TPTP Problem File: SLH0853^1.p
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%------------------------------------------------------------------------------
% 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_02139_073808__6211172_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 926 ( 413 unt; 156 typ; 0 def)
% Number of atoms : 2185 ( 824 equ; 0 cnn)
% Maximal formula atoms : 10 ( 2 avg)
% Number of connectives : 7272 ( 253 ~; 25 |; 197 &;6003 @)
% ( 0 <=>; 794 =>; 0 <=; 0 <~>)
% Maximal formula depth : 22 ( 6 avg)
% Number of types : 16 ( 15 usr)
% Number of type conns : 452 ( 452 >; 0 *; 0 +; 0 <<)
% Number of symbols : 144 ( 141 usr; 19 con; 0-9 aty)
% Number of variables : 2163 ( 243 ^;1850 !; 70 ?;2163 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 08:53:56.817
%------------------------------------------------------------------------------
% Could-be-implicit typings (15)
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sCC_Bloemen_is_scc_v: ( v > set_v ) > set_v > $o ).
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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_001t__Set__Oset_Itf__v_J,type,
sCC_Bl7907073126578335045_set_v: ( set_v > set_set_v ) > set_set_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,
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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 ).
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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_001t__Set__Oset_Itf__v_J,type,
sCC_Bl7354734129683093653_set_v: ( set_v > set_set_v ) > set_v > set_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_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_001t__Set__Oset_It__Product____Type__Oprod_Itf__v_Mtf__v_J_J,type,
collec8263177866097347122od_v_v: ( set_Product_prod_v_v > $o ) > set_se8455005133513928103od_v_v ).
thf(sy_c_Set_OCollect_001t__Set__Oset_Itf__v_J,type,
collect_set_v: ( set_v > $o ) > set_set_v ).
thf(sy_c_Set_OCollect_001tf__v,type,
collect_v: ( v > $o ) > set_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_It__Set__Oset_Itf__v_J_J,type,
insert_set_set_v: set_set_v > set_set_set_v > set_set_set_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_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_001t__Set__Oset_Itf__v_J,type,
the_elem_set_v: set_set_v > set_v ).
thf(sy_c_Set_Othe__elem_001tf__v,type,
the_elem_v: set_v > v ).
thf(sy_c_Sum__Type_OInl_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_In526841707622398774t_unit: produc5741669702376414499t_unit > sum_su8181647976486975269t_unit ).
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_fChoice_001tf__v,type,
fChoice_v: ( v > $o ) > v ).
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__Product____Type__Ounit,type,
member_Product_unit: product_unit > set_Product_unit > $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_It__Set__Oset_Itf__v_J_J,type,
member_set_set_v: set_set_v > set_set_set_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_cc____,type,
cc: set_v ).
thf(sy_v_e,type,
e: sCC_Bl1394983891496994913t_unit ).
thf(sy_v_e_H,type,
e2: sCC_Bl1394983891496994913t_unit ).
thf(sy_v_e_H_H,type,
e3: sCC_Bl1394983891496994913t_unit ).
thf(sy_v_pfx____,type,
pfx: list_v ).
thf(sy_v_successors,type,
successors: v > set_v ).
thf(sy_v_v,type,
v2: v ).
thf(sy_v_vertices,type,
vertices: set_v ).
thf(sy_v_w,type,
w: v ).
% Relevant facts (763)
thf(fact_0__092_060open_062_092_060forall_062n_Am_O_A_Im_A_092_060in_062_A_092_060S_062_Ae_H_H_An_J_A_061_A_I_092_060S_062_Ae_H_H_An_A_061_A_092_060S_062_Ae_H_H_Am_J_092_060close_062,axiom,
! [N: v,M: v] :
( ( member_v @ M @ ( sCC_Bl1280885523602775798t_unit @ e3 @ N ) )
= ( ( sCC_Bl1280885523602775798t_unit @ e3 @ N )
= ( sCC_Bl1280885523602775798t_unit @ e3 @ M ) ) ) ).
% \<open>\<forall>n m. (m \<in> \<S> e'' n) = (\<S> e'' n = \<S> e'' m)\<close>
thf(fact_1__092_060open_062distinct_A_Icstack_Ae_H_H_J_092_060close_062,axiom,
distinct_v @ ( sCC_Bl9201514103433284750t_unit @ e3 ) ).
% \<open>distinct (cstack e'')\<close>
thf(fact_2__092_060open_062distinct_A_Istack_Ae_H_H_J_092_060close_062,axiom,
distinct_v @ ( sCC_Bl8828226123343373779t_unit @ e3 ) ).
% \<open>distinct (stack e'')\<close>
thf(fact_3__092_060open_062_092_060Union_062_A_Isccs_Ae_H_H_J_A_061_Aexplored_Ae_H_H_092_060close_062,axiom,
( ( comple2307003700295860064_set_v @ ( sCC_Bl2536197123907397897t_unit @ e3 ) )
= ( sCC_Bl157864678168468314t_unit @ e3 ) ) ).
% \<open>\<Union> (sccs e'') = explored e''\<close>
thf(fact_4__092_060open_062_092_060forall_062n_Am_O_An_A_092_060preceq_062_Am_Ain_Astack_Ae_H_H_A_092_060longrightarrow_062_An_A_092_060preceq_062_Am_Ain_Acstack_Ae_H_H_092_060close_062,axiom,
! [N: v,M: v] :
( ( sCC_Bl4022239298816431255edes_v @ N @ M @ ( sCC_Bl8828226123343373779t_unit @ e3 ) )
=> ( sCC_Bl4022239298816431255edes_v @ N @ M @ ( sCC_Bl9201514103433284750t_unit @ e3 ) ) ) ).
% \<open>\<forall>n m. n \<preceq> m in stack e'' \<longrightarrow> n \<preceq> m in cstack e''\<close>
thf(fact_5__092_060open_062explored_Ae_H_H_A_092_060subseteq_062_Avisited_Ae_H_H_092_060close_062,axiom,
ord_less_eq_set_v @ ( sCC_Bl157864678168468314t_unit @ e3 ) @ ( sCC_Bl4645233313691564917t_unit @ e3 ) ).
% \<open>explored e'' \<subseteq> visited e''\<close>
thf(fact_6__092_060open_062_092_060forall_062n_O_An_A_092_060notin_062_Avisited_Ae_H_H_A_092_060longrightarrow_062_Avsuccs_Ae_H_H_An_A_061_A_123_125_092_060close_062,axiom,
! [N: v] :
( ~ ( member_v @ N @ ( sCC_Bl4645233313691564917t_unit @ e3 ) )
=> ( ( sCC_Bl3795065053823578884t_unit @ e3 @ N )
= bot_bot_set_v ) ) ).
% \<open>\<forall>n. n \<notin> visited e'' \<longrightarrow> vsuccs e'' n = {}\<close>
thf(fact_7__092_060open_062Ball_A_Isccs_Ae_H_H_J_Ais__scc_092_060close_062,axiom,
! [X: set_v] :
( ( member_set_v @ X @ ( sCC_Bl2536197123907397897t_unit @ e3 ) )
=> ( sCC_Bloemen_is_scc_v @ successors @ X ) ) ).
% \<open>Ball (sccs e'') is_scc\<close>
thf(fact_8__092_060open_062set_A_Icstack_Ae_H_H_J_A_092_060subseteq_062_Avisited_Ae_H_H_092_060close_062,axiom,
ord_less_eq_set_v @ ( set_v2 @ ( sCC_Bl9201514103433284750t_unit @ e3 ) ) @ ( sCC_Bl4645233313691564917t_unit @ e3 ) ).
% \<open>set (cstack e'') \<subseteq> visited e''\<close>
thf(fact_9__092_060open_062_092_060forall_062n_O_Ais__subscc_A_I_092_060S_062_Ae_H_H_An_J_092_060close_062,axiom,
! [N: v] : ( sCC_Bl5398416737448265317bscc_v @ successors @ ( sCC_Bl1280885523602775798t_unit @ e3 @ N ) ) ).
% \<open>\<forall>n. is_subscc (\<S> e'' n)\<close>
thf(fact_10__092_060open_062_092_060forall_062n_092_060in_062explored_Ae_H_H_O_Avsuccs_Ae_H_H_An_A_061_Asuccessors_An_092_060close_062,axiom,
! [X: v] :
( ( member_v @ X @ ( sCC_Bl157864678168468314t_unit @ e3 ) )
=> ( ( sCC_Bl3795065053823578884t_unit @ e3 @ X )
= ( successors @ X ) ) ) ).
% \<open>\<forall>n\<in>explored e''. vsuccs e'' n = successors n\<close>
thf(fact_11__092_060open_062_092_060forall_062n_092_060in_062explored_Ae_H_H_O_A_092_060forall_062m_O_Areachable_An_Am_A_092_060longrightarrow_062_Am_A_092_060in_062_Aexplored_Ae_H_H_092_060close_062,axiom,
! [X: v] :
( ( member_v @ X @ ( sCC_Bl157864678168468314t_unit @ e3 ) )
=> ! [M: v] :
( ( sCC_Bl649662514949026229able_v @ successors @ X @ M )
=> ( member_v @ M @ ( sCC_Bl157864678168468314t_unit @ e3 ) ) ) ) ).
% \<open>\<forall>n\<in>explored e''. \<forall>m. reachable n m \<longrightarrow> m \<in> explored e''\<close>
thf(fact_12_dfs__dfss__rel_Ocong,axiom,
sCC_Bl907557413677168252_rel_v = sCC_Bl907557413677168252_rel_v ).
% dfs_dfss_rel.cong
thf(fact_13__092_060open_062_092_060forall_062n_O_An_A_092_060notin_062_Avisited_Ae_H_H_A_092_060longrightarrow_062_A_092_060S_062_Ae_H_H_An_A_061_A_123n_125_092_060close_062,axiom,
! [N: v] :
( ~ ( member_v @ N @ ( sCC_Bl4645233313691564917t_unit @ e3 ) )
=> ( ( sCC_Bl1280885523602775798t_unit @ e3 @ N )
= ( insert_v @ N @ bot_bot_set_v ) ) ) ).
% \<open>\<forall>n. n \<notin> visited e'' \<longrightarrow> \<S> e'' n = {n}\<close>
thf(fact_14__092_060open_062_092_060forall_062n_092_060in_062visited_Ae_H_H_O_Areachable_A_Iroot_Ae_H_H_J_An_092_060close_062,axiom,
! [X: v] :
( ( member_v @ X @ ( sCC_Bl4645233313691564917t_unit @ e3 ) )
=> ( sCC_Bl649662514949026229able_v @ successors @ ( sCC_Bl1090238580953940555t_unit @ e3 ) @ X ) ) ).
% \<open>\<forall>n\<in>visited e''. reachable (root e'') n\<close>
thf(fact_15__092_060open_062_092_060forall_062n_Am_O_An_A_092_060preceq_062_Am_Ain_Astack_Ae_H_H_A_092_060longrightarrow_062_Areachable_Am_An_092_060close_062,axiom,
! [N: v,M: v] :
( ( sCC_Bl4022239298816431255edes_v @ N @ M @ ( sCC_Bl8828226123343373779t_unit @ e3 ) )
=> ( sCC_Bl649662514949026229able_v @ successors @ M @ N ) ) ).
% \<open>\<forall>n m. n \<preceq> m in stack e'' \<longrightarrow> reachable m n\<close>
thf(fact_16__092_060open_062_092_060forall_062n_092_060in_062set_A_Istack_Ae_H_H_J_O_A_092_060forall_062m_092_060in_062_092_060S_062_Ae_H_H_An_O_Am_A_092_060in_062_Aset_A_Icstack_Ae_H_H_J_A_092_060longrightarrow_062_Am_A_092_060preceq_062_An_Ain_Acstack_Ae_H_H_092_060close_062,axiom,
! [X: v] :
( ( member_v @ X @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ e3 ) ) )
=> ! [Xa: v] :
( ( member_v @ Xa @ ( sCC_Bl1280885523602775798t_unit @ e3 @ X ) )
=> ( ( member_v @ Xa @ ( set_v2 @ ( sCC_Bl9201514103433284750t_unit @ e3 ) ) )
=> ( sCC_Bl4022239298816431255edes_v @ Xa @ X @ ( sCC_Bl9201514103433284750t_unit @ e3 ) ) ) ) ) ).
% \<open>\<forall>n\<in>set (stack e''). \<forall>m\<in>\<S> e'' n. m \<in> set (cstack e'') \<longrightarrow> m \<preceq> n in cstack e''\<close>
thf(fact_17_ra__refl,axiom,
! [X2: v,E: set_Product_prod_v_v] : ( sCC_Bl4291963740693775144ding_v @ successors @ X2 @ X2 @ E ) ).
% ra_refl
thf(fact_18_ra__trans,axiom,
! [X2: v,Y: v,E: set_Product_prod_v_v,Z: v] :
( ( sCC_Bl4291963740693775144ding_v @ successors @ X2 @ Y @ E )
=> ( ( sCC_Bl4291963740693775144ding_v @ successors @ Y @ Z @ E )
=> ( sCC_Bl4291963740693775144ding_v @ successors @ X2 @ Z @ E ) ) ) ).
% ra_trans
thf(fact_19__092_060open_062_092_060forall_062n_O_Avsuccs_Ae_H_H_An_A_092_060subseteq_062_Asuccessors_An_A_092_060inter_062_Avisited_Ae_H_H_092_060close_062,axiom,
! [N: v] : ( ord_less_eq_set_v @ ( sCC_Bl3795065053823578884t_unit @ e3 @ N ) @ ( inf_inf_set_v @ ( successors @ N ) @ ( sCC_Bl4645233313691564917t_unit @ e3 ) ) ) ).
% \<open>\<forall>n. vsuccs e'' n \<subseteq> successors n \<inter> visited e''\<close>
thf(fact_20__092_060open_062_092_060forall_062n_092_060in_062set_A_Istack_Ae_H_H_J_O_A_092_060forall_062m_092_060in_062set_A_Istack_Ae_H_H_J_O_An_A_092_060noteq_062_Am_A_092_060longrightarrow_062_A_092_060S_062_Ae_H_H_An_A_092_060inter_062_A_092_060S_062_Ae_H_H_Am_A_061_A_123_125_092_060close_062,axiom,
! [X: v] :
( ( member_v @ X @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ e3 ) ) )
=> ! [Xa: v] :
( ( member_v @ Xa @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ e3 ) ) )
=> ( ( X != Xa )
=> ( ( inf_inf_set_v @ ( sCC_Bl1280885523602775798t_unit @ e3 @ X ) @ ( sCC_Bl1280885523602775798t_unit @ e3 @ Xa ) )
= bot_bot_set_v ) ) ) ) ).
% \<open>\<forall>n\<in>set (stack e''). \<forall>m\<in>set (stack e''). n \<noteq> m \<longrightarrow> \<S> e'' n \<inter> \<S> e'' m = {}\<close>
thf(fact_21_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_22_reachable__refl,axiom,
! [X2: v] : ( sCC_Bl649662514949026229able_v @ successors @ X2 @ X2 ) ).
% reachable_refl
thf(fact_23_reachable__succ,axiom,
! [Y: v,X2: v,Z: v] :
( ( member_v @ Y @ ( successors @ X2 ) )
=> ( ( sCC_Bl649662514949026229able_v @ successors @ Y @ Z )
=> ( sCC_Bl649662514949026229able_v @ successors @ X2 @ Z ) ) ) ).
% reachable_succ
thf(fact_24_reachable_Osimps,axiom,
! [A1: v,A2: v] :
( ( sCC_Bl649662514949026229able_v @ successors @ A1 @ A2 )
= ( ? [X3: v] :
( ( A1 = X3 )
& ( A2 = X3 ) )
| ? [X3: v,Y3: v,Z2: v] :
( ( A1 = X3 )
& ( A2 = Z2 )
& ( member_v @ Y3 @ ( successors @ X3 ) )
& ( sCC_Bl649662514949026229able_v @ successors @ Y3 @ Z2 ) ) ) ) ).
% reachable.simps
thf(fact_25_reachable__edge,axiom,
! [Y: v,X2: v] :
( ( member_v @ Y @ ( successors @ X2 ) )
=> ( sCC_Bl649662514949026229able_v @ successors @ X2 @ Y ) ) ).
% reachable_edge
thf(fact_26_reachable__end__induct,axiom,
! [X2: v,Y: v,P: v > v > $o] :
( ( sCC_Bl649662514949026229able_v @ successors @ X2 @ Y )
=> ( ! [X4: v] : ( P @ X4 @ X4 )
=> ( ! [X4: v,Y2: v,Z3: v] :
( ( P @ X4 @ Y2 )
=> ( ( member_v @ Z3 @ ( successors @ Y2 ) )
=> ( P @ X4 @ Z3 ) ) )
=> ( P @ X2 @ Y ) ) ) ) ).
% reachable_end_induct
thf(fact_27_reachable__trans,axiom,
! [X2: v,Y: v,Z: v] :
( ( sCC_Bl649662514949026229able_v @ successors @ X2 @ Y )
=> ( ( sCC_Bl649662514949026229able_v @ successors @ Y @ Z )
=> ( sCC_Bl649662514949026229able_v @ successors @ X2 @ Z ) ) ) ).
% reachable_trans
thf(fact_28_succ__reachable,axiom,
! [X2: v,Y: v,Z: v] :
( ( sCC_Bl649662514949026229able_v @ successors @ X2 @ Y )
=> ( ( member_v @ Z @ ( successors @ Y ) )
=> ( sCC_Bl649662514949026229able_v @ successors @ X2 @ Z ) ) ) ).
% succ_reachable
thf(fact_29__092_060open_062_092_060forall_062x_Ay_O_Ax_A_092_060preceq_062_Ay_Ain_Astack_Ae_H_H_A_092_060and_062_Ax_A_092_060noteq_062_Ay_A_092_060longrightarrow_062_A_I_092_060forall_062u_092_060in_062_092_060S_062_Ae_H_H_Ax_O_A_092_060not_062_Areachable__avoiding_Au_Ay_A_Iunvisited_Ae_H_H_Ax_J_J_092_060close_062,axiom,
! [X: v,Y4: v] :
( ( ( sCC_Bl4022239298816431255edes_v @ X @ Y4 @ ( sCC_Bl8828226123343373779t_unit @ e3 ) )
& ( X != Y4 ) )
=> ! [Xa: v] :
( ( member_v @ Xa @ ( sCC_Bl1280885523602775798t_unit @ e3 @ X ) )
=> ~ ( sCC_Bl4291963740693775144ding_v @ successors @ Xa @ Y4 @ ( sCC_Bl3123350270117520219t_unit @ successors @ e3 @ X ) ) ) ) ).
% \<open>\<forall>x y. x \<preceq> y in stack e'' \<and> x \<noteq> y \<longrightarrow> (\<forall>u\<in>\<S> e'' x. \<not> reachable_avoiding u y (unvisited e'' x))\<close>
thf(fact_30__092_060open_062_092_060forall_062n_092_060in_062visited_Ae_H_H_A_N_Aset_A_Icstack_Ae_H_H_J_O_Avsuccs_Ae_H_H_An_A_061_Asuccessors_An_092_060close_062,axiom,
! [X: v] :
( ( member_v @ X @ ( minus_minus_set_v @ ( sCC_Bl4645233313691564917t_unit @ e3 ) @ ( set_v2 @ ( sCC_Bl9201514103433284750t_unit @ e3 ) ) ) )
=> ( ( sCC_Bl3795065053823578884t_unit @ e3 @ X )
= ( successors @ X ) ) ) ).
% \<open>\<forall>n\<in>visited e'' - set (cstack e''). vsuccs e'' n = successors n\<close>
thf(fact_31_S__reflexive,axiom,
! [E2: sCC_Bl1394983891496994913t_unit,N2: v] :
( ( sCC_Bl9196236973127232072t_unit @ successors @ E2 )
=> ( member_v @ N2 @ ( sCC_Bl1280885523602775798t_unit @ E2 @ N2 ) ) ) ).
% S_reflexive
thf(fact_32_scc__partition,axiom,
! [S: set_v,S2: set_v,X2: v] :
( ( sCC_Bloemen_is_scc_v @ successors @ S )
=> ( ( sCC_Bloemen_is_scc_v @ successors @ S2 )
=> ( ( member_v @ X2 @ ( inf_inf_set_v @ S @ S2 ) )
=> ( S = S2 ) ) ) ) ).
% scc_partition
thf(fact_33_sccE,axiom,
! [S: set_v,X2: v,Y: v] :
( ( sCC_Bloemen_is_scc_v @ successors @ S )
=> ( ( member_v @ X2 @ S )
=> ( ( sCC_Bl649662514949026229able_v @ successors @ X2 @ Y )
=> ( ( sCC_Bl649662514949026229able_v @ successors @ Y @ X2 )
=> ( member_v @ Y @ S ) ) ) ) ) ).
% sccE
thf(fact_34_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_35_is__subscc__def,axiom,
! [S: set_v] :
( ( sCC_Bl5398416737448265317bscc_v @ successors @ S )
= ( ! [X3: v] :
( ( member_v @ X3 @ S )
=> ! [Y3: v] :
( ( member_v @ Y3 @ S )
=> ( sCC_Bl649662514949026229able_v @ successors @ X3 @ Y3 ) ) ) ) ) ).
% is_subscc_def
thf(fact_36_ra__reachable,axiom,
! [X2: v,Y: v,E: set_Product_prod_v_v] :
( ( sCC_Bl4291963740693775144ding_v @ successors @ X2 @ Y @ E )
=> ( sCC_Bl649662514949026229able_v @ successors @ X2 @ Y ) ) ).
% ra_reachable
thf(fact_37_graph_Owf__env_Ocong,axiom,
sCC_Bl9196236973127232072t_unit = sCC_Bl9196236973127232072t_unit ).
% graph.wf_env.cong
thf(fact_38_stack__unexplored,axiom,
! [E2: sCC_Bl1394983891496994913t_unit,N2: v] :
( ( sCC_Bl9196236973127232072t_unit @ successors @ E2 )
=> ( ( member_v @ N2 @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E2 ) ) )
=> ~ ( member_v @ N2 @ ( sCC_Bl157864678168468314t_unit @ E2 ) ) ) ) ).
% stack_unexplored
thf(fact_39_stack__visited,axiom,
! [E2: sCC_Bl1394983891496994913t_unit,N2: v] :
( ( sCC_Bl9196236973127232072t_unit @ successors @ E2 )
=> ( ( member_v @ N2 @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E2 ) ) )
=> ( member_v @ N2 @ ( sCC_Bl4645233313691564917t_unit @ E2 ) ) ) ) ).
% stack_visited
thf(fact_40_subscc__add,axiom,
! [S: set_v,X2: v,Y: v] :
( ( sCC_Bl5398416737448265317bscc_v @ successors @ S )
=> ( ( member_v @ X2 @ S )
=> ( ( sCC_Bl649662514949026229able_v @ successors @ X2 @ Y )
=> ( ( sCC_Bl649662514949026229able_v @ successors @ Y @ X2 )
=> ( sCC_Bl5398416737448265317bscc_v @ successors @ ( insert_v @ Y @ S ) ) ) ) ) ) ).
% subscc_add
thf(fact_41_sclosed,axiom,
! [X: v] :
( ( member_v @ X @ vertices )
=> ( ord_less_eq_set_v @ ( successors @ X ) @ vertices ) ) ).
% sclosed
thf(fact_42_is__scc__def,axiom,
! [S: set_v] :
( ( sCC_Bloemen_is_scc_v @ successors @ S )
= ( ( S != bot_bot_set_v )
& ( sCC_Bl5398416737448265317bscc_v @ successors @ S )
& ! [S3: set_v] :
( ( ( ord_less_eq_set_v @ S @ S3 )
& ( sCC_Bl5398416737448265317bscc_v @ successors @ S3 ) )
=> ( S3 = S ) ) ) ) ).
% is_scc_def
thf(fact_43_mem__Collect__eq,axiom,
! [A: v,P: v > $o] :
( ( member_v @ A @ ( collect_v @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_44_mem__Collect__eq,axiom,
! [A: product_prod_v_v,P: product_prod_v_v > $o] :
( ( member7453568604450474000od_v_v @ A @ ( collec140062887454715474od_v_v @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_45_mem__Collect__eq,axiom,
! [A: set_v,P: set_v > $o] :
( ( member_set_v @ A @ ( collect_set_v @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_46_Collect__mem__eq,axiom,
! [A3: set_v] :
( ( collect_v
@ ^ [X3: v] : ( member_v @ X3 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_47_Collect__mem__eq,axiom,
! [A3: set_Product_prod_v_v] :
( ( collec140062887454715474od_v_v
@ ^ [X3: product_prod_v_v] : ( member7453568604450474000od_v_v @ X3 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_48_Collect__mem__eq,axiom,
! [A3: set_set_v] :
( ( collect_set_v
@ ^ [X3: set_v] : ( member_set_v @ X3 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_49_Collect__cong,axiom,
! [P: set_v > $o,Q: set_v > $o] :
( ! [X4: set_v] :
( ( P @ X4 )
= ( Q @ X4 ) )
=> ( ( collect_set_v @ P )
= ( collect_set_v @ Q ) ) ) ).
% Collect_cong
thf(fact_50_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 )
=> ( ! [X4: v] :
( ( member_v @ X4 @ ( sCC_Bl4645233313691564917t_unit @ E2 ) )
=> ! [Xa2: v] :
( ( member_v @ Xa2 @ ( minus_minus_set_v @ ( successors @ X4 ) @ ( sCC_Bl3795065053823578884t_unit @ E2 @ X4 ) ) )
=> ( ( sCC_Bl649662514949026229able_v @ successors @ V @ X4 )
=> ~ ( sCC_Bl649662514949026229able_v @ successors @ Xa2 @ W ) ) ) )
=> ( member_v @ W @ ( sCC_Bl4645233313691564917t_unit @ E2 ) ) ) ) ) ) ).
% reachable_visited
thf(fact_51_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_52_re__refl,axiom,
! [X2: v] : ( sCC_Bl770211535891879572_end_v @ successors @ X2 @ X2 ) ).
% re_refl
thf(fact_53_re__succ,axiom,
! [X2: v,Y: v,Z: v] :
( ( sCC_Bl770211535891879572_end_v @ successors @ X2 @ Y )
=> ( ( member_v @ Z @ ( successors @ Y ) )
=> ( sCC_Bl770211535891879572_end_v @ successors @ X2 @ Z ) ) ) ).
% re_succ
thf(fact_54_reachable__end_Osimps,axiom,
! [A1: v,A2: v] :
( ( sCC_Bl770211535891879572_end_v @ successors @ A1 @ A2 )
= ( ? [X3: v] :
( ( A1 = X3 )
& ( A2 = X3 ) )
| ? [X3: v,Y3: v,Z2: v] :
( ( A1 = X3 )
& ( A2 = Z2 )
& ( sCC_Bl770211535891879572_end_v @ successors @ X3 @ Y3 )
& ( member_v @ Z2 @ ( successors @ Y3 ) ) ) ) ) ).
% reachable_end.simps
thf(fact_55_succ__re,axiom,
! [Y: v,X2: v,Z: v] :
( ( member_v @ Y @ ( successors @ X2 ) )
=> ( ( sCC_Bl770211535891879572_end_v @ successors @ Y @ Z )
=> ( sCC_Bl770211535891879572_end_v @ successors @ X2 @ Z ) ) ) ).
% succ_re
thf(fact_56_reachable__re,axiom,
! [X2: v,Y: v] :
( ( sCC_Bl649662514949026229able_v @ successors @ X2 @ Y )
=> ( sCC_Bl770211535891879572_end_v @ successors @ X2 @ Y ) ) ).
% reachable_re
thf(fact_57_re__reachable,axiom,
! [X2: v,Y: v] :
( ( sCC_Bl770211535891879572_end_v @ successors @ X2 @ Y )
=> ( sCC_Bl649662514949026229able_v @ successors @ X2 @ Y ) ) ).
% re_reachable
thf(fact_58_visited__unexplored,axiom,
! [E2: sCC_Bl1394983891496994913t_unit,M2: v] :
( ( sCC_Bl9196236973127232072t_unit @ successors @ E2 )
=> ( ( member_v @ M2 @ ( sCC_Bl4645233313691564917t_unit @ E2 ) )
=> ( ~ ( member_v @ M2 @ ( sCC_Bl157864678168468314t_unit @ E2 ) )
=> ~ ! [N3: v] :
( ( member_v @ N3 @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E2 ) ) )
=> ~ ( member_v @ M2 @ ( sCC_Bl1280885523602775798t_unit @ E2 @ N3 ) ) ) ) ) ) ).
% visited_unexplored
thf(fact_59_precedes__refl,axiom,
! [X2: product_prod_v_v,Xs: list_P7986770385144383213od_v_v] :
( ( sCC_Bl2026170059108282219od_v_v @ X2 @ X2 @ Xs )
= ( member7453568604450474000od_v_v @ X2 @ ( set_Product_prod_v_v2 @ Xs ) ) ) ).
% precedes_refl
thf(fact_60_precedes__refl,axiom,
! [X2: v,Xs: list_v] :
( ( sCC_Bl4022239298816431255edes_v @ X2 @ X2 @ Xs )
= ( member_v @ X2 @ ( set_v2 @ Xs ) ) ) ).
% precedes_refl
thf(fact_61_stack__class,axiom,
! [E2: sCC_Bl1394983891496994913t_unit,N2: v,M2: v] :
( ( sCC_Bl9196236973127232072t_unit @ successors @ E2 )
=> ( ( member_v @ N2 @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E2 ) ) )
=> ( ( member_v @ M2 @ ( sCC_Bl1280885523602775798t_unit @ E2 @ N2 ) )
=> ( member_v @ M2 @ ( minus_minus_set_v @ ( sCC_Bl4645233313691564917t_unit @ E2 ) @ ( sCC_Bl157864678168468314t_unit @ E2 ) ) ) ) ) ) ).
% stack_class
thf(fact_62_graph__axioms,axiom,
sCC_Bloemen_graph_v @ vertices @ successors ).
% graph_axioms
thf(fact_63_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 )
& ! [X3: v] :
( ( member_v @ X3 @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E2 ) ) )
=> ( sCC_Bl649662514949026229able_v @ successors @ X3 @ V ) ) ) ) ).
% pre_dfs_def
thf(fact_64_graph_Oreachable__end_Ocong,axiom,
sCC_Bl770211535891879572_end_v = sCC_Bl770211535891879572_end_v ).
% graph.reachable_end.cong
thf(fact_65_precedes__antisym,axiom,
! [X2: v,Y: v,Xs: list_v] :
( ( sCC_Bl4022239298816431255edes_v @ X2 @ Y @ Xs )
=> ( ( sCC_Bl4022239298816431255edes_v @ Y @ X2 @ Xs )
=> ( ( distinct_v @ Xs )
=> ( X2 = Y ) ) ) ) ).
% precedes_antisym
thf(fact_66_precedes__trans,axiom,
! [X2: v,Y: v,Xs: list_v,Z: v] :
( ( sCC_Bl4022239298816431255edes_v @ X2 @ Y @ Xs )
=> ( ( sCC_Bl4022239298816431255edes_v @ Y @ Z @ Xs )
=> ( ( distinct_v @ Xs )
=> ( sCC_Bl4022239298816431255edes_v @ X2 @ Z @ Xs ) ) ) ) ).
% precedes_trans
thf(fact_67_precedes__mem_I1_J,axiom,
! [X2: product_prod_v_v,Y: product_prod_v_v,Xs: list_P7986770385144383213od_v_v] :
( ( sCC_Bl2026170059108282219od_v_v @ X2 @ Y @ Xs )
=> ( member7453568604450474000od_v_v @ X2 @ ( set_Product_prod_v_v2 @ Xs ) ) ) ).
% precedes_mem(1)
thf(fact_68_precedes__mem_I1_J,axiom,
! [X2: v,Y: v,Xs: list_v] :
( ( sCC_Bl4022239298816431255edes_v @ X2 @ Y @ Xs )
=> ( member_v @ X2 @ ( set_v2 @ Xs ) ) ) ).
% precedes_mem(1)
thf(fact_69_precedes__mem_I2_J,axiom,
! [X2: product_prod_v_v,Y: product_prod_v_v,Xs: list_P7986770385144383213od_v_v] :
( ( sCC_Bl2026170059108282219od_v_v @ X2 @ Y @ Xs )
=> ( member7453568604450474000od_v_v @ Y @ ( set_Product_prod_v_v2 @ Xs ) ) ) ).
% precedes_mem(2)
thf(fact_70_precedes__mem_I2_J,axiom,
! [X2: v,Y: v,Xs: list_v] :
( ( sCC_Bl4022239298816431255edes_v @ X2 @ Y @ Xs )
=> ( member_v @ Y @ ( set_v2 @ Xs ) ) ) ).
% precedes_mem(2)
thf(fact_71_graph_Oreachable_Ocong,axiom,
sCC_Bl649662514949026229able_v = sCC_Bl649662514949026229able_v ).
% graph.reachable.cong
thf(fact_72_graph_Oreachable__avoiding_Ocong,axiom,
sCC_Bl4291963740693775144ding_v = sCC_Bl4291963740693775144ding_v ).
% graph.reachable_avoiding.cong
thf(fact_73_graph_Ois__subscc_Ocong,axiom,
sCC_Bl5398416737448265317bscc_v = sCC_Bl5398416737448265317bscc_v ).
% graph.is_subscc.cong
thf(fact_74_graph_Ois__scc_Ocong,axiom,
sCC_Bloemen_is_scc_v = sCC_Bloemen_is_scc_v ).
% graph.is_scc.cong
thf(fact_75_graph_Ounvisited_Ocong,axiom,
sCC_Bl3123350270117520219t_unit = sCC_Bl3123350270117520219t_unit ).
% graph.unvisited.cong
thf(fact_76_Diff__disjoint,axiom,
! [A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ A3 @ ( minus_4183494784930505774od_v_v @ B @ A3 ) )
= bot_bo723834152578015283od_v_v ) ).
% Diff_disjoint
thf(fact_77_Diff__disjoint,axiom,
! [A3: set_set_v,B: set_set_v] :
( ( inf_inf_set_set_v @ A3 @ ( minus_7228012346218142266_set_v @ B @ A3 ) )
= bot_bot_set_set_v ) ).
% Diff_disjoint
thf(fact_78_Diff__disjoint,axiom,
! [A3: set_v,B: set_v] :
( ( inf_inf_set_v @ A3 @ ( minus_minus_set_v @ B @ A3 ) )
= bot_bot_set_v ) ).
% Diff_disjoint
thf(fact_79_insert__Diff__single,axiom,
! [A: product_prod_v_v,A3: set_Product_prod_v_v] :
( ( insert1338601472111419319od_v_v @ A @ ( minus_4183494784930505774od_v_v @ A3 @ ( insert1338601472111419319od_v_v @ A @ bot_bo723834152578015283od_v_v ) ) )
= ( insert1338601472111419319od_v_v @ A @ A3 ) ) ).
% insert_Diff_single
thf(fact_80_insert__Diff__single,axiom,
! [A: set_v,A3: set_set_v] :
( ( insert_set_v @ A @ ( minus_7228012346218142266_set_v @ A3 @ ( insert_set_v @ A @ bot_bot_set_set_v ) ) )
= ( insert_set_v @ A @ A3 ) ) ).
% insert_Diff_single
thf(fact_81_insert__Diff__single,axiom,
! [A: v,A3: set_v] :
( ( insert_v @ A @ ( minus_minus_set_v @ A3 @ ( insert_v @ A @ bot_bot_set_v ) ) )
= ( insert_v @ A @ A3 ) ) ).
% insert_Diff_single
thf(fact_82_Diff__eq__empty__iff,axiom,
! [A3: set_set_v,B: set_set_v] :
( ( ( minus_7228012346218142266_set_v @ A3 @ B )
= bot_bot_set_set_v )
= ( ord_le5216385588623774835_set_v @ A3 @ B ) ) ).
% Diff_eq_empty_iff
thf(fact_83_Diff__eq__empty__iff,axiom,
! [A3: set_v,B: set_v] :
( ( ( minus_minus_set_v @ A3 @ B )
= bot_bot_set_v )
= ( ord_less_eq_set_v @ A3 @ B ) ) ).
% Diff_eq_empty_iff
thf(fact_84_Diff__eq__empty__iff,axiom,
! [A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ( minus_4183494784930505774od_v_v @ A3 @ B )
= bot_bo723834152578015283od_v_v )
= ( ord_le7336532860387713383od_v_v @ A3 @ B ) ) ).
% Diff_eq_empty_iff
thf(fact_85_disjoint__insert_I2_J,axiom,
! [A3: set_v,B2: v,B: set_v] :
( ( bot_bot_set_v
= ( inf_inf_set_v @ A3 @ ( insert_v @ B2 @ B ) ) )
= ( ~ ( member_v @ B2 @ A3 )
& ( bot_bot_set_v
= ( inf_inf_set_v @ A3 @ B ) ) ) ) ).
% disjoint_insert(2)
thf(fact_86_disjoint__insert_I2_J,axiom,
! [A3: set_Product_prod_v_v,B2: product_prod_v_v,B: set_Product_prod_v_v] :
( ( bot_bo723834152578015283od_v_v
= ( inf_in6271465464967711157od_v_v @ A3 @ ( insert1338601472111419319od_v_v @ B2 @ B ) ) )
= ( ~ ( member7453568604450474000od_v_v @ B2 @ A3 )
& ( bot_bo723834152578015283od_v_v
= ( inf_in6271465464967711157od_v_v @ A3 @ B ) ) ) ) ).
