ITP001 Axioms: ITP018_7.ax


%------------------------------------------------------------------------------
% File     : ITP018_7 : TPTP v9.0.0. Bugfixed v7.5.0.
% Domain   : Interactive Theorem Proving
% Axioms   : HOL4 syntactic export, chainy mode
% Version  : [BG+19] axioms.
% English  :

% Refs     : [BG+19] Brown et al. (2019), GRUNGE: A Grand Unified ATP Chall
%          : [Gau19] Gauthier (2019), Email to Geoff Sutcliffe
% Source   : [BG+19]
% Names    : numeral.ax [Gau19]
%          : HL4018_7.ax [TPAP]

% Status   : Satisfiable
% Syntax   : Number of formulae    :  212 (  59 unt; 105 typ;   0 def)
%            Number of atoms       :  304 ( 243 equ)
%            Maximal formula atoms :   40 (   1 avg)
%            Number of connectives :  206 (   9   ~;   3   |; 143   &)
%                                         (  28 <=>;  23  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   37 (   5 avg)
%            Maximal term depth    :   10 (   2 avg)
%            Number of types       :    3 (   2 usr)
%            Number of type conns  :   83 (  50   >;  33   *;   0   +;   0  <<)
%            Number of predicates  :    2 (   1 usr;   0 prp; 1-2 aty)
%            Number of functors    :  102 ( 102 usr;  40 con; 0-5 aty)
%            Number of variables   :  362 ( 324   !;   2   ?; 362   :)
%                                         (  36  !>;   0  ?*;   0  @-;   0  @+)
% SPC      : TF1_SAT_EQU_NAR

% Comments :
% Bugfixes : v7.5.0 - Fixes to the axioms.
%------------------------------------------------------------------------------
tff(tyop_2Emin_2Ebool,type,
    tyop_2Emin_2Ebool: $tType ).

tff(tyop_2Emin_2Efun,type,
    tyop_2Emin_2Efun: ( $tType * $tType ) > $tType ).

tff(tyop_2Enum_2Enum,type,
    tyop_2Enum_2Enum: $tType ).

tff(tyop_2Epair_2Eprod,type,
    tyop_2Epair_2Eprod: ( $tType * $tType ) > $tType ).

tff(app_2E2,type,
    app_2E2: 
      !>[A_27a: $tType,A_27b: $tType] : ( ( tyop_2Emin_2Efun(A_27a,A_27b) * A_27a ) > A_27b ) ).

tff(p,type,
    p: tyop_2Emin_2Ebool > $o ).

tff(combin_i_2E0,type,
    combin_i_2E0: 
      !>[A_27a: $tType] : tyop_2Emin_2Efun(A_27a,A_27a) ).

tff(combin_k_2E0,type,
    combin_k_2E0: 
      !>[A_27a: $tType,A_27b: $tType] : tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,A_27a)) ).

tff(combin_s_2E0,type,
    combin_s_2E0: 
      !>[A_27a: $tType,A_27b: $tType,A_27c: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,A_27c)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(A_27a,A_27c))) ).

tff(c_2Ebool_2E_21_2E0,type,
    c_2Ebool_2E_21_2E0: 
      !>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool) ).

tff(c_2Ebool_2E_21_2E1,type,
    c_2Ebool_2E_21_2E1: 
      !>[A_27a: $tType] : ( tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool) > tyop_2Emin_2Ebool ) ).

tff(c_2Earithmetic_2E_2A_2E0,type,
    c_2Earithmetic_2E_2A_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum)) ).

tff(c_2Earithmetic_2E_2A_2E2,type,
    c_2Earithmetic_2E_2A_2E2: ( tyop_2Enum_2Enum * tyop_2Enum_2Enum ) > tyop_2Enum_2Enum ).

tff(c_2Earithmetic_2E_2B_2E0,type,
    c_2Earithmetic_2E_2B_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum)) ).

tff(c_2Earithmetic_2E_2B_2E2,type,
    c_2Earithmetic_2E_2B_2E2: ( tyop_2Enum_2Enum * tyop_2Enum_2Enum ) > tyop_2Enum_2Enum ).

tff(c_2Epair_2E_2C_2E0,type,
    c_2Epair_2E_2C_2E0: 
      !>[A_27a: $tType,A_27b: $tType] : tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Epair_2Eprod(A_27a,A_27b))) ).

tff(c_2Epair_2E_2C_2E2,type,
    c_2Epair_2E_2C_2E2: 
      !>[A_27a: $tType,A_27b: $tType] : ( ( A_27a * A_27b ) > tyop_2Epair_2Eprod(A_27a,A_27b) ) ).

tff(c_2Earithmetic_2E_2D_2E0,type,
    c_2Earithmetic_2E_2D_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum)) ).

tff(c_2Earithmetic_2E_2D_2E2,type,
    c_2Earithmetic_2E_2D_2E2: ( tyop_2Enum_2Enum * tyop_2Enum_2Enum ) > tyop_2Enum_2Enum ).

tff(c_2Ebool_2E_2F_5C_2E0,type,
    c_2Ebool_2E_2F_5C_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Ebool)) ).

tff(c_2Ebool_2E_2F_5C_2E2,type,
    c_2Ebool_2E_2F_5C_2E2: ( tyop_2Emin_2Ebool * tyop_2Emin_2Ebool ) > tyop_2Emin_2Ebool ).

tff(c_2Enum_2E0_2E0,type,
    c_2Enum_2E0_2E0: tyop_2Enum_2Enum ).

tff(c_2Eprim__rec_2E_3C_2E0,type,
    c_2Eprim__rec_2E_3C_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)) ).

tff(c_2Eprim__rec_2E_3C_2E2,type,
    c_2Eprim__rec_2E_3C_2E2: ( tyop_2Enum_2Enum * tyop_2Enum_2Enum ) > tyop_2Emin_2Ebool ).

tff(c_2Earithmetic_2E_3C_3D_2E0,type,
    c_2Earithmetic_2E_3C_3D_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)) ).

tff(c_2Earithmetic_2E_3C_3D_2E2,type,
    c_2Earithmetic_2E_3C_3D_2E2: ( tyop_2Enum_2Enum * tyop_2Enum_2Enum ) > tyop_2Emin_2Ebool ).

tff(c_2Emin_2E_3D_2E0,type,
    c_2Emin_2E_3D_2E0: 
      !>[A_27a: $tType] : tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)) ).

tff(c_2Emin_2E_3D_2E2,type,
    c_2Emin_2E_3D_2E2: 
      !>[A_27a: $tType] : ( ( A_27a * A_27a ) > tyop_2Emin_2Ebool ) ).

tff(c_2Emin_2E_3D_3D_3E_2E0,type,
    c_2Emin_2E_3D_3D_3E_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Ebool)) ).

tff(c_2Emin_2E_3D_3D_3E_2E2,type,
    c_2Emin_2E_3D_3D_3E_2E2: ( tyop_2Emin_2Ebool * tyop_2Emin_2Ebool ) > tyop_2Emin_2Ebool ).

tff(c_2Earithmetic_2E_3E_2E0,type,
    c_2Earithmetic_2E_3E_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)) ).

tff(c_2Earithmetic_2E_3E_2E2,type,
    c_2Earithmetic_2E_3E_2E2: ( tyop_2Enum_2Enum * tyop_2Enum_2Enum ) > tyop_2Emin_2Ebool ).

tff(c_2Earithmetic_2E_3E_3D_2E0,type,
    c_2Earithmetic_2E_3E_3D_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)) ).

tff(c_2Earithmetic_2E_3E_3D_2E2,type,
    c_2Earithmetic_2E_3E_3D_2E2: ( tyop_2Enum_2Enum * tyop_2Enum_2Enum ) > tyop_2Emin_2Ebool ).

tff(c_2Ebool_2E_3F_2E0,type,
    c_2Ebool_2E_3F_2E0: 
      !>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool) ).

tff(c_2Ebool_2E_3F_2E1,type,
    c_2Ebool_2E_3F_2E1: 
      !>[A_27a: $tType] : ( tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool) > tyop_2Emin_2Ebool ) ).

tff(c_2Earithmetic_2EBIT1_2E0,type,
    c_2Earithmetic_2EBIT1_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum) ).

tff(c_2Earithmetic_2EBIT1_2E1,type,
    c_2Earithmetic_2EBIT1_2E1: tyop_2Enum_2Enum > tyop_2Enum_2Enum ).

tff(c_2Earithmetic_2EBIT2_2E0,type,
    c_2Earithmetic_2EBIT2_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum) ).

tff(c_2Earithmetic_2EBIT2_2E1,type,
    c_2Earithmetic_2EBIT2_2E1: tyop_2Enum_2Enum > tyop_2Enum_2Enum ).

tff(c_2Ebool_2ECOND_2E0,type,
    c_2Ebool_2ECOND_2E0: 
      !>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,A_27a))) ).

tff(c_2Ebool_2ECOND_2E3,type,
    c_2Ebool_2ECOND_2E3: 
      !>[A_27a: $tType] : ( ( tyop_2Emin_2Ebool * A_27a * A_27a ) > A_27a ) ).

tff(c_2Earithmetic_2EDIV_2E0,type,
    c_2Earithmetic_2EDIV_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum)) ).

tff(c_2Earithmetic_2EDIV_2E2,type,
    c_2Earithmetic_2EDIV_2E2: ( tyop_2Enum_2Enum * tyop_2Enum_2Enum ) > tyop_2Enum_2Enum ).

tff(c_2Earithmetic_2EDIV2_2E0,type,
    c_2Earithmetic_2EDIV2_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum) ).

tff(c_2Earithmetic_2EDIV2_2E1,type,
    c_2Earithmetic_2EDIV2_2E1: tyop_2Enum_2Enum > tyop_2Enum_2Enum ).

tff(c_2Earithmetic_2EDIVMOD_2E0,type,
    c_2Earithmetic_2EDIVMOD_2E0: tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum)),tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum)) ).

tff(c_2Earithmetic_2EDIVMOD_2E1,type,
    c_2Earithmetic_2EDIVMOD_2E1: tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum)) > tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum) ).

tff(c_2Earithmetic_2EEVEN_2E0,type,
    c_2Earithmetic_2EEVEN_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool) ).

tff(c_2Earithmetic_2EEVEN_2E1,type,
    c_2Earithmetic_2EEVEN_2E1: tyop_2Enum_2Enum > tyop_2Emin_2Ebool ).

tff(c_2Earithmetic_2EEXP_2E0,type,
    c_2Earithmetic_2EEXP_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum)) ).

tff(c_2Earithmetic_2EEXP_2E2,type,
    c_2Earithmetic_2EEXP_2E2: ( tyop_2Enum_2Enum * tyop_2Enum_2Enum ) > tyop_2Enum_2Enum ).

tff(c_2Ebool_2EF_2E0,type,
    c_2Ebool_2EF_2E0: tyop_2Emin_2Ebool ).

tff(c_2Earithmetic_2EFACT_2E0,type,
    c_2Earithmetic_2EFACT_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum) ).

tff(c_2Earithmetic_2EFACT_2E1,type,
    c_2Earithmetic_2EFACT_2E1: tyop_2Enum_2Enum > tyop_2Enum_2Enum ).

tff(c_2Epair_2EFST_2E0,type,
    c_2Epair_2EFST_2E0: 
      !>[A_27a: $tType,A_27b: $tType] : tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),A_27a) ).

tff(c_2Epair_2EFST_2E1,type,
    c_2Epair_2EFST_2E1: 
      !>[A_27a: $tType,A_27b: $tType] : ( tyop_2Epair_2Eprod(A_27a,A_27b) > A_27a ) ).

tff(c_2Earithmetic_2EFUNPOW_2E0,type,
    c_2Earithmetic_2EFUNPOW_2E0: 
      !>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27a),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(A_27a,A_27a))) ).

tff(c_2Earithmetic_2EFUNPOW_2E3,type,
    c_2Earithmetic_2EFUNPOW_2E3: 
      !>[A_27a: $tType] : ( ( tyop_2Emin_2Efun(A_27a,A_27a) * tyop_2Enum_2Enum * A_27a ) > A_27a ) ).

tff(c_2Ebool_2ELET_2E0,type,
    c_2Ebool_2ELET_2E0: 
      !>[A_27a: $tType,A_27b: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(A_27a,A_27b)) ).

tff(c_2Ebool_2ELET_2E2,type,
    c_2Ebool_2ELET_2E2: 
      !>[A_27a: $tType,A_27b: $tType] : ( ( tyop_2Emin_2Efun(A_27a,A_27b) * A_27a ) > A_27b ) ).

tff(c_2Earithmetic_2EMAX_2E0,type,
    c_2Earithmetic_2EMAX_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum)) ).

tff(c_2Earithmetic_2EMAX_2E2,type,
    c_2Earithmetic_2EMAX_2E2: ( tyop_2Enum_2Enum * tyop_2Enum_2Enum ) > tyop_2Enum_2Enum ).

tff(c_2Earithmetic_2EMIN_2E0,type,
    c_2Earithmetic_2EMIN_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum)) ).

tff(c_2Earithmetic_2EMIN_2E2,type,
    c_2Earithmetic_2EMIN_2E2: ( tyop_2Enum_2Enum * tyop_2Enum_2Enum ) > tyop_2Enum_2Enum ).

tff(c_2Earithmetic_2EMOD_2E0,type,
    c_2Earithmetic_2EMOD_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum)) ).

tff(c_2Earithmetic_2EMOD_2E2,type,
    c_2Earithmetic_2EMOD_2E2: ( tyop_2Enum_2Enum * tyop_2Enum_2Enum ) > tyop_2Enum_2Enum ).

tff(c_2Earithmetic_2ENUMERAL_2E0,type,
    c_2Earithmetic_2ENUMERAL_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum) ).

tff(c_2Earithmetic_2ENUMERAL_2E1,type,
    c_2Earithmetic_2ENUMERAL_2E1: tyop_2Enum_2Enum > tyop_2Enum_2Enum ).

tff(c_2Earithmetic_2EODD_2E0,type,
    c_2Earithmetic_2EODD_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool) ).

tff(c_2Earithmetic_2EODD_2E1,type,
    c_2Earithmetic_2EODD_2E1: tyop_2Enum_2Enum > tyop_2Emin_2Ebool ).

tff(c_2Eprim__rec_2EPRE_2E0,type,
    c_2Eprim__rec_2EPRE_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum) ).

tff(c_2Eprim__rec_2EPRE_2E1,type,
    c_2Eprim__rec_2EPRE_2E1: tyop_2Enum_2Enum > tyop_2Enum_2Enum ).

tff(c_2Epair_2ESND_2E0,type,
    c_2Epair_2ESND_2E0: 
      !>[A_27a: $tType,A_27b: $tType] : tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),A_27b) ).

