ITP001 Axioms: ITP022^5.ax
%------------------------------------------------------------------------------
% File : ITP022^5 : 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 : ind_type^2.ax [Gau20]
% : HL4022^5.ax [TPAP]
% Status : Satisfiable
% Syntax : Number of formulae : 90 ( 11 unt; 25 typ; 0 def)
% Number of atoms : 1004 ( 62 equ; 0 cnn)
% Maximal formula atoms : 40 ( 11 avg)
% Number of connectives : 1601 ( 2 ~; 1 |; 30 &;1455 @)
% ( 17 <=>; 96 =>; 0 <=; 0 <~>)
% Maximal formula depth : 26 ( 9 avg;1455 nst)
% Number of types : 1 ( 0 usr)
% Number of type conns : 21 ( 21 >; 0 *; 0 +; 0 <<)
% Number of symbols : 58 ( 57 usr; 38 con; 0-2 aty)
% Number of variables : 213 ( 23 ^ 181 !; 9 ?; 213 :)
% SPC : TH0_SAT_EQU_NAR
% Comments :
% Bugfixes : v7.5.0 - Fixes to the axioms.
%------------------------------------------------------------------------------
thf(tp_ty_2Eind__type_2Erecspace,type,
ty_2Eind__type_2Erecspace: del > del ).
thf(tp_c_2Eind__type_2EBOTTOM,type,
c_2Eind__type_2EBOTTOM: del > $i ).
thf(mem_c_2Eind__type_2EBOTTOM,axiom,
! [A_27a: del] : ( mem @ ( c_2Eind__type_2EBOTTOM @ A_27a ) @ ( ty_2Eind__type_2Erecspace @ A_27a ) ) ).
thf(tp_c_2Eind__type_2ECONSTR,type,
c_2Eind__type_2ECONSTR: del > $i ).
thf(mem_c_2Eind__type_2ECONSTR,axiom,
! [A_27a: del] : ( mem @ ( c_2Eind__type_2ECONSTR @ A_27a ) @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ ( arr @ ( arr @ ty_2Enum_2Enum @ ( ty_2Eind__type_2Erecspace @ A_27a ) ) @ ( ty_2Eind__type_2Erecspace @ A_27a ) ) ) ) ) ).
thf(tp_c_2Eind__type_2EFCONS,type,
c_2Eind__type_2EFCONS: del > $i ).
thf(mem_c_2Eind__type_2EFCONS,axiom,
! [A_27a: del] : ( mem @ ( c_2Eind__type_2EFCONS @ A_27a ) @ ( arr @ A_27a @ ( arr @ ( arr @ ty_2Enum_2Enum @ A_27a ) @ ( arr @ ty_2Enum_2Enum @ A_27a ) ) ) ) ).
thf(tp_c_2Eind__type_2EFNIL,type,
c_2Eind__type_2EFNIL: del > $i ).
thf(mem_c_2Eind__type_2EFNIL,axiom,
! [A_27a: del] : ( mem @ ( c_2Eind__type_2EFNIL @ A_27a ) @ ( arr @ ty_2Enum_2Enum @ A_27a ) ) ).
thf(tp_c_2Eind__type_2EINJA,type,
c_2Eind__type_2EINJA: del > $i ).
thf(mem_c_2Eind__type_2EINJA,axiom,
! [A_27a: del] : ( mem @ ( c_2Eind__type_2EINJA @ A_27a ) @ ( arr @ A_27a @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) ) ) ).
thf(tp_c_2Eind__type_2EINJF,type,
c_2Eind__type_2EINJF: del > $i ).
thf(mem_c_2Eind__type_2EINJF,axiom,
! [A_27a: del] : ( mem @ ( c_2Eind__type_2EINJF @ A_27a ) @ ( arr @ ( arr @ ty_2Enum_2Enum @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) ) @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) ) ) ).
thf(tp_c_2Eind__type_2EINJN,type,
c_2Eind__type_2EINJN: del > $i ).
thf(mem_c_2Eind__type_2EINJN,axiom,
! [A_27a: del] : ( mem @ ( c_2Eind__type_2EINJN @ A_27a ) @ ( arr @ ty_2Enum_2Enum @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) ) ) ).
thf(tp_c_2Eind__type_2EINJP,type,
c_2Eind__type_2EINJP: del > $i ).
thf(mem_c_2Eind__type_2EINJP,axiom,
! [A_27a: del] : ( mem @ ( c_2Eind__type_2EINJP @ A_27a ) @ ( arr @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) @ ( arr @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) ) ) ) ).
thf(tp_c_2Eind__type_2EISO,type,
c_2Eind__type_2EISO: del > del > $i ).
thf(mem_c_2Eind__type_2EISO,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2Eind__type_2EISO @ A_27a @ A_27b ) @ ( arr @ ( arr @ A_27a @ A_27b ) @ ( arr @ ( arr @ A_27b @ A_27a ) @ bool ) ) ) ).
thf(tp_c_2Eind__type_2ENUMFST,type,
c_2Eind__type_2ENUMFST: $i ).
thf(mem_c_2Eind__type_2ENUMFST,axiom,
mem @ c_2Eind__type_2ENUMFST @ ( arr @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) ).
thf(stp_fo_c_2Eind__type_2ENUMFST,type,
fo__c_2Eind__type_2ENUMFST: tp__ty_2Enum_2Enum > tp__ty_2Enum_2Enum ).
thf(stp_eq_fo_c_2Eind__type_2ENUMFST,axiom,
! [X0: tp__ty_2Enum_2Enum] :
( ( inj__ty_2Enum_2Enum @ ( fo__c_2Eind__type_2ENUMFST @ X0 ) )
= ( ap @ c_2Eind__type_2ENUMFST @ ( inj__ty_2Enum_2Enum @ X0 ) ) ) ).
thf(tp_c_2Eind__type_2ENUMLEFT,type,
c_2Eind__type_2ENUMLEFT: $i ).
