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 : 170 ( 58 unt; 59 typ; 0 def)
% Number of atoms : 228 ( 80 equ)
% Maximal formula atoms : 9 ( 1 avg)
% Number of connectives : 136 ( 19 ~; 9 |; 39 &)
% ( 31 <=>; 38 =>; 0 <=; 0 <~>)
% Maximal formula depth : 11 ( 5 avg)
% Maximal term depth : 6 ( 2 avg)
% Number of types : 4 ( 3 usr)
% Number of type conns : 43 ( 27 >; 16 *; 0 +; 0 <<)
% Number of predicates : 2 ( 1 usr; 0 prp; 1-2 aty)
% Number of functors : 55 ( 55 usr; 22 con; 0-4 aty)
% Number of variables : 318 ( 262 !; 38 ?; 318 :)
% ( 18 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TF1_SAT_EQU_NAR
% Comments :
% Bugfixes : v7.5.0 - Fixes to the axioms.
%------------------------------------------------------------------------------
tff(tyop_2Ehrat_2Ehrat,type,
tyop_2Ehrat_2Ehrat: $tType ).
tff(tyop_2Ehreal_2Ehreal,type,
tyop_2Ehreal_2Ehreal: $tType ).
tff(tyop_2Emin_2Ebool,type,
tyop_2Emin_2Ebool: $tType ).
tff(tyop_2Emin_2Efun,type,
tyop_2Emin_2Efun: ( $tType * $tType ) > $tType ).
tff(app_2E2,type,
app_2E2:
!>[A_27a: $tType,A_27b: $tType] : ( ( tyop_2Emin_2Efun(A_27a,A_27b) * A_27a ) > A_27b ) ).
tff(p,type,
p: tyop_2Emin_2Ebool > $o ).
tff(combin_i_2E0,type,
combin_i_2E0:
!>[A_27a: $tType] : tyop_2Emin_2Efun(A_27a,A_27a) ).
tff(combin_k_2E0,type,
combin_k_2E0:
!>[A_27a: $tType,A_27b: $tType] : tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,A_27a)) ).
tff(combin_s_2E0,type,
combin_s_2E0:
!>[A_27a: $tType,A_27b: $tType,A_27c: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,A_27c)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(A_27a,A_27c))) ).
tff(c_2Ebool_2E_21_2E0,type,
c_2Ebool_2E_21_2E0:
!>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool) ).
tff(c_2Ebool_2E_21_2E1,type,
c_2Ebool_2E_21_2E1:
!>[A_27a: $tType] : ( tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool) > tyop_2Emin_2Ebool ) ).
tff(c_2Ebool_2E_2F_5C_2E0,type,
c_2Ebool_2E_2F_5C_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Ebool)) ).
tff(c_2Ebool_2E_2F_5C_2E2,type,
c_2Ebool_2E_2F_5C_2E2: ( tyop_2Emin_2Ebool * tyop_2Emin_2Ebool ) > tyop_2Emin_2Ebool ).
tff(c_2Emin_2E_3D_2E0,type,
c_2Emin_2E_3D_2E0:
!>[A_27a: $tType] : tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)) ).
tff(c_2Emin_2E_3D_2E2,type,
c_2Emin_2E_3D_2E2:
!>[A_27a: $tType] : ( ( A_27a * A_27a ) > tyop_2Emin_2Ebool ) ).
tff(c_2Emin_2E_3D_3D_3E_2E0,type,
c_2Emin_2E_3D_3D_3E_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Ebool)) ).
tff(c_2Emin_2E_3D_3D_3E_2E2,type,
c_2Emin_2E_3D_3D_3E_2E2: ( tyop_2Emin_2Ebool * tyop_2Emin_2Ebool ) > tyop_2Emin_2Ebool ).
tff(c_2Ebool_2E_3F_2E0,type,
c_2Ebool_2E_3F_2E0:
!>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool) ).
tff(c_2Ebool_2E_3F_2E1,type,
c_2Ebool_2E_3F_2E1:
!>[A_27a: $tType] : ( tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool) > tyop_2Emin_2Ebool ) ).
tff(c_2Ebool_2EF_2E0,type,
c_2Ebool_2EF_2E0: tyop_2Emin_2Ebool ).
tff(c_2Ebool_2ET_2E0,type,
c_2Ebool_2ET_2E0: tyop_2Emin_2Ebool ).
tff(c_2Ebool_2ETYPE__DEFINITION_2E0,type,
c_2Ebool_2ETYPE__DEFINITION_2E0:
!>[A_27a: $tType,A_27b: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,A_27a),tyop_2Emin_2Ebool)) ).
tff(c_2Ebool_2ETYPE__DEFINITION_2E2,type,
c_2Ebool_2ETYPE__DEFINITION_2E2:
!>[A_27a: $tType,A_27b: $tType] : ( ( tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool) * tyop_2Emin_2Efun(A_27b,A_27a) ) > tyop_2Emin_2Ebool ) ).
tff(c_2Ebool_2E_5C_2F_2E0,type,
c_2Ebool_2E_5C_2F_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Ebool)) ).
tff(c_2Ebool_2E_5C_2F_2E2,type,
c_2Ebool_2E_5C_2F_2E2: ( tyop_2Emin_2Ebool * tyop_2Emin_2Ebool ) > tyop_2Emin_2Ebool ).
tff(c_2Ehreal_2Ecut_2E0,type,
c_2Ehreal_2Ecut_2E0: tyop_2Emin_2Efun(tyop_2Ehreal_2Ehreal,tyop_2Emin_2Efun(tyop_2Ehrat_2Ehrat,tyop_2Emin_2Ebool)) ).
tff(c_2Ehreal_2Ecut_2E1,type,
c_2Ehreal_2Ecut_2E1: tyop_2Ehreal_2Ehreal > tyop_2Emin_2Efun(tyop_2Ehrat_2Ehrat,tyop_2Emin_2Ebool) ).
tff(c_2Ehreal_2Ecut_2E2,type,
c_2Ehreal_2Ecut_2E2: ( tyop_2Ehreal_2Ehreal * tyop_2Ehrat_2Ehrat ) > tyop_2Emin_2Ebool ).
tff(c_2Ehreal_2Ecut__of__hrat_2E0,type,
c_2Ehreal_2Ecut__of__hrat_2E0: tyop_2Emin_2Efun(tyop_2Ehrat_2Ehrat,tyop_2Emin_2Efun(tyop_2Ehrat_2Ehrat,tyop_2Emin_2Ebool)) ).
tff(c_2Ehreal_2Ecut__of__hrat_2E1,type,
c_2Ehreal_2Ecut__of__hrat_2E1: tyop_2Ehrat_2Ehrat > tyop_2Emin_2Efun(tyop_2Ehrat_2Ehrat,tyop_2Emin_2Ebool) ).
tff(c_2Ehreal_2Ecut__of__hrat_2E2,type,
c_2Ehreal_2Ecut__of__hrat_2E2: ( tyop_2Ehrat_2Ehrat * tyop_2Ehrat_2Ehrat ) > tyop_2Emin_2Ebool ).
tff(c_2Ehrat_2Ehrat__1_2E0,type,
c_2Ehrat_2Ehrat__1_2E0: tyop_2Ehrat_2Ehrat ).
tff(c_2Ehrat_2Ehrat__add_2E0,type,
c_2Ehrat_2Ehrat__add_2E0: tyop_2Emin_2Efun(tyop_2Ehrat_2Ehrat,tyop_2Emin_2Efun(tyop_2Ehrat_2Ehrat,tyop_2Ehrat_2Ehrat)) ).
tff(c_2Ehrat_2Ehrat__add_2E2,type,
c_2Ehrat_2Ehrat__add_2E2: ( tyop_2Ehrat_2Ehrat * tyop_2Ehrat_2Ehrat ) > tyop_2Ehrat_2Ehrat ).
tff(c_2Ehrat_2Ehrat__inv_2E0,type,
c_2Ehrat_2Ehrat__inv_2E0: tyop_2Emin_2Efun(tyop_2Ehrat_2Ehrat,tyop_2Ehrat_2Ehrat) ).
tff(c_2Ehrat_2Ehrat__inv_2E1,type,
c_2Ehrat_2Ehrat__inv_2E1: tyop_2Ehrat_2Ehrat > tyop_2Ehrat_2Ehrat ).
tff(c_2Ehreal_2Ehrat__lt_2E0,type,
c_2Ehreal_2Ehrat__lt_2E0: tyop_2Emin_2Efun(tyop_2Ehrat_2Ehrat,tyop_2Emin_2Efun(tyop_2Ehrat_2Ehrat,tyop_2Emin_2Ebool)) ).
tff(c_2Ehreal_2Ehrat__lt_2E2,type,
c_2Ehreal_2Ehrat__lt_2E2: ( tyop_2Ehrat_2Ehrat * tyop_2Ehrat_2Ehrat ) > tyop_2Emin_2Ebool ).
tff(c_2Ehrat_2Ehrat__mul_2E0,type,
c_2Ehrat_2Ehrat__mul_2E0: tyop_2Emin_2Efun(tyop_2Ehrat_2Ehrat,tyop_2Emin_2Efun(tyop_2Ehrat_2Ehrat,tyop_2Ehrat_2Ehrat)) ).
tff(c_2Ehrat_2Ehrat__mul_2E2,type,
c_2Ehrat_2Ehrat__mul_2E2: ( tyop_2Ehrat_2Ehrat * tyop_2Ehrat_2Ehrat ) > tyop_2Ehrat_2Ehrat ).
tff(c_2Ehreal_2Ehreal_2E0,type,
c_2Ehreal_2Ehreal_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Ehrat_2Ehrat,tyop_2Emin_2Ebool),tyop_2Ehreal_2Ehreal) ).
tff(c_2Ehreal_2Ehreal_2E1,type,
c_2Ehreal_2Ehreal_2E1: tyop_2Emin_2Efun(tyop_2Ehrat_2Ehrat,tyop_2Emin_2Ebool) > tyop_2Ehreal_2Ehreal ).
tff(c_2Ehreal_2Ehreal__1_2E0,type,
c_2Ehreal_2Ehreal__1_2E0: tyop_2Ehreal_2Ehreal ).
tff(c_2Ehreal_2Ehreal__add_2E0,type,
c_2Ehreal_2Ehreal__add_2E0: tyop_2Emin_2Efun(tyop_2Ehreal_2Ehreal,tyop_2Emin_2Efun(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal)) ).
tff(c_2Ehreal_2Ehreal__add_2E2,type,
c_2Ehreal_2Ehreal__add_2E2: ( tyop_2Ehreal_2Ehreal * tyop_2Ehreal_2Ehreal ) > tyop_2Ehreal_2Ehreal ).
tff(c_2Ehreal_2Ehreal__inv_2E0,type,
c_2Ehreal_2Ehreal__inv_2E0: tyop_2Emin_2Efun(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal) ).
tff(c_2Ehreal_2Ehreal__inv_2E1,type,
c_2Ehreal_2Ehreal__inv_2E1: tyop_2Ehreal_2Ehreal > tyop_2Ehreal_2Ehreal ).
