TPTP Problem File: SLH0856^1.p
View Solutions
- Solve Problem
%------------------------------------------------------------------------------
% File : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain : Archive of Formal Proofs
% Problem :
% Version : Especial.
% English :
% Refs : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source : [Des23]
% Names : SCC_Bloemen_Sequential/0000_SCC_Bloemen_Sequential/prob_02828_097271__6502144_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1502 ( 642 unt; 216 typ; 0 def)
% Number of atoms : 3611 (1444 equ; 0 cnn)
% Maximal formula atoms : 10 ( 2 avg)
% Number of connectives : 12003 ( 501 ~; 62 |; 333 &;9574 @)
% ( 0 <=>;1533 =>; 0 <=; 0 <~>)
% Maximal formula depth : 21 ( 6 avg)
% Number of types : 29 ( 28 usr)
% Number of type conns : 549 ( 549 >; 0 *; 0 +; 0 <<)
% Number of symbols : 191 ( 188 usr; 17 con; 0-9 aty)
% Number of variables : 3645 ( 136 ^;3321 !; 188 ?;3645 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 08:54:19.729
%------------------------------------------------------------------------------
% Could-be-implicit typings (28)
thf(ty_n_t__Sum____Type__Osum_It__Product____Type__Oprod_Itf__v_Mt__SCC____Bloemen____Sequential__Oenv__Oenv____ext_Itf__v_Mt__Product____Type__Ounit_J_J_Mt__Product____Type__Oprod_Itf__v_Mt__SCC____Bloemen____Sequential__Oenv__Oenv____ext_Itf__v_Mt__Product____Type__Ounit_J_J_J,type,
sum_su8181647976486975269t_unit: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__List__Olist_Itf__v_J_Mt__List__Olist_It__SCC____Bloemen____Sequential__Oenv__Oenv____ext_Itf__v_Mt__Product____Type__Ounit_J_J_J_J,type,
set_Pr8903222484228591t_unit: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__List__Olist_It__Product____Type__Oprod_Itf__v_Mtf__v_J_J_Mt__List__Olist_It__Product____Type__Oprod_Itf__v_Mtf__v_J_J_J_J,type,
set_Pr7499474215547700295od_v_v: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__List__Olist_Itf__v_J_Mt__List__Olist_It__SCC____Bloemen____Sequential__Oenv__Oenv____ext_Itf__v_Mt__Product____Type__Ounit_J_J_J,type,
produc7473616835194958905t_unit: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__List__Olist_It__Product____Type__Oprod_Itf__v_Mtf__v_J_J_Mt__List__Olist_It__Product____Type__Oprod_Itf__v_Mtf__v_J_J_J,type,
produc1504107476793160551od_v_v: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_Itf__v_Mt__SCC____Bloemen____Sequential__Oenv__Oenv____ext_Itf__v_Mt__Product____Type__Ounit_J_J_J,type,
set_Pr6425124735969554649t_unit: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_Itf__v_Mtf__v_J_Mt__Product____Type__Oprod_Itf__v_Mtf__v_J_J_J,type,
set_Pr2149350503807050951od_v_v: $tType ).
thf(ty_n_t__SCC____Bloemen____Sequential__Oenv__Oenv____ext_It__Product____Type__Oprod_Itf__v_Mtf__v_J_Mt__Product____Type__Ounit_J,type,
sCC_Bl7326425374436813197t_unit: $tType ).
thf(ty_n_t__Product____Type__Oprod_Itf__v_Mt__SCC____Bloemen____Sequential__Oenv__Oenv____ext_Itf__v_Mt__Product____Type__Ounit_J_J,type,
produc5741669702376414499t_unit: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__v_Mtf__v_J_Mt__Product____Type__Oprod_Itf__v_Mtf__v_J_J,type,
produc206430290419586791od_v_v: $tType ).
thf(ty_n_t__List__Olist_It__SCC____Bloemen____Sequential__Oenv__Oenv____ext_Itf__v_Mt__Product____Type__Ounit_J_J,type,
list_S9195319530159730289t_unit: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__List__Olist_Itf__v_J_Mt__List__Olist_Itf__v_J_J_J,type,
set_Pr6206931691796273479list_v: $tType ).
thf(ty_n_t__Product____Type__Oprod_I_062_Itf__v_M_062_Itf__v_M_Eo_J_J_Mt__List__Olist_Itf__v_J_J,type,
produc8237170675765753490list_v: $tType ).
thf(ty_n_t__SCC____Bloemen____Sequential__Oenv__Oenv____ext_Itf__v_Mt__Product____Type__Ounit_J,type,
sCC_Bl1394983891496994913t_unit: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__List__Olist_Itf__v_J_Mt__List__Olist_Itf__v_J_J,type,
produc1391462591744249447list_v: $tType ).
thf(ty_n_t__List__Olist_It__List__Olist_It__Product____Type__Oprod_Itf__v_Mtf__v_J_J_J,type,
list_l4795378083388841843od_v_v: $tType ).
thf(ty_n_t__Set__Oset_It__List__Olist_It__Product____Type__Oprod_Itf__v_Mtf__v_J_J_J,type,
set_li2323639185124838733od_v_v: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Product____Type__Oprod_Itf__v_Mtf__v_J_J_J,type,
set_se8455005133513928103od_v_v: $tType ).
thf(ty_n_t__List__Olist_It__Product____Type__Oprod_Itf__v_Mtf__v_J_J,type,
list_P7986770385144383213od_v_v: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_Itf__v_Mtf__v_J_J,type,
set_Product_prod_v_v: $tType ).
thf(ty_n_t__Product____Type__Oprod_Itf__v_Mtf__v_J,type,
product_prod_v_v: $tType ).
thf(ty_n_t__List__Olist_It__List__Olist_Itf__v_J_J,type,
list_list_v: $tType ).
thf(ty_n_t__Set__Oset_It__List__Olist_Itf__v_J_J,type,
set_list_v: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_Itf__v_J_J,type,
set_set_v: $tType ).
thf(ty_n_t__Product____Type__Ounit,type,
product_unit: $tType ).
thf(ty_n_t__List__Olist_Itf__v_J,type,
list_v: $tType ).
thf(ty_n_t__Set__Oset_Itf__v_J,type,
set_v: $tType ).
thf(ty_n_tf__v,type,
v: $tType ).
% Explicit typings (188)
thf(sy_c_Finite__Set_Ofinite_001t__Product____Type__Oprod_Itf__v_Mtf__v_J,type,
finite3348123685078250256od_v_v: set_Product_prod_v_v > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_It__Product____Type__Oprod_Itf__v_Mtf__v_J_J,type,
finite6084192165098772208od_v_v: set_se8455005133513928103od_v_v > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_Itf__v_J,type,
finite_finite_set_v: set_set_v > $o ).
thf(sy_c_Finite__Set_Ofinite_001tf__v,type,
finite_finite_v: set_v > $o ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Product____Type__Oprod_Itf__v_Mtf__v_J_J,type,
minus_4183494784930505774od_v_v: set_Product_prod_v_v > set_Product_prod_v_v > set_Product_prod_v_v ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Set__Oset_It__Product____Type__Oprod_Itf__v_Mtf__v_J_J_J,type,
minus_7679383599658060814od_v_v: set_se8455005133513928103od_v_v > set_se8455005133513928103od_v_v > set_se8455005133513928103od_v_v ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Set__Oset_Itf__v_J_J,type,
minus_7228012346218142266_set_v: set_set_v > set_set_v > set_set_v ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_Itf__v_J,type,
minus_minus_set_v: set_v > set_v > set_v ).
thf(sy_c_If_001t__List__Olist_It__Product____Type__Oprod_Itf__v_Mtf__v_J_J,type,
if_lis7521272669439687347od_v_v: $o > list_P7986770385144383213od_v_v > list_P7986770385144383213od_v_v > list_P7986770385144383213od_v_v ).
thf(sy_c_If_001t__List__Olist_Itf__v_J,type,
if_list_v: $o > list_v > list_v > list_v ).
thf(sy_c_If_001t__Set__Oset_It__Product____Type__Oprod_Itf__v_Mtf__v_J_J,type,
if_set4279007504652509325od_v_v: $o > set_Product_prod_v_v > set_Product_prod_v_v > set_Product_prod_v_v ).
thf(sy_c_If_001t__Set__Oset_Itf__v_J,type,
if_set_v: $o > set_v > set_v > set_v ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Product____Type__Oprod_Itf__v_Mtf__v_J_J,type,
inf_in6271465464967711157od_v_v: set_Product_prod_v_v > set_Product_prod_v_v > set_Product_prod_v_v ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_Itf__v_J,type,
inf_inf_set_v: set_v > set_v > set_v ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_Itf__v_Mtf__v_J_Mt__Product____Type__Oprod_Itf__v_Mtf__v_J_J_J,type,
sup_su1742609618068805275od_v_v: set_Pr2149350503807050951od_v_v > set_Pr2149350503807050951od_v_v > set_Pr2149350503807050951od_v_v ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Product____Type__Oprod_Itf__v_Mtf__v_J_J,type,
sup_su414716646722978715od_v_v: set_Product_prod_v_v > set_Product_prod_v_v > set_Product_prod_v_v ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Set__Oset_It__Product____Type__Oprod_Itf__v_Mtf__v_J_J_J,type,
sup_su335656005089752955od_v_v: set_se8455005133513928103od_v_v > set_se8455005133513928103od_v_v > set_se8455005133513928103od_v_v ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Set__Oset_Itf__v_J_J,type,
sup_sup_set_set_v: set_set_v > set_set_v > set_set_v ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_Itf__v_J,type,
sup_sup_set_v: set_v > set_v > set_v ).
thf(sy_c_Lattices__Big_Osemilattice__inf__class_OInf__fin_001t__Set__Oset_It__Product____Type__Oprod_Itf__v_Mtf__v_J_J,type,
lattic4767070952889939172od_v_v: set_se8455005133513928103od_v_v > set_Product_prod_v_v ).
thf(sy_c_Lattices__Big_Osemilattice__inf__class_OInf__fin_001t__Set__Oset_Itf__v_J,type,
lattic8209813555532694032_set_v: set_set_v > set_v ).
thf(sy_c_Lattices__Big_Osemilattice__sup__class_OSup__fin_001t__Set__Oset_It__Product____Type__Oprod_Itf__v_Mtf__v_J_J,type,
lattic5151207300795964030od_v_v: set_se8455005133513928103od_v_v > set_Product_prod_v_v ).
thf(sy_c_Lattices__Big_Osemilattice__sup__class_OSup__fin_001t__Set__Oset_Itf__v_J,type,
lattic2918178447194608042_set_v: set_set_v > set_v ).
thf(sy_c_List_Oappend_001t__List__Olist_Itf__v_J,type,
append_list_v: list_list_v > list_list_v > list_list_v ).
thf(sy_c_List_Oappend_001t__Product____Type__Oprod_Itf__v_Mtf__v_J,type,
append2138873909117096322od_v_v: list_P7986770385144383213od_v_v > list_P7986770385144383213od_v_v > list_P7986770385144383213od_v_v ).
thf(sy_c_List_Oappend_001tf__v,type,
append_v: list_v > list_v > list_v ).
thf(sy_c_List_Obind_001tf__v_001tf__v,type,
bind_v_v: list_v > ( v > list_v ) > list_v ).
thf(sy_c_List_Oconcat_001t__Product____Type__Oprod_Itf__v_Mtf__v_J,type,
concat2875663619778446888od_v_v: list_l4795378083388841843od_v_v > list_P7986770385144383213od_v_v ).
thf(sy_c_List_Oconcat_001tf__v,type,
concat_v: list_list_v > list_v ).
thf(sy_c_List_Ocoset_001t__Product____Type__Oprod_Itf__v_Mtf__v_J,type,
coset_766761627116920666od_v_v: list_P7986770385144383213od_v_v > set_Product_prod_v_v ).
thf(sy_c_List_Ocoset_001tf__v,type,
coset_v: list_v > set_v ).
thf(sy_c_List_Odistinct_001t__List__Olist_It__Product____Type__Oprod_Itf__v_Mtf__v_J_J,type,
distin913317783593574886od_v_v: list_l4795378083388841843od_v_v > $o ).
thf(sy_c_List_Odistinct_001t__List__Olist_Itf__v_J,type,
distinct_list_v: list_list_v > $o ).
thf(sy_c_List_Odistinct_001t__Product____Type__Oprod_Itf__v_Mtf__v_J,type,
distin6159370996967099744od_v_v: list_P7986770385144383213od_v_v > $o ).
thf(sy_c_List_Odistinct_001tf__v,type,
distinct_v: list_v > $o ).
thf(sy_c_List_Oinsert_001t__Product____Type__Oprod_Itf__v_Mtf__v_J,type,
insert4539780211034306307od_v_v: product_prod_v_v > list_P7986770385144383213od_v_v > list_P7986770385144383213od_v_v ).
thf(sy_c_List_Oinsert_001tf__v,type,
insert_v: v > list_v > list_v ).
thf(sy_c_List_Olexord_001t__Product____Type__Oprod_Itf__v_Mtf__v_J,type,
lexord8601710409828808922od_v_v: set_Pr2149350503807050951od_v_v > set_Pr7499474215547700295od_v_v ).
thf(sy_c_List_Olexord_001tf__v,type,
lexord_v: set_Product_prod_v_v > set_Pr6206931691796273479list_v ).
thf(sy_c_List_Olist_OCons_001t__List__Olist_Itf__v_J,type,
cons_list_v: list_v > list_list_v > list_list_v ).
thf(sy_c_List_Olist_OCons_001t__Product____Type__Oprod_Itf__v_Mtf__v_J,type,
cons_P4120604216776828829od_v_v: product_prod_v_v > list_P7986770385144383213od_v_v > list_P7986770385144383213od_v_v ).
thf(sy_c_List_Olist_OCons_001t__SCC____Bloemen____Sequential__Oenv__Oenv____ext_Itf__v_Mt__Product____Type__Ounit_J,type,
cons_S4552210989952385579t_unit: sCC_Bl1394983891496994913t_unit > list_S9195319530159730289t_unit > list_S9195319530159730289t_unit ).
thf(sy_c_List_Olist_OCons_001tf__v,type,
cons_v: v > list_v > list_v ).
thf(sy_c_List_Olist_ONil_001t__List__Olist_Itf__v_J,type,
nil_list_v: list_list_v ).
thf(sy_c_List_Olist_ONil_001t__Product____Type__Oprod_Itf__v_Mtf__v_J,type,
nil_Product_prod_v_v: list_P7986770385144383213od_v_v ).
thf(sy_c_List_Olist_ONil_001t__SCC____Bloemen____Sequential__Oenv__Oenv____ext_Itf__v_Mt__Product____Type__Ounit_J,type,
nil_SC5400981268189614555t_unit: list_S9195319530159730289t_unit ).
thf(sy_c_List_Olist_ONil_001tf__v,type,
nil_v: list_v ).
thf(sy_c_List_Olist_Ohd_001t__List__Olist_Itf__v_J,type,
hd_list_v: list_list_v > list_v ).
thf(sy_c_List_Olist_Ohd_001t__Product____Type__Oprod_Itf__v_Mtf__v_J,type,
hd_Product_prod_v_v: list_P7986770385144383213od_v_v > product_prod_v_v ).
thf(sy_c_List_Olist_Ohd_001tf__v,type,
hd_v: list_v > v ).
thf(sy_c_List_Olist_Oset_001t__List__Olist_It__Product____Type__Oprod_Itf__v_Mtf__v_J_J,type,
set_li2340707408155270402od_v_v: list_l4795378083388841843od_v_v > set_li2323639185124838733od_v_v ).
thf(sy_c_List_Olist_Oset_001t__List__Olist_Itf__v_J,type,
set_list_v2: list_list_v > set_list_v ).
thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_Itf__v_Mtf__v_J,type,
set_Product_prod_v_v2: list_P7986770385144383213od_v_v > set_Product_prod_v_v ).
thf(sy_c_List_Olist_Oset_001tf__v,type,
set_v2: list_v > set_v ).
thf(sy_c_List_Olist_Otl_001t__Product____Type__Oprod_Itf__v_Mtf__v_J,type,
tl_Product_prod_v_v: list_P7986770385144383213od_v_v > list_P7986770385144383213od_v_v ).
thf(sy_c_List_Olist_Otl_001tf__v,type,
tl_v: list_v > list_v ).
thf(sy_c_List_Olistrel1_001tf__v,type,
listrel1_v: set_Product_prod_v_v > set_Pr6206931691796273479list_v ).
thf(sy_c_List_Olistrel_001tf__v_001t__SCC____Bloemen____Sequential__Oenv__Oenv____ext_Itf__v_Mt__Product____Type__Ounit_J,type,
listre2737131563223898179t_unit: set_Pr6425124735969554649t_unit > set_Pr8903222484228591t_unit ).
thf(sy_c_List_Olistrel_001tf__v_001tf__v,type,
listrel_v_v: set_Product_prod_v_v > set_Pr6206931691796273479list_v ).
thf(sy_c_List_Omaps_001tf__v_001tf__v,type,
maps_v_v: ( v > list_v ) > list_v > list_v ).
thf(sy_c_List_Onull_001tf__v,type,
null_v: list_v > $o ).
thf(sy_c_List_Opartition_001t__Product____Type__Oprod_Itf__v_Mtf__v_J,type,
partit5288610572509583718od_v_v: ( product_prod_v_v > $o ) > list_P7986770385144383213od_v_v > produc1504107476793160551od_v_v ).
thf(sy_c_List_Opartition_001tf__v,type,
partition_v: ( v > $o ) > list_v > produc1391462591744249447list_v ).
thf(sy_c_List_Oproduct__lists_001tf__v,type,
product_lists_v: list_list_v > list_list_v ).
thf(sy_c_List_Oremove1_001t__Product____Type__Oprod_Itf__v_Mtf__v_J,type,
remove333779696311199107od_v_v: product_prod_v_v > list_P7986770385144383213od_v_v > list_P7986770385144383213od_v_v ).
thf(sy_c_List_Oremove1_001tf__v,type,
remove1_v: v > list_v > list_v ).
thf(sy_c_List_OremoveAll_001t__List__Olist_It__Product____Type__Oprod_Itf__v_Mtf__v_J_J,type,
remove5095778601549809401od_v_v: list_P7986770385144383213od_v_v > list_l4795378083388841843od_v_v > list_l4795378083388841843od_v_v ).
thf(sy_c_List_OremoveAll_001t__List__Olist_Itf__v_J,type,
removeAll_list_v: list_v > list_list_v > list_list_v ).
thf(sy_c_List_OremoveAll_001t__Product____Type__Oprod_Itf__v_Mtf__v_J,type,
remove481895986417801203od_v_v: product_prod_v_v > list_P7986770385144383213od_v_v > list_P7986770385144383213od_v_v ).
thf(sy_c_List_OremoveAll_001tf__v,type,
removeAll_v: v > list_v > list_v ).
thf(sy_c_List_Orotate1_001tf__v,type,
rotate1_v: list_v > list_v ).
thf(sy_c_List_Osubseqs_001tf__v,type,
subseqs_v: list_v > list_list_v ).
thf(sy_c_List_Ounion_001t__Product____Type__Oprod_Itf__v_Mtf__v_J,type,
union_4602324378607836129od_v_v: list_P7986770385144383213od_v_v > list_P7986770385144383213od_v_v > list_P7986770385144383213od_v_v ).
thf(sy_c_List_Ounion_001tf__v,type,
union_v: list_v > list_v > list_v ).
thf(sy_c_Order__Relation_Olinear__order__on_001t__Product____Type__Oprod_Itf__v_Mtf__v_J,type,
order_6462556390437124636od_v_v: set_Product_prod_v_v > set_Pr2149350503807050951od_v_v > $o ).
thf(sy_c_Order__Relation_Olinear__order__on_001tf__v,type,
order_8768733634509060168r_on_v: set_v > set_Product_prod_v_v > $o ).
thf(sy_c_Order__Relation_OunderS_001t__Product____Type__Oprod_Itf__v_Mtf__v_J,type,
order_5211820470575790509od_v_v: set_Pr2149350503807050951od_v_v > product_prod_v_v > set_Product_prod_v_v ).
thf(sy_c_Order__Relation_OunderS_001tf__v,type,
order_underS_v: set_Product_prod_v_v > v > set_v ).
thf(sy_c_Order__Relation_Ounder_001t__Product____Type__Oprod_Itf__v_Mtf__v_J,type,
order_6892855479609198156od_v_v: set_Pr2149350503807050951od_v_v > product_prod_v_v > set_Product_prod_v_v ).
thf(sy_c_Order__Relation_Ounder_001tf__v,type,
order_under_v: set_Product_prod_v_v > v > set_v ).
thf(sy_c_Order__Relation_Owell__order__on_001t__Product____Type__Oprod_Itf__v_Mtf__v_J,type,
order_7541072052284126853od_v_v: set_Product_prod_v_v > set_Pr2149350503807050951od_v_v > $o ).
thf(sy_c_Order__Relation_Owell__order__on_001tf__v,type,
order_6972113574731384241r_on_v: set_v > set_Product_prod_v_v > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Product____Type__Oprod_Itf__v_Mtf__v_J_M_Eo_J,type,
bot_bo8461541820394803818_v_v_o: product_prod_v_v > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_Itf__v_M_Eo_J,type,
bot_bot_v_o: v > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_Itf__v_Mtf__v_J_Mt__Product____Type__Oprod_Itf__v_Mtf__v_J_J_J,type,
bot_bo3282589961317712691od_v_v: set_Pr2149350503807050951od_v_v ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Product____Type__Oprod_Itf__v_Mtf__v_J_J,type,
bot_bo723834152578015283od_v_v: set_Product_prod_v_v ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_It__Product____Type__Oprod_Itf__v_Mtf__v_J_J_J,type,
bot_bo3497076220358800403od_v_v: set_se8455005133513928103od_v_v ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_Itf__v_J_J,type,
bot_bot_set_set_v: set_set_v ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_Itf__v_J,type,
bot_bot_set_v: set_v ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_It__List__Olist_Itf__v_J_Mt__List__Olist_Itf__v_J_J_J,type,
ord_le791731619978752231list_v: set_Pr6206931691796273479list_v > set_Pr6206931691796273479list_v > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_Itf__v_Mtf__v_J_Mt__Product____Type__Oprod_Itf__v_Mtf__v_J_J_J,type,
ord_le6241436655786843239od_v_v: set_Pr2149350503807050951od_v_v > set_Pr2149350503807050951od_v_v > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_Itf__v_Mt__SCC____Bloemen____Sequential__Oenv__Oenv____ext_Itf__v_Mt__Product____Type__Ounit_J_J_J,type,
ord_le7290744839000465721t_unit: set_Pr6425124735969554649t_unit > set_Pr6425124735969554649t_unit > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_Itf__v_Mtf__v_J_J,type,
ord_le7336532860387713383od_v_v: set_Product_prod_v_v > set_Product_prod_v_v > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Product____Type__Oprod_Itf__v_Mtf__v_J_J_J,type,
ord_le4714265922333009223od_v_v: set_se8455005133513928103od_v_v > set_se8455005133513928103od_v_v > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_Itf__v_J_J,type,
ord_le5216385588623774835_set_v: set_set_v > set_set_v > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_Itf__v_J,type,
ord_less_eq_set_v: set_v > set_v > $o ).
thf(sy_c_Orderings_Oord__class_Omin_001t__Set__Oset_It__Product____Type__Oprod_Itf__v_Mtf__v_J_J,type,
ord_mi6996445931809003310od_v_v: set_Product_prod_v_v > set_Product_prod_v_v > set_Product_prod_v_v ).
thf(sy_c_Orderings_Oord__class_Omin_001t__Set__Oset_Itf__v_J,type,
ord_min_set_v: set_v > set_v > set_v ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Product____Type__Oprod_Itf__v_Mtf__v_J_J,type,
top_to5429829297380968215od_v_v: set_Product_prod_v_v ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_Itf__v_J,type,
top_top_set_v: set_v ).
thf(sy_c_Product__Type_OPair_001_062_Itf__v_M_062_Itf__v_M_Eo_J_J_001t__List__Olist_Itf__v_J,type,
produc601102195597853570list_v: ( v > v > $o ) > list_v > produc8237170675765753490list_v ).
thf(sy_c_Product__Type_OPair_001t__List__Olist_It__Product____Type__Oprod_Itf__v_Mtf__v_J_J_001t__List__Olist_It__Product____Type__Oprod_Itf__v_Mtf__v_J_J,type,
produc674067373767953879od_v_v: list_P7986770385144383213od_v_v > list_P7986770385144383213od_v_v > produc1504107476793160551od_v_v ).
thf(sy_c_Product__Type_OPair_001t__List__Olist_Itf__v_J_001t__List__Olist_It__SCC____Bloemen____Sequential__Oenv__Oenv____ext_Itf__v_Mt__Product____Type__Ounit_J_J,type,
produc2297727903729291827t_unit: list_v > list_S9195319530159730289t_unit > produc7473616835194958905t_unit ).
thf(sy_c_Product__Type_OPair_001t__List__Olist_Itf__v_J_001t__List__Olist_Itf__v_J,type,
produc6795410681906604247list_v: list_v > list_v > produc1391462591744249447list_v ).
thf(sy_c_Product__Type_OPair_001t__Product____Type__Oprod_Itf__v_Mtf__v_J_001t__Product____Type__Oprod_Itf__v_Mtf__v_J,type,
produc4031800376763917143od_v_v: product_prod_v_v > product_prod_v_v > produc206430290419586791od_v_v ).
thf(sy_c_Product__Type_OPair_001tf__v_001t__SCC____Bloemen____Sequential__Oenv__Oenv____ext_Itf__v_Mt__Product____Type__Ounit_J,type,
produc3862955338007567901t_unit: v > sCC_Bl1394983891496994913t_unit > produc5741669702376414499t_unit ).
thf(sy_c_Product__Type_OPair_001tf__v_001tf__v,type,
product_Pair_v_v: v > v > product_prod_v_v ).
thf(sy_c_Relation_OField_001t__Product____Type__Oprod_Itf__v_Mtf__v_J,type,
field_7153129647634986036od_v_v: set_Pr2149350503807050951od_v_v > set_Product_prod_v_v ).
thf(sy_c_Relation_OField_001tf__v,type,
field_v: set_Product_prod_v_v > set_v ).
thf(sy_c_Relation_Orefl__on_001t__Product____Type__Oprod_Itf__v_Mtf__v_J,type,
refl_o4548774019903118566od_v_v: set_Product_prod_v_v > set_Pr2149350503807050951od_v_v > $o ).
thf(sy_c_Relation_Orefl__on_001tf__v,type,
refl_on_v: set_v > set_Product_prod_v_v > $o ).
thf(sy_c_SCC__Bloemen__Sequential_Oenv_O_092_060S_062_001t__Product____Type__Oprod_Itf__v_Mtf__v_J_001t__Product____Type__Ounit,type,
sCC_Bl8440648026628373538t_unit: sCC_Bl7326425374436813197t_unit > product_prod_v_v > set_Product_prod_v_v ).
thf(sy_c_SCC__Bloemen__Sequential_Oenv_O_092_060S_062_001tf__v_001t__Product____Type__Ounit,type,
sCC_Bl1280885523602775798t_unit: sCC_Bl1394983891496994913t_unit > v > set_v ).
thf(sy_c_SCC__Bloemen__Sequential_Oenv_Ocstack_001tf__v_001t__Product____Type__Ounit,type,
sCC_Bl9201514103433284750t_unit: sCC_Bl1394983891496994913t_unit > list_v ).
thf(sy_c_SCC__Bloemen__Sequential_Oenv_Oenv__ext_001tf__v_001t__Product____Type__Ounit,type,
sCC_Bl8064756265740546429t_unit: v > ( v > set_v ) > set_v > set_v > ( v > set_v ) > set_set_v > list_v > list_v > product_unit > sCC_Bl1394983891496994913t_unit ).
thf(sy_c_SCC__Bloemen__Sequential_Oenv_Oexplored_001t__Product____Type__Oprod_Itf__v_Mtf__v_J_001t__Product____Type__Ounit,type,
sCC_Bl5094201334446601350t_unit: sCC_Bl7326425374436813197t_unit > set_Product_prod_v_v ).
thf(sy_c_SCC__Bloemen__Sequential_Oenv_Oexplored_001tf__v_001t__Product____Type__Ounit,type,
sCC_Bl157864678168468314t_unit: sCC_Bl1394983891496994913t_unit > set_v ).
thf(sy_c_SCC__Bloemen__Sequential_Oenv_Omore_001tf__v_001t__Product____Type__Ounit,type,
sCC_Bl3567736435408124606t_unit: sCC_Bl1394983891496994913t_unit > product_unit ).
thf(sy_c_SCC__Bloemen__Sequential_Oenv_Oroot_001tf__v_001t__Product____Type__Ounit,type,
sCC_Bl1090238580953940555t_unit: sCC_Bl1394983891496994913t_unit > v ).
thf(sy_c_SCC__Bloemen__Sequential_Oenv_Osccs_001tf__v_001t__Product____Type__Ounit,type,
sCC_Bl2536197123907397897t_unit: sCC_Bl1394983891496994913t_unit > set_set_v ).
thf(sy_c_SCC__Bloemen__Sequential_Oenv_Ostack_001t__Product____Type__Oprod_Itf__v_Mtf__v_J_001t__Product____Type__Ounit,type,
sCC_Bl2021302119412358655t_unit: sCC_Bl7326425374436813197t_unit > list_P7986770385144383213od_v_v ).
thf(sy_c_SCC__Bloemen__Sequential_Oenv_Ostack_001tf__v_001t__Product____Type__Ounit,type,
sCC_Bl8828226123343373779t_unit: sCC_Bl1394983891496994913t_unit > list_v ).
thf(sy_c_SCC__Bloemen__Sequential_Oenv_Ovisited_001t__Product____Type__Oprod_Itf__v_Mtf__v_J_001t__Product____Type__Ounit,type,
sCC_Bl5498988629518860705t_unit: sCC_Bl7326425374436813197t_unit > set_Product_prod_v_v ).
thf(sy_c_SCC__Bloemen__Sequential_Oenv_Ovisited_001tf__v_001t__Product____Type__Ounit,type,
sCC_Bl4645233313691564917t_unit: sCC_Bl1394983891496994913t_unit > set_v ).
thf(sy_c_SCC__Bloemen__Sequential_Oenv_Ovsuccs_001t__Product____Type__Oprod_Itf__v_Mtf__v_J_001t__Product____Type__Ounit,type,
sCC_Bl3878977043676959280t_unit: sCC_Bl7326425374436813197t_unit > product_prod_v_v > set_Product_prod_v_v ).
thf(sy_c_SCC__Bloemen__Sequential_Oenv_Ovsuccs_001tf__v_001t__Product____Type__Ounit,type,
sCC_Bl3795065053823578884t_unit: sCC_Bl1394983891496994913t_unit > v > set_v ).
thf(sy_c_SCC__Bloemen__Sequential_Ograph_001t__Product____Type__Oprod_Itf__v_Mtf__v_J,type,
sCC_Bl8307124943676871238od_v_v: set_Product_prod_v_v > ( product_prod_v_v > set_Product_prod_v_v ) > $o ).
thf(sy_c_SCC__Bloemen__Sequential_Ograph_001tf__v,type,
sCC_Bloemen_graph_v: set_v > ( v > set_v ) > $o ).
thf(sy_c_SCC__Bloemen__Sequential_Ograph_Odfs__dfss__rel_001tf__v,type,
sCC_Bl907557413677168252_rel_v: ( v > set_v ) > sum_su8181647976486975269t_unit > sum_su8181647976486975269t_unit > $o ).
thf(sy_c_SCC__Bloemen__Sequential_Ograph_Odfss_001tf__v,type,
sCC_Bloemen_dfss_v: ( v > set_v ) > v > sCC_Bl1394983891496994913t_unit > sCC_Bl1394983891496994913t_unit ).
thf(sy_c_SCC__Bloemen__Sequential_Ograph_Ois__scc_001t__Product____Type__Oprod_Itf__v_Mtf__v_J,type,
sCC_Bl6242042402218619277od_v_v: ( product_prod_v_v > set_Product_prod_v_v ) > set_Product_prod_v_v > $o ).
thf(sy_c_SCC__Bloemen__Sequential_Ograph_Ois__scc_001tf__v,type,
sCC_Bloemen_is_scc_v: ( v > set_v ) > set_v > $o ).
thf(sy_c_SCC__Bloemen__Sequential_Ograph_Ois__subscc_001t__Product____Type__Oprod_Itf__v_Mtf__v_J,type,
sCC_Bl2301996248249672505od_v_v: ( product_prod_v_v > set_Product_prod_v_v ) > set_Product_prod_v_v > $o ).
thf(sy_c_SCC__Bloemen__Sequential_Ograph_Ois__subscc_001tf__v,type,
sCC_Bl5398416737448265317bscc_v: ( v > set_v ) > set_v > $o ).
thf(sy_c_SCC__Bloemen__Sequential_Ograph_Opost__dfs_001tf__v_001t__Product____Type__Ounit_001t__Product____Type__Ounit,type,
sCC_Bl8953792750115413617t_unit: ( v > set_v ) > v > sCC_Bl1394983891496994913t_unit > sCC_Bl1394983891496994913t_unit > $o ).
thf(sy_c_SCC__Bloemen__Sequential_Ograph_Opost__dfss_001tf__v_001t__Product____Type__Ounit_001t__Product____Type__Ounit,type,
sCC_Bl6082031138996704384t_unit: ( v > set_v ) > v > sCC_Bl1394983891496994913t_unit > sCC_Bl1394983891496994913t_unit > $o ).
thf(sy_c_SCC__Bloemen__Sequential_Ograph_Opre__dfs_001tf__v_001t__Product____Type__Ounit,type,
sCC_Bl36166008131615352t_unit: ( v > set_v ) > v > sCC_Bl1394983891496994913t_unit > $o ).
thf(sy_c_SCC__Bloemen__Sequential_Ograph_Opre__dfss_001t__Product____Type__Oprod_Itf__v_Mtf__v_J_001t__Product____Type__Ounit,type,
sCC_Bl3607325323686918683t_unit: ( product_prod_v_v > set_Product_prod_v_v ) > product_prod_v_v > sCC_Bl7326425374436813197t_unit > $o ).
thf(sy_c_SCC__Bloemen__Sequential_Ograph_Opre__dfss_001tf__v_001t__Product____Type__Ounit,type,
sCC_Bl1748261141445803503t_unit: ( v > set_v ) > v > sCC_Bl1394983891496994913t_unit > $o ).
thf(sy_c_SCC__Bloemen__Sequential_Ograph_Oreachable_001t__Product____Type__Oprod_Itf__v_Mtf__v_J,type,
sCC_Bl4981926079593201289od_v_v: ( product_prod_v_v > set_Product_prod_v_v ) > product_prod_v_v > product_prod_v_v > $o ).
thf(sy_c_SCC__Bloemen__Sequential_Ograph_Oreachable_001tf__v,type,
sCC_Bl649662514949026229able_v: ( v > set_v ) > v > v > $o ).
thf(sy_c_SCC__Bloemen__Sequential_Ograph_Oreachable__avoiding_001t__Product____Type__Oprod_Itf__v_Mtf__v_J,type,
sCC_Bl5370300055464682748od_v_v: ( product_prod_v_v > set_Product_prod_v_v ) > product_prod_v_v > product_prod_v_v > set_Pr2149350503807050951od_v_v > $o ).
thf(sy_c_SCC__Bloemen__Sequential_Ograph_Oreachable__avoiding_001tf__v,type,
sCC_Bl4291963740693775144ding_v: ( v > set_v ) > v > v > set_Product_prod_v_v > $o ).
thf(sy_c_SCC__Bloemen__Sequential_Ograph_Oreachable__end_001t__Product____Type__Oprod_Itf__v_Mtf__v_J,type,
sCC_Bl4714988717384592488od_v_v: ( product_prod_v_v > set_Product_prod_v_v ) > product_prod_v_v > product_prod_v_v > $o ).
thf(sy_c_SCC__Bloemen__Sequential_Ograph_Oreachable__end_001tf__v,type,
sCC_Bl770211535891879572_end_v: ( v > set_v ) > v > v > $o ).
thf(sy_c_SCC__Bloemen__Sequential_Ograph_Osub__env_001t__Product____Type__Oprod_Itf__v_Mtf__v_J_001t__Product____Type__Ounit_001t__Product____Type__Ounit,type,
sCC_Bl7963838319573962697t_unit: sCC_Bl7326425374436813197t_unit > sCC_Bl7326425374436813197t_unit > $o ).
thf(sy_c_SCC__Bloemen__Sequential_Ograph_Osub__env_001tf__v_001t__Product____Type__Ounit_001t__Product____Type__Ounit,type,
sCC_Bl5768913643336123637t_unit: sCC_Bl1394983891496994913t_unit > sCC_Bl1394983891496994913t_unit > $o ).
thf(sy_c_SCC__Bloemen__Sequential_Ograph_Ounite_001t__Product____Type__Oprod_Itf__v_Mtf__v_J,type,
sCC_Bl4702006153222411093od_v_v: product_prod_v_v > product_prod_v_v > sCC_Bl7326425374436813197t_unit > sCC_Bl7326425374436813197t_unit ).
thf(sy_c_SCC__Bloemen__Sequential_Ograph_Ounite_001tf__v,type,
sCC_Bloemen_unite_v: v > v > sCC_Bl1394983891496994913t_unit > sCC_Bl1394983891496994913t_unit ).
thf(sy_c_SCC__Bloemen__Sequential_Ograph_Ounvisited_001tf__v_001t__Product____Type__Ounit,type,
sCC_Bl3123350270117520219t_unit: ( v > set_v ) > sCC_Bl1394983891496994913t_unit > v > set_Product_prod_v_v ).
thf(sy_c_SCC__Bloemen__Sequential_Ograph_Owf__env_001t__Product____Type__Oprod_Itf__v_Mtf__v_J_001t__Product____Type__Ounit,type,
sCC_Bl7798947040364291444t_unit: ( product_prod_v_v > set_Product_prod_v_v ) > sCC_Bl7326425374436813197t_unit > $o ).
thf(sy_c_SCC__Bloemen__Sequential_Ograph_Owf__env_001tf__v_001t__Product____Type__Ounit,type,
sCC_Bl9196236973127232072t_unit: ( v > set_v ) > sCC_Bl1394983891496994913t_unit > $o ).
thf(sy_c_SCC__Bloemen__Sequential_Oinit__env_001tf__v,type,
sCC_Bl7693227186847904995_env_v: v > sCC_Bl1394983891496994913t_unit ).
thf(sy_c_SCC__Bloemen__Sequential_Oprecedes_001t__Product____Type__Oprod_Itf__v_Mtf__v_J,type,
sCC_Bl2026170059108282219od_v_v: product_prod_v_v > product_prod_v_v > list_P7986770385144383213od_v_v > $o ).
thf(sy_c_SCC__Bloemen__Sequential_Oprecedes_001tf__v,type,
sCC_Bl4022239298816431255edes_v: v > v > list_v > $o ).
thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_Itf__v_Mtf__v_J,type,
collec140062887454715474od_v_v: ( product_prod_v_v > $o ) > set_Product_prod_v_v ).
thf(sy_c_Set_OCollect_001tf__v,type,
collect_v: ( v > $o ) > set_v ).
thf(sy_c_Set_Oinsert_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__v_Mtf__v_J_Mt__Product____Type__Oprod_Itf__v_Mtf__v_J_J,type,
insert5641704497130386615od_v_v: produc206430290419586791od_v_v > set_Pr2149350503807050951od_v_v > set_Pr2149350503807050951od_v_v ).
thf(sy_c_Set_Oinsert_001t__Product____Type__Oprod_Itf__v_Mtf__v_J,type,
insert1338601472111419319od_v_v: product_prod_v_v > set_Product_prod_v_v > set_Product_prod_v_v ).
thf(sy_c_Set_Oinsert_001t__Set__Oset_It__Product____Type__Oprod_Itf__v_Mtf__v_J_J,type,
insert7504383016908236695od_v_v: set_Product_prod_v_v > set_se8455005133513928103od_v_v > set_se8455005133513928103od_v_v ).
thf(sy_c_Set_Oinsert_001t__Set__Oset_Itf__v_J,type,
insert_set_v: set_v > set_set_v > set_set_v ).
thf(sy_c_Set_Oinsert_001tf__v,type,
insert_v2: v > set_v > set_v ).
thf(sy_c_Set_Ois__empty_001t__Product____Type__Oprod_Itf__v_Mtf__v_J,type,
is_emp8964507351669718201od_v_v: set_Product_prod_v_v > $o ).
thf(sy_c_Set_Ois__empty_001tf__v,type,
is_empty_v: set_v > $o ).
thf(sy_c_Set_Ois__singleton_001t__Product____Type__Oprod_Itf__v_Mtf__v_J,type,
is_sin9198872032823709915od_v_v: set_Product_prod_v_v > $o ).
thf(sy_c_Set_Ois__singleton_001tf__v,type,
is_singleton_v: set_v > $o ).
thf(sy_c_Set_Oremove_001t__Product____Type__Oprod_Itf__v_Mtf__v_J,type,
remove5001965847480235980od_v_v: product_prod_v_v > set_Product_prod_v_v > set_Product_prod_v_v ).
thf(sy_c_Set_Oremove_001tf__v,type,
remove_v: v > set_v > set_v ).
thf(sy_c_Set_Othe__elem_001t__Product____Type__Oprod_Itf__v_Mtf__v_J,type,
the_el5392834299063928540od_v_v: set_Product_prod_v_v > product_prod_v_v ).
thf(sy_c_Set_Othe__elem_001tf__v,type,
the_elem_v: set_v > v ).
thf(sy_c_Sum__Type_OInr_001t__Product____Type__Oprod_Itf__v_Mt__SCC____Bloemen____Sequential__Oenv__Oenv____ext_Itf__v_Mt__Product____Type__Ounit_J_J_001t__Product____Type__Oprod_Itf__v_Mt__SCC____Bloemen____Sequential__Oenv__Oenv____ext_Itf__v_Mt__Product____Type__Ounit_J_J,type,
sum_In5289330923152326972t_unit: produc5741669702376414499t_unit > sum_su8181647976486975269t_unit ).
thf(sy_c_Wellfounded_Oaccp_001t__Sum____Type__Osum_It__Product____Type__Oprod_Itf__v_Mt__SCC____Bloemen____Sequential__Oenv__Oenv____ext_Itf__v_Mt__Product____Type__Ounit_J_J_Mt__Product____Type__Oprod_Itf__v_Mt__SCC____Bloemen____Sequential__Oenv__Oenv____ext_Itf__v_Mt__Product____Type__Ounit_J_J_J,type,
accp_S2303753412255344476t_unit: ( sum_su8181647976486975269t_unit > sum_su8181647976486975269t_unit > $o ) > sum_su8181647976486975269t_unit > $o ).
thf(sy_c_member_001t__List__Olist_It__Product____Type__Oprod_Itf__v_Mtf__v_J_J,type,
member4190458934886417558od_v_v: list_P7986770385144383213od_v_v > set_li2323639185124838733od_v_v > $o ).
thf(sy_c_member_001t__List__Olist_Itf__v_J,type,
member_list_v: list_v > set_list_v > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__Product____Type__Oprod_Itf__v_Mtf__v_J_J_Mt__List__Olist_It__Product____Type__Oprod_Itf__v_Mtf__v_J_J_J,type,
member6382463057129219728od_v_v: produc1504107476793160551od_v_v > set_Pr7499474215547700295od_v_v > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_Itf__v_J_Mt__List__Olist_It__SCC____Bloemen____Sequential__Oenv__Oenv____ext_Itf__v_Mt__Product____Type__Ounit_J_J_J,type,
member5221658779932351824t_unit: produc7473616835194958905t_unit > set_Pr8903222484228591t_unit > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_Itf__v_J_Mt__List__Olist_Itf__v_J_J,type,
member418487059593946000list_v: produc1391462591744249447list_v > set_Pr6206931691796273479list_v > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__v_Mtf__v_J_Mt__Product____Type__Oprod_Itf__v_Mtf__v_J_J,type,
member3038538357316246288od_v_v: produc206430290419586791od_v_v > set_Pr2149350503807050951od_v_v > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_Itf__v_Mt__SCC____Bloemen____Sequential__Oenv__Oenv____ext_Itf__v_Mt__Product____Type__Ounit_J_J,type,
member7924940910754673978t_unit: produc5741669702376414499t_unit > set_Pr6425124735969554649t_unit > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_Itf__v_Mtf__v_J,type,
member7453568604450474000od_v_v: product_prod_v_v > set_Product_prod_v_v > $o ).
thf(sy_c_member_001t__Set__Oset_It__Product____Type__Oprod_Itf__v_Mtf__v_J_J,type,
member8406446414694345712od_v_v: set_Product_prod_v_v > set_se8455005133513928103od_v_v > $o ).
thf(sy_c_member_001t__Set__Oset_Itf__v_J,type,
member_set_v: set_v > set_set_v > $o ).
thf(sy_c_member_001tf__v,type,
member_v: v > set_v > $o ).
thf(sy_v_ea____,type,
ea: sCC_Bl1394983891496994913t_unit ).
thf(sy_v_n____,type,
n: v ).
thf(sy_v_successors,type,
successors: v > set_v ).
thf(sy_v_va____,type,
va: v ).
thf(sy_v_vertices,type,
vertices: set_v ).
% Relevant facts (1276)
thf(fact_0__092_060open_062v_A_092_060in_062_A_092_060S_062_Ae_Av_092_060close_062,axiom,
member_v @ va @ ( sCC_Bl1280885523602775798t_unit @ ea @ va ) ).
% \<open>v \<in> \<S> e v\<close>
thf(fact_1__092_060open_062v_A_092_060noteq_062_An_092_060close_062,axiom,
va != n ).
% \<open>v \<noteq> n\<close>
thf(fact_2__092_060open_062unvisited_Ae_Av_A_061_A_123_125_092_060close_062,axiom,
( ( sCC_Bl3123350270117520219t_unit @ successors @ ea @ va )
= bot_bo723834152578015283od_v_v ) ).
% \<open>unvisited e v = {}\<close>
thf(fact_3_dfs__dfss__rel_Ocong,axiom,
sCC_Bl907557413677168252_rel_v = sCC_Bl907557413677168252_rel_v ).
% dfs_dfss_rel.cong
thf(fact_4_ra__refl,axiom,
! [X: v,E: set_Product_prod_v_v] : ( sCC_Bl4291963740693775144ding_v @ successors @ X @ X @ E ) ).
% ra_refl
thf(fact_5_ra__trans,axiom,
! [X: v,Y: v,E: set_Product_prod_v_v,Z: v] :
( ( sCC_Bl4291963740693775144ding_v @ successors @ X @ Y @ E )
=> ( ( sCC_Bl4291963740693775144ding_v @ successors @ Y @ Z @ E )
=> ( sCC_Bl4291963740693775144ding_v @ successors @ X @ Z @ E ) ) ) ).
% ra_trans
thf(fact_6__092_060open_062v_A_092_060preceq_062_An_Ain_Astack_Ae_092_060close_062,axiom,
sCC_Bl4022239298816431255edes_v @ va @ n @ ( sCC_Bl8828226123343373779t_unit @ ea ) ).
% \<open>v \<preceq> n in stack e\<close>
thf(fact_7_sub__env__trans,axiom,
! [E2: sCC_Bl1394983891496994913t_unit,E3: sCC_Bl1394983891496994913t_unit,E4: sCC_Bl1394983891496994913t_unit] :
( ( sCC_Bl5768913643336123637t_unit @ E2 @ E3 )
=> ( ( sCC_Bl5768913643336123637t_unit @ E3 @ E4 )
=> ( sCC_Bl5768913643336123637t_unit @ E2 @ E4 ) ) ) ).
% sub_env_trans
thf(fact_8_asm_I3_J,axiom,
sCC_Bl649662514949026229able_v @ successors @ va @ n ).
% asm(3)
thf(fact_9_graph_Oreachable__avoiding_Ocong,axiom,
sCC_Bl4291963740693775144ding_v = sCC_Bl4291963740693775144ding_v ).
% graph.reachable_avoiding.cong
thf(fact_10_predfss,axiom,
sCC_Bl1748261141445803503t_unit @ successors @ va @ ea ).
% predfss
thf(fact_11__092_060open_062dfss_Av_Ae_A_061_Ae_092_060close_062,axiom,
( ( sCC_Bloemen_dfss_v @ successors @ va @ ea )
= ea ) ).
% \<open>dfss v e = e\<close>
thf(fact_12_ra__empty,axiom,
! [X: v,Y: v] :
( ( sCC_Bl4291963740693775144ding_v @ successors @ X @ Y @ bot_bo723834152578015283od_v_v )
= ( sCC_Bl649662514949026229able_v @ successors @ X @ Y ) ) ).
% ra_empty
thf(fact_13_ra__mono,axiom,
! [X: v,Y: v,E: set_Product_prod_v_v,E5: set_Product_prod_v_v] :
( ( sCC_Bl4291963740693775144ding_v @ successors @ X @ Y @ E )
=> ( ( ord_le7336532860387713383od_v_v @ E5 @ E )
=> ( sCC_Bl4291963740693775144ding_v @ successors @ X @ Y @ E5 ) ) ) ).
% ra_mono
thf(fact_14_empty__iff,axiom,
! [C: product_prod_v_v] :
~ ( member7453568604450474000od_v_v @ C @ bot_bo723834152578015283od_v_v ) ).
% empty_iff
thf(fact_15_empty__iff,axiom,
! [C: v] :
~ ( member_v @ C @ bot_bot_set_v ) ).
% empty_iff
thf(fact_16_all__not__in__conv,axiom,
! [A: set_Product_prod_v_v] :
( ( ! [X2: product_prod_v_v] :
~ ( member7453568604450474000od_v_v @ X2 @ A ) )
= ( A = bot_bo723834152578015283od_v_v ) ) ).
% all_not_in_conv
thf(fact_17_all__not__in__conv,axiom,
! [A: set_v] :
( ( ! [X2: v] :
~ ( member_v @ X2 @ A ) )
= ( A = bot_bot_set_v ) ) ).
% all_not_in_conv
thf(fact_18_Collect__empty__eq,axiom,
! [P: product_prod_v_v > $o] :
( ( ( collec140062887454715474od_v_v @ P )
= bot_bo723834152578015283od_v_v )
= ( ! [X2: product_prod_v_v] :
~ ( P @ X2 ) ) ) ).
% Collect_empty_eq
thf(fact_19_Collect__empty__eq,axiom,
! [P: v > $o] :
( ( ( collect_v @ P )
= bot_bot_set_v )
= ( ! [X2: v] :
~ ( P @ X2 ) ) ) ).
% Collect_empty_eq
thf(fact_20_empty__Collect__eq,axiom,
! [P: product_prod_v_v > $o] :
( ( bot_bo723834152578015283od_v_v
= ( collec140062887454715474od_v_v @ P ) )
= ( ! [X2: product_prod_v_v] :
~ ( P @ X2 ) ) ) ).
% empty_Collect_eq
thf(fact_21_empty__Collect__eq,axiom,
! [P: v > $o] :
( ( bot_bot_set_v
= ( collect_v @ P ) )
= ( ! [X2: v] :
~ ( P @ X2 ) ) ) ).
% empty_Collect_eq
thf(fact_22_reachable__avoiding_Ocases,axiom,
! [A1: v,A2: v,A3: set_Product_prod_v_v] :
( ( sCC_Bl4291963740693775144ding_v @ successors @ A1 @ A2 @ A3 )
=> ( ( A2 != A1 )
=> ~ ! [Y2: v] :
( ( sCC_Bl4291963740693775144ding_v @ successors @ A1 @ Y2 @ A3 )
=> ( ( member_v @ A2 @ ( successors @ Y2 ) )
=> ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ Y2 @ A2 ) @ A3 ) ) ) ) ) ).
% reachable_avoiding.cases
thf(fact_23_ra__succ,axiom,
! [X: v,Y: v,E: set_Product_prod_v_v,Z: v] :
( ( sCC_Bl4291963740693775144ding_v @ successors @ X @ Y @ E )
=> ( ( member_v @ Z @ ( successors @ Y ) )
=> ( ~ ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ Y @ Z ) @ E )
=> ( sCC_Bl4291963740693775144ding_v @ successors @ X @ Z @ E ) ) ) ) ).
% ra_succ
thf(fact_24__092_060open_062sub__env_Ae_Ae_092_060close_062,axiom,
sCC_Bl5768913643336123637t_unit @ ea @ ea ).
% \<open>sub_env e e\<close>
thf(fact_25_succ__reachable,axiom,
! [X: v,Y: v,Z: v] :
( ( sCC_Bl649662514949026229able_v @ successors @ X @ Y )
=> ( ( member_v @ Z @ ( successors @ Y ) )
=> ( sCC_Bl649662514949026229able_v @ successors @ X @ Z ) ) ) ).
% succ_reachable
thf(fact_26_reachable__trans,axiom,
! [X: v,Y: v,Z: v] :
( ( sCC_Bl649662514949026229able_v @ successors @ X @ Y )
=> ( ( sCC_Bl649662514949026229able_v @ successors @ Y @ Z )
=> ( sCC_Bl649662514949026229able_v @ successors @ X @ Z ) ) ) ).
% reachable_trans
thf(fact_27_reachable__end__induct,axiom,
! [X: v,Y: v,P: v > v > $o] :
( ( sCC_Bl649662514949026229able_v @ successors @ X @ Y )
=> ( ! [X3: v] : ( P @ X3 @ X3 )
=> ( ! [X3: v,Y2: v,Z2: v] :
( ( P @ X3 @ Y2 )
=> ( ( member_v @ Z2 @ ( successors @ Y2 ) )
=> ( P @ X3 @ Z2 ) ) )
=> ( P @ X @ Y ) ) ) ) ).
% reachable_end_induct
thf(fact_28_reachable__edge,axiom,
! [Y: v,X: v] :
( ( member_v @ Y @ ( successors @ X ) )
=> ( sCC_Bl649662514949026229able_v @ successors @ X @ Y ) ) ).
% reachable_edge
thf(fact_29_reachable_Osimps,axiom,
! [A1: v,A2: v] :
( ( sCC_Bl649662514949026229able_v @ successors @ A1 @ A2 )
= ( ? [X2: v] :
( ( A1 = X2 )
& ( A2 = X2 ) )
| ? [X2: v,Y3: v,Z3: v] :
( ( A1 = X2 )
& ( A2 = Z3 )
& ( member_v @ Y3 @ ( successors @ X2 ) )
& ( sCC_Bl649662514949026229able_v @ successors @ Y3 @ Z3 ) ) ) ) ).
% reachable.simps
thf(fact_30_reachable__succ,axiom,
! [Y: v,X: v,Z: v] :
( ( member_v @ Y @ ( successors @ X ) )
=> ( ( sCC_Bl649662514949026229able_v @ successors @ Y @ Z )
=> ( sCC_Bl649662514949026229able_v @ successors @ X @ Z ) ) ) ).
% reachable_succ
thf(fact_31_reachable__refl,axiom,
! [X: v] : ( sCC_Bl649662514949026229able_v @ successors @ X @ X ) ).
% reachable_refl
thf(fact_32_reachable_Ocases,axiom,
! [A1: v,A2: v] :
( ( sCC_Bl649662514949026229able_v @ successors @ A1 @ A2 )
=> ( ( A2 != A1 )
=> ~ ! [Y2: v] :
( ( member_v @ Y2 @ ( successors @ A1 ) )
=> ~ ( sCC_Bl649662514949026229able_v @ successors @ Y2 @ A2 ) ) ) ) ).
% reachable.cases
thf(fact_33_ra__reachable,axiom,
! [X: v,Y: v,E: set_Product_prod_v_v] :
( ( sCC_Bl4291963740693775144ding_v @ successors @ X @ Y @ E )
=> ( sCC_Bl649662514949026229able_v @ successors @ X @ Y ) ) ).
% ra_reachable
thf(fact_34_ra__cases,axiom,
! [X: v,Y: v,E: set_Product_prod_v_v] :
( ( sCC_Bl4291963740693775144ding_v @ successors @ X @ Y @ E )
=> ( ( X = Y )
| ? [Z2: v] :
( ( member_v @ Z2 @ ( successors @ X ) )
& ~ ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ X @ Z2 ) @ E )
& ( sCC_Bl4291963740693775144ding_v @ successors @ Z2 @ Y @ E ) ) ) ) ).
% ra_cases
thf(fact_35_edge__ra,axiom,
! [Y: v,X: v,E: set_Product_prod_v_v] :
( ( member_v @ Y @ ( successors @ X ) )
=> ( ~ ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ X @ Y ) @ E )
=> ( sCC_Bl4291963740693775144ding_v @ successors @ X @ Y @ E ) ) ) ).
% edge_ra
thf(fact_36_reachable__avoiding_Osimps,axiom,
! [A1: v,A2: v,A3: set_Product_prod_v_v] :
( ( sCC_Bl4291963740693775144ding_v @ successors @ A1 @ A2 @ A3 )
= ( ? [X2: v,E6: set_Product_prod_v_v] :
( ( A1 = X2 )
& ( A2 = X2 )
& ( A3 = E6 ) )
| ? [X2: v,Y3: v,E6: set_Product_prod_v_v,Z3: v] :
( ( A1 = X2 )
& ( A2 = Z3 )
& ( A3 = E6 )
& ( sCC_Bl4291963740693775144ding_v @ successors @ X2 @ Y3 @ E6 )
& ( member_v @ Z3 @ ( successors @ Y3 ) )
& ~ ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ Y3 @ Z3 ) @ E6 ) ) ) ) ).
% reachable_avoiding.simps
thf(fact_37_order__refl,axiom,
! [X: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ X @ X ) ).
% order_refl
thf(fact_38_order__refl,axiom,
! [X: set_v] : ( ord_less_eq_set_v @ X @ X ) ).
% order_refl
thf(fact_39_dual__order_Orefl,axiom,
! [A4: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ A4 @ A4 ) ).
% dual_order.refl
thf(fact_40_dual__order_Orefl,axiom,
! [A4: set_v] : ( ord_less_eq_set_v @ A4 @ A4 ) ).
% dual_order.refl
thf(fact_41_subsetI,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ! [X3: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X3 @ A )
=> ( member7453568604450474000od_v_v @ X3 @ B ) )
=> ( ord_le7336532860387713383od_v_v @ A @ B ) ) ).
% subsetI
thf(fact_42_subsetI,axiom,
! [A: set_v,B: set_v] :
( ! [X3: v] :
( ( member_v @ X3 @ A )
=> ( member_v @ X3 @ B ) )
=> ( ord_less_eq_set_v @ A @ B ) ) ).
% subsetI
thf(fact_43_subset__antisym,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ B )
=> ( ( ord_le7336532860387713383od_v_v @ B @ A )
=> ( A = B ) ) ) ).
% subset_antisym
thf(fact_44_subset__antisym,axiom,
! [A: set_v,B: set_v] :
( ( ord_less_eq_set_v @ A @ B )
=> ( ( ord_less_eq_set_v @ B @ A )
=> ( A = B ) ) ) ).
% subset_antisym
thf(fact_45_empty__subsetI,axiom,
! [A: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ bot_bo723834152578015283od_v_v @ A ) ).
% empty_subsetI
thf(fact_46_empty__subsetI,axiom,
! [A: set_v] : ( ord_less_eq_set_v @ bot_bot_set_v @ A ) ).
% empty_subsetI
thf(fact_47_subset__empty,axiom,
! [A: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ bot_bo723834152578015283od_v_v )
= ( A = bot_bo723834152578015283od_v_v ) ) ).
% subset_empty
thf(fact_48_subset__empty,axiom,
! [A: set_v] :
( ( ord_less_eq_set_v @ A @ bot_bot_set_v )
= ( A = bot_bot_set_v ) ) ).
% subset_empty
thf(fact_49_asm_I1_J,axiom,
( ( hd_v @ ( sCC_Bl8828226123343373779t_unit @ ea ) )
= va ) ).
% asm(1)
thf(fact_50_sccE,axiom,
! [S: set_v,X: v,Y: v] :
( ( sCC_Bloemen_is_scc_v @ successors @ S )
=> ( ( member_v @ X @ S )
=> ( ( sCC_Bl649662514949026229able_v @ successors @ X @ Y )
=> ( ( sCC_Bl649662514949026229able_v @ successors @ Y @ X )
=> ( member_v @ Y @ S ) ) ) ) ) ).
% sccE
thf(fact_51_mem__Collect__eq,axiom,
! [A4: v,P: v > $o] :
( ( member_v @ A4 @ ( collect_v @ P ) )
= ( P @ A4 ) ) ).
% mem_Collect_eq
thf(fact_52_mem__Collect__eq,axiom,
! [A4: product_prod_v_v,P: product_prod_v_v > $o] :
( ( member7453568604450474000od_v_v @ A4 @ ( collec140062887454715474od_v_v @ P ) )
= ( P @ A4 ) ) ).
% mem_Collect_eq
thf(fact_53_Collect__mem__eq,axiom,
! [A: set_v] :
( ( collect_v
@ ^ [X2: v] : ( member_v @ X2 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_54_Collect__mem__eq,axiom,
! [A: set_Product_prod_v_v] :
( ( collec140062887454715474od_v_v
@ ^ [X2: product_prod_v_v] : ( member7453568604450474000od_v_v @ X2 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_55_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_Product_prod_v_v,Z4: set_Product_prod_v_v] : ( Y4 = Z4 ) )
= ( ^ [X2: set_Product_prod_v_v,Y3: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X2 @ Y3 )
& ( ord_le7336532860387713383od_v_v @ Y3 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_56_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_v,Z4: set_v] : ( Y4 = Z4 ) )
= ( ^ [X2: set_v,Y3: set_v] :
( ( ord_less_eq_set_v @ X2 @ Y3 )
& ( ord_less_eq_set_v @ Y3 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_57_ord__eq__le__trans,axiom,
! [A4: set_Product_prod_v_v,B2: set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( A4 = B2 )
=> ( ( ord_le7336532860387713383od_v_v @ B2 @ C )
=> ( ord_le7336532860387713383od_v_v @ A4 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_58_ord__eq__le__trans,axiom,
! [A4: set_v,B2: set_v,C: set_v] :
( ( A4 = B2 )
=> ( ( ord_less_eq_set_v @ B2 @ C )
=> ( ord_less_eq_set_v @ A4 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_59_ord__le__eq__trans,axiom,
! [A4: set_Product_prod_v_v,B2: set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A4 @ B2 )
=> ( ( B2 = C )
=> ( ord_le7336532860387713383od_v_v @ A4 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_60_ord__le__eq__trans,axiom,
! [A4: set_v,B2: set_v,C: set_v] :
( ( ord_less_eq_set_v @ A4 @ B2 )
=> ( ( B2 = C )
=> ( ord_less_eq_set_v @ A4 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_61_order__antisym,axiom,
! [X: set_Product_prod_v_v,Y: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X @ Y )
=> ( ( ord_le7336532860387713383od_v_v @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_62_order__antisym,axiom,
! [X: set_v,Y: set_v] :
( ( ord_less_eq_set_v @ X @ Y )
=> ( ( ord_less_eq_set_v @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_63_order_Otrans,axiom,
! [A4: set_Product_prod_v_v,B2: set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A4 @ B2 )
=> ( ( ord_le7336532860387713383od_v_v @ B2 @ C )
=> ( ord_le7336532860387713383od_v_v @ A4 @ C ) ) ) ).
% order.trans
thf(fact_64_order_Otrans,axiom,
! [A4: set_v,B2: set_v,C: set_v] :
( ( ord_less_eq_set_v @ A4 @ B2 )
=> ( ( ord_less_eq_set_v @ B2 @ C )
=> ( ord_less_eq_set_v @ A4 @ C ) ) ) ).
% order.trans
thf(fact_65_order__trans,axiom,
! [X: set_Product_prod_v_v,Y: set_Product_prod_v_v,Z: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X @ Y )
=> ( ( ord_le7336532860387713383od_v_v @ Y @ Z )
=> ( ord_le7336532860387713383od_v_v @ X @ Z ) ) ) ).
% order_trans
thf(fact_66_order__trans,axiom,
! [X: set_v,Y: set_v,Z: set_v] :
( ( ord_less_eq_set_v @ X @ Y )
=> ( ( ord_less_eq_set_v @ Y @ Z )
=> ( ord_less_eq_set_v @ X @ Z ) ) ) ).
% order_trans
thf(fact_67_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: set_Product_prod_v_v,Z4: set_Product_prod_v_v] : ( Y4 = Z4 ) )
= ( ^ [A5: set_Product_prod_v_v,B3: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ B3 @ A5 )
& ( ord_le7336532860387713383od_v_v @ A5 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_68_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: set_v,Z4: set_v] : ( Y4 = Z4 ) )
= ( ^ [A5: set_v,B3: set_v] :
( ( ord_less_eq_set_v @ B3 @ A5 )
& ( ord_less_eq_set_v @ A5 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_69_dual__order_Oantisym,axiom,
! [B2: set_Product_prod_v_v,A4: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ B2 @ A4 )
=> ( ( ord_le7336532860387713383od_v_v @ A4 @ B2 )
=> ( A4 = B2 ) ) ) ).
% dual_order.antisym
thf(fact_70_dual__order_Oantisym,axiom,
! [B2: set_v,A4: set_v] :
( ( ord_less_eq_set_v @ B2 @ A4 )
=> ( ( ord_less_eq_set_v @ A4 @ B2 )
=> ( A4 = B2 ) ) ) ).
% dual_order.antisym
thf(fact_71_dual__order_Otrans,axiom,
! [B2: set_Product_prod_v_v,A4: set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ B2 @ A4 )
=> ( ( ord_le7336532860387713383od_v_v @ C @ B2 )
=> ( ord_le7336532860387713383od_v_v @ C @ A4 ) ) ) ).
% dual_order.trans
thf(fact_72_dual__order_Otrans,axiom,
! [B2: set_v,A4: set_v,C: set_v] :
( ( ord_less_eq_set_v @ B2 @ A4 )
=> ( ( ord_less_eq_set_v @ C @ B2 )
=> ( ord_less_eq_set_v @ C @ A4 ) ) ) ).
% dual_order.trans
thf(fact_73_graph_Oreachable_Ocong,axiom,
sCC_Bl649662514949026229able_v = sCC_Bl649662514949026229able_v ).
% graph.reachable.cong
thf(fact_74_in__mono,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v,X: product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ B )
=> ( ( member7453568604450474000od_v_v @ X @ A )
=> ( member7453568604450474000od_v_v @ X @ B ) ) ) ).
% in_mono
thf(fact_75_in__mono,axiom,
! [A: set_v,B: set_v,X: v] :
( ( ord_less_eq_set_v @ A @ B )
=> ( ( member_v @ X @ A )
=> ( member_v @ X @ B ) ) ) ).
% in_mono
thf(fact_76_subsetD,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v,C: product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ B )
=> ( ( member7453568604450474000od_v_v @ C @ A )
=> ( member7453568604450474000od_v_v @ C @ B ) ) ) ).
% subsetD
thf(fact_77_subsetD,axiom,
! [A: set_v,B: set_v,C: v] :
( ( ord_less_eq_set_v @ A @ B )
=> ( ( member_v @ C @ A )
=> ( member_v @ C @ B ) ) ) ).
% subsetD
thf(fact_78_equalityE,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( A = B )
=> ~ ( ( ord_le7336532860387713383od_v_v @ A @ B )
=> ~ ( ord_le7336532860387713383od_v_v @ B @ A ) ) ) ).
% equalityE
thf(fact_79_equalityE,axiom,
! [A: set_v,B: set_v] :
( ( A = B )
=> ~ ( ( ord_less_eq_set_v @ A @ B )
=> ~ ( ord_less_eq_set_v @ B @ A ) ) ) ).
