ITP001 Axioms: ITP084

ITP001 Axioms: ITP084_7.ax

%------------------------------------------------------------------------------
% File     : ITP084_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    : hrat.ax [Gau19]
%          : HL4084_7.ax [TPAP]

% Status   : Satisfiable
% Syntax   : Number of formulae    :  168 (  64 unt;  72 typ;   0 def)
%            Number of atoms       :  146 (  74 equ)
%            Maximal formula atoms :    4 (   0 avg)
%            Number of connectives :   55 (   5   ~;   7   |;  13   &)
%                                         (  18 <=>;  12  =>;   0  <=;   0 <~>)
%            Maximal formula depth :    7 (   4 avg)
%            Maximal term depth    :    8 (   2 avg)
%            Number of types       :    4 (   3 usr)
%            Number of type conns  :   51 (  33   >;  18   *;   0   +;   0  <<)
%            Number of predicates  :    2 (   1 usr;   0 prp; 1-2 aty)
%            Number of functors    :   68 (  68 usr;  26 con; 0-5 aty)
%            Number of variables   :  225 ( 184   !;  13   ?; 225   :)
%                                         (  28  !>;   0  ?*;   0  @-;   0  @+)
% SPC      : TF1_SAT_EQU_NAR

% Comments :
% Bugfixes : v7.5.0 - Fixes to the axioms.
%------------------------------------------------------------------------------
tff(tyop_2Ehrat_2Ehrat,type,
    tyop_2Ehrat_2Ehrat: $tType ).

tff(tyop_2Emin_2Ebool,type,
    tyop_2Emin_2Ebool: $tType ).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

tff(c_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_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_2Emin_2E_40_2E0,type,
    c_2Emin_2E_40_2E0: 
      !>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),A_27a) ).

tff(c_2Emin_2E_40_2E1,type,
    c_2Emin_2E_40_2E1: 
      !>[A_27a: $tType] : ( tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool) > A_27a ) ).

tff(c_2Ebool_2EF_2E0,type,
    c_2Ebool_2EF_2E0: tyop_2Emin_2Ebool ).

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

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

tff(c_2Equotient_2EQUOTIENT_2E0,type,
    c_2Equotient_2EQUOTIENT_2E0: 
      !>[A_27a: $tType,A_27b: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,A_27a),tyop_2Emin_2Ebool))) ).

tff(c_2Equotient_2EQUOTIENT_2E3,type,
    c_2Equotient_2EQUOTIENT_2E3: 
      !>[A_27a: $tType,A_27b: $tType] : ( ( tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)) * tyop_2Emin_2Efun(A_27a,A_27b) * tyop_2Emin_2Efun(A_27b,A_27a) ) > 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_2Ebool_2ETYPE__DEFINITION_2E0,type,
    c_2Ebool_2ETYPE__DEFINITION_2E0: 
      !>[A_27a: $tType,A_27b: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,A_27a),tyop_2Emin_2Ebool)) ).

tff(c_2Ebool_2ETYPE__DEFINITION_2E2,type,
    c_2Ebool_2ETYPE__DEFINITION_2E2: 
      !>[A_27a: $tType,A_27b: $tType] : ( ( tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool) * tyop_2Emin_2Efun(A_27b,A_27a) ) > tyop_2Emin_2Ebool ) ).

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_2Ehrat_2Ehrat__1_2E0,type,
    c_2Ehrat_2Ehrat__1_2E0: tyop_2Ehrat_2Ehrat ).

tff(c_2Ehrat_2Ehrat__ABS_2E0,type,
    c_2Ehrat_2Ehrat__ABS_2E0: tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum),tyop_2Ehrat_2Ehrat) ).

tff(c_2Ehrat_2Ehrat__ABS_2E1,type,
    c_2Ehrat_2Ehrat__ABS_2E1: tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum) > tyop_2Ehrat_2Ehrat ).

tff(c_2Ehrat_2Ehrat__ABS__CLASS_2E0,type,
    c_2Ehrat_2Ehrat__ABS__CLASS_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum),tyop_2Emin_2Ebool),tyop_2Ehrat_2Ehrat) ).

tff(c_2Ehrat_2Ehrat__ABS__CLASS_2E1,type,
    c_2Ehrat_2Ehrat__ABS__CLASS_2E1: tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum),tyop_2Emin_2Ebool) > tyop_2Ehrat_2Ehrat ).

tff(c_2Ehrat_2Ehrat__REP_2E0,type,
    c_2Ehrat_2Ehrat__REP_2E0: tyop_2Emin_2Efun(tyop_2Ehrat_2Ehrat,tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum)) ).

tff(c_2Ehrat_2Ehrat__REP_2E1,type,
    c_2Ehrat_2Ehrat__REP_2E1: tyop_2Ehrat_2Ehrat > tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum) ).

tff(c_2Ehrat_2Ehrat__REP__CLASS_2E0,type,
    c_2Ehrat_2Ehrat__REP__CLASS_2E0: tyop_2Emin_2Efun(tyop_2Ehrat_2Ehrat,tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum),tyop_2Emin_2Ebool)) ).

tff(c_2Ehrat_2Ehrat__REP__CLASS_2E1,type,
    c_2Ehrat_2Ehrat__REP__CLASS_2E1: tyop_2Ehrat_2Ehrat > tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum),tyop_2Emin_2Ebool) ).

tff(c_2Ehrat_2Ehrat__add_2E0,type,
    c_2Ehrat_2Ehrat__add_2E0: tyop_2Emin_2Efun(tyop_2Ehrat_2Ehrat,tyop_2Emin_2Efun(tyop_2Ehrat_2Ehrat,tyop_2Ehrat_2Ehrat)) ).

tff(c_2Ehrat_2Ehrat__add_2E2,type,
    c_2Ehrat_2Ehrat__add_2E2: ( tyop_2Ehrat_2Ehrat * tyop_2Ehrat_2Ehrat ) > tyop_2Ehrat_2Ehrat ).