tff(c_2Ehreal_2Ehreal__lt_2E0,type,
c_2Ehreal_2Ehreal__lt_2E0: tyop_2Emin_2Efun(tyop_2Ehreal_2Ehreal,tyop_2Emin_2Efun(tyop_2Ehreal_2Ehreal,tyop_2Emin_2Ebool)) ).
tff(c_2Ehreal_2Ehreal__lt_2E2,type,
c_2Ehreal_2Ehreal__lt_2E2: ( tyop_2Ehreal_2Ehreal * tyop_2Ehreal_2Ehreal ) > tyop_2Emin_2Ebool ).
tff(c_2Ehreal_2Ehreal__mul_2E0,type,
c_2Ehreal_2Ehreal__mul_2E0: tyop_2Emin_2Efun(tyop_2Ehreal_2Ehreal,tyop_2Emin_2Efun(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal)) ).
tff(c_2Ehreal_2Ehreal__mul_2E2,type,
c_2Ehreal_2Ehreal__mul_2E2: ( tyop_2Ehreal_2Ehreal * tyop_2Ehreal_2Ehreal ) > tyop_2Ehreal_2Ehreal ).
tff(c_2Ehreal_2Ehreal__sub_2E0,type,
c_2Ehreal_2Ehreal__sub_2E0: tyop_2Emin_2Efun(tyop_2Ehreal_2Ehreal,tyop_2Emin_2Efun(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal)) ).
tff(c_2Ehreal_2Ehreal__sub_2E2,type,
c_2Ehreal_2Ehreal__sub_2E2: ( tyop_2Ehreal_2Ehreal * tyop_2Ehreal_2Ehreal ) > tyop_2Ehreal_2Ehreal ).
tff(c_2Ehreal_2Ehreal__sup_2E0,type,
c_2Ehreal_2Ehreal__sup_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Ehreal_2Ehreal,tyop_2Emin_2Ebool),tyop_2Ehreal_2Ehreal) ).
tff(c_2Ehreal_2Ehreal__sup_2E1,type,
c_2Ehreal_2Ehreal__sup_2E1: tyop_2Emin_2Efun(tyop_2Ehreal_2Ehreal,tyop_2Emin_2Ebool) > tyop_2Ehreal_2Ehreal ).
tff(c_2Ehreal_2Eisacut_2E0,type,
c_2Ehreal_2Eisacut_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Ehrat_2Ehrat,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool) ).
tff(c_2Ehreal_2Eisacut_2E1,type,
c_2Ehreal_2Eisacut_2E1: tyop_2Emin_2Efun(tyop_2Ehrat_2Ehrat,tyop_2Emin_2Ebool) > tyop_2Emin_2Ebool ).
tff(c_2Ebool_2E_7E_2E0,type,
c_2Ebool_2E_7E_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Ebool) ).
tff(c_2Ebool_2E_7E_2E1,type,
c_2Ebool_2E_7E_2E1: tyop_2Emin_2Ebool > tyop_2Emin_2Ebool ).
tff(thm_2Eextra_2Dho_2Eeq__ext,axiom,
! [A_27a: $tType,A_27b: $tType,V0f_2E0: tyop_2Emin_2Efun(A_27a,A_27b),V1g_2E0: tyop_2Emin_2Efun(A_27a,A_27b)] :
( ! [V2x_2E0: A_27a] : ( app_2E2(A_27a,A_27b,V0f_2E0,V2x_2E0) = app_2E2(A_27a,A_27b,V1g_2E0,V2x_2E0) )
=> ( V0f_2E0 = V1g_2E0 ) ) ).
tff(thm_2Eextra_2Dho_2Eboolext,axiom,
! [V0_2E0: tyop_2Emin_2Ebool,V1_2E0: tyop_2Emin_2Ebool] :
( ( p(V0_2E0)
<=> p(V1_2E0) )
=> ( V0_2E0 = V1_2E0 ) ) ).
tff(thm_2Eextra_2Dho_2Etruth,axiom,
p(c_2Ebool_2ET_2E0) ).
tff(thm_2Eextra_2Dho_2Enotfalse,axiom,
~ p(c_2Ebool_2EF_2E0) ).
tff(thm_2Eextra_2Dho_2Ebool__cases__ax,axiom,
! [V0t_2E0: tyop_2Emin_2Ebool] :
( ( V0t_2E0 = c_2Ebool_2ET_2E0 )
| ( V0t_2E0 = c_2Ebool_2EF_2E0 ) ) ).
tff(thm_2Eextra_2Dho_2Ei__thm,axiom,
! [A_27a: $tType,V0x_2E0: A_27a] : ( app_2E2(A_27a,A_27a,combin_i_2E0(A_27a),V0x_2E0) = V0x_2E0 ) ).
tff(thm_2Eextra_2Dho_2Ek__thm,axiom,
! [A_27a: $tType,A_27b: $tType,V0x_2E0: A_27a,V1y_2E0: A_27b] : ( app_2E2(A_27b,A_27a,app_2E2(A_27a,tyop_2Emin_2Efun(A_27b,A_27a),combin_k_2E0(A_27a,A_27b),V0x_2E0),V1y_2E0) = V0x_2E0 ) ).
tff(thm_2Eextra_2Dho_2Es__thm,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,A_27c)),V1g_2E0: tyop_2Emin_2Efun(A_27a,A_27b),V2x_2E0: A_27a] : ( app_2E2(A_27a,A_27c,app_2E2(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(A_27a,A_27c),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,A_27c)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(A_27a,A_27c)),combin_s_2E0(A_27a,A_27b,A_27c),V0f_2E0),V1g_2E0),V2x_2E0) = app_2E2(A_27b,A_27c,app_2E2(A_27a,tyop_2Emin_2Efun(A_27b,A_27c),V0f_2E0,V2x_2E0),app_2E2(A_27a,A_27b,V1g_2E0,V2x_2E0)) ) ).
tff(logicdef_2E_2F_5C,axiom,
! [V0_2E0: tyop_2Emin_2Ebool,V1_2E0: tyop_2Emin_2Ebool] :
( p(c_2Ebool_2E_2F_5C_2E2(V0_2E0,V1_2E0))
<=> ( p(V0_2E0)
& p(V1_2E0) ) ) ).
tff(logicdef_2E_5C_2F,axiom,
! [V0_2E0: tyop_2Emin_2Ebool,V1_2E0: tyop_2Emin_2Ebool] :
( p(c_2Ebool_2E_5C_2F_2E2(V0_2E0,V1_2E0))
<=> ( p(V0_2E0)
| p(V1_2E0) ) ) ).
tff(logicdef_2E_7E,axiom,
! [V0_2E0: tyop_2Emin_2Ebool] :
( p(c_2Ebool_2E_7E_2E1(V0_2E0))
<=> ~ p(V0_2E0) ) ).
tff(logicdef_2E_3D_3D_3E,axiom,
! [V0_2E0: tyop_2Emin_2Ebool,V1_2E0: tyop_2Emin_2Ebool] :
( p(c_2Emin_2E_3D_3D_3E_2E2(V0_2E0,V1_2E0))
<=> ( p(V0_2E0)
=> p(V1_2E0) ) ) ).
tff(logicdef_2E_3D,axiom,
! [A_27a: $tType,V0_2E0: A_27a,V1_2E0: A_27a] :
( p(c_2Emin_2E_3D_2E2(A_27a,V0_2E0,V1_2E0))
<=> ( V0_2E0 = V1_2E0 ) ) ).
tff(quantdef_2E_21,axiom,
! [A_27a: $tType,V0f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] :
( p(c_2Ebool_2E_21_2E1(A_27a,V0f_2E0))
<=> ! [V1x_2E0: A_27a] : p(app_2E2(A_27a,tyop_2Emin_2Ebool,V0f_2E0,V1x_2E0)) ) ).
tff(quantdef_2E_3F,axiom,
! [A_27a: $tType,V0f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] :
( p(c_2Ebool_2E_3F_2E1(A_27a,V0f_2E0))
<=> ? [V1x_2E0: A_27a] : p(app_2E2(A_27a,tyop_2Emin_2Ebool,V0f_2E0,V1x_2E0)) ) ).
tff(arityeq1_2Ec_2Ebool_2E_21_2E1_2Emono_2EA_27a,axiom,
! [A_27a: $tType,X0_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] : ( c_2Ebool_2E_21_2E1(A_27a,X0_2E0) = app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,c_2Ebool_2E_21_2E0(A_27a),X0_2E0) ) ).
tff(arityeq2_2Ec_2Ebool_2E_2F_5C_2E2,axiom,
! [X0_2E0: tyop_2Emin_2Ebool,X1_2E0: tyop_2Emin_2Ebool] :
( ( p(X0_2E0)
& p(X1_2E0) )
<=> p(app_2E2(tyop_2Emin_2Ebool,tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Ebool,tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Ebool),c_2Ebool_2E_2F_5C_2E0,X0_2E0),X1_2E0)) ) ).
tff(arityeq2_2Ec_2Emin_2E_3D_2E2_2Emono_2EA_27a,axiom,
! [A_27a: $tType,X0_2E0: A_27a,X1_2E0: A_27a] :
( ( X0_2E0 = X1_2E0 )
<=> p(app_2E2(A_27a,tyop_2Emin_2Ebool,app_2E2(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),c_2Emin_2E_3D_2E0(A_27a),X0_2E0),X1_2E0)) ) ).
tff(arityeq2_2Ec_2Emin_2E_3D_3D_3E_2E2,axiom,
! [X0_2E0: tyop_2Emin_2Ebool,X1_2E0: tyop_2Emin_2Ebool] :
( ( p(X0_2E0)
=> p(X1_2E0) )
<=> p(app_2E2(tyop_2Emin_2Ebool,tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Ebool,tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Ebool),c_2Emin_2E_3D_3D_3E_2E0,X0_2E0),X1_2E0)) ) ).
tff(arityeq1_2Ec_2Ebool_2E_3F_2E1_2Emono_2EA_27a,axiom,
! [A_27a: $tType,X0_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] : ( c_2Ebool_2E_3F_2E1(A_27a,X0_2E0) = app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,c_2Ebool_2E_3F_2E0(A_27a),X0_2E0) ) ).
