TPTP Problem File: ITP016^3.p
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%------------------------------------------------------------------------------
% File : ITP016^3 : TPTP v8.2.0. Bugfixed v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : HOL4 syntactic export of thm_2Ereal_2ESUP__EPSILON.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_2Ereal_2ESUP__EPSILON.p [Gau19]
% : HL407501^3.p [TPAP]
% Status : Theorem
% Rating : 1.00 v7.5.0
% Syntax : Number of formulae : 102 ( 28 unt; 31 typ; 0 def)
% Number of atoms : 177 ( 46 equ; 53 cnn)
% Maximal formula atoms : 8 ( 2 avg)
% Number of connectives : 501 ( 53 ~; 44 |; 47 &; 250 @)
% ( 61 <=>; 46 =>; 0 <=; 0 <~>)
% Maximal formula depth : 11 ( 6 avg)
% Number of types : 4 ( 3 usr)
% Number of type conns : 48 ( 48 >; 0 *; 0 +; 0 <<)
% Number of symbols : 30 ( 28 usr; 5 con; 0-3 aty)
% Number of variables : 154 ( 0 ^; 140 !; 11 ?; 154 :)
% ( 3 !>; 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_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_2B,type,
c_2Earithmetic_2E_2B: tyop_2Enum_2Enum > tyop_2Enum_2Enum > tyop_2Enum_2Enum ).
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_2Ebool_2E_3F,type,
c_2Ebool_2E_3F:
!>[A_27a: $tType] : ( ( A_27a > $o ) > $o ) ).
thf(c_2Earithmetic_2EBIT1,type,
c_2Earithmetic_2EBIT1: tyop_2Enum_2Enum > tyop_2Enum_2Enum ).
thf(c_2Ebool_2EF,type,
c_2Ebool_2EF: $o ).
thf(c_2Ewhile_2ELEAST,type,
c_2Ewhile_2ELEAST: ( tyop_2Enum_2Enum > $o ) > tyop_2Enum_2Enum ).
thf(c_2Earithmetic_2ENUMERAL,type,
c_2Earithmetic_2ENUMERAL: tyop_2Enum_2Enum > tyop_2Enum_2Enum ).
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_2Erealax_2Ereal__add,type,
c_2Erealax_2Ereal__add: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal ).
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_2Ereal_2Esup,type,
c_2Ereal_2Esup: ( tyop_2Erealax_2Ereal > $o ) > 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_2Enum__CASES,axiom,
! [V0m: tyop_2Enum_2Enum] :
( ( V0m = c_2Enum_2E0 )
| ? [V1n: tyop_2Enum_2Enum] :
( V0m
= ( c_2Enum_2ESUC @ V1n ) ) ) ).
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_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_2Ebool_2EBOOL__CASES__AX,axiom,
! [V0t: $o] :
( ( V0t = c_2Ebool_2ET )
| ( V0t = c_2Ebool_2EF ) ) ).
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_2EIMP__F,axiom,
! [V0t: $o] :
( ( V0t
=> c_2Ebool_2EF )
=> ( (~) @ V0t ) ) ).
thf(thm_2Ebool_2EF__IMP,axiom,
! [V0t: $o] :
( ( (~) @ V0t )
=> ( V0t
=> c_2Ebool_2EF ) ) ).
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_2EOR__CLAUSES,axiom,
! [V0t: $o] :
( ( ( c_2Ebool_2ET
| V0t )
<=> c_2Ebool_2ET )
& ( ( V0t
| c_2Ebool_2ET )
<=> c_2Ebool_2ET )
& ( ( c_2Ebool_2EF
| V0t )
<=> V0t )
& ( ( V0t
| c_2Ebool_2EF )
<=> V0t )
& ( ( 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_2ENOT__EXISTS__THM,axiom,
! [A_27a: $tType,V0P: A_27a > $o] :
( ( (~)
@ ? [V1x: A_27a] : ( V0P @ V1x ) )
<=> ! [V2x: A_27a] : ( (~) @ ( V0P @ V2x ) ) ) ).
