TPTP Problem File: DAT177^1.p
View Solutions
- Solve Problem
%------------------------------------------------------------------------------
% File : DAT177^1 : TPTP v8.2.0. Released v7.0.0.
% Domain : Data Structures
% Problem : Lazy lists II 375
% Version : [Bla16] axioms : Especial.
% English :
% Refs : [Fri04] Friedrich (2004), Lazy Lists II
% : [RB15] Reynolds & Blanchette (2015), A Decision Procedure for
% : [Bla16] Blanchette (2016), Email to Geoff Sutcliffe
% Source : [Bla16]
% Names : llist2__375.p [Bla16]
% Status : Theorem
% Rating : 0.00 v7.2.0, 0.25 v7.1.0
% Syntax : Number of formulae : 326 ( 176 unt; 48 typ; 0 def)
% Number of atoms : 600 ( 301 equ; 0 cnn)
% Maximal formula atoms : 5 ( 2 avg)
% Number of connectives : 3790 ( 79 ~; 14 |; 54 &;3459 @)
% ( 0 <=>; 184 =>; 0 <=; 0 <~>)
% Maximal formula depth : 19 ( 6 avg)
% Number of types : 2 ( 1 usr)
% Number of type conns : 123 ( 123 >; 0 *; 0 +; 0 <<)
% Number of symbols : 48 ( 47 usr; 4 con; 0-5 aty)
% Number of variables : 892 ( 22 ^; 809 !; 20 ?; 892 :)
% ( 41 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2016-07-13 14:46:04.744
%------------------------------------------------------------------------------
%----Could-be-implicit typings (4)
thf(ty_t_Coinductive__List_Ollist,type,
coinductive_llist: $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 (44)
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_Orderings_Otop,type,
top:
!>[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__lattice__top,type,
bounded_lattice_top:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Topological__Spaces_Operfect__space,type,
topolo890362671_space:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Topological__Spaces_Ouniform__space,type,
topolo47006728_space:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Lattices_Obounded__semilattice__inf__top,type,
bounde1561333602nf_top:
!>[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_Ofinite__lprefix,type,
coindu328551480prefix:
!>[A: $tType] : ( ( coinductive_llist @ A ) > ( coinductive_llist @ 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_OlSup,type,
coinductive_lSup:
!>[A: $tType] : ( ( set @ ( coinductive_llist @ A ) ) > ( coinductive_llist @ 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_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_Olstrict__prefix,type,
coindu1478340336prefix:
!>[A: $tType] : ( ( coinductive_llist @ A ) > ( coinductive_llist @ A ) > $o ) ).
thf(sy_c_Groups_Ominus__class_Ominus,type,
minus_minus:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_LList2__Mirabelle__hamjzmohle_Oalllsts,type,
lList2435255213lllsts:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( coinductive_llist @ A ) ) ) ).
thf(sy_c_LList2__Mirabelle__hamjzmohle_Ofinlsts,type,
lList2236698231inlsts:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( coinductive_llist @ A ) ) ) ).
thf(sy_c_LList2__Mirabelle__hamjzmohle_Ofinlsts__rec,type,
lList21916056377ts_rec:
!>[B: $tType,A: $tType] : ( B > ( A > ( coinductive_llist @ A ) > B > B ) > ( coinductive_llist @ A ) > B ) ).
thf(sy_c_LList2__Mirabelle__hamjzmohle_Ofpslsts,type,
lList22096119349pslsts:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( coinductive_llist @ A ) ) ) ).
thf(sy_c_LList2__Mirabelle__hamjzmohle_Oinflsts,type,
lList21612149805nflsts:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( coinductive_llist @ A ) ) ) ).
thf(sy_c_LList2__Mirabelle__hamjzmohle_Oposlsts,type,
lList21148268032oslsts:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( coinductive_llist @ 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_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_OCollect,type,
collect:
!>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
thf(sy_c_Set_Oinsert,type,
insert:
!>[A: $tType] : ( A > ( set @ A ) > ( set @ A ) ) ).
thf(sy_c_Set_Ois__empty,type,
is_empty:
!>[A: $tType] : ( ( set @ A ) > $o ) ).
thf(sy_c_Set_Ois__singleton,type,
is_singleton:
!>[A: $tType] : ( ( set @ A ) > $o ) ).
thf(sy_c_Set_Opairwise,type,
pairwise:
!>[A: $tType] : ( ( A > A > $o ) > ( set @ A ) > $o ) ).
thf(sy_c_Set_Oremove,type,
remove:
!>[A: $tType] : ( A > ( set @ A ) > ( set @ A ) ) ).
thf(sy_c_Set_Othe__elem,type,
the_elem:
!>[A: $tType] : ( ( set @ A ) > A ) ).
thf(sy_c_Topological__Spaces_Ouniform__space__class_Ototally__bounded,type,
topolo406746546ounded:
!>[A: $tType] : ( ( set @ A ) > $o ) ).
thf(sy_c_member,type,
member:
!>[A: $tType] : ( A > ( set @ A ) > $o ) ).
thf(sy_v_A,type,
a2: set @ a ).
thf(sy_v_l______,type,
l: coinductive_llist @ a ).
thf(sy_v_r,type,
r: coinductive_llist @ a ).
thf(sy_v_sa____,type,
sa: coinductive_llist @ a ).
%----Relevant facts (256)
thf(fact_0_LNil__fin_Oprems,axiom,
member @ ( coinductive_llist @ a ) @ ( coinductive_lappend @ a @ l @ sa ) @ ( lList21612149805nflsts @ a @ a2 ) ).
% LNil_fin.prems
thf(fact_1_assms,axiom,
member @ ( coinductive_llist @ a ) @ r @ ( lList2236698231inlsts @ a @ a2 ) ).
% assms
thf(fact_2_lapp__inf,axiom,
! [A: $tType,S: coinductive_llist @ A,A2: set @ A,T: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ S @ ( lList21612149805nflsts @ A @ A2 ) )
=> ( ( coinductive_lappend @ A @ S @ T )
= S ) ) ).
% lapp_inf
thf(fact_3_infsubsetall,axiom,
! [A: $tType,X: coinductive_llist @ A,A2: set @ A] :
( ( member @ ( coinductive_llist @ A ) @ X @ ( lList21612149805nflsts @ A @ A2 ) )
=> ( member @ ( coinductive_llist @ A ) @ X @ ( lList2435255213lllsts @ A @ A2 ) ) ) ).
% infsubsetall
thf(fact_4_infT__simp,axiom,
! [A: $tType,R: coinductive_llist @ A,A2: set @ A] :
( ( member @ ( coinductive_llist @ A ) @ R @ ( lList21612149805nflsts @ A @ A2 ) )
=> ( member @ ( coinductive_llist @ A ) @ R @ ( lList21612149805nflsts @ A @ ( top_top @ ( set @ A ) ) ) ) ) ).
% infT_simp
thf(fact_5_inflstsI2,axiom,
! [A: $tType,A3: A,A2: set @ A,T: coinductive_llist @ A] :
( ( member @ A @ A3 @ A2 )
=> ( ( member @ ( coinductive_llist @ A ) @ T @ ( lList21612149805nflsts @ A @ A2 ) )
=> ( member @ ( coinductive_llist @ A ) @ ( coinductive_LCons @ A @ A3 @ T ) @ ( lList21612149805nflsts @ A @ A2 ) ) ) ) ).
% inflstsI2
thf(fact_6_inflsts__cases,axiom,
! [A: $tType,S: coinductive_llist @ A,A2: set @ A] :
( ( member @ ( coinductive_llist @ A ) @ S @ ( lList21612149805nflsts @ A @ A2 ) )
=> ~ ! [A4: A,L: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ L @ ( lList21612149805nflsts @ A @ A2 ) )
=> ( ( member @ A @ A4 @ A2 )
=> ( S
!= ( coinductive_LCons @ A @ A4 @ L ) ) ) ) ) ).
% inflsts_cases
thf(fact_7_lapp__fin__infT,axiom,
! [A: $tType,S: coinductive_llist @ A,A2: set @ A,T: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ S @ ( lList2236698231inlsts @ A @ A2 ) )
=> ( ( member @ ( coinductive_llist @ A ) @ T @ ( lList21612149805nflsts @ A @ A2 ) )
=> ( member @ ( coinductive_llist @ A ) @ ( coinductive_lappend @ A @ S @ T ) @ ( lList21612149805nflsts @ A @ A2 ) ) ) ) ).
% lapp_fin_infT
thf(fact_8_LNil__is__lappend__conv,axiom,
! [A: $tType,S: coinductive_llist @ A,T: coinductive_llist @ A] :
( ( ( coinductive_LNil @ A )
= ( coinductive_lappend @ A @ S @ T ) )
= ( ( S
= ( coinductive_LNil @ A ) )
& ( T
= ( coinductive_LNil @ A ) ) ) ) ).
% LNil_is_lappend_conv
thf(fact_9_lappend__is__LNil__conv,axiom,
! [A: $tType,S: coinductive_llist @ A,T: coinductive_llist @ A] :
( ( ( coinductive_lappend @ A @ S @ T )
= ( coinductive_LNil @ A ) )
= ( ( S
= ( coinductive_LNil @ A ) )
& ( T
= ( coinductive_LNil @ A ) ) ) ) ).
% lappend_is_LNil_conv
thf(fact_10_inflsts__empty,axiom,
! [A: $tType] :
( ( lList21612149805nflsts @ A @ ( bot_bot @ ( set @ A ) ) )
= ( bot_bot @ ( set @ ( coinductive_llist @ A ) ) ) ) ).
% inflsts_empty
thf(fact_11_notfin__inf,axiom,
! [A: $tType,X: coinductive_llist @ A] :
( ( ~ ( member @ ( coinductive_llist @ A ) @ X @ ( lList2236698231inlsts @ A @ ( top_top @ ( set @ A ) ) ) ) )
= ( member @ ( coinductive_llist @ A ) @ X @ ( lList21612149805nflsts @ A @ ( top_top @ ( set @ A ) ) ) ) ) ).
% notfin_inf
thf(fact_12_notinf__fin,axiom,
! [A: $tType,X: coinductive_llist @ A] :
( ( ~ ( member @ ( coinductive_llist @ A ) @ X @ ( lList21612149805nflsts @ A @ ( top_top @ ( set @ A ) ) ) ) )
= ( member @ ( coinductive_llist @ A ) @ X @ ( lList2236698231inlsts @ A @ ( top_top @ ( set @ A ) ) ) ) ) ).
% notinf_fin
thf(fact_13_LNil__fin_Ohyps,axiom,
( l
= ( coinductive_LNil @ a ) ) ).
% LNil_fin.hyps
thf(fact_14_alllsts__UNIV,axiom,
! [A: $tType,S: coinductive_llist @ A] : ( member @ ( coinductive_llist @ A ) @ S @ ( lList2435255213lllsts @ A @ ( top_top @ ( set @ A ) ) ) ) ).
% alllsts_UNIV
thf(fact_15_same__lappend__eq,axiom,
! [A: $tType,R: coinductive_llist @ A,A2: set @ A,S: coinductive_llist @ A,T: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ R @ ( lList2236698231inlsts @ A @ A2 ) )
=> ( ( ( coinductive_lappend @ A @ R @ S )
= ( coinductive_lappend @ A @ R @ T ) )
= ( S = T ) ) ) ).
% same_lappend_eq
thf(fact_16_lapp__fin__fin__iff,axiom,
! [A: $tType,R: coinductive_llist @ A,S: coinductive_llist @ A,A2: set @ A] :
( ( member @ ( coinductive_llist @ A ) @ ( coinductive_lappend @ A @ R @ S ) @ ( lList2236698231inlsts @ A @ A2 ) )
= ( ( member @ ( coinductive_llist @ A ) @ R @ ( lList2236698231inlsts @ A @ A2 ) )
& ( member @ ( coinductive_llist @ A ) @ S @ ( lList2236698231inlsts @ A @ A2 ) ) ) ) ).
% lapp_fin_fin_iff
thf(fact_17_LConsE,axiom,
! [A: $tType,X: A,Xs: coinductive_llist @ A,A2: set @ A] :
( ( member @ ( coinductive_llist @ A ) @ ( coinductive_LCons @ A @ X @ Xs ) @ ( lList2435255213lllsts @ A @ A2 ) )
= ( ( member @ A @ X @ A2 )
& ( member @ ( coinductive_llist @ A ) @ Xs @ ( lList2435255213lllsts @ A @ A2 ) ) ) ) ).
