ITP001 Axioms: ITP023^5.ax


%------------------------------------------------------------------------------
% File     : ITP023^5 : TPTP v9.0.0. 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    : gcd^2.ax [Gau20]
%          : HL4023^5.ax [TPAP]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   51 (  12 unt;   5 typ;   0 def)
%            Number of atoms       : 1533 (  49 equ;   0 cnn)
%            Maximal formula atoms :   68 (  30 avg)
%            Number of connectives : 1848 (   6   ~;   4   |;  46   &;1762   @)
%                                         (   3 <=>;  27  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   19 (   8 avg;1762 nst)
%            Number of types       :    1 (   0 usr)
%            Number of type conns  :    4 (   4   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :   35 (  34 usr;  32 con; 0-2 aty)
%            Number of variables   :  122 (   0   ^ 118   !;   4   ?; 122   :)
% SPC      : TH0_SAT_EQU_NAR

% Comments :
% Bugfixes : v7.5.0 - Fixes to the axioms.
%------------------------------------------------------------------------------
thf(tp_c_2Egcd_2Egcd,type,
    c_2Egcd_2Egcd: $i ).

thf(mem_c_2Egcd_2Egcd,axiom,
    mem @ c_2Egcd_2Egcd @ ( arr @ ty_2Enum_2Enum @ ( arr @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) ) ).

thf(stp_fo_c_2Egcd_2Egcd,type,
    fo__c_2Egcd_2Egcd: tp__ty_2Enum_2Enum > tp__ty_2Enum_2Enum > tp__ty_2Enum_2Enum ).

thf(stp_eq_fo_c_2Egcd_2Egcd,axiom,
    ! [X0: tp__ty_2Enum_2Enum,X1: tp__ty_2Enum_2Enum] :
      ( ( inj__ty_2Enum_2Enum @ ( fo__c_2Egcd_2Egcd @ X0 @ X1 ) )
      = ( ap @ ( ap @ c_2Egcd_2Egcd @ ( inj__ty_2Enum_2Enum @ X0 ) ) @ ( inj__ty_2Enum_2Enum @ X1 ) ) ) ).

thf(tp_c_2Egcd_2Eis__gcd,type,
    c_2Egcd_2Eis__gcd: $i ).

thf(mem_c_2Egcd_2Eis__gcd,axiom,
    mem @ c_2Egcd_2Eis__gcd @ ( arr @ ty_2Enum_2Enum @ ( arr @ ty_2Enum_2Enum @ ( arr @ ty_2Enum_2Enum @ bool ) ) ) ).

thf(tp_c_2Egcd_2Elcm,type,
    c_2Egcd_2Elcm: $i ).

thf(mem_c_2Egcd_2Elcm,axiom,
    mem @ c_2Egcd_2Elcm @ ( arr @ ty_2Enum_2Enum @ ( arr @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) ) ).

thf(stp_fo_c_2Egcd_2Elcm,type,
    fo__c_2Egcd_2Elcm: tp__ty_2Enum_2Enum > tp__ty_2Enum_2Enum > tp__ty_2Enum_2Enum ).

thf(stp_eq_fo_c_2Egcd_2Elcm,axiom,
    ! [X0: tp__ty_2Enum_2Enum,X1: tp__ty_2Enum_2Enum] :
      ( ( inj__ty_2Enum_2Enum @ ( fo__c_2Egcd_2Elcm @ X0 @ X1 ) )
      = ( ap @ ( ap @ c_2Egcd_2Elcm @ ( inj__ty_2Enum_2Enum @ X0 ) ) @ ( inj__ty_2Enum_2Enum @ X1 ) ) ) ).

thf(ax_thm_2Egcd_2Eis__gcd__def,axiom,
    ! [V0a: tp__ty_2Enum_2Enum,V1b: tp__ty_2Enum_2Enum,V2c: tp__ty_2Enum_2Enum] :
      ( ( p @ ( ap @ ( ap @ ( ap @ c_2Egcd_2Eis__gcd @ ( inj__ty_2Enum_2Enum @ V0a ) ) @ ( inj__ty_2Enum_2Enum @ V1b ) ) @ ( inj__ty_2Enum_2Enum @ V2c ) ) )
    <=> ( ( p @ ( ap @ ( ap @ c_2Edivides_2Edivides @ ( inj__ty_2Enum_2Enum @ V2c ) ) @ ( inj__ty_2Enum_2Enum @ V0a ) ) )
        & ( p @ ( ap @ ( ap @ c_2Edivides_2Edivides @ ( inj__ty_2Enum_2Enum @ V2c ) ) @ ( inj__ty_2Enum_2Enum @ V1b ) ) )
        & ! [V3d: tp__ty_2Enum_2Enum] :
            ( ( ( p @ ( ap @ ( ap @ c_2Edivides_2Edivides @ ( inj__ty_2Enum_2Enum @ V3d ) ) @ ( inj__ty_2Enum_2Enum @ V0a ) ) )
              & ( p @ ( ap @ ( ap @ c_2Edivides_2Edivides @ ( inj__ty_2Enum_2Enum @ V3d ) ) @ ( inj__ty_2Enum_2Enum @ V1b ) ) ) )
           => ( p @ ( ap @ ( ap @ c_2Edivides_2Edivides @ ( inj__ty_2Enum_2Enum @ V3d ) ) @ ( inj__ty_2Enum_2Enum @ V2c ) ) ) ) ) ) ).

