ITP001 Axioms: ITP016^7.ax
%------------------------------------------------------------------------------
% File : ITP016^7 : TPTP v8.2.0. Bugfixed v7.5.0.
% Domain : Interactive Theorem Proving
% Axioms : HOL4 syntactic export, chainy mode
% Version : [BG+19] axioms.
% English :
% Refs : [BG+19] Brown et al. (2019), GRUNGE: A Grand Unified ATP Chall
% : [Gau19] Gauthier (2019), Email to Geoff Sutcliffe
% Source : [BG+19]
% Names : arithmetic.ax [Gau19]
% : HL4016^7.ax [TPAP]
% Status : Satisfiable
% Syntax : Number of formulae : 470 ( 129 unt; 58 typ; 0 def)
% Number of atoms : 1020 ( 384 equ; 37 cnn)
% Maximal formula atoms : 8 ( 2 avg)
% Number of connectives : 4191 ( 37 ~; 54 |; 156 &;3623 @)
% ( 144 <=>; 177 =>; 0 <=; 0 <~>)
% Maximal formula depth : 15 ( 6 avg;3623 nst)
% Number of types : 3 ( 2 usr)
% Number of type conns : 193 ( 193 >; 0 *; 0 +; 0 <<)
% Number of symbols : 58 ( 56 usr; 5 con; 0-6 aty)
% Number of variables : 1086 ( 17 ^ 992 !; 47 ?;1086 :)
% ( 30 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_SAT_EQU_NAR
% Comments :
% Bugfixes : v7.5.0 - Fixes to the axioms.
%------------------------------------------------------------------------------
thf(tyop_2Emin_2Ebool,type,
tyop_2Emin_2Ebool: $tType ).
thf(tyop_2Emin_2Efun,type,
tyop_2Emin_2Efun: $tType > $tType > $tType ).
thf(tyop_2Enum_2Enum,type,
tyop_2Enum_2Enum: $tType ).
thf(tyop_2Epair_2Eprod,type,
tyop_2Epair_2Eprod: $tType > $tType > $tType ).
thf(c_2Ebool_2E_21,type,
c_2Ebool_2E_21:
!>[A_27a: $tType] : ( ( A_27a > $o ) > $o ) ).
thf(c_2Earithmetic_2E_2A,type,
c_2Earithmetic_2E_2A: tyop_2Enum_2Enum > tyop_2Enum_2Enum > tyop_2Enum_2Enum ).
thf(c_2Earithmetic_2E_2B,type,
c_2Earithmetic_2E_2B: tyop_2Enum_2Enum > tyop_2Enum_2Enum > tyop_2Enum_2Enum ).
thf(c_2Epair_2E_2C,type,
c_2Epair_2E_2C:
!>[A_27a: $tType,A_27b: $tType] : ( A_27a > A_27b > ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) ) ).
thf(c_2Earithmetic_2E_2D,type,
c_2Earithmetic_2E_2D: tyop_2Enum_2Enum > tyop_2Enum_2Enum > tyop_2Enum_2Enum ).
thf(c_2Ebool_2E_2F_5C,type,
c_2Ebool_2E_2F_5C: $o > $o > $o ).
thf(c_2Enum_2E0,type,
c_2Enum_2E0: tyop_2Enum_2Enum ).
thf(c_2Eprim__rec_2E_3C,type,
c_2Eprim__rec_2E_3C: tyop_2Enum_2Enum > tyop_2Enum_2Enum > $o ).
thf(c_2Earithmetic_2E_3C_3D,type,
c_2Earithmetic_2E_3C_3D: tyop_2Enum_2Enum > tyop_2Enum_2Enum > $o ).
thf(c_2Emin_2E_3D,type,
c_2Emin_2E_3D:
!>[A_27a: $tType] : ( A_27a > A_27a > $o ) ).
thf(c_2Emin_2E_3D_3D_3E,type,
c_2Emin_2E_3D_3D_3E: $o > $o > $o ).
thf(c_2Earithmetic_2E_3E,type,
c_2Earithmetic_2E_3E: tyop_2Enum_2Enum > tyop_2Enum_2Enum > $o ).
thf(c_2Earithmetic_2E_3E_3D,type,
c_2Earithmetic_2E_3E_3D: tyop_2Enum_2Enum > tyop_2Enum_2Enum > $o ).
thf(c_2Ebool_2E_3F,type,
c_2Ebool_2E_3F:
!>[A_27a: $tType] : ( ( A_27a > $o ) > $o ) ).
thf(c_2Earithmetic_2EABS__DIFF,type,
c_2Earithmetic_2EABS__DIFF: tyop_2Enum_2Enum > tyop_2Enum_2Enum > tyop_2Enum_2Enum ).
thf(c_2Earithmetic_2EBIT1,type,
c_2Earithmetic_2EBIT1: tyop_2Enum_2Enum > tyop_2Enum_2Enum ).
thf(c_2Earithmetic_2EBIT2,type,
c_2Earithmetic_2EBIT2: tyop_2Enum_2Enum > tyop_2Enum_2Enum ).
thf(c_2Ebool_2ECOND,type,
c_2Ebool_2ECOND:
!>[A_27a: $tType] : ( $o > A_27a > A_27a > A_27a ) ).
thf(c_2Ebool_2EDATATYPE,type,
c_2Ebool_2EDATATYPE:
!>[A_27a: $tType] : ( A_27a > $o ) ).
thf(c_2Earithmetic_2EDIV,type,
c_2Earithmetic_2EDIV: tyop_2Enum_2Enum > tyop_2Enum_2Enum > tyop_2Enum_2Enum ).
thf(c_2Earithmetic_2EDIV2,type,
c_2Earithmetic_2EDIV2: tyop_2Enum_2Enum > tyop_2Enum_2Enum ).
thf(c_2Earithmetic_2EDIVMOD,type,
c_2Earithmetic_2EDIVMOD: ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum ) ) > ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum ) ).
thf(c_2Earithmetic_2EEVEN,type,
c_2Earithmetic_2EEVEN: tyop_2Enum_2Enum > $o ).
thf(c_2Earithmetic_2EEXP,type,
c_2Earithmetic_2EEXP: tyop_2Enum_2Enum > tyop_2Enum_2Enum > tyop_2Enum_2Enum ).
thf(c_2Ebool_2EF,type,
c_2Ebool_2EF: $o ).
thf(c_2Earithmetic_2EFACT,type,
c_2Earithmetic_2EFACT: tyop_2Enum_2Enum > tyop_2Enum_2Enum ).
thf(c_2Epair_2EFST,type,
c_2Epair_2EFST:
!>[A_27a: $tType,A_27b: $tType] : ( ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) > A_27a ) ).
thf(c_2Earithmetic_2EFUNPOW,type,
c_2Earithmetic_2EFUNPOW:
!>[A_27a: $tType] : ( ( A_27a > A_27a ) > tyop_2Enum_2Enum > A_27a > A_27a ) ).
thf(c_2Ebool_2ELET,type,
c_2Ebool_2ELET:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > A_27b ) > A_27a > A_27b ) ).
thf(c_2Earithmetic_2EMAX,type,
c_2Earithmetic_2EMAX: tyop_2Enum_2Enum > tyop_2Enum_2Enum > tyop_2Enum_2Enum ).
thf(c_2Earithmetic_2EMIN,type,
c_2Earithmetic_2EMIN: tyop_2Enum_2Enum > tyop_2Enum_2Enum > tyop_2Enum_2Enum ).
thf(c_2Earithmetic_2EMOD,type,
c_2Earithmetic_2EMOD: tyop_2Enum_2Enum > tyop_2Enum_2Enum > tyop_2Enum_2Enum ).
thf(c_2Earithmetic_2EMODEQ,type,
c_2Earithmetic_2EMODEQ: tyop_2Enum_2Enum > tyop_2Enum_2Enum > tyop_2Enum_2Enum > $o ).
thf(c_2Earithmetic_2ENRC,type,
c_2Earithmetic_2ENRC:
!>[A_27a: $tType] : ( ( A_27a > A_27a > $o ) > tyop_2Enum_2Enum > A_27a > A_27a > $o ) ).
thf(c_2Earithmetic_2ENUMERAL,type,
c_2Earithmetic_2ENUMERAL: tyop_2Enum_2Enum > tyop_2Enum_2Enum ).
thf(c_2Earithmetic_2EODD,type,
c_2Earithmetic_2EODD: tyop_2Enum_2Enum > $o ).
thf(c_2Ebool_2EONE__ONE,type,
c_2Ebool_2EONE__ONE:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > A_27b ) > $o ) ).
thf(c_2Eprim__rec_2EPRE,type,
c_2Eprim__rec_2EPRE: tyop_2Enum_2Enum > tyop_2Enum_2Enum ).
thf(c_2Erelation_2ERTC,type,
c_2Erelation_2ERTC:
!>[A_27a: $tType] : ( ( A_27a > A_27a > $o ) > A_27a > A_27a > $o ) ).
thf(c_2Epair_2ESND,type,
c_2Epair_2ESND:
!>[A_27a: $tType,A_27b: $tType] : ( ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) > A_27b ) ).
thf(c_2Enum_2ESUC,type,
c_2Enum_2ESUC: tyop_2Enum_2Enum > tyop_2Enum_2Enum ).
thf(c_2Ebool_2ET,type,
c_2Ebool_2ET: $o ).
thf(c_2Erelation_2ETC,type,
c_2Erelation_2ETC:
!>[A_27a: $tType] : ( ( A_27a > A_27a > $o ) > A_27a > A_27a > $o ) ).
thf(c_2Epair_2EUNCURRY,type,
c_2Epair_2EUNCURRY:
!>[A_27a: $tType,A_27b: $tType,A_27c: $tType] : ( ( A_27a > A_27b > A_27c ) > ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) > A_27c ) ).
thf(c_2Erelation_2EWFREC,type,
c_2Erelation_2EWFREC:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > A_27a > $o ) > ( ( A_27a > A_27b ) > A_27a > A_27b ) > A_27a > A_27b ) ).
thf(c_2Earithmetic_2EZERO,type,
c_2Earithmetic_2EZERO: tyop_2Enum_2Enum ).
thf(c_2Ebool_2E_5C_2F,type,
c_2Ebool_2E_5C_2F: $o > $o > $o ).
thf(c_2Earithmetic_2Efindq,type,
c_2Earithmetic_2Efindq: ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum ) ) > tyop_2Enum_2Enum ).
thf(c_2Eprim__rec_2Emeasure,type,
c_2Eprim__rec_2Emeasure:
!>[A_27a: $tType] : ( ( A_27a > tyop_2Enum_2Enum ) > A_27a > A_27a > $o ) ).
thf(c_2Earithmetic_2Enat__elim____magic,type,
c_2Earithmetic_2Enat__elim____magic: tyop_2Enum_2Enum > tyop_2Enum_2Enum ).
thf(c_2Earithmetic_2Enum__CASE,type,
c_2Earithmetic_2Enum__CASE:
!>[A_27a: $tType] : ( tyop_2Enum_2Enum > A_27a > ( tyop_2Enum_2Enum > A_27a ) > A_27a ) ).
thf(c_2Ecombin_2Eo,type,
c_2Ecombin_2Eo:
!>[A_27a: $tType,A_27b: $tType,A_27c: $tType] : ( ( A_27c > A_27b ) > ( A_27a > A_27c ) > A_27a > A_27b ) ).
thf(c_2Erelation_2Etransitive,type,
c_2Erelation_2Etransitive:
!>[A_27a: $tType] : ( ( A_27a > A_27a > $o ) > $o ) ).
thf(c_2Ebool_2E_7E,type,
c_2Ebool_2E_7E: $o > $o ).
thf(logicdef_2E_2F_5C,axiom,
! [V0: $o,V1: $o] :
( ( c_2Ebool_2E_2F_5C @ V0 @ V1 )
<=> ( V0
& V1 ) ) ).
thf(logicdef_2E_5C_2F,axiom,
! [V0: $o,V1: $o] :
( ( c_2Ebool_2E_5C_2F @ V0 @ V1 )
<=> ( V0
| V1 ) ) ).
thf(logicdef_2E_7E,axiom,
! [V0: $o] :
( ( c_2Ebool_2E_7E @ V0 )
<=> ( (~) @ V0 ) ) ).
thf(logicdef_2E_3D_3D_3E,axiom,
! [V0: $o,V1: $o] :
( ( c_2Emin_2E_3D_3D_3E @ V0 @ V1 )
<=> ( V0
=> V1 ) ) ).
thf(logicdef_2E_3D,axiom,
! [A_27a: $tType,V0: A_27a,V1: A_27a] :
( ( c_2Emin_2E_3D @ A_27a @ V0 @ V1 )
<=> ( V0 = V1 ) ) ).
thf(quantdef_2E_21,axiom,
! [A_27a: $tType,V0f: A_27a > $o] :
( ( c_2Ebool_2E_21 @ A_27a @ V0f )
<=> ! [V1x: A_27a] : ( V0f @ V1x ) ) ).
thf(quantdef_2E_3F,axiom,
! [A_27a: $tType,V0f: A_27a > $o] :
( ( c_2Ebool_2E_3F @ A_27a @ V0f )
<=> ? [V1x: A_27a] : ( V0f @ V1x ) ) ).
thf(thm_2Earithmetic_2EADD,axiom,
( ! [V0n: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_2B @ c_2Enum_2E0 @ V0n )
= V0n )
& ! [V1m: tyop_2Enum_2Enum,V2n: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_2B @ ( c_2Enum_2ESUC @ V1m ) @ V2n )
= ( c_2Enum_2ESUC @ ( c_2Earithmetic_2E_2B @ V1m @ V2n ) ) ) ) ).
thf(thm_2Earithmetic_2ENUMERAL__DEF,axiom,
! [V0x: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2ENUMERAL @ V0x )
= V0x ) ).
thf(thm_2Earithmetic_2EALT__ZERO,axiom,
c_2Earithmetic_2EZERO = c_2Enum_2E0 ).
thf(thm_2Earithmetic_2EBIT1,axiom,
! [V0n: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2EBIT1 @ V0n )
= ( c_2Earithmetic_2E_2B @ V0n @ ( c_2Earithmetic_2E_2B @ V0n @ ( c_2Enum_2ESUC @ c_2Enum_2E0 ) ) ) ) ).
thf(thm_2Earithmetic_2EBIT2,axiom,
! [V0n: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2EBIT2 @ V0n )
= ( c_2Earithmetic_2E_2B @ V0n @ ( c_2Earithmetic_2E_2B @ V0n @ ( c_2Enum_2ESUC @ ( c_2Enum_2ESUC @ c_2Enum_2E0 ) ) ) ) ) ).
thf(thm_2Earithmetic_2Enat__elim____magic,axiom,
! [V0n: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2Enat__elim____magic @ V0n )
= V0n ) ).
thf(thm_2Earithmetic_2ESUB,axiom,
( ! [V0m: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_2D @ c_2Enum_2E0 @ V0m )
= c_2Enum_2E0 )
& ! [V1m: tyop_2Enum_2Enum,V2n: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_2D @ ( c_2Enum_2ESUC @ V1m ) @ V2n )
= ( c_2Ebool_2ECOND @ tyop_2Enum_2Enum @ ( c_2Eprim__rec_2E_3C @ V1m @ V2n ) @ c_2Enum_2E0 @ ( c_2Enum_2ESUC @ ( c_2Earithmetic_2E_2D @ V1m @ V2n ) ) ) ) ) ).
thf(thm_2Earithmetic_2EMULT,axiom,
( ! [V0n: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_2A @ c_2Enum_2E0 @ V0n )
= c_2Enum_2E0 )
& ! [V1m: tyop_2Enum_2Enum,V2n: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_2A @ ( c_2Enum_2ESUC @ V1m ) @ V2n )
= ( c_2Earithmetic_2E_2B @ ( c_2Earithmetic_2E_2A @ V1m @ V2n ) @ V2n ) ) ) ).
thf(thm_2Earithmetic_2EEXP,axiom,
( ! [V0m: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2EEXP @ V0m @ c_2Enum_2E0 )
= ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) )
& ! [V1m: tyop_2Enum_2Enum,V2n: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2EEXP @ V1m @ ( c_2Enum_2ESUC @ V2n ) )
= ( c_2Earithmetic_2E_2A @ V1m @ ( c_2Earithmetic_2EEXP @ V1m @ V2n ) ) ) ) ).
thf(thm_2Earithmetic_2EGREATER__DEF,axiom,
! [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_3E @ V0m @ V1n )
= ( c_2Eprim__rec_2E_3C @ V1n @ V0m ) ) ).
thf(thm_2Earithmetic_2ELESS__OR__EQ,axiom,
! [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_3C_3D @ V0m @ V1n )
<=> ( ( c_2Eprim__rec_2E_3C @ V0m @ V1n )
| ( V0m = V1n ) ) ) ).
thf(thm_2Earithmetic_2EGREATER__OR__EQ,axiom,
! [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_3E_3D @ V0m @ V1n )
<=> ( ( c_2Earithmetic_2E_3E @ V0m @ V1n )
| ( V0m = V1n ) ) ) ).
thf(thm_2Earithmetic_2EEVEN,axiom,
( ( ( c_2Earithmetic_2EEVEN @ c_2Enum_2E0 )
= c_2Ebool_2ET )
& ! [V0n: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2EEVEN @ ( c_2Enum_2ESUC @ V0n ) )
<=> ( (~) @ ( c_2Earithmetic_2EEVEN @ V0n ) ) ) ) ).
thf(thm_2Earithmetic_2EODD,axiom,
( ( ( c_2Earithmetic_2EODD @ c_2Enum_2E0 )
= c_2Ebool_2EF )
& ! [V0n: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2EODD @ ( c_2Enum_2ESUC @ V0n ) )
<=> ( (~) @ ( c_2Earithmetic_2EODD @ V0n ) ) ) ) ).
thf(thm_2Earithmetic_2Enum__case__def,axiom,
! [A_27a: $tType] :
( ! [V0v: A_27a,V1f: tyop_2Enum_2Enum > A_27a] :
( ( c_2Earithmetic_2Enum__CASE @ A_27a @ c_2Enum_2E0 @ V0v @ V1f )
= V0v )
& ! [V2n: tyop_2Enum_2Enum,V3v: A_27a,V4f: tyop_2Enum_2Enum > A_27a] :
( ( c_2Earithmetic_2Enum__CASE @ A_27a @ ( c_2Enum_2ESUC @ V2n ) @ V3v @ V4f )
= ( V4f @ V2n ) ) ) ).
thf(thm_2Earithmetic_2EFUNPOW,axiom,
! [A_27a: $tType] :
( ! [V0f: A_27a > A_27a,V1x: A_27a] :
( ( c_2Earithmetic_2EFUNPOW @ A_27a @ V0f @ c_2Enum_2E0 @ V1x )
= V1x )
& ! [V2f: A_27a > A_27a,V3n: tyop_2Enum_2Enum,V4x: A_27a] :
( ( c_2Earithmetic_2EFUNPOW @ A_27a @ V2f @ ( c_2Enum_2ESUC @ V3n ) @ V4x )
= ( c_2Earithmetic_2EFUNPOW @ A_27a @ V2f @ V3n @ ( V2f @ V4x ) ) ) ) ).
thf(thm_2Earithmetic_2ENRC,axiom,
! [A_27a: $tType] :
( ! [V0R: A_27a > A_27a > $o,V1x: A_27a,V2y: A_27a] :
( ( c_2Earithmetic_2ENRC @ A_27a @ V0R @ c_2Enum_2E0 @ V1x @ V2y )
<=> ( V1x = V2y ) )
& ! [V3R: A_27a > A_27a > $o,V4n: tyop_2Enum_2Enum,V5x: A_27a,V6y: A_27a] :
( ( c_2Earithmetic_2ENRC @ A_27a @ V3R @ ( c_2Enum_2ESUC @ V4n ) @ V5x @ V6y )
<=> ? [V7z: A_27a] :
( ( V3R @ V5x @ V7z )
& ( c_2Earithmetic_2ENRC @ A_27a @ V3R @ V4n @ V7z @ V6y ) ) ) ) ).
thf(thm_2Earithmetic_2EFACT,axiom,
( ( ( c_2Earithmetic_2EFACT @ c_2Enum_2E0 )
= ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) )
& ! [V0n: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2EFACT @ ( c_2Enum_2ESUC @ V0n ) )
= ( c_2Earithmetic_2E_2A @ ( c_2Enum_2ESUC @ V0n ) @ ( c_2Earithmetic_2EFACT @ V0n ) ) ) ) ).
thf(thm_2Earithmetic_2EDIVISION,axiom,
! [V0n: tyop_2Enum_2Enum] :
( ( c_2Eprim__rec_2E_3C @ c_2Enum_2E0 @ V0n )
=> ! [V1k: tyop_2Enum_2Enum] :
( ( V1k
= ( c_2Earithmetic_2E_2B @ ( c_2Earithmetic_2E_2A @ ( c_2Earithmetic_2EDIV @ V1k @ V0n ) @ V0n ) @ ( c_2Earithmetic_2EMOD @ V1k @ V0n ) ) )
& ( c_2Eprim__rec_2E_3C @ ( c_2Earithmetic_2EMOD @ V1k @ V0n ) @ V0n ) ) ) ).
thf(thm_2Earithmetic_2EDIV2__def,axiom,
! [V0n: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2EDIV2 @ V0n )
= ( c_2Earithmetic_2EDIV @ V0n @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) ) ).
thf(thm_2Earithmetic_2EMAX__DEF,axiom,
! [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2EMAX @ V0m @ V1n )
= ( c_2Ebool_2ECOND @ tyop_2Enum_2Enum @ ( c_2Eprim__rec_2E_3C @ V0m @ V1n ) @ V1n @ V0m ) ) ).
thf(thm_2Earithmetic_2EMIN__DEF,axiom,
! [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2EMIN @ V0m @ V1n )
= ( c_2Ebool_2ECOND @ tyop_2Enum_2Enum @ ( c_2Eprim__rec_2E_3C @ V0m @ V1n ) @ V0m @ V1n ) ) ).
thf(thm_2Earithmetic_2EABS__DIFF__def,axiom,
! [V0n: tyop_2Enum_2Enum,V1m: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2EABS__DIFF @ V0n @ V1m )
= ( c_2Ebool_2ECOND @ tyop_2Enum_2Enum @ ( c_2Eprim__rec_2E_3C @ V0n @ V1m ) @ ( c_2Earithmetic_2E_2D @ V1m @ V0n ) @ ( c_2Earithmetic_2E_2D @ V0n @ V1m ) ) ) ).
thf(thm_2Earithmetic_2Efindq__def,axiom,
( c_2Earithmetic_2Efindq
= ( c_2Erelation_2EWFREC @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum ) ) @ tyop_2Enum_2Enum
@ ( c_2Eprim__rec_2Emeasure @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum ) )
@ ( c_2Epair_2EUNCURRY @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum ) @ tyop_2Enum_2Enum
@ ^ [V0a: tyop_2Enum_2Enum] :
( c_2Epair_2EUNCURRY @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum
@ ^ [V1m: tyop_2Enum_2Enum,V2n: tyop_2Enum_2Enum] : ( c_2Earithmetic_2E_2D @ V1m @ V2n ) ) ) )
@ ^ [V3f: ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum ) ) > tyop_2Enum_2Enum] :
( c_2Epair_2EUNCURRY @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum ) @ tyop_2Enum_2Enum
@ ^ [V4a: tyop_2Enum_2Enum] :
( c_2Epair_2EUNCURRY @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum
@ ^ [V5m: tyop_2Enum_2Enum,V6n: tyop_2Enum_2Enum] :
( c_2Ebool_2ECOND @ tyop_2Enum_2Enum @ ( c_2Emin_2E_3D @ tyop_2Enum_2Enum @ V6n @ c_2Enum_2E0 ) @ V4a
@ ( c_2Ebool_2ELET @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum
@ ^ [V7d: tyop_2Enum_2Enum] : ( c_2Ebool_2ECOND @ tyop_2Enum_2Enum @ ( c_2Eprim__rec_2E_3C @ V5m @ V7d ) @ V4a @ ( V3f @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum ) @ ( c_2Earithmetic_2E_2A @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) @ V4a ) @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum @ V5m @ V7d ) ) ) )
@ ( c_2Earithmetic_2E_2A @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) @ V6n ) ) ) ) ) ) ) ).
thf(thm_2Earithmetic_2EDIVMOD__DEF,axiom,
( c_2Earithmetic_2EDIVMOD
= ( c_2Erelation_2EWFREC @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum ) ) @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum ) @ ( c_2Eprim__rec_2Emeasure @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum ) ) @ ( c_2Ecombin_2Eo @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum ) ) @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum ) @ ( c_2Epair_2EFST @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum ) @ ( c_2Epair_2ESND @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum ) ) ) )
@ ^ [V0f: ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum ) ) > ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum )] :
( c_2Epair_2EUNCURRY @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum ) @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum )
@ ^ [V1a: tyop_2Enum_2Enum] :
( c_2Epair_2EUNCURRY @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum )
@ ^ [V2m: tyop_2Enum_2Enum,V3n: tyop_2Enum_2Enum] :
( c_2Ebool_2ECOND @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum ) @ ( c_2Emin_2E_3D @ tyop_2Enum_2Enum @ V3n @ c_2Enum_2E0 ) @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum @ c_2Enum_2E0 @ c_2Enum_2E0 )
@ ( c_2Ebool_2ECOND @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum ) @ ( c_2Eprim__rec_2E_3C @ V2m @ V3n ) @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum @ V1a @ V2m )
@ ( c_2Ebool_2ELET @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum )
@ ^ [V4q: tyop_2Enum_2Enum] : ( V0f @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum ) @ ( c_2Earithmetic_2E_2B @ V1a @ V4q ) @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum @ ( c_2Earithmetic_2E_2D @ V2m @ ( c_2Earithmetic_2E_2A @ V3n @ V4q ) ) @ V3n ) ) )
@ ( c_2Earithmetic_2Efindq @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum ) @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum @ V2m @ V3n ) ) ) ) ) ) ) ) ) ) ).
