ITP001 Axioms: ITP018^5.ax
%------------------------------------------------------------------------------
% File : ITP018^5 : 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 : numeral^2.ax [Gau20]
% : HL4018^5.ax [TPAP]
% Status : Satisfiable
% Syntax : Number of formulae : 78 ( 17 unt; 18 typ; 0 def)
% Number of atoms : 3056 ( 158 equ; 0 cnn)
% Maximal formula atoms : 205 ( 39 avg)
% Number of connectives : 3455 ( 6 ~; 0 |; 142 &;3243 @)
% ( 39 <=>; 25 =>; 0 <=; 0 <~>)
% Maximal formula depth : 42 ( 7 avg;3243 nst)
% Number of types : 1 ( 0 usr)
% Number of type conns : 12 ( 12 >; 0 *; 0 +; 0 <<)
% Number of symbols : 68 ( 65 usr; 58 con; 0-2 aty)
% Number of variables : 180 ( 14 ^ 165 !; 1 ?; 180 :)
% SPC : TH0_SAT_EQU_NAR
% Comments :
% Bugfixes : v7.5.0 - Fixes to the axioms.
%------------------------------------------------------------------------------
thf(tp_c_2Enumeral_2Eexactlog,type,
c_2Enumeral_2Eexactlog: $i ).
thf(mem_c_2Enumeral_2Eexactlog,axiom,
mem @ c_2Enumeral_2Eexactlog @ ( arr @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) ).
thf(stp_fo_c_2Enumeral_2Eexactlog,type,
fo__c_2Enumeral_2Eexactlog: tp__ty_2Enum_2Enum > tp__ty_2Enum_2Enum ).
thf(stp_eq_fo_c_2Enumeral_2Eexactlog,axiom,
! [X0: tp__ty_2Enum_2Enum] :
( ( inj__ty_2Enum_2Enum @ ( fo__c_2Enumeral_2Eexactlog @ X0 ) )
= ( ap @ c_2Enumeral_2Eexactlog @ ( inj__ty_2Enum_2Enum @ X0 ) ) ) ).
thf(tp_c_2Enumeral_2EiBIT__cases,type,
c_2Enumeral_2EiBIT__cases: del > $i ).
thf(mem_c_2Enumeral_2EiBIT__cases,axiom,
! [A_27a: del] : ( mem @ ( c_2Enumeral_2EiBIT__cases @ A_27a ) @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ ( arr @ ( arr @ ty_2Enum_2Enum @ A_27a ) @ ( arr @ ( arr @ ty_2Enum_2Enum @ A_27a ) @ A_27a ) ) ) ) ) ).
thf(tp_c_2Enumeral_2EiDUB,type,
c_2Enumeral_2EiDUB: $i ).
thf(mem_c_2Enumeral_2EiDUB,axiom,
mem @ c_2Enumeral_2EiDUB @ ( arr @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) ).
thf(stp_fo_c_2Enumeral_2EiDUB,type,
fo__c_2Enumeral_2EiDUB: tp__ty_2Enum_2Enum > tp__ty_2Enum_2Enum ).
thf(stp_eq_fo_c_2Enumeral_2EiDUB,axiom,
! [X0: tp__ty_2Enum_2Enum] :
( ( inj__ty_2Enum_2Enum @ ( fo__c_2Enumeral_2EiDUB @ X0 ) )
= ( ap @ c_2Enumeral_2EiDUB @ ( inj__ty_2Enum_2Enum @ X0 ) ) ) ).
thf(tp_c_2Enumeral_2EiSQR,type,
c_2Enumeral_2EiSQR: $i ).
thf(mem_c_2Enumeral_2EiSQR,axiom,
mem @ c_2Enumeral_2EiSQR @ ( arr @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) ).
thf(stp_fo_c_2Enumeral_2EiSQR,type,
fo__c_2Enumeral_2EiSQR: tp__ty_2Enum_2Enum > tp__ty_2Enum_2Enum ).
thf(stp_eq_fo_c_2Enumeral_2EiSQR,axiom,
! [X0: tp__ty_2Enum_2Enum] :
( ( inj__ty_2Enum_2Enum @ ( fo__c_2Enumeral_2EiSQR @ X0 ) )
= ( ap @ c_2Enumeral_2EiSQR @ ( inj__ty_2Enum_2Enum @ X0 ) ) ) ).
thf(tp_c_2Enumeral_2EiSUB,type,
c_2Enumeral_2EiSUB: $i ).
thf(mem_c_2Enumeral_2EiSUB,axiom,
mem @ c_2Enumeral_2EiSUB @ ( arr @ bool @ ( arr @ ty_2Enum_2Enum @ ( arr @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) ) ) ).
thf(tp_c_2Enumeral_2EiZ,type,
c_2Enumeral_2EiZ: $i ).
thf(mem_c_2Enumeral_2EiZ,axiom,
mem @ c_2Enumeral_2EiZ @ ( arr @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) ).
thf(stp_fo_c_2Enumeral_2EiZ,type,
fo__c_2Enumeral_2EiZ: tp__ty_2Enum_2Enum > tp__ty_2Enum_2Enum ).
thf(stp_eq_fo_c_2Enumeral_2EiZ,axiom,
! [X0: tp__ty_2Enum_2Enum] :
( ( inj__ty_2Enum_2Enum @ ( fo__c_2Enumeral_2EiZ @ X0 ) )
= ( ap @ c_2Enumeral_2EiZ @ ( inj__ty_2Enum_2Enum @ X0 ) ) ) ).
thf(tp_c_2Enumeral_2EiiSUC,type,
c_2Enumeral_2EiiSUC: $i ).
thf(mem_c_2Enumeral_2EiiSUC,axiom,
mem @ c_2Enumeral_2EiiSUC @ ( arr @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) ).
thf(stp_fo_c_2Enumeral_2EiiSUC,type,
fo__c_2Enumeral_2EiiSUC: tp__ty_2Enum_2Enum > tp__ty_2Enum_2Enum ).
thf(stp_eq_fo_c_2Enumeral_2EiiSUC,axiom,
! [X0: tp__ty_2Enum_2Enum] :
( ( inj__ty_2Enum_2Enum @ ( fo__c_2Enumeral_2EiiSUC @ X0 ) )
= ( ap @ c_2Enumeral_2EiiSUC @ ( inj__ty_2Enum_2Enum @ X0 ) ) ) ).
thf(tp_c_2Enumeral_2Einternal__mult,type,
c_2Enumeral_2Einternal__mult: $i ).
thf(mem_c_2Enumeral_2Einternal__mult,axiom,
mem @ c_2Enumeral_2Einternal__mult @ ( arr @ ty_2Enum_2Enum @ ( arr @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) ) ).
thf(stp_fo_c_2Enumeral_2Einternal__mult,type,
fo__c_2Enumeral_2Einternal__mult: tp__ty_2Enum_2Enum > tp__ty_2Enum_2Enum > tp__ty_2Enum_2Enum ).
thf(stp_eq_fo_c_2Enumeral_2Einternal__mult,axiom,
! [X0: tp__ty_2Enum_2Enum,X1: tp__ty_2Enum_2Enum] :
( ( inj__ty_2Enum_2Enum @ ( fo__c_2Enumeral_2Einternal__mult @ X0 @ X1 ) )
= ( ap @ ( ap @ c_2Enumeral_2Einternal__mult @ ( inj__ty_2Enum_2Enum @ X0 ) ) @ ( inj__ty_2Enum_2Enum @ X1 ) ) ) ).
thf(tp_c_2Enumeral_2Eonecount,type,
c_2Enumeral_2Eonecount: $i ).
thf(mem_c_2Enumeral_2Eonecount,axiom,
mem @ c_2Enumeral_2Eonecount @ ( arr @ ty_2Enum_2Enum @ ( arr @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) ) ).
thf(stp_fo_c_2Enumeral_2Eonecount,type,
fo__c_2Enumeral_2Eonecount: tp__ty_2Enum_2Enum > tp__ty_2Enum_2Enum > tp__ty_2Enum_2Enum ).
thf(stp_eq_fo_c_2Enumeral_2Eonecount,axiom,
! [X0: tp__ty_2Enum_2Enum,X1: tp__ty_2Enum_2Enum] :
( ( inj__ty_2Enum_2Enum @ ( fo__c_2Enumeral_2Eonecount @ X0 @ X1 ) )
= ( ap @ ( ap @ c_2Enumeral_2Eonecount @ ( inj__ty_2Enum_2Enum @ X0 ) ) @ ( inj__ty_2Enum_2Enum @ X1 ) ) ) ).
