TPTP Problem File: DAT188^1.p
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%------------------------------------------------------------------------------
% File : DAT188^1 : TPTP v8.2.0. Released v7.0.0.
% Domain : Data Structures
% Problem : Lazy lists II 1288
% 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__1288.p [Bla16]
% Status : Theorem
% Rating : 0.67 v8.1.0, 0.50 v7.5.0, 0.33 v7.2.0, 0.50 v7.1.0
% Syntax : Number of formulae : 372 ( 123 unt; 68 typ; 0 def)
% Number of atoms : 791 ( 221 equ; 0 cnn)
% Maximal formula atoms : 7 ( 2 avg)
% Number of connectives : 4634 ( 76 ~; 14 |; 52 &;4137 @)
% ( 0 <=>; 355 =>; 0 <=; 0 <~>)
% Maximal formula depth : 19 ( 8 avg)
% Number of types : 3 ( 2 usr)
% Number of type conns : 156 ( 156 >; 0 *; 0 +; 0 <<)
% Number of symbols : 67 ( 66 usr; 4 con; 0-5 aty)
% Number of variables : 1043 ( 39 ^; 914 !; 31 ?;1043 :)
% ( 59 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2016-07-13 14:52:31.804
%------------------------------------------------------------------------------
%----Could-be-implicit typings (5)
thf(ty_t_Coinductive__List_Ollist,type,
coinductive_llist: $tType > $tType ).
thf(ty_t_Set_Oset,type,
set: $tType > $tType ).
thf(ty_t_Nat_Onat,type,
nat: $tType ).
thf(ty_t_itself,type,
itself: $tType > $tType ).
thf(ty_tf_a,type,
a: $tType ).
%----Explicit typings (63)
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_Groups_Ominus,type,
minus:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Oord,type,
ord:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Otop,type,
top:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Oorder,type,
order:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Ogroup__add,type,
group_add:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Lattices_Olattice,type,
lattice:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Ono__bot,type,
no_bot:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Ono__top,type,
no_top:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Olinorder,type,
linorder:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Opreorder,type,
preorder:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Oorder__top,type,
order_top:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Owellorder,type,
wellorder:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Odense__order,type,
dense_order:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Lattices_Osemilattice__sup,type,
semilattice_sup:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Oordered__ab__group__add,type,
ordered_ab_group_add:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Lattices_Obounded__lattice__top,type,
bounded_lattice_top:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Ocancel__ab__semigroup__add,type,
cancel146912293up_add:
!>[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_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_Ofinpref,type,
lList21202317876inpref:
!>[A: $tType] : ( ( set @ A ) > ( coinductive_llist @ A ) > ( set @ ( coinductive_llist @ A ) ) ) ).
thf(sy_c_LList2__Mirabelle__hamjzmohle_Ofpslsts,type,
lList22096119349pslsts:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( coinductive_llist @ A ) ) ) ).
thf(sy_c_LList2__Mirabelle__hamjzmohle_Oinfliveness,type,
lList21015763786veness:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( coinductive_llist @ A ) ) > $o ) ).
thf(sy_c_LList2__Mirabelle__hamjzmohle_Oinflsts,type,
lList21612149805nflsts:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( coinductive_llist @ A ) ) ) ).
thf(sy_c_LList2__Mirabelle__hamjzmohle_Oinfsafety,type,
lList21015939545safety:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( coinductive_llist @ A ) ) > $o ) ).
thf(sy_c_LList2__Mirabelle__hamjzmohle_Oinfsuff,type,
lList2649413865nfsuff:
!>[A: $tType] : ( ( set @ A ) > ( coinductive_llist @ A ) > ( set @ ( coinductive_llist @ A ) ) ) ).
thf(sy_c_LList2__Mirabelle__hamjzmohle_Olbutlast,type,
lList2370560421utlast:
!>[A: $tType] : ( ( coinductive_llist @ A ) > ( coinductive_llist @ A ) ) ).
thf(sy_c_LList2__Mirabelle__hamjzmohle_Oldrop,type,
lList2508575361_ldrop:
!>[A: $tType] : ( ( coinductive_llist @ A ) > nat > ( coinductive_llist @ A ) ) ).
thf(sy_c_LList2__Mirabelle__hamjzmohle_Oliveness,type,
lList21805353693veness:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( coinductive_llist @ A ) ) > $o ) ).
thf(sy_c_LList2__Mirabelle__hamjzmohle_Ollast,type,
lList2170638824_llast:
!>[A: $tType] : ( ( coinductive_llist @ A ) > A ) ).
thf(sy_c_LList2__Mirabelle__hamjzmohle_Ollength,type,
lList21232602520length:
!>[A: $tType] : ( ( coinductive_llist @ A ) > nat ) ).
thf(sy_c_LList2__Mirabelle__hamjzmohle_Olrev,type,
lList2281150353e_lrev:
!>[A: $tType] : ( ( coinductive_llist @ A ) > ( coinductive_llist @ A ) ) ).
thf(sy_c_LList2__Mirabelle__hamjzmohle_Oltake,type,
lList22119844313_ltake:
!>[A: $tType] : ( ( coinductive_llist @ A ) > nat > ( coinductive_llist @ A ) ) ).
thf(sy_c_LList2__Mirabelle__hamjzmohle_Opfinpref,type,
lList2467029176inpref:
!>[A: $tType] : ( ( set @ A ) > ( coinductive_llist @ A ) > ( set @ ( coinductive_llist @ A ) ) ) ).
thf(sy_c_LList2__Mirabelle__hamjzmohle_Oposliveness,type,
lList21952340509veness:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( coinductive_llist @ A ) ) > $o ) ).
thf(sy_c_LList2__Mirabelle__hamjzmohle_Oposlsts,type,
lList21148268032oslsts:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( coinductive_llist @ A ) ) ) ).
thf(sy_c_LList2__Mirabelle__hamjzmohle_Opossafety,type,
lList292406316safety:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( coinductive_llist @ A ) ) > $o ) ).
thf(sy_c_LList2__Mirabelle__hamjzmohle_Opprefix__closed,type,
lList21974196564closed:
!>[A: $tType] : ( ( set @ ( coinductive_llist @ A ) ) > $o ) ).
thf(sy_c_LList2__Mirabelle__hamjzmohle_Oprefix__closed,type,
lList21638733016closed:
!>[A: $tType] : ( ( set @ ( coinductive_llist @ A ) ) > $o ) ).
thf(sy_c_LList2__Mirabelle__hamjzmohle_Osafety,type,
lList21350011628safety:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( coinductive_llist @ A ) ) > $o ) ).
thf(sy_c_LList2__Mirabelle__hamjzmohle_Osuff,type,
lList21475143548e_suff:
!>[A: $tType] : ( ( set @ A ) > ( coinductive_llist @ A ) > ( set @ ( coinductive_llist @ A ) ) ) ).
thf(sy_c_Lattices_Osup__class_Osup,type,
sup_sup:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Nat_OSuc,type,
suc: nat > nat ).
thf(sy_c_Orderings_Oord__class_Oless,type,
ord_less:
!>[A: $tType] : ( A > A > $o ) ).
thf(sy_c_Orderings_Oord__class_Oless__eq,type,
ord_less_eq:
!>[A: $tType] : ( A > A > $o ) ).
thf(sy_c_Orderings_Otop__class_Otop,type,
top_top:
!>[A: $tType] : A ).
thf(sy_c_Pure_Otype,type,
type2:
!>[A: $tType] : ( itself @ A ) ).
thf(sy_c_Set_OCollect,type,
collect:
!>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
thf(sy_c_member,type,
member:
!>[A: $tType] : ( A > ( set @ A ) > $o ) ).
thf(sy_v_A,type,
a2: set @ a ).
thf(sy_v_P,type,
p: set @ ( coinductive_llist @ a ) ).
thf(sy_v_R,type,
r: $o ).
thf(sy_v_s,type,
s: coinductive_llist @ a ).
%----Relevant facts (256)
thf(fact_0_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_1_poslivenessI,axiom,
! [A: $tType,A2: set @ A,P: set @ ( coinductive_llist @ A )] :
( ! [S2: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ S2 @ ( lList22096119349pslsts @ A @ A2 ) )
=> ? [X: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ X @ ( lList2435255213lllsts @ A @ A2 ) )
& ( member @ ( coinductive_llist @ A ) @ ( coinductive_lappend @ A @ S2 @ X ) @ P ) ) )
=> ( lList21952340509veness @ A @ A2 @ P ) ) ).
% poslivenessI
thf(fact_2_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_3_posliveness__def,axiom,
! [A: $tType] :
( ( lList21952340509veness @ A )
= ( ^ [A3: set @ A,P2: set @ ( coinductive_llist @ A )] :
! [X2: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ X2 @ ( lList22096119349pslsts @ A @ A3 ) )
=> ? [Y: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ Y @ ( lList2435255213lllsts @ A @ A3 ) )
& ( member @ ( coinductive_llist @ A ) @ ( coinductive_lappend @ A @ X2 @ Y ) @ P2 ) ) ) ) ) ).
% posliveness_def
thf(fact_4_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_5_suff__all,axiom,
! [A: $tType,X3: coinductive_llist @ A,A2: set @ A,S: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ X3 @ ( lList21475143548e_suff @ A @ A2 @ S ) )
=> ( member @ ( coinductive_llist @ A ) @ X3 @ ( lList2435255213lllsts @ A @ A2 ) ) ) ).
% suff_all
thf(fact_6_safetyD,axiom,
! [A: $tType,A2: set @ A,P: set @ ( coinductive_llist @ A ),T: coinductive_llist @ A] :
( ( lList21350011628safety @ A @ A2 @ P )
=> ( ( member @ ( coinductive_llist @ A ) @ T @ ( lList2435255213lllsts @ A @ A2 ) )
=> ( ! [R2: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ R2 @ ( lList21202317876inpref @ A @ A2 @ T ) )
=> ? [X: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ X @ ( lList2435255213lllsts @ A @ A2 ) )
& ( member @ ( coinductive_llist @ A ) @ ( coinductive_lappend @ A @ R2 @ X ) @ P ) ) )
=> ( member @ ( coinductive_llist @ A ) @ T @ P ) ) ) ) ).
% safetyD
thf(fact_7_safetyE,axiom,
! [A: $tType,A2: set @ A,P: set @ ( coinductive_llist @ A )] :
( ( lList21350011628safety @ A @ A2 @ P )
=> ! [X: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ X @ ( lList2435255213lllsts @ A @ A2 ) )
=> ( ! [Xa: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ Xa @ ( lList21202317876inpref @ A @ A2 @ X ) )
=> ? [Xb: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ Xb @ ( lList2435255213lllsts @ A @ A2 ) )
& ( member @ ( coinductive_llist @ A ) @ ( coinductive_lappend @ A @ Xa @ Xb ) @ P ) ) )
=> ( member @ ( coinductive_llist @ A ) @ X @ P ) ) ) ) ).
% safetyE
thf(fact_8_safetyI,axiom,
! [A: $tType,A2: set @ A,P: set @ ( coinductive_llist @ A )] :
( ! [T2: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ T2 @ ( lList2435255213lllsts @ A @ A2 ) )
=> ( ! [X: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ X @ ( lList21202317876inpref @ A @ A2 @ T2 ) )
=> ? [Xa: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ Xa @ ( lList2435255213lllsts @ A @ A2 ) )
& ( member @ ( coinductive_llist @ A ) @ ( coinductive_lappend @ A @ X @ Xa ) @ P ) ) )
=> ( member @ ( coinductive_llist @ A ) @ T2 @ P ) ) )
=> ( lList21350011628safety @ A @ A2 @ P ) ) ).
% safetyI
thf(fact_9_safety__def,axiom,
! [A: $tType] :
( ( lList21350011628safety @ A )
= ( ^ [A3: set @ A,P2: set @ ( coinductive_llist @ A )] :
! [X2: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ X2 @ ( lList2435255213lllsts @ A @ A3 ) )
=> ( ! [Y: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ Y @ ( lList21202317876inpref @ A @ A3 @ X2 ) )
=> ? [Z: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ Z @ ( lList2435255213lllsts @ A @ A3 ) )
& ( member @ ( coinductive_llist @ A ) @ ( coinductive_lappend @ A @ Y @ Z ) @ P2 ) ) )
=> ( member @ ( coinductive_llist @ A ) @ X2 @ P2 ) ) ) ) ) ).
% safety_def
thf(fact_10_ldropT,axiom,
! [A: $tType,T: coinductive_llist @ A,A2: set @ A,I: nat] :
( ( member @ ( coinductive_llist @ A ) @ T @ ( lList2435255213lllsts @ A @ A2 ) )
=> ( member @ ( coinductive_llist @ A ) @ ( lList2508575361_ldrop @ A @ T @ I ) @ ( lList2435255213lllsts @ A @ A2 ) ) ) ).
% ldropT
thf(fact_11_alllsts__UNIV,axiom,
! [A: $tType,S: coinductive_llist @ A] : ( member @ ( coinductive_llist @ A ) @ S @ ( lList2435255213lllsts @ A @ ( top_top @ ( set @ A ) ) ) ) ).
% alllsts_UNIV
thf(fact_12_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_13_livenessE,axiom,
! [A: $tType,A2: set @ A,P: set @ ( coinductive_llist @ A ),S: coinductive_llist @ A] :
( ( lList21805353693veness @ A @ A2 @ P )
=> ( ! [T2: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ T2 @ ( lList2435255213lllsts @ A @ A2 ) )
=> ~ ( member @ ( coinductive_llist @ A ) @ ( coinductive_lappend @ A @ S @ T2 ) @ P ) )
=> ~ ( member @ ( coinductive_llist @ A ) @ S @ ( lList2236698231inlsts @ A @ A2 ) ) ) ) ).
