TPTP Problem File: DAT147^1.p
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%------------------------------------------------------------------------------
% File : DAT147^1 : TPTP v8.2.0. Released v7.0.0.
% Domain : Data Structures
% Problem : Coinductive stream 192
% 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__192.p [Bla16]
% Status : Theorem
% Rating : 0.00 v7.2.0, 0.25 v7.1.0
% Syntax : Number of formulae : 336 ( 133 unt; 54 typ; 0 def)
% Number of atoms : 636 ( 270 equ; 0 cnn)
% Maximal formula atoms : 8 ( 2 avg)
% Number of connectives : 3790 ( 112 ~; 17 |; 50 &;3350 @)
% ( 0 <=>; 261 =>; 0 <=; 0 <~>)
% Maximal formula depth : 19 ( 7 avg)
% Number of types : 2 ( 1 usr)
% Number of type conns : 211 ( 211 >; 0 *; 0 +; 0 <<)
% Number of symbols : 56 ( 53 usr; 5 con; 0-6 aty)
% Number of variables : 984 ( 48 ^; 862 !; 25 ?; 984 :)
% ( 49 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2016-07-13 15:13:31.164
%------------------------------------------------------------------------------
%----Could-be-implicit typings (5)
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_itself,type,
itself: $tType > $tType ).
thf(ty_tf_a,type,
a: $tType ).
%----Explicit typings (49)
thf(sy_cl_Lattices_Obounded__lattice,type,
bounded_lattice:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_HOL_Otype,type,
type:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Obot,type,
bot:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Lattices_Olattice,type,
lattice:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Lattices_Odistrib__lattice,type,
distrib_lattice:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Lattices_Osemilattice__inf,type,
semilattice_inf:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Lattices_Osemilattice__sup,type,
semilattice_sup:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Lattices_Obounded__lattice__bot,type,
bounded_lattice_bot:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Lattices_Obounded__semilattice__sup__bot,type,
bounde1808546759up_bot:
!>[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_Olappend,type,
coinductive_lappend:
!>[A: $tType] : ( ( coinductive_llist @ A ) > ( coinductive_llist @ A ) > ( coinductive_llist @ A ) ) ).
thf(sy_c_Coinductive__List_Olconcat,type,
coinductive_lconcat:
!>[A: $tType] : ( ( coinductive_llist @ ( coinductive_llist @ A ) ) > ( coinductive_llist @ A ) ) ).
thf(sy_c_Coinductive__List_Oldistinct,type,
coindu351974385stinct:
!>[A: $tType] : ( ( coinductive_llist @ A ) > $o ) ).
thf(sy_c_Coinductive__List_OldropWhile,type,
coindu218763757pWhile:
!>[A: $tType] : ( ( A > $o ) > ( coinductive_llist @ A ) > ( coinductive_llist @ A ) ) ).
thf(sy_c_Coinductive__List_Olfilter,type,
coinductive_lfilter:
!>[A: $tType] : ( ( A > $o ) > ( coinductive_llist @ A ) > ( coinductive_llist @ A ) ) ).
thf(sy_c_Coinductive__List_Olfinite,type,
coinductive_lfinite:
!>[A: $tType] : ( ( coinductive_llist @ A ) > $o ) ).
thf(sy_c_Coinductive__List_Ollast,type,
coinductive_llast:
!>[A: $tType] : ( ( coinductive_llist @ A ) > A ) ).
thf(sy_c_Coinductive__List_Ollist_OLCons,type,
coinductive_LCons:
!>[A: $tType] : ( A > ( coinductive_llist @ A ) > ( coinductive_llist @ A ) ) ).
thf(sy_c_Coinductive__List_Ollist_OLNil,type,
coinductive_LNil:
!>[A: $tType] : ( coinductive_llist @ A ) ).
thf(sy_c_Coinductive__List_Ollist_Ocase__llist,type,
coindu1381640503_llist:
!>[B: $tType,A: $tType] : ( B > ( A > ( coinductive_llist @ A ) > B ) > ( coinductive_llist @ A ) > B ) ).
thf(sy_c_Coinductive__List_Ollist_Olhd,type,
coinductive_lhd:
!>[A: $tType] : ( ( coinductive_llist @ A ) > A ) ).
thf(sy_c_Coinductive__List_Ollist_Olnull,type,
coinductive_lnull:
!>[A: $tType] : ( ( coinductive_llist @ A ) > $o ) ).
thf(sy_c_Coinductive__List_Ollist_Olset,type,
coinductive_lset:
!>[A: $tType] : ( ( coinductive_llist @ A ) > ( set @ A ) ) ).
thf(sy_c_Coinductive__List_Ollist_Oltl,type,
coinductive_ltl:
!>[A: $tType] : ( ( coinductive_llist @ A ) > ( coinductive_llist @ A ) ) ).
thf(sy_c_Coinductive__List_Olmember,type,
coinductive_lmember:
!>[A: $tType] : ( A > ( coinductive_llist @ A ) > $o ) ).
thf(sy_c_Coinductive__List_Olsetp,type,
coinductive_lsetp:
!>[A: $tType] : ( ( coinductive_llist @ A ) > A > $o ) ).
thf(sy_c_Coinductive__List_Olstrict__prefix,type,
coindu1478340336prefix:
!>[A: $tType] : ( ( coinductive_llist @ A ) > ( coinductive_llist @ A ) > $o ) ).
thf(sy_c_Coinductive__List_OltakeWhile,type,
coindu501562517eWhile:
!>[A: $tType] : ( ( A > $o ) > ( coinductive_llist @ A ) > ( coinductive_llist @ A ) ) ).
thf(sy_c_Coinductive__List_Ounfold__llist,type,
coindu1441602521_llist:
!>[A: $tType,B: $tType] : ( ( A > $o ) > ( A > B ) > ( A > A ) > A > ( coinductive_llist @ B ) ) ).
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_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_Coinductive__Stream__Mirabelle__dydkjoctes_Ounfold__stream,type,
coindu139217191stream:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( A > A ) > A > ( stream @ B ) ) ).
thf(sy_c_If,type,
if:
!>[A: $tType] : ( $o > A > A > A ) ).
thf(sy_c_Lattices_Oinf__class_Oinf,type,
inf_inf:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Lattices_Osup__class_Osup,type,
sup_sup:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Orderings_Obot__class_Obot,type,
bot_bot:
!>[A: $tType] : A ).
thf(sy_c_Pure_Otype,type,
type2:
!>[A: $tType] : ( itself @ A ) ).
thf(sy_c_Set_OCollect,type,
collect:
!>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
thf(sy_c_Set_Odisjnt,type,
disjnt:
!>[A: $tType] : ( ( set @ A ) > ( set @ A ) > $o ) ).
thf(sy_c_Set_Ois__empty,type,
is_empty:
!>[A: $tType] : ( ( set @ A ) > $o ) ).
thf(sy_c_Stream_Osinterleave,type,
sinterleave:
!>[A: $tType] : ( ( stream @ A ) > ( stream @ A ) > ( stream @ A ) ) ).
thf(sy_c_Stream_Osmember,type,
smember:
!>[A: $tType] : ( A > ( stream @ A ) > $o ) ).
thf(sy_c_Stream_Ostream_Oshd,type,
shd:
!>[A: $tType] : ( ( stream @ A ) > A ) ).
thf(sy_c_Stream_Ostream_Osset,type,
sset:
!>[A: $tType] : ( ( stream @ A ) > ( set @ A ) ) ).
thf(sy_c_member,type,
member:
!>[A: $tType] : ( A > ( set @ A ) > $o ) ).
thf(sy_v_x____,type,
x: a ).
thf(sy_v_xs,type,
xs: stream @ a ).
thf(sy_v_xsa____,type,
xsa: stream @ a ).
%----Relevant facts (254)
thf(fact_0__092_060open_062_092_060not_062_Alnull_A_Illist__of__stream_Axs_J_A_092_060Longrightarrow_062_Alhd_A_Illist__of__stream_Axs_J_A_092_060in_062_Alset_A_Illist__of__stream_Axs_J_092_060close_062,axiom,
( ~ ( coinductive_lnull @ a @ ( coindu1724414836stream @ a @ xsa ) )
=> ( member @ a @ ( coinductive_lhd @ a @ ( coindu1724414836stream @ a @ xsa ) ) @ ( coinductive_lset @ a @ ( coindu1724414836stream @ a @ xsa ) ) ) ) ).
% \<open>\<not> lnull (llist_of_stream xs) \<Longrightarrow> lhd (llist_of_stream xs) \<in> lset (llist_of_stream xs)\<close>
thf(fact_1_stream_ORep__inject,axiom,
! [A: $tType,X: stream @ A,Y: stream @ A] :
( ( ( coindu1724414836stream @ A @ X )
= ( coindu1724414836stream @ A @ Y ) )
= ( X = Y ) ) ).
% stream.Rep_inject
thf(fact_2_lhd__llist__of__stream,axiom,
! [A: $tType,Xs: stream @ A] :
( ( coinductive_lhd @ A @ ( coindu1724414836stream @ A @ Xs ) )
= ( shd @ A @ Xs ) ) ).
% lhd_llist_of_stream
thf(fact_3_lnull__llist__of__stream,axiom,
! [A: $tType,Xs: stream @ A] :
~ ( coinductive_lnull @ A @ ( coindu1724414836stream @ A @ Xs ) ) ).
% lnull_llist_of_stream
thf(fact_4_stream_ORep__inverse,axiom,
! [A: $tType,X: stream @ A] :
( ( coindu2010755910_llist @ A @ ( coindu1724414836stream @ A @ X ) )
= X ) ).
% stream.Rep_inverse
thf(fact_5_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_6_unfold__stream_Osimps_I1_J,axiom,
! [B: $tType,A: $tType,G1: A > B,G2: A > A,A2: A] :
( ( shd @ B @ ( coindu139217191stream @ A @ B @ G1 @ G2 @ A2 ) )
= ( G1 @ A2 ) ) ).
% unfold_stream.simps(1)
thf(fact_7_shd__stream__of__llist,axiom,
! [A: $tType,Xs: coinductive_llist @ A] :
( ( shd @ A @ ( coindu2010755910_llist @ A @ Xs ) )
= ( coinductive_lhd @ A @ Xs ) ) ).
% shd_stream_of_llist
thf(fact_8_lset__lmember,axiom,
! [A: $tType,X: A,Xs: coinductive_llist @ A] :
( ( member @ A @ X @ ( coinductive_lset @ A @ Xs ) )
= ( coinductive_lmember @ A @ X @ Xs ) ) ).
% lset_lmember
thf(fact_9_lfinite__llist__of__stream,axiom,
! [A: $tType,Xs: stream @ A] :
~ ( coinductive_lfinite @ A @ ( coindu1724414836stream @ A @ Xs ) ) ).
% lfinite_llist_of_stream
thf(fact_10__092_060open_062x_A_092_060in_062_Asset_Axs_092_060close_062,axiom,
member @ a @ x @ ( sset @ a @ xs ) ).
% \<open>x \<in> sset xs\<close>
thf(fact_11_llist_Oset__sel_I1_J,axiom,
! [A: $tType,A2: coinductive_llist @ A] :
( ~ ( coinductive_lnull @ A @ A2 )
=> ( member @ A @ ( coinductive_lhd @ A @ A2 ) @ ( coinductive_lset @ A @ A2 ) ) ) ).
% llist.set_sel(1)
thf(fact_12_sinterleave_Osimps_I1_J,axiom,
! [A: $tType,S1: stream @ A,S2: stream @ A] :
( ( shd @ A @ ( sinterleave @ A @ S1 @ S2 ) )
= ( shd @ A @ S1 ) ) ).
% sinterleave.simps(1)
thf(fact_13_Coinductive__List_Olset__into__lsetp,axiom,
! [A: $tType,X: A,Xs: coinductive_llist @ A] :
( ( member @ A @ X @ ( coinductive_lset @ A @ Xs ) )
=> ( coinductive_lsetp @ A @ Xs @ X ) ) ).
% Coinductive_List.lset_into_lsetp
thf(fact_14_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_15_lzip_Oexhaust,axiom,
! [A: $tType,B: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ B] :
( ~ ( ( coinductive_lnull @ A @ Xs )
| ( coinductive_lnull @ B @ Ys ) )
=> ~ ( ~ ( coinductive_lnull @ A @ Xs )
=> ( coinductive_lnull @ B @ Ys ) ) ) ).
% lzip.exhaust
thf(fact_16_lappend_Oexhaust,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( ( coinductive_lnull @ A @ Xs )
=> ~ ( coinductive_lnull @ A @ Ys ) )
=> ( ~ ( coinductive_lnull @ A @ Xs )
| ~ ( coinductive_lnull @ A @ Ys ) ) ) ).
% lappend.exhaust
thf(fact_17_lnull__imp__lfinite,axiom,
! [A: $tType,Xs: coinductive_llist @ A] :
( ( coinductive_lnull @ A @ Xs )
=> ( coinductive_lfinite @ A @ Xs ) ) ).
% lnull_imp_lfinite
thf(fact_18_ltakeWhile_Oexhaust,axiom,
! [A: $tType,Xs: coinductive_llist @ A,P: A > $o] :
( ~ ( ( coinductive_lnull @ A @ Xs )
| ~ ( P @ ( coinductive_lhd @ A @ Xs ) ) )
=> ~ ( ~ ( coinductive_lnull @ A @ Xs )
=> ~ ( P @ ( coinductive_lhd @ A @ Xs ) ) ) ) ).
% ltakeWhile.exhaust
thf(fact_19_shd__sset,axiom,
! [A: $tType,A2: stream @ A] : ( member @ A @ ( shd @ A @ A2 ) @ ( sset @ A @ A2 ) ) ).
% shd_sset
thf(fact_20_lsetp__into__lset,axiom,
! [A: $tType,Xs: coinductive_llist @ A,X: A] :
( ( coinductive_lsetp @ A @ Xs @ X )
=> ( member @ A @ X @ ( coinductive_lset @ A @ Xs ) ) ) ).