% LConsE
thf(fact_18_inflstsI,axiom,
! [A: $tType,X: coinductive_llist @ A,A2: set @ A] :
( ( member @ ( coinductive_llist @ A ) @ X @ ( lList2435255213lllsts @ A @ A2 ) )
=> ( ~ ( member @ ( coinductive_llist @ A ) @ X @ ( lList2236698231inlsts @ A @ ( top_top @ ( set @ A ) ) ) )
=> ( member @ ( coinductive_llist @ A ) @ X @ ( lList21612149805nflsts @ A @ A2 ) ) ) ) ).
% inflstsI
thf(fact_19_lappT,axiom,
! [A: $tType,S: coinductive_llist @ A,A2: set @ A,T: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ S @ ( lList2435255213lllsts @ A @ A2 ) )
=> ( ( member @ ( coinductive_llist @ A ) @ T @ ( lList2435255213lllsts @ A @ A2 ) )
=> ( member @ ( coinductive_llist @ A ) @ ( coinductive_lappend @ A @ S @ T ) @ ( lList2435255213lllsts @ A @ A2 ) ) ) ) ).
% lappT
thf(fact_20_llistE,axiom,
! [A: $tType,Y: coinductive_llist @ A] :
( ( Y
!= ( coinductive_LNil @ A ) )
=> ~ ! [X21: A,X22: coinductive_llist @ A] :
( Y
!= ( coinductive_LCons @ A @ X21 @ X22 ) ) ) ).
% llistE
thf(fact_21_finT__simp,axiom,
! [A: $tType,R: coinductive_llist @ A,A2: set @ A] :
( ( member @ ( coinductive_llist @ A ) @ R @ ( lList2236698231inlsts @ A @ A2 ) )
=> ( member @ ( coinductive_llist @ A ) @ R @ ( lList2236698231inlsts @ A @ ( top_top @ ( set @ A ) ) ) ) ) ).
% finT_simp
thf(fact_22_fin__finite,axiom,
! [A: $tType,R: coinductive_llist @ A,A2: set @ A] :
( ( member @ ( coinductive_llist @ A ) @ R @ ( lList2236698231inlsts @ A @ A2 ) )
=> ( member @ ( coinductive_llist @ A ) @ R @ ( lList2236698231inlsts @ A @ ( top_top @ ( set @ A ) ) ) ) ) ).
% fin_finite
thf(fact_23_finite__lemma,axiom,
! [A: $tType,X: coinductive_llist @ A,A2: set @ A,B2: set @ A] :
( ( member @ ( coinductive_llist @ A ) @ X @ ( lList2236698231inlsts @ A @ A2 ) )
=> ( ( member @ ( coinductive_llist @ A ) @ X @ ( lList2435255213lllsts @ A @ B2 ) )
=> ( member @ ( coinductive_llist @ A ) @ X @ ( lList2236698231inlsts @ A @ B2 ) ) ) ) ).
% finite_lemma
thf(fact_24_finsubsetall,axiom,
! [A: $tType,X: coinductive_llist @ A,A2: set @ A] :
( ( member @ ( coinductive_llist @ A ) @ X @ ( lList2236698231inlsts @ A @ A2 ) )
=> ( member @ ( coinductive_llist @ A ) @ X @ ( lList2435255213lllsts @ A @ A2 ) ) ) ).
% finsubsetall
thf(fact_25_lappfin__finT,axiom,
! [A: $tType,S: coinductive_llist @ A,A2: set @ A,T: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ S @ ( lList2236698231inlsts @ A @ A2 ) )
=> ( ( member @ ( coinductive_llist @ A ) @ T @ ( lList2236698231inlsts @ A @ A2 ) )
=> ( member @ ( coinductive_llist @ A ) @ ( coinductive_lappend @ A @ S @ T ) @ ( lList2236698231inlsts @ A @ A2 ) ) ) ) ).
% lappfin_finT
thf(fact_26_alllsts_Ocases,axiom,
! [A: $tType,A3: coinductive_llist @ A,A2: set @ A] :
( ( member @ ( coinductive_llist @ A ) @ A3 @ ( lList2435255213lllsts @ A @ A2 ) )
=> ( ( A3
!= ( coinductive_LNil @ A ) )
=> ~ ! [L: coinductive_llist @ A,A4: A] :
( ( A3
= ( coinductive_LCons @ A @ A4 @ L ) )
=> ( ( member @ ( coinductive_llist @ A ) @ L @ ( lList2435255213lllsts @ A @ A2 ) )
=> ~ ( member @ A @ A4 @ A2 ) ) ) ) ) ).
% alllsts.cases
thf(fact_27_alllsts_Osimps,axiom,
! [A: $tType,A3: coinductive_llist @ A,A2: set @ A] :
( ( member @ ( coinductive_llist @ A ) @ A3 @ ( lList2435255213lllsts @ A @ A2 ) )
= ( ( A3
= ( coinductive_LNil @ A ) )
| ? [L2: coinductive_llist @ A,A5: A] :
( ( A3
= ( coinductive_LCons @ A @ A5 @ L2 ) )
& ( member @ ( coinductive_llist @ A ) @ L2 @ ( lList2435255213lllsts @ A @ A2 ) )
& ( member @ A @ A5 @ A2 ) ) ) ) ).
% alllsts.simps
thf(fact_28_finlsts_Ocases,axiom,
! [A: $tType,A3: coinductive_llist @ A,A2: set @ A] :
( ( member @ ( coinductive_llist @ A ) @ A3 @ ( lList2236698231inlsts @ A @ A2 ) )
=> ( ( A3
!= ( coinductive_LNil @ A ) )
=> ~ ! [L: coinductive_llist @ A,A4: A] :
( ( A3
= ( coinductive_LCons @ A @ A4 @ L ) )
=> ( ( member @ ( coinductive_llist @ A ) @ L @ ( lList2236698231inlsts @ A @ A2 ) )
=> ~ ( member @ A @ A4 @ A2 ) ) ) ) ) ).
% finlsts.cases
thf(fact_29_finlsts_Osimps,axiom,
! [A: $tType,A3: coinductive_llist @ A,A2: set @ A] :
( ( member @ ( coinductive_llist @ A ) @ A3 @ ( lList2236698231inlsts @ A @ A2 ) )
= ( ( A3
= ( coinductive_LNil @ A ) )
| ? [L2: coinductive_llist @ A,A5: A] :
( ( A3
= ( coinductive_LCons @ A @ A5 @ L2 ) )
& ( member @ ( coinductive_llist @ A ) @ L2 @ ( lList2236698231inlsts @ A @ A2 ) )
& ( member @ A @ A5 @ A2 ) ) ) ) ).
% finlsts.simps
thf(fact_30_lapp__all__invT,axiom,
! [A: $tType,R: coinductive_llist @ A,S: coinductive_llist @ A,A2: set @ A] :
( ( member @ ( coinductive_llist @ A ) @ ( coinductive_lappend @ A @ R @ S ) @ ( lList2435255213lllsts @ A @ A2 ) )
=> ( member @ ( coinductive_llist @ A ) @ R @ ( lList2435255213lllsts @ A @ A2 ) ) ) ).
% lapp_all_invT
thf(fact_31_finlsts__induct,axiom,
! [A: $tType,X: coinductive_llist @ A,A2: set @ A,P: ( coinductive_llist @ A ) > $o] :
( ( member @ ( coinductive_llist @ A ) @ X @ ( lList2236698231inlsts @ A @ A2 ) )
=> ( ! [L: coinductive_llist @ A] :
( ( L
= ( coinductive_LNil @ A ) )
=> ( P @ L ) )
=> ( ! [A4: A,L: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ L @ ( lList2236698231inlsts @ A @ A2 ) )
=> ( ( P @ L )
=> ( ( member @ A @ A4 @ A2 )
=> ( P @ ( coinductive_LCons @ A @ A4 @ L ) ) ) ) )
=> ( P @ X ) ) ) ) ).
% finlsts_induct
thf(fact_32_finlsts_Oinducts,axiom,
! [A: $tType,X: coinductive_llist @ A,A2: set @ A,P: ( coinductive_llist @ A ) > $o] :
( ( member @ ( coinductive_llist @ A ) @ X @ ( lList2236698231inlsts @ A @ A2 ) )
=> ( ( P @ ( coinductive_LNil @ A ) )
=> ( ! [L: coinductive_llist @ A,A4: A] :
( ( member @ ( coinductive_llist @ A ) @ L @ ( lList2236698231inlsts @ A @ A2 ) )
=> ( ( P @ L )
=> ( ( member @ A @ A4 @ A2 )
=> ( P @ ( coinductive_LCons @ A @ A4 @ L ) ) ) ) )
=> ( P @ X ) ) ) ) ).
% finlsts.inducts
thf(fact_33_alllsts_OLNil__all,axiom,
! [A: $tType,A2: set @ A] : ( member @ ( coinductive_llist @ A ) @ ( coinductive_LNil @ A ) @ ( lList2435255213lllsts @ A @ A2 ) ) ).
% alllsts.LNil_all
thf(fact_34_alllsts_Ocoinduct,axiom,
! [A: $tType,X2: ( coinductive_llist @ A ) > $o,X: coinductive_llist @ A,A2: set @ A] :
( ( X2 @ X )
=> ( ! [X3: coinductive_llist @ A] :
( ( X2 @ X3 )
=> ( ( X3
= ( coinductive_LNil @ A ) )
| ? [L3: coinductive_llist @ A,A6: A] :
( ( X3
= ( coinductive_LCons @ A @ A6 @ L3 ) )
& ( ( X2 @ L3 )
| ( member @ ( coinductive_llist @ A ) @ L3 @ ( lList2435255213lllsts @ A @ A2 ) ) )
& ( member @ A @ A6 @ A2 ) ) ) )
=> ( member @ ( coinductive_llist @ A ) @ X @ ( lList2435255213lllsts @ A @ A2 ) ) ) ) ).
% alllsts.coinduct
thf(fact_35_finlsts_OLNil__fin,axiom,
! [A: $tType,A2: set @ A] : ( member @ ( coinductive_llist @ A ) @ ( coinductive_LNil @ A ) @ ( lList2236698231inlsts @ A @ A2 ) ) ).
% finlsts.LNil_fin
thf(fact_36_alllsts_OLCons__all,axiom,
! [A: $tType,L4: coinductive_llist @ A,A2: set @ A,A3: A] :
( ( member @ ( coinductive_llist @ A ) @ L4 @ ( lList2435255213lllsts @ A @ A2 ) )
=> ( ( member @ A @ A3 @ A2 )
=> ( member @ ( coinductive_llist @ A ) @ ( coinductive_LCons @ A @ A3 @ L4 ) @ ( lList2435255213lllsts @ A @ A2 ) ) ) ) ).
% alllsts.LCons_all
thf(fact_37_finlsts_OLCons__fin,axiom,
! [A: $tType,L4: coinductive_llist @ A,A2: set @ A,A3: A] :
( ( member @ ( coinductive_llist @ A ) @ L4 @ ( lList2236698231inlsts @ A @ A2 ) )
=> ( ( member @ A @ A3 @ A2 )
=> ( member @ ( coinductive_llist @ A ) @ ( coinductive_LCons @ A @ A3 @ L4 ) @ ( lList2236698231inlsts @ A @ A2 ) ) ) ) ).
% finlsts.LCons_fin
thf(fact_38_lapp__fin__fin__lemma,axiom,
! [A: $tType,R: coinductive_llist @ A,S: coinductive_llist @ A,A2: set @ A] :
( ( member @ ( coinductive_llist @ A ) @ ( coinductive_lappend @ A @ R @ S ) @ ( lList2236698231inlsts @ A @ A2 ) )
=> ( member @ ( coinductive_llist @ A ) @ R @ ( lList2236698231inlsts @ A @ A2 ) ) ) ).
