TPTP Problem File: ITP113^2.p
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%------------------------------------------------------------------------------
% File : ITP113^2 : TPTP v8.2.0. Released v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer Lower_Semicontinuous problem prob_663__6254006_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_663__6254006_1 [Des21]
% Status : Theorem
% Rating : 0.00 v7.5.0
% Syntax : Number of formulae : 368 ( 96 unt; 48 typ; 0 def)
% Number of atoms : 951 ( 174 equ; 0 cnn)
% Maximal formula atoms : 13 ( 2 avg)
% Number of connectives : 3688 ( 109 ~; 17 |; 82 &;2973 @)
% ( 0 <=>; 507 =>; 0 <=; 0 <~>)
% Maximal formula depth : 28 ( 8 avg)
% Number of types : 5 ( 4 usr)
% Number of type conns : 132 ( 132 >; 0 *; 0 +; 0 <<)
% Number of symbols : 45 ( 44 usr; 5 con; 0-5 aty)
% Number of variables : 1118 ( 61 ^; 978 !; 39 ?;1118 :)
% ( 40 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Sledgehammer 2021-02-23 16:27:16.540
%------------------------------------------------------------------------------
% Could-be-implicit typings (6)
thf(ty_t_Extended__Real_Oereal,type,
extended_ereal: $tType ).
thf(ty_t_Product__Type_Oprod,type,
product_prod: $tType > $tType > $tType ).
thf(ty_t_Real_Oreal,type,
real: $tType ).
thf(ty_t_Set_Oset,type,
set: $tType > $tType ).
thf(ty_t_Nat_Onat,type,
nat: $tType ).
thf(ty_tf_a,type,
a: $tType ).
% Explicit typings (42)
thf(sy_cl_Euclidean__Space_Oeuclidean__space,type,
euclid925273238_space:
!>[A: $tType] : $o ).
thf(sy_cl_HOL_Otype,type,
type:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ozero,type,
zero:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Oord,type,
ord:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Otop,type,
top:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Oorder,type,
order:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Ono__bot,type,
no_bot:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Ono__top,type,
no_top:
!>[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_Orderings_Oorder__top,type,
order_top:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Owellorder,type,
wellorder:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Odense__order,type,
dense_order:
!>[A: $tType] : $o ).
thf(sy_cl_Fields_Olinordered__field,type,
linordered_field:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Odense__linorder,type,
dense_linorder:
!>[A: $tType] : $o ).
thf(sy_cl_Real__Vector__Spaces_Ometric__space,type,
real_V2090557954_space:
!>[A: $tType] : $o ).
thf(sy_cl_Topological__Spaces_Operfect__space,type,
topolo890362671_space:
!>[A: $tType] : $o ).
thf(sy_cl_Real__Vector__Spaces_Ocomplete__space,type,
real_V1759440057_space:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Olinordered__nonzero__semiring,type,
linord1659791738miring:
!>[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_cl_Real__Vector__Spaces_Oreal__normed__vector,type,
real_V55928688vector:
!>[A: $tType] : $o ).
thf(sy_cl_Conditionally__Complete__Lattices_Olinear__continuum,type,
condit1656338222tinuum:
!>[A: $tType] : $o ).
thf(sy_cl_Conditionally__Complete__Lattices_Oconditionally__complete__linorder,type,
condit1037483654norder:
!>[A: $tType] : $o ).
thf(sy_c_Elementary__Topology_Oclosure,type,
elementary_closure:
!>[A: $tType] : ( ( set @ A ) > ( set @ A ) ) ).
thf(sy_c_Groups_Ozero__class_Ozero,type,
zero_zero:
!>[A: $tType] : A ).
thf(sy_c_Lower__Semicontinuous__Mirabelle__quczrylfpw_OEpigraph,type,
lower_261390618igraph:
!>[A: $tType] : ( ( set @ A ) > ( A > extended_ereal ) > ( set @ ( product_prod @ A @ real ) ) ) ).
thf(sy_c_Orderings_Oord__class_Oless,type,
ord_less:
!>[A: $tType] : ( A > A > $o ) ).
thf(sy_c_Orderings_Oord__class_Oless__eq,type,
ord_less_eq:
!>[A: $tType] : ( A > A > $o ) ).
thf(sy_c_Orderings_Otop__class_Otop,type,
top_top:
!>[A: $tType] : A ).
thf(sy_c_Product__Type_OPair,type,
product_Pair:
!>[A: $tType,B: $tType] : ( A > B > ( product_prod @ A @ B ) ) ).
thf(sy_c_Product__Type_Oold_Oprod_Orec__prod,type,
product_rec_prod:
!>[A: $tType,B: $tType,T: $tType] : ( ( A > B > T ) > ( product_prod @ A @ B ) > T ) ).
thf(sy_c_Real__Vector__Spaces_Odist__class_Odist,type,
real_V2000881966t_dist:
!>[A: $tType] : ( A > A > real ) ).
thf(sy_c_Set_OCollect,type,
collect:
!>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
thf(sy_c_Uniform__Limit_Ouniformly__Cauchy__on,type,
unifor760693226chy_on:
!>[A: $tType,B: $tType] : ( ( set @ A ) > ( nat > A > B ) > $o ) ).
thf(sy_c_member,type,
member:
!>[A: $tType] : ( A > ( set @ A ) > $o ) ).
thf(sy_v_e____,type,
e: real ).
thf(sy_v_f,type,
f: a > extended_ereal ).
thf(sy_v_thesis____,type,
thesis: $o ).
thf(sy_v_x____,type,
x: a ).
thf(sy_v_y____,type,
y: real ).
thf(sy_v_z____,type,
z: real ).
% Relevant facts (254)
thf(fact_0__092_060open_062_092_060exists_062ya_092_060in_062Epigraph_AUNIV_Af_O_Adist_Aya_A_Ix_M_Ay_J_A_060_Ae_092_060close_062,axiom,
? [X: product_prod @ a @ real] :
( ( member @ ( product_prod @ a @ real ) @ X @ ( lower_261390618igraph @ a @ ( top_top @ ( set @ a ) ) @ f ) )
& ( ord_less @ real @ ( real_V2000881966t_dist @ ( product_prod @ a @ real ) @ X @ ( product_Pair @ a @ real @ x @ y ) ) @ e ) ) ).
% \<open>\<exists>ya\<in>Epigraph UNIV f. dist ya (x, y) < e\<close>
thf(fact_1__092_060open_0620_A_060_Ae_092_060close_062,axiom,
ord_less @ real @ ( zero_zero @ real ) @ e ).
% \<open>0 < e\<close>
thf(fact_2_xy,axiom,
( ( member @ ( product_prod @ a @ real ) @ ( product_Pair @ a @ real @ x @ y ) @ ( elementary_closure @ ( product_prod @ a @ real ) @ ( lower_261390618igraph @ a @ ( top_top @ ( set @ a ) ) @ f ) ) )
& ( ord_less_eq @ real @ y @ z ) ) ).
% xy
thf(fact_3_UNIV__I,axiom,
! [A: $tType,X2: A] : ( member @ A @ X2 @ ( top_top @ ( set @ A ) ) ) ).
% UNIV_I
thf(fact_4_iso__tuple__UNIV__I,axiom,
! [A: $tType,X2: A] : ( member @ A @ X2 @ ( top_top @ ( set @ A ) ) ) ).
% iso_tuple_UNIV_I
thf(fact_5_prod_Oinject,axiom,
! [A: $tType,B: $tType,X1: A,X22: B,Y1: A,Y2: B] :
( ( ( product_Pair @ A @ B @ X1 @ X22 )
= ( product_Pair @ A @ B @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X22 = Y2 ) ) ) ).
% prod.inject
thf(fact_6_old_Oprod_Oinject,axiom,
! [A: $tType,B: $tType,A2: A,B2: B,A3: A,B3: B] :
( ( ( product_Pair @ A @ B @ A2 @ B2 )
= ( product_Pair @ A @ B @ A3 @ B3 ) )
= ( ( A2 = A3 )
& ( B2 = B3 ) ) ) ).
% old.prod.inject
thf(fact_7_top__apply,axiom,
! [C: $tType,D: $tType] :
( ( top @ C )
=> ( ( top_top @ ( D > C ) )
= ( ^ [X3: D] : ( top_top @ C ) ) ) ) ).
% top_apply
thf(fact_8_dist__commute__lessI,axiom,
! [A: $tType] :
( ( real_V2090557954_space @ A )
=> ! [Y: A,X2: A,E: real] :
( ( ord_less @ real @ ( real_V2000881966t_dist @ A @ Y @ X2 ) @ E )
=> ( ord_less @ real @ ( real_V2000881966t_dist @ A @ X2 @ Y ) @ E ) ) ) ).
% dist_commute_lessI
thf(fact_9_top__prod__def,axiom,
! [A: $tType,B: $tType] :
( ( ( top @ B )
& ( top @ A ) )
=> ( ( top_top @ ( product_prod @ A @ B ) )
= ( product_Pair @ A @ B @ ( top_top @ A ) @ ( top_top @ B ) ) ) ) ).
% top_prod_def
thf(fact_10_top_Oextremum__strict,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ! [A2: A] :
~ ( ord_less @ A @ ( top_top @ A ) @ A2 ) ) ).
% top.extremum_strict
thf(fact_11_top_Onot__eq__extremum,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ! [A2: A] :
( ( A2
!= ( top_top @ A ) )
= ( ord_less @ A @ A2 @ ( top_top @ A ) ) ) ) ).
% top.not_eq_extremum
thf(fact_12_epigraph__mono,axiom,
! [A: $tType] :
( ( real_V2090557954_space @ A )
=> ! [X2: A,Y: real,F: A > extended_ereal,Z: real] :
( ( ( member @ ( product_prod @ A @ real ) @ ( product_Pair @ A @ real @ X2 @ Y ) @ ( lower_261390618igraph @ A @ ( top_top @ ( set @ A ) ) @ F ) )
& ( ord_less_eq @ real @ Y @ Z ) )
=> ( member @ ( product_prod @ A @ real ) @ ( product_Pair @ A @ real @ X2 @ Z ) @ ( lower_261390618igraph @ A @ ( top_top @ ( set @ A ) ) @ F ) ) ) ) ).
% epigraph_mono
thf(fact_13_metric__eq__thm,axiom,
! [A4: $tType] :
( ( real_V2090557954_space @ A4 )
=> ! [X4: A4,S: set @ A4,Y3: A4] :
( ( member @ A4 @ X4 @ S )
=> ( ( member @ A4 @ Y3 @ S )
=> ( ( X4 = Y3 )
= ( ! [X3: A4] :
( ( member @ A4 @ X3 @ S )
=> ( ( real_V2000881966t_dist @ A4 @ X4 @ X3 )
= ( real_V2000881966t_dist @ A4 @ Y3 @ X3 ) ) ) ) ) ) ) ) ).
% metric_eq_thm
thf(fact_14_order__refl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X2: A] : ( ord_less_eq @ A @ X2 @ X2 ) ) ).
% order_refl
thf(fact_15_Pair__le,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( ord @ A ) )
=> ! [A2: A,B2: B,C2: A,D2: B] :
( ( ord_less_eq @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B2 ) @ ( product_Pair @ A @ B @ C2 @ D2 ) )
= ( ( ord_less_eq @ A @ A2 @ C2 )
& ( ord_less_eq @ B @ B2 @ D2 ) ) ) ) ).
% Pair_le
thf(fact_16_dist__eq__0__iff,axiom,
! [A: $tType] :
( ( real_V2090557954_space @ A )
=> ! [X2: A,Y: A] :
( ( ( real_V2000881966t_dist @ A @ X2 @ Y )
= ( zero_zero @ real ) )
= ( X2 = Y ) ) ) ).
% dist_eq_0_iff
thf(fact_17_dist__self,axiom,
! [A: $tType] :
( ( real_V2090557954_space @ A )
=> ! [X2: A] :
( ( real_V2000881966t_dist @ A @ X2 @ X2 )
= ( zero_zero @ real ) ) ) ).
