TPTP Problem File: DAT178^1.p
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%------------------------------------------------------------------------------
% File : DAT178^1 : TPTP v8.2.0. Released v7.0.0.
% Domain : Data Structures
% Problem : Lazy lists II 507
% 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__507.p [Bla16]
% Status : Theorem
% Rating : 0.00 v7.2.0, 0.25 v7.1.0
% Syntax : Number of formulae : 321 ( 131 unt; 48 typ; 0 def)
% Number of atoms : 665 ( 253 equ; 0 cnn)
% Maximal formula atoms : 7 ( 2 avg)
% Number of connectives : 3892 ( 92 ~; 30 |; 64 &;3425 @)
% ( 0 <=>; 281 =>; 0 <=; 0 <~>)
% Maximal formula depth : 19 ( 7 avg)
% Number of types : 2 ( 1 usr)
% Number of type conns : 191 ( 191 >; 0 *; 0 +; 0 <<)
% Number of symbols : 48 ( 47 usr; 4 con; 0-5 aty)
% Number of variables : 970 ( 33 ^; 859 !; 38 ?; 970 :)
% ( 40 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2016-07-13 14:46:49.184
%------------------------------------------------------------------------------
%----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_Groups_Ominus,type,
minus:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Otop,type,
top:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Lattices_Olattice,type,
lattice:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Lattices_Osemilattice__sup,type,
semilattice_sup:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Lattices_Obounded__lattice__top,type,
bounded_lattice_top:
!>[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_Olappend,type,
coinductive_lappend:
!>[A: $tType] : ( ( coinductive_llist @ A ) > ( coinductive_llist @ A ) > ( coinductive_llist @ A ) ) ).
thf(sy_c_Coinductive__List_Oldistinct,type,
coindu351974385stinct:
!>[A: $tType] : ( ( coinductive_llist @ A ) > $o ) ).
thf(sy_c_Coinductive__List_Olfilter,type,
coinductive_lfilter:
!>[A: $tType] : ( ( A > $o ) > ( coinductive_llist @ A ) > ( coinductive_llist @ A ) ) ).
thf(sy_c_Coinductive__List_Olfinite,type,
coinductive_lfinite:
!>[A: $tType] : ( ( coinductive_llist @ A ) > $o ) ).
thf(sy_c_Coinductive__List_Ollast,type,
coinductive_llast:
!>[A: $tType] : ( ( coinductive_llist @ A ) > A ) ).
thf(sy_c_Coinductive__List_Ollexord,type,
coinductive_llexord:
!>[A: $tType] : ( ( A > A > $o ) > ( coinductive_llist @ A ) > ( coinductive_llist @ A ) > $o ) ).
thf(sy_c_Coinductive__List_Ollist_OLCons,type,
coinductive_LCons:
!>[A: $tType] : ( A > ( coinductive_llist @ A ) > ( coinductive_llist @ A ) ) ).
thf(sy_c_Coinductive__List_Ollist_OLNil,type,
coinductive_LNil:
!>[A: $tType] : ( coinductive_llist @ A ) ).
thf(sy_c_Coinductive__List_Ollist_Olnull,type,
coinductive_lnull:
!>[A: $tType] : ( ( coinductive_llist @ A ) > $o ) ).
thf(sy_c_Coinductive__List_Ollist_Olset,type,
coinductive_lset:
!>[A: $tType] : ( ( coinductive_llist @ A ) > ( set @ A ) ) ).
thf(sy_c_Coinductive__List_Olmember,type,
coinductive_lmember:
!>[A: $tType] : ( A > ( coinductive_llist @ A ) > $o ) ).
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_HOL_Oundefined,type,
undefined:
!>[A: $tType] : A ).
thf(sy_c_LList2__Mirabelle__hamjzmohle_Oalllsts,type,
lList2435255213lllsts:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( coinductive_llist @ A ) ) ) ).
thf(sy_c_LList2__Mirabelle__hamjzmohle_Oalllstsp,type,
lList21511617539llstsp:
!>[A: $tType] : ( ( A > $o ) > ( coinductive_llist @ A ) > $o ) ).
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_Ofinlstsp,type,
lList2860480441nlstsp:
!>[A: $tType] : ( ( A > $o ) > ( coinductive_llist @ A ) > $o ) ).
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_Olbutlast,type,
lList2370560421utlast:
!>[A: $tType] : ( ( coinductive_llist @ A ) > ( coinductive_llist @ A ) ) ).
thf(sy_c_LList2__Mirabelle__hamjzmohle_Ollast,type,
lList2170638824_llast:
!>[A: $tType] : ( ( coinductive_llist @ A ) > A ) ).
thf(sy_c_LList2__Mirabelle__hamjzmohle_Olrev,type,
lList2281150353e_lrev:
!>[A: $tType] : ( ( coinductive_llist @ A ) > ( coinductive_llist @ A ) ) ).
thf(sy_c_LList2__Mirabelle__hamjzmohle_Oposlsts,type,
lList21148268032oslsts:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( coinductive_llist @ A ) ) ) ).
thf(sy_c_Lattices_Osup__class_Osup,type,
sup_sup:
!>[A: $tType] : ( A > A > 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_member,type,
member:
!>[A: $tType] : ( A > ( set @ A ) > $o ) ).
thf(sy_v_A,type,
a2: set @ a ).
thf(sy_v_P,type,
p: ( coinductive_llist @ a ) > $o ).
thf(sy_v_l____,type,
l: coinductive_llist @ a ).
thf(sy_v_la______,type,
la: coinductive_llist @ a ).
thf(sy_v_xs,type,
xs: coinductive_llist @ a ).
%----Relevant facts (254)
thf(fact_0_init,axiom,
p @ ( coinductive_LNil @ a ) ).
% init
thf(fact_1_LNil__fin_Ohyps,axiom,
( la
= ( coinductive_LNil @ a ) ) ).
% LNil_fin.hyps
thf(fact_2_l__def,axiom,
( l
= ( lList2281150353e_lrev @ a @ xs ) ) ).
% l_def
thf(fact_3_lrev__LNil,axiom,
! [A: $tType] :
( ( lList2281150353e_lrev @ A @ ( coinductive_LNil @ A ) )
= ( coinductive_LNil @ A ) ) ).
% lrev_LNil
thf(fact_4_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_5_local_Ostep,axiom,
! [Xs: coinductive_llist @ a,X: a] :
( ( member @ ( coinductive_llist @ a ) @ Xs @ ( lList2236698231inlsts @ a @ a2 ) )
=> ( ( p @ Xs )
=> ( ( member @ a @ X @ a2 )
=> ( p @ ( coinductive_lappend @ a @ Xs @ ( coinductive_LCons @ a @ X @ ( coinductive_LNil @ a ) ) ) ) ) ) ) ).
% local.step
thf(fact_6_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_7_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_8_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_9_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_10_lbutlast__LNil,axiom,
! [A: $tType] :
( ( lList2370560421utlast @ A @ ( coinductive_LNil @ A ) )
= ( coinductive_LNil @ A ) ) ).
% lbutlast_LNil
thf(fact_11_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_12_fin,axiom,
member @ ( coinductive_llist @ a ) @ xs @ ( lList2236698231inlsts @ a @ a2 ) ).
% fin
thf(fact_13__092_060open_062l_A_092_060in_062_AA_092_060_094sup_062_092_060star_062_092_060close_062,axiom,
member @ ( coinductive_llist @ a ) @ l @ ( lList2236698231inlsts @ a @ a2 ) ).
% \<open>l \<in> A\<^sup>\<star>\<close>
thf(fact_14_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_15_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_16_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_17_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_18_lbutlast__LCons,axiom,
! [A: $tType,R: coinductive_llist @ A,A2: set @ A,A3: A] :
( ( member @ ( coinductive_llist @ A ) @ R @ ( lList2236698231inlsts @ A @ A2 ) )
=> ( ( ( R
= ( coinductive_LNil @ A ) )
=> ( ( lList2370560421utlast @ A @ ( coinductive_LCons @ A @ A3 @ R ) )
= ( coinductive_LNil @ A ) ) )
& ( ( R
!= ( coinductive_LNil @ A ) )
=> ( ( lList2370560421utlast @ A @ ( coinductive_LCons @ A @ A3 @ R ) )
= ( coinductive_LCons @ A @ A3 @ ( lList2370560421utlast @ A @ R ) ) ) ) ) ) ).
