TPTP Problem File: SLH0858^1.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_01421_048403__5970772_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1210 ( 603 unt; 134 typ; 0 def)
% Number of atoms : 2949 (1230 equ; 0 cnn)
% Maximal formula atoms : 10 ( 2 avg)
% Number of connectives : 9603 ( 315 ~; 49 |; 303 &;7879 @)
% ( 0 <=>;1057 =>; 0 <=; 0 <~>)
% Maximal formula depth : 21 ( 6 avg)
% Number of types : 13 ( 12 usr)
% Number of type conns : 441 ( 441 >; 0 *; 0 +; 0 <<)
% Number of symbols : 125 ( 122 usr; 19 con; 0-9 aty)
% Number of variables : 3087 ( 272 ^;2668 !; 147 ?;3087 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 08:53:30.938
%------------------------------------------------------------------------------
% Could-be-implicit typings (12)
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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_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_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_001tf__v,type,
the_elem_v: set_v > 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__Set__Oset_It__Product____Type__Oprod_Itf__v_Mtf__v_J_J,type,
member8406446414694345712od_v_v: set_Product_prod_v_v > set_se8455005133513928103od_v_v > $o ).
thf(sy_c_member_001t__Set__Oset_Itf__v_J,type,
member_set_v: set_v > set_set_v > $o ).
thf(sy_c_member_001tf__v,type,
member_v: v > set_v > $o ).
thf(sy_v_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_l____,type,
l: v ).
thf(sy_v_m____,type,
m: v ).
thf(sy_v_n____,type,
n: v ).
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 (1072)
thf(fact_0__092_060open_062m_A_092_060preceq_062_Al_Ain_Acstack_Ae_092_060close_062,axiom,
sCC_Bl4022239298816431255edes_v @ m @ l @ ( sCC_Bl9201514103433284750t_unit @ e ) ).
% \<open>m \<preceq> l in cstack e\<close>
thf(fact_1__092_060open_062l_A_092_060preceq_062_An_Ain_Acstack_Ae_092_060close_062,axiom,
sCC_Bl4022239298816431255edes_v @ l @ n @ ( sCC_Bl9201514103433284750t_unit @ e ) ).
% \<open>l \<preceq> n in cstack e\<close>
thf(fact_2_sub__env__trans,axiom,
! [E: sCC_Bl1394983891496994913t_unit,E2: sCC_Bl1394983891496994913t_unit,E3: sCC_Bl1394983891496994913t_unit] :
( ( sCC_Bl5768913643336123637t_unit @ E @ E2 )
=> ( ( sCC_Bl5768913643336123637t_unit @ E2 @ E3 )
=> ( sCC_Bl5768913643336123637t_unit @ E @ E3 ) ) ) ).
% sub_env_trans
thf(fact_3__092_060open_062m_A_092_060in_062_A_092_060S_062_Ae_Al_092_060close_062,axiom,
member_v @ m @ ( sCC_Bl1280885523602775798t_unit @ e @ l ) ).
% \<open>m \<in> \<S> e l\<close>
thf(fact_4__092_060open_062m_A_092_060in_062_Acc_092_060close_062,axiom,
member_v @ m @ cc ).
% \<open>m \<in> cc\<close>
thf(fact_5__092_060open_062m_A_092_060in_062_Aset_A_Icstack_Ae_J_092_060close_062,axiom,
member_v @ m @ ( set_v2 @ ( sCC_Bl9201514103433284750t_unit @ e ) ) ).
% \<open>m \<in> set (cstack e)\<close>
thf(fact_6__092_060open_062l_A_092_060preceq_062_An_Ain_Astack_Ae_092_060close_062,axiom,
sCC_Bl4022239298816431255edes_v @ l @ n @ ( sCC_Bl8828226123343373779t_unit @ e ) ).
% \<open>l \<preceq> n in stack e\<close>
thf(fact_7_w_I4_J,axiom,
~ ( member_v @ w @ ( sCC_Bl157864678168468314t_unit @ e ) ) ).
% w(4)
thf(fact_8_w_I3_J,axiom,
member_v @ w @ ( sCC_Bl4645233313691564917t_unit @ e ) ).
% w(3)
thf(fact_9__092_060open_062m_A_092_060in_062_A_092_060S_062_Ae_H_An_092_060close_062,axiom,
member_v @ m @ ( sCC_Bl1280885523602775798t_unit @ e2 @ n ) ).
% \<open>m \<in> \<S> e' n\<close>
thf(fact_10_fold__congs_I8_J,axiom,
! [R: sCC_Bl1394983891496994913t_unit,R2: sCC_Bl1394983891496994913t_unit,V: list_v,F: list_v > list_v,F2: list_v > list_v] :
( ( R = R2 )
=> ( ( ( sCC_Bl9201514103433284750t_unit @ R2 )
= V )
=> ( ! [V2: list_v] :
( ( V = V2 )
=> ( ( F @ V2 )
= ( F2 @ V2 ) ) )
=> ( ( sCC_Bl7876664385711583351t_unit @ F @ R )
= ( sCC_Bl7876664385711583351t_unit @ F2 @ R2 ) ) ) ) ) ).
% fold_congs(8)
thf(fact_11_unfold__congs_I8_J,axiom,
! [R: sCC_Bl1394983891496994913t_unit,R2: sCC_Bl1394983891496994913t_unit,V: list_v,F: list_v > list_v,F2: list_v > list_v] :
( ( R = R2 )
=> ( ( ( sCC_Bl9201514103433284750t_unit @ R2 )
= V )
=> ( ! [V2: list_v] :
( ( V2 = V )
=> ( ( F @ V2 )
= ( F2 @ V2 ) ) )
=> ( ( sCC_Bl7876664385711583351t_unit @ F @ R )
= ( sCC_Bl7876664385711583351t_unit @ F2 @ R2 ) ) ) ) ) ).
% unfold_congs(8)
thf(fact_12__092_060open_062m_A_092_060in_062_Aset_A_Icstack_Ae_H_J_092_060close_062,axiom,
member_v @ m @ ( set_v2 @ ( sCC_Bl9201514103433284750t_unit @ e2 ) ) ).
% \<open>m \<in> set (cstack e')\<close>
thf(fact_13_w_I2_J,axiom,
~ ( member_v @ w @ ( sCC_Bl3795065053823578884t_unit @ e @ v2 ) ) ).
% w(2)
thf(fact_14_precedes__refl,axiom,
! [X: product_prod_v_v,Xs: list_P7986770385144383213od_v_v] :
( ( sCC_Bl2026170059108282219od_v_v @ X @ X @ Xs )
= ( member7453568604450474000od_v_v @ X @ ( set_Product_prod_v_v2 @ Xs ) ) ) ).
% precedes_refl
thf(fact_15_precedes__refl,axiom,
! [X: v,Xs: list_v] :
( ( sCC_Bl4022239298816431255edes_v @ X @ X @ Xs )
= ( member_v @ X @ ( set_v2 @ Xs ) ) ) ).
% precedes_refl
thf(fact_16__092_060open_062_092_060forall_062n_Am_O_A_Im_A_092_060in_062_A_092_060S_062_Ae_H_An_J_A_061_A_I_092_060S_062_Ae_H_An_A_061_A_092_060S_062_Ae_H_Am_J_092_060close_062,axiom,
! [N: v,M: v] :
( ( member_v @ M @ ( sCC_Bl1280885523602775798t_unit @ e2 @ N ) )
= ( ( sCC_Bl1280885523602775798t_unit @ e2 @ N )
= ( sCC_Bl1280885523602775798t_unit @ e2 @ M ) ) ) ).
% \<open>\<forall>n m. (m \<in> \<S> e' n) = (\<S> e' n = \<S> e' m)\<close>
thf(fact_17__092_060open_062n_A_092_060in_062_Aset_A_Istack_Ae_H_J_092_060close_062,axiom,
member_v @ n @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ e2 ) ) ).
% \<open>n \<in> set (stack e')\<close>
thf(fact_18__092_060open_062_092_060forall_062n_Am_O_An_A_092_060preceq_062_Am_Ain_Astack_Ae_H_A_092_060longrightarrow_062_An_A_092_060preceq_062_Am_Ain_Acstack_Ae_H_092_060close_062,axiom,
! [N: v,M: v] :
( ( sCC_Bl4022239298816431255edes_v @ N @ M @ ( sCC_Bl8828226123343373779t_unit @ e2 ) )
=> ( sCC_Bl4022239298816431255edes_v @ N @ M @ ( sCC_Bl9201514103433284750t_unit @ e2 ) ) ) ).
% \<open>\<forall>n m. n \<preceq> m in stack e' \<longrightarrow> n \<preceq> m in cstack e'\<close>
thf(fact_19__092_060open_062l_A_092_060in_062_Aset_A_Istack_Ae_J_092_060close_062,axiom,
member_v @ l @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ e ) ) ).
% \<open>l \<in> set (stack e)\<close>
thf(fact_20_Se_H,axiom,
! [X2: v] :
( ( ( member_v @ X2 @ cc )
=> ( ( sCC_Bl1280885523602775798t_unit @ e2 @ X2 )
= cc ) )
& ( ~ ( member_v @ X2 @ cc )
=> ( ( sCC_Bl1280885523602775798t_unit @ e2 @ X2 )
= ( sCC_Bl1280885523602775798t_unit @ e @ X2 ) ) ) ) ).
% Se'
thf(fact_21_pfx_I2_J,axiom,
( ( sCC_Bl8828226123343373779t_unit @ e2 )
!= nil_v ) ).
% pfx(2)
thf(fact_22_e_H__def,axiom,
( e2
= ( sCC_Bloemen_unite_v @ v2 @ w @ e ) ) ).
% e'_def
thf(fact_23_True,axiom,
( n
= ( hd_v @ ( sCC_Bl8828226123343373779t_unit @ e2 ) ) ) ).
% True
thf(fact_24_precedes__mem_I1_J,axiom,
! [X: product_prod_v_v,Y: product_prod_v_v,Xs: list_P7986770385144383213od_v_v] :
( ( sCC_Bl2026170059108282219od_v_v @ X @ Y @ Xs )
=> ( member7453568604450474000od_v_v @ X @ ( set_Product_prod_v_v2 @ Xs ) ) ) ).
% precedes_mem(1)
thf(fact_25_precedes__mem_I1_J,axiom,
! [X: v,Y: v,Xs: list_v] :
( ( sCC_Bl4022239298816431255edes_v @ X @ Y @ Xs )
=> ( member_v @ X @ ( set_v2 @ Xs ) ) ) ).
% precedes_mem(1)
thf(fact_26_precedes__mem_I2_J,axiom,
! [X: product_prod_v_v,Y: product_prod_v_v,Xs: list_P7986770385144383213od_v_v] :
( ( sCC_Bl2026170059108282219od_v_v @ X @ Y @ Xs )
=> ( member7453568604450474000od_v_v @ Y @ ( set_Product_prod_v_v2 @ Xs ) ) ) ).
% precedes_mem(2)
thf(fact_27_precedes__mem_I2_J,axiom,
! [X: v,Y: v,Xs: list_v] :
( ( sCC_Bl4022239298816431255edes_v @ X @ Y @ Xs )
=> ( member_v @ Y @ ( set_v2 @ Xs ) ) ) ).
% precedes_mem(2)
thf(fact_28__092_060open_062set_A_Icstack_Ae_H_J_A_092_060subseteq_062_Avisited_Ae_H_092_060close_062,axiom,
ord_less_eq_set_v @ ( set_v2 @ ( sCC_Bl9201514103433284750t_unit @ e2 ) ) @ ( sCC_Bl4645233313691564917t_unit @ e2 ) ).
% \<open>set (cstack e') \<subseteq> visited e'\<close>
thf(fact_29__092_060open_062_092_060forall_062n_O_An_A_092_060notin_062_Avisited_Ae_H_A_092_060longrightarrow_062_Avsuccs_Ae_H_An_A_061_A_123_125_092_060close_062,axiom,
! [N: v] :
( ~ ( member_v @ N @ ( sCC_Bl4645233313691564917t_unit @ e2 ) )
=> ( ( sCC_Bl3795065053823578884t_unit @ e2 @ N )
= bot_bot_set_v ) ) ).
% \<open>\<forall>n. n \<notin> visited e' \<longrightarrow> vsuccs e' n = {}\<close>
thf(fact_30_hd__cc,axiom,
( ( sCC_Bl1280885523602775798t_unit @ e2 @ ( hd_v @ ( sCC_Bl8828226123343373779t_unit @ e2 ) ) )
= cc ) ).
% hd_cc
thf(fact_31__092_060open_062explored_Ae_H_A_092_060subseteq_062_Avisited_Ae_H_092_060close_062,axiom,
ord_less_eq_set_v @ ( sCC_Bl157864678168468314t_unit @ e2 ) @ ( sCC_Bl4645233313691564917t_unit @ e2 ) ).
% \<open>explored e' \<subseteq> visited e'\<close>
thf(fact_32__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_33_w_I1_J,axiom,
member_v @ w @ ( successors @ v2 ) ).
% w(1)
thf(fact_34_pfx_I1_J,axiom,
( ( sCC_Bl8828226123343373779t_unit @ e )
= ( append_v @ pfx @ ( sCC_Bl8828226123343373779t_unit @ e2 ) ) ) ).
% pfx(1)
thf(fact_35__092_060open_062distinct_A_Icstack_Ae_H_J_092_060close_062,axiom,
distinct_v @ ( sCC_Bl9201514103433284750t_unit @ e2 ) ).
% \<open>distinct (cstack e')\<close>
thf(fact_36__092_060open_062distinct_A_Istack_Ae_H_J_092_060close_062,axiom,
distinct_v @ ( sCC_Bl8828226123343373779t_unit @ e2 ) ).
% \<open>distinct (stack e')\<close>
thf(fact_37__092_060open_062_092_060And_062n_Am_O_A_092_060lbrakk_062n_A_092_060in_062_Aset_A_Itl_A_Istack_Ae_H_J_J_059_Am_A_092_060in_062_A_092_060S_062_Ae_An_A_092_060inter_062_Acc_092_060rbrakk_062_A_092_060Longrightarrow_062_AFalse_092_060close_062,axiom,
! [N2: v,M2: v] :
( ( member_v @ N2 @ ( set_v2 @ ( tl_v @ ( sCC_Bl8828226123343373779t_unit @ e2 ) ) ) )
=> ~ ( member_v @ M2 @ ( inf_inf_set_v @ ( sCC_Bl1280885523602775798t_unit @ e @ N2 ) @ cc ) ) ) ).
% \<open>\<And>n m. \<lbrakk>n \<in> set (tl (stack e')); m \<in> \<S> e n \<inter> cc\<rbrakk> \<Longrightarrow> False\<close>
thf(fact_38__092_060open_062_092_060forall_062n_092_060in_062explored_Ae_H_O_Avsuccs_Ae_H_An_A_061_Asuccessors_An_092_060close_062,axiom,
! [X2: v] :
( ( member_v @ X2 @ ( sCC_Bl157864678168468314t_unit @ e2 ) )
=> ( ( sCC_Bl3795065053823578884t_unit @ e2 @ X2 )
= ( successors @ X2 ) ) ) ).
% \<open>\<forall>n\<in>explored e'. vsuccs e' n = successors n\<close>
thf(fact_39_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_40_pre,axiom,
sCC_Bl1748261141445803503t_unit @ successors @ v2 @ e ).
% pre
thf(fact_41__092_060open_062_092_060forall_062n_092_060in_062set_A_Istack_Ae_H_J_O_A_092_060forall_062m_092_060in_062set_A_Istack_Ae_H_J_O_An_A_092_060noteq_062_Am_A_092_060longrightarrow_062_A_092_060S_062_Ae_H_An_A_092_060inter_062_A_092_060S_062_Ae_H_Am_A_061_A_123_125_092_060close_062,axiom,
! [X2: v] :
( ( member_v @ X2 @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ e2 ) ) )
=> ! [Xa: v] :
( ( member_v @ Xa @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ e2 ) ) )
=> ( ( X2 != Xa )
=> ( ( inf_inf_set_v @ ( sCC_Bl1280885523602775798t_unit @ e2 @ X2 ) @ ( sCC_Bl1280885523602775798t_unit @ e2 @ 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_42__092_060open_062_092_060forall_062n_O_Avsuccs_Ae_H_An_A_092_060subseteq_062_Asuccessors_An_A_092_060inter_062_Avisited_Ae_H_092_060close_062,axiom,
! [N: v] : ( ord_less_eq_set_v @ ( sCC_Bl3795065053823578884t_unit @ e2 @ N ) @ ( inf_inf_set_v @ ( successors @ N ) @ ( sCC_Bl4645233313691564917t_unit @ e2 ) ) ) ).
% \<open>\<forall>n. vsuccs e' n \<subseteq> successors n \<inter> visited e'\<close>
thf(fact_43_tl__cc,axiom,
! [X2: v] :
( ( member_v @ X2 @ ( set_v2 @ ( tl_v @ ( sCC_Bl8828226123343373779t_unit @ e2 ) ) ) )
=> ( ( inf_inf_set_v @ ( sCC_Bl1280885523602775798t_unit @ e @ X2 ) @ cc )
= bot_bot_set_v ) ) ).
% tl_cc
thf(fact_44_sclosed,axiom,
! [X2: v] :
( ( member_v @ X2 @ vertices )
=> ( ord_less_eq_set_v @ ( successors @ X2 ) @ vertices ) ) ).
% sclosed
thf(fact_45_unite__sub__env,axiom,
! [V3: v,E: sCC_Bl1394983891496994913t_unit,W: v] :
( ( sCC_Bl1748261141445803503t_unit @ successors @ V3 @ E )
=> ( ( member_v @ W @ ( successors @ V3 ) )
=> ( ~ ( member_v @ W @ ( sCC_Bl3795065053823578884t_unit @ E @ V3 ) )
=> ( ( member_v @ W @ ( sCC_Bl4645233313691564917t_unit @ E ) )
=> ( ~ ( member_v @ W @ ( sCC_Bl157864678168468314t_unit @ E ) )
=> ( sCC_Bl5768913643336123637t_unit @ E @ ( sCC_Bloemen_unite_v @ V3 @ W @ E ) ) ) ) ) ) ) ).
% unite_sub_env
thf(fact_46_mem__Collect__eq,axiom,
! [A: v,P: v > $o] :
( ( member_v @ A @ ( collect_v @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_47_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_48_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_49_Collect__mem__eq,axiom,
! [A2: set_v] :
( ( collect_v
@ ^ [X3: v] : ( member_v @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_50_Collect__mem__eq,axiom,
! [A2: set_Product_prod_v_v] :
( ( collec140062887454715474od_v_v
@ ^ [X3: product_prod_v_v] : ( member7453568604450474000od_v_v @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_51_Collect__mem__eq,axiom,
! [A2: set_set_v] :
( ( collect_set_v
@ ^ [X3: set_v] : ( member_set_v @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_52_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_53_precedes__trans,axiom,
! [X: v,Y: v,Xs: list_v,Z: v] :
( ( sCC_Bl4022239298816431255edes_v @ X @ Y @ Xs )
=> ( ( sCC_Bl4022239298816431255edes_v @ Y @ Z @ Xs )
=> ( ( distinct_v @ Xs )
=> ( sCC_Bl4022239298816431255edes_v @ X @ Z @ Xs ) ) ) ) ).
% precedes_trans
thf(fact_54_precedes__antisym,axiom,
! [X: v,Y: v,Xs: list_v] :
( ( sCC_Bl4022239298816431255edes_v @ X @ Y @ Xs )
=> ( ( sCC_Bl4022239298816431255edes_v @ Y @ X @ Xs )
=> ( ( distinct_v @ Xs )
=> ( X = Y ) ) ) ) ).
% precedes_antisym
thf(fact_55_precedes__append__left,axiom,
! [X: v,Y: v,Xs: list_v,Ys: list_v] :
( ( sCC_Bl4022239298816431255edes_v @ X @ Y @ Xs )
=> ( sCC_Bl4022239298816431255edes_v @ X @ Y @ ( append_v @ Ys @ Xs ) ) ) ).
% precedes_append_left
thf(fact_56_precedes__append__right,axiom,
! [X: v,Y: v,Xs: list_v,Ys: list_v] :
( ( sCC_Bl4022239298816431255edes_v @ X @ Y @ Xs )
=> ( sCC_Bl4022239298816431255edes_v @ X @ Y @ ( append_v @ Xs @ Ys ) ) ) ).
% precedes_append_right
thf(fact_57_precedes__append__left__iff,axiom,
! [X: product_prod_v_v,Ys: list_P7986770385144383213od_v_v,Y: product_prod_v_v,Xs: list_P7986770385144383213od_v_v] :
( ~ ( member7453568604450474000od_v_v @ X @ ( set_Product_prod_v_v2 @ Ys ) )
=> ( ( sCC_Bl2026170059108282219od_v_v @ X @ Y @ ( append2138873909117096322od_v_v @ Ys @ Xs ) )
= ( sCC_Bl2026170059108282219od_v_v @ X @ Y @ Xs ) ) ) ).
% precedes_append_left_iff
thf(fact_58_precedes__append__left__iff,axiom,
! [X: v,Ys: list_v,Y: v,Xs: list_v] :
( ~ ( member_v @ X @ ( set_v2 @ Ys ) )
=> ( ( sCC_Bl4022239298816431255edes_v @ X @ Y @ ( append_v @ Ys @ Xs ) )
= ( sCC_Bl4022239298816431255edes_v @ X @ Y @ Xs ) ) ) ).
% precedes_append_left_iff
thf(fact_59_precedes__append__right__iff,axiom,
! [Y: product_prod_v_v,Ys: list_P7986770385144383213od_v_v,X: product_prod_v_v,Xs: list_P7986770385144383213od_v_v] :
( ~ ( member7453568604450474000od_v_v @ Y @ ( set_Product_prod_v_v2 @ Ys ) )
=> ( ( sCC_Bl2026170059108282219od_v_v @ X @ Y @ ( append2138873909117096322od_v_v @ Xs @ Ys ) )
= ( sCC_Bl2026170059108282219od_v_v @ X @ Y @ Xs ) ) ) ).
% precedes_append_right_iff
thf(fact_60_precedes__append__right__iff,axiom,
! [Y: v,Ys: list_v,X: v,Xs: list_v] :
( ~ ( member_v @ Y @ ( set_v2 @ Ys ) )
=> ( ( sCC_Bl4022239298816431255edes_v @ X @ Y @ ( append_v @ Xs @ Ys ) )
= ( sCC_Bl4022239298816431255edes_v @ X @ Y @ Xs ) ) ) ).
% precedes_append_right_iff
thf(fact_61_dfs__S__tl__stack_I2_J,axiom,
! [V3: v,E: sCC_Bl1394983891496994913t_unit,E2: sCC_Bl1394983891496994913t_unit] :
( ( sCC_Bl8953792750115413617t_unit @ successors @ V3 @ E @ E2 )
=> ( ( ( sCC_Bl8828226123343373779t_unit @ E )
!= nil_v )
=> ! [X2: v] :
( ( member_v @ X2 @ ( set_v2 @ ( tl_v @ ( sCC_Bl8828226123343373779t_unit @ E2 ) ) ) )
=> ( ( sCC_Bl1280885523602775798t_unit @ E2 @ X2 )
= ( sCC_Bl1280885523602775798t_unit @ E @ X2 ) ) ) ) ) ).
% dfs_S_tl_stack(2)
thf(fact_62__092_060open_062Ball_A_Isccs_Ae_H_J_Ais__scc_092_060close_062,axiom,
! [X2: set_v] :
( ( member_set_v @ X2 @ ( sCC_Bl2536197123907397897t_unit @ e2 ) )
=> ( sCC_Bloemen_is_scc_v @ successors @ X2 ) ) ).
% \<open>Ball (sccs e') is_scc\<close>
thf(fact_63_scc__partition,axiom,
! [S: set_v,S2: set_v,X: v] :
( ( sCC_Bloemen_is_scc_v @ successors @ S )
=> ( ( sCC_Bloemen_is_scc_v @ successors @ S2 )
=> ( ( member_v @ X @ ( inf_inf_set_v @ S @ S2 ) )
=> ( S = S2 ) ) ) ) ).
% scc_partition
thf(fact_64_dfs__S__tl__stack_I1_J,axiom,
! [V3: v,E: sCC_Bl1394983891496994913t_unit,E2: sCC_Bl1394983891496994913t_unit] :
( ( sCC_Bl8953792750115413617t_unit @ successors @ V3 @ E @ E2 )
=> ( ( ( sCC_Bl8828226123343373779t_unit @ E )
!= nil_v )
=> ( ( sCC_Bl8828226123343373779t_unit @ E2 )
!= nil_v ) ) ) ).
% dfs_S_tl_stack(1)
thf(fact_65_distinct__append,axiom,
! [Xs: list_v,Ys: list_v] :
( ( distinct_v @ ( append_v @ Xs @ Ys ) )
= ( ( distinct_v @ Xs )
& ( distinct_v @ Ys )
& ( ( inf_inf_set_v @ ( set_v2 @ Xs ) @ ( set_v2 @ Ys ) )
= bot_bot_set_v ) ) ) ).
% distinct_append
thf(fact_66_distinct__append,axiom,
! [Xs: list_P7986770385144383213od_v_v,Ys: list_P7986770385144383213od_v_v] :
( ( distin6159370996967099744od_v_v @ ( append2138873909117096322od_v_v @ Xs @ Ys ) )
= ( ( distin6159370996967099744od_v_v @ Xs )
& ( distin6159370996967099744od_v_v @ Ys )
& ( ( inf_in6271465464967711157od_v_v @ ( set_Product_prod_v_v2 @ Xs ) @ ( set_Product_prod_v_v2 @ Ys ) )
= bot_bo723834152578015283od_v_v ) ) ) ).
% distinct_append
thf(fact_67_unite__S__tl,axiom,
! [E: sCC_Bl1394983891496994913t_unit,W: v,V3: v,N2: v] :
( ( sCC_Bl9196236973127232072t_unit @ successors @ E )
=> ( ( member_v @ W @ ( successors @ V3 ) )
=> ( ~ ( member_v @ W @ ( sCC_Bl3795065053823578884t_unit @ E @ V3 ) )
=> ( ( member_v @ W @ ( sCC_Bl4645233313691564917t_unit @ E ) )
=> ( ~ ( member_v @ W @ ( sCC_Bl157864678168468314t_unit @ E ) )
=> ( ( member_v @ N2 @ ( set_v2 @ ( tl_v @ ( sCC_Bl8828226123343373779t_unit @ ( sCC_Bloemen_unite_v @ V3 @ W @ E ) ) ) ) )
=> ( ( sCC_Bl1280885523602775798t_unit @ ( sCC_Bloemen_unite_v @ V3 @ W @ E ) @ N2 )
= ( sCC_Bl1280885523602775798t_unit @ E @ N2 ) ) ) ) ) ) ) ) ).
% unite_S_tl
thf(fact_68_visited__unexplored,axiom,
! [E: sCC_Bl1394983891496994913t_unit,M2: v] :
( ( sCC_Bl9196236973127232072t_unit @ successors @ E )
=> ( ( member_v @ M2 @ ( sCC_Bl4645233313691564917t_unit @ E ) )
=> ( ~ ( member_v @ M2 @ ( sCC_Bl157864678168468314t_unit @ E ) )
=> ~ ! [N3: v] :
( ( member_v @ N3 @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E ) ) )
=> ~ ( member_v @ M2 @ ( sCC_Bl1280885523602775798t_unit @ E @ N3 ) ) ) ) ) ) ).
% visited_unexplored
thf(fact_69_tl__append2,axiom,
! [Xs: list_v,Ys: list_v] :
( ( Xs != nil_v )
=> ( ( tl_v @ ( append_v @ Xs @ Ys ) )
= ( append_v @ ( tl_v @ Xs ) @ Ys ) ) ) ).
% tl_append2
thf(fact_70_hd__append2,axiom,
! [Xs: list_v,Ys: list_v] :
( ( Xs != nil_v )
=> ( ( hd_v @ ( append_v @ Xs @ Ys ) )
= ( hd_v @ Xs ) ) ) ).
% hd_append2
thf(fact_71_set__empty,axiom,
! [Xs: list_v] :
( ( ( set_v2 @ Xs )
= bot_bot_set_v )
= ( Xs = nil_v ) ) ).
% set_empty
thf(fact_72_set__empty,axiom,
! [Xs: list_P7986770385144383213od_v_v] :
( ( ( set_Product_prod_v_v2 @ Xs )
= bot_bo723834152578015283od_v_v )
= ( Xs = nil_Product_prod_v_v ) ) ).
% set_empty
thf(fact_73_set__empty2,axiom,
! [Xs: list_v] :
( ( bot_bot_set_v
= ( set_v2 @ Xs ) )
= ( Xs = nil_v ) ) ).
% set_empty2
thf(fact_74_set__empty2,axiom,
! [Xs: list_P7986770385144383213od_v_v] :
( ( bot_bo723834152578015283od_v_v
= ( set_Product_prod_v_v2 @ Xs ) )
= ( Xs = nil_Product_prod_v_v ) ) ).
% set_empty2
thf(fact_75_S__reflexive,axiom,
! [E: sCC_Bl1394983891496994913t_unit,N2: v] :
( ( sCC_Bl9196236973127232072t_unit @ successors @ E )
=> ( member_v @ N2 @ ( sCC_Bl1280885523602775798t_unit @ E @ N2 ) ) ) ).
% S_reflexive
thf(fact_76_append_Oassoc,axiom,
! [A: list_v,B: list_v,C: list_v] :
( ( append_v @ ( append_v @ A @ B ) @ C )
= ( append_v @ A @ ( append_v @ B @ C ) ) ) ).
% append.assoc
thf(fact_77_append__assoc,axiom,
! [Xs: list_v,Ys: list_v,Zs: list_v] :
( ( append_v @ ( append_v @ Xs @ Ys ) @ Zs )
= ( append_v @ Xs @ ( append_v @ Ys @ Zs ) ) ) ).
% append_assoc
thf(fact_78_append__same__eq,axiom,
! [Ys: list_v,Xs: list_v,Zs: list_v] :
( ( ( append_v @ Ys @ Xs )
= ( append_v @ Zs @ Xs ) )
= ( Ys = Zs ) ) ).
% append_same_eq
thf(fact_79_same__append__eq,axiom,
! [Xs: list_v,Ys: list_v,Zs: list_v] :
( ( ( append_v @ Xs @ Ys )
= ( append_v @ Xs @ Zs ) )
= ( Ys = Zs ) ) ).
% same_append_eq
thf(fact_80_local_Owf,axiom,
sCC_Bl9196236973127232072t_unit @ successors @ e ).
% local.wf
thf(fact_81_stack__visited,axiom,
! [E: sCC_Bl1394983891496994913t_unit,N2: v] :
( ( sCC_Bl9196236973127232072t_unit @ successors @ E )
=> ( ( member_v @ N2 @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E ) ) )
=> ( member_v @ N2 @ ( sCC_Bl4645233313691564917t_unit @ E ) ) ) ) ).
% stack_visited
thf(fact_82_stack__unexplored,axiom,
! [E: sCC_Bl1394983891496994913t_unit,N2: v] :
( ( sCC_Bl9196236973127232072t_unit @ successors @ E )
=> ( ( member_v @ N2 @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E ) ) )
=> ~ ( member_v @ N2 @ ( sCC_Bl157864678168468314t_unit @ E ) ) ) ) ).
% stack_unexplored
thf(fact_83_append_Oright__neutral,axiom,
! [A: list_v] :
( ( append_v @ A @ nil_v )
= A ) ).
% append.right_neutral
thf(fact_84_append__Nil2,axiom,
! [Xs: list_v] :
( ( append_v @ Xs @ nil_v )
= Xs ) ).
% append_Nil2
thf(fact_85_append__self__conv,axiom,
! [Xs: list_v,Ys: list_v] :
( ( ( append_v @ Xs @ Ys )
= Xs )
= ( Ys = nil_v ) ) ).
% append_self_conv
thf(fact_86_self__append__conv,axiom,
! [Y: list_v,Ys: list_v] :
( ( Y
= ( append_v @ Y @ Ys ) )
= ( Ys = nil_v ) ) ).
% self_append_conv
thf(fact_87_append__self__conv2,axiom,
! [Xs: list_v,Ys: list_v] :
( ( ( append_v @ Xs @ Ys )
= Ys )
= ( Xs = nil_v ) ) ).
% append_self_conv2
thf(fact_88_self__append__conv2,axiom,
! [Y: list_v,Xs: list_v] :
( ( Y
= ( append_v @ Xs @ Y ) )
= ( Xs = nil_v ) ) ).
% self_append_conv2
thf(fact_89_Nil__is__append__conv,axiom,
! [Xs: list_v,Ys: list_v] :
( ( nil_v
= ( append_v @ Xs @ Ys ) )
= ( ( Xs = nil_v )
& ( Ys = nil_v ) ) ) ).
% Nil_is_append_conv
thf(fact_90_append__is__Nil__conv,axiom,
! [Xs: list_v,Ys: list_v] :
( ( ( append_v @ Xs @ Ys )
= nil_v )
= ( ( Xs = nil_v )
& ( Ys = nil_v ) ) ) ).
% append_is_Nil_conv
thf(fact_91_dfs__S__hd__stack_I1_J,axiom,
! [E: sCC_Bl1394983891496994913t_unit,V3: v,E2: sCC_Bl1394983891496994913t_unit,N2: v] :
( ( sCC_Bl9196236973127232072t_unit @ successors @ E )
=> ( ( sCC_Bl8953792750115413617t_unit @ successors @ V3 @ E @ E2 )
=> ( ( ( sCC_Bl8828226123343373779t_unit @ E )
!= nil_v )
=> ( ( member_v @ N2 @ ( sCC_Bl1280885523602775798t_unit @ E @ ( hd_v @ ( sCC_Bl8828226123343373779t_unit @ E ) ) ) )
=> ( ( sCC_Bl8828226123343373779t_unit @ E2 )
!= nil_v ) ) ) ) ) ).
% dfs_S_hd_stack(1)
thf(fact_92_dfs__S__hd__stack_I2_J,axiom,
! [E: sCC_Bl1394983891496994913t_unit,V3: v,E2: sCC_Bl1394983891496994913t_unit,N2: v] :
( ( sCC_Bl9196236973127232072t_unit @ successors @ E )
=> ( ( sCC_Bl8953792750115413617t_unit @ successors @ V3 @ E @ E2 )
=> ( ( ( sCC_Bl8828226123343373779t_unit @ E )
!= nil_v )
=> ( ( member_v @ N2 @ ( sCC_Bl1280885523602775798t_unit @ E @ ( hd_v @ ( sCC_Bl8828226123343373779t_unit @ E ) ) ) )
=> ( member_v @ N2 @ ( sCC_Bl1280885523602775798t_unit @ E2 @ ( hd_v @ ( sCC_Bl8828226123343373779t_unit @ E2 ) ) ) ) ) ) ) ) ).
% dfs_S_hd_stack(2)
thf(fact_93_graph__axioms,axiom,
sCC_Bloemen_graph_v @ vertices @ successors ).
% graph_axioms
thf(fact_94_graph_Owf__env_Ocong,axiom,
sCC_Bl9196236973127232072t_unit = sCC_Bl9196236973127232072t_unit ).
% graph.wf_env.cong
thf(fact_95_graph_Ois__scc_Ocong,axiom,
sCC_Bloemen_is_scc_v = sCC_Bloemen_is_scc_v ).
% graph.is_scc.cong
thf(fact_96_graph_Opre__dfss_Ocong,axiom,
sCC_Bl1748261141445803503t_unit = sCC_Bl1748261141445803503t_unit ).
% graph.pre_dfss.cong
thf(fact_97_append__eq__appendI,axiom,
! [Xs: list_v,Xs1: list_v,Zs: list_v,Ys: list_v,Us: list_v] :
( ( ( append_v @ Xs @ Xs1 )
= Zs )
=> ( ( Ys
= ( append_v @ Xs1 @ Us ) )
=> ( ( append_v @ Xs @ Ys )
= ( append_v @ Zs @ Us ) ) ) ) ).
% append_eq_appendI
thf(fact_98_append__eq__append__conv2,axiom,
! [Xs: list_v,Ys: list_v,Zs: list_v,Ts: list_v] :
( ( ( append_v @ Xs @ Ys )
= ( append_v @ Zs @ Ts ) )
= ( ? [Us2: list_v] :
( ( ( Xs
= ( append_v @ Zs @ Us2 ) )
& ( ( append_v @ Us2 @ Ys )
= Ts ) )
| ( ( ( append_v @ Xs @ Us2 )
= Zs )
& ( Ys
= ( append_v @ Us2 @ Ts ) ) ) ) ) ) ).
% append_eq_append_conv2
thf(fact_99_subset__code_I1_J,axiom,
! [Xs: list_v,B2: set_v] :
( ( ord_less_eq_set_v @ ( set_v2 @ Xs ) @ B2 )
= ( ! [X3: v] :
( ( member_v @ X3 @ ( set_v2 @ Xs ) )
=> ( member_v @ X3 @ B2 ) ) ) ) ).
% subset_code(1)
thf(fact_100_subset__code_I1_J,axiom,
! [Xs: list_P7986770385144383213od_v_v,B2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ ( set_Product_prod_v_v2 @ Xs ) @ B2 )
= ( ! [X3: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X3 @ ( set_Product_prod_v_v2 @ Xs ) )
=> ( member7453568604450474000od_v_v @ X3 @ B2 ) ) ) ) ).
% subset_code(1)
thf(fact_101_append__Nil,axiom,
! [Ys: list_v] :
( ( append_v @ nil_v @ Ys )
= Ys ) ).
% append_Nil
thf(fact_102_append_Oleft__neutral,axiom,
! [A: list_v] :
( ( append_v @ nil_v @ A )
= A ) ).
% append.left_neutral
thf(fact_103_eq__Nil__appendI,axiom,
! [Xs: list_v,Ys: list_v] :
( ( Xs = Ys )
=> ( Xs
= ( append_v @ nil_v @ Ys ) ) ) ).
% eq_Nil_appendI
thf(fact_104_distinct_Osimps_I1_J,axiom,
distinct_v @ nil_v ).
% distinct.simps(1)
thf(fact_105_list_Osel_I2_J,axiom,
( ( tl_v @ nil_v )
= nil_v ) ).
% list.sel(2)
thf(fact_106_distinct__tl,axiom,
! [Xs: list_v] :
( ( distinct_v @ Xs )
=> ( distinct_v @ ( tl_v @ Xs ) ) ) ).
% distinct_tl
thf(fact_107_empty__set,axiom,
( bot_bot_set_v
= ( set_v2 @ nil_v ) ) ).
% empty_set
thf(fact_108_empty__set,axiom,
( bot_bo723834152578015283od_v_v
= ( set_Product_prod_v_v2 @ nil_Product_prod_v_v ) ) ).
% empty_set
thf(fact_109_hd__in__set,axiom,
! [Xs: list_P7986770385144383213od_v_v] :
( ( Xs != nil_Product_prod_v_v )
=> ( member7453568604450474000od_v_v @ ( hd_Product_prod_v_v @ Xs ) @ ( set_Product_prod_v_v2 @ Xs ) ) ) ).
% hd_in_set
thf(fact_110_hd__in__set,axiom,
! [Xs: list_v] :
( ( Xs != nil_v )
=> ( member_v @ ( hd_v @ Xs ) @ ( set_v2 @ Xs ) ) ) ).
% hd_in_set
thf(fact_111_list_Oset__sel_I1_J,axiom,
! [A: list_P7986770385144383213od_v_v] :
( ( A != nil_Product_prod_v_v )
=> ( member7453568604450474000od_v_v @ ( hd_Product_prod_v_v @ A ) @ ( set_Product_prod_v_v2 @ A ) ) ) ).
% list.set_sel(1)
thf(fact_112_list_Oset__sel_I1_J,axiom,
! [A: list_v] :
( ( A != nil_v )
=> ( member_v @ ( hd_v @ A ) @ ( set_v2 @ A ) ) ) ).
% list.set_sel(1)
thf(fact_113_list_Oset__sel_I2_J,axiom,
! [A: list_P7986770385144383213od_v_v,X: product_prod_v_v] :
( ( A != nil_Product_prod_v_v )
=> ( ( member7453568604450474000od_v_v @ X @ ( set_Product_prod_v_v2 @ ( tl_Product_prod_v_v @ A ) ) )
=> ( member7453568604450474000od_v_v @ X @ ( set_Product_prod_v_v2 @ A ) ) ) ) ).
% list.set_sel(2)
thf(fact_114_list_Oset__sel_I2_J,axiom,
! [A: list_v,X: v] :
( ( A != nil_v )
=> ( ( member_v @ X @ ( set_v2 @ ( tl_v @ A ) ) )
=> ( member_v @ X @ ( set_v2 @ A ) ) ) ) ).
% list.set_sel(2)
thf(fact_115_hd__append,axiom,
! [Xs: list_v,Ys: list_v] :
( ( ( Xs = nil_v )
=> ( ( hd_v @ ( append_v @ Xs @ Ys ) )
= ( hd_v @ Ys ) ) )
& ( ( Xs != nil_v )
=> ( ( hd_v @ ( append_v @ Xs @ Ys ) )
= ( hd_v @ Xs ) ) ) ) ).
% hd_append
thf(fact_116_longest__common__prefix,axiom,
! [Xs: list_v,Ys: list_v] :
? [Ps: list_v,Xs2: list_v,Ys2: list_v] :
( ( Xs
= ( append_v @ Ps @ Xs2 ) )
& ( Ys
= ( append_v @ Ps @ Ys2 ) )
& ( ( Xs2 = nil_v )
| ( Ys2 = nil_v )
| ( ( hd_v @ Xs2 )
!= ( hd_v @ Ys2 ) ) ) ) ).
% longest_common_prefix
thf(fact_117_tl__append__if,axiom,
! [Xs: list_v,Ys: list_v] :
( ( ( Xs = nil_v )
=> ( ( tl_v @ ( append_v @ Xs @ Ys ) )
= ( tl_v @ Ys ) ) )
& ( ( Xs != nil_v )
=> ( ( tl_v @ ( append_v @ Xs @ Ys ) )
= ( append_v @ ( tl_v @ Xs ) @ Ys ) ) ) ) ).
% tl_append_if
thf(fact_118_list_Oexpand,axiom,
! [List: list_v,List2: list_v] :
( ( ( List = nil_v )
= ( List2 = nil_v ) )
=> ( ( ( List != nil_v )
=> ( ( List2 != nil_v )
=> ( ( ( hd_v @ List )
= ( hd_v @ List2 ) )
& ( ( tl_v @ List )
= ( tl_v @ List2 ) ) ) ) )
=> ( List = List2 ) ) ) ).
% list.expand
thf(fact_119_unite__subscc,axiom,
! [V3: v,E: sCC_Bl1394983891496994913t_unit,W: v] :
( ( sCC_Bl1748261141445803503t_unit @ successors @ V3 @ E )
=> ( ( member_v @ W @ ( successors @ V3 ) )
=> ( ~ ( member_v @ W @ ( sCC_Bl3795065053823578884t_unit @ E @ V3 ) )
=> ( ( member_v @ W @ ( sCC_Bl4645233313691564917t_unit @ E ) )
=> ( ~ ( member_v @ W @ ( sCC_Bl157864678168468314t_unit @ E ) )
=> ( sCC_Bl5398416737448265317bscc_v @ successors @ ( sCC_Bl1280885523602775798t_unit @ ( sCC_Bloemen_unite_v @ V3 @ W @ E ) @ ( hd_v @ ( sCC_Bl8828226123343373779t_unit @ ( sCC_Bloemen_unite_v @ V3 @ W @ E ) ) ) ) ) ) ) ) ) ) ).
% unite_subscc
thf(fact_120_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_121_post__dfs__def,axiom,
! [V3: v,E: sCC_Bl1394983891496994913t_unit,E2: sCC_Bl1394983891496994913t_unit] :
( ( sCC_Bl8953792750115413617t_unit @ successors @ V3 @ E @ E2 )
= ( ( sCC_Bl9196236973127232072t_unit @ successors @ E2 )
& ( member_v @ V3 @ ( sCC_Bl4645233313691564917t_unit @ E2 ) )
& ( sCC_Bl5768913643336123637t_unit @ E @ E2 )
& ( ( sCC_Bl3795065053823578884t_unit @ E2 @ V3 )
= ( successors @ V3 ) )
& ! [X3: v] :
( ( member_v @ X3 @ ( sCC_Bl4645233313691564917t_unit @ E ) )
=> ( ( sCC_Bl3795065053823578884t_unit @ E2 @ X3 )
= ( sCC_Bl3795065053823578884t_unit @ E @ X3 ) ) )
& ! [X3: v] :
( ( member_v @ X3 @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E2 ) ) )
=> ( sCC_Bl649662514949026229able_v @ successors @ X3 @ V3 ) )
& ? [Ns: list_v] :
( ( sCC_Bl8828226123343373779t_unit @ E )
= ( append_v @ Ns @ ( sCC_Bl8828226123343373779t_unit @ E2 ) ) )
& ( ( ( member_v @ V3 @ ( sCC_Bl157864678168468314t_unit @ E2 ) )
& ( ( sCC_Bl8828226123343373779t_unit @ E2 )
= ( sCC_Bl8828226123343373779t_unit @ E ) )
& ! [X3: v] :
( ( member_v @ X3 @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E2 ) ) )
=> ( ( sCC_Bl1280885523602775798t_unit @ E2 @ X3 )
= ( sCC_Bl1280885523602775798t_unit @ E @ X3 ) ) ) )
| ( ( ( sCC_Bl8828226123343373779t_unit @ E2 )
!= nil_v )
& ( member_v @ V3 @ ( sCC_Bl1280885523602775798t_unit @ E2 @ ( hd_v @ ( sCC_Bl8828226123343373779t_unit @ E2 ) ) ) )
& ! [X3: v] :
( ( member_v @ X3 @ ( set_v2 @ ( tl_v @ ( sCC_Bl8828226123343373779t_unit @ E2 ) ) ) )
=> ( ( sCC_Bl1280885523602775798t_unit @ E2 @ X3 )
= ( sCC_Bl1280885523602775798t_unit @ E @ X3 ) ) ) ) )
& ( ( sCC_Bl9201514103433284750t_unit @ E2 )
= ( sCC_Bl9201514103433284750t_unit @ E ) ) ) ) ).
% post_dfs_def
thf(fact_122_stack__class,axiom,
! [E: sCC_Bl1394983891496994913t_unit,N2: v,M2: v] :
( ( sCC_Bl9196236973127232072t_unit @ successors @ E )
=> ( ( member_v @ N2 @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E ) ) )
=> ( ( member_v @ M2 @ ( sCC_Bl1280885523602775798t_unit @ E @ N2 ) )
=> ( member_v @ M2 @ ( minus_minus_set_v @ ( sCC_Bl4645233313691564917t_unit @ E ) @ ( sCC_Bl157864678168468314t_unit @ E ) ) ) ) ) ) ).
% stack_class
thf(fact_123__092_060open_062_092_060forall_062n_092_060in_062visited_Ae_H_A_N_Aset_A_Icstack_Ae_H_J_O_Avsuccs_Ae_H_An_A_061_Asuccessors_An_092_060close_062,axiom,
! [X2: v] :
( ( member_v @ X2 @ ( minus_minus_set_v @ ( sCC_Bl4645233313691564917t_unit @ e2 ) @ ( set_v2 @ ( sCC_Bl9201514103433284750t_unit @ e2 ) ) ) )
=> ( ( sCC_Bl3795065053823578884t_unit @ e2 @ X2 )
= ( successors @ X2 ) ) ) ).