% dist_self
thf(fact_18_dist__le__zero__iff,axiom,
! [A: $tType] :
( ( real_V2090557954_space @ A )
=> ! [X2: A,Y: A] :
( ( ord_less_eq @ real @ ( real_V2000881966t_dist @ A @ X2 @ Y ) @ ( zero_zero @ real ) )
= ( X2 = Y ) ) ) ).
% dist_le_zero_iff
thf(fact_19_zero__less__dist__iff,axiom,
! [A: $tType] :
( ( real_V2090557954_space @ A )
=> ! [X2: A,Y: A] :
( ( ord_less @ real @ ( zero_zero @ real ) @ ( real_V2000881966t_dist @ A @ X2 @ Y ) )
= ( X2 != Y ) ) ) ).
% zero_less_dist_iff
thf(fact_20_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_21_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_22_le__funI,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ! [F: A > B,G: A > B] :
( ! [X: A] : ( ord_less_eq @ B @ ( F @ X ) @ ( G @ X ) )
=> ( ord_less_eq @ ( A > B ) @ F @ G ) ) ) ).
% le_funI
thf(fact_23_le__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ( ( ord_less_eq @ ( A > B ) )
= ( ^ [F2: A > B,G2: A > B] :
! [X3: A] : ( ord_less_eq @ B @ ( F2 @ X3 ) @ ( G2 @ X3 ) ) ) ) ) ).
% le_fun_def
thf(fact_24_order__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B )
& ( order @ A ) )
=> ! [A2: A,F: B > A,B2: B,C2: B] :
( ( ord_less_eq @ A @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq @ B @ B2 @ C2 )
=> ( ! [X: B,Y4: B] :
( ( ord_less_eq @ B @ X @ Y4 )
=> ( ord_less_eq @ A @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq @ A @ A2 @ ( F @ C2 ) ) ) ) ) ) ).
% order_subst1
thf(fact_25_order__subst2,axiom,
! [A: $tType,C: $tType] :
( ( ( order @ C )
& ( order @ A ) )
=> ! [A2: A,B2: A,F: A > C,C2: C] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ C @ ( F @ B2 ) @ C2 )
=> ( ! [X: A,Y4: A] :
( ( ord_less_eq @ A @ X @ Y4 )
=> ( ord_less_eq @ C @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq @ C @ ( F @ A2 ) @ C2 ) ) ) ) ) ).
% order_subst2
thf(fact_26_Pair__mono,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( ord @ A ) )
=> ! [X2: A,X5: A,Y: B,Y5: B] :
( ( ord_less_eq @ A @ X2 @ X5 )
=> ( ( ord_less_eq @ B @ Y @ Y5 )
=> ( ord_less_eq @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y ) @ ( product_Pair @ A @ B @ X5 @ Y5 ) ) ) ) ) ).
% Pair_mono
thf(fact_27_ord__eq__le__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( ord @ A ) )
=> ! [A2: A,F: B > A,B2: B,C2: B] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq @ B @ B2 @ C2 )
=> ( ! [X: B,Y4: B] :
( ( ord_less_eq @ B @ X @ Y4 )
=> ( ord_less_eq @ A @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq @ A @ A2 @ ( F @ C2 ) ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_28_ord__le__eq__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( ord @ A ) )
=> ! [A2: A,B2: A,F: A > B,C2: B] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C2 )
=> ( ! [X: A,Y4: A] :
( ( ord_less_eq @ A @ X @ Y4 )
=> ( ord_less_eq @ B @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq @ B @ ( F @ A2 ) @ C2 ) ) ) ) ) ).
% ord_le_eq_subst
thf(fact_29_eq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y6: A,Z2: A] : Y6 = Z2 )
= ( ^ [X3: A,Y7: A] :
( ( ord_less_eq @ A @ X3 @ Y7 )
& ( ord_less_eq @ A @ Y7 @ X3 ) ) ) ) ) ).
% eq_iff
thf(fact_30_antisym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X2: A,Y: A] :
( ( ord_less_eq @ A @ X2 @ Y )
=> ( ( ord_less_eq @ A @ Y @ X2 )
=> ( X2 = Y ) ) ) ) ).
% antisym
thf(fact_31_linear,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X2: A,Y: A] :
( ( ord_less_eq @ A @ X2 @ Y )
| ( ord_less_eq @ A @ Y @ X2 ) ) ) ).
% linear
thf(fact_32_eq__refl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X2: A,Y: A] :
( ( X2 = Y )
=> ( ord_less_eq @ A @ X2 @ Y ) ) ) ).
% eq_refl
thf(fact_33_le__cases,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X2: A,Y: A] :
( ~ ( ord_less_eq @ A @ X2 @ Y )
=> ( ord_less_eq @ A @ Y @ X2 ) ) ) ).
% le_cases
thf(fact_34_order_Otrans,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ B2 @ C2 )
=> ( ord_less_eq @ A @ A2 @ C2 ) ) ) ) ).
% order.trans
thf(fact_35_le__cases3,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X2: A,Y: A,Z: A] :
( ( ( ord_less_eq @ A @ X2 @ Y )
=> ~ ( ord_less_eq @ A @ Y @ Z ) )
=> ( ( ( ord_less_eq @ A @ Y @ X2 )
=> ~ ( ord_less_eq @ A @ X2 @ Z ) )
=> ( ( ( ord_less_eq @ A @ X2 @ Z )
=> ~ ( ord_less_eq @ A @ Z @ Y ) )
=> ( ( ( ord_less_eq @ A @ Z @ Y )
=> ~ ( ord_less_eq @ A @ Y @ X2 ) )
=> ( ( ( ord_less_eq @ A @ Y @ Z )
=> ~ ( ord_less_eq @ A @ Z @ X2 ) )
=> ~ ( ( ord_less_eq @ A @ Z @ X2 )
=> ~ ( ord_less_eq @ A @ X2 @ Y ) ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_36_antisym__conv,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [Y: A,X2: A] :
( ( ord_less_eq @ A @ Y @ X2 )
=> ( ( ord_less_eq @ A @ X2 @ Y )
= ( X2 = Y ) ) ) ) ).
% antisym_conv
thf(fact_37_order__class_Oorder_Oeq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y6: A,Z2: A] : Y6 = Z2 )
= ( ^ [A5: A,B4: A] :
( ( ord_less_eq @ A @ A5 @ B4 )
& ( ord_less_eq @ A @ B4 @ A5 ) ) ) ) ) ).
% order_class.order.eq_iff
thf(fact_38_ord__eq__le__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( A2 = B2 )
=> ( ( ord_less_eq @ A @ B2 @ C2 )
=> ( ord_less_eq @ A @ A2 @ C2 ) ) ) ) ).
% ord_eq_le_trans
thf(fact_39_ord__le__eq__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( B2 = C2 )
=> ( ord_less_eq @ A @ A2 @ C2 ) ) ) ) ).
% ord_le_eq_trans
thf(fact_40_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_41_order__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X2: A,Y: A,Z: A] :
( ( ord_less_eq @ A @ X2 @ Y )
=> ( ( ord_less_eq @ A @ Y @ Z )
=> ( ord_less_eq @ A @ X2 @ Z ) ) ) ) ).
% order_trans
thf(fact_42_dual__order_Orefl,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: A] : ( ord_less_eq @ A @ A2 @ A2 ) ) ).
% dual_order.refl
thf(fact_43_linorder__wlog,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > A > $o,A2: A,B2: A] :
( ! [A6: A,B5: A] :
( ( ord_less_eq @ A @ A6 @ B5 )
=> ( P @ A6 @ B5 ) )
=> ( ! [A6: A,B5: A] :
( ( P @ B5 @ A6 )
=> ( P @ A6 @ B5 ) )
=> ( P @ A2 @ B2 ) ) ) ) ).
% linorder_wlog
thf(fact_44_dual__order_Otrans,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B2: A,A2: A,C2: A] :
( ( ord_less_eq @ A @ B2 @ A2 )
=> ( ( ord_less_eq @ A @ C2 @ B2 )
=> ( ord_less_eq @ A @ C2 @ A2 ) ) ) ) ).
% dual_order.trans
thf(fact_45_mem__Collect__eq,axiom,
! [A: $tType,A2: A,P: A > $o] :
( ( member @ A @ A2 @ ( collect @ A @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_46_Collect__mem__eq,axiom,
! [A: $tType,A7: set @ A] :
( ( collect @ A
@ ^ [X3: A] : ( member @ A @ X3 @ A7 ) )
= A7 ) ).
% Collect_mem_eq
thf(fact_47_Collect__cong,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ! [X: A] :
( ( P @ X )
= ( Q @ X ) )
=> ( ( collect @ A @ P )
= ( collect @ A @ Q ) ) ) ).
% Collect_cong
thf(fact_48_ext,axiom,
! [B: $tType,A: $tType,F: A > B,G: A > B] :
( ! [X: A] :
( ( F @ X )
= ( G @ X ) )
=> ( F = G ) ) ).
% ext
thf(fact_49_dual__order_Oeq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y6: A,Z2: A] : Y6 = Z2 )
= ( ^ [A5: A,B4: A] :
( ( ord_less_eq @ A @ B4 @ A5 )
& ( ord_less_eq @ A @ A5 @ B4 ) ) ) ) ) ).
% dual_order.eq_iff
thf(fact_50_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_51_zero__le__dist,axiom,
! [A: $tType] :
( ( real_V2090557954_space @ A )
=> ! [X2: A,Y: A] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( real_V2000881966t_dist @ A @ X2 @ Y ) ) ) ).
% zero_le_dist
thf(fact_52_top__set__def,axiom,
! [A: $tType] :
( ( top_top @ ( set @ A ) )
= ( collect @ A @ ( top_top @ ( A > $o ) ) ) ) ).
% top_set_def
thf(fact_53_order_Onot__eq__order__implies__strict,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: A,B2: A] :
( ( A2 != B2 )
=> ( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ord_less @ A @ A2 @ B2 ) ) ) ) ).
% order.not_eq_order_implies_strict
thf(fact_54_dual__order_Ostrict__implies__order,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B2: A,A2: A] :
( ( ord_less @ A @ B2 @ A2 )
=> ( ord_less_eq @ A @ B2 @ A2 ) ) ) ).
% dual_order.strict_implies_order
thf(fact_55_dual__order_Ostrict__iff__order,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less @ A )
= ( ^ [B4: A,A5: A] :
( ( ord_less_eq @ A @ B4 @ A5 )
& ( A5 != B4 ) ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_56_dual__order_Oorder__iff__strict,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [B4: A,A5: A] :
( ( ord_less @ A @ B4 @ A5 )
| ( A5 = B4 ) ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_57_order_Ostrict__implies__order,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ord_less_eq @ A @ A2 @ B2 ) ) ) ).
% order.strict_implies_order
thf(fact_58_dense__le__bounded,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [X2: A,Y: A,Z: A] :
( ( ord_less @ A @ X2 @ Y )
=> ( ! [W: A] :
( ( ord_less @ A @ X2 @ W )
=> ( ( ord_less @ A @ W @ Y )
=> ( ord_less_eq @ A @ W @ Z ) ) )
=> ( ord_less_eq @ A @ Y @ Z ) ) ) ) ).
% dense_le_bounded
thf(fact_59_dense__ge__bounded,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [Z: A,X2: A,Y: A] :
( ( ord_less @ A @ Z @ X2 )
=> ( ! [W: A] :
( ( ord_less @ A @ Z @ W )
=> ( ( ord_less @ A @ W @ X2 )
=> ( ord_less_eq @ A @ Y @ W ) ) )
=> ( ord_less_eq @ A @ Y @ Z ) ) ) ) ).
% dense_ge_bounded
thf(fact_60_dual__order_Ostrict__trans2,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B2: A,A2: A,C2: A] :
( ( ord_less @ A @ B2 @ A2 )
=> ( ( ord_less_eq @ A @ C2 @ B2 )
=> ( ord_less @ A @ C2 @ A2 ) ) ) ) ).
% dual_order.strict_trans2
thf(fact_61_dual__order_Ostrict__trans1,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B2: A,A2: A,C2: A] :
( ( ord_less_eq @ A @ B2 @ A2 )
=> ( ( ord_less @ A @ C2 @ B2 )
=> ( ord_less @ A @ C2 @ A2 ) ) ) ) ).
