TPTP Problem File: DAT150^1.p
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%------------------------------------------------------------------------------
% File : DAT150^1 : TPTP v8.2.0. Released v7.0.0.
% Domain : Data Structures
% Problem : Coinductive stream 331
% Version : [Bla16] axioms : Especial.
% English :
% Refs : [Loc10] Lochbihler (2010), Coinductive
% : [RB15] Reynolds & Blanchette (2015), A Decision Procedure for
% : [Bla16] Blanchette (2016), Email to Geoff Sutcliffe
% Source : [Bla16]
% Names : coinductive_stream__331.p [Bla16]
% Status : Theorem
% Rating : 0.00 v7.2.0, 0.25 v7.1.0
% Syntax : Number of formulae : 381 ( 142 unt; 68 typ; 0 def)
% Number of atoms : 839 ( 246 equ; 0 cnn)
% Maximal formula atoms : 13 ( 2 avg)
% Number of connectives : 3986 ( 56 ~; 1 |; 31 &;3476 @)
% ( 0 <=>; 422 =>; 0 <=; 0 <~>)
% Maximal formula depth : 19 ( 8 avg)
% Number of types : 3 ( 2 usr)
% Number of type conns : 299 ( 299 >; 0 *; 0 +; 0 <<)
% Number of symbols : 68 ( 66 usr; 1 con; 0-6 aty)
% Number of variables : 1113 ( 40 ^; 980 !; 25 ?;1113 :)
% ( 68 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2016-07-13 15:14:17.934
%------------------------------------------------------------------------------
%----Could-be-implicit typings (6)
thf(ty_t_Coinductive__List_Ollist,type,
coinductive_llist: $tType > $tType ).
thf(ty_t_Stream_Ostream,type,
stream: $tType > $tType ).
thf(ty_t_Set_Oset,type,
set: $tType > $tType ).
thf(ty_t_Nat_Onat,type,
nat: $tType ).
thf(ty_t_itself,type,
itself: $tType > $tType ).
thf(ty_tf_a,type,
a: $tType ).
%----Explicit typings (62)
thf(sy_cl_Finite__Set_Ofinite,type,
finite_finite:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_HOL_Otype,type,
type:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Ozero,type,
zero:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Obot,type,
bot:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Oord,type,
ord:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Otop,type,
top:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Oorder,type,
order:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Ono__bot,type,
no_bot:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Omonoid__add,type,
monoid_add:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Olinorder,type,
linorder:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Opreorder,type,
preorder:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Countable_Ocountable,type,
countable:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Oorder__bot,type,
order_bot:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Oorder__top,type,
order_top:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Complete__Lattices_OSup,type,
complete_Sup:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Rings_Olinordered__semidom,type,
linordered_semidom:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Complete__Partial__Order_Occpo,type,
comple1141879883l_ccpo:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Oordered__ab__semigroup__add,type,
ordere779506340up_add:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Complete__Lattices_Ocomplete__lattice,type,
comple187826305attice:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Ocanonically__ordered__monoid__add,type,
canoni770627133id_add:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Countable__Complete__Lattices_Ocountable__complete__lattice,type,
counta840220525attice:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Conditionally__Complete__Lattices_Oconditionally__complete__lattice,type,
condit378418413attice:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_c_Coinductive__List_Ogen__lset,type,
coinductive_gen_lset:
!>[A: $tType] : ( ( set @ A ) > ( coinductive_llist @ A ) > ( set @ A ) ) ).
thf(sy_c_Coinductive__List_Oinf__llist,type,
coindu68654304_llist:
!>[A: $tType] : ( ( nat > A ) > ( coinductive_llist @ A ) ) ).
thf(sy_c_Coinductive__List_Olfinite,type,
coinductive_lfinite:
!>[A: $tType] : ( ( coinductive_llist @ A ) > $o ) ).
thf(sy_c_Coinductive__List_Ollist_Olmap,type,
coinductive_lmap:
!>[A: $tType,Aa: $tType] : ( ( A > Aa ) > ( coinductive_llist @ A ) > ( coinductive_llist @ Aa ) ) ).
thf(sy_c_Coinductive__List_Ollist_Olset,type,
coinductive_lset:
!>[A: $tType] : ( ( coinductive_llist @ A ) > ( set @ A ) ) ).
thf(sy_c_Coinductive__List_Ollist_Opred__llist,type,
coindu543516966_llist:
!>[A: $tType] : ( ( A > $o ) > ( coinductive_llist @ A ) > $o ) ).
thf(sy_c_Coinductive__List_Olmember,type,
coinductive_lmember:
!>[A: $tType] : ( A > ( coinductive_llist @ A ) > $o ) ).
thf(sy_c_Coinductive__List_Olnth,type,
coinductive_lnth:
!>[A: $tType] : ( ( coinductive_llist @ A ) > nat > A ) ).
thf(sy_c_Coinductive__List_Omonoid__add__class_Ollistsum,type,
coindu780009021istsum:
!>[A: $tType] : ( ( coinductive_llist @ A ) > A ) ).
thf(sy_c_Coinductive__Stream__Mirabelle__dydkjoctes_Ollist__of__stream,type,
coindu1724414836stream:
!>[A: $tType] : ( ( stream @ A ) > ( coinductive_llist @ A ) ) ).
thf(sy_c_Coinductive__Stream__Mirabelle__dydkjoctes_Oof__seq,type,
coindu176146587of_seq:
!>[A: $tType] : ( ( nat > A ) > ( stream @ A ) ) ).
thf(sy_c_Coinductive__Stream__Mirabelle__dydkjoctes_Ostream__from__llist__setup_Ocr__stream,type,
coindu1183105481stream:
!>[A: $tType] : ( ( coinductive_llist @ A ) > ( stream @ A ) > $o ) ).
thf(sy_c_Coinductive__Stream__Mirabelle__dydkjoctes_Ostream__of__llist,type,
coindu2010755910_llist:
!>[A: $tType] : ( ( coinductive_llist @ A ) > ( stream @ A ) ) ).
thf(sy_c_Complete__Lattices_OSup__class_OSup,type,
complete_Sup_Sup:
!>[A: $tType] : ( ( set @ A ) > A ) ).
thf(sy_c_Complete__Partial__Order2_Ocompact,type,
comple2143767107ompact:
!>[A: $tType] : ( ( ( set @ A ) > A ) > ( A > A > $o ) > A > $o ) ).
thf(sy_c_Conditionally__Complete__Lattices_Opreorder__class_Obdd__above,type,
condit1810911227_above:
!>[A: $tType] : ( ( set @ A ) > $o ) ).
thf(sy_c_Countable_Ofrom__nat,type,
from_nat:
!>[A: $tType] : ( nat > A ) ).
thf(sy_c_Fun_Oswap,type,
swap:
!>[A: $tType,B: $tType] : ( A > A > ( A > B ) > A > B ) ).
thf(sy_c_Groups_Ozero__class_Ozero,type,
zero_zero:
!>[A: $tType] : A ).
thf(sy_c_Orderings_Obot__class_Obot,type,
bot_bot:
!>[A: $tType] : A ).
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_Pure_Otype,type,
type2:
!>[A: $tType] : ( itself @ A ) ).
thf(sy_c_Set_OBall,type,
ball:
!>[A: $tType] : ( ( set @ A ) > ( A > $o ) > $o ) ).
thf(sy_c_Set_OCollect,type,
collect:
!>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
thf(sy_c_Set_OPow,type,
pow:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( set @ A ) ) ) ).
thf(sy_c_Set_Obind,type,
bind:
!>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > ( set @ B ) ) > ( set @ B ) ) ).
thf(sy_c_Set_Oimage,type,
image:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ A ) > ( set @ B ) ) ).
thf(sy_c_Set_Othe__elem,type,
the_elem:
!>[A: $tType] : ( ( set @ A ) > A ) ).
thf(sy_c_Stream_Osmap2,type,
smap2:
!>[A: $tType,B: $tType,C: $tType] : ( ( A > B > C ) > ( stream @ A ) > ( stream @ B ) > ( stream @ C ) ) ).
thf(sy_c_Stream_Osmember,type,
smember:
!>[A: $tType] : ( A > ( stream @ A ) > $o ) ).
thf(sy_c_Stream_Osmerge,type,
smerge:
!>[A: $tType] : ( ( stream @ ( stream @ A ) ) > ( stream @ A ) ) ).
thf(sy_c_Stream_Osnth,type,
snth:
!>[A: $tType] : ( ( stream @ A ) > nat > A ) ).
thf(sy_c_Stream_Ostream_Opred__stream,type,
pred_stream:
!>[A: $tType] : ( ( A > $o ) > ( stream @ A ) > $o ) ).
thf(sy_c_Stream_Ostream_Osmap,type,
smap:
!>[A: $tType,Aa: $tType] : ( ( A > Aa ) > ( stream @ A ) > ( stream @ Aa ) ) ).
thf(sy_c_Stream_Ostream_Osset,type,
sset:
!>[A: $tType] : ( ( stream @ A ) > ( set @ A ) ) ).
thf(sy_c_Stream_Ostream__all,type,
stream_all:
!>[A: $tType] : ( ( A > $o ) > ( stream @ A ) > $o ) ).
thf(sy_c_Stream_Ostreams,type,
streams:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( stream @ A ) ) ) ).
thf(sy_c_member,type,
member:
!>[A: $tType] : ( A > ( set @ A ) > $o ) ).
thf(sy_v_f,type,
f: nat > a ).
%----Relevant facts (255)
thf(fact_0_stream__all__def,axiom,
! [A: $tType] :
( ( stream_all @ A )
= ( ^ [P: A > $o,S: stream @ A] :
! [P2: nat] : ( P @ ( snth @ A @ S @ P2 ) ) ) ) ).
% stream_all_def
thf(fact_1_snth__smap2,axiom,
! [B: $tType,A: $tType,C: $tType,F: B > C > A,S1: stream @ B,S2: stream @ C,N: nat] :
( ( snth @ A @ ( smap2 @ B @ C @ A @ F @ S1 @ S2 ) @ N )
= ( F @ ( snth @ B @ S1 @ N ) @ ( snth @ C @ S2 @ N ) ) ) ).
% snth_smap2
thf(fact_2_smap2__alt,axiom,
! [A: $tType,B: $tType,C: $tType,F: B > C > A,S1: stream @ B,S2: stream @ C,S3: stream @ A] :
( ( ( smap2 @ B @ C @ A @ F @ S1 @ S2 )
= S3 )
= ( ! [N2: nat] :
( ( F @ ( snth @ B @ S1 @ N2 ) @ ( snth @ C @ S2 @ N2 ) )
= ( snth @ A @ S3 @ N2 ) ) ) ) ).
% smap2_alt
thf(fact_3_of__seq_Oabs__eq,axiom,
! [A: $tType] :
( ( coindu176146587of_seq @ A )
= ( ^ [X: nat > A] : ( coindu2010755910_llist @ A @ ( coindu68654304_llist @ A @ X ) ) ) ) ).
% of_seq.abs_eq
thf(fact_4_of__seq_Orep__eq,axiom,
! [A: $tType,X2: nat > A] :
( ( coindu1724414836stream @ A @ ( coindu176146587of_seq @ A @ X2 ) )
= ( coindu68654304_llist @ A @ X2 ) ) ).
% of_seq.rep_eq
thf(fact_5_snth__in,axiom,
! [A: $tType,S3: stream @ A,X3: set @ A,N: nat] :
( ( member @ ( stream @ A ) @ S3 @ ( streams @ A @ X3 ) )
=> ( member @ A @ ( snth @ A @ S3 @ N ) @ X3 ) ) ).
% snth_in
thf(fact_6_streams__iff__snth,axiom,
! [A: $tType,S3: stream @ A,X3: set @ A] :
( ( member @ ( stream @ A ) @ S3 @ ( streams @ A @ X3 ) )
= ( ! [N2: nat] : ( member @ A @ ( snth @ A @ S3 @ N2 ) @ X3 ) ) ) ).
% streams_iff_snth
thf(fact_7_snth__smap,axiom,
! [A: $tType,B: $tType,F: B > A,S3: stream @ B,N: nat] :
( ( snth @ A @ ( smap @ B @ A @ F @ S3 ) @ N )
= ( F @ ( snth @ B @ S3 @ N ) ) ) ).
% snth_smap
thf(fact_8_snth__sset,axiom,
! [A: $tType,S3: stream @ A,N: nat] : ( member @ A @ ( snth @ A @ S3 @ N ) @ ( sset @ A @ S3 ) ) ).
% snth_sset
thf(fact_9_lnth__list__of__stream,axiom,
! [A: $tType,Xs: stream @ A] :
( ( coinductive_lnth @ A @ ( coindu1724414836stream @ A @ Xs ) )
= ( snth @ A @ Xs ) ) ).
% lnth_list_of_stream
thf(fact_10_smap__alt,axiom,
! [A: $tType,B: $tType,F: B > A,S3: stream @ B,S4: stream @ A] :
( ( ( smap @ B @ A @ F @ S3 )
= S4 )
= ( ! [N2: nat] :
( ( F @ ( snth @ B @ S3 @ N2 ) )
= ( snth @ A @ S4 @ N2 ) ) ) ) ).
% smap_alt
thf(fact_11_snth__sset__smerge,axiom,
! [A: $tType,Ss: stream @ ( stream @ A ),N: nat,M: nat] : ( member @ A @ ( snth @ A @ ( snth @ ( stream @ A ) @ Ss @ N ) @ M ) @ ( sset @ A @ ( smerge @ A @ Ss ) ) ) ).
% snth_sset_smerge
thf(fact_12_stream_ORep__inverse,axiom,
! [A: $tType,X2: stream @ A] :
( ( coindu2010755910_llist @ A @ ( coindu1724414836stream @ A @ X2 ) )
= X2 ) ).
