TPTP Problem File: DAT219^1.p
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%------------------------------------------------------------------------------
% File : DAT219^1 : TPTP v8.2.0. Released v7.0.0.
% Domain : Data Structures
% Problem : Terminated lazy lists 54
% Version : [Bla16] axioms : Especial.
% English :
% Refs : [Loc10] Lochbihler (2010), Coinductive
% : [RB15] Reynolds & Blanchette (2015), A Decision Procedure for
% : [Bla16] Blanchette (2016), Email to Geoff Sutcliffe
% Source : [Bla16]
% Names : tllist__54.p [Bla16]
% Status : Theorem
% Rating : 0.00 v7.2.0, 0.25 v7.1.0
% Syntax : Number of formulae : 324 ( 147 unt; 48 typ; 0 def)
% Number of atoms : 746 ( 296 equ; 0 cnn)
% Maximal formula atoms : 13 ( 2 avg)
% Number of connectives : 4478 ( 103 ~; 8 |; 49 &;3972 @)
% ( 0 <=>; 346 =>; 0 <=; 0 <~>)
% Maximal formula depth : 28 ( 9 avg)
% Number of types : 3 ( 2 usr)
% Number of type conns : 402 ( 402 >; 0 *; 0 +; 0 <<)
% Number of symbols : 49 ( 46 usr; 4 con; 0-10 aty)
% Number of variables : 1285 ( 68 ^;1148 !; 5 ?;1285 :)
% ( 64 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2016-07-13 14:49:47.576
%------------------------------------------------------------------------------
%----Could-be-implicit typings (5)
thf(ty_t_TLList__Mirabelle__qhjoikztpd_Otllist,type,
tLList446370796tllist: $tType > $tType > $tType ).
thf(ty_t_Set_Oset,type,
set: $tType > $tType ).
thf(ty_t_itself,type,
itself: $tType > $tType ).
thf(ty_tf_b,type,
b: $tType ).
thf(ty_tf_a,type,
a: $tType ).
%----Explicit typings (43)
thf(sy_cl_HOL_Otype,type,
type:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Obot,type,
bot:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Oord,type,
ord:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Oorder,type,
order:
!>[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__bot,type,
order_bot:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_c_Fun_Ofcomp,type,
fcomp:
!>[A: $tType,B: $tType,C: $tType] : ( ( A > B ) > ( B > C ) > A > C ) ).
thf(sy_c_Fun_Oid,type,
id:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Fun_Ooverride__on,type,
override_on:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( A > B ) > ( set @ A ) > A > B ) ).
thf(sy_c_Groups_Ominus__class_Ominus,type,
minus_minus:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Groups_Ouminus__class_Ouminus,type,
uminus_uminus:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_HOL_Oundefined,type,
undefined:
!>[A: $tType] : A ).
thf(sy_c_If,type,
if:
!>[A: $tType] : ( $o > A > A > A ) ).
thf(sy_c_Lattices_Oinf__class_Oinf,type,
inf_inf:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Orderings_Obot__class_Obot,type,
bot_bot:
!>[A: $tType] : A ).
thf(sy_c_Orderings_Oord__class_Oless__eq,type,
ord_less_eq:
!>[A: $tType] : ( A > A > $o ) ).
thf(sy_c_Pure_Otype,type,
type2:
!>[A: $tType] : ( itself @ A ) ).
thf(sy_c_Relation_Oinv__imagep,type,
inv_imagep:
!>[B: $tType,A: $tType] : ( ( B > B > $o ) > ( A > B ) > A > A > $o ) ).
thf(sy_c_Set_OBall,type,
ball:
!>[A: $tType] : ( ( set @ A ) > ( A > $o ) > $o ) ).
thf(sy_c_Set_OCollect,type,
collect:
!>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
thf(sy_c_Set_Oinsert,type,
insert:
!>[A: $tType] : ( A > ( set @ A ) > ( set @ A ) ) ).
thf(sy_c_Set_Ois__empty,type,
is_empty:
!>[A: $tType] : ( ( set @ A ) > $o ) ).
thf(sy_c_Set_Ois__singleton,type,
is_singleton:
!>[A: $tType] : ( ( set @ A ) > $o ) ).
thf(sy_c_Set_Opairwise,type,
pairwise:
!>[A: $tType] : ( ( A > A > $o ) > ( set @ A ) > $o ) ).
thf(sy_c_Set_Othe__elem,type,
the_elem:
!>[A: $tType] : ( ( set @ A ) > A ) ).
thf(sy_c_TLList__Mirabelle__qhjoikztpd_Oterminal0,type,
tLList1825092077minal0:
!>[A: $tType] : A ).
thf(sy_c_TLList__Mirabelle__qhjoikztpd_Otllist_OTCons,type,
tLList1992840728_TCons:
!>[A: $tType,B: $tType] : ( A > ( tLList446370796tllist @ A @ B ) > ( tLList446370796tllist @ A @ B ) ) ).
thf(sy_c_TLList__Mirabelle__qhjoikztpd_Otllist_OTNil,type,
tLList57907754e_TNil:
!>[B: $tType,A: $tType] : ( B > ( tLList446370796tllist @ A @ B ) ) ).
thf(sy_c_TLList__Mirabelle__qhjoikztpd_Otllist_Ocase__tllist,type,
tLList200813139tllist:
!>[B: $tType,C: $tType,A: $tType] : ( ( B > C ) > ( A > ( tLList446370796tllist @ A @ B ) > C ) > ( tLList446370796tllist @ A @ B ) > C ) ).
thf(sy_c_TLList__Mirabelle__qhjoikztpd_Otllist_Ocorec__tllist,type,
tLList1614408749tllist:
!>[E: $tType,B: $tType,A: $tType] : ( ( E > $o ) > ( E > B ) > ( E > A ) > ( E > $o ) > ( E > ( tLList446370796tllist @ A @ B ) ) > ( E > E ) > E > ( tLList446370796tllist @ A @ B ) ) ).
thf(sy_c_TLList__Mirabelle__qhjoikztpd_Otllist_Ois__TNil,type,
tLList1968894004s_TNil:
!>[A: $tType,B: $tType] : ( ( tLList446370796tllist @ A @ B ) > $o ) ).
thf(sy_c_TLList__Mirabelle__qhjoikztpd_Otllist_Opred__tllist,type,
tLList11265572tllist:
!>[A: $tType,B: $tType] : ( ( A > $o ) > ( B > $o ) > ( tLList446370796tllist @ A @ B ) > $o ) ).
thf(sy_c_TLList__Mirabelle__qhjoikztpd_Otllist_Oset2__tllist,type,
tLList457864275tllist:
!>[A: $tType,B: $tType] : ( ( tLList446370796tllist @ A @ B ) > ( set @ B ) ) ).
thf(sy_c_TLList__Mirabelle__qhjoikztpd_Otllist_Oterminal,type,
tLList2110128105rminal:
!>[A: $tType,B: $tType] : ( ( tLList446370796tllist @ A @ B ) > B ) ).
thf(sy_c_TLList__Mirabelle__qhjoikztpd_Otllist_Othd,type,
tLList_Mirabelle_thd:
!>[A: $tType,B: $tType] : ( ( tLList446370796tllist @ A @ B ) > A ) ).
thf(sy_c_TLList__Mirabelle__qhjoikztpd_Otllist_Otllist__all2,type,
tLList1380991092t_all2:
!>[A: $tType,C: $tType,B: $tType,D: $tType] : ( ( A > C > $o ) > ( B > D > $o ) > ( tLList446370796tllist @ A @ B ) > ( tLList446370796tllist @ C @ D ) > $o ) ).
thf(sy_c_TLList__Mirabelle__qhjoikztpd_Otllist_Otmap,type,
tLList1669959861e_tmap:
!>[A: $tType,Aa: $tType,B: $tType,Ba: $tType] : ( ( A > Aa ) > ( B > Ba ) > ( tLList446370796tllist @ A @ B ) > ( tLList446370796tllist @ Aa @ Ba ) ) ).
thf(sy_c_TLList__Mirabelle__qhjoikztpd_Otllist_Otset,type,
tLList1719791291e_tset:
!>[A: $tType,B: $tType] : ( ( tLList446370796tllist @ A @ B ) > ( set @ A ) ) ).
thf(sy_c_TLList__Mirabelle__qhjoikztpd_Otllist_Ottl,type,
tLList_Mirabelle_ttl:
!>[A: $tType,B: $tType] : ( ( tLList446370796tllist @ A @ B ) > ( tLList446370796tllist @ A @ B ) ) ).
thf(sy_c_member,type,
member:
!>[A: $tType] : ( A > ( set @ A ) > $o ) ).
thf(sy_v_x,type,
x: tLList446370796tllist @ a @ b ).
thf(sy_v_x1,type,
x1: b ).
%----Relevant facts (255)
thf(fact_0_tllist_Oinject_I1_J,axiom,
! [A: $tType,B: $tType,X1: B,Y1: B] :
( ( ( tLList57907754e_TNil @ B @ A @ X1 )
= ( tLList57907754e_TNil @ B @ A @ Y1 ) )
= ( X1 = Y1 ) ) ).
% tllist.inject(1)
thf(fact_1_tllist_Ocollapse_I1_J,axiom,
! [B: $tType,A: $tType,Tllist: tLList446370796tllist @ A @ B] :
( ( tLList1968894004s_TNil @ A @ B @ Tllist )
=> ( ( tLList57907754e_TNil @ B @ A @ ( tLList2110128105rminal @ A @ B @ Tllist ) )
= Tllist ) ) ).
% tllist.collapse(1)
thf(fact_2_tllist_Osel_I4_J,axiom,
! [A: $tType,B: $tType,X1: B] :
( ( tLList_Mirabelle_ttl @ A @ B @ ( tLList57907754e_TNil @ B @ A @ X1 ) )
= ( tLList57907754e_TNil @ B @ A @ X1 ) ) ).
% tllist.sel(4)
thf(fact_3_tllist_Osel_I1_J,axiom,
! [Aa: $tType,A: $tType,X1: A] :
( ( tLList2110128105rminal @ Aa @ A @ ( tLList57907754e_TNil @ A @ Aa @ X1 ) )
= X1 ) ).
% tllist.sel(1)
thf(fact_4_tllist_Odisc_I1_J,axiom,
! [A: $tType,B: $tType,X1: B] : ( tLList1968894004s_TNil @ A @ B @ ( tLList57907754e_TNil @ B @ A @ X1 ) ) ).
% tllist.disc(1)
thf(fact_5_tllist_OdiscI_I1_J,axiom,
! [B: $tType,A: $tType,Tllist: tLList446370796tllist @ A @ B,X1: B] :
( ( Tllist
= ( tLList57907754e_TNil @ B @ A @ X1 ) )
=> ( tLList1968894004s_TNil @ A @ B @ Tllist ) ) ).
% tllist.discI(1)
thf(fact_6_tllist_Osimps_I5_J,axiom,
! [A: $tType,C: $tType,B: $tType,F1: B > C,F2: A > ( tLList446370796tllist @ A @ B ) > C,X1: B] :
( ( tLList200813139tllist @ B @ C @ A @ F1 @ F2 @ ( tLList57907754e_TNil @ B @ A @ X1 ) )
= ( F1 @ X1 ) ) ).
% tllist.simps(5)
thf(fact_7_terminal0_Osimps,axiom,
! [D: $tType,C: $tType] :
( ( tLList1825092077minal0 @ ( ( tLList446370796tllist @ C @ D ) > D ) )
= ( ^ [Xs: tLList446370796tllist @ C @ D] : ( if @ D @ ( tLList1968894004s_TNil @ C @ D @ Xs ) @ ( tLList200813139tllist @ D @ D @ C @ ( id @ D ) @ ( undefined @ ( C > ( tLList446370796tllist @ C @ D ) > D ) ) @ Xs ) @ ( tLList1825092077minal0 @ ( ( tLList446370796tllist @ C @ D ) > D ) @ ( tLList_Mirabelle_ttl @ C @ D @ Xs ) ) ) ) ) ).
% terminal0.simps
thf(fact_8_is__TNil__def,axiom,
! [B: $tType,A: $tType] :
( ( tLList1968894004s_TNil @ A @ B )
= ( ^ [Tllist2: tLList446370796tllist @ A @ B] :
? [X12: B] :
( Tllist2
= ( tLList57907754e_TNil @ B @ A @ X12 ) ) ) ) ).
% is_TNil_def
thf(fact_9_terminal0_Oraw__induct,axiom,
! [C: $tType,D: $tType,P: ( tLList446370796tllist @ C @ D ) > D > $o,Xs2: tLList446370796tllist @ C @ D,Y: D] :
( ! [Terminal0: ( tLList446370796tllist @ C @ D ) > D] :
( ! [X: tLList446370796tllist @ C @ D] :
( ( ( Terminal0 @ X )
!= ( undefined @ D ) )
=> ( P @ X @ ( Terminal0 @ X ) ) )
=> ! [Xs3: tLList446370796tllist @ C @ D,Xsa: D] :
( ( ( ( tLList1968894004s_TNil @ C @ D @ Xs3 )
=> ( ( tLList200813139tllist @ D @ D @ C @ ( id @ D ) @ ( undefined @ ( C > ( tLList446370796tllist @ C @ D ) > D ) ) @ Xs3 )
= Xsa ) )
& ( ~ ( tLList1968894004s_TNil @ C @ D @ Xs3 )
=> ( ( Terminal0 @ ( tLList_Mirabelle_ttl @ C @ D @ Xs3 ) )
= Xsa ) ) )
=> ( ( Xsa
!= ( undefined @ D ) )
=> ( P @ Xs3 @ Xsa ) ) ) )
=> ( ( ( tLList1825092077minal0 @ ( ( tLList446370796tllist @ C @ D ) > D ) @ Xs2 )
= Y )
=> ( ( Y
!= ( undefined @ D ) )
=> ( P @ Xs2 @ Y ) ) ) ) ).
% terminal0.raw_induct
thf(fact_10_id__apply,axiom,
! [A: $tType] :
( ( id @ A )
= ( ^ [X2: A] : X2 ) ) ).
% id_apply
thf(fact_11_tllist_Ocase__eq__if,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( tLList200813139tllist @ B @ C @ A )
= ( ^ [F12: B > C,F22: A > ( tLList446370796tllist @ A @ B ) > C,Tllist2: tLList446370796tllist @ A @ B] : ( if @ C @ ( tLList1968894004s_TNil @ A @ B @ Tllist2 ) @ ( F12 @ ( tLList2110128105rminal @ A @ B @ Tllist2 ) ) @ ( F22 @ ( tLList_Mirabelle_thd @ A @ B @ Tllist2 ) @ ( tLList_Mirabelle_ttl @ A @ B @ Tllist2 ) ) ) ) ) ).