% dual_order.strict_trans1
thf(fact_62_order_Ostrict__iff__order,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less @ A )
= ( ^ [A5: A,B4: A] :
( ( ord_less_eq @ A @ A5 @ B4 )
& ( A5 != B4 ) ) ) ) ) ).
% order.strict_iff_order
thf(fact_63_order_Oorder__iff__strict,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A5: A,B4: A] :
( ( ord_less @ A @ A5 @ B4 )
| ( A5 = B4 ) ) ) ) ) ).
% order.order_iff_strict
thf(fact_64_order_Ostrict__trans2,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ B2 @ C2 )
=> ( ord_less @ A @ A2 @ C2 ) ) ) ) ).
% order.strict_trans2
thf(fact_65_order_Ostrict__trans1,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less @ A @ B2 @ C2 )
=> ( ord_less @ A @ A2 @ C2 ) ) ) ) ).
% order.strict_trans1
thf(fact_66_not__le__imp__less,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Y: A,X2: A] :
( ~ ( ord_less_eq @ A @ Y @ X2 )
=> ( ord_less @ A @ X2 @ Y ) ) ) ).
% not_le_imp_less
thf(fact_67_less__le__not__le,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( ( ord_less @ A )
= ( ^ [X3: A,Y7: A] :
( ( ord_less_eq @ A @ X3 @ Y7 )
& ~ ( ord_less_eq @ A @ Y7 @ X3 ) ) ) ) ) ).
% less_le_not_le
thf(fact_68_le__imp__less__or__eq,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X2: A,Y: A] :
( ( ord_less_eq @ A @ X2 @ Y )
=> ( ( ord_less @ A @ X2 @ Y )
| ( X2 = Y ) ) ) ) ).
% le_imp_less_or_eq
thf(fact_69_le__less__linear,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X2: A,Y: A] :
( ( ord_less_eq @ A @ X2 @ Y )
| ( ord_less @ A @ Y @ X2 ) ) ) ).
% le_less_linear
thf(fact_70_dense__le,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [Y: A,Z: A] :
( ! [X: A] :
( ( ord_less @ A @ X @ Y )
=> ( ord_less_eq @ A @ X @ Z ) )
=> ( ord_less_eq @ A @ Y @ Z ) ) ) ).
% dense_le
thf(fact_71_dense__ge,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [Z: A,Y: A] :
( ! [X: A] :
( ( ord_less @ A @ Z @ X )
=> ( ord_less_eq @ A @ Y @ X ) )
=> ( ord_less_eq @ A @ Y @ Z ) ) ) ).
% dense_ge
thf(fact_72_less__le__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X2: A,Y: A,Z: A] :
( ( ord_less @ A @ X2 @ Y )
=> ( ( ord_less_eq @ A @ Y @ Z )
=> ( ord_less @ A @ X2 @ Z ) ) ) ) ).
% less_le_trans
thf(fact_73_le__less__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X2: A,Y: A,Z: A] :
( ( ord_less_eq @ A @ X2 @ Y )
=> ( ( ord_less @ A @ Y @ Z )
=> ( ord_less @ A @ X2 @ Z ) ) ) ) ).
% le_less_trans
thf(fact_74_less__imp__le,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X2: A,Y: A] :
( ( ord_less @ A @ X2 @ Y )
=> ( ord_less_eq @ A @ X2 @ Y ) ) ) ).
% less_imp_le
thf(fact_75_antisym__conv2,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X2: A,Y: A] :
( ( ord_less_eq @ A @ X2 @ Y )
=> ( ( ~ ( ord_less @ A @ X2 @ Y ) )
= ( X2 = Y ) ) ) ) ).
% antisym_conv2
thf(fact_76_antisym__conv1,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X2: A,Y: A] :
( ~ ( ord_less @ A @ X2 @ Y )
=> ( ( ord_less_eq @ A @ X2 @ Y )
= ( X2 = Y ) ) ) ) ).
% antisym_conv1
thf(fact_77_le__neq__trans,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( A2 != B2 )
=> ( ord_less @ A @ A2 @ B2 ) ) ) ) ).
% le_neq_trans
thf(fact_78_not__less,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X2: A,Y: A] :
( ( ~ ( ord_less @ A @ X2 @ Y ) )
= ( ord_less_eq @ A @ Y @ X2 ) ) ) ).
% not_less
thf(fact_79_not__le,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X2: A,Y: A] :
( ( ~ ( ord_less_eq @ A @ X2 @ Y ) )
= ( ord_less @ A @ Y @ X2 ) ) ) ).
% not_le
thf(fact_80_order__less__le__subst2,axiom,
! [A: $tType,C: $tType] :
( ( ( order @ C )
& ( order @ A ) )
=> ! [A2: A,B2: A,F: A > C,C2: C] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ C @ ( F @ B2 ) @ C2 )
=> ( ! [X: A,Y4: A] :
( ( ord_less @ A @ X @ Y4 )
=> ( ord_less @ C @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less @ C @ ( F @ A2 ) @ C2 ) ) ) ) ) ).
% order_less_le_subst2
thf(fact_81_order__less__le__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B )
& ( order @ A ) )
=> ! [A2: A,F: B > A,B2: B,C2: B] :
( ( ord_less @ A @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq @ B @ B2 @ C2 )
=> ( ! [X: B,Y4: B] :
( ( ord_less_eq @ B @ X @ Y4 )
=> ( ord_less_eq @ A @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less @ A @ A2 @ ( F @ C2 ) ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_82_order__le__less__subst2,axiom,
! [A: $tType,C: $tType] :
( ( ( order @ C )
& ( order @ A ) )
=> ! [A2: A,B2: A,F: A > C,C2: C] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less @ C @ ( F @ B2 ) @ C2 )
=> ( ! [X: A,Y4: A] :
( ( ord_less_eq @ A @ X @ Y4 )
=> ( ord_less_eq @ C @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less @ C @ ( F @ A2 ) @ C2 ) ) ) ) ) ).
% order_le_less_subst2
thf(fact_83_order__le__less__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B )
& ( order @ A ) )
=> ! [A2: A,F: B > A,B2: B,C2: B] :
( ( ord_less_eq @ A @ A2 @ ( F @ B2 ) )
=> ( ( ord_less @ B @ B2 @ C2 )
=> ( ! [X: B,Y4: B] :
( ( ord_less @ B @ X @ Y4 )
=> ( ord_less @ A @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less @ A @ A2 @ ( F @ C2 ) ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_84_less__le,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less @ A )
= ( ^ [X3: A,Y7: A] :
( ( ord_less_eq @ A @ X3 @ Y7 )
& ( X3 != Y7 ) ) ) ) ) ).
% less_le
thf(fact_85_le__less,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [X3: A,Y7: A] :
( ( ord_less @ A @ X3 @ Y7 )
| ( X3 = Y7 ) ) ) ) ) ).
% le_less
thf(fact_86_leI,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X2: A,Y: A] :
( ~ ( ord_less @ A @ X2 @ Y )
=> ( ord_less_eq @ A @ Y @ X2 ) ) ) ).
% leI
thf(fact_87_leD,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [Y: A,X2: A] :
( ( ord_less_eq @ A @ Y @ X2 )
=> ~ ( ord_less @ A @ X2 @ Y ) ) ) ).
% leD
thf(fact_88_top_Oextremum__uniqueI,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ! [A2: A] :
( ( ord_less_eq @ A @ ( top_top @ A ) @ A2 )
=> ( A2
= ( top_top @ A ) ) ) ) ).
% top.extremum_uniqueI
thf(fact_89_top_Oextremum__unique,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ! [A2: A] :
( ( ord_less_eq @ A @ ( top_top @ A ) @ A2 )
= ( A2
= ( top_top @ A ) ) ) ) ).
% top.extremum_unique
thf(fact_90_top__greatest,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ! [A2: A] : ( ord_less_eq @ A @ A2 @ ( top_top @ A ) ) ) ).
% top_greatest
thf(fact_91_dist__not__less__zero,axiom,
! [A: $tType] :
( ( real_V2090557954_space @ A )
=> ! [X2: A,Y: A] :
~ ( ord_less @ real @ ( real_V2000881966t_dist @ A @ X2 @ Y ) @ ( zero_zero @ real ) ) ) ).
% dist_not_less_zero
thf(fact_92_dist__pos__lt,axiom,
! [A: $tType] :
( ( real_V2090557954_space @ A )
=> ! [X2: A,Y: A] :
( ( X2 != Y )
=> ( ord_less @ real @ ( zero_zero @ real ) @ ( real_V2000881966t_dist @ A @ X2 @ Y ) ) ) ) ).
% dist_pos_lt
thf(fact_93_dual__order_Ostrict__implies__not__eq,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B2: A,A2: A] :
( ( ord_less @ A @ B2 @ A2 )
=> ( A2 != B2 ) ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_94_order_Ostrict__implies__not__eq,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( A2 != B2 ) ) ) ).
% order.strict_implies_not_eq
thf(fact_95_not__less__iff__gr__or__eq,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X2: A,Y: A] :
( ( ~ ( ord_less @ A @ X2 @ Y ) )
= ( ( ord_less @ A @ Y @ X2 )
| ( X2 = Y ) ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_96_dual__order_Ostrict__trans,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B2: A,A2: A,C2: A] :
( ( ord_less @ A @ B2 @ A2 )
=> ( ( ord_less @ A @ C2 @ B2 )
=> ( ord_less @ A @ C2 @ A2 ) ) ) ) ).
% dual_order.strict_trans
thf(fact_97_linorder__less__wlog,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > A > $o,A2: A,B2: A] :
( ! [A6: A,B5: A] :
( ( ord_less @ A @ A6 @ B5 )
=> ( P @ A6 @ B5 ) )
=> ( ! [A6: A] : ( P @ A6 @ A6 )
=> ( ! [A6: A,B5: A] :
( ( P @ B5 @ A6 )
=> ( P @ A6 @ B5 ) )
=> ( P @ A2 @ B2 ) ) ) ) ) ).
% linorder_less_wlog
thf(fact_98_exists__least__iff,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ( ( ^ [P2: A > $o] :
? [X6: A] : ( P2 @ X6 ) )
= ( ^ [P3: A > $o] :
? [N: A] :
( ( P3 @ N )
& ! [M: A] :
( ( ord_less @ A @ M @ N )
=> ~ ( P3 @ M ) ) ) ) ) ) ).
% exists_least_iff
thf(fact_99_less__imp__not__less,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X2: A,Y: A] :
( ( ord_less @ A @ X2 @ Y )
=> ~ ( ord_less @ A @ Y @ X2 ) ) ) ).
% less_imp_not_less
thf(fact_100_order_Ostrict__trans,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less @ A @ B2 @ C2 )
=> ( ord_less @ A @ A2 @ C2 ) ) ) ) ).
% order.strict_trans
thf(fact_101_dual__order_Oirrefl,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: A] :
~ ( ord_less @ A @ A2 @ A2 ) ) ).
% dual_order.irrefl
thf(fact_102_linorder__cases,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X2: A,Y: A] :
( ~ ( ord_less @ A @ X2 @ Y )
=> ( ( X2 != Y )
=> ( ord_less @ A @ Y @ X2 ) ) ) ) ).
% linorder_cases
thf(fact_103_less__imp__triv,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X2: A,Y: A,P: $o] :
( ( ord_less @ A @ X2 @ Y )
=> ( ( ord_less @ A @ Y @ X2 )
=> P ) ) ) ).
% less_imp_triv
thf(fact_104_less__imp__not__eq2,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X2: A,Y: A] :
( ( ord_less @ A @ X2 @ Y )
=> ( Y != X2 ) ) ) ).
% less_imp_not_eq2
thf(fact_105_antisym__conv3,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Y: A,X2: A] :
( ~ ( ord_less @ A @ Y @ X2 )
=> ( ( ~ ( ord_less @ A @ X2 @ Y ) )
= ( X2 = Y ) ) ) ) ).