% equalityE
thf(fact_80_subset__eq,axiom,
( ord_le7336532860387713383od_v_v
= ( ^ [A6: set_Product_prod_v_v,B4: set_Product_prod_v_v] :
! [X2: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X2 @ A6 )
=> ( member7453568604450474000od_v_v @ X2 @ B4 ) ) ) ) ).
% subset_eq
thf(fact_81_subset__eq,axiom,
( ord_less_eq_set_v
= ( ^ [A6: set_v,B4: set_v] :
! [X2: v] :
( ( member_v @ X2 @ A6 )
=> ( member_v @ X2 @ B4 ) ) ) ) ).
% subset_eq
thf(fact_82_equalityD1,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( A = B )
=> ( ord_le7336532860387713383od_v_v @ A @ B ) ) ).
% equalityD1
thf(fact_83_equalityD1,axiom,
! [A: set_v,B: set_v] :
( ( A = B )
=> ( ord_less_eq_set_v @ A @ B ) ) ).
% equalityD1
thf(fact_84_equalityD2,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( A = B )
=> ( ord_le7336532860387713383od_v_v @ B @ A ) ) ).
% equalityD2
thf(fact_85_equalityD2,axiom,
! [A: set_v,B: set_v] :
( ( A = B )
=> ( ord_less_eq_set_v @ B @ A ) ) ).
% equalityD2
thf(fact_86_subset__iff,axiom,
( ord_le7336532860387713383od_v_v
= ( ^ [A6: set_Product_prod_v_v,B4: set_Product_prod_v_v] :
! [T: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ T @ A6 )
=> ( member7453568604450474000od_v_v @ T @ B4 ) ) ) ) ).
% subset_iff
thf(fact_87_subset__iff,axiom,
( ord_less_eq_set_v
= ( ^ [A6: set_v,B4: set_v] :
! [T: v] :
( ( member_v @ T @ A6 )
=> ( member_v @ T @ B4 ) ) ) ) ).
% subset_iff
thf(fact_88_subset__refl,axiom,
! [A: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ A @ A ) ).
% subset_refl
thf(fact_89_subset__refl,axiom,
! [A: set_v] : ( ord_less_eq_set_v @ A @ A ) ).
% subset_refl
thf(fact_90_Collect__mono,axiom,
! [P: product_prod_v_v > $o,Q: product_prod_v_v > $o] :
( ! [X3: product_prod_v_v] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( ord_le7336532860387713383od_v_v @ ( collec140062887454715474od_v_v @ P ) @ ( collec140062887454715474od_v_v @ Q ) ) ) ).
% Collect_mono
thf(fact_91_Collect__mono,axiom,
! [P: v > $o,Q: v > $o] :
( ! [X3: v] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( ord_less_eq_set_v @ ( collect_v @ P ) @ ( collect_v @ Q ) ) ) ).
% Collect_mono
thf(fact_92_subset__trans,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v,C2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ B )
=> ( ( ord_le7336532860387713383od_v_v @ B @ C2 )
=> ( ord_le7336532860387713383od_v_v @ A @ C2 ) ) ) ).
% subset_trans
thf(fact_93_subset__trans,axiom,
! [A: set_v,B: set_v,C2: set_v] :
( ( ord_less_eq_set_v @ A @ B )
=> ( ( ord_less_eq_set_v @ B @ C2 )
=> ( ord_less_eq_set_v @ A @ C2 ) ) ) ).
% subset_trans
thf(fact_94_antisym,axiom,
! [A4: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A4 @ B2 )
=> ( ( ord_le7336532860387713383od_v_v @ B2 @ A4 )
=> ( A4 = B2 ) ) ) ).
% antisym
thf(fact_95_antisym,axiom,
! [A4: set_v,B2: set_v] :
( ( ord_less_eq_set_v @ A4 @ B2 )
=> ( ( ord_less_eq_set_v @ B2 @ A4 )
=> ( A4 = B2 ) ) ) ).
% antisym
thf(fact_96_set__eq__subset,axiom,
( ( ^ [Y4: set_Product_prod_v_v,Z4: set_Product_prod_v_v] : ( Y4 = Z4 ) )
= ( ^ [A6: set_Product_prod_v_v,B4: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A6 @ B4 )
& ( ord_le7336532860387713383od_v_v @ B4 @ A6 ) ) ) ) ).
% set_eq_subset
thf(fact_97_set__eq__subset,axiom,
( ( ^ [Y4: set_v,Z4: set_v] : ( Y4 = Z4 ) )
= ( ^ [A6: set_v,B4: set_v] :
( ( ord_less_eq_set_v @ A6 @ B4 )
& ( ord_less_eq_set_v @ B4 @ A6 ) ) ) ) ).
% set_eq_subset
thf(fact_98_Collect__mono__iff,axiom,
! [P: product_prod_v_v > $o,Q: product_prod_v_v > $o] :
( ( ord_le7336532860387713383od_v_v @ ( collec140062887454715474od_v_v @ P ) @ ( collec140062887454715474od_v_v @ Q ) )
= ( ! [X2: product_prod_v_v] :
( ( P @ X2 )
=> ( Q @ X2 ) ) ) ) ).
% Collect_mono_iff
thf(fact_99_Collect__mono__iff,axiom,
! [P: v > $o,Q: v > $o] :
( ( ord_less_eq_set_v @ ( collect_v @ P ) @ ( collect_v @ Q ) )
= ( ! [X2: v] :
( ( P @ X2 )
=> ( Q @ X2 ) ) ) ) ).
% Collect_mono_iff
thf(fact_100_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_Product_prod_v_v,Z4: set_Product_prod_v_v] : ( Y4 = Z4 ) )
= ( ^ [A5: set_Product_prod_v_v,B3: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A5 @ B3 )
& ( ord_le7336532860387713383od_v_v @ B3 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_101_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_v,Z4: set_v] : ( Y4 = Z4 ) )
= ( ^ [A5: set_v,B3: set_v] :
( ( ord_less_eq_set_v @ A5 @ B3 )
& ( ord_less_eq_set_v @ B3 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_102_order__subst1,axiom,
! [A4: set_Product_prod_v_v,F: set_Product_prod_v_v > set_Product_prod_v_v,B2: set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A4 @ ( F @ B2 ) )
=> ( ( ord_le7336532860387713383od_v_v @ B2 @ C )
=> ( ! [X3: set_Product_prod_v_v,Y2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X3 @ Y2 )
=> ( ord_le7336532860387713383od_v_v @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le7336532860387713383od_v_v @ A4 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_103_order__subst1,axiom,
! [A4: set_Product_prod_v_v,F: set_v > set_Product_prod_v_v,B2: set_v,C: set_v] :
( ( ord_le7336532860387713383od_v_v @ A4 @ ( F @ B2 ) )
=> ( ( ord_less_eq_set_v @ B2 @ C )
=> ( ! [X3: set_v,Y2: set_v] :
( ( ord_less_eq_set_v @ X3 @ Y2 )
=> ( ord_le7336532860387713383od_v_v @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le7336532860387713383od_v_v @ A4 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_104_order__subst1,axiom,
! [A4: set_v,F: set_Product_prod_v_v > set_v,B2: set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( ord_less_eq_set_v @ A4 @ ( F @ B2 ) )
=> ( ( ord_le7336532860387713383od_v_v @ B2 @ C )
=> ( ! [X3: set_Product_prod_v_v,Y2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X3 @ Y2 )
=> ( ord_less_eq_set_v @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_v @ A4 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_105_order__subst1,axiom,
! [A4: set_v,F: set_v > set_v,B2: set_v,C: set_v] :
( ( ord_less_eq_set_v @ A4 @ ( F @ B2 ) )
=> ( ( ord_less_eq_set_v @ B2 @ C )
=> ( ! [X3: set_v,Y2: set_v] :
( ( ord_less_eq_set_v @ X3 @ Y2 )
=> ( ord_less_eq_set_v @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_v @ A4 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_106_order__subst2,axiom,
! [A4: set_Product_prod_v_v,B2: set_Product_prod_v_v,F: set_Product_prod_v_v > set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A4 @ B2 )
=> ( ( ord_le7336532860387713383od_v_v @ ( F @ B2 ) @ C )
=> ( ! [X3: set_Product_prod_v_v,Y2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X3 @ Y2 )
=> ( ord_le7336532860387713383od_v_v @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le7336532860387713383od_v_v @ ( F @ A4 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_107_order__subst2,axiom,
! [A4: set_Product_prod_v_v,B2: set_Product_prod_v_v,F: set_Product_prod_v_v > set_v,C: set_v] :
( ( ord_le7336532860387713383od_v_v @ A4 @ B2 )
=> ( ( ord_less_eq_set_v @ ( F @ B2 ) @ C )
=> ( ! [X3: set_Product_prod_v_v,Y2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X3 @ Y2 )
=> ( ord_less_eq_set_v @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_v @ ( F @ A4 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_108_order__subst2,axiom,
! [A4: set_v,B2: set_v,F: set_v > set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( ord_less_eq_set_v @ A4 @ B2 )
=> ( ( ord_le7336532860387713383od_v_v @ ( F @ B2 ) @ C )
=> ( ! [X3: set_v,Y2: set_v] :
( ( ord_less_eq_set_v @ X3 @ Y2 )
=> ( ord_le7336532860387713383od_v_v @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le7336532860387713383od_v_v @ ( F @ A4 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_109_order__subst2,axiom,
! [A4: set_v,B2: set_v,F: set_v > set_v,C: set_v] :
( ( ord_less_eq_set_v @ A4 @ B2 )
=> ( ( ord_less_eq_set_v @ ( F @ B2 ) @ C )
=> ( ! [X3: set_v,Y2: set_v] :
( ( ord_less_eq_set_v @ X3 @ Y2 )
=> ( ord_less_eq_set_v @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_v @ ( F @ A4 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_110_order__eq__refl,axiom,
! [X: set_Product_prod_v_v,Y: set_Product_prod_v_v] :
( ( X = Y )
=> ( ord_le7336532860387713383od_v_v @ X @ Y ) ) ).
% order_eq_refl
thf(fact_111_order__eq__refl,axiom,
! [X: set_v,Y: set_v] :
( ( X = Y )
=> ( ord_less_eq_set_v @ X @ Y ) ) ).
% order_eq_refl
thf(fact_112_ord__eq__le__subst,axiom,
! [A4: set_Product_prod_v_v,F: set_Product_prod_v_v > set_Product_prod_v_v,B2: set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( A4
= ( F @ B2 ) )
=> ( ( ord_le7336532860387713383od_v_v @ B2 @ C )
=> ( ! [X3: set_Product_prod_v_v,Y2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X3 @ Y2 )
=> ( ord_le7336532860387713383od_v_v @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le7336532860387713383od_v_v @ A4 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_113_ord__eq__le__subst,axiom,
! [A4: set_v,F: set_Product_prod_v_v > set_v,B2: set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( A4
= ( F @ B2 ) )
=> ( ( ord_le7336532860387713383od_v_v @ B2 @ C )
=> ( ! [X3: set_Product_prod_v_v,Y2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X3 @ Y2 )
=> ( ord_less_eq_set_v @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_v @ A4 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_114_ord__eq__le__subst,axiom,
! [A4: set_Product_prod_v_v,F: set_v > set_Product_prod_v_v,B2: set_v,C: set_v] :
( ( A4
= ( F @ B2 ) )
=> ( ( ord_less_eq_set_v @ B2 @ C )
=> ( ! [X3: set_v,Y2: set_v] :
( ( ord_less_eq_set_v @ X3 @ Y2 )
=> ( ord_le7336532860387713383od_v_v @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le7336532860387713383od_v_v @ A4 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_115_ord__eq__le__subst,axiom,
! [A4: set_v,F: set_v > set_v,B2: set_v,C: set_v] :
( ( A4
= ( F @ B2 ) )
=> ( ( ord_less_eq_set_v @ B2 @ C )
=> ( ! [X3: set_v,Y2: set_v] :
( ( ord_less_eq_set_v @ X3 @ Y2 )
=> ( ord_less_eq_set_v @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_v @ A4 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_116_ord__le__eq__subst,axiom,
! [A4: set_Product_prod_v_v,B2: set_Product_prod_v_v,F: set_Product_prod_v_v > set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A4 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: set_Product_prod_v_v,Y2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X3 @ Y2 )
=> ( ord_le7336532860387713383od_v_v @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le7336532860387713383od_v_v @ ( F @ A4 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_117_ord__le__eq__subst,axiom,
! [A4: set_Product_prod_v_v,B2: set_Product_prod_v_v,F: set_Product_prod_v_v > set_v,C: set_v] :
( ( ord_le7336532860387713383od_v_v @ A4 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: set_Product_prod_v_v,Y2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X3 @ Y2 )
=> ( ord_less_eq_set_v @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_v @ ( F @ A4 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_118_ord__le__eq__subst,axiom,
! [A4: set_v,B2: set_v,F: set_v > set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( ord_less_eq_set_v @ A4 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: set_v,Y2: set_v] :
( ( ord_less_eq_set_v @ X3 @ Y2 )
=> ( ord_le7336532860387713383od_v_v @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le7336532860387713383od_v_v @ ( F @ A4 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_119_ord__le__eq__subst,axiom,
! [A4: set_v,B2: set_v,F: set_v > set_v,C: set_v] :
( ( ord_less_eq_set_v @ A4 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: set_v,Y2: set_v] :
( ( ord_less_eq_set_v @ X3 @ Y2 )
=> ( ord_less_eq_set_v @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_v @ ( F @ A4 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_120_order__antisym__conv,axiom,
! [Y: set_Product_prod_v_v,X: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ Y @ X )
=> ( ( ord_le7336532860387713383od_v_v @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_121_order__antisym__conv,axiom,
! [Y: set_v,X: set_v] :
( ( ord_less_eq_set_v @ Y @ X )
=> ( ( ord_less_eq_set_v @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_122_bot__set__def,axiom,
( bot_bo723834152578015283od_v_v
= ( collec140062887454715474od_v_v @ bot_bo8461541820394803818_v_v_o ) ) ).
% bot_set_def
thf(fact_123_bot__set__def,axiom,
( bot_bot_set_v
= ( collect_v @ bot_bot_v_o ) ) ).
% bot_set_def
thf(fact_124_graph_Opre__dfss_Ocong,axiom,
sCC_Bl1748261141445803503t_unit = sCC_Bl1748261141445803503t_unit ).
% graph.pre_dfss.cong
thf(fact_125_bot_Oextremum__uniqueI,axiom,
! [A4: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A4 @ bot_bo723834152578015283od_v_v )
=> ( A4 = bot_bo723834152578015283od_v_v ) ) ).
% bot.extremum_uniqueI
thf(fact_126_bot_Oextremum__uniqueI,axiom,
! [A4: set_v] :
( ( ord_less_eq_set_v @ A4 @ bot_bot_set_v )
=> ( A4 = bot_bot_set_v ) ) ).
% bot.extremum_uniqueI
thf(fact_127_bot_Oextremum__unique,axiom,
! [A4: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A4 @ bot_bo723834152578015283od_v_v )
= ( A4 = bot_bo723834152578015283od_v_v ) ) ).
% bot.extremum_unique
thf(fact_128_bot_Oextremum__unique,axiom,
! [A4: set_v] :
( ( ord_less_eq_set_v @ A4 @ bot_bot_set_v )
= ( A4 = bot_bot_set_v ) ) ).
% bot.extremum_unique
thf(fact_129_bot_Oextremum,axiom,
! [A4: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ bot_bo723834152578015283od_v_v @ A4 ) ).
% bot.extremum
thf(fact_130_bot_Oextremum,axiom,
! [A4: set_v] : ( ord_less_eq_set_v @ bot_bot_set_v @ A4 ) ).
% bot.extremum
thf(fact_131_graph_Odfss_Ocong,axiom,
sCC_Bloemen_dfss_v = sCC_Bloemen_dfss_v ).
% graph.dfss.cong
thf(fact_132_ex__in__conv,axiom,
! [A: set_Product_prod_v_v] :
( ( ? [X2: product_prod_v_v] : ( member7453568604450474000od_v_v @ X2 @ A ) )
= ( A != bot_bo723834152578015283od_v_v ) ) ).
% ex_in_conv
thf(fact_133_ex__in__conv,axiom,
! [A: set_v] :
( ( ? [X2: v] : ( member_v @ X2 @ A ) )
= ( A != bot_bot_set_v ) ) ).
% ex_in_conv
thf(fact_134_equals0I,axiom,
! [A: set_Product_prod_v_v] :
( ! [Y2: product_prod_v_v] :
~ ( member7453568604450474000od_v_v @ Y2 @ A )
=> ( A = bot_bo723834152578015283od_v_v ) ) ).
% equals0I
thf(fact_135_equals0I,axiom,
! [A: set_v] :
( ! [Y2: v] :
~ ( member_v @ Y2 @ A )
=> ( A = bot_bot_set_v ) ) ).
% equals0I
thf(fact_136_equals0D,axiom,
! [A: set_Product_prod_v_v,A4: product_prod_v_v] :
( ( A = bot_bo723834152578015283od_v_v )
=> ~ ( member7453568604450474000od_v_v @ A4 @ A ) ) ).
% equals0D
thf(fact_137_equals0D,axiom,
! [A: set_v,A4: v] :
( ( A = bot_bot_set_v )
=> ~ ( member_v @ A4 @ A ) ) ).
% equals0D
thf(fact_138_emptyE,axiom,
! [A4: product_prod_v_v] :
~ ( member7453568604450474000od_v_v @ A4 @ bot_bo723834152578015283od_v_v ) ).
% emptyE
thf(fact_139_emptyE,axiom,
! [A4: v] :
~ ( member_v @ A4 @ bot_bot_set_v ) ).
% emptyE
thf(fact_140_graph_Ounvisited_Ocong,axiom,
sCC_Bl3123350270117520219t_unit = sCC_Bl3123350270117520219t_unit ).
% graph.unvisited.cong
thf(fact_141_is__subscc__def,axiom,
! [S: set_v] :
( ( sCC_Bl5398416737448265317bscc_v @ successors @ S )
= ( ! [X2: v] :
( ( member_v @ X2 @ S )
=> ! [Y3: v] :
( ( member_v @ Y3 @ S )
=> ( sCC_Bl649662514949026229able_v @ successors @ X2 @ Y3 ) ) ) ) ) ).
% is_subscc_def
thf(fact_142__092_060open_062_092_060forall_062n_092_060in_062set_A_Istack_Ae_J_O_Areachable_An_Av_092_060close_062,axiom,
! [X4: v] :
( ( member_v @ X4 @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ ea ) ) )
=> ( sCC_Bl649662514949026229able_v @ successors @ X4 @ va ) ) ).
% \<open>\<forall>n\<in>set (stack e). reachable n v\<close>
thf(fact_143__092_060open_062v_A_092_060in_062_A_092_060S_062_Ae_A_Ihd_A_Istack_Ae_J_J_092_060close_062,axiom,
member_v @ va @ ( sCC_Bl1280885523602775798t_unit @ ea @ ( hd_v @ ( sCC_Bl8828226123343373779t_unit @ ea ) ) ) ).
% \<open>v \<in> \<S> e (hd (stack e))\<close>
thf(fact_144__092_060open_062vsuccs_Ae_Av_A_061_Asuccessors_Av_092_060close_062,axiom,
( ( sCC_Bl3795065053823578884t_unit @ ea @ va )
= ( successors @ va ) ) ).
% \<open>vsuccs e v = successors v\<close>
thf(fact_145_S__reflexive,axiom,
! [E2: sCC_Bl1394983891496994913t_unit,N: v] :
( ( sCC_Bl9196236973127232072t_unit @ successors @ E2 )
=> ( member_v @ N @ ( sCC_Bl1280885523602775798t_unit @ E2 @ N ) ) ) ).
% S_reflexive
thf(fact_146_reachable__re,axiom,
! [X: v,Y: v] :
( ( sCC_Bl649662514949026229able_v @ successors @ X @ Y )
=> ( sCC_Bl770211535891879572_end_v @ successors @ X @ Y ) ) ).
% reachable_re
thf(fact_147_re__reachable,axiom,
! [X: v,Y: v] :
( ( sCC_Bl770211535891879572_end_v @ successors @ X @ Y )
=> ( sCC_Bl649662514949026229able_v @ successors @ X @ Y ) ) ).
% re_reachable
thf(fact_148_local_Owf,axiom,
sCC_Bl9196236973127232072t_unit @ successors @ ea ).
% local.wf
thf(fact_149__092_060open_062v_A_092_060notin_062_Aexplored_Ae_092_060close_062,axiom,
~ ( member_v @ va @ ( sCC_Bl157864678168468314t_unit @ ea ) ) ).
% \<open>v \<notin> explored e\<close>
thf(fact_150__092_060open_062v_A_092_060in_062_Avisited_Ae_092_060close_062,axiom,
member_v @ va @ ( sCC_Bl4645233313691564917t_unit @ ea ) ).
% \<open>v \<in> visited e\<close>
thf(fact_151__092_060open_062stack_Ae_A_092_060noteq_062_A_091_093_092_060close_062,axiom,
( ( sCC_Bl8828226123343373779t_unit @ ea )
!= nil_v ) ).
% \<open>stack e \<noteq> []\<close>
thf(fact_152_reachable__end_Ocases,axiom,
! [A1: v,A2: v] :
( ( sCC_Bl770211535891879572_end_v @ successors @ A1 @ A2 )
=> ( ( A2 != A1 )
=> ~ ! [Y2: v] :
( ( sCC_Bl770211535891879572_end_v @ successors @ A1 @ Y2 )
=> ~ ( member_v @ A2 @ ( successors @ Y2 ) ) ) ) ) ).
% reachable_end.cases
thf(fact_153_re__refl,axiom,
! [X: v] : ( sCC_Bl770211535891879572_end_v @ successors @ X @ X ) ).
% re_refl
thf(fact_154_re__succ,axiom,
! [X: v,Y: v,Z: v] :
( ( sCC_Bl770211535891879572_end_v @ successors @ X @ Y )
=> ( ( member_v @ Z @ ( successors @ Y ) )
=> ( sCC_Bl770211535891879572_end_v @ successors @ X @ Z ) ) ) ).
% re_succ
thf(fact_155_reachable__end_Osimps,axiom,
! [A1: v,A2: v] :
( ( sCC_Bl770211535891879572_end_v @ successors @ A1 @ A2 )
= ( ? [X2: v] :
( ( A1 = X2 )
& ( A2 = X2 ) )
| ? [X2: v,Y3: v,Z3: v] :
( ( A1 = X2 )
& ( A2 = Z3 )
& ( sCC_Bl770211535891879572_end_v @ successors @ X2 @ Y3 )
& ( member_v @ Z3 @ ( successors @ Y3 ) ) ) ) ) ).
% reachable_end.simps
thf(fact_156_succ__re,axiom,
! [Y: v,X: v,Z: v] :
( ( member_v @ Y @ ( successors @ X ) )
=> ( ( sCC_Bl770211535891879572_end_v @ successors @ Y @ Z )
=> ( sCC_Bl770211535891879572_end_v @ successors @ X @ Z ) ) ) ).
% succ_re
thf(fact_157_is__scc__def,axiom,
! [S: set_v] :
( ( sCC_Bloemen_is_scc_v @ successors @ S )
= ( ( S != bot_bot_set_v )
& ( sCC_Bl5398416737448265317bscc_v @ successors @ S )
& ! [S2: set_v] :
( ( ( ord_less_eq_set_v @ S @ S2 )
& ( sCC_Bl5398416737448265317bscc_v @ successors @ S2 ) )
=> ( S2 = S ) ) ) ) ).
% is_scc_def
thf(fact_158_stack__visited,axiom,
! [E2: sCC_Bl1394983891496994913t_unit,N: v] :
( ( sCC_Bl9196236973127232072t_unit @ successors @ E2 )
=> ( ( member_v @ N @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E2 ) ) )
=> ( member_v @ N @ ( sCC_Bl4645233313691564917t_unit @ E2 ) ) ) ) ).
% stack_visited
thf(fact_159_stack__unexplored,axiom,
! [E2: sCC_Bl1394983891496994913t_unit,N: v] :
( ( sCC_Bl9196236973127232072t_unit @ successors @ E2 )
=> ( ( member_v @ N @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E2 ) ) )
=> ~ ( member_v @ N @ ( sCC_Bl157864678168468314t_unit @ E2 ) ) ) ) ).
% stack_unexplored
thf(fact_160_visited__unexplored,axiom,
! [E2: sCC_Bl1394983891496994913t_unit,M: v] :
( ( sCC_Bl9196236973127232072t_unit @ successors @ E2 )
=> ( ( member_v @ M @ ( sCC_Bl4645233313691564917t_unit @ E2 ) )
=> ( ~ ( member_v @ M @ ( sCC_Bl157864678168468314t_unit @ E2 ) )
=> ~ ! [N2: v] :
( ( member_v @ N2 @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E2 ) ) )
=> ~ ( member_v @ M @ ( sCC_Bl1280885523602775798t_unit @ E2 @ N2 ) ) ) ) ) ) ).
% visited_unexplored
thf(fact_161_precedes__refl,axiom,
! [X: product_prod_v_v,Xs: list_P7986770385144383213od_v_v] :
( ( sCC_Bl2026170059108282219od_v_v @ X @ X @ Xs )
= ( member7453568604450474000od_v_v @ X @ ( set_Product_prod_v_v2 @ Xs ) ) ) ).
% precedes_refl
thf(fact_162_precedes__refl,axiom,
! [X: v,Xs: list_v] :
( ( sCC_Bl4022239298816431255edes_v @ X @ X @ Xs )
= ( member_v @ X @ ( set_v2 @ Xs ) ) ) ).
% precedes_refl
thf(fact_163_sclosed,axiom,
! [X4: v] :
( ( member_v @ X4 @ vertices )
=> ( ord_less_eq_set_v @ ( successors @ X4 ) @ vertices ) ) ).
% sclosed
thf(fact_164_True,axiom,
( ( minus_minus_set_v @ ( successors @ va ) @ ( sCC_Bl3795065053823578884t_unit @ ea @ va ) )
= bot_bot_set_v ) ).
% True
thf(fact_165_asm_I2_J,axiom,
member_v @ n @ ( set_v2 @ ( tl_v @ ( sCC_Bl8828226123343373779t_unit @ ea ) ) ) ).
% asm(2)
thf(fact_166_pre__dfss__pre__dfs,axiom,
! [V: v,E2: sCC_Bl1394983891496994913t_unit,W: v] :
( ( sCC_Bl1748261141445803503t_unit @ successors @ V @ E2 )
=> ( ~ ( member_v @ W @ ( sCC_Bl4645233313691564917t_unit @ E2 ) )
=> ( ( member_v @ W @ ( successors @ V ) )
=> ( sCC_Bl36166008131615352t_unit @ successors @ W @ E2 ) ) ) ) ).
% pre_dfss_pre_dfs
thf(fact_167_graph_Oreachable__end_Ocong,axiom,
sCC_Bl770211535891879572_end_v = sCC_Bl770211535891879572_end_v ).
% graph.reachable_end.cong
thf(fact_168_graph_Owf__env_Ocong,axiom,
sCC_Bl9196236973127232072t_unit = sCC_Bl9196236973127232072t_unit ).
% graph.wf_env.cong
thf(fact_169_graph_Ois__subscc_Ocong,axiom,
sCC_Bl5398416737448265317bscc_v = sCC_Bl5398416737448265317bscc_v ).
% graph.is_subscc.cong
thf(fact_170_graph_Ois__scc_Ocong,axiom,
sCC_Bloemen_is_scc_v = sCC_Bloemen_is_scc_v ).
% graph.is_scc.cong
thf(fact_171_precedes__mem_I1_J,axiom,
! [X: product_prod_v_v,Y: product_prod_v_v,Xs: list_P7986770385144383213od_v_v] :
( ( sCC_Bl2026170059108282219od_v_v @ X @ Y @ Xs )
=> ( member7453568604450474000od_v_v @ X @ ( set_Product_prod_v_v2 @ Xs ) ) ) ).
% precedes_mem(1)
thf(fact_172_precedes__mem_I1_J,axiom,
! [X: v,Y: v,Xs: list_v] :
( ( sCC_Bl4022239298816431255edes_v @ X @ Y @ Xs )
=> ( member_v @ X @ ( set_v2 @ Xs ) ) ) ).
% precedes_mem(1)
thf(fact_173_precedes__mem_I2_J,axiom,
! [X: product_prod_v_v,Y: product_prod_v_v,Xs: list_P7986770385144383213od_v_v] :
( ( sCC_Bl2026170059108282219od_v_v @ X @ Y @ Xs )
=> ( member7453568604450474000od_v_v @ Y @ ( set_Product_prod_v_v2 @ Xs ) ) ) ).
% precedes_mem(2)
thf(fact_174_precedes__mem_I2_J,axiom,
! [X: v,Y: v,Xs: list_v] :
( ( sCC_Bl4022239298816431255edes_v @ X @ Y @ Xs )
=> ( member_v @ Y @ ( set_v2 @ Xs ) ) ) ).
% precedes_mem(2)
thf(fact_175_dfs__S__hd__stack_I2_J,axiom,
! [E2: sCC_Bl1394983891496994913t_unit,V: v,E3: sCC_Bl1394983891496994913t_unit,N: v] :
( ( sCC_Bl9196236973127232072t_unit @ successors @ E2 )
=> ( ( sCC_Bl8953792750115413617t_unit @ successors @ V @ E2 @ E3 )
=> ( ( ( sCC_Bl8828226123343373779t_unit @ E2 )
!= nil_v )
=> ( ( member_v @ N @ ( sCC_Bl1280885523602775798t_unit @ E2 @ ( hd_v @ ( sCC_Bl8828226123343373779t_unit @ E2 ) ) ) )
=> ( member_v @ N @ ( sCC_Bl1280885523602775798t_unit @ E3 @ ( hd_v @ ( sCC_Bl8828226123343373779t_unit @ E3 ) ) ) ) ) ) ) ) ).
% dfs_S_hd_stack(2)
thf(fact_176_dfs__S__hd__stack_I1_J,axiom,
! [E2: sCC_Bl1394983891496994913t_unit,V: v,E3: sCC_Bl1394983891496994913t_unit,N: v] :
( ( sCC_Bl9196236973127232072t_unit @ successors @ E2 )
=> ( ( sCC_Bl8953792750115413617t_unit @ successors @ V @ E2 @ E3 )
=> ( ( ( sCC_Bl8828226123343373779t_unit @ E2 )
!= nil_v )
=> ( ( member_v @ N @ ( sCC_Bl1280885523602775798t_unit @ E2 @ ( hd_v @ ( sCC_Bl8828226123343373779t_unit @ E2 ) ) ) )
=> ( ( sCC_Bl8828226123343373779t_unit @ E3 )
!= nil_v ) ) ) ) ) ).
% dfs_S_hd_stack(1)
thf(fact_177_dfs__S__tl__stack_I1_J,axiom,
! [V: v,E2: sCC_Bl1394983891496994913t_unit,E3: sCC_Bl1394983891496994913t_unit] :
( ( sCC_Bl8953792750115413617t_unit @ successors @ V @ E2 @ E3 )
=> ( ( ( sCC_Bl8828226123343373779t_unit @ E2 )
!= nil_v )
=> ( ( sCC_Bl8828226123343373779t_unit @ E3 )
!= nil_v ) ) ) ).
% dfs_S_tl_stack(1)
thf(fact_178_unite__subscc,axiom,
! [V: v,E2: sCC_Bl1394983891496994913t_unit,W: v] :
( ( sCC_Bl1748261141445803503t_unit @ successors @ V @ E2 )
=> ( ( member_v @ W @ ( successors @ V ) )
=> ( ~ ( member_v @ W @ ( sCC_Bl3795065053823578884t_unit @ E2 @ V ) )
=> ( ( member_v @ W @ ( sCC_Bl4645233313691564917t_unit @ E2 ) )
=> ( ~ ( member_v @ W @ ( sCC_Bl157864678168468314t_unit @ E2 ) )
=> ( sCC_Bl5398416737448265317bscc_v @ successors @ ( sCC_Bl1280885523602775798t_unit @ ( sCC_Bloemen_unite_v @ V @ W @ E2 ) @ ( hd_v @ ( sCC_Bl8828226123343373779t_unit @ ( sCC_Bloemen_unite_v @ V @ W @ E2 ) ) ) ) ) ) ) ) ) ) ).
% unite_subscc
thf(fact_179_unite__sub__env,axiom,
! [V: v,E2: sCC_Bl1394983891496994913t_unit,W: v] :
( ( sCC_Bl1748261141445803503t_unit @ successors @ V @ E2 )
=> ( ( member_v @ W @ ( successors @ V ) )
=> ( ~ ( member_v @ W @ ( sCC_Bl3795065053823578884t_unit @ E2 @ V ) )
=> ( ( member_v @ W @ ( sCC_Bl4645233313691564917t_unit @ E2 ) )
=> ( ~ ( member_v @ W @ ( sCC_Bl157864678168468314t_unit @ E2 ) )
=> ( sCC_Bl5768913643336123637t_unit @ E2 @ ( sCC_Bloemen_unite_v @ V @ W @ E2 ) ) ) ) ) ) ) ).
% unite_sub_env
thf(fact_180_unite__wf__env,axiom,
! [V: v,E2: sCC_Bl1394983891496994913t_unit,W: v] :
( ( sCC_Bl1748261141445803503t_unit @ successors @ V @ E2 )
=> ( ( member_v @ W @ ( successors @ V ) )
=> ( ~ ( member_v @ W @ ( sCC_Bl3795065053823578884t_unit @ E2 @ V ) )
=> ( ( member_v @ W @ ( sCC_Bl4645233313691564917t_unit @ E2 ) )
=> ( ~ ( member_v @ W @ ( sCC_Bl157864678168468314t_unit @ E2 ) )
=> ( sCC_Bl9196236973127232072t_unit @ successors @ ( sCC_Bloemen_unite_v @ V @ W @ E2 ) ) ) ) ) ) ) ).
% unite_wf_env
thf(fact_181_set__empty,axiom,
! [Xs: list_P7986770385144383213od_v_v] :
( ( ( set_Product_prod_v_v2 @ Xs )
= bot_bo723834152578015283od_v_v )
= ( Xs = nil_Product_prod_v_v ) ) ).
% set_empty
thf(fact_182_set__empty,axiom,
! [Xs: list_v] :
( ( ( set_v2 @ Xs )
= bot_bot_set_v )
= ( Xs = nil_v ) ) ).
% set_empty
thf(fact_183_set__empty2,axiom,
! [Xs: list_P7986770385144383213od_v_v] :
( ( bot_bo723834152578015283od_v_v
= ( set_Product_prod_v_v2 @ Xs ) )
= ( Xs = nil_Product_prod_v_v ) ) ).
% set_empty2
thf(fact_184_set__empty2,axiom,
! [Xs: list_v] :
( ( bot_bot_set_v
= ( set_v2 @ Xs ) )
= ( Xs = nil_v ) ) ).
% set_empty2
thf(fact_185__092_060open_062_092_060forall_062w_092_060in_062successors_Av_O_Aw_A_092_060in_062_Aexplored_Ae_A_092_060union_062_A_092_060S_062_Ae_A_Ihd_A_Istack_Ae_J_J_092_060close_062,axiom,
! [X4: v] :
( ( member_v @ X4 @ ( successors @ va ) )
=> ( member_v @ X4 @ ( sup_sup_set_v @ ( sCC_Bl157864678168468314t_unit @ ea ) @ ( sCC_Bl1280885523602775798t_unit @ ea @ ( hd_v @ ( sCC_Bl8828226123343373779t_unit @ ea ) ) ) ) ) ) ).
% \<open>\<forall>w\<in>successors v. w \<in> explored e \<union> \<S> e (hd (stack e))\<close>
thf(fact_186_init__env__pre__dfs,axiom,
! [V: v] : ( sCC_Bl36166008131615352t_unit @ successors @ V @ ( sCC_Bl7693227186847904995_env_v @ V ) ) ).
% init_env_pre_dfs
thf(fact_187_Un__iff,axiom,
! [C: v,A: set_v,B: set_v] :
( ( member_v @ C @ ( sup_sup_set_v @ A @ B ) )
= ( ( member_v @ C @ A )
| ( member_v @ C @ B ) ) ) ).
% Un_iff
thf(fact_188_Un__iff,axiom,
! [C: product_prod_v_v,A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ C @ ( sup_su414716646722978715od_v_v @ A @ B ) )
= ( ( member7453568604450474000od_v_v @ C @ A )
| ( member7453568604450474000od_v_v @ C @ B ) ) ) ).
% Un_iff
thf(fact_189_UnCI,axiom,
! [C: v,B: set_v,A: set_v] :
( ( ~ ( member_v @ C @ B )
=> ( member_v @ C @ A ) )
=> ( member_v @ C @ ( sup_sup_set_v @ A @ B ) ) ) ).
% UnCI
thf(fact_190_UnCI,axiom,
! [C: product_prod_v_v,B: set_Product_prod_v_v,A: set_Product_prod_v_v] :
( ( ~ ( member7453568604450474000od_v_v @ C @ B )
=> ( member7453568604450474000od_v_v @ C @ A ) )
=> ( member7453568604450474000od_v_v @ C @ ( sup_su414716646722978715od_v_v @ A @ B ) ) ) ).
% UnCI
thf(fact_191_Diff__idemp,axiom,
! [A: set_v,B: set_v] :
( ( minus_minus_set_v @ ( minus_minus_set_v @ A @ B ) @ B )
= ( minus_minus_set_v @ A @ B ) ) ).
% Diff_idemp
thf(fact_192_Diff__iff,axiom,
! [C: product_prod_v_v,A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ C @ ( minus_4183494784930505774od_v_v @ A @ B ) )
= ( ( member7453568604450474000od_v_v @ C @ A )
& ~ ( member7453568604450474000od_v_v @ C @ B ) ) ) ).
% Diff_iff
thf(fact_193_Diff__iff,axiom,
! [C: v,A: set_v,B: set_v] :
( ( member_v @ C @ ( minus_minus_set_v @ A @ B ) )
= ( ( member_v @ C @ A )
& ~ ( member_v @ C @ B ) ) ) ).
% Diff_iff
thf(fact_194_DiffI,axiom,
! [C: product_prod_v_v,A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ C @ A )
=> ( ~ ( member7453568604450474000od_v_v @ C @ B )
=> ( member7453568604450474000od_v_v @ C @ ( minus_4183494784930505774od_v_v @ A @ B ) ) ) ) ).
% DiffI
thf(fact_195_DiffI,axiom,
! [C: v,A: set_v,B: set_v] :
( ( member_v @ C @ A )
=> ( ~ ( member_v @ C @ B )
=> ( member_v @ C @ ( minus_minus_set_v @ A @ B ) ) ) ) ).
% DiffI
thf(fact_196_reachable__visited,axiom,
! [E2: sCC_Bl1394983891496994913t_unit,V: v,W: v] :
( ( sCC_Bl9196236973127232072t_unit @ successors @ E2 )
=> ( ( member_v @ V @ ( sCC_Bl4645233313691564917t_unit @ E2 ) )
=> ( ( sCC_Bl649662514949026229able_v @ successors @ V @ W )
=> ( ! [X3: v] :
( ( member_v @ X3 @ ( sCC_Bl4645233313691564917t_unit @ E2 ) )
=> ! [Xa: v] :
( ( member_v @ Xa @ ( minus_minus_set_v @ ( successors @ X3 ) @ ( sCC_Bl3795065053823578884t_unit @ E2 @ X3 ) ) )
=> ( ( sCC_Bl649662514949026229able_v @ successors @ V @ X3 )
=> ~ ( sCC_Bl649662514949026229able_v @ successors @ Xa @ W ) ) ) )
=> ( member_v @ W @ ( sCC_Bl4645233313691564917t_unit @ E2 ) ) ) ) ) ) ).
% reachable_visited
thf(fact_197_Un__empty,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ( sup_su414716646722978715od_v_v @ A @ B )
= bot_bo723834152578015283od_v_v )
= ( ( A = bot_bo723834152578015283od_v_v )
& ( B = bot_bo723834152578015283od_v_v ) ) ) ).
% Un_empty
thf(fact_198_Un__empty,axiom,
! [A: set_v,B: set_v] :
( ( ( sup_sup_set_v @ A @ B )
= bot_bot_set_v )
= ( ( A = bot_bot_set_v )
& ( B = bot_bot_set_v ) ) ) ).
% Un_empty
thf(fact_199_Diff__cancel,axiom,
! [A: set_Product_prod_v_v] :
( ( minus_4183494784930505774od_v_v @ A @ A )
= bot_bo723834152578015283od_v_v ) ).
% Diff_cancel
thf(fact_200_Diff__cancel,axiom,
! [A: set_v] :
( ( minus_minus_set_v @ A @ A )
= bot_bot_set_v ) ).
% Diff_cancel
thf(fact_201_empty__Diff,axiom,
! [A: set_Product_prod_v_v] :
( ( minus_4183494784930505774od_v_v @ bot_bo723834152578015283od_v_v @ A )
= bot_bo723834152578015283od_v_v ) ).
% empty_Diff
thf(fact_202_empty__Diff,axiom,
! [A: set_v] :
( ( minus_minus_set_v @ bot_bot_set_v @ A )
= bot_bot_set_v ) ).
% empty_Diff
thf(fact_203_Diff__empty,axiom,
! [A: set_Product_prod_v_v] :
( ( minus_4183494784930505774od_v_v @ A @ bot_bo723834152578015283od_v_v )
= A ) ).
% Diff_empty
thf(fact_204_Diff__empty,axiom,
! [A: set_v] :
( ( minus_minus_set_v @ A @ bot_bot_set_v )
= A ) ).
% Diff_empty
thf(fact_205_Un__subset__iff,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v,C2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ ( sup_su414716646722978715od_v_v @ A @ B ) @ C2 )
= ( ( ord_le7336532860387713383od_v_v @ A @ C2 )
& ( ord_le7336532860387713383od_v_v @ B @ C2 ) ) ) ).
% Un_subset_iff
thf(fact_206_Un__subset__iff,axiom,
! [A: set_v,B: set_v,C2: set_v] :
( ( ord_less_eq_set_v @ ( sup_sup_set_v @ A @ B ) @ C2 )
= ( ( ord_less_eq_set_v @ A @ C2 )
& ( ord_less_eq_set_v @ B @ C2 ) ) ) ).
% Un_subset_iff
thf(fact_207_Un__Diff__cancel2,axiom,
! [B: set_Product_prod_v_v,A: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ ( minus_4183494784930505774od_v_v @ B @ A ) @ A )
= ( sup_su414716646722978715od_v_v @ B @ A ) ) ).
% Un_Diff_cancel2
thf(fact_208_Un__Diff__cancel2,axiom,
! [B: set_v,A: set_v] :
( ( sup_sup_set_v @ ( minus_minus_set_v @ B @ A ) @ A )
= ( sup_sup_set_v @ B @ A ) ) ).
% Un_Diff_cancel2
thf(fact_209_Un__Diff__cancel,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ A @ ( minus_4183494784930505774od_v_v @ B @ A ) )
= ( sup_su414716646722978715od_v_v @ A @ B ) ) ).
% Un_Diff_cancel
thf(fact_210_Un__Diff__cancel,axiom,
! [A: set_v,B: set_v] :
( ( sup_sup_set_v @ A @ ( minus_minus_set_v @ B @ A ) )
= ( sup_sup_set_v @ A @ B ) ) ).
% Un_Diff_cancel
thf(fact_211__092_060open_062_092_060And_062b_Aa_O_A_092_060lbrakk_062a_A_092_060in_062_A_092_060S_062_Ae_Av_059_Ab_A_092_060in_062_Asuccessors_Aa_A_N_Avsuccs_Ae_Aa_092_060rbrakk_062_A_092_060Longrightarrow_062_AFalse_092_060close_062,axiom,
! [A4: v,B2: v] :
( ( member_v @ A4 @ ( sCC_Bl1280885523602775798t_unit @ ea @ va ) )
=> ~ ( member_v @ B2 @ ( minus_minus_set_v @ ( successors @ A4 ) @ ( sCC_Bl3795065053823578884t_unit @ ea @ A4 ) ) ) ) ).
% \<open>\<And>b a. \<lbrakk>a \<in> \<S> e v; b \<in> successors a - vsuccs e a\<rbrakk> \<Longrightarrow> False\<close>
thf(fact_212_dfs__S__tl__stack_I2_J,axiom,
! [V: v,E2: sCC_Bl1394983891496994913t_unit,E3: sCC_Bl1394983891496994913t_unit] :
( ( sCC_Bl8953792750115413617t_unit @ successors @ V @ E2 @ E3 )
=> ( ( ( sCC_Bl8828226123343373779t_unit @ E2 )
!= nil_v )
=> ! [X4: v] :
( ( member_v @ X4 @ ( set_v2 @ ( tl_v @ ( sCC_Bl8828226123343373779t_unit @ E3 ) ) ) )
=> ( ( sCC_Bl1280885523602775798t_unit @ E3 @ X4 )
= ( sCC_Bl1280885523602775798t_unit @ E2 @ X4 ) ) ) ) ) ).
% dfs_S_tl_stack(2)
thf(fact_213_stack__class,axiom,
! [E2: sCC_Bl1394983891496994913t_unit,N: v,M: v] :
( ( sCC_Bl9196236973127232072t_unit @ successors @ E2 )
=> ( ( member_v @ N @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E2 ) ) )
=> ( ( member_v @ M @ ( sCC_Bl1280885523602775798t_unit @ E2 @ N ) )
=> ( member_v @ M @ ( minus_minus_set_v @ ( sCC_Bl4645233313691564917t_unit @ E2 ) @ ( sCC_Bl157864678168468314t_unit @ E2 ) ) ) ) ) ) ).
% stack_class
thf(fact_214_Diff__eq__empty__iff,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ( minus_4183494784930505774od_v_v @ A @ B )
= bot_bo723834152578015283od_v_v )
= ( ord_le7336532860387713383od_v_v @ A @ B ) ) ).
% Diff_eq_empty_iff
thf(fact_215_Diff__eq__empty__iff,axiom,
! [A: set_v,B: set_v] :
( ( ( minus_minus_set_v @ A @ B )
= bot_bot_set_v )
= ( ord_less_eq_set_v @ A @ B ) ) ).
% Diff_eq_empty_iff
thf(fact_216_unite__S__tl,axiom,
! [E2: sCC_Bl1394983891496994913t_unit,W: v,V: v,N: v] :
( ( sCC_Bl9196236973127232072t_unit @ successors @ E2 )
=> ( ( member_v @ W @ ( successors @ V ) )
=> ( ~ ( member_v @ W @ ( sCC_Bl3795065053823578884t_unit @ E2 @ V ) )
=> ( ( member_v @ W @ ( sCC_Bl4645233313691564917t_unit @ E2 ) )
=> ( ~ ( member_v @ W @ ( sCC_Bl157864678168468314t_unit @ E2 ) )
=> ( ( member_v @ N @ ( set_v2 @ ( tl_v @ ( sCC_Bl8828226123343373779t_unit @ ( sCC_Bloemen_unite_v @ V @ W @ E2 ) ) ) ) )
=> ( ( sCC_Bl1280885523602775798t_unit @ ( sCC_Bloemen_unite_v @ V @ W @ E2 ) @ N )
= ( sCC_Bl1280885523602775798t_unit @ E2 @ N ) ) ) ) ) ) ) ) ).
% unite_S_tl
thf(fact_217_graph__axioms,axiom,
sCC_Bloemen_graph_v @ vertices @ successors ).
% graph_axioms
thf(fact_218_Diff__subset__conv,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v,C2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ ( minus_4183494784930505774od_v_v @ A @ B ) @ C2 )
= ( ord_le7336532860387713383od_v_v @ A @ ( sup_su414716646722978715od_v_v @ B @ C2 ) ) ) ).
% Diff_subset_conv
thf(fact_219_Diff__subset__conv,axiom,
! [A: set_v,B: set_v,C2: set_v] :
( ( ord_less_eq_set_v @ ( minus_minus_set_v @ A @ B ) @ C2 )
= ( ord_less_eq_set_v @ A @ ( sup_sup_set_v @ B @ C2 ) ) ) ).
% Diff_subset_conv
thf(fact_220_Diff__partition,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ B )
=> ( ( sup_su414716646722978715od_v_v @ A @ ( minus_4183494784930505774od_v_v @ B @ A ) )
= B ) ) ).
% Diff_partition
thf(fact_221_Diff__partition,axiom,
! [A: set_v,B: set_v] :
( ( ord_less_eq_set_v @ A @ B )
=> ( ( sup_sup_set_v @ A @ ( minus_minus_set_v @ B @ A ) )
= B ) ) ).
% Diff_partition
thf(fact_222_Un__left__commute,axiom,
! [A: set_v,B: set_v,C2: set_v] :
( ( sup_sup_set_v @ A @ ( sup_sup_set_v @ B @ C2 ) )
= ( sup_sup_set_v @ B @ ( sup_sup_set_v @ A @ C2 ) ) ) ).
% Un_left_commute
thf(fact_223_Un__left__commute,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v,C2: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ A @ ( sup_su414716646722978715od_v_v @ B @ C2 ) )
= ( sup_su414716646722978715od_v_v @ B @ ( sup_su414716646722978715od_v_v @ A @ C2 ) ) ) ).
% Un_left_commute
thf(fact_224_Un__left__absorb,axiom,
! [A: set_v,B: set_v] :
( ( sup_sup_set_v @ A @ ( sup_sup_set_v @ A @ B ) )
= ( sup_sup_set_v @ A @ B ) ) ).
% Un_left_absorb
thf(fact_225_Un__left__absorb,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ A @ ( sup_su414716646722978715od_v_v @ A @ B ) )
= ( sup_su414716646722978715od_v_v @ A @ B ) ) ).
% Un_left_absorb
thf(fact_226_Un__commute,axiom,
( sup_sup_set_v
= ( ^ [A6: set_v,B4: set_v] : ( sup_sup_set_v @ B4 @ A6 ) ) ) ).
% Un_commute
thf(fact_227_Un__commute,axiom,
( sup_su414716646722978715od_v_v
= ( ^ [A6: set_Product_prod_v_v,B4: set_Product_prod_v_v] : ( sup_su414716646722978715od_v_v @ B4 @ A6 ) ) ) ).
% Un_commute
thf(fact_228_Un__absorb,axiom,
! [A: set_v] :
( ( sup_sup_set_v @ A @ A )
= A ) ).
% Un_absorb
thf(fact_229_Un__absorb,axiom,
! [A: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ A @ A )
= A ) ).
% Un_absorb
thf(fact_230_Un__assoc,axiom,
! [A: set_v,B: set_v,C2: set_v] :
( ( sup_sup_set_v @ ( sup_sup_set_v @ A @ B ) @ C2 )
= ( sup_sup_set_v @ A @ ( sup_sup_set_v @ B @ C2 ) ) ) ).
% Un_assoc
thf(fact_231_Un__assoc,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v,C2: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ ( sup_su414716646722978715od_v_v @ A @ B ) @ C2 )
= ( sup_su414716646722978715od_v_v @ A @ ( sup_su414716646722978715od_v_v @ B @ C2 ) ) ) ).
% Un_assoc
thf(fact_232_ball__Un,axiom,
! [A: set_v,B: set_v,P: v > $o] :
( ( ! [X2: v] :
( ( member_v @ X2 @ ( sup_sup_set_v @ A @ B ) )
=> ( P @ X2 ) ) )
= ( ! [X2: v] :
( ( member_v @ X2 @ A )
=> ( P @ X2 ) )
& ! [X2: v] :
( ( member_v @ X2 @ B )
=> ( P @ X2 ) ) ) ) ).
% ball_Un
thf(fact_233_ball__Un,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v,P: product_prod_v_v > $o] :
( ( ! [X2: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X2 @ ( sup_su414716646722978715od_v_v @ A @ B ) )
=> ( P @ X2 ) ) )
= ( ! [X2: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X2 @ A )
=> ( P @ X2 ) )
& ! [X2: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X2 @ B )
=> ( P @ X2 ) ) ) ) ).
% ball_Un
thf(fact_234_Un__Diff,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v,C2: set_Product_prod_v_v] :
( ( minus_4183494784930505774od_v_v @ ( sup_su414716646722978715od_v_v @ A @ B ) @ C2 )
= ( sup_su414716646722978715od_v_v @ ( minus_4183494784930505774od_v_v @ A @ C2 ) @ ( minus_4183494784930505774od_v_v @ B @ C2 ) ) ) ).
% Un_Diff
thf(fact_235_Un__Diff,axiom,
! [A: set_v,B: set_v,C2: set_v] :
( ( minus_minus_set_v @ ( sup_sup_set_v @ A @ B ) @ C2 )
= ( sup_sup_set_v @ ( minus_minus_set_v @ A @ C2 ) @ ( minus_minus_set_v @ B @ C2 ) ) ) ).
% Un_Diff
thf(fact_236_bex__Un,axiom,
! [A: set_v,B: set_v,P: v > $o] :
( ( ? [X2: v] :
( ( member_v @ X2 @ ( sup_sup_set_v @ A @ B ) )
& ( P @ X2 ) ) )
= ( ? [X2: v] :
( ( member_v @ X2 @ A )
& ( P @ X2 ) )
| ? [X2: v] :
( ( member_v @ X2 @ B )
& ( P @ X2 ) ) ) ) ).
% bex_Un
thf(fact_237_bex__Un,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v,P: product_prod_v_v > $o] :
( ( ? [X2: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X2 @ ( sup_su414716646722978715od_v_v @ A @ B ) )
& ( P @ X2 ) ) )
= ( ? [X2: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X2 @ A )
& ( P @ X2 ) )
| ? [X2: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X2 @ B )
& ( P @ X2 ) ) ) ) ).
% bex_Un
thf(fact_238_DiffD2,axiom,
! [C: product_prod_v_v,A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ C @ ( minus_4183494784930505774od_v_v @ A @ B ) )
=> ~ ( member7453568604450474000od_v_v @ C @ B ) ) ).
% DiffD2
thf(fact_239_DiffD2,axiom,
! [C: v,A: set_v,B: set_v] :
( ( member_v @ C @ ( minus_minus_set_v @ A @ B ) )
=> ~ ( member_v @ C @ B ) ) ).
% DiffD2
thf(fact_240_DiffD1,axiom,
! [C: product_prod_v_v,A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ C @ ( minus_4183494784930505774od_v_v @ A @ B ) )
=> ( member7453568604450474000od_v_v @ C @ A ) ) ).
% DiffD1
thf(fact_241_DiffD1,axiom,
! [C: v,A: set_v,B: set_v] :
( ( member_v @ C @ ( minus_minus_set_v @ A @ B ) )
=> ( member_v @ C @ A ) ) ).
% DiffD1
thf(fact_242_DiffE,axiom,
! [C: product_prod_v_v,A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ C @ ( minus_4183494784930505774od_v_v @ A @ B ) )
=> ~ ( ( member7453568604450474000od_v_v @ C @ A )
=> ( member7453568604450474000od_v_v @ C @ B ) ) ) ).
% DiffE
thf(fact_243_DiffE,axiom,
! [C: v,A: set_v,B: set_v] :
( ( member_v @ C @ ( minus_minus_set_v @ A @ B ) )
=> ~ ( ( member_v @ C @ A )
=> ( member_v @ C @ B ) ) ) ).
% DiffE
thf(fact_244_UnI2,axiom,
! [C: v,B: set_v,A: set_v] :
( ( member_v @ C @ B )
=> ( member_v @ C @ ( sup_sup_set_v @ A @ B ) ) ) ).
% UnI2
thf(fact_245_UnI2,axiom,
! [C: product_prod_v_v,B: set_Product_prod_v_v,A: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ C @ B )
=> ( member7453568604450474000od_v_v @ C @ ( sup_su414716646722978715od_v_v @ A @ B ) ) ) ).
% UnI2
thf(fact_246_UnI1,axiom,
! [C: v,A: set_v,B: set_v] :
( ( member_v @ C @ A )
=> ( member_v @ C @ ( sup_sup_set_v @ A @ B ) ) ) ).
% UnI1
thf(fact_247_UnI1,axiom,
! [C: product_prod_v_v,A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ C @ A )
=> ( member7453568604450474000od_v_v @ C @ ( sup_su414716646722978715od_v_v @ A @ B ) ) ) ).
% UnI1
thf(fact_248_UnE,axiom,
! [C: v,A: set_v,B: set_v] :
( ( member_v @ C @ ( sup_sup_set_v @ A @ B ) )
=> ( ~ ( member_v @ C @ A )
=> ( member_v @ C @ B ) ) ) ).
% UnE
thf(fact_249_UnE,axiom,
! [C: product_prod_v_v,A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ C @ ( sup_su414716646722978715od_v_v @ A @ B ) )
=> ( ~ ( member7453568604450474000od_v_v @ C @ A )
=> ( member7453568604450474000od_v_v @ C @ B ) ) ) ).
% UnE
thf(fact_250_list_Osel_I2_J,axiom,
( ( tl_v @ nil_v )
= nil_v ) ).
% list.sel(2)
thf(fact_251_graph_Opre__dfs_Ocong,axiom,
sCC_Bl36166008131615352t_unit = sCC_Bl36166008131615352t_unit ).
% graph.pre_dfs.cong
thf(fact_252_Un__empty__right,axiom,
! [A: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ A @ bot_bo723834152578015283od_v_v )
= A ) ).
% Un_empty_right
thf(fact_253_Un__empty__right,axiom,
! [A: set_v] :
( ( sup_sup_set_v @ A @ bot_bot_set_v )
= A ) ).
% Un_empty_right
thf(fact_254_Un__empty__left,axiom,
! [B: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ bot_bo723834152578015283od_v_v @ B )
= B ) ).
% Un_empty_left
thf(fact_255_Un__empty__left,axiom,
! [B: set_v] :
( ( sup_sup_set_v @ bot_bot_set_v @ B )
= B ) ).
% Un_empty_left
thf(fact_256_Un__mono,axiom,
! [A: set_Product_prod_v_v,C2: set_Product_prod_v_v,B: set_Product_prod_v_v,D: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ C2 )
=> ( ( ord_le7336532860387713383od_v_v @ B @ D )
=> ( ord_le7336532860387713383od_v_v @ ( sup_su414716646722978715od_v_v @ A @ B ) @ ( sup_su414716646722978715od_v_v @ C2 @ D ) ) ) ) ).
% Un_mono
thf(fact_257_Un__mono,axiom,
! [A: set_v,C2: set_v,B: set_v,D: set_v] :
( ( ord_less_eq_set_v @ A @ C2 )
=> ( ( ord_less_eq_set_v @ B @ D )
=> ( ord_less_eq_set_v @ ( sup_sup_set_v @ A @ B ) @ ( sup_sup_set_v @ C2 @ D ) ) ) ) ).
% Un_mono
thf(fact_258_Un__least,axiom,
! [A: set_Product_prod_v_v,C2: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ C2 )
=> ( ( ord_le7336532860387713383od_v_v @ B @ C2 )
=> ( ord_le7336532860387713383od_v_v @ ( sup_su414716646722978715od_v_v @ A @ B ) @ C2 ) ) ) ).
% Un_least
thf(fact_259_Un__least,axiom,
! [A: set_v,C2: set_v,B: set_v] :
( ( ord_less_eq_set_v @ A @ C2 )
=> ( ( ord_less_eq_set_v @ B @ C2 )
=> ( ord_less_eq_set_v @ ( sup_sup_set_v @ A @ B ) @ C2 ) ) ) ).
% Un_least
thf(fact_260_Un__upper1,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ A @ ( sup_su414716646722978715od_v_v @ A @ B ) ) ).
% Un_upper1
thf(fact_261_Un__upper1,axiom,
! [A: set_v,B: set_v] : ( ord_less_eq_set_v @ A @ ( sup_sup_set_v @ A @ B ) ) ).
% Un_upper1
thf(fact_262_Un__upper2,axiom,
! [B: set_Product_prod_v_v,A: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ B @ ( sup_su414716646722978715od_v_v @ A @ B ) ) ).
% Un_upper2
thf(fact_263_Un__upper2,axiom,
! [B: set_v,A: set_v] : ( ord_less_eq_set_v @ B @ ( sup_sup_set_v @ A @ B ) ) ).
% Un_upper2
thf(fact_264_Un__absorb1,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ B )
=> ( ( sup_su414716646722978715od_v_v @ A @ B )
= B ) ) ).
% Un_absorb1
thf(fact_265_Un__absorb1,axiom,
! [A: set_v,B: set_v] :
( ( ord_less_eq_set_v @ A @ B )
=> ( ( sup_sup_set_v @ A @ B )
= B ) ) ).
% Un_absorb1
thf(fact_266_Un__absorb2,axiom,
! [B: set_Product_prod_v_v,A: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ B @ A )
=> ( ( sup_su414716646722978715od_v_v @ A @ B )
= A ) ) ).
% Un_absorb2
thf(fact_267_Un__absorb2,axiom,
! [B: set_v,A: set_v] :
( ( ord_less_eq_set_v @ B @ A )
=> ( ( sup_sup_set_v @ A @ B )
= A ) ) ).
% Un_absorb2
thf(fact_268_subset__UnE,axiom,
! [C2: set_Product_prod_v_v,A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ C2 @ ( sup_su414716646722978715od_v_v @ A @ B ) )
=> ~ ! [A7: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A7 @ A )
=> ! [B5: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ B5 @ B )
=> ( C2
!= ( sup_su414716646722978715od_v_v @ A7 @ B5 ) ) ) ) ) ).
% subset_UnE
thf(fact_269_subset__UnE,axiom,
! [C2: set_v,A: set_v,B: set_v] :
( ( ord_less_eq_set_v @ C2 @ ( sup_sup_set_v @ A @ B ) )
=> ~ ! [A7: set_v] :
( ( ord_less_eq_set_v @ A7 @ A )
=> ! [B5: set_v] :
( ( ord_less_eq_set_v @ B5 @ B )
=> ( C2
!= ( sup_sup_set_v @ A7 @ B5 ) ) ) ) ) ).
% subset_UnE
thf(fact_270_subset__Un__eq,axiom,
( ord_le7336532860387713383od_v_v
= ( ^ [A6: set_Product_prod_v_v,B4: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ A6 @ B4 )
= B4 ) ) ) ).
% subset_Un_eq
thf(fact_271_subset__Un__eq,axiom,
( ord_less_eq_set_v
= ( ^ [A6: set_v,B4: set_v] :
( ( sup_sup_set_v @ A6 @ B4 )
= B4 ) ) ) ).
% subset_Un_eq
thf(fact_272_Diff__mono,axiom,
! [A: set_Product_prod_v_v,C2: set_Product_prod_v_v,D: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ C2 )
=> ( ( ord_le7336532860387713383od_v_v @ D @ B )
=> ( ord_le7336532860387713383od_v_v @ ( minus_4183494784930505774od_v_v @ A @ B ) @ ( minus_4183494784930505774od_v_v @ C2 @ D ) ) ) ) ).
% Diff_mono
thf(fact_273_Diff__mono,axiom,
! [A: set_v,C2: set_v,D: set_v,B: set_v] :
( ( ord_less_eq_set_v @ A @ C2 )
=> ( ( ord_less_eq_set_v @ D @ B )
=> ( ord_less_eq_set_v @ ( minus_minus_set_v @ A @ B ) @ ( minus_minus_set_v @ C2 @ D ) ) ) ) ).
% Diff_mono
thf(fact_274_Diff__subset,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ ( minus_4183494784930505774od_v_v @ A @ B ) @ A ) ).
% Diff_subset
thf(fact_275_Diff__subset,axiom,
! [A: set_v,B: set_v] : ( ord_less_eq_set_v @ ( minus_minus_set_v @ A @ B ) @ A ) ).
% Diff_subset
thf(fact_276_double__diff,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v,C2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ B )
=> ( ( ord_le7336532860387713383od_v_v @ B @ C2 )
=> ( ( minus_4183494784930505774od_v_v @ B @ ( minus_4183494784930505774od_v_v @ C2 @ A ) )
= A ) ) ) ).
% double_diff
thf(fact_277_double__diff,axiom,
! [A: set_v,B: set_v,C2: set_v] :
( ( ord_less_eq_set_v @ A @ B )
=> ( ( ord_less_eq_set_v @ B @ C2 )
=> ( ( minus_minus_set_v @ B @ ( minus_minus_set_v @ C2 @ A ) )
= A ) ) ) ).
% double_diff
thf(fact_278_list_Oset__sel_I2_J,axiom,
! [A4: list_P7986770385144383213od_v_v,X: product_prod_v_v] :
( ( A4 != nil_Product_prod_v_v )
=> ( ( member7453568604450474000od_v_v @ X @ ( set_Product_prod_v_v2 @ ( tl_Product_prod_v_v @ A4 ) ) )
=> ( member7453568604450474000od_v_v @ X @ ( set_Product_prod_v_v2 @ A4 ) ) ) ) ).
% list.set_sel(2)
thf(fact_279_list_Oset__sel_I2_J,axiom,
! [A4: list_v,X: v] :
( ( A4 != nil_v )
=> ( ( member_v @ X @ ( set_v2 @ ( tl_v @ A4 ) ) )
=> ( member_v @ X @ ( set_v2 @ A4 ) ) ) ) ).
% list.set_sel(2)
thf(fact_280_list_Oexpand,axiom,
! [List: list_v,List2: list_v] :
( ( ( List = nil_v )
= ( List2 = nil_v ) )
=> ( ( ( List != nil_v )
=> ( ( List2 != nil_v )
=> ( ( ( hd_v @ List )
= ( hd_v @ List2 ) )
& ( ( tl_v @ List )
= ( tl_v @ List2 ) ) ) ) )
=> ( List = List2 ) ) ) ).
% list.expand
thf(fact_281_subset__code_I1_J,axiom,
! [Xs: list_P7986770385144383213od_v_v,B: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ ( set_Product_prod_v_v2 @ Xs ) @ B )
= ( ! [X2: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X2 @ ( set_Product_prod_v_v2 @ Xs ) )
=> ( member7453568604450474000od_v_v @ X2 @ B ) ) ) ) ).
% subset_code(1)
thf(fact_282_subset__code_I1_J,axiom,
! [Xs: list_v,B: set_v] :
( ( ord_less_eq_set_v @ ( set_v2 @ Xs ) @ B )
= ( ! [X2: v] :
( ( member_v @ X2 @ ( set_v2 @ Xs ) )
=> ( member_v @ X2 @ B ) ) ) ) ).
% subset_code(1)
thf(fact_283_empty__set,axiom,
( bot_bo723834152578015283od_v_v
= ( set_Product_prod_v_v2 @ nil_Product_prod_v_v ) ) ).
% empty_set
thf(fact_284_empty__set,axiom,
( bot_bot_set_v
= ( set_v2 @ nil_v ) ) ).
% empty_set
thf(fact_285_list_Oset__sel_I1_J,axiom,
! [A4: list_P7986770385144383213od_v_v] :
( ( A4 != nil_Product_prod_v_v )
=> ( member7453568604450474000od_v_v @ ( hd_Product_prod_v_v @ A4 ) @ ( set_Product_prod_v_v2 @ A4 ) ) ) ).
% list.set_sel(1)
thf(fact_286_list_Oset__sel_I1_J,axiom,
! [A4: list_v] :
( ( A4 != nil_v )
=> ( member_v @ ( hd_v @ A4 ) @ ( set_v2 @ A4 ) ) ) ).
% list.set_sel(1)
thf(fact_287_hd__in__set,axiom,
! [Xs: list_P7986770385144383213od_v_v] :
( ( Xs != nil_Product_prod_v_v )
=> ( member7453568604450474000od_v_v @ ( hd_Product_prod_v_v @ Xs ) @ ( set_Product_prod_v_v2 @ Xs ) ) ) ).
% hd_in_set
thf(fact_288_hd__in__set,axiom,
! [Xs: list_v] :
( ( Xs != nil_v )
=> ( member_v @ ( hd_v @ Xs ) @ ( set_v2 @ Xs ) ) ) ).