thf(tp_c_2Enumeral_2Etexp__help,type,
c_2Enumeral_2Etexp__help: $i ).
thf(mem_c_2Enumeral_2Etexp__help,axiom,
mem @ c_2Enumeral_2Etexp__help @ ( arr @ ty_2Enum_2Enum @ ( arr @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) ) ).
thf(stp_fo_c_2Enumeral_2Etexp__help,type,
fo__c_2Enumeral_2Etexp__help: tp__ty_2Enum_2Enum > tp__ty_2Enum_2Enum > tp__ty_2Enum_2Enum ).
thf(stp_eq_fo_c_2Enumeral_2Etexp__help,axiom,
! [X0: tp__ty_2Enum_2Enum,X1: tp__ty_2Enum_2Enum] :
( ( inj__ty_2Enum_2Enum @ ( fo__c_2Enumeral_2Etexp__help @ X0 @ X1 ) )
= ( ap @ ( ap @ c_2Enumeral_2Etexp__help @ ( inj__ty_2Enum_2Enum @ X0 ) ) @ ( inj__ty_2Enum_2Enum @ X1 ) ) ) ).
thf(conj_thm_2Enumeral_2Enumeral__suc,axiom,
( ( ( surj__ty_2Enum_2Enum @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) )
& ! [V0n: tp__ty_2Enum_2Enum] :
( ( surj__ty_2Enum_2Enum @ ( ap @ c_2Enum_2ESUC @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) )
& ! [V1n: tp__ty_2Enum_2Enum] :
( ( surj__ty_2Enum_2Enum @ ( ap @ c_2Enum_2ESUC @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ c_2Earithmetic_2EBIT1 @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) ) ) ) ).
thf(ax_thm_2Enumeral_2EiZ,axiom,
! [V0x: tp__ty_2Enum_2Enum] :
( ( surj__ty_2Enum_2Enum @ ( ap @ c_2Enumeral_2EiZ @ ( inj__ty_2Enum_2Enum @ V0x ) ) )
= V0x ) ).
thf(ax_thm_2Enumeral_2EiiSUC,axiom,
! [V0n: tp__ty_2Enum_2Enum] :
( ( surj__ty_2Enum_2Enum @ ( ap @ c_2Enumeral_2EiiSUC @ ( inj__ty_2Enum_2Enum @ V0n ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ c_2Enum_2ESUC @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) ) ) ).
thf(conj_thm_2Enumeral_2Enumeral__distrib,axiom,
( ! [V0n: tp__ty_2Enum_2Enum] :
( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) )
= V0n )
& ! [V1n: tp__ty_2Enum_2Enum] :
( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( inj__ty_2Enum_2Enum @ V1n ) ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) )
= V1n )
& ! [V2n: tp__ty_2Enum_2Enum,V3m: tp__ty_2Enum_2Enum] :
( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( inj__ty_2Enum_2Enum @ V3m ) ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Enumeral_2EiZ @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( inj__ty_2Enum_2Enum @ V2n ) ) @ ( inj__ty_2Enum_2Enum @ V3m ) ) ) ) ) )
& ! [V4n: tp__ty_2Enum_2Enum] :
( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Earithmetic_2E_2A @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( inj__ty_2Enum_2Enum @ V4n ) ) )
= fo__c_2Enum_2E0 )
& ! [V5n: tp__ty_2Enum_2Enum] :
( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Earithmetic_2E_2A @ ( inj__ty_2Enum_2Enum @ V5n ) ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) )
= fo__c_2Enum_2E0 )
& ! [V6n: tp__ty_2Enum_2Enum,V7m: tp__ty_2Enum_2Enum] :
( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Earithmetic_2E_2A @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( inj__ty_2Enum_2Enum @ V6n ) ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( inj__ty_2Enum_2Enum @ V7m ) ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ ( ap @ c_2Earithmetic_2E_2A @ ( inj__ty_2Enum_2Enum @ V6n ) ) @ ( inj__ty_2Enum_2Enum @ V7m ) ) ) ) )
& ! [V8n: tp__ty_2Enum_2Enum] :
( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( inj__ty_2Enum_2Enum @ V8n ) ) )
= fo__c_2Enum_2E0 )
& ! [V9n: tp__ty_2Enum_2Enum] :
( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ ( inj__ty_2Enum_2Enum @ V9n ) ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) )
= V9n )
& ! [V10n: tp__ty_2Enum_2Enum,V11m: tp__ty_2Enum_2Enum] :
( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( inj__ty_2Enum_2Enum @ V10n ) ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( inj__ty_2Enum_2Enum @ V11m ) ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ ( inj__ty_2Enum_2Enum @ V10n ) ) @ ( inj__ty_2Enum_2Enum @ V11m ) ) ) ) )
& ! [V12n: tp__ty_2Enum_2Enum] :
( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Earithmetic_2EEXP @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ V12n ) ) ) ) )
= fo__c_2Enum_2E0 )
& ! [V13n: tp__ty_2Enum_2Enum] :
( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Earithmetic_2EEXP @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ V13n ) ) ) ) )
= fo__c_2Enum_2E0 )
& ! [V14n: tp__ty_2Enum_2Enum] :
( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Earithmetic_2EEXP @ ( inj__ty_2Enum_2Enum @ V14n ) ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) )
& ! [V15n: tp__ty_2Enum_2Enum,V16m: tp__ty_2Enum_2Enum] :
( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Earithmetic_2EEXP @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( inj__ty_2Enum_2Enum @ V15n ) ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( inj__ty_2Enum_2Enum @ V16m ) ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ ( ap @ c_2Earithmetic_2EEXP @ ( inj__ty_2Enum_2Enum @ V15n ) ) @ ( inj__ty_2Enum_2Enum @ V16m ) ) ) ) )
& ( ( surj__ty_2Enum_2Enum @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) )
& ! [V17n: tp__ty_2Enum_2Enum] :
( ( surj__ty_2Enum_2Enum @ ( ap @ c_2Enum_2ESUC @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( inj__ty_2Enum_2Enum @ V17n ) ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V17n ) ) ) ) )
& ( ( surj__ty_2Enum_2Enum @ ( ap @ c_2Eprim__rec_2EPRE @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) )
= fo__c_2Enum_2E0 )
& ! [V18n: tp__ty_2Enum_2Enum] :
( ( surj__ty_2Enum_2Enum @ ( ap @ c_2Eprim__rec_2EPRE @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( inj__ty_2Enum_2Enum @ V18n ) ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Eprim__rec_2EPRE @ ( inj__ty_2Enum_2Enum @ V18n ) ) ) ) )
& ! [V19n: tp__ty_2Enum_2Enum] :
( ( ( surj__ty_2Enum_2Enum @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( inj__ty_2Enum_2Enum @ V19n ) ) )
= fo__c_2Enum_2E0 )
<=> ( V19n = fo__c_2Earithmetic_2EZERO ) )
& ! [V20n: tp__ty_2Enum_2Enum] :
( ( fo__c_2Enum_2E0
= ( surj__ty_2Enum_2Enum @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( inj__ty_2Enum_2Enum @ V20n ) ) ) )
<=> ( V20n = fo__c_2Earithmetic_2EZERO ) )
& ! [V21n: tp__ty_2Enum_2Enum,V22m: tp__ty_2Enum_2Enum] :
( ( ( surj__ty_2Enum_2Enum @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( inj__ty_2Enum_2Enum @ V21n ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( inj__ty_2Enum_2Enum @ V22m ) ) ) )
<=> ( V21n = V22m ) )
& ! [V23n: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V23n ) ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) )
<=> $false )
& ! [V24n: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( inj__ty_2Enum_2Enum @ V24n ) ) ) )
<=> ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) @ ( inj__ty_2Enum_2Enum @ V24n ) ) ) )
& ! [V25n: tp__ty_2Enum_2Enum,V26m: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( inj__ty_2Enum_2Enum @ V25n ) ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( inj__ty_2Enum_2Enum @ V26m ) ) ) )
<=> ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V25n ) ) @ ( inj__ty_2Enum_2Enum @ V26m ) ) ) )
& ! [V27n: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3E @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( inj__ty_2Enum_2Enum @ V27n ) ) )
<=> $false )
& ! [V28n: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3E @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( inj__ty_2Enum_2Enum @ V28n ) ) ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) )
<=> ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) @ ( inj__ty_2Enum_2Enum @ V28n ) ) ) )
& ! [V29n: tp__ty_2Enum_2Enum,V30m: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3E @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( inj__ty_2Enum_2Enum @ V29n ) ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( inj__ty_2Enum_2Enum @ V30m ) ) ) )
<=> ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V30m ) ) @ ( inj__ty_2Enum_2Enum @ V29n ) ) ) )
& ! [V31n: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( inj__ty_2Enum_2Enum @ V31n ) ) )
<=> $true )
& ! [V32n: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( inj__ty_2Enum_2Enum @ V32n ) ) ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) )
<=> ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ V32n ) ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) )
& ! [V33n: tp__ty_2Enum_2Enum,V34m: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( inj__ty_2Enum_2Enum @ V33n ) ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( inj__ty_2Enum_2Enum @ V34m ) ) ) )
<=> ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ V33n ) ) @ ( inj__ty_2Enum_2Enum @ V34m ) ) ) )
& ! [V35n: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3E_3D @ ( inj__ty_2Enum_2Enum @ V35n ) ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) )
<=> $true )
& ! [V36n: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3E_3D @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( inj__ty_2Enum_2Enum @ V36n ) ) )
<=> ( V36n = fo__c_2Enum_2E0 ) )
& ! [V37n: tp__ty_2Enum_2Enum,V38m: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3E_3D @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( inj__ty_2Enum_2Enum @ V37n ) ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( inj__ty_2Enum_2Enum @ V38m ) ) ) )
<=> ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ V38m ) ) @ ( inj__ty_2Enum_2Enum @ V37n ) ) ) )
& ! [V39n: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ c_2Earithmetic_2EODD @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( inj__ty_2Enum_2Enum @ V39n ) ) ) )
<=> ( p @ ( ap @ c_2Earithmetic_2EODD @ ( inj__ty_2Enum_2Enum @ V39n ) ) ) )
& ! [V40n: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ c_2Earithmetic_2EEVEN @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( inj__ty_2Enum_2Enum @ V40n ) ) ) )
<=> ( p @ ( ap @ c_2Earithmetic_2EEVEN @ ( inj__ty_2Enum_2Enum @ V40n ) ) ) )
& ~ ( p @ ( ap @ c_2Earithmetic_2EODD @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) )
& ( p @ ( ap @ c_2Earithmetic_2EEVEN @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ).