% antisym_conv3
thf(fact_106_less__induct,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [P: A > $o,A2: A] :
( ! [X: A] :
( ! [Y8: A] :
( ( ord_less @ A @ Y8 @ X )
=> ( P @ Y8 ) )
=> ( P @ X ) )
=> ( P @ A2 ) ) ) ).
% less_induct
thf(fact_107_less__not__sym,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X2: A,Y: A] :
( ( ord_less @ A @ X2 @ Y )
=> ~ ( ord_less @ A @ Y @ X2 ) ) ) ).
% less_not_sym
thf(fact_108_less__imp__not__eq,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X2: A,Y: A] :
( ( ord_less @ A @ X2 @ Y )
=> ( X2 != Y ) ) ) ).
% less_imp_not_eq
thf(fact_109_dual__order_Oasym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B2: A,A2: A] :
( ( ord_less @ A @ B2 @ A2 )
=> ~ ( ord_less @ A @ A2 @ B2 ) ) ) ).
% dual_order.asym
thf(fact_110_ord__less__eq__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( B2 = C2 )
=> ( ord_less @ A @ A2 @ C2 ) ) ) ) ).
% ord_less_eq_trans
thf(fact_111_ord__eq__less__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( A2 = B2 )
=> ( ( ord_less @ A @ B2 @ C2 )
=> ( ord_less @ A @ A2 @ C2 ) ) ) ) ).
% ord_eq_less_trans
thf(fact_112_less__irrefl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X2: A] :
~ ( ord_less @ A @ X2 @ X2 ) ) ).
% less_irrefl
thf(fact_113_less__linear,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X2: A,Y: A] :
( ( ord_less @ A @ X2 @ Y )
| ( X2 = Y )
| ( ord_less @ A @ Y @ X2 ) ) ) ).
% less_linear
thf(fact_114_less__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X2: A,Y: A,Z: A] :
( ( ord_less @ A @ X2 @ Y )
=> ( ( ord_less @ A @ Y @ Z )
=> ( ord_less @ A @ X2 @ Z ) ) ) ) ).
% less_trans
thf(fact_115_less__asym_H,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ~ ( ord_less @ A @ B2 @ A2 ) ) ) ).
% less_asym'
thf(fact_116_less__asym,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X2: A,Y: A] :
( ( ord_less @ A @ X2 @ Y )
=> ~ ( ord_less @ A @ Y @ X2 ) ) ) ).
% less_asym
thf(fact_117_less__imp__neq,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X2: A,Y: A] :
( ( ord_less @ A @ X2 @ Y )
=> ( X2 != Y ) ) ) ).
% less_imp_neq
thf(fact_118_dense,axiom,
! [A: $tType] :
( ( dense_order @ A )
=> ! [X2: A,Y: A] :
( ( ord_less @ A @ X2 @ Y )
=> ? [Z3: A] :
( ( ord_less @ A @ X2 @ Z3 )
& ( ord_less @ A @ Z3 @ Y ) ) ) ) ).
% dense
thf(fact_119_order_Oasym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ~ ( ord_less @ A @ B2 @ A2 ) ) ) ).
% order.asym
thf(fact_120_neq__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X2: A,Y: A] :
( ( X2 != Y )
= ( ( ord_less @ A @ X2 @ Y )
| ( ord_less @ A @ Y @ X2 ) ) ) ) ).
% neq_iff
thf(fact_121_neqE,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X2: A,Y: A] :
( ( X2 != Y )
=> ( ~ ( ord_less @ A @ X2 @ Y )
=> ( ord_less @ A @ Y @ X2 ) ) ) ) ).
% neqE
thf(fact_122_gt__ex,axiom,
! [A: $tType] :
( ( no_top @ A )
=> ! [X2: A] :
? [X_1: A] : ( ord_less @ A @ X2 @ X_1 ) ) ).
% gt_ex
thf(fact_123_lt__ex,axiom,
! [A: $tType] :
( ( no_bot @ A )
=> ! [X2: A] :
? [Y4: A] : ( ord_less @ A @ Y4 @ X2 ) ) ).
% lt_ex
thf(fact_124_order__less__subst2,axiom,
! [A: $tType,C: $tType] :
( ( ( order @ C )
& ( order @ A ) )
=> ! [A2: A,B2: A,F: A > C,C2: C] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less @ C @ ( F @ B2 ) @ C2 )
=> ( ! [X: A,Y4: A] :
( ( ord_less @ A @ X @ Y4 )
=> ( ord_less @ C @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less @ C @ ( F @ A2 ) @ C2 ) ) ) ) ) ).
% order_less_subst2
thf(fact_125_order__less__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B )
& ( order @ A ) )
=> ! [A2: A,F: B > A,B2: B,C2: B] :
( ( ord_less @ A @ A2 @ ( F @ B2 ) )
=> ( ( ord_less @ B @ B2 @ C2 )
=> ( ! [X: B,Y4: B] :
( ( ord_less @ B @ X @ Y4 )
=> ( ord_less @ A @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less @ A @ A2 @ ( F @ C2 ) ) ) ) ) ) ).
% order_less_subst1
thf(fact_126_ord__less__eq__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( ord @ A ) )
=> ! [A2: A,B2: A,F: A > B,C2: B] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C2 )
=> ( ! [X: A,Y4: A] :
( ( ord_less @ A @ X @ Y4 )
=> ( ord_less @ B @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less @ B @ ( F @ A2 ) @ C2 ) ) ) ) ) ).
% ord_less_eq_subst
thf(fact_127_ord__eq__less__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( ord @ A ) )
=> ! [A2: A,F: B > A,B2: B,C2: B] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less @ B @ B2 @ C2 )
=> ( ! [X: B,Y4: B] :
( ( ord_less @ B @ X @ Y4 )
=> ( ord_less @ A @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less @ A @ A2 @ ( F @ C2 ) ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_128_old_Oprod_Oinducts,axiom,
! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o,Prod: product_prod @ A @ B] :
( ! [A6: A,B5: B] : ( P @ ( product_Pair @ A @ B @ A6 @ B5 ) )
=> ( P @ Prod ) ) ).
% old.prod.inducts
thf(fact_129_old_Oprod_Oexhaust,axiom,
! [A: $tType,B: $tType,Y: product_prod @ A @ B] :
~ ! [A6: A,B5: B] :
( Y
!= ( product_Pair @ A @ B @ A6 @ B5 ) ) ).
% old.prod.exhaust
thf(fact_130_prod__induct7,axiom,
! [G3: $tType,F3: $tType,E2: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E2 @ ( product_prod @ F3 @ G3 ) ) ) ) ) ) > $o,X2: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E2 @ ( product_prod @ F3 @ G3 ) ) ) ) )] :
( ! [A6: A,B5: B,C3: C,D3: D,E3: E2,F4: F3,G4: G3] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E2 @ ( product_prod @ F3 @ G3 ) ) ) ) ) @ A6 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E2 @ ( product_prod @ F3 @ G3 ) ) ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E2 @ ( product_prod @ F3 @ G3 ) ) ) @ C3 @ ( product_Pair @ D @ ( product_prod @ E2 @ ( product_prod @ F3 @ G3 ) ) @ D3 @ ( product_Pair @ E2 @ ( product_prod @ F3 @ G3 ) @ E3 @ ( product_Pair @ F3 @ G3 @ F4 @ G4 ) ) ) ) ) ) )
=> ( P @ X2 ) ) ).
% prod_induct7
thf(fact_131_prod__induct6,axiom,
! [F3: $tType,E2: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E2 @ F3 ) ) ) ) ) > $o,X2: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E2 @ F3 ) ) ) )] :
( ! [A6: A,B5: B,C3: C,D3: D,E3: E2,F4: F3] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E2 @ F3 ) ) ) ) @ A6 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E2 @ F3 ) ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E2 @ F3 ) ) @ C3 @ ( product_Pair @ D @ ( product_prod @ E2 @ F3 ) @ D3 @ ( product_Pair @ E2 @ F3 @ E3 @ F4 ) ) ) ) ) )
=> ( P @ X2 ) ) ).
% prod_induct6
thf(fact_132_prod__induct5,axiom,
! [E2: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E2 ) ) ) ) > $o,X2: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E2 ) ) )] :
( ! [A6: A,B5: B,C3: C,D3: D,E3: E2] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E2 ) ) ) @ A6 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ E2 ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ E2 ) @ C3 @ ( product_Pair @ D @ E2 @ D3 @ E3 ) ) ) ) )
=> ( P @ X2 ) ) ).
% prod_induct5
thf(fact_133_prod__induct4,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) ) > $o,X2: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) )] :
( ! [A6: A,B5: B,C3: C,D3: D] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A6 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B5 @ ( product_Pair @ C @ D @ C3 @ D3 ) ) ) )
=> ( P @ X2 ) ) ).
% prod_induct4
thf(fact_134_prod__induct3,axiom,
! [C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ C ) ) > $o,X2: product_prod @ A @ ( product_prod @ B @ C )] :
( ! [A6: A,B5: B,C3: C] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ C ) @ A6 @ ( product_Pair @ B @ C @ B5 @ C3 ) ) )
=> ( P @ X2 ) ) ).
% prod_induct3
thf(fact_135_prod__cases7,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,E2: $tType,F3: $tType,G3: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E2 @ ( product_prod @ F3 @ G3 ) ) ) ) )] :
~ ! [A6: A,B5: B,C3: C,D3: D,E3: E2,F4: F3,G4: G3] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E2 @ ( product_prod @ F3 @ G3 ) ) ) ) ) @ A6 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E2 @ ( product_prod @ F3 @ G3 ) ) ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E2 @ ( product_prod @ F3 @ G3 ) ) ) @ C3 @ ( product_Pair @ D @ ( product_prod @ E2 @ ( product_prod @ F3 @ G3 ) ) @ D3 @ ( product_Pair @ E2 @ ( product_prod @ F3 @ G3 ) @ E3 @ ( product_Pair @ F3 @ G3 @ F4 @ G4 ) ) ) ) ) ) ) ).
% prod_cases7
thf(fact_136_prod__cases6,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,E2: $tType,F3: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E2 @ F3 ) ) ) )] :
~ ! [A6: A,B5: B,C3: C,D3: D,E3: E2,F4: F3] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E2 @ F3 ) ) ) ) @ A6 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E2 @ F3 ) ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E2 @ F3 ) ) @ C3 @ ( product_Pair @ D @ ( product_prod @ E2 @ F3 ) @ D3 @ ( product_Pair @ E2 @ F3 @ E3 @ F4 ) ) ) ) ) ) ).
% prod_cases6
thf(fact_137_prod__cases5,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,E2: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E2 ) ) )] :
~ ! [A6: A,B5: B,C3: C,D3: D,E3: E2] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E2 ) ) ) @ A6 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ E2 ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ E2 ) @ C3 @ ( product_Pair @ D @ E2 @ D3 @ E3 ) ) ) ) ) ).
% prod_cases5
thf(fact_138_prod__cases4,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) )] :
~ ! [A6: A,B5: B,C3: C,D3: D] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A6 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B5 @ ( product_Pair @ C @ D @ C3 @ D3 ) ) ) ) ).
% prod_cases4
thf(fact_139_prod__cases3,axiom,
! [A: $tType,B: $tType,C: $tType,Y: product_prod @ A @ ( product_prod @ B @ C )] :
~ ! [A6: A,B5: B,C3: C] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ C ) @ A6 @ ( product_Pair @ B @ C @ B5 @ C3 ) ) ) ).
% prod_cases3
thf(fact_140_Pair__inject,axiom,
! [A: $tType,B: $tType,A2: A,B2: B,A3: A,B3: B] :
( ( ( product_Pair @ A @ B @ A2 @ B2 )
= ( product_Pair @ A @ B @ A3 @ B3 ) )
=> ~ ( ( A2 = A3 )
=> ( B2 != B3 ) ) ) ).
% Pair_inject
thf(fact_141_prod__cases,axiom,
! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o,P4: product_prod @ A @ B] :
( ! [A6: A,B5: B] : ( P @ ( product_Pair @ A @ B @ A6 @ B5 ) )
=> ( P @ P4 ) ) ).