thf(mem_c_2Eind__type_2ENUMLEFT,axiom,
mem @ c_2Eind__type_2ENUMLEFT @ ( arr @ ty_2Enum_2Enum @ bool ) ).
thf(tp_c_2Eind__type_2ENUMPAIR,type,
c_2Eind__type_2ENUMPAIR: $i ).
thf(mem_c_2Eind__type_2ENUMPAIR,axiom,
mem @ c_2Eind__type_2ENUMPAIR @ ( arr @ ty_2Enum_2Enum @ ( arr @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) ) ).
thf(stp_fo_c_2Eind__type_2ENUMPAIR,type,
fo__c_2Eind__type_2ENUMPAIR: tp__ty_2Enum_2Enum > tp__ty_2Enum_2Enum > tp__ty_2Enum_2Enum ).
thf(stp_eq_fo_c_2Eind__type_2ENUMPAIR,axiom,
! [X0: tp__ty_2Enum_2Enum,X1: tp__ty_2Enum_2Enum] :
( ( inj__ty_2Enum_2Enum @ ( fo__c_2Eind__type_2ENUMPAIR @ X0 @ X1 ) )
= ( ap @ ( ap @ c_2Eind__type_2ENUMPAIR @ ( inj__ty_2Enum_2Enum @ X0 ) ) @ ( inj__ty_2Enum_2Enum @ X1 ) ) ) ).
thf(tp_c_2Eind__type_2ENUMRIGHT,type,
c_2Eind__type_2ENUMRIGHT: $i ).
thf(mem_c_2Eind__type_2ENUMRIGHT,axiom,
mem @ c_2Eind__type_2ENUMRIGHT @ ( arr @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) ).
thf(stp_fo_c_2Eind__type_2ENUMRIGHT,type,
fo__c_2Eind__type_2ENUMRIGHT: tp__ty_2Enum_2Enum > tp__ty_2Enum_2Enum ).
thf(stp_eq_fo_c_2Eind__type_2ENUMRIGHT,axiom,
! [X0: tp__ty_2Enum_2Enum] :
( ( inj__ty_2Enum_2Enum @ ( fo__c_2Eind__type_2ENUMRIGHT @ X0 ) )
= ( ap @ c_2Eind__type_2ENUMRIGHT @ ( inj__ty_2Enum_2Enum @ X0 ) ) ) ).
thf(tp_c_2Eind__type_2ENUMSND,type,
c_2Eind__type_2ENUMSND: $i ).
thf(mem_c_2Eind__type_2ENUMSND,axiom,
mem @ c_2Eind__type_2ENUMSND @ ( arr @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) ).
thf(stp_fo_c_2Eind__type_2ENUMSND,type,
fo__c_2Eind__type_2ENUMSND: tp__ty_2Enum_2Enum > tp__ty_2Enum_2Enum ).
thf(stp_eq_fo_c_2Eind__type_2ENUMSND,axiom,
! [X0: tp__ty_2Enum_2Enum] :
( ( inj__ty_2Enum_2Enum @ ( fo__c_2Eind__type_2ENUMSND @ X0 ) )
= ( ap @ c_2Eind__type_2ENUMSND @ ( inj__ty_2Enum_2Enum @ X0 ) ) ) ).
thf(tp_c_2Eind__type_2ENUMSUM,type,
c_2Eind__type_2ENUMSUM: $i ).
thf(mem_c_2Eind__type_2ENUMSUM,axiom,
mem @ c_2Eind__type_2ENUMSUM @ ( arr @ bool @ ( arr @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) ) ).
thf(tp_c_2Eind__type_2EZBOT,type,
c_2Eind__type_2EZBOT: del > $i ).
thf(mem_c_2Eind__type_2EZBOT,axiom,
! [A_27a: del] : ( mem @ ( c_2Eind__type_2EZBOT @ A_27a ) @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) ) ).
thf(tp_c_2Eind__type_2EZCONSTR,type,
c_2Eind__type_2EZCONSTR: del > $i ).
thf(mem_c_2Eind__type_2EZCONSTR,axiom,
! [A_27a: del] : ( mem @ ( c_2Eind__type_2EZCONSTR @ A_27a ) @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ ( arr @ ( arr @ ty_2Enum_2Enum @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) ) @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) ) ) ) ) ).
thf(tp_c_2Eind__type_2EZRECSPACE,type,
c_2Eind__type_2EZRECSPACE: del > $i ).
thf(mem_c_2Eind__type_2EZRECSPACE,axiom,
! [A_27a: del] : ( mem @ ( c_2Eind__type_2EZRECSPACE @ A_27a ) @ ( arr @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) @ bool ) ) ).
thf(tp_c_2Eind__type_2Edest__rec,type,
c_2Eind__type_2Edest__rec: del > $i ).
thf(mem_c_2Eind__type_2Edest__rec,axiom,
! [A_27a: del] : ( mem @ ( c_2Eind__type_2Edest__rec @ A_27a ) @ ( arr @ ( ty_2Eind__type_2Erecspace @ A_27a ) @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) ) ) ).
thf(tp_c_2Eind__type_2Emk__rec,type,
c_2Eind__type_2Emk__rec: del > $i ).
thf(mem_c_2Eind__type_2Emk__rec,axiom,
! [A_27a: del] : ( mem @ ( c_2Eind__type_2Emk__rec @ A_27a ) @ ( arr @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) @ ( ty_2Eind__type_2Erecspace @ A_27a ) ) ) ).
