ITP001 Axioms: ITP041^5.ax
%------------------------------------------------------------------------------
% File : ITP041^5 : 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 : numposrep^2.ax [Gau20]
% : HL4041^5.ax [TPAP]
% Status : Satisfiable
% Syntax : Number of formulae : 72 ( 20 unt; 18 typ; 0 def)
% Number of atoms : 1285 ( 47 equ; 0 cnn)
% Maximal formula atoms : 42 ( 17 avg)
% Number of connectives : 1437 ( 3 ~; 1 |; 19 &;1390 @)
% ( 2 <=>; 22 =>; 0 <=; 0 <~>)
% Maximal formula depth : 18 ( 6 avg;1390 nst)
% Number of types : 1 ( 0 usr)
% Number of type conns : 7 ( 7 >; 0 *; 0 +; 0 <<)
% Number of symbols : 81 ( 80 usr; 74 con; 0-2 aty)
% Number of variables : 70 ( 3 ^ 67 !; 0 ?; 70 :)
% SPC : TH0_SAT_EQU_NAR
% Comments :
% Bugfixes : v7.5.0 - Fixes to the axioms.
%------------------------------------------------------------------------------
thf(tp_c_2Enumposrep_2EBOOLIFY,type,
c_2Enumposrep_2EBOOLIFY: $i ).
thf(mem_c_2Enumposrep_2EBOOLIFY,axiom,
mem @ c_2Enumposrep_2EBOOLIFY @ ( arr @ ty_2Enum_2Enum @ ( arr @ ty_2Enum_2Enum @ ( arr @ ( ty_2Elist_2Elist @ bool ) @ ( ty_2Elist_2Elist @ bool ) ) ) ) ).
thf(tp_c_2Enumposrep_2El2n,type,
c_2Enumposrep_2El2n: $i ).
thf(mem_c_2Enumposrep_2El2n,axiom,
mem @ c_2Enumposrep_2El2n @ ( arr @ ty_2Enum_2Enum @ ( arr @ ( ty_2Elist_2Elist @ ty_2Enum_2Enum ) @ ty_2Enum_2Enum ) ) ).
thf(stp_fo_c_2Enumposrep_2El2n,type,
fo__c_2Enumposrep_2El2n: tp__ty_2Enum_2Enum > tp__c_ty_2Elist_2Elist_ty_2Enum_2Enum > tp__ty_2Enum_2Enum ).
thf(stp_eq_fo_c_2Enumposrep_2El2n,axiom,
! [X0: tp__ty_2Enum_2Enum,X1: tp__c_ty_2Elist_2Elist_ty_2Enum_2Enum] :
( ( inj__ty_2Enum_2Enum @ ( fo__c_2Enumposrep_2El2n @ X0 @ X1 ) )
= ( ap @ ( ap @ c_2Enumposrep_2El2n @ ( inj__ty_2Enum_2Enum @ X0 ) ) @ ( inj__c_ty_2Elist_2Elist_ty_2Enum_2Enum @ X1 ) ) ) ).
thf(tp_c_2Enumposrep_2El2n2,type,
c_2Enumposrep_2El2n2: $i ).
thf(mem_c_2Enumposrep_2El2n2,axiom,
mem @ c_2Enumposrep_2El2n2 @ ( arr @ ( ty_2Elist_2Elist @ ty_2Enum_2Enum ) @ ty_2Enum_2Enum ) ).
thf(stp_fo_c_2Enumposrep_2El2n2,type,
fo__c_2Enumposrep_2El2n2: tp__c_ty_2Elist_2Elist_ty_2Enum_2Enum > tp__ty_2Enum_2Enum ).
thf(stp_eq_fo_c_2Enumposrep_2El2n2,axiom,
! [X0: tp__c_ty_2Elist_2Elist_ty_2Enum_2Enum] :
( ( inj__ty_2Enum_2Enum @ ( fo__c_2Enumposrep_2El2n2 @ X0 ) )
= ( ap @ c_2Enumposrep_2El2n2 @ ( inj__c_ty_2Elist_2Elist_ty_2Enum_2Enum @ X0 ) ) ) ).
thf(tp_c_2Enumposrep_2En2l,type,
c_2Enumposrep_2En2l: $i ).
thf(mem_c_2Enumposrep_2En2l,axiom,
mem @ c_2Enumposrep_2En2l @ ( arr @ ty_2Enum_2Enum @ ( arr @ ty_2Enum_2Enum @ ( ty_2Elist_2Elist @ ty_2Enum_2Enum ) ) ) ).
thf(tp_c_2Enumposrep_2Enum__from__bin__list,type,
c_2Enumposrep_2Enum__from__bin__list: $i ).
thf(mem_c_2Enumposrep_2Enum__from__bin__list,axiom,
mem @ c_2Enumposrep_2Enum__from__bin__list @ ( arr @ ( ty_2Elist_2Elist @ ty_2Enum_2Enum ) @ ty_2Enum_2Enum ) ).
thf(stp_fo_c_2Enumposrep_2Enum__from__bin__list,type,
fo__c_2Enumposrep_2Enum__from__bin__list: tp__c_ty_2Elist_2Elist_ty_2Enum_2Enum > tp__ty_2Enum_2Enum ).
thf(stp_eq_fo_c_2Enumposrep_2Enum__from__bin__list,axiom,
! [X0: tp__c_ty_2Elist_2Elist_ty_2Enum_2Enum] :
( ( inj__ty_2Enum_2Enum @ ( fo__c_2Enumposrep_2Enum__from__bin__list @ X0 ) )
= ( ap @ c_2Enumposrep_2Enum__from__bin__list @ ( inj__c_ty_2Elist_2Elist_ty_2Enum_2Enum @ X0 ) ) ) ).