% livenessE
thf(fact_14_livenessI,axiom,
! [A: $tType,A2: set @ A,P: set @ ( coinductive_llist @ A )] :
( ! [S2: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ S2 @ ( lList2236698231inlsts @ A @ A2 ) )
=> ? [X: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ X @ ( lList2435255213lllsts @ A @ A2 ) )
& ( member @ ( coinductive_llist @ A ) @ ( coinductive_lappend @ A @ S2 @ X ) @ P ) ) )
=> ( lList21805353693veness @ A @ A2 @ P ) ) ).
% livenessI
thf(fact_15_liveness__def,axiom,
! [A: $tType] :
( ( lList21805353693veness @ A )
= ( ^ [A3: set @ A,P2: set @ ( coinductive_llist @ A )] :
! [X2: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ X2 @ ( lList2236698231inlsts @ A @ A3 ) )
=> ? [Y: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ Y @ ( lList2435255213lllsts @ A @ A3 ) )
& ( member @ ( coinductive_llist @ A ) @ ( coinductive_lappend @ A @ X2 @ Y ) @ P2 ) ) ) ) ) ).
% liveness_def
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_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_18_notfin__inf,axiom,
! [A: $tType,X3: coinductive_llist @ A] :
( ( ~ ( member @ ( coinductive_llist @ A ) @ X3 @ ( lList2236698231inlsts @ A @ ( top_top @ ( set @ A ) ) ) ) )
= ( member @ ( coinductive_llist @ A ) @ X3 @ ( lList21612149805nflsts @ A @ ( top_top @ ( set @ A ) ) ) ) ) ).
% notfin_inf
thf(fact_19_notinf__fin,axiom,
! [A: $tType,X3: coinductive_llist @ A] :
( ( ~ ( member @ ( coinductive_llist @ A ) @ X3 @ ( lList21612149805nflsts @ A @ ( top_top @ ( set @ A ) ) ) ) )
= ( member @ ( coinductive_llist @ A ) @ X3 @ ( lList2236698231inlsts @ A @ ( top_top @ ( set @ A ) ) ) ) ) ).
% notinf_fin
thf(fact_20_ldrop__fin__iffT,axiom,
! [A: $tType,T: coinductive_llist @ A,I: nat] :
( ( member @ ( coinductive_llist @ A ) @ ( lList2508575361_ldrop @ A @ T @ I ) @ ( lList2236698231inlsts @ A @ ( top_top @ ( set @ A ) ) ) )
= ( member @ ( coinductive_llist @ A ) @ T @ ( lList2236698231inlsts @ A @ ( top_top @ ( set @ A ) ) ) ) ) ).
% ldrop_fin_iffT
thf(fact_21_ldrop__inf__iffT,axiom,
! [A: $tType,T: coinductive_llist @ A,I: nat] :
( ( member @ ( coinductive_llist @ A ) @ ( lList2508575361_ldrop @ A @ T @ I ) @ ( lList21612149805nflsts @ A @ ( top_top @ ( set @ A ) ) ) )
= ( member @ ( coinductive_llist @ A ) @ T @ ( lList21612149805nflsts @ A @ ( top_top @ ( set @ A ) ) ) ) ) ).
% ldrop_inf_iffT
thf(fact_22_inflstsI,axiom,
! [A: $tType,X3: coinductive_llist @ A,A2: set @ A] :
( ( member @ ( coinductive_llist @ A ) @ X3 @ ( lList2435255213lllsts @ A @ A2 ) )
=> ( ~ ( member @ ( coinductive_llist @ A ) @ X3 @ ( lList2236698231inlsts @ A @ ( top_top @ ( set @ A ) ) ) )
=> ( member @ ( coinductive_llist @ A ) @ X3 @ ( lList21612149805nflsts @ A @ A2 ) ) ) ) ).
% inflstsI
thf(fact_23_inflstsE,axiom,
! [A: $tType,X3: coinductive_llist @ A,A2: set @ A] :
( ( member @ ( coinductive_llist @ A ) @ X3 @ ( lList21612149805nflsts @ A @ A2 ) )
=> ~ ( ( member @ ( coinductive_llist @ A ) @ X3 @ ( lList2435255213lllsts @ A @ A2 ) )
=> ( member @ ( coinductive_llist @ A ) @ X3 @ ( lList2236698231inlsts @ A @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% inflstsE
thf(fact_24_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_25_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_26_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_27_ldrop__finT,axiom,
! [A: $tType,T: coinductive_llist @ A,A2: set @ A,I: nat] :
( ( member @ ( coinductive_llist @ A ) @ T @ ( lList2236698231inlsts @ A @ A2 ) )
=> ( member @ ( coinductive_llist @ A ) @ ( lList2508575361_ldrop @ A @ T @ I ) @ ( lList2236698231inlsts @ A @ A2 ) ) ) ).
% ldrop_finT
thf(fact_28_ldrop__infT,axiom,
! [A: $tType,T: coinductive_llist @ A,A2: set @ A,I: nat] :
( ( member @ ( coinductive_llist @ A ) @ T @ ( lList21612149805nflsts @ A @ A2 ) )
=> ( member @ ( coinductive_llist @ A ) @ ( lList2508575361_ldrop @ A @ T @ I ) @ ( lList21612149805nflsts @ A @ A2 ) ) ) ).
% ldrop_infT
thf(fact_29_finpref__fin,axiom,
! [A: $tType,X3: coinductive_llist @ A,A2: set @ A,S: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ X3 @ ( lList21202317876inpref @ A @ A2 @ S ) )
=> ( member @ ( coinductive_llist @ A ) @ X3 @ ( lList2236698231inlsts @ A @ A2 ) ) ) ).
% finpref_fin
thf(fact_30_suff__finpref,axiom,
! [A: $tType,T: coinductive_llist @ A,A2: set @ A,R: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ T @ ( lList21475143548e_suff @ A @ A2 @ R ) )
=> ( ( member @ ( coinductive_llist @ A ) @ R @ ( lList2236698231inlsts @ A @ A2 ) )
=> ( member @ ( coinductive_llist @ A ) @ R @ ( lList21202317876inpref @ A @ A2 @ T ) ) ) ) ).
% suff_finpref
thf(fact_31_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_32_suff__finpref__iff,axiom,
! [A: $tType,R: coinductive_llist @ A,A2: set @ A,T: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ R @ ( lList2236698231inlsts @ A @ A2 ) )
=> ( ( member @ ( coinductive_llist @ A ) @ T @ ( lList2435255213lllsts @ A @ A2 ) )
=> ( ( member @ ( coinductive_llist @ A ) @ R @ ( lList21202317876inpref @ A @ A2 @ T ) )
= ( member @ ( coinductive_llist @ A ) @ T @ ( lList21475143548e_suff @ A @ A2 @ R ) ) ) ) ) ).
% suff_finpref_iff
thf(fact_33_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_34_lapp__inv2T,axiom,
! [A: $tType,R: coinductive_llist @ A,S: coinductive_llist @ A,A2: set @ A] :
( ( member @ ( coinductive_llist @ A ) @ ( coinductive_lappend @ A @ R @ S ) @ ( lList21612149805nflsts @ A @ A2 ) )
=> ( ( ( member @ ( coinductive_llist @ A ) @ R @ ( lList2236698231inlsts @ A @ A2 ) )
& ( member @ ( coinductive_llist @ A ) @ S @ ( lList21612149805nflsts @ A @ A2 ) ) )
| ( member @ ( coinductive_llist @ A ) @ R @ ( lList21612149805nflsts @ A @ A2 ) ) ) ) ).
% lapp_inv2T
thf(fact_35_lapp__infT,axiom,
! [A: $tType,R: coinductive_llist @ A,S: coinductive_llist @ A,A2: set @ A] :
( ( member @ ( coinductive_llist @ A ) @ ( coinductive_lappend @ A @ R @ S ) @ ( lList21612149805nflsts @ A @ A2 ) )
= ( ( ( member @ ( coinductive_llist @ A ) @ R @ ( lList2236698231inlsts @ A @ A2 ) )
& ( member @ ( coinductive_llist @ A ) @ S @ ( lList21612149805nflsts @ A @ A2 ) ) )
| ( member @ ( coinductive_llist @ A ) @ R @ ( lList21612149805nflsts @ A @ A2 ) ) ) ) ).
% lapp_infT
thf(fact_36_app__invT,axiom,
! [A: $tType,R: coinductive_llist @ A,A2: set @ A,S: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ R @ ( lList2236698231inlsts @ A @ A2 ) )
=> ( ( member @ ( coinductive_llist @ A ) @ ( coinductive_lappend @ A @ R @ S ) @ ( lList21612149805nflsts @ A @ A2 ) )
=> ( member @ ( coinductive_llist @ A ) @ S @ ( lList21612149805nflsts @ A @ A2 ) ) ) ) ).
% app_invT
thf(fact_37_alllstsE,axiom,
! [A: $tType,X3: coinductive_llist @ A,A2: set @ A] :
( ( member @ ( coinductive_llist @ A ) @ X3 @ ( lList2435255213lllsts @ A @ A2 ) )
=> ( ~ ( member @ ( coinductive_llist @ A ) @ X3 @ ( lList2236698231inlsts @ A @ A2 ) )
=> ( member @ ( coinductive_llist @ A ) @ X3 @ ( lList21612149805nflsts @ A @ A2 ) ) ) ) ).
% alllstsE
thf(fact_38_finpref__suff,axiom,
! [A: $tType,R: coinductive_llist @ A,A2: set @ A,T: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ R @ ( lList21202317876inpref @ A @ A2 @ T ) )
=> ( ( member @ ( coinductive_llist @ A ) @ T @ ( lList2435255213lllsts @ A @ A2 ) )
=> ( member @ ( coinductive_llist @ A ) @ T @ ( lList21475143548e_suff @ A @ A2 @ R ) ) ) ) ).
% finpref_suff
thf(fact_39_lapp__allT__iff,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 @ ( lList2236698231inlsts @ A @ A2 ) )
& ( member @ ( coinductive_llist @ A ) @ S @ ( lList2435255213lllsts @ A @ A2 ) ) )
| ( member @ ( coinductive_llist @ A ) @ R @ ( lList21612149805nflsts @ A @ A2 ) ) ) ) ).
% lapp_allT_iff
thf(fact_40_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_41_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_42_finsubsetall,axiom,
! [A: $tType,X3: coinductive_llist @ A,A2: set @ A] :
( ( member @ ( coinductive_llist @ A ) @ X3 @ ( lList2236698231inlsts @ A @ A2 ) )
=> ( member @ ( coinductive_llist @ A ) @ X3 @ ( lList2435255213lllsts @ A @ A2 ) ) ) ).
% finsubsetall
thf(fact_43_finite__lemma,axiom,
! [A: $tType,X3: coinductive_llist @ A,A2: set @ A,B2: set @ A] :
( ( member @ ( coinductive_llist @ A ) @ X3 @ ( lList2236698231inlsts @ A @ A2 ) )
=> ( ( member @ ( coinductive_llist @ A ) @ X3 @ ( lList2435255213lllsts @ A @ B2 ) )
=> ( member @ ( coinductive_llist @ A ) @ X3 @ ( lList2236698231inlsts @ A @ B2 ) ) ) ) ).
% finite_lemma
thf(fact_44_suff__appE,axiom,
! [A: $tType,R: coinductive_llist @ A,A2: set @ A,T: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ R @ ( lList2236698231inlsts @ A @ A2 ) )
=> ( ( member @ ( coinductive_llist @ A ) @ T @ ( lList21475143548e_suff @ A @ A2 @ R ) )
=> ~ ! [S2: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ S2 @ ( lList2435255213lllsts @ A @ A2 ) )
=> ( T
!= ( coinductive_lappend @ A @ R @ S2 ) ) ) ) ) ).
% suff_appE
thf(fact_45_mem__Collect__eq,axiom,
! [A: $tType,A4: A,P: A > $o] :
( ( member @ A @ A4 @ ( collect @ A @ P ) )
= ( P @ A4 ) ) ).
% mem_Collect_eq
thf(fact_46_Collect__mem__eq,axiom,
! [A: $tType,A2: set @ A] :
( ( collect @ A
@ ^ [X2: A] : ( member @ A @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_47_Collect__cong,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ! [X4: A] :
( ( P @ X4 )
= ( Q @ X4 ) )
=> ( ( collect @ A @ P )
= ( collect @ A @ Q ) ) ) ).
% Collect_cong
thf(fact_48_ext,axiom,
! [B: $tType,A: $tType,F: A > B,G: A > B] :
( ! [X4: A] :
( ( F @ X4 )
= ( G @ X4 ) )
=> ( F = G ) ) ).
% ext
thf(fact_49_infsubsetall,axiom,
! [A: $tType,X3: coinductive_llist @ A,A2: set @ A] :
( ( member @ ( coinductive_llist @ A ) @ X3 @ ( lList21612149805nflsts @ A @ A2 ) )
=> ( member @ ( coinductive_llist @ A ) @ X3 @ ( lList2435255213lllsts @ A @ A2 ) ) ) ).
% infsubsetall
thf(fact_50_infsafety__def,axiom,
! [A: $tType] :
( ( lList21015939545safety @ A )
= ( ^ [A3: set @ A,P2: set @ ( coinductive_llist @ A )] :
! [X2: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ X2 @ ( lList21612149805nflsts @ A @ A3 ) )
=> ( ! [Y: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ Y @ ( lList21202317876inpref @ A @ A3 @ X2 ) )
=> ? [Z: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ Z @ ( lList21612149805nflsts @ A @ A3 ) )
& ( member @ ( coinductive_llist @ A ) @ ( coinductive_lappend @ A @ Y @ Z ) @ P2 ) ) )
=> ( member @ ( coinductive_llist @ A ) @ X2 @ P2 ) ) ) ) ) ).