tff(arityeq2_2Ec_2Ebool_2ETYPE__DEFINITION_2E2_2Emono_2Etyop_2Emin_2Efun_28tyop_2Ehrat_2Ehrat_2Ctyop_2Emin_2Ebool_29_20mono_2Etyop_2Ehreal_2Ehreal,axiom,
! [X0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Ehrat_2Ehrat,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),X1_2E0: tyop_2Emin_2Efun(tyop_2Ehreal_2Ehreal,tyop_2Emin_2Efun(tyop_2Ehrat_2Ehrat,tyop_2Emin_2Ebool))] : ( c_2Ebool_2ETYPE__DEFINITION_2E2(tyop_2Emin_2Efun(tyop_2Ehrat_2Ehrat,tyop_2Emin_2Ebool),tyop_2Ehreal_2Ehreal,X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Ehreal_2Ehreal,tyop_2Emin_2Efun(tyop_2Ehrat_2Ehrat,tyop_2Emin_2Ebool)),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Ehrat_2Ehrat,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Ehreal_2Ehreal,tyop_2Emin_2Efun(tyop_2Ehrat_2Ehrat,tyop_2Emin_2Ebool)),tyop_2Emin_2Ebool),c_2Ebool_2ETYPE__DEFINITION_2E0(tyop_2Emin_2Efun(tyop_2Ehrat_2Ehrat,tyop_2Emin_2Ebool),tyop_2Ehreal_2Ehreal),X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Ebool_2E_5C_2F_2E2,axiom,
! [X0_2E0: tyop_2Emin_2Ebool,X1_2E0: tyop_2Emin_2Ebool] :
( ( p(X0_2E0)
| p(X1_2E0) )
<=> p(app_2E2(tyop_2Emin_2Ebool,tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Ebool,tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Ebool),c_2Ebool_2E_5C_2F_2E0,X0_2E0),X1_2E0)) ) ).
tff(arityeq1_2Ec_2Ehreal_2Ecut_2E1,axiom,
! [X0_2E0: tyop_2Ehreal_2Ehreal] : ( c_2Ehreal_2Ecut_2E1(X0_2E0) = app_2E2(tyop_2Ehreal_2Ehreal,tyop_2Emin_2Efun(tyop_2Ehrat_2Ehrat,tyop_2Emin_2Ebool),c_2Ehreal_2Ecut_2E0,X0_2E0) ) ).
tff(arityeq2_2Ec_2Ehreal_2Ecut_2E2,axiom,
! [X0_2E0: tyop_2Ehreal_2Ehreal,X1_2E0: tyop_2Ehrat_2Ehrat] : ( c_2Ehreal_2Ecut_2E2(X0_2E0,X1_2E0) = app_2E2(tyop_2Ehrat_2Ehrat,tyop_2Emin_2Ebool,app_2E2(tyop_2Ehreal_2Ehreal,tyop_2Emin_2Efun(tyop_2Ehrat_2Ehrat,tyop_2Emin_2Ebool),c_2Ehreal_2Ecut_2E0,X0_2E0),X1_2E0) ) ).
tff(arityeq1_2Ec_2Ehreal_2Ecut__of__hrat_2E1,axiom,
! [X0_2E0: tyop_2Ehrat_2Ehrat] : ( c_2Ehreal_2Ecut__of__hrat_2E1(X0_2E0) = app_2E2(tyop_2Ehrat_2Ehrat,tyop_2Emin_2Efun(tyop_2Ehrat_2Ehrat,tyop_2Emin_2Ebool),c_2Ehreal_2Ecut__of__hrat_2E0,X0_2E0) ) ).
tff(arityeq2_2Ec_2Ehreal_2Ecut__of__hrat_2E2,axiom,
! [X0_2E0: tyop_2Ehrat_2Ehrat,X1_2E0: tyop_2Ehrat_2Ehrat] : ( c_2Ehreal_2Ecut__of__hrat_2E2(X0_2E0,X1_2E0) = app_2E2(tyop_2Ehrat_2Ehrat,tyop_2Emin_2Ebool,app_2E2(tyop_2Ehrat_2Ehrat,tyop_2Emin_2Efun(tyop_2Ehrat_2Ehrat,tyop_2Emin_2Ebool),c_2Ehreal_2Ecut__of__hrat_2E0,X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Ehrat_2Ehrat__add_2E2,axiom,
! [X0_2E0: tyop_2Ehrat_2Ehrat,X1_2E0: tyop_2Ehrat_2Ehrat] : ( c_2Ehrat_2Ehrat__add_2E2(X0_2E0,X1_2E0) = app_2E2(tyop_2Ehrat_2Ehrat,tyop_2Ehrat_2Ehrat,app_2E2(tyop_2Ehrat_2Ehrat,tyop_2Emin_2Efun(tyop_2Ehrat_2Ehrat,tyop_2Ehrat_2Ehrat),c_2Ehrat_2Ehrat__add_2E0,X0_2E0),X1_2E0) ) ).
tff(arityeq1_2Ec_2Ehrat_2Ehrat__inv_2E1,axiom,
! [X0_2E0: tyop_2Ehrat_2Ehrat] : ( c_2Ehrat_2Ehrat__inv_2E1(X0_2E0) = app_2E2(tyop_2Ehrat_2Ehrat,tyop_2Ehrat_2Ehrat,c_2Ehrat_2Ehrat__inv_2E0,X0_2E0) ) ).
tff(arityeq2_2Ec_2Ehreal_2Ehrat__lt_2E2,axiom,
! [X0_2E0: tyop_2Ehrat_2Ehrat,X1_2E0: tyop_2Ehrat_2Ehrat] : ( c_2Ehreal_2Ehrat__lt_2E2(X0_2E0,X1_2E0) = app_2E2(tyop_2Ehrat_2Ehrat,tyop_2Emin_2Ebool,app_2E2(tyop_2Ehrat_2Ehrat,tyop_2Emin_2Efun(tyop_2Ehrat_2Ehrat,tyop_2Emin_2Ebool),c_2Ehreal_2Ehrat__lt_2E0,X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Ehrat_2Ehrat__mul_2E2,axiom,
! [X0_2E0: tyop_2Ehrat_2Ehrat,X1_2E0: tyop_2Ehrat_2Ehrat] : ( c_2Ehrat_2Ehrat__mul_2E2(X0_2E0,X1_2E0) = app_2E2(tyop_2Ehrat_2Ehrat,tyop_2Ehrat_2Ehrat,app_2E2(tyop_2Ehrat_2Ehrat,tyop_2Emin_2Efun(tyop_2Ehrat_2Ehrat,tyop_2Ehrat_2Ehrat),c_2Ehrat_2Ehrat__mul_2E0,X0_2E0),X1_2E0) ) ).
tff(arityeq1_2Ec_2Ehreal_2Ehreal_2E1,axiom,
! [X0_2E0: tyop_2Emin_2Efun(tyop_2Ehrat_2Ehrat,tyop_2Emin_2Ebool)] : ( c_2Ehreal_2Ehreal_2E1(X0_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Ehrat_2Ehrat,tyop_2Emin_2Ebool),tyop_2Ehreal_2Ehreal,c_2Ehreal_2Ehreal_2E0,X0_2E0) ) ).
tff(arityeq2_2Ec_2Ehreal_2Ehreal__add_2E2,axiom,
! [X0_2E0: tyop_2Ehreal_2Ehreal,X1_2E0: tyop_2Ehreal_2Ehreal] : ( c_2Ehreal_2Ehreal__add_2E2(X0_2E0,X1_2E0) = app_2E2(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal,app_2E2(tyop_2Ehreal_2Ehreal,tyop_2Emin_2Efun(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),c_2Ehreal_2Ehreal__add_2E0,X0_2E0),X1_2E0) ) ).
tff(arityeq1_2Ec_2Ehreal_2Ehreal__inv_2E1,axiom,
! [X0_2E0: tyop_2Ehreal_2Ehreal] : ( c_2Ehreal_2Ehreal__inv_2E1(X0_2E0) = app_2E2(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal,c_2Ehreal_2Ehreal__inv_2E0,X0_2E0) ) ).
tff(arityeq2_2Ec_2Ehreal_2Ehreal__lt_2E2,axiom,
! [X0_2E0: tyop_2Ehreal_2Ehreal,X1_2E0: tyop_2Ehreal_2Ehreal] : ( c_2Ehreal_2Ehreal__lt_2E2(X0_2E0,X1_2E0) = app_2E2(tyop_2Ehreal_2Ehreal,tyop_2Emin_2Ebool,app_2E2(tyop_2Ehreal_2Ehreal,tyop_2Emin_2Efun(tyop_2Ehreal_2Ehreal,tyop_2Emin_2Ebool),c_2Ehreal_2Ehreal__lt_2E0,X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Ehreal_2Ehreal__mul_2E2,axiom,
! [X0_2E0: tyop_2Ehreal_2Ehreal,X1_2E0: tyop_2Ehreal_2Ehreal] : ( c_2Ehreal_2Ehreal__mul_2E2(X0_2E0,X1_2E0) = app_2E2(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal,app_2E2(tyop_2Ehreal_2Ehreal,tyop_2Emin_2Efun(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),c_2Ehreal_2Ehreal__mul_2E0,X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Ehreal_2Ehreal__sub_2E2,axiom,
! [X0_2E0: tyop_2Ehreal_2Ehreal,X1_2E0: tyop_2Ehreal_2Ehreal] : ( c_2Ehreal_2Ehreal__sub_2E2(X0_2E0,X1_2E0) = app_2E2(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal,app_2E2(tyop_2Ehreal_2Ehreal,tyop_2Emin_2Efun(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),c_2Ehreal_2Ehreal__sub_2E0,X0_2E0),X1_2E0) ) ).
tff(arityeq1_2Ec_2Ehreal_2Ehreal__sup_2E1,axiom,
! [X0_2E0: tyop_2Emin_2Efun(tyop_2Ehreal_2Ehreal,tyop_2Emin_2Ebool)] : ( c_2Ehreal_2Ehreal__sup_2E1(X0_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Ehreal_2Ehreal,tyop_2Emin_2Ebool),tyop_2Ehreal_2Ehreal,c_2Ehreal_2Ehreal__sup_2E0,X0_2E0) ) ).
tff(arityeq1_2Ec_2Ehreal_2Eisacut_2E1,axiom,
! [X0_2E0: tyop_2Emin_2Efun(tyop_2Ehrat_2Ehrat,tyop_2Emin_2Ebool)] : ( c_2Ehreal_2Eisacut_2E1(X0_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Ehrat_2Ehrat,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,c_2Ehreal_2Eisacut_2E0,X0_2E0) ) ).
tff(arityeq1_2Ec_2Ebool_2E_7E_2E1,axiom,
! [X0_2E0: tyop_2Emin_2Ebool] :
( ~ p(X0_2E0)
<=> p(app_2E2(tyop_2Emin_2Ebool,tyop_2Emin_2Ebool,c_2Ebool_2E_7E_2E0,X0_2E0)) ) ).
tff(thm_2Ehreal_2Ehrat__lt,axiom,
! [V0x_2E0: tyop_2Ehrat_2Ehrat,V1y_2E0: tyop_2Ehrat_2Ehrat] :
( p(c_2Ehreal_2Ehrat__lt_2E2(V0x_2E0,V1y_2E0))
<=> ? [V2d_2E0: tyop_2Ehrat_2Ehrat] : ( V1y_2E0 = c_2Ehrat_2Ehrat__add_2E2(V0x_2E0,V2d_2E0) ) ) ).