tff(c_2Ehrat_2Ehrat__inv_2E0,type,
    c_2Ehrat_2Ehrat__inv_2E0: tyop_2Emin_2Efun(tyop_2Ehrat_2Ehrat,tyop_2Ehrat_2Ehrat) ).

tff(c_2Ehrat_2Ehrat__inv_2E1,type,
    c_2Ehrat_2Ehrat__inv_2E1: tyop_2Ehrat_2Ehrat > tyop_2Ehrat_2Ehrat ).

tff(c_2Ehrat_2Ehrat__mul_2E0,type,
    c_2Ehrat_2Ehrat__mul_2E0: tyop_2Emin_2Efun(tyop_2Ehrat_2Ehrat,tyop_2Emin_2Efun(tyop_2Ehrat_2Ehrat,tyop_2Ehrat_2Ehrat)) ).

tff(c_2Ehrat_2Ehrat__mul_2E2,type,
    c_2Ehrat_2Ehrat__mul_2E2: ( tyop_2Ehrat_2Ehrat * tyop_2Ehrat_2Ehrat ) > tyop_2Ehrat_2Ehrat ).

tff(c_2Ehrat_2Ehrat__sucint_2E0,type,
    c_2Ehrat_2Ehrat__sucint_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Ehrat_2Ehrat) ).

tff(c_2Ehrat_2Ehrat__sucint_2E1,type,
    c_2Ehrat_2Ehrat__sucint_2E1: tyop_2Enum_2Enum > tyop_2Ehrat_2Ehrat ).

tff(c_2Ehrat_2Etrat__1_2E0,type,
    c_2Ehrat_2Etrat__1_2E0: tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum) ).

tff(c_2Ehrat_2Etrat__add_2E0,type,
    c_2Ehrat_2Etrat__add_2E0: tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum),tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum))) ).

tff(c_2Ehrat_2Etrat__add_2E2,type,
    c_2Ehrat_2Etrat__add_2E2: ( tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum) * tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum) ) > tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum) ).

tff(c_2Ehrat_2Etrat__eq_2E0,type,
    c_2Ehrat_2Etrat__eq_2E0: tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum),tyop_2Emin_2Ebool)) ).

tff(c_2Ehrat_2Etrat__eq_2E1,type,
    c_2Ehrat_2Etrat__eq_2E1: tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum) > tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum),tyop_2Emin_2Ebool) ).

tff(c_2Ehrat_2Etrat__eq_2E2,type,
    c_2Ehrat_2Etrat__eq_2E2: ( tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum) * tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum) ) > tyop_2Emin_2Ebool ).

tff(c_2Ehrat_2Etrat__inv_2E0,type,
    c_2Ehrat_2Etrat__inv_2E0: tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum),tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum)) ).

tff(c_2Ehrat_2Etrat__inv_2E1,type,
    c_2Ehrat_2Etrat__inv_2E1: tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum) > tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum) ).

tff(c_2Ehrat_2Etrat__mul_2E0,type,
    c_2Ehrat_2Etrat__mul_2E0: tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum),tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum))) ).

tff(c_2Ehrat_2Etrat__mul_2E2,type,
    c_2Ehrat_2Etrat__mul_2E2: ( tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum) * tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum) ) > tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum) ).

tff(c_2Ehrat_2Etrat__sucint_2E0,type,
    c_2Ehrat_2Etrat__sucint_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum)) ).

tff(c_2Ehrat_2Etrat__sucint_2E1,type,
    c_2Ehrat_2Etrat__sucint_2E1: tyop_2Enum_2Enum > tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum) ).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

tff(arityeq2_2Ec_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_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_2Emin_2E_40_2E1_2Emono_2Etyop_2Epair_2Eprod_28tyop_2Enum_2Enum_2Ctyop_2Enum_2Enum_29,axiom,
    ! [X0_2E0: tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum),tyop_2Emin_2Ebool)] : ( c_2Emin_2E_40_2E1(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum),X0_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum),tyop_2Emin_2Ebool),tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum),c_2Emin_2E_40_2E0(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum)),X0_2E0) ) ).

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

tff(arityeq3_2Ec_2Equotient_2EQUOTIENT_2E3_2Emono_2Etyop_2Epair_2Eprod_28tyop_2Enum_2Enum_2Ctyop_2Enum_2Enum_29_20mono_2Etyop_2Ehrat_2Ehrat,axiom,
    ! [X0_2E0: tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum),tyop_2Emin_2Ebool)),X1_2E0: tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum),tyop_2Ehrat_2Ehrat),X2_2E0: tyop_2Emin_2Efun(tyop_2Ehrat_2Ehrat,tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum))] : ( c_2Equotient_2EQUOTIENT_2E3(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum),tyop_2Ehrat_2Ehrat,X0_2E0,X1_2E0,X2_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Ehrat_2Ehrat,tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum)),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum),tyop_2Ehrat_2Ehrat),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Ehrat_2Ehrat,tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum)),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum),tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum),tyop_2Ehrat_2Ehrat),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Ehrat_2Ehrat,tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum)),tyop_2Emin_2Ebool)),c_2Equotient_2EQUOTIENT_2E0(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum),tyop_2Ehrat_2Ehrat),X0_2E0),X1_2E0),X2_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_2ETYPE__DEFINITION_2E2_2Emono_2Etyop_2Emin_2Efun_28tyop_2Epair_2Eprod_28tyop_2Enum_2Enum_2Ctyop_2Enum_2Enum_29_2Ctyop_2Emin_2Ebool_29_20mono_2Etyop_2Ehrat_2Ehrat,axiom,
    ! [X0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum),tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),X1_2E0: tyop_2Emin_2Efun(tyop_2Ehrat_2Ehrat,tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum),tyop_2Emin_2Ebool))] : ( c_2Ebool_2ETYPE__DEFINITION_2E2(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum),tyop_2Emin_2Ebool),tyop_2Ehrat_2Ehrat,X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Ehrat_2Ehrat,tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum),tyop_2Emin_2Ebool)),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum),tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Ehrat_2Ehrat,tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum),tyop_2Emin_2Ebool)),tyop_2Emin_2Ebool),c_2Ebool_2ETYPE__DEFINITION_2E0(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum),tyop_2Emin_2Ebool),tyop_2Ehrat_2Ehrat),X0_2E0),X1_2E0) ) ).

