TPTP Problem File: ITP114^2.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : ITP114^2 : TPTP v8.2.0. Released v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer Lower_Semicontinuous problem prob_784__6255430_1
% Version : Especial.
% English :
% Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% : [Des21] Desharnais (2021), Email to Geoff Sutcliffe
% Source : [Des21]
% Names : Lower_Semicontinuous/prob_784__6255430_1 [Des21]
% Status : Theorem
% Rating : 0.33 v8.1.0, 0.50 v7.5.0
% Syntax : Number of formulae : 384 ( 153 unt; 59 typ; 0 def)
% Number of atoms : 888 ( 386 equ; 0 cnn)
% Maximal formula atoms : 81 ( 2 avg)
% Number of connectives : 2816 ( 98 ~; 11 |; 30 &;2243 @)
% ( 0 <=>; 434 =>; 0 <=; 0 <~>)
% Maximal formula depth : 34 ( 5 avg)
% Number of types : 7 ( 6 usr)
% Number of type conns : 118 ( 118 >; 0 *; 0 +; 0 <<)
% Number of symbols : 56 ( 53 usr; 5 con; 0-4 aty)
% Number of variables : 705 ( 29 ^; 602 !; 29 ?; 705 :)
% ( 45 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Sledgehammer 2021-02-23 16:27:42.435
%------------------------------------------------------------------------------
% Could-be-implicit typings (7)
thf(ty_t_Extended__Real_Oereal,type,
extended_ereal: $tType ).
thf(ty_t_Extended__Nat_Oenat,type,
extended_enat: $tType ).
thf(ty_t_Real_Oreal,type,
real: $tType ).
thf(ty_t_Set_Oset,type,
set: $tType > $tType ).
thf(ty_t_Num_Onum,type,
num: $tType ).
thf(ty_t_Nat_Onat,type,
nat: $tType ).
thf(ty_tf_a,type,
a: $tType ).
% Explicit typings (52)
thf(sy_cl_HOL_Otype,type,
type:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ozero,type,
zero:
!>[A: $tType] : $o ).
thf(sy_cl_Num_Onumeral,type,
numeral:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ouminus,type,
uminus:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Oord,type,
ord:
!>[A: $tType] : $o ).
thf(sy_cl_Nat_Oring__char__0,type,
ring_char_0:
!>[A: $tType] : $o ).
thf(sy_cl_Num_Oneg__numeral,type,
neg_numeral:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Oorder,type,
order:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ogroup__add,type,
group_add:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Olinorder,type,
linorder:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Opreorder,type,
preorder:
!>[A: $tType] : $o ).
thf(sy_cl_Nat_Osemiring__char__0,type,
semiring_char_0:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Olinordered__idom,type,
linordered_idom:
!>[A: $tType] : $o ).
thf(sy_cl_Fields_Olinordered__field,type,
linordered_field:
!>[A: $tType] : $o ).
thf(sy_cl_Lattices_Oboolean__algebra,type,
boolean_algebra:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oordered__ab__group__add,type,
ordered_ab_group_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Olinordered__ab__group__add,type,
linord219039673up_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oordered__ab__group__add__abs,type,
ordere142940540dd_abs:
!>[A: $tType] : $o ).
thf(sy_cl_Real__Vector__Spaces_Ometric__space,type,
real_V2090557954_space:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Olinordered__nonzero__semiring,type,
linord1659791738miring:
!>[A: $tType] : $o ).
thf(sy_cl_Topological__Spaces_Oorder__topology,type,
topolo259154727pology:
!>[A: $tType] : $o ).
thf(sy_cl_Topological__Spaces_Otopological__space,type,
topolo503727757_space:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ocanonically__ordered__monoid__add,type,
canoni770627133id_add:
!>[A: $tType] : $o ).
thf(sy_c_Extended__Nat_Oenat,type,
extended_enat2: nat > extended_enat ).
thf(sy_c_Extended__Nat_Oenat_Ocase__enat,type,
extended_case_enat:
!>[T: $tType] : ( ( nat > T ) > T > extended_enat > T ) ).
thf(sy_c_Extended__Nat_Oenat_Orec__enat,type,
extended_rec_enat:
!>[T: $tType] : ( ( nat > T ) > T > extended_enat > T ) ).
thf(sy_c_Extended__Nat_Oinfinity__class_Oinfinity,type,
extend1396239628finity:
!>[A: $tType] : A ).
thf(sy_c_Extended__Nat_Othe__enat,type,
extended_the_enat: extended_enat > nat ).
thf(sy_c_Extended__Real_Oereal_OMInfty,type,
extended_MInfty: extended_ereal ).
thf(sy_c_Extended__Real_Oereal_OPInfty,type,
extended_PInfty: extended_ereal ).
thf(sy_c_Extended__Real_Oereal_Oereal,type,
extended_ereal2: real > extended_ereal ).
thf(sy_c_Extended__Real_Oereal_Osize__ereal,type,
extended_size_ereal: extended_ereal > nat ).
thf(sy_c_Extended__Real_Oereal__of__enat,type,
extend1771934483f_enat: extended_enat > extended_ereal ).
thf(sy_c_Extended__Real_Oreal__of__ereal,type,
extend1716541707_ereal: extended_ereal > real ).
thf(sy_c_Groups_Oabs__class_Oabs,type,
abs_abs:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Groups_Ouminus__class_Ouminus,type,
uminus_uminus:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Groups_Ozero__class_Ozero,type,
zero_zero:
!>[A: $tType] : A ).
thf(sy_c_Lower__Semicontinuous__Mirabelle__quczrylfpw_Olsc,type,
lower_107104146le_lsc:
!>[A: $tType,B: $tType] : ( ( A > B ) > $o ) ).
thf(sy_c_Lower__Semicontinuous__Mirabelle__quczrylfpw_Olsc__at,type,
lower_582600101lsc_at:
!>[A: $tType,B: $tType] : ( A > ( A > B ) > $o ) ).
thf(sy_c_Lower__Semicontinuous__Mirabelle__quczrylfpw_Olsc__hull,type,
lower_1879122655c_hull:
!>[A: $tType] : ( ( A > extended_ereal ) > A > extended_ereal ) ).
thf(sy_c_Num_Oneg__numeral__class_Odbl,type,
neg_numeral_dbl:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Num_Oneg__numeral__class_Osub,type,
neg_numeral_sub:
!>[A: $tType] : ( num > num > A ) ).
thf(sy_c_Num_Onumeral__class_Onumeral,type,
numeral_numeral:
!>[A: $tType] : ( num > A ) ).
thf(sy_c_Orderings_Oord__class_Oless__eq,type,
ord_less_eq:
!>[A: $tType] : ( A > A > $o ) ).
thf(sy_c_Orderings_Oorder__class_OGreatest,type,
order_Greatest:
!>[A: $tType] : ( ( A > $o ) > A ) ).
thf(sy_c_Orderings_Oorder__class_Oantimono,type,
order_antimono:
!>[A: $tType,B: $tType] : ( ( A > B ) > $o ) ).
thf(sy_c_Orderings_Oorder__class_Ostrict__mono,type,
order_strict_mono:
!>[A: $tType,B: $tType] : ( ( A > B ) > $o ) ).
thf(sy_c_Rings_Odivide__class_Odivide,type,
divide_divide:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_member,type,
member:
!>[A: $tType] : ( A > ( set @ A ) > $o ) ).
thf(sy_v_f,type,
f: a > extended_ereal ).
thf(sy_v_g,type,
g: a > extended_ereal ).
thf(sy_v_x____,type,
x: a ).
% Relevant facts (252)
thf(fact_0_assms,axiom,
! [X: a] : ( ord_less_eq @ extended_ereal @ ( g @ X ) @ ( f @ X ) ) ).
% assms
thf(fact_1_calculation,axiom,
ord_less_eq @ extended_ereal @ ( lower_1879122655c_hull @ a @ g @ x ) @ ( g @ x ) ).
% calculation
thf(fact_2_order__refl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X2: A] : ( ord_less_eq @ A @ X2 @ X2 ) ) ).
% order_refl
thf(fact_3_ereal__complete__Inf,axiom,
! [S: set @ extended_ereal] :
? [X3: extended_ereal] :
( ! [Xa: extended_ereal] :
( ( member @ extended_ereal @ Xa @ S )
=> ( ord_less_eq @ extended_ereal @ X3 @ Xa ) )
& ! [Z: extended_ereal] :
( ! [Xa2: extended_ereal] :
( ( member @ extended_ereal @ Xa2 @ S )
=> ( ord_less_eq @ extended_ereal @ Z @ Xa2 ) )
=> ( ord_less_eq @ extended_ereal @ Z @ X3 ) ) ) ).
% ereal_complete_Inf
thf(fact_4_ereal__complete__Sup,axiom,
! [S: set @ extended_ereal] :
? [X3: extended_ereal] :
( ! [Xa: extended_ereal] :
( ( member @ extended_ereal @ Xa @ S )
=> ( ord_less_eq @ extended_ereal @ Xa @ X3 ) )
& ! [Z: extended_ereal] :
( ! [Xa2: extended_ereal] :
( ( member @ extended_ereal @ Xa2 @ S )
=> ( ord_less_eq @ extended_ereal @ Xa2 @ Z ) )
=> ( ord_less_eq @ extended_ereal @ X3 @ Z ) ) ) ).
% ereal_complete_Sup
thf(fact_5_le__funD,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ! [F: A > B,G: A > B,X2: A] :
( ( ord_less_eq @ ( A > B ) @ F @ G )
=> ( ord_less_eq @ B @ ( F @ X2 ) @ ( G @ X2 ) ) ) ) ).
% le_funD
thf(fact_6_le__funE,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ! [F: A > B,G: A > B,X2: A] :
( ( ord_less_eq @ ( A > B ) @ F @ G )
=> ( ord_less_eq @ B @ ( F @ X2 ) @ ( G @ X2 ) ) ) ) ).
% le_funE
thf(fact_7_le__funI,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ! [F: A > B,G: A > B] :
( ! [X3: A] : ( ord_less_eq @ B @ ( F @ X3 ) @ ( G @ X3 ) )
=> ( ord_less_eq @ ( A > B ) @ F @ G ) ) ) ).
% le_funI
thf(fact_8_le__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ( ( ord_less_eq @ ( A > B ) )
= ( ^ [F2: A > B,G2: A > B] :
! [X4: A] : ( ord_less_eq @ B @ ( F2 @ X4 ) @ ( G2 @ X4 ) ) ) ) ) ).
% le_fun_def
thf(fact_9_order__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B )
& ( order @ A ) )
=> ! [A2: A,F: B > A,B2: B,C: B] :
( ( ord_less_eq @ A @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq @ B @ B2 @ C )
=> ( ! [X3: B,Y: B] :
( ( ord_less_eq @ B @ X3 @ Y )
=> ( ord_less_eq @ A @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_eq @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% order_subst1
thf(fact_10_order__subst2,axiom,
! [A: $tType,C2: $tType] :
( ( ( order @ C2 )
& ( order @ A ) )
=> ! [A2: A,B2: A,F: A > C2,C: C2] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ C2 @ ( F @ B2 ) @ C )
=> ( ! [X3: A,Y: A] :
( ( ord_less_eq @ A @ X3 @ Y )
=> ( ord_less_eq @ C2 @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_eq @ C2 @ ( F @ A2 ) @ C ) ) ) ) ) ).
% order_subst2
thf(fact_11_verit__la__disequality,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,B2: A] :
( ( A2 = B2 )
| ~ ( ord_less_eq @ A @ A2 @ B2 )
| ~ ( ord_less_eq @ A @ B2 @ A2 ) ) ) ).
% verit_la_disequality
thf(fact_12_ord__eq__le__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( ord @ A ) )
=> ! [A2: A,F: B > A,B2: B,C: B] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq @ B @ B2 @ C )
=> ( ! [X3: B,Y: B] :
( ( ord_less_eq @ B @ X3 @ Y )
=> ( ord_less_eq @ A @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_eq @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_13_ord__le__eq__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( ord @ A ) )
=> ! [A2: A,B2: A,F: A > B,C: B] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: A,Y: A] :
( ( ord_less_eq @ A @ X3 @ Y )
=> ( ord_less_eq @ B @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_eq @ B @ ( F @ A2 ) @ C ) ) ) ) ) ).
% ord_le_eq_subst
thf(fact_14_lsc__hull__le,axiom,
! [A: $tType] :
( ( real_V2090557954_space @ A )
=> ! [F: A > extended_ereal,X2: A] : ( ord_less_eq @ extended_ereal @ ( lower_1879122655c_hull @ A @ F @ X2 ) @ ( F @ X2 ) ) ) ).
% lsc_hull_le
thf(fact_15_dual__order_Oantisym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B2: A,A2: A] :
( ( ord_less_eq @ A @ B2 @ A2 )
=> ( ( ord_less_eq @ A @ A2 @ B2 )
=> ( A2 = B2 ) ) ) ) ).