% tllist.case_eq_if
thf(fact_12_tllist_Oexpand,axiom,
! [B: $tType,A: $tType,Tllist: tLList446370796tllist @ A @ B,Tllist3: tLList446370796tllist @ A @ B] :
( ( ( tLList1968894004s_TNil @ A @ B @ Tllist )
= ( tLList1968894004s_TNil @ A @ B @ Tllist3 ) )
=> ( ( ( tLList1968894004s_TNil @ A @ B @ Tllist )
=> ( ( tLList1968894004s_TNil @ A @ B @ Tllist3 )
=> ( ( tLList2110128105rminal @ A @ B @ Tllist )
= ( tLList2110128105rminal @ A @ B @ Tllist3 ) ) ) )
=> ( ( ~ ( tLList1968894004s_TNil @ A @ B @ Tllist )
=> ( ~ ( tLList1968894004s_TNil @ A @ B @ Tllist3 )
=> ( ( ( tLList_Mirabelle_thd @ A @ B @ Tllist )
= ( tLList_Mirabelle_thd @ A @ B @ Tllist3 ) )
& ( ( tLList_Mirabelle_ttl @ A @ B @ Tllist )
= ( tLList_Mirabelle_ttl @ A @ B @ Tllist3 ) ) ) ) )
=> ( Tllist = Tllist3 ) ) ) ) ).
% tllist.expand
thf(fact_13_tllist_Ocoinduct,axiom,
! [B: $tType,A: $tType,R: ( tLList446370796tllist @ A @ B ) > ( tLList446370796tllist @ A @ B ) > $o,Tllist: tLList446370796tllist @ A @ B,Tllist3: tLList446370796tllist @ A @ B] :
( ( R @ Tllist @ Tllist3 )
=> ( ! [Tllist4: tLList446370796tllist @ A @ B,Tllist5: tLList446370796tllist @ A @ B] :
( ( R @ Tllist4 @ Tllist5 )
=> ( ( ( tLList1968894004s_TNil @ A @ B @ Tllist4 )
= ( tLList1968894004s_TNil @ A @ B @ Tllist5 ) )
& ( ( tLList1968894004s_TNil @ A @ B @ Tllist4 )
=> ( ( tLList1968894004s_TNil @ A @ B @ Tllist5 )
=> ( ( tLList2110128105rminal @ A @ B @ Tllist4 )
= ( tLList2110128105rminal @ A @ B @ Tllist5 ) ) ) )
& ( ~ ( tLList1968894004s_TNil @ A @ B @ Tllist4 )
=> ( ~ ( tLList1968894004s_TNil @ A @ B @ Tllist5 )
=> ( ( ( tLList_Mirabelle_thd @ A @ B @ Tllist4 )
= ( tLList_Mirabelle_thd @ A @ B @ Tllist5 ) )
& ( R @ ( tLList_Mirabelle_ttl @ A @ B @ Tllist4 ) @ ( tLList_Mirabelle_ttl @ A @ B @ Tllist5 ) ) ) ) ) ) )
=> ( Tllist = Tllist3 ) ) ) ).
% tllist.coinduct
thf(fact_14_tllist_Ocoinduct__strong,axiom,
! [B: $tType,A: $tType,R: ( tLList446370796tllist @ A @ B ) > ( tLList446370796tllist @ A @ B ) > $o,Tllist: tLList446370796tllist @ A @ B,Tllist3: tLList446370796tllist @ A @ B] :
( ( R @ Tllist @ Tllist3 )
=> ( ! [Tllist4: tLList446370796tllist @ A @ B,Tllist5: tLList446370796tllist @ A @ B] :
( ( R @ Tllist4 @ Tllist5 )
=> ( ( ( tLList1968894004s_TNil @ A @ B @ Tllist4 )
= ( tLList1968894004s_TNil @ A @ B @ Tllist5 ) )
& ( ( tLList1968894004s_TNil @ A @ B @ Tllist4 )
=> ( ( tLList1968894004s_TNil @ A @ B @ Tllist5 )
=> ( ( tLList2110128105rminal @ A @ B @ Tllist4 )
= ( tLList2110128105rminal @ A @ B @ Tllist5 ) ) ) )
& ( ~ ( tLList1968894004s_TNil @ A @ B @ Tllist4 )
=> ( ~ ( tLList1968894004s_TNil @ A @ B @ Tllist5 )
=> ( ( ( tLList_Mirabelle_thd @ A @ B @ Tllist4 )
= ( tLList_Mirabelle_thd @ A @ B @ Tllist5 ) )
& ( ( R @ ( tLList_Mirabelle_ttl @ A @ B @ Tllist4 ) @ ( tLList_Mirabelle_ttl @ A @ B @ Tllist5 ) )
| ( ( tLList_Mirabelle_ttl @ A @ B @ Tllist4 )
= ( tLList_Mirabelle_ttl @ A @ B @ Tllist5 ) ) ) ) ) ) ) )
=> ( Tllist = Tllist3 ) ) ) ).
% tllist.coinduct_strong
thf(fact_15_id__def,axiom,
! [A: $tType] :
( ( id @ A )
= ( ^ [X2: A] : X2 ) ) ).
% id_def
thf(fact_16_eq__id__iff,axiom,
! [A: $tType,F: A > A] :
( ( ! [X2: A] :
( ( F @ X2 )
= X2 ) )
= ( F
= ( id @ A ) ) ) ).
% eq_id_iff
thf(fact_17_tllist_Osel_I2_J,axiom,
! [A: $tType,Aa: $tType,X21: Aa,X22: tLList446370796tllist @ Aa @ A] :
( ( tLList2110128105rminal @ Aa @ A @ ( tLList1992840728_TCons @ Aa @ A @ X21 @ X22 ) )
= ( tLList1825092077minal0 @ ( ( tLList446370796tllist @ Aa @ A ) > A ) @ X22 ) ) ).
% tllist.sel(2)
thf(fact_18_tllist_Oset__sel_I3_J,axiom,
! [B: $tType,A: $tType,A2: tLList446370796tllist @ A @ B] :
( ( tLList1968894004s_TNil @ A @ B @ A2 )
=> ( member @ B @ ( tLList2110128105rminal @ A @ B @ A2 ) @ ( tLList457864275tllist @ A @ B @ A2 ) ) ) ).
% tllist.set_sel(3)
thf(fact_19_tllist_Oset__sel_I4_J,axiom,
! [B: $tType,A: $tType,A2: tLList446370796tllist @ A @ B,Xc: B] :
( ~ ( tLList1968894004s_TNil @ A @ B @ A2 )
=> ( ( member @ B @ Xc @ ( tLList457864275tllist @ A @ B @ ( tLList_Mirabelle_ttl @ A @ B @ A2 ) ) )
=> ( member @ B @ Xc @ ( tLList457864275tllist @ A @ B @ A2 ) ) ) ) ).
% tllist.set_sel(4)
thf(fact_20_tllist_Oset__sel_I2_J,axiom,
! [B: $tType,A: $tType,A2: tLList446370796tllist @ A @ B,X3: A] :
( ~ ( tLList1968894004s_TNil @ A @ B @ A2 )
=> ( ( member @ A @ X3 @ ( tLList1719791291e_tset @ A @ B @ ( tLList_Mirabelle_ttl @ A @ B @ A2 ) ) )
=> ( member @ A @ X3 @ ( tLList1719791291e_tset @ A @ B @ A2 ) ) ) ) ).
% tllist.set_sel(2)
thf(fact_21_tllist_Osplit__sel,axiom,
! [C: $tType,B: $tType,A: $tType,P: C > $o,F1: B > C,F2: A > ( tLList446370796tllist @ A @ B ) > C,Tllist: tLList446370796tllist @ A @ B] :
( ( P @ ( tLList200813139tllist @ B @ C @ A @ F1 @ F2 @ Tllist ) )
= ( ( ( Tllist
= ( tLList57907754e_TNil @ B @ A @ ( tLList2110128105rminal @ A @ B @ Tllist ) ) )
=> ( P @ ( F1 @ ( tLList2110128105rminal @ A @ B @ Tllist ) ) ) )
& ( ( Tllist
= ( tLList1992840728_TCons @ A @ B @ ( tLList_Mirabelle_thd @ A @ B @ Tllist ) @ ( tLList_Mirabelle_ttl @ A @ B @ Tllist ) ) )
=> ( P @ ( F2 @ ( tLList_Mirabelle_thd @ A @ B @ Tllist ) @ ( tLList_Mirabelle_ttl @ A @ B @ Tllist ) ) ) ) ) ) ).
% tllist.split_sel
thf(fact_22_tllist_Oinject_I2_J,axiom,
! [B: $tType,A: $tType,X21: A,X22: tLList446370796tllist @ A @ B,Y21: A,Y22: tLList446370796tllist @ A @ B] :
( ( ( tLList1992840728_TCons @ A @ B @ X21 @ X22 )
= ( tLList1992840728_TCons @ A @ B @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% tllist.inject(2)
thf(fact_23_tllist_Ocollapse_I2_J,axiom,
! [B: $tType,A: $tType,Tllist: tLList446370796tllist @ A @ B] :
( ~ ( tLList1968894004s_TNil @ A @ B @ Tllist )
=> ( ( tLList1992840728_TCons @ A @ B @ ( tLList_Mirabelle_thd @ A @ B @ Tllist ) @ ( tLList_Mirabelle_ttl @ A @ B @ Tllist ) )
= Tllist ) ) ).
% tllist.collapse(2)
thf(fact_24_tllist_Osel_I3_J,axiom,
! [B: $tType,A: $tType,X21: A,X22: tLList446370796tllist @ A @ B] :
( ( tLList_Mirabelle_thd @ A @ B @ ( tLList1992840728_TCons @ A @ B @ X21 @ X22 ) )
= X21 ) ).
% tllist.sel(3)
thf(fact_25_tllist_Osimps_I23_J,axiom,
! [B: $tType,A: $tType,X21: A,X22: tLList446370796tllist @ A @ B] :
( ( tLList457864275tllist @ A @ B @ ( tLList1992840728_TCons @ A @ B @ X21 @ X22 ) )
= ( tLList457864275tllist @ A @ B @ X22 ) ) ).
% tllist.simps(23)
thf(fact_26_tllist_Oset__cases_I1_J,axiom,
! [B: $tType,A: $tType,E2: A,A2: tLList446370796tllist @ A @ B] :
( ( member @ A @ E2 @ ( tLList1719791291e_tset @ A @ B @ A2 ) )
=> ( ! [Z2: tLList446370796tllist @ A @ B] :
( A2
!= ( tLList1992840728_TCons @ A @ B @ E2 @ Z2 ) )
=> ~ ! [Z1: A,Z2: tLList446370796tllist @ A @ B] :
( ( A2
= ( tLList1992840728_TCons @ A @ B @ Z1 @ Z2 ) )
=> ~ ( member @ A @ E2 @ ( tLList1719791291e_tset @ A @ B @ Z2 ) ) ) ) ) ).
% tllist.set_cases(1)
thf(fact_27_tllist_Oset__induct_I1_J,axiom,
! [B: $tType,A: $tType,X3: A,A2: tLList446370796tllist @ A @ B,P: A > ( tLList446370796tllist @ A @ B ) > $o] :
( ( member @ A @ X3 @ ( tLList1719791291e_tset @ A @ B @ A2 ) )
=> ( ! [Z1: A,Z2: tLList446370796tllist @ A @ B] : ( P @ Z1 @ ( tLList1992840728_TCons @ A @ B @ Z1 @ Z2 ) )
=> ( ! [Z1: A,Z2: tLList446370796tllist @ A @ B,Xa: A] :
( ( member @ A @ Xa @ ( tLList1719791291e_tset @ A @ B @ Z2 ) )
=> ( ( P @ Xa @ Z2 )
=> ( P @ Xa @ ( tLList1992840728_TCons @ A @ B @ Z1 @ Z2 ) ) ) )
=> ( P @ X3 @ A2 ) ) ) ) ).
% tllist.set_induct(1)
thf(fact_28_tllist_Oset__intros_I4_J,axiom,
! [B: $tType,A: $tType,Xc: B,A2a: tLList446370796tllist @ A @ B,A1a: A] :
( ( member @ B @ Xc @ ( tLList457864275tllist @ A @ B @ A2a ) )
=> ( member @ B @ Xc @ ( tLList457864275tllist @ A @ B @ ( tLList1992840728_TCons @ A @ B @ A1a @ A2a ) ) ) ) ).
% tllist.set_intros(4)
thf(fact_29_tllist_Oset__intros_I2_J,axiom,
! [B: $tType,A: $tType,X3: A,A22: tLList446370796tllist @ A @ B,A1: A] :
( ( member @ A @ X3 @ ( tLList1719791291e_tset @ A @ B @ A22 ) )
=> ( member @ A @ X3 @ ( tLList1719791291e_tset @ A @ B @ ( tLList1992840728_TCons @ A @ B @ A1 @ A22 ) ) ) ) ).
% tllist.set_intros(2)
thf(fact_30_tllist_Oset__intros_I1_J,axiom,
! [B: $tType,A: $tType,A1: A,A22: tLList446370796tllist @ A @ B] : ( member @ A @ A1 @ ( tLList1719791291e_tset @ A @ B @ ( tLList1992840728_TCons @ A @ B @ A1 @ A22 ) ) ) ).
% tllist.set_intros(1)
thf(fact_31_tllist_Oset__induct_I2_J,axiom,
! [A: $tType,Aa: $tType,X3: A,A2: tLList446370796tllist @ Aa @ A,P: A > ( tLList446370796tllist @ Aa @ A ) > $o] :
( ( member @ A @ X3 @ ( tLList457864275tllist @ Aa @ A @ A2 ) )
=> ( ! [Z: A] : ( P @ Z @ ( tLList57907754e_TNil @ A @ Aa @ Z ) )
=> ( ! [Z1: Aa,Z2: tLList446370796tllist @ Aa @ A,Xb: A] :
( ( member @ A @ Xb @ ( tLList457864275tllist @ Aa @ A @ Z2 ) )
=> ( ( P @ Xb @ Z2 )
=> ( P @ Xb @ ( tLList1992840728_TCons @ Aa @ A @ Z1 @ Z2 ) ) ) )
=> ( P @ X3 @ A2 ) ) ) ) ).
% tllist.set_induct(2)
thf(fact_32_tllist_Oset__cases_I2_J,axiom,
! [B: $tType,A: $tType,E2: B,A2: tLList446370796tllist @ A @ B] :
( ( member @ B @ E2 @ ( tLList457864275tllist @ A @ B @ A2 ) )
=> ( ( A2
!= ( tLList57907754e_TNil @ B @ A @ E2 ) )
=> ~ ! [Z1: A,Z2: tLList446370796tllist @ A @ B] :
( ( A2
= ( tLList1992840728_TCons @ A @ B @ Z1 @ Z2 ) )
=> ~ ( member @ B @ E2 @ ( tLList457864275tllist @ A @ B @ Z2 ) ) ) ) ) ).
% tllist.set_cases(2)
thf(fact_33_tllist_Oset__sel_I1_J,axiom,
! [B: $tType,A: $tType,A2: tLList446370796tllist @ A @ B] :
( ~ ( tLList1968894004s_TNil @ A @ B @ A2 )
=> ( member @ A @ ( tLList_Mirabelle_thd @ A @ B @ A2 ) @ ( tLList1719791291e_tset @ A @ B @ A2 ) ) ) ).
% tllist.set_sel(1)
thf(fact_34_tllist_OdiscI_I2_J,axiom,
! [B: $tType,A: $tType,Tllist: tLList446370796tllist @ A @ B,X21: A,X22: tLList446370796tllist @ A @ B] :
( ( Tllist
= ( tLList1992840728_TCons @ A @ B @ X21 @ X22 ) )
=> ~ ( tLList1968894004s_TNil @ A @ B @ Tllist ) ) ).