% stream.Rep_inverse
thf(fact_13_stream__of__llist__llist__of__stream,axiom,
! [A: $tType,Xs: stream @ A] :
( ( coindu2010755910_llist @ A @ ( coindu1724414836stream @ A @ Xs ) )
= Xs ) ).
% stream_of_llist_llist_of_stream
thf(fact_14_smap__streams,axiom,
! [A: $tType,B: $tType,S3: stream @ A,A2: set @ A,F: A > B,B2: set @ B] :
( ( member @ ( stream @ A ) @ S3 @ ( streams @ A @ A2 ) )
=> ( ! [X4: A] :
( ( member @ A @ X4 @ A2 )
=> ( member @ B @ ( F @ X4 ) @ B2 ) )
=> ( member @ ( stream @ B ) @ ( smap @ A @ B @ F @ S3 ) @ ( streams @ B @ B2 ) ) ) ) ).
% smap_streams
thf(fact_15_stream_Omap__cong,axiom,
! [B: $tType,A: $tType,X2: stream @ A,Ya: stream @ A,F: A > B,G: A > B] :
( ( X2 = Ya )
=> ( ! [Z: A] :
( ( member @ A @ Z @ ( sset @ A @ Ya ) )
=> ( ( F @ Z )
= ( G @ Z ) ) )
=> ( ( smap @ A @ B @ F @ X2 )
= ( smap @ A @ B @ G @ Ya ) ) ) ) ).
% stream.map_cong
thf(fact_16_stream_Omap__cong0,axiom,
! [B: $tType,A: $tType,X2: stream @ A,F: A > B,G: A > B] :
( ! [Z: A] :
( ( member @ A @ Z @ ( sset @ A @ X2 ) )
=> ( ( F @ Z )
= ( G @ Z ) ) )
=> ( ( smap @ A @ B @ F @ X2 )
= ( smap @ A @ B @ G @ X2 ) ) ) ).
% stream.map_cong0
thf(fact_17_stream_Oinj__map__strong,axiom,
! [B: $tType,A: $tType,X2: stream @ A,Xa: stream @ A,F: A > B,Fa: A > B] :
( ! [Z: A,Za: A] :
( ( member @ A @ Z @ ( sset @ A @ X2 ) )
=> ( ( member @ A @ Za @ ( sset @ A @ Xa ) )
=> ( ( ( F @ Z )
= ( Fa @ Za ) )
=> ( Z = Za ) ) ) )
=> ( ( ( smap @ A @ B @ F @ X2 )
= ( smap @ A @ B @ Fa @ Xa ) )
=> ( X2 = Xa ) ) ) ).
% stream.inj_map_strong
thf(fact_18_stream_ORep__inject,axiom,
! [A: $tType,X2: stream @ A,Y: stream @ A] :
( ( ( coindu1724414836stream @ A @ X2 )
= ( coindu1724414836stream @ A @ Y ) )
= ( X2 = Y ) ) ).
% stream.Rep_inject
thf(fact_19_lnth__inf__llist,axiom,
! [A: $tType,F: nat > A,N: nat] :
( ( coinductive_lnth @ A @ ( coindu68654304_llist @ A @ F ) @ N )
= ( F @ N ) ) ).
% lnth_inf_llist
thf(fact_20_inf__llist__inj,axiom,
! [A: $tType,F: nat > A,G: nat > A] :
( ( ( coindu68654304_llist @ A @ F )
= ( coindu68654304_llist @ A @ G ) )
= ( F = G ) ) ).
% inf_llist_inj
thf(fact_21_Stream_Osmember__def,axiom,
! [A: $tType] :
( ( smember @ A )
= ( ^ [X: A,S: stream @ A] : ( member @ A @ X @ ( sset @ A @ S ) ) ) ) ).
% Stream.smember_def
thf(fact_22_stream__all__iff,axiom,
! [A: $tType] :
( ( stream_all @ A )
= ( ^ [P: A > $o,S: stream @ A] :
! [X: A] :
( ( member @ A @ X @ ( sset @ A @ S ) )
=> ( P @ X ) ) ) ) ).
% stream_all_iff
thf(fact_23_llist__of__stream__stream__of__llist,axiom,
! [A: $tType,Xs: coinductive_llist @ A] :
( ~ ( coinductive_lfinite @ A @ Xs )
=> ( ( coindu1724414836stream @ A @ ( coindu2010755910_llist @ A @ Xs ) )
= Xs ) ) ).
% llist_of_stream_stream_of_llist
thf(fact_24_lset__llist__of__stream,axiom,
! [A: $tType,Xs: stream @ A] :
( ( coinductive_lset @ A @ ( coindu1724414836stream @ A @ Xs ) )
= ( sset @ A @ Xs ) ) ).
% lset_llist_of_stream
thf(fact_25_sset__streams,axiom,
! [A: $tType,S3: stream @ A,A2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( sset @ A @ S3 ) @ A2 )
=> ( member @ ( stream @ A ) @ S3 @ ( streams @ A @ A2 ) ) ) ).
% sset_streams
thf(fact_26_streams__sset,axiom,
! [A: $tType,S3: stream @ A,A2: set @ A] :
( ( member @ ( stream @ A ) @ S3 @ ( streams @ A @ A2 ) )
=> ( ord_less_eq @ ( set @ A ) @ ( sset @ A @ S3 ) @ A2 ) ) ).
% streams_sset
thf(fact_27_streams__iff__sset,axiom,
! [A: $tType,S3: stream @ A,A2: set @ A] :
( ( member @ ( stream @ A ) @ S3 @ ( streams @ A @ A2 ) )
= ( ord_less_eq @ ( set @ A ) @ ( sset @ A @ S3 ) @ A2 ) ) ).
% streams_iff_sset
thf(fact_28_lmap__llist__of__stream,axiom,
! [A: $tType,B: $tType,F: B > A,Xs: stream @ B] :
( ( coinductive_lmap @ B @ A @ F @ ( coindu1724414836stream @ B @ Xs ) )
= ( coindu1724414836stream @ A @ ( smap @ B @ A @ F @ Xs ) ) ) ).
% lmap_llist_of_stream
thf(fact_29_stream_Oset__map,axiom,
! [B: $tType,A: $tType,F: A > B,V: stream @ A] :
( ( sset @ B @ ( smap @ A @ B @ F @ V ) )
= ( image @ A @ B @ F @ ( sset @ A @ V ) ) ) ).
% stream.set_map
thf(fact_30_lfinite__lmap,axiom,
! [A: $tType,B: $tType,F: B > A,Xs: coinductive_llist @ B] :
( ( coinductive_lfinite @ A @ ( coinductive_lmap @ B @ A @ F @ Xs ) )
= ( coinductive_lfinite @ B @ Xs ) ) ).
% lfinite_lmap
thf(fact_31_llist_Oset__map,axiom,
! [B: $tType,A: $tType,F: A > B,V: coinductive_llist @ A] :
( ( coinductive_lset @ B @ ( coinductive_lmap @ A @ B @ F @ V ) )
= ( image @ A @ B @ F @ ( coinductive_lset @ A @ V ) ) ) ).
% llist.set_map
thf(fact_32_inf__llist__lnth,axiom,
! [A: $tType,Xs: coinductive_llist @ A] :
( ~ ( coinductive_lfinite @ A @ Xs )
=> ( ( coindu68654304_llist @ A @ ( coinductive_lnth @ A @ Xs ) )
= Xs ) ) ).
% inf_llist_lnth
thf(fact_33_sset__stream__of__llist,axiom,
! [A: $tType,Xs: coinductive_llist @ A] :
( ~ ( coinductive_lfinite @ A @ Xs )
=> ( ~ ( coinductive_lfinite @ A @ Xs )
=> ( ( sset @ A @ ( coindu2010755910_llist @ A @ Xs ) )
= ( coinductive_lset @ A @ Xs ) ) ) ) ).
% sset_stream_of_llist
thf(fact_34_smap__stream__of__llist,axiom,
! [B: $tType,A: $tType,Xs: coinductive_llist @ A,F: A > B] :
( ~ ( coinductive_lfinite @ A @ Xs )
=> ( ( smap @ A @ B @ F @ ( coindu2010755910_llist @ A @ Xs ) )
= ( coindu2010755910_llist @ B @ ( coinductive_lmap @ A @ B @ F @ Xs ) ) ) ) ).
% smap_stream_of_llist
thf(fact_35_llist_Oinj__map__strong,axiom,
! [B: $tType,A: $tType,X2: coinductive_llist @ A,Xa: coinductive_llist @ A,F: A > B,Fa: A > B] :
( ! [Z: A,Za: A] :
( ( member @ A @ Z @ ( coinductive_lset @ A @ X2 ) )
=> ( ( member @ A @ Za @ ( coinductive_lset @ A @ Xa ) )
=> ( ( ( F @ Z )
= ( Fa @ Za ) )
=> ( Z = Za ) ) ) )
=> ( ( ( coinductive_lmap @ A @ B @ F @ X2 )
= ( coinductive_lmap @ A @ B @ Fa @ Xa ) )
=> ( X2 = Xa ) ) ) ).
% llist.inj_map_strong
thf(fact_36_llist_Omap__cong0,axiom,
! [B: $tType,A: $tType,X2: coinductive_llist @ A,F: A > B,G: A > B] :
( ! [Z: A] :
( ( member @ A @ Z @ ( coinductive_lset @ A @ X2 ) )
=> ( ( F @ Z )
= ( G @ Z ) ) )
=> ( ( coinductive_lmap @ A @ B @ F @ X2 )
= ( coinductive_lmap @ A @ B @ G @ X2 ) ) ) ).
% llist.map_cong0
thf(fact_37_llist_Omap__cong,axiom,
! [B: $tType,A: $tType,X2: coinductive_llist @ A,Ya: coinductive_llist @ A,F: A > B,G: A > B] :
( ( X2 = Ya )
=> ( ! [Z: A] :
( ( member @ A @ Z @ ( coinductive_lset @ A @ Ya ) )
=> ( ( F @ Z )
= ( G @ Z ) ) )
=> ( ( coinductive_lmap @ A @ B @ F @ X2 )
= ( coinductive_lmap @ A @ B @ G @ Ya ) ) ) ) ).
% llist.map_cong
thf(fact_38_wlog__linorder__le,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [P3: A > A > $o,B3: A,A3: A] :
( ! [A4: A,B4: A] :
( ( ord_less_eq @ A @ A4 @ B4 )
=> ( P3 @ A4 @ B4 ) )
=> ( ( ( P3 @ B3 @ A3 )
=> ( P3 @ A3 @ B3 ) )
=> ( P3 @ A3 @ B3 ) ) ) ) ).
% wlog_linorder_le
thf(fact_39_streams__mono2,axiom,
! [A: $tType,S5: set @ A,T: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ S5 @ T )
=> ( ord_less_eq @ ( set @ ( stream @ A ) ) @ ( streams @ A @ S5 ) @ ( streams @ A @ T ) ) ) ).
% streams_mono2
thf(fact_40_lfinite__inf__llist,axiom,
! [A: $tType,F: nat > A] :
~ ( coinductive_lfinite @ A @ ( coindu68654304_llist @ A @ F ) ) ).
% lfinite_inf_llist
thf(fact_41_lfinite__llist__of__stream,axiom,
! [A: $tType,Xs: stream @ A] :
~ ( coinductive_lfinite @ A @ ( coindu1724414836stream @ A @ Xs ) ) ).
% lfinite_llist_of_stream
thf(fact_42_streams__mono,axiom,
! [A: $tType,S3: stream @ A,A2: set @ A,B2: set @ A] :
( ( member @ ( stream @ A ) @ S3 @ ( streams @ A @ A2 ) )
=> ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ( member @ ( stream @ A ) @ S3 @ ( streams @ A @ B2 ) ) ) ) ).
% streams_mono
thf(fact_43_not__False__in__image__Ball,axiom,
! [A: $tType,P3: A > $o,A2: set @ A] :
( ( ~ ( member @ $o @ $false @ ( image @ A @ $o @ P3 @ A2 ) ) )
= ( ! [X: A] :
( ( member @ A @ X @ A2 )
=> ( P3 @ X ) ) ) ) ).
% not_False_in_image_Ball
thf(fact_44_subset__antisym,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ( ( ord_less_eq @ ( set @ A ) @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% subset_antisym
thf(fact_45_mem__Collect__eq,axiom,
! [A: $tType,A3: A,P3: A > $o] :
( ( member @ A @ A3 @ ( collect @ A @ P3 ) )
= ( P3 @ A3 ) ) ).
% mem_Collect_eq
thf(fact_46_Collect__mem__eq,axiom,
! [A: $tType,A2: set @ A] :
( ( collect @ A
@ ^ [X: A] : ( member @ A @ X @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_47_Collect__cong,axiom,
! [A: $tType,P3: A > $o,Q: A > $o] :
( ! [X4: A] :
( ( P3 @ X4 )
= ( Q @ X4 ) )
=> ( ( collect @ A @ P3 )
= ( collect @ A @ Q ) ) ) ).
% Collect_cong
thf(fact_48_ext,axiom,
! [B: $tType,A: $tType,F: A > B,G: A > B] :
( ! [X4: A] :
( ( F @ X4 )
= ( G @ X4 ) )
=> ( F = G ) ) ).
% ext
thf(fact_49_subsetI,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] :
( ! [X4: A] :
( ( member @ A @ X4 @ A2 )
=> ( member @ A @ X4 @ B2 ) )
=> ( ord_less_eq @ ( set @ A ) @ A2 @ B2 ) ) ).