thf(conj_thm_2Egcd_2EIS__GCD__UNIQUE,axiom,
    ! [V0a: tp__ty_2Enum_2Enum,V1b: tp__ty_2Enum_2Enum,V2c: tp__ty_2Enum_2Enum,V3d: tp__ty_2Enum_2Enum] :
      ( ( ( p @ ( ap @ ( ap @ ( ap @ c_2Egcd_2Eis__gcd @ ( inj__ty_2Enum_2Enum @ V0a ) ) @ ( inj__ty_2Enum_2Enum @ V1b ) ) @ ( inj__ty_2Enum_2Enum @ V2c ) ) )
        & ( p @ ( ap @ ( ap @ ( ap @ c_2Egcd_2Eis__gcd @ ( inj__ty_2Enum_2Enum @ V0a ) ) @ ( inj__ty_2Enum_2Enum @ V1b ) ) @ ( inj__ty_2Enum_2Enum @ V3d ) ) ) )
     => ( V2c = V3d ) ) ).

thf(conj_thm_2Egcd_2EIS__GCD__REF,axiom,
    ! [V0a: tp__ty_2Enum_2Enum] : ( p @ ( ap @ ( ap @ ( ap @ c_2Egcd_2Eis__gcd @ ( inj__ty_2Enum_2Enum @ V0a ) ) @ ( inj__ty_2Enum_2Enum @ V0a ) ) @ ( inj__ty_2Enum_2Enum @ V0a ) ) ) ).

thf(conj_thm_2Egcd_2EIS__GCD__SYM,axiom,
    ! [V0a: tp__ty_2Enum_2Enum,V1b: tp__ty_2Enum_2Enum,V2c: tp__ty_2Enum_2Enum] :
      ( ( p @ ( ap @ ( ap @ ( ap @ c_2Egcd_2Eis__gcd @ ( inj__ty_2Enum_2Enum @ V0a ) ) @ ( inj__ty_2Enum_2Enum @ V1b ) ) @ ( inj__ty_2Enum_2Enum @ V2c ) ) )
    <=> ( p @ ( ap @ ( ap @ ( ap @ c_2Egcd_2Eis__gcd @ ( inj__ty_2Enum_2Enum @ V1b ) ) @ ( inj__ty_2Enum_2Enum @ V0a ) ) @ ( inj__ty_2Enum_2Enum @ V2c ) ) ) ) ).

thf(conj_thm_2Egcd_2EIS__GCD__0R,axiom,
    ! [V0a: tp__ty_2Enum_2Enum] : ( p @ ( ap @ ( ap @ ( ap @ c_2Egcd_2Eis__gcd @ ( inj__ty_2Enum_2Enum @ V0a ) ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( inj__ty_2Enum_2Enum @ V0a ) ) ) ).

thf(conj_thm_2Egcd_2EIS__GCD__0L,axiom,
    ! [V0a: tp__ty_2Enum_2Enum] : ( p @ ( ap @ ( ap @ ( ap @ c_2Egcd_2Eis__gcd @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( inj__ty_2Enum_2Enum @ V0a ) ) @ ( inj__ty_2Enum_2Enum @ V0a ) ) ) ).

thf(conj_thm_2Egcd_2EPRIME__IS__GCD,axiom,
    ! [V0p: tp__ty_2Enum_2Enum,V1b: tp__ty_2Enum_2Enum] :
      ( ( p @ ( ap @ c_2Edivides_2Eprime @ ( inj__ty_2Enum_2Enum @ V0p ) ) )
     => ( ( p @ ( ap @ ( ap @ c_2Edivides_2Edivides @ ( inj__ty_2Enum_2Enum @ V0p ) ) @ ( inj__ty_2Enum_2Enum @ V1b ) ) )
        | ( p @ ( ap @ ( ap @ ( ap @ c_2Egcd_2Eis__gcd @ ( inj__ty_2Enum_2Enum @ V0p ) ) @ ( inj__ty_2Enum_2Enum @ V1b ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) ).

thf(conj_thm_2Egcd_2EIS__GCD__MINUS__L,axiom,
    ! [V0a: tp__ty_2Enum_2Enum,V1b: tp__ty_2Enum_2Enum,V2c: tp__ty_2Enum_2Enum] :
      ( ( ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ V1b ) ) @ ( inj__ty_2Enum_2Enum @ V0a ) ) )
        & ( p @ ( ap @ ( ap @ ( ap @ c_2Egcd_2Eis__gcd @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ ( inj__ty_2Enum_2Enum @ V0a ) ) @ ( inj__ty_2Enum_2Enum @ V1b ) ) ) @ ( inj__ty_2Enum_2Enum @ V1b ) ) @ ( inj__ty_2Enum_2Enum @ V2c ) ) ) )
     => ( p @ ( ap @ ( ap @ ( ap @ c_2Egcd_2Eis__gcd @ ( inj__ty_2Enum_2Enum @ V0a ) ) @ ( inj__ty_2Enum_2Enum @ V1b ) ) @ ( inj__ty_2Enum_2Enum @ V2c ) ) ) ) ).