tff(c_2Epair_2ESND_2E1,type,
    c_2Epair_2ESND_2E1: 
      !>[A_27a: $tType,A_27b: $tType] : ( tyop_2Epair_2Eprod(A_27a,A_27b) > A_27b ) ).

tff(c_2Enum_2ESUC_2E0,type,
    c_2Enum_2ESUC_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum) ).

tff(c_2Enum_2ESUC_2E1,type,
    c_2Enum_2ESUC_2E1: tyop_2Enum_2Enum > tyop_2Enum_2Enum ).

tff(c_2Ebool_2ET_2E0,type,
    c_2Ebool_2ET_2E0: tyop_2Emin_2Ebool ).

tff(c_2Earithmetic_2EZERO_2E0,type,
    c_2Earithmetic_2EZERO_2E0: tyop_2Enum_2Enum ).

tff(c_2Ebool_2E_5C_2F_2E0,type,
    c_2Ebool_2E_5C_2F_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Ebool)) ).

tff(c_2Ebool_2E_5C_2F_2E2,type,
    c_2Ebool_2E_5C_2F_2E2: ( tyop_2Emin_2Ebool * tyop_2Emin_2Ebool ) > tyop_2Emin_2Ebool ).

tff(c_2Enumeral_2Eexactlog_2E0,type,
    c_2Enumeral_2Eexactlog_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum) ).

tff(c_2Enumeral_2Eexactlog_2E1,type,
    c_2Enumeral_2Eexactlog_2E1: tyop_2Enum_2Enum > tyop_2Enum_2Enum ).

tff(c_2Earithmetic_2Efindq_2E0,type,
    c_2Earithmetic_2Efindq_2E0: tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum)),tyop_2Enum_2Enum) ).

tff(c_2Earithmetic_2Efindq_2E1,type,
    c_2Earithmetic_2Efindq_2E1: tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum)) > tyop_2Enum_2Enum ).

tff(c_2Enumeral_2EiBIT__cases_2E0,type,
    c_2Enumeral_2EiBIT__cases_2E0: 
      !>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,A_27a),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,A_27a),A_27a)))) ).

tff(c_2Enumeral_2EiBIT__cases_2E4,type,
    c_2Enumeral_2EiBIT__cases_2E4: 
      !>[A_27a: $tType] : ( ( tyop_2Enum_2Enum * A_27a * tyop_2Emin_2Efun(tyop_2Enum_2Enum,A_27a) * tyop_2Emin_2Efun(tyop_2Enum_2Enum,A_27a) ) > A_27a ) ).

tff(c_2Enumeral_2EiDUB_2E0,type,
    c_2Enumeral_2EiDUB_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum) ).

tff(c_2Enumeral_2EiDUB_2E1,type,
    c_2Enumeral_2EiDUB_2E1: tyop_2Enum_2Enum > tyop_2Enum_2Enum ).

tff(c_2Enumeral_2EiSQR_2E0,type,
    c_2Enumeral_2EiSQR_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum) ).

tff(c_2Enumeral_2EiSQR_2E1,type,
    c_2Enumeral_2EiSQR_2E1: tyop_2Enum_2Enum > tyop_2Enum_2Enum ).

tff(c_2Enumeral_2EiSUB_2E0,type,
    c_2Enumeral_2EiSUB_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum))) ).

tff(c_2Enumeral_2EiSUB_2E3,type,
    c_2Enumeral_2EiSUB_2E3: ( tyop_2Emin_2Ebool * tyop_2Enum_2Enum * tyop_2Enum_2Enum ) > tyop_2Enum_2Enum ).

tff(c_2Enumeral_2EiZ_2E0,type,
    c_2Enumeral_2EiZ_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum) ).

tff(c_2Enumeral_2EiZ_2E1,type,
    c_2Enumeral_2EiZ_2E1: tyop_2Enum_2Enum > tyop_2Enum_2Enum ).

tff(c_2Enumeral_2EiiSUC_2E0,type,
    c_2Enumeral_2EiiSUC_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum) ).

tff(c_2Enumeral_2EiiSUC_2E1,type,
    c_2Enumeral_2EiiSUC_2E1: tyop_2Enum_2Enum > tyop_2Enum_2Enum ).

tff(c_2Enumeral_2Einternal__mult_2E0,type,
    c_2Enumeral_2Einternal__mult_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum)) ).

tff(c_2Enumeral_2Einternal__mult_2E2,type,
    c_2Enumeral_2Einternal__mult_2E2: ( tyop_2Enum_2Enum * tyop_2Enum_2Enum ) > tyop_2Enum_2Enum ).

tff(c_2Enumeral_2Eonecount_2E0,type,
    c_2Enumeral_2Eonecount_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum)) ).

tff(c_2Enumeral_2Eonecount_2E2,type,
    c_2Enumeral_2Eonecount_2E2: ( tyop_2Enum_2Enum * tyop_2Enum_2Enum ) > tyop_2Enum_2Enum ).

tff(c_2Enumeral_2Etexp__help_2E0,type,
    c_2Enumeral_2Etexp__help_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum)) ).

tff(c_2Enumeral_2Etexp__help_2E2,type,
    c_2Enumeral_2Etexp__help_2E2: ( tyop_2Enum_2Enum * tyop_2Enum_2Enum ) > tyop_2Enum_2Enum ).

tff(c_2Ebool_2E_7E_2E0,type,
    c_2Ebool_2E_7E_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Ebool) ).

tff(c_2Ebool_2E_7E_2E1,type,
    c_2Ebool_2E_7E_2E1: tyop_2Emin_2Ebool > tyop_2Emin_2Ebool ).

tff(thm_2Eextra_2Dho_2Eeq__ext,axiom,
    ! [A_27a: $tType,A_27b: $tType,V0f_2E0: tyop_2Emin_2Efun(A_27a,A_27b),V1g_2E0: tyop_2Emin_2Efun(A_27a,A_27b)] :
      ( ! [V2x_2E0: A_27a] : ( app_2E2(A_27a,A_27b,V0f_2E0,V2x_2E0) = app_2E2(A_27a,A_27b,V1g_2E0,V2x_2E0) )
     => ( V0f_2E0 = V1g_2E0 ) ) ).

tff(thm_2Eextra_2Dho_2Eboolext,axiom,
    ! [V0_2E0: tyop_2Emin_2Ebool,V1_2E0: tyop_2Emin_2Ebool] :
      ( ( p(V0_2E0)
      <=> p(V1_2E0) )
     => ( V0_2E0 = V1_2E0 ) ) ).

tff(thm_2Eextra_2Dho_2Etruth,axiom,
    p(c_2Ebool_2ET_2E0) ).

tff(thm_2Eextra_2Dho_2Enotfalse,axiom,
    ~ p(c_2Ebool_2EF_2E0) ).

tff(thm_2Eextra_2Dho_2Ebool__cases__ax,axiom,
    ! [V0t_2E0: tyop_2Emin_2Ebool] :
      ( ( V0t_2E0 = c_2Ebool_2ET_2E0 )
      | ( V0t_2E0 = c_2Ebool_2EF_2E0 ) ) ).

tff(thm_2Eextra_2Dho_2Ei__thm,axiom,
    ! [A_27a: $tType,V0x_2E0: A_27a] : ( app_2E2(A_27a,A_27a,combin_i_2E0(A_27a),V0x_2E0) = V0x_2E0 ) ).

tff(thm_2Eextra_2Dho_2Ek__thm,axiom,
    ! [A_27a: $tType,A_27b: $tType,V0x_2E0: A_27a,V1y_2E0: A_27b] : ( app_2E2(A_27b,A_27a,app_2E2(A_27a,tyop_2Emin_2Efun(A_27b,A_27a),combin_k_2E0(A_27a,A_27b),V0x_2E0),V1y_2E0) = V0x_2E0 ) ).

tff(thm_2Eextra_2Dho_2Es__thm,axiom,
    ! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,A_27c)),V1g_2E0: tyop_2Emin_2Efun(A_27a,A_27b),V2x_2E0: A_27a] : ( app_2E2(A_27a,A_27c,app_2E2(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(A_27a,A_27c),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,A_27c)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(A_27a,A_27c)),combin_s_2E0(A_27a,A_27b,A_27c),V0f_2E0),V1g_2E0),V2x_2E0) = app_2E2(A_27b,A_27c,app_2E2(A_27a,tyop_2Emin_2Efun(A_27b,A_27c),V0f_2E0,V2x_2E0),app_2E2(A_27a,A_27b,V1g_2E0,V2x_2E0)) ) ).

tff(logicdef_2E_2F_5C,axiom,
    ! [V0_2E0: tyop_2Emin_2Ebool,V1_2E0: tyop_2Emin_2Ebool] :
      ( p(c_2Ebool_2E_2F_5C_2E2(V0_2E0,V1_2E0))
    <=> ( p(V0_2E0)
        & p(V1_2E0) ) ) ).

tff(logicdef_2E_5C_2F,axiom,
    ! [V0_2E0: tyop_2Emin_2Ebool,V1_2E0: tyop_2Emin_2Ebool] :
      ( p(c_2Ebool_2E_5C_2F_2E2(V0_2E0,V1_2E0))
    <=> ( p(V0_2E0)
        | p(V1_2E0) ) ) ).

tff(logicdef_2E_7E,axiom,
    ! [V0_2E0: tyop_2Emin_2Ebool] :
      ( p(c_2Ebool_2E_7E_2E1(V0_2E0))
    <=> ~ p(V0_2E0) ) ).

tff(logicdef_2E_3D_3D_3E,axiom,
    ! [V0_2E0: tyop_2Emin_2Ebool,V1_2E0: tyop_2Emin_2Ebool] :
      ( p(c_2Emin_2E_3D_3D_3E_2E2(V0_2E0,V1_2E0))
    <=> ( p(V0_2E0)
       => p(V1_2E0) ) ) ).

tff(logicdef_2E_3D,axiom,
    ! [A_27a: $tType,V0_2E0: A_27a,V1_2E0: A_27a] :
      ( p(c_2Emin_2E_3D_2E2(A_27a,V0_2E0,V1_2E0))
    <=> ( V0_2E0 = V1_2E0 ) ) ).

tff(quantdef_2E_21,axiom,
    ! [A_27a: $tType,V0f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] :
      ( p(c_2Ebool_2E_21_2E1(A_27a,V0f_2E0))
    <=> ! [V1x_2E0: A_27a] : p(app_2E2(A_27a,tyop_2Emin_2Ebool,V0f_2E0,V1x_2E0)) ) ).

tff(quantdef_2E_3F,axiom,
    ! [A_27a: $tType,V0f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] :
      ( p(c_2Ebool_2E_3F_2E1(A_27a,V0f_2E0))
    <=> ? [V1x_2E0: A_27a] : p(app_2E2(A_27a,tyop_2Emin_2Ebool,V0f_2E0,V1x_2E0)) ) ).

tff(arityeq1_2Ec_2Ebool_2E_21_2E1_2Emono_2EA_27a,axiom,
    ! [A_27a: $tType,X0_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] : ( c_2Ebool_2E_21_2E1(A_27a,X0_2E0) = app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,c_2Ebool_2E_21_2E0(A_27a),X0_2E0) ) ).

tff(arityeq2_2Ec_2Earithmetic_2E_2A_2E2,axiom,
    ! [X0_2E0: tyop_2Enum_2Enum,X1_2E0: tyop_2Enum_2Enum] : ( c_2Earithmetic_2E_2A_2E2(X0_2E0,X1_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),c_2Earithmetic_2E_2A_2E0,X0_2E0),X1_2E0) ) ).

tff(arityeq2_2Ec_2Earithmetic_2E_2B_2E2,axiom,
    ! [X0_2E0: tyop_2Enum_2Enum,X1_2E0: tyop_2Enum_2Enum] : ( c_2Earithmetic_2E_2B_2E2(X0_2E0,X1_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),c_2Earithmetic_2E_2B_2E0,X0_2E0),X1_2E0) ) ).

tff(arityeq2_2Ec_2Epair_2E_2C_2E2_2Emono_2Etyop_2Enum_2Enum_20mono_2Etyop_2Enum_2Enum,axiom,
    ! [X0_2E0: tyop_2Enum_2Enum,X1_2E0: tyop_2Enum_2Enum] : ( c_2Epair_2E_2C_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,X0_2E0,X1_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum)),c_2Epair_2E_2C_2E0(tyop_2Enum_2Enum,tyop_2Enum_2Enum),X0_2E0),X1_2E0) ) ).

tff(arityeq2_2Ec_2Epair_2E_2C_2E2_2Emono_2Etyop_2Enum_2Enum_20mono_2Etyop_2Epair_2Eprod_28tyop_2Enum_2Enum_2Ctyop_2Enum_2Enum_29,axiom,
    ! [X0_2E0: tyop_2Enum_2Enum,X1_2E0: tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum)] : ( c_2Epair_2E_2C_2E2(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum),X0_2E0,X1_2E0) = app_2E2(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum),tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum)),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum),tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum))),c_2Epair_2E_2C_2E0(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum)),X0_2E0),X1_2E0) ) ).

tff(arityeq2_2Ec_2Earithmetic_2E_2D_2E2,axiom,
    ! [X0_2E0: tyop_2Enum_2Enum,X1_2E0: tyop_2Enum_2Enum] : ( c_2Earithmetic_2E_2D_2E2(X0_2E0,X1_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),c_2Earithmetic_2E_2D_2E0,X0_2E0),X1_2E0) ) ).

tff(arityeq2_2Ec_2Ebool_2E_2F_5C_2E2,axiom,
    ! [X0_2E0: tyop_2Emin_2Ebool,X1_2E0: tyop_2Emin_2Ebool] :
      ( ( p(X0_2E0)
        & p(X1_2E0) )
    <=> p(app_2E2(tyop_2Emin_2Ebool,tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Ebool,tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Ebool),c_2Ebool_2E_2F_5C_2E0,X0_2E0),X1_2E0)) ) ).

tff(arityeq2_2Ec_2Eprim__rec_2E_3C_2E2,axiom,
    ! [X0_2E0: tyop_2Enum_2Enum,X1_2E0: tyop_2Enum_2Enum] : ( c_2Eprim__rec_2E_3C_2E2(X0_2E0,X1_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),c_2Eprim__rec_2E_3C_2E0,X0_2E0),X1_2E0) ) ).

tff(arityeq2_2Ec_2Earithmetic_2E_3C_3D_2E2,axiom,
    ! [X0_2E0: tyop_2Enum_2Enum,X1_2E0: tyop_2Enum_2Enum] : ( c_2Earithmetic_2E_3C_3D_2E2(X0_2E0,X1_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),c_2Earithmetic_2E_3C_3D_2E0,X0_2E0),X1_2E0) ) ).

