ITP001 Axioms: ITP099^7.ax
%------------------------------------------------------------------------------
% File : ITP099^7 : TPTP v8.2.0. Bugfixed v7.5.0.
% Domain : Interactive Theorem Proving
% Axioms : HOL4 syntactic export, chainy mode
% Version : [BG+19] axioms.
% English :
% Refs : [BG+19] Brown et al. (2019), GRUNGE: A Grand Unified ATP Chall
% : [Gau19] Gauthier (2019), Email to Geoff Sutcliffe
% Source : [BG+19]
% Names : hreal.ax [Gau19]
% : HL4099^7.ax [TPAP]
% Status : Satisfiable
% Syntax : Number of formulae : 107 ( 38 unt; 28 typ; 0 def)
% Number of atoms : 231 ( 48 equ; 15 cnn)
% Maximal formula atoms : 8 ( 2 avg)
% Number of connectives : 598 ( 15 ~; 7 |; 24 &; 514 @)
% ( 15 <=>; 23 =>; 0 <=; 0 <~>)
% Maximal formula depth : 14 ( 6 avg; 514 nst)
% Number of types : 4 ( 3 usr)
% Number of type conns : 55 ( 55 >; 0 *; 0 +; 0 <<)
% Number of symbols : 27 ( 25 usr; 3 con; 0-4 aty)
% Number of variables : 218 ( 27 ^ 162 !; 24 ?; 218 :)
% ( 5 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_SAT_EQU_NAR
% Comments :
% Bugfixes : v7.5.0 - Fixes to the axioms.
%------------------------------------------------------------------------------
thf(tyop_2Ehrat_2Ehrat,type,
tyop_2Ehrat_2Ehrat: $tType ).
thf(tyop_2Ehreal_2Ehreal,type,
tyop_2Ehreal_2Ehreal: $tType ).
thf(tyop_2Emin_2Ebool,type,
tyop_2Emin_2Ebool: $tType ).
thf(tyop_2Emin_2Efun,type,
tyop_2Emin_2Efun: $tType > $tType > $tType ).
thf(c_2Ebool_2E_21,type,
c_2Ebool_2E_21:
!>[A_27a: $tType] : ( ( A_27a > $o ) > $o ) ).
thf(c_2Ebool_2E_2F_5C,type,
c_2Ebool_2E_2F_5C: $o > $o > $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_2Ebool_2ETYPE__DEFINITION,type,
c_2Ebool_2ETYPE__DEFINITION:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > $o ) > ( A_27b > A_27a ) > $o ) ).
thf(c_2Ebool_2E_5C_2F,type,
c_2Ebool_2E_5C_2F: $o > $o > $o ).
thf(c_2Ehreal_2Ecut,type,
c_2Ehreal_2Ecut: tyop_2Ehreal_2Ehreal > tyop_2Ehrat_2Ehrat > $o ).
thf(c_2Ehreal_2Ecut__of__hrat,type,
c_2Ehreal_2Ecut__of__hrat: tyop_2Ehrat_2Ehrat > tyop_2Ehrat_2Ehrat > $o ).
thf(c_2Ehrat_2Ehrat__1,type,
c_2Ehrat_2Ehrat__1: tyop_2Ehrat_2Ehrat ).
thf(c_2Ehrat_2Ehrat__add,type,
c_2Ehrat_2Ehrat__add: tyop_2Ehrat_2Ehrat > tyop_2Ehrat_2Ehrat > tyop_2Ehrat_2Ehrat ).
thf(c_2Ehrat_2Ehrat__inv,type,
c_2Ehrat_2Ehrat__inv: tyop_2Ehrat_2Ehrat > tyop_2Ehrat_2Ehrat ).
thf(c_2Ehreal_2Ehrat__lt,type,
c_2Ehreal_2Ehrat__lt: tyop_2Ehrat_2Ehrat > tyop_2Ehrat_2Ehrat > $o ).
thf(c_2Ehrat_2Ehrat__mul,type,
c_2Ehrat_2Ehrat__mul: tyop_2Ehrat_2Ehrat > tyop_2Ehrat_2Ehrat > tyop_2Ehrat_2Ehrat ).
thf(c_2Ehreal_2Ehreal,type,
c_2Ehreal_2Ehreal: ( tyop_2Ehrat_2Ehrat > $o ) > tyop_2Ehreal_2Ehreal ).
thf(c_2Ehreal_2Ehreal__1,type,
c_2Ehreal_2Ehreal__1: tyop_2Ehreal_2Ehreal ).
thf(c_2Ehreal_2Ehreal__add,type,
c_2Ehreal_2Ehreal__add: tyop_2Ehreal_2Ehreal > tyop_2Ehreal_2Ehreal > tyop_2Ehreal_2Ehreal ).
thf(c_2Ehreal_2Ehreal__inv,type,
c_2Ehreal_2Ehreal__inv: tyop_2Ehreal_2Ehreal > tyop_2Ehreal_2Ehreal ).
thf(c_2Ehreal_2Ehreal__lt,type,
c_2Ehreal_2Ehreal__lt: tyop_2Ehreal_2Ehreal > tyop_2Ehreal_2Ehreal > $o ).
thf(c_2Ehreal_2Ehreal__mul,type,
c_2Ehreal_2Ehreal__mul: tyop_2Ehreal_2Ehreal > tyop_2Ehreal_2Ehreal > tyop_2Ehreal_2Ehreal ).
thf(c_2Ehreal_2Ehreal__sub,type,
c_2Ehreal_2Ehreal__sub: tyop_2Ehreal_2Ehreal > tyop_2Ehreal_2Ehreal > tyop_2Ehreal_2Ehreal ).
thf(c_2Ehreal_2Ehreal__sup,type,
c_2Ehreal_2Ehreal__sup: ( tyop_2Ehreal_2Ehreal > $o ) > tyop_2Ehreal_2Ehreal ).