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

tff(arityeq1_2Ec_2Ehrat_2Ehrat__ABS_2E1,axiom,
    ! [X0_2E0: tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum)] : ( c_2Ehrat_2Ehrat__ABS_2E1(X0_2E0) = app_2E2(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum),tyop_2Ehrat_2Ehrat,c_2Ehrat_2Ehrat__ABS_2E0,X0_2E0) ) ).

tff(arityeq1_2Ec_2Ehrat_2Ehrat__ABS__CLASS_2E1,axiom,
    ! [X0_2E0: tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum),tyop_2Emin_2Ebool)] : ( c_2Ehrat_2Ehrat__ABS__CLASS_2E1(X0_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum),tyop_2Emin_2Ebool),tyop_2Ehrat_2Ehrat,c_2Ehrat_2Ehrat__ABS__CLASS_2E0,X0_2E0) ) ).

tff(arityeq1_2Ec_2Ehrat_2Ehrat__REP_2E1,axiom,
    ! [X0_2E0: tyop_2Ehrat_2Ehrat] : ( c_2Ehrat_2Ehrat__REP_2E1(X0_2E0) = app_2E2(tyop_2Ehrat_2Ehrat,tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum),c_2Ehrat_2Ehrat__REP_2E0,X0_2E0) ) ).

tff(arityeq1_2Ec_2Ehrat_2Ehrat__REP__CLASS_2E1,axiom,
    ! [X0_2E0: tyop_2Ehrat_2Ehrat] : ( c_2Ehrat_2Ehrat__REP__CLASS_2E1(X0_2E0) = app_2E2(tyop_2Ehrat_2Ehrat,tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum),tyop_2Emin_2Ebool),c_2Ehrat_2Ehrat__REP__CLASS_2E0,X0_2E0) ) ).

tff(arityeq2_2Ec_2Ehrat_2Ehrat__add_2E2,axiom,
    ! [X0_2E0: tyop_2Ehrat_2Ehrat,X1_2E0: tyop_2Ehrat_2Ehrat] : ( c_2Ehrat_2Ehrat__add_2E2(X0_2E0,X1_2E0) = app_2E2(tyop_2Ehrat_2Ehrat,tyop_2Ehrat_2Ehrat,app_2E2(tyop_2Ehrat_2Ehrat,tyop_2Emin_2Efun(tyop_2Ehrat_2Ehrat,tyop_2Ehrat_2Ehrat),c_2Ehrat_2Ehrat__add_2E0,X0_2E0),X1_2E0) ) ).

tff(arityeq1_2Ec_2Ehrat_2Ehrat__inv_2E1,axiom,
    ! [X0_2E0: tyop_2Ehrat_2Ehrat] : ( c_2Ehrat_2Ehrat__inv_2E1(X0_2E0) = app_2E2(tyop_2Ehrat_2Ehrat,tyop_2Ehrat_2Ehrat,c_2Ehrat_2Ehrat__inv_2E0,X0_2E0) ) ).

tff(arityeq2_2Ec_2Ehrat_2Ehrat__mul_2E2,axiom,
    ! [X0_2E0: tyop_2Ehrat_2Ehrat,X1_2E0: tyop_2Ehrat_2Ehrat] : ( c_2Ehrat_2Ehrat__mul_2E2(X0_2E0,X1_2E0) = app_2E2(tyop_2Ehrat_2Ehrat,tyop_2Ehrat_2Ehrat,app_2E2(tyop_2Ehrat_2Ehrat,tyop_2Emin_2Efun(tyop_2Ehrat_2Ehrat,tyop_2Ehrat_2Ehrat),c_2Ehrat_2Ehrat__mul_2E0,X0_2E0),X1_2E0) ) ).

tff(arityeq1_2Ec_2Ehrat_2Ehrat__sucint_2E1,axiom,
    ! [X0_2E0: tyop_2Enum_2Enum] : ( c_2Ehrat_2Ehrat__sucint_2E1(X0_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Ehrat_2Ehrat,c_2Ehrat_2Ehrat__sucint_2E0,X0_2E0) ) ).

tff(arityeq2_2Ec_2Ehrat_2Etrat__add_2E2,axiom,
    ! [X0_2E0: tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum),X1_2E0: tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum)] : ( c_2Ehrat_2Etrat__add_2E2(X0_2E0,X1_2E0) = app_2E2(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum),tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum),app_2E2(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum),tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum)),c_2Ehrat_2Etrat__add_2E0,X0_2E0),X1_2E0) ) ).

tff(arityeq1_2Ec_2Ehrat_2Etrat__eq_2E1,axiom,
    ! [X0_2E0: tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum)] : ( c_2Ehrat_2Etrat__eq_2E1(X0_2E0) = app_2E2(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum),tyop_2Emin_2Ebool),c_2Ehrat_2Etrat__eq_2E0,X0_2E0) ) ).

tff(arityeq2_2Ec_2Ehrat_2Etrat__eq_2E2,axiom,
    ! [X0_2E0: tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum),X1_2E0: tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum)] : ( c_2Ehrat_2Etrat__eq_2E2(X0_2E0,X1_2E0) = app_2E2(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum),tyop_2Emin_2Ebool,app_2E2(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum),tyop_2Emin_2Ebool),c_2Ehrat_2Etrat__eq_2E0,X0_2E0),X1_2E0) ) ).

tff(arityeq1_2Ec_2Ehrat_2Etrat__inv_2E1,axiom,
    ! [X0_2E0: tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum)] : ( c_2Ehrat_2Etrat__inv_2E1(X0_2E0) = app_2E2(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum),tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum),c_2Ehrat_2Etrat__inv_2E0,X0_2E0) ) ).