% \<open>\<forall>n\<in>visited e' - set (cstack e'). vsuccs e' n = successors n\<close>
thf(fact_124_Int__subset__iff,axiom,
! [C2: set_v,A2: set_v,B2: set_v] :
( ( ord_less_eq_set_v @ C2 @ ( inf_inf_set_v @ A2 @ B2 ) )
= ( ( ord_less_eq_set_v @ C2 @ A2 )
& ( ord_less_eq_set_v @ C2 @ B2 ) ) ) ).
% Int_subset_iff
thf(fact_125_Int__subset__iff,axiom,
! [C2: set_Product_prod_v_v,A2: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ C2 @ ( inf_in6271465464967711157od_v_v @ A2 @ B2 ) )
= ( ( ord_le7336532860387713383od_v_v @ C2 @ A2 )
& ( ord_le7336532860387713383od_v_v @ C2 @ B2 ) ) ) ).
% Int_subset_iff
thf(fact_126_inf__bot__left,axiom,
! [X: set_v] :
( ( inf_inf_set_v @ bot_bot_set_v @ X )
= bot_bot_set_v ) ).
% inf_bot_left
thf(fact_127_inf__bot__left,axiom,
! [X: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ bot_bo723834152578015283od_v_v @ X )
= bot_bo723834152578015283od_v_v ) ).
% inf_bot_left
thf(fact_128_inf__bot__right,axiom,
! [X: set_v] :
( ( inf_inf_set_v @ X @ bot_bot_set_v )
= bot_bot_set_v ) ).
% inf_bot_right
thf(fact_129_inf__bot__right,axiom,
! [X: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ X @ bot_bo723834152578015283od_v_v )
= bot_bo723834152578015283od_v_v ) ).
% inf_bot_right
thf(fact_130_boolean__algebra_Oconj__zero__left,axiom,
! [X: set_v] :
( ( inf_inf_set_v @ bot_bot_set_v @ X )
= bot_bot_set_v ) ).
% boolean_algebra.conj_zero_left
thf(fact_131_boolean__algebra_Oconj__zero__left,axiom,
! [X: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ bot_bo723834152578015283od_v_v @ X )
= bot_bo723834152578015283od_v_v ) ).
% boolean_algebra.conj_zero_left
thf(fact_132_boolean__algebra_Oconj__zero__right,axiom,
! [X: set_v] :
( ( inf_inf_set_v @ X @ bot_bot_set_v )
= bot_bot_set_v ) ).
% boolean_algebra.conj_zero_right
thf(fact_133_boolean__algebra_Oconj__zero__right,axiom,
! [X: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ X @ bot_bo723834152578015283od_v_v )
= bot_bo723834152578015283od_v_v ) ).
% boolean_algebra.conj_zero_right
thf(fact_134_succ__reachable,axiom,
! [X: v,Y: v,Z: v] :
( ( sCC_Bl649662514949026229able_v @ successors @ X @ Y )
=> ( ( member_v @ Z @ ( successors @ Y ) )
=> ( sCC_Bl649662514949026229able_v @ successors @ X @ Z ) ) ) ).
% succ_reachable
thf(fact_135_reachable__trans,axiom,
! [X: v,Y: v,Z: v] :
( ( sCC_Bl649662514949026229able_v @ successors @ X @ Y )
=> ( ( sCC_Bl649662514949026229able_v @ successors @ Y @ Z )
=> ( sCC_Bl649662514949026229able_v @ successors @ X @ Z ) ) ) ).
% reachable_trans
thf(fact_136_reachable__end__induct,axiom,
! [X: v,Y: v,P: v > v > $o] :
( ( sCC_Bl649662514949026229able_v @ successors @ X @ Y )
=> ( ! [X4: v] : ( P @ X4 @ X4 )
=> ( ! [X4: v,Y2: v,Z2: v] :
( ( P @ X4 @ Y2 )
=> ( ( member_v @ Z2 @ ( successors @ Y2 ) )
=> ( P @ X4 @ Z2 ) ) )
=> ( P @ X @ Y ) ) ) ) ).
% reachable_end_induct
thf(fact_137_reachable__edge,axiom,
! [Y: v,X: v] :
( ( member_v @ Y @ ( successors @ X ) )
=> ( sCC_Bl649662514949026229able_v @ successors @ X @ Y ) ) ).
% reachable_edge
thf(fact_138_reachable_Osimps,axiom,
! [A1: v,A22: v] :
( ( sCC_Bl649662514949026229able_v @ successors @ A1 @ A22 )
= ( ? [X3: v] :
( ( A1 = X3 )
& ( A22 = X3 ) )
| ? [X3: v,Y3: v,Z3: v] :
( ( A1 = X3 )
& ( A22 = Z3 )
& ( member_v @ Y3 @ ( successors @ X3 ) )
& ( sCC_Bl649662514949026229able_v @ successors @ Y3 @ Z3 ) ) ) ) ).
% reachable.simps
thf(fact_139_reachable__succ,axiom,
! [Y: v,X: v,Z: v] :
( ( member_v @ Y @ ( successors @ X ) )
=> ( ( sCC_Bl649662514949026229able_v @ successors @ Y @ Z )
=> ( sCC_Bl649662514949026229able_v @ successors @ X @ Z ) ) ) ).
% reachable_succ
thf(fact_140_reachable__refl,axiom,
! [X: v] : ( sCC_Bl649662514949026229able_v @ successors @ X @ X ) ).
% reachable_refl
thf(fact_141_reachable_Ocases,axiom,
! [A1: v,A22: v] :
( ( sCC_Bl649662514949026229able_v @ successors @ A1 @ A22 )
=> ( ( A22 != A1 )
=> ~ ! [Y2: v] :
( ( member_v @ Y2 @ ( successors @ A1 ) )
=> ~ ( sCC_Bl649662514949026229able_v @ successors @ Y2 @ A22 ) ) ) ) ).
% reachable.cases
thf(fact_142_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_143_sccE,axiom,
! [S: set_v,X: v,Y: v] :
( ( sCC_Bloemen_is_scc_v @ successors @ S )
=> ( ( member_v @ X @ S )
=> ( ( sCC_Bl649662514949026229able_v @ successors @ X @ Y )
=> ( ( sCC_Bl649662514949026229able_v @ successors @ Y @ X )
=> ( member_v @ Y @ S ) ) ) ) ) ).
% sccE
thf(fact_144_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_145_empty__Collect__eq,axiom,
! [P: v > $o] :
( ( bot_bot_set_v
= ( collect_v @ P ) )
= ( ! [X3: v] :
~ ( P @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_146_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_147_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_148_Collect__empty__eq,axiom,
! [P: v > $o] :
( ( ( collect_v @ P )
= bot_bot_set_v )
= ( ! [X3: v] :
~ ( P @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_149_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_150_all__not__in__conv,axiom,
! [A2: set_v] :
( ( ! [X3: v] :
~ ( member_v @ X3 @ A2 ) )
= ( A2 = bot_bot_set_v ) ) ).
% all_not_in_conv
thf(fact_151_all__not__in__conv,axiom,
! [A2: set_Product_prod_v_v] :
( ( ! [X3: product_prod_v_v] :
~ ( member7453568604450474000od_v_v @ X3 @ A2 ) )
= ( A2 = bot_bo723834152578015283od_v_v ) ) ).
% all_not_in_conv
thf(fact_152_empty__iff,axiom,
! [C: v] :
~ ( member_v @ C @ bot_bot_set_v ) ).
% empty_iff
thf(fact_153_empty__iff,axiom,
! [C: product_prod_v_v] :
~ ( member7453568604450474000od_v_v @ C @ bot_bo723834152578015283od_v_v ) ).
% empty_iff
thf(fact_154_subset__antisym,axiom,
! [A2: set_v,B2: set_v] :
( ( ord_less_eq_set_v @ A2 @ B2 )
=> ( ( ord_less_eq_set_v @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% subset_antisym
thf(fact_155_subset__antisym,axiom,
! [A2: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A2 @ B2 )
=> ( ( ord_le7336532860387713383od_v_v @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% subset_antisym
thf(fact_156_subsetI,axiom,
! [A2: set_v,B2: set_v] :
( ! [X4: v] :
( ( member_v @ X4 @ A2 )
=> ( member_v @ X4 @ B2 ) )
=> ( ord_less_eq_set_v @ A2 @ B2 ) ) ).
% subsetI
thf(fact_157_subsetI,axiom,
! [A2: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ! [X4: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X4 @ A2 )
=> ( member7453568604450474000od_v_v @ X4 @ B2 ) )
=> ( ord_le7336532860387713383od_v_v @ A2 @ B2 ) ) ).
% subsetI
thf(fact_158_inf__right__idem,axiom,
! [X: set_v,Y: set_v] :
( ( inf_inf_set_v @ ( inf_inf_set_v @ X @ Y ) @ Y )
= ( inf_inf_set_v @ X @ Y ) ) ).
% inf_right_idem
thf(fact_159_inf_Oright__idem,axiom,
! [A: set_v,B: set_v] :
( ( inf_inf_set_v @ ( inf_inf_set_v @ A @ B ) @ B )
= ( inf_inf_set_v @ A @ B ) ) ).
% inf.right_idem
thf(fact_160_inf__left__idem,axiom,
! [X: set_v,Y: set_v] :
( ( inf_inf_set_v @ X @ ( inf_inf_set_v @ X @ Y ) )
= ( inf_inf_set_v @ X @ Y ) ) ).
% inf_left_idem
thf(fact_161_inf_Oleft__idem,axiom,
! [A: set_v,B: set_v] :
( ( inf_inf_set_v @ A @ ( inf_inf_set_v @ A @ B ) )
= ( inf_inf_set_v @ A @ B ) ) ).
% inf.left_idem
thf(fact_162_inf__idem,axiom,
! [X: set_v] :
( ( inf_inf_set_v @ X @ X )
= X ) ).
% inf_idem
thf(fact_163_inf_Oidem,axiom,
! [A: set_v] :
( ( inf_inf_set_v @ A @ A )
= A ) ).
% inf.idem
thf(fact_164_Int__iff,axiom,
! [C: product_prod_v_v,A2: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ C @ ( inf_in6271465464967711157od_v_v @ A2 @ B2 ) )
= ( ( member7453568604450474000od_v_v @ C @ A2 )
& ( member7453568604450474000od_v_v @ C @ B2 ) ) ) ).
% Int_iff
thf(fact_165_Int__iff,axiom,
! [C: v,A2: set_v,B2: set_v] :
( ( member_v @ C @ ( inf_inf_set_v @ A2 @ B2 ) )
= ( ( member_v @ C @ A2 )
& ( member_v @ C @ B2 ) ) ) ).
% Int_iff
thf(fact_166_IntI,axiom,
! [C: product_prod_v_v,A2: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ C @ A2 )
=> ( ( member7453568604450474000od_v_v @ C @ B2 )
=> ( member7453568604450474000od_v_v @ C @ ( inf_in6271465464967711157od_v_v @ A2 @ B2 ) ) ) ) ).
% IntI
thf(fact_167_IntI,axiom,
! [C: v,A2: set_v,B2: set_v] :
( ( member_v @ C @ A2 )
=> ( ( member_v @ C @ B2 )
=> ( member_v @ C @ ( inf_inf_set_v @ A2 @ B2 ) ) ) ) ).
% IntI
thf(fact_168_Diff__idemp,axiom,
! [A2: set_v,B2: set_v] :
( ( minus_minus_set_v @ ( minus_minus_set_v @ A2 @ B2 ) @ B2 )
= ( minus_minus_set_v @ A2 @ B2 ) ) ).
% Diff_idemp
thf(fact_169_Diff__iff,axiom,
! [C: product_prod_v_v,A2: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ C @ ( minus_4183494784930505774od_v_v @ A2 @ B2 ) )
= ( ( member7453568604450474000od_v_v @ C @ A2 )
& ~ ( member7453568604450474000od_v_v @ C @ B2 ) ) ) ).
% Diff_iff
thf(fact_170_Diff__iff,axiom,
! [C: v,A2: set_v,B2: set_v] :
( ( member_v @ C @ ( minus_minus_set_v @ A2 @ B2 ) )
= ( ( member_v @ C @ A2 )
& ~ ( member_v @ C @ B2 ) ) ) ).
% Diff_iff
thf(fact_171_DiffI,axiom,
! [C: product_prod_v_v,A2: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ C @ A2 )
=> ( ~ ( member7453568604450474000od_v_v @ C @ B2 )
=> ( member7453568604450474000od_v_v @ C @ ( minus_4183494784930505774od_v_v @ A2 @ B2 ) ) ) ) ).
% DiffI
thf(fact_172_DiffI,axiom,
! [C: v,A2: set_v,B2: set_v] :
( ( member_v @ C @ A2 )
=> ( ~ ( member_v @ C @ B2 )
=> ( member_v @ C @ ( minus_minus_set_v @ A2 @ B2 ) ) ) ) ).
% DiffI
thf(fact_173__092_060open_062_092_060forall_062n_092_060in_062visited_Ae_H_O_Areachable_A_Iroot_Ae_H_J_An_092_060close_062,axiom,
! [X2: v] :
( ( member_v @ X2 @ ( sCC_Bl4645233313691564917t_unit @ e2 ) )
=> ( sCC_Bl649662514949026229able_v @ successors @ ( sCC_Bl1090238580953940555t_unit @ e2 ) @ X2 ) ) ).
% \<open>\<forall>n\<in>visited e'. reachable (root e') n\<close>
thf(fact_174__092_060open_062_092_060forall_062n_092_060in_062explored_Ae_H_O_A_092_060forall_062m_O_Areachable_An_Am_A_092_060longrightarrow_062_Am_A_092_060in_062_Aexplored_Ae_H_092_060close_062,axiom,
! [X2: v] :
( ( member_v @ X2 @ ( sCC_Bl157864678168468314t_unit @ e2 ) )
=> ! [M: v] :
( ( sCC_Bl649662514949026229able_v @ successors @ X2 @ M )
=> ( member_v @ M @ ( sCC_Bl157864678168468314t_unit @ e2 ) ) ) ) ).
% \<open>\<forall>n\<in>explored e'. \<forall>m. reachable n m \<longrightarrow> m \<in> explored e'\<close>
thf(fact_175__092_060open_062_092_060forall_062n_Am_O_An_A_092_060preceq_062_Am_Ain_Astack_Ae_H_A_092_060longrightarrow_062_Areachable_Am_An_092_060close_062,axiom,
! [N: v,M: v] :
( ( sCC_Bl4022239298816431255edes_v @ N @ M @ ( sCC_Bl8828226123343373779t_unit @ e2 ) )
=> ( sCC_Bl649662514949026229able_v @ successors @ M @ N ) ) ).
% \<open>\<forall>n m. n \<preceq> m in stack e' \<longrightarrow> reachable m n\<close>
thf(fact_176_reachable__visited,axiom,
! [E: sCC_Bl1394983891496994913t_unit,V3: v,W: v] :
( ( sCC_Bl9196236973127232072t_unit @ successors @ E )
=> ( ( member_v @ V3 @ ( sCC_Bl4645233313691564917t_unit @ E ) )
=> ( ( sCC_Bl649662514949026229able_v @ successors @ V3 @ W )
=> ( ! [X4: v] :
( ( member_v @ X4 @ ( sCC_Bl4645233313691564917t_unit @ E ) )
=> ! [Xa2: v] :
( ( member_v @ Xa2 @ ( minus_minus_set_v @ ( successors @ X4 ) @ ( sCC_Bl3795065053823578884t_unit @ E @ X4 ) ) )
=> ( ( sCC_Bl649662514949026229able_v @ successors @ V3 @ X4 )
=> ~ ( sCC_Bl649662514949026229able_v @ successors @ Xa2 @ W ) ) ) )
=> ( member_v @ W @ ( sCC_Bl4645233313691564917t_unit @ E ) ) ) ) ) ) ).
% reachable_visited
thf(fact_177_inf_Obounded__iff,axiom,
! [A: set_v,B: set_v,C: set_v] :
( ( ord_less_eq_set_v @ A @ ( inf_inf_set_v @ B @ C ) )
= ( ( ord_less_eq_set_v @ A @ B )
& ( ord_less_eq_set_v @ A @ C ) ) ) ).
% inf.bounded_iff
thf(fact_178_inf_Obounded__iff,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ ( inf_in6271465464967711157od_v_v @ B @ C ) )
= ( ( ord_le7336532860387713383od_v_v @ A @ B )
& ( ord_le7336532860387713383od_v_v @ A @ C ) ) ) ).
% inf.bounded_iff
thf(fact_179_le__inf__iff,axiom,
! [X: set_v,Y: set_v,Z: set_v] :
( ( ord_less_eq_set_v @ X @ ( inf_inf_set_v @ Y @ Z ) )
= ( ( ord_less_eq_set_v @ X @ Y )
& ( ord_less_eq_set_v @ X @ Z ) ) ) ).
% le_inf_iff
thf(fact_180_le__inf__iff,axiom,
! [X: set_Product_prod_v_v,Y: set_Product_prod_v_v,Z: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X @ ( inf_in6271465464967711157od_v_v @ Y @ Z ) )
= ( ( ord_le7336532860387713383od_v_v @ X @ Y )
& ( ord_le7336532860387713383od_v_v @ X @ Z ) ) ) ).
% le_inf_iff
thf(fact_181_empty__subsetI,axiom,
! [A2: set_v] : ( ord_less_eq_set_v @ bot_bot_set_v @ A2 ) ).
% empty_subsetI
thf(fact_182_empty__subsetI,axiom,
! [A2: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ bot_bo723834152578015283od_v_v @ A2 ) ).
% empty_subsetI
thf(fact_183_subset__empty,axiom,
! [A2: set_v] :
( ( ord_less_eq_set_v @ A2 @ bot_bot_set_v )
= ( A2 = bot_bot_set_v ) ) ).
% subset_empty
thf(fact_184_subset__empty,axiom,
! [A2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A2 @ bot_bo723834152578015283od_v_v )
= ( A2 = bot_bo723834152578015283od_v_v ) ) ).
% subset_empty
thf(fact_185_Diff__cancel,axiom,
! [A2: set_Product_prod_v_v] :
( ( minus_4183494784930505774od_v_v @ A2 @ A2 )
= bot_bo723834152578015283od_v_v ) ).
% Diff_cancel
thf(fact_186_Diff__cancel,axiom,
! [A2: set_v] :
( ( minus_minus_set_v @ A2 @ A2 )
= bot_bot_set_v ) ).
% Diff_cancel
thf(fact_187_empty__Diff,axiom,
! [A2: set_Product_prod_v_v] :
( ( minus_4183494784930505774od_v_v @ bot_bo723834152578015283od_v_v @ A2 )
= bot_bo723834152578015283od_v_v ) ).
% empty_Diff
thf(fact_188_empty__Diff,axiom,
! [A2: set_v] :
( ( minus_minus_set_v @ bot_bot_set_v @ A2 )
= bot_bot_set_v ) ).
% empty_Diff
thf(fact_189_Diff__empty,axiom,
! [A2: set_Product_prod_v_v] :
( ( minus_4183494784930505774od_v_v @ A2 @ bot_bo723834152578015283od_v_v )
= A2 ) ).
% Diff_empty
thf(fact_190_Diff__empty,axiom,
! [A2: set_v] :
( ( minus_minus_set_v @ A2 @ bot_bot_set_v )
= A2 ) ).
% Diff_empty
thf(fact_191_Diff__eq__empty__iff,axiom,
! [A2: set_v,B2: set_v] :
( ( ( minus_minus_set_v @ A2 @ B2 )
= bot_bot_set_v )
= ( ord_less_eq_set_v @ A2 @ B2 ) ) ).
% Diff_eq_empty_iff
thf(fact_192_Diff__eq__empty__iff,axiom,
! [A2: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( ( minus_4183494784930505774od_v_v @ A2 @ B2 )
= bot_bo723834152578015283od_v_v )
= ( ord_le7336532860387713383od_v_v @ A2 @ B2 ) ) ).
% Diff_eq_empty_iff
thf(fact_193_Diff__disjoint,axiom,
! [A2: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ A2 @ ( minus_4183494784930505774od_v_v @ B2 @ A2 ) )
= bot_bo723834152578015283od_v_v ) ).
% Diff_disjoint
thf(fact_194_Diff__disjoint,axiom,
! [A2: set_v,B2: set_v] :
( ( inf_inf_set_v @ A2 @ ( minus_minus_set_v @ B2 @ A2 ) )
= bot_bot_set_v ) ).
% Diff_disjoint
thf(fact_195_pre__dfs__def,axiom,
! [V3: v,E: sCC_Bl1394983891496994913t_unit] :
( ( sCC_Bl36166008131615352t_unit @ successors @ V3 @ E )
= ( ( sCC_Bl9196236973127232072t_unit @ successors @ E )
& ~ ( member_v @ V3 @ ( sCC_Bl4645233313691564917t_unit @ E ) )
& ( sCC_Bl649662514949026229able_v @ successors @ ( sCC_Bl1090238580953940555t_unit @ E ) @ V3 )
& ( ( sCC_Bl3795065053823578884t_unit @ E @ V3 )
= bot_bot_set_v )
& ! [X3: v] :
( ( member_v @ X3 @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E ) ) )
=> ( sCC_Bl649662514949026229able_v @ successors @ X3 @ V3 ) ) ) ) ).
% pre_dfs_def
thf(fact_196_DiffD2,axiom,
! [C: product_prod_v_v,A2: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ C @ ( minus_4183494784930505774od_v_v @ A2 @ B2 ) )
=> ~ ( member7453568604450474000od_v_v @ C @ B2 ) ) ).
% DiffD2
thf(fact_197_DiffD2,axiom,
! [C: v,A2: set_v,B2: set_v] :
( ( member_v @ C @ ( minus_minus_set_v @ A2 @ B2 ) )
=> ~ ( member_v @ C @ B2 ) ) ).
% DiffD2
thf(fact_198_DiffD1,axiom,
! [C: product_prod_v_v,A2: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ C @ ( minus_4183494784930505774od_v_v @ A2 @ B2 ) )
=> ( member7453568604450474000od_v_v @ C @ A2 ) ) ).
% DiffD1
thf(fact_199_DiffD1,axiom,
! [C: v,A2: set_v,B2: set_v] :
( ( member_v @ C @ ( minus_minus_set_v @ A2 @ B2 ) )
=> ( member_v @ C @ A2 ) ) ).
% DiffD1
thf(fact_200_DiffE,axiom,
! [C: product_prod_v_v,A2: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ C @ ( minus_4183494784930505774od_v_v @ A2 @ B2 ) )
=> ~ ( ( member7453568604450474000od_v_v @ C @ A2 )
=> ( member7453568604450474000od_v_v @ C @ B2 ) ) ) ).
% DiffE
thf(fact_201_DiffE,axiom,
! [C: v,A2: set_v,B2: set_v] :
( ( member_v @ C @ ( minus_minus_set_v @ A2 @ B2 ) )
=> ~ ( ( member_v @ C @ A2 )
=> ( member_v @ C @ B2 ) ) ) ).
% DiffE
thf(fact_202_graph_Oreachable__succ,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,Y: product_prod_v_v,X: product_prod_v_v,Z: product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( member7453568604450474000od_v_v @ Y @ ( Successors @ X ) )
=> ( ( sCC_Bl4981926079593201289od_v_v @ Successors @ Y @ Z )
=> ( sCC_Bl4981926079593201289od_v_v @ Successors @ X @ Z ) ) ) ) ).
% graph.reachable_succ
thf(fact_203_graph_Oreachable__succ,axiom,
! [Vertices: set_v,Successors: v > set_v,Y: v,X: v,Z: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( member_v @ Y @ ( Successors @ X ) )
=> ( ( sCC_Bl649662514949026229able_v @ Successors @ Y @ Z )
=> ( sCC_Bl649662514949026229able_v @ Successors @ X @ Z ) ) ) ) ).
% graph.reachable_succ
thf(fact_204_graph_Oreachable__refl,axiom,
! [Vertices: set_v,Successors: v > set_v,X: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( sCC_Bl649662514949026229able_v @ Successors @ X @ X ) ) ).
% graph.reachable_refl
thf(fact_205_graph_Oreachable__end__induct,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,X: product_prod_v_v,Y: product_prod_v_v,P: product_prod_v_v > product_prod_v_v > $o] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( sCC_Bl4981926079593201289od_v_v @ Successors @ X @ Y )
=> ( ! [X4: product_prod_v_v] : ( P @ X4 @ X4 )
=> ( ! [X4: product_prod_v_v,Y2: product_prod_v_v,Z2: product_prod_v_v] :
( ( P @ X4 @ Y2 )
=> ( ( member7453568604450474000od_v_v @ Z2 @ ( Successors @ Y2 ) )
=> ( P @ X4 @ Z2 ) ) )
=> ( P @ X @ Y ) ) ) ) ) ).
% graph.reachable_end_induct
thf(fact_206_graph_Oreachable__end__induct,axiom,
! [Vertices: set_v,Successors: v > set_v,X: v,Y: v,P: v > v > $o] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl649662514949026229able_v @ Successors @ X @ Y )
=> ( ! [X4: v] : ( P @ X4 @ X4 )
=> ( ! [X4: v,Y2: v,Z2: v] :
( ( P @ X4 @ Y2 )
=> ( ( member_v @ Z2 @ ( Successors @ Y2 ) )
=> ( P @ X4 @ Z2 ) ) )
=> ( P @ X @ Y ) ) ) ) ) ).
% graph.reachable_end_induct
thf(fact_207_graph_Oreachable__trans,axiom,
! [Vertices: set_v,Successors: v > set_v,X: v,Y: v,Z: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl649662514949026229able_v @ Successors @ X @ Y )
=> ( ( sCC_Bl649662514949026229able_v @ Successors @ Y @ Z )
=> ( sCC_Bl649662514949026229able_v @ Successors @ X @ Z ) ) ) ) ).
% graph.reachable_trans
thf(fact_208_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,A22: product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( sCC_Bl4981926079593201289od_v_v @ Successors @ A1 @ A22 )
= ( ? [X3: product_prod_v_v] :
( ( A1 = X3 )
& ( A22 = X3 ) )
| ? [X3: product_prod_v_v,Y3: product_prod_v_v,Z3: product_prod_v_v] :
( ( A1 = X3 )
& ( A22 = Z3 )
& ( member7453568604450474000od_v_v @ Y3 @ ( Successors @ X3 ) )
& ( sCC_Bl4981926079593201289od_v_v @ Successors @ Y3 @ Z3 ) ) ) ) ) ).
% graph.reachable.simps
thf(fact_209_graph_Oreachable_Osimps,axiom,
! [Vertices: set_v,Successors: v > set_v,A1: v,A22: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl649662514949026229able_v @ Successors @ A1 @ A22 )
= ( ? [X3: v] :
( ( A1 = X3 )
& ( A22 = X3 ) )
| ? [X3: v,Y3: v,Z3: v] :
( ( A1 = X3 )
& ( A22 = Z3 )
& ( member_v @ Y3 @ ( Successors @ X3 ) )
& ( sCC_Bl649662514949026229able_v @ Successors @ Y3 @ Z3 ) ) ) ) ) ).
% graph.reachable.simps
thf(fact_210_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,A22: product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( sCC_Bl4981926079593201289od_v_v @ Successors @ A1 @ A22 )
=> ( ( A22 != A1 )
=> ~ ! [Y2: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ Y2 @ ( Successors @ A1 ) )
=> ~ ( sCC_Bl4981926079593201289od_v_v @ Successors @ Y2 @ A22 ) ) ) ) ) ).
% graph.reachable.cases
thf(fact_211_graph_Oreachable_Ocases,axiom,
! [Vertices: set_v,Successors: v > set_v,A1: v,A22: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl649662514949026229able_v @ Successors @ A1 @ A22 )
=> ( ( A22 != A1 )
=> ~ ! [Y2: v] :
( ( member_v @ Y2 @ ( Successors @ A1 ) )
=> ~ ( sCC_Bl649662514949026229able_v @ Successors @ Y2 @ A22 ) ) ) ) ) ).
% graph.reachable.cases
thf(fact_212_graph_Osucc__reachable,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,X: product_prod_v_v,Y: product_prod_v_v,Z: product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( sCC_Bl4981926079593201289od_v_v @ Successors @ X @ Y )
=> ( ( member7453568604450474000od_v_v @ Z @ ( Successors @ Y ) )
=> ( sCC_Bl4981926079593201289od_v_v @ Successors @ X @ Z ) ) ) ) ).
% graph.succ_reachable
thf(fact_213_graph_Osucc__reachable,axiom,
! [Vertices: set_v,Successors: v > set_v,X: v,Y: v,Z: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl649662514949026229able_v @ Successors @ X @ Y )
=> ( ( member_v @ Z @ ( Successors @ Y ) )
=> ( sCC_Bl649662514949026229able_v @ Successors @ X @ Z ) ) ) ) ).
% graph.succ_reachable
thf(fact_214_graph_Oreachable__edge,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,Y: product_prod_v_v,X: product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( member7453568604450474000od_v_v @ Y @ ( Successors @ X ) )
=> ( sCC_Bl4981926079593201289od_v_v @ Successors @ X @ Y ) ) ) ).
% graph.reachable_edge
thf(fact_215_graph_Oreachable__edge,axiom,
! [Vertices: set_v,Successors: v > set_v,Y: v,X: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( member_v @ Y @ ( Successors @ X ) )
=> ( sCC_Bl649662514949026229able_v @ Successors @ X @ Y ) ) ) ).
% graph.reachable_edge
thf(fact_216_graph_Oreachable_Ocong,axiom,
sCC_Bl649662514949026229able_v = sCC_Bl649662514949026229able_v ).
% graph.reachable.cong
thf(fact_217_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_218_graph_Ois__subscc_Ocong,axiom,
sCC_Bl5398416737448265317bscc_v = sCC_Bl5398416737448265317bscc_v ).
% graph.is_subscc.cong
thf(fact_219_bot__set__def,axiom,
( bot_bot_set_set_v
= ( collect_set_v @ bot_bot_set_v_o ) ) ).
% bot_set_def
thf(fact_220_bot__set__def,axiom,
( bot_bot_set_v
= ( collect_v @ bot_bot_v_o ) ) ).
% bot_set_def
thf(fact_221_bot__set__def,axiom,
( bot_bo723834152578015283od_v_v
= ( collec140062887454715474od_v_v @ bot_bo8461541820394803818_v_v_o ) ) ).
% bot_set_def
thf(fact_222_double__diff,axiom,
! [A2: set_v,B2: set_v,C2: set_v] :
( ( ord_less_eq_set_v @ A2 @ B2 )
=> ( ( ord_less_eq_set_v @ B2 @ C2 )
=> ( ( minus_minus_set_v @ B2 @ ( minus_minus_set_v @ C2 @ A2 ) )
= A2 ) ) ) ).
% double_diff
thf(fact_223_double__diff,axiom,
! [A2: set_Product_prod_v_v,B2: set_Product_prod_v_v,C2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A2 @ B2 )
=> ( ( ord_le7336532860387713383od_v_v @ B2 @ C2 )
=> ( ( minus_4183494784930505774od_v_v @ B2 @ ( minus_4183494784930505774od_v_v @ C2 @ A2 ) )
= A2 ) ) ) ).
% double_diff
thf(fact_224_Diff__subset,axiom,
! [A2: set_v,B2: set_v] : ( ord_less_eq_set_v @ ( minus_minus_set_v @ A2 @ B2 ) @ A2 ) ).
% Diff_subset
thf(fact_225_Diff__subset,axiom,
! [A2: set_Product_prod_v_v,B2: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ ( minus_4183494784930505774od_v_v @ A2 @ B2 ) @ A2 ) ).
% Diff_subset
thf(fact_226_Diff__mono,axiom,
! [A2: set_v,C2: set_v,D: set_v,B2: set_v] :
( ( ord_less_eq_set_v @ A2 @ C2 )
=> ( ( ord_less_eq_set_v @ D @ B2 )
=> ( ord_less_eq_set_v @ ( minus_minus_set_v @ A2 @ B2 ) @ ( minus_minus_set_v @ C2 @ D ) ) ) ) ).
% Diff_mono
thf(fact_227_Diff__mono,axiom,
! [A2: set_Product_prod_v_v,C2: set_Product_prod_v_v,D: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A2 @ C2 )
=> ( ( ord_le7336532860387713383od_v_v @ D @ B2 )
=> ( ord_le7336532860387713383od_v_v @ ( minus_4183494784930505774od_v_v @ A2 @ B2 ) @ ( minus_4183494784930505774od_v_v @ C2 @ D ) ) ) ) ).
% Diff_mono
thf(fact_228_Diff__Int__distrib2,axiom,
! [A2: set_v,B2: set_v,C2: set_v] :
( ( inf_inf_set_v @ ( minus_minus_set_v @ A2 @ B2 ) @ C2 )
= ( minus_minus_set_v @ ( inf_inf_set_v @ A2 @ C2 ) @ ( inf_inf_set_v @ B2 @ C2 ) ) ) ).
% Diff_Int_distrib2
thf(fact_229_Diff__Int__distrib,axiom,
! [C2: set_v,A2: set_v,B2: set_v] :
( ( inf_inf_set_v @ C2 @ ( minus_minus_set_v @ A2 @ B2 ) )
= ( minus_minus_set_v @ ( inf_inf_set_v @ C2 @ A2 ) @ ( inf_inf_set_v @ C2 @ B2 ) ) ) ).
% Diff_Int_distrib
thf(fact_230_Diff__Diff__Int,axiom,
! [A2: set_v,B2: set_v] :
( ( minus_minus_set_v @ A2 @ ( minus_minus_set_v @ A2 @ B2 ) )
= ( inf_inf_set_v @ A2 @ B2 ) ) ).
% Diff_Diff_Int
thf(fact_231_Diff__Int2,axiom,
! [A2: set_v,C2: set_v,B2: set_v] :
( ( minus_minus_set_v @ ( inf_inf_set_v @ A2 @ C2 ) @ ( inf_inf_set_v @ B2 @ C2 ) )
= ( minus_minus_set_v @ ( inf_inf_set_v @ A2 @ C2 ) @ B2 ) ) ).
% Diff_Int2
thf(fact_232_Int__Diff,axiom,
! [A2: set_v,B2: set_v,C2: set_v] :
( ( minus_minus_set_v @ ( inf_inf_set_v @ A2 @ B2 ) @ C2 )
= ( inf_inf_set_v @ A2 @ ( minus_minus_set_v @ B2 @ C2 ) ) ) ).
% Int_Diff
thf(fact_233_graph_OsccE,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,S: set_Product_prod_v_v,X: product_prod_v_v,Y: product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( sCC_Bl6242042402218619277od_v_v @ Successors @ S )
=> ( ( member7453568604450474000od_v_v @ X @ S )
=> ( ( sCC_Bl4981926079593201289od_v_v @ Successors @ X @ Y )
=> ( ( sCC_Bl4981926079593201289od_v_v @ Successors @ Y @ X )
=> ( member7453568604450474000od_v_v @ Y @ S ) ) ) ) ) ) ).
% graph.sccE
thf(fact_234_graph_OsccE,axiom,
! [Vertices: set_v,Successors: v > set_v,S: set_v,X: v,Y: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bloemen_is_scc_v @ Successors @ S )
=> ( ( member_v @ X @ S )
=> ( ( sCC_Bl649662514949026229able_v @ Successors @ X @ Y )
=> ( ( sCC_Bl649662514949026229able_v @ Successors @ Y @ X )
=> ( member_v @ Y @ S ) ) ) ) ) ) ).
% graph.sccE
thf(fact_235_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 )
=> ! [X2: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X2 @ Vertices )
=> ( ord_le7336532860387713383od_v_v @ ( Successors @ X2 ) @ Vertices ) ) ) ).
% graph.sclosed
thf(fact_236_graph_Osclosed,axiom,
! [Vertices: set_v,Successors: v > set_v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ! [X2: v] :
( ( member_v @ X2 @ Vertices )
=> ( ord_less_eq_set_v @ ( Successors @ X2 ) @ Vertices ) ) ) ).
% graph.sclosed
thf(fact_237_graph_Oreachable__visited,axiom,
! [Vertices: set_v,Successors: v > set_v,E: sCC_Bl1394983891496994913t_unit,V3: v,W: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl9196236973127232072t_unit @ Successors @ E )
=> ( ( member_v @ V3 @ ( sCC_Bl4645233313691564917t_unit @ E ) )
=> ( ( sCC_Bl649662514949026229able_v @ Successors @ V3 @ W )
=> ( ! [X4: v] :
( ( member_v @ X4 @ ( sCC_Bl4645233313691564917t_unit @ E ) )
=> ! [Xa2: v] :
( ( member_v @ Xa2 @ ( minus_minus_set_v @ ( Successors @ X4 ) @ ( sCC_Bl3795065053823578884t_unit @ E @ X4 ) ) )
=> ( ( sCC_Bl649662514949026229able_v @ Successors @ V3 @ X4 )
=> ~ ( sCC_Bl649662514949026229able_v @ Successors @ Xa2 @ W ) ) ) )
=> ( member_v @ W @ ( sCC_Bl4645233313691564917t_unit @ E ) ) ) ) ) ) ) ).
% graph.reachable_visited
thf(fact_238_diff__shunt__var,axiom,
! [X: set_v,Y: set_v] :
( ( ( minus_minus_set_v @ X @ Y )
= bot_bot_set_v )
= ( ord_less_eq_set_v @ X @ Y ) ) ).
% diff_shunt_var
thf(fact_239_diff__shunt__var,axiom,
! [X: set_Product_prod_v_v,Y: set_Product_prod_v_v] :
( ( ( minus_4183494784930505774od_v_v @ X @ Y )
= bot_bo723834152578015283od_v_v )
= ( ord_le7336532860387713383od_v_v @ X @ Y ) ) ).
% diff_shunt_var
thf(fact_240_Int__Diff__disjoint,axiom,
! [A2: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ ( inf_in6271465464967711157od_v_v @ A2 @ B2 ) @ ( minus_4183494784930505774od_v_v @ A2 @ B2 ) )
= bot_bo723834152578015283od_v_v ) ).
% Int_Diff_disjoint
thf(fact_241_Int__Diff__disjoint,axiom,
! [A2: set_v,B2: set_v] :
( ( inf_inf_set_v @ ( inf_inf_set_v @ A2 @ B2 ) @ ( minus_minus_set_v @ A2 @ B2 ) )
= bot_bot_set_v ) ).
% Int_Diff_disjoint
thf(fact_242_Diff__triv,axiom,
! [A2: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( ( inf_in6271465464967711157od_v_v @ A2 @ B2 )
= bot_bo723834152578015283od_v_v )
=> ( ( minus_4183494784930505774od_v_v @ A2 @ B2 )
= A2 ) ) ).
% Diff_triv
thf(fact_243_Diff__triv,axiom,
! [A2: set_v,B2: set_v] :
( ( ( inf_inf_set_v @ A2 @ B2 )
= bot_bot_set_v )
=> ( ( minus_minus_set_v @ A2 @ B2 )
= A2 ) ) ).
% Diff_triv
thf(fact_244_graph_Osub__env__trans,axiom,
! [Vertices: set_v,Successors: v > set_v,E: sCC_Bl1394983891496994913t_unit,E2: sCC_Bl1394983891496994913t_unit,E3: sCC_Bl1394983891496994913t_unit] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl5768913643336123637t_unit @ E @ E2 )
=> ( ( sCC_Bl5768913643336123637t_unit @ E2 @ E3 )
=> ( sCC_Bl5768913643336123637t_unit @ E @ E3 ) ) ) ) ).
% graph.sub_env_trans
thf(fact_245_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_246_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_247_graph_OS__reflexive,axiom,
! [Vertices: set_v,Successors: v > set_v,E: sCC_Bl1394983891496994913t_unit,N2: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl9196236973127232072t_unit @ Successors @ E )
=> ( member_v @ N2 @ ( sCC_Bl1280885523602775798t_unit @ E @ N2 ) ) ) ) ).
% graph.S_reflexive
thf(fact_248_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,X: 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 @ X @ ( inf_in6271465464967711157od_v_v @ S @ S2 ) )
=> ( S = S2 ) ) ) ) ) ).
% graph.scc_partition
thf(fact_249_graph_Oscc__partition,axiom,
! [Vertices: set_v,Successors: v > set_v,S: set_v,S2: set_v,X: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bloemen_is_scc_v @ Successors @ S )
=> ( ( sCC_Bloemen_is_scc_v @ Successors @ S2 )
=> ( ( member_v @ X @ ( inf_inf_set_v @ S @ S2 ) )
=> ( S = S2 ) ) ) ) ) ).
% graph.scc_partition
thf(fact_250_graph_Ostack__class,axiom,
! [Vertices: set_v,Successors: v > set_v,E: sCC_Bl1394983891496994913t_unit,N2: v,M2: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl9196236973127232072t_unit @ Successors @ E )
=> ( ( member_v @ N2 @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E ) ) )
=> ( ( member_v @ M2 @ ( sCC_Bl1280885523602775798t_unit @ E @ N2 ) )
=> ( member_v @ M2 @ ( minus_minus_set_v @ ( sCC_Bl4645233313691564917t_unit @ E ) @ ( sCC_Bl157864678168468314t_unit @ E ) ) ) ) ) ) ) ).
% graph.stack_class
thf(fact_251_ex__in__conv,axiom,
! [A2: set_v] :
( ( ? [X3: v] : ( member_v @ X3 @ A2 ) )
= ( A2 != bot_bot_set_v ) ) ).
% ex_in_conv
thf(fact_252_ex__in__conv,axiom,
! [A2: set_Product_prod_v_v] :
( ( ? [X3: product_prod_v_v] : ( member7453568604450474000od_v_v @ X3 @ A2 ) )
= ( A2 != bot_bo723834152578015283od_v_v ) ) ).
% ex_in_conv
thf(fact_253_equals0I,axiom,
! [A2: set_v] :
( ! [Y2: v] :
~ ( member_v @ Y2 @ A2 )
=> ( A2 = bot_bot_set_v ) ) ).
% equals0I
thf(fact_254_equals0I,axiom,
! [A2: set_Product_prod_v_v] :
( ! [Y2: product_prod_v_v] :
~ ( member7453568604450474000od_v_v @ Y2 @ A2 )
=> ( A2 = bot_bo723834152578015283od_v_v ) ) ).
% equals0I
thf(fact_255_equals0D,axiom,
! [A2: set_v,A: v] :
( ( A2 = bot_bot_set_v )
=> ~ ( member_v @ A @ A2 ) ) ).
% equals0D
thf(fact_256_equals0D,axiom,
! [A2: set_Product_prod_v_v,A: product_prod_v_v] :
( ( A2 = bot_bo723834152578015283od_v_v )
=> ~ ( member7453568604450474000od_v_v @ A @ A2 ) ) ).
% equals0D
thf(fact_257_emptyE,axiom,
! [A: v] :
~ ( member_v @ A @ bot_bot_set_v ) ).
% emptyE
thf(fact_258_emptyE,axiom,
! [A: product_prod_v_v] :
~ ( member7453568604450474000od_v_v @ A @ bot_bo723834152578015283od_v_v ) ).
% emptyE
thf(fact_259_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_260_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_261_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_262_set__eq__subset,axiom,
( ( ^ [Y4: set_v,Z4: set_v] : ( Y4 = Z4 ) )
= ( ^ [A3: set_v,B3: set_v] :
( ( ord_less_eq_set_v @ A3 @ B3 )
& ( ord_less_eq_set_v @ B3 @ A3 ) ) ) ) ).
% set_eq_subset
thf(fact_263_set__eq__subset,axiom,
( ( ^ [Y4: set_Product_prod_v_v,Z4: set_Product_prod_v_v] : ( Y4 = Z4 ) )
= ( ^ [A3: set_Product_prod_v_v,B3: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A3 @ B3 )
& ( ord_le7336532860387713383od_v_v @ B3 @ A3 ) ) ) ) ).
% set_eq_subset
thf(fact_264_subset__trans,axiom,
! [A2: set_v,B2: set_v,C2: set_v] :
( ( ord_less_eq_set_v @ A2 @ B2 )
=> ( ( ord_less_eq_set_v @ B2 @ C2 )
=> ( ord_less_eq_set_v @ A2 @ C2 ) ) ) ).
% subset_trans
thf(fact_265_subset__trans,axiom,
! [A2: set_Product_prod_v_v,B2: set_Product_prod_v_v,C2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A2 @ B2 )
=> ( ( ord_le7336532860387713383od_v_v @ B2 @ C2 )
=> ( ord_le7336532860387713383od_v_v @ A2 @ C2 ) ) ) ).
% subset_trans
thf(fact_266_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_267_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_268_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_269_subset__refl,axiom,
! [A2: set_v] : ( ord_less_eq_set_v @ A2 @ A2 ) ).
% subset_refl
thf(fact_270_subset__refl,axiom,
! [A2: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ A2 @ A2 ) ).
% subset_refl
thf(fact_271_subset__iff,axiom,
( ord_less_eq_set_v
= ( ^ [A3: set_v,B3: set_v] :
! [T: v] :
( ( member_v @ T @ A3 )
=> ( member_v @ T @ B3 ) ) ) ) ).
% subset_iff
thf(fact_272_subset__iff,axiom,
( ord_le7336532860387713383od_v_v
= ( ^ [A3: set_Product_prod_v_v,B3: set_Product_prod_v_v] :
! [T: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ T @ A3 )
=> ( member7453568604450474000od_v_v @ T @ B3 ) ) ) ) ).
% subset_iff
thf(fact_273_equalityD2,axiom,
! [A2: set_v,B2: set_v] :
( ( A2 = B2 )
=> ( ord_less_eq_set_v @ B2 @ A2 ) ) ).
% equalityD2
thf(fact_274_equalityD2,axiom,
! [A2: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( A2 = B2 )
=> ( ord_le7336532860387713383od_v_v @ B2 @ A2 ) ) ).
% equalityD2
thf(fact_275_equalityD1,axiom,
! [A2: set_v,B2: set_v] :
( ( A2 = B2 )
=> ( ord_less_eq_set_v @ A2 @ B2 ) ) ).
% equalityD1
thf(fact_276_equalityD1,axiom,
! [A2: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( A2 = B2 )
=> ( ord_le7336532860387713383od_v_v @ A2 @ B2 ) ) ).
% equalityD1
thf(fact_277_subset__eq,axiom,
( ord_less_eq_set_v
= ( ^ [A3: set_v,B3: set_v] :
! [X3: v] :
( ( member_v @ X3 @ A3 )
=> ( member_v @ X3 @ B3 ) ) ) ) ).
% subset_eq
thf(fact_278_subset__eq,axiom,
( ord_le7336532860387713383od_v_v
= ( ^ [A3: set_Product_prod_v_v,B3: set_Product_prod_v_v] :
! [X3: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X3 @ A3 )
=> ( member7453568604450474000od_v_v @ X3 @ B3 ) ) ) ) ).
% subset_eq
thf(fact_279_equalityE,axiom,
! [A2: set_v,B2: set_v] :
( ( A2 = B2 )
=> ~ ( ( ord_less_eq_set_v @ A2 @ B2 )
=> ~ ( ord_less_eq_set_v @ B2 @ A2 ) ) ) ).
% equalityE
thf(fact_280_equalityE,axiom,
! [A2: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( A2 = B2 )
=> ~ ( ( ord_le7336532860387713383od_v_v @ A2 @ B2 )
=> ~ ( ord_le7336532860387713383od_v_v @ B2 @ A2 ) ) ) ).
% equalityE
thf(fact_281_subsetD,axiom,
! [A2: set_v,B2: set_v,C: v] :
( ( ord_less_eq_set_v @ A2 @ B2 )
=> ( ( member_v @ C @ A2 )
=> ( member_v @ C @ B2 ) ) ) ).
% subsetD
thf(fact_282_subsetD,axiom,
! [A2: set_Product_prod_v_v,B2: set_Product_prod_v_v,C: product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A2 @ B2 )
=> ( ( member7453568604450474000od_v_v @ C @ A2 )
=> ( member7453568604450474000od_v_v @ C @ B2 ) ) ) ).