thf(c_2Ehreal_2Eisacut,type,
c_2Ehreal_2Eisacut: ( tyop_2Ehrat_2Ehrat > $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_2Ehreal_2Ehrat__lt,axiom,
! [V0x: tyop_2Ehrat_2Ehrat,V1y: tyop_2Ehrat_2Ehrat] :
( ( c_2Ehreal_2Ehrat__lt @ V0x @ V1y )
<=> ? [V2d: tyop_2Ehrat_2Ehrat] :
( V1y
= ( c_2Ehrat_2Ehrat__add @ V0x @ V2d ) ) ) ).
thf(thm_2Ehreal_2Eisacut,axiom,
! [V0C: tyop_2Ehrat_2Ehrat > $o] :
( ( c_2Ehreal_2Eisacut @ V0C )
<=> ( ? [V1x: tyop_2Ehrat_2Ehrat] : ( V0C @ V1x )
& ? [V2x: tyop_2Ehrat_2Ehrat] : ( (~) @ ( V0C @ V2x ) )
& ! [V3x: tyop_2Ehrat_2Ehrat,V4y: tyop_2Ehrat_2Ehrat] :
( ( ( V0C @ V3x )
& ( c_2Ehreal_2Ehrat__lt @ V4y @ V3x ) )
=> ( V0C @ V4y ) )
& ! [V5x: tyop_2Ehrat_2Ehrat] :
( ( V0C @ V5x )
=> ? [V6y: tyop_2Ehrat_2Ehrat] :
( ( V0C @ V6y )
& ( c_2Ehreal_2Ehrat__lt @ V5x @ V6y ) ) ) ) ) ).
thf(thm_2Ehreal_2Ecut__of__hrat,axiom,
! [V0x: tyop_2Ehrat_2Ehrat] :
( ( c_2Ehreal_2Ecut__of__hrat @ V0x )
= ( ^ [V1y: tyop_2Ehrat_2Ehrat] : ( c_2Ehreal_2Ehrat__lt @ V1y @ V0x ) ) ) ).
thf(thm_2Ehreal_2Ehreal__TY__DEF,axiom,
? [V0rep: tyop_2Ehreal_2Ehreal > tyop_2Ehrat_2Ehrat > $o] : ( c_2Ebool_2ETYPE__DEFINITION @ ( tyop_2Ehrat_2Ehrat > $o ) @ tyop_2Ehreal_2Ehreal @ c_2Ehreal_2Eisacut @ V0rep ) ).
thf(thm_2Ehreal_2Ehreal__tybij,axiom,
( ! [V0a: tyop_2Ehreal_2Ehreal] :
( ( c_2Ehreal_2Ehreal @ ( c_2Ehreal_2Ecut @ V0a ) )
= V0a )
& ! [V1r: tyop_2Ehrat_2Ehrat > $o] :
( ( c_2Ehreal_2Eisacut @ V1r )
<=> ( ( c_2Ehreal_2Ecut @ ( c_2Ehreal_2Ehreal @ V1r ) )
= V1r ) ) ) ).
thf(thm_2Ehreal_2Ehreal__1,axiom,
( c_2Ehreal_2Ehreal__1
= ( c_2Ehreal_2Ehreal @ ( c_2Ehreal_2Ecut__of__hrat @ c_2Ehrat_2Ehrat__1 ) ) ) ).
thf(thm_2Ehreal_2Ehreal__add,axiom,
! [V0X: tyop_2Ehreal_2Ehreal,V1Y: tyop_2Ehreal_2Ehreal] :
( ( c_2Ehreal_2Ehreal__add @ V0X @ V1Y )
= ( c_2Ehreal_2Ehreal
@ ^ [V2w: tyop_2Ehrat_2Ehrat] :
( c_2Ebool_2E_3F @ tyop_2Ehrat_2Ehrat
@ ^ [V3x: tyop_2Ehrat_2Ehrat] :
( c_2Ebool_2E_3F @ tyop_2Ehrat_2Ehrat
@ ^ [V4y: tyop_2Ehrat_2Ehrat] : ( c_2Ebool_2E_2F_5C @ ( c_2Emin_2E_3D @ tyop_2Ehrat_2Ehrat @ V2w @ ( c_2Ehrat_2Ehrat__add @ V3x @ V4y ) ) @ ( c_2Ebool_2E_2F_5C @ ( c_2Ehreal_2Ecut @ V0X @ V3x ) @ ( c_2Ehreal_2Ecut @ V1Y @ V4y ) ) ) ) ) ) ) ).
thf(thm_2Ehreal_2Ehreal__mul,axiom,
! [V0X: tyop_2Ehreal_2Ehreal,V1Y: tyop_2Ehreal_2Ehreal] :
( ( c_2Ehreal_2Ehreal__mul @ V0X @ V1Y )
= ( c_2Ehreal_2Ehreal
@ ^ [V2w: tyop_2Ehrat_2Ehrat] :
( c_2Ebool_2E_3F @ tyop_2Ehrat_2Ehrat
@ ^ [V3x: tyop_2Ehrat_2Ehrat] :
( c_2Ebool_2E_3F @ tyop_2Ehrat_2Ehrat
@ ^ [V4y: tyop_2Ehrat_2Ehrat] : ( c_2Ebool_2E_2F_5C @ ( c_2Emin_2E_3D @ tyop_2Ehrat_2Ehrat @ V2w @ ( c_2Ehrat_2Ehrat__mul @ V3x @ V4y ) ) @ ( c_2Ebool_2E_2F_5C @ ( c_2Ehreal_2Ecut @ V0X @ V3x ) @ ( c_2Ehreal_2Ecut @ V1Y @ V4y ) ) ) ) ) ) ) ).