% lapp_fin_fin_lemma
thf(fact_39_inflstsE,axiom,
! [A: $tType,X: coinductive_llist @ A,A2: set @ A] :
( ( member @ ( coinductive_llist @ A ) @ X @ ( lList21612149805nflsts @ A @ A2 ) )
=> ~ ( ( member @ ( coinductive_llist @ A ) @ X @ ( lList2435255213lllsts @ A @ A2 ) )
=> ( member @ ( coinductive_llist @ A ) @ X @ ( lList2236698231inlsts @ A @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% inflstsE
thf(fact_40_fin__inf__cases,axiom,
! [A: $tType,R: coinductive_llist @ A] :
( ~ ( member @ ( coinductive_llist @ A ) @ R @ ( lList2236698231inlsts @ A @ ( top_top @ ( set @ A ) ) ) )
=> ( member @ ( coinductive_llist @ A ) @ R @ ( lList21612149805nflsts @ A @ ( top_top @ ( set @ A ) ) ) ) ) ).
% fin_inf_cases
thf(fact_41_alllstsE,axiom,
! [A: $tType,X: coinductive_llist @ A,A2: set @ A] :
( ( member @ ( coinductive_llist @ A ) @ X @ ( lList2435255213lllsts @ A @ A2 ) )
=> ( ~ ( member @ ( coinductive_llist @ A ) @ X @ ( lList2236698231inlsts @ A @ A2 ) )
=> ( member @ ( coinductive_llist @ A ) @ X @ ( lList21612149805nflsts @ A @ A2 ) ) ) ) ).
% alllstsE
thf(fact_42_lappend__code_I1_J,axiom,
! [A: $tType,Ys: coinductive_llist @ A] :
( ( coinductive_lappend @ A @ ( coinductive_LNil @ A ) @ Ys )
= Ys ) ).
% lappend_code(1)
thf(fact_43_lappend__LNil2,axiom,
! [A: $tType,Xs: coinductive_llist @ A] :
( ( coinductive_lappend @ A @ Xs @ ( coinductive_LNil @ A ) )
= Xs ) ).
% lappend_LNil2
thf(fact_44_LNil__eq__lappend__iff,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( ( coinductive_LNil @ A )
= ( coinductive_lappend @ A @ Xs @ Ys ) )
= ( ( Xs
= ( coinductive_LNil @ A ) )
& ( Ys
= ( coinductive_LNil @ A ) ) ) ) ).
% LNil_eq_lappend_iff
thf(fact_45_mem__Collect__eq,axiom,
! [A: $tType,A3: A,P: A > $o] :
( ( member @ A @ A3 @ ( collect @ A @ P ) )
= ( P @ A3 ) ) ).
% mem_Collect_eq
thf(fact_46_Collect__mem__eq,axiom,
! [A: $tType,A2: set @ A] :
( ( collect @ A
@ ^ [X4: A] : ( member @ A @ X4 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_47_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_48_ext,axiom,
! [B: $tType,A: $tType,F: A > B,G: A > B] :
( ! [X3: A] :
( ( F @ X3 )
= ( G @ X3 ) )
=> ( F = G ) ) ).
% ext
thf(fact_49_lappend__eq__LNil__iff,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( ( coinductive_lappend @ A @ Xs @ Ys )
= ( coinductive_LNil @ A ) )
= ( ( Xs
= ( coinductive_LNil @ A ) )
& ( Ys
= ( coinductive_LNil @ A ) ) ) ) ).
% lappend_eq_LNil_iff
thf(fact_50_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_51_lappend__snocL1__conv__LCons2,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Y: A,Ys: coinductive_llist @ A] :
( ( coinductive_lappend @ A @ ( coinductive_lappend @ A @ Xs @ ( coinductive_LCons @ A @ Y @ ( coinductive_LNil @ A ) ) ) @ Ys )
= ( coinductive_lappend @ A @ Xs @ ( coinductive_LCons @ A @ Y @ Ys ) ) ) ).
% lappend_snocL1_conv_LCons2
thf(fact_52_llist_Oinject,axiom,
! [A: $tType,X212: A,X222: coinductive_llist @ A,Y21: A,Y22: coinductive_llist @ A] :
( ( ( coinductive_LCons @ A @ X212 @ X222 )
= ( coinductive_LCons @ A @ Y21 @ Y22 ) )
= ( ( X212 = Y21 )
& ( X222 = Y22 ) ) ) ).
% llist.inject
thf(fact_53_UNIV__I,axiom,
! [A: $tType,X: A] : ( member @ A @ X @ ( top_top @ ( set @ A ) ) ) ).
% UNIV_I
thf(fact_54_iso__tuple__UNIV__I,axiom,
! [A: $tType,X: A] : ( member @ A @ X @ ( top_top @ ( set @ A ) ) ) ).
% iso_tuple_UNIV_I
thf(fact_55_empty__iff,axiom,
! [A: $tType,C: A] :
~ ( member @ A @ C @ ( bot_bot @ ( set @ A ) ) ) ).
% empty_iff
thf(fact_56_all__not__in__conv,axiom,
! [A: $tType,A2: set @ A] :
( ( ! [X4: A] :
~ ( member @ A @ X4 @ A2 ) )
= ( A2
= ( bot_bot @ ( set @ A ) ) ) ) ).
% all_not_in_conv
thf(fact_57_empty__Collect__eq,axiom,
! [A: $tType,P: A > $o] :
( ( ( bot_bot @ ( set @ A ) )
= ( collect @ A @ P ) )
= ( ! [X4: A] :
~ ( P @ X4 ) ) ) ).
% empty_Collect_eq
thf(fact_58_Collect__empty__eq,axiom,
! [A: $tType,P: A > $o] :
( ( ( collect @ A @ P )
= ( bot_bot @ ( set @ A ) ) )
= ( ! [X4: A] :
~ ( P @ X4 ) ) ) ).
% Collect_empty_eq
thf(fact_59_ex__in__conv,axiom,
! [A: $tType,A2: set @ A] :
( ( ? [X4: A] : ( member @ A @ X4 @ A2 ) )
= ( A2
!= ( bot_bot @ ( set @ A ) ) ) ) ).
% ex_in_conv
thf(fact_60_equals0I,axiom,
! [A: $tType,A2: set @ A] :
( ! [Y2: A] :
~ ( member @ A @ Y2 @ A2 )
=> ( A2
= ( bot_bot @ ( set @ A ) ) ) ) ).
% equals0I
thf(fact_61_equals0D,axiom,
! [A: $tType,A2: set @ A,A3: A] :
( ( A2
= ( bot_bot @ ( set @ A ) ) )
=> ~ ( member @ A @ A3 @ A2 ) ) ).
% equals0D
thf(fact_62_emptyE,axiom,
! [A: $tType,A3: A] :
~ ( member @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ).
% emptyE
thf(fact_63_UNIV__witness,axiom,
! [A: $tType] :
? [X3: A] : ( member @ A @ X3 @ ( top_top @ ( set @ A ) ) ) ).
% UNIV_witness
thf(fact_64_UNIV__eq__I,axiom,
! [A: $tType,A2: set @ A] :
( ! [X3: A] : ( member @ A @ X3 @ A2 )
=> ( ( top_top @ ( set @ A ) )
= A2 ) ) ).
% UNIV_eq_I
thf(fact_65_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_66_empty__not__UNIV,axiom,
! [A: $tType] :
( ( bot_bot @ ( set @ A ) )
!= ( top_top @ ( set @ A ) ) ) ).
% empty_not_UNIV
thf(fact_67_neq__LNil__conv,axiom,
! [A: $tType,Xs: coinductive_llist @ A] :
( ( Xs
!= ( coinductive_LNil @ A ) )
= ( ? [X4: A,Xs2: coinductive_llist @ A] :
( Xs
= ( coinductive_LCons @ A @ X4 @ Xs2 ) ) ) ) ).
% neq_LNil_conv
thf(fact_68_llist_Odistinct_I1_J,axiom,
! [A: $tType,X212: A,X222: coinductive_llist @ A] :
( ( coinductive_LNil @ A )
!= ( coinductive_LCons @ A @ X212 @ X222 ) ) ).
% llist.distinct(1)
thf(fact_69_lappend__LNil__LNil,axiom,
! [A: $tType] :
( ( coinductive_lappend @ A @ ( coinductive_LNil @ A ) @ ( coinductive_LNil @ A ) )
= ( coinductive_LNil @ A ) ) ).
% lappend_LNil_LNil
thf(fact_70_top__apply,axiom,
! [C2: $tType,D: $tType] :
( ( top @ C2 @ ( type2 @ C2 ) )
=> ( ( top_top @ ( D > C2 ) )
= ( ^ [X4: D] : ( top_top @ C2 ) ) ) ) ).
% top_apply
thf(fact_71_bot__apply,axiom,
! [C2: $tType,D: $tType] :
( ( bot @ C2 @ ( type2 @ C2 ) )
=> ( ( bot_bot @ ( D > C2 ) )
= ( ^ [X4: D] : ( bot_bot @ C2 ) ) ) ) ).
% bot_apply
thf(fact_72_poslsts__empty,axiom,
! [A: $tType] :
( ( lList21148268032oslsts @ A @ ( bot_bot @ ( set @ A ) ) )
= ( bot_bot @ ( set @ ( coinductive_llist @ A ) ) ) ) ).
% poslsts_empty
thf(fact_73_fpslsts__empty,axiom,
! [A: $tType] :
( ( lList22096119349pslsts @ A @ ( bot_bot @ ( set @ A ) ) )
= ( bot_bot @ ( set @ ( coinductive_llist @ A ) ) ) ) ).
% fpslsts_empty
thf(fact_74_poslsts__iff,axiom,
! [A: $tType,S: coinductive_llist @ A,A2: set @ A] :
( ( member @ ( coinductive_llist @ A ) @ S @ ( lList21148268032oslsts @ A @ A2 ) )
= ( ( member @ ( coinductive_llist @ A ) @ S @ ( lList2435255213lllsts @ A @ A2 ) )
& ( S
!= ( coinductive_LNil @ A ) ) ) ) ).
% poslsts_iff
thf(fact_75_fpslsts__iff,axiom,
! [A: $tType,S: coinductive_llist @ A,A2: set @ A] :
( ( member @ ( coinductive_llist @ A ) @ S @ ( lList22096119349pslsts @ A @ A2 ) )
= ( ( member @ ( coinductive_llist @ A ) @ S @ ( lList2236698231inlsts @ A @ A2 ) )
& ( S
!= ( coinductive_LNil @ A ) ) ) ) ).
% fpslsts_iff
thf(fact_76_poslsts__UNIV,axiom,
! [A: $tType,S: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ S @ ( lList21148268032oslsts @ A @ ( top_top @ ( set @ A ) ) ) )
= ( S
!= ( coinductive_LNil @ A ) ) ) ).
% poslsts_UNIV
thf(fact_77_bot__set__def,axiom,
! [A: $tType] :
( ( bot_bot @ ( set @ A ) )
= ( collect @ A @ ( bot_bot @ ( A > $o ) ) ) ) ).
% bot_set_def
thf(fact_78_top__set__def,axiom,
! [A: $tType] :
( ( top_top @ ( set @ A ) )
= ( collect @ A @ ( top_top @ ( A > $o ) ) ) ) ).
% top_set_def
thf(fact_79_bot__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( bot @ B @ ( type2 @ B ) )
=> ( ( bot_bot @ ( A > B ) )
= ( ^ [X4: A] : ( bot_bot @ B ) ) ) ) ).
% bot_fun_def
thf(fact_80_fpslsts__cases,axiom,
! [A: $tType,R: coinductive_llist @ A,A2: set @ A] :
( ( member @ ( coinductive_llist @ A ) @ R @ ( lList22096119349pslsts @ A @ A2 ) )
=> ~ ! [A4: A,Rs: coinductive_llist @ A] :
( ( R
= ( coinductive_LCons @ A @ A4 @ Rs ) )
=> ( ( member @ A @ A4 @ A2 )
=> ~ ( member @ ( coinductive_llist @ A ) @ Rs @ ( lList2236698231inlsts @ A @ A2 ) ) ) ) ) ).
% fpslsts_cases
thf(fact_81_alllsts__empty,axiom,
! [A: $tType] :
( ( lList2435255213lllsts @ A @ ( bot_bot @ ( set @ A ) ) )
= ( insert @ ( coinductive_llist @ A ) @ ( coinductive_LNil @ A ) @ ( bot_bot @ ( set @ ( coinductive_llist @ A ) ) ) ) ) ).