% prod_cases
thf(fact_142_surj__pair,axiom,
! [A: $tType,B: $tType,P4: product_prod @ A @ B] :
? [X: A,Y4: B] :
( P4
= ( product_Pair @ A @ B @ X @ Y4 ) ) ).
% surj_pair
thf(fact_143_UNIV__witness,axiom,
! [A: $tType] :
? [X: A] : ( member @ A @ X @ ( top_top @ ( set @ A ) ) ) ).
% UNIV_witness
thf(fact_144_UNIV__eq__I,axiom,
! [A: $tType,A7: set @ A] :
( ! [X: A] : ( member @ A @ X @ A7 )
=> ( ( top_top @ ( set @ A ) )
= A7 ) ) ).
% UNIV_eq_I
thf(fact_145_dist__commute,axiom,
! [A: $tType] :
( ( real_V2090557954_space @ A )
=> ( ( real_V2000881966t_dist @ A )
= ( ^ [X3: A,Y7: A] : ( real_V2000881966t_dist @ A @ Y7 @ X3 ) ) ) ) ).
% dist_commute
thf(fact_146_closure__UNIV,axiom,
! [A: $tType] :
( ( topolo503727757_space @ A )
=> ( ( elementary_closure @ A @ ( top_top @ ( set @ A ) ) )
= ( top_top @ ( set @ A ) ) ) ) ).
% closure_UNIV
thf(fact_147_closure__approachable__le,axiom,
! [A: $tType] :
( ( real_V2090557954_space @ A )
=> ! [X2: A,S2: set @ A] :
( ( member @ A @ X2 @ ( elementary_closure @ A @ S2 ) )
= ( ! [E4: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E4 )
=> ? [X3: A] :
( ( member @ A @ X3 @ S2 )
& ( ord_less_eq @ real @ ( real_V2000881966t_dist @ A @ X3 @ X2 ) @ E4 ) ) ) ) ) ) ).
% closure_approachable_le
thf(fact_148_not__gr__zero,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A )
=> ! [N2: A] :
( ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ N2 ) )
= ( N2
= ( zero_zero @ A ) ) ) ) ).
% not_gr_zero
thf(fact_149_le__zero__eq,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A )
=> ! [N2: A] :
( ( ord_less_eq @ A @ N2 @ ( zero_zero @ A ) )
= ( N2
= ( zero_zero @ A ) ) ) ) ).
% le_zero_eq
thf(fact_150_closure__approachableD,axiom,
! [A: $tType] :
( ( real_V2090557954_space @ A )
=> ! [X2: A,S2: set @ A,E: real] :
( ( member @ A @ X2 @ ( elementary_closure @ A @ S2 ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ E )
=> ? [X: A] :
( ( member @ A @ X @ S2 )
& ( ord_less @ real @ ( real_V2000881966t_dist @ A @ X2 @ X ) @ E ) ) ) ) ) ).
% closure_approachableD
thf(fact_151_closure__approachable,axiom,
! [A: $tType] :
( ( real_V2090557954_space @ A )
=> ! [X2: A,S2: set @ A] :
( ( member @ A @ X2 @ ( elementary_closure @ A @ S2 ) )
= ( ! [E4: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E4 )
=> ? [X3: A] :
( ( member @ A @ X3 @ S2 )
& ( ord_less @ real @ ( real_V2000881966t_dist @ A @ X3 @ X2 ) @ E4 ) ) ) ) ) ) ).
% closure_approachable
thf(fact_152_old_Oprod_Orec,axiom,
! [A: $tType,T: $tType,B: $tType,F1: A > B > T,A2: A,B2: B] :
( ( product_rec_prod @ A @ B @ T @ F1 @ ( product_Pair @ A @ B @ A2 @ B2 ) )
= ( F1 @ A2 @ B2 ) ) ).
% old.prod.rec
thf(fact_153_perfect__choose__dist,axiom,
! [A: $tType] :
( ( ( real_V2090557954_space @ A )
& ( topolo890362671_space @ A ) )
=> ! [R: real,X2: A] :
( ( ord_less @ real @ ( zero_zero @ real ) @ R )
=> ? [A6: A] :
( ( A6 != X2 )
& ( ord_less @ real @ ( real_V2000881966t_dist @ A @ A6 @ X2 ) @ R ) ) ) ) ).
% perfect_choose_dist
thf(fact_154_vector__choose__dist,axiom,
! [A: $tType] :
( ( ( real_V55928688vector @ A )
& ( topolo890362671_space @ A ) )
=> ! [C2: real,X2: A] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ C2 )
=> ~ ! [Y4: A] :
( ( real_V2000881966t_dist @ A @ X2 @ Y4 )
!= C2 ) ) ) ).
% vector_choose_dist
thf(fact_155_approachable__lt__le2,axiom,
! [A: $tType,Q: A > $o,F: A > real,P: A > $o] :
( ( ? [D4: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D4 )
& ! [X3: A] :
( ( Q @ X3 )
=> ( ( ord_less @ real @ ( F @ X3 ) @ D4 )
=> ( P @ X3 ) ) ) ) )
= ( ? [D4: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D4 )
& ! [X3: A] :
( ( ord_less_eq @ real @ ( F @ X3 ) @ D4 )
=> ( ( Q @ X3 )
=> ( P @ X3 ) ) ) ) ) ) ).
% approachable_lt_le2
thf(fact_156_closure__closure,axiom,
! [A: $tType] :
( ( topolo503727757_space @ A )
=> ! [S2: set @ A] :
( ( elementary_closure @ A @ ( elementary_closure @ A @ S2 ) )
= ( elementary_closure @ A @ S2 ) ) ) ).
% closure_closure
thf(fact_157_top1I,axiom,
! [A: $tType,X2: A] : ( top_top @ ( A > $o ) @ X2 ) ).
% top1I
thf(fact_158_less__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ( ( ord_less @ ( A > B ) )
= ( ^ [F2: A > B,G2: A > B] :
( ( ord_less_eq @ ( A > B ) @ F2 @ G2 )
& ~ ( ord_less_eq @ ( A > B ) @ G2 @ F2 ) ) ) ) ) ).
% less_fun_def
thf(fact_159_less__prod__def,axiom,
! [B: $tType,A: $tType] :
( ( ( ord @ A )
& ( ord @ B ) )
=> ( ( ord_less @ ( product_prod @ A @ B ) )
= ( ^ [X3: product_prod @ A @ B,Y7: product_prod @ A @ B] :
( ( ord_less_eq @ ( product_prod @ A @ B ) @ X3 @ Y7 )
& ~ ( ord_less_eq @ ( product_prod @ A @ B ) @ Y7 @ X3 ) ) ) ) ) ).
% less_prod_def
thf(fact_160_closure__subset,axiom,
! [A: $tType] :
( ( topolo503727757_space @ A )
=> ! [S2: set @ A] : ( ord_less_eq @ ( set @ A ) @ S2 @ ( elementary_closure @ A @ S2 ) ) ) ).
% closure_subset
thf(fact_161_closure__mono,axiom,
! [A: $tType] :
( ( topolo503727757_space @ A )
=> ! [S2: set @ A,T2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ S2 @ T2 )
=> ( ord_less_eq @ ( set @ A ) @ ( elementary_closure @ A @ S2 ) @ ( elementary_closure @ A @ T2 ) ) ) ) ).
% closure_mono
thf(fact_162_subset__UNIV,axiom,
! [A: $tType,A7: set @ A] : ( ord_less_eq @ ( set @ A ) @ A7 @ ( top_top @ ( set @ A ) ) ) ).
% subset_UNIV
thf(fact_163_zero__reorient,axiom,
! [A: $tType] :
( ( zero @ A )
=> ! [X2: A] :
( ( ( zero_zero @ A )
= X2 )
= ( X2
= ( zero_zero @ A ) ) ) ) ).
% zero_reorient
thf(fact_164_zero__le,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A )
=> ! [X2: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ X2 ) ) ).
% zero_le
thf(fact_165_gr__zeroI,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A )
=> ! [N2: A] :
( ( N2
!= ( zero_zero @ A ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ N2 ) ) ) ).
% gr_zeroI
thf(fact_166_not__less__zero,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A )
=> ! [N2: A] :
~ ( ord_less @ A @ N2 @ ( zero_zero @ A ) ) ) ).
% not_less_zero
thf(fact_167_gr__implies__not__zero,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A )
=> ! [M2: A,N2: A] :
( ( ord_less @ A @ M2 @ N2 )
=> ( N2
!= ( zero_zero @ A ) ) ) ) ).
% gr_implies_not_zero
thf(fact_168_zero__less__iff__neq__zero,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A )
=> ! [N2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ N2 )
= ( N2
!= ( zero_zero @ A ) ) ) ) ).
% zero_less_iff_neq_zero
thf(fact_169_uniformly__convergent__eq__cauchy,axiom,
! [A: $tType,B: $tType] :
( ( real_V1759440057_space @ A )
=> ! [P: B > $o,S3: nat > B > A] :
( ( ? [L: B > A] :
! [E4: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E4 )
=> ? [N3: nat] :
! [N: nat,X3: B] :
( ( ( ord_less_eq @ nat @ N3 @ N )
& ( P @ X3 ) )
=> ( ord_less @ real @ ( real_V2000881966t_dist @ A @ ( S3 @ N @ X3 ) @ ( L @ X3 ) ) @ E4 ) ) ) )
= ( ! [E4: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E4 )
=> ? [N3: nat] :
! [M: nat,N: nat,X3: B] :
( ( ( ord_less_eq @ nat @ N3 @ M )
& ( ord_less_eq @ nat @ N3 @ N )
& ( P @ X3 ) )
=> ( ord_less @ real @ ( real_V2000881966t_dist @ A @ ( S3 @ M @ X3 ) @ ( S3 @ N @ X3 ) ) @ E4 ) ) ) ) ) ) ).
% uniformly_convergent_eq_cauchy
thf(fact_170_uniformly__cauchy__imp__uniformly__convergent,axiom,
! [A: $tType,B: $tType] :
( ( real_V1759440057_space @ B )
=> ! [P: A > $o,S3: nat > A > B,L2: A > B] :
( ! [E3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E3 )
=> ? [N4: nat] :
! [M3: nat,N5: nat,X: A] :
( ( ( ord_less_eq @ nat @ N4 @ M3 )
& ( ord_less_eq @ nat @ N4 @ N5 )
& ( P @ X ) )
=> ( ord_less @ real @ ( real_V2000881966t_dist @ B @ ( S3 @ M3 @ X ) @ ( S3 @ N5 @ X ) ) @ E3 ) ) )
=> ( ! [X: A] :
( ( P @ X )
=> ! [E3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E3 )
=> ? [N4: nat] :
! [N5: nat] :
( ( ord_less_eq @ nat @ N4 @ N5 )
=> ( ord_less @ real @ ( real_V2000881966t_dist @ B @ ( S3 @ N5 @ X ) @ ( L2 @ X ) ) @ E3 ) ) ) )
=> ! [E5: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E5 )
=> ? [N6: nat] :
! [N7: nat,X7: A] :
( ( ( ord_less_eq @ nat @ N6 @ N7 )
& ( P @ X7 ) )
=> ( ord_less @ real @ ( real_V2000881966t_dist @ B @ ( S3 @ N7 @ X7 ) @ ( L2 @ X7 ) ) @ E5 ) ) ) ) ) ) ).
% uniformly_cauchy_imp_uniformly_convergent
thf(fact_171_top__empty__eq,axiom,
! [A: $tType] :
( ( top_top @ ( A > $o ) )
= ( ^ [X3: A] : ( member @ A @ X3 @ ( top_top @ ( set @ A ) ) ) ) ) ).
% top_empty_eq
thf(fact_172_subsetI,axiom,
! [A: $tType,A7: set @ A,B6: set @ A] :
( ! [X: A] :
( ( member @ A @ X @ A7 )
=> ( member @ A @ X @ B6 ) )
=> ( ord_less_eq @ ( set @ A ) @ A7 @ B6 ) ) ).
% subsetI
thf(fact_173_psubsetI,axiom,
! [A: $tType,A7: set @ A,B6: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A7 @ B6 )
=> ( ( A7 != B6 )
=> ( ord_less @ ( set @ A ) @ A7 @ B6 ) ) ) ).