thf(thm_2Ehreal_2Ehreal__inv,axiom,
! [V0X: tyop_2Ehreal_2Ehreal] :
( ( c_2Ehreal_2Ehreal__inv @ V0X )
= ( c_2Ehreal_2Ehreal
@ ^ [V1w: tyop_2Ehrat_2Ehrat] :
( c_2Ebool_2E_3F @ tyop_2Ehrat_2Ehrat
@ ^ [V2d: tyop_2Ehrat_2Ehrat] :
( c_2Ebool_2E_2F_5C @ ( c_2Ehreal_2Ehrat__lt @ V2d @ c_2Ehrat_2Ehrat__1 )
@ ( c_2Ebool_2E_21 @ tyop_2Ehrat_2Ehrat
@ ^ [V3x: tyop_2Ehrat_2Ehrat] : ( c_2Emin_2E_3D_3D_3E @ ( c_2Ehreal_2Ecut @ V0X @ V3x ) @ ( c_2Ehreal_2Ehrat__lt @ ( c_2Ehrat_2Ehrat__mul @ V1w @ V3x ) @ V2d ) ) ) ) ) ) ) ).
thf(thm_2Ehreal_2Ehreal__sup,axiom,
! [V0P: tyop_2Ehreal_2Ehreal > $o] :
( ( c_2Ehreal_2Ehreal__sup @ V0P )
= ( c_2Ehreal_2Ehreal
@ ^ [V1w: tyop_2Ehrat_2Ehrat] :
( c_2Ebool_2E_3F @ tyop_2Ehreal_2Ehreal
@ ^ [V2X: tyop_2Ehreal_2Ehreal] : ( c_2Ebool_2E_2F_5C @ ( V0P @ V2X ) @ ( c_2Ehreal_2Ecut @ V2X @ V1w ) ) ) ) ) ).
thf(thm_2Ehreal_2Ehreal__lt,axiom,
! [V0X: tyop_2Ehreal_2Ehreal,V1Y: tyop_2Ehreal_2Ehreal] :
( ( c_2Ehreal_2Ehreal__lt @ V0X @ V1Y )
<=> ( ( (~) @ ( V0X = V1Y ) )
& ! [V2x: tyop_2Ehrat_2Ehrat] :
( ( c_2Ehreal_2Ecut @ V0X @ V2x )
=> ( c_2Ehreal_2Ecut @ V1Y @ V2x ) ) ) ) ).
thf(thm_2Ehreal_2Ehreal__sub,axiom,
! [V0Y: tyop_2Ehreal_2Ehreal,V1X: tyop_2Ehreal_2Ehreal] :
( ( c_2Ehreal_2Ehreal__sub @ V0Y @ V1X )
= ( c_2Ehreal_2Ehreal
@ ^ [V2w: tyop_2Ehrat_2Ehrat] :
( c_2Ebool_2E_3F @ tyop_2Ehrat_2Ehrat
@ ^ [V3x: tyop_2Ehrat_2Ehrat] : ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2E_7E @ ( c_2Ehreal_2Ecut @ V1X @ V3x ) ) @ ( c_2Ehreal_2Ecut @ V0Y @ ( c_2Ehrat_2Ehrat__add @ V3x @ V2w ) ) ) ) ) ) ).
thf(thm_2Ehreal_2EHRAT__LT__REFL,axiom,
! [V0x: tyop_2Ehrat_2Ehrat] : ( (~) @ ( c_2Ehreal_2Ehrat__lt @ V0x @ V0x ) ) ).
thf(thm_2Ehreal_2EHRAT__LT__TRANS,axiom,
! [V0x: tyop_2Ehrat_2Ehrat,V1y: tyop_2Ehrat_2Ehrat,V2z: tyop_2Ehrat_2Ehrat] :
( ( ( c_2Ehreal_2Ehrat__lt @ V0x @ V1y )
& ( c_2Ehreal_2Ehrat__lt @ V1y @ V2z ) )
=> ( c_2Ehreal_2Ehrat__lt @ V0x @ V2z ) ) ).
thf(thm_2Ehreal_2EHRAT__LT__ANTISYM,axiom,
! [V0x: tyop_2Ehrat_2Ehrat,V1y: tyop_2Ehrat_2Ehrat] :
( (~)
@ ( ( c_2Ehreal_2Ehrat__lt @ V0x @ V1y )
& ( c_2Ehreal_2Ehrat__lt @ V1y @ V0x ) ) ) ).
thf(thm_2Ehreal_2EHRAT__LT__TOTAL,axiom,
! [V0x: tyop_2Ehrat_2Ehrat,V1y: tyop_2Ehrat_2Ehrat] :
( ( V0x = V1y )
| ( c_2Ehreal_2Ehrat__lt @ V0x @ V1y )
| ( c_2Ehreal_2Ehrat__lt @ V1y @ V0x ) ) ).
thf(thm_2Ehreal_2EHRAT__MUL__RID,axiom,
! [V0x: tyop_2Ehrat_2Ehrat] :
( ( c_2Ehrat_2Ehrat__mul @ V0x @ c_2Ehrat_2Ehrat__1 )
= V0x ) ).
thf(thm_2Ehreal_2EHRAT__MUL__RINV,axiom,
! [V0x: tyop_2Ehrat_2Ehrat] :
( ( c_2Ehrat_2Ehrat__mul @ V0x @ ( c_2Ehrat_2Ehrat__inv @ V0x ) )
= c_2Ehrat_2Ehrat__1 ) ).
