TPTP Problem File: ITP017^3.p
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%------------------------------------------------------------------------------
% File : ITP017^3 : TPTP v8.2.0. Bugfixed v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : HOL4 syntactic export of thm_2Eseq_2EBOLZANO__LEMMA.p, bushy 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 : thm_2Eseq_2EBOLZANO__LEMMA.p [Gau19]
% : HL408001^3.p [TPAP]
% Status : Theorem
% Rating : 1.00 v7.5.0
% Syntax : Number of formulae : 143 ( 52 unt; 50 typ; 0 def)
% Number of atoms : 305 ( 120 equ; 38 cnn)
% Maximal formula atoms : 41 ( 3 avg)
% Number of connectives : 936 ( 38 ~; 14 |; 86 &; 702 @)
% ( 48 <=>; 48 =>; 0 <=; 0 <~>)
% Maximal formula depth : 38 ( 5 avg)
% Number of types : 4 ( 3 usr)
% Number of type conns : 90 ( 90 >; 0 *; 0 +; 0 <<)
% Number of symbols : 49 ( 47 usr; 5 con; 0-4 aty)
% Number of variables : 255 ( 13 ^; 227 !; 4 ?; 255 :)
% ( 11 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments :
% Bugfixes : v7.5.0 - Bugfixes in axioms and export.
%------------------------------------------------------------------------------
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(tyop_2Erealax_2Ereal,type,
tyop_2Erealax_2Ereal: $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_2Eseq_2E_2D_2D_3E,type,
c_2Eseq_2E_2D_2D_3E: ( tyop_2Enum_2Enum > tyop_2Erealax_2Ereal ) > tyop_2Erealax_2Ereal > $o ).
thf(c_2Ereal_2E_2F,type,
c_2Ereal_2E_2F: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal ).
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_2Ebool_2E_3F_21,type,
c_2Ebool_2E_3F_21:
!>[A_27a: $tType] : ( ( A_27a > $o ) > $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_2Ebool_2ECOND,type,
c_2Ebool_2ECOND:
!>[A_27a: $tType] : ( $o > A_27a > A_27a > A_27a ) ).
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_2Epair_2EFST,type,
c_2Epair_2EFST:
!>[A_27a: $tType,A_27b: $tType] : ( ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) > A_27a ) ).
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_2Eprim__rec_2EPRE,type,
c_2Eprim__rec_2EPRE: tyop_2Enum_2Enum > tyop_2Enum_2Enum ).
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_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_2Ereal_2Eabs,type,
c_2Ereal_2Eabs: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal ).
thf(c_2Enumeral_2EiZ,type,
c_2Enumeral_2EiZ: tyop_2Enum_2Enum > tyop_2Enum_2Enum ).
thf(c_2Erealax_2Einv,type,
c_2Erealax_2Einv: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal ).
thf(c_2Ereal_2Epow,type,
c_2Ereal_2Epow: tyop_2Erealax_2Ereal > tyop_2Enum_2Enum > tyop_2Erealax_2Ereal ).
thf(c_2Erealax_2Ereal__add,type,
c_2Erealax_2Ereal__add: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal ).
thf(c_2Ereal_2Ereal__ge,type,
c_2Ereal_2Ereal__ge: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal > $o ).
thf(c_2Erealax_2Ereal__lt,type,
c_2Erealax_2Ereal__lt: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal > $o ).
thf(c_2Ereal_2Ereal__lte,type,
c_2Ereal_2Ereal__lte: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal > $o ).
thf(c_2Erealax_2Ereal__mul,type,
c_2Erealax_2Ereal__mul: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal ).
thf(c_2Erealax_2Ereal__neg,type,
c_2Erealax_2Ereal__neg: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal ).
thf(c_2Ereal_2Ereal__of__num,type,
c_2Ereal_2Ereal__of__num: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal ).
thf(c_2Ereal_2Ereal__sub,type,
c_2Ereal_2Ereal__sub: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal ).
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_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_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_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_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_2Ebool_2EEXISTS__UNIQUE__DEF,axiom,
! [A_27a: $tType] :
( ( c_2Ebool_2E_3F_21 @ A_27a )
= ( ^ [V0P: A_27a > $o] :
( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2E_3F @ A_27a @ V0P )
@ ( c_2Ebool_2E_21 @ A_27a
@ ^ [V1x: A_27a] :
( c_2Ebool_2E_21 @ A_27a
@ ^ [V2y: A_27a] : ( c_2Emin_2E_3D_3D_3E @ ( c_2Ebool_2E_2F_5C @ ( V0P @ V1x ) @ ( V0P @ V2y ) ) @ ( c_2Emin_2E_3D @ A_27a @ V1x @ V2y ) ) ) ) ) ) ) ).
thf(thm_2Ebool_2EETA__AX,axiom,
! [A_27a: $tType,A_27b: $tType,V0t: A_27a > A_27b] :
( ( ^ [V1x: A_27a] : ( V0t @ V1x ) )
= V0t ) ).
thf(thm_2Ebool_2ETRUTH,axiom,
c_2Ebool_2ET ).
thf(thm_2Ebool_2EIMP__ANTISYM__AX,axiom,
! [V0t1: $o,V1t2: $o] :
( ( V0t1
=> V1t2 )
=> ( ( V1t2
=> V0t1 )
=> ( V0t1 = V1t2 ) ) ) ).