% subsetI
thf(fact_50_image__eqI,axiom,
! [A: $tType,B: $tType,B3: A,F: B > A,X2: B,A2: set @ B] :
( ( B3
= ( F @ X2 ) )
=> ( ( member @ B @ X2 @ A2 )
=> ( member @ A @ B3 @ ( image @ B @ A @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_51_order__refl,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X2: A] : ( ord_less_eq @ A @ X2 @ X2 ) ) ).
% order_refl
thf(fact_52_Ball__Collect,axiom,
! [A: $tType] :
( ( ball @ A )
= ( ^ [A5: set @ A,P: A > $o] : ( ord_less_eq @ ( set @ A ) @ A5 @ ( collect @ A @ P ) ) ) ) ).
% Ball_Collect
thf(fact_53_subset__image__iff,axiom,
! [A: $tType,B: $tType,B2: set @ A,F: B > A,A2: set @ B] :
( ( ord_less_eq @ ( set @ A ) @ B2 @ ( image @ B @ A @ F @ A2 ) )
= ( ? [AA: set @ B] :
( ( ord_less_eq @ ( set @ B ) @ AA @ A2 )
& ( B2
= ( image @ B @ A @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_54_image__subset__iff,axiom,
! [A: $tType,B: $tType,F: B > A,A2: set @ B,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( image @ B @ A @ F @ A2 ) @ B2 )
= ( ! [X: B] :
( ( member @ B @ X @ A2 )
=> ( member @ A @ ( F @ X ) @ B2 ) ) ) ) ).
% image_subset_iff
thf(fact_55_subset__imageE,axiom,
! [A: $tType,B: $tType,B2: set @ A,F: B > A,A2: set @ B] :
( ( ord_less_eq @ ( set @ A ) @ B2 @ ( image @ B @ A @ F @ A2 ) )
=> ~ ! [C2: set @ B] :
( ( ord_less_eq @ ( set @ B ) @ C2 @ A2 )
=> ( B2
!= ( image @ B @ A @ F @ C2 ) ) ) ) ).
% subset_imageE
thf(fact_56_le__funD,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B @ ( type2 @ 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_57_le__funE,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B @ ( type2 @ 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_58_le__funI,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B @ ( type2 @ B ) )
=> ! [F: A > B,G: A > B] :
( ! [X4: A] : ( ord_less_eq @ B @ ( F @ X4 ) @ ( G @ X4 ) )
=> ( ord_less_eq @ ( A > B ) @ F @ G ) ) ) ).
% le_funI
thf(fact_59_le__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B @ ( type2 @ B ) )
=> ( ( ord_less_eq @ ( A > B ) )
= ( ^ [F2: A > B,G2: A > B] :
! [X: A] : ( ord_less_eq @ B @ ( F2 @ X ) @ ( G2 @ X ) ) ) ) ) ).
% le_fun_def
thf(fact_60_order__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B @ ( type2 @ B ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A3: A,F: B > A,B3: B,C3: B] :
( ( ord_less_eq @ A @ A3 @ ( F @ B3 ) )
=> ( ( ord_less_eq @ B @ B3 @ C3 )
=> ( ! [X4: B,Y2: B] :
( ( ord_less_eq @ B @ X4 @ Y2 )
=> ( ord_less_eq @ A @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq @ A @ A3 @ ( F @ C3 ) ) ) ) ) ) ).
% order_subst1
thf(fact_61_order__subst2,axiom,
! [A: $tType,C: $tType] :
( ( ( order @ C @ ( type2 @ C ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A3: A,B3: A,F: A > C,C3: C] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( ord_less_eq @ C @ ( F @ B3 ) @ C3 )
=> ( ! [X4: A,Y2: A] :
( ( ord_less_eq @ A @ X4 @ Y2 )
=> ( ord_less_eq @ C @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq @ C @ ( F @ A3 ) @ C3 ) ) ) ) ) ).
% order_subst2
thf(fact_62_ord__eq__le__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B @ ( type2 @ B ) )
& ( ord @ A @ ( type2 @ A ) ) )
=> ! [A3: A,F: B > A,B3: B,C3: B] :
( ( A3
= ( F @ B3 ) )
=> ( ( ord_less_eq @ B @ B3 @ C3 )
=> ( ! [X4: B,Y2: B] :
( ( ord_less_eq @ B @ X4 @ Y2 )
=> ( ord_less_eq @ A @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq @ A @ A3 @ ( F @ C3 ) ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_63_ord__le__eq__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B @ ( type2 @ B ) )
& ( ord @ A @ ( type2 @ A ) ) )
=> ! [A3: A,B3: A,F: A > B,C3: B] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( ( F @ B3 )
= C3 )
=> ( ! [X4: A,Y2: A] :
( ( ord_less_eq @ A @ X4 @ Y2 )
=> ( ord_less_eq @ B @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq @ B @ ( F @ A3 ) @ C3 ) ) ) ) ) ).
% ord_le_eq_subst
thf(fact_64_eq__iff,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ( ( ^ [Y3: A,Z2: A] : Y3 = Z2 )
= ( ^ [X: A,Y4: A] :
( ( ord_less_eq @ A @ X @ Y4 )
& ( ord_less_eq @ A @ Y4 @ X ) ) ) ) ) ).
% eq_iff
thf(fact_65_antisym,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [X2: A,Y: A] :
( ( ord_less_eq @ A @ X2 @ Y )
=> ( ( ord_less_eq @ A @ Y @ X2 )
=> ( X2 = Y ) ) ) ) ).
% antisym
thf(fact_66_linear,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X2: A,Y: A] :
( ( ord_less_eq @ A @ X2 @ Y )
| ( ord_less_eq @ A @ Y @ X2 ) ) ) ).
% linear
thf(fact_67_eq__refl,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X2: A,Y: A] :
( ( X2 = Y )
=> ( ord_less_eq @ A @ X2 @ Y ) ) ) ).
% eq_refl
thf(fact_68_le__cases,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X2: A,Y: A] :
( ~ ( ord_less_eq @ A @ X2 @ Y )
=> ( ord_less_eq @ A @ Y @ X2 ) ) ) ).
% le_cases
thf(fact_69_order_Otrans,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A3: A,B3: A,C3: A] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( ord_less_eq @ A @ B3 @ C3 )
=> ( ord_less_eq @ A @ A3 @ C3 ) ) ) ) ).
% order.trans
thf(fact_70_le__cases3,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X2: A,Y: A,Z3: A] :
( ( ( ord_less_eq @ A @ X2 @ Y )
=> ~ ( ord_less_eq @ A @ Y @ Z3 ) )
=> ( ( ( ord_less_eq @ A @ Y @ X2 )
=> ~ ( ord_less_eq @ A @ X2 @ Z3 ) )
=> ( ( ( ord_less_eq @ A @ X2 @ Z3 )
=> ~ ( ord_less_eq @ A @ Z3 @ Y ) )
=> ( ( ( ord_less_eq @ A @ Z3 @ Y )
=> ~ ( ord_less_eq @ A @ Y @ X2 ) )
=> ( ( ( ord_less_eq @ A @ Y @ Z3 )
=> ~ ( ord_less_eq @ A @ Z3 @ X2 ) )
=> ~ ( ( ord_less_eq @ A @ Z3 @ X2 )
=> ~ ( ord_less_eq @ A @ X2 @ Y ) ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_71_antisym__conv,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [Y: A,X2: A] :
( ( ord_less_eq @ A @ Y @ X2 )
=> ( ( ord_less_eq @ A @ X2 @ Y )
= ( X2 = Y ) ) ) ) ).
% antisym_conv
thf(fact_72_ord__eq__le__trans,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [A3: A,B3: A,C3: A] :
( ( A3 = B3 )
=> ( ( ord_less_eq @ A @ B3 @ C3 )
=> ( ord_less_eq @ A @ A3 @ C3 ) ) ) ) ).
% ord_eq_le_trans
thf(fact_73_ord__le__eq__trans,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [A3: A,B3: A,C3: A] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( B3 = C3 )
=> ( ord_less_eq @ A @ A3 @ C3 ) ) ) ) ).
% ord_le_eq_trans
thf(fact_74_order__class_Oorder_Oantisym,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A3: A,B3: A] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( ord_less_eq @ A @ B3 @ A3 )
=> ( A3 = B3 ) ) ) ) ).
% order_class.order.antisym
thf(fact_75_order__trans,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X2: A,Y: A,Z3: A] :
( ( ord_less_eq @ A @ X2 @ Y )
=> ( ( ord_less_eq @ A @ Y @ Z3 )
=> ( ord_less_eq @ A @ X2 @ Z3 ) ) ) ) ).
% order_trans
thf(fact_76_dual__order_Orefl,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A3: A] : ( ord_less_eq @ A @ A3 @ A3 ) ) ).
% dual_order.refl
thf(fact_77_linorder__wlog,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [P3: A > A > $o,A3: A,B3: A] :
( ! [A4: A,B4: A] :
( ( ord_less_eq @ A @ A4 @ B4 )
=> ( P3 @ A4 @ B4 ) )
=> ( ! [A4: A,B4: A] :
( ( P3 @ B4 @ A4 )
=> ( P3 @ A4 @ B4 ) )
=> ( P3 @ A3 @ B3 ) ) ) ) ).
% linorder_wlog
thf(fact_78_dual__order_Otrans,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [B3: A,A3: A,C3: A] :
( ( ord_less_eq @ A @ B3 @ A3 )
=> ( ( ord_less_eq @ A @ C3 @ B3 )
=> ( ord_less_eq @ A @ C3 @ A3 ) ) ) ) ).
% dual_order.trans
thf(fact_79_dual__order_Oantisym,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [B3: A,A3: A] :
( ( ord_less_eq @ A @ B3 @ A3 )
=> ( ( ord_less_eq @ A @ A3 @ B3 )
=> ( A3 = B3 ) ) ) ) ).
% dual_order.antisym
thf(fact_80_imageI,axiom,
! [B: $tType,A: $tType,X2: A,A2: set @ A,F: A > B] :
( ( member @ A @ X2 @ A2 )
=> ( member @ B @ ( F @ X2 ) @ ( image @ A @ B @ F @ A2 ) ) ) ).
% imageI
thf(fact_81_image__iff,axiom,
! [A: $tType,B: $tType,Z3: A,F: B > A,A2: set @ B] :
( ( member @ A @ Z3 @ ( image @ B @ A @ F @ A2 ) )
= ( ? [X: B] :
( ( member @ B @ X @ A2 )
& ( Z3
= ( F @ X ) ) ) ) ) ).
% image_iff
thf(fact_82_bex__imageD,axiom,
! [A: $tType,B: $tType,F: B > A,A2: set @ B,P3: A > $o] :
( ? [X5: A] :
( ( member @ A @ X5 @ ( image @ B @ A @ F @ A2 ) )
& ( P3 @ X5 ) )
=> ? [X4: B] :
( ( member @ B @ X4 @ A2 )
& ( P3 @ ( F @ X4 ) ) ) ) ).
% bex_imageD
thf(fact_83_image__cong,axiom,
! [B: $tType,A: $tType,M2: set @ A,N3: set @ A,F: A > B,G: A > B] :
( ( M2 = N3 )
=> ( ! [X4: A] :
( ( member @ A @ X4 @ N3 )
=> ( ( F @ X4 )
= ( G @ X4 ) ) )
=> ( ( image @ A @ B @ F @ M2 )
= ( image @ A @ B @ G @ N3 ) ) ) ) ).
% image_cong
thf(fact_84_ball__imageD,axiom,
! [A: $tType,B: $tType,F: B > A,A2: set @ B,P3: A > $o] :
( ! [X4: A] :
( ( member @ A @ X4 @ ( image @ B @ A @ F @ A2 ) )
=> ( P3 @ X4 ) )
=> ! [X5: B] :
( ( member @ B @ X5 @ A2 )
=> ( P3 @ ( F @ X5 ) ) ) ) ).
% ball_imageD
thf(fact_85_rev__image__eqI,axiom,
! [B: $tType,A: $tType,X2: A,A2: set @ A,B3: B,F: A > B] :
( ( member @ A @ X2 @ A2 )
=> ( ( B3
= ( F @ X2 ) )
=> ( member @ B @ B3 @ ( image @ A @ B @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_86_Inf_OINF__cong,axiom,
! [A: $tType,B: $tType,A2: set @ B,B2: set @ B,C4: B > A,D: B > A,Inf: ( set @ A ) > A] :
( ( A2 = B2 )
=> ( ! [X4: B] :
( ( member @ B @ X4 @ B2 )
=> ( ( C4 @ X4 )
= ( D @ X4 ) ) )
=> ( ( Inf @ ( image @ B @ A @ C4 @ A2 ) )
= ( Inf @ ( image @ B @ A @ D @ B2 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_87_Sup_OSUP__cong,axiom,
! [A: $tType,B: $tType,A2: set @ B,B2: set @ B,C4: B > A,D: B > A,Sup: ( set @ A ) > A] :
( ( A2 = B2 )
=> ( ! [X4: B] :
( ( member @ B @ X4 @ B2 )
=> ( ( C4 @ X4 )
= ( D @ X4 ) ) )
=> ( ( Sup @ ( image @ B @ A @ C4 @ A2 ) )
= ( Sup @ ( image @ B @ A @ D @ B2 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_88_set__mp,axiom,
! [A: $tType,A2: set @ A,B2: set @ A,X2: A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ( ( member @ A @ X2 @ A2 )
=> ( member @ A @ X2 @ B2 ) ) ) ).
% set_mp
thf(fact_89_in__mono,axiom,
! [A: $tType,A2: set @ A,B2: set @ A,X2: A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ( ( member @ A @ X2 @ A2 )
=> ( member @ A @ X2 @ B2 ) ) ) ).
% in_mono
thf(fact_90_subsetD,axiom,
! [A: $tType,A2: set @ A,B2: set @ A,C3: A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ( ( member @ A @ C3 @ A2 )
=> ( member @ A @ C3 @ B2 ) ) ) ).