% dual_order.antisym
thf(fact_16_dual__order_Oeq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y2: A,Z2: A] : Y2 = Z2 )
= ( ^ [A3: A,B3: A] :
( ( ord_less_eq @ A @ B3 @ A3 )
& ( ord_less_eq @ A @ A3 @ B3 ) ) ) ) ) ).
% dual_order.eq_iff
thf(fact_17_dual__order_Otrans,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B2: A,A2: A,C: A] :
( ( ord_less_eq @ A @ B2 @ A2 )
=> ( ( ord_less_eq @ A @ C @ B2 )
=> ( ord_less_eq @ A @ C @ A2 ) ) ) ) ).
% dual_order.trans
thf(fact_18_linorder__wlog,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > A > $o,A2: A,B2: A] :
( ! [A4: A,B4: A] :
( ( ord_less_eq @ A @ A4 @ B4 )
=> ( P @ A4 @ B4 ) )
=> ( ! [A4: A,B4: A] :
( ( P @ B4 @ A4 )
=> ( P @ A4 @ B4 ) )
=> ( P @ A2 @ B2 ) ) ) ) ).
% linorder_wlog
thf(fact_19_dual__order_Orefl,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: A] : ( ord_less_eq @ A @ A2 @ A2 ) ) ).
% dual_order.refl
thf(fact_20_order__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X2: A,Y3: A,Z3: A] :
( ( ord_less_eq @ A @ X2 @ Y3 )
=> ( ( ord_less_eq @ A @ Y3 @ Z3 )
=> ( ord_less_eq @ A @ X2 @ Z3 ) ) ) ) ).
% order_trans
thf(fact_21_order__class_Oorder_Oantisym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ) ).
% order_class.order.antisym
thf(fact_22_ord__le__eq__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A2: A,B2: A,C: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( B2 = C )
=> ( ord_less_eq @ A @ A2 @ C ) ) ) ) ).
% ord_le_eq_trans
thf(fact_23_ord__eq__le__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A2: A,B2: A,C: A] :
( ( A2 = B2 )
=> ( ( ord_less_eq @ A @ B2 @ C )
=> ( ord_less_eq @ A @ A2 @ C ) ) ) ) ).
% ord_eq_le_trans
thf(fact_24_order__class_Oorder_Oeq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y2: A,Z2: A] : Y2 = Z2 )
= ( ^ [A3: A,B3: A] :
( ( ord_less_eq @ A @ A3 @ B3 )
& ( ord_less_eq @ A @ B3 @ A3 ) ) ) ) ) ).
% order_class.order.eq_iff
thf(fact_25_antisym__conv,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [Y3: A,X2: A] :
( ( ord_less_eq @ A @ Y3 @ X2 )
=> ( ( ord_less_eq @ A @ X2 @ Y3 )
= ( X2 = Y3 ) ) ) ) ).
% antisym_conv
thf(fact_26_le__cases3,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X2: A,Y3: A,Z3: A] :
( ( ( ord_less_eq @ A @ X2 @ Y3 )
=> ~ ( ord_less_eq @ A @ Y3 @ Z3 ) )
=> ( ( ( ord_less_eq @ A @ Y3 @ X2 )
=> ~ ( ord_less_eq @ A @ X2 @ Z3 ) )
=> ( ( ( ord_less_eq @ A @ X2 @ Z3 )
=> ~ ( ord_less_eq @ A @ Z3 @ Y3 ) )
=> ( ( ( ord_less_eq @ A @ Z3 @ Y3 )
=> ~ ( ord_less_eq @ A @ Y3 @ X2 ) )
=> ( ( ( ord_less_eq @ A @ Y3 @ Z3 )
=> ~ ( ord_less_eq @ A @ Z3 @ X2 ) )
=> ~ ( ( ord_less_eq @ A @ Z3 @ X2 )
=> ~ ( ord_less_eq @ A @ X2 @ Y3 ) ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_27_order_Otrans,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: A,B2: A,C: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ B2 @ C )
=> ( ord_less_eq @ A @ A2 @ C ) ) ) ) ).
% order.trans
thf(fact_28_le__cases,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X2: A,Y3: A] :
( ~ ( ord_less_eq @ A @ X2 @ Y3 )
=> ( ord_less_eq @ A @ Y3 @ X2 ) ) ) ).
% le_cases
thf(fact_29_eq__refl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X2: A,Y3: A] :
( ( X2 = Y3 )
=> ( ord_less_eq @ A @ X2 @ Y3 ) ) ) ).
% eq_refl
thf(fact_30_linear,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X2: A,Y3: A] :
( ( ord_less_eq @ A @ X2 @ Y3 )
| ( ord_less_eq @ A @ Y3 @ X2 ) ) ) ).
% linear
thf(fact_31_antisym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X2: A,Y3: A] :
( ( ord_less_eq @ A @ X2 @ Y3 )
=> ( ( ord_less_eq @ A @ Y3 @ X2 )
=> ( X2 = Y3 ) ) ) ) ).
% antisym
thf(fact_32_eq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y2: A,Z2: A] : Y2 = Z2 )
= ( ^ [X4: A,Y4: A] :
( ( ord_less_eq @ A @ X4 @ Y4 )
& ( ord_less_eq @ A @ Y4 @ X4 ) ) ) ) ) ).
% eq_iff
thf(fact_33_lsc__hull__iff__greatest,axiom,
! [A: $tType] :
( ( real_V2090557954_space @ A )
=> ! [G: A > extended_ereal,F: A > extended_ereal] :
( ( G
= ( lower_1879122655c_hull @ A @ F ) )
= ( ( lower_107104146le_lsc @ A @ extended_ereal @ G )
& ! [X4: A] : ( ord_less_eq @ extended_ereal @ ( G @ X4 ) @ ( F @ X4 ) )
& ! [H: A > extended_ereal] :
( ( ( lower_107104146le_lsc @ A @ extended_ereal @ H )
& ! [X4: A] : ( ord_less_eq @ extended_ereal @ ( H @ X4 ) @ ( F @ X4 ) ) )
=> ! [X4: A] : ( ord_less_eq @ extended_ereal @ ( H @ X4 ) @ ( G @ X4 ) ) ) ) ) ) ).
% lsc_hull_iff_greatest
thf(fact_34_lsc__hull__greatest,axiom,
! [A: $tType] :
( ( real_V2090557954_space @ A )
=> ! [G: A > extended_ereal,F: A > extended_ereal] :
( ( lower_107104146le_lsc @ A @ extended_ereal @ G )
=> ( ! [X3: A] : ( ord_less_eq @ extended_ereal @ ( G @ X3 ) @ ( F @ X3 ) )
=> ! [X: A] : ( ord_less_eq @ extended_ereal @ ( G @ X ) @ ( lower_1879122655c_hull @ A @ F @ X ) ) ) ) ) ).
% lsc_hull_greatest
thf(fact_35_Greatest__equality,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [P: A > $o,X2: A] :
( ( P @ X2 )
=> ( ! [Y: A] :
( ( P @ Y )
=> ( ord_less_eq @ A @ Y @ X2 ) )
=> ( ( order_Greatest @ A @ P )
= X2 ) ) ) ) ).
% Greatest_equality
thf(fact_36_GreatestI2__order,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [P: A > $o,X2: A,Q: A > $o] :
( ( P @ X2 )
=> ( ! [Y: A] :
( ( P @ Y )
=> ( ord_less_eq @ A @ Y @ X2 ) )
=> ( ! [X3: A] :
( ( P @ X3 )
=> ( ! [Y5: A] :
( ( P @ Y5 )
=> ( ord_less_eq @ A @ Y5 @ X3 ) )
=> ( Q @ X3 ) ) )
=> ( Q @ ( order_Greatest @ A @ P ) ) ) ) ) ) ).
% GreatestI2_order
thf(fact_37_le__rel__bool__arg__iff,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( ord_less_eq @ ( $o > A ) )
= ( ^ [X5: $o > A,Y6: $o > A] :
( ( ord_less_eq @ A @ ( X5 @ $false ) @ ( Y6 @ $false ) )
& ( ord_less_eq @ A @ ( X5 @ $true ) @ ( Y6 @ $true ) ) ) ) ) ) ).
% le_rel_bool_arg_iff
thf(fact_38_lsc__lsc__hull,axiom,
! [A: $tType] :
( ( real_V2090557954_space @ A )
=> ! [F: A > extended_ereal] : ( lower_107104146le_lsc @ A @ extended_ereal @ ( lower_1879122655c_hull @ A @ F ) ) ) ).
% lsc_lsc_hull
thf(fact_39_ereal__of__enat__le__iff,axiom,
! [M: extended_enat,N: extended_enat] :
( ( ord_less_eq @ extended_ereal @ ( extend1771934483f_enat @ M ) @ ( extend1771934483f_enat @ N ) )
= ( ord_less_eq @ extended_enat @ M @ N ) ) ).
% ereal_of_enat_le_iff
thf(fact_40_antimono__def,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ B ) )
=> ( ( order_antimono @ A @ B )
= ( ^ [F2: A > B] :
! [X4: A,Y4: A] :
( ( ord_less_eq @ A @ X4 @ Y4 )
=> ( ord_less_eq @ B @ ( F2 @ Y4 ) @ ( F2 @ X4 ) ) ) ) ) ) ).
% antimono_def
thf(fact_41_antimonoI,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ B ) )
=> ! [F: A > B] :
( ! [X3: A,Y: A] :
( ( ord_less_eq @ A @ X3 @ Y )
=> ( ord_less_eq @ B @ ( F @ Y ) @ ( F @ X3 ) ) )
=> ( order_antimono @ A @ B @ F ) ) ) ).
% antimonoI
thf(fact_42_antimonoE,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ B ) )
=> ! [F: A > B,X2: A,Y3: A] :
( ( order_antimono @ A @ B @ F )
=> ( ( ord_less_eq @ A @ X2 @ Y3 )
=> ( ord_less_eq @ B @ ( F @ Y3 ) @ ( F @ X2 ) ) ) ) ) ).
% antimonoE
thf(fact_43_antimonoD,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ B ) )
=> ! [F: A > B,X2: A,Y3: A] :
( ( order_antimono @ A @ B @ F )
=> ( ( ord_less_eq @ A @ X2 @ Y3 )
=> ( ord_less_eq @ B @ ( F @ Y3 ) @ ( F @ X2 ) ) ) ) ) ).
% antimonoD
thf(fact_44_lsc__def,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo503727757_space @ A )
& ( topolo259154727pology @ B ) )
=> ( ( lower_107104146le_lsc @ A @ B )
= ( ^ [F2: A > B] :
! [X4: A] : ( lower_582600101lsc_at @ A @ B @ X4 @ F2 ) ) ) ) ).
% lsc_def
thf(fact_45_ext,axiom,
! [B: $tType,A: $tType,F: A > B,G: A > B] :
( ! [X3: A] :
( ( F @ X3 )
= ( G @ X3 ) )
=> ( F = G ) ) ).
% ext
thf(fact_46_decseqD,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [F: nat > A,I: nat,J: nat] :
( ( order_antimono @ nat @ A @ F )
=> ( ( ord_less_eq @ nat @ I @ J )
=> ( ord_less_eq @ A @ ( F @ J ) @ ( F @ I ) ) ) ) ) ).
% decseqD
thf(fact_47_decseq__def,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( order_antimono @ nat @ A )
= ( ^ [X5: nat > A] :
! [M2: nat,N2: nat] :
( ( ord_less_eq @ nat @ M2 @ N2 )
=> ( ord_less_eq @ A @ ( X5 @ N2 ) @ ( X5 @ M2 ) ) ) ) ) ) ).
% decseq_def
thf(fact_48_ereal__of__enat__nonneg,axiom,
! [N: extended_enat] : ( ord_less_eq @ extended_ereal @ ( zero_zero @ extended_ereal ) @ ( extend1771934483f_enat @ N ) ) ).
% ereal_of_enat_nonneg
thf(fact_49_ereal__of__enat__ge__zero__cancel__iff,axiom,
! [N: extended_enat] :
( ( ord_less_eq @ extended_ereal @ ( zero_zero @ extended_ereal ) @ ( extend1771934483f_enat @ N ) )
= ( ord_less_eq @ extended_enat @ ( zero_zero @ extended_enat ) @ N ) ) ).
% ereal_of_enat_ge_zero_cancel_iff
thf(fact_50_numeral__le__ereal__of__enat__iff,axiom,
! [M: num,N: extended_enat] :
( ( ord_less_eq @ extended_ereal @ ( numeral_numeral @ extended_ereal @ M ) @ ( extend1771934483f_enat @ N ) )
= ( ord_less_eq @ extended_enat @ ( numeral_numeral @ extended_enat @ M ) @ N ) ) ).
% numeral_le_ereal_of_enat_iff
thf(fact_51_ereal__infty__less__eq_I1_J,axiom,
! [X2: extended_ereal] :
( ( ord_less_eq @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) @ X2 )
= ( X2
= ( extend1396239628finity @ extended_ereal ) ) ) ).