thf(thm_2Ebool_2EFALSITY,axiom,
! [V0t: $o] :
( c_2Ebool_2EF
=> V0t ) ).
thf(thm_2Ebool_2EEXCLUDED__MIDDLE,axiom,
! [V0t: $o] :
( V0t
| ( (~) @ V0t ) ) ).
thf(thm_2Ebool_2EABS__SIMP,axiom,
! [A_27a: $tType,A_27b: $tType,V0t1: A_27a,V1t2: A_27b] :
( ( ^ [V2x: A_27b] : V0t1
@ V1t2 )
= V0t1 ) ).
thf(thm_2Ebool_2EFORALL__SIMP,axiom,
! [A_27a: $tType,V0t: $o] :
( ! [V1x: A_27a] : V0t
<=> V0t ) ).
thf(thm_2Ebool_2EAND__CLAUSES,axiom,
! [V0t: $o] :
( ( ( c_2Ebool_2ET
& V0t )
<=> V0t )
& ( ( V0t
& c_2Ebool_2ET )
<=> V0t )
& ( ( c_2Ebool_2EF
& V0t )
<=> c_2Ebool_2EF )
& ( ( V0t
& c_2Ebool_2EF )
<=> c_2Ebool_2EF )
& ( ( V0t
& V0t )
<=> V0t ) ) ).
thf(thm_2Ebool_2EIMP__CLAUSES,axiom,
! [V0t: $o] :
( ( ( c_2Ebool_2ET
=> V0t )
<=> V0t )
& ( ( V0t
=> c_2Ebool_2ET )
<=> c_2Ebool_2ET )
& ( ( c_2Ebool_2EF
=> V0t )
<=> c_2Ebool_2ET )
& ( ( V0t
=> V0t )
<=> c_2Ebool_2ET )
& ( ( V0t
=> c_2Ebool_2EF )
<=> ( (~) @ V0t ) ) ) ).
thf(thm_2Ebool_2ENOT__CLAUSES,axiom,
( ! [V0t: $o] :
( ( (~) @ ( (~) @ V0t ) )
<=> V0t )
& ( ( (~) @ c_2Ebool_2ET )
<=> c_2Ebool_2EF )
& ( ( (~) @ c_2Ebool_2EF )
<=> c_2Ebool_2ET ) ) ).
thf(thm_2Ebool_2EREFL__CLAUSE,axiom,
! [A_27a: $tType,V0x: A_27a] :
( ( V0x = V0x )
<=> c_2Ebool_2ET ) ).
thf(thm_2Ebool_2EEQ__SYM__EQ,axiom,
! [A_27a: $tType,V0x: A_27a,V1y: A_27a] :
( ( V0x = V1y )
<=> ( V1y = V0x ) ) ).
thf(thm_2Ebool_2EEQ__CLAUSES,axiom,
! [V0t: $o] :
( ( ( c_2Ebool_2ET = V0t )
<=> V0t )
& ( ( V0t = c_2Ebool_2ET )
<=> V0t )
& ( ( c_2Ebool_2EF = V0t )
<=> ( (~) @ V0t ) )
& ( ( V0t = c_2Ebool_2EF )
<=> ( (~) @ V0t ) ) ) ).
thf(thm_2Ebool_2ECOND__CLAUSES,axiom,
! [A_27a: $tType,V0t1: A_27a,V1t2: A_27a] :
( ( ( c_2Ebool_2ECOND @ A_27a @ c_2Ebool_2ET @ V0t1 @ V1t2 )
= V0t1 )
& ( ( c_2Ebool_2ECOND @ A_27a @ c_2Ebool_2EF @ V0t1 @ V1t2 )
= V1t2 ) ) ).
thf(thm_2Enum_2EINDUCTION,axiom,
! [V0P: tyop_2Enum_2Enum > $o] :
( ( ( V0P @ c_2Enum_2E0 )
& ! [V1n: tyop_2Enum_2Enum] :
( ( V0P @ V1n )
=> ( V0P @ ( c_2Enum_2ESUC @ V1n ) ) ) )
=> ! [V2n: tyop_2Enum_2Enum] : ( V0P @ V2n ) ) ).