thf(conj_thm_2Egcd_2EIS__GCD__MINUS__R,axiom,
    ! [V0a: tp__ty_2Enum_2Enum,V1b: tp__ty_2Enum_2Enum,V2c: tp__ty_2Enum_2Enum] :
      ( ( ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ V0a ) ) @ ( inj__ty_2Enum_2Enum @ V1b ) ) )
        & ( p @ ( ap @ ( ap @ ( ap @ c_2Egcd_2Eis__gcd @ ( inj__ty_2Enum_2Enum @ V0a ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ ( inj__ty_2Enum_2Enum @ V1b ) ) @ ( inj__ty_2Enum_2Enum @ V0a ) ) ) @ ( inj__ty_2Enum_2Enum @ V2c ) ) ) )
     => ( p @ ( ap @ ( ap @ ( ap @ c_2Egcd_2Eis__gcd @ ( inj__ty_2Enum_2Enum @ V0a ) ) @ ( inj__ty_2Enum_2Enum @ V1b ) ) @ ( inj__ty_2Enum_2Enum @ V2c ) ) ) ) ).

thf(conj_thm_2Egcd_2Egcd__ind,axiom,
    ! [V0P: $i] :
      ( ( mem @ V0P @ ( arr @ ty_2Enum_2Enum @ ( arr @ ty_2Enum_2Enum @ bool ) ) )
     => ( ( ! [V1y: tp__ty_2Enum_2Enum] : ( p @ ( ap @ ( ap @ V0P @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( inj__ty_2Enum_2Enum @ V1y ) ) )
          & ! [V2x: tp__ty_2Enum_2Enum] : ( p @ ( ap @ ( ap @ V0P @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V2x ) ) ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) )
          & ! [V3x: tp__ty_2Enum_2Enum,V4y: tp__ty_2Enum_2Enum] :
              ( ( ( ~ ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ V4y ) ) @ ( inj__ty_2Enum_2Enum @ V3x ) ) )
                 => ( p @ ( ap @ ( ap @ V0P @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V3x ) ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ ( inj__ty_2Enum_2Enum @ V4y ) ) @ ( inj__ty_2Enum_2Enum @ V3x ) ) ) ) )
                & ( ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ V4y ) ) @ ( inj__ty_2Enum_2Enum @ V3x ) ) )
                 => ( p @ ( ap @ ( ap @ V0P @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ ( inj__ty_2Enum_2Enum @ V3x ) ) @ ( inj__ty_2Enum_2Enum @ V4y ) ) ) @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V4y ) ) ) ) ) )
             => ( p @ ( ap @ ( ap @ V0P @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V3x ) ) ) @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V4y ) ) ) ) ) )
       => ! [V5v: tp__ty_2Enum_2Enum,V6v1: tp__ty_2Enum_2Enum] : ( p @ ( ap @ ( ap @ V0P @ ( inj__ty_2Enum_2Enum @ V5v ) ) @ ( inj__ty_2Enum_2Enum @ V6v1 ) ) ) ) ) ).

thf(conj_thm_2Egcd_2Egcd__def,axiom,
    ( ! [V0y: tp__ty_2Enum_2Enum] :
        ( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Egcd_2Egcd @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( inj__ty_2Enum_2Enum @ V0y ) ) )
        = V0y )
    & ! [V1x: tp__ty_2Enum_2Enum] :
        ( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Egcd_2Egcd @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V1x ) ) ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) )
        = ( surj__ty_2Enum_2Enum @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V1x ) ) ) )
    & ! [V2y: tp__ty_2Enum_2Enum,V3x: tp__ty_2Enum_2Enum] :
        ( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Egcd_2Egcd @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V3x ) ) ) @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V2y ) ) ) )
        = ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ V2y ) ) @ ( inj__ty_2Enum_2Enum @ V3x ) ) ) @ ( ap @ ( ap @ c_2Egcd_2Egcd @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ ( inj__ty_2Enum_2Enum @ V3x ) ) @ ( inj__ty_2Enum_2Enum @ V2y ) ) ) @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V2y ) ) ) ) @ ( ap @ ( ap @ c_2Egcd_2Egcd @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V3x ) ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ ( inj__ty_2Enum_2Enum @ V2y ) ) @ ( inj__ty_2Enum_2Enum @ V3x ) ) ) ) ) ) ) ).

thf(conj_thm_2Egcd_2Egcd__def__compute,axiom,
    ( ! [V0y: tp__ty_2Enum_2Enum] :
        ( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Egcd_2Egcd @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( inj__ty_2Enum_2Enum @ V0y ) ) )
        = V0y )
    & ! [V1x: tp__ty_2Enum_2Enum] :
        ( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Egcd_2Egcd @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ V1x ) ) ) ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) )
        = ( surj__ty_2Enum_2Enum @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ V1x ) ) ) ) )
    & ! [V2x: tp__ty_2Enum_2Enum] :
        ( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Egcd_2Egcd @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ V2x ) ) ) ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) )
        = ( surj__ty_2Enum_2Enum @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ V2x ) ) ) ) )
    & ! [V3y: tp__ty_2Enum_2Enum,V4x: tp__ty_2Enum_2Enum] :
        ( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Egcd_2Egcd @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ V4x ) ) ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ V3y ) ) ) ) )
        = ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ V3y ) ) ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ V4x ) ) ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) @ ( ap @ ( ap @ c_2Egcd_2Egcd @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ V4x ) ) ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ V3y ) ) ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ V3y ) ) ) ) ) @ ( ap @ ( ap @ c_2Egcd_2Egcd @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ V4x ) ) ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ V3y ) ) ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ V4x ) ) ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) ) )
    & ! [V5y: tp__ty_2Enum_2Enum,V6x: tp__ty_2Enum_2Enum] :
        ( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Egcd_2Egcd @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ V6x ) ) ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ V5y ) ) ) ) )
        = ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ V5y ) ) ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ V6x ) ) ) ) ) @ ( ap @ ( ap @ c_2Egcd_2Egcd @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ V6x ) ) ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ V5y ) ) ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ V5y ) ) ) ) ) @ ( ap @ ( ap @ c_2Egcd_2Egcd @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ V6x ) ) ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ V5y ) ) ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ V6x ) ) ) ) ) ) ) )
    & ! [V7y: tp__ty_2Enum_2Enum,V8x: tp__ty_2Enum_2Enum] :
        ( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Egcd_2Egcd @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ V8x ) ) ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ V7y ) ) ) ) )
        = ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ V7y ) ) ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ V8x ) ) ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) @ ( ap @ ( ap @ c_2Egcd_2Egcd @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ V8x ) ) ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ V7y ) ) ) ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ V7y ) ) ) ) ) @ ( ap @ ( ap @ c_2Egcd_2Egcd @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ V8x ) ) ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ V7y ) ) ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ V8x ) ) ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) ) )
    & ! [V9y: tp__ty_2Enum_2Enum,V10x: tp__ty_2Enum_2Enum] :
        ( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Egcd_2Egcd @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ V10x ) ) ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ V9y ) ) ) ) )
        = ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ V9y ) ) ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ V10x ) ) ) ) ) @ ( ap @ ( ap @ c_2Egcd_2Egcd @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ V10x ) ) ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ V9y ) ) ) ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ V9y ) ) ) ) ) @ ( ap @ ( ap @ c_2Egcd_2Egcd @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ V10x ) ) ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ V9y ) ) ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ V10x ) ) ) ) ) ) ) ) ) ).

