ITP001 Axioms: ITP023_7.ax
%------------------------------------------------------------------------------
% File : ITP023_7 : TPTP v8.2.0. 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 : gcd.ax [Gau19]
% : HL4023_7.ax [TPAP]
% Status : Satisfiable
% Syntax : Number of formulae : 146 ( 39 unt; 64 typ; 0 def)
% Number of atoms : 186 ( 82 equ)
% Maximal formula atoms : 9 ( 1 avg)
% Number of connectives : 113 ( 9 ~; 8 |; 48 &)
% ( 17 <=>; 31 =>; 0 <=; 0 <~>)
% Maximal formula depth : 11 ( 4 avg)
% Maximal term depth : 7 ( 2 avg)
% Number of types : 3 ( 2 usr)
% Number of type conns : 49 ( 29 >; 20 *; 0 +; 0 <<)
% Number of predicates : 2 ( 1 usr; 0 prp; 1-2 aty)
% Number of functors : 61 ( 61 usr; 26 con; 0-4 aty)
% Number of variables : 222 ( 201 !; 5 ?; 222 :)
% ( 16 !>; 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(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_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_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_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_2Ebool_2EF_2E0,type,
c_2Ebool_2EF_2E0: tyop_2Emin_2Ebool ).
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_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_2Edivides_2Edivides_2E0,type,
c_2Edivides_2Edivides_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)) ).
tff(c_2Edivides_2Edivides_2E2,type,
c_2Edivides_2Edivides_2E2: ( tyop_2Enum_2Enum * tyop_2Enum_2Enum ) > tyop_2Emin_2Ebool ).
tff(c_2Egcd_2Egcd_2E0,type,
c_2Egcd_2Egcd_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum)) ).
tff(c_2Egcd_2Egcd_2E2,type,
c_2Egcd_2Egcd_2E2: ( tyop_2Enum_2Enum * tyop_2Enum_2Enum ) > tyop_2Enum_2Enum ).
tff(c_2Egcd_2Eis__gcd_2E0,type,
c_2Egcd_2Eis__gcd_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool))) ).
tff(c_2Egcd_2Eis__gcd_2E3,type,
c_2Egcd_2Eis__gcd_2E3: ( tyop_2Enum_2Enum * tyop_2Enum_2Enum * tyop_2Enum_2Enum ) > tyop_2Emin_2Ebool ).
tff(c_2Egcd_2Elcm_2E0,type,
c_2Egcd_2Elcm_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum)) ).
tff(c_2Egcd_2Elcm_2E2,type,
c_2Egcd_2Elcm_2E2: ( tyop_2Enum_2Enum * tyop_2Enum_2Enum ) > tyop_2Enum_2Enum ).
tff(c_2Edivides_2Eprime_2E0,type,
c_2Edivides_2Eprime_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool) ).
tff(c_2Edivides_2Eprime_2E1,type,
c_2Edivides_2Eprime_2E1: tyop_2Enum_2Enum > tyop_2Emin_2Ebool ).
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_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(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(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_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_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_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(arityeq2_2Ec_2Edivides_2Edivides_2E2,axiom,
! [X0_2E0: tyop_2Enum_2Enum,X1_2E0: tyop_2Enum_2Enum] : ( c_2Edivides_2Edivides_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_2Edivides_2Edivides_2E0,X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Egcd_2Egcd_2E2,axiom,
! [X0_2E0: tyop_2Enum_2Enum,X1_2E0: tyop_2Enum_2Enum] : ( c_2Egcd_2Egcd_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_2Egcd_2Egcd_2E0,X0_2E0),X1_2E0) ) ).
tff(arityeq3_2Ec_2Egcd_2Eis__gcd_2E3,axiom,
! [X0_2E0: tyop_2Enum_2Enum,X1_2E0: tyop_2Enum_2Enum,X2_2E0: tyop_2Enum_2Enum] : ( c_2Egcd_2Eis__gcd_2E3(X0_2E0,X1_2E0,X2_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)),c_2Egcd_2Eis__gcd_2E0,X0_2E0),X1_2E0),X2_2E0) ) ).
tff(arityeq2_2Ec_2Egcd_2Elcm_2E2,axiom,
! [X0_2E0: tyop_2Enum_2Enum,X1_2E0: tyop_2Enum_2Enum] : ( c_2Egcd_2Elcm_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_2Egcd_2Elcm_2E0,X0_2E0),X1_2E0) ) ).