% alllsts_empty
thf(fact_82_finlsts__empty,axiom,
! [A: $tType] :
( ( lList2236698231inlsts @ A @ ( bot_bot @ ( set @ A ) ) )
= ( insert @ ( coinductive_llist @ A ) @ ( coinductive_LNil @ A ) @ ( bot_bot @ ( set @ ( coinductive_llist @ A ) ) ) ) ) ).
% finlsts_empty
thf(fact_83_Coinductive__List_Ofinite__lprefix__nitpick__simps_I3_J,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Y: A,Ys: coinductive_llist @ A] :
( ( coindu328551480prefix @ A @ Xs @ ( coinductive_LCons @ A @ Y @ Ys ) )
= ( ( Xs
= ( coinductive_LNil @ A ) )
| ? [Xs2: coinductive_llist @ A] :
( ( Xs
= ( coinductive_LCons @ A @ Y @ Xs2 ) )
& ( coindu328551480prefix @ A @ Xs2 @ Ys ) ) ) ) ).
% Coinductive_List.finite_lprefix_nitpick_simps(3)
thf(fact_84_finlsts__rec__LCons,axiom,
! [B: $tType,A: $tType,R: coinductive_llist @ A,A2: set @ A,C: B,D2: A > ( coinductive_llist @ A ) > B > B,A3: A] :
( ( member @ ( coinductive_llist @ A ) @ R @ ( lList2236698231inlsts @ A @ A2 ) )
=> ( ( lList21916056377ts_rec @ B @ A @ C @ D2 @ ( coinductive_LCons @ A @ A3 @ R ) )
= ( D2 @ A3 @ R @ ( lList21916056377ts_rec @ B @ A @ C @ D2 @ R ) ) ) ) ).
% finlsts_rec_LCons
thf(fact_85_finlsts__rec__LCons__def,axiom,
! [B: $tType,A: $tType,F: ( coinductive_llist @ A ) > B,C: B,D2: A > ( coinductive_llist @ A ) > B > B,R: coinductive_llist @ A,A2: set @ A,A3: A] :
( ( F
= ( lList21916056377ts_rec @ B @ A @ C @ D2 ) )
=> ( ( member @ ( coinductive_llist @ A ) @ R @ ( lList2236698231inlsts @ A @ A2 ) )
=> ( ( F @ ( coinductive_LCons @ A @ A3 @ R ) )
= ( D2 @ A3 @ R @ ( F @ R ) ) ) ) ) ).
% finlsts_rec_LCons_def
thf(fact_86_Set_Ois__empty__def,axiom,
! [A: $tType] :
( ( is_empty @ A )
= ( ^ [A7: set @ A] :
( A7
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Set.is_empty_def
thf(fact_87_insertCI,axiom,
! [A: $tType,A3: A,B2: set @ A,B3: A] :
( ( ~ ( member @ A @ A3 @ B2 )
=> ( A3 = B3 ) )
=> ( member @ A @ A3 @ ( insert @ A @ B3 @ B2 ) ) ) ).
% insertCI
thf(fact_88_insert__iff,axiom,
! [A: $tType,A3: A,B3: A,A2: set @ A] :
( ( member @ A @ A3 @ ( insert @ A @ B3 @ A2 ) )
= ( ( A3 = B3 )
| ( member @ A @ A3 @ A2 ) ) ) ).
% insert_iff
thf(fact_89_insert__absorb2,axiom,
! [A: $tType,X: A,A2: set @ A] :
( ( insert @ A @ X @ ( insert @ A @ X @ A2 ) )
= ( insert @ A @ X @ A2 ) ) ).
% insert_absorb2
thf(fact_90_top1I,axiom,
! [A: $tType,X: A] : ( top_top @ ( A > $o ) @ X ) ).
% top1I
thf(fact_91_singletonI,axiom,
! [A: $tType,A3: A] : ( member @ A @ A3 @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% singletonI
thf(fact_92_insertE,axiom,
! [A: $tType,A3: A,B3: A,A2: set @ A] :
( ( member @ A @ A3 @ ( insert @ A @ B3 @ A2 ) )
=> ( ( A3 != B3 )
=> ( member @ A @ A3 @ A2 ) ) ) ).
% insertE
thf(fact_93_insertI1,axiom,
! [A: $tType,A3: A,B2: set @ A] : ( member @ A @ A3 @ ( insert @ A @ A3 @ B2 ) ) ).
% insertI1
thf(fact_94_insertI2,axiom,
! [A: $tType,A3: A,B2: set @ A,B3: A] :
( ( member @ A @ A3 @ B2 )
=> ( member @ A @ A3 @ ( insert @ A @ B3 @ B2 ) ) ) ).
% insertI2
thf(fact_95_Set_Oset__insert,axiom,
! [A: $tType,X: A,A2: set @ A] :
( ( member @ A @ X @ A2 )
=> ~ ! [B4: set @ A] :
( ( A2
= ( insert @ A @ X @ B4 ) )
=> ( member @ A @ X @ B4 ) ) ) ).
% Set.set_insert
thf(fact_96_insert__ident,axiom,
! [A: $tType,X: A,A2: set @ A,B2: set @ A] :
( ~ ( member @ A @ X @ A2 )
=> ( ~ ( member @ A @ X @ B2 )
=> ( ( ( insert @ A @ X @ A2 )
= ( insert @ A @ X @ B2 ) )
= ( A2 = B2 ) ) ) ) ).
% insert_ident
thf(fact_97_insert__absorb,axiom,
! [A: $tType,A3: A,A2: set @ A] :
( ( member @ A @ A3 @ A2 )
=> ( ( insert @ A @ A3 @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_98_insert__eq__iff,axiom,
! [A: $tType,A3: A,A2: set @ A,B3: A,B2: set @ A] :
( ~ ( member @ A @ A3 @ A2 )
=> ( ~ ( member @ A @ B3 @ B2 )
=> ( ( ( insert @ A @ A3 @ A2 )
= ( insert @ A @ B3 @ B2 ) )
= ( ( ( A3 = B3 )
=> ( A2 = B2 ) )
& ( ( A3 != B3 )
=> ? [C3: set @ A] :
( ( A2
= ( insert @ A @ B3 @ C3 ) )
& ~ ( member @ A @ B3 @ C3 )
& ( B2
= ( insert @ A @ A3 @ C3 ) )
& ~ ( member @ A @ A3 @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_99_insert__commute,axiom,
! [A: $tType,X: A,Y: A,A2: set @ A] :
( ( insert @ A @ X @ ( insert @ A @ Y @ A2 ) )
= ( insert @ A @ Y @ ( insert @ A @ X @ A2 ) ) ) ).
% insert_commute
thf(fact_100_mk__disjoint__insert,axiom,
! [A: $tType,A3: A,A2: set @ A] :
( ( member @ A @ A3 @ A2 )
=> ? [B4: set @ A] :
( ( A2
= ( insert @ A @ A3 @ B4 ) )
& ~ ( member @ A @ A3 @ B4 ) ) ) ).
% mk_disjoint_insert
thf(fact_101_singleton__inject,axiom,
! [A: $tType,A3: A,B3: A] :
( ( ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) )
= ( insert @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) )
=> ( A3 = B3 ) ) ).
% singleton_inject
thf(fact_102_insert__not__empty,axiom,
! [A: $tType,A3: A,A2: set @ A] :
( ( insert @ A @ A3 @ A2 )
!= ( bot_bot @ ( set @ A ) ) ) ).
% insert_not_empty
thf(fact_103_doubleton__eq__iff,axiom,
! [A: $tType,A3: A,B3: A,C: A,D2: A] :
( ( ( insert @ A @ A3 @ ( insert @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) )
= ( insert @ A @ C @ ( insert @ A @ D2 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( ( ( A3 = C )
& ( B3 = D2 ) )
| ( ( A3 = D2 )
& ( B3 = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_104_singleton__iff,axiom,
! [A: $tType,B3: A,A3: A] :
( ( member @ A @ B3 @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) )
= ( B3 = A3 ) ) ).
% singleton_iff
thf(fact_105_singletonD,axiom,
! [A: $tType,B3: A,A3: A] :
( ( member @ A @ B3 @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) )
=> ( B3 = A3 ) ) ).
% singletonD
thf(fact_106_insert__UNIV,axiom,
! [A: $tType,X: A] :
( ( insert @ A @ X @ ( top_top @ ( set @ A ) ) )
= ( top_top @ ( set @ A ) ) ) ).
% insert_UNIV
thf(fact_107_finlsts__rec__LNil__def,axiom,
! [A: $tType,B: $tType,F: ( coinductive_llist @ A ) > B,C: B,D2: A > ( coinductive_llist @ A ) > B > B] :
( ( F
= ( lList21916056377ts_rec @ B @ A @ C @ D2 ) )
=> ( ( F @ ( coinductive_LNil @ A ) )
= C ) ) ).
% finlsts_rec_LNil_def
thf(fact_108_finlsts__rec__LNil,axiom,
! [B: $tType,A: $tType,C: A,D2: B > ( coinductive_llist @ B ) > A > A] :
( ( lList21916056377ts_rec @ A @ B @ C @ D2 @ ( coinductive_LNil @ B ) )
= C ) ).
% finlsts_rec_LNil
thf(fact_109_Coinductive__List_Ofinite__lprefix__nitpick__simps_I1_J,axiom,
! [A: $tType,Xs: coinductive_llist @ A] :
( ( coindu328551480prefix @ A @ Xs @ ( coinductive_LNil @ A ) )
= ( Xs
= ( coinductive_LNil @ A ) ) ) ).
% Coinductive_List.finite_lprefix_nitpick_simps(1)
thf(fact_110_Coinductive__List_Ofinite__lprefix__nitpick__simps_I2_J,axiom,
! [A: $tType,Xs: coinductive_llist @ A] : ( coindu328551480prefix @ A @ ( coinductive_LNil @ A ) @ Xs ) ).
% Coinductive_List.finite_lprefix_nitpick_simps(2)
thf(fact_111_Topological__Spaces_OUNIV__not__singleton,axiom,
! [A: $tType] :
( ( topolo890362671_space @ A @ ( type2 @ A ) )
=> ! [X: A] :
( ( top_top @ ( set @ A ) )
!= ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Topological_Spaces.UNIV_not_singleton
thf(fact_112_the__elem__eq,axiom,
! [A: $tType,X: A] :
( ( the_elem @ A @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= X ) ).
% the_elem_eq
thf(fact_113_top__empty__eq,axiom,
! [A: $tType] :
( ( top_top @ ( A > $o ) )
= ( ^ [X4: A] : ( member @ A @ X4 @ ( top_top @ ( set @ A ) ) ) ) ) ).
% top_empty_eq
thf(fact_114_bot__empty__eq,axiom,
! [A: $tType] :
( ( bot_bot @ ( A > $o ) )
= ( ^ [X4: A] : ( member @ A @ X4 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% bot_empty_eq
thf(fact_115_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_116_is__singletonI,axiom,
! [A: $tType,X: A] : ( is_singleton @ A @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ).
% is_singletonI
thf(fact_117_is__singleton__the__elem,axiom,
! [A: $tType] :
( ( is_singleton @ A )
= ( ^ [A7: set @ A] :
( A7
= ( insert @ A @ ( the_elem @ A @ A7 ) @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% is_singleton_the_elem
thf(fact_118_is__singletonI_H,axiom,
! [A: $tType,A2: set @ A] :
( ( A2
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X3: A,Y2: A] :
( ( member @ A @ X3 @ A2 )
=> ( ( member @ A @ Y2 @ A2 )
=> ( X3 = Y2 ) ) )
=> ( is_singleton @ A @ A2 ) ) ) ).
% is_singletonI'
thf(fact_119_is__singleton__def,axiom,
! [A: $tType] :
( ( is_singleton @ A )
= ( ^ [A7: set @ A] :
? [X4: A] :
( A7
= ( insert @ A @ X4 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% is_singleton_def
thf(fact_120_is__singletonE,axiom,
! [A: $tType,A2: set @ A] :
( ( is_singleton @ A @ A2 )
=> ~ ! [X3: A] :
( A2
!= ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% is_singletonE
thf(fact_121_poslsts__def,axiom,
! [A: $tType] :
( ( lList21148268032oslsts @ A )
= ( ^ [A7: set @ A] : ( minus_minus @ ( set @ ( coinductive_llist @ A ) ) @ ( lList2435255213lllsts @ A @ A7 ) @ ( insert @ ( coinductive_llist @ A ) @ ( coinductive_LNil @ A ) @ ( bot_bot @ ( set @ ( coinductive_llist @ A ) ) ) ) ) ) ) ).