tff(thm_2Ehreal_2Eisacut,axiom,
! [V0C_2E0: tyop_2Emin_2Efun(tyop_2Ehrat_2Ehrat,tyop_2Emin_2Ebool)] :
( p(c_2Ehreal_2Eisacut_2E1(V0C_2E0))
<=> ( ? [V1x_2E0: tyop_2Ehrat_2Ehrat] : p(app_2E2(tyop_2Ehrat_2Ehrat,tyop_2Emin_2Ebool,V0C_2E0,V1x_2E0))
& ? [V2x_2E0: tyop_2Ehrat_2Ehrat] : ~ p(app_2E2(tyop_2Ehrat_2Ehrat,tyop_2Emin_2Ebool,V0C_2E0,V2x_2E0))
& ! [V3x_2E0: tyop_2Ehrat_2Ehrat,V4y_2E0: tyop_2Ehrat_2Ehrat] :
( ( p(app_2E2(tyop_2Ehrat_2Ehrat,tyop_2Emin_2Ebool,V0C_2E0,V3x_2E0))
& p(c_2Ehreal_2Ehrat__lt_2E2(V4y_2E0,V3x_2E0)) )
=> p(app_2E2(tyop_2Ehrat_2Ehrat,tyop_2Emin_2Ebool,V0C_2E0,V4y_2E0)) )
& ! [V5x_2E0: tyop_2Ehrat_2Ehrat] :
( p(app_2E2(tyop_2Ehrat_2Ehrat,tyop_2Emin_2Ebool,V0C_2E0,V5x_2E0))
=> ? [V6y_2E0: tyop_2Ehrat_2Ehrat] :
( p(app_2E2(tyop_2Ehrat_2Ehrat,tyop_2Emin_2Ebool,V0C_2E0,V6y_2E0))
& p(c_2Ehreal_2Ehrat__lt_2E2(V5x_2E0,V6y_2E0)) ) ) ) ) ).
tff(thm_2Ehreal_2Ecut__of__hrat,axiom,
! [V0x_2E0: tyop_2Ehrat_2Ehrat,V1x_27_2E0: tyop_2Ehrat_2Ehrat] : ( c_2Ehreal_2Ecut__of__hrat_2E2(V0x_2E0,V1x_27_2E0) = c_2Ehreal_2Ehrat__lt_2E2(V1x_27_2E0,V0x_2E0) ) ).
tff(thm_2Ehreal_2Ehreal__TY__DEF,axiom,
? [V0rep_2E0: tyop_2Emin_2Efun(tyop_2Ehreal_2Ehreal,tyop_2Emin_2Efun(tyop_2Ehrat_2Ehrat,tyop_2Emin_2Ebool))] : p(c_2Ebool_2ETYPE__DEFINITION_2E2(tyop_2Emin_2Efun(tyop_2Ehrat_2Ehrat,tyop_2Emin_2Ebool),tyop_2Ehreal_2Ehreal,c_2Ehreal_2Eisacut_2E0,V0rep_2E0)) ).
tff(thm_2Ehreal_2Ehreal__tybij,axiom,
( ! [V0a_2E0: tyop_2Ehreal_2Ehreal] : ( c_2Ehreal_2Ehreal_2E1(c_2Ehreal_2Ecut_2E1(V0a_2E0)) = V0a_2E0 )
& ! [V1r_2E0: tyop_2Emin_2Efun(tyop_2Ehrat_2Ehrat,tyop_2Emin_2Ebool)] :
( p(c_2Ehreal_2Eisacut_2E1(V1r_2E0))
<=> ( c_2Ehreal_2Ecut_2E1(c_2Ehreal_2Ehreal_2E1(V1r_2E0)) = V1r_2E0 ) ) ) ).
tff(thm_2Ehreal_2Ehreal__1,axiom,
c_2Ehreal_2Ehreal__1_2E0 = c_2Ehreal_2Ehreal_2E1(c_2Ehreal_2Ecut__of__hrat_2E1(c_2Ehrat_2Ehrat__1_2E0)) ).
tff(thm_2Ehreal_2Ehreal__add,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Ehreal_2Ehreal,tyop_2Emin_2Efun(tyop_2Ehreal_2Ehreal,tyop_2Emin_2Efun(tyop_2Ehrat_2Ehrat,tyop_2Emin_2Ebool)))] :
( ! [V0X_2E0: tyop_2Ehreal_2Ehreal,V1Y_2E0: tyop_2Ehreal_2Ehreal,V2w_2E0: tyop_2Ehrat_2Ehrat] :
( p(app_2E2(tyop_2Ehrat_2Ehrat,tyop_2Emin_2Ebool,app_2E2(tyop_2Ehreal_2Ehreal,tyop_2Emin_2Efun(tyop_2Ehrat_2Ehrat,tyop_2Emin_2Ebool),app_2E2(tyop_2Ehreal_2Ehreal,tyop_2Emin_2Efun(tyop_2Ehreal_2Ehreal,tyop_2Emin_2Efun(tyop_2Ehrat_2Ehrat,tyop_2Emin_2Ebool)),F0_2E0,V0X_2E0),V1Y_2E0),V2w_2E0))
<=> ? [V3x_2E0: tyop_2Ehrat_2Ehrat,V4y_2E0: tyop_2Ehrat_2Ehrat] :
( ( V2w_2E0 = c_2Ehrat_2Ehrat__add_2E2(V3x_2E0,V4y_2E0) )
& p(c_2Ehreal_2Ecut_2E2(V0X_2E0,V3x_2E0))
& p(c_2Ehreal_2Ecut_2E2(V1Y_2E0,V4y_2E0)) ) )
=> ! [V0X_2E0: tyop_2Ehreal_2Ehreal,V1Y_2E0: tyop_2Ehreal_2Ehreal] : ( c_2Ehreal_2Ehreal__add_2E2(V0X_2E0,V1Y_2E0) = c_2Ehreal_2Ehreal_2E1(app_2E2(tyop_2Ehreal_2Ehreal,tyop_2Emin_2Efun(tyop_2Ehrat_2Ehrat,tyop_2Emin_2Ebool),app_2E2(tyop_2Ehreal_2Ehreal,tyop_2Emin_2Efun(tyop_2Ehreal_2Ehreal,tyop_2Emin_2Efun(tyop_2Ehrat_2Ehrat,tyop_2Emin_2Ebool)),F0_2E0,V0X_2E0),V1Y_2E0)) ) ) ).
tff(thm_2Ehreal_2Ehreal__mul,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Ehreal_2Ehreal,tyop_2Emin_2Efun(tyop_2Ehreal_2Ehreal,tyop_2Emin_2Efun(tyop_2Ehrat_2Ehrat,tyop_2Emin_2Ebool)))] :
( ! [V0X_2E0: tyop_2Ehreal_2Ehreal,V1Y_2E0: tyop_2Ehreal_2Ehreal,V2w_2E0: tyop_2Ehrat_2Ehrat] :
( p(app_2E2(tyop_2Ehrat_2Ehrat,tyop_2Emin_2Ebool,app_2E2(tyop_2Ehreal_2Ehreal,tyop_2Emin_2Efun(tyop_2Ehrat_2Ehrat,tyop_2Emin_2Ebool),app_2E2(tyop_2Ehreal_2Ehreal,tyop_2Emin_2Efun(tyop_2Ehreal_2Ehreal,tyop_2Emin_2Efun(tyop_2Ehrat_2Ehrat,tyop_2Emin_2Ebool)),F0_2E0,V0X_2E0),V1Y_2E0),V2w_2E0))
<=> ? [V3x_2E0: tyop_2Ehrat_2Ehrat,V4y_2E0: tyop_2Ehrat_2Ehrat] :
( ( V2w_2E0 = c_2Ehrat_2Ehrat__mul_2E2(V3x_2E0,V4y_2E0) )
& p(c_2Ehreal_2Ecut_2E2(V0X_2E0,V3x_2E0))
& p(c_2Ehreal_2Ecut_2E2(V1Y_2E0,V4y_2E0)) ) )
=> ! [V0X_2E0: tyop_2Ehreal_2Ehreal,V1Y_2E0: tyop_2Ehreal_2Ehreal] : ( c_2Ehreal_2Ehreal__mul_2E2(V0X_2E0,V1Y_2E0) = c_2Ehreal_2Ehreal_2E1(app_2E2(tyop_2Ehreal_2Ehreal,tyop_2Emin_2Efun(tyop_2Ehrat_2Ehrat,tyop_2Emin_2Ebool),app_2E2(tyop_2Ehreal_2Ehreal,tyop_2Emin_2Efun(tyop_2Ehreal_2Ehreal,tyop_2Emin_2Efun(tyop_2Ehrat_2Ehrat,tyop_2Emin_2Ebool)),F0_2E0,V0X_2E0),V1Y_2E0)) ) ) ).
tff(thm_2Ehreal_2Ehreal__inv,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Ehreal_2Ehreal,tyop_2Emin_2Efun(tyop_2Ehrat_2Ehrat,tyop_2Emin_2Ebool))] :
( ! [V0X_2E0: tyop_2Ehreal_2Ehreal,V1w_2E0: tyop_2Ehrat_2Ehrat] :
( p(app_2E2(tyop_2Ehrat_2Ehrat,tyop_2Emin_2Ebool,app_2E2(tyop_2Ehreal_2Ehreal,tyop_2Emin_2Efun(tyop_2Ehrat_2Ehrat,tyop_2Emin_2Ebool),F0_2E0,V0X_2E0),V1w_2E0))
<=> ? [V2d_2E0: tyop_2Ehrat_2Ehrat] :
( p(c_2Ehreal_2Ehrat__lt_2E2(V2d_2E0,c_2Ehrat_2Ehrat__1_2E0))
& ! [V3x_2E0: tyop_2Ehrat_2Ehrat] :
( p(c_2Ehreal_2Ecut_2E2(V0X_2E0,V3x_2E0))
=> p(c_2Ehreal_2Ehrat__lt_2E2(c_2Ehrat_2Ehrat__mul_2E2(V1w_2E0,V3x_2E0),V2d_2E0)) ) ) )
=> ! [V0X_2E0: tyop_2Ehreal_2Ehreal] : ( c_2Ehreal_2Ehreal__inv_2E1(V0X_2E0) = c_2Ehreal_2Ehreal_2E1(app_2E2(tyop_2Ehreal_2Ehreal,tyop_2Emin_2Efun(tyop_2Ehrat_2Ehrat,tyop_2Emin_2Ebool),F0_2E0,V0X_2E0)) ) ) ).
tff(thm_2Ehreal_2Ehreal__sup,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Ehreal_2Ehreal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Ehrat_2Ehrat,tyop_2Emin_2Ebool))] :
( ! [V0P_2E0: tyop_2Emin_2Efun(tyop_2Ehreal_2Ehreal,tyop_2Emin_2Ebool),V1w_2E0: tyop_2Ehrat_2Ehrat] :
( p(app_2E2(tyop_2Ehrat_2Ehrat,tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Ehreal_2Ehreal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Ehrat_2Ehrat,tyop_2Emin_2Ebool),F0_2E0,V0P_2E0),V1w_2E0))
<=> ? [V2X_2E0: tyop_2Ehreal_2Ehreal] :
( p(app_2E2(tyop_2Ehreal_2Ehreal,tyop_2Emin_2Ebool,V0P_2E0,V2X_2E0))
& p(c_2Ehreal_2Ecut_2E2(V2X_2E0,V1w_2E0)) ) )
=> ! [V0P_2E0: tyop_2Emin_2Efun(tyop_2Ehreal_2Ehreal,tyop_2Emin_2Ebool)] : ( c_2Ehreal_2Ehreal__sup_2E1(V0P_2E0) = c_2Ehreal_2Ehreal_2E1(app_2E2(tyop_2Emin_2Efun(tyop_2Ehreal_2Ehreal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Ehrat_2Ehrat,tyop_2Emin_2Ebool),F0_2E0,V0P_2E0)) ) ) ).