thf(tp_c_2Enumposrep_2Enum__from__dec__list,type,
c_2Enumposrep_2Enum__from__dec__list: $i ).
thf(mem_c_2Enumposrep_2Enum__from__dec__list,axiom,
mem @ c_2Enumposrep_2Enum__from__dec__list @ ( arr @ ( ty_2Elist_2Elist @ ty_2Enum_2Enum ) @ ty_2Enum_2Enum ) ).
thf(stp_fo_c_2Enumposrep_2Enum__from__dec__list,type,
fo__c_2Enumposrep_2Enum__from__dec__list: tp__c_ty_2Elist_2Elist_ty_2Enum_2Enum > tp__ty_2Enum_2Enum ).
thf(stp_eq_fo_c_2Enumposrep_2Enum__from__dec__list,axiom,
! [X0: tp__c_ty_2Elist_2Elist_ty_2Enum_2Enum] :
( ( inj__ty_2Enum_2Enum @ ( fo__c_2Enumposrep_2Enum__from__dec__list @ X0 ) )
= ( ap @ c_2Enumposrep_2Enum__from__dec__list @ ( inj__c_ty_2Elist_2Elist_ty_2Enum_2Enum @ X0 ) ) ) ).
thf(tp_c_2Enumposrep_2Enum__from__hex__list,type,
c_2Enumposrep_2Enum__from__hex__list: $i ).
thf(mem_c_2Enumposrep_2Enum__from__hex__list,axiom,
mem @ c_2Enumposrep_2Enum__from__hex__list @ ( arr @ ( ty_2Elist_2Elist @ ty_2Enum_2Enum ) @ ty_2Enum_2Enum ) ).
thf(stp_fo_c_2Enumposrep_2Enum__from__hex__list,type,
fo__c_2Enumposrep_2Enum__from__hex__list: tp__c_ty_2Elist_2Elist_ty_2Enum_2Enum > tp__ty_2Enum_2Enum ).
thf(stp_eq_fo_c_2Enumposrep_2Enum__from__hex__list,axiom,
! [X0: tp__c_ty_2Elist_2Elist_ty_2Enum_2Enum] :
( ( inj__ty_2Enum_2Enum @ ( fo__c_2Enumposrep_2Enum__from__hex__list @ X0 ) )
= ( ap @ c_2Enumposrep_2Enum__from__hex__list @ ( inj__c_ty_2Elist_2Elist_ty_2Enum_2Enum @ X0 ) ) ) ).
thf(tp_c_2Enumposrep_2Enum__from__oct__list,type,
c_2Enumposrep_2Enum__from__oct__list: $i ).
thf(mem_c_2Enumposrep_2Enum__from__oct__list,axiom,
mem @ c_2Enumposrep_2Enum__from__oct__list @ ( arr @ ( ty_2Elist_2Elist @ ty_2Enum_2Enum ) @ ty_2Enum_2Enum ) ).
thf(stp_fo_c_2Enumposrep_2Enum__from__oct__list,type,
fo__c_2Enumposrep_2Enum__from__oct__list: tp__c_ty_2Elist_2Elist_ty_2Enum_2Enum > tp__ty_2Enum_2Enum ).
thf(stp_eq_fo_c_2Enumposrep_2Enum__from__oct__list,axiom,
! [X0: tp__c_ty_2Elist_2Elist_ty_2Enum_2Enum] :
( ( inj__ty_2Enum_2Enum @ ( fo__c_2Enumposrep_2Enum__from__oct__list @ X0 ) )
= ( ap @ c_2Enumposrep_2Enum__from__oct__list @ ( inj__c_ty_2Elist_2Elist_ty_2Enum_2Enum @ X0 ) ) ) ).
thf(tp_c_2Enumposrep_2Enum__to__bin__list,type,
c_2Enumposrep_2Enum__to__bin__list: $i ).
thf(mem_c_2Enumposrep_2Enum__to__bin__list,axiom,
mem @ c_2Enumposrep_2Enum__to__bin__list @ ( arr @ ty_2Enum_2Enum @ ( ty_2Elist_2Elist @ ty_2Enum_2Enum ) ) ).
thf(tp_c_2Enumposrep_2Enum__to__dec__list,type,
c_2Enumposrep_2Enum__to__dec__list: $i ).
thf(mem_c_2Enumposrep_2Enum__to__dec__list,axiom,
mem @ c_2Enumposrep_2Enum__to__dec__list @ ( arr @ ty_2Enum_2Enum @ ( ty_2Elist_2Elist @ ty_2Enum_2Enum ) ) ).
thf(tp_c_2Enumposrep_2Enum__to__hex__list,type,
c_2Enumposrep_2Enum__to__hex__list: $i ).
thf(mem_c_2Enumposrep_2Enum__to__hex__list,axiom,
mem @ c_2Enumposrep_2Enum__to__hex__list @ ( arr @ ty_2Enum_2Enum @ ( ty_2Elist_2Elist @ ty_2Enum_2Enum ) ) ).
thf(tp_c_2Enumposrep_2Enum__to__oct__list,type,
c_2Enumposrep_2Enum__to__oct__list: $i ).
thf(mem_c_2Enumposrep_2Enum__to__oct__list,axiom,
mem @ c_2Enumposrep_2Enum__to__oct__list @ ( arr @ ty_2Enum_2Enum @ ( ty_2Elist_2Elist @ ty_2Enum_2Enum ) ) ).