tff(arityeq2_2Ec_2Emin_2E_3D_2E2_2Emono_2EA_27a,axiom,
    ! [A_27a: $tType,X0_2E0: A_27a,X1_2E0: A_27a] :
      ( ( X0_2E0 = X1_2E0 )
    <=> p(app_2E2(A_27a,tyop_2Emin_2Ebool,app_2E2(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),c_2Emin_2E_3D_2E0(A_27a),X0_2E0),X1_2E0)) ) ).

tff(arityeq2_2Ec_2Emin_2E_3D_3D_3E_2E2,axiom,
    ! [X0_2E0: tyop_2Emin_2Ebool,X1_2E0: tyop_2Emin_2Ebool] :
      ( ( p(X0_2E0)
       => p(X1_2E0) )
    <=> p(app_2E2(tyop_2Emin_2Ebool,tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Ebool,tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Ebool),c_2Emin_2E_3D_3D_3E_2E0,X0_2E0),X1_2E0)) ) ).

tff(arityeq2_2Ec_2Earithmetic_2E_3E_2E2,axiom,
    ! [X0_2E0: tyop_2Enum_2Enum,X1_2E0: tyop_2Enum_2Enum] : ( c_2Earithmetic_2E_3E_2E2(X0_2E0,X1_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),c_2Earithmetic_2E_3E_2E0,X0_2E0),X1_2E0) ) ).

tff(arityeq2_2Ec_2Earithmetic_2E_3E_3D_2E2,axiom,
    ! [X0_2E0: tyop_2Enum_2Enum,X1_2E0: tyop_2Enum_2Enum] : ( c_2Earithmetic_2E_3E_3D_2E2(X0_2E0,X1_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),c_2Earithmetic_2E_3E_3D_2E0,X0_2E0),X1_2E0) ) ).

tff(arityeq1_2Ec_2Ebool_2E_3F_2E1_2Emono_2EA_27a,axiom,
    ! [A_27a: $tType,X0_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] : ( c_2Ebool_2E_3F_2E1(A_27a,X0_2E0) = app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,c_2Ebool_2E_3F_2E0(A_27a),X0_2E0) ) ).

tff(arityeq1_2Ec_2Earithmetic_2EBIT1_2E1,axiom,
    ! [X0_2E0: tyop_2Enum_2Enum] : ( c_2Earithmetic_2EBIT1_2E1(X0_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,c_2Earithmetic_2EBIT1_2E0,X0_2E0) ) ).

tff(arityeq1_2Ec_2Earithmetic_2EBIT2_2E1,axiom,
    ! [X0_2E0: tyop_2Enum_2Enum] : ( c_2Earithmetic_2EBIT2_2E1(X0_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,c_2Earithmetic_2EBIT2_2E0,X0_2E0) ) ).

tff(arityeq3_2Ec_2Ebool_2ECOND_2E3_2Emono_2Etyop_2Enum_2Enum,axiom,
    ! [X0_2E0: tyop_2Emin_2Ebool,X1_2E0: tyop_2Enum_2Enum,X2_2E0: tyop_2Enum_2Enum] : ( c_2Ebool_2ECOND_2E3(tyop_2Enum_2Enum,X0_2E0,X1_2E0,X2_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),app_2E2(tyop_2Emin_2Ebool,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum)),c_2Ebool_2ECOND_2E0(tyop_2Enum_2Enum),X0_2E0),X1_2E0),X2_2E0) ) ).

tff(arityeq3_2Ec_2Ebool_2ECOND_2E3_2Emono_2Etyop_2Epair_2Eprod_28tyop_2Enum_2Enum_2Ctyop_2Enum_2Enum_29,axiom,
    ! [X0_2E0: tyop_2Emin_2Ebool,X1_2E0: tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum),X2_2E0: tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum)] : ( c_2Ebool_2ECOND_2E3(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum),X0_2E0,X1_2E0,X2_2E0) = app_2E2(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum),tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum),app_2E2(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum),tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum)),app_2E2(tyop_2Emin_2Ebool,tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum),tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum))),c_2Ebool_2ECOND_2E0(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum)),X0_2E0),X1_2E0),X2_2E0) ) ).

tff(arityeq2_2Ec_2Earithmetic_2EDIV_2E2,axiom,
    ! [X0_2E0: tyop_2Enum_2Enum,X1_2E0: tyop_2Enum_2Enum] : ( c_2Earithmetic_2EDIV_2E2(X0_2E0,X1_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),c_2Earithmetic_2EDIV_2E0,X0_2E0),X1_2E0) ) ).

tff(arityeq1_2Ec_2Earithmetic_2EDIV2_2E1,axiom,
    ! [X0_2E0: tyop_2Enum_2Enum] : ( c_2Earithmetic_2EDIV2_2E1(X0_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,c_2Earithmetic_2EDIV2_2E0,X0_2E0) ) ).

tff(arityeq1_2Ec_2Earithmetic_2EDIVMOD_2E1,axiom,
    ! [X0_2E0: tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum))] : ( c_2Earithmetic_2EDIVMOD_2E1(X0_2E0) = app_2E2(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum)),tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum),c_2Earithmetic_2EDIVMOD_2E0,X0_2E0) ) ).

tff(arityeq1_2Ec_2Earithmetic_2EEVEN_2E1,axiom,
    ! [X0_2E0: tyop_2Enum_2Enum] : ( c_2Earithmetic_2EEVEN_2E1(X0_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,c_2Earithmetic_2EEVEN_2E0,X0_2E0) ) ).

tff(arityeq2_2Ec_2Earithmetic_2EEXP_2E2,axiom,
    ! [X0_2E0: tyop_2Enum_2Enum,X1_2E0: tyop_2Enum_2Enum] : ( c_2Earithmetic_2EEXP_2E2(X0_2E0,X1_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),c_2Earithmetic_2EEXP_2E0,X0_2E0),X1_2E0) ) ).

tff(arityeq1_2Ec_2Earithmetic_2EFACT_2E1,axiom,
    ! [X0_2E0: tyop_2Enum_2Enum] : ( c_2Earithmetic_2EFACT_2E1(X0_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,c_2Earithmetic_2EFACT_2E0,X0_2E0) ) ).

tff(arityeq1_2Ec_2Epair_2EFST_2E1_2Emono_2Etyop_2Enum_2Enum_20mono_2Etyop_2Enum_2Enum,axiom,
    ! [X0_2E0: tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum)] : ( c_2Epair_2EFST_2E1(tyop_2Enum_2Enum,tyop_2Enum_2Enum,X0_2E0) = app_2E2(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum),tyop_2Enum_2Enum,c_2Epair_2EFST_2E0(tyop_2Enum_2Enum,tyop_2Enum_2Enum),X0_2E0) ) ).

tff(arityeq3_2Ec_2Earithmetic_2EFUNPOW_2E3_2Emono_2EA_27a,axiom,
    ! [A_27a: $tType,X0_2E0: tyop_2Emin_2Efun(A_27a,A_27a),X1_2E0: tyop_2Enum_2Enum,X2_2E0: A_27a] : ( c_2Earithmetic_2EFUNPOW_2E3(A_27a,X0_2E0,X1_2E0,X2_2E0) = app_2E2(A_27a,A_27a,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(A_27a,A_27a),app_2E2(tyop_2Emin_2Efun(A_27a,A_27a),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(A_27a,A_27a)),c_2Earithmetic_2EFUNPOW_2E0(A_27a),X0_2E0),X1_2E0),X2_2E0) ) ).

tff(arityeq2_2Ec_2Ebool_2ELET_2E2_2Emono_2Etyop_2Enum_2Enum_20mono_2Etyop_2Enum_2Enum,axiom,
    ! [X0_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),X1_2E0: tyop_2Enum_2Enum] : ( c_2Ebool_2ELET_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,X0_2E0,X1_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),c_2Ebool_2ELET_2E0(tyop_2Enum_2Enum,tyop_2Enum_2Enum),X0_2E0),X1_2E0) ) ).

tff(arityeq2_2Ec_2Ebool_2ELET_2E2_2Emono_2Etyop_2Enum_2Enum_20mono_2Etyop_2Epair_2Eprod_28tyop_2Enum_2Enum_2Ctyop_2Enum_2Enum_29,axiom,
    ! [X0_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum)),X1_2E0: tyop_2Enum_2Enum] : ( c_2Ebool_2ELET_2E2(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum),X0_2E0,X1_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum)),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum)),c_2Ebool_2ELET_2E0(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum)),X0_2E0),X1_2E0) ) ).

tff(arityeq2_2Ec_2Earithmetic_2EMAX_2E2,axiom,
    ! [X0_2E0: tyop_2Enum_2Enum,X1_2E0: tyop_2Enum_2Enum] : ( c_2Earithmetic_2EMAX_2E2(X0_2E0,X1_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),c_2Earithmetic_2EMAX_2E0,X0_2E0),X1_2E0) ) ).

tff(arityeq2_2Ec_2Earithmetic_2EMIN_2E2,axiom,
    ! [X0_2E0: tyop_2Enum_2Enum,X1_2E0: tyop_2Enum_2Enum] : ( c_2Earithmetic_2EMIN_2E2(X0_2E0,X1_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),c_2Earithmetic_2EMIN_2E0,X0_2E0),X1_2E0) ) ).

tff(arityeq2_2Ec_2Earithmetic_2EMOD_2E2,axiom,
    ! [X0_2E0: tyop_2Enum_2Enum,X1_2E0: tyop_2Enum_2Enum] : ( c_2Earithmetic_2EMOD_2E2(X0_2E0,X1_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),c_2Earithmetic_2EMOD_2E0,X0_2E0),X1_2E0) ) ).

tff(arityeq1_2Ec_2Earithmetic_2ENUMERAL_2E1,axiom,
    ! [X0_2E0: tyop_2Enum_2Enum] : ( c_2Earithmetic_2ENUMERAL_2E1(X0_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,c_2Earithmetic_2ENUMERAL_2E0,X0_2E0) ) ).

tff(arityeq1_2Ec_2Earithmetic_2EODD_2E1,axiom,
    ! [X0_2E0: tyop_2Enum_2Enum] : ( c_2Earithmetic_2EODD_2E1(X0_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,c_2Earithmetic_2EODD_2E0,X0_2E0) ) ).

tff(arityeq1_2Ec_2Eprim__rec_2EPRE_2E1,axiom,
    ! [X0_2E0: tyop_2Enum_2Enum] : ( c_2Eprim__rec_2EPRE_2E1(X0_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,c_2Eprim__rec_2EPRE_2E0,X0_2E0) ) ).

tff(arityeq1_2Ec_2Epair_2ESND_2E1_2Emono_2Etyop_2Enum_2Enum_20mono_2Etyop_2Enum_2Enum,axiom,
    ! [X0_2E0: tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum)] : ( c_2Epair_2ESND_2E1(tyop_2Enum_2Enum,tyop_2Enum_2Enum,X0_2E0) = app_2E2(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum),tyop_2Enum_2Enum,c_2Epair_2ESND_2E0(tyop_2Enum_2Enum,tyop_2Enum_2Enum),X0_2E0) ) ).

tff(arityeq1_2Ec_2Enum_2ESUC_2E1,axiom,
    ! [X0_2E0: tyop_2Enum_2Enum] : ( c_2Enum_2ESUC_2E1(X0_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,c_2Enum_2ESUC_2E0,X0_2E0) ) ).

tff(arityeq2_2Ec_2Ebool_2E_5C_2F_2E2,axiom,
    ! [X0_2E0: tyop_2Emin_2Ebool,X1_2E0: tyop_2Emin_2Ebool] :
      ( ( p(X0_2E0)
        | p(X1_2E0) )
    <=> p(app_2E2(tyop_2Emin_2Ebool,tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Ebool,tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Ebool),c_2Ebool_2E_5C_2F_2E0,X0_2E0),X1_2E0)) ) ).

tff(arityeq1_2Ec_2Enumeral_2Eexactlog_2E1,axiom,
    ! [X0_2E0: tyop_2Enum_2Enum] : ( c_2Enumeral_2Eexactlog_2E1(X0_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,c_2Enumeral_2Eexactlog_2E0,X0_2E0) ) ).

tff(arityeq1_2Ec_2Earithmetic_2Efindq_2E1,axiom,
    ! [X0_2E0: tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum))] : ( c_2Earithmetic_2Efindq_2E1(X0_2E0) = app_2E2(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum)),tyop_2Enum_2Enum,c_2Earithmetic_2Efindq_2E0,X0_2E0) ) ).

tff(arityeq4_2Ec_2Enumeral_2EiBIT__cases_2E4_2Emono_2EA_27a,axiom,
    ! [A_27a: $tType,X0_2E0: tyop_2Enum_2Enum,X1_2E0: A_27a,X2_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,A_27a),X3_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,A_27a)] : ( c_2Enumeral_2EiBIT__cases_2E4(A_27a,X0_2E0,X1_2E0,X2_2E0,X3_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,A_27a),A_27a,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,A_27a),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,A_27a),A_27a),app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,A_27a),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,A_27a),A_27a)),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,A_27a),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,A_27a),A_27a))),c_2Enumeral_2EiBIT__cases_2E0(A_27a),X0_2E0),X1_2E0),X2_2E0),X3_2E0) ) ).

tff(arityeq4_2Ec_2Enumeral_2EiBIT__cases_2E4_2Emono_2Etyop_2Enum_2Enum,axiom,
    ! [X0_2E0: tyop_2Enum_2Enum,X1_2E0: tyop_2Enum_2Enum,X2_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),X3_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum)] : ( c_2Enumeral_2EiBIT__cases_2E4(tyop_2Enum_2Enum,X0_2E0,X1_2E0,X2_2E0,X3_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),tyop_2Enum_2Enum,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),tyop_2Enum_2Enum),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),tyop_2Enum_2Enum)),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),tyop_2Enum_2Enum))),c_2Enumeral_2EiBIT__cases_2E0(tyop_2Enum_2Enum),X0_2E0),X1_2E0),X2_2E0),X3_2E0) ) ).

tff(arityeq1_2Ec_2Enumeral_2EiDUB_2E1,axiom,
    ! [X0_2E0: tyop_2Enum_2Enum] : ( c_2Enumeral_2EiDUB_2E1(X0_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,c_2Enumeral_2EiDUB_2E0,X0_2E0) ) ).

tff(arityeq1_2Ec_2Enumeral_2EiSQR_2E1,axiom,
    ! [X0_2E0: tyop_2Enum_2Enum] : ( c_2Enumeral_2EiSQR_2E1(X0_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,c_2Enumeral_2EiSQR_2E0,X0_2E0) ) ).

