ITP001 Axioms: ITP099+5.ax
%------------------------------------------------------------------------------
% File : ITP099+5 : TPTP v9.0.0. Bugfixed v7.5.0.
% Domain : Interactive Theorem Proving
% Axioms : HOL4 set theory export, chainy mode
% Version : [BG+19] axioms.
% English :
% Refs : [BG+19] Brown et al. (2019), GRUNGE: A Grand Unified ATP Chall
% : [Gau20] Gauthier (2020), Email to Geoff Sutcliffe
% Source : [BG+19]
% Names : hreal+2.ax [Gau20]
% : HL4099+5.ax [TPAP]
% Status : Satisfiable
% Syntax : Number of formulae : 103 ( 14 unt; 0 def)
% Number of atoms : 383 ( 55 equ)
% Maximal formula atoms : 16 ( 3 avg)
% Number of connectives : 294 ( 14 ~; 6 |; 46 &)
% ( 18 <=>; 210 =>; 0 <=; 0 <~>)
% Maximal formula depth : 13 ( 6 avg)
% Maximal term depth : 6 ( 2 avg)
% Number of predicates : 4 ( 3 usr; 0 prp; 1-2 aty)
% Number of functors : 46 ( 46 usr; 22 con; 0-4 aty)
% Number of variables : 229 ( 206 !; 23 ?)
% SPC : FOF_SAT_RFO_SEQ
% Comments :
% Bugfixes : v7.5.0 - Fixes to the axioms.
%------------------------------------------------------------------------------
fof(ne_ty_2Ehreal_2Ehreal,axiom,
ne(ty_2Ehreal_2Ehreal) ).
fof(mem_c_2Ehreal_2Ecut,axiom,
mem(c_2Ehreal_2Ecut,arr(ty_2Ehreal_2Ehreal,arr(ty_2Ehrat_2Ehrat,bool))) ).
fof(mem_c_2Ehreal_2Ecut__of__hrat,axiom,
mem(c_2Ehreal_2Ecut__of__hrat,arr(ty_2Ehrat_2Ehrat,arr(ty_2Ehrat_2Ehrat,bool))) ).
fof(mem_c_2Ehreal_2Ehrat__lt,axiom,
mem(c_2Ehreal_2Ehrat__lt,arr(ty_2Ehrat_2Ehrat,arr(ty_2Ehrat_2Ehrat,bool))) ).
fof(mem_c_2Ehreal_2Ehreal,axiom,
mem(c_2Ehreal_2Ehreal,arr(arr(ty_2Ehrat_2Ehrat,bool),ty_2Ehreal_2Ehreal)) ).
fof(mem_c_2Ehreal_2Ehreal__1,axiom,
mem(c_2Ehreal_2Ehreal__1,ty_2Ehreal_2Ehreal) ).
fof(mem_c_2Ehreal_2Ehreal__add,axiom,
mem(c_2Ehreal_2Ehreal__add,arr(ty_2Ehreal_2Ehreal,arr(ty_2Ehreal_2Ehreal,ty_2Ehreal_2Ehreal))) ).
fof(mem_c_2Ehreal_2Ehreal__inv,axiom,
mem(c_2Ehreal_2Ehreal__inv,arr(ty_2Ehreal_2Ehreal,ty_2Ehreal_2Ehreal)) ).
fof(mem_c_2Ehreal_2Ehreal__lt,axiom,
mem(c_2Ehreal_2Ehreal__lt,arr(ty_2Ehreal_2Ehreal,arr(ty_2Ehreal_2Ehreal,bool))) ).
fof(mem_c_2Ehreal_2Ehreal__mul,axiom,
mem(c_2Ehreal_2Ehreal__mul,arr(ty_2Ehreal_2Ehreal,arr(ty_2Ehreal_2Ehreal,ty_2Ehreal_2Ehreal))) ).
fof(mem_c_2Ehreal_2Ehreal__sub,axiom,
mem(c_2Ehreal_2Ehreal__sub,arr(ty_2Ehreal_2Ehreal,arr(ty_2Ehreal_2Ehreal,ty_2Ehreal_2Ehreal))) ).
fof(mem_c_2Ehreal_2Ehreal__sup,axiom,
mem(c_2Ehreal_2Ehreal__sup,arr(arr(ty_2Ehreal_2Ehreal,bool),ty_2Ehreal_2Ehreal)) ).
fof(mem_c_2Ehreal_2Eisacut,axiom,
mem(c_2Ehreal_2Eisacut,arr(arr(ty_2Ehrat_2Ehrat,bool),bool)) ).
fof(ax_thm_2Ehreal_2Ehrat__lt,axiom,
! [V0x] :
( mem(V0x,ty_2Ehrat_2Ehrat)
=> ! [V1y] :
( mem(V1y,ty_2Ehrat_2Ehrat)
=> ( p(ap(ap(c_2Ehreal_2Ehrat__lt,V0x),V1y))
<=> ? [V2d] :
( mem(V2d,ty_2Ehrat_2Ehrat)
& V1y = ap(ap(c_2Ehrat_2Ehrat__add,V0x),V2d) ) ) ) ) ).
fof(conj_thm_2Ehreal_2EHRAT__LT__REFL,axiom,
! [V0x] :
( mem(V0x,ty_2Ehrat_2Ehrat)
=> ~ p(ap(ap(c_2Ehreal_2Ehrat__lt,V0x),V0x)) ) ).
fof(conj_thm_2Ehreal_2EHRAT__LT__TRANS,axiom,
! [V0x] :
( mem(V0x,ty_2Ehrat_2Ehrat)
=> ! [V1y] :
( mem(V1y,ty_2Ehrat_2Ehrat)
=> ! [V2z] :
( mem(V2z,ty_2Ehrat_2Ehrat)
=> ( ( p(ap(ap(c_2Ehreal_2Ehrat__lt,V0x),V1y))
& p(ap(ap(c_2Ehreal_2Ehrat__lt,V1y),V2z)) )
=> p(ap(ap(c_2Ehreal_2Ehrat__lt,V0x),V2z)) ) ) ) ) ).
fof(conj_thm_2Ehreal_2EHRAT__LT__ANTISYM,axiom,
! [V0x] :
( mem(V0x,ty_2Ehrat_2Ehrat)
=> ! [V1y] :
( mem(V1y,ty_2Ehrat_2Ehrat)
=> ~ ( p(ap(ap(c_2Ehreal_2Ehrat__lt,V0x),V1y))
& p(ap(ap(c_2Ehreal_2Ehrat__lt,V1y),V0x)) ) ) ) ).
fof(conj_thm_2Ehreal_2EHRAT__LT__TOTAL,axiom,
! [V0x] :
( mem(V0x,ty_2Ehrat_2Ehrat)
=> ! [V1y] :
( mem(V1y,ty_2Ehrat_2Ehrat)
=> ( V0x = V1y
| p(ap(ap(c_2Ehreal_2Ehrat__lt,V0x),V1y))
| p(ap(ap(c_2Ehreal_2Ehrat__lt,V1y),V0x)) ) ) ) ).