% lbutlast_LCons
thf(fact_19_lrev__LCons,axiom,
! [A: $tType,R: coinductive_llist @ A,A2: set @ A,A3: A] :
( ( member @ ( coinductive_llist @ A ) @ R @ ( lList2236698231inlsts @ A @ A2 ) )
=> ( ( lList2281150353e_lrev @ A @ ( coinductive_LCons @ A @ A3 @ R ) )
= ( coinductive_lappend @ A @ ( lList2281150353e_lrev @ A @ R ) @ ( coinductive_LCons @ A @ A3 @ ( coinductive_LNil @ A ) ) ) ) ) ).
% lrev_LCons
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_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_24_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_25_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_26_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_27_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_28_finlsts_OLNil__fin,axiom,
! [A: $tType,A2: set @ A] : ( member @ ( coinductive_llist @ A ) @ ( coinductive_LNil @ A ) @ ( lList2236698231inlsts @ A @ A2 ) ) ).
% finlsts.LNil_fin
thf(fact_29_finlsts_OLCons__fin,axiom,
! [A: $tType,L3: coinductive_llist @ A,A2: set @ A,A3: A] :
( ( member @ ( coinductive_llist @ A ) @ L3 @ ( lList2236698231inlsts @ A @ A2 ) )
=> ( ( member @ A @ A3 @ A2 )
=> ( member @ ( coinductive_llist @ A ) @ ( coinductive_LCons @ A @ A3 @ L3 ) @ ( lList2236698231inlsts @ A @ A2 ) ) ) ) ).
% finlsts.LCons_fin
thf(fact_30_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_31_lappend__code_I1_J,axiom,
! [A: $tType,Ys: coinductive_llist @ A] :
( ( coinductive_lappend @ A @ ( coinductive_LNil @ A ) @ Ys )
= Ys ) ).
% lappend_code(1)
thf(fact_32_lappend__LNil2,axiom,
! [A: $tType,Xs: coinductive_llist @ A] :
( ( coinductive_lappend @ A @ Xs @ ( coinductive_LNil @ A ) )
= Xs ) ).
% lappend_LNil2
thf(fact_33_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_34_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_35_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_36_LList2__Mirabelle__hamjzmohle_Ollast__LCons,axiom,
! [B: $tType,R: coinductive_llist @ B,A2: set @ B,A3: B] :
( ( member @ ( coinductive_llist @ B ) @ R @ ( lList2236698231inlsts @ B @ A2 ) )
=> ( ( ( R
= ( coinductive_LNil @ B ) )
=> ( ( lList2170638824_llast @ B @ ( coinductive_LCons @ B @ A3 @ R ) )
= A3 ) )
& ( ( R
!= ( coinductive_LNil @ B ) )
=> ( ( lList2170638824_llast @ B @ ( coinductive_LCons @ B @ A3 @ R ) )
= ( lList2170638824_llast @ B @ R ) ) ) ) ) ).
% LList2_Mirabelle_hamjzmohle.llast_LCons
thf(fact_37_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_38_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_39_UNIV__I,axiom,
! [A: $tType,X: A] : ( member @ A @ X @ ( top_top @ ( set @ A ) ) ) ).
% UNIV_I
thf(fact_40_iso__tuple__UNIV__I,axiom,
! [A: $tType,X: A] : ( member @ A @ X @ ( top_top @ ( set @ A ) ) ) ).
% iso_tuple_UNIV_I
thf(fact_41_top__apply,axiom,
! [C: $tType,D: $tType] :
( ( top @ C @ ( type2 @ C ) )
=> ( ( top_top @ ( D > C ) )
= ( ^ [X2: D] : ( top_top @ C ) ) ) ) ).
% top_apply
thf(fact_42_top__set__def,axiom,
! [A: $tType] :
( ( top_top @ ( set @ A ) )
= ( collect @ A @ ( top_top @ ( A > $o ) ) ) ) ).
% top_set_def
thf(fact_43_UNIV__witness,axiom,
! [A: $tType] :
? [X3: A] : ( member @ A @ X3 @ ( top_top @ ( set @ A ) ) ) ).
% UNIV_witness
thf(fact_44_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_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
@ ^ [X2: A] : ( member @ A @ X2 @ 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__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_50_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_51_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_52_lappend__LNil__LNil,axiom,
! [A: $tType] :
( ( coinductive_lappend @ A @ ( coinductive_LNil @ A ) @ ( coinductive_LNil @ A ) )
= ( coinductive_LNil @ A ) ) ).
% lappend_LNil_LNil
thf(fact_53_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_54_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_55_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_56_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_57_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_58_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_59_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,A3: A] :
( ( F
= ( lList21916056377ts_rec @ B @ A @ C2 @ 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_60_finlsts__rec__LCons,axiom,
! [B: $tType,A: $tType,R: coinductive_llist @ A,A2: set @ A,C2: B,D2: A > ( coinductive_llist @ A ) > B > B,A3: A] :
( ( member @ ( coinductive_llist @ A ) @ R @ ( lList2236698231inlsts @ A @ A2 ) )
=> ( ( lList21916056377ts_rec @ B @ A @ C2 @ D2 @ ( coinductive_LCons @ A @ A3 @ R ) )
= ( D2 @ A3 @ R @ ( lList21916056377ts_rec @ B @ A @ C2 @ D2 @ R ) ) ) ) ).
% finlsts_rec_LCons
thf(fact_61_top1I,axiom,
! [A: $tType,X: A] : ( top_top @ ( A > $o ) @ X ) ).
% top1I
thf(fact_62_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_63_lstrict__prefix__code_I1_J,axiom,
! [A: $tType] :
~ ( coindu1478340336prefix @ A @ ( coinductive_LNil @ A ) @ ( coinductive_LNil @ A ) ) ).
% lstrict_prefix_code(1)
thf(fact_64_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_65_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_66_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_67_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_68_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_69_top__empty__eq,axiom,
! [A: $tType] :
( ( top_top @ ( A > $o ) )
= ( ^ [X2: A] : ( member @ A @ X2 @ ( top_top @ ( set @ A ) ) ) ) ) ).
% top_empty_eq
thf(fact_70_top__conj_I2_J,axiom,
! [A: $tType,P: $o,X: A] :
( ( P
& ( top_top @ ( A > $o ) @ X ) )
= P ) ).
% top_conj(2)
thf(fact_71_top__conj_I1_J,axiom,
! [A: $tType,X: A,P: $o] :
( ( ( top_top @ ( A > $o ) @ X )
& P )
= P ) ).
% top_conj(1)
thf(fact_72_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_73_llast__singleton,axiom,
! [A: $tType,X: A] :
( ( coinductive_llast @ A @ ( coinductive_LCons @ A @ X @ ( coinductive_LNil @ A ) ) )
= X ) ).
% llast_singleton
thf(fact_74_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_75_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_76_alllsts__UNIV,axiom,
! [A: $tType,S: coinductive_llist @ A] : ( member @ ( coinductive_llist @ A ) @ S @ ( lList2435255213lllsts @ A @ ( top_top @ ( set @ A ) ) ) ) ).
% alllsts_UNIV
thf(fact_77_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_78_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_79_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_80_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_81_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_82_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_83_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_84_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_85_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_86_alllsts_OLCons__all,axiom,
! [A: $tType,L3: coinductive_llist @ A,A2: set @ A,A3: A] :
( ( member @ ( coinductive_llist @ A ) @ L3 @ ( lList2435255213lllsts @ A @ A2 ) )
=> ( ( member @ A @ A3 @ A2 )
=> ( member @ ( coinductive_llist @ A ) @ ( coinductive_LCons @ A @ A3 @ L3 ) @ ( lList2435255213lllsts @ A @ A2 ) ) ) ) ).
% alllsts.LCons_all
thf(fact_87_alllsts_OLNil__all,axiom,
! [A: $tType,A2: set @ A] : ( member @ ( coinductive_llist @ A ) @ ( coinductive_LNil @ A ) @ ( lList2435255213lllsts @ A @ A2 ) ) ).