thf(conj_thm_2Enumeral_2Enumeral__iisuc,axiom,
! [V0n: tp__ty_2Enum_2Enum] :
( ( ( surj__ty_2Enum_2Enum @ ( ap @ c_2Enumeral_2EiiSUC @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) )
& ( ( surj__ty_2Enum_2Enum @ ( ap @ c_2Enumeral_2EiiSUC @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ c_2Earithmetic_2EBIT1 @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) ) )
& ( ( surj__ty_2Enum_2Enum @ ( ap @ c_2Enumeral_2EiiSUC @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ c_2Earithmetic_2EBIT2 @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) ) ) ) ).
thf(conj_thm_2Enumeral_2Enumeral__add,axiom,
! [V0n: tp__ty_2Enum_2Enum,V1m: tp__ty_2Enum_2Enum] :
( ( ( surj__ty_2Enum_2Enum @ ( ap @ c_2Enumeral_2EiZ @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) )
= V0n )
& ( ( surj__ty_2Enum_2Enum @ ( ap @ c_2Enumeral_2EiZ @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( inj__ty_2Enum_2Enum @ V0n ) ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) )
= V0n )
& ( ( surj__ty_2Enum_2Enum @ ( ap @ c_2Enumeral_2EiZ @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ V1m ) ) ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ c_2Earithmetic_2EBIT2 @ ( ap @ c_2Enumeral_2EiZ @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( inj__ty_2Enum_2Enum @ V0n ) ) @ ( inj__ty_2Enum_2Enum @ V1m ) ) ) ) ) )
& ( ( surj__ty_2Enum_2Enum @ ( ap @ c_2Enumeral_2EiZ @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ V1m ) ) ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ c_2Earithmetic_2EBIT1 @ ( ap @ c_2Enum_2ESUC @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( inj__ty_2Enum_2Enum @ V0n ) ) @ ( inj__ty_2Enum_2Enum @ V1m ) ) ) ) ) )
& ( ( surj__ty_2Enum_2Enum @ ( ap @ c_2Enumeral_2EiZ @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ V1m ) ) ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ c_2Earithmetic_2EBIT1 @ ( ap @ c_2Enum_2ESUC @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( inj__ty_2Enum_2Enum @ V0n ) ) @ ( inj__ty_2Enum_2Enum @ V1m ) ) ) ) ) )
& ( ( surj__ty_2Enum_2Enum @ ( ap @ c_2Enumeral_2EiZ @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ V1m ) ) ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ c_2Earithmetic_2EBIT2 @ ( ap @ c_2Enum_2ESUC @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( inj__ty_2Enum_2Enum @ V0n ) ) @ ( inj__ty_2Enum_2Enum @ V1m ) ) ) ) ) )
& ( ( surj__ty_2Enum_2Enum @ ( ap @ c_2Enum_2ESUC @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) )
& ( ( surj__ty_2Enum_2Enum @ ( ap @ c_2Enum_2ESUC @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( inj__ty_2Enum_2Enum @ V0n ) ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) )
& ( ( surj__ty_2Enum_2Enum @ ( ap @ c_2Enum_2ESUC @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ V1m ) ) ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ c_2Earithmetic_2EBIT1 @ ( ap @ c_2Enum_2ESUC @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( inj__ty_2Enum_2Enum @ V0n ) ) @ ( inj__ty_2Enum_2Enum @ V1m ) ) ) ) ) )
& ( ( surj__ty_2Enum_2Enum @ ( ap @ c_2Enum_2ESUC @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ V1m ) ) ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ c_2Earithmetic_2EBIT2 @ ( ap @ c_2Enum_2ESUC @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( inj__ty_2Enum_2Enum @ V0n ) ) @ ( inj__ty_2Enum_2Enum @ V1m ) ) ) ) ) )
& ( ( surj__ty_2Enum_2Enum @ ( ap @ c_2Enum_2ESUC @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ V1m ) ) ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ c_2Earithmetic_2EBIT2 @ ( ap @ c_2Enum_2ESUC @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( inj__ty_2Enum_2Enum @ V0n ) ) @ ( inj__ty_2Enum_2Enum @ V1m ) ) ) ) ) )
& ( ( surj__ty_2Enum_2Enum @ ( ap @ c_2Enum_2ESUC @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ V1m ) ) ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ c_2Earithmetic_2EBIT1 @ ( ap @ c_2Enumeral_2EiiSUC @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( inj__ty_2Enum_2Enum @ V0n ) ) @ ( inj__ty_2Enum_2Enum @ V1m ) ) ) ) ) )
& ( ( surj__ty_2Enum_2Enum @ ( ap @ c_2Enumeral_2EiiSUC @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ c_2Enumeral_2EiiSUC @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) )
& ( ( surj__ty_2Enum_2Enum @ ( ap @ c_2Enumeral_2EiiSUC @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( inj__ty_2Enum_2Enum @ V0n ) ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ c_2Enumeral_2EiiSUC @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) )
& ( ( surj__ty_2Enum_2Enum @ ( ap @ c_2Enumeral_2EiiSUC @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ V1m ) ) ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ c_2Earithmetic_2EBIT2 @ ( ap @ c_2Enum_2ESUC @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( inj__ty_2Enum_2Enum @ V0n ) ) @ ( inj__ty_2Enum_2Enum @ V1m ) ) ) ) ) )
& ( ( surj__ty_2Enum_2Enum @ ( ap @ c_2Enumeral_2EiiSUC @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ V1m ) ) ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ c_2Earithmetic_2EBIT1 @ ( ap @ c_2Enumeral_2EiiSUC @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( inj__ty_2Enum_2Enum @ V0n ) ) @ ( inj__ty_2Enum_2Enum @ V1m ) ) ) ) ) )
& ( ( surj__ty_2Enum_2Enum @ ( ap @ c_2Enumeral_2EiiSUC @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ V1m ) ) ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ c_2Earithmetic_2EBIT1 @ ( ap @ c_2Enumeral_2EiiSUC @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( inj__ty_2Enum_2Enum @ V0n ) ) @ ( inj__ty_2Enum_2Enum @ V1m ) ) ) ) ) )
& ( ( surj__ty_2Enum_2Enum @ ( ap @ c_2Enumeral_2EiiSUC @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ V1m ) ) ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ c_2Earithmetic_2EBIT2 @ ( ap @ c_2Enumeral_2EiiSUC @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( inj__ty_2Enum_2Enum @ V0n ) ) @ ( inj__ty_2Enum_2Enum @ V1m ) ) ) ) ) ) ) ).