% ereal_infty_less_eq(1)
thf(fact_52_strict__mono__less__eq,axiom,
! [B: $tType,A: $tType] :
( ( ( linorder @ A )
& ( order @ B ) )
=> ! [F: A > B,X2: A,Y3: A] :
( ( order_strict_mono @ A @ B @ F )
=> ( ( ord_less_eq @ B @ ( F @ X2 ) @ ( F @ Y3 ) )
= ( ord_less_eq @ A @ X2 @ Y3 ) ) ) ) ).
% strict_mono_less_eq
thf(fact_53_ereal__of__enat__zero,axiom,
( ( extend1771934483f_enat @ ( zero_zero @ extended_enat ) )
= ( zero_zero @ extended_ereal ) ) ).
% ereal_of_enat_zero
thf(fact_54_strict__mono__eq,axiom,
! [B: $tType,A: $tType] :
( ( ( linorder @ A )
& ( order @ B ) )
=> ! [F: A > B,X2: A,Y3: A] :
( ( order_strict_mono @ A @ B @ F )
=> ( ( ( F @ X2 )
= ( F @ Y3 ) )
= ( X2 = Y3 ) ) ) ) ).
% strict_mono_eq
thf(fact_55_Infty__neq__0_I1_J,axiom,
( ( extend1396239628finity @ extended_ereal )
!= ( zero_zero @ extended_ereal ) ) ).
% Infty_neq_0(1)
thf(fact_56_strict__mono__leD,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ B ) )
=> ! [R: A > B,M: A,N: A] :
( ( order_strict_mono @ A @ B @ R )
=> ( ( ord_less_eq @ A @ M @ N )
=> ( ord_less_eq @ B @ ( R @ M ) @ ( R @ N ) ) ) ) ) ).
% strict_mono_leD
thf(fact_57_neq__PInf__trans,axiom,
! [Y3: extended_ereal,X2: extended_ereal] :
( ( Y3
!= ( extend1396239628finity @ extended_ereal ) )
=> ( ( ord_less_eq @ extended_ereal @ X2 @ Y3 )
=> ( X2
!= ( extend1396239628finity @ extended_ereal ) ) ) ) ).
% neq_PInf_trans
thf(fact_58_ereal__infty__less__eq2_I1_J,axiom,
! [A2: extended_ereal,B2: extended_ereal] :
( ( ord_less_eq @ extended_ereal @ A2 @ B2 )
=> ( ( A2
= ( extend1396239628finity @ extended_ereal ) )
=> ( B2
= ( extend1396239628finity @ extended_ereal ) ) ) ) ).
% ereal_infty_less_eq2(1)
thf(fact_59_ereal__less__eq_I1_J,axiom,
! [X2: extended_ereal] : ( ord_less_eq @ extended_ereal @ X2 @ ( extend1396239628finity @ extended_ereal ) ) ).
% ereal_less_eq(1)
thf(fact_60_numeral__le__iff,axiom,
! [A: $tType] :
( ( linord1659791738miring @ A )
=> ! [M: num,N: num] :
( ( ord_less_eq @ A @ ( numeral_numeral @ A @ M ) @ ( numeral_numeral @ A @ N ) )
= ( ord_less_eq @ num @ M @ N ) ) ) ).
% numeral_le_iff
thf(fact_61_le__zero__eq,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A )
=> ! [N: A] :
( ( ord_less_eq @ A @ N @ ( zero_zero @ A ) )
= ( N
= ( zero_zero @ A ) ) ) ) ).
% le_zero_eq
thf(fact_62_zero__le__numeral,axiom,
! [A: $tType] :
( ( linord1659791738miring @ A )
=> ! [N: num] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ N ) ) ) ).
% zero_le_numeral
thf(fact_63_not__numeral__le__zero,axiom,
! [A: $tType] :
( ( linord1659791738miring @ A )
=> ! [N: num] :
~ ( ord_less_eq @ A @ ( numeral_numeral @ A @ N ) @ ( zero_zero @ A ) ) ) ).
% not_numeral_le_zero
thf(fact_64_numeral__eq__iff,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ! [M: num,N: num] :
( ( ( numeral_numeral @ A @ M )
= ( numeral_numeral @ A @ N ) )
= ( M = N ) ) ) ).
% numeral_eq_iff
thf(fact_65_ile0__eq,axiom,
! [N: extended_enat] :
( ( ord_less_eq @ extended_enat @ N @ ( zero_zero @ extended_enat ) )
= ( N
= ( zero_zero @ extended_enat ) ) ) ).
% ile0_eq
thf(fact_66_i0__lb,axiom,
! [N: extended_enat] : ( ord_less_eq @ extended_enat @ ( zero_zero @ extended_enat ) @ N ) ).
% i0_lb
thf(fact_67_enat__ord__number_I1_J,axiom,
! [M: num,N: num] :
( ( ord_less_eq @ extended_enat @ ( numeral_numeral @ extended_enat @ M ) @ ( numeral_numeral @ extended_enat @ N ) )
= ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ M ) @ ( numeral_numeral @ nat @ N ) ) ) ).
% enat_ord_number(1)
thf(fact_68_zero__reorient,axiom,
! [A: $tType] :
( ( zero @ A )
=> ! [X2: A] :
( ( ( zero_zero @ A )
= X2 )
= ( X2
= ( zero_zero @ A ) ) ) ) ).
% zero_reorient
thf(fact_69_zero__le,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A )
=> ! [X2: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ X2 ) ) ).
% zero_le
thf(fact_70_le__numeral__extra_I3_J,axiom,
! [A: $tType] :
( ( linord1659791738miring @ A )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( zero_zero @ A ) ) ) ).
% le_numeral_extra(3)
thf(fact_71_zero__neq__numeral,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ! [N: num] :
( ( zero_zero @ A )
!= ( numeral_numeral @ A @ N ) ) ) ).
% zero_neq_numeral
thf(fact_72_bot__nat__0_Oextremum,axiom,
! [A2: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ A2 ) ).
% bot_nat_0.extremum
thf(fact_73_le0,axiom,
! [N: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ N ) ).
% le0
thf(fact_74_neg__numeral__le__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [M: num,N: num] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) )
= ( ord_less_eq @ num @ N @ M ) ) ) ).
% neg_numeral_le_iff
thf(fact_75_numeral__le__enat__iff,axiom,
! [M: num,N: nat] :
( ( ord_less_eq @ extended_enat @ ( numeral_numeral @ extended_enat @ M ) @ ( extended_enat2 @ N ) )
= ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ M ) @ N ) ) ).
% numeral_le_enat_iff
thf(fact_76_sub__non__positive,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N: num,M: num] :
( ( ord_less_eq @ A @ ( neg_numeral_sub @ A @ N @ M ) @ ( zero_zero @ A ) )
= ( ord_less_eq @ num @ N @ M ) ) ) ).
% sub_non_positive
thf(fact_77_sub__non__negative,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N: num,M: num] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( neg_numeral_sub @ A @ N @ M ) )
= ( ord_less_eq @ num @ M @ N ) ) ) ).
% sub_non_negative
thf(fact_78_real__of__ereal__le__0,axiom,
! [X2: extended_ereal] :
( ( ord_less_eq @ real @ ( extend1716541707_ereal @ X2 ) @ ( zero_zero @ real ) )
= ( ( ord_less_eq @ extended_ereal @ X2 @ ( zero_zero @ extended_ereal ) )
| ( X2
= ( extend1396239628finity @ extended_ereal ) ) ) ) ).
% real_of_ereal_le_0
thf(fact_79_neg__equal__iff__equal,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A,B2: A] :
( ( ( uminus_uminus @ A @ A2 )
= ( uminus_uminus @ A @ B2 ) )
= ( A2 = B2 ) ) ) ).
% neg_equal_iff_equal
thf(fact_80_add_Oinverse__inverse,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A] :
( ( uminus_uminus @ A @ ( uminus_uminus @ A @ A2 ) )
= A2 ) ) ).
% add.inverse_inverse
thf(fact_81_verit__minus__simplify_I4_J,axiom,
! [B: $tType] :
( ( group_add @ B )
=> ! [B2: B] :
( ( uminus_uminus @ B @ ( uminus_uminus @ B @ B2 ) )
= B2 ) ) ).
% verit_minus_simplify(4)
thf(fact_82_ereal__uminus__zero__iff,axiom,
! [A2: extended_ereal] :
( ( ( uminus_uminus @ extended_ereal @ A2 )
= ( zero_zero @ extended_ereal ) )
= ( A2
= ( zero_zero @ extended_ereal ) ) ) ).
% ereal_uminus_zero_iff
thf(fact_83_ereal__uminus__zero,axiom,
( ( uminus_uminus @ extended_ereal @ ( zero_zero @ extended_ereal ) )
= ( zero_zero @ extended_ereal ) ) ).
% ereal_uminus_zero
thf(fact_84_ereal__minus__le__minus,axiom,
! [A2: extended_ereal,B2: extended_ereal] :
( ( ord_less_eq @ extended_ereal @ ( uminus_uminus @ extended_ereal @ A2 ) @ ( uminus_uminus @ extended_ereal @ B2 ) )
= ( ord_less_eq @ extended_ereal @ B2 @ A2 ) ) ).
% ereal_minus_le_minus
thf(fact_85_enat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( extended_enat2 @ Nat )
= ( extended_enat2 @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% enat.inject
thf(fact_86_real__of__ereal,axiom,
! [X2: extended_ereal] :
( ( extend1716541707_ereal @ ( uminus_uminus @ extended_ereal @ X2 ) )
= ( uminus_uminus @ real @ ( extend1716541707_ereal @ X2 ) ) ) ).
% real_of_ereal
thf(fact_87_neg__le__iff__le,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [B2: A,A2: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ B2 ) @ ( uminus_uminus @ A @ A2 ) )
= ( ord_less_eq @ A @ A2 @ B2 ) ) ) ).
% neg_le_iff_le
thf(fact_88_neg__equal__zero,axiom,
! [A: $tType] :
( ( linord219039673up_add @ A )
=> ! [A2: A] :
( ( ( uminus_uminus @ A @ A2 )
= A2 )
= ( A2
= ( zero_zero @ A ) ) ) ) ).
% neg_equal_zero
thf(fact_89_equal__neg__zero,axiom,
! [A: $tType] :
( ( linord219039673up_add @ A )
=> ! [A2: A] :
( ( A2
= ( uminus_uminus @ A @ A2 ) )
= ( A2
= ( zero_zero @ A ) ) ) ) ).
% equal_neg_zero
thf(fact_90_neg__equal__0__iff__equal,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A] :
( ( ( uminus_uminus @ A @ A2 )
= ( zero_zero @ A ) )
= ( A2
= ( zero_zero @ A ) ) ) ) ).
% neg_equal_0_iff_equal
thf(fact_91_neg__0__equal__iff__equal,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A] :
( ( ( zero_zero @ A )
= ( uminus_uminus @ A @ A2 ) )
= ( ( zero_zero @ A )
= A2 ) ) ) ).
% neg_0_equal_iff_equal
thf(fact_92_add_Oinverse__neutral,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ( ( uminus_uminus @ A @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% add.inverse_neutral
thf(fact_93_neg__numeral__eq__iff,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [M: num,N: num] :
( ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) )
= ( M = N ) ) ) ).
% neg_numeral_eq_iff
thf(fact_94_ereal__infty__less__eq_I2_J,axiom,
! [X2: extended_ereal] :
( ( ord_less_eq @ extended_ereal @ X2 @ ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
= ( X2
= ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) ) ).
% ereal_infty_less_eq(2)
thf(fact_95_ereal__uminus__le__0__iff,axiom,
! [A2: extended_ereal] :
( ( ord_less_eq @ extended_ereal @ ( uminus_uminus @ extended_ereal @ A2 ) @ ( zero_zero @ extended_ereal ) )
= ( ord_less_eq @ extended_ereal @ ( zero_zero @ extended_ereal ) @ A2 ) ) ).
% ereal_uminus_le_0_iff
thf(fact_96_ereal__0__le__uminus__iff,axiom,
! [A2: extended_ereal] :
( ( ord_less_eq @ extended_ereal @ ( zero_zero @ extended_ereal ) @ ( uminus_uminus @ extended_ereal @ A2 ) )
= ( ord_less_eq @ extended_ereal @ A2 @ ( zero_zero @ extended_ereal ) ) ) ).