% psubsetI
thf(fact_174_subset__antisym,axiom,
! [A: $tType,A7: set @ A,B6: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A7 @ B6 )
=> ( ( ord_less_eq @ ( set @ A ) @ B6 @ A7 )
=> ( A7 = B6 ) ) ) ).
% subset_antisym
thf(fact_175_in__mono,axiom,
! [A: $tType,A7: set @ A,B6: set @ A,X2: A] :
( ( ord_less_eq @ ( set @ A ) @ A7 @ B6 )
=> ( ( member @ A @ X2 @ A7 )
=> ( member @ A @ X2 @ B6 ) ) ) ).
% in_mono
thf(fact_176_subsetD,axiom,
! [A: $tType,A7: set @ A,B6: set @ A,C2: A] :
( ( ord_less_eq @ ( set @ A ) @ A7 @ B6 )
=> ( ( member @ A @ C2 @ A7 )
=> ( member @ A @ C2 @ B6 ) ) ) ).
% subsetD
thf(fact_177_psubsetE,axiom,
! [A: $tType,A7: set @ A,B6: set @ A] :
( ( ord_less @ ( set @ A ) @ A7 @ B6 )
=> ~ ( ( ord_less_eq @ ( set @ A ) @ A7 @ B6 )
=> ( ord_less_eq @ ( set @ A ) @ B6 @ A7 ) ) ) ).
% psubsetE
thf(fact_178_equalityE,axiom,
! [A: $tType,A7: set @ A,B6: set @ A] :
( ( A7 = B6 )
=> ~ ( ( ord_less_eq @ ( set @ A ) @ A7 @ B6 )
=> ~ ( ord_less_eq @ ( set @ A ) @ B6 @ A7 ) ) ) ).
% equalityE
thf(fact_179_subset__eq,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A8: set @ A,B7: set @ A] :
! [X3: A] :
( ( member @ A @ X3 @ A8 )
=> ( member @ A @ X3 @ B7 ) ) ) ) ).
% subset_eq
thf(fact_180_equalityD1,axiom,
! [A: $tType,A7: set @ A,B6: set @ A] :
( ( A7 = B6 )
=> ( ord_less_eq @ ( set @ A ) @ A7 @ B6 ) ) ).
% equalityD1
thf(fact_181_equalityD2,axiom,
! [A: $tType,A7: set @ A,B6: set @ A] :
( ( A7 = B6 )
=> ( ord_less_eq @ ( set @ A ) @ B6 @ A7 ) ) ).
% equalityD2
thf(fact_182_psubset__eq,axiom,
! [A: $tType] :
( ( ord_less @ ( set @ A ) )
= ( ^ [A8: set @ A,B7: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A8 @ B7 )
& ( A8 != B7 ) ) ) ) ).
% psubset_eq
thf(fact_183_subset__iff,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A8: set @ A,B7: set @ A] :
! [T3: A] :
( ( member @ A @ T3 @ A8 )
=> ( member @ A @ T3 @ B7 ) ) ) ) ).
% subset_iff
thf(fact_184_subset__refl,axiom,
! [A: $tType,A7: set @ A] : ( ord_less_eq @ ( set @ A ) @ A7 @ A7 ) ).
% subset_refl
thf(fact_185_Collect__mono,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ! [X: A] :
( ( P @ X )
=> ( Q @ X ) )
=> ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) ) ) ).
% Collect_mono
thf(fact_186_subset__trans,axiom,
! [A: $tType,A7: set @ A,B6: set @ A,C4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A7 @ B6 )
=> ( ( ord_less_eq @ ( set @ A ) @ B6 @ C4 )
=> ( ord_less_eq @ ( set @ A ) @ A7 @ C4 ) ) ) ).
% subset_trans
thf(fact_187_set__eq__subset,axiom,
! [A: $tType] :
( ( ^ [Y6: set @ A,Z2: set @ A] : Y6 = Z2 )
= ( ^ [A8: set @ A,B7: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A8 @ B7 )
& ( ord_less_eq @ ( set @ A ) @ B7 @ A8 ) ) ) ) ).
% set_eq_subset
thf(fact_188_Collect__mono__iff,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) )
= ( ! [X3: A] :
( ( P @ X3 )
=> ( Q @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_189_psubset__imp__subset,axiom,
! [A: $tType,A7: set @ A,B6: set @ A] :
( ( ord_less @ ( set @ A ) @ A7 @ B6 )
=> ( ord_less_eq @ ( set @ A ) @ A7 @ B6 ) ) ).
% psubset_imp_subset
thf(fact_190_psubset__subset__trans,axiom,
! [A: $tType,A7: set @ A,B6: set @ A,C4: set @ A] :
( ( ord_less @ ( set @ A ) @ A7 @ B6 )
=> ( ( ord_less_eq @ ( set @ A ) @ B6 @ C4 )
=> ( ord_less @ ( set @ A ) @ A7 @ C4 ) ) ) ).
% psubset_subset_trans
thf(fact_191_subset__not__subset__eq,axiom,
! [A: $tType] :
( ( ord_less @ ( set @ A ) )
= ( ^ [A8: set @ A,B7: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A8 @ B7 )
& ~ ( ord_less_eq @ ( set @ A ) @ B7 @ A8 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_192_subset__psubset__trans,axiom,
! [A: $tType,A7: set @ A,B6: set @ A,C4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A7 @ B6 )
=> ( ( ord_less @ ( set @ A ) @ B6 @ C4 )
=> ( ord_less @ ( set @ A ) @ A7 @ C4 ) ) ) ).
% subset_psubset_trans
thf(fact_193_subset__iff__psubset__eq,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A8: set @ A,B7: set @ A] :
( ( ord_less @ ( set @ A ) @ A8 @ B7 )
| ( A8 = B7 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_194_subrelI,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ B ),S3: set @ ( product_prod @ A @ B )] :
( ! [X: A,Y4: B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y4 ) @ R )
=> ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y4 ) @ S3 ) )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R @ S3 ) ) ).
% subrelI
thf(fact_195_seq__mono__lemma,axiom,
! [M2: nat,D2: nat > real,E: nat > real] :
( ! [N5: nat] :
( ( ord_less_eq @ nat @ M2 @ N5 )
=> ( ord_less @ real @ ( D2 @ N5 ) @ ( E @ N5 ) ) )
=> ( ! [N5: nat] :
( ( ord_less_eq @ nat @ M2 @ N5 )
=> ( ord_less_eq @ real @ ( E @ N5 ) @ ( E @ M2 ) ) )
=> ! [N7: nat] :
( ( ord_less_eq @ nat @ M2 @ N7 )
=> ( ord_less @ real @ ( D2 @ N7 ) @ ( E @ M2 ) ) ) ) ) ).
% seq_mono_lemma
thf(fact_196_le0,axiom,
! [N2: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ N2 ) ).
% le0
thf(fact_197_bot__nat__0_Oextremum,axiom,
! [A2: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ A2 ) ).
% bot_nat_0.extremum
thf(fact_198_neq0__conv,axiom,
! [N2: nat] :
( ( N2
!= ( zero_zero @ nat ) )
= ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) ) ).
% neq0_conv
thf(fact_199_less__nat__zero__code,axiom,
! [N2: nat] :
~ ( ord_less @ nat @ N2 @ ( zero_zero @ nat ) ) ).
% less_nat_zero_code
thf(fact_200_bot__nat__0_Onot__eq__extremum,axiom,
! [A2: nat] :
( ( A2
!= ( zero_zero @ nat ) )
= ( ord_less @ nat @ ( zero_zero @ nat ) @ A2 ) ) ).
% bot_nat_0.not_eq_extremum
thf(fact_201_gr0I,axiom,
! [N2: nat] :
( ( N2
!= ( zero_zero @ nat ) )
=> ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) ) ).
% gr0I
thf(fact_202_not__gr0,axiom,
! [N2: nat] :
( ( ~ ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) )
= ( N2
= ( zero_zero @ nat ) ) ) ).
% not_gr0
thf(fact_203_not__less0,axiom,
! [N2: nat] :
~ ( ord_less @ nat @ N2 @ ( zero_zero @ nat ) ) ).
% not_less0
thf(fact_204_less__zeroE,axiom,
! [N2: nat] :
~ ( ord_less @ nat @ N2 @ ( zero_zero @ nat ) ) ).
% less_zeroE
thf(fact_205_gr__implies__not0,axiom,
! [M2: nat,N2: nat] :
( ( ord_less @ nat @ M2 @ N2 )
=> ( N2
!= ( zero_zero @ nat ) ) ) ).
% gr_implies_not0
thf(fact_206_infinite__descent0,axiom,
! [P: nat > $o,N2: nat] :
( ( P @ ( zero_zero @ nat ) )
=> ( ! [N5: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N5 )
=> ( ~ ( P @ N5 )
=> ? [M4: nat] :
( ( ord_less @ nat @ M4 @ N5 )
& ~ ( P @ M4 ) ) ) )
=> ( P @ N2 ) ) ) ).
% infinite_descent0
thf(fact_207_bot__nat__0_Oextremum__strict,axiom,
! [A2: nat] :
~ ( ord_less @ nat @ A2 @ ( zero_zero @ nat ) ) ).
% bot_nat_0.extremum_strict
thf(fact_208_infinite__descent0__measure,axiom,
! [A: $tType,V: A > nat,P: A > $o,X2: A] :
( ! [X: A] :
( ( ( V @ X )
= ( zero_zero @ nat ) )
=> ( P @ X ) )
=> ( ! [X: A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( V @ X ) )
=> ( ~ ( P @ X )
=> ? [Y8: A] :
( ( ord_less @ nat @ ( V @ Y8 ) @ ( V @ X ) )
& ~ ( P @ Y8 ) ) ) )
=> ( P @ X2 ) ) ) ).
% infinite_descent0_measure
thf(fact_209_psubset__trans,axiom,
! [A: $tType,A7: set @ A,B6: set @ A,C4: set @ A] :
( ( ord_less @ ( set @ A ) @ A7 @ B6 )
=> ( ( ord_less @ ( set @ A ) @ B6 @ C4 )
=> ( ord_less @ ( set @ A ) @ A7 @ C4 ) ) ) ).
% psubset_trans
thf(fact_210_psubsetD,axiom,
! [A: $tType,A7: set @ A,B6: set @ A,C2: A] :
( ( ord_less @ ( set @ A ) @ A7 @ B6 )
=> ( ( member @ A @ C2 @ A7 )
=> ( member @ A @ C2 @ B6 ) ) ) ).
% psubsetD
thf(fact_211_measure__induct,axiom,
! [B: $tType,A: $tType] :
( ( wellorder @ B )
=> ! [F: A > B,P: A > $o,A2: A] :
( ! [X: A] :
( ! [Y8: A] :
( ( ord_less @ B @ ( F @ Y8 ) @ ( F @ X ) )
=> ( P @ Y8 ) )
=> ( P @ X ) )
=> ( P @ A2 ) ) ) ).
% measure_induct
thf(fact_212_measure__induct__rule,axiom,
! [B: $tType,A: $tType] :
( ( wellorder @ B )
=> ! [F: A > B,P: A > $o,A2: A] :
( ! [X: A] :
( ! [Y8: A] :
( ( ord_less @ B @ ( F @ Y8 ) @ ( F @ X ) )
=> ( P @ Y8 ) )
=> ( P @ X ) )
=> ( P @ A2 ) ) ) ).
% measure_induct_rule
thf(fact_213_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_214_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_215_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I: nat,J: nat] :
( ! [I2: nat,J2: nat] :
( ( ord_less @ nat @ I2 @ J2 )
=> ( ord_less @ nat @ ( F @ I2 ) @ ( F @ J2 ) ) )
=> ( ( ord_less_eq @ nat @ I @ J )
=> ( ord_less_eq @ nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_216_le__neq__implies__less,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq @ nat @ M2 @ N2 )
=> ( ( M2 != N2 )
=> ( ord_less @ nat @ M2 @ N2 ) ) ) ).