% subsetD
thf(fact_91_subsetCE,axiom,
! [A: $tType,A2: set @ A,B2: set @ A,C3: A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ( ( member @ A @ C3 @ A2 )
=> ( member @ A @ C3 @ B2 ) ) ) ).
% subsetCE
thf(fact_92_equalityE,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] :
( ( A2 = B2 )
=> ~ ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ~ ( ord_less_eq @ ( set @ A ) @ B2 @ A2 ) ) ) ).
% equalityE
thf(fact_93_subset__eq,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A5: set @ A,B5: set @ A] :
! [X: A] :
( ( member @ A @ X @ A5 )
=> ( member @ A @ X @ B5 ) ) ) ) ).
% subset_eq
thf(fact_94_equalityD1,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] :
( ( A2 = B2 )
=> ( ord_less_eq @ ( set @ A ) @ A2 @ B2 ) ) ).
% equalityD1
thf(fact_95_equalityD2,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] :
( ( A2 = B2 )
=> ( ord_less_eq @ ( set @ A ) @ B2 @ A2 ) ) ).
% equalityD2
thf(fact_96_set__rev__mp,axiom,
! [A: $tType,X2: A,A2: set @ A,B2: set @ A] :
( ( member @ A @ X2 @ A2 )
=> ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ( member @ A @ X2 @ B2 ) ) ) ).
% set_rev_mp
thf(fact_97_subset__iff,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A5: set @ A,B5: set @ A] :
! [T2: A] :
( ( member @ A @ T2 @ A5 )
=> ( member @ A @ T2 @ B5 ) ) ) ) ).
% subset_iff
thf(fact_98_rev__subsetD,axiom,
! [A: $tType,C3: A,A2: set @ A,B2: set @ A] :
( ( member @ A @ C3 @ A2 )
=> ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ( member @ A @ C3 @ B2 ) ) ) ).
% rev_subsetD
thf(fact_99_subset__refl,axiom,
! [A: $tType,A2: set @ A] : ( ord_less_eq @ ( set @ A ) @ A2 @ A2 ) ).
% subset_refl
thf(fact_100_Collect__mono,axiom,
! [A: $tType,P3: A > $o,Q: A > $o] :
( ! [X4: A] :
( ( P3 @ X4 )
=> ( Q @ X4 ) )
=> ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P3 ) @ ( collect @ A @ Q ) ) ) ).
% Collect_mono
thf(fact_101_subset__trans,axiom,
! [A: $tType,A2: set @ A,B2: set @ A,C4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ( ( ord_less_eq @ ( set @ A ) @ B2 @ C4 )
=> ( ord_less_eq @ ( set @ A ) @ A2 @ C4 ) ) ) ).
% subset_trans
thf(fact_102_set__eq__subset,axiom,
! [A: $tType] :
( ( ^ [Y3: set @ A,Z2: set @ A] : Y3 = Z2 )
= ( ^ [A5: set @ A,B5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ B5 )
& ( ord_less_eq @ ( set @ A ) @ B5 @ A5 ) ) ) ) ).
% set_eq_subset
thf(fact_103_contra__subsetD,axiom,
! [A: $tType,A2: set @ A,B2: set @ A,C3: A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ( ~ ( member @ A @ C3 @ B2 )
=> ~ ( member @ A @ C3 @ A2 ) ) ) ).
% contra_subsetD
thf(fact_104_Collect__mono__iff,axiom,
! [A: $tType,P3: A > $o,Q: A > $o] :
( ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P3 ) @ ( collect @ A @ Q ) )
= ( ! [X: A] :
( ( P3 @ X )
=> ( Q @ X ) ) ) ) ).
% Collect_mono_iff
thf(fact_105_Ball__def,axiom,
! [A: $tType] :
( ( ball @ A )
= ( ^ [A5: set @ A,P: A > $o] :
! [X: A] :
( ( member @ A @ X @ A5 )
=> ( P @ X ) ) ) ) ).
% Ball_def
thf(fact_106_image__mono,axiom,
! [B: $tType,A: $tType,A2: set @ A,B2: set @ A,F: A > B] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F @ A2 ) @ ( image @ A @ B @ F @ B2 ) ) ) ).
% image_mono
thf(fact_107_image__subsetI,axiom,
! [A: $tType,B: $tType,A2: set @ A,F: A > B,B2: set @ B] :
( ! [X4: A] :
( ( member @ A @ X4 @ A2 )
=> ( member @ B @ ( F @ X4 ) @ B2 ) )
=> ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F @ A2 ) @ B2 ) ) ).
% image_subsetI
thf(fact_108_cr__streamI,axiom,
! [A: $tType,Xs: coinductive_llist @ A] :
( ~ ( coinductive_lfinite @ A @ Xs )
=> ( coindu1183105481stream @ A @ Xs @ ( coindu2010755910_llist @ A @ Xs ) ) ) ).
% cr_streamI
thf(fact_109_llistsum__mono,axiom,
! [B: $tType,A: $tType] :
( ( ( monoid_add @ B @ ( type2 @ B ) )
& ( ordere779506340up_add @ B @ ( type2 @ B ) ) )
=> ! [Xs: coinductive_llist @ A,F: A > B,G: A > B] :
( ! [X4: A] :
( ( member @ A @ X4 @ ( coinductive_lset @ A @ Xs ) )
=> ( ord_less_eq @ B @ ( F @ X4 ) @ ( G @ X4 ) ) )
=> ( ord_less_eq @ B @ ( coindu780009021istsum @ B @ ( coinductive_lmap @ A @ B @ F @ Xs ) ) @ ( coindu780009021istsum @ B @ ( coinductive_lmap @ A @ B @ G @ Xs ) ) ) ) ) ).
% llistsum_mono
thf(fact_110_lset__lmember,axiom,
! [A: $tType,X2: A,Xs: coinductive_llist @ A] :
( ( member @ A @ X2 @ ( coinductive_lset @ A @ Xs ) )
= ( coinductive_lmember @ A @ X2 @ Xs ) ) ).
% lset_lmember
thf(fact_111_ball__reg,axiom,
! [A: $tType,R: set @ A,P3: A > $o,Q: A > $o] :
( ! [X4: A] :
( ( member @ A @ X4 @ R )
=> ( ( P3 @ X4 )
=> ( Q @ X4 ) ) )
=> ( ! [X4: A] :
( ( member @ A @ X4 @ R )
=> ( P3 @ X4 ) )
=> ! [X5: A] :
( ( member @ A @ X5 @ R )
=> ( Q @ X5 ) ) ) ) ).
% ball_reg
thf(fact_112_stream_Opred__set,axiom,
! [A: $tType] :
( ( pred_stream @ A )
= ( ^ [P: A > $o,X: stream @ A] :
! [Y4: A] :
( ( member @ A @ Y4 @ ( sset @ A @ X ) )
=> ( P @ Y4 ) ) ) ) ).
% stream.pred_set
thf(fact_113_lfp_Oleq__refl,axiom,
! [A: $tType] :
( ( comple187826305attice @ A @ ( type2 @ A ) )
=> ! [X2: A] : ( ord_less_eq @ A @ X2 @ X2 ) ) ).
% lfp.leq_refl
thf(fact_114_gfp_Oleq__trans,axiom,
! [A: $tType] :
( ( comple187826305attice @ A @ ( type2 @ A ) )
=> ! [Y: A,X2: A,Z3: A] :
( ( ord_less_eq @ A @ Y @ X2 )
=> ( ( ord_less_eq @ A @ Z3 @ Y )
=> ( ord_less_eq @ A @ Z3 @ X2 ) ) ) ) ).
% gfp.leq_trans
thf(fact_115_lfp_Oleq__trans,axiom,
! [A: $tType] :
( ( comple187826305attice @ A @ ( type2 @ A ) )
=> ! [X2: A,Y: A,Z3: A] :
( ( ord_less_eq @ A @ X2 @ Y )
=> ( ( ord_less_eq @ A @ Y @ Z3 )
=> ( ord_less_eq @ A @ X2 @ Z3 ) ) ) ) ).
% lfp.leq_trans
thf(fact_116_stream_Opred__mono__strong,axiom,
! [A: $tType,P3: A > $o,X2: stream @ A,Pa: A > $o] :
( ( pred_stream @ A @ P3 @ X2 )
=> ( ! [Z: A] :
( ( member @ A @ Z @ ( sset @ A @ X2 ) )
=> ( ( P3 @ Z )
=> ( Pa @ Z ) ) )
=> ( pred_stream @ A @ Pa @ X2 ) ) ) ).
% stream.pred_mono_strong
thf(fact_117_stream_Opred__cong,axiom,
! [A: $tType,X2: stream @ A,Ya: stream @ A,P3: A > $o,Pa: A > $o] :
( ( X2 = Ya )
=> ( ! [Z: A] :
( ( member @ A @ Z @ ( sset @ A @ Ya ) )
=> ( ( P3 @ Z )
= ( Pa @ Z ) ) )
=> ( ( pred_stream @ A @ P3 @ X2 )
= ( pred_stream @ A @ Pa @ Ya ) ) ) ) ).
% stream.pred_cong
thf(fact_118_lfp_Oleq__antisym,axiom,
! [A: $tType] :
( ( comple187826305attice @ A @ ( type2 @ A ) )
=> ! [X2: A,Y: A] :
( ( ord_less_eq @ A @ X2 @ Y )
=> ( ( ord_less_eq @ A @ Y @ X2 )
=> ( X2 = Y ) ) ) ) ).
% lfp.leq_antisym
thf(fact_119_gfp_Oleq__antisym,axiom,
! [A: $tType] :
( ( comple187826305attice @ A @ ( type2 @ A ) )
=> ! [Y: A,X2: A] :
( ( ord_less_eq @ A @ Y @ X2 )
=> ( ( ord_less_eq @ A @ X2 @ Y )
=> ( X2 = Y ) ) ) ) ).
% gfp.leq_antisym
thf(fact_120_llistsum__inf,axiom,
! [A: $tType] :
( ( monoid_add @ A @ ( type2 @ A ) )
=> ! [Xs: coinductive_llist @ A] :
( ~ ( coinductive_lfinite @ A @ Xs )
=> ( ( coindu780009021istsum @ A @ Xs )
= ( zero_zero @ A ) ) ) ) ).
% llistsum_inf
thf(fact_121_lset__inf__llist,axiom,
! [A: $tType,F: nat > A] :
( ( coinductive_lset @ A @ ( coindu68654304_llist @ A @ F ) )
= ( image @ nat @ A @ F @ ( top_top @ ( set @ nat ) ) ) ) ).
% lset_inf_llist
thf(fact_122_sset__smerge,axiom,
! [A: $tType,Ss: stream @ ( stream @ A )] :
( ( sset @ A @ ( smerge @ A @ Ss ) )
= ( complete_Sup_Sup @ ( set @ A ) @ ( image @ ( stream @ A ) @ ( set @ A ) @ ( sset @ A ) @ ( sset @ ( stream @ A ) @ Ss ) ) ) ) ).
% sset_smerge
thf(fact_123_llist_Opred__set,axiom,
! [A: $tType] :
( ( coindu543516966_llist @ A )
= ( ^ [P: A > $o,X: coinductive_llist @ A] :
! [Y4: A] :
( ( member @ A @ Y4 @ ( coinductive_lset @ A @ X ) )
=> ( P @ Y4 ) ) ) ) ).
% llist.pred_set
thf(fact_124_top__apply,axiom,
! [C: $tType,D2: $tType] :
( ( top @ C @ ( type2 @ C ) )
=> ( ( top_top @ ( D2 > C ) )
= ( ^ [X: D2] : ( top_top @ C ) ) ) ) ).
% top_apply
thf(fact_125_UNIV__I,axiom,
! [A: $tType,X2: A] : ( member @ A @ X2 @ ( top_top @ ( set @ A ) ) ) ).
% UNIV_I
thf(fact_126_Union__iff,axiom,
! [A: $tType,A2: A,C4: set @ ( set @ A )] :
( ( member @ A @ A2 @ ( complete_Sup_Sup @ ( set @ A ) @ C4 ) )
= ( ? [X: set @ A] :
( ( member @ ( set @ A ) @ X @ C4 )
& ( member @ A @ A2 @ X ) ) ) ) ).
% Union_iff
thf(fact_127_UnionI,axiom,
! [A: $tType,X3: set @ A,C4: set @ ( set @ A ),A2: A] :
( ( member @ ( set @ A ) @ X3 @ C4 )
=> ( ( member @ A @ A2 @ X3 )
=> ( member @ A @ A2 @ ( complete_Sup_Sup @ ( set @ A ) @ C4 ) ) ) ) ).
% UnionI
thf(fact_128_UN__ball__bex__simps_I1_J,axiom,
! [A: $tType,A2: set @ ( set @ A ),P3: A > $o] :
( ( ! [X: A] :
( ( member @ A @ X @ ( complete_Sup_Sup @ ( set @ A ) @ A2 ) )
=> ( P3 @ X ) ) )
= ( ! [X: set @ A] :
( ( member @ ( set @ A ) @ X @ A2 )
=> ! [Y4: A] :
( ( member @ A @ Y4 @ X )
=> ( P3 @ Y4 ) ) ) ) ) ).
% UN_ball_bex_simps(1)
thf(fact_129_UN__ball__bex__simps_I3_J,axiom,
! [D2: $tType,A2: set @ ( set @ D2 ),P3: D2 > $o] :
( ( ? [X: D2] :
( ( member @ D2 @ X @ ( complete_Sup_Sup @ ( set @ D2 ) @ A2 ) )
& ( P3 @ X ) ) )
= ( ? [X: set @ D2] :
( ( member @ ( set @ D2 ) @ X @ A2 )
& ? [Y4: D2] :
( ( member @ D2 @ Y4 @ X )
& ( P3 @ Y4 ) ) ) ) ) ).