% disjoint_insert(2)
thf(fact_87_disjoint__insert_I2_J,axiom,
! [A3: set_set_v,B2: set_v,B: set_set_v] :
( ( bot_bot_set_set_v
= ( inf_inf_set_set_v @ A3 @ ( insert_set_v @ B2 @ B ) ) )
= ( ~ ( member_set_v @ B2 @ A3 )
& ( bot_bot_set_set_v
= ( inf_inf_set_set_v @ A3 @ B ) ) ) ) ).
% disjoint_insert(2)
thf(fact_88_disjoint__insert_I1_J,axiom,
! [B: set_v,A: v,A3: set_v] :
( ( ( inf_inf_set_v @ B @ ( insert_v @ A @ A3 ) )
= bot_bot_set_v )
= ( ~ ( member_v @ A @ B )
& ( ( inf_inf_set_v @ B @ A3 )
= bot_bot_set_v ) ) ) ).
% disjoint_insert(1)
thf(fact_89_disjoint__insert_I1_J,axiom,
! [B: set_Product_prod_v_v,A: product_prod_v_v,A3: set_Product_prod_v_v] :
( ( ( inf_in6271465464967711157od_v_v @ B @ ( insert1338601472111419319od_v_v @ A @ A3 ) )
= bot_bo723834152578015283od_v_v )
= ( ~ ( member7453568604450474000od_v_v @ A @ B )
& ( ( inf_in6271465464967711157od_v_v @ B @ A3 )
= bot_bo723834152578015283od_v_v ) ) ) ).
% disjoint_insert(1)
thf(fact_90_disjoint__insert_I1_J,axiom,
! [B: set_set_v,A: set_v,A3: set_set_v] :
( ( ( inf_inf_set_set_v @ B @ ( insert_set_v @ A @ A3 ) )
= bot_bot_set_set_v )
= ( ~ ( member_set_v @ A @ B )
& ( ( inf_inf_set_set_v @ B @ A3 )
= bot_bot_set_set_v ) ) ) ).
% disjoint_insert(1)
thf(fact_91_insert__disjoint_I2_J,axiom,
! [A: v,A3: set_v,B: set_v] :
( ( bot_bot_set_v
= ( inf_inf_set_v @ ( insert_v @ A @ A3 ) @ B ) )
= ( ~ ( member_v @ A @ B )
& ( bot_bot_set_v
= ( inf_inf_set_v @ A3 @ B ) ) ) ) ).
% insert_disjoint(2)
thf(fact_92_insert__disjoint_I2_J,axiom,
! [A: product_prod_v_v,A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( bot_bo723834152578015283od_v_v
= ( inf_in6271465464967711157od_v_v @ ( insert1338601472111419319od_v_v @ A @ A3 ) @ B ) )
= ( ~ ( member7453568604450474000od_v_v @ A @ B )
& ( bot_bo723834152578015283od_v_v
= ( inf_in6271465464967711157od_v_v @ A3 @ B ) ) ) ) ).
% insert_disjoint(2)
thf(fact_93_insert__disjoint_I2_J,axiom,
! [A: set_v,A3: set_set_v,B: set_set_v] :
( ( bot_bot_set_set_v
= ( inf_inf_set_set_v @ ( insert_set_v @ A @ A3 ) @ B ) )
= ( ~ ( member_set_v @ A @ B )
& ( bot_bot_set_set_v
= ( inf_inf_set_set_v @ A3 @ B ) ) ) ) ).
% insert_disjoint(2)
thf(fact_94_insert__disjoint_I1_J,axiom,
! [A: v,A3: set_v,B: set_v] :
( ( ( inf_inf_set_v @ ( insert_v @ A @ A3 ) @ B )
= bot_bot_set_v )
= ( ~ ( member_v @ A @ B )
& ( ( inf_inf_set_v @ A3 @ B )
= bot_bot_set_v ) ) ) ).
% insert_disjoint(1)
thf(fact_95_insert__disjoint_I1_J,axiom,
! [A: product_prod_v_v,A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ( inf_in6271465464967711157od_v_v @ ( insert1338601472111419319od_v_v @ A @ A3 ) @ B )
= bot_bo723834152578015283od_v_v )
= ( ~ ( member7453568604450474000od_v_v @ A @ B )
& ( ( inf_in6271465464967711157od_v_v @ A3 @ B )
= bot_bo723834152578015283od_v_v ) ) ) ).
% insert_disjoint(1)
thf(fact_96_insert__disjoint_I1_J,axiom,
! [A: set_v,A3: set_set_v,B: set_set_v] :
( ( ( inf_inf_set_set_v @ ( insert_set_v @ A @ A3 ) @ B )
= bot_bot_set_set_v )
= ( ~ ( member_set_v @ A @ B )
& ( ( inf_inf_set_set_v @ A3 @ B )
= bot_bot_set_set_v ) ) ) ).
% insert_disjoint(1)
thf(fact_97_ccpo__Sup__singleton,axiom,
! [X2: set_v] :
( ( comple2307003700295860064_set_v @ ( insert_set_v @ X2 @ bot_bot_set_set_v ) )
= X2 ) ).
% ccpo_Sup_singleton
thf(fact_98_cSup__singleton,axiom,
! [X2: set_v] :
( ( comple2307003700295860064_set_v @ ( insert_set_v @ X2 @ bot_bot_set_set_v ) )
= X2 ) ).
% cSup_singleton
thf(fact_99_singleton__insert__inj__eq,axiom,
! [B2: set_v,A: set_v,A3: set_set_v] :
( ( ( insert_set_v @ B2 @ bot_bot_set_set_v )
= ( insert_set_v @ A @ A3 ) )
= ( ( A = B2 )
& ( ord_le5216385588623774835_set_v @ A3 @ ( insert_set_v @ B2 @ bot_bot_set_set_v ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_100_singleton__insert__inj__eq,axiom,
! [B2: v,A: v,A3: set_v] :
( ( ( insert_v @ B2 @ bot_bot_set_v )
= ( insert_v @ A @ A3 ) )
= ( ( A = B2 )
& ( ord_less_eq_set_v @ A3 @ ( insert_v @ B2 @ bot_bot_set_v ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_101_singleton__insert__inj__eq,axiom,
! [B2: product_prod_v_v,A: product_prod_v_v,A3: set_Product_prod_v_v] :
( ( ( insert1338601472111419319od_v_v @ B2 @ bot_bo723834152578015283od_v_v )
= ( insert1338601472111419319od_v_v @ A @ A3 ) )
= ( ( A = B2 )
& ( ord_le7336532860387713383od_v_v @ A3 @ ( insert1338601472111419319od_v_v @ B2 @ bot_bo723834152578015283od_v_v ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_102_singleton__insert__inj__eq_H,axiom,
! [A: set_v,A3: set_set_v,B2: set_v] :
( ( ( insert_set_v @ A @ A3 )
= ( insert_set_v @ B2 @ bot_bot_set_set_v ) )
= ( ( A = B2 )
& ( ord_le5216385588623774835_set_v @ A3 @ ( insert_set_v @ B2 @ bot_bot_set_set_v ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_103_singleton__insert__inj__eq_H,axiom,
! [A: v,A3: set_v,B2: v] :
( ( ( insert_v @ A @ A3 )
= ( insert_v @ B2 @ bot_bot_set_v ) )
= ( ( A = B2 )
& ( ord_less_eq_set_v @ A3 @ ( insert_v @ B2 @ bot_bot_set_v ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_104_singleton__insert__inj__eq_H,axiom,
! [A: product_prod_v_v,A3: set_Product_prod_v_v,B2: product_prod_v_v] :
( ( ( insert1338601472111419319od_v_v @ A @ A3 )
= ( insert1338601472111419319od_v_v @ B2 @ bot_bo723834152578015283od_v_v ) )
= ( ( A = B2 )
& ( ord_le7336532860387713383od_v_v @ A3 @ ( insert1338601472111419319od_v_v @ B2 @ bot_bo723834152578015283od_v_v ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_105_ra__mono,axiom,
! [X2: v,Y: v,E: set_Product_prod_v_v,E5: set_Product_prod_v_v] :
( ( sCC_Bl4291963740693775144ding_v @ successors @ X2 @ Y @ E )
=> ( ( ord_le7336532860387713383od_v_v @ E5 @ E )
=> ( sCC_Bl4291963740693775144ding_v @ successors @ X2 @ Y @ E5 ) ) ) ).
% ra_mono
thf(fact_106_init__env__pre__dfs,axiom,
! [V: v] : ( sCC_Bl36166008131615352t_unit @ successors @ V @ ( sCC_Bl7693227186847904995_env_v @ V ) ) ).
% init_env_pre_dfs
thf(fact_107_ra__empty,axiom,
! [X2: v,Y: v] :
( ( sCC_Bl4291963740693775144ding_v @ successors @ X2 @ Y @ bot_bo723834152578015283od_v_v )
= ( sCC_Bl649662514949026229able_v @ successors @ X2 @ Y ) ) ).
% ra_empty
thf(fact_108_empty__Collect__eq,axiom,
! [P: v > $o] :
( ( bot_bot_set_v
= ( collect_v @ P ) )
= ( ! [X3: v] :
~ ( P @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_109_empty__Collect__eq,axiom,
! [P: product_prod_v_v > $o] :
( ( bot_bo723834152578015283od_v_v
= ( collec140062887454715474od_v_v @ P ) )
= ( ! [X3: product_prod_v_v] :
~ ( P @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_110_empty__Collect__eq,axiom,
! [P: set_v > $o] :
( ( bot_bot_set_set_v
= ( collect_set_v @ P ) )
= ( ! [X3: set_v] :
~ ( P @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_111_Collect__empty__eq,axiom,
! [P: v > $o] :
( ( ( collect_v @ P )
= bot_bot_set_v )
= ( ! [X3: v] :
~ ( P @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_112_Collect__empty__eq,axiom,
! [P: product_prod_v_v > $o] :
( ( ( collec140062887454715474od_v_v @ P )
= bot_bo723834152578015283od_v_v )
= ( ! [X3: product_prod_v_v] :
~ ( P @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_113_Collect__empty__eq,axiom,
! [P: set_v > $o] :
( ( ( collect_set_v @ P )
= bot_bot_set_set_v )
= ( ! [X3: set_v] :
~ ( P @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_114_all__not__in__conv,axiom,
! [A3: set_v] :
( ( ! [X3: v] :
~ ( member_v @ X3 @ A3 ) )
= ( A3 = bot_bot_set_v ) ) ).
% all_not_in_conv
thf(fact_115_all__not__in__conv,axiom,
! [A3: set_Product_prod_v_v] :
( ( ! [X3: product_prod_v_v] :
~ ( member7453568604450474000od_v_v @ X3 @ A3 ) )
= ( A3 = bot_bo723834152578015283od_v_v ) ) ).
% all_not_in_conv
thf(fact_116_all__not__in__conv,axiom,
! [A3: set_set_v] :
( ( ! [X3: set_v] :
~ ( member_set_v @ X3 @ A3 ) )
= ( A3 = bot_bot_set_set_v ) ) ).
% all_not_in_conv
thf(fact_117_empty__iff,axiom,
! [C: v] :
~ ( member_v @ C @ bot_bot_set_v ) ).
% empty_iff
thf(fact_118_empty__iff,axiom,
! [C: product_prod_v_v] :
~ ( member7453568604450474000od_v_v @ C @ bot_bo723834152578015283od_v_v ) ).
% empty_iff
thf(fact_119_empty__iff,axiom,
! [C: set_v] :
~ ( member_set_v @ C @ bot_bot_set_set_v ) ).
% empty_iff
thf(fact_120_subset__antisym,axiom,
! [A3: set_v,B: set_v] :
( ( ord_less_eq_set_v @ A3 @ B )
=> ( ( ord_less_eq_set_v @ B @ A3 )
=> ( A3 = B ) ) ) ).
% subset_antisym
thf(fact_121_subset__antisym,axiom,
! [A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A3 @ B )
=> ( ( ord_le7336532860387713383od_v_v @ B @ A3 )
=> ( A3 = B ) ) ) ).
% subset_antisym
thf(fact_122_subsetI,axiom,
! [A3: set_v,B: set_v] :
( ! [X4: v] :
( ( member_v @ X4 @ A3 )
=> ( member_v @ X4 @ B ) )
=> ( ord_less_eq_set_v @ A3 @ B ) ) ).
% subsetI
thf(fact_123_subsetI,axiom,
! [A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ! [X4: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X4 @ A3 )
=> ( member7453568604450474000od_v_v @ X4 @ B ) )
=> ( ord_le7336532860387713383od_v_v @ A3 @ B ) ) ).
% subsetI
thf(fact_124_insert__absorb2,axiom,
! [X2: v,A3: set_v] :
( ( insert_v @ X2 @ ( insert_v @ X2 @ A3 ) )
= ( insert_v @ X2 @ A3 ) ) ).
% insert_absorb2
thf(fact_125_insert__absorb2,axiom,
! [X2: product_prod_v_v,A3: set_Product_prod_v_v] :
( ( insert1338601472111419319od_v_v @ X2 @ ( insert1338601472111419319od_v_v @ X2 @ A3 ) )
= ( insert1338601472111419319od_v_v @ X2 @ A3 ) ) ).
% insert_absorb2
thf(fact_126_insert__absorb2,axiom,
! [X2: set_v,A3: set_set_v] :
( ( insert_set_v @ X2 @ ( insert_set_v @ X2 @ A3 ) )
= ( insert_set_v @ X2 @ A3 ) ) ).
% insert_absorb2
thf(fact_127_insert__iff,axiom,
! [A: set_v,B2: set_v,A3: set_set_v] :
( ( member_set_v @ A @ ( insert_set_v @ B2 @ A3 ) )
= ( ( A = B2 )
| ( member_set_v @ A @ A3 ) ) ) ).
% insert_iff
thf(fact_128_insert__iff,axiom,
! [A: v,B2: v,A3: set_v] :
( ( member_v @ A @ ( insert_v @ B2 @ A3 ) )
= ( ( A = B2 )
| ( member_v @ A @ A3 ) ) ) ).
% insert_iff
thf(fact_129_insert__iff,axiom,
! [A: product_prod_v_v,B2: product_prod_v_v,A3: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ A @ ( insert1338601472111419319od_v_v @ B2 @ A3 ) )
= ( ( A = B2 )
| ( member7453568604450474000od_v_v @ A @ A3 ) ) ) ).
% insert_iff
thf(fact_130_insertCI,axiom,
! [A: set_v,B: set_set_v,B2: set_v] :
( ( ~ ( member_set_v @ A @ B )
=> ( A = B2 ) )
=> ( member_set_v @ A @ ( insert_set_v @ B2 @ B ) ) ) ).
% insertCI
thf(fact_131_insertCI,axiom,
! [A: v,B: set_v,B2: v] :
( ( ~ ( member_v @ A @ B )
=> ( A = B2 ) )
=> ( member_v @ A @ ( insert_v @ B2 @ B ) ) ) ).
% insertCI
thf(fact_132_insertCI,axiom,
! [A: product_prod_v_v,B: set_Product_prod_v_v,B2: product_prod_v_v] :
( ( ~ ( member7453568604450474000od_v_v @ A @ B )
=> ( A = B2 ) )
=> ( member7453568604450474000od_v_v @ A @ ( insert1338601472111419319od_v_v @ B2 @ B ) ) ) ).
% insertCI
thf(fact_133_Int__iff,axiom,
! [C: product_prod_v_v,A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ C @ ( inf_in6271465464967711157od_v_v @ A3 @ B ) )
= ( ( member7453568604450474000od_v_v @ C @ A3 )
& ( member7453568604450474000od_v_v @ C @ B ) ) ) ).
% Int_iff
thf(fact_134_Int__iff,axiom,
! [C: v,A3: set_v,B: set_v] :
( ( member_v @ C @ ( inf_inf_set_v @ A3 @ B ) )
= ( ( member_v @ C @ A3 )
& ( member_v @ C @ B ) ) ) ).
% Int_iff
thf(fact_135_IntI,axiom,
! [C: product_prod_v_v,A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ C @ A3 )
=> ( ( member7453568604450474000od_v_v @ C @ B )
=> ( member7453568604450474000od_v_v @ C @ ( inf_in6271465464967711157od_v_v @ A3 @ B ) ) ) ) ).
% IntI
thf(fact_136_IntI,axiom,
! [C: v,A3: set_v,B: set_v] :
( ( member_v @ C @ A3 )
=> ( ( member_v @ C @ B )
=> ( member_v @ C @ ( inf_inf_set_v @ A3 @ B ) ) ) ) ).
% IntI
thf(fact_137_Diff__idemp,axiom,
! [A3: set_v,B: set_v] :
( ( minus_minus_set_v @ ( minus_minus_set_v @ A3 @ B ) @ B )
= ( minus_minus_set_v @ A3 @ B ) ) ).
% Diff_idemp
thf(fact_138_Diff__iff,axiom,
! [C: product_prod_v_v,A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ C @ ( minus_4183494784930505774od_v_v @ A3 @ B ) )
= ( ( member7453568604450474000od_v_v @ C @ A3 )
& ~ ( member7453568604450474000od_v_v @ C @ B ) ) ) ).
% Diff_iff
thf(fact_139_Diff__iff,axiom,
! [C: v,A3: set_v,B: set_v] :
( ( member_v @ C @ ( minus_minus_set_v @ A3 @ B ) )
= ( ( member_v @ C @ A3 )
& ~ ( member_v @ C @ B ) ) ) ).
% Diff_iff
thf(fact_140_DiffI,axiom,
! [C: product_prod_v_v,A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ C @ A3 )
=> ( ~ ( member7453568604450474000od_v_v @ C @ B )
=> ( member7453568604450474000od_v_v @ C @ ( minus_4183494784930505774od_v_v @ A3 @ B ) ) ) ) ).
% DiffI
thf(fact_141_DiffI,axiom,
! [C: v,A3: set_v,B: set_v] :
( ( member_v @ C @ A3 )
=> ( ~ ( member_v @ C @ B )
=> ( member_v @ C @ ( minus_minus_set_v @ A3 @ B ) ) ) ) ).
% DiffI
thf(fact_142_empty__subsetI,axiom,
! [A3: set_set_v] : ( ord_le5216385588623774835_set_v @ bot_bot_set_set_v @ A3 ) ).
% empty_subsetI
thf(fact_143_empty__subsetI,axiom,
! [A3: set_v] : ( ord_less_eq_set_v @ bot_bot_set_v @ A3 ) ).
% empty_subsetI
thf(fact_144_empty__subsetI,axiom,
! [A3: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ bot_bo723834152578015283od_v_v @ A3 ) ).
% empty_subsetI
thf(fact_145_subset__empty,axiom,
! [A3: set_set_v] :
( ( ord_le5216385588623774835_set_v @ A3 @ bot_bot_set_set_v )
= ( A3 = bot_bot_set_set_v ) ) ).
% subset_empty
thf(fact_146_subset__empty,axiom,
! [A3: set_v] :
( ( ord_less_eq_set_v @ A3 @ bot_bot_set_v )
= ( A3 = bot_bot_set_v ) ) ).
% subset_empty
thf(fact_147_subset__empty,axiom,
! [A3: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A3 @ bot_bo723834152578015283od_v_v )
= ( A3 = bot_bo723834152578015283od_v_v ) ) ).
% subset_empty
thf(fact_148_singletonI,axiom,
! [A: v] : ( member_v @ A @ ( insert_v @ A @ bot_bot_set_v ) ) ).
% singletonI
thf(fact_149_singletonI,axiom,
! [A: product_prod_v_v] : ( member7453568604450474000od_v_v @ A @ ( insert1338601472111419319od_v_v @ A @ bot_bo723834152578015283od_v_v ) ) ).
% singletonI
thf(fact_150_singletonI,axiom,
! [A: set_v] : ( member_set_v @ A @ ( insert_set_v @ A @ bot_bot_set_set_v ) ) ).
% singletonI
thf(fact_151_insert__subset,axiom,
! [X2: set_v,A3: set_set_v,B: set_set_v] :
( ( ord_le5216385588623774835_set_v @ ( insert_set_v @ X2 @ A3 ) @ B )
= ( ( member_set_v @ X2 @ B )
& ( ord_le5216385588623774835_set_v @ A3 @ B ) ) ) ).
% insert_subset
thf(fact_152_insert__subset,axiom,
! [X2: v,A3: set_v,B: set_v] :
( ( ord_less_eq_set_v @ ( insert_v @ X2 @ A3 ) @ B )
= ( ( member_v @ X2 @ B )
& ( ord_less_eq_set_v @ A3 @ B ) ) ) ).
% insert_subset
thf(fact_153_insert__subset,axiom,
! [X2: product_prod_v_v,A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ ( insert1338601472111419319od_v_v @ X2 @ A3 ) @ B )
= ( ( member7453568604450474000od_v_v @ X2 @ B )
& ( ord_le7336532860387713383od_v_v @ A3 @ B ) ) ) ).
% insert_subset
thf(fact_154_Diff__cancel,axiom,
! [A3: set_Product_prod_v_v] :
( ( minus_4183494784930505774od_v_v @ A3 @ A3 )
= bot_bo723834152578015283od_v_v ) ).
% Diff_cancel
thf(fact_155_Diff__cancel,axiom,
! [A3: set_set_v] :
( ( minus_7228012346218142266_set_v @ A3 @ A3 )
= bot_bot_set_set_v ) ).
% Diff_cancel
thf(fact_156_Diff__cancel,axiom,
! [A3: set_v] :
( ( minus_minus_set_v @ A3 @ A3 )
= bot_bot_set_v ) ).
% Diff_cancel
thf(fact_157_empty__Diff,axiom,
! [A3: set_Product_prod_v_v] :
( ( minus_4183494784930505774od_v_v @ bot_bo723834152578015283od_v_v @ A3 )
= bot_bo723834152578015283od_v_v ) ).
% empty_Diff
thf(fact_158_empty__Diff,axiom,
! [A3: set_set_v] :
( ( minus_7228012346218142266_set_v @ bot_bot_set_set_v @ A3 )
= bot_bot_set_set_v ) ).
% empty_Diff
thf(fact_159_empty__Diff,axiom,
! [A3: set_v] :
( ( minus_minus_set_v @ bot_bot_set_v @ A3 )
= bot_bot_set_v ) ).
% empty_Diff
thf(fact_160_Diff__empty,axiom,
! [A3: set_Product_prod_v_v] :
( ( minus_4183494784930505774od_v_v @ A3 @ bot_bo723834152578015283od_v_v )
= A3 ) ).
% Diff_empty
thf(fact_161_Diff__empty,axiom,
! [A3: set_set_v] :
( ( minus_7228012346218142266_set_v @ A3 @ bot_bot_set_set_v )
= A3 ) ).
% Diff_empty
thf(fact_162_Diff__empty,axiom,
! [A3: set_v] :
( ( minus_minus_set_v @ A3 @ bot_bot_set_v )
= A3 ) ).
% Diff_empty
thf(fact_163_Int__subset__iff,axiom,
! [C2: set_v,A3: set_v,B: set_v] :
( ( ord_less_eq_set_v @ C2 @ ( inf_inf_set_v @ A3 @ B ) )
= ( ( ord_less_eq_set_v @ C2 @ A3 )
& ( ord_less_eq_set_v @ C2 @ B ) ) ) ).
% Int_subset_iff
thf(fact_164_Int__subset__iff,axiom,
! [C2: set_Product_prod_v_v,A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ C2 @ ( inf_in6271465464967711157od_v_v @ A3 @ B ) )
= ( ( ord_le7336532860387713383od_v_v @ C2 @ A3 )
& ( ord_le7336532860387713383od_v_v @ C2 @ B ) ) ) ).
% Int_subset_iff
thf(fact_165_Int__insert__right__if1,axiom,
! [A: set_v,A3: set_set_v,B: set_set_v] :
( ( member_set_v @ A @ A3 )
=> ( ( inf_inf_set_set_v @ A3 @ ( insert_set_v @ A @ B ) )
= ( insert_set_v @ A @ ( inf_inf_set_set_v @ A3 @ B ) ) ) ) ).
% Int_insert_right_if1
thf(fact_166_Int__insert__right__if1,axiom,
! [A: product_prod_v_v,A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ A @ A3 )
=> ( ( inf_in6271465464967711157od_v_v @ A3 @ ( insert1338601472111419319od_v_v @ A @ B ) )
= ( insert1338601472111419319od_v_v @ A @ ( inf_in6271465464967711157od_v_v @ A3 @ B ) ) ) ) ).
% Int_insert_right_if1
thf(fact_167_Int__insert__right__if1,axiom,
! [A: v,A3: set_v,B: set_v] :
( ( member_v @ A @ A3 )
=> ( ( inf_inf_set_v @ A3 @ ( insert_v @ A @ B ) )
= ( insert_v @ A @ ( inf_inf_set_v @ A3 @ B ) ) ) ) ).
% Int_insert_right_if1
thf(fact_168_Int__insert__right__if0,axiom,
! [A: set_v,A3: set_set_v,B: set_set_v] :
( ~ ( member_set_v @ A @ A3 )
=> ( ( inf_inf_set_set_v @ A3 @ ( insert_set_v @ A @ B ) )
= ( inf_inf_set_set_v @ A3 @ B ) ) ) ).
% Int_insert_right_if0
thf(fact_169_Int__insert__right__if0,axiom,
! [A: product_prod_v_v,A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ~ ( member7453568604450474000od_v_v @ A @ A3 )
=> ( ( inf_in6271465464967711157od_v_v @ A3 @ ( insert1338601472111419319od_v_v @ A @ B ) )
= ( inf_in6271465464967711157od_v_v @ A3 @ B ) ) ) ).
% Int_insert_right_if0
thf(fact_170_Int__insert__right__if0,axiom,
! [A: v,A3: set_v,B: set_v] :
( ~ ( member_v @ A @ A3 )
=> ( ( inf_inf_set_v @ A3 @ ( insert_v @ A @ B ) )
= ( inf_inf_set_v @ A3 @ B ) ) ) ).
% Int_insert_right_if0
thf(fact_171_insert__inter__insert,axiom,
! [A: product_prod_v_v,A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ ( insert1338601472111419319od_v_v @ A @ A3 ) @ ( insert1338601472111419319od_v_v @ A @ B ) )
= ( insert1338601472111419319od_v_v @ A @ ( inf_in6271465464967711157od_v_v @ A3 @ B ) ) ) ).
% insert_inter_insert
thf(fact_172_insert__inter__insert,axiom,
! [A: set_v,A3: set_set_v,B: set_set_v] :
( ( inf_inf_set_set_v @ ( insert_set_v @ A @ A3 ) @ ( insert_set_v @ A @ B ) )
= ( insert_set_v @ A @ ( inf_inf_set_set_v @ A3 @ B ) ) ) ).
% insert_inter_insert
thf(fact_173_insert__inter__insert,axiom,
! [A: v,A3: set_v,B: set_v] :
( ( inf_inf_set_v @ ( insert_v @ A @ A3 ) @ ( insert_v @ A @ B ) )
= ( insert_v @ A @ ( inf_inf_set_v @ A3 @ B ) ) ) ).
% insert_inter_insert
thf(fact_174_Int__insert__left__if1,axiom,
! [A: set_v,C2: set_set_v,B: set_set_v] :
( ( member_set_v @ A @ C2 )
=> ( ( inf_inf_set_set_v @ ( insert_set_v @ A @ B ) @ C2 )
= ( insert_set_v @ A @ ( inf_inf_set_set_v @ B @ C2 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_175_Int__insert__left__if1,axiom,
! [A: product_prod_v_v,C2: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ A @ C2 )
=> ( ( inf_in6271465464967711157od_v_v @ ( insert1338601472111419319od_v_v @ A @ B ) @ C2 )
= ( insert1338601472111419319od_v_v @ A @ ( inf_in6271465464967711157od_v_v @ B @ C2 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_176_Int__insert__left__if1,axiom,
! [A: v,C2: set_v,B: set_v] :
( ( member_v @ A @ C2 )
=> ( ( inf_inf_set_v @ ( insert_v @ A @ B ) @ C2 )
= ( insert_v @ A @ ( inf_inf_set_v @ B @ C2 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_177_Int__insert__left__if0,axiom,
! [A: set_v,C2: set_set_v,B: set_set_v] :
( ~ ( member_set_v @ A @ C2 )
=> ( ( inf_inf_set_set_v @ ( insert_set_v @ A @ B ) @ C2 )
= ( inf_inf_set_set_v @ B @ C2 ) ) ) ).
% Int_insert_left_if0
thf(fact_178_Int__insert__left__if0,axiom,
! [A: product_prod_v_v,C2: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ~ ( member7453568604450474000od_v_v @ A @ C2 )
=> ( ( inf_in6271465464967711157od_v_v @ ( insert1338601472111419319od_v_v @ A @ B ) @ C2 )
= ( inf_in6271465464967711157od_v_v @ B @ C2 ) ) ) ).
% Int_insert_left_if0
thf(fact_179_Int__insert__left__if0,axiom,
! [A: v,C2: set_v,B: set_v] :
( ~ ( member_v @ A @ C2 )
=> ( ( inf_inf_set_v @ ( insert_v @ A @ B ) @ C2 )
= ( inf_inf_set_v @ B @ C2 ) ) ) ).
% Int_insert_left_if0
thf(fact_180_insert__Diff1,axiom,
! [X2: set_v,B: set_set_v,A3: set_set_v] :
( ( member_set_v @ X2 @ B )
=> ( ( minus_7228012346218142266_set_v @ ( insert_set_v @ X2 @ A3 ) @ B )
= ( minus_7228012346218142266_set_v @ A3 @ B ) ) ) ).
% insert_Diff1
thf(fact_181_insert__Diff1,axiom,
! [X2: product_prod_v_v,B: set_Product_prod_v_v,A3: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X2 @ B )
=> ( ( minus_4183494784930505774od_v_v @ ( insert1338601472111419319od_v_v @ X2 @ A3 ) @ B )
= ( minus_4183494784930505774od_v_v @ A3 @ B ) ) ) ).
% insert_Diff1
thf(fact_182_insert__Diff1,axiom,
! [X2: v,B: set_v,A3: set_v] :
( ( member_v @ X2 @ B )
=> ( ( minus_minus_set_v @ ( insert_v @ X2 @ A3 ) @ B )
= ( minus_minus_set_v @ A3 @ B ) ) ) ).
% insert_Diff1
thf(fact_183_Diff__insert0,axiom,
! [X2: set_v,A3: set_set_v,B: set_set_v] :
( ~ ( member_set_v @ X2 @ A3 )
=> ( ( minus_7228012346218142266_set_v @ A3 @ ( insert_set_v @ X2 @ B ) )
= ( minus_7228012346218142266_set_v @ A3 @ B ) ) ) ).
% Diff_insert0
thf(fact_184_Diff__insert0,axiom,
! [X2: product_prod_v_v,A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ~ ( member7453568604450474000od_v_v @ X2 @ A3 )
=> ( ( minus_4183494784930505774od_v_v @ A3 @ ( insert1338601472111419319od_v_v @ X2 @ B ) )
= ( minus_4183494784930505774od_v_v @ A3 @ B ) ) ) ).
% Diff_insert0
thf(fact_185_Diff__insert0,axiom,
! [X2: v,A3: set_v,B: set_v] :
( ~ ( member_v @ X2 @ A3 )
=> ( ( minus_minus_set_v @ A3 @ ( insert_v @ X2 @ B ) )
= ( minus_minus_set_v @ A3 @ B ) ) ) ).
% Diff_insert0
thf(fact_186_graph_Ora__mono,axiom,
! [Vertices: set_v,Successors: v > set_v,X2: 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 @ X2 @ Y @ E )
=> ( ( ord_le7336532860387713383od_v_v @ E5 @ E )
=> ( sCC_Bl4291963740693775144ding_v @ Successors @ X2 @ Y @ E5 ) ) ) ) ).
% graph.ra_mono
thf(fact_187_graph_Opre__dfs_Ocong,axiom,
sCC_Bl36166008131615352t_unit = sCC_Bl36166008131615352t_unit ).
% graph.pre_dfs.cong
thf(fact_188_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 )
=> ! [X: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X @ Vertices )
=> ( ord_le7336532860387713383od_v_v @ ( Successors @ X ) @ Vertices ) ) ) ).
% graph.sclosed
thf(fact_189_graph_Osclosed,axiom,
! [Vertices: set_v,Successors: v > set_v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ! [X: v] :
( ( member_v @ X @ Vertices )
=> ( ord_less_eq_set_v @ ( Successors @ X ) @ Vertices ) ) ) ).
% graph.sclosed
thf(fact_190_graph_Ora__empty,axiom,
! [Vertices: set_v,Successors: v > set_v,X2: v,Y: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl4291963740693775144ding_v @ Successors @ X2 @ Y @ bot_bo723834152578015283od_v_v )
= ( sCC_Bl649662514949026229able_v @ Successors @ X2 @ Y ) ) ) ).
% graph.ra_empty
thf(fact_191_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,X2: product_prod_v_v,Z: product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( member7453568604450474000od_v_v @ Y @ ( Successors @ X2 ) )
=> ( ( sCC_Bl4981926079593201289od_v_v @ Successors @ Y @ Z )
=> ( sCC_Bl4981926079593201289od_v_v @ Successors @ X2 @ Z ) ) ) ) ).
% graph.reachable_succ
thf(fact_192_graph_Oreachable__succ,axiom,
! [Vertices: set_v,Successors: v > set_v,Y: v,X2: v,Z: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( member_v @ Y @ ( Successors @ X2 ) )
=> ( ( sCC_Bl649662514949026229able_v @ Successors @ Y @ Z )
=> ( sCC_Bl649662514949026229able_v @ Successors @ X2 @ Z ) ) ) ) ).
% graph.reachable_succ
thf(fact_193_graph_Oreachable__refl,axiom,
! [Vertices: set_v,Successors: v > set_v,X2: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( sCC_Bl649662514949026229able_v @ Successors @ X2 @ X2 ) ) ).
% graph.reachable_refl
thf(fact_194_graph_Oreachable__end__induct,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,X2: 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 @ X2 @ Y )
=> ( ! [X4: product_prod_v_v] : ( P @ X4 @ X4 )
=> ( ! [X4: product_prod_v_v,Y2: product_prod_v_v,Z3: product_prod_v_v] :
( ( P @ X4 @ Y2 )
=> ( ( member7453568604450474000od_v_v @ Z3 @ ( Successors @ Y2 ) )
=> ( P @ X4 @ Z3 ) ) )
=> ( P @ X2 @ Y ) ) ) ) ) ).
% graph.reachable_end_induct
thf(fact_195_graph_Oreachable__end__induct,axiom,
! [Vertices: set_v,Successors: v > set_v,X2: v,Y: v,P: v > v > $o] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl649662514949026229able_v @ Successors @ X2 @ Y )
=> ( ! [X4: v] : ( P @ X4 @ X4 )
=> ( ! [X4: v,Y2: v,Z3: v] :
( ( P @ X4 @ Y2 )
=> ( ( member_v @ Z3 @ ( Successors @ Y2 ) )
=> ( P @ X4 @ Z3 ) ) )
=> ( P @ X2 @ Y ) ) ) ) ) ).
% graph.reachable_end_induct
thf(fact_196_graph_Oreachable__trans,axiom,
! [Vertices: set_v,Successors: v > set_v,X2: v,Y: v,Z: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl649662514949026229able_v @ Successors @ X2 @ Y )
=> ( ( sCC_Bl649662514949026229able_v @ Successors @ Y @ Z )
=> ( sCC_Bl649662514949026229able_v @ Successors @ X2 @ Z ) ) ) ) ).
% graph.reachable_trans
thf(fact_197_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 )
= ( ? [X3: product_prod_v_v] :
( ( A1 = X3 )
& ( A2 = X3 ) )
| ? [X3: product_prod_v_v,Y3: product_prod_v_v,Z2: product_prod_v_v] :
( ( A1 = X3 )
& ( A2 = Z2 )
& ( member7453568604450474000od_v_v @ Y3 @ ( Successors @ X3 ) )
& ( sCC_Bl4981926079593201289od_v_v @ Successors @ Y3 @ Z2 ) ) ) ) ) ).
% graph.reachable.simps
thf(fact_198_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 )
= ( ? [X3: v] :
( ( A1 = X3 )
& ( A2 = X3 ) )
| ? [X3: v,Y3: v,Z2: v] :
( ( A1 = X3 )
& ( A2 = Z2 )
& ( member_v @ Y3 @ ( Successors @ X3 ) )
& ( sCC_Bl649662514949026229able_v @ Successors @ Y3 @ Z2 ) ) ) ) ) ).
% graph.reachable.simps
thf(fact_199_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_200_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_201_graph_Osucc__reachable,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,X2: 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 @ X2 @ Y )
=> ( ( member7453568604450474000od_v_v @ Z @ ( Successors @ Y ) )
=> ( sCC_Bl4981926079593201289od_v_v @ Successors @ X2 @ Z ) ) ) ) ).
% graph.succ_reachable
thf(fact_202_graph_Osucc__reachable,axiom,
! [Vertices: set_v,Successors: v > set_v,X2: v,Y: v,Z: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl649662514949026229able_v @ Successors @ X2 @ Y )
=> ( ( member_v @ Z @ ( Successors @ Y ) )
=> ( sCC_Bl649662514949026229able_v @ Successors @ X2 @ Z ) ) ) ) ).
% graph.succ_reachable
thf(fact_203_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,X2: product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( member7453568604450474000od_v_v @ Y @ ( Successors @ X2 ) )
=> ( sCC_Bl4981926079593201289od_v_v @ Successors @ X2 @ Y ) ) ) ).