thf(thm_2Earithmetic_2EMODEQ__DEF,axiom,
! [V0n: tyop_2Enum_2Enum,V1m1: tyop_2Enum_2Enum,V2m2: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2EMODEQ @ V0n @ V1m1 @ V2m2 )
<=> ? [V3a: tyop_2Enum_2Enum,V4b: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_2B @ ( c_2Earithmetic_2E_2A @ V3a @ V0n ) @ V1m1 )
= ( c_2Earithmetic_2E_2B @ ( c_2Earithmetic_2E_2A @ V4b @ V0n ) @ V2m2 ) ) ) ).
thf(thm_2Earithmetic_2EONE,axiom,
( ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) )
= ( c_2Enum_2ESUC @ c_2Enum_2E0 ) ) ).
thf(thm_2Earithmetic_2ETWO,axiom,
( ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) )
= ( c_2Enum_2ESUC @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ).
thf(thm_2Earithmetic_2ENORM__0,axiom,
( ( c_2Earithmetic_2ENUMERAL @ c_2Earithmetic_2EZERO )
= c_2Enum_2E0 ) ).
thf(thm_2Earithmetic_2Enum__case__compute,axiom,
! [A_27a: $tType,V0g: tyop_2Enum_2Enum > A_27a,V1f: A_27a,V2n: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2Enum__CASE @ A_27a @ V2n @ V1f @ V0g )
= ( c_2Ebool_2ECOND @ A_27a @ ( c_2Emin_2E_3D @ tyop_2Enum_2Enum @ V2n @ c_2Enum_2E0 ) @ V1f @ ( V0g @ ( c_2Eprim__rec_2EPRE @ V2n ) ) ) ) ).
thf(thm_2Earithmetic_2ESUC__NOT,axiom,
! [V0n: tyop_2Enum_2Enum] :
( (~)
@ ( c_2Enum_2E0
= ( c_2Enum_2ESUC @ V0n ) ) ) ).
thf(thm_2Earithmetic_2EADD__0,axiom,
! [V0m: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_2B @ V0m @ c_2Enum_2E0 )
= V0m ) ).
thf(thm_2Earithmetic_2EADD__SUC,axiom,
! [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum] :
( ( c_2Enum_2ESUC @ ( c_2Earithmetic_2E_2B @ V0m @ V1n ) )
= ( c_2Earithmetic_2E_2B @ V0m @ ( c_2Enum_2ESUC @ V1n ) ) ) ).
thf(thm_2Earithmetic_2EADD__CLAUSES,axiom,
! [V0n: tyop_2Enum_2Enum,V1m: tyop_2Enum_2Enum] :
( ( ( c_2Earithmetic_2E_2B @ c_2Enum_2E0 @ V1m )
= V1m )
& ( ( c_2Earithmetic_2E_2B @ V1m @ c_2Enum_2E0 )
= V1m )
& ( ( c_2Earithmetic_2E_2B @ ( c_2Enum_2ESUC @ V1m ) @ V0n )
= ( c_2Enum_2ESUC @ ( c_2Earithmetic_2E_2B @ V1m @ V0n ) ) )
& ( ( c_2Earithmetic_2E_2B @ V1m @ ( c_2Enum_2ESUC @ V0n ) )
= ( c_2Enum_2ESUC @ ( c_2Earithmetic_2E_2B @ V1m @ V0n ) ) ) ) ).
thf(thm_2Earithmetic_2EADD__SYM,axiom,
! [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_2B @ V0m @ V1n )
= ( c_2Earithmetic_2E_2B @ V1n @ V0m ) ) ).
thf(thm_2Earithmetic_2EADD__COMM,axiom,
! [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_2B @ V0m @ V1n )
= ( c_2Earithmetic_2E_2B @ V1n @ V0m ) ) ).
thf(thm_2Earithmetic_2EADD__ASSOC,axiom,
! [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum,V2p: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_2B @ V0m @ ( c_2Earithmetic_2E_2B @ V1n @ V2p ) )
= ( c_2Earithmetic_2E_2B @ ( c_2Earithmetic_2E_2B @ V0m @ V1n ) @ V2p ) ) ).
thf(thm_2Earithmetic_2Enum__CASES,axiom,
! [V0m: tyop_2Enum_2Enum] :
( ( V0m = c_2Enum_2E0 )
| ? [V1n: tyop_2Enum_2Enum] :
( V0m
= ( c_2Enum_2ESUC @ V1n ) ) ) ).
thf(thm_2Earithmetic_2ENOT__ZERO__LT__ZERO,axiom,
! [V0n: tyop_2Enum_2Enum] :
( ( (~) @ ( V0n = c_2Enum_2E0 ) )
<=> ( c_2Eprim__rec_2E_3C @ c_2Enum_2E0 @ V0n ) ) ).
thf(thm_2Earithmetic_2ENOT__LT__ZERO__EQ__ZERO,axiom,
! [V0n: tyop_2Enum_2Enum] :
( ( (~) @ ( c_2Eprim__rec_2E_3C @ c_2Enum_2E0 @ V0n ) )
<=> ( V0n = c_2Enum_2E0 ) ) ).
thf(thm_2Earithmetic_2ELESS__OR__EQ__ALT,axiom,
( c_2Earithmetic_2E_3C_3D
= ( c_2Erelation_2ERTC @ tyop_2Enum_2Enum
@ ^ [V0x: tyop_2Enum_2Enum,V1y: tyop_2Enum_2Enum] : ( c_2Emin_2E_3D @ tyop_2Enum_2Enum @ V1y @ ( c_2Enum_2ESUC @ V0x ) ) ) ) ).
thf(thm_2Earithmetic_2ELESS__ADD,axiom,
! [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum] :
( ( c_2Eprim__rec_2E_3C @ V1n @ V0m )
=> ? [V2p: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_2B @ V2p @ V1n )
= V0m ) ) ).
thf(thm_2Earithmetic_2Etransitive__LESS,axiom,
c_2Erelation_2Etransitive @ tyop_2Enum_2Enum @ c_2Eprim__rec_2E_3C ).
thf(thm_2Earithmetic_2ELESS__TRANS,axiom,
! [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum,V2p: tyop_2Enum_2Enum] :
( ( ( c_2Eprim__rec_2E_3C @ V0m @ V1n )
& ( c_2Eprim__rec_2E_3C @ V1n @ V2p ) )
=> ( c_2Eprim__rec_2E_3C @ V0m @ V2p ) ) ).
thf(thm_2Earithmetic_2ELESS__ANTISYM,axiom,
! [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum] :
( (~)
@ ( ( c_2Eprim__rec_2E_3C @ V0m @ V1n )
& ( c_2Eprim__rec_2E_3C @ V1n @ V0m ) ) ) ).
thf(thm_2Earithmetic_2ELESS__MONO__REV,axiom,
! [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum] :
( ( c_2Eprim__rec_2E_3C @ ( c_2Enum_2ESUC @ V0m ) @ ( c_2Enum_2ESUC @ V1n ) )
=> ( c_2Eprim__rec_2E_3C @ V0m @ V1n ) ) ).
thf(thm_2Earithmetic_2ELESS__MONO__EQ,axiom,
! [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum] :
( ( c_2Eprim__rec_2E_3C @ ( c_2Enum_2ESUC @ V0m ) @ ( c_2Enum_2ESUC @ V1n ) )
= ( c_2Eprim__rec_2E_3C @ V0m @ V1n ) ) ).
thf(thm_2Earithmetic_2ELESS__EQ__MONO,axiom,
! [V0n: tyop_2Enum_2Enum,V1m: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_3C_3D @ ( c_2Enum_2ESUC @ V0n ) @ ( c_2Enum_2ESUC @ V1m ) )
= ( c_2Earithmetic_2E_3C_3D @ V0n @ V1m ) ) ).
thf(thm_2Earithmetic_2ELESS__LESS__SUC,axiom,
! [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum] :
( (~)
@ ( ( c_2Eprim__rec_2E_3C @ V0m @ V1n )
& ( c_2Eprim__rec_2E_3C @ V1n @ ( c_2Enum_2ESUC @ V0m ) ) ) ) ).
thf(thm_2Earithmetic_2Etransitive__measure,axiom,
! [A_27a: $tType,V0f: A_27a > tyop_2Enum_2Enum] : ( c_2Erelation_2Etransitive @ A_27a @ ( c_2Eprim__rec_2Emeasure @ A_27a @ V0f ) ) ).
thf(thm_2Earithmetic_2ELESS__EQ,axiom,
! [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum] :
( ( c_2Eprim__rec_2E_3C @ V0m @ V1n )
= ( c_2Earithmetic_2E_3C_3D @ ( c_2Enum_2ESUC @ V0m ) @ V1n ) ) ).
thf(thm_2Earithmetic_2ELESS__OR,axiom,
! [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum] :
( ( c_2Eprim__rec_2E_3C @ V0m @ V1n )
=> ( c_2Earithmetic_2E_3C_3D @ ( c_2Enum_2ESUC @ V0m ) @ V1n ) ) ).
thf(thm_2Earithmetic_2EOR__LESS,axiom,
! [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_3C_3D @ ( c_2Enum_2ESUC @ V0m ) @ V1n )
=> ( c_2Eprim__rec_2E_3C @ V0m @ V1n ) ) ).
thf(thm_2Earithmetic_2ELESS__EQ__IFF__LESS__SUC,axiom,
! [V0n: tyop_2Enum_2Enum,V1m: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_3C_3D @ V0n @ V1m )
= ( c_2Eprim__rec_2E_3C @ V0n @ ( c_2Enum_2ESUC @ V1m ) ) ) ).
thf(thm_2Earithmetic_2ELESS__EQ__IMP__LESS__SUC,axiom,
! [V0n: tyop_2Enum_2Enum,V1m: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_3C_3D @ V0n @ V1m )
=> ( c_2Eprim__rec_2E_3C @ V0n @ ( c_2Enum_2ESUC @ V1m ) ) ) ).
thf(thm_2Earithmetic_2EZERO__LESS__EQ,axiom,
! [V0n: tyop_2Enum_2Enum] : ( c_2Earithmetic_2E_3C_3D @ c_2Enum_2E0 @ V0n ) ).
thf(thm_2Earithmetic_2ELESS__SUC__EQ__COR,axiom,
! [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum] :
( ( ( c_2Eprim__rec_2E_3C @ V0m @ V1n )
& ( (~)
@ ( ( c_2Enum_2ESUC @ V0m )
= V1n ) ) )
=> ( c_2Eprim__rec_2E_3C @ ( c_2Enum_2ESUC @ V0m ) @ V1n ) ) ).
thf(thm_2Earithmetic_2ELESS__NOT__SUC,axiom,
! [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum] :
( ( ( c_2Eprim__rec_2E_3C @ V0m @ V1n )
& ( (~)
@ ( V1n
= ( c_2Enum_2ESUC @ V0m ) ) ) )
=> ( c_2Eprim__rec_2E_3C @ ( c_2Enum_2ESUC @ V0m ) @ V1n ) ) ).
thf(thm_2Earithmetic_2ELESS__0__CASES,axiom,
! [V0m: tyop_2Enum_2Enum] :
( ( c_2Enum_2E0 = V0m )
| ( c_2Eprim__rec_2E_3C @ c_2Enum_2E0 @ V0m ) ) ).
thf(thm_2Earithmetic_2ELESS__CASES__IMP,axiom,
! [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum] :
( ( ( (~) @ ( c_2Eprim__rec_2E_3C @ V0m @ V1n ) )
& ( (~) @ ( V0m = V1n ) ) )
=> ( c_2Eprim__rec_2E_3C @ V1n @ V0m ) ) ).
thf(thm_2Earithmetic_2ELESS__CASES,axiom,
! [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum] :
( ( c_2Eprim__rec_2E_3C @ V0m @ V1n )
| ( c_2Earithmetic_2E_3C_3D @ V1n @ V0m ) ) ).
thf(thm_2Earithmetic_2EADD__INV__0,axiom,
! [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum] :
( ( ( c_2Earithmetic_2E_2B @ V0m @ V1n )
= V0m )
=> ( V1n = c_2Enum_2E0 ) ) ).
thf(thm_2Earithmetic_2ELESS__EQ__ADD,axiom,
! [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum] : ( c_2Earithmetic_2E_3C_3D @ V0m @ ( c_2Earithmetic_2E_2B @ V0m @ V1n ) ) ).
thf(thm_2Earithmetic_2ELESS__EQ__ADD__EXISTS,axiom,
! [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_3C_3D @ V1n @ V0m )
=> ? [V2p: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_2B @ V2p @ V1n )
= V0m ) ) ).
thf(thm_2Earithmetic_2ELESS__STRONG__ADD,axiom,
! [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum] :
( ( c_2Eprim__rec_2E_3C @ V1n @ V0m )
=> ? [V2p: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_2B @ ( c_2Enum_2ESUC @ V2p ) @ V1n )
= V0m ) ) ).
thf(thm_2Earithmetic_2ELESS__EQ__SUC__REFL,axiom,
! [V0m: tyop_2Enum_2Enum] : ( c_2Earithmetic_2E_3C_3D @ V0m @ ( c_2Enum_2ESUC @ V0m ) ) ).
thf(thm_2Earithmetic_2ELESS__ADD__NONZERO,axiom,
! [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum] :
( ( (~) @ ( V1n = c_2Enum_2E0 ) )
=> ( c_2Eprim__rec_2E_3C @ V0m @ ( c_2Earithmetic_2E_2B @ V0m @ V1n ) ) ) ).
thf(thm_2Earithmetic_2ENOT__SUC__LESS__EQ__0,axiom,
! [V0n: tyop_2Enum_2Enum] : ( (~) @ ( c_2Earithmetic_2E_3C_3D @ ( c_2Enum_2ESUC @ V0n ) @ c_2Enum_2E0 ) ) ).
thf(thm_2Earithmetic_2ENOT__LESS,axiom,
! [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum] :
( ( (~) @ ( c_2Eprim__rec_2E_3C @ V0m @ V1n ) )
<=> ( c_2Earithmetic_2E_3C_3D @ V1n @ V0m ) ) ).
thf(thm_2Earithmetic_2ENOT__LESS__EQUAL,axiom,
! [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum] :
( ( (~) @ ( c_2Earithmetic_2E_3C_3D @ V0m @ V1n ) )
<=> ( c_2Eprim__rec_2E_3C @ V1n @ V0m ) ) ).
thf(thm_2Earithmetic_2ELESS__EQ__ANTISYM,axiom,
! [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum] :
( (~)
@ ( ( c_2Eprim__rec_2E_3C @ V0m @ V1n )
& ( c_2Earithmetic_2E_3C_3D @ V1n @ V0m ) ) ) ).
thf(thm_2Earithmetic_2ELESS__EQ__0,axiom,
! [V0n: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_3C_3D @ V0n @ c_2Enum_2E0 )
<=> ( V0n = c_2Enum_2E0 ) ) ).
thf(thm_2Earithmetic_2ESUB__0,axiom,
! [V0m: tyop_2Enum_2Enum] :
( ( ( c_2Earithmetic_2E_2D @ c_2Enum_2E0 @ V0m )
= c_2Enum_2E0 )
& ( ( c_2Earithmetic_2E_2D @ V0m @ c_2Enum_2E0 )
= V0m ) ) ).
thf(thm_2Earithmetic_2ESUB__MONO__EQ,axiom,
! [V0n: tyop_2Enum_2Enum,V1m: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_2D @ ( c_2Enum_2ESUC @ V0n ) @ ( c_2Enum_2ESUC @ V1m ) )
= ( c_2Earithmetic_2E_2D @ V0n @ V1m ) ) ).
thf(thm_2Earithmetic_2ESUB__EQ__0,axiom,
! [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum] :
( ( ( c_2Earithmetic_2E_2D @ V0m @ V1n )
= c_2Enum_2E0 )
<=> ( c_2Earithmetic_2E_3C_3D @ V0m @ V1n ) ) ).
thf(thm_2Earithmetic_2EADD1,axiom,
! [V0m: tyop_2Enum_2Enum] :
( ( c_2Enum_2ESUC @ V0m )
= ( c_2Earithmetic_2E_2B @ V0m @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ).
thf(thm_2Earithmetic_2ESUC__SUB1,axiom,
! [V0m: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_2D @ ( c_2Enum_2ESUC @ V0m ) @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) )
= V0m ) ).
thf(thm_2Earithmetic_2EPRE__SUB1,axiom,
! [V0m: tyop_2Enum_2Enum] :
( ( c_2Eprim__rec_2EPRE @ V0m )
= ( c_2Earithmetic_2E_2D @ V0m @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ).
thf(thm_2Earithmetic_2EMULT__0,axiom,
! [V0m: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_2A @ V0m @ c_2Enum_2E0 )
= c_2Enum_2E0 ) ).
thf(thm_2Earithmetic_2EMULT__SUC,axiom,
! [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_2A @ V0m @ ( c_2Enum_2ESUC @ V1n ) )
= ( c_2Earithmetic_2E_2B @ V0m @ ( c_2Earithmetic_2E_2A @ V0m @ V1n ) ) ) ).
thf(thm_2Earithmetic_2EMULT__LEFT__1,axiom,
! [V0m: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_2A @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) @ V0m )
= V0m ) ).
thf(thm_2Earithmetic_2EMULT__RIGHT__1,axiom,
! [V0m: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_2A @ V0m @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) )
= V0m ) ).
thf(thm_2Earithmetic_2EMULT__CLAUSES,axiom,
! [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum] :
( ( ( c_2Earithmetic_2E_2A @ c_2Enum_2E0 @ V0m )
= c_2Enum_2E0 )
& ( ( c_2Earithmetic_2E_2A @ V0m @ c_2Enum_2E0 )
= c_2Enum_2E0 )
& ( ( c_2Earithmetic_2E_2A @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) @ V0m )
= V0m )
& ( ( c_2Earithmetic_2E_2A @ V0m @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) )
= V0m )
& ( ( c_2Earithmetic_2E_2A @ ( c_2Enum_2ESUC @ V0m ) @ V1n )
= ( c_2Earithmetic_2E_2B @ ( c_2Earithmetic_2E_2A @ V0m @ V1n ) @ V1n ) )
& ( ( c_2Earithmetic_2E_2A @ V0m @ ( c_2Enum_2ESUC @ V1n ) )
= ( c_2Earithmetic_2E_2B @ V0m @ ( c_2Earithmetic_2E_2A @ V0m @ V1n ) ) ) ) ).
thf(thm_2Earithmetic_2EMULT__SYM,axiom,
! [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_2A @ V0m @ V1n )
= ( c_2Earithmetic_2E_2A @ V1n @ V0m ) ) ).
thf(thm_2Earithmetic_2EMULT__COMM,axiom,
! [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_2A @ V0m @ V1n )
= ( c_2Earithmetic_2E_2A @ V1n @ V0m ) ) ).
thf(thm_2Earithmetic_2ERIGHT__ADD__DISTRIB,axiom,
! [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum,V2p: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_2A @ ( c_2Earithmetic_2E_2B @ V0m @ V1n ) @ V2p )
= ( c_2Earithmetic_2E_2B @ ( c_2Earithmetic_2E_2A @ V0m @ V2p ) @ ( c_2Earithmetic_2E_2A @ V1n @ V2p ) ) ) ).
thf(thm_2Earithmetic_2ELEFT__ADD__DISTRIB,axiom,
! [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum,V2p: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_2A @ V2p @ ( c_2Earithmetic_2E_2B @ V0m @ V1n ) )
= ( c_2Earithmetic_2E_2B @ ( c_2Earithmetic_2E_2A @ V2p @ V0m ) @ ( c_2Earithmetic_2E_2A @ V2p @ V1n ) ) ) ).
thf(thm_2Earithmetic_2EMULT__ASSOC,axiom,
! [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum,V2p: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_2A @ V0m @ ( c_2Earithmetic_2E_2A @ V1n @ V2p ) )
= ( c_2Earithmetic_2E_2A @ ( c_2Earithmetic_2E_2A @ V0m @ V1n ) @ V2p ) ) ).
thf(thm_2Earithmetic_2ESUB__ADD,axiom,
! [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_3C_3D @ V1n @ V0m )
=> ( ( c_2Earithmetic_2E_2B @ ( c_2Earithmetic_2E_2D @ V0m @ V1n ) @ V1n )
= V0m ) ) ).
thf(thm_2Earithmetic_2EPRE__SUB,axiom,
! [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum] :
( ( c_2Eprim__rec_2EPRE @ ( c_2Earithmetic_2E_2D @ V0m @ V1n ) )
= ( c_2Earithmetic_2E_2D @ ( c_2Eprim__rec_2EPRE @ V0m ) @ V1n ) ) ).
thf(thm_2Earithmetic_2EADD__EQ__0,axiom,
! [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum] :
( ( ( c_2Earithmetic_2E_2B @ V0m @ V1n )
= c_2Enum_2E0 )
<=> ( ( V0m = c_2Enum_2E0 )
& ( V1n = c_2Enum_2E0 ) ) ) ).
thf(thm_2Earithmetic_2EADD__EQ__1,axiom,
! [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum] :
( ( ( c_2Earithmetic_2E_2B @ V0m @ V1n )
= ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) )
<=> ( ( ( V0m
= ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) )
& ( V1n = c_2Enum_2E0 ) )
| ( ( V0m = c_2Enum_2E0 )
& ( V1n
= ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ) ) ).
thf(thm_2Earithmetic_2EADD__INV__0__EQ,axiom,
! [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum] :
( ( ( c_2Earithmetic_2E_2B @ V0m @ V1n )
= V0m )
<=> ( V1n = c_2Enum_2E0 ) ) ).
thf(thm_2Earithmetic_2EPRE__SUC__EQ,axiom,
! [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum] :
( ( c_2Eprim__rec_2E_3C @ c_2Enum_2E0 @ V1n )
=> ( ( V0m
= ( c_2Eprim__rec_2EPRE @ V1n ) )
<=> ( ( c_2Enum_2ESUC @ V0m )
= V1n ) ) ) ).
thf(thm_2Earithmetic_2EINV__PRE__EQ,axiom,
! [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum] :
( ( ( c_2Eprim__rec_2E_3C @ c_2Enum_2E0 @ V0m )
& ( c_2Eprim__rec_2E_3C @ c_2Enum_2E0 @ V1n ) )
=> ( ( ( c_2Eprim__rec_2EPRE @ V0m )
= ( c_2Eprim__rec_2EPRE @ V1n ) )
<=> ( V0m = V1n ) ) ) ).
thf(thm_2Earithmetic_2ELESS__SUC__NOT,axiom,
! [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum] :
( ( c_2Eprim__rec_2E_3C @ V0m @ V1n )
=> ( (~) @ ( c_2Eprim__rec_2E_3C @ V1n @ ( c_2Enum_2ESUC @ V0m ) ) ) ) ).
thf(thm_2Earithmetic_2EADD__EQ__SUB,axiom,
! [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum,V2p: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_3C_3D @ V1n @ V2p )
=> ( ( ( c_2Earithmetic_2E_2B @ V0m @ V1n )
= V2p )
<=> ( V0m
= ( c_2Earithmetic_2E_2D @ V2p @ V1n ) ) ) ) ).
thf(thm_2Earithmetic_2ELESS__MONO__ADD,axiom,
! [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum,V2p: tyop_2Enum_2Enum] :
( ( c_2Eprim__rec_2E_3C @ V0m @ V1n )
=> ( c_2Eprim__rec_2E_3C @ ( c_2Earithmetic_2E_2B @ V0m @ V2p ) @ ( c_2Earithmetic_2E_2B @ V1n @ V2p ) ) ) ).
thf(thm_2Earithmetic_2ELESS__MONO__ADD__INV,axiom,
! [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum,V2p: tyop_2Enum_2Enum] :
( ( c_2Eprim__rec_2E_3C @ ( c_2Earithmetic_2E_2B @ V0m @ V2p ) @ ( c_2Earithmetic_2E_2B @ V1n @ V2p ) )
=> ( c_2Eprim__rec_2E_3C @ V0m @ V1n ) ) ).
thf(thm_2Earithmetic_2ELESS__MONO__ADD__EQ,axiom,
! [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum,V2p: tyop_2Enum_2Enum] :
( ( c_2Eprim__rec_2E_3C @ ( c_2Earithmetic_2E_2B @ V0m @ V2p ) @ ( c_2Earithmetic_2E_2B @ V1n @ V2p ) )
= ( c_2Eprim__rec_2E_3C @ V0m @ V1n ) ) ).