thf(conj_thm_2Enumeral_2Enumeral__eq,axiom,
! [V0n: tp__ty_2Enum_2Enum,V1m: tp__ty_2Enum_2Enum] :
( ( ( fo__c_2Earithmetic_2EZERO
= ( surj__ty_2Enum_2Enum @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) )
<=> $false )
& ( ( ( surj__ty_2Enum_2Enum @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ V0n ) ) )
= fo__c_2Earithmetic_2EZERO )
<=> $false )
& ( ( fo__c_2Earithmetic_2EZERO
= ( surj__ty_2Enum_2Enum @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) )
<=> $false )
& ( ( ( surj__ty_2Enum_2Enum @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ V0n ) ) )
= fo__c_2Earithmetic_2EZERO )
<=> $false )
& ( ( ( surj__ty_2Enum_2Enum @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ V0n ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ V1m ) ) ) )
<=> $false )
& ( ( ( surj__ty_2Enum_2Enum @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ V0n ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ V1m ) ) ) )
<=> $false )
& ( ( ( surj__ty_2Enum_2Enum @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ V0n ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ V1m ) ) ) )
<=> ( V0n = V1m ) )
& ( ( ( surj__ty_2Enum_2Enum @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ V0n ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ V1m ) ) ) )
<=> ( V0n = V1m ) ) ) ).
thf(conj_thm_2Enumeral_2Enumeral__lt,axiom,
! [V0n: tp__ty_2Enum_2Enum,V1m: tp__ty_2Enum_2Enum] :
( ( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) )
<=> $true )
& ( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) )
<=> $true )
& ( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V0n ) ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) )
<=> $false )
& ( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ V1m ) ) ) )
<=> ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V0n ) ) @ ( inj__ty_2Enum_2Enum @ V1m ) ) ) )
& ( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ V1m ) ) ) )
<=> ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V0n ) ) @ ( inj__ty_2Enum_2Enum @ V1m ) ) ) )
& ( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ V1m ) ) ) )
<=> ~ ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V1m ) ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) )
& ( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ V1m ) ) ) )
<=> ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V0n ) ) @ ( inj__ty_2Enum_2Enum @ V1m ) ) ) ) ) ).
thf(conj_thm_2Enumeral_2Enumeral__lte,axiom,
! [V0n: tp__ty_2Enum_2Enum,V1m: tp__ty_2Enum_2Enum] :
( ( ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) )
<=> $true )
& ( ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) )
<=> $false )
& ( ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) )
<=> $false )
& ( ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ V1m ) ) ) )
<=> ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ V0n ) ) @ ( inj__ty_2Enum_2Enum @ V1m ) ) ) )
& ( ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ V1m ) ) ) )
<=> ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ V0n ) ) @ ( inj__ty_2Enum_2Enum @ V1m ) ) ) )
& ( ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ V1m ) ) ) )
<=> ~ ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ V1m ) ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) )
& ( ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ V1m ) ) ) )
<=> ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ V0n ) ) @ ( inj__ty_2Enum_2Enum @ V1m ) ) ) ) ) ).
thf(conj_thm_2Enumeral_2Enumeral__pre,axiom,
( ( ( surj__ty_2Enum_2Enum @ ( ap @ c_2Eprim__rec_2EPRE @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) )
= fo__c_2Earithmetic_2EZERO )
& ( ( surj__ty_2Enum_2Enum @ ( ap @ c_2Eprim__rec_2EPRE @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) )
= fo__c_2Earithmetic_2EZERO )
& ! [V0n: tp__ty_2Enum_2Enum] :
( ( surj__ty_2Enum_2Enum @ ( ap @ c_2Eprim__rec_2EPRE @ ( ap @ c_2Earithmetic_2EBIT1 @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ c_2Earithmetic_2EBIT2 @ ( ap @ c_2Eprim__rec_2EPRE @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) ) ) )
& ! [V1n: tp__ty_2Enum_2Enum] :
( ( surj__ty_2Enum_2Enum @ ( ap @ c_2Eprim__rec_2EPRE @ ( ap @ c_2Earithmetic_2EBIT1 @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ c_2Earithmetic_2EBIT2 @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) ) )
& ! [V2n: tp__ty_2Enum_2Enum] :
( ( surj__ty_2Enum_2Enum @ ( ap @ c_2Eprim__rec_2EPRE @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) ) ) ).
thf(conj_thm_2Enumeral_2Ebit__initiality,axiom,
! [A_27a: del,V0zf: $i] :
( ( mem @ V0zf @ A_27a )
=> ! [V1b1f: $i] :
( ( mem @ V1b1f @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ A_27a ) ) )
=> ! [V2b2f: $i] :
( ( mem @ V2b2f @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ A_27a ) ) )
=> ? [V3f: $i] :
( ( mem @ V3f @ ( arr @ ty_2Enum_2Enum @ A_27a ) )
& ( ( ap @ V3f @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) )
= V0zf )
& ! [V4n: tp__ty_2Enum_2Enum] :
( ( ap @ V3f @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ V4n ) ) )
= ( ap @ ( ap @ V1b1f @ ( inj__ty_2Enum_2Enum @ V4n ) ) @ ( ap @ V3f @ ( inj__ty_2Enum_2Enum @ V4n ) ) ) )
& ! [V5n: tp__ty_2Enum_2Enum] :
( ( ap @ V3f @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ V5n ) ) )
= ( ap @ ( ap @ V2b2f @ ( inj__ty_2Enum_2Enum @ V5n ) ) @ ( ap @ V3f @ ( inj__ty_2Enum_2Enum @ V5n ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Enumeral_2EiBIT__cases,axiom,
! [A_27a: del] :
( ! [V0zf: $i] :
( ( mem @ V0zf @ A_27a )
=> ! [V1bf1: $i] :
( ( mem @ V1bf1 @ ( arr @ ty_2Enum_2Enum @ A_27a ) )
=> ! [V2bf2: $i] :
( ( mem @ V2bf2 @ ( arr @ ty_2Enum_2Enum @ A_27a ) )
=> ( ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Enumeral_2EiBIT__cases @ A_27a ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) @ V0zf ) @ V1bf1 ) @ V2bf2 )
= V0zf ) ) ) )
& ! [V3n: tp__ty_2Enum_2Enum,V4zf: $i] :
( ( mem @ V4zf @ A_27a )
=> ! [V5bf1: $i] :
( ( mem @ V5bf1 @ ( arr @ ty_2Enum_2Enum @ A_27a ) )
=> ! [V6bf2: $i] :
( ( mem @ V6bf2 @ ( arr @ ty_2Enum_2Enum @ A_27a ) )
=> ( ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Enumeral_2EiBIT__cases @ A_27a ) @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) @ V4zf ) @ V5bf1 ) @ V6bf2 )
= ( ap @ V5bf1 @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) ) ) )
& ! [V7n: tp__ty_2Enum_2Enum,V8zf: $i] :
( ( mem @ V8zf @ A_27a )
=> ! [V9bf1: $i] :
( ( mem @ V9bf1 @ ( arr @ ty_2Enum_2Enum @ A_27a ) )
=> ! [V10bf2: $i] :
( ( mem @ V10bf2 @ ( arr @ ty_2Enum_2Enum @ A_27a ) )
=> ( ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Enumeral_2EiBIT__cases @ A_27a ) @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ V7n ) ) ) @ V8zf ) @ V9bf1 ) @ V10bf2 )
= ( ap @ V10bf2 @ ( inj__ty_2Enum_2Enum @ V7n ) ) ) ) ) ) ) ).
thf(ax_thm_2Enumeral_2EiDUB,axiom,
! [V0x: tp__ty_2Enum_2Enum] :
( ( surj__ty_2Enum_2Enum @ ( ap @ c_2Enumeral_2EiDUB @ ( inj__ty_2Enum_2Enum @ V0x ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( inj__ty_2Enum_2Enum @ V0x ) ) @ ( inj__ty_2Enum_2Enum @ V0x ) ) ) ) ).