tff(thm_2Ehreal_2Ehreal__lt,axiom,
! [V0X_2E0: tyop_2Ehreal_2Ehreal,V1Y_2E0: tyop_2Ehreal_2Ehreal] :
( p(c_2Ehreal_2Ehreal__lt_2E2(V0X_2E0,V1Y_2E0))
<=> ( ( V0X_2E0 != V1Y_2E0 )
& ! [V2x_2E0: tyop_2Ehrat_2Ehrat] :
( p(c_2Ehreal_2Ecut_2E2(V0X_2E0,V2x_2E0))
=> p(c_2Ehreal_2Ecut_2E2(V1Y_2E0,V2x_2E0)) ) ) ) ).
tff(thm_2Ehreal_2Ehreal__sub,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Ehreal_2Ehreal,tyop_2Emin_2Efun(tyop_2Ehreal_2Ehreal,tyop_2Emin_2Efun(tyop_2Ehrat_2Ehrat,tyop_2Emin_2Ebool)))] :
( ! [V1X_2E0: tyop_2Ehreal_2Ehreal,V0Y_2E0: tyop_2Ehreal_2Ehreal,V2w_2E0: tyop_2Ehrat_2Ehrat] :
( p(app_2E2(tyop_2Ehrat_2Ehrat,tyop_2Emin_2Ebool,app_2E2(tyop_2Ehreal_2Ehreal,tyop_2Emin_2Efun(tyop_2Ehrat_2Ehrat,tyop_2Emin_2Ebool),app_2E2(tyop_2Ehreal_2Ehreal,tyop_2Emin_2Efun(tyop_2Ehreal_2Ehreal,tyop_2Emin_2Efun(tyop_2Ehrat_2Ehrat,tyop_2Emin_2Ebool)),F0_2E0,V1X_2E0),V0Y_2E0),V2w_2E0))
<=> ? [V3x_2E0: tyop_2Ehrat_2Ehrat] :
( ~ p(c_2Ehreal_2Ecut_2E2(V1X_2E0,V3x_2E0))
& p(c_2Ehreal_2Ecut_2E2(V0Y_2E0,c_2Ehrat_2Ehrat__add_2E2(V3x_2E0,V2w_2E0))) ) )
=> ! [V0Y_2E0: tyop_2Ehreal_2Ehreal,V1X_2E0: tyop_2Ehreal_2Ehreal] : ( c_2Ehreal_2Ehreal__sub_2E2(V0Y_2E0,V1X_2E0) = c_2Ehreal_2Ehreal_2E1(app_2E2(tyop_2Ehreal_2Ehreal,tyop_2Emin_2Efun(tyop_2Ehrat_2Ehrat,tyop_2Emin_2Ebool),app_2E2(tyop_2Ehreal_2Ehreal,tyop_2Emin_2Efun(tyop_2Ehreal_2Ehreal,tyop_2Emin_2Efun(tyop_2Ehrat_2Ehrat,tyop_2Emin_2Ebool)),F0_2E0,V1X_2E0),V0Y_2E0)) ) ) ).
tff(thm_2Ehreal_2EHRAT__LT__REFL,axiom,
! [V0x_2E0: tyop_2Ehrat_2Ehrat] : ~ p(c_2Ehreal_2Ehrat__lt_2E2(V0x_2E0,V0x_2E0)) ).
tff(thm_2Ehreal_2EHRAT__LT__TRANS,axiom,
! [V0x_2E0: tyop_2Ehrat_2Ehrat,V1y_2E0: tyop_2Ehrat_2Ehrat,V2z_2E0: tyop_2Ehrat_2Ehrat] :
( ( p(c_2Ehreal_2Ehrat__lt_2E2(V0x_2E0,V1y_2E0))
& p(c_2Ehreal_2Ehrat__lt_2E2(V1y_2E0,V2z_2E0)) )
=> p(c_2Ehreal_2Ehrat__lt_2E2(V0x_2E0,V2z_2E0)) ) ).
tff(thm_2Ehreal_2EHRAT__LT__ANTISYM,axiom,
! [V0x_2E0: tyop_2Ehrat_2Ehrat,V1y_2E0: tyop_2Ehrat_2Ehrat] :
~ ( p(c_2Ehreal_2Ehrat__lt_2E2(V0x_2E0,V1y_2E0))
& p(c_2Ehreal_2Ehrat__lt_2E2(V1y_2E0,V0x_2E0)) ) ).
tff(thm_2Ehreal_2EHRAT__LT__TOTAL,axiom,
! [V0x_2E0: tyop_2Ehrat_2Ehrat,V1y_2E0: tyop_2Ehrat_2Ehrat] :
( ( V0x_2E0 = V1y_2E0 )
| p(c_2Ehreal_2Ehrat__lt_2E2(V0x_2E0,V1y_2E0))
| p(c_2Ehreal_2Ehrat__lt_2E2(V1y_2E0,V0x_2E0)) ) ).
tff(thm_2Ehreal_2EHRAT__MUL__RID,axiom,
! [V0x_2E0: tyop_2Ehrat_2Ehrat] : ( c_2Ehrat_2Ehrat__mul_2E2(V0x_2E0,c_2Ehrat_2Ehrat__1_2E0) = V0x_2E0 ) ).
tff(thm_2Ehreal_2EHRAT__MUL__RINV,axiom,
! [V0x_2E0: tyop_2Ehrat_2Ehrat] : ( c_2Ehrat_2Ehrat__mul_2E2(V0x_2E0,c_2Ehrat_2Ehrat__inv_2E1(V0x_2E0)) = c_2Ehrat_2Ehrat__1_2E0 ) ).
tff(thm_2Ehreal_2EHRAT__RDISTRIB,axiom,
! [V0x_2E0: tyop_2Ehrat_2Ehrat,V1y_2E0: tyop_2Ehrat_2Ehrat,V2z_2E0: tyop_2Ehrat_2Ehrat] : ( c_2Ehrat_2Ehrat__mul_2E2(c_2Ehrat_2Ehrat__add_2E2(V0x_2E0,V1y_2E0),V2z_2E0) = c_2Ehrat_2Ehrat__add_2E2(c_2Ehrat_2Ehrat__mul_2E2(V0x_2E0,V2z_2E0),c_2Ehrat_2Ehrat__mul_2E2(V1y_2E0,V2z_2E0)) ) ).
tff(thm_2Ehreal_2EHRAT__LT__ADDL,axiom,
! [V0x_2E0: tyop_2Ehrat_2Ehrat,V1y_2E0: tyop_2Ehrat_2Ehrat] : p(c_2Ehreal_2Ehrat__lt_2E2(V0x_2E0,c_2Ehrat_2Ehrat__add_2E2(V0x_2E0,V1y_2E0))) ).
tff(thm_2Ehreal_2EHRAT__LT__ADDR,axiom,
! [V0x_2E0: tyop_2Ehrat_2Ehrat,V1y_2E0: tyop_2Ehrat_2Ehrat] : p(c_2Ehreal_2Ehrat__lt_2E2(V1y_2E0,c_2Ehrat_2Ehrat__add_2E2(V0x_2E0,V1y_2E0))) ).
tff(thm_2Ehreal_2EHRAT__LT__GT,axiom,
! [V0x_2E0: tyop_2Ehrat_2Ehrat,V1y_2E0: tyop_2Ehrat_2Ehrat] :
( p(c_2Ehreal_2Ehrat__lt_2E2(V0x_2E0,V1y_2E0))
=> ~ p(c_2Ehreal_2Ehrat__lt_2E2(V1y_2E0,V0x_2E0)) ) ).
tff(thm_2Ehreal_2EHRAT__LT__NE,axiom,
! [V0x_2E0: tyop_2Ehrat_2Ehrat,V1y_2E0: tyop_2Ehrat_2Ehrat] :
( p(c_2Ehreal_2Ehrat__lt_2E2(V0x_2E0,V1y_2E0))
=> ( V0x_2E0 != V1y_2E0 ) ) ).
tff(thm_2Ehreal_2EHRAT__EQ__LADD,axiom,
! [V0x_2E0: tyop_2Ehrat_2Ehrat,V1y_2E0: tyop_2Ehrat_2Ehrat,V2z_2E0: tyop_2Ehrat_2Ehrat] :
( ( c_2Ehrat_2Ehrat__add_2E2(V0x_2E0,V1y_2E0) = c_2Ehrat_2Ehrat__add_2E2(V0x_2E0,V2z_2E0) )
<=> ( V1y_2E0 = V2z_2E0 ) ) ).
tff(thm_2Ehreal_2EHRAT__EQ__LMUL,axiom,
! [V0x_2E0: tyop_2Ehrat_2Ehrat,V1y_2E0: tyop_2Ehrat_2Ehrat,V2z_2E0: tyop_2Ehrat_2Ehrat] :
( ( c_2Ehrat_2Ehrat__mul_2E2(V0x_2E0,V1y_2E0) = c_2Ehrat_2Ehrat__mul_2E2(V0x_2E0,V2z_2E0) )
<=> ( V1y_2E0 = V2z_2E0 ) ) ).
tff(thm_2Ehreal_2EHRAT__LT__ADD2,axiom,
! [V0u_2E0: tyop_2Ehrat_2Ehrat,V1v_2E0: tyop_2Ehrat_2Ehrat,V2x_2E0: tyop_2Ehrat_2Ehrat,V3y_2E0: tyop_2Ehrat_2Ehrat] :
( ( p(c_2Ehreal_2Ehrat__lt_2E2(V0u_2E0,V2x_2E0))
& p(c_2Ehreal_2Ehrat__lt_2E2(V1v_2E0,V3y_2E0)) )
=> p(c_2Ehreal_2Ehrat__lt_2E2(c_2Ehrat_2Ehrat__add_2E2(V0u_2E0,V1v_2E0),c_2Ehrat_2Ehrat__add_2E2(V2x_2E0,V3y_2E0))) ) ).
tff(thm_2Ehreal_2EHRAT__LT__LADD,axiom,
! [V0x_2E0: tyop_2Ehrat_2Ehrat,V1y_2E0: tyop_2Ehrat_2Ehrat,V2z_2E0: tyop_2Ehrat_2Ehrat] : ( c_2Ehreal_2Ehrat__lt_2E2(c_2Ehrat_2Ehrat__add_2E2(V2z_2E0,V0x_2E0),c_2Ehrat_2Ehrat__add_2E2(V2z_2E0,V1y_2E0)) = c_2Ehreal_2Ehrat__lt_2E2(V0x_2E0,V1y_2E0) ) ).