% infsafety_def
thf(fact_51_infliveness__def,axiom,
! [A: $tType] :
( ( lList21015763786veness @ A )
= ( ^ [A3: set @ A,P2: set @ ( coinductive_llist @ A )] :
! [X2: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ X2 @ ( lList2236698231inlsts @ A @ A3 ) )
=> ? [Y: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ Y @ ( lList21612149805nflsts @ A @ A3 ) )
& ( member @ ( coinductive_llist @ A ) @ ( coinductive_lappend @ A @ X2 @ Y ) @ P2 ) ) ) ) ) ).
% infliveness_def
thf(fact_52_safety__prefix__closed,axiom,
! [A: $tType,P: set @ ( coinductive_llist @ A )] :
( ( lList21350011628safety @ A @ ( top_top @ ( set @ A ) ) @ P )
=> ( lList21638733016closed @ A @ P ) ) ).
% safety_prefix_closed
thf(fact_53_infsuff__finpref__iff,axiom,
! [A: $tType,R: coinductive_llist @ A,A2: set @ A,T: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ R @ ( lList2236698231inlsts @ A @ A2 ) )
=> ( ( member @ ( coinductive_llist @ A ) @ T @ ( lList21612149805nflsts @ A @ A2 ) )
=> ( ( member @ ( coinductive_llist @ A ) @ T @ ( lList21202317876inpref @ A @ A2 @ R ) )
= ( member @ ( coinductive_llist @ A ) @ R @ ( lList2649413865nfsuff @ A @ A2 @ T ) ) ) ) ) ).
% infsuff_finpref_iff
thf(fact_54_UNIV__I,axiom,
! [A: $tType,X3: A] : ( member @ A @ X3 @ ( top_top @ ( set @ A ) ) ) ).
% UNIV_I
thf(fact_55_iso__tuple__UNIV__I,axiom,
! [A: $tType,X3: A] : ( member @ A @ X3 @ ( top_top @ ( set @ A ) ) ) ).
% iso_tuple_UNIV_I
thf(fact_56_top__apply,axiom,
! [C: $tType,D: $tType] :
( ( top @ C @ ( type2 @ C ) )
=> ( ( top_top @ ( D > C ) )
= ( ^ [X2: D] : ( top_top @ C ) ) ) ) ).
% top_apply
thf(fact_57_lrev__lappend,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ Xs @ ( lList2236698231inlsts @ A @ ( top_top @ ( set @ A ) ) ) )
=> ( ( member @ ( coinductive_llist @ A ) @ Ys @ ( lList2236698231inlsts @ A @ ( top_top @ ( set @ A ) ) ) )
=> ( ( lList2281150353e_lrev @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) )
= ( coinductive_lappend @ A @ ( lList2281150353e_lrev @ A @ Ys ) @ ( lList2281150353e_lrev @ A @ Xs ) ) ) ) ) ).
% lrev_lappend
thf(fact_58_infsuff__appE,axiom,
! [A: $tType,R: coinductive_llist @ A,A2: set @ A,T: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ R @ ( lList2236698231inlsts @ A @ A2 ) )
=> ( ( member @ ( coinductive_llist @ A ) @ T @ ( lList2649413865nfsuff @ A @ A2 @ R ) )
=> ~ ! [S2: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ S2 @ ( lList21612149805nflsts @ A @ A2 ) )
=> ( T
!= ( coinductive_lappend @ A @ R @ S2 ) ) ) ) ) ).
% infsuff_appE
thf(fact_59_lapp__suff__llength,axiom,
! [A: $tType,R: coinductive_llist @ A,A2: set @ A,S: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ R @ ( lList2236698231inlsts @ A @ A2 ) )
=> ( ( lList2508575361_ldrop @ A @ ( coinductive_lappend @ A @ R @ S ) @ ( lList21232602520length @ A @ R ) )
= S ) ) ).
% lapp_suff_llength
thf(fact_60_lrevT,axiom,
! [A: $tType,Xs: coinductive_llist @ A,A2: set @ A] :
( ( member @ ( coinductive_llist @ A ) @ Xs @ ( lList2236698231inlsts @ A @ A2 ) )
=> ( member @ ( coinductive_llist @ A ) @ ( lList2281150353e_lrev @ A @ Xs ) @ ( lList2236698231inlsts @ A @ A2 ) ) ) ).
% lrevT
thf(fact_61_lrev__is__lrev__conv,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ Xs @ ( lList2236698231inlsts @ A @ ( top_top @ ( set @ A ) ) ) )
=> ( ( member @ ( coinductive_llist @ A ) @ Ys @ ( lList2236698231inlsts @ A @ ( top_top @ ( set @ A ) ) ) )
=> ( ( ( lList2281150353e_lrev @ A @ Xs )
= ( lList2281150353e_lrev @ A @ Ys ) )
= ( Xs = Ys ) ) ) ) ).
% lrev_is_lrev_conv
thf(fact_62_lrev__lrev__ident,axiom,
! [A: $tType,Xs: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ Xs @ ( lList2236698231inlsts @ A @ ( top_top @ ( set @ A ) ) ) )
=> ( ( lList2281150353e_lrev @ A @ ( lList2281150353e_lrev @ A @ Xs ) )
= Xs ) ) ).
% lrev_lrev_ident
thf(fact_63_top__set__def,axiom,
! [A: $tType] :
( ( top_top @ ( set @ A ) )
= ( collect @ A @ ( top_top @ ( A > $o ) ) ) ) ).
% top_set_def
thf(fact_64_infsuff__inf,axiom,
! [A: $tType,X3: coinductive_llist @ A,A2: set @ A,S: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ X3 @ ( lList2649413865nfsuff @ A @ A2 @ S ) )
=> ( member @ ( coinductive_llist @ A ) @ X3 @ ( lList21612149805nflsts @ A @ A2 ) ) ) ).
% infsuff_inf
thf(fact_65_UNIV__witness,axiom,
! [A: $tType] :
? [X4: A] : ( member @ A @ X4 @ ( top_top @ ( set @ A ) ) ) ).
% UNIV_witness
thf(fact_66_UNIV__eq__I,axiom,
! [A: $tType,A2: set @ A] :
( ! [X4: A] : ( member @ A @ X4 @ A2 )
=> ( ( top_top @ ( set @ A ) )
= A2 ) ) ).
% UNIV_eq_I
thf(fact_67_infsuff__finpref,axiom,
! [A: $tType,T: coinductive_llist @ A,A2: set @ A,R: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ T @ ( lList2649413865nfsuff @ A @ A2 @ R ) )
=> ( ( member @ ( coinductive_llist @ A ) @ R @ ( lList2236698231inlsts @ A @ A2 ) )
=> ( member @ ( coinductive_llist @ A ) @ R @ ( lList21202317876inpref @ A @ A2 @ T ) ) ) ) ).
% infsuff_finpref
thf(fact_68_finpref__infsuff,axiom,
! [A: $tType,R: coinductive_llist @ A,A2: set @ A,T: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ R @ ( lList21202317876inpref @ A @ A2 @ T ) )
=> ( ( member @ ( coinductive_llist @ A ) @ T @ ( lList21612149805nflsts @ A @ A2 ) )
=> ( member @ ( coinductive_llist @ A ) @ T @ ( lList2649413865nfsuff @ A @ A2 @ R ) ) ) ) ).
% finpref_infsuff
thf(fact_69_ltake__lappend__llength,axiom,
! [A: $tType,R: coinductive_llist @ A,A2: set @ A,S: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ R @ ( lList2236698231inlsts @ A @ A2 ) )
=> ( ( lList22119844313_ltake @ A @ ( coinductive_lappend @ A @ R @ S ) @ ( lList21232602520length @ A @ R ) )
= R ) ) ).
% ltake_lappend_llength
thf(fact_70_llength__take,axiom,
! [A: $tType,T: coinductive_llist @ A,A2: set @ A,I: nat] :
( ( member @ ( coinductive_llist @ A ) @ T @ ( lList21612149805nflsts @ A @ A2 ) )
=> ( ( lList21232602520length @ A @ ( lList22119844313_ltake @ A @ T @ I ) )
= I ) ) ).
% llength_take
thf(fact_71_take__fin,axiom,
! [A: $tType,T: coinductive_llist @ A,A2: set @ A,I: nat] :
( ( member @ ( coinductive_llist @ A ) @ T @ ( lList2435255213lllsts @ A @ A2 ) )
=> ( member @ ( coinductive_llist @ A ) @ ( lList22119844313_ltake @ A @ T @ I ) @ ( lList2236698231inlsts @ A @ A2 ) ) ) ).
% take_fin
thf(fact_72_ltake__fin,axiom,
! [A: $tType,R: coinductive_llist @ A,I: nat] : ( member @ ( coinductive_llist @ A ) @ ( lList22119844313_ltake @ A @ R @ I ) @ ( lList2236698231inlsts @ A @ ( top_top @ ( set @ A ) ) ) ) ).
% ltake_fin
thf(fact_73_inflsts__def,axiom,
! [A: $tType] :
( ( lList21612149805nflsts @ A )
= ( ^ [A3: set @ A] : ( minus_minus @ ( set @ ( coinductive_llist @ A ) ) @ ( lList2435255213lllsts @ A @ A3 ) @ ( lList2236698231inlsts @ A @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% inflsts_def
thf(fact_74_lrev__is__LNil__conv,axiom,
! [A: $tType,Xs: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ Xs @ ( lList2236698231inlsts @ A @ ( top_top @ ( set @ A ) ) ) )
=> ( ( ( lList2281150353e_lrev @ A @ Xs )
= ( coinductive_LNil @ A ) )
= ( Xs
= ( coinductive_LNil @ A ) ) ) ) ).
% lrev_is_LNil_conv
thf(fact_75_LNil__is__lrev__conv,axiom,
! [A: $tType,Xs: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ Xs @ ( lList2236698231inlsts @ A @ ( top_top @ ( set @ A ) ) ) )
=> ( ( ( coinductive_LNil @ A )
= ( lList2281150353e_lrev @ A @ Xs ) )
= ( Xs
= ( coinductive_LNil @ A ) ) ) ) ).
% LNil_is_lrev_conv
thf(fact_76_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_77_Diff__iff,axiom,
! [A: $tType,C2: A,A2: set @ A,B2: set @ A] :
( ( member @ A @ C2 @ ( minus_minus @ ( set @ A ) @ A2 @ B2 ) )
= ( ( member @ A @ C2 @ A2 )
& ~ ( member @ A @ C2 @ B2 ) ) ) ).
% Diff_iff
thf(fact_78_DiffI,axiom,
! [A: $tType,C2: A,A2: set @ A,B2: set @ A] :
( ( member @ A @ C2 @ A2 )
=> ( ~ ( member @ A @ C2 @ B2 )
=> ( member @ A @ C2 @ ( minus_minus @ ( set @ A ) @ A2 @ B2 ) ) ) ) ).
% DiffI
thf(fact_79_top1I,axiom,
! [A: $tType,X3: A] : ( top_top @ ( A > $o ) @ X3 ) ).
% top1I
thf(fact_80_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_81_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_82_lappend__LNil2,axiom,
! [A: $tType,Xs: coinductive_llist @ A] :
( ( coinductive_lappend @ A @ Xs @ ( coinductive_LNil @ A ) )
= Xs ) ).
% lappend_LNil2
thf(fact_83_lappend__code_I1_J,axiom,
! [A: $tType,Ys: coinductive_llist @ A] :
( ( coinductive_lappend @ A @ ( coinductive_LNil @ A ) @ Ys )
= Ys ) ).
% lappend_code(1)
thf(fact_84_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_85_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_86_LList2__Mirabelle__hamjzmohle_Oldrop__LNil,axiom,
! [A: $tType,I: nat] :
( ( lList2508575361_ldrop @ A @ ( coinductive_LNil @ A ) @ I )
= ( coinductive_LNil @ A ) ) ).
% LList2_Mirabelle_hamjzmohle.ldrop_LNil
thf(fact_87_LList2__Mirabelle__hamjzmohle_Oltake__LNil,axiom,
! [A: $tType,I: nat] :
( ( lList22119844313_ltake @ A @ ( coinductive_LNil @ A ) @ I )
= ( coinductive_LNil @ A ) ) ).
% LList2_Mirabelle_hamjzmohle.ltake_LNil
thf(fact_88_lrev__LNil,axiom,
! [A: $tType] :
( ( lList2281150353e_lrev @ A @ ( coinductive_LNil @ A ) )
= ( coinductive_LNil @ A ) ) ).
% lrev_LNil
thf(fact_89_LNil__suff,axiom,
! [A: $tType,A2: set @ A,S: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ ( coinductive_LNil @ A ) @ ( lList21475143548e_suff @ A @ A2 @ S ) )
= ( S
= ( coinductive_LNil @ A ) ) ) ).
% LNil_suff
thf(fact_90_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_91_suff__LNil,axiom,
! [A: $tType,A2: set @ A] :
( ( lList21475143548e_suff @ A @ A2 @ ( coinductive_LNil @ A ) )
= ( lList2435255213lllsts @ A @ A2 ) ) ).
% suff_LNil
thf(fact_92_infsuff__LNil,axiom,
! [A: $tType,A2: set @ A] :
( ( lList2649413865nfsuff @ A @ A2 @ ( coinductive_LNil @ A ) )
= ( lList21612149805nflsts @ A @ A2 ) ) ).