% ereal_0_le_uminus_iff
thf(fact_97_neg__0__le__iff__le,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( uminus_uminus @ A @ A2 ) )
= ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% neg_0_le_iff_le
thf(fact_98_neg__le__0__iff__le,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ A2 ) @ ( zero_zero @ A ) )
= ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% neg_le_0_iff_le
thf(fact_99_less__eq__neg__nonpos,axiom,
! [A: $tType] :
( ( linord219039673up_add @ A )
=> ! [A2: A] :
( ( ord_less_eq @ A @ A2 @ ( uminus_uminus @ A @ A2 ) )
= ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% less_eq_neg_nonpos
thf(fact_100_neg__less__eq__nonneg,axiom,
! [A: $tType] :
( ( linord219039673up_add @ A )
=> ! [A2: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ A2 ) @ A2 )
= ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% neg_less_eq_nonneg
thf(fact_101_real__of__ereal__0,axiom,
( ( extend1716541707_ereal @ ( zero_zero @ extended_ereal ) )
= ( zero_zero @ real ) ) ).
% real_of_ereal_0
thf(fact_102_enat__ord__simps_I1_J,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq @ extended_enat @ ( extended_enat2 @ M ) @ ( extended_enat2 @ N ) )
= ( ord_less_eq @ nat @ M @ N ) ) ).
% enat_ord_simps(1)
thf(fact_103_real__of__ereal_Osimps_I3_J,axiom,
( ( extend1716541707_ereal @ ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
= ( zero_zero @ real ) ) ).
% real_of_ereal.simps(3)
thf(fact_104_minus__equation__iff,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A,B2: A] :
( ( ( uminus_uminus @ A @ A2 )
= B2 )
= ( ( uminus_uminus @ A @ B2 )
= A2 ) ) ) ).
% minus_equation_iff
thf(fact_105_equation__minus__iff,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A,B2: A] :
( ( A2
= ( uminus_uminus @ A @ B2 ) )
= ( B2
= ( uminus_uminus @ A @ A2 ) ) ) ) ).
% equation_minus_iff
thf(fact_106_verit__negate__coefficient_I3_J,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: A,B2: A] :
( ( A2 = B2 )
=> ( ( uminus_uminus @ A @ A2 )
= ( uminus_uminus @ A @ B2 ) ) ) ) ).
% verit_negate_coefficient(3)
thf(fact_107_real__of__ereal__eq__0,axiom,
! [X2: extended_ereal] :
( ( ( extend1716541707_ereal @ X2 )
= ( zero_zero @ real ) )
= ( ( X2
= ( extend1396239628finity @ extended_ereal ) )
| ( X2
= ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
| ( X2
= ( zero_zero @ extended_ereal ) ) ) ) ).
% real_of_ereal_eq_0
thf(fact_108_real__of__ereal_Osimps_I2_J,axiom,
( ( extend1716541707_ereal @ ( extend1396239628finity @ extended_ereal ) )
= ( zero_zero @ real ) ) ).
% real_of_ereal.simps(2)
thf(fact_109_le__imp__neg__le,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ord_less_eq @ A @ ( uminus_uminus @ A @ B2 ) @ ( uminus_uminus @ A @ A2 ) ) ) ) ).
% le_imp_neg_le
thf(fact_110_minus__le__iff,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ A2 ) @ B2 )
= ( ord_less_eq @ A @ ( uminus_uminus @ A @ B2 ) @ A2 ) ) ) ).
% minus_le_iff
thf(fact_111_le__minus__iff,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ ( uminus_uminus @ A @ B2 ) )
= ( ord_less_eq @ A @ B2 @ ( uminus_uminus @ A @ A2 ) ) ) ) ).
% le_minus_iff
thf(fact_112_numeral__neq__neg__numeral,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [M: num,N: num] :
( ( numeral_numeral @ A @ M )
!= ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) ) ) ).
% numeral_neq_neg_numeral
thf(fact_113_neg__numeral__neq__numeral,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [M: num,N: num] :
( ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) )
!= ( numeral_numeral @ A @ N ) ) ) ).
% neg_numeral_neq_numeral
thf(fact_114_zero__enat__def,axiom,
( ( zero_zero @ extended_enat )
= ( extended_enat2 @ ( zero_zero @ nat ) ) ) ).
% zero_enat_def
thf(fact_115_enat__0__iff_I1_J,axiom,
! [X2: nat] :
( ( ( extended_enat2 @ X2 )
= ( zero_zero @ extended_enat ) )
= ( X2
= ( zero_zero @ nat ) ) ) ).
% enat_0_iff(1)
thf(fact_116_enat__0__iff_I2_J,axiom,
! [X2: nat] :
( ( ( zero_zero @ extended_enat )
= ( extended_enat2 @ X2 ) )
= ( X2
= ( zero_zero @ nat ) ) ) ).
% enat_0_iff(2)
thf(fact_117_lsc__at__MInfty,axiom,
! [A: $tType] :
( ( topolo503727757_space @ A )
=> ! [F: A > extended_ereal,X0: A] :
( ( ( F @ X0 )
= ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
=> ( lower_582600101lsc_at @ A @ extended_ereal @ X0 @ F ) ) ) ).
% lsc_at_MInfty
thf(fact_118_MInfty__neq__PInfty_I1_J,axiom,
( ( extend1396239628finity @ extended_ereal )
!= ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) ).
% MInfty_neq_PInfty(1)
thf(fact_119_ereal__uminus__le__reorder,axiom,
! [A2: extended_ereal,B2: extended_ereal] :
( ( ord_less_eq @ extended_ereal @ ( uminus_uminus @ extended_ereal @ A2 ) @ B2 )
= ( ord_less_eq @ extended_ereal @ ( uminus_uminus @ extended_ereal @ B2 ) @ A2 ) ) ).
% ereal_uminus_le_reorder
thf(fact_120_enat__ile,axiom,
! [N: extended_enat,M: nat] :
( ( ord_less_eq @ extended_enat @ N @ ( extended_enat2 @ M ) )
=> ? [K: nat] :
( N
= ( extended_enat2 @ K ) ) ) ).
% enat_ile
thf(fact_121_numeral__eq__enat,axiom,
( ( numeral_numeral @ extended_enat )
= ( ^ [K2: num] : ( extended_enat2 @ ( numeral_numeral @ nat @ K2 ) ) ) ) ).
% numeral_eq_enat
thf(fact_122_real__of__ereal__pos,axiom,
! [X2: extended_ereal] :
( ( ord_less_eq @ extended_ereal @ ( zero_zero @ extended_ereal ) @ X2 )
=> ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( extend1716541707_ereal @ X2 ) ) ) ).
% real_of_ereal_pos
thf(fact_123_neg__numeral__le__numeral,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [M: num,N: num] : ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) @ ( numeral_numeral @ A @ N ) ) ) ).
% neg_numeral_le_numeral
thf(fact_124_not__numeral__le__neg__numeral,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [M: num,N: num] :
~ ( ord_less_eq @ A @ ( numeral_numeral @ A @ M ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) ) ) ).
% not_numeral_le_neg_numeral
thf(fact_125_zero__neq__neg__numeral,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [N: num] :
( ( zero_zero @ A )
!= ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) ) ) ).
% zero_neq_neg_numeral
thf(fact_126_Infty__neq__0_I3_J,axiom,
( ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) )
!= ( zero_zero @ extended_ereal ) ) ).
% Infty_neq_0(3)
thf(fact_127_ereal__infty__less__eq2_I2_J,axiom,
! [A2: extended_ereal,B2: extended_ereal] :
( ( ord_less_eq @ extended_ereal @ A2 @ B2 )
=> ( ( B2
= ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
=> ( A2
= ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) ) ) ).
% ereal_infty_less_eq2(2)
thf(fact_128_ereal__less__eq_I2_J,axiom,
! [X2: extended_ereal] : ( ord_less_eq @ extended_ereal @ ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) @ X2 ) ).
% ereal_less_eq(2)
thf(fact_129_real__of__ereal__positive__mono,axiom,
! [X2: extended_ereal,Y3: extended_ereal] :
( ( ord_less_eq @ extended_ereal @ ( zero_zero @ extended_ereal ) @ X2 )
=> ( ( ord_less_eq @ extended_ereal @ X2 @ Y3 )
=> ( ( Y3
!= ( extend1396239628finity @ extended_ereal ) )
=> ( ord_less_eq @ real @ ( extend1716541707_ereal @ X2 ) @ ( extend1716541707_ereal @ Y3 ) ) ) ) ) ).
% real_of_ereal_positive_mono
thf(fact_130_neg__numeral__le__zero,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N: num] : ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) @ ( zero_zero @ A ) ) ) ).
% neg_numeral_le_zero
thf(fact_131_not__zero__le__neg__numeral,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N: num] :
~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) ) ) ).
% not_zero_le_neg_numeral
thf(fact_132_not__MInfty__nonneg,axiom,
! [X2: extended_ereal] :
( ( ord_less_eq @ extended_ereal @ ( zero_zero @ extended_ereal ) @ X2 )
=> ( X2
!= ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) ) ).
% not_MInfty_nonneg
thf(fact_133_le__refl,axiom,
! [N: nat] : ( ord_less_eq @ nat @ N @ N ) ).
% le_refl
thf(fact_134_le__trans,axiom,
! [I: nat,J: nat,K3: nat] :
( ( ord_less_eq @ nat @ I @ J )
=> ( ( ord_less_eq @ nat @ J @ K3 )
=> ( ord_less_eq @ nat @ I @ K3 ) ) ) ).
% le_trans
thf(fact_135_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq @ nat @ M @ N ) ) ).
% eq_imp_le
thf(fact_136_le__antisym,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( ord_less_eq @ nat @ N @ M )
=> ( M = N ) ) ) ).
% le_antisym
thf(fact_137_GreatestI__nat,axiom,
! [P: nat > $o,K3: nat,B2: nat] :
( ( P @ K3 )
=> ( ! [Y: nat] :
( ( P @ Y )
=> ( ord_less_eq @ nat @ Y @ B2 ) )
=> ( P @ ( order_Greatest @ nat @ P ) ) ) ) ).
% GreatestI_nat
thf(fact_138_nat__le__linear,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq @ nat @ M @ N )
| ( ord_less_eq @ nat @ N @ M ) ) ).
% nat_le_linear
thf(fact_139_Greatest__le__nat,axiom,
! [P: nat > $o,K3: nat,B2: nat] :
( ( P @ K3 )
=> ( ! [Y: nat] :
( ( P @ Y )
=> ( ord_less_eq @ nat @ Y @ B2 ) )
=> ( ord_less_eq @ nat @ K3 @ ( order_Greatest @ nat @ P ) ) ) ) ).
% Greatest_le_nat
thf(fact_140_GreatestI__ex__nat,axiom,
! [P: nat > $o,B2: nat] :
( ? [X_1: nat] : ( P @ X_1 )
=> ( ! [Y: nat] :
( ( P @ Y )
=> ( ord_less_eq @ nat @ Y @ B2 ) )
=> ( P @ ( order_Greatest @ nat @ P ) ) ) ) ).
% GreatestI_ex_nat
thf(fact_141_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K3: nat,B2: nat] :
( ( P @ K3 )
=> ( ! [Y: nat] :
( ( P @ Y )
=> ( ord_less_eq @ nat @ Y @ B2 ) )
=> ? [X3: nat] :
( ( P @ X3 )
& ! [Y5: nat] :
( ( P @ Y5 )
=> ( ord_less_eq @ nat @ Y5 @ X3 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_142_strict__mono__imp__increasing,axiom,
! [F: nat > nat,N: nat] :
( ( order_strict_mono @ nat @ nat @ F )
=> ( ord_less_eq @ nat @ N @ ( F @ N ) ) ) ).
% strict_mono_imp_increasing
thf(fact_143_bot__nat__0_Oextremum__uniqueI,axiom,
! [A2: nat] :
( ( ord_less_eq @ nat @ A2 @ ( zero_zero @ nat ) )
=> ( A2
= ( zero_zero @ nat ) ) ) ).
% bot_nat_0.extremum_uniqueI
thf(fact_144_bot__nat__0_Oextremum__unique,axiom,
! [A2: nat] :
( ( ord_less_eq @ nat @ A2 @ ( zero_zero @ nat ) )
= ( A2
= ( zero_zero @ nat ) ) ) ).
% bot_nat_0.extremum_unique
thf(fact_145_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq @ nat @ N @ ( zero_zero @ nat ) )
= ( N
= ( zero_zero @ nat ) ) ) ).
% le_0_eq
thf(fact_146_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ N ) ).
% less_eq_nat.simps(1)
thf(fact_147_compl__le__compl__iff,axiom,
! [A: $tType] :
( ( boolean_algebra @ A )
=> ! [X2: A,Y3: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ X2 ) @ ( uminus_uminus @ A @ Y3 ) )
= ( ord_less_eq @ A @ Y3 @ X2 ) ) ) ).
% compl_le_compl_iff
thf(fact_148_the__enat_Osimps,axiom,
! [N: nat] :
( ( extended_the_enat @ ( extended_enat2 @ N ) )
= N ) ).
% the_enat.simps
thf(fact_149_uminus__apply,axiom,
! [B: $tType,A: $tType] :
( ( uminus @ B )
=> ( ( uminus_uminus @ ( A > B ) )
= ( ^ [A5: A > B,X4: A] : ( uminus_uminus @ B @ ( A5 @ X4 ) ) ) ) ) ).
