TPTP Problem File: ITP003_7.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : ITP003_7 : TPTP v8.2.0. Bugfixed v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : HOL4 syntactic export of thm_2Earithmetic_2EMOD__2.p, 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 : thm_2Earithmetic_2EMOD__2.p [Gau20]
% : HL401001_7.p [TPAP]
% Status : Theorem
% Rating : 1.00 v7.5.0
% Syntax : Number of formulae : 2833 ( 895 unt; 872 typ; 0 def)
% Number of atoms : 4501 (1767 equ)
% Maximal formula atoms : 27 ( 1 avg)
% Number of connectives : 2783 ( 243 ~; 260 |; 680 &)
% ( 701 <=>; 899 =>; 0 <=; 0 <~>)
% Maximal formula depth : 26 ( 5 avg)
% Maximal term depth : 12 ( 2 avg)
% Number of types : 5 ( 4 usr)
% Number of type conns : 734 ( 423 >; 311 *; 0 +; 0 <<)
% Number of predicates : 2 ( 1 usr; 0 prp; 1-2 aty)
% Number of functors : 415 ( 415 usr; 45 con; 0-7 aty)
% Number of variables : 7687 (6588 !; 198 ?;7687 :)
% ( 901 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TF1_THM_EQU_NAR
% Comments :
% Bugfixes : v7.5.0 - Bugfixes in axioms and export.
%------------------------------------------------------------------------------
include('Axioms/ITP001/ITP002_7.ax').
include('Axioms/ITP001/ITP003_7.ax').
include('Axioms/ITP001/ITP004_7.ax').
include('Axioms/ITP001/ITP005_7.ax').
include('Axioms/ITP001/ITP006_7.ax').
include('Axioms/ITP001/ITP007_7.ax').
include('Axioms/ITP001/ITP008_7.ax').
include('Axioms/ITP001/ITP009_7.ax').
include('Axioms/ITP001/ITP010_7.ax').
include('Axioms/ITP001/ITP011_7.ax').
include('Axioms/ITP001/ITP012_7.ax').
include('Axioms/ITP001/ITP013_7.ax').
include('Axioms/ITP001/ITP014_7.ax').
include('Axioms/ITP001/ITP015_7.ax').
%------------------------------------------------------------------------------
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_2Earithmetic_2E_3E_2E0,type,
c_2Earithmetic_2E_3E_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)) ).
tff(c_2Earithmetic_2E_3E_2E2,type,
c_2Earithmetic_2E_3E_2E2: ( tyop_2Enum_2Enum * tyop_2Enum_2Enum ) > tyop_2Emin_2Ebool ).
tff(c_2Earithmetic_2E_3E_3D_2E0,type,
c_2Earithmetic_2E_3E_3D_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)) ).
tff(c_2Earithmetic_2E_3E_3D_2E2,type,
c_2Earithmetic_2E_3E_3D_2E2: ( tyop_2Enum_2Enum * tyop_2Enum_2Enum ) > tyop_2Emin_2Ebool ).
tff(c_2Ebool_2E_3F_2E0,type,
c_2Ebool_2E_3F_2E0:
!>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool) ).
tff(c_2Ebool_2E_3F_2E1,type,
c_2Ebool_2E_3F_2E1:
!>[A_27a: $tType] : ( tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool) > tyop_2Emin_2Ebool ) ).
tff(c_2Earithmetic_2EABS__DIFF_2E0,type,
c_2Earithmetic_2EABS__DIFF_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum)) ).
tff(c_2Earithmetic_2EABS__DIFF_2E2,type,
c_2Earithmetic_2EABS__DIFF_2E2: ( tyop_2Enum_2Enum * tyop_2Enum_2Enum ) > tyop_2Enum_2Enum ).
tff(c_2Earithmetic_2EBIT1_2E0,type,
c_2Earithmetic_2EBIT1_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum) ).
tff(c_2Earithmetic_2EBIT1_2E1,type,
c_2Earithmetic_2EBIT1_2E1: tyop_2Enum_2Enum > tyop_2Enum_2Enum ).
tff(c_2Earithmetic_2EBIT2_2E0,type,
c_2Earithmetic_2EBIT2_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum) ).
tff(c_2Earithmetic_2EBIT2_2E1,type,
c_2Earithmetic_2EBIT2_2E1: tyop_2Enum_2Enum > tyop_2Enum_2Enum ).
tff(c_2Ebool_2ECOND_2E0,type,
c_2Ebool_2ECOND_2E0:
!>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,A_27a))) ).
tff(c_2Ebool_2ECOND_2E3,type,
c_2Ebool_2ECOND_2E3:
!>[A_27a: $tType] : ( ( tyop_2Emin_2Ebool * A_27a * A_27a ) > A_27a ) ).
tff(c_2Earithmetic_2EDIV_2E0,type,
c_2Earithmetic_2EDIV_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum)) ).
tff(c_2Earithmetic_2EDIV_2E2,type,
c_2Earithmetic_2EDIV_2E2: ( tyop_2Enum_2Enum * tyop_2Enum_2Enum ) > tyop_2Enum_2Enum ).
tff(c_2Earithmetic_2EDIV2_2E0,type,
c_2Earithmetic_2EDIV2_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum) ).
tff(c_2Earithmetic_2EDIV2_2E1,type,
c_2Earithmetic_2EDIV2_2E1: tyop_2Enum_2Enum > tyop_2Enum_2Enum ).
tff(c_2Earithmetic_2EEVEN_2E0,type,
c_2Earithmetic_2EEVEN_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool) ).
tff(c_2Earithmetic_2EEVEN_2E1,type,
c_2Earithmetic_2EEVEN_2E1: tyop_2Enum_2Enum > tyop_2Emin_2Ebool ).
tff(c_2Earithmetic_2EEXP_2E0,type,
c_2Earithmetic_2EEXP_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum)) ).
tff(c_2Earithmetic_2EEXP_2E2,type,
c_2Earithmetic_2EEXP_2E2: ( tyop_2Enum_2Enum * tyop_2Enum_2Enum ) > tyop_2Enum_2Enum ).
tff(c_2Ebool_2EF_2E0,type,
c_2Ebool_2EF_2E0: tyop_2Emin_2Ebool ).
tff(c_2Earithmetic_2EFACT_2E0,type,
c_2Earithmetic_2EFACT_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum) ).
tff(c_2Earithmetic_2EFACT_2E1,type,
c_2Earithmetic_2EFACT_2E1: tyop_2Enum_2Enum > tyop_2Enum_2Enum ).
tff(c_2Earithmetic_2EFUNPOW_2E0,type,
c_2Earithmetic_2EFUNPOW_2E0:
!>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27a),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(A_27a,A_27a))) ).
tff(c_2Earithmetic_2EFUNPOW_2E3,type,
c_2Earithmetic_2EFUNPOW_2E3:
!>[A_27a: $tType] : ( ( tyop_2Emin_2Efun(A_27a,A_27a) * tyop_2Enum_2Enum * A_27a ) > A_27a ) ).
tff(c_2Earithmetic_2EMAX_2E0,type,
c_2Earithmetic_2EMAX_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum)) ).
tff(c_2Earithmetic_2EMAX_2E2,type,
c_2Earithmetic_2EMAX_2E2: ( tyop_2Enum_2Enum * tyop_2Enum_2Enum ) > tyop_2Enum_2Enum ).
tff(c_2Earithmetic_2EMIN_2E0,type,
c_2Earithmetic_2EMIN_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum)) ).
tff(c_2Earithmetic_2EMIN_2E2,type,
c_2Earithmetic_2EMIN_2E2: ( tyop_2Enum_2Enum * tyop_2Enum_2Enum ) > tyop_2Enum_2Enum ).
tff(c_2Earithmetic_2EMOD_2E0,type,
c_2Earithmetic_2EMOD_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum)) ).
tff(c_2Earithmetic_2EMOD_2E2,type,
c_2Earithmetic_2EMOD_2E2: ( tyop_2Enum_2Enum * tyop_2Enum_2Enum ) > tyop_2Enum_2Enum ).
tff(c_2Earithmetic_2ENRC_2E0,type,
c_2Earithmetic_2ENRC_2E0:
!>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)))) ).
tff(c_2Earithmetic_2ENRC_2E4,type,
c_2Earithmetic_2ENRC_2E4:
!>[A_27a: $tType] : ( ( tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)) * tyop_2Enum_2Enum * A_27a * A_27a ) > tyop_2Emin_2Ebool ) ).
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_2Ebool_2EONE__ONE_2E0,type,
c_2Ebool_2EONE__ONE_2E0:
!>[A_27a: $tType,A_27b: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Ebool) ).
tff(c_2Ebool_2EONE__ONE_2E1,type,
c_2Ebool_2EONE__ONE_2E1:
!>[A_27a: $tType,A_27b: $tType] : ( tyop_2Emin_2Efun(A_27a,A_27b) > 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_2Erelation_2ERTC_2E0,type,
c_2Erelation_2ERTC_2E0:
!>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))) ).
tff(c_2Erelation_2ERTC_2E1,type,
c_2Erelation_2ERTC_2E1:
!>[A_27a: $tType] : ( tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)) > tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)) ) ).
tff(c_2Erelation_2ERTC_2E3,type,
c_2Erelation_2ERTC_2E3:
!>[A_27a: $tType] : ( ( tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)) * A_27a * 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_2Erelation_2ETC_2E0,type,
c_2Erelation_2ETC_2E0:
!>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))) ).
tff(c_2Erelation_2ETC_2E3,type,
c_2Erelation_2ETC_2E3:
!>[A_27a: $tType] : ( ( tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)) * A_27a * A_27a ) > 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_2Eprim__rec_2Emeasure_2E0,type,
c_2Eprim__rec_2Emeasure_2E0:
!>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))) ).
tff(c_2Eprim__rec_2Emeasure_2E1,type,
c_2Eprim__rec_2Emeasure_2E1:
!>[A_27a: $tType] : ( tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum) > tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)) ) ).
tff(c_2Earithmetic_2Enat__elim____magic_2E0,type,
c_2Earithmetic_2Enat__elim____magic_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum) ).
tff(c_2Earithmetic_2Enat__elim____magic_2E1,type,
c_2Earithmetic_2Enat__elim____magic_2E1: tyop_2Enum_2Enum > tyop_2Enum_2Enum ).
tff(c_2Earithmetic_2Enum__CASE_2E0,type,
c_2Earithmetic_2Enum__CASE_2E0:
!>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,A_27a),A_27a))) ).
tff(c_2Earithmetic_2Enum__CASE_2E3,type,
c_2Earithmetic_2Enum__CASE_2E3:
!>[A_27a: $tType] : ( ( tyop_2Enum_2Enum * A_27a * tyop_2Emin_2Efun(tyop_2Enum_2Enum,A_27a) ) > A_27a ) ).
tff(c_2Erelation_2Etransitive_2E0,type,
c_2Erelation_2Etransitive_2E0:
!>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),tyop_2Emin_2Ebool) ).
tff(c_2Erelation_2Etransitive_2E1,type,
c_2Erelation_2Etransitive_2E1:
!>[A_27a: $tType] : ( tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)) > 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(arityeq2_2Ec_2Earithmetic_2E_3E_2E2,axiom,
! [X0_2E0: tyop_2Enum_2Enum,X1_2E0: tyop_2Enum_2Enum] : c_2Earithmetic_2E_3E_2E2(X0_2E0,X1_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),c_2Earithmetic_2E_3E_2E0,X0_2E0),X1_2E0) ).
tff(arityeq2_2Ec_2Earithmetic_2E_3E_3D_2E2,axiom,
! [X0_2E0: tyop_2Enum_2Enum,X1_2E0: tyop_2Enum_2Enum] : c_2Earithmetic_2E_3E_3D_2E2(X0_2E0,X1_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),c_2Earithmetic_2E_3E_3D_2E0,X0_2E0),X1_2E0) ).
tff(arityeq1_2Ec_2Ebool_2E_3F_2E1_2Emono_2EA_27a,axiom,
! [A_27a: $tType,X0_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] : c_2Ebool_2E_3F_2E1(A_27a,X0_2E0) = app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,c_2Ebool_2E_3F_2E0(A_27a),X0_2E0) ).
tff(arityeq2_2Ec_2Earithmetic_2EABS__DIFF_2E2,axiom,
! [X0_2E0: tyop_2Enum_2Enum,X1_2E0: tyop_2Enum_2Enum] : c_2Earithmetic_2EABS__DIFF_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_2EABS__DIFF_2E0,X0_2E0),X1_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_2EA_27a,axiom,
! [A_27a: $tType,X0_2E0: tyop_2Emin_2Ebool,X1_2E0: A_27a,X2_2E0: A_27a] : c_2Ebool_2ECOND_2E3(A_27a,X0_2E0,X1_2E0,X2_2E0) = app_2E2(A_27a,A_27a,app_2E2(A_27a,tyop_2Emin_2Efun(A_27a,A_27a),app_2E2(tyop_2Emin_2Ebool,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,A_27a)),c_2Ebool_2ECOND_2E0(A_27a),X0_2E0),X1_2E0),X2_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_2EDIV2_2E1,axiom,
! [X0_2E0: tyop_2Enum_2Enum] : c_2Earithmetic_2EDIV2_2E1(X0_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,c_2Earithmetic_2EDIV2_2E0,X0_2E0) ).
tff(arityeq1_2Ec_2Earithmetic_2EEVEN_2E1,axiom,
! [X0_2E0: tyop_2Enum_2Enum] : c_2Earithmetic_2EEVEN_2E1(X0_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,c_2Earithmetic_2EEVEN_2E0,X0_2E0) ).
tff(arityeq2_2Ec_2Earithmetic_2EEXP_2E2,axiom,
! [X0_2E0: tyop_2Enum_2Enum,X1_2E0: tyop_2Enum_2Enum] : c_2Earithmetic_2EEXP_2E2(X0_2E0,X1_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),c_2Earithmetic_2EEXP_2E0,X0_2E0),X1_2E0) ).
tff(arityeq1_2Ec_2Earithmetic_2EFACT_2E1,axiom,
! [X0_2E0: tyop_2Enum_2Enum] : c_2Earithmetic_2EFACT_2E1(X0_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,c_2Earithmetic_2EFACT_2E0,X0_2E0) ).
tff(arityeq3_2Ec_2Earithmetic_2EFUNPOW_2E3_2Emono_2EA_27a,axiom,
! [A_27a: $tType,X0_2E0: tyop_2Emin_2Efun(A_27a,A_27a),X1_2E0: tyop_2Enum_2Enum,X2_2E0: A_27a] : c_2Earithmetic_2EFUNPOW_2E3(A_27a,X0_2E0,X1_2E0,X2_2E0) = app_2E2(A_27a,A_27a,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(A_27a,A_27a),app_2E2(tyop_2Emin_2Efun(A_27a,A_27a),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(A_27a,A_27a)),c_2Earithmetic_2EFUNPOW_2E0(A_27a),X0_2E0),X1_2E0),X2_2E0) ).
tff(arityeq2_2Ec_2Earithmetic_2EMAX_2E2,axiom,
! [X0_2E0: tyop_2Enum_2Enum,X1_2E0: tyop_2Enum_2Enum] : c_2Earithmetic_2EMAX_2E2(X0_2E0,X1_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),c_2Earithmetic_2EMAX_2E0,X0_2E0),X1_2E0) ).
tff(arityeq2_2Ec_2Earithmetic_2EMIN_2E2,axiom,
! [X0_2E0: tyop_2Enum_2Enum,X1_2E0: tyop_2Enum_2Enum] : c_2Earithmetic_2EMIN_2E2(X0_2E0,X1_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),c_2Earithmetic_2EMIN_2E0,X0_2E0),X1_2E0) ).
tff(arityeq2_2Ec_2Earithmetic_2EMOD_2E2,axiom,
! [X0_2E0: tyop_2Enum_2Enum,X1_2E0: tyop_2Enum_2Enum] : c_2Earithmetic_2EMOD_2E2(X0_2E0,X1_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),c_2Earithmetic_2EMOD_2E0,X0_2E0),X1_2E0) ).
tff(arityeq4_2Ec_2Earithmetic_2ENRC_2E4_2Emono_2EA_27a,axiom,
! [A_27a: $tType,X0_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),X1_2E0: tyop_2Enum_2Enum,X2_2E0: A_27a,X3_2E0: A_27a] : c_2Earithmetic_2ENRC_2E4(A_27a,X0_2E0,X1_2E0,X2_2E0,X3_2E0) = app_2E2(A_27a,tyop_2Emin_2Ebool,app_2E2(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))),c_2Earithmetic_2ENRC_2E0(A_27a),X0_2E0),X1_2E0),X2_2E0),X3_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_2Ebool_2EONE__ONE_2E1_2Emono_2Etyop_2Enum_2Enum_20mono_2Etyop_2Enum_2Enum,axiom,
! [X0_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum)] : c_2Ebool_2EONE__ONE_2E1(tyop_2Enum_2Enum,tyop_2Enum_2Enum,X0_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),tyop_2Emin_2Ebool,c_2Ebool_2EONE__ONE_2E0(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_2Erelation_2ERTC_2E3_2Emono_2EA_27a,axiom,
! [A_27a: $tType,X0_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),X1_2E0: A_27a,X2_2E0: A_27a] : c_2Erelation_2ERTC_2E3(A_27a,X0_2E0,X1_2E0,X2_2E0) = app_2E2(A_27a,tyop_2Emin_2Ebool,app_2E2(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),c_2Erelation_2ERTC_2E0(A_27a),X0_2E0),X1_2E0),X2_2E0) ).
tff(arityeq1_2Ec_2Erelation_2ERTC_2E1_2Emono_2Etyop_2Enum_2Enum,axiom,
! [X0_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool))] : c_2Erelation_2ERTC_2E1(tyop_2Enum_2Enum,X0_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)),c_2Erelation_2ERTC_2E0(tyop_2Enum_2Enum),X0_2E0) ).
tff(arityeq1_2Ec_2Enum_2ESUC_2E1,axiom,
! [X0_2E0: tyop_2Enum_2Enum] : c_2Enum_2ESUC_2E1(X0_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,c_2Enum_2ESUC_2E0,X0_2E0) ).
tff(arityeq3_2Ec_2Erelation_2ETC_2E3_2Emono_2EA_27a,axiom,
! [A_27a: $tType,X0_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),X1_2E0: A_27a,X2_2E0: A_27a] : c_2Erelation_2ETC_2E3(A_27a,X0_2E0,X1_2E0,X2_2E0) = app_2E2(A_27a,tyop_2Emin_2Ebool,app_2E2(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),c_2Erelation_2ETC_2E0(A_27a),X0_2E0),X1_2E0),X2_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_2Eprim__rec_2Emeasure_2E1_2Emono_2EA_27a,axiom,
! [A_27a: $tType,X0_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum)] : c_2Eprim__rec_2Emeasure_2E1(A_27a,X0_2E0) = app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),c_2Eprim__rec_2Emeasure_2E0(A_27a),X0_2E0) ).
tff(arityeq1_2Ec_2Earithmetic_2Enat__elim____magic_2E1,axiom,
! [X0_2E0: tyop_2Enum_2Enum] : c_2Earithmetic_2Enat__elim____magic_2E1(X0_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,c_2Earithmetic_2Enat__elim____magic_2E0,X0_2E0) ).
tff(arityeq3_2Ec_2Earithmetic_2Enum__CASE_2E3_2Emono_2EA_27a,axiom,
! [A_27a: $tType,X0_2E0: tyop_2Enum_2Enum,X1_2E0: A_27a,X2_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,A_27a)] : c_2Earithmetic_2Enum__CASE_2E3(A_27a,X0_2E0,X1_2E0,X2_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,A_27a),A_27a,app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,A_27a),A_27a),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,A_27a),A_27a)),c_2Earithmetic_2Enum__CASE_2E0(A_27a),X0_2E0),X1_2E0),X2_2E0) ).
tff(arityeq1_2Ec_2Erelation_2Etransitive_2E1_2Emono_2EA_27a,axiom,
! [A_27a: $tType,X0_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))] : c_2Erelation_2Etransitive_2E1(A_27a,X0_2E0) = app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),tyop_2Emin_2Ebool,c_2Erelation_2Etransitive_2E0(A_27a),X0_2E0) ).
tff(arityeq1_2Ec_2Erelation_2Etransitive_2E1_2Emono_2Etyop_2Enum_2Enum,axiom,
! [X0_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool))] : c_2Erelation_2Etransitive_2E1(tyop_2Enum_2Enum,X0_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)),tyop_2Emin_2Ebool,c_2Erelation_2Etransitive_2E0(tyop_2Enum_2Enum),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_2Earithmetic_2EADD,axiom,
( ! [V0n_2E0: tyop_2Enum_2Enum] : c_2Earithmetic_2E_2B_2E2(c_2Enum_2E0_2E0,V0n_2E0) = V0n_2E0
& ! [V1m_2E0: tyop_2Enum_2Enum,V2n_2E0: tyop_2Enum_2Enum] : c_2Earithmetic_2E_2B_2E2(c_2Enum_2ESUC_2E1(V1m_2E0),V2n_2E0) = c_2Enum_2ESUC_2E1(c_2Earithmetic_2E_2B_2E2(V1m_2E0,V2n_2E0)) ) ).
tff(thm_2Earithmetic_2ENUMERAL__DEF,axiom,
! [V0x_2E0: tyop_2Enum_2Enum] : c_2Earithmetic_2ENUMERAL_2E1(V0x_2E0) = V0x_2E0 ).
tff(thm_2Earithmetic_2EALT__ZERO,axiom,
c_2Earithmetic_2EZERO_2E0 = c_2Enum_2E0_2E0 ).
tff(thm_2Earithmetic_2EBIT1,axiom,
! [V0n_2E0: tyop_2Enum_2Enum] : c_2Earithmetic_2EBIT1_2E1(V0n_2E0) = c_2Earithmetic_2E_2B_2E2(V0n_2E0,c_2Earithmetic_2E_2B_2E2(V0n_2E0,c_2Enum_2ESUC_2E1(c_2Enum_2E0_2E0))) ).
tff(thm_2Earithmetic_2EBIT2,axiom,
! [V0n_2E0: tyop_2Enum_2Enum] : c_2Earithmetic_2EBIT2_2E1(V0n_2E0) = c_2Earithmetic_2E_2B_2E2(V0n_2E0,c_2Earithmetic_2E_2B_2E2(V0n_2E0,c_2Enum_2ESUC_2E1(c_2Enum_2ESUC_2E1(c_2Enum_2E0_2E0)))) ).
tff(thm_2Earithmetic_2Enat__elim____magic,axiom,
! [V0n_2E0: tyop_2Enum_2Enum] : c_2Earithmetic_2Enat__elim____magic_2E1(V0n_2E0) = V0n_2E0 ).
tff(thm_2Earithmetic_2ESUB,axiom,
( ! [V0m_2E0: tyop_2Enum_2Enum] : c_2Earithmetic_2E_2D_2E2(c_2Enum_2E0_2E0,V0m_2E0) = c_2Enum_2E0_2E0
& ! [V1m_2E0: tyop_2Enum_2Enum,V2n_2E0: tyop_2Enum_2Enum] : c_2Earithmetic_2E_2D_2E2(c_2Enum_2ESUC_2E1(V1m_2E0),V2n_2E0) = c_2Ebool_2ECOND_2E3(tyop_2Enum_2Enum,c_2Eprim__rec_2E_3C_2E2(V1m_2E0,V2n_2E0),c_2Enum_2E0_2E0,c_2Enum_2ESUC_2E1(c_2Earithmetic_2E_2D_2E2(V1m_2E0,V2n_2E0))) ) ).
tff(thm_2Earithmetic_2EMULT,axiom,
( ! [V0n_2E0: tyop_2Enum_2Enum] : c_2Earithmetic_2E_2A_2E2(c_2Enum_2E0_2E0,V0n_2E0) = c_2Enum_2E0_2E0
& ! [V1m_2E0: tyop_2Enum_2Enum,V2n_2E0: tyop_2Enum_2Enum] : c_2Earithmetic_2E_2A_2E2(c_2Enum_2ESUC_2E1(V1m_2E0),V2n_2E0) = c_2Earithmetic_2E_2B_2E2(c_2Earithmetic_2E_2A_2E2(V1m_2E0,V2n_2E0),V2n_2E0) ) ).
tff(thm_2Earithmetic_2EEXP,axiom,
( ! [V0m_2E0: tyop_2Enum_2Enum] : c_2Earithmetic_2EEXP_2E2(V0m_2E0,c_2Enum_2E0_2E0) = c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))
& ! [V1m_2E0: tyop_2Enum_2Enum,V2n_2E0: tyop_2Enum_2Enum] : c_2Earithmetic_2EEXP_2E2(V1m_2E0,c_2Enum_2ESUC_2E1(V2n_2E0)) = c_2Earithmetic_2E_2A_2E2(V1m_2E0,c_2Earithmetic_2EEXP_2E2(V1m_2E0,V2n_2E0)) ) ).
tff(thm_2Earithmetic_2EGREATER__DEF,axiom,
! [V0m_2E0: tyop_2Enum_2Enum,V1n_2E0: tyop_2Enum_2Enum] : c_2Earithmetic_2E_3E_2E2(V0m_2E0,V1n_2E0) = c_2Eprim__rec_2E_3C_2E2(V1n_2E0,V0m_2E0) ).
tff(thm_2Earithmetic_2ELESS__OR__EQ,axiom,
! [V0m_2E0: tyop_2Enum_2Enum,V1n_2E0: tyop_2Enum_2Enum] :
( p(c_2Earithmetic_2E_3C_3D_2E2(V0m_2E0,V1n_2E0))
<=> ( p(c_2Eprim__rec_2E_3C_2E2(V0m_2E0,V1n_2E0))
| ( V0m_2E0 = V1n_2E0 ) ) ) ).
tff(thm_2Earithmetic_2EGREATER__OR__EQ,axiom,
! [V0m_2E0: tyop_2Enum_2Enum,V1n_2E0: tyop_2Enum_2Enum] :
( p(c_2Earithmetic_2E_3E_3D_2E2(V0m_2E0,V1n_2E0))
<=> ( p(c_2Earithmetic_2E_3E_2E2(V0m_2E0,V1n_2E0))
| ( V0m_2E0 = V1n_2E0 ) ) ) ).
tff(thm_2Earithmetic_2EEVEN,axiom,
( ( c_2Earithmetic_2EEVEN_2E1(c_2Enum_2E0_2E0) = c_2Ebool_2ET_2E0 )
& ! [V0n_2E0: tyop_2Enum_2Enum] :
( p(c_2Earithmetic_2EEVEN_2E1(c_2Enum_2ESUC_2E1(V0n_2E0)))
<=> ~ p(c_2Earithmetic_2EEVEN_2E1(V0n_2E0)) ) ) ).
tff(thm_2Earithmetic_2EODD,axiom,
( ( c_2Earithmetic_2EODD_2E1(c_2Enum_2E0_2E0) = c_2Ebool_2EF_2E0 )
& ! [V0n_2E0: tyop_2Enum_2Enum] :
( p(c_2Earithmetic_2EODD_2E1(c_2Enum_2ESUC_2E1(V0n_2E0)))
<=> ~ p(c_2Earithmetic_2EODD_2E1(V0n_2E0)) ) ) ).
tff(thm_2Earithmetic_2Enum__case__def,axiom,
! [A_27a: $tType] :
( ! [V0v_2E0: A_27a,V1f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,A_27a)] : c_2Earithmetic_2Enum__CASE_2E3(A_27a,c_2Enum_2E0_2E0,V0v_2E0,V1f_2E0) = V0v_2E0
& ! [V2n_2E0: tyop_2Enum_2Enum,V3v_2E0: A_27a,V4f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,A_27a)] : c_2Earithmetic_2Enum__CASE_2E3(A_27a,c_2Enum_2ESUC_2E1(V2n_2E0),V3v_2E0,V4f_2E0) = app_2E2(tyop_2Enum_2Enum,A_27a,V4f_2E0,V2n_2E0) ) ).
tff(thm_2Earithmetic_2EFUNPOW,axiom,
! [A_27a: $tType] :
( ! [V0f_2E0: tyop_2Emin_2Efun(A_27a,A_27a),V1x_2E0: A_27a] : c_2Earithmetic_2EFUNPOW_2E3(A_27a,V0f_2E0,c_2Enum_2E0_2E0,V1x_2E0) = V1x_2E0
& ! [V2f_2E0: tyop_2Emin_2Efun(A_27a,A_27a),V3n_2E0: tyop_2Enum_2Enum,V4x_2E0: A_27a] : c_2Earithmetic_2EFUNPOW_2E3(A_27a,V2f_2E0,c_2Enum_2ESUC_2E1(V3n_2E0),V4x_2E0) = c_2Earithmetic_2EFUNPOW_2E3(A_27a,V2f_2E0,V3n_2E0,app_2E2(A_27a,A_27a,V2f_2E0,V4x_2E0)) ) ).