% alllsts.LNil_all
thf(fact_88_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_89_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_90_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_91_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_92_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_93_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_94_alllsts_Ocoinduct,axiom,
! [A: $tType,X4: ( coinductive_llist @ A ) > $o,X: coinductive_llist @ A,A2: set @ A] :
( ( X4 @ X )
=> ( ! [X3: coinductive_llist @ A] :
( ( X4 @ X3 )
=> ( ( X3
= ( coinductive_LNil @ A ) )
| ? [L4: coinductive_llist @ A,A6: A] :
( ( X3
= ( coinductive_LCons @ A @ A6 @ L4 ) )
& ( ( X4 @ L4 )
| ( member @ ( coinductive_llist @ A ) @ L4 @ ( lList2435255213lllsts @ A @ A2 ) ) )
& ( member @ A @ A6 @ A2 ) ) ) )
=> ( member @ ( coinductive_llist @ A ) @ X @ ( lList2435255213lllsts @ A @ A2 ) ) ) ) ).
% alllsts.coinduct
thf(fact_95_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_96_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_97_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_98_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_99_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_100_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_101_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_102_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_103_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_104_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_105_alllstsp_Ocoinduct,axiom,
! [A: $tType,X4: ( coinductive_llist @ A ) > $o,X: coinductive_llist @ A,A2: A > $o] :
( ( X4 @ X )
=> ( ! [X3: coinductive_llist @ A] :
( ( X4 @ X3 )
=> ( ( X3
= ( coinductive_LNil @ A ) )
| ? [L4: coinductive_llist @ A,A6: A] :
( ( X3
= ( coinductive_LCons @ A @ A6 @ L4 ) )
& ( ( X4 @ L4 )
| ( lList21511617539llstsp @ A @ A2 @ L4 ) )
& ( A2 @ A6 ) ) ) )
=> ( lList21511617539llstsp @ A @ A2 @ X ) ) ) ).
% alllstsp.coinduct
thf(fact_106_finlstsp_Oinducts,axiom,
! [A: $tType,A2: A > $o,X: coinductive_llist @ A,P: ( coinductive_llist @ A ) > $o] :
( ( lList2860480441nlstsp @ A @ A2 @ X )
=> ( ( P @ ( coinductive_LNil @ A ) )
=> ( ! [L: coinductive_llist @ A,A4: A] :
( ( lList2860480441nlstsp @ A @ A2 @ L )
=> ( ( P @ L )
=> ( ( A2 @ A4 )
=> ( P @ ( coinductive_LCons @ A @ A4 @ L ) ) ) ) )
=> ( P @ X ) ) ) ) ).
% finlstsp.inducts
thf(fact_107_finlstsp_Osimps,axiom,
! [A: $tType] :
( ( lList2860480441nlstsp @ A )
= ( ^ [A7: A > $o,A5: coinductive_llist @ A] :
( ( A5
= ( coinductive_LNil @ A ) )
| ? [L2: coinductive_llist @ A,B3: A] :
( ( A5
= ( coinductive_LCons @ A @ B3 @ L2 ) )
& ( lList2860480441nlstsp @ A @ A7 @ L2 )
& ( A7 @ B3 ) ) ) ) ) ).
% finlstsp.simps
thf(fact_108_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_109_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_110_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_111_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_112_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_113_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_114_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_115_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_116_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_117_lfinite__code_I1_J,axiom,
! [A: $tType] : ( coinductive_lfinite @ A @ ( coinductive_LNil @ A ) ) ).
% lfinite_code(1)
thf(fact_118_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_119_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_120_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_121_Un__commute,axiom,
! [A: $tType] :
( ( sup_sup @ ( set @ A ) )
= ( ^ [A7: set @ A,B4: set @ A] : ( sup_sup @ ( set @ A ) @ B4 @ A7 ) ) ) ).
% Un_commute
thf(fact_122_Un__absorb,axiom,
! [A: $tType,A2: set @ A] :
( ( sup_sup @ ( set @ A ) @ A2 @ A2 )
= A2 ) ).
% Un_absorb
thf(fact_123_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_124_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_125_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_126_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_127_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_128_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_129_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_130_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_131_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_132_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_133_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_134_lfinite__LNil,axiom,
! [A: $tType] : ( coinductive_lfinite @ A @ ( coinductive_LNil @ A ) ) ).
% lfinite_LNil
thf(fact_135_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_136_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_137_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_138_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_139_alllstsp_OLCons__all,axiom,
! [A: $tType,A2: A > $o,L3: coinductive_llist @ A,A3: A] :
( ( lList21511617539llstsp @ A @ A2 @ L3 )
=> ( ( A2 @ A3 )
=> ( lList21511617539llstsp @ A @ A2 @ ( coinductive_LCons @ A @ A3 @ L3 ) ) ) ) ).
% alllstsp.LCons_all
thf(fact_140_finlstsp_OLCons__fin,axiom,
! [A: $tType,A2: A > $o,L3: coinductive_llist @ A,A3: A] :
( ( lList2860480441nlstsp @ A @ A2 @ L3 )
=> ( ( A2 @ A3 )
=> ( lList2860480441nlstsp @ A @ A2 @ ( coinductive_LCons @ A @ A3 @ L3 ) ) ) ) ).
% finlstsp.LCons_fin
thf(fact_141_alllstsp_OLNil__all,axiom,
! [A: $tType,A2: A > $o] : ( lList21511617539llstsp @ A @ A2 @ ( coinductive_LNil @ A ) ) ).
% alllstsp.LNil_all
thf(fact_142_finlstsp_OLNil__fin,axiom,
! [A: $tType,A2: A > $o] : ( lList2860480441nlstsp @ A @ A2 @ ( coinductive_LNil @ A ) ) ).
% finlstsp.LNil_fin
thf(fact_143_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_144_lfinite_Osimps,axiom,
! [A: $tType] :
( ( coinductive_lfinite @ A )
= ( ^ [A5: coinductive_llist @ A] :
( ( A5
= ( coinductive_LNil @ A ) )
| ? [Xs4: coinductive_llist @ A,X2: A] :
( ( A5
= ( coinductive_LCons @ A @ X2 @ Xs4 ) )
& ( coinductive_lfinite @ A @ Xs4 ) ) ) ) ) ).
% lfinite.simps
thf(fact_145_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_146_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_147_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_148_alllstsp_Ocases,axiom,
! [A: $tType,A2: A > $o,A3: coinductive_llist @ A] :
( ( lList21511617539llstsp @ A @ A2 @ A3 )
=> ( ( A3
!= ( coinductive_LNil @ A ) )
=> ~ ! [L: coinductive_llist @ A,A4: A] :
( ( A3
= ( coinductive_LCons @ A @ A4 @ L ) )
=> ( ( lList21511617539llstsp @ A @ A2 @ L )
=> ~ ( A2 @ A4 ) ) ) ) ) ).
% alllstsp.cases
thf(fact_149_alllstsp_Osimps,axiom,
! [A: $tType] :
( ( lList21511617539llstsp @ A )
= ( ^ [A7: A > $o,A5: coinductive_llist @ A] :
( ( A5
= ( coinductive_LNil @ A ) )
| ? [L2: coinductive_llist @ A,B3: A] :
( ( A5
= ( coinductive_LCons @ A @ B3 @ L2 ) )
& ( lList21511617539llstsp @ A @ A7 @ L2 )
& ( A7 @ B3 ) ) ) ) ) ).
% alllstsp.simps
thf(fact_150_finlstsp_Ocases,axiom,
! [A: $tType,A2: A > $o,A3: coinductive_llist @ A] :
( ( lList2860480441nlstsp @ A @ A2 @ A3 )
=> ( ( A3
!= ( coinductive_LNil @ A ) )
=> ~ ! [L: coinductive_llist @ A,A4: A] :
( ( A3
= ( coinductive_LCons @ A @ A4 @ L ) )
=> ( ( lList2860480441nlstsp @ A @ A2 @ L )
=> ~ ( A2 @ A4 ) ) ) ) ) ).
% finlstsp.cases
thf(fact_151_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_152_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_153_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_154_sup_Oidem,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [A3: A] :
( ( sup_sup @ A @ A3 @ A3 )
= A3 ) ) ).
% sup.idem
thf(fact_155_sup__idem,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [X: A] :
( ( sup_sup @ A @ X @ X )
= X ) ) ).
% sup_idem
thf(fact_156_minus__apply,axiom,
! [B: $tType,A: $tType] :
( ( minus @ B @ ( type2 @ B ) )
=> ( ( minus_minus @ ( A > B ) )
= ( ^ [A7: A > B,B4: A > B,X2: A] : ( minus_minus @ B @ ( A7 @ X2 ) @ ( B4 @ X2 ) ) ) ) ) ).
