ITP001 Axioms: ITP026^7.ax
%------------------------------------------------------------------------------
% File : ITP026^7 : TPTP v9.0.0. 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 : bit.ax [Gau19]
% : HL4026^7.ax [TPAP]
% Status : Satisfiable
% Syntax : Number of formulae : 190 ( 60 unt; 49 typ; 0 def)
% Number of atoms : 348 ( 126 equ; 29 cnn)
% Maximal formula atoms : 8 ( 1 avg)
% Number of connectives : 1927 ( 29 ~; 3 |; 30 &;1773 @)
% ( 26 <=>; 66 =>; 0 <=; 0 <~>)
% Maximal formula depth : 17 ( 7 avg;1773 nst)
% Number of types : 3 ( 2 usr)
% Number of type conns : 122 ( 122 >; 0 *; 0 +; 0 <<)
% Number of symbols : 49 ( 47 usr; 3 con; 0-4 aty)
% Number of variables : 392 ( 4 ^ 375 !; 5 ?; 392 :)
% ( 8 !>; 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_2Ebool_2E_3F,type,
c_2Ebool_2E_3F:
!>[A_27a: $tType] : ( ( A_27a > $o ) > $o ) ).
thf(c_2Ebit_2EBIT,type,
c_2Ebit_2EBIT: tyop_2Enum_2Enum > tyop_2Enum_2Enum > $o ).
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_2Ebit_2EBITS,type,
c_2Ebit_2EBITS: tyop_2Enum_2Enum > tyop_2Enum_2Enum > tyop_2Enum_2Enum > tyop_2Enum_2Enum ).
thf(c_2Ebit_2EBITV,type,
c_2Ebit_2EBITV: tyop_2Enum_2Enum > tyop_2Enum_2Enum > tyop_2Enum_2Enum ).
thf(c_2Ebit_2EBITWISE,type,
c_2Ebit_2EBITWISE: tyop_2Enum_2Enum > ( $o > $o > $o ) > tyop_2Enum_2Enum > tyop_2Enum_2Enum > tyop_2Enum_2Enum ).
thf(c_2Ebit_2EBIT__MODIFY,type,
c_2Ebit_2EBIT__MODIFY: tyop_2Enum_2Enum > ( tyop_2Enum_2Enum > $o > $o ) > tyop_2Enum_2Enum > tyop_2Enum_2Enum ).
thf(c_2Ebit_2EBIT__REVERSE,type,
c_2Ebit_2EBIT__REVERSE: tyop_2Enum_2Enum > 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_2Earithmetic_2EDIV,type,
c_2Earithmetic_2EDIV: tyop_2Enum_2Enum > tyop_2Enum_2Enum > tyop_2Enum_2Enum ).
thf(c_2Ebit_2EDIVMOD__2EXP,type,
c_2Ebit_2EDIVMOD__2EXP: tyop_2Enum_2Enum > tyop_2Enum_2Enum > ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum ) ).
thf(c_2Ebit_2EDIV__2EXP,type,
c_2Ebit_2EDIV__2EXP: tyop_2Enum_2Enum > tyop_2Enum_2Enum > tyop_2Enum_2Enum ).
thf(c_2Earithmetic_2EEXP,type,
c_2Earithmetic_2EEXP: tyop_2Enum_2Enum > tyop_2Enum_2Enum > tyop_2Enum_2Enum ).
thf(c_2Ewhile_2ELEAST,type,
c_2Ewhile_2ELEAST: ( tyop_2Enum_2Enum > $o ) > tyop_2Enum_2Enum ).
thf(c_2Ebool_2ELET,type,
c_2Ebool_2ELET:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > A_27b ) > A_27a > A_27b ) ).
thf(c_2Elogroot_2ELOG,type,
c_2Elogroot_2ELOG: tyop_2Enum_2Enum > tyop_2Enum_2Enum > tyop_2Enum_2Enum ).
thf(c_2Ebit_2ELOG2,type,
c_2Ebit_2ELOG2: tyop_2Enum_2Enum > tyop_2Enum_2Enum ).
thf(c_2Ebit_2ELOWEST__SET__BIT,type,
c_2Ebit_2ELOWEST__SET__BIT: 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_2Ebit_2EMOD__2EXP,type,
c_2Ebit_2EMOD__2EXP: tyop_2Enum_2Enum > tyop_2Enum_2Enum > tyop_2Enum_2Enum ).
thf(c_2Ebit_2EMOD__2EXP__EQ,type,
c_2Ebit_2EMOD__2EXP__EQ: tyop_2Enum_2Enum > tyop_2Enum_2Enum > tyop_2Enum_2Enum > $o ).
thf(c_2Ebit_2EMOD__2EXP__MAX,type,
c_2Ebit_2EMOD__2EXP__MAX: tyop_2Enum_2Enum > tyop_2Enum_2Enum > $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_2Ebit_2ESBIT,type,
c_2Ebit_2ESBIT: $o > tyop_2Enum_2Enum > tyop_2Enum_2Enum ).
thf(c_2Ebit_2ESIGN__EXTEND,type,
c_2Ebit_2ESIGN__EXTEND: tyop_2Enum_2Enum > tyop_2Enum_2Enum > tyop_2Enum_2Enum > tyop_2Enum_2Enum ).
thf(c_2Ebit_2ESLICE,type,
c_2Ebit_2ESLICE: tyop_2Enum_2Enum > tyop_2Enum_2Enum > tyop_2Enum_2Enum > tyop_2Enum_2Enum ).
thf(c_2Enum_2ESUC,type,
c_2Enum_2ESUC: tyop_2Enum_2Enum > tyop_2Enum_2Enum ).
thf(c_2Ebit_2ETIMES__2EXP,type,
c_2Ebit_2ETIMES__2EXP: tyop_2Enum_2Enum > tyop_2Enum_2Enum > tyop_2Enum_2Enum ).
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_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_2Ebit_2EMOD__2EXP__def,axiom,
! [V0x: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum] :
( ( c_2Ebit_2EMOD__2EXP @ V0x @ V1n )
= ( c_2Earithmetic_2EMOD @ V1n @ ( c_2Earithmetic_2EEXP @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) @ V0x ) ) ) ).
thf(thm_2Ebit_2EDIV__2EXP__def,axiom,
! [V0x: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum] :
( ( c_2Ebit_2EDIV__2EXP @ V0x @ V1n )
= ( c_2Earithmetic_2EDIV @ V1n @ ( c_2Earithmetic_2EEXP @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) @ V0x ) ) ) ).
thf(thm_2Ebit_2ETIMES__2EXP__def,axiom,
! [V0x: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum] :
( ( c_2Ebit_2ETIMES__2EXP @ V0x @ V1n )
= ( c_2Earithmetic_2E_2A @ V1n @ ( c_2Earithmetic_2EEXP @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) @ V0x ) ) ) ).
thf(thm_2Ebit_2EDIVMOD__2EXP__def,axiom,
! [V0x: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum] :
( ( c_2Ebit_2EDIVMOD__2EXP @ V0x @ V1n )
= ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum @ ( c_2Earithmetic_2EDIV @ V1n @ ( c_2Earithmetic_2EEXP @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) @ V0x ) ) @ ( c_2Earithmetic_2EMOD @ V1n @ ( c_2Earithmetic_2EEXP @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) @ V0x ) ) ) ) ).
thf(thm_2Ebit_2ESBIT__def,axiom,
! [V0b: $o,V1n: tyop_2Enum_2Enum] :
( ( c_2Ebit_2ESBIT @ V0b @ V1n )
= ( c_2Ebool_2ECOND @ tyop_2Enum_2Enum @ V0b @ ( c_2Earithmetic_2EEXP @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) @ V1n ) @ c_2Enum_2E0 ) ) ).
thf(thm_2Ebit_2EBITS__def,axiom,
! [V0h: tyop_2Enum_2Enum,V1l: tyop_2Enum_2Enum,V2n: tyop_2Enum_2Enum] :
( ( c_2Ebit_2EBITS @ V0h @ V1l @ V2n )
= ( c_2Ebit_2EMOD__2EXP @ ( c_2Earithmetic_2E_2D @ ( c_2Enum_2ESUC @ V0h ) @ V1l ) @ ( c_2Ebit_2EDIV__2EXP @ V1l @ V2n ) ) ) ).
thf(thm_2Ebit_2EBITV__def,axiom,
! [V0n: tyop_2Enum_2Enum,V1b: tyop_2Enum_2Enum] :
( ( c_2Ebit_2EBITV @ V0n @ V1b )
= ( c_2Ebit_2EBITS @ V1b @ V1b @ V0n ) ) ).
thf(thm_2Ebit_2EBIT__def,axiom,
! [V0b: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum] :
( ( c_2Ebit_2EBIT @ V0b @ V1n )
<=> ( ( c_2Ebit_2EBITS @ V0b @ V0b @ V1n )
= ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ).
thf(thm_2Ebit_2ESLICE__def,axiom,
! [V0h: tyop_2Enum_2Enum,V1l: tyop_2Enum_2Enum,V2n: tyop_2Enum_2Enum] :
( ( c_2Ebit_2ESLICE @ V0h @ V1l @ V2n )
= ( c_2Earithmetic_2E_2D @ ( c_2Ebit_2EMOD__2EXP @ ( c_2Enum_2ESUC @ V0h ) @ V2n ) @ ( c_2Ebit_2EMOD__2EXP @ V1l @ V2n ) ) ) ).
thf(thm_2Ebit_2ELOG2__def,axiom,
( c_2Ebit_2ELOG2
= ( c_2Elogroot_2ELOG @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) ) ).
thf(thm_2Ebit_2ELOWEST__SET__BIT__def,axiom,
! [V0n: tyop_2Enum_2Enum] :
( ( c_2Ebit_2ELOWEST__SET__BIT @ V0n )
= ( c_2Ewhile_2ELEAST
@ ^ [V1i: tyop_2Enum_2Enum] : ( c_2Ebit_2EBIT @ V1i @ V0n ) ) ) ).