thf(thm_2Ehreal_2EHRAT__RDISTRIB,axiom,
! [V0x: tyop_2Ehrat_2Ehrat,V1y: tyop_2Ehrat_2Ehrat,V2z: tyop_2Ehrat_2Ehrat] :
( ( c_2Ehrat_2Ehrat__mul @ ( c_2Ehrat_2Ehrat__add @ V0x @ V1y ) @ V2z )
= ( c_2Ehrat_2Ehrat__add @ ( c_2Ehrat_2Ehrat__mul @ V0x @ V2z ) @ ( c_2Ehrat_2Ehrat__mul @ V1y @ V2z ) ) ) ).
thf(thm_2Ehreal_2EHRAT__LT__ADDL,axiom,
! [V0x: tyop_2Ehrat_2Ehrat,V1y: tyop_2Ehrat_2Ehrat] : ( c_2Ehreal_2Ehrat__lt @ V0x @ ( c_2Ehrat_2Ehrat__add @ V0x @ V1y ) ) ).
thf(thm_2Ehreal_2EHRAT__LT__ADDR,axiom,
! [V0x: tyop_2Ehrat_2Ehrat,V1y: tyop_2Ehrat_2Ehrat] : ( c_2Ehreal_2Ehrat__lt @ V1y @ ( c_2Ehrat_2Ehrat__add @ V0x @ V1y ) ) ).
thf(thm_2Ehreal_2EHRAT__LT__GT,axiom,
! [V0x: tyop_2Ehrat_2Ehrat,V1y: tyop_2Ehrat_2Ehrat] :
( ( c_2Ehreal_2Ehrat__lt @ V0x @ V1y )
=> ( (~) @ ( c_2Ehreal_2Ehrat__lt @ V1y @ V0x ) ) ) ).
thf(thm_2Ehreal_2EHRAT__LT__NE,axiom,
! [V0x: tyop_2Ehrat_2Ehrat,V1y: tyop_2Ehrat_2Ehrat] :
( ( c_2Ehreal_2Ehrat__lt @ V0x @ V1y )
=> ( (~) @ ( V0x = V1y ) ) ) ).
thf(thm_2Ehreal_2EHRAT__EQ__LADD,axiom,
! [V0x: tyop_2Ehrat_2Ehrat,V1y: tyop_2Ehrat_2Ehrat,V2z: tyop_2Ehrat_2Ehrat] :
( ( ( c_2Ehrat_2Ehrat__add @ V0x @ V1y )
= ( c_2Ehrat_2Ehrat__add @ V0x @ V2z ) )
<=> ( V1y = V2z ) ) ).
thf(thm_2Ehreal_2EHRAT__EQ__LMUL,axiom,
! [V0x: tyop_2Ehrat_2Ehrat,V1y: tyop_2Ehrat_2Ehrat,V2z: tyop_2Ehrat_2Ehrat] :
( ( ( c_2Ehrat_2Ehrat__mul @ V0x @ V1y )
= ( c_2Ehrat_2Ehrat__mul @ V0x @ V2z ) )
<=> ( V1y = V2z ) ) ).
thf(thm_2Ehreal_2EHRAT__LT__ADD2,axiom,
! [V0u: tyop_2Ehrat_2Ehrat,V1v: tyop_2Ehrat_2Ehrat,V2x: tyop_2Ehrat_2Ehrat,V3y: tyop_2Ehrat_2Ehrat] :
( ( ( c_2Ehreal_2Ehrat__lt @ V0u @ V2x )
& ( c_2Ehreal_2Ehrat__lt @ V1v @ V3y ) )
=> ( c_2Ehreal_2Ehrat__lt @ ( c_2Ehrat_2Ehrat__add @ V0u @ V1v ) @ ( c_2Ehrat_2Ehrat__add @ V2x @ V3y ) ) ) ).
thf(thm_2Ehreal_2EHRAT__LT__LADD,axiom,
! [V0x: tyop_2Ehrat_2Ehrat,V1y: tyop_2Ehrat_2Ehrat,V2z: tyop_2Ehrat_2Ehrat] :
( ( c_2Ehreal_2Ehrat__lt @ ( c_2Ehrat_2Ehrat__add @ V2z @ V0x ) @ ( c_2Ehrat_2Ehrat__add @ V2z @ V1y ) )
= ( c_2Ehreal_2Ehrat__lt @ V0x @ V1y ) ) ).
thf(thm_2Ehreal_2EHRAT__LT__RADD,axiom,
! [V0x: tyop_2Ehrat_2Ehrat,V1y: tyop_2Ehrat_2Ehrat,V2z: tyop_2Ehrat_2Ehrat] :
( ( c_2Ehreal_2Ehrat__lt @ ( c_2Ehrat_2Ehrat__add @ V0x @ V2z ) @ ( c_2Ehrat_2Ehrat__add @ V1y @ V2z ) )
= ( c_2Ehreal_2Ehrat__lt @ V0x @ V1y ) ) ).
thf(thm_2Ehreal_2EHRAT__LT__MUL2,axiom,
! [V0u: tyop_2Ehrat_2Ehrat,V1v: tyop_2Ehrat_2Ehrat,V2x: tyop_2Ehrat_2Ehrat,V3y: tyop_2Ehrat_2Ehrat] :
( ( ( c_2Ehreal_2Ehrat__lt @ V0u @ V2x )
& ( c_2Ehreal_2Ehrat__lt @ V1v @ V3y ) )
=> ( c_2Ehreal_2Ehrat__lt @ ( c_2Ehrat_2Ehrat__mul @ V0u @ V1v ) @ ( c_2Ehrat_2Ehrat__mul @ V2x @ V3y ) ) ) ).
thf(thm_2Ehreal_2EHRAT__LT__LMUL,axiom,
! [V0x: tyop_2Ehrat_2Ehrat,V1y: tyop_2Ehrat_2Ehrat,V2z: tyop_2Ehrat_2Ehrat] :
( ( c_2Ehreal_2Ehrat__lt @ ( c_2Ehrat_2Ehrat__mul @ V2z @ V0x ) @ ( c_2Ehrat_2Ehrat__mul @ V2z @ V1y ) )
= ( c_2Ehreal_2Ehrat__lt @ V0x @ V1y ) ) ).