% le_neq_implies_less
thf(fact_217_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B2: nat] :
( ( P @ K )
=> ( ! [Y4: nat] :
( ( P @ Y4 )
=> ( ord_less_eq @ nat @ Y4 @ B2 ) )
=> ? [X: nat] :
( ( P @ X )
& ! [Y8: nat] :
( ( P @ Y8 )
=> ( ord_less_eq @ nat @ Y8 @ X ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_218_less__or__eq__imp__le,axiom,
! [M2: nat,N2: nat] :
( ( ( ord_less @ nat @ M2 @ N2 )
| ( M2 = N2 ) )
=> ( ord_less_eq @ nat @ M2 @ N2 ) ) ).
% less_or_eq_imp_le
thf(fact_219_le__eq__less__or__eq,axiom,
( ( ord_less_eq @ nat )
= ( ^ [M: nat,N: nat] :
( ( ord_less @ nat @ M @ N )
| ( M = N ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_220_less__imp__le__nat,axiom,
! [M2: nat,N2: nat] :
( ( ord_less @ nat @ M2 @ N2 )
=> ( ord_less_eq @ nat @ M2 @ N2 ) ) ).
% less_imp_le_nat
thf(fact_221_ex__least__nat__le,axiom,
! [P: nat > $o,N2: nat] :
( ( P @ N2 )
=> ( ~ ( P @ ( zero_zero @ nat ) )
=> ? [K2: nat] :
( ( ord_less_eq @ nat @ K2 @ N2 )
& ! [I3: nat] :
( ( ord_less @ nat @ I3 @ K2 )
=> ~ ( P @ I3 ) )
& ( P @ K2 ) ) ) ) ).
% ex_least_nat_le
thf(fact_222_nat__le__linear,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq @ nat @ M2 @ N2 )
| ( ord_less_eq @ nat @ N2 @ M2 ) ) ).
% nat_le_linear
thf(fact_223_nat__less__le,axiom,
( ( ord_less @ nat )
= ( ^ [M: nat,N: nat] :
( ( ord_less_eq @ nat @ M @ N )
& ( M != N ) ) ) ) ).
% nat_less_le
thf(fact_224_le__antisym,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq @ nat @ M2 @ N2 )
=> ( ( ord_less_eq @ nat @ N2 @ M2 )
=> ( M2 = N2 ) ) ) ).
% le_antisym
thf(fact_225_eq__imp__le,axiom,
! [M2: nat,N2: nat] :
( ( M2 = N2 )
=> ( ord_less_eq @ nat @ M2 @ N2 ) ) ).
% eq_imp_le
thf(fact_226_le__trans,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq @ nat @ I @ J )
=> ( ( ord_less_eq @ nat @ J @ K )
=> ( ord_less_eq @ nat @ I @ K ) ) ) ).
% le_trans
thf(fact_227_le__refl,axiom,
! [N2: nat] : ( ord_less_eq @ nat @ N2 @ N2 ) ).
% le_refl
thf(fact_228_le__0__eq,axiom,
! [N2: nat] :
( ( ord_less_eq @ nat @ N2 @ ( zero_zero @ nat ) )
= ( N2
= ( zero_zero @ nat ) ) ) ).
% le_0_eq
thf(fact_229_less__eq__nat_Osimps_I1_J,axiom,
! [N2: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ N2 ) ).
% less_eq_nat.simps(1)
thf(fact_230_uniformly__Cauchy__onI,axiom,
! [B: $tType,A: $tType] :
( ( real_V2090557954_space @ B )
=> ! [X8: set @ A,F: nat > A > B] :
( ! [E3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E3 )
=> ? [M5: nat] :
! [X: A] :
( ( member @ A @ X @ X8 )
=> ! [M3: nat] :
( ( ord_less_eq @ nat @ M5 @ M3 )
=> ! [N5: nat] :
( ( ord_less_eq @ nat @ M5 @ N5 )
=> ( ord_less @ real @ ( real_V2000881966t_dist @ B @ ( F @ M3 @ X ) @ ( F @ N5 @ X ) ) @ E3 ) ) ) ) )
=> ( unifor760693226chy_on @ A @ B @ X8 @ F ) ) ) ).
% uniformly_Cauchy_onI
thf(fact_231_uniformly__Cauchy__on__def,axiom,
! [B: $tType,A: $tType] :
( ( real_V2090557954_space @ B )
=> ( ( unifor760693226chy_on @ A @ B )
= ( ^ [X9: set @ A,F2: nat > A > B] :
! [E4: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E4 )
=> ? [M6: nat] :
! [X3: A] :
( ( member @ A @ X3 @ X9 )
=> ! [M: nat] :
( ( ord_less_eq @ nat @ M6 @ M )
=> ! [N: nat] :
( ( ord_less_eq @ nat @ M6 @ N )
=> ( ord_less @ real @ ( real_V2000881966t_dist @ B @ ( F2 @ M @ X3 ) @ ( F2 @ N @ X3 ) ) @ E4 ) ) ) ) ) ) ) ) ).
% uniformly_Cauchy_on_def
thf(fact_232_infinite__descent__measure,axiom,
! [A: $tType,P: A > $o,V: A > nat,X2: A] :
( ! [X: A] :
( ~ ( P @ X )
=> ? [Y8: A] :
( ( ord_less @ nat @ ( V @ Y8 ) @ ( V @ X ) )
& ~ ( P @ Y8 ) ) )
=> ( P @ X2 ) ) ).
% infinite_descent_measure
thf(fact_233_linorder__neqE__nat,axiom,
! [X2: nat,Y: nat] :
( ( X2 != Y )
=> ( ~ ( ord_less @ nat @ X2 @ Y )
=> ( ord_less @ nat @ Y @ X2 ) ) ) ).
% linorder_neqE_nat
thf(fact_234_infinite__descent,axiom,
! [P: nat > $o,N2: nat] :
( ! [N5: nat] :
( ~ ( P @ N5 )
=> ? [M4: nat] :
( ( ord_less @ nat @ M4 @ N5 )
& ~ ( P @ M4 ) ) )
=> ( P @ N2 ) ) ).
% infinite_descent
thf(fact_235_nat__less__induct,axiom,
! [P: nat > $o,N2: nat] :
( ! [N5: nat] :
( ! [M4: nat] :
( ( ord_less @ nat @ M4 @ N5 )
=> ( P @ M4 ) )
=> ( P @ N5 ) )
=> ( P @ N2 ) ) ).
% nat_less_induct
thf(fact_236_less__irrefl__nat,axiom,
! [N2: nat] :
~ ( ord_less @ nat @ N2 @ N2 ) ).
% less_irrefl_nat
thf(fact_237_less__not__refl3,axiom,
! [S3: nat,T4: nat] :
( ( ord_less @ nat @ S3 @ T4 )
=> ( S3 != T4 ) ) ).
% less_not_refl3
thf(fact_238_less__not__refl2,axiom,
! [N2: nat,M2: nat] :
( ( ord_less @ nat @ N2 @ M2 )
=> ( M2 != N2 ) ) ).
% less_not_refl2
thf(fact_239_less__not__refl,axiom,
! [N2: nat] :
~ ( ord_less @ nat @ N2 @ N2 ) ).
% less_not_refl
thf(fact_240_nat__neq__iff,axiom,
! [M2: nat,N2: nat] :
( ( M2 != N2 )
= ( ( ord_less @ nat @ M2 @ N2 )
| ( ord_less @ nat @ N2 @ M2 ) ) ) ).
% nat_neq_iff
thf(fact_241_uniformly__Cauchy__onI_H,axiom,
! [B: $tType,A: $tType] :
( ( real_V2090557954_space @ B )
=> ! [X8: set @ A,F: nat > A > B] :
( ! [E3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E3 )
=> ? [M5: nat] :
! [X: A] :
( ( member @ A @ X @ X8 )
=> ! [M3: nat] :
( ( ord_less_eq @ nat @ M5 @ M3 )
=> ! [N5: nat] :
( ( ord_less @ nat @ M3 @ N5 )
=> ( ord_less @ real @ ( real_V2000881966t_dist @ B @ ( F @ M3 @ X ) @ ( F @ N5 @ X ) ) @ E3 ) ) ) ) )
=> ( unifor760693226chy_on @ A @ B @ X8 @ F ) ) ) ).
% uniformly_Cauchy_onI'
thf(fact_242_nat__descend__induct,axiom,
! [N2: nat,P: nat > $o,M2: nat] :
( ! [K2: nat] :
( ( ord_less @ nat @ N2 @ K2 )
=> ( P @ K2 ) )
=> ( ! [K2: nat] :
( ( ord_less_eq @ nat @ K2 @ N2 )
=> ( ! [I3: nat] :
( ( ord_less @ nat @ K2 @ I3 )
=> ( P @ I3 ) )
=> ( P @ K2 ) ) )
=> ( P @ M2 ) ) ) ).
% nat_descend_induct
thf(fact_243_Lattices__Big_Oex__has__greatest__nat,axiom,
! [A: $tType,P: A > $o,K: A,F: A > nat,B2: nat] :
( ( P @ K )
=> ( ! [Y4: A] :
( ( P @ Y4 )
=> ( ord_less @ nat @ ( F @ Y4 ) @ B2 ) )
=> ? [X: A] :
( ( P @ X )
& ! [Y8: A] :
( ( P @ Y8 )
=> ( ord_less_eq @ nat @ ( F @ Y8 ) @ ( F @ X ) ) ) ) ) ) ).
% Lattices_Big.ex_has_greatest_nat
thf(fact_244_ex__has__least__nat,axiom,
! [A: $tType,P: A > $o,K: A,M2: A > nat] :
( ( P @ K )
=> ? [X: A] :
( ( P @ X )
& ! [Y8: A] :
( ( P @ Y8 )
=> ( ord_less_eq @ nat @ ( M2 @ X ) @ ( M2 @ Y8 ) ) ) ) ) ).
% ex_has_least_nat
thf(fact_245_less__eq__real__def,axiom,
( ( ord_less_eq @ real )
= ( ^ [X3: real,Y7: real] :
( ( ord_less @ real @ X3 @ Y7 )
| ( X3 = Y7 ) ) ) ) ).
% less_eq_real_def
thf(fact_246_zero__prod__def,axiom,
! [A: $tType,B: $tType] :
( ( ( zero @ B )
& ( zero @ A ) )
=> ( ( zero_zero @ ( product_prod @ A @ B ) )
= ( product_Pair @ A @ B @ ( zero_zero @ A ) @ ( zero_zero @ B ) ) ) ) ).
% zero_prod_def
thf(fact_247_complete__real,axiom,
! [S2: set @ real] :
( ? [X7: real] : ( member @ real @ X7 @ S2 )
=> ( ? [Z4: real] :
! [X: real] :
( ( member @ real @ X @ S2 )
=> ( ord_less_eq @ real @ X @ Z4 ) )
=> ? [Y4: real] :
( ! [X7: real] :
( ( member @ real @ X7 @ S2 )
=> ( ord_less_eq @ real @ X7 @ Y4 ) )
& ! [Z4: real] :
( ! [X: real] :
( ( member @ real @ X @ S2 )
=> ( ord_less_eq @ real @ X @ Z4 ) )
=> ( ord_less_eq @ real @ Y4 @ Z4 ) ) ) ) ) ).
% complete_real
thf(fact_248_field__lbound__gt__zero,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [D1: A,D22: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ D1 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ D22 )
=> ? [E3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ E3 )
& ( ord_less @ A @ E3 @ D1 )
& ( ord_less @ A @ E3 @ D22 ) ) ) ) ) ).
% field_lbound_gt_zero
thf(fact_249_less__numeral__extra_I3_J,axiom,
! [A: $tType] :
( ( linord1659791738miring @ A )
=> ~ ( ord_less @ A @ ( zero_zero @ A ) @ ( zero_zero @ A ) ) ) ).