thf(thm_2Ebit_2EBIT__REVERSE__def,axiom,
( ! [V0x: tyop_2Enum_2Enum] :
( ( c_2Ebit_2EBIT__REVERSE @ c_2Enum_2E0 @ V0x )
= c_2Enum_2E0 )
& ! [V1n: tyop_2Enum_2Enum,V2x: tyop_2Enum_2Enum] :
( ( c_2Ebit_2EBIT__REVERSE @ ( c_2Enum_2ESUC @ V1n ) @ V2x )
= ( c_2Earithmetic_2E_2B @ ( c_2Earithmetic_2E_2A @ ( c_2Ebit_2EBIT__REVERSE @ V1n @ V2x ) @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) @ ( c_2Ebit_2ESBIT @ ( c_2Ebit_2EBIT @ V1n @ V2x ) @ c_2Enum_2E0 ) ) ) ) ).
thf(thm_2Ebit_2EBITWISE__def,axiom,
( ! [V0op: $o > $o > $o,V1x: tyop_2Enum_2Enum,V2y: tyop_2Enum_2Enum] :
( ( c_2Ebit_2EBITWISE @ c_2Enum_2E0 @ V0op @ V1x @ V2y )
= c_2Enum_2E0 )
& ! [V3n: tyop_2Enum_2Enum,V4op: $o > $o > $o,V5x: tyop_2Enum_2Enum,V6y: tyop_2Enum_2Enum] :
( ( c_2Ebit_2EBITWISE @ ( c_2Enum_2ESUC @ V3n ) @ V4op @ V5x @ V6y )
= ( c_2Earithmetic_2E_2B @ ( c_2Ebit_2EBITWISE @ V3n @ V4op @ V5x @ V6y ) @ ( c_2Ebit_2ESBIT @ ( V4op @ ( c_2Ebit_2EBIT @ V3n @ V5x ) @ ( c_2Ebit_2EBIT @ V3n @ V6y ) ) @ V3n ) ) ) ) ).
thf(thm_2Ebit_2EBIT__MODIFY__def,axiom,
( ! [V0f: tyop_2Enum_2Enum > $o > $o,V1x: tyop_2Enum_2Enum] :
( ( c_2Ebit_2EBIT__MODIFY @ c_2Enum_2E0 @ V0f @ V1x )
= c_2Enum_2E0 )
& ! [V2n: tyop_2Enum_2Enum,V3f: tyop_2Enum_2Enum > $o > $o,V4x: tyop_2Enum_2Enum] :
( ( c_2Ebit_2EBIT__MODIFY @ ( c_2Enum_2ESUC @ V2n ) @ V3f @ V4x )
= ( c_2Earithmetic_2E_2B @ ( c_2Ebit_2EBIT__MODIFY @ V2n @ V3f @ V4x ) @ ( c_2Ebit_2ESBIT @ ( V3f @ V2n @ ( c_2Ebit_2EBIT @ V2n @ V4x ) ) @ V2n ) ) ) ) ).
thf(thm_2Ebit_2ESIGN__EXTEND__def,axiom,
! [V0l: tyop_2Enum_2Enum,V1h: tyop_2Enum_2Enum,V2n: tyop_2Enum_2Enum] :
( ( c_2Ebit_2ESIGN__EXTEND @ V0l @ V1h @ V2n )
= ( c_2Ebool_2ELET @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum
@ ^ [V3m: tyop_2Enum_2Enum] : ( c_2Ebool_2ECOND @ tyop_2Enum_2Enum @ ( c_2Ebit_2EBIT @ ( c_2Earithmetic_2E_2D @ V0l @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) @ V2n ) @ ( c_2Earithmetic_2E_2B @ ( c_2Earithmetic_2E_2D @ ( c_2Earithmetic_2EEXP @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) @ V1h ) @ ( c_2Earithmetic_2EEXP @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) @ V0l ) ) @ V3m ) @ V3m )
@ ( c_2Earithmetic_2EMOD @ V2n @ ( c_2Earithmetic_2EEXP @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) @ V0l ) ) ) ) ).
thf(thm_2Ebit_2EMOD__2EXP__EQ__def,axiom,
! [V0n: tyop_2Enum_2Enum,V1a: tyop_2Enum_2Enum,V2b: tyop_2Enum_2Enum] :
( ( c_2Ebit_2EMOD__2EXP__EQ @ V0n @ V1a @ V2b )
<=> ( ( c_2Ebit_2EMOD__2EXP @ V0n @ V1a )
= ( c_2Ebit_2EMOD__2EXP @ V0n @ V2b ) ) ) ).
thf(thm_2Ebit_2EMOD__2EXP__MAX__def,axiom,
! [V0n: tyop_2Enum_2Enum,V1a: tyop_2Enum_2Enum] :
( ( c_2Ebit_2EMOD__2EXP__MAX @ V0n @ V1a )
<=> ( ( c_2Ebit_2EMOD__2EXP @ V0n @ V1a )
= ( c_2Earithmetic_2E_2D @ ( c_2Earithmetic_2EEXP @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) @ V0n ) @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ) ).
thf(thm_2Ebit_2ELESS__MULT__MONO2,axiom,
! [V0a: tyop_2Enum_2Enum,V1b: tyop_2Enum_2Enum,V2x: tyop_2Enum_2Enum,V3y: tyop_2Enum_2Enum] :
( ( ( c_2Eprim__rec_2E_3C @ V0a @ V2x )
& ( c_2Eprim__rec_2E_3C @ V1b @ V3y ) )
=> ( c_2Eprim__rec_2E_3C @ ( c_2Earithmetic_2E_2A @ V0a @ V1b ) @ ( c_2Earithmetic_2E_2A @ V2x @ V3y ) ) ) ).
thf(thm_2Ebit_2ELOG2__UNIQUE,axiom,
! [V0n: tyop_2Enum_2Enum,V1p: tyop_2Enum_2Enum] :
( ( ( c_2Earithmetic_2E_3C_3D @ ( c_2Earithmetic_2EEXP @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) @ V1p ) @ V0n )
& ( c_2Eprim__rec_2E_3C @ V0n @ ( c_2Earithmetic_2EEXP @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) @ ( c_2Enum_2ESUC @ V1p ) ) ) )
=> ( ( c_2Ebit_2ELOG2 @ V0n )
= V1p ) ) ).
thf(thm_2Ebit_2ELOG2__TWOEXP,axiom,
! [V0n: tyop_2Enum_2Enum] :
( ( c_2Ebit_2ELOG2 @ ( c_2Earithmetic_2EEXP @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) @ V0n ) )
= V0n ) ).
thf(thm_2Ebit_2EDIVMOD__2EXP,axiom,
! [V0x: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum] :
( ( c_2Ebit_2EDIVMOD__2EXP @ V0x @ V1n )
= ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum @ ( c_2Ebit_2EDIV__2EXP @ V0x @ V1n ) @ ( c_2Ebit_2EMOD__2EXP @ V0x @ V1n ) ) ) ).
thf(thm_2Ebit_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_2Ebit_2EDIV__MULT__1,axiom,
! [V0r: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum] :
( ( c_2Eprim__rec_2E_3C @ V0r @ V1n )
=> ( ( c_2Earithmetic_2EDIV @ ( c_2Earithmetic_2E_2B @ V1n @ V0r ) @ V1n )
= ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ).
thf(thm_2Ebit_2ENOT__ZERO__ADD1,axiom,
! [V0m: tyop_2Enum_2Enum] :
( ( (~) @ ( V0m = c_2Enum_2E0 ) )
=> ? [V1p: tyop_2Enum_2Enum] :
( V0m
= ( c_2Enum_2ESUC @ V1p ) ) ) ).
thf(thm_2Ebit_2EZERO__LT__TWOEXP,axiom,
! [V0n: tyop_2Enum_2Enum] : ( c_2Eprim__rec_2E_3C @ c_2Enum_2E0 @ ( c_2Earithmetic_2EEXP @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) @ V0n ) ) ).
thf(thm_2Ebit_2EONE__LE__TWOEXP,axiom,
! [V0n: tyop_2Enum_2Enum] : ( c_2Earithmetic_2E_3C_3D @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) @ ( c_2Earithmetic_2EEXP @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) @ V0n ) ) ).
thf(thm_2Ebit_2ETWOEXP__NOT__ZERO,axiom,
! [V0n: tyop_2Enum_2Enum] :
( (~)
@ ( ( c_2Earithmetic_2EEXP @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) @ V0n )
= c_2Enum_2E0 ) ) ).
thf(thm_2Ebit_2EMOD__2EXP__LT,axiom,
! [V0n: tyop_2Enum_2Enum,V1k: tyop_2Enum_2Enum] : ( c_2Eprim__rec_2E_3C @ ( c_2Earithmetic_2EMOD @ V1k @ ( c_2Earithmetic_2EEXP @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) @ V0n ) ) @ ( c_2Earithmetic_2EEXP @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) @ V0n ) ) ).
thf(thm_2Ebit_2ETWOEXP__DIVISION,axiom,
! [V0n: tyop_2Enum_2Enum,V1k: tyop_2Enum_2Enum] :
( V1k
= ( c_2Earithmetic_2E_2B @ ( c_2Earithmetic_2E_2A @ ( c_2Earithmetic_2EDIV @ V1k @ ( c_2Earithmetic_2EEXP @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) @ V0n ) ) @ ( c_2Earithmetic_2EEXP @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) @ V0n ) ) @ ( c_2Earithmetic_2EMOD @ V1k @ ( c_2Earithmetic_2EEXP @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) @ V0n ) ) ) ) ).
thf(thm_2Ebit_2ETWOEXP__MONO,axiom,
! [V0a: tyop_2Enum_2Enum,V1b: tyop_2Enum_2Enum] :
( ( c_2Eprim__rec_2E_3C @ V0a @ V1b )
=> ( c_2Eprim__rec_2E_3C @ ( c_2Earithmetic_2EEXP @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) @ V0a ) @ ( c_2Earithmetic_2EEXP @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) @ V1b ) ) ) ).
thf(thm_2Ebit_2ETWOEXP__MONO2,axiom,
! [V0a: tyop_2Enum_2Enum,V1b: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_3C_3D @ V0a @ V1b )
=> ( c_2Earithmetic_2E_3C_3D @ ( c_2Earithmetic_2EEXP @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) @ V0a ) @ ( c_2Earithmetic_2EEXP @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) @ V1b ) ) ) ).