thf(thm_2Earithmetic_2ELT__ADD__RCANCEL,axiom,
! [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum,V2p: tyop_2Enum_2Enum] :
( ( c_2Eprim__rec_2E_3C @ ( c_2Earithmetic_2E_2B @ V0m @ V2p ) @ ( c_2Earithmetic_2E_2B @ V1n @ V2p ) )
= ( c_2Eprim__rec_2E_3C @ V0m @ V1n ) ) ).
thf(thm_2Earithmetic_2ELT__ADD__LCANCEL,axiom,
! [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum,V2p: tyop_2Enum_2Enum] :
( ( c_2Eprim__rec_2E_3C @ ( c_2Earithmetic_2E_2B @ V2p @ V0m ) @ ( c_2Earithmetic_2E_2B @ V2p @ V1n ) )
= ( c_2Eprim__rec_2E_3C @ V0m @ V1n ) ) ).
thf(thm_2Earithmetic_2EEQ__MONO__ADD__EQ,axiom,
! [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum,V2p: tyop_2Enum_2Enum] :
( ( ( c_2Earithmetic_2E_2B @ V0m @ V2p )
= ( c_2Earithmetic_2E_2B @ V1n @ V2p ) )
<=> ( V0m = V1n ) ) ).
thf(thm_2Earithmetic_2ELESS__EQ__MONO__ADD__EQ,axiom,
! [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum,V2p: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_3C_3D @ ( c_2Earithmetic_2E_2B @ V0m @ V2p ) @ ( c_2Earithmetic_2E_2B @ V1n @ V2p ) )
= ( c_2Earithmetic_2E_3C_3D @ V0m @ V1n ) ) ).
thf(thm_2Earithmetic_2ELESS__EQ__TRANS,axiom,
! [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum,V2p: tyop_2Enum_2Enum] :
( ( ( c_2Earithmetic_2E_3C_3D @ V0m @ V1n )
& ( c_2Earithmetic_2E_3C_3D @ V1n @ V2p ) )
=> ( c_2Earithmetic_2E_3C_3D @ V0m @ V2p ) ) ).
thf(thm_2Earithmetic_2ELESS__EQ__LESS__TRANS,axiom,
! [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum,V2p: tyop_2Enum_2Enum] :
( ( ( c_2Earithmetic_2E_3C_3D @ V0m @ V1n )
& ( c_2Eprim__rec_2E_3C @ V1n @ V2p ) )
=> ( c_2Eprim__rec_2E_3C @ V0m @ V2p ) ) ).
thf(thm_2Earithmetic_2ELESS__LESS__EQ__TRANS,axiom,
! [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum,V2p: tyop_2Enum_2Enum] :
( ( ( c_2Eprim__rec_2E_3C @ V0m @ V1n )
& ( c_2Earithmetic_2E_3C_3D @ V1n @ V2p ) )
=> ( c_2Eprim__rec_2E_3C @ V0m @ V2p ) ) ).
thf(thm_2Earithmetic_2ELESS__EQ__LESS__EQ__MONO,axiom,
! [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum,V2p: tyop_2Enum_2Enum,V3q: tyop_2Enum_2Enum] :
( ( ( c_2Earithmetic_2E_3C_3D @ V0m @ V2p )
& ( c_2Earithmetic_2E_3C_3D @ V1n @ V3q ) )
=> ( c_2Earithmetic_2E_3C_3D @ ( c_2Earithmetic_2E_2B @ V0m @ V1n ) @ ( c_2Earithmetic_2E_2B @ V2p @ V3q ) ) ) ).
thf(thm_2Earithmetic_2ELESS__EQ__REFL,axiom,
! [V0m: tyop_2Enum_2Enum] : ( c_2Earithmetic_2E_3C_3D @ V0m @ V0m ) ).
thf(thm_2Earithmetic_2ELESS__IMP__LESS__OR__EQ,axiom,
! [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum] :
( ( c_2Eprim__rec_2E_3C @ V0m @ V1n )
=> ( c_2Earithmetic_2E_3C_3D @ V0m @ V1n ) ) ).
thf(thm_2Earithmetic_2ELESS__MONO__MULT,axiom,
! [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum,V2p: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_3C_3D @ V0m @ V1n )
=> ( c_2Earithmetic_2E_3C_3D @ ( c_2Earithmetic_2E_2A @ V0m @ V2p ) @ ( c_2Earithmetic_2E_2A @ V1n @ V2p ) ) ) ).
thf(thm_2Earithmetic_2ELESS__MONO__MULT2,axiom,
! [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum,V2i: tyop_2Enum_2Enum,V3j: tyop_2Enum_2Enum] :
( ( ( c_2Earithmetic_2E_3C_3D @ V0m @ V2i )
& ( c_2Earithmetic_2E_3C_3D @ V1n @ V3j ) )
=> ( c_2Earithmetic_2E_3C_3D @ ( c_2Earithmetic_2E_2A @ V0m @ V1n ) @ ( c_2Earithmetic_2E_2A @ V2i @ V3j ) ) ) ).
thf(thm_2Earithmetic_2ERIGHT__SUB__DISTRIB,axiom,
! [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum,V2p: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_2A @ ( c_2Earithmetic_2E_2D @ V0m @ V1n ) @ V2p )
= ( c_2Earithmetic_2E_2D @ ( c_2Earithmetic_2E_2A @ V0m @ V2p ) @ ( c_2Earithmetic_2E_2A @ V1n @ V2p ) ) ) ).
thf(thm_2Earithmetic_2ELEFT__SUB__DISTRIB,axiom,
! [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum,V2p: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_2A @ V2p @ ( c_2Earithmetic_2E_2D @ V0m @ V1n ) )
= ( c_2Earithmetic_2E_2D @ ( c_2Earithmetic_2E_2A @ V2p @ V0m ) @ ( c_2Earithmetic_2E_2A @ V2p @ V1n ) ) ) ).
thf(thm_2Earithmetic_2ELESS__ADD__1,axiom,
! [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum] :
( ( c_2Eprim__rec_2E_3C @ V1n @ V0m )
=> ? [V2p: tyop_2Enum_2Enum] :
( V0m
= ( c_2Earithmetic_2E_2B @ V1n @ ( c_2Earithmetic_2E_2B @ V2p @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ) ) ).
thf(thm_2Earithmetic_2EEXP__ADD,axiom,
! [V0p: tyop_2Enum_2Enum,V1q: tyop_2Enum_2Enum,V2n: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2EEXP @ V2n @ ( c_2Earithmetic_2E_2B @ V0p @ V1q ) )
= ( c_2Earithmetic_2E_2A @ ( c_2Earithmetic_2EEXP @ V2n @ V0p ) @ ( c_2Earithmetic_2EEXP @ V2n @ V1q ) ) ) ).
thf(thm_2Earithmetic_2ENOT__ODD__EQ__EVEN,axiom,
! [V0n: tyop_2Enum_2Enum,V1m: tyop_2Enum_2Enum] :
( (~)
@ ( ( c_2Enum_2ESUC @ ( c_2Earithmetic_2E_2B @ V0n @ V0n ) )
= ( c_2Earithmetic_2E_2B @ V1m @ V1m ) ) ) ).
thf(thm_2Earithmetic_2ELESS__EQUAL__ANTISYM,axiom,
! [V0n: tyop_2Enum_2Enum,V1m: tyop_2Enum_2Enum] :
( ( ( c_2Earithmetic_2E_3C_3D @ V0n @ V1m )
& ( c_2Earithmetic_2E_3C_3D @ V1m @ V0n ) )
=> ( V0n = V1m ) ) ).
thf(thm_2Earithmetic_2ELESS__ADD__SUC,axiom,
! [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum] : ( c_2Eprim__rec_2E_3C @ V0m @ ( c_2Earithmetic_2E_2B @ V0m @ ( c_2Enum_2ESUC @ V1n ) ) ) ).
thf(thm_2Earithmetic_2ELESS__OR__EQ__ADD,axiom,
! [V0n: tyop_2Enum_2Enum,V1m: tyop_2Enum_2Enum] :
( ( c_2Eprim__rec_2E_3C @ V0n @ V1m )
| ? [V2p: tyop_2Enum_2Enum] :
( V0n
= ( c_2Earithmetic_2E_2B @ V2p @ V1m ) ) ) ).
thf(thm_2Earithmetic_2EWOP,axiom,
! [V0P: tyop_2Enum_2Enum > $o] :
( ? [V1n: tyop_2Enum_2Enum] : ( V0P @ V1n )
=> ? [V2n: tyop_2Enum_2Enum] :
( ( V0P @ V2n )
& ! [V3m: tyop_2Enum_2Enum] :
( ( c_2Eprim__rec_2E_3C @ V3m @ V2n )
=> ( (~) @ ( V0P @ V3m ) ) ) ) ) ).
thf(thm_2Earithmetic_2ECOMPLETE__INDUCTION,axiom,
! [V0P: tyop_2Enum_2Enum > $o] :
( ! [V1n: tyop_2Enum_2Enum] :
( ! [V2m: tyop_2Enum_2Enum] :
( ( c_2Eprim__rec_2E_3C @ V2m @ V1n )
=> ( V0P @ V2m ) )
=> ( V0P @ V1n ) )
=> ! [V3n: tyop_2Enum_2Enum] : ( V0P @ V3n ) ) ).
thf(thm_2Earithmetic_2EFORALL__NUM__THM,axiom,
! [V0P: tyop_2Enum_2Enum > $o] :
( ! [V1n: tyop_2Enum_2Enum] : ( V0P @ V1n )
<=> ( ( V0P @ c_2Enum_2E0 )
& ! [V2n: tyop_2Enum_2Enum] :
( ( V0P @ V2n )
=> ( V0P @ ( c_2Enum_2ESUC @ V2n ) ) ) ) ) ).
thf(thm_2Earithmetic_2ESUC__SUB,axiom,
! [V0a: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_2D @ ( c_2Enum_2ESUC @ V0a ) @ V0a )
= ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ).
thf(thm_2Earithmetic_2ESUB__PLUS,axiom,
! [V0a: tyop_2Enum_2Enum,V1b: tyop_2Enum_2Enum,V2c: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_2D @ V0a @ ( c_2Earithmetic_2E_2B @ V1b @ V2c ) )
= ( c_2Earithmetic_2E_2D @ ( c_2Earithmetic_2E_2D @ V0a @ V1b ) @ V2c ) ) ).
thf(thm_2Earithmetic_2EINV__PRE__LESS,axiom,
! [V0m: tyop_2Enum_2Enum] :
( ( c_2Eprim__rec_2E_3C @ c_2Enum_2E0 @ V0m )
=> ! [V1n: tyop_2Enum_2Enum] :
( ( c_2Eprim__rec_2E_3C @ ( c_2Eprim__rec_2EPRE @ V0m ) @ ( c_2Eprim__rec_2EPRE @ V1n ) )
= ( c_2Eprim__rec_2E_3C @ V0m @ V1n ) ) ) ).
thf(thm_2Earithmetic_2EINV__PRE__LESS__EQ,axiom,
! [V0n: tyop_2Enum_2Enum] :
( ( c_2Eprim__rec_2E_3C @ c_2Enum_2E0 @ V0n )
=> ! [V1m: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_3C_3D @ ( c_2Eprim__rec_2EPRE @ V1m ) @ ( c_2Eprim__rec_2EPRE @ V0n ) )
= ( c_2Earithmetic_2E_3C_3D @ V1m @ V0n ) ) ) ).
thf(thm_2Earithmetic_2EPRE__LESS__EQ,axiom,
! [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_3C_3D @ V0m @ V1n )
=> ( c_2Earithmetic_2E_3C_3D @ ( c_2Eprim__rec_2EPRE @ V0m ) @ ( c_2Eprim__rec_2EPRE @ V1n ) ) ) ).
thf(thm_2Earithmetic_2ESUB__LESS__EQ,axiom,
! [V0n: tyop_2Enum_2Enum,V1m: tyop_2Enum_2Enum] : ( c_2Earithmetic_2E_3C_3D @ ( c_2Earithmetic_2E_2D @ V0n @ V1m ) @ V0n ) ).
thf(thm_2Earithmetic_2ESUB__EQ__EQ__0,axiom,
! [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum] :
( ( ( c_2Earithmetic_2E_2D @ V0m @ V1n )
= V0m )
<=> ( ( V0m = c_2Enum_2E0 )
| ( V1n = c_2Enum_2E0 ) ) ) ).
thf(thm_2Earithmetic_2ESUB__LESS__0,axiom,
! [V0n: tyop_2Enum_2Enum,V1m: tyop_2Enum_2Enum] :
( ( c_2Eprim__rec_2E_3C @ V1m @ V0n )
= ( c_2Eprim__rec_2E_3C @ c_2Enum_2E0 @ ( c_2Earithmetic_2E_2D @ V0n @ V1m ) ) ) ).
thf(thm_2Earithmetic_2ESUB__LESS__OR,axiom,
! [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum] :
( ( c_2Eprim__rec_2E_3C @ V1n @ V0m )
=> ( c_2Earithmetic_2E_3C_3D @ V1n @ ( c_2Earithmetic_2E_2D @ V0m @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ) ).
thf(thm_2Earithmetic_2ELESS__SUB__ADD__LESS,axiom,
! [V0n: tyop_2Enum_2Enum,V1m: tyop_2Enum_2Enum,V2i: tyop_2Enum_2Enum] :
( ( c_2Eprim__rec_2E_3C @ V2i @ ( c_2Earithmetic_2E_2D @ V0n @ V1m ) )
=> ( c_2Eprim__rec_2E_3C @ ( c_2Earithmetic_2E_2B @ V2i @ V1m ) @ V0n ) ) ).
thf(thm_2Earithmetic_2ETIMES2,axiom,
! [V0n: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_2A @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) @ V0n )
= ( c_2Earithmetic_2E_2B @ V0n @ V0n ) ) ).
thf(thm_2Earithmetic_2ELESS__MULT__MONO,axiom,
! [V0m: tyop_2Enum_2Enum,V1i: tyop_2Enum_2Enum,V2n: tyop_2Enum_2Enum] :
( ( c_2Eprim__rec_2E_3C @ ( c_2Earithmetic_2E_2A @ ( c_2Enum_2ESUC @ V2n ) @ V0m ) @ ( c_2Earithmetic_2E_2A @ ( c_2Enum_2ESUC @ V2n ) @ V1i ) )
= ( c_2Eprim__rec_2E_3C @ V0m @ V1i ) ) ).
thf(thm_2Earithmetic_2EMULT__MONO__EQ,axiom,
! [V0m: tyop_2Enum_2Enum,V1i: tyop_2Enum_2Enum,V2n: tyop_2Enum_2Enum] :
( ( ( c_2Earithmetic_2E_2A @ ( c_2Enum_2ESUC @ V2n ) @ V0m )
= ( c_2Earithmetic_2E_2A @ ( c_2Enum_2ESUC @ V2n ) @ V1i ) )
<=> ( V0m = V1i ) ) ).
thf(thm_2Earithmetic_2EMULT__SUC__EQ,axiom,
! [V0p: tyop_2Enum_2Enum,V1m: tyop_2Enum_2Enum,V2n: tyop_2Enum_2Enum] :
( ( ( c_2Earithmetic_2E_2A @ V2n @ ( c_2Enum_2ESUC @ V0p ) )
= ( c_2Earithmetic_2E_2A @ V1m @ ( c_2Enum_2ESUC @ V0p ) ) )
<=> ( V2n = V1m ) ) ).
thf(thm_2Earithmetic_2EMULT__EXP__MONO,axiom,
! [V0p: tyop_2Enum_2Enum,V1q: tyop_2Enum_2Enum,V2n: tyop_2Enum_2Enum,V3m: tyop_2Enum_2Enum] :
( ( ( c_2Earithmetic_2E_2A @ V2n @ ( c_2Earithmetic_2EEXP @ ( c_2Enum_2ESUC @ V1q ) @ V0p ) )
= ( c_2Earithmetic_2E_2A @ V3m @ ( c_2Earithmetic_2EEXP @ ( c_2Enum_2ESUC @ V1q ) @ V0p ) ) )
<=> ( V2n = V3m ) ) ).
thf(thm_2Earithmetic_2EEQ__ADD__LCANCEL,axiom,
! [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum,V2p: tyop_2Enum_2Enum] :
( ( ( c_2Earithmetic_2E_2B @ V0m @ V1n )
= ( c_2Earithmetic_2E_2B @ V0m @ V2p ) )
<=> ( V1n = V2p ) ) ).
thf(thm_2Earithmetic_2EEQ__ADD__RCANCEL,axiom,
! [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum,V2p: tyop_2Enum_2Enum] :
( ( ( c_2Earithmetic_2E_2B @ V0m @ V2p )
= ( c_2Earithmetic_2E_2B @ V1n @ V2p ) )
<=> ( V0m = V1n ) ) ).
thf(thm_2Earithmetic_2EEQ__MULT__LCANCEL,axiom,
! [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum,V2p: tyop_2Enum_2Enum] :
( ( ( c_2Earithmetic_2E_2A @ V0m @ V1n )
= ( c_2Earithmetic_2E_2A @ V0m @ V2p ) )
<=> ( ( V0m = c_2Enum_2E0 )
| ( V1n = V2p ) ) ) ).
thf(thm_2Earithmetic_2EEQ__MULT__RCANCEL,axiom,
! [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum,V2p: tyop_2Enum_2Enum] :
( ( ( c_2Earithmetic_2E_2A @ V1n @ V0m )
= ( c_2Earithmetic_2E_2A @ V2p @ V0m ) )
<=> ( ( V0m = c_2Enum_2E0 )
| ( V1n = V2p ) ) ) ).
thf(thm_2Earithmetic_2EADD__SUB,axiom,
! [V0a: tyop_2Enum_2Enum,V1c: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_2D @ ( c_2Earithmetic_2E_2B @ V0a @ V1c ) @ V1c )
= V0a ) ).
thf(thm_2Earithmetic_2ELESS__EQ__ADD__SUB,axiom,
! [V0c: tyop_2Enum_2Enum,V1b: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_3C_3D @ V0c @ V1b )
=> ! [V2a: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_2D @ ( c_2Earithmetic_2E_2B @ V2a @ V1b ) @ V0c )
= ( c_2Earithmetic_2E_2B @ V2a @ ( c_2Earithmetic_2E_2D @ V1b @ V0c ) ) ) ) ).
thf(thm_2Earithmetic_2ESUB__EQUAL__0,axiom,
! [V0c: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_2D @ V0c @ V0c )
= c_2Enum_2E0 ) ).
thf(thm_2Earithmetic_2ELESS__EQ__SUB__LESS,axiom,
! [V0a: tyop_2Enum_2Enum,V1b: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_3C_3D @ V1b @ V0a )
=> ! [V2c: tyop_2Enum_2Enum] :
( ( c_2Eprim__rec_2E_3C @ ( c_2Earithmetic_2E_2D @ V0a @ V1b ) @ V2c )
= ( c_2Eprim__rec_2E_3C @ V0a @ ( c_2Earithmetic_2E_2B @ V1b @ V2c ) ) ) ) ).
thf(thm_2Earithmetic_2ENOT__SUC__LESS__EQ,axiom,
! [V0n: tyop_2Enum_2Enum,V1m: tyop_2Enum_2Enum] :
( ( (~) @ ( c_2Earithmetic_2E_3C_3D @ ( c_2Enum_2ESUC @ V0n ) @ V1m ) )
<=> ( c_2Earithmetic_2E_3C_3D @ V1m @ V0n ) ) ).
thf(thm_2Earithmetic_2ESUB__SUB,axiom,
! [V0b: tyop_2Enum_2Enum,V1c: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_3C_3D @ V1c @ V0b )
=> ! [V2a: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_2D @ V2a @ ( c_2Earithmetic_2E_2D @ V0b @ V1c ) )
= ( c_2Earithmetic_2E_2D @ ( c_2Earithmetic_2E_2B @ V2a @ V1c ) @ V0b ) ) ) ).
thf(thm_2Earithmetic_2ELESS__IMP__LESS__ADD,axiom,
! [V0n: tyop_2Enum_2Enum,V1m: tyop_2Enum_2Enum] :
( ( c_2Eprim__rec_2E_3C @ V0n @ V1m )
=> ! [V2p: tyop_2Enum_2Enum] : ( c_2Eprim__rec_2E_3C @ V0n @ ( c_2Earithmetic_2E_2B @ V1m @ V2p ) ) ) ).
thf(thm_2Earithmetic_2ESUB__LESS__EQ__ADD,axiom,
! [V0m: tyop_2Enum_2Enum,V1p: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_3C_3D @ V0m @ V1p )
=> ! [V2n: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_3C_3D @ ( c_2Earithmetic_2E_2D @ V1p @ V0m ) @ V2n )
= ( c_2Earithmetic_2E_3C_3D @ V1p @ ( c_2Earithmetic_2E_2B @ V0m @ V2n ) ) ) ) ).
thf(thm_2Earithmetic_2ESUB__LESS__SUC,axiom,
! [V0p: tyop_2Enum_2Enum,V1m: tyop_2Enum_2Enum] : ( c_2Eprim__rec_2E_3C @ ( c_2Earithmetic_2E_2D @ V0p @ V1m ) @ ( c_2Enum_2ESUC @ V0p ) ) ).
thf(thm_2Earithmetic_2ESUB__CANCEL,axiom,
! [V0p: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum,V2m: tyop_2Enum_2Enum] :
( ( ( c_2Earithmetic_2E_3C_3D @ V1n @ V0p )
& ( c_2Earithmetic_2E_3C_3D @ V2m @ V0p ) )
=> ( ( ( c_2Earithmetic_2E_2D @ V0p @ V1n )
= ( c_2Earithmetic_2E_2D @ V0p @ V2m ) )
<=> ( V1n = V2m ) ) ) ).
thf(thm_2Earithmetic_2ECANCEL__SUB,axiom,
! [V0p: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum,V2m: tyop_2Enum_2Enum] :
( ( ( c_2Earithmetic_2E_3C_3D @ V0p @ V1n )
& ( c_2Earithmetic_2E_3C_3D @ V0p @ V2m ) )
=> ( ( ( c_2Earithmetic_2E_2D @ V1n @ V0p )
= ( c_2Earithmetic_2E_2D @ V2m @ V0p ) )
<=> ( V1n = V2m ) ) ) ).
thf(thm_2Earithmetic_2ENOT__EXP__0,axiom,
! [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum] :
( (~)
@ ( ( c_2Earithmetic_2EEXP @ ( c_2Enum_2ESUC @ V1n ) @ V0m )
= c_2Enum_2E0 ) ) ).
thf(thm_2Earithmetic_2EZERO__LESS__EXP,axiom,
! [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum] : ( c_2Eprim__rec_2E_3C @ c_2Enum_2E0 @ ( c_2Earithmetic_2EEXP @ ( c_2Enum_2ESUC @ V1n ) @ V0m ) ) ).
thf(thm_2Earithmetic_2EODD__OR__EVEN,axiom,
! [V0n: tyop_2Enum_2Enum] :
? [V1m: tyop_2Enum_2Enum] :
( ( V0n
= ( c_2Earithmetic_2E_2A @ ( c_2Enum_2ESUC @ ( c_2Enum_2ESUC @ c_2Enum_2E0 ) ) @ V1m ) )
| ( V0n
= ( c_2Earithmetic_2E_2B @ ( c_2Earithmetic_2E_2A @ ( c_2Enum_2ESUC @ ( c_2Enum_2ESUC @ c_2Enum_2E0 ) ) @ V1m ) @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ) ).
thf(thm_2Earithmetic_2ELESS__EXP__SUC__MONO,axiom,
! [V0n: tyop_2Enum_2Enum,V1m: tyop_2Enum_2Enum] : ( c_2Eprim__rec_2E_3C @ ( c_2Earithmetic_2EEXP @ ( c_2Enum_2ESUC @ ( c_2Enum_2ESUC @ V1m ) ) @ V0n ) @ ( c_2Earithmetic_2EEXP @ ( c_2Enum_2ESUC @ ( c_2Enum_2ESUC @ V1m ) ) @ ( c_2Enum_2ESUC @ V0n ) ) ) ).
thf(thm_2Earithmetic_2ELESS__LESS__CASES,axiom,
! [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum] :
( ( V0m = V1n )
| ( c_2Eprim__rec_2E_3C @ V0m @ V1n )
| ( c_2Eprim__rec_2E_3C @ V1n @ V0m ) ) ).
thf(thm_2Earithmetic_2EGREATER__EQ,axiom,
! [V0n: tyop_2Enum_2Enum,V1m: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_3E_3D @ V0n @ V1m )
= ( c_2Earithmetic_2E_3C_3D @ V1m @ V0n ) ) ).
thf(thm_2Earithmetic_2ELESS__EQ__CASES,axiom,
! [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_3C_3D @ V0m @ V1n )
| ( c_2Earithmetic_2E_3C_3D @ V1n @ V0m ) ) ).
thf(thm_2Earithmetic_2ELESS__EQUAL__ADD,axiom,
! [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_3C_3D @ V0m @ V1n )
=> ? [V2p: tyop_2Enum_2Enum] :
( V1n
= ( c_2Earithmetic_2E_2B @ V0m @ V2p ) ) ) ).
thf(thm_2Earithmetic_2ELESS__EQ__EXISTS,axiom,
! [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_3C_3D @ V0m @ V1n )
<=> ? [V2p: tyop_2Enum_2Enum] :
( V1n
= ( c_2Earithmetic_2E_2B @ V0m @ V2p ) ) ) ).