% uminus_apply
thf(fact_150_double__compl,axiom,
! [A: $tType] :
( ( boolean_algebra @ A )
=> ! [X2: A] :
( ( uminus_uminus @ A @ ( uminus_uminus @ A @ X2 ) )
= X2 ) ) ).
% double_compl
thf(fact_151_compl__eq__compl__iff,axiom,
! [A: $tType] :
( ( boolean_algebra @ A )
=> ! [X2: A,Y3: A] :
( ( ( uminus_uminus @ A @ X2 )
= ( uminus_uminus @ A @ Y3 ) )
= ( X2 = Y3 ) ) ) ).
% compl_eq_compl_iff
thf(fact_152_enat_Osimps_I4_J,axiom,
! [T: $tType,F1: nat > T,F22: T,Nat: nat] :
( ( extended_case_enat @ T @ F1 @ F22 @ ( extended_enat2 @ Nat ) )
= ( F1 @ Nat ) ) ).
% enat.simps(4)
thf(fact_153_ereal__uminus__uminus,axiom,
! [A2: extended_ereal] :
( ( uminus_uminus @ extended_ereal @ ( uminus_uminus @ extended_ereal @ A2 ) )
= A2 ) ).
% ereal_uminus_uminus
thf(fact_154_ereal__uminus__eq__iff,axiom,
! [A2: extended_ereal,B2: extended_ereal] :
( ( ( uminus_uminus @ extended_ereal @ A2 )
= ( uminus_uminus @ extended_ereal @ B2 ) )
= ( A2 = B2 ) ) ).
% ereal_uminus_eq_iff
thf(fact_155_ereal__uminus__eq__reorder,axiom,
! [A2: extended_ereal,B2: extended_ereal] :
( ( ( uminus_uminus @ extended_ereal @ A2 )
= B2 )
= ( A2
= ( uminus_uminus @ extended_ereal @ B2 ) ) ) ).
% ereal_uminus_eq_reorder
thf(fact_156_fun__Compl__def,axiom,
! [B: $tType,A: $tType] :
( ( uminus @ B )
=> ( ( uminus_uminus @ ( A > B ) )
= ( ^ [A5: A > B,X4: A] : ( uminus_uminus @ B @ ( A5 @ X4 ) ) ) ) ) ).
% fun_Compl_def
thf(fact_157_compl__le__swap2,axiom,
! [A: $tType] :
( ( boolean_algebra @ A )
=> ! [Y3: A,X2: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ Y3 ) @ X2 )
=> ( ord_less_eq @ A @ ( uminus_uminus @ A @ X2 ) @ Y3 ) ) ) ).
% compl_le_swap2
thf(fact_158_compl__le__swap1,axiom,
! [A: $tType] :
( ( boolean_algebra @ A )
=> ! [Y3: A,X2: A] :
( ( ord_less_eq @ A @ Y3 @ ( uminus_uminus @ A @ X2 ) )
=> ( ord_less_eq @ A @ X2 @ ( uminus_uminus @ A @ Y3 ) ) ) ) ).
% compl_le_swap1
thf(fact_159_compl__mono,axiom,
! [A: $tType] :
( ( boolean_algebra @ A )
=> ! [X2: A,Y3: A] :
( ( ord_less_eq @ A @ X2 @ Y3 )
=> ( ord_less_eq @ A @ ( uminus_uminus @ A @ Y3 ) @ ( uminus_uminus @ A @ X2 ) ) ) ) ).
% compl_mono
thf(fact_160_real__eq__0__iff__le__ge__0,axiom,
! [X2: real] :
( ( X2
= ( zero_zero @ real ) )
= ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X2 )
& ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( uminus_uminus @ real @ X2 ) ) ) ) ).
% real_eq_0_iff_le_ge_0
thf(fact_161_enat_Osimps_I6_J,axiom,
! [T: $tType,F1: nat > T,F22: T,Nat: nat] :
( ( extended_rec_enat @ T @ F1 @ F22 @ ( extended_enat2 @ Nat ) )
= ( F1 @ Nat ) ) ).
% enat.simps(6)
thf(fact_162_dbl__simps_I1_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [K3: num] :
( ( neg_numeral_dbl @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ K3 ) ) )
= ( uminus_uminus @ A @ ( neg_numeral_dbl @ A @ ( numeral_numeral @ A @ K3 ) ) ) ) ) ).
% dbl_simps(1)
thf(fact_163_dbl__simps_I2_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ( ( neg_numeral_dbl @ A @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% dbl_simps(2)
thf(fact_164_ereal__less__eq_I5_J,axiom,
! [R: real] :
( ( ord_less_eq @ extended_ereal @ ( zero_zero @ extended_ereal ) @ ( extended_ereal2 @ R ) )
= ( ord_less_eq @ real @ ( zero_zero @ real ) @ R ) ) ).
% ereal_less_eq(5)
thf(fact_165_ereal__less__eq_I4_J,axiom,
! [R: real] :
( ( ord_less_eq @ extended_ereal @ ( extended_ereal2 @ R ) @ ( zero_zero @ extended_ereal ) )
= ( ord_less_eq @ real @ R @ ( zero_zero @ real ) ) ) ).
% ereal_less_eq(4)
thf(fact_166_MInfty__eq__minfinity,axiom,
( extended_MInfty
= ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) ).
% MInfty_eq_minfinity
thf(fact_167_ereal__cong,axiom,
! [X2: real,Y3: real] :
( ( X2 = Y3 )
=> ( ( extended_ereal2 @ X2 )
= ( extended_ereal2 @ Y3 ) ) ) ).
% ereal_cong
thf(fact_168_ereal_Oinject,axiom,
! [X1: real,Y1: real] :
( ( ( extended_ereal2 @ X1 )
= ( extended_ereal2 @ Y1 ) )
= ( X1 = Y1 ) ) ).
% ereal.inject
thf(fact_169_numeral__eq__ereal,axiom,
( ( numeral_numeral @ extended_ereal )
= ( ^ [W: num] : ( extended_ereal2 @ ( numeral_numeral @ real @ W ) ) ) ) ).
% numeral_eq_ereal
thf(fact_170_ereal__eq__0_I1_J,axiom,
! [R: real] :
( ( ( extended_ereal2 @ R )
= ( zero_zero @ extended_ereal ) )
= ( R
= ( zero_zero @ real ) ) ) ).
% ereal_eq_0(1)
thf(fact_171_ereal__eq__0_I2_J,axiom,
! [R: real] :
( ( ( zero_zero @ extended_ereal )
= ( extended_ereal2 @ R ) )
= ( R
= ( zero_zero @ real ) ) ) ).
% ereal_eq_0(2)
thf(fact_172_ereal__less__eq_I3_J,axiom,
! [R: real,P2: real] :
( ( ord_less_eq @ extended_ereal @ ( extended_ereal2 @ R ) @ ( extended_ereal2 @ P2 ) )
= ( ord_less_eq @ real @ R @ P2 ) ) ).
% ereal_less_eq(3)
thf(fact_173_ereal_Odistinct_I3_J,axiom,
! [X1: real] :
( ( extended_ereal2 @ X1 )
!= extended_MInfty ) ).
% ereal.distinct(3)
thf(fact_174_real__of__ereal_Osimps_I1_J,axiom,
! [R: real] :
( ( extend1716541707_ereal @ ( extended_ereal2 @ R ) )
= R ) ).
% real_of_ereal.simps(1)
thf(fact_175_ereal__le__real,axiom,
! [X2: extended_ereal,Y3: extended_ereal] :
( ! [Z4: real] :
( ( ord_less_eq @ extended_ereal @ X2 @ ( extended_ereal2 @ Z4 ) )
=> ( ord_less_eq @ extended_ereal @ Y3 @ ( extended_ereal2 @ Z4 ) ) )
=> ( ord_less_eq @ extended_ereal @ Y3 @ X2 ) ) ).
% ereal_le_real
thf(fact_176_PInfty__neq__ereal_I1_J,axiom,
! [R: real] :
( ( extended_ereal2 @ R )
!= ( extend1396239628finity @ extended_ereal ) ) ).
% PInfty_neq_ereal(1)
thf(fact_177_zero__ereal__def,axiom,
( ( zero_zero @ extended_ereal )
= ( extended_ereal2 @ ( zero_zero @ real ) ) ) ).
% zero_ereal_def
thf(fact_178_le__ereal__le,axiom,
! [A2: extended_ereal,X2: real,Y3: real] :
( ( ord_less_eq @ extended_ereal @ A2 @ ( extended_ereal2 @ X2 ) )
=> ( ( ord_less_eq @ real @ X2 @ Y3 )
=> ( ord_less_eq @ extended_ereal @ A2 @ ( extended_ereal2 @ Y3 ) ) ) ) ).
% le_ereal_le
thf(fact_179_ereal__le__le,axiom,
! [Y3: real,A2: extended_ereal,X2: real] :
( ( ord_less_eq @ extended_ereal @ ( extended_ereal2 @ Y3 ) @ A2 )
=> ( ( ord_less_eq @ real @ X2 @ Y3 )
=> ( ord_less_eq @ extended_ereal @ ( extended_ereal2 @ X2 ) @ A2 ) ) ) ).
% ereal_le_le
thf(fact_180_real__of__ereal_Oinduct,axiom,
! [P: extended_ereal > $o,A0: extended_ereal] :
( ! [R2: real] : ( P @ ( extended_ereal2 @ R2 ) )
=> ( ( P @ ( extend1396239628finity @ extended_ereal ) )
=> ( ( P @ ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
=> ( P @ A0 ) ) ) ) ).
% real_of_ereal.induct
thf(fact_181_real__of__ereal_Ocases,axiom,
! [X2: extended_ereal] :
( ! [R2: real] :
( X2
!= ( extended_ereal2 @ R2 ) )
=> ( ( X2
!= ( extend1396239628finity @ extended_ereal ) )
=> ( X2
= ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) ) ) ).
% real_of_ereal.cases
thf(fact_182_times__ereal_Oinduct,axiom,
! [P: extended_ereal > extended_ereal > $o,A0: extended_ereal,A1: extended_ereal] :
( ! [R2: real,P3: real] : ( P @ ( extended_ereal2 @ R2 ) @ ( extended_ereal2 @ P3 ) )
=> ( ! [R2: real] : ( P @ ( extended_ereal2 @ R2 ) @ ( extend1396239628finity @ extended_ereal ) )
=> ( ! [R2: real] : ( P @ ( extend1396239628finity @ extended_ereal ) @ ( extended_ereal2 @ R2 ) )
=> ( ! [R2: real] : ( P @ ( extended_ereal2 @ R2 ) @ ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
=> ( ! [R2: real] : ( P @ ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) @ ( extended_ereal2 @ R2 ) )
=> ( ( P @ ( extend1396239628finity @ extended_ereal ) @ ( extend1396239628finity @ extended_ereal ) )
=> ( ( P @ ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) @ ( extend1396239628finity @ extended_ereal ) )
=> ( ( P @ ( extend1396239628finity @ extended_ereal ) @ ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
=> ( ( P @ ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) @ ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
=> ( P @ A0 @ A1 ) ) ) ) ) ) ) ) ) ) ).
% times_ereal.induct
thf(fact_183_plus__ereal_Oinduct,axiom,
! [P: extended_ereal > extended_ereal > $o,A0: extended_ereal,A1: extended_ereal] :
( ! [R2: real,P3: real] : ( P @ ( extended_ereal2 @ R2 ) @ ( extended_ereal2 @ P3 ) )
=> ( ! [X_12: extended_ereal] : ( P @ ( extend1396239628finity @ extended_ereal ) @ X_12 )
=> ( ! [A4: extended_ereal] : ( P @ A4 @ ( extend1396239628finity @ extended_ereal ) )
=> ( ! [R2: real] : ( P @ ( extended_ereal2 @ R2 ) @ ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
=> ( ! [P3: real] : ( P @ ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) @ ( extended_ereal2 @ P3 ) )
=> ( ( P @ ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) @ ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
=> ( P @ A0 @ A1 ) ) ) ) ) ) ) ).
% plus_ereal.induct
thf(fact_184_less__ereal_Oinduct,axiom,
! [P: extended_ereal > extended_ereal > $o,A0: extended_ereal,A1: extended_ereal] :
( ! [X3: real,Y: real] : ( P @ ( extended_ereal2 @ X3 ) @ ( extended_ereal2 @ Y ) )
=> ( ! [X_12: extended_ereal] : ( P @ ( extend1396239628finity @ extended_ereal ) @ X_12 )
=> ( ! [A4: extended_ereal] : ( P @ A4 @ ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
=> ( ! [X3: real] : ( P @ ( extended_ereal2 @ X3 ) @ ( extend1396239628finity @ extended_ereal ) )
=> ( ! [R2: real] : ( P @ ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) @ ( extended_ereal2 @ R2 ) )
=> ( ( P @ ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) @ ( extend1396239628finity @ extended_ereal ) )
=> ( P @ A0 @ A1 ) ) ) ) ) ) ) ).