thf(ax_thm_2Enumeral_2EiSUB__DEF,axiom,
( ! [V0b: $i] :
( ( mem @ V0b @ bool )
=> ! [V1x: tp__ty_2Enum_2Enum] :
( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ ( ap @ c_2Enumeral_2EiSUB @ V0b ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) @ ( inj__ty_2Enum_2Enum @ V1x ) ) )
= fo__c_2Earithmetic_2EZERO ) )
& ! [V2b: $i] :
( ( mem @ V2b @ bool )
=> ! [V3n: tp__ty_2Enum_2Enum,V4x: tp__ty_2Enum_2Enum] :
( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ ( ap @ c_2Enumeral_2EiSUB @ V2b ) @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) @ ( inj__ty_2Enum_2Enum @ V4x ) ) )
= ( surj__ty_2Enum_2Enum
@ ( ap
@ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ty_2Enum_2Enum ) @ V2b )
@ ( ap
@ ( ap @ ( ap @ ( ap @ ( c_2Enumeral_2EiBIT__cases @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V4x ) ) @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ V3n ) ) )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V5m: $i] : ( ap @ c_2Enumeral_2EiDUB @ ( ap @ ( ap @ ( ap @ c_2Enumeral_2EiSUB @ c_2Ebool_2ET ) @ ( inj__ty_2Enum_2Enum @ V3n ) ) @ V5m ) ) ) )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V6m: $i] : ( ap @ c_2Earithmetic_2EBIT1 @ ( ap @ ( ap @ ( ap @ c_2Enumeral_2EiSUB @ c_2Ebool_2EF ) @ ( inj__ty_2Enum_2Enum @ V3n ) ) @ V6m ) ) ) ) )
@ ( ap
@ ( ap @ ( ap @ ( ap @ ( c_2Enumeral_2EiBIT__cases @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V4x ) ) @ ( ap @ c_2Enumeral_2EiDUB @ ( inj__ty_2Enum_2Enum @ V3n ) ) )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V7m: $i] : ( ap @ c_2Earithmetic_2EBIT1 @ ( ap @ ( ap @ ( ap @ c_2Enumeral_2EiSUB @ c_2Ebool_2EF ) @ ( inj__ty_2Enum_2Enum @ V3n ) ) @ V7m ) ) ) )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V8m: $i] : ( ap @ c_2Enumeral_2EiDUB @ ( ap @ ( ap @ ( ap @ c_2Enumeral_2EiSUB @ c_2Ebool_2EF ) @ ( inj__ty_2Enum_2Enum @ V3n ) ) @ V8m ) ) ) ) ) ) ) )
& ! [V9b: $i] :
( ( mem @ V9b @ bool )
=> ! [V10n: tp__ty_2Enum_2Enum,V11x: tp__ty_2Enum_2Enum] :
( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ ( ap @ c_2Enumeral_2EiSUB @ V9b ) @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ V10n ) ) ) @ ( inj__ty_2Enum_2Enum @ V11x ) ) )
= ( surj__ty_2Enum_2Enum
@ ( ap
@ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ty_2Enum_2Enum ) @ V9b )
@ ( ap
@ ( ap @ ( ap @ ( ap @ ( c_2Enumeral_2EiBIT__cases @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V11x ) ) @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ V10n ) ) )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V12m: $i] : ( ap @ c_2Earithmetic_2EBIT1 @ ( ap @ ( ap @ ( ap @ c_2Enumeral_2EiSUB @ c_2Ebool_2ET ) @ ( inj__ty_2Enum_2Enum @ V10n ) ) @ V12m ) ) ) )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V13m: $i] : ( ap @ c_2Enumeral_2EiDUB @ ( ap @ ( ap @ ( ap @ c_2Enumeral_2EiSUB @ c_2Ebool_2ET ) @ ( inj__ty_2Enum_2Enum @ V10n ) ) @ V13m ) ) ) ) )
@ ( ap
@ ( ap @ ( ap @ ( ap @ ( c_2Enumeral_2EiBIT__cases @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V11x ) ) @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ V10n ) ) )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V14m: $i] : ( ap @ c_2Enumeral_2EiDUB @ ( ap @ ( ap @ ( ap @ c_2Enumeral_2EiSUB @ c_2Ebool_2ET ) @ ( inj__ty_2Enum_2Enum @ V10n ) ) @ V14m ) ) ) )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V15m: $i] : ( ap @ c_2Earithmetic_2EBIT1 @ ( ap @ ( ap @ ( ap @ c_2Enumeral_2EiSUB @ c_2Ebool_2EF ) @ ( inj__ty_2Enum_2Enum @ V10n ) ) @ V15m ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Enumeral_2Ebit__induction,axiom,
! [V0P: $i] :
( ( mem @ V0P @ ( arr @ ty_2Enum_2Enum @ bool ) )
=> ( ( ( p @ ( ap @ V0P @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) )
& ! [V1n: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ V0P @ ( inj__ty_2Enum_2Enum @ V1n ) ) )
=> ( p @ ( ap @ V0P @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) ) )
& ! [V2n: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ V0P @ ( inj__ty_2Enum_2Enum @ V2n ) ) )
=> ( p @ ( ap @ V0P @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) ) ) )
=> ! [V3n: tp__ty_2Enum_2Enum] : ( p @ ( ap @ V0P @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) ) ) ).
thf(conj_thm_2Enumeral_2EiSUB__THM,axiom,
! [V0x: tp__ty_2Enum_2Enum,V1b: $i] :
( ( mem @ V1b @ bool )
=> ! [V2n: tp__ty_2Enum_2Enum,V3m: tp__ty_2Enum_2Enum] :
( ( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ ( ap @ c_2Enumeral_2EiSUB @ V1b ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) @ ( inj__ty_2Enum_2Enum @ V0x ) ) )
= fo__c_2Earithmetic_2EZERO )
& ( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ ( ap @ c_2Enumeral_2EiSUB @ c_2Ebool_2ET ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) )
= V2n )
& ( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ ( ap @ c_2Enumeral_2EiSUB @ c_2Ebool_2EF ) @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ c_2Enumeral_2EiDUB @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) )
& ( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ ( ap @ c_2Enumeral_2EiSUB @ c_2Ebool_2ET ) @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ V3m ) ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ c_2Enumeral_2EiDUB @ ( ap @ ( ap @ ( ap @ c_2Enumeral_2EiSUB @ c_2Ebool_2ET ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) @ ( inj__ty_2Enum_2Enum @ V3m ) ) ) ) )
& ( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ ( ap @ c_2Enumeral_2EiSUB @ c_2Ebool_2EF ) @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ V3m ) ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ c_2Earithmetic_2EBIT1 @ ( ap @ ( ap @ ( ap @ c_2Enumeral_2EiSUB @ c_2Ebool_2EF ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) @ ( inj__ty_2Enum_2Enum @ V3m ) ) ) ) )
& ( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ ( ap @ c_2Enumeral_2EiSUB @ c_2Ebool_2ET ) @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ V3m ) ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ c_2Earithmetic_2EBIT1 @ ( ap @ ( ap @ ( ap @ c_2Enumeral_2EiSUB @ c_2Ebool_2EF ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) @ ( inj__ty_2Enum_2Enum @ V3m ) ) ) ) )
& ( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ ( ap @ c_2Enumeral_2EiSUB @ c_2Ebool_2EF ) @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ V3m ) ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ c_2Enumeral_2EiDUB @ ( ap @ ( ap @ ( ap @ c_2Enumeral_2EiSUB @ c_2Ebool_2EF ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) @ ( inj__ty_2Enum_2Enum @ V3m ) ) ) ) )
& ( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ ( ap @ c_2Enumeral_2EiSUB @ c_2Ebool_2EF ) @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) )
& ( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ ( ap @ c_2Enumeral_2EiSUB @ c_2Ebool_2ET ) @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ V3m ) ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ c_2Earithmetic_2EBIT1 @ ( ap @ ( ap @ ( ap @ c_2Enumeral_2EiSUB @ c_2Ebool_2ET ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) @ ( inj__ty_2Enum_2Enum @ V3m ) ) ) ) )
& ( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ ( ap @ c_2Enumeral_2EiSUB @ c_2Ebool_2EF ) @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ V3m ) ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ c_2Enumeral_2EiDUB @ ( ap @ ( ap @ ( ap @ c_2Enumeral_2EiSUB @ c_2Ebool_2ET ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) @ ( inj__ty_2Enum_2Enum @ V3m ) ) ) ) )
& ( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ ( ap @ c_2Enumeral_2EiSUB @ c_2Ebool_2ET ) @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ V3m ) ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ c_2Enumeral_2EiDUB @ ( ap @ ( ap @ ( ap @ c_2Enumeral_2EiSUB @ c_2Ebool_2ET ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) @ ( inj__ty_2Enum_2Enum @ V3m ) ) ) ) )
& ( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ ( ap @ c_2Enumeral_2EiSUB @ c_2Ebool_2EF ) @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ V3m ) ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ c_2Earithmetic_2EBIT1 @ ( ap @ ( ap @ ( ap @ c_2Enumeral_2EiSUB @ c_2Ebool_2EF ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) @ ( inj__ty_2Enum_2Enum @ V3m ) ) ) ) ) ) ) ).