% subsetD
thf(fact_283_in__mono,axiom,
! [A2: set_v,B2: set_v,X: v] :
( ( ord_less_eq_set_v @ A2 @ B2 )
=> ( ( member_v @ X @ A2 )
=> ( member_v @ X @ B2 ) ) ) ).
% in_mono
thf(fact_284_in__mono,axiom,
! [A2: set_Product_prod_v_v,B2: set_Product_prod_v_v,X: product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A2 @ B2 )
=> ( ( member7453568604450474000od_v_v @ X @ A2 )
=> ( member7453568604450474000od_v_v @ X @ B2 ) ) ) ).
% in_mono
thf(fact_285_graph_Odfs__S__tl__stack_I1_J,axiom,
! [Vertices: set_v,Successors: v > set_v,V3: v,E: sCC_Bl1394983891496994913t_unit,E2: sCC_Bl1394983891496994913t_unit] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl8953792750115413617t_unit @ Successors @ V3 @ E @ E2 )
=> ( ( ( sCC_Bl8828226123343373779t_unit @ E )
!= nil_v )
=> ( ( sCC_Bl8828226123343373779t_unit @ E2 )
!= nil_v ) ) ) ) ).
% graph.dfs_S_tl_stack(1)
thf(fact_286_inf__left__commute,axiom,
! [X: set_v,Y: set_v,Z: set_v] :
( ( inf_inf_set_v @ X @ ( inf_inf_set_v @ Y @ Z ) )
= ( inf_inf_set_v @ Y @ ( inf_inf_set_v @ X @ Z ) ) ) ).
% inf_left_commute
thf(fact_287_inf_Oleft__commute,axiom,
! [B: set_v,A: set_v,C: set_v] :
( ( inf_inf_set_v @ B @ ( inf_inf_set_v @ A @ C ) )
= ( inf_inf_set_v @ A @ ( inf_inf_set_v @ B @ C ) ) ) ).
% inf.left_commute
thf(fact_288_boolean__algebra__cancel_Oinf2,axiom,
! [B2: set_v,K: set_v,B: set_v,A: set_v] :
( ( B2
= ( inf_inf_set_v @ K @ B ) )
=> ( ( inf_inf_set_v @ A @ B2 )
= ( inf_inf_set_v @ K @ ( inf_inf_set_v @ A @ B ) ) ) ) ).
% boolean_algebra_cancel.inf2
thf(fact_289_boolean__algebra__cancel_Oinf1,axiom,
! [A2: set_v,K: set_v,A: set_v,B: set_v] :
( ( A2
= ( inf_inf_set_v @ K @ A ) )
=> ( ( inf_inf_set_v @ A2 @ B )
= ( inf_inf_set_v @ K @ ( inf_inf_set_v @ A @ B ) ) ) ) ).
% boolean_algebra_cancel.inf1
thf(fact_290_inf__commute,axiom,
( inf_inf_set_v
= ( ^ [X3: set_v,Y3: set_v] : ( inf_inf_set_v @ Y3 @ X3 ) ) ) ).
% inf_commute
thf(fact_291_inf_Ocommute,axiom,
( inf_inf_set_v
= ( ^ [A4: set_v,B4: set_v] : ( inf_inf_set_v @ B4 @ A4 ) ) ) ).
% inf.commute
thf(fact_292_inf__assoc,axiom,
! [X: set_v,Y: set_v,Z: set_v] :
( ( inf_inf_set_v @ ( inf_inf_set_v @ X @ Y ) @ Z )
= ( inf_inf_set_v @ X @ ( inf_inf_set_v @ Y @ Z ) ) ) ).
% inf_assoc
thf(fact_293_inf_Oassoc,axiom,
! [A: set_v,B: set_v,C: set_v] :
( ( inf_inf_set_v @ ( inf_inf_set_v @ A @ B ) @ C )
= ( inf_inf_set_v @ A @ ( inf_inf_set_v @ B @ C ) ) ) ).
% inf.assoc
thf(fact_294_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_295_inf__sup__aci_I2_J,axiom,
! [X: set_v,Y: set_v,Z: set_v] :
( ( inf_inf_set_v @ ( inf_inf_set_v @ X @ Y ) @ Z )
= ( inf_inf_set_v @ X @ ( inf_inf_set_v @ Y @ Z ) ) ) ).
% inf_sup_aci(2)
thf(fact_296_inf__sup__aci_I3_J,axiom,
! [X: set_v,Y: set_v,Z: set_v] :
( ( inf_inf_set_v @ X @ ( inf_inf_set_v @ Y @ Z ) )
= ( inf_inf_set_v @ Y @ ( inf_inf_set_v @ X @ Z ) ) ) ).
% inf_sup_aci(3)
thf(fact_297_inf__sup__aci_I4_J,axiom,
! [X: set_v,Y: set_v] :
( ( inf_inf_set_v @ X @ ( inf_inf_set_v @ X @ Y ) )
= ( inf_inf_set_v @ X @ Y ) ) ).
% inf_sup_aci(4)
thf(fact_298_Int__left__commute,axiom,
! [A2: set_v,B2: set_v,C2: set_v] :
( ( inf_inf_set_v @ A2 @ ( inf_inf_set_v @ B2 @ C2 ) )
= ( inf_inf_set_v @ B2 @ ( inf_inf_set_v @ A2 @ C2 ) ) ) ).
% Int_left_commute
thf(fact_299_Int__left__absorb,axiom,
! [A2: set_v,B2: set_v] :
( ( inf_inf_set_v @ A2 @ ( inf_inf_set_v @ A2 @ B2 ) )
= ( inf_inf_set_v @ A2 @ B2 ) ) ).
% Int_left_absorb
thf(fact_300_Int__commute,axiom,
( inf_inf_set_v
= ( ^ [A3: set_v,B3: set_v] : ( inf_inf_set_v @ B3 @ A3 ) ) ) ).
% Int_commute
thf(fact_301_Int__absorb,axiom,
! [A2: set_v] :
( ( inf_inf_set_v @ A2 @ A2 )
= A2 ) ).
% Int_absorb
thf(fact_302_Int__assoc,axiom,
! [A2: set_v,B2: set_v,C2: set_v] :
( ( inf_inf_set_v @ ( inf_inf_set_v @ A2 @ B2 ) @ C2 )
= ( inf_inf_set_v @ A2 @ ( inf_inf_set_v @ B2 @ C2 ) ) ) ).
% Int_assoc
thf(fact_303_IntD2,axiom,
! [C: product_prod_v_v,A2: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ C @ ( inf_in6271465464967711157od_v_v @ A2 @ B2 ) )
=> ( member7453568604450474000od_v_v @ C @ B2 ) ) ).
% IntD2
thf(fact_304_IntD2,axiom,
! [C: v,A2: set_v,B2: set_v] :
( ( member_v @ C @ ( inf_inf_set_v @ A2 @ B2 ) )
=> ( member_v @ C @ B2 ) ) ).
% IntD2
thf(fact_305_IntD1,axiom,
! [C: product_prod_v_v,A2: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ C @ ( inf_in6271465464967711157od_v_v @ A2 @ B2 ) )
=> ( member7453568604450474000od_v_v @ C @ A2 ) ) ).
% IntD1
thf(fact_306_IntD1,axiom,
! [C: v,A2: set_v,B2: set_v] :
( ( member_v @ C @ ( inf_inf_set_v @ A2 @ B2 ) )
=> ( member_v @ C @ A2 ) ) ).
% IntD1
thf(fact_307_IntE,axiom,
! [C: product_prod_v_v,A2: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ C @ ( inf_in6271465464967711157od_v_v @ A2 @ B2 ) )
=> ~ ( ( member7453568604450474000od_v_v @ C @ A2 )
=> ~ ( member7453568604450474000od_v_v @ C @ B2 ) ) ) ).
% IntE
thf(fact_308_IntE,axiom,
! [C: v,A2: set_v,B2: set_v] :
( ( member_v @ C @ ( inf_inf_set_v @ A2 @ B2 ) )
=> ~ ( ( member_v @ C @ A2 )
=> ~ ( member_v @ C @ B2 ) ) ) ).
% IntE
thf(fact_309_graph_Ostack__visited,axiom,
! [Vertices: set_v,Successors: v > set_v,E: sCC_Bl1394983891496994913t_unit,N2: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl9196236973127232072t_unit @ Successors @ E )
=> ( ( member_v @ N2 @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E ) ) )
=> ( member_v @ N2 @ ( sCC_Bl4645233313691564917t_unit @ E ) ) ) ) ) ).
% graph.stack_visited
thf(fact_310_graph_Ostack__unexplored,axiom,
! [Vertices: set_v,Successors: v > set_v,E: sCC_Bl1394983891496994913t_unit,N2: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl9196236973127232072t_unit @ Successors @ E )
=> ( ( member_v @ N2 @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E ) ) )
=> ~ ( member_v @ N2 @ ( sCC_Bl157864678168468314t_unit @ E ) ) ) ) ) ).
% graph.stack_unexplored
thf(fact_311_graph_Ounite__subscc,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,V3: product_prod_v_v,E: sCC_Bl7326425374436813197t_unit,W: product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( sCC_Bl3607325323686918683t_unit @ Successors @ V3 @ E )
=> ( ( member7453568604450474000od_v_v @ W @ ( Successors @ V3 ) )
=> ( ~ ( member7453568604450474000od_v_v @ W @ ( sCC_Bl3878977043676959280t_unit @ E @ V3 ) )
=> ( ( member7453568604450474000od_v_v @ W @ ( sCC_Bl5498988629518860705t_unit @ E ) )
=> ( ~ ( member7453568604450474000od_v_v @ W @ ( sCC_Bl5094201334446601350t_unit @ E ) )
=> ( sCC_Bl2301996248249672505od_v_v @ Successors @ ( sCC_Bl8440648026628373538t_unit @ ( sCC_Bl4702006153222411093od_v_v @ V3 @ W @ E ) @ ( hd_Product_prod_v_v @ ( sCC_Bl2021302119412358655t_unit @ ( sCC_Bl4702006153222411093od_v_v @ V3 @ W @ E ) ) ) ) ) ) ) ) ) ) ) ).
% graph.unite_subscc
thf(fact_312_graph_Ounite__subscc,axiom,
! [Vertices: set_v,Successors: v > set_v,V3: v,E: sCC_Bl1394983891496994913t_unit,W: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl1748261141445803503t_unit @ Successors @ V3 @ E )
=> ( ( member_v @ W @ ( Successors @ V3 ) )
=> ( ~ ( member_v @ W @ ( sCC_Bl3795065053823578884t_unit @ E @ V3 ) )
=> ( ( member_v @ W @ ( sCC_Bl4645233313691564917t_unit @ E ) )
=> ( ~ ( member_v @ W @ ( sCC_Bl157864678168468314t_unit @ E ) )
=> ( sCC_Bl5398416737448265317bscc_v @ Successors @ ( sCC_Bl1280885523602775798t_unit @ ( sCC_Bloemen_unite_v @ V3 @ W @ E ) @ ( hd_v @ ( sCC_Bl8828226123343373779t_unit @ ( sCC_Bloemen_unite_v @ V3 @ W @ E ) ) ) ) ) ) ) ) ) ) ) ).
% graph.unite_subscc
thf(fact_313_graph_Odfs__S__tl__stack_I2_J,axiom,
! [Vertices: set_v,Successors: v > set_v,V3: v,E: sCC_Bl1394983891496994913t_unit,E2: sCC_Bl1394983891496994913t_unit] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl8953792750115413617t_unit @ Successors @ V3 @ E @ E2 )
=> ( ( ( sCC_Bl8828226123343373779t_unit @ E )
!= nil_v )
=> ! [X2: v] :
( ( member_v @ X2 @ ( set_v2 @ ( tl_v @ ( sCC_Bl8828226123343373779t_unit @ E2 ) ) ) )
=> ( ( sCC_Bl1280885523602775798t_unit @ E2 @ X2 )
= ( sCC_Bl1280885523602775798t_unit @ E @ X2 ) ) ) ) ) ) ).
% graph.dfs_S_tl_stack(2)
thf(fact_314_graph_Ovisited__unexplored,axiom,
! [Vertices: set_v,Successors: v > set_v,E: sCC_Bl1394983891496994913t_unit,M2: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl9196236973127232072t_unit @ Successors @ E )
=> ( ( member_v @ M2 @ ( sCC_Bl4645233313691564917t_unit @ E ) )
=> ( ~ ( member_v @ M2 @ ( sCC_Bl157864678168468314t_unit @ E ) )
=> ~ ! [N3: v] :
( ( member_v @ N3 @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E ) ) )
=> ~ ( member_v @ M2 @ ( sCC_Bl1280885523602775798t_unit @ E @ N3 ) ) ) ) ) ) ) ).
% graph.visited_unexplored
thf(fact_315_graph_Odfs__S__hd__stack_I1_J,axiom,
! [Vertices: set_v,Successors: v > set_v,E: sCC_Bl1394983891496994913t_unit,V3: v,E2: sCC_Bl1394983891496994913t_unit,N2: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl9196236973127232072t_unit @ Successors @ E )
=> ( ( sCC_Bl8953792750115413617t_unit @ Successors @ V3 @ E @ E2 )
=> ( ( ( sCC_Bl8828226123343373779t_unit @ E )
!= nil_v )
=> ( ( member_v @ N2 @ ( sCC_Bl1280885523602775798t_unit @ E @ ( hd_v @ ( sCC_Bl8828226123343373779t_unit @ E ) ) ) )
=> ( ( sCC_Bl8828226123343373779t_unit @ E2 )
!= nil_v ) ) ) ) ) ) ).
% graph.dfs_S_hd_stack(1)
thf(fact_316_graph_Odfs__S__hd__stack_I2_J,axiom,
! [Vertices: set_v,Successors: v > set_v,E: sCC_Bl1394983891496994913t_unit,V3: v,E2: sCC_Bl1394983891496994913t_unit,N2: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl9196236973127232072t_unit @ Successors @ E )
=> ( ( sCC_Bl8953792750115413617t_unit @ Successors @ V3 @ E @ E2 )
=> ( ( ( sCC_Bl8828226123343373779t_unit @ E )
!= nil_v )
=> ( ( member_v @ N2 @ ( sCC_Bl1280885523602775798t_unit @ E @ ( hd_v @ ( sCC_Bl8828226123343373779t_unit @ E ) ) ) )
=> ( member_v @ N2 @ ( sCC_Bl1280885523602775798t_unit @ E2 @ ( hd_v @ ( sCC_Bl8828226123343373779t_unit @ E2 ) ) ) ) ) ) ) ) ) ).
% graph.dfs_S_hd_stack(2)
thf(fact_317_inf_OcoboundedI2,axiom,
! [B: set_v,C: set_v,A: set_v] :
( ( ord_less_eq_set_v @ B @ C )
=> ( ord_less_eq_set_v @ ( inf_inf_set_v @ A @ B ) @ C ) ) ).
% inf.coboundedI2
thf(fact_318_inf_OcoboundedI2,axiom,
! [B: set_Product_prod_v_v,C: set_Product_prod_v_v,A: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ B @ C )
=> ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ A @ B ) @ C ) ) ).
% inf.coboundedI2
thf(fact_319_inf_OcoboundedI1,axiom,
! [A: set_v,C: set_v,B: set_v] :
( ( ord_less_eq_set_v @ A @ C )
=> ( ord_less_eq_set_v @ ( inf_inf_set_v @ A @ B ) @ C ) ) ).
% inf.coboundedI1
thf(fact_320_inf_OcoboundedI1,axiom,
! [A: set_Product_prod_v_v,C: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ C )
=> ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ A @ B ) @ C ) ) ).
% inf.coboundedI1
thf(fact_321_inf_Oabsorb__iff2,axiom,
( ord_less_eq_set_v
= ( ^ [B4: set_v,A4: set_v] :
( ( inf_inf_set_v @ A4 @ B4 )
= B4 ) ) ) ).
% inf.absorb_iff2
thf(fact_322_inf_Oabsorb__iff2,axiom,
( ord_le7336532860387713383od_v_v
= ( ^ [B4: set_Product_prod_v_v,A4: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ A4 @ B4 )
= B4 ) ) ) ).
% inf.absorb_iff2
thf(fact_323_inf_Oabsorb__iff1,axiom,
( ord_less_eq_set_v
= ( ^ [A4: set_v,B4: set_v] :
( ( inf_inf_set_v @ A4 @ B4 )
= A4 ) ) ) ).
% inf.absorb_iff1
thf(fact_324_inf_Oabsorb__iff1,axiom,
( ord_le7336532860387713383od_v_v
= ( ^ [A4: set_Product_prod_v_v,B4: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ A4 @ B4 )
= A4 ) ) ) ).
% inf.absorb_iff1
thf(fact_325_inf_Ocobounded2,axiom,
! [A: set_v,B: set_v] : ( ord_less_eq_set_v @ ( inf_inf_set_v @ A @ B ) @ B ) ).
% inf.cobounded2
thf(fact_326_inf_Ocobounded2,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ A @ B ) @ B ) ).
% inf.cobounded2
thf(fact_327_inf_Ocobounded1,axiom,
! [A: set_v,B: set_v] : ( ord_less_eq_set_v @ ( inf_inf_set_v @ A @ B ) @ A ) ).
% inf.cobounded1
thf(fact_328_inf_Ocobounded1,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ A @ B ) @ A ) ).
% inf.cobounded1
thf(fact_329_inf_Oorder__iff,axiom,
( ord_less_eq_set_v
= ( ^ [A4: set_v,B4: set_v] :
( A4
= ( inf_inf_set_v @ A4 @ B4 ) ) ) ) ).
% inf.order_iff
thf(fact_330_inf_Oorder__iff,axiom,
( ord_le7336532860387713383od_v_v
= ( ^ [A4: set_Product_prod_v_v,B4: set_Product_prod_v_v] :
( A4
= ( inf_in6271465464967711157od_v_v @ A4 @ B4 ) ) ) ) ).
% inf.order_iff
thf(fact_331_inf__greatest,axiom,
! [X: set_v,Y: set_v,Z: set_v] :
( ( ord_less_eq_set_v @ X @ Y )
=> ( ( ord_less_eq_set_v @ X @ Z )
=> ( ord_less_eq_set_v @ X @ ( inf_inf_set_v @ Y @ Z ) ) ) ) ).
% inf_greatest
thf(fact_332_inf__greatest,axiom,
! [X: set_Product_prod_v_v,Y: set_Product_prod_v_v,Z: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X @ Y )
=> ( ( ord_le7336532860387713383od_v_v @ X @ Z )
=> ( ord_le7336532860387713383od_v_v @ X @ ( inf_in6271465464967711157od_v_v @ Y @ Z ) ) ) ) ).
% inf_greatest
thf(fact_333_inf_OboundedI,axiom,
! [A: set_v,B: set_v,C: set_v] :
( ( ord_less_eq_set_v @ A @ B )
=> ( ( ord_less_eq_set_v @ A @ C )
=> ( ord_less_eq_set_v @ A @ ( inf_inf_set_v @ B @ C ) ) ) ) ).
% inf.boundedI
thf(fact_334_inf_OboundedI,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ B )
=> ( ( ord_le7336532860387713383od_v_v @ A @ C )
=> ( ord_le7336532860387713383od_v_v @ A @ ( inf_in6271465464967711157od_v_v @ B @ C ) ) ) ) ).
% inf.boundedI
thf(fact_335_inf_OboundedE,axiom,
! [A: set_v,B: set_v,C: set_v] :
( ( ord_less_eq_set_v @ A @ ( inf_inf_set_v @ B @ C ) )
=> ~ ( ( ord_less_eq_set_v @ A @ B )
=> ~ ( ord_less_eq_set_v @ A @ C ) ) ) ).
% inf.boundedE
thf(fact_336_inf_OboundedE,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ ( inf_in6271465464967711157od_v_v @ B @ C ) )
=> ~ ( ( ord_le7336532860387713383od_v_v @ A @ B )
=> ~ ( ord_le7336532860387713383od_v_v @ A @ C ) ) ) ).
% inf.boundedE
thf(fact_337_inf__absorb2,axiom,
! [Y: set_v,X: set_v] :
( ( ord_less_eq_set_v @ Y @ X )
=> ( ( inf_inf_set_v @ X @ Y )
= Y ) ) ).
% inf_absorb2
thf(fact_338_inf__absorb2,axiom,
! [Y: set_Product_prod_v_v,X: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ Y @ X )
=> ( ( inf_in6271465464967711157od_v_v @ X @ Y )
= Y ) ) ).
% inf_absorb2
thf(fact_339_inf__absorb1,axiom,
! [X: set_v,Y: set_v] :
( ( ord_less_eq_set_v @ X @ Y )
=> ( ( inf_inf_set_v @ X @ Y )
= X ) ) ).
% inf_absorb1
thf(fact_340_inf__absorb1,axiom,
! [X: set_Product_prod_v_v,Y: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X @ Y )
=> ( ( inf_in6271465464967711157od_v_v @ X @ Y )
= X ) ) ).
% inf_absorb1
thf(fact_341_inf_Oabsorb2,axiom,
! [B: set_v,A: set_v] :
( ( ord_less_eq_set_v @ B @ A )
=> ( ( inf_inf_set_v @ A @ B )
= B ) ) ).
% inf.absorb2
thf(fact_342_inf_Oabsorb2,axiom,
! [B: set_Product_prod_v_v,A: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ B @ A )
=> ( ( inf_in6271465464967711157od_v_v @ A @ B )
= B ) ) ).
% inf.absorb2
thf(fact_343_inf_Oabsorb1,axiom,
! [A: set_v,B: set_v] :
( ( ord_less_eq_set_v @ A @ B )
=> ( ( inf_inf_set_v @ A @ B )
= A ) ) ).
% inf.absorb1
thf(fact_344_inf_Oabsorb1,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ B )
=> ( ( inf_in6271465464967711157od_v_v @ A @ B )
= A ) ) ).
% inf.absorb1
thf(fact_345_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_346_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_347_inf__unique,axiom,
! [F: set_v > set_v > set_v,X: 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,Z2: set_v] :
( ( ord_less_eq_set_v @ X4 @ Y2 )
=> ( ( ord_less_eq_set_v @ X4 @ Z2 )
=> ( ord_less_eq_set_v @ X4 @ ( F @ Y2 @ Z2 ) ) ) )
=> ( ( inf_inf_set_v @ X @ Y )
= ( F @ X @ Y ) ) ) ) ) ).
% inf_unique
thf(fact_348_inf__unique,axiom,
! [F: set_Product_prod_v_v > set_Product_prod_v_v > set_Product_prod_v_v,X: set_Product_prod_v_v,Y: set_Product_prod_v_v] :
( ! [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,Z2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X4 @ Y2 )
=> ( ( ord_le7336532860387713383od_v_v @ X4 @ Z2 )
=> ( ord_le7336532860387713383od_v_v @ X4 @ ( F @ Y2 @ Z2 ) ) ) )
=> ( ( inf_in6271465464967711157od_v_v @ X @ Y )
= ( F @ X @ Y ) ) ) ) ) ).
% inf_unique
thf(fact_349_inf_OorderI,axiom,
! [A: set_v,B: set_v] :
( ( A
= ( inf_inf_set_v @ A @ B ) )
=> ( ord_less_eq_set_v @ A @ B ) ) ).
% inf.orderI
thf(fact_350_inf_OorderI,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( A
= ( inf_in6271465464967711157od_v_v @ A @ B ) )
=> ( ord_le7336532860387713383od_v_v @ A @ B ) ) ).
% inf.orderI
thf(fact_351_inf_OorderE,axiom,
! [A: set_v,B: set_v] :
( ( ord_less_eq_set_v @ A @ B )
=> ( A
= ( inf_inf_set_v @ A @ B ) ) ) ).
% inf.orderE
thf(fact_352_inf_OorderE,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ B )
=> ( A
= ( inf_in6271465464967711157od_v_v @ A @ B ) ) ) ).
% inf.orderE
thf(fact_353_le__infI2,axiom,
! [B: set_v,X: set_v,A: set_v] :
( ( ord_less_eq_set_v @ B @ X )
=> ( ord_less_eq_set_v @ ( inf_inf_set_v @ A @ B ) @ X ) ) ).
% le_infI2
thf(fact_354_le__infI2,axiom,
! [B: set_Product_prod_v_v,X: set_Product_prod_v_v,A: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ B @ X )
=> ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ A @ B ) @ X ) ) ).
% le_infI2
thf(fact_355_le__infI1,axiom,
! [A: set_v,X: set_v,B: set_v] :
( ( ord_less_eq_set_v @ A @ X )
=> ( ord_less_eq_set_v @ ( inf_inf_set_v @ A @ B ) @ X ) ) ).
% le_infI1
thf(fact_356_le__infI1,axiom,
! [A: set_Product_prod_v_v,X: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ X )
=> ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ A @ B ) @ X ) ) ).
% le_infI1
thf(fact_357_inf__mono,axiom,
! [A: set_v,C: set_v,B: set_v,D2: set_v] :
( ( ord_less_eq_set_v @ A @ C )
=> ( ( ord_less_eq_set_v @ B @ D2 )
=> ( ord_less_eq_set_v @ ( inf_inf_set_v @ A @ B ) @ ( inf_inf_set_v @ C @ D2 ) ) ) ) ).
% inf_mono
thf(fact_358_inf__mono,axiom,
! [A: set_Product_prod_v_v,C: set_Product_prod_v_v,B: set_Product_prod_v_v,D2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ C )
=> ( ( ord_le7336532860387713383od_v_v @ B @ D2 )
=> ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ A @ B ) @ ( inf_in6271465464967711157od_v_v @ C @ D2 ) ) ) ) ).
% inf_mono
thf(fact_359_le__infI,axiom,
! [X: set_v,A: set_v,B: set_v] :
( ( ord_less_eq_set_v @ X @ A )
=> ( ( ord_less_eq_set_v @ X @ B )
=> ( ord_less_eq_set_v @ X @ ( inf_inf_set_v @ A @ B ) ) ) ) ).
% le_infI
thf(fact_360_le__infI,axiom,
! [X: set_Product_prod_v_v,A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X @ A )
=> ( ( ord_le7336532860387713383od_v_v @ X @ B )
=> ( ord_le7336532860387713383od_v_v @ X @ ( inf_in6271465464967711157od_v_v @ A @ B ) ) ) ) ).
% le_infI
thf(fact_361_le__infE,axiom,
! [X: set_v,A: set_v,B: set_v] :
( ( ord_less_eq_set_v @ X @ ( inf_inf_set_v @ A @ B ) )
=> ~ ( ( ord_less_eq_set_v @ X @ A )
=> ~ ( ord_less_eq_set_v @ X @ B ) ) ) ).
% le_infE
thf(fact_362_le__infE,axiom,
! [X: set_Product_prod_v_v,A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X @ ( inf_in6271465464967711157od_v_v @ A @ B ) )
=> ~ ( ( ord_le7336532860387713383od_v_v @ X @ A )
=> ~ ( ord_le7336532860387713383od_v_v @ X @ B ) ) ) ).
% le_infE
thf(fact_363_inf__le2,axiom,
! [X: set_v,Y: set_v] : ( ord_less_eq_set_v @ ( inf_inf_set_v @ X @ Y ) @ Y ) ).
% inf_le2
thf(fact_364_inf__le2,axiom,
! [X: set_Product_prod_v_v,Y: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ X @ Y ) @ Y ) ).
% inf_le2
thf(fact_365_inf__le1,axiom,
! [X: set_v,Y: set_v] : ( ord_less_eq_set_v @ ( inf_inf_set_v @ X @ Y ) @ X ) ).
% inf_le1
thf(fact_366_inf__le1,axiom,
! [X: set_Product_prod_v_v,Y: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ X @ Y ) @ X ) ).
% inf_le1
thf(fact_367_inf__sup__ord_I1_J,axiom,
! [X: set_v,Y: set_v] : ( ord_less_eq_set_v @ ( inf_inf_set_v @ X @ Y ) @ X ) ).
% inf_sup_ord(1)
thf(fact_368_inf__sup__ord_I1_J,axiom,
! [X: set_Product_prod_v_v,Y: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ X @ Y ) @ X ) ).
% inf_sup_ord(1)
thf(fact_369_inf__sup__ord_I2_J,axiom,
! [X: set_v,Y: set_v] : ( ord_less_eq_set_v @ ( inf_inf_set_v @ X @ Y ) @ Y ) ).
% inf_sup_ord(2)
thf(fact_370_inf__sup__ord_I2_J,axiom,
! [X: set_Product_prod_v_v,Y: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ X @ Y ) @ Y ) ).
% inf_sup_ord(2)
thf(fact_371_disjoint__iff__not__equal,axiom,
! [A2: set_v,B2: set_v] :
( ( ( inf_inf_set_v @ A2 @ B2 )
= bot_bot_set_v )
= ( ! [X3: v] :
( ( member_v @ X3 @ A2 )
=> ! [Y3: v] :
( ( member_v @ Y3 @ B2 )
=> ( X3 != Y3 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_372_disjoint__iff__not__equal,axiom,
! [A2: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( ( inf_in6271465464967711157od_v_v @ A2 @ B2 )
= bot_bo723834152578015283od_v_v )
= ( ! [X3: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X3 @ A2 )
=> ! [Y3: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ Y3 @ B2 )
=> ( X3 != Y3 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_373_Int__empty__right,axiom,
! [A2: set_v] :
( ( inf_inf_set_v @ A2 @ bot_bot_set_v )
= bot_bot_set_v ) ).
% Int_empty_right
thf(fact_374_Int__empty__right,axiom,
! [A2: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ A2 @ bot_bo723834152578015283od_v_v )
= bot_bo723834152578015283od_v_v ) ).
% Int_empty_right
thf(fact_375_Int__empty__left,axiom,
! [B2: set_v] :
( ( inf_inf_set_v @ bot_bot_set_v @ B2 )
= bot_bot_set_v ) ).
% Int_empty_left
thf(fact_376_Int__empty__left,axiom,
! [B2: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ bot_bo723834152578015283od_v_v @ B2 )
= bot_bo723834152578015283od_v_v ) ).
% Int_empty_left
thf(fact_377_disjoint__iff,axiom,
! [A2: set_v,B2: set_v] :
( ( ( inf_inf_set_v @ A2 @ B2 )
= bot_bot_set_v )
= ( ! [X3: v] :
( ( member_v @ X3 @ A2 )
=> ~ ( member_v @ X3 @ B2 ) ) ) ) ).
% disjoint_iff
thf(fact_378_disjoint__iff,axiom,
! [A2: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( ( inf_in6271465464967711157od_v_v @ A2 @ B2 )
= bot_bo723834152578015283od_v_v )
= ( ! [X3: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X3 @ A2 )
=> ~ ( member7453568604450474000od_v_v @ X3 @ B2 ) ) ) ) ).
% disjoint_iff
thf(fact_379_Int__emptyI,axiom,
! [A2: set_v,B2: set_v] :
( ! [X4: v] :
( ( member_v @ X4 @ A2 )
=> ~ ( member_v @ X4 @ B2 ) )
=> ( ( inf_inf_set_v @ A2 @ B2 )
= bot_bot_set_v ) ) ).
% Int_emptyI
thf(fact_380_Int__emptyI,axiom,
! [A2: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ! [X4: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X4 @ A2 )
=> ~ ( member7453568604450474000od_v_v @ X4 @ B2 ) )
=> ( ( inf_in6271465464967711157od_v_v @ A2 @ B2 )
= bot_bo723834152578015283od_v_v ) ) ).
% Int_emptyI
thf(fact_381_Int__Collect__mono,axiom,
! [A2: set_set_v,B2: set_set_v,P: set_v > $o,Q: set_v > $o] :
( ( ord_le5216385588623774835_set_v @ A2 @ B2 )
=> ( ! [X4: set_v] :
( ( member_set_v @ X4 @ A2 )
=> ( ( P @ X4 )
=> ( Q @ X4 ) ) )
=> ( ord_le5216385588623774835_set_v @ ( inf_inf_set_set_v @ A2 @ ( collect_set_v @ P ) ) @ ( inf_inf_set_set_v @ B2 @ ( collect_set_v @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_382_Int__Collect__mono,axiom,
! [A2: set_v,B2: set_v,P: v > $o,Q: v > $o] :
( ( ord_less_eq_set_v @ A2 @ B2 )
=> ( ! [X4: v] :
( ( member_v @ X4 @ A2 )
=> ( ( P @ X4 )
=> ( Q @ X4 ) ) )
=> ( ord_less_eq_set_v @ ( inf_inf_set_v @ A2 @ ( collect_v @ P ) ) @ ( inf_inf_set_v @ B2 @ ( collect_v @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_383_Int__Collect__mono,axiom,
! [A2: set_Product_prod_v_v,B2: set_Product_prod_v_v,P: product_prod_v_v > $o,Q: product_prod_v_v > $o] :
( ( ord_le7336532860387713383od_v_v @ A2 @ B2 )
=> ( ! [X4: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X4 @ A2 )
=> ( ( P @ X4 )
=> ( Q @ X4 ) ) )
=> ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ A2 @ ( collec140062887454715474od_v_v @ P ) ) @ ( inf_in6271465464967711157od_v_v @ B2 @ ( collec140062887454715474od_v_v @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_384_Int__greatest,axiom,
! [C2: set_v,A2: set_v,B2: set_v] :
( ( ord_less_eq_set_v @ C2 @ A2 )
=> ( ( ord_less_eq_set_v @ C2 @ B2 )
=> ( ord_less_eq_set_v @ C2 @ ( inf_inf_set_v @ A2 @ B2 ) ) ) ) ).
% Int_greatest
thf(fact_385_Int__greatest,axiom,
! [C2: set_Product_prod_v_v,A2: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ C2 @ A2 )
=> ( ( ord_le7336532860387713383od_v_v @ C2 @ B2 )
=> ( ord_le7336532860387713383od_v_v @ C2 @ ( inf_in6271465464967711157od_v_v @ A2 @ B2 ) ) ) ) ).
% Int_greatest
thf(fact_386_Int__absorb2,axiom,
! [A2: set_v,B2: set_v] :
( ( ord_less_eq_set_v @ A2 @ B2 )
=> ( ( inf_inf_set_v @ A2 @ B2 )
= A2 ) ) ).
% Int_absorb2
thf(fact_387_Int__absorb2,axiom,
! [A2: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A2 @ B2 )
=> ( ( inf_in6271465464967711157od_v_v @ A2 @ B2 )
= A2 ) ) ).
% Int_absorb2
thf(fact_388_Int__absorb1,axiom,
! [B2: set_v,A2: set_v] :
( ( ord_less_eq_set_v @ B2 @ A2 )
=> ( ( inf_inf_set_v @ A2 @ B2 )
= B2 ) ) ).
% Int_absorb1
thf(fact_389_Int__absorb1,axiom,
! [B2: set_Product_prod_v_v,A2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ B2 @ A2 )
=> ( ( inf_in6271465464967711157od_v_v @ A2 @ B2 )
= B2 ) ) ).
% Int_absorb1
thf(fact_390_Int__lower2,axiom,
! [A2: set_v,B2: set_v] : ( ord_less_eq_set_v @ ( inf_inf_set_v @ A2 @ B2 ) @ B2 ) ).
% Int_lower2
thf(fact_391_Int__lower2,axiom,
! [A2: set_Product_prod_v_v,B2: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ A2 @ B2 ) @ B2 ) ).
% Int_lower2
thf(fact_392_Int__lower1,axiom,
! [A2: set_v,B2: set_v] : ( ord_less_eq_set_v @ ( inf_inf_set_v @ A2 @ B2 ) @ A2 ) ).
% Int_lower1
thf(fact_393_Int__lower1,axiom,
! [A2: set_Product_prod_v_v,B2: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ A2 @ B2 ) @ A2 ) ).
% Int_lower1
thf(fact_394_Int__mono,axiom,
! [A2: set_v,C2: set_v,B2: set_v,D: set_v] :
( ( ord_less_eq_set_v @ A2 @ C2 )
=> ( ( ord_less_eq_set_v @ B2 @ D )
=> ( ord_less_eq_set_v @ ( inf_inf_set_v @ A2 @ B2 ) @ ( inf_inf_set_v @ C2 @ D ) ) ) ) ).
% Int_mono
thf(fact_395_Int__mono,axiom,
! [A2: set_Product_prod_v_v,C2: set_Product_prod_v_v,B2: set_Product_prod_v_v,D: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A2 @ C2 )
=> ( ( ord_le7336532860387713383od_v_v @ B2 @ D )
=> ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ A2 @ B2 ) @ ( inf_in6271465464967711157od_v_v @ C2 @ D ) ) ) ) ).
% Int_mono
thf(fact_396_graph_Ounite__sub__env,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,V3: product_prod_v_v,E: sCC_Bl7326425374436813197t_unit,W: product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( sCC_Bl3607325323686918683t_unit @ Successors @ V3 @ E )
=> ( ( member7453568604450474000od_v_v @ W @ ( Successors @ V3 ) )
=> ( ~ ( member7453568604450474000od_v_v @ W @ ( sCC_Bl3878977043676959280t_unit @ E @ V3 ) )
=> ( ( member7453568604450474000od_v_v @ W @ ( sCC_Bl5498988629518860705t_unit @ E ) )
=> ( ~ ( member7453568604450474000od_v_v @ W @ ( sCC_Bl5094201334446601350t_unit @ E ) )
=> ( sCC_Bl7963838319573962697t_unit @ E @ ( sCC_Bl4702006153222411093od_v_v @ V3 @ W @ E ) ) ) ) ) ) ) ) ).
% graph.unite_sub_env
thf(fact_397_graph_Ounite__sub__env,axiom,
! [Vertices: set_v,Successors: v > set_v,V3: v,E: sCC_Bl1394983891496994913t_unit,W: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl1748261141445803503t_unit @ Successors @ V3 @ E )
=> ( ( member_v @ W @ ( Successors @ V3 ) )
=> ( ~ ( member_v @ W @ ( sCC_Bl3795065053823578884t_unit @ E @ V3 ) )
=> ( ( member_v @ W @ ( sCC_Bl4645233313691564917t_unit @ E ) )
=> ( ~ ( member_v @ W @ ( sCC_Bl157864678168468314t_unit @ E ) )
=> ( sCC_Bl5768913643336123637t_unit @ E @ ( sCC_Bloemen_unite_v @ V3 @ W @ E ) ) ) ) ) ) ) ) ).
% graph.unite_sub_env
thf(fact_398_graph_Ounite__S__tl,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,E: sCC_Bl7326425374436813197t_unit,W: product_prod_v_v,V3: product_prod_v_v,N2: product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( sCC_Bl7798947040364291444t_unit @ Successors @ E )
=> ( ( member7453568604450474000od_v_v @ W @ ( Successors @ V3 ) )
=> ( ~ ( member7453568604450474000od_v_v @ W @ ( sCC_Bl3878977043676959280t_unit @ E @ V3 ) )
=> ( ( member7453568604450474000od_v_v @ W @ ( sCC_Bl5498988629518860705t_unit @ E ) )
=> ( ~ ( member7453568604450474000od_v_v @ W @ ( sCC_Bl5094201334446601350t_unit @ E ) )
=> ( ( member7453568604450474000od_v_v @ N2 @ ( set_Product_prod_v_v2 @ ( tl_Product_prod_v_v @ ( sCC_Bl2021302119412358655t_unit @ ( sCC_Bl4702006153222411093od_v_v @ V3 @ W @ E ) ) ) ) )
=> ( ( sCC_Bl8440648026628373538t_unit @ ( sCC_Bl4702006153222411093od_v_v @ V3 @ W @ E ) @ N2 )
= ( sCC_Bl8440648026628373538t_unit @ E @ N2 ) ) ) ) ) ) ) ) ) ).
% graph.unite_S_tl
thf(fact_399_graph_Ounite__S__tl,axiom,
! [Vertices: set_v,Successors: v > set_v,E: sCC_Bl1394983891496994913t_unit,W: v,V3: v,N2: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl9196236973127232072t_unit @ Successors @ E )
=> ( ( member_v @ W @ ( Successors @ V3 ) )
=> ( ~ ( member_v @ W @ ( sCC_Bl3795065053823578884t_unit @ E @ V3 ) )
=> ( ( member_v @ W @ ( sCC_Bl4645233313691564917t_unit @ E ) )
=> ( ~ ( member_v @ W @ ( sCC_Bl157864678168468314t_unit @ E ) )
=> ( ( member_v @ N2 @ ( set_v2 @ ( tl_v @ ( sCC_Bl8828226123343373779t_unit @ ( sCC_Bloemen_unite_v @ V3 @ W @ E ) ) ) ) )
=> ( ( sCC_Bl1280885523602775798t_unit @ ( sCC_Bloemen_unite_v @ V3 @ W @ E ) @ N2 )
= ( sCC_Bl1280885523602775798t_unit @ E @ N2 ) ) ) ) ) ) ) ) ) ).
% graph.unite_S_tl
thf(fact_400_graph_Opre__dfs__def,axiom,
! [Vertices: set_v,Successors: v > set_v,V3: v,E: sCC_Bl1394983891496994913t_unit] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl36166008131615352t_unit @ Successors @ V3 @ E )
= ( ( sCC_Bl9196236973127232072t_unit @ Successors @ E )
& ~ ( member_v @ V3 @ ( sCC_Bl4645233313691564917t_unit @ E ) )
& ( sCC_Bl649662514949026229able_v @ Successors @ ( sCC_Bl1090238580953940555t_unit @ E ) @ V3 )
& ( ( sCC_Bl3795065053823578884t_unit @ E @ V3 )
= bot_bot_set_v )
& ! [X3: v] :
( ( member_v @ X3 @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E ) ) )
=> ( sCC_Bl649662514949026229able_v @ Successors @ X3 @ V3 ) ) ) ) ) ).
% graph.pre_dfs_def
thf(fact_401__092_060open_062_092_060forall_062n_O_An_A_092_060notin_062_Avisited_Ae_H_A_092_060longrightarrow_062_A_092_060S_062_Ae_H_An_A_061_A_123n_125_092_060close_062,axiom,
! [N: v] :
( ~ ( member_v @ N @ ( sCC_Bl4645233313691564917t_unit @ e2 ) )
=> ( ( sCC_Bl1280885523602775798t_unit @ e2 @ N )
= ( insert_v @ N @ bot_bot_set_v ) ) ) ).
% \<open>\<forall>n. n \<notin> visited e' \<longrightarrow> \<S> e' n = {n}\<close>
thf(fact_402__092_060open_062l_A_092_060in_062_Aset_Apfx_A_092_060union_062_A_123hd_A_Istack_Ae_H_J_125_092_060close_062,axiom,
member_v @ l @ ( sup_sup_set_v @ ( set_v2 @ pfx ) @ ( insert_v @ ( hd_v @ ( sCC_Bl8828226123343373779t_unit @ e2 ) ) @ bot_bot_set_v ) ) ).
% \<open>l \<in> set pfx \<union> {hd (stack e')}\<close>
thf(fact_403_reachable__re,axiom,
! [X: v,Y: v] :
( ( sCC_Bl649662514949026229able_v @ successors @ X @ Y )
=> ( sCC_Bl770211535891879572_end_v @ successors @ X @ Y ) ) ).
% reachable_re
thf(fact_404_re__reachable,axiom,
! [X: v,Y: v] :
( ( sCC_Bl770211535891879572_end_v @ successors @ X @ Y )
=> ( sCC_Bl649662514949026229able_v @ successors @ X @ Y ) ) ).
% re_reachable
thf(fact_405__092_060open_062_092_060Union_062_A_Isccs_Ae_H_J_A_061_Aexplored_Ae_H_092_060close_062,axiom,
( ( comple2307003700295860064_set_v @ ( sCC_Bl2536197123907397897t_unit @ e2 ) )
= ( sCC_Bl157864678168468314t_unit @ e2 ) ) ).
% \<open>\<Union> (sccs e') = explored e'\<close>
thf(fact_406_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_407_post__dfss__def,axiom,
! [V3: v,E: sCC_Bl1394983891496994913t_unit,E2: sCC_Bl1394983891496994913t_unit] :
( ( sCC_Bl6082031138996704384t_unit @ successors @ V3 @ E @ E2 )
= ( ( sCC_Bl9196236973127232072t_unit @ successors @ E2 )
& ( ( sCC_Bl3795065053823578884t_unit @ E2 @ V3 )
= ( successors @ V3 ) )
& ! [X3: v] :
( ( member_v @ X3 @ ( minus_minus_set_v @ ( sCC_Bl4645233313691564917t_unit @ E ) @ ( insert_v @ V3 @ bot_bot_set_v ) ) )
=> ( ( sCC_Bl3795065053823578884t_unit @ E2 @ X3 )
= ( sCC_Bl3795065053823578884t_unit @ E @ X3 ) ) )
& ( sCC_Bl5768913643336123637t_unit @ E @ E2 )
& ! [X3: v] :
( ( member_v @ X3 @ ( successors @ V3 ) )
=> ( member_v @ X3 @ ( sup_sup_set_v @ ( sCC_Bl157864678168468314t_unit @ E2 ) @ ( sCC_Bl1280885523602775798t_unit @ E2 @ ( hd_v @ ( sCC_Bl8828226123343373779t_unit @ E2 ) ) ) ) ) )
& ! [X3: v] :
( ( member_v @ X3 @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E2 ) ) )
=> ( sCC_Bl649662514949026229able_v @ successors @ X3 @ V3 ) )
& ( ( sCC_Bl8828226123343373779t_unit @ E2 )
!= nil_v )
& ? [Ns: list_v] :
( ( sCC_Bl8828226123343373779t_unit @ E )
= ( append_v @ Ns @ ( sCC_Bl8828226123343373779t_unit @ E2 ) ) )
& ( member_v @ V3 @ ( sCC_Bl1280885523602775798t_unit @ E2 @ ( hd_v @ ( sCC_Bl8828226123343373779t_unit @ E2 ) ) ) )
& ! [X3: v] :
( ( member_v @ X3 @ ( set_v2 @ ( tl_v @ ( sCC_Bl8828226123343373779t_unit @ E2 ) ) ) )
=> ( ( sCC_Bl1280885523602775798t_unit @ E2 @ X3 )
= ( sCC_Bl1280885523602775798t_unit @ E @ X3 ) ) )
& ( ( ( hd_v @ ( sCC_Bl8828226123343373779t_unit @ E2 ) )
= V3 )
=> ! [X3: v] :
( ( member_v @ X3 @ ( set_v2 @ ( tl_v @ ( sCC_Bl8828226123343373779t_unit @ E2 ) ) ) )
=> ~ ( sCC_Bl649662514949026229able_v @ successors @ V3 @ X3 ) ) )
& ( ( sCC_Bl9201514103433284750t_unit @ E2 )
= ( sCC_Bl9201514103433284750t_unit @ E ) ) ) ) ).
% post_dfss_def
thf(fact_408_pre__dfss__def,axiom,
! [V3: v,E: sCC_Bl1394983891496994913t_unit] :
( ( sCC_Bl1748261141445803503t_unit @ successors @ V3 @ E )
= ( ( sCC_Bl9196236973127232072t_unit @ successors @ E )
& ( member_v @ V3 @ ( sCC_Bl4645233313691564917t_unit @ E ) )
& ( ( sCC_Bl8828226123343373779t_unit @ E )
!= nil_v )
& ( member_v @ V3 @ ( sCC_Bl1280885523602775798t_unit @ E @ ( hd_v @ ( sCC_Bl8828226123343373779t_unit @ E ) ) ) )
& ! [X3: v] :
( ( member_v @ X3 @ ( sCC_Bl3795065053823578884t_unit @ E @ V3 ) )
=> ( member_v @ X3 @ ( sup_sup_set_v @ ( sCC_Bl157864678168468314t_unit @ E ) @ ( sCC_Bl1280885523602775798t_unit @ E @ ( hd_v @ ( sCC_Bl8828226123343373779t_unit @ E ) ) ) ) ) )
& ! [X3: v] :
( ( member_v @ X3 @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E ) ) )
=> ( sCC_Bl649662514949026229able_v @ successors @ X3 @ V3 ) )
& ? [Ns: list_v] :
( ( sCC_Bl9201514103433284750t_unit @ E )
= ( cons_v @ V3 @ Ns ) ) ) ) ).