tff(arityeq3_2Ec_2Enumeral_2EiSUB_2E3,axiom,
    ! [X0_2E0: tyop_2Emin_2Ebool,X1_2E0: tyop_2Enum_2Enum,X2_2E0: tyop_2Enum_2Enum] : ( c_2Enumeral_2EiSUB_2E3(X0_2E0,X1_2E0,X2_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),app_2E2(tyop_2Emin_2Ebool,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum)),c_2Enumeral_2EiSUB_2E0,X0_2E0),X1_2E0),X2_2E0) ) ).

tff(arityeq1_2Ec_2Enumeral_2EiZ_2E1,axiom,
    ! [X0_2E0: tyop_2Enum_2Enum] : ( c_2Enumeral_2EiZ_2E1(X0_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,c_2Enumeral_2EiZ_2E0,X0_2E0) ) ).

tff(arityeq1_2Ec_2Enumeral_2EiiSUC_2E1,axiom,
    ! [X0_2E0: tyop_2Enum_2Enum] : ( c_2Enumeral_2EiiSUC_2E1(X0_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,c_2Enumeral_2EiiSUC_2E0,X0_2E0) ) ).

tff(arityeq2_2Ec_2Enumeral_2Einternal__mult_2E2,axiom,
    ! [X0_2E0: tyop_2Enum_2Enum,X1_2E0: tyop_2Enum_2Enum] : ( c_2Enumeral_2Einternal__mult_2E2(X0_2E0,X1_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),c_2Enumeral_2Einternal__mult_2E0,X0_2E0),X1_2E0) ) ).

tff(arityeq2_2Ec_2Enumeral_2Eonecount_2E2,axiom,
    ! [X0_2E0: tyop_2Enum_2Enum,X1_2E0: tyop_2Enum_2Enum] : ( c_2Enumeral_2Eonecount_2E2(X0_2E0,X1_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),c_2Enumeral_2Eonecount_2E0,X0_2E0),X1_2E0) ) ).

tff(arityeq2_2Ec_2Enumeral_2Etexp__help_2E2,axiom,
    ! [X0_2E0: tyop_2Enum_2Enum,X1_2E0: tyop_2Enum_2Enum] : ( c_2Enumeral_2Etexp__help_2E2(X0_2E0,X1_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),c_2Enumeral_2Etexp__help_2E0,X0_2E0),X1_2E0) ) ).

tff(arityeq1_2Ec_2Ebool_2E_7E_2E1,axiom,
    ! [X0_2E0: tyop_2Emin_2Ebool] :
      ( ~ p(X0_2E0)
    <=> p(app_2E2(tyop_2Emin_2Ebool,tyop_2Emin_2Ebool,c_2Ebool_2E_7E_2E0,X0_2E0)) ) ).

tff(thm_2Enumeral_2EiZ,axiom,
    ! [V0x_2E0: tyop_2Enum_2Enum] : ( c_2Enumeral_2EiZ_2E1(V0x_2E0) = V0x_2E0 ) ).

tff(thm_2Enumeral_2EiiSUC,axiom,
    ! [V0n_2E0: tyop_2Enum_2Enum] : ( c_2Enumeral_2EiiSUC_2E1(V0n_2E0) = c_2Enum_2ESUC_2E1(c_2Enum_2ESUC_2E1(V0n_2E0)) ) ).

tff(thm_2Enumeral_2EiBIT__cases,axiom,
    ! [A_27a: $tType] :
      ( ! [V0zf_2E0: A_27a,V1bf1_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,A_27a),V2bf2_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,A_27a)] : ( c_2Enumeral_2EiBIT__cases_2E4(A_27a,c_2Earithmetic_2EZERO_2E0,V0zf_2E0,V1bf1_2E0,V2bf2_2E0) = V0zf_2E0 )
      & ! [V3n_2E0: tyop_2Enum_2Enum,V4zf_2E0: A_27a,V5bf1_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,A_27a),V6bf2_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,A_27a)] : ( c_2Enumeral_2EiBIT__cases_2E4(A_27a,c_2Earithmetic_2EBIT1_2E1(V3n_2E0),V4zf_2E0,V5bf1_2E0,V6bf2_2E0) = app_2E2(tyop_2Enum_2Enum,A_27a,V5bf1_2E0,V3n_2E0) )
      & ! [V7n_2E0: tyop_2Enum_2Enum,V8zf_2E0: A_27a,V9bf1_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,A_27a),V10bf2_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,A_27a)] : ( c_2Enumeral_2EiBIT__cases_2E4(A_27a,c_2Earithmetic_2EBIT2_2E1(V7n_2E0),V8zf_2E0,V9bf1_2E0,V10bf2_2E0) = app_2E2(tyop_2Enum_2Enum,A_27a,V10bf2_2E0,V7n_2E0) ) ) ).

tff(thm_2Enumeral_2EiDUB,axiom,
    ! [V0x_2E0: tyop_2Enum_2Enum] : ( c_2Enumeral_2EiDUB_2E1(V0x_2E0) = c_2Earithmetic_2E_2B_2E2(V0x_2E0,V0x_2E0) ) ).

tff(thm_2Enumeral_2EiSUB__DEF,axiom,
    ! [F3_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum)),F0_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum)),F4_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum)),F1_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum)),F5_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum)),F2_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum))] :
      ( ! [V10n_2E0: tyop_2Enum_2Enum,V13m_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),F3_2E0,V10n_2E0),V13m_2E0) = c_2Enumeral_2EiDUB_2E1(c_2Enumeral_2EiSUB_2E3(c_2Ebool_2ET_2E0,V10n_2E0,V13m_2E0)) )
     => ( ! [V3n_2E0: tyop_2Enum_2Enum,V5m_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),F0_2E0,V3n_2E0),V5m_2E0) = c_2Enumeral_2EiDUB_2E1(c_2Enumeral_2EiSUB_2E3(c_2Ebool_2ET_2E0,V3n_2E0,V5m_2E0)) )
       => ( ! [V3n_2E0: tyop_2Enum_2Enum,V8m_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),F4_2E0,V3n_2E0),V8m_2E0) = c_2Enumeral_2EiDUB_2E1(c_2Enumeral_2EiSUB_2E3(c_2Ebool_2EF_2E0,V3n_2E0,V8m_2E0)) )
         => ( ! [V10n_2E0: tyop_2Enum_2Enum,V12m_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),F1_2E0,V10n_2E0),V12m_2E0) = c_2Earithmetic_2EBIT1_2E1(c_2Enumeral_2EiSUB_2E3(c_2Ebool_2ET_2E0,V10n_2E0,V12m_2E0)) )
           => ( ! [V10n_2E0: tyop_2Enum_2Enum,V15m_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),F5_2E0,V10n_2E0),V15m_2E0) = c_2Earithmetic_2EBIT1_2E1(c_2Enumeral_2EiSUB_2E3(c_2Ebool_2EF_2E0,V10n_2E0,V15m_2E0)) )
             => ( ! [V3n_2E0: tyop_2Enum_2Enum,V6m_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),F2_2E0,V3n_2E0),V6m_2E0) = c_2Earithmetic_2EBIT1_2E1(c_2Enumeral_2EiSUB_2E3(c_2Ebool_2EF_2E0,V3n_2E0,V6m_2E0)) )
               => ( ! [V0b_2E0: tyop_2Emin_2Ebool,V1x_2E0: tyop_2Enum_2Enum] : ( c_2Enumeral_2EiSUB_2E3(V0b_2E0,c_2Earithmetic_2EZERO_2E0,V1x_2E0) = c_2Earithmetic_2EZERO_2E0 )
                  & ! [V2b_2E0: tyop_2Emin_2Ebool,V3n_2E0: tyop_2Enum_2Enum,V4x_2E0: tyop_2Enum_2Enum] : ( c_2Enumeral_2EiSUB_2E3(V2b_2E0,c_2Earithmetic_2EBIT1_2E1(V3n_2E0),V4x_2E0) = c_2Ebool_2ECOND_2E3(tyop_2Enum_2Enum,V2b_2E0,c_2Enumeral_2EiBIT__cases_2E4(tyop_2Enum_2Enum,V4x_2E0,c_2Earithmetic_2EBIT1_2E1(V3n_2E0),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),F0_2E0,V3n_2E0),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),F2_2E0,V3n_2E0)),c_2Enumeral_2EiBIT__cases_2E4(tyop_2Enum_2Enum,V4x_2E0,c_2Enumeral_2EiDUB_2E1(V3n_2E0),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),F2_2E0,V3n_2E0),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),F4_2E0,V3n_2E0))) )
                  & ! [V9b_2E0: tyop_2Emin_2Ebool,V10n_2E0: tyop_2Enum_2Enum,V11x_2E0: tyop_2Enum_2Enum] : ( c_2Enumeral_2EiSUB_2E3(V9b_2E0,c_2Earithmetic_2EBIT2_2E1(V10n_2E0),V11x_2E0) = c_2Ebool_2ECOND_2E3(tyop_2Enum_2Enum,V9b_2E0,c_2Enumeral_2EiBIT__cases_2E4(tyop_2Enum_2Enum,V11x_2E0,c_2Earithmetic_2EBIT2_2E1(V10n_2E0),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),F1_2E0,V10n_2E0),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),F3_2E0,V10n_2E0)),c_2Enumeral_2EiBIT__cases_2E4(tyop_2Enum_2Enum,V11x_2E0,c_2Earithmetic_2EBIT1_2E1(V10n_2E0),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),F3_2E0,V10n_2E0),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),F5_2E0,V10n_2E0))) ) ) ) ) ) ) ) ) ).

tff(thm_2Enumeral_2EiSQR,axiom,
    ! [V0x_2E0: tyop_2Enum_2Enum] : ( c_2Enumeral_2EiSQR_2E1(V0x_2E0) = c_2Earithmetic_2E_2A_2E2(V0x_2E0,V0x_2E0) ) ).

tff(thm_2Enumeral_2Etexp__help__def,axiom,
    ( ! [V0acc_2E0: tyop_2Enum_2Enum] : ( c_2Enumeral_2Etexp__help_2E2(c_2Enum_2E0_2E0,V0acc_2E0) = c_2Earithmetic_2EBIT2_2E1(V0acc_2E0) )
    & ! [V1n_2E0: tyop_2Enum_2Enum,V2acc_2E0: tyop_2Enum_2Enum] : ( c_2Enumeral_2Etexp__help_2E2(c_2Enum_2ESUC_2E1(V1n_2E0),V2acc_2E0) = c_2Enumeral_2Etexp__help_2E2(V1n_2E0,c_2Earithmetic_2EBIT1_2E1(V2acc_2E0)) ) ) ).

tff(thm_2Enumeral_2Eonecount__def,axiom,
    ( ! [V0x_2E0: tyop_2Enum_2Enum] : ( c_2Enumeral_2Eonecount_2E2(c_2Earithmetic_2EZERO_2E0,V0x_2E0) = V0x_2E0 )
    & ! [V1n_2E0: tyop_2Enum_2Enum,V2x_2E0: tyop_2Enum_2Enum] : ( c_2Enumeral_2Eonecount_2E2(c_2Earithmetic_2EBIT1_2E1(V1n_2E0),V2x_2E0) = c_2Enumeral_2Eonecount_2E2(V1n_2E0,c_2Enum_2ESUC_2E1(V2x_2E0)) )
    & ! [V3n_2E0: tyop_2Enum_2Enum,V4x_2E0: tyop_2Enum_2Enum] : ( c_2Enumeral_2Eonecount_2E2(c_2Earithmetic_2EBIT2_2E1(V3n_2E0),V4x_2E0) = c_2Earithmetic_2EZERO_2E0 ) ) ).

tff(thm_2Enumeral_2Eexactlog__def,axiom,
    ! [F1_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),F0_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum)] :
      ( ! [V2x_2E0: tyop_2Enum_2Enum] :
          ( p(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,F1_2E0,V2x_2E0))
        <=> ( V2x_2E0 = c_2Earithmetic_2EZERO_2E0 ) )
     => ( ! [V2x_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,F0_2E0,V2x_2E0) = c_2Ebool_2ECOND_2E3(tyop_2Enum_2Enum,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,F1_2E0,V2x_2E0),c_2Earithmetic_2EZERO_2E0,c_2Earithmetic_2EBIT1_2E1(V2x_2E0)) )
       => ( ( c_2Enumeral_2Eexactlog_2E1(c_2Earithmetic_2EZERO_2E0) = c_2Earithmetic_2EZERO_2E0 )
          & ! [V0n_2E0: tyop_2Enum_2Enum] : ( c_2Enumeral_2Eexactlog_2E1(c_2Earithmetic_2EBIT1_2E1(V0n_2E0)) = c_2Earithmetic_2EZERO_2E0 )
          & ! [V1n_2E0: tyop_2Enum_2Enum] : ( c_2Enumeral_2Eexactlog_2E1(c_2Earithmetic_2EBIT2_2E1(V1n_2E0)) = c_2Ebool_2ELET_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,F0_2E0,c_2Enumeral_2Eonecount_2E2(V1n_2E0,c_2Earithmetic_2EZERO_2E0)) ) ) ) ) ).

tff(thm_2Enumeral_2Einternal__mult__def,axiom,
    c_2Enumeral_2Einternal__mult_2E0 = c_2Earithmetic_2E_2A_2E0 ).

tff(thm_2Enumeral_2Enumeral__suc,axiom,
    ( ( c_2Enum_2ESUC_2E1(c_2Earithmetic_2EZERO_2E0) = c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0) )
    & ! [V0n_2E0: tyop_2Enum_2Enum] : ( c_2Enum_2ESUC_2E1(c_2Earithmetic_2EBIT1_2E1(V0n_2E0)) = c_2Earithmetic_2EBIT2_2E1(V0n_2E0) )
    & ! [V1n_2E0: tyop_2Enum_2Enum] : ( c_2Enum_2ESUC_2E1(c_2Earithmetic_2EBIT2_2E1(V1n_2E0)) = c_2Earithmetic_2EBIT1_2E1(c_2Enum_2ESUC_2E1(V1n_2E0)) ) ) ).