fof(conj_thm_2Ehreal_2EHRAT__MUL__RID,axiom,
! [V0x] :
( mem(V0x,ty_2Ehrat_2Ehrat)
=> ap(ap(c_2Ehrat_2Ehrat__mul,V0x),c_2Ehrat_2Ehrat__1) = V0x ) ).
fof(conj_thm_2Ehreal_2EHRAT__MUL__RINV,axiom,
! [V0x] :
( mem(V0x,ty_2Ehrat_2Ehrat)
=> ap(ap(c_2Ehrat_2Ehrat__mul,V0x),ap(c_2Ehrat_2Ehrat__inv,V0x)) = c_2Ehrat_2Ehrat__1 ) ).
fof(conj_thm_2Ehreal_2EHRAT__RDISTRIB,axiom,
! [V0x] :
( mem(V0x,ty_2Ehrat_2Ehrat)
=> ! [V1y] :
( mem(V1y,ty_2Ehrat_2Ehrat)
=> ! [V2z] :
( mem(V2z,ty_2Ehrat_2Ehrat)
=> ap(ap(c_2Ehrat_2Ehrat__mul,ap(ap(c_2Ehrat_2Ehrat__add,V0x),V1y)),V2z) = ap(ap(c_2Ehrat_2Ehrat__add,ap(ap(c_2Ehrat_2Ehrat__mul,V0x),V2z)),ap(ap(c_2Ehrat_2Ehrat__mul,V1y),V2z)) ) ) ) ).
fof(conj_thm_2Ehreal_2EHRAT__LT__ADDL,axiom,
! [V0x] :
( mem(V0x,ty_2Ehrat_2Ehrat)
=> ! [V1y] :
( mem(V1y,ty_2Ehrat_2Ehrat)
=> p(ap(ap(c_2Ehreal_2Ehrat__lt,V0x),ap(ap(c_2Ehrat_2Ehrat__add,V0x),V1y))) ) ) ).
fof(conj_thm_2Ehreal_2EHRAT__LT__ADDR,axiom,
! [V0x] :
( mem(V0x,ty_2Ehrat_2Ehrat)
=> ! [V1y] :
( mem(V1y,ty_2Ehrat_2Ehrat)
=> p(ap(ap(c_2Ehreal_2Ehrat__lt,V1y),ap(ap(c_2Ehrat_2Ehrat__add,V0x),V1y))) ) ) ).
fof(conj_thm_2Ehreal_2EHRAT__LT__GT,axiom,
! [V0x] :
( mem(V0x,ty_2Ehrat_2Ehrat)
=> ! [V1y] :
( mem(V1y,ty_2Ehrat_2Ehrat)
=> ( p(ap(ap(c_2Ehreal_2Ehrat__lt,V0x),V1y))
=> ~ p(ap(ap(c_2Ehreal_2Ehrat__lt,V1y),V0x)) ) ) ) ).
fof(conj_thm_2Ehreal_2EHRAT__LT__NE,axiom,
! [V0x] :
( mem(V0x,ty_2Ehrat_2Ehrat)
=> ! [V1y] :
( mem(V1y,ty_2Ehrat_2Ehrat)
=> ( p(ap(ap(c_2Ehreal_2Ehrat__lt,V0x),V1y))
=> V0x != V1y ) ) ) ).
fof(conj_thm_2Ehreal_2EHRAT__EQ__LADD,axiom,
! [V0x] :
( mem(V0x,ty_2Ehrat_2Ehrat)
=> ! [V1y] :
( mem(V1y,ty_2Ehrat_2Ehrat)
=> ! [V2z] :
( mem(V2z,ty_2Ehrat_2Ehrat)
=> ( ap(ap(c_2Ehrat_2Ehrat__add,V0x),V1y) = ap(ap(c_2Ehrat_2Ehrat__add,V0x),V2z)
<=> V1y = V2z ) ) ) ) ).
fof(conj_thm_2Ehreal_2EHRAT__EQ__LMUL,axiom,
! [V0x] :
( mem(V0x,ty_2Ehrat_2Ehrat)
=> ! [V1y] :
( mem(V1y,ty_2Ehrat_2Ehrat)
=> ! [V2z] :
( mem(V2z,ty_2Ehrat_2Ehrat)
=> ( ap(ap(c_2Ehrat_2Ehrat__mul,V0x),V1y) = ap(ap(c_2Ehrat_2Ehrat__mul,V0x),V2z)
<=> V1y = V2z ) ) ) ) ).
fof(conj_thm_2Ehreal_2EHRAT__LT__ADD2,axiom,
! [V0u] :
( mem(V0u,ty_2Ehrat_2Ehrat)
=> ! [V1v] :
( mem(V1v,ty_2Ehrat_2Ehrat)
=> ! [V2x] :
( mem(V2x,ty_2Ehrat_2Ehrat)
=> ! [V3y] :
( mem(V3y,ty_2Ehrat_2Ehrat)
=> ( ( p(ap(ap(c_2Ehreal_2Ehrat__lt,V0u),V2x))
& p(ap(ap(c_2Ehreal_2Ehrat__lt,V1v),V3y)) )
=> p(ap(ap(c_2Ehreal_2Ehrat__lt,ap(ap(c_2Ehrat_2Ehrat__add,V0u),V1v)),ap(ap(c_2Ehrat_2Ehrat__add,V2x),V3y))) ) ) ) ) ) ).
fof(conj_thm_2Ehreal_2EHRAT__LT__LADD,axiom,
! [V0x] :
( mem(V0x,ty_2Ehrat_2Ehrat)
=> ! [V1y] :
( mem(V1y,ty_2Ehrat_2Ehrat)
=> ! [V2z] :
( mem(V2z,ty_2Ehrat_2Ehrat)
=> ( p(ap(ap(c_2Ehreal_2Ehrat__lt,ap(ap(c_2Ehrat_2Ehrat__add,V2z),V0x)),ap(ap(c_2Ehrat_2Ehrat__add,V2z),V1y)))
<=> p(ap(ap(c_2Ehreal_2Ehrat__lt,V0x),V1y)) ) ) ) ) ).
fof(conj_thm_2Ehreal_2EHRAT__LT__RADD,axiom,
! [V0x] :
( mem(V0x,ty_2Ehrat_2Ehrat)
=> ! [V1y] :
( mem(V1y,ty_2Ehrat_2Ehrat)
=> ! [V2z] :
( mem(V2z,ty_2Ehrat_2Ehrat)
=> ( p(ap(ap(c_2Ehreal_2Ehrat__lt,ap(ap(c_2Ehrat_2Ehrat__add,V0x),V2z)),ap(ap(c_2Ehrat_2Ehrat__add,V1y),V2z)))
<=> p(ap(ap(c_2Ehreal_2Ehrat__lt,V0x),V1y)) ) ) ) ) ).