% minus_apply
thf(fact_157_sup_Oright__idem,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [A3: A,B5: A] :
( ( sup_sup @ A @ ( sup_sup @ A @ A3 @ B5 ) @ B5 )
= ( sup_sup @ A @ A3 @ B5 ) ) ) ).
% sup.right_idem
thf(fact_158_sup__left__idem,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( sup_sup @ A @ X @ ( sup_sup @ A @ X @ Y ) )
= ( sup_sup @ A @ X @ Y ) ) ) ).
% sup_left_idem
thf(fact_159_sup_Oleft__idem,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [A3: A,B5: A] :
( ( sup_sup @ A @ A3 @ ( sup_sup @ A @ A3 @ B5 ) )
= ( sup_sup @ A @ A3 @ B5 ) ) ) ).
% sup.left_idem
thf(fact_160_fun__diff__def,axiom,
! [B: $tType,A: $tType] :
( ( minus @ B @ ( type2 @ B ) )
=> ( ( minus_minus @ ( A > B ) )
= ( ^ [A7: A > B,B4: A > B,X2: A] : ( minus_minus @ B @ ( A7 @ X2 ) @ ( B4 @ X2 ) ) ) ) ) ).
% fun_diff_def
thf(fact_161_sup__left__commute,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A,Z: A] :
( ( sup_sup @ A @ X @ ( sup_sup @ A @ Y @ Z ) )
= ( sup_sup @ A @ Y @ ( sup_sup @ A @ X @ Z ) ) ) ) ).
% sup_left_commute
thf(fact_162_sup_Oleft__commute,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [B5: A,A3: A,C2: A] :
( ( sup_sup @ A @ B5 @ ( sup_sup @ A @ A3 @ C2 ) )
= ( sup_sup @ A @ A3 @ ( sup_sup @ A @ B5 @ C2 ) ) ) ) ).
% sup.left_commute
thf(fact_163_sup__commute,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ( ( sup_sup @ A )
= ( ^ [X2: A,Y2: A] : ( sup_sup @ A @ Y2 @ X2 ) ) ) ) ).
% sup_commute
thf(fact_164_sup_Ocommute,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ( ( sup_sup @ A )
= ( ^ [A5: A,B3: A] : ( sup_sup @ A @ B3 @ A5 ) ) ) ) ).
% sup.commute
thf(fact_165_sup__assoc,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A,Z: A] :
( ( sup_sup @ A @ ( sup_sup @ A @ X @ Y ) @ Z )
= ( sup_sup @ A @ X @ ( sup_sup @ A @ Y @ Z ) ) ) ) ).
% sup_assoc
thf(fact_166_sup_Oassoc,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [A3: A,B5: A,C2: A] :
( ( sup_sup @ A @ ( sup_sup @ A @ A3 @ B5 ) @ C2 )
= ( sup_sup @ A @ A3 @ ( sup_sup @ A @ B5 @ C2 ) ) ) ) ).
% sup.assoc
thf(fact_167_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_168_inf__sup__aci_I5_J,axiom,
! [A: $tType] :
( ( lattice @ A @ ( type2 @ A ) )
=> ( ( sup_sup @ A )
= ( ^ [X2: A,Y2: A] : ( sup_sup @ A @ Y2 @ X2 ) ) ) ) ).
% inf_sup_aci(5)
thf(fact_169_inf__sup__aci_I6_J,axiom,
! [A: $tType] :
( ( lattice @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A,Z: A] :
( ( sup_sup @ A @ ( sup_sup @ A @ X @ Y ) @ Z )
= ( sup_sup @ A @ X @ ( sup_sup @ A @ Y @ Z ) ) ) ) ).
% inf_sup_aci(6)
thf(fact_170_inf__sup__aci_I7_J,axiom,
! [A: $tType] :
( ( lattice @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A,Z: A] :
( ( sup_sup @ A @ X @ ( sup_sup @ A @ Y @ Z ) )
= ( sup_sup @ A @ Y @ ( sup_sup @ A @ X @ Z ) ) ) ) ).
% inf_sup_aci(7)
thf(fact_171_inf__sup__aci_I8_J,axiom,
! [A: $tType] :
( ( lattice @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( sup_sup @ A @ X @ ( sup_sup @ A @ X @ Y ) )
= ( sup_sup @ A @ X @ Y ) ) ) ).
% inf_sup_aci(8)
thf(fact_172_lset__lappend__lfinite,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( coinductive_lfinite @ A @ Xs )
=> ( ( coinductive_lset @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) )
= ( sup_sup @ ( set @ A ) @ ( coinductive_lset @ A @ Xs ) @ ( coinductive_lset @ A @ Ys ) ) ) ) ).
% lset_lappend_lfinite
thf(fact_173_lstrict__prefix__lappend__conv,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( coindu1478340336prefix @ A @ Xs @ ( coinductive_lappend @ A @ Xs @ Ys ) )
= ( ( coinductive_lfinite @ A @ Xs )
& ~ ( coinductive_lnull @ A @ Ys ) ) ) ).
% lstrict_prefix_lappend_conv
thf(fact_174_lmember__code_I1_J,axiom,
! [A: $tType,X: A] :
~ ( coinductive_lmember @ A @ X @ ( coinductive_LNil @ A ) ) ).
% lmember_code(1)
thf(fact_175_lnull__lappend,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( coinductive_lnull @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) )
= ( ( coinductive_lnull @ A @ Xs )
& ( coinductive_lnull @ A @ Ys ) ) ) ).
% lnull_lappend
thf(fact_176_lappend_Odisc__iff_I2_J,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( ~ ( coinductive_lnull @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) ) )
= ( ~ ( coinductive_lnull @ A @ Xs )
| ~ ( coinductive_lnull @ A @ Ys ) ) ) ).
% lappend.disc_iff(2)
thf(fact_177_lappend_Oexhaust,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( ( coinductive_lnull @ A @ Xs )
=> ~ ( coinductive_lnull @ A @ Ys ) )
=> ( ~ ( coinductive_lnull @ A @ Xs )
| ~ ( coinductive_lnull @ A @ Ys ) ) ) ).
% lappend.exhaust
thf(fact_178_lzip_Oexhaust,axiom,
! [A: $tType,B: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ B] :
( ~ ( ( coinductive_lnull @ A @ Xs )
| ( coinductive_lnull @ B @ Ys ) )
=> ~ ( ~ ( coinductive_lnull @ A @ Xs )
=> ( coinductive_lnull @ B @ Ys ) ) ) ).
% lzip.exhaust
thf(fact_179_lset__lmember,axiom,
! [A: $tType,X: A,Xs: coinductive_llist @ A] :
( ( member @ A @ X @ ( coinductive_lset @ A @ Xs ) )
= ( coinductive_lmember @ A @ X @ Xs ) ) ).
% lset_lmember
thf(fact_180_lappend__lnull2,axiom,
! [A: $tType,Ys: coinductive_llist @ A,Xs: coinductive_llist @ A] :
( ( coinductive_lnull @ A @ Ys )
=> ( ( coinductive_lappend @ A @ Xs @ Ys )
= Xs ) ) ).
% lappend_lnull2
thf(fact_181_lappend__lnull1,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( coinductive_lnull @ A @ Xs )
=> ( ( coinductive_lappend @ A @ Xs @ Ys )
= Ys ) ) ).
% lappend_lnull1
thf(fact_182_lappend_Odisc_I1_J,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( coinductive_lnull @ A @ Xs )
=> ( ( coinductive_lnull @ A @ Ys )
=> ( coinductive_lnull @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) ) ) ) ).
% lappend.disc(1)
thf(fact_183_lappend_Odisc_I2_J,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( ~ ( coinductive_lnull @ A @ Xs )
| ~ ( coinductive_lnull @ A @ Ys ) )
=> ~ ( coinductive_lnull @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) ) ) ).
% lappend.disc(2)
thf(fact_184_lnull__def,axiom,
! [A: $tType] :
( ( coinductive_lnull @ A )
= ( ^ [Llist: coinductive_llist @ A] :
( Llist
= ( coinductive_LNil @ A ) ) ) ) ).