thf(ax_thm_2Enumposrep_2El2n__def,axiom,
( ! [V0b: tp__ty_2Enum_2Enum] :
( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Enumposrep_2El2n @ ( inj__ty_2Enum_2Enum @ V0b ) ) @ ( c_2Elist_2ENIL @ ty_2Enum_2Enum ) ) )
= fo__c_2Enum_2E0 )
& ! [V1b: tp__ty_2Enum_2Enum,V2h: tp__ty_2Enum_2Enum,V3t: tp__c_ty_2Elist_2Elist_ty_2Enum_2Enum] :
( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Enumposrep_2El2n @ ( inj__ty_2Enum_2Enum @ V1b ) ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V2h ) ) @ ( inj__c_ty_2Elist_2Elist_ty_2Enum_2Enum @ V3t ) ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( ap @ ( ap @ c_2Earithmetic_2EMOD @ ( inj__ty_2Enum_2Enum @ V2h ) ) @ ( inj__ty_2Enum_2Enum @ V1b ) ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2A @ ( inj__ty_2Enum_2Enum @ V1b ) ) @ ( ap @ ( ap @ c_2Enumposrep_2El2n @ ( inj__ty_2Enum_2Enum @ V1b ) ) @ ( inj__c_ty_2Elist_2Elist_ty_2Enum_2Enum @ V3t ) ) ) ) ) ) ) ).
thf(conj_thm_2Enumposrep_2En2l__ind,axiom,
! [V0P: $i] :
( ( mem @ V0P @ ( arr @ ty_2Enum_2Enum @ ( arr @ ty_2Enum_2Enum @ bool ) ) )
=> ( ! [V1b: tp__ty_2Enum_2Enum,V2n: tp__ty_2Enum_2Enum] :
( ( ~ ( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V2n ) ) @ ( inj__ty_2Enum_2Enum @ V1b ) ) )
| ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V1b ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) )
=> ( p @ ( ap @ ( ap @ V0P @ ( inj__ty_2Enum_2Enum @ V1b ) ) @ ( ap @ ( ap @ c_2Earithmetic_2EDIV @ ( inj__ty_2Enum_2Enum @ V2n ) ) @ ( inj__ty_2Enum_2Enum @ V1b ) ) ) ) )
=> ( p @ ( ap @ ( ap @ V0P @ ( inj__ty_2Enum_2Enum @ V1b ) ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) )
=> ! [V3v: tp__ty_2Enum_2Enum,V4v1: tp__ty_2Enum_2Enum] : ( p @ ( ap @ ( ap @ V0P @ ( inj__ty_2Enum_2Enum @ V3v ) ) @ ( inj__ty_2Enum_2Enum @ V4v1 ) ) ) ) ) ).
thf(conj_thm_2Enumposrep_2En2l__def,axiom,
! [V0n: tp__ty_2Enum_2Enum,V1b: tp__ty_2Enum_2Enum] :
( ( surj__c_ty_2Elist_2Elist_ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Enumposrep_2En2l @ ( inj__ty_2Enum_2Enum @ V1b ) ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) )
= ( surj__c_ty_2Elist_2Elist_ty_2Enum_2Enum @ ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ( ty_2Elist_2Elist @ ty_2Enum_2Enum ) ) @ ( ap @ ( ap @ c_2Ebool_2E_5C_2F @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V0n ) ) @ ( inj__ty_2Enum_2Enum @ V1b ) ) ) @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V1b ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Earithmetic_2EMOD @ ( inj__ty_2Enum_2Enum @ V0n ) ) @ ( inj__ty_2Enum_2Enum @ V1b ) ) ) @ ( c_2Elist_2ENIL @ ty_2Enum_2Enum ) ) ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Earithmetic_2EMOD @ ( inj__ty_2Enum_2Enum @ V0n ) ) @ ( inj__ty_2Enum_2Enum @ V1b ) ) ) @ ( ap @ ( ap @ c_2Enumposrep_2En2l @ ( inj__ty_2Enum_2Enum @ V1b ) ) @ ( ap @ ( ap @ c_2Earithmetic_2EDIV @ ( inj__ty_2Enum_2Enum @ V0n ) ) @ ( inj__ty_2Enum_2Enum @ V1b ) ) ) ) ) ) ) ).
thf(ax_thm_2Enumposrep_2Enum__from__bin__list__def,axiom,
( c_2Enumposrep_2Enum__from__bin__list
= ( ap @ c_2Enumposrep_2El2n @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ).
thf(ax_thm_2Enumposrep_2Enum__from__oct__list__def,axiom,
( c_2Enumposrep_2Enum__from__oct__list
= ( ap @ c_2Enumposrep_2El2n @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( ap @ c_2Earithmetic_2EBIT1 @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) ).
thf(ax_thm_2Enumposrep_2Enum__from__dec__list__def,axiom,
( c_2Enumposrep_2Enum__from__dec__list
= ( ap @ c_2Enumposrep_2El2n @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( ap @ c_2Earithmetic_2EBIT2 @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) ).
thf(ax_thm_2Enumposrep_2Enum__from__hex__list__def,axiom,
( c_2Enumposrep_2Enum__from__hex__list
= ( ap @ c_2Enumposrep_2El2n @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( ap @ c_2Earithmetic_2EBIT1 @ ( ap @ c_2Earithmetic_2EBIT1 @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Enumposrep_2Enum__to__bin__list__def,axiom,
( c_2Enumposrep_2Enum__to__bin__list
= ( ap @ c_2Enumposrep_2En2l @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ).
thf(ax_thm_2Enumposrep_2Enum__to__oct__list__def,axiom,
( c_2Enumposrep_2Enum__to__oct__list
= ( ap @ c_2Enumposrep_2En2l @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( ap @ c_2Earithmetic_2EBIT1 @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) ).