tff(thm_2Earithmetic_2ENRC,axiom,
! [A_27a: $tType] :
( ! [V0R_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),V1x_2E0: A_27a,V2y_2E0: A_27a] :
( p(c_2Earithmetic_2ENRC_2E4(A_27a,V0R_2E0,c_2Enum_2E0_2E0,V1x_2E0,V2y_2E0))
<=> ( V1x_2E0 = V2y_2E0 ) )
& ! [V3R_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),V4n_2E0: tyop_2Enum_2Enum,V5x_2E0: A_27a,V6y_2E0: A_27a] :
( p(c_2Earithmetic_2ENRC_2E4(A_27a,V3R_2E0,c_2Enum_2ESUC_2E1(V4n_2E0),V5x_2E0,V6y_2E0))
<=> ? [V7z_2E0: A_27a] :
( p(app_2E2(A_27a,tyop_2Emin_2Ebool,app_2E2(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V3R_2E0,V5x_2E0),V7z_2E0))
& p(c_2Earithmetic_2ENRC_2E4(A_27a,V3R_2E0,V4n_2E0,V7z_2E0,V6y_2E0)) ) ) ) ).
tff(thm_2Earithmetic_2EFACT,axiom,
( ( c_2Earithmetic_2EFACT_2E1(c_2Enum_2E0_2E0) = c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)) )
& ! [V0n_2E0: tyop_2Enum_2Enum] : c_2Earithmetic_2EFACT_2E1(c_2Enum_2ESUC_2E1(V0n_2E0)) = c_2Earithmetic_2E_2A_2E2(c_2Enum_2ESUC_2E1(V0n_2E0),c_2Earithmetic_2EFACT_2E1(V0n_2E0)) ) ).
tff(thm_2Earithmetic_2EDIVISION,axiom,
! [V0n_2E0: tyop_2Enum_2Enum] :
( p(c_2Eprim__rec_2E_3C_2E2(c_2Enum_2E0_2E0,V0n_2E0))
=> ! [V1k_2E0: tyop_2Enum_2Enum] :
( ( V1k_2E0 = c_2Earithmetic_2E_2B_2E2(c_2Earithmetic_2E_2A_2E2(c_2Earithmetic_2EDIV_2E2(V1k_2E0,V0n_2E0),V0n_2E0),c_2Earithmetic_2EMOD_2E2(V1k_2E0,V0n_2E0)) )
& p(c_2Eprim__rec_2E_3C_2E2(c_2Earithmetic_2EMOD_2E2(V1k_2E0,V0n_2E0),V0n_2E0)) ) ) ).
tff(thm_2Earithmetic_2EDIV2__def,axiom,
! [V0n_2E0: tyop_2Enum_2Enum] : c_2Earithmetic_2EDIV2_2E1(V0n_2E0) = c_2Earithmetic_2EDIV_2E2(V0n_2E0,c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EZERO_2E0))) ).
tff(thm_2Earithmetic_2EMAX__DEF,axiom,
! [V0m_2E0: tyop_2Enum_2Enum,V1n_2E0: tyop_2Enum_2Enum] : c_2Earithmetic_2EMAX_2E2(V0m_2E0,V1n_2E0) = c_2Ebool_2ECOND_2E3(tyop_2Enum_2Enum,c_2Eprim__rec_2E_3C_2E2(V0m_2E0,V1n_2E0),V1n_2E0,V0m_2E0) ).
tff(thm_2Earithmetic_2EMIN__DEF,axiom,
! [V0m_2E0: tyop_2Enum_2Enum,V1n_2E0: tyop_2Enum_2Enum] : c_2Earithmetic_2EMIN_2E2(V0m_2E0,V1n_2E0) = c_2Ebool_2ECOND_2E3(tyop_2Enum_2Enum,c_2Eprim__rec_2E_3C_2E2(V0m_2E0,V1n_2E0),V0m_2E0,V1n_2E0) ).
tff(thm_2Earithmetic_2EABS__DIFF__def,axiom,
! [V0n_2E0: tyop_2Enum_2Enum,V1m_2E0: tyop_2Enum_2Enum] : c_2Earithmetic_2EABS__DIFF_2E2(V0n_2E0,V1m_2E0) = c_2Ebool_2ECOND_2E3(tyop_2Enum_2Enum,c_2Eprim__rec_2E_3C_2E2(V0n_2E0,V1m_2E0),c_2Earithmetic_2E_2D_2E2(V1m_2E0,V0n_2E0),c_2Earithmetic_2E_2D_2E2(V0n_2E0,V1m_2E0)) ).
tff(thm_2Earithmetic_2EONE,axiom,
c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)) = c_2Enum_2ESUC_2E1(c_2Enum_2E0_2E0) ).
tff(thm_2Earithmetic_2ETWO,axiom,
c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EZERO_2E0)) = c_2Enum_2ESUC_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))) ).
tff(thm_2Earithmetic_2ENORM__0,axiom,
c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EZERO_2E0) = c_2Enum_2E0_2E0 ).
tff(thm_2Earithmetic_2Enum__case__compute,axiom,
! [A_27a: $tType,F0_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)] :
( ! [V2n_2E0: tyop_2Enum_2Enum] :
( p(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,F0_2E0,V2n_2E0))
<=> ( V2n_2E0 = c_2Enum_2E0_2E0 ) )
=> ! [V0g_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,A_27a),V1f_2E0: A_27a,V2n_2E0: tyop_2Enum_2Enum] : c_2Earithmetic_2Enum__CASE_2E3(A_27a,V2n_2E0,V1f_2E0,V0g_2E0) = c_2Ebool_2ECOND_2E3(A_27a,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,F0_2E0,V2n_2E0),V1f_2E0,app_2E2(tyop_2Enum_2Enum,A_27a,V0g_2E0,c_2Eprim__rec_2EPRE_2E1(V2n_2E0))) ) ).
tff(thm_2Earithmetic_2ESUC__NOT,axiom,
! [V0n_2E0: tyop_2Enum_2Enum] : c_2Enum_2E0_2E0 != c_2Enum_2ESUC_2E1(V0n_2E0) ).
tff(thm_2Earithmetic_2EADD__0,axiom,
! [V0m_2E0: tyop_2Enum_2Enum] : c_2Earithmetic_2E_2B_2E2(V0m_2E0,c_2Enum_2E0_2E0) = V0m_2E0 ).
tff(thm_2Earithmetic_2EADD__SUC,axiom,
! [V0m_2E0: tyop_2Enum_2Enum,V1n_2E0: tyop_2Enum_2Enum] : c_2Enum_2ESUC_2E1(c_2Earithmetic_2E_2B_2E2(V0m_2E0,V1n_2E0)) = c_2Earithmetic_2E_2B_2E2(V0m_2E0,c_2Enum_2ESUC_2E1(V1n_2E0)) ).
tff(thm_2Earithmetic_2EADD__CLAUSES,axiom,
! [V0n_2E0: tyop_2Enum_2Enum,V1m_2E0: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_2B_2E2(c_2Enum_2E0_2E0,V1m_2E0) = V1m_2E0 )
& ( c_2Earithmetic_2E_2B_2E2(V1m_2E0,c_2Enum_2E0_2E0) = V1m_2E0 )
& ( c_2Earithmetic_2E_2B_2E2(c_2Enum_2ESUC_2E1(V1m_2E0),V0n_2E0) = c_2Enum_2ESUC_2E1(c_2Earithmetic_2E_2B_2E2(V1m_2E0,V0n_2E0)) )
& ( c_2Earithmetic_2E_2B_2E2(V1m_2E0,c_2Enum_2ESUC_2E1(V0n_2E0)) = c_2Enum_2ESUC_2E1(c_2Earithmetic_2E_2B_2E2(V1m_2E0,V0n_2E0)) ) ) ).
tff(thm_2Earithmetic_2EADD__SYM,axiom,
! [V0m_2E0: tyop_2Enum_2Enum,V1n_2E0: tyop_2Enum_2Enum] : c_2Earithmetic_2E_2B_2E2(V0m_2E0,V1n_2E0) = c_2Earithmetic_2E_2B_2E2(V1n_2E0,V0m_2E0) ).
tff(thm_2Earithmetic_2EADD__COMM,axiom,
! [V0m_2E0: tyop_2Enum_2Enum,V1n_2E0: tyop_2Enum_2Enum] : c_2Earithmetic_2E_2B_2E2(V0m_2E0,V1n_2E0) = c_2Earithmetic_2E_2B_2E2(V1n_2E0,V0m_2E0) ).
tff(thm_2Earithmetic_2EADD__ASSOC,axiom,
! [V0m_2E0: tyop_2Enum_2Enum,V1n_2E0: tyop_2Enum_2Enum,V2p_2E0: tyop_2Enum_2Enum] : c_2Earithmetic_2E_2B_2E2(V0m_2E0,c_2Earithmetic_2E_2B_2E2(V1n_2E0,V2p_2E0)) = c_2Earithmetic_2E_2B_2E2(c_2Earithmetic_2E_2B_2E2(V0m_2E0,V1n_2E0),V2p_2E0) ).
tff(thm_2Earithmetic_2Enum__CASES,axiom,
! [V0m_2E0: tyop_2Enum_2Enum] :
( ( V0m_2E0 = c_2Enum_2E0_2E0 )
| ? [V1n_2E0: tyop_2Enum_2Enum] : V0m_2E0 = c_2Enum_2ESUC_2E1(V1n_2E0) ) ).
tff(thm_2Earithmetic_2ENOT__ZERO__LT__ZERO,axiom,
! [V0n_2E0: tyop_2Enum_2Enum] :
( ( V0n_2E0 != c_2Enum_2E0_2E0 )
<=> p(c_2Eprim__rec_2E_3C_2E2(c_2Enum_2E0_2E0,V0n_2E0)) ) ).
tff(thm_2Earithmetic_2ENOT__LT__ZERO__EQ__ZERO,axiom,
! [V0n_2E0: tyop_2Enum_2Enum] :
( ~ p(c_2Eprim__rec_2E_3C_2E2(c_2Enum_2E0_2E0,V0n_2E0))
<=> ( V0n_2E0 = c_2Enum_2E0_2E0 ) ) ).
tff(thm_2Earithmetic_2ELESS__OR__EQ__ALT,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool))] :
( ! [V0x_2E0: tyop_2Enum_2Enum,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),F0_2E0,V0x_2E0),V1y_2E0))
<=> ( V1y_2E0 = c_2Enum_2ESUC_2E1(V0x_2E0) ) )
=> ( c_2Earithmetic_2E_3C_3D_2E0 = c_2Erelation_2ERTC_2E1(tyop_2Enum_2Enum,F0_2E0) ) ) ).
tff(thm_2Earithmetic_2ELESS__ADD,axiom,
! [V0m_2E0: tyop_2Enum_2Enum,V1n_2E0: tyop_2Enum_2Enum] :
( p(c_2Eprim__rec_2E_3C_2E2(V1n_2E0,V0m_2E0))
=> ? [V2p_2E0: tyop_2Enum_2Enum] : c_2Earithmetic_2E_2B_2E2(V2p_2E0,V1n_2E0) = V0m_2E0 ) ).
tff(thm_2Earithmetic_2Etransitive__LESS,axiom,
p(c_2Erelation_2Etransitive_2E1(tyop_2Enum_2Enum,c_2Eprim__rec_2E_3C_2E0)) ).
tff(thm_2Earithmetic_2ELESS__TRANS,axiom,
! [V0m_2E0: tyop_2Enum_2Enum,V1n_2E0: tyop_2Enum_2Enum,V2p_2E0: tyop_2Enum_2Enum] :
( ( p(c_2Eprim__rec_2E_3C_2E2(V0m_2E0,V1n_2E0))
& p(c_2Eprim__rec_2E_3C_2E2(V1n_2E0,V2p_2E0)) )
=> p(c_2Eprim__rec_2E_3C_2E2(V0m_2E0,V2p_2E0)) ) ).
tff(thm_2Earithmetic_2ELESS__ANTISYM,axiom,
! [V0m_2E0: tyop_2Enum_2Enum,V1n_2E0: tyop_2Enum_2Enum] :
~ ( p(c_2Eprim__rec_2E_3C_2E2(V0m_2E0,V1n_2E0))
& p(c_2Eprim__rec_2E_3C_2E2(V1n_2E0,V0m_2E0)) ) ).
tff(thm_2Earithmetic_2ELESS__MONO__REV,axiom,
! [V0m_2E0: tyop_2Enum_2Enum,V1n_2E0: tyop_2Enum_2Enum] :
( p(c_2Eprim__rec_2E_3C_2E2(c_2Enum_2ESUC_2E1(V0m_2E0),c_2Enum_2ESUC_2E1(V1n_2E0)))
=> p(c_2Eprim__rec_2E_3C_2E2(V0m_2E0,V1n_2E0)) ) ).
tff(thm_2Earithmetic_2ELESS__MONO__EQ,axiom,
! [V0m_2E0: tyop_2Enum_2Enum,V1n_2E0: tyop_2Enum_2Enum] : c_2Eprim__rec_2E_3C_2E2(c_2Enum_2ESUC_2E1(V0m_2E0),c_2Enum_2ESUC_2E1(V1n_2E0)) = c_2Eprim__rec_2E_3C_2E2(V0m_2E0,V1n_2E0) ).
tff(thm_2Earithmetic_2ELESS__EQ__MONO,axiom,
! [V0n_2E0: tyop_2Enum_2Enum,V1m_2E0: tyop_2Enum_2Enum] : c_2Earithmetic_2E_3C_3D_2E2(c_2Enum_2ESUC_2E1(V0n_2E0),c_2Enum_2ESUC_2E1(V1m_2E0)) = c_2Earithmetic_2E_3C_3D_2E2(V0n_2E0,V1m_2E0) ).
tff(thm_2Earithmetic_2ELESS__LESS__SUC,axiom,
! [V0m_2E0: tyop_2Enum_2Enum,V1n_2E0: tyop_2Enum_2Enum] :
~ ( p(c_2Eprim__rec_2E_3C_2E2(V0m_2E0,V1n_2E0))
& p(c_2Eprim__rec_2E_3C_2E2(V1n_2E0,c_2Enum_2ESUC_2E1(V0m_2E0))) ) ).
tff(thm_2Earithmetic_2Etransitive__measure,axiom,
! [A_27a: $tType,V0f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum)] : p(c_2Erelation_2Etransitive_2E1(A_27a,c_2Eprim__rec_2Emeasure_2E1(A_27a,V0f_2E0))) ).
tff(thm_2Earithmetic_2ELESS__EQ,axiom,
! [V0m_2E0: tyop_2Enum_2Enum,V1n_2E0: tyop_2Enum_2Enum] : c_2Eprim__rec_2E_3C_2E2(V0m_2E0,V1n_2E0) = c_2Earithmetic_2E_3C_3D_2E2(c_2Enum_2ESUC_2E1(V0m_2E0),V1n_2E0) ).
tff(thm_2Earithmetic_2ELESS__OR,axiom,
! [V0m_2E0: tyop_2Enum_2Enum,V1n_2E0: tyop_2Enum_2Enum] :
( p(c_2Eprim__rec_2E_3C_2E2(V0m_2E0,V1n_2E0))
=> p(c_2Earithmetic_2E_3C_3D_2E2(c_2Enum_2ESUC_2E1(V0m_2E0),V1n_2E0)) ) ).
tff(thm_2Earithmetic_2EOR__LESS,axiom,
! [V0m_2E0: tyop_2Enum_2Enum,V1n_2E0: tyop_2Enum_2Enum] :
( p(c_2Earithmetic_2E_3C_3D_2E2(c_2Enum_2ESUC_2E1(V0m_2E0),V1n_2E0))
=> p(c_2Eprim__rec_2E_3C_2E2(V0m_2E0,V1n_2E0)) ) ).
tff(thm_2Earithmetic_2ELESS__EQ__IFF__LESS__SUC,axiom,
! [V0n_2E0: tyop_2Enum_2Enum,V1m_2E0: tyop_2Enum_2Enum] : c_2Earithmetic_2E_3C_3D_2E2(V0n_2E0,V1m_2E0) = c_2Eprim__rec_2E_3C_2E2(V0n_2E0,c_2Enum_2ESUC_2E1(V1m_2E0)) ).
tff(thm_2Earithmetic_2ELESS__EQ__IMP__LESS__SUC,axiom,
! [V0n_2E0: tyop_2Enum_2Enum,V1m_2E0: tyop_2Enum_2Enum] :
( p(c_2Earithmetic_2E_3C_3D_2E2(V0n_2E0,V1m_2E0))
=> p(c_2Eprim__rec_2E_3C_2E2(V0n_2E0,c_2Enum_2ESUC_2E1(V1m_2E0))) ) ).
tff(thm_2Earithmetic_2EZERO__LESS__EQ,axiom,
! [V0n_2E0: tyop_2Enum_2Enum] : p(c_2Earithmetic_2E_3C_3D_2E2(c_2Enum_2E0_2E0,V0n_2E0)) ).
tff(thm_2Earithmetic_2ELESS__SUC__EQ__COR,axiom,
! [V0m_2E0: tyop_2Enum_2Enum,V1n_2E0: tyop_2Enum_2Enum] :
( ( p(c_2Eprim__rec_2E_3C_2E2(V0m_2E0,V1n_2E0))
& ( c_2Enum_2ESUC_2E1(V0m_2E0) != V1n_2E0 ) )
=> p(c_2Eprim__rec_2E_3C_2E2(c_2Enum_2ESUC_2E1(V0m_2E0),V1n_2E0)) ) ).
tff(thm_2Earithmetic_2ELESS__NOT__SUC,axiom,
! [V0m_2E0: tyop_2Enum_2Enum,V1n_2E0: tyop_2Enum_2Enum] :
( ( p(c_2Eprim__rec_2E_3C_2E2(V0m_2E0,V1n_2E0))
& ( V1n_2E0 != c_2Enum_2ESUC_2E1(V0m_2E0) ) )
=> p(c_2Eprim__rec_2E_3C_2E2(c_2Enum_2ESUC_2E1(V0m_2E0),V1n_2E0)) ) ).
tff(thm_2Earithmetic_2ELESS__0__CASES,axiom,
! [V0m_2E0: tyop_2Enum_2Enum] :
( ( c_2Enum_2E0_2E0 = V0m_2E0 )
| p(c_2Eprim__rec_2E_3C_2E2(c_2Enum_2E0_2E0,V0m_2E0)) ) ).
tff(thm_2Earithmetic_2ELESS__CASES__IMP,axiom,
! [V0m_2E0: tyop_2Enum_2Enum,V1n_2E0: tyop_2Enum_2Enum] :
( ( ~ p(c_2Eprim__rec_2E_3C_2E2(V0m_2E0,V1n_2E0))
& ( V0m_2E0 != V1n_2E0 ) )
=> p(c_2Eprim__rec_2E_3C_2E2(V1n_2E0,V0m_2E0)) ) ).
tff(thm_2Earithmetic_2ELESS__CASES,axiom,
! [V0m_2E0: tyop_2Enum_2Enum,V1n_2E0: tyop_2Enum_2Enum] :
( p(c_2Eprim__rec_2E_3C_2E2(V0m_2E0,V1n_2E0))
| p(c_2Earithmetic_2E_3C_3D_2E2(V1n_2E0,V0m_2E0)) ) ).
tff(thm_2Earithmetic_2EADD__INV__0,axiom,
! [V0m_2E0: tyop_2Enum_2Enum,V1n_2E0: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_2B_2E2(V0m_2E0,V1n_2E0) = V0m_2E0 )
=> ( V1n_2E0 = c_2Enum_2E0_2E0 ) ) ).
tff(thm_2Earithmetic_2ELESS__EQ__ADD,axiom,
! [V0m_2E0: tyop_2Enum_2Enum,V1n_2E0: tyop_2Enum_2Enum] : p(c_2Earithmetic_2E_3C_3D_2E2(V0m_2E0,c_2Earithmetic_2E_2B_2E2(V0m_2E0,V1n_2E0))) ).
tff(thm_2Earithmetic_2ELESS__EQ__ADD__EXISTS,axiom,
! [V0m_2E0: tyop_2Enum_2Enum,V1n_2E0: tyop_2Enum_2Enum] :
( p(c_2Earithmetic_2E_3C_3D_2E2(V1n_2E0,V0m_2E0))
=> ? [V2p_2E0: tyop_2Enum_2Enum] : c_2Earithmetic_2E_2B_2E2(V2p_2E0,V1n_2E0) = V0m_2E0 ) ).
tff(thm_2Earithmetic_2ELESS__STRONG__ADD,axiom,
! [V0m_2E0: tyop_2Enum_2Enum,V1n_2E0: tyop_2Enum_2Enum] :
( p(c_2Eprim__rec_2E_3C_2E2(V1n_2E0,V0m_2E0))
=> ? [V2p_2E0: tyop_2Enum_2Enum] : c_2Earithmetic_2E_2B_2E2(c_2Enum_2ESUC_2E1(V2p_2E0),V1n_2E0) = V0m_2E0 ) ).
tff(thm_2Earithmetic_2ELESS__EQ__SUC__REFL,axiom,
! [V0m_2E0: tyop_2Enum_2Enum] : p(c_2Earithmetic_2E_3C_3D_2E2(V0m_2E0,c_2Enum_2ESUC_2E1(V0m_2E0))) ).
tff(thm_2Earithmetic_2ELESS__ADD__NONZERO,axiom,
! [V0m_2E0: tyop_2Enum_2Enum,V1n_2E0: tyop_2Enum_2Enum] :
( ( V1n_2E0 != c_2Enum_2E0_2E0 )
=> p(c_2Eprim__rec_2E_3C_2E2(V0m_2E0,c_2Earithmetic_2E_2B_2E2(V0m_2E0,V1n_2E0))) ) ).
tff(thm_2Earithmetic_2ENOT__SUC__LESS__EQ__0,axiom,
! [V0n_2E0: tyop_2Enum_2Enum] : ~ p(c_2Earithmetic_2E_3C_3D_2E2(c_2Enum_2ESUC_2E1(V0n_2E0),c_2Enum_2E0_2E0)) ).
tff(thm_2Earithmetic_2ENOT__LESS,axiom,
! [V0m_2E0: tyop_2Enum_2Enum,V1n_2E0: tyop_2Enum_2Enum] :
( ~ p(c_2Eprim__rec_2E_3C_2E2(V0m_2E0,V1n_2E0))
<=> p(c_2Earithmetic_2E_3C_3D_2E2(V1n_2E0,V0m_2E0)) ) ).
tff(thm_2Earithmetic_2ENOT__LESS__EQUAL,axiom,
! [V0m_2E0: tyop_2Enum_2Enum,V1n_2E0: tyop_2Enum_2Enum] :
( ~ p(c_2Earithmetic_2E_3C_3D_2E2(V0m_2E0,V1n_2E0))
<=> p(c_2Eprim__rec_2E_3C_2E2(V1n_2E0,V0m_2E0)) ) ).
tff(thm_2Earithmetic_2ELESS__EQ__ANTISYM,axiom,
! [V0m_2E0: tyop_2Enum_2Enum,V1n_2E0: tyop_2Enum_2Enum] :
~ ( p(c_2Eprim__rec_2E_3C_2E2(V0m_2E0,V1n_2E0))
& p(c_2Earithmetic_2E_3C_3D_2E2(V1n_2E0,V0m_2E0)) ) ).
tff(thm_2Earithmetic_2ELESS__EQ__0,axiom,
! [V0n_2E0: tyop_2Enum_2Enum] :
( p(c_2Earithmetic_2E_3C_3D_2E2(V0n_2E0,c_2Enum_2E0_2E0))
<=> ( V0n_2E0 = c_2Enum_2E0_2E0 ) ) ).
tff(thm_2Earithmetic_2ESUB__0,axiom,
! [V0m_2E0: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_2D_2E2(c_2Enum_2E0_2E0,V0m_2E0) = c_2Enum_2E0_2E0 )
& ( c_2Earithmetic_2E_2D_2E2(V0m_2E0,c_2Enum_2E0_2E0) = V0m_2E0 ) ) ).
tff(thm_2Earithmetic_2ESUB__MONO__EQ,axiom,
! [V0n_2E0: tyop_2Enum_2Enum,V1m_2E0: tyop_2Enum_2Enum] : c_2Earithmetic_2E_2D_2E2(c_2Enum_2ESUC_2E1(V0n_2E0),c_2Enum_2ESUC_2E1(V1m_2E0)) = c_2Earithmetic_2E_2D_2E2(V0n_2E0,V1m_2E0) ).
tff(thm_2Earithmetic_2ESUB__EQ__0,axiom,
! [V0m_2E0: tyop_2Enum_2Enum,V1n_2E0: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_2D_2E2(V0m_2E0,V1n_2E0) = c_2Enum_2E0_2E0 )
<=> p(c_2Earithmetic_2E_3C_3D_2E2(V0m_2E0,V1n_2E0)) ) ).
tff(thm_2Earithmetic_2EADD1,axiom,
! [V0m_2E0: tyop_2Enum_2Enum] : c_2Enum_2ESUC_2E1(V0m_2E0) = c_2Earithmetic_2E_2B_2E2(V0m_2E0,c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))) ).
tff(thm_2Earithmetic_2ESUC__SUB1,axiom,
! [V0m_2E0: tyop_2Enum_2Enum] : c_2Earithmetic_2E_2D_2E2(c_2Enum_2ESUC_2E1(V0m_2E0),c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))) = V0m_2E0 ).
tff(thm_2Earithmetic_2EPRE__SUB1,axiom,
! [V0m_2E0: tyop_2Enum_2Enum] : c_2Eprim__rec_2EPRE_2E1(V0m_2E0) = c_2Earithmetic_2E_2D_2E2(V0m_2E0,c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))) ).
tff(thm_2Earithmetic_2EMULT__0,axiom,
! [V0m_2E0: tyop_2Enum_2Enum] : c_2Earithmetic_2E_2A_2E2(V0m_2E0,c_2Enum_2E0_2E0) = c_2Enum_2E0_2E0 ).
tff(thm_2Earithmetic_2EMULT__SUC,axiom,
! [V0m_2E0: tyop_2Enum_2Enum,V1n_2E0: tyop_2Enum_2Enum] : c_2Earithmetic_2E_2A_2E2(V0m_2E0,c_2Enum_2ESUC_2E1(V1n_2E0)) = c_2Earithmetic_2E_2B_2E2(V0m_2E0,c_2Earithmetic_2E_2A_2E2(V0m_2E0,V1n_2E0)) ).
tff(thm_2Earithmetic_2EMULT__LEFT__1,axiom,
! [V0m_2E0: tyop_2Enum_2Enum] : c_2Earithmetic_2E_2A_2E2(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)),V0m_2E0) = V0m_2E0 ).
tff(thm_2Earithmetic_2EMULT__RIGHT__1,axiom,
! [V0m_2E0: tyop_2Enum_2Enum] : c_2Earithmetic_2E_2A_2E2(V0m_2E0,c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))) = V0m_2E0 ).
tff(thm_2Earithmetic_2EMULT__CLAUSES,axiom,
! [V0m_2E0: tyop_2Enum_2Enum,V1n_2E0: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_2A_2E2(c_2Enum_2E0_2E0,V0m_2E0) = c_2Enum_2E0_2E0 )
& ( c_2Earithmetic_2E_2A_2E2(V0m_2E0,c_2Enum_2E0_2E0) = c_2Enum_2E0_2E0 )
& ( c_2Earithmetic_2E_2A_2E2(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)),V0m_2E0) = V0m_2E0 )
& ( c_2Earithmetic_2E_2A_2E2(V0m_2E0,c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))) = V0m_2E0 )
& ( c_2Earithmetic_2E_2A_2E2(c_2Enum_2ESUC_2E1(V0m_2E0),V1n_2E0) = c_2Earithmetic_2E_2B_2E2(c_2Earithmetic_2E_2A_2E2(V0m_2E0,V1n_2E0),V1n_2E0) )
& ( c_2Earithmetic_2E_2A_2E2(V0m_2E0,c_2Enum_2ESUC_2E1(V1n_2E0)) = c_2Earithmetic_2E_2B_2E2(V0m_2E0,c_2Earithmetic_2E_2A_2E2(V0m_2E0,V1n_2E0)) ) ) ).
tff(thm_2Earithmetic_2EMULT__SYM,axiom,
! [V0m_2E0: tyop_2Enum_2Enum,V1n_2E0: tyop_2Enum_2Enum] : c_2Earithmetic_2E_2A_2E2(V0m_2E0,V1n_2E0) = c_2Earithmetic_2E_2A_2E2(V1n_2E0,V0m_2E0) ).