% lnull_def
thf(fact_185_llist_Ocollapse_I1_J,axiom,
! [A: $tType,Llist2: coinductive_llist @ A] :
( ( coinductive_lnull @ A @ Llist2 )
=> ( Llist2
= ( coinductive_LNil @ A ) ) ) ).
% llist.collapse(1)
thf(fact_186_llist_OdiscI_I1_J,axiom,
! [A: $tType,Llist2: coinductive_llist @ A] :
( ( Llist2
= ( coinductive_LNil @ A ) )
=> ( coinductive_lnull @ A @ Llist2 ) ) ).
% llist.discI(1)
thf(fact_187_llist_Odisc_I1_J,axiom,
! [A: $tType] : ( coinductive_lnull @ A @ ( coinductive_LNil @ A ) ) ).
% llist.disc(1)
thf(fact_188_not__lnull__conv,axiom,
! [A: $tType,Xs: coinductive_llist @ A] :
( ( ~ ( coinductive_lnull @ A @ Xs ) )
= ( ? [X2: A,Xs2: coinductive_llist @ A] :
( Xs
= ( coinductive_LCons @ A @ X2 @ Xs2 ) ) ) ) ).
% not_lnull_conv
thf(fact_189_llist_OdiscI_I2_J,axiom,
! [A: $tType,Llist2: coinductive_llist @ A,X212: A,X222: coinductive_llist @ A] :
( ( Llist2
= ( coinductive_LCons @ A @ X212 @ X222 ) )
=> ~ ( coinductive_lnull @ A @ Llist2 ) ) ).
% llist.discI(2)
thf(fact_190_llist_Odisc_I2_J,axiom,
! [A: $tType,X212: A,X222: coinductive_llist @ A] :
~ ( coinductive_lnull @ A @ ( coinductive_LCons @ A @ X212 @ X222 ) ) ).
% llist.disc(2)
thf(fact_191_llist_Oset__induct,axiom,
! [A: $tType,X: A,A3: coinductive_llist @ A,P: A > ( coinductive_llist @ A ) > $o] :
( ( member @ A @ X @ ( coinductive_lset @ A @ A3 ) )
=> ( ! [Z1: A,Z2: coinductive_llist @ A] : ( P @ Z1 @ ( coinductive_LCons @ A @ Z1 @ Z2 ) )
=> ( ! [Z1: A,Z2: coinductive_llist @ A,Xa2: A] :
( ( member @ A @ Xa2 @ ( coinductive_lset @ A @ Z2 ) )
=> ( ( P @ Xa2 @ Z2 )
=> ( P @ Xa2 @ ( coinductive_LCons @ A @ Z1 @ Z2 ) ) ) )
=> ( P @ X @ A3 ) ) ) ) ).
% llist.set_induct
thf(fact_192_llist_Oset__cases,axiom,
! [A: $tType,E: A,A3: coinductive_llist @ A] :
( ( member @ A @ E @ ( coinductive_lset @ A @ A3 ) )
=> ( ! [Z2: coinductive_llist @ A] :
( A3
!= ( coinductive_LCons @ A @ E @ Z2 ) )
=> ~ ! [Z1: A,Z2: coinductive_llist @ A] :
( ( A3
= ( coinductive_LCons @ A @ Z1 @ Z2 ) )
=> ~ ( member @ A @ E @ ( coinductive_lset @ A @ Z2 ) ) ) ) ) ).
% llist.set_cases
thf(fact_193_lset__induct_H,axiom,
! [A: $tType,X: A,Xs: coinductive_llist @ A,P: ( coinductive_llist @ A ) > $o] :
( ( member @ A @ X @ ( coinductive_lset @ A @ Xs ) )
=> ( ! [Xs3: coinductive_llist @ A] : ( P @ ( coinductive_LCons @ A @ X @ Xs3 ) )
=> ( ! [X5: A,Xs3: coinductive_llist @ A] :
( ( member @ A @ X @ ( coinductive_lset @ A @ Xs3 ) )
=> ( ( P @ Xs3 )
=> ( P @ ( coinductive_LCons @ A @ X5 @ Xs3 ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% lset_induct'
thf(fact_194_lset__induct,axiom,
! [A: $tType,X: A,Xs: coinductive_llist @ A,P: ( coinductive_llist @ A ) > $o] :
( ( member @ A @ X @ ( coinductive_lset @ A @ Xs ) )
=> ( ! [Xs3: coinductive_llist @ A] : ( P @ ( coinductive_LCons @ A @ X @ Xs3 ) )
=> ( ! [X5: A,Xs3: coinductive_llist @ A] :
( ( member @ A @ X @ ( coinductive_lset @ A @ Xs3 ) )
=> ( ( X != X5 )
=> ( ( P @ Xs3 )
=> ( P @ ( coinductive_LCons @ A @ X5 @ Xs3 ) ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% lset_induct
thf(fact_195_lset__cases,axiom,
! [A: $tType,X: A,Xs: coinductive_llist @ A] :
( ( member @ A @ X @ ( coinductive_lset @ A @ Xs ) )
=> ( ! [Xs5: coinductive_llist @ A] :
( Xs
!= ( coinductive_LCons @ A @ X @ Xs5 ) )
=> ~ ! [X5: A,Xs5: coinductive_llist @ A] :
( ( Xs
= ( coinductive_LCons @ A @ X5 @ Xs5 ) )
=> ~ ( member @ A @ X @ ( coinductive_lset @ A @ Xs5 ) ) ) ) ) ).
% lset_cases
thf(fact_196_llist_Oset__intros_I1_J,axiom,
! [A: $tType,A1: A,A22: coinductive_llist @ A] : ( member @ A @ A1 @ ( coinductive_lset @ A @ ( coinductive_LCons @ A @ A1 @ A22 ) ) ) ).
% llist.set_intros(1)
thf(fact_197_llist_Oset__intros_I2_J,axiom,
! [A: $tType,X: A,A22: coinductive_llist @ A,A1: A] :
( ( member @ A @ X @ ( coinductive_lset @ A @ A22 ) )
=> ( member @ A @ X @ ( coinductive_lset @ A @ ( coinductive_LCons @ A @ A1 @ A22 ) ) ) ) ).
% llist.set_intros(2)
thf(fact_198_lset__intros_I1_J,axiom,
! [A: $tType,X: A,Xs: coinductive_llist @ A] : ( member @ A @ X @ ( coinductive_lset @ A @ ( coinductive_LCons @ A @ X @ Xs ) ) ) ).
% lset_intros(1)
thf(fact_199_lset__intros_I2_J,axiom,
! [A: $tType,X: A,Xs: coinductive_llist @ A,X6: A] :
( ( member @ A @ X @ ( coinductive_lset @ A @ Xs ) )
=> ( member @ A @ X @ ( coinductive_lset @ A @ ( coinductive_LCons @ A @ X6 @ Xs ) ) ) ) ).
% lset_intros(2)
thf(fact_200_lnull__imp__lfinite,axiom,
! [A: $tType,Xs: coinductive_llist @ A] :
( ( coinductive_lnull @ A @ Xs )
=> ( coinductive_lfinite @ A @ Xs ) ) ).
% lnull_imp_lfinite
thf(fact_201_in__lset__lappend__iff,axiom,
! [A: $tType,X: A,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( member @ A @ X @ ( coinductive_lset @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) ) )
= ( ( member @ A @ X @ ( coinductive_lset @ A @ Xs ) )
| ( ( coinductive_lfinite @ A @ Xs )
& ( member @ A @ X @ ( coinductive_lset @ A @ Ys ) ) ) ) ) ).
% in_lset_lappend_iff
thf(fact_202_lappend_Octr_I1_J,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( coinductive_lnull @ A @ Xs )
=> ( ( coinductive_lnull @ A @ Ys )
=> ( ( coinductive_lappend @ A @ Xs @ Ys )
= ( coinductive_LNil @ A ) ) ) ) ).
% lappend.ctr(1)
thf(fact_203_Coinductive__List_Ollast__LCons,axiom,
! [A: $tType,Xs: coinductive_llist @ A,X: A] :
( ( ( coinductive_lnull @ A @ Xs )
=> ( ( coinductive_llast @ A @ ( coinductive_LCons @ A @ X @ Xs ) )
= X ) )
& ( ~ ( coinductive_lnull @ A @ Xs )
=> ( ( coinductive_llast @ A @ ( coinductive_LCons @ A @ X @ Xs ) )
= ( coinductive_llast @ A @ Xs ) ) ) ) ).