tff(thm_2Ehreal_2EHRAT__LT__RADD,axiom,
! [V0x_2E0: tyop_2Ehrat_2Ehrat,V1y_2E0: tyop_2Ehrat_2Ehrat,V2z_2E0: tyop_2Ehrat_2Ehrat] : ( c_2Ehreal_2Ehrat__lt_2E2(c_2Ehrat_2Ehrat__add_2E2(V0x_2E0,V2z_2E0),c_2Ehrat_2Ehrat__add_2E2(V1y_2E0,V2z_2E0)) = c_2Ehreal_2Ehrat__lt_2E2(V0x_2E0,V1y_2E0) ) ).
tff(thm_2Ehreal_2EHRAT__LT__MUL2,axiom,
! [V0u_2E0: tyop_2Ehrat_2Ehrat,V1v_2E0: tyop_2Ehrat_2Ehrat,V2x_2E0: tyop_2Ehrat_2Ehrat,V3y_2E0: tyop_2Ehrat_2Ehrat] :
( ( p(c_2Ehreal_2Ehrat__lt_2E2(V0u_2E0,V2x_2E0))
& p(c_2Ehreal_2Ehrat__lt_2E2(V1v_2E0,V3y_2E0)) )
=> p(c_2Ehreal_2Ehrat__lt_2E2(c_2Ehrat_2Ehrat__mul_2E2(V0u_2E0,V1v_2E0),c_2Ehrat_2Ehrat__mul_2E2(V2x_2E0,V3y_2E0))) ) ).
tff(thm_2Ehreal_2EHRAT__LT__LMUL,axiom,
! [V0x_2E0: tyop_2Ehrat_2Ehrat,V1y_2E0: tyop_2Ehrat_2Ehrat,V2z_2E0: tyop_2Ehrat_2Ehrat] : ( c_2Ehreal_2Ehrat__lt_2E2(c_2Ehrat_2Ehrat__mul_2E2(V2z_2E0,V0x_2E0),c_2Ehrat_2Ehrat__mul_2E2(V2z_2E0,V1y_2E0)) = c_2Ehreal_2Ehrat__lt_2E2(V0x_2E0,V1y_2E0) ) ).
tff(thm_2Ehreal_2EHRAT__LT__RMUL,axiom,
! [V0x_2E0: tyop_2Ehrat_2Ehrat,V1y_2E0: tyop_2Ehrat_2Ehrat,V2z_2E0: tyop_2Ehrat_2Ehrat] : ( c_2Ehreal_2Ehrat__lt_2E2(c_2Ehrat_2Ehrat__mul_2E2(V0x_2E0,V2z_2E0),c_2Ehrat_2Ehrat__mul_2E2(V1y_2E0,V2z_2E0)) = c_2Ehreal_2Ehrat__lt_2E2(V0x_2E0,V1y_2E0) ) ).
tff(thm_2Ehreal_2EHRAT__LT__LMUL1,axiom,
! [V0x_2E0: tyop_2Ehrat_2Ehrat,V1y_2E0: tyop_2Ehrat_2Ehrat] : ( c_2Ehreal_2Ehrat__lt_2E2(c_2Ehrat_2Ehrat__mul_2E2(V0x_2E0,V1y_2E0),V1y_2E0) = c_2Ehreal_2Ehrat__lt_2E2(V0x_2E0,c_2Ehrat_2Ehrat__1_2E0) ) ).
tff(thm_2Ehreal_2EHRAT__LT__RMUL1,axiom,
! [V0x_2E0: tyop_2Ehrat_2Ehrat,V1y_2E0: tyop_2Ehrat_2Ehrat] : ( c_2Ehreal_2Ehrat__lt_2E2(c_2Ehrat_2Ehrat__mul_2E2(V0x_2E0,V1y_2E0),V0x_2E0) = c_2Ehreal_2Ehrat__lt_2E2(V1y_2E0,c_2Ehrat_2Ehrat__1_2E0) ) ).
tff(thm_2Ehreal_2EHRAT__GT__LMUL1,axiom,
! [V0x_2E0: tyop_2Ehrat_2Ehrat,V1y_2E0: tyop_2Ehrat_2Ehrat] : ( c_2Ehreal_2Ehrat__lt_2E2(V1y_2E0,c_2Ehrat_2Ehrat__mul_2E2(V0x_2E0,V1y_2E0)) = c_2Ehreal_2Ehrat__lt_2E2(c_2Ehrat_2Ehrat__1_2E0,V0x_2E0) ) ).
tff(thm_2Ehreal_2EHRAT__LT__L1,axiom,
! [V0x_2E0: tyop_2Ehrat_2Ehrat,V1y_2E0: tyop_2Ehrat_2Ehrat] : ( c_2Ehreal_2Ehrat__lt_2E2(c_2Ehrat_2Ehrat__mul_2E2(c_2Ehrat_2Ehrat__inv_2E1(V0x_2E0),V1y_2E0),c_2Ehrat_2Ehrat__1_2E0) = c_2Ehreal_2Ehrat__lt_2E2(V1y_2E0,V0x_2E0) ) ).
tff(thm_2Ehreal_2EHRAT__LT__R1,axiom,
! [V0x_2E0: tyop_2Ehrat_2Ehrat,V1y_2E0: tyop_2Ehrat_2Ehrat] : ( c_2Ehreal_2Ehrat__lt_2E2(c_2Ehrat_2Ehrat__mul_2E2(V0x_2E0,c_2Ehrat_2Ehrat__inv_2E1(V1y_2E0)),c_2Ehrat_2Ehrat__1_2E0) = c_2Ehreal_2Ehrat__lt_2E2(V0x_2E0,V1y_2E0) ) ).
tff(thm_2Ehreal_2EHRAT__GT__L1,axiom,
! [V0x_2E0: tyop_2Ehrat_2Ehrat,V1y_2E0: tyop_2Ehrat_2Ehrat] : ( c_2Ehreal_2Ehrat__lt_2E2(c_2Ehrat_2Ehrat__1_2E0,c_2Ehrat_2Ehrat__mul_2E2(c_2Ehrat_2Ehrat__inv_2E1(V0x_2E0),V1y_2E0)) = c_2Ehreal_2Ehrat__lt_2E2(V0x_2E0,V1y_2E0) ) ).
tff(thm_2Ehreal_2EHRAT__INV__MUL,axiom,
! [V0x_2E0: tyop_2Ehrat_2Ehrat,V1y_2E0: tyop_2Ehrat_2Ehrat] : ( c_2Ehrat_2Ehrat__inv_2E1(c_2Ehrat_2Ehrat__mul_2E2(V0x_2E0,V1y_2E0)) = c_2Ehrat_2Ehrat__mul_2E2(c_2Ehrat_2Ehrat__inv_2E1(V0x_2E0),c_2Ehrat_2Ehrat__inv_2E1(V1y_2E0)) ) ).
tff(thm_2Ehreal_2EHRAT__UP,axiom,
! [V0x_2E0: tyop_2Ehrat_2Ehrat] :
? [V1y_2E0: tyop_2Ehrat_2Ehrat] : p(c_2Ehreal_2Ehrat__lt_2E2(V0x_2E0,V1y_2E0)) ).
tff(thm_2Ehreal_2EHRAT__DOWN,axiom,
! [V0x_2E0: tyop_2Ehrat_2Ehrat] :
? [V1y_2E0: tyop_2Ehrat_2Ehrat] : p(c_2Ehreal_2Ehrat__lt_2E2(V1y_2E0,V0x_2E0)) ).
tff(thm_2Ehreal_2EHRAT__DOWN2,axiom,
! [V0x_2E0: tyop_2Ehrat_2Ehrat,V1y_2E0: tyop_2Ehrat_2Ehrat] :
? [V2z_2E0: tyop_2Ehrat_2Ehrat] :
( p(c_2Ehreal_2Ehrat__lt_2E2(V2z_2E0,V0x_2E0))
& p(c_2Ehreal_2Ehrat__lt_2E2(V2z_2E0,V1y_2E0)) ) ).
tff(thm_2Ehreal_2EHRAT__MEAN,axiom,
! [V0x_2E0: tyop_2Ehrat_2Ehrat,V1y_2E0: tyop_2Ehrat_2Ehrat] :
( p(c_2Ehreal_2Ehrat__lt_2E2(V0x_2E0,V1y_2E0))
=> ? [V2z_2E0: tyop_2Ehrat_2Ehrat] :
( p(c_2Ehreal_2Ehrat__lt_2E2(V0x_2E0,V2z_2E0))
& p(c_2Ehreal_2Ehrat__lt_2E2(V2z_2E0,V1y_2E0)) ) ) ).
tff(thm_2Ehreal_2EISACUT__HRAT,axiom,
! [V0h_2E0: tyop_2Ehrat_2Ehrat] : p(c_2Ehreal_2Eisacut_2E1(c_2Ehreal_2Ecut__of__hrat_2E1(V0h_2E0))) ).
tff(thm_2Ehreal_2EEQUAL__CUTS,axiom,
! [V0X_2E0: tyop_2Ehreal_2Ehreal,V1Y_2E0: tyop_2Ehreal_2Ehreal] :
( ( c_2Ehreal_2Ecut_2E1(V0X_2E0) = c_2Ehreal_2Ecut_2E1(V1Y_2E0) )
=> ( V0X_2E0 = V1Y_2E0 ) ) ).
tff(thm_2Ehreal_2ECUT__ISACUT,axiom,
! [V0X_2E0: tyop_2Ehreal_2Ehreal] : p(c_2Ehreal_2Eisacut_2E1(c_2Ehreal_2Ecut_2E1(V0X_2E0))) ).
tff(thm_2Ehreal_2ECUT__NONEMPTY,axiom,
! [V0X_2E0: tyop_2Ehreal_2Ehreal] :
? [V1x_2E0: tyop_2Ehrat_2Ehrat] : p(c_2Ehreal_2Ecut_2E2(V0X_2E0,V1x_2E0)) ).
tff(thm_2Ehreal_2ECUT__BOUNDED,axiom,
! [V0X_2E0: tyop_2Ehreal_2Ehreal] :
? [V1x_2E0: tyop_2Ehrat_2Ehrat] : ~ p(c_2Ehreal_2Ecut_2E2(V0X_2E0,V1x_2E0)) ).
tff(thm_2Ehreal_2ECUT__DOWN,axiom,
! [V0X_2E0: tyop_2Ehreal_2Ehreal,V1x_2E0: tyop_2Ehrat_2Ehrat,V2y_2E0: tyop_2Ehrat_2Ehrat] :
( ( p(c_2Ehreal_2Ecut_2E2(V0X_2E0,V1x_2E0))
& p(c_2Ehreal_2Ehrat__lt_2E2(V2y_2E0,V1x_2E0)) )
=> p(c_2Ehreal_2Ecut_2E2(V0X_2E0,V2y_2E0)) ) ).