thf(conj_thm_2Eind__type_2EINJ__INVERSE2,axiom,
! [A_27A: del,A_27B: del,A_27C: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27A @ ( arr @ A_27B @ A_27C ) ) )
=> ( ! [V1x1: $i] :
( ( mem @ V1x1 @ A_27A )
=> ! [V2y1: $i] :
( ( mem @ V2y1 @ A_27B )
=> ! [V3x2: $i] :
( ( mem @ V3x2 @ A_27A )
=> ! [V4y2: $i] :
( ( mem @ V4y2 @ A_27B )
=> ( ( ( ap @ ( ap @ V0P @ V1x1 ) @ V2y1 )
= ( ap @ ( ap @ V0P @ V3x2 ) @ V4y2 ) )
<=> ( ( V1x1 = V3x2 )
& ( V2y1 = V4y2 ) ) ) ) ) ) )
=> ? [V5X: $i] :
( ( mem @ V5X @ ( arr @ A_27C @ A_27A ) )
& ? [V6Y: $i] :
( ( mem @ V6Y @ ( arr @ A_27C @ A_27B ) )
& ! [V7x: $i] :
( ( mem @ V7x @ A_27A )
=> ! [V8y: $i] :
( ( mem @ V8y @ A_27B )
=> ( ( ( ap @ V5X @ ( ap @ ( ap @ V0P @ V7x ) @ V8y ) )
= V7x )
& ( ( ap @ V6Y @ ( ap @ ( ap @ V0P @ V7x ) @ V8y ) )
= V8y ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Eind__type_2ENUMPAIR,axiom,
! [V0x: tp__ty_2Enum_2Enum,V1y: tp__ty_2Enum_2Enum] :
( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Eind__type_2ENUMPAIR @ ( inj__ty_2Enum_2Enum @ V0x ) ) @ ( inj__ty_2Enum_2Enum @ V1y ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Earithmetic_2E_2A @ ( ap @ ( ap @ c_2Earithmetic_2EEXP @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) @ ( inj__ty_2Enum_2Enum @ V0x ) ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( ap @ ( ap @ c_2Earithmetic_2E_2A @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) @ ( inj__ty_2Enum_2Enum @ V1y ) ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) ).
thf(conj_thm_2Eind__type_2ENUMPAIR__INJ__LEMMA,axiom,
! [V0x1: tp__ty_2Enum_2Enum,V1y1: tp__ty_2Enum_2Enum,V2x2: tp__ty_2Enum_2Enum,V3y2: tp__ty_2Enum_2Enum] :
( ( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Eind__type_2ENUMPAIR @ ( inj__ty_2Enum_2Enum @ V0x1 ) ) @ ( inj__ty_2Enum_2Enum @ V1y1 ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Eind__type_2ENUMPAIR @ ( inj__ty_2Enum_2Enum @ V2x2 ) ) @ ( inj__ty_2Enum_2Enum @ V3y2 ) ) ) )
=> ( V0x1 = V2x2 ) ) ).
thf(conj_thm_2Eind__type_2ENUMPAIR__INJ,axiom,
! [V0x1: tp__ty_2Enum_2Enum,V1y1: tp__ty_2Enum_2Enum,V2x2: tp__ty_2Enum_2Enum,V3y2: tp__ty_2Enum_2Enum] :
( ( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Eind__type_2ENUMPAIR @ ( inj__ty_2Enum_2Enum @ V0x1 ) ) @ ( inj__ty_2Enum_2Enum @ V1y1 ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Eind__type_2ENUMPAIR @ ( inj__ty_2Enum_2Enum @ V2x2 ) ) @ ( inj__ty_2Enum_2Enum @ V3y2 ) ) ) )
<=> ( ( V0x1 = V2x2 )
& ( V1y1 = V3y2 ) ) ) ).
thf(ax_thm_2Eind__type_2ENUMPAIR__DEST,axiom,
! [V0x: tp__ty_2Enum_2Enum,V1y: tp__ty_2Enum_2Enum] :
( ( ( surj__ty_2Enum_2Enum @ ( ap @ c_2Eind__type_2ENUMFST @ ( ap @ ( ap @ c_2Eind__type_2ENUMPAIR @ ( inj__ty_2Enum_2Enum @ V0x ) ) @ ( inj__ty_2Enum_2Enum @ V1y ) ) ) )
= V0x )
& ( ( surj__ty_2Enum_2Enum @ ( ap @ c_2Eind__type_2ENUMSND @ ( ap @ ( ap @ c_2Eind__type_2ENUMPAIR @ ( inj__ty_2Enum_2Enum @ V0x ) ) @ ( inj__ty_2Enum_2Enum @ V1y ) ) ) )
= V1y ) ) ).
thf(ax_thm_2Eind__type_2ENUMSUM,axiom,
! [V0b: $i] :
( ( mem @ V0b @ bool )
=> ! [V1x: tp__ty_2Enum_2Enum] :
( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Eind__type_2ENUMSUM @ V0b ) @ ( inj__ty_2Enum_2Enum @ V1x ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ty_2Enum_2Enum ) @ V0b ) @ ( ap @ c_2Enum_2ESUC @ ( ap @ ( ap @ c_2Earithmetic_2E_2A @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) @ ( inj__ty_2Enum_2Enum @ V1x ) ) ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2A @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) @ ( inj__ty_2Enum_2Enum @ V1x ) ) ) ) ) ) ).
thf(conj_thm_2Eind__type_2ENUMSUM__INJ,axiom,
! [V0b1: $i] :
( ( mem @ V0b1 @ bool )
=> ! [V1x1: tp__ty_2Enum_2Enum,V2b2: $i] :
( ( mem @ V2b2 @ bool )
=> ! [V3x2: tp__ty_2Enum_2Enum] :
( ( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Eind__type_2ENUMSUM @ V0b1 ) @ ( inj__ty_2Enum_2Enum @ V1x1 ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Eind__type_2ENUMSUM @ V2b2 ) @ ( inj__ty_2Enum_2Enum @ V3x2 ) ) ) )
<=> ( ( ( p @ V0b1 )
<=> ( p @ V2b2 ) )
& ( V1x1 = V3x2 ) ) ) ) ) ).