% infsuff_LNil
thf(fact_93_DiffD2,axiom,
! [A: $tType,C2: A,A2: set @ A,B2: set @ A] :
( ( member @ A @ C2 @ ( minus_minus @ ( set @ A ) @ A2 @ B2 ) )
=> ~ ( member @ A @ C2 @ B2 ) ) ).
% DiffD2
thf(fact_94_DiffD1,axiom,
! [A: $tType,C2: A,A2: set @ A,B2: set @ A] :
( ( member @ A @ C2 @ ( minus_minus @ ( set @ A ) @ A2 @ B2 ) )
=> ( member @ A @ C2 @ A2 ) ) ).
% DiffD1
thf(fact_95_DiffE,axiom,
! [A: $tType,C2: A,A2: set @ A,B2: set @ A] :
( ( member @ A @ C2 @ ( minus_minus @ ( set @ A ) @ A2 @ B2 ) )
=> ~ ( ( member @ A @ C2 @ A2 )
=> ( member @ A @ C2 @ B2 ) ) ) ).
% DiffE
thf(fact_96_lappend__LNil__LNil,axiom,
! [A: $tType] :
( ( coinductive_lappend @ A @ ( coinductive_LNil @ A ) @ ( coinductive_LNil @ A ) )
= ( coinductive_LNil @ A ) ) ).
% lappend_LNil_LNil
thf(fact_97_finlsts_OLNil__fin,axiom,
! [A: $tType,A2: set @ A] : ( member @ ( coinductive_llist @ A ) @ ( coinductive_LNil @ A ) @ ( lList2236698231inlsts @ A @ A2 ) ) ).
% finlsts.LNil_fin
thf(fact_98_alllsts_OLNil__all,axiom,
! [A: $tType,A2: set @ A] : ( member @ ( coinductive_llist @ A ) @ ( coinductive_LNil @ A ) @ ( lList2435255213lllsts @ A @ A2 ) ) ).
% alllsts.LNil_all
thf(fact_99_drop__nonLNil,axiom,
! [A: $tType,T: coinductive_llist @ A,I: nat] :
( ( ( lList2508575361_ldrop @ A @ T @ I )
!= ( coinductive_LNil @ A ) )
=> ( T
!= ( coinductive_LNil @ A ) ) ) ).
% drop_nonLNil
thf(fact_100_llength__drop__take,axiom,
! [A: $tType,T: coinductive_llist @ A,I: nat] :
( ( ( lList2508575361_ldrop @ A @ T @ I )
!= ( coinductive_LNil @ A ) )
=> ( ( lList21232602520length @ A @ ( lList22119844313_ltake @ A @ T @ I ) )
= I ) ) ).
% llength_drop_take
thf(fact_101_ltake__ldrop__id,axiom,
! [A: $tType,X3: coinductive_llist @ A,I: nat] :
( ( coinductive_lappend @ A @ ( lList22119844313_ltake @ A @ X3 @ I ) @ ( lList2508575361_ldrop @ A @ X3 @ I ) )
= X3 ) ).
% ltake_ldrop_id
thf(fact_102_pfinpref__iff,axiom,
! [A: $tType,X3: coinductive_llist @ A,A2: set @ A,S: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ X3 @ ( lList2467029176inpref @ A @ A2 @ S ) )
= ( ( member @ ( coinductive_llist @ A ) @ X3 @ ( lList21202317876inpref @ A @ A2 @ S ) )
& ( X3
!= ( coinductive_LNil @ A ) ) ) ) ).
% pfinpref_iff
thf(fact_103_minus__apply,axiom,
! [B: $tType,A: $tType] :
( ( minus @ B @ ( type2 @ B ) )
=> ( ( minus_minus @ ( A > B ) )
= ( ^ [A3: A > B,B3: A > B,X2: A] : ( minus_minus @ B @ ( A3 @ X2 ) @ ( B3 @ X2 ) ) ) ) ) ).
% minus_apply
thf(fact_104_top__empty__eq,axiom,
! [A: $tType] :
( ( top_top @ ( A > $o ) )
= ( ^ [X2: A] : ( member @ A @ X2 @ ( top_top @ ( set @ A ) ) ) ) ) ).
% top_empty_eq
thf(fact_105_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_106_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_107_lrev__LCons,axiom,
! [A: $tType,R: coinductive_llist @ A,A2: set @ A,A4: A] :
( ( member @ ( coinductive_llist @ A ) @ R @ ( lList2236698231inlsts @ A @ A2 ) )
=> ( ( lList2281150353e_lrev @ A @ ( coinductive_LCons @ A @ A4 @ R ) )
= ( coinductive_lappend @ A @ ( lList2281150353e_lrev @ A @ R ) @ ( coinductive_LCons @ A @ A4 @ ( coinductive_LNil @ A ) ) ) ) ) ).
% lrev_LCons
thf(fact_108_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_109_lappend__code_I2_J,axiom,
! [A: $tType,Xa2: A,X3: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( coinductive_lappend @ A @ ( coinductive_LCons @ A @ Xa2 @ X3 ) @ Ys )
= ( coinductive_LCons @ A @ Xa2 @ ( coinductive_lappend @ A @ X3 @ Ys ) ) ) ).
% lappend_code(2)
thf(fact_110_LConsE,axiom,
! [A: $tType,X3: A,Xs: coinductive_llist @ A,A2: set @ A] :
( ( member @ ( coinductive_llist @ A ) @ ( coinductive_LCons @ A @ X3 @ Xs ) @ ( lList2435255213lllsts @ A @ A2 ) )
= ( ( member @ A @ X3 @ A2 )
& ( member @ ( coinductive_llist @ A ) @ Xs @ ( lList2435255213lllsts @ A @ A2 ) ) ) ) ).
% LConsE
thf(fact_111_llist_Odistinct_I1_J,axiom,
! [A: $tType,X21: A,X22: coinductive_llist @ A] :
( ( coinductive_LNil @ A )
!= ( coinductive_LCons @ A @ X21 @ X22 ) ) ).
% llist.distinct(1)
thf(fact_112_neq__LNil__conv,axiom,
! [A: $tType,Xs: coinductive_llist @ A] :
( ( Xs
!= ( coinductive_LNil @ A ) )
= ( ? [X2: A,Xs2: coinductive_llist @ A] :
( Xs
= ( coinductive_LCons @ A @ X2 @ Xs2 ) ) ) ) ).
% neq_LNil_conv
thf(fact_113_llistE,axiom,
! [A: $tType,Y2: coinductive_llist @ A] :
( ( Y2
!= ( coinductive_LNil @ A ) )
=> ~ ! [X212: A,X222: coinductive_llist @ A] :
( Y2
!= ( coinductive_LCons @ A @ X212 @ X222 ) ) ) ).
% llistE
thf(fact_114_finlsts_OLCons__fin,axiom,
! [A: $tType,L: coinductive_llist @ A,A2: set @ A,A4: A] :
( ( member @ ( coinductive_llist @ A ) @ L @ ( lList2236698231inlsts @ A @ A2 ) )
=> ( ( member @ A @ A4 @ A2 )
=> ( member @ ( coinductive_llist @ A ) @ ( coinductive_LCons @ A @ A4 @ L ) @ ( lList2236698231inlsts @ A @ A2 ) ) ) ) ).
% finlsts.LCons_fin
thf(fact_115_alllsts_OLCons__all,axiom,
! [A: $tType,L: coinductive_llist @ A,A2: set @ A,A4: A] :
( ( member @ ( coinductive_llist @ A ) @ L @ ( lList2435255213lllsts @ A @ A2 ) )
=> ( ( member @ A @ A4 @ A2 )
=> ( member @ ( coinductive_llist @ A ) @ ( coinductive_LCons @ A @ A4 @ L ) @ ( lList2435255213lllsts @ A @ A2 ) ) ) ) ).
% alllsts.LCons_all
thf(fact_116_inflsts__cases,axiom,
! [A: $tType,S: coinductive_llist @ A,A2: set @ A] :
( ( member @ ( coinductive_llist @ A ) @ S @ ( lList21612149805nflsts @ A @ A2 ) )
=> ~ ! [A5: A,L2: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ L2 @ ( lList21612149805nflsts @ A @ A2 ) )
=> ( ( member @ A @ A5 @ A2 )
=> ( S
!= ( coinductive_LCons @ A @ A5 @ L2 ) ) ) ) ) ).
% inflsts_cases
thf(fact_117_inflstsI2,axiom,
! [A: $tType,A4: A,A2: set @ A,T: coinductive_llist @ A] :
( ( member @ A @ A4 @ A2 )
=> ( ( member @ ( coinductive_llist @ A ) @ T @ ( lList21612149805nflsts @ A @ A2 ) )
=> ( member @ ( coinductive_llist @ A ) @ ( coinductive_LCons @ A @ A4 @ T ) @ ( lList21612149805nflsts @ A @ A2 ) ) ) ) ).
% inflstsI2
thf(fact_118_lappend__snocL1__conv__LCons2,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Y2: A,Ys: coinductive_llist @ A] :
( ( coinductive_lappend @ A @ ( coinductive_lappend @ A @ Xs @ ( coinductive_LCons @ A @ Y2 @ ( coinductive_LNil @ A ) ) ) @ Ys )
= ( coinductive_lappend @ A @ Xs @ ( coinductive_LCons @ A @ Y2 @ Ys ) ) ) ).
% lappend_snocL1_conv_LCons2
thf(fact_119_finlsts_Ocases,axiom,
! [A: $tType,A4: coinductive_llist @ A,A2: set @ A] :
( ( member @ ( coinductive_llist @ A ) @ A4 @ ( lList2236698231inlsts @ A @ A2 ) )
=> ( ( A4
!= ( coinductive_LNil @ A ) )
=> ~ ! [L2: coinductive_llist @ A,A5: A] :
( ( A4
= ( coinductive_LCons @ A @ A5 @ L2 ) )
=> ( ( member @ ( coinductive_llist @ A ) @ L2 @ ( lList2236698231inlsts @ A @ A2 ) )
=> ~ ( member @ A @ A5 @ A2 ) ) ) ) ) ).
% finlsts.cases
thf(fact_120_finlsts_Osimps,axiom,
! [A: $tType,A4: coinductive_llist @ A,A2: set @ A] :
( ( member @ ( coinductive_llist @ A ) @ A4 @ ( lList2236698231inlsts @ A @ A2 ) )
= ( ( A4
= ( coinductive_LNil @ A ) )
| ? [L3: coinductive_llist @ A,A6: A] :
( ( A4
= ( coinductive_LCons @ A @ A6 @ L3 ) )
& ( member @ ( coinductive_llist @ A ) @ L3 @ ( lList2236698231inlsts @ A @ A2 ) )
& ( member @ A @ A6 @ A2 ) ) ) ) ).
% finlsts.simps
thf(fact_121_finlsts__induct,axiom,
! [A: $tType,X3: coinductive_llist @ A,A2: set @ A,P: ( coinductive_llist @ A ) > $o] :
( ( member @ ( coinductive_llist @ A ) @ X3 @ ( lList2236698231inlsts @ A @ A2 ) )
=> ( ! [L2: coinductive_llist @ A] :
( ( L2
= ( coinductive_LNil @ A ) )
=> ( P @ L2 ) )
=> ( ! [A5: A,L2: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ L2 @ ( lList2236698231inlsts @ A @ A2 ) )
=> ( ( P @ L2 )
=> ( ( member @ A @ A5 @ A2 )
=> ( P @ ( coinductive_LCons @ A @ A5 @ L2 ) ) ) ) )
=> ( P @ X3 ) ) ) ) ).
% finlsts_induct
thf(fact_122_finlsts_Oinducts,axiom,
! [A: $tType,X3: coinductive_llist @ A,A2: set @ A,P: ( coinductive_llist @ A ) > $o] :
( ( member @ ( coinductive_llist @ A ) @ X3 @ ( lList2236698231inlsts @ A @ A2 ) )
=> ( ( P @ ( coinductive_LNil @ A ) )
=> ( ! [L2: coinductive_llist @ A,A5: A] :
( ( member @ ( coinductive_llist @ A ) @ L2 @ ( lList2236698231inlsts @ A @ A2 ) )
=> ( ( P @ L2 )
=> ( ( member @ A @ A5 @ A2 )
=> ( P @ ( coinductive_LCons @ A @ A5 @ L2 ) ) ) ) )
=> ( P @ X3 ) ) ) ) ).
% finlsts.inducts
thf(fact_123_alllsts_Ocoinduct,axiom,
! [A: $tType,X5: ( coinductive_llist @ A ) > $o,X3: coinductive_llist @ A,A2: set @ A] :
( ( X5 @ X3 )
=> ( ! [X4: coinductive_llist @ A] :
( ( X5 @ X4 )
=> ( ( X4
= ( coinductive_LNil @ A ) )
| ? [L4: coinductive_llist @ A,A7: A] :
( ( X4
= ( coinductive_LCons @ A @ A7 @ L4 ) )
& ( ( X5 @ L4 )
| ( member @ ( coinductive_llist @ A ) @ L4 @ ( lList2435255213lllsts @ A @ A2 ) ) )
& ( member @ A @ A7 @ A2 ) ) ) )
=> ( member @ ( coinductive_llist @ A ) @ X3 @ ( lList2435255213lllsts @ A @ A2 ) ) ) ) ).