% hd_in_set
thf(fact_289_pre__dfs__def,axiom,
! [V: v,E2: sCC_Bl1394983891496994913t_unit] :
( ( sCC_Bl36166008131615352t_unit @ successors @ V @ E2 )
= ( ( sCC_Bl9196236973127232072t_unit @ successors @ E2 )
& ~ ( member_v @ V @ ( sCC_Bl4645233313691564917t_unit @ E2 ) )
& ( sCC_Bl649662514949026229able_v @ successors @ ( sCC_Bl1090238580953940555t_unit @ E2 ) @ V )
& ( ( sCC_Bl3795065053823578884t_unit @ E2 @ V )
= bot_bot_set_v )
& ! [X2: v] :
( ( member_v @ X2 @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E2 ) ) )
=> ( sCC_Bl649662514949026229able_v @ successors @ X2 @ V ) ) ) ) ).
% pre_dfs_def
thf(fact_290_sup__bot_Oright__neutral,axiom,
! [A4: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ A4 @ bot_bo723834152578015283od_v_v )
= A4 ) ).
% sup_bot.right_neutral
thf(fact_291_sup__bot_Oright__neutral,axiom,
! [A4: set_v] :
( ( sup_sup_set_v @ A4 @ bot_bot_set_v )
= A4 ) ).
% sup_bot.right_neutral
thf(fact_292_sup__bot_Oneutr__eq__iff,axiom,
! [A4: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( bot_bo723834152578015283od_v_v
= ( sup_su414716646722978715od_v_v @ A4 @ B2 ) )
= ( ( A4 = bot_bo723834152578015283od_v_v )
& ( B2 = bot_bo723834152578015283od_v_v ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_293_sup__bot_Oneutr__eq__iff,axiom,
! [A4: set_v,B2: set_v] :
( ( bot_bot_set_v
= ( sup_sup_set_v @ A4 @ B2 ) )
= ( ( A4 = bot_bot_set_v )
& ( B2 = bot_bot_set_v ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_294_sup__bot_Oleft__neutral,axiom,
! [A4: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ bot_bo723834152578015283od_v_v @ A4 )
= A4 ) ).
% sup_bot.left_neutral
thf(fact_295_sup__bot_Oleft__neutral,axiom,
! [A4: set_v] :
( ( sup_sup_set_v @ bot_bot_set_v @ A4 )
= A4 ) ).
% sup_bot.left_neutral
thf(fact_296_sup__bot_Oeq__neutr__iff,axiom,
! [A4: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( ( sup_su414716646722978715od_v_v @ A4 @ B2 )
= bot_bo723834152578015283od_v_v )
= ( ( A4 = bot_bo723834152578015283od_v_v )
& ( B2 = bot_bo723834152578015283od_v_v ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_297_sup__bot_Oeq__neutr__iff,axiom,
! [A4: set_v,B2: set_v] :
( ( ( sup_sup_set_v @ A4 @ B2 )
= bot_bot_set_v )
= ( ( A4 = bot_bot_set_v )
& ( B2 = bot_bot_set_v ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_298_sup__eq__bot__iff,axiom,
! [X: set_Product_prod_v_v,Y: set_Product_prod_v_v] :
( ( ( sup_su414716646722978715od_v_v @ X @ Y )
= bot_bo723834152578015283od_v_v )
= ( ( X = bot_bo723834152578015283od_v_v )
& ( Y = bot_bo723834152578015283od_v_v ) ) ) ).
% sup_eq_bot_iff
thf(fact_299_sup__eq__bot__iff,axiom,
! [X: set_v,Y: set_v] :
( ( ( sup_sup_set_v @ X @ Y )
= bot_bot_set_v )
= ( ( X = bot_bot_set_v )
& ( Y = bot_bot_set_v ) ) ) ).
% sup_eq_bot_iff
thf(fact_300_bot__eq__sup__iff,axiom,
! [X: set_Product_prod_v_v,Y: set_Product_prod_v_v] :
( ( bot_bo723834152578015283od_v_v
= ( sup_su414716646722978715od_v_v @ X @ Y ) )
= ( ( X = bot_bo723834152578015283od_v_v )
& ( Y = bot_bo723834152578015283od_v_v ) ) ) ).
% bot_eq_sup_iff
thf(fact_301_bot__eq__sup__iff,axiom,
! [X: set_v,Y: set_v] :
( ( bot_bot_set_v
= ( sup_sup_set_v @ X @ Y ) )
= ( ( X = bot_bot_set_v )
& ( Y = bot_bot_set_v ) ) ) ).
% bot_eq_sup_iff
thf(fact_302_sup__bot__right,axiom,
! [X: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ X @ bot_bo723834152578015283od_v_v )
= X ) ).
% sup_bot_right
thf(fact_303_sup__bot__right,axiom,
! [X: set_v] :
( ( sup_sup_set_v @ X @ bot_bot_set_v )
= X ) ).
% sup_bot_right
thf(fact_304_sup__bot__left,axiom,
! [X: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ bot_bo723834152578015283od_v_v @ X )
= X ) ).
% sup_bot_left
thf(fact_305_sup__bot__left,axiom,
! [X: set_v] :
( ( sup_sup_set_v @ bot_bot_set_v @ X )
= X ) ).
% sup_bot_left
thf(fact_306_sup_Oright__idem,axiom,
! [A4: set_v,B2: set_v] :
( ( sup_sup_set_v @ ( sup_sup_set_v @ A4 @ B2 ) @ B2 )
= ( sup_sup_set_v @ A4 @ B2 ) ) ).
% sup.right_idem
thf(fact_307_sup_Oright__idem,axiom,
! [A4: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ ( sup_su414716646722978715od_v_v @ A4 @ B2 ) @ B2 )
= ( sup_su414716646722978715od_v_v @ A4 @ B2 ) ) ).
% sup.right_idem
thf(fact_308_sup__left__idem,axiom,
! [X: set_v,Y: set_v] :
( ( sup_sup_set_v @ X @ ( sup_sup_set_v @ X @ Y ) )
= ( sup_sup_set_v @ X @ Y ) ) ).
% sup_left_idem
thf(fact_309_sup__left__idem,axiom,
! [X: set_Product_prod_v_v,Y: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ X @ ( sup_su414716646722978715od_v_v @ X @ Y ) )
= ( sup_su414716646722978715od_v_v @ X @ Y ) ) ).
% sup_left_idem
thf(fact_310_sup_Oleft__idem,axiom,
! [A4: set_v,B2: set_v] :
( ( sup_sup_set_v @ A4 @ ( sup_sup_set_v @ A4 @ B2 ) )
= ( sup_sup_set_v @ A4 @ B2 ) ) ).
% sup.left_idem
thf(fact_311_sup_Oleft__idem,axiom,
! [A4: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ A4 @ ( sup_su414716646722978715od_v_v @ A4 @ B2 ) )
= ( sup_su414716646722978715od_v_v @ A4 @ B2 ) ) ).
% sup.left_idem
thf(fact_312_sup__idem,axiom,
! [X: set_v] :
( ( sup_sup_set_v @ X @ X )
= X ) ).
% sup_idem
thf(fact_313_sup__idem,axiom,
! [X: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ X @ X )
= X ) ).
% sup_idem
thf(fact_314_sup_Oidem,axiom,
! [A4: set_v] :
( ( sup_sup_set_v @ A4 @ A4 )
= A4 ) ).
% sup.idem
thf(fact_315_sup_Oidem,axiom,
! [A4: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ A4 @ A4 )
= A4 ) ).
% sup.idem
thf(fact_316_sup_Obounded__iff,axiom,
! [B2: set_Product_prod_v_v,C: set_Product_prod_v_v,A4: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ ( sup_su414716646722978715od_v_v @ B2 @ C ) @ A4 )
= ( ( ord_le7336532860387713383od_v_v @ B2 @ A4 )
& ( ord_le7336532860387713383od_v_v @ C @ A4 ) ) ) ).
% sup.bounded_iff
thf(fact_317_sup_Obounded__iff,axiom,
! [B2: set_v,C: set_v,A4: set_v] :
( ( ord_less_eq_set_v @ ( sup_sup_set_v @ B2 @ C ) @ A4 )
= ( ( ord_less_eq_set_v @ B2 @ A4 )
& ( ord_less_eq_set_v @ C @ A4 ) ) ) ).
% sup.bounded_iff
thf(fact_318_le__sup__iff,axiom,
! [X: set_Product_prod_v_v,Y: set_Product_prod_v_v,Z: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ ( sup_su414716646722978715od_v_v @ X @ Y ) @ Z )
= ( ( ord_le7336532860387713383od_v_v @ X @ Z )
& ( ord_le7336532860387713383od_v_v @ Y @ Z ) ) ) ).
% le_sup_iff
thf(fact_319_le__sup__iff,axiom,
! [X: set_v,Y: set_v,Z: set_v] :
( ( ord_less_eq_set_v @ ( sup_sup_set_v @ X @ Y ) @ Z )
= ( ( ord_less_eq_set_v @ X @ Z )
& ( ord_less_eq_set_v @ Y @ Z ) ) ) ).
% le_sup_iff
thf(fact_320_graph_Oinit__env__pre__dfs,axiom,
! [Vertices: set_v,Successors: v > set_v,V: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( sCC_Bl36166008131615352t_unit @ Successors @ V @ ( sCC_Bl7693227186847904995_env_v @ V ) ) ) ).
% graph.init_env_pre_dfs
thf(fact_321_graph_Osclosed,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ! [X4: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X4 @ Vertices )
=> ( ord_le7336532860387713383od_v_v @ ( Successors @ X4 ) @ Vertices ) ) ) ).
% graph.sclosed
thf(fact_322_graph_Osclosed,axiom,
! [Vertices: set_v,Successors: v > set_v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ! [X4: v] :
( ( member_v @ X4 @ Vertices )
=> ( ord_less_eq_set_v @ ( Successors @ X4 ) @ Vertices ) ) ) ).
% graph.sclosed
thf(fact_323_graph_Oreachable__succ,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,Y: product_prod_v_v,X: product_prod_v_v,Z: product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( member7453568604450474000od_v_v @ Y @ ( Successors @ X ) )
=> ( ( sCC_Bl4981926079593201289od_v_v @ Successors @ Y @ Z )
=> ( sCC_Bl4981926079593201289od_v_v @ Successors @ X @ Z ) ) ) ) ).
% graph.reachable_succ
thf(fact_324_graph_Oreachable__succ,axiom,
! [Vertices: set_v,Successors: v > set_v,Y: v,X: v,Z: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( member_v @ Y @ ( Successors @ X ) )
=> ( ( sCC_Bl649662514949026229able_v @ Successors @ Y @ Z )
=> ( sCC_Bl649662514949026229able_v @ Successors @ X @ Z ) ) ) ) ).
% graph.reachable_succ
thf(fact_325_graph_Oreachable__refl,axiom,
! [Vertices: set_v,Successors: v > set_v,X: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( sCC_Bl649662514949026229able_v @ Successors @ X @ X ) ) ).
% graph.reachable_refl
thf(fact_326_graph_Oreachable__end__induct,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,X: product_prod_v_v,Y: product_prod_v_v,P: product_prod_v_v > product_prod_v_v > $o] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( sCC_Bl4981926079593201289od_v_v @ Successors @ X @ Y )
=> ( ! [X3: product_prod_v_v] : ( P @ X3 @ X3 )
=> ( ! [X3: product_prod_v_v,Y2: product_prod_v_v,Z2: product_prod_v_v] :
( ( P @ X3 @ Y2 )
=> ( ( member7453568604450474000od_v_v @ Z2 @ ( Successors @ Y2 ) )
=> ( P @ X3 @ Z2 ) ) )
=> ( P @ X @ Y ) ) ) ) ) ).
% graph.reachable_end_induct
thf(fact_327_graph_Oreachable__end__induct,axiom,
! [Vertices: set_v,Successors: v > set_v,X: v,Y: v,P: v > v > $o] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl649662514949026229able_v @ Successors @ X @ Y )
=> ( ! [X3: v] : ( P @ X3 @ X3 )
=> ( ! [X3: v,Y2: v,Z2: v] :
( ( P @ X3 @ Y2 )
=> ( ( member_v @ Z2 @ ( Successors @ Y2 ) )
=> ( P @ X3 @ Z2 ) ) )
=> ( P @ X @ Y ) ) ) ) ) ).
% graph.reachable_end_induct
thf(fact_328_graph_Oreachable__trans,axiom,
! [Vertices: set_v,Successors: v > set_v,X: v,Y: v,Z: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl649662514949026229able_v @ Successors @ X @ Y )
=> ( ( sCC_Bl649662514949026229able_v @ Successors @ Y @ Z )
=> ( sCC_Bl649662514949026229able_v @ Successors @ X @ Z ) ) ) ) ).
% graph.reachable_trans
thf(fact_329_graph_Oreachable_Osimps,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,A1: product_prod_v_v,A2: product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( sCC_Bl4981926079593201289od_v_v @ Successors @ A1 @ A2 )
= ( ? [X2: product_prod_v_v] :
( ( A1 = X2 )
& ( A2 = X2 ) )
| ? [X2: product_prod_v_v,Y3: product_prod_v_v,Z3: product_prod_v_v] :
( ( A1 = X2 )
& ( A2 = Z3 )
& ( member7453568604450474000od_v_v @ Y3 @ ( Successors @ X2 ) )
& ( sCC_Bl4981926079593201289od_v_v @ Successors @ Y3 @ Z3 ) ) ) ) ) ).
% graph.reachable.simps
thf(fact_330_graph_Oreachable_Osimps,axiom,
! [Vertices: set_v,Successors: v > set_v,A1: v,A2: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl649662514949026229able_v @ Successors @ A1 @ A2 )
= ( ? [X2: v] :
( ( A1 = X2 )
& ( A2 = X2 ) )
| ? [X2: v,Y3: v,Z3: v] :
( ( A1 = X2 )
& ( A2 = Z3 )
& ( member_v @ Y3 @ ( Successors @ X2 ) )
& ( sCC_Bl649662514949026229able_v @ Successors @ Y3 @ Z3 ) ) ) ) ) ).
% graph.reachable.simps
thf(fact_331_graph_Oreachable_Ocases,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,A1: product_prod_v_v,A2: product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( sCC_Bl4981926079593201289od_v_v @ Successors @ A1 @ A2 )
=> ( ( A2 != A1 )
=> ~ ! [Y2: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ Y2 @ ( Successors @ A1 ) )
=> ~ ( sCC_Bl4981926079593201289od_v_v @ Successors @ Y2 @ A2 ) ) ) ) ) ).
% graph.reachable.cases
thf(fact_332_graph_Oreachable_Ocases,axiom,
! [Vertices: set_v,Successors: v > set_v,A1: v,A2: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl649662514949026229able_v @ Successors @ A1 @ A2 )
=> ( ( A2 != A1 )
=> ~ ! [Y2: v] :
( ( member_v @ Y2 @ ( Successors @ A1 ) )
=> ~ ( sCC_Bl649662514949026229able_v @ Successors @ Y2 @ A2 ) ) ) ) ) ).
% graph.reachable.cases
thf(fact_333_graph_Osucc__reachable,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,X: product_prod_v_v,Y: product_prod_v_v,Z: product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( sCC_Bl4981926079593201289od_v_v @ Successors @ X @ Y )
=> ( ( member7453568604450474000od_v_v @ Z @ ( Successors @ Y ) )
=> ( sCC_Bl4981926079593201289od_v_v @ Successors @ X @ Z ) ) ) ) ).
% graph.succ_reachable
thf(fact_334_graph_Osucc__reachable,axiom,
! [Vertices: set_v,Successors: v > set_v,X: v,Y: v,Z: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl649662514949026229able_v @ Successors @ X @ Y )
=> ( ( member_v @ Z @ ( Successors @ Y ) )
=> ( sCC_Bl649662514949026229able_v @ Successors @ X @ Z ) ) ) ) ).
% graph.succ_reachable
thf(fact_335_graph_Oreachable__edge,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,Y: product_prod_v_v,X: product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( member7453568604450474000od_v_v @ Y @ ( Successors @ X ) )
=> ( sCC_Bl4981926079593201289od_v_v @ Successors @ X @ Y ) ) ) ).
% graph.reachable_edge
thf(fact_336_graph_Oreachable__edge,axiom,
! [Vertices: set_v,Successors: v > set_v,Y: v,X: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( member_v @ Y @ ( Successors @ X ) )
=> ( sCC_Bl649662514949026229able_v @ Successors @ X @ Y ) ) ) ).
% graph.reachable_edge
thf(fact_337_graph_Ora__trans,axiom,
! [Vertices: set_v,Successors: v > set_v,X: v,Y: v,E: set_Product_prod_v_v,Z: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl4291963740693775144ding_v @ Successors @ X @ Y @ E )
=> ( ( sCC_Bl4291963740693775144ding_v @ Successors @ Y @ Z @ E )
=> ( sCC_Bl4291963740693775144ding_v @ Successors @ X @ Z @ E ) ) ) ) ).
% graph.ra_trans
thf(fact_338_graph_Ora__refl,axiom,
! [Vertices: set_v,Successors: v > set_v,X: v,E: set_Product_prod_v_v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( sCC_Bl4291963740693775144ding_v @ Successors @ X @ X @ E ) ) ).
% graph.ra_refl
thf(fact_339_graph_Ore__succ,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,X: product_prod_v_v,Y: product_prod_v_v,Z: product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( sCC_Bl4714988717384592488od_v_v @ Successors @ X @ Y )
=> ( ( member7453568604450474000od_v_v @ Z @ ( Successors @ Y ) )
=> ( sCC_Bl4714988717384592488od_v_v @ Successors @ X @ Z ) ) ) ) ).
% graph.re_succ
thf(fact_340_graph_Ore__succ,axiom,
! [Vertices: set_v,Successors: v > set_v,X: v,Y: v,Z: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl770211535891879572_end_v @ Successors @ X @ Y )
=> ( ( member_v @ Z @ ( Successors @ Y ) )
=> ( sCC_Bl770211535891879572_end_v @ Successors @ X @ Z ) ) ) ) ).
% graph.re_succ
thf(fact_341_graph_Ore__refl,axiom,
! [Vertices: set_v,Successors: v > set_v,X: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( sCC_Bl770211535891879572_end_v @ Successors @ X @ X ) ) ).
% graph.re_refl
thf(fact_342_graph_Oreachable__end_Osimps,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,A1: product_prod_v_v,A2: product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( sCC_Bl4714988717384592488od_v_v @ Successors @ A1 @ A2 )
= ( ? [X2: product_prod_v_v] :
( ( A1 = X2 )
& ( A2 = X2 ) )
| ? [X2: product_prod_v_v,Y3: product_prod_v_v,Z3: product_prod_v_v] :
( ( A1 = X2 )
& ( A2 = Z3 )
& ( sCC_Bl4714988717384592488od_v_v @ Successors @ X2 @ Y3 )
& ( member7453568604450474000od_v_v @ Z3 @ ( Successors @ Y3 ) ) ) ) ) ) ).
% graph.reachable_end.simps
thf(fact_343_graph_Oreachable__end_Osimps,axiom,
! [Vertices: set_v,Successors: v > set_v,A1: v,A2: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl770211535891879572_end_v @ Successors @ A1 @ A2 )
= ( ? [X2: v] :
( ( A1 = X2 )
& ( A2 = X2 ) )
| ? [X2: v,Y3: v,Z3: v] :
( ( A1 = X2 )
& ( A2 = Z3 )
& ( sCC_Bl770211535891879572_end_v @ Successors @ X2 @ Y3 )
& ( member_v @ Z3 @ ( Successors @ Y3 ) ) ) ) ) ) ).
% graph.reachable_end.simps
thf(fact_344_graph_Oreachable__end_Ocases,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,A1: product_prod_v_v,A2: product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( sCC_Bl4714988717384592488od_v_v @ Successors @ A1 @ A2 )
=> ( ( A2 != A1 )
=> ~ ! [Y2: product_prod_v_v] :
( ( sCC_Bl4714988717384592488od_v_v @ Successors @ A1 @ Y2 )
=> ~ ( member7453568604450474000od_v_v @ A2 @ ( Successors @ Y2 ) ) ) ) ) ) ).
% graph.reachable_end.cases
thf(fact_345_graph_Oreachable__end_Ocases,axiom,
! [Vertices: set_v,Successors: v > set_v,A1: v,A2: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl770211535891879572_end_v @ Successors @ A1 @ A2 )
=> ( ( A2 != A1 )
=> ~ ! [Y2: v] :
( ( sCC_Bl770211535891879572_end_v @ Successors @ A1 @ Y2 )
=> ~ ( member_v @ A2 @ ( Successors @ Y2 ) ) ) ) ) ) ).
% graph.reachable_end.cases
thf(fact_346_graph_Osucc__re,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,Y: product_prod_v_v,X: product_prod_v_v,Z: product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( member7453568604450474000od_v_v @ Y @ ( Successors @ X ) )
=> ( ( sCC_Bl4714988717384592488od_v_v @ Successors @ Y @ Z )
=> ( sCC_Bl4714988717384592488od_v_v @ Successors @ X @ Z ) ) ) ) ).
% graph.succ_re
thf(fact_347_graph_Osucc__re,axiom,
! [Vertices: set_v,Successors: v > set_v,Y: v,X: v,Z: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( member_v @ Y @ ( Successors @ X ) )
=> ( ( sCC_Bl770211535891879572_end_v @ Successors @ Y @ Z )
=> ( sCC_Bl770211535891879572_end_v @ Successors @ X @ Z ) ) ) ) ).
% graph.succ_re
thf(fact_348_graph_Osub__env__trans,axiom,
! [Vertices: set_v,Successors: v > set_v,E2: sCC_Bl1394983891496994913t_unit,E3: sCC_Bl1394983891496994913t_unit,E4: sCC_Bl1394983891496994913t_unit] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl5768913643336123637t_unit @ E2 @ E3 )
=> ( ( sCC_Bl5768913643336123637t_unit @ E3 @ E4 )
=> ( sCC_Bl5768913643336123637t_unit @ E2 @ E4 ) ) ) ) ).
% graph.sub_env_trans
thf(fact_349_graph_Oedge__ra,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,Y: product_prod_v_v,X: product_prod_v_v,E: set_Pr2149350503807050951od_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( member7453568604450474000od_v_v @ Y @ ( Successors @ X ) )
=> ( ~ ( member3038538357316246288od_v_v @ ( produc4031800376763917143od_v_v @ X @ Y ) @ E )
=> ( sCC_Bl5370300055464682748od_v_v @ Successors @ X @ Y @ E ) ) ) ) ).
% graph.edge_ra
thf(fact_350_graph_Oedge__ra,axiom,
! [Vertices: set_v,Successors: v > set_v,Y: v,X: v,E: set_Product_prod_v_v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( member_v @ Y @ ( Successors @ X ) )
=> ( ~ ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ X @ Y ) @ E )
=> ( sCC_Bl4291963740693775144ding_v @ Successors @ X @ Y @ E ) ) ) ) ).
% graph.edge_ra
thf(fact_351_graph_Ora__cases,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,X: product_prod_v_v,Y: product_prod_v_v,E: set_Pr2149350503807050951od_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( sCC_Bl5370300055464682748od_v_v @ Successors @ X @ Y @ E )
=> ( ( X = Y )
| ? [Z2: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ Z2 @ ( Successors @ X ) )
& ~ ( member3038538357316246288od_v_v @ ( produc4031800376763917143od_v_v @ X @ Z2 ) @ E )
& ( sCC_Bl5370300055464682748od_v_v @ Successors @ Z2 @ Y @ E ) ) ) ) ) ).
% graph.ra_cases
thf(fact_352_graph_Ora__cases,axiom,
! [Vertices: set_v,Successors: v > set_v,X: v,Y: v,E: set_Product_prod_v_v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl4291963740693775144ding_v @ Successors @ X @ Y @ E )
=> ( ( X = Y )
| ? [Z2: v] :
( ( member_v @ Z2 @ ( Successors @ X ) )
& ~ ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ X @ Z2 ) @ E )
& ( sCC_Bl4291963740693775144ding_v @ Successors @ Z2 @ Y @ E ) ) ) ) ) ).
% graph.ra_cases
thf(fact_353_graph_Oreachable__avoiding_Ocases,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,A1: product_prod_v_v,A2: product_prod_v_v,A3: set_Pr2149350503807050951od_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( sCC_Bl5370300055464682748od_v_v @ Successors @ A1 @ A2 @ A3 )
=> ( ( A2 != A1 )
=> ~ ! [Y2: product_prod_v_v] :
( ( sCC_Bl5370300055464682748od_v_v @ Successors @ A1 @ Y2 @ A3 )
=> ( ( member7453568604450474000od_v_v @ A2 @ ( Successors @ Y2 ) )
=> ( member3038538357316246288od_v_v @ ( produc4031800376763917143od_v_v @ Y2 @ A2 ) @ A3 ) ) ) ) ) ) ).
% graph.reachable_avoiding.cases
thf(fact_354_graph_Oreachable__avoiding_Ocases,axiom,
! [Vertices: set_v,Successors: v > set_v,A1: v,A2: v,A3: set_Product_prod_v_v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl4291963740693775144ding_v @ Successors @ A1 @ A2 @ A3 )
=> ( ( A2 != A1 )
=> ~ ! [Y2: v] :
( ( sCC_Bl4291963740693775144ding_v @ Successors @ A1 @ Y2 @ A3 )
=> ( ( member_v @ A2 @ ( Successors @ Y2 ) )
=> ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ Y2 @ A2 ) @ A3 ) ) ) ) ) ) ).
% graph.reachable_avoiding.cases
thf(fact_355_graph_Oreachable__avoiding_Osimps,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,A1: product_prod_v_v,A2: product_prod_v_v,A3: set_Pr2149350503807050951od_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( sCC_Bl5370300055464682748od_v_v @ Successors @ A1 @ A2 @ A3 )
= ( ? [X2: product_prod_v_v,E6: set_Pr2149350503807050951od_v_v] :
( ( A1 = X2 )
& ( A2 = X2 )
& ( A3 = E6 ) )
| ? [X2: product_prod_v_v,Y3: product_prod_v_v,E6: set_Pr2149350503807050951od_v_v,Z3: product_prod_v_v] :
( ( A1 = X2 )
& ( A2 = Z3 )
& ( A3 = E6 )
& ( sCC_Bl5370300055464682748od_v_v @ Successors @ X2 @ Y3 @ E6 )
& ( member7453568604450474000od_v_v @ Z3 @ ( Successors @ Y3 ) )
& ~ ( member3038538357316246288od_v_v @ ( produc4031800376763917143od_v_v @ Y3 @ Z3 ) @ E6 ) ) ) ) ) ).
% graph.reachable_avoiding.simps
thf(fact_356_graph_Oreachable__avoiding_Osimps,axiom,
! [Vertices: set_v,Successors: v > set_v,A1: v,A2: v,A3: set_Product_prod_v_v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl4291963740693775144ding_v @ Successors @ A1 @ A2 @ A3 )
= ( ? [X2: v,E6: set_Product_prod_v_v] :
( ( A1 = X2 )
& ( A2 = X2 )
& ( A3 = E6 ) )
| ? [X2: v,Y3: v,E6: set_Product_prod_v_v,Z3: v] :
( ( A1 = X2 )
& ( A2 = Z3 )
& ( A3 = E6 )
& ( sCC_Bl4291963740693775144ding_v @ Successors @ X2 @ Y3 @ E6 )
& ( member_v @ Z3 @ ( Successors @ Y3 ) )
& ~ ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ Y3 @ Z3 ) @ E6 ) ) ) ) ) ).
% graph.reachable_avoiding.simps
thf(fact_357_graph_Ora__succ,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,X: product_prod_v_v,Y: product_prod_v_v,E: set_Pr2149350503807050951od_v_v,Z: product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( sCC_Bl5370300055464682748od_v_v @ Successors @ X @ Y @ E )
=> ( ( member7453568604450474000od_v_v @ Z @ ( Successors @ Y ) )
=> ( ~ ( member3038538357316246288od_v_v @ ( produc4031800376763917143od_v_v @ Y @ Z ) @ E )
=> ( sCC_Bl5370300055464682748od_v_v @ Successors @ X @ Z @ E ) ) ) ) ) ).
% graph.ra_succ
thf(fact_358_graph_Ora__succ,axiom,
! [Vertices: set_v,Successors: v > set_v,X: v,Y: v,E: set_Product_prod_v_v,Z: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl4291963740693775144ding_v @ Successors @ X @ Y @ E )
=> ( ( member_v @ Z @ ( Successors @ Y ) )
=> ( ~ ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ Y @ Z ) @ E )
=> ( sCC_Bl4291963740693775144ding_v @ Successors @ X @ Z @ E ) ) ) ) ) ).
% graph.ra_succ
thf(fact_359_graph_Ora__mono,axiom,
! [Vertices: set_v,Successors: v > set_v,X: v,Y: v,E: set_Product_prod_v_v,E5: set_Product_prod_v_v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl4291963740693775144ding_v @ Successors @ X @ Y @ E )
=> ( ( ord_le7336532860387713383od_v_v @ E5 @ E )
=> ( sCC_Bl4291963740693775144ding_v @ Successors @ X @ Y @ E5 ) ) ) ) ).
% graph.ra_mono
thf(fact_360_graph_Ora__reachable,axiom,
! [Vertices: set_v,Successors: v > set_v,X: v,Y: v,E: set_Product_prod_v_v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl4291963740693775144ding_v @ Successors @ X @ Y @ E )
=> ( sCC_Bl649662514949026229able_v @ Successors @ X @ Y ) ) ) ).
% graph.ra_reachable
thf(fact_361_graph_OS__reflexive,axiom,
! [Vertices: set_v,Successors: v > set_v,E2: sCC_Bl1394983891496994913t_unit,N: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl9196236973127232072t_unit @ Successors @ E2 )
=> ( member_v @ N @ ( sCC_Bl1280885523602775798t_unit @ E2 @ N ) ) ) ) ).
% graph.S_reflexive
thf(fact_362_graph_Oreachable__re,axiom,
! [Vertices: set_v,Successors: v > set_v,X: v,Y: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl649662514949026229able_v @ Successors @ X @ Y )
=> ( sCC_Bl770211535891879572_end_v @ Successors @ X @ Y ) ) ) ).
% graph.reachable_re
thf(fact_363_graph_Ore__reachable,axiom,
! [Vertices: set_v,Successors: v > set_v,X: v,Y: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl770211535891879572_end_v @ Successors @ X @ Y )
=> ( sCC_Bl649662514949026229able_v @ Successors @ X @ Y ) ) ) ).
% graph.re_reachable
thf(fact_364_graph_Ois__subscc__def,axiom,
! [Vertices: set_v,Successors: v > set_v,S: set_v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl5398416737448265317bscc_v @ Successors @ S )
= ( ! [X2: v] :
( ( member_v @ X2 @ S )
=> ! [Y3: v] :
( ( member_v @ Y3 @ S )
=> ( sCC_Bl649662514949026229able_v @ Successors @ X2 @ Y3 ) ) ) ) ) ) ).
% graph.is_subscc_def
thf(fact_365_graph_OsccE,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,S: set_Product_prod_v_v,X: product_prod_v_v,Y: product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( sCC_Bl6242042402218619277od_v_v @ Successors @ S )
=> ( ( member7453568604450474000od_v_v @ X @ S )
=> ( ( sCC_Bl4981926079593201289od_v_v @ Successors @ X @ Y )
=> ( ( sCC_Bl4981926079593201289od_v_v @ Successors @ Y @ X )
=> ( member7453568604450474000od_v_v @ Y @ S ) ) ) ) ) ) ).
% graph.sccE
thf(fact_366_graph_OsccE,axiom,
! [Vertices: set_v,Successors: v > set_v,S: set_v,X: v,Y: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bloemen_is_scc_v @ Successors @ S )
=> ( ( member_v @ X @ S )
=> ( ( sCC_Bl649662514949026229able_v @ Successors @ X @ Y )
=> ( ( sCC_Bl649662514949026229able_v @ Successors @ Y @ X )
=> ( member_v @ Y @ S ) ) ) ) ) ) ).
% graph.sccE
thf(fact_367_graph_Odfs__S__tl__stack_I1_J,axiom,
! [Vertices: set_v,Successors: v > set_v,V: v,E2: sCC_Bl1394983891496994913t_unit,E3: sCC_Bl1394983891496994913t_unit] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl8953792750115413617t_unit @ Successors @ V @ E2 @ E3 )
=> ( ( ( sCC_Bl8828226123343373779t_unit @ E2 )
!= nil_v )
=> ( ( sCC_Bl8828226123343373779t_unit @ E3 )
!= nil_v ) ) ) ) ).
% graph.dfs_S_tl_stack(1)
thf(fact_368_graph_Ora__empty,axiom,
! [Vertices: set_v,Successors: v > set_v,X: v,Y: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl4291963740693775144ding_v @ Successors @ X @ Y @ bot_bo723834152578015283od_v_v )
= ( sCC_Bl649662514949026229able_v @ Successors @ X @ Y ) ) ) ).
% graph.ra_empty
thf(fact_369_sup__left__commute,axiom,
! [X: set_v,Y: set_v,Z: set_v] :
( ( sup_sup_set_v @ X @ ( sup_sup_set_v @ Y @ Z ) )
= ( sup_sup_set_v @ Y @ ( sup_sup_set_v @ X @ Z ) ) ) ).
% sup_left_commute
thf(fact_370_sup__left__commute,axiom,
! [X: set_Product_prod_v_v,Y: set_Product_prod_v_v,Z: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ X @ ( sup_su414716646722978715od_v_v @ Y @ Z ) )
= ( sup_su414716646722978715od_v_v @ Y @ ( sup_su414716646722978715od_v_v @ X @ Z ) ) ) ).
% sup_left_commute
thf(fact_371_sup_Oleft__commute,axiom,
! [B2: set_v,A4: set_v,C: set_v] :
( ( sup_sup_set_v @ B2 @ ( sup_sup_set_v @ A4 @ C ) )
= ( sup_sup_set_v @ A4 @ ( sup_sup_set_v @ B2 @ C ) ) ) ).
% sup.left_commute
thf(fact_372_sup_Oleft__commute,axiom,
! [B2: set_Product_prod_v_v,A4: set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ B2 @ ( sup_su414716646722978715od_v_v @ A4 @ C ) )
= ( sup_su414716646722978715od_v_v @ A4 @ ( sup_su414716646722978715od_v_v @ B2 @ C ) ) ) ).
% sup.left_commute
thf(fact_373_sup__commute,axiom,
( sup_sup_set_v
= ( ^ [X2: set_v,Y3: set_v] : ( sup_sup_set_v @ Y3 @ X2 ) ) ) ).
% sup_commute
thf(fact_374_sup__commute,axiom,
( sup_su414716646722978715od_v_v
= ( ^ [X2: set_Product_prod_v_v,Y3: set_Product_prod_v_v] : ( sup_su414716646722978715od_v_v @ Y3 @ X2 ) ) ) ).
% sup_commute
thf(fact_375_sup_Ocommute,axiom,
( sup_sup_set_v
= ( ^ [A5: set_v,B3: set_v] : ( sup_sup_set_v @ B3 @ A5 ) ) ) ).
% sup.commute
thf(fact_376_sup_Ocommute,axiom,
( sup_su414716646722978715od_v_v
= ( ^ [A5: set_Product_prod_v_v,B3: set_Product_prod_v_v] : ( sup_su414716646722978715od_v_v @ B3 @ A5 ) ) ) ).
% sup.commute
thf(fact_377_sup__assoc,axiom,
! [X: set_v,Y: set_v,Z: set_v] :
( ( sup_sup_set_v @ ( sup_sup_set_v @ X @ Y ) @ Z )
= ( sup_sup_set_v @ X @ ( sup_sup_set_v @ Y @ Z ) ) ) ).
% sup_assoc
thf(fact_378_sup__assoc,axiom,
! [X: set_Product_prod_v_v,Y: set_Product_prod_v_v,Z: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ ( sup_su414716646722978715od_v_v @ X @ Y ) @ Z )
= ( sup_su414716646722978715od_v_v @ X @ ( sup_su414716646722978715od_v_v @ Y @ Z ) ) ) ).
% sup_assoc
thf(fact_379_sup_Oassoc,axiom,
! [A4: set_v,B2: set_v,C: set_v] :
( ( sup_sup_set_v @ ( sup_sup_set_v @ A4 @ B2 ) @ C )
= ( sup_sup_set_v @ A4 @ ( sup_sup_set_v @ B2 @ C ) ) ) ).
% sup.assoc
thf(fact_380_sup_Oassoc,axiom,
! [A4: set_Product_prod_v_v,B2: set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ ( sup_su414716646722978715od_v_v @ A4 @ B2 ) @ C )
= ( sup_su414716646722978715od_v_v @ A4 @ ( sup_su414716646722978715od_v_v @ B2 @ C ) ) ) ).
% sup.assoc
thf(fact_381_inf__sup__aci_I5_J,axiom,
( sup_sup_set_v
= ( ^ [X2: set_v,Y3: set_v] : ( sup_sup_set_v @ Y3 @ X2 ) ) ) ).
% inf_sup_aci(5)
thf(fact_382_inf__sup__aci_I5_J,axiom,
( sup_su414716646722978715od_v_v
= ( ^ [X2: set_Product_prod_v_v,Y3: set_Product_prod_v_v] : ( sup_su414716646722978715od_v_v @ Y3 @ X2 ) ) ) ).
% inf_sup_aci(5)
thf(fact_383_inf__sup__aci_I6_J,axiom,
! [X: set_v,Y: set_v,Z: set_v] :
( ( sup_sup_set_v @ ( sup_sup_set_v @ X @ Y ) @ Z )
= ( sup_sup_set_v @ X @ ( sup_sup_set_v @ Y @ Z ) ) ) ).
% inf_sup_aci(6)
thf(fact_384_inf__sup__aci_I6_J,axiom,
! [X: set_Product_prod_v_v,Y: set_Product_prod_v_v,Z: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ ( sup_su414716646722978715od_v_v @ X @ Y ) @ Z )
= ( sup_su414716646722978715od_v_v @ X @ ( sup_su414716646722978715od_v_v @ Y @ Z ) ) ) ).
% inf_sup_aci(6)
thf(fact_385_inf__sup__aci_I7_J,axiom,
! [X: set_v,Y: set_v,Z: set_v] :
( ( sup_sup_set_v @ X @ ( sup_sup_set_v @ Y @ Z ) )
= ( sup_sup_set_v @ Y @ ( sup_sup_set_v @ X @ Z ) ) ) ).
% inf_sup_aci(7)
thf(fact_386_inf__sup__aci_I7_J,axiom,
! [X: set_Product_prod_v_v,Y: set_Product_prod_v_v,Z: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ X @ ( sup_su414716646722978715od_v_v @ Y @ Z ) )
= ( sup_su414716646722978715od_v_v @ Y @ ( sup_su414716646722978715od_v_v @ X @ Z ) ) ) ).
% inf_sup_aci(7)
thf(fact_387_inf__sup__aci_I8_J,axiom,
! [X: set_v,Y: set_v] :
( ( sup_sup_set_v @ X @ ( sup_sup_set_v @ X @ Y ) )
= ( sup_sup_set_v @ X @ Y ) ) ).
% inf_sup_aci(8)
thf(fact_388_inf__sup__aci_I8_J,axiom,
! [X: set_Product_prod_v_v,Y: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ X @ ( sup_su414716646722978715od_v_v @ X @ Y ) )
= ( sup_su414716646722978715od_v_v @ X @ Y ) ) ).
% inf_sup_aci(8)
thf(fact_389_graph_Opre__dfss__pre__dfs,axiom,
! [Vertices: set_v,Successors: v > set_v,V: v,E2: sCC_Bl1394983891496994913t_unit,W: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl1748261141445803503t_unit @ Successors @ V @ E2 )
=> ( ~ ( member_v @ W @ ( sCC_Bl4645233313691564917t_unit @ E2 ) )
=> ( ( member_v @ W @ ( Successors @ V ) )
=> ( sCC_Bl36166008131615352t_unit @ Successors @ W @ E2 ) ) ) ) ) ).
% graph.pre_dfss_pre_dfs
thf(fact_390_graph_Ostack__visited,axiom,
! [Vertices: set_v,Successors: v > set_v,E2: sCC_Bl1394983891496994913t_unit,N: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl9196236973127232072t_unit @ Successors @ E2 )
=> ( ( member_v @ N @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E2 ) ) )
=> ( member_v @ N @ ( sCC_Bl4645233313691564917t_unit @ E2 ) ) ) ) ) ).
% graph.stack_visited
thf(fact_391_graph_Ostack__unexplored,axiom,
! [Vertices: set_v,Successors: v > set_v,E2: sCC_Bl1394983891496994913t_unit,N: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl9196236973127232072t_unit @ Successors @ E2 )
=> ( ( member_v @ N @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E2 ) ) )
=> ~ ( member_v @ N @ ( sCC_Bl157864678168468314t_unit @ E2 ) ) ) ) ) ).
% graph.stack_unexplored
thf(fact_392_graph_Ois__scc__def,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,S: set_Product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( sCC_Bl6242042402218619277od_v_v @ Successors @ S )
= ( ( S != bot_bo723834152578015283od_v_v )
& ( sCC_Bl2301996248249672505od_v_v @ Successors @ S )
& ! [S2: set_Product_prod_v_v] :
( ( ( ord_le7336532860387713383od_v_v @ S @ S2 )
& ( sCC_Bl2301996248249672505od_v_v @ Successors @ S2 ) )
=> ( S2 = S ) ) ) ) ) ).
% graph.is_scc_def
thf(fact_393_graph_Ois__scc__def,axiom,
! [Vertices: set_v,Successors: v > set_v,S: set_v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bloemen_is_scc_v @ Successors @ S )
= ( ( S != bot_bot_set_v )
& ( sCC_Bl5398416737448265317bscc_v @ Successors @ S )
& ! [S2: set_v] :
( ( ( ord_less_eq_set_v @ S @ S2 )
& ( sCC_Bl5398416737448265317bscc_v @ Successors @ S2 ) )
=> ( S2 = S ) ) ) ) ) ).
% graph.is_scc_def
thf(fact_394_graph_Oreachable__visited,axiom,
! [Vertices: set_v,Successors: v > set_v,E2: sCC_Bl1394983891496994913t_unit,V: v,W: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl9196236973127232072t_unit @ Successors @ E2 )
=> ( ( member_v @ V @ ( sCC_Bl4645233313691564917t_unit @ E2 ) )
=> ( ( sCC_Bl649662514949026229able_v @ Successors @ V @ W )
=> ( ! [X3: v] :
( ( member_v @ X3 @ ( sCC_Bl4645233313691564917t_unit @ E2 ) )
=> ! [Xa: v] :
( ( member_v @ Xa @ ( minus_minus_set_v @ ( Successors @ X3 ) @ ( sCC_Bl3795065053823578884t_unit @ E2 @ X3 ) ) )
=> ( ( sCC_Bl649662514949026229able_v @ Successors @ V @ X3 )
=> ~ ( sCC_Bl649662514949026229able_v @ Successors @ Xa @ W ) ) ) )
=> ( member_v @ W @ ( sCC_Bl4645233313691564917t_unit @ E2 ) ) ) ) ) ) ) ).
% graph.reachable_visited
thf(fact_395_graph_Opre__dfs__def,axiom,
! [Vertices: set_v,Successors: v > set_v,V: v,E2: sCC_Bl1394983891496994913t_unit] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl36166008131615352t_unit @ Successors @ V @ E2 )
= ( ( sCC_Bl9196236973127232072t_unit @ Successors @ E2 )
& ~ ( member_v @ V @ ( sCC_Bl4645233313691564917t_unit @ E2 ) )
& ( sCC_Bl649662514949026229able_v @ Successors @ ( sCC_Bl1090238580953940555t_unit @ E2 ) @ V )
& ( ( sCC_Bl3795065053823578884t_unit @ E2 @ V )
= bot_bot_set_v )
& ! [X2: v] :
( ( member_v @ X2 @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E2 ) ) )
=> ( sCC_Bl649662514949026229able_v @ Successors @ X2 @ V ) ) ) ) ) ).
% graph.pre_dfs_def
thf(fact_396_graph_Odfs__S__tl__stack_I2_J,axiom,
! [Vertices: set_v,Successors: v > set_v,V: v,E2: sCC_Bl1394983891496994913t_unit,E3: sCC_Bl1394983891496994913t_unit] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl8953792750115413617t_unit @ Successors @ V @ E2 @ E3 )
=> ( ( ( sCC_Bl8828226123343373779t_unit @ E2 )
!= nil_v )
=> ! [X4: v] :
( ( member_v @ X4 @ ( set_v2 @ ( tl_v @ ( sCC_Bl8828226123343373779t_unit @ E3 ) ) ) )
=> ( ( sCC_Bl1280885523602775798t_unit @ E3 @ X4 )
= ( sCC_Bl1280885523602775798t_unit @ E2 @ X4 ) ) ) ) ) ) ).
% graph.dfs_S_tl_stack(2)
thf(fact_397_graph_Ovisited__unexplored,axiom,
! [Vertices: set_v,Successors: v > set_v,E2: sCC_Bl1394983891496994913t_unit,M: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl9196236973127232072t_unit @ Successors @ E2 )
=> ( ( member_v @ M @ ( sCC_Bl4645233313691564917t_unit @ E2 ) )
=> ( ~ ( member_v @ M @ ( sCC_Bl157864678168468314t_unit @ E2 ) )
=> ~ ! [N2: v] :
( ( member_v @ N2 @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E2 ) ) )
=> ~ ( member_v @ M @ ( sCC_Bl1280885523602775798t_unit @ E2 @ N2 ) ) ) ) ) ) ) ).
% graph.visited_unexplored
thf(fact_398_graph_Odfs__S__hd__stack_I2_J,axiom,
! [Vertices: set_v,Successors: v > set_v,E2: sCC_Bl1394983891496994913t_unit,V: v,E3: sCC_Bl1394983891496994913t_unit,N: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl9196236973127232072t_unit @ Successors @ E2 )
=> ( ( sCC_Bl8953792750115413617t_unit @ Successors @ V @ E2 @ E3 )
=> ( ( ( sCC_Bl8828226123343373779t_unit @ E2 )
!= nil_v )
=> ( ( member_v @ N @ ( sCC_Bl1280885523602775798t_unit @ E2 @ ( hd_v @ ( sCC_Bl8828226123343373779t_unit @ E2 ) ) ) )
=> ( member_v @ N @ ( sCC_Bl1280885523602775798t_unit @ E3 @ ( hd_v @ ( sCC_Bl8828226123343373779t_unit @ E3 ) ) ) ) ) ) ) ) ) ).
% graph.dfs_S_hd_stack(2)
thf(fact_399_graph_Odfs__S__hd__stack_I1_J,axiom,
! [Vertices: set_v,Successors: v > set_v,E2: sCC_Bl1394983891496994913t_unit,V: v,E3: sCC_Bl1394983891496994913t_unit,N: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl9196236973127232072t_unit @ Successors @ E2 )
=> ( ( sCC_Bl8953792750115413617t_unit @ Successors @ V @ E2 @ E3 )
=> ( ( ( sCC_Bl8828226123343373779t_unit @ E2 )
!= nil_v )
=> ( ( member_v @ N @ ( sCC_Bl1280885523602775798t_unit @ E2 @ ( hd_v @ ( sCC_Bl8828226123343373779t_unit @ E2 ) ) ) )
=> ( ( sCC_Bl8828226123343373779t_unit @ E3 )
!= nil_v ) ) ) ) ) ) ).
% graph.dfs_S_hd_stack(1)
thf(fact_400_inf__sup__ord_I4_J,axiom,
! [Y: set_Product_prod_v_v,X: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ Y @ ( sup_su414716646722978715od_v_v @ X @ Y ) ) ).
% inf_sup_ord(4)
thf(fact_401_inf__sup__ord_I4_J,axiom,
! [Y: set_v,X: set_v] : ( ord_less_eq_set_v @ Y @ ( sup_sup_set_v @ X @ Y ) ) ).
% inf_sup_ord(4)
thf(fact_402_inf__sup__ord_I3_J,axiom,
! [X: set_Product_prod_v_v,Y: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ X @ ( sup_su414716646722978715od_v_v @ X @ Y ) ) ).
% inf_sup_ord(3)
thf(fact_403_inf__sup__ord_I3_J,axiom,
! [X: set_v,Y: set_v] : ( ord_less_eq_set_v @ X @ ( sup_sup_set_v @ X @ Y ) ) ).
% inf_sup_ord(3)
thf(fact_404_le__supE,axiom,
! [A4: set_Product_prod_v_v,B2: set_Product_prod_v_v,X: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ ( sup_su414716646722978715od_v_v @ A4 @ B2 ) @ X )
=> ~ ( ( ord_le7336532860387713383od_v_v @ A4 @ X )
=> ~ ( ord_le7336532860387713383od_v_v @ B2 @ X ) ) ) ).
% le_supE
thf(fact_405_le__supE,axiom,
! [A4: set_v,B2: set_v,X: set_v] :
( ( ord_less_eq_set_v @ ( sup_sup_set_v @ A4 @ B2 ) @ X )
=> ~ ( ( ord_less_eq_set_v @ A4 @ X )
=> ~ ( ord_less_eq_set_v @ B2 @ X ) ) ) ).
% le_supE
thf(fact_406_le__supI,axiom,
! [A4: set_Product_prod_v_v,X: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A4 @ X )
=> ( ( ord_le7336532860387713383od_v_v @ B2 @ X )
=> ( ord_le7336532860387713383od_v_v @ ( sup_su414716646722978715od_v_v @ A4 @ B2 ) @ X ) ) ) ).
% le_supI
thf(fact_407_le__supI,axiom,
! [A4: set_v,X: set_v,B2: set_v] :
( ( ord_less_eq_set_v @ A4 @ X )
=> ( ( ord_less_eq_set_v @ B2 @ X )
=> ( ord_less_eq_set_v @ ( sup_sup_set_v @ A4 @ B2 ) @ X ) ) ) ).
% le_supI
thf(fact_408_sup__ge1,axiom,
! [X: set_Product_prod_v_v,Y: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ X @ ( sup_su414716646722978715od_v_v @ X @ Y ) ) ).
% sup_ge1
thf(fact_409_sup__ge1,axiom,
! [X: set_v,Y: set_v] : ( ord_less_eq_set_v @ X @ ( sup_sup_set_v @ X @ Y ) ) ).
% sup_ge1
thf(fact_410_sup__ge2,axiom,
! [Y: set_Product_prod_v_v,X: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ Y @ ( sup_su414716646722978715od_v_v @ X @ Y ) ) ).
% sup_ge2
thf(fact_411_sup__ge2,axiom,
! [Y: set_v,X: set_v] : ( ord_less_eq_set_v @ Y @ ( sup_sup_set_v @ X @ Y ) ) ).
% sup_ge2
thf(fact_412_le__supI1,axiom,
! [X: set_Product_prod_v_v,A4: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X @ A4 )
=> ( ord_le7336532860387713383od_v_v @ X @ ( sup_su414716646722978715od_v_v @ A4 @ B2 ) ) ) ).
% le_supI1
thf(fact_413_le__supI1,axiom,
! [X: set_v,A4: set_v,B2: set_v] :
( ( ord_less_eq_set_v @ X @ A4 )
=> ( ord_less_eq_set_v @ X @ ( sup_sup_set_v @ A4 @ B2 ) ) ) ).
% le_supI1
thf(fact_414_le__supI2,axiom,
! [X: set_Product_prod_v_v,B2: set_Product_prod_v_v,A4: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X @ B2 )
=> ( ord_le7336532860387713383od_v_v @ X @ ( sup_su414716646722978715od_v_v @ A4 @ B2 ) ) ) ).
% le_supI2
thf(fact_415_le__supI2,axiom,
! [X: set_v,B2: set_v,A4: set_v] :
( ( ord_less_eq_set_v @ X @ B2 )
=> ( ord_less_eq_set_v @ X @ ( sup_sup_set_v @ A4 @ B2 ) ) ) ).
% le_supI2
thf(fact_416_sup_Omono,axiom,
! [C: set_Product_prod_v_v,A4: set_Product_prod_v_v,D2: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ C @ A4 )
=> ( ( ord_le7336532860387713383od_v_v @ D2 @ B2 )
=> ( ord_le7336532860387713383od_v_v @ ( sup_su414716646722978715od_v_v @ C @ D2 ) @ ( sup_su414716646722978715od_v_v @ A4 @ B2 ) ) ) ) ).
% sup.mono
thf(fact_417_sup_Omono,axiom,
! [C: set_v,A4: set_v,D2: set_v,B2: set_v] :
( ( ord_less_eq_set_v @ C @ A4 )
=> ( ( ord_less_eq_set_v @ D2 @ B2 )
=> ( ord_less_eq_set_v @ ( sup_sup_set_v @ C @ D2 ) @ ( sup_sup_set_v @ A4 @ B2 ) ) ) ) ).
% sup.mono
thf(fact_418_sup__mono,axiom,
! [A4: set_Product_prod_v_v,C: set_Product_prod_v_v,B2: set_Product_prod_v_v,D2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A4 @ C )
=> ( ( ord_le7336532860387713383od_v_v @ B2 @ D2 )
=> ( ord_le7336532860387713383od_v_v @ ( sup_su414716646722978715od_v_v @ A4 @ B2 ) @ ( sup_su414716646722978715od_v_v @ C @ D2 ) ) ) ) ).
% sup_mono
thf(fact_419_sup__mono,axiom,
! [A4: set_v,C: set_v,B2: set_v,D2: set_v] :
( ( ord_less_eq_set_v @ A4 @ C )
=> ( ( ord_less_eq_set_v @ B2 @ D2 )
=> ( ord_less_eq_set_v @ ( sup_sup_set_v @ A4 @ B2 ) @ ( sup_sup_set_v @ C @ D2 ) ) ) ) ).
% sup_mono
thf(fact_420_sup__least,axiom,
! [Y: set_Product_prod_v_v,X: set_Product_prod_v_v,Z: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ Y @ X )
=> ( ( ord_le7336532860387713383od_v_v @ Z @ X )
=> ( ord_le7336532860387713383od_v_v @ ( sup_su414716646722978715od_v_v @ Y @ Z ) @ X ) ) ) ).
% sup_least
thf(fact_421_sup__least,axiom,
! [Y: set_v,X: set_v,Z: set_v] :
( ( ord_less_eq_set_v @ Y @ X )
=> ( ( ord_less_eq_set_v @ Z @ X )
=> ( ord_less_eq_set_v @ ( sup_sup_set_v @ Y @ Z ) @ X ) ) ) ).
% sup_least
thf(fact_422_le__iff__sup,axiom,
( ord_le7336532860387713383od_v_v
= ( ^ [X2: set_Product_prod_v_v,Y3: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ X2 @ Y3 )
= Y3 ) ) ) ).
% le_iff_sup
thf(fact_423_le__iff__sup,axiom,
( ord_less_eq_set_v
= ( ^ [X2: set_v,Y3: set_v] :
( ( sup_sup_set_v @ X2 @ Y3 )
= Y3 ) ) ) ).
% le_iff_sup
thf(fact_424_sup_OorderE,axiom,
! [B2: set_Product_prod_v_v,A4: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ B2 @ A4 )
=> ( A4
= ( sup_su414716646722978715od_v_v @ A4 @ B2 ) ) ) ).
% sup.orderE
thf(fact_425_sup_OorderE,axiom,
! [B2: set_v,A4: set_v] :
( ( ord_less_eq_set_v @ B2 @ A4 )
=> ( A4
= ( sup_sup_set_v @ A4 @ B2 ) ) ) ).
% sup.orderE
thf(fact_426_sup_OorderI,axiom,
! [A4: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( A4
= ( sup_su414716646722978715od_v_v @ A4 @ B2 ) )
=> ( ord_le7336532860387713383od_v_v @ B2 @ A4 ) ) ).
% sup.orderI
thf(fact_427_sup_OorderI,axiom,
! [A4: set_v,B2: set_v] :
( ( A4
= ( sup_sup_set_v @ A4 @ B2 ) )
=> ( ord_less_eq_set_v @ B2 @ A4 ) ) ).
% sup.orderI
thf(fact_428_sup__unique,axiom,
! [F: set_Product_prod_v_v > set_Product_prod_v_v > set_Product_prod_v_v,X: set_Product_prod_v_v,Y: set_Product_prod_v_v] :
( ! [X3: set_Product_prod_v_v,Y2: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ X3 @ ( F @ X3 @ Y2 ) )
=> ( ! [X3: set_Product_prod_v_v,Y2: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ Y2 @ ( F @ X3 @ Y2 ) )
=> ( ! [X3: set_Product_prod_v_v,Y2: set_Product_prod_v_v,Z2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ Y2 @ X3 )
=> ( ( ord_le7336532860387713383od_v_v @ Z2 @ X3 )
=> ( ord_le7336532860387713383od_v_v @ ( F @ Y2 @ Z2 ) @ X3 ) ) )
=> ( ( sup_su414716646722978715od_v_v @ X @ Y )
= ( F @ X @ Y ) ) ) ) ) ).
% sup_unique
thf(fact_429_sup__unique,axiom,
! [F: set_v > set_v > set_v,X: set_v,Y: set_v] :
( ! [X3: set_v,Y2: set_v] : ( ord_less_eq_set_v @ X3 @ ( F @ X3 @ Y2 ) )
=> ( ! [X3: set_v,Y2: set_v] : ( ord_less_eq_set_v @ Y2 @ ( F @ X3 @ Y2 ) )
=> ( ! [X3: set_v,Y2: set_v,Z2: set_v] :
( ( ord_less_eq_set_v @ Y2 @ X3 )
=> ( ( ord_less_eq_set_v @ Z2 @ X3 )
=> ( ord_less_eq_set_v @ ( F @ Y2 @ Z2 ) @ X3 ) ) )
=> ( ( sup_sup_set_v @ X @ Y )
= ( F @ X @ Y ) ) ) ) ) ).
% sup_unique
thf(fact_430_sup_Oabsorb1,axiom,
! [B2: set_Product_prod_v_v,A4: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ B2 @ A4 )
=> ( ( sup_su414716646722978715od_v_v @ A4 @ B2 )
= A4 ) ) ).
% sup.absorb1
thf(fact_431_sup_Oabsorb1,axiom,
! [B2: set_v,A4: set_v] :
( ( ord_less_eq_set_v @ B2 @ A4 )
=> ( ( sup_sup_set_v @ A4 @ B2 )
= A4 ) ) ).
% sup.absorb1
thf(fact_432_sup_Oabsorb2,axiom,
! [A4: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A4 @ B2 )
=> ( ( sup_su414716646722978715od_v_v @ A4 @ B2 )
= B2 ) ) ).
% sup.absorb2
thf(fact_433_sup_Oabsorb2,axiom,
! [A4: set_v,B2: set_v] :
( ( ord_less_eq_set_v @ A4 @ B2 )
=> ( ( sup_sup_set_v @ A4 @ B2 )
= B2 ) ) ).
% sup.absorb2
thf(fact_434_sup__absorb1,axiom,
! [Y: set_Product_prod_v_v,X: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ Y @ X )
=> ( ( sup_su414716646722978715od_v_v @ X @ Y )
= X ) ) ).
% sup_absorb1
thf(fact_435_sup__absorb1,axiom,
! [Y: set_v,X: set_v] :
( ( ord_less_eq_set_v @ Y @ X )
=> ( ( sup_sup_set_v @ X @ Y )
= X ) ) ).
% sup_absorb1
thf(fact_436_sup__absorb2,axiom,
! [X: set_Product_prod_v_v,Y: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X @ Y )
=> ( ( sup_su414716646722978715od_v_v @ X @ Y )
= Y ) ) ).
% sup_absorb2
thf(fact_437_sup__absorb2,axiom,
! [X: set_v,Y: set_v] :
( ( ord_less_eq_set_v @ X @ Y )
=> ( ( sup_sup_set_v @ X @ Y )
= Y ) ) ).
% sup_absorb2
thf(fact_438_sup_OboundedE,axiom,
! [B2: set_Product_prod_v_v,C: set_Product_prod_v_v,A4: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ ( sup_su414716646722978715od_v_v @ B2 @ C ) @ A4 )
=> ~ ( ( ord_le7336532860387713383od_v_v @ B2 @ A4 )
=> ~ ( ord_le7336532860387713383od_v_v @ C @ A4 ) ) ) ).
% sup.boundedE
thf(fact_439_sup_OboundedE,axiom,
! [B2: set_v,C: set_v,A4: set_v] :
( ( ord_less_eq_set_v @ ( sup_sup_set_v @ B2 @ C ) @ A4 )
=> ~ ( ( ord_less_eq_set_v @ B2 @ A4 )
=> ~ ( ord_less_eq_set_v @ C @ A4 ) ) ) ).
% sup.boundedE
thf(fact_440_sup_OboundedI,axiom,
! [B2: set_Product_prod_v_v,A4: set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ B2 @ A4 )
=> ( ( ord_le7336532860387713383od_v_v @ C @ A4 )
=> ( ord_le7336532860387713383od_v_v @ ( sup_su414716646722978715od_v_v @ B2 @ C ) @ A4 ) ) ) ).
% sup.boundedI
thf(fact_441_sup_OboundedI,axiom,
! [B2: set_v,A4: set_v,C: set_v] :
( ( ord_less_eq_set_v @ B2 @ A4 )
=> ( ( ord_less_eq_set_v @ C @ A4 )
=> ( ord_less_eq_set_v @ ( sup_sup_set_v @ B2 @ C ) @ A4 ) ) ) ).
% sup.boundedI
thf(fact_442_sup_Oorder__iff,axiom,
( ord_le7336532860387713383od_v_v
= ( ^ [B3: set_Product_prod_v_v,A5: set_Product_prod_v_v] :
( A5
= ( sup_su414716646722978715od_v_v @ A5 @ B3 ) ) ) ) ).
% sup.order_iff
thf(fact_443_sup_Oorder__iff,axiom,
( ord_less_eq_set_v
= ( ^ [B3: set_v,A5: set_v] :
( A5
= ( sup_sup_set_v @ A5 @ B3 ) ) ) ) ).
% sup.order_iff
thf(fact_444_sup_Ocobounded1,axiom,
! [A4: set_Product_prod_v_v,B2: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ A4 @ ( sup_su414716646722978715od_v_v @ A4 @ B2 ) ) ).
% sup.cobounded1
thf(fact_445_sup_Ocobounded1,axiom,
! [A4: set_v,B2: set_v] : ( ord_less_eq_set_v @ A4 @ ( sup_sup_set_v @ A4 @ B2 ) ) ).
% sup.cobounded1
thf(fact_446_sup_Ocobounded2,axiom,
! [B2: set_Product_prod_v_v,A4: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ B2 @ ( sup_su414716646722978715od_v_v @ A4 @ B2 ) ) ).
% sup.cobounded2
thf(fact_447_sup_Ocobounded2,axiom,
! [B2: set_v,A4: set_v] : ( ord_less_eq_set_v @ B2 @ ( sup_sup_set_v @ A4 @ B2 ) ) ).
% sup.cobounded2
thf(fact_448_sup_Oabsorb__iff1,axiom,
( ord_le7336532860387713383od_v_v
= ( ^ [B3: set_Product_prod_v_v,A5: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ A5 @ B3 )
= A5 ) ) ) ).
% sup.absorb_iff1
thf(fact_449_sup_Oabsorb__iff1,axiom,
( ord_less_eq_set_v
= ( ^ [B3: set_v,A5: set_v] :
( ( sup_sup_set_v @ A5 @ B3 )
= A5 ) ) ) ).
% sup.absorb_iff1
thf(fact_450_sup_Oabsorb__iff2,axiom,
( ord_le7336532860387713383od_v_v
= ( ^ [A5: set_Product_prod_v_v,B3: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ A5 @ B3 )
= B3 ) ) ) ).
% sup.absorb_iff2
thf(fact_451_sup_Oabsorb__iff2,axiom,
( ord_less_eq_set_v
= ( ^ [A5: set_v,B3: set_v] :
( ( sup_sup_set_v @ A5 @ B3 )
= B3 ) ) ) ).
% sup.absorb_iff2
thf(fact_452_sup_OcoboundedI1,axiom,
! [C: set_Product_prod_v_v,A4: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ C @ A4 )
=> ( ord_le7336532860387713383od_v_v @ C @ ( sup_su414716646722978715od_v_v @ A4 @ B2 ) ) ) ).
% sup.coboundedI1
thf(fact_453_sup_OcoboundedI1,axiom,
! [C: set_v,A4: set_v,B2: set_v] :
( ( ord_less_eq_set_v @ C @ A4 )
=> ( ord_less_eq_set_v @ C @ ( sup_sup_set_v @ A4 @ B2 ) ) ) ).
% sup.coboundedI1
thf(fact_454_sup_OcoboundedI2,axiom,
! [C: set_Product_prod_v_v,B2: set_Product_prod_v_v,A4: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ C @ B2 )
=> ( ord_le7336532860387713383od_v_v @ C @ ( sup_su414716646722978715od_v_v @ A4 @ B2 ) ) ) ).
% sup.coboundedI2
thf(fact_455_sup_OcoboundedI2,axiom,
! [C: set_v,B2: set_v,A4: set_v] :
( ( ord_less_eq_set_v @ C @ B2 )
=> ( ord_less_eq_set_v @ C @ ( sup_sup_set_v @ A4 @ B2 ) ) ) ).
% sup.coboundedI2
thf(fact_456_graph_Ostack__class,axiom,
! [Vertices: set_v,Successors: v > set_v,E2: sCC_Bl1394983891496994913t_unit,N: v,M: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl9196236973127232072t_unit @ Successors @ E2 )
=> ( ( member_v @ N @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E2 ) ) )
=> ( ( member_v @ M @ ( sCC_Bl1280885523602775798t_unit @ E2 @ N ) )
=> ( member_v @ M @ ( minus_minus_set_v @ ( sCC_Bl4645233313691564917t_unit @ E2 ) @ ( sCC_Bl157864678168468314t_unit @ E2 ) ) ) ) ) ) ) ).
% graph.stack_class
thf(fact_457_graph_Ounite__wf__env,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,V: product_prod_v_v,E2: sCC_Bl7326425374436813197t_unit,W: product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( sCC_Bl3607325323686918683t_unit @ Successors @ V @ E2 )
=> ( ( member7453568604450474000od_v_v @ W @ ( Successors @ V ) )
=> ( ~ ( member7453568604450474000od_v_v @ W @ ( sCC_Bl3878977043676959280t_unit @ E2 @ V ) )
=> ( ( member7453568604450474000od_v_v @ W @ ( sCC_Bl5498988629518860705t_unit @ E2 ) )
=> ( ~ ( member7453568604450474000od_v_v @ W @ ( sCC_Bl5094201334446601350t_unit @ E2 ) )
=> ( sCC_Bl7798947040364291444t_unit @ Successors @ ( sCC_Bl4702006153222411093od_v_v @ V @ W @ E2 ) ) ) ) ) ) ) ) ).
% graph.unite_wf_env
thf(fact_458_graph_Ounite__wf__env,axiom,
! [Vertices: set_v,Successors: v > set_v,V: v,E2: sCC_Bl1394983891496994913t_unit,W: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl1748261141445803503t_unit @ Successors @ V @ E2 )
=> ( ( member_v @ W @ ( Successors @ V ) )
=> ( ~ ( member_v @ W @ ( sCC_Bl3795065053823578884t_unit @ E2 @ V ) )
=> ( ( member_v @ W @ ( sCC_Bl4645233313691564917t_unit @ E2 ) )
=> ( ~ ( member_v @ W @ ( sCC_Bl157864678168468314t_unit @ E2 ) )
=> ( sCC_Bl9196236973127232072t_unit @ Successors @ ( sCC_Bloemen_unite_v @ V @ W @ E2 ) ) ) ) ) ) ) ) ).