% lsetp_into_lset
thf(fact_21_cr__streamI,axiom,
! [A: $tType,Xs: coinductive_llist @ A] :
( ~ ( coinductive_lfinite @ A @ Xs )
=> ( coindu1183105481stream @ A @ Xs @ ( coindu2010755910_llist @ A @ Xs ) ) ) ).
% cr_streamI
thf(fact_22_Stream_Osmember__def,axiom,
! [A: $tType] :
( ( smember @ A )
= ( ^ [X2: A,S: stream @ A] : ( member @ A @ X2 @ ( sset @ A @ S ) ) ) ) ).
% Stream.smember_def
thf(fact_23_llist__set__induct,axiom,
! [A: $tType,X: A,Xs: coinductive_llist @ A,P: A > ( coinductive_llist @ A ) > $o] :
( ( member @ A @ X @ ( coinductive_lset @ A @ Xs ) )
=> ( ! [Xs2: coinductive_llist @ A] :
( ~ ( coinductive_lnull @ A @ Xs2 )
=> ( P @ ( coinductive_lhd @ A @ Xs2 ) @ Xs2 ) )
=> ( ! [Xs2: coinductive_llist @ A,Y2: A] :
( ~ ( coinductive_lnull @ A @ Xs2 )
=> ( ( member @ A @ Y2 @ ( coinductive_lset @ A @ ( coinductive_ltl @ A @ Xs2 ) ) )
=> ( ( P @ Y2 @ ( coinductive_ltl @ A @ Xs2 ) )
=> ( P @ Y2 @ Xs2 ) ) ) )
=> ( P @ X @ Xs ) ) ) ) ).
% llist_set_induct
thf(fact_24_lset__eq__empty,axiom,
! [A: $tType,Xs: coinductive_llist @ A] :
( ( ( coinductive_lset @ A @ Xs )
= ( bot_bot @ ( set @ A ) ) )
= ( coinductive_lnull @ A @ Xs ) ) ).
% lset_eq_empty
thf(fact_25_lnull__ldropWhile,axiom,
! [A: $tType,P: A > $o,Xs: coinductive_llist @ A] :
( ( coinductive_lnull @ A @ ( coindu218763757pWhile @ A @ P @ Xs ) )
= ( ! [X2: A] :
( ( member @ A @ X2 @ ( coinductive_lset @ A @ Xs ) )
=> ( P @ X2 ) ) ) ) ).
% lnull_ldropWhile
thf(fact_26_lhd__lconcat,axiom,
! [A: $tType,Xss: coinductive_llist @ ( coinductive_llist @ A )] :
( ~ ( coinductive_lnull @ ( coinductive_llist @ A ) @ Xss )
=> ( ~ ( coinductive_lnull @ A @ ( coinductive_lhd @ ( coinductive_llist @ A ) @ Xss ) )
=> ( ( coinductive_lhd @ A @ ( coinductive_lconcat @ A @ Xss ) )
= ( coinductive_lhd @ A @ ( coinductive_lhd @ ( coinductive_llist @ A ) @ Xss ) ) ) ) ) ).
% lhd_lconcat
thf(fact_27_sset__sinterleave,axiom,
! [A: $tType,S1: stream @ A,S2: stream @ A] :
( ( sset @ A @ ( sinterleave @ A @ S1 @ S2 ) )
= ( sup_sup @ ( set @ A ) @ ( sset @ A @ S1 ) @ ( sset @ A @ S2 ) ) ) ).
% sset_sinterleave
thf(fact_28_ltakeWhile_Odisc__iff_I2_J,axiom,
! [A: $tType,P: A > $o,Xs: coinductive_llist @ A] :
( ( ~ ( coinductive_lnull @ A @ ( coindu501562517eWhile @ A @ P @ Xs ) ) )
= ( ~ ( coinductive_lnull @ A @ Xs )
& ( P @ ( coinductive_lhd @ A @ Xs ) ) ) ) ).
% ltakeWhile.disc_iff(2)
thf(fact_29_ltakeWhile_Odisc__iff_I1_J,axiom,
! [A: $tType,P: A > $o,Xs: coinductive_llist @ A] :
( ( coinductive_lnull @ A @ ( coindu501562517eWhile @ A @ P @ Xs ) )
= ( ( coinductive_lnull @ A @ Xs )
| ~ ( P @ ( coinductive_lhd @ A @ Xs ) ) ) ) ).
% ltakeWhile.disc_iff(1)
thf(fact_30_lnull__ltakeWhile,axiom,
! [A: $tType,P: A > $o,Xs: coinductive_llist @ A] :
( ( coinductive_lnull @ A @ ( coindu501562517eWhile @ A @ P @ Xs ) )
= ( ~ ( coinductive_lnull @ A @ Xs )
=> ~ ( P @ ( coinductive_lhd @ A @ Xs ) ) ) ) ).
% lnull_ltakeWhile
thf(fact_31_lnull__lfilter,axiom,
! [A: $tType,P: A > $o,Xs: coinductive_llist @ A] :
( ( coinductive_lnull @ A @ ( coinductive_lfilter @ A @ P @ Xs ) )
= ( ! [X2: A] :
( ( member @ A @ X2 @ ( coinductive_lset @ A @ Xs ) )
=> ~ ( P @ X2 ) ) ) ) ).
% lnull_lfilter
thf(fact_32_lfilter__idem,axiom,
! [A: $tType,P: A > $o,Xs: coinductive_llist @ A] :
( ( coinductive_lfilter @ A @ P @ ( coinductive_lfilter @ A @ P @ Xs ) )
= ( coinductive_lfilter @ A @ P @ Xs ) ) ).
% lfilter_idem
thf(fact_33_lfinite__ltl,axiom,
! [A: $tType,Xs: coinductive_llist @ A] :
( ( coinductive_lfinite @ A @ ( coinductive_ltl @ A @ Xs ) )
= ( coinductive_lfinite @ A @ Xs ) ) ).
% lfinite_ltl
thf(fact_34_ltakeWhile_Osimps_I4_J,axiom,
! [A: $tType,Xs: coinductive_llist @ A,P: A > $o] :
( ~ ( coinductive_lnull @ A @ Xs )
=> ( ( P @ ( coinductive_lhd @ A @ Xs ) )
=> ( ( coinductive_ltl @ A @ ( coindu501562517eWhile @ A @ P @ Xs ) )
= ( coindu501562517eWhile @ A @ P @ ( coinductive_ltl @ A @ Xs ) ) ) ) ) ).
% ltakeWhile.simps(4)
thf(fact_35_lnull__ltlI,axiom,
! [A: $tType,Xs: coinductive_llist @ A] :
( ( coinductive_lnull @ A @ Xs )
=> ( coinductive_lnull @ A @ ( coinductive_ltl @ A @ Xs ) ) ) ).
% lnull_ltlI
thf(fact_36_in__lset__ltlD,axiom,
! [A: $tType,X: A,Xs: coinductive_llist @ A] :
( ( member @ A @ X @ ( coinductive_lset @ A @ ( coinductive_ltl @ A @ Xs ) ) )
=> ( member @ A @ X @ ( coinductive_lset @ A @ Xs ) ) ) ).
% in_lset_ltlD
thf(fact_37_lfilter__cong,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A,P: A > $o,Q: A > $o] :
( ( Xs = Ys )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ ( coinductive_lset @ A @ Ys ) )
=> ( ( P @ X3 )
= ( Q @ X3 ) ) )
=> ( ( coinductive_lfilter @ A @ P @ Xs )
= ( coinductive_lfilter @ A @ Q @ Ys ) ) ) ) ).
% lfilter_cong
thf(fact_38_lfilter__id__conv,axiom,
! [A: $tType,P: A > $o,Xs: coinductive_llist @ A] :
( ( ( coinductive_lfilter @ A @ P @ Xs )
= Xs )
= ( ! [X2: A] :
( ( member @ A @ X2 @ ( coinductive_lset @ A @ Xs ) )
=> ( P @ X2 ) ) ) ) ).
% lfilter_id_conv
thf(fact_39_lfinite__lfilterI,axiom,
! [A: $tType,Xs: coinductive_llist @ A,P: A > $o] :
( ( coinductive_lfinite @ A @ Xs )
=> ( coinductive_lfinite @ A @ ( coinductive_lfilter @ A @ P @ Xs ) ) ) ).
% lfinite_lfilterI
thf(fact_40_ltakeWhile__all,axiom,
! [A: $tType,Xs: coinductive_llist @ A,P: A > $o] :
( ! [X3: A] :
( ( member @ A @ X3 @ ( coinductive_lset @ A @ Xs ) )
=> ( P @ X3 ) )
=> ( ( coindu501562517eWhile @ A @ P @ Xs )
= Xs ) ) ).
% ltakeWhile_all
thf(fact_41_ltakeWhile__cong,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A,P: A > $o,Q: A > $o] :
( ( Xs = Ys )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ ( coinductive_lset @ A @ Ys ) )
=> ( ( P @ X3 )
= ( Q @ X3 ) ) )
=> ( ( coindu501562517eWhile @ A @ P @ Xs )
= ( coindu501562517eWhile @ A @ Q @ Ys ) ) ) ) ).
% ltakeWhile_cong
thf(fact_42_lset__ltakeWhileD,axiom,
! [A: $tType,X: A,P: A > $o,Xs: coinductive_llist @ A] :
( ( member @ A @ X @ ( coinductive_lset @ A @ ( coindu501562517eWhile @ A @ P @ Xs ) ) )
=> ( ( member @ A @ X @ ( coinductive_lset @ A @ Xs ) )
& ( P @ X ) ) ) ).
% lset_ltakeWhileD
thf(fact_43_mem__Collect__eq,axiom,
! [A: $tType,A2: A,P: A > $o] :
( ( member @ A @ A2 @ ( collect @ A @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_44_Collect__mem__eq,axiom,
! [A: $tType,A3: set @ A] :
( ( collect @ A
@ ^ [X2: A] : ( member @ A @ X2 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_45_Collect__cong,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ! [X3: A] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collect @ A @ P )
= ( collect @ A @ Q ) ) ) ).
% Collect_cong
thf(fact_46_ext,axiom,
! [B: $tType,A: $tType,F: A > B,G: A > B] :
( ! [X3: A] :
( ( F @ X3 )
= ( G @ X3 ) )
=> ( F = G ) ) ).
% ext
thf(fact_47_ldropWhile__cong,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A,P: A > $o,Q: A > $o] :
( ( Xs = Ys )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ ( coinductive_lset @ A @ Ys ) )
=> ( ( P @ X3 )
= ( Q @ X3 ) ) )
=> ( ( coindu218763757pWhile @ A @ P @ Xs )
= ( coindu218763757pWhile @ A @ Q @ Ys ) ) ) ) ).
% ldropWhile_cong
thf(fact_48_in__lset__ldropWhileD,axiom,
! [A: $tType,X: A,P: A > $o,Xs: coinductive_llist @ A] :
( ( member @ A @ X @ ( coinductive_lset @ A @ ( coindu218763757pWhile @ A @ P @ Xs ) ) )
=> ( member @ A @ X @ ( coinductive_lset @ A @ Xs ) ) ) ).
% in_lset_ldropWhileD
thf(fact_49_sset__neq__empty,axiom,
! [A: $tType,Xs: stream @ A] :
( ( sset @ A @ Xs )
!= ( bot_bot @ ( set @ A ) ) ) ).
% sset_neq_empty
thf(fact_50_llist_Oset__sel_I2_J,axiom,
! [A: $tType,A2: coinductive_llist @ A,X: A] :
( ~ ( coinductive_lnull @ A @ A2 )
=> ( ( member @ A @ X @ ( coinductive_lset @ A @ ( coinductive_ltl @ A @ A2 ) ) )
=> ( member @ A @ X @ ( coinductive_lset @ A @ A2 ) ) ) ) ).
% llist.set_sel(2)
thf(fact_51_lfinite__induct,axiom,
! [A: $tType,Xs: coinductive_llist @ A,P: ( coinductive_llist @ A ) > $o] :
( ( coinductive_lfinite @ A @ Xs )
=> ( ! [Xs2: coinductive_llist @ A] :
( ( coinductive_lnull @ A @ Xs2 )
=> ( P @ Xs2 ) )
=> ( ! [Xs2: coinductive_llist @ A] :
( ( coinductive_lfinite @ A @ Xs2 )
=> ( ~ ( coinductive_lnull @ A @ Xs2 )
=> ( ( P @ ( coinductive_ltl @ A @ Xs2 ) )
=> ( P @ Xs2 ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% lfinite_induct
thf(fact_52_llist_Ocoinduct__strong,axiom,
! [A: $tType,R: ( coinductive_llist @ A ) > ( coinductive_llist @ A ) > $o,Llist: coinductive_llist @ A,Llist2: coinductive_llist @ A] :
( ( R @ Llist @ Llist2 )
=> ( ! [Llist3: coinductive_llist @ A,Llist4: coinductive_llist @ A] :
( ( R @ Llist3 @ Llist4 )
=> ( ( ( coinductive_lnull @ A @ Llist3 )
= ( coinductive_lnull @ A @ Llist4 ) )
& ( ~ ( coinductive_lnull @ A @ Llist3 )
=> ( ~ ( coinductive_lnull @ A @ Llist4 )
=> ( ( ( coinductive_lhd @ A @ Llist3 )
= ( coinductive_lhd @ A @ Llist4 ) )
& ( ( R @ ( coinductive_ltl @ A @ Llist3 ) @ ( coinductive_ltl @ A @ Llist4 ) )
| ( ( coinductive_ltl @ A @ Llist3 )
= ( coinductive_ltl @ A @ Llist4 ) ) ) ) ) ) ) )
=> ( Llist = Llist2 ) ) ) ).