% alllsts.coinduct
thf(fact_124_alllsts_Osimps,axiom,
! [A: $tType,A4: coinductive_llist @ A,A2: set @ A] :
( ( member @ ( coinductive_llist @ A ) @ A4 @ ( lList2435255213lllsts @ A @ A2 ) )
= ( ( A4
= ( coinductive_LNil @ A ) )
| ? [L3: coinductive_llist @ A,A6: A] :
( ( A4
= ( coinductive_LCons @ A @ A6 @ L3 ) )
& ( member @ ( coinductive_llist @ A ) @ L3 @ ( lList2435255213lllsts @ A @ A2 ) )
& ( member @ A @ A6 @ A2 ) ) ) ) ).
% alllsts.simps
thf(fact_125_alllsts_Ocases,axiom,
! [A: $tType,A4: coinductive_llist @ A,A2: set @ A] :
( ( member @ ( coinductive_llist @ A ) @ A4 @ ( lList2435255213lllsts @ A @ A2 ) )
=> ( ( A4
!= ( coinductive_LNil @ A ) )
=> ~ ! [L2: coinductive_llist @ A,A5: A] :
( ( A4
= ( coinductive_LCons @ A @ A5 @ L2 ) )
=> ( ( member @ ( coinductive_llist @ A ) @ L2 @ ( lList2435255213lllsts @ A @ A2 ) )
=> ~ ( member @ A @ A5 @ A2 ) ) ) ) ) ).
% alllsts.cases
thf(fact_126_fps__induct,axiom,
! [A: $tType,L: coinductive_llist @ A,A2: set @ A,P: ( coinductive_llist @ A ) > $o] :
( ( member @ ( coinductive_llist @ A ) @ L @ ( lList22096119349pslsts @ A @ A2 ) )
=> ( ! [A5: A] :
( ( member @ A @ A5 @ A2 )
=> ( P @ ( coinductive_LCons @ A @ A5 @ ( coinductive_LNil @ A ) ) ) )
=> ( ! [A5: A,L2: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ L2 @ ( lList22096119349pslsts @ A @ A2 ) )
=> ( ( P @ L2 )
=> ( ( member @ A @ A5 @ A2 )
=> ( P @ ( coinductive_LCons @ A @ A5 @ L2 ) ) ) ) )
=> ( P @ L ) ) ) ) ).
% fps_induct
thf(fact_127_fpslsts__cases,axiom,
! [A: $tType,R: coinductive_llist @ A,A2: set @ A] :
( ( member @ ( coinductive_llist @ A ) @ R @ ( lList22096119349pslsts @ A @ A2 ) )
=> ~ ! [A5: A,Rs: coinductive_llist @ A] :
( ( R
= ( coinductive_LCons @ A @ A5 @ Rs ) )
=> ( ( member @ A @ A5 @ A2 )
=> ~ ( member @ ( coinductive_llist @ A ) @ Rs @ ( lList2236698231inlsts @ A @ A2 ) ) ) ) ) ).
% fpslsts_cases
thf(fact_128_finlsts__rev__cases,axiom,
! [A: $tType,T: coinductive_llist @ A,A2: set @ A] :
( ( member @ ( coinductive_llist @ A ) @ T @ ( lList2236698231inlsts @ A @ A2 ) )
=> ( ( T
!= ( coinductive_LNil @ A ) )
=> ~ ! [A5: A,L2: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ L2 @ ( lList2236698231inlsts @ A @ A2 ) )
=> ( ( member @ A @ A5 @ A2 )
=> ( T
!= ( coinductive_lappend @ A @ L2 @ ( coinductive_LCons @ A @ A5 @ ( coinductive_LNil @ A ) ) ) ) ) ) ) ) ).
% finlsts_rev_cases
thf(fact_129_lrev__induct,axiom,
! [A: $tType,Xs: coinductive_llist @ A,A2: set @ A,P: ( coinductive_llist @ A ) > $o] :
( ( member @ ( coinductive_llist @ A ) @ Xs @ ( lList2236698231inlsts @ A @ A2 ) )
=> ( ( P @ ( coinductive_LNil @ A ) )
=> ( ! [X4: A,Xs3: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ Xs3 @ ( lList2236698231inlsts @ A @ A2 ) )
=> ( ( P @ Xs3 )
=> ( ( member @ A @ X4 @ A2 )
=> ( P @ ( coinductive_lappend @ A @ Xs3 @ ( coinductive_LCons @ A @ X4 @ ( coinductive_LNil @ A ) ) ) ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% lrev_induct
thf(fact_130_fun__diff__def,axiom,
! [B: $tType,A: $tType] :
( ( minus @ B @ ( type2 @ B ) )
=> ( ( minus_minus @ ( A > B ) )
= ( ^ [A3: A > B,B3: A > B,X2: A] : ( minus_minus @ B @ ( A3 @ X2 ) @ ( B3 @ X2 ) ) ) ) ) ).
% fun_diff_def
thf(fact_131_lbutlast__snoc,axiom,
! [A: $tType,Xs: coinductive_llist @ A,A2: set @ A,X3: A] :
( ( member @ ( coinductive_llist @ A ) @ Xs @ ( lList2236698231inlsts @ A @ A2 ) )
=> ( ( lList2370560421utlast @ A @ ( coinductive_lappend @ A @ Xs @ ( coinductive_LCons @ A @ X3 @ ( coinductive_LNil @ A ) ) ) )
= Xs ) ) ).
% lbutlast_snoc
thf(fact_132_possafetyD,axiom,
! [A: $tType,A2: set @ A,P: set @ ( coinductive_llist @ A ),T: coinductive_llist @ A] :
( ( lList292406316safety @ A @ A2 @ P )
=> ( ( member @ ( coinductive_llist @ A ) @ T @ ( lList21148268032oslsts @ A @ A2 ) )
=> ( ! [R2: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ R2 @ ( lList2467029176inpref @ A @ A2 @ T ) )
=> ? [X: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ X @ ( lList2435255213lllsts @ A @ A2 ) )
& ( member @ ( coinductive_llist @ A ) @ ( coinductive_lappend @ A @ R2 @ X ) @ P ) ) )
=> ( member @ ( coinductive_llist @ A ) @ T @ P ) ) ) ) ).
% possafetyD
thf(fact_133_possafetyE,axiom,
! [A: $tType,A2: set @ A,P: set @ ( coinductive_llist @ A )] :
( ( lList292406316safety @ A @ A2 @ P )
=> ! [X: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ X @ ( lList21148268032oslsts @ A @ A2 ) )
=> ( ! [Xa: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ Xa @ ( lList2467029176inpref @ A @ A2 @ X ) )
=> ? [Xb: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ Xb @ ( lList2435255213lllsts @ A @ A2 ) )
& ( member @ ( coinductive_llist @ A ) @ ( coinductive_lappend @ A @ Xa @ Xb ) @ P ) ) )
=> ( member @ ( coinductive_llist @ A ) @ X @ P ) ) ) ) ).
% possafetyE
thf(fact_134_possafetyI,axiom,
! [A: $tType,A2: set @ A,P: set @ ( coinductive_llist @ A )] :
( ! [T2: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ T2 @ ( lList21148268032oslsts @ A @ A2 ) )
=> ( ! [X: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ X @ ( lList2467029176inpref @ A @ A2 @ T2 ) )
=> ? [Xa: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ Xa @ ( lList2435255213lllsts @ A @ A2 ) )
& ( member @ ( coinductive_llist @ A ) @ ( coinductive_lappend @ A @ X @ Xa ) @ P ) ) )
=> ( member @ ( coinductive_llist @ A ) @ T2 @ P ) ) )
=> ( lList292406316safety @ A @ A2 @ P ) ) ).
% possafetyI
thf(fact_135_possafety__def,axiom,
! [A: $tType] :
( ( lList292406316safety @ A )
= ( ^ [A3: set @ A,P2: set @ ( coinductive_llist @ A )] :
! [X2: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ X2 @ ( lList21148268032oslsts @ A @ A3 ) )
=> ( ! [Y: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ Y @ ( lList2467029176inpref @ A @ A3 @ X2 ) )
=> ? [Z: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ Z @ ( lList2435255213lllsts @ A @ A3 ) )
& ( member @ ( coinductive_llist @ A ) @ ( coinductive_lappend @ A @ Y @ Z ) @ P2 ) ) )
=> ( member @ ( coinductive_llist @ A ) @ X2 @ P2 ) ) ) ) ) ).
% possafety_def
thf(fact_136_lbutlast__LNil,axiom,
! [A: $tType] :
( ( lList2370560421utlast @ A @ ( coinductive_LNil @ A ) )
= ( coinductive_LNil @ A ) ) ).
% lbutlast_LNil
thf(fact_137_lbutlast__LCons,axiom,
! [A: $tType,R: coinductive_llist @ A,A2: set @ A,A4: A] :
( ( member @ ( coinductive_llist @ A ) @ R @ ( lList2236698231inlsts @ A @ A2 ) )
=> ( ( ( R
= ( coinductive_LNil @ A ) )
=> ( ( lList2370560421utlast @ A @ ( coinductive_LCons @ A @ A4 @ R ) )
= ( coinductive_LNil @ A ) ) )
& ( ( R
!= ( coinductive_LNil @ A ) )
=> ( ( lList2370560421utlast @ A @ ( coinductive_LCons @ A @ A4 @ R ) )
= ( coinductive_LCons @ A @ A4 @ ( lList2370560421utlast @ A @ R ) ) ) ) ) ) ).
% lbutlast_LCons
thf(fact_138_llast__snoc,axiom,
! [A: $tType,Xs: coinductive_llist @ A,A2: set @ A,X3: A] :
( ( member @ ( coinductive_llist @ A ) @ Xs @ ( lList2236698231inlsts @ A @ A2 ) )
=> ( ( lList2170638824_llast @ A @ ( coinductive_lappend @ A @ Xs @ ( coinductive_LCons @ A @ X3 @ ( coinductive_LNil @ A ) ) ) )
= X3 ) ) ).
% llast_snoc
thf(fact_139_LList2__Mirabelle__hamjzmohle_Ollast__lappend,axiom,
! [A: $tType,X3: coinductive_llist @ A,Y2: coinductive_llist @ A,A4: A] :
( ( member @ ( coinductive_llist @ A ) @ X3 @ ( lList2236698231inlsts @ A @ ( top_top @ ( set @ A ) ) ) )
=> ( ( member @ ( coinductive_llist @ A ) @ Y2 @ ( lList2236698231inlsts @ A @ ( top_top @ ( set @ A ) ) ) )
=> ( ( lList2170638824_llast @ A @ ( coinductive_lappend @ A @ X3 @ ( coinductive_LCons @ A @ A4 @ Y2 ) ) )
= ( lList2170638824_llast @ A @ ( coinductive_LCons @ A @ A4 @ Y2 ) ) ) ) ) ).
% LList2_Mirabelle_hamjzmohle.llast_lappend
thf(fact_140_lbutlast__lapp__llast,axiom,
! [A: $tType,L: coinductive_llist @ A,A2: set @ A] :
( ( member @ ( coinductive_llist @ A ) @ L @ ( lList22096119349pslsts @ A @ A2 ) )
=> ( L
= ( coinductive_lappend @ A @ ( lList2370560421utlast @ A @ L ) @ ( coinductive_LCons @ A @ ( lList2170638824_llast @ A @ L ) @ ( coinductive_LNil @ A ) ) ) ) ) ).
% lbutlast_lapp_llast
thf(fact_141_possafety__pprefix__closed,axiom,
! [A: $tType,P: set @ ( coinductive_llist @ A )] :
( ( lList292406316safety @ A @ ( top_top @ ( set @ A ) ) @ P )
=> ( lList21974196564closed @ A @ P ) ) ).
% possafety_pprefix_closed
thf(fact_142_LList2__Mirabelle__hamjzmohle_Ollast__LCons,axiom,
! [B: $tType,R: coinductive_llist @ B,A2: set @ B,A4: B] :
( ( member @ ( coinductive_llist @ B ) @ R @ ( lList2236698231inlsts @ B @ A2 ) )
=> ( ( ( R
= ( coinductive_LNil @ B ) )
=> ( ( lList2170638824_llast @ B @ ( coinductive_LCons @ B @ A4 @ R ) )
= A4 ) )
& ( ( R
!= ( coinductive_LNil @ B ) )
=> ( ( lList2170638824_llast @ B @ ( coinductive_LCons @ B @ A4 @ R ) )
= ( lList2170638824_llast @ B @ R ) ) ) ) ) ).
% LList2_Mirabelle_hamjzmohle.llast_LCons
thf(fact_143_Coinductive__List_Ofinite__lprefix__nitpick__simps_I3_J,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Y2: A,Ys: coinductive_llist @ A] :
( ( coindu328551480prefix @ A @ Xs @ ( coinductive_LCons @ A @ Y2 @ Ys ) )
= ( ( Xs
= ( coinductive_LNil @ A ) )
| ? [Xs2: coinductive_llist @ A] :
( ( Xs
= ( coinductive_LCons @ A @ Y2 @ Xs2 ) )
& ( coindu328551480prefix @ A @ Xs2 @ Ys ) ) ) ) ).
% Coinductive_List.finite_lprefix_nitpick_simps(3)
thf(fact_144_top__conj_I1_J,axiom,
! [A: $tType,X3: A,P: $o] :
( ( ( top_top @ ( A > $o ) @ X3 )
& P )
= P ) ).
% top_conj(1)
thf(fact_145_top__conj_I2_J,axiom,
! [A: $tType,P: $o,X3: A] :
( ( P
& ( top_top @ ( A > $o ) @ X3 ) )
= P ) ).