thf(thm_2Enumeral_2Enumeral__distrib,axiom,
( ! [V0n: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_2B @ c_2Enum_2E0 @ V0n )
= V0n )
& ! [V1n: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_2B @ V1n @ c_2Enum_2E0 )
= V1n )
& ! [V2n: tyop_2Enum_2Enum,V3m: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_2B @ ( c_2Earithmetic_2ENUMERAL @ V2n ) @ ( c_2Earithmetic_2ENUMERAL @ V3m ) )
= ( c_2Earithmetic_2ENUMERAL @ ( c_2Enumeral_2EiZ @ ( c_2Earithmetic_2E_2B @ V2n @ V3m ) ) ) )
& ! [V4n: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_2A @ c_2Enum_2E0 @ V4n )
= c_2Enum_2E0 )
& ! [V5n: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_2A @ V5n @ c_2Enum_2E0 )
= c_2Enum_2E0 )
& ! [V6n: tyop_2Enum_2Enum,V7m: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_2A @ ( c_2Earithmetic_2ENUMERAL @ V6n ) @ ( c_2Earithmetic_2ENUMERAL @ V7m ) )
= ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2E_2A @ V6n @ V7m ) ) )
& ! [V8n: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_2D @ c_2Enum_2E0 @ V8n )
= c_2Enum_2E0 )
& ! [V9n: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_2D @ V9n @ c_2Enum_2E0 )
= V9n )
& ! [V10n: tyop_2Enum_2Enum,V11m: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_2D @ ( c_2Earithmetic_2ENUMERAL @ V10n ) @ ( c_2Earithmetic_2ENUMERAL @ V11m ) )
= ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2E_2D @ V10n @ V11m ) ) )
& ! [V12n: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2EEXP @ c_2Enum_2E0 @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ V12n ) ) )
= c_2Enum_2E0 )
& ! [V13n: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2EEXP @ c_2Enum_2E0 @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ V13n ) ) )
= c_2Enum_2E0 )
& ! [V14n: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2EEXP @ V14n @ c_2Enum_2E0 )
= ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) )
& ! [V15n: tyop_2Enum_2Enum,V16m: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2EEXP @ ( c_2Earithmetic_2ENUMERAL @ V15n ) @ ( c_2Earithmetic_2ENUMERAL @ V16m ) )
= ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EEXP @ V15n @ V16m ) ) )
& ( ( c_2Enum_2ESUC @ c_2Enum_2E0 )
= ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) )
& ! [V17n: tyop_2Enum_2Enum] :
( ( c_2Enum_2ESUC @ ( c_2Earithmetic_2ENUMERAL @ V17n ) )
= ( c_2Earithmetic_2ENUMERAL @ ( c_2Enum_2ESUC @ V17n ) ) )
& ( ( c_2Eprim__rec_2EPRE @ c_2Enum_2E0 )
= c_2Enum_2E0 )
& ! [V18n: tyop_2Enum_2Enum] :
( ( c_2Eprim__rec_2EPRE @ ( c_2Earithmetic_2ENUMERAL @ V18n ) )
= ( c_2Earithmetic_2ENUMERAL @ ( c_2Eprim__rec_2EPRE @ V18n ) ) )
& ! [V19n: tyop_2Enum_2Enum] :
( ( ( c_2Earithmetic_2ENUMERAL @ V19n )
= c_2Enum_2E0 )
<=> ( V19n = c_2Earithmetic_2EZERO ) )
& ! [V20n: tyop_2Enum_2Enum] :
( ( c_2Enum_2E0
= ( c_2Earithmetic_2ENUMERAL @ V20n ) )
<=> ( V20n = c_2Earithmetic_2EZERO ) )
& ! [V21n: tyop_2Enum_2Enum,V22m: tyop_2Enum_2Enum] :
( ( ( c_2Earithmetic_2ENUMERAL @ V21n )
= ( c_2Earithmetic_2ENUMERAL @ V22m ) )
<=> ( V21n = V22m ) )
& ! [V23n: tyop_2Enum_2Enum] :
( ( c_2Eprim__rec_2E_3C @ V23n @ c_2Enum_2E0 )
= c_2Ebool_2EF )
& ! [V24n: tyop_2Enum_2Enum] :
( ( c_2Eprim__rec_2E_3C @ c_2Enum_2E0 @ ( c_2Earithmetic_2ENUMERAL @ V24n ) )
= ( c_2Eprim__rec_2E_3C @ c_2Earithmetic_2EZERO @ V24n ) )
& ! [V25n: tyop_2Enum_2Enum,V26m: tyop_2Enum_2Enum] :
( ( c_2Eprim__rec_2E_3C @ ( c_2Earithmetic_2ENUMERAL @ V25n ) @ ( c_2Earithmetic_2ENUMERAL @ V26m ) )
= ( c_2Eprim__rec_2E_3C @ V25n @ V26m ) )
& ! [V27n: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_3E @ c_2Enum_2E0 @ V27n )
= c_2Ebool_2EF )
& ! [V28n: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_3E @ ( c_2Earithmetic_2ENUMERAL @ V28n ) @ c_2Enum_2E0 )
= ( c_2Eprim__rec_2E_3C @ c_2Earithmetic_2EZERO @ V28n ) )
& ! [V29n: tyop_2Enum_2Enum,V30m: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_3E @ ( c_2Earithmetic_2ENUMERAL @ V29n ) @ ( c_2Earithmetic_2ENUMERAL @ V30m ) )
= ( c_2Eprim__rec_2E_3C @ V30m @ V29n ) )
& ! [V31n: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_3C_3D @ c_2Enum_2E0 @ V31n )
= c_2Ebool_2ET )
& ! [V32n: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_3C_3D @ ( c_2Earithmetic_2ENUMERAL @ V32n ) @ c_2Enum_2E0 )
= ( c_2Earithmetic_2E_3C_3D @ V32n @ c_2Earithmetic_2EZERO ) )
& ! [V33n: tyop_2Enum_2Enum,V34m: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_3C_3D @ ( c_2Earithmetic_2ENUMERAL @ V33n ) @ ( c_2Earithmetic_2ENUMERAL @ V34m ) )
= ( c_2Earithmetic_2E_3C_3D @ V33n @ V34m ) )
& ! [V35n: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_3E_3D @ V35n @ c_2Enum_2E0 )
= c_2Ebool_2ET )
& ! [V36n: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_3E_3D @ c_2Enum_2E0 @ V36n )
<=> ( V36n = c_2Enum_2E0 ) )
& ! [V37n: tyop_2Enum_2Enum,V38m: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_3E_3D @ ( c_2Earithmetic_2ENUMERAL @ V37n ) @ ( c_2Earithmetic_2ENUMERAL @ V38m ) )
= ( c_2Earithmetic_2E_3C_3D @ V38m @ V37n ) )
& ! [V39n: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2EODD @ ( c_2Earithmetic_2ENUMERAL @ V39n ) )
= ( c_2Earithmetic_2EODD @ V39n ) )
& ! [V40n: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2EEVEN @ ( c_2Earithmetic_2ENUMERAL @ V40n ) )
= ( c_2Earithmetic_2EEVEN @ V40n ) )
& ( (~) @ ( c_2Earithmetic_2EODD @ c_2Enum_2E0 ) )
& ( c_2Earithmetic_2EEVEN @ c_2Enum_2E0 ) ) ).