% poslsts_def
thf(fact_122_fpslsts__def,axiom,
! [A: $tType] :
( ( lList22096119349pslsts @ A )
= ( ^ [A7: set @ A] : ( minus_minus @ ( set @ ( coinductive_llist @ A ) ) @ ( lList2236698231inlsts @ A @ A7 ) @ ( insert @ ( coinductive_llist @ A ) @ ( coinductive_LNil @ A ) @ ( bot_bot @ ( set @ ( coinductive_llist @ A ) ) ) ) ) ) ) ).
% fpslsts_def
thf(fact_123_gen__lset__code_I2_J,axiom,
! [A: $tType,A2: set @ A,X: A,Xs: coinductive_llist @ A] :
( ( coinductive_gen_lset @ A @ A2 @ ( coinductive_LCons @ A @ X @ Xs ) )
= ( coinductive_gen_lset @ A @ ( insert @ A @ X @ A2 ) @ Xs ) ) ).
% gen_lset_code(2)
thf(fact_124_top__conj_I2_J,axiom,
! [A: $tType,P: $o,X: A] :
( ( P
& ( top_top @ ( A > $o ) @ X ) )
= P ) ).
% top_conj(2)
thf(fact_125_DiffI,axiom,
! [A: $tType,C: A,A2: set @ A,B2: set @ A] :
( ( member @ A @ C @ A2 )
=> ( ~ ( member @ A @ C @ B2 )
=> ( member @ A @ C @ ( minus_minus @ ( set @ A ) @ A2 @ B2 ) ) ) ) ).
% DiffI
thf(fact_126_Diff__iff,axiom,
! [A: $tType,C: A,A2: set @ A,B2: set @ A] :
( ( member @ A @ C @ ( minus_minus @ ( set @ A ) @ A2 @ B2 ) )
= ( ( member @ A @ C @ A2 )
& ~ ( member @ A @ C @ B2 ) ) ) ).
% Diff_iff
thf(fact_127_Diff__idemp,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] :
( ( minus_minus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A2 @ B2 ) @ B2 )
= ( minus_minus @ ( set @ A ) @ A2 @ B2 ) ) ).
% Diff_idemp
thf(fact_128_Diff__cancel,axiom,
! [A: $tType,A2: set @ A] :
( ( minus_minus @ ( set @ A ) @ A2 @ A2 )
= ( bot_bot @ ( set @ A ) ) ) ).
% Diff_cancel
thf(fact_129_empty__Diff,axiom,
! [A: $tType,A2: set @ A] :
( ( minus_minus @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ A2 )
= ( bot_bot @ ( set @ A ) ) ) ).
% empty_Diff
thf(fact_130_Diff__empty,axiom,
! [A: $tType,A2: set @ A] :
( ( minus_minus @ ( set @ A ) @ A2 @ ( bot_bot @ ( set @ A ) ) )
= A2 ) ).
% Diff_empty
thf(fact_131_Diff__insert0,axiom,
! [A: $tType,X: A,A2: set @ A,B2: set @ A] :
( ~ ( member @ A @ X @ A2 )
=> ( ( minus_minus @ ( set @ A ) @ A2 @ ( insert @ A @ X @ B2 ) )
= ( minus_minus @ ( set @ A ) @ A2 @ B2 ) ) ) ).
% Diff_insert0
thf(fact_132_insert__Diff1,axiom,
! [A: $tType,X: A,B2: set @ A,A2: set @ A] :
( ( member @ A @ X @ B2 )
=> ( ( minus_minus @ ( set @ A ) @ ( insert @ A @ X @ A2 ) @ B2 )
= ( minus_minus @ ( set @ A ) @ A2 @ B2 ) ) ) ).
% insert_Diff1
thf(fact_133_Diff__UNIV,axiom,
! [A: $tType,A2: set @ A] :
( ( minus_minus @ ( set @ A ) @ A2 @ ( top_top @ ( set @ A ) ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% Diff_UNIV
thf(fact_134_insert__Diff__single,axiom,
! [A: $tType,A3: A,A2: set @ A] :
( ( insert @ A @ A3 @ ( minus_minus @ ( set @ A ) @ A2 @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( insert @ A @ A3 @ A2 ) ) ).
% insert_Diff_single
thf(fact_135_insert__Diff__if,axiom,
! [A: $tType,X: A,B2: set @ A,A2: set @ A] :
( ( ( member @ A @ X @ B2 )
=> ( ( minus_minus @ ( set @ A ) @ ( insert @ A @ X @ A2 ) @ B2 )
= ( minus_minus @ ( set @ A ) @ A2 @ B2 ) ) )
& ( ~ ( member @ A @ X @ B2 )
=> ( ( minus_minus @ ( set @ A ) @ ( insert @ A @ X @ A2 ) @ B2 )
= ( insert @ A @ X @ ( minus_minus @ ( set @ A ) @ A2 @ B2 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_136_DiffE,axiom,
! [A: $tType,C: A,A2: set @ A,B2: set @ A] :
( ( member @ A @ C @ ( minus_minus @ ( set @ A ) @ A2 @ B2 ) )
=> ~ ( ( member @ A @ C @ A2 )
=> ( member @ A @ C @ B2 ) ) ) ).
% DiffE
thf(fact_137_DiffD1,axiom,
! [A: $tType,C: A,A2: set @ A,B2: set @ A] :
( ( member @ A @ C @ ( minus_minus @ ( set @ A ) @ A2 @ B2 ) )
=> ( member @ A @ C @ A2 ) ) ).
% DiffD1
thf(fact_138_DiffD2,axiom,
! [A: $tType,C: A,A2: set @ A,B2: set @ A] :
( ( member @ A @ C @ ( minus_minus @ ( set @ A ) @ A2 @ B2 ) )
=> ~ ( member @ A @ C @ B2 ) ) ).
% DiffD2
thf(fact_139_Diff__insert,axiom,
! [A: $tType,A2: set @ A,A3: A,B2: set @ A] :
( ( minus_minus @ ( set @ A ) @ A2 @ ( insert @ A @ A3 @ B2 ) )
= ( minus_minus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A2 @ B2 ) @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Diff_insert
thf(fact_140_insert__Diff,axiom,
! [A: $tType,A3: A,A2: set @ A] :
( ( member @ A @ A3 @ A2 )
=> ( ( insert @ A @ A3 @ ( minus_minus @ ( set @ A ) @ A2 @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) )
= A2 ) ) ).
% insert_Diff
thf(fact_141_Diff__insert2,axiom,
! [A: $tType,A2: set @ A,A3: A,B2: set @ A] :
( ( minus_minus @ ( set @ A ) @ A2 @ ( insert @ A @ A3 @ B2 ) )
= ( minus_minus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A2 @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) @ B2 ) ) ).
% Diff_insert2
thf(fact_142_Diff__insert__absorb,axiom,
! [A: $tType,X: A,A2: set @ A] :
( ~ ( member @ A @ X @ A2 )
=> ( ( minus_minus @ ( set @ A ) @ ( insert @ A @ X @ A2 ) @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= A2 ) ) ).
% Diff_insert_absorb
thf(fact_143_gen__lset__code_I1_J,axiom,
! [A: $tType,A2: set @ A] :
( ( coinductive_gen_lset @ A @ A2 @ ( coinductive_LNil @ A ) )
= A2 ) ).
% gen_lset_code(1)
thf(fact_144_inflsts__def,axiom,
! [A: $tType] :
( ( lList21612149805nflsts @ A )
= ( ^ [A7: set @ A] : ( minus_minus @ ( set @ ( coinductive_llist @ A ) ) @ ( lList2435255213lllsts @ A @ A7 ) @ ( lList2236698231inlsts @ A @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% inflsts_def
thf(fact_145_top__conj_I1_J,axiom,
! [A: $tType,X: A,P: $o] :
( ( ( top_top @ ( A > $o ) @ X )
& P )
= P ) ).
% top_conj(1)
thf(fact_146_remove__def,axiom,
! [A: $tType] :
( ( remove @ A )
= ( ^ [X4: A,A7: set @ A] : ( minus_minus @ ( set @ A ) @ A7 @ ( insert @ A @ X4 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% remove_def
thf(fact_147_lSup__minus__LNil,axiom,
! [A: $tType,Y3: set @ ( coinductive_llist @ A )] :
( ( coinductive_lSup @ A @ ( minus_minus @ ( set @ ( coinductive_llist @ A ) ) @ Y3 @ ( insert @ ( coinductive_llist @ A ) @ ( coinductive_LNil @ A ) @ ( bot_bot @ ( set @ ( coinductive_llist @ A ) ) ) ) ) )
= ( coinductive_lSup @ A @ Y3 ) ) ).
% lSup_minus_LNil
thf(fact_148_lstrict__prefix__code_I2_J,axiom,
! [B: $tType,Y: B,Ys: coinductive_llist @ B] : ( coindu1478340336prefix @ B @ ( coinductive_LNil @ B ) @ ( coinductive_LCons @ B @ Y @ Ys ) ) ).
% lstrict_prefix_code(2)
thf(fact_149_lstrict__prefix__code_I3_J,axiom,
! [B: $tType,X: B,Xs: coinductive_llist @ B] :
~ ( coindu1478340336prefix @ B @ ( coinductive_LCons @ B @ X @ Xs ) @ ( coinductive_LNil @ B ) ) ).
% lstrict_prefix_code(3)
thf(fact_150_member__remove,axiom,
! [A: $tType,X: A,Y: A,A2: set @ A] :
( ( member @ A @ X @ ( remove @ A @ Y @ A2 ) )
= ( ( member @ A @ X @ A2 )
& ( X != Y ) ) ) ).
% member_remove
thf(fact_151_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_152_lstrict__prefix__code_I1_J,axiom,
! [A: $tType] :
~ ( coindu1478340336prefix @ A @ ( coinductive_LNil @ A ) @ ( coinductive_LNil @ A ) ) ).
% lstrict_prefix_code(1)
thf(fact_153_lSup__empty,axiom,
! [A: $tType] :
( ( coinductive_lSup @ A @ ( bot_bot @ ( set @ ( coinductive_llist @ A ) ) ) )
= ( coinductive_LNil @ A ) ) ).
% lSup_empty
thf(fact_154_lSup__singleton,axiom,
! [A: $tType,Xs: coinductive_llist @ A] :
( ( coinductive_lSup @ A @ ( insert @ ( coinductive_llist @ A ) @ Xs @ ( bot_bot @ ( set @ ( coinductive_llist @ A ) ) ) ) )
= Xs ) ).
% lSup_singleton
thf(fact_155_llist__less__induct,axiom,
! [A: $tType,P: ( coinductive_llist @ A ) > $o,Xs: coinductive_llist @ A] :
( ! [Xs3: coinductive_llist @ A] :
( ! [Ys2: coinductive_llist @ A] :
( ( coindu1478340336prefix @ A @ Ys2 @ Xs3 )
=> ( P @ Ys2 ) )
=> ( P @ Xs3 ) )
=> ( P @ Xs ) ) ).
% llist_less_induct
thf(fact_156_lSup__insert__LNil,axiom,
! [A: $tType,Y3: set @ ( coinductive_llist @ A )] :
( ( coinductive_lSup @ A @ ( insert @ ( coinductive_llist @ A ) @ ( coinductive_LNil @ A ) @ Y3 ) )
= ( coinductive_lSup @ A @ Y3 ) ) ).
% lSup_insert_LNil
thf(fact_157_llast__singleton,axiom,
! [A: $tType,X: A] :
( ( coinductive_llast @ A @ ( coinductive_LCons @ A @ X @ ( coinductive_LNil @ A ) ) )
= X ) ).
% llast_singleton
thf(fact_158_totally__bounded__empty,axiom,
! [A: $tType] :
( ( topolo47006728_space @ A @ ( type2 @ A ) )
=> ( topolo406746546ounded @ A @ ( bot_bot @ ( set @ A ) ) ) ) ).