tff(arityeq2_2Ec_2Ehrat_2Etrat__mul_2E2,axiom,
    ! [X0_2E0: tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum),X1_2E0: tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum)] : ( c_2Ehrat_2Etrat__mul_2E2(X0_2E0,X1_2E0) = app_2E2(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum),tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum),app_2E2(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum),tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum)),c_2Ehrat_2Etrat__mul_2E0,X0_2E0),X1_2E0) ) ).

tff(arityeq1_2Ec_2Ehrat_2Etrat__sucint_2E1,axiom,
    ! [X0_2E0: tyop_2Enum_2Enum] : ( c_2Ehrat_2Etrat__sucint_2E1(X0_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum),c_2Ehrat_2Etrat__sucint_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_2Ehrat_2Etrat__1,axiom,
    c_2Ehrat_2Etrat__1_2E0 = c_2Epair_2E_2C_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,c_2Enum_2E0_2E0,c_2Enum_2E0_2E0) ).

tff(thm_2Ehrat_2Etrat__inv,axiom,
    ! [V0x_2E0: tyop_2Enum_2Enum,V1y_2E0: tyop_2Enum_2Enum] : ( c_2Ehrat_2Etrat__inv_2E1(c_2Epair_2E_2C_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,V0x_2E0,V1y_2E0)) = c_2Epair_2E_2C_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,V1y_2E0,V0x_2E0) ) ).

tff(thm_2Ehrat_2Etrat__add,axiom,
    ! [V0x_2E0: tyop_2Enum_2Enum,V1y_2E0: tyop_2Enum_2Enum,V2x_27_2E0: tyop_2Enum_2Enum,V3y_27_2E0: tyop_2Enum_2Enum] : ( c_2Ehrat_2Etrat__add_2E2(c_2Epair_2E_2C_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,V0x_2E0,V1y_2E0),c_2Epair_2E_2C_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,V2x_27_2E0,V3y_27_2E0)) = c_2Epair_2E_2C_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,c_2Eprim__rec_2EPRE_2E1(c_2Earithmetic_2E_2B_2E2(c_2Earithmetic_2E_2A_2E2(c_2Enum_2ESUC_2E1(V0x_2E0),c_2Enum_2ESUC_2E1(V3y_27_2E0)),c_2Earithmetic_2E_2A_2E2(c_2Enum_2ESUC_2E1(V2x_27_2E0),c_2Enum_2ESUC_2E1(V1y_2E0)))),c_2Eprim__rec_2EPRE_2E1(c_2Earithmetic_2E_2A_2E2(c_2Enum_2ESUC_2E1(V1y_2E0),c_2Enum_2ESUC_2E1(V3y_27_2E0)))) ) ).

tff(thm_2Ehrat_2Etrat__mul,axiom,
    ! [V0x_2E0: tyop_2Enum_2Enum,V1y_2E0: tyop_2Enum_2Enum,V2x_27_2E0: tyop_2Enum_2Enum,V3y_27_2E0: tyop_2Enum_2Enum] : ( c_2Ehrat_2Etrat__mul_2E2(c_2Epair_2E_2C_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,V0x_2E0,V1y_2E0),c_2Epair_2E_2C_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,V2x_27_2E0,V3y_27_2E0)) = c_2Epair_2E_2C_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,c_2Eprim__rec_2EPRE_2E1(c_2Earithmetic_2E_2A_2E2(c_2Enum_2ESUC_2E1(V0x_2E0),c_2Enum_2ESUC_2E1(V2x_27_2E0))),c_2Eprim__rec_2EPRE_2E1(c_2Earithmetic_2E_2A_2E2(c_2Enum_2ESUC_2E1(V1y_2E0),c_2Enum_2ESUC_2E1(V3y_27_2E0)))) ) ).

tff(thm_2Ehrat_2Etrat__sucint,axiom,
    ( ( c_2Ehrat_2Etrat__sucint_2E1(c_2Enum_2E0_2E0) = c_2Ehrat_2Etrat__1_2E0 )
    & ! [V0n_2E0: tyop_2Enum_2Enum] : ( c_2Ehrat_2Etrat__sucint_2E1(c_2Enum_2ESUC_2E1(V0n_2E0)) = c_2Ehrat_2Etrat__add_2E2(c_2Ehrat_2Etrat__sucint_2E1(V0n_2E0),c_2Ehrat_2Etrat__1_2E0) ) ) ).

tff(thm_2Ehrat_2Etrat__eq,axiom,
    ! [V0x_2E0: tyop_2Enum_2Enum,V1y_2E0: tyop_2Enum_2Enum,V2x_27_2E0: tyop_2Enum_2Enum,V3y_27_2E0: tyop_2Enum_2Enum] :
      ( p(c_2Ehrat_2Etrat__eq_2E2(c_2Epair_2E_2C_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,V0x_2E0,V1y_2E0),c_2Epair_2E_2C_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,V2x_27_2E0,V3y_27_2E0)))
    <=> ( c_2Earithmetic_2E_2A_2E2(c_2Enum_2ESUC_2E1(V0x_2E0),c_2Enum_2ESUC_2E1(V3y_27_2E0)) = c_2Earithmetic_2E_2A_2E2(c_2Enum_2ESUC_2E1(V2x_27_2E0),c_2Enum_2ESUC_2E1(V1y_2E0)) ) ) ).