thf(conj_thm_2Egcd_2EGCD__IS__GCD,axiom,
    ! [V0a: tp__ty_2Enum_2Enum,V1b: tp__ty_2Enum_2Enum] : ( p @ ( ap @ ( ap @ ( ap @ c_2Egcd_2Eis__gcd @ ( inj__ty_2Enum_2Enum @ V0a ) ) @ ( inj__ty_2Enum_2Enum @ V1b ) ) @ ( ap @ ( ap @ c_2Egcd_2Egcd @ ( inj__ty_2Enum_2Enum @ V0a ) ) @ ( inj__ty_2Enum_2Enum @ V1b ) ) ) ) ).

thf(conj_thm_2Egcd_2EGCD__IS__GREATEST__COMMON__DIVISOR,axiom,
    ! [V0a: tp__ty_2Enum_2Enum,V1b: tp__ty_2Enum_2Enum] :
      ( ( p @ ( ap @ ( ap @ c_2Edivides_2Edivides @ ( ap @ ( ap @ c_2Egcd_2Egcd @ ( inj__ty_2Enum_2Enum @ V0a ) ) @ ( inj__ty_2Enum_2Enum @ V1b ) ) ) @ ( inj__ty_2Enum_2Enum @ V0a ) ) )
      & ( p @ ( ap @ ( ap @ c_2Edivides_2Edivides @ ( ap @ ( ap @ c_2Egcd_2Egcd @ ( inj__ty_2Enum_2Enum @ V0a ) ) @ ( inj__ty_2Enum_2Enum @ V1b ) ) ) @ ( inj__ty_2Enum_2Enum @ V1b ) ) )
      & ! [V2d: tp__ty_2Enum_2Enum] :
          ( ( ( p @ ( ap @ ( ap @ c_2Edivides_2Edivides @ ( inj__ty_2Enum_2Enum @ V2d ) ) @ ( inj__ty_2Enum_2Enum @ V0a ) ) )
            & ( p @ ( ap @ ( ap @ c_2Edivides_2Edivides @ ( inj__ty_2Enum_2Enum @ V2d ) ) @ ( inj__ty_2Enum_2Enum @ V1b ) ) ) )
         => ( p @ ( ap @ ( ap @ c_2Edivides_2Edivides @ ( inj__ty_2Enum_2Enum @ V2d ) ) @ ( ap @ ( ap @ c_2Egcd_2Egcd @ ( inj__ty_2Enum_2Enum @ V0a ) ) @ ( inj__ty_2Enum_2Enum @ V1b ) ) ) ) ) ) ).

thf(conj_thm_2Egcd_2EGCD__REF,axiom,
    ! [V0a: tp__ty_2Enum_2Enum] :
      ( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Egcd_2Egcd @ ( inj__ty_2Enum_2Enum @ V0a ) ) @ ( inj__ty_2Enum_2Enum @ V0a ) ) )
      = V0a ) ).

thf(conj_thm_2Egcd_2EGCD__SYM,axiom,
    ! [V0a: tp__ty_2Enum_2Enum,V1b: tp__ty_2Enum_2Enum] :
      ( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Egcd_2Egcd @ ( inj__ty_2Enum_2Enum @ V0a ) ) @ ( inj__ty_2Enum_2Enum @ V1b ) ) )
      = ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Egcd_2Egcd @ ( inj__ty_2Enum_2Enum @ V1b ) ) @ ( inj__ty_2Enum_2Enum @ V0a ) ) ) ) ).

thf(conj_thm_2Egcd_2EGCD__0R,axiom,
    ! [V0a: tp__ty_2Enum_2Enum] :
      ( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Egcd_2Egcd @ ( inj__ty_2Enum_2Enum @ V0a ) ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) )
      = V0a ) ).

thf(conj_thm_2Egcd_2EGCD__0L,axiom,
    ! [V0a: tp__ty_2Enum_2Enum] :
      ( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Egcd_2Egcd @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( inj__ty_2Enum_2Enum @ V0a ) ) )
      = V0a ) ).