tff(thm_2Earithmetic_2EMULT__COMM,axiom,
! [V0m_2E0: tyop_2Enum_2Enum,V1n_2E0: tyop_2Enum_2Enum] : c_2Earithmetic_2E_2A_2E2(V0m_2E0,V1n_2E0) = c_2Earithmetic_2E_2A_2E2(V1n_2E0,V0m_2E0) ).
tff(thm_2Earithmetic_2ERIGHT__ADD__DISTRIB,axiom,
! [V0m_2E0: tyop_2Enum_2Enum,V1n_2E0: tyop_2Enum_2Enum,V2p_2E0: tyop_2Enum_2Enum] : c_2Earithmetic_2E_2A_2E2(c_2Earithmetic_2E_2B_2E2(V0m_2E0,V1n_2E0),V2p_2E0) = c_2Earithmetic_2E_2B_2E2(c_2Earithmetic_2E_2A_2E2(V0m_2E0,V2p_2E0),c_2Earithmetic_2E_2A_2E2(V1n_2E0,V2p_2E0)) ).
tff(thm_2Earithmetic_2ELEFT__ADD__DISTRIB,axiom,
! [V0m_2E0: tyop_2Enum_2Enum,V1n_2E0: tyop_2Enum_2Enum,V2p_2E0: tyop_2Enum_2Enum] : c_2Earithmetic_2E_2A_2E2(V2p_2E0,c_2Earithmetic_2E_2B_2E2(V0m_2E0,V1n_2E0)) = c_2Earithmetic_2E_2B_2E2(c_2Earithmetic_2E_2A_2E2(V2p_2E0,V0m_2E0),c_2Earithmetic_2E_2A_2E2(V2p_2E0,V1n_2E0)) ).
tff(thm_2Earithmetic_2EMULT__ASSOC,axiom,
! [V0m_2E0: tyop_2Enum_2Enum,V1n_2E0: tyop_2Enum_2Enum,V2p_2E0: tyop_2Enum_2Enum] : c_2Earithmetic_2E_2A_2E2(V0m_2E0,c_2Earithmetic_2E_2A_2E2(V1n_2E0,V2p_2E0)) = c_2Earithmetic_2E_2A_2E2(c_2Earithmetic_2E_2A_2E2(V0m_2E0,V1n_2E0),V2p_2E0) ).
tff(thm_2Earithmetic_2ESUB__ADD,axiom,
! [V0m_2E0: tyop_2Enum_2Enum,V1n_2E0: tyop_2Enum_2Enum] :
( p(c_2Earithmetic_2E_3C_3D_2E2(V1n_2E0,V0m_2E0))
=> ( c_2Earithmetic_2E_2B_2E2(c_2Earithmetic_2E_2D_2E2(V0m_2E0,V1n_2E0),V1n_2E0) = V0m_2E0 ) ) ).
tff(thm_2Earithmetic_2EPRE__SUB,axiom,
! [V0m_2E0: tyop_2Enum_2Enum,V1n_2E0: tyop_2Enum_2Enum] : c_2Eprim__rec_2EPRE_2E1(c_2Earithmetic_2E_2D_2E2(V0m_2E0,V1n_2E0)) = c_2Earithmetic_2E_2D_2E2(c_2Eprim__rec_2EPRE_2E1(V0m_2E0),V1n_2E0) ).
tff(thm_2Earithmetic_2EADD__EQ__0,axiom,
! [V0m_2E0: tyop_2Enum_2Enum,V1n_2E0: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_2B_2E2(V0m_2E0,V1n_2E0) = c_2Enum_2E0_2E0 )
<=> ( ( V0m_2E0 = c_2Enum_2E0_2E0 )
& ( V1n_2E0 = c_2Enum_2E0_2E0 ) ) ) ).
tff(thm_2Earithmetic_2EADD__EQ__1,axiom,
! [V0m_2E0: tyop_2Enum_2Enum,V1n_2E0: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_2B_2E2(V0m_2E0,V1n_2E0) = c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)) )
<=> ( ( ( V0m_2E0 = c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)) )
& ( V1n_2E0 = c_2Enum_2E0_2E0 ) )
| ( ( V0m_2E0 = c_2Enum_2E0_2E0 )
& ( V1n_2E0 = c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)) ) ) ) ) ).
tff(thm_2Earithmetic_2EADD__INV__0__EQ,axiom,
! [V0m_2E0: tyop_2Enum_2Enum,V1n_2E0: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_2B_2E2(V0m_2E0,V1n_2E0) = V0m_2E0 )
<=> ( V1n_2E0 = c_2Enum_2E0_2E0 ) ) ).
tff(thm_2Earithmetic_2EPRE__SUC__EQ,axiom,
! [V0m_2E0: tyop_2Enum_2Enum,V1n_2E0: tyop_2Enum_2Enum] :
( p(c_2Eprim__rec_2E_3C_2E2(c_2Enum_2E0_2E0,V1n_2E0))
=> ( ( V0m_2E0 = c_2Eprim__rec_2EPRE_2E1(V1n_2E0) )
<=> ( c_2Enum_2ESUC_2E1(V0m_2E0) = V1n_2E0 ) ) ) ).
tff(thm_2Earithmetic_2EINV__PRE__EQ,axiom,
! [V0m_2E0: tyop_2Enum_2Enum,V1n_2E0: tyop_2Enum_2Enum] :
( ( p(c_2Eprim__rec_2E_3C_2E2(c_2Enum_2E0_2E0,V0m_2E0))
& p(c_2Eprim__rec_2E_3C_2E2(c_2Enum_2E0_2E0,V1n_2E0)) )
=> ( ( c_2Eprim__rec_2EPRE_2E1(V0m_2E0) = c_2Eprim__rec_2EPRE_2E1(V1n_2E0) )
<=> ( V0m_2E0 = V1n_2E0 ) ) ) ).
tff(thm_2Earithmetic_2ELESS__SUC__NOT,axiom,
! [V0m_2E0: tyop_2Enum_2Enum,V1n_2E0: tyop_2Enum_2Enum] :
( p(c_2Eprim__rec_2E_3C_2E2(V0m_2E0,V1n_2E0))
=> ~ p(c_2Eprim__rec_2E_3C_2E2(V1n_2E0,c_2Enum_2ESUC_2E1(V0m_2E0))) ) ).
tff(thm_2Earithmetic_2EADD__EQ__SUB,axiom,
! [V0m_2E0: tyop_2Enum_2Enum,V1n_2E0: tyop_2Enum_2Enum,V2p_2E0: tyop_2Enum_2Enum] :
( p(c_2Earithmetic_2E_3C_3D_2E2(V1n_2E0,V2p_2E0))
=> ( ( c_2Earithmetic_2E_2B_2E2(V0m_2E0,V1n_2E0) = V2p_2E0 )
<=> ( V0m_2E0 = c_2Earithmetic_2E_2D_2E2(V2p_2E0,V1n_2E0) ) ) ) ).
tff(thm_2Earithmetic_2ELESS__MONO__ADD,axiom,
! [V0m_2E0: tyop_2Enum_2Enum,V1n_2E0: tyop_2Enum_2Enum,V2p_2E0: tyop_2Enum_2Enum] :
( p(c_2Eprim__rec_2E_3C_2E2(V0m_2E0,V1n_2E0))
=> p(c_2Eprim__rec_2E_3C_2E2(c_2Earithmetic_2E_2B_2E2(V0m_2E0,V2p_2E0),c_2Earithmetic_2E_2B_2E2(V1n_2E0,V2p_2E0))) ) ).
tff(thm_2Earithmetic_2ELESS__MONO__ADD__INV,axiom,
! [V0m_2E0: tyop_2Enum_2Enum,V1n_2E0: tyop_2Enum_2Enum,V2p_2E0: tyop_2Enum_2Enum] :
( p(c_2Eprim__rec_2E_3C_2E2(c_2Earithmetic_2E_2B_2E2(V0m_2E0,V2p_2E0),c_2Earithmetic_2E_2B_2E2(V1n_2E0,V2p_2E0)))
=> p(c_2Eprim__rec_2E_3C_2E2(V0m_2E0,V1n_2E0)) ) ).
tff(thm_2Earithmetic_2ELESS__MONO__ADD__EQ,axiom,
! [V0m_2E0: tyop_2Enum_2Enum,V1n_2E0: tyop_2Enum_2Enum,V2p_2E0: tyop_2Enum_2Enum] : c_2Eprim__rec_2E_3C_2E2(c_2Earithmetic_2E_2B_2E2(V0m_2E0,V2p_2E0),c_2Earithmetic_2E_2B_2E2(V1n_2E0,V2p_2E0)) = c_2Eprim__rec_2E_3C_2E2(V0m_2E0,V1n_2E0) ).
tff(thm_2Earithmetic_2ELT__ADD__RCANCEL,axiom,
! [V0m_2E0: tyop_2Enum_2Enum,V1n_2E0: tyop_2Enum_2Enum,V2p_2E0: tyop_2Enum_2Enum] : c_2Eprim__rec_2E_3C_2E2(c_2Earithmetic_2E_2B_2E2(V0m_2E0,V2p_2E0),c_2Earithmetic_2E_2B_2E2(V1n_2E0,V2p_2E0)) = c_2Eprim__rec_2E_3C_2E2(V0m_2E0,V1n_2E0) ).
tff(thm_2Earithmetic_2ELT__ADD__LCANCEL,axiom,
! [V0m_2E0: tyop_2Enum_2Enum,V1n_2E0: tyop_2Enum_2Enum,V2p_2E0: tyop_2Enum_2Enum] : c_2Eprim__rec_2E_3C_2E2(c_2Earithmetic_2E_2B_2E2(V2p_2E0,V0m_2E0),c_2Earithmetic_2E_2B_2E2(V2p_2E0,V1n_2E0)) = c_2Eprim__rec_2E_3C_2E2(V0m_2E0,V1n_2E0) ).
tff(thm_2Earithmetic_2EEQ__MONO__ADD__EQ,axiom,
! [V0m_2E0: tyop_2Enum_2Enum,V1n_2E0: tyop_2Enum_2Enum,V2p_2E0: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_2B_2E2(V0m_2E0,V2p_2E0) = c_2Earithmetic_2E_2B_2E2(V1n_2E0,V2p_2E0) )
<=> ( V0m_2E0 = V1n_2E0 ) ) ).
tff(thm_2Earithmetic_2ELESS__EQ__MONO__ADD__EQ,axiom,
! [V0m_2E0: tyop_2Enum_2Enum,V1n_2E0: tyop_2Enum_2Enum,V2p_2E0: tyop_2Enum_2Enum] : c_2Earithmetic_2E_3C_3D_2E2(c_2Earithmetic_2E_2B_2E2(V0m_2E0,V2p_2E0),c_2Earithmetic_2E_2B_2E2(V1n_2E0,V2p_2E0)) = c_2Earithmetic_2E_3C_3D_2E2(V0m_2E0,V1n_2E0) ).
tff(thm_2Earithmetic_2ELESS__EQ__TRANS,axiom,
! [V0m_2E0: tyop_2Enum_2Enum,V1n_2E0: tyop_2Enum_2Enum,V2p_2E0: tyop_2Enum_2Enum] :
( ( p(c_2Earithmetic_2E_3C_3D_2E2(V0m_2E0,V1n_2E0))
& p(c_2Earithmetic_2E_3C_3D_2E2(V1n_2E0,V2p_2E0)) )
=> p(c_2Earithmetic_2E_3C_3D_2E2(V0m_2E0,V2p_2E0)) ) ).
tff(thm_2Earithmetic_2ELESS__EQ__LESS__TRANS,axiom,
! [V0m_2E0: tyop_2Enum_2Enum,V1n_2E0: tyop_2Enum_2Enum,V2p_2E0: tyop_2Enum_2Enum] :
( ( p(c_2Earithmetic_2E_3C_3D_2E2(V0m_2E0,V1n_2E0))
& p(c_2Eprim__rec_2E_3C_2E2(V1n_2E0,V2p_2E0)) )
=> p(c_2Eprim__rec_2E_3C_2E2(V0m_2E0,V2p_2E0)) ) ).
tff(thm_2Earithmetic_2ELESS__LESS__EQ__TRANS,axiom,
! [V0m_2E0: tyop_2Enum_2Enum,V1n_2E0: tyop_2Enum_2Enum,V2p_2E0: tyop_2Enum_2Enum] :
( ( p(c_2Eprim__rec_2E_3C_2E2(V0m_2E0,V1n_2E0))
& p(c_2Earithmetic_2E_3C_3D_2E2(V1n_2E0,V2p_2E0)) )
=> p(c_2Eprim__rec_2E_3C_2E2(V0m_2E0,V2p_2E0)) ) ).
tff(thm_2Earithmetic_2ELESS__EQ__LESS__EQ__MONO,axiom,
! [V0m_2E0: tyop_2Enum_2Enum,V1n_2E0: tyop_2Enum_2Enum,V2p_2E0: tyop_2Enum_2Enum,V3q_2E0: tyop_2Enum_2Enum] :
( ( p(c_2Earithmetic_2E_3C_3D_2E2(V0m_2E0,V2p_2E0))
& p(c_2Earithmetic_2E_3C_3D_2E2(V1n_2E0,V3q_2E0)) )
=> p(c_2Earithmetic_2E_3C_3D_2E2(c_2Earithmetic_2E_2B_2E2(V0m_2E0,V1n_2E0),c_2Earithmetic_2E_2B_2E2(V2p_2E0,V3q_2E0))) ) ).
tff(thm_2Earithmetic_2ELESS__EQ__REFL,axiom,
! [V0m_2E0: tyop_2Enum_2Enum] : p(c_2Earithmetic_2E_3C_3D_2E2(V0m_2E0,V0m_2E0)) ).
tff(thm_2Earithmetic_2ELESS__IMP__LESS__OR__EQ,axiom,
! [V0m_2E0: tyop_2Enum_2Enum,V1n_2E0: tyop_2Enum_2Enum] :
( p(c_2Eprim__rec_2E_3C_2E2(V0m_2E0,V1n_2E0))
=> p(c_2Earithmetic_2E_3C_3D_2E2(V0m_2E0,V1n_2E0)) ) ).
tff(thm_2Earithmetic_2ELESS__MONO__MULT,axiom,
! [V0m_2E0: tyop_2Enum_2Enum,V1n_2E0: tyop_2Enum_2Enum,V2p_2E0: tyop_2Enum_2Enum] :
( p(c_2Earithmetic_2E_3C_3D_2E2(V0m_2E0,V1n_2E0))
=> p(c_2Earithmetic_2E_3C_3D_2E2(c_2Earithmetic_2E_2A_2E2(V0m_2E0,V2p_2E0),c_2Earithmetic_2E_2A_2E2(V1n_2E0,V2p_2E0))) ) ).
tff(thm_2Earithmetic_2ELESS__MONO__MULT2,axiom,
! [V0m_2E0: tyop_2Enum_2Enum,V1n_2E0: tyop_2Enum_2Enum,V2i_2E0: tyop_2Enum_2Enum,V3j_2E0: tyop_2Enum_2Enum] :
( ( p(c_2Earithmetic_2E_3C_3D_2E2(V0m_2E0,V2i_2E0))
& p(c_2Earithmetic_2E_3C_3D_2E2(V1n_2E0,V3j_2E0)) )
=> p(c_2Earithmetic_2E_3C_3D_2E2(c_2Earithmetic_2E_2A_2E2(V0m_2E0,V1n_2E0),c_2Earithmetic_2E_2A_2E2(V2i_2E0,V3j_2E0))) ) ).
tff(thm_2Earithmetic_2ERIGHT__SUB__DISTRIB,axiom,
! [V0m_2E0: tyop_2Enum_2Enum,V1n_2E0: tyop_2Enum_2Enum,V2p_2E0: tyop_2Enum_2Enum] : c_2Earithmetic_2E_2A_2E2(c_2Earithmetic_2E_2D_2E2(V0m_2E0,V1n_2E0),V2p_2E0) = c_2Earithmetic_2E_2D_2E2(c_2Earithmetic_2E_2A_2E2(V0m_2E0,V2p_2E0),c_2Earithmetic_2E_2A_2E2(V1n_2E0,V2p_2E0)) ).
tff(thm_2Earithmetic_2ELEFT__SUB__DISTRIB,axiom,
! [V0m_2E0: tyop_2Enum_2Enum,V1n_2E0: tyop_2Enum_2Enum,V2p_2E0: tyop_2Enum_2Enum] : c_2Earithmetic_2E_2A_2E2(V2p_2E0,c_2Earithmetic_2E_2D_2E2(V0m_2E0,V1n_2E0)) = c_2Earithmetic_2E_2D_2E2(c_2Earithmetic_2E_2A_2E2(V2p_2E0,V0m_2E0),c_2Earithmetic_2E_2A_2E2(V2p_2E0,V1n_2E0)) ).
tff(thm_2Earithmetic_2ELESS__ADD__1,axiom,
! [V0m_2E0: tyop_2Enum_2Enum,V1n_2E0: tyop_2Enum_2Enum] :
( p(c_2Eprim__rec_2E_3C_2E2(V1n_2E0,V0m_2E0))
=> ? [V2p_2E0: tyop_2Enum_2Enum] : V0m_2E0 = c_2Earithmetic_2E_2B_2E2(V1n_2E0,c_2Earithmetic_2E_2B_2E2(V2p_2E0,c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)))) ) ).
tff(thm_2Earithmetic_2EEXP__ADD,axiom,
! [V0p_2E0: tyop_2Enum_2Enum,V1q_2E0: tyop_2Enum_2Enum,V2n_2E0: tyop_2Enum_2Enum] : c_2Earithmetic_2EEXP_2E2(V2n_2E0,c_2Earithmetic_2E_2B_2E2(V0p_2E0,V1q_2E0)) = c_2Earithmetic_2E_2A_2E2(c_2Earithmetic_2EEXP_2E2(V2n_2E0,V0p_2E0),c_2Earithmetic_2EEXP_2E2(V2n_2E0,V1q_2E0)) ).
tff(thm_2Earithmetic_2ENOT__ODD__EQ__EVEN,axiom,
! [V0n_2E0: tyop_2Enum_2Enum,V1m_2E0: tyop_2Enum_2Enum] : c_2Enum_2ESUC_2E1(c_2Earithmetic_2E_2B_2E2(V0n_2E0,V0n_2E0)) != c_2Earithmetic_2E_2B_2E2(V1m_2E0,V1m_2E0) ).
tff(thm_2Earithmetic_2ELESS__EQUAL__ANTISYM,axiom,
! [V0n_2E0: tyop_2Enum_2Enum,V1m_2E0: tyop_2Enum_2Enum] :
( ( p(c_2Earithmetic_2E_3C_3D_2E2(V0n_2E0,V1m_2E0))
& p(c_2Earithmetic_2E_3C_3D_2E2(V1m_2E0,V0n_2E0)) )
=> ( V0n_2E0 = V1m_2E0 ) ) ).
tff(thm_2Earithmetic_2ELESS__ADD__SUC,axiom,
! [V0m_2E0: tyop_2Enum_2Enum,V1n_2E0: tyop_2Enum_2Enum] : p(c_2Eprim__rec_2E_3C_2E2(V0m_2E0,c_2Earithmetic_2E_2B_2E2(V0m_2E0,c_2Enum_2ESUC_2E1(V1n_2E0)))) ).
tff(thm_2Earithmetic_2ELESS__OR__EQ__ADD,axiom,
! [V0n_2E0: tyop_2Enum_2Enum,V1m_2E0: tyop_2Enum_2Enum] :
( p(c_2Eprim__rec_2E_3C_2E2(V0n_2E0,V1m_2E0))
| ? [V2p_2E0: tyop_2Enum_2Enum] : V0n_2E0 = c_2Earithmetic_2E_2B_2E2(V2p_2E0,V1m_2E0) ) ).
tff(thm_2Earithmetic_2EWOP,axiom,
! [V0P_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)] :
( ? [V1n_2E0: tyop_2Enum_2Enum] : p(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,V0P_2E0,V1n_2E0))
=> ? [V2n_2E0: tyop_2Enum_2Enum] :
( p(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,V0P_2E0,V2n_2E0))
& ! [V3m_2E0: tyop_2Enum_2Enum] :
( p(c_2Eprim__rec_2E_3C_2E2(V3m_2E0,V2n_2E0))
=> ~ p(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,V0P_2E0,V3m_2E0)) ) ) ) ).
tff(thm_2Earithmetic_2ECOMPLETE__INDUCTION,axiom,
! [V0P_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)] :
( ! [V1n_2E0: tyop_2Enum_2Enum] :
( ! [V2m_2E0: tyop_2Enum_2Enum] :
( p(c_2Eprim__rec_2E_3C_2E2(V2m_2E0,V1n_2E0))
=> p(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,V0P_2E0,V2m_2E0)) )
=> p(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,V0P_2E0,V1n_2E0)) )
=> ! [V3n_2E0: tyop_2Enum_2Enum] : p(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,V0P_2E0,V3n_2E0)) ) ).
tff(thm_2Earithmetic_2EFORALL__NUM__THM,axiom,
! [V0P_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)] :
( ! [V1n_2E0: tyop_2Enum_2Enum] : p(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,V0P_2E0,V1n_2E0))
<=> ( p(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,V0P_2E0,c_2Enum_2E0_2E0))
& ! [V2n_2E0: tyop_2Enum_2Enum] :
( p(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,V0P_2E0,V2n_2E0))
=> p(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,V0P_2E0,c_2Enum_2ESUC_2E1(V2n_2E0))) ) ) ) ).
tff(thm_2Earithmetic_2ESUC__SUB,axiom,
! [V0a_2E0: tyop_2Enum_2Enum] : c_2Earithmetic_2E_2D_2E2(c_2Enum_2ESUC_2E1(V0a_2E0),V0a_2E0) = c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)) ).
tff(thm_2Earithmetic_2ESUB__PLUS,axiom,
! [V0a_2E0: tyop_2Enum_2Enum,V1b_2E0: tyop_2Enum_2Enum,V2c_2E0: tyop_2Enum_2Enum] : c_2Earithmetic_2E_2D_2E2(V0a_2E0,c_2Earithmetic_2E_2B_2E2(V1b_2E0,V2c_2E0)) = c_2Earithmetic_2E_2D_2E2(c_2Earithmetic_2E_2D_2E2(V0a_2E0,V1b_2E0),V2c_2E0) ).
tff(thm_2Earithmetic_2EINV__PRE__LESS,axiom,
! [V0m_2E0: tyop_2Enum_2Enum] :
( p(c_2Eprim__rec_2E_3C_2E2(c_2Enum_2E0_2E0,V0m_2E0))
=> ! [V1n_2E0: tyop_2Enum_2Enum] : c_2Eprim__rec_2E_3C_2E2(c_2Eprim__rec_2EPRE_2E1(V0m_2E0),c_2Eprim__rec_2EPRE_2E1(V1n_2E0)) = c_2Eprim__rec_2E_3C_2E2(V0m_2E0,V1n_2E0) ) ).
tff(thm_2Earithmetic_2EINV__PRE__LESS__EQ,axiom,
! [V0n_2E0: tyop_2Enum_2Enum] :
( p(c_2Eprim__rec_2E_3C_2E2(c_2Enum_2E0_2E0,V0n_2E0))
=> ! [V1m_2E0: tyop_2Enum_2Enum] : c_2Earithmetic_2E_3C_3D_2E2(c_2Eprim__rec_2EPRE_2E1(V1m_2E0),c_2Eprim__rec_2EPRE_2E1(V0n_2E0)) = c_2Earithmetic_2E_3C_3D_2E2(V1m_2E0,V0n_2E0) ) ).
tff(thm_2Earithmetic_2EPRE__LESS__EQ,axiom,
! [V0m_2E0: tyop_2Enum_2Enum,V1n_2E0: tyop_2Enum_2Enum] :
( p(c_2Earithmetic_2E_3C_3D_2E2(V0m_2E0,V1n_2E0))
=> p(c_2Earithmetic_2E_3C_3D_2E2(c_2Eprim__rec_2EPRE_2E1(V0m_2E0),c_2Eprim__rec_2EPRE_2E1(V1n_2E0))) ) ).
tff(thm_2Earithmetic_2ESUB__LESS__EQ,axiom,
! [V0n_2E0: tyop_2Enum_2Enum,V1m_2E0: tyop_2Enum_2Enum] : p(c_2Earithmetic_2E_3C_3D_2E2(c_2Earithmetic_2E_2D_2E2(V0n_2E0,V1m_2E0),V0n_2E0)) ).
tff(thm_2Earithmetic_2ESUB__EQ__EQ__0,axiom,
! [V0m_2E0: tyop_2Enum_2Enum,V1n_2E0: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_2D_2E2(V0m_2E0,V1n_2E0) = V0m_2E0 )
<=> ( ( V0m_2E0 = c_2Enum_2E0_2E0 )
| ( V1n_2E0 = c_2Enum_2E0_2E0 ) ) ) ).
tff(thm_2Earithmetic_2ESUB__LESS__0,axiom,
! [V0n_2E0: tyop_2Enum_2Enum,V1m_2E0: tyop_2Enum_2Enum] : c_2Eprim__rec_2E_3C_2E2(V1m_2E0,V0n_2E0) = c_2Eprim__rec_2E_3C_2E2(c_2Enum_2E0_2E0,c_2Earithmetic_2E_2D_2E2(V0n_2E0,V1m_2E0)) ).
tff(thm_2Earithmetic_2ESUB__LESS__OR,axiom,
! [V0m_2E0: tyop_2Enum_2Enum,V1n_2E0: tyop_2Enum_2Enum] :
( p(c_2Eprim__rec_2E_3C_2E2(V1n_2E0,V0m_2E0))
=> p(c_2Earithmetic_2E_3C_3D_2E2(V1n_2E0,c_2Earithmetic_2E_2D_2E2(V0m_2E0,c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))))) ) ).
tff(thm_2Earithmetic_2ELESS__SUB__ADD__LESS,axiom,
! [V0n_2E0: tyop_2Enum_2Enum,V1m_2E0: tyop_2Enum_2Enum,V2i_2E0: tyop_2Enum_2Enum] :
( p(c_2Eprim__rec_2E_3C_2E2(V2i_2E0,c_2Earithmetic_2E_2D_2E2(V0n_2E0,V1m_2E0)))
=> p(c_2Eprim__rec_2E_3C_2E2(c_2Earithmetic_2E_2B_2E2(V2i_2E0,V1m_2E0),V0n_2E0)) ) ).
tff(thm_2Earithmetic_2ETIMES2,axiom,
! [V0n_2E0: tyop_2Enum_2Enum] : c_2Earithmetic_2E_2A_2E2(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EZERO_2E0)),V0n_2E0) = c_2Earithmetic_2E_2B_2E2(V0n_2E0,V0n_2E0) ).
tff(thm_2Earithmetic_2ELESS__MULT__MONO,axiom,
! [V0m_2E0: tyop_2Enum_2Enum,V1i_2E0: tyop_2Enum_2Enum,V2n_2E0: tyop_2Enum_2Enum] : c_2Eprim__rec_2E_3C_2E2(c_2Earithmetic_2E_2A_2E2(c_2Enum_2ESUC_2E1(V2n_2E0),V0m_2E0),c_2Earithmetic_2E_2A_2E2(c_2Enum_2ESUC_2E1(V2n_2E0),V1i_2E0)) = c_2Eprim__rec_2E_3C_2E2(V0m_2E0,V1i_2E0) ).
tff(thm_2Earithmetic_2EMULT__MONO__EQ,axiom,
! [V0m_2E0: tyop_2Enum_2Enum,V1i_2E0: tyop_2Enum_2Enum,V2n_2E0: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_2A_2E2(c_2Enum_2ESUC_2E1(V2n_2E0),V0m_2E0) = c_2Earithmetic_2E_2A_2E2(c_2Enum_2ESUC_2E1(V2n_2E0),V1i_2E0) )
<=> ( V0m_2E0 = V1i_2E0 ) ) ).
tff(thm_2Earithmetic_2EMULT__SUC__EQ,axiom,
! [V0p_2E0: tyop_2Enum_2Enum,V1m_2E0: tyop_2Enum_2Enum,V2n_2E0: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_2A_2E2(V2n_2E0,c_2Enum_2ESUC_2E1(V0p_2E0)) = c_2Earithmetic_2E_2A_2E2(V1m_2E0,c_2Enum_2ESUC_2E1(V0p_2E0)) )
<=> ( V2n_2E0 = V1m_2E0 ) ) ).
tff(thm_2Earithmetic_2EMULT__EXP__MONO,axiom,
! [V0p_2E0: tyop_2Enum_2Enum,V1q_2E0: tyop_2Enum_2Enum,V2n_2E0: tyop_2Enum_2Enum,V3m_2E0: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_2A_2E2(V2n_2E0,c_2Earithmetic_2EEXP_2E2(c_2Enum_2ESUC_2E1(V1q_2E0),V0p_2E0)) = c_2Earithmetic_2E_2A_2E2(V3m_2E0,c_2Earithmetic_2EEXP_2E2(c_2Enum_2ESUC_2E1(V1q_2E0),V0p_2E0)) )
<=> ( V2n_2E0 = V3m_2E0 ) ) ).