thf(thm_2Ebit_2EEXP__SUB__LESS__EQ,axiom,
! [V0a: tyop_2Enum_2Enum,V1b: tyop_2Enum_2Enum] : ( c_2Earithmetic_2E_3C_3D @ ( c_2Earithmetic_2EEXP @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) @ ( c_2Earithmetic_2E_2D @ V0a @ V1b ) ) @ ( c_2Earithmetic_2EEXP @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) @ V0a ) ) ).
thf(thm_2Ebit_2EMOD__LEQ,axiom,
! [V0a: tyop_2Enum_2Enum,V1b: tyop_2Enum_2Enum] :
( ( c_2Eprim__rec_2E_3C @ c_2Enum_2E0 @ V1b )
=> ( c_2Earithmetic_2E_3C_3D @ ( c_2Earithmetic_2EMOD @ V0a @ V1b ) @ V0a ) ) ).
thf(thm_2Ebit_2EBITS__THM,axiom,
! [V0h: tyop_2Enum_2Enum,V1l: tyop_2Enum_2Enum,V2n: tyop_2Enum_2Enum] :
( ( c_2Ebit_2EBITS @ V0h @ V1l @ V2n )
= ( c_2Earithmetic_2EMOD @ ( c_2Earithmetic_2EDIV @ V2n @ ( c_2Earithmetic_2EEXP @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) @ V1l ) ) @ ( c_2Earithmetic_2EEXP @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) @ ( c_2Earithmetic_2E_2D @ ( c_2Enum_2ESUC @ V0h ) @ V1l ) ) ) ) ).
thf(thm_2Ebit_2EBITSLT__THM,axiom,
! [V0h: tyop_2Enum_2Enum,V1l: tyop_2Enum_2Enum,V2n: tyop_2Enum_2Enum] : ( c_2Eprim__rec_2E_3C @ ( c_2Ebit_2EBITS @ V0h @ V1l @ V2n ) @ ( c_2Earithmetic_2EEXP @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) @ ( c_2Earithmetic_2E_2D @ ( c_2Enum_2ESUC @ V0h ) @ V1l ) ) ) ).
thf(thm_2Ebit_2EBITSLT__THM2,axiom,
! [V0h: tyop_2Enum_2Enum,V1l: tyop_2Enum_2Enum,V2n: tyop_2Enum_2Enum] : ( c_2Eprim__rec_2E_3C @ ( c_2Ebit_2EBITS @ V0h @ V1l @ V2n ) @ ( c_2Earithmetic_2EEXP @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) @ ( c_2Enum_2ESUC @ V0h ) ) ) ).
thf(thm_2Ebit_2EBITS__THM2,axiom,
! [V0h: tyop_2Enum_2Enum,V1l: tyop_2Enum_2Enum,V2n: tyop_2Enum_2Enum] :
( ( c_2Ebit_2EBITS @ V0h @ V1l @ V2n )
= ( c_2Earithmetic_2EDIV @ ( c_2Earithmetic_2EMOD @ V2n @ ( c_2Earithmetic_2EEXP @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) @ ( c_2Enum_2ESUC @ V0h ) ) ) @ ( c_2Earithmetic_2EEXP @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) @ V1l ) ) ) ).
thf(thm_2Ebit_2EBITS__LEQ,axiom,
! [V0h: tyop_2Enum_2Enum,V1l: tyop_2Enum_2Enum,V2n: tyop_2Enum_2Enum] : ( c_2Earithmetic_2E_3C_3D @ ( c_2Ebit_2EBITS @ V0h @ V1l @ V2n ) @ V2n ) ).
thf(thm_2Ebit_2EBITS__COMP__THM,axiom,
! [V0h1: tyop_2Enum_2Enum,V1l1: tyop_2Enum_2Enum,V2h2: tyop_2Enum_2Enum,V3l2: tyop_2Enum_2Enum,V4n: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_3C_3D @ ( c_2Earithmetic_2E_2B @ V2h2 @ V1l1 ) @ V0h1 )
=> ( ( c_2Ebit_2EBITS @ V2h2 @ V3l2 @ ( c_2Ebit_2EBITS @ V0h1 @ V1l1 @ V4n ) )
= ( c_2Ebit_2EBITS @ ( c_2Earithmetic_2E_2B @ V2h2 @ V1l1 ) @ ( c_2Earithmetic_2E_2B @ V3l2 @ V1l1 ) @ V4n ) ) ) ).
thf(thm_2Ebit_2EBITS__DIV__THM,axiom,
! [V0h: tyop_2Enum_2Enum,V1l: tyop_2Enum_2Enum,V2x: tyop_2Enum_2Enum,V3n: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2EDIV @ ( c_2Ebit_2EBITS @ V0h @ V1l @ V2x ) @ ( c_2Earithmetic_2EEXP @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) @ V3n ) )
= ( c_2Ebit_2EBITS @ V0h @ ( c_2Earithmetic_2E_2B @ V1l @ V3n ) @ V2x ) ) ).
thf(thm_2Ebit_2EBITS__LT__HIGH,axiom,
! [V0h: tyop_2Enum_2Enum,V1l: tyop_2Enum_2Enum,V2n: tyop_2Enum_2Enum] :
( ( c_2Eprim__rec_2E_3C @ V2n @ ( c_2Earithmetic_2EEXP @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) @ ( c_2Enum_2ESUC @ V0h ) ) )
=> ( ( c_2Ebit_2EBITS @ V0h @ V1l @ V2n )
= ( c_2Earithmetic_2EDIV @ V2n @ ( c_2Earithmetic_2EEXP @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) @ V1l ) ) ) ) ).
thf(thm_2Ebit_2EBITS__ZERO,axiom,
! [V0h: tyop_2Enum_2Enum,V1l: tyop_2Enum_2Enum,V2n: tyop_2Enum_2Enum] :
( ( c_2Eprim__rec_2E_3C @ V0h @ V1l )
=> ( ( c_2Ebit_2EBITS @ V0h @ V1l @ V2n )
= c_2Enum_2E0 ) ) ).
thf(thm_2Ebit_2EBITS__ZERO2,axiom,
! [V0h: tyop_2Enum_2Enum,V1l: tyop_2Enum_2Enum] :
( ( c_2Ebit_2EBITS @ V0h @ V1l @ c_2Enum_2E0 )
= c_2Enum_2E0 ) ).
thf(thm_2Ebit_2EBITS__ZERO3,axiom,
! [V0h: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum] :
( ( c_2Ebit_2EBITS @ V0h @ c_2Enum_2E0 @ V1n )
= ( c_2Earithmetic_2EMOD @ V1n @ ( c_2Earithmetic_2EEXP @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) @ ( c_2Enum_2ESUC @ V0h ) ) ) ) ).
thf(thm_2Ebit_2EBITS__ZERO4,axiom,
! [V0h: tyop_2Enum_2Enum,V1l: tyop_2Enum_2Enum,V2a: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_3C_3D @ V1l @ V0h )
=> ( ( c_2Ebit_2EBITS @ V0h @ V1l @ ( c_2Earithmetic_2E_2A @ V2a @ ( c_2Earithmetic_2EEXP @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) @ V1l ) ) )
= ( c_2Ebit_2EBITS @ ( c_2Earithmetic_2E_2D @ V0h @ V1l ) @ c_2Enum_2E0 @ V2a ) ) ) ).
thf(thm_2Ebit_2EBITS__ZEROL,axiom,
! [V0h: tyop_2Enum_2Enum,V1a: tyop_2Enum_2Enum] :
( ( c_2Eprim__rec_2E_3C @ V1a @ ( c_2Earithmetic_2EEXP @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) @ ( c_2Enum_2ESUC @ V0h ) ) )
=> ( ( c_2Ebit_2EBITS @ V0h @ c_2Enum_2E0 @ V1a )
= V1a ) ) ).
thf(thm_2Ebit_2EBITS__LOG2__ZERO__ID,axiom,
! [V0n: tyop_2Enum_2Enum] :
( ( c_2Eprim__rec_2E_3C @ c_2Enum_2E0 @ V0n )
=> ( ( c_2Ebit_2EBITS @ ( c_2Ebit_2ELOG2 @ V0n ) @ c_2Enum_2E0 @ V0n )
= V0n ) ) ).
thf(thm_2Ebit_2EBITS__LT__LOW,axiom,
! [V0h: tyop_2Enum_2Enum,V1l: tyop_2Enum_2Enum,V2n: tyop_2Enum_2Enum] :
( ( c_2Eprim__rec_2E_3C @ V2n @ ( c_2Earithmetic_2EEXP @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) @ V1l ) )
=> ( ( c_2Ebit_2EBITS @ V0h @ V1l @ V2n )
= c_2Enum_2E0 ) ) ).
thf(thm_2Ebit_2EBIT__ZERO,axiom,
! [V0b: tyop_2Enum_2Enum] : ( (~) @ ( c_2Ebit_2EBIT @ V0b @ c_2Enum_2E0 ) ) ).
thf(thm_2Ebit_2EBIT__B,axiom,
! [V0b: tyop_2Enum_2Enum] : ( c_2Ebit_2EBIT @ V0b @ ( c_2Earithmetic_2EEXP @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) @ V0b ) ) ).
thf(thm_2Ebit_2EBIT__TWO__POW,axiom,
! [V0n: tyop_2Enum_2Enum,V1m: tyop_2Enum_2Enum] :
( ( c_2Ebit_2EBIT @ V0n @ ( c_2Earithmetic_2EEXP @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) @ V1m ) )
<=> ( V1m = V0n ) ) ).
thf(thm_2Ebit_2EBIT__B__NEQ,axiom,
! [V0a: tyop_2Enum_2Enum,V1b: tyop_2Enum_2Enum] :
( ( (~) @ ( V0a = V1b ) )
=> ( (~) @ ( c_2Ebit_2EBIT @ V0a @ ( c_2Earithmetic_2EEXP @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) @ V1b ) ) ) ) ).