thf(ax_thm_2Eind__type_2ENUMSUM__DEST,axiom,
! [V0x: $i] :
( ( mem @ V0x @ bool )
=> ! [V1y: tp__ty_2Enum_2Enum] :
( ( ( p @ ( ap @ c_2Eind__type_2ENUMLEFT @ ( ap @ ( ap @ c_2Eind__type_2ENUMSUM @ V0x ) @ ( inj__ty_2Enum_2Enum @ V1y ) ) ) )
<=> ( p @ V0x ) )
& ( ( surj__ty_2Enum_2Enum @ ( ap @ c_2Eind__type_2ENUMRIGHT @ ( ap @ ( ap @ c_2Eind__type_2ENUMSUM @ V0x ) @ ( inj__ty_2Enum_2Enum @ V1y ) ) ) )
= V1y ) ) ) ).
thf(ax_thm_2Eind__type_2EINJN,axiom,
! [A_27a: del,V0m: tp__ty_2Enum_2Enum] :
( ( ap @ ( c_2Eind__type_2EINJN @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V0m ) )
= ( lam @ ty_2Enum_2Enum
@ ^ [V1n: $i] :
( lam @ A_27a
@ ^ [V2a: $i] : ( ap @ ( ap @ ( c_2Emin_2E_3D @ ty_2Enum_2Enum ) @ V1n ) @ ( inj__ty_2Enum_2Enum @ V0m ) ) ) ) ) ).
thf(conj_thm_2Eind__type_2EINJN__INJ,axiom,
! [A_27a: del,V0n1: tp__ty_2Enum_2Enum,V1n2: tp__ty_2Enum_2Enum] :
( ( ( ap @ ( c_2Eind__type_2EINJN @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V0n1 ) )
= ( ap @ ( c_2Eind__type_2EINJN @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V1n2 ) ) )
<=> ( V0n1 = V1n2 ) ) ).
thf(ax_thm_2Eind__type_2EINJA,axiom,
! [A_27a: del,V0a: $i] :
( ( mem @ V0a @ A_27a )
=> ( ( ap @ ( c_2Eind__type_2EINJA @ A_27a ) @ V0a )
= ( lam @ ty_2Enum_2Enum
@ ^ [V1n: $i] :
( lam @ A_27a
@ ^ [V2b: $i] : ( ap @ ( ap @ ( c_2Emin_2E_3D @ A_27a ) @ V2b ) @ V0a ) ) ) ) ) ).
thf(conj_thm_2Eind__type_2EINJA__INJ,axiom,
! [A_27a: del,V0a1: $i] :
( ( mem @ V0a1 @ A_27a )
=> ! [V1a2: $i] :
( ( mem @ V1a2 @ A_27a )
=> ( ( ( ap @ ( c_2Eind__type_2EINJA @ A_27a ) @ V0a1 )
= ( ap @ ( c_2Eind__type_2EINJA @ A_27a ) @ V1a2 ) )
<=> ( V0a1 = V1a2 ) ) ) ) ).
thf(ax_thm_2Eind__type_2EINJF,axiom,
! [A_27a: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) ) )
=> ( ( ap @ ( c_2Eind__type_2EINJF @ A_27a ) @ V0f )
= ( lam @ ty_2Enum_2Enum
@ ^ [V1n: $i] : ( ap @ ( ap @ V0f @ ( ap @ c_2Eind__type_2ENUMFST @ V1n ) ) @ ( ap @ c_2Eind__type_2ENUMSND @ V1n ) ) ) ) ) ).
thf(conj_thm_2Eind__type_2EINJF__INJ,axiom,
! [A_27a: del,V0f1: $i] :
( ( mem @ V0f1 @ ( arr @ ty_2Enum_2Enum @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) ) )
=> ! [V1f2: $i] :
( ( mem @ V1f2 @ ( arr @ ty_2Enum_2Enum @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) ) )
=> ( ( ( ap @ ( c_2Eind__type_2EINJF @ A_27a ) @ V0f1 )
= ( ap @ ( c_2Eind__type_2EINJF @ A_27a ) @ V1f2 ) )
<=> ( V0f1 = V1f2 ) ) ) ) ).
thf(ax_thm_2Eind__type_2EINJP,axiom,
! [A_27a: del,V0f1: $i] :
( ( mem @ V0f1 @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) )
=> ! [V1f2: $i] :
( ( mem @ V1f2 @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) )
=> ( ( ap @ ( ap @ ( c_2Eind__type_2EINJP @ A_27a ) @ V0f1 ) @ V1f2 )
= ( lam @ ty_2Enum_2Enum
@ ^ [V2n: $i] :
( lam @ A_27a
@ ^ [V3a: $i] : ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ bool ) @ ( ap @ c_2Eind__type_2ENUMLEFT @ V2n ) ) @ ( ap @ ( ap @ V0f1 @ ( ap @ c_2Eind__type_2ENUMRIGHT @ V2n ) ) @ V3a ) ) @ ( ap @ ( ap @ V1f2 @ ( ap @ c_2Eind__type_2ENUMRIGHT @ V2n ) ) @ V3a ) ) ) ) ) ) ) ).
thf(conj_thm_2Eind__type_2EINJP__INJ,axiom,
! [A_27a: del,V0f1: $i] :
( ( mem @ V0f1 @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) )
=> ! [V1f1_27: $i] :
( ( mem @ V1f1_27 @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) )
=> ! [V2f2: $i] :
( ( mem @ V2f2 @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) )
=> ! [V3f2_27: $i] :
( ( mem @ V3f2_27 @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) )
=> ( ( ( ap @ ( ap @ ( c_2Eind__type_2EINJP @ A_27a ) @ V0f1 ) @ V2f2 )
= ( ap @ ( ap @ ( c_2Eind__type_2EINJP @ A_27a ) @ V1f1_27 ) @ V3f2_27 ) )
<=> ( ( V0f1 = V1f1_27 )
& ( V2f2 = V3f2_27 ) ) ) ) ) ) ) ).