tff(arityeq1_2Ec_2Edivides_2Eprime_2E1,axiom,
! [X0_2E0: tyop_2Enum_2Enum] : ( c_2Edivides_2Eprime_2E1(X0_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,c_2Edivides_2Eprime_2E0,X0_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_2Egcd_2Eis__gcd__def,axiom,
! [V0a_2E0: tyop_2Enum_2Enum,V1b_2E0: tyop_2Enum_2Enum,V2c_2E0: tyop_2Enum_2Enum] :
( p(c_2Egcd_2Eis__gcd_2E3(V0a_2E0,V1b_2E0,V2c_2E0))
<=> ( p(c_2Edivides_2Edivides_2E2(V2c_2E0,V0a_2E0))
& p(c_2Edivides_2Edivides_2E2(V2c_2E0,V1b_2E0))
& ! [V3d_2E0: tyop_2Enum_2Enum] :
( ( p(c_2Edivides_2Edivides_2E2(V3d_2E0,V0a_2E0))
& p(c_2Edivides_2Edivides_2E2(V3d_2E0,V1b_2E0)) )
=> p(c_2Edivides_2Edivides_2E2(V3d_2E0,V2c_2E0)) ) ) ) ).
tff(thm_2Egcd_2Elcm__def,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool))] :
( ! [V0m_2E0: tyop_2Enum_2Enum,V1n_2E0: tyop_2Enum_2Enum] :
( p(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),F0_2E0,V0m_2E0),V1n_2E0))
<=> ( ( V0m_2E0 = c_2Enum_2E0_2E0 )
| ( V1n_2E0 = c_2Enum_2E0_2E0 ) ) )
=> ! [V0m_2E0: tyop_2Enum_2Enum,V1n_2E0: tyop_2Enum_2Enum] : ( c_2Egcd_2Elcm_2E2(V0m_2E0,V1n_2E0) = c_2Ebool_2ECOND_2E3(tyop_2Enum_2Enum,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),F0_2E0,V0m_2E0),V1n_2E0),c_2Enum_2E0_2E0,c_2Earithmetic_2EDIV_2E2(c_2Earithmetic_2E_2A_2E2(V0m_2E0,V1n_2E0),c_2Egcd_2Egcd_2E2(V0m_2E0,V1n_2E0))) ) ) ).
tff(thm_2Egcd_2EIS__GCD__UNIQUE,axiom,
! [V0a_2E0: tyop_2Enum_2Enum,V1b_2E0: tyop_2Enum_2Enum,V2c_2E0: tyop_2Enum_2Enum,V3d_2E0: tyop_2Enum_2Enum] :
( ( p(c_2Egcd_2Eis__gcd_2E3(V0a_2E0,V1b_2E0,V2c_2E0))
& p(c_2Egcd_2Eis__gcd_2E3(V0a_2E0,V1b_2E0,V3d_2E0)) )
=> ( V2c_2E0 = V3d_2E0 ) ) ).
tff(thm_2Egcd_2EIS__GCD__REF,axiom,
! [V0a_2E0: tyop_2Enum_2Enum] : p(c_2Egcd_2Eis__gcd_2E3(V0a_2E0,V0a_2E0,V0a_2E0)) ).
tff(thm_2Egcd_2EIS__GCD__SYM,axiom,
! [V0a_2E0: tyop_2Enum_2Enum,V1b_2E0: tyop_2Enum_2Enum,V2c_2E0: tyop_2Enum_2Enum] : ( c_2Egcd_2Eis__gcd_2E3(V0a_2E0,V1b_2E0,V2c_2E0) = c_2Egcd_2Eis__gcd_2E3(V1b_2E0,V0a_2E0,V2c_2E0) ) ).
tff(thm_2Egcd_2EIS__GCD__0R,axiom,
! [V0a_2E0: tyop_2Enum_2Enum] : p(c_2Egcd_2Eis__gcd_2E3(V0a_2E0,c_2Enum_2E0_2E0,V0a_2E0)) ).
tff(thm_2Egcd_2EIS__GCD__0L,axiom,
! [V0a_2E0: tyop_2Enum_2Enum] : p(c_2Egcd_2Eis__gcd_2E3(c_2Enum_2E0_2E0,V0a_2E0,V0a_2E0)) ).
tff(thm_2Egcd_2EPRIME__IS__GCD,axiom,
! [V0p_2E0: tyop_2Enum_2Enum,V1b_2E0: tyop_2Enum_2Enum] :
( p(c_2Edivides_2Eprime_2E1(V0p_2E0))
=> ( p(c_2Edivides_2Edivides_2E2(V0p_2E0,V1b_2E0))
| p(c_2Egcd_2Eis__gcd_2E3(V0p_2E0,V1b_2E0,c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)))) ) ) ).
