
( ( mem @ V3f2 @ ( ty_2Efinite__map_2Efmap @ A_27c @ A_27d ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2EDISJOINT @ A_27c ) @ ( ap @ ( c_2Efinite__map_2EFDOM @ A_27c @ A_27d ) @ V2f1 ) ) @ ( ap @ ( c_2Efinite__map_2EFDOM @ A_27c @ A_27d ) @ V3f2 ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Efinite__map_2ESUBMAP @ A_27c @ A_27d ) @ V3f2 ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EFUNION @ A_27c @ A_27d ) @ V2f1 ) @ V3f2 ) ) ) ) ) ) ) ).
thf(conj_thm_2Efinite__map_2EFEMPTY__SUBMAP,axiom,
! [A_27a: del,A_27b: del,V0h: $i] :
( ( mem @ V0h @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ( ( p @ ( ap @ ( ap @ ( 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(conj_thm_2Efinite__map_2EFUNION__EQ,axiom,
! [A_27a: del,A_27b: del,V0f1: $i] :
( ( mem @ V0f1 @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ! [V1f2: $i] :
( ( mem @ V1f2 @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ! [V2f3: $i] :
( ( mem @ V2f3 @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2EDISJOINT @ A_27a ) @ ( ap @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b ) @ V0f1 ) ) @ ( ap @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b ) @ V1f2 ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2EDISJOINT @ A_27a ) @ ( ap @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b ) @ V0f1 ) ) @ ( ap @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b ) @ V2f3 ) ) ) )
=> ( ( ( ap @ ( ap @ ( c_2Efinite__map_2EFUNION @ A_27a @ A_27b ) @ V0f1 ) @ V1f2 )
= ( ap @ ( ap @ ( c_2Efinite__map_2EFUNION @ A_27a @ A_27b ) @ V0f1 ) @ V2f3 ) )
<=> ( V1f2 = V2f3 ) ) ) ) ) ) ).
thf(conj_thm_2Efinite__map_2EFUNION__EQ__IMPL,axiom,
! [A_27a: del,A_27b: del,V0f1: $i] :
( ( mem @ V0f1 @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ! [V1f2: $i] :
( ( mem @ V1f2 @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ! [V2f3: $i] :
( ( mem @ V2f3 @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2EDISJOINT @ A_27a ) @ ( ap @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b ) @ V0f1 ) ) @ ( ap @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b ) @ V1f2 ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2EDISJOINT @ A_27a ) @ ( ap @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b ) @ V0f1 ) ) @ ( ap @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b ) @ V2f3 ) ) )
& ( V1f2 = V2f3 ) )
=> ( ( ap @ ( ap @ ( c_2Efinite__map_2EFUNION @ A_27a @ A_27b ) @ V0f1 ) @ V1f2 )
= ( ap @ ( ap @ ( c_2Efinite__map_2EFUNION @ A_27a @ A_27b ) @ V0f1 ) @ V2f3 ) ) ) ) ) ) ).
thf(conj_thm_2Efinite__map_2EDOMSUB__FUNION,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ! [V2k: $i] :
( ( mem @ V2k @ A_27a )
=> ( ( ap @ ( ap @ ( c_2Efinite__map_2Efdomsub @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EFUNION @ A_27a @ A_27b ) @ V0f ) @ V1g ) ) @ V2k )
= ( ap @ ( ap @ ( c_2Efinite__map_2EFUNION @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Efinite__map_2Efdomsub @ A_27a @ A_27b ) @ V0f ) @ V2k ) ) @ ( ap @ ( ap @ ( c_2Efinite__map_2Efdomsub @ A_27a @ A_27b ) @ V1g ) @ V2k ) ) ) ) ) ) ).
thf(conj_thm_2Efinite__map_2EFUNION__COMM,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2EDISJOINT @ A_27a ) @ ( ap @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b ) @ V0f ) ) @ ( ap @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b ) @ V1g ) ) )
=> ( ( ap @ ( ap @ ( c_2Efinite__map_2EFUNION @ A_27a @ A_27b ) @ V0f ) @ V1g )
= ( ap @ ( ap @ ( c_2Efinite__map_2EFUNION @ A_27a @ A_27b ) @ V1g ) @ V0f ) ) ) ) ) ).
thf(conj_thm_2Efinite__map_2EFUNION__ASSOC,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ! [V2h: $i] :
( ( mem @ V2h @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ( ( ap @ ( ap @ ( c_2Efinite__map_2EFUNION @ A_27a @ A_27b ) @ V0f ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EFUNION @ A_27a @ A_27b ) @ V1g ) @ V2h ) )
= ( ap @ ( ap @ ( c_2Efinite__map_2EFUNION @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EFUNION @ A_27a @ A_27b ) @ V0f ) @ V1g ) ) @ V2h ) ) ) ) ) ).
thf(conj_thm_2Efinite__map_2EDRESTRICT__FUNION,axiom,
! [A_27a: del,A_27b: del,V0h: $i] :
( ( mem @ V0h @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ! [V1s1: $i] :
( ( mem @ V1s1 @ ( arr @ A_27a @ bool ) )
=> ! [V2s2: $i] :
( ( mem @ V2s2 @ ( arr @ A_27a @ bool ) )
=> ( ( ap @ ( ap @ ( c_2Efinite__map_2EFUNION @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EDRESTRICT @ A_27a @ A_27b ) @ V0h ) @ V1s1 ) ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EDRESTRICT @ A_27a @ A_27b ) @ V0h ) @ V2s2 ) )
= ( ap @ ( ap @ ( c_2Efinite__map_2EDRESTRICT @ A_27a @ A_27b ) @ V0h ) @ ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ A_27a ) @ V1s1 ) @ V2s2 ) ) ) ) ) ) ).
thf(conj_thm_2Efinite__map_2EDRESTRICT__EQ__FUNION,axiom,
! [A_27a: del,A_27b: del,V0h: $i] :
( ( mem @ V0h @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ! [V1h1: $i] :
( ( mem @ V1h1 @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ! [V2h2: $i] :
( ( mem @ V2h2 @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2EDISJOINT @ A_27a ) @ ( ap @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b ) @ V1h1 ) ) @ ( ap @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b ) @ V2h2 ) ) )
& ( ( ap @ ( ap @ ( c_2Efinite__map_2EFUNION @ A_27a @ A_27b ) @ V1h1 ) @ V2h2 )
= V0h ) )
=> ( V2h2
= ( ap @ ( ap @ ( c_2Efinite__map_2EDRESTRICT @ A_27a @ A_27b ) @ V0h ) @ ( ap @ ( c_2Epred__set_2ECOMPL @ A_27a ) @ ( ap @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b ) @ V1h1 ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Efinite__map_2EIN__FDOM__FOLDR__UNION,axiom,
! [A_27a: del,A_27b: del,V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ! [V1hL: $i] :
( ( mem @ V1hL @ ( ty_2Elist_2Elist @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V0x ) @ ( ap @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( ap @ ( c_2Elist_2EFOLDR @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) @ ( ty_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: $i] :
( ( mem @ V2h @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) ) @ V2h ) @ ( ap @ ( c_2Elist_2ELIST__TO__SET @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) ) @ V1hL ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V0x ) @ ( ap @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b ) @ V2h ) ) ) ) ) ) ) ).
thf(conj_thm_2Efinite__map_2EDRESTRICT__FUNION__DRESTRICT__COMPL,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ( ( ap @ ( ap @ ( c_2Efinite__map_2EFUNION @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EDRESTRICT @ A_27a @ A_27b ) @ V0f ) @ V1s ) ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EDRESTRICT @ A_27a @ A_27b ) @ V0f ) @ ( ap @ ( c_2Epred__set_2ECOMPL @ A_27a ) @ V1s ) ) )
= V0f ) ) ) ).
thf(conj_thm_2Efinite__map_2EDRESTRICT__IDEMPOT,axiom,
! [A_27a: del,A_27b: del,V0s: $i] :
( ( mem @ V0s @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ! [V1vs: $i] :
( ( mem @ V1vs @ ( arr @ A_27a @ bool ) )
=> ( ( ap @ ( ap @ ( c_2Efinite__map_2EDRESTRICT @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EDRESTRICT @ A_27a @ A_27b ) @ V0s ) @ V1vs ) ) @ V1vs )
= ( ap @ ( ap @ ( c_2Efinite__map_2EDRESTRICT @ A_27a @ A_27b ) @ V0s ) @ V1vs ) ) ) ) ).
thf(conj_thm_2Efinite__map_2ESUBMAP__FUNION__ABSORPTION,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Efinite__map_2ESUBMAP @ A_27a @ A_27b ) @ V0f ) @ V1g ) )
<=> ( ( ap @ ( ap @ ( c_2Efinite__map_2EFUNION @ A_27a @ A_27b ) @ V0f ) @ V1g )
= V1g ) ) ) ) ).
thf(conj_thm_2Efinite__map_2EMAP__KEYS__witness,axiom,
! [A_27a: del,A_27b: del,A_27c: del] :
( p
@ ( ap
@ ( ap @ ( c_2Ebool_2ELET @ ( arr @ ( arr @ A_27a @ A_27b ) @ ( arr @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27c ) @ ( ty_2Efinite__map_2Efmap @ A_27b @ A_27c ) ) ) @ bool )
@ ( lam @ ( arr @ ( arr @ A_27a @ A_27b ) @ ( arr @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27c ) @ ( ty_2Efinite__map_2Efmap @ A_27b @ A_27c ) ) )
@ ^ [V0m: $i] :
( ap @ ( c_2Ebool_2E_21 @ ( arr @ A_27a @ A_27b ) )
@ ( lam @ ( arr @ A_27a @ A_27b )
@ ^ [V1f: $i] :
( ap @ ( c_2Ebool_2E_21 @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27c ) )
@ ( lam @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27c )
@ ^ [V2fm: $i] :
( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ( arr @ A_27b @ bool ) ) @ ( ap @ ( c_2Efinite__map_2EFDOM @ A_27b @ A_27c ) @ ( ap @ ( ap @ V0m @ V1f ) @ V2fm ) ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27b ) @ V1f ) @ ( ap @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27c ) @ V2fm ) ) ) )
@ ( ap @ ( ap @ c_2Emin_2E_3D_3D_3E @ ( ap @ ( ap @ ( ap @ ( c_2Epred__set_2EINJ @ A_27a @ A_27b ) @ V1f ) @ ( ap @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27c ) @ V2fm ) ) @ ( c_2Epred__set_2EUNIV @ A_27b ) ) )
@ ( ap @ ( c_2Ebool_2E_21 @ A_27a )
@ ( lam @ A_27a
@ ^ [V3x: $i] : ( ap @ ( ap @ c_2Emin_2E_3D_3D_3E @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3x ) @ ( ap @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27c ) @ V2fm ) ) ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ A_27c ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EFAPPLY @ A_27b @ A_27c ) @ ( ap @ ( ap @ V0m @ V1f ) @ V2fm ) ) @ ( ap @ V1f @ V3x ) ) ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EFAPPLY @ A_27a @ A_27c ) @ V2fm ) @ V3x ) ) ) ) ) ) ) ) ) ) ) ) )
@ ( lam @ ( arr @ A_27a @ A_27b )
@ ^ [V4f: $i] :
( lam @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27c )
@ ^ [V5fm: $i] : ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ( ty_2Efinite__map_2Efmap @ A_27b @ A_27c ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Epred__set_2EINJ @ A_27a @ A_27b ) @ V4f ) @ ( ap @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27c ) @ V5fm ) ) @ ( c_2Epred__set_2EUNIV @ A_27b ) ) ) @ ( ap @ ( ap @ ( c_2Efinite__map_2Ef__o__f @ A_27b @ A_27a @ A_27c ) @ V5fm ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EFUN__FMAP @ A_27b @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2ELINV @ A_27a @ A_27b ) @ V4f ) @ ( ap @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27c ) @ V5fm ) ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27b ) @ V4f ) @ ( ap @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27c ) @ V5fm ) ) ) ) ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EFUN__FMAP @ A_27b @ A_27c ) @ ( c_2Ebool_2EARB @ ( arr @ A_27b @ A_27c ) ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27b ) @ V4f ) @ ( ap @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27c ) @ V5fm ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Efinite__map_2EMAP__KEYS__def,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27b ) )
=> ! [V1fm: $i] :
( ( mem @ V1fm @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27c ) )
=> ( ( ( ap @ ( c_2Efinite__map_2EFDOM @ A_27b @ A_27c ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EMAP__KEYS @ A_27a @ A_27b @ A_27c ) @ V0f ) @ V1fm ) )
= ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27b ) @ V0f ) @ ( ap @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27c ) @ V1fm ) ) )
& ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Epred__set_2EINJ @ A_27a @ A_27b ) @ V0f ) @ ( ap @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27c ) @ V1fm ) ) @ ( c_2Epred__set_2EUNIV @ A_27b ) ) )
=> ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x ) @ ( ap @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27c ) @ V1fm ) ) )
=> ( ( ap @ ( ap @ ( c_2Efinite__map_2EFAPPLY @ A_27b @ A_27c ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EMAP__KEYS @ A_27a @ A_27b @ A_27c ) @ V0f ) @ V1fm ) ) @ ( ap @ V0f @ V2x ) )
= ( ap @ ( ap @ ( c_2Efinite__map_2EFAPPLY @ A_27a @ A_27c ) @ V1fm ) @ V2x ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Efinite__map_2EMAP__KEYS__FEMPTY,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27b ) )
=> ( ( ap @ ( ap @ ( 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(conj_thm_2Efinite__map_2EMAP__KEYS__FUPDATE,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27b ) )
=> ! [V1fm: $i] :
( ( mem @ V1fm @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27c ) )
=> ! [V2k: $i] :
( ( mem @ V2k @ A_27a )
=> ! [V3v: $i] :
( ( mem @ V3v @ A_27c )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Epred__set_2EINJ @ A_27a @ A_27b ) @ V0f ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ A_27a ) @ V2k ) @ ( ap @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27c ) @ V1fm ) ) ) @ ( c_2Epred__set_2EUNIV @ A_27b ) ) )
=> ( ( ap @ ( ap @ ( c_2Efinite__map_2EMAP__KEYS @ A_27a @ A_27b @ A_27c ) @ V0f ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EFUPDATE @ A_27a @ A_27c ) @ V1fm ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27c ) @ V2k ) @ V3v ) ) )
= ( ap @ ( ap @ ( c_2Efinite__map_2EFUPDATE @ A_27b @ A_27c ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EMAP__KEYS @ A_27a @ A_27b @ A_27c ) @ V0f ) @ V1fm ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27b @ A_27c ) @ ( ap @ V0f @ V2k ) ) @ V3v ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Efinite__map_2EMAP__KEYS__using__LINV,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27b ) )
=> ! [V1fm: $i] :
( ( mem @ V1fm @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27c ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Epred__set_2EINJ @ A_27a @ A_27b ) @ V0f ) @ ( ap @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27c ) @ V1fm ) ) @ ( c_2Epred__set_2EUNIV @ A_27b ) ) )
=> ( ( ap @ ( ap @ ( c_2Efinite__map_2EMAP__KEYS @ A_27a @ A_27b @ A_27c ) @ V0f ) @ V1fm )
= ( ap @ ( ap @ ( c_2Efinite__map_2Ef__o__f @ A_27b @ A_27a @ A_27c ) @ V1fm ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EFUN__FMAP @ A_27b @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2ELINV @ A_27a @ A_27b ) @ V0f ) @ ( ap @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27c ) @ V1fm ) ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27b ) @ V0f ) @ ( ap @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27c ) @ V1fm ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Efinite__map_2EMAP__KEYS__BIJ__LINV,axiom,
! [A_27a: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( ty_2Efinite__map_2Efmap @ ty_2Enum_2Enum @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Epred__set_2EBIJ @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) @ V0f ) @ ( c_2Epred__set_2EUNIV @ ty_2Enum_2Enum ) ) @ ( c_2Epred__set_2EUNIV @ ty_2Enum_2Enum ) ) )
=> ( ( ap @ ( ap @ ( c_2Efinite__map_2EMAP__KEYS @ ty_2Enum_2Enum @ ty_2Enum_2Enum @ A_27a ) @ V0f ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EMAP__KEYS @ ty_2Enum_2Enum @ ty_2Enum_2Enum @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2ELINV @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) @ V0f ) @ ( c_2Epred__set_2EUNIV @ ty_2Enum_2Enum ) ) ) @ V1t ) )
= V1t ) ) ) ) ).