% tllist.discI(2)
thf(fact_35_tllist_Odisc_I2_J,axiom,
! [B: $tType,A: $tType,X21: A,X22: tLList446370796tllist @ A @ B] :
~ ( tLList1968894004s_TNil @ A @ B @ ( tLList1992840728_TCons @ A @ B @ X21 @ X22 ) ) ).
% tllist.disc(2)
thf(fact_36_tllist_Oexhaust,axiom,
! [B: $tType,A: $tType,Y: tLList446370796tllist @ A @ B] :
( ! [X13: B] :
( Y
!= ( tLList57907754e_TNil @ B @ A @ X13 ) )
=> ~ ! [X212: A,X222: tLList446370796tllist @ A @ B] :
( Y
!= ( tLList1992840728_TCons @ A @ B @ X212 @ X222 ) ) ) ).
% tllist.exhaust
thf(fact_37_tllist_Odistinct_I1_J,axiom,
! [B: $tType,A: $tType,X1: B,X21: A,X22: tLList446370796tllist @ A @ B] :
( ( tLList57907754e_TNil @ B @ A @ X1 )
!= ( tLList1992840728_TCons @ A @ B @ X21 @ X22 ) ) ).
% tllist.distinct(1)
thf(fact_38_tllist_Osel_I5_J,axiom,
! [B: $tType,A: $tType,X21: A,X22: tLList446370796tllist @ A @ B] :
( ( tLList_Mirabelle_ttl @ A @ B @ ( tLList1992840728_TCons @ A @ B @ X21 @ X22 ) )
= X22 ) ).
% tllist.sel(5)
thf(fact_39_tllist_Osimps_I6_J,axiom,
! [C: $tType,B: $tType,A: $tType,F1: B > C,F2: A > ( tLList446370796tllist @ A @ B ) > C,X21: A,X22: tLList446370796tllist @ A @ B] :
( ( tLList200813139tllist @ B @ C @ A @ F1 @ F2 @ ( tLList1992840728_TCons @ A @ B @ X21 @ X22 ) )
= ( F2 @ X21 @ X22 ) ) ).
% tllist.simps(6)
thf(fact_40_tllist_Oset__intros_I3_J,axiom,
! [A: $tType,B: $tType,Aa2: B] : ( member @ B @ Aa2 @ ( tLList457864275tllist @ A @ B @ ( tLList57907754e_TNil @ B @ A @ Aa2 ) ) ) ).
% tllist.set_intros(3)
thf(fact_41_tllist_Oexhaust__sel,axiom,
! [B: $tType,A: $tType,Tllist: tLList446370796tllist @ A @ B] :
( ( Tllist
!= ( tLList57907754e_TNil @ B @ A @ ( tLList2110128105rminal @ A @ B @ Tllist ) ) )
=> ( Tllist
= ( tLList1992840728_TCons @ A @ B @ ( tLList_Mirabelle_thd @ A @ B @ Tllist ) @ ( tLList_Mirabelle_ttl @ A @ B @ Tllist ) ) ) ) ).
% tllist.exhaust_sel
thf(fact_42_tllist_Osplit__sel__asm,axiom,
! [C: $tType,B: $tType,A: $tType,P: C > $o,F1: B > C,F2: A > ( tLList446370796tllist @ A @ B ) > C,Tllist: tLList446370796tllist @ A @ B] :
( ( P @ ( tLList200813139tllist @ B @ C @ A @ F1 @ F2 @ Tllist ) )
= ( ~ ( ( ( Tllist
= ( tLList57907754e_TNil @ B @ A @ ( tLList2110128105rminal @ A @ B @ Tllist ) ) )
& ~ ( P @ ( F1 @ ( tLList2110128105rminal @ A @ B @ Tllist ) ) ) )
| ( ( Tllist
= ( tLList1992840728_TCons @ A @ B @ ( tLList_Mirabelle_thd @ A @ B @ Tllist ) @ ( tLList_Mirabelle_ttl @ A @ B @ Tllist ) ) )
& ~ ( P @ ( F2 @ ( tLList_Mirabelle_thd @ A @ B @ Tllist ) @ ( tLList_Mirabelle_ttl @ A @ B @ Tllist ) ) ) ) ) ) ) ).
% tllist.split_sel_asm
thf(fact_43_tllist_Orel__coinduct,axiom,
! [A: $tType,B: $tType,D: $tType,C: $tType,P: ( tLList446370796tllist @ A @ B ) > ( tLList446370796tllist @ C @ D ) > $o,X3: tLList446370796tllist @ A @ B,Y: tLList446370796tllist @ C @ D,R2: B > D > $o,R1: A > C > $o] :
( ( P @ X3 @ Y )
=> ( ! [Tllist4: tLList446370796tllist @ A @ B,Tllist5: tLList446370796tllist @ C @ D] :
( ( P @ Tllist4 @ Tllist5 )
=> ( ( ( tLList1968894004s_TNil @ A @ B @ Tllist4 )
= ( tLList1968894004s_TNil @ C @ D @ Tllist5 ) )
& ( ( tLList1968894004s_TNil @ A @ B @ Tllist4 )
=> ( ( tLList1968894004s_TNil @ C @ D @ Tllist5 )
=> ( R2 @ ( tLList2110128105rminal @ A @ B @ Tllist4 ) @ ( tLList2110128105rminal @ C @ D @ Tllist5 ) ) ) )
& ( ~ ( tLList1968894004s_TNil @ A @ B @ Tllist4 )
=> ( ~ ( tLList1968894004s_TNil @ C @ D @ Tllist5 )
=> ( ( R1 @ ( tLList_Mirabelle_thd @ A @ B @ Tllist4 ) @ ( tLList_Mirabelle_thd @ C @ D @ Tllist5 ) )
& ( P @ ( tLList_Mirabelle_ttl @ A @ B @ Tllist4 ) @ ( tLList_Mirabelle_ttl @ C @ D @ Tllist5 ) ) ) ) ) ) )
=> ( tLList1380991092t_all2 @ A @ C @ B @ D @ R1 @ R2 @ X3 @ Y ) ) ) ).
% tllist.rel_coinduct
thf(fact_44_mem__Collect__eq,axiom,
! [A: $tType,A2: A,P: A > $o] :
( ( member @ A @ A2 @ ( collect @ A @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_45_Collect__mem__eq,axiom,
! [A: $tType,A3: set @ A] :
( ( collect @ A
@ ^ [X2: A] : ( member @ A @ X2 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_46_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_47_ext,axiom,
! [B: $tType,A: $tType,F: A > B,G: A > B] :
( ! [X4: A] :
( ( F @ X4 )
= ( G @ X4 ) )
=> ( F = G ) ) ).
% ext
thf(fact_48_tllist_Orel__sel,axiom,
! [B: $tType,D: $tType,C: $tType,A: $tType] :
( ( tLList1380991092t_all2 @ A @ C @ B @ D )
= ( ^ [R12: A > C > $o,R22: B > D > $o,A4: tLList446370796tllist @ A @ B,B2: tLList446370796tllist @ C @ D] :
( ( ( tLList1968894004s_TNil @ A @ B @ A4 )
= ( tLList1968894004s_TNil @ C @ D @ B2 ) )
& ( ( tLList1968894004s_TNil @ A @ B @ A4 )
=> ( ( tLList1968894004s_TNil @ C @ D @ B2 )
=> ( R22 @ ( tLList2110128105rminal @ A @ B @ A4 ) @ ( tLList2110128105rminal @ C @ D @ B2 ) ) ) )
& ( ~ ( tLList1968894004s_TNil @ A @ B @ A4 )
=> ( ~ ( tLList1968894004s_TNil @ C @ D @ B2 )
=> ( ( R12 @ ( tLList_Mirabelle_thd @ A @ B @ A4 ) @ ( tLList_Mirabelle_thd @ C @ D @ B2 ) )
& ( tLList1380991092t_all2 @ A @ C @ B @ D @ R12 @ R22 @ ( tLList_Mirabelle_ttl @ A @ B @ A4 ) @ ( tLList_Mirabelle_ttl @ C @ D @ B2 ) ) ) ) ) ) ) ) ).
% tllist.rel_sel
thf(fact_49_tllist_Opred__cong,axiom,
! [B: $tType,A: $tType,X3: tLList446370796tllist @ A @ B,Ya: tLList446370796tllist @ A @ B,P1: A > $o,P1a: A > $o,P2: B > $o,P2a: B > $o] :
( ( X3 = Ya )
=> ( ! [Z1: A] :
( ( member @ A @ Z1 @ ( tLList1719791291e_tset @ A @ B @ Ya ) )
=> ( ( P1 @ Z1 )
= ( P1a @ Z1 ) ) )
=> ( ! [Z2: B] :
( ( member @ B @ Z2 @ ( tLList457864275tllist @ A @ B @ Ya ) )
=> ( ( P2 @ Z2 )
= ( P2a @ Z2 ) ) )
=> ( ( tLList11265572tllist @ A @ B @ P1 @ P2 @ X3 )
= ( tLList11265572tllist @ A @ B @ P1a @ P2a @ Ya ) ) ) ) ) ).
% tllist.pred_cong
thf(fact_50_tllist_Opred__mono__strong,axiom,
! [B: $tType,A: $tType,P1: A > $o,P2: B > $o,X3: tLList446370796tllist @ A @ B,P1a: A > $o,P2a: B > $o] :
( ( tLList11265572tllist @ A @ B @ P1 @ P2 @ X3 )
=> ( ! [Z1: A] :
( ( member @ A @ Z1 @ ( tLList1719791291e_tset @ A @ B @ X3 ) )
=> ( ( P1 @ Z1 )
=> ( P1a @ Z1 ) ) )
=> ( ! [Z2: B] :
( ( member @ B @ Z2 @ ( tLList457864275tllist @ A @ B @ X3 ) )
=> ( ( P2 @ Z2 )
=> ( P2a @ Z2 ) ) )
=> ( tLList11265572tllist @ A @ B @ P1a @ P2a @ X3 ) ) ) ) ).
% tllist.pred_mono_strong
thf(fact_51_tllist_Ocorec__code,axiom,
! [B: $tType,A: $tType,E: $tType] :
( ( tLList1614408749tllist @ E @ B @ A )
= ( ^ [P3: E > $o,G1: E > B,G21: E > A,Q22: E > $o,G221: E > ( tLList446370796tllist @ A @ B ),G222: E > E,A4: E] : ( if @ ( tLList446370796tllist @ A @ B ) @ ( P3 @ A4 ) @ ( tLList57907754e_TNil @ B @ A @ ( G1 @ A4 ) ) @ ( tLList1992840728_TCons @ A @ B @ ( G21 @ A4 ) @ ( if @ ( tLList446370796tllist @ A @ B ) @ ( Q22 @ A4 ) @ ( G221 @ A4 ) @ ( tLList1614408749tllist @ E @ B @ A @ P3 @ G1 @ G21 @ Q22 @ G221 @ G222 @ ( G222 @ A4 ) ) ) ) ) ) ) ).
% tllist.corec_code
thf(fact_52_tllist_Osimps_I21_J,axiom,
! [B: $tType,A: $tType,X21: A,X22: tLList446370796tllist @ A @ B] :
( ( tLList1719791291e_tset @ A @ B @ ( tLList1992840728_TCons @ A @ B @ X21 @ X22 ) )
= ( insert @ A @ X21 @ ( tLList1719791291e_tset @ A @ B @ X22 ) ) ) ).
% tllist.simps(21)
thf(fact_53_tllist_Omap__cong,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType,X3: tLList446370796tllist @ A @ B,Ya: tLList446370796tllist @ A @ B,F1: A > C,G12: A > C,F2: B > D,G2: B > D] :
( ( X3 = Ya )
=> ( ! [Z1: A] :
( ( member @ A @ Z1 @ ( tLList1719791291e_tset @ A @ B @ Ya ) )
=> ( ( F1 @ Z1 )
= ( G12 @ Z1 ) ) )
=> ( ! [Z2: B] :
( ( member @ B @ Z2 @ ( tLList457864275tllist @ A @ B @ Ya ) )
=> ( ( F2 @ Z2 )
= ( G2 @ Z2 ) ) )
=> ( ( tLList1669959861e_tmap @ A @ C @ B @ D @ F1 @ F2 @ X3 )
= ( tLList1669959861e_tmap @ A @ C @ B @ D @ G12 @ G2 @ Ya ) ) ) ) ) ).
% tllist.map_cong
thf(fact_54_tllist_Omap__cong0,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType,X3: tLList446370796tllist @ A @ B,F1: A > C,G12: A > C,F2: B > D,G2: B > D] :
( ! [Z1: A] :
( ( member @ A @ Z1 @ ( tLList1719791291e_tset @ A @ B @ X3 ) )
=> ( ( F1 @ Z1 )
= ( G12 @ Z1 ) ) )
=> ( ! [Z2: B] :
( ( member @ B @ Z2 @ ( tLList457864275tllist @ A @ B @ X3 ) )
=> ( ( F2 @ Z2 )
= ( G2 @ Z2 ) ) )
=> ( ( tLList1669959861e_tmap @ A @ C @ B @ D @ F1 @ F2 @ X3 )
= ( tLList1669959861e_tmap @ A @ C @ B @ D @ G12 @ G2 @ X3 ) ) ) ) ).
% tllist.map_cong0
thf(fact_55_tllist_Oinj__map__strong,axiom,
! [C: $tType,D: $tType,B: $tType,A: $tType,X3: tLList446370796tllist @ A @ B,Xa2: tLList446370796tllist @ A @ B,F1: A > C,F1a: A > C,F2: B > D,F2a: B > D] :
( ! [Z1: A,Z1a: A] :
( ( member @ A @ Z1 @ ( tLList1719791291e_tset @ A @ B @ X3 ) )
=> ( ( member @ A @ Z1a @ ( tLList1719791291e_tset @ A @ B @ Xa2 ) )
=> ( ( ( F1 @ Z1 )
= ( F1a @ Z1a ) )
=> ( Z1 = Z1a ) ) ) )
=> ( ! [Z2: B,Z2a: B] :
( ( member @ B @ Z2 @ ( tLList457864275tllist @ A @ B @ X3 ) )
=> ( ( member @ B @ Z2a @ ( tLList457864275tllist @ A @ B @ Xa2 ) )
=> ( ( ( F2 @ Z2 )
= ( F2a @ Z2a ) )
=> ( Z2 = Z2a ) ) ) )
=> ( ( ( tLList1669959861e_tmap @ A @ C @ B @ D @ F1 @ F2 @ X3 )
= ( tLList1669959861e_tmap @ A @ C @ B @ D @ F1a @ F2a @ Xa2 ) )
=> ( X3 = Xa2 ) ) ) ) ).
% tllist.inj_map_strong
thf(fact_56_tllist_Omap__sel_I2_J,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType,A2: tLList446370796tllist @ A @ B,F1: A > C,F2: B > D] :
( ~ ( tLList1968894004s_TNil @ A @ B @ A2 )
=> ( ( tLList_Mirabelle_thd @ C @ D @ ( tLList1669959861e_tmap @ A @ C @ B @ D @ F1 @ F2 @ A2 ) )
= ( F1 @ ( tLList_Mirabelle_thd @ A @ B @ A2 ) ) ) ) ).