% totally_bounded_empty
thf(fact_159_pairwise__singleton,axiom,
! [A: $tType,P: A > A > $o,A2: A] : ( pairwise @ A @ P @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% pairwise_singleton
thf(fact_160_llast__LCons2,axiom,
! [A: $tType,X: A,Y: A,Xs: coinductive_llist @ A] :
( ( coinductive_llast @ A @ ( coinductive_LCons @ A @ X @ ( coinductive_LCons @ A @ Y @ Xs ) ) )
= ( coinductive_llast @ A @ ( coinductive_LCons @ A @ Y @ Xs ) ) ) ).
% llast_LCons2
thf(fact_161_pairwise__def,axiom,
! [A: $tType] :
( ( pairwise @ A )
= ( ^ [R2: A > A > $o,S2: set @ A] :
! [X4: A] :
( ( member @ A @ X4 @ S2 )
=> ! [Y4: A] :
( ( member @ A @ Y4 @ S2 )
=> ( ( X4 != Y4 )
=> ( R2 @ X4 @ Y4 ) ) ) ) ) ) ).
% pairwise_def
thf(fact_162_pairwise__empty,axiom,
! [A: $tType,P: A > A > $o] : ( pairwise @ A @ P @ ( bot_bot @ ( set @ A ) ) ) ).
% pairwise_empty
thf(fact_163_pairwise__insert,axiom,
! [A: $tType,R: A > A > $o,X: A,S: set @ A] :
( ( pairwise @ A @ R @ ( insert @ A @ X @ S ) )
= ( ! [Y4: A] :
( ( ( member @ A @ Y4 @ S )
& ( Y4 != X ) )
=> ( ( R @ X @ Y4 )
& ( R @ Y4 @ X ) ) )
& ( pairwise @ A @ R @ S ) ) ) ).
% pairwise_insert
thf(fact_164_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_165_fin__Int__inf,axiom,
! [A: $tType,A2: set @ A] :
( ( inf_inf @ ( set @ ( coinductive_llist @ A ) ) @ ( lList2236698231inlsts @ A @ A2 ) @ ( lList21612149805nflsts @ A @ A2 ) )
= ( bot_bot @ ( set @ ( coinductive_llist @ A ) ) ) ) ).
% fin_Int_inf
thf(fact_166_fin__Un__inf,axiom,
! [A: $tType,A2: set @ A] :
( ( sup_sup @ ( set @ ( coinductive_llist @ A ) ) @ ( lList2236698231inlsts @ A @ A2 ) @ ( lList21612149805nflsts @ A @ A2 ) )
= ( lList2435255213lllsts @ A @ A2 ) ) ).
% fin_Un_inf
thf(fact_167_Int__iff,axiom,
! [A: $tType,C: A,A2: set @ A,B2: set @ A] :
( ( member @ A @ C @ ( inf_inf @ ( set @ A ) @ A2 @ B2 ) )
= ( ( member @ A @ C @ A2 )
& ( member @ A @ C @ B2 ) ) ) ).
% Int_iff
thf(fact_168_IntI,axiom,
! [A: $tType,C: A,A2: set @ A,B2: set @ A] :
( ( member @ A @ C @ A2 )
=> ( ( member @ A @ C @ B2 )
=> ( member @ A @ C @ ( inf_inf @ ( set @ A ) @ A2 @ B2 ) ) ) ) ).
% IntI
thf(fact_169_Un__iff,axiom,
! [A: $tType,C: A,A2: set @ A,B2: set @ A] :
( ( member @ A @ C @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) )
= ( ( member @ A @ C @ A2 )
| ( member @ A @ C @ B2 ) ) ) ).
% Un_iff
thf(fact_170_UnCI,axiom,
! [A: $tType,C: A,B2: set @ A,A2: set @ A] :
( ( ~ ( member @ A @ C @ B2 )
=> ( member @ A @ C @ A2 ) )
=> ( member @ A @ C @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) ) ) ).
% UnCI
thf(fact_171_Int__UNIV,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] :
( ( ( inf_inf @ ( set @ A ) @ A2 @ B2 )
= ( top_top @ ( set @ A ) ) )
= ( ( A2
= ( top_top @ ( set @ A ) ) )
& ( B2
= ( top_top @ ( set @ A ) ) ) ) ) ).
% Int_UNIV
thf(fact_172_Un__empty,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] :
( ( ( sup_sup @ ( set @ A ) @ A2 @ B2 )
= ( bot_bot @ ( set @ A ) ) )
= ( ( A2
= ( bot_bot @ ( set @ A ) ) )
& ( B2
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Un_empty
thf(fact_173_Int__insert__left__if0,axiom,
! [A: $tType,A3: A,C4: set @ A,B2: set @ A] :
( ~ ( member @ A @ A3 @ C4 )
=> ( ( inf_inf @ ( set @ A ) @ ( insert @ A @ A3 @ B2 ) @ C4 )
= ( inf_inf @ ( set @ A ) @ B2 @ C4 ) ) ) ).
% Int_insert_left_if0
thf(fact_174_Int__insert__left__if1,axiom,
! [A: $tType,A3: A,C4: set @ A,B2: set @ A] :
( ( member @ A @ A3 @ C4 )
=> ( ( inf_inf @ ( set @ A ) @ ( insert @ A @ A3 @ B2 ) @ C4 )
= ( insert @ A @ A3 @ ( inf_inf @ ( set @ A ) @ B2 @ C4 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_175_insert__inter__insert,axiom,
! [A: $tType,A3: A,A2: set @ A,B2: set @ A] :
( ( inf_inf @ ( set @ A ) @ ( insert @ A @ A3 @ A2 ) @ ( insert @ A @ A3 @ B2 ) )
= ( insert @ A @ A3 @ ( inf_inf @ ( set @ A ) @ A2 @ B2 ) ) ) ).
% insert_inter_insert
thf(fact_176_Int__insert__right__if0,axiom,
! [A: $tType,A3: A,A2: set @ A,B2: set @ A] :
( ~ ( member @ A @ A3 @ A2 )
=> ( ( inf_inf @ ( set @ A ) @ A2 @ ( insert @ A @ A3 @ B2 ) )
= ( inf_inf @ ( set @ A ) @ A2 @ B2 ) ) ) ).
% Int_insert_right_if0
thf(fact_177_Int__insert__right__if1,axiom,
! [A: $tType,A3: A,A2: set @ A,B2: set @ A] :
( ( member @ A @ A3 @ A2 )
=> ( ( inf_inf @ ( set @ A ) @ A2 @ ( insert @ A @ A3 @ B2 ) )
= ( insert @ A @ A3 @ ( inf_inf @ ( set @ A ) @ A2 @ B2 ) ) ) ) ).
% Int_insert_right_if1
thf(fact_178_Un__insert__left,axiom,
! [A: $tType,A3: A,B2: set @ A,C4: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( insert @ A @ A3 @ B2 ) @ C4 )
= ( insert @ A @ A3 @ ( sup_sup @ ( set @ A ) @ B2 @ C4 ) ) ) ).
% Un_insert_left
thf(fact_179_Un__insert__right,axiom,
! [A: $tType,A2: set @ A,A3: A,B2: set @ A] :
( ( sup_sup @ ( set @ A ) @ A2 @ ( insert @ A @ A3 @ B2 ) )
= ( insert @ A @ A3 @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) ) ) ).
% Un_insert_right
thf(fact_180_Un__Diff__cancel,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] :
( ( sup_sup @ ( set @ A ) @ A2 @ ( minus_minus @ ( set @ A ) @ B2 @ A2 ) )
= ( sup_sup @ ( set @ A ) @ A2 @ B2 ) ) ).
% Un_Diff_cancel
thf(fact_181_Un__Diff__cancel2,axiom,
! [A: $tType,B2: set @ A,A2: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ B2 @ A2 ) @ A2 )
= ( sup_sup @ ( set @ A ) @ B2 @ A2 ) ) ).
% Un_Diff_cancel2
thf(fact_182_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_183_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_184_lfinite__code_I1_J,axiom,
! [A: $tType] : ( coinductive_lfinite @ A @ ( coinductive_LNil @ A ) ) ).
% lfinite_code(1)
thf(fact_185_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_186_insert__disjoint_I1_J,axiom,
! [A: $tType,A3: A,A2: set @ A,B2: set @ A] :
( ( ( inf_inf @ ( set @ A ) @ ( insert @ A @ A3 @ A2 ) @ B2 )
= ( bot_bot @ ( set @ A ) ) )
= ( ~ ( member @ A @ A3 @ B2 )
& ( ( inf_inf @ ( set @ A ) @ A2 @ B2 )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% insert_disjoint(1)
thf(fact_187_insert__disjoint_I2_J,axiom,
! [A: $tType,A3: A,A2: set @ A,B2: set @ A] :
( ( ( bot_bot @ ( set @ A ) )
= ( inf_inf @ ( set @ A ) @ ( insert @ A @ A3 @ A2 ) @ B2 ) )
= ( ~ ( member @ A @ A3 @ B2 )
& ( ( bot_bot @ ( set @ A ) )
= ( inf_inf @ ( set @ A ) @ A2 @ B2 ) ) ) ) ).
% insert_disjoint(2)
thf(fact_188_disjoint__insert_I1_J,axiom,
! [A: $tType,B2: set @ A,A3: A,A2: set @ A] :
( ( ( inf_inf @ ( set @ A ) @ B2 @ ( insert @ A @ A3 @ A2 ) )
= ( bot_bot @ ( set @ A ) ) )
= ( ~ ( member @ A @ A3 @ B2 )
& ( ( inf_inf @ ( set @ A ) @ B2 @ A2 )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% disjoint_insert(1)
thf(fact_189_disjoint__insert_I2_J,axiom,
! [A: $tType,A2: set @ A,B3: A,B2: set @ A] :
( ( ( bot_bot @ ( set @ A ) )
= ( inf_inf @ ( set @ A ) @ A2 @ ( insert @ A @ B3 @ B2 ) ) )
= ( ~ ( member @ A @ B3 @ A2 )
& ( ( bot_bot @ ( set @ A ) )
= ( inf_inf @ ( set @ A ) @ A2 @ B2 ) ) ) ) ).
% disjoint_insert(2)
thf(fact_190_Diff__disjoint,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] :
( ( inf_inf @ ( set @ A ) @ A2 @ ( minus_minus @ ( set @ A ) @ B2 @ A2 ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% Diff_disjoint
thf(fact_191_Int__left__commute,axiom,
! [A: $tType,A2: set @ A,B2: set @ A,C4: set @ A] :
( ( inf_inf @ ( set @ A ) @ A2 @ ( inf_inf @ ( set @ A ) @ B2 @ C4 ) )
= ( inf_inf @ ( set @ A ) @ B2 @ ( inf_inf @ ( set @ A ) @ A2 @ C4 ) ) ) ).
% Int_left_commute
thf(fact_192_Un__left__commute,axiom,
! [A: $tType,A2: set @ A,B2: set @ A,C4: set @ A] :
( ( sup_sup @ ( set @ A ) @ A2 @ ( sup_sup @ ( set @ A ) @ B2 @ C4 ) )
= ( sup_sup @ ( set @ A ) @ B2 @ ( sup_sup @ ( set @ A ) @ A2 @ C4 ) ) ) ).
% Un_left_commute
thf(fact_193_Un__Int__distrib2,axiom,
! [A: $tType,B2: set @ A,C4: set @ A,A2: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ B2 @ C4 ) @ A2 )
= ( inf_inf @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ B2 @ A2 ) @ ( sup_sup @ ( set @ A ) @ C4 @ A2 ) ) ) ).
% Un_Int_distrib2
thf(fact_194_Int__left__absorb,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] :
( ( inf_inf @ ( set @ A ) @ A2 @ ( inf_inf @ ( set @ A ) @ A2 @ B2 ) )
= ( inf_inf @ ( set @ A ) @ A2 @ B2 ) ) ).
% Int_left_absorb
thf(fact_195_Int__Un__distrib2,axiom,
! [A: $tType,B2: set @ A,C4: set @ A,A2: set @ A] :
( ( inf_inf @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ B2 @ C4 ) @ A2 )
= ( sup_sup @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ B2 @ A2 ) @ ( inf_inf @ ( set @ A ) @ C4 @ A2 ) ) ) ).