thf(thm_2Ebit_2EBITS__COMP__THM2,axiom,
! [V0h1: tyop_2Enum_2Enum,V1l1: tyop_2Enum_2Enum,V2h2: tyop_2Enum_2Enum,V3l2: tyop_2Enum_2Enum,V4n: tyop_2Enum_2Enum] :
( ( c_2Ebit_2EBITS @ V2h2 @ V3l2 @ ( c_2Ebit_2EBITS @ V0h1 @ V1l1 @ V4n ) )
= ( c_2Ebit_2EBITS @ ( c_2Earithmetic_2EMIN @ V0h1 @ ( c_2Earithmetic_2E_2B @ V2h2 @ V1l1 ) ) @ ( c_2Earithmetic_2E_2B @ V3l2 @ V1l1 ) @ V4n ) ) ).
thf(thm_2Ebit_2ENOT__MOD2__LEM,axiom,
! [V0n: tyop_2Enum_2Enum] :
( ( (~)
@ ( ( c_2Earithmetic_2EMOD @ V0n @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) )
= c_2Enum_2E0 ) )
<=> ( ( c_2Earithmetic_2EMOD @ V0n @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) )
= ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ).
thf(thm_2Ebit_2ENOT__MOD2__LEM2,axiom,
! [V0n: tyop_2Enum_2Enum] :
( ( (~)
@ ( ( c_2Earithmetic_2EMOD @ V0n @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) )
= ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) )
<=> ( ( c_2Earithmetic_2EMOD @ V0n @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) )
= c_2Enum_2E0 ) ) ).
thf(thm_2Ebit_2EODD__MOD2__LEM,axiom,
! [V0n: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2EODD @ V0n )
<=> ( ( c_2Earithmetic_2EMOD @ V0n @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) )
= ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ).
thf(thm_2Ebit_2EDIV__MULT__THM,axiom,
! [V0x: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_2A @ ( c_2Earithmetic_2EDIV @ V1n @ ( c_2Earithmetic_2EEXP @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) @ V0x ) ) @ ( c_2Earithmetic_2EEXP @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) @ V0x ) )
= ( c_2Earithmetic_2E_2D @ V1n @ ( c_2Earithmetic_2EMOD @ V1n @ ( c_2Earithmetic_2EEXP @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) @ V0x ) ) ) ) ).
thf(thm_2Ebit_2EDIV__MULT__THM2,axiom,
! [V0n: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_2A @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) @ ( c_2Earithmetic_2EDIV @ V0n @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) )
= ( c_2Earithmetic_2E_2D @ V0n @ ( c_2Earithmetic_2EMOD @ V0n @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) ) ) ).
thf(thm_2Ebit_2ELESS__EQ__EXP__MULT,axiom,
! [V0a: tyop_2Enum_2Enum,V1b: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_3C_3D @ V0a @ V1b )
=> ? [V2p: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2EEXP @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) @ V1b )
= ( c_2Earithmetic_2E_2A @ V2p @ ( c_2Earithmetic_2EEXP @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) @ V0a ) ) ) ) ).
thf(thm_2Ebit_2ESLICE__THM,axiom,
! [V0n: tyop_2Enum_2Enum,V1h: tyop_2Enum_2Enum,V2l: tyop_2Enum_2Enum] :
( ( c_2Ebit_2ESLICE @ V1h @ V2l @ V0n )
= ( c_2Earithmetic_2E_2A @ ( c_2Ebit_2EBITS @ V1h @ V2l @ V0n ) @ ( c_2Earithmetic_2EEXP @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) @ V2l ) ) ) ).
thf(thm_2Ebit_2ESLICELT__THM,axiom,
! [V0h: tyop_2Enum_2Enum,V1l: tyop_2Enum_2Enum,V2n: tyop_2Enum_2Enum] : ( c_2Eprim__rec_2E_3C @ ( c_2Ebit_2ESLICE @ V0h @ V1l @ V2n ) @ ( c_2Earithmetic_2EEXP @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) @ ( c_2Enum_2ESUC @ V0h ) ) ) ).
thf(thm_2Ebit_2EBITS__SLICE__THM,axiom,
! [V0h: tyop_2Enum_2Enum,V1l: tyop_2Enum_2Enum,V2n: tyop_2Enum_2Enum] :
( ( c_2Ebit_2EBITS @ V0h @ V1l @ ( c_2Ebit_2ESLICE @ V0h @ V1l @ V2n ) )
= ( c_2Ebit_2EBITS @ V0h @ V1l @ V2n ) ) ).
thf(thm_2Ebit_2EBITS__SLICE__THM2,axiom,
! [V0h: tyop_2Enum_2Enum,V1h2: tyop_2Enum_2Enum,V2l: tyop_2Enum_2Enum,V3n: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_3C_3D @ V0h @ V1h2 )
=> ( ( c_2Ebit_2EBITS @ V1h2 @ V2l @ ( c_2Ebit_2ESLICE @ V0h @ V2l @ V3n ) )
= ( c_2Ebit_2EBITS @ V0h @ V2l @ V3n ) ) ) ).
thf(thm_2Ebit_2ESLICE__ZERO__THM,axiom,
! [V0n: tyop_2Enum_2Enum,V1h: tyop_2Enum_2Enum] :
( ( c_2Ebit_2ESLICE @ V1h @ c_2Enum_2E0 @ V0n )
= ( c_2Ebit_2EBITS @ V1h @ c_2Enum_2E0 @ V0n ) ) ).
thf(thm_2Ebit_2EMOD__2EXP__MONO,axiom,
! [V0n: tyop_2Enum_2Enum,V1h: tyop_2Enum_2Enum,V2l: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_3C_3D @ V2l @ V1h )
=> ( c_2Earithmetic_2E_3C_3D @ ( c_2Earithmetic_2EMOD @ V0n @ ( c_2Earithmetic_2EEXP @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) @ V2l ) ) @ ( c_2Earithmetic_2EMOD @ V0n @ ( c_2Earithmetic_2EEXP @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) @ ( c_2Enum_2ESUC @ V1h ) ) ) ) ) ).
thf(thm_2Ebit_2ESLICE__COMP__THM,axiom,
! [V0h: tyop_2Enum_2Enum,V1m: tyop_2Enum_2Enum,V2l: tyop_2Enum_2Enum,V3n: tyop_2Enum_2Enum] :
( ( ( c_2Earithmetic_2E_3C_3D @ ( c_2Enum_2ESUC @ V1m ) @ V0h )
& ( c_2Earithmetic_2E_3C_3D @ V2l @ V1m ) )
=> ( ( c_2Earithmetic_2E_2B @ ( c_2Ebit_2ESLICE @ V0h @ ( c_2Enum_2ESUC @ V1m ) @ V3n ) @ ( c_2Ebit_2ESLICE @ V1m @ V2l @ V3n ) )
= ( c_2Ebit_2ESLICE @ V0h @ V2l @ V3n ) ) ) ).
thf(thm_2Ebit_2ESLICE__COMP__RWT,axiom,
! [V0h: tyop_2Enum_2Enum,V1m_27: tyop_2Enum_2Enum,V2m: tyop_2Enum_2Enum,V3l: tyop_2Enum_2Enum,V4n: tyop_2Enum_2Enum] :
( ( ( c_2Earithmetic_2E_3C_3D @ V3l @ V2m )
& ( V1m_27
= ( c_2Earithmetic_2E_2B @ V2m @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) )
& ( c_2Eprim__rec_2E_3C @ V2m @ V0h ) )
=> ( ( c_2Earithmetic_2E_2B @ ( c_2Ebit_2ESLICE @ V0h @ V1m_27 @ V4n ) @ ( c_2Ebit_2ESLICE @ V2m @ V3l @ V4n ) )
= ( c_2Ebit_2ESLICE @ V0h @ V3l @ V4n ) ) ) ).
thf(thm_2Ebit_2ESLICE__ZERO,axiom,
! [V0h: tyop_2Enum_2Enum,V1l: tyop_2Enum_2Enum,V2n: tyop_2Enum_2Enum] :
( ( c_2Eprim__rec_2E_3C @ V0h @ V1l )
=> ( ( c_2Ebit_2ESLICE @ V0h @ V1l @ V2n )
= c_2Enum_2E0 ) ) ).
thf(thm_2Ebit_2ESLICE__ZERO2,axiom,
! [V0l: tyop_2Enum_2Enum,V1h: tyop_2Enum_2Enum] :
( ( c_2Ebit_2ESLICE @ V1h @ V0l @ c_2Enum_2E0 )
= c_2Enum_2E0 ) ).
thf(thm_2Ebit_2EBITS__SUM,axiom,
! [V0h: tyop_2Enum_2Enum,V1l: tyop_2Enum_2Enum,V2a: tyop_2Enum_2Enum,V3b: tyop_2Enum_2Enum] :
( ( c_2Eprim__rec_2E_3C @ V3b @ ( c_2Earithmetic_2EEXP @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) @ V1l ) )
=> ( ( c_2Ebit_2EBITS @ V0h @ V1l @ ( c_2Earithmetic_2E_2B @ ( c_2Earithmetic_2E_2A @ V2a @ ( c_2Earithmetic_2EEXP @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) @ V1l ) ) @ V3b ) )
= ( c_2Ebit_2EBITS @ V0h @ V1l @ ( c_2Earithmetic_2E_2A @ V2a @ ( c_2Earithmetic_2EEXP @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) @ V1l ) ) ) ) ) ).
thf(thm_2Ebit_2EBITS__SUM2,axiom,
! [V0h: tyop_2Enum_2Enum,V1l: tyop_2Enum_2Enum,V2a: tyop_2Enum_2Enum,V3b: tyop_2Enum_2Enum] :
( ( c_2Ebit_2EBITS @ V0h @ V1l @ ( c_2Earithmetic_2E_2B @ ( c_2Earithmetic_2E_2A @ V2a @ ( c_2Earithmetic_2EEXP @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) @ ( c_2Enum_2ESUC @ V0h ) ) ) @ V3b ) )
= ( c_2Ebit_2EBITS @ V0h @ V1l @ V3b ) ) ).