% Coinductive_List.llast_LCons
thf(fact_204_lset__lappend__conv,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( ( coinductive_lfinite @ A @ Xs )
=> ( ( coinductive_lset @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) )
= ( sup_sup @ ( set @ A ) @ ( coinductive_lset @ A @ Xs ) @ ( coinductive_lset @ A @ Ys ) ) ) )
& ( ~ ( coinductive_lfinite @ A @ Xs )
=> ( ( coinductive_lset @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) )
= ( coinductive_lset @ A @ Xs ) ) ) ) ).
% lset_lappend_conv
thf(fact_205_split__llist,axiom,
! [A: $tType,X: A,Xs: coinductive_llist @ A] :
( ( member @ A @ X @ ( coinductive_lset @ A @ Xs ) )
=> ? [Ys3: coinductive_llist @ A,Zs2: coinductive_llist @ A] :
( ( Xs
= ( coinductive_lappend @ A @ Ys3 @ ( coinductive_LCons @ A @ X @ Zs2 ) ) )
& ( coinductive_lfinite @ A @ Ys3 ) ) ) ).
% split_llist
thf(fact_206_split__llist__first,axiom,
! [A: $tType,X: A,Xs: coinductive_llist @ A] :
( ( member @ A @ X @ ( coinductive_lset @ A @ Xs ) )
=> ? [Ys3: coinductive_llist @ A,Zs2: coinductive_llist @ A] :
( ( Xs
= ( coinductive_lappend @ A @ Ys3 @ ( coinductive_LCons @ A @ X @ Zs2 ) ) )
& ( coinductive_lfinite @ A @ Ys3 )
& ~ ( member @ A @ X @ ( coinductive_lset @ A @ Ys3 ) ) ) ) ).
% split_llist_first
thf(fact_207_lmember__code_I2_J,axiom,
! [A: $tType,X: A,Y: A,Ys: coinductive_llist @ A] :
( ( coinductive_lmember @ A @ X @ ( coinductive_LCons @ A @ Y @ Ys ) )
= ( ( X = Y )
| ( coinductive_lmember @ A @ X @ Ys ) ) ) ).
% lmember_code(2)
thf(fact_208_gen__lset__def,axiom,
! [A: $tType] :
( ( coinductive_gen_lset @ A )
= ( ^ [A7: set @ A,Xs4: coinductive_llist @ A] : ( sup_sup @ ( set @ A ) @ A7 @ ( coinductive_lset @ A @ Xs4 ) ) ) ) ).
% gen_lset_def
thf(fact_209_llast__lappend,axiom,
! [A: $tType,Ys: coinductive_llist @ A,Xs: coinductive_llist @ A] :
( ( ( coinductive_lnull @ A @ Ys )
=> ( ( coinductive_llast @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) )
= ( coinductive_llast @ A @ Xs ) ) )
& ( ~ ( coinductive_lnull @ A @ Ys )
=> ( ( ( coinductive_lfinite @ A @ Xs )
=> ( ( coinductive_llast @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) )
= ( coinductive_llast @ A @ Ys ) ) )
& ( ~ ( coinductive_lfinite @ A @ Xs )
=> ( ( coinductive_llast @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) )
= ( undefined @ A ) ) ) ) ) ) ).
% llast_lappend
thf(fact_210_lfilter__eq__LConsD,axiom,
! [A: $tType,P: A > $o,Ys: coinductive_llist @ A,X: A,Xs: coinductive_llist @ A] :
( ( ( coinductive_lfilter @ A @ P @ Ys )
= ( coinductive_LCons @ A @ X @ Xs ) )
=> ? [Us: coinductive_llist @ A,Vs: coinductive_llist @ A] :
( ( Ys
= ( coinductive_lappend @ A @ Us @ ( coinductive_LCons @ A @ X @ Vs ) ) )
& ( coinductive_lfinite @ A @ Us )
& ! [X7: A] :
( ( member @ A @ X7 @ ( coinductive_lset @ A @ Us ) )
=> ~ ( P @ X7 ) )
& ( P @ X )
& ( Xs
= ( coinductive_lfilter @ A @ P @ Vs ) ) ) ) ).
% lfilter_eq_LConsD
thf(fact_211_lfilter__idem,axiom,
! [A: $tType,P: A > $o,Xs: coinductive_llist @ A] :
( ( coinductive_lfilter @ A @ P @ ( coinductive_lfilter @ A @ P @ Xs ) )
= ( coinductive_lfilter @ A @ P @ Xs ) ) ).
% lfilter_idem
thf(fact_212_lfilter__LCons,axiom,
! [A: $tType,P: A > $o,X: A,Xs: coinductive_llist @ A] :
( ( ( P @ X )
=> ( ( coinductive_lfilter @ A @ P @ ( coinductive_LCons @ A @ X @ Xs ) )
= ( coinductive_LCons @ A @ X @ ( coinductive_lfilter @ A @ P @ Xs ) ) ) )
& ( ~ ( P @ X )
=> ( ( coinductive_lfilter @ A @ P @ ( coinductive_LCons @ A @ X @ Xs ) )
= ( coinductive_lfilter @ A @ P @ Xs ) ) ) ) ).
% lfilter_LCons
thf(fact_213_lfilter__LNil,axiom,
! [A: $tType,P: A > $o] :
( ( coinductive_lfilter @ A @ P @ ( coinductive_LNil @ A ) )
= ( coinductive_LNil @ A ) ) ).
% lfilter_LNil
thf(fact_214_diverge__lfilter__LNil,axiom,
! [A: $tType,Xs: coinductive_llist @ A,P: A > $o] :
( ! [X3: A] :
( ( member @ A @ X3 @ ( coinductive_lset @ A @ Xs ) )
=> ~ ( P @ X3 ) )
=> ( ( coinductive_lfilter @ A @ P @ Xs )
= ( coinductive_LNil @ A ) ) ) ).
% diverge_lfilter_LNil
thf(fact_215_lnull__lfilter,axiom,
! [A: $tType,P: A > $o,Xs: coinductive_llist @ A] :
( ( coinductive_lnull @ A @ ( coinductive_lfilter @ A @ P @ Xs ) )
= ( ! [X2: A] :
( ( member @ A @ X2 @ ( coinductive_lset @ A @ Xs ) )
=> ~ ( P @ X2 ) ) ) ) ).
% lnull_lfilter
thf(fact_216_lfilter__lappend__lfinite,axiom,
! [A: $tType,Xs: coinductive_llist @ A,P: A > $o,Ys: coinductive_llist @ A] :
( ( coinductive_lfinite @ A @ Xs )
=> ( ( coinductive_lfilter @ A @ P @ ( coinductive_lappend @ A @ Xs @ Ys ) )
= ( coinductive_lappend @ A @ ( coinductive_lfilter @ A @ P @ Xs ) @ ( coinductive_lfilter @ A @ P @ Ys ) ) ) ) ).
% lfilter_lappend_lfinite
thf(fact_217_LList2__Mirabelle__hamjzmohle_Ollast__LNil,axiom,
! [B: $tType] :
( ( lList2170638824_llast @ B @ ( coinductive_LNil @ B ) )
= ( undefined @ B ) ) ).
% LList2_Mirabelle_hamjzmohle.llast_LNil
thf(fact_218_lfilter__id__conv,axiom,
! [A: $tType,P: A > $o,Xs: coinductive_llist @ A] :
( ( ( coinductive_lfilter @ A @ P @ Xs )
= Xs )
= ( ! [X2: A] :
( ( member @ A @ X2 @ ( coinductive_lset @ A @ Xs ) )
=> ( P @ X2 ) ) ) ) ).
% lfilter_id_conv
thf(fact_219_lfilter__cong,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A,P: A > $o,Q: A > $o] :
( ( Xs = Ys )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ ( coinductive_lset @ A @ Ys ) )
=> ( ( P @ X3 )
= ( Q @ X3 ) ) )
=> ( ( coinductive_lfilter @ A @ P @ Xs )
= ( coinductive_lfilter @ A @ Q @ Ys ) ) ) ) ).
% lfilter_cong
thf(fact_220_lfinite__lfilterI,axiom,
! [A: $tType,Xs: coinductive_llist @ A,P: A > $o] :
( ( coinductive_lfinite @ A @ Xs )
=> ( coinductive_lfinite @ A @ ( coinductive_lfilter @ A @ P @ Xs ) ) ) ).