thf(ax_thm_2Eind__type_2EZCONSTR,axiom,
! [A_27a: del,V0c: tp__ty_2Enum_2Enum,V1i: $i] :
( ( mem @ V1i @ A_27a )
=> ! [V2r: $i] :
( ( mem @ V2r @ ( arr @ ty_2Enum_2Enum @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) ) )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Eind__type_2EZCONSTR @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V0c ) ) @ V1i ) @ V2r )
= ( ap @ ( ap @ ( c_2Eind__type_2EINJP @ A_27a ) @ ( ap @ ( c_2Eind__type_2EINJN @ A_27a ) @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V0c ) ) ) ) @ ( ap @ ( ap @ ( c_2Eind__type_2EINJP @ A_27a ) @ ( ap @ ( c_2Eind__type_2EINJA @ A_27a ) @ V1i ) ) @ ( ap @ ( c_2Eind__type_2EINJF @ A_27a ) @ V2r ) ) ) ) ) ) ).
thf(ax_thm_2Eind__type_2EZBOT,axiom,
! [A_27a: del] :
( ( c_2Eind__type_2EZBOT @ A_27a )
= ( ap @ ( ap @ ( c_2Eind__type_2EINJP @ A_27a ) @ ( ap @ ( c_2Eind__type_2EINJN @ A_27a ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) )
@ ( ap @ ( c_2Emin_2E_40 @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) )
@ ( lam @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) )
@ ^ [V0z: $i] : c_2Ebool_2ET ) ) ) ) ).
thf(conj_thm_2Eind__type_2EZCONSTR__ZBOT,axiom,
! [A_27a: del,V0c: tp__ty_2Enum_2Enum,V1i: $i] :
( ( mem @ V1i @ A_27a )
=> ! [V2r: $i] :
( ( mem @ V2r @ ( arr @ ty_2Enum_2Enum @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) ) )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Eind__type_2EZCONSTR @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V0c ) ) @ V1i ) @ V2r )
!= ( c_2Eind__type_2EZBOT @ A_27a ) ) ) ) ).
thf(ax_thm_2Eind__type_2EZRECSPACE__def,axiom,
! [A_27a: del] :
( ( c_2Eind__type_2EZRECSPACE @ A_27a )
= ( lam @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) )
@ ^ [V0a0: $i] :
( ap @ ( c_2Ebool_2E_21 @ ( arr @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) @ bool ) )
@ ( lam @ ( arr @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) @ bool )
@ ^ [V1ZRECSPACE_27: $i] :
( ap
@ ( ap @ c_2Emin_2E_3D_3D_3E
@ ( ap @ ( c_2Ebool_2E_21 @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) )
@ ( lam @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) )
@ ^ [V2a0: $i] :
( ap
@ ( ap @ c_2Emin_2E_3D_3D_3E
@ ( ap @ ( ap @ c_2Ebool_2E_5C_2F @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) ) @ V2a0 ) @ ( c_2Eind__type_2EZBOT @ A_27a ) ) )
@ ( ap @ ( c_2Ebool_2E_3F @ ty_2Enum_2Enum )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V3c: $i] :
( ap @ ( c_2Ebool_2E_3F @ A_27a )
@ ( lam @ A_27a
@ ^ [V4i: $i] :
( ap @ ( c_2Ebool_2E_3F @ ( arr @ ty_2Enum_2Enum @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) ) )
@ ( lam @ ( arr @ ty_2Enum_2Enum @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) )
@ ^ [V5r: $i] :
( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) ) @ V2a0 ) @ ( ap @ ( ap @ ( ap @ ( c_2Eind__type_2EZCONSTR @ A_27a ) @ V3c ) @ V4i ) @ V5r ) ) )
@ ( ap @ ( c_2Ebool_2E_21 @ ty_2Enum_2Enum )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V6n: $i] : ( ap @ V1ZRECSPACE_27 @ ( ap @ V5r @ V6n ) ) ) ) ) ) ) ) ) ) ) ) )
@ ( ap @ V1ZRECSPACE_27 @ V2a0 ) ) ) ) )
@ ( ap @ V1ZRECSPACE_27 @ V0a0 ) ) ) ) ) ) ).
thf(conj_thm_2Eind__type_2EZRECSPACE__rules,axiom,
! [A_27a: del] :
( ( p @ ( ap @ ( c_2Eind__type_2EZRECSPACE @ A_27a ) @ ( c_2Eind__type_2EZBOT @ A_27a ) ) )
& ! [V0c: tp__ty_2Enum_2Enum,V1i: $i] :
( ( mem @ V1i @ A_27a )
=> ! [V2r: $i] :
( ( mem @ V2r @ ( arr @ ty_2Enum_2Enum @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) ) )
=> ( ! [V3n: tp__ty_2Enum_2Enum] : ( p @ ( ap @ ( c_2Eind__type_2EZRECSPACE @ A_27a ) @ ( ap @ V2r @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) )
=> ( p @ ( ap @ ( c_2Eind__type_2EZRECSPACE @ A_27a ) @ ( ap @ ( ap @ ( ap @ ( c_2Eind__type_2EZCONSTR @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V0c ) ) @ V1i ) @ V2r ) ) ) ) ) ) ) ).
thf(conj_thm_2Eind__type_2EZRECSPACE__ind,axiom,
! [A_27a: del,V0ZRECSPACE_27: $i] :
( ( mem @ V0ZRECSPACE_27 @ ( arr @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) @ bool ) )
=> ( ( ( p @ ( ap @ V0ZRECSPACE_27 @ ( c_2Eind__type_2EZBOT @ A_27a ) ) )
& ! [V1c: tp__ty_2Enum_2Enum,V2i: $i] :
( ( mem @ V2i @ A_27a )
=> ! [V3r: $i] :
( ( mem @ V3r @ ( arr @ ty_2Enum_2Enum @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) ) )
=> ( ! [V4n: tp__ty_2Enum_2Enum] : ( p @ ( ap @ V0ZRECSPACE_27 @ ( ap @ V3r @ ( inj__ty_2Enum_2Enum @ V4n ) ) ) )
=> ( p @ ( ap @ V0ZRECSPACE_27 @ ( ap @ ( ap @ ( ap @ ( c_2Eind__type_2EZCONSTR @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V1c ) ) @ V2i ) @ V3r ) ) ) ) ) ) )
=> ! [V5a0: $i] :
( ( mem @ V5a0 @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) )
=> ( ( p @ ( ap @ ( c_2Eind__type_2EZRECSPACE @ A_27a ) @ V5a0 ) )
=> ( p @ ( ap @ V0ZRECSPACE_27 @ V5a0 ) ) ) ) ) ) ).