thf(conj_thm_2Efinite__map_2EFLOOKUP__MAP__KEYS,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27b ) )
=> ! [V1m: $i] :
( ( mem @ V1m @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27c ) )
=> ! [V2k: $i] :
( ( mem @ V2k @ A_27b )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Epred__set_2EINJ @ A_27a @ A_27b ) @ V0f ) @ ( ap @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27c ) @ V1m ) ) @ ( c_2Epred__set_2EUNIV @ A_27b ) ) )
=> ( ( ap @ ( ap @ ( c_2Efinite__map_2EFLOOKUP @ A_27b @ A_27c ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EMAP__KEYS @ A_27a @ A_27b @ A_27c ) @ V0f ) @ V1m ) ) @ V2k )
= ( ap
@ ( ap @ ( c_2Eoption_2EOPTION__BIND @ A_27c @ A_27a )
@ ( ap @ ( c_2Eoption_2Esome @ A_27a )
@ ( lam @ A_27a
@ ^ [V3x: $i] : ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ A_27b ) @ V2k ) @ ( ap @ V0f @ V3x ) ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3x ) @ ( ap @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27c ) @ V1m ) ) ) ) ) )
@ ( ap @ ( c_2Efinite__map_2EFLOOKUP @ A_27a @ A_27c ) @ V1m ) ) ) ) ) ) ) ).
thf(conj_thm_2Efinite__map_2EFLOOKUP__MAP__KEYS__MAPPED,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27b ) )
=> ! [V1m: $i] :
( ( mem @ V1m @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27c ) )
=> ! [V2k: $i] :
( ( mem @ V2k @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Epred__set_2EINJ @ A_27a @ A_27b ) @ V0f ) @ ( c_2Epred__set_2EUNIV @ A_27a ) ) @ ( c_2Epred__set_2EUNIV @ A_27b ) ) )
=> ( ( ap @ ( ap @ ( c_2Efinite__map_2EFLOOKUP @ A_27b @ A_27c ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EMAP__KEYS @ A_27a @ A_27b @ A_27c ) @ V0f ) @ V1m ) ) @ ( ap @ V0f @ V2k ) )
= ( ap @ ( ap @ ( c_2Efinite__map_2EFLOOKUP @ A_27a @ A_27c ) @ V1m ) @ V2k ) ) ) ) ) ) ).
thf(conj_thm_2Efinite__map_2EDRESTRICT__MAP__KEYS__IMAGE,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27b ) )
=> ! [V1fm: $i] :
( ( mem @ V1fm @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27c ) )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Epred__set_2EINJ @ A_27a @ A_27b ) @ V0f ) @ ( c_2Epred__set_2EUNIV @ A_27a ) ) @ ( c_2Epred__set_2EUNIV @ A_27b ) ) )
=> ( ( ap @ ( ap @ ( c_2Efinite__map_2EDRESTRICT @ A_27b @ A_27c ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EMAP__KEYS @ A_27a @ A_27b @ A_27c ) @ V0f ) @ V1fm ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27b ) @ V0f ) @ V2s ) )
= ( ap @ ( ap @ ( c_2Efinite__map_2EMAP__KEYS @ A_27a @ A_27b @ A_27c ) @ V0f ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EDRESTRICT @ A_27a @ A_27c ) @ V1fm ) @ V2s ) ) ) ) ) ) ) ).
thf(conj_thm_2Efinite__map_2EDOMSUB__MAP__KEYS,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27b ) )
=> ! [V1fm: $i] :
( ( mem @ V1fm @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27c ) )
=> ! [V2s: $i] :
( ( mem @ V2s @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Epred__set_2EBIJ @ A_27a @ A_27b ) @ V0f ) @ ( c_2Epred__set_2EUNIV @ A_27a ) ) @ ( c_2Epred__set_2EUNIV @ A_27b ) ) )
=> ( ( ap @ ( ap @ ( c_2Efinite__map_2Efdomsub @ A_27b @ A_27c ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EMAP__KEYS @ A_27a @ A_27b @ A_27c ) @ V0f ) @ V1fm ) ) @ ( ap @ V0f @ V2s ) )
= ( ap @ ( ap @ ( c_2Efinite__map_2EMAP__KEYS @ A_27a @ A_27b @ A_27c ) @ V0f ) @ ( ap @ ( ap @ ( c_2Efinite__map_2Efdomsub @ A_27a @ A_27c ) @ V1fm ) @ V2s ) ) ) ) ) ) ) ).
thf(ax_thm_2Efinite__map_2Efmap__rel__def,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) )
=> ! [V1f1: $i] :
( ( mem @ V1f1 @ ( ty_2Efinite__map_2Efmap @ A_27c @ A_27a ) )
=> ! [V2f2: $i] :
( ( mem @ V2f2 @ ( ty_2Efinite__map_2Efmap @ A_27c @ A_27b ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Efinite__map_2Efmap__rel @ A_27a @ A_27b @ A_27c ) @ V0R ) @ V1f1 ) @ V2f2 ) )
<=> ( ( ( ap @ ( c_2Efinite__map_2EFDOM @ A_27c @ A_27b ) @ V2f2 )
= ( ap @ ( c_2Efinite__map_2EFDOM @ A_27c @ A_27a ) @ V1f1 ) )
& ! [V3x: $i] :
( ( mem @ V3x @ A_27c )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27c ) @ V3x ) @ ( ap @ ( c_2Efinite__map_2EFDOM @ A_27c @ A_27a ) @ V1f1 ) ) )
=> ( p @ ( ap @ ( ap @ V0R @ ( ap @ ( ap @ ( c_2Efinite__map_2EFAPPLY @ A_27c @ A_27a ) @ V1f1 ) @ V3x ) ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EFAPPLY @ A_27c @ A_27b ) @ V2f2 ) @ V3x ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Efinite__map_2Efmap__rel__FUPDATE__same,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) )
=> ! [V1f1: $i] :
( ( mem @ V1f1 @ ( ty_2Efinite__map_2Efmap @ A_27c @ A_27a ) )
=> ! [V2f2: $i] :
( ( mem @ V2f2 @ ( ty_2Efinite__map_2Efmap @ A_27c @ A_27b ) )
=> ! [V3v1: $i] :
( ( mem @ V3v1 @ A_27a )
=> ! [V4v2: $i] :
( ( mem @ V4v2 @ A_27b )
=> ! [V5k: $i] :
( ( mem @ V5k @ A_27c )
=> ( ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Efinite__map_2Efmap__rel @ A_27a @ A_27b @ A_27c ) @ V0R ) @ V1f1 ) @ V2f2 ) )
& ( p @ ( ap @ ( ap @ V0R @ V3v1 ) @ V4v2 ) ) )
=> ( p @ ( ap @ ( ap @ ( ap @ ( c_2Efinite__map_2Efmap__rel @ A_27a @ A_27b @ A_27c ) @ V0R ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EFUPDATE @ A_27c @ A_27a ) @ V1f1 ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27c @ A_27a ) @ V5k ) @ V3v1 ) ) ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EFUPDATE @ A_27c @ A_27b ) @ V2f2 ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27c @ A_27b ) @ V5k ) @ V4v2 ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Efinite__map_2Efmap__rel__FUPDATE__LIST__same,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) )
=> ! [V1ls1: $i] :
( ( mem @ V1ls1 @ ( ty_2Elist_2Elist @ ( ty_2Epair_2Eprod @ A_27c @ A_27a ) ) )
=> ! [V2ls2: $i] :
( ( mem @ V2ls2 @ ( ty_2Elist_2Elist @ ( ty_2Epair_2Eprod @ A_27c @ A_27b ) ) )
=> ! [V3f1: $i] :
( ( mem @ V3f1 @ ( ty_2Efinite__map_2Efmap @ A_27c @ A_27a ) )
=> ! [V4f2: $i] :
( ( mem @ V4f2 @ ( ty_2Efinite__map_2Efmap @ A_27c @ A_27b ) )
=> ( ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Efinite__map_2Efmap__rel @ A_27a @ A_27b @ A_27c ) @ V0R ) @ V3f1 ) @ V4f2 ) )
& ( ( ap @ ( ap @ ( c_2Elist_2EMAP @ ( ty_2Epair_2Eprod @ A_27c @ A_27a ) @ A_27c ) @ ( c_2Epair_2EFST @ A_27c @ A_27a ) ) @ V1ls1 )
= ( ap @ ( ap @ ( c_2Elist_2EMAP @ ( ty_2Epair_2Eprod @ A_27c @ A_27b ) @ A_27c ) @ ( c_2Epair_2EFST @ A_27c @ A_27b ) ) @ V2ls2 ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( c_2Elist_2ELIST__REL @ A_27a @ A_27b ) @ V0R ) @ ( ap @ ( ap @ ( c_2Elist_2EMAP @ ( ty_2Epair_2Eprod @ A_27c @ A_27a ) @ A_27a ) @ ( c_2Epair_2ESND @ A_27c @ A_27a ) ) @ V1ls1 ) ) @ ( ap @ ( ap @ ( c_2Elist_2EMAP @ ( ty_2Epair_2Eprod @ A_27c @ A_27b ) @ A_27b ) @ ( c_2Epair_2ESND @ A_27c @ A_27b ) ) @ V2ls2 ) ) ) )
=> ( p @ ( ap @ ( ap @ ( ap @ ( c_2Efinite__map_2Efmap__rel @ A_27a @ A_27b @ A_27c ) @ V0R ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EFUPDATE__LIST @ A_27c @ A_27a ) @ V3f1 ) @ V1ls1 ) ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EFUPDATE__LIST @ A_27c @ A_27b ) @ V4f2 ) @ V2ls2 ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Efinite__map_2Efmap__rel__FEMPTY,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) )
=> ( p @ ( ap @ ( ap @ ( ap @ ( 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(conj_thm_2Efinite__map_2Efmap__rel__FEMPTY2,axiom,
! [A_27a: del,A_27b: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( ty_2Efinite__map_2Efmap @ A_27b @ A_27a ) )
=> ( ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Efinite__map_2Efmap__rel @ A_27a @ A_27a @ A_27b ) @ V0R ) @ ( c_2Efinite__map_2EFEMPTY @ A_27b @ A_27a ) ) @ V1f ) )
<=> ( V1f
= ( c_2Efinite__map_2EFEMPTY @ A_27b @ A_27a ) ) )
& ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Efinite__map_2Efmap__rel @ A_27a @ A_27a @ A_27b ) @ V0R ) @ V1f ) @ ( c_2Efinite__map_2EFEMPTY @ A_27b @ A_27a ) ) )
<=> ( V1f
= ( c_2Efinite__map_2EFEMPTY @ A_27b @ A_27a ) ) ) ) ) ) ).