% graph.unite_wf_env
thf(fact_459_graph_Ounite__sub__env,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,V: product_prod_v_v,E2: sCC_Bl7326425374436813197t_unit,W: product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( sCC_Bl3607325323686918683t_unit @ Successors @ V @ E2 )
=> ( ( member7453568604450474000od_v_v @ W @ ( Successors @ V ) )
=> ( ~ ( member7453568604450474000od_v_v @ W @ ( sCC_Bl3878977043676959280t_unit @ E2 @ V ) )
=> ( ( member7453568604450474000od_v_v @ W @ ( sCC_Bl5498988629518860705t_unit @ E2 ) )
=> ( ~ ( member7453568604450474000od_v_v @ W @ ( sCC_Bl5094201334446601350t_unit @ E2 ) )
=> ( sCC_Bl7963838319573962697t_unit @ E2 @ ( sCC_Bl4702006153222411093od_v_v @ V @ W @ E2 ) ) ) ) ) ) ) ) ).
% graph.unite_sub_env
thf(fact_460_graph_Ounite__sub__env,axiom,
! [Vertices: set_v,Successors: v > set_v,V: v,E2: sCC_Bl1394983891496994913t_unit,W: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl1748261141445803503t_unit @ Successors @ V @ E2 )
=> ( ( member_v @ W @ ( Successors @ V ) )
=> ( ~ ( member_v @ W @ ( sCC_Bl3795065053823578884t_unit @ E2 @ V ) )
=> ( ( member_v @ W @ ( sCC_Bl4645233313691564917t_unit @ E2 ) )
=> ( ~ ( member_v @ W @ ( sCC_Bl157864678168468314t_unit @ E2 ) )
=> ( sCC_Bl5768913643336123637t_unit @ E2 @ ( sCC_Bloemen_unite_v @ V @ W @ E2 ) ) ) ) ) ) ) ) ).
% graph.unite_sub_env
thf(fact_461_graph_Ounite__S__tl,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,E2: sCC_Bl7326425374436813197t_unit,W: product_prod_v_v,V: product_prod_v_v,N: product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( sCC_Bl7798947040364291444t_unit @ Successors @ E2 )
=> ( ( member7453568604450474000od_v_v @ W @ ( Successors @ V ) )
=> ( ~ ( member7453568604450474000od_v_v @ W @ ( sCC_Bl3878977043676959280t_unit @ E2 @ V ) )
=> ( ( member7453568604450474000od_v_v @ W @ ( sCC_Bl5498988629518860705t_unit @ E2 ) )
=> ( ~ ( member7453568604450474000od_v_v @ W @ ( sCC_Bl5094201334446601350t_unit @ E2 ) )
=> ( ( member7453568604450474000od_v_v @ N @ ( set_Product_prod_v_v2 @ ( tl_Product_prod_v_v @ ( sCC_Bl2021302119412358655t_unit @ ( sCC_Bl4702006153222411093od_v_v @ V @ W @ E2 ) ) ) ) )
=> ( ( sCC_Bl8440648026628373538t_unit @ ( sCC_Bl4702006153222411093od_v_v @ V @ W @ E2 ) @ N )
= ( sCC_Bl8440648026628373538t_unit @ E2 @ N ) ) ) ) ) ) ) ) ) ).
% graph.unite_S_tl
thf(fact_462_graph_Ounite__S__tl,axiom,
! [Vertices: set_v,Successors: v > set_v,E2: sCC_Bl1394983891496994913t_unit,W: v,V: v,N: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl9196236973127232072t_unit @ Successors @ E2 )
=> ( ( member_v @ W @ ( Successors @ V ) )
=> ( ~ ( member_v @ W @ ( sCC_Bl3795065053823578884t_unit @ E2 @ V ) )
=> ( ( member_v @ W @ ( sCC_Bl4645233313691564917t_unit @ E2 ) )
=> ( ~ ( member_v @ W @ ( sCC_Bl157864678168468314t_unit @ E2 ) )
=> ( ( member_v @ N @ ( set_v2 @ ( tl_v @ ( sCC_Bl8828226123343373779t_unit @ ( sCC_Bloemen_unite_v @ V @ W @ E2 ) ) ) ) )
=> ( ( sCC_Bl1280885523602775798t_unit @ ( sCC_Bloemen_unite_v @ V @ W @ E2 ) @ N )
= ( sCC_Bl1280885523602775798t_unit @ E2 @ N ) ) ) ) ) ) ) ) ) ).
% graph.unite_S_tl
thf(fact_463_graph_Ounite__subscc,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,V: product_prod_v_v,E2: sCC_Bl7326425374436813197t_unit,W: product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( sCC_Bl3607325323686918683t_unit @ Successors @ V @ E2 )
=> ( ( member7453568604450474000od_v_v @ W @ ( Successors @ V ) )
=> ( ~ ( member7453568604450474000od_v_v @ W @ ( sCC_Bl3878977043676959280t_unit @ E2 @ V ) )
=> ( ( member7453568604450474000od_v_v @ W @ ( sCC_Bl5498988629518860705t_unit @ E2 ) )
=> ( ~ ( member7453568604450474000od_v_v @ W @ ( sCC_Bl5094201334446601350t_unit @ E2 ) )
=> ( sCC_Bl2301996248249672505od_v_v @ Successors @ ( sCC_Bl8440648026628373538t_unit @ ( sCC_Bl4702006153222411093od_v_v @ V @ W @ E2 ) @ ( hd_Product_prod_v_v @ ( sCC_Bl2021302119412358655t_unit @ ( sCC_Bl4702006153222411093od_v_v @ V @ W @ E2 ) ) ) ) ) ) ) ) ) ) ) ).
% graph.unite_subscc
thf(fact_464_graph_Ounite__subscc,axiom,
! [Vertices: set_v,Successors: v > set_v,V: v,E2: sCC_Bl1394983891496994913t_unit,W: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl1748261141445803503t_unit @ Successors @ V @ E2 )
=> ( ( member_v @ W @ ( Successors @ V ) )
=> ( ~ ( member_v @ W @ ( sCC_Bl3795065053823578884t_unit @ E2 @ V ) )
=> ( ( member_v @ W @ ( sCC_Bl4645233313691564917t_unit @ E2 ) )
=> ( ~ ( member_v @ W @ ( sCC_Bl157864678168468314t_unit @ E2 ) )
=> ( sCC_Bl5398416737448265317bscc_v @ Successors @ ( sCC_Bl1280885523602775798t_unit @ ( sCC_Bloemen_unite_v @ V @ W @ E2 ) @ ( hd_v @ ( sCC_Bl8828226123343373779t_unit @ ( sCC_Bloemen_unite_v @ V @ W @ E2 ) ) ) ) ) ) ) ) ) ) ) ).
% graph.unite_subscc
thf(fact_465_avoiding__explored,axiom,
! [E2: sCC_Bl1394983891496994913t_unit,X: v,Y: v,E: set_Product_prod_v_v,W: v,V: v] :
( ( sCC_Bl9196236973127232072t_unit @ successors @ E2 )
=> ( ( sCC_Bl4291963740693775144ding_v @ successors @ X @ Y @ E )
=> ( ~ ( member_v @ Y @ ( sCC_Bl157864678168468314t_unit @ E2 ) )
=> ( ( member_v @ W @ ( sCC_Bl157864678168468314t_unit @ E2 ) )
=> ( sCC_Bl4291963740693775144ding_v @ successors @ X @ Y @ ( sup_su414716646722978715od_v_v @ E @ ( insert1338601472111419319od_v_v @ ( product_Pair_v_v @ V @ W ) @ bot_bo723834152578015283od_v_v ) ) ) ) ) ) ) ).
% avoiding_explored
thf(fact_466_post__dfs__def,axiom,
! [V: v,E2: sCC_Bl1394983891496994913t_unit,E3: sCC_Bl1394983891496994913t_unit] :
( ( sCC_Bl8953792750115413617t_unit @ successors @ V @ E2 @ E3 )
= ( ( sCC_Bl9196236973127232072t_unit @ successors @ E3 )
& ( member_v @ V @ ( sCC_Bl4645233313691564917t_unit @ E3 ) )
& ( sCC_Bl5768913643336123637t_unit @ E2 @ E3 )
& ( ( sCC_Bl3795065053823578884t_unit @ E3 @ V )
= ( successors @ V ) )
& ! [X2: v] :
( ( member_v @ X2 @ ( sCC_Bl4645233313691564917t_unit @ E2 ) )
=> ( ( sCC_Bl3795065053823578884t_unit @ E3 @ X2 )
= ( sCC_Bl3795065053823578884t_unit @ E2 @ X2 ) ) )
& ! [X2: v] :
( ( member_v @ X2 @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E3 ) ) )
=> ( sCC_Bl649662514949026229able_v @ successors @ X2 @ V ) )
& ? [Ns: list_v] :
( ( sCC_Bl8828226123343373779t_unit @ E2 )
= ( append_v @ Ns @ ( sCC_Bl8828226123343373779t_unit @ E3 ) ) )
& ( ( ( member_v @ V @ ( sCC_Bl157864678168468314t_unit @ E3 ) )
& ( ( sCC_Bl8828226123343373779t_unit @ E3 )
= ( sCC_Bl8828226123343373779t_unit @ E2 ) )
& ! [X2: v] :
( ( member_v @ X2 @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E3 ) ) )
=> ( ( sCC_Bl1280885523602775798t_unit @ E3 @ X2 )
= ( sCC_Bl1280885523602775798t_unit @ E2 @ X2 ) ) ) )
| ( ( ( sCC_Bl8828226123343373779t_unit @ E3 )
!= nil_v )
& ( member_v @ V @ ( sCC_Bl1280885523602775798t_unit @ E3 @ ( hd_v @ ( sCC_Bl8828226123343373779t_unit @ E3 ) ) ) )
& ! [X2: v] :
( ( member_v @ X2 @ ( set_v2 @ ( tl_v @ ( sCC_Bl8828226123343373779t_unit @ E3 ) ) ) )
=> ( ( sCC_Bl1280885523602775798t_unit @ E3 @ X2 )
= ( sCC_Bl1280885523602775798t_unit @ E2 @ X2 ) ) ) ) )
& ( ( sCC_Bl9201514103433284750t_unit @ E3 )
= ( sCC_Bl9201514103433284750t_unit @ E2 ) ) ) ) ).
% post_dfs_def
thf(fact_467_pre__dfss__def,axiom,
! [V: v,E2: sCC_Bl1394983891496994913t_unit] :
( ( sCC_Bl1748261141445803503t_unit @ successors @ V @ E2 )
= ( ( sCC_Bl9196236973127232072t_unit @ successors @ E2 )
& ( member_v @ V @ ( sCC_Bl4645233313691564917t_unit @ E2 ) )
& ( ( sCC_Bl8828226123343373779t_unit @ E2 )
!= nil_v )
& ( member_v @ V @ ( sCC_Bl1280885523602775798t_unit @ E2 @ ( hd_v @ ( sCC_Bl8828226123343373779t_unit @ E2 ) ) ) )
& ! [X2: v] :
( ( member_v @ X2 @ ( sCC_Bl3795065053823578884t_unit @ E2 @ V ) )
=> ( member_v @ X2 @ ( sup_sup_set_v @ ( sCC_Bl157864678168468314t_unit @ E2 ) @ ( sCC_Bl1280885523602775798t_unit @ E2 @ ( hd_v @ ( sCC_Bl8828226123343373779t_unit @ E2 ) ) ) ) ) )
& ! [X2: v] :
( ( member_v @ X2 @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E2 ) ) )
=> ( sCC_Bl649662514949026229able_v @ successors @ X2 @ V ) )
& ? [Ns: list_v] :
( ( sCC_Bl9201514103433284750t_unit @ E2 )
= ( cons_v @ V @ Ns ) ) ) ) ).
% pre_dfss_def
thf(fact_468_ra__add__edge,axiom,
! [X: v,Y: v,E: set_Product_prod_v_v,V: v,W: v] :
( ( sCC_Bl4291963740693775144ding_v @ successors @ X @ Y @ E )
=> ( ( sCC_Bl4291963740693775144ding_v @ successors @ X @ Y @ ( sup_su414716646722978715od_v_v @ E @ ( insert1338601472111419319od_v_v @ ( product_Pair_v_v @ V @ W ) @ bot_bo723834152578015283od_v_v ) ) )
| ( ( sCC_Bl4291963740693775144ding_v @ successors @ X @ V @ ( sup_su414716646722978715od_v_v @ E @ ( insert1338601472111419319od_v_v @ ( product_Pair_v_v @ V @ W ) @ bot_bo723834152578015283od_v_v ) ) )
& ( sCC_Bl4291963740693775144ding_v @ successors @ W @ Y @ ( sup_su414716646722978715od_v_v @ E @ ( insert1338601472111419319od_v_v @ ( product_Pair_v_v @ V @ W ) @ bot_bo723834152578015283od_v_v ) ) ) ) ) ) ).
% ra_add_edge
thf(fact_469_set__union,axiom,
! [Xs: list_v,Ys: list_v] :
( ( set_v2 @ ( union_v @ Xs @ Ys ) )
= ( sup_sup_set_v @ ( set_v2 @ Xs ) @ ( set_v2 @ Ys ) ) ) ).
% set_union
thf(fact_470_set__union,axiom,
! [Xs: list_P7986770385144383213od_v_v,Ys: list_P7986770385144383213od_v_v] :
( ( set_Product_prod_v_v2 @ ( union_4602324378607836129od_v_v @ Xs @ Ys ) )
= ( sup_su414716646722978715od_v_v @ ( set_Product_prod_v_v2 @ Xs ) @ ( set_Product_prod_v_v2 @ Ys ) ) ) ).
% set_union
thf(fact_471_subscc__add,axiom,
! [S: set_v,X: v,Y: v] :
( ( sCC_Bl5398416737448265317bscc_v @ successors @ S )
=> ( ( member_v @ X @ S )
=> ( ( sCC_Bl649662514949026229able_v @ successors @ X @ Y )
=> ( ( sCC_Bl649662514949026229able_v @ successors @ Y @ X )
=> ( sCC_Bl5398416737448265317bscc_v @ successors @ ( insert_v2 @ Y @ S ) ) ) ) ) ) ).
% subscc_add
thf(fact_472_diff__shunt__var,axiom,
! [X: set_Product_prod_v_v,Y: set_Product_prod_v_v] :
( ( ( minus_4183494784930505774od_v_v @ X @ Y )
= bot_bo723834152578015283od_v_v )
= ( ord_le7336532860387713383od_v_v @ X @ Y ) ) ).
% diff_shunt_var
thf(fact_473_diff__shunt__var,axiom,
! [X: set_v,Y: set_v] :
( ( ( minus_minus_set_v @ X @ Y )
= bot_bot_set_v )
= ( ord_less_eq_set_v @ X @ Y ) ) ).
% diff_shunt_var
thf(fact_474_insert__absorb2,axiom,
! [X: product_prod_v_v,A: set_Product_prod_v_v] :
( ( insert1338601472111419319od_v_v @ X @ ( insert1338601472111419319od_v_v @ X @ A ) )
= ( insert1338601472111419319od_v_v @ X @ A ) ) ).
% insert_absorb2
thf(fact_475_insert__absorb2,axiom,
! [X: v,A: set_v] :
( ( insert_v2 @ X @ ( insert_v2 @ X @ A ) )
= ( insert_v2 @ X @ A ) ) ).
% insert_absorb2
thf(fact_476_insert__iff,axiom,
! [A4: v,B2: v,A: set_v] :
( ( member_v @ A4 @ ( insert_v2 @ B2 @ A ) )
= ( ( A4 = B2 )
| ( member_v @ A4 @ A ) ) ) ).
% insert_iff
thf(fact_477_insert__iff,axiom,
! [A4: product_prod_v_v,B2: product_prod_v_v,A: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ A4 @ ( insert1338601472111419319od_v_v @ B2 @ A ) )
= ( ( A4 = B2 )
| ( member7453568604450474000od_v_v @ A4 @ A ) ) ) ).
% insert_iff
thf(fact_478_insertCI,axiom,
! [A4: v,B: set_v,B2: v] :
( ( ~ ( member_v @ A4 @ B )
=> ( A4 = B2 ) )
=> ( member_v @ A4 @ ( insert_v2 @ B2 @ B ) ) ) ).
% insertCI
thf(fact_479_insertCI,axiom,
! [A4: product_prod_v_v,B: set_Product_prod_v_v,B2: product_prod_v_v] :
( ( ~ ( member7453568604450474000od_v_v @ A4 @ B )
=> ( A4 = B2 ) )
=> ( member7453568604450474000od_v_v @ A4 @ ( insert1338601472111419319od_v_v @ B2 @ B ) ) ) ).
% insertCI
thf(fact_480_same__append__eq,axiom,
! [Xs: list_v,Ys: list_v,Zs: list_v] :
( ( ( append_v @ Xs @ Ys )
= ( append_v @ Xs @ Zs ) )
= ( Ys = Zs ) ) ).
% same_append_eq
thf(fact_481_append__same__eq,axiom,
! [Ys: list_v,Xs: list_v,Zs: list_v] :
( ( ( append_v @ Ys @ Xs )
= ( append_v @ Zs @ Xs ) )
= ( Ys = Zs ) ) ).
% append_same_eq
thf(fact_482_append__assoc,axiom,
! [Xs: list_v,Ys: list_v,Zs: list_v] :
( ( append_v @ ( append_v @ Xs @ Ys ) @ Zs )
= ( append_v @ Xs @ ( append_v @ Ys @ Zs ) ) ) ).
% append_assoc
thf(fact_483_append_Oassoc,axiom,
! [A4: list_v,B2: list_v,C: list_v] :
( ( append_v @ ( append_v @ A4 @ B2 ) @ C )
= ( append_v @ A4 @ ( append_v @ B2 @ C ) ) ) ).
% append.assoc
thf(fact_484__092_060open_062_092_060exists_062ns_O_Astack_Ae_A_061_Ans_A_064_Astack_Ae_092_060close_062,axiom,
? [Ns2: list_v] :
( ( sCC_Bl8828226123343373779t_unit @ ea )
= ( append_v @ Ns2 @ ( sCC_Bl8828226123343373779t_unit @ ea ) ) ) ).
% \<open>\<exists>ns. stack e = ns @ stack e\<close>
thf(fact_485_singletonI,axiom,
! [A4: product_prod_v_v] : ( member7453568604450474000od_v_v @ A4 @ ( insert1338601472111419319od_v_v @ A4 @ bot_bo723834152578015283od_v_v ) ) ).
% singletonI
thf(fact_486_singletonI,axiom,
! [A4: v] : ( member_v @ A4 @ ( insert_v2 @ A4 @ bot_bot_set_v ) ) ).
% singletonI
thf(fact_487_insert__subset,axiom,
! [X: product_prod_v_v,A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ ( insert1338601472111419319od_v_v @ X @ A ) @ B )
= ( ( member7453568604450474000od_v_v @ X @ B )
& ( ord_le7336532860387713383od_v_v @ A @ B ) ) ) ).
% insert_subset
thf(fact_488_insert__subset,axiom,
! [X: v,A: set_v,B: set_v] :
( ( ord_less_eq_set_v @ ( insert_v2 @ X @ A ) @ B )
= ( ( member_v @ X @ B )
& ( ord_less_eq_set_v @ A @ B ) ) ) ).
% insert_subset
thf(fact_489_Un__insert__right,axiom,
! [A: set_v,A4: v,B: set_v] :
( ( sup_sup_set_v @ A @ ( insert_v2 @ A4 @ B ) )
= ( insert_v2 @ A4 @ ( sup_sup_set_v @ A @ B ) ) ) ).
% Un_insert_right
thf(fact_490_Un__insert__right,axiom,
! [A: set_Product_prod_v_v,A4: product_prod_v_v,B: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ A @ ( insert1338601472111419319od_v_v @ A4 @ B ) )
= ( insert1338601472111419319od_v_v @ A4 @ ( sup_su414716646722978715od_v_v @ A @ B ) ) ) ).
% Un_insert_right
thf(fact_491_Un__insert__left,axiom,
! [A4: v,B: set_v,C2: set_v] :
( ( sup_sup_set_v @ ( insert_v2 @ A4 @ B ) @ C2 )
= ( insert_v2 @ A4 @ ( sup_sup_set_v @ B @ C2 ) ) ) ).
% Un_insert_left
thf(fact_492_Un__insert__left,axiom,
! [A4: product_prod_v_v,B: set_Product_prod_v_v,C2: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ ( insert1338601472111419319od_v_v @ A4 @ B ) @ C2 )
= ( insert1338601472111419319od_v_v @ A4 @ ( sup_su414716646722978715od_v_v @ B @ C2 ) ) ) ).
% Un_insert_left
thf(fact_493_insert__Diff1,axiom,
! [X: product_prod_v_v,B: set_Product_prod_v_v,A: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X @ B )
=> ( ( minus_4183494784930505774od_v_v @ ( insert1338601472111419319od_v_v @ X @ A ) @ B )
= ( minus_4183494784930505774od_v_v @ A @ B ) ) ) ).
% insert_Diff1
thf(fact_494_insert__Diff1,axiom,
! [X: v,B: set_v,A: set_v] :
( ( member_v @ X @ B )
=> ( ( minus_minus_set_v @ ( insert_v2 @ X @ A ) @ B )
= ( minus_minus_set_v @ A @ B ) ) ) ).
% insert_Diff1
thf(fact_495_Diff__insert0,axiom,
! [X: product_prod_v_v,A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ~ ( member7453568604450474000od_v_v @ X @ A )
=> ( ( minus_4183494784930505774od_v_v @ A @ ( insert1338601472111419319od_v_v @ X @ B ) )
= ( minus_4183494784930505774od_v_v @ A @ B ) ) ) ).
% Diff_insert0
thf(fact_496_Diff__insert0,axiom,
! [X: v,A: set_v,B: set_v] :
( ~ ( member_v @ X @ A )
=> ( ( minus_minus_set_v @ A @ ( insert_v2 @ X @ B ) )
= ( minus_minus_set_v @ A @ B ) ) ) ).
% Diff_insert0
thf(fact_497_append_Oright__neutral,axiom,
! [A4: list_v] :
( ( append_v @ A4 @ nil_v )
= A4 ) ).
% append.right_neutral
thf(fact_498_append__Nil2,axiom,
! [Xs: list_v] :
( ( append_v @ Xs @ nil_v )
= Xs ) ).
% append_Nil2
thf(fact_499_append__self__conv,axiom,
! [Xs: list_v,Ys: list_v] :
( ( ( append_v @ Xs @ Ys )
= Xs )
= ( Ys = nil_v ) ) ).
% append_self_conv
thf(fact_500_self__append__conv,axiom,
! [Y: list_v,Ys: list_v] :
( ( Y
= ( append_v @ Y @ Ys ) )
= ( Ys = nil_v ) ) ).
% self_append_conv
thf(fact_501_append__self__conv2,axiom,
! [Xs: list_v,Ys: list_v] :
( ( ( append_v @ Xs @ Ys )
= Ys )
= ( Xs = nil_v ) ) ).
% append_self_conv2
thf(fact_502_self__append__conv2,axiom,
! [Y: list_v,Xs: list_v] :
( ( Y
= ( append_v @ Xs @ Y ) )
= ( Xs = nil_v ) ) ).
% self_append_conv2
thf(fact_503_Nil__is__append__conv,axiom,
! [Xs: list_v,Ys: list_v] :
( ( nil_v
= ( append_v @ Xs @ Ys ) )
= ( ( Xs = nil_v )
& ( Ys = nil_v ) ) ) ).
% Nil_is_append_conv
thf(fact_504_append__is__Nil__conv,axiom,
! [Xs: list_v,Ys: list_v] :
( ( ( append_v @ Xs @ Ys )
= nil_v )
= ( ( Xs = nil_v )
& ( Ys = nil_v ) ) ) ).
% append_is_Nil_conv
thf(fact_505_singleton__insert__inj__eq,axiom,
! [B2: product_prod_v_v,A4: product_prod_v_v,A: set_Product_prod_v_v] :
( ( ( insert1338601472111419319od_v_v @ B2 @ bot_bo723834152578015283od_v_v )
= ( insert1338601472111419319od_v_v @ A4 @ A ) )
= ( ( A4 = B2 )
& ( ord_le7336532860387713383od_v_v @ A @ ( insert1338601472111419319od_v_v @ B2 @ bot_bo723834152578015283od_v_v ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_506_singleton__insert__inj__eq,axiom,
! [B2: v,A4: v,A: set_v] :
( ( ( insert_v2 @ B2 @ bot_bot_set_v )
= ( insert_v2 @ A4 @ A ) )
= ( ( A4 = B2 )
& ( ord_less_eq_set_v @ A @ ( insert_v2 @ B2 @ bot_bot_set_v ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_507_singleton__insert__inj__eq_H,axiom,
! [A4: product_prod_v_v,A: set_Product_prod_v_v,B2: product_prod_v_v] :
( ( ( insert1338601472111419319od_v_v @ A4 @ A )
= ( insert1338601472111419319od_v_v @ B2 @ bot_bo723834152578015283od_v_v ) )
= ( ( A4 = B2 )
& ( ord_le7336532860387713383od_v_v @ A @ ( insert1338601472111419319od_v_v @ B2 @ bot_bo723834152578015283od_v_v ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_508_singleton__insert__inj__eq_H,axiom,
! [A4: v,A: set_v,B2: v] :
( ( ( insert_v2 @ A4 @ A )
= ( insert_v2 @ B2 @ bot_bot_set_v ) )
= ( ( A4 = B2 )
& ( ord_less_eq_set_v @ A @ ( insert_v2 @ B2 @ bot_bot_set_v ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_509_list_Osimps_I15_J,axiom,
! [X21: product_prod_v_v,X22: list_P7986770385144383213od_v_v] :
( ( set_Product_prod_v_v2 @ ( cons_P4120604216776828829od_v_v @ X21 @ X22 ) )
= ( insert1338601472111419319od_v_v @ X21 @ ( set_Product_prod_v_v2 @ X22 ) ) ) ).
% list.simps(15)
thf(fact_510_list_Osimps_I15_J,axiom,
! [X21: v,X22: list_v] :
( ( set_v2 @ ( cons_v @ X21 @ X22 ) )
= ( insert_v2 @ X21 @ ( set_v2 @ X22 ) ) ) ).
% list.simps(15)
thf(fact_511_insert__Diff__single,axiom,
! [A4: product_prod_v_v,A: set_Product_prod_v_v] :
( ( insert1338601472111419319od_v_v @ A4 @ ( minus_4183494784930505774od_v_v @ A @ ( insert1338601472111419319od_v_v @ A4 @ bot_bo723834152578015283od_v_v ) ) )
= ( insert1338601472111419319od_v_v @ A4 @ A ) ) ).
% insert_Diff_single
thf(fact_512_insert__Diff__single,axiom,
! [A4: v,A: set_v] :
( ( insert_v2 @ A4 @ ( minus_minus_set_v @ A @ ( insert_v2 @ A4 @ bot_bot_set_v ) ) )
= ( insert_v2 @ A4 @ A ) ) ).
% insert_Diff_single
thf(fact_513_append1__eq__conv,axiom,
! [Xs: list_v,X: v,Ys: list_v,Y: v] :
( ( ( append_v @ Xs @ ( cons_v @ X @ nil_v ) )
= ( append_v @ Ys @ ( cons_v @ Y @ nil_v ) ) )
= ( ( Xs = Ys )
& ( X = Y ) ) ) ).
% append1_eq_conv
thf(fact_514_set__append,axiom,
! [Xs: list_v,Ys: list_v] :
( ( set_v2 @ ( append_v @ Xs @ Ys ) )
= ( sup_sup_set_v @ ( set_v2 @ Xs ) @ ( set_v2 @ Ys ) ) ) ).
% set_append
thf(fact_515_set__append,axiom,
! [Xs: list_P7986770385144383213od_v_v,Ys: list_P7986770385144383213od_v_v] :
( ( set_Product_prod_v_v2 @ ( append2138873909117096322od_v_v @ Xs @ Ys ) )
= ( sup_su414716646722978715od_v_v @ ( set_Product_prod_v_v2 @ Xs ) @ ( set_Product_prod_v_v2 @ Ys ) ) ) ).
% set_append
thf(fact_516_hd__append2,axiom,
! [Xs: list_v,Ys: list_v] :
( ( Xs != nil_v )
=> ( ( hd_v @ ( append_v @ Xs @ Ys ) )
= ( hd_v @ Xs ) ) ) ).
% hd_append2
thf(fact_517_tl__append2,axiom,
! [Xs: list_v,Ys: list_v] :
( ( Xs != nil_v )
=> ( ( tl_v @ ( append_v @ Xs @ Ys ) )
= ( append_v @ ( tl_v @ Xs ) @ Ys ) ) ) ).
% tl_append2
thf(fact_518_list_Ocollapse,axiom,
! [List: list_v] :
( ( List != nil_v )
=> ( ( cons_v @ ( hd_v @ List ) @ ( tl_v @ List ) )
= List ) ) ).
% list.collapse
thf(fact_519_hd__Cons__tl,axiom,
! [Xs: list_v] :
( ( Xs != nil_v )
=> ( ( cons_v @ ( hd_v @ Xs ) @ ( tl_v @ Xs ) )
= Xs ) ) ).
% hd_Cons_tl
thf(fact_520_post__dfss__def,axiom,
! [V: v,E2: sCC_Bl1394983891496994913t_unit,E3: sCC_Bl1394983891496994913t_unit] :
( ( sCC_Bl6082031138996704384t_unit @ successors @ V @ E2 @ E3 )
= ( ( sCC_Bl9196236973127232072t_unit @ successors @ E3 )
& ( ( sCC_Bl3795065053823578884t_unit @ E3 @ V )
= ( successors @ V ) )
& ! [X2: v] :
( ( member_v @ X2 @ ( minus_minus_set_v @ ( sCC_Bl4645233313691564917t_unit @ E2 ) @ ( insert_v2 @ V @ bot_bot_set_v ) ) )
=> ( ( sCC_Bl3795065053823578884t_unit @ E3 @ X2 )
= ( sCC_Bl3795065053823578884t_unit @ E2 @ X2 ) ) )
& ( sCC_Bl5768913643336123637t_unit @ E2 @ E3 )
& ! [X2: v] :
( ( member_v @ X2 @ ( successors @ V ) )
=> ( member_v @ X2 @ ( sup_sup_set_v @ ( sCC_Bl157864678168468314t_unit @ E3 ) @ ( sCC_Bl1280885523602775798t_unit @ E3 @ ( hd_v @ ( sCC_Bl8828226123343373779t_unit @ E3 ) ) ) ) ) )
& ! [X2: v] :
( ( member_v @ X2 @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E3 ) ) )
=> ( sCC_Bl649662514949026229able_v @ successors @ X2 @ V ) )
& ( ( sCC_Bl8828226123343373779t_unit @ E3 )
!= nil_v )
& ? [Ns: list_v] :
( ( sCC_Bl8828226123343373779t_unit @ E2 )
= ( append_v @ Ns @ ( sCC_Bl8828226123343373779t_unit @ E3 ) ) )
& ( member_v @ V @ ( sCC_Bl1280885523602775798t_unit @ E3 @ ( hd_v @ ( sCC_Bl8828226123343373779t_unit @ E3 ) ) ) )
& ! [X2: v] :
( ( member_v @ X2 @ ( set_v2 @ ( tl_v @ ( sCC_Bl8828226123343373779t_unit @ E3 ) ) ) )
=> ( ( sCC_Bl1280885523602775798t_unit @ E3 @ X2 )
= ( sCC_Bl1280885523602775798t_unit @ E2 @ X2 ) ) )
& ( ( ( hd_v @ ( sCC_Bl8828226123343373779t_unit @ E3 ) )
= V )
=> ! [X2: v] :
( ( member_v @ X2 @ ( set_v2 @ ( tl_v @ ( sCC_Bl8828226123343373779t_unit @ E3 ) ) ) )
=> ~ ( sCC_Bl649662514949026229able_v @ successors @ V @ X2 ) ) )
& ( ( sCC_Bl9201514103433284750t_unit @ E3 )
= ( sCC_Bl9201514103433284750t_unit @ E2 ) ) ) ) ).
% post_dfss_def
thf(fact_521_mk__disjoint__insert,axiom,
! [A4: v,A: set_v] :
( ( member_v @ A4 @ A )
=> ? [B6: set_v] :
( ( A
= ( insert_v2 @ A4 @ B6 ) )
& ~ ( member_v @ A4 @ B6 ) ) ) ).
% mk_disjoint_insert
thf(fact_522_mk__disjoint__insert,axiom,
! [A4: product_prod_v_v,A: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ A4 @ A )
=> ? [B6: set_Product_prod_v_v] :
( ( A
= ( insert1338601472111419319od_v_v @ A4 @ B6 ) )
& ~ ( member7453568604450474000od_v_v @ A4 @ B6 ) ) ) ).
% mk_disjoint_insert
thf(fact_523_insert__commute,axiom,
! [X: product_prod_v_v,Y: product_prod_v_v,A: set_Product_prod_v_v] :
( ( insert1338601472111419319od_v_v @ X @ ( insert1338601472111419319od_v_v @ Y @ A ) )
= ( insert1338601472111419319od_v_v @ Y @ ( insert1338601472111419319od_v_v @ X @ A ) ) ) ).
% insert_commute
thf(fact_524_insert__commute,axiom,
! [X: v,Y: v,A: set_v] :
( ( insert_v2 @ X @ ( insert_v2 @ Y @ A ) )
= ( insert_v2 @ Y @ ( insert_v2 @ X @ A ) ) ) ).
% insert_commute
thf(fact_525_insert__eq__iff,axiom,
! [A4: v,A: set_v,B2: v,B: set_v] :
( ~ ( member_v @ A4 @ A )
=> ( ~ ( member_v @ B2 @ B )
=> ( ( ( insert_v2 @ A4 @ A )
= ( insert_v2 @ B2 @ B ) )
= ( ( ( A4 = B2 )
=> ( A = B ) )
& ( ( A4 != B2 )
=> ? [C3: set_v] :
( ( A
= ( insert_v2 @ B2 @ C3 ) )
& ~ ( member_v @ B2 @ C3 )
& ( B
= ( insert_v2 @ A4 @ C3 ) )
& ~ ( member_v @ A4 @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_526_insert__eq__iff,axiom,
! [A4: product_prod_v_v,A: set_Product_prod_v_v,B2: product_prod_v_v,B: set_Product_prod_v_v] :
( ~ ( member7453568604450474000od_v_v @ A4 @ A )
=> ( ~ ( member7453568604450474000od_v_v @ B2 @ B )
=> ( ( ( insert1338601472111419319od_v_v @ A4 @ A )
= ( insert1338601472111419319od_v_v @ B2 @ B ) )
= ( ( ( A4 = B2 )
=> ( A = B ) )
& ( ( A4 != B2 )
=> ? [C3: set_Product_prod_v_v] :
( ( A
= ( insert1338601472111419319od_v_v @ B2 @ C3 ) )
& ~ ( member7453568604450474000od_v_v @ B2 @ C3 )
& ( B
= ( insert1338601472111419319od_v_v @ A4 @ C3 ) )
& ~ ( member7453568604450474000od_v_v @ A4 @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_527_insert__absorb,axiom,
! [A4: v,A: set_v] :
( ( member_v @ A4 @ A )
=> ( ( insert_v2 @ A4 @ A )
= A ) ) ).
% insert_absorb
thf(fact_528_insert__absorb,axiom,
! [A4: product_prod_v_v,A: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ A4 @ A )
=> ( ( insert1338601472111419319od_v_v @ A4 @ A )
= A ) ) ).
% insert_absorb
thf(fact_529_insert__ident,axiom,
! [X: v,A: set_v,B: set_v] :
( ~ ( member_v @ X @ A )
=> ( ~ ( member_v @ X @ B )
=> ( ( ( insert_v2 @ X @ A )
= ( insert_v2 @ X @ B ) )
= ( A = B ) ) ) ) ).
% insert_ident
thf(fact_530_insert__ident,axiom,
! [X: product_prod_v_v,A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ~ ( member7453568604450474000od_v_v @ X @ A )
=> ( ~ ( member7453568604450474000od_v_v @ X @ B )
=> ( ( ( insert1338601472111419319od_v_v @ X @ A )
= ( insert1338601472111419319od_v_v @ X @ B ) )
= ( A = B ) ) ) ) ).
% insert_ident
thf(fact_531_Set_Oset__insert,axiom,
! [X: v,A: set_v] :
( ( member_v @ X @ A )
=> ~ ! [B6: set_v] :
( ( A
= ( insert_v2 @ X @ B6 ) )
=> ( member_v @ X @ B6 ) ) ) ).
% Set.set_insert
thf(fact_532_Set_Oset__insert,axiom,
! [X: product_prod_v_v,A: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X @ A )
=> ~ ! [B6: set_Product_prod_v_v] :
( ( A
= ( insert1338601472111419319od_v_v @ X @ B6 ) )
=> ( member7453568604450474000od_v_v @ X @ B6 ) ) ) ).
% Set.set_insert
thf(fact_533_insertI2,axiom,
! [A4: v,B: set_v,B2: v] :
( ( member_v @ A4 @ B )
=> ( member_v @ A4 @ ( insert_v2 @ B2 @ B ) ) ) ).
% insertI2
thf(fact_534_insertI2,axiom,
! [A4: product_prod_v_v,B: set_Product_prod_v_v,B2: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ A4 @ B )
=> ( member7453568604450474000od_v_v @ A4 @ ( insert1338601472111419319od_v_v @ B2 @ B ) ) ) ).
% insertI2
thf(fact_535_insertI1,axiom,
! [A4: v,B: set_v] : ( member_v @ A4 @ ( insert_v2 @ A4 @ B ) ) ).
% insertI1
thf(fact_536_insertI1,axiom,
! [A4: product_prod_v_v,B: set_Product_prod_v_v] : ( member7453568604450474000od_v_v @ A4 @ ( insert1338601472111419319od_v_v @ A4 @ B ) ) ).
% insertI1
thf(fact_537_insertE,axiom,
! [A4: v,B2: v,A: set_v] :
( ( member_v @ A4 @ ( insert_v2 @ B2 @ A ) )
=> ( ( A4 != B2 )
=> ( member_v @ A4 @ A ) ) ) ).
% insertE
thf(fact_538_insertE,axiom,
! [A4: product_prod_v_v,B2: product_prod_v_v,A: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ A4 @ ( insert1338601472111419319od_v_v @ B2 @ A ) )
=> ( ( A4 != B2 )
=> ( member7453568604450474000od_v_v @ A4 @ A ) ) ) ).
% insertE
thf(fact_539_append__eq__append__conv2,axiom,
! [Xs: list_v,Ys: list_v,Zs: list_v,Ts: list_v] :
( ( ( append_v @ Xs @ Ys )
= ( append_v @ Zs @ Ts ) )
= ( ? [Us: list_v] :
( ( ( Xs
= ( append_v @ Zs @ Us ) )
& ( ( append_v @ Us @ Ys )
= Ts ) )
| ( ( ( append_v @ Xs @ Us )
= Zs )
& ( Ys
= ( append_v @ Us @ Ts ) ) ) ) ) ) ).
% append_eq_append_conv2
thf(fact_540_append__eq__appendI,axiom,
! [Xs: list_v,Xs1: list_v,Zs: list_v,Ys: list_v,Us2: list_v] :
( ( ( append_v @ Xs @ Xs1 )
= Zs )
=> ( ( Ys
= ( append_v @ Xs1 @ Us2 ) )
=> ( ( append_v @ Xs @ Ys )
= ( append_v @ Zs @ Us2 ) ) ) ) ).
% append_eq_appendI
thf(fact_541_Cons__eq__appendI,axiom,
! [X: v,Xs1: list_v,Ys: list_v,Xs: list_v,Zs: list_v] :
( ( ( cons_v @ X @ Xs1 )
= Ys )
=> ( ( Xs
= ( append_v @ Xs1 @ Zs ) )
=> ( ( cons_v @ X @ Xs )
= ( append_v @ Ys @ Zs ) ) ) ) ).
% Cons_eq_appendI
thf(fact_542_append__Cons,axiom,
! [X: v,Xs: list_v,Ys: list_v] :
( ( append_v @ ( cons_v @ X @ Xs ) @ Ys )
= ( cons_v @ X @ ( append_v @ Xs @ Ys ) ) ) ).
% append_Cons
thf(fact_543_transpose_Ocases,axiom,
! [X: list_list_v] :
( ( X != nil_list_v )
=> ( ! [Xss: list_list_v] :
( X
!= ( cons_list_v @ nil_v @ Xss ) )
=> ~ ! [X3: v,Xs2: list_v,Xss: list_list_v] :
( X
!= ( cons_list_v @ ( cons_v @ X3 @ Xs2 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_544_rev__induct,axiom,
! [P: list_v > $o,Xs: list_v] :
( ( P @ nil_v )
=> ( ! [X3: v,Xs2: list_v] :
( ( P @ Xs2 )
=> ( P @ ( append_v @ Xs2 @ ( cons_v @ X3 @ nil_v ) ) ) )
=> ( P @ Xs ) ) ) ).
% rev_induct
thf(fact_545_split__list,axiom,
! [X: product_prod_v_v,Xs: list_P7986770385144383213od_v_v] :
( ( member7453568604450474000od_v_v @ X @ ( set_Product_prod_v_v2 @ Xs ) )
=> ? [Ys2: list_P7986770385144383213od_v_v,Zs2: list_P7986770385144383213od_v_v] :
( Xs
= ( append2138873909117096322od_v_v @ Ys2 @ ( cons_P4120604216776828829od_v_v @ X @ Zs2 ) ) ) ) ).
% split_list
thf(fact_546_split__list,axiom,
! [X: v,Xs: list_v] :
( ( member_v @ X @ ( set_v2 @ Xs ) )
=> ? [Ys2: list_v,Zs2: list_v] :
( Xs
= ( append_v @ Ys2 @ ( cons_v @ X @ Zs2 ) ) ) ) ).
% split_list
thf(fact_547_rev__exhaust,axiom,
! [Xs: list_v] :
( ( Xs != nil_v )
=> ~ ! [Ys2: list_v,Y2: v] :
( Xs
!= ( append_v @ Ys2 @ ( cons_v @ Y2 @ nil_v ) ) ) ) ).
% rev_exhaust
thf(fact_548_split__list__last,axiom,
! [X: product_prod_v_v,Xs: list_P7986770385144383213od_v_v] :
( ( member7453568604450474000od_v_v @ X @ ( set_Product_prod_v_v2 @ Xs ) )
=> ? [Ys2: list_P7986770385144383213od_v_v,Zs2: list_P7986770385144383213od_v_v] :
( ( Xs
= ( append2138873909117096322od_v_v @ Ys2 @ ( cons_P4120604216776828829od_v_v @ X @ Zs2 ) ) )
& ~ ( member7453568604450474000od_v_v @ X @ ( set_Product_prod_v_v2 @ Zs2 ) ) ) ) ).
% split_list_last
thf(fact_549_split__list__last,axiom,
! [X: v,Xs: list_v] :
( ( member_v @ X @ ( set_v2 @ Xs ) )
=> ? [Ys2: list_v,Zs2: list_v] :
( ( Xs
= ( append_v @ Ys2 @ ( cons_v @ X @ Zs2 ) ) )
& ~ ( member_v @ X @ ( set_v2 @ Zs2 ) ) ) ) ).
% split_list_last
thf(fact_550_split__list__prop,axiom,
! [Xs: list_v,P: v > $o] :
( ? [X4: v] :
( ( member_v @ X4 @ ( set_v2 @ Xs ) )
& ( P @ X4 ) )
=> ? [Ys2: list_v,X3: v] :
( ? [Zs2: list_v] :
( Xs
= ( append_v @ Ys2 @ ( cons_v @ X3 @ Zs2 ) ) )
& ( P @ X3 ) ) ) ).
% split_list_prop
thf(fact_551_split__list__first,axiom,
! [X: product_prod_v_v,Xs: list_P7986770385144383213od_v_v] :
( ( member7453568604450474000od_v_v @ X @ ( set_Product_prod_v_v2 @ Xs ) )
=> ? [Ys2: list_P7986770385144383213od_v_v,Zs2: list_P7986770385144383213od_v_v] :
( ( Xs
= ( append2138873909117096322od_v_v @ Ys2 @ ( cons_P4120604216776828829od_v_v @ X @ Zs2 ) ) )
& ~ ( member7453568604450474000od_v_v @ X @ ( set_Product_prod_v_v2 @ Ys2 ) ) ) ) ).
% split_list_first
thf(fact_552_split__list__first,axiom,
! [X: v,Xs: list_v] :
( ( member_v @ X @ ( set_v2 @ Xs ) )
=> ? [Ys2: list_v,Zs2: list_v] :
( ( Xs
= ( append_v @ Ys2 @ ( cons_v @ X @ Zs2 ) ) )
& ~ ( member_v @ X @ ( set_v2 @ Ys2 ) ) ) ) ).
% split_list_first
thf(fact_553_split__list__propE,axiom,
! [Xs: list_v,P: v > $o] :
( ? [X4: v] :
( ( member_v @ X4 @ ( set_v2 @ Xs ) )
& ( P @ X4 ) )
=> ~ ! [Ys2: list_v,X3: v] :
( ? [Zs2: list_v] :
( Xs
= ( append_v @ Ys2 @ ( cons_v @ X3 @ Zs2 ) ) )
=> ~ ( P @ X3 ) ) ) ).
% split_list_propE
thf(fact_554_append__Cons__eq__iff,axiom,
! [X: product_prod_v_v,Xs: list_P7986770385144383213od_v_v,Ys: list_P7986770385144383213od_v_v,Xs3: list_P7986770385144383213od_v_v,Ys3: list_P7986770385144383213od_v_v] :
( ~ ( member7453568604450474000od_v_v @ X @ ( set_Product_prod_v_v2 @ Xs ) )
=> ( ~ ( member7453568604450474000od_v_v @ X @ ( set_Product_prod_v_v2 @ Ys ) )
=> ( ( ( append2138873909117096322od_v_v @ Xs @ ( cons_P4120604216776828829od_v_v @ X @ Ys ) )
= ( append2138873909117096322od_v_v @ Xs3 @ ( cons_P4120604216776828829od_v_v @ X @ Ys3 ) ) )
= ( ( Xs = Xs3 )
& ( Ys = Ys3 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_555_append__Cons__eq__iff,axiom,
! [X: v,Xs: list_v,Ys: list_v,Xs3: list_v,Ys3: list_v] :
( ~ ( member_v @ X @ ( set_v2 @ Xs ) )
=> ( ~ ( member_v @ X @ ( set_v2 @ Ys ) )
=> ( ( ( append_v @ Xs @ ( cons_v @ X @ Ys ) )
= ( append_v @ Xs3 @ ( cons_v @ X @ Ys3 ) ) )
= ( ( Xs = Xs3 )
& ( Ys = Ys3 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_556_in__set__conv__decomp,axiom,
! [X: product_prod_v_v,Xs: list_P7986770385144383213od_v_v] :
( ( member7453568604450474000od_v_v @ X @ ( set_Product_prod_v_v2 @ Xs ) )
= ( ? [Ys4: list_P7986770385144383213od_v_v,Zs3: list_P7986770385144383213od_v_v] :
( Xs
= ( append2138873909117096322od_v_v @ Ys4 @ ( cons_P4120604216776828829od_v_v @ X @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_557_in__set__conv__decomp,axiom,
! [X: v,Xs: list_v] :
( ( member_v @ X @ ( set_v2 @ Xs ) )
= ( ? [Ys4: list_v,Zs3: list_v] :
( Xs
= ( append_v @ Ys4 @ ( cons_v @ X @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_558_Cons__eq__append__conv,axiom,
! [X: v,Xs: list_v,Ys: list_v,Zs: list_v] :
( ( ( cons_v @ X @ Xs )
= ( append_v @ Ys @ Zs ) )
= ( ( ( Ys = nil_v )
& ( ( cons_v @ X @ Xs )
= Zs ) )
| ? [Ys5: list_v] :
( ( ( cons_v @ X @ Ys5 )
= Ys )
& ( Xs
= ( append_v @ Ys5 @ Zs ) ) ) ) ) ).
% Cons_eq_append_conv
thf(fact_559_append__eq__Cons__conv,axiom,
! [Ys: list_v,Zs: list_v,X: v,Xs: list_v] :
( ( ( append_v @ Ys @ Zs )
= ( cons_v @ X @ Xs ) )
= ( ( ( Ys = nil_v )
& ( Zs
= ( cons_v @ X @ Xs ) ) )
| ? [Ys5: list_v] :
( ( Ys
= ( cons_v @ X @ Ys5 ) )
& ( ( append_v @ Ys5 @ Zs )
= Xs ) ) ) ) ).
% append_eq_Cons_conv
thf(fact_560_rev__nonempty__induct,axiom,
! [Xs: list_v,P: list_v > $o] :
( ( Xs != nil_v )
=> ( ! [X3: v] : ( P @ ( cons_v @ X3 @ nil_v ) )
=> ( ! [X3: v,Xs2: list_v] :
( ( Xs2 != nil_v )
=> ( ( P @ Xs2 )
=> ( P @ ( append_v @ Xs2 @ ( cons_v @ X3 @ nil_v ) ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% rev_nonempty_induct
thf(fact_561_split__list__last__prop,axiom,
! [Xs: list_v,P: v > $o] :
( ? [X4: v] :
( ( member_v @ X4 @ ( set_v2 @ Xs ) )
& ( P @ X4 ) )
=> ? [Ys2: list_v,X3: v,Zs2: list_v] :
( ( Xs
= ( append_v @ Ys2 @ ( cons_v @ X3 @ Zs2 ) ) )
& ( P @ X3 )
& ! [Xa2: v] :
( ( member_v @ Xa2 @ ( set_v2 @ Zs2 ) )
=> ~ ( P @ Xa2 ) ) ) ) ).
% split_list_last_prop
thf(fact_562_split__list__first__prop,axiom,
! [Xs: list_v,P: v > $o] :
( ? [X4: v] :
( ( member_v @ X4 @ ( set_v2 @ Xs ) )
& ( P @ X4 ) )
=> ? [Ys2: list_v,X3: v] :
( ? [Zs2: list_v] :
( Xs
= ( append_v @ Ys2 @ ( cons_v @ X3 @ Zs2 ) ) )
& ( P @ X3 )
& ! [Xa2: v] :
( ( member_v @ Xa2 @ ( set_v2 @ Ys2 ) )
=> ~ ( P @ Xa2 ) ) ) ) ).
% split_list_first_prop
thf(fact_563_split__list__last__propE,axiom,
! [Xs: list_v,P: v > $o] :
( ? [X4: v] :
( ( member_v @ X4 @ ( set_v2 @ Xs ) )
& ( P @ X4 ) )
=> ~ ! [Ys2: list_v,X3: v,Zs2: list_v] :
( ( Xs
= ( append_v @ Ys2 @ ( cons_v @ X3 @ Zs2 ) ) )
=> ( ( P @ X3 )
=> ~ ! [Xa2: v] :
( ( member_v @ Xa2 @ ( set_v2 @ Zs2 ) )
=> ~ ( P @ Xa2 ) ) ) ) ) ).
% split_list_last_propE
thf(fact_564_split__list__first__propE,axiom,
! [Xs: list_v,P: v > $o] :
( ? [X4: v] :
( ( member_v @ X4 @ ( set_v2 @ Xs ) )
& ( P @ X4 ) )
=> ~ ! [Ys2: list_v,X3: v] :
( ? [Zs2: list_v] :
( Xs
= ( append_v @ Ys2 @ ( cons_v @ X3 @ Zs2 ) ) )
=> ( ( P @ X3 )
=> ~ ! [Xa2: v] :
( ( member_v @ Xa2 @ ( set_v2 @ Ys2 ) )
=> ~ ( P @ Xa2 ) ) ) ) ) ).
% split_list_first_propE
thf(fact_565_in__set__conv__decomp__last,axiom,
! [X: product_prod_v_v,Xs: list_P7986770385144383213od_v_v] :
( ( member7453568604450474000od_v_v @ X @ ( set_Product_prod_v_v2 @ Xs ) )
= ( ? [Ys4: list_P7986770385144383213od_v_v,Zs3: list_P7986770385144383213od_v_v] :
( ( Xs
= ( append2138873909117096322od_v_v @ Ys4 @ ( cons_P4120604216776828829od_v_v @ X @ Zs3 ) ) )
& ~ ( member7453568604450474000od_v_v @ X @ ( set_Product_prod_v_v2 @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_566_in__set__conv__decomp__last,axiom,
! [X: v,Xs: list_v] :
( ( member_v @ X @ ( set_v2 @ Xs ) )
= ( ? [Ys4: list_v,Zs3: list_v] :
( ( Xs
= ( append_v @ Ys4 @ ( cons_v @ X @ Zs3 ) ) )
& ~ ( member_v @ X @ ( set_v2 @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_567_in__set__conv__decomp__first,axiom,
! [X: product_prod_v_v,Xs: list_P7986770385144383213od_v_v] :
( ( member7453568604450474000od_v_v @ X @ ( set_Product_prod_v_v2 @ Xs ) )
= ( ? [Ys4: list_P7986770385144383213od_v_v,Zs3: list_P7986770385144383213od_v_v] :
( ( Xs
= ( append2138873909117096322od_v_v @ Ys4 @ ( cons_P4120604216776828829od_v_v @ X @ Zs3 ) ) )
& ~ ( member7453568604450474000od_v_v @ X @ ( set_Product_prod_v_v2 @ Ys4 ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_568_in__set__conv__decomp__first,axiom,
! [X: v,Xs: list_v] :
( ( member_v @ X @ ( set_v2 @ Xs ) )
= ( ? [Ys4: list_v,Zs3: list_v] :
( ( Xs
= ( append_v @ Ys4 @ ( cons_v @ X @ Zs3 ) ) )
& ~ ( member_v @ X @ ( set_v2 @ Ys4 ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_569_split__list__last__prop__iff,axiom,
! [Xs: list_v,P: v > $o] :
( ( ? [X2: v] :
( ( member_v @ X2 @ ( set_v2 @ Xs ) )
& ( P @ X2 ) ) )
= ( ? [Ys4: list_v,X2: v,Zs3: list_v] :
( ( Xs
= ( append_v @ Ys4 @ ( cons_v @ X2 @ Zs3 ) ) )
& ( P @ X2 )
& ! [Y3: v] :
( ( member_v @ Y3 @ ( set_v2 @ Zs3 ) )
=> ~ ( P @ Y3 ) ) ) ) ) ).
% split_list_last_prop_iff
thf(fact_570_split__list__first__prop__iff,axiom,
! [Xs: list_v,P: v > $o] :
( ( ? [X2: v] :
( ( member_v @ X2 @ ( set_v2 @ Xs ) )
& ( P @ X2 ) ) )
= ( ? [Ys4: list_v,X2: v] :
( ? [Zs3: list_v] :
( Xs
= ( append_v @ Ys4 @ ( cons_v @ X2 @ Zs3 ) ) )
& ( P @ X2 )
& ! [Y3: v] :
( ( member_v @ Y3 @ ( set_v2 @ Ys4 ) )
=> ~ ( P @ Y3 ) ) ) ) ) ).
% split_list_first_prop_iff
thf(fact_571_successively_Ocases,axiom,
! [X: produc8237170675765753490list_v] :
( ! [P2: v > v > $o] :
( X
!= ( produc601102195597853570list_v @ P2 @ nil_v ) )
=> ( ! [P2: v > v > $o,X3: v] :
( X
!= ( produc601102195597853570list_v @ P2 @ ( cons_v @ X3 @ nil_v ) ) )
=> ~ ! [P2: v > v > $o,X3: v,Y2: v,Xs2: list_v] :
( X
!= ( produc601102195597853570list_v @ P2 @ ( cons_v @ X3 @ ( cons_v @ Y2 @ Xs2 ) ) ) ) ) ) ).
% successively.cases
thf(fact_572_sorted__wrt_Ocases,axiom,
! [X: produc8237170675765753490list_v] :
( ! [P2: v > v > $o] :
( X
!= ( produc601102195597853570list_v @ P2 @ nil_v ) )
=> ~ ! [P2: v > v > $o,X3: v,Ys2: list_v] :
( X
!= ( produc601102195597853570list_v @ P2 @ ( cons_v @ X3 @ Ys2 ) ) ) ) ).
% sorted_wrt.cases
thf(fact_573_shuffles_Ocases,axiom,
! [X: produc1391462591744249447list_v] :
( ! [Ys2: list_v] :
( X
!= ( produc6795410681906604247list_v @ nil_v @ Ys2 ) )
=> ( ! [Xs2: list_v] :
( X
!= ( produc6795410681906604247list_v @ Xs2 @ nil_v ) )
=> ~ ! [X3: v,Xs2: list_v,Y2: v,Ys2: list_v] :
( X
!= ( produc6795410681906604247list_v @ ( cons_v @ X3 @ Xs2 ) @ ( cons_v @ Y2 @ Ys2 ) ) ) ) ) ).
% shuffles.cases
thf(fact_574_splice_Ocases,axiom,
! [X: produc1391462591744249447list_v] :
( ! [Ys2: list_v] :
( X
!= ( produc6795410681906604247list_v @ nil_v @ Ys2 ) )
=> ~ ! [X3: v,Xs2: list_v,Ys2: list_v] :
( X
!= ( produc6795410681906604247list_v @ ( cons_v @ X3 @ Xs2 ) @ Ys2 ) ) ) ).
% splice.cases
thf(fact_575_precedes__def,axiom,
( sCC_Bl2026170059108282219od_v_v
= ( ^ [X2: product_prod_v_v,Y3: product_prod_v_v,Xs4: list_P7986770385144383213od_v_v] :
? [L: list_P7986770385144383213od_v_v,R: list_P7986770385144383213od_v_v] :
( ( Xs4
= ( append2138873909117096322od_v_v @ L @ ( cons_P4120604216776828829od_v_v @ X2 @ R ) ) )
& ( member7453568604450474000od_v_v @ Y3 @ ( set_Product_prod_v_v2 @ ( cons_P4120604216776828829od_v_v @ X2 @ R ) ) ) ) ) ) ).
% precedes_def
thf(fact_576_precedes__def,axiom,
( sCC_Bl4022239298816431255edes_v
= ( ^ [X2: v,Y3: v,Xs4: list_v] :
? [L: list_v,R: list_v] :
( ( Xs4
= ( append_v @ L @ ( cons_v @ X2 @ R ) ) )
& ( member_v @ Y3 @ ( set_v2 @ ( cons_v @ X2 @ R ) ) ) ) ) ) ).
% precedes_def
thf(fact_577_append__Nil,axiom,
! [Ys: list_v] :
( ( append_v @ nil_v @ Ys )
= Ys ) ).
% append_Nil
thf(fact_578_append_Oleft__neutral,axiom,
! [A4: list_v] :
( ( append_v @ nil_v @ A4 )
= A4 ) ).
% append.left_neutral
thf(fact_579_eq__Nil__appendI,axiom,
! [Xs: list_v,Ys: list_v] :
( ( Xs = Ys )
=> ( Xs
= ( append_v @ nil_v @ Ys ) ) ) ).
% eq_Nil_appendI
thf(fact_580_singleton__inject,axiom,
! [A4: product_prod_v_v,B2: product_prod_v_v] :
( ( ( insert1338601472111419319od_v_v @ A4 @ bot_bo723834152578015283od_v_v )
= ( insert1338601472111419319od_v_v @ B2 @ bot_bo723834152578015283od_v_v ) )
=> ( A4 = B2 ) ) ).
% singleton_inject
thf(fact_581_singleton__inject,axiom,
! [A4: v,B2: v] :
( ( ( insert_v2 @ A4 @ bot_bot_set_v )
= ( insert_v2 @ B2 @ bot_bot_set_v ) )
=> ( A4 = B2 ) ) ).
% singleton_inject
thf(fact_582_insert__not__empty,axiom,
! [A4: product_prod_v_v,A: set_Product_prod_v_v] :
( ( insert1338601472111419319od_v_v @ A4 @ A )
!= bot_bo723834152578015283od_v_v ) ).
% insert_not_empty
thf(fact_583_insert__not__empty,axiom,
! [A4: v,A: set_v] :
( ( insert_v2 @ A4 @ A )
!= bot_bot_set_v ) ).
% insert_not_empty
thf(fact_584_doubleton__eq__iff,axiom,
! [A4: product_prod_v_v,B2: product_prod_v_v,C: product_prod_v_v,D2: product_prod_v_v] :
( ( ( insert1338601472111419319od_v_v @ A4 @ ( insert1338601472111419319od_v_v @ B2 @ bot_bo723834152578015283od_v_v ) )
= ( insert1338601472111419319od_v_v @ C @ ( insert1338601472111419319od_v_v @ D2 @ bot_bo723834152578015283od_v_v ) ) )
= ( ( ( A4 = C )
& ( B2 = D2 ) )
| ( ( A4 = D2 )
& ( B2 = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_585_doubleton__eq__iff,axiom,
! [A4: v,B2: v,C: v,D2: v] :
( ( ( insert_v2 @ A4 @ ( insert_v2 @ B2 @ bot_bot_set_v ) )
= ( insert_v2 @ C @ ( insert_v2 @ D2 @ bot_bot_set_v ) ) )
= ( ( ( A4 = C )
& ( B2 = D2 ) )
| ( ( A4 = D2 )
& ( B2 = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_586_singleton__iff,axiom,
! [B2: product_prod_v_v,A4: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ B2 @ ( insert1338601472111419319od_v_v @ A4 @ bot_bo723834152578015283od_v_v ) )
= ( B2 = A4 ) ) ).
% singleton_iff
thf(fact_587_singleton__iff,axiom,
! [B2: v,A4: v] :
( ( member_v @ B2 @ ( insert_v2 @ A4 @ bot_bot_set_v ) )
= ( B2 = A4 ) ) ).
% singleton_iff
thf(fact_588_singletonD,axiom,
! [B2: product_prod_v_v,A4: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ B2 @ ( insert1338601472111419319od_v_v @ A4 @ bot_bo723834152578015283od_v_v ) )
=> ( B2 = A4 ) ) ).
% singletonD
thf(fact_589_singletonD,axiom,
! [B2: v,A4: v] :
( ( member_v @ B2 @ ( insert_v2 @ A4 @ bot_bot_set_v ) )
=> ( B2 = A4 ) ) ).
% singletonD
thf(fact_590_insert__mono,axiom,
! [C2: set_Product_prod_v_v,D: set_Product_prod_v_v,A4: product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ C2 @ D )
=> ( ord_le7336532860387713383od_v_v @ ( insert1338601472111419319od_v_v @ A4 @ C2 ) @ ( insert1338601472111419319od_v_v @ A4 @ D ) ) ) ).
% insert_mono
thf(fact_591_insert__mono,axiom,
! [C2: set_v,D: set_v,A4: v] :
( ( ord_less_eq_set_v @ C2 @ D )
=> ( ord_less_eq_set_v @ ( insert_v2 @ A4 @ C2 ) @ ( insert_v2 @ A4 @ D ) ) ) ).
% insert_mono
thf(fact_592_subset__insert,axiom,
! [X: product_prod_v_v,A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ~ ( member7453568604450474000od_v_v @ X @ A )
=> ( ( ord_le7336532860387713383od_v_v @ A @ ( insert1338601472111419319od_v_v @ X @ B ) )
= ( ord_le7336532860387713383od_v_v @ A @ B ) ) ) ).
% subset_insert
thf(fact_593_subset__insert,axiom,
! [X: v,A: set_v,B: set_v] :
( ~ ( member_v @ X @ A )
=> ( ( ord_less_eq_set_v @ A @ ( insert_v2 @ X @ B ) )
= ( ord_less_eq_set_v @ A @ B ) ) ) ).
% subset_insert
thf(fact_594_subset__insertI,axiom,
! [B: set_Product_prod_v_v,A4: product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ B @ ( insert1338601472111419319od_v_v @ A4 @ B ) ) ).
% subset_insertI
thf(fact_595_subset__insertI,axiom,
! [B: set_v,A4: v] : ( ord_less_eq_set_v @ B @ ( insert_v2 @ A4 @ B ) ) ).
% subset_insertI
thf(fact_596_subset__insertI2,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v,B2: product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ B )
=> ( ord_le7336532860387713383od_v_v @ A @ ( insert1338601472111419319od_v_v @ B2 @ B ) ) ) ).
% subset_insertI2
thf(fact_597_subset__insertI2,axiom,
! [A: set_v,B: set_v,B2: v] :
( ( ord_less_eq_set_v @ A @ B )
=> ( ord_less_eq_set_v @ A @ ( insert_v2 @ B2 @ B ) ) ) ).
% subset_insertI2
thf(fact_598_list_Odistinct_I1_J,axiom,
! [X21: v,X22: list_v] :
( nil_v
!= ( cons_v @ X21 @ X22 ) ) ).
% list.distinct(1)
thf(fact_599_list_OdiscI,axiom,
! [List: list_v,X21: v,X22: list_v] :
( ( List
= ( cons_v @ X21 @ X22 ) )
=> ( List != nil_v ) ) ).
% list.discI
thf(fact_600_list_Oexhaust,axiom,
! [Y: list_v] :
( ( Y != nil_v )
=> ~ ! [X212: v,X222: list_v] :
( Y
!= ( cons_v @ X212 @ X222 ) ) ) ).
% list.exhaust
thf(fact_601_remdups__adj_Ocases,axiom,
! [X: list_v] :
( ( X != nil_v )
=> ( ! [X3: v] :
( X
!= ( cons_v @ X3 @ nil_v ) )
=> ~ ! [X3: v,Y2: v,Xs2: list_v] :
( X
!= ( cons_v @ X3 @ ( cons_v @ Y2 @ Xs2 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_602_neq__Nil__conv,axiom,
! [Xs: list_v] :
( ( Xs != nil_v )
= ( ? [Y3: v,Ys4: list_v] :
( Xs
= ( cons_v @ Y3 @ Ys4 ) ) ) ) ).
% neq_Nil_conv
thf(fact_603_list__induct2_H,axiom,
! [P: list_v > list_v > $o,Xs: list_v,Ys: list_v] :
( ( P @ nil_v @ nil_v )
=> ( ! [X3: v,Xs2: list_v] : ( P @ ( cons_v @ X3 @ Xs2 ) @ nil_v )
=> ( ! [Y2: v,Ys2: list_v] : ( P @ nil_v @ ( cons_v @ Y2 @ Ys2 ) )
=> ( ! [X3: v,Xs2: list_v,Y2: v,Ys2: list_v] :
( ( P @ Xs2 @ Ys2 )
=> ( P @ ( cons_v @ X3 @ Xs2 ) @ ( cons_v @ Y2 @ Ys2 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_604_list__nonempty__induct,axiom,
! [Xs: list_v,P: list_v > $o] :
( ( Xs != nil_v )
=> ( ! [X3: v] : ( P @ ( cons_v @ X3 @ nil_v ) )
=> ( ! [X3: v,Xs2: list_v] :
( ( Xs2 != nil_v )
=> ( ( P @ Xs2 )
=> ( P @ ( cons_v @ X3 @ Xs2 ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_605_list_Oset__intros_I2_J,axiom,
! [Y: product_prod_v_v,X22: list_P7986770385144383213od_v_v,X21: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ Y @ ( set_Product_prod_v_v2 @ X22 ) )
=> ( member7453568604450474000od_v_v @ Y @ ( set_Product_prod_v_v2 @ ( cons_P4120604216776828829od_v_v @ X21 @ X22 ) ) ) ) ).
% list.set_intros(2)
thf(fact_606_list_Oset__intros_I2_J,axiom,
! [Y: v,X22: list_v,X21: v] :
( ( member_v @ Y @ ( set_v2 @ X22 ) )
=> ( member_v @ Y @ ( set_v2 @ ( cons_v @ X21 @ X22 ) ) ) ) ).