thf(thm_2Ebit_2ESLICE__COMP__THM2,axiom,
! [V0h: tyop_2Enum_2Enum,V1l: tyop_2Enum_2Enum,V2x: tyop_2Enum_2Enum,V3y: tyop_2Enum_2Enum,V4n: tyop_2Enum_2Enum] :
( ( ( c_2Earithmetic_2E_3C_3D @ V0h @ V2x )
& ( c_2Earithmetic_2E_3C_3D @ V3y @ V1l ) )
=> ( ( c_2Ebit_2ESLICE @ V0h @ V1l @ ( c_2Ebit_2ESLICE @ V2x @ V3y @ V4n ) )
= ( c_2Ebit_2ESLICE @ V0h @ V1l @ V4n ) ) ) ).
thf(thm_2Ebit_2EBITS__SUM3,axiom,
! [V0h: tyop_2Enum_2Enum,V1a: tyop_2Enum_2Enum,V2b: tyop_2Enum_2Enum] :
( ( c_2Ebit_2EBITS @ V0h @ c_2Enum_2E0 @ ( c_2Earithmetic_2E_2B @ ( c_2Ebit_2EBITS @ V0h @ c_2Enum_2E0 @ V1a ) @ ( c_2Ebit_2EBITS @ V0h @ c_2Enum_2E0 @ V2b ) ) )
= ( c_2Ebit_2EBITS @ V0h @ c_2Enum_2E0 @ ( c_2Earithmetic_2E_2B @ V1a @ V2b ) ) ) ).
thf(thm_2Ebit_2EBITS__MUL,axiom,
! [V0h: tyop_2Enum_2Enum,V1a: tyop_2Enum_2Enum,V2b: tyop_2Enum_2Enum] :
( ( c_2Ebit_2EBITS @ V0h @ c_2Enum_2E0 @ ( c_2Earithmetic_2E_2A @ ( c_2Ebit_2EBITS @ V0h @ c_2Enum_2E0 @ V1a ) @ ( c_2Ebit_2EBITS @ V0h @ c_2Enum_2E0 @ V2b ) ) )
= ( c_2Ebit_2EBITS @ V0h @ c_2Enum_2E0 @ ( c_2Earithmetic_2E_2A @ V1a @ V2b ) ) ) ).
thf(thm_2Ebit_2EBIT__COMP__THM3,axiom,
! [V0h: tyop_2Enum_2Enum,V1m: tyop_2Enum_2Enum,V2l: tyop_2Enum_2Enum,V3n: tyop_2Enum_2Enum] :
( ( ( c_2Earithmetic_2E_3C_3D @ ( c_2Enum_2ESUC @ V1m ) @ V0h )
& ( c_2Earithmetic_2E_3C_3D @ V2l @ V1m ) )
=> ( ( c_2Earithmetic_2E_2B @ ( c_2Earithmetic_2E_2A @ ( c_2Ebit_2EBITS @ V0h @ ( c_2Enum_2ESUC @ V1m ) @ V3n ) @ ( c_2Earithmetic_2EEXP @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) @ ( c_2Earithmetic_2E_2D @ ( c_2Enum_2ESUC @ V1m ) @ V2l ) ) ) @ ( c_2Ebit_2EBITS @ V1m @ V2l @ V3n ) )
= ( c_2Ebit_2EBITS @ V0h @ V2l @ V3n ) ) ) ).
thf(thm_2Ebit_2ENOT__BIT,axiom,
! [V0n: tyop_2Enum_2Enum,V1a: tyop_2Enum_2Enum] :
( ( (~) @ ( c_2Ebit_2EBIT @ V0n @ V1a ) )
<=> ( ( c_2Ebit_2EBITS @ V0n @ V0n @ V1a )
= c_2Enum_2E0 ) ) ).
thf(thm_2Ebit_2ENOT__BITS,axiom,
! [V0n: tyop_2Enum_2Enum,V1a: tyop_2Enum_2Enum] :
( ( (~)
@ ( ( c_2Ebit_2EBITS @ V0n @ V0n @ V1a )
= c_2Enum_2E0 ) )
<=> ( ( c_2Ebit_2EBITS @ V0n @ V0n @ V1a )
= ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ).
thf(thm_2Ebit_2ENOT__BITS2,axiom,
! [V0n: tyop_2Enum_2Enum,V1a: tyop_2Enum_2Enum] :
( ( (~)
@ ( ( c_2Ebit_2EBITS @ V0n @ V0n @ V1a )
= ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) )
<=> ( ( c_2Ebit_2EBITS @ V0n @ V0n @ V1a )
= c_2Enum_2E0 ) ) ).
thf(thm_2Ebit_2EBIT__SLICE,axiom,
! [V0n: tyop_2Enum_2Enum,V1a: tyop_2Enum_2Enum,V2b: tyop_2Enum_2Enum] :
( ( ( c_2Ebit_2EBIT @ V0n @ V1a )
= ( c_2Ebit_2EBIT @ V0n @ V2b ) )
<=> ( ( c_2Ebit_2ESLICE @ V0n @ V0n @ V1a )
= ( c_2Ebit_2ESLICE @ V0n @ V0n @ V2b ) ) ) ).
thf(thm_2Ebit_2EBIT__SLICE__LEM,axiom,
! [V0y: tyop_2Enum_2Enum,V1x: tyop_2Enum_2Enum,V2n: tyop_2Enum_2Enum] :
( ( c_2Ebit_2ESBIT @ ( c_2Ebit_2EBIT @ V1x @ V2n ) @ ( c_2Earithmetic_2E_2B @ V1x @ V0y ) )
= ( c_2Earithmetic_2E_2A @ ( c_2Ebit_2ESLICE @ V1x @ V1x @ V2n ) @ ( c_2Earithmetic_2EEXP @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) @ V0y ) ) ) ).
thf(thm_2Ebit_2EBIT__SLICE__THM,axiom,
! [V0x: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum] :
( ( c_2Ebit_2ESBIT @ ( c_2Ebit_2EBIT @ V0x @ V1n ) @ V0x )
= ( c_2Ebit_2ESLICE @ V0x @ V0x @ V1n ) ) ).
thf(thm_2Ebit_2EBIT__SLICE__THM2,axiom,
! [V0b: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum] :
( ( c_2Ebit_2EBIT @ V0b @ V1n )
<=> ( ( c_2Ebit_2ESLICE @ V0b @ V0b @ V1n )
= ( c_2Earithmetic_2EEXP @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) @ V0b ) ) ) ).
thf(thm_2Ebit_2EBIT__SLICE__THM3,axiom,
! [V0b: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum] :
( ( (~) @ ( c_2Ebit_2EBIT @ V0b @ V1n ) )
<=> ( ( c_2Ebit_2ESLICE @ V0b @ V0b @ V1n )
= c_2Enum_2E0 ) ) ).
thf(thm_2Ebit_2EBIT__SLICE__THM4,axiom,
! [V0b: tyop_2Enum_2Enum,V1h: tyop_2Enum_2Enum,V2l: tyop_2Enum_2Enum,V3n: tyop_2Enum_2Enum] :
( ( c_2Ebit_2EBIT @ V0b @ ( c_2Ebit_2ESLICE @ V1h @ V2l @ V3n ) )
<=> ( ( c_2Earithmetic_2E_3C_3D @ V2l @ V0b )
& ( c_2Earithmetic_2E_3C_3D @ V0b @ V1h )
& ( c_2Ebit_2EBIT @ V0b @ V3n ) ) ) ).
thf(thm_2Ebit_2ESBIT__DIV,axiom,
! [V0b: $o,V1m: tyop_2Enum_2Enum,V2n: tyop_2Enum_2Enum] :
( ( c_2Eprim__rec_2E_3C @ V2n @ V1m )
=> ( ( c_2Ebit_2ESBIT @ V0b @ ( c_2Earithmetic_2E_2D @ V1m @ V2n ) )
= ( c_2Earithmetic_2EDIV @ ( c_2Ebit_2ESBIT @ V0b @ V1m ) @ ( c_2Earithmetic_2EEXP @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) @ V2n ) ) ) ) ).
thf(thm_2Ebit_2EBITS__SUC,axiom,
! [V0h: tyop_2Enum_2Enum,V1l: tyop_2Enum_2Enum,V2n: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_3C_3D @ V1l @ ( c_2Enum_2ESUC @ V0h ) )
=> ( ( c_2Earithmetic_2E_2B @ ( c_2Ebit_2ESBIT @ ( c_2Ebit_2EBIT @ ( c_2Enum_2ESUC @ V0h ) @ V2n ) @ ( c_2Earithmetic_2E_2D @ ( c_2Enum_2ESUC @ V0h ) @ V1l ) ) @ ( c_2Ebit_2EBITS @ V0h @ V1l @ V2n ) )
= ( c_2Ebit_2EBITS @ ( c_2Enum_2ESUC @ V0h ) @ V1l @ V2n ) ) ) ).
thf(thm_2Ebit_2EBITS__SUC__THM,axiom,
! [V0h: tyop_2Enum_2Enum,V1l: tyop_2Enum_2Enum,V2n: tyop_2Enum_2Enum] :
( ( c_2Ebit_2EBITS @ ( c_2Enum_2ESUC @ V0h ) @ V1l @ V2n )
= ( c_2Ebool_2ECOND @ tyop_2Enum_2Enum @ ( c_2Eprim__rec_2E_3C @ ( c_2Enum_2ESUC @ V0h ) @ V1l ) @ c_2Enum_2E0 @ ( c_2Earithmetic_2E_2B @ ( c_2Ebit_2ESBIT @ ( c_2Ebit_2EBIT @ ( c_2Enum_2ESUC @ V0h ) @ V2n ) @ ( c_2Earithmetic_2E_2D @ ( c_2Enum_2ESUC @ V0h ) @ V1l ) ) @ ( c_2Ebit_2EBITS @ V0h @ V1l @ V2n ) ) ) ) ).
thf(thm_2Ebit_2EBIT__BITS__THM,axiom,
! [V0h: tyop_2Enum_2Enum,V1l: tyop_2Enum_2Enum,V2a: tyop_2Enum_2Enum,V3b: tyop_2Enum_2Enum] :
( ! [V4x: tyop_2Enum_2Enum] :
( ( ( c_2Earithmetic_2E_3C_3D @ V1l @ V4x )
& ( c_2Earithmetic_2E_3C_3D @ V4x @ V0h ) )
=> ( ( c_2Ebit_2EBIT @ V4x @ V2a )
= ( c_2Ebit_2EBIT @ V4x @ V3b ) ) )
<=> ( ( c_2Ebit_2EBITS @ V0h @ V1l @ V2a )
= ( c_2Ebit_2EBITS @ V0h @ V1l @ V3b ) ) ) ).