% less_ereal.induct
thf(fact_185_abs__ereal_Oinduct,axiom,
! [P: extended_ereal > $o,A0: extended_ereal] :
( ! [R2: real] : ( P @ ( extended_ereal2 @ R2 ) )
=> ( ( P @ ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
=> ( ( P @ ( extend1396239628finity @ extended_ereal ) )
=> ( P @ A0 ) ) ) ) ).
% abs_ereal.induct
thf(fact_186_ereal__all__split,axiom,
( ( ^ [P4: extended_ereal > $o] :
! [X6: extended_ereal] : ( P4 @ X6 ) )
= ( ^ [P5: extended_ereal > $o] :
( ( P5 @ ( extend1396239628finity @ extended_ereal ) )
& ! [X4: real] : ( P5 @ ( extended_ereal2 @ X4 ) )
& ( P5 @ ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) ) ) ) ).
% ereal_all_split
thf(fact_187_abs__ereal_Ocases,axiom,
! [X2: extended_ereal] :
( ! [R2: real] :
( X2
!= ( extended_ereal2 @ R2 ) )
=> ( ( X2
!= ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
=> ( X2
= ( extend1396239628finity @ extended_ereal ) ) ) ) ).
% abs_ereal.cases
thf(fact_188_ereal__ex__split,axiom,
( ( ^ [P4: extended_ereal > $o] :
? [X6: extended_ereal] : ( P4 @ X6 ) )
= ( ^ [P5: extended_ereal > $o] :
( ( P5 @ ( extend1396239628finity @ extended_ereal ) )
| ? [X4: real] : ( P5 @ ( extended_ereal2 @ X4 ) )
| ( P5 @ ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) ) ) ) ).
% ereal_ex_split
thf(fact_189_ereal3__cases,axiom,
! [X2: extended_ereal,Xa3: extended_ereal,Xb: extended_ereal] :
( ( ? [R2: real] :
( X2
= ( extended_ereal2 @ R2 ) )
=> ( ? [Ra: real] :
( Xa3
= ( extended_ereal2 @ Ra ) )
=> ! [Rb: real] :
( Xb
!= ( extended_ereal2 @ Rb ) ) ) )
=> ( ( ? [R2: real] :
( X2
= ( extended_ereal2 @ R2 ) )
=> ( ? [Ra: real] :
( Xa3
= ( extended_ereal2 @ Ra ) )
=> ( Xb
!= ( extend1396239628finity @ extended_ereal ) ) ) )
=> ( ( ? [R2: real] :
( X2
= ( extended_ereal2 @ R2 ) )
=> ( ? [Ra: real] :
( Xa3
= ( extended_ereal2 @ Ra ) )
=> ( Xb
!= ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) ) )
=> ( ( ? [R2: real] :
( X2
= ( extended_ereal2 @ R2 ) )
=> ( ( Xa3
= ( extend1396239628finity @ extended_ereal ) )
=> ! [Ra: real] :
( Xb
!= ( extended_ereal2 @ Ra ) ) ) )
=> ( ( ? [R2: real] :
( X2
= ( extended_ereal2 @ R2 ) )
=> ( ( Xa3
= ( extend1396239628finity @ extended_ereal ) )
=> ( Xb
!= ( extend1396239628finity @ extended_ereal ) ) ) )
=> ( ( ? [R2: real] :
( X2
= ( extended_ereal2 @ R2 ) )
=> ( ( Xa3
= ( extend1396239628finity @ extended_ereal ) )
=> ( Xb
!= ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) ) )
=> ( ( ? [R2: real] :
( X2
= ( extended_ereal2 @ R2 ) )
=> ( ( Xa3
= ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
=> ! [Ra: real] :
( Xb
!= ( extended_ereal2 @ Ra ) ) ) )
=> ( ( ? [R2: real] :
( X2
= ( extended_ereal2 @ R2 ) )
=> ( ( Xa3
= ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
=> ( Xb
!= ( extend1396239628finity @ extended_ereal ) ) ) )
=> ( ( ? [R2: real] :
( X2
= ( extended_ereal2 @ R2 ) )
=> ( ( Xa3
= ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
=> ( Xb
!= ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) ) )
=> ( ( ( X2
= ( extend1396239628finity @ extended_ereal ) )
=> ( ? [R2: real] :
( Xa3
= ( extended_ereal2 @ R2 ) )
=> ! [Ra: real] :
( Xb
!= ( extended_ereal2 @ Ra ) ) ) )
=> ( ( ( X2
= ( extend1396239628finity @ extended_ereal ) )
=> ( ? [R2: real] :
( Xa3
= ( extended_ereal2 @ R2 ) )
=> ( Xb
!= ( extend1396239628finity @ extended_ereal ) ) ) )
=> ( ( ( X2
= ( extend1396239628finity @ extended_ereal ) )
=> ( ? [R2: real] :
( Xa3
= ( extended_ereal2 @ R2 ) )
=> ( Xb
!= ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) ) )
=> ( ( ( X2
= ( extend1396239628finity @ extended_ereal ) )
=> ( ( Xa3
= ( extend1396239628finity @ extended_ereal ) )
=> ! [R2: real] :
( Xb
!= ( extended_ereal2 @ R2 ) ) ) )
=> ( ( ( X2
= ( extend1396239628finity @ extended_ereal ) )
=> ( ( Xa3
= ( extend1396239628finity @ extended_ereal ) )
=> ( Xb
!= ( extend1396239628finity @ extended_ereal ) ) ) )
=> ( ( ( X2
= ( extend1396239628finity @ extended_ereal ) )
=> ( ( Xa3
= ( extend1396239628finity @ extended_ereal ) )
=> ( Xb
!= ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) ) )
=> ( ( ( X2
= ( extend1396239628finity @ extended_ereal ) )
=> ( ( Xa3
= ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
=> ! [R2: real] :
( Xb
!= ( extended_ereal2 @ R2 ) ) ) )
=> ( ( ( X2
= ( extend1396239628finity @ extended_ereal ) )
=> ( ( Xa3
= ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
=> ( Xb
!= ( extend1396239628finity @ extended_ereal ) ) ) )
=> ( ( ( X2
= ( extend1396239628finity @ extended_ereal ) )
=> ( ( Xa3
= ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
=> ( Xb
!= ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) ) )
=> ( ( ( X2
= ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
=> ( ? [R2: real] :
( Xa3
= ( extended_ereal2 @ R2 ) )
=> ! [Ra: real] :
( Xb
!= ( extended_ereal2 @ Ra ) ) ) )
=> ( ( ( X2
= ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
=> ( ? [R2: real] :
( Xa3
= ( extended_ereal2 @ R2 ) )
=> ( Xb
!= ( extend1396239628finity @ extended_ereal ) ) ) )
=> ( ( ( X2
= ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
=> ( ? [R2: real] :
( Xa3
= ( extended_ereal2 @ R2 ) )
=> ( Xb
!= ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) ) )
=> ( ( ( X2
= ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
=> ( ( Xa3
= ( extend1396239628finity @ extended_ereal ) )
=> ! [R2: real] :
( Xb
!= ( extended_ereal2 @ R2 ) ) ) )
=> ( ( ( X2
= ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
=> ( ( Xa3
= ( extend1396239628finity @ extended_ereal ) )
=> ( Xb
!= ( extend1396239628finity @ extended_ereal ) ) ) )
=> ( ( ( X2
= ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
=> ( ( Xa3
= ( extend1396239628finity @ extended_ereal ) )
=> ( Xb
!= ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) ) )
=> ( ( ( X2
= ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
=> ( ( Xa3
= ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
=> ! [R2: real] :
( Xb
!= ( extended_ereal2 @ R2 ) ) ) )
=> ( ( ( X2
= ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
=> ( ( Xa3
= ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
=> ( Xb
!= ( extend1396239628finity @ extended_ereal ) ) ) )
=> ~ ( ( X2
= ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
=> ( ( Xa3
= ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
=> ( Xb
!= ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% ereal3_cases
thf(fact_190_ereal2__cases,axiom,
! [X2: extended_ereal,Xa3: extended_ereal] :
( ( ? [R2: real] :
( X2
= ( extended_ereal2 @ R2 ) )
=> ! [Ra: real] :
( Xa3
!= ( extended_ereal2 @ Ra ) ) )
=> ( ( ? [R2: real] :
( X2
= ( extended_ereal2 @ R2 ) )
=> ( Xa3
!= ( extend1396239628finity @ extended_ereal ) ) )
=> ( ( ? [R2: real] :
( X2
= ( extended_ereal2 @ R2 ) )
=> ( Xa3
!= ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) )
=> ( ( ( X2
= ( extend1396239628finity @ extended_ereal ) )
=> ! [R2: real] :
( Xa3
!= ( extended_ereal2 @ R2 ) ) )
=> ( ( ( X2
= ( extend1396239628finity @ extended_ereal ) )
=> ( Xa3
!= ( extend1396239628finity @ extended_ereal ) ) )
=> ( ( ( X2
= ( extend1396239628finity @ extended_ereal ) )
=> ( Xa3
!= ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) )
=> ( ( ( X2
= ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
=> ! [R2: real] :
( Xa3
!= ( extended_ereal2 @ R2 ) ) )
=> ( ( ( X2
= ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
=> ( Xa3
!= ( extend1396239628finity @ extended_ereal ) ) )
=> ~ ( ( X2
= ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
=> ( Xa3
!= ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) ) ) ) ) ) ) ) ) ) ).
% ereal2_cases
thf(fact_191_ereal__cases,axiom,
! [X2: extended_ereal] :
( ! [R2: real] :
( X2
!= ( extended_ereal2 @ R2 ) )
=> ( ( X2
!= ( extend1396239628finity @ extended_ereal ) )
=> ( X2
= ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) ) ) ).
% ereal_cases
thf(fact_192_MInfty__neq__ereal_I1_J,axiom,
! [R: real] :
( ( extended_ereal2 @ R )
!= ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) ).
% MInfty_neq_ereal(1)
thf(fact_193_ereal__top,axiom,
! [X2: extended_ereal] :
( ! [B5: real] : ( ord_less_eq @ extended_ereal @ ( extended_ereal2 @ B5 ) @ X2 )
=> ( X2
= ( extend1396239628finity @ extended_ereal ) ) ) ).
% ereal_top
thf(fact_194_uminus__ereal_Osimps_I1_J,axiom,
! [R: real] :
( ( uminus_uminus @ extended_ereal @ ( extended_ereal2 @ R ) )
= ( extended_ereal2 @ ( uminus_uminus @ real @ R ) ) ) ).
% uminus_ereal.simps(1)
thf(fact_195_ereal__bot,axiom,
! [X2: extended_ereal] :
( ! [B5: real] : ( ord_less_eq @ extended_ereal @ X2 @ ( extended_ereal2 @ B5 ) )
=> ( X2
= ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) ) ).
% ereal_bot
thf(fact_196_real__of__ereal_Oelims,axiom,
! [X2: extended_ereal,Y3: real] :
( ( ( extend1716541707_ereal @ X2 )
= Y3 )
=> ( ! [R2: real] :
( ( X2
= ( extended_ereal2 @ R2 ) )
=> ( Y3 != R2 ) )
=> ( ( ( X2
= ( extend1396239628finity @ extended_ereal ) )
=> ( Y3
!= ( zero_zero @ real ) ) )
=> ~ ( ( X2
= ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
=> ( Y3
!= ( zero_zero @ real ) ) ) ) ) ) ).
% real_of_ereal.elims
thf(fact_197_ereal_Osize__gen_I3_J,axiom,
( ( extended_size_ereal @ extended_MInfty )
= ( zero_zero @ nat ) ) ).
% ereal.size_gen(3)
thf(fact_198_ereal_Osize__gen_I1_J,axiom,
! [X1: real] :
( ( extended_size_ereal @ ( extended_ereal2 @ X1 ) )
= ( zero_zero @ nat ) ) ).
% ereal.size_gen(1)
thf(fact_199_uminus__ereal_Oelims,axiom,
! [X2: extended_ereal,Y3: extended_ereal] :
( ( ( uminus_uminus @ extended_ereal @ X2 )
= Y3 )
=> ( ! [R2: real] :
( ( X2
= ( extended_ereal2 @ R2 ) )
=> ( Y3
!= ( extended_ereal2 @ ( uminus_uminus @ real @ R2 ) ) ) )
=> ( ( ( X2 = extended_PInfty )
=> ( Y3 != extended_MInfty ) )
=> ~ ( ( X2 = extended_MInfty )
=> ( Y3 != extended_PInfty ) ) ) ) ) ).
% uminus_ereal.elims
thf(fact_200_ereal_Odistinct_I1_J,axiom,
! [X1: real] :
( ( extended_ereal2 @ X1 )
!= extended_PInfty ) ).