thf(thm_2Ebool_2EDISJ__ASSOC,axiom,
! [V0A: $o,V1B: $o,V2C: $o] :
( ( V0A
| V1B
| V2C )
<=> ( V0A
| V1B
| V2C ) ) ).
thf(thm_2Ebool_2EDISJ__SYM,axiom,
! [V0A: $o,V1B: $o] :
( ( V0A
| V1B )
<=> ( V1B
| V0A ) ) ).
thf(thm_2Ebool_2EDE__MORGAN__THM,axiom,
! [V0A: $o,V1B: $o] :
( ( ( (~)
@ ( V0A
& V1B ) )
<=> ( ( (~) @ V0A )
| ( (~) @ V1B ) ) )
& ( ( (~)
@ ( V0A
| V1B ) )
<=> ( ( (~) @ V0A )
& ( (~) @ V1B ) ) ) ) ).
thf(thm_2Ebool_2EIMP__DISJ__THM,axiom,
! [V0A: $o,V1B: $o] :
( ( V0A
=> V1B )
<=> ( ( (~) @ V0A )
| V1B ) ) ).
thf(thm_2Ebool_2EAND__IMP__INTRO,axiom,
! [V0t1: $o,V1t2: $o,V2t3: $o] :
( ( V0t1
=> ( V1t2
=> V2t3 ) )
<=> ( ( V0t1
& V1t2 )
=> V2t3 ) ) ).
thf(thm_2Ebool_2EIMP__CONG,axiom,
! [V0x: $o,V1x_27: $o,V2y: $o,V3y_27: $o] :
( ( ( V0x = V1x_27 )
& ( V1x_27
=> ( V2y = V3y_27 ) ) )
=> ( ( V0x
=> V2y )
<=> ( V1x_27
=> V3y_27 ) ) ) ).
thf(thm_2Eprim__rec_2ELESS__SUC__REFL,axiom,
! [V0n: tyop_2Enum_2Enum] : ( c_2Eprim__rec_2E_3C @ V0n @ ( c_2Enum_2ESUC @ V0n ) ) ).
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__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__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__EQ__LADD,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal,V2z: tyop_2Erealax_2Ereal] :
( ( ( c_2Erealax_2Ereal__add @ V0x @ V1y )
= ( c_2Erealax_2Ereal__add @ V0x @ V2z ) )
<=> ( V1y = V2z ) ) ).
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__LZERO,axiom,
! [V0x: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__mul @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V0x )
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ).
thf(thm_2Ereal_2EREAL__NEGNEG,axiom,
! [V0x: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__neg @ ( c_2Erealax_2Ereal__neg @ V0x ) )
= V0x ) ).
thf(thm_2Ereal_2EREAL__NOT__LT,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal] :
( ( (~) @ ( c_2Erealax_2Ereal__lt @ V0x @ V1y ) )
<=> ( c_2Ereal_2Ereal__lte @ V1y @ V0x ) ) ).
thf(thm_2Ereal_2EREAL__LT__LE,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ V0x @ V1y )
<=> ( ( c_2Ereal_2Ereal__lte @ V0x @ V1y )
& ( (~) @ ( V0x = V1y ) ) ) ) ).
thf(thm_2Ereal_2EREAL__LE__TRANS,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal,V2z: tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal_2Ereal__lte @ V0x @ V1y )
& ( c_2Ereal_2Ereal__lte @ V1y @ V2z ) )
=> ( c_2Ereal_2Ereal__lte @ V0x @ V2z ) ) ).
thf(thm_2Ereal_2EREAL__LE__RADD,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal,V2z: tyop_2Erealax_2Ereal] :
( ( c_2Ereal_2Ereal__lte @ ( c_2Erealax_2Ereal__add @ V0x @ V2z ) @ ( c_2Erealax_2Ereal__add @ V1y @ V2z ) )
= ( c_2Ereal_2Ereal__lte @ V0x @ V1y ) ) ).