thf(conj_thm_2Eind__type_2EZRECSPACE__strongind,axiom,
! [A_27a: del,V0ZRECSPACE_27: $i] :
( ( mem @ V0ZRECSPACE_27 @ ( arr @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) @ bool ) )
=> ( ( ( p @ ( ap @ V0ZRECSPACE_27 @ ( c_2Eind__type_2EZBOT @ A_27a ) ) )
& ! [V1c: tp__ty_2Enum_2Enum,V2i: $i] :
( ( mem @ V2i @ A_27a )
=> ! [V3r: $i] :
( ( mem @ V3r @ ( arr @ ty_2Enum_2Enum @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) ) )
=> ( ! [V4n: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( c_2Eind__type_2EZRECSPACE @ A_27a ) @ ( ap @ V3r @ ( inj__ty_2Enum_2Enum @ V4n ) ) ) )
& ( p @ ( ap @ V0ZRECSPACE_27 @ ( ap @ V3r @ ( inj__ty_2Enum_2Enum @ V4n ) ) ) ) )
=> ( p @ ( ap @ V0ZRECSPACE_27 @ ( ap @ ( ap @ ( ap @ ( c_2Eind__type_2EZCONSTR @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V1c ) ) @ V2i ) @ V3r ) ) ) ) ) ) )
=> ! [V5a0: $i] :
( ( mem @ V5a0 @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) )
=> ( ( p @ ( ap @ ( c_2Eind__type_2EZRECSPACE @ A_27a ) @ V5a0 ) )
=> ( p @ ( ap @ V0ZRECSPACE_27 @ V5a0 ) ) ) ) ) ) ).
thf(conj_thm_2Eind__type_2EZRECSPACE__cases,axiom,
! [A_27a: del,V0a0: $i] :
( ( mem @ V0a0 @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) )
=> ( ( p @ ( ap @ ( c_2Eind__type_2EZRECSPACE @ A_27a ) @ V0a0 ) )
<=> ( ( V0a0
= ( c_2Eind__type_2EZBOT @ A_27a ) )
| ? [V1c: tp__ty_2Enum_2Enum,V2i: $i] :
( ( mem @ V2i @ A_27a )
& ? [V3r: $i] :
( ( mem @ V3r @ ( arr @ ty_2Enum_2Enum @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) ) )
& ( V0a0
= ( ap @ ( ap @ ( ap @ ( c_2Eind__type_2EZCONSTR @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V1c ) ) @ V2i ) @ V3r ) )
& ! [V4n: tp__ty_2Enum_2Enum] : ( p @ ( ap @ ( c_2Eind__type_2EZRECSPACE @ A_27a ) @ ( ap @ V3r @ ( inj__ty_2Enum_2Enum @ V4n ) ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Eind__type_2Erecspace__TY__DEF,axiom,
! [A_27a: del] :
? [V0rep: $i] :
( ( mem @ V0rep @ ( arr @ ( ty_2Eind__type_2Erecspace @ A_27a ) @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2ETYPE__DEFINITION @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) @ ( ty_2Eind__type_2Erecspace @ A_27a ) ) @ ( c_2Eind__type_2EZRECSPACE @ A_27a ) ) @ V0rep ) ) ) ).
thf(ax_thm_2Eind__type_2Erecspace__repfns,axiom,
! [A_27a: del] :
( ! [V0a: $i] :
( ( mem @ V0a @ ( ty_2Eind__type_2Erecspace @ A_27a ) )
=> ( ( ap @ ( c_2Eind__type_2Emk__rec @ A_27a ) @ ( ap @ ( c_2Eind__type_2Edest__rec @ A_27a ) @ V0a ) )
= V0a ) )
& ! [V1r: $i] :
( ( mem @ V1r @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) )
=> ( ( p @ ( ap @ ( c_2Eind__type_2EZRECSPACE @ A_27a ) @ V1r ) )
<=> ( ( ap @ ( c_2Eind__type_2Edest__rec @ A_27a ) @ ( ap @ ( c_2Eind__type_2Emk__rec @ A_27a ) @ V1r ) )
= V1r ) ) ) ) ).
thf(ax_thm_2Eind__type_2EBOTTOM,axiom,
! [A_27a: del] :
( ( c_2Eind__type_2EBOTTOM @ A_27a )
= ( ap @ ( c_2Eind__type_2Emk__rec @ A_27a ) @ ( c_2Eind__type_2EZBOT @ A_27a ) ) ) ).