thf(thm_2Earithmetic_2EMULT__EQ__0,axiom,
! [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum] :
( ( ( c_2Earithmetic_2E_2A @ V0m @ V1n )
= c_2Enum_2E0 )
<=> ( ( V0m = c_2Enum_2E0 )
| ( V1n = c_2Enum_2E0 ) ) ) ).
thf(thm_2Earithmetic_2EMULT__EQ__1,axiom,
! [V0x: tyop_2Enum_2Enum,V1y: tyop_2Enum_2Enum] :
( ( ( c_2Earithmetic_2E_2A @ V0x @ V1y )
= ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) )
<=> ( ( V0x
= ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) )
& ( V1y
= ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ) ).
thf(thm_2Earithmetic_2EMULT__EQ__ID,axiom,
! [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum] :
( ( ( c_2Earithmetic_2E_2A @ V0m @ V1n )
= V1n )
<=> ( ( V0m
= ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) )
| ( V1n = c_2Enum_2E0 ) ) ) ).
thf(thm_2Earithmetic_2ELESS__MULT2,axiom,
! [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum] :
( ( ( c_2Eprim__rec_2E_3C @ c_2Enum_2E0 @ V0m )
& ( c_2Eprim__rec_2E_3C @ c_2Enum_2E0 @ V1n ) )
=> ( c_2Eprim__rec_2E_3C @ c_2Enum_2E0 @ ( c_2Earithmetic_2E_2A @ V0m @ V1n ) ) ) ).
thf(thm_2Earithmetic_2EZERO__LESS__MULT,axiom,
! [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum] :
( ( c_2Eprim__rec_2E_3C @ c_2Enum_2E0 @ ( c_2Earithmetic_2E_2A @ V0m @ V1n ) )
<=> ( ( c_2Eprim__rec_2E_3C @ c_2Enum_2E0 @ V0m )
& ( c_2Eprim__rec_2E_3C @ c_2Enum_2E0 @ V1n ) ) ) ).
thf(thm_2Earithmetic_2EZERO__LESS__ADD,axiom,
! [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum] :
( ( c_2Eprim__rec_2E_3C @ c_2Enum_2E0 @ ( c_2Earithmetic_2E_2B @ V0m @ V1n ) )
<=> ( ( c_2Eprim__rec_2E_3C @ c_2Enum_2E0 @ V0m )
| ( c_2Eprim__rec_2E_3C @ c_2Enum_2E0 @ V1n ) ) ) ).
thf(thm_2Earithmetic_2EFACT__LESS,axiom,
! [V0n: tyop_2Enum_2Enum] : ( c_2Eprim__rec_2E_3C @ c_2Enum_2E0 @ ( c_2Earithmetic_2EFACT @ V0n ) ) ).
thf(thm_2Earithmetic_2EEVEN__ODD,axiom,
! [V0n: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2EEVEN @ V0n )
<=> ( (~) @ ( c_2Earithmetic_2EODD @ V0n ) ) ) ).
thf(thm_2Earithmetic_2EODD__EVEN,axiom,
! [V0n: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2EODD @ V0n )
<=> ( (~) @ ( c_2Earithmetic_2EEVEN @ V0n ) ) ) ).
thf(thm_2Earithmetic_2EEVEN__OR__ODD,axiom,
! [V0n: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2EEVEN @ V0n )
| ( c_2Earithmetic_2EODD @ V0n ) ) ).
thf(thm_2Earithmetic_2EEVEN__AND__ODD,axiom,
! [V0n: tyop_2Enum_2Enum] :
( (~)
@ ( ( c_2Earithmetic_2EEVEN @ V0n )
& ( c_2Earithmetic_2EODD @ V0n ) ) ) ).
thf(thm_2Earithmetic_2EEVEN__ADD,axiom,
! [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2EEVEN @ ( c_2Earithmetic_2E_2B @ V0m @ V1n ) )
<=> ( ( c_2Earithmetic_2EEVEN @ V0m )
= ( c_2Earithmetic_2EEVEN @ V1n ) ) ) ).
thf(thm_2Earithmetic_2EEVEN__MULT,axiom,
! [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2EEVEN @ ( c_2Earithmetic_2E_2A @ V0m @ V1n ) )
<=> ( ( c_2Earithmetic_2EEVEN @ V0m )
| ( c_2Earithmetic_2EEVEN @ V1n ) ) ) ).
thf(thm_2Earithmetic_2EODD__ADD,axiom,
! [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2EODD @ ( c_2Earithmetic_2E_2B @ V0m @ V1n ) )
<=> ( (~)
@ ( ( c_2Earithmetic_2EODD @ V0m )
= ( c_2Earithmetic_2EODD @ V1n ) ) ) ) ).
thf(thm_2Earithmetic_2EODD__MULT,axiom,
! [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2EODD @ ( c_2Earithmetic_2E_2A @ V0m @ V1n ) )
<=> ( ( c_2Earithmetic_2EODD @ V0m )
& ( c_2Earithmetic_2EODD @ V1n ) ) ) ).
thf(thm_2Earithmetic_2EEVEN__DOUBLE,axiom,
! [V0n: tyop_2Enum_2Enum] : ( c_2Earithmetic_2EEVEN @ ( c_2Earithmetic_2E_2A @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) @ V0n ) ) ).
thf(thm_2Earithmetic_2EODD__DOUBLE,axiom,
! [V0n: tyop_2Enum_2Enum] : ( c_2Earithmetic_2EODD @ ( c_2Enum_2ESUC @ ( c_2Earithmetic_2E_2A @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) @ V0n ) ) ) ).
thf(thm_2Earithmetic_2EEVEN__ODD__EXISTS,axiom,
! [V0n: tyop_2Enum_2Enum] :
( ( ( c_2Earithmetic_2EEVEN @ V0n )
=> ? [V1m: tyop_2Enum_2Enum] :
( V0n
= ( c_2Earithmetic_2E_2A @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) @ V1m ) ) )
& ( ( c_2Earithmetic_2EODD @ V0n )
=> ? [V2m: tyop_2Enum_2Enum] :
( V0n
= ( c_2Enum_2ESUC @ ( c_2Earithmetic_2E_2A @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) @ V2m ) ) ) ) ) ).
thf(thm_2Earithmetic_2EEVEN__EXISTS,axiom,
! [V0n: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2EEVEN @ V0n )
<=> ? [V1m: tyop_2Enum_2Enum] :
( V0n
= ( c_2Earithmetic_2E_2A @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) @ V1m ) ) ) ).
thf(thm_2Earithmetic_2EODD__EXISTS,axiom,
! [V0n: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2EODD @ V0n )
<=> ? [V1m: tyop_2Enum_2Enum] :
( V0n
= ( c_2Enum_2ESUC @ ( c_2Earithmetic_2E_2A @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) @ V1m ) ) ) ) ).
thf(thm_2Earithmetic_2EEVEN__EXP__IFF,axiom,
! [V0n: tyop_2Enum_2Enum,V1m: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2EEVEN @ ( c_2Earithmetic_2EEXP @ V1m @ V0n ) )
<=> ( ( c_2Eprim__rec_2E_3C @ c_2Enum_2E0 @ V0n )
& ( c_2Earithmetic_2EEVEN @ V1m ) ) ) ).
thf(thm_2Earithmetic_2EEVEN__EXP,axiom,
! [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum] :
( ( ( c_2Eprim__rec_2E_3C @ c_2Enum_2E0 @ V1n )
& ( c_2Earithmetic_2EEVEN @ V0m ) )
=> ( c_2Earithmetic_2EEVEN @ ( c_2Earithmetic_2EEXP @ V0m @ V1n ) ) ) ).
thf(thm_2Earithmetic_2EODD__EXP__IFF,axiom,
! [V0n: tyop_2Enum_2Enum,V1m: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2EODD @ ( c_2Earithmetic_2EEXP @ V1m @ V0n ) )
<=> ( ( V0n = c_2Enum_2E0 )
| ( c_2Earithmetic_2EODD @ V1m ) ) ) ).
thf(thm_2Earithmetic_2EODD__EXP,axiom,
! [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum] :
( ( ( c_2Eprim__rec_2E_3C @ c_2Enum_2E0 @ V1n )
& ( c_2Earithmetic_2EODD @ V0m ) )
=> ( c_2Earithmetic_2EODD @ ( c_2Earithmetic_2EEXP @ V0m @ V1n ) ) ) ).
thf(thm_2Earithmetic_2EEQ__LESS__EQ,axiom,
! [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum] :
( ( V0m = V1n )
<=> ( ( c_2Earithmetic_2E_3C_3D @ V0m @ V1n )
& ( c_2Earithmetic_2E_3C_3D @ V1n @ V0m ) ) ) ).
thf(thm_2Earithmetic_2EADD__MONO__LESS__EQ,axiom,
! [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum,V2p: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_3C_3D @ ( c_2Earithmetic_2E_2B @ V0m @ V1n ) @ ( c_2Earithmetic_2E_2B @ V0m @ V2p ) )
= ( c_2Earithmetic_2E_3C_3D @ V1n @ V2p ) ) ).
thf(thm_2Earithmetic_2ELE__ADD__LCANCEL,axiom,
! [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum,V2p: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_3C_3D @ ( c_2Earithmetic_2E_2B @ V0m @ V1n ) @ ( c_2Earithmetic_2E_2B @ V0m @ V2p ) )
= ( c_2Earithmetic_2E_3C_3D @ V1n @ V2p ) ) ).
thf(thm_2Earithmetic_2ELE__ADD__RCANCEL,axiom,
! [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum,V2p: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_3C_3D @ ( c_2Earithmetic_2E_2B @ V1n @ V0m ) @ ( c_2Earithmetic_2E_2B @ V2p @ V0m ) )
= ( c_2Earithmetic_2E_3C_3D @ V1n @ V2p ) ) ).
thf(thm_2Earithmetic_2ENOT__LEQ,axiom,
! [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum] :
( ( (~) @ ( c_2Earithmetic_2E_3C_3D @ V0m @ V1n ) )
<=> ( c_2Earithmetic_2E_3C_3D @ ( c_2Enum_2ESUC @ V1n ) @ V0m ) ) ).
thf(thm_2Earithmetic_2ENOT__NUM__EQ,axiom,
! [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum] :
( ( (~) @ ( V0m = V1n ) )
<=> ( ( c_2Earithmetic_2E_3C_3D @ ( c_2Enum_2ESUC @ V0m ) @ V1n )
| ( c_2Earithmetic_2E_3C_3D @ ( c_2Enum_2ESUC @ V1n ) @ V0m ) ) ) ).
thf(thm_2Earithmetic_2ENOT__GREATER,axiom,
! [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum] :
( ( (~) @ ( c_2Earithmetic_2E_3E @ V0m @ V1n ) )
<=> ( c_2Earithmetic_2E_3C_3D @ V0m @ V1n ) ) ).
thf(thm_2Earithmetic_2ENOT__GREATER__EQ,axiom,
! [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum] :
( ( (~) @ ( c_2Earithmetic_2E_3E_3D @ V0m @ V1n ) )
<=> ( c_2Earithmetic_2E_3C_3D @ ( c_2Enum_2ESUC @ V0m ) @ V1n ) ) ).
thf(thm_2Earithmetic_2ESUC__ONE__ADD,axiom,
! [V0n: tyop_2Enum_2Enum] :
( ( c_2Enum_2ESUC @ V0n )
= ( c_2Earithmetic_2E_2B @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) @ V0n ) ) ).
thf(thm_2Earithmetic_2ESUC__ADD__SYM,axiom,
! [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum] :
( ( c_2Enum_2ESUC @ ( c_2Earithmetic_2E_2B @ V0m @ V1n ) )
= ( c_2Earithmetic_2E_2B @ ( c_2Enum_2ESUC @ V1n ) @ V0m ) ) ).
thf(thm_2Earithmetic_2ENOT__SUC__ADD__LESS__EQ,axiom,
! [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum] : ( (~) @ ( c_2Earithmetic_2E_3C_3D @ ( c_2Enum_2ESUC @ ( c_2Earithmetic_2E_2B @ V0m @ V1n ) ) @ V0m ) ) ).
thf(thm_2Earithmetic_2EMULT__LESS__EQ__SUC,axiom,
! [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum,V2p: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_3C_3D @ V0m @ V1n )
= ( c_2Earithmetic_2E_3C_3D @ ( c_2Earithmetic_2E_2A @ ( c_2Enum_2ESUC @ V2p ) @ V0m ) @ ( c_2Earithmetic_2E_2A @ ( c_2Enum_2ESUC @ V2p ) @ V1n ) ) ) ).
thf(thm_2Earithmetic_2ELE__MULT__LCANCEL,axiom,
! [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum,V2p: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_3C_3D @ ( c_2Earithmetic_2E_2A @ V0m @ V1n ) @ ( c_2Earithmetic_2E_2A @ V0m @ V2p ) )
<=> ( ( V0m = c_2Enum_2E0 )
| ( c_2Earithmetic_2E_3C_3D @ V1n @ V2p ) ) ) ).
thf(thm_2Earithmetic_2ELE__MULT__RCANCEL,axiom,
! [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum,V2p: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_3C_3D @ ( c_2Earithmetic_2E_2A @ V0m @ V1n ) @ ( c_2Earithmetic_2E_2A @ V2p @ V1n ) )
<=> ( ( V1n = c_2Enum_2E0 )
| ( c_2Earithmetic_2E_3C_3D @ V0m @ V2p ) ) ) ).
thf(thm_2Earithmetic_2ELT__MULT__LCANCEL,axiom,
! [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum,V2p: tyop_2Enum_2Enum] :
( ( c_2Eprim__rec_2E_3C @ ( c_2Earithmetic_2E_2A @ V0m @ V1n ) @ ( c_2Earithmetic_2E_2A @ V0m @ V2p ) )
<=> ( ( c_2Eprim__rec_2E_3C @ c_2Enum_2E0 @ V0m )
& ( c_2Eprim__rec_2E_3C @ V1n @ V2p ) ) ) ).
thf(thm_2Earithmetic_2ELT__MULT__RCANCEL,axiom,
! [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum,V2p: tyop_2Enum_2Enum] :
( ( c_2Eprim__rec_2E_3C @ ( c_2Earithmetic_2E_2A @ V0m @ V1n ) @ ( c_2Earithmetic_2E_2A @ V2p @ V1n ) )
<=> ( ( c_2Eprim__rec_2E_3C @ c_2Enum_2E0 @ V1n )
& ( c_2Eprim__rec_2E_3C @ V0m @ V2p ) ) ) ).
thf(thm_2Earithmetic_2ELT__MULT__CANCEL__LBARE,axiom,
! [V0n: tyop_2Enum_2Enum,V1m: tyop_2Enum_2Enum] :
( ( ( c_2Eprim__rec_2E_3C @ V1m @ ( c_2Earithmetic_2E_2A @ V1m @ V0n ) )
<=> ( ( c_2Eprim__rec_2E_3C @ c_2Enum_2E0 @ V1m )
& ( c_2Eprim__rec_2E_3C @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) @ V0n ) ) )
& ( ( c_2Eprim__rec_2E_3C @ V1m @ ( c_2Earithmetic_2E_2A @ V0n @ V1m ) )
<=> ( ( c_2Eprim__rec_2E_3C @ c_2Enum_2E0 @ V1m )
& ( c_2Eprim__rec_2E_3C @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) @ V0n ) ) ) ) ).
thf(thm_2Earithmetic_2ELT__MULT__CANCEL__RBARE,axiom,
! [V0n: tyop_2Enum_2Enum,V1m: tyop_2Enum_2Enum] :
( ( ( c_2Eprim__rec_2E_3C @ ( c_2Earithmetic_2E_2A @ V1m @ V0n ) @ V1m )
<=> ( ( c_2Eprim__rec_2E_3C @ c_2Enum_2E0 @ V1m )
& ( V0n = c_2Enum_2E0 ) ) )
& ( ( c_2Eprim__rec_2E_3C @ ( c_2Earithmetic_2E_2A @ V1m @ V0n ) @ V0n )
<=> ( ( c_2Eprim__rec_2E_3C @ c_2Enum_2E0 @ V0n )
& ( V1m = c_2Enum_2E0 ) ) ) ) ).
thf(thm_2Earithmetic_2ELE__MULT__CANCEL__LBARE,axiom,
! [V0n: tyop_2Enum_2Enum,V1m: tyop_2Enum_2Enum] :
( ( ( c_2Earithmetic_2E_3C_3D @ V1m @ ( c_2Earithmetic_2E_2A @ V1m @ V0n ) )
<=> ( ( V1m = c_2Enum_2E0 )
| ( c_2Eprim__rec_2E_3C @ c_2Enum_2E0 @ V0n ) ) )
& ( ( c_2Earithmetic_2E_3C_3D @ V1m @ ( c_2Earithmetic_2E_2A @ V0n @ V1m ) )
<=> ( ( V1m = c_2Enum_2E0 )
| ( c_2Eprim__rec_2E_3C @ c_2Enum_2E0 @ V0n ) ) ) ) ).
thf(thm_2Earithmetic_2ELE__MULT__CANCEL__RBARE,axiom,
! [V0n: tyop_2Enum_2Enum,V1m: tyop_2Enum_2Enum] :
( ( ( c_2Earithmetic_2E_3C_3D @ ( c_2Earithmetic_2E_2A @ V1m @ V0n ) @ V1m )
<=> ( ( V1m = c_2Enum_2E0 )
| ( c_2Earithmetic_2E_3C_3D @ V0n @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) )
& ( ( c_2Earithmetic_2E_3C_3D @ ( c_2Earithmetic_2E_2A @ V1m @ V0n ) @ V0n )
<=> ( ( V0n = c_2Enum_2E0 )
| ( c_2Earithmetic_2E_3C_3D @ V1m @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ) ) ).
thf(thm_2Earithmetic_2ESUB__LEFT__ADD,axiom,
! [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum,V2p: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_2B @ V0m @ ( c_2Earithmetic_2E_2D @ V1n @ V2p ) )
= ( c_2Ebool_2ECOND @ tyop_2Enum_2Enum @ ( c_2Earithmetic_2E_3C_3D @ V1n @ V2p ) @ V0m @ ( c_2Earithmetic_2E_2D @ ( c_2Earithmetic_2E_2B @ V0m @ V1n ) @ V2p ) ) ) ).
thf(thm_2Earithmetic_2ESUB__RIGHT__ADD,axiom,
! [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum,V2p: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_2B @ ( c_2Earithmetic_2E_2D @ V0m @ V1n ) @ V2p )
= ( c_2Ebool_2ECOND @ tyop_2Enum_2Enum @ ( c_2Earithmetic_2E_3C_3D @ V0m @ V1n ) @ V2p @ ( c_2Earithmetic_2E_2D @ ( c_2Earithmetic_2E_2B @ V0m @ V2p ) @ V1n ) ) ) ).
thf(thm_2Earithmetic_2ESUB__LEFT__SUB,axiom,
! [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum,V2p: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_2D @ V0m @ ( c_2Earithmetic_2E_2D @ V1n @ V2p ) )
= ( c_2Ebool_2ECOND @ tyop_2Enum_2Enum @ ( c_2Earithmetic_2E_3C_3D @ V1n @ V2p ) @ V0m @ ( c_2Earithmetic_2E_2D @ ( c_2Earithmetic_2E_2B @ V0m @ V2p ) @ V1n ) ) ) ).
thf(thm_2Earithmetic_2ESUB__RIGHT__SUB,axiom,
! [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum,V2p: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_2D @ ( c_2Earithmetic_2E_2D @ V0m @ V1n ) @ V2p )
= ( c_2Earithmetic_2E_2D @ V0m @ ( c_2Earithmetic_2E_2B @ V1n @ V2p ) ) ) ).
thf(thm_2Earithmetic_2ESUB__LEFT__SUC,axiom,
! [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum] :
( ( c_2Enum_2ESUC @ ( c_2Earithmetic_2E_2D @ V0m @ V1n ) )
= ( c_2Ebool_2ECOND @ tyop_2Enum_2Enum @ ( c_2Earithmetic_2E_3C_3D @ V0m @ V1n ) @ ( c_2Enum_2ESUC @ c_2Enum_2E0 ) @ ( c_2Earithmetic_2E_2D @ ( c_2Enum_2ESUC @ V0m ) @ V1n ) ) ) ).
thf(thm_2Earithmetic_2ESUB__LEFT__LESS__EQ,axiom,
! [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum,V2p: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_3C_3D @ V0m @ ( c_2Earithmetic_2E_2D @ V1n @ V2p ) )
<=> ( ( c_2Earithmetic_2E_3C_3D @ ( c_2Earithmetic_2E_2B @ V0m @ V2p ) @ V1n )
| ( c_2Earithmetic_2E_3C_3D @ V0m @ c_2Enum_2E0 ) ) ) ).
thf(thm_2Earithmetic_2ESUB__RIGHT__LESS__EQ,axiom,
! [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum,V2p: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_3C_3D @ ( c_2Earithmetic_2E_2D @ V0m @ V1n ) @ V2p )
= ( c_2Earithmetic_2E_3C_3D @ V0m @ ( c_2Earithmetic_2E_2B @ V1n @ V2p ) ) ) ).
thf(thm_2Earithmetic_2ESUB__LEFT__LESS,axiom,
! [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum,V2p: tyop_2Enum_2Enum] :
( ( c_2Eprim__rec_2E_3C @ V0m @ ( c_2Earithmetic_2E_2D @ V1n @ V2p ) )
= ( c_2Eprim__rec_2E_3C @ ( c_2Earithmetic_2E_2B @ V0m @ V2p ) @ V1n ) ) ).
thf(thm_2Earithmetic_2ESUB__RIGHT__LESS,axiom,
! [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum,V2p: tyop_2Enum_2Enum] :
( ( c_2Eprim__rec_2E_3C @ ( c_2Earithmetic_2E_2D @ V0m @ V1n ) @ V2p )
<=> ( ( c_2Eprim__rec_2E_3C @ V0m @ ( c_2Earithmetic_2E_2B @ V1n @ V2p ) )
& ( c_2Eprim__rec_2E_3C @ c_2Enum_2E0 @ V2p ) ) ) ).
thf(thm_2Earithmetic_2ESUB__LEFT__GREATER__EQ,axiom,
! [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum,V2p: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_3E_3D @ V0m @ ( c_2Earithmetic_2E_2D @ V1n @ V2p ) )
= ( c_2Earithmetic_2E_3E_3D @ ( c_2Earithmetic_2E_2B @ V0m @ V2p ) @ V1n ) ) ).
thf(thm_2Earithmetic_2ESUB__RIGHT__GREATER__EQ,axiom,
! [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum,V2p: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_3E_3D @ ( c_2Earithmetic_2E_2D @ V0m @ V1n ) @ V2p )
<=> ( ( c_2Earithmetic_2E_3E_3D @ V0m @ ( c_2Earithmetic_2E_2B @ V1n @ V2p ) )
| ( c_2Earithmetic_2E_3E_3D @ c_2Enum_2E0 @ V2p ) ) ) ).
thf(thm_2Earithmetic_2ESUB__LEFT__GREATER,axiom,
! [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum,V2p: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_3E @ V0m @ ( c_2Earithmetic_2E_2D @ V1n @ V2p ) )
<=> ( ( c_2Earithmetic_2E_3E @ ( c_2Earithmetic_2E_2B @ V0m @ V2p ) @ V1n )
& ( c_2Earithmetic_2E_3E @ V0m @ c_2Enum_2E0 ) ) ) ).
thf(thm_2Earithmetic_2ESUB__RIGHT__GREATER,axiom,
! [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum,V2p: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_3E @ ( c_2Earithmetic_2E_2D @ V0m @ V1n ) @ V2p )
= ( c_2Earithmetic_2E_3E @ V0m @ ( c_2Earithmetic_2E_2B @ V1n @ V2p ) ) ) ).
thf(thm_2Earithmetic_2ESUB__LEFT__EQ,axiom,
! [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum,V2p: tyop_2Enum_2Enum] :
( ( V0m
= ( c_2Earithmetic_2E_2D @ V1n @ V2p ) )
<=> ( ( ( c_2Earithmetic_2E_2B @ V0m @ V2p )
= V1n )
| ( ( c_2Earithmetic_2E_3C_3D @ V0m @ c_2Enum_2E0 )
& ( c_2Earithmetic_2E_3C_3D @ V1n @ V2p ) ) ) ) ).
thf(thm_2Earithmetic_2ESUB__RIGHT__EQ,axiom,
! [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum,V2p: tyop_2Enum_2Enum] :
( ( ( c_2Earithmetic_2E_2D @ V0m @ V1n )
= V2p )
<=> ( ( V0m
= ( c_2Earithmetic_2E_2B @ V1n @ V2p ) )
| ( ( c_2Earithmetic_2E_3C_3D @ V0m @ V1n )
& ( c_2Earithmetic_2E_3C_3D @ V2p @ c_2Enum_2E0 ) ) ) ) ).
thf(thm_2Earithmetic_2ELE,axiom,
( ! [V0n: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_3C_3D @ V0n @ c_2Enum_2E0 )
<=> ( V0n = c_2Enum_2E0 ) )
& ! [V1m: tyop_2Enum_2Enum,V2n: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_3C_3D @ V1m @ ( c_2Enum_2ESUC @ V2n ) )
<=> ( ( V1m
= ( c_2Enum_2ESUC @ V2n ) )
| ( c_2Earithmetic_2E_3C_3D @ V1m @ V2n ) ) ) ) ).