% less_numeral_extra(3)
thf(fact_250_complete__interval,axiom,
! [A: $tType] :
( ( condit1037483654norder @ A )
=> ! [A2: A,B2: A,P: A > $o] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( P @ A2 )
=> ( ~ ( P @ B2 )
=> ? [C3: A] :
( ( ord_less_eq @ A @ A2 @ C3 )
& ( ord_less_eq @ A @ C3 @ B2 )
& ! [X7: A] :
( ( ( ord_less_eq @ A @ A2 @ X7 )
& ( ord_less @ A @ X7 @ C3 ) )
=> ( P @ X7 ) )
& ! [D5: A] :
( ! [X: A] :
( ( ( ord_less_eq @ A @ A2 @ X )
& ( ord_less @ A @ X @ D5 ) )
=> ( P @ X ) )
=> ( ord_less_eq @ A @ D5 @ C3 ) ) ) ) ) ) ) ).
% complete_interval
thf(fact_251_ex__gt__or__lt,axiom,
! [A: $tType] :
( ( condit1656338222tinuum @ A )
=> ! [A2: A] :
? [B5: A] :
( ( ord_less @ A @ A2 @ B5 )
| ( ord_less @ A @ B5 @ A2 ) ) ) ).
% ex_gt_or_lt
thf(fact_252_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_253_minf_I8_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T4: A] :
? [Z3: A] :
! [X7: A] :
( ( ord_less @ A @ X7 @ Z3 )
=> ~ ( ord_less_eq @ A @ T4 @ X7 ) ) ) ).
% minf(8)
% 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 (61)
thf(tcon_Product__Type_Oprod___Euclidean__Space_Oeuclidean__space,axiom,
! [A4: $tType,A9: $tType] :
( ( ( euclid925273238_space @ A4 )
& ( euclid925273238_space @ A9 ) )
=> ( euclid925273238_space @ ( product_prod @ A4 @ A9 ) ) ) ).
thf(tcon_Real_Oreal___Euclidean__Space_Oeuclidean__space_1,axiom,
euclid925273238_space @ real ).
thf(tcon_fun___Topological__Spaces_Otopological__space,axiom,
! [A4: $tType,A9: $tType] :
( ( topolo503727757_space @ A9 )
=> ( topolo503727757_space @ ( A4 > A9 ) ) ) ).
thf(tcon_fun___Orderings_Oorder__top,axiom,
! [A4: $tType,A9: $tType] :
( ( order_top @ A9 )
=> ( order_top @ ( A4 > A9 ) ) ) ).
thf(tcon_fun___Orderings_Opreorder,axiom,
! [A4: $tType,A9: $tType] :
( ( preorder @ A9 )
=> ( preorder @ ( A4 > A9 ) ) ) ).
thf(tcon_fun___Orderings_Oorder,axiom,
! [A4: $tType,A9: $tType] :
( ( order @ A9 )
=> ( order @ ( A4 > A9 ) ) ) ).
thf(tcon_fun___Orderings_Otop,axiom,
! [A4: $tType,A9: $tType] :
( ( top @ A9 )
=> ( top @ ( A4 > A9 ) ) ) ).
thf(tcon_fun___Orderings_Oord,axiom,
! [A4: $tType,A9: $tType] :
( ( ord @ A9 )
=> ( ord @ ( A4 > A9 ) ) ) ).
thf(tcon_Nat_Onat___Conditionally__Complete__Lattices_Oconditionally__complete__linorder,axiom,
condit1037483654norder @ nat ).
thf(tcon_Nat_Onat___Groups_Ocanonically__ordered__monoid__add,axiom,
canoni770627133id_add @ nat ).
thf(tcon_Nat_Onat___Topological__Spaces_Otopological__space_2,axiom,
topolo503727757_space @ nat ).
thf(tcon_Nat_Onat___Rings_Olinordered__nonzero__semiring,axiom,
linord1659791738miring @ nat ).
thf(tcon_Nat_Onat___Orderings_Owellorder,axiom,
wellorder @ nat ).
thf(tcon_Nat_Onat___Orderings_Opreorder_3,axiom,
preorder @ nat ).
thf(tcon_Nat_Onat___Orderings_Olinorder,axiom,
linorder @ nat ).
thf(tcon_Nat_Onat___Orderings_Ono__top,axiom,
no_top @ nat ).
thf(tcon_Nat_Onat___Orderings_Oorder_4,axiom,
order @ nat ).
thf(tcon_Nat_Onat___Orderings_Oord_5,axiom,
ord @ nat ).
thf(tcon_Nat_Onat___Groups_Ozero,axiom,
zero @ nat ).
thf(tcon_Set_Oset___Orderings_Oorder__top_6,axiom,
! [A4: $tType] : ( order_top @ ( set @ A4 ) ) ).
thf(tcon_Set_Oset___Orderings_Opreorder_7,axiom,
! [A4: $tType] : ( preorder @ ( set @ A4 ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder_8,axiom,
! [A4: $tType] : ( order @ ( set @ A4 ) ) ).
thf(tcon_Set_Oset___Orderings_Otop_9,axiom,
! [A4: $tType] : ( top @ ( set @ A4 ) ) ).
thf(tcon_Set_Oset___Orderings_Oord_10,axiom,
! [A4: $tType] : ( ord @ ( set @ A4 ) ) ).
thf(tcon_Set_Oset___Groups_Ozero_11,axiom,
! [A4: $tType] :
( ( zero @ A4 )
=> ( zero @ ( set @ A4 ) ) ) ).
thf(tcon_HOL_Obool___Topological__Spaces_Otopological__space_12,axiom,
topolo503727757_space @ $o ).
thf(tcon_HOL_Obool___Orderings_Oorder__top_13,axiom,
order_top @ $o ).
thf(tcon_HOL_Obool___Orderings_Opreorder_14,axiom,
preorder @ $o ).
thf(tcon_HOL_Obool___Orderings_Olinorder_15,axiom,
linorder @ $o ).
thf(tcon_HOL_Obool___Orderings_Oorder_16,axiom,
order @ $o ).
thf(tcon_HOL_Obool___Orderings_Otop_17,axiom,
top @ $o ).
thf(tcon_HOL_Obool___Orderings_Oord_18,axiom,
ord @ $o ).
thf(tcon_Real_Oreal___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_19,axiom,
condit1037483654norder @ real ).
thf(tcon_Real_Oreal___Conditionally__Complete__Lattices_Olinear__continuum,axiom,
condit1656338222tinuum @ real ).
thf(tcon_Real_Oreal___Real__Vector__Spaces_Oreal__normed__vector,axiom,
real_V55928688vector @ real ).
thf(tcon_Real_Oreal___Topological__Spaces_Otopological__space_20,axiom,
topolo503727757_space @ real ).
thf(tcon_Real_Oreal___Rings_Olinordered__nonzero__semiring_21,axiom,
linord1659791738miring @ real ).
thf(tcon_Real_Oreal___Real__Vector__Spaces_Ocomplete__space,axiom,
real_V1759440057_space @ real ).
thf(tcon_Real_Oreal___Topological__Spaces_Operfect__space,axiom,
topolo890362671_space @ real ).
thf(tcon_Real_Oreal___Real__Vector__Spaces_Ometric__space,axiom,
real_V2090557954_space @ real ).
thf(tcon_Real_Oreal___Orderings_Odense__linorder,axiom,
dense_linorder @ real ).
thf(tcon_Real_Oreal___Fields_Olinordered__field,axiom,
linordered_field @ real ).
thf(tcon_Real_Oreal___Orderings_Odense__order,axiom,
dense_order @ real ).
thf(tcon_Real_Oreal___Orderings_Opreorder_22,axiom,
preorder @ real ).
thf(tcon_Real_Oreal___Orderings_Olinorder_23,axiom,
linorder @ real ).
thf(tcon_Real_Oreal___Orderings_Ono__top_24,axiom,
no_top @ real ).
thf(tcon_Real_Oreal___Orderings_Ono__bot,axiom,
no_bot @ real ).
thf(tcon_Real_Oreal___Orderings_Oorder_25,axiom,
order @ real ).
thf(tcon_Real_Oreal___Orderings_Oord_26,axiom,
ord @ real ).
thf(tcon_Real_Oreal___Groups_Ozero_27,axiom,
zero @ real ).
thf(tcon_Product__Type_Oprod___Real__Vector__Spaces_Oreal__normed__vector_28,axiom,
! [A4: $tType,A9: $tType] :
( ( ( real_V55928688vector @ A4 )
& ( real_V55928688vector @ A9 ) )
=> ( real_V55928688vector @ ( product_prod @ A4 @ A9 ) ) ) ).
thf(tcon_Product__Type_Oprod___Topological__Spaces_Otopological__space_29,axiom,
! [A4: $tType,A9: $tType] :
( ( ( topolo503727757_space @ A4 )
& ( topolo503727757_space @ A9 ) )
=> ( topolo503727757_space @ ( product_prod @ A4 @ A9 ) ) ) ).
thf(tcon_Product__Type_Oprod___Real__Vector__Spaces_Ocomplete__space_30,axiom,
! [A4: $tType,A9: $tType] :
( ( ( real_V1759440057_space @ A4 )
& ( real_V1759440057_space @ A9 ) )
=> ( real_V1759440057_space @ ( product_prod @ A4 @ A9 ) ) ) ).
thf(tcon_Product__Type_Oprod___Topological__Spaces_Operfect__space_31,axiom,
! [A4: $tType,A9: $tType] :
( ( ( euclid925273238_space @ A4 )
& ( euclid925273238_space @ A9 ) )
=> ( topolo890362671_space @ ( product_prod @ A4 @ A9 ) ) ) ).
thf(tcon_Product__Type_Oprod___Real__Vector__Spaces_Ometric__space_32,axiom,
! [A4: $tType,A9: $tType] :
( ( ( real_V2090557954_space @ A4 )
& ( real_V2090557954_space @ A9 ) )
=> ( real_V2090557954_space @ ( product_prod @ A4 @ A9 ) ) ) ).
thf(tcon_Product__Type_Oprod___Orderings_Oorder__top_33,axiom,
! [A4: $tType,A9: $tType] :
( ( ( order_top @ A4 )
& ( order_top @ A9 ) )
=> ( order_top @ ( product_prod @ A4 @ A9 ) ) ) ).
thf(tcon_Product__Type_Oprod___Orderings_Opreorder_34,axiom,
! [A4: $tType,A9: $tType] :
( ( ( preorder @ A4 )
& ( preorder @ A9 ) )
=> ( preorder @ ( product_prod @ A4 @ A9 ) ) ) ).
thf(tcon_Product__Type_Oprod___Orderings_Oorder_35,axiom,
! [A4: $tType,A9: $tType] :
( ( ( order @ A4 )
& ( order @ A9 ) )
=> ( order @ ( product_prod @ A4 @ A9 ) ) ) ).
thf(tcon_Product__Type_Oprod___Orderings_Otop_36,axiom,
! [A4: $tType,A9: $tType] :
( ( ( top @ A4 )
& ( top @ A9 ) )
=> ( top @ ( product_prod @ A4 @ A9 ) ) ) ).
thf(tcon_Product__Type_Oprod___Orderings_Oord_37,axiom,
! [A4: $tType,A9: $tType] :
( ( ( ord @ A4 )
& ( ord @ A9 ) )
=> ( ord @ ( product_prod @ A4 @ A9 ) ) ) ).
thf(tcon_Product__Type_Oprod___Groups_Ozero_38,axiom,
! [A4: $tType,A9: $tType] :
( ( ( zero @ A4 )
& ( zero @ A9 ) )
=> ( zero @ ( product_prod @ A4 @ A9 ) ) ) ).
% Free types (1)
thf(tfree_0,hypothesis,
real_V2090557954_space @ a ).
% Conjectures (2)
thf(conj_0,hypothesis,
! [A10: a,B8: real] :
( ( ( member @ ( product_prod @ a @ real ) @ ( product_Pair @ a @ real @ A10 @ B8 ) @ ( lower_261390618igraph @ a @ ( top_top @ ( set @ a ) ) @ f ) )
& ( ord_less @ real @ ( real_V2000881966t_dist @ ( product_prod @ a @ real ) @ ( product_Pair @ a @ real @ A10 @ B8 ) @ ( product_Pair @ a @ real @ x @ y ) ) @ e ) )
=> thesis ) ).
thf(conj_1,conjecture,
thesis ).
%------------------------------------------------------------------------------