thf(ax_thm_2Eind__type_2ECONSTR,axiom,
! [A_27a: del,V0c: tp__ty_2Enum_2Enum,V1i: $i] :
( ( mem @ V1i @ A_27a )
=> ! [V2r: $i] :
( ( mem @ V2r @ ( arr @ ty_2Enum_2Enum @ ( ty_2Eind__type_2Erecspace @ A_27a ) ) )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Eind__type_2ECONSTR @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V0c ) ) @ V1i ) @ V2r )
= ( ap @ ( c_2Eind__type_2Emk__rec @ A_27a )
@ ( ap @ ( ap @ ( ap @ ( c_2Eind__type_2EZCONSTR @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V0c ) ) @ V1i )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V3n: $i] : ( ap @ ( c_2Eind__type_2Edest__rec @ A_27a ) @ ( ap @ V2r @ V3n ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eind__type_2EMK__REC__INJ,axiom,
! [A_27a: del,V0x: $i] :
( ( mem @ V0x @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) )
=> ! [V1y: $i] :
( ( mem @ V1y @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) )
=> ( ( ( ap @ ( c_2Eind__type_2Emk__rec @ A_27a ) @ V0x )
= ( ap @ ( c_2Eind__type_2Emk__rec @ A_27a ) @ V1y ) )
=> ( ( ( p @ ( ap @ ( c_2Eind__type_2EZRECSPACE @ A_27a ) @ V0x ) )
& ( p @ ( ap @ ( c_2Eind__type_2EZRECSPACE @ A_27a ) @ V1y ) ) )
=> ( V0x = V1y ) ) ) ) ) ).
thf(conj_thm_2Eind__type_2EDEST__REC__INJ,axiom,
! [A_27a: del,V0x: $i] :
( ( mem @ V0x @ ( ty_2Eind__type_2Erecspace @ A_27a ) )
=> ! [V1y: $i] :
( ( mem @ V1y @ ( ty_2Eind__type_2Erecspace @ A_27a ) )
=> ( ( ( ap @ ( c_2Eind__type_2Edest__rec @ A_27a ) @ V0x )
= ( ap @ ( c_2Eind__type_2Edest__rec @ A_27a ) @ V1y ) )
<=> ( V0x = V1y ) ) ) ) ).
thf(conj_thm_2Eind__type_2ECONSTR__BOT,axiom,
! [A_27a: del,V0c: tp__ty_2Enum_2Enum,V1i: $i] :
( ( mem @ V1i @ A_27a )
=> ! [V2r: $i] :
( ( mem @ V2r @ ( arr @ ty_2Enum_2Enum @ ( ty_2Eind__type_2Erecspace @ A_27a ) ) )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Eind__type_2ECONSTR @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V0c ) ) @ V1i ) @ V2r )
!= ( c_2Eind__type_2EBOTTOM @ A_27a ) ) ) ) ).
thf(conj_thm_2Eind__type_2ECONSTR__INJ,axiom,
! [A_27a: del,V0c1: tp__ty_2Enum_2Enum,V1i1: $i] :
( ( mem @ V1i1 @ A_27a )
=> ! [V2r1: $i] :
( ( mem @ V2r1 @ ( arr @ ty_2Enum_2Enum @ ( ty_2Eind__type_2Erecspace @ A_27a ) ) )
=> ! [V3c2: tp__ty_2Enum_2Enum,V4i2: $i] :
( ( mem @ V4i2 @ A_27a )
=> ! [V5r2: $i] :
( ( mem @ V5r2 @ ( arr @ ty_2Enum_2Enum @ ( ty_2Eind__type_2Erecspace @ A_27a ) ) )
=> ( ( ( ap @ ( ap @ ( ap @ ( c_2Eind__type_2ECONSTR @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V0c1 ) ) @ V1i1 ) @ V2r1 )
= ( ap @ ( ap @ ( ap @ ( c_2Eind__type_2ECONSTR @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V3c2 ) ) @ V4i2 ) @ V5r2 ) )
<=> ( ( V0c1 = V3c2 )
& ( V1i1 = V4i2 )
& ( V2r1 = V5r2 ) ) ) ) ) ) ) ).
thf(conj_thm_2Eind__type_2ECONSTR__IND,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ ( ty_2Eind__type_2Erecspace @ A_27a ) @ bool ) )
=> ( ( ( p @ ( ap @ V0P @ ( c_2Eind__type_2EBOTTOM @ A_27a ) ) )
& ! [V1c: tp__ty_2Enum_2Enum,V2i: $i] :
( ( mem @ V2i @ A_27a )
=> ! [V3r: $i] :
( ( mem @ V3r @ ( arr @ ty_2Enum_2Enum @ ( ty_2Eind__type_2Erecspace @ A_27a ) ) )
=> ( ! [V4n: tp__ty_2Enum_2Enum] : ( p @ ( ap @ V0P @ ( ap @ V3r @ ( inj__ty_2Enum_2Enum @ V4n ) ) ) )
=> ( p @ ( ap @ V0P @ ( ap @ ( ap @ ( ap @ ( c_2Eind__type_2ECONSTR @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V1c ) ) @ V2i ) @ V3r ) ) ) ) ) ) )
=> ! [V5x: $i] :
( ( mem @ V5x @ ( ty_2Eind__type_2Erecspace @ A_27a ) )
=> ( p @ ( ap @ V0P @ V5x ) ) ) ) ) ).
thf(conj_thm_2Eind__type_2ECONSTR__REC,axiom,
! [A_27a: del,A_27b: del,V0Fn: $i] :
( ( mem @ V0Fn @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ ( arr @ ( arr @ ty_2Enum_2Enum @ ( ty_2Eind__type_2Erecspace @ A_27a ) ) @ ( arr @ ( arr @ ty_2Enum_2Enum @ A_27b ) @ A_27b ) ) ) ) )
=> ? [V1f: $i] :
( ( mem @ V1f @ ( arr @ ( ty_2Eind__type_2Erecspace @ A_27a ) @ A_27b ) )
& ! [V2c: tp__ty_2Enum_2Enum,V3i: $i] :
( ( mem @ V3i @ A_27a )
=> ! [V4r: $i] :
( ( mem @ V4r @ ( arr @ ty_2Enum_2Enum @ ( ty_2Eind__type_2Erecspace @ A_27a ) ) )
=> ( ( ap @ V1f @ ( ap @ ( ap @ ( ap @ ( c_2Eind__type_2ECONSTR @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V2c ) ) @ V3i ) @ V4r ) )
= ( ap @ ( ap @ ( ap @ ( ap @ V0Fn @ ( inj__ty_2Enum_2Enum @ V2c ) ) @ V3i ) @ V4r )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V5n: $i] : ( ap @ V1f @ ( ap @ V4r @ V5n ) ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Eind__type_2EFCONS,axiom,
! [A_27a: del] :
( ! [V0a: $i] :
( ( mem @ V0a @ A_27a )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ ty_2Enum_2Enum @ A_27a ) )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Eind__type_2EFCONS @ A_27a ) @ V0a ) @ V1f ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) )
= V0a ) ) )
& ! [V2a: $i] :
( ( mem @ V2a @ A_27a )
=> ! [V3f: $i] :
( ( mem @ V3f @ ( arr @ ty_2Enum_2Enum @ A_27a ) )
=> ! [V4n: tp__ty_2Enum_2Enum] :
( ( ap @ ( ap @ ( ap @ ( c_2Eind__type_2EFCONS @ A_27a ) @ V2a ) @ V3f ) @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V4n ) ) )
= ( ap @ V3f @ ( inj__ty_2Enum_2Enum @ V4n ) ) ) ) ) ) ).