thf(thm_2Enumeral_2Enumeral__eq,axiom,
! [V0n: tyop_2Enum_2Enum,V1m: tyop_2Enum_2Enum] :
( ( ( c_2Earithmetic_2EZERO
= ( c_2Earithmetic_2EBIT1 @ V0n ) )
<=> c_2Ebool_2EF )
& ( ( ( c_2Earithmetic_2EBIT1 @ V0n )
= c_2Earithmetic_2EZERO )
<=> c_2Ebool_2EF )
& ( ( c_2Earithmetic_2EZERO
= ( c_2Earithmetic_2EBIT2 @ V0n ) )
<=> c_2Ebool_2EF )
& ( ( ( c_2Earithmetic_2EBIT2 @ V0n )
= c_2Earithmetic_2EZERO )
<=> c_2Ebool_2EF )
& ( ( ( c_2Earithmetic_2EBIT1 @ V0n )
= ( c_2Earithmetic_2EBIT2 @ V1m ) )
<=> c_2Ebool_2EF )
& ( ( ( c_2Earithmetic_2EBIT2 @ V0n )
= ( c_2Earithmetic_2EBIT1 @ V1m ) )
<=> c_2Ebool_2EF )
& ( ( ( c_2Earithmetic_2EBIT1 @ V0n )
= ( c_2Earithmetic_2EBIT1 @ V1m ) )
<=> ( V0n = V1m ) )
& ( ( ( c_2Earithmetic_2EBIT2 @ V0n )
= ( c_2Earithmetic_2EBIT2 @ V1m ) )
<=> ( V0n = V1m ) ) ) ).
thf(thm_2Epair_2EPAIR,axiom,
! [A_27a: $tType,A_27b: $tType,V0x: tyop_2Epair_2Eprod @ A_27a @ A_27b] :
( ( c_2Epair_2E_2C @ A_27a @ A_27b @ ( c_2Epair_2EFST @ A_27a @ A_27b @ V0x ) @ ( c_2Epair_2ESND @ A_27a @ A_27b @ V0x ) )
= V0x ) ).
thf(thm_2Epair_2EFST,axiom,
! [A_27a: $tType,A_27b: $tType,V0x: A_27a,V1y: A_27b] :
( ( c_2Epair_2EFST @ A_27a @ A_27b @ ( c_2Epair_2E_2C @ A_27a @ A_27b @ V0x @ V1y ) )
= V0x ) ).
thf(thm_2Epair_2ESND,axiom,
! [A_27a: $tType,A_27b: $tType,V0x: A_27a,V1y: A_27b] :
( ( c_2Epair_2ESND @ A_27a @ A_27b @ ( c_2Epair_2E_2C @ A_27a @ A_27b @ V0x @ V1y ) )
= V1y ) ).
thf(thm_2Eprim__rec_2ELESS__0,axiom,
! [V0n: tyop_2Enum_2Enum] : ( c_2Eprim__rec_2E_3C @ c_2Enum_2E0 @ ( c_2Enum_2ESUC @ V0n ) ) ).
thf(thm_2Eprim__rec_2ELESS__SUC__REFL,axiom,
! [V0n: tyop_2Enum_2Enum] : ( c_2Eprim__rec_2E_3C @ V0n @ ( c_2Enum_2ESUC @ V0n ) ) ).
thf(thm_2Eprim__rec_2Enum__Axiom__old,axiom,
! [A_27a: $tType,V0e: A_27a,V1f: A_27a > tyop_2Enum_2Enum > A_27a] :
( c_2Ebool_2E_3F_21 @ ( tyop_2Enum_2Enum > A_27a )
@ ^ [V2fn1: tyop_2Enum_2Enum > A_27a] :
( c_2Ebool_2E_2F_5C @ ( c_2Emin_2E_3D @ A_27a @ ( V2fn1 @ c_2Enum_2E0 ) @ V0e )
@ ( c_2Ebool_2E_21 @ tyop_2Enum_2Enum
@ ^ [V3n: tyop_2Enum_2Enum] : ( c_2Emin_2E_3D @ A_27a @ ( V2fn1 @ ( c_2Enum_2ESUC @ V3n ) ) @ ( V1f @ ( V2fn1 @ V3n ) @ V3n ) ) ) ) ) ).