thf(thm_2Ehreal_2EHRAT__LT__RMUL,axiom,
! [V0x: tyop_2Ehrat_2Ehrat,V1y: tyop_2Ehrat_2Ehrat,V2z: tyop_2Ehrat_2Ehrat] :
( ( c_2Ehreal_2Ehrat__lt @ ( c_2Ehrat_2Ehrat__mul @ V0x @ V2z ) @ ( c_2Ehrat_2Ehrat__mul @ V1y @ V2z ) )
= ( c_2Ehreal_2Ehrat__lt @ V0x @ V1y ) ) ).
thf(thm_2Ehreal_2EHRAT__LT__LMUL1,axiom,
! [V0x: tyop_2Ehrat_2Ehrat,V1y: tyop_2Ehrat_2Ehrat] :
( ( c_2Ehreal_2Ehrat__lt @ ( c_2Ehrat_2Ehrat__mul @ V0x @ V1y ) @ V1y )
= ( c_2Ehreal_2Ehrat__lt @ V0x @ c_2Ehrat_2Ehrat__1 ) ) ).
thf(thm_2Ehreal_2EHRAT__LT__RMUL1,axiom,
! [V0x: tyop_2Ehrat_2Ehrat,V1y: tyop_2Ehrat_2Ehrat] :
( ( c_2Ehreal_2Ehrat__lt @ ( c_2Ehrat_2Ehrat__mul @ V0x @ V1y ) @ V0x )
= ( c_2Ehreal_2Ehrat__lt @ V1y @ c_2Ehrat_2Ehrat__1 ) ) ).
thf(thm_2Ehreal_2EHRAT__GT__LMUL1,axiom,
! [V0x: tyop_2Ehrat_2Ehrat,V1y: tyop_2Ehrat_2Ehrat] :
( ( c_2Ehreal_2Ehrat__lt @ V1y @ ( c_2Ehrat_2Ehrat__mul @ V0x @ V1y ) )
= ( c_2Ehreal_2Ehrat__lt @ c_2Ehrat_2Ehrat__1 @ V0x ) ) ).
thf(thm_2Ehreal_2EHRAT__LT__L1,axiom,
! [V0x: tyop_2Ehrat_2Ehrat,V1y: tyop_2Ehrat_2Ehrat] :
( ( c_2Ehreal_2Ehrat__lt @ ( c_2Ehrat_2Ehrat__mul @ ( c_2Ehrat_2Ehrat__inv @ V0x ) @ V1y ) @ c_2Ehrat_2Ehrat__1 )
= ( c_2Ehreal_2Ehrat__lt @ V1y @ V0x ) ) ).
thf(thm_2Ehreal_2EHRAT__LT__R1,axiom,
! [V0x: tyop_2Ehrat_2Ehrat,V1y: tyop_2Ehrat_2Ehrat] :
( ( c_2Ehreal_2Ehrat__lt @ ( c_2Ehrat_2Ehrat__mul @ V0x @ ( c_2Ehrat_2Ehrat__inv @ V1y ) ) @ c_2Ehrat_2Ehrat__1 )
= ( c_2Ehreal_2Ehrat__lt @ V0x @ V1y ) ) ).
thf(thm_2Ehreal_2EHRAT__GT__L1,axiom,
! [V0x: tyop_2Ehrat_2Ehrat,V1y: tyop_2Ehrat_2Ehrat] :
( ( c_2Ehreal_2Ehrat__lt @ c_2Ehrat_2Ehrat__1 @ ( c_2Ehrat_2Ehrat__mul @ ( c_2Ehrat_2Ehrat__inv @ V0x ) @ V1y ) )
= ( c_2Ehreal_2Ehrat__lt @ V0x @ V1y ) ) ).
thf(thm_2Ehreal_2EHRAT__INV__MUL,axiom,
! [V0x: tyop_2Ehrat_2Ehrat,V1y: tyop_2Ehrat_2Ehrat] :
( ( c_2Ehrat_2Ehrat__inv @ ( c_2Ehrat_2Ehrat__mul @ V0x @ V1y ) )
= ( c_2Ehrat_2Ehrat__mul @ ( c_2Ehrat_2Ehrat__inv @ V0x ) @ ( c_2Ehrat_2Ehrat__inv @ V1y ) ) ) ).
thf(thm_2Ehreal_2EHRAT__UP,axiom,
! [V0x: tyop_2Ehrat_2Ehrat] :
? [V1y: tyop_2Ehrat_2Ehrat] : ( c_2Ehreal_2Ehrat__lt @ V0x @ V1y ) ).
thf(thm_2Ehreal_2EHRAT__DOWN,axiom,
! [V0x: tyop_2Ehrat_2Ehrat] :
? [V1y: tyop_2Ehrat_2Ehrat] : ( c_2Ehreal_2Ehrat__lt @ V1y @ V0x ) ).
thf(thm_2Ehreal_2EHRAT__DOWN2,axiom,
! [V0x: tyop_2Ehrat_2Ehrat,V1y: tyop_2Ehrat_2Ehrat] :
? [V2z: tyop_2Ehrat_2Ehrat] :
( ( c_2Ehreal_2Ehrat__lt @ V2z @ V0x )
& ( c_2Ehreal_2Ehrat__lt @ V2z @ V1y ) ) ).