thf(ax_thm_2Enumposrep_2Enum__to__dec__list__def,axiom,
( c_2Enumposrep_2Enum__to__dec__list
= ( ap @ c_2Enumposrep_2En2l @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( ap @ c_2Earithmetic_2EBIT2 @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) ).
thf(ax_thm_2Enumposrep_2Enum__to__hex__list__def,axiom,
( c_2Enumposrep_2Enum__to__hex__list
= ( ap @ c_2Enumposrep_2En2l @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( ap @ c_2Earithmetic_2EBIT1 @ ( ap @ c_2Earithmetic_2EBIT1 @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Enumposrep_2EBOOLIFY__def,axiom,
( ! [V0m: tp__ty_2Enum_2Enum,V1a: tp__c_ty_2Elist_2Elist_o] :
( ( surj__c_ty_2Elist_2Elist_o @ ( ap @ ( ap @ ( ap @ c_2Enumposrep_2EBOOLIFY @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( inj__ty_2Enum_2Enum @ V0m ) ) @ ( inj__c_ty_2Elist_2Elist_o @ V1a ) ) )
= V1a )
& ! [V2n: tp__ty_2Enum_2Enum,V3m: tp__ty_2Enum_2Enum,V4a: tp__c_ty_2Elist_2Elist_o] :
( ( surj__c_ty_2Elist_2Elist_o @ ( ap @ ( ap @ ( ap @ c_2Enumposrep_2EBOOLIFY @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) @ ( inj__ty_2Enum_2Enum @ V3m ) ) @ ( inj__c_ty_2Elist_2Elist_o @ V4a ) ) )
= ( surj__c_ty_2Elist_2Elist_o @ ( ap @ ( ap @ ( ap @ c_2Enumposrep_2EBOOLIFY @ ( inj__ty_2Enum_2Enum @ V2n ) ) @ ( ap @ c_2Earithmetic_2EDIV2 @ ( inj__ty_2Enum_2Enum @ V3m ) ) ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ bool ) @ ( ap @ c_2Earithmetic_2EODD @ ( inj__ty_2Enum_2Enum @ V3m ) ) ) @ ( inj__c_ty_2Elist_2Elist_o @ V4a ) ) ) ) ) ) ).
thf(conj_thm_2Enumposrep_2ELENGTH__n2l,axiom,
! [V0b: tp__ty_2Enum_2Enum,V1n: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) @ ( inj__ty_2Enum_2Enum @ V0b ) ) )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Elist_2ELENGTH @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Enumposrep_2En2l @ ( inj__ty_2Enum_2Enum @ V0b ) ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) @ ( ap @ c_2Enum_2ESUC @ ( ap @ ( ap @ c_2Elogroot_2ELOG @ ( inj__ty_2Enum_2Enum @ V0b ) ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) ) ) ) ) ).
thf(conj_thm_2Enumposrep_2El2n__n2l,axiom,
! [V0b: tp__ty_2Enum_2Enum,V1n: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) @ ( inj__ty_2Enum_2Enum @ V0b ) ) )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Enumposrep_2El2n @ ( inj__ty_2Enum_2Enum @ V0b ) ) @ ( ap @ ( ap @ c_2Enumposrep_2En2l @ ( inj__ty_2Enum_2Enum @ V0b ) ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) )
= V1n ) ) ).
thf(conj_thm_2Enumposrep_2El2n__lt,axiom,
! [V0l: tp__c_ty_2Elist_2Elist_ty_2Enum_2Enum,V1b: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( inj__ty_2Enum_2Enum @ V1b ) ) )
=> ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( ap @ ( ap @ c_2Enumposrep_2El2n @ ( inj__ty_2Enum_2Enum @ V1b ) ) @ ( inj__c_ty_2Elist_2Elist_ty_2Enum_2Enum @ V0l ) ) ) @ ( ap @ ( ap @ c_2Earithmetic_2EEXP @ ( inj__ty_2Enum_2Enum @ V1b ) ) @ ( ap @ ( c_2Elist_2ELENGTH @ ty_2Enum_2Enum ) @ ( inj__c_ty_2Elist_2Elist_ty_2Enum_2Enum @ V0l ) ) ) ) ) ) ).
thf(conj_thm_2Enumposrep_2ELENGTH__l2n,axiom,
! [V0b: tp__ty_2Enum_2Enum,V1l: tp__c_ty_2Elist_2Elist_ty_2Enum_2Enum] :
( ( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) @ ( inj__ty_2Enum_2Enum @ V0b ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Elist_2EEVERY @ ty_2Enum_2Enum ) @ ( ap @ c_2Earithmetic_2E_3E @ ( inj__ty_2Enum_2Enum @ V0b ) ) ) @ ( inj__c_ty_2Elist_2Elist_ty_2Enum_2Enum @ V1l ) ) )
& ( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Enumposrep_2El2n @ ( inj__ty_2Enum_2Enum @ V0b ) ) @ ( inj__c_ty_2Elist_2Elist_ty_2Enum_2Enum @ V1l ) ) )
!= fo__c_2Enum_2E0 ) )
=> ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( ap @ c_2Enum_2ESUC @ ( ap @ ( ap @ c_2Elogroot_2ELOG @ ( inj__ty_2Enum_2Enum @ V0b ) ) @ ( ap @ ( ap @ c_2Enumposrep_2El2n @ ( inj__ty_2Enum_2Enum @ V0b ) ) @ ( inj__c_ty_2Elist_2Elist_ty_2Enum_2Enum @ V1l ) ) ) ) ) @ ( ap @ ( c_2Elist_2ELENGTH @ ty_2Enum_2Enum ) @ ( inj__c_ty_2Elist_2Elist_ty_2Enum_2Enum @ V1l ) ) ) ) ) ).