thf(conj_thm_2Enumeral_2Enumeral__sub,axiom,
! [V0n: tp__ty_2Enum_2Enum,V1m: tp__ty_2Enum_2Enum] :
( ( surj__ty_2Enum_2Enum @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ ( inj__ty_2Enum_2Enum @ V0n ) ) @ ( inj__ty_2Enum_2Enum @ V1m ) ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V1m ) ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ ( ap @ ( ap @ c_2Enumeral_2EiSUB @ c_2Ebool_2ET ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) @ ( inj__ty_2Enum_2Enum @ V1m ) ) ) ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ).
thf(conj_thm_2Enumeral_2EiDUB__removal,axiom,
! [V0n: tp__ty_2Enum_2Enum] :
( ( ( surj__ty_2Enum_2Enum @ ( ap @ c_2Enumeral_2EiDUB @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ c_2Earithmetic_2EBIT2 @ ( ap @ c_2Enumeral_2EiDUB @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) ) )
& ( ( surj__ty_2Enum_2Enum @ ( ap @ c_2Enumeral_2EiDUB @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ c_2Earithmetic_2EBIT2 @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) ) )
& ( ( surj__ty_2Enum_2Enum @ ( ap @ c_2Enumeral_2EiDUB @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) )
= fo__c_2Earithmetic_2EZERO ) ) ).
thf(conj_thm_2Enumeral_2Enumeral__mult,axiom,
! [V0n: tp__ty_2Enum_2Enum,V1m: tp__ty_2Enum_2Enum] :
( ( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Earithmetic_2E_2A @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) )
= fo__c_2Earithmetic_2EZERO )
& ( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Earithmetic_2E_2A @ ( inj__ty_2Enum_2Enum @ V0n ) ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) )
= fo__c_2Earithmetic_2EZERO )
& ( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Earithmetic_2E_2A @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) @ ( inj__ty_2Enum_2Enum @ V1m ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ c_2Enumeral_2EiZ @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( ap @ c_2Enumeral_2EiDUB @ ( ap @ ( ap @ c_2Earithmetic_2E_2A @ ( inj__ty_2Enum_2Enum @ V0n ) ) @ ( inj__ty_2Enum_2Enum @ V1m ) ) ) ) @ ( inj__ty_2Enum_2Enum @ V1m ) ) ) ) )
& ( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Earithmetic_2E_2A @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) @ ( inj__ty_2Enum_2Enum @ V1m ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ c_2Enumeral_2EiDUB @ ( ap @ c_2Enumeral_2EiZ @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( ap @ ( ap @ c_2Earithmetic_2E_2A @ ( inj__ty_2Enum_2Enum @ V0n ) ) @ ( inj__ty_2Enum_2Enum @ V1m ) ) ) @ ( inj__ty_2Enum_2Enum @ V1m ) ) ) ) ) ) ) ).
thf(ax_thm_2Enumeral_2EiSQR,axiom,
! [V0x: tp__ty_2Enum_2Enum] :
( ( surj__ty_2Enum_2Enum @ ( ap @ c_2Enumeral_2EiSQR @ ( inj__ty_2Enum_2Enum @ V0x ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Earithmetic_2E_2A @ ( inj__ty_2Enum_2Enum @ V0x ) ) @ ( inj__ty_2Enum_2Enum @ V0x ) ) ) ) ).
thf(conj_thm_2Enumeral_2Enumeral__exp,axiom,
( ! [V0n: tp__ty_2Enum_2Enum] :
( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Earithmetic_2EEXP @ ( inj__ty_2Enum_2Enum @ V0n ) ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) )
& ! [V1n: tp__ty_2Enum_2Enum,V2m: tp__ty_2Enum_2Enum] :
( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Earithmetic_2EEXP @ ( inj__ty_2Enum_2Enum @ V1n ) ) @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ V2m ) ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Earithmetic_2E_2A @ ( inj__ty_2Enum_2Enum @ V1n ) ) @ ( ap @ c_2Enumeral_2EiSQR @ ( ap @ ( ap @ c_2Earithmetic_2EEXP @ ( inj__ty_2Enum_2Enum @ V1n ) ) @ ( inj__ty_2Enum_2Enum @ V2m ) ) ) ) ) )
& ! [V3n: tp__ty_2Enum_2Enum,V4m: tp__ty_2Enum_2Enum] :
( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Earithmetic_2EEXP @ ( inj__ty_2Enum_2Enum @ V3n ) ) @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ V4m ) ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Earithmetic_2E_2A @ ( ap @ c_2Enumeral_2EiSQR @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) @ ( ap @ c_2Enumeral_2EiSQR @ ( ap @ ( ap @ c_2Earithmetic_2EEXP @ ( inj__ty_2Enum_2Enum @ V3n ) ) @ ( inj__ty_2Enum_2Enum @ V4m ) ) ) ) ) ) ) ).
thf(conj_thm_2Enumeral_2Enumeral__evenodd,axiom,
! [V0n: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ c_2Earithmetic_2EEVEN @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) )
& ( p @ ( ap @ c_2Earithmetic_2EEVEN @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) )
& ~ ( p @ ( ap @ c_2Earithmetic_2EEVEN @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) )
& ~ ( p @ ( ap @ c_2Earithmetic_2EODD @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) )
& ~ ( p @ ( ap @ c_2Earithmetic_2EODD @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) )
& ( p @ ( ap @ c_2Earithmetic_2EODD @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) ) ) ).
thf(conj_thm_2Enumeral_2Enumeral__fact,axiom,
( ( ( surj__ty_2Enum_2Enum @ ( ap @ c_2Earithmetic_2EFACT @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) )
& ! [V0n: tp__ty_2Enum_2Enum] :
( ( surj__ty_2Enum_2Enum @ ( ap @ c_2Earithmetic_2EFACT @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Earithmetic_2E_2A @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) ) @ ( ap @ c_2Earithmetic_2EFACT @ ( ap @ c_2Eprim__rec_2EPRE @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) ) ) ) ) )
& ! [V1n: tp__ty_2Enum_2Enum] :
( ( surj__ty_2Enum_2Enum @ ( ap @ c_2Earithmetic_2EFACT @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Earithmetic_2E_2A @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) ) @ ( ap @ c_2Earithmetic_2EFACT @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Enumeral_2Enumeral__funpow,axiom,
! [A_27a: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27a ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ! [V2n: tp__ty_2Enum_2Enum] :
( ( ( ap @ ( ap @ ( ap @ ( c_2Earithmetic_2EFUNPOW @ A_27a ) @ V0f ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ V1x )
= V1x )
& ( ( ap @ ( ap @ ( ap @ ( c_2Earithmetic_2EFUNPOW @ A_27a ) @ V0f ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) ) @ V1x )
= ( ap @ ( ap @ ( ap @ ( c_2Earithmetic_2EFUNPOW @ A_27a ) @ V0f ) @ ( ap @ c_2Eprim__rec_2EPRE @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) ) ) @ ( ap @ V0f @ V1x ) ) )
& ( ( ap @ ( ap @ ( ap @ ( c_2Earithmetic_2EFUNPOW @ A_27a ) @ V0f ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) ) @ V1x )
= ( ap @ ( ap @ ( ap @ ( c_2Earithmetic_2EFUNPOW @ A_27a ) @ V0f ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) ) @ ( ap @ V0f @ V1x ) ) ) ) ) ) ).
thf(conj_thm_2Enumeral_2Enumeral__MIN,axiom,
! [V0x: tp__ty_2Enum_2Enum,V1y: tp__ty_2Enum_2Enum] :
( ( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Earithmetic_2EMIN @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( inj__ty_2Enum_2Enum @ V0x ) ) )
= fo__c_2Enum_2E0 )
& ( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Earithmetic_2EMIN @ ( inj__ty_2Enum_2Enum @ V0x ) ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) )
= fo__c_2Enum_2E0 )
& ( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Earithmetic_2EMIN @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( inj__ty_2Enum_2Enum @ V0x ) ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( inj__ty_2Enum_2Enum @ V1y ) ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V0x ) ) @ ( inj__ty_2Enum_2Enum @ V1y ) ) ) @ ( inj__ty_2Enum_2Enum @ V0x ) ) @ ( inj__ty_2Enum_2Enum @ V1y ) ) ) ) ) ) ).