fof(conj_thm_2Ehreal_2EHRAT__LT__MUL2,axiom,
! [V0u] :
( mem(V0u,ty_2Ehrat_2Ehrat)
=> ! [V1v] :
( mem(V1v,ty_2Ehrat_2Ehrat)
=> ! [V2x] :
( mem(V2x,ty_2Ehrat_2Ehrat)
=> ! [V3y] :
( mem(V3y,ty_2Ehrat_2Ehrat)
=> ( ( p(ap(ap(c_2Ehreal_2Ehrat__lt,V0u),V2x))
& p(ap(ap(c_2Ehreal_2Ehrat__lt,V1v),V3y)) )
=> p(ap(ap(c_2Ehreal_2Ehrat__lt,ap(ap(c_2Ehrat_2Ehrat__mul,V0u),V1v)),ap(ap(c_2Ehrat_2Ehrat__mul,V2x),V3y))) ) ) ) ) ) ).
fof(conj_thm_2Ehreal_2EHRAT__LT__LMUL,axiom,
! [V0x] :
( mem(V0x,ty_2Ehrat_2Ehrat)
=> ! [V1y] :
( mem(V1y,ty_2Ehrat_2Ehrat)
=> ! [V2z] :
( mem(V2z,ty_2Ehrat_2Ehrat)
=> ( p(ap(ap(c_2Ehreal_2Ehrat__lt,ap(ap(c_2Ehrat_2Ehrat__mul,V2z),V0x)),ap(ap(c_2Ehrat_2Ehrat__mul,V2z),V1y)))
<=> p(ap(ap(c_2Ehreal_2Ehrat__lt,V0x),V1y)) ) ) ) ) ).
fof(conj_thm_2Ehreal_2EHRAT__LT__RMUL,axiom,
! [V0x] :
( mem(V0x,ty_2Ehrat_2Ehrat)
=> ! [V1y] :
( mem(V1y,ty_2Ehrat_2Ehrat)
=> ! [V2z] :
( mem(V2z,ty_2Ehrat_2Ehrat)
=> ( p(ap(ap(c_2Ehreal_2Ehrat__lt,ap(ap(c_2Ehrat_2Ehrat__mul,V0x),V2z)),ap(ap(c_2Ehrat_2Ehrat__mul,V1y),V2z)))
<=> p(ap(ap(c_2Ehreal_2Ehrat__lt,V0x),V1y)) ) ) ) ) ).
fof(conj_thm_2Ehreal_2EHRAT__LT__LMUL1,axiom,
! [V0x] :
( mem(V0x,ty_2Ehrat_2Ehrat)
=> ! [V1y] :
( mem(V1y,ty_2Ehrat_2Ehrat)
=> ( p(ap(ap(c_2Ehreal_2Ehrat__lt,ap(ap(c_2Ehrat_2Ehrat__mul,V0x),V1y)),V1y))
<=> p(ap(ap(c_2Ehreal_2Ehrat__lt,V0x),c_2Ehrat_2Ehrat__1)) ) ) ) ).
fof(conj_thm_2Ehreal_2EHRAT__LT__RMUL1,axiom,
! [V0x] :
( mem(V0x,ty_2Ehrat_2Ehrat)
=> ! [V1y] :
( mem(V1y,ty_2Ehrat_2Ehrat)
=> ( p(ap(ap(c_2Ehreal_2Ehrat__lt,ap(ap(c_2Ehrat_2Ehrat__mul,V0x),V1y)),V0x))
<=> p(ap(ap(c_2Ehreal_2Ehrat__lt,V1y),c_2Ehrat_2Ehrat__1)) ) ) ) ).
fof(conj_thm_2Ehreal_2EHRAT__GT__LMUL1,axiom,
! [V0x] :
( mem(V0x,ty_2Ehrat_2Ehrat)
=> ! [V1y] :
( mem(V1y,ty_2Ehrat_2Ehrat)
=> ( p(ap(ap(c_2Ehreal_2Ehrat__lt,V1y),ap(ap(c_2Ehrat_2Ehrat__mul,V0x),V1y)))
<=> p(ap(ap(c_2Ehreal_2Ehrat__lt,c_2Ehrat_2Ehrat__1),V0x)) ) ) ) ).
fof(conj_thm_2Ehreal_2EHRAT__LT__L1,axiom,
! [V0x] :
( mem(V0x,ty_2Ehrat_2Ehrat)
=> ! [V1y] :
( mem(V1y,ty_2Ehrat_2Ehrat)
=> ( p(ap(ap(c_2Ehreal_2Ehrat__lt,ap(ap(c_2Ehrat_2Ehrat__mul,ap(c_2Ehrat_2Ehrat__inv,V0x)),V1y)),c_2Ehrat_2Ehrat__1))
<=> p(ap(ap(c_2Ehreal_2Ehrat__lt,V1y),V0x)) ) ) ) ).
fof(conj_thm_2Ehreal_2EHRAT__LT__R1,axiom,
! [V0x] :
( mem(V0x,ty_2Ehrat_2Ehrat)
=> ! [V1y] :
( mem(V1y,ty_2Ehrat_2Ehrat)
=> ( p(ap(ap(c_2Ehreal_2Ehrat__lt,ap(ap(c_2Ehrat_2Ehrat__mul,V0x),ap(c_2Ehrat_2Ehrat__inv,V1y))),c_2Ehrat_2Ehrat__1))
<=> p(ap(ap(c_2Ehreal_2Ehrat__lt,V0x),V1y)) ) ) ) ).
fof(conj_thm_2Ehreal_2EHRAT__GT__L1,axiom,
! [V0x] :
( mem(V0x,ty_2Ehrat_2Ehrat)
=> ! [V1y] :
( mem(V1y,ty_2Ehrat_2Ehrat)
=> ( p(ap(ap(c_2Ehreal_2Ehrat__lt,c_2Ehrat_2Ehrat__1),ap(ap(c_2Ehrat_2Ehrat__mul,ap(c_2Ehrat_2Ehrat__inv,V0x)),V1y)))
<=> p(ap(ap(c_2Ehreal_2Ehrat__lt,V0x),V1y)) ) ) ) ).
fof(conj_thm_2Ehreal_2EHRAT__INV__MUL,axiom,
! [V0x] :
( mem(V0x,ty_2Ehrat_2Ehrat)
=> ! [V1y] :
( mem(V1y,ty_2Ehrat_2Ehrat)
=> ap(c_2Ehrat_2Ehrat__inv,ap(ap(c_2Ehrat_2Ehrat__mul,V0x),V1y)) = ap(ap(c_2Ehrat_2Ehrat__mul,ap(c_2Ehrat_2Ehrat__inv,V0x)),ap(c_2Ehrat_2Ehrat__inv,V1y)) ) ) ).