% list.set_intros(2)
thf(fact_607_list_Oset__intros_I1_J,axiom,
! [X21: product_prod_v_v,X22: list_P7986770385144383213od_v_v] : ( member7453568604450474000od_v_v @ X21 @ ( set_Product_prod_v_v2 @ ( cons_P4120604216776828829od_v_v @ X21 @ X22 ) ) ) ).
% list.set_intros(1)
thf(fact_608_list_Oset__intros_I1_J,axiom,
! [X21: v,X22: list_v] : ( member_v @ X21 @ ( set_v2 @ ( cons_v @ X21 @ X22 ) ) ) ).
% list.set_intros(1)
thf(fact_609_list_Oset__cases,axiom,
! [E2: product_prod_v_v,A4: list_P7986770385144383213od_v_v] :
( ( member7453568604450474000od_v_v @ E2 @ ( set_Product_prod_v_v2 @ A4 ) )
=> ( ! [Z22: list_P7986770385144383213od_v_v] :
( A4
!= ( cons_P4120604216776828829od_v_v @ E2 @ Z22 ) )
=> ~ ! [Z1: product_prod_v_v,Z22: list_P7986770385144383213od_v_v] :
( ( A4
= ( cons_P4120604216776828829od_v_v @ Z1 @ Z22 ) )
=> ~ ( member7453568604450474000od_v_v @ E2 @ ( set_Product_prod_v_v2 @ Z22 ) ) ) ) ) ).
% list.set_cases
thf(fact_610_list_Oset__cases,axiom,
! [E2: v,A4: list_v] :
( ( member_v @ E2 @ ( set_v2 @ A4 ) )
=> ( ! [Z22: list_v] :
( A4
!= ( cons_v @ E2 @ Z22 ) )
=> ~ ! [Z1: v,Z22: list_v] :
( ( A4
= ( cons_v @ Z1 @ Z22 ) )
=> ~ ( member_v @ E2 @ ( set_v2 @ Z22 ) ) ) ) ) ).
% list.set_cases
thf(fact_611_set__ConsD,axiom,
! [Y: product_prod_v_v,X: product_prod_v_v,Xs: list_P7986770385144383213od_v_v] :
( ( member7453568604450474000od_v_v @ Y @ ( set_Product_prod_v_v2 @ ( cons_P4120604216776828829od_v_v @ X @ Xs ) ) )
=> ( ( Y = X )
| ( member7453568604450474000od_v_v @ Y @ ( set_Product_prod_v_v2 @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_612_set__ConsD,axiom,
! [Y: v,X: v,Xs: list_v] :
( ( member_v @ Y @ ( set_v2 @ ( cons_v @ X @ Xs ) ) )
=> ( ( Y = X )
| ( member_v @ Y @ ( set_v2 @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_613_insert__Diff__if,axiom,
! [X: product_prod_v_v,B: set_Product_prod_v_v,A: set_Product_prod_v_v] :
( ( ( member7453568604450474000od_v_v @ X @ B )
=> ( ( minus_4183494784930505774od_v_v @ ( insert1338601472111419319od_v_v @ X @ A ) @ B )
= ( minus_4183494784930505774od_v_v @ A @ B ) ) )
& ( ~ ( member7453568604450474000od_v_v @ X @ B )
=> ( ( minus_4183494784930505774od_v_v @ ( insert1338601472111419319od_v_v @ X @ A ) @ B )
= ( insert1338601472111419319od_v_v @ X @ ( minus_4183494784930505774od_v_v @ A @ B ) ) ) ) ) ).
% insert_Diff_if
thf(fact_614_insert__Diff__if,axiom,
! [X: v,B: set_v,A: set_v] :
( ( ( member_v @ X @ B )
=> ( ( minus_minus_set_v @ ( insert_v2 @ X @ A ) @ B )
= ( minus_minus_set_v @ A @ B ) ) )
& ( ~ ( member_v @ X @ B )
=> ( ( minus_minus_set_v @ ( insert_v2 @ X @ A ) @ B )
= ( insert_v2 @ X @ ( minus_minus_set_v @ A @ B ) ) ) ) ) ).
% insert_Diff_if
thf(fact_615_list_Osel_I1_J,axiom,
! [X21: v,X22: list_v] :
( ( hd_v @ ( cons_v @ X21 @ X22 ) )
= X21 ) ).
% list.sel(1)
thf(fact_616_list_Osel_I3_J,axiom,
! [X21: v,X22: list_v] :
( ( tl_v @ ( cons_v @ X21 @ X22 ) )
= X22 ) ).
% list.sel(3)
thf(fact_617_graph_Odfss_Ocases,axiom,
! [Vertices: set_v,Successors: v > set_v,X: produc5741669702376414499t_unit] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ~ ! [V2: v,E7: sCC_Bl1394983891496994913t_unit] :
( X
!= ( produc3862955338007567901t_unit @ V2 @ E7 ) ) ) ).
% graph.dfss.cases
thf(fact_618_precedes__append__left,axiom,
! [X: v,Y: v,Xs: list_v,Ys: list_v] :
( ( sCC_Bl4022239298816431255edes_v @ X @ Y @ Xs )
=> ( sCC_Bl4022239298816431255edes_v @ X @ Y @ ( append_v @ Ys @ Xs ) ) ) ).
% precedes_append_left
thf(fact_619_precedes__append__right,axiom,
! [X: v,Y: v,Xs: list_v,Ys: list_v] :
( ( sCC_Bl4022239298816431255edes_v @ X @ Y @ Xs )
=> ( sCC_Bl4022239298816431255edes_v @ X @ Y @ ( append_v @ Xs @ Ys ) ) ) ).
% precedes_append_right
thf(fact_620_precedes__in__tail,axiom,
! [X: v,Z: v,Y: v,Zs: list_v] :
( ( X != Z )
=> ( ( sCC_Bl4022239298816431255edes_v @ X @ Y @ ( cons_v @ Z @ Zs ) )
= ( sCC_Bl4022239298816431255edes_v @ X @ Y @ Zs ) ) ) ).
% precedes_in_tail
thf(fact_621_split__list__precedes,axiom,
! [Y: product_prod_v_v,Ys: list_P7986770385144383213od_v_v,X: product_prod_v_v,Xs: list_P7986770385144383213od_v_v] :
( ( member7453568604450474000od_v_v @ Y @ ( set_Product_prod_v_v2 @ ( append2138873909117096322od_v_v @ Ys @ ( cons_P4120604216776828829od_v_v @ X @ nil_Product_prod_v_v ) ) ) )
=> ( sCC_Bl2026170059108282219od_v_v @ Y @ X @ ( append2138873909117096322od_v_v @ Ys @ ( cons_P4120604216776828829od_v_v @ X @ Xs ) ) ) ) ).
% split_list_precedes
thf(fact_622_split__list__precedes,axiom,
! [Y: v,Ys: list_v,X: v,Xs: list_v] :
( ( member_v @ Y @ ( set_v2 @ ( append_v @ Ys @ ( cons_v @ X @ nil_v ) ) ) )
=> ( sCC_Bl4022239298816431255edes_v @ Y @ X @ ( append_v @ Ys @ ( cons_v @ X @ Xs ) ) ) ) ).
% split_list_precedes
thf(fact_623_subset__singletonD,axiom,
! [A: set_Product_prod_v_v,X: product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v ) )
=> ( ( A = bot_bo723834152578015283od_v_v )
| ( A
= ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v ) ) ) ) ).
% subset_singletonD
thf(fact_624_subset__singletonD,axiom,
! [A: set_v,X: v] :
( ( ord_less_eq_set_v @ A @ ( insert_v2 @ X @ bot_bot_set_v ) )
=> ( ( A = bot_bot_set_v )
| ( A
= ( insert_v2 @ X @ bot_bot_set_v ) ) ) ) ).
% subset_singletonD
thf(fact_625_subset__singleton__iff,axiom,
! [X5: set_Product_prod_v_v,A4: product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X5 @ ( insert1338601472111419319od_v_v @ A4 @ bot_bo723834152578015283od_v_v ) )
= ( ( X5 = bot_bo723834152578015283od_v_v )
| ( X5
= ( insert1338601472111419319od_v_v @ A4 @ bot_bo723834152578015283od_v_v ) ) ) ) ).
% subset_singleton_iff
thf(fact_626_subset__singleton__iff,axiom,
! [X5: set_v,A4: v] :
( ( ord_less_eq_set_v @ X5 @ ( insert_v2 @ A4 @ bot_bot_set_v ) )
= ( ( X5 = bot_bot_set_v )
| ( X5
= ( insert_v2 @ A4 @ bot_bot_set_v ) ) ) ) ).
% subset_singleton_iff
thf(fact_627_singleton__Un__iff,axiom,
! [X: product_prod_v_v,A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v )
= ( sup_su414716646722978715od_v_v @ A @ B ) )
= ( ( ( A = bot_bo723834152578015283od_v_v )
& ( B
= ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v ) ) )
| ( ( A
= ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v ) )
& ( B = bot_bo723834152578015283od_v_v ) )
| ( ( A
= ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v ) )
& ( B
= ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_628_singleton__Un__iff,axiom,
! [X: v,A: set_v,B: set_v] :
( ( ( insert_v2 @ X @ bot_bot_set_v )
= ( sup_sup_set_v @ A @ B ) )
= ( ( ( A = bot_bot_set_v )
& ( B
= ( insert_v2 @ X @ bot_bot_set_v ) ) )
| ( ( A
= ( insert_v2 @ X @ bot_bot_set_v ) )
& ( B = bot_bot_set_v ) )
| ( ( A
= ( insert_v2 @ X @ bot_bot_set_v ) )
& ( B
= ( insert_v2 @ X @ bot_bot_set_v ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_629_Un__singleton__iff,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v,X: product_prod_v_v] :
( ( ( sup_su414716646722978715od_v_v @ A @ B )
= ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v ) )
= ( ( ( A = bot_bo723834152578015283od_v_v )
& ( B
= ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v ) ) )
| ( ( A
= ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v ) )
& ( B = bot_bo723834152578015283od_v_v ) )
| ( ( A
= ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v ) )
& ( B
= ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_630_Un__singleton__iff,axiom,
! [A: set_v,B: set_v,X: v] :
( ( ( sup_sup_set_v @ A @ B )
= ( insert_v2 @ X @ bot_bot_set_v ) )
= ( ( ( A = bot_bot_set_v )
& ( B
= ( insert_v2 @ X @ bot_bot_set_v ) ) )
| ( ( A
= ( insert_v2 @ X @ bot_bot_set_v ) )
& ( B = bot_bot_set_v ) )
| ( ( A
= ( insert_v2 @ X @ bot_bot_set_v ) )
& ( B
= ( insert_v2 @ X @ bot_bot_set_v ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_631_insert__is__Un,axiom,
( insert1338601472111419319od_v_v
= ( ^ [A5: product_prod_v_v] : ( sup_su414716646722978715od_v_v @ ( insert1338601472111419319od_v_v @ A5 @ bot_bo723834152578015283od_v_v ) ) ) ) ).
% insert_is_Un
thf(fact_632_insert__is__Un,axiom,
( insert_v2
= ( ^ [A5: v] : ( sup_sup_set_v @ ( insert_v2 @ A5 @ bot_bot_set_v ) ) ) ) ).
% insert_is_Un
thf(fact_633_Diff__insert__absorb,axiom,
! [X: product_prod_v_v,A: set_Product_prod_v_v] :
( ~ ( member7453568604450474000od_v_v @ X @ A )
=> ( ( minus_4183494784930505774od_v_v @ ( insert1338601472111419319od_v_v @ X @ A ) @ ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v ) )
= A ) ) ).
% Diff_insert_absorb
thf(fact_634_Diff__insert__absorb,axiom,
! [X: v,A: set_v] :
( ~ ( member_v @ X @ A )
=> ( ( minus_minus_set_v @ ( insert_v2 @ X @ A ) @ ( insert_v2 @ X @ bot_bot_set_v ) )
= A ) ) ).
% Diff_insert_absorb
thf(fact_635_Diff__insert2,axiom,
! [A: set_Product_prod_v_v,A4: product_prod_v_v,B: set_Product_prod_v_v] :
( ( minus_4183494784930505774od_v_v @ A @ ( insert1338601472111419319od_v_v @ A4 @ B ) )
= ( minus_4183494784930505774od_v_v @ ( minus_4183494784930505774od_v_v @ A @ ( insert1338601472111419319od_v_v @ A4 @ bot_bo723834152578015283od_v_v ) ) @ B ) ) ).
% Diff_insert2
thf(fact_636_Diff__insert2,axiom,
! [A: set_v,A4: v,B: set_v] :
( ( minus_minus_set_v @ A @ ( insert_v2 @ A4 @ B ) )
= ( minus_minus_set_v @ ( minus_minus_set_v @ A @ ( insert_v2 @ A4 @ bot_bot_set_v ) ) @ B ) ) ).
% Diff_insert2
thf(fact_637_insert__Diff,axiom,
! [A4: product_prod_v_v,A: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ A4 @ A )
=> ( ( insert1338601472111419319od_v_v @ A4 @ ( minus_4183494784930505774od_v_v @ A @ ( insert1338601472111419319od_v_v @ A4 @ bot_bo723834152578015283od_v_v ) ) )
= A ) ) ).
% insert_Diff
thf(fact_638_insert__Diff,axiom,
! [A4: v,A: set_v] :
( ( member_v @ A4 @ A )
=> ( ( insert_v2 @ A4 @ ( minus_minus_set_v @ A @ ( insert_v2 @ A4 @ bot_bot_set_v ) ) )
= A ) ) ).
% insert_Diff
thf(fact_639_Diff__insert,axiom,
! [A: set_Product_prod_v_v,A4: product_prod_v_v,B: set_Product_prod_v_v] :
( ( minus_4183494784930505774od_v_v @ A @ ( insert1338601472111419319od_v_v @ A4 @ B ) )
= ( minus_4183494784930505774od_v_v @ ( minus_4183494784930505774od_v_v @ A @ B ) @ ( insert1338601472111419319od_v_v @ A4 @ bot_bo723834152578015283od_v_v ) ) ) ).
% Diff_insert
thf(fact_640_Diff__insert,axiom,
! [A: set_v,A4: v,B: set_v] :
( ( minus_minus_set_v @ A @ ( insert_v2 @ A4 @ B ) )
= ( minus_minus_set_v @ ( minus_minus_set_v @ A @ B ) @ ( insert_v2 @ A4 @ bot_bot_set_v ) ) ) ).
% Diff_insert
thf(fact_641_set__subset__Cons,axiom,
! [Xs: list_P7986770385144383213od_v_v,X: product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ ( set_Product_prod_v_v2 @ Xs ) @ ( set_Product_prod_v_v2 @ ( cons_P4120604216776828829od_v_v @ X @ Xs ) ) ) ).
% set_subset_Cons
thf(fact_642_set__subset__Cons,axiom,
! [Xs: list_v,X: v] : ( ord_less_eq_set_v @ ( set_v2 @ Xs ) @ ( set_v2 @ ( cons_v @ X @ Xs ) ) ) ).
% set_subset_Cons
thf(fact_643_subset__Diff__insert,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v,X: product_prod_v_v,C2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ ( minus_4183494784930505774od_v_v @ B @ ( insert1338601472111419319od_v_v @ X @ C2 ) ) )
= ( ( ord_le7336532860387713383od_v_v @ A @ ( minus_4183494784930505774od_v_v @ B @ C2 ) )
& ~ ( member7453568604450474000od_v_v @ X @ A ) ) ) ).
% subset_Diff_insert
thf(fact_644_subset__Diff__insert,axiom,
! [A: set_v,B: set_v,X: v,C2: set_v] :
( ( ord_less_eq_set_v @ A @ ( minus_minus_set_v @ B @ ( insert_v2 @ X @ C2 ) ) )
= ( ( ord_less_eq_set_v @ A @ ( minus_minus_set_v @ B @ C2 ) )
& ~ ( member_v @ X @ A ) ) ) ).
% subset_Diff_insert
thf(fact_645_hd__append,axiom,
! [Xs: list_v,Ys: list_v] :
( ( ( Xs = nil_v )
=> ( ( hd_v @ ( append_v @ Xs @ Ys ) )
= ( hd_v @ Ys ) ) )
& ( ( Xs != nil_v )
=> ( ( hd_v @ ( append_v @ Xs @ Ys ) )
= ( hd_v @ Xs ) ) ) ) ).
% hd_append
thf(fact_646_longest__common__prefix,axiom,
! [Xs: list_v,Ys: list_v] :
? [Ps: list_v,Xs5: list_v,Ys6: list_v] :
( ( Xs
= ( append_v @ Ps @ Xs5 ) )
& ( Ys
= ( append_v @ Ps @ Ys6 ) )
& ( ( Xs5 = nil_v )
| ( Ys6 = nil_v )
| ( ( hd_v @ Xs5 )
!= ( hd_v @ Ys6 ) ) ) ) ).
% longest_common_prefix
thf(fact_647_tl__append__if,axiom,
! [Xs: list_v,Ys: list_v] :
( ( ( Xs = nil_v )
=> ( ( tl_v @ ( append_v @ Xs @ Ys ) )
= ( tl_v @ Ys ) ) )
& ( ( Xs != nil_v )
=> ( ( tl_v @ ( append_v @ Xs @ Ys ) )
= ( append_v @ ( tl_v @ Xs ) @ Ys ) ) ) ) ).
% tl_append_if
thf(fact_648_tl__Nil,axiom,
! [Xs: list_v] :
( ( ( tl_v @ Xs )
= nil_v )
= ( ( Xs = nil_v )
| ? [X2: v] :
( Xs
= ( cons_v @ X2 @ nil_v ) ) ) ) ).
% tl_Nil
thf(fact_649_Nil__tl,axiom,
! [Xs: list_v] :
( ( nil_v
= ( tl_v @ Xs ) )
= ( ( Xs = nil_v )
| ? [X2: v] :
( Xs
= ( cons_v @ X2 @ nil_v ) ) ) ) ).
% Nil_tl
thf(fact_650_precedes__append__left__iff,axiom,
! [X: product_prod_v_v,Ys: list_P7986770385144383213od_v_v,Y: product_prod_v_v,Xs: list_P7986770385144383213od_v_v] :
( ~ ( member7453568604450474000od_v_v @ X @ ( set_Product_prod_v_v2 @ Ys ) )
=> ( ( sCC_Bl2026170059108282219od_v_v @ X @ Y @ ( append2138873909117096322od_v_v @ Ys @ Xs ) )
= ( sCC_Bl2026170059108282219od_v_v @ X @ Y @ Xs ) ) ) ).
% precedes_append_left_iff
thf(fact_651_precedes__append__left__iff,axiom,
! [X: v,Ys: list_v,Y: v,Xs: list_v] :
( ~ ( member_v @ X @ ( set_v2 @ Ys ) )
=> ( ( sCC_Bl4022239298816431255edes_v @ X @ Y @ ( append_v @ Ys @ Xs ) )
= ( sCC_Bl4022239298816431255edes_v @ X @ Y @ Xs ) ) ) ).
% precedes_append_left_iff
thf(fact_652_precedes__append__right__iff,axiom,
! [Y: product_prod_v_v,Ys: list_P7986770385144383213od_v_v,X: product_prod_v_v,Xs: list_P7986770385144383213od_v_v] :
( ~ ( member7453568604450474000od_v_v @ Y @ ( set_Product_prod_v_v2 @ Ys ) )
=> ( ( sCC_Bl2026170059108282219od_v_v @ X @ Y @ ( append2138873909117096322od_v_v @ Xs @ Ys ) )
= ( sCC_Bl2026170059108282219od_v_v @ X @ Y @ Xs ) ) ) ).
% precedes_append_right_iff
thf(fact_653_precedes__append__right__iff,axiom,
! [Y: v,Ys: list_v,X: v,Xs: list_v] :
( ~ ( member_v @ Y @ ( set_v2 @ Ys ) )
=> ( ( sCC_Bl4022239298816431255edes_v @ X @ Y @ ( append_v @ Xs @ Ys ) )
= ( sCC_Bl4022239298816431255edes_v @ X @ Y @ Xs ) ) ) ).
% precedes_append_right_iff
thf(fact_654_head__precedes,axiom,
! [Y: product_prod_v_v,X: product_prod_v_v,Xs: list_P7986770385144383213od_v_v] :
( ( member7453568604450474000od_v_v @ Y @ ( set_Product_prod_v_v2 @ ( cons_P4120604216776828829od_v_v @ X @ Xs ) ) )
=> ( sCC_Bl2026170059108282219od_v_v @ X @ Y @ ( cons_P4120604216776828829od_v_v @ X @ Xs ) ) ) ).
% head_precedes
thf(fact_655_head__precedes,axiom,
! [Y: v,X: v,Xs: list_v] :
( ( member_v @ Y @ ( set_v2 @ ( cons_v @ X @ Xs ) ) )
=> ( sCC_Bl4022239298816431255edes_v @ X @ Y @ ( cons_v @ X @ Xs ) ) ) ).
% head_precedes
thf(fact_656_tail__not__precedes,axiom,
! [Y: product_prod_v_v,X: product_prod_v_v,Xs: list_P7986770385144383213od_v_v] :
( ( sCC_Bl2026170059108282219od_v_v @ Y @ X @ ( cons_P4120604216776828829od_v_v @ X @ Xs ) )
=> ( ~ ( member7453568604450474000od_v_v @ X @ ( set_Product_prod_v_v2 @ Xs ) )
=> ( X = Y ) ) ) ).
% tail_not_precedes
thf(fact_657_tail__not__precedes,axiom,
! [Y: v,X: v,Xs: list_v] :
( ( sCC_Bl4022239298816431255edes_v @ Y @ X @ ( cons_v @ X @ Xs ) )
=> ( ~ ( member_v @ X @ ( set_v2 @ Xs ) )
=> ( X = Y ) ) ) ).
% tail_not_precedes
thf(fact_658_subset__insert__iff,axiom,
! [A: set_Product_prod_v_v,X: product_prod_v_v,B: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ ( insert1338601472111419319od_v_v @ X @ B ) )
= ( ( ( member7453568604450474000od_v_v @ X @ A )
=> ( ord_le7336532860387713383od_v_v @ ( minus_4183494784930505774od_v_v @ A @ ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v ) ) @ B ) )
& ( ~ ( member7453568604450474000od_v_v @ X @ A )
=> ( ord_le7336532860387713383od_v_v @ A @ B ) ) ) ) ).
% subset_insert_iff
thf(fact_659_subset__insert__iff,axiom,
! [A: set_v,X: v,B: set_v] :
( ( ord_less_eq_set_v @ A @ ( insert_v2 @ X @ B ) )
= ( ( ( member_v @ X @ A )
=> ( ord_less_eq_set_v @ ( minus_minus_set_v @ A @ ( insert_v2 @ X @ bot_bot_set_v ) ) @ B ) )
& ( ~ ( member_v @ X @ A )
=> ( ord_less_eq_set_v @ A @ B ) ) ) ) ).
% subset_insert_iff
thf(fact_660_Diff__single__insert,axiom,
! [A: set_Product_prod_v_v,X: product_prod_v_v,B: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ ( minus_4183494784930505774od_v_v @ A @ ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v ) ) @ B )
=> ( ord_le7336532860387713383od_v_v @ A @ ( insert1338601472111419319od_v_v @ X @ B ) ) ) ).
% Diff_single_insert
thf(fact_661_Diff__single__insert,axiom,
! [A: set_v,X: v,B: set_v] :
( ( ord_less_eq_set_v @ ( minus_minus_set_v @ A @ ( insert_v2 @ X @ bot_bot_set_v ) ) @ B )
=> ( ord_less_eq_set_v @ A @ ( insert_v2 @ X @ B ) ) ) ).
% Diff_single_insert
thf(fact_662_list_Oexhaust__sel,axiom,
! [List: list_v] :
( ( List != nil_v )
=> ( List
= ( cons_v @ ( hd_v @ List ) @ ( tl_v @ List ) ) ) ) ).
% list.exhaust_sel
thf(fact_663_boolean__algebra__cancel_Osup1,axiom,
! [A: set_v,K: set_v,A4: set_v,B2: set_v] :
( ( A
= ( sup_sup_set_v @ K @ A4 ) )
=> ( ( sup_sup_set_v @ A @ B2 )
= ( sup_sup_set_v @ K @ ( sup_sup_set_v @ A4 @ B2 ) ) ) ) ).
% boolean_algebra_cancel.sup1
thf(fact_664_boolean__algebra__cancel_Osup1,axiom,
! [A: set_Product_prod_v_v,K: set_Product_prod_v_v,A4: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( A
= ( sup_su414716646722978715od_v_v @ K @ A4 ) )
=> ( ( sup_su414716646722978715od_v_v @ A @ B2 )
= ( sup_su414716646722978715od_v_v @ K @ ( sup_su414716646722978715od_v_v @ A4 @ B2 ) ) ) ) ).
% boolean_algebra_cancel.sup1
thf(fact_665_boolean__algebra__cancel_Osup2,axiom,
! [B: set_v,K: set_v,B2: set_v,A4: set_v] :
( ( B
= ( sup_sup_set_v @ K @ B2 ) )
=> ( ( sup_sup_set_v @ A4 @ B )
= ( sup_sup_set_v @ K @ ( sup_sup_set_v @ A4 @ B2 ) ) ) ) ).
% boolean_algebra_cancel.sup2
thf(fact_666_boolean__algebra__cancel_Osup2,axiom,
! [B: set_Product_prod_v_v,K: set_Product_prod_v_v,B2: set_Product_prod_v_v,A4: set_Product_prod_v_v] :
( ( B
= ( sup_su414716646722978715od_v_v @ K @ B2 ) )
=> ( ( sup_su414716646722978715od_v_v @ A4 @ B )
= ( sup_su414716646722978715od_v_v @ K @ ( sup_su414716646722978715od_v_v @ A4 @ B2 ) ) ) ) ).
% boolean_algebra_cancel.sup2
thf(fact_667_graph_Osubscc__add,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,S: set_Product_prod_v_v,X: product_prod_v_v,Y: product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( sCC_Bl2301996248249672505od_v_v @ Successors @ S )
=> ( ( member7453568604450474000od_v_v @ X @ S )
=> ( ( sCC_Bl4981926079593201289od_v_v @ Successors @ X @ Y )
=> ( ( sCC_Bl4981926079593201289od_v_v @ Successors @ Y @ X )
=> ( sCC_Bl2301996248249672505od_v_v @ Successors @ ( insert1338601472111419319od_v_v @ Y @ S ) ) ) ) ) ) ) ).
% graph.subscc_add
thf(fact_668_graph_Osubscc__add,axiom,
! [Vertices: set_v,Successors: v > set_v,S: set_v,X: v,Y: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl5398416737448265317bscc_v @ Successors @ S )
=> ( ( member_v @ X @ S )
=> ( ( sCC_Bl649662514949026229able_v @ Successors @ X @ Y )
=> ( ( sCC_Bl649662514949026229able_v @ Successors @ Y @ X )
=> ( sCC_Bl5398416737448265317bscc_v @ Successors @ ( insert_v2 @ Y @ S ) ) ) ) ) ) ) ).
% graph.subscc_add
thf(fact_669_graph_Ora__add__edge,axiom,
! [Vertices: set_v,Successors: v > set_v,X: v,Y: v,E: set_Product_prod_v_v,V: v,W: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl4291963740693775144ding_v @ Successors @ X @ Y @ E )
=> ( ( sCC_Bl4291963740693775144ding_v @ Successors @ X @ Y @ ( sup_su414716646722978715od_v_v @ E @ ( insert1338601472111419319od_v_v @ ( product_Pair_v_v @ V @ W ) @ bot_bo723834152578015283od_v_v ) ) )
| ( ( sCC_Bl4291963740693775144ding_v @ Successors @ X @ V @ ( sup_su414716646722978715od_v_v @ E @ ( insert1338601472111419319od_v_v @ ( product_Pair_v_v @ V @ W ) @ bot_bo723834152578015283od_v_v ) ) )
& ( sCC_Bl4291963740693775144ding_v @ Successors @ W @ Y @ ( sup_su414716646722978715od_v_v @ E @ ( insert1338601472111419319od_v_v @ ( product_Pair_v_v @ V @ W ) @ bot_bo723834152578015283od_v_v ) ) ) ) ) ) ) ).
% graph.ra_add_edge
thf(fact_670_graph_Oavoiding__explored,axiom,
! [Vertices: set_v,Successors: v > set_v,E2: sCC_Bl1394983891496994913t_unit,X: v,Y: v,E: set_Product_prod_v_v,W: v,V: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl9196236973127232072t_unit @ Successors @ E2 )
=> ( ( sCC_Bl4291963740693775144ding_v @ Successors @ X @ Y @ E )
=> ( ~ ( member_v @ Y @ ( sCC_Bl157864678168468314t_unit @ E2 ) )
=> ( ( member_v @ W @ ( sCC_Bl157864678168468314t_unit @ E2 ) )
=> ( sCC_Bl4291963740693775144ding_v @ Successors @ X @ Y @ ( sup_su414716646722978715od_v_v @ E @ ( insert1338601472111419319od_v_v @ ( product_Pair_v_v @ V @ W ) @ bot_bo723834152578015283od_v_v ) ) ) ) ) ) ) ) ).
% graph.avoiding_explored
thf(fact_671_boolean__algebra_Odisj__zero__right,axiom,
! [X: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ X @ bot_bo723834152578015283od_v_v )
= X ) ).
% boolean_algebra.disj_zero_right
thf(fact_672_boolean__algebra_Odisj__zero__right,axiom,
! [X: set_v] :
( ( sup_sup_set_v @ X @ bot_bot_set_v )
= X ) ).
% boolean_algebra.disj_zero_right
thf(fact_673_graph_Opre__dfss__def,axiom,
! [Vertices: set_v,Successors: v > set_v,V: v,E2: sCC_Bl1394983891496994913t_unit] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl1748261141445803503t_unit @ Successors @ V @ E2 )
= ( ( sCC_Bl9196236973127232072t_unit @ Successors @ E2 )
& ( member_v @ V @ ( sCC_Bl4645233313691564917t_unit @ E2 ) )
& ( ( sCC_Bl8828226123343373779t_unit @ E2 )
!= nil_v )
& ( member_v @ V @ ( sCC_Bl1280885523602775798t_unit @ E2 @ ( hd_v @ ( sCC_Bl8828226123343373779t_unit @ E2 ) ) ) )
& ! [X2: v] :
( ( member_v @ X2 @ ( sCC_Bl3795065053823578884t_unit @ E2 @ V ) )
=> ( member_v @ X2 @ ( sup_sup_set_v @ ( sCC_Bl157864678168468314t_unit @ E2 ) @ ( sCC_Bl1280885523602775798t_unit @ E2 @ ( hd_v @ ( sCC_Bl8828226123343373779t_unit @ E2 ) ) ) ) ) )
& ! [X2: v] :
( ( member_v @ X2 @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E2 ) ) )
=> ( sCC_Bl649662514949026229able_v @ Successors @ X2 @ V ) )
& ? [Ns: list_v] :
( ( sCC_Bl9201514103433284750t_unit @ E2 )
= ( cons_v @ V @ Ns ) ) ) ) ) ).
% graph.pre_dfss_def
thf(fact_674_equality,axiom,
! [R2: sCC_Bl1394983891496994913t_unit,R3: sCC_Bl1394983891496994913t_unit] :
( ( ( sCC_Bl1090238580953940555t_unit @ R2 )
= ( sCC_Bl1090238580953940555t_unit @ R3 ) )
=> ( ( ( sCC_Bl1280885523602775798t_unit @ R2 )
= ( sCC_Bl1280885523602775798t_unit @ R3 ) )
=> ( ( ( sCC_Bl157864678168468314t_unit @ R2 )
= ( sCC_Bl157864678168468314t_unit @ R3 ) )
=> ( ( ( sCC_Bl4645233313691564917t_unit @ R2 )
= ( sCC_Bl4645233313691564917t_unit @ R3 ) )
=> ( ( ( sCC_Bl3795065053823578884t_unit @ R2 )
= ( sCC_Bl3795065053823578884t_unit @ R3 ) )
=> ( ( ( sCC_Bl2536197123907397897t_unit @ R2 )
= ( sCC_Bl2536197123907397897t_unit @ R3 ) )
=> ( ( ( sCC_Bl8828226123343373779t_unit @ R2 )
= ( sCC_Bl8828226123343373779t_unit @ R3 ) )
=> ( ( ( sCC_Bl9201514103433284750t_unit @ R2 )
= ( sCC_Bl9201514103433284750t_unit @ R3 ) )
=> ( ( ( sCC_Bl3567736435408124606t_unit @ R2 )
= ( sCC_Bl3567736435408124606t_unit @ R3 ) )
=> ( R2 = R3 ) ) ) ) ) ) ) ) ) ) ).
% equality
thf(fact_675_scc__partition,axiom,
! [S: set_v,S3: set_v,X: v] :
( ( sCC_Bloemen_is_scc_v @ successors @ S )
=> ( ( sCC_Bloemen_is_scc_v @ successors @ S3 )
=> ( ( member_v @ X @ ( inf_inf_set_v @ S @ S3 ) )
=> ( S = S3 ) ) ) ) ).
% scc_partition
thf(fact_676_the__elem__eq,axiom,
! [X: product_prod_v_v] :
( ( the_el5392834299063928540od_v_v @ ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v ) )
= X ) ).
% the_elem_eq
thf(fact_677_the__elem__eq,axiom,
! [X: v] :
( ( the_elem_v @ ( insert_v2 @ X @ bot_bot_set_v ) )
= X ) ).
% the_elem_eq
thf(fact_678_prod_Oinject,axiom,
! [X1: v,X23: v,Y1: v,Y22: v] :
( ( ( product_Pair_v_v @ X1 @ X23 )
= ( product_Pair_v_v @ Y1 @ Y22 ) )
= ( ( X1 = Y1 )
& ( X23 = Y22 ) ) ) ).
% prod.inject
thf(fact_679_prod_Oinject,axiom,
! [X1: v,X23: sCC_Bl1394983891496994913t_unit,Y1: v,Y22: sCC_Bl1394983891496994913t_unit] :
( ( ( produc3862955338007567901t_unit @ X1 @ X23 )
= ( produc3862955338007567901t_unit @ Y1 @ Y22 ) )
= ( ( X1 = Y1 )
& ( X23 = Y22 ) ) ) ).
% prod.inject
thf(fact_680_old_Oprod_Oinject,axiom,
! [A4: v,B2: v,A8: v,B7: v] :
( ( ( product_Pair_v_v @ A4 @ B2 )
= ( product_Pair_v_v @ A8 @ B7 ) )
= ( ( A4 = A8 )
& ( B2 = B7 ) ) ) ).
% old.prod.inject
thf(fact_681_old_Oprod_Oinject,axiom,
! [A4: v,B2: sCC_Bl1394983891496994913t_unit,A8: v,B7: sCC_Bl1394983891496994913t_unit] :
( ( ( produc3862955338007567901t_unit @ A4 @ B2 )
= ( produc3862955338007567901t_unit @ A8 @ B7 ) )
= ( ( A4 = A8 )
& ( B2 = B7 ) ) ) ).
% old.prod.inject
thf(fact_682_inf_Oidem,axiom,
! [A4: set_v] :
( ( inf_inf_set_v @ A4 @ A4 )
= A4 ) ).
% inf.idem
thf(fact_683_inf__idem,axiom,
! [X: set_v] :
( ( inf_inf_set_v @ X @ X )
= X ) ).
% inf_idem
thf(fact_684_inf_Oleft__idem,axiom,
! [A4: set_v,B2: set_v] :
( ( inf_inf_set_v @ A4 @ ( inf_inf_set_v @ A4 @ B2 ) )
= ( inf_inf_set_v @ A4 @ B2 ) ) ).
% inf.left_idem
thf(fact_685_inf__left__idem,axiom,
! [X: set_v,Y: set_v] :
( ( inf_inf_set_v @ X @ ( inf_inf_set_v @ X @ Y ) )
= ( inf_inf_set_v @ X @ Y ) ) ).
% inf_left_idem
thf(fact_686_inf_Oright__idem,axiom,
! [A4: set_v,B2: set_v] :
( ( inf_inf_set_v @ ( inf_inf_set_v @ A4 @ B2 ) @ B2 )
= ( inf_inf_set_v @ A4 @ B2 ) ) ).
% inf.right_idem
thf(fact_687_inf__right__idem,axiom,
! [X: set_v,Y: set_v] :
( ( inf_inf_set_v @ ( inf_inf_set_v @ X @ Y ) @ Y )
= ( inf_inf_set_v @ X @ Y ) ) ).
% inf_right_idem
thf(fact_688_IntI,axiom,
! [C: product_prod_v_v,A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ C @ A )
=> ( ( member7453568604450474000od_v_v @ C @ B )
=> ( member7453568604450474000od_v_v @ C @ ( inf_in6271465464967711157od_v_v @ A @ B ) ) ) ) ).
% IntI
thf(fact_689_IntI,axiom,
! [C: v,A: set_v,B: set_v] :
( ( member_v @ C @ A )
=> ( ( member_v @ C @ B )
=> ( member_v @ C @ ( inf_inf_set_v @ A @ B ) ) ) ) ).
% IntI
thf(fact_690_Int__iff,axiom,
! [C: product_prod_v_v,A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ C @ ( inf_in6271465464967711157od_v_v @ A @ B ) )
= ( ( member7453568604450474000od_v_v @ C @ A )
& ( member7453568604450474000od_v_v @ C @ B ) ) ) ).
% Int_iff
thf(fact_691_Int__iff,axiom,
! [C: v,A: set_v,B: set_v] :
( ( member_v @ C @ ( inf_inf_set_v @ A @ B ) )
= ( ( member_v @ C @ A )
& ( member_v @ C @ B ) ) ) ).
% Int_iff
thf(fact_692_inf_Obounded__iff,axiom,
! [A4: set_Product_prod_v_v,B2: set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A4 @ ( inf_in6271465464967711157od_v_v @ B2 @ C ) )
= ( ( ord_le7336532860387713383od_v_v @ A4 @ B2 )
& ( ord_le7336532860387713383od_v_v @ A4 @ C ) ) ) ).
% inf.bounded_iff
thf(fact_693_inf_Obounded__iff,axiom,
! [A4: set_v,B2: set_v,C: set_v] :
( ( ord_less_eq_set_v @ A4 @ ( inf_inf_set_v @ B2 @ C ) )
= ( ( ord_less_eq_set_v @ A4 @ B2 )
& ( ord_less_eq_set_v @ A4 @ C ) ) ) ).
% inf.bounded_iff
thf(fact_694_le__inf__iff,axiom,
! [X: set_Product_prod_v_v,Y: set_Product_prod_v_v,Z: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X @ ( inf_in6271465464967711157od_v_v @ Y @ Z ) )
= ( ( ord_le7336532860387713383od_v_v @ X @ Y )
& ( ord_le7336532860387713383od_v_v @ X @ Z ) ) ) ).
% le_inf_iff
thf(fact_695_le__inf__iff,axiom,
! [X: set_v,Y: set_v,Z: set_v] :
( ( ord_less_eq_set_v @ X @ ( inf_inf_set_v @ Y @ Z ) )
= ( ( ord_less_eq_set_v @ X @ Y )
& ( ord_less_eq_set_v @ X @ Z ) ) ) ).
% le_inf_iff
thf(fact_696_inf__bot__left,axiom,
! [X: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ bot_bo723834152578015283od_v_v @ X )
= bot_bo723834152578015283od_v_v ) ).
% inf_bot_left
thf(fact_697_inf__bot__left,axiom,
! [X: set_v] :
( ( inf_inf_set_v @ bot_bot_set_v @ X )
= bot_bot_set_v ) ).
% inf_bot_left
thf(fact_698_inf__bot__right,axiom,
! [X: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ X @ bot_bo723834152578015283od_v_v )
= bot_bo723834152578015283od_v_v ) ).
% inf_bot_right
thf(fact_699_inf__bot__right,axiom,
! [X: set_v] :
( ( inf_inf_set_v @ X @ bot_bot_set_v )
= bot_bot_set_v ) ).
% inf_bot_right
thf(fact_700_boolean__algebra_Oconj__zero__right,axiom,
! [X: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ X @ bot_bo723834152578015283od_v_v )
= bot_bo723834152578015283od_v_v ) ).
% boolean_algebra.conj_zero_right
thf(fact_701_boolean__algebra_Oconj__zero__right,axiom,
! [X: set_v] :
( ( inf_inf_set_v @ X @ bot_bot_set_v )
= bot_bot_set_v ) ).
% boolean_algebra.conj_zero_right
thf(fact_702_boolean__algebra_Oconj__zero__left,axiom,
! [X: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ bot_bo723834152578015283od_v_v @ X )
= bot_bo723834152578015283od_v_v ) ).
% boolean_algebra.conj_zero_left
thf(fact_703_boolean__algebra_Oconj__zero__left,axiom,
! [X: set_v] :
( ( inf_inf_set_v @ bot_bot_set_v @ X )
= bot_bot_set_v ) ).
% boolean_algebra.conj_zero_left
thf(fact_704_sup__inf__absorb,axiom,
! [X: set_v,Y: set_v] :
( ( sup_sup_set_v @ X @ ( inf_inf_set_v @ X @ Y ) )
= X ) ).
% sup_inf_absorb
thf(fact_705_sup__inf__absorb,axiom,
! [X: set_Product_prod_v_v,Y: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ X @ ( inf_in6271465464967711157od_v_v @ X @ Y ) )
= X ) ).
% sup_inf_absorb
thf(fact_706_inf__sup__absorb,axiom,
! [X: set_v,Y: set_v] :
( ( inf_inf_set_v @ X @ ( sup_sup_set_v @ X @ Y ) )
= X ) ).
% inf_sup_absorb
thf(fact_707_inf__sup__absorb,axiom,
! [X: set_Product_prod_v_v,Y: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ X @ ( sup_su414716646722978715od_v_v @ X @ Y ) )
= X ) ).
% inf_sup_absorb
thf(fact_708_Int__subset__iff,axiom,
! [C2: set_Product_prod_v_v,A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ C2 @ ( inf_in6271465464967711157od_v_v @ A @ B ) )
= ( ( ord_le7336532860387713383od_v_v @ C2 @ A )
& ( ord_le7336532860387713383od_v_v @ C2 @ B ) ) ) ).
% Int_subset_iff
thf(fact_709_Int__subset__iff,axiom,
! [C2: set_v,A: set_v,B: set_v] :
( ( ord_less_eq_set_v @ C2 @ ( inf_inf_set_v @ A @ B ) )
= ( ( ord_less_eq_set_v @ C2 @ A )
& ( ord_less_eq_set_v @ C2 @ B ) ) ) ).
% Int_subset_iff
thf(fact_710_Int__insert__left__if0,axiom,
! [A4: product_prod_v_v,C2: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ~ ( member7453568604450474000od_v_v @ A4 @ C2 )
=> ( ( inf_in6271465464967711157od_v_v @ ( insert1338601472111419319od_v_v @ A4 @ B ) @ C2 )
= ( inf_in6271465464967711157od_v_v @ B @ C2 ) ) ) ).
% Int_insert_left_if0
thf(fact_711_Int__insert__left__if0,axiom,
! [A4: v,C2: set_v,B: set_v] :
( ~ ( member_v @ A4 @ C2 )
=> ( ( inf_inf_set_v @ ( insert_v2 @ A4 @ B ) @ C2 )
= ( inf_inf_set_v @ B @ C2 ) ) ) ).
% Int_insert_left_if0
thf(fact_712_Int__insert__left__if1,axiom,
! [A4: product_prod_v_v,C2: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ A4 @ C2 )
=> ( ( inf_in6271465464967711157od_v_v @ ( insert1338601472111419319od_v_v @ A4 @ B ) @ C2 )
= ( insert1338601472111419319od_v_v @ A4 @ ( inf_in6271465464967711157od_v_v @ B @ C2 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_713_Int__insert__left__if1,axiom,
! [A4: v,C2: set_v,B: set_v] :
( ( member_v @ A4 @ C2 )
=> ( ( inf_inf_set_v @ ( insert_v2 @ A4 @ B ) @ C2 )
= ( insert_v2 @ A4 @ ( inf_inf_set_v @ B @ C2 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_714_insert__inter__insert,axiom,
! [A4: product_prod_v_v,A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ ( insert1338601472111419319od_v_v @ A4 @ A ) @ ( insert1338601472111419319od_v_v @ A4 @ B ) )
= ( insert1338601472111419319od_v_v @ A4 @ ( inf_in6271465464967711157od_v_v @ A @ B ) ) ) ).
% insert_inter_insert
thf(fact_715_insert__inter__insert,axiom,
! [A4: v,A: set_v,B: set_v] :
( ( inf_inf_set_v @ ( insert_v2 @ A4 @ A ) @ ( insert_v2 @ A4 @ B ) )
= ( insert_v2 @ A4 @ ( inf_inf_set_v @ A @ B ) ) ) ).
% insert_inter_insert
thf(fact_716_Int__insert__right__if0,axiom,
! [A4: product_prod_v_v,A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ~ ( member7453568604450474000od_v_v @ A4 @ A )
=> ( ( inf_in6271465464967711157od_v_v @ A @ ( insert1338601472111419319od_v_v @ A4 @ B ) )
= ( inf_in6271465464967711157od_v_v @ A @ B ) ) ) ).
% Int_insert_right_if0
thf(fact_717_Int__insert__right__if0,axiom,
! [A4: v,A: set_v,B: set_v] :
( ~ ( member_v @ A4 @ A )
=> ( ( inf_inf_set_v @ A @ ( insert_v2 @ A4 @ B ) )
= ( inf_inf_set_v @ A @ B ) ) ) ).
% Int_insert_right_if0
thf(fact_718_Int__insert__right__if1,axiom,
! [A4: product_prod_v_v,A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ A4 @ A )
=> ( ( inf_in6271465464967711157od_v_v @ A @ ( insert1338601472111419319od_v_v @ A4 @ B ) )
= ( insert1338601472111419319od_v_v @ A4 @ ( inf_in6271465464967711157od_v_v @ A @ B ) ) ) ) ).
% Int_insert_right_if1
thf(fact_719_Int__insert__right__if1,axiom,
! [A4: v,A: set_v,B: set_v] :
( ( member_v @ A4 @ A )
=> ( ( inf_inf_set_v @ A @ ( insert_v2 @ A4 @ B ) )
= ( insert_v2 @ A4 @ ( inf_inf_set_v @ A @ B ) ) ) ) ).
% Int_insert_right_if1
thf(fact_720_Int__Un__eq_I4_J,axiom,
! [T2: set_v,S: set_v] :
( ( sup_sup_set_v @ T2 @ ( inf_inf_set_v @ S @ T2 ) )
= T2 ) ).
% Int_Un_eq(4)
thf(fact_721_Int__Un__eq_I4_J,axiom,
! [T2: set_Product_prod_v_v,S: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ T2 @ ( inf_in6271465464967711157od_v_v @ S @ T2 ) )
= T2 ) ).
% Int_Un_eq(4)
thf(fact_722_Int__Un__eq_I3_J,axiom,
! [S: set_v,T2: set_v] :
( ( sup_sup_set_v @ S @ ( inf_inf_set_v @ S @ T2 ) )
= S ) ).
% Int_Un_eq(3)
thf(fact_723_Int__Un__eq_I3_J,axiom,
! [S: set_Product_prod_v_v,T2: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ S @ ( inf_in6271465464967711157od_v_v @ S @ T2 ) )
= S ) ).
% Int_Un_eq(3)
thf(fact_724_Int__Un__eq_I2_J,axiom,
! [S: set_v,T2: set_v] :
( ( sup_sup_set_v @ ( inf_inf_set_v @ S @ T2 ) @ T2 )
= T2 ) ).
% Int_Un_eq(2)
thf(fact_725_Int__Un__eq_I2_J,axiom,
! [S: set_Product_prod_v_v,T2: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ S @ T2 ) @ T2 )
= T2 ) ).
% Int_Un_eq(2)
thf(fact_726_Int__Un__eq_I1_J,axiom,
! [S: set_v,T2: set_v] :
( ( sup_sup_set_v @ ( inf_inf_set_v @ S @ T2 ) @ S )
= S ) ).
% Int_Un_eq(1)
thf(fact_727_Int__Un__eq_I1_J,axiom,
! [S: set_Product_prod_v_v,T2: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ S @ T2 ) @ S )
= S ) ).
% Int_Un_eq(1)
thf(fact_728_Un__Int__eq_I4_J,axiom,
! [T2: set_v,S: set_v] :
( ( inf_inf_set_v @ T2 @ ( sup_sup_set_v @ S @ T2 ) )
= T2 ) ).
% Un_Int_eq(4)
thf(fact_729_Un__Int__eq_I4_J,axiom,
! [T2: set_Product_prod_v_v,S: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ T2 @ ( sup_su414716646722978715od_v_v @ S @ T2 ) )
= T2 ) ).
% Un_Int_eq(4)
thf(fact_730_Un__Int__eq_I3_J,axiom,
! [S: set_v,T2: set_v] :
( ( inf_inf_set_v @ S @ ( sup_sup_set_v @ S @ T2 ) )
= S ) ).
% Un_Int_eq(3)
thf(fact_731_Un__Int__eq_I3_J,axiom,
! [S: set_Product_prod_v_v,T2: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ S @ ( sup_su414716646722978715od_v_v @ S @ T2 ) )
= S ) ).
% Un_Int_eq(3)
thf(fact_732_Un__Int__eq_I2_J,axiom,
! [S: set_v,T2: set_v] :
( ( inf_inf_set_v @ ( sup_sup_set_v @ S @ T2 ) @ T2 )
= T2 ) ).
% Un_Int_eq(2)
thf(fact_733_Un__Int__eq_I2_J,axiom,
! [S: set_Product_prod_v_v,T2: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ ( sup_su414716646722978715od_v_v @ S @ T2 ) @ T2 )
= T2 ) ).
% Un_Int_eq(2)
thf(fact_734_Un__Int__eq_I1_J,axiom,
! [S: set_v,T2: set_v] :
( ( inf_inf_set_v @ ( sup_sup_set_v @ S @ T2 ) @ S )
= S ) ).
% Un_Int_eq(1)
thf(fact_735_Un__Int__eq_I1_J,axiom,
! [S: set_Product_prod_v_v,T2: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ ( sup_su414716646722978715od_v_v @ S @ T2 ) @ S )
= S ) ).
% Un_Int_eq(1)
thf(fact_736_insert__disjoint_I1_J,axiom,
! [A4: product_prod_v_v,A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ( inf_in6271465464967711157od_v_v @ ( insert1338601472111419319od_v_v @ A4 @ A ) @ B )
= bot_bo723834152578015283od_v_v )
= ( ~ ( member7453568604450474000od_v_v @ A4 @ B )
& ( ( inf_in6271465464967711157od_v_v @ A @ B )
= bot_bo723834152578015283od_v_v ) ) ) ).
% insert_disjoint(1)
thf(fact_737_insert__disjoint_I1_J,axiom,
! [A4: v,A: set_v,B: set_v] :
( ( ( inf_inf_set_v @ ( insert_v2 @ A4 @ A ) @ B )
= bot_bot_set_v )
= ( ~ ( member_v @ A4 @ B )
& ( ( inf_inf_set_v @ A @ B )
= bot_bot_set_v ) ) ) ).
% insert_disjoint(1)
thf(fact_738_insert__disjoint_I2_J,axiom,
! [A4: product_prod_v_v,A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( bot_bo723834152578015283od_v_v
= ( inf_in6271465464967711157od_v_v @ ( insert1338601472111419319od_v_v @ A4 @ A ) @ B ) )
= ( ~ ( member7453568604450474000od_v_v @ A4 @ B )
& ( bot_bo723834152578015283od_v_v
= ( inf_in6271465464967711157od_v_v @ A @ B ) ) ) ) ).
% insert_disjoint(2)
thf(fact_739_insert__disjoint_I2_J,axiom,
! [A4: v,A: set_v,B: set_v] :
( ( bot_bot_set_v
= ( inf_inf_set_v @ ( insert_v2 @ A4 @ A ) @ B ) )
= ( ~ ( member_v @ A4 @ B )
& ( bot_bot_set_v
= ( inf_inf_set_v @ A @ B ) ) ) ) ).
% insert_disjoint(2)
thf(fact_740_disjoint__insert_I1_J,axiom,
! [B: set_Product_prod_v_v,A4: product_prod_v_v,A: set_Product_prod_v_v] :
( ( ( inf_in6271465464967711157od_v_v @ B @ ( insert1338601472111419319od_v_v @ A4 @ A ) )
= bot_bo723834152578015283od_v_v )
= ( ~ ( member7453568604450474000od_v_v @ A4 @ B )
& ( ( inf_in6271465464967711157od_v_v @ B @ A )
= bot_bo723834152578015283od_v_v ) ) ) ).
% disjoint_insert(1)
thf(fact_741_disjoint__insert_I1_J,axiom,
! [B: set_v,A4: v,A: set_v] :
( ( ( inf_inf_set_v @ B @ ( insert_v2 @ A4 @ A ) )
= bot_bot_set_v )
= ( ~ ( member_v @ A4 @ B )
& ( ( inf_inf_set_v @ B @ A )
= bot_bot_set_v ) ) ) ).
% disjoint_insert(1)
thf(fact_742_disjoint__insert_I2_J,axiom,
! [A: set_Product_prod_v_v,B2: product_prod_v_v,B: set_Product_prod_v_v] :
( ( bot_bo723834152578015283od_v_v
= ( inf_in6271465464967711157od_v_v @ A @ ( insert1338601472111419319od_v_v @ B2 @ B ) ) )
= ( ~ ( member7453568604450474000od_v_v @ B2 @ A )
& ( bot_bo723834152578015283od_v_v
= ( inf_in6271465464967711157od_v_v @ A @ B ) ) ) ) ).
% disjoint_insert(2)
thf(fact_743_disjoint__insert_I2_J,axiom,
! [A: set_v,B2: v,B: set_v] :
( ( bot_bot_set_v
= ( inf_inf_set_v @ A @ ( insert_v2 @ B2 @ B ) ) )
= ( ~ ( member_v @ B2 @ A )
& ( bot_bot_set_v
= ( inf_inf_set_v @ A @ B ) ) ) ) ).
% disjoint_insert(2)
thf(fact_744_Diff__disjoint,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ A @ ( minus_4183494784930505774od_v_v @ B @ A ) )
= bot_bo723834152578015283od_v_v ) ).
% Diff_disjoint
thf(fact_745_Diff__disjoint,axiom,
! [A: set_v,B: set_v] :
( ( inf_inf_set_v @ A @ ( minus_minus_set_v @ B @ A ) )
= bot_bot_set_v ) ).
% Diff_disjoint
thf(fact_746_boolean__algebra__cancel_Oinf1,axiom,
! [A: set_v,K: set_v,A4: set_v,B2: set_v] :
( ( A
= ( inf_inf_set_v @ K @ A4 ) )
=> ( ( inf_inf_set_v @ A @ B2 )
= ( inf_inf_set_v @ K @ ( inf_inf_set_v @ A4 @ B2 ) ) ) ) ).
% boolean_algebra_cancel.inf1
thf(fact_747_boolean__algebra__cancel_Oinf2,axiom,
! [B: set_v,K: set_v,B2: set_v,A4: set_v] :
( ( B
= ( inf_inf_set_v @ K @ B2 ) )
=> ( ( inf_inf_set_v @ A4 @ B )
= ( inf_inf_set_v @ K @ ( inf_inf_set_v @ A4 @ B2 ) ) ) ) ).
% boolean_algebra_cancel.inf2
thf(fact_748_IntE,axiom,
! [C: product_prod_v_v,A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ C @ ( inf_in6271465464967711157od_v_v @ A @ B ) )
=> ~ ( ( member7453568604450474000od_v_v @ C @ A )
=> ~ ( member7453568604450474000od_v_v @ C @ B ) ) ) ).
% IntE
thf(fact_749_IntE,axiom,
! [C: v,A: set_v,B: set_v] :
( ( member_v @ C @ ( inf_inf_set_v @ A @ B ) )
=> ~ ( ( member_v @ C @ A )
=> ~ ( member_v @ C @ B ) ) ) ).
% IntE
thf(fact_750_IntD1,axiom,
! [C: product_prod_v_v,A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ C @ ( inf_in6271465464967711157od_v_v @ A @ B ) )
=> ( member7453568604450474000od_v_v @ C @ A ) ) ).
% IntD1
thf(fact_751_IntD1,axiom,
! [C: v,A: set_v,B: set_v] :
( ( member_v @ C @ ( inf_inf_set_v @ A @ B ) )
=> ( member_v @ C @ A ) ) ).
% IntD1
thf(fact_752_IntD2,axiom,
! [C: product_prod_v_v,A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ C @ ( inf_in6271465464967711157od_v_v @ A @ B ) )
=> ( member7453568604450474000od_v_v @ C @ B ) ) ).
% IntD2
thf(fact_753_IntD2,axiom,
! [C: v,A: set_v,B: set_v] :
( ( member_v @ C @ ( inf_inf_set_v @ A @ B ) )
=> ( member_v @ C @ B ) ) ).
% IntD2
thf(fact_754_Int__assoc,axiom,
! [A: set_v,B: set_v,C2: set_v] :
( ( inf_inf_set_v @ ( inf_inf_set_v @ A @ B ) @ C2 )
= ( inf_inf_set_v @ A @ ( inf_inf_set_v @ B @ C2 ) ) ) ).
% Int_assoc
thf(fact_755_Int__absorb,axiom,
! [A: set_v] :
( ( inf_inf_set_v @ A @ A )
= A ) ).
% Int_absorb
thf(fact_756_Int__commute,axiom,
( inf_inf_set_v
= ( ^ [A6: set_v,B4: set_v] : ( inf_inf_set_v @ B4 @ A6 ) ) ) ).
% Int_commute
thf(fact_757_Int__left__absorb,axiom,
! [A: set_v,B: set_v] :
( ( inf_inf_set_v @ A @ ( inf_inf_set_v @ A @ B ) )
= ( inf_inf_set_v @ A @ B ) ) ).
% Int_left_absorb
thf(fact_758_Int__left__commute,axiom,
! [A: set_v,B: set_v,C2: set_v] :
( ( inf_inf_set_v @ A @ ( inf_inf_set_v @ B @ C2 ) )
= ( inf_inf_set_v @ B @ ( inf_inf_set_v @ A @ C2 ) ) ) ).
% Int_left_commute
thf(fact_759_inf__sup__aci_I4_J,axiom,
! [X: set_v,Y: set_v] :
( ( inf_inf_set_v @ X @ ( inf_inf_set_v @ X @ Y ) )
= ( inf_inf_set_v @ X @ Y ) ) ).
% inf_sup_aci(4)
thf(fact_760_inf__sup__aci_I3_J,axiom,
! [X: set_v,Y: set_v,Z: set_v] :
( ( inf_inf_set_v @ X @ ( inf_inf_set_v @ Y @ Z ) )
= ( inf_inf_set_v @ Y @ ( inf_inf_set_v @ X @ Z ) ) ) ).
% inf_sup_aci(3)
thf(fact_761_inf__sup__aci_I2_J,axiom,
! [X: set_v,Y: set_v,Z: set_v] :
( ( inf_inf_set_v @ ( inf_inf_set_v @ X @ Y ) @ Z )
= ( inf_inf_set_v @ X @ ( inf_inf_set_v @ Y @ Z ) ) ) ).
% inf_sup_aci(2)
thf(fact_762_inf__sup__aci_I1_J,axiom,
( inf_inf_set_v
= ( ^ [X2: set_v,Y3: set_v] : ( inf_inf_set_v @ Y3 @ X2 ) ) ) ).
% inf_sup_aci(1)
thf(fact_763_inf_Oassoc,axiom,
! [A4: set_v,B2: set_v,C: set_v] :
( ( inf_inf_set_v @ ( inf_inf_set_v @ A4 @ B2 ) @ C )
= ( inf_inf_set_v @ A4 @ ( inf_inf_set_v @ B2 @ C ) ) ) ).
% inf.assoc
thf(fact_764_inf__assoc,axiom,
! [X: set_v,Y: set_v,Z: set_v] :
( ( inf_inf_set_v @ ( inf_inf_set_v @ X @ Y ) @ Z )
= ( inf_inf_set_v @ X @ ( inf_inf_set_v @ Y @ Z ) ) ) ).
% inf_assoc
thf(fact_765_inf_Ocommute,axiom,
( inf_inf_set_v
= ( ^ [A5: set_v,B3: set_v] : ( inf_inf_set_v @ B3 @ A5 ) ) ) ).
% inf.commute
thf(fact_766_inf__commute,axiom,
( inf_inf_set_v
= ( ^ [X2: set_v,Y3: set_v] : ( inf_inf_set_v @ Y3 @ X2 ) ) ) ).
% inf_commute
thf(fact_767_inf_Oleft__commute,axiom,
! [B2: set_v,A4: set_v,C: set_v] :
( ( inf_inf_set_v @ B2 @ ( inf_inf_set_v @ A4 @ C ) )
= ( inf_inf_set_v @ A4 @ ( inf_inf_set_v @ B2 @ C ) ) ) ).
% inf.left_commute
thf(fact_768_inf__left__commute,axiom,
! [X: set_v,Y: set_v,Z: set_v] :
( ( inf_inf_set_v @ X @ ( inf_inf_set_v @ Y @ Z ) )
= ( inf_inf_set_v @ Y @ ( inf_inf_set_v @ X @ Z ) ) ) ).
% inf_left_commute
thf(fact_769_inf_OcoboundedI2,axiom,
! [B2: set_Product_prod_v_v,C: set_Product_prod_v_v,A4: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ B2 @ C )
=> ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ A4 @ B2 ) @ C ) ) ).
% inf.coboundedI2
thf(fact_770_inf_OcoboundedI2,axiom,
! [B2: set_v,C: set_v,A4: set_v] :
( ( ord_less_eq_set_v @ B2 @ C )
=> ( ord_less_eq_set_v @ ( inf_inf_set_v @ A4 @ B2 ) @ C ) ) ).
% inf.coboundedI2
thf(fact_771_inf_OcoboundedI1,axiom,
! [A4: set_Product_prod_v_v,C: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A4 @ C )
=> ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ A4 @ B2 ) @ C ) ) ).
% inf.coboundedI1
thf(fact_772_inf_OcoboundedI1,axiom,
! [A4: set_v,C: set_v,B2: set_v] :
( ( ord_less_eq_set_v @ A4 @ C )
=> ( ord_less_eq_set_v @ ( inf_inf_set_v @ A4 @ B2 ) @ C ) ) ).
% inf.coboundedI1
thf(fact_773_inf_Oabsorb__iff2,axiom,
( ord_le7336532860387713383od_v_v
= ( ^ [B3: set_Product_prod_v_v,A5: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ A5 @ B3 )
= B3 ) ) ) ).
% inf.absorb_iff2
thf(fact_774_inf_Oabsorb__iff2,axiom,
( ord_less_eq_set_v
= ( ^ [B3: set_v,A5: set_v] :
( ( inf_inf_set_v @ A5 @ B3 )
= B3 ) ) ) ).
% inf.absorb_iff2
thf(fact_775_inf_Oabsorb__iff1,axiom,
( ord_le7336532860387713383od_v_v
= ( ^ [A5: set_Product_prod_v_v,B3: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ A5 @ B3 )
= A5 ) ) ) ).
% inf.absorb_iff1
thf(fact_776_inf_Oabsorb__iff1,axiom,
( ord_less_eq_set_v
= ( ^ [A5: set_v,B3: set_v] :
( ( inf_inf_set_v @ A5 @ B3 )
= A5 ) ) ) ).
% inf.absorb_iff1
thf(fact_777_inf_Ocobounded2,axiom,
! [A4: set_Product_prod_v_v,B2: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ A4 @ B2 ) @ B2 ) ).
% inf.cobounded2
thf(fact_778_inf_Ocobounded2,axiom,
! [A4: set_v,B2: set_v] : ( ord_less_eq_set_v @ ( inf_inf_set_v @ A4 @ B2 ) @ B2 ) ).
% inf.cobounded2
thf(fact_779_inf_Ocobounded1,axiom,
! [A4: set_Product_prod_v_v,B2: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ A4 @ B2 ) @ A4 ) ).
% inf.cobounded1
thf(fact_780_inf_Ocobounded1,axiom,
! [A4: set_v,B2: set_v] : ( ord_less_eq_set_v @ ( inf_inf_set_v @ A4 @ B2 ) @ A4 ) ).
% inf.cobounded1
thf(fact_781_inf_Oorder__iff,axiom,
( ord_le7336532860387713383od_v_v
= ( ^ [A5: set_Product_prod_v_v,B3: set_Product_prod_v_v] :
( A5
= ( inf_in6271465464967711157od_v_v @ A5 @ B3 ) ) ) ) ).
% inf.order_iff
thf(fact_782_inf_Oorder__iff,axiom,
( ord_less_eq_set_v
= ( ^ [A5: set_v,B3: set_v] :
( A5
= ( inf_inf_set_v @ A5 @ B3 ) ) ) ) ).
% inf.order_iff
thf(fact_783_inf__greatest,axiom,
! [X: set_Product_prod_v_v,Y: set_Product_prod_v_v,Z: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X @ Y )
=> ( ( ord_le7336532860387713383od_v_v @ X @ Z )
=> ( ord_le7336532860387713383od_v_v @ X @ ( inf_in6271465464967711157od_v_v @ Y @ Z ) ) ) ) ).
% inf_greatest
thf(fact_784_inf__greatest,axiom,
! [X: set_v,Y: set_v,Z: set_v] :
( ( ord_less_eq_set_v @ X @ Y )
=> ( ( ord_less_eq_set_v @ X @ Z )
=> ( ord_less_eq_set_v @ X @ ( inf_inf_set_v @ Y @ Z ) ) ) ) ).
% inf_greatest
thf(fact_785_inf_OboundedI,axiom,
! [A4: set_Product_prod_v_v,B2: set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A4 @ B2 )
=> ( ( ord_le7336532860387713383od_v_v @ A4 @ C )
=> ( ord_le7336532860387713383od_v_v @ A4 @ ( inf_in6271465464967711157od_v_v @ B2 @ C ) ) ) ) ).
% inf.boundedI
thf(fact_786_inf_OboundedI,axiom,
! [A4: set_v,B2: set_v,C: set_v] :
( ( ord_less_eq_set_v @ A4 @ B2 )
=> ( ( ord_less_eq_set_v @ A4 @ C )
=> ( ord_less_eq_set_v @ A4 @ ( inf_inf_set_v @ B2 @ C ) ) ) ) ).
% inf.boundedI
thf(fact_787_inf_OboundedE,axiom,
! [A4: set_Product_prod_v_v,B2: set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A4 @ ( inf_in6271465464967711157od_v_v @ B2 @ C ) )
=> ~ ( ( ord_le7336532860387713383od_v_v @ A4 @ B2 )
=> ~ ( ord_le7336532860387713383od_v_v @ A4 @ C ) ) ) ).
% inf.boundedE
thf(fact_788_inf_OboundedE,axiom,
! [A4: set_v,B2: set_v,C: set_v] :
( ( ord_less_eq_set_v @ A4 @ ( inf_inf_set_v @ B2 @ C ) )
=> ~ ( ( ord_less_eq_set_v @ A4 @ B2 )
=> ~ ( ord_less_eq_set_v @ A4 @ C ) ) ) ).
% inf.boundedE
thf(fact_789_inf__absorb2,axiom,
! [Y: set_Product_prod_v_v,X: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ Y @ X )
=> ( ( inf_in6271465464967711157od_v_v @ X @ Y )
= Y ) ) ).