tff(thm_2Earithmetic_2EEQ__ADD__LCANCEL,axiom,
! [V0m_2E0: tyop_2Enum_2Enum,V1n_2E0: tyop_2Enum_2Enum,V2p_2E0: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_2B_2E2(V0m_2E0,V1n_2E0) = c_2Earithmetic_2E_2B_2E2(V0m_2E0,V2p_2E0) )
<=> ( V1n_2E0 = V2p_2E0 ) ) ).
tff(thm_2Earithmetic_2EEQ__ADD__RCANCEL,axiom,
! [V0m_2E0: tyop_2Enum_2Enum,V1n_2E0: tyop_2Enum_2Enum,V2p_2E0: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_2B_2E2(V0m_2E0,V2p_2E0) = c_2Earithmetic_2E_2B_2E2(V1n_2E0,V2p_2E0) )
<=> ( V0m_2E0 = V1n_2E0 ) ) ).
tff(thm_2Earithmetic_2EEQ__MULT__LCANCEL,axiom,
! [V0m_2E0: tyop_2Enum_2Enum,V1n_2E0: tyop_2Enum_2Enum,V2p_2E0: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_2A_2E2(V0m_2E0,V1n_2E0) = c_2Earithmetic_2E_2A_2E2(V0m_2E0,V2p_2E0) )
<=> ( ( V0m_2E0 = c_2Enum_2E0_2E0 )
| ( V1n_2E0 = V2p_2E0 ) ) ) ).
tff(thm_2Earithmetic_2EEQ__MULT__RCANCEL,axiom,
! [V0m_2E0: tyop_2Enum_2Enum,V1n_2E0: tyop_2Enum_2Enum,V2p_2E0: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_2A_2E2(V1n_2E0,V0m_2E0) = c_2Earithmetic_2E_2A_2E2(V2p_2E0,V0m_2E0) )
<=> ( ( V0m_2E0 = c_2Enum_2E0_2E0 )
| ( V1n_2E0 = V2p_2E0 ) ) ) ).
tff(thm_2Earithmetic_2EADD__SUB,axiom,
! [V0a_2E0: tyop_2Enum_2Enum,V1c_2E0: tyop_2Enum_2Enum] : c_2Earithmetic_2E_2D_2E2(c_2Earithmetic_2E_2B_2E2(V0a_2E0,V1c_2E0),V1c_2E0) = V0a_2E0 ).
tff(thm_2Earithmetic_2ELESS__EQ__ADD__SUB,axiom,
! [V0c_2E0: tyop_2Enum_2Enum,V1b_2E0: tyop_2Enum_2Enum] :
( p(c_2Earithmetic_2E_3C_3D_2E2(V0c_2E0,V1b_2E0))
=> ! [V2a_2E0: tyop_2Enum_2Enum] : c_2Earithmetic_2E_2D_2E2(c_2Earithmetic_2E_2B_2E2(V2a_2E0,V1b_2E0),V0c_2E0) = c_2Earithmetic_2E_2B_2E2(V2a_2E0,c_2Earithmetic_2E_2D_2E2(V1b_2E0,V0c_2E0)) ) ).
tff(thm_2Earithmetic_2ESUB__EQUAL__0,axiom,
! [V0c_2E0: tyop_2Enum_2Enum] : c_2Earithmetic_2E_2D_2E2(V0c_2E0,V0c_2E0) = c_2Enum_2E0_2E0 ).
tff(thm_2Earithmetic_2ELESS__EQ__SUB__LESS,axiom,
! [V0a_2E0: tyop_2Enum_2Enum,V1b_2E0: tyop_2Enum_2Enum] :
( p(c_2Earithmetic_2E_3C_3D_2E2(V1b_2E0,V0a_2E0))
=> ! [V2c_2E0: tyop_2Enum_2Enum] : c_2Eprim__rec_2E_3C_2E2(c_2Earithmetic_2E_2D_2E2(V0a_2E0,V1b_2E0),V2c_2E0) = c_2Eprim__rec_2E_3C_2E2(V0a_2E0,c_2Earithmetic_2E_2B_2E2(V1b_2E0,V2c_2E0)) ) ).
tff(thm_2Earithmetic_2ENOT__SUC__LESS__EQ,axiom,
! [V0n_2E0: tyop_2Enum_2Enum,V1m_2E0: tyop_2Enum_2Enum] :
( ~ p(c_2Earithmetic_2E_3C_3D_2E2(c_2Enum_2ESUC_2E1(V0n_2E0),V1m_2E0))
<=> p(c_2Earithmetic_2E_3C_3D_2E2(V1m_2E0,V0n_2E0)) ) ).
tff(thm_2Earithmetic_2ESUB__SUB,axiom,
! [V0b_2E0: tyop_2Enum_2Enum,V1c_2E0: tyop_2Enum_2Enum] :
( p(c_2Earithmetic_2E_3C_3D_2E2(V1c_2E0,V0b_2E0))
=> ! [V2a_2E0: tyop_2Enum_2Enum] : c_2Earithmetic_2E_2D_2E2(V2a_2E0,c_2Earithmetic_2E_2D_2E2(V0b_2E0,V1c_2E0)) = c_2Earithmetic_2E_2D_2E2(c_2Earithmetic_2E_2B_2E2(V2a_2E0,V1c_2E0),V0b_2E0) ) ).
tff(thm_2Earithmetic_2ELESS__IMP__LESS__ADD,axiom,
! [V0n_2E0: tyop_2Enum_2Enum,V1m_2E0: tyop_2Enum_2Enum] :
( p(c_2Eprim__rec_2E_3C_2E2(V0n_2E0,V1m_2E0))
=> ! [V2p_2E0: tyop_2Enum_2Enum] : p(c_2Eprim__rec_2E_3C_2E2(V0n_2E0,c_2Earithmetic_2E_2B_2E2(V1m_2E0,V2p_2E0))) ) ).
tff(thm_2Earithmetic_2ESUB__LESS__EQ__ADD,axiom,
! [V0m_2E0: tyop_2Enum_2Enum,V1p_2E0: tyop_2Enum_2Enum] :
( p(c_2Earithmetic_2E_3C_3D_2E2(V0m_2E0,V1p_2E0))
=> ! [V2n_2E0: tyop_2Enum_2Enum] : c_2Earithmetic_2E_3C_3D_2E2(c_2Earithmetic_2E_2D_2E2(V1p_2E0,V0m_2E0),V2n_2E0) = c_2Earithmetic_2E_3C_3D_2E2(V1p_2E0,c_2Earithmetic_2E_2B_2E2(V0m_2E0,V2n_2E0)) ) ).
tff(thm_2Earithmetic_2ESUB__LESS__SUC,axiom,
! [V0p_2E0: tyop_2Enum_2Enum,V1m_2E0: tyop_2Enum_2Enum] : p(c_2Eprim__rec_2E_3C_2E2(c_2Earithmetic_2E_2D_2E2(V0p_2E0,V1m_2E0),c_2Enum_2ESUC_2E1(V0p_2E0))) ).
tff(thm_2Earithmetic_2ESUB__CANCEL,axiom,
! [V0p_2E0: tyop_2Enum_2Enum,V1n_2E0: tyop_2Enum_2Enum,V2m_2E0: tyop_2Enum_2Enum] :
( ( p(c_2Earithmetic_2E_3C_3D_2E2(V1n_2E0,V0p_2E0))
& p(c_2Earithmetic_2E_3C_3D_2E2(V2m_2E0,V0p_2E0)) )
=> ( ( c_2Earithmetic_2E_2D_2E2(V0p_2E0,V1n_2E0) = c_2Earithmetic_2E_2D_2E2(V0p_2E0,V2m_2E0) )
<=> ( V1n_2E0 = V2m_2E0 ) ) ) ).
tff(thm_2Earithmetic_2ECANCEL__SUB,axiom,
! [V0p_2E0: tyop_2Enum_2Enum,V1n_2E0: tyop_2Enum_2Enum,V2m_2E0: tyop_2Enum_2Enum] :
( ( p(c_2Earithmetic_2E_3C_3D_2E2(V0p_2E0,V1n_2E0))
& p(c_2Earithmetic_2E_3C_3D_2E2(V0p_2E0,V2m_2E0)) )
=> ( ( c_2Earithmetic_2E_2D_2E2(V1n_2E0,V0p_2E0) = c_2Earithmetic_2E_2D_2E2(V2m_2E0,V0p_2E0) )
<=> ( V1n_2E0 = V2m_2E0 ) ) ) ).
tff(thm_2Earithmetic_2ENOT__EXP__0,axiom,
! [V0m_2E0: tyop_2Enum_2Enum,V1n_2E0: tyop_2Enum_2Enum] : c_2Earithmetic_2EEXP_2E2(c_2Enum_2ESUC_2E1(V1n_2E0),V0m_2E0) != c_2Enum_2E0_2E0 ).
tff(thm_2Earithmetic_2EZERO__LESS__EXP,axiom,
! [V0m_2E0: tyop_2Enum_2Enum,V1n_2E0: tyop_2Enum_2Enum] : p(c_2Eprim__rec_2E_3C_2E2(c_2Enum_2E0_2E0,c_2Earithmetic_2EEXP_2E2(c_2Enum_2ESUC_2E1(V1n_2E0),V0m_2E0))) ).
tff(thm_2Earithmetic_2EODD__OR__EVEN,axiom,
! [V0n_2E0: tyop_2Enum_2Enum] :
? [V1m_2E0: tyop_2Enum_2Enum] :
( ( V0n_2E0 = c_2Earithmetic_2E_2A_2E2(c_2Enum_2ESUC_2E1(c_2Enum_2ESUC_2E1(c_2Enum_2E0_2E0)),V1m_2E0) )
| ( V0n_2E0 = c_2Earithmetic_2E_2B_2E2(c_2Earithmetic_2E_2A_2E2(c_2Enum_2ESUC_2E1(c_2Enum_2ESUC_2E1(c_2Enum_2E0_2E0)),V1m_2E0),c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))) ) ) ).
tff(thm_2Earithmetic_2ELESS__EXP__SUC__MONO,axiom,
! [V0n_2E0: tyop_2Enum_2Enum,V1m_2E0: tyop_2Enum_2Enum] : p(c_2Eprim__rec_2E_3C_2E2(c_2Earithmetic_2EEXP_2E2(c_2Enum_2ESUC_2E1(c_2Enum_2ESUC_2E1(V1m_2E0)),V0n_2E0),c_2Earithmetic_2EEXP_2E2(c_2Enum_2ESUC_2E1(c_2Enum_2ESUC_2E1(V1m_2E0)),c_2Enum_2ESUC_2E1(V0n_2E0)))) ).
tff(thm_2Earithmetic_2ELESS__LESS__CASES,axiom,
! [V0m_2E0: tyop_2Enum_2Enum,V1n_2E0: tyop_2Enum_2Enum] :
( ( V0m_2E0 = V1n_2E0 )
| p(c_2Eprim__rec_2E_3C_2E2(V0m_2E0,V1n_2E0))
| p(c_2Eprim__rec_2E_3C_2E2(V1n_2E0,V0m_2E0)) ) ).
tff(thm_2Earithmetic_2EGREATER__EQ,axiom,
! [V0n_2E0: tyop_2Enum_2Enum,V1m_2E0: tyop_2Enum_2Enum] : c_2Earithmetic_2E_3E_3D_2E2(V0n_2E0,V1m_2E0) = c_2Earithmetic_2E_3C_3D_2E2(V1m_2E0,V0n_2E0) ).
tff(thm_2Earithmetic_2ELESS__EQ__CASES,axiom,
! [V0m_2E0: tyop_2Enum_2Enum,V1n_2E0: tyop_2Enum_2Enum] :
( p(c_2Earithmetic_2E_3C_3D_2E2(V0m_2E0,V1n_2E0))
| p(c_2Earithmetic_2E_3C_3D_2E2(V1n_2E0,V0m_2E0)) ) ).
tff(thm_2Earithmetic_2ELESS__EQUAL__ADD,axiom,
! [V0m_2E0: tyop_2Enum_2Enum,V1n_2E0: tyop_2Enum_2Enum] :
( p(c_2Earithmetic_2E_3C_3D_2E2(V0m_2E0,V1n_2E0))
=> ? [V2p_2E0: tyop_2Enum_2Enum] : V1n_2E0 = c_2Earithmetic_2E_2B_2E2(V0m_2E0,V2p_2E0) ) ).
tff(thm_2Earithmetic_2ELESS__EQ__EXISTS,axiom,
! [V0m_2E0: tyop_2Enum_2Enum,V1n_2E0: tyop_2Enum_2Enum] :
( p(c_2Earithmetic_2E_3C_3D_2E2(V0m_2E0,V1n_2E0))
<=> ? [V2p_2E0: tyop_2Enum_2Enum] : V1n_2E0 = c_2Earithmetic_2E_2B_2E2(V0m_2E0,V2p_2E0) ) ).
tff(thm_2Earithmetic_2EMULT__EQ__0,axiom,
! [V0m_2E0: tyop_2Enum_2Enum,V1n_2E0: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_2A_2E2(V0m_2E0,V1n_2E0) = c_2Enum_2E0_2E0 )
<=> ( ( V0m_2E0 = c_2Enum_2E0_2E0 )
| ( V1n_2E0 = c_2Enum_2E0_2E0 ) ) ) ).
tff(thm_2Earithmetic_2EMULT__EQ__1,axiom,
! [V0x_2E0: tyop_2Enum_2Enum,V1y_2E0: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_2A_2E2(V0x_2E0,V1y_2E0) = 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)) )
& ( V1y_2E0 = c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)) ) ) ) ).
tff(thm_2Earithmetic_2EMULT__EQ__ID,axiom,
! [V0m_2E0: tyop_2Enum_2Enum,V1n_2E0: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_2A_2E2(V0m_2E0,V1n_2E0) = V1n_2E0 )
<=> ( ( V0m_2E0 = c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)) )
| ( V1n_2E0 = c_2Enum_2E0_2E0 ) ) ) ).
tff(thm_2Earithmetic_2ELESS__MULT2,axiom,
! [V0m_2E0: tyop_2Enum_2Enum,V1n_2E0: tyop_2Enum_2Enum] :
( ( p(c_2Eprim__rec_2E_3C_2E2(c_2Enum_2E0_2E0,V0m_2E0))
& p(c_2Eprim__rec_2E_3C_2E2(c_2Enum_2E0_2E0,V1n_2E0)) )
=> p(c_2Eprim__rec_2E_3C_2E2(c_2Enum_2E0_2E0,c_2Earithmetic_2E_2A_2E2(V0m_2E0,V1n_2E0))) ) ).
tff(thm_2Earithmetic_2EZERO__LESS__MULT,axiom,
! [V0m_2E0: tyop_2Enum_2Enum,V1n_2E0: tyop_2Enum_2Enum] :
( p(c_2Eprim__rec_2E_3C_2E2(c_2Enum_2E0_2E0,c_2Earithmetic_2E_2A_2E2(V0m_2E0,V1n_2E0)))
<=> ( p(c_2Eprim__rec_2E_3C_2E2(c_2Enum_2E0_2E0,V0m_2E0))
& p(c_2Eprim__rec_2E_3C_2E2(c_2Enum_2E0_2E0,V1n_2E0)) ) ) ).
tff(thm_2Earithmetic_2EZERO__LESS__ADD,axiom,
! [V0m_2E0: tyop_2Enum_2Enum,V1n_2E0: tyop_2Enum_2Enum] :
( p(c_2Eprim__rec_2E_3C_2E2(c_2Enum_2E0_2E0,c_2Earithmetic_2E_2B_2E2(V0m_2E0,V1n_2E0)))
<=> ( p(c_2Eprim__rec_2E_3C_2E2(c_2Enum_2E0_2E0,V0m_2E0))
| p(c_2Eprim__rec_2E_3C_2E2(c_2Enum_2E0_2E0,V1n_2E0)) ) ) ).
tff(thm_2Earithmetic_2EFACT__LESS,axiom,
! [V0n_2E0: tyop_2Enum_2Enum] : p(c_2Eprim__rec_2E_3C_2E2(c_2Enum_2E0_2E0,c_2Earithmetic_2EFACT_2E1(V0n_2E0))) ).
tff(thm_2Earithmetic_2EEVEN__ODD,axiom,
! [V0n_2E0: tyop_2Enum_2Enum] :
( p(c_2Earithmetic_2EEVEN_2E1(V0n_2E0))
<=> ~ p(c_2Earithmetic_2EODD_2E1(V0n_2E0)) ) ).
tff(thm_2Earithmetic_2EODD__EVEN,axiom,
! [V0n_2E0: tyop_2Enum_2Enum] :
( p(c_2Earithmetic_2EODD_2E1(V0n_2E0))
<=> ~ p(c_2Earithmetic_2EEVEN_2E1(V0n_2E0)) ) ).
tff(thm_2Earithmetic_2EEVEN__OR__ODD,axiom,
! [V0n_2E0: tyop_2Enum_2Enum] :
( p(c_2Earithmetic_2EEVEN_2E1(V0n_2E0))
| p(c_2Earithmetic_2EODD_2E1(V0n_2E0)) ) ).
tff(thm_2Earithmetic_2EEVEN__AND__ODD,axiom,
! [V0n_2E0: tyop_2Enum_2Enum] :
~ ( p(c_2Earithmetic_2EEVEN_2E1(V0n_2E0))
& p(c_2Earithmetic_2EODD_2E1(V0n_2E0)) ) ).
tff(thm_2Earithmetic_2EEVEN__ADD,axiom,
! [V0m_2E0: tyop_2Enum_2Enum,V1n_2E0: tyop_2Enum_2Enum] :
( p(c_2Earithmetic_2EEVEN_2E1(c_2Earithmetic_2E_2B_2E2(V0m_2E0,V1n_2E0)))
<=> ( c_2Earithmetic_2EEVEN_2E1(V0m_2E0) = c_2Earithmetic_2EEVEN_2E1(V1n_2E0) ) ) ).
tff(thm_2Earithmetic_2EEVEN__MULT,axiom,
! [V0m_2E0: tyop_2Enum_2Enum,V1n_2E0: tyop_2Enum_2Enum] :
( p(c_2Earithmetic_2EEVEN_2E1(c_2Earithmetic_2E_2A_2E2(V0m_2E0,V1n_2E0)))
<=> ( p(c_2Earithmetic_2EEVEN_2E1(V0m_2E0))
| p(c_2Earithmetic_2EEVEN_2E1(V1n_2E0)) ) ) ).
tff(thm_2Earithmetic_2EODD__ADD,axiom,
! [V0m_2E0: tyop_2Enum_2Enum,V1n_2E0: tyop_2Enum_2Enum] :
( p(c_2Earithmetic_2EODD_2E1(c_2Earithmetic_2E_2B_2E2(V0m_2E0,V1n_2E0)))
<=> ( c_2Earithmetic_2EODD_2E1(V0m_2E0) != c_2Earithmetic_2EODD_2E1(V1n_2E0) ) ) ).
tff(thm_2Earithmetic_2EODD__MULT,axiom,
! [V0m_2E0: tyop_2Enum_2Enum,V1n_2E0: tyop_2Enum_2Enum] :
( p(c_2Earithmetic_2EODD_2E1(c_2Earithmetic_2E_2A_2E2(V0m_2E0,V1n_2E0)))
<=> ( p(c_2Earithmetic_2EODD_2E1(V0m_2E0))
& p(c_2Earithmetic_2EODD_2E1(V1n_2E0)) ) ) ).
tff(thm_2Earithmetic_2EEVEN__DOUBLE,axiom,
! [V0n_2E0: tyop_2Enum_2Enum] : p(c_2Earithmetic_2EEVEN_2E1(c_2Earithmetic_2E_2A_2E2(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EZERO_2E0)),V0n_2E0))) ).
tff(thm_2Earithmetic_2EODD__DOUBLE,axiom,
! [V0n_2E0: tyop_2Enum_2Enum] : p(c_2Earithmetic_2EODD_2E1(c_2Enum_2ESUC_2E1(c_2Earithmetic_2E_2A_2E2(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EZERO_2E0)),V0n_2E0)))) ).
tff(thm_2Earithmetic_2EEVEN__ODD__EXISTS,axiom,
! [V0n_2E0: tyop_2Enum_2Enum] :
( ( p(c_2Earithmetic_2EEVEN_2E1(V0n_2E0))
=> ? [V1m_2E0: tyop_2Enum_2Enum] : V0n_2E0 = c_2Earithmetic_2E_2A_2E2(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EZERO_2E0)),V1m_2E0) )
& ( p(c_2Earithmetic_2EODD_2E1(V0n_2E0))
=> ? [V2m_2E0: tyop_2Enum_2Enum] : V0n_2E0 = c_2Enum_2ESUC_2E1(c_2Earithmetic_2E_2A_2E2(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EZERO_2E0)),V2m_2E0)) ) ) ).
tff(thm_2Earithmetic_2EEVEN__EXISTS,axiom,
! [V0n_2E0: tyop_2Enum_2Enum] :
( p(c_2Earithmetic_2EEVEN_2E1(V0n_2E0))
<=> ? [V1m_2E0: tyop_2Enum_2Enum] : V0n_2E0 = c_2Earithmetic_2E_2A_2E2(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EZERO_2E0)),V1m_2E0) ) ).
tff(thm_2Earithmetic_2EODD__EXISTS,axiom,
! [V0n_2E0: tyop_2Enum_2Enum] :
( p(c_2Earithmetic_2EODD_2E1(V0n_2E0))
<=> ? [V1m_2E0: tyop_2Enum_2Enum] : V0n_2E0 = c_2Enum_2ESUC_2E1(c_2Earithmetic_2E_2A_2E2(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EZERO_2E0)),V1m_2E0)) ) ).
tff(thm_2Earithmetic_2EEVEN__EXP__IFF,axiom,
! [V0n_2E0: tyop_2Enum_2Enum,V1m_2E0: tyop_2Enum_2Enum] :
( p(c_2Earithmetic_2EEVEN_2E1(c_2Earithmetic_2EEXP_2E2(V1m_2E0,V0n_2E0)))
<=> ( p(c_2Eprim__rec_2E_3C_2E2(c_2Enum_2E0_2E0,V0n_2E0))
& p(c_2Earithmetic_2EEVEN_2E1(V1m_2E0)) ) ) ).
tff(thm_2Earithmetic_2EEVEN__EXP,axiom,
! [V0m_2E0: tyop_2Enum_2Enum,V1n_2E0: tyop_2Enum_2Enum] :
( ( p(c_2Eprim__rec_2E_3C_2E2(c_2Enum_2E0_2E0,V1n_2E0))
& p(c_2Earithmetic_2EEVEN_2E1(V0m_2E0)) )
=> p(c_2Earithmetic_2EEVEN_2E1(c_2Earithmetic_2EEXP_2E2(V0m_2E0,V1n_2E0))) ) ).
tff(thm_2Earithmetic_2EODD__EXP__IFF,axiom,
! [V0n_2E0: tyop_2Enum_2Enum,V1m_2E0: tyop_2Enum_2Enum] :
( p(c_2Earithmetic_2EODD_2E1(c_2Earithmetic_2EEXP_2E2(V1m_2E0,V0n_2E0)))
<=> ( ( V0n_2E0 = c_2Enum_2E0_2E0 )
| p(c_2Earithmetic_2EODD_2E1(V1m_2E0)) ) ) ).
tff(thm_2Earithmetic_2EODD__EXP,axiom,
! [V0m_2E0: tyop_2Enum_2Enum,V1n_2E0: tyop_2Enum_2Enum] :
( ( p(c_2Eprim__rec_2E_3C_2E2(c_2Enum_2E0_2E0,V1n_2E0))
& p(c_2Earithmetic_2EODD_2E1(V0m_2E0)) )
=> p(c_2Earithmetic_2EODD_2E1(c_2Earithmetic_2EEXP_2E2(V0m_2E0,V1n_2E0))) ) ).
tff(thm_2Earithmetic_2EEQ__LESS__EQ,axiom,
! [V0m_2E0: tyop_2Enum_2Enum,V1n_2E0: tyop_2Enum_2Enum] :
( ( V0m_2E0 = V1n_2E0 )
<=> ( p(c_2Earithmetic_2E_3C_3D_2E2(V0m_2E0,V1n_2E0))
& p(c_2Earithmetic_2E_3C_3D_2E2(V1n_2E0,V0m_2E0)) ) ) ).
tff(thm_2Earithmetic_2EADD__MONO__LESS__EQ,axiom,
! [V0m_2E0: tyop_2Enum_2Enum,V1n_2E0: tyop_2Enum_2Enum,V2p_2E0: tyop_2Enum_2Enum] : c_2Earithmetic_2E_3C_3D_2E2(c_2Earithmetic_2E_2B_2E2(V0m_2E0,V1n_2E0),c_2Earithmetic_2E_2B_2E2(V0m_2E0,V2p_2E0)) = c_2Earithmetic_2E_3C_3D_2E2(V1n_2E0,V2p_2E0) ).
tff(thm_2Earithmetic_2ELE__ADD__LCANCEL,axiom,
! [V0m_2E0: tyop_2Enum_2Enum,V1n_2E0: tyop_2Enum_2Enum,V2p_2E0: tyop_2Enum_2Enum] : c_2Earithmetic_2E_3C_3D_2E2(c_2Earithmetic_2E_2B_2E2(V0m_2E0,V1n_2E0),c_2Earithmetic_2E_2B_2E2(V0m_2E0,V2p_2E0)) = c_2Earithmetic_2E_3C_3D_2E2(V1n_2E0,V2p_2E0) ).
tff(thm_2Earithmetic_2ELE__ADD__RCANCEL,axiom,
! [V0m_2E0: tyop_2Enum_2Enum,V1n_2E0: tyop_2Enum_2Enum,V2p_2E0: tyop_2Enum_2Enum] : c_2Earithmetic_2E_3C_3D_2E2(c_2Earithmetic_2E_2B_2E2(V1n_2E0,V0m_2E0),c_2Earithmetic_2E_2B_2E2(V2p_2E0,V0m_2E0)) = c_2Earithmetic_2E_3C_3D_2E2(V1n_2E0,V2p_2E0) ).
tff(thm_2Earithmetic_2ENOT__LEQ,axiom,
! [V0m_2E0: tyop_2Enum_2Enum,V1n_2E0: tyop_2Enum_2Enum] :
( ~ p(c_2Earithmetic_2E_3C_3D_2E2(V0m_2E0,V1n_2E0))
<=> p(c_2Earithmetic_2E_3C_3D_2E2(c_2Enum_2ESUC_2E1(V1n_2E0),V0m_2E0)) ) ).
tff(thm_2Earithmetic_2ENOT__NUM__EQ,axiom,
! [V0m_2E0: tyop_2Enum_2Enum,V1n_2E0: tyop_2Enum_2Enum] :
( ( V0m_2E0 != V1n_2E0 )
<=> ( p(c_2Earithmetic_2E_3C_3D_2E2(c_2Enum_2ESUC_2E1(V0m_2E0),V1n_2E0))
| p(c_2Earithmetic_2E_3C_3D_2E2(c_2Enum_2ESUC_2E1(V1n_2E0),V0m_2E0)) ) ) ).
tff(thm_2Earithmetic_2ENOT__GREATER,axiom,
! [V0m_2E0: tyop_2Enum_2Enum,V1n_2E0: tyop_2Enum_2Enum] :
( ~ p(c_2Earithmetic_2E_3E_2E2(V0m_2E0,V1n_2E0))
<=> p(c_2Earithmetic_2E_3C_3D_2E2(V0m_2E0,V1n_2E0)) ) ).
tff(thm_2Earithmetic_2ENOT__GREATER__EQ,axiom,
! [V0m_2E0: tyop_2Enum_2Enum,V1n_2E0: tyop_2Enum_2Enum] :
( ~ p(c_2Earithmetic_2E_3E_3D_2E2(V0m_2E0,V1n_2E0))
<=> p(c_2Earithmetic_2E_3C_3D_2E2(c_2Enum_2ESUC_2E1(V0m_2E0),V1n_2E0)) ) ).
tff(thm_2Earithmetic_2ESUC__ONE__ADD,axiom,
! [V0n_2E0: tyop_2Enum_2Enum] : c_2Enum_2ESUC_2E1(V0n_2E0) = c_2Earithmetic_2E_2B_2E2(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)),V0n_2E0) ).