fof(conj_thm_2Ehreal_2EHRAT__UP,axiom,
! [V0x] :
( mem(V0x,ty_2Ehrat_2Ehrat)
=> ? [V1y] :
( mem(V1y,ty_2Ehrat_2Ehrat)
& p(ap(ap(c_2Ehreal_2Ehrat__lt,V0x),V1y)) ) ) ).
fof(conj_thm_2Ehreal_2EHRAT__DOWN,axiom,
! [V0x] :
( mem(V0x,ty_2Ehrat_2Ehrat)
=> ? [V1y] :
( mem(V1y,ty_2Ehrat_2Ehrat)
& p(ap(ap(c_2Ehreal_2Ehrat__lt,V1y),V0x)) ) ) ).
fof(conj_thm_2Ehreal_2EHRAT__DOWN2,axiom,
! [V0x] :
( mem(V0x,ty_2Ehrat_2Ehrat)
=> ! [V1y] :
( mem(V1y,ty_2Ehrat_2Ehrat)
=> ? [V2z] :
( mem(V2z,ty_2Ehrat_2Ehrat)
& p(ap(ap(c_2Ehreal_2Ehrat__lt,V2z),V0x))
& p(ap(ap(c_2Ehreal_2Ehrat__lt,V2z),V1y)) ) ) ) ).
fof(conj_thm_2Ehreal_2EHRAT__MEAN,axiom,
! [V0x] :
( mem(V0x,ty_2Ehrat_2Ehrat)
=> ! [V1y] :
( mem(V1y,ty_2Ehrat_2Ehrat)
=> ( p(ap(ap(c_2Ehreal_2Ehrat__lt,V0x),V1y))
=> ? [V2z] :
( mem(V2z,ty_2Ehrat_2Ehrat)
& p(ap(ap(c_2Ehreal_2Ehrat__lt,V0x),V2z))
& p(ap(ap(c_2Ehreal_2Ehrat__lt,V2z),V1y)) ) ) ) ) ).
fof(ax_thm_2Ehreal_2Eisacut,axiom,
! [V0C] :
( mem(V0C,arr(ty_2Ehrat_2Ehrat,bool))
=> ( p(ap(c_2Ehreal_2Eisacut,V0C))
<=> ( ? [V1x] :
( mem(V1x,ty_2Ehrat_2Ehrat)
& p(ap(V0C,V1x)) )
& ? [V2x] :
( mem(V2x,ty_2Ehrat_2Ehrat)
& ~ p(ap(V0C,V2x)) )
& ! [V3x] :
( mem(V3x,ty_2Ehrat_2Ehrat)
=> ! [V4y] :
( mem(V4y,ty_2Ehrat_2Ehrat)
=> ( ( p(ap(V0C,V3x))
& p(ap(ap(c_2Ehreal_2Ehrat__lt,V4y),V3x)) )
=> p(ap(V0C,V4y)) ) ) )
& ! [V5x] :
( mem(V5x,ty_2Ehrat_2Ehrat)
=> ( p(ap(V0C,V5x))
=> ? [V6y] :
( mem(V6y,ty_2Ehrat_2Ehrat)
& p(ap(V0C,V6y))
& p(ap(ap(c_2Ehreal_2Ehrat__lt,V5x),V6y)) ) ) ) ) ) ) ).
fof(lameq_f2571,axiom,
! [V0x] :
( mem(V0x,ty_2Ehrat_2Ehrat)
=> ! [V1y] : ap(f2571(V0x),V1y) = ap(ap(c_2Ehreal_2Ehrat__lt,V1y),V0x) ) ).
fof(ax_thm_2Ehreal_2Ecut__of__hrat,axiom,
! [V0x] :
( mem(V0x,ty_2Ehrat_2Ehrat)
=> ap(c_2Ehreal_2Ecut__of__hrat,V0x) = f2571(V0x) ) ).
fof(conj_thm_2Ehreal_2EISACUT__HRAT,axiom,
! [V0h] :
( mem(V0h,ty_2Ehrat_2Ehrat)
=> p(ap(c_2Ehreal_2Eisacut,ap(c_2Ehreal_2Ecut__of__hrat,V0h))) ) ).
fof(ax_thm_2Ehreal_2Ehreal__TY__DEF,axiom,
? [V0rep] :
( mem(V0rep,arr(ty_2Ehreal_2Ehreal,arr(ty_2Ehrat_2Ehrat,bool)))
& p(ap(ap(c_2Ebool_2ETYPE__DEFINITION(arr(ty_2Ehrat_2Ehrat,bool),ty_2Ehreal_2Ehreal),c_2Ehreal_2Eisacut),V0rep)) ) ).
fof(ax_thm_2Ehreal_2Ehreal__tybij,axiom,
( ! [V0a] :
( mem(V0a,ty_2Ehreal_2Ehreal)
=> ap(c_2Ehreal_2Ehreal,ap(c_2Ehreal_2Ecut,V0a)) = V0a )
& ! [V1r] :
( mem(V1r,arr(ty_2Ehrat_2Ehrat,bool))
=> ( p(ap(c_2Ehreal_2Eisacut,V1r))
<=> ap(c_2Ehreal_2Ecut,ap(c_2Ehreal_2Ehreal,V1r)) = V1r ) ) ) ).
fof(conj_thm_2Ehreal_2EEQUAL__CUTS,axiom,
! [V0X] :
( mem(V0X,ty_2Ehreal_2Ehreal)
=> ! [V1Y] :
( mem(V1Y,ty_2Ehreal_2Ehreal)
=> ( ap(c_2Ehreal_2Ecut,V0X) = ap(c_2Ehreal_2Ecut,V1Y)
=> V0X = V1Y ) ) ) ).
fof(conj_thm_2Ehreal_2ECUT__ISACUT,axiom,
! [V0X] :
( mem(V0X,ty_2Ehreal_2Ehreal)
=> p(ap(c_2Ehreal_2Eisacut,ap(c_2Ehreal_2Ecut,V0X))) ) ).
fof(conj_thm_2Ehreal_2ECUT__NONEMPTY,axiom,
! [V0X] :
( mem(V0X,ty_2Ehreal_2Ehreal)
=> ? [V1x] :
( mem(V1x,ty_2Ehrat_2Ehrat)
& p(ap(ap(c_2Ehreal_2Ecut,V0X),V1x)) ) ) ).
fof(conj_thm_2Ehreal_2ECUT__BOUNDED,axiom,
! [V0X] :
( mem(V0X,ty_2Ehreal_2Ehreal)
=> ? [V1x] :
( mem(V1x,ty_2Ehrat_2Ehrat)
& ~ p(ap(ap(c_2Ehreal_2Ecut,V0X),V1x)) ) ) ).