% top_conj(2)
thf(fact_146_lstrict__prefix__code_I2_J,axiom,
! [B: $tType,Y2: B,Ys: coinductive_llist @ B] : ( coindu1478340336prefix @ B @ ( coinductive_LNil @ B ) @ ( coinductive_LCons @ B @ Y2 @ Ys ) ) ).
% lstrict_prefix_code(2)
thf(fact_147_lstrict__prefix__code_I4_J,axiom,
! [B: $tType,X3: B,Xs: coinductive_llist @ B,Y2: B,Ys: coinductive_llist @ B] :
( ( coindu1478340336prefix @ B @ ( coinductive_LCons @ B @ X3 @ Xs ) @ ( coinductive_LCons @ B @ Y2 @ Ys ) )
= ( ( X3 = Y2 )
& ( coindu1478340336prefix @ B @ Xs @ Ys ) ) ) ).
% lstrict_prefix_code(4)
thf(fact_148_lstrict__prefix__code_I1_J,axiom,
! [A: $tType] :
~ ( coindu1478340336prefix @ A @ ( coinductive_LNil @ A ) @ ( coinductive_LNil @ A ) ) ).
% lstrict_prefix_code(1)
thf(fact_149_lstrict__prefix__code_I3_J,axiom,
! [B: $tType,X3: B,Xs: coinductive_llist @ B] :
~ ( coindu1478340336prefix @ B @ ( coinductive_LCons @ B @ X3 @ Xs ) @ ( coinductive_LNil @ B ) ) ).
% lstrict_prefix_code(3)
thf(fact_150_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_151_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_152_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_153_finlsts__rec__LCons__def,axiom,
! [B: $tType,A: $tType,F: ( coinductive_llist @ A ) > B,C2: B,D2: A > ( coinductive_llist @ A ) > B > B,R: coinductive_llist @ A,A2: set @ A,A4: A] :
( ( F
= ( lList21916056377ts_rec @ B @ A @ C2 @ D2 ) )
=> ( ( member @ ( coinductive_llist @ A ) @ R @ ( lList2236698231inlsts @ A @ A2 ) )
=> ( ( F @ ( coinductive_LCons @ A @ A4 @ R ) )
= ( D2 @ A4 @ R @ ( F @ R ) ) ) ) ) ).
% finlsts_rec_LCons_def
thf(fact_154_finlsts__rec__LCons,axiom,
! [B: $tType,A: $tType,R: coinductive_llist @ A,A2: set @ A,C2: B,D2: A > ( coinductive_llist @ A ) > B > B,A4: A] :
( ( member @ ( coinductive_llist @ A ) @ R @ ( lList2236698231inlsts @ A @ A2 ) )
=> ( ( lList21916056377ts_rec @ B @ A @ C2 @ D2 @ ( coinductive_LCons @ A @ A4 @ R ) )
= ( D2 @ A4 @ R @ ( lList21916056377ts_rec @ B @ A @ C2 @ D2 @ R ) ) ) ) ).
% finlsts_rec_LCons
thf(fact_155_LList2__Mirabelle__hamjzmohle_Ollength__LCons,axiom,
! [A: $tType,R: coinductive_llist @ A,A2: set @ A,A4: A] :
( ( member @ ( coinductive_llist @ A ) @ R @ ( lList2236698231inlsts @ A @ A2 ) )
=> ( ( lList21232602520length @ A @ ( coinductive_LCons @ A @ A4 @ R ) )
= ( suc @ ( lList21232602520length @ A @ R ) ) ) ) ).
% LList2_Mirabelle_hamjzmohle.llength_LCons
thf(fact_156_finlsts__rec__LNil__def,axiom,
! [A: $tType,B: $tType,F: ( coinductive_llist @ A ) > B,C2: B,D2: A > ( coinductive_llist @ A ) > B > B] :
( ( F
= ( lList21916056377ts_rec @ B @ A @ C2 @ D2 ) )
=> ( ( F @ ( coinductive_LNil @ A ) )
= C2 ) ) ).
% finlsts_rec_LNil_def
thf(fact_157_finlsts__rec__LNil,axiom,
! [B: $tType,A: $tType,C2: A,D2: B > ( coinductive_llist @ B ) > A > A] :
( ( lList21916056377ts_rec @ A @ B @ C2 @ D2 @ ( coinductive_LNil @ B ) )
= C2 ) ).
% finlsts_rec_LNil
thf(fact_158_ltake__LCons__Suc,axiom,
! [A: $tType,A4: A,L: coinductive_llist @ A,I: nat] :
( ( lList22119844313_ltake @ A @ ( coinductive_LCons @ A @ A4 @ L ) @ ( suc @ I ) )
= ( coinductive_LCons @ A @ A4 @ ( lList22119844313_ltake @ A @ L @ I ) ) ) ).
% ltake_LCons_Suc
thf(fact_159_llast__singleton,axiom,
! [A: $tType,X3: A] :
( ( coinductive_llast @ A @ ( coinductive_LCons @ A @ X3 @ ( coinductive_LNil @ A ) ) )
= X3 ) ).
% llast_singleton
thf(fact_160_take__inf__less,axiom,
! [A: $tType,T: coinductive_llist @ A,I: nat] :
( ( member @ ( coinductive_llist @ A ) @ T @ ( lList21612149805nflsts @ A @ ( top_top @ ( set @ A ) ) ) )
=> ( ord_less @ ( coinductive_llist @ A ) @ ( lList22119844313_ltake @ A @ T @ I ) @ T ) ) ).
% take_inf_less
thf(fact_161_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_162_sup_Oright__idem,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [A4: A,B4: A] :
( ( sup_sup @ A @ ( sup_sup @ A @ A4 @ B4 ) @ B4 )
= ( sup_sup @ A @ A4 @ B4 ) ) ) ).
% sup.right_idem
thf(fact_163_sup__left__idem,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [X3: A,Y2: A] :
( ( sup_sup @ A @ X3 @ ( sup_sup @ A @ X3 @ Y2 ) )
= ( sup_sup @ A @ X3 @ Y2 ) ) ) ).
% sup_left_idem
thf(fact_164_sup_Oleft__idem,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [A4: A,B4: A] :
( ( sup_sup @ A @ A4 @ ( sup_sup @ A @ A4 @ B4 ) )
= ( sup_sup @ A @ A4 @ B4 ) ) ) ).
% sup.left_idem
thf(fact_165_sup__idem,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [X3: A] :
( ( sup_sup @ A @ X3 @ X3 )
= X3 ) ) ).
% sup_idem
thf(fact_166_sup_Oidem,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [A4: A] :
( ( sup_sup @ A @ A4 @ A4 )
= A4 ) ) ).
% sup.idem
thf(fact_167_sup__apply,axiom,
! [B: $tType,A: $tType] :
( ( semilattice_sup @ B @ ( type2 @ B ) )
=> ( ( sup_sup @ ( A > B ) )
= ( ^ [F2: A > B,G2: A > B,X2: A] : ( sup_sup @ B @ ( F2 @ X2 ) @ ( G2 @ X2 ) ) ) ) ) ).
% sup_apply
thf(fact_168_Un__iff,axiom,
! [A: $tType,C2: A,A2: set @ A,B2: set @ A] :
( ( member @ A @ C2 @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) )
= ( ( member @ A @ C2 @ A2 )
| ( member @ A @ C2 @ B2 ) ) ) ).
% Un_iff
thf(fact_169_UnCI,axiom,
! [A: $tType,C2: A,B2: set @ A,A2: set @ A] :
( ( ~ ( member @ A @ C2 @ B2 )
=> ( member @ A @ C2 @ A2 ) )
=> ( member @ A @ C2 @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) ) ) ).
% UnCI
thf(fact_170_sup__top__right,axiom,
! [A: $tType] :
( ( bounded_lattice_top @ A @ ( type2 @ A ) )
=> ! [X3: A] :
( ( sup_sup @ A @ X3 @ ( top_top @ A ) )
= ( top_top @ A ) ) ) ).
% sup_top_right
thf(fact_171_sup__top__left,axiom,
! [A: $tType] :
( ( bounded_lattice_top @ A @ ( type2 @ A ) )
=> ! [X3: A] :
( ( sup_sup @ A @ ( top_top @ A ) @ X3 )
= ( top_top @ A ) ) ) ).
% sup_top_left
thf(fact_172_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_173_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_174_llast__LCons2,axiom,
! [A: $tType,X3: A,Y2: A,Xs: coinductive_llist @ A] :
( ( coinductive_llast @ A @ ( coinductive_LCons @ A @ X3 @ ( coinductive_LCons @ A @ Y2 @ Xs ) ) )
= ( coinductive_llast @ A @ ( coinductive_LCons @ A @ Y2 @ Xs ) ) ) ).
% llast_LCons2
thf(fact_175_less__LCons,axiom,
! [A: $tType,A4: A,R: coinductive_llist @ A,B4: A,T: coinductive_llist @ A] :
( ( ord_less @ ( coinductive_llist @ A ) @ ( coinductive_LCons @ A @ A4 @ R ) @ ( coinductive_LCons @ A @ B4 @ T ) )
= ( ( A4 = B4 )
& ( ord_less @ ( coinductive_llist @ A ) @ R @ T ) ) ) ).
% less_LCons
thf(fact_176_llist__less__finT,axiom,
! [A: $tType,R: coinductive_llist @ A,S: coinductive_llist @ A,A2: set @ A] :
( ( ord_less @ ( coinductive_llist @ A ) @ R @ S )
=> ( ( member @ ( coinductive_llist @ A ) @ S @ ( lList2236698231inlsts @ A @ A2 ) )
=> ( member @ ( coinductive_llist @ A ) @ R @ ( lList2236698231inlsts @ A @ A2 ) ) ) ) ).
% llist_less_finT
thf(fact_177_LNil__less__LCons,axiom,
! [A: $tType,A4: A,T: coinductive_llist @ A] : ( ord_less @ ( coinductive_llist @ A ) @ ( coinductive_LNil @ A ) @ ( coinductive_LCons @ A @ A4 @ T ) ) ).
% LNil_less_LCons
thf(fact_178_sup_Ostrict__coboundedI2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [C2: A,B4: A,A4: A] :
( ( ord_less @ A @ C2 @ B4 )
=> ( ord_less @ A @ C2 @ ( sup_sup @ A @ A4 @ B4 ) ) ) ) ).
% sup.strict_coboundedI2
thf(fact_179_sup_Ostrict__coboundedI1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [C2: A,A4: A,B4: A] :
( ( ord_less @ A @ C2 @ A4 )
=> ( ord_less @ A @ C2 @ ( sup_sup @ A @ A4 @ B4 ) ) ) ) ).
% sup.strict_coboundedI1
thf(fact_180_sup_Ostrict__order__iff,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ( ( ord_less @ A )
= ( ^ [B5: A,A6: A] :
( ( A6
= ( sup_sup @ A @ A6 @ B5 ) )
& ( A6 != B5 ) ) ) ) ) ).
% sup.strict_order_iff
thf(fact_181_sup_Ostrict__boundedE,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [B4: A,C2: A,A4: A] :
( ( ord_less @ A @ ( sup_sup @ A @ B4 @ C2 ) @ A4 )
=> ~ ( ( ord_less @ A @ B4 @ A4 )
=> ~ ( ord_less @ A @ C2 @ A4 ) ) ) ) ).
% sup.strict_boundedE
thf(fact_182_sup__left__commute,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [X3: A,Y2: A,Z2: A] :
( ( sup_sup @ A @ X3 @ ( sup_sup @ A @ Y2 @ Z2 ) )
= ( sup_sup @ A @ Y2 @ ( sup_sup @ A @ X3 @ Z2 ) ) ) ) ).
% sup_left_commute
thf(fact_183_sup_Oleft__commute,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [B4: A,A4: A,C2: A] :
( ( sup_sup @ A @ B4 @ ( sup_sup @ A @ A4 @ C2 ) )
= ( sup_sup @ A @ A4 @ ( sup_sup @ A @ B4 @ C2 ) ) ) ) ).
% sup.left_commute
thf(fact_184_sup__commute,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ( ( sup_sup @ A )
= ( ^ [X2: A,Y: A] : ( sup_sup @ A @ Y @ X2 ) ) ) ) ).
% sup_commute
thf(fact_185_sup_Ocommute,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ( ( sup_sup @ A )
= ( ^ [A6: A,B5: A] : ( sup_sup @ A @ B5 @ A6 ) ) ) ) ).
% sup.commute
thf(fact_186_less__supI2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [X3: A,B4: A,A4: A] :
( ( ord_less @ A @ X3 @ B4 )
=> ( ord_less @ A @ X3 @ ( sup_sup @ A @ A4 @ B4 ) ) ) ) ).
% less_supI2
thf(fact_187_less__supI1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [X3: A,A4: A,B4: A] :
( ( ord_less @ A @ X3 @ A4 )
=> ( ord_less @ A @ X3 @ ( sup_sup @ A @ A4 @ B4 ) ) ) ) ).
% less_supI1
thf(fact_188_sup__assoc,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [X3: A,Y2: A,Z2: A] :
( ( sup_sup @ A @ ( sup_sup @ A @ X3 @ Y2 ) @ Z2 )
= ( sup_sup @ A @ X3 @ ( sup_sup @ A @ Y2 @ Z2 ) ) ) ) ).
% sup_assoc
thf(fact_189_sup_Oassoc,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [A4: A,B4: A,C2: A] :
( ( sup_sup @ A @ ( sup_sup @ A @ A4 @ B4 ) @ C2 )
= ( sup_sup @ A @ A4 @ ( sup_sup @ A @ B4 @ C2 ) ) ) ) ).