thf(thm_2Ehreal_2EHRAT__MEAN,axiom,
! [V0x: tyop_2Ehrat_2Ehrat,V1y: tyop_2Ehrat_2Ehrat] :
( ( c_2Ehreal_2Ehrat__lt @ V0x @ V1y )
=> ? [V2z: tyop_2Ehrat_2Ehrat] :
( ( c_2Ehreal_2Ehrat__lt @ V0x @ V2z )
& ( c_2Ehreal_2Ehrat__lt @ V2z @ V1y ) ) ) ).
thf(thm_2Ehreal_2EISACUT__HRAT,axiom,
! [V0h: tyop_2Ehrat_2Ehrat] : ( c_2Ehreal_2Eisacut @ ( c_2Ehreal_2Ecut__of__hrat @ V0h ) ) ).
thf(thm_2Ehreal_2EEQUAL__CUTS,axiom,
! [V0X: tyop_2Ehreal_2Ehreal,V1Y: tyop_2Ehreal_2Ehreal] :
( ( ( c_2Ehreal_2Ecut @ V0X )
= ( c_2Ehreal_2Ecut @ V1Y ) )
=> ( V0X = V1Y ) ) ).
thf(thm_2Ehreal_2ECUT__ISACUT,axiom,
! [V0X: tyop_2Ehreal_2Ehreal] : ( c_2Ehreal_2Eisacut @ ( c_2Ehreal_2Ecut @ V0X ) ) ).
thf(thm_2Ehreal_2ECUT__NONEMPTY,axiom,
! [V0X: tyop_2Ehreal_2Ehreal] :
? [V1x: tyop_2Ehrat_2Ehrat] : ( c_2Ehreal_2Ecut @ V0X @ V1x ) ).
thf(thm_2Ehreal_2ECUT__BOUNDED,axiom,
! [V0X: tyop_2Ehreal_2Ehreal] :
? [V1x: tyop_2Ehrat_2Ehrat] : ( (~) @ ( c_2Ehreal_2Ecut @ V0X @ V1x ) ) ).
thf(thm_2Ehreal_2ECUT__DOWN,axiom,
! [V0X: tyop_2Ehreal_2Ehreal,V1x: tyop_2Ehrat_2Ehrat,V2y: tyop_2Ehrat_2Ehrat] :
( ( ( c_2Ehreal_2Ecut @ V0X @ V1x )
& ( c_2Ehreal_2Ehrat__lt @ V2y @ V1x ) )
=> ( c_2Ehreal_2Ecut @ V0X @ V2y ) ) ).
thf(thm_2Ehreal_2ECUT__UP,axiom,
! [V0X: tyop_2Ehreal_2Ehreal,V1x: tyop_2Ehrat_2Ehrat] :
( ( c_2Ehreal_2Ecut @ V0X @ V1x )
=> ? [V2y: tyop_2Ehrat_2Ehrat] :
( ( c_2Ehreal_2Ecut @ V0X @ V2y )
& ( c_2Ehreal_2Ehrat__lt @ V1x @ V2y ) ) ) ).
thf(thm_2Ehreal_2ECUT__UBOUND,axiom,
! [V0X: tyop_2Ehreal_2Ehreal,V1x: tyop_2Ehrat_2Ehrat,V2y: tyop_2Ehrat_2Ehrat] :
( ( ( (~) @ ( c_2Ehreal_2Ecut @ V0X @ V1x ) )
& ( c_2Ehreal_2Ehrat__lt @ V1x @ V2y ) )
=> ( (~) @ ( c_2Ehreal_2Ecut @ V0X @ V2y ) ) ) ).
thf(thm_2Ehreal_2ECUT__STRADDLE,axiom,
! [V0X: tyop_2Ehreal_2Ehreal,V1x: tyop_2Ehrat_2Ehrat,V2y: tyop_2Ehrat_2Ehrat] :
( ( ( c_2Ehreal_2Ecut @ V0X @ V1x )
& ( (~) @ ( c_2Ehreal_2Ecut @ V0X @ V2y ) ) )
=> ( c_2Ehreal_2Ehrat__lt @ V1x @ V2y ) ) ).
thf(thm_2Ehreal_2ECUT__NEARTOP__ADD,axiom,
! [V0X: tyop_2Ehreal_2Ehreal,V1e: tyop_2Ehrat_2Ehrat] :
? [V2x: tyop_2Ehrat_2Ehrat] :
( ( c_2Ehreal_2Ecut @ V0X @ V2x )
& ( (~) @ ( c_2Ehreal_2Ecut @ V0X @ ( c_2Ehrat_2Ehrat__add @ V2x @ V1e ) ) ) ) ).
thf(thm_2Ehreal_2ECUT__NEARTOP__MUL,axiom,
! [V0X: tyop_2Ehreal_2Ehreal,V1u: tyop_2Ehrat_2Ehrat] :
( ( c_2Ehreal_2Ehrat__lt @ c_2Ehrat_2Ehrat__1 @ V1u )
=> ? [V2x: tyop_2Ehrat_2Ehrat] :
( ( c_2Ehreal_2Ecut @ V0X @ V2x )
& ( (~) @ ( c_2Ehreal_2Ecut @ V0X @ ( c_2Ehrat_2Ehrat__mul @ V1u @ V2x ) ) ) ) ) ).
thf(thm_2Ehreal_2EHREAL__INV__ISACUT,axiom,
! [V0X: tyop_2Ehreal_2Ehreal] :
( c_2Ehreal_2Eisacut
@ ^ [V1w: tyop_2Ehrat_2Ehrat] :
( c_2Ebool_2E_3F @ tyop_2Ehrat_2Ehrat
@ ^ [V2d: tyop_2Ehrat_2Ehrat] :
( c_2Ebool_2E_2F_5C @ ( c_2Ehreal_2Ehrat__lt @ V2d @ c_2Ehrat_2Ehrat__1 )
@ ( c_2Ebool_2E_21 @ tyop_2Ehrat_2Ehrat
@ ^ [V3x: tyop_2Ehrat_2Ehrat] : ( c_2Emin_2E_3D_3D_3E @ ( c_2Ehreal_2Ecut @ V0X @ V3x ) @ ( c_2Ehreal_2Ehrat__lt @ ( c_2Ehrat_2Ehrat__mul @ V1w @ V3x ) @ V2d ) ) ) ) ) ) ).