% inf_absorb2
thf(fact_790_inf__absorb2,axiom,
! [Y: set_v,X: set_v] :
( ( ord_less_eq_set_v @ Y @ X )
=> ( ( inf_inf_set_v @ X @ Y )
= Y ) ) ).
% inf_absorb2
thf(fact_791_inf__absorb1,axiom,
! [X: set_Product_prod_v_v,Y: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X @ Y )
=> ( ( inf_in6271465464967711157od_v_v @ X @ Y )
= X ) ) ).
% inf_absorb1
thf(fact_792_inf__absorb1,axiom,
! [X: set_v,Y: set_v] :
( ( ord_less_eq_set_v @ X @ Y )
=> ( ( inf_inf_set_v @ X @ Y )
= X ) ) ).
% inf_absorb1
thf(fact_793_inf_Oabsorb2,axiom,
! [B2: set_Product_prod_v_v,A4: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ B2 @ A4 )
=> ( ( inf_in6271465464967711157od_v_v @ A4 @ B2 )
= B2 ) ) ).
% inf.absorb2
thf(fact_794_inf_Oabsorb2,axiom,
! [B2: set_v,A4: set_v] :
( ( ord_less_eq_set_v @ B2 @ A4 )
=> ( ( inf_inf_set_v @ A4 @ B2 )
= B2 ) ) ).
% inf.absorb2
thf(fact_795_inf_Oabsorb1,axiom,
! [A4: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A4 @ B2 )
=> ( ( inf_in6271465464967711157od_v_v @ A4 @ B2 )
= A4 ) ) ).
% inf.absorb1
thf(fact_796_inf_Oabsorb1,axiom,
! [A4: set_v,B2: set_v] :
( ( ord_less_eq_set_v @ A4 @ B2 )
=> ( ( inf_inf_set_v @ A4 @ B2 )
= A4 ) ) ).
% inf.absorb1
thf(fact_797_le__iff__inf,axiom,
( ord_le7336532860387713383od_v_v
= ( ^ [X2: set_Product_prod_v_v,Y3: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ X2 @ Y3 )
= X2 ) ) ) ).
% le_iff_inf
thf(fact_798_le__iff__inf,axiom,
( ord_less_eq_set_v
= ( ^ [X2: set_v,Y3: set_v] :
( ( inf_inf_set_v @ X2 @ Y3 )
= X2 ) ) ) ).
% le_iff_inf
thf(fact_799_inf__unique,axiom,
! [F: set_Product_prod_v_v > set_Product_prod_v_v > set_Product_prod_v_v,X: set_Product_prod_v_v,Y: set_Product_prod_v_v] :
( ! [X3: set_Product_prod_v_v,Y2: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ ( F @ X3 @ Y2 ) @ X3 )
=> ( ! [X3: set_Product_prod_v_v,Y2: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ ( F @ X3 @ Y2 ) @ Y2 )
=> ( ! [X3: set_Product_prod_v_v,Y2: set_Product_prod_v_v,Z2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X3 @ Y2 )
=> ( ( ord_le7336532860387713383od_v_v @ X3 @ Z2 )
=> ( ord_le7336532860387713383od_v_v @ X3 @ ( F @ Y2 @ Z2 ) ) ) )
=> ( ( inf_in6271465464967711157od_v_v @ X @ Y )
= ( F @ X @ Y ) ) ) ) ) ).
% inf_unique
thf(fact_800_inf__unique,axiom,
! [F: set_v > set_v > set_v,X: set_v,Y: set_v] :
( ! [X3: set_v,Y2: set_v] : ( ord_less_eq_set_v @ ( F @ X3 @ Y2 ) @ X3 )
=> ( ! [X3: set_v,Y2: set_v] : ( ord_less_eq_set_v @ ( F @ X3 @ Y2 ) @ Y2 )
=> ( ! [X3: set_v,Y2: set_v,Z2: set_v] :
( ( ord_less_eq_set_v @ X3 @ Y2 )
=> ( ( ord_less_eq_set_v @ X3 @ Z2 )
=> ( ord_less_eq_set_v @ X3 @ ( F @ Y2 @ Z2 ) ) ) )
=> ( ( inf_inf_set_v @ X @ Y )
= ( F @ X @ Y ) ) ) ) ) ).
% inf_unique
thf(fact_801_inf_OorderI,axiom,
! [A4: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( A4
= ( inf_in6271465464967711157od_v_v @ A4 @ B2 ) )
=> ( ord_le7336532860387713383od_v_v @ A4 @ B2 ) ) ).
% inf.orderI
thf(fact_802_inf_OorderI,axiom,
! [A4: set_v,B2: set_v] :
( ( A4
= ( inf_inf_set_v @ A4 @ B2 ) )
=> ( ord_less_eq_set_v @ A4 @ B2 ) ) ).
% inf.orderI
thf(fact_803_inf_OorderE,axiom,
! [A4: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A4 @ B2 )
=> ( A4
= ( inf_in6271465464967711157od_v_v @ A4 @ B2 ) ) ) ).
% inf.orderE
thf(fact_804_inf_OorderE,axiom,
! [A4: set_v,B2: set_v] :
( ( ord_less_eq_set_v @ A4 @ B2 )
=> ( A4
= ( inf_inf_set_v @ A4 @ B2 ) ) ) ).
% inf.orderE
thf(fact_805_le__infI2,axiom,
! [B2: set_Product_prod_v_v,X: set_Product_prod_v_v,A4: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ B2 @ X )
=> ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ A4 @ B2 ) @ X ) ) ).
% le_infI2
thf(fact_806_le__infI2,axiom,
! [B2: set_v,X: set_v,A4: set_v] :
( ( ord_less_eq_set_v @ B2 @ X )
=> ( ord_less_eq_set_v @ ( inf_inf_set_v @ A4 @ B2 ) @ X ) ) ).
% le_infI2
thf(fact_807_le__infI1,axiom,
! [A4: set_Product_prod_v_v,X: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A4 @ X )
=> ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ A4 @ B2 ) @ X ) ) ).
% le_infI1
thf(fact_808_le__infI1,axiom,
! [A4: set_v,X: set_v,B2: set_v] :
( ( ord_less_eq_set_v @ A4 @ X )
=> ( ord_less_eq_set_v @ ( inf_inf_set_v @ A4 @ B2 ) @ X ) ) ).
% le_infI1
thf(fact_809_inf__mono,axiom,
! [A4: set_Product_prod_v_v,C: set_Product_prod_v_v,B2: set_Product_prod_v_v,D2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A4 @ C )
=> ( ( ord_le7336532860387713383od_v_v @ B2 @ D2 )
=> ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ A4 @ B2 ) @ ( inf_in6271465464967711157od_v_v @ C @ D2 ) ) ) ) ).
% inf_mono
thf(fact_810_inf__mono,axiom,
! [A4: set_v,C: set_v,B2: set_v,D2: set_v] :
( ( ord_less_eq_set_v @ A4 @ C )
=> ( ( ord_less_eq_set_v @ B2 @ D2 )
=> ( ord_less_eq_set_v @ ( inf_inf_set_v @ A4 @ B2 ) @ ( inf_inf_set_v @ C @ D2 ) ) ) ) ).
% inf_mono
thf(fact_811_le__infI,axiom,
! [X: set_Product_prod_v_v,A4: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X @ A4 )
=> ( ( ord_le7336532860387713383od_v_v @ X @ B2 )
=> ( ord_le7336532860387713383od_v_v @ X @ ( inf_in6271465464967711157od_v_v @ A4 @ B2 ) ) ) ) ).
% le_infI
thf(fact_812_le__infI,axiom,
! [X: set_v,A4: set_v,B2: set_v] :
( ( ord_less_eq_set_v @ X @ A4 )
=> ( ( ord_less_eq_set_v @ X @ B2 )
=> ( ord_less_eq_set_v @ X @ ( inf_inf_set_v @ A4 @ B2 ) ) ) ) ).
% le_infI
thf(fact_813_le__infE,axiom,
! [X: set_Product_prod_v_v,A4: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X @ ( inf_in6271465464967711157od_v_v @ A4 @ B2 ) )
=> ~ ( ( ord_le7336532860387713383od_v_v @ X @ A4 )
=> ~ ( ord_le7336532860387713383od_v_v @ X @ B2 ) ) ) ).
% le_infE
thf(fact_814_le__infE,axiom,
! [X: set_v,A4: set_v,B2: set_v] :
( ( ord_less_eq_set_v @ X @ ( inf_inf_set_v @ A4 @ B2 ) )
=> ~ ( ( ord_less_eq_set_v @ X @ A4 )
=> ~ ( ord_less_eq_set_v @ X @ B2 ) ) ) ).
% le_infE
thf(fact_815_inf__le2,axiom,
! [X: set_Product_prod_v_v,Y: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ X @ Y ) @ Y ) ).
% inf_le2
thf(fact_816_inf__le2,axiom,
! [X: set_v,Y: set_v] : ( ord_less_eq_set_v @ ( inf_inf_set_v @ X @ Y ) @ Y ) ).
% inf_le2
thf(fact_817_inf__le1,axiom,
! [X: set_Product_prod_v_v,Y: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ X @ Y ) @ X ) ).
% inf_le1
thf(fact_818_inf__le1,axiom,
! [X: set_v,Y: set_v] : ( ord_less_eq_set_v @ ( inf_inf_set_v @ X @ Y ) @ X ) ).
% inf_le1
thf(fact_819_inf__sup__ord_I1_J,axiom,
! [X: set_Product_prod_v_v,Y: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ X @ Y ) @ X ) ).
% inf_sup_ord(1)
thf(fact_820_inf__sup__ord_I1_J,axiom,
! [X: set_v,Y: set_v] : ( ord_less_eq_set_v @ ( inf_inf_set_v @ X @ Y ) @ X ) ).
% inf_sup_ord(1)
thf(fact_821_inf__sup__ord_I2_J,axiom,
! [X: set_Product_prod_v_v,Y: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ X @ Y ) @ Y ) ).
% inf_sup_ord(2)
thf(fact_822_inf__sup__ord_I2_J,axiom,
! [X: set_v,Y: set_v] : ( ord_less_eq_set_v @ ( inf_inf_set_v @ X @ Y ) @ Y ) ).
% inf_sup_ord(2)
thf(fact_823_distrib__imp1,axiom,
! [X: set_v,Y: set_v,Z: set_v] :
( ! [X3: set_v,Y2: set_v,Z2: set_v] :
( ( inf_inf_set_v @ X3 @ ( sup_sup_set_v @ Y2 @ Z2 ) )
= ( sup_sup_set_v @ ( inf_inf_set_v @ X3 @ Y2 ) @ ( inf_inf_set_v @ X3 @ Z2 ) ) )
=> ( ( sup_sup_set_v @ X @ ( inf_inf_set_v @ Y @ Z ) )
= ( inf_inf_set_v @ ( sup_sup_set_v @ X @ Y ) @ ( sup_sup_set_v @ X @ Z ) ) ) ) ).
% distrib_imp1
thf(fact_824_distrib__imp1,axiom,
! [X: set_Product_prod_v_v,Y: set_Product_prod_v_v,Z: set_Product_prod_v_v] :
( ! [X3: set_Product_prod_v_v,Y2: set_Product_prod_v_v,Z2: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ X3 @ ( sup_su414716646722978715od_v_v @ Y2 @ Z2 ) )
= ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ X3 @ Y2 ) @ ( inf_in6271465464967711157od_v_v @ X3 @ Z2 ) ) )
=> ( ( sup_su414716646722978715od_v_v @ X @ ( inf_in6271465464967711157od_v_v @ Y @ Z ) )
= ( inf_in6271465464967711157od_v_v @ ( sup_su414716646722978715od_v_v @ X @ Y ) @ ( sup_su414716646722978715od_v_v @ X @ Z ) ) ) ) ).
% distrib_imp1
thf(fact_825_distrib__imp2,axiom,
! [X: set_v,Y: set_v,Z: set_v] :
( ! [X3: set_v,Y2: set_v,Z2: set_v] :
( ( sup_sup_set_v @ X3 @ ( inf_inf_set_v @ Y2 @ Z2 ) )
= ( inf_inf_set_v @ ( sup_sup_set_v @ X3 @ Y2 ) @ ( sup_sup_set_v @ X3 @ Z2 ) ) )
=> ( ( inf_inf_set_v @ X @ ( sup_sup_set_v @ Y @ Z ) )
= ( sup_sup_set_v @ ( inf_inf_set_v @ X @ Y ) @ ( inf_inf_set_v @ X @ Z ) ) ) ) ).
% distrib_imp2
thf(fact_826_distrib__imp2,axiom,
! [X: set_Product_prod_v_v,Y: set_Product_prod_v_v,Z: set_Product_prod_v_v] :
( ! [X3: set_Product_prod_v_v,Y2: set_Product_prod_v_v,Z2: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ X3 @ ( inf_in6271465464967711157od_v_v @ Y2 @ Z2 ) )
= ( inf_in6271465464967711157od_v_v @ ( sup_su414716646722978715od_v_v @ X3 @ Y2 ) @ ( sup_su414716646722978715od_v_v @ X3 @ Z2 ) ) )
=> ( ( inf_in6271465464967711157od_v_v @ X @ ( sup_su414716646722978715od_v_v @ Y @ Z ) )
= ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ X @ Y ) @ ( inf_in6271465464967711157od_v_v @ X @ Z ) ) ) ) ).
% distrib_imp2
thf(fact_827_inf__sup__distrib1,axiom,
! [X: set_v,Y: set_v,Z: set_v] :
( ( inf_inf_set_v @ X @ ( sup_sup_set_v @ Y @ Z ) )
= ( sup_sup_set_v @ ( inf_inf_set_v @ X @ Y ) @ ( inf_inf_set_v @ X @ Z ) ) ) ).
% inf_sup_distrib1
thf(fact_828_inf__sup__distrib1,axiom,
! [X: set_Product_prod_v_v,Y: set_Product_prod_v_v,Z: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ X @ ( sup_su414716646722978715od_v_v @ Y @ Z ) )
= ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ X @ Y ) @ ( inf_in6271465464967711157od_v_v @ X @ Z ) ) ) ).
% inf_sup_distrib1
thf(fact_829_inf__sup__distrib2,axiom,
! [Y: set_v,Z: set_v,X: set_v] :
( ( inf_inf_set_v @ ( sup_sup_set_v @ Y @ Z ) @ X )
= ( sup_sup_set_v @ ( inf_inf_set_v @ Y @ X ) @ ( inf_inf_set_v @ Z @ X ) ) ) ).
% inf_sup_distrib2
thf(fact_830_inf__sup__distrib2,axiom,
! [Y: set_Product_prod_v_v,Z: set_Product_prod_v_v,X: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ ( sup_su414716646722978715od_v_v @ Y @ Z ) @ X )
= ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ Y @ X ) @ ( inf_in6271465464967711157od_v_v @ Z @ X ) ) ) ).
% inf_sup_distrib2
thf(fact_831_sup__inf__distrib1,axiom,
! [X: set_v,Y: set_v,Z: set_v] :
( ( sup_sup_set_v @ X @ ( inf_inf_set_v @ Y @ Z ) )
= ( inf_inf_set_v @ ( sup_sup_set_v @ X @ Y ) @ ( sup_sup_set_v @ X @ Z ) ) ) ).
% sup_inf_distrib1
thf(fact_832_sup__inf__distrib1,axiom,
! [X: set_Product_prod_v_v,Y: set_Product_prod_v_v,Z: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ X @ ( inf_in6271465464967711157od_v_v @ Y @ Z ) )
= ( inf_in6271465464967711157od_v_v @ ( sup_su414716646722978715od_v_v @ X @ Y ) @ ( sup_su414716646722978715od_v_v @ X @ Z ) ) ) ).
% sup_inf_distrib1
thf(fact_833_sup__inf__distrib2,axiom,
! [Y: set_v,Z: set_v,X: set_v] :
( ( sup_sup_set_v @ ( inf_inf_set_v @ Y @ Z ) @ X )
= ( inf_inf_set_v @ ( sup_sup_set_v @ Y @ X ) @ ( sup_sup_set_v @ Z @ X ) ) ) ).
% sup_inf_distrib2
thf(fact_834_sup__inf__distrib2,axiom,
! [Y: set_Product_prod_v_v,Z: set_Product_prod_v_v,X: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ Y @ Z ) @ X )
= ( inf_in6271465464967711157od_v_v @ ( sup_su414716646722978715od_v_v @ Y @ X ) @ ( sup_su414716646722978715od_v_v @ Z @ X ) ) ) ).
% sup_inf_distrib2
thf(fact_835_boolean__algebra_Odisj__conj__distrib2,axiom,
! [Y: set_v,Z: set_v,X: set_v] :
( ( sup_sup_set_v @ ( inf_inf_set_v @ Y @ Z ) @ X )
= ( inf_inf_set_v @ ( sup_sup_set_v @ Y @ X ) @ ( sup_sup_set_v @ Z @ X ) ) ) ).
% boolean_algebra.disj_conj_distrib2
thf(fact_836_boolean__algebra_Odisj__conj__distrib2,axiom,
! [Y: set_Product_prod_v_v,Z: set_Product_prod_v_v,X: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ Y @ Z ) @ X )
= ( inf_in6271465464967711157od_v_v @ ( sup_su414716646722978715od_v_v @ Y @ X ) @ ( sup_su414716646722978715od_v_v @ Z @ X ) ) ) ).
% boolean_algebra.disj_conj_distrib2
thf(fact_837_boolean__algebra_Oconj__disj__distrib2,axiom,
! [Y: set_v,Z: set_v,X: set_v] :
( ( inf_inf_set_v @ ( sup_sup_set_v @ Y @ Z ) @ X )
= ( sup_sup_set_v @ ( inf_inf_set_v @ Y @ X ) @ ( inf_inf_set_v @ Z @ X ) ) ) ).
% boolean_algebra.conj_disj_distrib2
thf(fact_838_boolean__algebra_Oconj__disj__distrib2,axiom,
! [Y: set_Product_prod_v_v,Z: set_Product_prod_v_v,X: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ ( sup_su414716646722978715od_v_v @ Y @ Z ) @ X )
= ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ Y @ X ) @ ( inf_in6271465464967711157od_v_v @ Z @ X ) ) ) ).
% boolean_algebra.conj_disj_distrib2
thf(fact_839_boolean__algebra_Odisj__conj__distrib,axiom,
! [X: set_v,Y: set_v,Z: set_v] :
( ( sup_sup_set_v @ X @ ( inf_inf_set_v @ Y @ Z ) )
= ( inf_inf_set_v @ ( sup_sup_set_v @ X @ Y ) @ ( sup_sup_set_v @ X @ Z ) ) ) ).
% boolean_algebra.disj_conj_distrib
thf(fact_840_boolean__algebra_Odisj__conj__distrib,axiom,
! [X: set_Product_prod_v_v,Y: set_Product_prod_v_v,Z: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ X @ ( inf_in6271465464967711157od_v_v @ Y @ Z ) )
= ( inf_in6271465464967711157od_v_v @ ( sup_su414716646722978715od_v_v @ X @ Y ) @ ( sup_su414716646722978715od_v_v @ X @ Z ) ) ) ).
% boolean_algebra.disj_conj_distrib
thf(fact_841_boolean__algebra_Oconj__disj__distrib,axiom,
! [X: set_v,Y: set_v,Z: set_v] :
( ( inf_inf_set_v @ X @ ( sup_sup_set_v @ Y @ Z ) )
= ( sup_sup_set_v @ ( inf_inf_set_v @ X @ Y ) @ ( inf_inf_set_v @ X @ Z ) ) ) ).
% boolean_algebra.conj_disj_distrib
thf(fact_842_boolean__algebra_Oconj__disj__distrib,axiom,
! [X: set_Product_prod_v_v,Y: set_Product_prod_v_v,Z: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ X @ ( sup_su414716646722978715od_v_v @ Y @ Z ) )
= ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ X @ Y ) @ ( inf_in6271465464967711157od_v_v @ X @ Z ) ) ) ).
% boolean_algebra.conj_disj_distrib
thf(fact_843_Int__emptyI,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ! [X3: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X3 @ A )
=> ~ ( member7453568604450474000od_v_v @ X3 @ B ) )
=> ( ( inf_in6271465464967711157od_v_v @ A @ B )
= bot_bo723834152578015283od_v_v ) ) ).
% Int_emptyI
thf(fact_844_Int__emptyI,axiom,
! [A: set_v,B: set_v] :
( ! [X3: v] :
( ( member_v @ X3 @ A )
=> ~ ( member_v @ X3 @ B ) )
=> ( ( inf_inf_set_v @ A @ B )
= bot_bot_set_v ) ) ).
% Int_emptyI
thf(fact_845_disjoint__iff,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ( inf_in6271465464967711157od_v_v @ A @ B )
= bot_bo723834152578015283od_v_v )
= ( ! [X2: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X2 @ A )
=> ~ ( member7453568604450474000od_v_v @ X2 @ B ) ) ) ) ).
% disjoint_iff
thf(fact_846_disjoint__iff,axiom,
! [A: set_v,B: set_v] :
( ( ( inf_inf_set_v @ A @ B )
= bot_bot_set_v )
= ( ! [X2: v] :
( ( member_v @ X2 @ A )
=> ~ ( member_v @ X2 @ B ) ) ) ) ).
% disjoint_iff
thf(fact_847_Int__empty__left,axiom,
! [B: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ bot_bo723834152578015283od_v_v @ B )
= bot_bo723834152578015283od_v_v ) ).
% Int_empty_left
thf(fact_848_Int__empty__left,axiom,
! [B: set_v] :
( ( inf_inf_set_v @ bot_bot_set_v @ B )
= bot_bot_set_v ) ).
% Int_empty_left
thf(fact_849_Int__empty__right,axiom,
! [A: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ A @ bot_bo723834152578015283od_v_v )
= bot_bo723834152578015283od_v_v ) ).
% Int_empty_right
thf(fact_850_Int__empty__right,axiom,
! [A: set_v] :
( ( inf_inf_set_v @ A @ bot_bot_set_v )
= bot_bot_set_v ) ).
% Int_empty_right
thf(fact_851_disjoint__iff__not__equal,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ( inf_in6271465464967711157od_v_v @ A @ B )
= bot_bo723834152578015283od_v_v )
= ( ! [X2: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X2 @ A )
=> ! [Y3: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ Y3 @ B )
=> ( X2 != Y3 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_852_disjoint__iff__not__equal,axiom,
! [A: set_v,B: set_v] :
( ( ( inf_inf_set_v @ A @ B )
= bot_bot_set_v )
= ( ! [X2: v] :
( ( member_v @ X2 @ A )
=> ! [Y3: v] :
( ( member_v @ Y3 @ B )
=> ( X2 != Y3 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_853_Int__Collect__mono,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v,P: product_prod_v_v > $o,Q: product_prod_v_v > $o] :
( ( ord_le7336532860387713383od_v_v @ A @ B )
=> ( ! [X3: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X3 @ A )
=> ( ( P @ X3 )
=> ( Q @ X3 ) ) )
=> ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ A @ ( collec140062887454715474od_v_v @ P ) ) @ ( inf_in6271465464967711157od_v_v @ B @ ( collec140062887454715474od_v_v @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_854_Int__Collect__mono,axiom,
! [A: set_v,B: set_v,P: v > $o,Q: v > $o] :
( ( ord_less_eq_set_v @ A @ B )
=> ( ! [X3: v] :
( ( member_v @ X3 @ A )
=> ( ( P @ X3 )
=> ( Q @ X3 ) ) )
=> ( ord_less_eq_set_v @ ( inf_inf_set_v @ A @ ( collect_v @ P ) ) @ ( inf_inf_set_v @ B @ ( collect_v @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_855_Int__greatest,axiom,
! [C2: set_Product_prod_v_v,A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ C2 @ A )
=> ( ( ord_le7336532860387713383od_v_v @ C2 @ B )
=> ( ord_le7336532860387713383od_v_v @ C2 @ ( inf_in6271465464967711157od_v_v @ A @ B ) ) ) ) ).
% Int_greatest
thf(fact_856_Int__greatest,axiom,
! [C2: set_v,A: set_v,B: set_v] :
( ( ord_less_eq_set_v @ C2 @ A )
=> ( ( ord_less_eq_set_v @ C2 @ B )
=> ( ord_less_eq_set_v @ C2 @ ( inf_inf_set_v @ A @ B ) ) ) ) ).
% Int_greatest
thf(fact_857_Int__absorb2,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ B )
=> ( ( inf_in6271465464967711157od_v_v @ A @ B )
= A ) ) ).
% Int_absorb2
thf(fact_858_Int__absorb2,axiom,
! [A: set_v,B: set_v] :
( ( ord_less_eq_set_v @ A @ B )
=> ( ( inf_inf_set_v @ A @ B )
= A ) ) ).
% Int_absorb2
thf(fact_859_Int__absorb1,axiom,
! [B: set_Product_prod_v_v,A: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ B @ A )
=> ( ( inf_in6271465464967711157od_v_v @ A @ B )
= B ) ) ).
% Int_absorb1
thf(fact_860_Int__absorb1,axiom,
! [B: set_v,A: set_v] :
( ( ord_less_eq_set_v @ B @ A )
=> ( ( inf_inf_set_v @ A @ B )
= B ) ) ).
% Int_absorb1
thf(fact_861_Int__lower2,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ A @ B ) @ B ) ).
% Int_lower2
thf(fact_862_Int__lower2,axiom,
! [A: set_v,B: set_v] : ( ord_less_eq_set_v @ ( inf_inf_set_v @ A @ B ) @ B ) ).
% Int_lower2
thf(fact_863_Int__lower1,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ A @ B ) @ A ) ).
% Int_lower1
thf(fact_864_Int__lower1,axiom,
! [A: set_v,B: set_v] : ( ord_less_eq_set_v @ ( inf_inf_set_v @ A @ B ) @ A ) ).
% Int_lower1
thf(fact_865_Int__mono,axiom,
! [A: set_Product_prod_v_v,C2: set_Product_prod_v_v,B: set_Product_prod_v_v,D: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ C2 )
=> ( ( ord_le7336532860387713383od_v_v @ B @ D )
=> ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ A @ B ) @ ( inf_in6271465464967711157od_v_v @ C2 @ D ) ) ) ) ).
% Int_mono
thf(fact_866_Int__mono,axiom,
! [A: set_v,C2: set_v,B: set_v,D: set_v] :
( ( ord_less_eq_set_v @ A @ C2 )
=> ( ( ord_less_eq_set_v @ B @ D )
=> ( ord_less_eq_set_v @ ( inf_inf_set_v @ A @ B ) @ ( inf_inf_set_v @ C2 @ D ) ) ) ) ).
% Int_mono
thf(fact_867_Int__insert__left,axiom,
! [A4: product_prod_v_v,C2: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ( member7453568604450474000od_v_v @ A4 @ C2 )
=> ( ( inf_in6271465464967711157od_v_v @ ( insert1338601472111419319od_v_v @ A4 @ B ) @ C2 )
= ( insert1338601472111419319od_v_v @ A4 @ ( inf_in6271465464967711157od_v_v @ B @ C2 ) ) ) )
& ( ~ ( member7453568604450474000od_v_v @ A4 @ C2 )
=> ( ( inf_in6271465464967711157od_v_v @ ( insert1338601472111419319od_v_v @ A4 @ B ) @ C2 )
= ( inf_in6271465464967711157od_v_v @ B @ C2 ) ) ) ) ).
% Int_insert_left
thf(fact_868_Int__insert__left,axiom,
! [A4: v,C2: set_v,B: set_v] :
( ( ( member_v @ A4 @ C2 )
=> ( ( inf_inf_set_v @ ( insert_v2 @ A4 @ B ) @ C2 )
= ( insert_v2 @ A4 @ ( inf_inf_set_v @ B @ C2 ) ) ) )
& ( ~ ( member_v @ A4 @ C2 )
=> ( ( inf_inf_set_v @ ( insert_v2 @ A4 @ B ) @ C2 )
= ( inf_inf_set_v @ B @ C2 ) ) ) ) ).
% Int_insert_left
thf(fact_869_Int__insert__right,axiom,
! [A4: product_prod_v_v,A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ( member7453568604450474000od_v_v @ A4 @ A )
=> ( ( inf_in6271465464967711157od_v_v @ A @ ( insert1338601472111419319od_v_v @ A4 @ B ) )
= ( insert1338601472111419319od_v_v @ A4 @ ( inf_in6271465464967711157od_v_v @ A @ B ) ) ) )
& ( ~ ( member7453568604450474000od_v_v @ A4 @ A )
=> ( ( inf_in6271465464967711157od_v_v @ A @ ( insert1338601472111419319od_v_v @ A4 @ B ) )
= ( inf_in6271465464967711157od_v_v @ A @ B ) ) ) ) ).
% Int_insert_right
thf(fact_870_Int__insert__right,axiom,
! [A4: v,A: set_v,B: set_v] :
( ( ( member_v @ A4 @ A )
=> ( ( inf_inf_set_v @ A @ ( insert_v2 @ A4 @ B ) )
= ( insert_v2 @ A4 @ ( inf_inf_set_v @ A @ B ) ) ) )
& ( ~ ( member_v @ A4 @ A )
=> ( ( inf_inf_set_v @ A @ ( insert_v2 @ A4 @ B ) )
= ( inf_inf_set_v @ A @ B ) ) ) ) ).
% Int_insert_right
thf(fact_871_Un__Int__crazy,axiom,
! [A: set_v,B: set_v,C2: set_v] :
( ( sup_sup_set_v @ ( sup_sup_set_v @ ( inf_inf_set_v @ A @ B ) @ ( inf_inf_set_v @ B @ C2 ) ) @ ( inf_inf_set_v @ C2 @ A ) )
= ( inf_inf_set_v @ ( inf_inf_set_v @ ( sup_sup_set_v @ A @ B ) @ ( sup_sup_set_v @ B @ C2 ) ) @ ( sup_sup_set_v @ C2 @ A ) ) ) ).
% Un_Int_crazy
thf(fact_872_Un__Int__crazy,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v,C2: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ A @ B ) @ ( inf_in6271465464967711157od_v_v @ B @ C2 ) ) @ ( inf_in6271465464967711157od_v_v @ C2 @ A ) )
= ( inf_in6271465464967711157od_v_v @ ( inf_in6271465464967711157od_v_v @ ( sup_su414716646722978715od_v_v @ A @ B ) @ ( sup_su414716646722978715od_v_v @ B @ C2 ) ) @ ( sup_su414716646722978715od_v_v @ C2 @ A ) ) ) ).
% Un_Int_crazy
thf(fact_873_Int__Un__distrib,axiom,
! [A: set_v,B: set_v,C2: set_v] :
( ( inf_inf_set_v @ A @ ( sup_sup_set_v @ B @ C2 ) )
= ( sup_sup_set_v @ ( inf_inf_set_v @ A @ B ) @ ( inf_inf_set_v @ A @ C2 ) ) ) ).
% Int_Un_distrib
thf(fact_874_Int__Un__distrib,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v,C2: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ A @ ( sup_su414716646722978715od_v_v @ B @ C2 ) )
= ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ A @ B ) @ ( inf_in6271465464967711157od_v_v @ A @ C2 ) ) ) ).
% Int_Un_distrib
thf(fact_875_Un__Int__distrib,axiom,
! [A: set_v,B: set_v,C2: set_v] :
( ( sup_sup_set_v @ A @ ( inf_inf_set_v @ B @ C2 ) )
= ( inf_inf_set_v @ ( sup_sup_set_v @ A @ B ) @ ( sup_sup_set_v @ A @ C2 ) ) ) ).
% Un_Int_distrib
thf(fact_876_Un__Int__distrib,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v,C2: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ A @ ( inf_in6271465464967711157od_v_v @ B @ C2 ) )
= ( inf_in6271465464967711157od_v_v @ ( sup_su414716646722978715od_v_v @ A @ B ) @ ( sup_su414716646722978715od_v_v @ A @ C2 ) ) ) ).
% Un_Int_distrib
thf(fact_877_Int__Un__distrib2,axiom,
! [B: set_v,C2: set_v,A: set_v] :
( ( inf_inf_set_v @ ( sup_sup_set_v @ B @ C2 ) @ A )
= ( sup_sup_set_v @ ( inf_inf_set_v @ B @ A ) @ ( inf_inf_set_v @ C2 @ A ) ) ) ).
% Int_Un_distrib2
thf(fact_878_Int__Un__distrib2,axiom,
! [B: set_Product_prod_v_v,C2: set_Product_prod_v_v,A: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ ( sup_su414716646722978715od_v_v @ B @ C2 ) @ A )
= ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ B @ A ) @ ( inf_in6271465464967711157od_v_v @ C2 @ A ) ) ) ).
% Int_Un_distrib2
thf(fact_879_Un__Int__distrib2,axiom,
! [B: set_v,C2: set_v,A: set_v] :
( ( sup_sup_set_v @ ( inf_inf_set_v @ B @ C2 ) @ A )
= ( inf_inf_set_v @ ( sup_sup_set_v @ B @ A ) @ ( sup_sup_set_v @ C2 @ A ) ) ) ).
% Un_Int_distrib2
thf(fact_880_Un__Int__distrib2,axiom,
! [B: set_Product_prod_v_v,C2: set_Product_prod_v_v,A: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ B @ C2 ) @ A )
= ( inf_in6271465464967711157od_v_v @ ( sup_su414716646722978715od_v_v @ B @ A ) @ ( sup_su414716646722978715od_v_v @ C2 @ A ) ) ) ).
% Un_Int_distrib2
thf(fact_881_Int__Diff,axiom,
! [A: set_v,B: set_v,C2: set_v] :
( ( minus_minus_set_v @ ( inf_inf_set_v @ A @ B ) @ C2 )
= ( inf_inf_set_v @ A @ ( minus_minus_set_v @ B @ C2 ) ) ) ).
% Int_Diff
thf(fact_882_Diff__Int2,axiom,
! [A: set_v,C2: set_v,B: set_v] :
( ( minus_minus_set_v @ ( inf_inf_set_v @ A @ C2 ) @ ( inf_inf_set_v @ B @ C2 ) )
= ( minus_minus_set_v @ ( inf_inf_set_v @ A @ C2 ) @ B ) ) ).
% Diff_Int2
thf(fact_883_Diff__Diff__Int,axiom,
! [A: set_v,B: set_v] :
( ( minus_minus_set_v @ A @ ( minus_minus_set_v @ A @ B ) )
= ( inf_inf_set_v @ A @ B ) ) ).
% Diff_Diff_Int
thf(fact_884_Diff__Int__distrib,axiom,
! [C2: set_v,A: set_v,B: set_v] :
( ( inf_inf_set_v @ C2 @ ( minus_minus_set_v @ A @ B ) )
= ( minus_minus_set_v @ ( inf_inf_set_v @ C2 @ A ) @ ( inf_inf_set_v @ C2 @ B ) ) ) ).
% Diff_Int_distrib
thf(fact_885_Diff__Int__distrib2,axiom,
! [A: set_v,B: set_v,C2: set_v] :
( ( inf_inf_set_v @ ( minus_minus_set_v @ A @ B ) @ C2 )
= ( minus_minus_set_v @ ( inf_inf_set_v @ A @ C2 ) @ ( inf_inf_set_v @ B @ C2 ) ) ) ).
% Diff_Int_distrib2
thf(fact_886_distrib__sup__le,axiom,
! [X: set_Product_prod_v_v,Y: set_Product_prod_v_v,Z: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ ( sup_su414716646722978715od_v_v @ X @ ( inf_in6271465464967711157od_v_v @ Y @ Z ) ) @ ( inf_in6271465464967711157od_v_v @ ( sup_su414716646722978715od_v_v @ X @ Y ) @ ( sup_su414716646722978715od_v_v @ X @ Z ) ) ) ).
% distrib_sup_le
thf(fact_887_distrib__sup__le,axiom,
! [X: set_v,Y: set_v,Z: set_v] : ( ord_less_eq_set_v @ ( sup_sup_set_v @ X @ ( inf_inf_set_v @ Y @ Z ) ) @ ( inf_inf_set_v @ ( sup_sup_set_v @ X @ Y ) @ ( sup_sup_set_v @ X @ Z ) ) ) ).
% distrib_sup_le
thf(fact_888_distrib__inf__le,axiom,
! [X: set_Product_prod_v_v,Y: set_Product_prod_v_v,Z: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ X @ Y ) @ ( inf_in6271465464967711157od_v_v @ X @ Z ) ) @ ( inf_in6271465464967711157od_v_v @ X @ ( sup_su414716646722978715od_v_v @ Y @ Z ) ) ) ).
% distrib_inf_le
thf(fact_889_distrib__inf__le,axiom,
! [X: set_v,Y: set_v,Z: set_v] : ( ord_less_eq_set_v @ ( sup_sup_set_v @ ( inf_inf_set_v @ X @ Y ) @ ( inf_inf_set_v @ X @ Z ) ) @ ( inf_inf_set_v @ X @ ( sup_sup_set_v @ Y @ Z ) ) ) ).
% distrib_inf_le
thf(fact_890_Diff__triv,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ( inf_in6271465464967711157od_v_v @ A @ B )
= bot_bo723834152578015283od_v_v )
=> ( ( minus_4183494784930505774od_v_v @ A @ B )
= A ) ) ).
% Diff_triv
thf(fact_891_Diff__triv,axiom,
! [A: set_v,B: set_v] :
( ( ( inf_inf_set_v @ A @ B )
= bot_bot_set_v )
=> ( ( minus_minus_set_v @ A @ B )
= A ) ) ).
% Diff_triv
thf(fact_892_Int__Diff__disjoint,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ ( inf_in6271465464967711157od_v_v @ A @ B ) @ ( minus_4183494784930505774od_v_v @ A @ B ) )
= bot_bo723834152578015283od_v_v ) ).
% Int_Diff_disjoint
thf(fact_893_Int__Diff__disjoint,axiom,
! [A: set_v,B: set_v] :
( ( inf_inf_set_v @ ( inf_inf_set_v @ A @ B ) @ ( minus_minus_set_v @ A @ B ) )
= bot_bot_set_v ) ).
% Int_Diff_disjoint
thf(fact_894_Un__Int__assoc__eq,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v,C2: set_Product_prod_v_v] :
( ( ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ A @ B ) @ C2 )
= ( inf_in6271465464967711157od_v_v @ A @ ( sup_su414716646722978715od_v_v @ B @ C2 ) ) )
= ( ord_le7336532860387713383od_v_v @ C2 @ A ) ) ).
% Un_Int_assoc_eq
thf(fact_895_Un__Int__assoc__eq,axiom,
! [A: set_v,B: set_v,C2: set_v] :
( ( ( sup_sup_set_v @ ( inf_inf_set_v @ A @ B ) @ C2 )
= ( inf_inf_set_v @ A @ ( sup_sup_set_v @ B @ C2 ) ) )
= ( ord_less_eq_set_v @ C2 @ A ) ) ).
% Un_Int_assoc_eq
thf(fact_896_Diff__Un,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v,C2: set_Product_prod_v_v] :
( ( minus_4183494784930505774od_v_v @ A @ ( sup_su414716646722978715od_v_v @ B @ C2 ) )
= ( inf_in6271465464967711157od_v_v @ ( minus_4183494784930505774od_v_v @ A @ B ) @ ( minus_4183494784930505774od_v_v @ A @ C2 ) ) ) ).
% Diff_Un
thf(fact_897_Diff__Un,axiom,
! [A: set_v,B: set_v,C2: set_v] :
( ( minus_minus_set_v @ A @ ( sup_sup_set_v @ B @ C2 ) )
= ( inf_inf_set_v @ ( minus_minus_set_v @ A @ B ) @ ( minus_minus_set_v @ A @ C2 ) ) ) ).
% Diff_Un
thf(fact_898_Diff__Int,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v,C2: set_Product_prod_v_v] :
( ( minus_4183494784930505774od_v_v @ A @ ( inf_in6271465464967711157od_v_v @ B @ C2 ) )
= ( sup_su414716646722978715od_v_v @ ( minus_4183494784930505774od_v_v @ A @ B ) @ ( minus_4183494784930505774od_v_v @ A @ C2 ) ) ) ).
% Diff_Int
thf(fact_899_Diff__Int,axiom,
! [A: set_v,B: set_v,C2: set_v] :
( ( minus_minus_set_v @ A @ ( inf_inf_set_v @ B @ C2 ) )
= ( sup_sup_set_v @ ( minus_minus_set_v @ A @ B ) @ ( minus_minus_set_v @ A @ C2 ) ) ) ).
% Diff_Int
thf(fact_900_Int__Diff__Un,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ A @ B ) @ ( minus_4183494784930505774od_v_v @ A @ B ) )
= A ) ).
% Int_Diff_Un
thf(fact_901_Int__Diff__Un,axiom,
! [A: set_v,B: set_v] :
( ( sup_sup_set_v @ ( inf_inf_set_v @ A @ B ) @ ( minus_minus_set_v @ A @ B ) )
= A ) ).
% Int_Diff_Un
thf(fact_902_Un__Diff__Int,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ ( minus_4183494784930505774od_v_v @ A @ B ) @ ( inf_in6271465464967711157od_v_v @ A @ B ) )
= A ) ).
% Un_Diff_Int
thf(fact_903_Un__Diff__Int,axiom,
! [A: set_v,B: set_v] :
( ( sup_sup_set_v @ ( minus_minus_set_v @ A @ B ) @ ( inf_inf_set_v @ A @ B ) )
= A ) ).
% Un_Diff_Int
thf(fact_904_graph_Oscc__partition,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,S: set_Product_prod_v_v,S3: set_Product_prod_v_v,X: product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( sCC_Bl6242042402218619277od_v_v @ Successors @ S )
=> ( ( sCC_Bl6242042402218619277od_v_v @ Successors @ S3 )
=> ( ( member7453568604450474000od_v_v @ X @ ( inf_in6271465464967711157od_v_v @ S @ S3 ) )
=> ( S = S3 ) ) ) ) ) ).
% graph.scc_partition
thf(fact_905_graph_Oscc__partition,axiom,
! [Vertices: set_v,Successors: v > set_v,S: set_v,S3: set_v,X: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bloemen_is_scc_v @ Successors @ S )
=> ( ( sCC_Bloemen_is_scc_v @ Successors @ S3 )
=> ( ( member_v @ X @ ( inf_inf_set_v @ S @ S3 ) )
=> ( S = S3 ) ) ) ) ) ).
% graph.scc_partition
thf(fact_906_Pair__inject,axiom,
! [A4: v,B2: v,A8: v,B7: v] :
( ( ( product_Pair_v_v @ A4 @ B2 )
= ( product_Pair_v_v @ A8 @ B7 ) )
=> ~ ( ( A4 = A8 )
=> ( B2 != B7 ) ) ) ).
% Pair_inject
thf(fact_907_Pair__inject,axiom,
! [A4: v,B2: sCC_Bl1394983891496994913t_unit,A8: v,B7: sCC_Bl1394983891496994913t_unit] :
( ( ( produc3862955338007567901t_unit @ A4 @ B2 )
= ( produc3862955338007567901t_unit @ A8 @ B7 ) )
=> ~ ( ( A4 = A8 )
=> ( B2 != B7 ) ) ) ).
% Pair_inject
thf(fact_908_prod__cases,axiom,
! [P: product_prod_v_v > $o,P3: product_prod_v_v] :
( ! [A9: v,B8: v] : ( P @ ( product_Pair_v_v @ A9 @ B8 ) )
=> ( P @ P3 ) ) ).
% prod_cases
thf(fact_909_prod__cases,axiom,
! [P: produc5741669702376414499t_unit > $o,P3: produc5741669702376414499t_unit] :
( ! [A9: v,B8: sCC_Bl1394983891496994913t_unit] : ( P @ ( produc3862955338007567901t_unit @ A9 @ B8 ) )
=> ( P @ P3 ) ) ).
% prod_cases
thf(fact_910_surj__pair,axiom,
! [P3: product_prod_v_v] :
? [X3: v,Y2: v] :
( P3
= ( product_Pair_v_v @ X3 @ Y2 ) ) ).
% surj_pair
thf(fact_911_surj__pair,axiom,
! [P3: produc5741669702376414499t_unit] :
? [X3: v,Y2: sCC_Bl1394983891496994913t_unit] :
( P3
= ( produc3862955338007567901t_unit @ X3 @ Y2 ) ) ).
% surj_pair
thf(fact_912_old_Oprod_Oexhaust,axiom,
! [Y: product_prod_v_v] :
~ ! [A9: v,B8: v] :
( Y
!= ( product_Pair_v_v @ A9 @ B8 ) ) ).
% old.prod.exhaust
thf(fact_913_old_Oprod_Oexhaust,axiom,
! [Y: produc5741669702376414499t_unit] :
~ ! [A9: v,B8: sCC_Bl1394983891496994913t_unit] :
( Y
!= ( produc3862955338007567901t_unit @ A9 @ B8 ) ) ).
% old.prod.exhaust
thf(fact_914_the__elem__set,axiom,
! [X: v] :
( ( the_elem_v @ ( set_v2 @ ( cons_v @ X @ nil_v ) ) )
= X ) ).
% the_elem_set
thf(fact_915_surjective,axiom,
! [R2: sCC_Bl1394983891496994913t_unit] :
( R2
= ( sCC_Bl8064756265740546429t_unit @ ( sCC_Bl1090238580953940555t_unit @ R2 ) @ ( sCC_Bl1280885523602775798t_unit @ R2 ) @ ( sCC_Bl157864678168468314t_unit @ R2 ) @ ( sCC_Bl4645233313691564917t_unit @ R2 ) @ ( sCC_Bl3795065053823578884t_unit @ R2 ) @ ( sCC_Bl2536197123907397897t_unit @ R2 ) @ ( sCC_Bl8828226123343373779t_unit @ R2 ) @ ( sCC_Bl9201514103433284750t_unit @ R2 ) @ ( sCC_Bl3567736435408124606t_unit @ R2 ) ) ) ).
% surjective
thf(fact_916_bot__empty__eq,axiom,
( bot_bo8461541820394803818_v_v_o
= ( ^ [X2: product_prod_v_v] : ( member7453568604450474000od_v_v @ X2 @ bot_bo723834152578015283od_v_v ) ) ) ).
% bot_empty_eq
thf(fact_917_bot__empty__eq,axiom,
( bot_bot_v_o
= ( ^ [X2: v] : ( member_v @ X2 @ bot_bot_set_v ) ) ) ).
% bot_empty_eq
thf(fact_918_Collect__empty__eq__bot,axiom,
! [P: product_prod_v_v > $o] :
( ( ( collec140062887454715474od_v_v @ P )
= bot_bo723834152578015283od_v_v )
= ( P = bot_bo8461541820394803818_v_v_o ) ) ).
% Collect_empty_eq_bot
thf(fact_919_Collect__empty__eq__bot,axiom,
! [P: v > $o] :
( ( ( collect_v @ P )
= bot_bot_set_v )
= ( P = bot_bot_v_o ) ) ).
% Collect_empty_eq_bot
thf(fact_920_select__convs_I7_J,axiom,
! [Root: v,S4: v > set_v,Explored: set_v,Visited: set_v,Vsuccs: v > set_v,Sccs: set_set_v,Stack: list_v,Cstack: list_v,More: product_unit] :
( ( sCC_Bl8828226123343373779t_unit @ ( sCC_Bl8064756265740546429t_unit @ Root @ S4 @ Explored @ Visited @ Vsuccs @ Sccs @ Stack @ Cstack @ More ) )
= Stack ) ).
% select_convs(7)
thf(fact_921_select__convs_I2_J,axiom,
! [Root: v,S4: v > set_v,Explored: set_v,Visited: set_v,Vsuccs: v > set_v,Sccs: set_set_v,Stack: list_v,Cstack: list_v,More: product_unit] :
( ( sCC_Bl1280885523602775798t_unit @ ( sCC_Bl8064756265740546429t_unit @ Root @ S4 @ Explored @ Visited @ Vsuccs @ Sccs @ Stack @ Cstack @ More ) )
= S4 ) ).
% select_convs(2)
thf(fact_922_select__convs_I5_J,axiom,
! [Root: v,S4: v > set_v,Explored: set_v,Visited: set_v,Vsuccs: v > set_v,Sccs: set_set_v,Stack: list_v,Cstack: list_v,More: product_unit] :
( ( sCC_Bl3795065053823578884t_unit @ ( sCC_Bl8064756265740546429t_unit @ Root @ S4 @ Explored @ Visited @ Vsuccs @ Sccs @ Stack @ Cstack @ More ) )
= Vsuccs ) ).
% select_convs(5)
thf(fact_923_select__convs_I4_J,axiom,
! [Root: v,S4: v > set_v,Explored: set_v,Visited: set_v,Vsuccs: v > set_v,Sccs: set_set_v,Stack: list_v,Cstack: list_v,More: product_unit] :
( ( sCC_Bl4645233313691564917t_unit @ ( sCC_Bl8064756265740546429t_unit @ Root @ S4 @ Explored @ Visited @ Vsuccs @ Sccs @ Stack @ Cstack @ More ) )
= Visited ) ).
% select_convs(4)
thf(fact_924_select__convs_I3_J,axiom,
! [Root: v,S4: v > set_v,Explored: set_v,Visited: set_v,Vsuccs: v > set_v,Sccs: set_set_v,Stack: list_v,Cstack: list_v,More: product_unit] :
( ( sCC_Bl157864678168468314t_unit @ ( sCC_Bl8064756265740546429t_unit @ Root @ S4 @ Explored @ Visited @ Vsuccs @ Sccs @ Stack @ Cstack @ More ) )
= Explored ) ).
% select_convs(3)
thf(fact_925_subrelI,axiom,
! [R2: set_Pr6425124735969554649t_unit,S5: set_Pr6425124735969554649t_unit] :
( ! [X3: v,Y2: sCC_Bl1394983891496994913t_unit] :
( ( member7924940910754673978t_unit @ ( produc3862955338007567901t_unit @ X3 @ Y2 ) @ R2 )
=> ( member7924940910754673978t_unit @ ( produc3862955338007567901t_unit @ X3 @ Y2 ) @ S5 ) )
=> ( ord_le7290744839000465721t_unit @ R2 @ S5 ) ) ).
% subrelI
thf(fact_926_subrelI,axiom,
! [R2: set_Product_prod_v_v,S5: set_Product_prod_v_v] :
( ! [X3: v,Y2: v] :
( ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ X3 @ Y2 ) @ R2 )
=> ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ X3 @ Y2 ) @ S5 ) )
=> ( ord_le7336532860387713383od_v_v @ R2 @ S5 ) ) ).
% subrelI
thf(fact_927_is__singleton__the__elem,axiom,
( is_sin9198872032823709915od_v_v
= ( ^ [A6: set_Product_prod_v_v] :
( A6
= ( insert1338601472111419319od_v_v @ ( the_el5392834299063928540od_v_v @ A6 ) @ bot_bo723834152578015283od_v_v ) ) ) ) ).
% is_singleton_the_elem
thf(fact_928_is__singleton__the__elem,axiom,
( is_singleton_v
= ( ^ [A6: set_v] :
( A6
= ( insert_v2 @ ( the_elem_v @ A6 ) @ bot_bot_set_v ) ) ) ) ).
% is_singleton_the_elem
thf(fact_929_less__by__empty,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( A = bot_bo723834152578015283od_v_v )
=> ( ord_le7336532860387713383od_v_v @ A @ B ) ) ).
% less_by_empty
thf(fact_930_is__singletonI,axiom,
! [X: product_prod_v_v] : ( is_sin9198872032823709915od_v_v @ ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v ) ) ).
% is_singletonI
thf(fact_931_is__singletonI,axiom,
! [X: v] : ( is_singleton_v @ ( insert_v2 @ X @ bot_bot_set_v ) ) ).
% is_singletonI
thf(fact_932_is__singletonI_H,axiom,
! [A: set_Product_prod_v_v] :
( ( A != bot_bo723834152578015283od_v_v )
=> ( ! [X3: product_prod_v_v,Y2: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X3 @ A )
=> ( ( member7453568604450474000od_v_v @ Y2 @ A )
=> ( X3 = Y2 ) ) )
=> ( is_sin9198872032823709915od_v_v @ A ) ) ) ).
% is_singletonI'
thf(fact_933_is__singletonI_H,axiom,
! [A: set_v] :
( ( A != bot_bot_set_v )
=> ( ! [X3: v,Y2: v] :
( ( member_v @ X3 @ A )
=> ( ( member_v @ Y2 @ A )
=> ( X3 = Y2 ) ) )
=> ( is_singleton_v @ A ) ) ) ).
% is_singletonI'
thf(fact_934_is__singletonE,axiom,
! [A: set_Product_prod_v_v] :
( ( is_sin9198872032823709915od_v_v @ A )
=> ~ ! [X3: product_prod_v_v] :
( A
!= ( insert1338601472111419319od_v_v @ X3 @ bot_bo723834152578015283od_v_v ) ) ) ).
% is_singletonE
thf(fact_935_is__singletonE,axiom,
! [A: set_v] :
( ( is_singleton_v @ A )
=> ~ ! [X3: v] :
( A
!= ( insert_v2 @ X3 @ bot_bot_set_v ) ) ) ).
% is_singletonE
thf(fact_936_is__singleton__def,axiom,
( is_sin9198872032823709915od_v_v
= ( ^ [A6: set_Product_prod_v_v] :
? [X2: product_prod_v_v] :
( A6
= ( insert1338601472111419319od_v_v @ X2 @ bot_bo723834152578015283od_v_v ) ) ) ) ).
% is_singleton_def
thf(fact_937_is__singleton__def,axiom,
( is_singleton_v
= ( ^ [A6: set_v] :
? [X2: v] :
( A6
= ( insert_v2 @ X2 @ bot_bot_set_v ) ) ) ) ).
% is_singleton_def
thf(fact_938_vfin,axiom,
finite_finite_v @ vertices ).
% vfin
thf(fact_939_rotate1__hd__tl,axiom,
! [Xs: list_v] :
( ( Xs != nil_v )
=> ( ( rotate1_v @ Xs )
= ( append_v @ ( tl_v @ Xs ) @ ( cons_v @ ( hd_v @ Xs ) @ nil_v ) ) ) ) ).
% rotate1_hd_tl
thf(fact_940_set__removeAll,axiom,
! [X: product_prod_v_v,Xs: list_P7986770385144383213od_v_v] :
( ( set_Product_prod_v_v2 @ ( remove481895986417801203od_v_v @ X @ Xs ) )
= ( minus_4183494784930505774od_v_v @ ( set_Product_prod_v_v2 @ Xs ) @ ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v ) ) ) ).
% set_removeAll
thf(fact_941_set__removeAll,axiom,
! [X: v,Xs: list_v] :
( ( set_v2 @ ( removeAll_v @ X @ Xs ) )
= ( minus_minus_set_v @ ( set_v2 @ Xs ) @ ( insert_v2 @ X @ bot_bot_set_v ) ) ) ).
% set_removeAll
thf(fact_942_List_Ofinite__set,axiom,
! [Xs: list_v] : ( finite_finite_v @ ( set_v2 @ Xs ) ) ).
% List.finite_set
thf(fact_943_removeAll__id,axiom,
! [X: product_prod_v_v,Xs: list_P7986770385144383213od_v_v] :
( ~ ( member7453568604450474000od_v_v @ X @ ( set_Product_prod_v_v2 @ Xs ) )
=> ( ( remove481895986417801203od_v_v @ X @ Xs )
= Xs ) ) ).
% removeAll_id
thf(fact_944_removeAll__id,axiom,
! [X: v,Xs: list_v] :
( ~ ( member_v @ X @ ( set_v2 @ Xs ) )
=> ( ( removeAll_v @ X @ Xs )
= Xs ) ) ).
% removeAll_id
thf(fact_945_rotate1__is__Nil__conv,axiom,
! [Xs: list_v] :
( ( ( rotate1_v @ Xs )
= nil_v )
= ( Xs = nil_v ) ) ).
% rotate1_is_Nil_conv
thf(fact_946_set__rotate1,axiom,
! [Xs: list_v] :
( ( set_v2 @ ( rotate1_v @ Xs ) )
= ( set_v2 @ Xs ) ) ).
% set_rotate1
thf(fact_947_removeAll__append,axiom,
! [X: v,Xs: list_v,Ys: list_v] :
( ( removeAll_v @ X @ ( append_v @ Xs @ Ys ) )
= ( append_v @ ( removeAll_v @ X @ Xs ) @ ( removeAll_v @ X @ Ys ) ) ) ).
% removeAll_append
thf(fact_948_finite__list,axiom,
! [A: set_v] :
( ( finite_finite_v @ A )
=> ? [Xs2: list_v] :
( ( set_v2 @ Xs2 )
= A ) ) ).
% finite_list
thf(fact_949_graph_Ovfin,axiom,
! [Vertices: set_v,Successors: v > set_v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( finite_finite_v @ Vertices ) ) ).
% graph.vfin
thf(fact_950_rotate1_Osimps_I1_J,axiom,
( ( rotate1_v @ nil_v )
= nil_v ) ).
% rotate1.simps(1)
thf(fact_951_removeAll_Osimps_I1_J,axiom,
! [X: v] :
( ( removeAll_v @ X @ nil_v )
= nil_v ) ).
% removeAll.simps(1)
thf(fact_952_graph_Ointro,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v] :
( ( finite3348123685078250256od_v_v @ Vertices )
=> ( ! [X3: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X3 @ Vertices )
=> ( ord_le7336532860387713383od_v_v @ ( Successors @ X3 ) @ Vertices ) )
=> ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors ) ) ) ).
% graph.intro
thf(fact_953_graph_Ointro,axiom,
! [Vertices: set_v,Successors: v > set_v] :
( ( finite_finite_v @ Vertices )
=> ( ! [X3: v] :
( ( member_v @ X3 @ Vertices )
=> ( ord_less_eq_set_v @ ( Successors @ X3 ) @ Vertices ) )
=> ( sCC_Bloemen_graph_v @ Vertices @ Successors ) ) ) ).
% graph.intro
thf(fact_954_SCC__Bloemen__Sequential_Ograph__def,axiom,
( sCC_Bl8307124943676871238od_v_v
= ( ^ [Vertices2: set_Product_prod_v_v,Successors2: product_prod_v_v > set_Product_prod_v_v] :
( ( finite3348123685078250256od_v_v @ Vertices2 )
& ! [X2: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X2 @ Vertices2 )
=> ( ord_le7336532860387713383od_v_v @ ( Successors2 @ X2 ) @ Vertices2 ) ) ) ) ) ).
% SCC_Bloemen_Sequential.graph_def
thf(fact_955_SCC__Bloemen__Sequential_Ograph__def,axiom,
( sCC_Bloemen_graph_v
= ( ^ [Vertices2: set_v,Successors2: v > set_v] :
( ( finite_finite_v @ Vertices2 )
& ! [X2: v] :
( ( member_v @ X2 @ Vertices2 )
=> ( ord_less_eq_set_v @ ( Successors2 @ X2 ) @ Vertices2 ) ) ) ) ) ).
% SCC_Bloemen_Sequential.graph_def
thf(fact_956_rotate1_Osimps_I2_J,axiom,
! [X: v,Xs: list_v] :
( ( rotate1_v @ ( cons_v @ X @ Xs ) )
= ( append_v @ Xs @ ( cons_v @ X @ nil_v ) ) ) ).
% rotate1.simps(2)
thf(fact_957_finite__Diff__insert,axiom,
! [A: set_Product_prod_v_v,A4: product_prod_v_v,B: set_Product_prod_v_v] :
( ( finite3348123685078250256od_v_v @ ( minus_4183494784930505774od_v_v @ A @ ( insert1338601472111419319od_v_v @ A4 @ B ) ) )
= ( finite3348123685078250256od_v_v @ ( minus_4183494784930505774od_v_v @ A @ B ) ) ) ).
% finite_Diff_insert
thf(fact_958_finite__Diff__insert,axiom,
! [A: set_v,A4: v,B: set_v] :
( ( finite_finite_v @ ( minus_minus_set_v @ A @ ( insert_v2 @ A4 @ B ) ) )
= ( finite_finite_v @ ( minus_minus_set_v @ A @ B ) ) ) ).
% finite_Diff_insert
thf(fact_959_finite__Diff,axiom,
! [A: set_v,B: set_v] :
( ( finite_finite_v @ A )
=> ( finite_finite_v @ ( minus_minus_set_v @ A @ B ) ) ) ).
% finite_Diff
thf(fact_960_finite__Diff2,axiom,
! [B: set_v,A: set_v] :
( ( finite_finite_v @ B )
=> ( ( finite_finite_v @ ( minus_minus_set_v @ A @ B ) )
= ( finite_finite_v @ A ) ) ) ).
% finite_Diff2
thf(fact_961_finite__insert,axiom,
! [A4: product_prod_v_v,A: set_Product_prod_v_v] :
( ( finite3348123685078250256od_v_v @ ( insert1338601472111419319od_v_v @ A4 @ A ) )
= ( finite3348123685078250256od_v_v @ A ) ) ).
% finite_insert
thf(fact_962_finite__insert,axiom,
! [A4: v,A: set_v] :
( ( finite_finite_v @ ( insert_v2 @ A4 @ A ) )
= ( finite_finite_v @ A ) ) ).
% finite_insert
thf(fact_963_finite__Un,axiom,
! [F2: set_v,G: set_v] :
( ( finite_finite_v @ ( sup_sup_set_v @ F2 @ G ) )
= ( ( finite_finite_v @ F2 )
& ( finite_finite_v @ G ) ) ) ).
% finite_Un
thf(fact_964_finite__Un,axiom,
! [F2: set_Product_prod_v_v,G: set_Product_prod_v_v] :
( ( finite3348123685078250256od_v_v @ ( sup_su414716646722978715od_v_v @ F2 @ G ) )
= ( ( finite3348123685078250256od_v_v @ F2 )
& ( finite3348123685078250256od_v_v @ G ) ) ) ).
% finite_Un
thf(fact_965_finite__has__minimal2,axiom,
! [A: set_se8455005133513928103od_v_v,A4: set_Product_prod_v_v] :
( ( finite6084192165098772208od_v_v @ A )
=> ( ( member8406446414694345712od_v_v @ A4 @ A )
=> ? [X3: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ X3 @ A )
& ( ord_le7336532860387713383od_v_v @ X3 @ A4 )
& ! [Xa2: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ Xa2 @ A )
=> ( ( ord_le7336532860387713383od_v_v @ Xa2 @ X3 )
=> ( X3 = Xa2 ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_966_finite__has__minimal2,axiom,
! [A: set_set_v,A4: set_v] :
( ( finite_finite_set_v @ A )
=> ( ( member_set_v @ A4 @ A )
=> ? [X3: set_v] :
( ( member_set_v @ X3 @ A )
& ( ord_less_eq_set_v @ X3 @ A4 )
& ! [Xa2: set_v] :
( ( member_set_v @ Xa2 @ A )
=> ( ( ord_less_eq_set_v @ Xa2 @ X3 )
=> ( X3 = Xa2 ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_967_finite__has__maximal2,axiom,
! [A: set_se8455005133513928103od_v_v,A4: set_Product_prod_v_v] :
( ( finite6084192165098772208od_v_v @ A )
=> ( ( member8406446414694345712od_v_v @ A4 @ A )
=> ? [X3: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ X3 @ A )
& ( ord_le7336532860387713383od_v_v @ A4 @ X3 )
& ! [Xa2: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ Xa2 @ A )
=> ( ( ord_le7336532860387713383od_v_v @ X3 @ Xa2 )
=> ( X3 = Xa2 ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_968_finite__has__maximal2,axiom,
! [A: set_set_v,A4: set_v] :
( ( finite_finite_set_v @ A )
=> ( ( member_set_v @ A4 @ A )
=> ? [X3: set_v] :
( ( member_set_v @ X3 @ A )
& ( ord_less_eq_set_v @ A4 @ X3 )
& ! [Xa2: set_v] :
( ( member_set_v @ Xa2 @ A )
=> ( ( ord_less_eq_set_v @ X3 @ Xa2 )
=> ( X3 = Xa2 ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_969_finite_OemptyI,axiom,
finite3348123685078250256od_v_v @ bot_bo723834152578015283od_v_v ).
% finite.emptyI
thf(fact_970_finite_OemptyI,axiom,
finite_finite_v @ bot_bot_set_v ).
% finite.emptyI
thf(fact_971_infinite__imp__nonempty,axiom,
! [S: set_Product_prod_v_v] :
( ~ ( finite3348123685078250256od_v_v @ S )
=> ( S != bot_bo723834152578015283od_v_v ) ) ).
% infinite_imp_nonempty
thf(fact_972_infinite__imp__nonempty,axiom,
! [S: set_v] :
( ~ ( finite_finite_v @ S )
=> ( S != bot_bot_set_v ) ) ).
% infinite_imp_nonempty
thf(fact_973_rev__finite__subset,axiom,
! [B: set_Product_prod_v_v,A: set_Product_prod_v_v] :
( ( finite3348123685078250256od_v_v @ B )
=> ( ( ord_le7336532860387713383od_v_v @ A @ B )
=> ( finite3348123685078250256od_v_v @ A ) ) ) ).
% rev_finite_subset
thf(fact_974_rev__finite__subset,axiom,
! [B: set_v,A: set_v] :
( ( finite_finite_v @ B )
=> ( ( ord_less_eq_set_v @ A @ B )
=> ( finite_finite_v @ A ) ) ) ).
% rev_finite_subset
thf(fact_975_infinite__super,axiom,
! [S: set_Product_prod_v_v,T2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ S @ T2 )
=> ( ~ ( finite3348123685078250256od_v_v @ S )
=> ~ ( finite3348123685078250256od_v_v @ T2 ) ) ) ).
% infinite_super
thf(fact_976_infinite__super,axiom,
! [S: set_v,T2: set_v] :
( ( ord_less_eq_set_v @ S @ T2 )
=> ( ~ ( finite_finite_v @ S )
=> ~ ( finite_finite_v @ T2 ) ) ) ).
% infinite_super
thf(fact_977_finite__subset,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ B )
=> ( ( finite3348123685078250256od_v_v @ B )
=> ( finite3348123685078250256od_v_v @ A ) ) ) ).
% finite_subset
thf(fact_978_finite__subset,axiom,
! [A: set_v,B: set_v] :
( ( ord_less_eq_set_v @ A @ B )
=> ( ( finite_finite_v @ B )
=> ( finite_finite_v @ A ) ) ) ).
% finite_subset
thf(fact_979_finite_OinsertI,axiom,
! [A: set_Product_prod_v_v,A4: product_prod_v_v] :
( ( finite3348123685078250256od_v_v @ A )
=> ( finite3348123685078250256od_v_v @ ( insert1338601472111419319od_v_v @ A4 @ A ) ) ) ).
% finite.insertI
thf(fact_980_finite_OinsertI,axiom,
! [A: set_v,A4: v] :
( ( finite_finite_v @ A )
=> ( finite_finite_v @ ( insert_v2 @ A4 @ A ) ) ) ).
% finite.insertI
thf(fact_981_finite__UnI,axiom,
! [F2: set_v,G: set_v] :
( ( finite_finite_v @ F2 )
=> ( ( finite_finite_v @ G )
=> ( finite_finite_v @ ( sup_sup_set_v @ F2 @ G ) ) ) ) ).
% finite_UnI
thf(fact_982_finite__UnI,axiom,
! [F2: set_Product_prod_v_v,G: set_Product_prod_v_v] :
( ( finite3348123685078250256od_v_v @ F2 )
=> ( ( finite3348123685078250256od_v_v @ G )
=> ( finite3348123685078250256od_v_v @ ( sup_su414716646722978715od_v_v @ F2 @ G ) ) ) ) ).
% finite_UnI
thf(fact_983_Un__infinite,axiom,
! [S: set_v,T2: set_v] :
( ~ ( finite_finite_v @ S )
=> ~ ( finite_finite_v @ ( sup_sup_set_v @ S @ T2 ) ) ) ).