tff(thm_2Ehrat_2Ehrat__TY__DEF,axiom,
    ! [F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum),tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)] :
      ( ! [V1c_2E0: tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum),tyop_2Emin_2Ebool)] :
          ( p(app_2E2(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum),tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,F0_2E0,V1c_2E0))
        <=> ? [V2r_2E0: tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum)] :
              ( p(c_2Ehrat_2Etrat__eq_2E2(V2r_2E0,V2r_2E0))
              & ( V1c_2E0 = c_2Ehrat_2Etrat__eq_2E1(V2r_2E0) ) ) )
     => ? [V0rep_2E0: tyop_2Emin_2Efun(tyop_2Ehrat_2Ehrat,tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum),tyop_2Emin_2Ebool))] : p(c_2Ebool_2ETYPE__DEFINITION_2E2(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum),tyop_2Emin_2Ebool),tyop_2Ehrat_2Ehrat,F0_2E0,V0rep_2E0)) ) ).

tff(thm_2Ehrat_2Ehrat__bijections,axiom,
    ( ! [V0a_2E0: tyop_2Ehrat_2Ehrat] : ( c_2Ehrat_2Ehrat__ABS__CLASS_2E1(c_2Ehrat_2Ehrat__REP__CLASS_2E1(V0a_2E0)) = V0a_2E0 )
    & ! [V1r_2E0: tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum),tyop_2Emin_2Ebool)] :
        ( ? [V2r_2E0: tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum)] :
            ( p(c_2Ehrat_2Etrat__eq_2E2(V2r_2E0,V2r_2E0))
            & ( V1r_2E0 = c_2Ehrat_2Etrat__eq_2E1(V2r_2E0) ) )
      <=> ( c_2Ehrat_2Ehrat__REP__CLASS_2E1(c_2Ehrat_2Ehrat__ABS__CLASS_2E1(V1r_2E0)) = V1r_2E0 ) ) ) ).

tff(thm_2Ehrat_2Ehrat__REP__def,axiom,
    ! [V0a_2E0: tyop_2Ehrat_2Ehrat] : ( c_2Ehrat_2Ehrat__REP_2E1(V0a_2E0) = c_2Emin_2E_40_2E1(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum),c_2Ehrat_2Ehrat__REP__CLASS_2E1(V0a_2E0)) ) ).

tff(thm_2Ehrat_2Ehrat__ABS__def,axiom,
    ! [V0r_2E0: tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum)] : ( c_2Ehrat_2Ehrat__ABS_2E1(V0r_2E0) = c_2Ehrat_2Ehrat__ABS__CLASS_2E1(c_2Ehrat_2Etrat__eq_2E1(V0r_2E0)) ) ).

tff(thm_2Ehrat_2Ehrat__1,axiom,
    c_2Ehrat_2Ehrat__1_2E0 = c_2Ehrat_2Ehrat__ABS_2E1(c_2Ehrat_2Etrat__1_2E0) ).

tff(thm_2Ehrat_2Ehrat__inv,axiom,
    ! [V0T1_2E0: tyop_2Ehrat_2Ehrat] : ( c_2Ehrat_2Ehrat__inv_2E1(V0T1_2E0) = c_2Ehrat_2Ehrat__ABS_2E1(c_2Ehrat_2Etrat__inv_2E1(c_2Ehrat_2Ehrat__REP_2E1(V0T1_2E0))) ) ).

tff(thm_2Ehrat_2Ehrat__add,axiom,
    ! [V0T1_2E0: tyop_2Ehrat_2Ehrat,V1T2_2E0: tyop_2Ehrat_2Ehrat] : ( c_2Ehrat_2Ehrat__add_2E2(V0T1_2E0,V1T2_2E0) = c_2Ehrat_2Ehrat__ABS_2E1(c_2Ehrat_2Etrat__add_2E2(c_2Ehrat_2Ehrat__REP_2E1(V0T1_2E0),c_2Ehrat_2Ehrat__REP_2E1(V1T2_2E0))) ) ).

tff(thm_2Ehrat_2Ehrat__mul,axiom,
    ! [V0T1_2E0: tyop_2Ehrat_2Ehrat,V1T2_2E0: tyop_2Ehrat_2Ehrat] : ( c_2Ehrat_2Ehrat__mul_2E2(V0T1_2E0,V1T2_2E0) = c_2Ehrat_2Ehrat__ABS_2E1(c_2Ehrat_2Etrat__mul_2E2(c_2Ehrat_2Ehrat__REP_2E1(V0T1_2E0),c_2Ehrat_2Ehrat__REP_2E1(V1T2_2E0))) ) ).

tff(thm_2Ehrat_2Ehrat__sucint,axiom,
    ! [V0T1_2E0: tyop_2Enum_2Enum] : ( c_2Ehrat_2Ehrat__sucint_2E1(V0T1_2E0) = c_2Ehrat_2Ehrat__ABS_2E1(c_2Ehrat_2Etrat__sucint_2E1(V0T1_2E0)) ) ).

tff(thm_2Ehrat_2ETRAT__EQ__REFL,axiom,
    ! [V0p_2E0: tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum)] : p(c_2Ehrat_2Etrat__eq_2E2(V0p_2E0,V0p_2E0)) ).

tff(thm_2Ehrat_2ETRAT__EQ__SYM,axiom,
    ! [V0p_2E0: tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum),V1q_2E0: tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum)] : ( c_2Ehrat_2Etrat__eq_2E2(V0p_2E0,V1q_2E0) = c_2Ehrat_2Etrat__eq_2E2(V1q_2E0,V0p_2E0) ) ).

tff(thm_2Ehrat_2ETRAT__EQ__TRANS,axiom,
    ! [V0p_2E0: tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum),V1q_2E0: tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum),V2r_2E0: tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum)] :
      ( ( p(c_2Ehrat_2Etrat__eq_2E2(V0p_2E0,V1q_2E0))
        & p(c_2Ehrat_2Etrat__eq_2E2(V1q_2E0,V2r_2E0)) )
     => p(c_2Ehrat_2Etrat__eq_2E2(V0p_2E0,V2r_2E0)) ) ).

tff(thm_2Ehrat_2ETRAT__EQ__AP,axiom,
    ! [V0p_2E0: tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum),V1q_2E0: tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum)] :
      ( ( V0p_2E0 = V1q_2E0 )
     => p(c_2Ehrat_2Etrat__eq_2E2(V0p_2E0,V1q_2E0)) ) ).