% lfinite_lfilterI
thf(fact_221_lfilter__LCons__seek,axiom,
! [A: $tType,P2: A > $o,X: A,L3: coinductive_llist @ A] :
( ~ ( P2 @ X )
=> ( ( coinductive_lfilter @ A @ P2 @ ( coinductive_LCons @ A @ X @ L3 ) )
= ( coinductive_lfilter @ A @ P2 @ L3 ) ) ) ).
% lfilter_LCons_seek
thf(fact_222_lfilter__LCons__found,axiom,
! [A: $tType,P: A > $o,X: A,Xs: coinductive_llist @ A] :
( ( P @ X )
=> ( ( coinductive_lfilter @ A @ P @ ( coinductive_LCons @ A @ X @ Xs ) )
= ( coinductive_LCons @ A @ X @ ( coinductive_lfilter @ A @ P @ Xs ) ) ) ) ).
% lfilter_LCons_found
thf(fact_223_lfilter__empty__conv,axiom,
! [A: $tType,P: A > $o,Xs: coinductive_llist @ A] :
( ( ( coinductive_lfilter @ A @ P @ Xs )
= ( coinductive_LNil @ A ) )
= ( ! [X2: A] :
( ( member @ A @ X2 @ ( coinductive_lset @ A @ Xs ) )
=> ~ ( P @ X2 ) ) ) ) ).
% lfilter_empty_conv
thf(fact_224_lfilter__eq__lappend__lfiniteD,axiom,
! [A: $tType,P: A > $o,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A,Zs: coinductive_llist @ A] :
( ( ( coinductive_lfilter @ A @ P @ Xs )
= ( coinductive_lappend @ A @ Ys @ Zs ) )
=> ( ( coinductive_lfinite @ A @ Ys )
=> ? [Us: coinductive_llist @ A,Vs: coinductive_llist @ A] :
( ( Xs
= ( coinductive_lappend @ A @ Us @ Vs ) )
& ( coinductive_lfinite @ A @ Us )
& ( Ys
= ( coinductive_lfilter @ A @ P @ Us ) )
& ( Zs
= ( coinductive_lfilter @ A @ P @ Vs ) ) ) ) ) ).
% lfilter_eq_lappend_lfiniteD
thf(fact_225_Coinductive__List_Ollast__LNil,axiom,
! [A: $tType] :
( ( coinductive_llast @ A @ ( coinductive_LNil @ A ) )
= ( undefined @ A ) ) ).
% Coinductive_List.llast_LNil
thf(fact_226_llast__linfinite,axiom,
! [A: $tType,Xs: coinductive_llist @ A] :
( ~ ( coinductive_lfinite @ A @ Xs )
=> ( ( coinductive_llast @ A @ Xs )
= ( undefined @ A ) ) ) ).
% llast_linfinite
thf(fact_227_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_228_ldistinct_Ocases,axiom,
! [A: $tType,A3: coinductive_llist @ A] :
( ( coindu351974385stinct @ A @ A3 )
=> ( ( A3
!= ( coinductive_LNil @ A ) )
=> ~ ! [X3: A,Xs3: coinductive_llist @ A] :
( ( A3
= ( coinductive_LCons @ A @ X3 @ Xs3 ) )
=> ( ~ ( member @ A @ X3 @ ( coinductive_lset @ A @ Xs3 ) )
=> ~ ( coindu351974385stinct @ A @ Xs3 ) ) ) ) ) ).
% ldistinct.cases
thf(fact_229_ldistinct_Osimps,axiom,
! [A: $tType] :
( ( coindu351974385stinct @ A )
= ( ^ [A5: coinductive_llist @ A] :
( ( A5
= ( coinductive_LNil @ A ) )
| ? [X2: A,Xs4: coinductive_llist @ A] :
( ( A5
= ( coinductive_LCons @ A @ X2 @ Xs4 ) )
& ~ ( member @ A @ X2 @ ( coinductive_lset @ A @ Xs4 ) )
& ( coindu351974385stinct @ A @ Xs4 ) ) ) ) ) ).
% ldistinct.simps
thf(fact_230_ldistinct__LNil__code,axiom,
! [A: $tType] : ( coindu351974385stinct @ A @ ( coinductive_LNil @ A ) ) ).
% ldistinct_LNil_code
thf(fact_231_ldistinct__LCons,axiom,
! [A: $tType,X: A,Xs: coinductive_llist @ A] :
( ( coindu351974385stinct @ A @ ( coinductive_LCons @ A @ X @ Xs ) )
= ( ~ ( member @ A @ X @ ( coinductive_lset @ A @ Xs ) )
& ( coindu351974385stinct @ A @ Xs ) ) ) ).
% ldistinct_LCons
thf(fact_232_ldistinct__lfilterI,axiom,
! [A: $tType,Xs: coinductive_llist @ A,P: A > $o] :
( ( coindu351974385stinct @ A @ Xs )
=> ( coindu351974385stinct @ A @ ( coinductive_lfilter @ A @ P @ Xs ) ) ) ).
% ldistinct_lfilterI
thf(fact_233_ldistinct_OLCons,axiom,
! [A: $tType,X: A,Xs: coinductive_llist @ A] :
( ~ ( member @ A @ X @ ( coinductive_lset @ A @ Xs ) )
=> ( ( coindu351974385stinct @ A @ Xs )
=> ( coindu351974385stinct @ A @ ( coinductive_LCons @ A @ X @ Xs ) ) ) ) ).
% ldistinct.LCons
thf(fact_234_ldistinct_OLNil,axiom,
! [A: $tType] : ( coindu351974385stinct @ A @ ( coinductive_LNil @ A ) ) ).
% ldistinct.LNil
thf(fact_235_ldistinct_Ocoinduct,axiom,
! [A: $tType,X4: ( coinductive_llist @ A ) > $o,X: coinductive_llist @ A] :
( ( X4 @ X )
=> ( ! [X3: coinductive_llist @ A] :
( ( X4 @ X3 )
=> ( ( X3
= ( coinductive_LNil @ A ) )
| ? [Xa3: A,Xs6: coinductive_llist @ A] :
( ( X3
= ( coinductive_LCons @ A @ Xa3 @ Xs6 ) )
& ~ ( member @ A @ Xa3 @ ( coinductive_lset @ A @ Xs6 ) )
& ( ( X4 @ Xs6 )
| ( coindu351974385stinct @ A @ Xs6 ) ) ) ) )
=> ( coindu351974385stinct @ A @ X ) ) ) ).
% ldistinct.coinduct
thf(fact_236_llexord__lappend__leftD,axiom,
! [A: $tType,R: A > A > $o,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A,Zs: coinductive_llist @ A] :
( ( coinductive_llexord @ A @ R @ ( coinductive_lappend @ A @ Xs @ Ys ) @ ( coinductive_lappend @ A @ Xs @ Zs ) )
=> ( ( coinductive_lfinite @ A @ Xs )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ ( coinductive_lset @ A @ Xs ) )
=> ~ ( R @ X3 @ X3 ) )
=> ( coinductive_llexord @ A @ R @ Ys @ Zs ) ) ) ) ).
% llexord_lappend_leftD
thf(fact_237_llexord__lappend__left,axiom,
! [A: $tType,Xs: coinductive_llist @ A,R: A > A > $o,Ys: coinductive_llist @ A,Zs: coinductive_llist @ A] :
( ( coinductive_lfinite @ A @ Xs )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ ( coinductive_lset @ A @ Xs ) )
=> ~ ( R @ X3 @ X3 ) )
=> ( ( coinductive_llexord @ A @ R @ ( coinductive_lappend @ A @ Xs @ Ys ) @ ( coinductive_lappend @ A @ Xs @ Zs ) )
= ( coinductive_llexord @ A @ R @ Ys @ Zs ) ) ) ) ).
% llexord_lappend_left
thf(fact_238_llexord__refl,axiom,
! [A: $tType,R: A > A > $o,Xs: coinductive_llist @ A] : ( coinductive_llexord @ A @ R @ Xs @ Xs ) ).