thf(thm_2Ehreal_2EHREAL__ADD__ISACUT,axiom,
! [V0X: tyop_2Ehreal_2Ehreal,V1Y: tyop_2Ehreal_2Ehreal] :
( c_2Ehreal_2Eisacut
@ ^ [V2w: tyop_2Ehrat_2Ehrat] :
( c_2Ebool_2E_3F @ tyop_2Ehrat_2Ehrat
@ ^ [V3x: tyop_2Ehrat_2Ehrat] :
( c_2Ebool_2E_3F @ tyop_2Ehrat_2Ehrat
@ ^ [V4y: tyop_2Ehrat_2Ehrat] : ( c_2Ebool_2E_2F_5C @ ( c_2Emin_2E_3D @ tyop_2Ehrat_2Ehrat @ V2w @ ( c_2Ehrat_2Ehrat__add @ V3x @ V4y ) ) @ ( c_2Ebool_2E_2F_5C @ ( c_2Ehreal_2Ecut @ V0X @ V3x ) @ ( c_2Ehreal_2Ecut @ V1Y @ V4y ) ) ) ) ) ) ).
thf(thm_2Ehreal_2EHREAL__MUL__ISACUT,axiom,
! [V0X: tyop_2Ehreal_2Ehreal,V1Y: tyop_2Ehreal_2Ehreal] :
( c_2Ehreal_2Eisacut
@ ^ [V2w: tyop_2Ehrat_2Ehrat] :
( c_2Ebool_2E_3F @ tyop_2Ehrat_2Ehrat
@ ^ [V3x: tyop_2Ehrat_2Ehrat] :
( c_2Ebool_2E_3F @ tyop_2Ehrat_2Ehrat
@ ^ [V4y: tyop_2Ehrat_2Ehrat] : ( c_2Ebool_2E_2F_5C @ ( c_2Emin_2E_3D @ tyop_2Ehrat_2Ehrat @ V2w @ ( c_2Ehrat_2Ehrat__mul @ V3x @ V4y ) ) @ ( c_2Ebool_2E_2F_5C @ ( c_2Ehreal_2Ecut @ V0X @ V3x ) @ ( c_2Ehreal_2Ecut @ V1Y @ V4y ) ) ) ) ) ) ).
thf(thm_2Ehreal_2EHREAL__ADD__SYM,axiom,
! [V0X: tyop_2Ehreal_2Ehreal,V1Y: tyop_2Ehreal_2Ehreal] :
( ( c_2Ehreal_2Ehreal__add @ V0X @ V1Y )
= ( c_2Ehreal_2Ehreal__add @ V1Y @ V0X ) ) ).
thf(thm_2Ehreal_2EHREAL__MUL__SYM,axiom,
! [V0X: tyop_2Ehreal_2Ehreal,V1Y: tyop_2Ehreal_2Ehreal] :
( ( c_2Ehreal_2Ehreal__mul @ V0X @ V1Y )
= ( c_2Ehreal_2Ehreal__mul @ V1Y @ V0X ) ) ).
thf(thm_2Ehreal_2EHREAL__ADD__ASSOC,axiom,
! [V0X: tyop_2Ehreal_2Ehreal,V1Y: tyop_2Ehreal_2Ehreal,V2Z: tyop_2Ehreal_2Ehreal] :
( ( c_2Ehreal_2Ehreal__add @ V0X @ ( c_2Ehreal_2Ehreal__add @ V1Y @ V2Z ) )
= ( c_2Ehreal_2Ehreal__add @ ( c_2Ehreal_2Ehreal__add @ V0X @ V1Y ) @ V2Z ) ) ).
thf(thm_2Ehreal_2EHREAL__MUL__ASSOC,axiom,
! [V0X: tyop_2Ehreal_2Ehreal,V1Y: tyop_2Ehreal_2Ehreal,V2Z: tyop_2Ehreal_2Ehreal] :
( ( c_2Ehreal_2Ehreal__mul @ V0X @ ( c_2Ehreal_2Ehreal__mul @ V1Y @ V2Z ) )
= ( c_2Ehreal_2Ehreal__mul @ ( c_2Ehreal_2Ehreal__mul @ V0X @ V1Y ) @ V2Z ) ) ).
thf(thm_2Ehreal_2EHREAL__LDISTRIB,axiom,
! [V0X: tyop_2Ehreal_2Ehreal,V1Y: tyop_2Ehreal_2Ehreal,V2Z: tyop_2Ehreal_2Ehreal] :
( ( c_2Ehreal_2Ehreal__mul @ V0X @ ( c_2Ehreal_2Ehreal__add @ V1Y @ V2Z ) )
= ( c_2Ehreal_2Ehreal__add @ ( c_2Ehreal_2Ehreal__mul @ V0X @ V1Y ) @ ( c_2Ehreal_2Ehreal__mul @ V0X @ V2Z ) ) ) ).
thf(thm_2Ehreal_2EHREAL__MUL__LID,axiom,
! [V0X: tyop_2Ehreal_2Ehreal] :
( ( c_2Ehreal_2Ehreal__mul @ c_2Ehreal_2Ehreal__1 @ V0X )
= V0X ) ).
thf(thm_2Ehreal_2EHREAL__MUL__LINV,axiom,
! [V0X: tyop_2Ehreal_2Ehreal] :
( ( c_2Ehreal_2Ehreal__mul @ ( c_2Ehreal_2Ehreal__inv @ V0X ) @ V0X )
= c_2Ehreal_2Ehreal__1 ) ).