% UN_ball_bex_simps(3)
thf(fact_130_UN__ball__bex__simps_I4_J,axiom,
! [F3: $tType,E: $tType,B2: E > ( set @ F3 ),A2: set @ E,P3: F3 > $o] :
( ( ? [X: F3] :
( ( member @ F3 @ X @ ( complete_Sup_Sup @ ( set @ F3 ) @ ( image @ E @ ( set @ F3 ) @ B2 @ A2 ) ) )
& ( P3 @ X ) ) )
= ( ? [X: E] :
( ( member @ E @ X @ A2 )
& ? [Y4: F3] :
( ( member @ F3 @ Y4 @ ( B2 @ X ) )
& ( P3 @ Y4 ) ) ) ) ) ).
% UN_ball_bex_simps(4)
thf(fact_131_UN__ball__bex__simps_I2_J,axiom,
! [C: $tType,B: $tType,B2: B > ( set @ C ),A2: set @ B,P3: C > $o] :
( ( ! [X: C] :
( ( member @ C @ X @ ( complete_Sup_Sup @ ( set @ C ) @ ( image @ B @ ( set @ C ) @ B2 @ A2 ) ) )
=> ( P3 @ X ) ) )
= ( ! [X: B] :
( ( member @ B @ X @ A2 )
=> ! [Y4: C] :
( ( member @ C @ Y4 @ ( B2 @ X ) )
=> ( P3 @ Y4 ) ) ) ) ) ).
% UN_ball_bex_simps(2)
thf(fact_132_bex__UN,axiom,
! [A: $tType,B: $tType,B2: B > ( set @ A ),A2: set @ B,P3: A > $o] :
( ( ? [X: A] :
( ( member @ A @ X @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ B2 @ A2 ) ) )
& ( P3 @ X ) ) )
= ( ? [X: B] :
( ( member @ B @ X @ A2 )
& ? [Y4: A] :
( ( member @ A @ Y4 @ ( B2 @ X ) )
& ( P3 @ Y4 ) ) ) ) ) ).
% bex_UN
thf(fact_133_ball__UN,axiom,
! [A: $tType,B: $tType,B2: B > ( set @ A ),A2: set @ B,P3: A > $o] :
( ( ! [X: A] :
( ( member @ A @ X @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ B2 @ A2 ) ) )
=> ( P3 @ X ) ) )
= ( ! [X: B] :
( ( member @ B @ X @ A2 )
=> ! [Y4: A] :
( ( member @ A @ Y4 @ ( B2 @ X ) )
=> ( P3 @ Y4 ) ) ) ) ) ).
% ball_UN
thf(fact_134_streams__UNIV,axiom,
! [A: $tType] :
( ( streams @ A @ ( top_top @ ( set @ A ) ) )
= ( top_top @ ( set @ ( stream @ A ) ) ) ) ).
% streams_UNIV
thf(fact_135_Sup__UNIV,axiom,
! [A: $tType] :
( ( comple187826305attice @ A @ ( type2 @ A ) )
=> ( ( complete_Sup_Sup @ A @ ( top_top @ ( set @ A ) ) )
= ( top_top @ A ) ) ) ).
% Sup_UNIV
thf(fact_136_Sup__eqI,axiom,
! [A: $tType] :
( ( comple187826305attice @ A @ ( type2 @ A ) )
=> ! [A2: set @ A,X2: A] :
( ! [Y2: A] :
( ( member @ A @ Y2 @ A2 )
=> ( ord_less_eq @ A @ Y2 @ X2 ) )
=> ( ! [Y2: A] :
( ! [Z4: A] :
( ( member @ A @ Z4 @ A2 )
=> ( ord_less_eq @ A @ Z4 @ Y2 ) )
=> ( ord_less_eq @ A @ X2 @ Y2 ) )
=> ( ( complete_Sup_Sup @ A @ A2 )
= X2 ) ) ) ) ).
% Sup_eqI
thf(fact_137_complete__lattice__class_OSup__mono,axiom,
! [A: $tType] :
( ( comple187826305attice @ A @ ( type2 @ A ) )
=> ! [A2: set @ A,B2: set @ A] :
( ! [A4: A] :
( ( member @ A @ A4 @ A2 )
=> ? [X5: A] :
( ( member @ A @ X5 @ B2 )
& ( ord_less_eq @ A @ A4 @ X5 ) ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ A2 ) @ ( complete_Sup_Sup @ A @ B2 ) ) ) ) ).
% complete_lattice_class.Sup_mono
thf(fact_138_Sup__least,axiom,
! [A: $tType] :
( ( comple187826305attice @ A @ ( type2 @ A ) )
=> ! [A2: set @ A,Z3: A] :
( ! [X4: A] :
( ( member @ A @ X4 @ A2 )
=> ( ord_less_eq @ A @ X4 @ Z3 ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ A2 ) @ Z3 ) ) ) ).
% Sup_least
thf(fact_139_Sup__upper,axiom,
! [A: $tType] :
( ( comple187826305attice @ A @ ( type2 @ A ) )
=> ! [X2: A,A2: set @ A] :
( ( member @ A @ X2 @ A2 )
=> ( ord_less_eq @ A @ X2 @ ( complete_Sup_Sup @ A @ A2 ) ) ) ) ).
% Sup_upper
thf(fact_140_Sup__le__iff,axiom,
! [A: $tType] :
( ( comple187826305attice @ A @ ( type2 @ A ) )
=> ! [A2: set @ A,B3: A] :
( ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ A2 ) @ B3 )
= ( ! [X: A] :
( ( member @ A @ X @ A2 )
=> ( ord_less_eq @ A @ X @ B3 ) ) ) ) ) ).
% Sup_le_iff
thf(fact_141_Sup__upper2,axiom,
! [A: $tType] :
( ( comple187826305attice @ A @ ( type2 @ A ) )
=> ! [U: A,A2: set @ A,V: A] :
( ( member @ A @ U @ A2 )
=> ( ( ord_less_eq @ A @ V @ U )
=> ( ord_less_eq @ A @ V @ ( complete_Sup_Sup @ A @ A2 ) ) ) ) ) ).
% Sup_upper2
thf(fact_142_top__greatest,axiom,
! [A: $tType] :
( ( order_top @ A @ ( type2 @ A ) )
=> ! [A3: A] : ( ord_less_eq @ A @ A3 @ ( top_top @ A ) ) ) ).
% top_greatest
thf(fact_143_top_Oextremum__unique,axiom,
! [A: $tType] :
( ( order_top @ A @ ( type2 @ A ) )
=> ! [A3: A] :
( ( ord_less_eq @ A @ ( top_top @ A ) @ A3 )
= ( A3
= ( top_top @ A ) ) ) ) ).
% top.extremum_unique
thf(fact_144_top_Oextremum__uniqueI,axiom,
! [A: $tType] :
( ( order_top @ A @ ( type2 @ A ) )
=> ! [A3: A] :
( ( ord_less_eq @ A @ ( top_top @ A ) @ A3 )
=> ( A3
= ( top_top @ A ) ) ) ) ).
% top.extremum_uniqueI
thf(fact_145_SUP__cong,axiom,
! [A: $tType,B: $tType] :
( ( complete_Sup @ A @ ( type2 @ A ) )
=> ! [A2: set @ B,B2: set @ B,C4: B > A,D: B > A] :
( ( A2 = B2 )
=> ( ! [X4: B] :
( ( member @ B @ X4 @ B2 )
=> ( ( C4 @ X4 )
= ( D @ X4 ) ) )
=> ( ( complete_Sup_Sup @ A @ ( image @ B @ A @ C4 @ A2 ) )
= ( complete_Sup_Sup @ A @ ( image @ B @ A @ D @ B2 ) ) ) ) ) ) ).
% SUP_cong
thf(fact_146_Union__upper,axiom,
! [A: $tType,B2: set @ A,A2: set @ ( set @ A )] :
( ( member @ ( set @ A ) @ B2 @ A2 )
=> ( ord_less_eq @ ( set @ A ) @ B2 @ ( complete_Sup_Sup @ ( set @ A ) @ A2 ) ) ) ).
% Union_upper
thf(fact_147_Union__least,axiom,
! [A: $tType,A2: set @ ( set @ A ),C4: set @ A] :
( ! [X6: set @ A] :
( ( member @ ( set @ A ) @ X6 @ A2 )
=> ( ord_less_eq @ ( set @ A ) @ X6 @ C4 ) )
=> ( ord_less_eq @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ A2 ) @ C4 ) ) ).
% Union_least
thf(fact_148_Union__mono,axiom,
! [A: $tType,A2: set @ ( set @ A ),B2: set @ ( set @ A )] :
( ( ord_less_eq @ ( set @ ( set @ A ) ) @ A2 @ B2 )
=> ( ord_less_eq @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ A2 ) @ ( complete_Sup_Sup @ ( set @ A ) @ B2 ) ) ) ).
% Union_mono
thf(fact_149_range__eqI,axiom,
! [A: $tType,B: $tType,B3: A,F: B > A,X2: B] :
( ( B3
= ( F @ X2 ) )
=> ( member @ A @ B3 @ ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) ) ) ) ).
% range_eqI
thf(fact_150_rangeI,axiom,
! [A: $tType,B: $tType,F: B > A,X2: B] : ( member @ A @ ( F @ X2 ) @ ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) ) ) ).
% rangeI
thf(fact_151_subset__UNIV,axiom,
! [A: $tType,A2: set @ A] : ( ord_less_eq @ ( set @ A ) @ A2 @ ( top_top @ ( set @ A ) ) ) ).
% subset_UNIV
thf(fact_152_Union__UNIV,axiom,
! [A: $tType] :
( ( complete_Sup_Sup @ ( set @ A ) @ ( top_top @ ( set @ ( set @ A ) ) ) )
= ( top_top @ ( set @ A ) ) ) ).
% Union_UNIV
thf(fact_153_UnionE,axiom,
! [A: $tType,A2: A,C4: set @ ( set @ A )] :
( ( member @ A @ A2 @ ( complete_Sup_Sup @ ( set @ A ) @ C4 ) )
=> ~ ! [X6: set @ A] :
( ( member @ A @ A2 @ X6 )
=> ~ ( member @ ( set @ A ) @ X6 @ C4 ) ) ) ).
% UnionE
thf(fact_154_UNIV__witness,axiom,
! [A: $tType] :
? [X4: A] : ( member @ A @ X4 @ ( top_top @ ( set @ A ) ) ) ).
% UNIV_witness
thf(fact_155_UNIV__eq__I,axiom,
! [A: $tType,A2: set @ A] :
( ! [X4: A] : ( member @ A @ X4 @ A2 )
=> ( ( top_top @ ( set @ A ) )
= A2 ) ) ).
% UNIV_eq_I
thf(fact_156_llist_Opred__cong,axiom,
! [A: $tType,X2: coinductive_llist @ A,Ya: coinductive_llist @ A,P3: A > $o,Pa: A > $o] :
( ( X2 = Ya )
=> ( ! [Z: A] :
( ( member @ A @ Z @ ( coinductive_lset @ A @ Ya ) )
=> ( ( P3 @ Z )
= ( Pa @ Z ) ) )
=> ( ( coindu543516966_llist @ A @ P3 @ X2 )
= ( coindu543516966_llist @ A @ Pa @ Ya ) ) ) ) ).
% llist.pred_cong
thf(fact_157_llist_Opred__mono__strong,axiom,
! [A: $tType,P3: A > $o,X2: coinductive_llist @ A,Pa: A > $o] :
( ( coindu543516966_llist @ A @ P3 @ X2 )
=> ( ! [Z: A] :
( ( member @ A @ Z @ ( coinductive_lset @ A @ X2 ) )
=> ( ( P3 @ Z )
=> ( Pa @ Z ) ) )
=> ( coindu543516966_llist @ A @ Pa @ X2 ) ) ) ).
% llist.pred_mono_strong
thf(fact_158_SUP__eq,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( comple187826305attice @ A @ ( type2 @ A ) )
=> ! [A2: set @ B,B2: set @ C,F: B > A,G: C > A] :
( ! [I: B] :
( ( member @ B @ I @ A2 )
=> ? [X5: C] :
( ( member @ C @ X5 @ B2 )
& ( ord_less_eq @ A @ ( F @ I ) @ ( G @ X5 ) ) ) )
=> ( ! [J: C] :
( ( member @ C @ J @ B2 )
=> ? [X5: B] :
( ( member @ B @ X5 @ A2 )
& ( ord_less_eq @ A @ ( G @ J ) @ ( F @ X5 ) ) ) )
=> ( ( complete_Sup_Sup @ A @ ( image @ B @ A @ F @ A2 ) )
= ( complete_Sup_Sup @ A @ ( image @ C @ A @ G @ B2 ) ) ) ) ) ) ).
% SUP_eq
thf(fact_159_Sup__subset__mono,axiom,
! [A: $tType] :
( ( comple187826305attice @ A @ ( type2 @ A ) )
=> ! [A2: set @ A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ A2 ) @ ( complete_Sup_Sup @ A @ B2 ) ) ) ) ).
% Sup_subset_mono
thf(fact_160_sset__range,axiom,
! [A: $tType] :
( ( sset @ A )
= ( ^ [S: stream @ A] : ( image @ nat @ A @ ( snth @ A @ S ) @ ( top_top @ ( set @ nat ) ) ) ) ) ).
% sset_range
thf(fact_161_llistsum__infllist,axiom,
! [A: $tType] :
( ( monoid_add @ A @ ( type2 @ A ) )
=> ! [F: nat > A] :
( ( coindu780009021istsum @ A @ ( coindu68654304_llist @ A @ F ) )
= ( zero_zero @ A ) ) ) ).