% graph.reachable_edge
thf(fact_204_graph_Oreachable__edge,axiom,
! [Vertices: set_v,Successors: v > set_v,Y: v,X2: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( member_v @ Y @ ( Successors @ X2 ) )
=> ( sCC_Bl649662514949026229able_v @ Successors @ X2 @ Y ) ) ) ).
% graph.reachable_edge
thf(fact_205_graph_Ora__refl,axiom,
! [Vertices: set_v,Successors: v > set_v,X2: v,E: set_Product_prod_v_v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( sCC_Bl4291963740693775144ding_v @ Successors @ X2 @ X2 @ E ) ) ).
% graph.ra_refl
thf(fact_206_graph_Ora__trans,axiom,
! [Vertices: set_v,Successors: v > set_v,X2: v,Y: v,E: set_Product_prod_v_v,Z: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl4291963740693775144ding_v @ Successors @ X2 @ Y @ E )
=> ( ( sCC_Bl4291963740693775144ding_v @ Successors @ Y @ Z @ E )
=> ( sCC_Bl4291963740693775144ding_v @ Successors @ X2 @ Z @ E ) ) ) ) ).
% graph.ra_trans
thf(fact_207_graph_Ore__succ,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,X2: 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 @ X2 @ Y )
=> ( ( member7453568604450474000od_v_v @ Z @ ( Successors @ Y ) )
=> ( sCC_Bl4714988717384592488od_v_v @ Successors @ X2 @ Z ) ) ) ) ).
% graph.re_succ
thf(fact_208_graph_Ore__succ,axiom,
! [Vertices: set_v,Successors: v > set_v,X2: v,Y: v,Z: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl770211535891879572_end_v @ Successors @ X2 @ Y )
=> ( ( member_v @ Z @ ( Successors @ Y ) )
=> ( sCC_Bl770211535891879572_end_v @ Successors @ X2 @ Z ) ) ) ) ).
% graph.re_succ
thf(fact_209_graph_Ore__refl,axiom,
! [Vertices: set_v,Successors: v > set_v,X2: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( sCC_Bl770211535891879572_end_v @ Successors @ X2 @ X2 ) ) ).
% graph.re_refl
thf(fact_210_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 )
= ( ? [X3: product_prod_v_v] :
( ( A1 = X3 )
& ( A2 = X3 ) )
| ? [X3: product_prod_v_v,Y3: product_prod_v_v,Z2: product_prod_v_v] :
( ( A1 = X3 )
& ( A2 = Z2 )
& ( sCC_Bl4714988717384592488od_v_v @ Successors @ X3 @ Y3 )
& ( member7453568604450474000od_v_v @ Z2 @ ( Successors @ Y3 ) ) ) ) ) ) ).
% graph.reachable_end.simps
thf(fact_211_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 )
= ( ? [X3: v] :
( ( A1 = X3 )
& ( A2 = X3 ) )
| ? [X3: v,Y3: v,Z2: v] :
( ( A1 = X3 )
& ( A2 = Z2 )
& ( sCC_Bl770211535891879572_end_v @ Successors @ X3 @ Y3 )
& ( member_v @ Z2 @ ( Successors @ Y3 ) ) ) ) ) ) ).
% graph.reachable_end.simps
thf(fact_212_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_213_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_214_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,X2: product_prod_v_v,Z: product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( member7453568604450474000od_v_v @ Y @ ( Successors @ X2 ) )
=> ( ( sCC_Bl4714988717384592488od_v_v @ Successors @ Y @ Z )
=> ( sCC_Bl4714988717384592488od_v_v @ Successors @ X2 @ Z ) ) ) ) ).
% graph.succ_re
thf(fact_215_graph_Osucc__re,axiom,
! [Vertices: set_v,Successors: v > set_v,Y: v,X2: v,Z: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( member_v @ Y @ ( Successors @ X2 ) )
=> ( ( sCC_Bl770211535891879572_end_v @ Successors @ Y @ Z )
=> ( sCC_Bl770211535891879572_end_v @ Successors @ X2 @ Z ) ) ) ) ).
% graph.succ_re
thf(fact_216_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_217_graph_Ora__reachable,axiom,
! [Vertices: set_v,Successors: v > set_v,X2: v,Y: v,E: set_Product_prod_v_v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl4291963740693775144ding_v @ Successors @ X2 @ Y @ E )
=> ( sCC_Bl649662514949026229able_v @ Successors @ X2 @ Y ) ) ) ).
% graph.ra_reachable
thf(fact_218_graph_OS__reflexive,axiom,
! [Vertices: set_v,Successors: v > set_v,E2: sCC_Bl1394983891496994913t_unit,N2: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl9196236973127232072t_unit @ Successors @ E2 )
=> ( member_v @ N2 @ ( sCC_Bl1280885523602775798t_unit @ E2 @ N2 ) ) ) ) ).
% graph.S_reflexive
thf(fact_219_graph_Oreachable__re,axiom,
! [Vertices: set_v,Successors: v > set_v,X2: v,Y: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl649662514949026229able_v @ Successors @ X2 @ Y )
=> ( sCC_Bl770211535891879572_end_v @ Successors @ X2 @ Y ) ) ) ).
% graph.reachable_re
thf(fact_220_graph_Ore__reachable,axiom,
! [Vertices: set_v,Successors: v > set_v,X2: v,Y: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl770211535891879572_end_v @ Successors @ X2 @ Y )
=> ( sCC_Bl649662514949026229able_v @ Successors @ X2 @ Y ) ) ) ).
% graph.re_reachable
thf(fact_221_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,S2: set_Product_prod_v_v,X2: product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( sCC_Bl6242042402218619277od_v_v @ Successors @ S )
=> ( ( sCC_Bl6242042402218619277od_v_v @ Successors @ S2 )
=> ( ( member7453568604450474000od_v_v @ X2 @ ( inf_in6271465464967711157od_v_v @ S @ S2 ) )
=> ( S = S2 ) ) ) ) ) ).
% graph.scc_partition
thf(fact_222_graph_Oscc__partition,axiom,
! [Vertices: set_v,Successors: v > set_v,S: set_v,S2: set_v,X2: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bloemen_is_scc_v @ Successors @ S )
=> ( ( sCC_Bloemen_is_scc_v @ Successors @ S2 )
=> ( ( member_v @ X2 @ ( inf_inf_set_v @ S @ S2 ) )
=> ( S = S2 ) ) ) ) ) ).
% graph.scc_partition
thf(fact_223_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 )
= ( ! [X3: v] :
( ( member_v @ X3 @ S )
=> ! [Y3: v] :
( ( member_v @ Y3 @ S )
=> ( sCC_Bl649662514949026229able_v @ Successors @ X3 @ Y3 ) ) ) ) ) ) ).
% graph.is_subscc_def
thf(fact_224_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,X2: product_prod_v_v,Y: product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( sCC_Bl6242042402218619277od_v_v @ Successors @ S )
=> ( ( member7453568604450474000od_v_v @ X2 @ S )
=> ( ( sCC_Bl4981926079593201289od_v_v @ Successors @ X2 @ Y )
=> ( ( sCC_Bl4981926079593201289od_v_v @ Successors @ Y @ X2 )
=> ( member7453568604450474000od_v_v @ Y @ S ) ) ) ) ) ) ).
% graph.sccE
thf(fact_225_graph_OsccE,axiom,
! [Vertices: set_v,Successors: v > set_v,S: set_v,X2: v,Y: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bloemen_is_scc_v @ Successors @ S )
=> ( ( member_v @ X2 @ S )
=> ( ( sCC_Bl649662514949026229able_v @ Successors @ X2 @ Y )
=> ( ( sCC_Bl649662514949026229able_v @ Successors @ Y @ X2 )
=> ( member_v @ Y @ S ) ) ) ) ) ) ).
% graph.sccE
thf(fact_226_graph_Osubscc__add,axiom,
! [Vertices: set_set_v,Successors: set_v > set_set_v,S: set_set_v,X2: set_v,Y: set_v] :
( ( sCC_Bl5810666556806954322_set_v @ Vertices @ Successors )
=> ( ( sCC_Bl7907073126578335045_set_v @ Successors @ S )
=> ( ( member_set_v @ X2 @ S )
=> ( ( sCC_Bl7354734129683093653_set_v @ Successors @ X2 @ Y )
=> ( ( sCC_Bl7354734129683093653_set_v @ Successors @ Y @ X2 )
=> ( sCC_Bl7907073126578335045_set_v @ Successors @ ( insert_set_v @ Y @ S ) ) ) ) ) ) ) ).
% graph.subscc_add
thf(fact_227_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,X2: product_prod_v_v,Y: product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( sCC_Bl2301996248249672505od_v_v @ Successors @ S )
=> ( ( member7453568604450474000od_v_v @ X2 @ S )
=> ( ( sCC_Bl4981926079593201289od_v_v @ Successors @ X2 @ Y )
=> ( ( sCC_Bl4981926079593201289od_v_v @ Successors @ Y @ X2 )
=> ( sCC_Bl2301996248249672505od_v_v @ Successors @ ( insert1338601472111419319od_v_v @ Y @ S ) ) ) ) ) ) ) ).
% graph.subscc_add
thf(fact_228_graph_Osubscc__add,axiom,
! [Vertices: set_v,Successors: v > set_v,S: set_v,X2: v,Y: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl5398416737448265317bscc_v @ Successors @ S )
=> ( ( member_v @ X2 @ S )
=> ( ( sCC_Bl649662514949026229able_v @ Successors @ X2 @ Y )
=> ( ( sCC_Bl649662514949026229able_v @ Successors @ Y @ X2 )
=> ( sCC_Bl5398416737448265317bscc_v @ Successors @ ( insert_v @ Y @ S ) ) ) ) ) ) ) ).
% graph.subscc_add
thf(fact_229_ex__in__conv,axiom,
! [A3: set_v] :
( ( ? [X3: v] : ( member_v @ X3 @ A3 ) )
= ( A3 != bot_bot_set_v ) ) ).
% ex_in_conv
thf(fact_230_ex__in__conv,axiom,
! [A3: set_Product_prod_v_v] :
( ( ? [X3: product_prod_v_v] : ( member7453568604450474000od_v_v @ X3 @ A3 ) )
= ( A3 != bot_bo723834152578015283od_v_v ) ) ).
% ex_in_conv
thf(fact_231_ex__in__conv,axiom,
! [A3: set_set_v] :
( ( ? [X3: set_v] : ( member_set_v @ X3 @ A3 ) )
= ( A3 != bot_bot_set_set_v ) ) ).
% ex_in_conv
thf(fact_232_equals0I,axiom,
! [A3: set_v] :
( ! [Y2: v] :
~ ( member_v @ Y2 @ A3 )
=> ( A3 = bot_bot_set_v ) ) ).
% equals0I
thf(fact_233_equals0I,axiom,
! [A3: set_Product_prod_v_v] :
( ! [Y2: product_prod_v_v] :
~ ( member7453568604450474000od_v_v @ Y2 @ A3 )
=> ( A3 = bot_bo723834152578015283od_v_v ) ) ).
% equals0I
thf(fact_234_equals0I,axiom,
! [A3: set_set_v] :
( ! [Y2: set_v] :
~ ( member_set_v @ Y2 @ A3 )
=> ( A3 = bot_bot_set_set_v ) ) ).
% equals0I
thf(fact_235_equals0D,axiom,
! [A3: set_v,A: v] :
( ( A3 = bot_bot_set_v )
=> ~ ( member_v @ A @ A3 ) ) ).
% equals0D
thf(fact_236_equals0D,axiom,
! [A3: set_Product_prod_v_v,A: product_prod_v_v] :
( ( A3 = bot_bo723834152578015283od_v_v )
=> ~ ( member7453568604450474000od_v_v @ A @ A3 ) ) ).
% equals0D
thf(fact_237_equals0D,axiom,
! [A3: set_set_v,A: set_v] :
( ( A3 = bot_bot_set_set_v )
=> ~ ( member_set_v @ A @ A3 ) ) ).
% equals0D
thf(fact_238_emptyE,axiom,
! [A: v] :
~ ( member_v @ A @ bot_bot_set_v ) ).
% emptyE
thf(fact_239_emptyE,axiom,
! [A: product_prod_v_v] :
~ ( member7453568604450474000od_v_v @ A @ bot_bo723834152578015283od_v_v ) ).
% emptyE
thf(fact_240_emptyE,axiom,
! [A: set_v] :
~ ( member_set_v @ A @ bot_bot_set_set_v ) ).
% emptyE
thf(fact_241_Collect__mono__iff,axiom,
! [P: set_v > $o,Q: set_v > $o] :
( ( ord_le5216385588623774835_set_v @ ( collect_set_v @ P ) @ ( collect_set_v @ Q ) )
= ( ! [X3: set_v] :
( ( P @ X3 )
=> ( Q @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_242_Collect__mono__iff,axiom,
! [P: v > $o,Q: v > $o] :
( ( ord_less_eq_set_v @ ( collect_v @ P ) @ ( collect_v @ Q ) )
= ( ! [X3: v] :
( ( P @ X3 )
=> ( Q @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_243_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 ) )
= ( ! [X3: product_prod_v_v] :
( ( P @ X3 )
=> ( Q @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_244_set__eq__subset,axiom,
( ( ^ [Y5: set_v,Z4: set_v] : ( Y5 = Z4 ) )
= ( ^ [A4: set_v,B3: set_v] :
( ( ord_less_eq_set_v @ A4 @ B3 )
& ( ord_less_eq_set_v @ B3 @ A4 ) ) ) ) ).
% set_eq_subset
thf(fact_245_set__eq__subset,axiom,
( ( ^ [Y5: set_Product_prod_v_v,Z4: set_Product_prod_v_v] : ( Y5 = Z4 ) )
= ( ^ [A4: set_Product_prod_v_v,B3: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A4 @ B3 )
& ( ord_le7336532860387713383od_v_v @ B3 @ A4 ) ) ) ) ).
% set_eq_subset
thf(fact_246_subset__trans,axiom,
! [A3: set_v,B: set_v,C2: set_v] :
( ( ord_less_eq_set_v @ A3 @ B )
=> ( ( ord_less_eq_set_v @ B @ C2 )
=> ( ord_less_eq_set_v @ A3 @ C2 ) ) ) ).
% subset_trans
thf(fact_247_subset__trans,axiom,
! [A3: set_Product_prod_v_v,B: set_Product_prod_v_v,C2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A3 @ B )
=> ( ( ord_le7336532860387713383od_v_v @ B @ C2 )
=> ( ord_le7336532860387713383od_v_v @ A3 @ C2 ) ) ) ).
% subset_trans
thf(fact_248_Collect__mono,axiom,
! [P: set_v > $o,Q: set_v > $o] :
( ! [X4: set_v] :
( ( P @ X4 )
=> ( Q @ X4 ) )
=> ( ord_le5216385588623774835_set_v @ ( collect_set_v @ P ) @ ( collect_set_v @ Q ) ) ) ).
% Collect_mono
thf(fact_249_Collect__mono,axiom,
! [P: v > $o,Q: v > $o] :
( ! [X4: v] :
( ( P @ X4 )
=> ( Q @ X4 ) )
=> ( ord_less_eq_set_v @ ( collect_v @ P ) @ ( collect_v @ Q ) ) ) ).
% Collect_mono
thf(fact_250_Collect__mono,axiom,
! [P: product_prod_v_v > $o,Q: product_prod_v_v > $o] :
( ! [X4: product_prod_v_v] :
( ( P @ X4 )
=> ( Q @ X4 ) )
=> ( ord_le7336532860387713383od_v_v @ ( collec140062887454715474od_v_v @ P ) @ ( collec140062887454715474od_v_v @ Q ) ) ) ).
% Collect_mono
thf(fact_251_subset__refl,axiom,
! [A3: set_v] : ( ord_less_eq_set_v @ A3 @ A3 ) ).
% subset_refl
thf(fact_252_subset__refl,axiom,
! [A3: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ A3 @ A3 ) ).
% subset_refl
thf(fact_253_subset__iff,axiom,
( ord_less_eq_set_v
= ( ^ [A4: set_v,B3: set_v] :
! [T: v] :
( ( member_v @ T @ A4 )
=> ( member_v @ T @ B3 ) ) ) ) ).
% subset_iff
thf(fact_254_subset__iff,axiom,
( ord_le7336532860387713383od_v_v
= ( ^ [A4: set_Product_prod_v_v,B3: set_Product_prod_v_v] :
! [T: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ T @ A4 )
=> ( member7453568604450474000od_v_v @ T @ B3 ) ) ) ) ).
% subset_iff
thf(fact_255_equalityD2,axiom,
! [A3: set_v,B: set_v] :
( ( A3 = B )
=> ( ord_less_eq_set_v @ B @ A3 ) ) ).
% equalityD2
thf(fact_256_equalityD2,axiom,
! [A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( A3 = B )
=> ( ord_le7336532860387713383od_v_v @ B @ A3 ) ) ).
% equalityD2
thf(fact_257_equalityD1,axiom,
! [A3: set_v,B: set_v] :
( ( A3 = B )
=> ( ord_less_eq_set_v @ A3 @ B ) ) ).
% equalityD1
thf(fact_258_equalityD1,axiom,
! [A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( A3 = B )
=> ( ord_le7336532860387713383od_v_v @ A3 @ B ) ) ).
% equalityD1
thf(fact_259_subset__eq,axiom,
( ord_less_eq_set_v
= ( ^ [A4: set_v,B3: set_v] :
! [X3: v] :
( ( member_v @ X3 @ A4 )
=> ( member_v @ X3 @ B3 ) ) ) ) ).
% subset_eq
thf(fact_260_subset__eq,axiom,
( ord_le7336532860387713383od_v_v
= ( ^ [A4: set_Product_prod_v_v,B3: set_Product_prod_v_v] :
! [X3: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X3 @ A4 )
=> ( member7453568604450474000od_v_v @ X3 @ B3 ) ) ) ) ).
% subset_eq
thf(fact_261_equalityE,axiom,
! [A3: set_v,B: set_v] :
( ( A3 = B )
=> ~ ( ( ord_less_eq_set_v @ A3 @ B )
=> ~ ( ord_less_eq_set_v @ B @ A3 ) ) ) ).
% equalityE
thf(fact_262_equalityE,axiom,
! [A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( A3 = B )
=> ~ ( ( ord_le7336532860387713383od_v_v @ A3 @ B )
=> ~ ( ord_le7336532860387713383od_v_v @ B @ A3 ) ) ) ).
% equalityE
thf(fact_263_subsetD,axiom,
! [A3: set_v,B: set_v,C: v] :
( ( ord_less_eq_set_v @ A3 @ B )
=> ( ( member_v @ C @ A3 )
=> ( member_v @ C @ B ) ) ) ).
% subsetD
thf(fact_264_subsetD,axiom,
! [A3: set_Product_prod_v_v,B: set_Product_prod_v_v,C: product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A3 @ B )
=> ( ( member7453568604450474000od_v_v @ C @ A3 )
=> ( member7453568604450474000od_v_v @ C @ B ) ) ) ).
% subsetD
thf(fact_265_in__mono,axiom,
! [A3: set_v,B: set_v,X2: v] :
( ( ord_less_eq_set_v @ A3 @ B )
=> ( ( member_v @ X2 @ A3 )
=> ( member_v @ X2 @ B ) ) ) ).
% in_mono
thf(fact_266_in__mono,axiom,
! [A3: set_Product_prod_v_v,B: set_Product_prod_v_v,X2: product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A3 @ B )
=> ( ( member7453568604450474000od_v_v @ X2 @ A3 )
=> ( member7453568604450474000od_v_v @ X2 @ B ) ) ) ).
% in_mono
thf(fact_267_mk__disjoint__insert,axiom,
! [A: set_v,A3: set_set_v] :
( ( member_set_v @ A @ A3 )
=> ? [B4: set_set_v] :
( ( A3
= ( insert_set_v @ A @ B4 ) )
& ~ ( member_set_v @ A @ B4 ) ) ) ).
% mk_disjoint_insert
thf(fact_268_mk__disjoint__insert,axiom,
! [A: v,A3: set_v] :
( ( member_v @ A @ A3 )
=> ? [B4: set_v] :
( ( A3
= ( insert_v @ A @ B4 ) )
& ~ ( member_v @ A @ B4 ) ) ) ).
% mk_disjoint_insert
thf(fact_269_mk__disjoint__insert,axiom,
! [A: product_prod_v_v,A3: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ A @ A3 )
=> ? [B4: set_Product_prod_v_v] :
( ( A3
= ( insert1338601472111419319od_v_v @ A @ B4 ) )
& ~ ( member7453568604450474000od_v_v @ A @ B4 ) ) ) ).
% mk_disjoint_insert
thf(fact_270_insert__commute,axiom,
! [X2: v,Y: v,A3: set_v] :
( ( insert_v @ X2 @ ( insert_v @ Y @ A3 ) )
= ( insert_v @ Y @ ( insert_v @ X2 @ A3 ) ) ) ).
% insert_commute
thf(fact_271_insert__commute,axiom,
! [X2: product_prod_v_v,Y: product_prod_v_v,A3: set_Product_prod_v_v] :
( ( insert1338601472111419319od_v_v @ X2 @ ( insert1338601472111419319od_v_v @ Y @ A3 ) )
= ( insert1338601472111419319od_v_v @ Y @ ( insert1338601472111419319od_v_v @ X2 @ A3 ) ) ) ).
% insert_commute
thf(fact_272_insert__commute,axiom,
! [X2: set_v,Y: set_v,A3: set_set_v] :
( ( insert_set_v @ X2 @ ( insert_set_v @ Y @ A3 ) )
= ( insert_set_v @ Y @ ( insert_set_v @ X2 @ A3 ) ) ) ).
% insert_commute
thf(fact_273_insert__eq__iff,axiom,
! [A: set_v,A3: set_set_v,B2: set_v,B: set_set_v] :
( ~ ( member_set_v @ A @ A3 )
=> ( ~ ( member_set_v @ B2 @ B )
=> ( ( ( insert_set_v @ A @ A3 )
= ( insert_set_v @ B2 @ B ) )
= ( ( ( A = B2 )
=> ( A3 = B ) )
& ( ( A != B2 )
=> ? [C3: set_set_v] :
( ( A3
= ( insert_set_v @ B2 @ C3 ) )
& ~ ( member_set_v @ B2 @ C3 )
& ( B
= ( insert_set_v @ A @ C3 ) )
& ~ ( member_set_v @ A @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_274_insert__eq__iff,axiom,
! [A: v,A3: set_v,B2: v,B: set_v] :
( ~ ( member_v @ A @ A3 )
=> ( ~ ( member_v @ B2 @ B )
=> ( ( ( insert_v @ A @ A3 )
= ( insert_v @ B2 @ B ) )
= ( ( ( A = B2 )
=> ( A3 = B ) )
& ( ( A != B2 )
=> ? [C3: set_v] :
( ( A3
= ( insert_v @ B2 @ C3 ) )
& ~ ( member_v @ B2 @ C3 )
& ( B
= ( insert_v @ A @ C3 ) )
& ~ ( member_v @ A @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_275_insert__eq__iff,axiom,
! [A: product_prod_v_v,A3: set_Product_prod_v_v,B2: product_prod_v_v,B: set_Product_prod_v_v] :
( ~ ( member7453568604450474000od_v_v @ A @ A3 )
=> ( ~ ( member7453568604450474000od_v_v @ B2 @ B )
=> ( ( ( insert1338601472111419319od_v_v @ A @ A3 )
= ( insert1338601472111419319od_v_v @ B2 @ B ) )
= ( ( ( A = B2 )
=> ( A3 = B ) )
& ( ( A != B2 )
=> ? [C3: set_Product_prod_v_v] :
( ( A3
= ( insert1338601472111419319od_v_v @ B2 @ C3 ) )
& ~ ( member7453568604450474000od_v_v @ B2 @ C3 )
& ( B
= ( insert1338601472111419319od_v_v @ A @ C3 ) )
& ~ ( member7453568604450474000od_v_v @ A @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_276_insert__absorb,axiom,
! [A: set_v,A3: set_set_v] :
( ( member_set_v @ A @ A3 )
=> ( ( insert_set_v @ A @ A3 )
= A3 ) ) ).
% insert_absorb
thf(fact_277_insert__absorb,axiom,
! [A: v,A3: set_v] :
( ( member_v @ A @ A3 )
=> ( ( insert_v @ A @ A3 )
= A3 ) ) ).
% insert_absorb
thf(fact_278_insert__absorb,axiom,
! [A: product_prod_v_v,A3: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ A @ A3 )
=> ( ( insert1338601472111419319od_v_v @ A @ A3 )
= A3 ) ) ).
% insert_absorb
thf(fact_279_insert__ident,axiom,
! [X2: set_v,A3: set_set_v,B: set_set_v] :
( ~ ( member_set_v @ X2 @ A3 )
=> ( ~ ( member_set_v @ X2 @ B )
=> ( ( ( insert_set_v @ X2 @ A3 )
= ( insert_set_v @ X2 @ B ) )
= ( A3 = B ) ) ) ) ).
% insert_ident
thf(fact_280_insert__ident,axiom,
! [X2: v,A3: set_v,B: set_v] :
( ~ ( member_v @ X2 @ A3 )
=> ( ~ ( member_v @ X2 @ B )
=> ( ( ( insert_v @ X2 @ A3 )
= ( insert_v @ X2 @ B ) )
= ( A3 = B ) ) ) ) ).
% insert_ident
thf(fact_281_insert__ident,axiom,
! [X2: product_prod_v_v,A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ~ ( member7453568604450474000od_v_v @ X2 @ A3 )
=> ( ~ ( member7453568604450474000od_v_v @ X2 @ B )
=> ( ( ( insert1338601472111419319od_v_v @ X2 @ A3 )
= ( insert1338601472111419319od_v_v @ X2 @ B ) )
= ( A3 = B ) ) ) ) ).
% insert_ident
thf(fact_282_Set_Oset__insert,axiom,
! [X2: set_v,A3: set_set_v] :
( ( member_set_v @ X2 @ A3 )
=> ~ ! [B4: set_set_v] :
( ( A3
= ( insert_set_v @ X2 @ B4 ) )
=> ( member_set_v @ X2 @ B4 ) ) ) ).
% Set.set_insert
thf(fact_283_Set_Oset__insert,axiom,
! [X2: v,A3: set_v] :
( ( member_v @ X2 @ A3 )
=> ~ ! [B4: set_v] :
( ( A3
= ( insert_v @ X2 @ B4 ) )
=> ( member_v @ X2 @ B4 ) ) ) ).
% Set.set_insert
thf(fact_284_Set_Oset__insert,axiom,
! [X2: product_prod_v_v,A3: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X2 @ A3 )
=> ~ ! [B4: set_Product_prod_v_v] :
( ( A3
= ( insert1338601472111419319od_v_v @ X2 @ B4 ) )
=> ( member7453568604450474000od_v_v @ X2 @ B4 ) ) ) ).
% Set.set_insert
thf(fact_285_insertI2,axiom,
! [A: set_v,B: set_set_v,B2: set_v] :
( ( member_set_v @ A @ B )
=> ( member_set_v @ A @ ( insert_set_v @ B2 @ B ) ) ) ).
% insertI2
thf(fact_286_insertI2,axiom,
! [A: v,B: set_v,B2: v] :
( ( member_v @ A @ B )
=> ( member_v @ A @ ( insert_v @ B2 @ B ) ) ) ).
% insertI2
thf(fact_287_insertI2,axiom,
! [A: product_prod_v_v,B: set_Product_prod_v_v,B2: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ A @ B )
=> ( member7453568604450474000od_v_v @ A @ ( insert1338601472111419319od_v_v @ B2 @ B ) ) ) ).
% insertI2
thf(fact_288_insertI1,axiom,
! [A: set_v,B: set_set_v] : ( member_set_v @ A @ ( insert_set_v @ A @ B ) ) ).
% insertI1
thf(fact_289_insertI1,axiom,
! [A: v,B: set_v] : ( member_v @ A @ ( insert_v @ A @ B ) ) ).
% insertI1
thf(fact_290_insertI1,axiom,
! [A: product_prod_v_v,B: set_Product_prod_v_v] : ( member7453568604450474000od_v_v @ A @ ( insert1338601472111419319od_v_v @ A @ B ) ) ).
% insertI1
thf(fact_291_insertE,axiom,
! [A: set_v,B2: set_v,A3: set_set_v] :
( ( member_set_v @ A @ ( insert_set_v @ B2 @ A3 ) )
=> ( ( A != B2 )
=> ( member_set_v @ A @ A3 ) ) ) ).
% insertE
thf(fact_292_insertE,axiom,
! [A: v,B2: v,A3: set_v] :
( ( member_v @ A @ ( insert_v @ B2 @ A3 ) )
=> ( ( A != B2 )
=> ( member_v @ A @ A3 ) ) ) ).
% insertE
thf(fact_293_insertE,axiom,
! [A: product_prod_v_v,B2: product_prod_v_v,A3: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ A @ ( insert1338601472111419319od_v_v @ B2 @ A3 ) )
=> ( ( A != B2 )
=> ( member7453568604450474000od_v_v @ A @ A3 ) ) ) ).
% insertE
thf(fact_294_Int__left__commute,axiom,
! [A3: set_v,B: set_v,C2: set_v] :
( ( inf_inf_set_v @ A3 @ ( inf_inf_set_v @ B @ C2 ) )
= ( inf_inf_set_v @ B @ ( inf_inf_set_v @ A3 @ C2 ) ) ) ).
% Int_left_commute
thf(fact_295_Int__left__absorb,axiom,
! [A3: set_v,B: set_v] :
( ( inf_inf_set_v @ A3 @ ( inf_inf_set_v @ A3 @ B ) )
= ( inf_inf_set_v @ A3 @ B ) ) ).
% Int_left_absorb
thf(fact_296_Int__commute,axiom,
( inf_inf_set_v
= ( ^ [A4: set_v,B3: set_v] : ( inf_inf_set_v @ B3 @ A4 ) ) ) ).
% Int_commute
thf(fact_297_Int__absorb,axiom,
! [A3: set_v] :
( ( inf_inf_set_v @ A3 @ A3 )
= A3 ) ).
% Int_absorb
thf(fact_298_Int__assoc,axiom,
! [A3: set_v,B: set_v,C2: set_v] :
( ( inf_inf_set_v @ ( inf_inf_set_v @ A3 @ B ) @ C2 )
= ( inf_inf_set_v @ A3 @ ( inf_inf_set_v @ B @ C2 ) ) ) ).
% Int_assoc
thf(fact_299_IntD2,axiom,
! [C: product_prod_v_v,A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ C @ ( inf_in6271465464967711157od_v_v @ A3 @ B ) )
=> ( member7453568604450474000od_v_v @ C @ B ) ) ).
% IntD2
thf(fact_300_IntD2,axiom,
! [C: v,A3: set_v,B: set_v] :
( ( member_v @ C @ ( inf_inf_set_v @ A3 @ B ) )
=> ( member_v @ C @ B ) ) ).
% IntD2
thf(fact_301_IntD1,axiom,
! [C: product_prod_v_v,A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ C @ ( inf_in6271465464967711157od_v_v @ A3 @ B ) )
=> ( member7453568604450474000od_v_v @ C @ A3 ) ) ).
% IntD1
thf(fact_302_IntD1,axiom,
! [C: v,A3: set_v,B: set_v] :
( ( member_v @ C @ ( inf_inf_set_v @ A3 @ B ) )
=> ( member_v @ C @ A3 ) ) ).
% IntD1
thf(fact_303_IntE,axiom,
! [C: product_prod_v_v,A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ C @ ( inf_in6271465464967711157od_v_v @ A3 @ B ) )
=> ~ ( ( member7453568604450474000od_v_v @ C @ A3 )
=> ~ ( member7453568604450474000od_v_v @ C @ B ) ) ) ).
% IntE
thf(fact_304_IntE,axiom,
! [C: v,A3: set_v,B: set_v] :
( ( member_v @ C @ ( inf_inf_set_v @ A3 @ B ) )
=> ~ ( ( member_v @ C @ A3 )
=> ~ ( member_v @ C @ B ) ) ) ).
% IntE
thf(fact_305_DiffD2,axiom,
! [C: product_prod_v_v,A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ C @ ( minus_4183494784930505774od_v_v @ A3 @ B ) )
=> ~ ( member7453568604450474000od_v_v @ C @ B ) ) ).
% DiffD2
thf(fact_306_DiffD2,axiom,
! [C: v,A3: set_v,B: set_v] :
( ( member_v @ C @ ( minus_minus_set_v @ A3 @ B ) )
=> ~ ( member_v @ C @ B ) ) ).
% DiffD2
thf(fact_307_DiffD1,axiom,
! [C: product_prod_v_v,A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ C @ ( minus_4183494784930505774od_v_v @ A3 @ B ) )
=> ( member7453568604450474000od_v_v @ C @ A3 ) ) ).
% DiffD1
thf(fact_308_DiffD1,axiom,
! [C: v,A3: set_v,B: set_v] :
( ( member_v @ C @ ( minus_minus_set_v @ A3 @ B ) )
=> ( member_v @ C @ A3 ) ) ).
% DiffD1
thf(fact_309_DiffE,axiom,
! [C: product_prod_v_v,A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ C @ ( minus_4183494784930505774od_v_v @ A3 @ B ) )
=> ~ ( ( member7453568604450474000od_v_v @ C @ A3 )
=> ( member7453568604450474000od_v_v @ C @ B ) ) ) ).
% DiffE
thf(fact_310_DiffE,axiom,
! [C: v,A3: set_v,B: set_v] :
( ( member_v @ C @ ( minus_minus_set_v @ A3 @ B ) )
=> ~ ( ( member_v @ C @ A3 )
=> ( member_v @ C @ B ) ) ) ).
% DiffE
thf(fact_311_graph_Ostack__visited,axiom,
! [Vertices: set_v,Successors: v > set_v,E2: sCC_Bl1394983891496994913t_unit,N2: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl9196236973127232072t_unit @ Successors @ E2 )
=> ( ( member_v @ N2 @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E2 ) ) )
=> ( member_v @ N2 @ ( sCC_Bl4645233313691564917t_unit @ E2 ) ) ) ) ) ).
% graph.stack_visited
thf(fact_312_graph_Ostack__unexplored,axiom,
! [Vertices: set_v,Successors: v > set_v,E2: sCC_Bl1394983891496994913t_unit,N2: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl9196236973127232072t_unit @ Successors @ E2 )
=> ( ( member_v @ N2 @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E2 ) ) )
=> ~ ( member_v @ N2 @ ( sCC_Bl157864678168468314t_unit @ E2 ) ) ) ) ) ).
% graph.stack_unexplored
thf(fact_313_graph_Ois__scc__def,axiom,
! [Vertices: set_set_v,Successors: set_v > set_set_v,S: set_set_v] :
( ( sCC_Bl5810666556806954322_set_v @ Vertices @ Successors )
=> ( ( sCC_Bl1515522642333523865_set_v @ Successors @ S )
= ( ( S != bot_bot_set_set_v )
& ( sCC_Bl7907073126578335045_set_v @ Successors @ S )
& ! [S3: set_set_v] :
( ( ( ord_le5216385588623774835_set_v @ S @ S3 )
& ( sCC_Bl7907073126578335045_set_v @ Successors @ S3 ) )
=> ( S3 = S ) ) ) ) ) ).
% graph.is_scc_def
thf(fact_314_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 )
& ! [S3: set_Product_prod_v_v] :
( ( ( ord_le7336532860387713383od_v_v @ S @ S3 )
& ( sCC_Bl2301996248249672505od_v_v @ Successors @ S3 ) )
=> ( S3 = S ) ) ) ) ) ).
% graph.is_scc_def
thf(fact_315_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 )
& ! [S3: set_v] :
( ( ( ord_less_eq_set_v @ S @ S3 )
& ( sCC_Bl5398416737448265317bscc_v @ Successors @ S3 ) )
=> ( S3 = S ) ) ) ) ) ).
% graph.is_scc_def
thf(fact_316_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 )
=> ( ! [X4: v] :
( ( member_v @ X4 @ ( sCC_Bl4645233313691564917t_unit @ E2 ) )
=> ! [Xa2: v] :
( ( member_v @ Xa2 @ ( minus_minus_set_v @ ( Successors @ X4 ) @ ( sCC_Bl3795065053823578884t_unit @ E2 @ X4 ) ) )
=> ( ( sCC_Bl649662514949026229able_v @ Successors @ V @ X4 )
=> ~ ( sCC_Bl649662514949026229able_v @ Successors @ Xa2 @ W ) ) ) )
=> ( member_v @ W @ ( sCC_Bl4645233313691564917t_unit @ E2 ) ) ) ) ) ) ) ).
% graph.reachable_visited
thf(fact_317_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 )
& ! [X3: v] :
( ( member_v @ X3 @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E2 ) ) )
=> ( sCC_Bl649662514949026229able_v @ Successors @ X3 @ V ) ) ) ) ) ).
% graph.pre_dfs_def
thf(fact_318_graph_Ovisited__unexplored,axiom,
! [Vertices: set_v,Successors: v > set_v,E2: sCC_Bl1394983891496994913t_unit,M2: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl9196236973127232072t_unit @ Successors @ E2 )
=> ( ( member_v @ M2 @ ( sCC_Bl4645233313691564917t_unit @ E2 ) )
=> ( ~ ( member_v @ M2 @ ( sCC_Bl157864678168468314t_unit @ E2 ) )
=> ~ ! [N3: v] :
( ( member_v @ N3 @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E2 ) ) )
=> ~ ( member_v @ M2 @ ( sCC_Bl1280885523602775798t_unit @ E2 @ N3 ) ) ) ) ) ) ) ).