% tllist.map_sel(2)
thf(fact_57_tllist_Orel__inject_I2_J,axiom,
! [A: $tType,B: $tType,D: $tType,C: $tType,R1: A > C > $o,R2: B > D > $o,X21: A,X22: tLList446370796tllist @ A @ B,Y21: C,Y22: tLList446370796tllist @ C @ D] :
( ( tLList1380991092t_all2 @ A @ C @ B @ D @ R1 @ R2 @ ( tLList1992840728_TCons @ A @ B @ X21 @ X22 ) @ ( tLList1992840728_TCons @ C @ D @ Y21 @ Y22 ) )
= ( ( R1 @ X21 @ Y21 )
& ( tLList1380991092t_all2 @ A @ C @ B @ D @ R1 @ R2 @ X22 @ Y22 ) ) ) ).
% tllist.rel_inject(2)
thf(fact_58_tllist_Omap__disc__iff,axiom,
! [C: $tType,D: $tType,B: $tType,A: $tType,F1: A > C,F2: B > D,A2: tLList446370796tllist @ A @ B] :
( ( tLList1968894004s_TNil @ C @ D @ ( tLList1669959861e_tmap @ A @ C @ B @ D @ F1 @ F2 @ A2 ) )
= ( tLList1968894004s_TNil @ A @ B @ A2 ) ) ).
% tllist.map_disc_iff
thf(fact_59_tllist_Orel__inject_I1_J,axiom,
! [C: $tType,A: $tType,B: $tType,D: $tType,R1: A > C > $o,R2: B > D > $o,X1: B,Y1: D] :
( ( tLList1380991092t_all2 @ A @ C @ B @ D @ R1 @ R2 @ ( tLList57907754e_TNil @ B @ A @ X1 ) @ ( tLList57907754e_TNil @ D @ C @ Y1 ) )
= ( R2 @ X1 @ Y1 ) ) ).
% tllist.rel_inject(1)
thf(fact_60_tllist_Ocorec__disc__iff_I2_J,axiom,
! [A: $tType,B: $tType,E: $tType,P4: E > $o,G12: E > B,G212: E > A,Q222: E > $o,G2212: E > ( tLList446370796tllist @ A @ B ),G2222: E > E,A2: E] :
( ( ~ ( tLList1968894004s_TNil @ A @ B @ ( tLList1614408749tllist @ E @ B @ A @ P4 @ G12 @ G212 @ Q222 @ G2212 @ G2222 @ A2 ) ) )
= ( ~ ( P4 @ A2 ) ) ) ).
% tllist.corec_disc_iff(2)
thf(fact_61_tllist_Ocorec__disc__iff_I1_J,axiom,
! [A: $tType,B: $tType,E: $tType,P4: E > $o,G12: E > B,G212: E > A,Q222: E > $o,G2212: E > ( tLList446370796tllist @ A @ B ),G2222: E > E,A2: E] :
( ( tLList1968894004s_TNil @ A @ B @ ( tLList1614408749tllist @ E @ B @ A @ P4 @ G12 @ G212 @ Q222 @ G2212 @ G2222 @ A2 ) )
= ( P4 @ A2 ) ) ).
% tllist.corec_disc_iff(1)
thf(fact_62_tllist_Opred__inject_I2_J,axiom,
! [B: $tType,A: $tType,P1: A > $o,P2: B > $o,A2: A,Aa2: tLList446370796tllist @ A @ B] :
( ( tLList11265572tllist @ A @ B @ P1 @ P2 @ ( tLList1992840728_TCons @ A @ B @ A2 @ Aa2 ) )
= ( ( P1 @ A2 )
& ( tLList11265572tllist @ A @ B @ P1 @ P2 @ Aa2 ) ) ) ).
% tllist.pred_inject(2)
thf(fact_63_tllist_Opred__inject_I1_J,axiom,
! [A: $tType,B: $tType,P1: A > $o,P2: B > $o,A2: B] :
( ( tLList11265572tllist @ A @ B @ P1 @ P2 @ ( tLList57907754e_TNil @ B @ A @ A2 ) )
= ( P2 @ A2 ) ) ).
% tllist.pred_inject(1)
thf(fact_64_tllist_Orel__refl,axiom,
! [D: $tType,C: $tType,R1a: C > C > $o,R2a: D > D > $o,X3: tLList446370796tllist @ C @ D] :
( ! [X4: C] : ( R1a @ X4 @ X4 )
=> ( ! [X4: D] : ( R2a @ X4 @ X4 )
=> ( tLList1380991092t_all2 @ C @ C @ D @ D @ R1a @ R2a @ X3 @ X3 ) ) ) ).
% tllist.rel_refl
thf(fact_65_tllist_Orel__eq,axiom,
! [B: $tType,A: $tType] :
( ( tLList1380991092t_all2 @ A @ A @ B @ B
@ ^ [Y2: A,Z3: A] : Y2 = Z3
@ ^ [Y2: B,Z3: B] : Y2 = Z3 )
= ( ^ [Y2: tLList446370796tllist @ A @ B,Z3: tLList446370796tllist @ A @ B] : Y2 = Z3 ) ) ).
% tllist.rel_eq
thf(fact_66_tllist_Orel__intros_I2_J,axiom,
! [A: $tType,B: $tType,D: $tType,C: $tType,R1: A > C > $o,X21: A,Y21: C,R2: B > D > $o,X22: tLList446370796tllist @ A @ B,Y22: tLList446370796tllist @ C @ D] :
( ( R1 @ X21 @ Y21 )
=> ( ( tLList1380991092t_all2 @ A @ C @ B @ D @ R1 @ R2 @ X22 @ Y22 )
=> ( tLList1380991092t_all2 @ A @ C @ B @ D @ R1 @ R2 @ ( tLList1992840728_TCons @ A @ B @ X21 @ X22 ) @ ( tLList1992840728_TCons @ C @ D @ Y21 @ Y22 ) ) ) ) ).
% tllist.rel_intros(2)
thf(fact_67_tllist_Orel__intros_I1_J,axiom,
! [B: $tType,A: $tType,C: $tType,D: $tType,R2: B > D > $o,X1: B,Y1: D,R1: A > C > $o] :
( ( R2 @ X1 @ Y1 )
=> ( tLList1380991092t_all2 @ A @ C @ B @ D @ R1 @ R2 @ ( tLList57907754e_TNil @ B @ A @ X1 ) @ ( tLList57907754e_TNil @ D @ C @ Y1 ) ) ) ).
% tllist.rel_intros(1)
thf(fact_68_tllist_Omap__id,axiom,
! [B: $tType,A: $tType,T: tLList446370796tllist @ A @ B] :
( ( tLList1669959861e_tmap @ A @ A @ B @ B @ ( id @ A ) @ ( id @ B ) @ T )
= T ) ).
% tllist.map_id
thf(fact_69_tllist_Omap__id0,axiom,
! [B: $tType,A: $tType] :
( ( tLList1669959861e_tmap @ A @ A @ B @ B @ ( id @ A ) @ ( id @ B ) )
= ( id @ ( tLList446370796tllist @ A @ B ) ) ) ).
% tllist.map_id0
thf(fact_70_tllist_Osimps_I15_J,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType,F1: A > C,F2: B > D,X21: A,X22: tLList446370796tllist @ A @ B] :
( ( tLList1669959861e_tmap @ A @ C @ B @ D @ F1 @ F2 @ ( tLList1992840728_TCons @ A @ B @ X21 @ X22 ) )
= ( tLList1992840728_TCons @ C @ D @ ( F1 @ X21 ) @ ( tLList1669959861e_tmap @ A @ C @ B @ D @ F1 @ F2 @ X22 ) ) ) ).
% tllist.simps(15)
thf(fact_71_tllist_Osimps_I14_J,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,F1: A > C,F2: B > D,X1: B] :
( ( tLList1669959861e_tmap @ A @ C @ B @ D @ F1 @ F2 @ ( tLList57907754e_TNil @ B @ A @ X1 ) )
= ( tLList57907754e_TNil @ D @ C @ ( F2 @ X1 ) ) ) ).
% tllist.simps(14)
thf(fact_72_tllist_Ocorec__disc_I2_J,axiom,
! [B: $tType,A: $tType,E: $tType,P4: E > $o,A2: E,G12: E > B,G212: E > A,Q222: E > $o,G2212: E > ( tLList446370796tllist @ A @ B ),G2222: E > E] :
( ~ ( P4 @ A2 )
=> ~ ( tLList1968894004s_TNil @ A @ B @ ( tLList1614408749tllist @ E @ B @ A @ P4 @ G12 @ G212 @ Q222 @ G2212 @ G2222 @ A2 ) ) ) ).
% tllist.corec_disc(2)
thf(fact_73_tllist_Ocorec__disc_I1_J,axiom,
! [B: $tType,A: $tType,E: $tType,P4: E > $o,A2: E,G12: E > B,G212: E > A,Q222: E > $o,G2212: E > ( tLList446370796tllist @ A @ B ),G2222: E > E] :
( ( P4 @ A2 )
=> ( tLList1968894004s_TNil @ A @ B @ ( tLList1614408749tllist @ E @ B @ A @ P4 @ G12 @ G212 @ Q222 @ G2212 @ G2222 @ A2 ) ) ) ).
% tllist.corec_disc(1)
thf(fact_74_tllist_Ocorec_I2_J,axiom,
! [B: $tType,A: $tType,E: $tType,P4: E > $o,A2: E,G12: E > B,G212: E > A,Q222: E > $o,G2212: E > ( tLList446370796tllist @ A @ B ),G2222: E > E] :
( ~ ( P4 @ A2 )
=> ( ( tLList1614408749tllist @ E @ B @ A @ P4 @ G12 @ G212 @ Q222 @ G2212 @ G2222 @ A2 )
= ( tLList1992840728_TCons @ A @ B @ ( G212 @ A2 ) @ ( if @ ( tLList446370796tllist @ A @ B ) @ ( Q222 @ A2 ) @ ( G2212 @ A2 ) @ ( tLList1614408749tllist @ E @ B @ A @ P4 @ G12 @ G212 @ Q222 @ G2212 @ G2222 @ ( G2222 @ A2 ) ) ) ) ) ) ).
% tllist.corec(2)
thf(fact_75_tllist_Ocorec_I1_J,axiom,
! [A: $tType,B: $tType,E: $tType,P4: E > $o,A2: E,G12: E > B,G212: E > A,Q222: E > $o,G2212: E > ( tLList446370796tllist @ A @ B ),G2222: E > E] :
( ( P4 @ A2 )
=> ( ( tLList1614408749tllist @ E @ B @ A @ P4 @ G12 @ G212 @ Q222 @ G2212 @ G2222 @ A2 )
= ( tLList57907754e_TNil @ B @ A @ ( G12 @ A2 ) ) ) ) ).
% tllist.corec(1)
thf(fact_76_tllist_Ocorec__sel_I3_J,axiom,
! [B: $tType,A: $tType,E: $tType,P4: E > $o,A2: E,Q222: E > $o,G12: E > B,G212: E > A,G2212: E > ( tLList446370796tllist @ A @ B ),G2222: E > E] :
( ~ ( P4 @ A2 )
=> ( ( ( Q222 @ A2 )
=> ( ( tLList_Mirabelle_ttl @ A @ B @ ( tLList1614408749tllist @ E @ B @ A @ P4 @ G12 @ G212 @ Q222 @ G2212 @ G2222 @ A2 ) )
= ( G2212 @ A2 ) ) )
& ( ~ ( Q222 @ A2 )
=> ( ( tLList_Mirabelle_ttl @ A @ B @ ( tLList1614408749tllist @ E @ B @ A @ P4 @ G12 @ G212 @ Q222 @ G2212 @ G2222 @ A2 ) )
= ( tLList1614408749tllist @ E @ B @ A @ P4 @ G12 @ G212 @ Q222 @ G2212 @ G2222 @ ( G2222 @ A2 ) ) ) ) ) ) ).
% tllist.corec_sel(3)
thf(fact_77_tllist_Ocorec__sel_I1_J,axiom,
! [Aa: $tType,A: $tType,E: $tType,P4: E > $o,A2: E,G12: E > A,G212: E > Aa,Q222: E > $o,G2212: E > ( tLList446370796tllist @ Aa @ A ),G2222: E > E] :
( ( P4 @ A2 )
=> ( ( tLList2110128105rminal @ Aa @ A @ ( tLList1614408749tllist @ E @ A @ Aa @ P4 @ G12 @ G212 @ Q222 @ G2212 @ G2222 @ A2 ) )
= ( G12 @ A2 ) ) ) ).
% tllist.corec_sel(1)
thf(fact_78_tllist_Ocorec__sel_I2_J,axiom,
! [B: $tType,A: $tType,E: $tType,P4: E > $o,A2: E,G12: E > B,G212: E > A,Q222: E > $o,G2212: E > ( tLList446370796tllist @ A @ B ),G2222: E > E] :
( ~ ( P4 @ A2 )
=> ( ( tLList_Mirabelle_thd @ A @ B @ ( tLList1614408749tllist @ E @ B @ A @ P4 @ G12 @ G212 @ Q222 @ G2212 @ G2222 @ A2 ) )
= ( G212 @ A2 ) ) ) ).
% tllist.corec_sel(2)
thf(fact_79_tllist_Orel__distinct_I2_J,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,R1: A > C > $o,R2: B > D > $o,Y21: A,Y22: tLList446370796tllist @ A @ B,X1: D] :
~ ( tLList1380991092t_all2 @ A @ C @ B @ D @ R1 @ R2 @ ( tLList1992840728_TCons @ A @ B @ Y21 @ Y22 ) @ ( tLList57907754e_TNil @ D @ C @ X1 ) ) ).
% tllist.rel_distinct(2)
thf(fact_80_tllist_Orel__distinct_I1_J,axiom,
! [B: $tType,A: $tType,D: $tType,C: $tType,R1: A > C > $o,R2: B > D > $o,X1: B,Y21: C,Y22: tLList446370796tllist @ C @ D] :
~ ( tLList1380991092t_all2 @ A @ C @ B @ D @ R1 @ R2 @ ( tLList57907754e_TNil @ B @ A @ X1 ) @ ( tLList1992840728_TCons @ C @ D @ Y21 @ Y22 ) ) ).
% tllist.rel_distinct(1)
thf(fact_81_tllist_Orel__cases,axiom,
! [A: $tType,B: $tType,D: $tType,C: $tType,R1: A > C > $o,R2: B > D > $o,A2: tLList446370796tllist @ A @ B,B3: tLList446370796tllist @ C @ D] :
( ( tLList1380991092t_all2 @ A @ C @ B @ D @ R1 @ R2 @ A2 @ B3 )
=> ( ! [X4: B] :
( ( A2
= ( tLList57907754e_TNil @ B @ A @ X4 ) )
=> ! [Y3: D] :
( ( B3
= ( tLList57907754e_TNil @ D @ C @ Y3 ) )
=> ~ ( R2 @ X4 @ Y3 ) ) )
=> ~ ! [X1a: A,X23: tLList446370796tllist @ A @ B] :
( ( A2
= ( tLList1992840728_TCons @ A @ B @ X1a @ X23 ) )
=> ! [Y1a: C,Y23: tLList446370796tllist @ C @ D] :
( ( B3
= ( tLList1992840728_TCons @ C @ D @ Y1a @ Y23 ) )
=> ( ( R1 @ X1a @ Y1a )
=> ~ ( tLList1380991092t_all2 @ A @ C @ B @ D @ R1 @ R2 @ X23 @ Y23 ) ) ) ) ) ) ).