% llistsum_infllist
thf(fact_162_le__zero__eq,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A @ ( type2 @ A ) )
=> ! [N: A] :
( ( ord_less_eq @ A @ N @ ( zero_zero @ A ) )
= ( N
= ( zero_zero @ A ) ) ) ) ).
% le_zero_eq
thf(fact_163_iso__tuple__UNIV__I,axiom,
! [A: $tType,X2: A] : ( member @ A @ X2 @ ( top_top @ ( set @ A ) ) ) ).
% iso_tuple_UNIV_I
thf(fact_164_surj__def,axiom,
! [B: $tType,A: $tType,F: B > A] :
( ( ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) )
= ( ! [Y4: A] :
? [X: B] :
( Y4
= ( F @ X ) ) ) ) ).
% surj_def
thf(fact_165_top__set__def,axiom,
! [A: $tType] :
( ( top_top @ ( set @ A ) )
= ( collect @ A @ ( top_top @ ( A > $o ) ) ) ) ).
% top_set_def
thf(fact_166_zero__reorient,axiom,
! [A: $tType] :
( ( zero @ A @ ( type2 @ A ) )
=> ! [X2: A] :
( ( ( zero_zero @ A )
= X2 )
= ( X2
= ( zero_zero @ A ) ) ) ) ).
% zero_reorient
thf(fact_167_zero__le,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A @ ( type2 @ A ) )
=> ! [X2: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ X2 ) ) ).
% zero_le
thf(fact_168_surjD,axiom,
! [A: $tType,B: $tType,F: B > A,Y: A] :
( ( ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) )
=> ? [X4: B] :
( Y
= ( F @ X4 ) ) ) ).
% surjD
thf(fact_169_surjE,axiom,
! [A: $tType,B: $tType,F: B > A,Y: A] :
( ( ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) )
=> ~ ! [X4: B] :
( Y
!= ( F @ X4 ) ) ) ).
% surjE
thf(fact_170_surjI,axiom,
! [B: $tType,A: $tType,G: B > A,F: A > B] :
( ! [X4: A] :
( ( G @ ( F @ X4 ) )
= X4 )
=> ( ( image @ B @ A @ G @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) ) ) ).
% surjI
thf(fact_171_cSup__eq,axiom,
! [A: $tType] :
( ( ( condit378418413attice @ A @ ( type2 @ A ) )
& ( no_bot @ A @ ( type2 @ A ) ) )
=> ! [X3: set @ A,A3: A] :
( ! [X4: A] :
( ( member @ A @ X4 @ X3 )
=> ( ord_less_eq @ A @ X4 @ A3 ) )
=> ( ! [Y2: A] :
( ! [X5: A] :
( ( member @ A @ X5 @ X3 )
=> ( ord_less_eq @ A @ X5 @ Y2 ) )
=> ( ord_less_eq @ A @ A3 @ Y2 ) )
=> ( ( complete_Sup_Sup @ A @ X3 )
= A3 ) ) ) ) ).
% cSup_eq
thf(fact_172_cSup__eq__maximum,axiom,
! [A: $tType] :
( ( condit378418413attice @ A @ ( type2 @ A ) )
=> ! [Z3: A,X3: set @ A] :
( ( member @ A @ Z3 @ X3 )
=> ( ! [X4: A] :
( ( member @ A @ X4 @ X3 )
=> ( ord_less_eq @ A @ X4 @ Z3 ) )
=> ( ( complete_Sup_Sup @ A @ X3 )
= Z3 ) ) ) ) ).
% cSup_eq_maximum
thf(fact_173_le__numeral__extra_I3_J,axiom,
! [A: $tType] :
( ( linordered_semidom @ A @ ( type2 @ A ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( zero_zero @ A ) ) ) ).
% le_numeral_extra(3)
thf(fact_174_top1I,axiom,
! [A: $tType,X2: A] : ( top_top @ ( A > $o ) @ X2 ) ).
% top1I
thf(fact_175_top__empty__eq,axiom,
! [A: $tType] :
( ( top_top @ ( A > $o ) )
= ( ^ [X: A] : ( member @ A @ X @ ( top_top @ ( set @ A ) ) ) ) ) ).
% top_empty_eq
thf(fact_176_top__conj_I2_J,axiom,
! [A: $tType,P3: $o,X2: A] :
( ( P3
& ( top_top @ ( A > $o ) @ X2 ) )
= P3 ) ).
% top_conj(2)
thf(fact_177_top__conj_I1_J,axiom,
! [A: $tType,X2: A,P3: $o] :
( ( ( top_top @ ( A > $o ) @ X2 )
& P3 )
= P3 ) ).
% top_conj(1)
thf(fact_178_Sup__SUP__eq,axiom,
! [A: $tType] :
( ( complete_Sup_Sup @ ( A > $o ) )
= ( ^ [S6: set @ ( A > $o ),X: A] : ( member @ A @ X @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ ( A > $o ) @ ( set @ A ) @ ( collect @ A ) @ S6 ) ) ) ) ) ).
% Sup_SUP_eq
thf(fact_179_surj__from__nat,axiom,
! [A: $tType] :
( ( countable @ A @ ( type2 @ A ) )
=> ( ( image @ nat @ A @ ( from_nat @ A ) @ ( top_top @ ( set @ nat ) ) )
= ( top_top @ ( set @ A ) ) ) ) ).
% surj_from_nat
thf(fact_180_member__bind,axiom,
! [A: $tType,B: $tType,X2: A,P3: set @ B,F: B > ( set @ A )] :
( ( member @ A @ X2 @ ( bind @ B @ A @ P3 @ F ) )
= ( member @ A @ X2 @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ F @ P3 ) ) ) ) ).
% member_bind
thf(fact_181_image__Pow__mono,axiom,
! [B: $tType,A: $tType,F: B > A,A2: set @ B,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( image @ B @ A @ F @ A2 ) @ B2 )
=> ( ord_less_eq @ ( set @ ( set @ A ) ) @ ( image @ ( set @ B ) @ ( set @ A ) @ ( image @ B @ A @ F ) @ ( pow @ B @ A2 ) ) @ ( pow @ A @ B2 ) ) ) ).
% image_Pow_mono
thf(fact_182_Sup1__I,axiom,
! [A: $tType,P3: A > $o,A2: set @ ( A > $o ),A3: A] :
( ( member @ ( A > $o ) @ P3 @ A2 )
=> ( ( P3 @ A3 )
=> ( complete_Sup_Sup @ ( A > $o ) @ A2 @ A3 ) ) ) ).
% Sup1_I
thf(fact_183_Pow__iff,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] :
( ( member @ ( set @ A ) @ A2 @ ( pow @ A @ B2 ) )
= ( ord_less_eq @ ( set @ A ) @ A2 @ B2 ) ) ).
% Pow_iff
thf(fact_184_PowI,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ( member @ ( set @ A ) @ A2 @ ( pow @ A @ B2 ) ) ) ).
% PowI
thf(fact_185_Union__Pow__eq,axiom,
! [A: $tType,A2: set @ A] :
( ( complete_Sup_Sup @ ( set @ A ) @ ( pow @ A @ A2 ) )
= A2 ) ).
% Union_Pow_eq
thf(fact_186_Pow__UNIV,axiom,
! [A: $tType] :
( ( pow @ A @ ( top_top @ ( set @ A ) ) )
= ( top_top @ ( set @ ( set @ A ) ) ) ) ).
% Pow_UNIV
thf(fact_187_image__Pow__surj,axiom,
! [B: $tType,A: $tType,F: B > A,A2: set @ B,B2: set @ A] :
( ( ( image @ B @ A @ F @ A2 )
= B2 )
=> ( ( image @ ( set @ B ) @ ( set @ A ) @ ( image @ B @ A @ F ) @ ( pow @ B @ A2 ) )
= ( pow @ A @ B2 ) ) ) ).
% image_Pow_surj
thf(fact_188_Sup1__E,axiom,
! [A: $tType,A2: set @ ( A > $o ),A3: A] :
( ( complete_Sup_Sup @ ( A > $o ) @ A2 @ A3 )
=> ~ ! [P4: A > $o] :
( ( member @ ( A > $o ) @ P4 @ A2 )
=> ~ ( P4 @ A3 ) ) ) ).
% Sup1_E
thf(fact_189_Cantors__paradox,axiom,
! [A: $tType,A2: set @ A] :
~ ? [X5: A > ( set @ A )] :
( ( image @ A @ ( set @ A ) @ X5 @ A2 )
= ( pow @ A @ A2 ) ) ).
% Cantors_paradox
thf(fact_190_PowD,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] :
( ( member @ ( set @ A ) @ A2 @ ( pow @ A @ B2 ) )
=> ( ord_less_eq @ ( set @ A ) @ A2 @ B2 ) ) ).
% PowD
thf(fact_191_Pow__top,axiom,
! [A: $tType,A2: set @ A] : ( member @ ( set @ A ) @ A2 @ ( pow @ A @ A2 ) ) ).
% Pow_top
thf(fact_192_Pow__mono,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ( ord_less_eq @ ( set @ ( set @ A ) ) @ ( pow @ A @ A2 ) @ ( pow @ A @ B2 ) ) ) ).
% Pow_mono
thf(fact_193_subset__Pow__Union,axiom,
! [A: $tType,A2: set @ ( set @ A )] : ( ord_less_eq @ ( set @ ( set @ A ) ) @ A2 @ ( pow @ A @ ( complete_Sup_Sup @ ( set @ A ) @ A2 ) ) ) ).
% subset_Pow_Union
thf(fact_194_bind__UNION,axiom,
! [A: $tType,B: $tType] :
( ( bind @ B @ A )
= ( ^ [A5: set @ B,F2: B > ( set @ A )] : ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ F2 @ A5 ) ) ) ) ).
% bind_UNION
thf(fact_195_surj__swap__iff,axiom,
! [B: $tType,A: $tType,A3: B,B3: B,F: B > A] :
( ( ( image @ B @ A @ ( swap @ B @ A @ A3 @ B3 @ F ) @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) )
= ( ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) ) ) ).
% surj_swap_iff
thf(fact_196_cSUP__least,axiom,
! [B: $tType,A: $tType] :
( ( condit378418413attice @ A @ ( type2 @ A ) )
=> ! [A2: set @ B,F: B > A,M2: A] :
( ( A2
!= ( bot_bot @ ( set @ B ) ) )
=> ( ! [X4: B] :
( ( member @ B @ X4 @ A2 )
=> ( ord_less_eq @ A @ ( F @ X4 ) @ M2 ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F @ A2 ) ) @ M2 ) ) ) ) ).
% cSUP_least
thf(fact_197_SUP__eq__iff,axiom,
! [A: $tType,B: $tType] :
( ( comple187826305attice @ A @ ( type2 @ A ) )
=> ! [I2: set @ B,C3: A,F: B > A] :
( ( I2
!= ( bot_bot @ ( set @ B ) ) )
=> ( ! [I: B] :
( ( member @ B @ I @ I2 )
=> ( ord_less_eq @ A @ C3 @ ( F @ I ) ) )
=> ( ( ( complete_Sup_Sup @ A @ ( image @ B @ A @ F @ I2 ) )
= C3 )
= ( ! [X: B] :
( ( member @ B @ X @ I2 )
=> ( ( F @ X )
= C3 ) ) ) ) ) ) ) ).
% SUP_eq_iff
thf(fact_198_bot__apply,axiom,
! [C: $tType,D2: $tType] :
( ( bot @ C @ ( type2 @ C ) )
=> ( ( bot_bot @ ( D2 > C ) )
= ( ^ [X: D2] : ( bot_bot @ C ) ) ) ) ).
% bot_apply
thf(fact_199_empty__iff,axiom,
! [A: $tType,C3: A] :
~ ( member @ A @ C3 @ ( bot_bot @ ( set @ A ) ) ) ).
% empty_iff
thf(fact_200_all__not__in__conv,axiom,
! [A: $tType,A2: set @ A] :
( ( ! [X: A] :
~ ( member @ A @ X @ A2 ) )
= ( A2
= ( bot_bot @ ( set @ A ) ) ) ) ).
% all_not_in_conv
thf(fact_201_Collect__empty__eq,axiom,
! [A: $tType,P3: A > $o] :
( ( ( collect @ A @ P3 )
= ( bot_bot @ ( set @ A ) ) )
= ( ! [X: A] :
~ ( P3 @ X ) ) ) ).
% Collect_empty_eq
thf(fact_202_empty__Collect__eq,axiom,
! [A: $tType,P3: A > $o] :
( ( ( bot_bot @ ( set @ A ) )
= ( collect @ A @ P3 ) )
= ( ! [X: A] :
~ ( P3 @ X ) ) ) ).
% empty_Collect_eq
thf(fact_203_image__is__empty,axiom,
! [A: $tType,B: $tType,F: B > A,A2: set @ B] :
( ( ( image @ B @ A @ F @ A2 )
= ( bot_bot @ ( set @ A ) ) )
= ( A2
= ( bot_bot @ ( set @ B ) ) ) ) ).
% image_is_empty
thf(fact_204_empty__is__image,axiom,
! [A: $tType,B: $tType,F: B > A,A2: set @ B] :
( ( ( bot_bot @ ( set @ A ) )
= ( image @ B @ A @ F @ A2 ) )
= ( A2
= ( bot_bot @ ( set @ B ) ) ) ) ).
% empty_is_image
thf(fact_205_image__empty,axiom,
! [B: $tType,A: $tType,F: B > A] :
( ( image @ B @ A @ F @ ( bot_bot @ ( set @ B ) ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% image_empty
thf(fact_206_Sup__bot__conv_I1_J,axiom,
! [A: $tType] :
( ( comple187826305attice @ A @ ( type2 @ A ) )
=> ! [A2: set @ A] :
( ( ( complete_Sup_Sup @ A @ A2 )
= ( bot_bot @ A ) )
= ( ! [X: A] :
( ( member @ A @ X @ A2 )
=> ( X
= ( bot_bot @ A ) ) ) ) ) ) ).