thf(thm_2Ereal_2EREAL__ADD__SYM,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__add @ V0x @ V1y )
= ( c_2Erealax_2Ereal__add @ V1y @ V0x ) ) ).
thf(thm_2Ereal_2EREAL__ADD__ASSOC,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal,V2z: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__add @ V0x @ ( c_2Erealax_2Ereal__add @ V1y @ V2z ) )
= ( c_2Erealax_2Ereal__add @ ( c_2Erealax_2Ereal__add @ V0x @ V1y ) @ V2z ) ) ).
thf(thm_2Ereal_2EREAL__ADD__LID,axiom,
! [V0x: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__add @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V0x )
= V0x ) ).
thf(thm_2Ereal_2EREAL__ADD__LINV,axiom,
! [V0x: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__add @ ( c_2Erealax_2Ereal__neg @ V0x ) @ V0x )
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ).
thf(thm_2Ereal_2EREAL__MUL__SYM,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__mul @ V0x @ V1y )
= ( c_2Erealax_2Ereal__mul @ V1y @ V0x ) ) ).
thf(thm_2Ereal_2EREAL__MUL__ASSOC,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal,V2z: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__mul @ V0x @ ( c_2Erealax_2Ereal__mul @ V1y @ V2z ) )
= ( c_2Erealax_2Ereal__mul @ ( c_2Erealax_2Ereal__mul @ V0x @ V1y ) @ V2z ) ) ).
thf(thm_2Ereal_2EREAL__MUL__LID,axiom,
! [V0x: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__mul @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) @ V0x )
= V0x ) ).
thf(thm_2Ereal_2EREAL__MUL__LINV,axiom,
! [V0x: tyop_2Erealax_2Ereal] :
( ( (~)
@ ( V0x
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) )
=> ( ( c_2Erealax_2Ereal__mul @ ( c_2Erealax_2Einv @ V0x ) @ V0x )
= ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ) ).
thf(thm_2Ereal_2Ereal__sub,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal] :
( ( c_2Ereal_2Ereal__sub @ V0x @ V1y )
= ( c_2Erealax_2Ereal__add @ V0x @ ( c_2Erealax_2Ereal__neg @ V1y ) ) ) ).
thf(thm_2Ereal_2Ereal__ge,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal] :
( ( c_2Ereal_2Ereal__ge @ V0x @ V1y )
= ( c_2Ereal_2Ereal__lte @ V1y @ V0x ) ) ).
thf(thm_2Ereal_2Ereal__div,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal] :
( ( c_2Ereal_2E_2F @ V0x @ V1y )
= ( c_2Erealax_2Ereal__mul @ V0x @ ( c_2Erealax_2Einv @ V1y ) ) ) ).
thf(thm_2Ereal_2EREAL__ADD__RID,axiom,
! [V0x: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__add @ V0x @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) )
= V0x ) ).
thf(thm_2Ereal_2EREAL__MUL__RID,axiom,
! [V0x: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__mul @ V0x @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) )
= V0x ) ).
thf(thm_2Ereal_2EREAL__MUL__RINV,axiom,
! [V0x: tyop_2Erealax_2Ereal] :
( ( (~)
@ ( V0x
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) )
=> ( ( c_2Erealax_2Ereal__mul @ V0x @ ( c_2Erealax_2Einv @ V0x ) )
= ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ) ).
thf(thm_2Ereal_2EREAL__NEG__ADD,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__neg @ ( c_2Erealax_2Ereal__add @ V0x @ V1y ) )
= ( c_2Erealax_2Ereal__add @ ( c_2Erealax_2Ereal__neg @ V0x ) @ ( c_2Erealax_2Ereal__neg @ V1y ) ) ) ).
thf(thm_2Ereal_2EREAL__MUL__RZERO,axiom,
! [V0x: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__mul @ V0x @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) )
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ).
thf(thm_2Ereal_2EREAL__LE__REFL,axiom,
! [V0x: tyop_2Erealax_2Ereal] : ( c_2Ereal_2Ereal__lte @ V0x @ V0x ) ).
thf(thm_2Ereal_2EREAL__LET__TRANS,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal,V2z: tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal_2Ereal__lte @ V0x @ V1y )
& ( c_2Erealax_2Ereal__lt @ V1y @ V2z ) )
=> ( c_2Erealax_2Ereal__lt @ V0x @ V2z ) ) ).
thf(thm_2Ereal_2EREAL__LT__ADD2,axiom,
! [V0w: tyop_2Erealax_2Ereal,V1x: tyop_2Erealax_2Ereal,V2y: tyop_2Erealax_2Ereal,V3z: tyop_2Erealax_2Ereal] :
( ( ( c_2Erealax_2Ereal__lt @ V0w @ V1x )
& ( c_2Erealax_2Ereal__lt @ V2y @ V3z ) )
=> ( c_2Erealax_2Ereal__lt @ ( c_2Erealax_2Ereal__add @ V0w @ V2y ) @ ( c_2Erealax_2Ereal__add @ V1x @ V3z ) ) ) ).
thf(thm_2Ereal_2EREAL__NEG__0,axiom,
( ( c_2Erealax_2Ereal__neg @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) )
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ).
thf(thm_2Ereal_2EREAL__NEG__SUB,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__neg @ ( c_2Ereal_2Ereal__sub @ V0x @ V1y ) )
= ( c_2Ereal_2Ereal__sub @ V1y @ V0x ) ) ).