thf(conj_thm_2Efinite__map_2Efmap__rel__refl,axiom,
! [A_27a: del,A_27b: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ ( ty_2Efinite__map_2Efmap @ A_27b @ A_27a ) )
=> ( ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ( p @ ( ap @ ( ap @ V0R @ V2x ) @ V2x ) ) )
=> ( p @ ( ap @ ( ap @ ( ap @ ( c_2Efinite__map_2Efmap__rel @ A_27a @ A_27a @ A_27b ) @ V0R ) @ V1x ) @ V1x ) ) ) ) ) ).
thf(conj_thm_2Efinite__map_2Efmap__rel__FUNION__rels,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) )
=> ! [V1f1: $i] :
( ( mem @ V1f1 @ ( ty_2Efinite__map_2Efmap @ A_27c @ A_27a ) )
=> ! [V2f2: $i] :
( ( mem @ V2f2 @ ( ty_2Efinite__map_2Efmap @ A_27c @ A_27b ) )
=> ! [V3f3: $i] :
( ( mem @ V3f3 @ ( ty_2Efinite__map_2Efmap @ A_27c @ A_27a ) )
=> ! [V4f4: $i] :
( ( mem @ V4f4 @ ( ty_2Efinite__map_2Efmap @ A_27c @ A_27b ) )
=> ( ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Efinite__map_2Efmap__rel @ A_27a @ A_27b @ A_27c ) @ V0R ) @ V1f1 ) @ V2f2 ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( c_2Efinite__map_2Efmap__rel @ A_27a @ A_27b @ A_27c ) @ V0R ) @ V3f3 ) @ V4f4 ) ) )
=> ( p @ ( ap @ ( ap @ ( ap @ ( c_2Efinite__map_2Efmap__rel @ A_27a @ A_27b @ A_27c ) @ V0R ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EFUNION @ A_27c @ A_27a ) @ V1f1 ) @ V3f3 ) ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EFUNION @ A_27c @ A_27b ) @ V2f2 ) @ V4f4 ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Efinite__map_2Efmap__rel__FUPDATE__I,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) )
=> ! [V1f1: $i] :
( ( mem @ V1f1 @ ( ty_2Efinite__map_2Efmap @ A_27c @ A_27a ) )
=> ! [V2k: $i] :
( ( mem @ V2k @ A_27c )
=> ! [V3f2: $i] :
( ( mem @ V3f2 @ ( ty_2Efinite__map_2Efmap @ A_27c @ A_27b ) )
=> ! [V4v1: $i] :
( ( mem @ V4v1 @ A_27a )
=> ! [V5v2: $i] :
( ( mem @ V5v2 @ A_27b )
=> ( ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Efinite__map_2Efmap__rel @ A_27a @ A_27b @ A_27c ) @ V0R ) @ ( ap @ ( ap @ ( c_2Efinite__map_2Efdomsub @ A_27c @ A_27a ) @ V1f1 ) @ V2k ) ) @ ( ap @ ( ap @ ( c_2Efinite__map_2Efdomsub @ A_27c @ A_27b ) @ V3f2 ) @ V2k ) ) )
& ( p @ ( ap @ ( ap @ V0R @ V4v1 ) @ V5v2 ) ) )
=> ( p @ ( ap @ ( ap @ ( ap @ ( c_2Efinite__map_2Efmap__rel @ A_27a @ A_27b @ A_27c ) @ V0R ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EFUPDATE @ A_27c @ A_27a ) @ V1f1 ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27c @ A_27a ) @ V2k ) @ V4v1 ) ) ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EFUPDATE @ A_27c @ A_27b ) @ V3f2 ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27c @ A_27b ) @ V2k ) @ V5v2 ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Efinite__map_2Efmap__rel__mono,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0R1: $i] :
( ( mem @ V0R1 @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) )
=> ! [V1R2: $i] :
( ( mem @ V1R2 @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) )
=> ! [V2f1: $i] :
( ( mem @ V2f1 @ ( ty_2Efinite__map_2Efmap @ A_27c @ A_27a ) )
=> ! [V3f2: $i] :
( ( mem @ V3f2 @ ( ty_2Efinite__map_2Efmap @ A_27c @ A_27b ) )
=> ( ! [V4x: $i] :
( ( mem @ V4x @ A_27a )
=> ! [V5y: $i] :
( ( mem @ V5y @ A_27b )
=> ( ( p @ ( ap @ ( ap @ V0R1 @ V4x ) @ V5y ) )
=> ( p @ ( ap @ ( ap @ V1R2 @ V4x ) @ V5y ) ) ) ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Efinite__map_2Efmap__rel @ A_27a @ A_27b @ A_27c ) @ V0R1 ) @ V2f1 ) @ V3f2 ) )
=> ( p @ ( ap @ ( ap @ ( ap @ ( c_2Efinite__map_2Efmap__rel @ A_27a @ A_27b @ A_27c ) @ V1R2 ) @ V2f1 ) @ V3f2 ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Efinite__map_2Efmap__rel__OPTREL__FLOOKUP,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) )
=> ! [V1f1: $i] :
( ( mem @ V1f1 @ ( ty_2Efinite__map_2Efmap @ A_27c @ A_27a ) )
=> ! [V2f2: $i] :
( ( mem @ V2f2 @ ( ty_2Efinite__map_2Efmap @ A_27c @ A_27b ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Efinite__map_2Efmap__rel @ A_27a @ A_27b @ A_27c ) @ V0R ) @ V1f1 ) @ V2f2 ) )
<=> ! [V3k: $i] :
( ( mem @ V3k @ A_27c )
=> ( p @ ( ap @ ( ap @ ( ap @ ( c_2Eoption_2EOPTREL @ A_27a @ A_27b ) @ V0R ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EFLOOKUP @ A_27c @ A_27a ) @ V1f1 ) @ V3k ) ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EFLOOKUP @ A_27c @ A_27b ) @ V2f2 ) @ V3k ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Efinite__map_2Efmap__rel__FLOOKUP__imp,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) )
=> ! [V1f1: $i] :
( ( mem @ V1f1 @ ( ty_2Efinite__map_2Efmap @ A_27c @ A_27a ) )
=> ! [V2f2: $i] :
( ( mem @ V2f2 @ ( ty_2Efinite__map_2Efmap @ A_27c @ A_27b ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Efinite__map_2Efmap__rel @ A_27a @ A_27b @ A_27c ) @ V0R ) @ V1f1 ) @ V2f2 ) )
=> ( ! [V3k: $i] :
( ( mem @ V3k @ A_27c )
=> ( ( ( ap @ ( ap @ ( c_2Efinite__map_2EFLOOKUP @ A_27c @ A_27a ) @ V1f1 ) @ V3k )
= ( c_2Eoption_2ENONE @ A_27a ) )
=> ( ( ap @ ( ap @ ( c_2Efinite__map_2EFLOOKUP @ A_27c @ A_27b ) @ V2f2 ) @ V3k )
= ( c_2Eoption_2ENONE @ A_27b ) ) ) )
& ! [V4k: $i] :
( ( mem @ V4k @ A_27c )
=> ! [V5v1: $i] :
( ( mem @ V5v1 @ A_27a )
=> ( ( ( ap @ ( ap @ ( c_2Efinite__map_2EFLOOKUP @ A_27c @ A_27a ) @ V1f1 ) @ V4k )
= ( ap @ ( c_2Eoption_2ESOME @ A_27a ) @ V5v1 ) )
=> ? [V6v2: $i] :
( ( mem @ V6v2 @ A_27b )
& ( ( ap @ ( ap @ ( c_2Efinite__map_2EFLOOKUP @ A_27c @ A_27b ) @ V2f2 ) @ V4k )
= ( ap @ ( c_2Eoption_2ESOME @ A_27b ) @ V6v2 ) )
& ( p @ ( ap @ ( ap @ V0R @ V5v1 ) @ V6v2 ) ) ) ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Efinite__map_2Efmap__EQ__UPTO__def,axiom,
! [A_27a: del,A_27b: del,V0f1: $i] :
( ( mem @ V0f1 @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ! [V1f2: $i] :
( ( mem @ V1f2 @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ! [V2vs: $i] :
( ( mem @ V2vs @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Efinite__map_2Efmap__EQ__UPTO @ A_27a @ A_27b ) @ V0f1 ) @ V1f2 ) @ V2vs ) )
<=> ( ( ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ A_27a ) @ ( ap @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b ) @ V0f1 ) ) @ ( ap @ ( c_2Epred__set_2ECOMPL @ A_27a ) @ V2vs ) )
= ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ A_27a ) @ ( ap @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b ) @ V1f2 ) ) @ ( ap @ ( c_2Epred__set_2ECOMPL @ A_27a ) @ V2vs ) ) )
& ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3x ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ A_27a ) @ ( ap @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b ) @ V0f1 ) ) @ ( ap @ ( c_2Epred__set_2ECOMPL @ A_27a ) @ V2vs ) ) ) )
=> ( ( ap @ ( ap @ ( c_2Efinite__map_2EFAPPLY @ A_27a @ A_27b ) @ V0f1 ) @ V3x )
= ( ap @ ( ap @ ( c_2Efinite__map_2EFAPPLY @ A_27a @ A_27b ) @ V1f2 ) @ V3x ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Efinite__map_2Efmap__EQ__UPTO______EMPTY,axiom,
! [A_27a: del,A_27b: del,V0f1: $i] :
( ( mem @ V0f1 @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ! [V1f2: $i] :
( ( mem @ V1f2 @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Efinite__map_2Efmap__EQ__UPTO @ A_27a @ A_27b ) @ V0f1 ) @ V1f2 ) @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) )
<=> ( V0f1 = V1f2 ) ) ) ) ).
thf(conj_thm_2Efinite__map_2Efmap__EQ__UPTO______EQ,axiom,
! [A_27a: del,A_27b: del,V0vs: $i] :
( ( mem @ V0vs @ ( arr @ A_27a @ bool ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ( p @ ( ap @ ( ap @ ( ap @ ( c_2Efinite__map_2Efmap__EQ__UPTO @ A_27a @ A_27b ) @ V1f ) @ V1f ) @ V0vs ) ) ) ) ).