thf(conj_thm_2Egcd_2EGCD__ADD__R,axiom,
    ! [V0a: tp__ty_2Enum_2Enum,V1b: tp__ty_2Enum_2Enum] :
      ( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Egcd_2Egcd @ ( inj__ty_2Enum_2Enum @ V0a ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( inj__ty_2Enum_2Enum @ V0a ) ) @ ( inj__ty_2Enum_2Enum @ V1b ) ) ) )
      = ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Egcd_2Egcd @ ( inj__ty_2Enum_2Enum @ V0a ) ) @ ( inj__ty_2Enum_2Enum @ V1b ) ) ) ) ).

thf(conj_thm_2Egcd_2EGCD__ADD__R__THM,axiom,
    ( ! [V0a: tp__ty_2Enum_2Enum,V1b: tp__ty_2Enum_2Enum] :
        ( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Egcd_2Egcd @ ( inj__ty_2Enum_2Enum @ V0a ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( inj__ty_2Enum_2Enum @ V0a ) ) @ ( inj__ty_2Enum_2Enum @ V1b ) ) ) )
        = ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Egcd_2Egcd @ ( inj__ty_2Enum_2Enum @ V0a ) ) @ ( inj__ty_2Enum_2Enum @ V1b ) ) ) )
    & ! [V2a: tp__ty_2Enum_2Enum,V3b: tp__ty_2Enum_2Enum] :
        ( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Egcd_2Egcd @ ( inj__ty_2Enum_2Enum @ V2a ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( inj__ty_2Enum_2Enum @ V3b ) ) @ ( inj__ty_2Enum_2Enum @ V2a ) ) ) )
        = ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Egcd_2Egcd @ ( inj__ty_2Enum_2Enum @ V2a ) ) @ ( inj__ty_2Enum_2Enum @ V3b ) ) ) ) ) ).

thf(conj_thm_2Egcd_2EGCD__ADD__L,axiom,
    ! [V0a: tp__ty_2Enum_2Enum,V1b: tp__ty_2Enum_2Enum] :
      ( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Egcd_2Egcd @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( inj__ty_2Enum_2Enum @ V0a ) ) @ ( inj__ty_2Enum_2Enum @ V1b ) ) ) @ ( inj__ty_2Enum_2Enum @ V0a ) ) )
      = ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Egcd_2Egcd @ ( inj__ty_2Enum_2Enum @ V0a ) ) @ ( inj__ty_2Enum_2Enum @ V1b ) ) ) ) ).

thf(conj_thm_2Egcd_2EGCD__ADD__L__THM,axiom,
    ( ! [V0a: tp__ty_2Enum_2Enum,V1b: tp__ty_2Enum_2Enum] :
        ( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Egcd_2Egcd @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( inj__ty_2Enum_2Enum @ V0a ) ) @ ( inj__ty_2Enum_2Enum @ V1b ) ) ) @ ( inj__ty_2Enum_2Enum @ V0a ) ) )
        = ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Egcd_2Egcd @ ( inj__ty_2Enum_2Enum @ V0a ) ) @ ( inj__ty_2Enum_2Enum @ V1b ) ) ) )
    & ! [V2a: tp__ty_2Enum_2Enum,V3b: tp__ty_2Enum_2Enum] :
        ( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Egcd_2Egcd @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( inj__ty_2Enum_2Enum @ V3b ) ) @ ( inj__ty_2Enum_2Enum @ V2a ) ) ) @ ( inj__ty_2Enum_2Enum @ V2a ) ) )
        = ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Egcd_2Egcd @ ( inj__ty_2Enum_2Enum @ V2a ) ) @ ( inj__ty_2Enum_2Enum @ V3b ) ) ) ) ) ).

thf(conj_thm_2Egcd_2EGCD__EQ__0,axiom,
    ! [V0n: tp__ty_2Enum_2Enum,V1m: tp__ty_2Enum_2Enum] :
      ( ( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Egcd_2Egcd @ ( inj__ty_2Enum_2Enum @ V0n ) ) @ ( inj__ty_2Enum_2Enum @ V1m ) ) )
        = fo__c_2Enum_2E0 )
    <=> ( ( V0n = fo__c_2Enum_2E0 )
        & ( V1m = fo__c_2Enum_2E0 ) ) ) ).

thf(conj_thm_2Egcd_2EGCD__1,axiom,
    ! [V0x: tp__ty_2Enum_2Enum] :
      ( ( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Egcd_2Egcd @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) @ ( inj__ty_2Enum_2Enum @ V0x ) ) )
        = ( surj__ty_2Enum_2Enum @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) )
      & ( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Egcd_2Egcd @ ( inj__ty_2Enum_2Enum @ V0x ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) )
        = ( surj__ty_2Enum_2Enum @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ).

thf(conj_thm_2Egcd_2EPRIME__GCD,axiom,
    ! [V0p: tp__ty_2Enum_2Enum,V1b: tp__ty_2Enum_2Enum] :
      ( ( p @ ( ap @ c_2Edivides_2Eprime @ ( inj__ty_2Enum_2Enum @ V0p ) ) )
     => ( ( p @ ( ap @ ( ap @ c_2Edivides_2Edivides @ ( inj__ty_2Enum_2Enum @ V0p ) ) @ ( inj__ty_2Enum_2Enum @ V1b ) ) )
        | ( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Egcd_2Egcd @ ( inj__ty_2Enum_2Enum @ V0p ) ) @ ( inj__ty_2Enum_2Enum @ V1b ) ) )
          = ( surj__ty_2Enum_2Enum @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) ).