tff(thm_2Earithmetic_2ESUC__ADD__SYM,axiom,
! [V0m_2E0: tyop_2Enum_2Enum,V1n_2E0: tyop_2Enum_2Enum] : c_2Enum_2ESUC_2E1(c_2Earithmetic_2E_2B_2E2(V0m_2E0,V1n_2E0)) = c_2Earithmetic_2E_2B_2E2(c_2Enum_2ESUC_2E1(V1n_2E0),V0m_2E0) ).
tff(thm_2Earithmetic_2ENOT__SUC__ADD__LESS__EQ,axiom,
! [V0m_2E0: tyop_2Enum_2Enum,V1n_2E0: tyop_2Enum_2Enum] : ~ p(c_2Earithmetic_2E_3C_3D_2E2(c_2Enum_2ESUC_2E1(c_2Earithmetic_2E_2B_2E2(V0m_2E0,V1n_2E0)),V0m_2E0)) ).
tff(thm_2Earithmetic_2EMULT__LESS__EQ__SUC,axiom,
! [V0m_2E0: tyop_2Enum_2Enum,V1n_2E0: tyop_2Enum_2Enum,V2p_2E0: tyop_2Enum_2Enum] : c_2Earithmetic_2E_3C_3D_2E2(V0m_2E0,V1n_2E0) = c_2Earithmetic_2E_3C_3D_2E2(c_2Earithmetic_2E_2A_2E2(c_2Enum_2ESUC_2E1(V2p_2E0),V0m_2E0),c_2Earithmetic_2E_2A_2E2(c_2Enum_2ESUC_2E1(V2p_2E0),V1n_2E0)) ).
tff(thm_2Earithmetic_2ELE__MULT__LCANCEL,axiom,
! [V0m_2E0: tyop_2Enum_2Enum,V1n_2E0: tyop_2Enum_2Enum,V2p_2E0: tyop_2Enum_2Enum] :
( p(c_2Earithmetic_2E_3C_3D_2E2(c_2Earithmetic_2E_2A_2E2(V0m_2E0,V1n_2E0),c_2Earithmetic_2E_2A_2E2(V0m_2E0,V2p_2E0)))
<=> ( ( V0m_2E0 = c_2Enum_2E0_2E0 )
| p(c_2Earithmetic_2E_3C_3D_2E2(V1n_2E0,V2p_2E0)) ) ) ).
tff(thm_2Earithmetic_2ELE__MULT__RCANCEL,axiom,
! [V0m_2E0: tyop_2Enum_2Enum,V1n_2E0: tyop_2Enum_2Enum,V2p_2E0: tyop_2Enum_2Enum] :
( p(c_2Earithmetic_2E_3C_3D_2E2(c_2Earithmetic_2E_2A_2E2(V0m_2E0,V1n_2E0),c_2Earithmetic_2E_2A_2E2(V2p_2E0,V1n_2E0)))
<=> ( ( V1n_2E0 = c_2Enum_2E0_2E0 )
| p(c_2Earithmetic_2E_3C_3D_2E2(V0m_2E0,V2p_2E0)) ) ) ).
tff(thm_2Earithmetic_2ELT__MULT__LCANCEL,axiom,
! [V0m_2E0: tyop_2Enum_2Enum,V1n_2E0: tyop_2Enum_2Enum,V2p_2E0: tyop_2Enum_2Enum] :
( p(c_2Eprim__rec_2E_3C_2E2(c_2Earithmetic_2E_2A_2E2(V0m_2E0,V1n_2E0),c_2Earithmetic_2E_2A_2E2(V0m_2E0,V2p_2E0)))
<=> ( p(c_2Eprim__rec_2E_3C_2E2(c_2Enum_2E0_2E0,V0m_2E0))
& p(c_2Eprim__rec_2E_3C_2E2(V1n_2E0,V2p_2E0)) ) ) ).
tff(thm_2Earithmetic_2ELT__MULT__RCANCEL,axiom,
! [V0m_2E0: tyop_2Enum_2Enum,V1n_2E0: tyop_2Enum_2Enum,V2p_2E0: tyop_2Enum_2Enum] :
( p(c_2Eprim__rec_2E_3C_2E2(c_2Earithmetic_2E_2A_2E2(V0m_2E0,V1n_2E0),c_2Earithmetic_2E_2A_2E2(V2p_2E0,V1n_2E0)))
<=> ( p(c_2Eprim__rec_2E_3C_2E2(c_2Enum_2E0_2E0,V1n_2E0))
& p(c_2Eprim__rec_2E_3C_2E2(V0m_2E0,V2p_2E0)) ) ) ).
tff(thm_2Earithmetic_2ELT__MULT__CANCEL__LBARE,axiom,
! [V0n_2E0: tyop_2Enum_2Enum,V1m_2E0: tyop_2Enum_2Enum] :
( ( p(c_2Eprim__rec_2E_3C_2E2(V1m_2E0,c_2Earithmetic_2E_2A_2E2(V1m_2E0,V0n_2E0)))
<=> ( p(c_2Eprim__rec_2E_3C_2E2(c_2Enum_2E0_2E0,V1m_2E0))
& p(c_2Eprim__rec_2E_3C_2E2(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)),V0n_2E0)) ) )
& ( p(c_2Eprim__rec_2E_3C_2E2(V1m_2E0,c_2Earithmetic_2E_2A_2E2(V0n_2E0,V1m_2E0)))
<=> ( p(c_2Eprim__rec_2E_3C_2E2(c_2Enum_2E0_2E0,V1m_2E0))
& p(c_2Eprim__rec_2E_3C_2E2(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)),V0n_2E0)) ) ) ) ).
tff(thm_2Earithmetic_2ELT__MULT__CANCEL__RBARE,axiom,
! [V0n_2E0: tyop_2Enum_2Enum,V1m_2E0: tyop_2Enum_2Enum] :
( ( p(c_2Eprim__rec_2E_3C_2E2(c_2Earithmetic_2E_2A_2E2(V1m_2E0,V0n_2E0),V1m_2E0))
<=> ( p(c_2Eprim__rec_2E_3C_2E2(c_2Enum_2E0_2E0,V1m_2E0))
& ( V0n_2E0 = c_2Enum_2E0_2E0 ) ) )
& ( p(c_2Eprim__rec_2E_3C_2E2(c_2Earithmetic_2E_2A_2E2(V1m_2E0,V0n_2E0),V0n_2E0))
<=> ( p(c_2Eprim__rec_2E_3C_2E2(c_2Enum_2E0_2E0,V0n_2E0))
& ( V1m_2E0 = c_2Enum_2E0_2E0 ) ) ) ) ).
tff(thm_2Earithmetic_2ELE__MULT__CANCEL__LBARE,axiom,
! [V0n_2E0: tyop_2Enum_2Enum,V1m_2E0: tyop_2Enum_2Enum] :
( ( p(c_2Earithmetic_2E_3C_3D_2E2(V1m_2E0,c_2Earithmetic_2E_2A_2E2(V1m_2E0,V0n_2E0)))
<=> ( ( V1m_2E0 = c_2Enum_2E0_2E0 )
| p(c_2Eprim__rec_2E_3C_2E2(c_2Enum_2E0_2E0,V0n_2E0)) ) )
& ( p(c_2Earithmetic_2E_3C_3D_2E2(V1m_2E0,c_2Earithmetic_2E_2A_2E2(V0n_2E0,V1m_2E0)))
<=> ( ( V1m_2E0 = c_2Enum_2E0_2E0 )
| p(c_2Eprim__rec_2E_3C_2E2(c_2Enum_2E0_2E0,V0n_2E0)) ) ) ) ).
tff(thm_2Earithmetic_2ELE__MULT__CANCEL__RBARE,axiom,
! [V0n_2E0: tyop_2Enum_2Enum,V1m_2E0: tyop_2Enum_2Enum] :
( ( p(c_2Earithmetic_2E_3C_3D_2E2(c_2Earithmetic_2E_2A_2E2(V1m_2E0,V0n_2E0),V1m_2E0))
<=> ( ( V1m_2E0 = c_2Enum_2E0_2E0 )
| p(c_2Earithmetic_2E_3C_3D_2E2(V0n_2E0,c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)))) ) )
& ( p(c_2Earithmetic_2E_3C_3D_2E2(c_2Earithmetic_2E_2A_2E2(V1m_2E0,V0n_2E0),V0n_2E0))
<=> ( ( V0n_2E0 = c_2Enum_2E0_2E0 )
| p(c_2Earithmetic_2E_3C_3D_2E2(V1m_2E0,c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)))) ) ) ) ).
tff(thm_2Earithmetic_2ESUB__LEFT__ADD,axiom,
! [V0m_2E0: tyop_2Enum_2Enum,V1n_2E0: tyop_2Enum_2Enum,V2p_2E0: tyop_2Enum_2Enum] : c_2Earithmetic_2E_2B_2E2(V0m_2E0,c_2Earithmetic_2E_2D_2E2(V1n_2E0,V2p_2E0)) = c_2Ebool_2ECOND_2E3(tyop_2Enum_2Enum,c_2Earithmetic_2E_3C_3D_2E2(V1n_2E0,V2p_2E0),V0m_2E0,c_2Earithmetic_2E_2D_2E2(c_2Earithmetic_2E_2B_2E2(V0m_2E0,V1n_2E0),V2p_2E0)) ).
tff(thm_2Earithmetic_2ESUB__RIGHT__ADD,axiom,
! [V0m_2E0: tyop_2Enum_2Enum,V1n_2E0: tyop_2Enum_2Enum,V2p_2E0: tyop_2Enum_2Enum] : c_2Earithmetic_2E_2B_2E2(c_2Earithmetic_2E_2D_2E2(V0m_2E0,V1n_2E0),V2p_2E0) = c_2Ebool_2ECOND_2E3(tyop_2Enum_2Enum,c_2Earithmetic_2E_3C_3D_2E2(V0m_2E0,V1n_2E0),V2p_2E0,c_2Earithmetic_2E_2D_2E2(c_2Earithmetic_2E_2B_2E2(V0m_2E0,V2p_2E0),V1n_2E0)) ).
tff(thm_2Earithmetic_2ESUB__LEFT__SUB,axiom,
! [V0m_2E0: tyop_2Enum_2Enum,V1n_2E0: tyop_2Enum_2Enum,V2p_2E0: tyop_2Enum_2Enum] : c_2Earithmetic_2E_2D_2E2(V0m_2E0,c_2Earithmetic_2E_2D_2E2(V1n_2E0,V2p_2E0)) = c_2Ebool_2ECOND_2E3(tyop_2Enum_2Enum,c_2Earithmetic_2E_3C_3D_2E2(V1n_2E0,V2p_2E0),V0m_2E0,c_2Earithmetic_2E_2D_2E2(c_2Earithmetic_2E_2B_2E2(V0m_2E0,V2p_2E0),V1n_2E0)) ).
tff(thm_2Earithmetic_2ESUB__RIGHT__SUB,axiom,
! [V0m_2E0: tyop_2Enum_2Enum,V1n_2E0: tyop_2Enum_2Enum,V2p_2E0: tyop_2Enum_2Enum] : c_2Earithmetic_2E_2D_2E2(c_2Earithmetic_2E_2D_2E2(V0m_2E0,V1n_2E0),V2p_2E0) = c_2Earithmetic_2E_2D_2E2(V0m_2E0,c_2Earithmetic_2E_2B_2E2(V1n_2E0,V2p_2E0)) ).
tff(thm_2Earithmetic_2ESUB__LEFT__SUC,axiom,
! [V0m_2E0: tyop_2Enum_2Enum,V1n_2E0: tyop_2Enum_2Enum] : c_2Enum_2ESUC_2E1(c_2Earithmetic_2E_2D_2E2(V0m_2E0,V1n_2E0)) = c_2Ebool_2ECOND_2E3(tyop_2Enum_2Enum,c_2Earithmetic_2E_3C_3D_2E2(V0m_2E0,V1n_2E0),c_2Enum_2ESUC_2E1(c_2Enum_2E0_2E0),c_2Earithmetic_2E_2D_2E2(c_2Enum_2ESUC_2E1(V0m_2E0),V1n_2E0)) ).
tff(thm_2Earithmetic_2ESUB__LEFT__LESS__EQ,axiom,
! [V0m_2E0: tyop_2Enum_2Enum,V1n_2E0: tyop_2Enum_2Enum,V2p_2E0: tyop_2Enum_2Enum] :
( p(c_2Earithmetic_2E_3C_3D_2E2(V0m_2E0,c_2Earithmetic_2E_2D_2E2(V1n_2E0,V2p_2E0)))
<=> ( p(c_2Earithmetic_2E_3C_3D_2E2(c_2Earithmetic_2E_2B_2E2(V0m_2E0,V2p_2E0),V1n_2E0))
| p(c_2Earithmetic_2E_3C_3D_2E2(V0m_2E0,c_2Enum_2E0_2E0)) ) ) ).
tff(thm_2Earithmetic_2ESUB__RIGHT__LESS__EQ,axiom,
! [V0m_2E0: tyop_2Enum_2Enum,V1n_2E0: tyop_2Enum_2Enum,V2p_2E0: tyop_2Enum_2Enum] : c_2Earithmetic_2E_3C_3D_2E2(c_2Earithmetic_2E_2D_2E2(V0m_2E0,V1n_2E0),V2p_2E0) = c_2Earithmetic_2E_3C_3D_2E2(V0m_2E0,c_2Earithmetic_2E_2B_2E2(V1n_2E0,V2p_2E0)) ).
tff(thm_2Earithmetic_2ESUB__LEFT__LESS,axiom,
! [V0m_2E0: tyop_2Enum_2Enum,V1n_2E0: tyop_2Enum_2Enum,V2p_2E0: tyop_2Enum_2Enum] : c_2Eprim__rec_2E_3C_2E2(V0m_2E0,c_2Earithmetic_2E_2D_2E2(V1n_2E0,V2p_2E0)) = c_2Eprim__rec_2E_3C_2E2(c_2Earithmetic_2E_2B_2E2(V0m_2E0,V2p_2E0),V1n_2E0) ).
tff(thm_2Earithmetic_2ESUB__RIGHT__LESS,axiom,
! [V0m_2E0: tyop_2Enum_2Enum,V1n_2E0: tyop_2Enum_2Enum,V2p_2E0: tyop_2Enum_2Enum] :
( p(c_2Eprim__rec_2E_3C_2E2(c_2Earithmetic_2E_2D_2E2(V0m_2E0,V1n_2E0),V2p_2E0))
<=> ( p(c_2Eprim__rec_2E_3C_2E2(V0m_2E0,c_2Earithmetic_2E_2B_2E2(V1n_2E0,V2p_2E0)))
& p(c_2Eprim__rec_2E_3C_2E2(c_2Enum_2E0_2E0,V2p_2E0)) ) ) ).
tff(thm_2Earithmetic_2ESUB__LEFT__GREATER__EQ,axiom,
! [V0m_2E0: tyop_2Enum_2Enum,V1n_2E0: tyop_2Enum_2Enum,V2p_2E0: tyop_2Enum_2Enum] : c_2Earithmetic_2E_3E_3D_2E2(V0m_2E0,c_2Earithmetic_2E_2D_2E2(V1n_2E0,V2p_2E0)) = c_2Earithmetic_2E_3E_3D_2E2(c_2Earithmetic_2E_2B_2E2(V0m_2E0,V2p_2E0),V1n_2E0) ).
tff(thm_2Earithmetic_2ESUB__RIGHT__GREATER__EQ,axiom,
! [V0m_2E0: tyop_2Enum_2Enum,V1n_2E0: tyop_2Enum_2Enum,V2p_2E0: tyop_2Enum_2Enum] :
( p(c_2Earithmetic_2E_3E_3D_2E2(c_2Earithmetic_2E_2D_2E2(V0m_2E0,V1n_2E0),V2p_2E0))
<=> ( p(c_2Earithmetic_2E_3E_3D_2E2(V0m_2E0,c_2Earithmetic_2E_2B_2E2(V1n_2E0,V2p_2E0)))
| p(c_2Earithmetic_2E_3E_3D_2E2(c_2Enum_2E0_2E0,V2p_2E0)) ) ) ).
tff(thm_2Earithmetic_2ESUB__LEFT__GREATER,axiom,
! [V0m_2E0: tyop_2Enum_2Enum,V1n_2E0: tyop_2Enum_2Enum,V2p_2E0: tyop_2Enum_2Enum] :
( p(c_2Earithmetic_2E_3E_2E2(V0m_2E0,c_2Earithmetic_2E_2D_2E2(V1n_2E0,V2p_2E0)))
<=> ( p(c_2Earithmetic_2E_3E_2E2(c_2Earithmetic_2E_2B_2E2(V0m_2E0,V2p_2E0),V1n_2E0))
& p(c_2Earithmetic_2E_3E_2E2(V0m_2E0,c_2Enum_2E0_2E0)) ) ) ).
tff(thm_2Earithmetic_2ESUB__RIGHT__GREATER,axiom,
! [V0m_2E0: tyop_2Enum_2Enum,V1n_2E0: tyop_2Enum_2Enum,V2p_2E0: tyop_2Enum_2Enum] : c_2Earithmetic_2E_3E_2E2(c_2Earithmetic_2E_2D_2E2(V0m_2E0,V1n_2E0),V2p_2E0) = c_2Earithmetic_2E_3E_2E2(V0m_2E0,c_2Earithmetic_2E_2B_2E2(V1n_2E0,V2p_2E0)) ).
tff(thm_2Earithmetic_2ESUB__LEFT__EQ,axiom,
! [V0m_2E0: tyop_2Enum_2Enum,V1n_2E0: tyop_2Enum_2Enum,V2p_2E0: tyop_2Enum_2Enum] :
( ( V0m_2E0 = c_2Earithmetic_2E_2D_2E2(V1n_2E0,V2p_2E0) )
<=> ( ( c_2Earithmetic_2E_2B_2E2(V0m_2E0,V2p_2E0) = V1n_2E0 )
| ( p(c_2Earithmetic_2E_3C_3D_2E2(V0m_2E0,c_2Enum_2E0_2E0))
& p(c_2Earithmetic_2E_3C_3D_2E2(V1n_2E0,V2p_2E0)) ) ) ) ).
tff(thm_2Earithmetic_2ESUB__RIGHT__EQ,axiom,
! [V0m_2E0: tyop_2Enum_2Enum,V1n_2E0: tyop_2Enum_2Enum,V2p_2E0: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_2D_2E2(V0m_2E0,V1n_2E0) = V2p_2E0 )
<=> ( ( V0m_2E0 = c_2Earithmetic_2E_2B_2E2(V1n_2E0,V2p_2E0) )
| ( p(c_2Earithmetic_2E_3C_3D_2E2(V0m_2E0,V1n_2E0))
& p(c_2Earithmetic_2E_3C_3D_2E2(V2p_2E0,c_2Enum_2E0_2E0)) ) ) ) ).
tff(thm_2Earithmetic_2ELE,axiom,
( ! [V0n_2E0: tyop_2Enum_2Enum] :
( p(c_2Earithmetic_2E_3C_3D_2E2(V0n_2E0,c_2Enum_2E0_2E0))
<=> ( V0n_2E0 = c_2Enum_2E0_2E0 ) )
& ! [V1m_2E0: tyop_2Enum_2Enum,V2n_2E0: tyop_2Enum_2Enum] :
( p(c_2Earithmetic_2E_3C_3D_2E2(V1m_2E0,c_2Enum_2ESUC_2E1(V2n_2E0)))
<=> ( ( V1m_2E0 = c_2Enum_2ESUC_2E1(V2n_2E0) )
| p(c_2Earithmetic_2E_3C_3D_2E2(V1m_2E0,V2n_2E0)) ) ) ) ).
tff(thm_2Earithmetic_2EDA,axiom,
! [V0k_2E0: tyop_2Enum_2Enum,V1n_2E0: tyop_2Enum_2Enum] :
( p(c_2Eprim__rec_2E_3C_2E2(c_2Enum_2E0_2E0,V1n_2E0))
=> ? [V2r_2E0: tyop_2Enum_2Enum,V3q_2E0: tyop_2Enum_2Enum] :
( ( V0k_2E0 = c_2Earithmetic_2E_2B_2E2(c_2Earithmetic_2E_2A_2E2(V3q_2E0,V1n_2E0),V2r_2E0) )
& p(c_2Eprim__rec_2E_3C_2E2(V2r_2E0,V1n_2E0)) ) ) ).
tff(thm_2Earithmetic_2EMOD__ONE,axiom,
! [V0k_2E0: tyop_2Enum_2Enum] : c_2Earithmetic_2EMOD_2E2(V0k_2E0,c_2Enum_2ESUC_2E1(c_2Enum_2E0_2E0)) = c_2Enum_2E0_2E0 ).
tff(thm_2Earithmetic_2EMOD__1,axiom,
! [V0k_2E0: tyop_2Enum_2Enum] : c_2Earithmetic_2EMOD_2E2(V0k_2E0,c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))) = c_2Enum_2E0_2E0 ).
tff(thm_2Earithmetic_2EDIV__LESS__EQ,axiom,
! [V0n_2E0: tyop_2Enum_2Enum] :
( p(c_2Eprim__rec_2E_3C_2E2(c_2Enum_2E0_2E0,V0n_2E0))
=> ! [V1k_2E0: tyop_2Enum_2Enum] : p(c_2Earithmetic_2E_3C_3D_2E2(c_2Earithmetic_2EDIV_2E2(V1k_2E0,V0n_2E0),V1k_2E0)) ) ).
tff(thm_2Earithmetic_2EDIV__UNIQUE,axiom,
! [V0n_2E0: tyop_2Enum_2Enum,V1k_2E0: tyop_2Enum_2Enum,V2q_2E0: tyop_2Enum_2Enum] :
( ? [V3r_2E0: tyop_2Enum_2Enum] :
( ( V1k_2E0 = c_2Earithmetic_2E_2B_2E2(c_2Earithmetic_2E_2A_2E2(V2q_2E0,V0n_2E0),V3r_2E0) )
& p(c_2Eprim__rec_2E_3C_2E2(V3r_2E0,V0n_2E0)) )
=> ( c_2Earithmetic_2EDIV_2E2(V1k_2E0,V0n_2E0) = V2q_2E0 ) ) ).
tff(thm_2Earithmetic_2EMOD__UNIQUE,axiom,
! [V0n_2E0: tyop_2Enum_2Enum,V1k_2E0: tyop_2Enum_2Enum,V2r_2E0: tyop_2Enum_2Enum] :
( ? [V3q_2E0: tyop_2Enum_2Enum] :
( ( V1k_2E0 = c_2Earithmetic_2E_2B_2E2(c_2Earithmetic_2E_2A_2E2(V3q_2E0,V0n_2E0),V2r_2E0) )
& p(c_2Eprim__rec_2E_3C_2E2(V2r_2E0,V0n_2E0)) )
=> ( c_2Earithmetic_2EMOD_2E2(V1k_2E0,V0n_2E0) = V2r_2E0 ) ) ).
tff(thm_2Earithmetic_2EDIV__MULT,axiom,
! [V0n_2E0: tyop_2Enum_2Enum,V1r_2E0: tyop_2Enum_2Enum] :
( p(c_2Eprim__rec_2E_3C_2E2(V1r_2E0,V0n_2E0))
=> ! [V2q_2E0: tyop_2Enum_2Enum] : c_2Earithmetic_2EDIV_2E2(c_2Earithmetic_2E_2B_2E2(c_2Earithmetic_2E_2A_2E2(V2q_2E0,V0n_2E0),V1r_2E0),V0n_2E0) = V2q_2E0 ) ).
tff(thm_2Earithmetic_2ELESS__MOD,axiom,
! [V0n_2E0: tyop_2Enum_2Enum,V1k_2E0: tyop_2Enum_2Enum] :
( p(c_2Eprim__rec_2E_3C_2E2(V1k_2E0,V0n_2E0))
=> ( c_2Earithmetic_2EMOD_2E2(V1k_2E0,V0n_2E0) = V1k_2E0 ) ) ).
tff(thm_2Earithmetic_2EMOD__EQ__0,axiom,
! [V0n_2E0: tyop_2Enum_2Enum] :
( p(c_2Eprim__rec_2E_3C_2E2(c_2Enum_2E0_2E0,V0n_2E0))
=> ! [V1k_2E0: tyop_2Enum_2Enum] : c_2Earithmetic_2EMOD_2E2(c_2Earithmetic_2E_2A_2E2(V1k_2E0,V0n_2E0),V0n_2E0) = c_2Enum_2E0_2E0 ) ).
tff(thm_2Earithmetic_2EZERO__MOD,axiom,
! [V0n_2E0: tyop_2Enum_2Enum] :
( p(c_2Eprim__rec_2E_3C_2E2(c_2Enum_2E0_2E0,V0n_2E0))
=> ( c_2Earithmetic_2EMOD_2E2(c_2Enum_2E0_2E0,V0n_2E0) = c_2Enum_2E0_2E0 ) ) ).
tff(thm_2Earithmetic_2EZERO__DIV,axiom,
! [V0n_2E0: tyop_2Enum_2Enum] :
( p(c_2Eprim__rec_2E_3C_2E2(c_2Enum_2E0_2E0,V0n_2E0))
=> ( c_2Earithmetic_2EDIV_2E2(c_2Enum_2E0_2E0,V0n_2E0) = c_2Enum_2E0_2E0 ) ) ).
tff(thm_2Earithmetic_2EMOD__MULT,axiom,
! [V0n_2E0: tyop_2Enum_2Enum,V1r_2E0: tyop_2Enum_2Enum] :
( p(c_2Eprim__rec_2E_3C_2E2(V1r_2E0,V0n_2E0))
=> ! [V2q_2E0: tyop_2Enum_2Enum] : c_2Earithmetic_2EMOD_2E2(c_2Earithmetic_2E_2B_2E2(c_2Earithmetic_2E_2A_2E2(V2q_2E0,V0n_2E0),V1r_2E0),V0n_2E0) = V1r_2E0 ) ).
tff(thm_2Earithmetic_2EMOD__TIMES,axiom,
! [V0n_2E0: tyop_2Enum_2Enum] :
( p(c_2Eprim__rec_2E_3C_2E2(c_2Enum_2E0_2E0,V0n_2E0))
=> ! [V1q_2E0: tyop_2Enum_2Enum,V2r_2E0: tyop_2Enum_2Enum] : c_2Earithmetic_2EMOD_2E2(c_2Earithmetic_2E_2B_2E2(c_2Earithmetic_2E_2A_2E2(V1q_2E0,V0n_2E0),V2r_2E0),V0n_2E0) = c_2Earithmetic_2EMOD_2E2(V2r_2E0,V0n_2E0) ) ).
tff(thm_2Earithmetic_2EMOD__TIMES__SUB,axiom,
! [V0n_2E0: tyop_2Enum_2Enum,V1q_2E0: tyop_2Enum_2Enum,V2r_2E0: tyop_2Enum_2Enum] :
( ( p(c_2Eprim__rec_2E_3C_2E2(c_2Enum_2E0_2E0,V0n_2E0))
& p(c_2Eprim__rec_2E_3C_2E2(c_2Enum_2E0_2E0,V1q_2E0))
& p(c_2Earithmetic_2E_3C_3D_2E2(V2r_2E0,V0n_2E0)) )
=> ( c_2Earithmetic_2EMOD_2E2(c_2Earithmetic_2E_2D_2E2(c_2Earithmetic_2E_2A_2E2(V1q_2E0,V0n_2E0),V2r_2E0),V0n_2E0) = c_2Earithmetic_2EMOD_2E2(c_2Earithmetic_2E_2D_2E2(V0n_2E0,V2r_2E0),V0n_2E0) ) ) ).
tff(thm_2Earithmetic_2EMOD__PLUS,axiom,
! [V0n_2E0: tyop_2Enum_2Enum] :
( p(c_2Eprim__rec_2E_3C_2E2(c_2Enum_2E0_2E0,V0n_2E0))
=> ! [V1j_2E0: tyop_2Enum_2Enum,V2k_2E0: tyop_2Enum_2Enum] : c_2Earithmetic_2EMOD_2E2(c_2Earithmetic_2E_2B_2E2(c_2Earithmetic_2EMOD_2E2(V1j_2E0,V0n_2E0),c_2Earithmetic_2EMOD_2E2(V2k_2E0,V0n_2E0)),V0n_2E0) = c_2Earithmetic_2EMOD_2E2(c_2Earithmetic_2E_2B_2E2(V1j_2E0,V2k_2E0),V0n_2E0) ) ).
tff(thm_2Earithmetic_2EMOD__MOD,axiom,
! [V0n_2E0: tyop_2Enum_2Enum] :
( p(c_2Eprim__rec_2E_3C_2E2(c_2Enum_2E0_2E0,V0n_2E0))
=> ! [V1k_2E0: tyop_2Enum_2Enum] : c_2Earithmetic_2EMOD_2E2(c_2Earithmetic_2EMOD_2E2(V1k_2E0,V0n_2E0),V0n_2E0) = c_2Earithmetic_2EMOD_2E2(V1k_2E0,V0n_2E0) ) ).