thf(thm_2Ebit_2EBITS__ZERO5,axiom,
! [V0n: tyop_2Enum_2Enum,V1m: tyop_2Enum_2Enum] :
( ! [V2i: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_3C_3D @ V2i @ V0n )
=> ( (~) @ ( c_2Ebit_2EBIT @ V2i @ V1m ) ) )
=> ( ( c_2Ebit_2EBITS @ V0n @ c_2Enum_2E0 @ V1m )
= c_2Enum_2E0 ) ) ).
thf(thm_2Ebit_2EBIT0__ODD,axiom,
( ( c_2Ebit_2EBIT @ c_2Enum_2E0 )
= c_2Earithmetic_2EODD ) ).
thf(thm_2Ebit_2EBITV__THM,axiom,
! [V0b: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum] :
( ( c_2Ebit_2EBITV @ V1n @ V0b )
= ( c_2Ebit_2ESBIT @ ( c_2Ebit_2EBIT @ V0b @ V1n ) @ c_2Enum_2E0 ) ) ).
thf(thm_2Ebit_2EADD__BIT0,axiom,
! [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum] :
( ( c_2Ebit_2EBIT @ c_2Enum_2E0 @ ( c_2Earithmetic_2E_2B @ V0m @ V1n ) )
<=> ( (~)
@ ( ( c_2Ebit_2EBIT @ c_2Enum_2E0 @ V0m )
= ( c_2Ebit_2EBIT @ c_2Enum_2E0 @ V1n ) ) ) ) ).
thf(thm_2Ebit_2EADD__BITS__SUC,axiom,
! [V0n: tyop_2Enum_2Enum,V1a: tyop_2Enum_2Enum,V2b: tyop_2Enum_2Enum] :
( ( c_2Ebit_2EBITS @ ( c_2Enum_2ESUC @ V0n ) @ ( c_2Enum_2ESUC @ V0n ) @ ( c_2Earithmetic_2E_2B @ V1a @ V2b ) )
= ( c_2Earithmetic_2EMOD @ ( c_2Earithmetic_2E_2B @ ( c_2Earithmetic_2E_2B @ ( c_2Ebit_2EBITS @ ( c_2Enum_2ESUC @ V0n ) @ ( c_2Enum_2ESUC @ V0n ) @ V1a ) @ ( c_2Ebit_2EBITS @ ( c_2Enum_2ESUC @ V0n ) @ ( c_2Enum_2ESUC @ V0n ) @ V2b ) ) @ ( c_2Ebit_2EBITS @ ( c_2Enum_2ESUC @ V0n ) @ ( c_2Enum_2ESUC @ V0n ) @ ( c_2Earithmetic_2E_2B @ ( c_2Ebit_2EBITS @ V0n @ c_2Enum_2E0 @ V1a ) @ ( c_2Ebit_2EBITS @ V0n @ c_2Enum_2E0 @ V2b ) ) ) ) @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) ) ).
thf(thm_2Ebit_2EADD__BIT__SUC,axiom,
! [V0n: tyop_2Enum_2Enum,V1a: tyop_2Enum_2Enum,V2b: tyop_2Enum_2Enum] :
( ( c_2Ebit_2EBIT @ ( c_2Enum_2ESUC @ V0n ) @ ( c_2Earithmetic_2E_2B @ V1a @ V2b ) )
= ( c_2Ebool_2ECOND @ $o @ ( c_2Ebit_2EBIT @ ( c_2Enum_2ESUC @ V0n ) @ ( c_2Earithmetic_2E_2B @ ( c_2Ebit_2EBITS @ V0n @ c_2Enum_2E0 @ V1a ) @ ( c_2Ebit_2EBITS @ V0n @ c_2Enum_2E0 @ V2b ) ) ) @ ( c_2Emin_2E_3D @ $o @ ( c_2Ebit_2EBIT @ ( c_2Enum_2ESUC @ V0n ) @ V1a ) @ ( c_2Ebit_2EBIT @ ( c_2Enum_2ESUC @ V0n ) @ V2b ) ) @ ( c_2Ebool_2E_7E @ ( c_2Emin_2E_3D @ $o @ ( c_2Ebit_2EBIT @ ( c_2Enum_2ESUC @ V0n ) @ V1a ) @ ( c_2Ebit_2EBIT @ ( c_2Enum_2ESUC @ V0n ) @ V2b ) ) ) ) ) ).
thf(thm_2Ebit_2EBITWISE__LT__2EXP,axiom,
! [V0n: tyop_2Enum_2Enum,V1op: $o > $o > $o,V2a: tyop_2Enum_2Enum,V3b: tyop_2Enum_2Enum] : ( c_2Eprim__rec_2E_3C @ ( c_2Ebit_2EBITWISE @ V0n @ V1op @ V2a @ V3b ) @ ( c_2Earithmetic_2EEXP @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) @ V0n ) ) ).
thf(thm_2Ebit_2EBITWISE__THM,axiom,
! [V0x: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum,V2op: $o > $o > $o,V3a: tyop_2Enum_2Enum,V4b: tyop_2Enum_2Enum] :
( ( c_2Eprim__rec_2E_3C @ V0x @ V1n )
=> ( ( c_2Ebit_2EBIT @ V0x @ ( c_2Ebit_2EBITWISE @ V1n @ V2op @ V3a @ V4b ) )
= ( V2op @ ( c_2Ebit_2EBIT @ V0x @ V3a ) @ ( c_2Ebit_2EBIT @ V0x @ V4b ) ) ) ) ).
thf(thm_2Ebit_2EBITWISE__COR,axiom,
! [V0x: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum,V2op: $o > $o > $o,V3a: tyop_2Enum_2Enum,V4b: tyop_2Enum_2Enum] :
( ( c_2Eprim__rec_2E_3C @ V0x @ V1n )
=> ( ( V2op @ ( c_2Ebit_2EBIT @ V0x @ V3a ) @ ( c_2Ebit_2EBIT @ V0x @ V4b ) )
=> ( ( c_2Earithmetic_2EMOD @ ( c_2Earithmetic_2EDIV @ ( c_2Ebit_2EBITWISE @ V1n @ V2op @ V3a @ V4b ) @ ( c_2Earithmetic_2EEXP @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) @ V0x ) ) @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) )
= ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ) ).
thf(thm_2Ebit_2EBITWISE__NOT__COR,axiom,
! [V0x: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum,V2op: $o > $o > $o,V3a: tyop_2Enum_2Enum,V4b: tyop_2Enum_2Enum] :
( ( c_2Eprim__rec_2E_3C @ V0x @ V1n )
=> ( ( (~) @ ( V2op @ ( c_2Ebit_2EBIT @ V0x @ V3a ) @ ( c_2Ebit_2EBIT @ V0x @ V4b ) ) )
=> ( ( c_2Earithmetic_2EMOD @ ( c_2Earithmetic_2EDIV @ ( c_2Ebit_2EBITWISE @ V1n @ V2op @ V3a @ V4b ) @ ( c_2Earithmetic_2EEXP @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) @ V0x ) ) @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) )
= c_2Enum_2E0 ) ) ) ).
thf(thm_2Ebit_2EBITWISE__BITS,axiom,
! [V0wl: tyop_2Enum_2Enum,V1op: $o > $o > $o,V2a: tyop_2Enum_2Enum,V3b: tyop_2Enum_2Enum] :
( ( c_2Ebit_2EBITWISE @ ( c_2Enum_2ESUC @ V0wl ) @ V1op @ ( c_2Ebit_2EBITS @ V0wl @ c_2Enum_2E0 @ V2a ) @ ( c_2Ebit_2EBITS @ V0wl @ c_2Enum_2E0 @ V3b ) )
= ( c_2Ebit_2EBITWISE @ ( c_2Enum_2ESUC @ V0wl ) @ V1op @ V2a @ V3b ) ) ).
thf(thm_2Ebit_2ENOT__BIT__GT__TWOEXP,axiom,
! [V0i: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum] :
( ( c_2Eprim__rec_2E_3C @ V1n @ ( c_2Earithmetic_2EEXP @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) @ V0i ) )
=> ( (~) @ ( c_2Ebit_2EBIT @ V0i @ V1n ) ) ) ).
thf(thm_2Ebit_2EBIT__IMP__GE__TWOEXP,axiom,
! [V0i: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum] :
( ( c_2Ebit_2EBIT @ V0i @ V1n )
=> ( c_2Earithmetic_2E_3C_3D @ ( c_2Earithmetic_2EEXP @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) @ V0i ) @ V1n ) ) ).
thf(thm_2Ebit_2EBITWISE__ONE__COMP__LEM,axiom,
! [V0n: tyop_2Enum_2Enum,V1a: tyop_2Enum_2Enum,V2b: tyop_2Enum_2Enum] :
( ( c_2Ebit_2EBITWISE @ ( c_2Enum_2ESUC @ V0n )
@ ^ [V3x: $o,V4y: $o] : ( c_2Ebool_2E_7E @ V3x )
@ V1a
@ V2b )
= ( c_2Earithmetic_2E_2D @ ( c_2Earithmetic_2E_2D @ ( c_2Earithmetic_2EEXP @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) @ ( c_2Enum_2ESUC @ V0n ) ) @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) @ ( c_2Ebit_2EBITS @ V0n @ c_2Enum_2E0 @ V1a ) ) ) ).