tff(thm_2Egcd_2EIS__GCD__MINUS__L,axiom,
! [V0a_2E0: tyop_2Enum_2Enum,V1b_2E0: tyop_2Enum_2Enum,V2c_2E0: tyop_2Enum_2Enum] :
( ( p(c_2Earithmetic_2E_3C_3D_2E2(V1b_2E0,V0a_2E0))
& p(c_2Egcd_2Eis__gcd_2E3(c_2Earithmetic_2E_2D_2E2(V0a_2E0,V1b_2E0),V1b_2E0,V2c_2E0)) )
=> p(c_2Egcd_2Eis__gcd_2E3(V0a_2E0,V1b_2E0,V2c_2E0)) ) ).
tff(thm_2Egcd_2EIS__GCD__MINUS__R,axiom,
! [V0a_2E0: tyop_2Enum_2Enum,V1b_2E0: tyop_2Enum_2Enum,V2c_2E0: tyop_2Enum_2Enum] :
( ( p(c_2Earithmetic_2E_3C_3D_2E2(V0a_2E0,V1b_2E0))
& p(c_2Egcd_2Eis__gcd_2E3(V0a_2E0,c_2Earithmetic_2E_2D_2E2(V1b_2E0,V0a_2E0),V2c_2E0)) )
=> p(c_2Egcd_2Eis__gcd_2E3(V0a_2E0,V1b_2E0,V2c_2E0)) ) ).
tff(thm_2Egcd_2Egcd__ind,axiom,
! [V0P_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool))] :
( ( ! [V1y_2E0: tyop_2Enum_2Enum] : p(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),V0P_2E0,c_2Enum_2E0_2E0),V1y_2E0))
& ! [V2x_2E0: tyop_2Enum_2Enum] : p(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),V0P_2E0,c_2Enum_2ESUC_2E1(V2x_2E0)),c_2Enum_2E0_2E0))
& ! [V3x_2E0: tyop_2Enum_2Enum,V4y_2E0: tyop_2Enum_2Enum] :
( ( ( ~ p(c_2Earithmetic_2E_3C_3D_2E2(V4y_2E0,V3x_2E0))
=> p(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),V0P_2E0,c_2Enum_2ESUC_2E1(V3x_2E0)),c_2Earithmetic_2E_2D_2E2(V4y_2E0,V3x_2E0))) )
& ( p(c_2Earithmetic_2E_3C_3D_2E2(V4y_2E0,V3x_2E0))
=> p(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),V0P_2E0,c_2Earithmetic_2E_2D_2E2(V3x_2E0,V4y_2E0)),c_2Enum_2ESUC_2E1(V4y_2E0))) ) )
=> p(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),V0P_2E0,c_2Enum_2ESUC_2E1(V3x_2E0)),c_2Enum_2ESUC_2E1(V4y_2E0))) ) )
=> ! [V5v_2E0: tyop_2Enum_2Enum,V6v1_2E0: tyop_2Enum_2Enum] : p(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),V0P_2E0,V5v_2E0),V6v1_2E0)) ) ).
tff(thm_2Egcd_2Egcd__def,axiom,
( ! [V0y_2E0: tyop_2Enum_2Enum] : ( c_2Egcd_2Egcd_2E2(c_2Enum_2E0_2E0,V0y_2E0) = V0y_2E0 )
& ! [V1x_2E0: tyop_2Enum_2Enum] : ( c_2Egcd_2Egcd_2E2(c_2Enum_2ESUC_2E1(V1x_2E0),c_2Enum_2E0_2E0) = c_2Enum_2ESUC_2E1(V1x_2E0) )
& ! [V2y_2E0: tyop_2Enum_2Enum,V3x_2E0: tyop_2Enum_2Enum] : ( c_2Egcd_2Egcd_2E2(c_2Enum_2ESUC_2E1(V3x_2E0),c_2Enum_2ESUC_2E1(V2y_2E0)) = c_2Ebool_2ECOND_2E3(tyop_2Enum_2Enum,c_2Earithmetic_2E_3C_3D_2E2(V2y_2E0,V3x_2E0),c_2Egcd_2Egcd_2E2(c_2Earithmetic_2E_2D_2E2(V3x_2E0,V2y_2E0),c_2Enum_2ESUC_2E1(V2y_2E0)),c_2Egcd_2Egcd_2E2(c_2Enum_2ESUC_2E1(V3x_2E0),c_2Earithmetic_2E_2D_2E2(V2y_2E0,V3x_2E0))) ) ) ).