% pre_dfss_def
thf(fact_409_reachable__end_Ocases,axiom,
! [A1: v,A22: v] :
( ( sCC_Bl770211535891879572_end_v @ successors @ A1 @ A22 )
=> ( ( A22 != A1 )
=> ~ ! [Y2: v] :
( ( sCC_Bl770211535891879572_end_v @ successors @ A1 @ Y2 )
=> ~ ( member_v @ A22 @ ( successors @ Y2 ) ) ) ) ) ).
% reachable_end.cases
thf(fact_410_re__refl,axiom,
! [X: v] : ( sCC_Bl770211535891879572_end_v @ successors @ X @ X ) ).
% re_refl
thf(fact_411_re__succ,axiom,
! [X: v,Y: v,Z: v] :
( ( sCC_Bl770211535891879572_end_v @ successors @ X @ Y )
=> ( ( member_v @ Z @ ( successors @ Y ) )
=> ( sCC_Bl770211535891879572_end_v @ successors @ X @ Z ) ) ) ).
% re_succ
thf(fact_412_reachable__end_Osimps,axiom,
! [A1: v,A22: v] :
( ( sCC_Bl770211535891879572_end_v @ successors @ A1 @ A22 )
= ( ? [X3: v] :
( ( A1 = X3 )
& ( A22 = X3 ) )
| ? [X3: v,Y3: v,Z3: v] :
( ( A1 = X3 )
& ( A22 = Z3 )
& ( sCC_Bl770211535891879572_end_v @ successors @ X3 @ Y3 )
& ( member_v @ Z3 @ ( successors @ Y3 ) ) ) ) ) ).
% reachable_end.simps
thf(fact_413_succ__re,axiom,
! [Y: v,X: v,Z: v] :
( ( member_v @ Y @ ( successors @ X ) )
=> ( ( sCC_Bl770211535891879572_end_v @ successors @ Y @ Z )
=> ( sCC_Bl770211535891879572_end_v @ successors @ X @ Z ) ) ) ).
% succ_re
thf(fact_414_init__env__pre__dfs,axiom,
! [V3: v] : ( sCC_Bl36166008131615352t_unit @ successors @ V3 @ ( sCC_Bl7693227186847904995_env_v @ V3 ) ) ).
% init_env_pre_dfs
thf(fact_415_insert__absorb2,axiom,
! [X: v,A2: set_v] :
( ( insert_v @ X @ ( insert_v @ X @ A2 ) )
= ( insert_v @ X @ A2 ) ) ).
% insert_absorb2
thf(fact_416_insert__absorb2,axiom,
! [X: product_prod_v_v,A2: set_Product_prod_v_v] :
( ( insert1338601472111419319od_v_v @ X @ ( insert1338601472111419319od_v_v @ X @ A2 ) )
= ( insert1338601472111419319od_v_v @ X @ A2 ) ) ).
% insert_absorb2
thf(fact_417_insert__iff,axiom,
! [A: v,B: v,A2: set_v] :
( ( member_v @ A @ ( insert_v @ B @ A2 ) )
= ( ( A = B )
| ( member_v @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_418_insert__iff,axiom,
! [A: product_prod_v_v,B: product_prod_v_v,A2: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ A @ ( insert1338601472111419319od_v_v @ B @ A2 ) )
= ( ( A = B )
| ( member7453568604450474000od_v_v @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_419_insertCI,axiom,
! [A: v,B2: set_v,B: v] :
( ( ~ ( member_v @ A @ B2 )
=> ( A = B ) )
=> ( member_v @ A @ ( insert_v @ B @ B2 ) ) ) ).
% insertCI
thf(fact_420_insertCI,axiom,
! [A: product_prod_v_v,B2: set_Product_prod_v_v,B: product_prod_v_v] :
( ( ~ ( member7453568604450474000od_v_v @ A @ B2 )
=> ( A = B ) )
=> ( member7453568604450474000od_v_v @ A @ ( insert1338601472111419319od_v_v @ B @ B2 ) ) ) ).
% insertCI
thf(fact_421_sup_Oright__idem,axiom,
! [A: set_v,B: set_v] :
( ( sup_sup_set_v @ ( sup_sup_set_v @ A @ B ) @ B )
= ( sup_sup_set_v @ A @ B ) ) ).
% sup.right_idem
thf(fact_422_sup_Oright__idem,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ ( sup_su414716646722978715od_v_v @ A @ B ) @ B )
= ( sup_su414716646722978715od_v_v @ A @ B ) ) ).
% sup.right_idem
thf(fact_423_sup__left__idem,axiom,
! [X: set_v,Y: set_v] :
( ( sup_sup_set_v @ X @ ( sup_sup_set_v @ X @ Y ) )
= ( sup_sup_set_v @ X @ Y ) ) ).
% sup_left_idem
thf(fact_424_sup__left__idem,axiom,
! [X: set_Product_prod_v_v,Y: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ X @ ( sup_su414716646722978715od_v_v @ X @ Y ) )
= ( sup_su414716646722978715od_v_v @ X @ Y ) ) ).
% sup_left_idem
thf(fact_425_sup_Oleft__idem,axiom,
! [A: set_v,B: set_v] :
( ( sup_sup_set_v @ A @ ( sup_sup_set_v @ A @ B ) )
= ( sup_sup_set_v @ A @ B ) ) ).
% sup.left_idem
thf(fact_426_sup_Oleft__idem,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ A @ ( sup_su414716646722978715od_v_v @ A @ B ) )
= ( sup_su414716646722978715od_v_v @ A @ B ) ) ).
% sup.left_idem
thf(fact_427_sup__idem,axiom,
! [X: set_v] :
( ( sup_sup_set_v @ X @ X )
= X ) ).
% sup_idem
thf(fact_428_sup__idem,axiom,
! [X: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ X @ X )
= X ) ).
% sup_idem
thf(fact_429_sup_Oidem,axiom,
! [A: set_v] :
( ( sup_sup_set_v @ A @ A )
= A ) ).
% sup.idem
thf(fact_430_sup_Oidem,axiom,
! [A: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ A @ A )
= A ) ).
% sup.idem
thf(fact_431_Un__iff,axiom,
! [C: v,A2: set_v,B2: set_v] :
( ( member_v @ C @ ( sup_sup_set_v @ A2 @ B2 ) )
= ( ( member_v @ C @ A2 )
| ( member_v @ C @ B2 ) ) ) ).
% Un_iff
thf(fact_432_Un__iff,axiom,
! [C: product_prod_v_v,A2: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ C @ ( sup_su414716646722978715od_v_v @ A2 @ B2 ) )
= ( ( member7453568604450474000od_v_v @ C @ A2 )
| ( member7453568604450474000od_v_v @ C @ B2 ) ) ) ).
% Un_iff
thf(fact_433_UnCI,axiom,
! [C: v,B2: set_v,A2: set_v] :
( ( ~ ( member_v @ C @ B2 )
=> ( member_v @ C @ A2 ) )
=> ( member_v @ C @ ( sup_sup_set_v @ A2 @ B2 ) ) ) ).
% UnCI
thf(fact_434_UnCI,axiom,
! [C: product_prod_v_v,B2: set_Product_prod_v_v,A2: set_Product_prod_v_v] :
( ( ~ ( member7453568604450474000od_v_v @ C @ B2 )
=> ( member7453568604450474000od_v_v @ C @ A2 ) )
=> ( member7453568604450474000od_v_v @ C @ ( sup_su414716646722978715od_v_v @ A2 @ B2 ) ) ) ).
% UnCI
thf(fact_435_subscc__add,axiom,
! [S: set_v,X: v,Y: v] :
( ( sCC_Bl5398416737448265317bscc_v @ successors @ S )
=> ( ( member_v @ X @ S )
=> ( ( sCC_Bl649662514949026229able_v @ successors @ X @ Y )
=> ( ( sCC_Bl649662514949026229able_v @ successors @ Y @ X )
=> ( sCC_Bl5398416737448265317bscc_v @ successors @ ( insert_v @ Y @ S ) ) ) ) ) ) ).
% subscc_add
thf(fact_436_le__sup__iff,axiom,
! [X: set_v,Y: set_v,Z: set_v] :
( ( ord_less_eq_set_v @ ( sup_sup_set_v @ X @ Y ) @ Z )
= ( ( ord_less_eq_set_v @ X @ Z )
& ( ord_less_eq_set_v @ Y @ Z ) ) ) ).
% le_sup_iff
thf(fact_437_le__sup__iff,axiom,
! [X: set_Product_prod_v_v,Y: set_Product_prod_v_v,Z: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ ( sup_su414716646722978715od_v_v @ X @ Y ) @ Z )
= ( ( ord_le7336532860387713383od_v_v @ X @ Z )
& ( ord_le7336532860387713383od_v_v @ Y @ Z ) ) ) ).
% le_sup_iff
thf(fact_438_sup_Obounded__iff,axiom,
! [B: set_v,C: set_v,A: set_v] :
( ( ord_less_eq_set_v @ ( sup_sup_set_v @ B @ C ) @ A )
= ( ( ord_less_eq_set_v @ B @ A )
& ( ord_less_eq_set_v @ C @ A ) ) ) ).
% sup.bounded_iff
thf(fact_439_sup_Obounded__iff,axiom,
! [B: set_Product_prod_v_v,C: set_Product_prod_v_v,A: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ ( sup_su414716646722978715od_v_v @ B @ C ) @ A )
= ( ( ord_le7336532860387713383od_v_v @ B @ A )
& ( ord_le7336532860387713383od_v_v @ C @ A ) ) ) ).
% sup.bounded_iff
thf(fact_440_sup__bot__left,axiom,
! [X: set_v] :
( ( sup_sup_set_v @ bot_bot_set_v @ X )
= X ) ).
% sup_bot_left
thf(fact_441_sup__bot__left,axiom,
! [X: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ bot_bo723834152578015283od_v_v @ X )
= X ) ).
% sup_bot_left
thf(fact_442_sup__bot__right,axiom,
! [X: set_v] :
( ( sup_sup_set_v @ X @ bot_bot_set_v )
= X ) ).
% sup_bot_right
thf(fact_443_sup__bot__right,axiom,
! [X: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ X @ bot_bo723834152578015283od_v_v )
= X ) ).
% sup_bot_right
thf(fact_444_bot__eq__sup__iff,axiom,
! [X: set_v,Y: set_v] :
( ( bot_bot_set_v
= ( sup_sup_set_v @ X @ Y ) )
= ( ( X = bot_bot_set_v )
& ( Y = bot_bot_set_v ) ) ) ).
% bot_eq_sup_iff
thf(fact_445_bot__eq__sup__iff,axiom,
! [X: set_Product_prod_v_v,Y: set_Product_prod_v_v] :
( ( bot_bo723834152578015283od_v_v
= ( sup_su414716646722978715od_v_v @ X @ Y ) )
= ( ( X = bot_bo723834152578015283od_v_v )
& ( Y = bot_bo723834152578015283od_v_v ) ) ) ).
% bot_eq_sup_iff
thf(fact_446_sup__eq__bot__iff,axiom,
! [X: set_v,Y: set_v] :
( ( ( sup_sup_set_v @ X @ Y )
= bot_bot_set_v )
= ( ( X = bot_bot_set_v )
& ( Y = bot_bot_set_v ) ) ) ).
% sup_eq_bot_iff
thf(fact_447_sup__eq__bot__iff,axiom,
! [X: set_Product_prod_v_v,Y: set_Product_prod_v_v] :
( ( ( sup_su414716646722978715od_v_v @ X @ Y )
= bot_bo723834152578015283od_v_v )
= ( ( X = bot_bo723834152578015283od_v_v )
& ( Y = bot_bo723834152578015283od_v_v ) ) ) ).
% sup_eq_bot_iff
thf(fact_448_sup__bot_Oeq__neutr__iff,axiom,
! [A: set_v,B: set_v] :
( ( ( sup_sup_set_v @ A @ B )
= bot_bot_set_v )
= ( ( A = bot_bot_set_v )
& ( B = bot_bot_set_v ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_449_sup__bot_Oeq__neutr__iff,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ( sup_su414716646722978715od_v_v @ A @ B )
= bot_bo723834152578015283od_v_v )
= ( ( A = bot_bo723834152578015283od_v_v )
& ( B = bot_bo723834152578015283od_v_v ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_450_sup__bot_Oleft__neutral,axiom,
! [A: set_v] :
( ( sup_sup_set_v @ bot_bot_set_v @ A )
= A ) ).
% sup_bot.left_neutral
thf(fact_451_sup__bot_Oleft__neutral,axiom,
! [A: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ bot_bo723834152578015283od_v_v @ A )
= A ) ).
% sup_bot.left_neutral
thf(fact_452_sup__bot_Oneutr__eq__iff,axiom,
! [A: set_v,B: set_v] :
( ( bot_bot_set_v
= ( sup_sup_set_v @ A @ B ) )
= ( ( A = bot_bot_set_v )
& ( B = bot_bot_set_v ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_453_sup__bot_Oneutr__eq__iff,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( bot_bo723834152578015283od_v_v
= ( sup_su414716646722978715od_v_v @ A @ B ) )
= ( ( A = bot_bo723834152578015283od_v_v )
& ( B = bot_bo723834152578015283od_v_v ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_454_sup__bot_Oright__neutral,axiom,
! [A: set_v] :
( ( sup_sup_set_v @ A @ bot_bot_set_v )
= A ) ).
% sup_bot.right_neutral
thf(fact_455_sup__bot_Oright__neutral,axiom,
! [A: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ A @ bot_bo723834152578015283od_v_v )
= A ) ).
% sup_bot.right_neutral
thf(fact_456_singletonI,axiom,
! [A: v] : ( member_v @ A @ ( insert_v @ A @ bot_bot_set_v ) ) ).
% singletonI
thf(fact_457_singletonI,axiom,
! [A: product_prod_v_v] : ( member7453568604450474000od_v_v @ A @ ( insert1338601472111419319od_v_v @ A @ bot_bo723834152578015283od_v_v ) ) ).
% singletonI
thf(fact_458_insert__subset,axiom,
! [X: v,A2: set_v,B2: set_v] :
( ( ord_less_eq_set_v @ ( insert_v @ X @ A2 ) @ B2 )
= ( ( member_v @ X @ B2 )
& ( ord_less_eq_set_v @ A2 @ B2 ) ) ) ).
% insert_subset
thf(fact_459_insert__subset,axiom,
! [X: product_prod_v_v,A2: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ ( insert1338601472111419319od_v_v @ X @ A2 ) @ B2 )
= ( ( member7453568604450474000od_v_v @ X @ B2 )
& ( ord_le7336532860387713383od_v_v @ A2 @ B2 ) ) ) ).
% insert_subset
thf(fact_460_inf__sup__absorb,axiom,
! [X: set_v,Y: set_v] :
( ( inf_inf_set_v @ X @ ( sup_sup_set_v @ X @ Y ) )
= X ) ).
% inf_sup_absorb
thf(fact_461_inf__sup__absorb,axiom,
! [X: set_Product_prod_v_v,Y: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ X @ ( sup_su414716646722978715od_v_v @ X @ Y ) )
= X ) ).
% inf_sup_absorb
thf(fact_462_sup__inf__absorb,axiom,
! [X: set_v,Y: set_v] :
( ( sup_sup_set_v @ X @ ( inf_inf_set_v @ X @ Y ) )
= X ) ).
% sup_inf_absorb
thf(fact_463_sup__inf__absorb,axiom,
! [X: set_Product_prod_v_v,Y: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ X @ ( inf_in6271465464967711157od_v_v @ X @ Y ) )
= X ) ).
% sup_inf_absorb
thf(fact_464_Un__empty,axiom,
! [A2: set_v,B2: set_v] :
( ( ( sup_sup_set_v @ A2 @ B2 )
= bot_bot_set_v )
= ( ( A2 = bot_bot_set_v )
& ( B2 = bot_bot_set_v ) ) ) ).
% Un_empty
thf(fact_465_Un__empty,axiom,
! [A2: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( ( sup_su414716646722978715od_v_v @ A2 @ B2 )
= bot_bo723834152578015283od_v_v )
= ( ( A2 = bot_bo723834152578015283od_v_v )
& ( B2 = bot_bo723834152578015283od_v_v ) ) ) ).
% Un_empty
thf(fact_466_Int__insert__left__if0,axiom,
! [A: product_prod_v_v,C2: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ~ ( member7453568604450474000od_v_v @ A @ C2 )
=> ( ( inf_in6271465464967711157od_v_v @ ( insert1338601472111419319od_v_v @ A @ B2 ) @ C2 )
= ( inf_in6271465464967711157od_v_v @ B2 @ C2 ) ) ) ).
% Int_insert_left_if0
thf(fact_467_Int__insert__left__if0,axiom,
! [A: v,C2: set_v,B2: set_v] :
( ~ ( member_v @ A @ C2 )
=> ( ( inf_inf_set_v @ ( insert_v @ A @ B2 ) @ C2 )
= ( inf_inf_set_v @ B2 @ C2 ) ) ) ).
% Int_insert_left_if0
thf(fact_468_Int__insert__left__if1,axiom,
! [A: product_prod_v_v,C2: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ A @ C2 )
=> ( ( inf_in6271465464967711157od_v_v @ ( insert1338601472111419319od_v_v @ A @ B2 ) @ C2 )
= ( insert1338601472111419319od_v_v @ A @ ( inf_in6271465464967711157od_v_v @ B2 @ C2 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_469_Int__insert__left__if1,axiom,
! [A: v,C2: set_v,B2: set_v] :
( ( member_v @ A @ C2 )
=> ( ( inf_inf_set_v @ ( insert_v @ A @ B2 ) @ C2 )
= ( insert_v @ A @ ( inf_inf_set_v @ B2 @ C2 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_470_insert__inter__insert,axiom,
! [A: product_prod_v_v,A2: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ ( insert1338601472111419319od_v_v @ A @ A2 ) @ ( insert1338601472111419319od_v_v @ A @ B2 ) )
= ( insert1338601472111419319od_v_v @ A @ ( inf_in6271465464967711157od_v_v @ A2 @ B2 ) ) ) ).
% insert_inter_insert
thf(fact_471_insert__inter__insert,axiom,
! [A: v,A2: set_v,B2: set_v] :
( ( inf_inf_set_v @ ( insert_v @ A @ A2 ) @ ( insert_v @ A @ B2 ) )
= ( insert_v @ A @ ( inf_inf_set_v @ A2 @ B2 ) ) ) ).
% insert_inter_insert
thf(fact_472_Int__insert__right__if0,axiom,
! [A: product_prod_v_v,A2: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ~ ( member7453568604450474000od_v_v @ A @ A2 )
=> ( ( inf_in6271465464967711157od_v_v @ A2 @ ( insert1338601472111419319od_v_v @ A @ B2 ) )
= ( inf_in6271465464967711157od_v_v @ A2 @ B2 ) ) ) ).
% Int_insert_right_if0
thf(fact_473_Int__insert__right__if0,axiom,
! [A: v,A2: set_v,B2: set_v] :
( ~ ( member_v @ A @ A2 )
=> ( ( inf_inf_set_v @ A2 @ ( insert_v @ A @ B2 ) )
= ( inf_inf_set_v @ A2 @ B2 ) ) ) ).
% Int_insert_right_if0
thf(fact_474_Int__insert__right__if1,axiom,
! [A: product_prod_v_v,A2: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ A @ A2 )
=> ( ( inf_in6271465464967711157od_v_v @ A2 @ ( insert1338601472111419319od_v_v @ A @ B2 ) )
= ( insert1338601472111419319od_v_v @ A @ ( inf_in6271465464967711157od_v_v @ A2 @ B2 ) ) ) ) ).
% Int_insert_right_if1
thf(fact_475_Int__insert__right__if1,axiom,
! [A: v,A2: set_v,B2: set_v] :
( ( member_v @ A @ A2 )
=> ( ( inf_inf_set_v @ A2 @ ( insert_v @ A @ B2 ) )
= ( insert_v @ A @ ( inf_inf_set_v @ A2 @ B2 ) ) ) ) ).
% Int_insert_right_if1
thf(fact_476_Un__subset__iff,axiom,
! [A2: set_v,B2: set_v,C2: set_v] :
( ( ord_less_eq_set_v @ ( sup_sup_set_v @ A2 @ B2 ) @ C2 )
= ( ( ord_less_eq_set_v @ A2 @ C2 )
& ( ord_less_eq_set_v @ B2 @ C2 ) ) ) ).
% Un_subset_iff
thf(fact_477_Un__subset__iff,axiom,
! [A2: set_Product_prod_v_v,B2: set_Product_prod_v_v,C2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ ( sup_su414716646722978715od_v_v @ A2 @ B2 ) @ C2 )
= ( ( ord_le7336532860387713383od_v_v @ A2 @ C2 )
& ( ord_le7336532860387713383od_v_v @ B2 @ C2 ) ) ) ).
% Un_subset_iff
thf(fact_478_Un__insert__right,axiom,
! [A2: set_v,A: v,B2: set_v] :
( ( sup_sup_set_v @ A2 @ ( insert_v @ A @ B2 ) )
= ( insert_v @ A @ ( sup_sup_set_v @ A2 @ B2 ) ) ) ).
% Un_insert_right
thf(fact_479_Un__insert__right,axiom,
! [A2: set_Product_prod_v_v,A: product_prod_v_v,B2: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ A2 @ ( insert1338601472111419319od_v_v @ A @ B2 ) )
= ( insert1338601472111419319od_v_v @ A @ ( sup_su414716646722978715od_v_v @ A2 @ B2 ) ) ) ).
% Un_insert_right
thf(fact_480_Un__insert__left,axiom,
! [A: v,B2: set_v,C2: set_v] :
( ( sup_sup_set_v @ ( insert_v @ A @ B2 ) @ C2 )
= ( insert_v @ A @ ( sup_sup_set_v @ B2 @ C2 ) ) ) ).
% Un_insert_left
thf(fact_481_Un__insert__left,axiom,
! [A: product_prod_v_v,B2: set_Product_prod_v_v,C2: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ ( insert1338601472111419319od_v_v @ A @ B2 ) @ C2 )
= ( insert1338601472111419319od_v_v @ A @ ( sup_su414716646722978715od_v_v @ B2 @ C2 ) ) ) ).
% Un_insert_left
thf(fact_482_Diff__insert0,axiom,
! [X: product_prod_v_v,A2: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ~ ( member7453568604450474000od_v_v @ X @ A2 )
=> ( ( minus_4183494784930505774od_v_v @ A2 @ ( insert1338601472111419319od_v_v @ X @ B2 ) )
= ( minus_4183494784930505774od_v_v @ A2 @ B2 ) ) ) ).
% Diff_insert0
thf(fact_483_Diff__insert0,axiom,
! [X: v,A2: set_v,B2: set_v] :
( ~ ( member_v @ X @ A2 )
=> ( ( minus_minus_set_v @ A2 @ ( insert_v @ X @ B2 ) )
= ( minus_minus_set_v @ A2 @ B2 ) ) ) ).
% Diff_insert0
thf(fact_484_insert__Diff1,axiom,
! [X: product_prod_v_v,B2: set_Product_prod_v_v,A2: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X @ B2 )
=> ( ( minus_4183494784930505774od_v_v @ ( insert1338601472111419319od_v_v @ X @ A2 ) @ B2 )
= ( minus_4183494784930505774od_v_v @ A2 @ B2 ) ) ) ).
% insert_Diff1
thf(fact_485_insert__Diff1,axiom,
! [X: v,B2: set_v,A2: set_v] :
( ( member_v @ X @ B2 )
=> ( ( minus_minus_set_v @ ( insert_v @ X @ A2 ) @ B2 )
= ( minus_minus_set_v @ A2 @ B2 ) ) ) ).
% insert_Diff1
thf(fact_486_Int__Un__eq_I4_J,axiom,
! [T2: set_v,S: set_v] :
( ( sup_sup_set_v @ T2 @ ( inf_inf_set_v @ S @ T2 ) )
= T2 ) ).
% Int_Un_eq(4)
thf(fact_487_Int__Un__eq_I4_J,axiom,
! [T2: set_Product_prod_v_v,S: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ T2 @ ( inf_in6271465464967711157od_v_v @ S @ T2 ) )
= T2 ) ).
% Int_Un_eq(4)
thf(fact_488_Int__Un__eq_I3_J,axiom,
! [S: set_v,T2: set_v] :
( ( sup_sup_set_v @ S @ ( inf_inf_set_v @ S @ T2 ) )
= S ) ).
% Int_Un_eq(3)
thf(fact_489_Int__Un__eq_I3_J,axiom,
! [S: set_Product_prod_v_v,T2: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ S @ ( inf_in6271465464967711157od_v_v @ S @ T2 ) )
= S ) ).
% Int_Un_eq(3)
thf(fact_490_Int__Un__eq_I2_J,axiom,
! [S: set_v,T2: set_v] :
( ( sup_sup_set_v @ ( inf_inf_set_v @ S @ T2 ) @ T2 )
= T2 ) ).
% Int_Un_eq(2)
thf(fact_491_Int__Un__eq_I2_J,axiom,
! [S: set_Product_prod_v_v,T2: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ S @ T2 ) @ T2 )
= T2 ) ).
% Int_Un_eq(2)
thf(fact_492_Int__Un__eq_I1_J,axiom,
! [S: set_v,T2: set_v] :
( ( sup_sup_set_v @ ( inf_inf_set_v @ S @ T2 ) @ S )
= S ) ).
% Int_Un_eq(1)
thf(fact_493_Int__Un__eq_I1_J,axiom,
! [S: set_Product_prod_v_v,T2: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ S @ T2 ) @ S )
= S ) ).
% Int_Un_eq(1)
thf(fact_494_Un__Int__eq_I4_J,axiom,
! [T2: set_v,S: set_v] :
( ( inf_inf_set_v @ T2 @ ( sup_sup_set_v @ S @ T2 ) )
= T2 ) ).
% Un_Int_eq(4)
thf(fact_495_Un__Int__eq_I4_J,axiom,
! [T2: set_Product_prod_v_v,S: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ T2 @ ( sup_su414716646722978715od_v_v @ S @ T2 ) )
= T2 ) ).
% Un_Int_eq(4)
thf(fact_496_Un__Int__eq_I3_J,axiom,
! [S: set_v,T2: set_v] :
( ( inf_inf_set_v @ S @ ( sup_sup_set_v @ S @ T2 ) )
= S ) ).
% Un_Int_eq(3)
thf(fact_497_Un__Int__eq_I3_J,axiom,
! [S: set_Product_prod_v_v,T2: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ S @ ( sup_su414716646722978715od_v_v @ S @ T2 ) )
= S ) ).
% Un_Int_eq(3)
thf(fact_498_Un__Int__eq_I2_J,axiom,
! [S: set_v,T2: set_v] :
( ( inf_inf_set_v @ ( sup_sup_set_v @ S @ T2 ) @ T2 )
= T2 ) ).
% Un_Int_eq(2)
thf(fact_499_Un__Int__eq_I2_J,axiom,
! [S: set_Product_prod_v_v,T2: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ ( sup_su414716646722978715od_v_v @ S @ T2 ) @ T2 )
= T2 ) ).
% Un_Int_eq(2)
thf(fact_500_Un__Int__eq_I1_J,axiom,
! [S: set_v,T2: set_v] :
( ( inf_inf_set_v @ ( sup_sup_set_v @ S @ T2 ) @ S )
= S ) ).
% Un_Int_eq(1)
thf(fact_501_Un__Int__eq_I1_J,axiom,
! [S: set_Product_prod_v_v,T2: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ ( sup_su414716646722978715od_v_v @ S @ T2 ) @ S )
= S ) ).
% Un_Int_eq(1)
thf(fact_502_Un__Diff__cancel,axiom,
! [A2: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ A2 @ ( minus_4183494784930505774od_v_v @ B2 @ A2 ) )
= ( sup_su414716646722978715od_v_v @ A2 @ B2 ) ) ).
% Un_Diff_cancel
thf(fact_503_Un__Diff__cancel,axiom,
! [A2: set_v,B2: set_v] :
( ( sup_sup_set_v @ A2 @ ( minus_minus_set_v @ B2 @ A2 ) )
= ( sup_sup_set_v @ A2 @ B2 ) ) ).
% Un_Diff_cancel
thf(fact_504_Un__Diff__cancel2,axiom,
! [B2: set_Product_prod_v_v,A2: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ ( minus_4183494784930505774od_v_v @ B2 @ A2 ) @ A2 )
= ( sup_su414716646722978715od_v_v @ B2 @ A2 ) ) ).
% Un_Diff_cancel2
thf(fact_505_Un__Diff__cancel2,axiom,
! [B2: set_v,A2: set_v] :
( ( sup_sup_set_v @ ( minus_minus_set_v @ B2 @ A2 ) @ A2 )
= ( sup_sup_set_v @ B2 @ A2 ) ) ).
% Un_Diff_cancel2
thf(fact_506_singleton__insert__inj__eq,axiom,
! [B: v,A: v,A2: set_v] :
( ( ( insert_v @ B @ bot_bot_set_v )
= ( insert_v @ A @ A2 ) )
= ( ( A = B )
& ( ord_less_eq_set_v @ A2 @ ( insert_v @ B @ bot_bot_set_v ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_507_singleton__insert__inj__eq,axiom,
! [B: product_prod_v_v,A: product_prod_v_v,A2: set_Product_prod_v_v] :
( ( ( insert1338601472111419319od_v_v @ B @ bot_bo723834152578015283od_v_v )
= ( insert1338601472111419319od_v_v @ A @ A2 ) )
= ( ( A = B )
& ( ord_le7336532860387713383od_v_v @ A2 @ ( insert1338601472111419319od_v_v @ B @ bot_bo723834152578015283od_v_v ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_508_singleton__insert__inj__eq_H,axiom,
! [A: v,A2: set_v,B: v] :
( ( ( insert_v @ A @ A2 )
= ( insert_v @ B @ bot_bot_set_v ) )
= ( ( A = B )
& ( ord_less_eq_set_v @ A2 @ ( insert_v @ B @ bot_bot_set_v ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_509_singleton__insert__inj__eq_H,axiom,
! [A: product_prod_v_v,A2: set_Product_prod_v_v,B: product_prod_v_v] :
( ( ( insert1338601472111419319od_v_v @ A @ A2 )
= ( insert1338601472111419319od_v_v @ B @ bot_bo723834152578015283od_v_v ) )
= ( ( A = B )
& ( ord_le7336532860387713383od_v_v @ A2 @ ( insert1338601472111419319od_v_v @ B @ bot_bo723834152578015283od_v_v ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_510_list_Osimps_I15_J,axiom,
! [X21: product_prod_v_v,X22: list_P7986770385144383213od_v_v] :
( ( set_Product_prod_v_v2 @ ( cons_P4120604216776828829od_v_v @ X21 @ X22 ) )
= ( insert1338601472111419319od_v_v @ X21 @ ( set_Product_prod_v_v2 @ X22 ) ) ) ).
% list.simps(15)
thf(fact_511_list_Osimps_I15_J,axiom,
! [X21: v,X22: list_v] :
( ( set_v2 @ ( cons_v @ X21 @ X22 ) )
= ( insert_v @ X21 @ ( set_v2 @ X22 ) ) ) ).
% list.simps(15)
thf(fact_512_disjoint__insert_I2_J,axiom,
! [A2: set_v,B: v,B2: set_v] :
( ( bot_bot_set_v
= ( inf_inf_set_v @ A2 @ ( insert_v @ B @ B2 ) ) )
= ( ~ ( member_v @ B @ A2 )
& ( bot_bot_set_v
= ( inf_inf_set_v @ A2 @ B2 ) ) ) ) ).
% disjoint_insert(2)
thf(fact_513_disjoint__insert_I2_J,axiom,
! [A2: set_Product_prod_v_v,B: product_prod_v_v,B2: set_Product_prod_v_v] :
( ( bot_bo723834152578015283od_v_v
= ( inf_in6271465464967711157od_v_v @ A2 @ ( insert1338601472111419319od_v_v @ B @ B2 ) ) )
= ( ~ ( member7453568604450474000od_v_v @ B @ A2 )
& ( bot_bo723834152578015283od_v_v
= ( inf_in6271465464967711157od_v_v @ A2 @ B2 ) ) ) ) ).
% disjoint_insert(2)
thf(fact_514_disjoint__insert_I1_J,axiom,
! [B2: set_v,A: v,A2: set_v] :
( ( ( inf_inf_set_v @ B2 @ ( insert_v @ A @ A2 ) )
= bot_bot_set_v )
= ( ~ ( member_v @ A @ B2 )
& ( ( inf_inf_set_v @ B2 @ A2 )
= bot_bot_set_v ) ) ) ).
% disjoint_insert(1)
thf(fact_515_disjoint__insert_I1_J,axiom,
! [B2: set_Product_prod_v_v,A: product_prod_v_v,A2: set_Product_prod_v_v] :
( ( ( inf_in6271465464967711157od_v_v @ B2 @ ( insert1338601472111419319od_v_v @ A @ A2 ) )
= bot_bo723834152578015283od_v_v )
= ( ~ ( member7453568604450474000od_v_v @ A @ B2 )
& ( ( inf_in6271465464967711157od_v_v @ B2 @ A2 )
= bot_bo723834152578015283od_v_v ) ) ) ).
% disjoint_insert(1)
thf(fact_516_insert__disjoint_I2_J,axiom,
! [A: v,A2: set_v,B2: set_v] :
( ( bot_bot_set_v
= ( inf_inf_set_v @ ( insert_v @ A @ A2 ) @ B2 ) )
= ( ~ ( member_v @ A @ B2 )
& ( bot_bot_set_v
= ( inf_inf_set_v @ A2 @ B2 ) ) ) ) ).
% insert_disjoint(2)
thf(fact_517_insert__disjoint_I2_J,axiom,
! [A: product_prod_v_v,A2: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( bot_bo723834152578015283od_v_v
= ( inf_in6271465464967711157od_v_v @ ( insert1338601472111419319od_v_v @ A @ A2 ) @ B2 ) )
= ( ~ ( member7453568604450474000od_v_v @ A @ B2 )
& ( bot_bo723834152578015283od_v_v
= ( inf_in6271465464967711157od_v_v @ A2 @ B2 ) ) ) ) ).
% insert_disjoint(2)
thf(fact_518_insert__disjoint_I1_J,axiom,
! [A: v,A2: set_v,B2: set_v] :
( ( ( inf_inf_set_v @ ( insert_v @ A @ A2 ) @ B2 )
= bot_bot_set_v )
= ( ~ ( member_v @ A @ B2 )
& ( ( inf_inf_set_v @ A2 @ B2 )
= bot_bot_set_v ) ) ) ).
% insert_disjoint(1)
thf(fact_519_insert__disjoint_I1_J,axiom,
! [A: product_prod_v_v,A2: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( ( inf_in6271465464967711157od_v_v @ ( insert1338601472111419319od_v_v @ A @ A2 ) @ B2 )
= bot_bo723834152578015283od_v_v )
= ( ~ ( member7453568604450474000od_v_v @ A @ B2 )
& ( ( inf_in6271465464967711157od_v_v @ A2 @ B2 )
= bot_bo723834152578015283od_v_v ) ) ) ).
% insert_disjoint(1)
thf(fact_520_insert__Diff__single,axiom,
! [A: product_prod_v_v,A2: set_Product_prod_v_v] :
( ( insert1338601472111419319od_v_v @ A @ ( minus_4183494784930505774od_v_v @ A2 @ ( insert1338601472111419319od_v_v @ A @ bot_bo723834152578015283od_v_v ) ) )
= ( insert1338601472111419319od_v_v @ A @ A2 ) ) ).
% insert_Diff_single
thf(fact_521_insert__Diff__single,axiom,
! [A: v,A2: set_v] :
( ( insert_v @ A @ ( minus_minus_set_v @ A2 @ ( insert_v @ A @ bot_bot_set_v ) ) )
= ( insert_v @ A @ A2 ) ) ).
% insert_Diff_single
thf(fact_522_append1__eq__conv,axiom,
! [Xs: list_v,X: v,Ys: list_v,Y: v] :
( ( ( append_v @ Xs @ ( cons_v @ X @ nil_v ) )
= ( append_v @ Ys @ ( cons_v @ Y @ nil_v ) ) )
= ( ( Xs = Ys )
& ( X = Y ) ) ) ).
% append1_eq_conv
thf(fact_523_set__append,axiom,
! [Xs: list_v,Ys: list_v] :
( ( set_v2 @ ( append_v @ Xs @ Ys ) )
= ( sup_sup_set_v @ ( set_v2 @ Xs ) @ ( set_v2 @ Ys ) ) ) ).
% set_append
thf(fact_524_set__append,axiom,
! [Xs: list_P7986770385144383213od_v_v,Ys: list_P7986770385144383213od_v_v] :
( ( set_Product_prod_v_v2 @ ( append2138873909117096322od_v_v @ Xs @ Ys ) )
= ( sup_su414716646722978715od_v_v @ ( set_Product_prod_v_v2 @ Xs ) @ ( set_Product_prod_v_v2 @ Ys ) ) ) ).
% set_append
thf(fact_525_list_Ocollapse,axiom,
! [List: list_v] :
( ( List != nil_v )
=> ( ( cons_v @ ( hd_v @ List ) @ ( tl_v @ List ) )
= List ) ) ).
% list.collapse
thf(fact_526_hd__Cons__tl,axiom,
! [Xs: list_v] :
( ( Xs != nil_v )
=> ( ( cons_v @ ( hd_v @ Xs ) @ ( tl_v @ Xs ) )
= Xs ) ) ).
% hd_Cons_tl
thf(fact_527__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062l_O_A_092_060lbrakk_062l_A_092_060in_062_Aset_Apfx_A_092_060union_062_A_123hd_A_Istack_Ae_H_J_125_059_Am_A_092_060in_062_A_092_060S_062_Ae_Al_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [L: v] :
( ( member_v @ L @ ( sup_sup_set_v @ ( set_v2 @ pfx ) @ ( insert_v @ ( hd_v @ ( sCC_Bl8828226123343373779t_unit @ e2 ) ) @ bot_bot_set_v ) ) )
=> ~ ( member_v @ m @ ( sCC_Bl1280885523602775798t_unit @ e @ L ) ) ) ).
% \<open>\<And>thesis. (\<And>l. \<lbrakk>l \<in> set pfx \<union> {hd (stack e')}; m \<in> \<S> e l\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_528_singleton__Un__iff,axiom,
! [X: v,A2: set_v,B2: set_v] :
( ( ( insert_v @ X @ bot_bot_set_v )
= ( sup_sup_set_v @ A2 @ B2 ) )
= ( ( ( A2 = bot_bot_set_v )
& ( B2
= ( insert_v @ X @ bot_bot_set_v ) ) )
| ( ( A2
= ( insert_v @ X @ bot_bot_set_v ) )
& ( B2 = bot_bot_set_v ) )
| ( ( A2
= ( insert_v @ X @ bot_bot_set_v ) )
& ( B2
= ( insert_v @ X @ bot_bot_set_v ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_529_singleton__Un__iff,axiom,
! [X: product_prod_v_v,A2: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v )
= ( sup_su414716646722978715od_v_v @ A2 @ B2 ) )
= ( ( ( A2 = bot_bo723834152578015283od_v_v )
& ( B2
= ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v ) ) )
| ( ( A2
= ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v ) )
& ( B2 = bot_bo723834152578015283od_v_v ) )
| ( ( A2
= ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v ) )
& ( B2
= ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_530_Un__singleton__iff,axiom,
! [A2: set_v,B2: set_v,X: v] :
( ( ( sup_sup_set_v @ A2 @ B2 )
= ( insert_v @ X @ bot_bot_set_v ) )
= ( ( ( A2 = bot_bot_set_v )
& ( B2
= ( insert_v @ X @ bot_bot_set_v ) ) )
| ( ( A2
= ( insert_v @ X @ bot_bot_set_v ) )
& ( B2 = bot_bot_set_v ) )
| ( ( A2
= ( insert_v @ X @ bot_bot_set_v ) )
& ( B2
= ( insert_v @ X @ bot_bot_set_v ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_531_Un__singleton__iff,axiom,
! [A2: set_Product_prod_v_v,B2: set_Product_prod_v_v,X: product_prod_v_v] :
( ( ( sup_su414716646722978715od_v_v @ A2 @ B2 )
= ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v ) )
= ( ( ( A2 = bot_bo723834152578015283od_v_v )
& ( B2
= ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v ) ) )
| ( ( A2
= ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v ) )
& ( B2 = bot_bo723834152578015283od_v_v ) )
| ( ( A2
= ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v ) )
& ( B2
= ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_532_insert__is__Un,axiom,
( insert_v
= ( ^ [A4: v] : ( sup_sup_set_v @ ( insert_v @ A4 @ bot_bot_set_v ) ) ) ) ).
% insert_is_Un
thf(fact_533_insert__is__Un,axiom,
( insert1338601472111419319od_v_v
= ( ^ [A4: product_prod_v_v] : ( sup_su414716646722978715od_v_v @ ( insert1338601472111419319od_v_v @ A4 @ bot_bo723834152578015283od_v_v ) ) ) ) ).
% insert_is_Un
thf(fact_534_mk__disjoint__insert,axiom,
! [A: v,A2: set_v] :
( ( member_v @ A @ A2 )
=> ? [B5: set_v] :
( ( A2
= ( insert_v @ A @ B5 ) )
& ~ ( member_v @ A @ B5 ) ) ) ).
% mk_disjoint_insert
thf(fact_535_mk__disjoint__insert,axiom,
! [A: product_prod_v_v,A2: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ A @ A2 )
=> ? [B5: set_Product_prod_v_v] :
( ( A2
= ( insert1338601472111419319od_v_v @ A @ B5 ) )
& ~ ( member7453568604450474000od_v_v @ A @ B5 ) ) ) ).
% mk_disjoint_insert
thf(fact_536_Un__left__commute,axiom,
! [A2: set_v,B2: set_v,C2: set_v] :
( ( sup_sup_set_v @ A2 @ ( sup_sup_set_v @ B2 @ C2 ) )
= ( sup_sup_set_v @ B2 @ ( sup_sup_set_v @ A2 @ C2 ) ) ) ).
% Un_left_commute
thf(fact_537_Un__left__commute,axiom,
! [A2: set_Product_prod_v_v,B2: set_Product_prod_v_v,C2: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ A2 @ ( sup_su414716646722978715od_v_v @ B2 @ C2 ) )
= ( sup_su414716646722978715od_v_v @ B2 @ ( sup_su414716646722978715od_v_v @ A2 @ C2 ) ) ) ).
% Un_left_commute
thf(fact_538_insert__commute,axiom,
! [X: v,Y: v,A2: set_v] :
( ( insert_v @ X @ ( insert_v @ Y @ A2 ) )
= ( insert_v @ Y @ ( insert_v @ X @ A2 ) ) ) ).
% insert_commute
thf(fact_539_insert__commute,axiom,
! [X: product_prod_v_v,Y: product_prod_v_v,A2: set_Product_prod_v_v] :
( ( insert1338601472111419319od_v_v @ X @ ( insert1338601472111419319od_v_v @ Y @ A2 ) )
= ( insert1338601472111419319od_v_v @ Y @ ( insert1338601472111419319od_v_v @ X @ A2 ) ) ) ).
% insert_commute
thf(fact_540_Un__left__absorb,axiom,
! [A2: set_v,B2: set_v] :
( ( sup_sup_set_v @ A2 @ ( sup_sup_set_v @ A2 @ B2 ) )
= ( sup_sup_set_v @ A2 @ B2 ) ) ).
% Un_left_absorb
thf(fact_541_Un__left__absorb,axiom,
! [A2: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ A2 @ ( sup_su414716646722978715od_v_v @ A2 @ B2 ) )
= ( sup_su414716646722978715od_v_v @ A2 @ B2 ) ) ).
% Un_left_absorb
thf(fact_542_insert__eq__iff,axiom,
! [A: v,A2: set_v,B: v,B2: set_v] :
( ~ ( member_v @ A @ A2 )
=> ( ~ ( member_v @ B @ B2 )
=> ( ( ( insert_v @ A @ A2 )
= ( insert_v @ B @ B2 ) )
= ( ( ( A = B )
=> ( A2 = B2 ) )
& ( ( A != B )
=> ? [C3: set_v] :
( ( A2
= ( insert_v @ B @ C3 ) )
& ~ ( member_v @ B @ C3 )
& ( B2
= ( insert_v @ A @ C3 ) )
& ~ ( member_v @ A @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_543_insert__eq__iff,axiom,
! [A: product_prod_v_v,A2: set_Product_prod_v_v,B: product_prod_v_v,B2: set_Product_prod_v_v] :
( ~ ( member7453568604450474000od_v_v @ A @ A2 )
=> ( ~ ( member7453568604450474000od_v_v @ B @ B2 )
=> ( ( ( insert1338601472111419319od_v_v @ A @ A2 )
= ( insert1338601472111419319od_v_v @ B @ B2 ) )
= ( ( ( A = B )
=> ( A2 = B2 ) )
& ( ( A != B )
=> ? [C3: set_Product_prod_v_v] :
( ( A2
= ( insert1338601472111419319od_v_v @ B @ C3 ) )
& ~ ( member7453568604450474000od_v_v @ B @ C3 )
& ( B2
= ( insert1338601472111419319od_v_v @ A @ C3 ) )
& ~ ( member7453568604450474000od_v_v @ A @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_544_insert__absorb,axiom,
! [A: v,A2: set_v] :
( ( member_v @ A @ A2 )
=> ( ( insert_v @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_545_insert__absorb,axiom,
! [A: product_prod_v_v,A2: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ A @ A2 )
=> ( ( insert1338601472111419319od_v_v @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_546_insert__ident,axiom,
! [X: v,A2: set_v,B2: set_v] :
( ~ ( member_v @ X @ A2 )
=> ( ~ ( member_v @ X @ B2 )
=> ( ( ( insert_v @ X @ A2 )
= ( insert_v @ X @ B2 ) )
= ( A2 = B2 ) ) ) ) ).
% insert_ident
thf(fact_547_insert__ident,axiom,
! [X: product_prod_v_v,A2: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ~ ( member7453568604450474000od_v_v @ X @ A2 )
=> ( ~ ( member7453568604450474000od_v_v @ X @ B2 )
=> ( ( ( insert1338601472111419319od_v_v @ X @ A2 )
= ( insert1338601472111419319od_v_v @ X @ B2 ) )
= ( A2 = B2 ) ) ) ) ).
% insert_ident
thf(fact_548_Set_Oset__insert,axiom,
! [X: v,A2: set_v] :
( ( member_v @ X @ A2 )
=> ~ ! [B5: set_v] :
( ( A2
= ( insert_v @ X @ B5 ) )
=> ( member_v @ X @ B5 ) ) ) ).
% Set.set_insert
thf(fact_549_Set_Oset__insert,axiom,
! [X: product_prod_v_v,A2: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X @ A2 )
=> ~ ! [B5: set_Product_prod_v_v] :
( ( A2
= ( insert1338601472111419319od_v_v @ X @ B5 ) )
=> ( member7453568604450474000od_v_v @ X @ B5 ) ) ) ).
% Set.set_insert
thf(fact_550_Un__commute,axiom,
( sup_sup_set_v
= ( ^ [A3: set_v,B3: set_v] : ( sup_sup_set_v @ B3 @ A3 ) ) ) ).
% Un_commute
thf(fact_551_Un__commute,axiom,
( sup_su414716646722978715od_v_v
= ( ^ [A3: set_Product_prod_v_v,B3: set_Product_prod_v_v] : ( sup_su414716646722978715od_v_v @ B3 @ A3 ) ) ) ).