thf(conj_thm_2Enumposrep_2El2n__DIGIT,axiom,
! [V0b: tp__ty_2Enum_2Enum,V1l: tp__c_ty_2Elist_2Elist_ty_2Enum_2Enum,V2x: tp__ty_2Enum_2Enum] :
( ( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) @ ( inj__ty_2Enum_2Enum @ V0b ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Elist_2EEVERY @ ty_2Enum_2Enum ) @ ( ap @ c_2Earithmetic_2E_3E @ ( inj__ty_2Enum_2Enum @ V0b ) ) ) @ ( inj__c_ty_2Elist_2Elist_ty_2Enum_2Enum @ V1l ) ) )
& ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V2x ) ) @ ( ap @ ( c_2Elist_2ELENGTH @ ty_2Enum_2Enum ) @ ( inj__c_ty_2Elist_2Elist_ty_2Enum_2Enum @ V1l ) ) ) ) )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Earithmetic_2EMOD @ ( ap @ ( ap @ c_2Earithmetic_2EDIV @ ( ap @ ( ap @ c_2Enumposrep_2El2n @ ( inj__ty_2Enum_2Enum @ V0b ) ) @ ( inj__c_ty_2Elist_2Elist_ty_2Enum_2Enum @ V1l ) ) ) @ ( ap @ ( ap @ c_2Earithmetic_2EEXP @ ( inj__ty_2Enum_2Enum @ V0b ) ) @ ( inj__ty_2Enum_2Enum @ V2x ) ) ) ) @ ( inj__ty_2Enum_2Enum @ V0b ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ ( c_2Elist_2EEL @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V2x ) ) @ ( inj__c_ty_2Elist_2Elist_ty_2Enum_2Enum @ V1l ) ) ) ) ) ).
thf(conj_thm_2Enumposrep_2EDIV__0__IMP__LT,axiom,
! [V0b: tp__ty_2Enum_2Enum,V1n: tp__ty_2Enum_2Enum] :
( ( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) @ ( inj__ty_2Enum_2Enum @ V0b ) ) )
& ( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Earithmetic_2EDIV @ ( inj__ty_2Enum_2Enum @ V1n ) ) @ ( inj__ty_2Enum_2Enum @ V0b ) ) )
= fo__c_2Enum_2E0 ) )
=> ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V1n ) ) @ ( inj__ty_2Enum_2Enum @ V0b ) ) ) ) ).
thf(conj_thm_2Enumposrep_2EEL__n2l,axiom,
! [V0b: tp__ty_2Enum_2Enum,V1x: tp__ty_2Enum_2Enum,V2n: tp__ty_2Enum_2Enum] :
( ( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) @ ( inj__ty_2Enum_2Enum @ V0b ) ) )
& ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V1x ) ) @ ( ap @ ( c_2Elist_2ELENGTH @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Enumposrep_2En2l @ ( inj__ty_2Enum_2Enum @ V0b ) ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) ) ) )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ ( c_2Elist_2EEL @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V1x ) ) @ ( ap @ ( ap @ c_2Enumposrep_2En2l @ ( inj__ty_2Enum_2Enum @ V0b ) ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Earithmetic_2EMOD @ ( ap @ ( ap @ c_2Earithmetic_2EDIV @ ( inj__ty_2Enum_2Enum @ V2n ) ) @ ( ap @ ( ap @ c_2Earithmetic_2EEXP @ ( inj__ty_2Enum_2Enum @ V0b ) ) @ ( inj__ty_2Enum_2Enum @ V1x ) ) ) ) @ ( inj__ty_2Enum_2Enum @ V0b ) ) ) ) ) ).
thf(conj_thm_2Enumposrep_2En2l__l2n,axiom,
! [V0b: tp__ty_2Enum_2Enum,V1l: tp__c_ty_2Elist_2Elist_ty_2Enum_2Enum] :
( ( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) @ ( inj__ty_2Enum_2Enum @ V0b ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Elist_2EEVERY @ ty_2Enum_2Enum ) @ ( ap @ c_2Earithmetic_2E_3E @ ( inj__ty_2Enum_2Enum @ V0b ) ) ) @ ( inj__c_ty_2Elist_2Elist_ty_2Enum_2Enum @ V1l ) ) ) )
=> ( ( surj__c_ty_2Elist_2Elist_ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Enumposrep_2En2l @ ( inj__ty_2Enum_2Enum @ V0b ) ) @ ( ap @ ( ap @ c_2Enumposrep_2El2n @ ( inj__ty_2Enum_2Enum @ V0b ) ) @ ( inj__c_ty_2Elist_2Elist_ty_2Enum_2Enum @ V1l ) ) ) )
= ( surj__c_ty_2Elist_2Elist_ty_2Enum_2Enum @ ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ( ty_2Elist_2Elist @ ty_2Enum_2Enum ) ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Enumposrep_2El2n @ ( inj__ty_2Enum_2Enum @ V0b ) ) @ ( inj__c_ty_2Elist_2Elist_ty_2Enum_2Enum @ V1l ) ) ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( c_2Elist_2ENIL @ ty_2Enum_2Enum ) ) ) @ ( ap @ ( ap @ ( c_2Elist_2ETAKE @ ty_2Enum_2Enum ) @ ( ap @ c_2Enum_2ESUC @ ( ap @ ( ap @ c_2Elogroot_2ELOG @ ( inj__ty_2Enum_2Enum @ V0b ) ) @ ( ap @ ( ap @ c_2Enumposrep_2El2n @ ( inj__ty_2Enum_2Enum @ V0b ) ) @ ( inj__c_ty_2Elist_2Elist_ty_2Enum_2Enum @ V1l ) ) ) ) ) @ ( inj__c_ty_2Elist_2Elist_ty_2Enum_2Enum @ V1l ) ) ) ) ) ) ).