% sup.assoc
thf(fact_190_sup__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( semilattice_sup @ B @ ( type2 @ B ) )
=> ( ( sup_sup @ ( A > B ) )
= ( ^ [F2: A > B,G2: A > B,X2: A] : ( sup_sup @ B @ ( F2 @ X2 ) @ ( G2 @ X2 ) ) ) ) ) ).
% sup_fun_def
thf(fact_191_inf__sup__aci_I5_J,axiom,
! [A: $tType] :
( ( lattice @ A @ ( type2 @ A ) )
=> ( ( sup_sup @ A )
= ( ^ [X2: A,Y: A] : ( sup_sup @ A @ Y @ X2 ) ) ) ) ).
% inf_sup_aci(5)
thf(fact_192_inf__sup__aci_I6_J,axiom,
! [A: $tType] :
( ( lattice @ A @ ( type2 @ A ) )
=> ! [X3: A,Y2: A,Z2: A] :
( ( sup_sup @ A @ ( sup_sup @ A @ X3 @ Y2 ) @ Z2 )
= ( sup_sup @ A @ X3 @ ( sup_sup @ A @ Y2 @ Z2 ) ) ) ) ).
% inf_sup_aci(6)
thf(fact_193_inf__sup__aci_I7_J,axiom,
! [A: $tType] :
( ( lattice @ A @ ( type2 @ A ) )
=> ! [X3: A,Y2: A,Z2: A] :
( ( sup_sup @ A @ X3 @ ( sup_sup @ A @ Y2 @ Z2 ) )
= ( sup_sup @ A @ Y2 @ ( sup_sup @ A @ X3 @ Z2 ) ) ) ) ).
% inf_sup_aci(7)
thf(fact_194_inf__sup__aci_I8_J,axiom,
! [A: $tType] :
( ( lattice @ A @ ( type2 @ A ) )
=> ! [X3: A,Y2: A] :
( ( sup_sup @ A @ X3 @ ( sup_sup @ A @ X3 @ Y2 ) )
= ( sup_sup @ A @ X3 @ Y2 ) ) ) ).
% inf_sup_aci(8)
thf(fact_195_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_196_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_197_top_Onot__eq__extremum,axiom,
! [A: $tType] :
( ( order_top @ A @ ( type2 @ A ) )
=> ! [A4: A] :
( ( A4
!= ( top_top @ A ) )
= ( ord_less @ A @ A4 @ ( top_top @ A ) ) ) ) ).
% top.not_eq_extremum
thf(fact_198_top_Oextremum__strict,axiom,
! [A: $tType] :
( ( order_top @ A @ ( type2 @ A ) )
=> ! [A4: A] :
~ ( ord_less @ A @ ( top_top @ A ) @ A4 ) ) ).
% top.extremum_strict
thf(fact_199_Un__Diff,axiom,
! [A: $tType,A2: set @ A,B2: set @ A,C3: set @ A] :
( ( minus_minus @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) @ C3 )
= ( sup_sup @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A2 @ C3 ) @ ( minus_minus @ ( set @ A ) @ B2 @ C3 ) ) ) ).
% Un_Diff
thf(fact_200_not__less__LNil,axiom,
! [A: $tType,R: coinductive_llist @ A] :
~ ( ord_less @ ( coinductive_llist @ A ) @ R @ ( coinductive_LNil @ A ) ) ).
% not_less_LNil
thf(fact_201_dual__order_Ostrict__implies__not__eq,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [B4: A,A4: A] :
( ( ord_less @ A @ B4 @ A4 )
=> ( A4 != B4 ) ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_202_order_Ostrict__implies__not__eq,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A4: A,B4: A] :
( ( ord_less @ A @ A4 @ B4 )
=> ( A4 != B4 ) ) ) ).
% order.strict_implies_not_eq
thf(fact_203_not__less__iff__gr__or__eq,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X3: A,Y2: A] :
( ( ~ ( ord_less @ A @ X3 @ Y2 ) )
= ( ( ord_less @ A @ Y2 @ X3 )
| ( X3 = Y2 ) ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_204_dual__order_Ostrict__trans,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [B4: A,A4: A,C2: A] :
( ( ord_less @ A @ B4 @ A4 )
=> ( ( ord_less @ A @ C2 @ B4 )
=> ( ord_less @ A @ C2 @ A4 ) ) ) ) ).
% dual_order.strict_trans
thf(fact_205_less__imp__not__less,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X3: A,Y2: A] :
( ( ord_less @ A @ X3 @ Y2 )
=> ~ ( ord_less @ A @ Y2 @ X3 ) ) ) ).
% less_imp_not_less
thf(fact_206_order_Ostrict__trans,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A4: A,B4: A,C2: A] :
( ( ord_less @ A @ A4 @ B4 )
=> ( ( ord_less @ A @ B4 @ C2 )
=> ( ord_less @ A @ A4 @ C2 ) ) ) ) ).
% order.strict_trans
thf(fact_207_dual__order_Oirrefl,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A4: A] :
~ ( ord_less @ A @ A4 @ A4 ) ) ).
% dual_order.irrefl
thf(fact_208_linorder__cases,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X3: A,Y2: A] :
( ~ ( ord_less @ A @ X3 @ Y2 )
=> ( ( X3 != Y2 )
=> ( ord_less @ A @ Y2 @ X3 ) ) ) ) ).
% linorder_cases
thf(fact_209_less__imp__triv,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X3: A,Y2: A,P: $o] :
( ( ord_less @ A @ X3 @ Y2 )
=> ( ( ord_less @ A @ Y2 @ X3 )
=> P ) ) ) ).
% less_imp_triv
thf(fact_210_less__imp__not__eq2,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [X3: A,Y2: A] :
( ( ord_less @ A @ X3 @ Y2 )
=> ( Y2 != X3 ) ) ) ).
% less_imp_not_eq2
thf(fact_211_antisym__conv3,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [Y2: A,X3: A] :
( ~ ( ord_less @ A @ Y2 @ X3 )
=> ( ( ~ ( ord_less @ A @ X3 @ Y2 ) )
= ( X3 = Y2 ) ) ) ) ).
% antisym_conv3
thf(fact_212_less__induct,axiom,
! [A: $tType] :
( ( wellorder @ A @ ( type2 @ A ) )
=> ! [P: A > $o,A4: A] :
( ! [X4: A] :
( ! [Y3: A] :
( ( ord_less @ A @ Y3 @ X4 )
=> ( P @ Y3 ) )
=> ( P @ X4 ) )
=> ( P @ A4 ) ) ) ).
% less_induct
thf(fact_213_less__not__sym,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X3: A,Y2: A] :
( ( ord_less @ A @ X3 @ Y2 )
=> ~ ( ord_less @ A @ Y2 @ X3 ) ) ) ).
% less_not_sym
thf(fact_214_less__imp__not__eq,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [X3: A,Y2: A] :
( ( ord_less @ A @ X3 @ Y2 )
=> ( X3 != Y2 ) ) ) ).
% less_imp_not_eq
thf(fact_215_dual__order_Oasym,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [B4: A,A4: A] :
( ( ord_less @ A @ B4 @ A4 )
=> ~ ( ord_less @ A @ A4 @ B4 ) ) ) ).
% dual_order.asym
thf(fact_216_ord__less__eq__trans,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [A4: A,B4: A,C2: A] :
( ( ord_less @ A @ A4 @ B4 )
=> ( ( B4 = C2 )
=> ( ord_less @ A @ A4 @ C2 ) ) ) ) ).
% ord_less_eq_trans
thf(fact_217_ord__eq__less__trans,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [A4: A,B4: A,C2: A] :
( ( A4 = B4 )
=> ( ( ord_less @ A @ B4 @ C2 )
=> ( ord_less @ A @ A4 @ C2 ) ) ) ) ).
% ord_eq_less_trans
thf(fact_218_less__irrefl,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X3: A] :
~ ( ord_less @ A @ X3 @ X3 ) ) ).
% less_irrefl
thf(fact_219_less__linear,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X3: A,Y2: A] :
( ( ord_less @ A @ X3 @ Y2 )
| ( X3 = Y2 )
| ( ord_less @ A @ Y2 @ X3 ) ) ) ).
% less_linear
thf(fact_220_less__trans,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X3: A,Y2: A,Z2: A] :
( ( ord_less @ A @ X3 @ Y2 )
=> ( ( ord_less @ A @ Y2 @ Z2 )
=> ( ord_less @ A @ X3 @ Z2 ) ) ) ) ).
% less_trans
thf(fact_221_less__asym_H,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [A4: A,B4: A] :
( ( ord_less @ A @ A4 @ B4 )
=> ~ ( ord_less @ A @ B4 @ A4 ) ) ) ).
% less_asym'
thf(fact_222_less__asym,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X3: A,Y2: A] :
( ( ord_less @ A @ X3 @ Y2 )
=> ~ ( ord_less @ A @ Y2 @ X3 ) ) ) ).
% less_asym
thf(fact_223_less__imp__neq,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [X3: A,Y2: A] :
( ( ord_less @ A @ X3 @ Y2 )
=> ( X3 != Y2 ) ) ) ).
% less_imp_neq
thf(fact_224_dense,axiom,
! [A: $tType] :
( ( dense_order @ A @ ( type2 @ A ) )
=> ! [X3: A,Y2: A] :
( ( ord_less @ A @ X3 @ Y2 )
=> ? [Z3: A] :
( ( ord_less @ A @ X3 @ Z3 )
& ( ord_less @ A @ Z3 @ Y2 ) ) ) ) ).
% dense
thf(fact_225_order_Oasym,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A4: A,B4: A] :
( ( ord_less @ A @ A4 @ B4 )
=> ~ ( ord_less @ A @ B4 @ A4 ) ) ) ).
% order.asym
thf(fact_226_neq__iff,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X3: A,Y2: A] :
( ( X3 != Y2 )
= ( ( ord_less @ A @ X3 @ Y2 )
| ( ord_less @ A @ Y2 @ X3 ) ) ) ) ).
% neq_iff
thf(fact_227_neqE,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X3: A,Y2: A] :
( ( X3 != Y2 )
=> ( ~ ( ord_less @ A @ X3 @ Y2 )
=> ( ord_less @ A @ Y2 @ X3 ) ) ) ) ).
% neqE
thf(fact_228_gt__ex,axiom,
! [A: $tType] :
( ( no_top @ A @ ( type2 @ A ) )
=> ! [X3: A] :
? [X1: A] : ( ord_less @ A @ X3 @ X1 ) ) ).
% gt_ex
thf(fact_229_lt__ex,axiom,
! [A: $tType] :
( ( no_bot @ A @ ( type2 @ A ) )
=> ! [X3: A] :
? [Y4: A] : ( ord_less @ A @ Y4 @ X3 ) ) ).
% lt_ex
thf(fact_230_order__less__subst2,axiom,
! [A: $tType,C: $tType] :
( ( ( order @ C @ ( type2 @ C ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A4: A,B4: A,F: A > C,C2: C] :
( ( ord_less @ A @ A4 @ B4 )
=> ( ( ord_less @ C @ ( F @ B4 ) @ C2 )
=> ( ! [X4: A,Y4: A] :
( ( ord_less @ A @ X4 @ Y4 )
=> ( ord_less @ C @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less @ C @ ( F @ A4 ) @ C2 ) ) ) ) ) ).
% order_less_subst2
thf(fact_231_order__less__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B @ ( type2 @ B ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A4: A,F: B > A,B4: B,C2: B] :
( ( ord_less @ A @ A4 @ ( F @ B4 ) )
=> ( ( ord_less @ B @ B4 @ C2 )
=> ( ! [X4: B,Y4: B] :
( ( ord_less @ B @ X4 @ Y4 )
=> ( ord_less @ A @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less @ A @ A4 @ ( F @ C2 ) ) ) ) ) ) ).
% order_less_subst1
thf(fact_232_ord__less__eq__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B @ ( type2 @ B ) )
& ( ord @ A @ ( type2 @ A ) ) )
=> ! [A4: A,B4: A,F: A > B,C2: B] :
( ( ord_less @ A @ A4 @ B4 )
=> ( ( ( F @ B4 )
= C2 )
=> ( ! [X4: A,Y4: A] :
( ( ord_less @ A @ X4 @ Y4 )
=> ( ord_less @ B @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less @ B @ ( F @ A4 ) @ C2 ) ) ) ) ) ).
% ord_less_eq_subst
thf(fact_233_ord__eq__less__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B @ ( type2 @ B ) )
& ( ord @ A @ ( type2 @ A ) ) )
=> ! [A4: A,F: B > A,B4: B,C2: B] :
( ( A4
= ( F @ B4 ) )
=> ( ( ord_less @ B @ B4 @ C2 )
=> ( ! [X4: B,Y4: B] :
( ( ord_less @ B @ X4 @ Y4 )
=> ( ord_less @ A @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less @ A @ A4 @ ( F @ C2 ) ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_234_Un__left__commute,axiom,
! [A: $tType,A2: set @ A,B2: set @ A,C3: set @ A] :
( ( sup_sup @ ( set @ A ) @ A2 @ ( sup_sup @ ( set @ A ) @ B2 @ C3 ) )
= ( sup_sup @ ( set @ A ) @ B2 @ ( sup_sup @ ( set @ A ) @ A2 @ C3 ) ) ) ).
% Un_left_commute
thf(fact_235_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_236_Un__commute,axiom,
! [A: $tType] :
( ( sup_sup @ ( set @ A ) )
= ( ^ [A3: set @ A,B3: set @ A] : ( sup_sup @ ( set @ A ) @ B3 @ A3 ) ) ) ).