thf(conj_thm_2Egcd_2EL__EUCLIDES,axiom,
    ! [V0a: tp__ty_2Enum_2Enum,V1b: tp__ty_2Enum_2Enum,V2c: tp__ty_2Enum_2Enum] :
      ( ( ( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Egcd_2Egcd @ ( inj__ty_2Enum_2Enum @ V0a ) ) @ ( inj__ty_2Enum_2Enum @ V1b ) ) )
          = ( surj__ty_2Enum_2Enum @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) )
        & ( p @ ( ap @ ( ap @ c_2Edivides_2Edivides @ ( inj__ty_2Enum_2Enum @ V1b ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2A @ ( inj__ty_2Enum_2Enum @ V0a ) ) @ ( inj__ty_2Enum_2Enum @ V2c ) ) ) ) )
     => ( p @ ( ap @ ( ap @ c_2Edivides_2Edivides @ ( inj__ty_2Enum_2Enum @ V1b ) ) @ ( inj__ty_2Enum_2Enum @ V2c ) ) ) ) ).

thf(conj_thm_2Egcd_2EP__EUCLIDES,axiom,
    ! [V0p: tp__ty_2Enum_2Enum,V1a: tp__ty_2Enum_2Enum,V2b: tp__ty_2Enum_2Enum] :
      ( ( ( p @ ( ap @ c_2Edivides_2Eprime @ ( inj__ty_2Enum_2Enum @ V0p ) ) )
        & ( p @ ( ap @ ( ap @ c_2Edivides_2Edivides @ ( inj__ty_2Enum_2Enum @ V0p ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2A @ ( inj__ty_2Enum_2Enum @ V1a ) ) @ ( inj__ty_2Enum_2Enum @ V2b ) ) ) ) )
     => ( ( p @ ( ap @ ( ap @ c_2Edivides_2Edivides @ ( inj__ty_2Enum_2Enum @ V0p ) ) @ ( inj__ty_2Enum_2Enum @ V1a ) ) )
        | ( p @ ( ap @ ( ap @ c_2Edivides_2Edivides @ ( inj__ty_2Enum_2Enum @ V0p ) ) @ ( inj__ty_2Enum_2Enum @ V2b ) ) ) ) ) ).

thf(conj_thm_2Egcd_2EFACTOR__OUT__GCD,axiom,
    ! [V0n: tp__ty_2Enum_2Enum,V1m: tp__ty_2Enum_2Enum] :
      ( ( ( V0n != fo__c_2Enum_2E0 )
        & ( V1m != fo__c_2Enum_2E0 ) )
     => ? [V2p: tp__ty_2Enum_2Enum,V3q: tp__ty_2Enum_2Enum] :
          ( ( V0n
            = ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Earithmetic_2E_2A @ ( inj__ty_2Enum_2Enum @ V2p ) ) @ ( ap @ ( ap @ c_2Egcd_2Egcd @ ( inj__ty_2Enum_2Enum @ V0n ) ) @ ( inj__ty_2Enum_2Enum @ V1m ) ) ) ) )
          & ( V1m
            = ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Earithmetic_2E_2A @ ( inj__ty_2Enum_2Enum @ V3q ) ) @ ( ap @ ( ap @ c_2Egcd_2Egcd @ ( inj__ty_2Enum_2Enum @ V0n ) ) @ ( inj__ty_2Enum_2Enum @ V1m ) ) ) ) )
          & ( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Egcd_2Egcd @ ( inj__ty_2Enum_2Enum @ V2p ) ) @ ( inj__ty_2Enum_2Enum @ V3q ) ) )
            = ( surj__ty_2Enum_2Enum @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) ).

thf(conj_thm_2Egcd_2EGCD__SUCfree__ind,axiom,
    ! [V0P: $i] :
      ( ( mem @ V0P @ ( arr @ ty_2Enum_2Enum @ ( arr @ ty_2Enum_2Enum @ bool ) ) )
     => ( ( ! [V1y: tp__ty_2Enum_2Enum] : ( p @ ( ap @ ( ap @ V0P @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( inj__ty_2Enum_2Enum @ V1y ) ) )
          & ! [V2x: tp__ty_2Enum_2Enum,V3y: tp__ty_2Enum_2Enum] :
              ( ( p @ ( ap @ ( ap @ V0P @ ( inj__ty_2Enum_2Enum @ V2x ) ) @ ( inj__ty_2Enum_2Enum @ V3y ) ) )
             => ( p @ ( ap @ ( ap @ V0P @ ( inj__ty_2Enum_2Enum @ V3y ) ) @ ( inj__ty_2Enum_2Enum @ V2x ) ) ) )
          & ! [V4x: tp__ty_2Enum_2Enum] : ( p @ ( ap @ ( ap @ V0P @ ( inj__ty_2Enum_2Enum @ V4x ) ) @ ( inj__ty_2Enum_2Enum @ V4x ) ) )
          & ! [V5x: tp__ty_2Enum_2Enum,V6y: tp__ty_2Enum_2Enum] :
              ( ( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( inj__ty_2Enum_2Enum @ V5x ) ) )
                & ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( inj__ty_2Enum_2Enum @ V6y ) ) )
                & ( p @ ( ap @ ( ap @ V0P @ ( inj__ty_2Enum_2Enum @ V5x ) ) @ ( inj__ty_2Enum_2Enum @ V6y ) ) ) )
             => ( p @ ( ap @ ( ap @ V0P @ ( inj__ty_2Enum_2Enum @ V5x ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( inj__ty_2Enum_2Enum @ V5x ) ) @ ( inj__ty_2Enum_2Enum @ V6y ) ) ) ) ) )
       => ! [V7m: tp__ty_2Enum_2Enum,V8n: tp__ty_2Enum_2Enum] : ( p @ ( ap @ ( ap @ V0P @ ( inj__ty_2Enum_2Enum @ V7m ) ) @ ( inj__ty_2Enum_2Enum @ V8n ) ) ) ) ) ).