tff(thm_2Ehreal_2ECUT__UP,axiom,
! [V0X_2E0: tyop_2Ehreal_2Ehreal,V1x_2E0: tyop_2Ehrat_2Ehrat] :
( p(c_2Ehreal_2Ecut_2E2(V0X_2E0,V1x_2E0))
=> ? [V2y_2E0: tyop_2Ehrat_2Ehrat] :
( p(c_2Ehreal_2Ecut_2E2(V0X_2E0,V2y_2E0))
& p(c_2Ehreal_2Ehrat__lt_2E2(V1x_2E0,V2y_2E0)) ) ) ).
tff(thm_2Ehreal_2ECUT__UBOUND,axiom,
! [V0X_2E0: tyop_2Ehreal_2Ehreal,V1x_2E0: tyop_2Ehrat_2Ehrat,V2y_2E0: tyop_2Ehrat_2Ehrat] :
( ( ~ p(c_2Ehreal_2Ecut_2E2(V0X_2E0,V1x_2E0))
& p(c_2Ehreal_2Ehrat__lt_2E2(V1x_2E0,V2y_2E0)) )
=> ~ p(c_2Ehreal_2Ecut_2E2(V0X_2E0,V2y_2E0)) ) ).
tff(thm_2Ehreal_2ECUT__STRADDLE,axiom,
! [V0X_2E0: tyop_2Ehreal_2Ehreal,V1x_2E0: tyop_2Ehrat_2Ehrat,V2y_2E0: tyop_2Ehrat_2Ehrat] :
( ( p(c_2Ehreal_2Ecut_2E2(V0X_2E0,V1x_2E0))
& ~ p(c_2Ehreal_2Ecut_2E2(V0X_2E0,V2y_2E0)) )
=> p(c_2Ehreal_2Ehrat__lt_2E2(V1x_2E0,V2y_2E0)) ) ).
tff(thm_2Ehreal_2ECUT__NEARTOP__ADD,axiom,
! [V0X_2E0: tyop_2Ehreal_2Ehreal,V1e_2E0: tyop_2Ehrat_2Ehrat] :
? [V2x_2E0: tyop_2Ehrat_2Ehrat] :
( p(c_2Ehreal_2Ecut_2E2(V0X_2E0,V2x_2E0))
& ~ p(c_2Ehreal_2Ecut_2E2(V0X_2E0,c_2Ehrat_2Ehrat__add_2E2(V2x_2E0,V1e_2E0))) ) ).
tff(thm_2Ehreal_2ECUT__NEARTOP__MUL,axiom,
! [V0X_2E0: tyop_2Ehreal_2Ehreal,V1u_2E0: tyop_2Ehrat_2Ehrat] :
( p(c_2Ehreal_2Ehrat__lt_2E2(c_2Ehrat_2Ehrat__1_2E0,V1u_2E0))
=> ? [V2x_2E0: tyop_2Ehrat_2Ehrat] :
( p(c_2Ehreal_2Ecut_2E2(V0X_2E0,V2x_2E0))
& ~ p(c_2Ehreal_2Ecut_2E2(V0X_2E0,c_2Ehrat_2Ehrat__mul_2E2(V1u_2E0,V2x_2E0))) ) ) ).
tff(thm_2Ehreal_2EHREAL__INV__ISACUT,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Ehreal_2Ehreal,tyop_2Emin_2Efun(tyop_2Ehrat_2Ehrat,tyop_2Emin_2Ebool))] :
( ! [V0X_2E0: tyop_2Ehreal_2Ehreal,V1w_2E0: tyop_2Ehrat_2Ehrat] :
( p(app_2E2(tyop_2Ehrat_2Ehrat,tyop_2Emin_2Ebool,app_2E2(tyop_2Ehreal_2Ehreal,tyop_2Emin_2Efun(tyop_2Ehrat_2Ehrat,tyop_2Emin_2Ebool),F0_2E0,V0X_2E0),V1w_2E0))
<=> ? [V2d_2E0: tyop_2Ehrat_2Ehrat] :
( p(c_2Ehreal_2Ehrat__lt_2E2(V2d_2E0,c_2Ehrat_2Ehrat__1_2E0))
& ! [V3x_2E0: tyop_2Ehrat_2Ehrat] :
( p(c_2Ehreal_2Ecut_2E2(V0X_2E0,V3x_2E0))
=> p(c_2Ehreal_2Ehrat__lt_2E2(c_2Ehrat_2Ehrat__mul_2E2(V1w_2E0,V3x_2E0),V2d_2E0)) ) ) )
=> ! [V0X_2E0: tyop_2Ehreal_2Ehreal] : p(c_2Ehreal_2Eisacut_2E1(app_2E2(tyop_2Ehreal_2Ehreal,tyop_2Emin_2Efun(tyop_2Ehrat_2Ehrat,tyop_2Emin_2Ebool),F0_2E0,V0X_2E0))) ) ).
tff(thm_2Ehreal_2EHREAL__ADD__ISACUT,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Ehreal_2Ehreal,tyop_2Emin_2Efun(tyop_2Ehreal_2Ehreal,tyop_2Emin_2Efun(tyop_2Ehrat_2Ehrat,tyop_2Emin_2Ebool)))] :
( ! [V0X_2E0: tyop_2Ehreal_2Ehreal,V1Y_2E0: tyop_2Ehreal_2Ehreal,V2w_2E0: tyop_2Ehrat_2Ehrat] :
( p(app_2E2(tyop_2Ehrat_2Ehrat,tyop_2Emin_2Ebool,app_2E2(tyop_2Ehreal_2Ehreal,tyop_2Emin_2Efun(tyop_2Ehrat_2Ehrat,tyop_2Emin_2Ebool),app_2E2(tyop_2Ehreal_2Ehreal,tyop_2Emin_2Efun(tyop_2Ehreal_2Ehreal,tyop_2Emin_2Efun(tyop_2Ehrat_2Ehrat,tyop_2Emin_2Ebool)),F0_2E0,V0X_2E0),V1Y_2E0),V2w_2E0))
<=> ? [V3x_2E0: tyop_2Ehrat_2Ehrat,V4y_2E0: tyop_2Ehrat_2Ehrat] :
( ( V2w_2E0 = c_2Ehrat_2Ehrat__add_2E2(V3x_2E0,V4y_2E0) )
& p(c_2Ehreal_2Ecut_2E2(V0X_2E0,V3x_2E0))
& p(c_2Ehreal_2Ecut_2E2(V1Y_2E0,V4y_2E0)) ) )
=> ! [V0X_2E0: tyop_2Ehreal_2Ehreal,V1Y_2E0: tyop_2Ehreal_2Ehreal] : p(c_2Ehreal_2Eisacut_2E1(app_2E2(tyop_2Ehreal_2Ehreal,tyop_2Emin_2Efun(tyop_2Ehrat_2Ehrat,tyop_2Emin_2Ebool),app_2E2(tyop_2Ehreal_2Ehreal,tyop_2Emin_2Efun(tyop_2Ehreal_2Ehreal,tyop_2Emin_2Efun(tyop_2Ehrat_2Ehrat,tyop_2Emin_2Ebool)),F0_2E0,V0X_2E0),V1Y_2E0))) ) ).
tff(thm_2Ehreal_2EHREAL__MUL__ISACUT,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Ehreal_2Ehreal,tyop_2Emin_2Efun(tyop_2Ehreal_2Ehreal,tyop_2Emin_2Efun(tyop_2Ehrat_2Ehrat,tyop_2Emin_2Ebool)))] :
( ! [V0X_2E0: tyop_2Ehreal_2Ehreal,V1Y_2E0: tyop_2Ehreal_2Ehreal,V2w_2E0: tyop_2Ehrat_2Ehrat] :
( p(app_2E2(tyop_2Ehrat_2Ehrat,tyop_2Emin_2Ebool,app_2E2(tyop_2Ehreal_2Ehreal,tyop_2Emin_2Efun(tyop_2Ehrat_2Ehrat,tyop_2Emin_2Ebool),app_2E2(tyop_2Ehreal_2Ehreal,tyop_2Emin_2Efun(tyop_2Ehreal_2Ehreal,tyop_2Emin_2Efun(tyop_2Ehrat_2Ehrat,tyop_2Emin_2Ebool)),F0_2E0,V0X_2E0),V1Y_2E0),V2w_2E0))
<=> ? [V3x_2E0: tyop_2Ehrat_2Ehrat,V4y_2E0: tyop_2Ehrat_2Ehrat] :
( ( V2w_2E0 = c_2Ehrat_2Ehrat__mul_2E2(V3x_2E0,V4y_2E0) )
& p(c_2Ehreal_2Ecut_2E2(V0X_2E0,V3x_2E0))
& p(c_2Ehreal_2Ecut_2E2(V1Y_2E0,V4y_2E0)) ) )
=> ! [V0X_2E0: tyop_2Ehreal_2Ehreal,V1Y_2E0: tyop_2Ehreal_2Ehreal] : p(c_2Ehreal_2Eisacut_2E1(app_2E2(tyop_2Ehreal_2Ehreal,tyop_2Emin_2Efun(tyop_2Ehrat_2Ehrat,tyop_2Emin_2Ebool),app_2E2(tyop_2Ehreal_2Ehreal,tyop_2Emin_2Efun(tyop_2Ehreal_2Ehreal,tyop_2Emin_2Efun(tyop_2Ehrat_2Ehrat,tyop_2Emin_2Ebool)),F0_2E0,V0X_2E0),V1Y_2E0))) ) ).
tff(thm_2Ehreal_2EHREAL__ADD__SYM,axiom,
! [V0X_2E0: tyop_2Ehreal_2Ehreal,V1Y_2E0: tyop_2Ehreal_2Ehreal] : ( c_2Ehreal_2Ehreal__add_2E2(V0X_2E0,V1Y_2E0) = c_2Ehreal_2Ehreal__add_2E2(V1Y_2E0,V0X_2E0) ) ).
tff(thm_2Ehreal_2EHREAL__MUL__SYM,axiom,
! [V0X_2E0: tyop_2Ehreal_2Ehreal,V1Y_2E0: tyop_2Ehreal_2Ehreal] : ( c_2Ehreal_2Ehreal__mul_2E2(V0X_2E0,V1Y_2E0) = c_2Ehreal_2Ehreal__mul_2E2(V1Y_2E0,V0X_2E0) ) ).
tff(thm_2Ehreal_2EHREAL__ADD__ASSOC,axiom,
! [V0X_2E0: tyop_2Ehreal_2Ehreal,V1Y_2E0: tyop_2Ehreal_2Ehreal,V2Z_2E0: tyop_2Ehreal_2Ehreal] : ( c_2Ehreal_2Ehreal__add_2E2(V0X_2E0,c_2Ehreal_2Ehreal__add_2E2(V1Y_2E0,V2Z_2E0)) = c_2Ehreal_2Ehreal__add_2E2(c_2Ehreal_2Ehreal__add_2E2(V0X_2E0,V1Y_2E0),V2Z_2E0) ) ).