thf(thm_2Earithmetic_2EDA,axiom,
! [V0k: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum] :
( ( c_2Eprim__rec_2E_3C @ c_2Enum_2E0 @ V1n )
=> ? [V2r: tyop_2Enum_2Enum,V3q: tyop_2Enum_2Enum] :
( ( V0k
= ( c_2Earithmetic_2E_2B @ ( c_2Earithmetic_2E_2A @ V3q @ V1n ) @ V2r ) )
& ( c_2Eprim__rec_2E_3C @ V2r @ V1n ) ) ) ).
thf(thm_2Earithmetic_2EMOD__ONE,axiom,
! [V0k: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2EMOD @ V0k @ ( c_2Enum_2ESUC @ c_2Enum_2E0 ) )
= c_2Enum_2E0 ) ).
thf(thm_2Earithmetic_2EMOD__1,axiom,
! [V0k: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2EMOD @ V0k @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) )
= c_2Enum_2E0 ) ).
thf(thm_2Earithmetic_2EDIV__LESS__EQ,axiom,
! [V0n: tyop_2Enum_2Enum] :
( ( c_2Eprim__rec_2E_3C @ c_2Enum_2E0 @ V0n )
=> ! [V1k: tyop_2Enum_2Enum] : ( c_2Earithmetic_2E_3C_3D @ ( c_2Earithmetic_2EDIV @ V1k @ V0n ) @ V1k ) ) ).
thf(thm_2Earithmetic_2EDIV__UNIQUE,axiom,
! [V0n: tyop_2Enum_2Enum,V1k: tyop_2Enum_2Enum,V2q: tyop_2Enum_2Enum] :
( ? [V3r: tyop_2Enum_2Enum] :
( ( V1k
= ( c_2Earithmetic_2E_2B @ ( c_2Earithmetic_2E_2A @ V2q @ V0n ) @ V3r ) )
& ( c_2Eprim__rec_2E_3C @ V3r @ V0n ) )
=> ( ( c_2Earithmetic_2EDIV @ V1k @ V0n )
= V2q ) ) ).
thf(thm_2Earithmetic_2EMOD__UNIQUE,axiom,
! [V0n: tyop_2Enum_2Enum,V1k: tyop_2Enum_2Enum,V2r: tyop_2Enum_2Enum] :
( ? [V3q: tyop_2Enum_2Enum] :
( ( V1k
= ( c_2Earithmetic_2E_2B @ ( c_2Earithmetic_2E_2A @ V3q @ V0n ) @ V2r ) )
& ( c_2Eprim__rec_2E_3C @ V2r @ V0n ) )
=> ( ( c_2Earithmetic_2EMOD @ V1k @ V0n )
= V2r ) ) ).
thf(thm_2Earithmetic_2EDIV__MULT,axiom,
! [V0n: tyop_2Enum_2Enum,V1r: tyop_2Enum_2Enum] :
( ( c_2Eprim__rec_2E_3C @ V1r @ V0n )
=> ! [V2q: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2EDIV @ ( c_2Earithmetic_2E_2B @ ( c_2Earithmetic_2E_2A @ V2q @ V0n ) @ V1r ) @ V0n )
= V2q ) ) ).
thf(thm_2Earithmetic_2ELESS__MOD,axiom,
! [V0n: tyop_2Enum_2Enum,V1k: tyop_2Enum_2Enum] :
( ( c_2Eprim__rec_2E_3C @ V1k @ V0n )
=> ( ( c_2Earithmetic_2EMOD @ V1k @ V0n )
= V1k ) ) ).
thf(thm_2Earithmetic_2EMOD__EQ__0,axiom,
! [V0n: tyop_2Enum_2Enum] :
( ( c_2Eprim__rec_2E_3C @ c_2Enum_2E0 @ V0n )
=> ! [V1k: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2EMOD @ ( c_2Earithmetic_2E_2A @ V1k @ V0n ) @ V0n )
= c_2Enum_2E0 ) ) ).
thf(thm_2Earithmetic_2EZERO__MOD,axiom,
! [V0n: tyop_2Enum_2Enum] :
( ( c_2Eprim__rec_2E_3C @ c_2Enum_2E0 @ V0n )
=> ( ( c_2Earithmetic_2EMOD @ c_2Enum_2E0 @ V0n )
= c_2Enum_2E0 ) ) ).
thf(thm_2Earithmetic_2EZERO__DIV,axiom,
! [V0n: tyop_2Enum_2Enum] :
( ( c_2Eprim__rec_2E_3C @ c_2Enum_2E0 @ V0n )
=> ( ( c_2Earithmetic_2EDIV @ c_2Enum_2E0 @ V0n )
= c_2Enum_2E0 ) ) ).
thf(thm_2Earithmetic_2EMOD__MULT,axiom,
! [V0n: tyop_2Enum_2Enum,V1r: tyop_2Enum_2Enum] :
( ( c_2Eprim__rec_2E_3C @ V1r @ V0n )
=> ! [V2q: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2EMOD @ ( c_2Earithmetic_2E_2B @ ( c_2Earithmetic_2E_2A @ V2q @ V0n ) @ V1r ) @ V0n )
= V1r ) ) ).
thf(thm_2Earithmetic_2EMOD__TIMES,axiom,
! [V0n: tyop_2Enum_2Enum] :
( ( c_2Eprim__rec_2E_3C @ c_2Enum_2E0 @ V0n )
=> ! [V1q: tyop_2Enum_2Enum,V2r: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2EMOD @ ( c_2Earithmetic_2E_2B @ ( c_2Earithmetic_2E_2A @ V1q @ V0n ) @ V2r ) @ V0n )
= ( c_2Earithmetic_2EMOD @ V2r @ V0n ) ) ) ).
thf(thm_2Earithmetic_2EMOD__TIMES__SUB,axiom,
! [V0n: tyop_2Enum_2Enum,V1q: tyop_2Enum_2Enum,V2r: tyop_2Enum_2Enum] :
( ( ( c_2Eprim__rec_2E_3C @ c_2Enum_2E0 @ V0n )
& ( c_2Eprim__rec_2E_3C @ c_2Enum_2E0 @ V1q )
& ( c_2Earithmetic_2E_3C_3D @ V2r @ V0n ) )
=> ( ( c_2Earithmetic_2EMOD @ ( c_2Earithmetic_2E_2D @ ( c_2Earithmetic_2E_2A @ V1q @ V0n ) @ V2r ) @ V0n )
= ( c_2Earithmetic_2EMOD @ ( c_2Earithmetic_2E_2D @ V0n @ V2r ) @ V0n ) ) ) ).
thf(thm_2Earithmetic_2EMOD__PLUS,axiom,
! [V0n: tyop_2Enum_2Enum] :
( ( c_2Eprim__rec_2E_3C @ c_2Enum_2E0 @ V0n )
=> ! [V1j: tyop_2Enum_2Enum,V2k: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2EMOD @ ( c_2Earithmetic_2E_2B @ ( c_2Earithmetic_2EMOD @ V1j @ V0n ) @ ( c_2Earithmetic_2EMOD @ V2k @ V0n ) ) @ V0n )
= ( c_2Earithmetic_2EMOD @ ( c_2Earithmetic_2E_2B @ V1j @ V2k ) @ V0n ) ) ) ).
thf(thm_2Earithmetic_2EMOD__MOD,axiom,
! [V0n: tyop_2Enum_2Enum] :
( ( c_2Eprim__rec_2E_3C @ c_2Enum_2E0 @ V0n )
=> ! [V1k: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2EMOD @ ( c_2Earithmetic_2EMOD @ V1k @ V0n ) @ V0n )
= ( c_2Earithmetic_2EMOD @ V1k @ V0n ) ) ) ).
thf(thm_2Earithmetic_2ELESS__DIV__EQ__ZERO,axiom,
! [V0r: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum] :
( ( c_2Eprim__rec_2E_3C @ V0r @ V1n )
=> ( ( c_2Earithmetic_2EDIV @ V0r @ V1n )
= c_2Enum_2E0 ) ) ).
thf(thm_2Earithmetic_2EMULT__DIV,axiom,
! [V0n: tyop_2Enum_2Enum,V1q: tyop_2Enum_2Enum] :
( ( c_2Eprim__rec_2E_3C @ c_2Enum_2E0 @ V0n )
=> ( ( c_2Earithmetic_2EDIV @ ( c_2Earithmetic_2E_2A @ V1q @ V0n ) @ V0n )
= V1q ) ) ).
thf(thm_2Earithmetic_2EADD__DIV__ADD__DIV,axiom,
! [V0n: tyop_2Enum_2Enum] :
( ( c_2Eprim__rec_2E_3C @ c_2Enum_2E0 @ V0n )
=> ! [V1x: tyop_2Enum_2Enum,V2r: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2EDIV @ ( c_2Earithmetic_2E_2B @ ( c_2Earithmetic_2E_2A @ V1x @ V0n ) @ V2r ) @ V0n )
= ( c_2Earithmetic_2E_2B @ V1x @ ( c_2Earithmetic_2EDIV @ V2r @ V0n ) ) ) ) ).
thf(thm_2Earithmetic_2EADD__DIV__RWT,axiom,
! [V0n: tyop_2Enum_2Enum] :
( ( c_2Eprim__rec_2E_3C @ c_2Enum_2E0 @ V0n )
=> ! [V1m: tyop_2Enum_2Enum,V2p: tyop_2Enum_2Enum] :
( ( ( ( c_2Earithmetic_2EMOD @ V1m @ V0n )
= c_2Enum_2E0 )
| ( ( c_2Earithmetic_2EMOD @ V2p @ V0n )
= c_2Enum_2E0 ) )
=> ( ( c_2Earithmetic_2EDIV @ ( c_2Earithmetic_2E_2B @ V1m @ V2p ) @ V0n )
= ( c_2Earithmetic_2E_2B @ ( c_2Earithmetic_2EDIV @ V1m @ V0n ) @ ( c_2Earithmetic_2EDIV @ V2p @ V0n ) ) ) ) ) ).
thf(thm_2Earithmetic_2EMOD__MULT__MOD,axiom,
! [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum] :
( ( ( c_2Eprim__rec_2E_3C @ c_2Enum_2E0 @ V1n )
& ( c_2Eprim__rec_2E_3C @ c_2Enum_2E0 @ V0m ) )
=> ! [V2x: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2EMOD @ ( c_2Earithmetic_2EMOD @ V2x @ ( c_2Earithmetic_2E_2A @ V1n @ V0m ) ) @ V1n )
= ( c_2Earithmetic_2EMOD @ V2x @ V1n ) ) ) ).
thf(thm_2Earithmetic_2EDIV__ONE,axiom,
! [V0q: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2EDIV @ V0q @ ( c_2Enum_2ESUC @ c_2Enum_2E0 ) )
= V0q ) ).
thf(thm_2Earithmetic_2EDIV__1,axiom,
! [V0q: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2EDIV @ V0q @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) )
= V0q ) ).
thf(thm_2Earithmetic_2EDIVMOD__ID,axiom,
! [V0n: tyop_2Enum_2Enum] :
( ( c_2Eprim__rec_2E_3C @ c_2Enum_2E0 @ V0n )
=> ( ( ( c_2Earithmetic_2EDIV @ V0n @ V0n )
= ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) )
& ( ( c_2Earithmetic_2EMOD @ V0n @ V0n )
= c_2Enum_2E0 ) ) ) ).
thf(thm_2Earithmetic_2EDIV__DIV__DIV__MULT,axiom,
! [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum] :
( ( ( c_2Eprim__rec_2E_3C @ c_2Enum_2E0 @ V0m )
& ( c_2Eprim__rec_2E_3C @ c_2Enum_2E0 @ V1n ) )
=> ! [V2x: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2EDIV @ ( c_2Earithmetic_2EDIV @ V2x @ V0m ) @ V1n )
= ( c_2Earithmetic_2EDIV @ V2x @ ( c_2Earithmetic_2E_2A @ V0m @ V1n ) ) ) ) ).
thf(thm_2Earithmetic_2ESUC__PRE,axiom,
! [V0m: tyop_2Enum_2Enum] :
( ( c_2Eprim__rec_2E_3C @ c_2Enum_2E0 @ V0m )
<=> ( ( c_2Enum_2ESUC @ ( c_2Eprim__rec_2EPRE @ V0m ) )
= V0m ) ) ).
thf(thm_2Earithmetic_2EDIV__LESS,axiom,
! [V0n: tyop_2Enum_2Enum,V1d: tyop_2Enum_2Enum] :
( ( ( c_2Eprim__rec_2E_3C @ c_2Enum_2E0 @ V0n )
& ( c_2Eprim__rec_2E_3C @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) @ V1d ) )
=> ( c_2Eprim__rec_2E_3C @ ( c_2Earithmetic_2EDIV @ V0n @ V1d ) @ V0n ) ) ).
thf(thm_2Earithmetic_2EMOD__LESS,axiom,
! [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum] :
( ( c_2Eprim__rec_2E_3C @ c_2Enum_2E0 @ V1n )
=> ( c_2Eprim__rec_2E_3C @ ( c_2Earithmetic_2EMOD @ V0m @ V1n ) @ V1n ) ) ).
thf(thm_2Earithmetic_2EADD__MODULUS,axiom,
( ! [V0n: tyop_2Enum_2Enum,V1x: tyop_2Enum_2Enum] :
( ( c_2Eprim__rec_2E_3C @ c_2Enum_2E0 @ V0n )
=> ( ( c_2Earithmetic_2EMOD @ ( c_2Earithmetic_2E_2B @ V1x @ V0n ) @ V0n )
= ( c_2Earithmetic_2EMOD @ V1x @ V0n ) ) )
& ! [V2n: tyop_2Enum_2Enum,V3x: tyop_2Enum_2Enum] :
( ( c_2Eprim__rec_2E_3C @ c_2Enum_2E0 @ V2n )
=> ( ( c_2Earithmetic_2EMOD @ ( c_2Earithmetic_2E_2B @ V2n @ V3x ) @ V2n )
= ( c_2Earithmetic_2EMOD @ V3x @ V2n ) ) ) ) ).
thf(thm_2Earithmetic_2EADD__MODULUS__LEFT,axiom,
! [V0n: tyop_2Enum_2Enum,V1x: tyop_2Enum_2Enum] :
( ( c_2Eprim__rec_2E_3C @ c_2Enum_2E0 @ V0n )
=> ( ( c_2Earithmetic_2EMOD @ ( c_2Earithmetic_2E_2B @ V1x @ V0n ) @ V0n )
= ( c_2Earithmetic_2EMOD @ V1x @ V0n ) ) ) ).
thf(thm_2Earithmetic_2EADD__MODULUS__RIGHT,axiom,
! [V0n: tyop_2Enum_2Enum,V1x: tyop_2Enum_2Enum] :
( ( c_2Eprim__rec_2E_3C @ c_2Enum_2E0 @ V0n )
=> ( ( c_2Earithmetic_2EMOD @ ( c_2Earithmetic_2E_2B @ V0n @ V1x ) @ V0n )
= ( c_2Earithmetic_2EMOD @ V1x @ V0n ) ) ) ).
thf(thm_2Earithmetic_2EDIV__P,axiom,
! [V0P: tyop_2Enum_2Enum > $o,V1p: tyop_2Enum_2Enum,V2q: tyop_2Enum_2Enum] :
( ( c_2Eprim__rec_2E_3C @ c_2Enum_2E0 @ V2q )
=> ( ( V0P @ ( c_2Earithmetic_2EDIV @ V1p @ V2q ) )
<=> ? [V3k: tyop_2Enum_2Enum,V4r: tyop_2Enum_2Enum] :
( ( V1p
= ( c_2Earithmetic_2E_2B @ ( c_2Earithmetic_2E_2A @ V3k @ V2q ) @ V4r ) )
& ( c_2Eprim__rec_2E_3C @ V4r @ V2q )
& ( V0P @ V3k ) ) ) ) ).
thf(thm_2Earithmetic_2EDIV__P__UNIV,axiom,
! [V0P: tyop_2Enum_2Enum > $o,V1m: tyop_2Enum_2Enum,V2n: tyop_2Enum_2Enum] :
( ( c_2Eprim__rec_2E_3C @ c_2Enum_2E0 @ V2n )
=> ( ( V0P @ ( c_2Earithmetic_2EDIV @ V1m @ V2n ) )
<=> ! [V3q: tyop_2Enum_2Enum,V4r: tyop_2Enum_2Enum] :
( ( ( V1m
= ( c_2Earithmetic_2E_2B @ ( c_2Earithmetic_2E_2A @ V3q @ V2n ) @ V4r ) )
& ( c_2Eprim__rec_2E_3C @ V4r @ V2n ) )
=> ( V0P @ V3q ) ) ) ) ).
thf(thm_2Earithmetic_2EMOD__P,axiom,
! [V0P: tyop_2Enum_2Enum > $o,V1p: tyop_2Enum_2Enum,V2q: tyop_2Enum_2Enum] :
( ( c_2Eprim__rec_2E_3C @ c_2Enum_2E0 @ V2q )
=> ( ( V0P @ ( c_2Earithmetic_2EMOD @ V1p @ V2q ) )
<=> ? [V3k: tyop_2Enum_2Enum,V4r: tyop_2Enum_2Enum] :
( ( V1p
= ( c_2Earithmetic_2E_2B @ ( c_2Earithmetic_2E_2A @ V3k @ V2q ) @ V4r ) )
& ( c_2Eprim__rec_2E_3C @ V4r @ V2q )
& ( V0P @ V4r ) ) ) ) ).
thf(thm_2Earithmetic_2EMOD__P__UNIV,axiom,
! [V0P: tyop_2Enum_2Enum > $o,V1m: tyop_2Enum_2Enum,V2n: tyop_2Enum_2Enum] :
( ( c_2Eprim__rec_2E_3C @ c_2Enum_2E0 @ V2n )
=> ( ( V0P @ ( c_2Earithmetic_2EMOD @ V1m @ V2n ) )
<=> ! [V3q: tyop_2Enum_2Enum,V4r: tyop_2Enum_2Enum] :
( ( ( V1m
= ( c_2Earithmetic_2E_2B @ ( c_2Earithmetic_2E_2A @ V3q @ V2n ) @ V4r ) )
& ( c_2Eprim__rec_2E_3C @ V4r @ V2n ) )
=> ( V0P @ V4r ) ) ) ) ).
thf(thm_2Earithmetic_2EMOD__TIMES2,axiom,
! [V0n: tyop_2Enum_2Enum] :
( ( c_2Eprim__rec_2E_3C @ c_2Enum_2E0 @ V0n )
=> ! [V1j: tyop_2Enum_2Enum,V2k: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2EMOD @ ( c_2Earithmetic_2E_2A @ ( c_2Earithmetic_2EMOD @ V1j @ V0n ) @ ( c_2Earithmetic_2EMOD @ V2k @ V0n ) ) @ V0n )
= ( c_2Earithmetic_2EMOD @ ( c_2Earithmetic_2E_2A @ V1j @ V2k ) @ V0n ) ) ) ).
thf(thm_2Earithmetic_2EMOD__COMMON__FACTOR,axiom,
! [V0n: tyop_2Enum_2Enum,V1p: tyop_2Enum_2Enum,V2q: tyop_2Enum_2Enum] :
( ( ( c_2Eprim__rec_2E_3C @ c_2Enum_2E0 @ V0n )
& ( c_2Eprim__rec_2E_3C @ c_2Enum_2E0 @ V2q ) )
=> ( ( c_2Earithmetic_2E_2A @ V0n @ ( c_2Earithmetic_2EMOD @ V1p @ V2q ) )
= ( c_2Earithmetic_2EMOD @ ( c_2Earithmetic_2E_2A @ V0n @ V1p ) @ ( c_2Earithmetic_2E_2A @ V0n @ V2q ) ) ) ) ).
thf(thm_2Earithmetic_2EX__MOD__Y__EQ__X,axiom,
! [V0x: tyop_2Enum_2Enum,V1y: tyop_2Enum_2Enum] :
( ( c_2Eprim__rec_2E_3C @ c_2Enum_2E0 @ V1y )
=> ( ( ( c_2Earithmetic_2EMOD @ V0x @ V1y )
= V0x )
<=> ( c_2Eprim__rec_2E_3C @ V0x @ V1y ) ) ) ).
thf(thm_2Earithmetic_2EDIV__LE__MONOTONE,axiom,
! [V0n: tyop_2Enum_2Enum,V1x: tyop_2Enum_2Enum,V2y: tyop_2Enum_2Enum] :
( ( ( c_2Eprim__rec_2E_3C @ c_2Enum_2E0 @ V0n )
& ( c_2Earithmetic_2E_3C_3D @ V1x @ V2y ) )
=> ( c_2Earithmetic_2E_3C_3D @ ( c_2Earithmetic_2EDIV @ V1x @ V0n ) @ ( c_2Earithmetic_2EDIV @ V2y @ V0n ) ) ) ).
thf(thm_2Earithmetic_2ELE__LT1,axiom,
! [V0x: tyop_2Enum_2Enum,V1y: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_3C_3D @ V0x @ V1y )
= ( c_2Eprim__rec_2E_3C @ V0x @ ( c_2Earithmetic_2E_2B @ V1y @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ) ).
thf(thm_2Earithmetic_2EX__LE__DIV,axiom,
! [V0x: tyop_2Enum_2Enum,V1y: tyop_2Enum_2Enum,V2z: tyop_2Enum_2Enum] :
( ( c_2Eprim__rec_2E_3C @ c_2Enum_2E0 @ V2z )
=> ( ( c_2Earithmetic_2E_3C_3D @ V0x @ ( c_2Earithmetic_2EDIV @ V1y @ V2z ) )
= ( c_2Earithmetic_2E_3C_3D @ ( c_2Earithmetic_2E_2A @ V0x @ V2z ) @ V1y ) ) ) ).
thf(thm_2Earithmetic_2EX__LT__DIV,axiom,
! [V0x: tyop_2Enum_2Enum,V1y: tyop_2Enum_2Enum,V2z: tyop_2Enum_2Enum] :
( ( c_2Eprim__rec_2E_3C @ c_2Enum_2E0 @ V2z )
=> ( ( c_2Eprim__rec_2E_3C @ V0x @ ( c_2Earithmetic_2EDIV @ V1y @ V2z ) )
= ( c_2Earithmetic_2E_3C_3D @ ( c_2Earithmetic_2E_2A @ ( c_2Earithmetic_2E_2B @ V0x @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) @ V2z ) @ V1y ) ) ) ).
thf(thm_2Earithmetic_2EDIV__LT__X,axiom,
! [V0x: tyop_2Enum_2Enum,V1y: tyop_2Enum_2Enum,V2z: tyop_2Enum_2Enum] :
( ( c_2Eprim__rec_2E_3C @ c_2Enum_2E0 @ V2z )
=> ( ( c_2Eprim__rec_2E_3C @ ( c_2Earithmetic_2EDIV @ V1y @ V2z ) @ V0x )
= ( c_2Eprim__rec_2E_3C @ V1y @ ( c_2Earithmetic_2E_2A @ V0x @ V2z ) ) ) ) ).
thf(thm_2Earithmetic_2EDIV__LE__X,axiom,
! [V0x: tyop_2Enum_2Enum,V1y: tyop_2Enum_2Enum,V2z: tyop_2Enum_2Enum] :
( ( c_2Eprim__rec_2E_3C @ c_2Enum_2E0 @ V2z )
=> ( ( c_2Earithmetic_2E_3C_3D @ ( c_2Earithmetic_2EDIV @ V1y @ V2z ) @ V0x )
= ( c_2Eprim__rec_2E_3C @ V1y @ ( c_2Earithmetic_2E_2A @ ( c_2Earithmetic_2E_2B @ V0x @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) @ V2z ) ) ) ) ).
thf(thm_2Earithmetic_2EDIV__EQ__X,axiom,
! [V0x: tyop_2Enum_2Enum,V1y: tyop_2Enum_2Enum,V2z: tyop_2Enum_2Enum] :
( ( c_2Eprim__rec_2E_3C @ c_2Enum_2E0 @ V2z )
=> ( ( ( c_2Earithmetic_2EDIV @ V1y @ V2z )
= V0x )
<=> ( ( c_2Earithmetic_2E_3C_3D @ ( c_2Earithmetic_2E_2A @ V0x @ V2z ) @ V1y )
& ( c_2Eprim__rec_2E_3C @ V1y @ ( c_2Earithmetic_2E_2A @ ( c_2Enum_2ESUC @ V0x ) @ V2z ) ) ) ) ) ).
thf(thm_2Earithmetic_2EDIV__MOD__MOD__DIV,axiom,
! [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum,V2k: tyop_2Enum_2Enum] :
( ( ( c_2Eprim__rec_2E_3C @ c_2Enum_2E0 @ V1n )
& ( c_2Eprim__rec_2E_3C @ c_2Enum_2E0 @ V2k ) )
=> ( ( c_2Earithmetic_2EMOD @ ( c_2Earithmetic_2EDIV @ V0m @ V1n ) @ V2k )
= ( c_2Earithmetic_2EDIV @ ( c_2Earithmetic_2EMOD @ V0m @ ( c_2Earithmetic_2E_2A @ V1n @ V2k ) ) @ V1n ) ) ) ).