% Un_commute
thf(fact_237_Un__absorb,axiom,
! [A: $tType,A2: set @ A] :
( ( sup_sup @ ( set @ A ) @ A2 @ A2 )
= A2 ) ).
% Un_absorb
thf(fact_238_Un__assoc,axiom,
! [A: $tType,A2: set @ A,B2: set @ A,C3: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) @ C3 )
= ( sup_sup @ ( set @ A ) @ A2 @ ( sup_sup @ ( set @ A ) @ B2 @ C3 ) ) ) ).
% Un_assoc
thf(fact_239_ball__Un,axiom,
! [A: $tType,A2: set @ A,B2: set @ A,P: A > $o] :
( ( ! [X2: A] :
( ( member @ A @ X2 @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) )
=> ( P @ X2 ) ) )
= ( ! [X2: A] :
( ( member @ A @ X2 @ A2 )
=> ( P @ X2 ) )
& ! [X2: A] :
( ( member @ A @ X2 @ B2 )
=> ( P @ X2 ) ) ) ) ).
% ball_Un
thf(fact_240_bex__Un,axiom,
! [A: $tType,A2: set @ A,B2: set @ A,P: A > $o] :
( ( ? [X2: A] :
( ( member @ A @ X2 @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) )
& ( P @ X2 ) ) )
= ( ? [X2: A] :
( ( member @ A @ X2 @ A2 )
& ( P @ X2 ) )
| ? [X2: A] :
( ( member @ A @ X2 @ B2 )
& ( P @ X2 ) ) ) ) ).
% bex_Un
thf(fact_241_UnI2,axiom,
! [A: $tType,C2: A,B2: set @ A,A2: set @ A] :
( ( member @ A @ C2 @ B2 )
=> ( member @ A @ C2 @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) ) ) ).
% UnI2
thf(fact_242_UnI1,axiom,
! [A: $tType,C2: A,A2: set @ A,B2: set @ A] :
( ( member @ A @ C2 @ A2 )
=> ( member @ A @ C2 @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) ) ) ).
% UnI1
thf(fact_243_UnE,axiom,
! [A: $tType,C2: A,A2: set @ A,B2: set @ A] :
( ( member @ A @ C2 @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) )
=> ( ~ ( member @ A @ C2 @ A2 )
=> ( member @ A @ C2 @ B2 ) ) ) ).
% UnE
thf(fact_244_diff__strict__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
=> ! [A4: A,B4: A,D2: A,C2: A] :
( ( ord_less @ A @ A4 @ B4 )
=> ( ( ord_less @ A @ D2 @ C2 )
=> ( ord_less @ A @ ( minus_minus @ A @ A4 @ C2 ) @ ( minus_minus @ A @ B4 @ D2 ) ) ) ) ) ).
% diff_strict_mono
thf(fact_245_diff__eq__diff__less,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
=> ! [A4: A,B4: A,C2: A,D2: A] :
( ( ( minus_minus @ A @ A4 @ B4 )
= ( minus_minus @ A @ C2 @ D2 ) )
=> ( ( ord_less @ A @ A4 @ B4 )
= ( ord_less @ A @ C2 @ D2 ) ) ) ) ).
% diff_eq_diff_less
thf(fact_246_diff__strict__left__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
=> ! [B4: A,A4: A,C2: A] :
( ( ord_less @ A @ B4 @ A4 )
=> ( ord_less @ A @ ( minus_minus @ A @ C2 @ A4 ) @ ( minus_minus @ A @ C2 @ B4 ) ) ) ) ).
% diff_strict_left_mono
thf(fact_247_LList2__Mirabelle__hamjzmohle_Ollength__mono,axiom,
! [A: $tType,R: coinductive_llist @ A,A2: set @ A,S: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ R @ ( lList2236698231inlsts @ A @ A2 ) )
=> ( ( ord_less @ ( coinductive_llist @ A ) @ S @ R )
=> ( ord_less @ nat @ ( lList21232602520length @ A @ S ) @ ( lList21232602520length @ A @ R ) ) ) ) ).
% LList2_Mirabelle_hamjzmohle.llength_mono
thf(fact_248_psubset__imp__ex__mem,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] :
( ( ord_less @ ( set @ A ) @ A2 @ B2 )
=> ? [B6: A] : ( member @ A @ B6 @ ( minus_minus @ ( set @ A ) @ B2 @ A2 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_249_diff__right__commute,axiom,
! [A: $tType] :
( ( cancel146912293up_add @ A @ ( type2 @ A ) )
=> ! [A4: A,C2: A,B4: A] :
( ( minus_minus @ A @ ( minus_minus @ A @ A4 @ C2 ) @ B4 )
= ( minus_minus @ A @ ( minus_minus @ A @ A4 @ B4 ) @ C2 ) ) ) ).
% diff_right_commute
thf(fact_250_diff__eq__diff__eq,axiom,
! [A: $tType] :
( ( group_add @ A @ ( type2 @ A ) )
=> ! [A4: A,B4: A,C2: A,D2: A] :
( ( ( minus_minus @ A @ A4 @ B4 )
= ( minus_minus @ A @ C2 @ D2 ) )
=> ( ( A4 = B4 )
= ( C2 = D2 ) ) ) ) ).
% diff_eq_diff_eq
thf(fact_251_lapp__take__less,axiom,
! [A: $tType,I: nat,R: coinductive_llist @ A,S: coinductive_llist @ A] :
( ( ord_less @ nat @ I @ ( lList21232602520length @ A @ R ) )
=> ( ord_less @ ( coinductive_llist @ A ) @ ( lList22119844313_ltake @ A @ ( coinductive_lappend @ A @ R @ S ) @ I ) @ R ) ) ).
% lapp_take_less
thf(fact_252_diff__strict__right__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
=> ! [A4: A,B4: A,C2: A] :
( ( ord_less @ A @ A4 @ B4 )
=> ( ord_less @ A @ ( minus_minus @ A @ A4 @ C2 ) @ ( minus_minus @ A @ B4 @ C2 ) ) ) ) ).
% diff_strict_right_mono
thf(fact_253_llast__lappend__LCons,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Y2: A,Ys: coinductive_llist @ A] :
( ( coinductive_lfinite @ A @ Xs )
=> ( ( coinductive_llast @ A @ ( coinductive_lappend @ A @ Xs @ ( coinductive_LCons @ A @ Y2 @ Ys ) ) )
= ( coinductive_llast @ A @ ( coinductive_LCons @ A @ Y2 @ Ys ) ) ) ) ).
% llast_lappend_LCons
thf(fact_254_pprefix__closed__def,axiom,
! [A: $tType] :
( ( lList21974196564closed @ A )
= ( ^ [A3: set @ ( coinductive_llist @ A )] :
! [X2: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ X2 @ A3 )
=> ! [S3: coinductive_llist @ A] :
( ( ( ord_less_eq @ ( coinductive_llist @ A ) @ S3 @ X2 )
& ( S3
!= ( coinductive_LNil @ A ) ) )
=> ( member @ ( coinductive_llist @ A ) @ S3 @ A3 ) ) ) ) ) ).
% pprefix_closed_def
thf(fact_255_order__refl,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X3: A] : ( ord_less_eq @ A @ X3 @ X3 ) ) ).
% order_refl
%----Type constructors (44)
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__lattice__top,axiom,
! [A8: $tType,A9: $tType] :
( ( bounded_lattice @ A9 @ ( type2 @ A9 ) )
=> ( bounded_lattice_top @ ( A8 > A9 ) @ ( type2 @ ( A8 > A9 ) ) ) ) ).
thf(tcon_fun___Lattices_Osemilattice__sup,axiom,
! [A8: $tType,A9: $tType] :
( ( semilattice_sup @ A9 @ ( type2 @ A9 ) )
=> ( semilattice_sup @ ( A8 > A9 ) @ ( type2 @ ( A8 > A9 ) ) ) ) ).
thf(tcon_fun___Orderings_Oorder__top,axiom,
! [A8: $tType,A9: $tType] :
( ( order_top @ A9 @ ( type2 @ A9 ) )
=> ( order_top @ ( A8 > A9 ) @ ( type2 @ ( A8 > A9 ) ) ) ) ).
thf(tcon_fun___Orderings_Opreorder,axiom,
! [A8: $tType,A9: $tType] :
( ( preorder @ A9 @ ( type2 @ A9 ) )
=> ( preorder @ ( A8 > A9 ) @ ( type2 @ ( A8 > A9 ) ) ) ) ).
thf(tcon_fun___Lattices_Olattice,axiom,
! [A8: $tType,A9: $tType] :
( ( lattice @ A9 @ ( type2 @ A9 ) )
=> ( lattice @ ( A8 > A9 ) @ ( type2 @ ( A8 > A9 ) ) ) ) ).
thf(tcon_fun___Orderings_Oorder,axiom,
! [A8: $tType,A9: $tType] :
( ( order @ A9 @ ( type2 @ A9 ) )
=> ( order @ ( 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_Oord,axiom,
! [A8: $tType,A9: $tType] :
( ( ord @ A9 @ ( type2 @ A9 ) )
=> ( ord @ ( A8 > A9 ) @ ( type2 @ ( A8 > A9 ) ) ) ) ).
thf(tcon_fun___Groups_Ominus,axiom,
! [A8: $tType,A9: $tType] :
( ( minus @ A9 @ ( type2 @ A9 ) )
=> ( minus @ ( A8 > A9 ) @ ( type2 @ ( A8 > A9 ) ) ) ) ).
thf(tcon_Nat_Onat___Groups_Ocancel__ab__semigroup__add,axiom,
cancel146912293up_add @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Lattices_Osemilattice__sup_3,axiom,
semilattice_sup @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Orderings_Owellorder,axiom,
wellorder @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Orderings_Opreorder_4,axiom,
preorder @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Orderings_Olinorder,axiom,
linorder @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Orderings_Ono__top,axiom,
no_top @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Lattices_Olattice_5,axiom,
lattice @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Orderings_Oorder_6,axiom,
order @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Orderings_Oord_7,axiom,
ord @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Groups_Ominus_8,axiom,
minus @ nat @ ( type2 @ nat ) ).
thf(tcon_Set_Oset___Lattices_Obounded__lattice__top_9,axiom,
! [A8: $tType] : ( bounded_lattice_top @ ( set @ A8 ) @ ( type2 @ ( set @ A8 ) ) ) ).
thf(tcon_Set_Oset___Lattices_Osemilattice__sup_10,axiom,
! [A8: $tType] : ( semilattice_sup @ ( set @ A8 ) @ ( type2 @ ( set @ A8 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder__top_11,axiom,
! [A8: $tType] : ( order_top @ ( set @ A8 ) @ ( type2 @ ( set @ A8 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Opreorder_12,axiom,
! [A8: $tType] : ( preorder @ ( set @ A8 ) @ ( type2 @ ( set @ A8 ) ) ) ).
thf(tcon_Set_Oset___Lattices_Olattice_13,axiom,
! [A8: $tType] : ( lattice @ ( set @ A8 ) @ ( type2 @ ( set @ A8 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder_14,axiom,
! [A8: $tType] : ( order @ ( set @ A8 ) @ ( type2 @ ( set @ A8 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Otop_15,axiom,
! [A8: $tType] : ( top @ ( set @ A8 ) @ ( type2 @ ( set @ A8 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Oord_16,axiom,
! [A8: $tType] : ( ord @ ( set @ A8 ) @ ( type2 @ ( set @ A8 ) ) ) ).
thf(tcon_Set_Oset___Groups_Ominus_17,axiom,
! [A8: $tType] : ( minus @ ( set @ A8 ) @ ( type2 @ ( set @ A8 ) ) ) ).
thf(tcon_HOL_Obool___Lattices_Obounded__lattice__top_18,axiom,
bounded_lattice_top @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Lattices_Osemilattice__sup_19,axiom,
semilattice_sup @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Oorder__top_20,axiom,
order_top @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Opreorder_21,axiom,
preorder @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Olinorder_22,axiom,
linorder @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Lattices_Olattice_23,axiom,
lattice @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Oorder_24,axiom,
order @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Otop_25,axiom,
top @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Oord_26,axiom,
ord @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Groups_Ominus_27,axiom,
minus @ $o @ ( type2 @ $o ) ).
thf(tcon_Coinductive__List_Ollist___Orderings_Opreorder_28,axiom,
! [A8: $tType] : ( preorder @ ( coinductive_llist @ A8 ) @ ( type2 @ ( coinductive_llist @ A8 ) ) ) ).
thf(tcon_Coinductive__List_Ollist___Orderings_Oorder_29,axiom,
! [A8: $tType] : ( order @ ( coinductive_llist @ A8 ) @ ( type2 @ ( coinductive_llist @ A8 ) ) ) ).
thf(tcon_Coinductive__List_Ollist___Orderings_Oord_30,axiom,
! [A8: $tType] : ( ord @ ( coinductive_llist @ A8 ) @ ( type2 @ ( coinductive_llist @ A8 ) ) ) ).
%----Conjectures (4)
thf(conj_0,hypothesis,
lList21952340509veness @ a @ a2 @ p ).
thf(conj_1,hypothesis,
! [T3: coinductive_llist @ a] :
( ( member @ ( coinductive_llist @ a ) @ T3 @ ( lList2435255213lllsts @ a @ a2 ) )
=> ( ( member @ ( coinductive_llist @ a ) @ ( coinductive_lappend @ a @ s @ T3 ) @ p )
=> r ) ) ).
thf(conj_2,hypothesis,
( ~ ( member @ ( coinductive_llist @ a ) @ s @ ( lList22096119349pslsts @ a @ a2 ) )
=> r ) ).
thf(conj_3,conjecture,
r ).
%------------------------------------------------------------------------------