thf(conj_thm_2Efinite__map_2Efmap__EQ__UPTO______FUPDATE__BOTH,axiom,
! [A_27a: del,A_27b: del,V0f1: $i] :
( ( mem @ V0f1 @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ! [V1f2: $i] :
( ( mem @ V1f2 @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ! [V2ks: $i] :
( ( mem @ V2ks @ ( arr @ A_27a @ bool ) )
=> ! [V3k: $i] :
( ( mem @ V3k @ A_27a )
=> ! [V4v: $i] :
( ( mem @ V4v @ A_27b )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Efinite__map_2Efmap__EQ__UPTO @ A_27a @ A_27b ) @ V0f1 ) @ V1f2 ) @ V2ks ) )
=> ( p @ ( ap @ ( ap @ ( ap @ ( c_2Efinite__map_2Efmap__EQ__UPTO @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EFUPDATE @ A_27a @ A_27b ) @ V0f1 ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27b ) @ V3k ) @ V4v ) ) ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EFUPDATE @ A_27a @ A_27b ) @ V1f2 ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27b ) @ V3k ) @ V4v ) ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EDELETE @ A_27a ) @ V2ks ) @ V3k ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Efinite__map_2Efmap__EQ__UPTO______FUPDATE__BOTH______NO__DELETE,axiom,
! [A_27a: del,A_27b: del,V0f1: $i] :
( ( mem @ V0f1 @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ! [V1f2: $i] :
( ( mem @ V1f2 @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ! [V2ks: $i] :
( ( mem @ V2ks @ ( arr @ A_27a @ bool ) )
=> ! [V3k: $i] :
( ( mem @ V3k @ A_27a )
=> ! [V4v: $i] :
( ( mem @ V4v @ A_27b )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Efinite__map_2Efmap__EQ__UPTO @ A_27a @ A_27b ) @ V0f1 ) @ V1f2 ) @ V2ks ) )
=> ( p @ ( ap @ ( ap @ ( ap @ ( c_2Efinite__map_2Efmap__EQ__UPTO @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EFUPDATE @ A_27a @ A_27b ) @ V0f1 ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27b ) @ V3k ) @ V4v ) ) ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EFUPDATE @ A_27a @ A_27b ) @ V1f2 ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27b ) @ V3k ) @ V4v ) ) ) @ V2ks ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Efinite__map_2Efmap__EQ__UPTO______FUPDATE__SING,axiom,
! [A_27a: del,A_27b: del,V0f1: $i] :
( ( mem @ V0f1 @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ! [V1f2: $i] :
( ( mem @ V1f2 @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ! [V2ks: $i] :
( ( mem @ V2ks @ ( arr @ A_27a @ bool ) )
=> ! [V3k: $i] :
( ( mem @ V3k @ A_27a )
=> ! [V4v: $i] :
( ( mem @ V4v @ A_27b )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Efinite__map_2Efmap__EQ__UPTO @ A_27a @ A_27b ) @ V0f1 ) @ V1f2 ) @ V2ks ) )
=> ( p @ ( ap @ ( ap @ ( ap @ ( c_2Efinite__map_2Efmap__EQ__UPTO @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EFUPDATE @ A_27a @ A_27b ) @ V0f1 ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27b ) @ V3k ) @ V4v ) ) ) @ V1f2 ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ A_27a ) @ V3k ) @ V2ks ) ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Efinite__map_2Efmap__size__def,axiom,
! [A_27a: del,A_27b: del,V0kz: $i] :
( ( mem @ V0kz @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V1vz: $i] :
( ( mem @ V1vz @ ( arr @ A_27b @ ty_2Enum_2Enum ) )
=> ! [V2fm: $i] :
( ( mem @ V2fm @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ ( ap @ ( c_2Efinite__map_2Efmap__size @ A_27a @ A_27b ) @ V0kz ) @ V1vz ) @ V2fm ) )
= ( surj__ty_2Enum_2Enum
@ ( ap
@ ( ap @ ( c_2Epred__set_2ESUM__IMAGE @ A_27a )
@ ( lam @ A_27a
@ ^ [V3k: $i] : ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( ap @ V0kz @ V3k ) ) @ ( ap @ V1vz @ ( ap @ ( ap @ ( c_2Efinite__map_2EFAPPLY @ A_27a @ A_27b ) @ V2fm ) @ V3k ) ) ) ) )
@ ( ap @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b ) @ V2fm ) ) ) ) ) ) ) ).
thf(conj_thm_2Efinite__map_2Eo__f__FUNION,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27c @ A_27b ) )
=> ! [V1f1: $i] :
( ( mem @ V1f1 @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27c ) )
=> ! [V2f2: $i] :
( ( mem @ V2f2 @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27c ) )
=> ( ( ap @ ( ap @ ( c_2Efinite__map_2Eo__f @ A_27a @ A_27c @ A_27b ) @ V0f ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EFUNION @ A_27a @ A_27c ) @ V1f1 ) @ V2f2 ) )
= ( ap @ ( ap @ ( c_2Efinite__map_2EFUNION @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Efinite__map_2Eo__f @ A_27a @ A_27c @ A_27b ) @ V0f ) @ V1f1 ) ) @ ( ap @ ( ap @ ( c_2Efinite__map_2Eo__f @ A_27a @ A_27c @ A_27b ) @ V0f ) @ V2f2 ) ) ) ) ) ) ).
thf(conj_thm_2Efinite__map_2EFDOM__FOLDR__DOMSUB,axiom,
! [A_27a: del,A_27b: del,V0ls: $i] :
( ( mem @ V0ls @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V1fm: $i] :
( ( mem @ V1fm @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ( ( ap @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b )
@ ( ap
@ ( ap
@ ( ap @ ( c_2Elist_2EFOLDR @ A_27a @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
@ ( lam @ A_27a
@ ^ [V2k: $i] :
( lam @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b )
@ ^ [V3m: $i] : ( ap @ ( ap @ ( c_2Efinite__map_2Efdomsub @ A_27a @ A_27b ) @ V3m ) @ V2k ) ) ) )
@ V1fm )
@ V0ls ) )
= ( ap @ ( ap @ ( c_2Epred__set_2EDIFF @ A_27a ) @ ( ap @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b ) @ V1fm ) ) @ ( ap @ ( c_2Elist_2ELIST__TO__SET @ A_27a ) @ V0ls ) ) ) ) ) ).
thf(conj_thm_2Efinite__map_2EFEVERY__SUBMAP,axiom,
! [A_27a: del,A_27b: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ bool ) )
=> ! [V1fm: $i] :
( ( mem @ V1fm @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ! [V2fm0: $i] :
( ( mem @ V2fm0 @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Efinite__map_2EFEVERY @ A_27a @ A_27b ) @ V0P ) @ V1fm ) )
& ( p @ ( ap @ ( ap @ ( c_2Efinite__map_2ESUBMAP @ A_27a @ A_27b ) @ V2fm0 ) @ V1fm ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Efinite__map_2EFEVERY @ A_27a @ A_27b ) @ V0P ) @ V2fm0 ) ) ) ) ) ) ).
thf(conj_thm_2Efinite__map_2EFEVERY__ALL__FLOOKUP,axiom,
! [A_27a: del,A_27b: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ bool ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Efinite__map_2EFEVERY @ A_27a @ A_27b ) @ V0P ) @ V1f ) )
<=> ! [V2k: $i] :
( ( mem @ V2k @ A_27a )
=> ! [V3v: $i] :
( ( mem @ V3v @ A_27b )
=> ( ( ( ap @ ( ap @ ( c_2Efinite__map_2EFLOOKUP @ A_27a @ A_27b ) @ V1f ) @ V2k )
= ( ap @ ( c_2Eoption_2ESOME @ A_27b ) @ V3v ) )
=> ( p @ ( ap @ V0P @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27b ) @ V2k ) @ V3v ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Efinite__map_2EFUPDATE__LIST__CANCEL,axiom,
! [A_27a: del,A_27b: del,V0ls1: $i] :
( ( mem @ V0ls1 @ ( ty_2Elist_2Elist @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) ) )
=> ! [V1fm: $i] :
( ( mem @ V1fm @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ! [V2ls2: $i] :
( ( mem @ V2ls2 @ ( ty_2Elist_2Elist @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) ) )
=> ( ! [V3k: $i] :
( ( mem @ V3k @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3k ) @ ( ap @ ( c_2Elist_2ELIST__TO__SET @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2EMAP @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ A_27a ) @ ( c_2Epair_2EFST @ A_27a @ A_27b ) ) @ V0ls1 ) ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3k ) @ ( ap @ ( c_2Elist_2ELIST__TO__SET @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2EMAP @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ A_27a ) @ ( c_2Epair_2EFST @ A_27a @ A_27b ) ) @ V2ls2 ) ) ) ) ) )
=> ( ( ap @ ( ap @ ( c_2Efinite__map_2EFUPDATE__LIST @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EFUPDATE__LIST @ A_27a @ A_27b ) @ V1fm ) @ V0ls1 ) ) @ V2ls2 )
= ( ap @ ( ap @ ( c_2Efinite__map_2EFUPDATE__LIST @ A_27a @ A_27b ) @ V1fm ) @ V2ls2 ) ) ) ) ) ) ).
thf(conj_thm_2Efinite__map_2EFUPDATE__EQ__FUNION,axiom,
! [A_27a: del,A_27b: del,V0fm: $i] :
( ( mem @ V0fm @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ! [V1kv: $i] :
( ( mem @ V1kv @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) )
=> ( ( ap @ ( ap @ ( c_2Efinite__map_2EFUPDATE @ A_27a @ A_27b ) @ V0fm ) @ V1kv )
= ( ap @ ( ap @ ( c_2Efinite__map_2EFUNION @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EFUPDATE @ A_27a @ A_27b ) @ ( c_2Efinite__map_2EFEMPTY @ A_27a @ A_27b ) ) @ V1kv ) ) @ V0fm ) ) ) ) ).
thf(conj_thm_2Efinite__map_2EFUPDATE__LIST__APPEND__COMMUTES,axiom,
! [A_27a: del,A_27b: del,V0l1: $i] :
( ( mem @ V0l1 @ ( ty_2Elist_2Elist @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) ) )
=> ! [V1l2: $i] :
( ( mem @ V1l2 @ ( ty_2Elist_2Elist @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) ) )
=> ! [V2fm: $i] :
( ( mem @ V2fm @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2EDISJOINT @ A_27a ) @ ( ap @ ( c_2Elist_2ELIST__TO__SET @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2EMAP @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ A_27a ) @ ( c_2Epair_2EFST @ A_27a @ A_27b ) ) @ V0l1 ) ) ) @ ( ap @ ( c_2Elist_2ELIST__TO__SET @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2EMAP @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ A_27a ) @ ( c_2Epair_2EFST @ A_27a @ A_27b ) ) @ V1l2 ) ) ) )
=> ( ( ap @ ( ap @ ( c_2Efinite__map_2EFUPDATE__LIST @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EFUPDATE__LIST @ A_27a @ A_27b ) @ V2fm ) @ V0l1 ) ) @ V1l2 )
= ( ap @ ( ap @ ( c_2Efinite__map_2EFUPDATE__LIST @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EFUPDATE__LIST @ A_27a @ A_27b ) @ V2fm ) @ V1l2 ) ) @ V0l1 ) ) ) ) ) ) ).
thf(conj_thm_2Efinite__map_2EIMAGE__FRANGE,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27b ) )
=> ! [V1fm: $i] :
( ( mem @ V1fm @ ( ty_2Efinite__map_2Efmap @ A_27c @ A_27a ) )
=> ( ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27b ) @ V0f ) @ ( ap @ ( c_2Efinite__map_2EFRANGE @ A_27c @ A_27a ) @ V1fm ) )
= ( ap @ ( c_2Efinite__map_2EFRANGE @ A_27c @ A_27b ) @ ( ap @ ( ap @ ( c_2Efinite__map_2Eo__f @ A_27c @ A_27a @ A_27b ) @ V0f ) @ V1fm ) ) ) ) ) ).
thf(conj_thm_2Efinite__map_2ESUBMAP__mono__FUPDATE,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ! [V3y: $i] :
( ( mem @ V3y @ A_27b )
=> ( ( p @ ( ap @ ( ap @ ( c_2Efinite__map_2ESUBMAP @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Efinite__map_2Efdomsub @ A_27a @ A_27b ) @ V0f ) @ V2x ) ) @ ( ap @ ( ap @ ( c_2Efinite__map_2Efdomsub @ A_27a @ A_27b ) @ V1g ) @ V2x ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Efinite__map_2ESUBMAP @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EFUPDATE @ A_27a @ A_27b ) @ V0f ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27b ) @ V2x ) @ V3y ) ) ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EFUPDATE @ A_27a @ A_27b ) @ V1g ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27b ) @ V2x ) @ V3y ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Efinite__map_2ESUBMAP__DOMSUB__gen,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ! [V2k: $i] :
( ( mem @ V2k @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Efinite__map_2ESUBMAP @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Efinite__map_2Efdomsub @ A_27a @ A_27b ) @ V0f ) @ V2k ) ) @ V1g ) )
<=> ( p @ ( ap @ ( ap @ ( c_2Efinite__map_2ESUBMAP @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Efinite__map_2Efdomsub @ A_27a @ A_27b ) @ V0f ) @ V2k ) ) @ ( ap @ ( ap @ ( c_2Efinite__map_2Efdomsub @ A_27a @ A_27b ) @ V1g ) @ V2k ) ) ) ) ) ) ) ).
thf(conj_thm_2Efinite__map_2EDOMSUB__SUBMAP,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Efinite__map_2ESUBMAP @ A_27a @ A_27b ) @ V0f ) @ V1g ) )
& ~ ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x ) @ ( ap @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b ) @ V0f ) ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Efinite__map_2ESUBMAP @ A_27a @ A_27b ) @ V0f ) @ ( ap @ ( ap @ ( c_2Efinite__map_2Efdomsub @ A_27a @ A_27b ) @ V1g ) @ V2x ) ) ) ) ) ) ) ).
thf(conj_thm_2Efinite__map_2EDRESTRICT__DOMSUB,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ! [V2k: $i] :
( ( mem @ V2k @ A_27a )
=> ( ( ap @ ( ap @ ( c_2Efinite__map_2Efdomsub @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EDRESTRICT @ A_27a @ A_27b ) @ V0f ) @ V1s ) ) @ V2k )
= ( ap @ ( ap @ ( c_2Efinite__map_2EDRESTRICT @ A_27a @ A_27b ) @ V0f ) @ ( ap @ ( ap @ ( c_2Epred__set_2EDELETE @ A_27a ) @ V1s ) @ V2k ) ) ) ) ) ) ).