thf(conj_thm_2Enumposrep_2El2n__eq__0,axiom,
! [V0b: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( inj__ty_2Enum_2Enum @ V0b ) ) )
=> ! [V1l: tp__c_ty_2Elist_2Elist_ty_2Enum_2Enum] :
( ( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Enumposrep_2El2n @ ( inj__ty_2Enum_2Enum @ V0b ) ) @ ( inj__c_ty_2Elist_2Elist_ty_2Enum_2Enum @ V1l ) ) )
= fo__c_2Enum_2E0 )
<=> ( p @ ( ap @ ( ap @ ( c_2Elist_2EEVERY @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ ty_2Enum_2Enum @ bool @ ty_2Enum_2Enum ) @ ( ap @ ( c_2Emin_2E_3D @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ ( ap @ ( c_2Ecombin_2EC @ ty_2Enum_2Enum @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) @ c_2Earithmetic_2EMOD ) @ ( inj__ty_2Enum_2Enum @ V0b ) ) ) ) @ ( inj__c_ty_2Elist_2Elist_ty_2Enum_2Enum @ V1l ) ) ) ) ) ).
thf(conj_thm_2Enumposrep_2El2n__SNOC__0,axiom,
! [V0b: tp__ty_2Enum_2Enum,V1ls: tp__c_ty_2Elist_2Elist_ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( inj__ty_2Enum_2Enum @ V0b ) ) )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Enumposrep_2El2n @ ( inj__ty_2Enum_2Enum @ V0b ) ) @ ( ap @ ( ap @ ( c_2Elist_2ESNOC @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( inj__c_ty_2Elist_2Elist_ty_2Enum_2Enum @ V1ls ) ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Enumposrep_2El2n @ ( inj__ty_2Enum_2Enum @ V0b ) ) @ ( inj__c_ty_2Elist_2Elist_ty_2Enum_2Enum @ V1ls ) ) ) ) ) ).
thf(conj_thm_2Enumposrep_2ELOG__l2n,axiom,
! [V0b: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) @ ( inj__ty_2Enum_2Enum @ V0b ) ) )
=> ! [V1l: tp__c_ty_2Elist_2Elist_ty_2Enum_2Enum] :
( ( ( V1l
!= ( surj__c_ty_2Elist_2Elist_ty_2Enum_2Enum @ ( c_2Elist_2ENIL @ ty_2Enum_2Enum ) ) )
& ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( ap @ ( c_2Elist_2ELAST @ ty_2Enum_2Enum ) @ ( inj__c_ty_2Elist_2Elist_ty_2Enum_2Enum @ V1l ) ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Elist_2EEVERY @ ty_2Enum_2Enum ) @ ( ap @ c_2Earithmetic_2E_3E @ ( inj__ty_2Enum_2Enum @ V0b ) ) ) @ ( inj__c_ty_2Elist_2Elist_ty_2Enum_2Enum @ V1l ) ) ) )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Elogroot_2ELOG @ ( inj__ty_2Enum_2Enum @ V0b ) ) @ ( ap @ ( ap @ c_2Enumposrep_2El2n @ ( inj__ty_2Enum_2Enum @ V0b ) ) @ ( inj__c_ty_2Elist_2Elist_ty_2Enum_2Enum @ V1l ) ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ c_2Eprim__rec_2EPRE @ ( ap @ ( c_2Elist_2ELENGTH @ ty_2Enum_2Enum ) @ ( inj__c_ty_2Elist_2Elist_ty_2Enum_2Enum @ V1l ) ) ) ) ) ) ) ).
thf(conj_thm_2Enumposrep_2El2n__dropWhile__0,axiom,
! [V0b: tp__ty_2Enum_2Enum,V1ls: tp__c_ty_2Elist_2Elist_ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( inj__ty_2Enum_2Enum @ V0b ) ) )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Enumposrep_2El2n @ ( inj__ty_2Enum_2Enum @ V0b ) ) @ ( ap @ ( c_2Elist_2EREVERSE @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ ( c_2Elist_2EdropWhile @ ty_2Enum_2Enum ) @ ( ap @ ( c_2Emin_2E_3D @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ ( c_2Elist_2EREVERSE @ ty_2Enum_2Enum ) @ ( inj__c_ty_2Elist_2Elist_ty_2Enum_2Enum @ V1ls ) ) ) ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Enumposrep_2El2n @ ( inj__ty_2Enum_2Enum @ V0b ) ) @ ( inj__c_ty_2Elist_2Elist_ty_2Enum_2Enum @ V1ls ) ) ) ) ) ).