% Un_commute
thf(fact_552_Un__absorb,axiom,
! [A2: set_v] :
( ( sup_sup_set_v @ A2 @ A2 )
= A2 ) ).
% Un_absorb
thf(fact_553_Un__absorb,axiom,
! [A2: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ A2 @ A2 )
= A2 ) ).
% Un_absorb
thf(fact_554_insertI2,axiom,
! [A: v,B2: set_v,B: v] :
( ( member_v @ A @ B2 )
=> ( member_v @ A @ ( insert_v @ B @ B2 ) ) ) ).
% insertI2
thf(fact_555_insertI2,axiom,
! [A: product_prod_v_v,B2: set_Product_prod_v_v,B: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ A @ B2 )
=> ( member7453568604450474000od_v_v @ A @ ( insert1338601472111419319od_v_v @ B @ B2 ) ) ) ).
% insertI2
thf(fact_556_insertI1,axiom,
! [A: v,B2: set_v] : ( member_v @ A @ ( insert_v @ A @ B2 ) ) ).
% insertI1
thf(fact_557_insertI1,axiom,
! [A: product_prod_v_v,B2: set_Product_prod_v_v] : ( member7453568604450474000od_v_v @ A @ ( insert1338601472111419319od_v_v @ A @ B2 ) ) ).
% insertI1
thf(fact_558_Un__assoc,axiom,
! [A2: set_v,B2: set_v,C2: set_v] :
( ( sup_sup_set_v @ ( sup_sup_set_v @ A2 @ B2 ) @ C2 )
= ( sup_sup_set_v @ A2 @ ( sup_sup_set_v @ B2 @ C2 ) ) ) ).
% Un_assoc
thf(fact_559_Un__assoc,axiom,
! [A2: set_Product_prod_v_v,B2: set_Product_prod_v_v,C2: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ ( sup_su414716646722978715od_v_v @ A2 @ B2 ) @ C2 )
= ( sup_su414716646722978715od_v_v @ A2 @ ( sup_su414716646722978715od_v_v @ B2 @ C2 ) ) ) ).
% Un_assoc
thf(fact_560_insertE,axiom,
! [A: v,B: v,A2: set_v] :
( ( member_v @ A @ ( insert_v @ B @ A2 ) )
=> ( ( A != B )
=> ( member_v @ A @ A2 ) ) ) ).
% insertE
thf(fact_561_insertE,axiom,
! [A: product_prod_v_v,B: product_prod_v_v,A2: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ A @ ( insert1338601472111419319od_v_v @ B @ A2 ) )
=> ( ( A != B )
=> ( member7453568604450474000od_v_v @ A @ A2 ) ) ) ).
% insertE
thf(fact_562_ball__Un,axiom,
! [A2: set_v,B2: set_v,P: v > $o] :
( ( ! [X3: v] :
( ( member_v @ X3 @ ( sup_sup_set_v @ A2 @ B2 ) )
=> ( P @ X3 ) ) )
= ( ! [X3: v] :
( ( member_v @ X3 @ A2 )
=> ( P @ X3 ) )
& ! [X3: v] :
( ( member_v @ X3 @ B2 )
=> ( P @ X3 ) ) ) ) ).
% ball_Un
thf(fact_563_ball__Un,axiom,
! [A2: set_Product_prod_v_v,B2: set_Product_prod_v_v,P: product_prod_v_v > $o] :
( ( ! [X3: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X3 @ ( sup_su414716646722978715od_v_v @ A2 @ B2 ) )
=> ( P @ X3 ) ) )
= ( ! [X3: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X3 @ A2 )
=> ( P @ X3 ) )
& ! [X3: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X3 @ B2 )
=> ( P @ X3 ) ) ) ) ).
% ball_Un
thf(fact_564_bex__Un,axiom,
! [A2: set_v,B2: set_v,P: v > $o] :
( ( ? [X3: v] :
( ( member_v @ X3 @ ( sup_sup_set_v @ A2 @ B2 ) )
& ( P @ X3 ) ) )
= ( ? [X3: v] :
( ( member_v @ X3 @ A2 )
& ( P @ X3 ) )
| ? [X3: v] :
( ( member_v @ X3 @ B2 )
& ( P @ X3 ) ) ) ) ).
% bex_Un
thf(fact_565_bex__Un,axiom,
! [A2: set_Product_prod_v_v,B2: set_Product_prod_v_v,P: product_prod_v_v > $o] :
( ( ? [X3: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X3 @ ( sup_su414716646722978715od_v_v @ A2 @ B2 ) )
& ( P @ X3 ) ) )
= ( ? [X3: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X3 @ A2 )
& ( P @ X3 ) )
| ? [X3: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X3 @ B2 )
& ( P @ X3 ) ) ) ) ).
% bex_Un
thf(fact_566_UnI2,axiom,
! [C: v,B2: set_v,A2: set_v] :
( ( member_v @ C @ B2 )
=> ( member_v @ C @ ( sup_sup_set_v @ A2 @ B2 ) ) ) ).
% UnI2
thf(fact_567_UnI2,axiom,
! [C: product_prod_v_v,B2: set_Product_prod_v_v,A2: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ C @ B2 )
=> ( member7453568604450474000od_v_v @ C @ ( sup_su414716646722978715od_v_v @ A2 @ B2 ) ) ) ).
% UnI2
thf(fact_568_UnI1,axiom,
! [C: v,A2: set_v,B2: set_v] :
( ( member_v @ C @ A2 )
=> ( member_v @ C @ ( sup_sup_set_v @ A2 @ B2 ) ) ) ).
% UnI1
thf(fact_569_UnI1,axiom,
! [C: product_prod_v_v,A2: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ C @ A2 )
=> ( member7453568604450474000od_v_v @ C @ ( sup_su414716646722978715od_v_v @ A2 @ B2 ) ) ) ).
% UnI1
thf(fact_570_UnE,axiom,
! [C: v,A2: set_v,B2: set_v] :
( ( member_v @ C @ ( sup_sup_set_v @ A2 @ B2 ) )
=> ( ~ ( member_v @ C @ A2 )
=> ( member_v @ C @ B2 ) ) ) ).
% UnE
thf(fact_571_UnE,axiom,
! [C: product_prod_v_v,A2: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ C @ ( sup_su414716646722978715od_v_v @ A2 @ B2 ) )
=> ( ~ ( member7453568604450474000od_v_v @ C @ A2 )
=> ( member7453568604450474000od_v_v @ C @ B2 ) ) ) ).
% UnE
thf(fact_572_sup__left__commute,axiom,
! [X: set_v,Y: set_v,Z: set_v] :
( ( sup_sup_set_v @ X @ ( sup_sup_set_v @ Y @ Z ) )
= ( sup_sup_set_v @ Y @ ( sup_sup_set_v @ X @ Z ) ) ) ).
% sup_left_commute
thf(fact_573_sup__left__commute,axiom,
! [X: set_Product_prod_v_v,Y: set_Product_prod_v_v,Z: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ X @ ( sup_su414716646722978715od_v_v @ Y @ Z ) )
= ( sup_su414716646722978715od_v_v @ Y @ ( sup_su414716646722978715od_v_v @ X @ Z ) ) ) ).
% sup_left_commute
thf(fact_574_sup_Oleft__commute,axiom,
! [B: set_v,A: set_v,C: set_v] :
( ( sup_sup_set_v @ B @ ( sup_sup_set_v @ A @ C ) )
= ( sup_sup_set_v @ A @ ( sup_sup_set_v @ B @ C ) ) ) ).
% sup.left_commute
thf(fact_575_sup_Oleft__commute,axiom,
! [B: set_Product_prod_v_v,A: set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ B @ ( sup_su414716646722978715od_v_v @ A @ C ) )
= ( sup_su414716646722978715od_v_v @ A @ ( sup_su414716646722978715od_v_v @ B @ C ) ) ) ).
% sup.left_commute
thf(fact_576_boolean__algebra__cancel_Osup2,axiom,
! [B2: set_v,K: set_v,B: set_v,A: set_v] :
( ( B2
= ( sup_sup_set_v @ K @ B ) )
=> ( ( sup_sup_set_v @ A @ B2 )
= ( sup_sup_set_v @ K @ ( sup_sup_set_v @ A @ B ) ) ) ) ).
% boolean_algebra_cancel.sup2
thf(fact_577_boolean__algebra__cancel_Osup2,axiom,
! [B2: set_Product_prod_v_v,K: set_Product_prod_v_v,B: set_Product_prod_v_v,A: set_Product_prod_v_v] :
( ( B2
= ( sup_su414716646722978715od_v_v @ K @ B ) )
=> ( ( sup_su414716646722978715od_v_v @ A @ B2 )
= ( sup_su414716646722978715od_v_v @ K @ ( sup_su414716646722978715od_v_v @ A @ B ) ) ) ) ).
% boolean_algebra_cancel.sup2
thf(fact_578_boolean__algebra__cancel_Osup1,axiom,
! [A2: set_v,K: set_v,A: set_v,B: set_v] :
( ( A2
= ( sup_sup_set_v @ K @ A ) )
=> ( ( sup_sup_set_v @ A2 @ B )
= ( sup_sup_set_v @ K @ ( sup_sup_set_v @ A @ B ) ) ) ) ).
% boolean_algebra_cancel.sup1
thf(fact_579_boolean__algebra__cancel_Osup1,axiom,
! [A2: set_Product_prod_v_v,K: set_Product_prod_v_v,A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( A2
= ( sup_su414716646722978715od_v_v @ K @ A ) )
=> ( ( sup_su414716646722978715od_v_v @ A2 @ B )
= ( sup_su414716646722978715od_v_v @ K @ ( sup_su414716646722978715od_v_v @ A @ B ) ) ) ) ).
% boolean_algebra_cancel.sup1
thf(fact_580_sup__commute,axiom,
( sup_sup_set_v
= ( ^ [X3: set_v,Y3: set_v] : ( sup_sup_set_v @ Y3 @ X3 ) ) ) ).
% sup_commute
thf(fact_581_sup__commute,axiom,
( sup_su414716646722978715od_v_v
= ( ^ [X3: set_Product_prod_v_v,Y3: set_Product_prod_v_v] : ( sup_su414716646722978715od_v_v @ Y3 @ X3 ) ) ) ).
% sup_commute
thf(fact_582_sup_Ocommute,axiom,
( sup_sup_set_v
= ( ^ [A4: set_v,B4: set_v] : ( sup_sup_set_v @ B4 @ A4 ) ) ) ).
% sup.commute
thf(fact_583_sup_Ocommute,axiom,
( sup_su414716646722978715od_v_v
= ( ^ [A4: set_Product_prod_v_v,B4: set_Product_prod_v_v] : ( sup_su414716646722978715od_v_v @ B4 @ A4 ) ) ) ).
% sup.commute
thf(fact_584_sup__assoc,axiom,
! [X: set_v,Y: set_v,Z: set_v] :
( ( sup_sup_set_v @ ( sup_sup_set_v @ X @ Y ) @ Z )
= ( sup_sup_set_v @ X @ ( sup_sup_set_v @ Y @ Z ) ) ) ).
% sup_assoc
thf(fact_585_sup__assoc,axiom,
! [X: set_Product_prod_v_v,Y: set_Product_prod_v_v,Z: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ ( sup_su414716646722978715od_v_v @ X @ Y ) @ Z )
= ( sup_su414716646722978715od_v_v @ X @ ( sup_su414716646722978715od_v_v @ Y @ Z ) ) ) ).
% sup_assoc
thf(fact_586_sup_Oassoc,axiom,
! [A: set_v,B: set_v,C: set_v] :
( ( sup_sup_set_v @ ( sup_sup_set_v @ A @ B ) @ C )
= ( sup_sup_set_v @ A @ ( sup_sup_set_v @ B @ C ) ) ) ).
% sup.assoc
thf(fact_587_sup_Oassoc,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ ( sup_su414716646722978715od_v_v @ A @ B ) @ C )
= ( sup_su414716646722978715od_v_v @ A @ ( sup_su414716646722978715od_v_v @ B @ C ) ) ) ).
% sup.assoc
thf(fact_588_inf__sup__aci_I5_J,axiom,
( sup_sup_set_v
= ( ^ [X3: set_v,Y3: set_v] : ( sup_sup_set_v @ Y3 @ X3 ) ) ) ).
% inf_sup_aci(5)
thf(fact_589_inf__sup__aci_I5_J,axiom,
( sup_su414716646722978715od_v_v
= ( ^ [X3: set_Product_prod_v_v,Y3: set_Product_prod_v_v] : ( sup_su414716646722978715od_v_v @ Y3 @ X3 ) ) ) ).
% inf_sup_aci(5)
thf(fact_590_inf__sup__aci_I6_J,axiom,
! [X: set_v,Y: set_v,Z: set_v] :
( ( sup_sup_set_v @ ( sup_sup_set_v @ X @ Y ) @ Z )
= ( sup_sup_set_v @ X @ ( sup_sup_set_v @ Y @ Z ) ) ) ).
% inf_sup_aci(6)
thf(fact_591_inf__sup__aci_I6_J,axiom,
! [X: set_Product_prod_v_v,Y: set_Product_prod_v_v,Z: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ ( sup_su414716646722978715od_v_v @ X @ Y ) @ Z )
= ( sup_su414716646722978715od_v_v @ X @ ( sup_su414716646722978715od_v_v @ Y @ Z ) ) ) ).
% inf_sup_aci(6)
thf(fact_592_inf__sup__aci_I7_J,axiom,
! [X: set_v,Y: set_v,Z: set_v] :
( ( sup_sup_set_v @ X @ ( sup_sup_set_v @ Y @ Z ) )
= ( sup_sup_set_v @ Y @ ( sup_sup_set_v @ X @ Z ) ) ) ).
% inf_sup_aci(7)
thf(fact_593_inf__sup__aci_I7_J,axiom,
! [X: set_Product_prod_v_v,Y: set_Product_prod_v_v,Z: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ X @ ( sup_su414716646722978715od_v_v @ Y @ Z ) )
= ( sup_su414716646722978715od_v_v @ Y @ ( sup_su414716646722978715od_v_v @ X @ Z ) ) ) ).
% inf_sup_aci(7)
thf(fact_594_inf__sup__aci_I8_J,axiom,
! [X: set_v,Y: set_v] :
( ( sup_sup_set_v @ X @ ( sup_sup_set_v @ X @ Y ) )
= ( sup_sup_set_v @ X @ Y ) ) ).
% inf_sup_aci(8)
thf(fact_595_inf__sup__aci_I8_J,axiom,
! [X: set_Product_prod_v_v,Y: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ X @ ( sup_su414716646722978715od_v_v @ X @ Y ) )
= ( sup_su414716646722978715od_v_v @ X @ Y ) ) ).
% inf_sup_aci(8)
thf(fact_596_transpose_Ocases,axiom,
! [X: list_list_v] :
( ( X != nil_list_v )
=> ( ! [Xss: list_list_v] :
( X
!= ( cons_list_v @ nil_v @ Xss ) )
=> ~ ! [X4: v,Xs3: list_v,Xss: list_list_v] :
( X
!= ( cons_list_v @ ( cons_v @ X4 @ Xs3 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_597_graph_Oreachable__end_Ocong,axiom,
sCC_Bl770211535891879572_end_v = sCC_Bl770211535891879572_end_v ).
% graph.reachable_end.cong
thf(fact_598_graph_Opre__dfs_Ocong,axiom,
sCC_Bl36166008131615352t_unit = sCC_Bl36166008131615352t_unit ).
% graph.pre_dfs.cong
thf(fact_599_inf__sup__ord_I4_J,axiom,
! [Y: set_v,X: set_v] : ( ord_less_eq_set_v @ Y @ ( sup_sup_set_v @ X @ Y ) ) ).
% inf_sup_ord(4)
thf(fact_600_inf__sup__ord_I4_J,axiom,
! [Y: set_Product_prod_v_v,X: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ Y @ ( sup_su414716646722978715od_v_v @ X @ Y ) ) ).
% inf_sup_ord(4)
thf(fact_601_inf__sup__ord_I3_J,axiom,
! [X: set_v,Y: set_v] : ( ord_less_eq_set_v @ X @ ( sup_sup_set_v @ X @ Y ) ) ).
% inf_sup_ord(3)
thf(fact_602_inf__sup__ord_I3_J,axiom,
! [X: set_Product_prod_v_v,Y: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ X @ ( sup_su414716646722978715od_v_v @ X @ Y ) ) ).
% inf_sup_ord(3)
thf(fact_603_le__supE,axiom,
! [A: set_v,B: set_v,X: set_v] :
( ( ord_less_eq_set_v @ ( sup_sup_set_v @ A @ B ) @ X )
=> ~ ( ( ord_less_eq_set_v @ A @ X )
=> ~ ( ord_less_eq_set_v @ B @ X ) ) ) ).
% le_supE
thf(fact_604_le__supE,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v,X: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ ( sup_su414716646722978715od_v_v @ A @ B ) @ X )
=> ~ ( ( ord_le7336532860387713383od_v_v @ A @ X )
=> ~ ( ord_le7336532860387713383od_v_v @ B @ X ) ) ) ).
% le_supE
thf(fact_605_le__supI,axiom,
! [A: set_v,X: set_v,B: set_v] :
( ( ord_less_eq_set_v @ A @ X )
=> ( ( ord_less_eq_set_v @ B @ X )
=> ( ord_less_eq_set_v @ ( sup_sup_set_v @ A @ B ) @ X ) ) ) ).
% le_supI
thf(fact_606_le__supI,axiom,
! [A: set_Product_prod_v_v,X: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ X )
=> ( ( ord_le7336532860387713383od_v_v @ B @ X )
=> ( ord_le7336532860387713383od_v_v @ ( sup_su414716646722978715od_v_v @ A @ B ) @ X ) ) ) ).
% le_supI
thf(fact_607_sup__ge1,axiom,
! [X: set_v,Y: set_v] : ( ord_less_eq_set_v @ X @ ( sup_sup_set_v @ X @ Y ) ) ).
% sup_ge1
thf(fact_608_sup__ge1,axiom,
! [X: set_Product_prod_v_v,Y: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ X @ ( sup_su414716646722978715od_v_v @ X @ Y ) ) ).
% sup_ge1
thf(fact_609_sup__ge2,axiom,
! [Y: set_v,X: set_v] : ( ord_less_eq_set_v @ Y @ ( sup_sup_set_v @ X @ Y ) ) ).
% sup_ge2
thf(fact_610_sup__ge2,axiom,
! [Y: set_Product_prod_v_v,X: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ Y @ ( sup_su414716646722978715od_v_v @ X @ Y ) ) ).
% sup_ge2
thf(fact_611_le__supI1,axiom,
! [X: set_v,A: set_v,B: set_v] :
( ( ord_less_eq_set_v @ X @ A )
=> ( ord_less_eq_set_v @ X @ ( sup_sup_set_v @ A @ B ) ) ) ).
% le_supI1
thf(fact_612_le__supI1,axiom,
! [X: set_Product_prod_v_v,A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X @ A )
=> ( ord_le7336532860387713383od_v_v @ X @ ( sup_su414716646722978715od_v_v @ A @ B ) ) ) ).
% le_supI1
thf(fact_613_le__supI2,axiom,
! [X: set_v,B: set_v,A: set_v] :
( ( ord_less_eq_set_v @ X @ B )
=> ( ord_less_eq_set_v @ X @ ( sup_sup_set_v @ A @ B ) ) ) ).
% le_supI2
thf(fact_614_le__supI2,axiom,
! [X: set_Product_prod_v_v,B: set_Product_prod_v_v,A: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X @ B )
=> ( ord_le7336532860387713383od_v_v @ X @ ( sup_su414716646722978715od_v_v @ A @ B ) ) ) ).
% le_supI2
thf(fact_615_sup_Omono,axiom,
! [C: set_v,A: set_v,D2: set_v,B: set_v] :
( ( ord_less_eq_set_v @ C @ A )
=> ( ( ord_less_eq_set_v @ D2 @ B )
=> ( ord_less_eq_set_v @ ( sup_sup_set_v @ C @ D2 ) @ ( sup_sup_set_v @ A @ B ) ) ) ) ).
% sup.mono
thf(fact_616_sup_Omono,axiom,
! [C: set_Product_prod_v_v,A: set_Product_prod_v_v,D2: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ C @ A )
=> ( ( ord_le7336532860387713383od_v_v @ D2 @ B )
=> ( ord_le7336532860387713383od_v_v @ ( sup_su414716646722978715od_v_v @ C @ D2 ) @ ( sup_su414716646722978715od_v_v @ A @ B ) ) ) ) ).
% sup.mono
thf(fact_617_sup__mono,axiom,
! [A: set_v,C: set_v,B: set_v,D2: set_v] :
( ( ord_less_eq_set_v @ A @ C )
=> ( ( ord_less_eq_set_v @ B @ D2 )
=> ( ord_less_eq_set_v @ ( sup_sup_set_v @ A @ B ) @ ( sup_sup_set_v @ C @ D2 ) ) ) ) ).
% sup_mono
thf(fact_618_sup__mono,axiom,
! [A: set_Product_prod_v_v,C: set_Product_prod_v_v,B: set_Product_prod_v_v,D2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ C )
=> ( ( ord_le7336532860387713383od_v_v @ B @ D2 )
=> ( ord_le7336532860387713383od_v_v @ ( sup_su414716646722978715od_v_v @ A @ B ) @ ( sup_su414716646722978715od_v_v @ C @ D2 ) ) ) ) ).
% sup_mono
thf(fact_619_sup__least,axiom,
! [Y: set_v,X: set_v,Z: set_v] :
( ( ord_less_eq_set_v @ Y @ X )
=> ( ( ord_less_eq_set_v @ Z @ X )
=> ( ord_less_eq_set_v @ ( sup_sup_set_v @ Y @ Z ) @ X ) ) ) ).
% sup_least
thf(fact_620_sup__least,axiom,
! [Y: set_Product_prod_v_v,X: set_Product_prod_v_v,Z: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ Y @ X )
=> ( ( ord_le7336532860387713383od_v_v @ Z @ X )
=> ( ord_le7336532860387713383od_v_v @ ( sup_su414716646722978715od_v_v @ Y @ Z ) @ X ) ) ) ).
% sup_least
thf(fact_621_le__iff__sup,axiom,
( ord_less_eq_set_v
= ( ^ [X3: set_v,Y3: set_v] :
( ( sup_sup_set_v @ X3 @ Y3 )
= Y3 ) ) ) ).
% le_iff_sup
thf(fact_622_le__iff__sup,axiom,
( ord_le7336532860387713383od_v_v
= ( ^ [X3: set_Product_prod_v_v,Y3: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ X3 @ Y3 )
= Y3 ) ) ) ).
% le_iff_sup
thf(fact_623_sup_OorderE,axiom,
! [B: set_v,A: set_v] :
( ( ord_less_eq_set_v @ B @ A )
=> ( A
= ( sup_sup_set_v @ A @ B ) ) ) ).
% sup.orderE
thf(fact_624_sup_OorderE,axiom,
! [B: set_Product_prod_v_v,A: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ B @ A )
=> ( A
= ( sup_su414716646722978715od_v_v @ A @ B ) ) ) ).
% sup.orderE
thf(fact_625_sup_OorderI,axiom,
! [A: set_v,B: set_v] :
( ( A
= ( sup_sup_set_v @ A @ B ) )
=> ( ord_less_eq_set_v @ B @ A ) ) ).
% sup.orderI
thf(fact_626_sup_OorderI,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( A
= ( sup_su414716646722978715od_v_v @ A @ B ) )
=> ( ord_le7336532860387713383od_v_v @ B @ A ) ) ).
% sup.orderI
thf(fact_627_sup__unique,axiom,
! [F: set_v > set_v > set_v,X: set_v,Y: set_v] :
( ! [X4: set_v,Y2: set_v] : ( ord_less_eq_set_v @ X4 @ ( F @ X4 @ Y2 ) )
=> ( ! [X4: set_v,Y2: set_v] : ( ord_less_eq_set_v @ Y2 @ ( F @ X4 @ Y2 ) )
=> ( ! [X4: set_v,Y2: set_v,Z2: set_v] :
( ( ord_less_eq_set_v @ Y2 @ X4 )
=> ( ( ord_less_eq_set_v @ Z2 @ X4 )
=> ( ord_less_eq_set_v @ ( F @ Y2 @ Z2 ) @ X4 ) ) )
=> ( ( sup_sup_set_v @ X @ Y )
= ( F @ X @ Y ) ) ) ) ) ).
% sup_unique
thf(fact_628_sup__unique,axiom,
! [F: set_Product_prod_v_v > set_Product_prod_v_v > set_Product_prod_v_v,X: set_Product_prod_v_v,Y: set_Product_prod_v_v] :
( ! [X4: set_Product_prod_v_v,Y2: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ X4 @ ( F @ X4 @ Y2 ) )
=> ( ! [X4: set_Product_prod_v_v,Y2: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ Y2 @ ( F @ X4 @ Y2 ) )
=> ( ! [X4: set_Product_prod_v_v,Y2: set_Product_prod_v_v,Z2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ Y2 @ X4 )
=> ( ( ord_le7336532860387713383od_v_v @ Z2 @ X4 )
=> ( ord_le7336532860387713383od_v_v @ ( F @ Y2 @ Z2 ) @ X4 ) ) )
=> ( ( sup_su414716646722978715od_v_v @ X @ Y )
= ( F @ X @ Y ) ) ) ) ) ).
% sup_unique
thf(fact_629_sup_Oabsorb1,axiom,
! [B: set_v,A: set_v] :
( ( ord_less_eq_set_v @ B @ A )
=> ( ( sup_sup_set_v @ A @ B )
= A ) ) ).
% sup.absorb1
thf(fact_630_sup_Oabsorb1,axiom,
! [B: set_Product_prod_v_v,A: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ B @ A )
=> ( ( sup_su414716646722978715od_v_v @ A @ B )
= A ) ) ).
% sup.absorb1
thf(fact_631_sup_Oabsorb2,axiom,
! [A: set_v,B: set_v] :
( ( ord_less_eq_set_v @ A @ B )
=> ( ( sup_sup_set_v @ A @ B )
= B ) ) ).
% sup.absorb2
thf(fact_632_sup_Oabsorb2,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ B )
=> ( ( sup_su414716646722978715od_v_v @ A @ B )
= B ) ) ).
% sup.absorb2
thf(fact_633_sup__absorb1,axiom,
! [Y: set_v,X: set_v] :
( ( ord_less_eq_set_v @ Y @ X )
=> ( ( sup_sup_set_v @ X @ Y )
= X ) ) ).
% sup_absorb1
thf(fact_634_sup__absorb1,axiom,
! [Y: set_Product_prod_v_v,X: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ Y @ X )
=> ( ( sup_su414716646722978715od_v_v @ X @ Y )
= X ) ) ).
% sup_absorb1
thf(fact_635_sup__absorb2,axiom,
! [X: set_v,Y: set_v] :
( ( ord_less_eq_set_v @ X @ Y )
=> ( ( sup_sup_set_v @ X @ Y )
= Y ) ) ).
% sup_absorb2
thf(fact_636_sup__absorb2,axiom,
! [X: set_Product_prod_v_v,Y: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X @ Y )
=> ( ( sup_su414716646722978715od_v_v @ X @ Y )
= Y ) ) ).
% sup_absorb2
thf(fact_637_sup_OboundedE,axiom,
! [B: set_v,C: set_v,A: set_v] :
( ( ord_less_eq_set_v @ ( sup_sup_set_v @ B @ C ) @ A )
=> ~ ( ( ord_less_eq_set_v @ B @ A )
=> ~ ( ord_less_eq_set_v @ C @ A ) ) ) ).
% sup.boundedE
thf(fact_638_sup_OboundedE,axiom,
! [B: set_Product_prod_v_v,C: set_Product_prod_v_v,A: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ ( sup_su414716646722978715od_v_v @ B @ C ) @ A )
=> ~ ( ( ord_le7336532860387713383od_v_v @ B @ A )
=> ~ ( ord_le7336532860387713383od_v_v @ C @ A ) ) ) ).
% sup.boundedE
thf(fact_639_sup_OboundedI,axiom,
! [B: set_v,A: set_v,C: set_v] :
( ( ord_less_eq_set_v @ B @ A )
=> ( ( ord_less_eq_set_v @ C @ A )
=> ( ord_less_eq_set_v @ ( sup_sup_set_v @ B @ C ) @ A ) ) ) ).
% sup.boundedI
thf(fact_640_sup_OboundedI,axiom,
! [B: set_Product_prod_v_v,A: set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ B @ A )
=> ( ( ord_le7336532860387713383od_v_v @ C @ A )
=> ( ord_le7336532860387713383od_v_v @ ( sup_su414716646722978715od_v_v @ B @ C ) @ A ) ) ) ).
% sup.boundedI
thf(fact_641_sup_Oorder__iff,axiom,
( ord_less_eq_set_v
= ( ^ [B4: set_v,A4: set_v] :
( A4
= ( sup_sup_set_v @ A4 @ B4 ) ) ) ) ).
% sup.order_iff
thf(fact_642_sup_Oorder__iff,axiom,
( ord_le7336532860387713383od_v_v
= ( ^ [B4: set_Product_prod_v_v,A4: set_Product_prod_v_v] :
( A4
= ( sup_su414716646722978715od_v_v @ A4 @ B4 ) ) ) ) ).
% sup.order_iff
thf(fact_643_sup_Ocobounded1,axiom,
! [A: set_v,B: set_v] : ( ord_less_eq_set_v @ A @ ( sup_sup_set_v @ A @ B ) ) ).
% sup.cobounded1
thf(fact_644_sup_Ocobounded1,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ A @ ( sup_su414716646722978715od_v_v @ A @ B ) ) ).
% sup.cobounded1
thf(fact_645_sup_Ocobounded2,axiom,
! [B: set_v,A: set_v] : ( ord_less_eq_set_v @ B @ ( sup_sup_set_v @ A @ B ) ) ).
% sup.cobounded2
thf(fact_646_sup_Ocobounded2,axiom,
! [B: set_Product_prod_v_v,A: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ B @ ( sup_su414716646722978715od_v_v @ A @ B ) ) ).
% sup.cobounded2
thf(fact_647_sup_Oabsorb__iff1,axiom,
( ord_less_eq_set_v
= ( ^ [B4: set_v,A4: set_v] :
( ( sup_sup_set_v @ A4 @ B4 )
= A4 ) ) ) ).
% sup.absorb_iff1
thf(fact_648_sup_Oabsorb__iff1,axiom,
( ord_le7336532860387713383od_v_v
= ( ^ [B4: set_Product_prod_v_v,A4: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ A4 @ B4 )
= A4 ) ) ) ).
% sup.absorb_iff1
thf(fact_649_sup_Oabsorb__iff2,axiom,
( ord_less_eq_set_v
= ( ^ [A4: set_v,B4: set_v] :
( ( sup_sup_set_v @ A4 @ B4 )
= B4 ) ) ) ).
% sup.absorb_iff2
thf(fact_650_sup_Oabsorb__iff2,axiom,
( ord_le7336532860387713383od_v_v
= ( ^ [A4: set_Product_prod_v_v,B4: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ A4 @ B4 )
= B4 ) ) ) ).
% sup.absorb_iff2
thf(fact_651_sup_OcoboundedI1,axiom,
! [C: set_v,A: set_v,B: set_v] :
( ( ord_less_eq_set_v @ C @ A )
=> ( ord_less_eq_set_v @ C @ ( sup_sup_set_v @ A @ B ) ) ) ).
% sup.coboundedI1
thf(fact_652_sup_OcoboundedI1,axiom,
! [C: set_Product_prod_v_v,A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ C @ A )
=> ( ord_le7336532860387713383od_v_v @ C @ ( sup_su414716646722978715od_v_v @ A @ B ) ) ) ).
% sup.coboundedI1
thf(fact_653_sup_OcoboundedI2,axiom,
! [C: set_v,B: set_v,A: set_v] :
( ( ord_less_eq_set_v @ C @ B )
=> ( ord_less_eq_set_v @ C @ ( sup_sup_set_v @ A @ B ) ) ) ).
% sup.coboundedI2
thf(fact_654_sup_OcoboundedI2,axiom,
! [C: set_Product_prod_v_v,B: set_Product_prod_v_v,A: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ C @ B )
=> ( ord_le7336532860387713383od_v_v @ C @ ( sup_su414716646722978715od_v_v @ A @ B ) ) ) ).
% sup.coboundedI2
thf(fact_655_boolean__algebra_Odisj__zero__right,axiom,
! [X: set_v] :
( ( sup_sup_set_v @ X @ bot_bot_set_v )
= X ) ).
% boolean_algebra.disj_zero_right
thf(fact_656_boolean__algebra_Odisj__zero__right,axiom,
! [X: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ X @ bot_bo723834152578015283od_v_v )
= X ) ).
% boolean_algebra.disj_zero_right
thf(fact_657_distrib__imp1,axiom,
! [X: set_v,Y: set_v,Z: set_v] :
( ! [X4: set_v,Y2: set_v,Z2: set_v] :
( ( inf_inf_set_v @ X4 @ ( sup_sup_set_v @ Y2 @ Z2 ) )
= ( sup_sup_set_v @ ( inf_inf_set_v @ X4 @ Y2 ) @ ( inf_inf_set_v @ X4 @ Z2 ) ) )
=> ( ( sup_sup_set_v @ X @ ( inf_inf_set_v @ Y @ Z ) )
= ( inf_inf_set_v @ ( sup_sup_set_v @ X @ Y ) @ ( sup_sup_set_v @ X @ Z ) ) ) ) ).
% distrib_imp1
thf(fact_658_distrib__imp1,axiom,
! [X: set_Product_prod_v_v,Y: set_Product_prod_v_v,Z: set_Product_prod_v_v] :
( ! [X4: set_Product_prod_v_v,Y2: set_Product_prod_v_v,Z2: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ X4 @ ( sup_su414716646722978715od_v_v @ Y2 @ Z2 ) )
= ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ X4 @ Y2 ) @ ( inf_in6271465464967711157od_v_v @ X4 @ Z2 ) ) )
=> ( ( sup_su414716646722978715od_v_v @ X @ ( inf_in6271465464967711157od_v_v @ Y @ Z ) )
= ( inf_in6271465464967711157od_v_v @ ( sup_su414716646722978715od_v_v @ X @ Y ) @ ( sup_su414716646722978715od_v_v @ X @ Z ) ) ) ) ).
% distrib_imp1
thf(fact_659_distrib__imp2,axiom,
! [X: set_v,Y: set_v,Z: set_v] :
( ! [X4: set_v,Y2: set_v,Z2: set_v] :
( ( sup_sup_set_v @ X4 @ ( inf_inf_set_v @ Y2 @ Z2 ) )
= ( inf_inf_set_v @ ( sup_sup_set_v @ X4 @ Y2 ) @ ( sup_sup_set_v @ X4 @ Z2 ) ) )
=> ( ( inf_inf_set_v @ X @ ( sup_sup_set_v @ Y @ Z ) )
= ( sup_sup_set_v @ ( inf_inf_set_v @ X @ Y ) @ ( inf_inf_set_v @ X @ Z ) ) ) ) ).
% distrib_imp2
thf(fact_660_distrib__imp2,axiom,
! [X: set_Product_prod_v_v,Y: set_Product_prod_v_v,Z: set_Product_prod_v_v] :
( ! [X4: set_Product_prod_v_v,Y2: set_Product_prod_v_v,Z2: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ X4 @ ( inf_in6271465464967711157od_v_v @ Y2 @ Z2 ) )
= ( inf_in6271465464967711157od_v_v @ ( sup_su414716646722978715od_v_v @ X4 @ Y2 ) @ ( sup_su414716646722978715od_v_v @ X4 @ Z2 ) ) )
=> ( ( inf_in6271465464967711157od_v_v @ X @ ( sup_su414716646722978715od_v_v @ Y @ Z ) )
= ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ X @ Y ) @ ( inf_in6271465464967711157od_v_v @ X @ Z ) ) ) ) ).
% distrib_imp2
thf(fact_661_inf__sup__distrib1,axiom,
! [X: set_v,Y: set_v,Z: set_v] :
( ( inf_inf_set_v @ X @ ( sup_sup_set_v @ Y @ Z ) )
= ( sup_sup_set_v @ ( inf_inf_set_v @ X @ Y ) @ ( inf_inf_set_v @ X @ Z ) ) ) ).
% inf_sup_distrib1
thf(fact_662_inf__sup__distrib1,axiom,
! [X: set_Product_prod_v_v,Y: set_Product_prod_v_v,Z: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ X @ ( sup_su414716646722978715od_v_v @ Y @ Z ) )
= ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ X @ Y ) @ ( inf_in6271465464967711157od_v_v @ X @ Z ) ) ) ).
% inf_sup_distrib1
thf(fact_663_inf__sup__distrib2,axiom,
! [Y: set_v,Z: set_v,X: set_v] :
( ( inf_inf_set_v @ ( sup_sup_set_v @ Y @ Z ) @ X )
= ( sup_sup_set_v @ ( inf_inf_set_v @ Y @ X ) @ ( inf_inf_set_v @ Z @ X ) ) ) ).
% inf_sup_distrib2
thf(fact_664_inf__sup__distrib2,axiom,
! [Y: set_Product_prod_v_v,Z: set_Product_prod_v_v,X: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ ( sup_su414716646722978715od_v_v @ Y @ Z ) @ X )
= ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ Y @ X ) @ ( inf_in6271465464967711157od_v_v @ Z @ X ) ) ) ).
% inf_sup_distrib2
thf(fact_665_sup__inf__distrib1,axiom,
! [X: set_v,Y: set_v,Z: set_v] :
( ( sup_sup_set_v @ X @ ( inf_inf_set_v @ Y @ Z ) )
= ( inf_inf_set_v @ ( sup_sup_set_v @ X @ Y ) @ ( sup_sup_set_v @ X @ Z ) ) ) ).
% sup_inf_distrib1
thf(fact_666_sup__inf__distrib1,axiom,
! [X: set_Product_prod_v_v,Y: set_Product_prod_v_v,Z: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ X @ ( inf_in6271465464967711157od_v_v @ Y @ Z ) )
= ( inf_in6271465464967711157od_v_v @ ( sup_su414716646722978715od_v_v @ X @ Y ) @ ( sup_su414716646722978715od_v_v @ X @ Z ) ) ) ).
% sup_inf_distrib1
thf(fact_667_sup__inf__distrib2,axiom,
! [Y: set_v,Z: set_v,X: set_v] :
( ( sup_sup_set_v @ ( inf_inf_set_v @ Y @ Z ) @ X )
= ( inf_inf_set_v @ ( sup_sup_set_v @ Y @ X ) @ ( sup_sup_set_v @ Z @ X ) ) ) ).
% sup_inf_distrib2
thf(fact_668_sup__inf__distrib2,axiom,
! [Y: set_Product_prod_v_v,Z: set_Product_prod_v_v,X: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ Y @ Z ) @ X )
= ( inf_in6271465464967711157od_v_v @ ( sup_su414716646722978715od_v_v @ Y @ X ) @ ( sup_su414716646722978715od_v_v @ Z @ X ) ) ) ).
% sup_inf_distrib2
thf(fact_669_boolean__algebra_Oconj__disj__distrib,axiom,
! [X: set_v,Y: set_v,Z: set_v] :
( ( inf_inf_set_v @ X @ ( sup_sup_set_v @ Y @ Z ) )
= ( sup_sup_set_v @ ( inf_inf_set_v @ X @ Y ) @ ( inf_inf_set_v @ X @ Z ) ) ) ).
% boolean_algebra.conj_disj_distrib
thf(fact_670_boolean__algebra_Oconj__disj__distrib,axiom,
! [X: set_Product_prod_v_v,Y: set_Product_prod_v_v,Z: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ X @ ( sup_su414716646722978715od_v_v @ Y @ Z ) )
= ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ X @ Y ) @ ( inf_in6271465464967711157od_v_v @ X @ Z ) ) ) ).
% boolean_algebra.conj_disj_distrib
thf(fact_671_boolean__algebra_Odisj__conj__distrib,axiom,
! [X: set_v,Y: set_v,Z: set_v] :
( ( sup_sup_set_v @ X @ ( inf_inf_set_v @ Y @ Z ) )
= ( inf_inf_set_v @ ( sup_sup_set_v @ X @ Y ) @ ( sup_sup_set_v @ X @ Z ) ) ) ).
% boolean_algebra.disj_conj_distrib
thf(fact_672_boolean__algebra_Odisj__conj__distrib,axiom,
! [X: set_Product_prod_v_v,Y: set_Product_prod_v_v,Z: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ X @ ( inf_in6271465464967711157od_v_v @ Y @ Z ) )
= ( inf_in6271465464967711157od_v_v @ ( sup_su414716646722978715od_v_v @ X @ Y ) @ ( sup_su414716646722978715od_v_v @ X @ Z ) ) ) ).
% boolean_algebra.disj_conj_distrib
thf(fact_673_boolean__algebra_Oconj__disj__distrib2,axiom,
! [Y: set_v,Z: set_v,X: set_v] :
( ( inf_inf_set_v @ ( sup_sup_set_v @ Y @ Z ) @ X )
= ( sup_sup_set_v @ ( inf_inf_set_v @ Y @ X ) @ ( inf_inf_set_v @ Z @ X ) ) ) ).
% boolean_algebra.conj_disj_distrib2
thf(fact_674_boolean__algebra_Oconj__disj__distrib2,axiom,
! [Y: set_Product_prod_v_v,Z: set_Product_prod_v_v,X: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ ( sup_su414716646722978715od_v_v @ Y @ Z ) @ X )
= ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ Y @ X ) @ ( inf_in6271465464967711157od_v_v @ Z @ X ) ) ) ).
% boolean_algebra.conj_disj_distrib2
thf(fact_675_boolean__algebra_Odisj__conj__distrib2,axiom,
! [Y: set_v,Z: set_v,X: set_v] :
( ( sup_sup_set_v @ ( inf_inf_set_v @ Y @ Z ) @ X )
= ( inf_inf_set_v @ ( sup_sup_set_v @ Y @ X ) @ ( sup_sup_set_v @ Z @ X ) ) ) ).
% boolean_algebra.disj_conj_distrib2
thf(fact_676_boolean__algebra_Odisj__conj__distrib2,axiom,
! [Y: set_Product_prod_v_v,Z: set_Product_prod_v_v,X: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ Y @ Z ) @ X )
= ( inf_in6271465464967711157od_v_v @ ( sup_su414716646722978715od_v_v @ Y @ X ) @ ( sup_su414716646722978715od_v_v @ Z @ X ) ) ) ).
% boolean_algebra.disj_conj_distrib2
thf(fact_677_Un__empty__left,axiom,
! [B2: set_v] :
( ( sup_sup_set_v @ bot_bot_set_v @ B2 )
= B2 ) ).
% Un_empty_left
thf(fact_678_Un__empty__left,axiom,
! [B2: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ bot_bo723834152578015283od_v_v @ B2 )
= B2 ) ).
% Un_empty_left
thf(fact_679_Un__empty__right,axiom,
! [A2: set_v] :
( ( sup_sup_set_v @ A2 @ bot_bot_set_v )
= A2 ) ).
% Un_empty_right
thf(fact_680_Un__empty__right,axiom,
! [A2: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ A2 @ bot_bo723834152578015283od_v_v )
= A2 ) ).
% Un_empty_right
thf(fact_681_singletonD,axiom,
! [B: v,A: v] :
( ( member_v @ B @ ( insert_v @ A @ bot_bot_set_v ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_682_singletonD,axiom,
! [B: product_prod_v_v,A: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ B @ ( insert1338601472111419319od_v_v @ A @ bot_bo723834152578015283od_v_v ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_683_singleton__iff,axiom,
! [B: v,A: v] :
( ( member_v @ B @ ( insert_v @ A @ bot_bot_set_v ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_684_singleton__iff,axiom,
! [B: product_prod_v_v,A: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ B @ ( insert1338601472111419319od_v_v @ A @ bot_bo723834152578015283od_v_v ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_685_doubleton__eq__iff,axiom,
! [A: v,B: v,C: v,D2: v] :
( ( ( insert_v @ A @ ( insert_v @ B @ bot_bot_set_v ) )
= ( insert_v @ C @ ( insert_v @ D2 @ bot_bot_set_v ) ) )
= ( ( ( A = C )
& ( B = D2 ) )
| ( ( A = D2 )
& ( B = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_686_doubleton__eq__iff,axiom,
! [A: product_prod_v_v,B: product_prod_v_v,C: product_prod_v_v,D2: product_prod_v_v] :
( ( ( insert1338601472111419319od_v_v @ A @ ( insert1338601472111419319od_v_v @ B @ bot_bo723834152578015283od_v_v ) )
= ( insert1338601472111419319od_v_v @ C @ ( insert1338601472111419319od_v_v @ D2 @ bot_bo723834152578015283od_v_v ) ) )
= ( ( ( A = C )
& ( B = D2 ) )
| ( ( A = D2 )
& ( B = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_687_insert__not__empty,axiom,
! [A: v,A2: set_v] :
( ( insert_v @ A @ A2 )
!= bot_bot_set_v ) ).
% insert_not_empty
thf(fact_688_insert__not__empty,axiom,
! [A: product_prod_v_v,A2: set_Product_prod_v_v] :
( ( insert1338601472111419319od_v_v @ A @ A2 )
!= bot_bo723834152578015283od_v_v ) ).
% insert_not_empty
thf(fact_689_singleton__inject,axiom,
! [A: v,B: v] :
( ( ( insert_v @ A @ bot_bot_set_v )
= ( insert_v @ B @ bot_bot_set_v ) )
=> ( A = B ) ) ).
% singleton_inject
thf(fact_690_singleton__inject,axiom,
! [A: product_prod_v_v,B: product_prod_v_v] :
( ( ( insert1338601472111419319od_v_v @ A @ bot_bo723834152578015283od_v_v )
= ( insert1338601472111419319od_v_v @ B @ bot_bo723834152578015283od_v_v ) )
=> ( A = B ) ) ).
% singleton_inject
thf(fact_691_Un__mono,axiom,
! [A2: set_v,C2: set_v,B2: set_v,D: set_v] :
( ( ord_less_eq_set_v @ A2 @ C2 )
=> ( ( ord_less_eq_set_v @ B2 @ D )
=> ( ord_less_eq_set_v @ ( sup_sup_set_v @ A2 @ B2 ) @ ( sup_sup_set_v @ C2 @ D ) ) ) ) ).
% Un_mono
thf(fact_692_Un__mono,axiom,
! [A2: set_Product_prod_v_v,C2: set_Product_prod_v_v,B2: set_Product_prod_v_v,D: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A2 @ C2 )
=> ( ( ord_le7336532860387713383od_v_v @ B2 @ D )
=> ( ord_le7336532860387713383od_v_v @ ( sup_su414716646722978715od_v_v @ A2 @ B2 ) @ ( sup_su414716646722978715od_v_v @ C2 @ D ) ) ) ) ).
% Un_mono
thf(fact_693_Un__least,axiom,
! [A2: set_v,C2: set_v,B2: set_v] :
( ( ord_less_eq_set_v @ A2 @ C2 )
=> ( ( ord_less_eq_set_v @ B2 @ C2 )
=> ( ord_less_eq_set_v @ ( sup_sup_set_v @ A2 @ B2 ) @ C2 ) ) ) ).
% Un_least
thf(fact_694_Un__least,axiom,
! [A2: set_Product_prod_v_v,C2: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A2 @ C2 )
=> ( ( ord_le7336532860387713383od_v_v @ B2 @ C2 )
=> ( ord_le7336532860387713383od_v_v @ ( sup_su414716646722978715od_v_v @ A2 @ B2 ) @ C2 ) ) ) ).
% Un_least
thf(fact_695_Un__upper1,axiom,
! [A2: set_v,B2: set_v] : ( ord_less_eq_set_v @ A2 @ ( sup_sup_set_v @ A2 @ B2 ) ) ).