thf(conj_thm_2Efinite__map_2EDRESTRICT__SUBSET__SUBMAP__gen,axiom,
! [A_27a: del,A_27b: del,V0f1: $i] :
( ( mem @ V0f1 @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ! [V1f2: $i] :
( ( mem @ V1f2 @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ A_27a @ bool ) )
=> ! [V3t: $i] :
( ( mem @ V3t @ ( arr @ A_27a @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Efinite__map_2ESUBMAP @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EDRESTRICT @ A_27a @ A_27b ) @ V0f1 ) @ V2s ) ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EDRESTRICT @ A_27a @ A_27b ) @ V1f2 ) @ V2s ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ V3t ) @ V2s ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Efinite__map_2ESUBMAP @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EDRESTRICT @ A_27a @ A_27b ) @ V0f1 ) @ V3t ) ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EDRESTRICT @ A_27a @ A_27b ) @ V1f2 ) @ V3t ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Efinite__map_2EDRESTRICT__FUNION__SAME,axiom,
! [A_27a: del,A_27b: del,V0fm: $i] :
( ( mem @ V0fm @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ( ( ap @ ( ap @ ( c_2Efinite__map_2EFUNION @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EDRESTRICT @ A_27a @ A_27b ) @ V0fm ) @ V1s ) ) @ V0fm )
= V0fm ) ) ) ).
thf(conj_thm_2Efinite__map_2EDRESTRICT__EQ__DRESTRICT__SAME,axiom,
! [A_27a: del,A_27b: del,V0f1: $i] :
( ( mem @ V0f1 @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ! [V2f2: $i] :
( ( mem @ V2f2 @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ( ( ( ap @ ( ap @ ( c_2Efinite__map_2EDRESTRICT @ A_27a @ A_27b ) @ V0f1 ) @ V1s )
= ( ap @ ( ap @ ( c_2Efinite__map_2EDRESTRICT @ A_27a @ A_27b ) @ V2f2 ) @ V1s ) )
<=> ( ( ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ A_27a ) @ V1s ) @ ( ap @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b ) @ V0f1 ) )
= ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ A_27a ) @ V1s ) @ ( ap @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b ) @ V2f2 ) ) )
& ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3x ) @ ( ap @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b ) @ V0f1 ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3x ) @ V1s ) ) )
=> ( ( ap @ ( ap @ ( c_2Efinite__map_2EFAPPLY @ A_27a @ A_27b ) @ V0f1 ) @ V3x )
= ( ap @ ( ap @ ( c_2Efinite__map_2EFAPPLY @ A_27a @ A_27b ) @ V2f2 ) @ V3x ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Efinite__map_2EFOLDL2__FUPDATE__LIST,axiom,
! [A_27a: del,A_27b: del,A_27c: del,A_27d: del,V0f1: $i] :
( ( mem @ V0f1 @ ( arr @ A_27a @ ( arr @ A_27b @ A_27c ) ) )
=> ! [V1f2: $i] :
( ( mem @ V1f2 @ ( arr @ A_27a @ ( arr @ A_27b @ A_27d ) ) )
=> ! [V2bs: $i] :
( ( mem @ V2bs @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V3cs: $i] :
( ( mem @ V3cs @ ( ty_2Elist_2Elist @ A_27b ) )
=> ! [V4a: $i] :
( ( mem @ V4a @ ( ty_2Efinite__map_2Efmap @ A_27c @ A_27d ) )
=> ( ( ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ V2bs ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Elist_2ELENGTH @ A_27b ) @ V3cs ) ) )
=> ( ( ap
@ ( ap
@ ( ap
@ ( ap @ ( c_2Elist_2EFOLDL2 @ ( ty_2Efinite__map_2Efmap @ A_27c @ A_27d ) @ A_27a @ A_27b )
@ ( lam @ ( ty_2Efinite__map_2Efmap @ A_27c @ A_27d )
@ ^ [V5fm: $i] :
( lam @ A_27a
@ ^ [V6b: $i] :
( lam @ A_27b
@ ^ [V7c: $i] : ( ap @ ( ap @ ( c_2Efinite__map_2EFUPDATE @ A_27c @ A_27d ) @ V5fm ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27c @ A_27d ) @ ( ap @ ( ap @ V0f1 @ V6b ) @ V7c ) ) @ ( ap @ ( ap @ V1f2 @ V6b ) @ V7c ) ) ) ) ) ) )
@ V4a )
@ V2bs )
@ V3cs )
= ( ap @ ( ap @ ( c_2Efinite__map_2EFUPDATE__LIST @ A_27c @ A_27d ) @ V4a ) @ ( ap @ ( c_2Elist_2EZIP @ A_27c @ A_27d ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Elist_2Elist @ A_27c ) @ ( ty_2Elist_2Elist @ A_27d ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Elist_2EMAP2 @ A_27c @ A_27a @ A_27b ) @ V0f1 ) @ V2bs ) @ V3cs ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Elist_2EMAP2 @ A_27d @ A_27a @ A_27b ) @ V1f2 ) @ V2bs ) @ V3cs ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Efinite__map_2EFOLDL2__FUPDATE__LIST__paired,axiom,
! [A_27a: del,A_27b: del,A_27c: del,A_27d: del,A_27e: del,V0f1: $i] :
( ( mem @ V0f1 @ ( arr @ A_27a @ ( arr @ A_27b @ ( arr @ A_27c @ A_27d ) ) ) )
=> ! [V1f2: $i] :
( ( mem @ V1f2 @ ( arr @ A_27a @ ( arr @ A_27b @ ( arr @ A_27c @ A_27e ) ) ) )
=> ! [V2bs: $i] :
( ( mem @ V2bs @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V3cs: $i] :
( ( mem @ V3cs @ ( ty_2Elist_2Elist @ ( ty_2Epair_2Eprod @ A_27b @ A_27c ) ) )
=> ! [V4a: $i] :
( ( mem @ V4a @ ( ty_2Efinite__map_2Efmap @ A_27d @ A_27e ) )
=> ( ( ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ V2bs ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Elist_2ELENGTH @ ( ty_2Epair_2Eprod @ A_27b @ A_27c ) ) @ V3cs ) ) )
=> ( ( ap
@ ( ap
@ ( ap
@ ( ap @ ( c_2Elist_2EFOLDL2 @ ( ty_2Efinite__map_2Efmap @ A_27d @ A_27e ) @ A_27a @ ( ty_2Epair_2Eprod @ A_27b @ A_27c ) )
@ ( lam @ ( ty_2Efinite__map_2Efmap @ A_27d @ A_27e )
@ ^ [V5fm: $i] :
( lam @ A_27a
@ ^ [V6b: $i] :
( ap @ ( c_2Epair_2EUNCURRY @ A_27b @ A_27c @ ( ty_2Efinite__map_2Efmap @ A_27d @ A_27e ) )
@ ( lam @ A_27b
@ ^ [V7c: $i] :
( lam @ A_27c
@ ^ [V8d: $i] : ( ap @ ( ap @ ( c_2Efinite__map_2EFUPDATE @ A_27d @ A_27e ) @ V5fm ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27d @ A_27e ) @ ( ap @ ( ap @ ( ap @ V0f1 @ V6b ) @ V7c ) @ V8d ) ) @ ( ap @ ( ap @ ( ap @ V1f2 @ V6b ) @ V7c ) @ V8d ) ) ) ) ) ) ) ) )
@ V4a )
@ V2bs )
@ V3cs )
= ( ap @ ( ap @ ( c_2Efinite__map_2EFUPDATE__LIST @ A_27d @ A_27e ) @ V4a )
@ ( ap @ ( c_2Elist_2EZIP @ A_27d @ A_27e )
@ ( ap
@ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Elist_2Elist @ A_27d ) @ ( ty_2Elist_2Elist @ A_27e ) )
@ ( ap
@ ( ap
@ ( ap @ ( c_2Elist_2EMAP2 @ A_27d @ A_27a @ ( ty_2Epair_2Eprod @ A_27b @ A_27c ) )
@ ( lam @ A_27a
@ ^ [V9b: $i] : ( ap @ ( c_2Epair_2EUNCURRY @ A_27b @ A_27c @ A_27d ) @ ( ap @ V0f1 @ V9b ) ) ) )
@ V2bs )
@ V3cs ) )
@ ( ap
@ ( ap
@ ( ap @ ( c_2Elist_2EMAP2 @ A_27e @ A_27a @ ( ty_2Epair_2Eprod @ A_27b @ A_27c ) )
@ ( lam @ A_27a
@ ^ [V10b: $i] : ( ap @ ( c_2Epair_2EUNCURRY @ A_27b @ A_27c @ A_27e ) @ ( ap @ V1f2 @ V10b ) ) ) )
@ V2bs )
@ V3cs ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Efinite__map_2EDRESTRICT__FUNION__SUBSET,axiom,
! [A_27a: del,A_27b: del,V0s2: $i] :
( ( mem @ V0s2 @ ( arr @ A_27a @ bool ) )
=> ! [V1s1: $i] :
( ( mem @ V1s1 @ ( arr @ A_27a @ bool ) )
=> ! [V2f: $i] :
( ( mem @ V2f @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ! [V3g: $i] :
( ( mem @ V3g @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ V0s2 ) @ V1s1 ) )
=> ? [V4h: $i] :
( ( mem @ V4h @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
& ( ( ap @ ( ap @ ( c_2Efinite__map_2EFUNION @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EDRESTRICT @ A_27a @ A_27b ) @ V2f ) @ V1s1 ) ) @ V3g )
= ( ap @ ( ap @ ( c_2Efinite__map_2EFUNION @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EDRESTRICT @ A_27a @ A_27b ) @ V2f ) @ V0s2 ) ) @ V4h ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Efinite__map_2EFUPDATE__LIST__APPLY__NOT__MEM__matchable,axiom,
! [A_27a: del,A_27b: del,V0kvl: $i] :
( ( mem @ V0kvl @ ( ty_2Elist_2Elist @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ! [V2k: $i] :
( ( mem @ V2k @ A_27a )
=> ! [V3v: $i] :
( ( mem @ V3v @ A_27b )
=> ( ( ~ ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2k ) @ ( ap @ ( c_2Elist_2ELIST__TO__SET @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2EMAP @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ A_27a ) @ ( c_2Epair_2EFST @ A_27a @ A_27b ) ) @ V0kvl ) ) ) )
& ( V3v
= ( ap @ ( ap @ ( c_2Efinite__map_2EFAPPLY @ A_27a @ A_27b ) @ V1f ) @ V2k ) ) )
=> ( ( ap @ ( ap @ ( c_2Efinite__map_2EFAPPLY @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EFUPDATE__LIST @ A_27a @ A_27b ) @ V1f ) @ V0kvl ) ) @ V2k )
= V3v ) ) ) ) ) ) ).
thf(conj_thm_2Efinite__map_2EFUPDATE__LIST__APPLY__HO__THM,axiom,
! [A_27a: del,A_27b: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( ty_2Efinite__map_2Efmap @ A_27b @ A_27a ) )
=> ! [V2kvl: $i] :
( ( mem @ V2kvl @ ( ty_2Elist_2Elist @ ( ty_2Epair_2Eprod @ A_27b @ A_27a ) ) )
=> ! [V3k: $i] :
( ( mem @ V3k @ A_27b )
=> ( ( ? [V4n: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V4n ) ) @ ( ap @ ( c_2Elist_2ELENGTH @ ( ty_2Epair_2Eprod @ A_27b @ A_27a ) ) @ V2kvl ) ) )
& ( V3k
= ( ap @ ( ap @ ( c_2Elist_2EEL @ A_27b ) @ ( inj__ty_2Enum_2Enum @ V4n ) ) @ ( ap @ ( ap @ ( c_2Elist_2EMAP @ ( ty_2Epair_2Eprod @ A_27b @ A_27a ) @ A_27b ) @ ( c_2Epair_2EFST @ A_27b @ A_27a ) ) @ V2kvl ) ) )
& ( p @ ( ap @ V0P @ ( ap @ ( ap @ ( c_2Elist_2EEL @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V4n ) ) @ ( ap @ ( ap @ ( c_2Elist_2EMAP @ ( ty_2Epair_2Eprod @ A_27b @ A_27a ) @ A_27a ) @ ( c_2Epair_2ESND @ A_27b @ A_27a ) ) @ V2kvl ) ) ) )
& ! [V5m: tp__ty_2Enum_2Enum] :
( ( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V4n ) ) @ ( inj__ty_2Enum_2Enum @ V5m ) ) )
& ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V5m ) ) @ ( ap @ ( c_2Elist_2ELENGTH @ ( ty_2Epair_2Eprod @ A_27b @ A_27a ) ) @ V2kvl ) ) ) )
=> ( ( ap @ ( ap @ ( c_2Elist_2EEL @ A_27b ) @ ( inj__ty_2Enum_2Enum @ V5m ) ) @ ( ap @ ( ap @ ( c_2Elist_2EMAP @ ( ty_2Epair_2Eprod @ A_27b @ A_27a ) @ A_27b ) @ ( c_2Epair_2EFST @ A_27b @ A_27a ) ) @ V2kvl ) )
!= V3k ) ) )
| ( ~ ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27b ) @ V3k ) @ ( ap @ ( c_2Elist_2ELIST__TO__SET @ A_27b ) @ ( ap @ ( ap @ ( c_2Elist_2EMAP @ ( ty_2Epair_2Eprod @ A_27b @ A_27a ) @ A_27b ) @ ( c_2Epair_2EFST @ A_27b @ A_27a ) ) @ V2kvl ) ) ) )
& ( p @ ( ap @ V0P @ ( ap @ ( ap @ ( c_2Efinite__map_2EFAPPLY @ A_27b @ A_27a ) @ V1f ) @ V3k ) ) ) ) )
=> ( p @ ( ap @ V0P @ ( ap @ ( ap @ ( c_2Efinite__map_2EFAPPLY @ A_27b @ A_27a ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EFUPDATE__LIST @ A_27b @ A_27a ) @ V1f ) @ V2kvl ) ) @ V3k ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Efinite__map_2EFUPDATE__SAME__APPLY,axiom,
! [A_27a: del,A_27b: del,V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ! [V1kv: $i] :
( ( mem @ V1kv @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) )
=> ! [V2fm1: $i] :
( ( mem @ V2fm1 @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ! [V3fm2: $i] :
( ( mem @ V3fm2 @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ( ( ( V0x
= ( ap @ ( c_2Epair_2EFST @ A_27a @ A_27b ) @ V1kv ) )
| ( ( ap @ ( ap @ ( c_2Efinite__map_2EFAPPLY @ A_27a @ A_27b ) @ V2fm1 ) @ V0x )
= ( ap @ ( ap @ ( c_2Efinite__map_2EFAPPLY @ A_27a @ A_27b ) @ V3fm2 ) @ V0x ) ) )
=> ( ( ap @ ( ap @ ( c_2Efinite__map_2EFAPPLY @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EFUPDATE @ A_27a @ A_27b ) @ V2fm1 ) @ V1kv ) ) @ V0x )
= ( ap @ ( ap @ ( c_2Efinite__map_2EFAPPLY @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EFUPDATE @ A_27a @ A_27b ) @ V3fm2 ) @ V1kv ) ) @ V0x ) ) ) ) ) ) ) ).