% Sup_bot_conv(1)
thf(fact_207_Sup__bot__conv_I2_J,axiom,
! [A: $tType] :
( ( comple187826305attice @ A @ ( type2 @ A ) )
=> ! [A2: set @ A] :
( ( ( bot_bot @ A )
= ( complete_Sup_Sup @ A @ A2 ) )
= ( ! [X: A] :
( ( member @ A @ X @ A2 )
=> ( X
= ( bot_bot @ A ) ) ) ) ) ) ).
% Sup_bot_conv(2)
thf(fact_208_subset__empty,axiom,
! [A: $tType,A2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ ( bot_bot @ ( set @ A ) ) )
= ( A2
= ( bot_bot @ ( set @ A ) ) ) ) ).
% subset_empty
thf(fact_209_empty__subsetI,axiom,
! [A: $tType,A2: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ A2 ) ).
% empty_subsetI
thf(fact_210_ball__empty,axiom,
! [A: $tType,P3: A > $o,X5: A] :
( ( member @ A @ X5 @ ( bot_bot @ ( set @ A ) ) )
=> ( P3 @ X5 ) ) ).
% ball_empty
thf(fact_211_swap__image__eq,axiom,
! [B: $tType,A: $tType,A3: A,A2: set @ A,B3: A,F: A > B] :
( ( member @ A @ A3 @ A2 )
=> ( ( member @ A @ B3 @ A2 )
=> ( ( image @ A @ B @ ( swap @ A @ B @ A3 @ B3 @ F ) @ A2 )
= ( image @ A @ B @ F @ A2 ) ) ) ) ).
% swap_image_eq
thf(fact_212_empty__bind,axiom,
! [B: $tType,A: $tType,F: B > ( set @ A )] :
( ( bind @ B @ A @ ( bot_bot @ ( set @ B ) ) @ F )
= ( bot_bot @ ( set @ A ) ) ) ).
% empty_bind
thf(fact_213_Sup__empty,axiom,
! [A: $tType] :
( ( comple187826305attice @ A @ ( type2 @ A ) )
=> ( ( complete_Sup_Sup @ A @ ( bot_bot @ ( set @ A ) ) )
= ( bot_bot @ A ) ) ) ).
% Sup_empty
thf(fact_214_empty__Union__conv,axiom,
! [A: $tType,A2: set @ ( set @ A )] :
( ( ( bot_bot @ ( set @ A ) )
= ( complete_Sup_Sup @ ( set @ A ) @ A2 ) )
= ( ! [X: set @ A] :
( ( member @ ( set @ A ) @ X @ A2 )
=> ( X
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% empty_Union_conv
thf(fact_215_Union__empty__conv,axiom,
! [A: $tType,A2: set @ ( set @ A )] :
( ( ( complete_Sup_Sup @ ( set @ A ) @ A2 )
= ( bot_bot @ ( set @ A ) ) )
= ( ! [X: set @ A] :
( ( member @ ( set @ A ) @ X @ A2 )
=> ( X
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% Union_empty_conv
thf(fact_216_Union__empty,axiom,
! [A: $tType] :
( ( complete_Sup_Sup @ ( set @ A ) @ ( bot_bot @ ( set @ ( set @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% Union_empty
thf(fact_217_Pow__not__empty,axiom,
! [A: $tType,A2: set @ A] :
( ( pow @ A @ A2 )
!= ( bot_bot @ ( set @ ( set @ A ) ) ) ) ).
% Pow_not_empty
thf(fact_218_Pow__bottom,axiom,
! [A: $tType,B2: set @ A] : ( member @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ ( pow @ A @ B2 ) ) ).
% Pow_bottom
thf(fact_219_empty__not__UNIV,axiom,
! [A: $tType] :
( ( bot_bot @ ( set @ A ) )
!= ( top_top @ ( set @ A ) ) ) ).
% empty_not_UNIV
thf(fact_220_sset__neq__empty,axiom,
! [A: $tType,Xs: stream @ A] :
( ( sset @ A @ Xs )
!= ( bot_bot @ ( set @ A ) ) ) ).
% sset_neq_empty
thf(fact_221_emptyE,axiom,
! [A: $tType,A3: A] :
~ ( member @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ).
% emptyE
thf(fact_222_equals0D,axiom,
! [A: $tType,A2: set @ A,A3: A] :
( ( A2
= ( bot_bot @ ( set @ A ) ) )
=> ~ ( member @ A @ A3 @ A2 ) ) ).
% equals0D
thf(fact_223_equals0I,axiom,
! [A: $tType,A2: set @ A] :
( ! [Y2: A] :
~ ( member @ A @ Y2 @ A2 )
=> ( A2
= ( bot_bot @ ( set @ A ) ) ) ) ).
% equals0I
thf(fact_224_ex__in__conv,axiom,
! [A: $tType,A2: set @ A] :
( ( ? [X: A] : ( member @ A @ X @ A2 ) )
= ( A2
!= ( bot_bot @ ( set @ A ) ) ) ) ).
% ex_in_conv
thf(fact_225_bot__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( bot @ B @ ( type2 @ B ) )
=> ( ( bot_bot @ ( A > B ) )
= ( ^ [X: A] : ( bot_bot @ B ) ) ) ) ).
% bot_fun_def
thf(fact_226_streams__empty,axiom,
! [A: $tType] :
( ( streams @ A @ ( bot_bot @ ( set @ A ) ) )
= ( bot_bot @ ( set @ ( stream @ A ) ) ) ) ).
% streams_empty
thf(fact_227_bot_Oextremum,axiom,
! [A: $tType] :
( ( order_bot @ A @ ( type2 @ A ) )
=> ! [A3: A] : ( ord_less_eq @ A @ ( bot_bot @ A ) @ A3 ) ) ).
% bot.extremum
thf(fact_228_bot_Oextremum__unique,axiom,
! [A: $tType] :
( ( order_bot @ A @ ( type2 @ A ) )
=> ! [A3: A] :
( ( ord_less_eq @ A @ A3 @ ( bot_bot @ A ) )
= ( A3
= ( bot_bot @ A ) ) ) ) ).
% bot.extremum_unique
thf(fact_229_bot_Oextremum__uniqueI,axiom,
! [A: $tType] :
( ( order_bot @ A @ ( type2 @ A ) )
=> ! [A3: A] :
( ( ord_less_eq @ A @ A3 @ ( bot_bot @ A ) )
=> ( A3
= ( bot_bot @ A ) ) ) ) ).
% bot.extremum_uniqueI
thf(fact_230_less__eq__Sup,axiom,
! [A: $tType] :
( ( comple187826305attice @ A @ ( type2 @ A ) )
=> ! [A2: set @ A,U: A] :
( ! [V2: A] :
( ( member @ A @ V2 @ A2 )
=> ( ord_less_eq @ A @ U @ V2 ) )
=> ( ( A2
!= ( bot_bot @ ( set @ A ) ) )
=> ( ord_less_eq @ A @ U @ ( complete_Sup_Sup @ A @ A2 ) ) ) ) ) ).
% less_eq_Sup
thf(fact_231_cSup__least,axiom,
! [A: $tType] :
( ( condit378418413attice @ A @ ( type2 @ A ) )
=> ! [X3: set @ A,Z3: A] :
( ( X3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X4: A] :
( ( member @ A @ X4 @ X3 )
=> ( ord_less_eq @ A @ X4 @ Z3 ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ X3 ) @ Z3 ) ) ) ) ).
% cSup_least
thf(fact_232_cSup__eq__non__empty,axiom,
! [A: $tType] :
( ( condit378418413attice @ A @ ( type2 @ A ) )
=> ! [X3: set @ A,A3: A] :
( ( X3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X4: A] :
( ( member @ A @ X4 @ X3 )
=> ( ord_less_eq @ A @ X4 @ A3 ) )
=> ( ! [Y2: A] :
( ! [X5: A] :
( ( member @ A @ X5 @ X3 )
=> ( ord_less_eq @ A @ X5 @ Y2 ) )
=> ( ord_less_eq @ A @ A3 @ Y2 ) )
=> ( ( complete_Sup_Sup @ A @ X3 )
= A3 ) ) ) ) ) ).
% cSup_eq_non_empty
thf(fact_233_SUP__eq__const,axiom,
! [B: $tType,A: $tType] :
( ( comple187826305attice @ A @ ( type2 @ A ) )
=> ! [I2: set @ B,F: B > A,X2: A] :
( ( I2
!= ( bot_bot @ ( set @ B ) ) )
=> ( ! [I: B] :
( ( member @ B @ I @ I2 )
=> ( ( F @ I )
= X2 ) )
=> ( ( complete_Sup_Sup @ A @ ( image @ B @ A @ F @ I2 ) )
= X2 ) ) ) ) ).
% SUP_eq_const
thf(fact_234_surj__imp__surj__swap,axiom,
! [B: $tType,A: $tType,F: B > A,A3: B,B3: B] :
( ( ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) )
=> ( ( image @ B @ A @ ( swap @ B @ A @ A3 @ B3 @ F ) @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) ) ) ).
% surj_imp_surj_swap
thf(fact_235_ccSup__empty,axiom,
! [A: $tType] :
( ( counta840220525attice @ A @ ( type2 @ A ) )
=> ( ( complete_Sup_Sup @ A @ ( bot_bot @ ( set @ A ) ) )
= ( bot_bot @ A ) ) ) ).
% ccSup_empty
thf(fact_236_subset__emptyI,axiom,
! [A: $tType,A2: set @ A] :
( ! [X4: A] :
~ ( member @ A @ X4 @ A2 )
=> ( ord_less_eq @ ( set @ A ) @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% subset_emptyI
thf(fact_237_bot__set__def,axiom,
! [A: $tType] :
( ( bot_bot @ ( set @ A ) )
= ( collect @ A @ ( bot_bot @ ( A > $o ) ) ) ) ).
% bot_set_def
thf(fact_238_lset__code,axiom,
! [A: $tType] :
( ( coinductive_lset @ A )
= ( coinductive_gen_lset @ A @ ( bot_bot @ ( set @ A ) ) ) ) ).
% lset_code
thf(fact_239_the__elem__image__unique,axiom,
! [B: $tType,A: $tType,A2: set @ A,F: A > B,X2: A] :
( ( A2
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [Y2: A] :
( ( member @ A @ Y2 @ A2 )
=> ( ( F @ Y2 )
= ( F @ X2 ) ) )
=> ( ( the_elem @ B @ ( image @ A @ B @ F @ A2 ) )
= ( F @ X2 ) ) ) ) ).
% the_elem_image_unique
thf(fact_240_cSUP__subset__mono,axiom,
! [A: $tType,B: $tType] :
( ( condit378418413attice @ A @ ( type2 @ A ) )
=> ! [A2: set @ B,G: B > A,B2: set @ B,F: B > A] :
( ( A2
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit1810911227_above @ A @ ( image @ B @ A @ G @ B2 ) )
=> ( ( ord_less_eq @ ( set @ B ) @ A2 @ B2 )
=> ( ! [X4: B] :
( ( member @ B @ X4 @ B2 )
=> ( ord_less_eq @ A @ ( F @ X4 ) @ ( G @ X4 ) ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F @ A2 ) ) @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ G @ B2 ) ) ) ) ) ) ) ) ).
% cSUP_subset_mono
thf(fact_241_compact__bot,axiom,
! [A: $tType] :
( ( comple1141879883l_ccpo @ A @ ( type2 @ A ) )
=> ! [X2: A] :
( ( X2
= ( complete_Sup_Sup @ A @ ( bot_bot @ ( set @ A ) ) ) )
=> ( comple2143767107ompact @ A @ ( complete_Sup_Sup @ A ) @ ( ord_less_eq @ A ) @ X2 ) ) ) ).
% compact_bot
thf(fact_242_bdd__aboveI,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [A2: set @ A,M2: A] :
( ! [X4: A] :
( ( member @ A @ X4 @ A2 )
=> ( ord_less_eq @ A @ X4 @ M2 ) )
=> ( condit1810911227_above @ A @ A2 ) ) ) ).
% bdd_aboveI
thf(fact_243_cSup__le__iff,axiom,
! [A: $tType] :
( ( condit378418413attice @ A @ ( type2 @ A ) )
=> ! [S5: set @ A,A3: A] :
( ( S5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit1810911227_above @ A @ S5 )
=> ( ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ S5 ) @ A3 )
= ( ! [X: A] :
( ( member @ A @ X @ S5 )
=> ( ord_less_eq @ A @ X @ A3 ) ) ) ) ) ) ) ).
% cSup_le_iff
thf(fact_244_cSup__mono,axiom,
! [A: $tType] :
( ( condit378418413attice @ A @ ( type2 @ A ) )
=> ! [B2: set @ A,A2: set @ A] :
( ( B2
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit1810911227_above @ A @ A2 )
=> ( ! [B4: A] :
( ( member @ A @ B4 @ B2 )
=> ? [X5: A] :
( ( member @ A @ X5 @ A2 )
& ( ord_less_eq @ A @ B4 @ X5 ) ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ B2 ) @ ( complete_Sup_Sup @ A @ A2 ) ) ) ) ) ) ).
% cSup_mono
thf(fact_245_bdd__above__def,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ( ( condit1810911227_above @ A )
= ( ^ [A5: set @ A] :
? [M3: A] :
! [X: A] :
( ( member @ A @ X @ A5 )
=> ( ord_less_eq @ A @ X @ M3 ) ) ) ) ) ).