tff(thm_2Egcd_2Egcd__def__compute,axiom,
( ! [V0y_2E0: tyop_2Enum_2Enum] : ( c_2Egcd_2Egcd_2E2(c_2Enum_2E0_2E0,V0y_2E0) = V0y_2E0 )
& ! [V1x_2E0: tyop_2Enum_2Enum] : ( c_2Egcd_2Egcd_2E2(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(V1x_2E0)),c_2Enum_2E0_2E0) = c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(V1x_2E0)) )
& ! [V2x_2E0: tyop_2Enum_2Enum] : ( c_2Egcd_2Egcd_2E2(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(V2x_2E0)),c_2Enum_2E0_2E0) = c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(V2x_2E0)) )
& ! [V3y_2E0: tyop_2Enum_2Enum,V4x_2E0: tyop_2Enum_2Enum] : ( c_2Egcd_2Egcd_2E2(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(V4x_2E0)),c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(V3y_2E0))) = c_2Ebool_2ECOND_2E3(tyop_2Enum_2Enum,c_2Earithmetic_2E_3C_3D_2E2(c_2Earithmetic_2E_2D_2E2(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(V3y_2E0)),c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))),c_2Earithmetic_2E_2D_2E2(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(V4x_2E0)),c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)))),c_2Egcd_2Egcd_2E2(c_2Earithmetic_2E_2D_2E2(c_2Earithmetic_2E_2D_2E2(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(V4x_2E0)),c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))),c_2Earithmetic_2E_2D_2E2(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(V3y_2E0)),c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)))),c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(V3y_2E0))),c_2Egcd_2Egcd_2E2(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(V4x_2E0)),c_2Earithmetic_2E_2D_2E2(c_2Earithmetic_2E_2D_2E2(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(V3y_2E0)),c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))),c_2Earithmetic_2E_2D_2E2(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(V4x_2E0)),c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)))))) )
& ! [V5y_2E0: tyop_2Enum_2Enum,V6x_2E0: tyop_2Enum_2Enum] : ( c_2Egcd_2Egcd_2E2(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(V6x_2E0)),c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(V5y_2E0))) = c_2Ebool_2ECOND_2E3(tyop_2Enum_2Enum,c_2Earithmetic_2E_3C_3D_2E2(c_2Earithmetic_2E_2D_2E2(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(V5y_2E0)),c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))),c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(V6x_2E0))),c_2Egcd_2Egcd_2E2(c_2Earithmetic_2E_2D_2E2(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(V6x_2E0)),c_2Earithmetic_2E_2D_2E2(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(V5y_2E0)),c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)))),c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(V5y_2E0))),c_2Egcd_2Egcd_2E2(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(V6x_2E0)),c_2Earithmetic_2E_2D_2E2(c_2Earithmetic_2E_2D_2E2(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(V5y_2E0)),c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))),c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(V6x_2E0))))) )
& ! [V7y_2E0: tyop_2Enum_2Enum,V8x_2E0: tyop_2Enum_2Enum] : ( c_2Egcd_2Egcd_2E2(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(V8x_2E0)),c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(V7y_2E0))) = c_2Ebool_2ECOND_2E3(tyop_2Enum_2Enum,c_2Earithmetic_2E_3C_3D_2E2(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(V7y_2E0)),c_2Earithmetic_2E_2D_2E2(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(V8x_2E0)),c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)))),c_2Egcd_2Egcd_2E2(c_2Earithmetic_2E_2D_2E2(c_2Earithmetic_2E_2D_2E2(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(V8x_2E0)),c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))),c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(V7y_2E0))),c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(V7y_2E0))),c_2Egcd_2Egcd_2E2(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(V8x_2E0)),c_2Earithmetic_2E_2D_2E2(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(V7y_2E0)),c_2Earithmetic_2E_2D_2E2(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(V8x_2E0)),c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)))))) )
& ! [V9y_2E0: tyop_2Enum_2Enum,V10x_2E0: tyop_2Enum_2Enum] : ( c_2Egcd_2Egcd_2E2(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(V10x_2E0)),c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(V9y_2E0))) = c_2Ebool_2ECOND_2E3(tyop_2Enum_2Enum,c_2Earithmetic_2E_3C_3D_2E2(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(V9y_2E0)),c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(V10x_2E0))),c_2Egcd_2Egcd_2E2(c_2Earithmetic_2E_2D_2E2(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(V10x_2E0)),c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(V9y_2E0))),c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(V9y_2E0))),c_2Egcd_2Egcd_2E2(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(V10x_2E0)),c_2Earithmetic_2E_2D_2E2(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(V9y_2E0)),c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(V10x_2E0))))) ) ) ).