% llist.coinduct_strong
thf(fact_53_llist_Ocoinduct,axiom,
! [A: $tType,R: ( coinductive_llist @ A ) > ( coinductive_llist @ A ) > $o,Llist: coinductive_llist @ A,Llist2: coinductive_llist @ A] :
( ( R @ Llist @ Llist2 )
=> ( ! [Llist3: coinductive_llist @ A,Llist4: coinductive_llist @ A] :
( ( R @ Llist3 @ Llist4 )
=> ( ( ( coinductive_lnull @ A @ Llist3 )
= ( coinductive_lnull @ A @ Llist4 ) )
& ( ~ ( coinductive_lnull @ A @ Llist3 )
=> ( ~ ( coinductive_lnull @ A @ Llist4 )
=> ( ( ( coinductive_lhd @ A @ Llist3 )
= ( coinductive_lhd @ A @ Llist4 ) )
& ( R @ ( coinductive_ltl @ A @ Llist3 ) @ ( coinductive_ltl @ A @ Llist4 ) ) ) ) ) ) )
=> ( Llist = Llist2 ) ) ) ).
% llist.coinduct
thf(fact_54_llist_Oexpand,axiom,
! [A: $tType,Llist: coinductive_llist @ A,Llist2: coinductive_llist @ A] :
( ( ( coinductive_lnull @ A @ Llist )
= ( coinductive_lnull @ A @ Llist2 ) )
=> ( ( ~ ( coinductive_lnull @ A @ Llist )
=> ( ~ ( coinductive_lnull @ A @ Llist2 )
=> ( ( ( coinductive_lhd @ A @ Llist )
= ( coinductive_lhd @ A @ Llist2 ) )
& ( ( coinductive_ltl @ A @ Llist )
= ( coinductive_ltl @ A @ Llist2 ) ) ) ) )
=> ( Llist = Llist2 ) ) ) ).
% llist.expand
thf(fact_55_lhd__ltakeWhile,axiom,
! [A: $tType,Xs: coinductive_llist @ A,P: A > $o] :
( ~ ( coinductive_lnull @ A @ Xs )
=> ( ( P @ ( coinductive_lhd @ A @ Xs ) )
=> ( ( coinductive_lhd @ A @ ( coindu501562517eWhile @ A @ P @ Xs ) )
= ( coinductive_lhd @ A @ Xs ) ) ) ) ).
% lhd_ltakeWhile
thf(fact_56_ltakeWhile_Odisc_I1_J,axiom,
! [A: $tType,Xs: coinductive_llist @ A,P: A > $o] :
( ( ( coinductive_lnull @ A @ Xs )
| ~ ( P @ ( coinductive_lhd @ A @ Xs ) ) )
=> ( coinductive_lnull @ A @ ( coindu501562517eWhile @ A @ P @ Xs ) ) ) ).
% ltakeWhile.disc(1)
thf(fact_57_ltakeWhile_Odisc_I2_J,axiom,
! [A: $tType,Xs: coinductive_llist @ A,P: A > $o] :
( ~ ( coinductive_lnull @ A @ Xs )
=> ( ( P @ ( coinductive_lhd @ A @ Xs ) )
=> ~ ( coinductive_lnull @ A @ ( coindu501562517eWhile @ A @ P @ Xs ) ) ) ) ).
% ltakeWhile.disc(2)
thf(fact_58_lfinite__ltakeWhile,axiom,
! [A: $tType,P: A > $o,Xs: coinductive_llist @ A] :
( ( coinductive_lfinite @ A @ ( coindu501562517eWhile @ A @ P @ Xs ) )
= ( ( coinductive_lfinite @ A @ Xs )
| ? [X2: A] :
( ( member @ A @ X2 @ ( coinductive_lset @ A @ Xs ) )
& ~ ( P @ X2 ) ) ) ) ).
% lfinite_ltakeWhile
thf(fact_59_stream__of__llist__def,axiom,
! [A: $tType] :
( ( coindu2010755910_llist @ A )
= ( coindu139217191stream @ ( coinductive_llist @ A ) @ A @ ( coinductive_lhd @ A ) @ ( coinductive_ltl @ A ) ) ) ).
% stream_of_llist_def
thf(fact_60_lset__lnull,axiom,
! [A: $tType,Xs: coinductive_llist @ A] :
( ( coinductive_lnull @ A @ Xs )
=> ( ( coinductive_lset @ A @ Xs )
= ( bot_bot @ ( set @ A ) ) ) ) ).
% lset_lnull
thf(fact_61_lfinite__ldropWhile,axiom,
! [A: $tType,P: A > $o,Xs: coinductive_llist @ A] :
( ( coinductive_lfinite @ A @ ( coindu218763757pWhile @ A @ P @ Xs ) )
= ( ? [X2: A] :
( ( member @ A @ X2 @ ( coinductive_lset @ A @ Xs ) )
& ~ ( P @ X2 ) )
=> ( coinductive_lfinite @ A @ Xs ) ) ) ).
% lfinite_ldropWhile
thf(fact_62_lhd__ldropWhile,axiom,
! [A: $tType,Xs: coinductive_llist @ A,P: A > $o] :
( ? [X4: A] :
( ( member @ A @ X4 @ ( coinductive_lset @ A @ Xs ) )
& ~ ( P @ X4 ) )
=> ~ ( P @ ( coinductive_lhd @ A @ ( coindu218763757pWhile @ A @ P @ Xs ) ) ) ) ).
% lhd_ldropWhile
thf(fact_63_lhd__ldropWhile__in__lset,axiom,
! [A: $tType,Xs: coinductive_llist @ A,P: A > $o] :
( ? [X4: A] :
( ( member @ A @ X4 @ ( coinductive_lset @ A @ Xs ) )
& ~ ( P @ X4 ) )
=> ( member @ A @ ( coinductive_lhd @ A @ ( coindu218763757pWhile @ A @ P @ Xs ) ) @ ( coinductive_lset @ A @ Xs ) ) ) ).
% lhd_ldropWhile_in_lset
thf(fact_64_Un__empty,axiom,
! [A: $tType,A3: set @ A,B2: set @ A] :
( ( ( sup_sup @ ( set @ A ) @ A3 @ B2 )
= ( bot_bot @ ( set @ A ) ) )
= ( ( A3
= ( bot_bot @ ( set @ A ) ) )
& ( B2
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Un_empty
thf(fact_65_sup__bot_Oright__neutral,axiom,
! [A: $tType] :
( ( bounde1808546759up_bot @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( sup_sup @ A @ A2 @ ( bot_bot @ A ) )
= A2 ) ) ).
% sup_bot.right_neutral
thf(fact_66_sup__bot_Oleft__neutral,axiom,
! [A: $tType] :
( ( bounde1808546759up_bot @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( sup_sup @ A @ ( bot_bot @ A ) @ A2 )
= A2 ) ) ).
% sup_bot.left_neutral
thf(fact_67_sup__eq__bot__iff,axiom,
! [A: $tType] :
( ( bounded_lattice_bot @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( ( sup_sup @ A @ X @ Y )
= ( bot_bot @ A ) )
= ( ( X
= ( bot_bot @ A ) )
& ( Y
= ( bot_bot @ A ) ) ) ) ) ).
% sup_eq_bot_iff
thf(fact_68_bot__eq__sup__iff,axiom,
! [A: $tType] :
( ( bounded_lattice_bot @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( ( bot_bot @ A )
= ( sup_sup @ A @ X @ Y ) )
= ( ( X
= ( bot_bot @ A ) )
& ( Y
= ( bot_bot @ A ) ) ) ) ) ).
% bot_eq_sup_iff
thf(fact_69_Un__iff,axiom,
! [A: $tType,C: A,A3: set @ A,B2: set @ A] :
( ( member @ A @ C @ ( sup_sup @ ( set @ A ) @ A3 @ B2 ) )
= ( ( member @ A @ C @ A3 )
| ( member @ A @ C @ B2 ) ) ) ).
% Un_iff
thf(fact_70_UnCI,axiom,
! [A: $tType,C: A,B2: set @ A,A3: set @ A] :
( ( ~ ( member @ A @ C @ B2 )
=> ( member @ A @ C @ A3 ) )
=> ( member @ A @ C @ ( sup_sup @ ( set @ A ) @ A3 @ B2 ) ) ) ).
% UnCI
thf(fact_71_sup_Oright__idem,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [A2: A,B3: A] :
( ( sup_sup @ A @ ( sup_sup @ A @ A2 @ B3 ) @ B3 )
= ( sup_sup @ A @ A2 @ B3 ) ) ) ).
% sup.right_idem
thf(fact_72_sup__left__idem,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( sup_sup @ A @ X @ ( sup_sup @ A @ X @ Y ) )
= ( sup_sup @ A @ X @ Y ) ) ) ).
% sup_left_idem
thf(fact_73_empty__iff,axiom,
! [A: $tType,C: A] :
~ ( member @ A @ C @ ( bot_bot @ ( set @ A ) ) ) ).
% empty_iff
thf(fact_74_all__not__in__conv,axiom,
! [A: $tType,A3: set @ A] :
( ( ! [X2: A] :
~ ( member @ A @ X2 @ A3 ) )
= ( A3
= ( bot_bot @ ( set @ A ) ) ) ) ).
% all_not_in_conv
thf(fact_75_Collect__empty__eq,axiom,
! [A: $tType,P: A > $o] :
( ( ( collect @ A @ P )
= ( bot_bot @ ( set @ A ) ) )
= ( ! [X2: A] :
~ ( P @ X2 ) ) ) ).
% Collect_empty_eq
thf(fact_76_empty__Collect__eq,axiom,
! [A: $tType,P: A > $o] :
( ( ( bot_bot @ ( set @ A ) )
= ( collect @ A @ P ) )
= ( ! [X2: A] :
~ ( P @ X2 ) ) ) ).
% empty_Collect_eq
thf(fact_77_sup__apply,axiom,
! [B: $tType,A: $tType] :
( ( semilattice_sup @ B @ ( type2 @ B ) )
=> ( ( sup_sup @ ( A > B ) )
= ( ^ [F2: A > B,G3: A > B,X2: A] : ( sup_sup @ B @ ( F2 @ X2 ) @ ( G3 @ X2 ) ) ) ) ) ).
% sup_apply
thf(fact_78_sup_Oidem,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( sup_sup @ A @ A2 @ A2 )
= A2 ) ) ).
% sup.idem
thf(fact_79_sup__idem,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [X: A] :
( ( sup_sup @ A @ X @ X )
= X ) ) ).
% sup_idem
thf(fact_80_sup_Oleft__idem,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [A2: A,B3: A] :
( ( sup_sup @ A @ A2 @ ( sup_sup @ A @ A2 @ B3 ) )
= ( sup_sup @ A @ A2 @ B3 ) ) ) ).
% sup.left_idem
thf(fact_81_bot__set__def,axiom,
! [A: $tType] :
( ( bot_bot @ ( set @ A ) )
= ( collect @ A @ ( bot_bot @ ( A > $o ) ) ) ) ).
% bot_set_def
thf(fact_82_emptyE,axiom,
! [A: $tType,A2: A] :
~ ( member @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ).
% emptyE
thf(fact_83_equals0D,axiom,
! [A: $tType,A3: set @ A,A2: A] :
( ( A3
= ( bot_bot @ ( set @ A ) ) )
=> ~ ( member @ A @ A2 @ A3 ) ) ).
% equals0D
thf(fact_84_equals0I,axiom,
! [A: $tType,A3: set @ A] :
( ! [Y2: A] :
~ ( member @ A @ Y2 @ A3 )
=> ( A3
= ( bot_bot @ ( set @ A ) ) ) ) ).
% equals0I
thf(fact_85_ex__in__conv,axiom,
! [A: $tType,A3: set @ A] :
( ( ? [X2: A] : ( member @ A @ X2 @ A3 ) )
= ( A3
!= ( bot_bot @ ( set @ A ) ) ) ) ).
% ex_in_conv
thf(fact_86_inf__sup__aci_I8_J,axiom,
! [A: $tType] :
( ( lattice @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( sup_sup @ A @ X @ ( sup_sup @ A @ X @ Y ) )
= ( sup_sup @ A @ X @ Y ) ) ) ).
% inf_sup_aci(8)
thf(fact_87_inf__sup__aci_I7_J,axiom,
! [A: $tType] :
( ( lattice @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A,Z: A] :
( ( sup_sup @ A @ X @ ( sup_sup @ A @ Y @ Z ) )
= ( sup_sup @ A @ Y @ ( sup_sup @ A @ X @ Z ) ) ) ) ).
% inf_sup_aci(7)
thf(fact_88_inf__sup__aci_I6_J,axiom,
! [A: $tType] :
( ( lattice @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A,Z: A] :
( ( sup_sup @ A @ ( sup_sup @ A @ X @ Y ) @ Z )
= ( sup_sup @ A @ X @ ( sup_sup @ A @ Y @ Z ) ) ) ) ).
% inf_sup_aci(6)
thf(fact_89_inf__sup__aci_I5_J,axiom,
! [A: $tType] :
( ( lattice @ A @ ( type2 @ A ) )
=> ( ( sup_sup @ A )
= ( ^ [X2: A,Y3: A] : ( sup_sup @ A @ Y3 @ X2 ) ) ) ) ).
% inf_sup_aci(5)
thf(fact_90_sup__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( semilattice_sup @ B @ ( type2 @ B ) )
=> ( ( sup_sup @ ( A > B ) )
= ( ^ [F2: A > B,G3: A > B,X2: A] : ( sup_sup @ B @ ( F2 @ X2 ) @ ( G3 @ X2 ) ) ) ) ) ).
% sup_fun_def
thf(fact_91_sup_Oassoc,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [A2: A,B3: A,C: A] :
( ( sup_sup @ A @ ( sup_sup @ A @ A2 @ B3 ) @ C )
= ( sup_sup @ A @ A2 @ ( sup_sup @ A @ B3 @ C ) ) ) ) ).
% sup.assoc
thf(fact_92_sup__assoc,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A,Z: A] :
( ( sup_sup @ A @ ( sup_sup @ A @ X @ Y ) @ Z )
= ( sup_sup @ A @ X @ ( sup_sup @ A @ Y @ Z ) ) ) ) ).
% sup_assoc
thf(fact_93_sup_Ocommute,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ( ( sup_sup @ A )
= ( ^ [A4: A,B4: A] : ( sup_sup @ A @ B4 @ A4 ) ) ) ) ).