tff(thm_2Ehrat_2ETRAT__ADD__SYM__EQ,axiom,
    ! [V0h_2E0: tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum),V1i_2E0: tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum)] : ( c_2Ehrat_2Etrat__add_2E2(V0h_2E0,V1i_2E0) = c_2Ehrat_2Etrat__add_2E2(V1i_2E0,V0h_2E0) ) ).

tff(thm_2Ehrat_2ETRAT__MUL__SYM__EQ,axiom,
    ! [V0h_2E0: tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum),V1i_2E0: tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum)] : ( c_2Ehrat_2Etrat__mul_2E2(V0h_2E0,V1i_2E0) = c_2Ehrat_2Etrat__mul_2E2(V1i_2E0,V0h_2E0) ) ).

tff(thm_2Ehrat_2ETRAT__INV__WELLDEFINED,axiom,
    ! [V0p_2E0: tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum),V1q_2E0: tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum)] :
      ( p(c_2Ehrat_2Etrat__eq_2E2(V0p_2E0,V1q_2E0))
     => p(c_2Ehrat_2Etrat__eq_2E2(c_2Ehrat_2Etrat__inv_2E1(V0p_2E0),c_2Ehrat_2Etrat__inv_2E1(V1q_2E0))) ) ).

tff(thm_2Ehrat_2ETRAT__ADD__WELLDEFINED,axiom,
    ! [V0p_2E0: tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum),V1q_2E0: tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum),V2r_2E0: tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum)] :
      ( p(c_2Ehrat_2Etrat__eq_2E2(V0p_2E0,V1q_2E0))
     => p(c_2Ehrat_2Etrat__eq_2E2(c_2Ehrat_2Etrat__add_2E2(V0p_2E0,V2r_2E0),c_2Ehrat_2Etrat__add_2E2(V1q_2E0,V2r_2E0))) ) ).

tff(thm_2Ehrat_2ETRAT__ADD__WELLDEFINED2,axiom,
    ! [V0p1_2E0: tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum),V1p2_2E0: tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum),V2q1_2E0: tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum),V3q2_2E0: tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum)] :
      ( ( p(c_2Ehrat_2Etrat__eq_2E2(V0p1_2E0,V1p2_2E0))
        & p(c_2Ehrat_2Etrat__eq_2E2(V2q1_2E0,V3q2_2E0)) )
     => p(c_2Ehrat_2Etrat__eq_2E2(c_2Ehrat_2Etrat__add_2E2(V0p1_2E0,V2q1_2E0),c_2Ehrat_2Etrat__add_2E2(V1p2_2E0,V3q2_2E0))) ) ).

tff(thm_2Ehrat_2ETRAT__MUL__WELLDEFINED,axiom,
    ! [V0p_2E0: tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum),V1q_2E0: tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum),V2r_2E0: tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum)] :
      ( p(c_2Ehrat_2Etrat__eq_2E2(V0p_2E0,V1q_2E0))
     => p(c_2Ehrat_2Etrat__eq_2E2(c_2Ehrat_2Etrat__mul_2E2(V0p_2E0,V2r_2E0),c_2Ehrat_2Etrat__mul_2E2(V1q_2E0,V2r_2E0))) ) ).

tff(thm_2Ehrat_2ETRAT__MUL__WELLDEFINED2,axiom,
    ! [V0p1_2E0: tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum),V1p2_2E0: tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum),V2q1_2E0: tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum),V3q2_2E0: tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum)] :
      ( ( p(c_2Ehrat_2Etrat__eq_2E2(V0p1_2E0,V1p2_2E0))
        & p(c_2Ehrat_2Etrat__eq_2E2(V2q1_2E0,V3q2_2E0)) )
     => p(c_2Ehrat_2Etrat__eq_2E2(c_2Ehrat_2Etrat__mul_2E2(V0p1_2E0,V2q1_2E0),c_2Ehrat_2Etrat__mul_2E2(V1p2_2E0,V3q2_2E0))) ) ).

tff(thm_2Ehrat_2ETRAT__ADD__SYM,axiom,
    ! [V0h_2E0: tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum),V1i_2E0: tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum)] : p(c_2Ehrat_2Etrat__eq_2E2(c_2Ehrat_2Etrat__add_2E2(V0h_2E0,V1i_2E0),c_2Ehrat_2Etrat__add_2E2(V1i_2E0,V0h_2E0))) ).

tff(thm_2Ehrat_2ETRAT__ADD__ASSOC,axiom,
    ! [V0h_2E0: tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum),V1i_2E0: tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum),V2j_2E0: tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum)] : p(c_2Ehrat_2Etrat__eq_2E2(c_2Ehrat_2Etrat__add_2E2(V0h_2E0,c_2Ehrat_2Etrat__add_2E2(V1i_2E0,V2j_2E0)),c_2Ehrat_2Etrat__add_2E2(c_2Ehrat_2Etrat__add_2E2(V0h_2E0,V1i_2E0),V2j_2E0))) ).

tff(thm_2Ehrat_2ETRAT__MUL__SYM,axiom,
    ! [V0h_2E0: tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum),V1i_2E0: tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum)] : p(c_2Ehrat_2Etrat__eq_2E2(c_2Ehrat_2Etrat__mul_2E2(V0h_2E0,V1i_2E0),c_2Ehrat_2Etrat__mul_2E2(V1i_2E0,V0h_2E0))) ).

tff(thm_2Ehrat_2ETRAT__MUL__ASSOC,axiom,
    ! [V0h_2E0: tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum),V1i_2E0: tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum),V2j_2E0: tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum)] : p(c_2Ehrat_2Etrat__eq_2E2(c_2Ehrat_2Etrat__mul_2E2(V0h_2E0,c_2Ehrat_2Etrat__mul_2E2(V1i_2E0,V2j_2E0)),c_2Ehrat_2Etrat__mul_2E2(c_2Ehrat_2Etrat__mul_2E2(V0h_2E0,V1i_2E0),V2j_2E0))) ).