tff(thm_2Egcd_2EGCD__IS__GCD,axiom,
! [V0a_2E0: tyop_2Enum_2Enum,V1b_2E0: tyop_2Enum_2Enum] : p(c_2Egcd_2Eis__gcd_2E3(V0a_2E0,V1b_2E0,c_2Egcd_2Egcd_2E2(V0a_2E0,V1b_2E0))) ).
tff(thm_2Egcd_2EGCD__IS__GREATEST__COMMON__DIVISOR,axiom,
! [V0a_2E0: tyop_2Enum_2Enum,V1b_2E0: tyop_2Enum_2Enum] :
( p(c_2Edivides_2Edivides_2E2(c_2Egcd_2Egcd_2E2(V0a_2E0,V1b_2E0),V0a_2E0))
& p(c_2Edivides_2Edivides_2E2(c_2Egcd_2Egcd_2E2(V0a_2E0,V1b_2E0),V1b_2E0))
& ! [V2d_2E0: tyop_2Enum_2Enum] :
( ( p(c_2Edivides_2Edivides_2E2(V2d_2E0,V0a_2E0))
& p(c_2Edivides_2Edivides_2E2(V2d_2E0,V1b_2E0)) )
=> p(c_2Edivides_2Edivides_2E2(V2d_2E0,c_2Egcd_2Egcd_2E2(V0a_2E0,V1b_2E0))) ) ) ).
tff(thm_2Egcd_2EGCD__REF,axiom,
! [V0a_2E0: tyop_2Enum_2Enum] : ( c_2Egcd_2Egcd_2E2(V0a_2E0,V0a_2E0) = V0a_2E0 ) ).
tff(thm_2Egcd_2EGCD__SYM,axiom,
! [V0a_2E0: tyop_2Enum_2Enum,V1b_2E0: tyop_2Enum_2Enum] : ( c_2Egcd_2Egcd_2E2(V0a_2E0,V1b_2E0) = c_2Egcd_2Egcd_2E2(V1b_2E0,V0a_2E0) ) ).
tff(thm_2Egcd_2EGCD__0R,axiom,
! [V0a_2E0: tyop_2Enum_2Enum] : ( c_2Egcd_2Egcd_2E2(V0a_2E0,c_2Enum_2E0_2E0) = V0a_2E0 ) ).
tff(thm_2Egcd_2EGCD__0L,axiom,
! [V0a_2E0: tyop_2Enum_2Enum] : ( c_2Egcd_2Egcd_2E2(c_2Enum_2E0_2E0,V0a_2E0) = V0a_2E0 ) ).
tff(thm_2Egcd_2EGCD__ADD__R,axiom,
! [V0a_2E0: tyop_2Enum_2Enum,V1b_2E0: tyop_2Enum_2Enum] : ( c_2Egcd_2Egcd_2E2(V0a_2E0,c_2Earithmetic_2E_2B_2E2(V0a_2E0,V1b_2E0)) = c_2Egcd_2Egcd_2E2(V0a_2E0,V1b_2E0) ) ).
tff(thm_2Egcd_2EGCD__ADD__R__THM,axiom,
( ! [V0a_2E0: tyop_2Enum_2Enum,V1b_2E0: tyop_2Enum_2Enum] : ( c_2Egcd_2Egcd_2E2(V0a_2E0,c_2Earithmetic_2E_2B_2E2(V0a_2E0,V1b_2E0)) = c_2Egcd_2Egcd_2E2(V0a_2E0,V1b_2E0) )
& ! [V2a_2E0: tyop_2Enum_2Enum,V3b_2E0: tyop_2Enum_2Enum] : ( c_2Egcd_2Egcd_2E2(V2a_2E0,c_2Earithmetic_2E_2B_2E2(V3b_2E0,V2a_2E0)) = c_2Egcd_2Egcd_2E2(V2a_2E0,V3b_2E0) ) ) ).
tff(thm_2Egcd_2EGCD__ADD__L,axiom,
! [V0a_2E0: tyop_2Enum_2Enum,V1b_2E0: tyop_2Enum_2Enum] : ( c_2Egcd_2Egcd_2E2(c_2Earithmetic_2E_2B_2E2(V0a_2E0,V1b_2E0),V0a_2E0) = c_2Egcd_2Egcd_2E2(V0a_2E0,V1b_2E0) ) ).
tff(thm_2Egcd_2EGCD__ADD__L__THM,axiom,
( ! [V0a_2E0: tyop_2Enum_2Enum,V1b_2E0: tyop_2Enum_2Enum] : ( c_2Egcd_2Egcd_2E2(c_2Earithmetic_2E_2B_2E2(V0a_2E0,V1b_2E0),V0a_2E0) = c_2Egcd_2Egcd_2E2(V0a_2E0,V1b_2E0) )
& ! [V2a_2E0: tyop_2Enum_2Enum,V3b_2E0: tyop_2Enum_2Enum] : ( c_2Egcd_2Egcd_2E2(c_2Earithmetic_2E_2B_2E2(V3b_2E0,V2a_2E0),V2a_2E0) = c_2Egcd_2Egcd_2E2(V2a_2E0,V3b_2E0) ) ) ).