tff(thm_2Earithmetic_2ELESS__DIV__EQ__ZERO,axiom,
! [V0r_2E0: tyop_2Enum_2Enum,V1n_2E0: tyop_2Enum_2Enum] :
( p(c_2Eprim__rec_2E_3C_2E2(V0r_2E0,V1n_2E0))
=> ( c_2Earithmetic_2EDIV_2E2(V0r_2E0,V1n_2E0) = c_2Enum_2E0_2E0 ) ) ).
tff(thm_2Earithmetic_2EMULT__DIV,axiom,
! [V0n_2E0: tyop_2Enum_2Enum,V1q_2E0: tyop_2Enum_2Enum] :
( p(c_2Eprim__rec_2E_3C_2E2(c_2Enum_2E0_2E0,V0n_2E0))
=> ( c_2Earithmetic_2EDIV_2E2(c_2Earithmetic_2E_2A_2E2(V1q_2E0,V0n_2E0),V0n_2E0) = V1q_2E0 ) ) ).
tff(thm_2Earithmetic_2EADD__DIV__ADD__DIV,axiom,
! [V0n_2E0: tyop_2Enum_2Enum] :
( p(c_2Eprim__rec_2E_3C_2E2(c_2Enum_2E0_2E0,V0n_2E0))
=> ! [V1x_2E0: tyop_2Enum_2Enum,V2r_2E0: tyop_2Enum_2Enum] : c_2Earithmetic_2EDIV_2E2(c_2Earithmetic_2E_2B_2E2(c_2Earithmetic_2E_2A_2E2(V1x_2E0,V0n_2E0),V2r_2E0),V0n_2E0) = c_2Earithmetic_2E_2B_2E2(V1x_2E0,c_2Earithmetic_2EDIV_2E2(V2r_2E0,V0n_2E0)) ) ).
tff(thm_2Earithmetic_2EADD__DIV__RWT,axiom,
! [V0n_2E0: tyop_2Enum_2Enum] :
( p(c_2Eprim__rec_2E_3C_2E2(c_2Enum_2E0_2E0,V0n_2E0))
=> ! [V1m_2E0: tyop_2Enum_2Enum,V2p_2E0: tyop_2Enum_2Enum] :
( ( ( c_2Earithmetic_2EMOD_2E2(V1m_2E0,V0n_2E0) = c_2Enum_2E0_2E0 )
| ( c_2Earithmetic_2EMOD_2E2(V2p_2E0,V0n_2E0) = c_2Enum_2E0_2E0 ) )
=> ( c_2Earithmetic_2EDIV_2E2(c_2Earithmetic_2E_2B_2E2(V1m_2E0,V2p_2E0),V0n_2E0) = c_2Earithmetic_2E_2B_2E2(c_2Earithmetic_2EDIV_2E2(V1m_2E0,V0n_2E0),c_2Earithmetic_2EDIV_2E2(V2p_2E0,V0n_2E0)) ) ) ) ).
tff(thm_2Earithmetic_2EMOD__MULT__MOD,axiom,
! [V0m_2E0: tyop_2Enum_2Enum,V1n_2E0: tyop_2Enum_2Enum] :
( ( p(c_2Eprim__rec_2E_3C_2E2(c_2Enum_2E0_2E0,V1n_2E0))
& p(c_2Eprim__rec_2E_3C_2E2(c_2Enum_2E0_2E0,V0m_2E0)) )
=> ! [V2x_2E0: tyop_2Enum_2Enum] : c_2Earithmetic_2EMOD_2E2(c_2Earithmetic_2EMOD_2E2(V2x_2E0,c_2Earithmetic_2E_2A_2E2(V1n_2E0,V0m_2E0)),V1n_2E0) = c_2Earithmetic_2EMOD_2E2(V2x_2E0,V1n_2E0) ) ).
tff(thm_2Earithmetic_2EDIV__ONE,axiom,
! [V0q_2E0: tyop_2Enum_2Enum] : c_2Earithmetic_2EDIV_2E2(V0q_2E0,c_2Enum_2ESUC_2E1(c_2Enum_2E0_2E0)) = V0q_2E0 ).
tff(thm_2Earithmetic_2EDIV__1,axiom,
! [V0q_2E0: tyop_2Enum_2Enum] : c_2Earithmetic_2EDIV_2E2(V0q_2E0,c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))) = V0q_2E0 ).
tff(thm_2Earithmetic_2EDIVMOD__ID,axiom,
! [V0n_2E0: tyop_2Enum_2Enum] :
( p(c_2Eprim__rec_2E_3C_2E2(c_2Enum_2E0_2E0,V0n_2E0))
=> ( ( c_2Earithmetic_2EDIV_2E2(V0n_2E0,V0n_2E0) = c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)) )
& ( c_2Earithmetic_2EMOD_2E2(V0n_2E0,V0n_2E0) = c_2Enum_2E0_2E0 ) ) ) ).
tff(thm_2Earithmetic_2EDIV__DIV__DIV__MULT,axiom,
! [V0m_2E0: tyop_2Enum_2Enum,V1n_2E0: tyop_2Enum_2Enum] :
( ( p(c_2Eprim__rec_2E_3C_2E2(c_2Enum_2E0_2E0,V0m_2E0))
& p(c_2Eprim__rec_2E_3C_2E2(c_2Enum_2E0_2E0,V1n_2E0)) )
=> ! [V2x_2E0: tyop_2Enum_2Enum] : c_2Earithmetic_2EDIV_2E2(c_2Earithmetic_2EDIV_2E2(V2x_2E0,V0m_2E0),V1n_2E0) = c_2Earithmetic_2EDIV_2E2(V2x_2E0,c_2Earithmetic_2E_2A_2E2(V0m_2E0,V1n_2E0)) ) ).
tff(thm_2Earithmetic_2ESUC__PRE,axiom,
! [V0m_2E0: tyop_2Enum_2Enum] :
( p(c_2Eprim__rec_2E_3C_2E2(c_2Enum_2E0_2E0,V0m_2E0))
<=> ( c_2Enum_2ESUC_2E1(c_2Eprim__rec_2EPRE_2E1(V0m_2E0)) = V0m_2E0 ) ) ).
tff(thm_2Earithmetic_2EDIV__LESS,axiom,
! [V0n_2E0: tyop_2Enum_2Enum,V1d_2E0: tyop_2Enum_2Enum] :
( ( p(c_2Eprim__rec_2E_3C_2E2(c_2Enum_2E0_2E0,V0n_2E0))
& p(c_2Eprim__rec_2E_3C_2E2(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)),V1d_2E0)) )
=> p(c_2Eprim__rec_2E_3C_2E2(c_2Earithmetic_2EDIV_2E2(V0n_2E0,V1d_2E0),V0n_2E0)) ) ).
tff(thm_2Earithmetic_2EMOD__LESS,axiom,
! [V0m_2E0: tyop_2Enum_2Enum,V1n_2E0: tyop_2Enum_2Enum] :
( p(c_2Eprim__rec_2E_3C_2E2(c_2Enum_2E0_2E0,V1n_2E0))
=> p(c_2Eprim__rec_2E_3C_2E2(c_2Earithmetic_2EMOD_2E2(V0m_2E0,V1n_2E0),V1n_2E0)) ) ).
tff(thm_2Earithmetic_2EADD__MODULUS,axiom,
( ! [V0n_2E0: tyop_2Enum_2Enum,V1x_2E0: tyop_2Enum_2Enum] :
( p(c_2Eprim__rec_2E_3C_2E2(c_2Enum_2E0_2E0,V0n_2E0))
=> ( c_2Earithmetic_2EMOD_2E2(c_2Earithmetic_2E_2B_2E2(V1x_2E0,V0n_2E0),V0n_2E0) = c_2Earithmetic_2EMOD_2E2(V1x_2E0,V0n_2E0) ) )
& ! [V2n_2E0: tyop_2Enum_2Enum,V3x_2E0: tyop_2Enum_2Enum] :
( p(c_2Eprim__rec_2E_3C_2E2(c_2Enum_2E0_2E0,V2n_2E0))
=> ( c_2Earithmetic_2EMOD_2E2(c_2Earithmetic_2E_2B_2E2(V2n_2E0,V3x_2E0),V2n_2E0) = c_2Earithmetic_2EMOD_2E2(V3x_2E0,V2n_2E0) ) ) ) ).
tff(thm_2Earithmetic_2EADD__MODULUS__LEFT,axiom,
! [V0n_2E0: tyop_2Enum_2Enum,V1x_2E0: tyop_2Enum_2Enum] :
( p(c_2Eprim__rec_2E_3C_2E2(c_2Enum_2E0_2E0,V0n_2E0))
=> ( c_2Earithmetic_2EMOD_2E2(c_2Earithmetic_2E_2B_2E2(V1x_2E0,V0n_2E0),V0n_2E0) = c_2Earithmetic_2EMOD_2E2(V1x_2E0,V0n_2E0) ) ) ).
tff(thm_2Earithmetic_2EADD__MODULUS__RIGHT,axiom,
! [V0n_2E0: tyop_2Enum_2Enum,V1x_2E0: tyop_2Enum_2Enum] :
( p(c_2Eprim__rec_2E_3C_2E2(c_2Enum_2E0_2E0,V0n_2E0))
=> ( c_2Earithmetic_2EMOD_2E2(c_2Earithmetic_2E_2B_2E2(V0n_2E0,V1x_2E0),V0n_2E0) = c_2Earithmetic_2EMOD_2E2(V1x_2E0,V0n_2E0) ) ) ).
tff(thm_2Earithmetic_2EDIV__P,axiom,
! [V0P_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),V1p_2E0: tyop_2Enum_2Enum,V2q_2E0: tyop_2Enum_2Enum] :
( p(c_2Eprim__rec_2E_3C_2E2(c_2Enum_2E0_2E0,V2q_2E0))
=> ( p(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,V0P_2E0,c_2Earithmetic_2EDIV_2E2(V1p_2E0,V2q_2E0)))
<=> ? [V3k_2E0: tyop_2Enum_2Enum,V4r_2E0: tyop_2Enum_2Enum] :
( ( V1p_2E0 = c_2Earithmetic_2E_2B_2E2(c_2Earithmetic_2E_2A_2E2(V3k_2E0,V2q_2E0),V4r_2E0) )
& p(c_2Eprim__rec_2E_3C_2E2(V4r_2E0,V2q_2E0))
& p(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,V0P_2E0,V3k_2E0)) ) ) ) ).
tff(thm_2Earithmetic_2EDIV__P__UNIV,axiom,
! [V0P_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),V1m_2E0: tyop_2Enum_2Enum,V2n_2E0: tyop_2Enum_2Enum] :
( p(c_2Eprim__rec_2E_3C_2E2(c_2Enum_2E0_2E0,V2n_2E0))
=> ( p(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,V0P_2E0,c_2Earithmetic_2EDIV_2E2(V1m_2E0,V2n_2E0)))
<=> ! [V3q_2E0: tyop_2Enum_2Enum,V4r_2E0: tyop_2Enum_2Enum] :
( ( ( V1m_2E0 = c_2Earithmetic_2E_2B_2E2(c_2Earithmetic_2E_2A_2E2(V3q_2E0,V2n_2E0),V4r_2E0) )
& p(c_2Eprim__rec_2E_3C_2E2(V4r_2E0,V2n_2E0)) )
=> p(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,V0P_2E0,V3q_2E0)) ) ) ) ).
tff(thm_2Earithmetic_2EMOD__P,axiom,
! [V0P_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),V1p_2E0: tyop_2Enum_2Enum,V2q_2E0: tyop_2Enum_2Enum] :
( p(c_2Eprim__rec_2E_3C_2E2(c_2Enum_2E0_2E0,V2q_2E0))
=> ( p(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,V0P_2E0,c_2Earithmetic_2EMOD_2E2(V1p_2E0,V2q_2E0)))
<=> ? [V3k_2E0: tyop_2Enum_2Enum,V4r_2E0: tyop_2Enum_2Enum] :
( ( V1p_2E0 = c_2Earithmetic_2E_2B_2E2(c_2Earithmetic_2E_2A_2E2(V3k_2E0,V2q_2E0),V4r_2E0) )
& p(c_2Eprim__rec_2E_3C_2E2(V4r_2E0,V2q_2E0))
& p(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,V0P_2E0,V4r_2E0)) ) ) ) ).
tff(thm_2Earithmetic_2EMOD__P__UNIV,axiom,
! [V0P_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),V1m_2E0: tyop_2Enum_2Enum,V2n_2E0: tyop_2Enum_2Enum] :
( p(c_2Eprim__rec_2E_3C_2E2(c_2Enum_2E0_2E0,V2n_2E0))
=> ( p(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,V0P_2E0,c_2Earithmetic_2EMOD_2E2(V1m_2E0,V2n_2E0)))
<=> ! [V3q_2E0: tyop_2Enum_2Enum,V4r_2E0: tyop_2Enum_2Enum] :
( ( ( V1m_2E0 = c_2Earithmetic_2E_2B_2E2(c_2Earithmetic_2E_2A_2E2(V3q_2E0,V2n_2E0),V4r_2E0) )
& p(c_2Eprim__rec_2E_3C_2E2(V4r_2E0,V2n_2E0)) )
=> p(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,V0P_2E0,V4r_2E0)) ) ) ) ).
tff(thm_2Earithmetic_2EMOD__TIMES2,axiom,
! [V0n_2E0: tyop_2Enum_2Enum] :
( p(c_2Eprim__rec_2E_3C_2E2(c_2Enum_2E0_2E0,V0n_2E0))
=> ! [V1j_2E0: tyop_2Enum_2Enum,V2k_2E0: tyop_2Enum_2Enum] : c_2Earithmetic_2EMOD_2E2(c_2Earithmetic_2E_2A_2E2(c_2Earithmetic_2EMOD_2E2(V1j_2E0,V0n_2E0),c_2Earithmetic_2EMOD_2E2(V2k_2E0,V0n_2E0)),V0n_2E0) = c_2Earithmetic_2EMOD_2E2(c_2Earithmetic_2E_2A_2E2(V1j_2E0,V2k_2E0),V0n_2E0) ) ).
tff(thm_2Earithmetic_2EMOD__COMMON__FACTOR,axiom,
! [V0n_2E0: tyop_2Enum_2Enum,V1p_2E0: tyop_2Enum_2Enum,V2q_2E0: tyop_2Enum_2Enum] :
( ( p(c_2Eprim__rec_2E_3C_2E2(c_2Enum_2E0_2E0,V0n_2E0))
& p(c_2Eprim__rec_2E_3C_2E2(c_2Enum_2E0_2E0,V2q_2E0)) )
=> ( c_2Earithmetic_2E_2A_2E2(V0n_2E0,c_2Earithmetic_2EMOD_2E2(V1p_2E0,V2q_2E0)) = c_2Earithmetic_2EMOD_2E2(c_2Earithmetic_2E_2A_2E2(V0n_2E0,V1p_2E0),c_2Earithmetic_2E_2A_2E2(V0n_2E0,V2q_2E0)) ) ) ).
tff(thm_2Earithmetic_2EX__MOD__Y__EQ__X,axiom,
! [V0x_2E0: tyop_2Enum_2Enum,V1y_2E0: tyop_2Enum_2Enum] :
( p(c_2Eprim__rec_2E_3C_2E2(c_2Enum_2E0_2E0,V1y_2E0))
=> ( ( c_2Earithmetic_2EMOD_2E2(V0x_2E0,V1y_2E0) = V0x_2E0 )
<=> p(c_2Eprim__rec_2E_3C_2E2(V0x_2E0,V1y_2E0)) ) ) ).
tff(thm_2Earithmetic_2EDIV__LE__MONOTONE,axiom,
! [V0n_2E0: tyop_2Enum_2Enum,V1x_2E0: tyop_2Enum_2Enum,V2y_2E0: tyop_2Enum_2Enum] :
( ( p(c_2Eprim__rec_2E_3C_2E2(c_2Enum_2E0_2E0,V0n_2E0))
& p(c_2Earithmetic_2E_3C_3D_2E2(V1x_2E0,V2y_2E0)) )
=> p(c_2Earithmetic_2E_3C_3D_2E2(c_2Earithmetic_2EDIV_2E2(V1x_2E0,V0n_2E0),c_2Earithmetic_2EDIV_2E2(V2y_2E0,V0n_2E0))) ) ).
tff(thm_2Earithmetic_2ELE__LT1,axiom,
! [V0x_2E0: tyop_2Enum_2Enum,V1y_2E0: tyop_2Enum_2Enum] : c_2Earithmetic_2E_3C_3D_2E2(V0x_2E0,V1y_2E0) = c_2Eprim__rec_2E_3C_2E2(V0x_2E0,c_2Earithmetic_2E_2B_2E2(V1y_2E0,c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)))) ).
tff(thm_2Earithmetic_2EX__LE__DIV,axiom,
! [V0x_2E0: tyop_2Enum_2Enum,V1y_2E0: tyop_2Enum_2Enum,V2z_2E0: tyop_2Enum_2Enum] :
( p(c_2Eprim__rec_2E_3C_2E2(c_2Enum_2E0_2E0,V2z_2E0))
=> ( c_2Earithmetic_2E_3C_3D_2E2(V0x_2E0,c_2Earithmetic_2EDIV_2E2(V1y_2E0,V2z_2E0)) = c_2Earithmetic_2E_3C_3D_2E2(c_2Earithmetic_2E_2A_2E2(V0x_2E0,V2z_2E0),V1y_2E0) ) ) ).
tff(thm_2Earithmetic_2EX__LT__DIV,axiom,
! [V0x_2E0: tyop_2Enum_2Enum,V1y_2E0: tyop_2Enum_2Enum,V2z_2E0: tyop_2Enum_2Enum] :
( p(c_2Eprim__rec_2E_3C_2E2(c_2Enum_2E0_2E0,V2z_2E0))
=> ( c_2Eprim__rec_2E_3C_2E2(V0x_2E0,c_2Earithmetic_2EDIV_2E2(V1y_2E0,V2z_2E0)) = c_2Earithmetic_2E_3C_3D_2E2(c_2Earithmetic_2E_2A_2E2(c_2Earithmetic_2E_2B_2E2(V0x_2E0,c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))),V2z_2E0),V1y_2E0) ) ) ).
tff(thm_2Earithmetic_2EDIV__LT__X,axiom,
! [V0x_2E0: tyop_2Enum_2Enum,V1y_2E0: tyop_2Enum_2Enum,V2z_2E0: tyop_2Enum_2Enum] :
( p(c_2Eprim__rec_2E_3C_2E2(c_2Enum_2E0_2E0,V2z_2E0))
=> ( c_2Eprim__rec_2E_3C_2E2(c_2Earithmetic_2EDIV_2E2(V1y_2E0,V2z_2E0),V0x_2E0) = c_2Eprim__rec_2E_3C_2E2(V1y_2E0,c_2Earithmetic_2E_2A_2E2(V0x_2E0,V2z_2E0)) ) ) ).
tff(thm_2Earithmetic_2EDIV__LE__X,axiom,
! [V0x_2E0: tyop_2Enum_2Enum,V1y_2E0: tyop_2Enum_2Enum,V2z_2E0: tyop_2Enum_2Enum] :
( p(c_2Eprim__rec_2E_3C_2E2(c_2Enum_2E0_2E0,V2z_2E0))
=> ( c_2Earithmetic_2E_3C_3D_2E2(c_2Earithmetic_2EDIV_2E2(V1y_2E0,V2z_2E0),V0x_2E0) = c_2Eprim__rec_2E_3C_2E2(V1y_2E0,c_2Earithmetic_2E_2A_2E2(c_2Earithmetic_2E_2B_2E2(V0x_2E0,c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))),V2z_2E0)) ) ) ).
tff(thm_2Earithmetic_2EDIV__EQ__X,axiom,
! [V0x_2E0: tyop_2Enum_2Enum,V1y_2E0: tyop_2Enum_2Enum,V2z_2E0: tyop_2Enum_2Enum] :
( p(c_2Eprim__rec_2E_3C_2E2(c_2Enum_2E0_2E0,V2z_2E0))
=> ( ( c_2Earithmetic_2EDIV_2E2(V1y_2E0,V2z_2E0) = V0x_2E0 )
<=> ( p(c_2Earithmetic_2E_3C_3D_2E2(c_2Earithmetic_2E_2A_2E2(V0x_2E0,V2z_2E0),V1y_2E0))
& p(c_2Eprim__rec_2E_3C_2E2(V1y_2E0,c_2Earithmetic_2E_2A_2E2(c_2Enum_2ESUC_2E1(V0x_2E0),V2z_2E0))) ) ) ) ).
tff(thm_2Earithmetic_2EDIV__MOD__MOD__DIV,axiom,
! [V0m_2E0: tyop_2Enum_2Enum,V1n_2E0: tyop_2Enum_2Enum,V2k_2E0: tyop_2Enum_2Enum] :
( ( p(c_2Eprim__rec_2E_3C_2E2(c_2Enum_2E0_2E0,V1n_2E0))
& p(c_2Eprim__rec_2E_3C_2E2(c_2Enum_2E0_2E0,V2k_2E0)) )
=> ( c_2Earithmetic_2EMOD_2E2(c_2Earithmetic_2EDIV_2E2(V0m_2E0,V1n_2E0),V2k_2E0) = c_2Earithmetic_2EDIV_2E2(c_2Earithmetic_2EMOD_2E2(V0m_2E0,c_2Earithmetic_2E_2A_2E2(V1n_2E0,V2k_2E0)),V1n_2E0) ) ) ).
tff(thm_2Earithmetic_2EMULT__EQ__DIV,axiom,
! [V0z_2E0: tyop_2Enum_2Enum,V1y_2E0: tyop_2Enum_2Enum,V2x_2E0: tyop_2Enum_2Enum] :
( p(c_2Eprim__rec_2E_3C_2E2(c_2Enum_2E0_2E0,V2x_2E0))
=> ( ( c_2Earithmetic_2E_2A_2E2(V2x_2E0,V1y_2E0) = V0z_2E0 )
<=> ( ( V1y_2E0 = c_2Earithmetic_2EDIV_2E2(V0z_2E0,V2x_2E0) )
& ( c_2Earithmetic_2EMOD_2E2(V0z_2E0,V2x_2E0) = c_2Enum_2E0_2E0 ) ) ) ) ).
tff(thm_2Earithmetic_2ENUMERAL__MULT__EQ__DIV,axiom,
! [V0z_2E0: tyop_2Enum_2Enum,V1y_2E0: tyop_2Enum_2Enum,V2x_2E0: tyop_2Enum_2Enum] :
( ( ( c_2Earithmetic_2E_2A_2E2(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(V2x_2E0)),V1y_2E0) = c_2Earithmetic_2ENUMERAL_2E1(V0z_2E0) )
<=> ( ( V1y_2E0 = c_2Earithmetic_2EDIV_2E2(c_2Earithmetic_2ENUMERAL_2E1(V0z_2E0),c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(V2x_2E0))) )
& ( c_2Earithmetic_2EMOD_2E2(c_2Earithmetic_2ENUMERAL_2E1(V0z_2E0),c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(V2x_2E0))) = c_2Enum_2E0_2E0 ) ) )
& ( ( c_2Earithmetic_2E_2A_2E2(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(V2x_2E0)),V1y_2E0) = c_2Earithmetic_2ENUMERAL_2E1(V0z_2E0) )
<=> ( ( V1y_2E0 = c_2Earithmetic_2EDIV_2E2(c_2Earithmetic_2ENUMERAL_2E1(V0z_2E0),c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(V2x_2E0))) )
& ( c_2Earithmetic_2EMOD_2E2(c_2Earithmetic_2ENUMERAL_2E1(V0z_2E0),c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(V2x_2E0))) = c_2Enum_2E0_2E0 ) ) ) ) ).
tff(thm_2Earithmetic_2EMOD__EQ__0__DIVISOR,axiom,
! [V0n_2E0: tyop_2Enum_2Enum,V1k_2E0: tyop_2Enum_2Enum] :
( p(c_2Eprim__rec_2E_3C_2E2(c_2Enum_2E0_2E0,V0n_2E0))
=> ( ( c_2Earithmetic_2EMOD_2E2(V1k_2E0,V0n_2E0) = c_2Enum_2E0_2E0 )
<=> ? [V2d_2E0: tyop_2Enum_2Enum] : V1k_2E0 = c_2Earithmetic_2E_2A_2E2(V2d_2E0,V0n_2E0) ) ) ).
tff(thm_2Earithmetic_2EMOD__SUC,axiom,
! [V0y_2E0: tyop_2Enum_2Enum,V1x_2E0: tyop_2Enum_2Enum] :
( ( p(c_2Eprim__rec_2E_3C_2E2(c_2Enum_2E0_2E0,V0y_2E0))
& ( c_2Enum_2ESUC_2E1(V1x_2E0) != c_2Earithmetic_2E_2A_2E2(c_2Enum_2ESUC_2E1(c_2Earithmetic_2EDIV_2E2(V1x_2E0,V0y_2E0)),V0y_2E0) ) )
=> ( c_2Earithmetic_2EMOD_2E2(c_2Enum_2ESUC_2E1(V1x_2E0),V0y_2E0) = c_2Enum_2ESUC_2E1(c_2Earithmetic_2EMOD_2E2(V1x_2E0,V0y_2E0)) ) ) ).
tff(thm_2Earithmetic_2EMOD__SUC__IFF,axiom,
! [V0y_2E0: tyop_2Enum_2Enum,V1x_2E0: tyop_2Enum_2Enum] :
( p(c_2Eprim__rec_2E_3C_2E2(c_2Enum_2E0_2E0,V0y_2E0))
=> ( ( c_2Earithmetic_2EMOD_2E2(c_2Enum_2ESUC_2E1(V1x_2E0),V0y_2E0) = c_2Enum_2ESUC_2E1(c_2Earithmetic_2EMOD_2E2(V1x_2E0,V0y_2E0)) )
<=> ( c_2Enum_2ESUC_2E1(V1x_2E0) != c_2Earithmetic_2E_2A_2E2(c_2Enum_2ESUC_2E1(c_2Earithmetic_2EDIV_2E2(V1x_2E0,V0y_2E0)),V0y_2E0) ) ) ) ).
tff(thm_2Earithmetic_2EONE__MOD,axiom,
! [V0n_2E0: tyop_2Enum_2Enum] :
( p(c_2Eprim__rec_2E_3C_2E2(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)),V0n_2E0))
=> ( c_2Earithmetic_2EMOD_2E2(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)),V0n_2E0) = c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)) ) ) ).
tff(thm_2Earithmetic_2EONE__MOD__IFF,axiom,
! [V0n_2E0: tyop_2Enum_2Enum] :
( p(c_2Eprim__rec_2E_3C_2E2(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)),V0n_2E0))
<=> ( p(c_2Eprim__rec_2E_3C_2E2(c_2Enum_2E0_2E0,V0n_2E0))
& ( c_2Earithmetic_2EMOD_2E2(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)),V0n_2E0) = c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)) ) ) ) ).
tff(thm_2Earithmetic_2EMOD__LESS__EQ,axiom,
! [V0y_2E0: tyop_2Enum_2Enum,V1x_2E0: tyop_2Enum_2Enum] :
( p(c_2Eprim__rec_2E_3C_2E2(c_2Enum_2E0_2E0,V0y_2E0))
=> p(c_2Earithmetic_2E_3C_3D_2E2(c_2Earithmetic_2EMOD_2E2(V1x_2E0,V0y_2E0),V1x_2E0)) ) ).
tff(thm_2Earithmetic_2EMOD__LIFT__PLUS,axiom,
! [V0x_2E0: tyop_2Enum_2Enum,V1n_2E0: tyop_2Enum_2Enum,V2k_2E0: tyop_2Enum_2Enum] :
( ( p(c_2Eprim__rec_2E_3C_2E2(c_2Enum_2E0_2E0,V1n_2E0))
& p(c_2Eprim__rec_2E_3C_2E2(V2k_2E0,c_2Earithmetic_2E_2D_2E2(V1n_2E0,c_2Earithmetic_2EMOD_2E2(V0x_2E0,V1n_2E0)))) )
=> ( c_2Earithmetic_2EMOD_2E2(c_2Earithmetic_2E_2B_2E2(V0x_2E0,V2k_2E0),V1n_2E0) = c_2Earithmetic_2E_2B_2E2(c_2Earithmetic_2EMOD_2E2(V0x_2E0,V1n_2E0),V2k_2E0) ) ) ).