tff(thm_2Ehrat_2ETRAT__LDISTRIB,axiom,
    ! [V0h_2E0: tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum),V1i_2E0: tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum),V2j_2E0: tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum)] : p(c_2Ehrat_2Etrat__eq_2E2(c_2Ehrat_2Etrat__mul_2E2(V0h_2E0,c_2Ehrat_2Etrat__add_2E2(V1i_2E0,V2j_2E0)),c_2Ehrat_2Etrat__add_2E2(c_2Ehrat_2Etrat__mul_2E2(V0h_2E0,V1i_2E0),c_2Ehrat_2Etrat__mul_2E2(V0h_2E0,V2j_2E0)))) ).

tff(thm_2Ehrat_2ETRAT__MUL__LID,axiom,
    ! [V0h_2E0: tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum)] : p(c_2Ehrat_2Etrat__eq_2E2(c_2Ehrat_2Etrat__mul_2E2(c_2Ehrat_2Etrat__1_2E0,V0h_2E0),V0h_2E0)) ).

tff(thm_2Ehrat_2ETRAT__MUL__LINV,axiom,
    ! [V0h_2E0: tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum)] : p(c_2Ehrat_2Etrat__eq_2E2(c_2Ehrat_2Etrat__mul_2E2(c_2Ehrat_2Etrat__inv_2E1(V0h_2E0),V0h_2E0),c_2Ehrat_2Etrat__1_2E0)) ).

tff(thm_2Ehrat_2ETRAT__NOZERO,axiom,
    ! [V0h_2E0: tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum),V1i_2E0: tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum)] : ~ p(c_2Ehrat_2Etrat__eq_2E2(c_2Ehrat_2Etrat__add_2E2(V0h_2E0,V1i_2E0),V0h_2E0)) ).

tff(thm_2Ehrat_2ETRAT__ADD__TOTAL,axiom,
    ! [V0h_2E0: tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum),V1i_2E0: tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum)] :
      ( p(c_2Ehrat_2Etrat__eq_2E2(V0h_2E0,V1i_2E0))
      | ? [V2d_2E0: tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum)] : p(c_2Ehrat_2Etrat__eq_2E2(V0h_2E0,c_2Ehrat_2Etrat__add_2E2(V1i_2E0,V2d_2E0)))
      | ? [V3d_2E0: tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum)] : p(c_2Ehrat_2Etrat__eq_2E2(V1i_2E0,c_2Ehrat_2Etrat__add_2E2(V0h_2E0,V3d_2E0))) ) ).

tff(thm_2Ehrat_2ETRAT__SUCINT__0,axiom,
    ! [V0n_2E0: tyop_2Enum_2Enum] : p(c_2Ehrat_2Etrat__eq_2E2(c_2Ehrat_2Etrat__sucint_2E1(V0n_2E0),c_2Epair_2E_2C_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,V0n_2E0,c_2Enum_2E0_2E0))) ).

tff(thm_2Ehrat_2ETRAT__ARCH,axiom,
    ! [V0h_2E0: tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum)] :
    ? [V1n_2E0: tyop_2Enum_2Enum,V2d_2E0: tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum)] : p(c_2Ehrat_2Etrat__eq_2E2(c_2Ehrat_2Etrat__sucint_2E1(V1n_2E0),c_2Ehrat_2Etrat__add_2E2(V0h_2E0,V2d_2E0))) ).

tff(thm_2Ehrat_2ETRAT__SUCINT,axiom,
    ( p(c_2Ehrat_2Etrat__eq_2E2(c_2Ehrat_2Etrat__sucint_2E1(c_2Enum_2E0_2E0),c_2Ehrat_2Etrat__1_2E0))
    & ! [V0n_2E0: tyop_2Enum_2Enum] : p(c_2Ehrat_2Etrat__eq_2E2(c_2Ehrat_2Etrat__sucint_2E1(c_2Enum_2ESUC_2E1(V0n_2E0)),c_2Ehrat_2Etrat__add_2E2(c_2Ehrat_2Etrat__sucint_2E1(V0n_2E0),c_2Ehrat_2Etrat__1_2E0))) ) ).

tff(thm_2Ehrat_2ETRAT__EQ__EQUIV,axiom,
    ! [V0p_2E0: tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum),V1q_2E0: tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum)] :
      ( p(c_2Ehrat_2Etrat__eq_2E2(V0p_2E0,V1q_2E0))
    <=> ( c_2Ehrat_2Etrat__eq_2E1(V0p_2E0) = c_2Ehrat_2Etrat__eq_2E1(V1q_2E0) ) ) ).

tff(thm_2Ehrat_2Ehrat__ABS__REP__CLASS,axiom,
    ( ! [V0a_2E0: tyop_2Ehrat_2Ehrat] : ( c_2Ehrat_2Ehrat__ABS__CLASS_2E1(c_2Ehrat_2Ehrat__REP__CLASS_2E1(V0a_2E0)) = V0a_2E0 )
    & ! [V1c_2E0: tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum),tyop_2Emin_2Ebool)] :
        ( ? [V2r_2E0: tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum)] :
            ( p(c_2Ehrat_2Etrat__eq_2E2(V2r_2E0,V2r_2E0))
            & ( V1c_2E0 = c_2Ehrat_2Etrat__eq_2E1(V2r_2E0) ) )
      <=> ( c_2Ehrat_2Ehrat__REP__CLASS_2E1(c_2Ehrat_2Ehrat__ABS__CLASS_2E1(V1c_2E0)) = V1c_2E0 ) ) ) ).

tff(thm_2Ehrat_2Ehrat__QUOTIENT,axiom,
    p(c_2Equotient_2EQUOTIENT_2E3(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum),tyop_2Ehrat_2Ehrat,c_2Ehrat_2Etrat__eq_2E0,c_2Ehrat_2Ehrat__ABS_2E0,c_2Ehrat_2Ehrat__REP_2E0)) ).