% graph.visited_unexplored
thf(fact_319_cSup__eq__maximum,axiom,
! [Z: set_Product_prod_v_v,X5: set_se8455005133513928103od_v_v] :
( ( member8406446414694345712od_v_v @ Z @ X5 )
=> ( ! [X4: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ X4 @ X5 )
=> ( ord_le7336532860387713383od_v_v @ X4 @ Z ) )
=> ( ( comple5788137035815166516od_v_v @ X5 )
= Z ) ) ) ).
% cSup_eq_maximum
thf(fact_320_cSup__eq__maximum,axiom,
! [Z: set_v,X5: set_set_v] :
( ( member_set_v @ Z @ X5 )
=> ( ! [X4: set_v] :
( ( member_set_v @ X4 @ X5 )
=> ( ord_less_eq_set_v @ X4 @ Z ) )
=> ( ( comple2307003700295860064_set_v @ X5 )
= Z ) ) ) ).
% cSup_eq_maximum
thf(fact_321_singleton__inject,axiom,
! [A: v,B2: v] :
( ( ( insert_v @ A @ bot_bot_set_v )
= ( insert_v @ B2 @ bot_bot_set_v ) )
=> ( A = B2 ) ) ).
% singleton_inject
thf(fact_322_singleton__inject,axiom,
! [A: product_prod_v_v,B2: product_prod_v_v] :
( ( ( insert1338601472111419319od_v_v @ A @ bot_bo723834152578015283od_v_v )
= ( insert1338601472111419319od_v_v @ B2 @ bot_bo723834152578015283od_v_v ) )
=> ( A = B2 ) ) ).
% singleton_inject
thf(fact_323_singleton__inject,axiom,
! [A: set_v,B2: set_v] :
( ( ( insert_set_v @ A @ bot_bot_set_set_v )
= ( insert_set_v @ B2 @ bot_bot_set_set_v ) )
=> ( A = B2 ) ) ).
% singleton_inject
thf(fact_324_insert__not__empty,axiom,
! [A: v,A3: set_v] :
( ( insert_v @ A @ A3 )
!= bot_bot_set_v ) ).
% insert_not_empty
thf(fact_325_insert__not__empty,axiom,
! [A: product_prod_v_v,A3: set_Product_prod_v_v] :
( ( insert1338601472111419319od_v_v @ A @ A3 )
!= bot_bo723834152578015283od_v_v ) ).
% insert_not_empty
thf(fact_326_insert__not__empty,axiom,
! [A: set_v,A3: set_set_v] :
( ( insert_set_v @ A @ A3 )
!= bot_bot_set_set_v ) ).
% insert_not_empty
thf(fact_327_doubleton__eq__iff,axiom,
! [A: v,B2: v,C: v,D: v] :
( ( ( insert_v @ A @ ( insert_v @ B2 @ bot_bot_set_v ) )
= ( insert_v @ C @ ( insert_v @ D @ bot_bot_set_v ) ) )
= ( ( ( A = C )
& ( B2 = D ) )
| ( ( A = D )
& ( B2 = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_328_doubleton__eq__iff,axiom,
! [A: product_prod_v_v,B2: product_prod_v_v,C: product_prod_v_v,D: product_prod_v_v] :
( ( ( insert1338601472111419319od_v_v @ A @ ( insert1338601472111419319od_v_v @ B2 @ bot_bo723834152578015283od_v_v ) )
= ( insert1338601472111419319od_v_v @ C @ ( insert1338601472111419319od_v_v @ D @ bot_bo723834152578015283od_v_v ) ) )
= ( ( ( A = C )
& ( B2 = D ) )
| ( ( A = D )
& ( B2 = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_329_doubleton__eq__iff,axiom,
! [A: set_v,B2: set_v,C: set_v,D: set_v] :
( ( ( insert_set_v @ A @ ( insert_set_v @ B2 @ bot_bot_set_set_v ) )
= ( insert_set_v @ C @ ( insert_set_v @ D @ bot_bot_set_set_v ) ) )
= ( ( ( A = C )
& ( B2 = D ) )
| ( ( A = D )
& ( B2 = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_330_singleton__iff,axiom,
! [B2: v,A: v] :
( ( member_v @ B2 @ ( insert_v @ A @ bot_bot_set_v ) )
= ( B2 = A ) ) ).
% singleton_iff
thf(fact_331_singleton__iff,axiom,
! [B2: product_prod_v_v,A: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ B2 @ ( insert1338601472111419319od_v_v @ A @ bot_bo723834152578015283od_v_v ) )
= ( B2 = A ) ) ).
% singleton_iff
thf(fact_332_singleton__iff,axiom,
! [B2: set_v,A: set_v] :
( ( member_set_v @ B2 @ ( insert_set_v @ A @ bot_bot_set_set_v ) )
= ( B2 = A ) ) ).
% singleton_iff
thf(fact_333_singletonD,axiom,
! [B2: v,A: v] :
( ( member_v @ B2 @ ( insert_v @ A @ bot_bot_set_v ) )
=> ( B2 = A ) ) ).
% singletonD
thf(fact_334_singletonD,axiom,
! [B2: product_prod_v_v,A: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ B2 @ ( insert1338601472111419319od_v_v @ A @ bot_bo723834152578015283od_v_v ) )
=> ( B2 = A ) ) ).
% singletonD
thf(fact_335_singletonD,axiom,
! [B2: set_v,A: set_v] :
( ( member_set_v @ B2 @ ( insert_set_v @ A @ bot_bot_set_set_v ) )
=> ( B2 = A ) ) ).
% singletonD
thf(fact_336_subset__insertI2,axiom,
! [A3: set_set_v,B: set_set_v,B2: set_v] :
( ( ord_le5216385588623774835_set_v @ A3 @ B )
=> ( ord_le5216385588623774835_set_v @ A3 @ ( insert_set_v @ B2 @ B ) ) ) ).
% subset_insertI2
thf(fact_337_subset__insertI2,axiom,
! [A3: set_v,B: set_v,B2: v] :
( ( ord_less_eq_set_v @ A3 @ B )
=> ( ord_less_eq_set_v @ A3 @ ( insert_v @ B2 @ B ) ) ) ).
% subset_insertI2
thf(fact_338_subset__insertI2,axiom,
! [A3: set_Product_prod_v_v,B: set_Product_prod_v_v,B2: product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A3 @ B )
=> ( ord_le7336532860387713383od_v_v @ A3 @ ( insert1338601472111419319od_v_v @ B2 @ B ) ) ) ).
% subset_insertI2
thf(fact_339_subset__insertI,axiom,
! [B: set_set_v,A: set_v] : ( ord_le5216385588623774835_set_v @ B @ ( insert_set_v @ A @ B ) ) ).
% subset_insertI
thf(fact_340_subset__insertI,axiom,
! [B: set_v,A: v] : ( ord_less_eq_set_v @ B @ ( insert_v @ A @ B ) ) ).
% subset_insertI
thf(fact_341_subset__insertI,axiom,
! [B: set_Product_prod_v_v,A: product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ B @ ( insert1338601472111419319od_v_v @ A @ B ) ) ).
% subset_insertI
thf(fact_342_subset__insert,axiom,
! [X2: set_v,A3: set_set_v,B: set_set_v] :
( ~ ( member_set_v @ X2 @ A3 )
=> ( ( ord_le5216385588623774835_set_v @ A3 @ ( insert_set_v @ X2 @ B ) )
= ( ord_le5216385588623774835_set_v @ A3 @ B ) ) ) ).
% subset_insert
thf(fact_343_subset__insert,axiom,
! [X2: v,A3: set_v,B: set_v] :
( ~ ( member_v @ X2 @ A3 )
=> ( ( ord_less_eq_set_v @ A3 @ ( insert_v @ X2 @ B ) )
= ( ord_less_eq_set_v @ A3 @ B ) ) ) ).
% subset_insert
thf(fact_344_subset__insert,axiom,
! [X2: product_prod_v_v,A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ~ ( member7453568604450474000od_v_v @ X2 @ A3 )
=> ( ( ord_le7336532860387713383od_v_v @ A3 @ ( insert1338601472111419319od_v_v @ X2 @ B ) )
= ( ord_le7336532860387713383od_v_v @ A3 @ B ) ) ) ).
% subset_insert
thf(fact_345_insert__mono,axiom,
! [C2: set_set_v,D2: set_set_v,A: set_v] :
( ( ord_le5216385588623774835_set_v @ C2 @ D2 )
=> ( ord_le5216385588623774835_set_v @ ( insert_set_v @ A @ C2 ) @ ( insert_set_v @ A @ D2 ) ) ) ).
% insert_mono
thf(fact_346_insert__mono,axiom,
! [C2: set_v,D2: set_v,A: v] :
( ( ord_less_eq_set_v @ C2 @ D2 )
=> ( ord_less_eq_set_v @ ( insert_v @ A @ C2 ) @ ( insert_v @ A @ D2 ) ) ) ).
% insert_mono
thf(fact_347_insert__mono,axiom,
! [C2: set_Product_prod_v_v,D2: set_Product_prod_v_v,A: product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ C2 @ D2 )
=> ( ord_le7336532860387713383od_v_v @ ( insert1338601472111419319od_v_v @ A @ C2 ) @ ( insert1338601472111419319od_v_v @ A @ D2 ) ) ) ).
% insert_mono
thf(fact_348_disjoint__iff__not__equal,axiom,
! [A3: set_v,B: set_v] :
( ( ( inf_inf_set_v @ A3 @ B )
= bot_bot_set_v )
= ( ! [X3: v] :
( ( member_v @ X3 @ A3 )
=> ! [Y3: v] :
( ( member_v @ Y3 @ B )
=> ( X3 != Y3 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_349_disjoint__iff__not__equal,axiom,
! [A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ( inf_in6271465464967711157od_v_v @ A3 @ B )
= bot_bo723834152578015283od_v_v )
= ( ! [X3: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X3 @ A3 )
=> ! [Y3: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ Y3 @ B )
=> ( X3 != Y3 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_350_disjoint__iff__not__equal,axiom,
! [A3: set_set_v,B: set_set_v] :
( ( ( inf_inf_set_set_v @ A3 @ B )
= bot_bot_set_set_v )
= ( ! [X3: set_v] :
( ( member_set_v @ X3 @ A3 )
=> ! [Y3: set_v] :
( ( member_set_v @ Y3 @ B )
=> ( X3 != Y3 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_351_Int__empty__right,axiom,
! [A3: set_v] :
( ( inf_inf_set_v @ A3 @ bot_bot_set_v )
= bot_bot_set_v ) ).
% Int_empty_right
thf(fact_352_Int__empty__right,axiom,
! [A3: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ A3 @ bot_bo723834152578015283od_v_v )
= bot_bo723834152578015283od_v_v ) ).
% Int_empty_right
thf(fact_353_Int__empty__right,axiom,
! [A3: set_set_v] :
( ( inf_inf_set_set_v @ A3 @ bot_bot_set_set_v )
= bot_bot_set_set_v ) ).
% Int_empty_right
thf(fact_354_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_355_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_356_Int__empty__left,axiom,
! [B: set_set_v] :
( ( inf_inf_set_set_v @ bot_bot_set_set_v @ B )
= bot_bot_set_set_v ) ).
% Int_empty_left
thf(fact_357_disjoint__iff,axiom,
! [A3: set_v,B: set_v] :
( ( ( inf_inf_set_v @ A3 @ B )
= bot_bot_set_v )
= ( ! [X3: v] :
( ( member_v @ X3 @ A3 )
=> ~ ( member_v @ X3 @ B ) ) ) ) ).
% disjoint_iff
thf(fact_358_disjoint__iff,axiom,
! [A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ( inf_in6271465464967711157od_v_v @ A3 @ B )
= bot_bo723834152578015283od_v_v )
= ( ! [X3: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X3 @ A3 )
=> ~ ( member7453568604450474000od_v_v @ X3 @ B ) ) ) ) ).
% disjoint_iff
thf(fact_359_disjoint__iff,axiom,
! [A3: set_set_v,B: set_set_v] :
( ( ( inf_inf_set_set_v @ A3 @ B )
= bot_bot_set_set_v )
= ( ! [X3: set_v] :
( ( member_set_v @ X3 @ A3 )
=> ~ ( member_set_v @ X3 @ B ) ) ) ) ).
% disjoint_iff
thf(fact_360_Int__emptyI,axiom,
! [A3: set_v,B: set_v] :
( ! [X4: v] :
( ( member_v @ X4 @ A3 )
=> ~ ( member_v @ X4 @ B ) )
=> ( ( inf_inf_set_v @ A3 @ B )
= bot_bot_set_v ) ) ).
% Int_emptyI
thf(fact_361_Int__emptyI,axiom,
! [A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ! [X4: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X4 @ A3 )
=> ~ ( member7453568604450474000od_v_v @ X4 @ B ) )
=> ( ( inf_in6271465464967711157od_v_v @ A3 @ B )
= bot_bo723834152578015283od_v_v ) ) ).
% Int_emptyI
thf(fact_362_Int__emptyI,axiom,
! [A3: set_set_v,B: set_set_v] :
( ! [X4: set_v] :
( ( member_set_v @ X4 @ A3 )
=> ~ ( member_set_v @ X4 @ B ) )
=> ( ( inf_inf_set_set_v @ A3 @ B )
= bot_bot_set_set_v ) ) ).
% Int_emptyI
thf(fact_363_Int__Collect__mono,axiom,
! [A3: set_set_v,B: set_set_v,P: set_v > $o,Q: set_v > $o] :
( ( ord_le5216385588623774835_set_v @ A3 @ B )
=> ( ! [X4: set_v] :
( ( member_set_v @ X4 @ A3 )
=> ( ( P @ X4 )
=> ( Q @ X4 ) ) )
=> ( ord_le5216385588623774835_set_v @ ( inf_inf_set_set_v @ A3 @ ( collect_set_v @ P ) ) @ ( inf_inf_set_set_v @ B @ ( collect_set_v @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_364_Int__Collect__mono,axiom,
! [A3: set_v,B: set_v,P: v > $o,Q: v > $o] :
( ( ord_less_eq_set_v @ A3 @ B )
=> ( ! [X4: v] :
( ( member_v @ X4 @ A3 )
=> ( ( P @ X4 )
=> ( Q @ X4 ) ) )
=> ( ord_less_eq_set_v @ ( inf_inf_set_v @ A3 @ ( collect_v @ P ) ) @ ( inf_inf_set_v @ B @ ( collect_v @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_365_Int__Collect__mono,axiom,
! [A3: 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 @ A3 @ B )
=> ( ! [X4: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X4 @ A3 )
=> ( ( P @ X4 )
=> ( Q @ X4 ) ) )
=> ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ A3 @ ( collec140062887454715474od_v_v @ P ) ) @ ( inf_in6271465464967711157od_v_v @ B @ ( collec140062887454715474od_v_v @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_366_Int__greatest,axiom,
! [C2: set_v,A3: set_v,B: set_v] :
( ( ord_less_eq_set_v @ C2 @ A3 )
=> ( ( ord_less_eq_set_v @ C2 @ B )
=> ( ord_less_eq_set_v @ C2 @ ( inf_inf_set_v @ A3 @ B ) ) ) ) ).
% Int_greatest
thf(fact_367_Int__greatest,axiom,
! [C2: set_Product_prod_v_v,A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ C2 @ A3 )
=> ( ( ord_le7336532860387713383od_v_v @ C2 @ B )
=> ( ord_le7336532860387713383od_v_v @ C2 @ ( inf_in6271465464967711157od_v_v @ A3 @ B ) ) ) ) ).
% Int_greatest
thf(fact_368_Int__absorb2,axiom,
! [A3: set_v,B: set_v] :
( ( ord_less_eq_set_v @ A3 @ B )
=> ( ( inf_inf_set_v @ A3 @ B )
= A3 ) ) ).
% Int_absorb2
thf(fact_369_Int__absorb2,axiom,
! [A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A3 @ B )
=> ( ( inf_in6271465464967711157od_v_v @ A3 @ B )
= A3 ) ) ).
% Int_absorb2
thf(fact_370_Int__absorb1,axiom,
! [B: set_v,A3: set_v] :
( ( ord_less_eq_set_v @ B @ A3 )
=> ( ( inf_inf_set_v @ A3 @ B )
= B ) ) ).
% Int_absorb1
thf(fact_371_Int__absorb1,axiom,
! [B: set_Product_prod_v_v,A3: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ B @ A3 )
=> ( ( inf_in6271465464967711157od_v_v @ A3 @ B )
= B ) ) ).
% Int_absorb1
thf(fact_372_Int__lower2,axiom,
! [A3: set_v,B: set_v] : ( ord_less_eq_set_v @ ( inf_inf_set_v @ A3 @ B ) @ B ) ).
% Int_lower2
thf(fact_373_Int__lower2,axiom,
! [A3: set_Product_prod_v_v,B: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ A3 @ B ) @ B ) ).
% Int_lower2
thf(fact_374_Int__lower1,axiom,
! [A3: set_v,B: set_v] : ( ord_less_eq_set_v @ ( inf_inf_set_v @ A3 @ B ) @ A3 ) ).
% Int_lower1
thf(fact_375_Int__lower1,axiom,
! [A3: set_Product_prod_v_v,B: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ A3 @ B ) @ A3 ) ).
% Int_lower1
thf(fact_376_Int__mono,axiom,
! [A3: set_v,C2: set_v,B: set_v,D2: set_v] :
( ( ord_less_eq_set_v @ A3 @ C2 )
=> ( ( ord_less_eq_set_v @ B @ D2 )
=> ( ord_less_eq_set_v @ ( inf_inf_set_v @ A3 @ B ) @ ( inf_inf_set_v @ C2 @ D2 ) ) ) ) ).
% Int_mono
thf(fact_377_Int__mono,axiom,
! [A3: set_Product_prod_v_v,C2: set_Product_prod_v_v,B: set_Product_prod_v_v,D2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A3 @ C2 )
=> ( ( ord_le7336532860387713383od_v_v @ B @ D2 )
=> ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ A3 @ B ) @ ( inf_in6271465464967711157od_v_v @ C2 @ D2 ) ) ) ) ).
% Int_mono
thf(fact_378_Int__insert__right,axiom,
! [A: set_v,A3: set_set_v,B: set_set_v] :
( ( ( member_set_v @ A @ A3 )
=> ( ( inf_inf_set_set_v @ A3 @ ( insert_set_v @ A @ B ) )
= ( insert_set_v @ A @ ( inf_inf_set_set_v @ A3 @ B ) ) ) )
& ( ~ ( member_set_v @ A @ A3 )
=> ( ( inf_inf_set_set_v @ A3 @ ( insert_set_v @ A @ B ) )
= ( inf_inf_set_set_v @ A3 @ B ) ) ) ) ).
% Int_insert_right
thf(fact_379_Int__insert__right,axiom,
! [A: product_prod_v_v,A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ( member7453568604450474000od_v_v @ A @ A3 )
=> ( ( inf_in6271465464967711157od_v_v @ A3 @ ( insert1338601472111419319od_v_v @ A @ B ) )
= ( insert1338601472111419319od_v_v @ A @ ( inf_in6271465464967711157od_v_v @ A3 @ B ) ) ) )
& ( ~ ( member7453568604450474000od_v_v @ A @ A3 )
=> ( ( inf_in6271465464967711157od_v_v @ A3 @ ( insert1338601472111419319od_v_v @ A @ B ) )
= ( inf_in6271465464967711157od_v_v @ A3 @ B ) ) ) ) ).
% Int_insert_right
thf(fact_380_Int__insert__right,axiom,
! [A: v,A3: set_v,B: set_v] :
( ( ( member_v @ A @ A3 )
=> ( ( inf_inf_set_v @ A3 @ ( insert_v @ A @ B ) )
= ( insert_v @ A @ ( inf_inf_set_v @ A3 @ B ) ) ) )
& ( ~ ( member_v @ A @ A3 )
=> ( ( inf_inf_set_v @ A3 @ ( insert_v @ A @ B ) )
= ( inf_inf_set_v @ A3 @ B ) ) ) ) ).
% Int_insert_right
thf(fact_381_Int__insert__left,axiom,
! [A: set_v,C2: set_set_v,B: set_set_v] :
( ( ( member_set_v @ A @ C2 )
=> ( ( inf_inf_set_set_v @ ( insert_set_v @ A @ B ) @ C2 )
= ( insert_set_v @ A @ ( inf_inf_set_set_v @ B @ C2 ) ) ) )
& ( ~ ( member_set_v @ A @ C2 )
=> ( ( inf_inf_set_set_v @ ( insert_set_v @ A @ B ) @ C2 )
= ( inf_inf_set_set_v @ B @ C2 ) ) ) ) ).
% Int_insert_left
thf(fact_382_Int__insert__left,axiom,
! [A: product_prod_v_v,C2: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ( member7453568604450474000od_v_v @ A @ C2 )
=> ( ( inf_in6271465464967711157od_v_v @ ( insert1338601472111419319od_v_v @ A @ B ) @ C2 )
= ( insert1338601472111419319od_v_v @ A @ ( inf_in6271465464967711157od_v_v @ B @ C2 ) ) ) )
& ( ~ ( member7453568604450474000od_v_v @ A @ C2 )
=> ( ( inf_in6271465464967711157od_v_v @ ( insert1338601472111419319od_v_v @ A @ B ) @ C2 )
= ( inf_in6271465464967711157od_v_v @ B @ C2 ) ) ) ) ).
% Int_insert_left
thf(fact_383_Int__insert__left,axiom,
! [A: v,C2: set_v,B: set_v] :
( ( ( member_v @ A @ C2 )
=> ( ( inf_inf_set_v @ ( insert_v @ A @ B ) @ C2 )
= ( insert_v @ A @ ( inf_inf_set_v @ B @ C2 ) ) ) )
& ( ~ ( member_v @ A @ C2 )
=> ( ( inf_inf_set_v @ ( insert_v @ A @ B ) @ C2 )
= ( inf_inf_set_v @ B @ C2 ) ) ) ) ).
% Int_insert_left
thf(fact_384_double__diff,axiom,
! [A3: set_v,B: set_v,C2: set_v] :
( ( ord_less_eq_set_v @ A3 @ B )
=> ( ( ord_less_eq_set_v @ B @ C2 )
=> ( ( minus_minus_set_v @ B @ ( minus_minus_set_v @ C2 @ A3 ) )
= A3 ) ) ) ).
% double_diff
thf(fact_385_double__diff,axiom,
! [A3: set_Product_prod_v_v,B: set_Product_prod_v_v,C2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A3 @ B )
=> ( ( ord_le7336532860387713383od_v_v @ B @ C2 )
=> ( ( minus_4183494784930505774od_v_v @ B @ ( minus_4183494784930505774od_v_v @ C2 @ A3 ) )
= A3 ) ) ) ).
% double_diff
thf(fact_386_Diff__subset,axiom,
! [A3: set_v,B: set_v] : ( ord_less_eq_set_v @ ( minus_minus_set_v @ A3 @ B ) @ A3 ) ).
% Diff_subset
thf(fact_387_Diff__subset,axiom,
! [A3: set_Product_prod_v_v,B: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ ( minus_4183494784930505774od_v_v @ A3 @ B ) @ A3 ) ).
% Diff_subset
thf(fact_388_Diff__mono,axiom,
! [A3: set_v,C2: set_v,D2: set_v,B: set_v] :
( ( ord_less_eq_set_v @ A3 @ C2 )
=> ( ( ord_less_eq_set_v @ D2 @ B )
=> ( ord_less_eq_set_v @ ( minus_minus_set_v @ A3 @ B ) @ ( minus_minus_set_v @ C2 @ D2 ) ) ) ) ).
% Diff_mono
thf(fact_389_Diff__mono,axiom,
! [A3: set_Product_prod_v_v,C2: set_Product_prod_v_v,D2: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A3 @ C2 )
=> ( ( ord_le7336532860387713383od_v_v @ D2 @ B )
=> ( ord_le7336532860387713383od_v_v @ ( minus_4183494784930505774od_v_v @ A3 @ B ) @ ( minus_4183494784930505774od_v_v @ C2 @ D2 ) ) ) ) ).
% Diff_mono
thf(fact_390_insert__Diff__if,axiom,
! [X2: set_v,B: set_set_v,A3: set_set_v] :
( ( ( member_set_v @ X2 @ B )
=> ( ( minus_7228012346218142266_set_v @ ( insert_set_v @ X2 @ A3 ) @ B )
= ( minus_7228012346218142266_set_v @ A3 @ B ) ) )
& ( ~ ( member_set_v @ X2 @ B )
=> ( ( minus_7228012346218142266_set_v @ ( insert_set_v @ X2 @ A3 ) @ B )
= ( insert_set_v @ X2 @ ( minus_7228012346218142266_set_v @ A3 @ B ) ) ) ) ) ).
% insert_Diff_if
thf(fact_391_insert__Diff__if,axiom,
! [X2: product_prod_v_v,B: set_Product_prod_v_v,A3: set_Product_prod_v_v] :
( ( ( member7453568604450474000od_v_v @ X2 @ B )
=> ( ( minus_4183494784930505774od_v_v @ ( insert1338601472111419319od_v_v @ X2 @ A3 ) @ B )
= ( minus_4183494784930505774od_v_v @ A3 @ B ) ) )
& ( ~ ( member7453568604450474000od_v_v @ X2 @ B )
=> ( ( minus_4183494784930505774od_v_v @ ( insert1338601472111419319od_v_v @ X2 @ A3 ) @ B )
= ( insert1338601472111419319od_v_v @ X2 @ ( minus_4183494784930505774od_v_v @ A3 @ B ) ) ) ) ) ).
% insert_Diff_if
thf(fact_392_insert__Diff__if,axiom,
! [X2: v,B: set_v,A3: set_v] :
( ( ( member_v @ X2 @ B )
=> ( ( minus_minus_set_v @ ( insert_v @ X2 @ A3 ) @ B )
= ( minus_minus_set_v @ A3 @ B ) ) )
& ( ~ ( member_v @ X2 @ B )
=> ( ( minus_minus_set_v @ ( insert_v @ X2 @ A3 ) @ B )
= ( insert_v @ X2 @ ( minus_minus_set_v @ A3 @ B ) ) ) ) ) ).
% insert_Diff_if
thf(fact_393_Diff__Int__distrib2,axiom,
! [A3: set_v,B: set_v,C2: set_v] :
( ( inf_inf_set_v @ ( minus_minus_set_v @ A3 @ B ) @ C2 )
= ( minus_minus_set_v @ ( inf_inf_set_v @ A3 @ C2 ) @ ( inf_inf_set_v @ B @ C2 ) ) ) ).
% Diff_Int_distrib2
thf(fact_394_Diff__Int__distrib,axiom,
! [C2: set_v,A3: set_v,B: set_v] :
( ( inf_inf_set_v @ C2 @ ( minus_minus_set_v @ A3 @ B ) )
= ( minus_minus_set_v @ ( inf_inf_set_v @ C2 @ A3 ) @ ( inf_inf_set_v @ C2 @ B ) ) ) ).
% Diff_Int_distrib
thf(fact_395_Diff__Diff__Int,axiom,
! [A3: set_v,B: set_v] :
( ( minus_minus_set_v @ A3 @ ( minus_minus_set_v @ A3 @ B ) )
= ( inf_inf_set_v @ A3 @ B ) ) ).
% Diff_Diff_Int
thf(fact_396_Diff__Int2,axiom,
! [A3: set_v,C2: set_v,B: set_v] :
( ( minus_minus_set_v @ ( inf_inf_set_v @ A3 @ C2 ) @ ( inf_inf_set_v @ B @ C2 ) )
= ( minus_minus_set_v @ ( inf_inf_set_v @ A3 @ C2 ) @ B ) ) ).
% Diff_Int2
thf(fact_397_Int__Diff,axiom,
! [A3: set_v,B: set_v,C2: set_v] :
( ( minus_minus_set_v @ ( inf_inf_set_v @ A3 @ B ) @ C2 )
= ( inf_inf_set_v @ A3 @ ( minus_minus_set_v @ B @ C2 ) ) ) ).
% Int_Diff
thf(fact_398_graph_Ostack__class,axiom,
! [Vertices: set_v,Successors: v > set_v,E2: sCC_Bl1394983891496994913t_unit,N2: v,M2: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl9196236973127232072t_unit @ Successors @ E2 )
=> ( ( member_v @ N2 @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E2 ) ) )
=> ( ( member_v @ M2 @ ( sCC_Bl1280885523602775798t_unit @ E2 @ N2 ) )
=> ( member_v @ M2 @ ( minus_minus_set_v @ ( sCC_Bl4645233313691564917t_unit @ E2 ) @ ( sCC_Bl157864678168468314t_unit @ E2 ) ) ) ) ) ) ) ).
% graph.stack_class
thf(fact_399_cSup__eq__non__empty,axiom,
! [X5: set_se8455005133513928103od_v_v,A: set_Product_prod_v_v] :
( ( X5 != bot_bo3497076220358800403od_v_v )
=> ( ! [X4: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ X4 @ X5 )
=> ( ord_le7336532860387713383od_v_v @ X4 @ A ) )
=> ( ! [Y2: set_Product_prod_v_v] :
( ! [X: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ X @ X5 )
=> ( ord_le7336532860387713383od_v_v @ X @ Y2 ) )
=> ( ord_le7336532860387713383od_v_v @ A @ Y2 ) )
=> ( ( comple5788137035815166516od_v_v @ X5 )
= A ) ) ) ) ).
% cSup_eq_non_empty
thf(fact_400_cSup__eq__non__empty,axiom,
! [X5: set_set_v,A: set_v] :
( ( X5 != bot_bot_set_set_v )
=> ( ! [X4: set_v] :
( ( member_set_v @ X4 @ X5 )
=> ( ord_less_eq_set_v @ X4 @ A ) )
=> ( ! [Y2: set_v] :
( ! [X: set_v] :
( ( member_set_v @ X @ X5 )
=> ( ord_less_eq_set_v @ X @ Y2 ) )
=> ( ord_less_eq_set_v @ A @ Y2 ) )
=> ( ( comple2307003700295860064_set_v @ X5 )
= A ) ) ) ) ).
% cSup_eq_non_empty
thf(fact_401_cSup__least,axiom,
! [X5: set_se8455005133513928103od_v_v,Z: set_Product_prod_v_v] :
( ( X5 != bot_bo3497076220358800403od_v_v )
=> ( ! [X4: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ X4 @ X5 )
=> ( ord_le7336532860387713383od_v_v @ X4 @ Z ) )
=> ( ord_le7336532860387713383od_v_v @ ( comple5788137035815166516od_v_v @ X5 ) @ Z ) ) ) ).
% cSup_least
thf(fact_402_cSup__least,axiom,
! [X5: set_set_v,Z: set_v] :
( ( X5 != bot_bot_set_set_v )
=> ( ! [X4: set_v] :
( ( member_set_v @ X4 @ X5 )
=> ( ord_less_eq_set_v @ X4 @ Z ) )
=> ( ord_less_eq_set_v @ ( comple2307003700295860064_set_v @ X5 ) @ Z ) ) ) ).
% cSup_least
thf(fact_403_subset__singleton__iff,axiom,
! [X5: set_set_v,A: set_v] :
( ( ord_le5216385588623774835_set_v @ X5 @ ( insert_set_v @ A @ bot_bot_set_set_v ) )
= ( ( X5 = bot_bot_set_set_v )
| ( X5
= ( insert_set_v @ A @ bot_bot_set_set_v ) ) ) ) ).
% subset_singleton_iff
thf(fact_404_subset__singleton__iff,axiom,
! [X5: set_v,A: v] :
( ( ord_less_eq_set_v @ X5 @ ( insert_v @ A @ bot_bot_set_v ) )
= ( ( X5 = bot_bot_set_v )
| ( X5
= ( insert_v @ A @ bot_bot_set_v ) ) ) ) ).
% subset_singleton_iff
thf(fact_405_subset__singleton__iff,axiom,
! [X5: set_Product_prod_v_v,A: product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X5 @ ( insert1338601472111419319od_v_v @ A @ bot_bo723834152578015283od_v_v ) )
= ( ( X5 = bot_bo723834152578015283od_v_v )
| ( X5
= ( insert1338601472111419319od_v_v @ A @ bot_bo723834152578015283od_v_v ) ) ) ) ).
% subset_singleton_iff
thf(fact_406_subset__singletonD,axiom,
! [A3: set_set_v,X2: set_v] :
( ( ord_le5216385588623774835_set_v @ A3 @ ( insert_set_v @ X2 @ bot_bot_set_set_v ) )
=> ( ( A3 = bot_bot_set_set_v )
| ( A3
= ( insert_set_v @ X2 @ bot_bot_set_set_v ) ) ) ) ).
% subset_singletonD
thf(fact_407_subset__singletonD,axiom,
! [A3: set_v,X2: v] :
( ( ord_less_eq_set_v @ A3 @ ( insert_v @ X2 @ bot_bot_set_v ) )
=> ( ( A3 = bot_bot_set_v )
| ( A3
= ( insert_v @ X2 @ bot_bot_set_v ) ) ) ) ).
% subset_singletonD
thf(fact_408_subset__singletonD,axiom,
! [A3: set_Product_prod_v_v,X2: product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A3 @ ( insert1338601472111419319od_v_v @ X2 @ bot_bo723834152578015283od_v_v ) )
=> ( ( A3 = bot_bo723834152578015283od_v_v )
| ( A3
= ( insert1338601472111419319od_v_v @ X2 @ bot_bo723834152578015283od_v_v ) ) ) ) ).
% subset_singletonD
thf(fact_409_Diff__insert__absorb,axiom,
! [X2: product_prod_v_v,A3: set_Product_prod_v_v] :
( ~ ( member7453568604450474000od_v_v @ X2 @ A3 )
=> ( ( minus_4183494784930505774od_v_v @ ( insert1338601472111419319od_v_v @ X2 @ A3 ) @ ( insert1338601472111419319od_v_v @ X2 @ bot_bo723834152578015283od_v_v ) )
= A3 ) ) ).
% Diff_insert_absorb
thf(fact_410_Diff__insert__absorb,axiom,
! [X2: set_v,A3: set_set_v] :
( ~ ( member_set_v @ X2 @ A3 )
=> ( ( minus_7228012346218142266_set_v @ ( insert_set_v @ X2 @ A3 ) @ ( insert_set_v @ X2 @ bot_bot_set_set_v ) )
= A3 ) ) ).
% Diff_insert_absorb
thf(fact_411_Diff__insert__absorb,axiom,
! [X2: v,A3: set_v] :
( ~ ( member_v @ X2 @ A3 )
=> ( ( minus_minus_set_v @ ( insert_v @ X2 @ A3 ) @ ( insert_v @ X2 @ bot_bot_set_v ) )
= A3 ) ) ).
% Diff_insert_absorb
thf(fact_412_Diff__insert2,axiom,
! [A3: set_Product_prod_v_v,A: product_prod_v_v,B: set_Product_prod_v_v] :
( ( minus_4183494784930505774od_v_v @ A3 @ ( insert1338601472111419319od_v_v @ A @ B ) )
= ( minus_4183494784930505774od_v_v @ ( minus_4183494784930505774od_v_v @ A3 @ ( insert1338601472111419319od_v_v @ A @ bot_bo723834152578015283od_v_v ) ) @ B ) ) ).
% Diff_insert2
thf(fact_413_Diff__insert2,axiom,
! [A3: set_set_v,A: set_v,B: set_set_v] :
( ( minus_7228012346218142266_set_v @ A3 @ ( insert_set_v @ A @ B ) )
= ( minus_7228012346218142266_set_v @ ( minus_7228012346218142266_set_v @ A3 @ ( insert_set_v @ A @ bot_bot_set_set_v ) ) @ B ) ) ).
% Diff_insert2
thf(fact_414_Diff__insert2,axiom,
! [A3: set_v,A: v,B: set_v] :
( ( minus_minus_set_v @ A3 @ ( insert_v @ A @ B ) )
= ( minus_minus_set_v @ ( minus_minus_set_v @ A3 @ ( insert_v @ A @ bot_bot_set_v ) ) @ B ) ) ).
% Diff_insert2
thf(fact_415_insert__Diff,axiom,
! [A: product_prod_v_v,A3: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ A @ A3 )
=> ( ( insert1338601472111419319od_v_v @ A @ ( minus_4183494784930505774od_v_v @ A3 @ ( insert1338601472111419319od_v_v @ A @ bot_bo723834152578015283od_v_v ) ) )
= A3 ) ) ).
% insert_Diff
thf(fact_416_insert__Diff,axiom,
! [A: set_v,A3: set_set_v] :
( ( member_set_v @ A @ A3 )
=> ( ( insert_set_v @ A @ ( minus_7228012346218142266_set_v @ A3 @ ( insert_set_v @ A @ bot_bot_set_set_v ) ) )
= A3 ) ) ).
% insert_Diff
thf(fact_417_insert__Diff,axiom,
! [A: v,A3: set_v] :
( ( member_v @ A @ A3 )
=> ( ( insert_v @ A @ ( minus_minus_set_v @ A3 @ ( insert_v @ A @ bot_bot_set_v ) ) )
= A3 ) ) ).
% insert_Diff
thf(fact_418_Diff__insert,axiom,
! [A3: set_Product_prod_v_v,A: product_prod_v_v,B: set_Product_prod_v_v] :
( ( minus_4183494784930505774od_v_v @ A3 @ ( insert1338601472111419319od_v_v @ A @ B ) )
= ( minus_4183494784930505774od_v_v @ ( minus_4183494784930505774od_v_v @ A3 @ B ) @ ( insert1338601472111419319od_v_v @ A @ bot_bo723834152578015283od_v_v ) ) ) ).