% tllist.rel_cases
thf(fact_82_tllist_Orel__refl__strong,axiom,
! [B: $tType,A: $tType,X3: tLList446370796tllist @ A @ B,R1a: A > A > $o,R2a: B > B > $o] :
( ! [Z1: A] :
( ( member @ A @ Z1 @ ( tLList1719791291e_tset @ A @ B @ X3 ) )
=> ( R1a @ Z1 @ Z1 ) )
=> ( ! [Z2: B] :
( ( member @ B @ Z2 @ ( tLList457864275tllist @ A @ B @ X3 ) )
=> ( R2a @ Z2 @ Z2 ) )
=> ( tLList1380991092t_all2 @ A @ A @ B @ B @ R1a @ R2a @ X3 @ X3 ) ) ) ).
% tllist.rel_refl_strong
thf(fact_83_tllist_Orel__mono__strong,axiom,
! [A: $tType,B: $tType,D: $tType,C: $tType,R1: A > C > $o,R2: B > D > $o,X3: tLList446370796tllist @ A @ B,Y: tLList446370796tllist @ C @ D,R1a: A > C > $o,R2a: B > D > $o] :
( ( tLList1380991092t_all2 @ A @ C @ B @ D @ R1 @ R2 @ X3 @ Y )
=> ( ! [Z1: A,Y12: C] :
( ( member @ A @ Z1 @ ( tLList1719791291e_tset @ A @ B @ X3 ) )
=> ( ( member @ C @ Y12 @ ( tLList1719791291e_tset @ C @ D @ Y ) )
=> ( ( R1 @ Z1 @ Y12 )
=> ( R1a @ Z1 @ Y12 ) ) ) )
=> ( ! [Z2: B,Y23: D] :
( ( member @ B @ Z2 @ ( tLList457864275tllist @ A @ B @ X3 ) )
=> ( ( member @ D @ Y23 @ ( tLList457864275tllist @ C @ D @ Y ) )
=> ( ( R2 @ Z2 @ Y23 )
=> ( R2a @ Z2 @ Y23 ) ) ) )
=> ( tLList1380991092t_all2 @ A @ C @ B @ D @ R1a @ R2a @ X3 @ Y ) ) ) ) ).
% tllist.rel_mono_strong
thf(fact_84_tllist_Orel__cong,axiom,
! [A: $tType,B: $tType,D: $tType,C: $tType,X3: tLList446370796tllist @ A @ B,Ya: tLList446370796tllist @ A @ B,Y: tLList446370796tllist @ C @ D,Xa2: tLList446370796tllist @ C @ D,R1: A > C > $o,R1a: A > C > $o,R2: B > D > $o,R2a: B > D > $o] :
( ( X3 = Ya )
=> ( ( Y = Xa2 )
=> ( ! [Z1: A,Y12: C] :
( ( member @ A @ Z1 @ ( tLList1719791291e_tset @ A @ B @ Ya ) )
=> ( ( member @ C @ Y12 @ ( tLList1719791291e_tset @ C @ D @ Xa2 ) )
=> ( ( R1 @ Z1 @ Y12 )
= ( R1a @ Z1 @ Y12 ) ) ) )
=> ( ! [Z2: B,Y23: D] :
( ( member @ B @ Z2 @ ( tLList457864275tllist @ A @ B @ Ya ) )
=> ( ( member @ D @ Y23 @ ( tLList457864275tllist @ C @ D @ Xa2 ) )
=> ( ( R2 @ Z2 @ Y23 )
= ( R2a @ Z2 @ Y23 ) ) ) )
=> ( ( tLList1380991092t_all2 @ A @ C @ B @ D @ R1 @ R2 @ X3 @ Y )
= ( tLList1380991092t_all2 @ A @ C @ B @ D @ R1a @ R2a @ Ya @ Xa2 ) ) ) ) ) ) ).
% tllist.rel_cong
thf(fact_85_tllist_Omap__sel_I3_J,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType,A2: tLList446370796tllist @ A @ B,F1: A > C,F2: B > D] :
( ~ ( tLList1968894004s_TNil @ A @ B @ A2 )
=> ( ( tLList_Mirabelle_ttl @ C @ D @ ( tLList1669959861e_tmap @ A @ C @ B @ D @ F1 @ F2 @ A2 ) )
= ( tLList1669959861e_tmap @ A @ C @ B @ D @ F1 @ F2 @ ( tLList_Mirabelle_ttl @ A @ B @ A2 ) ) ) ) ).
% tllist.map_sel(3)
thf(fact_86_tllist_Omap__sel_I1_J,axiom,
! [C: $tType,D: $tType,B: $tType,A: $tType,A2: tLList446370796tllist @ A @ B,F1: A > C,F2: B > D] :
( ( tLList1968894004s_TNil @ A @ B @ A2 )
=> ( ( tLList2110128105rminal @ C @ D @ ( tLList1669959861e_tmap @ A @ C @ B @ D @ F1 @ F2 @ A2 ) )
= ( F2 @ ( tLList2110128105rminal @ A @ B @ A2 ) ) ) ) ).
% tllist.map_sel(1)
thf(fact_87_insert__absorb2,axiom,
! [A: $tType,X3: A,A3: set @ A] :
( ( insert @ A @ X3 @ ( insert @ A @ X3 @ A3 ) )
= ( insert @ A @ X3 @ A3 ) ) ).
% insert_absorb2
thf(fact_88_insert__iff,axiom,
! [A: $tType,A2: A,B3: A,A3: set @ A] :
( ( member @ A @ A2 @ ( insert @ A @ B3 @ A3 ) )
= ( ( A2 = B3 )
| ( member @ A @ A2 @ A3 ) ) ) ).
% insert_iff
thf(fact_89_insertCI,axiom,
! [A: $tType,A2: A,B4: set @ A,B3: A] :
( ( ~ ( member @ A @ A2 @ B4 )
=> ( A2 = B3 ) )
=> ( member @ A @ A2 @ ( insert @ A @ B3 @ B4 ) ) ) ).
% insertCI
thf(fact_90_tllist_Osimps_I22_J,axiom,
! [A: $tType,B: $tType,X1: B] :
( ( tLList457864275tllist @ A @ B @ ( tLList57907754e_TNil @ B @ A @ X1 ) )
= ( insert @ B @ X1 @ ( bot_bot @ ( set @ B ) ) ) ) ).
% tllist.simps(22)
thf(fact_91_tllist_Opred__set,axiom,
! [B: $tType,A: $tType] :
( ( tLList11265572tllist @ A @ B )
= ( ^ [P12: A > $o,P22: B > $o,X2: tLList446370796tllist @ A @ B] :
( ! [Y4: A] :
( ( member @ A @ Y4 @ ( tLList1719791291e_tset @ A @ B @ X2 ) )
=> ( P12 @ Y4 ) )
& ! [Y4: B] :
( ( member @ B @ Y4 @ ( tLList457864275tllist @ A @ B @ X2 ) )
=> ( P22 @ Y4 ) ) ) ) ) ).
% tllist.pred_set
thf(fact_92_mk__disjoint__insert,axiom,
! [A: $tType,A2: A,A3: set @ A] :
( ( member @ A @ A2 @ A3 )
=> ? [B5: set @ A] :
( ( A3
= ( insert @ A @ A2 @ B5 ) )
& ~ ( member @ A @ A2 @ B5 ) ) ) ).
% mk_disjoint_insert
thf(fact_93_insert__commute,axiom,
! [A: $tType,X3: A,Y: A,A3: set @ A] :
( ( insert @ A @ X3 @ ( insert @ A @ Y @ A3 ) )
= ( insert @ A @ Y @ ( insert @ A @ X3 @ A3 ) ) ) ).
% insert_commute
thf(fact_94_empty__iff,axiom,
! [A: $tType,C2: A] :
~ ( member @ A @ C2 @ ( bot_bot @ ( set @ A ) ) ) ).
% empty_iff
thf(fact_95_all__not__in__conv,axiom,
! [A: $tType,A3: set @ A] :
( ( ! [X2: A] :
~ ( member @ A @ X2 @ A3 ) )
= ( A3
= ( bot_bot @ ( set @ A ) ) ) ) ).
% all_not_in_conv
thf(fact_96_Collect__empty__eq,axiom,
! [A: $tType,P: A > $o] :
( ( ( collect @ A @ P )
= ( bot_bot @ ( set @ A ) ) )
= ( ! [X2: A] :
~ ( P @ X2 ) ) ) ).
% Collect_empty_eq
thf(fact_97_empty__Collect__eq,axiom,
! [A: $tType,P: A > $o] :
( ( ( bot_bot @ ( set @ A ) )
= ( collect @ A @ P ) )
= ( ! [X2: A] :
~ ( P @ X2 ) ) ) ).
% empty_Collect_eq
thf(fact_98_singletonI,axiom,
! [A: $tType,A2: A] : ( member @ A @ A2 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% singletonI
thf(fact_99_ball__empty,axiom,
! [A: $tType,P: A > $o,X: A] :
( ( member @ A @ X @ ( bot_bot @ ( set @ A ) ) )
=> ( P @ X ) ) ).
% ball_empty
thf(fact_100_singleton__inject,axiom,
! [A: $tType,A2: A,B3: A] :
( ( ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) )
= ( insert @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) )
=> ( A2 = B3 ) ) ).
% singleton_inject
thf(fact_101_insert__not__empty,axiom,
! [A: $tType,A2: A,A3: set @ A] :
( ( insert @ A @ A2 @ A3 )
!= ( bot_bot @ ( set @ A ) ) ) ).
% insert_not_empty
thf(fact_102_doubleton__eq__iff,axiom,
! [A: $tType,A2: A,B3: A,C2: A,D2: A] :
( ( ( insert @ A @ A2 @ ( insert @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) )
= ( insert @ A @ C2 @ ( insert @ A @ D2 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( ( ( A2 = C2 )
& ( B3 = D2 ) )
| ( ( A2 = D2 )
& ( B3 = C2 ) ) ) ) ).
% doubleton_eq_iff
thf(fact_103_singleton__iff,axiom,
! [A: $tType,B3: A,A2: A] :
( ( member @ A @ B3 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) )
= ( B3 = A2 ) ) ).
% singleton_iff
thf(fact_104_singletonD,axiom,
! [A: $tType,B3: A,A2: A] :
( ( member @ A @ B3 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) )
=> ( B3 = A2 ) ) ).
% singletonD
thf(fact_105_emptyE,axiom,
! [A: $tType,A2: A] :
~ ( member @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ).
% emptyE
thf(fact_106_Ball__def,axiom,
! [A: $tType] :
( ( ball @ A )
= ( ^ [A5: set @ A,P5: A > $o] :
! [X2: A] :
( ( member @ A @ X2 @ A5 )
=> ( P5 @ X2 ) ) ) ) ).
% Ball_def
thf(fact_107_equals0D,axiom,
! [A: $tType,A3: set @ A,A2: A] :
( ( A3
= ( bot_bot @ ( set @ A ) ) )
=> ~ ( member @ A @ A2 @ A3 ) ) ).
% equals0D
thf(fact_108_equals0I,axiom,
! [A: $tType,A3: set @ A] :
( ! [Y3: A] :
~ ( member @ A @ Y3 @ A3 )
=> ( A3
= ( bot_bot @ ( set @ A ) ) ) ) ).
% equals0I
thf(fact_109_ex__in__conv,axiom,
! [A: $tType,A3: set @ A] :
( ( ? [X2: A] : ( member @ A @ X2 @ A3 ) )
= ( A3
!= ( bot_bot @ ( set @ A ) ) ) ) ).
% ex_in_conv
thf(fact_110_tllist_Osimps_I20_J,axiom,
! [B: $tType,A: $tType,X1: B] :
( ( tLList1719791291e_tset @ A @ B @ ( tLList57907754e_TNil @ B @ A @ X1 ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% tllist.simps(20)
thf(fact_111_insertE,axiom,
! [A: $tType,A2: A,B3: A,A3: set @ A] :
( ( member @ A @ A2 @ ( insert @ A @ B3 @ A3 ) )
=> ( ( A2 != B3 )
=> ( member @ A @ A2 @ A3 ) ) ) ).
% insertE
thf(fact_112_insertI1,axiom,
! [A: $tType,A2: A,B4: set @ A] : ( member @ A @ A2 @ ( insert @ A @ A2 @ B4 ) ) ).
% insertI1
thf(fact_113_insertI2,axiom,
! [A: $tType,A2: A,B4: set @ A,B3: A] :
( ( member @ A @ A2 @ B4 )
=> ( member @ A @ A2 @ ( insert @ A @ B3 @ B4 ) ) ) ).
% insertI2
thf(fact_114_Set_Oset__insert,axiom,
! [A: $tType,X3: A,A3: set @ A] :
( ( member @ A @ X3 @ A3 )
=> ~ ! [B5: set @ A] :
( ( A3
= ( insert @ A @ X3 @ B5 ) )
=> ( member @ A @ X3 @ B5 ) ) ) ).
% Set.set_insert
thf(fact_115_insert__ident,axiom,
! [A: $tType,X3: A,A3: set @ A,B4: set @ A] :
( ~ ( member @ A @ X3 @ A3 )
=> ( ~ ( member @ A @ X3 @ B4 )
=> ( ( ( insert @ A @ X3 @ A3 )
= ( insert @ A @ X3 @ B4 ) )
= ( A3 = B4 ) ) ) ) ).
% insert_ident
thf(fact_116_insert__absorb,axiom,
! [A: $tType,A2: A,A3: set @ A] :
( ( member @ A @ A2 @ A3 )
=> ( ( insert @ A @ A2 @ A3 )
= A3 ) ) ).
% insert_absorb
thf(fact_117_insert__eq__iff,axiom,
! [A: $tType,A2: A,A3: set @ A,B3: A,B4: set @ A] :
( ~ ( member @ A @ A2 @ A3 )
=> ( ~ ( member @ A @ B3 @ B4 )
=> ( ( ( insert @ A @ A2 @ A3 )
= ( insert @ A @ B3 @ B4 ) )
= ( ( ( A2 = B3 )
=> ( A3 = B4 ) )
& ( ( A2 != B3 )
=> ? [C3: set @ A] :
( ( A3
= ( insert @ A @ B3 @ C3 ) )
& ~ ( member @ A @ B3 @ C3 )
& ( B4
= ( insert @ A @ A2 @ C3 ) )
& ~ ( member @ A @ A2 @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_118_the__elem__eq,axiom,
! [A: $tType,X3: A] :
( ( the_elem @ A @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
= X3 ) ).
% the_elem_eq
thf(fact_119_bot__apply,axiom,
! [C: $tType,D: $tType] :
( ( bot @ C @ ( type2 @ C ) )
=> ( ( bot_bot @ ( D > C ) )
= ( ^ [X2: D] : ( bot_bot @ C ) ) ) ) ).
% bot_apply
thf(fact_120_is__singletonI,axiom,
! [A: $tType,X3: A] : ( is_singleton @ A @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% is_singletonI
thf(fact_121_override__on__emptyset,axiom,
! [B: $tType,A: $tType,F: A > B,G: A > B] :
( ( override_on @ A @ B @ F @ G @ ( bot_bot @ ( set @ A ) ) )
= F ) ).
