ITP001 Axioms: ITP023^5.ax
%------------------------------------------------------------------------------
% File : ITP023^5 : TPTP v8.2.0. Bugfixed v7.5.0.
% Domain : Interactive Theorem Proving
% Axioms : HOL4 set theory export, chainy mode
% Version : [BG+19] axioms.
% English :
% Refs : [BG+19] Brown et al. (2019), GRUNGE: A Grand Unified ATP Chall
% : [Gau20] Gauthier (2020), Email to Geoff Sutcliffe
% Source : [BG+19]
% Names : 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 ) ) ) ) ) ) ).
%------------------------------------------------------------------------------