% Int_Un_distrib2
thf(fact_196_Un__left__absorb,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] :
( ( sup_sup @ ( set @ A ) @ A2 @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) )
= ( sup_sup @ ( set @ A ) @ A2 @ B2 ) ) ).
% Un_left_absorb
thf(fact_197_Un__Int__distrib,axiom,
! [A: $tType,A2: set @ A,B2: set @ A,C4: set @ A] :
( ( sup_sup @ ( set @ A ) @ A2 @ ( inf_inf @ ( set @ A ) @ B2 @ C4 ) )
= ( inf_inf @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) @ ( sup_sup @ ( set @ A ) @ A2 @ C4 ) ) ) ).
% Un_Int_distrib
thf(fact_198_Int__Un__distrib,axiom,
! [A: $tType,A2: set @ A,B2: set @ A,C4: set @ A] :
( ( inf_inf @ ( set @ A ) @ A2 @ ( sup_sup @ ( set @ A ) @ B2 @ C4 ) )
= ( sup_sup @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A2 @ B2 ) @ ( inf_inf @ ( set @ A ) @ A2 @ C4 ) ) ) ).
% Int_Un_distrib
thf(fact_199_Un__Int__crazy,axiom,
! [A: $tType,A2: set @ A,B2: set @ A,C4: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A2 @ B2 ) @ ( inf_inf @ ( set @ A ) @ B2 @ C4 ) ) @ ( inf_inf @ ( set @ A ) @ C4 @ A2 ) )
= ( inf_inf @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) @ ( sup_sup @ ( set @ A ) @ B2 @ C4 ) ) @ ( sup_sup @ ( set @ A ) @ C4 @ A2 ) ) ) ).
% Un_Int_crazy
thf(fact_200_Int__commute,axiom,
! [A: $tType] :
( ( inf_inf @ ( set @ A ) )
= ( ^ [A7: set @ A,B5: set @ A] : ( inf_inf @ ( set @ A ) @ B5 @ A7 ) ) ) ).
% Int_commute
thf(fact_201_Un__commute,axiom,
! [A: $tType] :
( ( sup_sup @ ( set @ A ) )
= ( ^ [A7: set @ A,B5: set @ A] : ( sup_sup @ ( set @ A ) @ B5 @ A7 ) ) ) ).
% Un_commute
thf(fact_202_Int__absorb,axiom,
! [A: $tType,A2: set @ A] :
( ( inf_inf @ ( set @ A ) @ A2 @ A2 )
= A2 ) ).
% Int_absorb
thf(fact_203_Un__absorb,axiom,
! [A: $tType,A2: set @ A] :
( ( sup_sup @ ( set @ A ) @ A2 @ A2 )
= A2 ) ).
% Un_absorb
thf(fact_204_Int__assoc,axiom,
! [A: $tType,A2: set @ A,B2: set @ A,C4: set @ A] :
( ( inf_inf @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A2 @ B2 ) @ C4 )
= ( inf_inf @ ( set @ A ) @ A2 @ ( inf_inf @ ( set @ A ) @ B2 @ C4 ) ) ) ).
% Int_assoc
thf(fact_205_Un__assoc,axiom,
! [A: $tType,A2: set @ A,B2: set @ A,C4: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) @ C4 )
= ( sup_sup @ ( set @ A ) @ A2 @ ( sup_sup @ ( set @ A ) @ B2 @ C4 ) ) ) ).
% Un_assoc
thf(fact_206_ball__Un,axiom,
! [A: $tType,A2: set @ A,B2: set @ A,P: A > $o] :
( ( ! [X4: A] :
( ( member @ A @ X4 @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) )
=> ( P @ X4 ) ) )
= ( ! [X4: A] :
( ( member @ A @ X4 @ A2 )
=> ( P @ X4 ) )
& ! [X4: A] :
( ( member @ A @ X4 @ B2 )
=> ( P @ X4 ) ) ) ) ).
% ball_Un
thf(fact_207_bex__Un,axiom,
! [A: $tType,A2: set @ A,B2: set @ A,P: A > $o] :
( ( ? [X4: A] :
( ( member @ A @ X4 @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) )
& ( P @ X4 ) ) )
= ( ? [X4: A] :
( ( member @ A @ X4 @ A2 )
& ( P @ X4 ) )
| ? [X4: A] :
( ( member @ A @ X4 @ B2 )
& ( P @ X4 ) ) ) ) ).
% bex_Un
thf(fact_208_IntD2,axiom,
! [A: $tType,C: A,A2: set @ A,B2: set @ A] :
( ( member @ A @ C @ ( inf_inf @ ( set @ A ) @ A2 @ B2 ) )
=> ( member @ A @ C @ B2 ) ) ).
% IntD2
thf(fact_209_IntD1,axiom,
! [A: $tType,C: A,A2: set @ A,B2: set @ A] :
( ( member @ A @ C @ ( inf_inf @ ( set @ A ) @ A2 @ B2 ) )
=> ( member @ A @ C @ A2 ) ) ).
% IntD1
thf(fact_210_UnI2,axiom,
! [A: $tType,C: A,B2: set @ A,A2: set @ A] :
( ( member @ A @ C @ B2 )
=> ( member @ A @ C @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) ) ) ).
% UnI2
thf(fact_211_UnI1,axiom,
! [A: $tType,C: A,A2: set @ A,B2: set @ A] :
( ( member @ A @ C @ A2 )
=> ( member @ A @ C @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) ) ) ).
% UnI1
thf(fact_212_IntE,axiom,
! [A: $tType,C: A,A2: set @ A,B2: set @ A] :
( ( member @ A @ C @ ( inf_inf @ ( set @ A ) @ A2 @ B2 ) )
=> ~ ( ( member @ A @ C @ A2 )
=> ~ ( member @ A @ C @ B2 ) ) ) ).
% IntE
thf(fact_213_UnE,axiom,
! [A: $tType,C: A,A2: set @ A,B2: set @ A] :
( ( member @ A @ C @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) )
=> ( ~ ( member @ A @ C @ A2 )
=> ( member @ A @ C @ B2 ) ) ) ).
% UnE
thf(fact_214_Int__insert__left,axiom,
! [A: $tType,A3: A,C4: set @ A,B2: set @ A] :
( ( ( member @ A @ A3 @ C4 )
=> ( ( inf_inf @ ( set @ A ) @ ( insert @ A @ A3 @ B2 ) @ C4 )
= ( insert @ A @ A3 @ ( inf_inf @ ( set @ A ) @ B2 @ C4 ) ) ) )
& ( ~ ( member @ A @ A3 @ C4 )
=> ( ( inf_inf @ ( set @ A ) @ ( insert @ A @ A3 @ B2 ) @ C4 )
= ( inf_inf @ ( set @ A ) @ B2 @ C4 ) ) ) ) ).
% Int_insert_left
thf(fact_215_Int__insert__right,axiom,
! [A: $tType,A3: A,A2: set @ A,B2: set @ A] :
( ( ( member @ A @ A3 @ A2 )
=> ( ( inf_inf @ ( set @ A ) @ A2 @ ( insert @ A @ A3 @ B2 ) )
= ( insert @ A @ A3 @ ( inf_inf @ ( set @ A ) @ A2 @ B2 ) ) ) )
& ( ~ ( member @ A @ A3 @ A2 )
=> ( ( inf_inf @ ( set @ A ) @ A2 @ ( insert @ A @ A3 @ B2 ) )
= ( inf_inf @ ( set @ A ) @ A2 @ B2 ) ) ) ) ).
% Int_insert_right
thf(fact_216_Int__UNIV__right,axiom,
! [A: $tType,A2: set @ A] :
( ( inf_inf @ ( set @ A ) @ A2 @ ( top_top @ ( set @ A ) ) )
= A2 ) ).
% Int_UNIV_right
thf(fact_217_Int__UNIV__left,axiom,
! [A: $tType,B2: set @ A] :
( ( inf_inf @ ( set @ A ) @ ( top_top @ ( set @ A ) ) @ B2 )
= B2 ) ).
% Int_UNIV_left
thf(fact_218_disjoint__iff__not__equal,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] :
( ( ( inf_inf @ ( set @ A ) @ A2 @ B2 )
= ( bot_bot @ ( set @ A ) ) )
= ( ! [X4: A] :
( ( member @ A @ X4 @ A2 )
=> ! [Y4: A] :
( ( member @ A @ Y4 @ B2 )
=> ( X4 != Y4 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_219_Int__empty__right,axiom,
! [A: $tType,A2: set @ A] :
( ( inf_inf @ ( set @ A ) @ A2 @ ( bot_bot @ ( set @ A ) ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% Int_empty_right
thf(fact_220_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_221_Int__emptyI,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] :
( ! [X3: A] :
( ( member @ A @ X3 @ A2 )
=> ~ ( member @ A @ X3 @ B2 ) )
=> ( ( inf_inf @ ( set @ A ) @ A2 @ B2 )
= ( bot_bot @ ( set @ A ) ) ) ) ).
% Int_emptyI
thf(fact_222_Un__UNIV__right,axiom,
! [A: $tType,A2: set @ A] :
( ( sup_sup @ ( set @ A ) @ A2 @ ( top_top @ ( set @ A ) ) )
= ( top_top @ ( set @ A ) ) ) ).
% Un_UNIV_right
thf(fact_223_Un__UNIV__left,axiom,
! [A: $tType,B2: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( top_top @ ( set @ A ) ) @ B2 )
= ( top_top @ ( set @ A ) ) ) ).
% Un_UNIV_left
thf(fact_224_Un__empty__right,axiom,
! [A: $tType,A2: set @ A] :
( ( sup_sup @ ( set @ A ) @ A2 @ ( bot_bot @ ( set @ A ) ) )
= A2 ) ).
% Un_empty_right
thf(fact_225_Un__empty__left,axiom,
! [A: $tType,B2: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ B2 )
= B2 ) ).
% Un_empty_left
thf(fact_226_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_227_lfinite__LNil,axiom,
! [A: $tType] : ( coinductive_lfinite @ A @ ( coinductive_LNil @ A ) ) ).
% lfinite_LNil
thf(fact_228_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_229_lfinite__lSupD,axiom,
! [A: $tType,A2: set @ ( coinductive_llist @ A )] :
( ( coinductive_lfinite @ A @ ( coinductive_lSup @ A @ A2 ) )
=> ! [X5: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ X5 @ A2 )
=> ( coinductive_lfinite @ A @ X5 ) ) ) ).
% lfinite_lSupD
thf(fact_230_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_231_Diff__Int__distrib2,axiom,
! [A: $tType,A2: set @ A,B2: set @ A,C4: set @ A] :
( ( inf_inf @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A2 @ B2 ) @ C4 )
= ( minus_minus @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A2 @ C4 ) @ ( inf_inf @ ( set @ A ) @ B2 @ C4 ) ) ) ).
% Diff_Int_distrib2
thf(fact_232_Diff__Int__distrib,axiom,
! [A: $tType,C4: set @ A,A2: set @ A,B2: set @ A] :
( ( inf_inf @ ( set @ A ) @ C4 @ ( minus_minus @ ( set @ A ) @ A2 @ B2 ) )
= ( minus_minus @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ C4 @ A2 ) @ ( inf_inf @ ( set @ A ) @ C4 @ B2 ) ) ) ).
% Diff_Int_distrib
thf(fact_233_Diff__Diff__Int,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] :
( ( minus_minus @ ( set @ A ) @ A2 @ ( minus_minus @ ( set @ A ) @ A2 @ B2 ) )
= ( inf_inf @ ( set @ A ) @ A2 @ B2 ) ) ).
% Diff_Diff_Int
thf(fact_234_Un__Diff__Int,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A2 @ B2 ) @ ( inf_inf @ ( set @ A ) @ A2 @ B2 ) )
= A2 ) ).
% Un_Diff_Int
thf(fact_235_Diff__Int2,axiom,
! [A: $tType,A2: set @ A,C4: set @ A,B2: set @ A] :
( ( minus_minus @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A2 @ C4 ) @ ( inf_inf @ ( set @ A ) @ B2 @ C4 ) )
= ( minus_minus @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A2 @ C4 ) @ B2 ) ) ).