% Diff_insert
thf(fact_419_Diff__insert,axiom,
! [A3: set_set_v,A: set_v,B: set_set_v] :
( ( minus_7228012346218142266_set_v @ A3 @ ( insert_set_v @ A @ B ) )
= ( minus_7228012346218142266_set_v @ ( minus_7228012346218142266_set_v @ A3 @ B ) @ ( insert_set_v @ A @ bot_bot_set_set_v ) ) ) ).
% Diff_insert
thf(fact_420_Diff__insert,axiom,
! [A3: set_v,A: v,B: set_v] :
( ( minus_minus_set_v @ A3 @ ( insert_v @ A @ B ) )
= ( minus_minus_set_v @ ( minus_minus_set_v @ A3 @ B ) @ ( insert_v @ A @ bot_bot_set_v ) ) ) ).
% Diff_insert
thf(fact_421_subset__Diff__insert,axiom,
! [A3: set_set_v,B: set_set_v,X2: set_v,C2: set_set_v] :
( ( ord_le5216385588623774835_set_v @ A3 @ ( minus_7228012346218142266_set_v @ B @ ( insert_set_v @ X2 @ C2 ) ) )
= ( ( ord_le5216385588623774835_set_v @ A3 @ ( minus_7228012346218142266_set_v @ B @ C2 ) )
& ~ ( member_set_v @ X2 @ A3 ) ) ) ).
% subset_Diff_insert
thf(fact_422_subset__Diff__insert,axiom,
! [A3: set_v,B: set_v,X2: v,C2: set_v] :
( ( ord_less_eq_set_v @ A3 @ ( minus_minus_set_v @ B @ ( insert_v @ X2 @ C2 ) ) )
= ( ( ord_less_eq_set_v @ A3 @ ( minus_minus_set_v @ B @ C2 ) )
& ~ ( member_v @ X2 @ A3 ) ) ) ).
% subset_Diff_insert
thf(fact_423_subset__Diff__insert,axiom,
! [A3: set_Product_prod_v_v,B: set_Product_prod_v_v,X2: product_prod_v_v,C2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A3 @ ( minus_4183494784930505774od_v_v @ B @ ( insert1338601472111419319od_v_v @ X2 @ C2 ) ) )
= ( ( ord_le7336532860387713383od_v_v @ A3 @ ( minus_4183494784930505774od_v_v @ B @ C2 ) )
& ~ ( member7453568604450474000od_v_v @ X2 @ A3 ) ) ) ).
% subset_Diff_insert
thf(fact_424_Int__Diff__disjoint,axiom,
! [A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ ( inf_in6271465464967711157od_v_v @ A3 @ B ) @ ( minus_4183494784930505774od_v_v @ A3 @ B ) )
= bot_bo723834152578015283od_v_v ) ).
% Int_Diff_disjoint
thf(fact_425_Int__Diff__disjoint,axiom,
! [A3: set_set_v,B: set_set_v] :
( ( inf_inf_set_set_v @ ( inf_inf_set_set_v @ A3 @ B ) @ ( minus_7228012346218142266_set_v @ A3 @ B ) )
= bot_bot_set_set_v ) ).
% Int_Diff_disjoint
thf(fact_426_Int__Diff__disjoint,axiom,
! [A3: set_v,B: set_v] :
( ( inf_inf_set_v @ ( inf_inf_set_v @ A3 @ B ) @ ( minus_minus_set_v @ A3 @ B ) )
= bot_bot_set_v ) ).
% Int_Diff_disjoint
thf(fact_427_Diff__triv,axiom,
! [A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ( inf_in6271465464967711157od_v_v @ A3 @ B )
= bot_bo723834152578015283od_v_v )
=> ( ( minus_4183494784930505774od_v_v @ A3 @ B )
= A3 ) ) ).
% Diff_triv
thf(fact_428_Diff__triv,axiom,
! [A3: set_set_v,B: set_set_v] :
( ( ( inf_inf_set_set_v @ A3 @ B )
= bot_bot_set_set_v )
=> ( ( minus_7228012346218142266_set_v @ A3 @ B )
= A3 ) ) ).
% Diff_triv
thf(fact_429_Diff__triv,axiom,
! [A3: set_v,B: set_v] :
( ( ( inf_inf_set_v @ A3 @ B )
= bot_bot_set_v )
=> ( ( minus_minus_set_v @ A3 @ B )
= A3 ) ) ).
% Diff_triv
thf(fact_430_Diff__single__insert,axiom,
! [A3: set_set_v,X2: set_v,B: set_set_v] :
( ( ord_le5216385588623774835_set_v @ ( minus_7228012346218142266_set_v @ A3 @ ( insert_set_v @ X2 @ bot_bot_set_set_v ) ) @ B )
=> ( ord_le5216385588623774835_set_v @ A3 @ ( insert_set_v @ X2 @ B ) ) ) ).
% Diff_single_insert
thf(fact_431_Diff__single__insert,axiom,
! [A3: set_v,X2: v,B: set_v] :
( ( ord_less_eq_set_v @ ( minus_minus_set_v @ A3 @ ( insert_v @ X2 @ bot_bot_set_v ) ) @ B )
=> ( ord_less_eq_set_v @ A3 @ ( insert_v @ X2 @ B ) ) ) ).
% Diff_single_insert
thf(fact_432_Diff__single__insert,axiom,
! [A3: set_Product_prod_v_v,X2: product_prod_v_v,B: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ ( minus_4183494784930505774od_v_v @ A3 @ ( insert1338601472111419319od_v_v @ X2 @ bot_bo723834152578015283od_v_v ) ) @ B )
=> ( ord_le7336532860387713383od_v_v @ A3 @ ( insert1338601472111419319od_v_v @ X2 @ B ) ) ) ).
% Diff_single_insert
thf(fact_433_subset__insert__iff,axiom,
! [A3: set_set_v,X2: set_v,B: set_set_v] :
( ( ord_le5216385588623774835_set_v @ A3 @ ( insert_set_v @ X2 @ B ) )
= ( ( ( member_set_v @ X2 @ A3 )
=> ( ord_le5216385588623774835_set_v @ ( minus_7228012346218142266_set_v @ A3 @ ( insert_set_v @ X2 @ bot_bot_set_set_v ) ) @ B ) )
& ( ~ ( member_set_v @ X2 @ A3 )
=> ( ord_le5216385588623774835_set_v @ A3 @ B ) ) ) ) ).
% subset_insert_iff
thf(fact_434_subset__insert__iff,axiom,
! [A3: set_v,X2: v,B: set_v] :
( ( ord_less_eq_set_v @ A3 @ ( insert_v @ X2 @ B ) )
= ( ( ( member_v @ X2 @ A3 )
=> ( ord_less_eq_set_v @ ( minus_minus_set_v @ A3 @ ( insert_v @ X2 @ bot_bot_set_v ) ) @ B ) )
& ( ~ ( member_v @ X2 @ A3 )
=> ( ord_less_eq_set_v @ A3 @ B ) ) ) ) ).
% subset_insert_iff
thf(fact_435_subset__insert__iff,axiom,
! [A3: set_Product_prod_v_v,X2: product_prod_v_v,B: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A3 @ ( insert1338601472111419319od_v_v @ X2 @ B ) )
= ( ( ( member7453568604450474000od_v_v @ X2 @ A3 )
=> ( ord_le7336532860387713383od_v_v @ ( minus_4183494784930505774od_v_v @ A3 @ ( insert1338601472111419319od_v_v @ X2 @ bot_bo723834152578015283od_v_v ) ) @ B ) )
& ( ~ ( member7453568604450474000od_v_v @ X2 @ A3 )
=> ( ord_le7336532860387713383od_v_v @ A3 @ B ) ) ) ) ).
% subset_insert_iff
thf(fact_436_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_437_Sup__empty,axiom,
( ( comple5788137035815166516od_v_v @ bot_bo3497076220358800403od_v_v )
= bot_bo723834152578015283od_v_v ) ).
% Sup_empty
thf(fact_438_Sup__empty,axiom,
( ( comple4687483117567863418t_unit @ bot_bo3957492148770167129t_unit )
= bot_bot_Product_unit ) ).
% Sup_empty
thf(fact_439_Sup__empty,axiom,
( ( comple5450237519782573632_set_v @ bot_bo5775917114081396255_set_v )
= bot_bot_set_set_v ) ).
% Sup_empty
thf(fact_440_Sup__empty,axiom,
( ( comple2307003700295860064_set_v @ bot_bot_set_set_v )
= bot_bot_set_v ) ).
% Sup_empty
thf(fact_441_equality,axiom,
! [R: sCC_Bl1394983891496994913t_unit,R2: sCC_Bl1394983891496994913t_unit] :
( ( ( sCC_Bl1090238580953940555t_unit @ R )
= ( sCC_Bl1090238580953940555t_unit @ R2 ) )
=> ( ( ( sCC_Bl1280885523602775798t_unit @ R )
= ( sCC_Bl1280885523602775798t_unit @ R2 ) )
=> ( ( ( sCC_Bl157864678168468314t_unit @ R )
= ( sCC_Bl157864678168468314t_unit @ R2 ) )
=> ( ( ( sCC_Bl4645233313691564917t_unit @ R )
= ( sCC_Bl4645233313691564917t_unit @ R2 ) )
=> ( ( ( sCC_Bl3795065053823578884t_unit @ R )
= ( sCC_Bl3795065053823578884t_unit @ R2 ) )
=> ( ( ( sCC_Bl2536197123907397897t_unit @ R )
= ( sCC_Bl2536197123907397897t_unit @ R2 ) )
=> ( ( ( sCC_Bl8828226123343373779t_unit @ R )
= ( sCC_Bl8828226123343373779t_unit @ R2 ) )
=> ( ( ( sCC_Bl9201514103433284750t_unit @ R )
= ( sCC_Bl9201514103433284750t_unit @ R2 ) )
=> ( ( ( sCC_Bl3567736435408124606t_unit @ R )
= ( sCC_Bl3567736435408124606t_unit @ R2 ) )
=> ( R = R2 ) ) ) ) ) ) ) ) ) ) ).
% equality
thf(fact_442_Sup__bot__conv_I1_J,axiom,
! [A3: set_se8455005133513928103od_v_v] :
( ( ( comple5788137035815166516od_v_v @ A3 )
= bot_bo723834152578015283od_v_v )
= ( ! [X3: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ X3 @ A3 )
=> ( X3 = bot_bo723834152578015283od_v_v ) ) ) ) ).
% Sup_bot_conv(1)
thf(fact_443_Sup__bot__conv_I1_J,axiom,
! [A3: set_Product_unit] :
( ( ( comple4687483117567863418t_unit @ A3 )
= bot_bot_Product_unit )
= ( ! [X3: product_unit] :
( ( member_Product_unit @ X3 @ A3 )
=> ( X3 = bot_bot_Product_unit ) ) ) ) ).
% Sup_bot_conv(1)
thf(fact_444_Sup__bot__conv_I1_J,axiom,
! [A3: set_set_set_v] :
( ( ( comple5450237519782573632_set_v @ A3 )
= bot_bot_set_set_v )
= ( ! [X3: set_set_v] :
( ( member_set_set_v @ X3 @ A3 )
=> ( X3 = bot_bot_set_set_v ) ) ) ) ).
% Sup_bot_conv(1)
thf(fact_445_Sup__bot__conv_I1_J,axiom,
! [A3: set_set_v] :
( ( ( comple2307003700295860064_set_v @ A3 )
= bot_bot_set_v )
= ( ! [X3: set_v] :
( ( member_set_v @ X3 @ A3 )
=> ( X3 = bot_bot_set_v ) ) ) ) ).
% Sup_bot_conv(1)
thf(fact_446_Sup__bot__conv_I2_J,axiom,
! [A3: set_se8455005133513928103od_v_v] :
( ( bot_bo723834152578015283od_v_v
= ( comple5788137035815166516od_v_v @ A3 ) )
= ( ! [X3: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ X3 @ A3 )
=> ( X3 = bot_bo723834152578015283od_v_v ) ) ) ) ).
% Sup_bot_conv(2)
thf(fact_447_Sup__bot__conv_I2_J,axiom,
! [A3: set_Product_unit] :
( ( bot_bot_Product_unit
= ( comple4687483117567863418t_unit @ A3 ) )
= ( ! [X3: product_unit] :
( ( member_Product_unit @ X3 @ A3 )
=> ( X3 = bot_bot_Product_unit ) ) ) ) ).
% Sup_bot_conv(2)
thf(fact_448_Sup__bot__conv_I2_J,axiom,
! [A3: set_set_set_v] :
( ( bot_bot_set_set_v
= ( comple5450237519782573632_set_v @ A3 ) )
= ( ! [X3: set_set_v] :
( ( member_set_set_v @ X3 @ A3 )
=> ( X3 = bot_bot_set_set_v ) ) ) ) ).
% Sup_bot_conv(2)
thf(fact_449_Sup__bot__conv_I2_J,axiom,
! [A3: set_set_v] :
( ( bot_bot_set_v
= ( comple2307003700295860064_set_v @ A3 ) )
= ( ! [X3: set_v] :
( ( member_set_v @ X3 @ A3 )
=> ( X3 = bot_bot_set_v ) ) ) ) ).
% Sup_bot_conv(2)
thf(fact_450_boolean__algebra_Oconj__zero__right,axiom,
! [X2: set_v] :
( ( inf_inf_set_v @ X2 @ bot_bot_set_v )
= bot_bot_set_v ) ).
% boolean_algebra.conj_zero_right
thf(fact_451_boolean__algebra_Oconj__zero__right,axiom,
! [X2: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ X2 @ bot_bo723834152578015283od_v_v )
= bot_bo723834152578015283od_v_v ) ).
% boolean_algebra.conj_zero_right
thf(fact_452_boolean__algebra_Oconj__zero__right,axiom,
! [X2: product_unit] :
( ( inf_inf_Product_unit @ X2 @ bot_bot_Product_unit )
= bot_bot_Product_unit ) ).
% boolean_algebra.conj_zero_right
thf(fact_453_boolean__algebra_Oconj__zero__right,axiom,
! [X2: set_set_v] :
( ( inf_inf_set_set_v @ X2 @ bot_bot_set_set_v )
= bot_bot_set_set_v ) ).
% boolean_algebra.conj_zero_right
thf(fact_454_boolean__algebra_Oconj__zero__left,axiom,
! [X2: set_v] :
( ( inf_inf_set_v @ bot_bot_set_v @ X2 )
= bot_bot_set_v ) ).
% boolean_algebra.conj_zero_left
thf(fact_455_boolean__algebra_Oconj__zero__left,axiom,
! [X2: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ bot_bo723834152578015283od_v_v @ X2 )
= bot_bo723834152578015283od_v_v ) ).
% boolean_algebra.conj_zero_left
thf(fact_456_boolean__algebra_Oconj__zero__left,axiom,
! [X2: product_unit] :
( ( inf_inf_Product_unit @ bot_bot_Product_unit @ X2 )
= bot_bot_Product_unit ) ).
% boolean_algebra.conj_zero_left
thf(fact_457_boolean__algebra_Oconj__zero__left,axiom,
! [X2: set_set_v] :
( ( inf_inf_set_set_v @ bot_bot_set_set_v @ X2 )
= bot_bot_set_set_v ) ).
% boolean_algebra.conj_zero_left
thf(fact_458_inf__bot__right,axiom,
! [X2: set_v] :
( ( inf_inf_set_v @ X2 @ bot_bot_set_v )
= bot_bot_set_v ) ).
% inf_bot_right
thf(fact_459_inf__bot__right,axiom,
! [X2: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ X2 @ bot_bo723834152578015283od_v_v )
= bot_bo723834152578015283od_v_v ) ).
% inf_bot_right
thf(fact_460_inf__bot__right,axiom,
! [X2: product_unit] :
( ( inf_inf_Product_unit @ X2 @ bot_bot_Product_unit )
= bot_bot_Product_unit ) ).
% inf_bot_right
thf(fact_461_inf__bot__right,axiom,
! [X2: set_set_v] :
( ( inf_inf_set_set_v @ X2 @ bot_bot_set_set_v )
= bot_bot_set_set_v ) ).
% inf_bot_right
thf(fact_462_inf__bot__left,axiom,
! [X2: set_v] :
( ( inf_inf_set_v @ bot_bot_set_v @ X2 )
= bot_bot_set_v ) ).
% inf_bot_left
thf(fact_463_inf__bot__left,axiom,
! [X2: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ bot_bo723834152578015283od_v_v @ X2 )
= bot_bo723834152578015283od_v_v ) ).
% inf_bot_left
thf(fact_464_inf__bot__left,axiom,
! [X2: product_unit] :
( ( inf_inf_Product_unit @ bot_bot_Product_unit @ X2 )
= bot_bot_Product_unit ) ).
% inf_bot_left
thf(fact_465_inf__bot__left,axiom,
! [X2: set_set_v] :
( ( inf_inf_set_set_v @ bot_bot_set_set_v @ X2 )
= bot_bot_set_set_v ) ).
% inf_bot_left
thf(fact_466_inf_Obounded__iff,axiom,
! [A: product_unit,B2: product_unit,C: product_unit] :
( ( ord_le3221252021190050221t_unit @ A @ ( inf_inf_Product_unit @ B2 @ C ) )
= ( ( ord_le3221252021190050221t_unit @ A @ B2 )
& ( ord_le3221252021190050221t_unit @ A @ C ) ) ) ).
% inf.bounded_iff
thf(fact_467_inf_Obounded__iff,axiom,
! [A: set_v,B2: set_v,C: set_v] :
( ( ord_less_eq_set_v @ A @ ( inf_inf_set_v @ B2 @ C ) )
= ( ( ord_less_eq_set_v @ A @ B2 )
& ( ord_less_eq_set_v @ A @ C ) ) ) ).
% inf.bounded_iff
thf(fact_468_inf_Obounded__iff,axiom,
! [A: set_Product_prod_v_v,B2: set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ ( inf_in6271465464967711157od_v_v @ B2 @ C ) )
= ( ( ord_le7336532860387713383od_v_v @ A @ B2 )
& ( ord_le7336532860387713383od_v_v @ A @ C ) ) ) ).
% inf.bounded_iff
thf(fact_469_inf_Oidem,axiom,
! [A: set_v] :
( ( inf_inf_set_v @ A @ A )
= A ) ).
% inf.idem
thf(fact_470_inf_Oidem,axiom,
! [A: product_unit] :
( ( inf_inf_Product_unit @ A @ A )
= A ) ).
% inf.idem
thf(fact_471_inf__idem,axiom,
! [X2: set_v] :
( ( inf_inf_set_v @ X2 @ X2 )
= X2 ) ).
% inf_idem
thf(fact_472_inf__idem,axiom,
! [X2: product_unit] :
( ( inf_inf_Product_unit @ X2 @ X2 )
= X2 ) ).
% inf_idem
thf(fact_473_inf_Oleft__idem,axiom,
! [A: set_v,B2: set_v] :
( ( inf_inf_set_v @ A @ ( inf_inf_set_v @ A @ B2 ) )
= ( inf_inf_set_v @ A @ B2 ) ) ).
% inf.left_idem
thf(fact_474_inf_Oleft__idem,axiom,
! [A: product_unit,B2: product_unit] :
( ( inf_inf_Product_unit @ A @ ( inf_inf_Product_unit @ A @ B2 ) )
= ( inf_inf_Product_unit @ A @ B2 ) ) ).
% inf.left_idem
thf(fact_475_inf__left__idem,axiom,
! [X2: set_v,Y: set_v] :
( ( inf_inf_set_v @ X2 @ ( inf_inf_set_v @ X2 @ Y ) )
= ( inf_inf_set_v @ X2 @ Y ) ) ).
% inf_left_idem
thf(fact_476_inf__left__idem,axiom,
! [X2: product_unit,Y: product_unit] :
( ( inf_inf_Product_unit @ X2 @ ( inf_inf_Product_unit @ X2 @ Y ) )
= ( inf_inf_Product_unit @ X2 @ Y ) ) ).
% inf_left_idem
thf(fact_477_inf_Oright__idem,axiom,
! [A: set_v,B2: set_v] :
( ( inf_inf_set_v @ ( inf_inf_set_v @ A @ B2 ) @ B2 )
= ( inf_inf_set_v @ A @ B2 ) ) ).
% inf.right_idem
thf(fact_478_inf_Oright__idem,axiom,
! [A: product_unit,B2: product_unit] :
( ( inf_inf_Product_unit @ ( inf_inf_Product_unit @ A @ B2 ) @ B2 )
= ( inf_inf_Product_unit @ A @ B2 ) ) ).
% inf.right_idem
thf(fact_479_inf__right__idem,axiom,
! [X2: set_v,Y: set_v] :
( ( inf_inf_set_v @ ( inf_inf_set_v @ X2 @ Y ) @ Y )
= ( inf_inf_set_v @ X2 @ Y ) ) ).
% inf_right_idem
thf(fact_480_inf__right__idem,axiom,
! [X2: product_unit,Y: product_unit] :
( ( inf_inf_Product_unit @ ( inf_inf_Product_unit @ X2 @ Y ) @ Y )
= ( inf_inf_Product_unit @ X2 @ Y ) ) ).
% inf_right_idem
thf(fact_481_UN__ball__bex__simps_I3_J,axiom,
! [A3: set_set_v,P: v > $o] :
( ( ? [X3: v] :
( ( member_v @ X3 @ ( comple2307003700295860064_set_v @ A3 ) )
& ( P @ X3 ) ) )
= ( ? [X3: set_v] :
( ( member_set_v @ X3 @ A3 )
& ? [Y3: v] :
( ( member_v @ Y3 @ X3 )
& ( P @ Y3 ) ) ) ) ) ).
% UN_ball_bex_simps(3)
thf(fact_482_UN__ball__bex__simps_I1_J,axiom,
! [A3: set_set_v,P: v > $o] :
( ( ! [X3: v] :
( ( member_v @ X3 @ ( comple2307003700295860064_set_v @ A3 ) )
=> ( P @ X3 ) ) )
= ( ! [X3: set_v] :
( ( member_set_v @ X3 @ A3 )
=> ! [Y3: v] :
( ( member_v @ Y3 @ X3 )
=> ( P @ Y3 ) ) ) ) ) ).
% UN_ball_bex_simps(1)
thf(fact_483_UnionI,axiom,
! [X5: set_Product_prod_v_v,C2: set_se8455005133513928103od_v_v,A3: product_prod_v_v] :
( ( member8406446414694345712od_v_v @ X5 @ C2 )
=> ( ( member7453568604450474000od_v_v @ A3 @ X5 )
=> ( member7453568604450474000od_v_v @ A3 @ ( comple5788137035815166516od_v_v @ C2 ) ) ) ) ).
% UnionI
thf(fact_484_UnionI,axiom,
! [X5: set_v,C2: set_set_v,A3: v] :
( ( member_set_v @ X5 @ C2 )
=> ( ( member_v @ A3 @ X5 )
=> ( member_v @ A3 @ ( comple2307003700295860064_set_v @ C2 ) ) ) ) ).
% UnionI
thf(fact_485_Union__iff,axiom,
! [A3: product_prod_v_v,C2: set_se8455005133513928103od_v_v] :
( ( member7453568604450474000od_v_v @ A3 @ ( comple5788137035815166516od_v_v @ C2 ) )
= ( ? [X3: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ X3 @ C2 )
& ( member7453568604450474000od_v_v @ A3 @ X3 ) ) ) ) ).
% Union_iff
thf(fact_486_Union__iff,axiom,
! [A3: v,C2: set_set_v] :
( ( member_v @ A3 @ ( comple2307003700295860064_set_v @ C2 ) )
= ( ? [X3: set_v] :
( ( member_set_v @ X3 @ C2 )
& ( member_v @ A3 @ X3 ) ) ) ) ).
% Union_iff
thf(fact_487_le__inf__iff,axiom,
! [X2: product_unit,Y: product_unit,Z: product_unit] :
( ( ord_le3221252021190050221t_unit @ X2 @ ( inf_inf_Product_unit @ Y @ Z ) )
= ( ( ord_le3221252021190050221t_unit @ X2 @ Y )
& ( ord_le3221252021190050221t_unit @ X2 @ Z ) ) ) ).
% le_inf_iff
thf(fact_488_le__inf__iff,axiom,
! [X2: set_v,Y: set_v,Z: set_v] :
( ( ord_less_eq_set_v @ X2 @ ( inf_inf_set_v @ Y @ Z ) )
= ( ( ord_less_eq_set_v @ X2 @ Y )
& ( ord_less_eq_set_v @ X2 @ Z ) ) ) ).
% le_inf_iff
thf(fact_489_le__inf__iff,axiom,
! [X2: set_Product_prod_v_v,Y: set_Product_prod_v_v,Z: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X2 @ ( inf_in6271465464967711157od_v_v @ Y @ Z ) )
= ( ( ord_le7336532860387713383od_v_v @ X2 @ Y )
& ( ord_le7336532860387713383od_v_v @ X2 @ Z ) ) ) ).
% le_inf_iff
thf(fact_490_graph_Opre__dfss_Ocong,axiom,
sCC_Bl1748261141445803503t_unit = sCC_Bl1748261141445803503t_unit ).
% graph.pre_dfss.cong
thf(fact_491_bot__set__def,axiom,
( bot_bot_set_v
= ( collect_v @ bot_bot_v_o ) ) ).
% bot_set_def
thf(fact_492_bot__set__def,axiom,
( bot_bo723834152578015283od_v_v
= ( collec140062887454715474od_v_v @ bot_bo8461541820394803818_v_v_o ) ) ).
% bot_set_def
thf(fact_493_bot__set__def,axiom,
( bot_bot_set_set_v
= ( collect_set_v @ bot_bot_set_v_o ) ) ).
% bot_set_def
thf(fact_494_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_495_inf__sup__aci_I4_J,axiom,
! [X2: set_v,Y: set_v] :
( ( inf_inf_set_v @ X2 @ ( inf_inf_set_v @ X2 @ Y ) )
= ( inf_inf_set_v @ X2 @ Y ) ) ).
% inf_sup_aci(4)
thf(fact_496_inf__sup__aci_I4_J,axiom,
! [X2: product_unit,Y: product_unit] :
( ( inf_inf_Product_unit @ X2 @ ( inf_inf_Product_unit @ X2 @ Y ) )
= ( inf_inf_Product_unit @ X2 @ Y ) ) ).
% inf_sup_aci(4)
thf(fact_497_inf__sup__aci_I3_J,axiom,
! [X2: set_v,Y: set_v,Z: set_v] :
( ( inf_inf_set_v @ X2 @ ( inf_inf_set_v @ Y @ Z ) )
= ( inf_inf_set_v @ Y @ ( inf_inf_set_v @ X2 @ Z ) ) ) ).
% inf_sup_aci(3)
thf(fact_498_inf__sup__aci_I3_J,axiom,
! [X2: product_unit,Y: product_unit,Z: product_unit] :
( ( inf_inf_Product_unit @ X2 @ ( inf_inf_Product_unit @ Y @ Z ) )
= ( inf_inf_Product_unit @ Y @ ( inf_inf_Product_unit @ X2 @ Z ) ) ) ).
% inf_sup_aci(3)
thf(fact_499_inf__sup__aci_I2_J,axiom,
! [X2: set_v,Y: set_v,Z: set_v] :
( ( inf_inf_set_v @ ( inf_inf_set_v @ X2 @ Y ) @ Z )
= ( inf_inf_set_v @ X2 @ ( inf_inf_set_v @ Y @ Z ) ) ) ).
% inf_sup_aci(2)
thf(fact_500_inf__sup__aci_I2_J,axiom,
! [X2: product_unit,Y: product_unit,Z: product_unit] :
( ( inf_inf_Product_unit @ ( inf_inf_Product_unit @ X2 @ Y ) @ Z )
= ( inf_inf_Product_unit @ X2 @ ( inf_inf_Product_unit @ Y @ Z ) ) ) ).
% inf_sup_aci(2)
thf(fact_501_inf__sup__aci_I1_J,axiom,
( inf_inf_set_v
= ( ^ [X3: set_v,Y3: set_v] : ( inf_inf_set_v @ Y3 @ X3 ) ) ) ).
% inf_sup_aci(1)
thf(fact_502_inf__sup__aci_I1_J,axiom,
( inf_inf_Product_unit
= ( ^ [X3: product_unit,Y3: product_unit] : ( inf_inf_Product_unit @ Y3 @ X3 ) ) ) ).
% inf_sup_aci(1)
thf(fact_503_inf_Oassoc,axiom,
! [A: set_v,B2: set_v,C: set_v] :
( ( inf_inf_set_v @ ( inf_inf_set_v @ A @ B2 ) @ C )
= ( inf_inf_set_v @ A @ ( inf_inf_set_v @ B2 @ C ) ) ) ).
% inf.assoc
thf(fact_504_inf_Oassoc,axiom,
! [A: product_unit,B2: product_unit,C: product_unit] :
( ( inf_inf_Product_unit @ ( inf_inf_Product_unit @ A @ B2 ) @ C )
= ( inf_inf_Product_unit @ A @ ( inf_inf_Product_unit @ B2 @ C ) ) ) ).
% inf.assoc
thf(fact_505_inf__assoc,axiom,
! [X2: set_v,Y: set_v,Z: set_v] :
( ( inf_inf_set_v @ ( inf_inf_set_v @ X2 @ Y ) @ Z )
= ( inf_inf_set_v @ X2 @ ( inf_inf_set_v @ Y @ Z ) ) ) ).
% inf_assoc
thf(fact_506_inf__assoc,axiom,
! [X2: product_unit,Y: product_unit,Z: product_unit] :
( ( inf_inf_Product_unit @ ( inf_inf_Product_unit @ X2 @ Y ) @ Z )
= ( inf_inf_Product_unit @ X2 @ ( inf_inf_Product_unit @ Y @ Z ) ) ) ).
% inf_assoc
thf(fact_507_inf_Ocommute,axiom,
( inf_inf_set_v
= ( ^ [A5: set_v,B5: set_v] : ( inf_inf_set_v @ B5 @ A5 ) ) ) ).
% inf.commute
thf(fact_508_inf_Ocommute,axiom,
( inf_inf_Product_unit
= ( ^ [A5: product_unit,B5: product_unit] : ( inf_inf_Product_unit @ B5 @ A5 ) ) ) ).
% inf.commute
thf(fact_509_inf__commute,axiom,
( inf_inf_set_v
= ( ^ [X3: set_v,Y3: set_v] : ( inf_inf_set_v @ Y3 @ X3 ) ) ) ).
% inf_commute
thf(fact_510_inf__commute,axiom,
( inf_inf_Product_unit
= ( ^ [X3: product_unit,Y3: product_unit] : ( inf_inf_Product_unit @ Y3 @ X3 ) ) ) ).
% inf_commute
thf(fact_511_boolean__algebra__cancel_Oinf1,axiom,
! [A3: set_v,K: set_v,A: set_v,B2: set_v] :
( ( A3
= ( inf_inf_set_v @ K @ A ) )
=> ( ( inf_inf_set_v @ A3 @ B2 )
= ( inf_inf_set_v @ K @ ( inf_inf_set_v @ A @ B2 ) ) ) ) ).
% boolean_algebra_cancel.inf1
thf(fact_512_boolean__algebra__cancel_Oinf1,axiom,
! [A3: product_unit,K: product_unit,A: product_unit,B2: product_unit] :
( ( A3
= ( inf_inf_Product_unit @ K @ A ) )
=> ( ( inf_inf_Product_unit @ A3 @ B2 )
= ( inf_inf_Product_unit @ K @ ( inf_inf_Product_unit @ A @ B2 ) ) ) ) ).
% boolean_algebra_cancel.inf1
thf(fact_513_boolean__algebra__cancel_Oinf2,axiom,
! [B: set_v,K: set_v,B2: set_v,A: set_v] :
( ( B
= ( inf_inf_set_v @ K @ B2 ) )
=> ( ( inf_inf_set_v @ A @ B )
= ( inf_inf_set_v @ K @ ( inf_inf_set_v @ A @ B2 ) ) ) ) ).
% boolean_algebra_cancel.inf2
thf(fact_514_boolean__algebra__cancel_Oinf2,axiom,
! [B: product_unit,K: product_unit,B2: product_unit,A: product_unit] :
( ( B
= ( inf_inf_Product_unit @ K @ B2 ) )
=> ( ( inf_inf_Product_unit @ A @ B )
= ( inf_inf_Product_unit @ K @ ( inf_inf_Product_unit @ A @ B2 ) ) ) ) ).
% boolean_algebra_cancel.inf2
thf(fact_515_inf_Oleft__commute,axiom,
! [B2: set_v,A: set_v,C: set_v] :
( ( inf_inf_set_v @ B2 @ ( inf_inf_set_v @ A @ C ) )
= ( inf_inf_set_v @ A @ ( inf_inf_set_v @ B2 @ C ) ) ) ).
% inf.left_commute
thf(fact_516_inf_Oleft__commute,axiom,
! [B2: product_unit,A: product_unit,C: product_unit] :
( ( inf_inf_Product_unit @ B2 @ ( inf_inf_Product_unit @ A @ C ) )
= ( inf_inf_Product_unit @ A @ ( inf_inf_Product_unit @ B2 @ C ) ) ) ).
% inf.left_commute
thf(fact_517_inf__left__commute,axiom,
! [X2: set_v,Y: set_v,Z: set_v] :
( ( inf_inf_set_v @ X2 @ ( inf_inf_set_v @ Y @ Z ) )
= ( inf_inf_set_v @ Y @ ( inf_inf_set_v @ X2 @ Z ) ) ) ).
% inf_left_commute
thf(fact_518_inf__left__commute,axiom,
! [X2: product_unit,Y: product_unit,Z: product_unit] :
( ( inf_inf_Product_unit @ X2 @ ( inf_inf_Product_unit @ Y @ Z ) )
= ( inf_inf_Product_unit @ Y @ ( inf_inf_Product_unit @ X2 @ Z ) ) ) ).
% inf_left_commute
thf(fact_519_UnionE,axiom,
! [A3: product_prod_v_v,C2: set_se8455005133513928103od_v_v] :
( ( member7453568604450474000od_v_v @ A3 @ ( comple5788137035815166516od_v_v @ C2 ) )
=> ~ ! [X6: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ A3 @ X6 )
=> ~ ( member8406446414694345712od_v_v @ X6 @ C2 ) ) ) ).
% UnionE
thf(fact_520_UnionE,axiom,
! [A3: v,C2: set_set_v] :
( ( member_v @ A3 @ ( comple2307003700295860064_set_v @ C2 ) )
=> ~ ! [X6: set_v] :
( ( member_v @ A3 @ X6 )
=> ~ ( member_set_v @ X6 @ C2 ) ) ) ).
% UnionE
thf(fact_521_inf__sup__ord_I2_J,axiom,
! [X2: product_unit,Y: product_unit] : ( ord_le3221252021190050221t_unit @ ( inf_inf_Product_unit @ X2 @ Y ) @ Y ) ).
% inf_sup_ord(2)
thf(fact_522_inf__sup__ord_I2_J,axiom,
! [X2: set_v,Y: set_v] : ( ord_less_eq_set_v @ ( inf_inf_set_v @ X2 @ Y ) @ Y ) ).
% inf_sup_ord(2)
thf(fact_523_inf__sup__ord_I2_J,axiom,
! [X2: set_Product_prod_v_v,Y: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ X2 @ Y ) @ Y ) ).
% inf_sup_ord(2)
thf(fact_524_inf__sup__ord_I1_J,axiom,
! [X2: product_unit,Y: product_unit] : ( ord_le3221252021190050221t_unit @ ( inf_inf_Product_unit @ X2 @ Y ) @ X2 ) ).
% inf_sup_ord(1)
thf(fact_525_inf__sup__ord_I1_J,axiom,
! [X2: set_v,Y: set_v] : ( ord_less_eq_set_v @ ( inf_inf_set_v @ X2 @ Y ) @ X2 ) ).
% inf_sup_ord(1)
thf(fact_526_inf__sup__ord_I1_J,axiom,
! [X2: set_Product_prod_v_v,Y: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ X2 @ Y ) @ X2 ) ).
% inf_sup_ord(1)
thf(fact_527_inf__le1,axiom,
! [X2: product_unit,Y: product_unit] : ( ord_le3221252021190050221t_unit @ ( inf_inf_Product_unit @ X2 @ Y ) @ X2 ) ).
% inf_le1
thf(fact_528_inf__le1,axiom,
! [X2: set_v,Y: set_v] : ( ord_less_eq_set_v @ ( inf_inf_set_v @ X2 @ Y ) @ X2 ) ).
% inf_le1
thf(fact_529_inf__le1,axiom,
! [X2: set_Product_prod_v_v,Y: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ X2 @ Y ) @ X2 ) ).
% inf_le1
thf(fact_530_inf__le2,axiom,
! [X2: product_unit,Y: product_unit] : ( ord_le3221252021190050221t_unit @ ( inf_inf_Product_unit @ X2 @ Y ) @ Y ) ).
% inf_le2
thf(fact_531_inf__le2,axiom,
! [X2: set_v,Y: set_v] : ( ord_less_eq_set_v @ ( inf_inf_set_v @ X2 @ Y ) @ Y ) ).
% inf_le2
thf(fact_532_inf__le2,axiom,
! [X2: set_Product_prod_v_v,Y: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ X2 @ Y ) @ Y ) ).