thf(thm_2Ereal_2EREAL__SUB__LE,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal] :
( ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ ( c_2Ereal_2Ereal__sub @ V0x @ V1y ) )
= ( c_2Ereal_2Ereal__lte @ V1y @ V0x ) ) ).
thf(thm_2Ereal_2EREAL__SUB__LDISTRIB,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal,V2z: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__mul @ V0x @ ( c_2Ereal_2Ereal__sub @ V1y @ V2z ) )
= ( c_2Ereal_2Ereal__sub @ ( c_2Erealax_2Ereal__mul @ V0x @ V1y ) @ ( c_2Erealax_2Ereal__mul @ V0x @ V2z ) ) ) ).
thf(thm_2Ereal_2EREAL__LT__RMUL,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal,V2z: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V2z )
=> ( ( c_2Erealax_2Ereal__lt @ ( c_2Erealax_2Ereal__mul @ V0x @ V2z ) @ ( c_2Erealax_2Ereal__mul @ V1y @ V2z ) )
= ( c_2Erealax_2Ereal__lt @ V0x @ V1y ) ) ) ).
thf(thm_2Ereal_2EREAL__LT,axiom,
! [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ V0m ) @ ( c_2Ereal_2Ereal__of__num @ V1n ) )
= ( c_2Eprim__rec_2E_3C @ V0m @ V1n ) ) ).
thf(thm_2Ereal_2EREAL__INJ,axiom,
! [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum] :
( ( ( c_2Ereal_2Ereal__of__num @ V0m )
= ( c_2Ereal_2Ereal__of__num @ V1n ) )
<=> ( V0m = V1n ) ) ).
thf(thm_2Ereal_2EREAL__INV1,axiom,
( ( c_2Erealax_2Einv @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) )
= ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ).
thf(thm_2Ereal_2EREAL__LT__HALF1,axiom,
! [V0d: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ ( c_2Ereal_2E_2F @ V0d @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) ) )
= ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V0d ) ) ).
thf(thm_2Ereal_2EREAL__DOUBLE,axiom,
! [V0x: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__add @ V0x @ V0x )
= ( c_2Erealax_2Ereal__mul @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) @ V0x ) ) ).
thf(thm_2Ereal_2EREAL__DIV__LMUL,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal] :
( ( (~)
@ ( V1y
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) )
=> ( ( c_2Erealax_2Ereal__mul @ V1y @ ( c_2Ereal_2E_2F @ V0x @ V1y ) )
= V0x ) ) ).
thf(thm_2Ereal_2EREAL__HALF__DOUBLE,axiom,
! [V0x: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__add @ ( c_2Ereal_2E_2F @ V0x @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) ) @ ( c_2Ereal_2E_2F @ V0x @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) ) )
= V0x ) ).
thf(thm_2Ereal_2EREAL__INV__MUL,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal] :
( ( ( (~)
@ ( V0x
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) )
& ( (~)
@ ( V1y
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ) )
=> ( ( c_2Erealax_2Einv @ ( c_2Erealax_2Ereal__mul @ V0x @ V1y ) )
= ( c_2Erealax_2Ereal__mul @ ( c_2Erealax_2Einv @ V0x ) @ ( c_2Erealax_2Einv @ V1y ) ) ) ) ).
thf(thm_2Ereal_2EREAL__EQ__LMUL__IMP,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal,V2z: tyop_2Erealax_2Ereal] :
( ( ( (~)
@ ( V0x
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) )
& ( ( c_2Erealax_2Ereal__mul @ V0x @ V1y )
= ( c_2Erealax_2Ereal__mul @ V0x @ V2z ) ) )
=> ( V1y = V2z ) ) ).
thf(thm_2Ereal_2EREAL__MIDDLE1,axiom,
! [V0a: tyop_2Erealax_2Ereal,V1b: tyop_2Erealax_2Ereal] :
( ( c_2Ereal_2Ereal__lte @ V0a @ V1b )
=> ( c_2Ereal_2Ereal__lte @ V0a @ ( c_2Ereal_2E_2F @ ( c_2Erealax_2Ereal__add @ V0a @ V1b ) @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) ) ) ) ).
thf(thm_2Ereal_2EREAL__MIDDLE2,axiom,
! [V0a: tyop_2Erealax_2Ereal,V1b: tyop_2Erealax_2Ereal] :
( ( c_2Ereal_2Ereal__lte @ V0a @ V1b )
=> ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2E_2F @ ( c_2Erealax_2Ereal__add @ V0a @ V1b ) @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) ) @ V1b ) ) ).
thf(thm_2Ereal_2Eabs,axiom,
! [V0x: tyop_2Erealax_2Ereal] :
( ( c_2Ereal_2Eabs @ V0x )
= ( c_2Ebool_2ECOND @ tyop_2Erealax_2Ereal @ ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V0x ) @ V0x @ ( c_2Erealax_2Ereal__neg @ V0x ) ) ) ).
thf(thm_2Ereal_2EABS__SUB,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal] :
( ( c_2Ereal_2Eabs @ ( c_2Ereal_2Ereal__sub @ V0x @ V1y ) )
= ( c_2Ereal_2Eabs @ ( c_2Ereal_2Ereal__sub @ V1y @ V0x ) ) ) ).