thf(conj_thm_2Enumeral_2Enumeral__MAX,axiom,
! [V0x: tp__ty_2Enum_2Enum,V1y: tp__ty_2Enum_2Enum] :
( ( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Earithmetic_2EMAX @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( inj__ty_2Enum_2Enum @ V0x ) ) )
= V0x )
& ( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Earithmetic_2EMAX @ ( inj__ty_2Enum_2Enum @ V0x ) ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) )
= V0x )
& ( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Earithmetic_2EMAX @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( inj__ty_2Enum_2Enum @ V0x ) ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( inj__ty_2Enum_2Enum @ V1y ) ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V0x ) ) @ ( inj__ty_2Enum_2Enum @ V1y ) ) ) @ ( inj__ty_2Enum_2Enum @ V1y ) ) @ ( inj__ty_2Enum_2Enum @ V0x ) ) ) ) ) ) ).
thf(conj_thm_2Enumeral_2Edivmod__POS,axiom,
! [V0a: tp__ty_2Enum_2Enum,V1m: tp__ty_2Enum_2Enum,V2n: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) )
=> ( ( surj__c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Enum_2Enum @ ( ap @ c_2Earithmetic_2EDIVMOD @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) ) @ ( inj__ty_2Enum_2Enum @ V0a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V1m ) ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) ) )
= ( surj__c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Enum_2Enum
@ ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) ) @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V1m ) ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V0a ) ) @ ( inj__ty_2Enum_2Enum @ V1m ) ) )
@ ( ap
@ ( ap @ ( c_2Ebool_2ELET @ ty_2Enum_2Enum @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V3q: $i] : ( ap @ c_2Earithmetic_2EDIVMOD @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( inj__ty_2Enum_2Enum @ V0a ) ) @ V3q ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ ( inj__ty_2Enum_2Enum @ V1m ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2A @ ( inj__ty_2Enum_2Enum @ V2n ) ) @ V3q ) ) ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) ) ) )
@ ( ap @ c_2Earithmetic_2Efindq @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V1m ) ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Enumeral_2EDIVMOD__NUMERAL__CALC,axiom,
( ! [V0m: tp__ty_2Enum_2Enum,V1n: tp__ty_2Enum_2Enum] :
( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Earithmetic_2EDIV @ ( inj__ty_2Enum_2Enum @ V0m ) ) @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Epair_2EFST @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) @ ( ap @ c_2Earithmetic_2EDIVMOD @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V0m ) ) @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) ) ) ) ) )
& ! [V2m: tp__ty_2Enum_2Enum,V3n: tp__ty_2Enum_2Enum] :
( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Earithmetic_2EDIV @ ( inj__ty_2Enum_2Enum @ V2m ) ) @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Epair_2EFST @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) @ ( ap @ c_2Earithmetic_2EDIVMOD @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V2m ) ) @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) ) ) ) ) )
& ! [V4m: tp__ty_2Enum_2Enum,V5n: tp__ty_2Enum_2Enum] :
( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Earithmetic_2EMOD @ ( inj__ty_2Enum_2Enum @ V4m ) ) @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ V5n ) ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Epair_2ESND @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) @ ( ap @ c_2Earithmetic_2EDIVMOD @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V4m ) ) @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ V5n ) ) ) ) ) ) ) )
& ! [V6m: tp__ty_2Enum_2Enum,V7n: tp__ty_2Enum_2Enum] :
( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Earithmetic_2EMOD @ ( inj__ty_2Enum_2Enum @ V6m ) ) @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ V7n ) ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Epair_2ESND @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) @ ( ap @ c_2Earithmetic_2EDIVMOD @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V6m ) ) @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ V7n ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Enumeral_2Enumeral__div2,axiom,
( ( ( surj__ty_2Enum_2Enum @ ( ap @ c_2Earithmetic_2EDIV2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) )
= fo__c_2Enum_2E0 )
& ! [V0n: tp__ty_2Enum_2Enum] :
( ( surj__ty_2Enum_2Enum @ ( ap @ c_2Earithmetic_2EDIV2 @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) )
& ! [V1n: tp__ty_2Enum_2Enum] :
( ( surj__ty_2Enum_2Enum @ ( ap @ c_2Earithmetic_2EDIV2 @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) ) ) ) ).
thf(ax_thm_2Enumeral_2Etexp__help__def,axiom,
( ! [V0acc: tp__ty_2Enum_2Enum] :
( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Enumeral_2Etexp__help @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( inj__ty_2Enum_2Enum @ V0acc ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ V0acc ) ) ) )
& ! [V1n: tp__ty_2Enum_2Enum,V2acc: tp__ty_2Enum_2Enum] :
( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Enumeral_2Etexp__help @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) @ ( inj__ty_2Enum_2Enum @ V2acc ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Enumeral_2Etexp__help @ ( inj__ty_2Enum_2Enum @ V1n ) ) @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ V2acc ) ) ) ) ) ) ).
thf(conj_thm_2Enumeral_2Etexp__help__thm,axiom,
! [V0n: tp__ty_2Enum_2Enum,V1a: tp__ty_2Enum_2Enum] :
( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Enumeral_2Etexp__help @ ( inj__ty_2Enum_2Enum @ V0n ) ) @ ( inj__ty_2Enum_2Enum @ V1a ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Earithmetic_2E_2A @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( inj__ty_2Enum_2Enum @ V1a ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( ap @ ( ap @ c_2Earithmetic_2EEXP @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( inj__ty_2Enum_2Enum @ V0n ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Enumeral_2Etexp__help0,axiom,
! [V0n: tp__ty_2Enum_2Enum] :
( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Enumeral_2Etexp__help @ ( inj__ty_2Enum_2Enum @ V0n ) ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Earithmetic_2EEXP @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( inj__ty_2Enum_2Enum @ V0n ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) ).
thf(conj_thm_2Enumeral_2Enumeral__texp__help,axiom,
! [V0acc: tp__ty_2Enum_2Enum,V1n: tp__ty_2Enum_2Enum] :
( ( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Enumeral_2Etexp__help @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) @ ( inj__ty_2Enum_2Enum @ V0acc ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ V0acc ) ) ) )
& ( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Enumeral_2Etexp__help @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) @ ( inj__ty_2Enum_2Enum @ V0acc ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Enumeral_2Etexp__help @ ( ap @ c_2Eprim__rec_2EPRE @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) ) @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ V0acc ) ) ) ) )
& ( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Enumeral_2Etexp__help @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) @ ( inj__ty_2Enum_2Enum @ V0acc ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Enumeral_2Etexp__help @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ V0acc ) ) ) ) ) ) ).
thf(conj_thm_2Enumeral_2ETWO__EXP__THM,axiom,
! [V0n: tp__ty_2Enum_2Enum] :
( ( ( surj__ty_2Enum_2Enum @ ( 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 @ fo__c_2Enum_2E0 ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) )
& ( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Earithmetic_2EEXP @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ ( ap @ c_2Enumeral_2Etexp__help @ ( ap @ c_2Eprim__rec_2EPRE @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) )
& ( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Earithmetic_2EEXP @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ ( ap @ c_2Enumeral_2Etexp__help @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ).
thf(ax_thm_2Enumeral_2Eonecount__def,axiom,
( ! [V0x: tp__ty_2Enum_2Enum] :
( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Enumeral_2Eonecount @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) @ ( inj__ty_2Enum_2Enum @ V0x ) ) )
= V0x )
& ! [V1n: tp__ty_2Enum_2Enum,V2x: tp__ty_2Enum_2Enum] :
( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Enumeral_2Eonecount @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) @ ( inj__ty_2Enum_2Enum @ V2x ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Enumeral_2Eonecount @ ( inj__ty_2Enum_2Enum @ V1n ) ) @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V2x ) ) ) ) )
& ! [V3n: tp__ty_2Enum_2Enum,V4x: tp__ty_2Enum_2Enum] :
( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Enumeral_2Eonecount @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) @ ( inj__ty_2Enum_2Enum @ V4x ) ) )
= fo__c_2Earithmetic_2EZERO ) ) ).