thf(thm_2Ereal_2EREAL__EQ__RMUL,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal,V2z: tyop_2Erealax_2Ereal] :
( ( ( c_2Erealax_2Ereal__mul @ V0x @ V2z )
= ( c_2Erealax_2Ereal__mul @ V1y @ V2z ) )
<=> ( ( V2z
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) )
| ( V0x = V1y ) ) ) ).
thf(thm_2Ereal_2EREAL__LE,axiom,
! [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum] :
( ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Ereal__of__num @ V0m ) @ ( c_2Ereal_2Ereal__of__num @ V1n ) )
= ( c_2Earithmetic_2E_3C_3D @ 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__ADD,axiom,
! [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum] :
( ( c_2Erealax_2Ereal__add @ ( c_2Ereal_2Ereal__of__num @ V0m ) @ ( c_2Ereal_2Ereal__of__num @ V1n ) )
= ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2E_2B @ V0m @ V1n ) ) ) ).
thf(thm_2Ereal_2EREAL__DIV__RMUL,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal] :
( ( (~)
@ ( V1y
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) )
=> ( ( c_2Erealax_2Ereal__mul @ ( c_2Ereal_2E_2F @ V0x @ V1y ) @ V1y )
= V0x ) ) ).
thf(thm_2Ereal_2EREAL__LE__SUB__RADD,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal,V2z: tyop_2Erealax_2Ereal] :
( ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Ereal__sub @ V0x @ V1y ) @ V2z )
= ( c_2Ereal_2Ereal__lte @ V0x @ ( c_2Erealax_2Ereal__add @ V2z @ V1y ) ) ) ).
thf(thm_2Ereal_2EREAL__SUB__RZERO,axiom,
! [V0x: tyop_2Erealax_2Ereal] :
( ( c_2Ereal_2Ereal__sub @ V0x @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) )
= V0x ) ).
thf(thm_2Ereal_2EREAL__EQ__SUB__LADD,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal,V2z: tyop_2Erealax_2Ereal] :
( ( V0x
= ( c_2Ereal_2Ereal__sub @ V1y @ V2z ) )
<=> ( ( c_2Erealax_2Ereal__add @ V0x @ V2z )
= V1y ) ) ).
thf(thm_2Ereal_2EREAL__LE__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_2Ereal_2Ereal__lte @ ( c_2Erealax_2Ereal__mul @ V0x @ V2z ) @ ( c_2Erealax_2Ereal__mul @ V1y @ V2z ) )
= ( c_2Ereal_2Ereal__lte @ V0x @ V1y ) ) ) ).
thf(thm_2Ereal_2EREAL__EQ__NEG,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal] :
( ( ( c_2Erealax_2Ereal__neg @ V0x )
= ( c_2Erealax_2Ereal__neg @ V1y ) )
<=> ( V0x = V1y ) ) ).
thf(thm_2Ereal_2EREAL__SUP__LE,axiom,
! [V0P: tyop_2Erealax_2Ereal > $o] :
( ( ? [V1x: tyop_2Erealax_2Ereal] : ( V0P @ V1x )
& ? [V2z: tyop_2Erealax_2Ereal] :
! [V3x: tyop_2Erealax_2Ereal] :
( ( V0P @ V3x )
=> ( c_2Ereal_2Ereal__lte @ V3x @ V2z ) ) )
=> ! [V4y: tyop_2Erealax_2Ereal] :
( ? [V5x: tyop_2Erealax_2Ereal] :
( ( V0P @ V5x )
& ( c_2Erealax_2Ereal__lt @ V4y @ V5x ) )
<=> ( c_2Erealax_2Ereal__lt @ V4y @ ( c_2Ereal_2Esup @ V0P ) ) ) ) ).
thf(thm_2Ereal_2EREAL__MUL__LNEG,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__mul @ ( c_2Erealax_2Ereal__neg @ V0x ) @ V1y )
= ( c_2Erealax_2Ereal__neg @ ( c_2Erealax_2Ereal__mul @ V0x @ V1y ) ) ) ).