% Un_infinite
thf(fact_984_Un__infinite,axiom,
! [S: set_Product_prod_v_v,T2: set_Product_prod_v_v] :
( ~ ( finite3348123685078250256od_v_v @ S )
=> ~ ( finite3348123685078250256od_v_v @ ( sup_su414716646722978715od_v_v @ S @ T2 ) ) ) ).
% Un_infinite
thf(fact_985_infinite__Un,axiom,
! [S: set_v,T2: set_v] :
( ( ~ ( finite_finite_v @ ( sup_sup_set_v @ S @ T2 ) ) )
= ( ~ ( finite_finite_v @ S )
| ~ ( finite_finite_v @ T2 ) ) ) ).
% infinite_Un
thf(fact_986_infinite__Un,axiom,
! [S: set_Product_prod_v_v,T2: set_Product_prod_v_v] :
( ( ~ ( finite3348123685078250256od_v_v @ ( sup_su414716646722978715od_v_v @ S @ T2 ) ) )
= ( ~ ( finite3348123685078250256od_v_v @ S )
| ~ ( finite3348123685078250256od_v_v @ T2 ) ) ) ).
% infinite_Un
thf(fact_987_Diff__infinite__finite,axiom,
! [T2: set_v,S: set_v] :
( ( finite_finite_v @ T2 )
=> ( ~ ( finite_finite_v @ S )
=> ~ ( finite_finite_v @ ( minus_minus_set_v @ S @ T2 ) ) ) ) ).
% Diff_infinite_finite
thf(fact_988_finite__has__maximal,axiom,
! [A: set_se8455005133513928103od_v_v] :
( ( finite6084192165098772208od_v_v @ A )
=> ( ( A != bot_bo3497076220358800403od_v_v )
=> ? [X3: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ X3 @ A )
& ! [Xa2: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ Xa2 @ A )
=> ( ( ord_le7336532860387713383od_v_v @ X3 @ Xa2 )
=> ( X3 = Xa2 ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_989_finite__has__maximal,axiom,
! [A: set_set_v] :
( ( finite_finite_set_v @ A )
=> ( ( A != bot_bot_set_set_v )
=> ? [X3: set_v] :
( ( member_set_v @ X3 @ A )
& ! [Xa2: set_v] :
( ( member_set_v @ Xa2 @ A )
=> ( ( ord_less_eq_set_v @ X3 @ Xa2 )
=> ( X3 = Xa2 ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_990_finite__has__minimal,axiom,
! [A: set_se8455005133513928103od_v_v] :
( ( finite6084192165098772208od_v_v @ A )
=> ( ( A != bot_bo3497076220358800403od_v_v )
=> ? [X3: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ X3 @ A )
& ! [Xa2: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ Xa2 @ A )
=> ( ( ord_le7336532860387713383od_v_v @ Xa2 @ X3 )
=> ( X3 = Xa2 ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_991_finite__has__minimal,axiom,
! [A: set_set_v] :
( ( finite_finite_set_v @ A )
=> ( ( A != bot_bot_set_set_v )
=> ? [X3: set_v] :
( ( member_set_v @ X3 @ A )
& ! [Xa2: set_v] :
( ( member_set_v @ Xa2 @ A )
=> ( ( ord_less_eq_set_v @ Xa2 @ X3 )
=> ( X3 = Xa2 ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_992_infinite__finite__induct,axiom,
! [P: set_Product_prod_v_v > $o,A: set_Product_prod_v_v] :
( ! [A10: set_Product_prod_v_v] :
( ~ ( finite3348123685078250256od_v_v @ A10 )
=> ( P @ A10 ) )
=> ( ( P @ bot_bo723834152578015283od_v_v )
=> ( ! [X3: product_prod_v_v,F3: set_Product_prod_v_v] :
( ( finite3348123685078250256od_v_v @ F3 )
=> ( ~ ( member7453568604450474000od_v_v @ X3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert1338601472111419319od_v_v @ X3 @ F3 ) ) ) ) )
=> ( P @ A ) ) ) ) ).
% infinite_finite_induct
thf(fact_993_infinite__finite__induct,axiom,
! [P: set_v > $o,A: set_v] :
( ! [A10: set_v] :
( ~ ( finite_finite_v @ A10 )
=> ( P @ A10 ) )
=> ( ( P @ bot_bot_set_v )
=> ( ! [X3: v,F3: set_v] :
( ( finite_finite_v @ F3 )
=> ( ~ ( member_v @ X3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_v2 @ X3 @ F3 ) ) ) ) )
=> ( P @ A ) ) ) ) ).
% infinite_finite_induct
thf(fact_994_finite__ne__induct,axiom,
! [F2: set_Product_prod_v_v,P: set_Product_prod_v_v > $o] :
( ( finite3348123685078250256od_v_v @ F2 )
=> ( ( F2 != bot_bo723834152578015283od_v_v )
=> ( ! [X3: product_prod_v_v] : ( P @ ( insert1338601472111419319od_v_v @ X3 @ bot_bo723834152578015283od_v_v ) )
=> ( ! [X3: product_prod_v_v,F3: set_Product_prod_v_v] :
( ( finite3348123685078250256od_v_v @ F3 )
=> ( ( F3 != bot_bo723834152578015283od_v_v )
=> ( ~ ( member7453568604450474000od_v_v @ X3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert1338601472111419319od_v_v @ X3 @ F3 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_995_finite__ne__induct,axiom,
! [F2: set_v,P: set_v > $o] :
( ( finite_finite_v @ F2 )
=> ( ( F2 != bot_bot_set_v )
=> ( ! [X3: v] : ( P @ ( insert_v2 @ X3 @ bot_bot_set_v ) )
=> ( ! [X3: v,F3: set_v] :
( ( finite_finite_v @ F3 )
=> ( ( F3 != bot_bot_set_v )
=> ( ~ ( member_v @ X3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_v2 @ X3 @ F3 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_996_finite__induct,axiom,
! [F2: set_Product_prod_v_v,P: set_Product_prod_v_v > $o] :
( ( finite3348123685078250256od_v_v @ F2 )
=> ( ( P @ bot_bo723834152578015283od_v_v )
=> ( ! [X3: product_prod_v_v,F3: set_Product_prod_v_v] :
( ( finite3348123685078250256od_v_v @ F3 )
=> ( ~ ( member7453568604450474000od_v_v @ X3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert1338601472111419319od_v_v @ X3 @ F3 ) ) ) ) )
=> ( P @ F2 ) ) ) ) ).
% finite_induct
thf(fact_997_finite__induct,axiom,
! [F2: set_v,P: set_v > $o] :
( ( finite_finite_v @ F2 )
=> ( ( P @ bot_bot_set_v )
=> ( ! [X3: v,F3: set_v] :
( ( finite_finite_v @ F3 )
=> ( ~ ( member_v @ X3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_v2 @ X3 @ F3 ) ) ) ) )
=> ( P @ F2 ) ) ) ) ).
% finite_induct
thf(fact_998_finite_Osimps,axiom,
( finite3348123685078250256od_v_v
= ( ^ [A5: set_Product_prod_v_v] :
( ( A5 = bot_bo723834152578015283od_v_v )
| ? [A6: set_Product_prod_v_v,B3: product_prod_v_v] :
( ( A5
= ( insert1338601472111419319od_v_v @ B3 @ A6 ) )
& ( finite3348123685078250256od_v_v @ A6 ) ) ) ) ) ).
% finite.simps
thf(fact_999_finite_Osimps,axiom,
( finite_finite_v
= ( ^ [A5: set_v] :
( ( A5 = bot_bot_set_v )
| ? [A6: set_v,B3: v] :
( ( A5
= ( insert_v2 @ B3 @ A6 ) )
& ( finite_finite_v @ A6 ) ) ) ) ) ).
% finite.simps
thf(fact_1000_finite_Ocases,axiom,
! [A4: set_Product_prod_v_v] :
( ( finite3348123685078250256od_v_v @ A4 )
=> ( ( A4 != bot_bo723834152578015283od_v_v )
=> ~ ! [A10: set_Product_prod_v_v] :
( ? [A9: product_prod_v_v] :
( A4
= ( insert1338601472111419319od_v_v @ A9 @ A10 ) )
=> ~ ( finite3348123685078250256od_v_v @ A10 ) ) ) ) ).
% finite.cases
thf(fact_1001_finite_Ocases,axiom,
! [A4: set_v] :
( ( finite_finite_v @ A4 )
=> ( ( A4 != bot_bot_set_v )
=> ~ ! [A10: set_v] :
( ? [A9: v] :
( A4
= ( insert_v2 @ A9 @ A10 ) )
=> ~ ( finite_finite_v @ A10 ) ) ) ) ).
% finite.cases
thf(fact_1002_finite__subset__induct,axiom,
! [F2: set_Product_prod_v_v,A: set_Product_prod_v_v,P: set_Product_prod_v_v > $o] :
( ( finite3348123685078250256od_v_v @ F2 )
=> ( ( ord_le7336532860387713383od_v_v @ F2 @ A )
=> ( ( P @ bot_bo723834152578015283od_v_v )
=> ( ! [A9: product_prod_v_v,F3: set_Product_prod_v_v] :
( ( finite3348123685078250256od_v_v @ F3 )
=> ( ( member7453568604450474000od_v_v @ A9 @ A )
=> ( ~ ( member7453568604450474000od_v_v @ A9 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert1338601472111419319od_v_v @ A9 @ F3 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_1003_finite__subset__induct,axiom,
! [F2: set_v,A: set_v,P: set_v > $o] :
( ( finite_finite_v @ F2 )
=> ( ( ord_less_eq_set_v @ F2 @ A )
=> ( ( P @ bot_bot_set_v )
=> ( ! [A9: v,F3: set_v] :
( ( finite_finite_v @ F3 )
=> ( ( member_v @ A9 @ A )
=> ( ~ ( member_v @ A9 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_v2 @ A9 @ F3 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_1004_finite__subset__induct_H,axiom,
! [F2: set_Product_prod_v_v,A: set_Product_prod_v_v,P: set_Product_prod_v_v > $o] :
( ( finite3348123685078250256od_v_v @ F2 )
=> ( ( ord_le7336532860387713383od_v_v @ F2 @ A )
=> ( ( P @ bot_bo723834152578015283od_v_v )
=> ( ! [A9: product_prod_v_v,F3: set_Product_prod_v_v] :
( ( finite3348123685078250256od_v_v @ F3 )
=> ( ( member7453568604450474000od_v_v @ A9 @ A )
=> ( ( ord_le7336532860387713383od_v_v @ F3 @ A )
=> ( ~ ( member7453568604450474000od_v_v @ A9 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert1338601472111419319od_v_v @ A9 @ F3 ) ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_1005_finite__subset__induct_H,axiom,
! [F2: set_v,A: set_v,P: set_v > $o] :
( ( finite_finite_v @ F2 )
=> ( ( ord_less_eq_set_v @ F2 @ A )
=> ( ( P @ bot_bot_set_v )
=> ( ! [A9: v,F3: set_v] :
( ( finite_finite_v @ F3 )
=> ( ( member_v @ A9 @ A )
=> ( ( ord_less_eq_set_v @ F3 @ A )
=> ( ~ ( member_v @ A9 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_v2 @ A9 @ F3 ) ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_1006_infinite__remove,axiom,
! [S: set_Product_prod_v_v,A4: product_prod_v_v] :
( ~ ( finite3348123685078250256od_v_v @ S )
=> ~ ( finite3348123685078250256od_v_v @ ( minus_4183494784930505774od_v_v @ S @ ( insert1338601472111419319od_v_v @ A4 @ bot_bo723834152578015283od_v_v ) ) ) ) ).
% infinite_remove
thf(fact_1007_infinite__remove,axiom,
! [S: set_v,A4: v] :
( ~ ( finite_finite_v @ S )
=> ~ ( finite_finite_v @ ( minus_minus_set_v @ S @ ( insert_v2 @ A4 @ bot_bot_set_v ) ) ) ) ).
% infinite_remove
thf(fact_1008_infinite__coinduct,axiom,
! [X5: set_Product_prod_v_v > $o,A: set_Product_prod_v_v] :
( ( X5 @ A )
=> ( ! [A10: set_Product_prod_v_v] :
( ( X5 @ A10 )
=> ? [X4: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X4 @ A10 )
& ( ( X5 @ ( minus_4183494784930505774od_v_v @ A10 @ ( insert1338601472111419319od_v_v @ X4 @ bot_bo723834152578015283od_v_v ) ) )
| ~ ( finite3348123685078250256od_v_v @ ( minus_4183494784930505774od_v_v @ A10 @ ( insert1338601472111419319od_v_v @ X4 @ bot_bo723834152578015283od_v_v ) ) ) ) ) )
=> ~ ( finite3348123685078250256od_v_v @ A ) ) ) ).
% infinite_coinduct
thf(fact_1009_infinite__coinduct,axiom,
! [X5: set_v > $o,A: set_v] :
( ( X5 @ A )
=> ( ! [A10: set_v] :
( ( X5 @ A10 )
=> ? [X4: v] :
( ( member_v @ X4 @ A10 )
& ( ( X5 @ ( minus_minus_set_v @ A10 @ ( insert_v2 @ X4 @ bot_bot_set_v ) ) )
| ~ ( finite_finite_v @ ( minus_minus_set_v @ A10 @ ( insert_v2 @ X4 @ bot_bot_set_v ) ) ) ) ) )
=> ~ ( finite_finite_v @ A ) ) ) ).
% infinite_coinduct
thf(fact_1010_finite__empty__induct,axiom,
! [A: set_Product_prod_v_v,P: set_Product_prod_v_v > $o] :
( ( finite3348123685078250256od_v_v @ A )
=> ( ( P @ A )
=> ( ! [A9: product_prod_v_v,A10: set_Product_prod_v_v] :
( ( finite3348123685078250256od_v_v @ A10 )
=> ( ( member7453568604450474000od_v_v @ A9 @ A10 )
=> ( ( P @ A10 )
=> ( P @ ( minus_4183494784930505774od_v_v @ A10 @ ( insert1338601472111419319od_v_v @ A9 @ bot_bo723834152578015283od_v_v ) ) ) ) ) )
=> ( P @ bot_bo723834152578015283od_v_v ) ) ) ) ).
% finite_empty_induct
thf(fact_1011_finite__empty__induct,axiom,
! [A: set_v,P: set_v > $o] :
( ( finite_finite_v @ A )
=> ( ( P @ A )
=> ( ! [A9: v,A10: set_v] :
( ( finite_finite_v @ A10 )
=> ( ( member_v @ A9 @ A10 )
=> ( ( P @ A10 )
=> ( P @ ( minus_minus_set_v @ A10 @ ( insert_v2 @ A9 @ bot_bot_set_v ) ) ) ) ) )
=> ( P @ bot_bot_set_v ) ) ) ) ).
% finite_empty_induct
thf(fact_1012_finite__remove__induct,axiom,
! [B: set_Product_prod_v_v,P: set_Product_prod_v_v > $o] :
( ( finite3348123685078250256od_v_v @ B )
=> ( ( P @ bot_bo723834152578015283od_v_v )
=> ( ! [A10: set_Product_prod_v_v] :
( ( finite3348123685078250256od_v_v @ A10 )
=> ( ( A10 != bot_bo723834152578015283od_v_v )
=> ( ( ord_le7336532860387713383od_v_v @ A10 @ B )
=> ( ! [X4: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X4 @ A10 )
=> ( P @ ( minus_4183494784930505774od_v_v @ A10 @ ( insert1338601472111419319od_v_v @ X4 @ bot_bo723834152578015283od_v_v ) ) ) )
=> ( P @ A10 ) ) ) ) )
=> ( P @ B ) ) ) ) ).
% finite_remove_induct
thf(fact_1013_finite__remove__induct,axiom,
! [B: set_v,P: set_v > $o] :
( ( finite_finite_v @ B )
=> ( ( P @ bot_bot_set_v )
=> ( ! [A10: set_v] :
( ( finite_finite_v @ A10 )
=> ( ( A10 != bot_bot_set_v )
=> ( ( ord_less_eq_set_v @ A10 @ B )
=> ( ! [X4: v] :
( ( member_v @ X4 @ A10 )
=> ( P @ ( minus_minus_set_v @ A10 @ ( insert_v2 @ X4 @ bot_bot_set_v ) ) ) )
=> ( P @ A10 ) ) ) ) )
=> ( P @ B ) ) ) ) ).
% finite_remove_induct
thf(fact_1014_remove__induct,axiom,
! [P: set_Product_prod_v_v > $o,B: set_Product_prod_v_v] :
( ( P @ bot_bo723834152578015283od_v_v )
=> ( ( ~ ( finite3348123685078250256od_v_v @ B )
=> ( P @ B ) )
=> ( ! [A10: set_Product_prod_v_v] :
( ( finite3348123685078250256od_v_v @ A10 )
=> ( ( A10 != bot_bo723834152578015283od_v_v )
=> ( ( ord_le7336532860387713383od_v_v @ A10 @ B )
=> ( ! [X4: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X4 @ A10 )
=> ( P @ ( minus_4183494784930505774od_v_v @ A10 @ ( insert1338601472111419319od_v_v @ X4 @ bot_bo723834152578015283od_v_v ) ) ) )
=> ( P @ A10 ) ) ) ) )
=> ( P @ B ) ) ) ) ).
% remove_induct
thf(fact_1015_remove__induct,axiom,
! [P: set_v > $o,B: set_v] :
( ( P @ bot_bot_set_v )
=> ( ( ~ ( finite_finite_v @ B )
=> ( P @ B ) )
=> ( ! [A10: set_v] :
( ( finite_finite_v @ A10 )
=> ( ( A10 != bot_bot_set_v )
=> ( ( ord_less_eq_set_v @ A10 @ B )
=> ( ! [X4: v] :
( ( member_v @ X4 @ A10 )
=> ( P @ ( minus_minus_set_v @ A10 @ ( insert_v2 @ X4 @ bot_bot_set_v ) ) ) )
=> ( P @ A10 ) ) ) ) )
=> ( P @ B ) ) ) ) ).
% remove_induct
thf(fact_1016_insert__subsetI,axiom,
! [X: product_prod_v_v,A: set_Product_prod_v_v,X5: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X @ A )
=> ( ( ord_le7336532860387713383od_v_v @ X5 @ A )
=> ( ord_le7336532860387713383od_v_v @ ( insert1338601472111419319od_v_v @ X @ X5 ) @ A ) ) ) ).
% insert_subsetI
thf(fact_1017_insert__subsetI,axiom,
! [X: v,A: set_v,X5: set_v] :
( ( member_v @ X @ A )
=> ( ( ord_less_eq_set_v @ X5 @ A )
=> ( ord_less_eq_set_v @ ( insert_v2 @ X @ X5 ) @ A ) ) ) ).
% insert_subsetI
thf(fact_1018_subset__emptyI,axiom,
! [A: set_Product_prod_v_v] :
( ! [X3: product_prod_v_v] :
~ ( member7453568604450474000od_v_v @ X3 @ A )
=> ( ord_le7336532860387713383od_v_v @ A @ bot_bo723834152578015283od_v_v ) ) ).
% subset_emptyI
thf(fact_1019_subset__emptyI,axiom,
! [A: set_v] :
( ! [X3: v] :
~ ( member_v @ X3 @ A )
=> ( ord_less_eq_set_v @ A @ bot_bot_set_v ) ) ).
% subset_emptyI
thf(fact_1020_ssubst__Pair__rhs,axiom,
! [R2: v,S5: v,R4: set_Product_prod_v_v,S6: v] :
( ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ R2 @ S5 ) @ R4 )
=> ( ( S6 = S5 )
=> ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ R2 @ S6 ) @ R4 ) ) ) ).
% ssubst_Pair_rhs
thf(fact_1021_ssubst__Pair__rhs,axiom,
! [R2: v,S5: sCC_Bl1394983891496994913t_unit,R4: set_Pr6425124735969554649t_unit,S6: sCC_Bl1394983891496994913t_unit] :
( ( member7924940910754673978t_unit @ ( produc3862955338007567901t_unit @ R2 @ S5 ) @ R4 )
=> ( ( S6 = S5 )
=> ( member7924940910754673978t_unit @ ( produc3862955338007567901t_unit @ R2 @ S6 ) @ R4 ) ) ) ).
% ssubst_Pair_rhs
thf(fact_1022_product__lists_Osimps_I1_J,axiom,
( ( product_lists_v @ nil_list_v )
= ( cons_list_v @ nil_v @ nil_list_v ) ) ).
% product_lists.simps(1)
thf(fact_1023_Field__insert,axiom,
! [A4: product_prod_v_v,B2: product_prod_v_v,R2: set_Pr2149350503807050951od_v_v] :
( ( field_7153129647634986036od_v_v @ ( insert5641704497130386615od_v_v @ ( produc4031800376763917143od_v_v @ A4 @ B2 ) @ R2 ) )
= ( sup_su414716646722978715od_v_v @ ( insert1338601472111419319od_v_v @ A4 @ ( insert1338601472111419319od_v_v @ B2 @ bot_bo723834152578015283od_v_v ) ) @ ( field_7153129647634986036od_v_v @ R2 ) ) ) ).
% Field_insert
thf(fact_1024_Field__insert,axiom,
! [A4: v,B2: v,R2: set_Product_prod_v_v] :
( ( field_v @ ( insert1338601472111419319od_v_v @ ( product_Pair_v_v @ A4 @ B2 ) @ R2 ) )
= ( sup_sup_set_v @ ( insert_v2 @ A4 @ ( insert_v2 @ B2 @ bot_bot_set_v ) ) @ ( field_v @ R2 ) ) ) ).
% Field_insert
thf(fact_1025_remove__def,axiom,
( remove5001965847480235980od_v_v
= ( ^ [X2: product_prod_v_v,A6: set_Product_prod_v_v] : ( minus_4183494784930505774od_v_v @ A6 @ ( insert1338601472111419319od_v_v @ X2 @ bot_bo723834152578015283od_v_v ) ) ) ) ).
% remove_def
thf(fact_1026_remove__def,axiom,
( remove_v
= ( ^ [X2: v,A6: set_v] : ( minus_minus_set_v @ A6 @ ( insert_v2 @ X2 @ bot_bot_set_v ) ) ) ) ).
% remove_def
thf(fact_1027_member__remove,axiom,
! [X: v,Y: v,A: set_v] :
( ( member_v @ X @ ( remove_v @ Y @ A ) )
= ( ( member_v @ X @ A )
& ( X != Y ) ) ) ).
% member_remove
thf(fact_1028_member__remove,axiom,
! [X: product_prod_v_v,Y: product_prod_v_v,A: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X @ ( remove5001965847480235980od_v_v @ Y @ A ) )
= ( ( member7453568604450474000od_v_v @ X @ A )
& ( X != Y ) ) ) ).
% member_remove
thf(fact_1029_bind__simps_I1_J,axiom,
! [F: v > list_v] :
( ( bind_v_v @ nil_v @ F )
= nil_v ) ).
% bind_simps(1)
thf(fact_1030_Field__empty,axiom,
( ( field_7153129647634986036od_v_v @ bot_bo3282589961317712691od_v_v )
= bot_bo723834152578015283od_v_v ) ).
% Field_empty
thf(fact_1031_Field__empty,axiom,
( ( field_v @ bot_bo723834152578015283od_v_v )
= bot_bot_set_v ) ).
% Field_empty
thf(fact_1032_Field__Un,axiom,
! [R2: set_Pr2149350503807050951od_v_v,S5: set_Pr2149350503807050951od_v_v] :
( ( field_7153129647634986036od_v_v @ ( sup_su1742609618068805275od_v_v @ R2 @ S5 ) )
= ( sup_su414716646722978715od_v_v @ ( field_7153129647634986036od_v_v @ R2 ) @ ( field_7153129647634986036od_v_v @ S5 ) ) ) ).
% Field_Un
thf(fact_1033_Field__Un,axiom,
! [R2: set_Product_prod_v_v,S5: set_Product_prod_v_v] :
( ( field_v @ ( sup_su414716646722978715od_v_v @ R2 @ S5 ) )
= ( sup_sup_set_v @ ( field_v @ R2 ) @ ( field_v @ S5 ) ) ) ).
% Field_Un
thf(fact_1034_FieldI2,axiom,
! [I: product_prod_v_v,J: product_prod_v_v,R4: set_Pr2149350503807050951od_v_v] :
( ( member3038538357316246288od_v_v @ ( produc4031800376763917143od_v_v @ I @ J ) @ R4 )
=> ( member7453568604450474000od_v_v @ J @ ( field_7153129647634986036od_v_v @ R4 ) ) ) ).
% FieldI2
thf(fact_1035_FieldI2,axiom,
! [I: v,J: v,R4: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ I @ J ) @ R4 )
=> ( member_v @ J @ ( field_v @ R4 ) ) ) ).
% FieldI2
thf(fact_1036_FieldI1,axiom,
! [I: product_prod_v_v,J: product_prod_v_v,R4: set_Pr2149350503807050951od_v_v] :
( ( member3038538357316246288od_v_v @ ( produc4031800376763917143od_v_v @ I @ J ) @ R4 )
=> ( member7453568604450474000od_v_v @ I @ ( field_7153129647634986036od_v_v @ R4 ) ) ) ).
% FieldI1
thf(fact_1037_FieldI1,axiom,
! [I: v,J: v,R4: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ I @ J ) @ R4 )
=> ( member_v @ I @ ( field_v @ R4 ) ) ) ).
% FieldI1
thf(fact_1038_mono__Field,axiom,
! [R2: set_Pr2149350503807050951od_v_v,S5: set_Pr2149350503807050951od_v_v] :
( ( ord_le6241436655786843239od_v_v @ R2 @ S5 )
=> ( ord_le7336532860387713383od_v_v @ ( field_7153129647634986036od_v_v @ R2 ) @ ( field_7153129647634986036od_v_v @ S5 ) ) ) ).
% mono_Field
thf(fact_1039_mono__Field,axiom,
! [R2: set_Product_prod_v_v,S5: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ R2 @ S5 )
=> ( ord_less_eq_set_v @ ( field_v @ R2 ) @ ( field_v @ S5 ) ) ) ).
% mono_Field
thf(fact_1040_remove__code_I1_J,axiom,
! [X: v,Xs: list_v] :
( ( remove_v @ X @ ( set_v2 @ Xs ) )
= ( set_v2 @ ( removeAll_v @ X @ Xs ) ) ) ).
% remove_code(1)
thf(fact_1041_dom,axiom,
accp_S2303753412255344476t_unit @ ( sCC_Bl907557413677168252_rel_v @ successors ) @ ( sum_In5289330923152326972t_unit @ ( produc3862955338007567901t_unit @ va @ ea ) ) ).
% dom
thf(fact_1042_Sup__fin_Oremove,axiom,
! [A: set_set_v,X: set_v] :
( ( finite_finite_set_v @ A )
=> ( ( member_set_v @ X @ A )
=> ( ( ( ( minus_7228012346218142266_set_v @ A @ ( insert_set_v @ X @ bot_bot_set_set_v ) )
= bot_bot_set_set_v )
=> ( ( lattic2918178447194608042_set_v @ A )
= X ) )
& ( ( ( minus_7228012346218142266_set_v @ A @ ( insert_set_v @ X @ bot_bot_set_set_v ) )
!= bot_bot_set_set_v )
=> ( ( lattic2918178447194608042_set_v @ A )
= ( sup_sup_set_v @ X @ ( lattic2918178447194608042_set_v @ ( minus_7228012346218142266_set_v @ A @ ( insert_set_v @ X @ bot_bot_set_set_v ) ) ) ) ) ) ) ) ) ).
% Sup_fin.remove
thf(fact_1043_Sup__fin_Oremove,axiom,
! [A: set_se8455005133513928103od_v_v,X: set_Product_prod_v_v] :
( ( finite6084192165098772208od_v_v @ A )
=> ( ( member8406446414694345712od_v_v @ X @ A )
=> ( ( ( ( minus_7679383599658060814od_v_v @ A @ ( insert7504383016908236695od_v_v @ X @ bot_bo3497076220358800403od_v_v ) )
= bot_bo3497076220358800403od_v_v )
=> ( ( lattic5151207300795964030od_v_v @ A )
= X ) )
& ( ( ( minus_7679383599658060814od_v_v @ A @ ( insert7504383016908236695od_v_v @ X @ bot_bo3497076220358800403od_v_v ) )
!= bot_bo3497076220358800403od_v_v )
=> ( ( lattic5151207300795964030od_v_v @ A )
= ( sup_su414716646722978715od_v_v @ X @ ( lattic5151207300795964030od_v_v @ ( minus_7679383599658060814od_v_v @ A @ ( insert7504383016908236695od_v_v @ X @ bot_bo3497076220358800403od_v_v ) ) ) ) ) ) ) ) ) ).
% Sup_fin.remove
thf(fact_1044_Sup__fin_Oinsert__remove,axiom,
! [A: set_set_v,X: set_v] :
( ( finite_finite_set_v @ A )
=> ( ( ( ( minus_7228012346218142266_set_v @ A @ ( insert_set_v @ X @ bot_bot_set_set_v ) )
= bot_bot_set_set_v )
=> ( ( lattic2918178447194608042_set_v @ ( insert_set_v @ X @ A ) )
= X ) )
& ( ( ( minus_7228012346218142266_set_v @ A @ ( insert_set_v @ X @ bot_bot_set_set_v ) )
!= bot_bot_set_set_v )
=> ( ( lattic2918178447194608042_set_v @ ( insert_set_v @ X @ A ) )
= ( sup_sup_set_v @ X @ ( lattic2918178447194608042_set_v @ ( minus_7228012346218142266_set_v @ A @ ( insert_set_v @ X @ bot_bot_set_set_v ) ) ) ) ) ) ) ) ).
% Sup_fin.insert_remove
thf(fact_1045_Sup__fin_Oinsert__remove,axiom,
! [A: set_se8455005133513928103od_v_v,X: set_Product_prod_v_v] :
( ( finite6084192165098772208od_v_v @ A )
=> ( ( ( ( minus_7679383599658060814od_v_v @ A @ ( insert7504383016908236695od_v_v @ X @ bot_bo3497076220358800403od_v_v ) )
= bot_bo3497076220358800403od_v_v )
=> ( ( lattic5151207300795964030od_v_v @ ( insert7504383016908236695od_v_v @ X @ A ) )
= X ) )
& ( ( ( minus_7679383599658060814od_v_v @ A @ ( insert7504383016908236695od_v_v @ X @ bot_bo3497076220358800403od_v_v ) )
!= bot_bo3497076220358800403od_v_v )
=> ( ( lattic5151207300795964030od_v_v @ ( insert7504383016908236695od_v_v @ X @ A ) )
= ( sup_su414716646722978715od_v_v @ X @ ( lattic5151207300795964030od_v_v @ ( minus_7679383599658060814od_v_v @ A @ ( insert7504383016908236695od_v_v @ X @ bot_bo3497076220358800403od_v_v ) ) ) ) ) ) ) ) ).
% Sup_fin.insert_remove
thf(fact_1046_dfss_Ocases,axiom,
! [X: produc5741669702376414499t_unit] :
~ ! [V2: v,E7: sCC_Bl1394983891496994913t_unit] :
( X
!= ( produc3862955338007567901t_unit @ V2 @ E7 ) ) ).
% dfss.cases
thf(fact_1047_Sup__fin_Oinsert,axiom,
! [A: set_set_v,X: set_v] :
( ( finite_finite_set_v @ A )
=> ( ( A != bot_bot_set_set_v )
=> ( ( lattic2918178447194608042_set_v @ ( insert_set_v @ X @ A ) )
= ( sup_sup_set_v @ X @ ( lattic2918178447194608042_set_v @ A ) ) ) ) ) ).
% Sup_fin.insert
thf(fact_1048_Sup__fin_Oinsert,axiom,
! [A: set_se8455005133513928103od_v_v,X: set_Product_prod_v_v] :
( ( finite6084192165098772208od_v_v @ A )
=> ( ( A != bot_bo3497076220358800403od_v_v )
=> ( ( lattic5151207300795964030od_v_v @ ( insert7504383016908236695od_v_v @ X @ A ) )
= ( sup_su414716646722978715od_v_v @ X @ ( lattic5151207300795964030od_v_v @ A ) ) ) ) ) ).
% Sup_fin.insert
thf(fact_1049_Sup__fin_OcoboundedI,axiom,
! [A: set_se8455005133513928103od_v_v,A4: set_Product_prod_v_v] :
( ( finite6084192165098772208od_v_v @ A )
=> ( ( member8406446414694345712od_v_v @ A4 @ A )
=> ( ord_le7336532860387713383od_v_v @ A4 @ ( lattic5151207300795964030od_v_v @ A ) ) ) ) ).
% Sup_fin.coboundedI
thf(fact_1050_Sup__fin_OcoboundedI,axiom,
! [A: set_set_v,A4: set_v] :
( ( finite_finite_set_v @ A )
=> ( ( member_set_v @ A4 @ A )
=> ( ord_less_eq_set_v @ A4 @ ( lattic2918178447194608042_set_v @ A ) ) ) ) ).
% Sup_fin.coboundedI
thf(fact_1051_Sup__fin_Oin__idem,axiom,
! [A: set_set_v,X: set_v] :
( ( finite_finite_set_v @ A )
=> ( ( member_set_v @ X @ A )
=> ( ( sup_sup_set_v @ X @ ( lattic2918178447194608042_set_v @ A ) )
= ( lattic2918178447194608042_set_v @ A ) ) ) ) ).
% Sup_fin.in_idem
thf(fact_1052_Sup__fin_Oin__idem,axiom,
! [A: set_se8455005133513928103od_v_v,X: set_Product_prod_v_v] :
( ( finite6084192165098772208od_v_v @ A )
=> ( ( member8406446414694345712od_v_v @ X @ A )
=> ( ( sup_su414716646722978715od_v_v @ X @ ( lattic5151207300795964030od_v_v @ A ) )
= ( lattic5151207300795964030od_v_v @ A ) ) ) ) ).
% Sup_fin.in_idem
thf(fact_1053_Sup__fin_OboundedE,axiom,
! [A: set_se8455005133513928103od_v_v,X: set_Product_prod_v_v] :
( ( finite6084192165098772208od_v_v @ A )
=> ( ( A != bot_bo3497076220358800403od_v_v )
=> ( ( ord_le7336532860387713383od_v_v @ ( lattic5151207300795964030od_v_v @ A ) @ X )
=> ! [A11: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ A11 @ A )
=> ( ord_le7336532860387713383od_v_v @ A11 @ X ) ) ) ) ) ).
% Sup_fin.boundedE
thf(fact_1054_Sup__fin_OboundedE,axiom,
! [A: set_set_v,X: set_v] :
( ( finite_finite_set_v @ A )
=> ( ( A != bot_bot_set_set_v )
=> ( ( ord_less_eq_set_v @ ( lattic2918178447194608042_set_v @ A ) @ X )
=> ! [A11: set_v] :
( ( member_set_v @ A11 @ A )
=> ( ord_less_eq_set_v @ A11 @ X ) ) ) ) ) ).
% Sup_fin.boundedE
thf(fact_1055_Sup__fin_OboundedI,axiom,
! [A: set_se8455005133513928103od_v_v,X: set_Product_prod_v_v] :
( ( finite6084192165098772208od_v_v @ A )
=> ( ( A != bot_bo3497076220358800403od_v_v )
=> ( ! [A9: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ A9 @ A )
=> ( ord_le7336532860387713383od_v_v @ A9 @ X ) )
=> ( ord_le7336532860387713383od_v_v @ ( lattic5151207300795964030od_v_v @ A ) @ X ) ) ) ) ).
% Sup_fin.boundedI
thf(fact_1056_Sup__fin_OboundedI,axiom,
! [A: set_set_v,X: set_v] :
( ( finite_finite_set_v @ A )
=> ( ( A != bot_bot_set_set_v )
=> ( ! [A9: set_v] :
( ( member_set_v @ A9 @ A )
=> ( ord_less_eq_set_v @ A9 @ X ) )
=> ( ord_less_eq_set_v @ ( lattic2918178447194608042_set_v @ A ) @ X ) ) ) ) ).
% Sup_fin.boundedI
thf(fact_1057_Sup__fin_Obounded__iff,axiom,
! [A: set_se8455005133513928103od_v_v,X: set_Product_prod_v_v] :
( ( finite6084192165098772208od_v_v @ A )
=> ( ( A != bot_bo3497076220358800403od_v_v )
=> ( ( ord_le7336532860387713383od_v_v @ ( lattic5151207300795964030od_v_v @ A ) @ X )
= ( ! [X2: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ X2 @ A )
=> ( ord_le7336532860387713383od_v_v @ X2 @ X ) ) ) ) ) ) ).
% Sup_fin.bounded_iff
thf(fact_1058_Sup__fin_Obounded__iff,axiom,
! [A: set_set_v,X: set_v] :
( ( finite_finite_set_v @ A )
=> ( ( A != bot_bot_set_set_v )
=> ( ( ord_less_eq_set_v @ ( lattic2918178447194608042_set_v @ A ) @ X )
= ( ! [X2: set_v] :
( ( member_set_v @ X2 @ A )
=> ( ord_less_eq_set_v @ X2 @ X ) ) ) ) ) ) ).
% Sup_fin.bounded_iff
thf(fact_1059_Sup__fin_Osubset__imp,axiom,
! [A: set_se8455005133513928103od_v_v,B: set_se8455005133513928103od_v_v] :
( ( ord_le4714265922333009223od_v_v @ A @ B )
=> ( ( A != bot_bo3497076220358800403od_v_v )
=> ( ( finite6084192165098772208od_v_v @ B )
=> ( ord_le7336532860387713383od_v_v @ ( lattic5151207300795964030od_v_v @ A ) @ ( lattic5151207300795964030od_v_v @ B ) ) ) ) ) ).
% Sup_fin.subset_imp
thf(fact_1060_Sup__fin_Osubset__imp,axiom,
! [A: set_set_v,B: set_set_v] :
( ( ord_le5216385588623774835_set_v @ A @ B )
=> ( ( A != bot_bot_set_set_v )
=> ( ( finite_finite_set_v @ B )
=> ( ord_less_eq_set_v @ ( lattic2918178447194608042_set_v @ A ) @ ( lattic2918178447194608042_set_v @ B ) ) ) ) ) ).
% Sup_fin.subset_imp
thf(fact_1061_Sup__fin_Osubset,axiom,
! [A: set_set_v,B: set_set_v] :
( ( finite_finite_set_v @ A )
=> ( ( B != bot_bot_set_set_v )
=> ( ( ord_le5216385588623774835_set_v @ B @ A )
=> ( ( sup_sup_set_v @ ( lattic2918178447194608042_set_v @ B ) @ ( lattic2918178447194608042_set_v @ A ) )
= ( lattic2918178447194608042_set_v @ A ) ) ) ) ) ).
% Sup_fin.subset
thf(fact_1062_Sup__fin_Osubset,axiom,
! [A: set_se8455005133513928103od_v_v,B: set_se8455005133513928103od_v_v] :
( ( finite6084192165098772208od_v_v @ A )
=> ( ( B != bot_bo3497076220358800403od_v_v )
=> ( ( ord_le4714265922333009223od_v_v @ B @ A )
=> ( ( sup_su414716646722978715od_v_v @ ( lattic5151207300795964030od_v_v @ B ) @ ( lattic5151207300795964030od_v_v @ A ) )
= ( lattic5151207300795964030od_v_v @ A ) ) ) ) ) ).
% Sup_fin.subset
thf(fact_1063_Sup__fin_Oinsert__not__elem,axiom,
! [A: set_set_v,X: set_v] :
( ( finite_finite_set_v @ A )
=> ( ~ ( member_set_v @ X @ A )
=> ( ( A != bot_bot_set_set_v )
=> ( ( lattic2918178447194608042_set_v @ ( insert_set_v @ X @ A ) )
= ( sup_sup_set_v @ X @ ( lattic2918178447194608042_set_v @ A ) ) ) ) ) ) ).
% Sup_fin.insert_not_elem
thf(fact_1064_Sup__fin_Oinsert__not__elem,axiom,
! [A: set_se8455005133513928103od_v_v,X: set_Product_prod_v_v] :
( ( finite6084192165098772208od_v_v @ A )
=> ( ~ ( member8406446414694345712od_v_v @ X @ A )
=> ( ( A != bot_bo3497076220358800403od_v_v )
=> ( ( lattic5151207300795964030od_v_v @ ( insert7504383016908236695od_v_v @ X @ A ) )
= ( sup_su414716646722978715od_v_v @ X @ ( lattic5151207300795964030od_v_v @ A ) ) ) ) ) ) ).
% Sup_fin.insert_not_elem
thf(fact_1065_Sup__fin_Oclosed,axiom,
! [A: set_set_v] :
( ( finite_finite_set_v @ A )
=> ( ( A != bot_bot_set_set_v )
=> ( ! [X3: set_v,Y2: set_v] : ( member_set_v @ ( sup_sup_set_v @ X3 @ Y2 ) @ ( insert_set_v @ X3 @ ( insert_set_v @ Y2 @ bot_bot_set_set_v ) ) )
=> ( member_set_v @ ( lattic2918178447194608042_set_v @ A ) @ A ) ) ) ) ).
% Sup_fin.closed
thf(fact_1066_Sup__fin_Oclosed,axiom,
! [A: set_se8455005133513928103od_v_v] :
( ( finite6084192165098772208od_v_v @ A )
=> ( ( A != bot_bo3497076220358800403od_v_v )
=> ( ! [X3: set_Product_prod_v_v,Y2: set_Product_prod_v_v] : ( member8406446414694345712od_v_v @ ( sup_su414716646722978715od_v_v @ X3 @ Y2 ) @ ( insert7504383016908236695od_v_v @ X3 @ ( insert7504383016908236695od_v_v @ Y2 @ bot_bo3497076220358800403od_v_v ) ) )
=> ( member8406446414694345712od_v_v @ ( lattic5151207300795964030od_v_v @ A ) @ A ) ) ) ) ).
% Sup_fin.closed
thf(fact_1067_Sup__fin_Ounion,axiom,
! [A: set_set_v,B: set_set_v] :
( ( finite_finite_set_v @ A )
=> ( ( A != bot_bot_set_set_v )
=> ( ( finite_finite_set_v @ B )
=> ( ( B != bot_bot_set_set_v )
=> ( ( lattic2918178447194608042_set_v @ ( sup_sup_set_set_v @ A @ B ) )
= ( sup_sup_set_v @ ( lattic2918178447194608042_set_v @ A ) @ ( lattic2918178447194608042_set_v @ B ) ) ) ) ) ) ) ).
% Sup_fin.union
thf(fact_1068_Sup__fin_Ounion,axiom,
! [A: set_se8455005133513928103od_v_v,B: set_se8455005133513928103od_v_v] :
( ( finite6084192165098772208od_v_v @ A )
=> ( ( A != bot_bo3497076220358800403od_v_v )
=> ( ( finite6084192165098772208od_v_v @ B )
=> ( ( B != bot_bo3497076220358800403od_v_v )
=> ( ( lattic5151207300795964030od_v_v @ ( sup_su335656005089752955od_v_v @ A @ B ) )
= ( sup_su414716646722978715od_v_v @ ( lattic5151207300795964030od_v_v @ A ) @ ( lattic5151207300795964030od_v_v @ B ) ) ) ) ) ) ) ).
% Sup_fin.union
thf(fact_1069_Inf__fin_Oremove,axiom,
! [A: set_set_v,X: set_v] :
( ( finite_finite_set_v @ A )
=> ( ( member_set_v @ X @ A )
=> ( ( ( ( minus_7228012346218142266_set_v @ A @ ( insert_set_v @ X @ bot_bot_set_set_v ) )
= bot_bot_set_set_v )
=> ( ( lattic8209813555532694032_set_v @ A )
= X ) )
& ( ( ( minus_7228012346218142266_set_v @ A @ ( insert_set_v @ X @ bot_bot_set_set_v ) )
!= bot_bot_set_set_v )
=> ( ( lattic8209813555532694032_set_v @ A )
= ( inf_inf_set_v @ X @ ( lattic8209813555532694032_set_v @ ( minus_7228012346218142266_set_v @ A @ ( insert_set_v @ X @ bot_bot_set_set_v ) ) ) ) ) ) ) ) ) ).
% Inf_fin.remove
thf(fact_1070_Inf__fin_Oinsert__remove,axiom,
! [A: set_set_v,X: set_v] :
( ( finite_finite_set_v @ A )
=> ( ( ( ( minus_7228012346218142266_set_v @ A @ ( insert_set_v @ X @ bot_bot_set_set_v ) )
= bot_bot_set_set_v )
=> ( ( lattic8209813555532694032_set_v @ ( insert_set_v @ X @ A ) )
= X ) )
& ( ( ( minus_7228012346218142266_set_v @ A @ ( insert_set_v @ X @ bot_bot_set_set_v ) )
!= bot_bot_set_set_v )
=> ( ( lattic8209813555532694032_set_v @ ( insert_set_v @ X @ A ) )
= ( inf_inf_set_v @ X @ ( lattic8209813555532694032_set_v @ ( minus_7228012346218142266_set_v @ A @ ( insert_set_v @ X @ bot_bot_set_set_v ) ) ) ) ) ) ) ) ).
% Inf_fin.insert_remove
thf(fact_1071_Inf__fin_Oinsert,axiom,
! [A: set_set_v,X: set_v] :
( ( finite_finite_set_v @ A )
=> ( ( A != bot_bot_set_set_v )
=> ( ( lattic8209813555532694032_set_v @ ( insert_set_v @ X @ A ) )
= ( inf_inf_set_v @ X @ ( lattic8209813555532694032_set_v @ A ) ) ) ) ) ).
% Inf_fin.insert
thf(fact_1072_sup__Inf__absorb,axiom,
! [A: set_set_v,A4: set_v] :
( ( finite_finite_set_v @ A )
=> ( ( member_set_v @ A4 @ A )
=> ( ( sup_sup_set_v @ ( lattic8209813555532694032_set_v @ A ) @ A4 )
= A4 ) ) ) ).
% sup_Inf_absorb
thf(fact_1073_sup__Inf__absorb,axiom,
! [A: set_se8455005133513928103od_v_v,A4: set_Product_prod_v_v] :
( ( finite6084192165098772208od_v_v @ A )
=> ( ( member8406446414694345712od_v_v @ A4 @ A )
=> ( ( sup_su414716646722978715od_v_v @ ( lattic4767070952889939172od_v_v @ A ) @ A4 )
= A4 ) ) ) ).
% sup_Inf_absorb
thf(fact_1074_Inf__fin_OcoboundedI,axiom,
! [A: set_se8455005133513928103od_v_v,A4: set_Product_prod_v_v] :
( ( finite6084192165098772208od_v_v @ A )
=> ( ( member8406446414694345712od_v_v @ A4 @ A )
=> ( ord_le7336532860387713383od_v_v @ ( lattic4767070952889939172od_v_v @ A ) @ A4 ) ) ) ).
% Inf_fin.coboundedI
thf(fact_1075_Inf__fin_OcoboundedI,axiom,
! [A: set_set_v,A4: set_v] :
( ( finite_finite_set_v @ A )
=> ( ( member_set_v @ A4 @ A )
=> ( ord_less_eq_set_v @ ( lattic8209813555532694032_set_v @ A ) @ A4 ) ) ) ).
% Inf_fin.coboundedI
thf(fact_1076_Inf__fin_Obounded__iff,axiom,
! [A: set_se8455005133513928103od_v_v,X: set_Product_prod_v_v] :
( ( finite6084192165098772208od_v_v @ A )
=> ( ( A != bot_bo3497076220358800403od_v_v )
=> ( ( ord_le7336532860387713383od_v_v @ X @ ( lattic4767070952889939172od_v_v @ A ) )
= ( ! [X2: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ X2 @ A )
=> ( ord_le7336532860387713383od_v_v @ X @ X2 ) ) ) ) ) ) ).
% Inf_fin.bounded_iff
thf(fact_1077_Inf__fin_Obounded__iff,axiom,
! [A: set_set_v,X: set_v] :
( ( finite_finite_set_v @ A )
=> ( ( A != bot_bot_set_set_v )
=> ( ( ord_less_eq_set_v @ X @ ( lattic8209813555532694032_set_v @ A ) )
= ( ! [X2: set_v] :
( ( member_set_v @ X2 @ A )
=> ( ord_less_eq_set_v @ X @ X2 ) ) ) ) ) ) ).
% Inf_fin.bounded_iff
thf(fact_1078_Inf__fin_OboundedI,axiom,
! [A: set_se8455005133513928103od_v_v,X: set_Product_prod_v_v] :
( ( finite6084192165098772208od_v_v @ A )
=> ( ( A != bot_bo3497076220358800403od_v_v )
=> ( ! [A9: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ A9 @ A )
=> ( ord_le7336532860387713383od_v_v @ X @ A9 ) )
=> ( ord_le7336532860387713383od_v_v @ X @ ( lattic4767070952889939172od_v_v @ A ) ) ) ) ) ).
% Inf_fin.boundedI
thf(fact_1079_Inf__fin_OboundedI,axiom,
! [A: set_set_v,X: set_v] :
( ( finite_finite_set_v @ A )
=> ( ( A != bot_bot_set_set_v )
=> ( ! [A9: set_v] :
( ( member_set_v @ A9 @ A )
=> ( ord_less_eq_set_v @ X @ A9 ) )
=> ( ord_less_eq_set_v @ X @ ( lattic8209813555532694032_set_v @ A ) ) ) ) ) ).
% Inf_fin.boundedI
thf(fact_1080_Inf__fin_OboundedE,axiom,
! [A: set_se8455005133513928103od_v_v,X: set_Product_prod_v_v] :
( ( finite6084192165098772208od_v_v @ A )
=> ( ( A != bot_bo3497076220358800403od_v_v )
=> ( ( ord_le7336532860387713383od_v_v @ X @ ( lattic4767070952889939172od_v_v @ A ) )
=> ! [A11: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ A11 @ A )
=> ( ord_le7336532860387713383od_v_v @ X @ A11 ) ) ) ) ) ).
% Inf_fin.boundedE
thf(fact_1081_Inf__fin_OboundedE,axiom,
! [A: set_set_v,X: set_v] :
( ( finite_finite_set_v @ A )
=> ( ( A != bot_bot_set_set_v )
=> ( ( ord_less_eq_set_v @ X @ ( lattic8209813555532694032_set_v @ A ) )
=> ! [A11: set_v] :
( ( member_set_v @ A11 @ A )
=> ( ord_less_eq_set_v @ X @ A11 ) ) ) ) ) ).
% Inf_fin.boundedE
thf(fact_1082_Inf__fin_Osubset__imp,axiom,
! [A: set_se8455005133513928103od_v_v,B: set_se8455005133513928103od_v_v] :
( ( ord_le4714265922333009223od_v_v @ A @ B )
=> ( ( A != bot_bo3497076220358800403od_v_v )
=> ( ( finite6084192165098772208od_v_v @ B )
=> ( ord_le7336532860387713383od_v_v @ ( lattic4767070952889939172od_v_v @ B ) @ ( lattic4767070952889939172od_v_v @ A ) ) ) ) ) ).
% Inf_fin.subset_imp
thf(fact_1083_Inf__fin_Osubset__imp,axiom,
! [A: set_set_v,B: set_set_v] :
( ( ord_le5216385588623774835_set_v @ A @ B )
=> ( ( A != bot_bot_set_set_v )
=> ( ( finite_finite_set_v @ B )
=> ( ord_less_eq_set_v @ ( lattic8209813555532694032_set_v @ B ) @ ( lattic8209813555532694032_set_v @ A ) ) ) ) ) ).
% Inf_fin.subset_imp
thf(fact_1084_Inf__fin_Osubset,axiom,
! [A: set_set_v,B: set_set_v] :
( ( finite_finite_set_v @ A )
=> ( ( B != bot_bot_set_set_v )
=> ( ( ord_le5216385588623774835_set_v @ B @ A )
=> ( ( inf_inf_set_v @ ( lattic8209813555532694032_set_v @ B ) @ ( lattic8209813555532694032_set_v @ A ) )
= ( lattic8209813555532694032_set_v @ A ) ) ) ) ) ).
% Inf_fin.subset
thf(fact_1085_Inf__fin_Oinsert__not__elem,axiom,
! [A: set_set_v,X: set_v] :
( ( finite_finite_set_v @ A )
=> ( ~ ( member_set_v @ X @ A )
=> ( ( A != bot_bot_set_set_v )
=> ( ( lattic8209813555532694032_set_v @ ( insert_set_v @ X @ A ) )
= ( inf_inf_set_v @ X @ ( lattic8209813555532694032_set_v @ A ) ) ) ) ) ) ).
% Inf_fin.insert_not_elem
thf(fact_1086_Inf__fin_Oclosed,axiom,
! [A: set_set_v] :
( ( finite_finite_set_v @ A )
=> ( ( A != bot_bot_set_set_v )
=> ( ! [X3: set_v,Y2: set_v] : ( member_set_v @ ( inf_inf_set_v @ X3 @ Y2 ) @ ( insert_set_v @ X3 @ ( insert_set_v @ Y2 @ bot_bot_set_set_v ) ) )
=> ( member_set_v @ ( lattic8209813555532694032_set_v @ A ) @ A ) ) ) ) ).
% Inf_fin.closed
thf(fact_1087_Inf__fin_Ounion,axiom,
! [A: set_set_v,B: set_set_v] :
( ( finite_finite_set_v @ A )
=> ( ( A != bot_bot_set_set_v )
=> ( ( finite_finite_set_v @ B )
=> ( ( B != bot_bot_set_set_v )
=> ( ( lattic8209813555532694032_set_v @ ( sup_sup_set_set_v @ A @ B ) )
= ( inf_inf_set_v @ ( lattic8209813555532694032_set_v @ A ) @ ( lattic8209813555532694032_set_v @ B ) ) ) ) ) ) ) ).
% Inf_fin.union
thf(fact_1088_Inf__fin__le__Sup__fin,axiom,
! [A: set_se8455005133513928103od_v_v] :
( ( finite6084192165098772208od_v_v @ A )
=> ( ( A != bot_bo3497076220358800403od_v_v )
=> ( ord_le7336532860387713383od_v_v @ ( lattic4767070952889939172od_v_v @ A ) @ ( lattic5151207300795964030od_v_v @ A ) ) ) ) ).
% Inf_fin_le_Sup_fin
thf(fact_1089_Inf__fin__le__Sup__fin,axiom,
! [A: set_set_v] :
( ( finite_finite_set_v @ A )
=> ( ( A != bot_bot_set_set_v )
=> ( ord_less_eq_set_v @ ( lattic8209813555532694032_set_v @ A ) @ ( lattic2918178447194608042_set_v @ A ) ) ) ) ).
% Inf_fin_le_Sup_fin
thf(fact_1090_refl__on__singleton,axiom,
! [X: product_prod_v_v] : ( refl_o4548774019903118566od_v_v @ ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v ) @ ( insert5641704497130386615od_v_v @ ( produc4031800376763917143od_v_v @ X @ X ) @ bot_bo3282589961317712691od_v_v ) ) ).
% refl_on_singleton
thf(fact_1091_refl__on__singleton,axiom,
! [X: v] : ( refl_on_v @ ( insert_v2 @ X @ bot_bot_set_v ) @ ( insert1338601472111419319od_v_v @ ( product_Pair_v_v @ X @ X ) @ bot_bo723834152578015283od_v_v ) ) ).
% refl_on_singleton
thf(fact_1092_subseqs_Osimps_I1_J,axiom,
( ( subseqs_v @ nil_v )
= ( cons_list_v @ nil_v @ nil_list_v ) ) ).
% subseqs.simps(1)
thf(fact_1093_snoc__listrel1__snoc__iff,axiom,
! [Xs: list_v,X: v,Ys: list_v,Y: v,R2: set_Product_prod_v_v] :
( ( member418487059593946000list_v @ ( produc6795410681906604247list_v @ ( append_v @ Xs @ ( cons_v @ X @ nil_v ) ) @ ( append_v @ Ys @ ( cons_v @ Y @ nil_v ) ) ) @ ( listrel1_v @ R2 ) )
= ( ( ( member418487059593946000list_v @ ( produc6795410681906604247list_v @ Xs @ Ys ) @ ( listrel1_v @ R2 ) )
& ( X = Y ) )
| ( ( Xs = Ys )
& ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ X @ Y ) @ R2 ) ) ) ) ).
% snoc_listrel1_snoc_iff
thf(fact_1094_Cons__listrel1__Cons,axiom,
! [X: v,Xs: list_v,Y: v,Ys: list_v,R2: set_Product_prod_v_v] :
( ( member418487059593946000list_v @ ( produc6795410681906604247list_v @ ( cons_v @ X @ Xs ) @ ( cons_v @ Y @ Ys ) ) @ ( listrel1_v @ R2 ) )
= ( ( ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ X @ Y ) @ R2 )
& ( Xs = Ys ) )
| ( ( X = Y )
& ( member418487059593946000list_v @ ( produc6795410681906604247list_v @ Xs @ Ys ) @ ( listrel1_v @ R2 ) ) ) ) ) ).
% Cons_listrel1_Cons
thf(fact_1095_listrel1__mono,axiom,
! [R2: set_Product_prod_v_v,S5: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ R2 @ S5 )
=> ( ord_le791731619978752231list_v @ ( listrel1_v @ R2 ) @ ( listrel1_v @ S5 ) ) ) ).
% listrel1_mono
thf(fact_1096_refl__onD2,axiom,
! [A: set_Product_prod_v_v,R2: set_Pr2149350503807050951od_v_v,X: product_prod_v_v,Y: product_prod_v_v] :
( ( refl_o4548774019903118566od_v_v @ A @ R2 )
=> ( ( member3038538357316246288od_v_v @ ( produc4031800376763917143od_v_v @ X @ Y ) @ R2 )
=> ( member7453568604450474000od_v_v @ Y @ A ) ) ) ).
% refl_onD2
thf(fact_1097_refl__onD2,axiom,
! [A: set_v,R2: set_Product_prod_v_v,X: v,Y: v] :
( ( refl_on_v @ A @ R2 )
=> ( ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ X @ Y ) @ R2 )
=> ( member_v @ Y @ A ) ) ) ).
% refl_onD2
thf(fact_1098_refl__onD1,axiom,
! [A: set_Product_prod_v_v,R2: set_Pr2149350503807050951od_v_v,X: product_prod_v_v,Y: product_prod_v_v] :
( ( refl_o4548774019903118566od_v_v @ A @ R2 )
=> ( ( member3038538357316246288od_v_v @ ( produc4031800376763917143od_v_v @ X @ Y ) @ R2 )
=> ( member7453568604450474000od_v_v @ X @ A ) ) ) ).
% refl_onD1
thf(fact_1099_refl__onD1,axiom,
! [A: set_v,R2: set_Product_prod_v_v,X: v,Y: v] :
( ( refl_on_v @ A @ R2 )
=> ( ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ X @ Y ) @ R2 )
=> ( member_v @ X @ A ) ) ) ).
% refl_onD1
thf(fact_1100_refl__onD,axiom,
! [A: set_Product_prod_v_v,R2: set_Pr2149350503807050951od_v_v,A4: product_prod_v_v] :
( ( refl_o4548774019903118566od_v_v @ A @ R2 )
=> ( ( member7453568604450474000od_v_v @ A4 @ A )
=> ( member3038538357316246288od_v_v @ ( produc4031800376763917143od_v_v @ A4 @ A4 ) @ R2 ) ) ) ).
% refl_onD
thf(fact_1101_refl__onD,axiom,
! [A: set_v,R2: set_Product_prod_v_v,A4: v] :
( ( refl_on_v @ A @ R2 )
=> ( ( member_v @ A4 @ A )
=> ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ A4 @ A4 ) @ R2 ) ) ) ).
% refl_onD
thf(fact_1102_refl__on__empty,axiom,
refl_o4548774019903118566od_v_v @ bot_bo723834152578015283od_v_v @ bot_bo3282589961317712691od_v_v ).
% refl_on_empty
thf(fact_1103_refl__on__empty,axiom,
refl_on_v @ bot_bot_set_v @ bot_bo723834152578015283od_v_v ).
% refl_on_empty
thf(fact_1104_not__listrel1__Nil,axiom,
! [Xs: list_v,R2: set_Product_prod_v_v] :
~ ( member418487059593946000list_v @ ( produc6795410681906604247list_v @ Xs @ nil_v ) @ ( listrel1_v @ R2 ) ) ).
% not_listrel1_Nil
thf(fact_1105_not__Nil__listrel1,axiom,
! [Xs: list_v,R2: set_Product_prod_v_v] :
~ ( member418487059593946000list_v @ ( produc6795410681906604247list_v @ nil_v @ Xs ) @ ( listrel1_v @ R2 ) ) ).
% not_Nil_listrel1
thf(fact_1106_refl__on__Un,axiom,
! [A: set_v,R2: set_Product_prod_v_v,B: set_v,S5: set_Product_prod_v_v] :
( ( refl_on_v @ A @ R2 )
=> ( ( refl_on_v @ B @ S5 )
=> ( refl_on_v @ ( sup_sup_set_v @ A @ B ) @ ( sup_su414716646722978715od_v_v @ R2 @ S5 ) ) ) ) ).
% refl_on_Un
thf(fact_1107_refl__on__Un,axiom,
! [A: set_Product_prod_v_v,R2: set_Pr2149350503807050951od_v_v,B: set_Product_prod_v_v,S5: set_Pr2149350503807050951od_v_v] :
( ( refl_o4548774019903118566od_v_v @ A @ R2 )
=> ( ( refl_o4548774019903118566od_v_v @ B @ S5 )
=> ( refl_o4548774019903118566od_v_v @ ( sup_su414716646722978715od_v_v @ A @ B ) @ ( sup_su1742609618068805275od_v_v @ R2 @ S5 ) ) ) ) ).
% refl_on_Un
thf(fact_1108_append__listrel1I,axiom,
! [Xs: list_v,Ys: list_v,R2: set_Product_prod_v_v,Us2: list_v,Vs: list_v] :
( ( ( ( member418487059593946000list_v @ ( produc6795410681906604247list_v @ Xs @ Ys ) @ ( listrel1_v @ R2 ) )
& ( Us2 = Vs ) )
| ( ( Xs = Ys )
& ( member418487059593946000list_v @ ( produc6795410681906604247list_v @ Us2 @ Vs ) @ ( listrel1_v @ R2 ) ) ) )
=> ( member418487059593946000list_v @ ( produc6795410681906604247list_v @ ( append_v @ Xs @ Us2 ) @ ( append_v @ Ys @ Vs ) ) @ ( listrel1_v @ R2 ) ) ) ).
% append_listrel1I
thf(fact_1109_listrel1I1,axiom,
! [X: v,Y: v,R2: set_Product_prod_v_v,Xs: list_v] :
( ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ X @ Y ) @ R2 )
=> ( member418487059593946000list_v @ ( produc6795410681906604247list_v @ ( cons_v @ X @ Xs ) @ ( cons_v @ Y @ Xs ) ) @ ( listrel1_v @ R2 ) ) ) ).
% listrel1I1
thf(fact_1110_Cons__listrel1E1,axiom,
! [X: v,Xs: list_v,Ys: list_v,R2: set_Product_prod_v_v] :
( ( member418487059593946000list_v @ ( produc6795410681906604247list_v @ ( cons_v @ X @ Xs ) @ Ys ) @ ( listrel1_v @ R2 ) )
=> ( ! [Y2: v] :
( ( Ys
= ( cons_v @ Y2 @ Xs ) )
=> ~ ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ X @ Y2 ) @ R2 ) )
=> ~ ! [Zs2: list_v] :
( ( Ys
= ( cons_v @ X @ Zs2 ) )
=> ~ ( member418487059593946000list_v @ ( produc6795410681906604247list_v @ Xs @ Zs2 ) @ ( listrel1_v @ R2 ) ) ) ) ) ).
% Cons_listrel1E1
thf(fact_1111_Cons__listrel1E2,axiom,
! [Xs: list_v,Y: v,Ys: list_v,R2: set_Product_prod_v_v] :
( ( member418487059593946000list_v @ ( produc6795410681906604247list_v @ Xs @ ( cons_v @ Y @ Ys ) ) @ ( listrel1_v @ R2 ) )
=> ( ! [X3: v] :
( ( Xs
= ( cons_v @ X3 @ Ys ) )
=> ~ ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ X3 @ Y ) @ R2 ) )
=> ~ ! [Zs2: list_v] :
( ( Xs
= ( cons_v @ Y @ Zs2 ) )
=> ~ ( member418487059593946000list_v @ ( produc6795410681906604247list_v @ Zs2 @ Ys ) @ ( listrel1_v @ R2 ) ) ) ) ) ).
% Cons_listrel1E2
thf(fact_1112_listrel1E,axiom,
! [Xs: list_v,Ys: list_v,R2: set_Product_prod_v_v] :
( ( member418487059593946000list_v @ ( produc6795410681906604247list_v @ Xs @ Ys ) @ ( listrel1_v @ R2 ) )
=> ~ ! [X3: v,Y2: v] :
( ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ X3 @ Y2 ) @ R2 )
=> ! [Us3: list_v,Vs2: list_v] :
( ( Xs
= ( append_v @ Us3 @ ( cons_v @ X3 @ Vs2 ) ) )
=> ( Ys
!= ( append_v @ Us3 @ ( cons_v @ Y2 @ Vs2 ) ) ) ) ) ) ).
% listrel1E
thf(fact_1113_listrel1I,axiom,
! [X: v,Y: v,R2: set_Product_prod_v_v,Xs: list_v,Us2: list_v,Vs: list_v,Ys: list_v] :
( ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ X @ Y ) @ R2 )
=> ( ( Xs
= ( append_v @ Us2 @ ( cons_v @ X @ Vs ) ) )
=> ( ( Ys
= ( append_v @ Us2 @ ( cons_v @ Y @ Vs ) ) )
=> ( member418487059593946000list_v @ ( produc6795410681906604247list_v @ Xs @ Ys ) @ ( listrel1_v @ R2 ) ) ) ) ) ).
% listrel1I
thf(fact_1114_refl__on__domain,axiom,
! [A: set_Product_prod_v_v,R2: set_Pr2149350503807050951od_v_v,A4: product_prod_v_v,B2: product_prod_v_v] :
( ( refl_o4548774019903118566od_v_v @ A @ R2 )
=> ( ( member3038538357316246288od_v_v @ ( produc4031800376763917143od_v_v @ A4 @ B2 ) @ R2 )
=> ( ( member7453568604450474000od_v_v @ A4 @ A )
& ( member7453568604450474000od_v_v @ B2 @ A ) ) ) ) ).
% refl_on_domain
thf(fact_1115_refl__on__domain,axiom,
! [A: set_v,R2: set_Product_prod_v_v,A4: v,B2: v] :
( ( refl_on_v @ A @ R2 )
=> ( ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ A4 @ B2 ) @ R2 )
=> ( ( member_v @ A4 @ A )
& ( member_v @ B2 @ A ) ) ) ) ).
% refl_on_domain
thf(fact_1116_partition__set,axiom,
! [P: v > $o,Xs: list_v,Yes: list_v,No: list_v] :
( ( ( partition_v @ P @ Xs )
= ( produc6795410681906604247list_v @ Yes @ No ) )
=> ( ( sup_sup_set_v @ ( set_v2 @ Yes ) @ ( set_v2 @ No ) )
= ( set_v2 @ Xs ) ) ) ).
% partition_set
thf(fact_1117_partition__set,axiom,
! [P: product_prod_v_v > $o,Xs: list_P7986770385144383213od_v_v,Yes: list_P7986770385144383213od_v_v,No: list_P7986770385144383213od_v_v] :
( ( ( partit5288610572509583718od_v_v @ P @ Xs )
= ( produc674067373767953879od_v_v @ Yes @ No ) )
=> ( ( sup_su414716646722978715od_v_v @ ( set_Product_prod_v_v2 @ Yes ) @ ( set_Product_prod_v_v2 @ No ) )
= ( set_Product_prod_v_v2 @ Xs ) ) ) ).