% ereal.distinct(1)
thf(fact_201_ereal_Osize__gen_I2_J,axiom,
( ( extended_size_ereal @ extended_PInfty )
= ( zero_zero @ nat ) ) ).
% ereal.size_gen(2)
thf(fact_202_infinity__ereal__def,axiom,
( ( extend1396239628finity @ extended_ereal )
= extended_PInfty ) ).
% infinity_ereal_def
thf(fact_203_ereal_Odistinct_I5_J,axiom,
extended_PInfty != extended_MInfty ).
% ereal.distinct(5)
thf(fact_204_uminus__ereal_Oinduct,axiom,
! [P: extended_ereal > $o,A0: extended_ereal] :
( ! [R2: real] : ( P @ ( extended_ereal2 @ R2 ) )
=> ( ( P @ extended_PInfty )
=> ( ( P @ extended_MInfty )
=> ( P @ A0 ) ) ) ) ).
% uminus_ereal.induct
thf(fact_205_uminus__ereal_Ocases,axiom,
! [X2: extended_ereal] :
( ! [R2: real] :
( X2
!= ( extended_ereal2 @ R2 ) )
=> ( ( X2 != extended_PInfty )
=> ( X2 = extended_MInfty ) ) ) ).
% uminus_ereal.cases
thf(fact_206_ereal_Oexhaust,axiom,
! [Y3: extended_ereal] :
( ! [X12: real] :
( Y3
!= ( extended_ereal2 @ X12 ) )
=> ( ( Y3 != extended_PInfty )
=> ( Y3 = extended_MInfty ) ) ) ).
% ereal.exhaust
thf(fact_207_ereal_Oinduct,axiom,
! [P: extended_ereal > $o,Ereal: extended_ereal] :
( ! [X3: real] : ( P @ ( extended_ereal2 @ X3 ) )
=> ( ( P @ extended_PInfty )
=> ( ( P @ extended_MInfty )
=> ( P @ Ereal ) ) ) ) ).
% ereal.induct
thf(fact_208_uminus__ereal_Osimps_I2_J,axiom,
( ( uminus_uminus @ extended_ereal @ extended_PInfty )
= extended_MInfty ) ).
% uminus_ereal.simps(2)
thf(fact_209_uminus__ereal_Osimps_I3_J,axiom,
( ( uminus_uminus @ extended_ereal @ extended_MInfty )
= extended_PInfty ) ).
% uminus_ereal.simps(3)
thf(fact_210_ereal__divide__ereal,axiom,
! [R: real] :
( ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ R )
=> ( ( divide_divide @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) @ ( extended_ereal2 @ R ) )
= ( extend1396239628finity @ extended_ereal ) ) )
& ( ~ ( ord_less_eq @ real @ ( zero_zero @ real ) @ R )
=> ( ( divide_divide @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) @ ( extended_ereal2 @ R ) )
= ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) ) ) ).
% ereal_divide_ereal
thf(fact_211_ereal__le__real__iff,axiom,
! [X2: real,Y3: extended_ereal] :
( ( ord_less_eq @ real @ X2 @ ( extend1716541707_ereal @ Y3 ) )
= ( ( ( ( abs_abs @ extended_ereal @ Y3 )
!= ( extend1396239628finity @ extended_ereal ) )
=> ( ord_less_eq @ extended_ereal @ ( extended_ereal2 @ X2 ) @ Y3 ) )
& ( ( ( abs_abs @ extended_ereal @ Y3 )
= ( extend1396239628finity @ extended_ereal ) )
=> ( ord_less_eq @ real @ X2 @ ( zero_zero @ real ) ) ) ) ) ).
% ereal_le_real_iff
thf(fact_212_abs__idempotent,axiom,
! [A: $tType] :
( ( ordere142940540dd_abs @ A )
=> ! [A2: A] :
( ( abs_abs @ A @ ( abs_abs @ A @ A2 ) )
= ( abs_abs @ A @ A2 ) ) ) ).
% abs_idempotent
thf(fact_213_abs__zero,axiom,
! [A: $tType] :
( ( ordere142940540dd_abs @ A )
=> ( ( abs_abs @ A @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% abs_zero
thf(fact_214_abs__eq__0,axiom,
! [A: $tType] :
( ( ordere142940540dd_abs @ A )
=> ! [A2: A] :
( ( ( abs_abs @ A @ A2 )
= ( zero_zero @ A ) )
= ( A2
= ( zero_zero @ A ) ) ) ) ).
% abs_eq_0
thf(fact_215_abs__0__eq,axiom,
! [A: $tType] :
( ( ordere142940540dd_abs @ A )
=> ! [A2: A] :
( ( ( zero_zero @ A )
= ( abs_abs @ A @ A2 ) )
= ( A2
= ( zero_zero @ A ) ) ) ) ).
% abs_0_eq
thf(fact_216_abs__numeral,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N: num] :
( ( abs_abs @ A @ ( numeral_numeral @ A @ N ) )
= ( numeral_numeral @ A @ N ) ) ) ).
% abs_numeral
thf(fact_217_abs__minus__cancel,axiom,
! [A: $tType] :
( ( ordere142940540dd_abs @ A )
=> ! [A2: A] :
( ( abs_abs @ A @ ( uminus_uminus @ A @ A2 ) )
= ( abs_abs @ A @ A2 ) ) ) ).
% abs_minus_cancel
thf(fact_218_abs__ereal__uminus,axiom,
! [X2: extended_ereal] :
( ( abs_abs @ extended_ereal @ ( uminus_uminus @ extended_ereal @ X2 ) )
= ( abs_abs @ extended_ereal @ X2 ) ) ).
% abs_ereal_uminus
thf(fact_219_abs__ereal__zero,axiom,
( ( abs_abs @ extended_ereal @ ( zero_zero @ extended_ereal ) )
= ( zero_zero @ extended_ereal ) ) ).
% abs_ereal_zero
thf(fact_220_ereal__uminus__divide,axiom,
! [X2: extended_ereal,Y3: extended_ereal] :
( ( divide_divide @ extended_ereal @ ( uminus_uminus @ extended_ereal @ X2 ) @ Y3 )
= ( uminus_uminus @ extended_ereal @ ( divide_divide @ extended_ereal @ X2 @ Y3 ) ) ) ).
% ereal_uminus_divide
thf(fact_221_ereal__divide__zero__left,axiom,
! [A2: extended_ereal] :
( ( divide_divide @ extended_ereal @ ( zero_zero @ extended_ereal ) @ A2 )
= ( zero_zero @ extended_ereal ) ) ).
% ereal_divide_zero_left
thf(fact_222_abs__of__nonneg,axiom,
! [A: $tType] :
( ( ordere142940540dd_abs @ A )
=> ! [A2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( abs_abs @ A @ A2 )
= A2 ) ) ) ).
% abs_of_nonneg
thf(fact_223_abs__le__self__iff,axiom,
! [A: $tType] :
( ( ordere142940540dd_abs @ A )
=> ! [A2: A] :
( ( ord_less_eq @ A @ ( abs_abs @ A @ A2 ) @ A2 )
= ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% abs_le_self_iff
thf(fact_224_abs__le__zero__iff,axiom,
! [A: $tType] :
( ( ordere142940540dd_abs @ A )
=> ! [A2: A] :
( ( ord_less_eq @ A @ ( abs_abs @ A @ A2 ) @ ( zero_zero @ A ) )
= ( A2
= ( zero_zero @ A ) ) ) ) ).
% abs_le_zero_iff
thf(fact_225_abs__neg__numeral,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N: num] :
( ( abs_abs @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) )
= ( numeral_numeral @ A @ N ) ) ) ).
% abs_neg_numeral
thf(fact_226_abs__ereal__ge0,axiom,
! [X2: extended_ereal] :
( ( ord_less_eq @ extended_ereal @ ( zero_zero @ extended_ereal ) @ X2 )
=> ( ( abs_abs @ extended_ereal @ X2 )
= X2 ) ) ).
% abs_ereal_ge0
thf(fact_227_ereal__divide__Infty_I1_J,axiom,
! [X2: extended_ereal] :
( ( divide_divide @ extended_ereal @ X2 @ ( extend1396239628finity @ extended_ereal ) )
= ( zero_zero @ extended_ereal ) ) ).
% ereal_divide_Infty(1)
thf(fact_228_abs__of__nonpos,axiom,
! [A: $tType] :
( ( ordere142940540dd_abs @ A )
=> ! [A2: A] :
( ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) )
=> ( ( abs_abs @ A @ A2 )
= ( uminus_uminus @ A @ A2 ) ) ) ) ).
% abs_of_nonpos
thf(fact_229_ereal__divide__Infty_I2_J,axiom,
! [X2: extended_ereal] :
( ( divide_divide @ extended_ereal @ X2 @ ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
= ( zero_zero @ extended_ereal ) ) ).
% ereal_divide_Infty(2)
thf(fact_230_not__infty__ereal,axiom,
! [X2: extended_ereal] :
( ( ( abs_abs @ extended_ereal @ X2 )
!= ( extend1396239628finity @ extended_ereal ) )
= ( ? [X7: real] :
( X2
= ( extended_ereal2 @ X7 ) ) ) ) ).
% not_infty_ereal
thf(fact_231_abs__neq__infinity__cases,axiom,
! [X2: extended_ereal] :
( ( ( abs_abs @ extended_ereal @ X2 )
!= ( extend1396239628finity @ extended_ereal ) )
=> ~ ! [R2: real] :
( X2
!= ( extended_ereal2 @ R2 ) ) ) ).
% abs_neq_infinity_cases
thf(fact_232_abs__ereal_Osimps_I2_J,axiom,
( ( abs_abs @ extended_ereal @ ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
= ( extend1396239628finity @ extended_ereal ) ) ).
% abs_ereal.simps(2)
thf(fact_233_ereal__infinity__cases,axiom,
! [A2: extended_ereal] :
( ( A2
!= ( extend1396239628finity @ extended_ereal ) )
=> ( ( A2
!= ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) )
=> ( ( abs_abs @ extended_ereal @ A2 )
!= ( extend1396239628finity @ extended_ereal ) ) ) ) ).
% ereal_infinity_cases
thf(fact_234_abs__eq__infinity__cases,axiom,
! [X2: extended_ereal] :
( ( ( abs_abs @ extended_ereal @ X2 )
= ( extend1396239628finity @ extended_ereal ) )
=> ( ( X2
!= ( extend1396239628finity @ extended_ereal ) )
=> ( X2
= ( uminus_uminus @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) ) ) ) ) ).
% abs_eq_infinity_cases
thf(fact_235_ereal__abs__leI,axiom,
! [X2: extended_ereal,Y3: extended_ereal] :
( ( ord_less_eq @ extended_ereal @ X2 @ Y3 )
=> ( ( ord_less_eq @ extended_ereal @ ( uminus_uminus @ extended_ereal @ X2 ) @ Y3 )
=> ( ord_less_eq @ extended_ereal @ ( abs_abs @ extended_ereal @ X2 ) @ Y3 ) ) ) ).
% ereal_abs_leI
thf(fact_236_abs__ereal__pos,axiom,
! [X2: extended_ereal] : ( ord_less_eq @ extended_ereal @ ( zero_zero @ extended_ereal ) @ ( abs_abs @ extended_ereal @ X2 ) ) ).
% abs_ereal_pos
thf(fact_237_zero__le__divide__ereal,axiom,
! [A2: extended_ereal,B2: extended_ereal] :
( ( ord_less_eq @ extended_ereal @ ( zero_zero @ extended_ereal ) @ A2 )
=> ( ( ord_less_eq @ extended_ereal @ ( zero_zero @ extended_ereal ) @ B2 )
=> ( ord_less_eq @ extended_ereal @ ( zero_zero @ extended_ereal ) @ ( divide_divide @ extended_ereal @ A2 @ B2 ) ) ) ) ).
% zero_le_divide_ereal
thf(fact_238_abs__ge__zero,axiom,
! [A: $tType] :
( ( ordere142940540dd_abs @ A )
=> ! [A2: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( abs_abs @ A @ A2 ) ) ) ).
% abs_ge_zero
thf(fact_239_abs__leI,axiom,
! [A: $tType] :
( ( ordere142940540dd_abs @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ A2 ) @ B2 )
=> ( ord_less_eq @ A @ ( abs_abs @ A @ A2 ) @ B2 ) ) ) ) ).
% abs_leI
thf(fact_240_abs__le__D2,axiom,
! [A: $tType] :
( ( ordere142940540dd_abs @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ ( abs_abs @ A @ A2 ) @ B2 )
=> ( ord_less_eq @ A @ ( uminus_uminus @ A @ A2 ) @ B2 ) ) ) ).
% abs_le_D2
thf(fact_241_abs__le__iff,axiom,
! [A: $tType] :
( ( ordere142940540dd_abs @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ ( abs_abs @ A @ A2 ) @ B2 )
= ( ( ord_less_eq @ A @ A2 @ B2 )
& ( ord_less_eq @ A @ ( uminus_uminus @ A @ A2 ) @ B2 ) ) ) ) ).