fof(conj_thm_2Ehreal_2ECUT__DOWN,axiom,
! [V0X] :
( mem(V0X,ty_2Ehreal_2Ehreal)
=> ! [V1x] :
( mem(V1x,ty_2Ehrat_2Ehrat)
=> ! [V2y] :
( mem(V2y,ty_2Ehrat_2Ehrat)
=> ( ( p(ap(ap(c_2Ehreal_2Ecut,V0X),V1x))
& p(ap(ap(c_2Ehreal_2Ehrat__lt,V2y),V1x)) )
=> p(ap(ap(c_2Ehreal_2Ecut,V0X),V2y)) ) ) ) ) ).
fof(conj_thm_2Ehreal_2ECUT__UP,axiom,
! [V0X] :
( mem(V0X,ty_2Ehreal_2Ehreal)
=> ! [V1x] :
( mem(V1x,ty_2Ehrat_2Ehrat)
=> ( p(ap(ap(c_2Ehreal_2Ecut,V0X),V1x))
=> ? [V2y] :
( mem(V2y,ty_2Ehrat_2Ehrat)
& p(ap(ap(c_2Ehreal_2Ecut,V0X),V2y))
& p(ap(ap(c_2Ehreal_2Ehrat__lt,V1x),V2y)) ) ) ) ) ).
fof(conj_thm_2Ehreal_2ECUT__UBOUND,axiom,
! [V0X] :
( mem(V0X,ty_2Ehreal_2Ehreal)
=> ! [V1x] :
( mem(V1x,ty_2Ehrat_2Ehrat)
=> ! [V2y] :
( mem(V2y,ty_2Ehrat_2Ehrat)
=> ( ( ~ p(ap(ap(c_2Ehreal_2Ecut,V0X),V1x))
& p(ap(ap(c_2Ehreal_2Ehrat__lt,V1x),V2y)) )
=> ~ p(ap(ap(c_2Ehreal_2Ecut,V0X),V2y)) ) ) ) ) ).
fof(conj_thm_2Ehreal_2ECUT__STRADDLE,axiom,
! [V0X] :
( mem(V0X,ty_2Ehreal_2Ehreal)
=> ! [V1x] :
( mem(V1x,ty_2Ehrat_2Ehrat)
=> ! [V2y] :
( mem(V2y,ty_2Ehrat_2Ehrat)
=> ( ( p(ap(ap(c_2Ehreal_2Ecut,V0X),V1x))
& ~ p(ap(ap(c_2Ehreal_2Ecut,V0X),V2y)) )
=> p(ap(ap(c_2Ehreal_2Ehrat__lt,V1x),V2y)) ) ) ) ) ).
fof(conj_thm_2Ehreal_2ECUT__NEARTOP__ADD,axiom,
! [V0X] :
( mem(V0X,ty_2Ehreal_2Ehreal)
=> ! [V1e] :
( mem(V1e,ty_2Ehrat_2Ehrat)
=> ? [V2x] :
( mem(V2x,ty_2Ehrat_2Ehrat)
& p(ap(ap(c_2Ehreal_2Ecut,V0X),V2x))
& ~ p(ap(ap(c_2Ehreal_2Ecut,V0X),ap(ap(c_2Ehrat_2Ehrat__add,V2x),V1e))) ) ) ) ).
fof(conj_thm_2Ehreal_2ECUT__NEARTOP__MUL,axiom,
! [V0X] :
( mem(V0X,ty_2Ehreal_2Ehreal)
=> ! [V1u] :
( mem(V1u,ty_2Ehrat_2Ehrat)
=> ( p(ap(ap(c_2Ehreal_2Ehrat__lt,c_2Ehrat_2Ehrat__1),V1u))
=> ? [V2x] :
( mem(V2x,ty_2Ehrat_2Ehrat)
& p(ap(ap(c_2Ehreal_2Ecut,V0X),V2x))
& ~ p(ap(ap(c_2Ehreal_2Ecut,V0X),ap(ap(c_2Ehrat_2Ehrat__mul,V1u),V2x))) ) ) ) ) ).
fof(ax_thm_2Ehreal_2Ehreal__1,axiom,
c_2Ehreal_2Ehreal__1 = ap(c_2Ehreal_2Ehreal,ap(c_2Ehreal_2Ecut__of__hrat,c_2Ehrat_2Ehrat__1)) ).
fof(lameq_f2572,axiom,
! [V1Y] :
( mem(V1Y,ty_2Ehreal_2Ehreal)
=> ! [V0X] :
( mem(V0X,ty_2Ehreal_2Ehreal)
=> ! [V2w] :
( mem(V2w,ty_2Ehrat_2Ehrat)
=> ! [V3x] :
( mem(V3x,ty_2Ehrat_2Ehrat)
=> ! [V4y] : ap(f2572(V1Y,V0X,V2w,V3x),V4y) = ap(ap(c_2Ebool_2E_2F_5C,ap(ap(c_2Emin_2E_3D(ty_2Ehrat_2Ehrat),V2w),ap(ap(c_2Ehrat_2Ehrat__add,V3x),V4y))),ap(ap(c_2Ebool_2E_2F_5C,ap(ap(c_2Ehreal_2Ecut,V0X),V3x)),ap(ap(c_2Ehreal_2Ecut,V1Y),V4y))) ) ) ) ) ).
fof(lameq_f2573,axiom,
! [V2w] :
( mem(V2w,ty_2Ehrat_2Ehrat)
=> ! [V0X] :
( mem(V0X,ty_2Ehreal_2Ehreal)
=> ! [V1Y] :
( mem(V1Y,ty_2Ehreal_2Ehreal)
=> ! [V3x] : ap(f2573(V2w,V0X,V1Y),V3x) = ap(c_2Ebool_2E_3F(ty_2Ehrat_2Ehrat),f2572(V1Y,V0X,V2w,V3x)) ) ) ) ).
fof(lameq_f2574,axiom,
! [V1Y] :
( mem(V1Y,ty_2Ehreal_2Ehreal)
=> ! [V0X] :
( mem(V0X,ty_2Ehreal_2Ehreal)
=> ! [V2w] : ap(f2574(V1Y,V0X),V2w) = ap(c_2Ebool_2E_3F(ty_2Ehrat_2Ehrat),f2573(V2w,V0X,V1Y)) ) ) ).
fof(ax_thm_2Ehreal_2Ehreal__add,axiom,
! [V0X] :
( mem(V0X,ty_2Ehreal_2Ehreal)
=> ! [V1Y] :
( mem(V1Y,ty_2Ehreal_2Ehreal)
=> ap(ap(c_2Ehreal_2Ehreal__add,V0X),V1Y) = ap(c_2Ehreal_2Ehreal,f2574(V1Y,V0X)) ) ) ).