tff(thm_2Ehrat_2EHRAT__ADD__SYM,axiom,
    ! [V0h_2E0: tyop_2Ehrat_2Ehrat,V1i_2E0: tyop_2Ehrat_2Ehrat] : ( c_2Ehrat_2Ehrat__add_2E2(V0h_2E0,V1i_2E0) = c_2Ehrat_2Ehrat__add_2E2(V1i_2E0,V0h_2E0) ) ).

tff(thm_2Ehrat_2EHRAT__ADD__ASSOC,axiom,
    ! [V0h_2E0: tyop_2Ehrat_2Ehrat,V1i_2E0: tyop_2Ehrat_2Ehrat,V2j_2E0: tyop_2Ehrat_2Ehrat] : ( c_2Ehrat_2Ehrat__add_2E2(V0h_2E0,c_2Ehrat_2Ehrat__add_2E2(V1i_2E0,V2j_2E0)) = c_2Ehrat_2Ehrat__add_2E2(c_2Ehrat_2Ehrat__add_2E2(V0h_2E0,V1i_2E0),V2j_2E0) ) ).

tff(thm_2Ehrat_2EHRAT__MUL__SYM,axiom,
    ! [V0h_2E0: tyop_2Ehrat_2Ehrat,V1i_2E0: tyop_2Ehrat_2Ehrat] : ( c_2Ehrat_2Ehrat__mul_2E2(V0h_2E0,V1i_2E0) = c_2Ehrat_2Ehrat__mul_2E2(V1i_2E0,V0h_2E0) ) ).

tff(thm_2Ehrat_2EHRAT__MUL__ASSOC,axiom,
    ! [V0h_2E0: tyop_2Ehrat_2Ehrat,V1i_2E0: tyop_2Ehrat_2Ehrat,V2j_2E0: tyop_2Ehrat_2Ehrat] : ( c_2Ehrat_2Ehrat__mul_2E2(V0h_2E0,c_2Ehrat_2Ehrat__mul_2E2(V1i_2E0,V2j_2E0)) = c_2Ehrat_2Ehrat__mul_2E2(c_2Ehrat_2Ehrat__mul_2E2(V0h_2E0,V1i_2E0),V2j_2E0) ) ).

tff(thm_2Ehrat_2EHRAT__LDISTRIB,axiom,
    ! [V0h_2E0: tyop_2Ehrat_2Ehrat,V1i_2E0: tyop_2Ehrat_2Ehrat,V2j_2E0: tyop_2Ehrat_2Ehrat] : ( c_2Ehrat_2Ehrat__mul_2E2(V0h_2E0,c_2Ehrat_2Ehrat__add_2E2(V1i_2E0,V2j_2E0)) = c_2Ehrat_2Ehrat__add_2E2(c_2Ehrat_2Ehrat__mul_2E2(V0h_2E0,V1i_2E0),c_2Ehrat_2Ehrat__mul_2E2(V0h_2E0,V2j_2E0)) ) ).

tff(thm_2Ehrat_2EHRAT__MUL__LID,axiom,
    ! [V0h_2E0: tyop_2Ehrat_2Ehrat] : ( c_2Ehrat_2Ehrat__mul_2E2(c_2Ehrat_2Ehrat__1_2E0,V0h_2E0) = V0h_2E0 ) ).

tff(thm_2Ehrat_2EHRAT__MUL__LINV,axiom,
    ! [V0h_2E0: tyop_2Ehrat_2Ehrat] : ( c_2Ehrat_2Ehrat__mul_2E2(c_2Ehrat_2Ehrat__inv_2E1(V0h_2E0),V0h_2E0) = c_2Ehrat_2Ehrat__1_2E0 ) ).

tff(thm_2Ehrat_2EHRAT__NOZERO,axiom,
    ! [V0h_2E0: tyop_2Ehrat_2Ehrat,V1i_2E0: tyop_2Ehrat_2Ehrat] : ( c_2Ehrat_2Ehrat__add_2E2(V0h_2E0,V1i_2E0) != V0h_2E0 ) ).

tff(thm_2Ehrat_2EHRAT__ADD__TOTAL,axiom,
    ! [V0h_2E0: tyop_2Ehrat_2Ehrat,V1i_2E0: tyop_2Ehrat_2Ehrat] :
      ( ( V0h_2E0 = V1i_2E0 )
      | ? [V2d_2E0: tyop_2Ehrat_2Ehrat] : ( V0h_2E0 = c_2Ehrat_2Ehrat__add_2E2(V1i_2E0,V2d_2E0) )
      | ? [V3d_2E0: tyop_2Ehrat_2Ehrat] : ( V1i_2E0 = c_2Ehrat_2Ehrat__add_2E2(V0h_2E0,V3d_2E0) ) ) ).

tff(thm_2Ehrat_2EHRAT__ARCH,axiom,
    ! [V0h_2E0: tyop_2Ehrat_2Ehrat] :
    ? [V1n_2E0: tyop_2Enum_2Enum,V2d_2E0: tyop_2Ehrat_2Ehrat] : ( c_2Ehrat_2Ehrat__sucint_2E1(V1n_2E0) = c_2Ehrat_2Ehrat__add_2E2(V0h_2E0,V2d_2E0) ) ).

tff(thm_2Ehrat_2EHRAT__SUCINT,axiom,
    ( ( c_2Ehrat_2Ehrat__sucint_2E1(c_2Enum_2E0_2E0) = c_2Ehrat_2Ehrat__1_2E0 )
    & ! [V0n_2E0: tyop_2Enum_2Enum] : ( c_2Ehrat_2Ehrat__sucint_2E1(c_2Enum_2ESUC_2E1(V0n_2E0)) = c_2Ehrat_2Ehrat__add_2E2(c_2Ehrat_2Ehrat__sucint_2E1(V0n_2E0),c_2Ehrat_2Ehrat__1_2E0) ) ) ).

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