% override_on_emptyset
thf(fact_122_Set_Ois__empty__def,axiom,
! [A: $tType] :
( ( is_empty @ A )
= ( ^ [A5: set @ A] :
( A5
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Set.is_empty_def
thf(fact_123_override__on__apply__notin,axiom,
! [B: $tType,A: $tType,A2: A,A3: set @ A,F: A > B,G: A > B] :
( ~ ( member @ A @ A2 @ A3 )
=> ( ( override_on @ A @ B @ F @ G @ A3 @ A2 )
= ( F @ A2 ) ) ) ).
% override_on_apply_notin
thf(fact_124_override__on__apply__in,axiom,
! [B: $tType,A: $tType,A2: A,A3: set @ A,F: A > B,G: A > B] :
( ( member @ A @ A2 @ A3 )
=> ( ( override_on @ A @ B @ F @ G @ A3 @ A2 )
= ( G @ A2 ) ) ) ).
% override_on_apply_in
thf(fact_125_bot__set__def,axiom,
! [A: $tType] :
( ( bot_bot @ ( set @ A ) )
= ( collect @ A @ ( bot_bot @ ( A > $o ) ) ) ) ).
% bot_set_def
thf(fact_126_override__on__def,axiom,
! [B: $tType,A: $tType] :
( ( override_on @ A @ B )
= ( ^ [F3: A > B,G3: A > B,A5: set @ A,A4: A] : ( if @ B @ ( member @ A @ A4 @ A5 ) @ ( G3 @ A4 ) @ ( F3 @ A4 ) ) ) ) ).
% override_on_def
thf(fact_127_is__singleton__the__elem,axiom,
! [A: $tType] :
( ( is_singleton @ A )
= ( ^ [A5: set @ A] :
( A5
= ( insert @ A @ ( the_elem @ A @ A5 ) @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% is_singleton_the_elem
thf(fact_128_is__singletonI_H,axiom,
! [A: $tType,A3: set @ A] :
( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X4: A,Y3: A] :
( ( member @ A @ X4 @ A3 )
=> ( ( member @ A @ Y3 @ A3 )
=> ( X4 = Y3 ) ) )
=> ( is_singleton @ A @ A3 ) ) ) ).
% is_singletonI'
thf(fact_129_bot__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( bot @ B @ ( type2 @ B ) )
=> ( ( bot_bot @ ( A > B ) )
= ( ^ [X2: A] : ( bot_bot @ B ) ) ) ) ).
% bot_fun_def
thf(fact_130_is__singletonE,axiom,
! [A: $tType,A3: set @ A] :
( ( is_singleton @ A @ A3 )
=> ~ ! [X4: A] :
( A3
!= ( insert @ A @ X4 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% is_singletonE
thf(fact_131_is__singleton__def,axiom,
! [A: $tType] :
( ( is_singleton @ A )
= ( ^ [A5: set @ A] :
? [X2: A] :
( A5
= ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% is_singleton_def
thf(fact_132_pairwise__singleton,axiom,
! [A: $tType,P: A > A > $o,A3: A] : ( pairwise @ A @ P @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% pairwise_singleton
thf(fact_133_ball__reg,axiom,
! [A: $tType,R: set @ A,P: A > $o,Q: A > $o] :
( ! [X4: A] :
( ( member @ A @ X4 @ R )
=> ( ( P @ X4 )
=> ( Q @ X4 ) ) )
=> ( ! [X4: A] :
( ( member @ A @ X4 @ R )
=> ( P @ X4 ) )
=> ! [X: A] :
( ( member @ A @ X @ R )
=> ( Q @ X ) ) ) ) ).
% ball_reg
thf(fact_134_bot__empty__eq,axiom,
! [A: $tType] :
( ( bot_bot @ ( A > $o ) )
= ( ^ [X2: A] : ( member @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% bot_empty_eq
thf(fact_135_Collect__empty__eq__bot,axiom,
! [A: $tType,P: A > $o] :
( ( ( collect @ A @ P )
= ( bot_bot @ ( set @ A ) ) )
= ( P
= ( bot_bot @ ( A > $o ) ) ) ) ).
% Collect_empty_eq_bot
thf(fact_136_pairwise__def,axiom,
! [A: $tType] :
( ( pairwise @ A )
= ( ^ [R3: A > A > $o,S: set @ A] :
! [X2: A] :
( ( member @ A @ X2 @ S )
=> ! [Y4: A] :
( ( member @ A @ Y4 @ S )
=> ( ( X2 != Y4 )
=> ( R3 @ X2 @ Y4 ) ) ) ) ) ) ).
% pairwise_def
thf(fact_137_pairwise__empty,axiom,
! [A: $tType,P: A > A > $o] : ( pairwise @ A @ P @ ( bot_bot @ ( set @ A ) ) ) ).
% pairwise_empty
thf(fact_138_pairwise__insert,axiom,
! [A: $tType,R4: A > A > $o,X3: A,S2: set @ A] :
( ( pairwise @ A @ R4 @ ( insert @ A @ X3 @ S2 ) )
= ( ! [Y4: A] :
( ( ( member @ A @ Y4 @ S2 )
& ( Y4 != X3 ) )
=> ( ( R4 @ X3 @ Y4 )
& ( R4 @ Y4 @ X3 ) ) )
& ( pairwise @ A @ R4 @ S2 ) ) ) ).
% pairwise_insert
thf(fact_139_in__inv__imagep,axiom,
! [B: $tType,A: $tType] :
( ( inv_imagep @ A @ B )
= ( ^ [R5: A > A > $o,F3: B > A,X2: B,Y4: B] : ( R5 @ ( F3 @ X2 ) @ ( F3 @ Y4 ) ) ) ) ).
% in_inv_imagep
thf(fact_140_id__fcomp,axiom,
! [B: $tType,A: $tType,G: A > B] :
( ( fcomp @ A @ A @ B @ ( id @ A ) @ G )
= G ) ).
% id_fcomp
thf(fact_141_fcomp__id,axiom,
! [B: $tType,A: $tType,F: A > B] :
( ( fcomp @ A @ B @ B @ F @ ( id @ B ) )
= F ) ).
% fcomp_id
thf(fact_142_singleton__insert__inj__eq_H,axiom,
! [A: $tType,A2: A,A3: set @ A,B3: A] :
( ( ( insert @ A @ A2 @ A3 )
= ( insert @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) )
= ( ( A2 = B3 )
& ( ord_less_eq @ ( set @ A ) @ A3 @ ( insert @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_143_order__refl,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X3: A] : ( ord_less_eq @ A @ X3 @ X3 ) ) ).
% order_refl
thf(fact_144_subsetI,axiom,
! [A: $tType,A3: set @ A,B4: set @ A] :
( ! [X4: A] :
( ( member @ A @ X4 @ A3 )
=> ( member @ A @ X4 @ B4 ) )
=> ( ord_less_eq @ ( set @ A ) @ A3 @ B4 ) ) ).
% subsetI
thf(fact_145_subset__antisym,axiom,
! [A: $tType,A3: set @ A,B4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B4 )
=> ( ( ord_less_eq @ ( set @ A ) @ B4 @ A3 )
=> ( A3 = B4 ) ) ) ).
% subset_antisym
thf(fact_146_fcomp__apply,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( fcomp @ B @ C @ A )
= ( ^ [F3: B > C,G3: C > A,X2: B] : ( G3 @ ( F3 @ X2 ) ) ) ) ).
% fcomp_apply
thf(fact_147_subset__empty,axiom,
! [A: $tType,A3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ ( bot_bot @ ( set @ A ) ) )
= ( A3
= ( bot_bot @ ( set @ A ) ) ) ) ).
% subset_empty
thf(fact_148_empty__subsetI,axiom,
! [A: $tType,A3: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ A3 ) ).
% empty_subsetI
thf(fact_149_insert__subset,axiom,
! [A: $tType,X3: A,A3: set @ A,B4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ X3 @ A3 ) @ B4 )
= ( ( member @ A @ X3 @ B4 )
& ( ord_less_eq @ ( set @ A ) @ A3 @ B4 ) ) ) ).
% insert_subset
thf(fact_150_singleton__insert__inj__eq,axiom,
! [A: $tType,B3: A,A2: A,A3: set @ A] :
( ( ( insert @ A @ B3 @ ( bot_bot @ ( set @ A ) ) )
= ( insert @ A @ A2 @ A3 ) )
= ( ( A2 = B3 )
& ( ord_less_eq @ ( set @ A ) @ A3 @ ( insert @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_151_bot_Oextremum,axiom,
! [A: $tType] :
( ( order_bot @ A @ ( type2 @ A ) )
=> ! [A2: A] : ( ord_less_eq @ A @ ( bot_bot @ A ) @ A2 ) ) ).
% bot.extremum
thf(fact_152_bot_Oextremum__unique,axiom,
! [A: $tType] :
( ( order_bot @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( ord_less_eq @ A @ A2 @ ( bot_bot @ A ) )
= ( A2
= ( bot_bot @ A ) ) ) ) ).
% bot.extremum_unique
thf(fact_153_bot_Oextremum__uniqueI,axiom,
! [A: $tType] :
( ( order_bot @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( ord_less_eq @ A @ A2 @ ( bot_bot @ A ) )
=> ( A2
= ( bot_bot @ A ) ) ) ) ).
% bot.extremum_uniqueI
thf(fact_154_set__mp,axiom,
! [A: $tType,A3: set @ A,B4: set @ A,X3: A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B4 )
=> ( ( member @ A @ X3 @ A3 )
=> ( member @ A @ X3 @ B4 ) ) ) ).
% set_mp
thf(fact_155_in__mono,axiom,
! [A: $tType,A3: set @ A,B4: set @ A,X3: A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B4 )
=> ( ( member @ A @ X3 @ A3 )
=> ( member @ A @ X3 @ B4 ) ) ) ).
% in_mono
thf(fact_156_subsetD,axiom,
! [A: $tType,A3: set @ A,B4: set @ A,C2: A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B4 )
=> ( ( member @ A @ C2 @ A3 )
=> ( member @ A @ C2 @ B4 ) ) ) ).
% subsetD
thf(fact_157_subsetCE,axiom,
! [A: $tType,A3: set @ A,B4: set @ A,C2: A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B4 )
=> ( ( member @ A @ C2 @ A3 )
=> ( member @ A @ C2 @ B4 ) ) ) ).
% subsetCE
thf(fact_158_equalityE,axiom,
! [A: $tType,A3: set @ A,B4: set @ A] :
( ( A3 = B4 )
=> ~ ( ( ord_less_eq @ ( set @ A ) @ A3 @ B4 )
=> ~ ( ord_less_eq @ ( set @ A ) @ B4 @ A3 ) ) ) ).
% equalityE
thf(fact_159_subset__eq,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A5: set @ A,B6: set @ A] :
! [X2: A] :
( ( member @ A @ X2 @ A5 )
=> ( member @ A @ X2 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_160_equalityD1,axiom,
! [A: $tType,A3: set @ A,B4: set @ A] :
( ( A3 = B4 )
=> ( ord_less_eq @ ( set @ A ) @ A3 @ B4 ) ) ).
% equalityD1
thf(fact_161_equalityD2,axiom,
! [A: $tType,A3: set @ A,B4: set @ A] :
( ( A3 = B4 )
=> ( ord_less_eq @ ( set @ A ) @ B4 @ A3 ) ) ).
% equalityD2
thf(fact_162_set__rev__mp,axiom,
! [A: $tType,X3: A,A3: set @ A,B4: set @ A] :
( ( member @ A @ X3 @ A3 )
=> ( ( ord_less_eq @ ( set @ A ) @ A3 @ B4 )
=> ( member @ A @ X3 @ B4 ) ) ) ).
% set_rev_mp
thf(fact_163_subset__iff,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A5: set @ A,B6: set @ A] :
! [T2: A] :
( ( member @ A @ T2 @ A5 )
=> ( member @ A @ T2 @ B6 ) ) ) ) ).
% subset_iff
thf(fact_164_rev__subsetD,axiom,
! [A: $tType,C2: A,A3: set @ A,B4: set @ A] :
( ( member @ A @ C2 @ A3 )
=> ( ( ord_less_eq @ ( set @ A ) @ A3 @ B4 )
=> ( member @ A @ C2 @ B4 ) ) ) ).
% rev_subsetD
thf(fact_165_subset__refl,axiom,
! [A: $tType,A3: set @ A] : ( ord_less_eq @ ( set @ A ) @ A3 @ A3 ) ).
% subset_refl
thf(fact_166_Collect__mono,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ! [X4: A] :
( ( P @ X4 )
=> ( Q @ X4 ) )
=> ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) ) ) ).
% Collect_mono
thf(fact_167_subset__trans,axiom,
! [A: $tType,A3: set @ A,B4: set @ A,C4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B4 )
=> ( ( ord_less_eq @ ( set @ A ) @ B4 @ C4 )
=> ( ord_less_eq @ ( set @ A ) @ A3 @ C4 ) ) ) ).
% subset_trans
thf(fact_168_set__eq__subset,axiom,
! [A: $tType] :
( ( ^ [Y2: set @ A,Z3: set @ A] : Y2 = Z3 )
= ( ^ [A5: set @ A,B6: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ B6 )
& ( ord_less_eq @ ( set @ A ) @ B6 @ A5 ) ) ) ) ).
% set_eq_subset
thf(fact_169_contra__subsetD,axiom,
! [A: $tType,A3: set @ A,B4: set @ A,C2: A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B4 )
=> ( ~ ( member @ A @ C2 @ B4 )
=> ~ ( member @ A @ C2 @ A3 ) ) ) ).
% contra_subsetD
thf(fact_170_Collect__mono__iff,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) )
= ( ! [X2: A] :
( ( P @ X2 )
=> ( Q @ X2 ) ) ) ) ).
% Collect_mono_iff
thf(fact_171_fcomp__def,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( fcomp @ A @ B @ C )
= ( ^ [F3: A > B,G3: B > C,X2: A] : ( G3 @ ( F3 @ X2 ) ) ) ) ).
% fcomp_def
thf(fact_172_fcomp__assoc,axiom,
! [A: $tType,D: $tType,B: $tType,C: $tType,F: A > D,G: D > C,H: C > B] :
( ( fcomp @ A @ C @ B @ ( fcomp @ A @ D @ C @ F @ G ) @ H )
= ( fcomp @ A @ D @ B @ F @ ( fcomp @ D @ C @ B @ G @ H ) ) ) ).
% fcomp_assoc
thf(fact_173_subset__insertI2,axiom,
! [A: $tType,A3: set @ A,B4: set @ A,B3: A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B4 )
=> ( ord_less_eq @ ( set @ A ) @ A3 @ ( insert @ A @ B3 @ B4 ) ) ) ).
% subset_insertI2
thf(fact_174_subset__insertI,axiom,
! [A: $tType,B4: set @ A,A2: A] : ( ord_less_eq @ ( set @ A ) @ B4 @ ( insert @ A @ A2 @ B4 ) ) ).
% subset_insertI
thf(fact_175_subset__insert,axiom,
! [A: $tType,X3: A,A3: set @ A,B4: set @ A] :
( ~ ( member @ A @ X3 @ A3 )
=> ( ( ord_less_eq @ ( set @ A ) @ A3 @ ( insert @ A @ X3 @ B4 ) )
= ( ord_less_eq @ ( set @ A ) @ A3 @ B4 ) ) ) ).