% abs_le_iff
thf(fact_242_abs__ge__minus__self,axiom,
! [A: $tType] :
( ( ordere142940540dd_abs @ A )
=> ! [A2: A] : ( ord_less_eq @ A @ ( uminus_uminus @ A @ A2 ) @ ( abs_abs @ A @ A2 ) ) ) ).
% abs_ge_minus_self
thf(fact_243_ereal__divide__eq__0__iff,axiom,
! [X2: extended_ereal,Y3: extended_ereal] :
( ( ( divide_divide @ extended_ereal @ X2 @ Y3 )
= ( zero_zero @ extended_ereal ) )
= ( ( X2
= ( zero_zero @ extended_ereal ) )
| ( ( abs_abs @ extended_ereal @ Y3 )
= ( extend1396239628finity @ extended_ereal ) ) ) ) ).
% ereal_divide_eq_0_iff
thf(fact_244_abs__ereal_Osimps_I3_J,axiom,
( ( abs_abs @ extended_ereal @ ( extend1396239628finity @ extended_ereal ) )
= ( extend1396239628finity @ extended_ereal ) ) ).
% abs_ereal.simps(3)
thf(fact_245_abs__ge__self,axiom,
! [A: $tType] :
( ( ordere142940540dd_abs @ A )
=> ! [A2: A] : ( ord_less_eq @ A @ A2 @ ( abs_abs @ A @ A2 ) ) ) ).
% abs_ge_self
thf(fact_246_abs__le__D1,axiom,
! [A: $tType] :
( ( ordere142940540dd_abs @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ ( abs_abs @ A @ A2 ) @ B2 )
=> ( ord_less_eq @ A @ A2 @ B2 ) ) ) ).
% abs_le_D1
thf(fact_247_abs__minus__le__zero,axiom,
! [A: $tType] :
( ( ordere142940540dd_abs @ A )
=> ! [A2: A] : ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( abs_abs @ A @ A2 ) ) @ ( zero_zero @ A ) ) ) ).
% abs_minus_le_zero
thf(fact_248_ereal__real_H,axiom,
! [X2: extended_ereal] :
( ( ( abs_abs @ extended_ereal @ X2 )
!= ( extend1396239628finity @ extended_ereal ) )
=> ( ( extended_ereal2 @ ( extend1716541707_ereal @ X2 ) )
= X2 ) ) ).
% ereal_real'
thf(fact_249_ereal__real,axiom,
! [X2: extended_ereal] :
( ( ( ( abs_abs @ extended_ereal @ X2 )
= ( extend1396239628finity @ extended_ereal ) )
=> ( ( extended_ereal2 @ ( extend1716541707_ereal @ X2 ) )
= ( zero_zero @ extended_ereal ) ) )
& ( ( ( abs_abs @ extended_ereal @ X2 )
!= ( extend1396239628finity @ extended_ereal ) )
=> ( ( extended_ereal2 @ ( extend1716541707_ereal @ X2 ) )
= X2 ) ) ) ).
% ereal_real
thf(fact_250_real__le__ereal__iff,axiom,
! [Y3: extended_ereal,X2: real] :
( ( ord_less_eq @ real @ ( extend1716541707_ereal @ Y3 ) @ X2 )
= ( ( ( ( abs_abs @ extended_ereal @ Y3 )
!= ( extend1396239628finity @ extended_ereal ) )
=> ( ord_less_eq @ extended_ereal @ Y3 @ ( extended_ereal2 @ X2 ) ) )
& ( ( ( abs_abs @ extended_ereal @ Y3 )
= ( extend1396239628finity @ extended_ereal ) )
=> ( ord_less_eq @ real @ ( zero_zero @ real ) @ X2 ) ) ) ) ).
% real_le_ereal_iff
thf(fact_251_divide__le__0__abs__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ ( divide_divide @ A @ A2 @ ( abs_abs @ A @ B2 ) ) @ ( zero_zero @ A ) )
= ( ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) )
| ( B2
= ( zero_zero @ A ) ) ) ) ) ).
% divide_le_0_abs_iff
% Subclasses (2)
thf(subcl_Real__Vector__Spaces_Ometric__space___HOL_Otype,axiom,
! [A: $tType] :
( ( real_V2090557954_space @ A )
=> ( type @ A ) ) ).
thf(subcl_Real__Vector__Spaces_Ometric__space___Topological__Spaces_Otopological__space,axiom,
! [A: $tType] :
( ( real_V2090557954_space @ A )
=> ( topolo503727757_space @ A ) ) ).
% Type constructors (69)
thf(tcon_fun___Topological__Spaces_Otopological__space,axiom,
! [A6: $tType,A7: $tType] :
( ( topolo503727757_space @ A7 )
=> ( topolo503727757_space @ ( A6 > A7 ) ) ) ).
thf(tcon_fun___Lattices_Oboolean__algebra,axiom,
! [A6: $tType,A7: $tType] :
( ( boolean_algebra @ A7 )
=> ( boolean_algebra @ ( A6 > A7 ) ) ) ).
thf(tcon_fun___Orderings_Opreorder,axiom,
! [A6: $tType,A7: $tType] :
( ( preorder @ A7 )
=> ( preorder @ ( A6 > A7 ) ) ) ).
thf(tcon_fun___Orderings_Oorder,axiom,
! [A6: $tType,A7: $tType] :
( ( order @ A7 )
=> ( order @ ( A6 > A7 ) ) ) ).
thf(tcon_fun___Orderings_Oord,axiom,
! [A6: $tType,A7: $tType] :
( ( ord @ A7 )
=> ( ord @ ( A6 > A7 ) ) ) ).
thf(tcon_fun___Groups_Ouminus,axiom,
! [A6: $tType,A7: $tType] :
( ( uminus @ A7 )
=> ( uminus @ ( A6 > A7 ) ) ) ).
thf(tcon_Nat_Onat___Groups_Ocanonically__ordered__monoid__add,axiom,
canoni770627133id_add @ nat ).
thf(tcon_Nat_Onat___Topological__Spaces_Otopological__space_1,axiom,
topolo503727757_space @ nat ).
thf(tcon_Nat_Onat___Topological__Spaces_Oorder__topology,axiom,
topolo259154727pology @ nat ).
thf(tcon_Nat_Onat___Rings_Olinordered__nonzero__semiring,axiom,
linord1659791738miring @ nat ).
thf(tcon_Nat_Onat___Nat_Osemiring__char__0,axiom,
semiring_char_0 @ nat ).
thf(tcon_Nat_Onat___Orderings_Opreorder_2,axiom,
preorder @ nat ).
thf(tcon_Nat_Onat___Orderings_Olinorder,axiom,
linorder @ nat ).
thf(tcon_Nat_Onat___Orderings_Oorder_3,axiom,
order @ nat ).
thf(tcon_Nat_Onat___Orderings_Oord_4,axiom,
ord @ nat ).
thf(tcon_Nat_Onat___Num_Onumeral,axiom,
numeral @ nat ).
thf(tcon_Nat_Onat___Groups_Ozero,axiom,
zero @ nat ).
thf(tcon_Num_Onum___Orderings_Opreorder_5,axiom,
preorder @ num ).
thf(tcon_Num_Onum___Orderings_Olinorder_6,axiom,
linorder @ num ).
thf(tcon_Num_Onum___Orderings_Oorder_7,axiom,
order @ num ).
thf(tcon_Num_Onum___Orderings_Oord_8,axiom,
ord @ num ).
thf(tcon_HOL_Obool___Topological__Spaces_Otopological__space_9,axiom,
topolo503727757_space @ $o ).
thf(tcon_HOL_Obool___Topological__Spaces_Oorder__topology_10,axiom,
topolo259154727pology @ $o ).
thf(tcon_HOL_Obool___Lattices_Oboolean__algebra_11,axiom,
boolean_algebra @ $o ).
thf(tcon_HOL_Obool___Orderings_Opreorder_12,axiom,
preorder @ $o ).
thf(tcon_HOL_Obool___Orderings_Olinorder_13,axiom,
linorder @ $o ).
thf(tcon_HOL_Obool___Orderings_Oorder_14,axiom,
order @ $o ).
thf(tcon_HOL_Obool___Orderings_Oord_15,axiom,
ord @ $o ).
thf(tcon_HOL_Obool___Groups_Ouminus_16,axiom,
uminus @ $o ).
thf(tcon_Real_Oreal___Topological__Spaces_Otopological__space_17,axiom,
topolo503727757_space @ real ).
thf(tcon_Real_Oreal___Topological__Spaces_Oorder__topology_18,axiom,
topolo259154727pology @ real ).
thf(tcon_Real_Oreal___Rings_Olinordered__nonzero__semiring_19,axiom,
linord1659791738miring @ real ).
thf(tcon_Real_Oreal___Real__Vector__Spaces_Ometric__space,axiom,
real_V2090557954_space @ real ).
thf(tcon_Real_Oreal___Groups_Oordered__ab__group__add__abs,axiom,
ordere142940540dd_abs @ real ).
thf(tcon_Real_Oreal___Groups_Olinordered__ab__group__add,axiom,
linord219039673up_add @ real ).
thf(tcon_Real_Oreal___Groups_Oordered__ab__group__add,axiom,
ordered_ab_group_add @ real ).
thf(tcon_Real_Oreal___Fields_Olinordered__field,axiom,
linordered_field @ real ).
thf(tcon_Real_Oreal___Rings_Olinordered__idom,axiom,
linordered_idom @ real ).
thf(tcon_Real_Oreal___Nat_Osemiring__char__0_20,axiom,
semiring_char_0 @ real ).
thf(tcon_Real_Oreal___Orderings_Opreorder_21,axiom,
preorder @ real ).
thf(tcon_Real_Oreal___Orderings_Olinorder_22,axiom,
linorder @ real ).
thf(tcon_Real_Oreal___Groups_Ogroup__add,axiom,
group_add @ real ).
thf(tcon_Real_Oreal___Orderings_Oorder_23,axiom,
order @ real ).
thf(tcon_Real_Oreal___Num_Oneg__numeral,axiom,
neg_numeral @ real ).
thf(tcon_Real_Oreal___Nat_Oring__char__0,axiom,
ring_char_0 @ real ).
thf(tcon_Real_Oreal___Orderings_Oord_24,axiom,
ord @ real ).
thf(tcon_Real_Oreal___Groups_Ouminus_25,axiom,
uminus @ real ).
thf(tcon_Real_Oreal___Num_Onumeral_26,axiom,
numeral @ real ).
thf(tcon_Real_Oreal___Groups_Ozero_27,axiom,
zero @ real ).
thf(tcon_Extended__Nat_Oenat___Groups_Ocanonically__ordered__monoid__add_28,axiom,
canoni770627133id_add @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Topological__Spaces_Otopological__space_29,axiom,
topolo503727757_space @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Topological__Spaces_Oorder__topology_30,axiom,
topolo259154727pology @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Rings_Olinordered__nonzero__semiring_31,axiom,
linord1659791738miring @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Nat_Osemiring__char__0_32,axiom,
semiring_char_0 @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Orderings_Opreorder_33,axiom,
preorder @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Orderings_Olinorder_34,axiom,
linorder @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Orderings_Oorder_35,axiom,
order @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Orderings_Oord_36,axiom,
ord @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Num_Onumeral_37,axiom,
numeral @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Ozero_38,axiom,
zero @ extended_enat ).
thf(tcon_Extended__Real_Oereal___Topological__Spaces_Otopological__space_39,axiom,
topolo503727757_space @ extended_ereal ).
thf(tcon_Extended__Real_Oereal___Topological__Spaces_Oorder__topology_40,axiom,
topolo259154727pology @ extended_ereal ).
thf(tcon_Extended__Real_Oereal___Orderings_Opreorder_41,axiom,
preorder @ extended_ereal ).
thf(tcon_Extended__Real_Oereal___Orderings_Olinorder_42,axiom,
linorder @ extended_ereal ).
thf(tcon_Extended__Real_Oereal___Orderings_Oorder_43,axiom,
order @ extended_ereal ).
thf(tcon_Extended__Real_Oereal___Orderings_Oord_44,axiom,
ord @ extended_ereal ).
thf(tcon_Extended__Real_Oereal___Groups_Ouminus_45,axiom,
uminus @ extended_ereal ).
thf(tcon_Extended__Real_Oereal___Num_Onumeral_46,axiom,
numeral @ extended_ereal ).
thf(tcon_Extended__Real_Oereal___Groups_Ozero_47,axiom,
zero @ extended_ereal ).
% Free types (1)
thf(tfree_0,hypothesis,
real_V2090557954_space @ a ).
% Conjectures (1)
thf(conj_0,conjecture,
ord_less_eq @ extended_ereal @ ( g @ x ) @ ( f @ x ) ).
%------------------------------------------------------------------------------