thf(conj_thm_2Enumposrep_2ELOG__l2n__dropWhile,axiom,
! [V0b: tp__ty_2Enum_2Enum,V1l: tp__c_ty_2Elist_2Elist_ty_2Enum_2Enum] :
( ( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) @ ( inj__ty_2Enum_2Enum @ V0b ) ) )
& ( p
@ ( ap
@ ( ap @ ( c_2Elist_2EEXISTS @ ty_2Enum_2Enum )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V2y: $i] : ( ap @ c_2Ebool_2E_7E @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ V2y ) ) ) )
@ ( inj__c_ty_2Elist_2Elist_ty_2Enum_2Enum @ V1l ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Elist_2EEVERY @ ty_2Enum_2Enum ) @ ( ap @ c_2Earithmetic_2E_3E @ ( inj__ty_2Enum_2Enum @ V0b ) ) ) @ ( inj__c_ty_2Elist_2Elist_ty_2Enum_2Enum @ V1l ) ) ) )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Elogroot_2ELOG @ ( inj__ty_2Enum_2Enum @ V0b ) ) @ ( ap @ ( ap @ c_2Enumposrep_2El2n @ ( inj__ty_2Enum_2Enum @ V0b ) ) @ ( inj__c_ty_2Elist_2Elist_ty_2Enum_2Enum @ V1l ) ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ c_2Eprim__rec_2EPRE @ ( ap @ ( c_2Elist_2ELENGTH @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ ( c_2Elist_2EdropWhile @ ty_2Enum_2Enum ) @ ( ap @ ( c_2Emin_2E_3D @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ ( c_2Elist_2EREVERSE @ ty_2Enum_2Enum ) @ ( inj__c_ty_2Elist_2Elist_ty_2Enum_2Enum @ V1l ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Enumposrep_2En2l__BOUND,axiom,
! [V0b: tp__ty_2Enum_2Enum,V1n: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( inj__ty_2Enum_2Enum @ V0b ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Elist_2EEVERY @ ty_2Enum_2Enum ) @ ( ap @ c_2Earithmetic_2E_3E @ ( inj__ty_2Enum_2Enum @ V0b ) ) ) @ ( ap @ ( ap @ c_2Enumposrep_2En2l @ ( inj__ty_2Enum_2Enum @ V0b ) ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) ) ) ).
thf(conj_thm_2Enumposrep_2El2n__pow2__compute,axiom,
( ! [V0p: tp__ty_2Enum_2Enum] :
( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Enumposrep_2El2n @ ( 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 @ V0p ) ) ) @ ( c_2Elist_2ENIL @ ty_2Enum_2Enum ) ) )
= fo__c_2Enum_2E0 )
& ! [V1p: tp__ty_2Enum_2Enum,V2h: tp__ty_2Enum_2Enum,V3t: tp__c_ty_2Elist_2Elist_ty_2Enum_2Enum] :
( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Enumposrep_2El2n @ ( 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 @ V1p ) ) ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V2h ) ) @ ( inj__c_ty_2Elist_2Elist_ty_2Enum_2Enum @ V3t ) ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( ap @ ( ap @ c_2Ebit_2EMOD__2EXP @ ( inj__ty_2Enum_2Enum @ V1p ) ) @ ( inj__ty_2Enum_2Enum @ V2h ) ) ) @ ( ap @ ( ap @ c_2Ebit_2ETIMES__2EXP @ ( inj__ty_2Enum_2Enum @ V1p ) ) @ ( ap @ ( ap @ c_2Enumposrep_2El2n @ ( 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 @ V1p ) ) ) @ ( inj__c_ty_2Elist_2Elist_ty_2Enum_2Enum @ V3t ) ) ) ) ) ) ) ).
thf(conj_thm_2Enumposrep_2En2l__pow2__compute,axiom,
! [V0p: tp__ty_2Enum_2Enum,V1n: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( inj__ty_2Enum_2Enum @ V0p ) ) )
=> ( ( surj__c_ty_2Elist_2Elist_ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Enumposrep_2En2l @ ( 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 @ V0p ) ) ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) )
= ( surj__c_ty_2Elist_2Elist_ty_2Enum_2Enum
@ ( ap
@ ( ap @ ( c_2Ebool_2ELET @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) @ ( ty_2Elist_2Elist @ ty_2Enum_2Enum ) )
@ ( ap @ ( c_2Epair_2EUNCURRY @ ty_2Enum_2Enum @ ty_2Enum_2Enum @ ( ty_2Elist_2Elist @ ty_2Enum_2Enum ) )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V2q: $i] :
( lam @ ty_2Enum_2Enum
@ ^ [V3r: $i] : ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ( ty_2Elist_2Elist @ ty_2Enum_2Enum ) ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ty_2Enum_2Enum ) @ V2q ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ty_2Enum_2Enum ) @ V3r ) @ ( c_2Elist_2ENIL @ ty_2Enum_2Enum ) ) ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ty_2Enum_2Enum ) @ V3r ) @ ( ap @ ( ap @ c_2Enumposrep_2En2l @ ( 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 @ V0p ) ) ) @ V2q ) ) ) ) ) ) )
@ ( ap @ ( ap @ c_2Ebit_2EDIVMOD__2EXP @ ( inj__ty_2Enum_2Enum @ V0p ) ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) ) ) ) ).
thf(ax_thm_2Enumposrep_2El2n2,axiom,
( c_2Enumposrep_2El2n2
= ( ap @ c_2Enumposrep_2El2n @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ).