% inf_le2
thf(fact_533_le__infE,axiom,
! [X2: product_unit,A: product_unit,B2: product_unit] :
( ( ord_le3221252021190050221t_unit @ X2 @ ( inf_inf_Product_unit @ A @ B2 ) )
=> ~ ( ( ord_le3221252021190050221t_unit @ X2 @ A )
=> ~ ( ord_le3221252021190050221t_unit @ X2 @ B2 ) ) ) ).
% le_infE
thf(fact_534_le__infE,axiom,
! [X2: set_v,A: set_v,B2: set_v] :
( ( ord_less_eq_set_v @ X2 @ ( inf_inf_set_v @ A @ B2 ) )
=> ~ ( ( ord_less_eq_set_v @ X2 @ A )
=> ~ ( ord_less_eq_set_v @ X2 @ B2 ) ) ) ).
% le_infE
thf(fact_535_le__infE,axiom,
! [X2: set_Product_prod_v_v,A: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X2 @ ( inf_in6271465464967711157od_v_v @ A @ B2 ) )
=> ~ ( ( ord_le7336532860387713383od_v_v @ X2 @ A )
=> ~ ( ord_le7336532860387713383od_v_v @ X2 @ B2 ) ) ) ).
% le_infE
thf(fact_536_le__infI,axiom,
! [X2: product_unit,A: product_unit,B2: product_unit] :
( ( ord_le3221252021190050221t_unit @ X2 @ A )
=> ( ( ord_le3221252021190050221t_unit @ X2 @ B2 )
=> ( ord_le3221252021190050221t_unit @ X2 @ ( inf_inf_Product_unit @ A @ B2 ) ) ) ) ).
% le_infI
thf(fact_537_le__infI,axiom,
! [X2: set_v,A: set_v,B2: set_v] :
( ( ord_less_eq_set_v @ X2 @ A )
=> ( ( ord_less_eq_set_v @ X2 @ B2 )
=> ( ord_less_eq_set_v @ X2 @ ( inf_inf_set_v @ A @ B2 ) ) ) ) ).
% le_infI
thf(fact_538_le__infI,axiom,
! [X2: set_Product_prod_v_v,A: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X2 @ A )
=> ( ( ord_le7336532860387713383od_v_v @ X2 @ B2 )
=> ( ord_le7336532860387713383od_v_v @ X2 @ ( inf_in6271465464967711157od_v_v @ A @ B2 ) ) ) ) ).
% le_infI
thf(fact_539_inf__mono,axiom,
! [A: product_unit,C: product_unit,B2: product_unit,D: product_unit] :
( ( ord_le3221252021190050221t_unit @ A @ C )
=> ( ( ord_le3221252021190050221t_unit @ B2 @ D )
=> ( ord_le3221252021190050221t_unit @ ( inf_inf_Product_unit @ A @ B2 ) @ ( inf_inf_Product_unit @ C @ D ) ) ) ) ).
% inf_mono
thf(fact_540_inf__mono,axiom,
! [A: set_v,C: set_v,B2: set_v,D: set_v] :
( ( ord_less_eq_set_v @ A @ C )
=> ( ( ord_less_eq_set_v @ B2 @ D )
=> ( ord_less_eq_set_v @ ( inf_inf_set_v @ A @ B2 ) @ ( inf_inf_set_v @ C @ D ) ) ) ) ).
% inf_mono
thf(fact_541_inf__mono,axiom,
! [A: set_Product_prod_v_v,C: set_Product_prod_v_v,B2: set_Product_prod_v_v,D: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ C )
=> ( ( ord_le7336532860387713383od_v_v @ B2 @ D )
=> ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ A @ B2 ) @ ( inf_in6271465464967711157od_v_v @ C @ D ) ) ) ) ).
% inf_mono
thf(fact_542_le__infI1,axiom,
! [A: product_unit,X2: product_unit,B2: product_unit] :
( ( ord_le3221252021190050221t_unit @ A @ X2 )
=> ( ord_le3221252021190050221t_unit @ ( inf_inf_Product_unit @ A @ B2 ) @ X2 ) ) ).
% le_infI1
thf(fact_543_le__infI1,axiom,
! [A: set_v,X2: set_v,B2: set_v] :
( ( ord_less_eq_set_v @ A @ X2 )
=> ( ord_less_eq_set_v @ ( inf_inf_set_v @ A @ B2 ) @ X2 ) ) ).
% le_infI1
thf(fact_544_le__infI1,axiom,
! [A: set_Product_prod_v_v,X2: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ X2 )
=> ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ A @ B2 ) @ X2 ) ) ).
% le_infI1
thf(fact_545_le__infI2,axiom,
! [B2: product_unit,X2: product_unit,A: product_unit] :
( ( ord_le3221252021190050221t_unit @ B2 @ X2 )
=> ( ord_le3221252021190050221t_unit @ ( inf_inf_Product_unit @ A @ B2 ) @ X2 ) ) ).
% le_infI2
thf(fact_546_le__infI2,axiom,
! [B2: set_v,X2: set_v,A: set_v] :
( ( ord_less_eq_set_v @ B2 @ X2 )
=> ( ord_less_eq_set_v @ ( inf_inf_set_v @ A @ B2 ) @ X2 ) ) ).
% le_infI2
thf(fact_547_le__infI2,axiom,
! [B2: set_Product_prod_v_v,X2: set_Product_prod_v_v,A: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ B2 @ X2 )
=> ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ A @ B2 ) @ X2 ) ) ).
% le_infI2
thf(fact_548_inf_OorderE,axiom,
! [A: product_unit,B2: product_unit] :
( ( ord_le3221252021190050221t_unit @ A @ B2 )
=> ( A
= ( inf_inf_Product_unit @ A @ B2 ) ) ) ).
% inf.orderE
thf(fact_549_inf_OorderE,axiom,
! [A: set_v,B2: set_v] :
( ( ord_less_eq_set_v @ A @ B2 )
=> ( A
= ( inf_inf_set_v @ A @ B2 ) ) ) ).
% inf.orderE
thf(fact_550_inf_OorderE,axiom,
! [A: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ B2 )
=> ( A
= ( inf_in6271465464967711157od_v_v @ A @ B2 ) ) ) ).
% inf.orderE
thf(fact_551_inf_OorderI,axiom,
! [A: product_unit,B2: product_unit] :
( ( A
= ( inf_inf_Product_unit @ A @ B2 ) )
=> ( ord_le3221252021190050221t_unit @ A @ B2 ) ) ).
% inf.orderI
thf(fact_552_inf_OorderI,axiom,
! [A: set_v,B2: set_v] :
( ( A
= ( inf_inf_set_v @ A @ B2 ) )
=> ( ord_less_eq_set_v @ A @ B2 ) ) ).
% inf.orderI
thf(fact_553_inf_OorderI,axiom,
! [A: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( A
= ( inf_in6271465464967711157od_v_v @ A @ B2 ) )
=> ( ord_le7336532860387713383od_v_v @ A @ B2 ) ) ).
% inf.orderI
thf(fact_554_inf__unique,axiom,
! [F: product_unit > product_unit > product_unit,X2: product_unit,Y: product_unit] :
( ! [X4: product_unit,Y2: product_unit] : ( ord_le3221252021190050221t_unit @ ( F @ X4 @ Y2 ) @ X4 )
=> ( ! [X4: product_unit,Y2: product_unit] : ( ord_le3221252021190050221t_unit @ ( F @ X4 @ Y2 ) @ Y2 )
=> ( ! [X4: product_unit,Y2: product_unit,Z3: product_unit] :
( ( ord_le3221252021190050221t_unit @ X4 @ Y2 )
=> ( ( ord_le3221252021190050221t_unit @ X4 @ Z3 )
=> ( ord_le3221252021190050221t_unit @ X4 @ ( F @ Y2 @ Z3 ) ) ) )
=> ( ( inf_inf_Product_unit @ X2 @ Y )
= ( F @ X2 @ Y ) ) ) ) ) ).
% inf_unique
thf(fact_555_inf__unique,axiom,
! [F: set_v > set_v > set_v,X2: set_v,Y: set_v] :
( ! [X4: set_v,Y2: set_v] : ( ord_less_eq_set_v @ ( F @ X4 @ Y2 ) @ X4 )
=> ( ! [X4: set_v,Y2: set_v] : ( ord_less_eq_set_v @ ( F @ X4 @ Y2 ) @ Y2 )
=> ( ! [X4: set_v,Y2: set_v,Z3: set_v] :
( ( ord_less_eq_set_v @ X4 @ Y2 )
=> ( ( ord_less_eq_set_v @ X4 @ Z3 )
=> ( ord_less_eq_set_v @ X4 @ ( F @ Y2 @ Z3 ) ) ) )
=> ( ( inf_inf_set_v @ X2 @ Y )
= ( F @ X2 @ Y ) ) ) ) ) ).
% inf_unique
thf(fact_556_inf__unique,axiom,
! [F: set_Product_prod_v_v > set_Product_prod_v_v > set_Product_prod_v_v,X2: set_Product_prod_v_v,Y: set_Product_prod_v_v] :
( ! [X4: set_Product_prod_v_v,Y2: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ ( F @ X4 @ Y2 ) @ X4 )
=> ( ! [X4: set_Product_prod_v_v,Y2: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ ( F @ X4 @ Y2 ) @ Y2 )
=> ( ! [X4: set_Product_prod_v_v,Y2: set_Product_prod_v_v,Z3: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X4 @ Y2 )
=> ( ( ord_le7336532860387713383od_v_v @ X4 @ Z3 )
=> ( ord_le7336532860387713383od_v_v @ X4 @ ( F @ Y2 @ Z3 ) ) ) )
=> ( ( inf_in6271465464967711157od_v_v @ X2 @ Y )
= ( F @ X2 @ Y ) ) ) ) ) ).
% inf_unique
thf(fact_557_le__iff__inf,axiom,
( ord_le3221252021190050221t_unit
= ( ^ [X3: product_unit,Y3: product_unit] :
( ( inf_inf_Product_unit @ X3 @ Y3 )
= X3 ) ) ) ).
% le_iff_inf
thf(fact_558_le__iff__inf,axiom,
( ord_less_eq_set_v
= ( ^ [X3: set_v,Y3: set_v] :
( ( inf_inf_set_v @ X3 @ Y3 )
= X3 ) ) ) ).
% le_iff_inf
thf(fact_559_le__iff__inf,axiom,
( ord_le7336532860387713383od_v_v
= ( ^ [X3: set_Product_prod_v_v,Y3: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ X3 @ Y3 )
= X3 ) ) ) ).
% le_iff_inf
thf(fact_560_inf_Oabsorb1,axiom,
! [A: product_unit,B2: product_unit] :
( ( ord_le3221252021190050221t_unit @ A @ B2 )
=> ( ( inf_inf_Product_unit @ A @ B2 )
= A ) ) ).
% inf.absorb1
thf(fact_561_inf_Oabsorb1,axiom,
! [A: set_v,B2: set_v] :
( ( ord_less_eq_set_v @ A @ B2 )
=> ( ( inf_inf_set_v @ A @ B2 )
= A ) ) ).
% inf.absorb1
thf(fact_562_inf_Oabsorb1,axiom,
! [A: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ B2 )
=> ( ( inf_in6271465464967711157od_v_v @ A @ B2 )
= A ) ) ).
% inf.absorb1
thf(fact_563_inf_Oabsorb2,axiom,
! [B2: product_unit,A: product_unit] :
( ( ord_le3221252021190050221t_unit @ B2 @ A )
=> ( ( inf_inf_Product_unit @ A @ B2 )
= B2 ) ) ).
% inf.absorb2
thf(fact_564_inf_Oabsorb2,axiom,
! [B2: set_v,A: set_v] :
( ( ord_less_eq_set_v @ B2 @ A )
=> ( ( inf_inf_set_v @ A @ B2 )
= B2 ) ) ).
% inf.absorb2
thf(fact_565_inf_Oabsorb2,axiom,
! [B2: set_Product_prod_v_v,A: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ B2 @ A )
=> ( ( inf_in6271465464967711157od_v_v @ A @ B2 )
= B2 ) ) ).
% inf.absorb2
thf(fact_566_inf__absorb1,axiom,
! [X2: product_unit,Y: product_unit] :
( ( ord_le3221252021190050221t_unit @ X2 @ Y )
=> ( ( inf_inf_Product_unit @ X2 @ Y )
= X2 ) ) ).
% inf_absorb1
thf(fact_567_inf__absorb1,axiom,
! [X2: set_v,Y: set_v] :
( ( ord_less_eq_set_v @ X2 @ Y )
=> ( ( inf_inf_set_v @ X2 @ Y )
= X2 ) ) ).
% inf_absorb1
thf(fact_568_inf__absorb1,axiom,
! [X2: set_Product_prod_v_v,Y: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X2 @ Y )
=> ( ( inf_in6271465464967711157od_v_v @ X2 @ Y )
= X2 ) ) ).
% inf_absorb1
thf(fact_569_inf__absorb2,axiom,
! [Y: product_unit,X2: product_unit] :
( ( ord_le3221252021190050221t_unit @ Y @ X2 )
=> ( ( inf_inf_Product_unit @ X2 @ Y )
= Y ) ) ).
% inf_absorb2
thf(fact_570_inf__absorb2,axiom,
! [Y: set_v,X2: set_v] :
( ( ord_less_eq_set_v @ Y @ X2 )
=> ( ( inf_inf_set_v @ X2 @ Y )
= Y ) ) ).
% inf_absorb2
thf(fact_571_inf__absorb2,axiom,
! [Y: set_Product_prod_v_v,X2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ Y @ X2 )
=> ( ( inf_in6271465464967711157od_v_v @ X2 @ Y )
= Y ) ) ).
% inf_absorb2
thf(fact_572_inf_OboundedE,axiom,
! [A: product_unit,B2: product_unit,C: product_unit] :
( ( ord_le3221252021190050221t_unit @ A @ ( inf_inf_Product_unit @ B2 @ C ) )
=> ~ ( ( ord_le3221252021190050221t_unit @ A @ B2 )
=> ~ ( ord_le3221252021190050221t_unit @ A @ C ) ) ) ).
% inf.boundedE
thf(fact_573_inf_OboundedE,axiom,
! [A: set_v,B2: set_v,C: set_v] :
( ( ord_less_eq_set_v @ A @ ( inf_inf_set_v @ B2 @ C ) )
=> ~ ( ( ord_less_eq_set_v @ A @ B2 )
=> ~ ( ord_less_eq_set_v @ A @ C ) ) ) ).
% inf.boundedE
thf(fact_574_inf_OboundedE,axiom,
! [A: set_Product_prod_v_v,B2: set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ ( inf_in6271465464967711157od_v_v @ B2 @ C ) )
=> ~ ( ( ord_le7336532860387713383od_v_v @ A @ B2 )
=> ~ ( ord_le7336532860387713383od_v_v @ A @ C ) ) ) ).
% inf.boundedE
thf(fact_575_inf_OboundedI,axiom,
! [A: product_unit,B2: product_unit,C: product_unit] :
( ( ord_le3221252021190050221t_unit @ A @ B2 )
=> ( ( ord_le3221252021190050221t_unit @ A @ C )
=> ( ord_le3221252021190050221t_unit @ A @ ( inf_inf_Product_unit @ B2 @ C ) ) ) ) ).
% inf.boundedI
thf(fact_576_inf_OboundedI,axiom,
! [A: set_v,B2: set_v,C: set_v] :
( ( ord_less_eq_set_v @ A @ B2 )
=> ( ( ord_less_eq_set_v @ A @ C )
=> ( ord_less_eq_set_v @ A @ ( inf_inf_set_v @ B2 @ C ) ) ) ) ).
% inf.boundedI
thf(fact_577_inf_OboundedI,axiom,
! [A: set_Product_prod_v_v,B2: set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ B2 )
=> ( ( ord_le7336532860387713383od_v_v @ A @ C )
=> ( ord_le7336532860387713383od_v_v @ A @ ( inf_in6271465464967711157od_v_v @ B2 @ C ) ) ) ) ).
% inf.boundedI
thf(fact_578_inf__greatest,axiom,
! [X2: product_unit,Y: product_unit,Z: product_unit] :
( ( ord_le3221252021190050221t_unit @ X2 @ Y )
=> ( ( ord_le3221252021190050221t_unit @ X2 @ Z )
=> ( ord_le3221252021190050221t_unit @ X2 @ ( inf_inf_Product_unit @ Y @ Z ) ) ) ) ).
% inf_greatest
thf(fact_579_inf__greatest,axiom,
! [X2: set_v,Y: set_v,Z: set_v] :
( ( ord_less_eq_set_v @ X2 @ Y )
=> ( ( ord_less_eq_set_v @ X2 @ Z )
=> ( ord_less_eq_set_v @ X2 @ ( inf_inf_set_v @ Y @ Z ) ) ) ) ).
% inf_greatest
thf(fact_580_inf__greatest,axiom,
! [X2: set_Product_prod_v_v,Y: set_Product_prod_v_v,Z: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X2 @ Y )
=> ( ( ord_le7336532860387713383od_v_v @ X2 @ Z )
=> ( ord_le7336532860387713383od_v_v @ X2 @ ( inf_in6271465464967711157od_v_v @ Y @ Z ) ) ) ) ).
% inf_greatest
thf(fact_581_inf_Oorder__iff,axiom,
( ord_le3221252021190050221t_unit
= ( ^ [A5: product_unit,B5: product_unit] :
( A5
= ( inf_inf_Product_unit @ A5 @ B5 ) ) ) ) ).
% inf.order_iff
thf(fact_582_inf_Oorder__iff,axiom,
( ord_less_eq_set_v
= ( ^ [A5: set_v,B5: set_v] :
( A5
= ( inf_inf_set_v @ A5 @ B5 ) ) ) ) ).
% inf.order_iff
thf(fact_583_inf_Oorder__iff,axiom,
( ord_le7336532860387713383od_v_v
= ( ^ [A5: set_Product_prod_v_v,B5: set_Product_prod_v_v] :
( A5
= ( inf_in6271465464967711157od_v_v @ A5 @ B5 ) ) ) ) ).
% inf.order_iff
thf(fact_584_inf_Ocobounded1,axiom,
! [A: product_unit,B2: product_unit] : ( ord_le3221252021190050221t_unit @ ( inf_inf_Product_unit @ A @ B2 ) @ A ) ).
% inf.cobounded1
thf(fact_585_inf_Ocobounded1,axiom,
! [A: set_v,B2: set_v] : ( ord_less_eq_set_v @ ( inf_inf_set_v @ A @ B2 ) @ A ) ).
% inf.cobounded1
thf(fact_586_inf_Ocobounded1,axiom,
! [A: set_Product_prod_v_v,B2: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ A @ B2 ) @ A ) ).
% inf.cobounded1
thf(fact_587_inf_Ocobounded2,axiom,
! [A: product_unit,B2: product_unit] : ( ord_le3221252021190050221t_unit @ ( inf_inf_Product_unit @ A @ B2 ) @ B2 ) ).
% inf.cobounded2
thf(fact_588_inf_Ocobounded2,axiom,
! [A: set_v,B2: set_v] : ( ord_less_eq_set_v @ ( inf_inf_set_v @ A @ B2 ) @ B2 ) ).
% inf.cobounded2
thf(fact_589_inf_Ocobounded2,axiom,
! [A: set_Product_prod_v_v,B2: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ A @ B2 ) @ B2 ) ).
% inf.cobounded2
thf(fact_590_inf_Oabsorb__iff1,axiom,
( ord_le3221252021190050221t_unit
= ( ^ [A5: product_unit,B5: product_unit] :
( ( inf_inf_Product_unit @ A5 @ B5 )
= A5 ) ) ) ).
% inf.absorb_iff1
thf(fact_591_inf_Oabsorb__iff1,axiom,
( ord_less_eq_set_v
= ( ^ [A5: set_v,B5: set_v] :
( ( inf_inf_set_v @ A5 @ B5 )
= A5 ) ) ) ).
% inf.absorb_iff1
thf(fact_592_inf_Oabsorb__iff1,axiom,
( ord_le7336532860387713383od_v_v
= ( ^ [A5: set_Product_prod_v_v,B5: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ A5 @ B5 )
= A5 ) ) ) ).
% inf.absorb_iff1
thf(fact_593_inf_Oabsorb__iff2,axiom,
( ord_le3221252021190050221t_unit
= ( ^ [B5: product_unit,A5: product_unit] :
( ( inf_inf_Product_unit @ A5 @ B5 )
= B5 ) ) ) ).
% inf.absorb_iff2
thf(fact_594_inf_Oabsorb__iff2,axiom,
( ord_less_eq_set_v
= ( ^ [B5: set_v,A5: set_v] :
( ( inf_inf_set_v @ A5 @ B5 )
= B5 ) ) ) ).
% inf.absorb_iff2
thf(fact_595_inf_Oabsorb__iff2,axiom,
( ord_le7336532860387713383od_v_v
= ( ^ [B5: set_Product_prod_v_v,A5: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ A5 @ B5 )
= B5 ) ) ) ).
% inf.absorb_iff2
thf(fact_596_inf_OcoboundedI1,axiom,
! [A: product_unit,C: product_unit,B2: product_unit] :
( ( ord_le3221252021190050221t_unit @ A @ C )
=> ( ord_le3221252021190050221t_unit @ ( inf_inf_Product_unit @ A @ B2 ) @ C ) ) ).
% inf.coboundedI1
thf(fact_597_inf_OcoboundedI1,axiom,
! [A: set_v,C: set_v,B2: set_v] :
( ( ord_less_eq_set_v @ A @ C )
=> ( ord_less_eq_set_v @ ( inf_inf_set_v @ A @ B2 ) @ C ) ) ).
% inf.coboundedI1
thf(fact_598_inf_OcoboundedI1,axiom,
! [A: set_Product_prod_v_v,C: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ C )
=> ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ A @ B2 ) @ C ) ) ).
% inf.coboundedI1
thf(fact_599_inf_OcoboundedI2,axiom,
! [B2: product_unit,C: product_unit,A: product_unit] :
( ( ord_le3221252021190050221t_unit @ B2 @ C )
=> ( ord_le3221252021190050221t_unit @ ( inf_inf_Product_unit @ A @ B2 ) @ C ) ) ).
% inf.coboundedI2
thf(fact_600_inf_OcoboundedI2,axiom,
! [B2: set_v,C: set_v,A: set_v] :
( ( ord_less_eq_set_v @ B2 @ C )
=> ( ord_less_eq_set_v @ ( inf_inf_set_v @ A @ B2 ) @ C ) ) ).
% inf.coboundedI2
thf(fact_601_inf_OcoboundedI2,axiom,
! [B2: set_Product_prod_v_v,C: set_Product_prod_v_v,A: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ B2 @ C )
=> ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ A @ B2 ) @ C ) ) ).
% inf.coboundedI2
thf(fact_602_Sup__eqI,axiom,
! [A3: set_se8455005133513928103od_v_v,X2: set_Product_prod_v_v] :
( ! [Y2: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ Y2 @ A3 )
=> ( ord_le7336532860387713383od_v_v @ Y2 @ X2 ) )
=> ( ! [Y2: set_Product_prod_v_v] :
( ! [Z5: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ Z5 @ A3 )
=> ( ord_le7336532860387713383od_v_v @ Z5 @ Y2 ) )
=> ( ord_le7336532860387713383od_v_v @ X2 @ Y2 ) )
=> ( ( comple5788137035815166516od_v_v @ A3 )
= X2 ) ) ) ).
% Sup_eqI
thf(fact_603_Sup__eqI,axiom,
! [A3: set_set_v,X2: set_v] :
( ! [Y2: set_v] :
( ( member_set_v @ Y2 @ A3 )
=> ( ord_less_eq_set_v @ Y2 @ X2 ) )
=> ( ! [Y2: set_v] :
( ! [Z5: set_v] :
( ( member_set_v @ Z5 @ A3 )
=> ( ord_less_eq_set_v @ Z5 @ Y2 ) )
=> ( ord_less_eq_set_v @ X2 @ Y2 ) )
=> ( ( comple2307003700295860064_set_v @ A3 )
= X2 ) ) ) ).
% Sup_eqI
thf(fact_604_Sup__mono,axiom,
! [A3: set_se8455005133513928103od_v_v,B: set_se8455005133513928103od_v_v] :
( ! [A6: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ A6 @ A3 )
=> ? [X: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ X @ B )
& ( ord_le7336532860387713383od_v_v @ A6 @ X ) ) )
=> ( ord_le7336532860387713383od_v_v @ ( comple5788137035815166516od_v_v @ A3 ) @ ( comple5788137035815166516od_v_v @ B ) ) ) ).
% Sup_mono
thf(fact_605_Sup__mono,axiom,
! [A3: set_set_v,B: set_set_v] :
( ! [A6: set_v] :
( ( member_set_v @ A6 @ A3 )
=> ? [X: set_v] :
( ( member_set_v @ X @ B )
& ( ord_less_eq_set_v @ A6 @ X ) ) )
=> ( ord_less_eq_set_v @ ( comple2307003700295860064_set_v @ A3 ) @ ( comple2307003700295860064_set_v @ B ) ) ) ).
% Sup_mono
thf(fact_606_Sup__least,axiom,
! [A3: set_se8455005133513928103od_v_v,Z: set_Product_prod_v_v] :
( ! [X4: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ X4 @ A3 )
=> ( ord_le7336532860387713383od_v_v @ X4 @ Z ) )
=> ( ord_le7336532860387713383od_v_v @ ( comple5788137035815166516od_v_v @ A3 ) @ Z ) ) ).
% Sup_least
thf(fact_607_Sup__least,axiom,
! [A3: set_set_v,Z: set_v] :
( ! [X4: set_v] :
( ( member_set_v @ X4 @ A3 )
=> ( ord_less_eq_set_v @ X4 @ Z ) )
=> ( ord_less_eq_set_v @ ( comple2307003700295860064_set_v @ A3 ) @ Z ) ) ).
% Sup_least
thf(fact_608_Sup__upper,axiom,
! [X2: set_Product_prod_v_v,A3: set_se8455005133513928103od_v_v] :
( ( member8406446414694345712od_v_v @ X2 @ A3 )
=> ( ord_le7336532860387713383od_v_v @ X2 @ ( comple5788137035815166516od_v_v @ A3 ) ) ) ).
% Sup_upper
thf(fact_609_Sup__upper,axiom,
! [X2: set_v,A3: set_set_v] :
( ( member_set_v @ X2 @ A3 )
=> ( ord_less_eq_set_v @ X2 @ ( comple2307003700295860064_set_v @ A3 ) ) ) ).
% Sup_upper
thf(fact_610_Sup__le__iff,axiom,
! [A3: set_se8455005133513928103od_v_v,B2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ ( comple5788137035815166516od_v_v @ A3 ) @ B2 )
= ( ! [X3: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ X3 @ A3 )
=> ( ord_le7336532860387713383od_v_v @ X3 @ B2 ) ) ) ) ).
% Sup_le_iff
thf(fact_611_Sup__le__iff,axiom,
! [A3: set_set_v,B2: set_v] :
( ( ord_less_eq_set_v @ ( comple2307003700295860064_set_v @ A3 ) @ B2 )
= ( ! [X3: set_v] :
( ( member_set_v @ X3 @ A3 )
=> ( ord_less_eq_set_v @ X3 @ B2 ) ) ) ) ).
% Sup_le_iff
thf(fact_612_Sup__upper2,axiom,
! [U: set_Product_prod_v_v,A3: set_se8455005133513928103od_v_v,V: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ U @ A3 )
=> ( ( ord_le7336532860387713383od_v_v @ V @ U )
=> ( ord_le7336532860387713383od_v_v @ V @ ( comple5788137035815166516od_v_v @ A3 ) ) ) ) ).
% Sup_upper2
thf(fact_613_Sup__upper2,axiom,
! [U: set_v,A3: set_set_v,V: set_v] :
( ( member_set_v @ U @ A3 )
=> ( ( ord_less_eq_set_v @ V @ U )
=> ( ord_less_eq_set_v @ V @ ( comple2307003700295860064_set_v @ A3 ) ) ) ) ).
% Sup_upper2
thf(fact_614_Union__empty,axiom,
( ( comple5788137035815166516od_v_v @ bot_bo3497076220358800403od_v_v )
= bot_bo723834152578015283od_v_v ) ).
% Union_empty
thf(fact_615_Union__empty,axiom,
( ( comple5450237519782573632_set_v @ bot_bo5775917114081396255_set_v )
= bot_bot_set_set_v ) ).
% Union_empty
thf(fact_616_Union__empty,axiom,
( ( comple2307003700295860064_set_v @ bot_bot_set_set_v )
= bot_bot_set_v ) ).
% Union_empty
thf(fact_617_Union__empty__conv,axiom,
! [A3: set_se8455005133513928103od_v_v] :
( ( ( comple5788137035815166516od_v_v @ A3 )
= bot_bo723834152578015283od_v_v )
= ( ! [X3: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ X3 @ A3 )
=> ( X3 = bot_bo723834152578015283od_v_v ) ) ) ) ).
% Union_empty_conv
thf(fact_618_Union__empty__conv,axiom,
! [A3: set_set_set_v] :
( ( ( comple5450237519782573632_set_v @ A3 )
= bot_bot_set_set_v )
= ( ! [X3: set_set_v] :
( ( member_set_set_v @ X3 @ A3 )
=> ( X3 = bot_bot_set_set_v ) ) ) ) ).
% Union_empty_conv
thf(fact_619_Union__empty__conv,axiom,
! [A3: set_set_v] :
( ( ( comple2307003700295860064_set_v @ A3 )
= bot_bot_set_v )
= ( ! [X3: set_v] :
( ( member_set_v @ X3 @ A3 )
=> ( X3 = bot_bot_set_v ) ) ) ) ).
% Union_empty_conv
thf(fact_620_empty__Union__conv,axiom,
! [A3: set_se8455005133513928103od_v_v] :
( ( bot_bo723834152578015283od_v_v
= ( comple5788137035815166516od_v_v @ A3 ) )
= ( ! [X3: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ X3 @ A3 )
=> ( X3 = bot_bo723834152578015283od_v_v ) ) ) ) ).
% empty_Union_conv
thf(fact_621_empty__Union__conv,axiom,
! [A3: set_set_set_v] :
( ( bot_bot_set_set_v
= ( comple5450237519782573632_set_v @ A3 ) )
= ( ! [X3: set_set_v] :
( ( member_set_set_v @ X3 @ A3 )
=> ( X3 = bot_bot_set_set_v ) ) ) ) ).
% empty_Union_conv
thf(fact_622_empty__Union__conv,axiom,
! [A3: set_set_v] :
( ( bot_bot_set_v
= ( comple2307003700295860064_set_v @ A3 ) )
= ( ! [X3: set_v] :
( ( member_set_v @ X3 @ A3 )
=> ( X3 = bot_bot_set_v ) ) ) ) ).
% empty_Union_conv
thf(fact_623_Union__mono,axiom,
! [A3: set_se8455005133513928103od_v_v,B: set_se8455005133513928103od_v_v] :
( ( ord_le4714265922333009223od_v_v @ A3 @ B )
=> ( ord_le7336532860387713383od_v_v @ ( comple5788137035815166516od_v_v @ A3 ) @ ( comple5788137035815166516od_v_v @ B ) ) ) ).
% Union_mono
thf(fact_624_Union__mono,axiom,
! [A3: set_set_v,B: set_set_v] :
( ( ord_le5216385588623774835_set_v @ A3 @ B )
=> ( ord_less_eq_set_v @ ( comple2307003700295860064_set_v @ A3 ) @ ( comple2307003700295860064_set_v @ B ) ) ) ).
% Union_mono
thf(fact_625_Union__least,axiom,
! [A3: set_se8455005133513928103od_v_v,C2: set_Product_prod_v_v] :
( ! [X6: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ X6 @ A3 )
=> ( ord_le7336532860387713383od_v_v @ X6 @ C2 ) )
=> ( ord_le7336532860387713383od_v_v @ ( comple5788137035815166516od_v_v @ A3 ) @ C2 ) ) ).
% Union_least
thf(fact_626_Union__least,axiom,
! [A3: set_set_v,C2: set_v] :
( ! [X6: set_v] :
( ( member_set_v @ X6 @ A3 )
=> ( ord_less_eq_set_v @ X6 @ C2 ) )
=> ( ord_less_eq_set_v @ ( comple2307003700295860064_set_v @ A3 ) @ C2 ) ) ).
% Union_least
thf(fact_627_Union__upper,axiom,
! [B: set_Product_prod_v_v,A3: set_se8455005133513928103od_v_v] :
( ( member8406446414694345712od_v_v @ B @ A3 )
=> ( ord_le7336532860387713383od_v_v @ B @ ( comple5788137035815166516od_v_v @ A3 ) ) ) ).
% Union_upper
thf(fact_628_Union__upper,axiom,
! [B: set_v,A3: set_set_v] :
( ( member_set_v @ B @ A3 )
=> ( ord_less_eq_set_v @ B @ ( comple2307003700295860064_set_v @ A3 ) ) ) ).
% Union_upper
thf(fact_629_Union__subsetI,axiom,
! [A3: set_se8455005133513928103od_v_v,B: set_se8455005133513928103od_v_v] :
( ! [X4: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ X4 @ A3 )
=> ? [Y4: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ Y4 @ B )
& ( ord_le7336532860387713383od_v_v @ X4 @ Y4 ) ) )
=> ( ord_le7336532860387713383od_v_v @ ( comple5788137035815166516od_v_v @ A3 ) @ ( comple5788137035815166516od_v_v @ B ) ) ) ).
% Union_subsetI
thf(fact_630_Union__subsetI,axiom,
! [A3: set_set_v,B: set_set_v] :
( ! [X4: set_v] :
( ( member_set_v @ X4 @ A3 )
=> ? [Y4: set_v] :
( ( member_set_v @ Y4 @ B )
& ( ord_less_eq_set_v @ X4 @ Y4 ) ) )
=> ( ord_less_eq_set_v @ ( comple2307003700295860064_set_v @ A3 ) @ ( comple2307003700295860064_set_v @ B ) ) ) ).
% Union_subsetI
thf(fact_631_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_632_diff__shunt__var,axiom,
! [X2: product_unit,Y: product_unit] :
( ( ( minus_3524152463667985524t_unit @ X2 @ Y )
= bot_bot_Product_unit )
= ( ord_le3221252021190050221t_unit @ X2 @ Y ) ) ).
% diff_shunt_var
thf(fact_633_diff__shunt__var,axiom,
! [X2: set_set_v,Y: set_set_v] :
( ( ( minus_7228012346218142266_set_v @ X2 @ Y )
= bot_bot_set_set_v )
= ( ord_le5216385588623774835_set_v @ X2 @ Y ) ) ).
% diff_shunt_var
thf(fact_634_diff__shunt__var,axiom,
! [X2: set_v,Y: set_v] :
( ( ( minus_minus_set_v @ X2 @ Y )
= bot_bot_set_v )
= ( ord_less_eq_set_v @ X2 @ Y ) ) ).
% diff_shunt_var
thf(fact_635_diff__shunt__var,axiom,
! [X2: set_Product_prod_v_v,Y: set_Product_prod_v_v] :
( ( ( minus_4183494784930505774od_v_v @ X2 @ Y )
= bot_bo723834152578015283od_v_v )
= ( ord_le7336532860387713383od_v_v @ X2 @ Y ) ) ).
% diff_shunt_var
thf(fact_636_less__eq__Sup,axiom,
! [A3: set_se8455005133513928103od_v_v,U: set_Product_prod_v_v] :
( ! [V2: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ V2 @ A3 )
=> ( ord_le7336532860387713383od_v_v @ U @ V2 ) )
=> ( ( A3 != bot_bo3497076220358800403od_v_v )
=> ( ord_le7336532860387713383od_v_v @ U @ ( comple5788137035815166516od_v_v @ A3 ) ) ) ) ).
% less_eq_Sup
thf(fact_637_less__eq__Sup,axiom,
! [A3: set_set_v,U: set_v] :
( ! [V2: set_v] :
( ( member_set_v @ V2 @ A3 )
=> ( ord_less_eq_set_v @ U @ V2 ) )
=> ( ( A3 != bot_bot_set_set_v )
=> ( ord_less_eq_set_v @ U @ ( comple2307003700295860064_set_v @ A3 ) ) ) ) ).
% less_eq_Sup
thf(fact_638_Sup__subset__mono,axiom,
! [A3: set_se8455005133513928103od_v_v,B: set_se8455005133513928103od_v_v] :
( ( ord_le4714265922333009223od_v_v @ A3 @ B )
=> ( ord_le7336532860387713383od_v_v @ ( comple5788137035815166516od_v_v @ A3 ) @ ( comple5788137035815166516od_v_v @ B ) ) ) ).
% Sup_subset_mono
thf(fact_639_Sup__subset__mono,axiom,
! [A3: set_set_v,B: set_set_v] :
( ( ord_le5216385588623774835_set_v @ A3 @ B )
=> ( ord_less_eq_set_v @ ( comple2307003700295860064_set_v @ A3 ) @ ( comple2307003700295860064_set_v @ B ) ) ) ).
% Sup_subset_mono
thf(fact_640_Union__disjoint,axiom,
! [C2: set_se8455005133513928103od_v_v,A3: set_Product_prod_v_v] :
( ( ( inf_in6271465464967711157od_v_v @ ( comple5788137035815166516od_v_v @ C2 ) @ A3 )
= bot_bo723834152578015283od_v_v )
= ( ! [X3: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ X3 @ C2 )
=> ( ( inf_in6271465464967711157od_v_v @ X3 @ A3 )
= bot_bo723834152578015283od_v_v ) ) ) ) ).