thf(conj_thm_2Efinite__map_2EFUPDATE__SAME__LIST__APPLY,axiom,
! [A_27a: del,A_27b: del,V0kvl: $i] :
( ( mem @ V0kvl @ ( ty_2Elist_2Elist @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) ) )
=> ! [V1fm1: $i] :
( ( mem @ V1fm1 @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ! [V2fm2: $i] :
( ( mem @ V2fm2 @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3x ) @ ( ap @ ( c_2Elist_2ELIST__TO__SET @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2EMAP @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ A_27a ) @ ( c_2Epair_2EFST @ A_27a @ A_27b ) ) @ V0kvl ) ) ) )
=> ( ( ap @ ( ap @ ( c_2Efinite__map_2EFAPPLY @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EFUPDATE__LIST @ A_27a @ A_27b ) @ V1fm1 ) @ V0kvl ) ) @ V3x )
= ( ap @ ( ap @ ( c_2Efinite__map_2EFAPPLY @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EFUPDATE__LIST @ A_27a @ A_27b ) @ V2fm2 ) @ V0kvl ) ) @ V3x ) ) ) ) ) ) ) ).
thf(conj_thm_2Efinite__map_2EFUPDATE__LIST__ALL__DISTINCT__APPLY__MEM,axiom,
! [A_27a: del,A_27b: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ! [V1ls: $i] :
( ( mem @ V1ls @ ( ty_2Elist_2Elist @ ( ty_2Epair_2Eprod @ A_27b @ A_27a ) ) )
=> ! [V2k: $i] :
( ( mem @ V2k @ A_27b )
=> ! [V3v: $i] :
( ( mem @ V3v @ A_27a )
=> ! [V4fm: $i] :
( ( mem @ V4fm @ ( ty_2Efinite__map_2Efmap @ A_27b @ A_27a ) )
=> ( ( ( p @ ( ap @ ( c_2Elist_2EALL__DISTINCT @ A_27b ) @ ( ap @ ( ap @ ( c_2Elist_2EMAP @ ( ty_2Epair_2Eprod @ A_27b @ A_27a ) @ A_27b ) @ ( c_2Epair_2EFST @ A_27b @ A_27a ) ) @ V1ls ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27b @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27b @ A_27a ) @ V2k ) @ V3v ) ) @ ( ap @ ( c_2Elist_2ELIST__TO__SET @ ( ty_2Epair_2Eprod @ A_27b @ A_27a ) ) @ V1ls ) ) )
& ( p @ ( ap @ V0P @ V3v ) ) )
=> ( p @ ( ap @ V0P @ ( ap @ ( ap @ ( c_2Efinite__map_2EFAPPLY @ A_27b @ A_27a ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EFUPDATE__LIST @ A_27b @ A_27a ) @ V4fm ) @ V1ls ) ) @ V2k ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Efinite__map_2EFUPDATE__LIST__ALL__DISTINCT__REVERSE,axiom,
! [A_27a: del,A_27b: del,V0ls: $i] :
( ( mem @ V0ls @ ( ty_2Elist_2Elist @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) ) )
=> ( ( p @ ( ap @ ( c_2Elist_2EALL__DISTINCT @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2EMAP @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ A_27a ) @ ( c_2Epair_2EFST @ A_27a @ A_27b ) ) @ V0ls ) ) )
=> ! [V1fm: $i] :
( ( mem @ V1fm @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ( ( ap @ ( ap @ ( c_2Efinite__map_2EFUPDATE__LIST @ A_27a @ A_27b ) @ V1fm ) @ ( ap @ ( c_2Elist_2EREVERSE @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) ) @ V0ls ) )
= ( ap @ ( ap @ ( c_2Efinite__map_2EFUPDATE__LIST @ A_27a @ A_27b ) @ V1fm ) @ V0ls ) ) ) ) ) ).
thf(conj_thm_2Efinite__map_2EIN__FRANGE,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ! [V1v: $i] :
( ( mem @ V1v @ A_27b )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27b ) @ V1v ) @ ( ap @ ( c_2Efinite__map_2EFRANGE @ A_27a @ A_27b ) @ V0f ) ) )
<=> ? [V2k: $i] :
( ( mem @ V2k @ A_27a )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2k ) @ ( ap @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b ) @ V0f ) ) )
& ( ( ap @ ( ap @ ( c_2Efinite__map_2EFAPPLY @ A_27a @ A_27b ) @ V0f ) @ V2k )
= V1v ) ) ) ) ) ).
thf(conj_thm_2Efinite__map_2EIN__FRANGE__FLOOKUP,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ! [V1v: $i] :
( ( mem @ V1v @ A_27b )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27b ) @ V1v ) @ ( ap @ ( c_2Efinite__map_2EFRANGE @ A_27a @ A_27b ) @ V0f ) ) )
<=> ? [V2k: $i] :
( ( mem @ V2k @ A_27a )
& ( ( ap @ ( ap @ ( c_2Efinite__map_2EFLOOKUP @ A_27a @ A_27b ) @ V0f ) @ V2k )
= ( ap @ ( c_2Eoption_2ESOME @ A_27b ) @ V1v ) ) ) ) ) ) ).
thf(conj_thm_2Efinite__map_2EFRANGE__FUPDATE__LIST__SUBSET,axiom,
! [A_27a: del,A_27b: del,V0ls: $i] :
( ( mem @ V0ls @ ( ty_2Elist_2Elist @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) ) )
=> ! [V1fm: $i] :
( ( mem @ V1fm @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27b ) @ ( ap @ ( c_2Efinite__map_2EFRANGE @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EFUPDATE__LIST @ A_27a @ A_27b ) @ V1fm ) @ V0ls ) ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ A_27b ) @ ( ap @ ( c_2Efinite__map_2EFRANGE @ A_27a @ A_27b ) @ V1fm ) ) @ ( ap @ ( c_2Elist_2ELIST__TO__SET @ A_27b ) @ ( ap @ ( ap @ ( c_2Elist_2EMAP @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ A_27b ) @ ( c_2Epair_2ESND @ A_27a @ A_27b ) ) @ V0ls ) ) ) ) ) ) ) ).
thf(conj_thm_2Efinite__map_2EIN__FRANGE__FUPDATE__LIST__suff,axiom,
! [A_27a: del,A_27b: del,V0fm: $i] :
( ( mem @ V0fm @ ( ty_2Efinite__map_2Efmap @ A_27b @ A_27a ) )
=> ! [V1P: $i] :
( ( mem @ V1P @ ( arr @ A_27a @ bool ) )
=> ! [V2ls: $i] :
( ( mem @ V2ls @ ( ty_2Elist_2Elist @ ( ty_2Epair_2Eprod @ A_27b @ A_27a ) ) )
=> ( ( ! [V3v: $i] :
( ( mem @ V3v @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3v ) @ ( ap @ ( c_2Efinite__map_2EFRANGE @ A_27b @ A_27a ) @ V0fm ) ) )
=> ( p @ ( ap @ V1P @ V3v ) ) ) )
& ! [V4v: $i] :
( ( mem @ V4v @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V4v ) @ ( ap @ ( c_2Elist_2ELIST__TO__SET @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2EMAP @ ( ty_2Epair_2Eprod @ A_27b @ A_27a ) @ A_27a ) @ ( c_2Epair_2ESND @ A_27b @ A_27a ) ) @ V2ls ) ) ) )
=> ( p @ ( ap @ V1P @ V4v ) ) ) ) )
=> ! [V5v: $i] :
( ( mem @ V5v @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V5v ) @ ( ap @ ( c_2Efinite__map_2EFRANGE @ A_27b @ A_27a ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EFUPDATE__LIST @ A_27b @ A_27a ) @ V0fm ) @ V2ls ) ) ) )
=> ( p @ ( ap @ V1P @ V5v ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Efinite__map_2EFRANGE__FUNION__SUBSET,axiom,
! [A_27a: del,A_27b: del,V0f1: $i] :
( ( mem @ V0f1 @ ( ty_2Efinite__map_2Efmap @ A_27b @ A_27a ) )
=> ! [V1f2: $i] :
( ( mem @ V1f2 @ ( ty_2Efinite__map_2Efmap @ A_27b @ A_27a ) )
=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ ( ap @ ( c_2Efinite__map_2EFRANGE @ A_27b @ A_27a ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EFUNION @ A_27b @ A_27a ) @ V0f1 ) @ V1f2 ) ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ A_27a ) @ ( ap @ ( c_2Efinite__map_2EFRANGE @ A_27b @ A_27a ) @ V0f1 ) ) @ ( ap @ ( c_2Efinite__map_2EFRANGE @ A_27b @ A_27a ) @ V1f2 ) ) ) ) ) ) ).
thf(conj_thm_2Efinite__map_2EIN__FRANGE__FUNION__suff,axiom,
! [A_27a: del,A_27b: del,V0f1: $i] :
( ( mem @ V0f1 @ ( ty_2Efinite__map_2Efmap @ A_27b @ A_27a ) )
=> ! [V1P: $i] :
( ( mem @ V1P @ ( arr @ A_27a @ bool ) )
=> ! [V2f2: $i] :
( ( mem @ V2f2 @ ( ty_2Efinite__map_2Efmap @ A_27b @ A_27a ) )
=> ( ( ! [V3v: $i] :
( ( mem @ V3v @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3v ) @ ( ap @ ( c_2Efinite__map_2EFRANGE @ A_27b @ A_27a ) @ V0f1 ) ) )
=> ( p @ ( ap @ V1P @ V3v ) ) ) )
& ! [V4v: $i] :
( ( mem @ V4v @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V4v ) @ ( ap @ ( c_2Efinite__map_2EFRANGE @ A_27b @ A_27a ) @ V2f2 ) ) )
=> ( p @ ( ap @ V1P @ V4v ) ) ) ) )
=> ! [V5v: $i] :
( ( mem @ V5v @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V5v ) @ ( ap @ ( c_2Efinite__map_2EFRANGE @ A_27b @ A_27a ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EFUNION @ A_27b @ A_27a ) @ V0f1 ) @ V2f2 ) ) ) )
=> ( p @ ( ap @ V1P @ V5v ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Efinite__map_2EFRANGE__DOMSUB__SUBSET,axiom,
! [A_27a: del,A_27b: del,V0fm: $i] :
( ( mem @ V0fm @ ( ty_2Efinite__map_2Efmap @ A_27b @ A_27a ) )
=> ! [V1k: $i] :
( ( mem @ V1k @ A_27b )
=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ ( ap @ ( c_2Efinite__map_2EFRANGE @ A_27b @ A_27a ) @ ( ap @ ( ap @ ( c_2Efinite__map_2Efdomsub @ A_27b @ A_27a ) @ V0fm ) @ V1k ) ) ) @ ( ap @ ( c_2Efinite__map_2EFRANGE @ A_27b @ A_27a ) @ V0fm ) ) ) ) ) ).