tff(thm_2Enumeral_2Enumeral__distrib,axiom,
    ( ! [V0n_2E0: tyop_2Enum_2Enum] : ( c_2Earithmetic_2E_2B_2E2(c_2Enum_2E0_2E0,V0n_2E0) = V0n_2E0 )
    & ! [V1n_2E0: tyop_2Enum_2Enum] : ( c_2Earithmetic_2E_2B_2E2(V1n_2E0,c_2Enum_2E0_2E0) = V1n_2E0 )
    & ! [V2n_2E0: tyop_2Enum_2Enum,V3m_2E0: tyop_2Enum_2Enum] : ( c_2Earithmetic_2E_2B_2E2(c_2Earithmetic_2ENUMERAL_2E1(V2n_2E0),c_2Earithmetic_2ENUMERAL_2E1(V3m_2E0)) = c_2Earithmetic_2ENUMERAL_2E1(c_2Enumeral_2EiZ_2E1(c_2Earithmetic_2E_2B_2E2(V2n_2E0,V3m_2E0))) )
    & ! [V4n_2E0: tyop_2Enum_2Enum] : ( c_2Earithmetic_2E_2A_2E2(c_2Enum_2E0_2E0,V4n_2E0) = c_2Enum_2E0_2E0 )
    & ! [V5n_2E0: tyop_2Enum_2Enum] : ( c_2Earithmetic_2E_2A_2E2(V5n_2E0,c_2Enum_2E0_2E0) = c_2Enum_2E0_2E0 )
    & ! [V6n_2E0: tyop_2Enum_2Enum,V7m_2E0: tyop_2Enum_2Enum] : ( c_2Earithmetic_2E_2A_2E2(c_2Earithmetic_2ENUMERAL_2E1(V6n_2E0),c_2Earithmetic_2ENUMERAL_2E1(V7m_2E0)) = c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2E_2A_2E2(V6n_2E0,V7m_2E0)) )
    & ! [V8n_2E0: tyop_2Enum_2Enum] : ( c_2Earithmetic_2E_2D_2E2(c_2Enum_2E0_2E0,V8n_2E0) = c_2Enum_2E0_2E0 )
    & ! [V9n_2E0: tyop_2Enum_2Enum] : ( c_2Earithmetic_2E_2D_2E2(V9n_2E0,c_2Enum_2E0_2E0) = V9n_2E0 )
    & ! [V10n_2E0: tyop_2Enum_2Enum,V11m_2E0: tyop_2Enum_2Enum] : ( c_2Earithmetic_2E_2D_2E2(c_2Earithmetic_2ENUMERAL_2E1(V10n_2E0),c_2Earithmetic_2ENUMERAL_2E1(V11m_2E0)) = c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2E_2D_2E2(V10n_2E0,V11m_2E0)) )
    & ! [V12n_2E0: tyop_2Enum_2Enum] : ( c_2Earithmetic_2EEXP_2E2(c_2Enum_2E0_2E0,c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(V12n_2E0))) = c_2Enum_2E0_2E0 )
    & ! [V13n_2E0: tyop_2Enum_2Enum] : ( c_2Earithmetic_2EEXP_2E2(c_2Enum_2E0_2E0,c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(V13n_2E0))) = c_2Enum_2E0_2E0 )
    & ! [V14n_2E0: tyop_2Enum_2Enum] : ( c_2Earithmetic_2EEXP_2E2(V14n_2E0,c_2Enum_2E0_2E0) = c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)) )
    & ! [V15n_2E0: tyop_2Enum_2Enum,V16m_2E0: tyop_2Enum_2Enum] : ( c_2Earithmetic_2EEXP_2E2(c_2Earithmetic_2ENUMERAL_2E1(V15n_2E0),c_2Earithmetic_2ENUMERAL_2E1(V16m_2E0)) = c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EEXP_2E2(V15n_2E0,V16m_2E0)) )
    & ( c_2Enum_2ESUC_2E1(c_2Enum_2E0_2E0) = c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)) )
    & ! [V17n_2E0: tyop_2Enum_2Enum] : ( c_2Enum_2ESUC_2E1(c_2Earithmetic_2ENUMERAL_2E1(V17n_2E0)) = c_2Earithmetic_2ENUMERAL_2E1(c_2Enum_2ESUC_2E1(V17n_2E0)) )
    & ( c_2Eprim__rec_2EPRE_2E1(c_2Enum_2E0_2E0) = c_2Enum_2E0_2E0 )
    & ! [V18n_2E0: tyop_2Enum_2Enum] : ( c_2Eprim__rec_2EPRE_2E1(c_2Earithmetic_2ENUMERAL_2E1(V18n_2E0)) = c_2Earithmetic_2ENUMERAL_2E1(c_2Eprim__rec_2EPRE_2E1(V18n_2E0)) )
    & ! [V19n_2E0: tyop_2Enum_2Enum] :
        ( ( c_2Earithmetic_2ENUMERAL_2E1(V19n_2E0) = c_2Enum_2E0_2E0 )
      <=> ( V19n_2E0 = c_2Earithmetic_2EZERO_2E0 ) )
    & ! [V20n_2E0: tyop_2Enum_2Enum] :
        ( ( c_2Enum_2E0_2E0 = c_2Earithmetic_2ENUMERAL_2E1(V20n_2E0) )
      <=> ( V20n_2E0 = c_2Earithmetic_2EZERO_2E0 ) )
    & ! [V21n_2E0: tyop_2Enum_2Enum,V22m_2E0: tyop_2Enum_2Enum] :
        ( ( c_2Earithmetic_2ENUMERAL_2E1(V21n_2E0) = c_2Earithmetic_2ENUMERAL_2E1(V22m_2E0) )
      <=> ( V21n_2E0 = V22m_2E0 ) )
    & ! [V23n_2E0: tyop_2Enum_2Enum] : ( c_2Eprim__rec_2E_3C_2E2(V23n_2E0,c_2Enum_2E0_2E0) = c_2Ebool_2EF_2E0 )
    & ! [V24n_2E0: tyop_2Enum_2Enum] : ( c_2Eprim__rec_2E_3C_2E2(c_2Enum_2E0_2E0,c_2Earithmetic_2ENUMERAL_2E1(V24n_2E0)) = c_2Eprim__rec_2E_3C_2E2(c_2Earithmetic_2EZERO_2E0,V24n_2E0) )
    & ! [V25n_2E0: tyop_2Enum_2Enum,V26m_2E0: tyop_2Enum_2Enum] : ( c_2Eprim__rec_2E_3C_2E2(c_2Earithmetic_2ENUMERAL_2E1(V25n_2E0),c_2Earithmetic_2ENUMERAL_2E1(V26m_2E0)) = c_2Eprim__rec_2E_3C_2E2(V25n_2E0,V26m_2E0) )
    & ! [V27n_2E0: tyop_2Enum_2Enum] : ( c_2Earithmetic_2E_3E_2E2(c_2Enum_2E0_2E0,V27n_2E0) = c_2Ebool_2EF_2E0 )
    & ! [V28n_2E0: tyop_2Enum_2Enum] : ( c_2Earithmetic_2E_3E_2E2(c_2Earithmetic_2ENUMERAL_2E1(V28n_2E0),c_2Enum_2E0_2E0) = c_2Eprim__rec_2E_3C_2E2(c_2Earithmetic_2EZERO_2E0,V28n_2E0) )
    & ! [V29n_2E0: tyop_2Enum_2Enum,V30m_2E0: tyop_2Enum_2Enum] : ( c_2Earithmetic_2E_3E_2E2(c_2Earithmetic_2ENUMERAL_2E1(V29n_2E0),c_2Earithmetic_2ENUMERAL_2E1(V30m_2E0)) = c_2Eprim__rec_2E_3C_2E2(V30m_2E0,V29n_2E0) )
    & ! [V31n_2E0: tyop_2Enum_2Enum] : ( c_2Earithmetic_2E_3C_3D_2E2(c_2Enum_2E0_2E0,V31n_2E0) = c_2Ebool_2ET_2E0 )
    & ! [V32n_2E0: tyop_2Enum_2Enum] : ( c_2Earithmetic_2E_3C_3D_2E2(c_2Earithmetic_2ENUMERAL_2E1(V32n_2E0),c_2Enum_2E0_2E0) = c_2Earithmetic_2E_3C_3D_2E2(V32n_2E0,c_2Earithmetic_2EZERO_2E0) )
    & ! [V33n_2E0: tyop_2Enum_2Enum,V34m_2E0: tyop_2Enum_2Enum] : ( c_2Earithmetic_2E_3C_3D_2E2(c_2Earithmetic_2ENUMERAL_2E1(V33n_2E0),c_2Earithmetic_2ENUMERAL_2E1(V34m_2E0)) = c_2Earithmetic_2E_3C_3D_2E2(V33n_2E0,V34m_2E0) )
    & ! [V35n_2E0: tyop_2Enum_2Enum] : ( c_2Earithmetic_2E_3E_3D_2E2(V35n_2E0,c_2Enum_2E0_2E0) = c_2Ebool_2ET_2E0 )
    & ! [V36n_2E0: tyop_2Enum_2Enum] :
        ( p(c_2Earithmetic_2E_3E_3D_2E2(c_2Enum_2E0_2E0,V36n_2E0))
      <=> ( V36n_2E0 = c_2Enum_2E0_2E0 ) )
    & ! [V37n_2E0: tyop_2Enum_2Enum,V38m_2E0: tyop_2Enum_2Enum] : ( c_2Earithmetic_2E_3E_3D_2E2(c_2Earithmetic_2ENUMERAL_2E1(V37n_2E0),c_2Earithmetic_2ENUMERAL_2E1(V38m_2E0)) = c_2Earithmetic_2E_3C_3D_2E2(V38m_2E0,V37n_2E0) )
    & ! [V39n_2E0: tyop_2Enum_2Enum] : ( c_2Earithmetic_2EODD_2E1(c_2Earithmetic_2ENUMERAL_2E1(V39n_2E0)) = c_2Earithmetic_2EODD_2E1(V39n_2E0) )
    & ! [V40n_2E0: tyop_2Enum_2Enum] : ( c_2Earithmetic_2EEVEN_2E1(c_2Earithmetic_2ENUMERAL_2E1(V40n_2E0)) = c_2Earithmetic_2EEVEN_2E1(V40n_2E0) )
    & ~ p(c_2Earithmetic_2EODD_2E1(c_2Enum_2E0_2E0))
    & p(c_2Earithmetic_2EEVEN_2E1(c_2Enum_2E0_2E0)) ) ).

tff(thm_2Enumeral_2Enumeral__iisuc,axiom,
    ! [V0n_2E0: tyop_2Enum_2Enum] :
      ( ( c_2Enumeral_2EiiSUC_2E1(c_2Earithmetic_2EZERO_2E0) = c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EZERO_2E0) )
      & ( c_2Enumeral_2EiiSUC_2E1(c_2Earithmetic_2EBIT1_2E1(V0n_2E0)) = c_2Earithmetic_2EBIT1_2E1(c_2Enum_2ESUC_2E1(V0n_2E0)) )
      & ( c_2Enumeral_2EiiSUC_2E1(c_2Earithmetic_2EBIT2_2E1(V0n_2E0)) = c_2Earithmetic_2EBIT2_2E1(c_2Enum_2ESUC_2E1(V0n_2E0)) ) ) ).

tff(thm_2Enumeral_2Enumeral__add,axiom,
    ! [V0n_2E0: tyop_2Enum_2Enum,V1m_2E0: tyop_2Enum_2Enum] :
      ( ( c_2Enumeral_2EiZ_2E1(c_2Earithmetic_2E_2B_2E2(c_2Earithmetic_2EZERO_2E0,V0n_2E0)) = V0n_2E0 )
      & ( c_2Enumeral_2EiZ_2E1(c_2Earithmetic_2E_2B_2E2(V0n_2E0,c_2Earithmetic_2EZERO_2E0)) = V0n_2E0 )
      & ( c_2Enumeral_2EiZ_2E1(c_2Earithmetic_2E_2B_2E2(c_2Earithmetic_2EBIT1_2E1(V0n_2E0),c_2Earithmetic_2EBIT1_2E1(V1m_2E0))) = c_2Earithmetic_2EBIT2_2E1(c_2Enumeral_2EiZ_2E1(c_2Earithmetic_2E_2B_2E2(V0n_2E0,V1m_2E0))) )
      & ( c_2Enumeral_2EiZ_2E1(c_2Earithmetic_2E_2B_2E2(c_2Earithmetic_2EBIT1_2E1(V0n_2E0),c_2Earithmetic_2EBIT2_2E1(V1m_2E0))) = c_2Earithmetic_2EBIT1_2E1(c_2Enum_2ESUC_2E1(c_2Earithmetic_2E_2B_2E2(V0n_2E0,V1m_2E0))) )
      & ( c_2Enumeral_2EiZ_2E1(c_2Earithmetic_2E_2B_2E2(c_2Earithmetic_2EBIT2_2E1(V0n_2E0),c_2Earithmetic_2EBIT1_2E1(V1m_2E0))) = c_2Earithmetic_2EBIT1_2E1(c_2Enum_2ESUC_2E1(c_2Earithmetic_2E_2B_2E2(V0n_2E0,V1m_2E0))) )
      & ( c_2Enumeral_2EiZ_2E1(c_2Earithmetic_2E_2B_2E2(c_2Earithmetic_2EBIT2_2E1(V0n_2E0),c_2Earithmetic_2EBIT2_2E1(V1m_2E0))) = c_2Earithmetic_2EBIT2_2E1(c_2Enum_2ESUC_2E1(c_2Earithmetic_2E_2B_2E2(V0n_2E0,V1m_2E0))) )
      & ( c_2Enum_2ESUC_2E1(c_2Earithmetic_2E_2B_2E2(c_2Earithmetic_2EZERO_2E0,V0n_2E0)) = c_2Enum_2ESUC_2E1(V0n_2E0) )
      & ( c_2Enum_2ESUC_2E1(c_2Earithmetic_2E_2B_2E2(V0n_2E0,c_2Earithmetic_2EZERO_2E0)) = c_2Enum_2ESUC_2E1(V0n_2E0) )
      & ( c_2Enum_2ESUC_2E1(c_2Earithmetic_2E_2B_2E2(c_2Earithmetic_2EBIT1_2E1(V0n_2E0),c_2Earithmetic_2EBIT1_2E1(V1m_2E0))) = c_2Earithmetic_2EBIT1_2E1(c_2Enum_2ESUC_2E1(c_2Earithmetic_2E_2B_2E2(V0n_2E0,V1m_2E0))) )
      & ( c_2Enum_2ESUC_2E1(c_2Earithmetic_2E_2B_2E2(c_2Earithmetic_2EBIT1_2E1(V0n_2E0),c_2Earithmetic_2EBIT2_2E1(V1m_2E0))) = c_2Earithmetic_2EBIT2_2E1(c_2Enum_2ESUC_2E1(c_2Earithmetic_2E_2B_2E2(V0n_2E0,V1m_2E0))) )
      & ( c_2Enum_2ESUC_2E1(c_2Earithmetic_2E_2B_2E2(c_2Earithmetic_2EBIT2_2E1(V0n_2E0),c_2Earithmetic_2EBIT1_2E1(V1m_2E0))) = c_2Earithmetic_2EBIT2_2E1(c_2Enum_2ESUC_2E1(c_2Earithmetic_2E_2B_2E2(V0n_2E0,V1m_2E0))) )
      & ( c_2Enum_2ESUC_2E1(c_2Earithmetic_2E_2B_2E2(c_2Earithmetic_2EBIT2_2E1(V0n_2E0),c_2Earithmetic_2EBIT2_2E1(V1m_2E0))) = c_2Earithmetic_2EBIT1_2E1(c_2Enumeral_2EiiSUC_2E1(c_2Earithmetic_2E_2B_2E2(V0n_2E0,V1m_2E0))) )
      & ( c_2Enumeral_2EiiSUC_2E1(c_2Earithmetic_2E_2B_2E2(c_2Earithmetic_2EZERO_2E0,V0n_2E0)) = c_2Enumeral_2EiiSUC_2E1(V0n_2E0) )
      & ( c_2Enumeral_2EiiSUC_2E1(c_2Earithmetic_2E_2B_2E2(V0n_2E0,c_2Earithmetic_2EZERO_2E0)) = c_2Enumeral_2EiiSUC_2E1(V0n_2E0) )
      & ( c_2Enumeral_2EiiSUC_2E1(c_2Earithmetic_2E_2B_2E2(c_2Earithmetic_2EBIT1_2E1(V0n_2E0),c_2Earithmetic_2EBIT1_2E1(V1m_2E0))) = c_2Earithmetic_2EBIT2_2E1(c_2Enum_2ESUC_2E1(c_2Earithmetic_2E_2B_2E2(V0n_2E0,V1m_2E0))) )
      & ( c_2Enumeral_2EiiSUC_2E1(c_2Earithmetic_2E_2B_2E2(c_2Earithmetic_2EBIT1_2E1(V0n_2E0),c_2Earithmetic_2EBIT2_2E1(V1m_2E0))) = c_2Earithmetic_2EBIT1_2E1(c_2Enumeral_2EiiSUC_2E1(c_2Earithmetic_2E_2B_2E2(V0n_2E0,V1m_2E0))) )
      & ( c_2Enumeral_2EiiSUC_2E1(c_2Earithmetic_2E_2B_2E2(c_2Earithmetic_2EBIT2_2E1(V0n_2E0),c_2Earithmetic_2EBIT1_2E1(V1m_2E0))) = c_2Earithmetic_2EBIT1_2E1(c_2Enumeral_2EiiSUC_2E1(c_2Earithmetic_2E_2B_2E2(V0n_2E0,V1m_2E0))) )
      & ( c_2Enumeral_2EiiSUC_2E1(c_2Earithmetic_2E_2B_2E2(c_2Earithmetic_2EBIT2_2E1(V0n_2E0),c_2Earithmetic_2EBIT2_2E1(V1m_2E0))) = c_2Earithmetic_2EBIT2_2E1(c_2Enumeral_2EiiSUC_2E1(c_2Earithmetic_2E_2B_2E2(V0n_2E0,V1m_2E0))) ) ) ).