thf(thm_2Ehreal_2EHREAL__NOZERO,axiom,
! [V0X: tyop_2Ehreal_2Ehreal,V1Y: tyop_2Ehreal_2Ehreal] :
( (~)
@ ( ( c_2Ehreal_2Ehreal__add @ V0X @ V1Y )
= V0X ) ) ).
thf(thm_2Ehreal_2EHREAL__LT__LEMMA,axiom,
! [V0X: tyop_2Ehreal_2Ehreal,V1Y: tyop_2Ehreal_2Ehreal] :
( ( c_2Ehreal_2Ehreal__lt @ V0X @ V1Y )
=> ? [V2x: tyop_2Ehrat_2Ehrat] :
( ( (~) @ ( c_2Ehreal_2Ecut @ V0X @ V2x ) )
& ( c_2Ehreal_2Ecut @ V1Y @ V2x ) ) ) ).
thf(thm_2Ehreal_2EHREAL__SUB__ISACUT,axiom,
! [V0X: tyop_2Ehreal_2Ehreal,V1Y: tyop_2Ehreal_2Ehreal] :
( ( c_2Ehreal_2Ehreal__lt @ V0X @ V1Y )
=> ( c_2Ehreal_2Eisacut
@ ^ [V2w: tyop_2Ehrat_2Ehrat] :
( c_2Ebool_2E_3F @ tyop_2Ehrat_2Ehrat
@ ^ [V3x: tyop_2Ehrat_2Ehrat] : ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2E_7E @ ( c_2Ehreal_2Ecut @ V0X @ V3x ) ) @ ( c_2Ehreal_2Ecut @ V1Y @ ( c_2Ehrat_2Ehrat__add @ V3x @ V2w ) ) ) ) ) ) ).
thf(thm_2Ehreal_2EHREAL__SUB__ADD,axiom,
! [V0X: tyop_2Ehreal_2Ehreal,V1Y: tyop_2Ehreal_2Ehreal] :
( ( c_2Ehreal_2Ehreal__lt @ V0X @ V1Y )
=> ( ( c_2Ehreal_2Ehreal__add @ ( c_2Ehreal_2Ehreal__sub @ V1Y @ V0X ) @ V0X )
= V1Y ) ) ).
thf(thm_2Ehreal_2EHREAL__LT__TOTAL,axiom,
! [V0X: tyop_2Ehreal_2Ehreal,V1Y: tyop_2Ehreal_2Ehreal] :
( ( V0X = V1Y )
| ( c_2Ehreal_2Ehreal__lt @ V0X @ V1Y )
| ( c_2Ehreal_2Ehreal__lt @ V1Y @ V0X ) ) ).
thf(thm_2Ehreal_2EHREAL__LT,axiom,
! [V0X: tyop_2Ehreal_2Ehreal,V1Y: tyop_2Ehreal_2Ehreal] :
( ( c_2Ehreal_2Ehreal__lt @ V0X @ V1Y )
<=> ? [V2D: tyop_2Ehreal_2Ehreal] :
( V1Y
= ( c_2Ehreal_2Ehreal__add @ V0X @ V2D ) ) ) ).
thf(thm_2Ehreal_2EHREAL__ADD__TOTAL,axiom,
! [V0X: tyop_2Ehreal_2Ehreal,V1Y: tyop_2Ehreal_2Ehreal] :
( ( V0X = V1Y )
| ? [V2D: tyop_2Ehreal_2Ehreal] :
( V1Y
= ( c_2Ehreal_2Ehreal__add @ V0X @ V2D ) )
| ? [V3D: tyop_2Ehreal_2Ehreal] :
( V0X
= ( c_2Ehreal_2Ehreal__add @ V1Y @ V3D ) ) ) ).
thf(thm_2Ehreal_2EHREAL__SUP__ISACUT,axiom,
! [V0P: tyop_2Ehreal_2Ehreal > $o] :
( ( ? [V1X: tyop_2Ehreal_2Ehreal] : ( V0P @ V1X )
& ? [V2Y: tyop_2Ehreal_2Ehreal] :
! [V3X: tyop_2Ehreal_2Ehreal] :
( ( V0P @ V3X )
=> ( c_2Ehreal_2Ehreal__lt @ V3X @ V2Y ) ) )
=> ( c_2Ehreal_2Eisacut
@ ^ [V4w: tyop_2Ehrat_2Ehrat] :
( c_2Ebool_2E_3F @ tyop_2Ehreal_2Ehreal
@ ^ [V5X: tyop_2Ehreal_2Ehreal] : ( c_2Ebool_2E_2F_5C @ ( V0P @ V5X ) @ ( c_2Ehreal_2Ecut @ V5X @ V4w ) ) ) ) ) ).
thf(thm_2Ehreal_2EHREAL__SUP,axiom,
! [V0P: tyop_2Ehreal_2Ehreal > $o] :
( ( ? [V1X: tyop_2Ehreal_2Ehreal] : ( V0P @ V1X )
& ? [V2Y: tyop_2Ehreal_2Ehreal] :
! [V3X: tyop_2Ehreal_2Ehreal] :
( ( V0P @ V3X )
=> ( c_2Ehreal_2Ehreal__lt @ V3X @ V2Y ) ) )
=> ! [V4Y: tyop_2Ehreal_2Ehreal] :
( ? [V5X: tyop_2Ehreal_2Ehreal] :
( ( V0P @ V5X )
& ( c_2Ehreal_2Ehreal__lt @ V4Y @ V5X ) )
<=> ( c_2Ehreal_2Ehreal__lt @ V4Y @ ( c_2Ehreal_2Ehreal__sup @ V0P ) ) ) ) ).
%------------------------------------------------------------------------------