thf(conj_thm_2Efinite__map_2EIN__FRANGE__DOMSUB__suff,axiom,
! [A_27a: del,A_27b: del,V0fm: $i] :
( ( mem @ V0fm @ ( ty_2Efinite__map_2Efmap @ A_27b @ A_27a ) )
=> ! [V1P: $i] :
( ( mem @ V1P @ ( arr @ A_27a @ bool ) )
=> ! [V2k: $i] :
( ( mem @ V2k @ A_27b )
=> ( ! [V3v: $i] :
( ( mem @ V3v @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3v ) @ ( ap @ ( c_2Efinite__map_2EFRANGE @ A_27b @ A_27a ) @ V0fm ) ) )
=> ( p @ ( ap @ V1P @ V3v ) ) ) )
=> ! [V4v: $i] :
( ( mem @ V4v @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V4v ) @ ( ap @ ( c_2Efinite__map_2EFRANGE @ A_27b @ A_27a ) @ ( ap @ ( ap @ ( c_2Efinite__map_2Efdomsub @ A_27b @ A_27a ) @ V0fm ) @ V2k ) ) ) )
=> ( p @ ( ap @ V1P @ V4v ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Efinite__map_2EIN__FRANGE__DRESTRICT__suff,axiom,
! [A_27a: del,A_27b: del,V0fm: $i] :
( ( mem @ V0fm @ ( ty_2Efinite__map_2Efmap @ A_27b @ A_27a ) )
=> ! [V1P: $i] :
( ( mem @ V1P @ ( arr @ A_27a @ bool ) )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ A_27b @ bool ) )
=> ( ! [V3v: $i] :
( ( mem @ V3v @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3v ) @ ( ap @ ( c_2Efinite__map_2EFRANGE @ A_27b @ A_27a ) @ V0fm ) ) )
=> ( p @ ( ap @ V1P @ V3v ) ) ) )
=> ! [V4v: $i] :
( ( mem @ V4v @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V4v ) @ ( ap @ ( c_2Efinite__map_2EFRANGE @ A_27b @ A_27a ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EDRESTRICT @ A_27b @ A_27a ) @ V0fm ) @ V2s ) ) ) )
=> ( p @ ( ap @ V1P @ V4v ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Efinite__map_2EFRANGE__FUPDATE__SUBSET,axiom,
! [A_27a: del,A_27b: del,V0fm: $i] :
( ( mem @ V0fm @ ( ty_2Efinite__map_2Efmap @ A_27b @ A_27a ) )
=> ! [V1kv: $i] :
( ( mem @ V1kv @ ( ty_2Epair_2Eprod @ A_27b @ A_27a ) )
=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ ( ap @ ( c_2Efinite__map_2EFRANGE @ A_27b @ A_27a ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EFUPDATE @ A_27b @ A_27a ) @ V0fm ) @ V1kv ) ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ A_27a ) @ ( ap @ ( c_2Efinite__map_2EFRANGE @ A_27b @ A_27a ) @ V0fm ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ A_27a ) @ ( ap @ ( c_2Epair_2ESND @ A_27b @ A_27a ) @ V1kv ) ) @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) ) ) ) ) ) ).
thf(conj_thm_2Efinite__map_2EIN__FRANGE__FUPDATE__suff,axiom,
! [A_27a: del,A_27b: del,V0fm: $i] :
( ( mem @ V0fm @ ( ty_2Efinite__map_2Efmap @ A_27b @ A_27a ) )
=> ! [V1P: $i] :
( ( mem @ V1P @ ( arr @ A_27a @ bool ) )
=> ! [V2kv: $i] :
( ( mem @ V2kv @ ( ty_2Epair_2Eprod @ A_27b @ A_27a ) )
=> ( ( ! [V3v: $i] :
( ( mem @ V3v @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3v ) @ ( ap @ ( c_2Efinite__map_2EFRANGE @ A_27b @ A_27a ) @ V0fm ) ) )
=> ( p @ ( ap @ V1P @ V3v ) ) ) )
& ( p @ ( ap @ V1P @ ( ap @ ( c_2Epair_2ESND @ A_27b @ A_27a ) @ V2kv ) ) ) )
=> ! [V4v: $i] :
( ( mem @ V4v @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V4v ) @ ( ap @ ( c_2Efinite__map_2EFRANGE @ A_27b @ A_27a ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EFUPDATE @ A_27b @ A_27a ) @ V0fm ) @ V2kv ) ) ) )
=> ( p @ ( ap @ V1P @ V4v ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Efinite__map_2EIN__FRANGE__o__f__suff,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0fm: $i] :
( ( mem @ V0fm @ ( ty_2Efinite__map_2Efmap @ A_27b @ A_27a ) )
=> ! [V1P: $i] :
( ( mem @ V1P @ ( arr @ A_27c @ bool ) )
=> ! [V2f: $i] :
( ( mem @ V2f @ ( arr @ A_27a @ A_27c ) )
=> ( ! [V3v: $i] :
( ( mem @ V3v @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3v ) @ ( ap @ ( c_2Efinite__map_2EFRANGE @ A_27b @ A_27a ) @ V0fm ) ) )
=> ( p @ ( ap @ V1P @ ( ap @ V2f @ V3v ) ) ) ) )
=> ! [V4v: $i] :
( ( mem @ V4v @ A_27c )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27c ) @ V4v ) @ ( ap @ ( c_2Efinite__map_2EFRANGE @ A_27b @ A_27c ) @ ( ap @ ( ap @ ( c_2Efinite__map_2Eo__f @ A_27b @ A_27a @ A_27c ) @ V2f ) @ V0fm ) ) ) )
=> ( p @ ( ap @ V1P @ V4v ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Efinite__map_2EDRESTRICT__SUBMAP__gen,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ! [V2P: $i] :
( ( mem @ V2P @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Efinite__map_2ESUBMAP @ A_27a @ A_27b ) @ V0f ) @ V1g ) )
=> ( p @ ( ap @ ( ap @ ( c_2Efinite__map_2ESUBMAP @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EDRESTRICT @ A_27a @ A_27b ) @ V0f ) @ V2P ) ) @ V1g ) ) ) ) ) ) ).
thf(conj_thm_2Efinite__map_2EDRESTRICT__SUBSET__SUBMAP,axiom,
! [A_27a: del,A_27b: del,V0s1: $i] :
( ( mem @ V0s1 @ ( arr @ A_27a @ bool ) )
=> ! [V1s2: $i] :
( ( mem @ V1s2 @ ( arr @ A_27a @ bool ) )
=> ! [V2f: $i] :
( ( mem @ V2f @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ V0s1 ) @ V1s2 ) )
=> ( p @ ( ap @ ( ap @ ( c_2Efinite__map_2ESUBMAP @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EDRESTRICT @ A_27a @ A_27b ) @ V2f ) @ V0s1 ) ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EDRESTRICT @ A_27a @ A_27b ) @ V2f ) @ V1s2 ) ) ) ) ) ) ) ).
thf(conj_thm_2Efinite__map_2EDRESTRICTED__FUNION,axiom,
! [A_27a: del,A_27b: del,V0f1: $i] :
( ( mem @ V0f1 @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ! [V1f2: $i] :
( ( mem @ V1f2 @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ A_27a @ bool ) )
=> ( ( ap @ ( ap @ ( c_2Efinite__map_2EDRESTRICT @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EFUNION @ A_27a @ A_27b ) @ V0f1 ) @ V1f2 ) ) @ V2s )
= ( ap @ ( ap @ ( c_2Efinite__map_2EFUNION @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EDRESTRICT @ A_27a @ A_27b ) @ V0f1 ) @ V2s ) ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EDRESTRICT @ A_27a @ A_27b ) @ V1f2 ) @ ( ap @ ( ap @ ( c_2Epred__set_2EDIFF @ A_27a ) @ V2s ) @ ( ap @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b ) @ V0f1 ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Efinite__map_2EFRANGE__DRESTRICT__SUBSET,axiom,
! [A_27a: del,A_27b: del,V0fm: $i] :
( ( mem @ V0fm @ ( ty_2Efinite__map_2Efmap @ A_27b @ A_27a ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27b @ bool ) )
=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ ( ap @ ( c_2Efinite__map_2EFRANGE @ A_27b @ A_27a ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EDRESTRICT @ A_27b @ A_27a ) @ V0fm ) @ V1s ) ) ) @ ( ap @ ( c_2Efinite__map_2EFRANGE @ A_27b @ A_27a ) @ V0fm ) ) ) ) ) ).
thf(conj_thm_2Efinite__map_2EDRESTRICT__FDOM,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ( ( ap @ ( ap @ ( c_2Efinite__map_2EDRESTRICT @ A_27a @ A_27b ) @ V0f ) @ ( ap @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b ) @ V0f ) )
= V0f ) ) ).
thf(conj_thm_2Efinite__map_2EDRESTRICT__SUBSET,axiom,
! [A_27a: del,A_27b: del,V0f1: $i] :
( ( mem @ V0f1 @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ! [V1f2: $i] :
( ( mem @ V1f2 @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ A_27a @ bool ) )
=> ! [V3t: $i] :
( ( mem @ V3t @ ( arr @ A_27a @ bool ) )
=> ( ( ( ( ap @ ( ap @ ( c_2Efinite__map_2EDRESTRICT @ A_27a @ A_27b ) @ V0f1 ) @ V2s )
= ( ap @ ( ap @ ( c_2Efinite__map_2EDRESTRICT @ A_27a @ A_27b ) @ V1f2 ) @ V2s ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ V3t ) @ V2s ) ) )
=> ( ( ap @ ( ap @ ( c_2Efinite__map_2EDRESTRICT @ A_27a @ A_27b ) @ V0f1 ) @ V3t )
= ( ap @ ( ap @ ( c_2Efinite__map_2EDRESTRICT @ A_27a @ A_27b ) @ V1f2 ) @ V3t ) ) ) ) ) ) ) ).
thf(conj_thm_2Efinite__map_2Ef__o__f__FUPDATE__compose,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0f1: $i] :
( ( mem @ V0f1 @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ! [V1f2: $i] :
( ( mem @ V1f2 @ ( ty_2Efinite__map_2Efmap @ A_27c @ A_27a ) )
=> ! [V2k: $i] :
( ( mem @ V2k @ A_27c )
=> ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ! [V4v: $i] :
( ( mem @ V4v @ A_27b )
=> ( ( ~ ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3x ) @ ( ap @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b ) @ V0f1 ) ) )
& ~ ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3x ) @ ( ap @ ( c_2Efinite__map_2EFRANGE @ A_27c @ A_27a ) @ V1f2 ) ) ) )
=> ( ( ap @ ( ap @ ( c_2Efinite__map_2Ef__o__f @ A_27c @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EFUPDATE @ A_27a @ A_27b ) @ V0f1 ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27b ) @ V3x ) @ V4v ) ) ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EFUPDATE @ A_27c @ A_27a ) @ V1f2 ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27c @ A_27a ) @ V2k ) @ V3x ) ) )
= ( ap @ ( ap @ ( c_2Efinite__map_2EFUPDATE @ A_27c @ A_27b ) @ ( ap @ ( ap @ ( c_2Efinite__map_2Ef__o__f @ A_27c @ A_27a @ A_27b ) @ V0f1 ) @ V1f2 ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27c @ A_27b ) @ V2k ) @ V4v ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Efinite__map_2Efmap__rel__trans,axiom,
! [A_27a: del,A_27b: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ! [V2y: $i] :
( ( mem @ V2y @ A_27a )
=> ! [V3z: $i] :
( ( mem @ V3z @ A_27a )
=> ( ( ( p @ ( ap @ ( ap @ V0R @ V1x ) @ V2y ) )
& ( p @ ( ap @ ( ap @ V0R @ V2y ) @ V3z ) ) )
=> ( p @ ( ap @ ( ap @ V0R @ V1x ) @ V3z ) ) ) ) ) )
=> ! [V4x: $i] :
( ( mem @ V4x @ ( ty_2Efinite__map_2Efmap @ A_27b @ A_27a ) )
=> ! [V5y: $i] :
( ( mem @ V5y @ ( ty_2Efinite__map_2Efmap @ A_27b @ A_27a ) )
=> ! [V6z: $i] :
( ( mem @ V6z @ ( ty_2Efinite__map_2Efmap @ A_27b @ A_27a ) )
=> ( ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Efinite__map_2Efmap__rel @ A_27a @ A_27a @ A_27b ) @ V0R ) @ V4x ) @ V5y ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( c_2Efinite__map_2Efmap__rel @ A_27a @ A_27a @ A_27b ) @ V0R ) @ V5y ) @ V6z ) ) )
=> ( p @ ( ap @ ( ap @ ( ap @ ( c_2Efinite__map_2Efmap__rel @ A_27a @ A_27a @ A_27b ) @ V0R ) @ V4x ) @ V6z ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Efinite__map_2Efmap__rel__sym,axiom,
! [A_27a: del,A_27b: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ! [V2y: $i] :
( ( mem @ V2y @ A_27a )
=> ( ( p @ ( ap @ ( ap @ V0R @ V1x ) @ V2y ) )
=> ( p @ ( ap @ ( ap @ V0R @ V2y ) @ V1x ) ) ) ) )
=> ! [V3x: $i] :
( ( mem @ V3x @ ( ty_2Efinite__map_2Efmap @ A_27b @ A_27a ) )
=> ! [V4y: $i] :
( ( mem @ V4y @ ( ty_2Efinite__map_2Efmap @ A_27b @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Efinite__map_2Efmap__rel @ A_27a @ A_27a @ A_27b ) @ V0R ) @ V3x ) @ V4y ) )
=> ( p @ ( ap @ ( ap @ ( ap @ ( c_2Efinite__map_2Efmap__rel @ A_27a @ A_27a @ A_27b ) @ V0R ) @ V4y ) @ V3x ) ) ) ) ) ) ) ).