tff(thm_2Ehreal_2EHREAL__MUL__ASSOC,axiom,
! [V0X_2E0: tyop_2Ehreal_2Ehreal,V1Y_2E0: tyop_2Ehreal_2Ehreal,V2Z_2E0: tyop_2Ehreal_2Ehreal] : ( c_2Ehreal_2Ehreal__mul_2E2(V0X_2E0,c_2Ehreal_2Ehreal__mul_2E2(V1Y_2E0,V2Z_2E0)) = c_2Ehreal_2Ehreal__mul_2E2(c_2Ehreal_2Ehreal__mul_2E2(V0X_2E0,V1Y_2E0),V2Z_2E0) ) ).
tff(thm_2Ehreal_2EHREAL__LDISTRIB,axiom,
! [V0X_2E0: tyop_2Ehreal_2Ehreal,V1Y_2E0: tyop_2Ehreal_2Ehreal,V2Z_2E0: tyop_2Ehreal_2Ehreal] : ( c_2Ehreal_2Ehreal__mul_2E2(V0X_2E0,c_2Ehreal_2Ehreal__add_2E2(V1Y_2E0,V2Z_2E0)) = c_2Ehreal_2Ehreal__add_2E2(c_2Ehreal_2Ehreal__mul_2E2(V0X_2E0,V1Y_2E0),c_2Ehreal_2Ehreal__mul_2E2(V0X_2E0,V2Z_2E0)) ) ).
tff(thm_2Ehreal_2EHREAL__MUL__LID,axiom,
! [V0X_2E0: tyop_2Ehreal_2Ehreal] : ( c_2Ehreal_2Ehreal__mul_2E2(c_2Ehreal_2Ehreal__1_2E0,V0X_2E0) = V0X_2E0 ) ).
tff(thm_2Ehreal_2EHREAL__MUL__LINV,axiom,
! [V0X_2E0: tyop_2Ehreal_2Ehreal] : ( c_2Ehreal_2Ehreal__mul_2E2(c_2Ehreal_2Ehreal__inv_2E1(V0X_2E0),V0X_2E0) = c_2Ehreal_2Ehreal__1_2E0 ) ).
tff(thm_2Ehreal_2EHREAL__NOZERO,axiom,
! [V0X_2E0: tyop_2Ehreal_2Ehreal,V1Y_2E0: tyop_2Ehreal_2Ehreal] : ( c_2Ehreal_2Ehreal__add_2E2(V0X_2E0,V1Y_2E0) != V0X_2E0 ) ).
tff(thm_2Ehreal_2EHREAL__LT__LEMMA,axiom,
! [V0X_2E0: tyop_2Ehreal_2Ehreal,V1Y_2E0: tyop_2Ehreal_2Ehreal] :
( p(c_2Ehreal_2Ehreal__lt_2E2(V0X_2E0,V1Y_2E0))
=> ? [V2x_2E0: tyop_2Ehrat_2Ehrat] :
( ~ p(c_2Ehreal_2Ecut_2E2(V0X_2E0,V2x_2E0))
& p(c_2Ehreal_2Ecut_2E2(V1Y_2E0,V2x_2E0)) ) ) ).
tff(thm_2Ehreal_2EHREAL__SUB__ISACUT,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Ehreal_2Ehreal,tyop_2Emin_2Efun(tyop_2Ehreal_2Ehreal,tyop_2Emin_2Efun(tyop_2Ehrat_2Ehrat,tyop_2Emin_2Ebool)))] :
( ! [V0X_2E0: tyop_2Ehreal_2Ehreal,V1Y_2E0: tyop_2Ehreal_2Ehreal,V2w_2E0: tyop_2Ehrat_2Ehrat] :
( p(app_2E2(tyop_2Ehrat_2Ehrat,tyop_2Emin_2Ebool,app_2E2(tyop_2Ehreal_2Ehreal,tyop_2Emin_2Efun(tyop_2Ehrat_2Ehrat,tyop_2Emin_2Ebool),app_2E2(tyop_2Ehreal_2Ehreal,tyop_2Emin_2Efun(tyop_2Ehreal_2Ehreal,tyop_2Emin_2Efun(tyop_2Ehrat_2Ehrat,tyop_2Emin_2Ebool)),F0_2E0,V0X_2E0),V1Y_2E0),V2w_2E0))
<=> ? [V3x_2E0: tyop_2Ehrat_2Ehrat] :
( ~ p(c_2Ehreal_2Ecut_2E2(V0X_2E0,V3x_2E0))
& p(c_2Ehreal_2Ecut_2E2(V1Y_2E0,c_2Ehrat_2Ehrat__add_2E2(V3x_2E0,V2w_2E0))) ) )
=> ! [V0X_2E0: tyop_2Ehreal_2Ehreal,V1Y_2E0: tyop_2Ehreal_2Ehreal] :
( p(c_2Ehreal_2Ehreal__lt_2E2(V0X_2E0,V1Y_2E0))
=> p(c_2Ehreal_2Eisacut_2E1(app_2E2(tyop_2Ehreal_2Ehreal,tyop_2Emin_2Efun(tyop_2Ehrat_2Ehrat,tyop_2Emin_2Ebool),app_2E2(tyop_2Ehreal_2Ehreal,tyop_2Emin_2Efun(tyop_2Ehreal_2Ehreal,tyop_2Emin_2Efun(tyop_2Ehrat_2Ehrat,tyop_2Emin_2Ebool)),F0_2E0,V0X_2E0),V1Y_2E0))) ) ) ).
tff(thm_2Ehreal_2EHREAL__SUB__ADD,axiom,
! [V0X_2E0: tyop_2Ehreal_2Ehreal,V1Y_2E0: tyop_2Ehreal_2Ehreal] :
( p(c_2Ehreal_2Ehreal__lt_2E2(V0X_2E0,V1Y_2E0))
=> ( c_2Ehreal_2Ehreal__add_2E2(c_2Ehreal_2Ehreal__sub_2E2(V1Y_2E0,V0X_2E0),V0X_2E0) = V1Y_2E0 ) ) ).
tff(thm_2Ehreal_2EHREAL__LT__TOTAL,axiom,
! [V0X_2E0: tyop_2Ehreal_2Ehreal,V1Y_2E0: tyop_2Ehreal_2Ehreal] :
( ( V0X_2E0 = V1Y_2E0 )
| p(c_2Ehreal_2Ehreal__lt_2E2(V0X_2E0,V1Y_2E0))
| p(c_2Ehreal_2Ehreal__lt_2E2(V1Y_2E0,V0X_2E0)) ) ).
tff(thm_2Ehreal_2EHREAL__LT,axiom,
! [V0X_2E0: tyop_2Ehreal_2Ehreal,V1Y_2E0: tyop_2Ehreal_2Ehreal] :
( p(c_2Ehreal_2Ehreal__lt_2E2(V0X_2E0,V1Y_2E0))
<=> ? [V2D_2E0: tyop_2Ehreal_2Ehreal] : ( V1Y_2E0 = c_2Ehreal_2Ehreal__add_2E2(V0X_2E0,V2D_2E0) ) ) ).
tff(thm_2Ehreal_2EHREAL__ADD__TOTAL,axiom,
! [V0X_2E0: tyop_2Ehreal_2Ehreal,V1Y_2E0: tyop_2Ehreal_2Ehreal] :
( ( V0X_2E0 = V1Y_2E0 )
| ? [V2D_2E0: tyop_2Ehreal_2Ehreal] : ( V1Y_2E0 = c_2Ehreal_2Ehreal__add_2E2(V0X_2E0,V2D_2E0) )
| ? [V3D_2E0: tyop_2Ehreal_2Ehreal] : ( V0X_2E0 = c_2Ehreal_2Ehreal__add_2E2(V1Y_2E0,V3D_2E0) ) ) ).
tff(thm_2Ehreal_2EHREAL__SUP__ISACUT,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Ehreal_2Ehreal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Ehrat_2Ehrat,tyop_2Emin_2Ebool))] :
( ! [V0P_2E0: tyop_2Emin_2Efun(tyop_2Ehreal_2Ehreal,tyop_2Emin_2Ebool),V4w_2E0: tyop_2Ehrat_2Ehrat] :
( p(app_2E2(tyop_2Ehrat_2Ehrat,tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Ehreal_2Ehreal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Ehrat_2Ehrat,tyop_2Emin_2Ebool),F0_2E0,V0P_2E0),V4w_2E0))
<=> ? [V5X_2E0: tyop_2Ehreal_2Ehreal] :
( p(app_2E2(tyop_2Ehreal_2Ehreal,tyop_2Emin_2Ebool,V0P_2E0,V5X_2E0))
& p(c_2Ehreal_2Ecut_2E2(V5X_2E0,V4w_2E0)) ) )
=> ! [V0P_2E0: tyop_2Emin_2Efun(tyop_2Ehreal_2Ehreal,tyop_2Emin_2Ebool)] :
( ( ? [V1X_2E0: tyop_2Ehreal_2Ehreal] : p(app_2E2(tyop_2Ehreal_2Ehreal,tyop_2Emin_2Ebool,V0P_2E0,V1X_2E0))
& ? [V2Y_2E0: tyop_2Ehreal_2Ehreal] :
! [V3X_2E0: tyop_2Ehreal_2Ehreal] :
( p(app_2E2(tyop_2Ehreal_2Ehreal,tyop_2Emin_2Ebool,V0P_2E0,V3X_2E0))
=> p(c_2Ehreal_2Ehreal__lt_2E2(V3X_2E0,V2Y_2E0)) ) )
=> p(c_2Ehreal_2Eisacut_2E1(app_2E2(tyop_2Emin_2Efun(tyop_2Ehreal_2Ehreal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Ehrat_2Ehrat,tyop_2Emin_2Ebool),F0_2E0,V0P_2E0))) ) ) ).
tff(thm_2Ehreal_2EHREAL__SUP,axiom,
! [V0P_2E0: tyop_2Emin_2Efun(tyop_2Ehreal_2Ehreal,tyop_2Emin_2Ebool)] :
( ( ? [V1X_2E0: tyop_2Ehreal_2Ehreal] : p(app_2E2(tyop_2Ehreal_2Ehreal,tyop_2Emin_2Ebool,V0P_2E0,V1X_2E0))
& ? [V2Y_2E0: tyop_2Ehreal_2Ehreal] :
! [V3X_2E0: tyop_2Ehreal_2Ehreal] :
( p(app_2E2(tyop_2Ehreal_2Ehreal,tyop_2Emin_2Ebool,V0P_2E0,V3X_2E0))
=> p(c_2Ehreal_2Ehreal__lt_2E2(V3X_2E0,V2Y_2E0)) ) )
=> ! [V4Y_2E0: tyop_2Ehreal_2Ehreal] :
( ? [V5X_2E0: tyop_2Ehreal_2Ehreal] :
( p(app_2E2(tyop_2Ehreal_2Ehreal,tyop_2Emin_2Ebool,V0P_2E0,V5X_2E0))
& p(c_2Ehreal_2Ehreal__lt_2E2(V4Y_2E0,V5X_2E0)) )
<=> p(c_2Ehreal_2Ehreal__lt_2E2(V4Y_2E0,c_2Ehreal_2Ehreal__sup_2E1(V0P_2E0))) ) ) ).
%------------------------------------------------------------------------------