thf(ax_thm_2Eind__type_2EFNIL,axiom,
! [A_27a: del,V0n: tp__ty_2Enum_2Enum] :
( ( ap @ ( c_2Eind__type_2EFNIL @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V0n ) )
= ( c_2Ebool_2EARB @ A_27a ) ) ).
thf(conj_thm_2Eind__type_2EFCONS__DEST,axiom,
! [A_27a: del,V0a: $i] :
( ( mem @ V0a @ A_27a )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ ty_2Enum_2Enum @ A_27a ) )
=> ! [V2n: tp__ty_2Enum_2Enum] :
( ( ap @ ( ap @ ( ap @ ( c_2Eind__type_2EFCONS @ A_27a ) @ V0a ) @ V1f ) @ ( inj__ty_2Enum_2Enum @ V2n ) )
= ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ A_27a ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ V0a ) @ ( ap @ V1f @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ ( inj__ty_2Enum_2Enum @ V2n ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Eind__type_2EISO,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27b ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ A_27b @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Eind__type_2EISO @ A_27a @ A_27b ) @ V0f ) @ V1g ) )
<=> ( ! [V2x: $i] :
( ( mem @ V2x @ A_27b )
=> ( ( ap @ V0f @ ( ap @ V1g @ V2x ) )
= V2x ) )
& ! [V3y: $i] :
( ( mem @ V3y @ A_27a )
=> ( ( ap @ V1g @ ( ap @ V0f @ V3y ) )
= V3y ) ) ) ) ) ) ).
thf(conj_thm_2Eind__type_2EISO__REFL,axiom,
! [A_27a: del] :
( p
@ ( ap
@ ( ap @ ( c_2Eind__type_2EISO @ A_27a @ A_27a )
@ ( lam @ A_27a
@ ^ [V0x: $i] : V0x ) )
@ ( lam @ A_27a
@ ^ [V1x: $i] : V1x ) ) ) ).
thf(conj_thm_2Eind__type_2EISO__FUN,axiom,
! [A_27a: del,A_27b: del,A_27c: del,A_27d: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27c ) )
=> ! [V1f_27: $i] :
( ( mem @ V1f_27 @ ( arr @ A_27c @ A_27a ) )
=> ! [V2g: $i] :
( ( mem @ V2g @ ( arr @ A_27b @ A_27d ) )
=> ! [V3g_27: $i] :
( ( mem @ V3g_27 @ ( arr @ A_27d @ A_27b ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Eind__type_2EISO @ A_27a @ A_27c ) @ V0f ) @ V1f_27 ) )
& ( p @ ( ap @ ( ap @ ( c_2Eind__type_2EISO @ A_27b @ A_27d ) @ V2g ) @ V3g_27 ) ) )
=> ( p
@ ( ap
@ ( ap @ ( c_2Eind__type_2EISO @ ( arr @ A_27a @ A_27b ) @ ( arr @ A_27c @ A_27d ) )
@ ( lam @ ( arr @ A_27a @ A_27b )
@ ^ [V4h: $i] :
( lam @ A_27c
@ ^ [V5a_27: $i] : ( ap @ V2g @ ( ap @ V4h @ ( ap @ V1f_27 @ V5a_27 ) ) ) ) ) )
@ ( lam @ ( arr @ A_27c @ A_27d )
@ ^ [V6h: $i] :
( lam @ A_27a
@ ^ [V7a: $i] : ( ap @ V3g_27 @ ( ap @ V6h @ ( ap @ V0f @ V7a ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eind__type_2EISO__USAGE,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27b ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ A_27b @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Eind__type_2EISO @ A_27a @ A_27b ) @ V0f ) @ V1g ) )
=> ( ! [V2P: $i] :
( ( mem @ V2P @ ( arr @ A_27a @ bool ) )
=> ( ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ( p @ ( ap @ V2P @ V3x ) ) )
<=> ! [V4x: $i] :
( ( mem @ V4x @ A_27b )
=> ( p @ ( ap @ V2P @ ( ap @ V1g @ V4x ) ) ) ) ) )
& ! [V5P: $i] :
( ( mem @ V5P @ ( arr @ A_27a @ bool ) )
=> ( ? [V6x: $i] :
( ( mem @ V6x @ A_27a )
& ( p @ ( ap @ V5P @ V6x ) ) )
<=> ? [V7x: $i] :
( ( mem @ V7x @ A_27b )
& ( p @ ( ap @ V5P @ ( ap @ V1g @ V7x ) ) ) ) ) )
& ! [V8a: $i] :
( ( mem @ V8a @ A_27a )
=> ! [V9b: $i] :
( ( mem @ V9b @ A_27b )
=> ( ( V8a
= ( ap @ V1g @ V9b ) )
<=> ( ( ap @ V0f @ V8a )
= V9b ) ) ) ) ) ) ) ) ).
%------------------------------------------------------------------------------