tff(thm_2Egcd_2EGCD__EQ__0,axiom,
! [V0n_2E0: tyop_2Enum_2Enum,V1m_2E0: tyop_2Enum_2Enum] :
( ( c_2Egcd_2Egcd_2E2(V0n_2E0,V1m_2E0) = c_2Enum_2E0_2E0 )
<=> ( ( V0n_2E0 = c_2Enum_2E0_2E0 )
& ( V1m_2E0 = c_2Enum_2E0_2E0 ) ) ) ).
tff(thm_2Egcd_2EGCD__1,axiom,
! [V0x_2E0: tyop_2Enum_2Enum] :
( ( c_2Egcd_2Egcd_2E2(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)),V0x_2E0) = c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)) )
& ( c_2Egcd_2Egcd_2E2(V0x_2E0,c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))) = c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)) ) ) ).
tff(thm_2Egcd_2EPRIME__GCD,axiom,
! [V0p_2E0: tyop_2Enum_2Enum,V1b_2E0: tyop_2Enum_2Enum] :
( p(c_2Edivides_2Eprime_2E1(V0p_2E0))
=> ( p(c_2Edivides_2Edivides_2E2(V0p_2E0,V1b_2E0))
| ( c_2Egcd_2Egcd_2E2(V0p_2E0,V1b_2E0) = c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)) ) ) ) ).
tff(thm_2Egcd_2EL__EUCLIDES,axiom,
! [V0a_2E0: tyop_2Enum_2Enum,V1b_2E0: tyop_2Enum_2Enum,V2c_2E0: tyop_2Enum_2Enum] :
( ( ( c_2Egcd_2Egcd_2E2(V0a_2E0,V1b_2E0) = c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)) )
& p(c_2Edivides_2Edivides_2E2(V1b_2E0,c_2Earithmetic_2E_2A_2E2(V0a_2E0,V2c_2E0))) )
=> p(c_2Edivides_2Edivides_2E2(V1b_2E0,V2c_2E0)) ) ).
tff(thm_2Egcd_2EP__EUCLIDES,axiom,
! [V0p_2E0: tyop_2Enum_2Enum,V1a_2E0: tyop_2Enum_2Enum,V2b_2E0: tyop_2Enum_2Enum] :
( ( p(c_2Edivides_2Eprime_2E1(V0p_2E0))
& p(c_2Edivides_2Edivides_2E2(V0p_2E0,c_2Earithmetic_2E_2A_2E2(V1a_2E0,V2b_2E0))) )
=> ( p(c_2Edivides_2Edivides_2E2(V0p_2E0,V1a_2E0))
| p(c_2Edivides_2Edivides_2E2(V0p_2E0,V2b_2E0)) ) ) ).
tff(thm_2Egcd_2EFACTOR__OUT__GCD,axiom,
! [V0n_2E0: tyop_2Enum_2Enum,V1m_2E0: tyop_2Enum_2Enum] :
( ( ( V0n_2E0 != c_2Enum_2E0_2E0 )
& ( V1m_2E0 != c_2Enum_2E0_2E0 ) )
=> ? [V2p_2E0: tyop_2Enum_2Enum,V3q_2E0: tyop_2Enum_2Enum] :
( ( V0n_2E0 = c_2Earithmetic_2E_2A_2E2(V2p_2E0,c_2Egcd_2Egcd_2E2(V0n_2E0,V1m_2E0)) )
& ( V1m_2E0 = c_2Earithmetic_2E_2A_2E2(V3q_2E0,c_2Egcd_2Egcd_2E2(V0n_2E0,V1m_2E0)) )
& ( c_2Egcd_2Egcd_2E2(V2p_2E0,V3q_2E0) = c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)) ) ) ) ).