thf(conj_thm_2Efinite__map_2Efupdate__list__map,axiom,
! [A_27a: del,A_27b: del,A_27c: del,A_27d: del,V0l: $i] :
( ( mem @ V0l @ ( ty_2Elist_2Elist @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27b @ A_27c ) )
=> ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ! [V3y: $i] :
( ( mem @ V3y @ A_27d )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x ) @ ( ap @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EFUPDATE__LIST @ A_27a @ A_27b ) @ ( c_2Efinite__map_2EFEMPTY @ A_27a @ A_27b ) ) @ V0l ) ) ) )
=> ( ( ap
@ ( ap @ ( c_2Efinite__map_2EFAPPLY @ A_27a @ A_27c )
@ ( ap @ ( ap @ ( c_2Efinite__map_2EFUPDATE__LIST @ A_27a @ A_27c ) @ ( c_2Efinite__map_2EFEMPTY @ A_27a @ A_27c ) )
@ ( ap
@ ( ap @ ( c_2Elist_2EMAP @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ ( ty_2Epair_2Eprod @ A_27a @ A_27c ) )
@ ( ap @ ( c_2Epair_2EUNCURRY @ A_27a @ A_27b @ ( ty_2Epair_2Eprod @ A_27a @ A_27c ) )
@ ( lam @ A_27a
@ ^ [V4a: $i] :
( lam @ A_27b
@ ^ [V5b: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27c ) @ V4a ) @ ( ap @ V1f @ V5b ) ) ) ) ) )
@ V0l ) ) )
@ V2x )
= ( ap @ V1f @ ( ap @ ( ap @ ( c_2Efinite__map_2EFAPPLY @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EFUPDATE__LIST @ A_27a @ A_27b ) @ ( c_2Efinite__map_2EFEMPTY @ A_27a @ A_27b ) ) @ V0l ) ) @ V2x ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Efinite__map_2Efdom__fupdate__list2,axiom,
! [A_27a: del,A_27b: del,V0kvl: $i] :
( ( mem @ V0kvl @ ( ty_2Elist_2Elist @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) ) )
=> ! [V1fm: $i] :
( ( mem @ V1fm @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ( ( ap @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EFUPDATE__LIST @ A_27a @ A_27b ) @ V1fm ) @ V0kvl ) )
= ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EDIFF @ A_27a ) @ ( ap @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b ) @ V1fm ) ) @ ( ap @ ( c_2Elist_2ELIST__TO__SET @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2EMAP @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ A_27a ) @ ( c_2Epair_2EFST @ A_27a @ A_27b ) ) @ V0kvl ) ) ) ) @ ( ap @ ( c_2Elist_2ELIST__TO__SET @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2EMAP @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ A_27a ) @ ( c_2Epair_2EFST @ A_27a @ A_27b ) ) @ V0kvl ) ) ) ) ) ) ).
thf(conj_thm_2Efinite__map_2Eflookup__thm,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ! [V2v: $i] :
( ( mem @ V2v @ A_27b )
=> ( ( ( ( ap @ ( ap @ ( c_2Efinite__map_2EFLOOKUP @ A_27a @ A_27b ) @ V0f ) @ V1x )
= ( c_2Eoption_2ENONE @ A_27b ) )
<=> ~ ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V1x ) @ ( ap @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b ) @ V0f ) ) ) )
& ( ( ( ap @ ( ap @ ( c_2Efinite__map_2EFLOOKUP @ A_27a @ A_27b ) @ V0f ) @ V1x )
= ( ap @ ( c_2Eoption_2ESOME @ A_27b ) @ V2v ) )
<=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V1x ) @ ( ap @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b ) @ V0f ) ) )
& ( ( ap @ ( ap @ ( c_2Efinite__map_2EFAPPLY @ A_27a @ A_27b ) @ V0f ) @ V1x )
= V2v ) ) ) ) ) ) ) ).
thf(conj_thm_2Efinite__map_2EFUPDATE__EQ__FUPDATE__LIST,axiom,
! [A_27a: del,A_27b: del,V0fm: $i] :
( ( mem @ V0fm @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ! [V1kv: $i] :
( ( mem @ V1kv @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) )
=> ( ( ap @ ( ap @ ( c_2Efinite__map_2EFUPDATE @ A_27a @ A_27b ) @ V0fm ) @ V1kv )
= ( ap @ ( ap @ ( c_2Efinite__map_2EFUPDATE__LIST @ A_27a @ A_27b ) @ V0fm ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) ) @ V1kv ) @ ( c_2Elist_2ENIL @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) ) ) ) ) ) ) ).
thf(ax_thm_2Efinite__map_2Efmap__inverse__def,axiom,
! [A_27a: del,A_27b: del,V0m1: $i] :
( ( mem @ V0m1 @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ! [V1m2: $i] :
( ( mem @ V1m2 @ ( ty_2Efinite__map_2Efmap @ A_27b @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Efinite__map_2Efmap__inverse @ A_27a @ A_27b ) @ V0m1 ) @ V1m2 ) )
<=> ! [V2k: $i] :
( ( mem @ V2k @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2k ) @ ( ap @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b ) @ V0m1 ) ) )
=> ? [V3v: $i] :
( ( mem @ V3v @ A_27b )
& ( ( ap @ ( ap @ ( c_2Efinite__map_2EFLOOKUP @ A_27a @ A_27b ) @ V0m1 ) @ V2k )
= ( ap @ ( c_2Eoption_2ESOME @ A_27b ) @ V3v ) )
& ( ( ap @ ( ap @ ( c_2Efinite__map_2EFLOOKUP @ A_27b @ A_27a ) @ V1m2 ) @ V3v )
= ( ap @ ( c_2Eoption_2ESOME @ A_27a ) @ V2k ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Efinite__map_2Efupdate__list__foldr,axiom,
! [A_27a: del,A_27b: del,V0m: $i] :
( ( mem @ V0m @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ! [V1l: $i] :
( ( mem @ V1l @ ( ty_2Elist_2Elist @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) ) )
=> ( ( ap
@ ( ap
@ ( ap @ ( c_2Elist_2EFOLDR @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
@ ( ap @ ( c_2Epair_2EUNCURRY @ A_27a @ A_27b @ ( arr @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) ) )
@ ( lam @ A_27a
@ ^ [V2k: $i] :
( lam @ A_27b
@ ^ [V3v: $i] :
( lam @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b )
@ ^ [V4env: $i] : ( ap @ ( ap @ ( c_2Efinite__map_2EFUPDATE @ A_27a @ A_27b ) @ V4env ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27b ) @ V2k ) @ V3v ) ) ) ) ) ) )
@ V0m )
@ V1l )
= ( ap @ ( ap @ ( c_2Efinite__map_2EFUPDATE__LIST @ A_27a @ A_27b ) @ V0m ) @ ( ap @ ( c_2Elist_2EREVERSE @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) ) @ V1l ) ) ) ) ) ).
thf(conj_thm_2Efinite__map_2Efupdate__list__foldl,axiom,
! [A_27a: del,A_27b: del,V0m: $i] :
( ( mem @ V0m @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ! [V1l: $i] :
( ( mem @ V1l @ ( ty_2Elist_2Elist @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) ) )
=> ( ( ap
@ ( ap
@ ( ap @ ( c_2Elist_2EFOLDL @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
@ ( lam @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b )
@ ^ [V2env: $i] :
( ap @ ( c_2Epair_2EUNCURRY @ A_27a @ A_27b @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
@ ( lam @ A_27a
@ ^ [V3k: $i] :
( lam @ A_27b
@ ^ [V4v: $i] : ( ap @ ( ap @ ( c_2Efinite__map_2EFUPDATE @ A_27a @ A_27b ) @ V2env ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27b ) @ V3k ) @ V4v ) ) ) ) ) ) )
@ V0m )
@ V1l )
= ( ap @ ( ap @ ( c_2Efinite__map_2EFUPDATE__LIST @ A_27a @ A_27b ) @ V0m ) @ V1l ) ) ) ) ).
thf(conj_thm_2Efinite__map_2Efmap__to__list,axiom,
! [A_27a: del,A_27b: del,V0m: $i] :
( ( mem @ V0m @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ? [V1l: $i] :
( ( mem @ V1l @ ( ty_2Elist_2Elist @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) ) )
& ( p @ ( ap @ ( c_2Elist_2EALL__DISTINCT @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2EMAP @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ A_27a ) @ ( c_2Epair_2EFST @ A_27a @ A_27b ) ) @ V1l ) ) )
& ( V0m
= ( ap @ ( ap @ ( c_2Efinite__map_2EFUPDATE__LIST @ A_27a @ A_27b ) @ ( c_2Efinite__map_2EFEMPTY @ A_27a @ A_27b ) ) @ V1l ) ) ) ) ).
thf(conj_thm_2Efinite__map_2Edisjoint__drestrict,axiom,
! [A_27a: del,A_27b: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1m: $i] :
( ( mem @ V1m @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2EDISJOINT @ A_27a ) @ V0s ) @ ( ap @ ( c_2Efinite__map_2EFDOM @ A_27a @ A_27b ) @ V1m ) ) )
=> ( ( ap @ ( ap @ ( c_2Efinite__map_2EDRESTRICT @ A_27a @ A_27b ) @ V1m ) @ ( ap @ ( c_2Epred__set_2ECOMPL @ A_27a ) @ V0s ) )
= V1m ) ) ) ) ).
thf(conj_thm_2Efinite__map_2Edrestrict__iter__list,axiom,
! [A_27a: del,A_27b: del,V0m: $i] :
( ( mem @ V0m @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ! [V1l: $i] :
( ( mem @ V1l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( ap
@ ( ap
@ ( ap @ ( c_2Elist_2EFOLDR @ A_27a @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
@ ( lam @ A_27a
@ ^ [V2k: $i] :
( lam @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b )
@ ^ [V3m: $i] : ( ap @ ( ap @ ( c_2Efinite__map_2Efdomsub @ A_27a @ A_27b ) @ V3m ) @ V2k ) ) ) )
@ V0m )
@ V1l )
= ( ap @ ( ap @ ( c_2Efinite__map_2EDRESTRICT @ A_27a @ A_27b ) @ V0m ) @ ( ap @ ( c_2Epred__set_2ECOMPL @ A_27a ) @ ( ap @ ( c_2Elist_2ELIST__TO__SET @ A_27a ) @ V1l ) ) ) ) ) ) ).
thf(conj_thm_2Efinite__map_2Efevery__funion,axiom,
! [A_27a: del,A_27b: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ bool ) )
=> ! [V1m1: $i] :
( ( mem @ V1m1 @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ! [V2m2: $i] :
( ( mem @ V2m2 @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Efinite__map_2EFEVERY @ A_27a @ A_27b ) @ V0P ) @ V1m1 ) )
& ( p @ ( ap @ ( ap @ ( c_2Efinite__map_2EFEVERY @ A_27a @ A_27b ) @ V0P ) @ V2m2 ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Efinite__map_2EFEVERY @ A_27a @ A_27b ) @ V0P ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EFUNION @ A_27a @ A_27b ) @ V1m1 ) @ V2m2 ) ) ) ) ) ) ) ).
thf(conj_thm_2Efinite__map_2EFUPDATE__LIST__ALL__DISTINCT__PERM,axiom,
! [A_27a: del,A_27b: del,V0ls: $i] :
( ( mem @ V0ls @ ( ty_2Elist_2Elist @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) ) )
=> ! [V1ls_27: $i] :
( ( mem @ V1ls_27 @ ( ty_2Elist_2Elist @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) ) )
=> ! [V2fm: $i] :
( ( mem @ V2fm @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ( ( ( p @ ( ap @ ( c_2Elist_2EALL__DISTINCT @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2EMAP @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ A_27a ) @ ( c_2Epair_2EFST @ A_27a @ A_27b ) ) @ V0ls ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Esorting_2EPERM @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) ) @ V0ls ) @ V1ls_27 ) ) )
=> ( ( ap @ ( ap @ ( c_2Efinite__map_2EFUPDATE__LIST @ A_27a @ A_27b ) @ V2fm ) @ V0ls )
= ( ap @ ( ap @ ( c_2Efinite__map_2EFUPDATE__LIST @ A_27a @ A_27b ) @ V2fm ) @ V1ls_27 ) ) ) ) ) ) ).
%------------------------------------------------------------------------------