% Un_upper1
thf(fact_696_Un__upper1,axiom,
! [A2: set_Product_prod_v_v,B2: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ A2 @ ( sup_su414716646722978715od_v_v @ A2 @ B2 ) ) ).
% Un_upper1
thf(fact_697_Un__upper2,axiom,
! [B2: set_v,A2: set_v] : ( ord_less_eq_set_v @ B2 @ ( sup_sup_set_v @ A2 @ B2 ) ) ).
% Un_upper2
thf(fact_698_Un__upper2,axiom,
! [B2: set_Product_prod_v_v,A2: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ B2 @ ( sup_su414716646722978715od_v_v @ A2 @ B2 ) ) ).
% Un_upper2
thf(fact_699_Un__absorb1,axiom,
! [A2: set_v,B2: set_v] :
( ( ord_less_eq_set_v @ A2 @ B2 )
=> ( ( sup_sup_set_v @ A2 @ B2 )
= B2 ) ) ).
% Un_absorb1
thf(fact_700_Un__absorb1,axiom,
! [A2: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A2 @ B2 )
=> ( ( sup_su414716646722978715od_v_v @ A2 @ B2 )
= B2 ) ) ).
% Un_absorb1
thf(fact_701_Un__absorb2,axiom,
! [B2: set_v,A2: set_v] :
( ( ord_less_eq_set_v @ B2 @ A2 )
=> ( ( sup_sup_set_v @ A2 @ B2 )
= A2 ) ) ).
% Un_absorb2
thf(fact_702_Un__absorb2,axiom,
! [B2: set_Product_prod_v_v,A2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ B2 @ A2 )
=> ( ( sup_su414716646722978715od_v_v @ A2 @ B2 )
= A2 ) ) ).
% Un_absorb2
thf(fact_703_subset__UnE,axiom,
! [C2: set_v,A2: set_v,B2: set_v] :
( ( ord_less_eq_set_v @ C2 @ ( sup_sup_set_v @ A2 @ B2 ) )
=> ~ ! [A5: set_v] :
( ( ord_less_eq_set_v @ A5 @ A2 )
=> ! [B6: set_v] :
( ( ord_less_eq_set_v @ B6 @ B2 )
=> ( C2
!= ( sup_sup_set_v @ A5 @ B6 ) ) ) ) ) ).
% subset_UnE
thf(fact_704_subset__UnE,axiom,
! [C2: set_Product_prod_v_v,A2: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ C2 @ ( sup_su414716646722978715od_v_v @ A2 @ B2 ) )
=> ~ ! [A5: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A5 @ A2 )
=> ! [B6: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ B6 @ B2 )
=> ( C2
!= ( sup_su414716646722978715od_v_v @ A5 @ B6 ) ) ) ) ) ).
% subset_UnE
thf(fact_705_subset__Un__eq,axiom,
( ord_less_eq_set_v
= ( ^ [A3: set_v,B3: set_v] :
( ( sup_sup_set_v @ A3 @ B3 )
= B3 ) ) ) ).
% subset_Un_eq
thf(fact_706_subset__Un__eq,axiom,
( ord_le7336532860387713383od_v_v
= ( ^ [A3: set_Product_prod_v_v,B3: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ A3 @ B3 )
= B3 ) ) ) ).
% subset_Un_eq
thf(fact_707_insert__mono,axiom,
! [C2: set_v,D: set_v,A: v] :
( ( ord_less_eq_set_v @ C2 @ D )
=> ( ord_less_eq_set_v @ ( insert_v @ A @ C2 ) @ ( insert_v @ A @ D ) ) ) ).
% insert_mono
thf(fact_708_insert__mono,axiom,
! [C2: set_Product_prod_v_v,D: set_Product_prod_v_v,A: product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ C2 @ D )
=> ( ord_le7336532860387713383od_v_v @ ( insert1338601472111419319od_v_v @ A @ C2 ) @ ( insert1338601472111419319od_v_v @ A @ D ) ) ) ).
% insert_mono
thf(fact_709_subset__insert,axiom,
! [X: v,A2: set_v,B2: set_v] :
( ~ ( member_v @ X @ A2 )
=> ( ( ord_less_eq_set_v @ A2 @ ( insert_v @ X @ B2 ) )
= ( ord_less_eq_set_v @ A2 @ B2 ) ) ) ).
% subset_insert
thf(fact_710_subset__insert,axiom,
! [X: product_prod_v_v,A2: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ~ ( member7453568604450474000od_v_v @ X @ A2 )
=> ( ( ord_le7336532860387713383od_v_v @ A2 @ ( insert1338601472111419319od_v_v @ X @ B2 ) )
= ( ord_le7336532860387713383od_v_v @ A2 @ B2 ) ) ) ).
% subset_insert
thf(fact_711_subset__insertI,axiom,
! [B2: set_v,A: v] : ( ord_less_eq_set_v @ B2 @ ( insert_v @ A @ B2 ) ) ).
% subset_insertI
thf(fact_712_subset__insertI,axiom,
! [B2: set_Product_prod_v_v,A: product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ B2 @ ( insert1338601472111419319od_v_v @ A @ B2 ) ) ).
% subset_insertI
thf(fact_713_subset__insertI2,axiom,
! [A2: set_v,B2: set_v,B: v] :
( ( ord_less_eq_set_v @ A2 @ B2 )
=> ( ord_less_eq_set_v @ A2 @ ( insert_v @ B @ B2 ) ) ) ).
% subset_insertI2
thf(fact_714_subset__insertI2,axiom,
! [A2: set_Product_prod_v_v,B2: set_Product_prod_v_v,B: product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A2 @ B2 )
=> ( ord_le7336532860387713383od_v_v @ A2 @ ( insert1338601472111419319od_v_v @ B @ B2 ) ) ) ).
% subset_insertI2
thf(fact_715_list_Odistinct_I1_J,axiom,
! [X21: v,X22: list_v] :
( nil_v
!= ( cons_v @ X21 @ X22 ) ) ).
% list.distinct(1)
thf(fact_716_list_OdiscI,axiom,
! [List: list_v,X21: v,X22: list_v] :
( ( List
= ( cons_v @ X21 @ X22 ) )
=> ( List != nil_v ) ) ).
% list.discI
thf(fact_717_list_Oexhaust,axiom,
! [Y: list_v] :
( ( Y != nil_v )
=> ~ ! [X212: v,X222: list_v] :
( Y
!= ( cons_v @ X212 @ X222 ) ) ) ).
% list.exhaust
thf(fact_718_remdups__adj_Ocases,axiom,
! [X: list_v] :
( ( X != nil_v )
=> ( ! [X4: v] :
( X
!= ( cons_v @ X4 @ nil_v ) )
=> ~ ! [X4: v,Y2: v,Xs3: list_v] :
( X
!= ( cons_v @ X4 @ ( cons_v @ Y2 @ Xs3 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_719_neq__Nil__conv,axiom,
! [Xs: list_v] :
( ( Xs != nil_v )
= ( ? [Y3: v,Ys3: list_v] :
( Xs
= ( cons_v @ Y3 @ Ys3 ) ) ) ) ).
% neq_Nil_conv
thf(fact_720_list__induct2_H,axiom,
! [P: list_v > list_v > $o,Xs: list_v,Ys: list_v] :
( ( P @ nil_v @ nil_v )
=> ( ! [X4: v,Xs3: list_v] : ( P @ ( cons_v @ X4 @ Xs3 ) @ nil_v )
=> ( ! [Y2: v,Ys4: list_v] : ( P @ nil_v @ ( cons_v @ Y2 @ Ys4 ) )
=> ( ! [X4: v,Xs3: list_v,Y2: v,Ys4: list_v] :
( ( P @ Xs3 @ Ys4 )
=> ( P @ ( cons_v @ X4 @ Xs3 ) @ ( cons_v @ Y2 @ Ys4 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_721_list__nonempty__induct,axiom,
! [Xs: list_v,P: list_v > $o] :
( ( Xs != nil_v )
=> ( ! [X4: v] : ( P @ ( cons_v @ X4 @ nil_v ) )
=> ( ! [X4: v,Xs3: list_v] :
( ( Xs3 != nil_v )
=> ( ( P @ Xs3 )
=> ( P @ ( cons_v @ X4 @ Xs3 ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_722_Un__Int__crazy,axiom,
! [A2: set_v,B2: set_v,C2: set_v] :
( ( sup_sup_set_v @ ( sup_sup_set_v @ ( inf_inf_set_v @ A2 @ B2 ) @ ( inf_inf_set_v @ B2 @ C2 ) ) @ ( inf_inf_set_v @ C2 @ A2 ) )
= ( inf_inf_set_v @ ( inf_inf_set_v @ ( sup_sup_set_v @ A2 @ B2 ) @ ( sup_sup_set_v @ B2 @ C2 ) ) @ ( sup_sup_set_v @ C2 @ A2 ) ) ) ).
% Un_Int_crazy
thf(fact_723_Un__Int__crazy,axiom,
! [A2: set_Product_prod_v_v,B2: set_Product_prod_v_v,C2: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ A2 @ B2 ) @ ( inf_in6271465464967711157od_v_v @ B2 @ C2 ) ) @ ( inf_in6271465464967711157od_v_v @ C2 @ A2 ) )
= ( inf_in6271465464967711157od_v_v @ ( inf_in6271465464967711157od_v_v @ ( sup_su414716646722978715od_v_v @ A2 @ B2 ) @ ( sup_su414716646722978715od_v_v @ B2 @ C2 ) ) @ ( sup_su414716646722978715od_v_v @ C2 @ A2 ) ) ) ).
% Un_Int_crazy
thf(fact_724_Int__Un__distrib,axiom,
! [A2: set_v,B2: set_v,C2: set_v] :
( ( inf_inf_set_v @ A2 @ ( sup_sup_set_v @ B2 @ C2 ) )
= ( sup_sup_set_v @ ( inf_inf_set_v @ A2 @ B2 ) @ ( inf_inf_set_v @ A2 @ C2 ) ) ) ).
% Int_Un_distrib
thf(fact_725_Int__Un__distrib,axiom,
! [A2: set_Product_prod_v_v,B2: set_Product_prod_v_v,C2: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ A2 @ ( sup_su414716646722978715od_v_v @ B2 @ C2 ) )
= ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ A2 @ B2 ) @ ( inf_in6271465464967711157od_v_v @ A2 @ C2 ) ) ) ).
% Int_Un_distrib
thf(fact_726_Un__Int__distrib,axiom,
! [A2: set_v,B2: set_v,C2: set_v] :
( ( sup_sup_set_v @ A2 @ ( inf_inf_set_v @ B2 @ C2 ) )
= ( inf_inf_set_v @ ( sup_sup_set_v @ A2 @ B2 ) @ ( sup_sup_set_v @ A2 @ C2 ) ) ) ).
% Un_Int_distrib
thf(fact_727_Un__Int__distrib,axiom,
! [A2: set_Product_prod_v_v,B2: set_Product_prod_v_v,C2: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ A2 @ ( inf_in6271465464967711157od_v_v @ B2 @ C2 ) )
= ( inf_in6271465464967711157od_v_v @ ( sup_su414716646722978715od_v_v @ A2 @ B2 ) @ ( sup_su414716646722978715od_v_v @ A2 @ C2 ) ) ) ).
% Un_Int_distrib
thf(fact_728_Int__Un__distrib2,axiom,
! [B2: set_v,C2: set_v,A2: set_v] :
( ( inf_inf_set_v @ ( sup_sup_set_v @ B2 @ C2 ) @ A2 )
= ( sup_sup_set_v @ ( inf_inf_set_v @ B2 @ A2 ) @ ( inf_inf_set_v @ C2 @ A2 ) ) ) ).
% Int_Un_distrib2
thf(fact_729_Int__Un__distrib2,axiom,
! [B2: set_Product_prod_v_v,C2: set_Product_prod_v_v,A2: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ ( sup_su414716646722978715od_v_v @ B2 @ C2 ) @ A2 )
= ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ B2 @ A2 ) @ ( inf_in6271465464967711157od_v_v @ C2 @ A2 ) ) ) ).
% Int_Un_distrib2
thf(fact_730_Un__Int__distrib2,axiom,
! [B2: set_v,C2: set_v,A2: set_v] :
( ( sup_sup_set_v @ ( inf_inf_set_v @ B2 @ C2 ) @ A2 )
= ( inf_inf_set_v @ ( sup_sup_set_v @ B2 @ A2 ) @ ( sup_sup_set_v @ C2 @ A2 ) ) ) ).
% Un_Int_distrib2
thf(fact_731_Un__Int__distrib2,axiom,
! [B2: set_Product_prod_v_v,C2: set_Product_prod_v_v,A2: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ B2 @ C2 ) @ A2 )
= ( inf_in6271465464967711157od_v_v @ ( sup_su414716646722978715od_v_v @ B2 @ A2 ) @ ( sup_su414716646722978715od_v_v @ C2 @ A2 ) ) ) ).
% Un_Int_distrib2
thf(fact_732_list_Oset__intros_I2_J,axiom,
! [Y: product_prod_v_v,X22: list_P7986770385144383213od_v_v,X21: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ Y @ ( set_Product_prod_v_v2 @ X22 ) )
=> ( member7453568604450474000od_v_v @ Y @ ( set_Product_prod_v_v2 @ ( cons_P4120604216776828829od_v_v @ X21 @ X22 ) ) ) ) ).
% list.set_intros(2)
thf(fact_733_list_Oset__intros_I2_J,axiom,
! [Y: v,X22: list_v,X21: v] :
( ( member_v @ Y @ ( set_v2 @ X22 ) )
=> ( member_v @ Y @ ( set_v2 @ ( cons_v @ X21 @ X22 ) ) ) ) ).
% list.set_intros(2)
thf(fact_734_list_Oset__intros_I1_J,axiom,
! [X21: product_prod_v_v,X22: list_P7986770385144383213od_v_v] : ( member7453568604450474000od_v_v @ X21 @ ( set_Product_prod_v_v2 @ ( cons_P4120604216776828829od_v_v @ X21 @ X22 ) ) ) ).
% list.set_intros(1)
thf(fact_735_list_Oset__intros_I1_J,axiom,
! [X21: v,X22: list_v] : ( member_v @ X21 @ ( set_v2 @ ( cons_v @ X21 @ X22 ) ) ) ).
% list.set_intros(1)
thf(fact_736_list_Oset__cases,axiom,
! [E: product_prod_v_v,A: list_P7986770385144383213od_v_v] :
( ( member7453568604450474000od_v_v @ E @ ( set_Product_prod_v_v2 @ A ) )
=> ( ! [Z22: list_P7986770385144383213od_v_v] :
( A
!= ( cons_P4120604216776828829od_v_v @ E @ Z22 ) )
=> ~ ! [Z1: product_prod_v_v,Z22: list_P7986770385144383213od_v_v] :
( ( A
= ( cons_P4120604216776828829od_v_v @ Z1 @ Z22 ) )
=> ~ ( member7453568604450474000od_v_v @ E @ ( set_Product_prod_v_v2 @ Z22 ) ) ) ) ) ).
% list.set_cases
thf(fact_737_list_Oset__cases,axiom,
! [E: v,A: list_v] :
( ( member_v @ E @ ( set_v2 @ A ) )
=> ( ! [Z22: list_v] :
( A
!= ( cons_v @ E @ Z22 ) )
=> ~ ! [Z1: v,Z22: list_v] :
( ( A
= ( cons_v @ Z1 @ Z22 ) )
=> ~ ( member_v @ E @ ( set_v2 @ Z22 ) ) ) ) ) ).
% list.set_cases
thf(fact_738_set__ConsD,axiom,
! [Y: product_prod_v_v,X: product_prod_v_v,Xs: list_P7986770385144383213od_v_v] :
( ( member7453568604450474000od_v_v @ Y @ ( set_Product_prod_v_v2 @ ( cons_P4120604216776828829od_v_v @ X @ Xs ) ) )
=> ( ( Y = X )
| ( member7453568604450474000od_v_v @ Y @ ( set_Product_prod_v_v2 @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_739_set__ConsD,axiom,
! [Y: v,X: v,Xs: list_v] :
( ( member_v @ Y @ ( set_v2 @ ( cons_v @ X @ Xs ) ) )
=> ( ( Y = X )
| ( member_v @ Y @ ( set_v2 @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_740_Int__insert__left,axiom,
! [A: product_prod_v_v,C2: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( ( member7453568604450474000od_v_v @ A @ C2 )
=> ( ( inf_in6271465464967711157od_v_v @ ( insert1338601472111419319od_v_v @ A @ B2 ) @ C2 )
= ( insert1338601472111419319od_v_v @ A @ ( inf_in6271465464967711157od_v_v @ B2 @ C2 ) ) ) )
& ( ~ ( member7453568604450474000od_v_v @ A @ C2 )
=> ( ( inf_in6271465464967711157od_v_v @ ( insert1338601472111419319od_v_v @ A @ B2 ) @ C2 )
= ( inf_in6271465464967711157od_v_v @ B2 @ C2 ) ) ) ) ).
% Int_insert_left
thf(fact_741_Int__insert__left,axiom,
! [A: v,C2: set_v,B2: set_v] :
( ( ( member_v @ A @ C2 )
=> ( ( inf_inf_set_v @ ( insert_v @ A @ B2 ) @ C2 )
= ( insert_v @ A @ ( inf_inf_set_v @ B2 @ C2 ) ) ) )
& ( ~ ( member_v @ A @ C2 )
=> ( ( inf_inf_set_v @ ( insert_v @ A @ B2 ) @ C2 )
= ( inf_inf_set_v @ B2 @ C2 ) ) ) ) ).
% Int_insert_left
thf(fact_742_Int__insert__right,axiom,
! [A: product_prod_v_v,A2: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( ( member7453568604450474000od_v_v @ A @ A2 )
=> ( ( inf_in6271465464967711157od_v_v @ A2 @ ( insert1338601472111419319od_v_v @ A @ B2 ) )
= ( insert1338601472111419319od_v_v @ A @ ( inf_in6271465464967711157od_v_v @ A2 @ B2 ) ) ) )
& ( ~ ( member7453568604450474000od_v_v @ A @ A2 )
=> ( ( inf_in6271465464967711157od_v_v @ A2 @ ( insert1338601472111419319od_v_v @ A @ B2 ) )
= ( inf_in6271465464967711157od_v_v @ A2 @ B2 ) ) ) ) ).
% Int_insert_right
thf(fact_743_Int__insert__right,axiom,
! [A: v,A2: set_v,B2: set_v] :
( ( ( member_v @ A @ A2 )
=> ( ( inf_inf_set_v @ A2 @ ( insert_v @ A @ B2 ) )
= ( insert_v @ A @ ( inf_inf_set_v @ A2 @ B2 ) ) ) )
& ( ~ ( member_v @ A @ A2 )
=> ( ( inf_inf_set_v @ A2 @ ( insert_v @ A @ B2 ) )
= ( inf_inf_set_v @ A2 @ B2 ) ) ) ) ).
% Int_insert_right
thf(fact_744_Un__Diff,axiom,
! [A2: set_Product_prod_v_v,B2: set_Product_prod_v_v,C2: set_Product_prod_v_v] :
( ( minus_4183494784930505774od_v_v @ ( sup_su414716646722978715od_v_v @ A2 @ B2 ) @ C2 )
= ( sup_su414716646722978715od_v_v @ ( minus_4183494784930505774od_v_v @ A2 @ C2 ) @ ( minus_4183494784930505774od_v_v @ B2 @ C2 ) ) ) ).
% Un_Diff
thf(fact_745_Un__Diff,axiom,
! [A2: set_v,B2: set_v,C2: set_v] :
( ( minus_minus_set_v @ ( sup_sup_set_v @ A2 @ B2 ) @ C2 )
= ( sup_sup_set_v @ ( minus_minus_set_v @ A2 @ C2 ) @ ( minus_minus_set_v @ B2 @ C2 ) ) ) ).
% Un_Diff
thf(fact_746_insert__Diff__if,axiom,
! [X: product_prod_v_v,B2: set_Product_prod_v_v,A2: set_Product_prod_v_v] :
( ( ( member7453568604450474000od_v_v @ X @ B2 )
=> ( ( minus_4183494784930505774od_v_v @ ( insert1338601472111419319od_v_v @ X @ A2 ) @ B2 )
= ( minus_4183494784930505774od_v_v @ A2 @ B2 ) ) )
& ( ~ ( member7453568604450474000od_v_v @ X @ B2 )
=> ( ( minus_4183494784930505774od_v_v @ ( insert1338601472111419319od_v_v @ X @ A2 ) @ B2 )
= ( insert1338601472111419319od_v_v @ X @ ( minus_4183494784930505774od_v_v @ A2 @ B2 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_747_insert__Diff__if,axiom,
! [X: v,B2: set_v,A2: set_v] :
( ( ( member_v @ X @ B2 )
=> ( ( minus_minus_set_v @ ( insert_v @ X @ A2 ) @ B2 )
= ( minus_minus_set_v @ A2 @ B2 ) ) )
& ( ~ ( member_v @ X @ B2 )
=> ( ( minus_minus_set_v @ ( insert_v @ X @ A2 ) @ B2 )
= ( insert_v @ X @ ( minus_minus_set_v @ A2 @ B2 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_748_append__Cons,axiom,
! [X: v,Xs: list_v,Ys: list_v] :
( ( append_v @ ( cons_v @ X @ Xs ) @ Ys )
= ( cons_v @ X @ ( append_v @ Xs @ Ys ) ) ) ).
% append_Cons
thf(fact_749_Cons__eq__appendI,axiom,
! [X: v,Xs1: list_v,Ys: list_v,Xs: list_v,Zs: list_v] :
( ( ( cons_v @ X @ Xs1 )
= Ys )
=> ( ( Xs
= ( append_v @ Xs1 @ Zs ) )
=> ( ( cons_v @ X @ Xs )
= ( append_v @ Ys @ Zs ) ) ) ) ).
% Cons_eq_appendI
thf(fact_750_distinct__length__2__or__more,axiom,
! [A: v,B: v,Xs: list_v] :
( ( distinct_v @ ( cons_v @ A @ ( cons_v @ B @ Xs ) ) )
= ( ( A != B )
& ( distinct_v @ ( cons_v @ A @ Xs ) )
& ( distinct_v @ ( cons_v @ B @ Xs ) ) ) ) ).
% distinct_length_2_or_more
thf(fact_751_list_Osel_I1_J,axiom,
! [X21: v,X22: list_v] :
( ( hd_v @ ( cons_v @ X21 @ X22 ) )
= X21 ) ).
% list.sel(1)
thf(fact_752_list_Osel_I3_J,axiom,
! [X21: v,X22: list_v] :
( ( tl_v @ ( cons_v @ X21 @ X22 ) )
= X22 ) ).
% list.sel(3)
thf(fact_753_precedes__in__tail,axiom,
! [X: v,Z: v,Y: v,Zs: list_v] :
( ( X != Z )
=> ( ( sCC_Bl4022239298816431255edes_v @ X @ Y @ ( cons_v @ Z @ Zs ) )
= ( sCC_Bl4022239298816431255edes_v @ X @ Y @ Zs ) ) ) ).
% precedes_in_tail
thf(fact_754_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_755_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_756_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_757_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_758_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_759_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_760_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_761_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_762_graph_Osucc__re,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,Y: product_prod_v_v,X: product_prod_v_v,Z: product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( member7453568604450474000od_v_v @ Y @ ( Successors @ X ) )
=> ( ( sCC_Bl4714988717384592488od_v_v @ Successors @ Y @ Z )
=> ( sCC_Bl4714988717384592488od_v_v @ Successors @ X @ Z ) ) ) ) ).
% graph.succ_re
thf(fact_763_graph_Osucc__re,axiom,
! [Vertices: set_v,Successors: v > set_v,Y: v,X: v,Z: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( member_v @ Y @ ( Successors @ X ) )
=> ( ( sCC_Bl770211535891879572_end_v @ Successors @ Y @ Z )
=> ( sCC_Bl770211535891879572_end_v @ Successors @ X @ Z ) ) ) ) ).
% graph.succ_re
thf(fact_764_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,A22: product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( sCC_Bl4714988717384592488od_v_v @ Successors @ A1 @ A22 )
=> ( ( A22 != A1 )
=> ~ ! [Y2: product_prod_v_v] :
( ( sCC_Bl4714988717384592488od_v_v @ Successors @ A1 @ Y2 )
=> ~ ( member7453568604450474000od_v_v @ A22 @ ( Successors @ Y2 ) ) ) ) ) ) ).
% graph.reachable_end.cases
thf(fact_765_graph_Oreachable__end_Ocases,axiom,
! [Vertices: set_v,Successors: v > set_v,A1: v,A22: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl770211535891879572_end_v @ Successors @ A1 @ A22 )
=> ( ( A22 != A1 )
=> ~ ! [Y2: v] :
( ( sCC_Bl770211535891879572_end_v @ Successors @ A1 @ Y2 )
=> ~ ( member_v @ A22 @ ( Successors @ Y2 ) ) ) ) ) ) ).
% graph.reachable_end.cases
thf(fact_766_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,A22: product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( sCC_Bl4714988717384592488od_v_v @ Successors @ A1 @ A22 )
= ( ? [X3: product_prod_v_v] :
( ( A1 = X3 )
& ( A22 = X3 ) )
| ? [X3: product_prod_v_v,Y3: product_prod_v_v,Z3: product_prod_v_v] :
( ( A1 = X3 )
& ( A22 = Z3 )
& ( sCC_Bl4714988717384592488od_v_v @ Successors @ X3 @ Y3 )
& ( member7453568604450474000od_v_v @ Z3 @ ( Successors @ Y3 ) ) ) ) ) ) ).
% graph.reachable_end.simps
thf(fact_767_graph_Oreachable__end_Osimps,axiom,
! [Vertices: set_v,Successors: v > set_v,A1: v,A22: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl770211535891879572_end_v @ Successors @ A1 @ A22 )
= ( ? [X3: v] :
( ( A1 = X3 )
& ( A22 = X3 ) )
| ? [X3: v,Y3: v,Z3: v] :
( ( A1 = X3 )
& ( A22 = Z3 )
& ( sCC_Bl770211535891879572_end_v @ Successors @ X3 @ Y3 )
& ( member_v @ Z3 @ ( Successors @ Y3 ) ) ) ) ) ) ).
% graph.reachable_end.simps
thf(fact_768_graph_Ore__refl,axiom,
! [Vertices: set_v,Successors: v > set_v,X: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( sCC_Bl770211535891879572_end_v @ Successors @ X @ X ) ) ).
% graph.re_refl
thf(fact_769_graph_Ore__succ,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,X: product_prod_v_v,Y: product_prod_v_v,Z: product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( sCC_Bl4714988717384592488od_v_v @ Successors @ X @ Y )
=> ( ( member7453568604450474000od_v_v @ Z @ ( Successors @ Y ) )
=> ( sCC_Bl4714988717384592488od_v_v @ Successors @ X @ Z ) ) ) ) ).
% graph.re_succ
thf(fact_770_graph_Ore__succ,axiom,
! [Vertices: set_v,Successors: v > set_v,X: v,Y: v,Z: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl770211535891879572_end_v @ Successors @ X @ Y )
=> ( ( member_v @ Z @ ( Successors @ Y ) )
=> ( sCC_Bl770211535891879572_end_v @ Successors @ X @ Z ) ) ) ) ).
% graph.re_succ
thf(fact_771_distrib__inf__le,axiom,
! [X: set_v,Y: set_v,Z: set_v] : ( ord_less_eq_set_v @ ( sup_sup_set_v @ ( inf_inf_set_v @ X @ Y ) @ ( inf_inf_set_v @ X @ Z ) ) @ ( inf_inf_set_v @ X @ ( sup_sup_set_v @ Y @ Z ) ) ) ).
% distrib_inf_le
thf(fact_772_distrib__inf__le,axiom,
! [X: set_Product_prod_v_v,Y: set_Product_prod_v_v,Z: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ X @ Y ) @ ( inf_in6271465464967711157od_v_v @ X @ Z ) ) @ ( inf_in6271465464967711157od_v_v @ X @ ( sup_su414716646722978715od_v_v @ Y @ Z ) ) ) ).
% distrib_inf_le
thf(fact_773_distrib__sup__le,axiom,
! [X: set_v,Y: set_v,Z: set_v] : ( ord_less_eq_set_v @ ( sup_sup_set_v @ X @ ( inf_inf_set_v @ Y @ Z ) ) @ ( inf_inf_set_v @ ( sup_sup_set_v @ X @ Y ) @ ( sup_sup_set_v @ X @ Z ) ) ) ).
% distrib_sup_le
thf(fact_774_distrib__sup__le,axiom,
! [X: set_Product_prod_v_v,Y: set_Product_prod_v_v,Z: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ ( sup_su414716646722978715od_v_v @ X @ ( inf_in6271465464967711157od_v_v @ Y @ Z ) ) @ ( inf_in6271465464967711157od_v_v @ ( sup_su414716646722978715od_v_v @ X @ Y ) @ ( sup_su414716646722978715od_v_v @ X @ Z ) ) ) ).
% distrib_sup_le
thf(fact_775_subset__singletonD,axiom,
! [A2: set_v,X: v] :
( ( ord_less_eq_set_v @ A2 @ ( insert_v @ X @ bot_bot_set_v ) )
=> ( ( A2 = bot_bot_set_v )
| ( A2
= ( insert_v @ X @ bot_bot_set_v ) ) ) ) ).
% subset_singletonD
thf(fact_776_subset__singletonD,axiom,
! [A2: set_Product_prod_v_v,X: product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A2 @ ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v ) )
=> ( ( A2 = bot_bo723834152578015283od_v_v )
| ( A2
= ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v ) ) ) ) ).
% subset_singletonD
thf(fact_777_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_778_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_779_Un__Int__assoc__eq,axiom,
! [A2: set_v,B2: set_v,C2: set_v] :
( ( ( sup_sup_set_v @ ( inf_inf_set_v @ A2 @ B2 ) @ C2 )
= ( inf_inf_set_v @ A2 @ ( sup_sup_set_v @ B2 @ C2 ) ) )
= ( ord_less_eq_set_v @ C2 @ A2 ) ) ).
% Un_Int_assoc_eq
thf(fact_780_Un__Int__assoc__eq,axiom,
! [A2: set_Product_prod_v_v,B2: set_Product_prod_v_v,C2: set_Product_prod_v_v] :
( ( ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ A2 @ B2 ) @ C2 )
= ( inf_in6271465464967711157od_v_v @ A2 @ ( sup_su414716646722978715od_v_v @ B2 @ C2 ) ) )
= ( ord_le7336532860387713383od_v_v @ C2 @ A2 ) ) ).
% Un_Int_assoc_eq
thf(fact_781_Diff__subset__conv,axiom,
! [A2: set_v,B2: set_v,C2: set_v] :
( ( ord_less_eq_set_v @ ( minus_minus_set_v @ A2 @ B2 ) @ C2 )
= ( ord_less_eq_set_v @ A2 @ ( sup_sup_set_v @ B2 @ C2 ) ) ) ).
% Diff_subset_conv
thf(fact_782_Diff__subset__conv,axiom,
! [A2: set_Product_prod_v_v,B2: set_Product_prod_v_v,C2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ ( minus_4183494784930505774od_v_v @ A2 @ B2 ) @ C2 )
= ( ord_le7336532860387713383od_v_v @ A2 @ ( sup_su414716646722978715od_v_v @ B2 @ C2 ) ) ) ).
% Diff_subset_conv
thf(fact_783_Diff__partition,axiom,
! [A2: set_v,B2: set_v] :
( ( ord_less_eq_set_v @ A2 @ B2 )
=> ( ( sup_sup_set_v @ A2 @ ( minus_minus_set_v @ B2 @ A2 ) )
= B2 ) ) ).
% Diff_partition
thf(fact_784_Diff__partition,axiom,
! [A2: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A2 @ B2 )
=> ( ( sup_su414716646722978715od_v_v @ A2 @ ( minus_4183494784930505774od_v_v @ B2 @ A2 ) )
= B2 ) ) ).
% Diff_partition
thf(fact_785_Diff__insert,axiom,
! [A2: set_Product_prod_v_v,A: product_prod_v_v,B2: set_Product_prod_v_v] :
( ( minus_4183494784930505774od_v_v @ A2 @ ( insert1338601472111419319od_v_v @ A @ B2 ) )
= ( minus_4183494784930505774od_v_v @ ( minus_4183494784930505774od_v_v @ A2 @ B2 ) @ ( insert1338601472111419319od_v_v @ A @ bot_bo723834152578015283od_v_v ) ) ) ).
% Diff_insert
thf(fact_786_Diff__insert,axiom,
! [A2: set_v,A: v,B2: set_v] :
( ( minus_minus_set_v @ A2 @ ( insert_v @ A @ B2 ) )
= ( minus_minus_set_v @ ( minus_minus_set_v @ A2 @ B2 ) @ ( insert_v @ A @ bot_bot_set_v ) ) ) ).
% Diff_insert
thf(fact_787_insert__Diff,axiom,
! [A: product_prod_v_v,A2: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ A @ A2 )
=> ( ( insert1338601472111419319od_v_v @ A @ ( minus_4183494784930505774od_v_v @ A2 @ ( insert1338601472111419319od_v_v @ A @ bot_bo723834152578015283od_v_v ) ) )
= A2 ) ) ).
% insert_Diff
thf(fact_788_insert__Diff,axiom,
! [A: v,A2: set_v] :
( ( member_v @ A @ A2 )
=> ( ( insert_v @ A @ ( minus_minus_set_v @ A2 @ ( insert_v @ A @ bot_bot_set_v ) ) )
= A2 ) ) ).
% insert_Diff
thf(fact_789_Diff__insert2,axiom,
! [A2: set_Product_prod_v_v,A: product_prod_v_v,B2: set_Product_prod_v_v] :
( ( minus_4183494784930505774od_v_v @ A2 @ ( insert1338601472111419319od_v_v @ A @ B2 ) )
= ( minus_4183494784930505774od_v_v @ ( minus_4183494784930505774od_v_v @ A2 @ ( insert1338601472111419319od_v_v @ A @ bot_bo723834152578015283od_v_v ) ) @ B2 ) ) ).
% Diff_insert2
thf(fact_790_Diff__insert2,axiom,
! [A2: set_v,A: v,B2: set_v] :
( ( minus_minus_set_v @ A2 @ ( insert_v @ A @ B2 ) )
= ( minus_minus_set_v @ ( minus_minus_set_v @ A2 @ ( insert_v @ A @ bot_bot_set_v ) ) @ B2 ) ) ).
% Diff_insert2
thf(fact_791_Diff__insert__absorb,axiom,
! [X: product_prod_v_v,A2: set_Product_prod_v_v] :
( ~ ( member7453568604450474000od_v_v @ X @ A2 )
=> ( ( minus_4183494784930505774od_v_v @ ( insert1338601472111419319od_v_v @ X @ A2 ) @ ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v ) )
= A2 ) ) ).
% Diff_insert_absorb
thf(fact_792_Diff__insert__absorb,axiom,
! [X: v,A2: set_v] :
( ~ ( member_v @ X @ A2 )
=> ( ( minus_minus_set_v @ ( insert_v @ X @ A2 ) @ ( insert_v @ X @ bot_bot_set_v ) )
= A2 ) ) ).
% Diff_insert_absorb
thf(fact_793_set__subset__Cons,axiom,
! [Xs: list_v,X: v] : ( ord_less_eq_set_v @ ( set_v2 @ Xs ) @ ( set_v2 @ ( cons_v @ X @ Xs ) ) ) ).
% set_subset_Cons
thf(fact_794_set__subset__Cons,axiom,
! [Xs: list_P7986770385144383213od_v_v,X: product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ ( set_Product_prod_v_v2 @ Xs ) @ ( set_Product_prod_v_v2 @ ( cons_P4120604216776828829od_v_v @ X @ Xs ) ) ) ).
% set_subset_Cons
thf(fact_795_subset__Diff__insert,axiom,
! [A2: set_v,B2: set_v,X: v,C2: set_v] :
( ( ord_less_eq_set_v @ A2 @ ( minus_minus_set_v @ B2 @ ( insert_v @ X @ C2 ) ) )
= ( ( ord_less_eq_set_v @ A2 @ ( minus_minus_set_v @ B2 @ C2 ) )
& ~ ( member_v @ X @ A2 ) ) ) ).
% subset_Diff_insert
thf(fact_796_subset__Diff__insert,axiom,
! [A2: set_Product_prod_v_v,B2: set_Product_prod_v_v,X: product_prod_v_v,C2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A2 @ ( minus_4183494784930505774od_v_v @ B2 @ ( insert1338601472111419319od_v_v @ X @ C2 ) ) )
= ( ( ord_le7336532860387713383od_v_v @ A2 @ ( minus_4183494784930505774od_v_v @ B2 @ C2 ) )
& ~ ( member7453568604450474000od_v_v @ X @ A2 ) ) ) ).
% subset_Diff_insert
thf(fact_797_Diff__Un,axiom,
! [A2: set_Product_prod_v_v,B2: set_Product_prod_v_v,C2: set_Product_prod_v_v] :
( ( minus_4183494784930505774od_v_v @ A2 @ ( sup_su414716646722978715od_v_v @ B2 @ C2 ) )
= ( inf_in6271465464967711157od_v_v @ ( minus_4183494784930505774od_v_v @ A2 @ B2 ) @ ( minus_4183494784930505774od_v_v @ A2 @ C2 ) ) ) ).
% Diff_Un
thf(fact_798_Diff__Un,axiom,
! [A2: set_v,B2: set_v,C2: set_v] :
( ( minus_minus_set_v @ A2 @ ( sup_sup_set_v @ B2 @ C2 ) )
= ( inf_inf_set_v @ ( minus_minus_set_v @ A2 @ B2 ) @ ( minus_minus_set_v @ A2 @ C2 ) ) ) ).
% Diff_Un
thf(fact_799_Diff__Int,axiom,
! [A2: set_Product_prod_v_v,B2: set_Product_prod_v_v,C2: set_Product_prod_v_v] :
( ( minus_4183494784930505774od_v_v @ A2 @ ( inf_in6271465464967711157od_v_v @ B2 @ C2 ) )
= ( sup_su414716646722978715od_v_v @ ( minus_4183494784930505774od_v_v @ A2 @ B2 ) @ ( minus_4183494784930505774od_v_v @ A2 @ C2 ) ) ) ).
% Diff_Int
thf(fact_800_Diff__Int,axiom,
! [A2: set_v,B2: set_v,C2: set_v] :
( ( minus_minus_set_v @ A2 @ ( inf_inf_set_v @ B2 @ C2 ) )
= ( sup_sup_set_v @ ( minus_minus_set_v @ A2 @ B2 ) @ ( minus_minus_set_v @ A2 @ C2 ) ) ) ).
% Diff_Int
thf(fact_801_Int__Diff__Un,axiom,
! [A2: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ A2 @ B2 ) @ ( minus_4183494784930505774od_v_v @ A2 @ B2 ) )
= A2 ) ).
% Int_Diff_Un
thf(fact_802_Int__Diff__Un,axiom,
! [A2: set_v,B2: set_v] :
( ( sup_sup_set_v @ ( inf_inf_set_v @ A2 @ B2 ) @ ( minus_minus_set_v @ A2 @ B2 ) )
= A2 ) ).
% Int_Diff_Un
thf(fact_803_Un__Diff__Int,axiom,
! [A2: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ ( minus_4183494784930505774od_v_v @ A2 @ B2 ) @ ( inf_in6271465464967711157od_v_v @ A2 @ B2 ) )
= A2 ) ).
% Un_Diff_Int
thf(fact_804_Un__Diff__Int,axiom,
! [A2: set_v,B2: set_v] :
( ( sup_sup_set_v @ ( minus_minus_set_v @ A2 @ B2 ) @ ( inf_inf_set_v @ A2 @ B2 ) )
= A2 ) ).
% Un_Diff_Int
thf(fact_805_rev__induct,axiom,
! [P: list_v > $o,Xs: list_v] :
( ( P @ nil_v )
=> ( ! [X4: v,Xs3: list_v] :
( ( P @ Xs3 )
=> ( P @ ( append_v @ Xs3 @ ( cons_v @ X4 @ nil_v ) ) ) )
=> ( P @ Xs ) ) ) ).
% rev_induct
thf(fact_806_rev__exhaust,axiom,
! [Xs: list_v] :
( ( Xs != nil_v )
=> ~ ! [Ys4: list_v,Y2: v] :
( Xs
!= ( append_v @ Ys4 @ ( cons_v @ Y2 @ nil_v ) ) ) ) ).
% rev_exhaust
thf(fact_807_Cons__eq__append__conv,axiom,
! [X: v,Xs: list_v,Ys: list_v,Zs: list_v] :
( ( ( cons_v @ X @ Xs )
= ( append_v @ Ys @ Zs ) )
= ( ( ( Ys = nil_v )
& ( ( cons_v @ X @ Xs )
= Zs ) )
| ? [Ys5: list_v] :
( ( ( cons_v @ X @ Ys5 )
= Ys )
& ( Xs
= ( append_v @ Ys5 @ Zs ) ) ) ) ) ).
% Cons_eq_append_conv
thf(fact_808_append__eq__Cons__conv,axiom,
! [Ys: list_v,Zs: list_v,X: v,Xs: list_v] :
( ( ( append_v @ Ys @ Zs )
= ( cons_v @ X @ Xs ) )
= ( ( ( Ys = nil_v )
& ( Zs
= ( cons_v @ X @ Xs ) ) )
| ? [Ys5: list_v] :
( ( Ys
= ( cons_v @ X @ Ys5 ) )
& ( ( append_v @ Ys5 @ Zs )
= Xs ) ) ) ) ).
% append_eq_Cons_conv
thf(fact_809_rev__nonempty__induct,axiom,
! [Xs: list_v,P: list_v > $o] :
( ( Xs != nil_v )
=> ( ! [X4: v] : ( P @ ( cons_v @ X4 @ nil_v ) )
=> ( ! [X4: v,Xs3: list_v] :
( ( Xs3 != nil_v )
=> ( ( P @ Xs3 )
=> ( P @ ( append_v @ Xs3 @ ( cons_v @ X4 @ nil_v ) ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% rev_nonempty_induct
thf(fact_810_split__list__first__prop__iff,axiom,
! [Xs: list_v,P: v > $o] :
( ( ? [X3: v] :
( ( member_v @ X3 @ ( set_v2 @ Xs ) )
& ( P @ X3 ) ) )
= ( ? [Ys3: list_v,X3: v] :
( ? [Zs2: list_v] :
( Xs
= ( append_v @ Ys3 @ ( cons_v @ X3 @ Zs2 ) ) )
& ( P @ X3 )
& ! [Y3: v] :
( ( member_v @ Y3 @ ( set_v2 @ Ys3 ) )
=> ~ ( P @ Y3 ) ) ) ) ) ).
% split_list_first_prop_iff
thf(fact_811_split__list__last__prop__iff,axiom,
! [Xs: list_v,P: v > $o] :
( ( ? [X3: v] :
( ( member_v @ X3 @ ( set_v2 @ Xs ) )
& ( P @ X3 ) ) )
= ( ? [Ys3: list_v,X3: v,Zs2: list_v] :
( ( Xs
= ( append_v @ Ys3 @ ( cons_v @ X3 @ Zs2 ) ) )
& ( P @ X3 )
& ! [Y3: v] :
( ( member_v @ Y3 @ ( set_v2 @ Zs2 ) )
=> ~ ( P @ Y3 ) ) ) ) ) ).
% split_list_last_prop_iff
thf(fact_812_in__set__conv__decomp__first,axiom,
! [X: product_prod_v_v,Xs: list_P7986770385144383213od_v_v] :
( ( member7453568604450474000od_v_v @ X @ ( set_Product_prod_v_v2 @ Xs ) )
= ( ? [Ys3: list_P7986770385144383213od_v_v,Zs2: list_P7986770385144383213od_v_v] :
( ( Xs
= ( append2138873909117096322od_v_v @ Ys3 @ ( cons_P4120604216776828829od_v_v @ X @ Zs2 ) ) )
& ~ ( member7453568604450474000od_v_v @ X @ ( set_Product_prod_v_v2 @ Ys3 ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_813_in__set__conv__decomp__first,axiom,
! [X: v,Xs: list_v] :
( ( member_v @ X @ ( set_v2 @ Xs ) )
= ( ? [Ys3: list_v,Zs2: list_v] :
( ( Xs
= ( append_v @ Ys3 @ ( cons_v @ X @ Zs2 ) ) )
& ~ ( member_v @ X @ ( set_v2 @ Ys3 ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_814_in__set__conv__decomp__last,axiom,
! [X: product_prod_v_v,Xs: list_P7986770385144383213od_v_v] :
( ( member7453568604450474000od_v_v @ X @ ( set_Product_prod_v_v2 @ Xs ) )
= ( ? [Ys3: list_P7986770385144383213od_v_v,Zs2: list_P7986770385144383213od_v_v] :
( ( Xs
= ( append2138873909117096322od_v_v @ Ys3 @ ( cons_P4120604216776828829od_v_v @ X @ Zs2 ) ) )
& ~ ( member7453568604450474000od_v_v @ X @ ( set_Product_prod_v_v2 @ Zs2 ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_815_in__set__conv__decomp__last,axiom,
! [X: v,Xs: list_v] :
( ( member_v @ X @ ( set_v2 @ Xs ) )
= ( ? [Ys3: list_v,Zs2: list_v] :
( ( Xs
= ( append_v @ Ys3 @ ( cons_v @ X @ Zs2 ) ) )
& ~ ( member_v @ X @ ( set_v2 @ Zs2 ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_816_split__list__first__propE,axiom,
! [Xs: list_v,P: v > $o] :
( ? [X2: v] :
( ( member_v @ X2 @ ( set_v2 @ Xs ) )
& ( P @ X2 ) )
=> ~ ! [Ys4: list_v,X4: v] :
( ? [Zs3: list_v] :
( Xs
= ( append_v @ Ys4 @ ( cons_v @ X4 @ Zs3 ) ) )
=> ( ( P @ X4 )
=> ~ ! [Xa: v] :
( ( member_v @ Xa @ ( set_v2 @ Ys4 ) )
=> ~ ( P @ Xa ) ) ) ) ) ).
% split_list_first_propE
thf(fact_817_split__list__last__propE,axiom,
! [Xs: list_v,P: v > $o] :
( ? [X2: v] :
( ( member_v @ X2 @ ( set_v2 @ Xs ) )
& ( P @ X2 ) )
=> ~ ! [Ys4: list_v,X4: v,Zs3: list_v] :
( ( Xs
= ( append_v @ Ys4 @ ( cons_v @ X4 @ Zs3 ) ) )
=> ( ( P @ X4 )
=> ~ ! [Xa: v] :
( ( member_v @ Xa @ ( set_v2 @ Zs3 ) )
=> ~ ( P @ Xa ) ) ) ) ) ).
% split_list_last_propE
thf(fact_818_split__list__first__prop,axiom,
! [Xs: list_v,P: v > $o] :
( ? [X2: v] :
( ( member_v @ X2 @ ( set_v2 @ Xs ) )
& ( P @ X2 ) )
=> ? [Ys4: list_v,X4: v] :
( ? [Zs3: list_v] :
( Xs
= ( append_v @ Ys4 @ ( cons_v @ X4 @ Zs3 ) ) )
& ( P @ X4 )
& ! [Xa: v] :
( ( member_v @ Xa @ ( set_v2 @ Ys4 ) )
=> ~ ( P @ Xa ) ) ) ) ).