fof(lameq_f2575,axiom,
! [V1Y] :
( mem(V1Y,ty_2Ehreal_2Ehreal)
=> ! [V0X] :
( mem(V0X,ty_2Ehreal_2Ehreal)
=> ! [V2w] :
( mem(V2w,ty_2Ehrat_2Ehrat)
=> ! [V3x] :
( mem(V3x,ty_2Ehrat_2Ehrat)
=> ! [V4y] : ap(f2575(V1Y,V0X,V2w,V3x),V4y) = ap(ap(c_2Ebool_2E_2F_5C,ap(ap(c_2Emin_2E_3D(ty_2Ehrat_2Ehrat),V2w),ap(ap(c_2Ehrat_2Ehrat__mul,V3x),V4y))),ap(ap(c_2Ebool_2E_2F_5C,ap(ap(c_2Ehreal_2Ecut,V0X),V3x)),ap(ap(c_2Ehreal_2Ecut,V1Y),V4y))) ) ) ) ) ).
fof(lameq_f2576,axiom,
! [V2w] :
( mem(V2w,ty_2Ehrat_2Ehrat)
=> ! [V0X] :
( mem(V0X,ty_2Ehreal_2Ehreal)
=> ! [V1Y] :
( mem(V1Y,ty_2Ehreal_2Ehreal)
=> ! [V3x] : ap(f2576(V2w,V0X,V1Y),V3x) = ap(c_2Ebool_2E_3F(ty_2Ehrat_2Ehrat),f2575(V1Y,V0X,V2w,V3x)) ) ) ) ).
fof(lameq_f2577,axiom,
! [V1Y] :
( mem(V1Y,ty_2Ehreal_2Ehreal)
=> ! [V0X] :
( mem(V0X,ty_2Ehreal_2Ehreal)
=> ! [V2w] : ap(f2577(V1Y,V0X),V2w) = ap(c_2Ebool_2E_3F(ty_2Ehrat_2Ehrat),f2576(V2w,V0X,V1Y)) ) ) ).
fof(ax_thm_2Ehreal_2Ehreal__mul,axiom,
! [V0X] :
( mem(V0X,ty_2Ehreal_2Ehreal)
=> ! [V1Y] :
( mem(V1Y,ty_2Ehreal_2Ehreal)
=> ap(ap(c_2Ehreal_2Ehreal__mul,V0X),V1Y) = ap(c_2Ehreal_2Ehreal,f2577(V1Y,V0X)) ) ) ).
fof(lameq_f2578,axiom,
! [V2d] :
( mem(V2d,ty_2Ehrat_2Ehrat)
=> ! [V1w] :
( mem(V1w,ty_2Ehrat_2Ehrat)
=> ! [V0X] :
( mem(V0X,ty_2Ehreal_2Ehreal)
=> ! [V3x] : ap(f2578(V2d,V1w,V0X),V3x) = ap(ap(c_2Emin_2E_3D_3D_3E,ap(ap(c_2Ehreal_2Ecut,V0X),V3x)),ap(ap(c_2Ehreal_2Ehrat__lt,ap(ap(c_2Ehrat_2Ehrat__mul,V1w),V3x)),V2d)) ) ) ) ).
fof(lameq_f2579,axiom,
! [V1w] :
( mem(V1w,ty_2Ehrat_2Ehrat)
=> ! [V0X] :
( mem(V0X,ty_2Ehreal_2Ehreal)
=> ! [V2d] : ap(f2579(V1w,V0X),V2d) = ap(ap(c_2Ebool_2E_2F_5C,ap(ap(c_2Ehreal_2Ehrat__lt,V2d),c_2Ehrat_2Ehrat__1)),ap(c_2Ebool_2E_21(ty_2Ehrat_2Ehrat),f2578(V2d,V1w,V0X))) ) ) ).
fof(lameq_f2580,axiom,
! [V0X] :
( mem(V0X,ty_2Ehreal_2Ehreal)
=> ! [V1w] : ap(f2580(V0X),V1w) = ap(c_2Ebool_2E_3F(ty_2Ehrat_2Ehrat),f2579(V1w,V0X)) ) ).
fof(ax_thm_2Ehreal_2Ehreal__inv,axiom,
! [V0X] :
( mem(V0X,ty_2Ehreal_2Ehreal)
=> ap(c_2Ehreal_2Ehreal__inv,V0X) = ap(c_2Ehreal_2Ehreal,f2580(V0X)) ) ).
fof(lameq_f2581,axiom,
! [V0P] :
( mem(V0P,arr(ty_2Ehreal_2Ehreal,bool))
=> ! [V1w] :
( mem(V1w,ty_2Ehrat_2Ehrat)
=> ! [V2X] : ap(f2581(V0P,V1w),V2X) = ap(ap(c_2Ebool_2E_2F_5C,ap(V0P,V2X)),ap(ap(c_2Ehreal_2Ecut,V2X),V1w)) ) ) ).
fof(lameq_f2582,axiom,
! [V0P] :
( mem(V0P,arr(ty_2Ehreal_2Ehreal,bool))
=> ! [V1w] : ap(f2582(V0P),V1w) = ap(c_2Ebool_2E_3F(ty_2Ehreal_2Ehreal),f2581(V0P,V1w)) ) ).
fof(ax_thm_2Ehreal_2Ehreal__sup,axiom,
! [V0P] :
( mem(V0P,arr(ty_2Ehreal_2Ehreal,bool))
=> ap(c_2Ehreal_2Ehreal__sup,V0P) = ap(c_2Ehreal_2Ehreal,f2582(V0P)) ) ).
fof(ax_thm_2Ehreal_2Ehreal__lt,axiom,
! [V0X] :
( mem(V0X,ty_2Ehreal_2Ehreal)
=> ! [V1Y] :
( mem(V1Y,ty_2Ehreal_2Ehreal)
=> ( p(ap(ap(c_2Ehreal_2Ehreal__lt,V0X),V1Y))
<=> ( V0X != V1Y
& ! [V2x] :
( mem(V2x,ty_2Ehrat_2Ehrat)
=> ( p(ap(ap(c_2Ehreal_2Ecut,V0X),V2x))
=> p(ap(ap(c_2Ehreal_2Ecut,V1Y),V2x)) ) ) ) ) ) ) ).
fof(conj_thm_2Ehreal_2EHREAL__INV__ISACUT,axiom,
! [V0X] :
( mem(V0X,ty_2Ehreal_2Ehreal)
=> p(ap(c_2Ehreal_2Eisacut,f2580(V0X))) ) ).
fof(conj_thm_2Ehreal_2EHREAL__ADD__ISACUT,axiom,
! [V0X] :
( mem(V0X,ty_2Ehreal_2Ehreal)
=> ! [V1Y] :
( mem(V1Y,ty_2Ehreal_2Ehreal)
=> p(ap(c_2Ehreal_2Eisacut,f2574(V1Y,V0X))) ) ) ).