% sup.commute
thf(fact_94_sup__commute,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ( ( sup_sup @ A )
= ( ^ [X2: A,Y3: A] : ( sup_sup @ A @ Y3 @ X2 ) ) ) ) ).
% sup_commute
thf(fact_95_sup_Oleft__commute,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [B3: A,A2: A,C: A] :
( ( sup_sup @ A @ B3 @ ( sup_sup @ A @ A2 @ C ) )
= ( sup_sup @ A @ A2 @ ( sup_sup @ A @ B3 @ C ) ) ) ) ).
% sup.left_commute
thf(fact_96_sup__left__commute,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A,Z: A] :
( ( sup_sup @ A @ X @ ( sup_sup @ A @ Y @ Z ) )
= ( sup_sup @ A @ Y @ ( sup_sup @ A @ X @ Z ) ) ) ) ).
% sup_left_commute
thf(fact_97_UnE,axiom,
! [A: $tType,C: A,A3: set @ A,B2: set @ A] :
( ( member @ A @ C @ ( sup_sup @ ( set @ A ) @ A3 @ B2 ) )
=> ( ~ ( member @ A @ C @ A3 )
=> ( member @ A @ C @ B2 ) ) ) ).
% UnE
thf(fact_98_UnI1,axiom,
! [A: $tType,C: A,A3: set @ A,B2: set @ A] :
( ( member @ A @ C @ A3 )
=> ( member @ A @ C @ ( sup_sup @ ( set @ A ) @ A3 @ B2 ) ) ) ).
% UnI1
thf(fact_99_UnI2,axiom,
! [A: $tType,C: A,B2: set @ A,A3: set @ A] :
( ( member @ A @ C @ B2 )
=> ( member @ A @ C @ ( sup_sup @ ( set @ A ) @ A3 @ B2 ) ) ) ).
% UnI2
thf(fact_100_bex__Un,axiom,
! [A: $tType,A3: set @ A,B2: set @ A,P: A > $o] :
( ( ? [X2: A] :
( ( member @ A @ X2 @ ( sup_sup @ ( set @ A ) @ A3 @ B2 ) )
& ( P @ X2 ) ) )
= ( ? [X2: A] :
( ( member @ A @ X2 @ A3 )
& ( P @ X2 ) )
| ? [X2: A] :
( ( member @ A @ X2 @ B2 )
& ( P @ X2 ) ) ) ) ).
% bex_Un
thf(fact_101_ball__Un,axiom,
! [A: $tType,A3: set @ A,B2: set @ A,P: A > $o] :
( ( ! [X2: A] :
( ( member @ A @ X2 @ ( sup_sup @ ( set @ A ) @ A3 @ B2 ) )
=> ( P @ X2 ) ) )
= ( ! [X2: A] :
( ( member @ A @ X2 @ A3 )
=> ( P @ X2 ) )
& ! [X2: A] :
( ( member @ A @ X2 @ B2 )
=> ( P @ X2 ) ) ) ) ).
% ball_Un
thf(fact_102_Un__assoc,axiom,
! [A: $tType,A3: set @ A,B2: set @ A,C2: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A3 @ B2 ) @ C2 )
= ( sup_sup @ ( set @ A ) @ A3 @ ( sup_sup @ ( set @ A ) @ B2 @ C2 ) ) ) ).
% Un_assoc
thf(fact_103_Un__absorb,axiom,
! [A: $tType,A3: set @ A] :
( ( sup_sup @ ( set @ A ) @ A3 @ A3 )
= A3 ) ).
% Un_absorb
thf(fact_104_Un__commute,axiom,
! [A: $tType] :
( ( sup_sup @ ( set @ A ) )
= ( ^ [A5: set @ A,B5: set @ A] : ( sup_sup @ ( set @ A ) @ B5 @ A5 ) ) ) ).
% Un_commute
thf(fact_105_Un__left__absorb,axiom,
! [A: $tType,A3: set @ A,B2: set @ A] :
( ( sup_sup @ ( set @ A ) @ A3 @ ( sup_sup @ ( set @ A ) @ A3 @ B2 ) )
= ( sup_sup @ ( set @ A ) @ A3 @ B2 ) ) ).
% Un_left_absorb
thf(fact_106_Un__left__commute,axiom,
! [A: $tType,A3: set @ A,B2: set @ A,C2: set @ A] :
( ( sup_sup @ ( set @ A ) @ A3 @ ( sup_sup @ ( set @ A ) @ B2 @ C2 ) )
= ( sup_sup @ ( set @ A ) @ B2 @ ( sup_sup @ ( set @ A ) @ A3 @ C2 ) ) ) ).
% Un_left_commute
thf(fact_107_sup__bot__left,axiom,
! [A: $tType] :
( ( bounded_lattice_bot @ A @ ( type2 @ A ) )
=> ! [X: A] :
( ( sup_sup @ A @ ( bot_bot @ A ) @ X )
= X ) ) ).
% sup_bot_left
thf(fact_108_sup__bot__right,axiom,
! [A: $tType] :
( ( bounded_lattice_bot @ A @ ( type2 @ A ) )
=> ! [X: A] :
( ( sup_sup @ A @ X @ ( bot_bot @ A ) )
= X ) ) ).
% sup_bot_right
thf(fact_109_Un__empty__left,axiom,
! [A: $tType,B2: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ B2 )
= B2 ) ).
% Un_empty_left
thf(fact_110_Un__empty__right,axiom,
! [A: $tType,A3: set @ A] :
( ( sup_sup @ ( set @ A ) @ A3 @ ( bot_bot @ ( set @ A ) ) )
= A3 ) ).
% Un_empty_right
thf(fact_111_bot__apply,axiom,
! [C3: $tType,D: $tType] :
( ( bot @ C3 @ ( type2 @ C3 ) )
=> ( ( bot_bot @ ( D > C3 ) )
= ( ^ [X2: D] : ( bot_bot @ C3 ) ) ) ) ).
% bot_apply
thf(fact_112_ltl__lconcat,axiom,
! [A: $tType,Xss: coinductive_llist @ ( coinductive_llist @ A )] :
( ~ ( coinductive_lnull @ ( coinductive_llist @ A ) @ Xss )
=> ( ~ ( coinductive_lnull @ A @ ( coinductive_lhd @ ( coinductive_llist @ A ) @ Xss ) )
=> ( ( coinductive_ltl @ A @ ( coinductive_lconcat @ A @ Xss ) )
= ( coinductive_lappend @ A @ ( coinductive_ltl @ A @ ( coinductive_lhd @ ( coinductive_llist @ A ) @ Xss ) ) @ ( coinductive_lconcat @ A @ ( coinductive_ltl @ ( coinductive_llist @ A ) @ Xss ) ) ) ) ) ) ).
% ltl_lconcat
thf(fact_113_gen__lset__def,axiom,
! [A: $tType] :
( ( coinductive_gen_lset @ A )
= ( ^ [A5: set @ A,Xs3: coinductive_llist @ A] : ( sup_sup @ ( set @ A ) @ A5 @ ( coinductive_lset @ A @ Xs3 ) ) ) ) ).
% gen_lset_def
thf(fact_114_lset__code,axiom,
! [A: $tType] :
( ( coinductive_lset @ A )
= ( coinductive_gen_lset @ A @ ( bot_bot @ ( set @ A ) ) ) ) ).
% lset_code
thf(fact_115_unfold__llist__id,axiom,
! [A: $tType,Xs: coinductive_llist @ A] :
( ( coindu1441602521_llist @ ( coinductive_llist @ A ) @ A @ ( coinductive_lnull @ A ) @ ( coinductive_lhd @ A ) @ ( coinductive_ltl @ A ) @ Xs )
= Xs ) ).
% unfold_llist_id
thf(fact_116_Set_Ois__empty__def,axiom,
! [A: $tType] :
( ( is_empty @ A )
= ( ^ [A5: set @ A] :
( A5
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Set.is_empty_def
thf(fact_117_lnull__lappend,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( coinductive_lnull @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) )
= ( ( coinductive_lnull @ A @ Xs )
& ( coinductive_lnull @ A @ Ys ) ) ) ).
% lnull_lappend
thf(fact_118_lappend_Odisc__iff_I2_J,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( ~ ( coinductive_lnull @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) ) )
= ( ~ ( coinductive_lnull @ A @ Xs )
| ~ ( coinductive_lnull @ A @ Ys ) ) ) ).
% lappend.disc_iff(2)
thf(fact_119_lfinite__lappend,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( coinductive_lfinite @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) )
= ( ( coinductive_lfinite @ A @ Xs )
& ( coinductive_lfinite @ A @ Ys ) ) ) ).
% lfinite_lappend
thf(fact_120_lconcat__lappend,axiom,
! [A: $tType,Xss: coinductive_llist @ ( coinductive_llist @ A ),Yss: coinductive_llist @ ( coinductive_llist @ A )] :
( ( coinductive_lfinite @ ( coinductive_llist @ A ) @ Xss )
=> ( ( coinductive_lconcat @ A @ ( coinductive_lappend @ ( coinductive_llist @ A ) @ Xss @ Yss ) )
= ( coinductive_lappend @ A @ ( coinductive_lconcat @ A @ Xss ) @ ( coinductive_lconcat @ A @ Yss ) ) ) ) ).
% lconcat_lappend
thf(fact_121_unfold__llist_Odisc__iff_I1_J,axiom,
! [B: $tType,A: $tType,P2: A > $o,G21: A > B,G22: A > A,A2: A] :
( ( coinductive_lnull @ B @ ( coindu1441602521_llist @ A @ B @ P2 @ G21 @ G22 @ A2 ) )
= ( P2 @ A2 ) ) ).
% unfold_llist.disc_iff(1)
thf(fact_122_unfold__llist_Odisc__iff_I2_J,axiom,
! [B: $tType,A: $tType,P2: A > $o,G21: A > B,G22: A > A,A2: A] :
( ( ~ ( coinductive_lnull @ B @ ( coindu1441602521_llist @ A @ B @ P2 @ G21 @ G22 @ A2 ) ) )
= ( ~ ( P2 @ A2 ) ) ) ).
% unfold_llist.disc_iff(2)
thf(fact_123_ltl__lappend,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( ( coinductive_lnull @ A @ Xs )
=> ( ( coinductive_ltl @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) )
= ( coinductive_ltl @ A @ Ys ) ) )
& ( ~ ( coinductive_lnull @ A @ Xs )
=> ( ( coinductive_ltl @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) )
= ( coinductive_lappend @ A @ ( coinductive_ltl @ A @ Xs ) @ Ys ) ) ) ) ).
% ltl_lappend
thf(fact_124_lhd__lappend,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( ( coinductive_lnull @ A @ Xs )
=> ( ( coinductive_lhd @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) )
= ( coinductive_lhd @ A @ Ys ) ) )
& ( ~ ( coinductive_lnull @ A @ Xs )
=> ( ( coinductive_lhd @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) )
= ( coinductive_lhd @ A @ Xs ) ) ) ) ).
% lhd_lappend
thf(fact_125_lfilter__lappend__lfinite,axiom,
! [A: $tType,Xs: coinductive_llist @ A,P: A > $o,Ys: coinductive_llist @ A] :
( ( coinductive_lfinite @ A @ Xs )
=> ( ( coinductive_lfilter @ A @ P @ ( coinductive_lappend @ A @ Xs @ Ys ) )
= ( coinductive_lappend @ A @ ( coinductive_lfilter @ A @ P @ Xs ) @ ( coinductive_lfilter @ A @ P @ Ys ) ) ) ) ).
% lfilter_lappend_lfinite
thf(fact_126_lappend__ltakeWhile__ldropWhile,axiom,
! [A: $tType,P: A > $o,Xs: coinductive_llist @ A] :
( ( coinductive_lappend @ A @ ( coindu501562517eWhile @ A @ P @ Xs ) @ ( coindu218763757pWhile @ A @ P @ Xs ) )
= Xs ) ).
% lappend_ltakeWhile_ldropWhile
thf(fact_127_lset__lappend__lfinite,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( coinductive_lfinite @ A @ Xs )
=> ( ( coinductive_lset @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) )
= ( sup_sup @ ( set @ A ) @ ( coinductive_lset @ A @ Xs ) @ ( coinductive_lset @ A @ Ys ) ) ) ) ).
% lset_lappend_lfinite
thf(fact_128_lappend__assoc,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A,Zs: coinductive_llist @ A] :
( ( coinductive_lappend @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) @ Zs )
= ( coinductive_lappend @ A @ Xs @ ( coinductive_lappend @ A @ Ys @ Zs ) ) ) ).
% lappend_assoc
thf(fact_129_lappend__lnull2,axiom,
! [A: $tType,Ys: coinductive_llist @ A,Xs: coinductive_llist @ A] :
( ( coinductive_lnull @ A @ Ys )
=> ( ( coinductive_lappend @ A @ Xs @ Ys )
= Xs ) ) ).
% lappend_lnull2
thf(fact_130_lappend__lnull1,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( coinductive_lnull @ A @ Xs )
=> ( ( coinductive_lappend @ A @ Xs @ Ys )
= Ys ) ) ).
% lappend_lnull1
thf(fact_131_lappend_Odisc_I1_J,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( coinductive_lnull @ A @ Xs )
=> ( ( coinductive_lnull @ A @ Ys )
=> ( coinductive_lnull @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) ) ) ) ).
% lappend.disc(1)
thf(fact_132_lappend_Odisc_I2_J,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( ~ ( coinductive_lnull @ A @ Xs )
| ~ ( coinductive_lnull @ A @ Ys ) )
=> ~ ( coinductive_lnull @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) ) ) ).
% lappend.disc(2)
thf(fact_133_lappend__inf,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ~ ( coinductive_lfinite @ A @ Xs )
=> ( ( coinductive_lappend @ A @ Xs @ Ys )
= Xs ) ) ).
% lappend_inf
thf(fact_134_unfold__llist_Odisc_I1_J,axiom,
! [B: $tType,A: $tType,P2: A > $o,A2: A,G21: A > B,G22: A > A] :
( ( P2 @ A2 )
=> ( coinductive_lnull @ B @ ( coindu1441602521_llist @ A @ B @ P2 @ G21 @ G22 @ A2 ) ) ) ).