tff(thm_2Earithmetic_2EMOD__LIFT__PLUS__IFF,axiom,
! [V0x_2E0: tyop_2Enum_2Enum,V1n_2E0: tyop_2Enum_2Enum,V2k_2E0: tyop_2Enum_2Enum] :
( p(c_2Eprim__rec_2E_3C_2E2(c_2Enum_2E0_2E0,V1n_2E0))
=> ( ( c_2Earithmetic_2EMOD_2E2(c_2Earithmetic_2E_2B_2E2(V0x_2E0,V2k_2E0),V1n_2E0) = c_2Earithmetic_2E_2B_2E2(c_2Earithmetic_2EMOD_2E2(V0x_2E0,V1n_2E0),V2k_2E0) )
<=> p(c_2Eprim__rec_2E_3C_2E2(V2k_2E0,c_2Earithmetic_2E_2D_2E2(V1n_2E0,c_2Earithmetic_2EMOD_2E2(V0x_2E0,V1n_2E0)))) ) ) ).
tff(thm_2Earithmetic_2Enum__case__cong,axiom,
! [A_27a: $tType,V0v_27_2E0: A_27a,V1f_27_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,A_27a),V2M_2E0: tyop_2Enum_2Enum,V3M_27_2E0: tyop_2Enum_2Enum,V4v_2E0: A_27a,V5f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,A_27a)] :
( ( ( V2M_2E0 = V3M_27_2E0 )
& ( ( V3M_27_2E0 = c_2Enum_2E0_2E0 )
=> ( V4v_2E0 = V0v_27_2E0 ) )
& ! [V6n_2E0: tyop_2Enum_2Enum] :
( ( V3M_27_2E0 = c_2Enum_2ESUC_2E1(V6n_2E0) )
=> ( app_2E2(tyop_2Enum_2Enum,A_27a,V5f_2E0,V6n_2E0) = app_2E2(tyop_2Enum_2Enum,A_27a,V1f_27_2E0,V6n_2E0) ) ) )
=> ( c_2Earithmetic_2Enum__CASE_2E3(A_27a,V2M_2E0,V4v_2E0,V5f_2E0) = c_2Earithmetic_2Enum__CASE_2E3(A_27a,V3M_27_2E0,V0v_27_2E0,V1f_27_2E0) ) ) ).
tff(thm_2Earithmetic_2ESUC__ELIM__THM,axiom,
! [V0P_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool))] :
( ! [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),V0P_2E0,c_2Enum_2ESUC_2E1(V1n_2E0)),V1n_2E0))
<=> ! [V2n_2E0: tyop_2Enum_2Enum] :
( p(c_2Eprim__rec_2E_3C_2E2(c_2Enum_2E0_2E0,V2n_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,V2n_2E0),c_2Earithmetic_2E_2D_2E2(V2n_2E0,c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))))) ) ) ).
tff(thm_2Earithmetic_2ESUC__ELIM__NUMERALS,axiom,
! [A_27a: $tType,V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,A_27a)),V1g_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,A_27a)] :
( ! [V2n_2E0: tyop_2Enum_2Enum] : app_2E2(tyop_2Enum_2Enum,A_27a,V1g_2E0,c_2Enum_2ESUC_2E1(V2n_2E0)) = app_2E2(tyop_2Enum_2Enum,A_27a,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,A_27a),V0f_2E0,V2n_2E0),c_2Enum_2ESUC_2E1(V2n_2E0))
<=> ( ! [V3n_2E0: tyop_2Enum_2Enum] : app_2E2(tyop_2Enum_2Enum,A_27a,V1g_2E0,c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(V3n_2E0))) = app_2E2(tyop_2Enum_2Enum,A_27a,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,A_27a),V0f_2E0,c_2Earithmetic_2E_2D_2E2(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(V3n_2E0)),c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)))),c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(V3n_2E0)))
& ! [V4n_2E0: tyop_2Enum_2Enum] : app_2E2(tyop_2Enum_2Enum,A_27a,V1g_2E0,c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(V4n_2E0))) = app_2E2(tyop_2Enum_2Enum,A_27a,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,A_27a),V0f_2E0,c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(V4n_2E0))),c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(V4n_2E0))) ) ) ).
tff(thm_2Earithmetic_2ESUB__ELIM__THM,axiom,
! [V0b_2E0: tyop_2Enum_2Enum,V1a_2E0: tyop_2Enum_2Enum,V2P_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)] :
( p(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,V2P_2E0,c_2Earithmetic_2E_2D_2E2(V1a_2E0,V0b_2E0)))
<=> ! [V3d_2E0: tyop_2Enum_2Enum] :
( ( ( V0b_2E0 = c_2Earithmetic_2E_2B_2E2(V1a_2E0,V3d_2E0) )
=> p(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,V2P_2E0,c_2Enum_2E0_2E0)) )
& ( ( V1a_2E0 = c_2Earithmetic_2E_2B_2E2(V0b_2E0,V3d_2E0) )
=> p(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,V2P_2E0,V3d_2E0)) ) ) ) ).
tff(thm_2Earithmetic_2EPRE__ELIM__THM,axiom,
! [V0n_2E0: tyop_2Enum_2Enum,V1P_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)] :
( p(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,V1P_2E0,c_2Eprim__rec_2EPRE_2E1(V0n_2E0)))
<=> ! [V2m_2E0: tyop_2Enum_2Enum] :
( ( ( V0n_2E0 = c_2Enum_2E0_2E0 )
=> p(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,V1P_2E0,c_2Enum_2E0_2E0)) )
& ( ( V0n_2E0 = c_2Enum_2ESUC_2E1(V2m_2E0) )
=> p(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,V1P_2E0,V2m_2E0)) ) ) ) ).
tff(thm_2Earithmetic_2EMULT__INCREASES,axiom,
! [V0m_2E0: tyop_2Enum_2Enum,V1n_2E0: tyop_2Enum_2Enum] :
( ( p(c_2Eprim__rec_2E_3C_2E2(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)),V0m_2E0))
& p(c_2Eprim__rec_2E_3C_2E2(c_2Enum_2E0_2E0,V1n_2E0)) )
=> p(c_2Earithmetic_2E_3C_3D_2E2(c_2Enum_2ESUC_2E1(V1n_2E0),c_2Earithmetic_2E_2A_2E2(V0m_2E0,V1n_2E0))) ) ).
tff(thm_2Earithmetic_2EEXP__ALWAYS__BIG__ENOUGH,axiom,
! [V0b_2E0: tyop_2Enum_2Enum] :
( p(c_2Eprim__rec_2E_3C_2E2(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)),V0b_2E0))
=> ! [V1n_2E0: tyop_2Enum_2Enum] :
? [V2m_2E0: tyop_2Enum_2Enum] : p(c_2Earithmetic_2E_3C_3D_2E2(V1n_2E0,c_2Earithmetic_2EEXP_2E2(V0b_2E0,V2m_2E0))) ) ).
tff(thm_2Earithmetic_2EEXP__EQ__0,axiom,
! [V0n_2E0: tyop_2Enum_2Enum,V1m_2E0: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2EEXP_2E2(V0n_2E0,V1m_2E0) = c_2Enum_2E0_2E0 )
<=> ( ( V0n_2E0 = c_2Enum_2E0_2E0 )
& p(c_2Eprim__rec_2E_3C_2E2(c_2Enum_2E0_2E0,V1m_2E0)) ) ) ).
tff(thm_2Earithmetic_2EZERO__LT__EXP,axiom,
! [V0y_2E0: tyop_2Enum_2Enum,V1x_2E0: tyop_2Enum_2Enum] :
( p(c_2Eprim__rec_2E_3C_2E2(c_2Enum_2E0_2E0,c_2Earithmetic_2EEXP_2E2(V1x_2E0,V0y_2E0)))
<=> ( p(c_2Eprim__rec_2E_3C_2E2(c_2Enum_2E0_2E0,V1x_2E0))
| ( V0y_2E0 = c_2Enum_2E0_2E0 ) ) ) ).
tff(thm_2Earithmetic_2EEXP__1,axiom,
! [V0n_2E0: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2EEXP_2E2(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)),V0n_2E0) = c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)) )
& ( c_2Earithmetic_2EEXP_2E2(V0n_2E0,c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))) = V0n_2E0 ) ) ).
tff(thm_2Earithmetic_2EEXP__EQ__1,axiom,
! [V0n_2E0: tyop_2Enum_2Enum,V1m_2E0: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2EEXP_2E2(V0n_2E0,V1m_2E0) = c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)) )
<=> ( ( V0n_2E0 = c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)) )
| ( V1m_2E0 = c_2Enum_2E0_2E0 ) ) ) ).
tff(thm_2Earithmetic_2EEXP__BASE__LE__MONO,axiom,
! [V0b_2E0: tyop_2Enum_2Enum] :
( p(c_2Eprim__rec_2E_3C_2E2(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)),V0b_2E0))
=> ! [V1n_2E0: tyop_2Enum_2Enum,V2m_2E0: tyop_2Enum_2Enum] : c_2Earithmetic_2E_3C_3D_2E2(c_2Earithmetic_2EEXP_2E2(V0b_2E0,V2m_2E0),c_2Earithmetic_2EEXP_2E2(V0b_2E0,V1n_2E0)) = c_2Earithmetic_2E_3C_3D_2E2(V2m_2E0,V1n_2E0) ) ).
tff(thm_2Earithmetic_2EEXP__BASE__LT__MONO,axiom,
! [V0b_2E0: tyop_2Enum_2Enum] :
( p(c_2Eprim__rec_2E_3C_2E2(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)),V0b_2E0))
=> ! [V1n_2E0: tyop_2Enum_2Enum,V2m_2E0: tyop_2Enum_2Enum] : c_2Eprim__rec_2E_3C_2E2(c_2Earithmetic_2EEXP_2E2(V0b_2E0,V2m_2E0),c_2Earithmetic_2EEXP_2E2(V0b_2E0,V1n_2E0)) = c_2Eprim__rec_2E_3C_2E2(V2m_2E0,V1n_2E0) ) ).
tff(thm_2Earithmetic_2EEXP__BASE__INJECTIVE,axiom,
! [V0b_2E0: tyop_2Enum_2Enum] :
( p(c_2Eprim__rec_2E_3C_2E2(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)),V0b_2E0))
=> ! [V1n_2E0: tyop_2Enum_2Enum,V2m_2E0: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2EEXP_2E2(V0b_2E0,V1n_2E0) = c_2Earithmetic_2EEXP_2E2(V0b_2E0,V2m_2E0) )
<=> ( V1n_2E0 = V2m_2E0 ) ) ) ).
tff(thm_2Earithmetic_2EEXP__BASE__LEQ__MONO__IMP,axiom,
! [V0n_2E0: tyop_2Enum_2Enum,V1m_2E0: tyop_2Enum_2Enum,V2b_2E0: tyop_2Enum_2Enum] :
( ( p(c_2Eprim__rec_2E_3C_2E2(c_2Enum_2E0_2E0,V2b_2E0))
& p(c_2Earithmetic_2E_3C_3D_2E2(V1m_2E0,V0n_2E0)) )
=> p(c_2Earithmetic_2E_3C_3D_2E2(c_2Earithmetic_2EEXP_2E2(V2b_2E0,V1m_2E0),c_2Earithmetic_2EEXP_2E2(V2b_2E0,V0n_2E0))) ) ).
tff(thm_2Earithmetic_2EEXP__BASE__LEQ__MONO__SUC__IMP,axiom,
! [V0n_2E0: tyop_2Enum_2Enum,V1m_2E0: tyop_2Enum_2Enum,V2b_2E0: tyop_2Enum_2Enum] :
( p(c_2Earithmetic_2E_3C_3D_2E2(V1m_2E0,V0n_2E0))
=> p(c_2Earithmetic_2E_3C_3D_2E2(c_2Earithmetic_2EEXP_2E2(c_2Enum_2ESUC_2E1(V2b_2E0),V1m_2E0),c_2Earithmetic_2EEXP_2E2(c_2Enum_2ESUC_2E1(V2b_2E0),V0n_2E0))) ) ).
tff(thm_2Earithmetic_2EEXP__BASE__LE__IFF,axiom,
! [V0n_2E0: tyop_2Enum_2Enum,V1m_2E0: tyop_2Enum_2Enum,V2b_2E0: tyop_2Enum_2Enum] :
( p(c_2Earithmetic_2E_3C_3D_2E2(c_2Earithmetic_2EEXP_2E2(V2b_2E0,V1m_2E0),c_2Earithmetic_2EEXP_2E2(V2b_2E0,V0n_2E0)))
<=> ( ( ( V2b_2E0 = c_2Enum_2E0_2E0 )
& ( V0n_2E0 = c_2Enum_2E0_2E0 ) )
| ( ( V2b_2E0 = c_2Enum_2E0_2E0 )
& p(c_2Eprim__rec_2E_3C_2E2(c_2Enum_2E0_2E0,V1m_2E0)) )
| ( V2b_2E0 = c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)) )
| ( p(c_2Eprim__rec_2E_3C_2E2(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)),V2b_2E0))
& p(c_2Earithmetic_2E_3C_3D_2E2(V1m_2E0,V0n_2E0)) ) ) ) ).
tff(thm_2Earithmetic_2EX__LE__X__EXP,axiom,
! [V0x_2E0: tyop_2Enum_2Enum,V1n_2E0: tyop_2Enum_2Enum] :
( p(c_2Eprim__rec_2E_3C_2E2(c_2Enum_2E0_2E0,V1n_2E0))
=> p(c_2Earithmetic_2E_3C_3D_2E2(V0x_2E0,c_2Earithmetic_2EEXP_2E2(V0x_2E0,V1n_2E0))) ) ).
tff(thm_2Earithmetic_2EX__LT__EXP__X,axiom,
! [V0x_2E0: tyop_2Enum_2Enum,V1b_2E0: tyop_2Enum_2Enum] :
( p(c_2Eprim__rec_2E_3C_2E2(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)),V1b_2E0))
=> p(c_2Eprim__rec_2E_3C_2E2(V0x_2E0,c_2Earithmetic_2EEXP_2E2(V1b_2E0,V0x_2E0))) ) ).
tff(thm_2Earithmetic_2EZERO__EXP,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)] :
( ! [V0x_2E0: tyop_2Enum_2Enum] :
( p(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,F0_2E0,V0x_2E0))
<=> ( V0x_2E0 = c_2Enum_2E0_2E0 ) )
=> ! [V0x_2E0: tyop_2Enum_2Enum] : c_2Earithmetic_2EEXP_2E2(c_2Enum_2E0_2E0,V0x_2E0) = c_2Ebool_2ECOND_2E3(tyop_2Enum_2Enum,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,F0_2E0,V0x_2E0),c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)),c_2Enum_2E0_2E0) ) ).
tff(thm_2Earithmetic_2EX__LT__EXP__X__IFF,axiom,
! [V0x_2E0: tyop_2Enum_2Enum,V1b_2E0: tyop_2Enum_2Enum] :
( p(c_2Eprim__rec_2E_3C_2E2(V0x_2E0,c_2Earithmetic_2EEXP_2E2(V1b_2E0,V0x_2E0)))
<=> ( p(c_2Eprim__rec_2E_3C_2E2(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)),V1b_2E0))
| ( V0x_2E0 = c_2Enum_2E0_2E0 ) ) ) ).
tff(thm_2Earithmetic_2EEXP__EXP__LT__MONO,axiom,
! [V0n_2E0: tyop_2Enum_2Enum,V1a_2E0: tyop_2Enum_2Enum,V2b_2E0: tyop_2Enum_2Enum] :
( p(c_2Eprim__rec_2E_3C_2E2(c_2Earithmetic_2EEXP_2E2(V1a_2E0,V0n_2E0),c_2Earithmetic_2EEXP_2E2(V2b_2E0,V0n_2E0)))
<=> ( p(c_2Eprim__rec_2E_3C_2E2(V1a_2E0,V2b_2E0))
& p(c_2Eprim__rec_2E_3C_2E2(c_2Enum_2E0_2E0,V0n_2E0)) ) ) ).
tff(thm_2Earithmetic_2EEXP__EXP__LE__MONO,axiom,
! [V0n_2E0: tyop_2Enum_2Enum,V1a_2E0: tyop_2Enum_2Enum,V2b_2E0: tyop_2Enum_2Enum] :
( p(c_2Earithmetic_2E_3C_3D_2E2(c_2Earithmetic_2EEXP_2E2(V1a_2E0,V0n_2E0),c_2Earithmetic_2EEXP_2E2(V2b_2E0,V0n_2E0)))
<=> ( p(c_2Earithmetic_2E_3C_3D_2E2(V1a_2E0,V2b_2E0))
| ( V0n_2E0 = c_2Enum_2E0_2E0 ) ) ) ).
tff(thm_2Earithmetic_2EEXP__EXP__INJECTIVE,axiom,
! [V0b1_2E0: tyop_2Enum_2Enum,V1b2_2E0: tyop_2Enum_2Enum,V2x_2E0: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2EEXP_2E2(V0b1_2E0,V2x_2E0) = c_2Earithmetic_2EEXP_2E2(V1b2_2E0,V2x_2E0) )
<=> ( ( V2x_2E0 = c_2Enum_2E0_2E0 )
| ( V0b1_2E0 = V1b2_2E0 ) ) ) ).
tff(thm_2Earithmetic_2EEXP__SUB,axiom,
! [V0p_2E0: tyop_2Enum_2Enum,V1q_2E0: tyop_2Enum_2Enum,V2n_2E0: tyop_2Enum_2Enum] :
( ( p(c_2Eprim__rec_2E_3C_2E2(c_2Enum_2E0_2E0,V2n_2E0))
& p(c_2Earithmetic_2E_3C_3D_2E2(V1q_2E0,V0p_2E0)) )
=> ( c_2Earithmetic_2EEXP_2E2(V2n_2E0,c_2Earithmetic_2E_2D_2E2(V0p_2E0,V1q_2E0)) = c_2Earithmetic_2EDIV_2E2(c_2Earithmetic_2EEXP_2E2(V2n_2E0,V0p_2E0),c_2Earithmetic_2EEXP_2E2(V2n_2E0,V1q_2E0)) ) ) ).
tff(thm_2Earithmetic_2EEXP__SUB__NUMERAL,axiom,
! [V0x_2E0: tyop_2Enum_2Enum,V1n_2E0: tyop_2Enum_2Enum] :
( p(c_2Eprim__rec_2E_3C_2E2(c_2Enum_2E0_2E0,V1n_2E0))
=> ( ( c_2Earithmetic_2EDIV_2E2(c_2Earithmetic_2EEXP_2E2(V1n_2E0,c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(V0x_2E0))),V1n_2E0) = c_2Earithmetic_2EEXP_2E2(V1n_2E0,c_2Earithmetic_2E_2D_2E2(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(V0x_2E0)),c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)))) )
& ( c_2Earithmetic_2EDIV_2E2(c_2Earithmetic_2EEXP_2E2(V1n_2E0,c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(V0x_2E0))),V1n_2E0) = c_2Earithmetic_2EEXP_2E2(V1n_2E0,c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(V0x_2E0))) ) ) ) ).
tff(thm_2Earithmetic_2EEXP__BASE__MULT,axiom,
! [V0z_2E0: tyop_2Enum_2Enum,V1x_2E0: tyop_2Enum_2Enum,V2y_2E0: tyop_2Enum_2Enum] : c_2Earithmetic_2EEXP_2E2(c_2Earithmetic_2E_2A_2E2(V1x_2E0,V2y_2E0),V0z_2E0) = c_2Earithmetic_2E_2A_2E2(c_2Earithmetic_2EEXP_2E2(V1x_2E0,V0z_2E0),c_2Earithmetic_2EEXP_2E2(V2y_2E0,V0z_2E0)) ).
tff(thm_2Earithmetic_2EEXP__EXP__MULT,axiom,
! [V0z_2E0: tyop_2Enum_2Enum,V1x_2E0: tyop_2Enum_2Enum,V2y_2E0: tyop_2Enum_2Enum] : c_2Earithmetic_2EEXP_2E2(V1x_2E0,c_2Earithmetic_2E_2A_2E2(V2y_2E0,V0z_2E0)) = c_2Earithmetic_2EEXP_2E2(c_2Earithmetic_2EEXP_2E2(V1x_2E0,V2y_2E0),V0z_2E0) ).
tff(thm_2Earithmetic_2EMAX__COMM,axiom,
! [V0m_2E0: tyop_2Enum_2Enum,V1n_2E0: tyop_2Enum_2Enum] : c_2Earithmetic_2EMAX_2E2(V0m_2E0,V1n_2E0) = c_2Earithmetic_2EMAX_2E2(V1n_2E0,V0m_2E0) ).
tff(thm_2Earithmetic_2EMIN__COMM,axiom,
! [V0m_2E0: tyop_2Enum_2Enum,V1n_2E0: tyop_2Enum_2Enum] : c_2Earithmetic_2EMIN_2E2(V0m_2E0,V1n_2E0) = c_2Earithmetic_2EMIN_2E2(V1n_2E0,V0m_2E0) ).
tff(thm_2Earithmetic_2EMAX__ASSOC,axiom,
! [V0m_2E0: tyop_2Enum_2Enum,V1n_2E0: tyop_2Enum_2Enum,V2p_2E0: tyop_2Enum_2Enum] : c_2Earithmetic_2EMAX_2E2(V0m_2E0,c_2Earithmetic_2EMAX_2E2(V1n_2E0,V2p_2E0)) = c_2Earithmetic_2EMAX_2E2(c_2Earithmetic_2EMAX_2E2(V0m_2E0,V1n_2E0),V2p_2E0) ).
tff(thm_2Earithmetic_2EMIN__ASSOC,axiom,
! [V0m_2E0: tyop_2Enum_2Enum,V1n_2E0: tyop_2Enum_2Enum,V2p_2E0: tyop_2Enum_2Enum] : c_2Earithmetic_2EMIN_2E2(V0m_2E0,c_2Earithmetic_2EMIN_2E2(V1n_2E0,V2p_2E0)) = c_2Earithmetic_2EMIN_2E2(c_2Earithmetic_2EMIN_2E2(V0m_2E0,V1n_2E0),V2p_2E0) ).
tff(thm_2Earithmetic_2EMIN__MAX__EQ,axiom,
! [V0m_2E0: tyop_2Enum_2Enum,V1n_2E0: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2EMIN_2E2(V0m_2E0,V1n_2E0) = c_2Earithmetic_2EMAX_2E2(V0m_2E0,V1n_2E0) )
<=> ( V0m_2E0 = V1n_2E0 ) ) ).
tff(thm_2Earithmetic_2EMIN__MAX__LT,axiom,
! [V0m_2E0: tyop_2Enum_2Enum,V1n_2E0: tyop_2Enum_2Enum] :
( p(c_2Eprim__rec_2E_3C_2E2(c_2Earithmetic_2EMIN_2E2(V0m_2E0,V1n_2E0),c_2Earithmetic_2EMAX_2E2(V0m_2E0,V1n_2E0)))
<=> ( V0m_2E0 != V1n_2E0 ) ) ).
tff(thm_2Earithmetic_2EMIN__MAX__LE,axiom,
! [V0m_2E0: tyop_2Enum_2Enum,V1n_2E0: tyop_2Enum_2Enum] : p(c_2Earithmetic_2E_3C_3D_2E2(c_2Earithmetic_2EMIN_2E2(V0m_2E0,V1n_2E0),c_2Earithmetic_2EMAX_2E2(V0m_2E0,V1n_2E0))) ).
tff(thm_2Earithmetic_2EMIN__MAX__PRED,axiom,
! [V0P_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),V1m_2E0: tyop_2Enum_2Enum,V2n_2E0: tyop_2Enum_2Enum] :
( ( p(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,V0P_2E0,V1m_2E0))
& p(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,V0P_2E0,V2n_2E0)) )
=> ( p(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,V0P_2E0,c_2Earithmetic_2EMIN_2E2(V1m_2E0,V2n_2E0)))
& p(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,V0P_2E0,c_2Earithmetic_2EMAX_2E2(V1m_2E0,V2n_2E0))) ) ) ).
tff(thm_2Earithmetic_2EMIN__LT,axiom,
! [V0n_2E0: tyop_2Enum_2Enum,V1m_2E0: tyop_2Enum_2Enum,V2p_2E0: tyop_2Enum_2Enum] :
( ( p(c_2Eprim__rec_2E_3C_2E2(c_2Earithmetic_2EMIN_2E2(V1m_2E0,V0n_2E0),V2p_2E0))
<=> ( p(c_2Eprim__rec_2E_3C_2E2(V1m_2E0,V2p_2E0))
| p(c_2Eprim__rec_2E_3C_2E2(V0n_2E0,V2p_2E0)) ) )
& ( p(c_2Eprim__rec_2E_3C_2E2(V2p_2E0,c_2Earithmetic_2EMIN_2E2(V1m_2E0,V0n_2E0)))
<=> ( p(c_2Eprim__rec_2E_3C_2E2(V2p_2E0,V1m_2E0))
& p(c_2Eprim__rec_2E_3C_2E2(V2p_2E0,V0n_2E0)) ) ) ) ).
tff(thm_2Earithmetic_2EMAX__LT,axiom,
! [V0n_2E0: tyop_2Enum_2Enum,V1m_2E0: tyop_2Enum_2Enum,V2p_2E0: tyop_2Enum_2Enum] :
( ( p(c_2Eprim__rec_2E_3C_2E2(V2p_2E0,c_2Earithmetic_2EMAX_2E2(V1m_2E0,V0n_2E0)))
<=> ( p(c_2Eprim__rec_2E_3C_2E2(V2p_2E0,V1m_2E0))
| p(c_2Eprim__rec_2E_3C_2E2(V2p_2E0,V0n_2E0)) ) )
& ( p(c_2Eprim__rec_2E_3C_2E2(c_2Earithmetic_2EMAX_2E2(V1m_2E0,V0n_2E0),V2p_2E0))
<=> ( p(c_2Eprim__rec_2E_3C_2E2(V1m_2E0,V2p_2E0))
& p(c_2Eprim__rec_2E_3C_2E2(V0n_2E0,V2p_2E0)) ) ) ) ).
tff(thm_2Earithmetic_2EMIN__LE,axiom,
! [V0n_2E0: tyop_2Enum_2Enum,V1m_2E0: tyop_2Enum_2Enum,V2p_2E0: tyop_2Enum_2Enum] :
( ( p(c_2Earithmetic_2E_3C_3D_2E2(c_2Earithmetic_2EMIN_2E2(V1m_2E0,V0n_2E0),V2p_2E0))
<=> ( p(c_2Earithmetic_2E_3C_3D_2E2(V1m_2E0,V2p_2E0))
| p(c_2Earithmetic_2E_3C_3D_2E2(V0n_2E0,V2p_2E0)) ) )
& ( p(c_2Earithmetic_2E_3C_3D_2E2(V2p_2E0,c_2Earithmetic_2EMIN_2E2(V1m_2E0,V0n_2E0)))
<=> ( p(c_2Earithmetic_2E_3C_3D_2E2(V2p_2E0,V1m_2E0))
& p(c_2Earithmetic_2E_3C_3D_2E2(V2p_2E0,V0n_2E0)) ) ) ) ).
tff(thm_2Earithmetic_2EMAX__LE,axiom,
! [V0n_2E0: tyop_2Enum_2Enum,V1m_2E0: tyop_2Enum_2Enum,V2p_2E0: tyop_2Enum_2Enum] :
( ( p(c_2Earithmetic_2E_3C_3D_2E2(V2p_2E0,c_2Earithmetic_2EMAX_2E2(V1m_2E0,V0n_2E0)))
<=> ( p(c_2Earithmetic_2E_3C_3D_2E2(V2p_2E0,V1m_2E0))
| p(c_2Earithmetic_2E_3C_3D_2E2(V2p_2E0,V0n_2E0)) ) )
& ( p(c_2Earithmetic_2E_3C_3D_2E2(c_2Earithmetic_2EMAX_2E2(V1m_2E0,V0n_2E0),V2p_2E0))
<=> ( p(c_2Earithmetic_2E_3C_3D_2E2(V1m_2E0,V2p_2E0))
& p(c_2Earithmetic_2E_3C_3D_2E2(V0n_2E0,V2p_2E0)) ) ) ) ).
tff(thm_2Earithmetic_2EMIN__0,axiom,
! [V0n_2E0: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2EMIN_2E2(V0n_2E0,c_2Enum_2E0_2E0) = c_2Enum_2E0_2E0 )
& ( c_2Earithmetic_2EMIN_2E2(c_2Enum_2E0_2E0,V0n_2E0) = c_2Enum_2E0_2E0 ) ) ).
tff(thm_2Earithmetic_2EMAX__0,axiom,
! [V0n_2E0: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2EMAX_2E2(V0n_2E0,c_2Enum_2E0_2E0) = V0n_2E0 )
& ( c_2Earithmetic_2EMAX_2E2(c_2Enum_2E0_2E0,V0n_2E0) = V0n_2E0 ) ) ).