% Diff_Int2
thf(fact_236_Int__Diff,axiom,
! [A: $tType,A2: set @ A,B2: set @ A,C4: set @ A] :
( ( minus_minus @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A2 @ B2 ) @ C4 )
= ( inf_inf @ ( set @ A ) @ A2 @ ( minus_minus @ ( set @ A ) @ B2 @ C4 ) ) ) ).
% Int_Diff
thf(fact_237_Diff__Int,axiom,
! [A: $tType,A2: set @ A,B2: set @ A,C4: set @ A] :
( ( minus_minus @ ( set @ A ) @ A2 @ ( inf_inf @ ( set @ A ) @ B2 @ C4 ) )
= ( sup_sup @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A2 @ B2 ) @ ( minus_minus @ ( set @ A ) @ A2 @ C4 ) ) ) ).
% Diff_Int
thf(fact_238_Un__Diff,axiom,
! [A: $tType,A2: set @ A,B2: set @ A,C4: set @ A] :
( ( minus_minus @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) @ C4 )
= ( sup_sup @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A2 @ C4 ) @ ( minus_minus @ ( set @ A ) @ B2 @ C4 ) ) ) ).
% Un_Diff
thf(fact_239_Diff__Un,axiom,
! [A: $tType,A2: set @ A,B2: set @ A,C4: set @ A] :
( ( minus_minus @ ( set @ A ) @ A2 @ ( sup_sup @ ( set @ A ) @ B2 @ C4 ) )
= ( inf_inf @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A2 @ B2 ) @ ( minus_minus @ ( set @ A ) @ A2 @ C4 ) ) ) ).
% Diff_Un
thf(fact_240_lfinite_Ocases,axiom,
! [A: $tType,A3: coinductive_llist @ A] :
( ( coinductive_lfinite @ A @ A3 )
=> ( ( A3
!= ( coinductive_LNil @ A ) )
=> ~ ! [Xs3: coinductive_llist @ A] :
( ? [X3: A] :
( A3
= ( coinductive_LCons @ A @ X3 @ Xs3 ) )
=> ~ ( coinductive_lfinite @ A @ Xs3 ) ) ) ) ).
% lfinite.cases
thf(fact_241_lfinite_Osimps,axiom,
! [A: $tType] :
( ( coinductive_lfinite @ A )
= ( ^ [A5: coinductive_llist @ A] :
( ( A5
= ( coinductive_LNil @ A ) )
| ? [Xs4: coinductive_llist @ A,X4: A] :
( ( A5
= ( coinductive_LCons @ A @ X4 @ Xs4 ) )
& ( coinductive_lfinite @ A @ Xs4 ) ) ) ) ) ).
% lfinite.simps
thf(fact_242_lfinite_Oinducts,axiom,
! [A: $tType,X: coinductive_llist @ A,P: ( coinductive_llist @ A ) > $o] :
( ( coinductive_lfinite @ A @ X )
=> ( ( P @ ( coinductive_LNil @ A ) )
=> ( ! [Xs3: coinductive_llist @ A,X3: A] :
( ( coinductive_lfinite @ A @ Xs3 )
=> ( ( P @ Xs3 )
=> ( P @ ( coinductive_LCons @ A @ X3 @ Xs3 ) ) ) )
=> ( P @ X ) ) ) ) ).
% lfinite.inducts
thf(fact_243_singleton__Un__iff,axiom,
! [A: $tType,X: A,A2: set @ A,B2: set @ A] :
( ( ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) )
= ( sup_sup @ ( set @ A ) @ A2 @ B2 ) )
= ( ( ( A2
= ( bot_bot @ ( set @ A ) ) )
& ( B2
= ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) )
| ( ( A2
= ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
& ( B2
= ( bot_bot @ ( set @ A ) ) ) )
| ( ( A2
= ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
& ( B2
= ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_244_Un__singleton__iff,axiom,
! [A: $tType,A2: set @ A,B2: set @ A,X: A] :
( ( ( sup_sup @ ( set @ A ) @ A2 @ B2 )
= ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= ( ( ( A2
= ( bot_bot @ ( set @ A ) ) )
& ( B2
= ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) )
| ( ( A2
= ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
& ( B2
= ( bot_bot @ ( set @ A ) ) ) )
| ( ( A2
= ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
& ( B2
= ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_245_insert__is__Un,axiom,
! [A: $tType] :
( ( insert @ A )
= ( ^ [A5: A] : ( sup_sup @ ( set @ A ) @ ( insert @ A @ A5 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% insert_is_Un
thf(fact_246_Diff__triv,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] :
( ( ( inf_inf @ ( set @ A ) @ A2 @ B2 )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( minus_minus @ ( set @ A ) @ A2 @ B2 )
= A2 ) ) ).
% Diff_triv
thf(fact_247_lfinite__rev__induct,axiom,
! [A: $tType,Xs: coinductive_llist @ A,P: ( coinductive_llist @ A ) > $o] :
( ( coinductive_lfinite @ A @ Xs )
=> ( ( P @ ( coinductive_LNil @ A ) )
=> ( ! [X3: A,Xs3: coinductive_llist @ A] :
( ( coinductive_lfinite @ A @ Xs3 )
=> ( ( P @ Xs3 )
=> ( P @ ( coinductive_lappend @ A @ Xs3 @ ( coinductive_LCons @ A @ X3 @ ( coinductive_LNil @ A ) ) ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% lfinite_rev_induct
thf(fact_248_llimit__induct,axiom,
! [A: $tType,P: ( coinductive_llist @ A ) > $o,Xs: coinductive_llist @ A] :
( ( P @ ( coinductive_LNil @ A ) )
=> ( ! [X3: A,Xs3: coinductive_llist @ A] :
( ( coinductive_lfinite @ A @ Xs3 )
=> ( ( P @ Xs3 )
=> ( P @ ( coinductive_LCons @ A @ X3 @ Xs3 ) ) ) )
=> ( ( ! [Ys2: coinductive_llist @ A] :
( ( coindu1478340336prefix @ A @ Ys2 @ Xs )
=> ( P @ Ys2 ) )
=> ( P @ Xs ) )
=> ( P @ Xs ) ) ) ) ).
% llimit_induct
thf(fact_249_sup__top__right,axiom,
! [A: $tType] :
( ( bounded_lattice_top @ A @ ( type2 @ A ) )
=> ! [X: A] :
( ( sup_sup @ A @ X @ ( top_top @ A ) )
= ( top_top @ A ) ) ) ).
% sup_top_right
thf(fact_250_sup__top__left,axiom,
! [A: $tType] :
( ( bounded_lattice_top @ A @ ( type2 @ A ) )
=> ! [X: A] :
( ( sup_sup @ A @ ( top_top @ A ) @ X )
= ( top_top @ A ) ) ) ).
% sup_top_left
thf(fact_251_sup__bot_Oright__neutral,axiom,
! [A: $tType] :
( ( bounde1808546759up_bot @ A @ ( type2 @ A ) )
=> ! [A3: A] :
( ( sup_sup @ A @ A3 @ ( bot_bot @ A ) )
= A3 ) ) ).
% sup_bot.right_neutral
thf(fact_252_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_253_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_254_inf__eq__top__iff,axiom,
! [A: $tType] :
( ( bounded_lattice_top @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( ( inf_inf @ A @ X @ Y )
= ( top_top @ A ) )
= ( ( X
= ( top_top @ A ) )
& ( Y
= ( top_top @ A ) ) ) ) ) ).
% inf_eq_top_iff
thf(fact_255_inf__top_Oleft__neutral,axiom,
! [A: $tType] :
( ( bounde1561333602nf_top @ A @ ( type2 @ A ) )
=> ! [A3: A] :
( ( inf_inf @ A @ ( top_top @ A ) @ A3 )
= A3 ) ) ).
% inf_top.left_neutral
%----Type constructors (21)
thf(tcon_HOL_Obool___Lattices_Obounded__lattice,axiom,
bounded_lattice @ $o @ ( type2 @ $o ) ).
thf(tcon_Set_Oset___Lattices_Obounded__lattice_1,axiom,
! [A8: $tType] : ( bounded_lattice @ ( set @ A8 ) @ ( type2 @ ( set @ A8 ) ) ) ).
thf(tcon_fun___Lattices_Obounded__lattice_2,axiom,
! [A8: $tType,A9: $tType] :
( ( bounded_lattice @ A9 @ ( type2 @ A9 ) )
=> ( bounded_lattice @ ( A8 > A9 ) @ ( type2 @ ( A8 > A9 ) ) ) ) ).
thf(tcon_fun___Lattices_Obounded__semilattice__sup__bot,axiom,
! [A8: $tType,A9: $tType] :
( ( bounded_lattice @ A9 @ ( type2 @ A9 ) )
=> ( bounde1808546759up_bot @ ( A8 > A9 ) @ ( type2 @ ( A8 > A9 ) ) ) ) ).
thf(tcon_fun___Lattices_Obounded__semilattice__inf__top,axiom,
! [A8: $tType,A9: $tType] :
( ( bounded_lattice @ A9 @ ( type2 @ A9 ) )
=> ( bounde1561333602nf_top @ ( A8 > A9 ) @ ( type2 @ ( A8 > A9 ) ) ) ) ).
thf(tcon_fun___Lattices_Obounded__lattice__top,axiom,
! [A8: $tType,A9: $tType] :
( ( bounded_lattice @ A9 @ ( type2 @ A9 ) )
=> ( bounded_lattice_top @ ( A8 > A9 ) @ ( type2 @ ( A8 > A9 ) ) ) ) ).
thf(tcon_fun___Lattices_Obounded__lattice__bot,axiom,
! [A8: $tType,A9: $tType] :
( ( bounded_lattice @ A9 @ ( type2 @ A9 ) )
=> ( bounded_lattice_bot @ ( A8 > A9 ) @ ( type2 @ ( A8 > A9 ) ) ) ) ).
thf(tcon_fun___Orderings_Otop,axiom,
! [A8: $tType,A9: $tType] :
( ( top @ A9 @ ( type2 @ A9 ) )
=> ( top @ ( A8 > A9 ) @ ( type2 @ ( A8 > A9 ) ) ) ) ).
thf(tcon_fun___Orderings_Obot,axiom,
! [A8: $tType,A9: $tType] :
( ( bot @ A9 @ ( type2 @ A9 ) )
=> ( bot @ ( A8 > A9 ) @ ( type2 @ ( A8 > A9 ) ) ) ) ).
thf(tcon_Set_Oset___Lattices_Obounded__semilattice__sup__bot_3,axiom,
! [A8: $tType] : ( bounde1808546759up_bot @ ( set @ A8 ) @ ( type2 @ ( set @ A8 ) ) ) ).
thf(tcon_Set_Oset___Lattices_Obounded__semilattice__inf__top_4,axiom,
! [A8: $tType] : ( bounde1561333602nf_top @ ( set @ A8 ) @ ( type2 @ ( set @ A8 ) ) ) ).
thf(tcon_Set_Oset___Lattices_Obounded__lattice__top_5,axiom,
! [A8: $tType] : ( bounded_lattice_top @ ( set @ A8 ) @ ( type2 @ ( set @ A8 ) ) ) ).
thf(tcon_Set_Oset___Lattices_Obounded__lattice__bot_6,axiom,
! [A8: $tType] : ( bounded_lattice_bot @ ( set @ A8 ) @ ( type2 @ ( set @ A8 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Otop_7,axiom,
! [A8: $tType] : ( top @ ( set @ A8 ) @ ( type2 @ ( set @ A8 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Obot_8,axiom,
! [A8: $tType] : ( bot @ ( set @ A8 ) @ ( type2 @ ( set @ A8 ) ) ) ).
thf(tcon_HOL_Obool___Lattices_Obounded__semilattice__sup__bot_9,axiom,
bounde1808546759up_bot @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Lattices_Obounded__semilattice__inf__top_10,axiom,
bounde1561333602nf_top @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Lattices_Obounded__lattice__top_11,axiom,
bounded_lattice_top @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Lattices_Obounded__lattice__bot_12,axiom,
bounded_lattice_bot @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Otop_13,axiom,
top @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Obot_14,axiom,
bot @ $o @ ( type2 @ $o ) ).
%----Conjectures (1)
thf(conj_0,conjecture,
member @ ( coinductive_llist @ a ) @ sa @ ( lList21612149805nflsts @ a @ a2 ) ).
%------------------------------------------------------------------------------