% unfold_llist.disc(1)
thf(fact_135_unfold__llist_Odisc_I2_J,axiom,
! [B: $tType,A: $tType,P2: A > $o,A2: A,G21: A > B,G22: A > A] :
( ~ ( P2 @ A2 )
=> ~ ( coinductive_lnull @ B @ ( coindu1441602521_llist @ A @ B @ P2 @ G21 @ G22 @ A2 ) ) ) ).
% unfold_llist.disc(2)
thf(fact_136_unfold__llist_Osimps_I4_J,axiom,
! [B: $tType,A: $tType,P2: A > $o,A2: A,G21: A > B,G22: A > A] :
( ~ ( P2 @ A2 )
=> ( ( coinductive_ltl @ B @ ( coindu1441602521_llist @ A @ B @ P2 @ G21 @ G22 @ A2 ) )
= ( coindu1441602521_llist @ A @ B @ P2 @ G21 @ G22 @ ( G22 @ A2 ) ) ) ) ).
% unfold_llist.simps(4)
thf(fact_137_unfold__llist_Osimps_I3_J,axiom,
! [B: $tType,A: $tType,P2: A > $o,A2: A,G21: A > B,G22: A > A] :
( ~ ( P2 @ A2 )
=> ( ( coinductive_lhd @ B @ ( coindu1441602521_llist @ A @ B @ P2 @ G21 @ G22 @ A2 ) )
= ( G21 @ A2 ) ) ) ).
% unfold_llist.simps(3)
thf(fact_138_lappend__ltl,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ~ ( coinductive_lnull @ A @ Xs )
=> ( ( coinductive_lappend @ A @ ( coinductive_ltl @ A @ Xs ) @ Ys )
= ( coinductive_ltl @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) ) ) ) ).
% lappend_ltl
thf(fact_139_in__lset__lappend__iff,axiom,
! [A: $tType,X: A,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( member @ A @ X @ ( coinductive_lset @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) ) )
= ( ( member @ A @ X @ ( coinductive_lset @ A @ Xs ) )
| ( ( coinductive_lfinite @ A @ Xs )
& ( member @ A @ X @ ( coinductive_lset @ A @ Ys ) ) ) ) ) ).
% in_lset_lappend_iff
thf(fact_140_ltakeWhile__lappend1,axiom,
! [A: $tType,X: A,Xs: coinductive_llist @ A,P: A > $o,Ys: coinductive_llist @ A] :
( ( member @ A @ X @ ( coinductive_lset @ A @ Xs ) )
=> ( ~ ( P @ X )
=> ( ( coindu501562517eWhile @ A @ P @ ( coinductive_lappend @ A @ Xs @ Ys ) )
= ( coindu501562517eWhile @ A @ P @ Xs ) ) ) ) ).
% ltakeWhile_lappend1
thf(fact_141_lfilter__eq__lappend__lfiniteD,axiom,
! [A: $tType,P: A > $o,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A,Zs: coinductive_llist @ A] :
( ( ( coinductive_lfilter @ A @ P @ Xs )
= ( coinductive_lappend @ A @ Ys @ Zs ) )
=> ( ( coinductive_lfinite @ A @ Ys )
=> ? [Us: coinductive_llist @ A,Vs: coinductive_llist @ A] :
( ( Xs
= ( coinductive_lappend @ A @ Us @ Vs ) )
& ( coinductive_lfinite @ A @ Us )
& ( Ys
= ( coinductive_lfilter @ A @ P @ Us ) )
& ( Zs
= ( coinductive_lfilter @ A @ P @ Vs ) ) ) ) ) ).
% lfilter_eq_lappend_lfiniteD
thf(fact_142_lset__lappend__conv,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( ( coinductive_lfinite @ A @ Xs )
=> ( ( coinductive_lset @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) )
= ( sup_sup @ ( set @ A ) @ ( coinductive_lset @ A @ Xs ) @ ( coinductive_lset @ A @ Ys ) ) ) )
& ( ~ ( coinductive_lfinite @ A @ Xs )
=> ( ( coinductive_lset @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) )
= ( coinductive_lset @ A @ Xs ) ) ) ) ).
% lset_lappend_conv
thf(fact_143_bot__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( bot @ B @ ( type2 @ B ) )
=> ( ( bot_bot @ ( A > B ) )
= ( ^ [X2: A] : ( bot_bot @ B ) ) ) ) ).
% bot_fun_def
thf(fact_144_bot__empty__eq,axiom,
! [A: $tType] :
( ( bot_bot @ ( A > $o ) )
= ( ^ [X2: A] : ( member @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% bot_empty_eq
thf(fact_145_Collect__empty__eq__bot,axiom,
! [A: $tType,P: A > $o] :
( ( ( collect @ A @ P )
= ( bot_bot @ ( set @ A ) ) )
= ( P
= ( bot_bot @ ( A > $o ) ) ) ) ).
% Collect_empty_eq_bot
thf(fact_146_lstrict__prefix__lappend__conv,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( coindu1478340336prefix @ A @ Xs @ ( coinductive_lappend @ A @ Xs @ Ys ) )
= ( ( coinductive_lfinite @ A @ Xs )
& ~ ( coinductive_lnull @ A @ Ys ) ) ) ).
% lstrict_prefix_lappend_conv
thf(fact_147_ldistinct__coinduct,axiom,
! [A: $tType,X5: ( coinductive_llist @ A ) > $o,Xs: coinductive_llist @ A] :
( ( X5 @ Xs )
=> ( ! [Xs2: coinductive_llist @ A] :
( ( X5 @ Xs2 )
=> ( ~ ( coinductive_lnull @ A @ Xs2 )
=> ( ~ ( member @ A @ ( coinductive_lhd @ A @ Xs2 ) @ ( coinductive_lset @ A @ ( coinductive_ltl @ A @ Xs2 ) ) )
& ( ( X5 @ ( coinductive_ltl @ A @ Xs2 ) )
| ( coindu351974385stinct @ A @ ( coinductive_ltl @ A @ Xs2 ) ) ) ) ) )
=> ( coindu351974385stinct @ A @ Xs ) ) ) ).
% ldistinct_coinduct
thf(fact_148_llist__less__induct,axiom,
! [A: $tType,P: ( coinductive_llist @ A ) > $o,Xs: coinductive_llist @ A] :
( ! [Xs2: coinductive_llist @ A] :
( ! [Ys2: coinductive_llist @ A] :
( ( coindu1478340336prefix @ A @ Ys2 @ Xs2 )
=> ( P @ Ys2 ) )
=> ( P @ Xs2 ) )
=> ( P @ Xs ) ) ).
% llist_less_induct
thf(fact_149_ldistinct__ltlI,axiom,
! [A: $tType,Xs: coinductive_llist @ A] :
( ( coindu351974385stinct @ A @ Xs )
=> ( coindu351974385stinct @ A @ ( coinductive_ltl @ A @ Xs ) ) ) ).
% ldistinct_ltlI
thf(fact_150_ldistinct__lfilterI,axiom,
! [A: $tType,Xs: coinductive_llist @ A,P: A > $o] :
( ( coindu351974385stinct @ A @ Xs )
=> ( coindu351974385stinct @ A @ ( coinductive_lfilter @ A @ P @ Xs ) ) ) ).
% ldistinct_lfilterI
thf(fact_151_lstrict__prefix__lfinite1,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( coindu1478340336prefix @ A @ Xs @ Ys )
=> ( coinductive_lfinite @ A @ Xs ) ) ).
% lstrict_prefix_lfinite1
thf(fact_152_ldistinct__lhdD,axiom,
! [A: $tType,Xs: coinductive_llist @ A] :
( ( coindu351974385stinct @ A @ Xs )
=> ( ~ ( coinductive_lnull @ A @ Xs )
=> ~ ( member @ A @ ( coinductive_lhd @ A @ Xs ) @ ( coinductive_lset @ A @ ( coinductive_ltl @ A @ Xs ) ) ) ) ) ).
% ldistinct_lhdD
thf(fact_153_ldistinct__lappend,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( coindu351974385stinct @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) )
= ( ( coindu351974385stinct @ A @ Xs )
& ( ( coinductive_lfinite @ A @ Xs )
=> ( ( coindu351974385stinct @ A @ Ys )
& ( ( inf_inf @ ( set @ A ) @ ( coinductive_lset @ A @ Xs ) @ ( coinductive_lset @ A @ Ys ) )
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% ldistinct_lappend
thf(fact_154_ldistinct__lconcat,axiom,
! [A: $tType,Xss: coinductive_llist @ ( coinductive_llist @ A )] :
( ( coindu351974385stinct @ ( coinductive_llist @ A ) @ Xss )
=> ( ! [Ys3: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ Ys3 @ ( coinductive_lset @ ( coinductive_llist @ A ) @ Xss ) )
=> ( coindu351974385stinct @ A @ Ys3 ) )
=> ( ! [Ys3: coinductive_llist @ A,Zs2: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ Ys3 @ ( coinductive_lset @ ( coinductive_llist @ A ) @ Xss ) )
=> ( ( member @ ( coinductive_llist @ A ) @ Zs2 @ ( coinductive_lset @ ( coinductive_llist @ A ) @ Xss ) )
=> ( ( Ys3 != Zs2 )
=> ( ( inf_inf @ ( set @ A ) @ ( coinductive_lset @ A @ Ys3 ) @ ( coinductive_lset @ A @ Zs2 ) )
= ( bot_bot @ ( set @ A ) ) ) ) ) )
=> ( coindu351974385stinct @ A @ ( coinductive_lconcat @ A @ Xss ) ) ) ) ) ).
% ldistinct_lconcat
thf(fact_155_llist_Ocase__eq__if,axiom,
! [A: $tType,B: $tType] :
( ( coindu1381640503_llist @ B @ A )
= ( ^ [F1: B,F22: A > ( coinductive_llist @ A ) > B,Llist5: coinductive_llist @ A] : ( if @ B @ ( coinductive_lnull @ A @ Llist5 ) @ F1 @ ( F22 @ ( coinductive_lhd @ A @ Llist5 ) @ ( coinductive_ltl @ A @ Llist5 ) ) ) ) ) ).
% llist.case_eq_if
thf(fact_156_inf__apply,axiom,
! [B: $tType,A: $tType] :
( ( semilattice_inf @ B @ ( type2 @ B ) )
=> ( ( inf_inf @ ( A > B ) )
= ( ^ [F2: A > B,G3: A > B,X2: A] : ( inf_inf @ B @ ( F2 @ X2 ) @ ( G3 @ X2 ) ) ) ) ) ).
% inf_apply
thf(fact_157_inf_Oidem,axiom,
! [A: $tType] :
( ( semilattice_inf @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( inf_inf @ A @ A2 @ A2 )
= A2 ) ) ).
% inf.idem
thf(fact_158_inf__idem,axiom,
! [A: $tType] :
( ( semilattice_inf @ A @ ( type2 @ A ) )
=> ! [X: A] :
( ( inf_inf @ A @ X @ X )
= X ) ) ).
% inf_idem
thf(fact_159_inf_Oleft__idem,axiom,
! [A: $tType] :
( ( semilattice_inf @ A @ ( type2 @ A ) )
=> ! [A2: A,B3: A] :
( ( inf_inf @ A @ A2 @ ( inf_inf @ A @ A2 @ B3 ) )
= ( inf_inf @ A @ A2 @ B3 ) ) ) ).
% inf.left_idem
thf(fact_160_inf__left__idem,axiom,
! [A: $tType] :
( ( semilattice_inf @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( inf_inf @ A @ X @ ( inf_inf @ A @ X @ Y ) )
= ( inf_inf @ A @ X @ Y ) ) ) ).
% inf_left_idem
thf(fact_161_inf_Oright__idem,axiom,
! [A: $tType] :
( ( semilattice_inf @ A @ ( type2 @ A ) )
=> ! [A2: A,B3: A] :
( ( inf_inf @ A @ ( inf_inf @ A @ A2 @ B3 ) @ B3 )
= ( inf_inf @ A @ A2 @ B3 ) ) ) ).
% inf.right_idem
thf(fact_162_inf__right__idem,axiom,
! [A: $tType] :
( ( semilattice_inf @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( inf_inf @ A @ ( inf_inf @ A @ X @ Y ) @ Y )
= ( inf_inf @ A @ X @ Y ) ) ) ).
% inf_right_idem
thf(fact_163_IntI,axiom,
! [A: $tType,C: A,A3: set @ A,B2: set @ A] :
( ( member @ A @ C @ A3 )
=> ( ( member @ A @ C @ B2 )
=> ( member @ A @ C @ ( inf_inf @ ( set @ A ) @ A3 @ B2 ) ) ) ) ).
% IntI
thf(fact_164_Int__iff,axiom,
! [A: $tType,C: A,A3: set @ A,B2: set @ A] :
( ( member @ A @ C @ ( inf_inf @ ( set @ A ) @ A3 @ B2 ) )
= ( ( member @ A @ C @ A3 )
& ( member @ A @ C @ B2 ) ) ) ).
% Int_iff
thf(fact_165_inf__bot__left,axiom,
! [A: $tType] :
( ( bounded_lattice_bot @ A @ ( type2 @ A ) )
=> ! [X: A] :
( ( inf_inf @ A @ ( bot_bot @ A ) @ X )
= ( bot_bot @ A ) ) ) ).
% inf_bot_left
thf(fact_166_inf__bot__right,axiom,
! [A: $tType] :
( ( bounded_lattice_bot @ A @ ( type2 @ A ) )
=> ! [X: A] :
( ( inf_inf @ A @ X @ ( bot_bot @ A ) )
= ( bot_bot @ A ) ) ) ).
% inf_bot_right
thf(fact_167_inf__sup__absorb,axiom,
! [A: $tType] :
( ( lattice @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( inf_inf @ A @ X @ ( sup_sup @ A @ X @ Y ) )
= X ) ) ).