thf(conj_thm_2Egcd_2ELINEAR__GCD,axiom,
    ! [V0n: tp__ty_2Enum_2Enum,V1m: tp__ty_2Enum_2Enum] :
      ( ( V0n != fo__c_2Enum_2E0 )
     => ? [V2p: tp__ty_2Enum_2Enum,V3q: tp__ty_2Enum_2Enum] :
          ( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Earithmetic_2E_2A @ ( inj__ty_2Enum_2Enum @ V2p ) ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) )
          = ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( ap @ ( ap @ c_2Earithmetic_2E_2A @ ( inj__ty_2Enum_2Enum @ V3q ) ) @ ( inj__ty_2Enum_2Enum @ V1m ) ) ) @ ( ap @ ( ap @ c_2Egcd_2Egcd @ ( inj__ty_2Enum_2Enum @ V1m ) ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) ) ) ) ).

thf(conj_thm_2Egcd_2EGCD__EFFICIENTLY,axiom,
    ! [V0a: tp__ty_2Enum_2Enum,V1b: tp__ty_2Enum_2Enum] :
      ( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Egcd_2Egcd @ ( inj__ty_2Enum_2Enum @ V0a ) ) @ ( inj__ty_2Enum_2Enum @ V1b ) ) )
      = ( 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 @ V0a ) ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Enum_2Enum @ V1b ) ) @ ( ap @ ( ap @ c_2Egcd_2Egcd @ ( ap @ ( ap @ c_2Earithmetic_2EMOD @ ( inj__ty_2Enum_2Enum @ V1b ) ) @ ( inj__ty_2Enum_2Enum @ V0a ) ) ) @ ( inj__ty_2Enum_2Enum @ V0a ) ) ) ) ) ).

thf(ax_thm_2Egcd_2Elcm__def,axiom,
    ! [V0m: tp__ty_2Enum_2Enum,V1n: tp__ty_2Enum_2Enum] :
      ( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Egcd_2Elcm @ ( inj__ty_2Enum_2Enum @ V0m ) ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) )
      = ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Ebool_2E_5C_2F @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V0m ) ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( ap @ ( ap @ c_2Earithmetic_2EDIV @ ( ap @ ( ap @ c_2Earithmetic_2E_2A @ ( inj__ty_2Enum_2Enum @ V0m ) ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) @ ( ap @ ( ap @ c_2Egcd_2Egcd @ ( inj__ty_2Enum_2Enum @ V0m ) ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) ) ) ) ).

thf(conj_thm_2Egcd_2ELCM__IS__LEAST__COMMON__MULTIPLE,axiom,
    ! [V0m: tp__ty_2Enum_2Enum,V1n: tp__ty_2Enum_2Enum] :
      ( ( p @ ( ap @ ( ap @ c_2Edivides_2Edivides @ ( inj__ty_2Enum_2Enum @ V0m ) ) @ ( ap @ ( ap @ c_2Egcd_2Elcm @ ( inj__ty_2Enum_2Enum @ V0m ) ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) )
      & ( p @ ( ap @ ( ap @ c_2Edivides_2Edivides @ ( inj__ty_2Enum_2Enum @ V1n ) ) @ ( ap @ ( ap @ c_2Egcd_2Elcm @ ( inj__ty_2Enum_2Enum @ V0m ) ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) )
      & ! [V2p: tp__ty_2Enum_2Enum] :
          ( ( ( p @ ( ap @ ( ap @ c_2Edivides_2Edivides @ ( inj__ty_2Enum_2Enum @ V0m ) ) @ ( inj__ty_2Enum_2Enum @ V2p ) ) )
            & ( p @ ( ap @ ( ap @ c_2Edivides_2Edivides @ ( inj__ty_2Enum_2Enum @ V1n ) ) @ ( inj__ty_2Enum_2Enum @ V2p ) ) ) )
         => ( p @ ( ap @ ( ap @ c_2Edivides_2Edivides @ ( ap @ ( ap @ c_2Egcd_2Elcm @ ( inj__ty_2Enum_2Enum @ V0m ) ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) @ ( inj__ty_2Enum_2Enum @ V2p ) ) ) ) ) ).

thf(conj_thm_2Egcd_2ELCM__0,axiom,
    ! [V0x: tp__ty_2Enum_2Enum] :
      ( ( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Egcd_2Elcm @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( inj__ty_2Enum_2Enum @ V0x ) ) )
        = fo__c_2Enum_2E0 )
      & ( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Egcd_2Elcm @ ( inj__ty_2Enum_2Enum @ V0x ) ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) )
        = fo__c_2Enum_2E0 ) ) ).

thf(conj_thm_2Egcd_2ELCM__1,axiom,
    ! [V0x: tp__ty_2Enum_2Enum] :
      ( ( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Egcd_2Elcm @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) @ ( inj__ty_2Enum_2Enum @ V0x ) ) )
        = V0x )
      & ( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Egcd_2Elcm @ ( inj__ty_2Enum_2Enum @ V0x ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) )
        = V0x ) ) ).