% Union_disjoint
thf(fact_641_Union__disjoint,axiom,
! [C2: set_set_set_v,A3: set_set_v] :
( ( ( inf_inf_set_set_v @ ( comple5450237519782573632_set_v @ C2 ) @ A3 )
= bot_bot_set_set_v )
= ( ! [X3: set_set_v] :
( ( member_set_set_v @ X3 @ C2 )
=> ( ( inf_inf_set_set_v @ X3 @ A3 )
= bot_bot_set_set_v ) ) ) ) ).
% Union_disjoint
thf(fact_642_Union__disjoint,axiom,
! [C2: set_set_v,A3: set_v] :
( ( ( inf_inf_set_v @ ( comple2307003700295860064_set_v @ C2 ) @ A3 )
= bot_bot_set_v )
= ( ! [X3: set_v] :
( ( member_set_v @ X3 @ C2 )
=> ( ( inf_inf_set_v @ X3 @ A3 )
= bot_bot_set_v ) ) ) ) ).
% Union_disjoint
thf(fact_643_Union__Int__subset,axiom,
! [A3: set_se8455005133513928103od_v_v,B: set_se8455005133513928103od_v_v] : ( ord_le7336532860387713383od_v_v @ ( comple5788137035815166516od_v_v @ ( inf_in6058586722421118357od_v_v @ A3 @ B ) ) @ ( inf_in6271465464967711157od_v_v @ ( comple5788137035815166516od_v_v @ A3 ) @ ( comple5788137035815166516od_v_v @ B ) ) ) ).
% Union_Int_subset
thf(fact_644_Union__Int__subset,axiom,
! [A3: set_set_v,B: set_set_v] : ( ord_less_eq_set_v @ ( comple2307003700295860064_set_v @ ( inf_inf_set_set_v @ A3 @ B ) ) @ ( inf_inf_set_v @ ( comple2307003700295860064_set_v @ A3 ) @ ( comple2307003700295860064_set_v @ B ) ) ) ).
% Union_Int_subset
thf(fact_645_Sup__inter__less__eq,axiom,
! [A3: set_Product_unit,B: set_Product_unit] : ( ord_le3221252021190050221t_unit @ ( comple4687483117567863418t_unit @ ( inf_in4660618365625256667t_unit @ A3 @ B ) ) @ ( inf_inf_Product_unit @ ( comple4687483117567863418t_unit @ A3 ) @ ( comple4687483117567863418t_unit @ B ) ) ) ).
% Sup_inter_less_eq
thf(fact_646_Sup__inter__less__eq,axiom,
! [A3: set_se8455005133513928103od_v_v,B: set_se8455005133513928103od_v_v] : ( ord_le7336532860387713383od_v_v @ ( comple5788137035815166516od_v_v @ ( inf_in6058586722421118357od_v_v @ A3 @ B ) ) @ ( inf_in6271465464967711157od_v_v @ ( comple5788137035815166516od_v_v @ A3 ) @ ( comple5788137035815166516od_v_v @ B ) ) ) ).
% Sup_inter_less_eq
thf(fact_647_Sup__inter__less__eq,axiom,
! [A3: set_set_v,B: set_set_v] : ( ord_less_eq_set_v @ ( comple2307003700295860064_set_v @ ( inf_inf_set_set_v @ A3 @ B ) ) @ ( inf_inf_set_v @ ( comple2307003700295860064_set_v @ A3 ) @ ( comple2307003700295860064_set_v @ B ) ) ) ).
% Sup_inter_less_eq
thf(fact_648_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_649_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_650__092_060open_062_092_060Union_062_A_123_092_060S_062_Ae_H_H_An_A_124n_O_An_A_092_060in_062_Aset_A_Istack_Ae_H_H_J_125_A_061_Avisited_Ae_H_H_A_N_Aexplored_Ae_H_H_092_060close_062,axiom,
( ( comple2307003700295860064_set_v
@ ( collect_set_v
@ ^ [Uu: set_v] :
? [N4: v] :
( ( Uu
= ( sCC_Bl1280885523602775798t_unit @ e3 @ N4 ) )
& ( member_v @ N4 @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ e3 ) ) ) ) ) )
= ( minus_minus_set_v @ ( sCC_Bl4645233313691564917t_unit @ e3 ) @ ( sCC_Bl157864678168468314t_unit @ e3 ) ) ) ).
% \<open>\<Union> {\<S> e'' n |n. n \<in> set (stack e'')} = visited e'' - explored e''\<close>
thf(fact_651_surjective,axiom,
! [R: sCC_Bl1394983891496994913t_unit] :
( R
= ( sCC_Bl8064756265740546429t_unit @ ( sCC_Bl1090238580953940555t_unit @ R ) @ ( sCC_Bl1280885523602775798t_unit @ R ) @ ( sCC_Bl157864678168468314t_unit @ R ) @ ( sCC_Bl4645233313691564917t_unit @ R ) @ ( sCC_Bl3795065053823578884t_unit @ R ) @ ( sCC_Bl2536197123907397897t_unit @ R ) @ ( sCC_Bl8828226123343373779t_unit @ R ) @ ( sCC_Bl9201514103433284750t_unit @ R ) @ ( sCC_Bl3567736435408124606t_unit @ R ) ) ) ).
% surjective
thf(fact_652_insert__partition,axiom,
! [X2: set_Product_prod_v_v,F2: set_se8455005133513928103od_v_v] :
( ~ ( member8406446414694345712od_v_v @ X2 @ F2 )
=> ( ! [X4: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ X4 @ ( insert7504383016908236695od_v_v @ X2 @ F2 ) )
=> ! [Xa2: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ Xa2 @ ( insert7504383016908236695od_v_v @ X2 @ F2 ) )
=> ( ( X4 != Xa2 )
=> ( ( inf_in6271465464967711157od_v_v @ X4 @ Xa2 )
= bot_bo723834152578015283od_v_v ) ) ) )
=> ( ( inf_in6271465464967711157od_v_v @ X2 @ ( comple5788137035815166516od_v_v @ F2 ) )
= bot_bo723834152578015283od_v_v ) ) ) ).
% insert_partition
thf(fact_653_insert__partition,axiom,
! [X2: set_set_v,F2: set_set_set_v] :
( ~ ( member_set_set_v @ X2 @ F2 )
=> ( ! [X4: set_set_v] :
( ( member_set_set_v @ X4 @ ( insert_set_set_v @ X2 @ F2 ) )
=> ! [Xa2: set_set_v] :
( ( member_set_set_v @ Xa2 @ ( insert_set_set_v @ X2 @ F2 ) )
=> ( ( X4 != Xa2 )
=> ( ( inf_inf_set_set_v @ X4 @ Xa2 )
= bot_bot_set_set_v ) ) ) )
=> ( ( inf_inf_set_set_v @ X2 @ ( comple5450237519782573632_set_v @ F2 ) )
= bot_bot_set_set_v ) ) ) ).
% insert_partition
thf(fact_654_insert__partition,axiom,
! [X2: set_v,F2: set_set_v] :
( ~ ( member_set_v @ X2 @ F2 )
=> ( ! [X4: set_v] :
( ( member_set_v @ X4 @ ( insert_set_v @ X2 @ F2 ) )
=> ! [Xa2: set_v] :
( ( member_set_v @ Xa2 @ ( insert_set_v @ X2 @ F2 ) )
=> ( ( X4 != Xa2 )
=> ( ( inf_inf_set_v @ X4 @ Xa2 )
= bot_bot_set_v ) ) ) )
=> ( ( inf_inf_set_v @ X2 @ ( comple2307003700295860064_set_v @ F2 ) )
= bot_bot_set_v ) ) ) ).
% insert_partition
thf(fact_655_Sup__inf__eq__bot__iff,axiom,
! [B: set_se8455005133513928103od_v_v,A: set_Product_prod_v_v] :
( ( ( inf_in6271465464967711157od_v_v @ ( comple5788137035815166516od_v_v @ B ) @ A )
= bot_bo723834152578015283od_v_v )
= ( ! [X3: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ X3 @ B )
=> ( ( inf_in6271465464967711157od_v_v @ X3 @ A )
= bot_bo723834152578015283od_v_v ) ) ) ) ).
% Sup_inf_eq_bot_iff
thf(fact_656_Sup__inf__eq__bot__iff,axiom,
! [B: set_Product_unit,A: product_unit] :
( ( ( inf_inf_Product_unit @ ( comple4687483117567863418t_unit @ B ) @ A )
= bot_bot_Product_unit )
= ( ! [X3: product_unit] :
( ( member_Product_unit @ X3 @ B )
=> ( ( inf_inf_Product_unit @ X3 @ A )
= bot_bot_Product_unit ) ) ) ) ).
% Sup_inf_eq_bot_iff
thf(fact_657_Sup__inf__eq__bot__iff,axiom,
! [B: set_set_set_v,A: set_set_v] :
( ( ( inf_inf_set_set_v @ ( comple5450237519782573632_set_v @ B ) @ A )
= bot_bot_set_set_v )
= ( ! [X3: set_set_v] :
( ( member_set_set_v @ X3 @ B )
=> ( ( inf_inf_set_set_v @ X3 @ A )
= bot_bot_set_set_v ) ) ) ) ).
% Sup_inf_eq_bot_iff
thf(fact_658_Sup__inf__eq__bot__iff,axiom,
! [B: set_set_v,A: set_v] :
( ( ( inf_inf_set_v @ ( comple2307003700295860064_set_v @ B ) @ A )
= bot_bot_set_v )
= ( ! [X3: set_v] :
( ( member_set_v @ X3 @ B )
=> ( ( inf_inf_set_v @ X3 @ A )
= bot_bot_set_v ) ) ) ) ).
% Sup_inf_eq_bot_iff
thf(fact_659_the__elem__eq,axiom,
! [X2: v] :
( ( the_elem_v @ ( insert_v @ X2 @ bot_bot_set_v ) )
= X2 ) ).
% the_elem_eq
thf(fact_660_the__elem__eq,axiom,
! [X2: product_prod_v_v] :
( ( the_el5392834299063928540od_v_v @ ( insert1338601472111419319od_v_v @ X2 @ bot_bo723834152578015283od_v_v ) )
= X2 ) ).
% the_elem_eq
thf(fact_661_the__elem__eq,axiom,
! [X2: set_v] :
( ( the_elem_set_v @ ( insert_set_v @ X2 @ bot_bot_set_set_v ) )
= X2 ) ).
% the_elem_eq
thf(fact_662_singleton__conv2,axiom,
! [A: v] :
( ( collect_v
@ ( ^ [Y5: v,Z4: v] : ( Y5 = Z4 )
@ A ) )
= ( insert_v @ A @ bot_bot_set_v ) ) ).
% singleton_conv2
thf(fact_663_singleton__conv2,axiom,
! [A: product_prod_v_v] :
( ( collec140062887454715474od_v_v
@ ( ^ [Y5: product_prod_v_v,Z4: product_prod_v_v] : ( Y5 = Z4 )
@ A ) )
= ( insert1338601472111419319od_v_v @ A @ bot_bo723834152578015283od_v_v ) ) ).
% singleton_conv2
thf(fact_664_singleton__conv2,axiom,
! [A: set_v] :
( ( collect_set_v
@ ( ^ [Y5: set_v,Z4: set_v] : ( Y5 = Z4 )
@ A ) )
= ( insert_set_v @ A @ bot_bot_set_set_v ) ) ).
% singleton_conv2
thf(fact_665_singleton__conv,axiom,
! [A: v] :
( ( collect_v
@ ^ [X3: v] : ( X3 = A ) )
= ( insert_v @ A @ bot_bot_set_v ) ) ).
% singleton_conv
thf(fact_666_singleton__conv,axiom,
! [A: product_prod_v_v] :
( ( collec140062887454715474od_v_v
@ ^ [X3: product_prod_v_v] : ( X3 = A ) )
= ( insert1338601472111419319od_v_v @ A @ bot_bo723834152578015283od_v_v ) ) ).
% singleton_conv
thf(fact_667_singleton__conv,axiom,
! [A: set_v] :
( ( collect_set_v
@ ^ [X3: set_v] : ( X3 = A ) )
= ( insert_set_v @ A @ bot_bot_set_set_v ) ) ).
% singleton_conv
thf(fact_668_sub__env__def,axiom,
( sCC_Bl5768913643336123637t_unit
= ( ^ [E6: sCC_Bl1394983891496994913t_unit,E7: sCC_Bl1394983891496994913t_unit] :
( ( ( sCC_Bl1090238580953940555t_unit @ E7 )
= ( sCC_Bl1090238580953940555t_unit @ E6 ) )
& ( ord_less_eq_set_v @ ( sCC_Bl4645233313691564917t_unit @ E6 ) @ ( sCC_Bl4645233313691564917t_unit @ E7 ) )
& ( ord_less_eq_set_v @ ( sCC_Bl157864678168468314t_unit @ E6 ) @ ( sCC_Bl157864678168468314t_unit @ E7 ) )
& ! [V3: v] : ( ord_less_eq_set_v @ ( sCC_Bl3795065053823578884t_unit @ E6 @ V3 ) @ ( sCC_Bl3795065053823578884t_unit @ E7 @ V3 ) )
& ! [V3: v] : ( ord_less_eq_set_v @ ( sCC_Bl1280885523602775798t_unit @ E6 @ V3 ) @ ( sCC_Bl1280885523602775798t_unit @ E7 @ V3 ) )
& ( ord_less_eq_set_v
@ ( comple2307003700295860064_set_v
@ ( collect_set_v
@ ^ [Uu: set_v] :
? [V3: v] :
( ( Uu
= ( sCC_Bl1280885523602775798t_unit @ E6 @ V3 ) )
& ( member_v @ V3 @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E6 ) ) ) ) ) )
@ ( comple2307003700295860064_set_v
@ ( collect_set_v
@ ^ [Uu: set_v] :
? [V3: v] :
( ( Uu
= ( sCC_Bl1280885523602775798t_unit @ E7 @ V3 ) )
& ( member_v @ V3 @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E7 ) ) ) ) ) ) ) ) ) ) ).
% sub_env_def
thf(fact_669_unite__S__equal,axiom,
! [E2: sCC_Bl1394983891496994913t_unit,W: v,V: 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 ) )
=> ( ( comple2307003700295860064_set_v
@ ( collect_set_v
@ ^ [Uu: set_v] :
? [N4: v] :
( ( Uu
= ( sCC_Bl1280885523602775798t_unit @ ( sCC_Bloemen_unite_v @ V @ W @ E2 ) @ N4 ) )
& ( member_v @ N4 @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ ( sCC_Bloemen_unite_v @ V @ W @ E2 ) ) ) ) ) ) )
= ( comple2307003700295860064_set_v
@ ( collect_set_v
@ ^ [Uu: set_v] :
? [N4: v] :
( ( Uu
= ( sCC_Bl1280885523602775798t_unit @ E2 @ N4 ) )
& ( member_v @ N4 @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E2 ) ) ) ) ) ) ) ) ) ) ) ) ).
% unite_S_equal
thf(fact_670_Int__def,axiom,
( inf_in6271465464967711157od_v_v
= ( ^ [A4: set_Product_prod_v_v,B3: set_Product_prod_v_v] :
( collec140062887454715474od_v_v
@ ^ [X3: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X3 @ A4 )
& ( member7453568604450474000od_v_v @ X3 @ B3 ) ) ) ) ) ).
% Int_def
thf(fact_671_Int__def,axiom,
( inf_inf_set_set_v
= ( ^ [A4: set_set_v,B3: set_set_v] :
( collect_set_v
@ ^ [X3: set_v] :
( ( member_set_v @ X3 @ A4 )
& ( member_set_v @ X3 @ B3 ) ) ) ) ) ).
% Int_def
thf(fact_672_Int__def,axiom,
( inf_inf_set_v
= ( ^ [A4: set_v,B3: set_v] :
( collect_v
@ ^ [X3: v] :
( ( member_v @ X3 @ A4 )
& ( member_v @ X3 @ B3 ) ) ) ) ) ).
% Int_def
thf(fact_673_empty__def,axiom,
( bot_bot_set_v
= ( collect_v
@ ^ [X3: v] : $false ) ) ).
% empty_def
thf(fact_674_empty__def,axiom,
( bot_bo723834152578015283od_v_v
= ( collec140062887454715474od_v_v
@ ^ [X3: product_prod_v_v] : $false ) ) ).
% empty_def
thf(fact_675_empty__def,axiom,
( bot_bot_set_set_v
= ( collect_set_v
@ ^ [X3: set_v] : $false ) ) ).
% empty_def
thf(fact_676_Int__Collect,axiom,
! [X2: product_prod_v_v,A3: set_Product_prod_v_v,P: product_prod_v_v > $o] :
( ( member7453568604450474000od_v_v @ X2 @ ( inf_in6271465464967711157od_v_v @ A3 @ ( collec140062887454715474od_v_v @ P ) ) )
= ( ( member7453568604450474000od_v_v @ X2 @ A3 )
& ( P @ X2 ) ) ) ).
% Int_Collect
thf(fact_677_Int__Collect,axiom,
! [X2: set_v,A3: set_set_v,P: set_v > $o] :
( ( member_set_v @ X2 @ ( inf_inf_set_set_v @ A3 @ ( collect_set_v @ P ) ) )
= ( ( member_set_v @ X2 @ A3 )
& ( P @ X2 ) ) ) ).
% Int_Collect
thf(fact_678_Int__Collect,axiom,
! [X2: v,A3: set_v,P: v > $o] :
( ( member_v @ X2 @ ( inf_inf_set_v @ A3 @ ( collect_v @ P ) ) )
= ( ( member_v @ X2 @ A3 )
& ( P @ X2 ) ) ) ).
% Int_Collect
thf(fact_679_inf__set__def,axiom,
( inf_in6271465464967711157od_v_v
= ( ^ [A4: set_Product_prod_v_v,B3: set_Product_prod_v_v] :
( collec140062887454715474od_v_v
@ ( inf_in6860806757119575912_v_v_o
@ ^ [X3: product_prod_v_v] : ( member7453568604450474000od_v_v @ X3 @ A4 )
@ ^ [X3: product_prod_v_v] : ( member7453568604450474000od_v_v @ X3 @ B3 ) ) ) ) ) ).
% inf_set_def
thf(fact_680_inf__set__def,axiom,
( inf_inf_set_set_v
= ( ^ [A4: set_set_v,B3: set_set_v] :
( collect_set_v
@ ( inf_inf_set_v_o
@ ^ [X3: set_v] : ( member_set_v @ X3 @ A4 )
@ ^ [X3: set_v] : ( member_set_v @ X3 @ B3 ) ) ) ) ) ).
% inf_set_def
thf(fact_681_inf__set__def,axiom,
( inf_inf_set_v
= ( ^ [A4: set_v,B3: set_v] :
( collect_v
@ ( inf_inf_v_o
@ ^ [X3: v] : ( member_v @ X3 @ A4 )
@ ^ [X3: v] : ( member_v @ X3 @ B3 ) ) ) ) ) ).
% inf_set_def
thf(fact_682_Collect__subset,axiom,
! [A3: set_set_v,P: set_v > $o] :
( ord_le5216385588623774835_set_v
@ ( collect_set_v
@ ^ [X3: set_v] :
( ( member_set_v @ X3 @ A3 )
& ( P @ X3 ) ) )
@ A3 ) ).
% Collect_subset
thf(fact_683_Collect__subset,axiom,
! [A3: set_v,P: v > $o] :
( ord_less_eq_set_v
@ ( collect_v
@ ^ [X3: v] :
( ( member_v @ X3 @ A3 )
& ( P @ X3 ) ) )
@ A3 ) ).
% Collect_subset
thf(fact_684_Collect__subset,axiom,
! [A3: set_Product_prod_v_v,P: product_prod_v_v > $o] :
( ord_le7336532860387713383od_v_v
@ ( collec140062887454715474od_v_v
@ ^ [X3: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X3 @ A3 )
& ( P @ X3 ) ) )
@ A3 ) ).
% Collect_subset
thf(fact_685_Collect__conj__eq,axiom,
! [P: set_v > $o,Q: set_v > $o] :
( ( collect_set_v
@ ^ [X3: set_v] :
( ( P @ X3 )
& ( Q @ X3 ) ) )
= ( inf_inf_set_set_v @ ( collect_set_v @ P ) @ ( collect_set_v @ Q ) ) ) ).
% Collect_conj_eq
thf(fact_686_Collect__conj__eq,axiom,
! [P: v > $o,Q: v > $o] :
( ( collect_v
@ ^ [X3: v] :
( ( P @ X3 )
& ( Q @ X3 ) ) )
= ( inf_inf_set_v @ ( collect_v @ P ) @ ( collect_v @ Q ) ) ) ).
% Collect_conj_eq
thf(fact_687_less__eq__set__def,axiom,
( ord_less_eq_set_v
= ( ^ [A4: set_v,B3: set_v] :
( ord_less_eq_v_o
@ ^ [X3: v] : ( member_v @ X3 @ A4 )
@ ^ [X3: v] : ( member_v @ X3 @ B3 ) ) ) ) ).
% less_eq_set_def
thf(fact_688_less__eq__set__def,axiom,
( ord_le7336532860387713383od_v_v
= ( ^ [A4: set_Product_prod_v_v,B3: set_Product_prod_v_v] :
( ord_le5892402249245633078_v_v_o
@ ^ [X3: product_prod_v_v] : ( member7453568604450474000od_v_v @ X3 @ A4 )
@ ^ [X3: product_prod_v_v] : ( member7453568604450474000od_v_v @ X3 @ B3 ) ) ) ) ).
% less_eq_set_def
thf(fact_689_insert__Collect,axiom,
! [A: v,P: v > $o] :
( ( insert_v @ A @ ( collect_v @ P ) )
= ( collect_v
@ ^ [U2: v] :
( ( U2 != A )
=> ( P @ U2 ) ) ) ) ).
% insert_Collect
thf(fact_690_insert__Collect,axiom,
! [A: product_prod_v_v,P: product_prod_v_v > $o] :
( ( insert1338601472111419319od_v_v @ A @ ( collec140062887454715474od_v_v @ P ) )
= ( collec140062887454715474od_v_v
@ ^ [U2: product_prod_v_v] :
( ( U2 != A )
=> ( P @ U2 ) ) ) ) ).
% insert_Collect
thf(fact_691_insert__Collect,axiom,
! [A: set_v,P: set_v > $o] :
( ( insert_set_v @ A @ ( collect_set_v @ P ) )
= ( collect_set_v
@ ^ [U2: set_v] :
( ( U2 != A )
=> ( P @ U2 ) ) ) ) ).
% insert_Collect
thf(fact_692_minus__set__def,axiom,
( minus_4183494784930505774od_v_v
= ( ^ [A4: set_Product_prod_v_v,B3: set_Product_prod_v_v] :
( collec140062887454715474od_v_v
@ ( minus_9095120230875558447_v_v_o
@ ^ [X3: product_prod_v_v] : ( member7453568604450474000od_v_v @ X3 @ A4 )
@ ^ [X3: product_prod_v_v] : ( member7453568604450474000od_v_v @ X3 @ B3 ) ) ) ) ) ).
% minus_set_def
thf(fact_693_minus__set__def,axiom,
( minus_7228012346218142266_set_v
= ( ^ [A4: set_set_v,B3: set_set_v] :
( collect_set_v
@ ( minus_minus_set_v_o
@ ^ [X3: set_v] : ( member_set_v @ X3 @ A4 )
@ ^ [X3: set_v] : ( member_set_v @ X3 @ B3 ) ) ) ) ) ).
% minus_set_def
thf(fact_694_minus__set__def,axiom,
( minus_minus_set_v
= ( ^ [A4: set_v,B3: set_v] :
( collect_v
@ ( minus_minus_v_o
@ ^ [X3: v] : ( member_v @ X3 @ A4 )
@ ^ [X3: v] : ( member_v @ X3 @ B3 ) ) ) ) ) ).
% minus_set_def
thf(fact_695_insert__compr,axiom,
( insert_v
= ( ^ [A5: v,B3: set_v] :
( collect_v
@ ^ [X3: v] :
( ( X3 = A5 )
| ( member_v @ X3 @ B3 ) ) ) ) ) ).
% insert_compr
thf(fact_696_insert__compr,axiom,
( insert1338601472111419319od_v_v
= ( ^ [A5: product_prod_v_v,B3: set_Product_prod_v_v] :
( collec140062887454715474od_v_v
@ ^ [X3: product_prod_v_v] :
( ( X3 = A5 )
| ( member7453568604450474000od_v_v @ X3 @ B3 ) ) ) ) ) ).
% insert_compr
thf(fact_697_insert__compr,axiom,
( insert_set_v
= ( ^ [A5: set_v,B3: set_set_v] :
( collect_set_v
@ ^ [X3: set_v] :
( ( X3 = A5 )
| ( member_set_v @ X3 @ B3 ) ) ) ) ) ).
% insert_compr
thf(fact_698_set__diff__eq,axiom,
( minus_4183494784930505774od_v_v
= ( ^ [A4: set_Product_prod_v_v,B3: set_Product_prod_v_v] :
( collec140062887454715474od_v_v
@ ^ [X3: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X3 @ A4 )
& ~ ( member7453568604450474000od_v_v @ X3 @ B3 ) ) ) ) ) ).
% set_diff_eq
thf(fact_699_set__diff__eq,axiom,
( minus_7228012346218142266_set_v
= ( ^ [A4: set_set_v,B3: set_set_v] :
( collect_set_v
@ ^ [X3: set_v] :
( ( member_set_v @ X3 @ A4 )
& ~ ( member_set_v @ X3 @ B3 ) ) ) ) ) ).
% set_diff_eq
thf(fact_700_set__diff__eq,axiom,
( minus_minus_set_v
= ( ^ [A4: set_v,B3: set_v] :
( collect_v
@ ^ [X3: v] :
( ( member_v @ X3 @ A4 )
& ~ ( member_v @ X3 @ B3 ) ) ) ) ) ).
% set_diff_eq
thf(fact_701_Collect__conv__if2,axiom,
! [P: v > $o,A: v] :
( ( ( P @ A )
=> ( ( collect_v
@ ^ [X3: v] :
( ( A = X3 )
& ( P @ X3 ) ) )
= ( insert_v @ A @ bot_bot_set_v ) ) )
& ( ~ ( P @ A )
=> ( ( collect_v
@ ^ [X3: v] :
( ( A = X3 )
& ( P @ X3 ) ) )
= bot_bot_set_v ) ) ) ).
% Collect_conv_if2
thf(fact_702_Collect__conv__if2,axiom,
! [P: product_prod_v_v > $o,A: product_prod_v_v] :
( ( ( P @ A )
=> ( ( collec140062887454715474od_v_v
@ ^ [X3: product_prod_v_v] :
( ( A = X3 )
& ( P @ X3 ) ) )
= ( insert1338601472111419319od_v_v @ A @ bot_bo723834152578015283od_v_v ) ) )
& ( ~ ( P @ A )
=> ( ( collec140062887454715474od_v_v
@ ^ [X3: product_prod_v_v] :
( ( A = X3 )
& ( P @ X3 ) ) )
= bot_bo723834152578015283od_v_v ) ) ) ).
% Collect_conv_if2
thf(fact_703_Collect__conv__if2,axiom,
! [P: set_v > $o,A: set_v] :
( ( ( P @ A )
=> ( ( collect_set_v
@ ^ [X3: set_v] :
( ( A = X3 )
& ( P @ X3 ) ) )
= ( insert_set_v @ A @ bot_bot_set_set_v ) ) )
& ( ~ ( P @ A )
=> ( ( collect_set_v
@ ^ [X3: set_v] :
( ( A = X3 )
& ( P @ X3 ) ) )
= bot_bot_set_set_v ) ) ) ).
% Collect_conv_if2
thf(fact_704_Collect__conv__if,axiom,
! [P: v > $o,A: v] :
( ( ( P @ A )
=> ( ( collect_v
@ ^ [X3: v] :
( ( X3 = A )
& ( P @ X3 ) ) )
= ( insert_v @ A @ bot_bot_set_v ) ) )
& ( ~ ( P @ A )
=> ( ( collect_v
@ ^ [X3: v] :
( ( X3 = A )
& ( P @ X3 ) ) )
= bot_bot_set_v ) ) ) ).
% Collect_conv_if
thf(fact_705_Collect__conv__if,axiom,
! [P: product_prod_v_v > $o,A: product_prod_v_v] :
( ( ( P @ A )
=> ( ( collec140062887454715474od_v_v
@ ^ [X3: product_prod_v_v] :
( ( X3 = A )
& ( P @ X3 ) ) )
= ( insert1338601472111419319od_v_v @ A @ bot_bo723834152578015283od_v_v ) ) )
& ( ~ ( P @ A )
=> ( ( collec140062887454715474od_v_v
@ ^ [X3: product_prod_v_v] :
( ( X3 = A )
& ( P @ X3 ) ) )
= bot_bo723834152578015283od_v_v ) ) ) ).
% Collect_conv_if
thf(fact_706_Collect__conv__if,axiom,
! [P: set_v > $o,A: set_v] :
( ( ( P @ A )
=> ( ( collect_set_v
@ ^ [X3: set_v] :
( ( X3 = A )
& ( P @ X3 ) ) )
= ( insert_set_v @ A @ bot_bot_set_set_v ) ) )
& ( ~ ( P @ A )
=> ( ( collect_set_v
@ ^ [X3: set_v] :
( ( X3 = A )
& ( P @ X3 ) ) )
= bot_bot_set_set_v ) ) ) ).
% Collect_conv_if
thf(fact_707_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_708_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_709_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_710_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_711_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_712_select__convs_I8_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_Bl9201514103433284750t_unit @ ( sCC_Bl8064756265740546429t_unit @ Root @ S4 @ Explored @ Visited @ Vsuccs @ Sccs @ Stack @ Cstack @ More ) )
= Cstack ) ).
% select_convs(8)
thf(fact_713_select__convs_I1_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_Bl1090238580953940555t_unit @ ( sCC_Bl8064756265740546429t_unit @ Root @ S4 @ Explored @ Visited @ Vsuccs @ Sccs @ Stack @ Cstack @ More ) )
= Root ) ).
% select_convs(1)
thf(fact_714_select__convs_I6_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_Bl2536197123907397897t_unit @ ( sCC_Bl8064756265740546429t_unit @ Root @ S4 @ Explored @ Visited @ Vsuccs @ Sccs @ Stack @ Cstack @ More ) )
= Sccs ) ).
% select_convs(6)
thf(fact_715_graph_Ounite__S__equal,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] :
( ( 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 ) )
=> ( ( comple5788137035815166516od_v_v
@ ( collec8263177866097347122od_v_v
@ ^ [Uu: set_Product_prod_v_v] :
? [N4: product_prod_v_v] :
( ( Uu
= ( sCC_Bl8440648026628373538t_unit @ ( sCC_Bl4702006153222411093od_v_v @ V @ W @ E2 ) @ N4 ) )
& ( member7453568604450474000od_v_v @ N4 @ ( set_Product_prod_v_v2 @ ( sCC_Bl2021302119412358655t_unit @ ( sCC_Bl4702006153222411093od_v_v @ V @ W @ E2 ) ) ) ) ) ) )
= ( comple5788137035815166516od_v_v
@ ( collec8263177866097347122od_v_v
@ ^ [Uu: set_Product_prod_v_v] :
? [N4: product_prod_v_v] :
( ( Uu
= ( sCC_Bl8440648026628373538t_unit @ E2 @ N4 ) )
& ( member7453568604450474000od_v_v @ N4 @ ( set_Product_prod_v_v2 @ ( sCC_Bl2021302119412358655t_unit @ E2 ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% graph.unite_S_equal
thf(fact_716_graph_Ounite__S__equal,axiom,
! [Vertices: set_v,Successors: v > set_v,E2: sCC_Bl1394983891496994913t_unit,W: v,V: 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 ) )
=> ( ( comple2307003700295860064_set_v
@ ( collect_set_v
@ ^ [Uu: set_v] :
? [N4: v] :
( ( Uu
= ( sCC_Bl1280885523602775798t_unit @ ( sCC_Bloemen_unite_v @ V @ W @ E2 ) @ N4 ) )
& ( member_v @ N4 @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ ( sCC_Bloemen_unite_v @ V @ W @ E2 ) ) ) ) ) ) )
= ( comple2307003700295860064_set_v
@ ( collect_set_v
@ ^ [Uu: set_v] :
? [N4: v] :
( ( Uu
= ( sCC_Bl1280885523602775798t_unit @ E2 @ N4 ) )
& ( member_v @ N4 @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E2 ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% graph.unite_S_equal
thf(fact_717_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_718_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_719_graph_Osub__env__def,axiom,
! [Vertices: set_v,Successors: v > set_v,E2: sCC_Bl1394983891496994913t_unit,E3: sCC_Bl1394983891496994913t_unit] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl5768913643336123637t_unit @ E2 @ E3 )
= ( ( ( sCC_Bl1090238580953940555t_unit @ E3 )
= ( sCC_Bl1090238580953940555t_unit @ E2 ) )
& ( ord_less_eq_set_v @ ( sCC_Bl4645233313691564917t_unit @ E2 ) @ ( sCC_Bl4645233313691564917t_unit @ E3 ) )
& ( ord_less_eq_set_v @ ( sCC_Bl157864678168468314t_unit @ E2 ) @ ( sCC_Bl157864678168468314t_unit @ E3 ) )
& ! [V3: v] : ( ord_less_eq_set_v @ ( sCC_Bl3795065053823578884t_unit @ E2 @ V3 ) @ ( sCC_Bl3795065053823578884t_unit @ E3 @ V3 ) )
& ! [V3: v] : ( ord_less_eq_set_v @ ( sCC_Bl1280885523602775798t_unit @ E2 @ V3 ) @ ( sCC_Bl1280885523602775798t_unit @ E3 @ V3 ) )
& ( ord_less_eq_set_v
@ ( comple2307003700295860064_set_v
@ ( collect_set_v
@ ^ [Uu: set_v] :
? [V3: v] :
( ( Uu
= ( sCC_Bl1280885523602775798t_unit @ E2 @ V3 ) )
& ( member_v @ V3 @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E2 ) ) ) ) ) )
@ ( comple2307003700295860064_set_v
@ ( collect_set_v
@ ^ [Uu: set_v] :
? [V3: v] :
( ( Uu
= ( sCC_Bl1280885523602775798t_unit @ E3 @ V3 ) )
& ( member_v @ V3 @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E3 ) ) ) ) ) ) ) ) ) ) ).
% graph.sub_env_def
thf(fact_720_unite__S__tl,axiom,
! [E2: sCC_Bl1394983891496994913t_unit,W: v,V: v,N2: 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 @ N2 @ ( set_v2 @ ( tl_v @ ( sCC_Bl8828226123343373779t_unit @ ( sCC_Bloemen_unite_v @ V @ W @ E2 ) ) ) ) )
=> ( ( sCC_Bl1280885523602775798t_unit @ ( sCC_Bloemen_unite_v @ V @ W @ E2 ) @ N2 )
= ( sCC_Bl1280885523602775798t_unit @ E2 @ N2 ) ) ) ) ) ) ) ) ).
% unite_S_tl
thf(fact_721_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_722_ra__cases,axiom,
! [X2: v,Y: v,E: set_Product_prod_v_v] :
( ( sCC_Bl4291963740693775144ding_v @ successors @ X2 @ Y @ E )
=> ( ( X2 = Y )
| ? [Z3: v] :
( ( member_v @ Z3 @ ( successors @ X2 ) )
& ~ ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ X2 @ Z3 ) @ E )
& ( sCC_Bl4291963740693775144ding_v @ successors @ Z3 @ Y @ E ) ) ) ) ).
% ra_cases
thf(fact_723_edge__ra,axiom,
! [Y: v,X2: v,E: set_Product_prod_v_v] :
( ( member_v @ Y @ ( successors @ X2 ) )
=> ( ~ ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ X2 @ Y ) @ E )
=> ( sCC_Bl4291963740693775144ding_v @ successors @ X2 @ Y @ E ) ) ) ).
% edge_ra
thf(fact_724_reachable__avoiding_Osimps,axiom,
! [A1: v,A2: v,A32: set_Product_prod_v_v] :
( ( sCC_Bl4291963740693775144ding_v @ successors @ A1 @ A2 @ A32 )
= ( ? [X3: v,E8: set_Product_prod_v_v] :
( ( A1 = X3 )
& ( A2 = X3 )
& ( A32 = E8 ) )
| ? [X3: v,Y3: v,E8: set_Product_prod_v_v,Z2: v] :
( ( A1 = X3 )
& ( A2 = Z2 )
& ( A32 = E8 )
& ( sCC_Bl4291963740693775144ding_v @ successors @ X3 @ Y3 @ E8 )
& ( member_v @ Z2 @ ( successors @ Y3 ) )
& ~ ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ Y3 @ Z2 ) @ E8 ) ) ) ) ).
% reachable_avoiding.simps
thf(fact_725_ra__succ,axiom,
! [X2: v,Y: v,E: set_Product_prod_v_v,Z: v] :
( ( sCC_Bl4291963740693775144ding_v @ successors @ X2 @ Y @ E )
=> ( ( member_v @ Z @ ( successors @ Y ) )
=> ( ~ ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ Y @ Z ) @ E )
=> ( sCC_Bl4291963740693775144ding_v @ successors @ X2 @ Z @ E ) ) ) ) ).
% ra_succ
thf(fact_726_reachable__avoiding_Ocases,axiom,
! [A1: v,A2: v,A32: set_Product_prod_v_v] :
( ( sCC_Bl4291963740693775144ding_v @ successors @ A1 @ A2 @ A32 )
=> ( ( A2 != A1 )
=> ~ ! [Y2: v] :
( ( sCC_Bl4291963740693775144ding_v @ successors @ A1 @ Y2 @ A32 )
=> ( ( member_v @ A2 @ ( successors @ Y2 ) )
=> ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ Y2 @ A2 ) @ A32 ) ) ) ) ) ).