tff(thm_2Earithmetic_2EMAX__EQ__0,axiom,
! [V0n_2E0: tyop_2Enum_2Enum,V1m_2E0: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2EMAX_2E2(V1m_2E0,V0n_2E0) = c_2Enum_2E0_2E0 )
<=> ( ( V1m_2E0 = c_2Enum_2E0_2E0 )
& ( V0n_2E0 = c_2Enum_2E0_2E0 ) ) ) ).
tff(thm_2Earithmetic_2EMIN__EQ__0,axiom,
! [V0n_2E0: tyop_2Enum_2Enum,V1m_2E0: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2EMIN_2E2(V1m_2E0,V0n_2E0) = c_2Enum_2E0_2E0 )
<=> ( ( V1m_2E0 = c_2Enum_2E0_2E0 )
| ( V0n_2E0 = c_2Enum_2E0_2E0 ) ) ) ).
tff(thm_2Earithmetic_2EMIN__IDEM,axiom,
! [V0n_2E0: tyop_2Enum_2Enum] : c_2Earithmetic_2EMIN_2E2(V0n_2E0,V0n_2E0) = V0n_2E0 ).
tff(thm_2Earithmetic_2EMAX__IDEM,axiom,
! [V0n_2E0: tyop_2Enum_2Enum] : c_2Earithmetic_2EMAX_2E2(V0n_2E0,V0n_2E0) = V0n_2E0 ).
tff(thm_2Earithmetic_2EEXISTS__GREATEST,axiom,
! [V0P_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)] :
( ( ? [V1x_2E0: tyop_2Enum_2Enum] : p(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,V0P_2E0,V1x_2E0))
& ? [V2x_2E0: tyop_2Enum_2Enum] :
! [V3y_2E0: tyop_2Enum_2Enum] :
( p(c_2Earithmetic_2E_3E_2E2(V3y_2E0,V2x_2E0))
=> ~ p(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,V0P_2E0,V3y_2E0)) ) )
<=> ? [V4x_2E0: tyop_2Enum_2Enum] :
( p(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,V0P_2E0,V4x_2E0))
& ! [V5y_2E0: tyop_2Enum_2Enum] :
( p(c_2Earithmetic_2E_3E_2E2(V5y_2E0,V4x_2E0))
=> ~ p(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,V0P_2E0,V5y_2E0)) ) ) ) ).
tff(thm_2Earithmetic_2EEXISTS__NUM,axiom,
! [V0P_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)] :
( ? [V1n_2E0: tyop_2Enum_2Enum] : p(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,V0P_2E0,V1n_2E0))
<=> ( p(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,V0P_2E0,c_2Enum_2E0_2E0))
| ? [V2m_2E0: tyop_2Enum_2Enum] : p(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,V0P_2E0,c_2Enum_2ESUC_2E1(V2m_2E0))) ) ) ).
tff(thm_2Earithmetic_2EFORALL__NUM,axiom,
! [V0P_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)] :
( ! [V1n_2E0: tyop_2Enum_2Enum] : p(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,V0P_2E0,V1n_2E0))
<=> ( p(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,V0P_2E0,c_2Enum_2E0_2E0))
& ! [V2n_2E0: tyop_2Enum_2Enum] : p(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,V0P_2E0,c_2Enum_2ESUC_2E1(V2n_2E0))) ) ) ).
tff(thm_2Earithmetic_2EBOUNDED__FORALL__THM,axiom,
! [V0P_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),V1c_2E0: tyop_2Enum_2Enum] :
( p(c_2Eprim__rec_2E_3C_2E2(c_2Enum_2E0_2E0,V1c_2E0))
=> ( ! [V2n_2E0: tyop_2Enum_2Enum] :
( p(c_2Eprim__rec_2E_3C_2E2(V2n_2E0,V1c_2E0))
=> p(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,V0P_2E0,V2n_2E0)) )
<=> ( p(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,V0P_2E0,c_2Earithmetic_2E_2D_2E2(V1c_2E0,c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)))))
& ! [V3n_2E0: tyop_2Enum_2Enum] :
( p(c_2Eprim__rec_2E_3C_2E2(V3n_2E0,c_2Earithmetic_2E_2D_2E2(V1c_2E0,c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)))))
=> p(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,V0P_2E0,V3n_2E0)) ) ) ) ) ).
tff(thm_2Earithmetic_2EBOUNDED__EXISTS__THM,axiom,
! [V0P_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),V1c_2E0: tyop_2Enum_2Enum] :
( p(c_2Eprim__rec_2E_3C_2E2(c_2Enum_2E0_2E0,V1c_2E0))
=> ( ? [V2n_2E0: tyop_2Enum_2Enum] :
( p(c_2Eprim__rec_2E_3C_2E2(V2n_2E0,V1c_2E0))
& p(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,V0P_2E0,V2n_2E0)) )
<=> ( p(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,V0P_2E0,c_2Earithmetic_2E_2D_2E2(V1c_2E0,c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)))))
| ? [V3n_2E0: tyop_2Enum_2Enum] :
( p(c_2Eprim__rec_2E_3C_2E2(V3n_2E0,c_2Earithmetic_2E_2D_2E2(V1c_2E0,c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)))))
& p(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,V0P_2E0,V3n_2E0)) ) ) ) ) ).
tff(thm_2Earithmetic_2Etransitive__monotone,axiom,
! [A_27a: $tType,V0R_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),V1f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,A_27a)] :
( ( p(c_2Erelation_2Etransitive_2E1(A_27a,V0R_2E0))
& ! [V2n_2E0: tyop_2Enum_2Enum] : p(app_2E2(A_27a,tyop_2Emin_2Ebool,app_2E2(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V0R_2E0,app_2E2(tyop_2Enum_2Enum,A_27a,V1f_2E0,V2n_2E0)),app_2E2(tyop_2Enum_2Enum,A_27a,V1f_2E0,c_2Enum_2ESUC_2E1(V2n_2E0)))) )
=> ! [V3m_2E0: tyop_2Enum_2Enum,V4n_2E0: tyop_2Enum_2Enum] :
( p(c_2Eprim__rec_2E_3C_2E2(V3m_2E0,V4n_2E0))
=> p(app_2E2(A_27a,tyop_2Emin_2Ebool,app_2E2(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V0R_2E0,app_2E2(tyop_2Enum_2Enum,A_27a,V1f_2E0,V3m_2E0)),app_2E2(tyop_2Enum_2Enum,A_27a,V1f_2E0,V4n_2E0))) ) ) ).
tff(thm_2Earithmetic_2ESTRICTLY__INCREASING__TC,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum)] :
( ! [V1n_2E0: tyop_2Enum_2Enum] : p(c_2Eprim__rec_2E_3C_2E2(app_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,V0f_2E0,V1n_2E0),app_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,V0f_2E0,c_2Enum_2ESUC_2E1(V1n_2E0))))
=> ! [V2m_2E0: tyop_2Enum_2Enum,V3n_2E0: tyop_2Enum_2Enum] :
( p(c_2Eprim__rec_2E_3C_2E2(V2m_2E0,V3n_2E0))
=> p(c_2Eprim__rec_2E_3C_2E2(app_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,V0f_2E0,V2m_2E0),app_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,V0f_2E0,V3n_2E0))) ) ) ).
tff(thm_2Earithmetic_2ESTRICTLY__INCREASING__ONE__ONE,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum)] :
( ! [V1n_2E0: tyop_2Enum_2Enum] : p(c_2Eprim__rec_2E_3C_2E2(app_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,V0f_2E0,V1n_2E0),app_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,V0f_2E0,c_2Enum_2ESUC_2E1(V1n_2E0))))
=> p(c_2Ebool_2EONE__ONE_2E1(tyop_2Enum_2Enum,tyop_2Enum_2Enum,V0f_2E0)) ) ).
tff(thm_2Earithmetic_2EONE__ONE__INV__IMAGE__BOUNDED,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum)] :
( p(c_2Ebool_2EONE__ONE_2E1(tyop_2Enum_2Enum,tyop_2Enum_2Enum,V0f_2E0))
=> ! [V1b_2E0: tyop_2Enum_2Enum] :
? [V2a_2E0: tyop_2Enum_2Enum] :
! [V3x_2E0: tyop_2Enum_2Enum] :
( p(c_2Earithmetic_2E_3C_3D_2E2(app_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,V0f_2E0,V3x_2E0),V1b_2E0))
=> p(c_2Earithmetic_2E_3C_3D_2E2(V3x_2E0,V2a_2E0)) ) ) ).
tff(thm_2Earithmetic_2EONE__ONE__UNBOUNDED,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum)] :
( p(c_2Ebool_2EONE__ONE_2E1(tyop_2Enum_2Enum,tyop_2Enum_2Enum,V0f_2E0))
=> ! [V1b_2E0: tyop_2Enum_2Enum] :
? [V2n_2E0: tyop_2Enum_2Enum] : p(c_2Eprim__rec_2E_3C_2E2(V1b_2E0,app_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,V0f_2E0,V2n_2E0))) ) ).
tff(thm_2Earithmetic_2ESTRICTLY__INCREASING__UNBOUNDED,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum)] :
( ! [V1n_2E0: tyop_2Enum_2Enum] : p(c_2Eprim__rec_2E_3C_2E2(app_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,V0f_2E0,V1n_2E0),app_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,V0f_2E0,c_2Enum_2ESUC_2E1(V1n_2E0))))
=> ! [V2b_2E0: tyop_2Enum_2Enum] :
? [V3n_2E0: tyop_2Enum_2Enum] : p(c_2Eprim__rec_2E_3C_2E2(V2b_2E0,app_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,V0f_2E0,V3n_2E0))) ) ).
tff(thm_2Earithmetic_2ENOT__STRICTLY__DECREASING,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum)] :
~ ! [V1n_2E0: tyop_2Enum_2Enum] : p(c_2Eprim__rec_2E_3C_2E2(app_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,V0f_2E0,c_2Enum_2ESUC_2E1(V1n_2E0)),app_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,V0f_2E0,V1n_2E0))) ).
tff(thm_2Earithmetic_2EABS__DIFF__SYM,axiom,
! [V0n_2E0: tyop_2Enum_2Enum,V1m_2E0: tyop_2Enum_2Enum] : c_2Earithmetic_2EABS__DIFF_2E2(V0n_2E0,V1m_2E0) = c_2Earithmetic_2EABS__DIFF_2E2(V1m_2E0,V0n_2E0) ).
tff(thm_2Earithmetic_2EABS__DIFF__COMM,axiom,
! [V0n_2E0: tyop_2Enum_2Enum,V1m_2E0: tyop_2Enum_2Enum] : c_2Earithmetic_2EABS__DIFF_2E2(V0n_2E0,V1m_2E0) = c_2Earithmetic_2EABS__DIFF_2E2(V1m_2E0,V0n_2E0) ).
tff(thm_2Earithmetic_2EABS__DIFF__EQS,axiom,
! [V0n_2E0: tyop_2Enum_2Enum] : c_2Earithmetic_2EABS__DIFF_2E2(V0n_2E0,V0n_2E0) = c_2Enum_2E0_2E0 ).
tff(thm_2Earithmetic_2EABS__DIFF__EQ__0,axiom,
! [V0n_2E0: tyop_2Enum_2Enum,V1m_2E0: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2EABS__DIFF_2E2(V0n_2E0,V1m_2E0) = c_2Enum_2E0_2E0 )
<=> ( V0n_2E0 = V1m_2E0 ) ) ).
tff(thm_2Earithmetic_2EABS__DIFF__ZERO,axiom,
! [V0n_2E0: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2EABS__DIFF_2E2(V0n_2E0,c_2Enum_2E0_2E0) = V0n_2E0 )
& ( c_2Earithmetic_2EABS__DIFF_2E2(c_2Enum_2E0_2E0,V0n_2E0) = V0n_2E0 ) ) ).
tff(thm_2Earithmetic_2EABS__DIFF__SUC,axiom,
! [V0n_2E0: tyop_2Enum_2Enum,V1m_2E0: tyop_2Enum_2Enum] : c_2Earithmetic_2EABS__DIFF_2E2(c_2Enum_2ESUC_2E1(V0n_2E0),c_2Enum_2ESUC_2E1(V1m_2E0)) = c_2Earithmetic_2EABS__DIFF_2E2(V0n_2E0,V1m_2E0) ).
tff(thm_2Earithmetic_2EABS__DIFF__SUC__LE,axiom,
! [V0x_2E0: tyop_2Enum_2Enum,V1z_2E0: tyop_2Enum_2Enum] : p(c_2Earithmetic_2E_3C_3D_2E2(c_2Earithmetic_2EABS__DIFF_2E2(V0x_2E0,c_2Enum_2ESUC_2E1(V1z_2E0)),c_2Enum_2ESUC_2E1(c_2Earithmetic_2EABS__DIFF_2E2(V0x_2E0,V1z_2E0)))) ).
tff(thm_2Earithmetic_2EABS__DIFF__PLUS__LE,axiom,
! [V0x_2E0: tyop_2Enum_2Enum,V1z_2E0: tyop_2Enum_2Enum,V2y_2E0: tyop_2Enum_2Enum] : p(c_2Earithmetic_2E_3C_3D_2E2(c_2Earithmetic_2EABS__DIFF_2E2(V0x_2E0,c_2Earithmetic_2E_2B_2E2(V2y_2E0,V1z_2E0)),c_2Earithmetic_2E_2B_2E2(V2y_2E0,c_2Earithmetic_2EABS__DIFF_2E2(V0x_2E0,V1z_2E0)))) ).
tff(thm_2Earithmetic_2EABS__DIFF__LE__SUM,axiom,
! [V0z_2E0: tyop_2Enum_2Enum,V1x_2E0: tyop_2Enum_2Enum] : p(c_2Earithmetic_2E_3C_3D_2E2(c_2Earithmetic_2EABS__DIFF_2E2(V1x_2E0,V0z_2E0),c_2Earithmetic_2E_2B_2E2(V1x_2E0,V0z_2E0))) ).
tff(thm_2Earithmetic_2EABS__DIFF__TRIANGLE__lem,axiom,
! [V0x_2E0: tyop_2Enum_2Enum,V1y_2E0: tyop_2Enum_2Enum] : p(c_2Earithmetic_2E_3C_3D_2E2(V0x_2E0,c_2Earithmetic_2E_2B_2E2(c_2Earithmetic_2EABS__DIFF_2E2(V0x_2E0,V1y_2E0),V1y_2E0))) ).
tff(thm_2Earithmetic_2EABS__DIFF__TRIANGLE,axiom,
! [V0x_2E0: tyop_2Enum_2Enum,V1y_2E0: tyop_2Enum_2Enum,V2z_2E0: tyop_2Enum_2Enum] : p(c_2Earithmetic_2E_3C_3D_2E2(c_2Earithmetic_2EABS__DIFF_2E2(V0x_2E0,V2z_2E0),c_2Earithmetic_2E_2B_2E2(c_2Earithmetic_2EABS__DIFF_2E2(V0x_2E0,V1y_2E0),c_2Earithmetic_2EABS__DIFF_2E2(V1y_2E0,V2z_2E0)))) ).
tff(thm_2Earithmetic_2EABS__DIFF__ADD__SAME,axiom,
! [V0n_2E0: tyop_2Enum_2Enum,V1m_2E0: tyop_2Enum_2Enum,V2p_2E0: tyop_2Enum_2Enum] : c_2Earithmetic_2EABS__DIFF_2E2(c_2Earithmetic_2E_2B_2E2(V0n_2E0,V2p_2E0),c_2Earithmetic_2E_2B_2E2(V1m_2E0,V2p_2E0)) = c_2Earithmetic_2EABS__DIFF_2E2(V0n_2E0,V1m_2E0) ).
tff(thm_2Earithmetic_2ELE__SUB__RCANCEL,axiom,
! [V0m_2E0: tyop_2Enum_2Enum,V1n_2E0: tyop_2Enum_2Enum,V2p_2E0: tyop_2Enum_2Enum] :
( p(c_2Earithmetic_2E_3C_3D_2E2(c_2Earithmetic_2E_2D_2E2(V1n_2E0,V0m_2E0),c_2Earithmetic_2E_2D_2E2(V2p_2E0,V0m_2E0)))
<=> ( p(c_2Earithmetic_2E_3C_3D_2E2(V1n_2E0,V0m_2E0))
| p(c_2Earithmetic_2E_3C_3D_2E2(V1n_2E0,V2p_2E0)) ) ) ).
tff(thm_2Earithmetic_2ELT__SUB__RCANCEL,axiom,
! [V0m_2E0: tyop_2Enum_2Enum,V1n_2E0: tyop_2Enum_2Enum,V2p_2E0: tyop_2Enum_2Enum] :
( p(c_2Eprim__rec_2E_3C_2E2(c_2Earithmetic_2E_2D_2E2(V1n_2E0,V0m_2E0),c_2Earithmetic_2E_2D_2E2(V2p_2E0,V0m_2E0)))
<=> ( p(c_2Eprim__rec_2E_3C_2E2(V1n_2E0,V2p_2E0))
& p(c_2Eprim__rec_2E_3C_2E2(V0m_2E0,V2p_2E0)) ) ) ).
tff(thm_2Earithmetic_2ELE__SUB__LCANCEL,axiom,
! [V0z_2E0: tyop_2Enum_2Enum,V1y_2E0: tyop_2Enum_2Enum,V2x_2E0: tyop_2Enum_2Enum] :
( p(c_2Earithmetic_2E_3C_3D_2E2(c_2Earithmetic_2E_2D_2E2(V2x_2E0,V1y_2E0),c_2Earithmetic_2E_2D_2E2(V2x_2E0,V0z_2E0)))
<=> ( p(c_2Earithmetic_2E_3C_3D_2E2(V0z_2E0,V1y_2E0))
| p(c_2Earithmetic_2E_3C_3D_2E2(V2x_2E0,V1y_2E0)) ) ) ).
tff(thm_2Earithmetic_2ELT__SUB__LCANCEL,axiom,
! [V0z_2E0: tyop_2Enum_2Enum,V1y_2E0: tyop_2Enum_2Enum,V2x_2E0: tyop_2Enum_2Enum] :
( p(c_2Eprim__rec_2E_3C_2E2(c_2Earithmetic_2E_2D_2E2(V2x_2E0,V1y_2E0),c_2Earithmetic_2E_2D_2E2(V2x_2E0,V0z_2E0)))
<=> ( p(c_2Eprim__rec_2E_3C_2E2(V0z_2E0,V1y_2E0))
& p(c_2Eprim__rec_2E_3C_2E2(V0z_2E0,V2x_2E0)) ) ) ).
tff(thm_2Earithmetic_2EABS__DIFF__SUMS,axiom,
! [V0n1_2E0: tyop_2Enum_2Enum,V1n2_2E0: tyop_2Enum_2Enum,V2m1_2E0: tyop_2Enum_2Enum,V3m2_2E0: tyop_2Enum_2Enum] : p(c_2Earithmetic_2E_3C_3D_2E2(c_2Earithmetic_2EABS__DIFF_2E2(c_2Earithmetic_2E_2B_2E2(V0n1_2E0,V1n2_2E0),c_2Earithmetic_2E_2B_2E2(V2m1_2E0,V3m2_2E0)),c_2Earithmetic_2E_2B_2E2(c_2Earithmetic_2EABS__DIFF_2E2(V0n1_2E0,V2m1_2E0),c_2Earithmetic_2EABS__DIFF_2E2(V1n2_2E0,V3m2_2E0)))) ).
tff(thm_2Earithmetic_2EFUNPOW__SUC,axiom,
! [A_27a: $tType,V0f_2E0: tyop_2Emin_2Efun(A_27a,A_27a),V1n_2E0: tyop_2Enum_2Enum,V2x_2E0: A_27a] : c_2Earithmetic_2EFUNPOW_2E3(A_27a,V0f_2E0,c_2Enum_2ESUC_2E1(V1n_2E0),V2x_2E0) = app_2E2(A_27a,A_27a,V0f_2E0,c_2Earithmetic_2EFUNPOW_2E3(A_27a,V0f_2E0,V1n_2E0,V2x_2E0)) ).
tff(thm_2Earithmetic_2EFUNPOW__0,axiom,
! [A_27a: $tType,V0x_2E0: A_27a,V1f_2E0: tyop_2Emin_2Efun(A_27a,A_27a)] : c_2Earithmetic_2EFUNPOW_2E3(A_27a,V1f_2E0,c_2Enum_2E0_2E0,V0x_2E0) = V0x_2E0 ).
tff(thm_2Earithmetic_2EFUNPOW__ADD,axiom,
! [A_27a: $tType,V0x_2E0: A_27a,V1f_2E0: tyop_2Emin_2Efun(A_27a,A_27a),V2m_2E0: tyop_2Enum_2Enum,V3n_2E0: tyop_2Enum_2Enum] : c_2Earithmetic_2EFUNPOW_2E3(A_27a,V1f_2E0,c_2Earithmetic_2E_2B_2E2(V2m_2E0,V3n_2E0),V0x_2E0) = c_2Earithmetic_2EFUNPOW_2E3(A_27a,V1f_2E0,V2m_2E0,c_2Earithmetic_2EFUNPOW_2E3(A_27a,V1f_2E0,V3n_2E0,V0x_2E0)) ).
tff(thm_2Earithmetic_2EFUNPOW__1,axiom,
! [A_27a: $tType,V0x_2E0: A_27a,V1f_2E0: tyop_2Emin_2Efun(A_27a,A_27a)] : c_2Earithmetic_2EFUNPOW_2E3(A_27a,V1f_2E0,c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)),V0x_2E0) = app_2E2(A_27a,A_27a,V1f_2E0,V0x_2E0) ).
tff(thm_2Earithmetic_2ENRC__0,axiom,
! [A_27a: $tType,V0R_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),V1x_2E0: A_27a,V2y_2E0: A_27a] :
( p(c_2Earithmetic_2ENRC_2E4(A_27a,V0R_2E0,c_2Enum_2E0_2E0,V1x_2E0,V2y_2E0))
<=> ( V1x_2E0 = V2y_2E0 ) ) ).
tff(thm_2Earithmetic_2ENRC__1,axiom,
! [A_27a: $tType,V0y_2E0: A_27a,V1x_2E0: A_27a,V2R_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))] : c_2Earithmetic_2ENRC_2E4(A_27a,V2R_2E0,c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)),V1x_2E0,V0y_2E0) = app_2E2(A_27a,tyop_2Emin_2Ebool,app_2E2(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V2R_2E0,V1x_2E0),V0y_2E0) ).
tff(thm_2Earithmetic_2ENRC__ADD__I,axiom,
! [A_27a: $tType,V0R_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),V1m_2E0: tyop_2Enum_2Enum,V2n_2E0: tyop_2Enum_2Enum,V3x_2E0: A_27a,V4y_2E0: A_27a,V5z_2E0: A_27a] :
( ( p(c_2Earithmetic_2ENRC_2E4(A_27a,V0R_2E0,V1m_2E0,V3x_2E0,V4y_2E0))
& p(c_2Earithmetic_2ENRC_2E4(A_27a,V0R_2E0,V2n_2E0,V4y_2E0,V5z_2E0)) )
=> p(c_2Earithmetic_2ENRC_2E4(A_27a,V0R_2E0,c_2Earithmetic_2E_2B_2E2(V1m_2E0,V2n_2E0),V3x_2E0,V5z_2E0)) ) ).
tff(thm_2Earithmetic_2ENRC__ADD__E,axiom,
! [A_27a: $tType,V0R_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),V1m_2E0: tyop_2Enum_2Enum,V2n_2E0: tyop_2Enum_2Enum,V3x_2E0: A_27a,V4z_2E0: A_27a] :
( p(c_2Earithmetic_2ENRC_2E4(A_27a,V0R_2E0,c_2Earithmetic_2E_2B_2E2(V1m_2E0,V2n_2E0),V3x_2E0,V4z_2E0))
=> ? [V5y_2E0: A_27a] :
( p(c_2Earithmetic_2ENRC_2E4(A_27a,V0R_2E0,V1m_2E0,V3x_2E0,V5y_2E0))
& p(c_2Earithmetic_2ENRC_2E4(A_27a,V0R_2E0,V2n_2E0,V5y_2E0,V4z_2E0)) ) ) ).
tff(thm_2Earithmetic_2ENRC__ADD__EQN,axiom,
! [A_27a: $tType,V0z_2E0: A_27a,V1x_2E0: A_27a,V2n_2E0: tyop_2Enum_2Enum,V3m_2E0: tyop_2Enum_2Enum,V4R_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))] :
( p(c_2Earithmetic_2ENRC_2E4(A_27a,V4R_2E0,c_2Earithmetic_2E_2B_2E2(V3m_2E0,V2n_2E0),V1x_2E0,V0z_2E0))
<=> ? [V5y_2E0: A_27a] :
( p(c_2Earithmetic_2ENRC_2E4(A_27a,V4R_2E0,V3m_2E0,V1x_2E0,V5y_2E0))
& p(c_2Earithmetic_2ENRC_2E4(A_27a,V4R_2E0,V2n_2E0,V5y_2E0,V0z_2E0)) ) ) ).
tff(thm_2Earithmetic_2ENRC__SUC__RECURSE__LEFT,axiom,
! [A_27a: $tType,V0y_2E0: A_27a,V1x_2E0: A_27a,V2n_2E0: tyop_2Enum_2Enum,V3R_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))] :
( p(c_2Earithmetic_2ENRC_2E4(A_27a,V3R_2E0,c_2Enum_2ESUC_2E1(V2n_2E0),V1x_2E0,V0y_2E0))
<=> ? [V4z_2E0: A_27a] :
( p(c_2Earithmetic_2ENRC_2E4(A_27a,V3R_2E0,V2n_2E0,V1x_2E0,V4z_2E0))
& p(app_2E2(A_27a,tyop_2Emin_2Ebool,app_2E2(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V3R_2E0,V4z_2E0),V0y_2E0)) ) ) ).
tff(thm_2Earithmetic_2ENRC__RTC,axiom,
! [A_27a: $tType,V0R_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),V1n_2E0: tyop_2Enum_2Enum,V2x_2E0: A_27a,V3y_2E0: A_27a] :
( p(c_2Earithmetic_2ENRC_2E4(A_27a,V0R_2E0,V1n_2E0,V2x_2E0,V3y_2E0))
=> p(c_2Erelation_2ERTC_2E3(A_27a,V0R_2E0,V2x_2E0,V3y_2E0)) ) ).
tff(thm_2Earithmetic_2ERTC__NRC,axiom,
! [A_27a: $tType,V0R_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),V1x_2E0: A_27a,V2y_2E0: A_27a] :
( p(c_2Erelation_2ERTC_2E3(A_27a,V0R_2E0,V1x_2E0,V2y_2E0))
=> ? [V3n_2E0: tyop_2Enum_2Enum] : p(c_2Earithmetic_2ENRC_2E4(A_27a,V0R_2E0,V3n_2E0,V1x_2E0,V2y_2E0)) ) ).
tff(thm_2Earithmetic_2ERTC__eq__NRC,axiom,
! [A_27a: $tType,V0R_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),V1x_2E0: A_27a,V2y_2E0: A_27a] :
( p(c_2Erelation_2ERTC_2E3(A_27a,V0R_2E0,V1x_2E0,V2y_2E0))
<=> ? [V3n_2E0: tyop_2Enum_2Enum] : p(c_2Earithmetic_2ENRC_2E4(A_27a,V0R_2E0,V3n_2E0,V1x_2E0,V2y_2E0)) ) ).
tff(thm_2Earithmetic_2ETC__eq__NRC,axiom,
! [A_27a: $tType,V0R_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),V1x_2E0: A_27a,V2y_2E0: A_27a] :
( p(c_2Erelation_2ETC_2E3(A_27a,V0R_2E0,V1x_2E0,V2y_2E0))
<=> ? [V3n_2E0: tyop_2Enum_2Enum] : p(c_2Earithmetic_2ENRC_2E4(A_27a,V0R_2E0,c_2Enum_2ESUC_2E1(V3n_2E0),V1x_2E0,V2y_2E0)) ) ).
tff(thm_2Earithmetic_2ELESS__EQUAL__DIFF,axiom,
! [V0m_2E0: tyop_2Enum_2Enum,V1n_2E0: tyop_2Enum_2Enum] :
( p(c_2Earithmetic_2E_3C_3D_2E2(V0m_2E0,V1n_2E0))
=> ? [V2k_2E0: tyop_2Enum_2Enum] : V0m_2E0 = c_2Earithmetic_2E_2D_2E2(V1n_2E0,V2k_2E0) ) ).
tff(thm_2Earithmetic_2EMOD__2,conjecture,
! [V0n_2E0: tyop_2Enum_2Enum] : c_2Earithmetic_2EMOD_2E2(V0n_2E0,c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EZERO_2E0))) = c_2Ebool_2ECOND_2E3(tyop_2Enum_2Enum,c_2Earithmetic_2EEVEN_2E1(V0n_2E0),c_2Enum_2E0_2E0,c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))) ).
%------------------------------------------------------------------------------