% subset_insert
thf(fact_176_Set_Oinsert__mono,axiom,
! [A: $tType,C4: set @ A,D3: set @ A,A2: A] :
( ( ord_less_eq @ ( set @ A ) @ C4 @ D3 )
=> ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ A2 @ C4 ) @ ( insert @ A @ A2 @ D3 ) ) ) ).
% Set.insert_mono
thf(fact_177_le__funD,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B @ ( type2 @ B ) )
=> ! [F: A > B,G: A > B,X3: A] :
( ( ord_less_eq @ ( A > B ) @ F @ G )
=> ( ord_less_eq @ B @ ( F @ X3 ) @ ( G @ X3 ) ) ) ) ).
% le_funD
thf(fact_178_le__funE,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B @ ( type2 @ B ) )
=> ! [F: A > B,G: A > B,X3: A] :
( ( ord_less_eq @ ( A > B ) @ F @ G )
=> ( ord_less_eq @ B @ ( F @ X3 ) @ ( G @ X3 ) ) ) ) ).
% le_funE
thf(fact_179_le__funI,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B @ ( type2 @ B ) )
=> ! [F: A > B,G: A > B] :
( ! [X4: A] : ( ord_less_eq @ B @ ( F @ X4 ) @ ( G @ X4 ) )
=> ( ord_less_eq @ ( A > B ) @ F @ G ) ) ) ).
% le_funI
thf(fact_180_le__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B @ ( type2 @ B ) )
=> ( ( ord_less_eq @ ( A > B ) )
= ( ^ [F3: A > B,G3: A > B] :
! [X2: A] : ( ord_less_eq @ B @ ( F3 @ X2 ) @ ( G3 @ X2 ) ) ) ) ) ).
% le_fun_def
thf(fact_181_order__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B @ ( type2 @ B ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A2: A,F: B > A,B3: B,C2: B] :
( ( ord_less_eq @ A @ A2 @ ( F @ B3 ) )
=> ( ( ord_less_eq @ B @ B3 @ C2 )
=> ( ! [X4: B,Y3: B] :
( ( ord_less_eq @ B @ X4 @ Y3 )
=> ( ord_less_eq @ A @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq @ A @ A2 @ ( F @ C2 ) ) ) ) ) ) ).
% order_subst1
thf(fact_182_order__subst2,axiom,
! [A: $tType,C: $tType] :
( ( ( order @ C @ ( type2 @ C ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A2: A,B3: A,F: A > C,C2: C] :
( ( ord_less_eq @ A @ A2 @ B3 )
=> ( ( ord_less_eq @ C @ ( F @ B3 ) @ C2 )
=> ( ! [X4: A,Y3: A] :
( ( ord_less_eq @ A @ X4 @ Y3 )
=> ( ord_less_eq @ C @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq @ C @ ( F @ A2 ) @ C2 ) ) ) ) ) ).
% order_subst2
thf(fact_183_ord__eq__le__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B @ ( type2 @ B ) )
& ( ord @ A @ ( type2 @ A ) ) )
=> ! [A2: A,F: B > A,B3: B,C2: B] :
( ( A2
= ( F @ B3 ) )
=> ( ( ord_less_eq @ B @ B3 @ C2 )
=> ( ! [X4: B,Y3: B] :
( ( ord_less_eq @ B @ X4 @ Y3 )
=> ( ord_less_eq @ A @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq @ A @ A2 @ ( F @ C2 ) ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_184_ord__le__eq__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B @ ( type2 @ B ) )
& ( ord @ A @ ( type2 @ A ) ) )
=> ! [A2: A,B3: A,F: A > B,C2: B] :
( ( ord_less_eq @ A @ A2 @ B3 )
=> ( ( ( F @ B3 )
= C2 )
=> ( ! [X4: A,Y3: A] :
( ( ord_less_eq @ A @ X4 @ Y3 )
=> ( ord_less_eq @ B @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq @ B @ ( F @ A2 ) @ C2 ) ) ) ) ) ).
% ord_le_eq_subst
thf(fact_185_eq__iff,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ( ( ^ [Y2: A,Z3: A] : Y2 = Z3 )
= ( ^ [X2: A,Y4: A] :
( ( ord_less_eq @ A @ X2 @ Y4 )
& ( ord_less_eq @ A @ Y4 @ X2 ) ) ) ) ) ).
% eq_iff
thf(fact_186_antisym,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [X3: A,Y: A] :
( ( ord_less_eq @ A @ X3 @ Y )
=> ( ( ord_less_eq @ A @ Y @ X3 )
=> ( X3 = Y ) ) ) ) ).
% antisym
thf(fact_187_linear,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X3: A,Y: A] :
( ( ord_less_eq @ A @ X3 @ Y )
| ( ord_less_eq @ A @ Y @ X3 ) ) ) ).
% linear
thf(fact_188_eq__refl,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X3: A,Y: A] :
( ( X3 = Y )
=> ( ord_less_eq @ A @ X3 @ Y ) ) ) ).
% eq_refl
thf(fact_189_le__cases,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X3: A,Y: A] :
( ~ ( ord_less_eq @ A @ X3 @ Y )
=> ( ord_less_eq @ A @ Y @ X3 ) ) ) ).
% le_cases
thf(fact_190_order_Otrans,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A2: A,B3: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ B3 )
=> ( ( ord_less_eq @ A @ B3 @ C2 )
=> ( ord_less_eq @ A @ A2 @ C2 ) ) ) ) ).
% order.trans
thf(fact_191_le__cases3,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X3: A,Y: A,Z4: A] :
( ( ( ord_less_eq @ A @ X3 @ Y )
=> ~ ( ord_less_eq @ A @ Y @ Z4 ) )
=> ( ( ( ord_less_eq @ A @ Y @ X3 )
=> ~ ( ord_less_eq @ A @ X3 @ Z4 ) )
=> ( ( ( ord_less_eq @ A @ X3 @ Z4 )
=> ~ ( ord_less_eq @ A @ Z4 @ Y ) )
=> ( ( ( ord_less_eq @ A @ Z4 @ Y )
=> ~ ( ord_less_eq @ A @ Y @ X3 ) )
=> ( ( ( ord_less_eq @ A @ Y @ Z4 )
=> ~ ( ord_less_eq @ A @ Z4 @ X3 ) )
=> ~ ( ( ord_less_eq @ A @ Z4 @ X3 )
=> ~ ( ord_less_eq @ A @ X3 @ Y ) ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_192_antisym__conv,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [Y: A,X3: A] :
( ( ord_less_eq @ A @ Y @ X3 )
=> ( ( ord_less_eq @ A @ X3 @ Y )
= ( X3 = Y ) ) ) ) ).
% antisym_conv
thf(fact_193_ord__eq__le__trans,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [A2: A,B3: A,C2: A] :
( ( A2 = B3 )
=> ( ( ord_less_eq @ A @ B3 @ C2 )
=> ( ord_less_eq @ A @ A2 @ C2 ) ) ) ) ).
% ord_eq_le_trans
thf(fact_194_ord__le__eq__trans,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [A2: A,B3: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ B3 )
=> ( ( B3 = C2 )
=> ( ord_less_eq @ A @ A2 @ C2 ) ) ) ) ).
% ord_le_eq_trans
thf(fact_195_order__class_Oorder_Oantisym,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A2: A,B3: A] :
( ( ord_less_eq @ A @ A2 @ B3 )
=> ( ( ord_less_eq @ A @ B3 @ A2 )
=> ( A2 = B3 ) ) ) ) ).
% order_class.order.antisym
thf(fact_196_order__trans,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X3: A,Y: A,Z4: A] :
( ( ord_less_eq @ A @ X3 @ Y )
=> ( ( ord_less_eq @ A @ Y @ Z4 )
=> ( ord_less_eq @ A @ X3 @ Z4 ) ) ) ) ).
% order_trans
thf(fact_197_dual__order_Orefl,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A2: A] : ( ord_less_eq @ A @ A2 @ A2 ) ) ).
% dual_order.refl
thf(fact_198_linorder__wlog,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [P: A > A > $o,A2: A,B3: A] :
( ! [A6: A,B7: A] :
( ( ord_less_eq @ A @ A6 @ B7 )
=> ( P @ A6 @ B7 ) )
=> ( ! [A6: A,B7: A] :
( ( P @ B7 @ A6 )
=> ( P @ A6 @ B7 ) )
=> ( P @ A2 @ B3 ) ) ) ) ).
% linorder_wlog
thf(fact_199_dual__order_Otrans,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [B3: A,A2: A,C2: A] :
( ( ord_less_eq @ A @ B3 @ A2 )
=> ( ( ord_less_eq @ A @ C2 @ B3 )
=> ( ord_less_eq @ A @ C2 @ A2 ) ) ) ) ).
% dual_order.trans
thf(fact_200_dual__order_Oantisym,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [B3: A,A2: A] :
( ( ord_less_eq @ A @ B3 @ A2 )
=> ( ( ord_less_eq @ A @ A2 @ B3 )
=> ( A2 = B3 ) ) ) ) ).
% dual_order.antisym
thf(fact_201_pairwise__subset,axiom,
! [A: $tType,P: A > A > $o,S3: set @ A,T3: set @ A] :
( ( pairwise @ A @ P @ S3 )
=> ( ( ord_less_eq @ ( set @ A ) @ T3 @ S3 )
=> ( pairwise @ A @ P @ T3 ) ) ) ).
% pairwise_subset
thf(fact_202_subset__singletonD,axiom,
! [A: $tType,A3: set @ A,X3: A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
=> ( ( A3
= ( bot_bot @ ( set @ A ) ) )
| ( A3
= ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% subset_singletonD
thf(fact_203_subset__singleton__iff,axiom,
! [A: $tType,X5: set @ A,A2: A] :
( ( ord_less_eq @ ( set @ A ) @ X5 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) )
= ( ( X5
= ( bot_bot @ ( set @ A ) ) )
| ( X5
= ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% subset_singleton_iff
thf(fact_204_Ball__Collect,axiom,
! [A: $tType] :
( ( ball @ A )
= ( ^ [A5: set @ A,P5: A > $o] : ( ord_less_eq @ ( set @ A ) @ A5 @ ( collect @ A @ P5 ) ) ) ) ).
% Ball_Collect
thf(fact_205_insert__subsetI,axiom,
! [A: $tType,X3: A,A3: set @ A,X5: set @ A] :
( ( member @ A @ X3 @ A3 )
=> ( ( ord_less_eq @ ( set @ A ) @ X5 @ A3 )
=> ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ X3 @ X5 ) @ A3 ) ) ) ).
% insert_subsetI
thf(fact_206_subset__emptyI,axiom,
! [A: $tType,A3: set @ A] :
( ! [X4: A] :
~ ( member @ A @ X4 @ A3 )
=> ( ord_less_eq @ ( set @ A ) @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% subset_emptyI
thf(fact_207_subset__Compl__singleton,axiom,
! [A: $tType,A3: set @ A,B3: A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ ( uminus_uminus @ ( set @ A ) @ ( insert @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( ~ ( member @ A @ B3 @ A3 ) ) ) ).
% subset_Compl_singleton
thf(fact_208_Compl__eq__Compl__iff,axiom,
! [A: $tType,A3: set @ A,B4: set @ A] :
( ( ( uminus_uminus @ ( set @ A ) @ A3 )
= ( uminus_uminus @ ( set @ A ) @ B4 ) )
= ( A3 = B4 ) ) ).
% Compl_eq_Compl_iff
thf(fact_209_Compl__iff,axiom,
! [A: $tType,C2: A,A3: set @ A] :
( ( member @ A @ C2 @ ( uminus_uminus @ ( set @ A ) @ A3 ) )
= ( ~ ( member @ A @ C2 @ A3 ) ) ) ).
% Compl_iff
thf(fact_210_ComplI,axiom,
! [A: $tType,C2: A,A3: set @ A] :
( ~ ( member @ A @ C2 @ A3 )
=> ( member @ A @ C2 @ ( uminus_uminus @ ( set @ A ) @ A3 ) ) ) ).
% ComplI
thf(fact_211_Compl__anti__mono,axiom,
! [A: $tType,A3: set @ A,B4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B4 )
=> ( ord_less_eq @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ B4 ) @ ( uminus_uminus @ ( set @ A ) @ A3 ) ) ) ).
% Compl_anti_mono
thf(fact_212_Compl__subset__Compl__iff,axiom,
! [A: $tType,A3: set @ A,B4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A3 ) @ ( uminus_uminus @ ( set @ A ) @ B4 ) )
= ( ord_less_eq @ ( set @ A ) @ B4 @ A3 ) ) ).
% Compl_subset_Compl_iff
thf(fact_213_tllist_Orel__mono,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,R1: A > C > $o,R1a: A > C > $o,R2: B > D > $o,R2a: B > D > $o] :
( ( ord_less_eq @ ( A > C > $o ) @ R1 @ R1a )
=> ( ( ord_less_eq @ ( B > D > $o ) @ R2 @ R2a )
=> ( ord_less_eq @ ( ( tLList446370796tllist @ A @ B ) > ( tLList446370796tllist @ C @ D ) > $o ) @ ( tLList1380991092t_all2 @ A @ C @ B @ D @ R1 @ R2 ) @ ( tLList1380991092t_all2 @ A @ C @ B @ D @ R1a @ R2a ) ) ) ) ).
% tllist.rel_mono
thf(fact_214_double__complement,axiom,
! [A: $tType,A3: set @ A] :
( ( uminus_uminus @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A3 ) )
= A3 ) ).
% double_complement
thf(fact_215_ComplD,axiom,
! [A: $tType,C2: A,A3: set @ A] :
( ( member @ A @ C2 @ ( uminus_uminus @ ( set @ A ) @ A3 ) )
=> ~ ( member @ A @ C2 @ A3 ) ) ).
% ComplD
thf(fact_216_subset__Compl__self__eq,axiom,
! [A: $tType,A3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ ( uminus_uminus @ ( set @ A ) @ A3 ) )
= ( A3
= ( bot_bot @ ( set @ A ) ) ) ) ).
% subset_Compl_self_eq
thf(fact_217_Compl__insert,axiom,
! [A: $tType,X3: A,A3: set @ A] :
( ( uminus_uminus @ ( set @ A ) @ ( insert @ A @ X3 @ A3 ) )
= ( minus_minus @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A3 ) @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Compl_insert
thf(fact_218_disjoint__eq__subset__Compl,axiom,
! [A: $tType,A3: set @ A,B4: set @ A] :
( ( ( inf_inf @ ( set @ A ) @ A3 @ B4 )
= ( bot_bot @ ( set @ A ) ) )
= ( ord_less_eq @ ( set @ A ) @ A3 @ ( uminus_uminus @ ( set @ A ) @ B4 ) ) ) ).
% disjoint_eq_subset_Compl
thf(fact_219_IntI,axiom,
! [A: $tType,C2: A,A3: set @ A,B4: set @ A] :
( ( member @ A @ C2 @ A3 )
=> ( ( member @ A @ C2 @ B4 )
=> ( member @ A @ C2 @ ( inf_inf @ ( set @ A ) @ A3 @ B4 ) ) ) ) ).
% IntI
thf(fact_220_Int__iff,axiom,
! [A: $tType,C2: A,A3: set @ A,B4: set @ A] :
( ( member @ A @ C2 @ ( inf_inf @ ( set @ A ) @ A3 @ B4 ) )
= ( ( member @ A @ C2 @ A3 )
& ( member @ A @ C2 @ B4 ) ) ) ).