thf(thm_2Earithmetic_2EMULT__EQ__DIV,axiom,
! [V0z: tyop_2Enum_2Enum,V1y: tyop_2Enum_2Enum,V2x: tyop_2Enum_2Enum] :
( ( c_2Eprim__rec_2E_3C @ c_2Enum_2E0 @ V2x )
=> ( ( ( c_2Earithmetic_2E_2A @ V2x @ V1y )
= V0z )
<=> ( ( V1y
= ( c_2Earithmetic_2EDIV @ V0z @ V2x ) )
& ( ( c_2Earithmetic_2EMOD @ V0z @ V2x )
= c_2Enum_2E0 ) ) ) ) ).
thf(thm_2Earithmetic_2ENUMERAL__MULT__EQ__DIV,axiom,
! [V0z: tyop_2Enum_2Enum,V1y: tyop_2Enum_2Enum,V2x: tyop_2Enum_2Enum] :
( ( ( ( c_2Earithmetic_2E_2A @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ V2x ) ) @ V1y )
= ( c_2Earithmetic_2ENUMERAL @ V0z ) )
<=> ( ( V1y
= ( c_2Earithmetic_2EDIV @ ( c_2Earithmetic_2ENUMERAL @ V0z ) @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ V2x ) ) ) )
& ( ( c_2Earithmetic_2EMOD @ ( c_2Earithmetic_2ENUMERAL @ V0z ) @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ V2x ) ) )
= c_2Enum_2E0 ) ) )
& ( ( ( c_2Earithmetic_2E_2A @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ V2x ) ) @ V1y )
= ( c_2Earithmetic_2ENUMERAL @ V0z ) )
<=> ( ( V1y
= ( c_2Earithmetic_2EDIV @ ( c_2Earithmetic_2ENUMERAL @ V0z ) @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ V2x ) ) ) )
& ( ( c_2Earithmetic_2EMOD @ ( c_2Earithmetic_2ENUMERAL @ V0z ) @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ V2x ) ) )
= c_2Enum_2E0 ) ) ) ) ).
thf(thm_2Earithmetic_2EMOD__EQ__0__DIVISOR,axiom,
! [V0n: tyop_2Enum_2Enum,V1k: tyop_2Enum_2Enum] :
( ( c_2Eprim__rec_2E_3C @ c_2Enum_2E0 @ V0n )
=> ( ( ( c_2Earithmetic_2EMOD @ V1k @ V0n )
= c_2Enum_2E0 )
<=> ? [V2d: tyop_2Enum_2Enum] :
( V1k
= ( c_2Earithmetic_2E_2A @ V2d @ V0n ) ) ) ) ).
thf(thm_2Earithmetic_2EMOD__SUC,axiom,
! [V0y: tyop_2Enum_2Enum,V1x: tyop_2Enum_2Enum] :
( ( ( c_2Eprim__rec_2E_3C @ c_2Enum_2E0 @ V0y )
& ( (~)
@ ( ( c_2Enum_2ESUC @ V1x )
= ( c_2Earithmetic_2E_2A @ ( c_2Enum_2ESUC @ ( c_2Earithmetic_2EDIV @ V1x @ V0y ) ) @ V0y ) ) ) )
=> ( ( c_2Earithmetic_2EMOD @ ( c_2Enum_2ESUC @ V1x ) @ V0y )
= ( c_2Enum_2ESUC @ ( c_2Earithmetic_2EMOD @ V1x @ V0y ) ) ) ) ).
thf(thm_2Earithmetic_2EMOD__SUC__IFF,axiom,
! [V0y: tyop_2Enum_2Enum,V1x: tyop_2Enum_2Enum] :
( ( c_2Eprim__rec_2E_3C @ c_2Enum_2E0 @ V0y )
=> ( ( ( c_2Earithmetic_2EMOD @ ( c_2Enum_2ESUC @ V1x ) @ V0y )
= ( c_2Enum_2ESUC @ ( c_2Earithmetic_2EMOD @ V1x @ V0y ) ) )
<=> ( (~)
@ ( ( c_2Enum_2ESUC @ V1x )
= ( c_2Earithmetic_2E_2A @ ( c_2Enum_2ESUC @ ( c_2Earithmetic_2EDIV @ V1x @ V0y ) ) @ V0y ) ) ) ) ) ).
thf(thm_2Earithmetic_2EONE__MOD,axiom,
! [V0n: tyop_2Enum_2Enum] :
( ( c_2Eprim__rec_2E_3C @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) @ V0n )
=> ( ( c_2Earithmetic_2EMOD @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) @ V0n )
= ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ).
thf(thm_2Earithmetic_2EONE__MOD__IFF,axiom,
! [V0n: tyop_2Enum_2Enum] :
( ( c_2Eprim__rec_2E_3C @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) @ V0n )
<=> ( ( c_2Eprim__rec_2E_3C @ c_2Enum_2E0 @ V0n )
& ( ( c_2Earithmetic_2EMOD @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) @ V0n )
= ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ) ).
thf(thm_2Earithmetic_2EMOD__LESS__EQ,axiom,
! [V0y: tyop_2Enum_2Enum,V1x: tyop_2Enum_2Enum] :
( ( c_2Eprim__rec_2E_3C @ c_2Enum_2E0 @ V0y )
=> ( c_2Earithmetic_2E_3C_3D @ ( c_2Earithmetic_2EMOD @ V1x @ V0y ) @ V1x ) ) ).
thf(thm_2Earithmetic_2EMOD__LIFT__PLUS,axiom,
! [V0x: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum,V2k: tyop_2Enum_2Enum] :
( ( ( c_2Eprim__rec_2E_3C @ c_2Enum_2E0 @ V1n )
& ( c_2Eprim__rec_2E_3C @ V2k @ ( c_2Earithmetic_2E_2D @ V1n @ ( c_2Earithmetic_2EMOD @ V0x @ V1n ) ) ) )
=> ( ( c_2Earithmetic_2EMOD @ ( c_2Earithmetic_2E_2B @ V0x @ V2k ) @ V1n )
= ( c_2Earithmetic_2E_2B @ ( c_2Earithmetic_2EMOD @ V0x @ V1n ) @ V2k ) ) ) ).
thf(thm_2Earithmetic_2EMOD__LIFT__PLUS__IFF,axiom,
! [V0x: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum,V2k: tyop_2Enum_2Enum] :
( ( c_2Eprim__rec_2E_3C @ c_2Enum_2E0 @ V1n )
=> ( ( ( c_2Earithmetic_2EMOD @ ( c_2Earithmetic_2E_2B @ V0x @ V2k ) @ V1n )
= ( c_2Earithmetic_2E_2B @ ( c_2Earithmetic_2EMOD @ V0x @ V1n ) @ V2k ) )
<=> ( c_2Eprim__rec_2E_3C @ V2k @ ( c_2Earithmetic_2E_2D @ V1n @ ( c_2Earithmetic_2EMOD @ V0x @ V1n ) ) ) ) ) ).
thf(thm_2Earithmetic_2Enum__case__cong,axiom,
! [A_27a: $tType,V0v_27: A_27a,V1f_27: tyop_2Enum_2Enum > A_27a,V2M: tyop_2Enum_2Enum,V3M_27: tyop_2Enum_2Enum,V4v: A_27a,V5f: tyop_2Enum_2Enum > A_27a] :
( ( ( V2M = V3M_27 )
& ( ( V3M_27 = c_2Enum_2E0 )
=> ( V4v = V0v_27 ) )
& ! [V6n: tyop_2Enum_2Enum] :
( ( V3M_27
= ( c_2Enum_2ESUC @ V6n ) )
=> ( ( V5f @ V6n )
= ( V1f_27 @ V6n ) ) ) )
=> ( ( c_2Earithmetic_2Enum__CASE @ A_27a @ V2M @ V4v @ V5f )
= ( c_2Earithmetic_2Enum__CASE @ A_27a @ V3M_27 @ V0v_27 @ V1f_27 ) ) ) ).
thf(thm_2Earithmetic_2ESUC__ELIM__THM,axiom,
! [V0P: tyop_2Enum_2Enum > tyop_2Enum_2Enum > $o] :
( ! [V1n: tyop_2Enum_2Enum] : ( V0P @ ( c_2Enum_2ESUC @ V1n ) @ V1n )
<=> ! [V2n: tyop_2Enum_2Enum] :
( ( c_2Eprim__rec_2E_3C @ c_2Enum_2E0 @ V2n )
=> ( V0P @ V2n @ ( c_2Earithmetic_2E_2D @ V2n @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ) ) ).
thf(thm_2Earithmetic_2ESUC__ELIM__NUMERALS,axiom,
! [A_27a: $tType,V0f: tyop_2Enum_2Enum > tyop_2Enum_2Enum > A_27a,V1g: tyop_2Enum_2Enum > A_27a] :
( ! [V2n: tyop_2Enum_2Enum] :
( ( V1g @ ( c_2Enum_2ESUC @ V2n ) )
= ( V0f @ V2n @ ( c_2Enum_2ESUC @ V2n ) ) )
<=> ( ! [V3n: tyop_2Enum_2Enum] :
( ( V1g @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ V3n ) ) )
= ( V0f @ ( c_2Earithmetic_2E_2D @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ V3n ) ) @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ V3n ) ) ) )
& ! [V4n: tyop_2Enum_2Enum] :
( ( V1g @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ V4n ) ) )
= ( V0f @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ V4n ) ) @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ V4n ) ) ) ) ) ) ).
thf(thm_2Earithmetic_2ESUB__ELIM__THM,axiom,
! [V0b: tyop_2Enum_2Enum,V1a: tyop_2Enum_2Enum,V2P: tyop_2Enum_2Enum > $o] :
( ( V2P @ ( c_2Earithmetic_2E_2D @ V1a @ V0b ) )
<=> ! [V3d: tyop_2Enum_2Enum] :
( ( ( V0b
= ( c_2Earithmetic_2E_2B @ V1a @ V3d ) )
=> ( V2P @ c_2Enum_2E0 ) )
& ( ( V1a
= ( c_2Earithmetic_2E_2B @ V0b @ V3d ) )
=> ( V2P @ V3d ) ) ) ) ).
thf(thm_2Earithmetic_2EPRE__ELIM__THM,axiom,
! [V0n: tyop_2Enum_2Enum,V1P: tyop_2Enum_2Enum > $o] :
( ( V1P @ ( c_2Eprim__rec_2EPRE @ V0n ) )
<=> ! [V2m: tyop_2Enum_2Enum] :
( ( ( V0n = c_2Enum_2E0 )
=> ( V1P @ c_2Enum_2E0 ) )
& ( ( V0n
= ( c_2Enum_2ESUC @ V2m ) )
=> ( V1P @ V2m ) ) ) ) ).
thf(thm_2Earithmetic_2EMULT__INCREASES,axiom,
! [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum] :
( ( ( c_2Eprim__rec_2E_3C @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) @ V0m )
& ( c_2Eprim__rec_2E_3C @ c_2Enum_2E0 @ V1n ) )
=> ( c_2Earithmetic_2E_3C_3D @ ( c_2Enum_2ESUC @ V1n ) @ ( c_2Earithmetic_2E_2A @ V0m @ V1n ) ) ) ).
thf(thm_2Earithmetic_2EEXP__ALWAYS__BIG__ENOUGH,axiom,
! [V0b: tyop_2Enum_2Enum] :
( ( c_2Eprim__rec_2E_3C @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) @ V0b )
=> ! [V1n: tyop_2Enum_2Enum] :
? [V2m: tyop_2Enum_2Enum] : ( c_2Earithmetic_2E_3C_3D @ V1n @ ( c_2Earithmetic_2EEXP @ V0b @ V2m ) ) ) ).
thf(thm_2Earithmetic_2EEXP__EQ__0,axiom,
! [V0n: tyop_2Enum_2Enum,V1m: tyop_2Enum_2Enum] :
( ( ( c_2Earithmetic_2EEXP @ V0n @ V1m )
= c_2Enum_2E0 )
<=> ( ( V0n = c_2Enum_2E0 )
& ( c_2Eprim__rec_2E_3C @ c_2Enum_2E0 @ V1m ) ) ) ).
thf(thm_2Earithmetic_2EZERO__LT__EXP,axiom,
! [V0y: tyop_2Enum_2Enum,V1x: tyop_2Enum_2Enum] :
( ( c_2Eprim__rec_2E_3C @ c_2Enum_2E0 @ ( c_2Earithmetic_2EEXP @ V1x @ V0y ) )
<=> ( ( c_2Eprim__rec_2E_3C @ c_2Enum_2E0 @ V1x )
| ( V0y = c_2Enum_2E0 ) ) ) ).
thf(thm_2Earithmetic_2EEXP__1,axiom,
! [V0n: tyop_2Enum_2Enum] :
( ( ( c_2Earithmetic_2EEXP @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) @ V0n )
= ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) )
& ( ( c_2Earithmetic_2EEXP @ V0n @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) )
= V0n ) ) ).
thf(thm_2Earithmetic_2EEXP__EQ__1,axiom,
! [V0n: tyop_2Enum_2Enum,V1m: tyop_2Enum_2Enum] :
( ( ( c_2Earithmetic_2EEXP @ V0n @ V1m )
= ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) )
<=> ( ( V0n
= ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) )
| ( V1m = c_2Enum_2E0 ) ) ) ).
thf(thm_2Earithmetic_2EEXP__BASE__LE__MONO,axiom,
! [V0b: tyop_2Enum_2Enum] :
( ( c_2Eprim__rec_2E_3C @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) @ V0b )
=> ! [V1n: tyop_2Enum_2Enum,V2m: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_3C_3D @ ( c_2Earithmetic_2EEXP @ V0b @ V2m ) @ ( c_2Earithmetic_2EEXP @ V0b @ V1n ) )
= ( c_2Earithmetic_2E_3C_3D @ V2m @ V1n ) ) ) ).
thf(thm_2Earithmetic_2EEXP__BASE__LT__MONO,axiom,
! [V0b: tyop_2Enum_2Enum] :
( ( c_2Eprim__rec_2E_3C @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) @ V0b )
=> ! [V1n: tyop_2Enum_2Enum,V2m: tyop_2Enum_2Enum] :
( ( c_2Eprim__rec_2E_3C @ ( c_2Earithmetic_2EEXP @ V0b @ V2m ) @ ( c_2Earithmetic_2EEXP @ V0b @ V1n ) )
= ( c_2Eprim__rec_2E_3C @ V2m @ V1n ) ) ) ).
thf(thm_2Earithmetic_2EEXP__BASE__INJECTIVE,axiom,
! [V0b: tyop_2Enum_2Enum] :
( ( c_2Eprim__rec_2E_3C @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) @ V0b )
=> ! [V1n: tyop_2Enum_2Enum,V2m: tyop_2Enum_2Enum] :
( ( ( c_2Earithmetic_2EEXP @ V0b @ V1n )
= ( c_2Earithmetic_2EEXP @ V0b @ V2m ) )
<=> ( V1n = V2m ) ) ) ).
thf(thm_2Earithmetic_2EEXP__BASE__LEQ__MONO__IMP,axiom,
! [V0n: tyop_2Enum_2Enum,V1m: tyop_2Enum_2Enum,V2b: tyop_2Enum_2Enum] :
( ( ( c_2Eprim__rec_2E_3C @ c_2Enum_2E0 @ V2b )
& ( c_2Earithmetic_2E_3C_3D @ V1m @ V0n ) )
=> ( c_2Earithmetic_2E_3C_3D @ ( c_2Earithmetic_2EEXP @ V2b @ V1m ) @ ( c_2Earithmetic_2EEXP @ V2b @ V0n ) ) ) ).
thf(thm_2Earithmetic_2EEXP__BASE__LEQ__MONO__SUC__IMP,axiom,
! [V0n: tyop_2Enum_2Enum,V1m: tyop_2Enum_2Enum,V2b: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_3C_3D @ V1m @ V0n )
=> ( c_2Earithmetic_2E_3C_3D @ ( c_2Earithmetic_2EEXP @ ( c_2Enum_2ESUC @ V2b ) @ V1m ) @ ( c_2Earithmetic_2EEXP @ ( c_2Enum_2ESUC @ V2b ) @ V0n ) ) ) ).
thf(thm_2Earithmetic_2EEXP__BASE__LE__IFF,axiom,
! [V0n: tyop_2Enum_2Enum,V1m: tyop_2Enum_2Enum,V2b: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_3C_3D @ ( c_2Earithmetic_2EEXP @ V2b @ V1m ) @ ( c_2Earithmetic_2EEXP @ V2b @ V0n ) )
<=> ( ( ( V2b = c_2Enum_2E0 )
& ( V0n = c_2Enum_2E0 ) )
| ( ( V2b = c_2Enum_2E0 )
& ( c_2Eprim__rec_2E_3C @ c_2Enum_2E0 @ V1m ) )
| ( V2b
= ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) )
| ( ( c_2Eprim__rec_2E_3C @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) @ V2b )
& ( c_2Earithmetic_2E_3C_3D @ V1m @ V0n ) ) ) ) ).
thf(thm_2Earithmetic_2EX__LE__X__EXP,axiom,
! [V0x: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum] :
( ( c_2Eprim__rec_2E_3C @ c_2Enum_2E0 @ V1n )
=> ( c_2Earithmetic_2E_3C_3D @ V0x @ ( c_2Earithmetic_2EEXP @ V0x @ V1n ) ) ) ).
thf(thm_2Earithmetic_2EX__LT__EXP__X,axiom,
! [V0x: tyop_2Enum_2Enum,V1b: tyop_2Enum_2Enum] :
( ( c_2Eprim__rec_2E_3C @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) @ V1b )
=> ( c_2Eprim__rec_2E_3C @ V0x @ ( c_2Earithmetic_2EEXP @ V1b @ V0x ) ) ) ).
thf(thm_2Earithmetic_2EZERO__EXP,axiom,
! [V0x: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2EEXP @ c_2Enum_2E0 @ V0x )
= ( c_2Ebool_2ECOND @ tyop_2Enum_2Enum @ ( c_2Emin_2E_3D @ tyop_2Enum_2Enum @ V0x @ c_2Enum_2E0 ) @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) @ c_2Enum_2E0 ) ) ).
thf(thm_2Earithmetic_2EX__LT__EXP__X__IFF,axiom,
! [V0x: tyop_2Enum_2Enum,V1b: tyop_2Enum_2Enum] :
( ( c_2Eprim__rec_2E_3C @ V0x @ ( c_2Earithmetic_2EEXP @ V1b @ V0x ) )
<=> ( ( c_2Eprim__rec_2E_3C @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) @ V1b )
| ( V0x = c_2Enum_2E0 ) ) ) ).
thf(thm_2Earithmetic_2EEXP__EXP__LT__MONO,axiom,
! [V0n: tyop_2Enum_2Enum,V1a: tyop_2Enum_2Enum,V2b: tyop_2Enum_2Enum] :
( ( c_2Eprim__rec_2E_3C @ ( c_2Earithmetic_2EEXP @ V1a @ V0n ) @ ( c_2Earithmetic_2EEXP @ V2b @ V0n ) )
<=> ( ( c_2Eprim__rec_2E_3C @ V1a @ V2b )
& ( c_2Eprim__rec_2E_3C @ c_2Enum_2E0 @ V0n ) ) ) ).
thf(thm_2Earithmetic_2EEXP__EXP__LE__MONO,axiom,
! [V0n: tyop_2Enum_2Enum,V1a: tyop_2Enum_2Enum,V2b: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_3C_3D @ ( c_2Earithmetic_2EEXP @ V1a @ V0n ) @ ( c_2Earithmetic_2EEXP @ V2b @ V0n ) )
<=> ( ( c_2Earithmetic_2E_3C_3D @ V1a @ V2b )
| ( V0n = c_2Enum_2E0 ) ) ) ).
thf(thm_2Earithmetic_2EEXP__EXP__INJECTIVE,axiom,
! [V0b1: tyop_2Enum_2Enum,V1b2: tyop_2Enum_2Enum,V2x: tyop_2Enum_2Enum] :
( ( ( c_2Earithmetic_2EEXP @ V0b1 @ V2x )
= ( c_2Earithmetic_2EEXP @ V1b2 @ V2x ) )
<=> ( ( V2x = c_2Enum_2E0 )
| ( V0b1 = V1b2 ) ) ) ).
thf(thm_2Earithmetic_2EEXP__SUB,axiom,
! [V0p: tyop_2Enum_2Enum,V1q: tyop_2Enum_2Enum,V2n: tyop_2Enum_2Enum] :
( ( ( c_2Eprim__rec_2E_3C @ c_2Enum_2E0 @ V2n )
& ( c_2Earithmetic_2E_3C_3D @ V1q @ V0p ) )
=> ( ( c_2Earithmetic_2EEXP @ V2n @ ( c_2Earithmetic_2E_2D @ V0p @ V1q ) )
= ( c_2Earithmetic_2EDIV @ ( c_2Earithmetic_2EEXP @ V2n @ V0p ) @ ( c_2Earithmetic_2EEXP @ V2n @ V1q ) ) ) ) ).
thf(thm_2Earithmetic_2EEXP__SUB__NUMERAL,axiom,
! [V0x: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum] :
( ( c_2Eprim__rec_2E_3C @ c_2Enum_2E0 @ V1n )
=> ( ( ( c_2Earithmetic_2EDIV @ ( c_2Earithmetic_2EEXP @ V1n @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ V0x ) ) ) @ V1n )
= ( c_2Earithmetic_2EEXP @ V1n @ ( c_2Earithmetic_2E_2D @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ V0x ) ) @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) )
& ( ( c_2Earithmetic_2EDIV @ ( c_2Earithmetic_2EEXP @ V1n @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ V0x ) ) ) @ V1n )
= ( c_2Earithmetic_2EEXP @ V1n @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ V0x ) ) ) ) ) ) ).
thf(thm_2Earithmetic_2EEXP__BASE__MULT,axiom,
! [V0z: tyop_2Enum_2Enum,V1x: tyop_2Enum_2Enum,V2y: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2EEXP @ ( c_2Earithmetic_2E_2A @ V1x @ V2y ) @ V0z )
= ( c_2Earithmetic_2E_2A @ ( c_2Earithmetic_2EEXP @ V1x @ V0z ) @ ( c_2Earithmetic_2EEXP @ V2y @ V0z ) ) ) ).
thf(thm_2Earithmetic_2EEXP__EXP__MULT,axiom,
! [V0z: tyop_2Enum_2Enum,V1x: tyop_2Enum_2Enum,V2y: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2EEXP @ V1x @ ( c_2Earithmetic_2E_2A @ V2y @ V0z ) )
= ( c_2Earithmetic_2EEXP @ ( c_2Earithmetic_2EEXP @ V1x @ V2y ) @ V0z ) ) ).
thf(thm_2Earithmetic_2EMAX__COMM,axiom,
! [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2EMAX @ V0m @ V1n )
= ( c_2Earithmetic_2EMAX @ V1n @ V0m ) ) ).
thf(thm_2Earithmetic_2EMIN__COMM,axiom,
! [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2EMIN @ V0m @ V1n )
= ( c_2Earithmetic_2EMIN @ V1n @ V0m ) ) ).
thf(thm_2Earithmetic_2EMAX__ASSOC,axiom,
! [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum,V2p: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2EMAX @ V0m @ ( c_2Earithmetic_2EMAX @ V1n @ V2p ) )
= ( c_2Earithmetic_2EMAX @ ( c_2Earithmetic_2EMAX @ V0m @ V1n ) @ V2p ) ) ).
thf(thm_2Earithmetic_2EMIN__ASSOC,axiom,
! [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum,V2p: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2EMIN @ V0m @ ( c_2Earithmetic_2EMIN @ V1n @ V2p ) )
= ( c_2Earithmetic_2EMIN @ ( c_2Earithmetic_2EMIN @ V0m @ V1n ) @ V2p ) ) ).
thf(thm_2Earithmetic_2EMIN__MAX__EQ,axiom,
! [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum] :
( ( ( c_2Earithmetic_2EMIN @ V0m @ V1n )
= ( c_2Earithmetic_2EMAX @ V0m @ V1n ) )
<=> ( V0m = V1n ) ) ).
thf(thm_2Earithmetic_2EMIN__MAX__LT,axiom,
! [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum] :
( ( c_2Eprim__rec_2E_3C @ ( c_2Earithmetic_2EMIN @ V0m @ V1n ) @ ( c_2Earithmetic_2EMAX @ V0m @ V1n ) )
<=> ( (~) @ ( V0m = V1n ) ) ) ).
thf(thm_2Earithmetic_2EMIN__MAX__LE,axiom,
! [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum] : ( c_2Earithmetic_2E_3C_3D @ ( c_2Earithmetic_2EMIN @ V0m @ V1n ) @ ( c_2Earithmetic_2EMAX @ V0m @ V1n ) ) ).
thf(thm_2Earithmetic_2EMIN__MAX__PRED,axiom,
! [V0P: tyop_2Enum_2Enum > $o,V1m: tyop_2Enum_2Enum,V2n: tyop_2Enum_2Enum] :
( ( ( V0P @ V1m )
& ( V0P @ V2n ) )
=> ( ( V0P @ ( c_2Earithmetic_2EMIN @ V1m @ V2n ) )
& ( V0P @ ( c_2Earithmetic_2EMAX @ V1m @ V2n ) ) ) ) ).