tff(thm_2Egcd_2EGCD__SUCfree__ind,axiom,
! [V0P_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool))] :
( ( ! [V1y_2E0: tyop_2Enum_2Enum] : p(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),V0P_2E0,c_2Enum_2E0_2E0),V1y_2E0))
& ! [V2x_2E0: tyop_2Enum_2Enum,V3y_2E0: tyop_2Enum_2Enum] :
( p(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),V0P_2E0,V2x_2E0),V3y_2E0))
=> p(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),V0P_2E0,V3y_2E0),V2x_2E0)) )
& ! [V4x_2E0: tyop_2Enum_2Enum] : p(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),V0P_2E0,V4x_2E0),V4x_2E0))
& ! [V5x_2E0: tyop_2Enum_2Enum,V6y_2E0: tyop_2Enum_2Enum] :
( ( p(c_2Eprim__rec_2E_3C_2E2(c_2Enum_2E0_2E0,V5x_2E0))
& p(c_2Eprim__rec_2E_3C_2E2(c_2Enum_2E0_2E0,V6y_2E0))
& p(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),V0P_2E0,V5x_2E0),V6y_2E0)) )
=> p(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),V0P_2E0,V5x_2E0),c_2Earithmetic_2E_2B_2E2(V5x_2E0,V6y_2E0))) ) )
=> ! [V7m_2E0: tyop_2Enum_2Enum,V8n_2E0: tyop_2Enum_2Enum] : p(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),V0P_2E0,V7m_2E0),V8n_2E0)) ) ).
tff(thm_2Egcd_2ELINEAR__GCD,axiom,
! [V0n_2E0: tyop_2Enum_2Enum,V1m_2E0: tyop_2Enum_2Enum] :
( ( V0n_2E0 != c_2Enum_2E0_2E0 )
=> ? [V2p_2E0: tyop_2Enum_2Enum,V3q_2E0: tyop_2Enum_2Enum] : ( c_2Earithmetic_2E_2A_2E2(V2p_2E0,V0n_2E0) = c_2Earithmetic_2E_2B_2E2(c_2Earithmetic_2E_2A_2E2(V3q_2E0,V1m_2E0),c_2Egcd_2Egcd_2E2(V1m_2E0,V0n_2E0)) ) ) ).
tff(thm_2Egcd_2EGCD__EFFICIENTLY,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)] :
( ! [V0a_2E0: tyop_2Enum_2Enum] :
( p(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,F0_2E0,V0a_2E0))
<=> ( V0a_2E0 = c_2Enum_2E0_2E0 ) )
=> ! [V0a_2E0: tyop_2Enum_2Enum,V1b_2E0: tyop_2Enum_2Enum] : ( c_2Egcd_2Egcd_2E2(V0a_2E0,V1b_2E0) = c_2Ebool_2ECOND_2E3(tyop_2Enum_2Enum,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,F0_2E0,V0a_2E0),V1b_2E0,c_2Egcd_2Egcd_2E2(c_2Earithmetic_2EMOD_2E2(V1b_2E0,V0a_2E0),V0a_2E0)) ) ) ).
tff(thm_2Egcd_2ELCM__IS__LEAST__COMMON__MULTIPLE,axiom,
! [V0n_2E0: tyop_2Enum_2Enum,V1m_2E0: tyop_2Enum_2Enum] :
( p(c_2Edivides_2Edivides_2E2(V1m_2E0,c_2Egcd_2Elcm_2E2(V1m_2E0,V0n_2E0)))
& p(c_2Edivides_2Edivides_2E2(V0n_2E0,c_2Egcd_2Elcm_2E2(V1m_2E0,V0n_2E0)))
& ! [V2p_2E0: tyop_2Enum_2Enum] :
( ( p(c_2Edivides_2Edivides_2E2(V1m_2E0,V2p_2E0))
& p(c_2Edivides_2Edivides_2E2(V0n_2E0,V2p_2E0)) )
=> p(c_2Edivides_2Edivides_2E2(c_2Egcd_2Elcm_2E2(V1m_2E0,V0n_2E0),V2p_2E0)) ) ) ).
tff(thm_2Egcd_2ELCM__0,axiom,
! [V0x_2E0: tyop_2Enum_2Enum] :
( ( c_2Egcd_2Elcm_2E2(c_2Enum_2E0_2E0,V0x_2E0) = c_2Enum_2E0_2E0 )
& ( c_2Egcd_2Elcm_2E2(V0x_2E0,c_2Enum_2E0_2E0) = c_2Enum_2E0_2E0 ) ) ).
tff(thm_2Egcd_2ELCM__1,axiom,
! [V0x_2E0: tyop_2Enum_2Enum] :
( ( c_2Egcd_2Elcm_2E2(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)),V0x_2E0) = V0x_2E0 )
& ( c_2Egcd_2Elcm_2E2(V0x_2E0,c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))) = V0x_2E0 ) ) ).
tff(thm_2Egcd_2ELCM__COMM,axiom,
! [V0b_2E0: tyop_2Enum_2Enum,V1a_2E0: tyop_2Enum_2Enum] : ( c_2Egcd_2Elcm_2E2(V1a_2E0,V0b_2E0) = c_2Egcd_2Elcm_2E2(V0b_2E0,V1a_2E0) ) ).