% inf_sup_absorb
thf(fact_168_sup__inf__absorb,axiom,
! [A: $tType] :
( ( lattice @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( sup_sup @ A @ X @ ( inf_inf @ A @ X @ Y ) )
= X ) ) ).
% sup_inf_absorb
thf(fact_169_inf__sup__aci_I4_J,axiom,
! [A: $tType] :
( ( lattice @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( inf_inf @ A @ X @ ( inf_inf @ A @ X @ Y ) )
= ( inf_inf @ A @ X @ Y ) ) ) ).
% inf_sup_aci(4)
thf(fact_170_inf__sup__aci_I3_J,axiom,
! [A: $tType] :
( ( lattice @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A,Z: A] :
( ( inf_inf @ A @ X @ ( inf_inf @ A @ Y @ Z ) )
= ( inf_inf @ A @ Y @ ( inf_inf @ A @ X @ Z ) ) ) ) ).
% inf_sup_aci(3)
thf(fact_171_inf__sup__aci_I2_J,axiom,
! [A: $tType] :
( ( lattice @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A,Z: A] :
( ( inf_inf @ A @ ( inf_inf @ A @ X @ Y ) @ Z )
= ( inf_inf @ A @ X @ ( inf_inf @ A @ Y @ Z ) ) ) ) ).
% inf_sup_aci(2)
thf(fact_172_inf__sup__aci_I1_J,axiom,
! [A: $tType] :
( ( lattice @ A @ ( type2 @ A ) )
=> ( ( inf_inf @ A )
= ( ^ [X2: A,Y3: A] : ( inf_inf @ A @ Y3 @ X2 ) ) ) ) ).
% inf_sup_aci(1)
thf(fact_173_IntE,axiom,
! [A: $tType,C: A,A3: set @ A,B2: set @ A] :
( ( member @ A @ C @ ( inf_inf @ ( set @ A ) @ A3 @ B2 ) )
=> ~ ( ( member @ A @ C @ A3 )
=> ~ ( member @ A @ C @ B2 ) ) ) ).
% IntE
thf(fact_174_IntD1,axiom,
! [A: $tType,C: A,A3: set @ A,B2: set @ A] :
( ( member @ A @ C @ ( inf_inf @ ( set @ A ) @ A3 @ B2 ) )
=> ( member @ A @ C @ A3 ) ) ).
% IntD1
thf(fact_175_IntD2,axiom,
! [A: $tType,C: A,A3: set @ A,B2: set @ A] :
( ( member @ A @ C @ ( inf_inf @ ( set @ A ) @ A3 @ B2 ) )
=> ( member @ A @ C @ B2 ) ) ).
% IntD2
thf(fact_176_Int__assoc,axiom,
! [A: $tType,A3: set @ A,B2: set @ A,C2: set @ A] :
( ( inf_inf @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A3 @ B2 ) @ C2 )
= ( inf_inf @ ( set @ A ) @ A3 @ ( inf_inf @ ( set @ A ) @ B2 @ C2 ) ) ) ).
% Int_assoc
thf(fact_177_Int__absorb,axiom,
! [A: $tType,A3: set @ A] :
( ( inf_inf @ ( set @ A ) @ A3 @ A3 )
= A3 ) ).
% Int_absorb
thf(fact_178_Int__commute,axiom,
! [A: $tType] :
( ( inf_inf @ ( set @ A ) )
= ( ^ [A5: set @ A,B5: set @ A] : ( inf_inf @ ( set @ A ) @ B5 @ A5 ) ) ) ).
% Int_commute
thf(fact_179_Int__left__absorb,axiom,
! [A: $tType,A3: set @ A,B2: set @ A] :
( ( inf_inf @ ( set @ A ) @ A3 @ ( inf_inf @ ( set @ A ) @ A3 @ B2 ) )
= ( inf_inf @ ( set @ A ) @ A3 @ B2 ) ) ).
% Int_left_absorb
thf(fact_180_inf__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( semilattice_inf @ B @ ( type2 @ B ) )
=> ( ( inf_inf @ ( A > B ) )
= ( ^ [F2: A > B,G3: A > B,X2: A] : ( inf_inf @ B @ ( F2 @ X2 ) @ ( G3 @ X2 ) ) ) ) ) ).
% inf_fun_def
thf(fact_181_Int__left__commute,axiom,
! [A: $tType,A3: set @ A,B2: set @ A,C2: set @ A] :
( ( inf_inf @ ( set @ A ) @ A3 @ ( inf_inf @ ( set @ A ) @ B2 @ C2 ) )
= ( inf_inf @ ( set @ A ) @ B2 @ ( inf_inf @ ( set @ A ) @ A3 @ C2 ) ) ) ).
% Int_left_commute
thf(fact_182_inf_Oassoc,axiom,
! [A: $tType] :
( ( semilattice_inf @ A @ ( type2 @ A ) )
=> ! [A2: A,B3: A,C: A] :
( ( inf_inf @ A @ ( inf_inf @ A @ A2 @ B3 ) @ C )
= ( inf_inf @ A @ A2 @ ( inf_inf @ A @ B3 @ C ) ) ) ) ).
% inf.assoc
thf(fact_183_inf__assoc,axiom,
! [A: $tType] :
( ( semilattice_inf @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A,Z: A] :
( ( inf_inf @ A @ ( inf_inf @ A @ X @ Y ) @ Z )
= ( inf_inf @ A @ X @ ( inf_inf @ A @ Y @ Z ) ) ) ) ).
% inf_assoc
thf(fact_184_inf_Ocommute,axiom,
! [A: $tType] :
( ( semilattice_inf @ A @ ( type2 @ A ) )
=> ( ( inf_inf @ A )
= ( ^ [A4: A,B4: A] : ( inf_inf @ A @ B4 @ A4 ) ) ) ) ).
% inf.commute
thf(fact_185_inf__commute,axiom,
! [A: $tType] :
( ( semilattice_inf @ A @ ( type2 @ A ) )
=> ( ( inf_inf @ A )
= ( ^ [X2: A,Y3: A] : ( inf_inf @ A @ Y3 @ X2 ) ) ) ) ).
% inf_commute
thf(fact_186_inf_Oleft__commute,axiom,
! [A: $tType] :
( ( semilattice_inf @ A @ ( type2 @ A ) )
=> ! [B3: A,A2: A,C: A] :
( ( inf_inf @ A @ B3 @ ( inf_inf @ A @ A2 @ C ) )
= ( inf_inf @ A @ A2 @ ( inf_inf @ A @ B3 @ C ) ) ) ) ).
% inf.left_commute
thf(fact_187_inf__left__commute,axiom,
! [A: $tType] :
( ( semilattice_inf @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A,Z: A] :
( ( inf_inf @ A @ X @ ( inf_inf @ A @ Y @ Z ) )
= ( inf_inf @ A @ Y @ ( inf_inf @ A @ X @ Z ) ) ) ) ).
% inf_left_commute
thf(fact_188_Un__Int__distrib2,axiom,
! [A: $tType,B2: set @ A,C2: set @ A,A3: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ B2 @ C2 ) @ A3 )
= ( inf_inf @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ B2 @ A3 ) @ ( sup_sup @ ( set @ A ) @ C2 @ A3 ) ) ) ).
% Un_Int_distrib2
thf(fact_189_Int__Un__distrib2,axiom,
! [A: $tType,B2: set @ A,C2: set @ A,A3: set @ A] :
( ( inf_inf @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ B2 @ C2 ) @ A3 )
= ( sup_sup @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ B2 @ A3 ) @ ( inf_inf @ ( set @ A ) @ C2 @ A3 ) ) ) ).
% Int_Un_distrib2
thf(fact_190_Un__Int__distrib,axiom,
! [A: $tType,A3: set @ A,B2: set @ A,C2: set @ A] :
( ( sup_sup @ ( set @ A ) @ A3 @ ( inf_inf @ ( set @ A ) @ B2 @ C2 ) )
= ( inf_inf @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A3 @ B2 ) @ ( sup_sup @ ( set @ A ) @ A3 @ C2 ) ) ) ).
% Un_Int_distrib
thf(fact_191_Int__Un__distrib,axiom,
! [A: $tType,A3: set @ A,B2: set @ A,C2: set @ A] :
( ( inf_inf @ ( set @ A ) @ A3 @ ( sup_sup @ ( set @ A ) @ B2 @ C2 ) )
= ( sup_sup @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A3 @ B2 ) @ ( inf_inf @ ( set @ A ) @ A3 @ C2 ) ) ) ).
% Int_Un_distrib
thf(fact_192_Un__Int__crazy,axiom,
! [A: $tType,A3: set @ A,B2: set @ A,C2: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A3 @ B2 ) @ ( inf_inf @ ( set @ A ) @ B2 @ C2 ) ) @ ( inf_inf @ ( set @ A ) @ C2 @ A3 ) )
= ( inf_inf @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A3 @ B2 ) @ ( sup_sup @ ( set @ A ) @ B2 @ C2 ) ) @ ( sup_sup @ ( set @ A ) @ C2 @ A3 ) ) ) ).
% Un_Int_crazy
thf(fact_193_disjoint__iff__not__equal,axiom,
! [A: $tType,A3: set @ A,B2: set @ A] :
( ( ( inf_inf @ ( set @ A ) @ A3 @ B2 )
= ( bot_bot @ ( set @ A ) ) )
= ( ! [X2: A] :
( ( member @ A @ X2 @ A3 )
=> ! [Y3: A] :
( ( member @ A @ Y3 @ B2 )
=> ( X2 != Y3 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_194_Int__empty__right,axiom,
! [A: $tType,A3: set @ A] :
( ( inf_inf @ ( set @ A ) @ A3 @ ( bot_bot @ ( set @ A ) ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% Int_empty_right
thf(fact_195_Int__empty__left,axiom,
! [A: $tType,B2: set @ A] :
( ( inf_inf @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ B2 )
= ( bot_bot @ ( set @ A ) ) ) ).
% Int_empty_left
thf(fact_196_Int__emptyI,axiom,
! [A: $tType,A3: set @ A,B2: set @ A] :
( ! [X3: A] :
( ( member @ A @ X3 @ A3 )
=> ~ ( member @ A @ X3 @ B2 ) )
=> ( ( inf_inf @ ( set @ A ) @ A3 @ B2 )
= ( bot_bot @ ( set @ A ) ) ) ) ).
% Int_emptyI
thf(fact_197_sup__inf__distrib2,axiom,
! [A: $tType] :
( ( distrib_lattice @ A @ ( type2 @ A ) )
=> ! [Y: A,Z: A,X: A] :
( ( sup_sup @ A @ ( inf_inf @ A @ Y @ Z ) @ X )
= ( inf_inf @ A @ ( sup_sup @ A @ Y @ X ) @ ( sup_sup @ A @ Z @ X ) ) ) ) ).
% sup_inf_distrib2
thf(fact_198_sup__inf__distrib1,axiom,
! [A: $tType] :
( ( distrib_lattice @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A,Z: A] :
( ( sup_sup @ A @ X @ ( inf_inf @ A @ Y @ Z ) )
= ( inf_inf @ A @ ( sup_sup @ A @ X @ Y ) @ ( sup_sup @ A @ X @ Z ) ) ) ) ).
% sup_inf_distrib1
thf(fact_199_inf__sup__distrib2,axiom,
! [A: $tType] :
( ( distrib_lattice @ A @ ( type2 @ A ) )
=> ! [Y: A,Z: A,X: A] :
( ( inf_inf @ A @ ( sup_sup @ A @ Y @ Z ) @ X )
= ( sup_sup @ A @ ( inf_inf @ A @ Y @ X ) @ ( inf_inf @ A @ Z @ X ) ) ) ) ).
% inf_sup_distrib2
thf(fact_200_inf__sup__distrib1,axiom,
! [A: $tType] :
( ( distrib_lattice @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A,Z: A] :
( ( inf_inf @ A @ X @ ( sup_sup @ A @ Y @ Z ) )
= ( sup_sup @ A @ ( inf_inf @ A @ X @ Y ) @ ( inf_inf @ A @ X @ Z ) ) ) ) ).
% inf_sup_distrib1
thf(fact_201_distrib__imp2,axiom,
! [A: $tType] :
( ( lattice @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A,Z: A] :
( ! [X3: A,Y2: A,Z2: A] :
( ( sup_sup @ A @ X3 @ ( inf_inf @ A @ Y2 @ Z2 ) )
= ( inf_inf @ A @ ( sup_sup @ A @ X3 @ Y2 ) @ ( sup_sup @ A @ X3 @ Z2 ) ) )
=> ( ( inf_inf @ A @ X @ ( sup_sup @ A @ Y @ Z ) )
= ( sup_sup @ A @ ( inf_inf @ A @ X @ Y ) @ ( inf_inf @ A @ X @ Z ) ) ) ) ) ).
% distrib_imp2
thf(fact_202_distrib__imp1,axiom,
! [A: $tType] :
( ( lattice @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A,Z: A] :
( ! [X3: A,Y2: A,Z2: A] :
( ( inf_inf @ A @ X3 @ ( sup_sup @ A @ Y2 @ Z2 ) )
= ( sup_sup @ A @ ( inf_inf @ A @ X3 @ Y2 ) @ ( inf_inf @ A @ X3 @ Z2 ) ) )
=> ( ( sup_sup @ A @ X @ ( inf_inf @ A @ Y @ Z ) )
= ( inf_inf @ A @ ( sup_sup @ A @ X @ Y ) @ ( sup_sup @ A @ X @ Z ) ) ) ) ) ).
% distrib_imp1
thf(fact_203_disjnt__def,axiom,
! [A: $tType] :
( ( disjnt @ A )
= ( ^ [A5: set @ A,B5: set @ A] :
( ( inf_inf @ ( set @ A ) @ A5 @ B5 )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% disjnt_def
thf(fact_204_lfilter__eq__LConsD,axiom,
! [A: $tType,P: A > $o,Ys: coinductive_llist @ A,X: A,Xs: coinductive_llist @ A] :
( ( ( coinductive_lfilter @ A @ P @ Ys )
= ( coinductive_LCons @ A @ X @ Xs ) )
=> ? [Us: coinductive_llist @ A,Vs: coinductive_llist @ A] :
( ( Ys
= ( coinductive_lappend @ A @ Us @ ( coinductive_LCons @ A @ X @ Vs ) ) )
& ( coinductive_lfinite @ A @ Us )
& ! [X4: A] :
( ( member @ A @ X4 @ ( coinductive_lset @ A @ Us ) )
=> ~ ( P @ X4 ) )
& ( P @ X )
& ( Xs
= ( coinductive_lfilter @ A @ P @ Vs ) ) ) ) ).