thf(thm_2Ebit_2EBIT__COMPLEMENT,axiom,
! [V0n: tyop_2Enum_2Enum,V1i: tyop_2Enum_2Enum,V2a: tyop_2Enum_2Enum] :
( ( c_2Ebit_2EBIT @ V1i @ ( c_2Earithmetic_2E_2D @ ( c_2Earithmetic_2EEXP @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) @ V0n ) @ ( c_2Earithmetic_2EMOD @ V2a @ ( c_2Earithmetic_2EEXP @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) @ V0n ) ) ) )
<=> ( ( ( ( c_2Earithmetic_2EMOD @ V2a @ ( c_2Earithmetic_2EEXP @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) @ V0n ) )
= c_2Enum_2E0 )
& ( V1i = V0n ) )
| ( ( (~)
@ ( ( c_2Earithmetic_2EMOD @ V2a @ ( c_2Earithmetic_2EEXP @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) @ V0n ) )
= c_2Enum_2E0 ) )
& ( c_2Eprim__rec_2E_3C @ V1i @ V0n )
& ( (~) @ ( c_2Ebit_2EBIT @ V1i @ ( c_2Earithmetic_2E_2D @ ( c_2Earithmetic_2EMOD @ V2a @ ( c_2Earithmetic_2EEXP @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) @ V0n ) ) @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ) ) ) ) ).
thf(thm_2Ebit_2EBIT__OF__BITS__THM,axiom,
! [V0n: tyop_2Enum_2Enum,V1h: tyop_2Enum_2Enum,V2l: tyop_2Enum_2Enum,V3a: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_3C_3D @ ( c_2Earithmetic_2E_2B @ V2l @ V0n ) @ V1h )
=> ( ( c_2Ebit_2EBIT @ V0n @ ( c_2Ebit_2EBITS @ V1h @ V2l @ V3a ) )
= ( c_2Ebit_2EBIT @ ( c_2Earithmetic_2E_2B @ V2l @ V0n ) @ V3a ) ) ) ).
thf(thm_2Ebit_2EBIT__SHIFT__THM,axiom,
! [V0n: tyop_2Enum_2Enum,V1a: tyop_2Enum_2Enum,V2s: tyop_2Enum_2Enum] :
( ( c_2Ebit_2EBIT @ ( c_2Earithmetic_2E_2B @ V0n @ V2s ) @ ( c_2Earithmetic_2E_2A @ V1a @ ( c_2Earithmetic_2EEXP @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) @ V2s ) ) )
= ( c_2Ebit_2EBIT @ V0n @ V1a ) ) ).
thf(thm_2Ebit_2EBIT__SHIFT__THM2,axiom,
! [V0n: tyop_2Enum_2Enum,V1a: tyop_2Enum_2Enum,V2s: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_3C_3D @ V2s @ V0n )
=> ( ( c_2Ebit_2EBIT @ V0n @ ( c_2Earithmetic_2E_2A @ V1a @ ( c_2Earithmetic_2EEXP @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) @ V2s ) ) )
= ( c_2Ebit_2EBIT @ ( c_2Earithmetic_2E_2D @ V0n @ V2s ) @ V1a ) ) ) ).
thf(thm_2Ebit_2EBIT__SHIFT__THM3,axiom,
! [V0n: tyop_2Enum_2Enum,V1a: tyop_2Enum_2Enum,V2s: tyop_2Enum_2Enum] :
( ( c_2Eprim__rec_2E_3C @ V0n @ V2s )
=> ( (~) @ ( c_2Ebit_2EBIT @ V0n @ ( c_2Earithmetic_2E_2A @ V1a @ ( c_2Earithmetic_2EEXP @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) @ V2s ) ) ) ) ) ).
thf(thm_2Ebit_2EBIT__OF__BITS__THM2,axiom,
! [V0h: tyop_2Enum_2Enum,V1l: tyop_2Enum_2Enum,V2x: tyop_2Enum_2Enum,V3n: tyop_2Enum_2Enum] :
( ( c_2Eprim__rec_2E_3C @ V0h @ ( c_2Earithmetic_2E_2B @ V1l @ V2x ) )
=> ( (~) @ ( c_2Ebit_2EBIT @ V2x @ ( c_2Ebit_2EBITS @ V0h @ V1l @ V3n ) ) ) ) ).
thf(thm_2Ebit_2EBIT__DIV2,axiom,
! [V0n: tyop_2Enum_2Enum,V1i: tyop_2Enum_2Enum] :
( ( c_2Ebit_2EBIT @ V0n @ ( c_2Earithmetic_2EDIV @ V1i @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) )
= ( c_2Ebit_2EBIT @ ( c_2Enum_2ESUC @ V0n ) @ V1i ) ) ).
thf(thm_2Ebit_2EBIT__SHIFT__THM4,axiom,
! [V0n: tyop_2Enum_2Enum,V1i: tyop_2Enum_2Enum,V2a: tyop_2Enum_2Enum] :
( ( c_2Ebit_2EBIT @ V1i @ ( c_2Earithmetic_2EDIV @ V2a @ ( c_2Earithmetic_2EEXP @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) @ V0n ) ) )
= ( c_2Ebit_2EBIT @ ( c_2Earithmetic_2E_2B @ V1i @ V0n ) @ V2a ) ) ).
thf(thm_2Ebit_2EDIV__LT,axiom,
! [V0n: tyop_2Enum_2Enum,V1m: tyop_2Enum_2Enum,V2a: tyop_2Enum_2Enum] :
( ( ( c_2Eprim__rec_2E_3C @ V0n @ V1m )
& ( c_2Eprim__rec_2E_3C @ V2a @ ( c_2Earithmetic_2EEXP @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) @ V1m ) ) )
=> ( c_2Eprim__rec_2E_3C @ ( c_2Earithmetic_2EDIV @ V2a @ ( c_2Earithmetic_2EEXP @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) @ V0n ) ) @ ( c_2Earithmetic_2EEXP @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) @ V1m ) ) ) ).
thf(thm_2Ebit_2EMOD__ZERO__GT,axiom,
! [V0n: tyop_2Enum_2Enum,V1a: tyop_2Enum_2Enum] :
( ( ( (~) @ ( V1a = c_2Enum_2E0 ) )
& ( ( c_2Earithmetic_2EMOD @ V1a @ ( c_2Earithmetic_2EEXP @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) @ V0n ) )
= c_2Enum_2E0 ) )
=> ( c_2Earithmetic_2E_3C_3D @ ( c_2Earithmetic_2EEXP @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) @ V0n ) @ V1a ) ) ).
thf(thm_2Ebit_2EDIV__GT0,axiom,
! [V0a: tyop_2Enum_2Enum,V1b: tyop_2Enum_2Enum] :
( ( ( c_2Earithmetic_2E_3C_3D @ V1b @ V0a )
& ( c_2Eprim__rec_2E_3C @ c_2Enum_2E0 @ V1b ) )
=> ( c_2Eprim__rec_2E_3C @ c_2Enum_2E0 @ ( c_2Earithmetic_2EDIV @ V0a @ V1b ) ) ) ).
thf(thm_2Ebit_2EDIV__SUB1,axiom,
! [V0a: tyop_2Enum_2Enum,V1b: tyop_2Enum_2Enum] :
( ( ( c_2Earithmetic_2E_3C_3D @ ( c_2Earithmetic_2EEXP @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) @ V1b ) @ V0a )
& ( ( c_2Earithmetic_2EMOD @ V0a @ ( c_2Earithmetic_2EEXP @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) @ V1b ) )
= c_2Enum_2E0 ) )
=> ( ( c_2Earithmetic_2E_2D @ ( c_2Earithmetic_2EDIV @ V0a @ ( c_2Earithmetic_2EEXP @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) @ V1b ) ) @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) )
= ( c_2Earithmetic_2EDIV @ ( c_2Earithmetic_2E_2D @ V0a @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) @ ( c_2Earithmetic_2EEXP @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) @ V1b ) ) ) ) ).
thf(thm_2Ebit_2EBIT__EXP__SUB1,axiom,
! [V0b: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum] :
( ( c_2Ebit_2EBIT @ V0b @ ( c_2Earithmetic_2E_2D @ ( c_2Earithmetic_2EEXP @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) @ V1n ) @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) )
= ( c_2Eprim__rec_2E_3C @ V0b @ V1n ) ) ).
thf(thm_2Ebit_2EBIT__SHIFT__THM5,axiom,
! [V0n: tyop_2Enum_2Enum,V1m: tyop_2Enum_2Enum,V2i: tyop_2Enum_2Enum,V3a: tyop_2Enum_2Enum] :
( ( ( c_2Eprim__rec_2E_3C @ ( c_2Earithmetic_2E_2B @ V2i @ V0n ) @ V1m )
& ( c_2Eprim__rec_2E_3C @ V3a @ ( c_2Earithmetic_2EEXP @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) @ V1m ) ) )
=> ( ( c_2Ebit_2EBIT @ V2i @ ( c_2Earithmetic_2E_2D @ ( c_2Earithmetic_2EEXP @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) @ V1m ) @ ( c_2Earithmetic_2EMOD @ ( c_2Earithmetic_2E_2B @ ( c_2Earithmetic_2EDIV @ V3a @ ( c_2Earithmetic_2EEXP @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) @ V0n ) ) @ ( c_2Ebool_2ECOND @ tyop_2Enum_2Enum @ ( c_2Emin_2E_3D @ tyop_2Enum_2Enum @ ( c_2Earithmetic_2EMOD @ V3a @ ( c_2Earithmetic_2EEXP @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) @ V0n ) ) @ c_2Enum_2E0 ) @ c_2Enum_2E0 @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) @ ( c_2Earithmetic_2EEXP @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) @ V1m ) ) ) )
= ( c_2Ebit_2EBIT @ ( c_2Earithmetic_2E_2B @ V2i @ V0n ) @ ( c_2Earithmetic_2E_2D @ ( c_2Earithmetic_2EEXP @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) @ V1m ) @ ( c_2Earithmetic_2EMOD @ V3a @ ( c_2Earithmetic_2EEXP @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) @ V1m ) ) ) ) ) ) ).
thf(thm_2Ebit_2ESBIT__MULT,axiom,
! [V0b: $o,V1m: tyop_2Enum_2Enum,V2n: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_2A @ ( c_2Ebit_2ESBIT @ V0b @ V2n ) @ ( c_2Earithmetic_2EEXP @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) @ V1m ) )
= ( c_2Ebit_2ESBIT @ V0b @ ( c_2Earithmetic_2E_2B @ V2n @ V1m ) ) ) ).