% llexord_refl
thf(fact_239_llexord__LCons__LCons,axiom,
! [A: $tType,R: A > A > $o,X: A,Xs: coinductive_llist @ A,Y: A,Ys: coinductive_llist @ A] :
( ( coinductive_llexord @ A @ R @ ( coinductive_LCons @ A @ X @ Xs ) @ ( coinductive_LCons @ A @ Y @ Ys ) )
= ( ( ( X = Y )
& ( coinductive_llexord @ A @ R @ Xs @ Ys ) )
| ( R @ X @ Y ) ) ) ).
% llexord_LCons_LCons
thf(fact_240_llexord__code_I1_J,axiom,
! [A: $tType,R: A > A > $o,Ys: coinductive_llist @ A] : ( coinductive_llexord @ A @ R @ ( coinductive_LNil @ A ) @ Ys ) ).
% llexord_code(1)
thf(fact_241_llexord__LNil__right,axiom,
! [A: $tType,Ys: coinductive_llist @ A,R: A > A > $o,Xs: coinductive_llist @ A] :
( ( coinductive_lnull @ A @ Ys )
=> ( ( coinductive_llexord @ A @ R @ Xs @ Ys )
= ( coinductive_lnull @ A @ Xs ) ) ) ).
% llexord_LNil_right
thf(fact_242_lnull__llexord,axiom,
! [A: $tType,Xs: coinductive_llist @ A,R: A > A > $o,Ys: coinductive_llist @ A] :
( ( coinductive_lnull @ A @ Xs )
=> ( coinductive_llexord @ A @ R @ Xs @ Ys ) ) ).
% lnull_llexord
thf(fact_243_llexord__trans,axiom,
! [A: $tType,R: A > A > $o,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A,Zs: coinductive_llist @ A] :
( ( coinductive_llexord @ A @ R @ Xs @ Ys )
=> ( ( coinductive_llexord @ A @ R @ Ys @ Zs )
=> ( ! [A4: A,B6: A,C4: A] :
( ( R @ A4 @ B6 )
=> ( ( R @ B6 @ C4 )
=> ( R @ A4 @ C4 ) ) )
=> ( coinductive_llexord @ A @ R @ Xs @ Zs ) ) ) ) ).
% llexord_trans
thf(fact_244_llexord__linear,axiom,
! [A: $tType,R: A > A > $o,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ! [X3: A,Y3: A] :
( ( R @ X3 @ Y3 )
| ( X3 = Y3 )
| ( R @ Y3 @ X3 ) )
=> ( ( coinductive_llexord @ A @ R @ Xs @ Ys )
| ( coinductive_llexord @ A @ R @ Ys @ Xs ) ) ) ).
% llexord_linear
thf(fact_245_llexord__antisym,axiom,
! [A: $tType,R: A > A > $o,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( coinductive_llexord @ A @ R @ Xs @ Ys )
=> ( ( coinductive_llexord @ A @ R @ Ys @ Xs )
=> ( ! [A4: A,B6: A] :
( ( R @ A4 @ B6 )
=> ~ ( R @ B6 @ A4 ) )
=> ( Xs = Ys ) ) ) ) ).
% llexord_antisym
thf(fact_246_llexord__append__right,axiom,
! [A: $tType,R: A > A > $o,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] : ( coinductive_llexord @ A @ R @ Xs @ ( coinductive_lappend @ A @ Xs @ Ys ) ) ).
% llexord_append_right
thf(fact_247_llexord__lappend__leftI,axiom,
! [A: $tType,R: A > A > $o,Ys: coinductive_llist @ A,Zs: coinductive_llist @ A,Xs: coinductive_llist @ A] :
( ( coinductive_llexord @ A @ R @ Ys @ Zs )
=> ( coinductive_llexord @ A @ R @ ( coinductive_lappend @ A @ Xs @ Ys ) @ ( coinductive_lappend @ A @ Xs @ Zs ) ) ) ).
% llexord_lappend_leftI
thf(fact_248_llexord__LNil,axiom,
! [A: $tType,R: A > A > $o,Ys: coinductive_llist @ A] : ( coinductive_llexord @ A @ R @ ( coinductive_LNil @ A ) @ Ys ) ).
% llexord_LNil
thf(fact_249_llexord__code_I3_J,axiom,
! [A: $tType,R: A > A > $o,X: A,Xs: coinductive_llist @ A,Y: A,Ys: coinductive_llist @ A] :
( ( coinductive_llexord @ A @ R @ ( coinductive_LCons @ A @ X @ Xs ) @ ( coinductive_LCons @ A @ Y @ Ys ) )
= ( ( R @ X @ Y )
| ( ( X = Y )
& ( coinductive_llexord @ A @ R @ Xs @ Ys ) ) ) ) ).
% llexord_code(3)
thf(fact_250_llexord__LCons__left,axiom,
! [A: $tType,R: A > A > $o,X: A,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( coinductive_llexord @ A @ R @ ( coinductive_LCons @ A @ X @ Xs ) @ Ys )
= ( ? [Y2: A,Ys4: coinductive_llist @ A] :
( ( Ys
= ( coinductive_LCons @ A @ Y2 @ Ys4 ) )
& ( ( ( X = Y2 )
& ( coinductive_llexord @ A @ R @ Xs @ Ys4 ) )
| ( R @ X @ Y2 ) ) ) ) ) ).
% llexord_LCons_left
thf(fact_251_llexord__LCons__eq,axiom,
! [A: $tType,R: A > A > $o,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A,X: A] :
( ( coinductive_llexord @ A @ R @ Xs @ Ys )
=> ( coinductive_llexord @ A @ R @ ( coinductive_LCons @ A @ X @ Xs ) @ ( coinductive_LCons @ A @ X @ Ys ) ) ) ).
% llexord_LCons_eq
thf(fact_252_llexord__LCons__less,axiom,
! [A: $tType,R: A > A > $o,X: A,Y: A,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( R @ X @ Y )
=> ( coinductive_llexord @ A @ R @ ( coinductive_LCons @ A @ X @ Xs ) @ ( coinductive_LCons @ A @ Y @ Ys ) ) ) ).
% llexord_LCons_less
thf(fact_253_llexord__code_I2_J,axiom,
! [A: $tType,R: A > A > $o,X: A,Xs: coinductive_llist @ A] :
~ ( coinductive_llexord @ A @ R @ ( coinductive_LCons @ A @ X @ Xs ) @ ( coinductive_LNil @ A ) ) ).
% llexord_code(2)
%----Type constructors (18)
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___Lattices_Olattice,axiom,
! [A8: $tType,A9: $tType] :
( ( lattice @ A9 @ ( type2 @ A9 ) )
=> ( lattice @ ( 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___Groups_Ominus,axiom,
! [A8: $tType,A9: $tType] :
( ( minus @ A9 @ ( type2 @ A9 ) )
=> ( minus @ ( A8 > A9 ) @ ( type2 @ ( A8 > A9 ) ) ) ) ).
thf(tcon_Set_Oset___Lattices_Obounded__lattice__top_3,axiom,
! [A8: $tType] : ( bounded_lattice_top @ ( set @ A8 ) @ ( type2 @ ( set @ A8 ) ) ) ).
thf(tcon_Set_Oset___Lattices_Osemilattice__sup_4,axiom,
! [A8: $tType] : ( semilattice_sup @ ( set @ A8 ) @ ( type2 @ ( set @ A8 ) ) ) ).
thf(tcon_Set_Oset___Lattices_Olattice_5,axiom,
! [A8: $tType] : ( lattice @ ( set @ A8 ) @ ( type2 @ ( set @ A8 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Otop_6,axiom,
! [A8: $tType] : ( top @ ( set @ A8 ) @ ( type2 @ ( set @ A8 ) ) ) ).
thf(tcon_Set_Oset___Groups_Ominus_7,axiom,
! [A8: $tType] : ( minus @ ( set @ A8 ) @ ( type2 @ ( set @ A8 ) ) ) ).
thf(tcon_HOL_Obool___Lattices_Obounded__lattice__top_8,axiom,
bounded_lattice_top @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Lattices_Osemilattice__sup_9,axiom,
semilattice_sup @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Lattices_Olattice_10,axiom,
lattice @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Otop_11,axiom,
top @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Groups_Ominus_12,axiom,
minus @ $o @ ( type2 @ $o ) ).
%----Conjectures (1)
thf(conj_0,conjecture,
p @ ( lList2281150353e_lrev @ a @ la ) ).
%------------------------------------------------------------------------------