thf(thm_2Ereal_2Ereal__lt,axiom,
! [V0y: tyop_2Erealax_2Ereal,V1x: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ V1x @ V0y )
<=> ( (~) @ ( c_2Ereal_2Ereal__lte @ V0y @ V1x ) ) ) ).
thf(thm_2Ereal_2EREAL__ADD__RDISTRIB,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal,V2z: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__mul @ ( c_2Erealax_2Ereal__add @ V0x @ V1y ) @ V2z )
= ( c_2Erealax_2Ereal__add @ ( c_2Erealax_2Ereal__mul @ V0x @ V2z ) @ ( c_2Erealax_2Ereal__mul @ V1y @ V2z ) ) ) ).
thf(thm_2Ereal_2EREAL__BIGNUM,axiom,
! [V0r: tyop_2Erealax_2Ereal] :
? [V1n: tyop_2Enum_2Enum] : ( c_2Erealax_2Ereal__lt @ V0r @ ( c_2Ereal_2Ereal__of__num @ 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__conj,axiom,
! [V0r: $o,V1q: $o,V2p: $o] :
( ( V2p
<=> ( V1q
& V0r ) )
<=> ( ( V2p
| ( (~) @ V1q )
| ( (~) @ V0r ) )
& ( V1q
| ( (~) @ V2p ) )
& ( V0r
| ( (~) @ V2p ) ) ) ) ).
thf(thm_2Esat_2Edc__disj,axiom,
! [V0r: $o,V1q: $o,V2p: $o] :
( ( V2p
<=> ( V1q
| V0r ) )
<=> ( ( V2p
| ( (~) @ V1q ) )
& ( V2p
| ( (~) @ V0r ) )
& ( V1q
| V0r
| ( (~) @ V2p ) ) ) ) ).
thf(thm_2Esat_2Edc__imp,axiom,
! [V0r: $o,V1q: $o,V2p: $o] :
( ( V2p
<=> ( V1q
=> V0r ) )
<=> ( ( V2p
| V1q )
& ( V2p
| ( (~) @ V0r ) )
& ( ( (~) @ V1q )
| V0r
| ( (~) @ V2p ) ) ) ) ).
thf(thm_2Esat_2Edc__neg,axiom,
! [V0q: $o,V1p: $o] :
( ( V1p
<=> ( (~) @ V0q ) )
<=> ( ( V1p
| V0q )
& ( ( (~) @ V0q )
| ( (~) @ V1p ) ) ) ) ).
thf(thm_2Ewhile_2ELEAST__EXISTS__IMP,axiom,
! [V0p: tyop_2Enum_2Enum > $o] :
( ? [V1n: tyop_2Enum_2Enum] : ( V0p @ V1n )
=> ( ( V0p @ ( c_2Ewhile_2ELEAST @ V0p ) )
& ! [V2n: tyop_2Enum_2Enum] :
( ( c_2Eprim__rec_2E_3C @ V2n @ ( c_2Ewhile_2ELEAST @ V0p ) )
=> ( (~) @ ( V0p @ V2n ) ) ) ) ) ).
thf(thm_2Ereal_2ESUP__EPSILON,conjecture,
! [V0p: tyop_2Erealax_2Ereal > $o,V1e: tyop_2Erealax_2Ereal] :
( ( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V1e )
& ? [V2x: tyop_2Erealax_2Ereal] : ( V0p @ V2x )
& ? [V3z: tyop_2Erealax_2Ereal] :
! [V4x: tyop_2Erealax_2Ereal] :
( ( V0p @ V4x )
=> ( c_2Ereal_2Ereal__lte @ V4x @ V3z ) ) )
=> ? [V5x: tyop_2Erealax_2Ereal] :
( ( V0p @ V5x )
& ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Esup @ V0p ) @ ( c_2Erealax_2Ereal__add @ V5x @ V1e ) ) ) ) ).
%------------------------------------------------------------------------------