tff(thm_2Egcd_2ELCM__LE,axiom,
! [V0n_2E0: tyop_2Enum_2Enum,V1m_2E0: tyop_2Enum_2Enum] :
( ( p(c_2Eprim__rec_2E_3C_2E2(c_2Enum_2E0_2E0,V1m_2E0))
& p(c_2Eprim__rec_2E_3C_2E2(c_2Enum_2E0_2E0,V0n_2E0)) )
=> ( p(c_2Earithmetic_2E_3C_3D_2E2(V1m_2E0,c_2Egcd_2Elcm_2E2(V1m_2E0,V0n_2E0)))
& p(c_2Earithmetic_2E_3C_3D_2E2(V1m_2E0,c_2Egcd_2Elcm_2E2(V0n_2E0,V1m_2E0))) ) ) ).
tff(thm_2Egcd_2ELCM__LEAST,axiom,
! [V0n_2E0: tyop_2Enum_2Enum,V1m_2E0: tyop_2Enum_2Enum] :
( ( p(c_2Eprim__rec_2E_3C_2E2(c_2Enum_2E0_2E0,V1m_2E0))
& p(c_2Eprim__rec_2E_3C_2E2(c_2Enum_2E0_2E0,V0n_2E0)) )
=> ! [V2p_2E0: tyop_2Enum_2Enum] :
( ( p(c_2Eprim__rec_2E_3C_2E2(c_2Enum_2E0_2E0,V2p_2E0))
& p(c_2Eprim__rec_2E_3C_2E2(V2p_2E0,c_2Egcd_2Elcm_2E2(V1m_2E0,V0n_2E0))) )
=> ( ~ p(c_2Edivides_2Edivides_2E2(V1m_2E0,V2p_2E0))
| ~ p(c_2Edivides_2Edivides_2E2(V0n_2E0,V2p_2E0)) ) ) ) ).
tff(thm_2Egcd_2EGCD__COMMON__FACTOR,axiom,
! [V0m_2E0: tyop_2Enum_2Enum,V1n_2E0: tyop_2Enum_2Enum,V2k_2E0: tyop_2Enum_2Enum] : ( c_2Egcd_2Egcd_2E2(c_2Earithmetic_2E_2A_2E2(V2k_2E0,V0m_2E0),c_2Earithmetic_2E_2A_2E2(V2k_2E0,V1n_2E0)) = c_2Earithmetic_2E_2A_2E2(V2k_2E0,c_2Egcd_2Egcd_2E2(V0m_2E0,V1n_2E0)) ) ).
tff(thm_2Egcd_2EGCD__CANCEL__MULT,axiom,
! [V0m_2E0: tyop_2Enum_2Enum,V1n_2E0: tyop_2Enum_2Enum,V2k_2E0: tyop_2Enum_2Enum] :
( ( c_2Egcd_2Egcd_2E2(V0m_2E0,V2k_2E0) = c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)) )
=> ( c_2Egcd_2Egcd_2E2(V0m_2E0,c_2Earithmetic_2E_2A_2E2(V2k_2E0,V1n_2E0)) = c_2Egcd_2Egcd_2E2(V0m_2E0,V1n_2E0) ) ) ).
tff(thm_2Egcd_2EBINARY__GCD,axiom,
! [V0m_2E0: tyop_2Enum_2Enum,V1n_2E0: tyop_2Enum_2Enum] :
( ( ( p(c_2Earithmetic_2EEVEN_2E1(V0m_2E0))
& p(c_2Earithmetic_2EEVEN_2E1(V1n_2E0)) )
=> ( c_2Egcd_2Egcd_2E2(V0m_2E0,V1n_2E0) = c_2Earithmetic_2E_2A_2E2(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EZERO_2E0)),c_2Egcd_2Egcd_2E2(c_2Earithmetic_2EDIV_2E2(V0m_2E0,c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EZERO_2E0))),c_2Earithmetic_2EDIV_2E2(V1n_2E0,c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EZERO_2E0))))) ) )
& ( ( p(c_2Earithmetic_2EEVEN_2E1(V0m_2E0))
& p(c_2Earithmetic_2EODD_2E1(V1n_2E0)) )
=> ( c_2Egcd_2Egcd_2E2(V0m_2E0,V1n_2E0) = c_2Egcd_2Egcd_2E2(c_2Earithmetic_2EDIV_2E2(V0m_2E0,c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EZERO_2E0))),V1n_2E0) ) ) ) ).
%------------------------------------------------------------------------------