% Int_iff
thf(fact_221_DiffI,axiom,
! [A: $tType,C2: A,A3: set @ A,B4: set @ A] :
( ( member @ A @ C2 @ A3 )
=> ( ~ ( member @ A @ C2 @ B4 )
=> ( member @ A @ C2 @ ( minus_minus @ ( set @ A ) @ A3 @ B4 ) ) ) ) ).
% DiffI
thf(fact_222_Diff__iff,axiom,
! [A: $tType,C2: A,A3: set @ A,B4: set @ A] :
( ( member @ A @ C2 @ ( minus_minus @ ( set @ A ) @ A3 @ B4 ) )
= ( ( member @ A @ C2 @ A3 )
& ~ ( member @ A @ C2 @ B4 ) ) ) ).
% Diff_iff
thf(fact_223_Diff__idemp,axiom,
! [A: $tType,A3: set @ A,B4: set @ A] :
( ( minus_minus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A3 @ B4 ) @ B4 )
= ( minus_minus @ ( set @ A ) @ A3 @ B4 ) ) ).
% Diff_idemp
thf(fact_224_predicate2I,axiom,
! [B: $tType,A: $tType,P: A > B > $o,Q: A > B > $o] :
( ! [X4: A,Y3: B] :
( ( P @ X4 @ Y3 )
=> ( Q @ X4 @ Y3 ) )
=> ( ord_less_eq @ ( A > B > $o ) @ P @ Q ) ) ).
% predicate2I
thf(fact_225_Int__subset__iff,axiom,
! [A: $tType,C4: set @ A,A3: set @ A,B4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ C4 @ ( inf_inf @ ( set @ A ) @ A3 @ B4 ) )
= ( ( ord_less_eq @ ( set @ A ) @ C4 @ A3 )
& ( ord_less_eq @ ( set @ A ) @ C4 @ B4 ) ) ) ).
% Int_subset_iff
thf(fact_226_Diff__empty,axiom,
! [A: $tType,A3: set @ A] :
( ( minus_minus @ ( set @ A ) @ A3 @ ( bot_bot @ ( set @ A ) ) )
= A3 ) ).
% Diff_empty
thf(fact_227_empty__Diff,axiom,
! [A: $tType,A3: set @ A] :
( ( minus_minus @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ A3 )
= ( bot_bot @ ( set @ A ) ) ) ).
% empty_Diff
thf(fact_228_Diff__cancel,axiom,
! [A: $tType,A3: set @ A] :
( ( minus_minus @ ( set @ A ) @ A3 @ A3 )
= ( bot_bot @ ( set @ A ) ) ) ).
% Diff_cancel
thf(fact_229_Int__insert__right__if1,axiom,
! [A: $tType,A2: A,A3: set @ A,B4: set @ A] :
( ( member @ A @ A2 @ A3 )
=> ( ( inf_inf @ ( set @ A ) @ A3 @ ( insert @ A @ A2 @ B4 ) )
= ( insert @ A @ A2 @ ( inf_inf @ ( set @ A ) @ A3 @ B4 ) ) ) ) ).
% Int_insert_right_if1
thf(fact_230_Int__insert__right__if0,axiom,
! [A: $tType,A2: A,A3: set @ A,B4: set @ A] :
( ~ ( member @ A @ A2 @ A3 )
=> ( ( inf_inf @ ( set @ A ) @ A3 @ ( insert @ A @ A2 @ B4 ) )
= ( inf_inf @ ( set @ A ) @ A3 @ B4 ) ) ) ).
% Int_insert_right_if0
thf(fact_231_insert__inter__insert,axiom,
! [A: $tType,A2: A,A3: set @ A,B4: set @ A] :
( ( inf_inf @ ( set @ A ) @ ( insert @ A @ A2 @ A3 ) @ ( insert @ A @ A2 @ B4 ) )
= ( insert @ A @ A2 @ ( inf_inf @ ( set @ A ) @ A3 @ B4 ) ) ) ).
% insert_inter_insert
thf(fact_232_Int__insert__left__if1,axiom,
! [A: $tType,A2: A,C4: set @ A,B4: set @ A] :
( ( member @ A @ A2 @ C4 )
=> ( ( inf_inf @ ( set @ A ) @ ( insert @ A @ A2 @ B4 ) @ C4 )
= ( insert @ A @ A2 @ ( inf_inf @ ( set @ A ) @ B4 @ C4 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_233_Int__insert__left__if0,axiom,
! [A: $tType,A2: A,C4: set @ A,B4: set @ A] :
( ~ ( member @ A @ A2 @ C4 )
=> ( ( inf_inf @ ( set @ A ) @ ( insert @ A @ A2 @ B4 ) @ C4 )
= ( inf_inf @ ( set @ A ) @ B4 @ C4 ) ) ) ).
% Int_insert_left_if0
thf(fact_234_insert__Diff1,axiom,
! [A: $tType,X3: A,B4: set @ A,A3: set @ A] :
( ( member @ A @ X3 @ B4 )
=> ( ( minus_minus @ ( set @ A ) @ ( insert @ A @ X3 @ A3 ) @ B4 )
= ( minus_minus @ ( set @ A ) @ A3 @ B4 ) ) ) ).
% insert_Diff1
thf(fact_235_Diff__insert0,axiom,
! [A: $tType,X3: A,A3: set @ A,B4: set @ A] :
( ~ ( member @ A @ X3 @ A3 )
=> ( ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ X3 @ B4 ) )
= ( minus_minus @ ( set @ A ) @ A3 @ B4 ) ) ) ).
% Diff_insert0
thf(fact_236_insert__disjoint_I1_J,axiom,
! [A: $tType,A2: A,A3: set @ A,B4: set @ A] :
( ( ( inf_inf @ ( set @ A ) @ ( insert @ A @ A2 @ A3 ) @ B4 )
= ( bot_bot @ ( set @ A ) ) )
= ( ~ ( member @ A @ A2 @ B4 )
& ( ( inf_inf @ ( set @ A ) @ A3 @ B4 )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% insert_disjoint(1)
thf(fact_237_insert__disjoint_I2_J,axiom,
! [A: $tType,A2: A,A3: set @ A,B4: set @ A] :
( ( ( bot_bot @ ( set @ A ) )
= ( inf_inf @ ( set @ A ) @ ( insert @ A @ A2 @ A3 ) @ B4 ) )
= ( ~ ( member @ A @ A2 @ B4 )
& ( ( bot_bot @ ( set @ A ) )
= ( inf_inf @ ( set @ A ) @ A3 @ B4 ) ) ) ) ).
% insert_disjoint(2)
thf(fact_238_disjoint__insert_I1_J,axiom,
! [A: $tType,B4: set @ A,A2: A,A3: set @ A] :
( ( ( inf_inf @ ( set @ A ) @ B4 @ ( insert @ A @ A2 @ A3 ) )
= ( bot_bot @ ( set @ A ) ) )
= ( ~ ( member @ A @ A2 @ B4 )
& ( ( inf_inf @ ( set @ A ) @ B4 @ A3 )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% disjoint_insert(1)
thf(fact_239_disjoint__insert_I2_J,axiom,
! [A: $tType,A3: set @ A,B3: A,B4: set @ A] :
( ( ( bot_bot @ ( set @ A ) )
= ( inf_inf @ ( set @ A ) @ A3 @ ( insert @ A @ B3 @ B4 ) ) )
= ( ~ ( member @ A @ B3 @ A3 )
& ( ( bot_bot @ ( set @ A ) )
= ( inf_inf @ ( set @ A ) @ A3 @ B4 ) ) ) ) ).
% disjoint_insert(2)
thf(fact_240_Diff__eq__empty__iff,axiom,
! [A: $tType,A3: set @ A,B4: set @ A] :
( ( ( minus_minus @ ( set @ A ) @ A3 @ B4 )
= ( bot_bot @ ( set @ A ) ) )
= ( ord_less_eq @ ( set @ A ) @ A3 @ B4 ) ) ).
% Diff_eq_empty_iff
thf(fact_241_insert__Diff__single,axiom,
! [A: $tType,A2: A,A3: set @ A] :
( ( insert @ A @ A2 @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( insert @ A @ A2 @ A3 ) ) ).
% insert_Diff_single
thf(fact_242_Diff__disjoint,axiom,
! [A: $tType,A3: set @ A,B4: set @ A] :
( ( inf_inf @ ( set @ A ) @ A3 @ ( minus_minus @ ( set @ A ) @ B4 @ A3 ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% Diff_disjoint
thf(fact_243_Compl__disjoint,axiom,
! [A: $tType,A3: set @ A] :
( ( inf_inf @ ( set @ A ) @ A3 @ ( uminus_uminus @ ( set @ A ) @ A3 ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% Compl_disjoint
thf(fact_244_Compl__disjoint2,axiom,
! [A: $tType,A3: set @ A] :
( ( inf_inf @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A3 ) @ A3 )
= ( bot_bot @ ( set @ A ) ) ) ).
% Compl_disjoint2
thf(fact_245_Diff__Compl,axiom,
! [A: $tType,A3: set @ A,B4: set @ A] :
( ( minus_minus @ ( set @ A ) @ A3 @ ( uminus_uminus @ ( set @ A ) @ B4 ) )
= ( inf_inf @ ( set @ A ) @ A3 @ B4 ) ) ).
% Diff_Compl
thf(fact_246_Int__mono,axiom,
! [A: $tType,A3: set @ A,C4: set @ A,B4: set @ A,D3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ C4 )
=> ( ( ord_less_eq @ ( set @ A ) @ B4 @ D3 )
=> ( ord_less_eq @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A3 @ B4 ) @ ( inf_inf @ ( set @ A ) @ C4 @ D3 ) ) ) ) ).
% Int_mono
thf(fact_247_Diff__mono,axiom,
! [A: $tType,A3: set @ A,C4: set @ A,D3: set @ A,B4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ C4 )
=> ( ( ord_less_eq @ ( set @ A ) @ D3 @ B4 )
=> ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A3 @ B4 ) @ ( minus_minus @ ( set @ A ) @ C4 @ D3 ) ) ) ) ).
% Diff_mono
thf(fact_248_Int__lower1,axiom,
! [A: $tType,A3: set @ A,B4: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A3 @ B4 ) @ A3 ) ).
% Int_lower1
thf(fact_249_Int__lower2,axiom,
! [A: $tType,A3: set @ A,B4: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A3 @ B4 ) @ B4 ) ).
% Int_lower2
thf(fact_250_Diff__subset,axiom,
! [A: $tType,A3: set @ A,B4: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A3 @ B4 ) @ A3 ) ).
% Diff_subset
thf(fact_251_Int__absorb1,axiom,
! [A: $tType,B4: set @ A,A3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ B4 @ A3 )
=> ( ( inf_inf @ ( set @ A ) @ A3 @ B4 )
= B4 ) ) ).
% Int_absorb1
thf(fact_252_Int__absorb2,axiom,
! [A: $tType,A3: set @ A,B4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B4 )
=> ( ( inf_inf @ ( set @ A ) @ A3 @ B4 )
= A3 ) ) ).
% Int_absorb2
thf(fact_253_double__diff,axiom,
! [A: $tType,A3: set @ A,B4: set @ A,C4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B4 )
=> ( ( ord_less_eq @ ( set @ A ) @ B4 @ C4 )
=> ( ( minus_minus @ ( set @ A ) @ B4 @ ( minus_minus @ ( set @ A ) @ C4 @ A3 ) )
= A3 ) ) ) ).
% double_diff
thf(fact_254_Int__greatest,axiom,
! [A: $tType,C4: set @ A,A3: set @ A,B4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ C4 @ A3 )
=> ( ( ord_less_eq @ ( set @ A ) @ C4 @ B4 )
=> ( ord_less_eq @ ( set @ A ) @ C4 @ ( inf_inf @ ( set @ A ) @ A3 @ B4 ) ) ) ) ).
% Int_greatest
%----Type constructors (16)
thf(tcon_fun___Orderings_Oorder__bot,axiom,
! [A7: $tType,A8: $tType] :
( ( order_bot @ A8 @ ( type2 @ A8 ) )
=> ( order_bot @ ( A7 > A8 ) @ ( type2 @ ( A7 > A8 ) ) ) ) ).
thf(tcon_fun___Orderings_Opreorder,axiom,
! [A7: $tType,A8: $tType] :
( ( preorder @ A8 @ ( type2 @ A8 ) )
=> ( preorder @ ( A7 > A8 ) @ ( type2 @ ( A7 > A8 ) ) ) ) ).
thf(tcon_fun___Orderings_Oorder,axiom,
! [A7: $tType,A8: $tType] :
( ( order @ A8 @ ( type2 @ A8 ) )
=> ( order @ ( A7 > A8 ) @ ( type2 @ ( A7 > A8 ) ) ) ) ).
thf(tcon_fun___Orderings_Oord,axiom,
! [A7: $tType,A8: $tType] :
( ( ord @ A8 @ ( type2 @ A8 ) )
=> ( ord @ ( A7 > A8 ) @ ( type2 @ ( A7 > A8 ) ) ) ) ).
thf(tcon_fun___Orderings_Obot,axiom,
! [A7: $tType,A8: $tType] :
( ( bot @ A8 @ ( type2 @ A8 ) )
=> ( bot @ ( A7 > A8 ) @ ( type2 @ ( A7 > A8 ) ) ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder__bot_1,axiom,
! [A7: $tType] : ( order_bot @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Opreorder_2,axiom,
! [A7: $tType] : ( preorder @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder_3,axiom,
! [A7: $tType] : ( order @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Oord_4,axiom,
! [A7: $tType] : ( ord @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Obot_5,axiom,
! [A7: $tType] : ( bot @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
thf(tcon_HOL_Obool___Orderings_Oorder__bot_6,axiom,
order_bot @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Opreorder_7,axiom,
preorder @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Olinorder,axiom,
linorder @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Oorder_8,axiom,
order @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Oord_9,axiom,
ord @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Obot_10,axiom,
bot @ $o @ ( type2 @ $o ) ).
%----Helper facts (3)
thf(help_If_3_1_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_T,axiom,
! [A: $tType,X3: A,Y: A] :
( ( if @ A @ $false @ X3 @ Y )
= Y ) ).
thf(help_If_1_1_T,axiom,
! [A: $tType,X3: A,Y: A] :
( ( if @ A @ $true @ X3 @ Y )
= X3 ) ).
%----Conjectures (2)
thf(conj_0,hypothesis,
( x
= ( tLList57907754e_TNil @ b @ a @ x1 ) ) ).
thf(conj_1,conjecture,
( ( ( tLList1968894004s_TNil @ a @ b @ x )
=> ( ( tLList200813139tllist @ b @ b @ a @ ( id @ b ) @ ( undefined @ ( a > ( tLList446370796tllist @ a @ b ) > b ) ) @ x )
= ( tLList2110128105rminal @ a @ b @ x ) ) )
& ( ~ ( tLList1968894004s_TNil @ a @ b @ x )
=> ( ( tLList1825092077minal0 @ ( ( tLList446370796tllist @ a @ b ) > b ) @ ( tLList_Mirabelle_ttl @ a @ b @ x ) )
= ( tLList2110128105rminal @ a @ b @ x ) ) ) ) ).
%------------------------------------------------------------------------------