thf(conj_thm_2Enumposrep_2El2n__2__thms,axiom,
( ! [V0t: tp__c_ty_2Elist_2Elist_ty_2Enum_2Enum] :
( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Enumposrep_2El2n @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( inj__c_ty_2Elist_2Elist_ty_2Enum_2Enum @ V0t ) ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Enumposrep_2El2n2 @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( inj__c_ty_2Elist_2Elist_ty_2Enum_2Enum @ V0t ) ) ) ) ) )
& ! [V1t: tp__c_ty_2Elist_2Elist_ty_2Enum_2Enum] :
( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Enumposrep_2El2n @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ty_2Enum_2Enum ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) @ ( inj__c_ty_2Elist_2Elist_ty_2Enum_2Enum @ V1t ) ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Enumposrep_2El2n2 @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ty_2Enum_2Enum ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) @ ( inj__c_ty_2Elist_2Elist_ty_2Enum_2Enum @ V1t ) ) ) ) ) )
& ( ( surj__ty_2Enum_2Enum @ ( ap @ c_2Enumposrep_2El2n2 @ ( c_2Elist_2ENIL @ ty_2Enum_2Enum ) ) )
= fo__c_2Earithmetic_2EZERO )
& ! [V2t: tp__c_ty_2Elist_2Elist_ty_2Enum_2Enum] :
( ( surj__ty_2Enum_2Enum @ ( ap @ c_2Enumposrep_2El2n2 @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( inj__c_ty_2Elist_2Elist_ty_2Enum_2Enum @ V2t ) ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ c_2Enumeral_2EiDUB @ ( ap @ c_2Enumposrep_2El2n2 @ ( inj__c_ty_2Elist_2Elist_ty_2Enum_2Enum @ V2t ) ) ) ) )
& ! [V3t: tp__c_ty_2Elist_2Elist_ty_2Enum_2Enum] :
( ( surj__ty_2Enum_2Enum @ ( ap @ c_2Enumposrep_2El2n2 @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ty_2Enum_2Enum ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) @ ( inj__c_ty_2Elist_2Elist_ty_2Enum_2Enum @ V3t ) ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ c_2Earithmetic_2EBIT1 @ ( ap @ c_2Enumposrep_2El2n2 @ ( inj__c_ty_2Elist_2Elist_ty_2Enum_2Enum @ V3t ) ) ) ) ) ) ).
thf(conj_thm_2Enumposrep_2EBIT__num__from__bin__list,axiom,
! [V0x: tp__ty_2Enum_2Enum,V1l: tp__c_ty_2Elist_2Elist_ty_2Enum_2Enum] :
( ( ( p @ ( ap @ ( ap @ ( c_2Elist_2EEVERY @ ty_2Enum_2Enum ) @ ( ap @ c_2Earithmetic_2E_3E @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( inj__c_ty_2Elist_2Elist_ty_2Enum_2Enum @ V1l ) ) )
& ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V0x ) ) @ ( ap @ ( c_2Elist_2ELENGTH @ ty_2Enum_2Enum ) @ ( inj__c_ty_2Elist_2Elist_ty_2Enum_2Enum @ V1l ) ) ) ) )
=> ( ( p @ ( ap @ ( ap @ c_2Ebit_2EBIT @ ( inj__ty_2Enum_2Enum @ V0x ) ) @ ( ap @ c_2Enumposrep_2Enum__from__bin__list @ ( inj__c_ty_2Elist_2Elist_ty_2Enum_2Enum @ V1l ) ) ) )
<=> ( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ ( c_2Elist_2EEL @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V0x ) ) @ ( inj__c_ty_2Elist_2Elist_ty_2Enum_2Enum @ V1l ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) ).
thf(conj_thm_2Enumposrep_2EEL__num__to__bin__list,axiom,
! [V0x: tp__ty_2Enum_2Enum,V1n: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V0x ) ) @ ( ap @ ( c_2Elist_2ELENGTH @ ty_2Enum_2Enum ) @ ( ap @ c_2Enumposrep_2Enum__to__bin__list @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) ) )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ ( c_2Elist_2EEL @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V0x ) ) @ ( ap @ c_2Enumposrep_2Enum__to__bin__list @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Ebit_2EBITV @ ( inj__ty_2Enum_2Enum @ V1n ) ) @ ( inj__ty_2Enum_2Enum @ V0x ) ) ) ) ) ).
thf(conj_thm_2Enumposrep_2Enum__bin__list,axiom,
( ( ap @ ( ap @ ( c_2Ecombin_2Eo @ ty_2Enum_2Enum @ ty_2Enum_2Enum @ ( ty_2Elist_2Elist @ ty_2Enum_2Enum ) ) @ c_2Enumposrep_2Enum__from__bin__list ) @ c_2Enumposrep_2Enum__to__bin__list )
= ( c_2Ecombin_2EI @ ty_2Enum_2Enum ) ) ).
thf(conj_thm_2Enumposrep_2Enum__oct__list,axiom,
( ( ap @ ( ap @ ( c_2Ecombin_2Eo @ ty_2Enum_2Enum @ ty_2Enum_2Enum @ ( ty_2Elist_2Elist @ ty_2Enum_2Enum ) ) @ c_2Enumposrep_2Enum__from__oct__list ) @ c_2Enumposrep_2Enum__to__oct__list )
= ( c_2Ecombin_2EI @ ty_2Enum_2Enum ) ) ).
thf(conj_thm_2Enumposrep_2Enum__dec__list,axiom,
( ( ap @ ( ap @ ( c_2Ecombin_2Eo @ ty_2Enum_2Enum @ ty_2Enum_2Enum @ ( ty_2Elist_2Elist @ ty_2Enum_2Enum ) ) @ c_2Enumposrep_2Enum__from__dec__list ) @ c_2Enumposrep_2Enum__to__dec__list )
= ( c_2Ecombin_2EI @ ty_2Enum_2Enum ) ) ).
thf(conj_thm_2Enumposrep_2Enum__hex__list,axiom,
( ( ap @ ( ap @ ( c_2Ecombin_2Eo @ ty_2Enum_2Enum @ ty_2Enum_2Enum @ ( ty_2Elist_2Elist @ ty_2Enum_2Enum ) ) @ c_2Enumposrep_2Enum__from__hex__list ) @ c_2Enumposrep_2Enum__to__hex__list )
= ( c_2Ecombin_2EI @ ty_2Enum_2Enum ) ) ).
%------------------------------------------------------------------------------