% bdd_above_def
thf(fact_246_bdd__above__mono,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [B2: set @ A,A2: set @ A] :
( ( condit1810911227_above @ A @ B2 )
=> ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ( condit1810911227_above @ A @ A2 ) ) ) ) ).
% bdd_above_mono
thf(fact_247_cSup__upper2,axiom,
! [A: $tType] :
( ( condit378418413attice @ A @ ( type2 @ A ) )
=> ! [X2: A,X3: set @ A,Y: A] :
( ( member @ A @ X2 @ X3 )
=> ( ( ord_less_eq @ A @ Y @ X2 )
=> ( ( condit1810911227_above @ A @ X3 )
=> ( ord_less_eq @ A @ Y @ ( complete_Sup_Sup @ A @ X3 ) ) ) ) ) ) ).
% cSup_upper2
thf(fact_248_cSup__upper,axiom,
! [A: $tType] :
( ( condit378418413attice @ A @ ( type2 @ A ) )
=> ! [X2: A,X3: set @ A] :
( ( member @ A @ X2 @ X3 )
=> ( ( condit1810911227_above @ A @ X3 )
=> ( ord_less_eq @ A @ X2 @ ( complete_Sup_Sup @ A @ X3 ) ) ) ) ) ).
% cSup_upper
thf(fact_249_bdd__aboveI2,axiom,
! [A: $tType,B: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [A2: set @ B,F: B > A,M2: A] :
( ! [X4: B] :
( ( member @ B @ X4 @ A2 )
=> ( ord_less_eq @ A @ ( F @ X4 ) @ M2 ) )
=> ( condit1810911227_above @ A @ ( image @ B @ A @ F @ A2 ) ) ) ) ).
% bdd_aboveI2
thf(fact_250_cSUP__upper2,axiom,
! [A: $tType,B: $tType] :
( ( condit378418413attice @ A @ ( type2 @ A ) )
=> ! [F: B > A,A2: set @ B,X2: B,U: A] :
( ( condit1810911227_above @ A @ ( image @ B @ A @ F @ A2 ) )
=> ( ( member @ B @ X2 @ A2 )
=> ( ( ord_less_eq @ A @ U @ ( F @ X2 ) )
=> ( ord_less_eq @ A @ U @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F @ A2 ) ) ) ) ) ) ) ).
% cSUP_upper2
thf(fact_251_cSUP__upper,axiom,
! [A: $tType,B: $tType] :
( ( condit378418413attice @ A @ ( type2 @ A ) )
=> ! [X2: B,A2: set @ B,F: B > A] :
( ( member @ B @ X2 @ A2 )
=> ( ( condit1810911227_above @ A @ ( image @ B @ A @ F @ A2 ) )
=> ( ord_less_eq @ A @ ( F @ X2 ) @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F @ A2 ) ) ) ) ) ) ).
% cSUP_upper
thf(fact_252_cSUP__le__iff,axiom,
! [A: $tType,B: $tType] :
( ( condit378418413attice @ A @ ( type2 @ A ) )
=> ! [A2: set @ B,F: B > A,U: A] :
( ( A2
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit1810911227_above @ A @ ( image @ B @ A @ F @ A2 ) )
=> ( ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F @ A2 ) ) @ U )
= ( ! [X: B] :
( ( member @ B @ X @ A2 )
=> ( ord_less_eq @ A @ ( F @ X ) @ U ) ) ) ) ) ) ) ).
% cSUP_le_iff
thf(fact_253_cSUP__mono,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( condit378418413attice @ A @ ( type2 @ A ) )
=> ! [A2: set @ B,G: C > A,B2: set @ C,F: B > A] :
( ( A2
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit1810911227_above @ A @ ( image @ C @ A @ G @ B2 ) )
=> ( ! [N4: B] :
( ( member @ B @ N4 @ A2 )
=> ? [X5: C] :
( ( member @ C @ X5 @ B2 )
& ( ord_less_eq @ A @ ( F @ N4 ) @ ( G @ X5 ) ) ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F @ A2 ) ) @ ( complete_Sup_Sup @ A @ ( image @ C @ A @ G @ B2 ) ) ) ) ) ) ) ).
% cSUP_mono
thf(fact_254_cSup__subset__mono,axiom,
! [A: $tType] :
( ( condit378418413attice @ A @ ( type2 @ A ) )
=> ! [A2: set @ A,B2: set @ A] :
( ( A2
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit1810911227_above @ A @ B2 )
=> ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ A2 ) @ ( complete_Sup_Sup @ A @ B2 ) ) ) ) ) ) ).
% cSup_subset_mono
%----Type constructors (57)
thf(tcon_HOL_Obool___Finite__Set_Ofinite,axiom,
finite_finite @ $o @ ( type2 @ $o ) ).
thf(tcon_Set_Oset___Finite__Set_Ofinite_1,axiom,
! [A6: $tType] :
( ( finite_finite @ A6 @ ( type2 @ A6 ) )
=> ( finite_finite @ ( set @ A6 ) @ ( type2 @ ( set @ A6 ) ) ) ) ).
thf(tcon_fun___Finite__Set_Ofinite_2,axiom,
! [A6: $tType,A7: $tType] :
( ( ( finite_finite @ A6 @ ( type2 @ A6 ) )
& ( finite_finite @ A7 @ ( type2 @ A7 ) ) )
=> ( finite_finite @ ( A6 > A7 ) @ ( type2 @ ( A6 > A7 ) ) ) ) ).
thf(tcon_fun___Conditionally__Complete__Lattices_Oconditionally__complete__lattice,axiom,
! [A6: $tType,A7: $tType] :
( ( comple187826305attice @ A7 @ ( type2 @ A7 ) )
=> ( condit378418413attice @ ( A6 > A7 ) @ ( type2 @ ( A6 > A7 ) ) ) ) ).
thf(tcon_fun___Countable__Complete__Lattices_Ocountable__complete__lattice,axiom,
! [A6: $tType,A7: $tType] :
( ( counta840220525attice @ A7 @ ( type2 @ A7 ) )
=> ( counta840220525attice @ ( A6 > A7 ) @ ( type2 @ ( A6 > A7 ) ) ) ) ).
thf(tcon_fun___Complete__Lattices_Ocomplete__lattice,axiom,
! [A6: $tType,A7: $tType] :
( ( comple187826305attice @ A7 @ ( type2 @ A7 ) )
=> ( comple187826305attice @ ( A6 > A7 ) @ ( type2 @ ( A6 > A7 ) ) ) ) ).
thf(tcon_fun___Complete__Partial__Order_Occpo,axiom,
! [A6: $tType,A7: $tType] :
( ( comple187826305attice @ A7 @ ( type2 @ A7 ) )
=> ( comple1141879883l_ccpo @ ( A6 > A7 ) @ ( type2 @ ( A6 > A7 ) ) ) ) ).
thf(tcon_fun___Complete__Lattices_OSup,axiom,
! [A6: $tType,A7: $tType] :
( ( complete_Sup @ A7 @ ( type2 @ A7 ) )
=> ( complete_Sup @ ( A6 > A7 ) @ ( type2 @ ( A6 > A7 ) ) ) ) ).
thf(tcon_fun___Orderings_Oorder__top,axiom,
! [A6: $tType,A7: $tType] :
( ( order_top @ A7 @ ( type2 @ A7 ) )
=> ( order_top @ ( A6 > A7 ) @ ( type2 @ ( A6 > A7 ) ) ) ) ).
thf(tcon_fun___Orderings_Oorder__bot,axiom,
! [A6: $tType,A7: $tType] :
( ( order_bot @ A7 @ ( type2 @ A7 ) )
=> ( order_bot @ ( A6 > A7 ) @ ( type2 @ ( A6 > A7 ) ) ) ) ).
thf(tcon_fun___Countable_Ocountable,axiom,
! [A6: $tType,A7: $tType] :
( ( ( finite_finite @ A6 @ ( type2 @ A6 ) )
& ( countable @ A7 @ ( type2 @ A7 ) ) )
=> ( countable @ ( A6 > A7 ) @ ( type2 @ ( A6 > A7 ) ) ) ) ).
thf(tcon_fun___Orderings_Opreorder,axiom,
! [A6: $tType,A7: $tType] :
( ( preorder @ A7 @ ( type2 @ A7 ) )
=> ( preorder @ ( A6 > A7 ) @ ( type2 @ ( A6 > A7 ) ) ) ) ).
thf(tcon_fun___Orderings_Oorder,axiom,
! [A6: $tType,A7: $tType] :
( ( order @ A7 @ ( type2 @ A7 ) )
=> ( order @ ( A6 > A7 ) @ ( type2 @ ( A6 > A7 ) ) ) ) ).
thf(tcon_fun___Orderings_Otop,axiom,
! [A6: $tType,A7: $tType] :
( ( top @ A7 @ ( type2 @ A7 ) )
=> ( top @ ( A6 > A7 ) @ ( type2 @ ( A6 > A7 ) ) ) ) ).
thf(tcon_fun___Orderings_Oord,axiom,
! [A6: $tType,A7: $tType] :
( ( ord @ A7 @ ( type2 @ A7 ) )
=> ( ord @ ( A6 > A7 ) @ ( type2 @ ( A6 > A7 ) ) ) ) ).
thf(tcon_fun___Orderings_Obot,axiom,
! [A6: $tType,A7: $tType] :
( ( bot @ A7 @ ( type2 @ A7 ) )
=> ( bot @ ( A6 > A7 ) @ ( type2 @ ( A6 > A7 ) ) ) ) ).
thf(tcon_Nat_Onat___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_3,axiom,
condit378418413attice @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Groups_Ocanonically__ordered__monoid__add,axiom,
canoni770627133id_add @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add,axiom,
ordere779506340up_add @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Rings_Olinordered__semidom,axiom,
linordered_semidom @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Complete__Lattices_OSup_4,axiom,
complete_Sup @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Orderings_Oorder__bot_5,axiom,
order_bot @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Countable_Ocountable_6,axiom,
countable @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Orderings_Opreorder_7,axiom,
preorder @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Orderings_Olinorder,axiom,
linorder @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Groups_Omonoid__add,axiom,
monoid_add @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Orderings_Oorder_8,axiom,
order @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Orderings_Oord_9,axiom,
ord @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Orderings_Obot_10,axiom,
bot @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Groups_Ozero,axiom,
zero @ nat @ ( type2 @ nat ) ).
thf(tcon_Set_Oset___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_11,axiom,
! [A6: $tType] : ( condit378418413attice @ ( set @ A6 ) @ ( type2 @ ( set @ A6 ) ) ) ).
thf(tcon_Set_Oset___Countable__Complete__Lattices_Ocountable__complete__lattice_12,axiom,
! [A6: $tType] : ( counta840220525attice @ ( set @ A6 ) @ ( type2 @ ( set @ A6 ) ) ) ).
thf(tcon_Set_Oset___Complete__Lattices_Ocomplete__lattice_13,axiom,
! [A6: $tType] : ( comple187826305attice @ ( set @ A6 ) @ ( type2 @ ( set @ A6 ) ) ) ).
thf(tcon_Set_Oset___Complete__Partial__Order_Occpo_14,axiom,
! [A6: $tType] : ( comple1141879883l_ccpo @ ( set @ A6 ) @ ( type2 @ ( set @ A6 ) ) ) ).
thf(tcon_Set_Oset___Complete__Lattices_OSup_15,axiom,
! [A6: $tType] : ( complete_Sup @ ( set @ A6 ) @ ( type2 @ ( set @ A6 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder__top_16,axiom,
! [A6: $tType] : ( order_top @ ( set @ A6 ) @ ( type2 @ ( set @ A6 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder__bot_17,axiom,
! [A6: $tType] : ( order_bot @ ( set @ A6 ) @ ( type2 @ ( set @ A6 ) ) ) ).
thf(tcon_Set_Oset___Countable_Ocountable_18,axiom,
! [A6: $tType] :
( ( finite_finite @ A6 @ ( type2 @ A6 ) )
=> ( countable @ ( set @ A6 ) @ ( type2 @ ( set @ A6 ) ) ) ) ).
thf(tcon_Set_Oset___Orderings_Opreorder_19,axiom,
! [A6: $tType] : ( preorder @ ( set @ A6 ) @ ( type2 @ ( set @ A6 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder_20,axiom,
! [A6: $tType] : ( order @ ( set @ A6 ) @ ( type2 @ ( set @ A6 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Otop_21,axiom,
! [A6: $tType] : ( top @ ( set @ A6 ) @ ( type2 @ ( set @ A6 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Oord_22,axiom,
! [A6: $tType] : ( ord @ ( set @ A6 ) @ ( type2 @ ( set @ A6 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Obot_23,axiom,
! [A6: $tType] : ( bot @ ( set @ A6 ) @ ( type2 @ ( set @ A6 ) ) ) ).
thf(tcon_HOL_Obool___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_24,axiom,
condit378418413attice @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Countable__Complete__Lattices_Ocountable__complete__lattice_25,axiom,
counta840220525attice @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Complete__Lattices_Ocomplete__lattice_26,axiom,
comple187826305attice @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Complete__Partial__Order_Occpo_27,axiom,
comple1141879883l_ccpo @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Complete__Lattices_OSup_28,axiom,
complete_Sup @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Oorder__top_29,axiom,
order_top @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Oorder__bot_30,axiom,
order_bot @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Countable_Ocountable_31,axiom,
countable @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Opreorder_32,axiom,
preorder @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Olinorder_33,axiom,
linorder @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Oorder_34,axiom,
order @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Otop_35,axiom,
top @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Oord_36,axiom,
ord @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Obot_37,axiom,
bot @ $o @ ( type2 @ $o ) ).
%----Conjectures (1)
thf(conj_0,conjecture,
( ( snth @ a @ ( coindu176146587of_seq @ a @ f ) )
= f ) ).
%------------------------------------------------------------------------------