thf(thm_2Ebit_2EBITWISE__EVAL,axiom,
! [V0n: tyop_2Enum_2Enum,V1op: $o > $o > $o,V2a: tyop_2Enum_2Enum,V3b: tyop_2Enum_2Enum] :
( ( c_2Ebit_2EBITWISE @ ( c_2Enum_2ESUC @ V0n ) @ V1op @ V2a @ V3b )
= ( c_2Earithmetic_2E_2B @ ( c_2Earithmetic_2E_2A @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) @ ( c_2Ebit_2EBITWISE @ V0n @ V1op @ ( c_2Earithmetic_2EDIV @ V2a @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) @ ( c_2Earithmetic_2EDIV @ V3b @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) ) ) @ ( c_2Ebit_2ESBIT @ ( V1op @ ( c_2Earithmetic_2EODD @ V2a ) @ ( c_2Earithmetic_2EODD @ V3b ) ) @ c_2Enum_2E0 ) ) ) ).
thf(thm_2Ebit_2EMOD__PLUS__RIGHT,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 @ V1j @ ( c_2Earithmetic_2EMOD @ V2k @ V0n ) ) @ V0n )
= ( c_2Earithmetic_2EMOD @ ( c_2Earithmetic_2E_2B @ V1j @ V2k ) @ V0n ) ) ) ).
thf(thm_2Ebit_2EMOD__PLUS__LEFT,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 @ V2k @ V0n ) @ V1j ) @ V0n )
= ( c_2Earithmetic_2EMOD @ ( c_2Earithmetic_2E_2B @ V2k @ V1j ) @ V0n ) ) ) ).
thf(thm_2Ebit_2EMOD__PLUS__1,axiom,
! [V0n: tyop_2Enum_2Enum] :
( ( c_2Eprim__rec_2E_3C @ c_2Enum_2E0 @ V0n )
=> ! [V1x: tyop_2Enum_2Enum] :
( ( ( c_2Earithmetic_2EMOD @ ( c_2Earithmetic_2E_2B @ V1x @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) @ V0n )
= c_2Enum_2E0 )
<=> ( ( c_2Earithmetic_2E_2B @ ( c_2Earithmetic_2EMOD @ V1x @ V0n ) @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) )
= V0n ) ) ) ).
thf(thm_2Ebit_2EMOD__ADD__1,axiom,
! [V0n: tyop_2Enum_2Enum] :
( ( c_2Eprim__rec_2E_3C @ c_2Enum_2E0 @ V0n )
=> ! [V1x: tyop_2Enum_2Enum] :
( ( (~)
@ ( ( c_2Earithmetic_2EMOD @ ( c_2Earithmetic_2E_2B @ V1x @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) @ V0n )
= c_2Enum_2E0 ) )
=> ( ( c_2Earithmetic_2EMOD @ ( c_2Earithmetic_2E_2B @ V1x @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) @ V0n )
= ( c_2Earithmetic_2E_2B @ ( c_2Earithmetic_2EMOD @ V1x @ V0n ) @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ) ) ).
thf(thm_2Ebit_2EBIT__REVERSE__THM,axiom,
! [V0x: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum,V2a: tyop_2Enum_2Enum] :
( ( c_2Eprim__rec_2E_3C @ V0x @ V1n )
=> ( ( c_2Ebit_2EBIT @ V0x @ ( c_2Ebit_2EBIT__REVERSE @ V1n @ V2a ) )
= ( c_2Ebit_2EBIT @ ( c_2Earithmetic_2E_2D @ ( c_2Earithmetic_2E_2D @ V1n @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) @ V0x ) @ V2a ) ) ) ).
thf(thm_2Ebit_2ELOG2__LE__MONO,axiom,
! [V0x: tyop_2Enum_2Enum,V1y: tyop_2Enum_2Enum] :
( ( c_2Eprim__rec_2E_3C @ c_2Enum_2E0 @ V0x )
=> ( ( c_2Earithmetic_2E_3C_3D @ V0x @ V1y )
=> ( c_2Earithmetic_2E_3C_3D @ ( c_2Ebit_2ELOG2 @ V0x ) @ ( c_2Ebit_2ELOG2 @ V1y ) ) ) ) ).
thf(thm_2Ebit_2ETWOEXP__LE__IMP__LE__LOG2,axiom,
( ! [V0x: tyop_2Enum_2Enum,V1y: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_3C_3D @ ( c_2Earithmetic_2EEXP @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) @ V0x ) @ V1y )
=> ( c_2Earithmetic_2E_3C_3D @ V0x @ ( c_2Ebit_2ELOG2 @ V1y ) ) )
& ! [V2y: tyop_2Enum_2Enum,V3x: tyop_2Enum_2Enum] :
( ( c_2Eprim__rec_2E_3C @ c_2Enum_2E0 @ V3x )
=> ( ( c_2Earithmetic_2E_3C_3D @ V3x @ ( c_2Earithmetic_2EEXP @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) @ V2y ) )
=> ( c_2Earithmetic_2E_3C_3D @ ( c_2Ebit_2ELOG2 @ V3x ) @ V2y ) ) ) ) ).
thf(thm_2Ebit_2ENOT__BIT__GT__LOG2,axiom,
! [V0i: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum] :
( ( c_2Eprim__rec_2E_3C @ ( c_2Ebit_2ELOG2 @ V1n ) @ V0i )
=> ( (~) @ ( c_2Ebit_2EBIT @ V0i @ V1n ) ) ) ).
thf(thm_2Ebit_2ENOT__BIT__GT__BITWISE,axiom,
! [V0i: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum,V2op: $o > $o > $o,V3a: tyop_2Enum_2Enum,V4b: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_3C_3D @ V1n @ V0i )
=> ( (~) @ ( c_2Ebit_2EBIT @ V0i @ ( c_2Ebit_2EBITWISE @ V1n @ V2op @ V3a @ V4b ) ) ) ) ).
thf(thm_2Ebit_2ELT__TWOEXP,axiom,
! [V0x: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum] :
( ( c_2Eprim__rec_2E_3C @ V0x @ ( c_2Earithmetic_2EEXP @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) @ V1n ) )
<=> ( ( V0x = c_2Enum_2E0 )
| ( c_2Eprim__rec_2E_3C @ ( c_2Ebit_2ELOG2 @ V0x ) @ V1n ) ) ) ).
thf(thm_2Ebit_2EBIT__MODIFY__THM,axiom,
! [V0x: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum,V2f: tyop_2Enum_2Enum > $o > $o,V3a: tyop_2Enum_2Enum] :
( ( c_2Eprim__rec_2E_3C @ V0x @ V1n )
=> ( ( c_2Ebit_2EBIT @ V0x @ ( c_2Ebit_2EBIT__MODIFY @ V1n @ V2f @ V3a ) )
= ( V2f @ V0x @ ( c_2Ebit_2EBIT @ V0x @ V3a ) ) ) ) ).
thf(thm_2Ebit_2EBIT__SIGN__EXTEND,axiom,
! [V0l: tyop_2Enum_2Enum,V1h: tyop_2Enum_2Enum,V2n: tyop_2Enum_2Enum,V3i: tyop_2Enum_2Enum] :
( ( (~) @ ( V0l = c_2Enum_2E0 ) )
=> ( ( c_2Ebit_2EBIT @ V3i @ ( c_2Ebit_2ESIGN__EXTEND @ V0l @ V1h @ V2n ) )
= ( c_2Ebool_2ECOND @ $o @ ( c_2Emin_2E_3D_3D_3E @ ( c_2Earithmetic_2E_3C_3D @ V0l @ V1h ) @ ( c_2Eprim__rec_2E_3C @ V3i @ V0l ) ) @ ( c_2Ebit_2EBIT @ V3i @ ( c_2Earithmetic_2EMOD @ V2n @ ( c_2Earithmetic_2EEXP @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) @ V0l ) ) ) @ ( c_2Ebool_2E_2F_5C @ ( c_2Eprim__rec_2E_3C @ V3i @ V1h ) @ ( c_2Ebit_2EBIT @ ( c_2Earithmetic_2E_2D @ V0l @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) @ V2n ) ) ) ) ) ).
thf(thm_2Ebit_2EBIT__LOG2,axiom,
! [V0n: tyop_2Enum_2Enum] :
( ( (~) @ ( V0n = c_2Enum_2E0 ) )
=> ( c_2Ebit_2EBIT @ ( c_2Ebit_2ELOG2 @ V0n ) @ V0n ) ) ).
thf(thm_2Ebit_2EEXISTS__BIT__IN__RANGE,axiom,
! [V0a: tyop_2Enum_2Enum,V1b: tyop_2Enum_2Enum,V2n: tyop_2Enum_2Enum] :
( ( ( (~) @ ( V2n = c_2Enum_2E0 ) )
& ( c_2Earithmetic_2E_3C_3D @ ( c_2Earithmetic_2EEXP @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) @ V0a ) @ V2n )
& ( c_2Eprim__rec_2E_3C @ V2n @ ( c_2Earithmetic_2EEXP @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) @ V1b ) ) )
=> ? [V3i: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_3C_3D @ V0a @ V3i )
& ( c_2Eprim__rec_2E_3C @ V3i @ V1b )
& ( c_2Ebit_2EBIT @ V3i @ V2n ) ) ) ).
thf(thm_2Ebit_2EEXISTS__BIT__LT,axiom,
! [V0b: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum] :
( ( ( (~) @ ( V1n = c_2Enum_2E0 ) )
& ( c_2Eprim__rec_2E_3C @ V1n @ ( c_2Earithmetic_2EEXP @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) @ V0b ) ) )
=> ? [V2i: tyop_2Enum_2Enum] :
( ( c_2Eprim__rec_2E_3C @ V2i @ V0b )
& ( c_2Ebit_2EBIT @ V2i @ V1n ) ) ) ).
thf(thm_2Ebit_2ELEAST__THM,axiom,
! [V0n: tyop_2Enum_2Enum,V1P: tyop_2Enum_2Enum > $o] :
( ( ! [V2m: tyop_2Enum_2Enum] :
( ( c_2Eprim__rec_2E_3C @ V2m @ V0n )
=> ( (~) @ ( V1P @ V2m ) ) )
& ( V1P @ V0n ) )
=> ( ( c_2Ewhile_2ELEAST @ V1P )
= V0n ) ) ).
%------------------------------------------------------------------------------