thf(thm_2Earithmetic_2EMIN__LT,axiom,
! [V0n: tyop_2Enum_2Enum,V1m: tyop_2Enum_2Enum,V2p: tyop_2Enum_2Enum] :
( ( ( c_2Eprim__rec_2E_3C @ ( c_2Earithmetic_2EMIN @ V1m @ V0n ) @ V2p )
<=> ( ( c_2Eprim__rec_2E_3C @ V1m @ V2p )
| ( c_2Eprim__rec_2E_3C @ V0n @ V2p ) ) )
& ( ( c_2Eprim__rec_2E_3C @ V2p @ ( c_2Earithmetic_2EMIN @ V1m @ V0n ) )
<=> ( ( c_2Eprim__rec_2E_3C @ V2p @ V1m )
& ( c_2Eprim__rec_2E_3C @ V2p @ V0n ) ) ) ) ).
thf(thm_2Earithmetic_2EMAX__LT,axiom,
! [V0n: tyop_2Enum_2Enum,V1m: tyop_2Enum_2Enum,V2p: tyop_2Enum_2Enum] :
( ( ( c_2Eprim__rec_2E_3C @ V2p @ ( c_2Earithmetic_2EMAX @ V1m @ V0n ) )
<=> ( ( c_2Eprim__rec_2E_3C @ V2p @ V1m )
| ( c_2Eprim__rec_2E_3C @ V2p @ V0n ) ) )
& ( ( c_2Eprim__rec_2E_3C @ ( c_2Earithmetic_2EMAX @ V1m @ V0n ) @ V2p )
<=> ( ( c_2Eprim__rec_2E_3C @ V1m @ V2p )
& ( c_2Eprim__rec_2E_3C @ V0n @ V2p ) ) ) ) ).
thf(thm_2Earithmetic_2EMIN__LE,axiom,
! [V0n: tyop_2Enum_2Enum,V1m: tyop_2Enum_2Enum,V2p: tyop_2Enum_2Enum] :
( ( ( c_2Earithmetic_2E_3C_3D @ ( c_2Earithmetic_2EMIN @ V1m @ V0n ) @ V2p )
<=> ( ( c_2Earithmetic_2E_3C_3D @ V1m @ V2p )
| ( c_2Earithmetic_2E_3C_3D @ V0n @ V2p ) ) )
& ( ( c_2Earithmetic_2E_3C_3D @ V2p @ ( c_2Earithmetic_2EMIN @ V1m @ V0n ) )
<=> ( ( c_2Earithmetic_2E_3C_3D @ V2p @ V1m )
& ( c_2Earithmetic_2E_3C_3D @ V2p @ V0n ) ) ) ) ).
thf(thm_2Earithmetic_2EMAX__LE,axiom,
! [V0n: tyop_2Enum_2Enum,V1m: tyop_2Enum_2Enum,V2p: tyop_2Enum_2Enum] :
( ( ( c_2Earithmetic_2E_3C_3D @ V2p @ ( c_2Earithmetic_2EMAX @ V1m @ V0n ) )
<=> ( ( c_2Earithmetic_2E_3C_3D @ V2p @ V1m )
| ( c_2Earithmetic_2E_3C_3D @ V2p @ V0n ) ) )
& ( ( c_2Earithmetic_2E_3C_3D @ ( c_2Earithmetic_2EMAX @ V1m @ V0n ) @ V2p )
<=> ( ( c_2Earithmetic_2E_3C_3D @ V1m @ V2p )
& ( c_2Earithmetic_2E_3C_3D @ V0n @ V2p ) ) ) ) ).
thf(thm_2Earithmetic_2EMIN__0,axiom,
! [V0n: tyop_2Enum_2Enum] :
( ( ( c_2Earithmetic_2EMIN @ V0n @ c_2Enum_2E0 )
= c_2Enum_2E0 )
& ( ( c_2Earithmetic_2EMIN @ c_2Enum_2E0 @ V0n )
= c_2Enum_2E0 ) ) ).
thf(thm_2Earithmetic_2EMAX__0,axiom,
! [V0n: tyop_2Enum_2Enum] :
( ( ( c_2Earithmetic_2EMAX @ V0n @ c_2Enum_2E0 )
= V0n )
& ( ( c_2Earithmetic_2EMAX @ c_2Enum_2E0 @ V0n )
= V0n ) ) ).
thf(thm_2Earithmetic_2EMAX__EQ__0,axiom,
! [V0n: tyop_2Enum_2Enum,V1m: tyop_2Enum_2Enum] :
( ( ( c_2Earithmetic_2EMAX @ V1m @ V0n )
= c_2Enum_2E0 )
<=> ( ( V1m = c_2Enum_2E0 )
& ( V0n = c_2Enum_2E0 ) ) ) ).
thf(thm_2Earithmetic_2EMIN__EQ__0,axiom,
! [V0n: tyop_2Enum_2Enum,V1m: tyop_2Enum_2Enum] :
( ( ( c_2Earithmetic_2EMIN @ V1m @ V0n )
= c_2Enum_2E0 )
<=> ( ( V1m = c_2Enum_2E0 )
| ( V0n = c_2Enum_2E0 ) ) ) ).
thf(thm_2Earithmetic_2EMIN__IDEM,axiom,
! [V0n: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2EMIN @ V0n @ V0n )
= V0n ) ).
thf(thm_2Earithmetic_2EMAX__IDEM,axiom,
! [V0n: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2EMAX @ V0n @ V0n )
= V0n ) ).
thf(thm_2Earithmetic_2EEXISTS__GREATEST,axiom,
! [V0P: tyop_2Enum_2Enum > $o] :
( ( ? [V1x: tyop_2Enum_2Enum] : ( V0P @ V1x )
& ? [V2x: tyop_2Enum_2Enum] :
! [V3y: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_3E @ V3y @ V2x )
=> ( (~) @ ( V0P @ V3y ) ) ) )
<=> ? [V4x: tyop_2Enum_2Enum] :
( ( V0P @ V4x )
& ! [V5y: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_3E @ V5y @ V4x )
=> ( (~) @ ( V0P @ V5y ) ) ) ) ) ).
thf(thm_2Earithmetic_2EEXISTS__NUM,axiom,
! [V0P: tyop_2Enum_2Enum > $o] :
( ? [V1n: tyop_2Enum_2Enum] : ( V0P @ V1n )
<=> ( ( V0P @ c_2Enum_2E0 )
| ? [V2m: tyop_2Enum_2Enum] : ( V0P @ ( c_2Enum_2ESUC @ V2m ) ) ) ) ).
thf(thm_2Earithmetic_2EFORALL__NUM,axiom,
! [V0P: tyop_2Enum_2Enum > $o] :
( ! [V1n: tyop_2Enum_2Enum] : ( V0P @ V1n )
<=> ( ( V0P @ c_2Enum_2E0 )
& ! [V2n: tyop_2Enum_2Enum] : ( V0P @ ( c_2Enum_2ESUC @ V2n ) ) ) ) ).
thf(thm_2Earithmetic_2EBOUNDED__FORALL__THM,axiom,
! [V0P: tyop_2Enum_2Enum > $o,V1c: tyop_2Enum_2Enum] :
( ( c_2Eprim__rec_2E_3C @ c_2Enum_2E0 @ V1c )
=> ( ! [V2n: tyop_2Enum_2Enum] :
( ( c_2Eprim__rec_2E_3C @ V2n @ V1c )
=> ( V0P @ V2n ) )
<=> ( ( V0P @ ( c_2Earithmetic_2E_2D @ V1c @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) )
& ! [V3n: tyop_2Enum_2Enum] :
( ( c_2Eprim__rec_2E_3C @ V3n @ ( c_2Earithmetic_2E_2D @ V1c @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) )
=> ( V0P @ V3n ) ) ) ) ) ).
thf(thm_2Earithmetic_2EBOUNDED__EXISTS__THM,axiom,
! [V0P: tyop_2Enum_2Enum > $o,V1c: tyop_2Enum_2Enum] :
( ( c_2Eprim__rec_2E_3C @ c_2Enum_2E0 @ V1c )
=> ( ? [V2n: tyop_2Enum_2Enum] :
( ( c_2Eprim__rec_2E_3C @ V2n @ V1c )
& ( V0P @ V2n ) )
<=> ( ( V0P @ ( c_2Earithmetic_2E_2D @ V1c @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) )
| ? [V3n: tyop_2Enum_2Enum] :
( ( c_2Eprim__rec_2E_3C @ V3n @ ( c_2Earithmetic_2E_2D @ V1c @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) )
& ( V0P @ V3n ) ) ) ) ) ).
thf(thm_2Earithmetic_2Etransitive__monotone,axiom,
! [A_27a: $tType,V0R: A_27a > A_27a > $o,V1f: tyop_2Enum_2Enum > A_27a] :
( ( ( c_2Erelation_2Etransitive @ A_27a @ V0R )
& ! [V2n: tyop_2Enum_2Enum] : ( V0R @ ( V1f @ V2n ) @ ( V1f @ ( c_2Enum_2ESUC @ V2n ) ) ) )
=> ! [V3m: tyop_2Enum_2Enum,V4n: tyop_2Enum_2Enum] :
( ( c_2Eprim__rec_2E_3C @ V3m @ V4n )
=> ( V0R @ ( V1f @ V3m ) @ ( V1f @ V4n ) ) ) ) ).
thf(thm_2Earithmetic_2ESTRICTLY__INCREASING__TC,axiom,
! [V0f: tyop_2Enum_2Enum > tyop_2Enum_2Enum] :
( ! [V1n: tyop_2Enum_2Enum] : ( c_2Eprim__rec_2E_3C @ ( V0f @ V1n ) @ ( V0f @ ( c_2Enum_2ESUC @ V1n ) ) )
=> ! [V2m: tyop_2Enum_2Enum,V3n: tyop_2Enum_2Enum] :
( ( c_2Eprim__rec_2E_3C @ V2m @ V3n )
=> ( c_2Eprim__rec_2E_3C @ ( V0f @ V2m ) @ ( V0f @ V3n ) ) ) ) ).
thf(thm_2Earithmetic_2ESTRICTLY__INCREASING__ONE__ONE,axiom,
! [V0f: tyop_2Enum_2Enum > tyop_2Enum_2Enum] :
( ! [V1n: tyop_2Enum_2Enum] : ( c_2Eprim__rec_2E_3C @ ( V0f @ V1n ) @ ( V0f @ ( c_2Enum_2ESUC @ V1n ) ) )
=> ( c_2Ebool_2EONE__ONE @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum @ V0f ) ) ).
thf(thm_2Earithmetic_2EONE__ONE__INV__IMAGE__BOUNDED,axiom,
! [V0f: tyop_2Enum_2Enum > tyop_2Enum_2Enum] :
( ( c_2Ebool_2EONE__ONE @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum @ V0f )
=> ! [V1b: tyop_2Enum_2Enum] :
? [V2a: tyop_2Enum_2Enum] :
! [V3x: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_3C_3D @ ( V0f @ V3x ) @ V1b )
=> ( c_2Earithmetic_2E_3C_3D @ V3x @ V2a ) ) ) ).
thf(thm_2Earithmetic_2EONE__ONE__UNBOUNDED,axiom,
! [V0f: tyop_2Enum_2Enum > tyop_2Enum_2Enum] :
( ( c_2Ebool_2EONE__ONE @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum @ V0f )
=> ! [V1b: tyop_2Enum_2Enum] :
? [V2n: tyop_2Enum_2Enum] : ( c_2Eprim__rec_2E_3C @ V1b @ ( V0f @ V2n ) ) ) ).
thf(thm_2Earithmetic_2ESTRICTLY__INCREASING__UNBOUNDED,axiom,
! [V0f: tyop_2Enum_2Enum > tyop_2Enum_2Enum] :
( ! [V1n: tyop_2Enum_2Enum] : ( c_2Eprim__rec_2E_3C @ ( V0f @ V1n ) @ ( V0f @ ( c_2Enum_2ESUC @ V1n ) ) )
=> ! [V2b: tyop_2Enum_2Enum] :
? [V3n: tyop_2Enum_2Enum] : ( c_2Eprim__rec_2E_3C @ V2b @ ( V0f @ V3n ) ) ) ).
thf(thm_2Earithmetic_2ENOT__STRICTLY__DECREASING,axiom,
! [V0f: tyop_2Enum_2Enum > tyop_2Enum_2Enum] :
( (~)
@ ! [V1n: tyop_2Enum_2Enum] : ( c_2Eprim__rec_2E_3C @ ( V0f @ ( c_2Enum_2ESUC @ V1n ) ) @ ( V0f @ V1n ) ) ) ).
thf(thm_2Earithmetic_2EABS__DIFF__SYM,axiom,
! [V0n: tyop_2Enum_2Enum,V1m: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2EABS__DIFF @ V0n @ V1m )
= ( c_2Earithmetic_2EABS__DIFF @ V1m @ V0n ) ) ).
thf(thm_2Earithmetic_2EABS__DIFF__COMM,axiom,
! [V0n: tyop_2Enum_2Enum,V1m: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2EABS__DIFF @ V0n @ V1m )
= ( c_2Earithmetic_2EABS__DIFF @ V1m @ V0n ) ) ).
thf(thm_2Earithmetic_2EABS__DIFF__EQS,axiom,
! [V0n: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2EABS__DIFF @ V0n @ V0n )
= c_2Enum_2E0 ) ).
thf(thm_2Earithmetic_2EABS__DIFF__EQ__0,axiom,
! [V0n: tyop_2Enum_2Enum,V1m: tyop_2Enum_2Enum] :
( ( ( c_2Earithmetic_2EABS__DIFF @ V0n @ V1m )
= c_2Enum_2E0 )
<=> ( V0n = V1m ) ) ).
thf(thm_2Earithmetic_2EABS__DIFF__ZERO,axiom,
! [V0n: tyop_2Enum_2Enum] :
( ( ( c_2Earithmetic_2EABS__DIFF @ V0n @ c_2Enum_2E0 )
= V0n )
& ( ( c_2Earithmetic_2EABS__DIFF @ c_2Enum_2E0 @ V0n )
= V0n ) ) ).
thf(thm_2Earithmetic_2EABS__DIFF__SUC,axiom,
! [V0n: tyop_2Enum_2Enum,V1m: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2EABS__DIFF @ ( c_2Enum_2ESUC @ V0n ) @ ( c_2Enum_2ESUC @ V1m ) )
= ( c_2Earithmetic_2EABS__DIFF @ V0n @ V1m ) ) ).
thf(thm_2Earithmetic_2EABS__DIFF__SUC__LE,axiom,
! [V0x: tyop_2Enum_2Enum,V1z: tyop_2Enum_2Enum] : ( c_2Earithmetic_2E_3C_3D @ ( c_2Earithmetic_2EABS__DIFF @ V0x @ ( c_2Enum_2ESUC @ V1z ) ) @ ( c_2Enum_2ESUC @ ( c_2Earithmetic_2EABS__DIFF @ V0x @ V1z ) ) ) ).
thf(thm_2Earithmetic_2EABS__DIFF__PLUS__LE,axiom,
! [V0x: tyop_2Enum_2Enum,V1z: tyop_2Enum_2Enum,V2y: tyop_2Enum_2Enum] : ( c_2Earithmetic_2E_3C_3D @ ( c_2Earithmetic_2EABS__DIFF @ V0x @ ( c_2Earithmetic_2E_2B @ V2y @ V1z ) ) @ ( c_2Earithmetic_2E_2B @ V2y @ ( c_2Earithmetic_2EABS__DIFF @ V0x @ V1z ) ) ) ).
thf(thm_2Earithmetic_2EABS__DIFF__LE__SUM,axiom,
! [V0z: tyop_2Enum_2Enum,V1x: tyop_2Enum_2Enum] : ( c_2Earithmetic_2E_3C_3D @ ( c_2Earithmetic_2EABS__DIFF @ V1x @ V0z ) @ ( c_2Earithmetic_2E_2B @ V1x @ V0z ) ) ).
thf(thm_2Earithmetic_2EABS__DIFF__TRIANGLE__lem,axiom,
! [V0x: tyop_2Enum_2Enum,V1y: tyop_2Enum_2Enum] : ( c_2Earithmetic_2E_3C_3D @ V0x @ ( c_2Earithmetic_2E_2B @ ( c_2Earithmetic_2EABS__DIFF @ V0x @ V1y ) @ V1y ) ) ).
thf(thm_2Earithmetic_2EABS__DIFF__TRIANGLE,axiom,
! [V0x: tyop_2Enum_2Enum,V1y: tyop_2Enum_2Enum,V2z: tyop_2Enum_2Enum] : ( c_2Earithmetic_2E_3C_3D @ ( c_2Earithmetic_2EABS__DIFF @ V0x @ V2z ) @ ( c_2Earithmetic_2E_2B @ ( c_2Earithmetic_2EABS__DIFF @ V0x @ V1y ) @ ( c_2Earithmetic_2EABS__DIFF @ V1y @ V2z ) ) ) ).
thf(thm_2Earithmetic_2EABS__DIFF__ADD__SAME,axiom,
! [V0n: tyop_2Enum_2Enum,V1m: tyop_2Enum_2Enum,V2p: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2EABS__DIFF @ ( c_2Earithmetic_2E_2B @ V0n @ V2p ) @ ( c_2Earithmetic_2E_2B @ V1m @ V2p ) )
= ( c_2Earithmetic_2EABS__DIFF @ V0n @ V1m ) ) ).
thf(thm_2Earithmetic_2ELE__SUB__RCANCEL,axiom,
! [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum,V2p: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_3C_3D @ ( c_2Earithmetic_2E_2D @ V1n @ V0m ) @ ( c_2Earithmetic_2E_2D @ V2p @ V0m ) )
<=> ( ( c_2Earithmetic_2E_3C_3D @ V1n @ V0m )
| ( c_2Earithmetic_2E_3C_3D @ V1n @ V2p ) ) ) ).
thf(thm_2Earithmetic_2ELT__SUB__RCANCEL,axiom,
! [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum,V2p: tyop_2Enum_2Enum] :
( ( c_2Eprim__rec_2E_3C @ ( c_2Earithmetic_2E_2D @ V1n @ V0m ) @ ( c_2Earithmetic_2E_2D @ V2p @ V0m ) )
<=> ( ( c_2Eprim__rec_2E_3C @ V1n @ V2p )
& ( c_2Eprim__rec_2E_3C @ V0m @ V2p ) ) ) ).
thf(thm_2Earithmetic_2ELE__SUB__LCANCEL,axiom,
! [V0z: tyop_2Enum_2Enum,V1y: tyop_2Enum_2Enum,V2x: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_3C_3D @ ( c_2Earithmetic_2E_2D @ V2x @ V1y ) @ ( c_2Earithmetic_2E_2D @ V2x @ V0z ) )
<=> ( ( c_2Earithmetic_2E_3C_3D @ V0z @ V1y )
| ( c_2Earithmetic_2E_3C_3D @ V2x @ V1y ) ) ) ).
thf(thm_2Earithmetic_2ELT__SUB__LCANCEL,axiom,
! [V0z: tyop_2Enum_2Enum,V1y: tyop_2Enum_2Enum,V2x: tyop_2Enum_2Enum] :
( ( c_2Eprim__rec_2E_3C @ ( c_2Earithmetic_2E_2D @ V2x @ V1y ) @ ( c_2Earithmetic_2E_2D @ V2x @ V0z ) )
<=> ( ( c_2Eprim__rec_2E_3C @ V0z @ V1y )
& ( c_2Eprim__rec_2E_3C @ V0z @ V2x ) ) ) ).
thf(thm_2Earithmetic_2EABS__DIFF__SUMS,axiom,
! [V0n1: tyop_2Enum_2Enum,V1n2: tyop_2Enum_2Enum,V2m1: tyop_2Enum_2Enum,V3m2: tyop_2Enum_2Enum] : ( c_2Earithmetic_2E_3C_3D @ ( c_2Earithmetic_2EABS__DIFF @ ( c_2Earithmetic_2E_2B @ V0n1 @ V1n2 ) @ ( c_2Earithmetic_2E_2B @ V2m1 @ V3m2 ) ) @ ( c_2Earithmetic_2E_2B @ ( c_2Earithmetic_2EABS__DIFF @ V0n1 @ V2m1 ) @ ( c_2Earithmetic_2EABS__DIFF @ V1n2 @ V3m2 ) ) ) ).
thf(thm_2Earithmetic_2EFUNPOW__SUC,axiom,
! [A_27a: $tType,V0f: A_27a > A_27a,V1n: tyop_2Enum_2Enum,V2x: A_27a] :
( ( c_2Earithmetic_2EFUNPOW @ A_27a @ V0f @ ( c_2Enum_2ESUC @ V1n ) @ V2x )
= ( V0f @ ( c_2Earithmetic_2EFUNPOW @ A_27a @ V0f @ V1n @ V2x ) ) ) ).
thf(thm_2Earithmetic_2EFUNPOW__0,axiom,
! [A_27a: $tType,V0x: A_27a,V1f: A_27a > A_27a] :
( ( c_2Earithmetic_2EFUNPOW @ A_27a @ V1f @ c_2Enum_2E0 @ V0x )
= V0x ) ).
thf(thm_2Earithmetic_2EFUNPOW__ADD,axiom,
! [A_27a: $tType,V0x: A_27a,V1f: A_27a > A_27a,V2m: tyop_2Enum_2Enum,V3n: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2EFUNPOW @ A_27a @ V1f @ ( c_2Earithmetic_2E_2B @ V2m @ V3n ) @ V0x )
= ( c_2Earithmetic_2EFUNPOW @ A_27a @ V1f @ V2m @ ( c_2Earithmetic_2EFUNPOW @ A_27a @ V1f @ V3n @ V0x ) ) ) ).
thf(thm_2Earithmetic_2EFUNPOW__1,axiom,
! [A_27a: $tType,V0x: A_27a,V1f: A_27a > A_27a] :
( ( c_2Earithmetic_2EFUNPOW @ A_27a @ V1f @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) @ V0x )
= ( V1f @ V0x ) ) ).
thf(thm_2Earithmetic_2ENRC__0,axiom,
! [A_27a: $tType,V0R: A_27a > A_27a > $o,V1x: A_27a,V2y: A_27a] :
( ( c_2Earithmetic_2ENRC @ A_27a @ V0R @ c_2Enum_2E0 @ V1x @ V2y )
<=> ( V1x = V2y ) ) ).
thf(thm_2Earithmetic_2ENRC__1,axiom,
! [A_27a: $tType,V0y: A_27a,V1x: A_27a,V2R: A_27a > A_27a > $o] :
( ( c_2Earithmetic_2ENRC @ A_27a @ V2R @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) @ V1x @ V0y )
= ( V2R @ V1x @ V0y ) ) ).
thf(thm_2Earithmetic_2ENRC__ADD__I,axiom,
! [A_27a: $tType,V0R: A_27a > A_27a > $o,V1m: tyop_2Enum_2Enum,V2n: tyop_2Enum_2Enum,V3x: A_27a,V4y: A_27a,V5z: A_27a] :
( ( ( c_2Earithmetic_2ENRC @ A_27a @ V0R @ V1m @ V3x @ V4y )
& ( c_2Earithmetic_2ENRC @ A_27a @ V0R @ V2n @ V4y @ V5z ) )
=> ( c_2Earithmetic_2ENRC @ A_27a @ V0R @ ( c_2Earithmetic_2E_2B @ V1m @ V2n ) @ V3x @ V5z ) ) ).
thf(thm_2Earithmetic_2ENRC__ADD__E,axiom,
! [A_27a: $tType,V0R: A_27a > A_27a > $o,V1m: tyop_2Enum_2Enum,V2n: tyop_2Enum_2Enum,V3x: A_27a,V4z: A_27a] :
( ( c_2Earithmetic_2ENRC @ A_27a @ V0R @ ( c_2Earithmetic_2E_2B @ V1m @ V2n ) @ V3x @ V4z )
=> ? [V5y: A_27a] :
( ( c_2Earithmetic_2ENRC @ A_27a @ V0R @ V1m @ V3x @ V5y )
& ( c_2Earithmetic_2ENRC @ A_27a @ V0R @ V2n @ V5y @ V4z ) ) ) ).
thf(thm_2Earithmetic_2ENRC__ADD__EQN,axiom,
! [A_27a: $tType,V0z: A_27a,V1x: A_27a,V2n: tyop_2Enum_2Enum,V3m: tyop_2Enum_2Enum,V4R: A_27a > A_27a > $o] :
( ( c_2Earithmetic_2ENRC @ A_27a @ V4R @ ( c_2Earithmetic_2E_2B @ V3m @ V2n ) @ V1x @ V0z )
<=> ? [V5y: A_27a] :
( ( c_2Earithmetic_2ENRC @ A_27a @ V4R @ V3m @ V1x @ V5y )
& ( c_2Earithmetic_2ENRC @ A_27a @ V4R @ V2n @ V5y @ V0z ) ) ) ).