tff(thm_2Enumeral_2Enumeral__eq,axiom,
    ! [V0n_2E0: tyop_2Enum_2Enum,V1m_2E0: tyop_2Enum_2Enum] :
      ( ( ( c_2Earithmetic_2EZERO_2E0 = c_2Earithmetic_2EBIT1_2E1(V0n_2E0) )
      <=> p(c_2Ebool_2EF_2E0) )
      & ( ( c_2Earithmetic_2EBIT1_2E1(V0n_2E0) = c_2Earithmetic_2EZERO_2E0 )
      <=> p(c_2Ebool_2EF_2E0) )
      & ( ( c_2Earithmetic_2EZERO_2E0 = c_2Earithmetic_2EBIT2_2E1(V0n_2E0) )
      <=> p(c_2Ebool_2EF_2E0) )
      & ( ( c_2Earithmetic_2EBIT2_2E1(V0n_2E0) = c_2Earithmetic_2EZERO_2E0 )
      <=> p(c_2Ebool_2EF_2E0) )
      & ( ( c_2Earithmetic_2EBIT1_2E1(V0n_2E0) = c_2Earithmetic_2EBIT2_2E1(V1m_2E0) )
      <=> p(c_2Ebool_2EF_2E0) )
      & ( ( c_2Earithmetic_2EBIT2_2E1(V0n_2E0) = c_2Earithmetic_2EBIT1_2E1(V1m_2E0) )
      <=> p(c_2Ebool_2EF_2E0) )
      & ( ( c_2Earithmetic_2EBIT1_2E1(V0n_2E0) = c_2Earithmetic_2EBIT1_2E1(V1m_2E0) )
      <=> ( V0n_2E0 = V1m_2E0 ) )
      & ( ( c_2Earithmetic_2EBIT2_2E1(V0n_2E0) = c_2Earithmetic_2EBIT2_2E1(V1m_2E0) )
      <=> ( V0n_2E0 = V1m_2E0 ) ) ) ).

tff(thm_2Enumeral_2Enumeral__lt,axiom,
    ! [V0n_2E0: tyop_2Enum_2Enum,V1m_2E0: tyop_2Enum_2Enum] :
      ( ( c_2Eprim__rec_2E_3C_2E2(c_2Earithmetic_2EZERO_2E0,c_2Earithmetic_2EBIT1_2E1(V0n_2E0)) = c_2Ebool_2ET_2E0 )
      & ( c_2Eprim__rec_2E_3C_2E2(c_2Earithmetic_2EZERO_2E0,c_2Earithmetic_2EBIT2_2E1(V0n_2E0)) = c_2Ebool_2ET_2E0 )
      & ( c_2Eprim__rec_2E_3C_2E2(V0n_2E0,c_2Earithmetic_2EZERO_2E0) = c_2Ebool_2EF_2E0 )
      & ( c_2Eprim__rec_2E_3C_2E2(c_2Earithmetic_2EBIT1_2E1(V0n_2E0),c_2Earithmetic_2EBIT1_2E1(V1m_2E0)) = c_2Eprim__rec_2E_3C_2E2(V0n_2E0,V1m_2E0) )
      & ( c_2Eprim__rec_2E_3C_2E2(c_2Earithmetic_2EBIT2_2E1(V0n_2E0),c_2Earithmetic_2EBIT2_2E1(V1m_2E0)) = c_2Eprim__rec_2E_3C_2E2(V0n_2E0,V1m_2E0) )
      & ( p(c_2Eprim__rec_2E_3C_2E2(c_2Earithmetic_2EBIT1_2E1(V0n_2E0),c_2Earithmetic_2EBIT2_2E1(V1m_2E0)))
      <=> ~ p(c_2Eprim__rec_2E_3C_2E2(V1m_2E0,V0n_2E0)) )
      & ( c_2Eprim__rec_2E_3C_2E2(c_2Earithmetic_2EBIT2_2E1(V0n_2E0),c_2Earithmetic_2EBIT1_2E1(V1m_2E0)) = c_2Eprim__rec_2E_3C_2E2(V0n_2E0,V1m_2E0) ) ) ).

tff(thm_2Enumeral_2Enumeral__lte,axiom,
    ! [V0n_2E0: tyop_2Enum_2Enum,V1m_2E0: tyop_2Enum_2Enum] :
      ( ( c_2Earithmetic_2E_3C_3D_2E2(c_2Earithmetic_2EZERO_2E0,V0n_2E0) = c_2Ebool_2ET_2E0 )
      & ( c_2Earithmetic_2E_3C_3D_2E2(c_2Earithmetic_2EBIT1_2E1(V0n_2E0),c_2Earithmetic_2EZERO_2E0) = c_2Ebool_2EF_2E0 )
      & ( c_2Earithmetic_2E_3C_3D_2E2(c_2Earithmetic_2EBIT2_2E1(V0n_2E0),c_2Earithmetic_2EZERO_2E0) = c_2Ebool_2EF_2E0 )
      & ( c_2Earithmetic_2E_3C_3D_2E2(c_2Earithmetic_2EBIT1_2E1(V0n_2E0),c_2Earithmetic_2EBIT1_2E1(V1m_2E0)) = c_2Earithmetic_2E_3C_3D_2E2(V0n_2E0,V1m_2E0) )
      & ( c_2Earithmetic_2E_3C_3D_2E2(c_2Earithmetic_2EBIT1_2E1(V0n_2E0),c_2Earithmetic_2EBIT2_2E1(V1m_2E0)) = c_2Earithmetic_2E_3C_3D_2E2(V0n_2E0,V1m_2E0) )
      & ( p(c_2Earithmetic_2E_3C_3D_2E2(c_2Earithmetic_2EBIT2_2E1(V0n_2E0),c_2Earithmetic_2EBIT1_2E1(V1m_2E0)))
      <=> ~ p(c_2Earithmetic_2E_3C_3D_2E2(V1m_2E0,V0n_2E0)) )
      & ( c_2Earithmetic_2E_3C_3D_2E2(c_2Earithmetic_2EBIT2_2E1(V0n_2E0),c_2Earithmetic_2EBIT2_2E1(V1m_2E0)) = c_2Earithmetic_2E_3C_3D_2E2(V0n_2E0,V1m_2E0) ) ) ).

tff(thm_2Enumeral_2Enumeral__pre,axiom,
    ( ( c_2Eprim__rec_2EPRE_2E1(c_2Earithmetic_2EZERO_2E0) = c_2Earithmetic_2EZERO_2E0 )
    & ( c_2Eprim__rec_2EPRE_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)) = c_2Earithmetic_2EZERO_2E0 )
    & ! [V0n_2E0: tyop_2Enum_2Enum] : ( c_2Eprim__rec_2EPRE_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EBIT1_2E1(V0n_2E0))) = c_2Earithmetic_2EBIT2_2E1(c_2Eprim__rec_2EPRE_2E1(c_2Earithmetic_2EBIT1_2E1(V0n_2E0))) )
    & ! [V1n_2E0: tyop_2Enum_2Enum] : ( c_2Eprim__rec_2EPRE_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EBIT2_2E1(V1n_2E0))) = c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EBIT1_2E1(V1n_2E0)) )
    & ! [V2n_2E0: tyop_2Enum_2Enum] : ( c_2Eprim__rec_2EPRE_2E1(c_2Earithmetic_2EBIT2_2E1(V2n_2E0)) = c_2Earithmetic_2EBIT1_2E1(V2n_2E0) ) ) ).

tff(thm_2Enumeral_2Ebit__initiality,axiom,
    ! [A_27a: $tType,V0zf_2E0: A_27a,V1b1f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(A_27a,A_27a)),V2b2f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(A_27a,A_27a))] :
    ? [V3f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,A_27a)] :
      ( ( app_2E2(tyop_2Enum_2Enum,A_27a,V3f_2E0,c_2Earithmetic_2EZERO_2E0) = V0zf_2E0 )
      & ! [V4n_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,A_27a,V3f_2E0,c_2Earithmetic_2EBIT1_2E1(V4n_2E0)) = app_2E2(A_27a,A_27a,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(A_27a,A_27a),V1b1f_2E0,V4n_2E0),app_2E2(tyop_2Enum_2Enum,A_27a,V3f_2E0,V4n_2E0)) )
      & ! [V5n_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,A_27a,V3f_2E0,c_2Earithmetic_2EBIT2_2E1(V5n_2E0)) = app_2E2(A_27a,A_27a,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(A_27a,A_27a),V2b2f_2E0,V5n_2E0),app_2E2(tyop_2Enum_2Enum,A_27a,V3f_2E0,V5n_2E0)) ) ) ).

tff(thm_2Enumeral_2Ebit__induction,axiom,
    ! [V0P_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)] :
      ( ( p(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,V0P_2E0,c_2Earithmetic_2EZERO_2E0))
        & ! [V1n_2E0: tyop_2Enum_2Enum] :
            ( p(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,V0P_2E0,V1n_2E0))
           => p(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,V0P_2E0,c_2Earithmetic_2EBIT1_2E1(V1n_2E0))) )
        & ! [V2n_2E0: tyop_2Enum_2Enum] :
            ( p(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,V0P_2E0,V2n_2E0))
           => p(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,V0P_2E0,c_2Earithmetic_2EBIT2_2E1(V2n_2E0))) ) )
     => ! [V3n_2E0: tyop_2Enum_2Enum] : p(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,V0P_2E0,V3n_2E0)) ) ).