fof(conj_thm_2Ehreal_2EHREAL__MUL__ISACUT,axiom,
! [V0X] :
( mem(V0X,ty_2Ehreal_2Ehreal)
=> ! [V1Y] :
( mem(V1Y,ty_2Ehreal_2Ehreal)
=> p(ap(c_2Ehreal_2Eisacut,f2577(V1Y,V0X))) ) ) ).
fof(conj_thm_2Ehreal_2EHREAL__ADD__SYM,axiom,
! [V0X] :
( mem(V0X,ty_2Ehreal_2Ehreal)
=> ! [V1Y] :
( mem(V1Y,ty_2Ehreal_2Ehreal)
=> ap(ap(c_2Ehreal_2Ehreal__add,V0X),V1Y) = ap(ap(c_2Ehreal_2Ehreal__add,V1Y),V0X) ) ) ).
fof(conj_thm_2Ehreal_2EHREAL__MUL__SYM,axiom,
! [V0X] :
( mem(V0X,ty_2Ehreal_2Ehreal)
=> ! [V1Y] :
( mem(V1Y,ty_2Ehreal_2Ehreal)
=> ap(ap(c_2Ehreal_2Ehreal__mul,V0X),V1Y) = ap(ap(c_2Ehreal_2Ehreal__mul,V1Y),V0X) ) ) ).
fof(conj_thm_2Ehreal_2EHREAL__ADD__ASSOC,axiom,
! [V0X] :
( mem(V0X,ty_2Ehreal_2Ehreal)
=> ! [V1Y] :
( mem(V1Y,ty_2Ehreal_2Ehreal)
=> ! [V2Z] :
( mem(V2Z,ty_2Ehreal_2Ehreal)
=> ap(ap(c_2Ehreal_2Ehreal__add,V0X),ap(ap(c_2Ehreal_2Ehreal__add,V1Y),V2Z)) = ap(ap(c_2Ehreal_2Ehreal__add,ap(ap(c_2Ehreal_2Ehreal__add,V0X),V1Y)),V2Z) ) ) ) ).
fof(conj_thm_2Ehreal_2EHREAL__MUL__ASSOC,axiom,
! [V0X] :
( mem(V0X,ty_2Ehreal_2Ehreal)
=> ! [V1Y] :
( mem(V1Y,ty_2Ehreal_2Ehreal)
=> ! [V2Z] :
( mem(V2Z,ty_2Ehreal_2Ehreal)
=> ap(ap(c_2Ehreal_2Ehreal__mul,V0X),ap(ap(c_2Ehreal_2Ehreal__mul,V1Y),V2Z)) = ap(ap(c_2Ehreal_2Ehreal__mul,ap(ap(c_2Ehreal_2Ehreal__mul,V0X),V1Y)),V2Z) ) ) ) ).
fof(conj_thm_2Ehreal_2EHREAL__LDISTRIB,axiom,
! [V0X] :
( mem(V0X,ty_2Ehreal_2Ehreal)
=> ! [V1Y] :
( mem(V1Y,ty_2Ehreal_2Ehreal)
=> ! [V2Z] :
( mem(V2Z,ty_2Ehreal_2Ehreal)
=> ap(ap(c_2Ehreal_2Ehreal__mul,V0X),ap(ap(c_2Ehreal_2Ehreal__add,V1Y),V2Z)) = ap(ap(c_2Ehreal_2Ehreal__add,ap(ap(c_2Ehreal_2Ehreal__mul,V0X),V1Y)),ap(ap(c_2Ehreal_2Ehreal__mul,V0X),V2Z)) ) ) ) ).
fof(conj_thm_2Ehreal_2EHREAL__MUL__LID,axiom,
! [V0X] :
( mem(V0X,ty_2Ehreal_2Ehreal)
=> ap(ap(c_2Ehreal_2Ehreal__mul,c_2Ehreal_2Ehreal__1),V0X) = V0X ) ).
fof(conj_thm_2Ehreal_2EHREAL__MUL__LINV,axiom,
! [V0X] :
( mem(V0X,ty_2Ehreal_2Ehreal)
=> ap(ap(c_2Ehreal_2Ehreal__mul,ap(c_2Ehreal_2Ehreal__inv,V0X)),V0X) = c_2Ehreal_2Ehreal__1 ) ).
fof(conj_thm_2Ehreal_2EHREAL__NOZERO,axiom,
! [V0X] :
( mem(V0X,ty_2Ehreal_2Ehreal)
=> ! [V1Y] :
( mem(V1Y,ty_2Ehreal_2Ehreal)
=> ap(ap(c_2Ehreal_2Ehreal__add,V0X),V1Y) != V0X ) ) ).
fof(lameq_f2583,axiom,
! [V0Y] :
( mem(V0Y,ty_2Ehreal_2Ehreal)
=> ! [V2w] :
( mem(V2w,ty_2Ehrat_2Ehrat)
=> ! [V1X] :
( mem(V1X,ty_2Ehreal_2Ehreal)
=> ! [V3x] : ap(f2583(V0Y,V2w,V1X),V3x) = ap(ap(c_2Ebool_2E_2F_5C,ap(c_2Ebool_2E_7E,ap(ap(c_2Ehreal_2Ecut,V1X),V3x))),ap(ap(c_2Ehreal_2Ecut,V0Y),ap(ap(c_2Ehrat_2Ehrat__add,V3x),V2w))) ) ) ) ).
fof(lameq_f2584,axiom,
! [V1X] :
( mem(V1X,ty_2Ehreal_2Ehreal)
=> ! [V0Y] :
( mem(V0Y,ty_2Ehreal_2Ehreal)
=> ! [V2w] : ap(f2584(V1X,V0Y),V2w) = ap(c_2Ebool_2E_3F(ty_2Ehrat_2Ehrat),f2583(V0Y,V2w,V1X)) ) ) ).
fof(ax_thm_2Ehreal_2Ehreal__sub,axiom,
! [V0Y] :
( mem(V0Y,ty_2Ehreal_2Ehreal)
=> ! [V1X] :
( mem(V1X,ty_2Ehreal_2Ehreal)
=> ap(ap(c_2Ehreal_2Ehreal__sub,V0Y),V1X) = ap(c_2Ehreal_2Ehreal,f2584(V1X,V0Y)) ) ) ).
fof(conj_thm_2Ehreal_2EHREAL__LT__LEMMA,axiom,
! [V0X] :
( mem(V0X,ty_2Ehreal_2Ehreal)
=> ! [V1Y] :
( mem(V1Y,ty_2Ehreal_2Ehreal)
=> ( p(ap(ap(c_2Ehreal_2Ehreal__lt,V0X),V1Y))
=> ? [V2x] :
( mem(V2x,ty_2Ehrat_2Ehrat)
& ~ p(ap(ap(c_2Ehreal_2Ecut,V0X),V2x))
& p(ap(ap(c_2Ehreal_2Ecut,V1Y),V2x)) ) ) ) ) ).