thf(thm_2Ereal_2EABS__INV,axiom,
! [V0x: tyop_2Erealax_2Ereal] :
( ( (~)
@ ( V0x
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) )
=> ( ( c_2Ereal_2Eabs @ ( c_2Erealax_2Einv @ V0x ) )
= ( c_2Erealax_2Einv @ ( c_2Ereal_2Eabs @ V0x ) ) ) ) ).
thf(thm_2Ereal_2EABS__N,axiom,
! [V0n: tyop_2Enum_2Enum] :
( ( c_2Ereal_2Eabs @ ( c_2Ereal_2Ereal__of__num @ V0n ) )
= ( c_2Ereal_2Ereal__of__num @ V0n ) ) ).
thf(thm_2Ereal_2Epow,axiom,
( ! [V0x: tyop_2Erealax_2Ereal] :
( ( c_2Ereal_2Epow @ V0x @ c_2Enum_2E0 )
= ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) )
& ! [V1x: tyop_2Erealax_2Ereal,V2n: tyop_2Enum_2Enum] :
( ( c_2Ereal_2Epow @ V1x @ ( c_2Enum_2ESUC @ V2n ) )
= ( c_2Erealax_2Ereal__mul @ V1x @ ( c_2Ereal_2Epow @ V1x @ V2n ) ) ) ) ).
thf(thm_2Ereal_2EPOW__NZ,axiom,
! [V0c: tyop_2Erealax_2Ereal,V1n: tyop_2Enum_2Enum] :
( ( (~)
@ ( V0c
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) )
=> ( (~)
@ ( ( c_2Ereal_2Epow @ V0c @ V1n )
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ) ) ).
thf(thm_2Ereal_2EPOW__INV,axiom,
! [V0c: tyop_2Erealax_2Ereal] :
( ( (~)
@ ( V0c
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) )
=> ! [V1n: tyop_2Enum_2Enum] :
( ( c_2Erealax_2Einv @ ( c_2Ereal_2Epow @ V0c @ V1n ) )
= ( c_2Ereal_2Epow @ ( c_2Erealax_2Einv @ V0c ) @ V1n ) ) ) ).
thf(thm_2Ereal_2EPOW__ABS,axiom,
! [V0c: tyop_2Erealax_2Ereal,V1n: tyop_2Enum_2Enum] :
( ( c_2Ereal_2Epow @ ( c_2Ereal_2Eabs @ V0c ) @ V1n )
= ( c_2Ereal_2Eabs @ ( c_2Ereal_2Epow @ V0c @ V1n ) ) ) ).
thf(thm_2Esat_2ENOT__NOT,axiom,
! [V0t: $o] :
( ( (~) @ ( (~) @ V0t ) )
<=> V0t ) ).
thf(thm_2Esat_2EAND__INV__IMP,axiom,
! [V0A: $o] :
( V0A
=> ( ( (~) @ V0A )
=> c_2Ebool_2EF ) ) ).
thf(thm_2Esat_2EOR__DUAL2,axiom,
! [V0B: $o,V1A: $o] :
( ( ( (~)
@ ( V1A
| V0B ) )
=> c_2Ebool_2EF )
<=> ( ( V1A
=> c_2Ebool_2EF )
=> ( ( (~) @ V0B )
=> c_2Ebool_2EF ) ) ) ).
thf(thm_2Esat_2EOR__DUAL3,axiom,
! [V0B: $o,V1A: $o] :
( ( ( (~)
@ ( ( (~) @ V1A )
| V0B ) )
=> c_2Ebool_2EF )
<=> ( V1A
=> ( ( (~) @ V0B )
=> c_2Ebool_2EF ) ) ) ).
thf(thm_2Esat_2EAND__INV2,axiom,
! [V0A: $o] :
( ( ( (~) @ V0A )
=> c_2Ebool_2EF )
=> ( ( V0A
=> c_2Ebool_2EF )
=> c_2Ebool_2EF ) ) ).
thf(thm_2Esat_2Edc__eq,axiom,
! [V0r: $o,V1q: $o,V2p: $o] :
( ( V2p
<=> ( V1q = V0r ) )
<=> ( ( V2p
| V1q
| V0r )
& ( V2p
| ( (~) @ V0r )
| ( (~) @ V1q ) )
& ( V1q
| ( (~) @ V0r )
| ( (~) @ V2p ) )
& ( V0r
| ( (~) @ V1q )
| ( (~) @ V2p ) ) ) ) ).
thf(thm_2Esat_2Edc__neg,axiom,
! [V0q: $o,V1p: $o] :
( ( V1p
<=> ( (~) @ V0q ) )
<=> ( ( V1p
| V0q )
& ( ( (~) @ V0q )
| ( (~) @ V1p ) ) ) ) ).
thf(thm_2Eseq_2ESEQ,axiom,
! [V0x: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V1x0: tyop_2Erealax_2Ereal] :
( ( c_2Eseq_2E_2D_2D_3E @ V0x @ V1x0 )
<=> ! [V2e: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V2e )
=> ? [V3N: tyop_2Enum_2Enum] :
! [V4n: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_3E_3D @ V4n @ V3N )
=> ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Eabs @ ( c_2Ereal_2Ereal__sub @ ( V0x @ V4n ) @ V1x0 ) ) @ V2e ) ) ) ) ).