tff(thm_2Enumeral_2EiSUB__THM,axiom,
    ! [V0x_2E0: tyop_2Enum_2Enum,V1b_2E0: tyop_2Emin_2Ebool,V2n_2E0: tyop_2Enum_2Enum,V3m_2E0: tyop_2Enum_2Enum] :
      ( ( c_2Enumeral_2EiSUB_2E3(V1b_2E0,c_2Earithmetic_2EZERO_2E0,V0x_2E0) = c_2Earithmetic_2EZERO_2E0 )
      & ( c_2Enumeral_2EiSUB_2E3(c_2Ebool_2ET_2E0,V2n_2E0,c_2Earithmetic_2EZERO_2E0) = V2n_2E0 )
      & ( c_2Enumeral_2EiSUB_2E3(c_2Ebool_2EF_2E0,c_2Earithmetic_2EBIT1_2E1(V2n_2E0),c_2Earithmetic_2EZERO_2E0) = c_2Enumeral_2EiDUB_2E1(V2n_2E0) )
      & ( c_2Enumeral_2EiSUB_2E3(c_2Ebool_2ET_2E0,c_2Earithmetic_2EBIT1_2E1(V2n_2E0),c_2Earithmetic_2EBIT1_2E1(V3m_2E0)) = c_2Enumeral_2EiDUB_2E1(c_2Enumeral_2EiSUB_2E3(c_2Ebool_2ET_2E0,V2n_2E0,V3m_2E0)) )
      & ( c_2Enumeral_2EiSUB_2E3(c_2Ebool_2EF_2E0,c_2Earithmetic_2EBIT1_2E1(V2n_2E0),c_2Earithmetic_2EBIT1_2E1(V3m_2E0)) = c_2Earithmetic_2EBIT1_2E1(c_2Enumeral_2EiSUB_2E3(c_2Ebool_2EF_2E0,V2n_2E0,V3m_2E0)) )
      & ( c_2Enumeral_2EiSUB_2E3(c_2Ebool_2ET_2E0,c_2Earithmetic_2EBIT1_2E1(V2n_2E0),c_2Earithmetic_2EBIT2_2E1(V3m_2E0)) = c_2Earithmetic_2EBIT1_2E1(c_2Enumeral_2EiSUB_2E3(c_2Ebool_2EF_2E0,V2n_2E0,V3m_2E0)) )
      & ( c_2Enumeral_2EiSUB_2E3(c_2Ebool_2EF_2E0,c_2Earithmetic_2EBIT1_2E1(V2n_2E0),c_2Earithmetic_2EBIT2_2E1(V3m_2E0)) = c_2Enumeral_2EiDUB_2E1(c_2Enumeral_2EiSUB_2E3(c_2Ebool_2EF_2E0,V2n_2E0,V3m_2E0)) )
      & ( c_2Enumeral_2EiSUB_2E3(c_2Ebool_2EF_2E0,c_2Earithmetic_2EBIT2_2E1(V2n_2E0),c_2Earithmetic_2EZERO_2E0) = c_2Earithmetic_2EBIT1_2E1(V2n_2E0) )
      & ( c_2Enumeral_2EiSUB_2E3(c_2Ebool_2ET_2E0,c_2Earithmetic_2EBIT2_2E1(V2n_2E0),c_2Earithmetic_2EBIT1_2E1(V3m_2E0)) = c_2Earithmetic_2EBIT1_2E1(c_2Enumeral_2EiSUB_2E3(c_2Ebool_2ET_2E0,V2n_2E0,V3m_2E0)) )
      & ( c_2Enumeral_2EiSUB_2E3(c_2Ebool_2EF_2E0,c_2Earithmetic_2EBIT2_2E1(V2n_2E0),c_2Earithmetic_2EBIT1_2E1(V3m_2E0)) = c_2Enumeral_2EiDUB_2E1(c_2Enumeral_2EiSUB_2E3(c_2Ebool_2ET_2E0,V2n_2E0,V3m_2E0)) )
      & ( c_2Enumeral_2EiSUB_2E3(c_2Ebool_2ET_2E0,c_2Earithmetic_2EBIT2_2E1(V2n_2E0),c_2Earithmetic_2EBIT2_2E1(V3m_2E0)) = c_2Enumeral_2EiDUB_2E1(c_2Enumeral_2EiSUB_2E3(c_2Ebool_2ET_2E0,V2n_2E0,V3m_2E0)) )
      & ( c_2Enumeral_2EiSUB_2E3(c_2Ebool_2EF_2E0,c_2Earithmetic_2EBIT2_2E1(V2n_2E0),c_2Earithmetic_2EBIT2_2E1(V3m_2E0)) = c_2Earithmetic_2EBIT1_2E1(c_2Enumeral_2EiSUB_2E3(c_2Ebool_2EF_2E0,V2n_2E0,V3m_2E0)) ) ) ).

tff(thm_2Enumeral_2Enumeral__sub,axiom,
    ! [V0n_2E0: tyop_2Enum_2Enum,V1m_2E0: tyop_2Enum_2Enum] : ( c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2E_2D_2E2(V0n_2E0,V1m_2E0)) = c_2Ebool_2ECOND_2E3(tyop_2Enum_2Enum,c_2Eprim__rec_2E_3C_2E2(V1m_2E0,V0n_2E0),c_2Earithmetic_2ENUMERAL_2E1(c_2Enumeral_2EiSUB_2E3(c_2Ebool_2ET_2E0,V0n_2E0,V1m_2E0)),c_2Enum_2E0_2E0) ) ).

tff(thm_2Enumeral_2EiDUB__removal,axiom,
    ! [V0n_2E0: tyop_2Enum_2Enum] :
      ( ( c_2Enumeral_2EiDUB_2E1(c_2Earithmetic_2EBIT1_2E1(V0n_2E0)) = c_2Earithmetic_2EBIT2_2E1(c_2Enumeral_2EiDUB_2E1(V0n_2E0)) )
      & ( c_2Enumeral_2EiDUB_2E1(c_2Earithmetic_2EBIT2_2E1(V0n_2E0)) = c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EBIT1_2E1(V0n_2E0)) )
      & ( c_2Enumeral_2EiDUB_2E1(c_2Earithmetic_2EZERO_2E0) = c_2Earithmetic_2EZERO_2E0 ) ) ).

tff(thm_2Enumeral_2Enumeral__mult,axiom,
    ! [V0n_2E0: tyop_2Enum_2Enum,V1m_2E0: tyop_2Enum_2Enum] :
      ( ( c_2Earithmetic_2E_2A_2E2(c_2Earithmetic_2EZERO_2E0,V0n_2E0) = c_2Earithmetic_2EZERO_2E0 )
      & ( c_2Earithmetic_2E_2A_2E2(V0n_2E0,c_2Earithmetic_2EZERO_2E0) = c_2Earithmetic_2EZERO_2E0 )
      & ( c_2Earithmetic_2E_2A_2E2(c_2Earithmetic_2EBIT1_2E1(V0n_2E0),V1m_2E0) = c_2Enumeral_2EiZ_2E1(c_2Earithmetic_2E_2B_2E2(c_2Enumeral_2EiDUB_2E1(c_2Earithmetic_2E_2A_2E2(V0n_2E0,V1m_2E0)),V1m_2E0)) )
      & ( c_2Earithmetic_2E_2A_2E2(c_2Earithmetic_2EBIT2_2E1(V0n_2E0),V1m_2E0) = c_2Enumeral_2EiDUB_2E1(c_2Enumeral_2EiZ_2E1(c_2Earithmetic_2E_2B_2E2(c_2Earithmetic_2E_2A_2E2(V0n_2E0,V1m_2E0),V1m_2E0))) ) ) ).

tff(thm_2Enumeral_2Enumeral__exp,axiom,
    ( ! [V0n_2E0: tyop_2Enum_2Enum] : ( c_2Earithmetic_2EEXP_2E2(V0n_2E0,c_2Earithmetic_2EZERO_2E0) = c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0) )
    & ! [V1n_2E0: tyop_2Enum_2Enum,V2m_2E0: tyop_2Enum_2Enum] : ( c_2Earithmetic_2EEXP_2E2(V1n_2E0,c_2Earithmetic_2EBIT1_2E1(V2m_2E0)) = c_2Earithmetic_2E_2A_2E2(V1n_2E0,c_2Enumeral_2EiSQR_2E1(c_2Earithmetic_2EEXP_2E2(V1n_2E0,V2m_2E0))) )
    & ! [V3n_2E0: tyop_2Enum_2Enum,V4m_2E0: tyop_2Enum_2Enum] : ( c_2Earithmetic_2EEXP_2E2(V3n_2E0,c_2Earithmetic_2EBIT2_2E1(V4m_2E0)) = c_2Earithmetic_2E_2A_2E2(c_2Enumeral_2EiSQR_2E1(V3n_2E0),c_2Enumeral_2EiSQR_2E1(c_2Earithmetic_2EEXP_2E2(V3n_2E0,V4m_2E0))) ) ) ).

tff(thm_2Enumeral_2Enumeral__evenodd,axiom,
    ! [V0n_2E0: tyop_2Enum_2Enum] :
      ( p(c_2Earithmetic_2EEVEN_2E1(c_2Earithmetic_2EZERO_2E0))
      & p(c_2Earithmetic_2EEVEN_2E1(c_2Earithmetic_2EBIT2_2E1(V0n_2E0)))
      & ~ p(c_2Earithmetic_2EEVEN_2E1(c_2Earithmetic_2EBIT1_2E1(V0n_2E0)))
      & ~ p(c_2Earithmetic_2EODD_2E1(c_2Earithmetic_2EZERO_2E0))
      & ~ p(c_2Earithmetic_2EODD_2E1(c_2Earithmetic_2EBIT2_2E1(V0n_2E0)))
      & p(c_2Earithmetic_2EODD_2E1(c_2Earithmetic_2EBIT1_2E1(V0n_2E0))) ) ).

tff(thm_2Enumeral_2Enumeral__fact,axiom,
    ( ( c_2Earithmetic_2EFACT_2E1(c_2Enum_2E0_2E0) = c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)) )
    & ! [V0n_2E0: tyop_2Enum_2Enum] : ( c_2Earithmetic_2EFACT_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(V0n_2E0))) = c_2Earithmetic_2E_2A_2E2(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(V0n_2E0)),c_2Earithmetic_2EFACT_2E1(c_2Eprim__rec_2EPRE_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(V0n_2E0))))) )
    & ! [V1n_2E0: tyop_2Enum_2Enum] : ( c_2Earithmetic_2EFACT_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(V1n_2E0))) = c_2Earithmetic_2E_2A_2E2(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(V1n_2E0)),c_2Earithmetic_2EFACT_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(V1n_2E0)))) ) ) ).

tff(thm_2Enumeral_2Enumeral__funpow,axiom,
    ! [A_27a: $tType,V0x_2E0: A_27a,V1n_2E0: tyop_2Enum_2Enum,V2f_2E0: tyop_2Emin_2Efun(A_27a,A_27a)] :
      ( ( c_2Earithmetic_2EFUNPOW_2E3(A_27a,V2f_2E0,c_2Enum_2E0_2E0,V0x_2E0) = V0x_2E0 )
      & ( c_2Earithmetic_2EFUNPOW_2E3(A_27a,V2f_2E0,c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(V1n_2E0)),V0x_2E0) = c_2Earithmetic_2EFUNPOW_2E3(A_27a,V2f_2E0,c_2Eprim__rec_2EPRE_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(V1n_2E0))),app_2E2(A_27a,A_27a,V2f_2E0,V0x_2E0)) )
      & ( c_2Earithmetic_2EFUNPOW_2E3(A_27a,V2f_2E0,c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(V1n_2E0)),V0x_2E0) = c_2Earithmetic_2EFUNPOW_2E3(A_27a,V2f_2E0,c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(V1n_2E0)),app_2E2(A_27a,A_27a,V2f_2E0,V0x_2E0)) ) ) ).

tff(thm_2Enumeral_2Enumeral__MIN,axiom,
    ! [V0y_2E0: tyop_2Enum_2Enum,V1x_2E0: tyop_2Enum_2Enum] :
      ( ( c_2Earithmetic_2EMIN_2E2(c_2Enum_2E0_2E0,V1x_2E0) = c_2Enum_2E0_2E0 )
      & ( c_2Earithmetic_2EMIN_2E2(V1x_2E0,c_2Enum_2E0_2E0) = c_2Enum_2E0_2E0 )
      & ( c_2Earithmetic_2EMIN_2E2(c_2Earithmetic_2ENUMERAL_2E1(V1x_2E0),c_2Earithmetic_2ENUMERAL_2E1(V0y_2E0)) = c_2Earithmetic_2ENUMERAL_2E1(c_2Ebool_2ECOND_2E3(tyop_2Enum_2Enum,c_2Eprim__rec_2E_3C_2E2(V1x_2E0,V0y_2E0),V1x_2E0,V0y_2E0)) ) ) ).

tff(thm_2Enumeral_2Enumeral__MAX,axiom,
    ! [V0y_2E0: tyop_2Enum_2Enum,V1x_2E0: tyop_2Enum_2Enum] :
      ( ( c_2Earithmetic_2EMAX_2E2(c_2Enum_2E0_2E0,V1x_2E0) = V1x_2E0 )
      & ( c_2Earithmetic_2EMAX_2E2(V1x_2E0,c_2Enum_2E0_2E0) = V1x_2E0 )
      & ( c_2Earithmetic_2EMAX_2E2(c_2Earithmetic_2ENUMERAL_2E1(V1x_2E0),c_2Earithmetic_2ENUMERAL_2E1(V0y_2E0)) = c_2Earithmetic_2ENUMERAL_2E1(c_2Ebool_2ECOND_2E3(tyop_2Enum_2Enum,c_2Eprim__rec_2E_3C_2E2(V1x_2E0,V0y_2E0),V0y_2E0,V1x_2E0)) ) ) ).

tff(thm_2Enumeral_2Edivmod__POS,axiom,
    ! [F0_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum)))))] :
      ( ! [V1a_2E0: tyop_2Enum_2Enum,V0m_2E0: tyop_2Enum_2Enum,V2n_2E0: tyop_2Enum_2Enum,V3q_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum)),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum))),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum)))),F0_2E0,V1a_2E0),V0m_2E0),V2n_2E0),V3q_2E0) = c_2Earithmetic_2EDIVMOD_2E1(c_2Epair_2E_2C_2E2(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum),c_2Earithmetic_2E_2B_2E2(V1a_2E0,V3q_2E0),c_2Epair_2E_2C_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,c_2Earithmetic_2E_2D_2E2(V0m_2E0,c_2Earithmetic_2E_2A_2E2(V2n_2E0,V3q_2E0)),V2n_2E0))) )
     => ! [V0m_2E0: tyop_2Enum_2Enum,V1a_2E0: tyop_2Enum_2Enum,V2n_2E0: tyop_2Enum_2Enum] :
          ( p(c_2Eprim__rec_2E_3C_2E2(c_2Enum_2E0_2E0,V2n_2E0))
         => ( c_2Earithmetic_2EDIVMOD_2E1(c_2Epair_2E_2C_2E2(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum),V1a_2E0,c_2Epair_2E_2C_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,V0m_2E0,V2n_2E0))) = c_2Ebool_2ECOND_2E3(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum),c_2Eprim__rec_2E_3C_2E2(V0m_2E0,V2n_2E0),c_2Epair_2E_2C_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,V1a_2E0,V0m_2E0),c_2Ebool_2ELET_2E2(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum)),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum))),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum)))),F0_2E0,V1a_2E0),V0m_2E0),V2n_2E0),c_2Earithmetic_2Efindq_2E1(c_2Epair_2E_2C_2E2(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum),c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)),c_2Epair_2E_2C_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,V0m_2E0,V2n_2E0))))) ) ) ) ).