% reachable_avoiding.cases
thf(fact_727_ra__add__edge,axiom,
! [X2: v,Y: v,E: set_Product_prod_v_v,V: v,W: v] :
( ( sCC_Bl4291963740693775144ding_v @ successors @ X2 @ Y @ E )
=> ( ( sCC_Bl4291963740693775144ding_v @ successors @ X2 @ Y @ ( sup_su414716646722978715od_v_v @ E @ ( insert1338601472111419319od_v_v @ ( product_Pair_v_v @ V @ W ) @ bot_bo723834152578015283od_v_v ) ) )
| ( ( sCC_Bl4291963740693775144ding_v @ successors @ X2 @ 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_728_pre__dfss__post__dfs__pre__dfss,axiom,
! [V: v,E2: sCC_Bl1394983891496994913t_unit,W: v] :
( ( sCC_Bl1748261141445803503t_unit @ successors @ V @ E2 )
=> ( ( member_v @ W @ ( successors @ V ) )
=> ( ~ ( member_v @ W @ ( sCC_Bl4645233313691564917t_unit @ E2 ) )
=> ( ( sCC_Bl8953792750115413617t_unit @ successors @ W @ E2 @ ( sCC_Bloemen_dfs_v @ successors @ W @ E2 ) )
=> ( sCC_Bl1748261141445803503t_unit @ successors @ V
@ ( sCC_Bl48393358579903213t_unit
@ ^ [Uu: v > set_v,X3: v] : ( if_set_v @ ( X3 = V ) @ ( sup_sup_set_v @ ( sCC_Bl3795065053823578884t_unit @ ( sCC_Bloemen_dfs_v @ successors @ W @ E2 ) @ V ) @ ( insert_v @ W @ bot_bot_set_v ) ) @ ( sCC_Bl3795065053823578884t_unit @ ( sCC_Bloemen_dfs_v @ successors @ W @ E2 ) @ X3 ) )
@ ( sCC_Bloemen_dfs_v @ successors @ W @ E2 ) ) ) ) ) ) ) ).
% pre_dfss_post_dfs_pre_dfss
thf(fact_729_pfx_I2_J,axiom,
( ( sCC_Bl8828226123343373779t_unit @ e2 )
!= nil_v ) ).
% pfx(2)
thf(fact_730_wf_H,axiom,
sCC_Bl9196236973127232072t_unit @ successors @ e2 ).
% wf'
thf(fact_731_pfx_I4_J,axiom,
member_v @ w @ ( sCC_Bl1280885523602775798t_unit @ e2 @ ( hd_v @ ( sCC_Bl8828226123343373779t_unit @ e2 ) ) ) ).
% pfx(4)
thf(fact_732__092_060open_062hd_A_Istack_Ae_H_J_A_092_060in_062_Acc_092_060close_062,axiom,
member_v @ ( hd_v @ ( sCC_Bl8828226123343373779t_unit @ e2 ) ) @ cc ).
% \<open>hd (stack e') \<in> cc\<close>
thf(fact_733_pfx_I3_J,axiom,
( ( sCC_Bl1280885523602775798t_unit @ e2 )
= ( ^ [X3: v] :
( if_set_v
@ ( member_v @ X3
@ ( comple2307003700295860064_set_v
@ ( collect_set_v
@ ^ [Uu: set_v] :
? [N4: v] :
( ( Uu
= ( sCC_Bl1280885523602775798t_unit @ e @ N4 ) )
& ( member_v @ N4 @ ( sup_sup_set_v @ ( set_v2 @ pfx ) @ ( insert_v @ ( hd_v @ ( sCC_Bl8828226123343373779t_unit @ e2 ) ) @ bot_bot_set_v ) ) ) ) ) ) )
@ ( comple2307003700295860064_set_v
@ ( collect_set_v
@ ^ [Uu: set_v] :
? [N4: v] :
( ( Uu
= ( sCC_Bl1280885523602775798t_unit @ e @ N4 ) )
& ( member_v @ N4 @ ( sup_sup_set_v @ ( set_v2 @ pfx ) @ ( insert_v @ ( hd_v @ ( sCC_Bl8828226123343373779t_unit @ e2 ) ) @ bot_bot_set_v ) ) ) ) ) )
@ ( sCC_Bl1280885523602775798t_unit @ e @ X3 ) ) ) ) ).
% pfx(3)
thf(fact_734__092_060open_062v_A_092_060in_062_A_092_060S_062_Ae_H_A_Ihd_A_Istack_Ae_H_J_J_092_060close_062,axiom,
member_v @ v2 @ ( sCC_Bl1280885523602775798t_unit @ e2 @ ( hd_v @ ( sCC_Bl8828226123343373779t_unit @ e2 ) ) ) ).
% \<open>v \<in> \<S> e' (hd (stack e'))\<close>
thf(fact_735_w_I3_J,axiom,
member_v @ w @ ( sCC_Bl4645233313691564917t_unit @ e ) ).
% w(3)
thf(fact_736_w_I4_J,axiom,
~ ( member_v @ w @ ( sCC_Bl157864678168468314t_unit @ e ) ) ).
% w(4)
thf(fact_737_w_I1_J,axiom,
member_v @ w @ ( successors @ v2 ) ).
% w(1)
thf(fact_738_pre,axiom,
sCC_Bl1748261141445803503t_unit @ successors @ v2 @ e ).
% pre
thf(fact_739_w_I2_J,axiom,
~ ( member_v @ w @ ( sCC_Bl3795065053823578884t_unit @ e @ v2 ) ) ).
% w(2)
thf(fact_740_e_H__def,axiom,
( e2
= ( sCC_Bloemen_unite_v @ v2 @ w @ e ) ) ).
% e'_def
thf(fact_741_pfx_I1_J,axiom,
( ( sCC_Bl8828226123343373779t_unit @ e )
= ( append_v @ pfx @ ( sCC_Bl8828226123343373779t_unit @ e2 ) ) ) ).
% pfx(1)
thf(fact_742__092_060open_062v_A_092_060in_062_Acc_092_060close_062,axiom,
member_v @ v2 @ cc ).
% \<open>v \<in> cc\<close>
thf(fact_743_local_Owf,axiom,
sCC_Bl9196236973127232072t_unit @ successors @ e ).
% local.wf
thf(fact_744__092_060open_062v_A_092_060in_062_Avisited_Ae_092_060close_062,axiom,
member_v @ v2 @ ( sCC_Bl4645233313691564917t_unit @ e ) ).
% \<open>v \<in> visited e\<close>
thf(fact_745__092_060open_062v_A_092_060notin_062_Aexplored_Ae_092_060close_062,axiom,
~ ( member_v @ v2 @ ( sCC_Bl157864678168468314t_unit @ e ) ) ).
% \<open>v \<notin> explored e\<close>
thf(fact_746__092_060open_062sub__env_Ae_Ae_H_092_060close_062,axiom,
sCC_Bl5768913643336123637t_unit @ e @ e2 ).
% \<open>sub_env e e'\<close>
thf(fact_747__092_060open_062hd_A_Istack_Ae_H_J_A_092_060in_062_A_092_060S_062_Ae_A_Ihd_A_Istack_Ae_H_J_J_092_060close_062,axiom,
member_v @ ( hd_v @ ( sCC_Bl8828226123343373779t_unit @ e2 ) ) @ ( sCC_Bl1280885523602775798t_unit @ e @ ( hd_v @ ( sCC_Bl8828226123343373779t_unit @ e2 ) ) ) ).
% \<open>hd (stack e') \<in> \<S> e (hd (stack e'))\<close>
thf(fact_748_e_H_H__def,axiom,
( e3
= ( sCC_Bl48393358579903213t_unit
@ ^ [Uu: v > set_v,X3: v] : ( if_set_v @ ( X3 = v2 ) @ ( sup_sup_set_v @ ( sCC_Bl3795065053823578884t_unit @ e2 @ v2 ) @ ( insert_v @ w @ bot_bot_set_v ) ) @ ( sCC_Bl3795065053823578884t_unit @ e2 @ X3 ) )
@ e2 ) ) ).
% e''_def
thf(fact_749__092_060open_062sub__env_Ae_A_Iunite_Av_Aw_Ae_J_092_060close_062,axiom,
sCC_Bl5768913643336123637t_unit @ e @ ( sCC_Bloemen_unite_v @ v2 @ w @ e ) ).
% \<open>sub_env e (unite v w e)\<close>
thf(fact_750__092_060open_062wf__env_A_Iunite_Av_Aw_Ae_J_092_060close_062,axiom,
sCC_Bl9196236973127232072t_unit @ successors @ ( sCC_Bloemen_unite_v @ v2 @ w @ e ) ).
% \<open>wf_env (unite v w e)\<close>
thf(fact_751_cc__def,axiom,
( cc
= ( comple2307003700295860064_set_v
@ ( collect_set_v
@ ^ [Uu: set_v] :
? [N4: v] :
( ( Uu
= ( sCC_Bl1280885523602775798t_unit @ e @ N4 ) )
& ( member_v @ N4 @ ( sup_sup_set_v @ ( set_v2 @ pfx ) @ ( insert_v @ ( hd_v @ ( sCC_Bl8828226123343373779t_unit @ e2 ) ) @ bot_bot_set_v ) ) ) ) ) ) ) ).
% cc_def
thf(fact_752__092_060open_062hd_A_Istack_Ae_J_A_092_060in_062_Aset_Apfx_A_092_060union_062_A_123hd_A_Istack_Ae_H_J_125_092_060close_062,axiom,
member_v @ ( hd_v @ ( sCC_Bl8828226123343373779t_unit @ e ) ) @ ( sup_sup_set_v @ ( set_v2 @ pfx ) @ ( insert_v @ ( hd_v @ ( sCC_Bl8828226123343373779t_unit @ e2 ) ) @ bot_bot_set_v ) ) ).
% \<open>hd (stack e) \<in> set pfx \<union> {hd (stack e')}\<close>
thf(fact_753__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062pfx_O_A_092_060lbrakk_062stack_Ae_A_061_Apfx_A_064_Astack_Ae_H_059_Astack_Ae_H_A_092_060noteq_062_A_091_093_059_Alet_Acc_A_061_A_092_060Union_062_A_123_092_060S_062_Ae_An_A_124n_O_An_A_092_060in_062_Aset_Apfx_A_092_060union_062_A_123hd_A_Istack_Ae_H_J_125_125_Ain_A_092_060S_062_Ae_H_A_061_A_I_092_060lambda_062x_O_Aif_Ax_A_092_060in_062_Acc_Athen_Acc_Aelse_A_092_060S_062_Ae_Ax_J_059_Aw_A_092_060in_062_A_092_060S_062_Ae_H_A_Ihd_A_Istack_Ae_H_J_J_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [Pfx: list_v] :
( ( ( sCC_Bl8828226123343373779t_unit @ e )
= ( append_v @ Pfx @ ( sCC_Bl8828226123343373779t_unit @ e2 ) ) )
=> ( ( ( sCC_Bl8828226123343373779t_unit @ e2 )
!= nil_v )
=> ( ( ( sCC_Bl1280885523602775798t_unit @ e2 )
= ( ^ [X3: v] :
( if_set_v
@ ( member_v @ X3
@ ( comple2307003700295860064_set_v
@ ( collect_set_v
@ ^ [Uu: set_v] :
? [N4: v] :
( ( Uu
= ( sCC_Bl1280885523602775798t_unit @ e @ N4 ) )
& ( member_v @ N4 @ ( sup_sup_set_v @ ( set_v2 @ Pfx ) @ ( insert_v @ ( hd_v @ ( sCC_Bl8828226123343373779t_unit @ e2 ) ) @ bot_bot_set_v ) ) ) ) ) ) )
@ ( comple2307003700295860064_set_v
@ ( collect_set_v
@ ^ [Uu: set_v] :
? [N4: v] :
( ( Uu
= ( sCC_Bl1280885523602775798t_unit @ e @ N4 ) )
& ( member_v @ N4 @ ( sup_sup_set_v @ ( set_v2 @ Pfx ) @ ( insert_v @ ( hd_v @ ( sCC_Bl8828226123343373779t_unit @ e2 ) ) @ bot_bot_set_v ) ) ) ) ) )
@ ( sCC_Bl1280885523602775798t_unit @ e @ X3 ) ) ) )
=> ~ ( member_v @ w @ ( sCC_Bl1280885523602775798t_unit @ e2 @ ( hd_v @ ( sCC_Bl8828226123343373779t_unit @ e2 ) ) ) ) ) ) ) ).
% \<open>\<And>thesis. (\<And>pfx. \<lbrakk>stack e = pfx @ stack e'; stack e' \<noteq> []; let cc = \<Union> {\<S> e n |n. n \<in> set pfx \<union> {hd (stack e')}} in \<S> e' = (\<lambda>x. if x \<in> cc then cc else \<S> e x); w \<in> \<S> e' (hd (stack e'))\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_754__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062pfx_O_A_092_060lbrakk_062stack_Ae_A_061_Apfx_A_064_Astack_A_Iunite_Av_Aw_Ae_J_059_Astack_A_Iunite_Av_Aw_Ae_J_A_092_060noteq_062_A_091_093_059_Alet_Acc_A_061_A_092_060Union_062_A_123_092_060S_062_Ae_An_A_124n_O_An_A_092_060in_062_Aset_Apfx_A_092_060union_062_A_123hd_A_Istack_A_Iunite_Av_Aw_Ae_J_J_125_125_Ain_A_092_060S_062_A_Iunite_Av_Aw_Ae_J_A_061_A_I_092_060lambda_062x_O_Aif_Ax_A_092_060in_062_Acc_Athen_Acc_Aelse_A_092_060S_062_Ae_Ax_J_059_Aw_A_092_060in_062_A_092_060S_062_A_Iunite_Av_Aw_Ae_J_A_Ihd_A_Istack_A_Iunite_Av_Aw_Ae_J_J_J_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [Pfx: list_v] :
( ( ( sCC_Bl8828226123343373779t_unit @ e )
= ( append_v @ Pfx @ ( sCC_Bl8828226123343373779t_unit @ ( sCC_Bloemen_unite_v @ v2 @ w @ e ) ) ) )
=> ( ( ( sCC_Bl8828226123343373779t_unit @ ( sCC_Bloemen_unite_v @ v2 @ w @ e ) )
!= nil_v )
=> ( ( ( sCC_Bl1280885523602775798t_unit @ ( sCC_Bloemen_unite_v @ v2 @ w @ e ) )
= ( ^ [X3: v] :
( if_set_v
@ ( member_v @ X3
@ ( comple2307003700295860064_set_v
@ ( collect_set_v
@ ^ [Uu: set_v] :
? [N4: v] :
( ( Uu
= ( sCC_Bl1280885523602775798t_unit @ e @ N4 ) )
& ( member_v @ N4 @ ( sup_sup_set_v @ ( set_v2 @ Pfx ) @ ( insert_v @ ( hd_v @ ( sCC_Bl8828226123343373779t_unit @ ( sCC_Bloemen_unite_v @ v2 @ w @ e ) ) ) @ bot_bot_set_v ) ) ) ) ) ) )
@ ( comple2307003700295860064_set_v
@ ( collect_set_v
@ ^ [Uu: set_v] :
? [N4: v] :
( ( Uu
= ( sCC_Bl1280885523602775798t_unit @ e @ N4 ) )
& ( member_v @ N4 @ ( sup_sup_set_v @ ( set_v2 @ Pfx ) @ ( insert_v @ ( hd_v @ ( sCC_Bl8828226123343373779t_unit @ ( sCC_Bloemen_unite_v @ v2 @ w @ e ) ) ) @ bot_bot_set_v ) ) ) ) ) )
@ ( sCC_Bl1280885523602775798t_unit @ e @ X3 ) ) ) )
=> ~ ( member_v @ w @ ( sCC_Bl1280885523602775798t_unit @ ( sCC_Bloemen_unite_v @ v2 @ w @ e ) @ ( hd_v @ ( sCC_Bl8828226123343373779t_unit @ ( sCC_Bloemen_unite_v @ v2 @ w @ e ) ) ) ) ) ) ) ) ).
% \<open>\<And>thesis. (\<And>pfx. \<lbrakk>stack e = pfx @ stack (unite v w e); stack (unite v w e) \<noteq> []; let cc = \<Union> {\<S> e n |n. n \<in> set pfx \<union> {hd (stack (unite v w e))}} in \<S> (unite v w e) = (\<lambda>x. if x \<in> cc then cc else \<S> e x); w \<in> \<S> (unite v w e) (hd (stack (unite v w e)))\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_755_vfin,axiom,
finite_finite_v @ vertices ).
% vfin
thf(fact_756_bot__unit__def,axiom,
bot_bot_Product_unit = product_Unity ).
% bot_unit_def
thf(fact_757_inf__unit__def,axiom,
( inf_inf_Product_unit
= ( ^ [Uu2: product_unit,Uv: product_unit] : product_Unity ) ) ).
% inf_unit_def
thf(fact_758_dfs_Opsimps,axiom,
! [V: v,E2: sCC_Bl1394983891496994913t_unit] :
( ( accp_S2303753412255344476t_unit @ ( sCC_Bl907557413677168252_rel_v @ successors ) @ ( sum_In526841707622398774t_unit @ ( produc3862955338007567901t_unit @ V @ E2 ) ) )
=> ( ( sCC_Bloemen_dfs_v @ successors @ V @ E2 )
= ( if_SCC4926449794303880475t_unit
@ ( V
= ( hd_v
@ ( sCC_Bl8828226123343373779t_unit
@ ( sCC_Bloemen_dfss_v @ successors @ V
@ ( sCC_Bl7876664385711583351t_unit
@ ^ [Uu: list_v] : ( cons_v @ V @ ( sCC_Bl9201514103433284750t_unit @ E2 ) )
@ ( sCC_Bl349061681862590396t_unit
@ ^ [Uu: list_v] : ( cons_v @ V @ ( sCC_Bl8828226123343373779t_unit @ E2 ) )
@ ( sCC_Bl7870604408699998558t_unit
@ ^ [Uu: set_v] : ( sup_sup_set_v @ ( sCC_Bl4645233313691564917t_unit @ E2 ) @ ( insert_v @ V @ bot_bot_set_v ) )
@ E2 ) ) ) ) ) ) )
@ ( sCC_Bl7876664385711583351t_unit
@ ^ [Uu: list_v] :
( tl_v
@ ( sCC_Bl9201514103433284750t_unit
@ ( sCC_Bloemen_dfss_v @ successors @ V
@ ( sCC_Bl7876664385711583351t_unit
@ ^ [Uv2: list_v] : ( cons_v @ V @ ( sCC_Bl9201514103433284750t_unit @ E2 ) )
@ ( sCC_Bl349061681862590396t_unit
@ ^ [Uv2: list_v] : ( cons_v @ V @ ( sCC_Bl8828226123343373779t_unit @ E2 ) )
@ ( sCC_Bl7870604408699998558t_unit
@ ^ [Uv2: set_v] : ( sup_sup_set_v @ ( sCC_Bl4645233313691564917t_unit @ E2 ) @ ( insert_v @ V @ bot_bot_set_v ) )
@ E2 ) ) ) ) ) )
@ ( sCC_Bl349061681862590396t_unit
@ ^ [Uu: list_v] :
( tl_v
@ ( sCC_Bl8828226123343373779t_unit
@ ( sCC_Bloemen_dfss_v @ successors @ V
@ ( sCC_Bl7876664385711583351t_unit
@ ^ [Uv2: list_v] : ( cons_v @ V @ ( sCC_Bl9201514103433284750t_unit @ E2 ) )
@ ( sCC_Bl349061681862590396t_unit
@ ^ [Uv2: list_v] : ( cons_v @ V @ ( sCC_Bl8828226123343373779t_unit @ E2 ) )
@ ( sCC_Bl7870604408699998558t_unit
@ ^ [Uv2: set_v] : ( sup_sup_set_v @ ( sCC_Bl4645233313691564917t_unit @ E2 ) @ ( insert_v @ V @ bot_bot_set_v ) )
@ E2 ) ) ) ) ) )
@ ( sCC_Bl2708505634401380163t_unit
@ ^ [Uu: set_v] :
( sup_sup_set_v
@ ( sCC_Bl157864678168468314t_unit
@ ( sCC_Bloemen_dfss_v @ successors @ V
@ ( sCC_Bl7876664385711583351t_unit
@ ^ [Uv2: list_v] : ( cons_v @ V @ ( sCC_Bl9201514103433284750t_unit @ E2 ) )
@ ( sCC_Bl349061681862590396t_unit
@ ^ [Uv2: list_v] : ( cons_v @ V @ ( sCC_Bl8828226123343373779t_unit @ E2 ) )
@ ( sCC_Bl7870604408699998558t_unit
@ ^ [Uv2: set_v] : ( sup_sup_set_v @ ( sCC_Bl4645233313691564917t_unit @ E2 ) @ ( insert_v @ V @ bot_bot_set_v ) )
@ E2 ) ) ) ) )
@ ( sCC_Bl1280885523602775798t_unit
@ ( sCC_Bloemen_dfss_v @ successors @ V
@ ( sCC_Bl7876664385711583351t_unit
@ ^ [Uv2: list_v] : ( cons_v @ V @ ( sCC_Bl9201514103433284750t_unit @ E2 ) )
@ ( sCC_Bl349061681862590396t_unit
@ ^ [Uv2: list_v] : ( cons_v @ V @ ( sCC_Bl8828226123343373779t_unit @ E2 ) )
@ ( sCC_Bl7870604408699998558t_unit
@ ^ [Uv2: set_v] : ( sup_sup_set_v @ ( sCC_Bl4645233313691564917t_unit @ E2 ) @ ( insert_v @ V @ bot_bot_set_v ) )
@ E2 ) ) ) )
@ V ) )
@ ( sCC_Bl6816368539212994290t_unit
@ ^ [Uu: set_set_v] :
( sup_sup_set_set_v
@ ( sCC_Bl2536197123907397897t_unit
@ ( sCC_Bloemen_dfss_v @ successors @ V
@ ( sCC_Bl7876664385711583351t_unit
@ ^ [Uv2: list_v] : ( cons_v @ V @ ( sCC_Bl9201514103433284750t_unit @ E2 ) )
@ ( sCC_Bl349061681862590396t_unit
@ ^ [Uv2: list_v] : ( cons_v @ V @ ( sCC_Bl8828226123343373779t_unit @ E2 ) )
@ ( sCC_Bl7870604408699998558t_unit
@ ^ [Uv2: set_v] : ( sup_sup_set_v @ ( sCC_Bl4645233313691564917t_unit @ E2 ) @ ( insert_v @ V @ bot_bot_set_v ) )
@ E2 ) ) ) ) )
@ ( insert_set_v
@ ( sCC_Bl1280885523602775798t_unit
@ ( sCC_Bloemen_dfss_v @ successors @ V
@ ( sCC_Bl7876664385711583351t_unit
@ ^ [Uv2: list_v] : ( cons_v @ V @ ( sCC_Bl9201514103433284750t_unit @ E2 ) )
@ ( sCC_Bl349061681862590396t_unit
@ ^ [Uv2: list_v] : ( cons_v @ V @ ( sCC_Bl8828226123343373779t_unit @ E2 ) )
@ ( sCC_Bl7870604408699998558t_unit
@ ^ [Uv2: set_v] : ( sup_sup_set_v @ ( sCC_Bl4645233313691564917t_unit @ E2 ) @ ( insert_v @ V @ bot_bot_set_v ) )
@ E2 ) ) ) )
@ V )
@ bot_bot_set_set_v ) )
@ ( sCC_Bloemen_dfss_v @ successors @ V
@ ( sCC_Bl7876664385711583351t_unit
@ ^ [Uu: list_v] : ( cons_v @ V @ ( sCC_Bl9201514103433284750t_unit @ E2 ) )
@ ( sCC_Bl349061681862590396t_unit
@ ^ [Uu: list_v] : ( cons_v @ V @ ( sCC_Bl8828226123343373779t_unit @ E2 ) )
@ ( sCC_Bl7870604408699998558t_unit
@ ^ [Uu: set_v] : ( sup_sup_set_v @ ( sCC_Bl4645233313691564917t_unit @ E2 ) @ ( insert_v @ V @ bot_bot_set_v ) )
@ E2 ) ) ) ) ) ) ) )
@ ( sCC_Bl7876664385711583351t_unit
@ ^ [Uu: list_v] :
( tl_v
@ ( sCC_Bl9201514103433284750t_unit
@ ( sCC_Bloemen_dfss_v @ successors @ V
@ ( sCC_Bl7876664385711583351t_unit
@ ^ [Uv2: list_v] : ( cons_v @ V @ ( sCC_Bl9201514103433284750t_unit @ E2 ) )
@ ( sCC_Bl349061681862590396t_unit
@ ^ [Uv2: list_v] : ( cons_v @ V @ ( sCC_Bl8828226123343373779t_unit @ E2 ) )
@ ( sCC_Bl7870604408699998558t_unit
@ ^ [Uv2: set_v] : ( sup_sup_set_v @ ( sCC_Bl4645233313691564917t_unit @ E2 ) @ ( insert_v @ V @ bot_bot_set_v ) )
@ E2 ) ) ) ) ) )
@ ( sCC_Bloemen_dfss_v @ successors @ V
@ ( sCC_Bl7876664385711583351t_unit
@ ^ [Uu: list_v] : ( cons_v @ V @ ( sCC_Bl9201514103433284750t_unit @ E2 ) )
@ ( sCC_Bl349061681862590396t_unit
@ ^ [Uu: list_v] : ( cons_v @ V @ ( sCC_Bl8828226123343373779t_unit @ E2 ) )
@ ( sCC_Bl7870604408699998558t_unit
@ ^ [Uu: set_v] : ( sup_sup_set_v @ ( sCC_Bl4645233313691564917t_unit @ E2 ) @ ( insert_v @ V @ bot_bot_set_v ) )
@ E2 ) ) ) ) ) ) ) ) ).
% dfs.psimps
thf(fact_759_dfs__dfss_Odomintros_I1_J,axiom,
! [V: v,E2: sCC_Bl1394983891496994913t_unit] :
( ( accp_S2303753412255344476t_unit @ ( sCC_Bl907557413677168252_rel_v @ successors )
@ ( sum_In5289330923152326972t_unit
@ ( produc3862955338007567901t_unit @ V
@ ( sCC_Bl7876664385711583351t_unit
@ ^ [Uu: list_v] : ( cons_v @ V @ ( sCC_Bl9201514103433284750t_unit @ E2 ) )
@ ( sCC_Bl349061681862590396t_unit
@ ^ [Uu: list_v] : ( cons_v @ V @ ( sCC_Bl8828226123343373779t_unit @ E2 ) )
@ ( sCC_Bl7870604408699998558t_unit
@ ^ [Uu: set_v] : ( insert_v @ V @ ( sCC_Bl4645233313691564917t_unit @ E2 ) )
@ E2 ) ) ) ) ) )
=> ( accp_S2303753412255344476t_unit @ ( sCC_Bl907557413677168252_rel_v @ successors ) @ ( sum_In526841707622398774t_unit @ ( produc3862955338007567901t_unit @ V @ E2 ) ) ) ) ).
% dfs_dfss.domintros(1)
thf(fact_760_dfss_Ocases,axiom,
! [X2: produc5741669702376414499t_unit] :
~ ! [V2: v,E9: sCC_Bl1394983891496994913t_unit] :
( X2
!= ( produc3862955338007567901t_unit @ V2 @ E9 ) ) ).
% dfss.cases
thf(fact_761_dfss_Opsimps,axiom,
! [V: v,E2: sCC_Bl1394983891496994913t_unit] :
( ( accp_S2303753412255344476t_unit @ ( sCC_Bl907557413677168252_rel_v @ successors ) @ ( sum_In5289330923152326972t_unit @ ( produc3862955338007567901t_unit @ V @ E2 ) ) )
=> ( ( sCC_Bloemen_dfss_v @ successors @ V @ E2 )
= ( if_SCC4926449794303880475t_unit
@ ( ( minus_minus_set_v @ ( successors @ V ) @ ( sCC_Bl3795065053823578884t_unit @ E2 @ V ) )
= bot_bot_set_v )
@ E2
@ ( sCC_Bloemen_dfss_v @ successors @ V
@ ( sCC_Bl48393358579903213t_unit
@ ^ [Uu: v > set_v,X3: v] :
( if_set_v @ ( X3 = V )
@ ( sup_sup_set_v
@ ( sCC_Bl3795065053823578884t_unit
@ ( if_SCC4926449794303880475t_unit
@ ( member_v
@ ( fChoice_v
@ ^ [Y3: v] : ( member_v @ Y3 @ ( minus_minus_set_v @ ( successors @ V ) @ ( sCC_Bl3795065053823578884t_unit @ E2 @ V ) ) ) )
@ ( sCC_Bl157864678168468314t_unit @ E2 ) )
@ E2
@ ( if_SCC4926449794303880475t_unit
@ ~ ( member_v
@ ( fChoice_v
@ ^ [Y3: v] : ( member_v @ Y3 @ ( minus_minus_set_v @ ( successors @ V ) @ ( sCC_Bl3795065053823578884t_unit @ E2 @ V ) ) ) )
@ ( sCC_Bl4645233313691564917t_unit @ E2 ) )
@ ( sCC_Bloemen_dfs_v @ successors
@ ( fChoice_v
@ ^ [Y3: v] : ( member_v @ Y3 @ ( minus_minus_set_v @ ( successors @ V ) @ ( sCC_Bl3795065053823578884t_unit @ E2 @ V ) ) ) )
@ E2 )
@ ( sCC_Bloemen_unite_v @ V
@ ( fChoice_v
@ ^ [Y3: v] : ( member_v @ Y3 @ ( minus_minus_set_v @ ( successors @ V ) @ ( sCC_Bl3795065053823578884t_unit @ E2 @ V ) ) ) )
@ E2 ) ) )
@ V )
@ ( insert_v
@ ( fChoice_v
@ ^ [Y3: v] : ( member_v @ Y3 @ ( minus_minus_set_v @ ( successors @ V ) @ ( sCC_Bl3795065053823578884t_unit @ E2 @ V ) ) ) )
@ bot_bot_set_v ) )
@ ( sCC_Bl3795065053823578884t_unit
@ ( if_SCC4926449794303880475t_unit
@ ( member_v
@ ( fChoice_v
@ ^ [Y3: v] : ( member_v @ Y3 @ ( minus_minus_set_v @ ( successors @ V ) @ ( sCC_Bl3795065053823578884t_unit @ E2 @ V ) ) ) )
@ ( sCC_Bl157864678168468314t_unit @ E2 ) )
@ E2
@ ( if_SCC4926449794303880475t_unit
@ ~ ( member_v
@ ( fChoice_v
@ ^ [Y3: v] : ( member_v @ Y3 @ ( minus_minus_set_v @ ( successors @ V ) @ ( sCC_Bl3795065053823578884t_unit @ E2 @ V ) ) ) )
@ ( sCC_Bl4645233313691564917t_unit @ E2 ) )
@ ( sCC_Bloemen_dfs_v @ successors
@ ( fChoice_v
@ ^ [Y3: v] : ( member_v @ Y3 @ ( minus_minus_set_v @ ( successors @ V ) @ ( sCC_Bl3795065053823578884t_unit @ E2 @ V ) ) ) )
@ E2 )
@ ( sCC_Bloemen_unite_v @ V
@ ( fChoice_v
@ ^ [Y3: v] : ( member_v @ Y3 @ ( minus_minus_set_v @ ( successors @ V ) @ ( sCC_Bl3795065053823578884t_unit @ E2 @ V ) ) ) )
@ E2 ) ) )
@ X3 ) )
@ ( if_SCC4926449794303880475t_unit
@ ( member_v
@ ( fChoice_v
@ ^ [X3: v] : ( member_v @ X3 @ ( minus_minus_set_v @ ( successors @ V ) @ ( sCC_Bl3795065053823578884t_unit @ E2 @ V ) ) ) )
@ ( sCC_Bl157864678168468314t_unit @ E2 ) )
@ E2
@ ( if_SCC4926449794303880475t_unit
@ ~ ( member_v
@ ( fChoice_v
@ ^ [X3: v] : ( member_v @ X3 @ ( minus_minus_set_v @ ( successors @ V ) @ ( sCC_Bl3795065053823578884t_unit @ E2 @ V ) ) ) )
@ ( sCC_Bl4645233313691564917t_unit @ E2 ) )
@ ( sCC_Bloemen_dfs_v @ successors
@ ( fChoice_v
@ ^ [X3: v] : ( member_v @ X3 @ ( minus_minus_set_v @ ( successors @ V ) @ ( sCC_Bl3795065053823578884t_unit @ E2 @ V ) ) ) )
@ E2 )
@ ( sCC_Bloemen_unite_v @ V
@ ( fChoice_v
@ ^ [X3: v] : ( member_v @ X3 @ ( minus_minus_set_v @ ( successors @ V ) @ ( sCC_Bl3795065053823578884t_unit @ E2 @ V ) ) ) )
@ E2 ) ) ) ) ) ) ) ) ).
% dfss.psimps
thf(fact_762_unite__def,axiom,
( sCC_Bloemen_unite_v
= ( ^ [V3: v,W2: v,E6: sCC_Bl1394983891496994913t_unit] :
( sCC_Bl349061681862590396t_unit
@ ^ [Uu: list_v] :
( dropWhile_v
@ ^ [X3: v] :
~ ( member_v @ W2 @ ( sCC_Bl1280885523602775798t_unit @ E6 @ X3 ) )
@ ( sCC_Bl8828226123343373779t_unit @ E6 ) )
@ ( sCC_Bl3155122997657187039t_unit
@ ^ [Uu: v > set_v,X3: v] :
( if_set_v
@ ( member_v @ X3
@ ( comple2307003700295860064_set_v
@ ( collect_set_v
@ ^ [Uv2: set_v] :
? [Y3: v] :
( ( Uv2
= ( sCC_Bl1280885523602775798t_unit @ E6 @ Y3 ) )
& ( member_v @ Y3
@ ( sup_sup_set_v
@ ( set_v2
@ ( takeWhile_v
@ ^ [Z2: v] :
~ ( member_v @ W2 @ ( sCC_Bl1280885523602775798t_unit @ E6 @ Z2 ) )
@ ( sCC_Bl8828226123343373779t_unit @ E6 ) ) )
@ ( insert_v
@ ( hd_v
@ ( dropWhile_v
@ ^ [Z2: v] :
~ ( member_v @ W2 @ ( sCC_Bl1280885523602775798t_unit @ E6 @ Z2 ) )
@ ( sCC_Bl8828226123343373779t_unit @ E6 ) ) )
@ bot_bot_set_v ) ) ) ) ) ) )
@ ( comple2307003700295860064_set_v
@ ( collect_set_v
@ ^ [Uv2: set_v] :
? [Y3: v] :
( ( Uv2
= ( sCC_Bl1280885523602775798t_unit @ E6 @ Y3 ) )
& ( member_v @ Y3
@ ( sup_sup_set_v
@ ( set_v2
@ ( takeWhile_v
@ ^ [Z2: v] :
~ ( member_v @ W2 @ ( sCC_Bl1280885523602775798t_unit @ E6 @ Z2 ) )
@ ( sCC_Bl8828226123343373779t_unit @ E6 ) ) )
@ ( insert_v
@ ( hd_v
@ ( dropWhile_v
@ ^ [Z2: v] :
~ ( member_v @ W2 @ ( sCC_Bl1280885523602775798t_unit @ E6 @ Z2 ) )
@ ( sCC_Bl8828226123343373779t_unit @ E6 ) ) )
@ bot_bot_set_v ) ) ) ) ) )
@ ( sCC_Bl1280885523602775798t_unit @ E6 @ X3 ) )
@ E6 ) ) ) ) ).
% unite_def
% Helper facts (6)
thf(help_fChoice_1_1_fChoice_001tf__v_T,axiom,
! [P: v > $o] :
( ( P @ ( fChoice_v @ P ) )
= ( ? [X7: v] : ( P @ X7 ) ) ) ).
thf(help_If_2_1_If_001t__Set__Oset_Itf__v_J_T,axiom,
! [X2: set_v,Y: set_v] :
( ( if_set_v @ $false @ X2 @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Set__Oset_Itf__v_J_T,axiom,
! [X2: set_v,Y: set_v] :
( ( if_set_v @ $true @ X2 @ Y )
= X2 ) ).
thf(help_If_3_1_If_001t__SCC____Bloemen____Sequential__Oenv__Oenv____ext_Itf__v_Mt__Product____Type__Ounit_J_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__SCC____Bloemen____Sequential__Oenv__Oenv____ext_Itf__v_Mt__Product____Type__Ounit_J_T,axiom,
! [X2: sCC_Bl1394983891496994913t_unit,Y: sCC_Bl1394983891496994913t_unit] :
( ( if_SCC4926449794303880475t_unit @ $false @ X2 @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__SCC____Bloemen____Sequential__Oenv__Oenv____ext_Itf__v_Mt__Product____Type__Ounit_J_T,axiom,
! [X2: sCC_Bl1394983891496994913t_unit,Y: sCC_Bl1394983891496994913t_unit] :
( ( if_SCC4926449794303880475t_unit @ $true @ X2 @ Y )
= X2 ) ).
% Conjectures (1)
thf(conj_0,conjecture,
sCC_Bl9196236973127232072t_unit @ successors @ e3 ).
%------------------------------------------------------------------------------