fof(lameq_f2585,axiom,
! [V1Y] :
( mem(V1Y,ty_2Ehreal_2Ehreal)
=> ! [V2w] :
( mem(V2w,ty_2Ehrat_2Ehrat)
=> ! [V0X] :
( mem(V0X,ty_2Ehreal_2Ehreal)
=> ! [V3x] : ap(f2585(V1Y,V2w,V0X),V3x) = ap(ap(c_2Ebool_2E_2F_5C,ap(c_2Ebool_2E_7E,ap(ap(c_2Ehreal_2Ecut,V0X),V3x))),ap(ap(c_2Ehreal_2Ecut,V1Y),ap(ap(c_2Ehrat_2Ehrat__add,V3x),V2w))) ) ) ) ).
fof(lameq_f2586,axiom,
! [V0X] :
( mem(V0X,ty_2Ehreal_2Ehreal)
=> ! [V1Y] :
( mem(V1Y,ty_2Ehreal_2Ehreal)
=> ! [V2w] : ap(f2586(V0X,V1Y),V2w) = ap(c_2Ebool_2E_3F(ty_2Ehrat_2Ehrat),f2585(V1Y,V2w,V0X)) ) ) ).
fof(conj_thm_2Ehreal_2EHREAL__SUB__ISACUT,axiom,
! [V0X] :
( mem(V0X,ty_2Ehreal_2Ehreal)
=> ! [V1Y] :
( mem(V1Y,ty_2Ehreal_2Ehreal)
=> ( p(ap(ap(c_2Ehreal_2Ehreal__lt,V0X),V1Y))
=> p(ap(c_2Ehreal_2Eisacut,f2586(V0X,V1Y))) ) ) ) ).
fof(conj_thm_2Ehreal_2EHREAL__SUB__ADD,axiom,
! [V0X] :
( mem(V0X,ty_2Ehreal_2Ehreal)
=> ! [V1Y] :
( mem(V1Y,ty_2Ehreal_2Ehreal)
=> ( p(ap(ap(c_2Ehreal_2Ehreal__lt,V0X),V1Y))
=> ap(ap(c_2Ehreal_2Ehreal__add,ap(ap(c_2Ehreal_2Ehreal__sub,V1Y),V0X)),V0X) = V1Y ) ) ) ).
fof(conj_thm_2Ehreal_2EHREAL__LT__TOTAL,axiom,
! [V0X] :
( mem(V0X,ty_2Ehreal_2Ehreal)
=> ! [V1Y] :
( mem(V1Y,ty_2Ehreal_2Ehreal)
=> ( V0X = V1Y
| p(ap(ap(c_2Ehreal_2Ehreal__lt,V0X),V1Y))
| p(ap(ap(c_2Ehreal_2Ehreal__lt,V1Y),V0X)) ) ) ) ).
fof(conj_thm_2Ehreal_2EHREAL__LT,axiom,
! [V0X] :
( mem(V0X,ty_2Ehreal_2Ehreal)
=> ! [V1Y] :
( mem(V1Y,ty_2Ehreal_2Ehreal)
=> ( p(ap(ap(c_2Ehreal_2Ehreal__lt,V0X),V1Y))
<=> ? [V2D] :
( mem(V2D,ty_2Ehreal_2Ehreal)
& V1Y = ap(ap(c_2Ehreal_2Ehreal__add,V0X),V2D) ) ) ) ) ).
fof(conj_thm_2Ehreal_2EHREAL__ADD__TOTAL,axiom,
! [V0X] :
( mem(V0X,ty_2Ehreal_2Ehreal)
=> ! [V1Y] :
( mem(V1Y,ty_2Ehreal_2Ehreal)
=> ( V0X = V1Y
| ? [V2D] :
( mem(V2D,ty_2Ehreal_2Ehreal)
& V1Y = ap(ap(c_2Ehreal_2Ehreal__add,V0X),V2D) )
| ? [V3D] :
( mem(V3D,ty_2Ehreal_2Ehreal)
& V0X = ap(ap(c_2Ehreal_2Ehreal__add,V1Y),V3D) ) ) ) ) ).
fof(lameq_f2587,axiom,
! [V0P] :
( mem(V0P,arr(ty_2Ehreal_2Ehreal,bool))
=> ! [V4w] :
( mem(V4w,ty_2Ehrat_2Ehrat)
=> ! [V5X] : ap(f2587(V0P,V4w),V5X) = ap(ap(c_2Ebool_2E_2F_5C,ap(V0P,V5X)),ap(ap(c_2Ehreal_2Ecut,V5X),V4w)) ) ) ).
fof(lameq_f2588,axiom,
! [V0P] :
( mem(V0P,arr(ty_2Ehreal_2Ehreal,bool))
=> ! [V4w] : ap(f2588(V0P),V4w) = ap(c_2Ebool_2E_3F(ty_2Ehreal_2Ehreal),f2587(V0P,V4w)) ) ).
fof(conj_thm_2Ehreal_2EHREAL__SUP__ISACUT,axiom,
! [V0P] :
( mem(V0P,arr(ty_2Ehreal_2Ehreal,bool))
=> ( ( ? [V1X] :
( mem(V1X,ty_2Ehreal_2Ehreal)
& p(ap(V0P,V1X)) )
& ? [V2Y] :
( mem(V2Y,ty_2Ehreal_2Ehreal)
& ! [V3X] :
( mem(V3X,ty_2Ehreal_2Ehreal)
=> ( p(ap(V0P,V3X))
=> p(ap(ap(c_2Ehreal_2Ehreal__lt,V3X),V2Y)) ) ) ) )
=> p(ap(c_2Ehreal_2Eisacut,f2588(V0P))) ) ) ).
fof(conj_thm_2Ehreal_2EHREAL__SUP,axiom,
! [V0P] :
( mem(V0P,arr(ty_2Ehreal_2Ehreal,bool))
=> ( ( ? [V1X] :
( mem(V1X,ty_2Ehreal_2Ehreal)
& p(ap(V0P,V1X)) )
& ? [V2Y] :
( mem(V2Y,ty_2Ehreal_2Ehreal)
& ! [V3X] :
( mem(V3X,ty_2Ehreal_2Ehreal)
=> ( p(ap(V0P,V3X))
=> p(ap(ap(c_2Ehreal_2Ehreal__lt,V3X),V2Y)) ) ) ) )
=> ! [V4Y] :
( mem(V4Y,ty_2Ehreal_2Ehreal)
=> ( ? [V5X] :
( mem(V5X,ty_2Ehreal_2Ehreal)
& p(ap(V0P,V5X))
& p(ap(ap(c_2Ehreal_2Ehreal__lt,V4Y),V5X)) )
<=> p(ap(ap(c_2Ehreal_2Ehreal__lt,V4Y),ap(c_2Ehreal_2Ehreal__sup,V0P))) ) ) ) ) ).
%------------------------------------------------------------------------------