thf(thm_2Earithmetic_2ENRC__SUC__RECURSE__LEFT,axiom,
! [A_27a: $tType,V0y: A_27a,V1x: A_27a,V2n: tyop_2Enum_2Enum,V3R: A_27a > A_27a > $o] :
( ( c_2Earithmetic_2ENRC @ A_27a @ V3R @ ( c_2Enum_2ESUC @ V2n ) @ V1x @ V0y )
<=> ? [V4z: A_27a] :
( ( c_2Earithmetic_2ENRC @ A_27a @ V3R @ V2n @ V1x @ V4z )
& ( V3R @ V4z @ V0y ) ) ) ).
thf(thm_2Earithmetic_2ENRC__RTC,axiom,
! [A_27a: $tType,V0R: A_27a > A_27a > $o,V1n: tyop_2Enum_2Enum,V2x: A_27a,V3y: A_27a] :
( ( c_2Earithmetic_2ENRC @ A_27a @ V0R @ V1n @ V2x @ V3y )
=> ( c_2Erelation_2ERTC @ A_27a @ V0R @ V2x @ V3y ) ) ).
thf(thm_2Earithmetic_2ERTC__NRC,axiom,
! [A_27a: $tType,V0R: A_27a > A_27a > $o,V1x: A_27a,V2y: A_27a] :
( ( c_2Erelation_2ERTC @ A_27a @ V0R @ V1x @ V2y )
=> ? [V3n: tyop_2Enum_2Enum] : ( c_2Earithmetic_2ENRC @ A_27a @ V0R @ V3n @ V1x @ V2y ) ) ).
thf(thm_2Earithmetic_2ERTC__eq__NRC,axiom,
! [A_27a: $tType,V0R: A_27a > A_27a > $o,V1x: A_27a,V2y: A_27a] :
( ( c_2Erelation_2ERTC @ A_27a @ V0R @ V1x @ V2y )
<=> ? [V3n: tyop_2Enum_2Enum] : ( c_2Earithmetic_2ENRC @ A_27a @ V0R @ V3n @ V1x @ V2y ) ) ).
thf(thm_2Earithmetic_2ETC__eq__NRC,axiom,
! [A_27a: $tType,V0R: A_27a > A_27a > $o,V1x: A_27a,V2y: A_27a] :
( ( c_2Erelation_2ETC @ A_27a @ V0R @ V1x @ V2y )
<=> ? [V3n: tyop_2Enum_2Enum] : ( c_2Earithmetic_2ENRC @ A_27a @ V0R @ ( c_2Enum_2ESUC @ V3n ) @ V1x @ V2y ) ) ).
thf(thm_2Earithmetic_2ELESS__EQUAL__DIFF,axiom,
! [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_3C_3D @ V0m @ V1n )
=> ? [V2k: tyop_2Enum_2Enum] :
( V0m
= ( c_2Earithmetic_2E_2D @ V1n @ V2k ) ) ) ).
thf(thm_2Earithmetic_2EMOD__2,axiom,
! [V0n: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2EMOD @ V0n @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) )
= ( c_2Ebool_2ECOND @ tyop_2Enum_2Enum @ ( c_2Earithmetic_2EEVEN @ V0n ) @ c_2Enum_2E0 @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ).
thf(thm_2Earithmetic_2EEVEN__MOD2,axiom,
! [V0x: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2EEVEN @ V0x )
<=> ( ( c_2Earithmetic_2EMOD @ V0x @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) )
= c_2Enum_2E0 ) ) ).
thf(thm_2Earithmetic_2ESUC__MOD,axiom,
! [V0n: tyop_2Enum_2Enum,V1a: tyop_2Enum_2Enum,V2b: tyop_2Enum_2Enum] :
( ( c_2Eprim__rec_2E_3C @ c_2Enum_2E0 @ V0n )
=> ( ( ( c_2Earithmetic_2EMOD @ ( c_2Enum_2ESUC @ V1a ) @ V0n )
= ( c_2Earithmetic_2EMOD @ ( c_2Enum_2ESUC @ V2b ) @ V0n ) )
<=> ( ( c_2Earithmetic_2EMOD @ V1a @ V0n )
= ( c_2Earithmetic_2EMOD @ V2b @ V0n ) ) ) ) ).
thf(thm_2Earithmetic_2EADD__MOD,axiom,
! [V0n: tyop_2Enum_2Enum,V1a: tyop_2Enum_2Enum,V2b: tyop_2Enum_2Enum,V3p: tyop_2Enum_2Enum] :
( ( c_2Eprim__rec_2E_3C @ c_2Enum_2E0 @ V0n )
=> ( ( ( c_2Earithmetic_2EMOD @ ( c_2Earithmetic_2E_2B @ V1a @ V3p ) @ V0n )
= ( c_2Earithmetic_2EMOD @ ( c_2Earithmetic_2E_2B @ V2b @ V3p ) @ V0n ) )
<=> ( ( c_2Earithmetic_2EMOD @ V1a @ V0n )
= ( c_2Earithmetic_2EMOD @ V2b @ V0n ) ) ) ) ).
thf(thm_2Earithmetic_2EMOD__ELIM,axiom,
! [V0P: tyop_2Enum_2Enum > $o,V1x: tyop_2Enum_2Enum,V2n: tyop_2Enum_2Enum] :
( ( ( c_2Eprim__rec_2E_3C @ c_2Enum_2E0 @ V2n )
& ( V0P @ V1x )
& ! [V3y: tyop_2Enum_2Enum] :
( ( V0P @ ( c_2Earithmetic_2E_2B @ V3y @ V2n ) )
=> ( V0P @ V3y ) ) )
=> ( V0P @ ( c_2Earithmetic_2EMOD @ V1x @ V2n ) ) ) ).
thf(thm_2Earithmetic_2EDOUBLE__LT,axiom,
! [V0p: tyop_2Enum_2Enum,V1q: tyop_2Enum_2Enum] :
( ( c_2Eprim__rec_2E_3C @ ( c_2Earithmetic_2E_2B @ ( c_2Earithmetic_2E_2A @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) @ V0p ) @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) @ ( c_2Earithmetic_2E_2A @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) @ V1q ) )
= ( c_2Eprim__rec_2E_3C @ ( c_2Earithmetic_2E_2A @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) @ V0p ) @ ( c_2Earithmetic_2E_2A @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) @ V1q ) ) ) ).
thf(thm_2Earithmetic_2EEXP2__LT,axiom,
! [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum] :
( ( c_2Eprim__rec_2E_3C @ ( c_2Earithmetic_2EDIV @ V1n @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) @ ( c_2Earithmetic_2EEXP @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) @ V0m ) )
= ( c_2Eprim__rec_2E_3C @ V1n @ ( c_2Earithmetic_2EEXP @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) @ ( c_2Enum_2ESUC @ V0m ) ) ) ) ).
thf(thm_2Earithmetic_2ESUB__LESS,axiom,
! [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum] :
( ( ( c_2Eprim__rec_2E_3C @ c_2Enum_2E0 @ V1n )
& ( c_2Earithmetic_2E_3C_3D @ V1n @ V0m ) )
=> ( c_2Eprim__rec_2E_3C @ ( c_2Earithmetic_2E_2D @ V0m @ V1n ) @ V0m ) ) ).
thf(thm_2Earithmetic_2ESUB__MOD,axiom,
! [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum] :
( ( ( c_2Eprim__rec_2E_3C @ c_2Enum_2E0 @ V1n )
& ( c_2Earithmetic_2E_3C_3D @ V1n @ V0m ) )
=> ( ( c_2Earithmetic_2EMOD @ ( c_2Earithmetic_2E_2D @ V0m @ V1n ) @ V1n )
= ( c_2Earithmetic_2EMOD @ V0m @ V1n ) ) ) ).
thf(thm_2Earithmetic_2EONE__LT__MULT__IMP,axiom,
! [V0p: tyop_2Enum_2Enum,V1q: tyop_2Enum_2Enum] :
( ( ( c_2Eprim__rec_2E_3C @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) @ V0p )
& ( c_2Eprim__rec_2E_3C @ c_2Enum_2E0 @ V1q ) )
=> ( c_2Eprim__rec_2E_3C @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) @ ( c_2Earithmetic_2E_2A @ V0p @ V1q ) ) ) ).
thf(thm_2Earithmetic_2EONE__LT__MULT,axiom,
! [V0x: tyop_2Enum_2Enum,V1y: tyop_2Enum_2Enum] :
( ( c_2Eprim__rec_2E_3C @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) @ ( c_2Earithmetic_2E_2A @ V0x @ V1y ) )
<=> ( ( ( c_2Eprim__rec_2E_3C @ c_2Enum_2E0 @ V0x )
& ( c_2Eprim__rec_2E_3C @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) @ V1y ) )
| ( ( c_2Eprim__rec_2E_3C @ c_2Enum_2E0 @ V1y )
& ( c_2Eprim__rec_2E_3C @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) @ V0x ) ) ) ) ).
thf(thm_2Earithmetic_2EONE__LT__EXP,axiom,
! [V0x: tyop_2Enum_2Enum,V1y: tyop_2Enum_2Enum] :
( ( c_2Eprim__rec_2E_3C @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) @ ( c_2Earithmetic_2EEXP @ V0x @ V1y ) )
<=> ( ( c_2Eprim__rec_2E_3C @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) @ V0x )
& ( c_2Eprim__rec_2E_3C @ c_2Enum_2E0 @ V1y ) ) ) ).
thf(thm_2Earithmetic_2Efindq__thm,axiom,
! [V0n: tyop_2Enum_2Enum,V1m: tyop_2Enum_2Enum,V2a: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2Efindq @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum ) @ V2a @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum @ V1m @ V0n ) ) )
= ( c_2Ebool_2ECOND @ tyop_2Enum_2Enum @ ( c_2Emin_2E_3D @ tyop_2Enum_2Enum @ V0n @ c_2Enum_2E0 ) @ V2a
@ ( c_2Ebool_2ELET @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum
@ ^ [V3d: tyop_2Enum_2Enum] : ( c_2Ebool_2ECOND @ tyop_2Enum_2Enum @ ( c_2Eprim__rec_2E_3C @ V1m @ V3d ) @ V2a @ ( c_2Earithmetic_2Efindq @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum ) @ ( c_2Earithmetic_2E_2A @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) @ V2a ) @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum @ V1m @ V3d ) ) ) )
@ ( c_2Earithmetic_2E_2A @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) @ V0n ) ) ) ) ).
thf(thm_2Earithmetic_2Efindq__eq__0,axiom,
! [V0a: tyop_2Enum_2Enum,V1m: tyop_2Enum_2Enum,V2n: tyop_2Enum_2Enum] :
( ( ( c_2Earithmetic_2Efindq @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum ) @ V0a @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum @ V1m @ V2n ) ) )
= c_2Enum_2E0 )
<=> ( V0a = c_2Enum_2E0 ) ) ).
thf(thm_2Earithmetic_2Efindq__divisor,axiom,
! [V0n: tyop_2Enum_2Enum,V1m: tyop_2Enum_2Enum,V2a: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_3C_3D @ V0n @ V1m )
=> ( c_2Earithmetic_2E_3C_3D @ ( c_2Earithmetic_2E_2A @ ( c_2Earithmetic_2Efindq @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum ) @ V2a @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum @ V1m @ V0n ) ) ) @ V0n ) @ ( c_2Earithmetic_2E_2A @ V2a @ V1m ) ) ) ).
thf(thm_2Earithmetic_2EDIVMOD__THM,axiom,
! [V0n: tyop_2Enum_2Enum,V1m: tyop_2Enum_2Enum,V2a: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2EDIVMOD @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum ) @ V2a @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum @ V1m @ V0n ) ) )
= ( c_2Ebool_2ECOND @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum ) @ ( c_2Emin_2E_3D @ tyop_2Enum_2Enum @ V0n @ c_2Enum_2E0 ) @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum @ c_2Enum_2E0 @ c_2Enum_2E0 )
@ ( c_2Ebool_2ECOND @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum ) @ ( c_2Eprim__rec_2E_3C @ V1m @ V0n ) @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum @ V2a @ V1m )
@ ( c_2Ebool_2ELET @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum )
@ ^ [V3q: tyop_2Enum_2Enum] : ( c_2Earithmetic_2EDIVMOD @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum ) @ ( c_2Earithmetic_2E_2B @ V2a @ V3q ) @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum @ ( c_2Earithmetic_2E_2D @ V1m @ ( c_2Earithmetic_2E_2A @ V0n @ V3q ) ) @ V0n ) ) )
@ ( c_2Earithmetic_2Efindq @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum ) @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum @ V1m @ V0n ) ) ) ) ) ) ) ).
thf(thm_2Earithmetic_2EMOD__SUB,axiom,
! [V0q: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum,V2m: tyop_2Enum_2Enum] :
( ( ( c_2Eprim__rec_2E_3C @ c_2Enum_2E0 @ V1n )
& ( c_2Earithmetic_2E_3C_3D @ ( c_2Earithmetic_2E_2A @ V1n @ V0q ) @ V2m ) )
=> ( ( c_2Earithmetic_2EMOD @ ( c_2Earithmetic_2E_2D @ V2m @ ( c_2Earithmetic_2E_2A @ V1n @ V0q ) ) @ V1n )
= ( c_2Earithmetic_2EMOD @ V2m @ V1n ) ) ) ).
thf(thm_2Earithmetic_2EDIV__SUB,axiom,
! [V0q: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum,V2m: tyop_2Enum_2Enum] :
( ( ( c_2Eprim__rec_2E_3C @ c_2Enum_2E0 @ V1n )
& ( c_2Earithmetic_2E_3C_3D @ ( c_2Earithmetic_2E_2A @ V1n @ V0q ) @ V2m ) )
=> ( ( c_2Earithmetic_2EDIV @ ( c_2Earithmetic_2E_2D @ V2m @ ( c_2Earithmetic_2E_2A @ V1n @ V0q ) ) @ V1n )
= ( c_2Earithmetic_2E_2D @ ( c_2Earithmetic_2EDIV @ V2m @ V1n ) @ V0q ) ) ) ).
thf(thm_2Earithmetic_2EDIVMOD__CORRECT,axiom,
! [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum,V2a: tyop_2Enum_2Enum] :
( ( c_2Eprim__rec_2E_3C @ c_2Enum_2E0 @ V1n )
=> ( ( c_2Earithmetic_2EDIVMOD @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum ) @ V2a @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum @ V0m @ V1n ) ) )
= ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum @ ( c_2Earithmetic_2E_2B @ V2a @ ( c_2Earithmetic_2EDIV @ V0m @ V1n ) ) @ ( c_2Earithmetic_2EMOD @ V0m @ V1n ) ) ) ) ).
thf(thm_2Earithmetic_2EDIVMOD__CALC,axiom,
( ! [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum] :
( ( c_2Eprim__rec_2E_3C @ c_2Enum_2E0 @ V1n )
=> ( ( c_2Earithmetic_2EDIV @ V0m @ V1n )
= ( c_2Epair_2EFST @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum @ ( c_2Earithmetic_2EDIVMOD @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum ) @ c_2Enum_2E0 @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum @ V0m @ V1n ) ) ) ) ) )
& ! [V2m: tyop_2Enum_2Enum,V3n: tyop_2Enum_2Enum] :
( ( c_2Eprim__rec_2E_3C @ c_2Enum_2E0 @ V3n )
=> ( ( c_2Earithmetic_2EMOD @ V2m @ V3n )
= ( c_2Epair_2ESND @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum @ ( c_2Earithmetic_2EDIVMOD @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum ) @ c_2Enum_2E0 @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum @ V2m @ V3n ) ) ) ) ) ) ) ).
thf(thm_2Earithmetic_2EMODEQ__0__CONG,axiom,
! [V0m2: tyop_2Enum_2Enum,V1m1: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2EMODEQ @ c_2Enum_2E0 @ V1m1 @ V0m2 )
<=> ( V1m1 = V0m2 ) ) ).
thf(thm_2Earithmetic_2EMODEQ__NONZERO__MODEQUALITY,axiom,
! [V0n: tyop_2Enum_2Enum,V1m2: tyop_2Enum_2Enum,V2m1: tyop_2Enum_2Enum] :
( ( c_2Eprim__rec_2E_3C @ c_2Enum_2E0 @ V0n )
=> ( ( c_2Earithmetic_2EMODEQ @ V0n @ V2m1 @ V1m2 )
<=> ( ( c_2Earithmetic_2EMOD @ V2m1 @ V0n )
= ( c_2Earithmetic_2EMOD @ V1m2 @ V0n ) ) ) ) ).
thf(thm_2Earithmetic_2EMODEQ__THM,axiom,
! [V0n: tyop_2Enum_2Enum,V1m2: tyop_2Enum_2Enum,V2m1: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2EMODEQ @ V0n @ V2m1 @ V1m2 )
<=> ( ( ( V0n = c_2Enum_2E0 )
& ( V2m1 = V1m2 ) )
| ( ( c_2Eprim__rec_2E_3C @ c_2Enum_2E0 @ V0n )
& ( ( c_2Earithmetic_2EMOD @ V2m1 @ V0n )
= ( c_2Earithmetic_2EMOD @ V1m2 @ V0n ) ) ) ) ) ).
thf(thm_2Earithmetic_2EMODEQ__INTRO__CONG,axiom,
! [V0n: tyop_2Enum_2Enum,V1e1: tyop_2Enum_2Enum,V2e0: tyop_2Enum_2Enum] :
( ( c_2Eprim__rec_2E_3C @ c_2Enum_2E0 @ V0n )
=> ( ( c_2Earithmetic_2EMODEQ @ V0n @ V2e0 @ V1e1 )
=> ( ( c_2Earithmetic_2EMOD @ V2e0 @ V0n )
= ( c_2Earithmetic_2EMOD @ V1e1 @ V0n ) ) ) ) ).
thf(thm_2Earithmetic_2EMODEQ__PLUS__CONG,axiom,
! [V0y1: tyop_2Enum_2Enum,V1y0: tyop_2Enum_2Enum,V2x1: tyop_2Enum_2Enum,V3x0: tyop_2Enum_2Enum,V4n: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2EMODEQ @ V4n @ V3x0 @ V2x1 )
=> ( ( c_2Earithmetic_2EMODEQ @ V4n @ V1y0 @ V0y1 )
=> ( c_2Earithmetic_2EMODEQ @ V4n @ ( c_2Earithmetic_2E_2B @ V3x0 @ V1y0 ) @ ( c_2Earithmetic_2E_2B @ V2x1 @ V0y1 ) ) ) ) ).
thf(thm_2Earithmetic_2EMODEQ__MULT__CONG,axiom,
! [V0y1: tyop_2Enum_2Enum,V1y0: tyop_2Enum_2Enum,V2x1: tyop_2Enum_2Enum,V3x0: tyop_2Enum_2Enum,V4n: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2EMODEQ @ V4n @ V3x0 @ V2x1 )
=> ( ( c_2Earithmetic_2EMODEQ @ V4n @ V1y0 @ V0y1 )
=> ( c_2Earithmetic_2EMODEQ @ V4n @ ( c_2Earithmetic_2E_2A @ V3x0 @ V1y0 ) @ ( c_2Earithmetic_2E_2A @ V2x1 @ V0y1 ) ) ) ) ).
thf(thm_2Earithmetic_2EMODEQ__REFL,axiom,
! [V0n: tyop_2Enum_2Enum,V1x: tyop_2Enum_2Enum] : ( c_2Earithmetic_2EMODEQ @ V0n @ V1x @ V1x ) ).
thf(thm_2Earithmetic_2EMODEQ__SUC__CONG,axiom,
! [V0y: tyop_2Enum_2Enum,V1x: tyop_2Enum_2Enum,V2n: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2EMODEQ @ V2n @ V1x @ V0y )
=> ( c_2Earithmetic_2EMODEQ @ V2n @ ( c_2Enum_2ESUC @ V1x ) @ ( c_2Enum_2ESUC @ V0y ) ) ) ).
thf(thm_2Earithmetic_2EMODEQ__EXP__CONG,axiom,
! [V0y: tyop_2Enum_2Enum,V1x: tyop_2Enum_2Enum,V2n: tyop_2Enum_2Enum,V3e: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2EMODEQ @ V2n @ V1x @ V0y )
=> ( c_2Earithmetic_2EMODEQ @ V2n @ ( c_2Earithmetic_2EEXP @ V1x @ V3e ) @ ( c_2Earithmetic_2EEXP @ V0y @ V3e ) ) ) ).
thf(thm_2Earithmetic_2EEXP__MOD,axiom,
! [V0x: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum,V2e: tyop_2Enum_2Enum] :
( ( c_2Eprim__rec_2E_3C @ c_2Enum_2E0 @ V1n )
=> ( ( c_2Earithmetic_2EMOD @ ( c_2Earithmetic_2EEXP @ ( c_2Earithmetic_2EMOD @ V0x @ V1n ) @ V2e ) @ V1n )
= ( c_2Earithmetic_2EMOD @ ( c_2Earithmetic_2EEXP @ V0x @ V2e ) @ V1n ) ) ) ).
thf(thm_2Earithmetic_2EMODEQ__SYM,axiom,
! [V0y: tyop_2Enum_2Enum,V1x: tyop_2Enum_2Enum,V2n: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2EMODEQ @ V2n @ V1x @ V0y )
= ( c_2Earithmetic_2EMODEQ @ V2n @ V0y @ V1x ) ) ).
thf(thm_2Earithmetic_2EMODEQ__TRANS,axiom,
! [V0n: tyop_2Enum_2Enum,V1x: tyop_2Enum_2Enum,V2y: tyop_2Enum_2Enum,V3z: tyop_2Enum_2Enum] :
( ( ( c_2Earithmetic_2EMODEQ @ V0n @ V1x @ V2y )
& ( c_2Earithmetic_2EMODEQ @ V0n @ V2y @ V3z ) )
=> ( c_2Earithmetic_2EMODEQ @ V0n @ V1x @ V3z ) ) ).
thf(thm_2Earithmetic_2EMODEQ__NUMERAL,axiom,
! [V0n: tyop_2Enum_2Enum,V1m: tyop_2Enum_2Enum] :
( ( ( c_2Earithmetic_2E_3C_3D @ ( c_2Earithmetic_2ENUMERAL @ V0n ) @ ( c_2Earithmetic_2ENUMERAL @ V1m ) )
=> ( c_2Earithmetic_2EMODEQ @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ V0n ) ) @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ V1m ) ) @ ( c_2Earithmetic_2EMOD @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ V1m ) ) @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ V0n ) ) ) ) )
& ( ( c_2Earithmetic_2E_3C_3D @ ( c_2Earithmetic_2ENUMERAL @ V0n ) @ ( c_2Earithmetic_2ENUMERAL @ V1m ) )
=> ( c_2Earithmetic_2EMODEQ @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ V0n ) ) @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ V1m ) ) @ ( c_2Earithmetic_2EMOD @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ V1m ) ) @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ V0n ) ) ) ) )
& ( ( c_2Earithmetic_2E_3C_3D @ ( c_2Earithmetic_2ENUMERAL @ V0n ) @ ( c_2Earithmetic_2ENUMERAL @ V1m ) )
=> ( c_2Earithmetic_2EMODEQ @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ V0n ) ) @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ V1m ) ) @ ( c_2Earithmetic_2EMOD @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ V1m ) ) @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ V0n ) ) ) ) )
& ( ( c_2Eprim__rec_2E_3C @ ( c_2Earithmetic_2ENUMERAL @ V0n ) @ ( c_2Earithmetic_2ENUMERAL @ V1m ) )
=> ( c_2Earithmetic_2EMODEQ @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ V0n ) ) @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ V1m ) ) @ ( c_2Earithmetic_2EMOD @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ V1m ) ) @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ V0n ) ) ) ) ) ) ).
thf(thm_2Earithmetic_2EMODEQ__MOD,axiom,
! [V0x: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum] :
( ( c_2Eprim__rec_2E_3C @ c_2Enum_2E0 @ V1n )
=> ( c_2Earithmetic_2EMODEQ @ V1n @ ( c_2Earithmetic_2EMOD @ V0x @ V1n ) @ V0x ) ) ).
thf(thm_2Earithmetic_2EMODEQ__0,axiom,
! [V0n: tyop_2Enum_2Enum] :
( ( c_2Eprim__rec_2E_3C @ c_2Enum_2E0 @ V0n )
=> ( c_2Earithmetic_2EMODEQ @ V0n @ V0n @ c_2Enum_2E0 ) ) ).
thf(thm_2Earithmetic_2Enum__case__eq,axiom,
! [A_27a: $tType,V0zc: A_27a,V1v: A_27a,V2sc: tyop_2Enum_2Enum > A_27a,V3n: tyop_2Enum_2Enum] :
( ( ( c_2Earithmetic_2Enum__CASE @ A_27a @ V3n @ V0zc @ V2sc )
= V1v )
<=> ( ( ( V3n = c_2Enum_2E0 )
& ( V0zc = V1v ) )
| ? [V4x: tyop_2Enum_2Enum] :
( ( V3n
= ( c_2Enum_2ESUC @ V4x ) )
& ( ( V2sc @ V4x )
= V1v ) ) ) ) ).
thf(thm_2Earithmetic_2Edatatype__num,axiom,
! [A_27a: $tType,V0num: tyop_2Enum_2Enum > ( tyop_2Enum_2Enum > tyop_2Enum_2Enum ) > A_27a] : ( c_2Ebool_2EDATATYPE @ A_27a @ ( V0num @ c_2Enum_2E0 @ c_2Enum_2ESUC ) ) ).
%------------------------------------------------------------------------------