% lfilter_eq_LConsD
thf(fact_205_ltakeWhile_Octr_I2_J,axiom,
! [A: $tType,Xs: coinductive_llist @ A,P: A > $o] :
( ~ ( coinductive_lnull @ A @ Xs )
=> ( ( P @ ( coinductive_lhd @ A @ Xs ) )
=> ( ( coindu501562517eWhile @ A @ P @ Xs )
= ( coinductive_LCons @ A @ ( coinductive_lhd @ A @ Xs ) @ ( coindu501562517eWhile @ A @ P @ ( coinductive_ltl @ A @ Xs ) ) ) ) ) ) ).
% ltakeWhile.ctr(2)
thf(fact_206_llist_Oinject,axiom,
! [A: $tType,X21: A,X22: coinductive_llist @ A,Y21: A,Y22: coinductive_llist @ A] :
( ( ( coinductive_LCons @ A @ X21 @ X22 )
= ( coinductive_LCons @ A @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% llist.inject
thf(fact_207_lfinite__LCons,axiom,
! [A: $tType,X: A,Xs: coinductive_llist @ A] :
( ( coinductive_lfinite @ A @ ( coinductive_LCons @ A @ X @ Xs ) )
= ( coinductive_lfinite @ A @ Xs ) ) ).
% lfinite_LCons
thf(fact_208_lfinite__code_I2_J,axiom,
! [B: $tType,X: B,Xs: coinductive_llist @ B] :
( ( coinductive_lfinite @ B @ ( coinductive_LCons @ B @ X @ Xs ) )
= ( coinductive_lfinite @ B @ Xs ) ) ).
% lfinite_code(2)
thf(fact_209_lappend__code_I2_J,axiom,
! [A: $tType,Xa: A,X: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( coinductive_lappend @ A @ ( coinductive_LCons @ A @ Xa @ X ) @ Ys )
= ( coinductive_LCons @ A @ Xa @ ( coinductive_lappend @ A @ X @ Ys ) ) ) ).
% lappend_code(2)
thf(fact_210_lfilter__LCons,axiom,
! [A: $tType,P: A > $o,X: A,Xs: coinductive_llist @ A] :
( ( ( P @ X )
=> ( ( coinductive_lfilter @ A @ P @ ( coinductive_LCons @ A @ X @ Xs ) )
= ( coinductive_LCons @ A @ X @ ( coinductive_lfilter @ A @ P @ Xs ) ) ) )
& ( ~ ( P @ X )
=> ( ( coinductive_lfilter @ A @ P @ ( coinductive_LCons @ A @ X @ Xs ) )
= ( coinductive_lfilter @ A @ P @ Xs ) ) ) ) ).
% lfilter_LCons
thf(fact_211_ldropWhile__LCons,axiom,
! [A: $tType,P: A > $o,X: A,Xs: coinductive_llist @ A] :
( ( ( P @ X )
=> ( ( coindu218763757pWhile @ A @ P @ ( coinductive_LCons @ A @ X @ Xs ) )
= ( coindu218763757pWhile @ A @ P @ Xs ) ) )
& ( ~ ( P @ X )
=> ( ( coindu218763757pWhile @ A @ P @ ( coinductive_LCons @ A @ X @ Xs ) )
= ( coinductive_LCons @ A @ X @ Xs ) ) ) ) ).
% ldropWhile_LCons
thf(fact_212_lconcat__LCons,axiom,
! [B: $tType,Xs: coinductive_llist @ B,Xss: coinductive_llist @ ( coinductive_llist @ B )] :
( ( coinductive_lconcat @ B @ ( coinductive_LCons @ ( coinductive_llist @ B ) @ Xs @ Xss ) )
= ( coinductive_lappend @ B @ Xs @ ( coinductive_lconcat @ B @ Xss ) ) ) ).
% lconcat_LCons
thf(fact_213_unfold__llist__eq__LCons,axiom,
! [A: $tType,B: $tType,IS_LNIL: B > $o,LHD: B > A,LTL: B > B,B3: B,X: A,Xs: coinductive_llist @ A] :
( ( ( coindu1441602521_llist @ B @ A @ IS_LNIL @ LHD @ LTL @ B3 )
= ( coinductive_LCons @ A @ X @ Xs ) )
= ( ~ ( IS_LNIL @ B3 )
& ( X
= ( LHD @ B3 ) )
& ( Xs
= ( coindu1441602521_llist @ B @ A @ IS_LNIL @ LHD @ LTL @ ( LTL @ B3 ) ) ) ) ) ).
% unfold_llist_eq_LCons
thf(fact_214_lstrict__prefix__code_I4_J,axiom,
! [B: $tType,X: B,Xs: coinductive_llist @ B,Y: B,Ys: coinductive_llist @ B] :
( ( coindu1478340336prefix @ B @ ( coinductive_LCons @ B @ X @ Xs ) @ ( coinductive_LCons @ B @ Y @ Ys ) )
= ( ( X = Y )
& ( coindu1478340336prefix @ B @ Xs @ Ys ) ) ) ).
% lstrict_prefix_code(4)
thf(fact_215_ldistinct__LCons,axiom,
! [A: $tType,X: A,Xs: coinductive_llist @ A] :
( ( coindu351974385stinct @ A @ ( coinductive_LCons @ A @ X @ Xs ) )
= ( ~ ( member @ A @ X @ ( coinductive_lset @ A @ Xs ) )
& ( coindu351974385stinct @ A @ Xs ) ) ) ).
% ldistinct_LCons
thf(fact_216_lhd__LCons__ltl,axiom,
! [A: $tType,Llist: coinductive_llist @ A] :
( ~ ( coinductive_lnull @ A @ Llist )
=> ( ( coinductive_LCons @ A @ ( coinductive_lhd @ A @ Llist ) @ ( coinductive_ltl @ A @ Llist ) )
= Llist ) ) ).
% lhd_LCons_ltl
thf(fact_217_llist_Osimps_I5_J,axiom,
! [B: $tType,A: $tType,F12: B,F23: A > ( coinductive_llist @ A ) > B,X21: A,X22: coinductive_llist @ A] :
( ( coindu1381640503_llist @ B @ A @ F12 @ F23 @ ( coinductive_LCons @ A @ X21 @ X22 ) )
= ( F23 @ X21 @ X22 ) ) ).
% llist.simps(5)
thf(fact_218_llist_Oset__induct,axiom,
! [A: $tType,X: A,A2: coinductive_llist @ A,P: A > ( coinductive_llist @ A ) > $o] :
( ( member @ A @ X @ ( coinductive_lset @ A @ A2 ) )
=> ( ! [Z1: A,Z22: coinductive_llist @ A] : ( P @ Z1 @ ( coinductive_LCons @ A @ Z1 @ Z22 ) )
=> ( ! [Z1: A,Z22: coinductive_llist @ A,Xa2: A] :
( ( member @ A @ Xa2 @ ( coinductive_lset @ A @ Z22 ) )
=> ( ( P @ Xa2 @ Z22 )
=> ( P @ Xa2 @ ( coinductive_LCons @ A @ Z1 @ Z22 ) ) ) )
=> ( P @ X @ A2 ) ) ) ) ).
% llist.set_induct
thf(fact_219_llist_Oset__cases,axiom,
! [A: $tType,E: A,A2: coinductive_llist @ A] :
( ( member @ A @ E @ ( coinductive_lset @ A @ A2 ) )
=> ( ! [Z22: coinductive_llist @ A] :
( A2
!= ( coinductive_LCons @ A @ E @ Z22 ) )
=> ~ ! [Z1: A,Z22: coinductive_llist @ A] :
( ( A2
= ( coinductive_LCons @ A @ Z1 @ Z22 ) )
=> ~ ( member @ A @ E @ ( coinductive_lset @ A @ Z22 ) ) ) ) ) ).
% llist.set_cases
thf(fact_220_lset__induct_H,axiom,
! [A: $tType,X: A,Xs: coinductive_llist @ A,P: ( coinductive_llist @ A ) > $o] :
( ( member @ A @ X @ ( coinductive_lset @ A @ Xs ) )
=> ( ! [Xs2: coinductive_llist @ A] : ( P @ ( coinductive_LCons @ A @ X @ Xs2 ) )
=> ( ! [X6: A,Xs2: coinductive_llist @ A] :
( ( member @ A @ X @ ( coinductive_lset @ A @ Xs2 ) )
=> ( ( P @ Xs2 )
=> ( P @ ( coinductive_LCons @ A @ X6 @ Xs2 ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% lset_induct'
thf(fact_221_lset__induct,axiom,
! [A: $tType,X: A,Xs: coinductive_llist @ A,P: ( coinductive_llist @ A ) > $o] :
( ( member @ A @ X @ ( coinductive_lset @ A @ Xs ) )
=> ( ! [Xs2: coinductive_llist @ A] : ( P @ ( coinductive_LCons @ A @ X @ Xs2 ) )
=> ( ! [X6: A,Xs2: coinductive_llist @ A] :
( ( member @ A @ X @ ( coinductive_lset @ A @ Xs2 ) )
=> ( ( X != X6 )
=> ( ( P @ Xs2 )
=> ( P @ ( coinductive_LCons @ A @ X6 @ Xs2 ) ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% lset_induct
thf(fact_222_lset__cases,axiom,
! [A: $tType,X: A,Xs: coinductive_llist @ A] :
( ( member @ A @ X @ ( coinductive_lset @ A @ Xs ) )
=> ( ! [Xs4: coinductive_llist @ A] :
( Xs
!= ( coinductive_LCons @ A @ X @ Xs4 ) )
=> ~ ! [X6: A,Xs4: coinductive_llist @ A] :
( ( Xs
= ( coinductive_LCons @ A @ X6 @ Xs4 ) )
=> ~ ( member @ A @ X @ ( coinductive_lset @ A @ Xs4 ) ) ) ) ) ).
% lset_cases
thf(fact_223_llist_Oset__intros_I1_J,axiom,
! [A: $tType,A1: A,A22: coinductive_llist @ A] : ( member @ A @ A1 @ ( coinductive_lset @ A @ ( coinductive_LCons @ A @ A1 @ A22 ) ) ) ).
% llist.set_intros(1)
thf(fact_224_llist_Oset__intros_I2_J,axiom,
! [A: $tType,X: A,A22: coinductive_llist @ A,A1: A] :
( ( member @ A @ X @ ( coinductive_lset @ A @ A22 ) )
=> ( member @ A @ X @ ( coinductive_lset @ A @ ( coinductive_LCons @ A @ A1 @ A22 ) ) ) ) ).
% llist.set_intros(2)
thf(fact_225_lset__intros_I1_J,axiom,
! [A: $tType,X: A,Xs: coinductive_llist @ A] : ( member @ A @ X @ ( coinductive_lset @ A @ ( coinductive_LCons @ A @ X @ Xs ) ) ) ).
% lset_intros(1)
thf(fact_226_lset__intros_I2_J,axiom,
! [A: $tType,X: A,Xs: coinductive_llist @ A,X7: A] :
( ( member @ A @ X @ ( coinductive_lset @ A @ Xs ) )
=> ( member @ A @ X @ ( coinductive_lset @ A @ ( coinductive_LCons @ A @ X7 @ Xs ) ) ) ) ).
% lset_intros(2)
thf(fact_227_lmember__code_I2_J,axiom,
! [A: $tType,X: A,Y: A,Ys: coinductive_llist @ A] :
( ( coinductive_lmember @ A @ X @ ( coinductive_LCons @ A @ Y @ Ys ) )
= ( ( X = Y )
| ( coinductive_lmember @ A @ X @ Ys ) ) ) ).
% lmember_code(2)
thf(fact_228_not__lnull__conv,axiom,
! [A: $tType,Xs: coinductive_llist @ A] :
( ( ~ ( coinductive_lnull @ A @ Xs ) )
= ( ? [X2: A,Xs5: coinductive_llist @ A] :
( Xs
= ( coinductive_LCons @ A @ X2 @ Xs5 ) ) ) ) ).
% not_lnull_conv
thf(fact_229_llist_OdiscI_I2_J,axiom,
! [A: $tType,Llist: coinductive_llist @ A,X21: A,X22: coinductive_llist @ A] :
( ( Llist
= ( coinductive_LCons @ A @ X21 @ X22 ) )
=> ~ ( coinductive_lnull @ A @ Llist ) ) ).
% llist.discI(2)
thf(fact_230_llist_Odisc_I2_J,axiom,
! [A: $tType,X21: A,X22: coinductive_llist @ A] :
~ ( coinductive_lnull @ A @ ( coinductive_LCons @ A @ X21 @ X22 ) ) ).
% llist.disc(2)
thf(fact_231_ltl__simps_I2_J,axiom,
! [A: $tType,X21: A,X22: coinductive_llist @ A] :
( ( coinductive_ltl @ A @ ( coinductive_LCons @ A @ X21 @ X22 ) )
= X22 ) ).
% ltl_simps(2)
thf(fact_232_lhd__LCons,axiom,
! [A: $tType,X21: A,X22: coinductive_llist @ A] :
( ( coinductive_lhd @ A @ ( coinductive_LCons @ A @ X21 @ X22 ) )
= X21 ) ).
% lhd_LCons
thf(fact_233_lfilter__LCons__seek,axiom,
! [A: $tType,P2: A > $o,X: A,L: coinductive_llist @ A] :
( ~ ( P2 @ X )
=> ( ( coinductive_lfilter @ A @ P2 @ ( coinductive_LCons @ A @ X @ L ) )
= ( coinductive_lfilter @ A @ P2 @ L ) ) ) ).