tff(thm_2Enumeral_2EDIVMOD__NUMERAL__CALC,axiom,
    ( ! [V0m_2E0: tyop_2Enum_2Enum,V1n_2E0: tyop_2Enum_2Enum] : ( c_2Earithmetic_2EDIV_2E2(V0m_2E0,c_2Earithmetic_2EBIT1_2E1(V1n_2E0)) = c_2Epair_2EFST_2E1(tyop_2Enum_2Enum,tyop_2Enum_2Enum,c_2Earithmetic_2EDIVMOD_2E1(c_2Epair_2E_2C_2E2(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum),c_2Earithmetic_2EZERO_2E0,c_2Epair_2E_2C_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,V0m_2E0,c_2Earithmetic_2EBIT1_2E1(V1n_2E0))))) )
    & ! [V2m_2E0: tyop_2Enum_2Enum,V3n_2E0: tyop_2Enum_2Enum] : ( c_2Earithmetic_2EDIV_2E2(V2m_2E0,c_2Earithmetic_2EBIT2_2E1(V3n_2E0)) = c_2Epair_2EFST_2E1(tyop_2Enum_2Enum,tyop_2Enum_2Enum,c_2Earithmetic_2EDIVMOD_2E1(c_2Epair_2E_2C_2E2(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum),c_2Earithmetic_2EZERO_2E0,c_2Epair_2E_2C_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,V2m_2E0,c_2Earithmetic_2EBIT2_2E1(V3n_2E0))))) )
    & ! [V4m_2E0: tyop_2Enum_2Enum,V5n_2E0: tyop_2Enum_2Enum] : ( c_2Earithmetic_2EMOD_2E2(V4m_2E0,c_2Earithmetic_2EBIT1_2E1(V5n_2E0)) = c_2Epair_2ESND_2E1(tyop_2Enum_2Enum,tyop_2Enum_2Enum,c_2Earithmetic_2EDIVMOD_2E1(c_2Epair_2E_2C_2E2(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum),c_2Earithmetic_2EZERO_2E0,c_2Epair_2E_2C_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,V4m_2E0,c_2Earithmetic_2EBIT1_2E1(V5n_2E0))))) )
    & ! [V6m_2E0: tyop_2Enum_2Enum,V7n_2E0: tyop_2Enum_2Enum] : ( c_2Earithmetic_2EMOD_2E2(V6m_2E0,c_2Earithmetic_2EBIT2_2E1(V7n_2E0)) = c_2Epair_2ESND_2E1(tyop_2Enum_2Enum,tyop_2Enum_2Enum,c_2Earithmetic_2EDIVMOD_2E1(c_2Epair_2E_2C_2E2(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum),c_2Earithmetic_2EZERO_2E0,c_2Epair_2E_2C_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,V6m_2E0,c_2Earithmetic_2EBIT2_2E1(V7n_2E0))))) ) ) ).

tff(thm_2Enumeral_2Enumeral__div2,axiom,
    ( ( c_2Earithmetic_2EDIV2_2E1(c_2Enum_2E0_2E0) = c_2Enum_2E0_2E0 )
    & ! [V0n_2E0: tyop_2Enum_2Enum] : ( c_2Earithmetic_2EDIV2_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(V0n_2E0))) = c_2Earithmetic_2ENUMERAL_2E1(V0n_2E0) )
    & ! [V1n_2E0: tyop_2Enum_2Enum] : ( c_2Earithmetic_2EDIV2_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(V1n_2E0))) = c_2Earithmetic_2ENUMERAL_2E1(c_2Enum_2ESUC_2E1(V1n_2E0)) ) ) ).

tff(thm_2Enumeral_2Etexp__help__thm,axiom,
    ! [V0n_2E0: tyop_2Enum_2Enum,V1a_2E0: tyop_2Enum_2Enum] : ( c_2Enumeral_2Etexp__help_2E2(V0n_2E0,V1a_2E0) = c_2Earithmetic_2E_2A_2E2(c_2Earithmetic_2E_2B_2E2(V1a_2E0,c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))),c_2Earithmetic_2EEXP_2E2(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EZERO_2E0)),c_2Earithmetic_2E_2B_2E2(V0n_2E0,c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))))) ) ).

tff(thm_2Enumeral_2Etexp__help0,axiom,
    ! [V0n_2E0: tyop_2Enum_2Enum] : ( c_2Enumeral_2Etexp__help_2E2(V0n_2E0,c_2Enum_2E0_2E0) = c_2Earithmetic_2EEXP_2E2(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EZERO_2E0)),c_2Earithmetic_2E_2B_2E2(V0n_2E0,c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)))) ) ).

tff(thm_2Enumeral_2Enumeral__texp__help,axiom,
    ! [V0n_2E0: tyop_2Enum_2Enum,V1acc_2E0: tyop_2Enum_2Enum] :
      ( ( c_2Enumeral_2Etexp__help_2E2(c_2Earithmetic_2EZERO_2E0,V1acc_2E0) = c_2Earithmetic_2EBIT2_2E1(V1acc_2E0) )
      & ( c_2Enumeral_2Etexp__help_2E2(c_2Earithmetic_2EBIT1_2E1(V0n_2E0),V1acc_2E0) = c_2Enumeral_2Etexp__help_2E2(c_2Eprim__rec_2EPRE_2E1(c_2Earithmetic_2EBIT1_2E1(V0n_2E0)),c_2Earithmetic_2EBIT1_2E1(V1acc_2E0)) )
      & ( c_2Enumeral_2Etexp__help_2E2(c_2Earithmetic_2EBIT2_2E1(V0n_2E0),V1acc_2E0) = c_2Enumeral_2Etexp__help_2E2(c_2Earithmetic_2EBIT1_2E1(V0n_2E0),c_2Earithmetic_2EBIT1_2E1(V1acc_2E0)) ) ) ).

tff(thm_2Enumeral_2ETWO__EXP__THM,axiom,
    ! [V0n_2E0: tyop_2Enum_2Enum] :
      ( ( c_2Earithmetic_2EEXP_2E2(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EZERO_2E0)),c_2Enum_2E0_2E0) = c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)) )
      & ( c_2Earithmetic_2EEXP_2E2(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EZERO_2E0)),c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(V0n_2E0))) = c_2Earithmetic_2ENUMERAL_2E1(c_2Enumeral_2Etexp__help_2E2(c_2Eprim__rec_2EPRE_2E1(c_2Earithmetic_2EBIT1_2E1(V0n_2E0)),c_2Earithmetic_2EZERO_2E0)) )
      & ( c_2Earithmetic_2EEXP_2E2(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EZERO_2E0)),c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(V0n_2E0))) = c_2Earithmetic_2ENUMERAL_2E1(c_2Enumeral_2Etexp__help_2E2(c_2Earithmetic_2EBIT1_2E1(V0n_2E0),c_2Earithmetic_2EZERO_2E0)) ) ) ).

tff(thm_2Enumeral_2Eonecount__characterisation,axiom,
    ! [V0n_2E0: tyop_2Enum_2Enum,V1a_2E0: tyop_2Enum_2Enum] :
      ( ( p(c_2Eprim__rec_2E_3C_2E2(c_2Enum_2E0_2E0,c_2Enumeral_2Eonecount_2E2(V0n_2E0,V1a_2E0)))
        & p(c_2Eprim__rec_2E_3C_2E2(c_2Enum_2E0_2E0,V0n_2E0)) )
     => ( V0n_2E0 = c_2Earithmetic_2E_2D_2E2(c_2Earithmetic_2EEXP_2E2(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EZERO_2E0)),c_2Earithmetic_2E_2D_2E2(c_2Enumeral_2Eonecount_2E2(V0n_2E0,V1a_2E0),V1a_2E0)),c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))) ) ) ).

tff(thm_2Enumeral_2Eexactlog__characterisation,axiom,
    ! [V0n_2E0: tyop_2Enum_2Enum,V1m_2E0: tyop_2Enum_2Enum] :
      ( ( c_2Enumeral_2Eexactlog_2E1(V0n_2E0) = c_2Earithmetic_2EBIT1_2E1(V1m_2E0) )
     => ( V0n_2E0 = c_2Earithmetic_2EEXP_2E2(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EZERO_2E0)),c_2Earithmetic_2E_2B_2E2(V1m_2E0,c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)))) ) ) ).

tff(thm_2Enumeral_2EDIV2__BIT1,axiom,
    ! [V0x_2E0: tyop_2Enum_2Enum] : ( c_2Earithmetic_2EDIV2_2E1(c_2Earithmetic_2EBIT1_2E1(V0x_2E0)) = V0x_2E0 ) ).

tff(thm_2Enumeral_2Eenumeral__mult,axiom,
    ! [F1_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum))),F0_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum))),F3_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum)))),F2_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum)))] :
      ( ! [V0y_2E0: tyop_2Enum_2Enum,V1x_2E0: tyop_2Enum_2Enum,V4m_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum)),F1_2E0,V0y_2E0),V1x_2E0),V4m_2E0) = c_2Ebool_2ECOND_2E3(tyop_2Enum_2Enum,c_2Earithmetic_2EODD_2E1(V4m_2E0),c_2Enumeral_2Etexp__help_2E2(c_2Earithmetic_2EDIV2_2E1(V4m_2E0),c_2Eprim__rec_2EPRE_2E1(c_2Earithmetic_2EBIT1_2E1(V0y_2E0))),c_2Enumeral_2Einternal__mult_2E2(c_2Earithmetic_2EBIT2_2E1(V1x_2E0),c_2Earithmetic_2EBIT1_2E1(V0y_2E0))) )
     => ( ! [V1x_2E0: tyop_2Enum_2Enum,V0y_2E0: tyop_2Enum_2Enum,V3n_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum)),F0_2E0,V1x_2E0),V0y_2E0),V3n_2E0) = c_2Ebool_2ECOND_2E3(tyop_2Enum_2Enum,c_2Earithmetic_2EODD_2E1(V3n_2E0),c_2Enumeral_2Etexp__help_2E2(c_2Earithmetic_2EDIV2_2E1(V3n_2E0),c_2Eprim__rec_2EPRE_2E1(c_2Earithmetic_2EBIT1_2E1(V1x_2E0))),c_2Enumeral_2Einternal__mult_2E2(c_2Earithmetic_2EBIT1_2E1(V1x_2E0),c_2Earithmetic_2EBIT2_2E1(V0y_2E0))) )
       => ( ! [V5m_2E0: tyop_2Enum_2Enum,V0y_2E0: tyop_2Enum_2Enum,V1x_2E0: tyop_2Enum_2Enum,V6n_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum)),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum))),F3_2E0,V5m_2E0),V0y_2E0),V1x_2E0),V6n_2E0) = c_2Ebool_2ECOND_2E3(tyop_2Enum_2Enum,c_2Earithmetic_2EODD_2E1(V5m_2E0),c_2Enumeral_2Etexp__help_2E2(c_2Earithmetic_2EDIV2_2E1(V5m_2E0),c_2Eprim__rec_2EPRE_2E1(c_2Earithmetic_2EBIT2_2E1(V0y_2E0))),c_2Ebool_2ECOND_2E3(tyop_2Enum_2Enum,c_2Earithmetic_2EODD_2E1(V6n_2E0),c_2Enumeral_2Etexp__help_2E2(c_2Earithmetic_2EDIV2_2E1(V6n_2E0),c_2Eprim__rec_2EPRE_2E1(c_2Earithmetic_2EBIT2_2E1(V1x_2E0))),c_2Enumeral_2Einternal__mult_2E2(c_2Earithmetic_2EBIT2_2E1(V1x_2E0),c_2Earithmetic_2EBIT2_2E1(V0y_2E0)))) )
         => ( ! [V0y_2E0: tyop_2Enum_2Enum,V1x_2E0: tyop_2Enum_2Enum,V5m_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum)),F2_2E0,V0y_2E0),V1x_2E0),V5m_2E0) = c_2Ebool_2ELET_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum)),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum))),F3_2E0,V5m_2E0),V0y_2E0),V1x_2E0),c_2Enumeral_2Eexactlog_2E1(c_2Earithmetic_2EBIT2_2E1(V0y_2E0))) )
           => ! [V0y_2E0: tyop_2Enum_2Enum,V1x_2E0: tyop_2Enum_2Enum,V2n_2E0: tyop_2Enum_2Enum] :
                ( ( c_2Earithmetic_2E_2A_2E2(c_2Earithmetic_2EZERO_2E0,V2n_2E0) = c_2Earithmetic_2EZERO_2E0 )
                & ( c_2Earithmetic_2E_2A_2E2(V2n_2E0,c_2Earithmetic_2EZERO_2E0) = c_2Earithmetic_2EZERO_2E0 )
                & ( c_2Earithmetic_2E_2A_2E2(c_2Earithmetic_2EBIT1_2E1(V1x_2E0),c_2Earithmetic_2EBIT1_2E1(V0y_2E0)) = c_2Enumeral_2Einternal__mult_2E2(c_2Earithmetic_2EBIT1_2E1(V1x_2E0),c_2Earithmetic_2EBIT1_2E1(V0y_2E0)) )
                & ( c_2Earithmetic_2E_2A_2E2(c_2Earithmetic_2EBIT1_2E1(V1x_2E0),c_2Earithmetic_2EBIT2_2E1(V0y_2E0)) = c_2Ebool_2ELET_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum)),F0_2E0,V1x_2E0),V0y_2E0),c_2Enumeral_2Eexactlog_2E1(c_2Earithmetic_2EBIT2_2E1(V0y_2E0))) )
                & ( c_2Earithmetic_2E_2A_2E2(c_2Earithmetic_2EBIT2_2E1(V1x_2E0),c_2Earithmetic_2EBIT1_2E1(V0y_2E0)) = c_2Ebool_2ELET_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum)),F1_2E0,V0y_2E0),V1x_2E0),c_2Enumeral_2Eexactlog_2E1(c_2Earithmetic_2EBIT2_2E1(V1x_2E0))) )
                & ( c_2Earithmetic_2E_2A_2E2(c_2Earithmetic_2EBIT2_2E1(V1x_2E0),c_2Earithmetic_2EBIT2_2E1(V0y_2E0)) = c_2Ebool_2ELET_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum)),F2_2E0,V0y_2E0),V1x_2E0),c_2Enumeral_2Eexactlog_2E1(c_2Earithmetic_2EBIT2_2E1(V1x_2E0))) ) ) ) ) ) ) ).

tff(thm_2Enumeral_2Einternal__mult__characterisation,axiom,
    ! [V0n_2E0: tyop_2Enum_2Enum,V1m_2E0: tyop_2Enum_2Enum] :
      ( ( c_2Enumeral_2Einternal__mult_2E2(c_2Earithmetic_2EZERO_2E0,V0n_2E0) = c_2Earithmetic_2EZERO_2E0 )
      & ( c_2Enumeral_2Einternal__mult_2E2(V0n_2E0,c_2Earithmetic_2EZERO_2E0) = c_2Earithmetic_2EZERO_2E0 )
      & ( c_2Enumeral_2Einternal__mult_2E2(c_2Earithmetic_2EBIT1_2E1(V0n_2E0),V1m_2E0) = c_2Enumeral_2EiZ_2E1(c_2Earithmetic_2E_2B_2E2(c_2Enumeral_2EiDUB_2E1(c_2Enumeral_2Einternal__mult_2E2(V0n_2E0,V1m_2E0)),V1m_2E0)) )
      & ( c_2Enumeral_2Einternal__mult_2E2(c_2Earithmetic_2EBIT2_2E1(V0n_2E0),V1m_2E0) = c_2Enumeral_2EiDUB_2E1(c_2Enumeral_2EiZ_2E1(c_2Earithmetic_2E_2B_2E2(c_2Enumeral_2Einternal__mult_2E2(V0n_2E0,V1m_2E0),V1m_2E0))) ) ) ).

%------------------------------------------------------------------------------