thf(thm_2Eseq_2ESEQ__CONST,axiom,
! [V0k: tyop_2Erealax_2Ereal] :
( c_2Eseq_2E_2D_2D_3E
@ ^ [V1x: tyop_2Enum_2Enum] : V0k
@ V0k ) ).
thf(thm_2Eseq_2ESEQ__MUL,axiom,
! [V0x: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V1x0: tyop_2Erealax_2Ereal,V2y: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V3y0: tyop_2Erealax_2Ereal] :
( ( ( c_2Eseq_2E_2D_2D_3E @ V0x @ V1x0 )
& ( c_2Eseq_2E_2D_2D_3E @ V2y @ V3y0 ) )
=> ( c_2Eseq_2E_2D_2D_3E
@ ^ [V4n: tyop_2Enum_2Enum] : ( c_2Erealax_2Ereal__mul @ ( V0x @ V4n ) @ ( V2y @ V4n ) )
@ ( c_2Erealax_2Ereal__mul @ V1x0 @ V3y0 ) ) ) ).
thf(thm_2Eseq_2ESEQ__NEG,axiom,
! [V0x: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V1x0: tyop_2Erealax_2Ereal] :
( ( c_2Eseq_2E_2D_2D_3E @ V0x @ V1x0 )
= ( c_2Eseq_2E_2D_2D_3E
@ ^ [V2n: tyop_2Enum_2Enum] : ( c_2Erealax_2Ereal__neg @ ( V0x @ V2n ) )
@ ( c_2Erealax_2Ereal__neg @ V1x0 ) ) ) ).
thf(thm_2Eseq_2ESEQ__ABS,axiom,
! [V0f: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal] :
( ( c_2Eseq_2E_2D_2D_3E
@ ^ [V1n: tyop_2Enum_2Enum] : ( c_2Ereal_2Eabs @ ( V0f @ V1n ) )
@ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) )
= ( c_2Eseq_2E_2D_2D_3E @ V0f @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ) ).
thf(thm_2Eseq_2ESEQ__POWER__ABS,axiom,
! [V0c: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Eabs @ V0c ) @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) )
=> ( c_2Eseq_2E_2D_2D_3E
@ ^ [V1n: tyop_2Enum_2Enum] : ( c_2Ereal_2Epow @ ( c_2Ereal_2Eabs @ V0c ) @ V1n )
@ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ) ).
thf(thm_2Eseq_2ENEST__LEMMA__UNIQ,axiom,
! [V0f: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V1g: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal] :
( ( ! [V2n: tyop_2Enum_2Enum] : ( c_2Ereal_2Ereal__ge @ ( V0f @ ( c_2Enum_2ESUC @ V2n ) ) @ ( V0f @ V2n ) )
& ! [V3n: tyop_2Enum_2Enum] : ( c_2Ereal_2Ereal__lte @ ( V1g @ ( c_2Enum_2ESUC @ V3n ) ) @ ( V1g @ V3n ) )
& ! [V4n: tyop_2Enum_2Enum] : ( c_2Ereal_2Ereal__lte @ ( V0f @ V4n ) @ ( V1g @ V4n ) )
& ( c_2Eseq_2E_2D_2D_3E
@ ^ [V5n: tyop_2Enum_2Enum] : ( c_2Ereal_2Ereal__sub @ ( V0f @ V5n ) @ ( V1g @ V5n ) )
@ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) )
=> ? [V6l: tyop_2Erealax_2Ereal] :
( ! [V7n: tyop_2Enum_2Enum] : ( c_2Ereal_2Ereal__lte @ ( V0f @ V7n ) @ V6l )
& ( c_2Eseq_2E_2D_2D_3E @ V0f @ V6l )
& ! [V8n: tyop_2Enum_2Enum] : ( c_2Ereal_2Ereal__lte @ V6l @ ( V1g @ V8n ) )
& ( c_2Eseq_2E_2D_2D_3E @ V1g @ V6l ) ) ) ).
thf(thm_2Eseq_2EBOLZANO__LEMMA,conjecture,
! [V0P: ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) > $o] :
( ( ! [V1a: tyop_2Erealax_2Ereal,V2b: tyop_2Erealax_2Ereal,V3c: tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal_2Ereal__lte @ V1a @ V2b )
& ( c_2Ereal_2Ereal__lte @ V2b @ V3c )
& ( V0P @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V1a @ V2b ) )
& ( V0P @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V2b @ V3c ) ) )
=> ( V0P @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V1a @ V3c ) ) )
& ! [V4x: tyop_2Erealax_2Ereal] :
? [V5d: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V5d )
& ! [V6a: tyop_2Erealax_2Ereal,V7b: tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal_2Ereal__lte @ V6a @ V4x )
& ( c_2Ereal_2Ereal__lte @ V4x @ V7b )
& ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__sub @ V7b @ V6a ) @ V5d ) )
=> ( V0P @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V6a @ V7b ) ) ) ) )
=> ! [V8a: tyop_2Erealax_2Ereal,V9b: tyop_2Erealax_2Ereal] :
( ( c_2Ereal_2Ereal__lte @ V8a @ V9b )
=> ( V0P @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V8a @ V9b ) ) ) ) ).
%------------------------------------------------------------------------------