% split_list_first_prop
thf(fact_819_split__list__last__prop,axiom,
! [Xs: list_v,P: v > $o] :
( ? [X2: v] :
( ( member_v @ X2 @ ( set_v2 @ Xs ) )
& ( P @ X2 ) )
=> ? [Ys4: list_v,X4: v,Zs3: list_v] :
( ( Xs
= ( append_v @ Ys4 @ ( cons_v @ X4 @ Zs3 ) ) )
& ( P @ X4 )
& ! [Xa: v] :
( ( member_v @ Xa @ ( set_v2 @ Zs3 ) )
=> ~ ( P @ Xa ) ) ) ) ).
% split_list_last_prop
thf(fact_820_in__set__conv__decomp,axiom,
! [X: product_prod_v_v,Xs: list_P7986770385144383213od_v_v] :
( ( member7453568604450474000od_v_v @ X @ ( set_Product_prod_v_v2 @ Xs ) )
= ( ? [Ys3: list_P7986770385144383213od_v_v,Zs2: list_P7986770385144383213od_v_v] :
( Xs
= ( append2138873909117096322od_v_v @ Ys3 @ ( cons_P4120604216776828829od_v_v @ X @ Zs2 ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_821_in__set__conv__decomp,axiom,
! [X: v,Xs: list_v] :
( ( member_v @ X @ ( set_v2 @ Xs ) )
= ( ? [Ys3: list_v,Zs2: list_v] :
( Xs
= ( append_v @ Ys3 @ ( cons_v @ X @ Zs2 ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_822_append__Cons__eq__iff,axiom,
! [X: product_prod_v_v,Xs: list_P7986770385144383213od_v_v,Ys: list_P7986770385144383213od_v_v,Xs4: list_P7986770385144383213od_v_v,Ys6: list_P7986770385144383213od_v_v] :
( ~ ( member7453568604450474000od_v_v @ X @ ( set_Product_prod_v_v2 @ Xs ) )
=> ( ~ ( member7453568604450474000od_v_v @ X @ ( set_Product_prod_v_v2 @ Ys ) )
=> ( ( ( append2138873909117096322od_v_v @ Xs @ ( cons_P4120604216776828829od_v_v @ X @ Ys ) )
= ( append2138873909117096322od_v_v @ Xs4 @ ( cons_P4120604216776828829od_v_v @ X @ Ys6 ) ) )
= ( ( Xs = Xs4 )
& ( Ys = Ys6 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_823_append__Cons__eq__iff,axiom,
! [X: v,Xs: list_v,Ys: list_v,Xs4: list_v,Ys6: list_v] :
( ~ ( member_v @ X @ ( set_v2 @ Xs ) )
=> ( ~ ( member_v @ X @ ( set_v2 @ Ys ) )
=> ( ( ( append_v @ Xs @ ( cons_v @ X @ Ys ) )
= ( append_v @ Xs4 @ ( cons_v @ X @ Ys6 ) ) )
= ( ( Xs = Xs4 )
& ( Ys = Ys6 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_824_split__list__propE,axiom,
! [Xs: list_v,P: v > $o] :
( ? [X2: v] :
( ( member_v @ X2 @ ( set_v2 @ Xs ) )
& ( P @ X2 ) )
=> ~ ! [Ys4: list_v,X4: v] :
( ? [Zs3: list_v] :
( Xs
= ( append_v @ Ys4 @ ( cons_v @ X4 @ Zs3 ) ) )
=> ~ ( P @ X4 ) ) ) ).
% split_list_propE
thf(fact_825_split__list__first,axiom,
! [X: product_prod_v_v,Xs: list_P7986770385144383213od_v_v] :
( ( member7453568604450474000od_v_v @ X @ ( set_Product_prod_v_v2 @ Xs ) )
=> ? [Ys4: list_P7986770385144383213od_v_v,Zs3: list_P7986770385144383213od_v_v] :
( ( Xs
= ( append2138873909117096322od_v_v @ Ys4 @ ( cons_P4120604216776828829od_v_v @ X @ Zs3 ) ) )
& ~ ( member7453568604450474000od_v_v @ X @ ( set_Product_prod_v_v2 @ Ys4 ) ) ) ) ).
% split_list_first
thf(fact_826_split__list__first,axiom,
! [X: v,Xs: list_v] :
( ( member_v @ X @ ( set_v2 @ Xs ) )
=> ? [Ys4: list_v,Zs3: list_v] :
( ( Xs
= ( append_v @ Ys4 @ ( cons_v @ X @ Zs3 ) ) )
& ~ ( member_v @ X @ ( set_v2 @ Ys4 ) ) ) ) ).
% split_list_first
thf(fact_827_split__list__prop,axiom,
! [Xs: list_v,P: v > $o] :
( ? [X2: v] :
( ( member_v @ X2 @ ( set_v2 @ Xs ) )
& ( P @ X2 ) )
=> ? [Ys4: list_v,X4: v] :
( ? [Zs3: list_v] :
( Xs
= ( append_v @ Ys4 @ ( cons_v @ X4 @ Zs3 ) ) )
& ( P @ X4 ) ) ) ).
% split_list_prop
thf(fact_828_split__list__last,axiom,
! [X: product_prod_v_v,Xs: list_P7986770385144383213od_v_v] :
( ( member7453568604450474000od_v_v @ X @ ( set_Product_prod_v_v2 @ Xs ) )
=> ? [Ys4: list_P7986770385144383213od_v_v,Zs3: list_P7986770385144383213od_v_v] :
( ( Xs
= ( append2138873909117096322od_v_v @ Ys4 @ ( cons_P4120604216776828829od_v_v @ X @ Zs3 ) ) )
& ~ ( member7453568604450474000od_v_v @ X @ ( set_Product_prod_v_v2 @ Zs3 ) ) ) ) ).
% split_list_last
thf(fact_829_split__list__last,axiom,
! [X: v,Xs: list_v] :
( ( member_v @ X @ ( set_v2 @ Xs ) )
=> ? [Ys4: list_v,Zs3: list_v] :
( ( Xs
= ( append_v @ Ys4 @ ( cons_v @ X @ Zs3 ) ) )
& ~ ( member_v @ X @ ( set_v2 @ Zs3 ) ) ) ) ).
% split_list_last
thf(fact_830_split__list,axiom,
! [X: product_prod_v_v,Xs: list_P7986770385144383213od_v_v] :
( ( member7453568604450474000od_v_v @ X @ ( set_Product_prod_v_v2 @ Xs ) )
=> ? [Ys4: list_P7986770385144383213od_v_v,Zs3: list_P7986770385144383213od_v_v] :
( Xs
= ( append2138873909117096322od_v_v @ Ys4 @ ( cons_P4120604216776828829od_v_v @ X @ Zs3 ) ) ) ) ).
% split_list
thf(fact_831_split__list,axiom,
! [X: v,Xs: list_v] :
( ( member_v @ X @ ( set_v2 @ Xs ) )
=> ? [Ys4: list_v,Zs3: list_v] :
( Xs
= ( append_v @ Ys4 @ ( cons_v @ X @ Zs3 ) ) ) ) ).
% split_list
thf(fact_832_distinct__singleton,axiom,
! [X: v] : ( distinct_v @ ( cons_v @ X @ nil_v ) ) ).
% distinct_singleton
thf(fact_833_distinct_Osimps_I2_J,axiom,
! [X: product_prod_v_v,Xs: list_P7986770385144383213od_v_v] :
( ( distin6159370996967099744od_v_v @ ( cons_P4120604216776828829od_v_v @ X @ Xs ) )
= ( ~ ( member7453568604450474000od_v_v @ X @ ( set_Product_prod_v_v2 @ Xs ) )
& ( distin6159370996967099744od_v_v @ Xs ) ) ) ).
% distinct.simps(2)
thf(fact_834_distinct_Osimps_I2_J,axiom,
! [X: v,Xs: list_v] :
( ( distinct_v @ ( cons_v @ X @ Xs ) )
= ( ~ ( member_v @ X @ ( set_v2 @ Xs ) )
& ( distinct_v @ Xs ) ) ) ).
% distinct.simps(2)
thf(fact_835_Nil__tl,axiom,
! [Xs: list_v] :
( ( nil_v
= ( tl_v @ Xs ) )
= ( ( Xs = nil_v )
| ? [X3: v] :
( Xs
= ( cons_v @ X3 @ nil_v ) ) ) ) ).
% Nil_tl
thf(fact_836_tl__Nil,axiom,
! [Xs: list_v] :
( ( ( tl_v @ Xs )
= nil_v )
= ( ( Xs = nil_v )
| ? [X3: v] :
( Xs
= ( cons_v @ X3 @ nil_v ) ) ) ) ).
% tl_Nil
thf(fact_837_tail__not__precedes,axiom,
! [Y: product_prod_v_v,X: product_prod_v_v,Xs: list_P7986770385144383213od_v_v] :
( ( sCC_Bl2026170059108282219od_v_v @ Y @ X @ ( cons_P4120604216776828829od_v_v @ X @ Xs ) )
=> ( ~ ( member7453568604450474000od_v_v @ X @ ( set_Product_prod_v_v2 @ Xs ) )
=> ( X = Y ) ) ) ).
% tail_not_precedes
thf(fact_838_tail__not__precedes,axiom,
! [Y: v,X: v,Xs: list_v] :
( ( sCC_Bl4022239298816431255edes_v @ Y @ X @ ( cons_v @ X @ Xs ) )
=> ( ~ ( member_v @ X @ ( set_v2 @ Xs ) )
=> ( X = Y ) ) ) ).
% tail_not_precedes
thf(fact_839_head__precedes,axiom,
! [Y: product_prod_v_v,X: product_prod_v_v,Xs: list_P7986770385144383213od_v_v] :
( ( member7453568604450474000od_v_v @ Y @ ( set_Product_prod_v_v2 @ ( cons_P4120604216776828829od_v_v @ X @ Xs ) ) )
=> ( sCC_Bl2026170059108282219od_v_v @ X @ Y @ ( cons_P4120604216776828829od_v_v @ X @ Xs ) ) ) ).
% head_precedes
thf(fact_840_head__precedes,axiom,
! [Y: v,X: v,Xs: list_v] :
( ( member_v @ Y @ ( set_v2 @ ( cons_v @ X @ Xs ) ) )
=> ( sCC_Bl4022239298816431255edes_v @ X @ Y @ ( cons_v @ X @ Xs ) ) ) ).
% head_precedes
thf(fact_841_subset__insert__iff,axiom,
! [A2: set_v,X: v,B2: set_v] :
( ( ord_less_eq_set_v @ A2 @ ( insert_v @ X @ B2 ) )
= ( ( ( member_v @ X @ A2 )
=> ( ord_less_eq_set_v @ ( minus_minus_set_v @ A2 @ ( insert_v @ X @ bot_bot_set_v ) ) @ B2 ) )
& ( ~ ( member_v @ X @ A2 )
=> ( ord_less_eq_set_v @ A2 @ B2 ) ) ) ) ).
% subset_insert_iff
thf(fact_842_subset__insert__iff,axiom,
! [A2: set_Product_prod_v_v,X: product_prod_v_v,B2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A2 @ ( insert1338601472111419319od_v_v @ X @ B2 ) )
= ( ( ( member7453568604450474000od_v_v @ X @ A2 )
=> ( ord_le7336532860387713383od_v_v @ ( minus_4183494784930505774od_v_v @ A2 @ ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v ) ) @ B2 ) )
& ( ~ ( member7453568604450474000od_v_v @ X @ A2 )
=> ( ord_le7336532860387713383od_v_v @ A2 @ B2 ) ) ) ) ).
% subset_insert_iff
thf(fact_843_Diff__single__insert,axiom,
! [A2: set_v,X: v,B2: set_v] :
( ( ord_less_eq_set_v @ ( minus_minus_set_v @ A2 @ ( insert_v @ X @ bot_bot_set_v ) ) @ B2 )
=> ( ord_less_eq_set_v @ A2 @ ( insert_v @ X @ B2 ) ) ) ).
% Diff_single_insert
thf(fact_844_Diff__single__insert,axiom,
! [A2: set_Product_prod_v_v,X: product_prod_v_v,B2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ ( minus_4183494784930505774od_v_v @ A2 @ ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v ) ) @ B2 )
=> ( ord_le7336532860387713383od_v_v @ A2 @ ( insert1338601472111419319od_v_v @ X @ B2 ) ) ) ).
% Diff_single_insert
thf(fact_845_not__distinct__decomp,axiom,
! [Ws: list_v] :
( ~ ( distinct_v @ Ws )
=> ? [Xs3: list_v,Ys4: list_v,Zs3: list_v,Y2: v] :
( Ws
= ( append_v @ Xs3 @ ( append_v @ ( cons_v @ Y2 @ nil_v ) @ ( append_v @ Ys4 @ ( append_v @ ( cons_v @ Y2 @ nil_v ) @ Zs3 ) ) ) ) ) ) ).
% not_distinct_decomp
thf(fact_846_not__distinct__conv__prefix,axiom,
! [As: list_P7986770385144383213od_v_v] :
( ( ~ ( distin6159370996967099744od_v_v @ As ) )
= ( ? [Xs5: list_P7986770385144383213od_v_v,Y3: product_prod_v_v,Ys3: list_P7986770385144383213od_v_v] :
( ( member7453568604450474000od_v_v @ Y3 @ ( set_Product_prod_v_v2 @ Xs5 ) )
& ( distin6159370996967099744od_v_v @ Xs5 )
& ( As
= ( append2138873909117096322od_v_v @ Xs5 @ ( cons_P4120604216776828829od_v_v @ Y3 @ Ys3 ) ) ) ) ) ) ).
% not_distinct_conv_prefix
thf(fact_847_not__distinct__conv__prefix,axiom,
! [As: list_v] :
( ( ~ ( distinct_v @ As ) )
= ( ? [Xs5: list_v,Y3: v,Ys3: list_v] :
( ( member_v @ Y3 @ ( set_v2 @ Xs5 ) )
& ( distinct_v @ Xs5 )
& ( As
= ( append_v @ Xs5 @ ( cons_v @ Y3 @ Ys3 ) ) ) ) ) ) ).
% not_distinct_conv_prefix
thf(fact_848_list_Oexhaust__sel,axiom,
! [List: list_v] :
( ( List != nil_v )
=> ( List
= ( cons_v @ ( hd_v @ List ) @ ( tl_v @ List ) ) ) ) ).
% list.exhaust_sel
thf(fact_849_precedes__def,axiom,
( sCC_Bl2026170059108282219od_v_v
= ( ^ [X3: product_prod_v_v,Y3: product_prod_v_v,Xs5: list_P7986770385144383213od_v_v] :
? [L2: list_P7986770385144383213od_v_v,R3: list_P7986770385144383213od_v_v] :
( ( Xs5
= ( append2138873909117096322od_v_v @ L2 @ ( cons_P4120604216776828829od_v_v @ X3 @ R3 ) ) )
& ( member7453568604450474000od_v_v @ Y3 @ ( set_Product_prod_v_v2 @ ( cons_P4120604216776828829od_v_v @ X3 @ R3 ) ) ) ) ) ) ).
% precedes_def
thf(fact_850_precedes__def,axiom,
( sCC_Bl4022239298816431255edes_v
= ( ^ [X3: v,Y3: v,Xs5: list_v] :
? [L2: list_v,R3: list_v] :
( ( Xs5
= ( append_v @ L2 @ ( cons_v @ X3 @ R3 ) ) )
& ( member_v @ Y3 @ ( set_v2 @ ( cons_v @ X3 @ R3 ) ) ) ) ) ) ).
% precedes_def
thf(fact_851_graph_Ore__reachable,axiom,
! [Vertices: set_v,Successors: v > set_v,X: v,Y: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl770211535891879572_end_v @ Successors @ X @ Y )
=> ( sCC_Bl649662514949026229able_v @ Successors @ X @ Y ) ) ) ).
% graph.re_reachable
thf(fact_852_graph_Oreachable__re,axiom,
! [Vertices: set_v,Successors: v > set_v,X: v,Y: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl649662514949026229able_v @ Successors @ X @ Y )
=> ( sCC_Bl770211535891879572_end_v @ Successors @ X @ Y ) ) ) ).
% graph.reachable_re
thf(fact_853_graph_Osubscc__add,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,S: set_Product_prod_v_v,X: product_prod_v_v,Y: product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( sCC_Bl2301996248249672505od_v_v @ Successors @ S )
=> ( ( member7453568604450474000od_v_v @ X @ S )
=> ( ( sCC_Bl4981926079593201289od_v_v @ Successors @ X @ Y )
=> ( ( sCC_Bl4981926079593201289od_v_v @ Successors @ Y @ X )
=> ( sCC_Bl2301996248249672505od_v_v @ Successors @ ( insert1338601472111419319od_v_v @ Y @ S ) ) ) ) ) ) ) ).
% graph.subscc_add
thf(fact_854_graph_Osubscc__add,axiom,
! [Vertices: set_v,Successors: v > set_v,S: set_v,X: v,Y: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl5398416737448265317bscc_v @ Successors @ S )
=> ( ( member_v @ X @ S )
=> ( ( sCC_Bl649662514949026229able_v @ Successors @ X @ Y )
=> ( ( sCC_Bl649662514949026229able_v @ Successors @ Y @ X )
=> ( sCC_Bl5398416737448265317bscc_v @ Successors @ ( insert_v @ Y @ S ) ) ) ) ) ) ) ).
% graph.subscc_add
thf(fact_855_split__list__precedes,axiom,
! [Y: product_prod_v_v,Ys: list_P7986770385144383213od_v_v,X: product_prod_v_v,Xs: list_P7986770385144383213od_v_v] :
( ( member7453568604450474000od_v_v @ Y @ ( set_Product_prod_v_v2 @ ( append2138873909117096322od_v_v @ Ys @ ( cons_P4120604216776828829od_v_v @ X @ nil_Product_prod_v_v ) ) ) )
=> ( sCC_Bl2026170059108282219od_v_v @ Y @ X @ ( append2138873909117096322od_v_v @ Ys @ ( cons_P4120604216776828829od_v_v @ X @ Xs ) ) ) ) ).
% split_list_precedes
thf(fact_856_split__list__precedes,axiom,
! [Y: v,Ys: list_v,X: v,Xs: list_v] :
( ( member_v @ Y @ ( set_v2 @ ( append_v @ Ys @ ( cons_v @ X @ nil_v ) ) ) )
=> ( sCC_Bl4022239298816431255edes_v @ Y @ X @ ( append_v @ Ys @ ( cons_v @ X @ Xs ) ) ) ) ).
% split_list_precedes
thf(fact_857_graph_Opre__dfss__def,axiom,
! [Vertices: set_v,Successors: v > set_v,V3: v,E: sCC_Bl1394983891496994913t_unit] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl1748261141445803503t_unit @ Successors @ V3 @ E )
= ( ( sCC_Bl9196236973127232072t_unit @ Successors @ E )
& ( member_v @ V3 @ ( sCC_Bl4645233313691564917t_unit @ E ) )
& ( ( sCC_Bl8828226123343373779t_unit @ E )
!= nil_v )
& ( member_v @ V3 @ ( sCC_Bl1280885523602775798t_unit @ E @ ( hd_v @ ( sCC_Bl8828226123343373779t_unit @ E ) ) ) )
& ! [X3: v] :
( ( member_v @ X3 @ ( sCC_Bl3795065053823578884t_unit @ E @ V3 ) )
=> ( member_v @ X3 @ ( sup_sup_set_v @ ( sCC_Bl157864678168468314t_unit @ E ) @ ( sCC_Bl1280885523602775798t_unit @ E @ ( hd_v @ ( sCC_Bl8828226123343373779t_unit @ E ) ) ) ) ) )
& ! [X3: v] :
( ( member_v @ X3 @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E ) ) )
=> ( sCC_Bl649662514949026229able_v @ Successors @ X3 @ V3 ) )
& ? [Ns: list_v] :
( ( sCC_Bl9201514103433284750t_unit @ E )
= ( cons_v @ V3 @ Ns ) ) ) ) ) ).
% graph.pre_dfss_def
thf(fact_858_Sup__insert,axiom,
! [A: set_Product_prod_v_v,A2: set_se8455005133513928103od_v_v] :
( ( comple5788137035815166516od_v_v @ ( insert7504383016908236695od_v_v @ A @ A2 ) )
= ( sup_su414716646722978715od_v_v @ A @ ( comple5788137035815166516od_v_v @ A2 ) ) ) ).
% Sup_insert
thf(fact_859_Sup__insert,axiom,
! [A: set_v,A2: set_set_v] :
( ( comple2307003700295860064_set_v @ ( insert_set_v @ A @ A2 ) )
= ( sup_sup_set_v @ A @ ( comple2307003700295860064_set_v @ A2 ) ) ) ).
% Sup_insert
thf(fact_860_cSup__singleton,axiom,
! [X: set_v] :
( ( comple2307003700295860064_set_v @ ( insert_set_v @ X @ bot_bot_set_set_v ) )
= X ) ).
% cSup_singleton
thf(fact_861_ccpo__Sup__singleton,axiom,
! [X: set_v] :
( ( comple2307003700295860064_set_v @ ( insert_set_v @ X @ bot_bot_set_set_v ) )
= X ) ).
% ccpo_Sup_singleton
thf(fact_862_Sup__empty,axiom,
( ( comple5788137035815166516od_v_v @ bot_bo3497076220358800403od_v_v )
= bot_bo723834152578015283od_v_v ) ).
% Sup_empty
thf(fact_863_Sup__empty,axiom,
( ( comple2307003700295860064_set_v @ bot_bot_set_set_v )
= bot_bot_set_v ) ).
% Sup_empty
thf(fact_864_Union__Un__distrib,axiom,
! [A2: set_se8455005133513928103od_v_v,B2: set_se8455005133513928103od_v_v] :
( ( comple5788137035815166516od_v_v @ ( sup_su335656005089752955od_v_v @ A2 @ B2 ) )
= ( sup_su414716646722978715od_v_v @ ( comple5788137035815166516od_v_v @ A2 ) @ ( comple5788137035815166516od_v_v @ B2 ) ) ) ).
% Union_Un_distrib
thf(fact_865_Union__Un__distrib,axiom,
! [A2: set_set_v,B2: set_set_v] :
( ( comple2307003700295860064_set_v @ ( sup_sup_set_set_v @ A2 @ B2 ) )
= ( sup_sup_set_v @ ( comple2307003700295860064_set_v @ A2 ) @ ( comple2307003700295860064_set_v @ B2 ) ) ) ).
% Union_Un_distrib
thf(fact_866_dual__order_Orefl,axiom,
! [A: set_v] : ( ord_less_eq_set_v @ A @ A ) ).
% dual_order.refl
thf(fact_867_dual__order_Orefl,axiom,
! [A: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ A @ A ) ).
% dual_order.refl
thf(fact_868_order__refl,axiom,
! [X: set_v] : ( ord_less_eq_set_v @ X @ X ) ).
% order_refl
thf(fact_869_order__refl,axiom,
! [X: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ X @ X ) ).
% order_refl
thf(fact_870_Union__iff,axiom,
! [A2: product_prod_v_v,C2: set_se8455005133513928103od_v_v] :
( ( member7453568604450474000od_v_v @ A2 @ ( comple5788137035815166516od_v_v @ C2 ) )
= ( ? [X3: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ X3 @ C2 )
& ( member7453568604450474000od_v_v @ A2 @ X3 ) ) ) ) ).
% Union_iff
thf(fact_871_Union__iff,axiom,
! [A2: v,C2: set_set_v] :
( ( member_v @ A2 @ ( comple2307003700295860064_set_v @ C2 ) )
= ( ? [X3: set_v] :
( ( member_set_v @ X3 @ C2 )
& ( member_v @ A2 @ X3 ) ) ) ) ).
% Union_iff
thf(fact_872_UnionI,axiom,
! [X5: set_Product_prod_v_v,C2: set_se8455005133513928103od_v_v,A2: product_prod_v_v] :
( ( member8406446414694345712od_v_v @ X5 @ C2 )
=> ( ( member7453568604450474000od_v_v @ A2 @ X5 )
=> ( member7453568604450474000od_v_v @ A2 @ ( comple5788137035815166516od_v_v @ C2 ) ) ) ) ).
% UnionI
thf(fact_873_UnionI,axiom,
! [X5: set_v,C2: set_set_v,A2: v] :
( ( member_set_v @ X5 @ C2 )
=> ( ( member_v @ A2 @ X5 )
=> ( member_v @ A2 @ ( comple2307003700295860064_set_v @ C2 ) ) ) ) ).
% UnionI
thf(fact_874_UN__ball__bex__simps_I1_J,axiom,
! [A2: set_set_v,P: v > $o] :
( ( ! [X3: v] :
( ( member_v @ X3 @ ( comple2307003700295860064_set_v @ A2 ) )
=> ( P @ X3 ) ) )
= ( ! [X3: set_v] :
( ( member_set_v @ X3 @ A2 )
=> ! [Y3: v] :
( ( member_v @ Y3 @ X3 )
=> ( P @ Y3 ) ) ) ) ) ).
% UN_ball_bex_simps(1)
thf(fact_875_UN__ball__bex__simps_I3_J,axiom,
! [A2: set_set_v,P: v > $o] :
( ( ? [X3: v] :
( ( member_v @ X3 @ ( comple2307003700295860064_set_v @ A2 ) )
& ( P @ X3 ) ) )
= ( ? [X3: set_v] :
( ( member_set_v @ X3 @ A2 )
& ? [Y3: v] :
( ( member_v @ Y3 @ X3 )
& ( P @ Y3 ) ) ) ) ) ).
% UN_ball_bex_simps(3)
thf(fact_876_Sup__bot__conv_I1_J,axiom,
! [A2: set_se8455005133513928103od_v_v] :
( ( ( comple5788137035815166516od_v_v @ A2 )
= bot_bo723834152578015283od_v_v )
= ( ! [X3: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ X3 @ A2 )
=> ( X3 = bot_bo723834152578015283od_v_v ) ) ) ) ).
% Sup_bot_conv(1)
thf(fact_877_Sup__bot__conv_I1_J,axiom,
! [A2: set_set_v] :
( ( ( comple2307003700295860064_set_v @ A2 )
= bot_bot_set_v )
= ( ! [X3: set_v] :
( ( member_set_v @ X3 @ A2 )
=> ( X3 = bot_bot_set_v ) ) ) ) ).
% Sup_bot_conv(1)
thf(fact_878_Sup__bot__conv_I2_J,axiom,
! [A2: set_se8455005133513928103od_v_v] :
( ( bot_bo723834152578015283od_v_v
= ( comple5788137035815166516od_v_v @ A2 ) )
= ( ! [X3: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ X3 @ A2 )
=> ( X3 = bot_bo723834152578015283od_v_v ) ) ) ) ).
% Sup_bot_conv(2)
thf(fact_879_Sup__bot__conv_I2_J,axiom,
! [A2: set_set_v] :
( ( bot_bot_set_v
= ( comple2307003700295860064_set_v @ A2 ) )
= ( ! [X3: set_v] :
( ( member_set_v @ X3 @ A2 )
=> ( X3 = bot_bot_set_v ) ) ) ) ).
% Sup_bot_conv(2)
thf(fact_880_Union__insert,axiom,
! [A: set_Product_prod_v_v,B2: set_se8455005133513928103od_v_v] :
( ( comple5788137035815166516od_v_v @ ( insert7504383016908236695od_v_v @ A @ B2 ) )
= ( sup_su414716646722978715od_v_v @ A @ ( comple5788137035815166516od_v_v @ B2 ) ) ) ).
% Union_insert
thf(fact_881_Union__insert,axiom,
! [A: set_v,B2: set_set_v] :
( ( comple2307003700295860064_set_v @ ( insert_set_v @ A @ B2 ) )
= ( sup_sup_set_v @ A @ ( comple2307003700295860064_set_v @ B2 ) ) ) ).
% Union_insert
thf(fact_882_order__antisym__conv,axiom,
! [Y: set_v,X: set_v] :
( ( ord_less_eq_set_v @ Y @ X )
=> ( ( ord_less_eq_set_v @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_883_order__antisym__conv,axiom,
! [Y: set_Product_prod_v_v,X: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ Y @ X )
=> ( ( ord_le7336532860387713383od_v_v @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_884_ord__le__eq__subst,axiom,
! [A: set_v,B: set_v,F: set_v > set_v,C: set_v] :
( ( ord_less_eq_set_v @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: set_v,Y2: set_v] :
( ( ord_less_eq_set_v @ X4 @ Y2 )
=> ( ord_less_eq_set_v @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_v @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_885_ord__le__eq__subst,axiom,
! [A: set_v,B: set_v,F: set_v > set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( ord_less_eq_set_v @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: set_v,Y2: set_v] :
( ( ord_less_eq_set_v @ X4 @ Y2 )
=> ( ord_le7336532860387713383od_v_v @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_le7336532860387713383od_v_v @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_886_ord__le__eq__subst,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v,F: set_Product_prod_v_v > set_v,C: set_v] :
( ( ord_le7336532860387713383od_v_v @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: set_Product_prod_v_v,Y2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X4 @ Y2 )
=> ( ord_less_eq_set_v @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_v @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_887_ord__le__eq__subst,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v,F: set_Product_prod_v_v > set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: set_Product_prod_v_v,Y2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X4 @ Y2 )
=> ( ord_le7336532860387713383od_v_v @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_le7336532860387713383od_v_v @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_888_ord__eq__le__subst,axiom,
! [A: set_v,F: set_v > set_v,B: set_v,C: set_v] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_set_v @ B @ C )
=> ( ! [X4: set_v,Y2: set_v] :
( ( ord_less_eq_set_v @ X4 @ Y2 )
=> ( ord_less_eq_set_v @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_v @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_889_ord__eq__le__subst,axiom,
! [A: set_Product_prod_v_v,F: set_v > set_Product_prod_v_v,B: set_v,C: set_v] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_set_v @ B @ C )
=> ( ! [X4: set_v,Y2: set_v] :
( ( ord_less_eq_set_v @ X4 @ Y2 )
=> ( ord_le7336532860387713383od_v_v @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_le7336532860387713383od_v_v @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_890_ord__eq__le__subst,axiom,
! [A: set_v,F: set_Product_prod_v_v > set_v,B: set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( A
= ( F @ B ) )
=> ( ( ord_le7336532860387713383od_v_v @ B @ C )
=> ( ! [X4: set_Product_prod_v_v,Y2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X4 @ Y2 )
=> ( ord_less_eq_set_v @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_v @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_891_ord__eq__le__subst,axiom,
! [A: set_Product_prod_v_v,F: set_Product_prod_v_v > set_Product_prod_v_v,B: set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( A
= ( F @ B ) )
=> ( ( ord_le7336532860387713383od_v_v @ B @ C )
=> ( ! [X4: set_Product_prod_v_v,Y2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X4 @ Y2 )
=> ( ord_le7336532860387713383od_v_v @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_le7336532860387713383od_v_v @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_892_order__eq__refl,axiom,
! [X: set_v,Y: set_v] :
( ( X = Y )
=> ( ord_less_eq_set_v @ X @ Y ) ) ).
% order_eq_refl
thf(fact_893_order__eq__refl,axiom,
! [X: set_Product_prod_v_v,Y: set_Product_prod_v_v] :
( ( X = Y )
=> ( ord_le7336532860387713383od_v_v @ X @ Y ) ) ).
% order_eq_refl
thf(fact_894_order__subst2,axiom,
! [A: set_v,B: set_v,F: set_v > set_v,C: set_v] :
( ( ord_less_eq_set_v @ A @ B )
=> ( ( ord_less_eq_set_v @ ( F @ B ) @ C )
=> ( ! [X4: set_v,Y2: set_v] :
( ( ord_less_eq_set_v @ X4 @ Y2 )
=> ( ord_less_eq_set_v @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_v @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_895_order__subst2,axiom,
! [A: set_v,B: set_v,F: set_v > set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( ord_less_eq_set_v @ A @ B )
=> ( ( ord_le7336532860387713383od_v_v @ ( F @ B ) @ C )
=> ( ! [X4: set_v,Y2: set_v] :
( ( ord_less_eq_set_v @ X4 @ Y2 )
=> ( ord_le7336532860387713383od_v_v @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_le7336532860387713383od_v_v @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_896_order__subst2,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v,F: set_Product_prod_v_v > set_v,C: set_v] :
( ( ord_le7336532860387713383od_v_v @ A @ B )
=> ( ( ord_less_eq_set_v @ ( F @ B ) @ C )
=> ( ! [X4: set_Product_prod_v_v,Y2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X4 @ Y2 )
=> ( ord_less_eq_set_v @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_v @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_897_order__subst2,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v,F: set_Product_prod_v_v > set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ B )
=> ( ( ord_le7336532860387713383od_v_v @ ( F @ B ) @ C )
=> ( ! [X4: set_Product_prod_v_v,Y2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X4 @ Y2 )
=> ( ord_le7336532860387713383od_v_v @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_le7336532860387713383od_v_v @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_898_order__subst1,axiom,
! [A: set_v,F: set_v > set_v,B: set_v,C: set_v] :
( ( ord_less_eq_set_v @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_v @ B @ C )
=> ( ! [X4: set_v,Y2: set_v] :
( ( ord_less_eq_set_v @ X4 @ Y2 )
=> ( ord_less_eq_set_v @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_v @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_899_order__subst1,axiom,
! [A: set_v,F: set_Product_prod_v_v > set_v,B: set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( ord_less_eq_set_v @ A @ ( F @ B ) )
=> ( ( ord_le7336532860387713383od_v_v @ B @ C )
=> ( ! [X4: set_Product_prod_v_v,Y2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X4 @ Y2 )
=> ( ord_less_eq_set_v @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_v @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_900_order__subst1,axiom,
! [A: set_Product_prod_v_v,F: set_v > set_Product_prod_v_v,B: set_v,C: set_v] :
( ( ord_le7336532860387713383od_v_v @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_v @ B @ C )
=> ( ! [X4: set_v,Y2: set_v] :
( ( ord_less_eq_set_v @ X4 @ Y2 )
=> ( ord_le7336532860387713383od_v_v @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_le7336532860387713383od_v_v @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_901_order__subst1,axiom,
! [A: set_Product_prod_v_v,F: set_Product_prod_v_v > set_Product_prod_v_v,B: set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ ( F @ B ) )
=> ( ( ord_le7336532860387713383od_v_v @ B @ C )
=> ( ! [X4: set_Product_prod_v_v,Y2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X4 @ Y2 )
=> ( ord_le7336532860387713383od_v_v @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_le7336532860387713383od_v_v @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_902_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_v,Z4: set_v] : ( Y4 = Z4 ) )
= ( ^ [A4: set_v,B4: set_v] :
( ( ord_less_eq_set_v @ A4 @ B4 )
& ( ord_less_eq_set_v @ B4 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_903_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_Product_prod_v_v,Z4: set_Product_prod_v_v] : ( Y4 = Z4 ) )
= ( ^ [A4: set_Product_prod_v_v,B4: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A4 @ B4 )
& ( ord_le7336532860387713383od_v_v @ B4 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_904_antisym,axiom,
! [A: set_v,B: set_v] :
( ( ord_less_eq_set_v @ A @ B )
=> ( ( ord_less_eq_set_v @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_905_antisym,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ B )
=> ( ( ord_le7336532860387713383od_v_v @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_906_dual__order_Otrans,axiom,
! [B: set_v,A: set_v,C: set_v] :
( ( ord_less_eq_set_v @ B @ A )
=> ( ( ord_less_eq_set_v @ C @ B )
=> ( ord_less_eq_set_v @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_907_dual__order_Otrans,axiom,
! [B: set_Product_prod_v_v,A: set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ B @ A )
=> ( ( ord_le7336532860387713383od_v_v @ C @ B )
=> ( ord_le7336532860387713383od_v_v @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_908_dual__order_Oantisym,axiom,
! [B: set_v,A: set_v] :
( ( ord_less_eq_set_v @ B @ A )
=> ( ( ord_less_eq_set_v @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_909_dual__order_Oantisym,axiom,
! [B: set_Product_prod_v_v,A: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ B @ A )
=> ( ( ord_le7336532860387713383od_v_v @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_910_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: set_v,Z4: set_v] : ( Y4 = Z4 ) )
= ( ^ [A4: set_v,B4: set_v] :
( ( ord_less_eq_set_v @ B4 @ A4 )
& ( ord_less_eq_set_v @ A4 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_911_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: set_Product_prod_v_v,Z4: set_Product_prod_v_v] : ( Y4 = Z4 ) )
= ( ^ [A4: set_Product_prod_v_v,B4: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ B4 @ A4 )
& ( ord_le7336532860387713383od_v_v @ A4 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_912_order__trans,axiom,
! [X: set_v,Y: set_v,Z: set_v] :
( ( ord_less_eq_set_v @ X @ Y )
=> ( ( ord_less_eq_set_v @ Y @ Z )
=> ( ord_less_eq_set_v @ X @ Z ) ) ) ).
% order_trans
thf(fact_913_order__trans,axiom,
! [X: set_Product_prod_v_v,Y: set_Product_prod_v_v,Z: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X @ Y )
=> ( ( ord_le7336532860387713383od_v_v @ Y @ Z )
=> ( ord_le7336532860387713383od_v_v @ X @ Z ) ) ) ).
% order_trans
thf(fact_914_order_Otrans,axiom,
! [A: set_v,B: set_v,C: set_v] :
( ( ord_less_eq_set_v @ A @ B )
=> ( ( ord_less_eq_set_v @ B @ C )
=> ( ord_less_eq_set_v @ A @ C ) ) ) ).
% order.trans
thf(fact_915_order_Otrans,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ B )
=> ( ( ord_le7336532860387713383od_v_v @ B @ C )
=> ( ord_le7336532860387713383od_v_v @ A @ C ) ) ) ).
% order.trans
thf(fact_916_order__antisym,axiom,
! [X: set_v,Y: set_v] :
( ( ord_less_eq_set_v @ X @ Y )
=> ( ( ord_less_eq_set_v @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_917_order__antisym,axiom,
! [X: set_Product_prod_v_v,Y: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X @ Y )
=> ( ( ord_le7336532860387713383od_v_v @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_918_ord__le__eq__trans,axiom,
! [A: set_v,B: set_v,C: set_v] :
( ( ord_less_eq_set_v @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_set_v @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_919_ord__le__eq__trans,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ B )
=> ( ( B = C )
=> ( ord_le7336532860387713383od_v_v @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_920_ord__eq__le__trans,axiom,
! [A: set_v,B: set_v,C: set_v] :
( ( A = B )
=> ( ( ord_less_eq_set_v @ B @ C )
=> ( ord_less_eq_set_v @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_921_ord__eq__le__trans,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( A = B )
=> ( ( ord_le7336532860387713383od_v_v @ B @ C )
=> ( ord_le7336532860387713383od_v_v @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_922_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_v,Z4: set_v] : ( Y4 = Z4 ) )
= ( ^ [X3: set_v,Y3: set_v] :
( ( ord_less_eq_set_v @ X3 @ Y3 )
& ( ord_less_eq_set_v @ Y3 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_923_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_Product_prod_v_v,Z4: set_Product_prod_v_v] : ( Y4 = Z4 ) )
= ( ^ [X3: set_Product_prod_v_v,Y3: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X3 @ Y3 )
& ( ord_le7336532860387713383od_v_v @ Y3 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_924_UnionE,axiom,
! [A2: product_prod_v_v,C2: set_se8455005133513928103od_v_v] :
( ( member7453568604450474000od_v_v @ A2 @ ( comple5788137035815166516od_v_v @ C2 ) )
=> ~ ! [X6: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ A2 @ X6 )
=> ~ ( member8406446414694345712od_v_v @ X6 @ C2 ) ) ) ).
% UnionE
thf(fact_925_UnionE,axiom,
! [A2: v,C2: set_set_v] :
( ( member_v @ A2 @ ( comple2307003700295860064_set_v @ C2 ) )
=> ~ ! [X6: set_v] :
( ( member_v @ A2 @ X6 )
=> ~ ( member_set_v @ X6 @ C2 ) ) ) ).
% UnionE
thf(fact_926_graph_Oinit__env__pre__dfs,axiom,
! [Vertices: set_v,Successors: v > set_v,V3: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( sCC_Bl36166008131615352t_unit @ Successors @ V3 @ ( sCC_Bl7693227186847904995_env_v @ V3 ) ) ) ).
% graph.init_env_pre_dfs
thf(fact_927_bot_Oextremum__uniqueI,axiom,
! [A: set_v] :
( ( ord_less_eq_set_v @ A @ bot_bot_set_v )
=> ( A = bot_bot_set_v ) ) ).
% bot.extremum_uniqueI
thf(fact_928_bot_Oextremum__uniqueI,axiom,
! [A: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ bot_bo723834152578015283od_v_v )
=> ( A = bot_bo723834152578015283od_v_v ) ) ).
% bot.extremum_uniqueI
thf(fact_929_bot_Oextremum__unique,axiom,
! [A: set_v] :
( ( ord_less_eq_set_v @ A @ bot_bot_set_v )
= ( A = bot_bot_set_v ) ) ).
% bot.extremum_unique
thf(fact_930_bot_Oextremum__unique,axiom,
! [A: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ bot_bo723834152578015283od_v_v )
= ( A = bot_bo723834152578015283od_v_v ) ) ).
% bot.extremum_unique
thf(fact_931_bot_Oextremum,axiom,
! [A: set_v] : ( ord_less_eq_set_v @ bot_bot_set_v @ A ) ).
% bot.extremum
thf(fact_932_bot_Oextremum,axiom,
! [A: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ bot_bo723834152578015283od_v_v @ A ) ).
% bot.extremum
thf(fact_933_Sup__eqI,axiom,
! [A2: set_se8455005133513928103od_v_v,X: set_Product_prod_v_v] :
( ! [Y2: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ Y2 @ A2 )
=> ( ord_le7336532860387713383od_v_v @ Y2 @ X ) )
=> ( ! [Y2: set_Product_prod_v_v] :
( ! [Z5: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ Z5 @ A2 )
=> ( ord_le7336532860387713383od_v_v @ Z5 @ Y2 ) )
=> ( ord_le7336532860387713383od_v_v @ X @ Y2 ) )
=> ( ( comple5788137035815166516od_v_v @ A2 )
= X ) ) ) ).
% Sup_eqI
thf(fact_934_Sup__eqI,axiom,
! [A2: set_set_v,X: set_v] :
( ! [Y2: set_v] :
( ( member_set_v @ Y2 @ A2 )
=> ( ord_less_eq_set_v @ Y2 @ X ) )
=> ( ! [Y2: set_v] :
( ! [Z5: set_v] :
( ( member_set_v @ Z5 @ A2 )
=> ( ord_less_eq_set_v @ Z5 @ Y2 ) )
=> ( ord_less_eq_set_v @ X @ Y2 ) )
=> ( ( comple2307003700295860064_set_v @ A2 )
= X ) ) ) ).
% Sup_eqI
thf(fact_935_Sup__mono,axiom,
! [A2: set_se8455005133513928103od_v_v,B2: set_se8455005133513928103od_v_v] :
( ! [A6: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ A6 @ A2 )
=> ? [X2: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ X2 @ B2 )
& ( ord_le7336532860387713383od_v_v @ A6 @ X2 ) ) )
=> ( ord_le7336532860387713383od_v_v @ ( comple5788137035815166516od_v_v @ A2 ) @ ( comple5788137035815166516od_v_v @ B2 ) ) ) ).
% Sup_mono
thf(fact_936_Sup__mono,axiom,
! [A2: set_set_v,B2: set_set_v] :
( ! [A6: set_v] :
( ( member_set_v @ A6 @ A2 )
=> ? [X2: set_v] :
( ( member_set_v @ X2 @ B2 )
& ( ord_less_eq_set_v @ A6 @ X2 ) ) )
=> ( ord_less_eq_set_v @ ( comple2307003700295860064_set_v @ A2 ) @ ( comple2307003700295860064_set_v @ B2 ) ) ) ).
% Sup_mono
thf(fact_937_Sup__least,axiom,
! [A2: set_se8455005133513928103od_v_v,Z: set_Product_prod_v_v] :
( ! [X4: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ X4 @ A2 )
=> ( ord_le7336532860387713383od_v_v @ X4 @ Z ) )
=> ( ord_le7336532860387713383od_v_v @ ( comple5788137035815166516od_v_v @ A2 ) @ Z ) ) ).
% Sup_least
thf(fact_938_Sup__least,axiom,
! [A2: set_set_v,Z: set_v] :
( ! [X4: set_v] :
( ( member_set_v @ X4 @ A2 )
=> ( ord_less_eq_set_v @ X4 @ Z ) )
=> ( ord_less_eq_set_v @ ( comple2307003700295860064_set_v @ A2 ) @ Z ) ) ).
% Sup_least
thf(fact_939_Sup__upper,axiom,
! [X: set_Product_prod_v_v,A2: set_se8455005133513928103od_v_v] :
( ( member8406446414694345712od_v_v @ X @ A2 )
=> ( ord_le7336532860387713383od_v_v @ X @ ( comple5788137035815166516od_v_v @ A2 ) ) ) ).
% Sup_upper
thf(fact_940_Sup__upper,axiom,
! [X: set_v,A2: set_set_v] :
( ( member_set_v @ X @ A2 )
=> ( ord_less_eq_set_v @ X @ ( comple2307003700295860064_set_v @ A2 ) ) ) ).
% Sup_upper
thf(fact_941_Sup__le__iff,axiom,
! [A2: set_se8455005133513928103od_v_v,B: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ ( comple5788137035815166516od_v_v @ A2 ) @ B )
= ( ! [X3: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ X3 @ A2 )
=> ( ord_le7336532860387713383od_v_v @ X3 @ B ) ) ) ) ).
% Sup_le_iff
thf(fact_942_Sup__le__iff,axiom,
! [A2: set_set_v,B: set_v] :
( ( ord_less_eq_set_v @ ( comple2307003700295860064_set_v @ A2 ) @ B )
= ( ! [X3: set_v] :
( ( member_set_v @ X3 @ A2 )
=> ( ord_less_eq_set_v @ X3 @ B ) ) ) ) ).
% Sup_le_iff
thf(fact_943_Sup__upper2,axiom,
! [U: set_Product_prod_v_v,A2: set_se8455005133513928103od_v_v,V3: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ U @ A2 )
=> ( ( ord_le7336532860387713383od_v_v @ V3 @ U )
=> ( ord_le7336532860387713383od_v_v @ V3 @ ( comple5788137035815166516od_v_v @ A2 ) ) ) ) ).
% Sup_upper2
thf(fact_944_Sup__upper2,axiom,
! [U: set_v,A2: set_set_v,V3: set_v] :
( ( member_set_v @ U @ A2 )
=> ( ( ord_less_eq_set_v @ V3 @ U )
=> ( ord_less_eq_set_v @ V3 @ ( comple2307003700295860064_set_v @ A2 ) ) ) ) ).
% Sup_upper2
thf(fact_945_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_946_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_947_empty__Union__conv,axiom,
! [A2: set_se8455005133513928103od_v_v] :
( ( bot_bo723834152578015283od_v_v
= ( comple5788137035815166516od_v_v @ A2 ) )
= ( ! [X3: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ X3 @ A2 )
=> ( X3 = bot_bo723834152578015283od_v_v ) ) ) ) ).
% empty_Union_conv
thf(fact_948_empty__Union__conv,axiom,
! [A2: set_set_v] :
( ( bot_bot_set_v
= ( comple2307003700295860064_set_v @ A2 ) )
= ( ! [X3: set_v] :
( ( member_set_v @ X3 @ A2 )
=> ( X3 = bot_bot_set_v ) ) ) ) ).
% empty_Union_conv
thf(fact_949_Union__empty__conv,axiom,
! [A2: set_se8455005133513928103od_v_v] :
( ( ( comple5788137035815166516od_v_v @ A2 )
= bot_bo723834152578015283od_v_v )
= ( ! [X3: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ X3 @ A2 )
=> ( X3 = bot_bo723834152578015283od_v_v ) ) ) ) ).