% partition_set
thf(fact_1118_partition_Osimps_I1_J,axiom,
! [P: v > $o] :
( ( partition_v @ P @ nil_v )
= ( produc6795410681906604247list_v @ nil_v @ nil_v ) ) ).
% partition.simps(1)
thf(fact_1119_partition__P,axiom,
! [P: v > $o,Xs: list_v,Yes: list_v,No: list_v] :
( ( ( partition_v @ P @ Xs )
= ( produc6795410681906604247list_v @ Yes @ No ) )
=> ( ! [X4: v] :
( ( member_v @ X4 @ ( set_v2 @ Yes ) )
=> ( P @ X4 ) )
& ! [X4: v] :
( ( member_v @ X4 @ ( set_v2 @ No ) )
=> ~ ( P @ X4 ) ) ) ) ).
% partition_P
thf(fact_1120_linear__order__on__singleton,axiom,
! [X: product_prod_v_v] : ( order_6462556390437124636od_v_v @ ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v ) @ ( insert5641704497130386615od_v_v @ ( produc4031800376763917143od_v_v @ X @ X ) @ bot_bo3282589961317712691od_v_v ) ) ).
% linear_order_on_singleton
thf(fact_1121_linear__order__on__singleton,axiom,
! [X: v] : ( order_8768733634509060168r_on_v @ ( insert_v2 @ X @ bot_bot_set_v ) @ ( insert1338601472111419319od_v_v @ ( product_Pair_v_v @ X @ X ) @ bot_bo723834152578015283od_v_v ) ) ).
% linear_order_on_singleton
thf(fact_1122_lnear__order__on__empty,axiom,
order_6462556390437124636od_v_v @ bot_bo723834152578015283od_v_v @ bot_bo3282589961317712691od_v_v ).
% lnear_order_on_empty
thf(fact_1123_lnear__order__on__empty,axiom,
order_8768733634509060168r_on_v @ bot_bot_set_v @ bot_bo723834152578015283od_v_v ).
% lnear_order_on_empty
thf(fact_1124_in__set__remove1,axiom,
! [A4: product_prod_v_v,B2: product_prod_v_v,Xs: list_P7986770385144383213od_v_v] :
( ( A4 != B2 )
=> ( ( member7453568604450474000od_v_v @ A4 @ ( set_Product_prod_v_v2 @ ( remove333779696311199107od_v_v @ B2 @ Xs ) ) )
= ( member7453568604450474000od_v_v @ A4 @ ( set_Product_prod_v_v2 @ Xs ) ) ) ) ).
% in_set_remove1
thf(fact_1125_in__set__remove1,axiom,
! [A4: v,B2: v,Xs: list_v] :
( ( A4 != B2 )
=> ( ( member_v @ A4 @ ( set_v2 @ ( remove1_v @ B2 @ Xs ) ) )
= ( member_v @ A4 @ ( set_v2 @ Xs ) ) ) ) ).
% in_set_remove1
thf(fact_1126_remove1_Osimps_I1_J,axiom,
! [X: v] :
( ( remove1_v @ X @ nil_v )
= nil_v ) ).
% remove1.simps(1)
thf(fact_1127_notin__set__remove1,axiom,
! [X: product_prod_v_v,Xs: list_P7986770385144383213od_v_v,Y: product_prod_v_v] :
( ~ ( member7453568604450474000od_v_v @ X @ ( set_Product_prod_v_v2 @ Xs ) )
=> ~ ( member7453568604450474000od_v_v @ X @ ( set_Product_prod_v_v2 @ ( remove333779696311199107od_v_v @ Y @ Xs ) ) ) ) ).
% notin_set_remove1
thf(fact_1128_notin__set__remove1,axiom,
! [X: v,Xs: list_v,Y: v] :
( ~ ( member_v @ X @ ( set_v2 @ Xs ) )
=> ~ ( member_v @ X @ ( set_v2 @ ( remove1_v @ Y @ Xs ) ) ) ) ).
% notin_set_remove1
thf(fact_1129_remove1__idem,axiom,
! [X: product_prod_v_v,Xs: list_P7986770385144383213od_v_v] :
( ~ ( member7453568604450474000od_v_v @ X @ ( set_Product_prod_v_v2 @ Xs ) )
=> ( ( remove333779696311199107od_v_v @ X @ Xs )
= Xs ) ) ).
% remove1_idem
thf(fact_1130_remove1__idem,axiom,
! [X: v,Xs: list_v] :
( ~ ( member_v @ X @ ( set_v2 @ Xs ) )
=> ( ( remove1_v @ X @ Xs )
= Xs ) ) ).
% remove1_idem
thf(fact_1131_set__remove1__subset,axiom,
! [X: product_prod_v_v,Xs: list_P7986770385144383213od_v_v] : ( ord_le7336532860387713383od_v_v @ ( set_Product_prod_v_v2 @ ( remove333779696311199107od_v_v @ X @ Xs ) ) @ ( set_Product_prod_v_v2 @ Xs ) ) ).
% set_remove1_subset
thf(fact_1132_set__remove1__subset,axiom,
! [X: v,Xs: list_v] : ( ord_less_eq_set_v @ ( set_v2 @ ( remove1_v @ X @ Xs ) ) @ ( set_v2 @ Xs ) ) ).
% set_remove1_subset
thf(fact_1133_remove1__append,axiom,
! [X: product_prod_v_v,Xs: list_P7986770385144383213od_v_v,Ys: list_P7986770385144383213od_v_v] :
( ( ( member7453568604450474000od_v_v @ X @ ( set_Product_prod_v_v2 @ Xs ) )
=> ( ( remove333779696311199107od_v_v @ X @ ( append2138873909117096322od_v_v @ Xs @ Ys ) )
= ( append2138873909117096322od_v_v @ ( remove333779696311199107od_v_v @ X @ Xs ) @ Ys ) ) )
& ( ~ ( member7453568604450474000od_v_v @ X @ ( set_Product_prod_v_v2 @ Xs ) )
=> ( ( remove333779696311199107od_v_v @ X @ ( append2138873909117096322od_v_v @ Xs @ Ys ) )
= ( append2138873909117096322od_v_v @ Xs @ ( remove333779696311199107od_v_v @ X @ Ys ) ) ) ) ) ).
% remove1_append
thf(fact_1134_remove1__append,axiom,
! [X: v,Xs: list_v,Ys: list_v] :
( ( ( member_v @ X @ ( set_v2 @ Xs ) )
=> ( ( remove1_v @ X @ ( append_v @ Xs @ Ys ) )
= ( append_v @ ( remove1_v @ X @ Xs ) @ Ys ) ) )
& ( ~ ( member_v @ X @ ( set_v2 @ Xs ) )
=> ( ( remove1_v @ X @ ( append_v @ Xs @ Ys ) )
= ( append_v @ Xs @ ( remove1_v @ X @ Ys ) ) ) ) ) ).
% remove1_append
thf(fact_1135_remove1__split,axiom,
! [A4: product_prod_v_v,Xs: list_P7986770385144383213od_v_v,Ys: list_P7986770385144383213od_v_v] :
( ( member7453568604450474000od_v_v @ A4 @ ( set_Product_prod_v_v2 @ Xs ) )
=> ( ( ( remove333779696311199107od_v_v @ A4 @ Xs )
= Ys )
= ( ? [Ls: list_P7986770385144383213od_v_v,Rs: list_P7986770385144383213od_v_v] :
( ( Xs
= ( append2138873909117096322od_v_v @ Ls @ ( cons_P4120604216776828829od_v_v @ A4 @ Rs ) ) )
& ~ ( member7453568604450474000od_v_v @ A4 @ ( set_Product_prod_v_v2 @ Ls ) )
& ( Ys
= ( append2138873909117096322od_v_v @ Ls @ Rs ) ) ) ) ) ) ).
% remove1_split
thf(fact_1136_remove1__split,axiom,
! [A4: v,Xs: list_v,Ys: list_v] :
( ( member_v @ A4 @ ( set_v2 @ Xs ) )
=> ( ( ( remove1_v @ A4 @ Xs )
= Ys )
= ( ? [Ls: list_v,Rs: list_v] :
( ( Xs
= ( append_v @ Ls @ ( cons_v @ A4 @ Rs ) ) )
& ~ ( member_v @ A4 @ ( set_v2 @ Ls ) )
& ( Ys
= ( append_v @ Ls @ Rs ) ) ) ) ) ) ).
% remove1_split
thf(fact_1137_set__remove1__eq,axiom,
! [Xs: list_P7986770385144383213od_v_v,X: product_prod_v_v] :
( ( distin6159370996967099744od_v_v @ Xs )
=> ( ( set_Product_prod_v_v2 @ ( remove333779696311199107od_v_v @ X @ Xs ) )
= ( minus_4183494784930505774od_v_v @ ( set_Product_prod_v_v2 @ Xs ) @ ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v ) ) ) ) ).
% set_remove1_eq
thf(fact_1138_set__remove1__eq,axiom,
! [Xs: list_v,X: v] :
( ( distinct_v @ Xs )
=> ( ( set_v2 @ ( remove1_v @ X @ Xs ) )
= ( minus_minus_set_v @ ( set_v2 @ Xs ) @ ( insert_v2 @ X @ bot_bot_set_v ) ) ) ) ).
% set_remove1_eq
thf(fact_1139_subset__code_I3_J,axiom,
~ ( ord_le7336532860387713383od_v_v @ ( coset_766761627116920666od_v_v @ nil_Product_prod_v_v ) @ ( set_Product_prod_v_v2 @ nil_Product_prod_v_v ) ) ).
% subset_code(3)
thf(fact_1140_subset__code_I3_J,axiom,
~ ( ord_less_eq_set_v @ ( coset_v @ nil_v ) @ ( set_v2 @ nil_v ) ) ).
% subset_code(3)
thf(fact_1141_distinct__append,axiom,
! [Xs: list_P7986770385144383213od_v_v,Ys: list_P7986770385144383213od_v_v] :
( ( distin6159370996967099744od_v_v @ ( append2138873909117096322od_v_v @ Xs @ Ys ) )
= ( ( distin6159370996967099744od_v_v @ Xs )
& ( distin6159370996967099744od_v_v @ Ys )
& ( ( inf_in6271465464967711157od_v_v @ ( set_Product_prod_v_v2 @ Xs ) @ ( set_Product_prod_v_v2 @ Ys ) )
= bot_bo723834152578015283od_v_v ) ) ) ).
% distinct_append
thf(fact_1142_distinct__append,axiom,
! [Xs: list_v,Ys: list_v] :
( ( distinct_v @ ( append_v @ Xs @ Ys ) )
= ( ( distinct_v @ Xs )
& ( distinct_v @ Ys )
& ( ( inf_inf_set_v @ ( set_v2 @ Xs ) @ ( set_v2 @ Ys ) )
= bot_bot_set_v ) ) ) ).
% distinct_append
thf(fact_1143_finite__distinct__list,axiom,
! [A: set_v] :
( ( finite_finite_v @ A )
=> ? [Xs2: list_v] :
( ( ( set_v2 @ Xs2 )
= A )
& ( distinct_v @ Xs2 ) ) ) ).
% finite_distinct_list
thf(fact_1144_distinct_Osimps_I2_J,axiom,
! [X: product_prod_v_v,Xs: list_P7986770385144383213od_v_v] :
( ( distin6159370996967099744od_v_v @ ( cons_P4120604216776828829od_v_v @ X @ Xs ) )
= ( ~ ( member7453568604450474000od_v_v @ X @ ( set_Product_prod_v_v2 @ Xs ) )
& ( distin6159370996967099744od_v_v @ Xs ) ) ) ).
% distinct.simps(2)
thf(fact_1145_distinct_Osimps_I2_J,axiom,
! [X: v,Xs: list_v] :
( ( distinct_v @ ( cons_v @ X @ Xs ) )
= ( ~ ( member_v @ X @ ( set_v2 @ Xs ) )
& ( distinct_v @ Xs ) ) ) ).
% distinct.simps(2)
thf(fact_1146_distinct_Osimps_I1_J,axiom,
distinct_v @ nil_v ).
% distinct.simps(1)
thf(fact_1147_distinct__tl,axiom,
! [Xs: list_v] :
( ( distinct_v @ Xs )
=> ( distinct_v @ ( tl_v @ Xs ) ) ) ).
% distinct_tl
thf(fact_1148_precedes__antisym,axiom,
! [X: v,Y: v,Xs: list_v] :
( ( sCC_Bl4022239298816431255edes_v @ X @ Y @ Xs )
=> ( ( sCC_Bl4022239298816431255edes_v @ Y @ X @ Xs )
=> ( ( distinct_v @ Xs )
=> ( X = Y ) ) ) ) ).
% precedes_antisym
thf(fact_1149_precedes__trans,axiom,
! [X: v,Y: v,Xs: list_v,Z: v] :
( ( sCC_Bl4022239298816431255edes_v @ X @ Y @ Xs )
=> ( ( sCC_Bl4022239298816431255edes_v @ Y @ Z @ Xs )
=> ( ( distinct_v @ Xs )
=> ( sCC_Bl4022239298816431255edes_v @ X @ Z @ Xs ) ) ) ) ).
% precedes_trans
thf(fact_1150_distinct__singleton,axiom,
! [X: v] : ( distinct_v @ ( cons_v @ X @ nil_v ) ) ).
% distinct_singleton
thf(fact_1151_not__distinct__decomp,axiom,
! [Ws: list_v] :
( ~ ( distinct_v @ Ws )
=> ? [Xs2: list_v,Ys2: list_v,Zs2: list_v,Y2: v] :
( Ws
= ( append_v @ Xs2 @ ( append_v @ ( cons_v @ Y2 @ nil_v ) @ ( append_v @ Ys2 @ ( append_v @ ( cons_v @ Y2 @ nil_v ) @ Zs2 ) ) ) ) ) ) ).
% not_distinct_decomp
thf(fact_1152_not__distinct__conv__prefix,axiom,
! [As: list_P7986770385144383213od_v_v] :
( ( ~ ( distin6159370996967099744od_v_v @ As ) )
= ( ? [Xs4: list_P7986770385144383213od_v_v,Y3: product_prod_v_v,Ys4: list_P7986770385144383213od_v_v] :
( ( member7453568604450474000od_v_v @ Y3 @ ( set_Product_prod_v_v2 @ Xs4 ) )
& ( distin6159370996967099744od_v_v @ Xs4 )
& ( As
= ( append2138873909117096322od_v_v @ Xs4 @ ( cons_P4120604216776828829od_v_v @ Y3 @ Ys4 ) ) ) ) ) ) ).
% not_distinct_conv_prefix
thf(fact_1153_not__distinct__conv__prefix,axiom,
! [As: list_v] :
( ( ~ ( distinct_v @ As ) )
= ( ? [Xs4: list_v,Y3: v,Ys4: list_v] :
( ( member_v @ Y3 @ ( set_v2 @ Xs4 ) )
& ( distinct_v @ Xs4 )
& ( As
= ( append_v @ Xs4 @ ( cons_v @ Y3 @ Ys4 ) ) ) ) ) ) ).
% not_distinct_conv_prefix
thf(fact_1154_subset__code_I2_J,axiom,
! [A: set_Product_prod_v_v,Ys: list_P7986770385144383213od_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ ( coset_766761627116920666od_v_v @ Ys ) )
= ( ! [X2: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X2 @ ( set_Product_prod_v_v2 @ Ys ) )
=> ~ ( member7453568604450474000od_v_v @ X2 @ A ) ) ) ) ).
% subset_code(2)
thf(fact_1155_subset__code_I2_J,axiom,
! [A: set_v,Ys: list_v] :
( ( ord_less_eq_set_v @ A @ ( coset_v @ Ys ) )
= ( ! [X2: v] :
( ( member_v @ X2 @ ( set_v2 @ Ys ) )
=> ~ ( member_v @ X2 @ A ) ) ) ) ).
% subset_code(2)
thf(fact_1156_insert__code_I2_J,axiom,
! [X: product_prod_v_v,Xs: list_P7986770385144383213od_v_v] :
( ( insert1338601472111419319od_v_v @ X @ ( coset_766761627116920666od_v_v @ Xs ) )
= ( coset_766761627116920666od_v_v @ ( remove481895986417801203od_v_v @ X @ Xs ) ) ) ).
% insert_code(2)
thf(fact_1157_insert__code_I2_J,axiom,
! [X: v,Xs: list_v] :
( ( insert_v2 @ X @ ( coset_v @ Xs ) )
= ( coset_v @ ( removeAll_v @ X @ Xs ) ) ) ).
% insert_code(2)
thf(fact_1158_distinct__concat__iff,axiom,
! [Xs: list_l4795378083388841843od_v_v] :
( ( distin6159370996967099744od_v_v @ ( concat2875663619778446888od_v_v @ Xs ) )
= ( ( distin913317783593574886od_v_v @ ( remove5095778601549809401od_v_v @ nil_Product_prod_v_v @ Xs ) )
& ! [Ys4: list_P7986770385144383213od_v_v] :
( ( member4190458934886417558od_v_v @ Ys4 @ ( set_li2340707408155270402od_v_v @ Xs ) )
=> ( distin6159370996967099744od_v_v @ Ys4 ) )
& ! [Ys4: list_P7986770385144383213od_v_v,Zs3: list_P7986770385144383213od_v_v] :
( ( ( member4190458934886417558od_v_v @ Ys4 @ ( set_li2340707408155270402od_v_v @ Xs ) )
& ( member4190458934886417558od_v_v @ Zs3 @ ( set_li2340707408155270402od_v_v @ Xs ) )
& ( Ys4 != Zs3 ) )
=> ( ( inf_in6271465464967711157od_v_v @ ( set_Product_prod_v_v2 @ Ys4 ) @ ( set_Product_prod_v_v2 @ Zs3 ) )
= bot_bo723834152578015283od_v_v ) ) ) ) ).
% distinct_concat_iff
thf(fact_1159_distinct__concat__iff,axiom,
! [Xs: list_list_v] :
( ( distinct_v @ ( concat_v @ Xs ) )
= ( ( distinct_list_v @ ( removeAll_list_v @ nil_v @ Xs ) )
& ! [Ys4: list_v] :
( ( member_list_v @ Ys4 @ ( set_list_v2 @ Xs ) )
=> ( distinct_v @ Ys4 ) )
& ! [Ys4: list_v,Zs3: list_v] :
( ( ( member_list_v @ Ys4 @ ( set_list_v2 @ Xs ) )
& ( member_list_v @ Zs3 @ ( set_list_v2 @ Xs ) )
& ( Ys4 != Zs3 ) )
=> ( ( inf_inf_set_v @ ( set_v2 @ Ys4 ) @ ( set_v2 @ Zs3 ) )
= bot_bot_set_v ) ) ) ) ).
% distinct_concat_iff
thf(fact_1160_distinct__concat,axiom,
! [Xs: list_l4795378083388841843od_v_v] :
( ( distin913317783593574886od_v_v @ Xs )
=> ( ! [Ys2: list_P7986770385144383213od_v_v] :
( ( member4190458934886417558od_v_v @ Ys2 @ ( set_li2340707408155270402od_v_v @ Xs ) )
=> ( distin6159370996967099744od_v_v @ Ys2 ) )
=> ( ! [Ys2: list_P7986770385144383213od_v_v,Zs2: list_P7986770385144383213od_v_v] :
( ( member4190458934886417558od_v_v @ Ys2 @ ( set_li2340707408155270402od_v_v @ Xs ) )
=> ( ( member4190458934886417558od_v_v @ Zs2 @ ( set_li2340707408155270402od_v_v @ Xs ) )
=> ( ( Ys2 != Zs2 )
=> ( ( inf_in6271465464967711157od_v_v @ ( set_Product_prod_v_v2 @ Ys2 ) @ ( set_Product_prod_v_v2 @ Zs2 ) )
= bot_bo723834152578015283od_v_v ) ) ) )
=> ( distin6159370996967099744od_v_v @ ( concat2875663619778446888od_v_v @ Xs ) ) ) ) ) ).
% distinct_concat
thf(fact_1161_distinct__concat,axiom,
! [Xs: list_list_v] :
( ( distinct_list_v @ Xs )
=> ( ! [Ys2: list_v] :
( ( member_list_v @ Ys2 @ ( set_list_v2 @ Xs ) )
=> ( distinct_v @ Ys2 ) )
=> ( ! [Ys2: list_v,Zs2: list_v] :
( ( member_list_v @ Ys2 @ ( set_list_v2 @ Xs ) )
=> ( ( member_list_v @ Zs2 @ ( set_list_v2 @ Xs ) )
=> ( ( Ys2 != Zs2 )
=> ( ( inf_inf_set_v @ ( set_v2 @ Ys2 ) @ ( set_v2 @ Zs2 ) )
= bot_bot_set_v ) ) ) )
=> ( distinct_v @ ( concat_v @ Xs ) ) ) ) ) ).
% distinct_concat
thf(fact_1162_concat__append,axiom,
! [Xs: list_list_v,Ys: list_list_v] :
( ( concat_v @ ( append_list_v @ Xs @ Ys ) )
= ( append_v @ ( concat_v @ Xs ) @ ( concat_v @ Ys ) ) ) ).
% concat_append
thf(fact_1163_concat__eq__Nil__conv,axiom,
! [Xss2: list_list_v] :
( ( ( concat_v @ Xss2 )
= nil_v )
= ( ! [X2: list_v] :
( ( member_list_v @ X2 @ ( set_list_v2 @ Xss2 ) )
=> ( X2 = nil_v ) ) ) ) ).
% concat_eq_Nil_conv
thf(fact_1164_Nil__eq__concat__conv,axiom,
! [Xss2: list_list_v] :
( ( nil_v
= ( concat_v @ Xss2 ) )
= ( ! [X2: list_v] :
( ( member_list_v @ X2 @ ( set_list_v2 @ Xss2 ) )
=> ( X2 = nil_v ) ) ) ) ).
% Nil_eq_concat_conv
thf(fact_1165_concat_Osimps_I1_J,axiom,
( ( concat_v @ nil_list_v )
= nil_v ) ).
% concat.simps(1)
thf(fact_1166_concat_Osimps_I2_J,axiom,
! [X: list_v,Xs: list_list_v] :
( ( concat_v @ ( cons_list_v @ X @ Xs ) )
= ( append_v @ X @ ( concat_v @ Xs ) ) ) ).
% concat.simps(2)
thf(fact_1167_hd__concat,axiom,
! [Xs: list_list_v] :
( ( Xs != nil_list_v )
=> ( ( ( hd_list_v @ Xs )
!= nil_v )
=> ( ( hd_v @ ( concat_v @ Xs ) )
= ( hd_v @ ( hd_list_v @ Xs ) ) ) ) ) ).
% hd_concat
thf(fact_1168_concat__eq__appendD,axiom,
! [Xss2: list_list_v,Ys: list_v,Zs: list_v] :
( ( ( concat_v @ Xss2 )
= ( append_v @ Ys @ Zs ) )
=> ( ( Xss2 != nil_list_v )
=> ? [Xss1: list_list_v,Xs2: list_v,Xs5: list_v,Xss22: list_list_v] :
( ( Xss2
= ( append_list_v @ Xss1 @ ( cons_list_v @ ( append_v @ Xs2 @ Xs5 ) @ Xss22 ) ) )
& ( Ys
= ( append_v @ ( concat_v @ Xss1 ) @ Xs2 ) )
& ( Zs
= ( append_v @ Xs5 @ ( concat_v @ Xss22 ) ) ) ) ) ) ).
% concat_eq_appendD
thf(fact_1169_concat__eq__append__conv,axiom,
! [Xss2: list_list_v,Ys: list_v,Zs: list_v] :
( ( ( concat_v @ Xss2 )
= ( append_v @ Ys @ Zs ) )
= ( ( ( Xss2 = nil_list_v )
=> ( ( Ys = nil_v )
& ( Zs = nil_v ) ) )
& ( ( Xss2 != nil_list_v )
=> ? [Xss12: list_list_v,Xs4: list_v,Xs6: list_v,Xss23: list_list_v] :
( ( Xss2
= ( append_list_v @ Xss12 @ ( cons_list_v @ ( append_v @ Xs4 @ Xs6 ) @ Xss23 ) ) )
& ( Ys
= ( append_v @ ( concat_v @ Xss12 ) @ Xs4 ) )
& ( Zs
= ( append_v @ Xs6 @ ( concat_v @ Xss23 ) ) ) ) ) ) ) ).
% concat_eq_append_conv
thf(fact_1170_List_Oset__insert,axiom,
! [X: product_prod_v_v,Xs: list_P7986770385144383213od_v_v] :
( ( set_Product_prod_v_v2 @ ( insert4539780211034306307od_v_v @ X @ Xs ) )
= ( insert1338601472111419319od_v_v @ X @ ( set_Product_prod_v_v2 @ Xs ) ) ) ).
% List.set_insert
thf(fact_1171_List_Oset__insert,axiom,
! [X: v,Xs: list_v] :
( ( set_v2 @ ( insert_v @ X @ Xs ) )
= ( insert_v2 @ X @ ( set_v2 @ Xs ) ) ) ).
% List.set_insert
thf(fact_1172_in__set__insert,axiom,
! [X: product_prod_v_v,Xs: list_P7986770385144383213od_v_v] :
( ( member7453568604450474000od_v_v @ X @ ( set_Product_prod_v_v2 @ Xs ) )
=> ( ( insert4539780211034306307od_v_v @ X @ Xs )
= Xs ) ) ).
% in_set_insert
thf(fact_1173_in__set__insert,axiom,
! [X: v,Xs: list_v] :
( ( member_v @ X @ ( set_v2 @ Xs ) )
=> ( ( insert_v @ X @ Xs )
= Xs ) ) ).
% in_set_insert
thf(fact_1174_insert__Nil,axiom,
! [X: v] :
( ( insert_v @ X @ nil_v )
= ( cons_v @ X @ nil_v ) ) ).
% insert_Nil
thf(fact_1175_not__in__set__insert,axiom,
! [X: product_prod_v_v,Xs: list_P7986770385144383213od_v_v] :
( ~ ( member7453568604450474000od_v_v @ X @ ( set_Product_prod_v_v2 @ Xs ) )
=> ( ( insert4539780211034306307od_v_v @ X @ Xs )
= ( cons_P4120604216776828829od_v_v @ X @ Xs ) ) ) ).
% not_in_set_insert
thf(fact_1176_not__in__set__insert,axiom,
! [X: v,Xs: list_v] :
( ~ ( member_v @ X @ ( set_v2 @ Xs ) )
=> ( ( insert_v @ X @ Xs )
= ( cons_v @ X @ Xs ) ) ) ).
% not_in_set_insert
thf(fact_1177_maps__simps_I2_J,axiom,
! [F: v > list_v] :
( ( maps_v_v @ F @ nil_v )
= nil_v ) ).
% maps_simps(2)
thf(fact_1178_List_Oinsert__def,axiom,
( insert4539780211034306307od_v_v
= ( ^ [X2: product_prod_v_v,Xs4: list_P7986770385144383213od_v_v] : ( if_lis7521272669439687347od_v_v @ ( member7453568604450474000od_v_v @ X2 @ ( set_Product_prod_v_v2 @ Xs4 ) ) @ Xs4 @ ( cons_P4120604216776828829od_v_v @ X2 @ Xs4 ) ) ) ) ).
% List.insert_def
thf(fact_1179_List_Oinsert__def,axiom,
( insert_v
= ( ^ [X2: v,Xs4: list_v] : ( if_list_v @ ( member_v @ X2 @ ( set_v2 @ Xs4 ) ) @ Xs4 @ ( cons_v @ X2 @ Xs4 ) ) ) ) ).
% List.insert_def
thf(fact_1180_Set_Ois__empty__def,axiom,
( is_emp8964507351669718201od_v_v
= ( ^ [A6: set_Product_prod_v_v] : ( A6 = bot_bo723834152578015283od_v_v ) ) ) ).
% Set.is_empty_def
thf(fact_1181_Set_Ois__empty__def,axiom,
( is_empty_v
= ( ^ [A6: set_v] : ( A6 = bot_bot_set_v ) ) ) ).
% Set.is_empty_def
thf(fact_1182_Linear__order__Well__order__iff,axiom,
! [R2: set_Pr2149350503807050951od_v_v] :
( ( order_6462556390437124636od_v_v @ ( field_7153129647634986036od_v_v @ R2 ) @ R2 )
=> ( ( order_7541072052284126853od_v_v @ ( field_7153129647634986036od_v_v @ R2 ) @ R2 )
= ( ! [A6: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A6 @ ( field_7153129647634986036od_v_v @ R2 ) )
=> ( ( A6 != bot_bo723834152578015283od_v_v )
=> ? [X2: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X2 @ A6 )
& ! [Y3: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ Y3 @ A6 )
=> ( member3038538357316246288od_v_v @ ( produc4031800376763917143od_v_v @ X2 @ Y3 ) @ R2 ) ) ) ) ) ) ) ) ).
% Linear_order_Well_order_iff
thf(fact_1183_Linear__order__Well__order__iff,axiom,
! [R2: set_Product_prod_v_v] :
( ( order_8768733634509060168r_on_v @ ( field_v @ R2 ) @ R2 )
=> ( ( order_6972113574731384241r_on_v @ ( field_v @ R2 ) @ R2 )
= ( ! [A6: set_v] :
( ( ord_less_eq_set_v @ A6 @ ( field_v @ R2 ) )
=> ( ( A6 != bot_bot_set_v )
=> ? [X2: v] :
( ( member_v @ X2 @ A6 )
& ! [Y3: v] :
( ( member_v @ Y3 @ A6 )
=> ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ X2 @ Y3 ) @ R2 ) ) ) ) ) ) ) ) ).
% Linear_order_Well_order_iff
thf(fact_1184_well__order__on__empty,axiom,
order_7541072052284126853od_v_v @ bot_bo723834152578015283od_v_v @ bot_bo3282589961317712691od_v_v ).
% well_order_on_empty
thf(fact_1185_well__order__on__empty,axiom,
order_6972113574731384241r_on_v @ bot_bot_set_v @ bot_bo723834152578015283od_v_v ).
% well_order_on_empty
thf(fact_1186_well__order__on__domain,axiom,
! [A: set_Product_prod_v_v,R2: set_Pr2149350503807050951od_v_v,A4: product_prod_v_v,B2: product_prod_v_v] :
( ( order_7541072052284126853od_v_v @ A @ R2 )
=> ( ( member3038538357316246288od_v_v @ ( produc4031800376763917143od_v_v @ A4 @ B2 ) @ R2 )
=> ( ( member7453568604450474000od_v_v @ A4 @ A )
& ( member7453568604450474000od_v_v @ B2 @ A ) ) ) ) ).
% well_order_on_domain
thf(fact_1187_well__order__on__domain,axiom,
! [A: set_v,R2: set_Product_prod_v_v,A4: v,B2: v] :
( ( order_6972113574731384241r_on_v @ A @ R2 )
=> ( ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ A4 @ B2 ) @ R2 )
=> ( ( member_v @ A4 @ A )
& ( member_v @ B2 @ A ) ) ) ) ).
% well_order_on_domain
thf(fact_1188_is__empty__set,axiom,
! [Xs: list_v] :
( ( is_empty_v @ ( set_v2 @ Xs ) )
= ( null_v @ Xs ) ) ).
% is_empty_set
thf(fact_1189_min__bot,axiom,
! [X: set_Product_prod_v_v] :
( ( ord_mi6996445931809003310od_v_v @ bot_bo723834152578015283od_v_v @ X )
= bot_bo723834152578015283od_v_v ) ).
% min_bot
thf(fact_1190_min__bot,axiom,
! [X: set_v] :
( ( ord_min_set_v @ bot_bot_set_v @ X )
= bot_bot_set_v ) ).
% min_bot
thf(fact_1191_min__bot2,axiom,
! [X: set_Product_prod_v_v] :
( ( ord_mi6996445931809003310od_v_v @ X @ bot_bo723834152578015283od_v_v )
= bot_bo723834152578015283od_v_v ) ).
% min_bot2
thf(fact_1192_min__bot2,axiom,
! [X: set_v] :
( ( ord_min_set_v @ X @ bot_bot_set_v )
= bot_bot_set_v ) ).
% min_bot2
thf(fact_1193_min__def,axiom,
( ord_mi6996445931809003310od_v_v
= ( ^ [A5: set_Product_prod_v_v,B3: set_Product_prod_v_v] : ( if_set4279007504652509325od_v_v @ ( ord_le7336532860387713383od_v_v @ A5 @ B3 ) @ A5 @ B3 ) ) ) ).
% min_def
thf(fact_1194_min__def,axiom,
( ord_min_set_v
= ( ^ [A5: set_v,B3: set_v] : ( if_set_v @ ( ord_less_eq_set_v @ A5 @ B3 ) @ A5 @ B3 ) ) ) ).
% min_def
thf(fact_1195_min__absorb1,axiom,
! [X: set_Product_prod_v_v,Y: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X @ Y )
=> ( ( ord_mi6996445931809003310od_v_v @ X @ Y )
= X ) ) ).
% min_absorb1
thf(fact_1196_min__absorb1,axiom,
! [X: set_v,Y: set_v] :
( ( ord_less_eq_set_v @ X @ Y )
=> ( ( ord_min_set_v @ X @ Y )
= X ) ) ).
% min_absorb1
thf(fact_1197_min__absorb2,axiom,
! [Y: set_Product_prod_v_v,X: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ Y @ X )
=> ( ( ord_mi6996445931809003310od_v_v @ X @ Y )
= Y ) ) ).
% min_absorb2
thf(fact_1198_min__absorb2,axiom,
! [Y: set_v,X: set_v] :
( ( ord_less_eq_set_v @ Y @ X )
=> ( ( ord_min_set_v @ X @ Y )
= Y ) ) ).
% min_absorb2
thf(fact_1199_null__rec_I2_J,axiom,
null_v @ nil_v ).
% null_rec(2)
thf(fact_1200_eq__Nil__null,axiom,
! [Xs: list_v] :
( ( Xs = nil_v )
= ( null_v @ Xs ) ) ).
% eq_Nil_null
thf(fact_1201_lexord__same__pref__iff,axiom,
! [Xs: list_v,Ys: list_v,Zs: list_v,R2: set_Product_prod_v_v] :
( ( member418487059593946000list_v @ ( produc6795410681906604247list_v @ ( append_v @ Xs @ Ys ) @ ( append_v @ Xs @ Zs ) ) @ ( lexord_v @ R2 ) )
= ( ? [X2: v] :
( ( member_v @ X2 @ ( set_v2 @ Xs ) )
& ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ X2 @ X2 ) @ R2 ) )
| ( member418487059593946000list_v @ ( produc6795410681906604247list_v @ Ys @ Zs ) @ ( lexord_v @ R2 ) ) ) ) ).
% lexord_same_pref_iff
thf(fact_1202_lexord__append__left__rightI,axiom,
! [A4: v,B2: v,R2: set_Product_prod_v_v,U: list_v,X: list_v,Y: list_v] :
( ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ A4 @ B2 ) @ R2 )
=> ( member418487059593946000list_v @ ( produc6795410681906604247list_v @ ( append_v @ U @ ( cons_v @ A4 @ X ) ) @ ( append_v @ U @ ( cons_v @ B2 @ Y ) ) ) @ ( lexord_v @ R2 ) ) ) ).
% lexord_append_left_rightI
thf(fact_1203_lexord__cons__cons,axiom,
! [A4: v,X: list_v,B2: v,Y: list_v,R2: set_Product_prod_v_v] :
( ( member418487059593946000list_v @ ( produc6795410681906604247list_v @ ( cons_v @ A4 @ X ) @ ( cons_v @ B2 @ Y ) ) @ ( lexord_v @ R2 ) )
= ( ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ A4 @ B2 ) @ R2 )
| ( ( A4 = B2 )
& ( member418487059593946000list_v @ ( produc6795410681906604247list_v @ X @ Y ) @ ( lexord_v @ R2 ) ) ) ) ) ).
% lexord_cons_cons
thf(fact_1204_lexord__Nil__left,axiom,
! [Y: list_v,R2: set_Product_prod_v_v] :
( ( member418487059593946000list_v @ ( produc6795410681906604247list_v @ nil_v @ Y ) @ ( lexord_v @ R2 ) )
= ( ? [A5: v,X2: list_v] :
( Y
= ( cons_v @ A5 @ X2 ) ) ) ) ).
% lexord_Nil_left
thf(fact_1205_lexord__append__leftI,axiom,
! [U: list_v,V: list_v,R2: set_Product_prod_v_v,X: list_v] :
( ( member418487059593946000list_v @ ( produc6795410681906604247list_v @ U @ V ) @ ( lexord_v @ R2 ) )
=> ( member418487059593946000list_v @ ( produc6795410681906604247list_v @ ( append_v @ X @ U ) @ ( append_v @ X @ V ) ) @ ( lexord_v @ R2 ) ) ) ).
% lexord_append_leftI
thf(fact_1206_lexord__Nil__right,axiom,
! [X: list_v,R2: set_Product_prod_v_v] :
~ ( member418487059593946000list_v @ ( produc6795410681906604247list_v @ X @ nil_v ) @ ( lexord_v @ R2 ) ) ).
% lexord_Nil_right
thf(fact_1207_lexord__linear,axiom,
! [R2: set_Product_prod_v_v,X: list_v,Y: list_v] :
( ! [A9: v,B8: v] :
( ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ A9 @ B8 ) @ R2 )
| ( A9 = B8 )
| ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ B8 @ A9 ) @ R2 ) )
=> ( ( member418487059593946000list_v @ ( produc6795410681906604247list_v @ X @ Y ) @ ( lexord_v @ R2 ) )
| ( X = Y )
| ( member418487059593946000list_v @ ( produc6795410681906604247list_v @ Y @ X ) @ ( lexord_v @ R2 ) ) ) ) ).
% lexord_linear
thf(fact_1208_lexord__irreflexive,axiom,
! [R2: set_Product_prod_v_v,Xs: list_v] :
( ! [X3: v] :
~ ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ X3 @ X3 ) @ R2 )
=> ~ ( member418487059593946000list_v @ ( produc6795410681906604247list_v @ Xs @ Xs ) @ ( lexord_v @ R2 ) ) ) ).
% lexord_irreflexive
thf(fact_1209_lexord__partial__trans,axiom,
! [Xs: list_P7986770385144383213od_v_v,R2: set_Pr2149350503807050951od_v_v,Ys: list_P7986770385144383213od_v_v,Zs: list_P7986770385144383213od_v_v] :
( ! [X3: product_prod_v_v,Y2: product_prod_v_v,Z2: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X3 @ ( set_Product_prod_v_v2 @ Xs ) )
=> ( ( member3038538357316246288od_v_v @ ( produc4031800376763917143od_v_v @ X3 @ Y2 ) @ R2 )
=> ( ( member3038538357316246288od_v_v @ ( produc4031800376763917143od_v_v @ Y2 @ Z2 ) @ R2 )
=> ( member3038538357316246288od_v_v @ ( produc4031800376763917143od_v_v @ X3 @ Z2 ) @ R2 ) ) ) )
=> ( ( member6382463057129219728od_v_v @ ( produc674067373767953879od_v_v @ Xs @ Ys ) @ ( lexord8601710409828808922od_v_v @ R2 ) )
=> ( ( member6382463057129219728od_v_v @ ( produc674067373767953879od_v_v @ Ys @ Zs ) @ ( lexord8601710409828808922od_v_v @ R2 ) )
=> ( member6382463057129219728od_v_v @ ( produc674067373767953879od_v_v @ Xs @ Zs ) @ ( lexord8601710409828808922od_v_v @ R2 ) ) ) ) ) ).
% lexord_partial_trans
thf(fact_1210_lexord__partial__trans,axiom,
! [Xs: list_v,R2: set_Product_prod_v_v,Ys: list_v,Zs: list_v] :
( ! [X3: v,Y2: v,Z2: v] :
( ( member_v @ X3 @ ( set_v2 @ Xs ) )
=> ( ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ X3 @ Y2 ) @ R2 )
=> ( ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ Y2 @ Z2 ) @ R2 )
=> ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ X3 @ Z2 ) @ R2 ) ) ) )
=> ( ( member418487059593946000list_v @ ( produc6795410681906604247list_v @ Xs @ Ys ) @ ( lexord_v @ R2 ) )
=> ( ( member418487059593946000list_v @ ( produc6795410681906604247list_v @ Ys @ Zs ) @ ( lexord_v @ R2 ) )
=> ( member418487059593946000list_v @ ( produc6795410681906604247list_v @ Xs @ Zs ) @ ( lexord_v @ R2 ) ) ) ) ) ).
% lexord_partial_trans
thf(fact_1211_lexord__append__leftD,axiom,
! [X: list_v,U: list_v,V: list_v,R2: set_Product_prod_v_v] :
( ( member418487059593946000list_v @ ( produc6795410681906604247list_v @ ( append_v @ X @ U ) @ ( append_v @ X @ V ) ) @ ( lexord_v @ R2 ) )
=> ( ! [A9: v] :
~ ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ A9 @ A9 ) @ R2 )
=> ( member418487059593946000list_v @ ( produc6795410681906604247list_v @ U @ V ) @ ( lexord_v @ R2 ) ) ) ) ).
% lexord_append_leftD
thf(fact_1212_lexord__append__rightI,axiom,
! [Y: list_v,X: list_v,R2: set_Product_prod_v_v] :
( ? [B9: v,Z5: list_v] :
( Y
= ( cons_v @ B9 @ Z5 ) )
=> ( member418487059593946000list_v @ ( produc6795410681906604247list_v @ X @ ( append_v @ X @ Y ) ) @ ( lexord_v @ R2 ) ) ) ).
% lexord_append_rightI
thf(fact_1213_underS__incl__iff,axiom,
! [R2: set_Pr2149350503807050951od_v_v,A4: product_prod_v_v,B2: product_prod_v_v] :
( ( order_6462556390437124636od_v_v @ ( field_7153129647634986036od_v_v @ R2 ) @ R2 )
=> ( ( member7453568604450474000od_v_v @ A4 @ ( field_7153129647634986036od_v_v @ R2 ) )
=> ( ( member7453568604450474000od_v_v @ B2 @ ( field_7153129647634986036od_v_v @ R2 ) )
=> ( ( ord_le7336532860387713383od_v_v @ ( order_5211820470575790509od_v_v @ R2 @ A4 ) @ ( order_5211820470575790509od_v_v @ R2 @ B2 ) )
= ( member3038538357316246288od_v_v @ ( produc4031800376763917143od_v_v @ A4 @ B2 ) @ R2 ) ) ) ) ) ).
% underS_incl_iff
thf(fact_1214_underS__incl__iff,axiom,
! [R2: set_Product_prod_v_v,A4: v,B2: v] :
( ( order_8768733634509060168r_on_v @ ( field_v @ R2 ) @ R2 )
=> ( ( member_v @ A4 @ ( field_v @ R2 ) )
=> ( ( member_v @ B2 @ ( field_v @ R2 ) )
=> ( ( ord_less_eq_set_v @ ( order_underS_v @ R2 @ A4 ) @ ( order_underS_v @ R2 @ B2 ) )
= ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ A4 @ B2 ) @ R2 ) ) ) ) ) ).
% underS_incl_iff
thf(fact_1215_listrel_Ocases,axiom,
! [A1: list_v,A2: list_v,R2: set_Product_prod_v_v] :
( ( member418487059593946000list_v @ ( produc6795410681906604247list_v @ A1 @ A2 ) @ ( listrel_v_v @ R2 ) )
=> ( ( ( A1 = nil_v )
=> ( A2 != nil_v ) )
=> ~ ! [X3: v,Y2: v,Xs2: list_v] :
( ( A1
= ( cons_v @ X3 @ Xs2 ) )
=> ! [Ys2: list_v] :
( ( A2
= ( cons_v @ Y2 @ Ys2 ) )
=> ( ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ X3 @ Y2 ) @ R2 )
=> ~ ( member418487059593946000list_v @ ( produc6795410681906604247list_v @ Xs2 @ Ys2 ) @ ( listrel_v_v @ R2 ) ) ) ) ) ) ) ).
% listrel.cases
thf(fact_1216_listrel_Ocases,axiom,
! [A1: list_v,A2: list_S9195319530159730289t_unit,R2: set_Pr6425124735969554649t_unit] :
( ( member5221658779932351824t_unit @ ( produc2297727903729291827t_unit @ A1 @ A2 ) @ ( listre2737131563223898179t_unit @ R2 ) )
=> ( ( ( A1 = nil_v )
=> ( A2 != nil_SC5400981268189614555t_unit ) )
=> ~ ! [X3: v,Y2: sCC_Bl1394983891496994913t_unit,Xs2: list_v] :
( ( A1
= ( cons_v @ X3 @ Xs2 ) )
=> ! [Ys2: list_S9195319530159730289t_unit] :
( ( A2
= ( cons_S4552210989952385579t_unit @ Y2 @ Ys2 ) )
=> ( ( member7924940910754673978t_unit @ ( produc3862955338007567901t_unit @ X3 @ Y2 ) @ R2 )
=> ~ ( member5221658779932351824t_unit @ ( produc2297727903729291827t_unit @ Xs2 @ Ys2 ) @ ( listre2737131563223898179t_unit @ R2 ) ) ) ) ) ) ) ).
% listrel.cases
thf(fact_1217_listrel_ONil,axiom,
! [R2: set_Product_prod_v_v] : ( member418487059593946000list_v @ ( produc6795410681906604247list_v @ nil_v @ nil_v ) @ ( listrel_v_v @ R2 ) ) ).
% listrel.Nil
thf(fact_1218_listrel__Nil1,axiom,
! [Xs: list_v,R2: set_Product_prod_v_v] :
( ( member418487059593946000list_v @ ( produc6795410681906604247list_v @ nil_v @ Xs ) @ ( listrel_v_v @ R2 ) )
=> ( Xs = nil_v ) ) ).
% listrel_Nil1
thf(fact_1219_listrel__Nil2,axiom,
! [Xs: list_v,R2: set_Product_prod_v_v] :
( ( member418487059593946000list_v @ ( produc6795410681906604247list_v @ Xs @ nil_v ) @ ( listrel_v_v @ R2 ) )
=> ( Xs = nil_v ) ) ).
% listrel_Nil2
thf(fact_1220_underS__empty,axiom,
! [A4: product_prod_v_v,R2: set_Pr2149350503807050951od_v_v] :
( ~ ( member7453568604450474000od_v_v @ A4 @ ( field_7153129647634986036od_v_v @ R2 ) )
=> ( ( order_5211820470575790509od_v_v @ R2 @ A4 )
= bot_bo723834152578015283od_v_v ) ) ).
% underS_empty
thf(fact_1221_underS__empty,axiom,
! [A4: v,R2: set_Product_prod_v_v] :
( ~ ( member_v @ A4 @ ( field_v @ R2 ) )
=> ( ( order_underS_v @ R2 @ A4 )
= bot_bot_set_v ) ) ).
% underS_empty
thf(fact_1222_underS__E,axiom,
! [I: product_prod_v_v,R4: set_Pr2149350503807050951od_v_v,J: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ I @ ( order_5211820470575790509od_v_v @ R4 @ J ) )
=> ( ( I != J )
& ( member3038538357316246288od_v_v @ ( produc4031800376763917143od_v_v @ I @ J ) @ R4 ) ) ) ).
% underS_E
thf(fact_1223_underS__E,axiom,
! [I: v,R4: set_Product_prod_v_v,J: v] :
( ( member_v @ I @ ( order_underS_v @ R4 @ J ) )
=> ( ( I != J )
& ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ I @ J ) @ R4 ) ) ) ).
% underS_E
thf(fact_1224_underS__I,axiom,
! [I: product_prod_v_v,J: product_prod_v_v,R4: set_Pr2149350503807050951od_v_v] :
( ( I != J )
=> ( ( member3038538357316246288od_v_v @ ( produc4031800376763917143od_v_v @ I @ J ) @ R4 )
=> ( member7453568604450474000od_v_v @ I @ ( order_5211820470575790509od_v_v @ R4 @ J ) ) ) ) ).
% underS_I
thf(fact_1225_underS__I,axiom,
! [I: v,J: v,R4: set_Product_prod_v_v] :
( ( I != J )
=> ( ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ I @ J ) @ R4 )
=> ( member_v @ I @ ( order_underS_v @ R4 @ J ) ) ) ) ).
% underS_I
thf(fact_1226_Order__Relation_OunderS__Field,axiom,
! [R2: set_Pr2149350503807050951od_v_v,A4: product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ ( order_5211820470575790509od_v_v @ R2 @ A4 ) @ ( field_7153129647634986036od_v_v @ R2 ) ) ).
% Order_Relation.underS_Field
thf(fact_1227_Order__Relation_OunderS__Field,axiom,
! [R2: set_Product_prod_v_v,A4: v] : ( ord_less_eq_set_v @ ( order_underS_v @ R2 @ A4 ) @ ( field_v @ R2 ) ) ).
% Order_Relation.underS_Field
thf(fact_1228_listrel__mono,axiom,
! [R2: set_Product_prod_v_v,S5: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ R2 @ S5 )
=> ( ord_le791731619978752231list_v @ ( listrel_v_v @ R2 ) @ ( listrel_v_v @ S5 ) ) ) ).
% listrel_mono
thf(fact_1229_listrel_OCons,axiom,
! [X: v,Y: v,R2: set_Product_prod_v_v,Xs: list_v,Ys: list_v] :
( ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ X @ Y ) @ R2 )
=> ( ( member418487059593946000list_v @ ( produc6795410681906604247list_v @ Xs @ Ys ) @ ( listrel_v_v @ R2 ) )
=> ( member418487059593946000list_v @ ( produc6795410681906604247list_v @ ( cons_v @ X @ Xs ) @ ( cons_v @ Y @ Ys ) ) @ ( listrel_v_v @ R2 ) ) ) ) ).
% listrel.Cons
thf(fact_1230_listrel_OCons,axiom,
! [X: v,Y: sCC_Bl1394983891496994913t_unit,R2: set_Pr6425124735969554649t_unit,Xs: list_v,Ys: list_S9195319530159730289t_unit] :
( ( member7924940910754673978t_unit @ ( produc3862955338007567901t_unit @ X @ Y ) @ R2 )
=> ( ( member5221658779932351824t_unit @ ( produc2297727903729291827t_unit @ Xs @ Ys ) @ ( listre2737131563223898179t_unit @ R2 ) )
=> ( member5221658779932351824t_unit @ ( produc2297727903729291827t_unit @ ( cons_v @ X @ Xs ) @ ( cons_S4552210989952385579t_unit @ Y @ Ys ) ) @ ( listre2737131563223898179t_unit @ R2 ) ) ) ) ).
% listrel.Cons
thf(fact_1231_listrel__Cons1,axiom,
! [Y: v,Ys: list_v,Xs: list_v,R2: set_Product_prod_v_v] :
( ( member418487059593946000list_v @ ( produc6795410681906604247list_v @ ( cons_v @ Y @ Ys ) @ Xs ) @ ( listrel_v_v @ R2 ) )
=> ~ ! [Y2: v,Ys2: list_v] :
( ( Xs
= ( cons_v @ Y2 @ Ys2 ) )
=> ( ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ Y @ Y2 ) @ R2 )
=> ~ ( member418487059593946000list_v @ ( produc6795410681906604247list_v @ Ys @ Ys2 ) @ ( listrel_v_v @ R2 ) ) ) ) ) ).
% listrel_Cons1
thf(fact_1232_listrel__Cons1,axiom,
! [Y: v,Ys: list_v,Xs: list_S9195319530159730289t_unit,R2: set_Pr6425124735969554649t_unit] :
( ( member5221658779932351824t_unit @ ( produc2297727903729291827t_unit @ ( cons_v @ Y @ Ys ) @ Xs ) @ ( listre2737131563223898179t_unit @ R2 ) )
=> ~ ! [Y2: sCC_Bl1394983891496994913t_unit,Ys2: list_S9195319530159730289t_unit] :
( ( Xs
= ( cons_S4552210989952385579t_unit @ Y2 @ Ys2 ) )
=> ( ( member7924940910754673978t_unit @ ( produc3862955338007567901t_unit @ Y @ Y2 ) @ R2 )
=> ~ ( member5221658779932351824t_unit @ ( produc2297727903729291827t_unit @ Ys @ Ys2 ) @ ( listre2737131563223898179t_unit @ R2 ) ) ) ) ) ).
% listrel_Cons1
thf(fact_1233_listrel__Cons2,axiom,
! [Xs: list_v,Y: v,Ys: list_v,R2: set_Product_prod_v_v] :
( ( member418487059593946000list_v @ ( produc6795410681906604247list_v @ Xs @ ( cons_v @ Y @ Ys ) ) @ ( listrel_v_v @ R2 ) )
=> ~ ! [X3: v,Xs2: list_v] :
( ( Xs
= ( cons_v @ X3 @ Xs2 ) )
=> ( ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ X3 @ Y ) @ R2 )
=> ~ ( member418487059593946000list_v @ ( produc6795410681906604247list_v @ Xs2 @ Ys ) @ ( listrel_v_v @ R2 ) ) ) ) ) ).
% listrel_Cons2
thf(fact_1234_listrel__Cons2,axiom,
! [Xs: list_v,Y: sCC_Bl1394983891496994913t_unit,Ys: list_S9195319530159730289t_unit,R2: set_Pr6425124735969554649t_unit] :
( ( member5221658779932351824t_unit @ ( produc2297727903729291827t_unit @ Xs @ ( cons_S4552210989952385579t_unit @ Y @ Ys ) ) @ ( listre2737131563223898179t_unit @ R2 ) )
=> ~ ! [X3: v,Xs2: list_v] :
( ( Xs
= ( cons_v @ X3 @ Xs2 ) )
=> ( ( member7924940910754673978t_unit @ ( produc3862955338007567901t_unit @ X3 @ Y ) @ R2 )
=> ~ ( member5221658779932351824t_unit @ ( produc2297727903729291827t_unit @ Xs2 @ Ys ) @ ( listre2737131563223898179t_unit @ R2 ) ) ) ) ) ).
% listrel_Cons2
thf(fact_1235_listrel_Osimps,axiom,
! [A1: list_v,A2: list_v,R2: set_Product_prod_v_v] :
( ( member418487059593946000list_v @ ( produc6795410681906604247list_v @ A1 @ A2 ) @ ( listrel_v_v @ R2 ) )
= ( ( ( A1 = nil_v )
& ( A2 = nil_v ) )
| ? [X2: v,Y3: v,Xs4: list_v,Ys4: list_v] :
( ( A1
= ( cons_v @ X2 @ Xs4 ) )
& ( A2
= ( cons_v @ Y3 @ Ys4 ) )
& ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ X2 @ Y3 ) @ R2 )
& ( member418487059593946000list_v @ ( produc6795410681906604247list_v @ Xs4 @ Ys4 ) @ ( listrel_v_v @ R2 ) ) ) ) ) ).
% listrel.simps
thf(fact_1236_listrel_Osimps,axiom,
! [A1: list_v,A2: list_S9195319530159730289t_unit,R2: set_Pr6425124735969554649t_unit] :
( ( member5221658779932351824t_unit @ ( produc2297727903729291827t_unit @ A1 @ A2 ) @ ( listre2737131563223898179t_unit @ R2 ) )
= ( ( ( A1 = nil_v )
& ( A2 = nil_SC5400981268189614555t_unit ) )
| ? [X2: v,Y3: sCC_Bl1394983891496994913t_unit,Xs4: list_v,Ys4: list_S9195319530159730289t_unit] :
( ( A1
= ( cons_v @ X2 @ Xs4 ) )
& ( A2
= ( cons_S4552210989952385579t_unit @ Y3 @ Ys4 ) )
& ( member7924940910754673978t_unit @ ( produc3862955338007567901t_unit @ X2 @ Y3 ) @ R2 )
& ( member5221658779932351824t_unit @ ( produc2297727903729291827t_unit @ Xs4 @ Ys4 ) @ ( listre2737131563223898179t_unit @ R2 ) ) ) ) ) ).
% listrel.simps
thf(fact_1237_Refl__under__underS,axiom,
! [R2: set_Pr2149350503807050951od_v_v,A4: product_prod_v_v] :
( ( refl_o4548774019903118566od_v_v @ ( field_7153129647634986036od_v_v @ R2 ) @ R2 )
=> ( ( member7453568604450474000od_v_v @ A4 @ ( field_7153129647634986036od_v_v @ R2 ) )
=> ( ( order_6892855479609198156od_v_v @ R2 @ A4 )
= ( sup_su414716646722978715od_v_v @ ( order_5211820470575790509od_v_v @ R2 @ A4 ) @ ( insert1338601472111419319od_v_v @ A4 @ bot_bo723834152578015283od_v_v ) ) ) ) ) ).
% Refl_under_underS
thf(fact_1238_Refl__under__underS,axiom,
! [R2: set_Product_prod_v_v,A4: v] :
( ( refl_on_v @ ( field_v @ R2 ) @ R2 )
=> ( ( member_v @ A4 @ ( field_v @ R2 ) )
=> ( ( order_under_v @ R2 @ A4 )
= ( sup_sup_set_v @ ( order_underS_v @ R2 @ A4 ) @ ( insert_v2 @ A4 @ bot_bot_set_v ) ) ) ) ) ).
% Refl_under_underS
thf(fact_1239_listrel1__subset__listrel,axiom,
! [R2: set_Product_prod_v_v,R3: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ R2 @ R3 )
=> ( ( refl_on_v @ top_top_set_v @ R3 )
=> ( ord_le791731619978752231list_v @ ( listrel1_v @ R2 ) @ ( listrel_v_v @ R3 ) ) ) ) ).
% listrel1_subset_listrel
thf(fact_1240_UNIV__I,axiom,
! [X: v] : ( member_v @ X @ top_top_set_v ) ).
% UNIV_I
thf(fact_1241_UNIV__I,axiom,
! [X: product_prod_v_v] : ( member7453568604450474000od_v_v @ X @ top_to5429829297380968215od_v_v ) ).
% UNIV_I
thf(fact_1242_inf__top_Oright__neutral,axiom,
! [A4: set_v] :
( ( inf_inf_set_v @ A4 @ top_top_set_v )
= A4 ) ).
% inf_top.right_neutral
thf(fact_1243_inf__top_Oneutr__eq__iff,axiom,
! [A4: set_v,B2: set_v] :
( ( top_top_set_v
= ( inf_inf_set_v @ A4 @ B2 ) )
= ( ( A4 = top_top_set_v )
& ( B2 = top_top_set_v ) ) ) ).
% inf_top.neutr_eq_iff
thf(fact_1244_inf__top_Oleft__neutral,axiom,
! [A4: set_v] :
( ( inf_inf_set_v @ top_top_set_v @ A4 )
= A4 ) ).
% inf_top.left_neutral
thf(fact_1245_inf__top_Oeq__neutr__iff,axiom,
! [A4: set_v,B2: set_v] :
( ( ( inf_inf_set_v @ A4 @ B2 )
= top_top_set_v )
= ( ( A4 = top_top_set_v )
& ( B2 = top_top_set_v ) ) ) ).
% inf_top.eq_neutr_iff
thf(fact_1246_top__eq__inf__iff,axiom,
! [X: set_v,Y: set_v] :
( ( top_top_set_v
= ( inf_inf_set_v @ X @ Y ) )
= ( ( X = top_top_set_v )
& ( Y = top_top_set_v ) ) ) ).
% top_eq_inf_iff
thf(fact_1247_inf__eq__top__iff,axiom,
! [X: set_v,Y: set_v] :
( ( ( inf_inf_set_v @ X @ Y )
= top_top_set_v )
= ( ( X = top_top_set_v )
& ( Y = top_top_set_v ) ) ) ).
% inf_eq_top_iff
thf(fact_1248_inf__top__right,axiom,
! [X: set_v] :
( ( inf_inf_set_v @ X @ top_top_set_v )
= X ) ).
% inf_top_right
thf(fact_1249_inf__top__left,axiom,
! [X: set_v] :
( ( inf_inf_set_v @ top_top_set_v @ X )
= X ) ).
% inf_top_left
thf(fact_1250_boolean__algebra_Odisj__one__right,axiom,
! [X: set_v] :
( ( sup_sup_set_v @ X @ top_top_set_v )
= top_top_set_v ) ).
% boolean_algebra.disj_one_right
thf(fact_1251_boolean__algebra_Odisj__one__right,axiom,
! [X: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ X @ top_to5429829297380968215od_v_v )
= top_to5429829297380968215od_v_v ) ).
% boolean_algebra.disj_one_right
thf(fact_1252_boolean__algebra_Odisj__one__left,axiom,
! [X: set_v] :
( ( sup_sup_set_v @ top_top_set_v @ X )
= top_top_set_v ) ).
% boolean_algebra.disj_one_left
thf(fact_1253_boolean__algebra_Odisj__one__left,axiom,
! [X: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ top_to5429829297380968215od_v_v @ X )
= top_to5429829297380968215od_v_v ) ).
% boolean_algebra.disj_one_left
thf(fact_1254_sup__top__left,axiom,
! [X: set_v] :
( ( sup_sup_set_v @ top_top_set_v @ X )
= top_top_set_v ) ).
% sup_top_left
thf(fact_1255_sup__top__left,axiom,
! [X: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ top_to5429829297380968215od_v_v @ X )
= top_to5429829297380968215od_v_v ) ).
% sup_top_left
thf(fact_1256_sup__top__right,axiom,
! [X: set_v] :
( ( sup_sup_set_v @ X @ top_top_set_v )
= top_top_set_v ) ).
% sup_top_right
thf(fact_1257_sup__top__right,axiom,
! [X: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ X @ top_to5429829297380968215od_v_v )
= top_to5429829297380968215od_v_v ) ).
% sup_top_right
thf(fact_1258_Int__UNIV,axiom,
! [A: set_v,B: set_v] :
( ( ( inf_inf_set_v @ A @ B )
= top_top_set_v )
= ( ( A = top_top_set_v )
& ( B = top_top_set_v ) ) ) ).
% Int_UNIV
thf(fact_1259_Diff__UNIV,axiom,
! [A: set_Product_prod_v_v] :
( ( minus_4183494784930505774od_v_v @ A @ top_to5429829297380968215od_v_v )
= bot_bo723834152578015283od_v_v ) ).
% Diff_UNIV
thf(fact_1260_Diff__UNIV,axiom,
! [A: set_v] :
( ( minus_minus_set_v @ A @ top_top_set_v )
= bot_bot_set_v ) ).
% Diff_UNIV
thf(fact_1261_underS__subset__under,axiom,
! [R2: set_Pr2149350503807050951od_v_v,A4: product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ ( order_5211820470575790509od_v_v @ R2 @ A4 ) @ ( order_6892855479609198156od_v_v @ R2 @ A4 ) ) ).
% underS_subset_under
thf(fact_1262_underS__subset__under,axiom,
! [R2: set_Product_prod_v_v,A4: v] : ( ord_less_eq_set_v @ ( order_underS_v @ R2 @ A4 ) @ ( order_under_v @ R2 @ A4 ) ) ).
% underS_subset_under
thf(fact_1263_boolean__algebra_Oconj__one__right,axiom,
! [X: set_v] :
( ( inf_inf_set_v @ X @ top_top_set_v )
= X ) ).
% boolean_algebra.conj_one_right
thf(fact_1264_Int__UNIV__left,axiom,
! [B: set_v] :
( ( inf_inf_set_v @ top_top_set_v @ B )
= B ) ).
% Int_UNIV_left
thf(fact_1265_Int__UNIV__right,axiom,
! [A: set_v] :
( ( inf_inf_set_v @ A @ top_top_set_v )
= A ) ).
% Int_UNIV_right
thf(fact_1266_UNIV__eq__I,axiom,
! [A: set_v] :
( ! [X3: v] : ( member_v @ X3 @ A )
=> ( top_top_set_v = A ) ) ).
% UNIV_eq_I
thf(fact_1267_UNIV__eq__I,axiom,
! [A: set_Product_prod_v_v] :
( ! [X3: product_prod_v_v] : ( member7453568604450474000od_v_v @ X3 @ A )
=> ( top_to5429829297380968215od_v_v = A ) ) ).
% UNIV_eq_I
thf(fact_1268_UNIV__witness,axiom,
? [X3: v] : ( member_v @ X3 @ top_top_set_v ) ).
% UNIV_witness
thf(fact_1269_UNIV__witness,axiom,
? [X3: product_prod_v_v] : ( member7453568604450474000od_v_v @ X3 @ top_to5429829297380968215od_v_v ) ).
% UNIV_witness
thf(fact_1270_Un__UNIV__right,axiom,
! [A: set_v] :
( ( sup_sup_set_v @ A @ top_top_set_v )
= top_top_set_v ) ).
% Un_UNIV_right
thf(fact_1271_Un__UNIV__right,axiom,
! [A: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ A @ top_to5429829297380968215od_v_v )
= top_to5429829297380968215od_v_v ) ).
% Un_UNIV_right
thf(fact_1272_Un__UNIV__left,axiom,
! [B: set_v] :
( ( sup_sup_set_v @ top_top_set_v @ B )
= top_top_set_v ) ).
% Un_UNIV_left
thf(fact_1273_Un__UNIV__left,axiom,
! [B: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ top_to5429829297380968215od_v_v @ B )
= top_to5429829297380968215od_v_v ) ).
% Un_UNIV_left
thf(fact_1274_insert__UNIV,axiom,
! [X: product_prod_v_v] :
( ( insert1338601472111419319od_v_v @ X @ top_to5429829297380968215od_v_v )
= top_to5429829297380968215od_v_v ) ).
% insert_UNIV
thf(fact_1275_insert__UNIV,axiom,
! [X: v] :
( ( insert_v2 @ X @ top_top_set_v )
= top_top_set_v ) ).
% insert_UNIV
% Helper facts (9)
thf(help_If_2_1_If_001t__Set__Oset_Itf__v_J_T,axiom,
! [X: set_v,Y: set_v] :
( ( if_set_v @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Set__Oset_Itf__v_J_T,axiom,
! [X: set_v,Y: set_v] :
( ( if_set_v @ $true @ X @ Y )
= X ) ).
thf(help_If_2_1_If_001t__List__Olist_Itf__v_J_T,axiom,
! [X: list_v,Y: list_v] :
( ( if_list_v @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__List__Olist_Itf__v_J_T,axiom,
! [X: list_v,Y: list_v] :
( ( if_list_v @ $true @ X @ Y )
= X ) ).
thf(help_If_2_1_If_001t__Set__Oset_It__Product____Type__Oprod_Itf__v_Mtf__v_J_J_T,axiom,
! [X: set_Product_prod_v_v,Y: set_Product_prod_v_v] :
( ( if_set4279007504652509325od_v_v @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Set__Oset_It__Product____Type__Oprod_Itf__v_Mtf__v_J_J_T,axiom,
! [X: set_Product_prod_v_v,Y: set_Product_prod_v_v] :
( ( if_set4279007504652509325od_v_v @ $true @ X @ Y )
= X ) ).
thf(help_If_3_1_If_001t__List__Olist_It__Product____Type__Oprod_Itf__v_Mtf__v_J_J_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__List__Olist_It__Product____Type__Oprod_Itf__v_Mtf__v_J_J_T,axiom,
! [X: list_P7986770385144383213od_v_v,Y: list_P7986770385144383213od_v_v] :
( ( if_lis7521272669439687347od_v_v @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__List__Olist_It__Product____Type__Oprod_Itf__v_Mtf__v_J_J_T,axiom,
! [X: list_P7986770385144383213od_v_v,Y: list_P7986770385144383213od_v_v] :
( ( if_lis7521272669439687347od_v_v @ $true @ X @ Y )
= X ) ).
% Conjectures (1)
thf(conj_0,conjecture,
~ ( sCC_Bl4291963740693775144ding_v @ successors @ va @ n @ bot_bo723834152578015283od_v_v ) ).
%------------------------------------------------------------------------------