% lfilter_LCons_seek
thf(fact_234_lfilter__LCons__found,axiom,
! [A: $tType,P: A > $o,X: A,Xs: coinductive_llist @ A] :
( ( P @ X )
=> ( ( coinductive_lfilter @ A @ P @ ( coinductive_LCons @ A @ X @ Xs ) )
= ( coinductive_LCons @ A @ X @ ( coinductive_lfilter @ A @ P @ Xs ) ) ) ) ).
% lfilter_LCons_found
thf(fact_235_disjnt__iff,axiom,
! [A: $tType] :
( ( disjnt @ A )
= ( ^ [A5: set @ A,B5: set @ A] :
! [X2: A] :
~ ( ( member @ A @ X2 @ A5 )
& ( member @ A @ X2 @ B5 ) ) ) ) ).
% disjnt_iff
thf(fact_236_lsetp_Oinducts,axiom,
! [A: $tType,X1: coinductive_llist @ A,X23: A,P: ( coinductive_llist @ A ) > A > $o] :
( ( coinductive_lsetp @ A @ X1 @ X23 )
=> ( ! [X3: A,Xs2: coinductive_llist @ A] : ( P @ ( coinductive_LCons @ A @ X3 @ Xs2 ) @ X3 )
=> ( ! [Xs2: coinductive_llist @ A,X3: A,X6: A] :
( ( coinductive_lsetp @ A @ Xs2 @ X3 )
=> ( ( P @ Xs2 @ X3 )
=> ( P @ ( coinductive_LCons @ A @ X6 @ Xs2 ) @ X3 ) ) )
=> ( P @ X1 @ X23 ) ) ) ) ).
% lsetp.inducts
thf(fact_237_lsetp_Osimps,axiom,
! [A: $tType] :
( ( coinductive_lsetp @ A )
= ( ^ [A12: coinductive_llist @ A,A23: A] :
( ? [X2: A,Xs3: coinductive_llist @ A] :
( ( A12
= ( coinductive_LCons @ A @ X2 @ Xs3 ) )
& ( A23 = X2 ) )
| ? [Xs3: coinductive_llist @ A,X2: A,X8: A] :
( ( A12
= ( coinductive_LCons @ A @ X8 @ Xs3 ) )
& ( A23 = X2 )
& ( coinductive_lsetp @ A @ Xs3 @ X2 ) ) ) ) ) ).
% lsetp.simps
thf(fact_238_lsetp_Ocases,axiom,
! [A: $tType,A1: coinductive_llist @ A,A22: A] :
( ( coinductive_lsetp @ A @ A1 @ A22 )
=> ( ! [X3: A] :
( ? [Xs2: coinductive_llist @ A] :
( A1
= ( coinductive_LCons @ A @ X3 @ Xs2 ) )
=> ( A22 != X3 ) )
=> ~ ! [Xs2: coinductive_llist @ A] :
( ? [X6: A] :
( A1
= ( coinductive_LCons @ A @ X6 @ Xs2 ) )
=> ~ ( coinductive_lsetp @ A @ Xs2 @ A22 ) ) ) ) ).
% lsetp.cases
thf(fact_239_lsetp_Ointros_I1_J,axiom,
! [A: $tType,X: A,Xs: coinductive_llist @ A] : ( coinductive_lsetp @ A @ ( coinductive_LCons @ A @ X @ Xs ) @ X ) ).
% lsetp.intros(1)
thf(fact_240_lsetp_Ointros_I2_J,axiom,
! [A: $tType,Xs: coinductive_llist @ A,X: A,X7: A] :
( ( coinductive_lsetp @ A @ Xs @ X )
=> ( coinductive_lsetp @ A @ ( coinductive_LCons @ A @ X7 @ Xs ) @ X ) ) ).
% lsetp.intros(2)
thf(fact_241_lfinite__LConsI,axiom,
! [A: $tType,Xs: coinductive_llist @ A,X: A] :
( ( coinductive_lfinite @ A @ Xs )
=> ( coinductive_lfinite @ A @ ( coinductive_LCons @ A @ X @ Xs ) ) ) ).
% lfinite_LConsI
thf(fact_242_unfold__llist_Octr_I2_J,axiom,
! [B: $tType,A: $tType,P2: A > $o,A2: A,G21: A > B,G22: A > A] :
( ~ ( P2 @ A2 )
=> ( ( coindu1441602521_llist @ A @ B @ P2 @ G21 @ G22 @ A2 )
= ( coinductive_LCons @ B @ ( G21 @ A2 ) @ ( coindu1441602521_llist @ A @ B @ P2 @ G21 @ G22 @ ( G22 @ A2 ) ) ) ) ) ).
% unfold_llist.ctr(2)
thf(fact_243_ldistinct_OLCons,axiom,
! [A: $tType,X: A,Xs: coinductive_llist @ A] :
( ~ ( member @ A @ X @ ( coinductive_lset @ A @ Xs ) )
=> ( ( coindu351974385stinct @ A @ Xs )
=> ( coindu351974385stinct @ A @ ( coinductive_LCons @ A @ X @ Xs ) ) ) ) ).
% ldistinct.LCons
thf(fact_244_split__llist,axiom,
! [A: $tType,X: A,Xs: coinductive_llist @ A] :
( ( member @ A @ X @ ( coinductive_lset @ A @ Xs ) )
=> ? [Ys3: coinductive_llist @ A,Zs2: coinductive_llist @ A] :
( ( Xs
= ( coinductive_lappend @ A @ Ys3 @ ( coinductive_LCons @ A @ X @ Zs2 ) ) )
& ( coinductive_lfinite @ A @ Ys3 ) ) ) ).
% split_llist
thf(fact_245_split__llist__first,axiom,
! [A: $tType,X: A,Xs: coinductive_llist @ A] :
( ( member @ A @ X @ ( coinductive_lset @ A @ Xs ) )
=> ? [Ys3: coinductive_llist @ A,Zs2: coinductive_llist @ A] :
( ( Xs
= ( coinductive_lappend @ A @ Ys3 @ ( coinductive_LCons @ A @ X @ Zs2 ) ) )
& ( coinductive_lfinite @ A @ Ys3 )
& ~ ( member @ A @ X @ ( coinductive_lset @ A @ Ys3 ) ) ) ) ).
% split_llist_first
thf(fact_246_llast__lappend__LCons,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Y: A,Ys: coinductive_llist @ A] :
( ( coinductive_lfinite @ A @ Xs )
=> ( ( coinductive_llast @ A @ ( coinductive_lappend @ A @ Xs @ ( coinductive_LCons @ A @ Y @ Ys ) ) )
= ( coinductive_llast @ A @ ( coinductive_LCons @ A @ Y @ Ys ) ) ) ) ).
% llast_lappend_LCons
thf(fact_247_ltakeWhile_Ocode,axiom,
! [A: $tType] :
( ( coindu501562517eWhile @ A )
= ( ^ [P3: A > $o,Xs3: coinductive_llist @ A] :
( if @ ( coinductive_llist @ A )
@ ( ( coinductive_lnull @ A @ Xs3 )
| ~ ( P3 @ ( coinductive_lhd @ A @ Xs3 ) ) )
@ ( coinductive_LNil @ A )
@ ( coinductive_LCons @ A @ ( coinductive_lhd @ A @ Xs3 ) @ ( coindu501562517eWhile @ A @ P3 @ ( coinductive_ltl @ A @ Xs3 ) ) ) ) ) ) ).
% ltakeWhile.code
thf(fact_248_lfinite__code_I1_J,axiom,
! [A: $tType] : ( coinductive_lfinite @ A @ ( coinductive_LNil @ A ) ) ).
% lfinite_code(1)
thf(fact_249_lappend__LNil2,axiom,
! [A: $tType,Xs: coinductive_llist @ A] :
( ( coinductive_lappend @ A @ Xs @ ( coinductive_LNil @ A ) )
= Xs ) ).
% lappend_LNil2
thf(fact_250_lappend__code_I1_J,axiom,
! [A: $tType,Ys: coinductive_llist @ A] :
( ( coinductive_lappend @ A @ ( coinductive_LNil @ A ) @ Ys )
= Ys ) ).
% lappend_code(1)
thf(fact_251_lfilter__LNil,axiom,
! [A: $tType,P: A > $o] :
( ( coinductive_lfilter @ A @ P @ ( coinductive_LNil @ A ) )
= ( coinductive_LNil @ A ) ) ).
% lfilter_LNil
thf(fact_252_ltakeWhile__LNil,axiom,
! [A: $tType,P: A > $o] :
( ( coindu501562517eWhile @ A @ P @ ( coinductive_LNil @ A ) )
= ( coinductive_LNil @ A ) ) ).
% ltakeWhile_LNil
thf(fact_253_lconcat__LNil,axiom,
! [A: $tType] :
( ( coinductive_lconcat @ A @ ( coinductive_LNil @ ( coinductive_llist @ A ) ) )
= ( coinductive_LNil @ A ) ) ).
% lconcat_LNil
%----Type constructors (24)
thf(tcon_HOL_Obool___Lattices_Obounded__lattice,axiom,
bounded_lattice @ $o @ ( type2 @ $o ) ).
thf(tcon_Set_Oset___Lattices_Obounded__lattice_1,axiom,
! [A6: $tType] : ( bounded_lattice @ ( set @ A6 ) @ ( type2 @ ( set @ A6 ) ) ) ).
thf(tcon_fun___Lattices_Obounded__lattice_2,axiom,
! [A6: $tType,A7: $tType] :
( ( bounded_lattice @ A7 @ ( type2 @ A7 ) )
=> ( bounded_lattice @ ( A6 > A7 ) @ ( type2 @ ( A6 > A7 ) ) ) ) ).
thf(tcon_fun___Lattices_Obounded__semilattice__sup__bot,axiom,
! [A6: $tType,A7: $tType] :
( ( bounded_lattice @ A7 @ ( type2 @ A7 ) )
=> ( bounde1808546759up_bot @ ( A6 > A7 ) @ ( type2 @ ( A6 > A7 ) ) ) ) ).
thf(tcon_fun___Lattices_Obounded__lattice__bot,axiom,
! [A6: $tType,A7: $tType] :
( ( bounded_lattice @ A7 @ ( type2 @ A7 ) )
=> ( bounded_lattice_bot @ ( A6 > A7 ) @ ( type2 @ ( A6 > A7 ) ) ) ) ).
thf(tcon_fun___Lattices_Osemilattice__sup,axiom,
! [A6: $tType,A7: $tType] :
( ( semilattice_sup @ A7 @ ( type2 @ A7 ) )
=> ( semilattice_sup @ ( A6 > A7 ) @ ( type2 @ ( A6 > A7 ) ) ) ) ).
thf(tcon_fun___Lattices_Osemilattice__inf,axiom,
! [A6: $tType,A7: $tType] :
( ( semilattice_inf @ A7 @ ( type2 @ A7 ) )
=> ( semilattice_inf @ ( A6 > A7 ) @ ( type2 @ ( A6 > A7 ) ) ) ) ).
thf(tcon_fun___Lattices_Odistrib__lattice,axiom,
! [A6: $tType,A7: $tType] :
( ( distrib_lattice @ A7 @ ( type2 @ A7 ) )
=> ( distrib_lattice @ ( A6 > A7 ) @ ( type2 @ ( A6 > A7 ) ) ) ) ).
thf(tcon_fun___Lattices_Olattice,axiom,
! [A6: $tType,A7: $tType] :
( ( lattice @ A7 @ ( type2 @ A7 ) )
=> ( lattice @ ( 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_Set_Oset___Lattices_Obounded__semilattice__sup__bot_3,axiom,
! [A6: $tType] : ( bounde1808546759up_bot @ ( set @ A6 ) @ ( type2 @ ( set @ A6 ) ) ) ).
thf(tcon_Set_Oset___Lattices_Obounded__lattice__bot_4,axiom,
! [A6: $tType] : ( bounded_lattice_bot @ ( set @ A6 ) @ ( type2 @ ( set @ A6 ) ) ) ).
thf(tcon_Set_Oset___Lattices_Osemilattice__sup_5,axiom,
! [A6: $tType] : ( semilattice_sup @ ( set @ A6 ) @ ( type2 @ ( set @ A6 ) ) ) ).
thf(tcon_Set_Oset___Lattices_Osemilattice__inf_6,axiom,
! [A6: $tType] : ( semilattice_inf @ ( set @ A6 ) @ ( type2 @ ( set @ A6 ) ) ) ).
thf(tcon_Set_Oset___Lattices_Odistrib__lattice_7,axiom,
! [A6: $tType] : ( distrib_lattice @ ( set @ A6 ) @ ( type2 @ ( set @ A6 ) ) ) ).
thf(tcon_Set_Oset___Lattices_Olattice_8,axiom,
! [A6: $tType] : ( lattice @ ( set @ A6 ) @ ( type2 @ ( set @ A6 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Obot_9,axiom,
! [A6: $tType] : ( bot @ ( set @ A6 ) @ ( type2 @ ( set @ A6 ) ) ) ).
thf(tcon_HOL_Obool___Lattices_Obounded__semilattice__sup__bot_10,axiom,
bounde1808546759up_bot @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Lattices_Obounded__lattice__bot_11,axiom,
bounded_lattice_bot @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Lattices_Osemilattice__sup_12,axiom,
semilattice_sup @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Lattices_Osemilattice__inf_13,axiom,
semilattice_inf @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Lattices_Odistrib__lattice_14,axiom,
distrib_lattice @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Lattices_Olattice_15,axiom,
lattice @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Obot_16,axiom,
bot @ $o @ ( type2 @ $o ) ).
%----Helper facts (3)
thf(help_If_3_1_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_T,axiom,
! [A: $tType,X: A,Y: A] :
( ( if @ A @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_T,axiom,
! [A: $tType,X: A,Y: A] :
( ( if @ A @ $true @ X @ Y )
= X ) ).
%----Conjectures (1)
thf(conj_0,conjecture,
member @ a @ ( shd @ a @ xsa ) @ ( coinductive_lset @ a @ ( coindu1724414836stream @ a @ xsa ) ) ).
%------------------------------------------------------------------------------