thf(ax_thm_2Enumeral_2Eexactlog__def,axiom,
( ( ( surj__ty_2Enum_2Enum @ ( ap @ c_2Enumeral_2Eexactlog @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) )
= fo__c_2Earithmetic_2EZERO )
& ! [V0n: tp__ty_2Enum_2Enum] :
( ( surj__ty_2Enum_2Enum @ ( ap @ c_2Enumeral_2Eexactlog @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) )
= fo__c_2Earithmetic_2EZERO )
& ! [V1n: tp__ty_2Enum_2Enum] :
( ( surj__ty_2Enum_2Enum @ ( ap @ c_2Enumeral_2Eexactlog @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) )
= ( surj__ty_2Enum_2Enum
@ ( ap
@ ( ap @ ( c_2Ebool_2ELET @ ty_2Enum_2Enum @ ty_2Enum_2Enum )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V2x: $i] : ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ty_2Enum_2Enum ) @ V2x ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) @ ( ap @ c_2Earithmetic_2EBIT1 @ V2x ) ) ) )
@ ( ap @ ( ap @ c_2Enumeral_2Eonecount @ ( inj__ty_2Enum_2Enum @ V1n ) ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ).
thf(conj_thm_2Enumeral_2Eonecount__characterisation,axiom,
! [V0n: tp__ty_2Enum_2Enum,V1a: tp__ty_2Enum_2Enum] :
( ( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( ap @ ( ap @ c_2Enumeral_2Eonecount @ ( inj__ty_2Enum_2Enum @ V0n ) ) @ ( inj__ty_2Enum_2Enum @ V1a ) ) ) )
& ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) )
=> ( V0n
= ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ ( ap @ ( ap @ c_2Earithmetic_2EEXP @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ ( ap @ ( ap @ c_2Enumeral_2Eonecount @ ( inj__ty_2Enum_2Enum @ V0n ) ) @ ( inj__ty_2Enum_2Enum @ V1a ) ) ) @ ( inj__ty_2Enum_2Enum @ V1a ) ) ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) ).
thf(conj_thm_2Enumeral_2Eexactlog__characterisation,axiom,
! [V0n: tp__ty_2Enum_2Enum,V1m: tp__ty_2Enum_2Enum] :
( ( ( surj__ty_2Enum_2Enum @ ( ap @ c_2Enumeral_2Eexactlog @ ( inj__ty_2Enum_2Enum @ V0n ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ V1m ) ) ) )
=> ( V0n
= ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Earithmetic_2EEXP @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( inj__ty_2Enum_2Enum @ V1m ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Enumeral_2Einternal__mult__def,axiom,
c_2Enumeral_2Einternal__mult = c_2Earithmetic_2E_2A ).
thf(conj_thm_2Enumeral_2EDIV2__BIT1,axiom,
! [V0x: tp__ty_2Enum_2Enum] :
( ( surj__ty_2Enum_2Enum @ ( ap @ c_2Earithmetic_2EDIV2 @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ V0x ) ) ) )
= V0x ) ).
thf(conj_thm_2Enumeral_2Eenumeral__mult,axiom,
! [V0n: tp__ty_2Enum_2Enum,V1x: tp__ty_2Enum_2Enum,V2y: tp__ty_2Enum_2Enum] :
( ( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Earithmetic_2E_2A @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) )
= fo__c_2Earithmetic_2EZERO )
& ( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Earithmetic_2E_2A @ ( inj__ty_2Enum_2Enum @ V0n ) ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) )
= fo__c_2Earithmetic_2EZERO )
& ( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Earithmetic_2E_2A @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ V1x ) ) ) @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ V2y ) ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Enumeral_2Einternal__mult @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ V1x ) ) ) @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ V2y ) ) ) ) )
& ( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Earithmetic_2E_2A @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ V1x ) ) ) @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ V2y ) ) ) )
= ( surj__ty_2Enum_2Enum
@ ( ap
@ ( ap @ ( c_2Ebool_2ELET @ ty_2Enum_2Enum @ ty_2Enum_2Enum )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V3n: $i] : ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ty_2Enum_2Enum ) @ ( ap @ c_2Earithmetic_2EODD @ V3n ) ) @ ( ap @ ( ap @ c_2Enumeral_2Etexp__help @ ( ap @ c_2Earithmetic_2EDIV2 @ V3n ) ) @ ( ap @ c_2Eprim__rec_2EPRE @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ V1x ) ) ) ) ) @ ( ap @ ( ap @ c_2Enumeral_2Einternal__mult @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ V1x ) ) ) @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ V2y ) ) ) ) ) )
@ ( ap @ c_2Enumeral_2Eexactlog @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ V2y ) ) ) ) ) )
& ( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Earithmetic_2E_2A @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ V1x ) ) ) @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ V2y ) ) ) )
= ( surj__ty_2Enum_2Enum
@ ( ap
@ ( ap @ ( c_2Ebool_2ELET @ ty_2Enum_2Enum @ ty_2Enum_2Enum )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V4m: $i] : ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ty_2Enum_2Enum ) @ ( ap @ c_2Earithmetic_2EODD @ V4m ) ) @ ( ap @ ( ap @ c_2Enumeral_2Etexp__help @ ( ap @ c_2Earithmetic_2EDIV2 @ V4m ) ) @ ( ap @ c_2Eprim__rec_2EPRE @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ V2y ) ) ) ) ) @ ( ap @ ( ap @ c_2Enumeral_2Einternal__mult @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ V1x ) ) ) @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ V2y ) ) ) ) ) )
@ ( ap @ c_2Enumeral_2Eexactlog @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ V1x ) ) ) ) ) )
& ( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Earithmetic_2E_2A @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ V1x ) ) ) @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ V2y ) ) ) )
= ( surj__ty_2Enum_2Enum
@ ( ap
@ ( ap @ ( c_2Ebool_2ELET @ ty_2Enum_2Enum @ ty_2Enum_2Enum )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V5m: $i] :
( ap
@ ( ap @ ( c_2Ebool_2ELET @ ty_2Enum_2Enum @ ty_2Enum_2Enum )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V6n: $i] : ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ty_2Enum_2Enum ) @ ( ap @ c_2Earithmetic_2EODD @ V5m ) ) @ ( ap @ ( ap @ c_2Enumeral_2Etexp__help @ ( ap @ c_2Earithmetic_2EDIV2 @ V5m ) ) @ ( ap @ c_2Eprim__rec_2EPRE @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ V2y ) ) ) ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ty_2Enum_2Enum ) @ ( ap @ c_2Earithmetic_2EODD @ V6n ) ) @ ( ap @ ( ap @ c_2Enumeral_2Etexp__help @ ( ap @ c_2Earithmetic_2EDIV2 @ V6n ) ) @ ( ap @ c_2Eprim__rec_2EPRE @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ V1x ) ) ) ) ) @ ( ap @ ( ap @ c_2Enumeral_2Einternal__mult @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ V1x ) ) ) @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ V2y ) ) ) ) ) ) )
@ ( ap @ c_2Enumeral_2Eexactlog @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ V2y ) ) ) ) ) )
@ ( ap @ c_2Enumeral_2Eexactlog @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ V1x ) ) ) ) ) ) ) ).
thf(conj_thm_2Enumeral_2Einternal__mult__characterisation,axiom,
! [V0n: tp__ty_2Enum_2Enum,V1m: tp__ty_2Enum_2Enum] :
( ( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Enumeral_2Einternal__mult @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) )
= fo__c_2Earithmetic_2EZERO )
& ( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Enumeral_2Einternal__mult @ ( inj__ty_2Enum_2Enum @ V0n ) ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) )
= fo__c_2Earithmetic_2EZERO )
& ( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Enumeral_2Einternal__mult @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) @ ( inj__ty_2Enum_2Enum @ V1m ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ c_2Enumeral_2EiZ @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( ap @ c_2Enumeral_2EiDUB @ ( ap @ ( ap @ c_2Enumeral_2Einternal__mult @ ( inj__ty_2Enum_2Enum @ V0n ) ) @ ( inj__ty_2Enum_2Enum @ V1m ) ) ) ) @ ( inj__ty_2Enum_2Enum @ V1m ) ) ) ) )
& ( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Enumeral_2Einternal__mult @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) @ ( inj__ty_2Enum_2Enum @ V1m ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ c_2Enumeral_2EiDUB @ ( ap @ c_2Enumeral_2EiZ @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( ap @ ( ap @ c_2Enumeral_2Einternal__mult @ ( inj__ty_2Enum_2Enum @ V0n ) ) @ ( inj__ty_2Enum_2Enum @ V1m ) ) ) @ ( inj__ty_2Enum_2Enum @ V1m ) ) ) ) ) ) ) ).
%------------------------------------------------------------------------------