thf(conj_thm_2Egcd_2ELCM__COMM,axiom,
    ! [V0a: tp__ty_2Enum_2Enum,V1b: tp__ty_2Enum_2Enum] :
      ( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Egcd_2Elcm @ ( inj__ty_2Enum_2Enum @ V0a ) ) @ ( inj__ty_2Enum_2Enum @ V1b ) ) )
      = ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Egcd_2Elcm @ ( inj__ty_2Enum_2Enum @ V1b ) ) @ ( inj__ty_2Enum_2Enum @ V0a ) ) ) ) ).

thf(conj_thm_2Egcd_2ELCM__LE,axiom,
    ! [V0m: 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 @ V0m ) ) )
        & ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) )
     => ( ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ V0m ) ) @ ( ap @ ( ap @ c_2Egcd_2Elcm @ ( inj__ty_2Enum_2Enum @ V0m ) ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) )
        & ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ V0m ) ) @ ( ap @ ( ap @ c_2Egcd_2Elcm @ ( inj__ty_2Enum_2Enum @ V1n ) ) @ ( inj__ty_2Enum_2Enum @ V0m ) ) ) ) ) ) ).

thf(conj_thm_2Egcd_2ELCM__LEAST,axiom,
    ! [V0m: 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 @ V0m ) ) )
        & ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) )
     => ! [V2p: tp__ty_2Enum_2Enum] :
          ( ( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( inj__ty_2Enum_2Enum @ V2p ) ) )
            & ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V2p ) ) @ ( ap @ ( ap @ c_2Egcd_2Elcm @ ( inj__ty_2Enum_2Enum @ V0m ) ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) ) )
         => ( ~ ( p @ ( ap @ ( ap @ c_2Edivides_2Edivides @ ( inj__ty_2Enum_2Enum @ V0m ) ) @ ( inj__ty_2Enum_2Enum @ V2p ) ) )
            | ~ ( p @ ( ap @ ( ap @ c_2Edivides_2Edivides @ ( inj__ty_2Enum_2Enum @ V1n ) ) @ ( inj__ty_2Enum_2Enum @ V2p ) ) ) ) ) ) ).

thf(conj_thm_2Egcd_2EGCD__COMMON__FACTOR,axiom,
    ! [V0m: tp__ty_2Enum_2Enum,V1n: tp__ty_2Enum_2Enum,V2k: tp__ty_2Enum_2Enum] :
      ( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Egcd_2Egcd @ ( ap @ ( ap @ c_2Earithmetic_2E_2A @ ( inj__ty_2Enum_2Enum @ V2k ) ) @ ( inj__ty_2Enum_2Enum @ V0m ) ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2A @ ( inj__ty_2Enum_2Enum @ V2k ) ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) )
      = ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Earithmetic_2E_2A @ ( inj__ty_2Enum_2Enum @ V2k ) ) @ ( ap @ ( ap @ c_2Egcd_2Egcd @ ( inj__ty_2Enum_2Enum @ V0m ) ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) ) ) ).

thf(conj_thm_2Egcd_2EGCD__CANCEL__MULT,axiom,
    ! [V0m: tp__ty_2Enum_2Enum,V1n: tp__ty_2Enum_2Enum,V2k: tp__ty_2Enum_2Enum] :
      ( ( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Egcd_2Egcd @ ( inj__ty_2Enum_2Enum @ V0m ) ) @ ( inj__ty_2Enum_2Enum @ V2k ) ) )
        = ( surj__ty_2Enum_2Enum @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) )
     => ( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Egcd_2Egcd @ ( inj__ty_2Enum_2Enum @ V0m ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2A @ ( inj__ty_2Enum_2Enum @ V2k ) ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) )
        = ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Egcd_2Egcd @ ( inj__ty_2Enum_2Enum @ V0m ) ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) ) ) ).

thf(conj_thm_2Egcd_2EBINARY__GCD,axiom,
    ! [V0m: tp__ty_2Enum_2Enum,V1n: tp__ty_2Enum_2Enum] :
      ( ( ( ( p @ ( ap @ c_2Earithmetic_2EEVEN @ ( inj__ty_2Enum_2Enum @ V0m ) ) )
          & ( p @ ( ap @ c_2Earithmetic_2EEVEN @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) )
       => ( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Egcd_2Egcd @ ( inj__ty_2Enum_2Enum @ V0m ) ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) )
          = ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Earithmetic_2E_2A @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) @ ( ap @ ( ap @ c_2Egcd_2Egcd @ ( ap @ ( ap @ c_2Earithmetic_2EDIV @ ( inj__ty_2Enum_2Enum @ V0m ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( ap @ ( ap @ c_2Earithmetic_2EDIV @ ( inj__ty_2Enum_2Enum @ V1n ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) ) )
      & ( ( ( p @ ( ap @ c_2Earithmetic_2EEVEN @ ( inj__ty_2Enum_2Enum @ V0m ) ) )
          & ( p @ ( ap @ c_2Earithmetic_2EODD @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) )
       => ( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Egcd_2Egcd @ ( inj__ty_2Enum_2Enum @ V0m ) ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) )
          = ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Egcd_2Egcd @ ( ap @ ( ap @ c_2Earithmetic_2EDIV @ ( inj__ty_2Enum_2Enum @ V0m ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) ) ) ) ).

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