% Union_empty_conv
thf(fact_950_Union__empty__conv,axiom,
! [A2: set_set_v] :
( ( ( comple2307003700295860064_set_v @ A2 )
= bot_bot_set_v )
= ( ! [X3: set_v] :
( ( member_set_v @ X3 @ A2 )
=> ( X3 = bot_bot_set_v ) ) ) ) ).
% Union_empty_conv
thf(fact_951_Union__empty,axiom,
( ( comple5788137035815166516od_v_v @ bot_bo3497076220358800403od_v_v )
= bot_bo723834152578015283od_v_v ) ).
% Union_empty
thf(fact_952_Union__empty,axiom,
( ( comple2307003700295860064_set_v @ bot_bot_set_set_v )
= bot_bot_set_v ) ).
% Union_empty
thf(fact_953_Union__subsetI,axiom,
! [A2: set_se8455005133513928103od_v_v,B2: set_se8455005133513928103od_v_v] :
( ! [X4: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ X4 @ A2 )
=> ? [Y5: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ Y5 @ B2 )
& ( ord_le7336532860387713383od_v_v @ X4 @ Y5 ) ) )
=> ( ord_le7336532860387713383od_v_v @ ( comple5788137035815166516od_v_v @ A2 ) @ ( comple5788137035815166516od_v_v @ B2 ) ) ) ).
% Union_subsetI
thf(fact_954_Union__subsetI,axiom,
! [A2: set_set_v,B2: set_set_v] :
( ! [X4: set_v] :
( ( member_set_v @ X4 @ A2 )
=> ? [Y5: set_v] :
( ( member_set_v @ Y5 @ B2 )
& ( ord_less_eq_set_v @ X4 @ Y5 ) ) )
=> ( ord_less_eq_set_v @ ( comple2307003700295860064_set_v @ A2 ) @ ( comple2307003700295860064_set_v @ B2 ) ) ) ).
% Union_subsetI
thf(fact_955_Union__upper,axiom,
! [B2: set_Product_prod_v_v,A2: set_se8455005133513928103od_v_v] :
( ( member8406446414694345712od_v_v @ B2 @ A2 )
=> ( ord_le7336532860387713383od_v_v @ B2 @ ( comple5788137035815166516od_v_v @ A2 ) ) ) ).
% Union_upper
thf(fact_956_Union__upper,axiom,
! [B2: set_v,A2: set_set_v] :
( ( member_set_v @ B2 @ A2 )
=> ( ord_less_eq_set_v @ B2 @ ( comple2307003700295860064_set_v @ A2 ) ) ) ).
% Union_upper
thf(fact_957_Union__least,axiom,
! [A2: set_se8455005133513928103od_v_v,C2: set_Product_prod_v_v] :
( ! [X6: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ X6 @ A2 )
=> ( ord_le7336532860387713383od_v_v @ X6 @ C2 ) )
=> ( ord_le7336532860387713383od_v_v @ ( comple5788137035815166516od_v_v @ A2 ) @ C2 ) ) ).
% Union_least
thf(fact_958_Union__least,axiom,
! [A2: set_set_v,C2: set_v] :
( ! [X6: set_v] :
( ( member_set_v @ X6 @ A2 )
=> ( ord_less_eq_set_v @ X6 @ C2 ) )
=> ( ord_less_eq_set_v @ ( comple2307003700295860064_set_v @ A2 ) @ C2 ) ) ).
% Union_least
thf(fact_959_Union__mono,axiom,
! [A2: set_se8455005133513928103od_v_v,B2: set_se8455005133513928103od_v_v] :
( ( ord_le4714265922333009223od_v_v @ A2 @ B2 )
=> ( ord_le7336532860387713383od_v_v @ ( comple5788137035815166516od_v_v @ A2 ) @ ( comple5788137035815166516od_v_v @ B2 ) ) ) ).
% Union_mono
thf(fact_960_Union__mono,axiom,
! [A2: set_set_v,B2: set_set_v] :
( ( ord_le5216385588623774835_set_v @ A2 @ B2 )
=> ( ord_less_eq_set_v @ ( comple2307003700295860064_set_v @ A2 ) @ ( comple2307003700295860064_set_v @ B2 ) ) ) ).
% Union_mono
thf(fact_961_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] :
( ! [X2: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ X2 @ X5 )
=> ( ord_le7336532860387713383od_v_v @ X2 @ Y2 ) )
=> ( ord_le7336532860387713383od_v_v @ A @ Y2 ) )
=> ( ( comple5788137035815166516od_v_v @ X5 )
= A ) ) ) ) ).
% cSup_eq_non_empty
thf(fact_962_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] :
( ! [X2: set_v] :
( ( member_set_v @ X2 @ X5 )
=> ( ord_less_eq_set_v @ X2 @ Y2 ) )
=> ( ord_less_eq_set_v @ A @ Y2 ) )
=> ( ( comple2307003700295860064_set_v @ X5 )
= A ) ) ) ) ).
% cSup_eq_non_empty
thf(fact_963_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_964_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_965_less__eq__Sup,axiom,
! [A2: set_se8455005133513928103od_v_v,U: set_Product_prod_v_v] :
( ! [V2: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ V2 @ A2 )
=> ( ord_le7336532860387713383od_v_v @ U @ V2 ) )
=> ( ( A2 != bot_bo3497076220358800403od_v_v )
=> ( ord_le7336532860387713383od_v_v @ U @ ( comple5788137035815166516od_v_v @ A2 ) ) ) ) ).
% less_eq_Sup
thf(fact_966_less__eq__Sup,axiom,
! [A2: set_set_v,U: set_v] :
( ! [V2: set_v] :
( ( member_set_v @ V2 @ A2 )
=> ( ord_less_eq_set_v @ U @ V2 ) )
=> ( ( A2 != bot_bot_set_set_v )
=> ( ord_less_eq_set_v @ U @ ( comple2307003700295860064_set_v @ A2 ) ) ) ) ).
% less_eq_Sup
thf(fact_967_Sup__subset__mono,axiom,
! [A2: set_se8455005133513928103od_v_v,B2: set_se8455005133513928103od_v_v] :
( ( ord_le4714265922333009223od_v_v @ A2 @ B2 )
=> ( ord_le7336532860387713383od_v_v @ ( comple5788137035815166516od_v_v @ A2 ) @ ( comple5788137035815166516od_v_v @ B2 ) ) ) ).
% Sup_subset_mono
thf(fact_968_Sup__subset__mono,axiom,
! [A2: set_set_v,B2: set_set_v] :
( ( ord_le5216385588623774835_set_v @ A2 @ B2 )
=> ( ord_less_eq_set_v @ ( comple2307003700295860064_set_v @ A2 ) @ ( comple2307003700295860064_set_v @ B2 ) ) ) ).
% Sup_subset_mono
thf(fact_969_Sup__union__distrib,axiom,
! [A2: set_se8455005133513928103od_v_v,B2: set_se8455005133513928103od_v_v] :
( ( comple5788137035815166516od_v_v @ ( sup_su335656005089752955od_v_v @ A2 @ B2 ) )
= ( sup_su414716646722978715od_v_v @ ( comple5788137035815166516od_v_v @ A2 ) @ ( comple5788137035815166516od_v_v @ B2 ) ) ) ).
% Sup_union_distrib
thf(fact_970_Sup__union__distrib,axiom,
! [A2: set_set_v,B2: set_set_v] :
( ( comple2307003700295860064_set_v @ ( sup_sup_set_set_v @ A2 @ B2 ) )
= ( sup_sup_set_v @ ( comple2307003700295860064_set_v @ A2 ) @ ( comple2307003700295860064_set_v @ B2 ) ) ) ).
% Sup_union_distrib
thf(fact_971_Union__disjoint,axiom,
! [C2: set_se8455005133513928103od_v_v,A2: set_Product_prod_v_v] :
( ( ( inf_in6271465464967711157od_v_v @ ( comple5788137035815166516od_v_v @ C2 ) @ A2 )
= bot_bo723834152578015283od_v_v )
= ( ! [X3: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ X3 @ C2 )
=> ( ( inf_in6271465464967711157od_v_v @ X3 @ A2 )
= bot_bo723834152578015283od_v_v ) ) ) ) ).
% Union_disjoint
thf(fact_972_Union__disjoint,axiom,
! [C2: set_set_v,A2: set_v] :
( ( ( inf_inf_set_v @ ( comple2307003700295860064_set_v @ C2 ) @ A2 )
= bot_bot_set_v )
= ( ! [X3: set_v] :
( ( member_set_v @ X3 @ C2 )
=> ( ( inf_inf_set_v @ X3 @ A2 )
= bot_bot_set_v ) ) ) ) ).
% Union_disjoint
thf(fact_973_Union__Int__subset,axiom,
! [A2: set_se8455005133513928103od_v_v,B2: set_se8455005133513928103od_v_v] : ( ord_le7336532860387713383od_v_v @ ( comple5788137035815166516od_v_v @ ( inf_in6058586722421118357od_v_v @ A2 @ B2 ) ) @ ( inf_in6271465464967711157od_v_v @ ( comple5788137035815166516od_v_v @ A2 ) @ ( comple5788137035815166516od_v_v @ B2 ) ) ) ).
% Union_Int_subset
thf(fact_974_Union__Int__subset,axiom,
! [A2: set_set_v,B2: set_set_v] : ( ord_less_eq_set_v @ ( comple2307003700295860064_set_v @ ( inf_inf_set_set_v @ A2 @ B2 ) ) @ ( inf_inf_set_v @ ( comple2307003700295860064_set_v @ A2 ) @ ( comple2307003700295860064_set_v @ B2 ) ) ) ).
% Union_Int_subset
thf(fact_975_Sup__inter__less__eq,axiom,
! [A2: set_se8455005133513928103od_v_v,B2: set_se8455005133513928103od_v_v] : ( ord_le7336532860387713383od_v_v @ ( comple5788137035815166516od_v_v @ ( inf_in6058586722421118357od_v_v @ A2 @ B2 ) ) @ ( inf_in6271465464967711157od_v_v @ ( comple5788137035815166516od_v_v @ A2 ) @ ( comple5788137035815166516od_v_v @ B2 ) ) ) ).
% Sup_inter_less_eq
thf(fact_976_Sup__inter__less__eq,axiom,
! [A2: set_set_v,B2: set_set_v] : ( ord_less_eq_set_v @ ( comple2307003700295860064_set_v @ ( inf_inf_set_set_v @ A2 @ B2 ) ) @ ( inf_inf_set_v @ ( comple2307003700295860064_set_v @ A2 ) @ ( comple2307003700295860064_set_v @ B2 ) ) ) ).
% Sup_inter_less_eq
thf(fact_977_set__union,axiom,
! [Xs: list_v,Ys: list_v] :
( ( set_v2 @ ( union_v @ Xs @ Ys ) )
= ( sup_sup_set_v @ ( set_v2 @ Xs ) @ ( set_v2 @ Ys ) ) ) ).
% set_union
thf(fact_978_set__union,axiom,
! [Xs: list_P7986770385144383213od_v_v,Ys: list_P7986770385144383213od_v_v] :
( ( set_Product_prod_v_v2 @ ( union_4602324378607836129od_v_v @ Xs @ Ys ) )
= ( sup_su414716646722978715od_v_v @ ( set_Product_prod_v_v2 @ Xs ) @ ( set_Product_prod_v_v2 @ Ys ) ) ) ).
% set_union
thf(fact_979_insert__partition,axiom,
! [X: set_Product_prod_v_v,F3: set_se8455005133513928103od_v_v] :
( ~ ( member8406446414694345712od_v_v @ X @ F3 )
=> ( ! [X4: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ X4 @ ( insert7504383016908236695od_v_v @ X @ F3 ) )
=> ! [Xa2: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ Xa2 @ ( insert7504383016908236695od_v_v @ X @ F3 ) )
=> ( ( X4 != Xa2 )
=> ( ( inf_in6271465464967711157od_v_v @ X4 @ Xa2 )
= bot_bo723834152578015283od_v_v ) ) ) )
=> ( ( inf_in6271465464967711157od_v_v @ X @ ( comple5788137035815166516od_v_v @ F3 ) )
= bot_bo723834152578015283od_v_v ) ) ) ).
% insert_partition
thf(fact_980_insert__partition,axiom,
! [X: set_v,F3: set_set_v] :
( ~ ( member_set_v @ X @ F3 )
=> ( ! [X4: set_v] :
( ( member_set_v @ X4 @ ( insert_set_v @ X @ F3 ) )
=> ! [Xa2: set_v] :
( ( member_set_v @ Xa2 @ ( insert_set_v @ X @ F3 ) )
=> ( ( X4 != Xa2 )
=> ( ( inf_inf_set_v @ X4 @ Xa2 )
= bot_bot_set_v ) ) ) )
=> ( ( inf_inf_set_v @ X @ ( comple2307003700295860064_set_v @ F3 ) )
= bot_bot_set_v ) ) ) ).
% insert_partition
thf(fact_981_Sup__inf__eq__bot__iff,axiom,
! [B2: set_se8455005133513928103od_v_v,A: set_Product_prod_v_v] :
( ( ( inf_in6271465464967711157od_v_v @ ( comple5788137035815166516od_v_v @ B2 ) @ A )
= bot_bo723834152578015283od_v_v )
= ( ! [X3: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ X3 @ B2 )
=> ( ( inf_in6271465464967711157od_v_v @ X3 @ A )
= bot_bo723834152578015283od_v_v ) ) ) ) ).
% Sup_inf_eq_bot_iff
thf(fact_982_Sup__inf__eq__bot__iff,axiom,
! [B2: set_set_v,A: set_v] :
( ( ( inf_inf_set_v @ ( comple2307003700295860064_set_v @ B2 ) @ A )
= bot_bot_set_v )
= ( ! [X3: set_v] :
( ( member_set_v @ X3 @ B2 )
=> ( ( inf_inf_set_v @ X3 @ A )
= bot_bot_set_v ) ) ) ) ).
% Sup_inf_eq_bot_iff
thf(fact_983_distinct__union,axiom,
! [Xs: list_v,Ys: list_v] :
( ( distinct_v @ ( union_v @ Xs @ Ys ) )
= ( distinct_v @ Ys ) ) ).
% distinct_union
thf(fact_984_the__elem__eq,axiom,
! [X: v] :
( ( the_elem_v @ ( insert_v @ X @ bot_bot_set_v ) )
= X ) ).
% the_elem_eq
thf(fact_985_the__elem__eq,axiom,
! [X: product_prod_v_v] :
( ( the_el5392834299063928540od_v_v @ ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v ) )
= X ) ).
% the_elem_eq
thf(fact_986_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_987_bot__empty__eq,axiom,
( bot_bot_v_o
= ( ^ [X3: v] : ( member_v @ X3 @ bot_bot_set_v ) ) ) ).
% bot_empty_eq
thf(fact_988_bot__empty__eq,axiom,
( bot_bo8461541820394803818_v_v_o
= ( ^ [X3: product_prod_v_v] : ( member7453568604450474000od_v_v @ X3 @ bot_bo723834152578015283od_v_v ) ) ) ).
% bot_empty_eq
thf(fact_989_cc__Un,axiom,
( cc
= ( comple2307003700295860064_set_v
@ ( collect_set_v
@ ^ [Uu: set_v] :
? [X3: v] :
( ( Uu
= ( sCC_Bl1280885523602775798t_unit @ e @ X3 ) )
& ( member_v @ X3 @ cc ) ) ) ) ) ).
% cc_Un
thf(fact_990_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_991_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_992_singleton__conv,axiom,
! [A: v] :
( ( collect_v
@ ^ [X3: v] : ( X3 = A ) )
= ( insert_v @ A @ bot_bot_set_v ) ) ).
% singleton_conv
thf(fact_993_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_994_singleton__conv2,axiom,
! [A: set_v] :
( ( collect_set_v
@ ( ^ [Y4: set_v,Z4: set_v] : ( Y4 = Z4 )
@ A ) )
= ( insert_set_v @ A @ bot_bot_set_set_v ) ) ).
% singleton_conv2
thf(fact_995_singleton__conv2,axiom,
! [A: v] :
( ( collect_v
@ ( ^ [Y4: v,Z4: v] : ( Y4 = Z4 )
@ A ) )
= ( insert_v @ A @ bot_bot_set_v ) ) ).
% singleton_conv2
thf(fact_996_singleton__conv2,axiom,
! [A: product_prod_v_v] :
( ( collec140062887454715474od_v_v
@ ( ^ [Y4: product_prod_v_v,Z4: product_prod_v_v] : ( Y4 = Z4 )
@ A ) )
= ( insert1338601472111419319od_v_v @ A @ bot_bo723834152578015283od_v_v ) ) ).
% singleton_conv2
thf(fact_997__092_060open_062_092_060Union_062_A_123_092_060S_062_Ae_H_An_A_124n_O_An_A_092_060in_062_Aset_A_Istack_Ae_H_J_125_A_061_Avisited_Ae_H_A_N_Aexplored_Ae_H_092_060close_062,axiom,
( ( 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 ) ) ) ) ) )
= ( minus_minus_set_v @ ( sCC_Bl4645233313691564917t_unit @ e2 ) @ ( sCC_Bl157864678168468314t_unit @ e2 ) ) ) ).
% \<open>\<Union> {\<S> e' n |n. n \<in> set (stack e')} = visited e' - explored e'\<close>
thf(fact_998_sub__env__def,axiom,
( sCC_Bl5768913643336123637t_unit
= ( ^ [E4: sCC_Bl1394983891496994913t_unit,E5: sCC_Bl1394983891496994913t_unit] :
( ( ( sCC_Bl1090238580953940555t_unit @ E5 )
= ( sCC_Bl1090238580953940555t_unit @ E4 ) )
& ( ord_less_eq_set_v @ ( sCC_Bl4645233313691564917t_unit @ E4 ) @ ( sCC_Bl4645233313691564917t_unit @ E5 ) )
& ( ord_less_eq_set_v @ ( sCC_Bl157864678168468314t_unit @ E4 ) @ ( sCC_Bl157864678168468314t_unit @ E5 ) )
& ! [V4: v] : ( ord_less_eq_set_v @ ( sCC_Bl3795065053823578884t_unit @ E4 @ V4 ) @ ( sCC_Bl3795065053823578884t_unit @ E5 @ V4 ) )
& ! [V4: v] : ( ord_less_eq_set_v @ ( sCC_Bl1280885523602775798t_unit @ E4 @ V4 ) @ ( sCC_Bl1280885523602775798t_unit @ E5 @ V4 ) )
& ( ord_less_eq_set_v
@ ( comple2307003700295860064_set_v
@ ( collect_set_v
@ ^ [Uu: set_v] :
? [V4: v] :
( ( Uu
= ( sCC_Bl1280885523602775798t_unit @ E4 @ V4 ) )
& ( member_v @ V4 @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E4 ) ) ) ) ) )
@ ( comple2307003700295860064_set_v
@ ( collect_set_v
@ ^ [Uu: set_v] :
? [V4: v] :
( ( Uu
= ( sCC_Bl1280885523602775798t_unit @ E5 @ V4 ) )
& ( member_v @ V4 @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E5 ) ) ) ) ) ) ) ) ) ) ).
% sub_env_def
thf(fact_999__092_060open_062_092_060Union_062_A_123_092_060S_062_Ae_H_An_A_124n_O_An_A_092_060in_062_Aset_A_Istack_Ae_H_J_125_A_061_Avisited_Ae_A_N_Aexplored_Ae_092_060close_062,axiom,
( ( 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 ) ) ) ) ) )
= ( minus_minus_set_v @ ( sCC_Bl4645233313691564917t_unit @ e ) @ ( sCC_Bl157864678168468314t_unit @ e ) ) ) ).
% \<open>\<Union> {\<S> e' n |n. n \<in> set (stack e')} = visited e - explored e\<close>
thf(fact_1000_unite__S__equal,axiom,
! [E: sCC_Bl1394983891496994913t_unit,W: v,V3: v] :
( ( sCC_Bl9196236973127232072t_unit @ successors @ E )
=> ( ( member_v @ W @ ( successors @ V3 ) )
=> ( ~ ( member_v @ W @ ( sCC_Bl3795065053823578884t_unit @ E @ V3 ) )
=> ( ( member_v @ W @ ( sCC_Bl4645233313691564917t_unit @ E ) )
=> ( ~ ( member_v @ W @ ( sCC_Bl157864678168468314t_unit @ E ) )
=> ( ( comple2307003700295860064_set_v
@ ( collect_set_v
@ ^ [Uu: set_v] :
? [N4: v] :
( ( Uu
= ( sCC_Bl1280885523602775798t_unit @ ( sCC_Bloemen_unite_v @ V3 @ W @ E ) @ N4 ) )
& ( member_v @ N4 @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ ( sCC_Bloemen_unite_v @ V3 @ W @ E ) ) ) ) ) ) )
= ( comple2307003700295860064_set_v
@ ( collect_set_v
@ ^ [Uu: set_v] :
? [N4: v] :
( ( Uu
= ( sCC_Bl1280885523602775798t_unit @ E @ N4 ) )
& ( member_v @ N4 @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E ) ) ) ) ) ) ) ) ) ) ) ) ).
% unite_S_equal
thf(fact_1001__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_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 ) ) ) ) ) ) ).
% \<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)\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_1002_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_1003_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_1004_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_1005_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_1006_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_1007_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_1008_empty__def,axiom,
( bot_bot_set_set_v
= ( collect_set_v
@ ^ [X3: set_v] : $false ) ) ).
% empty_def
thf(fact_1009_empty__def,axiom,
( bot_bot_set_v
= ( collect_v
@ ^ [X3: v] : $false ) ) ).
% empty_def
thf(fact_1010_empty__def,axiom,
( bot_bo723834152578015283od_v_v
= ( collec140062887454715474od_v_v
@ ^ [X3: product_prod_v_v] : $false ) ) ).
% empty_def
thf(fact_1011_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_1012_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_1013_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_1014_insert__compr,axiom,
( insert_v
= ( ^ [A4: v,B3: set_v] :
( collect_v
@ ^ [X3: v] :
( ( X3 = A4 )
| ( member_v @ X3 @ B3 ) ) ) ) ) ).
% insert_compr
thf(fact_1015_insert__compr,axiom,
( insert1338601472111419319od_v_v
= ( ^ [A4: product_prod_v_v,B3: set_Product_prod_v_v] :
( collec140062887454715474od_v_v
@ ^ [X3: product_prod_v_v] :
( ( X3 = A4 )
| ( member7453568604450474000od_v_v @ X3 @ B3 ) ) ) ) ) ).
% insert_compr
thf(fact_1016_insert__compr,axiom,
( insert_set_v
= ( ^ [A4: set_v,B3: set_set_v] :
( collect_set_v
@ ^ [X3: set_v] :
( ( X3 = A4 )
| ( member_set_v @ X3 @ B3 ) ) ) ) ) ).
% insert_compr
thf(fact_1017_insert__def,axiom,
( insert_set_v
= ( ^ [A4: set_v] :
( sup_sup_set_set_v
@ ( collect_set_v
@ ^ [X3: set_v] : ( X3 = A4 ) ) ) ) ) ).
% insert_def
thf(fact_1018_insert__def,axiom,
( insert_v
= ( ^ [A4: v] :
( sup_sup_set_v
@ ( collect_v
@ ^ [X3: v] : ( X3 = A4 ) ) ) ) ) ).
% insert_def
thf(fact_1019_insert__def,axiom,
( insert1338601472111419319od_v_v
= ( ^ [A4: product_prod_v_v] :
( sup_su414716646722978715od_v_v
@ ( collec140062887454715474od_v_v
@ ^ [X3: product_prod_v_v] : ( X3 = A4 ) ) ) ) ) ).
% insert_def
thf(fact_1020_pred__subset__eq,axiom,
! [R4: set_v,S: set_v] :
( ( ord_less_eq_v_o
@ ^ [X3: v] : ( member_v @ X3 @ R4 )
@ ^ [X3: v] : ( member_v @ X3 @ S ) )
= ( ord_less_eq_set_v @ R4 @ S ) ) ).
% pred_subset_eq
thf(fact_1021_pred__subset__eq,axiom,
! [R4: set_Product_prod_v_v,S: set_Product_prod_v_v] :
( ( ord_le5892402249245633078_v_v_o
@ ^ [X3: product_prod_v_v] : ( member7453568604450474000od_v_v @ X3 @ R4 )
@ ^ [X3: product_prod_v_v] : ( member7453568604450474000od_v_v @ X3 @ S ) )
= ( ord_le7336532860387713383od_v_v @ R4 @ S ) ) ).
% pred_subset_eq
thf(fact_1022_less__eq__set__def,axiom,
( ord_less_eq_set_v
= ( ^ [A3: set_v,B3: set_v] :
( ord_less_eq_v_o
@ ^ [X3: v] : ( member_v @ X3 @ A3 )
@ ^ [X3: v] : ( member_v @ X3 @ B3 ) ) ) ) ).
% less_eq_set_def
thf(fact_1023_less__eq__set__def,axiom,
( ord_le7336532860387713383od_v_v
= ( ^ [A3: 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 @ A3 )
@ ^ [X3: product_prod_v_v] : ( member7453568604450474000od_v_v @ X3 @ B3 ) ) ) ) ).
% less_eq_set_def
thf(fact_1024_Collect__subset,axiom,
! [A2: set_set_v,P: set_v > $o] :
( ord_le5216385588623774835_set_v
@ ( collect_set_v
@ ^ [X3: set_v] :
( ( member_set_v @ X3 @ A2 )
& ( P @ X3 ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_1025_Collect__subset,axiom,
! [A2: set_v,P: v > $o] :
( ord_less_eq_set_v
@ ( collect_v
@ ^ [X3: v] :
( ( member_v @ X3 @ A2 )
& ( P @ X3 ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_1026_Collect__subset,axiom,
! [A2: 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 @ A2 )
& ( P @ X3 ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_1027_Int__def,axiom,
( inf_in6271465464967711157od_v_v
= ( ^ [A3: set_Product_prod_v_v,B3: set_Product_prod_v_v] :
( collec140062887454715474od_v_v
@ ^ [X3: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X3 @ A3 )
& ( member7453568604450474000od_v_v @ X3 @ B3 ) ) ) ) ) ).
% Int_def
thf(fact_1028_Int__def,axiom,
( inf_inf_set_set_v
= ( ^ [A3: set_set_v,B3: set_set_v] :
( collect_set_v
@ ^ [X3: set_v] :
( ( member_set_v @ X3 @ A3 )
& ( member_set_v @ X3 @ B3 ) ) ) ) ) ).
% Int_def
thf(fact_1029_Int__def,axiom,
( inf_inf_set_v
= ( ^ [A3: set_v,B3: set_v] :
( collect_v
@ ^ [X3: v] :
( ( member_v @ X3 @ A3 )
& ( member_v @ X3 @ B3 ) ) ) ) ) ).
% Int_def
thf(fact_1030_Int__Collect,axiom,
! [X: product_prod_v_v,A2: set_Product_prod_v_v,P: product_prod_v_v > $o] :
( ( member7453568604450474000od_v_v @ X @ ( inf_in6271465464967711157od_v_v @ A2 @ ( collec140062887454715474od_v_v @ P ) ) )
= ( ( member7453568604450474000od_v_v @ X @ A2 )
& ( P @ X ) ) ) ).
% Int_Collect
thf(fact_1031_Int__Collect,axiom,
! [X: set_v,A2: set_set_v,P: set_v > $o] :
( ( member_set_v @ X @ ( inf_inf_set_set_v @ A2 @ ( collect_set_v @ P ) ) )
= ( ( member_set_v @ X @ A2 )
& ( P @ X ) ) ) ).
% Int_Collect
thf(fact_1032_Int__Collect,axiom,
! [X: v,A2: set_v,P: v > $o] :
( ( member_v @ X @ ( inf_inf_set_v @ A2 @ ( collect_v @ P ) ) )
= ( ( member_v @ X @ A2 )
& ( P @ X ) ) ) ).
% Int_Collect
thf(fact_1033_inf__set__def,axiom,
( inf_in6271465464967711157od_v_v
= ( ^ [A3: 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 @ A3 )
@ ^ [X3: product_prod_v_v] : ( member7453568604450474000od_v_v @ X3 @ B3 ) ) ) ) ) ).
% inf_set_def
thf(fact_1034_inf__set__def,axiom,
( inf_inf_set_set_v
= ( ^ [A3: set_set_v,B3: set_set_v] :
( collect_set_v
@ ( inf_inf_set_v_o
@ ^ [X3: set_v] : ( member_set_v @ X3 @ A3 )
@ ^ [X3: set_v] : ( member_set_v @ X3 @ B3 ) ) ) ) ) ).
% inf_set_def
thf(fact_1035_inf__set__def,axiom,
( inf_inf_set_v
= ( ^ [A3: set_v,B3: set_v] :
( collect_v
@ ( inf_inf_v_o
@ ^ [X3: v] : ( member_v @ X3 @ A3 )
@ ^ [X3: v] : ( member_v @ X3 @ B3 ) ) ) ) ) ).
% inf_set_def
thf(fact_1036_inf__Int__eq,axiom,
! [R4: set_Product_prod_v_v,S: set_Product_prod_v_v] :
( ( inf_in6860806757119575912_v_v_o
@ ^ [X3: product_prod_v_v] : ( member7453568604450474000od_v_v @ X3 @ R4 )
@ ^ [X3: product_prod_v_v] : ( member7453568604450474000od_v_v @ X3 @ S ) )
= ( ^ [X3: product_prod_v_v] : ( member7453568604450474000od_v_v @ X3 @ ( inf_in6271465464967711157od_v_v @ R4 @ S ) ) ) ) ).
% inf_Int_eq
thf(fact_1037_inf__Int__eq,axiom,
! [R4: set_v,S: set_v] :
( ( inf_inf_v_o
@ ^ [X3: v] : ( member_v @ X3 @ R4 )
@ ^ [X3: v] : ( member_v @ X3 @ S ) )
= ( ^ [X3: v] : ( member_v @ X3 @ ( inf_inf_set_v @ R4 @ S ) ) ) ) ).
% inf_Int_eq
thf(fact_1038_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_1039_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_1040_minus__set__def,axiom,
( minus_4183494784930505774od_v_v
= ( ^ [A3: 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 @ A3 )
@ ^ [X3: product_prod_v_v] : ( member7453568604450474000od_v_v @ X3 @ B3 ) ) ) ) ) ).
% minus_set_def
thf(fact_1041_minus__set__def,axiom,
( minus_7228012346218142266_set_v
= ( ^ [A3: set_set_v,B3: set_set_v] :
( collect_set_v
@ ( minus_minus_set_v_o
@ ^ [X3: set_v] : ( member_set_v @ X3 @ A3 )
@ ^ [X3: set_v] : ( member_set_v @ X3 @ B3 ) ) ) ) ) ).
% minus_set_def
thf(fact_1042_minus__set__def,axiom,
( minus_minus_set_v
= ( ^ [A3: set_v,B3: set_v] :
( collect_v
@ ( minus_minus_v_o
@ ^ [X3: v] : ( member_v @ X3 @ A3 )
@ ^ [X3: v] : ( member_v @ X3 @ B3 ) ) ) ) ) ).
% minus_set_def
thf(fact_1043_set__diff__eq,axiom,
( minus_4183494784930505774od_v_v
= ( ^ [A3: set_Product_prod_v_v,B3: set_Product_prod_v_v] :
( collec140062887454715474od_v_v
@ ^ [X3: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X3 @ A3 )
& ~ ( member7453568604450474000od_v_v @ X3 @ B3 ) ) ) ) ) ).
% set_diff_eq
thf(fact_1044_set__diff__eq,axiom,
( minus_7228012346218142266_set_v
= ( ^ [A3: set_set_v,B3: set_set_v] :
( collect_set_v
@ ^ [X3: set_v] :
( ( member_set_v @ X3 @ A3 )
& ~ ( member_set_v @ X3 @ B3 ) ) ) ) ) ).
% set_diff_eq
thf(fact_1045_set__diff__eq,axiom,
( minus_minus_set_v
= ( ^ [A3: set_v,B3: set_v] :
( collect_v
@ ^ [X3: v] :
( ( member_v @ X3 @ A3 )
& ~ ( member_v @ X3 @ B3 ) ) ) ) ) ).
% set_diff_eq
thf(fact_1046_sup__Un__eq,axiom,
! [R4: set_v,S: set_v] :
( ( sup_sup_v_o
@ ^ [X3: v] : ( member_v @ X3 @ R4 )
@ ^ [X3: v] : ( member_v @ X3 @ S ) )
= ( ^ [X3: v] : ( member_v @ X3 @ ( sup_sup_set_v @ R4 @ S ) ) ) ) ).
% sup_Un_eq
thf(fact_1047_sup__Un__eq,axiom,
! [R4: set_Product_prod_v_v,S: set_Product_prod_v_v] :
( ( sup_su5941406310530359554_v_v_o
@ ^ [X3: product_prod_v_v] : ( member7453568604450474000od_v_v @ X3 @ R4 )
@ ^ [X3: product_prod_v_v] : ( member7453568604450474000od_v_v @ X3 @ S ) )
= ( ^ [X3: product_prod_v_v] : ( member7453568604450474000od_v_v @ X3 @ ( sup_su414716646722978715od_v_v @ R4 @ S ) ) ) ) ).
% sup_Un_eq
thf(fact_1048_Un__def,axiom,
( sup_sup_set_set_v
= ( ^ [A3: set_set_v,B3: set_set_v] :
( collect_set_v
@ ^ [X3: set_v] :
( ( member_set_v @ X3 @ A3 )
| ( member_set_v @ X3 @ B3 ) ) ) ) ) ).
% Un_def
thf(fact_1049_Un__def,axiom,
( sup_sup_set_v
= ( ^ [A3: set_v,B3: set_v] :
( collect_v
@ ^ [X3: v] :
( ( member_v @ X3 @ A3 )
| ( member_v @ X3 @ B3 ) ) ) ) ) ).
% Un_def
thf(fact_1050_Un__def,axiom,
( sup_su414716646722978715od_v_v
= ( ^ [A3: set_Product_prod_v_v,B3: set_Product_prod_v_v] :
( collec140062887454715474od_v_v
@ ^ [X3: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X3 @ A3 )
| ( member7453568604450474000od_v_v @ X3 @ B3 ) ) ) ) ) ).
% Un_def
thf(fact_1051_sup__set__def,axiom,
( sup_sup_set_set_v
= ( ^ [A3: set_set_v,B3: set_set_v] :
( collect_set_v
@ ( sup_sup_set_v_o
@ ^ [X3: set_v] : ( member_set_v @ X3 @ A3 )
@ ^ [X3: set_v] : ( member_set_v @ X3 @ B3 ) ) ) ) ) ).
% sup_set_def
thf(fact_1052_sup__set__def,axiom,
( sup_sup_set_v
= ( ^ [A3: set_v,B3: set_v] :
( collect_v
@ ( sup_sup_v_o
@ ^ [X3: v] : ( member_v @ X3 @ A3 )
@ ^ [X3: v] : ( member_v @ X3 @ B3 ) ) ) ) ) ).
% sup_set_def
thf(fact_1053_sup__set__def,axiom,
( sup_su414716646722978715od_v_v
= ( ^ [A3: set_Product_prod_v_v,B3: set_Product_prod_v_v] :
( collec140062887454715474od_v_v
@ ( sup_su5941406310530359554_v_v_o
@ ^ [X3: product_prod_v_v] : ( member7453568604450474000od_v_v @ X3 @ A3 )
@ ^ [X3: product_prod_v_v] : ( member7453568604450474000od_v_v @ X3 @ B3 ) ) ) ) ) ).
% sup_set_def
thf(fact_1054_Collect__disj__eq,axiom,
! [P: set_v > $o,Q: set_v > $o] :
( ( collect_set_v
@ ^ [X3: set_v] :
( ( P @ X3 )
| ( Q @ X3 ) ) )
= ( sup_sup_set_set_v @ ( collect_set_v @ P ) @ ( collect_set_v @ Q ) ) ) ).
% Collect_disj_eq
thf(fact_1055_Collect__disj__eq,axiom,
! [P: v > $o,Q: v > $o] :
( ( collect_v
@ ^ [X3: v] :
( ( P @ X3 )
| ( Q @ X3 ) ) )
= ( sup_sup_set_v @ ( collect_v @ P ) @ ( collect_v @ Q ) ) ) ).
% Collect_disj_eq
thf(fact_1056_Collect__disj__eq,axiom,
! [P: product_prod_v_v > $o,Q: product_prod_v_v > $o] :
( ( collec140062887454715474od_v_v
@ ^ [X3: product_prod_v_v] :
( ( P @ X3 )
| ( Q @ X3 ) ) )
= ( sup_su414716646722978715od_v_v @ ( collec140062887454715474od_v_v @ P ) @ ( collec140062887454715474od_v_v @ Q ) ) ) ).
% Collect_disj_eq
thf(fact_1057_graph_Ounite__S__equal,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,E: sCC_Bl7326425374436813197t_unit,W: product_prod_v_v,V3: product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( sCC_Bl7798947040364291444t_unit @ Successors @ E )
=> ( ( member7453568604450474000od_v_v @ W @ ( Successors @ V3 ) )
=> ( ~ ( member7453568604450474000od_v_v @ W @ ( sCC_Bl3878977043676959280t_unit @ E @ V3 ) )
=> ( ( member7453568604450474000od_v_v @ W @ ( sCC_Bl5498988629518860705t_unit @ E ) )
=> ( ~ ( member7453568604450474000od_v_v @ W @ ( sCC_Bl5094201334446601350t_unit @ E ) )
=> ( ( 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 @ V3 @ W @ E ) @ N4 ) )
& ( member7453568604450474000od_v_v @ N4 @ ( set_Product_prod_v_v2 @ ( sCC_Bl2021302119412358655t_unit @ ( sCC_Bl4702006153222411093od_v_v @ V3 @ W @ E ) ) ) ) ) ) )
= ( comple5788137035815166516od_v_v
@ ( collec8263177866097347122od_v_v
@ ^ [Uu: set_Product_prod_v_v] :
? [N4: product_prod_v_v] :
( ( Uu
= ( sCC_Bl8440648026628373538t_unit @ E @ N4 ) )
& ( member7453568604450474000od_v_v @ N4 @ ( set_Product_prod_v_v2 @ ( sCC_Bl2021302119412358655t_unit @ E ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% graph.unite_S_equal
thf(fact_1058_graph_Ounite__S__equal,axiom,
! [Vertices: set_v,Successors: v > set_v,E: sCC_Bl1394983891496994913t_unit,W: v,V3: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl9196236973127232072t_unit @ Successors @ E )
=> ( ( member_v @ W @ ( Successors @ V3 ) )
=> ( ~ ( member_v @ W @ ( sCC_Bl3795065053823578884t_unit @ E @ V3 ) )
=> ( ( member_v @ W @ ( sCC_Bl4645233313691564917t_unit @ E ) )
=> ( ~ ( member_v @ W @ ( sCC_Bl157864678168468314t_unit @ E ) )
=> ( ( comple2307003700295860064_set_v
@ ( collect_set_v
@ ^ [Uu: set_v] :
? [N4: v] :
( ( Uu
= ( sCC_Bl1280885523602775798t_unit @ ( sCC_Bloemen_unite_v @ V3 @ W @ E ) @ N4 ) )
& ( member_v @ N4 @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ ( sCC_Bloemen_unite_v @ V3 @ W @ E ) ) ) ) ) ) )
= ( comple2307003700295860064_set_v
@ ( collect_set_v
@ ^ [Uu: set_v] :
? [N4: v] :
( ( Uu
= ( sCC_Bl1280885523602775798t_unit @ E @ N4 ) )
& ( member_v @ N4 @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% graph.unite_S_equal
thf(fact_1059_graph_Osub__env__def,axiom,
! [Vertices: set_v,Successors: v > set_v,E: sCC_Bl1394983891496994913t_unit,E2: sCC_Bl1394983891496994913t_unit] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl5768913643336123637t_unit @ E @ E2 )
= ( ( ( sCC_Bl1090238580953940555t_unit @ E2 )
= ( sCC_Bl1090238580953940555t_unit @ E ) )
& ( ord_less_eq_set_v @ ( sCC_Bl4645233313691564917t_unit @ E ) @ ( sCC_Bl4645233313691564917t_unit @ E2 ) )
& ( ord_less_eq_set_v @ ( sCC_Bl157864678168468314t_unit @ E ) @ ( sCC_Bl157864678168468314t_unit @ E2 ) )
& ! [V4: v] : ( ord_less_eq_set_v @ ( sCC_Bl3795065053823578884t_unit @ E @ V4 ) @ ( sCC_Bl3795065053823578884t_unit @ E2 @ V4 ) )
& ! [V4: v] : ( ord_less_eq_set_v @ ( sCC_Bl1280885523602775798t_unit @ E @ V4 ) @ ( sCC_Bl1280885523602775798t_unit @ E2 @ V4 ) )
& ( ord_less_eq_set_v
@ ( comple2307003700295860064_set_v
@ ( collect_set_v
@ ^ [Uu: set_v] :
? [V4: v] :
( ( Uu
= ( sCC_Bl1280885523602775798t_unit @ E @ V4 ) )
& ( member_v @ V4 @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E ) ) ) ) ) )
@ ( comple2307003700295860064_set_v
@ ( collect_set_v
@ ^ [Uu: set_v] :
? [V4: v] :
( ( Uu
= ( sCC_Bl1280885523602775798t_unit @ E2 @ V4 ) )
& ( member_v @ V4 @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E2 ) ) ) ) ) ) ) ) ) ) ).
% graph.sub_env_def
thf(fact_1060_ra__cases,axiom,
! [X: v,Y: v,E6: set_Product_prod_v_v] :
( ( sCC_Bl4291963740693775144ding_v @ successors @ X @ Y @ E6 )
=> ( ( X = Y )
| ? [Z2: v] :
( ( member_v @ Z2 @ ( successors @ X ) )
& ~ ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ X @ Z2 ) @ E6 )
& ( sCC_Bl4291963740693775144ding_v @ successors @ Z2 @ Y @ E6 ) ) ) ) ).
% ra_cases
thf(fact_1061_edge__ra,axiom,
! [Y: v,X: v,E6: set_Product_prod_v_v] :
( ( member_v @ Y @ ( successors @ X ) )
=> ( ~ ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ X @ Y ) @ E6 )
=> ( sCC_Bl4291963740693775144ding_v @ successors @ X @ Y @ E6 ) ) ) ).
% edge_ra
thf(fact_1062_ra__mono,axiom,
! [X: v,Y: v,E6: set_Product_prod_v_v,E7: set_Product_prod_v_v] :
( ( sCC_Bl4291963740693775144ding_v @ successors @ X @ Y @ E6 )
=> ( ( ord_le7336532860387713383od_v_v @ E7 @ E6 )
=> ( sCC_Bl4291963740693775144ding_v @ successors @ X @ Y @ E7 ) ) ) ).
% ra_mono
thf(fact_1063_ra__trans,axiom,
! [X: v,Y: v,E6: set_Product_prod_v_v,Z: v] :
( ( sCC_Bl4291963740693775144ding_v @ successors @ X @ Y @ E6 )
=> ( ( sCC_Bl4291963740693775144ding_v @ successors @ Y @ Z @ E6 )
=> ( sCC_Bl4291963740693775144ding_v @ successors @ X @ Z @ E6 ) ) ) ).
% ra_trans
thf(fact_1064_ra__refl,axiom,
! [X: v,E6: set_Product_prod_v_v] : ( sCC_Bl4291963740693775144ding_v @ successors @ X @ X @ E6 ) ).
% ra_refl
thf(fact_1065_ra__reachable,axiom,
! [X: v,Y: v,E6: set_Product_prod_v_v] :
( ( sCC_Bl4291963740693775144ding_v @ successors @ X @ Y @ E6 )
=> ( sCC_Bl649662514949026229able_v @ successors @ X @ Y ) ) ).
% ra_reachable
thf(fact_1066_reachable__avoiding_Ocases,axiom,
! [A1: v,A22: v,A32: set_Product_prod_v_v] :
( ( sCC_Bl4291963740693775144ding_v @ successors @ A1 @ A22 @ A32 )
=> ( ( A22 != A1 )
=> ~ ! [Y2: v] :
( ( sCC_Bl4291963740693775144ding_v @ successors @ A1 @ Y2 @ A32 )
=> ( ( member_v @ A22 @ ( successors @ Y2 ) )
=> ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ Y2 @ A22 ) @ A32 ) ) ) ) ) ).
% reachable_avoiding.cases
thf(fact_1067_ra__succ,axiom,
! [X: v,Y: v,E6: set_Product_prod_v_v,Z: v] :
( ( sCC_Bl4291963740693775144ding_v @ successors @ X @ Y @ E6 )
=> ( ( member_v @ Z @ ( successors @ Y ) )
=> ( ~ ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ Y @ Z ) @ E6 )
=> ( sCC_Bl4291963740693775144ding_v @ successors @ X @ Z @ E6 ) ) ) ) ).
% ra_succ
thf(fact_1068_reachable__avoiding_Osimps,axiom,
! [A1: v,A22: v,A32: set_Product_prod_v_v] :
( ( sCC_Bl4291963740693775144ding_v @ successors @ A1 @ A22 @ A32 )
= ( ? [X3: v,E8: set_Product_prod_v_v] :
( ( A1 = X3 )
& ( A22 = X3 )
& ( A32 = E8 ) )
| ? [X3: v,Y3: v,E8: set_Product_prod_v_v,Z3: v] :
( ( A1 = X3 )
& ( A22 = Z3 )
& ( A32 = E8 )
& ( sCC_Bl4291963740693775144ding_v @ successors @ X3 @ Y3 @ E8 )
& ( member_v @ Z3 @ ( successors @ Y3 ) )
& ~ ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ Y3 @ Z3 ) @ E8 ) ) ) ) ).
% reachable_avoiding.simps
thf(fact_1069_ra__empty,axiom,
! [X: v,Y: v] :
( ( sCC_Bl4291963740693775144ding_v @ successors @ X @ Y @ bot_bo723834152578015283od_v_v )
= ( sCC_Bl649662514949026229able_v @ successors @ X @ Y ) ) ).
% ra_empty
thf(fact_1070_ra__add__edge,axiom,
! [X: v,Y: v,E6: set_Product_prod_v_v,V3: v,W: v] :
( ( sCC_Bl4291963740693775144ding_v @ successors @ X @ Y @ E6 )
=> ( ( sCC_Bl4291963740693775144ding_v @ successors @ X @ Y @ ( sup_su414716646722978715od_v_v @ E6 @ ( insert1338601472111419319od_v_v @ ( product_Pair_v_v @ V3 @ W ) @ bot_bo723834152578015283od_v_v ) ) )
| ( ( sCC_Bl4291963740693775144ding_v @ successors @ X @ V3 @ ( sup_su414716646722978715od_v_v @ E6 @ ( insert1338601472111419319od_v_v @ ( product_Pair_v_v @ V3 @ W ) @ bot_bo723834152578015283od_v_v ) ) )
& ( sCC_Bl4291963740693775144ding_v @ successors @ W @ Y @ ( sup_su414716646722978715od_v_v @ E6 @ ( insert1338601472111419319od_v_v @ ( product_Pair_v_v @ V3 @ W ) @ bot_bo723834152578015283od_v_v ) ) ) ) ) ) ).
% ra_add_edge
thf(fact_1071_vfin,axiom,
finite_finite_v @ vertices ).
% vfin
% Helper facts (3)
thf(help_If_3_1_If_001t__Set__Oset_Itf__v_J_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__Set__Oset_Itf__v_J_T,axiom,
! [X: set_v,Y: set_v] :
( ( if_set_v @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Set__Oset_Itf__v_J_T,axiom,
! [X: set_v,Y: set_v] :
( ( if_set_v @ $true @ X @ Y )
= X ) ).
% Conjectures (1)
thf(conj_0,conjecture,
sCC_Bl4022239298816431255edes_v @ m @ n @ ( sCC_Bl9201514103433284750t_unit @ e ) ).
%------------------------------------------------------------------------------