TPTP Problem File: SLH0860^1.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_02490_085696__6362640_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1445 ( 751 unt; 161 typ; 0 def)
% Number of atoms : 3209 (1422 equ; 0 cnn)
% Maximal formula atoms : 9 ( 2 avg)
% Number of connectives : 12048 ( 329 ~; 58 |; 261 &;10302 @)
% ( 0 <=>;1098 =>; 0 <=; 0 <~>)
% Maximal formula depth : 23 ( 6 avg)
% Number of types : 20 ( 19 usr)
% Number of type conns : 532 ( 532 >; 0 *; 0 +; 0 <<)
% Number of symbols : 145 ( 142 usr; 21 con; 0-4 aty)
% Number of variables : 3766 ( 547 ^;3141 !; 78 ?;3766 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 08:54:07.084
%------------------------------------------------------------------------------
% Could-be-implicit typings (19)
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thf(sy_c_SCC__Bloemen__Sequential_Ograph_Odfss_001tf__v,type,
sCC_Bloemen_dfss_v: ( v > set_v ) > v > sCC_Bl1394983891496994913t_unit > sCC_Bl1394983891496994913t_unit ).
thf(sy_c_SCC__Bloemen__Sequential_Ograph_Ois__scc_001t__Product____Type__Oprod_Itf__v_Mtf__v_J,type,
sCC_Bl6242042402218619277od_v_v: ( product_prod_v_v > set_Product_prod_v_v ) > set_Product_prod_v_v > $o ).
thf(sy_c_SCC__Bloemen__Sequential_Ograph_Ois__scc_001t__Set__Oset_Itf__v_J,type,
sCC_Bl1515522642333523865_set_v: ( set_v > set_set_v ) > set_set_v > $o ).
thf(sy_c_SCC__Bloemen__Sequential_Ograph_Ois__scc_001tf__v,type,
sCC_Bloemen_is_scc_v: ( v > set_v ) > set_v > $o ).
thf(sy_c_SCC__Bloemen__Sequential_Ograph_Ois__subscc_001t__Product____Type__Oprod_Itf__v_Mtf__v_J,type,
sCC_Bl2301996248249672505od_v_v: ( product_prod_v_v > set_Product_prod_v_v ) > set_Product_prod_v_v > $o ).
thf(sy_c_SCC__Bloemen__Sequential_Ograph_Ois__subscc_001t__Set__Oset_Itf__v_J,type,
sCC_Bl7907073126578335045_set_v: ( set_v > set_set_v ) > set_set_v > $o ).
thf(sy_c_SCC__Bloemen__Sequential_Ograph_Ois__subscc_001tf__v,type,
sCC_Bl5398416737448265317bscc_v: ( v > set_v ) > set_v > $o ).
thf(sy_c_SCC__Bloemen__Sequential_Ograph_Opost__dfs_001tf__v_001t__Product____Type__Ounit_001t__Product____Type__Ounit,type,
sCC_Bl8953792750115413617t_unit: ( v > set_v ) > v > sCC_Bl1394983891496994913t_unit > sCC_Bl1394983891496994913t_unit > $o ).
thf(sy_c_SCC__Bloemen__Sequential_Ograph_Opost__dfss_001tf__v_001t__Product____Type__Ounit_001t__Product____Type__Ounit,type,
sCC_Bl6082031138996704384t_unit: ( v > set_v ) > v > sCC_Bl1394983891496994913t_unit > sCC_Bl1394983891496994913t_unit > $o ).
thf(sy_c_SCC__Bloemen__Sequential_Ograph_Opre__dfs_001tf__v_001t__Product____Type__Ounit,type,
sCC_Bl36166008131615352t_unit: ( v > set_v ) > v > sCC_Bl1394983891496994913t_unit > $o ).
thf(sy_c_SCC__Bloemen__Sequential_Ograph_Opre__dfss_001t__Product____Type__Oprod_Itf__v_Mtf__v_J_001t__Product____Type__Ounit,type,
sCC_Bl3607325323686918683t_unit: ( product_prod_v_v > set_Product_prod_v_v ) > product_prod_v_v > sCC_Bl7326425374436813197t_unit > $o ).
thf(sy_c_SCC__Bloemen__Sequential_Ograph_Opre__dfss_001tf__v_001t__Product____Type__Ounit,type,
sCC_Bl1748261141445803503t_unit: ( v > set_v ) > v > sCC_Bl1394983891496994913t_unit > $o ).
thf(sy_c_SCC__Bloemen__Sequential_Ograph_Oreachable_001t__Product____Type__Oprod_Itf__v_Mtf__v_J,type,
sCC_Bl4981926079593201289od_v_v: ( product_prod_v_v > set_Product_prod_v_v ) > product_prod_v_v > product_prod_v_v > $o ).
thf(sy_c_SCC__Bloemen__Sequential_Ograph_Oreachable_001t__Set__Oset_Itf__v_J,type,
sCC_Bl7354734129683093653_set_v: ( set_v > set_set_v ) > set_v > set_v > $o ).
thf(sy_c_SCC__Bloemen__Sequential_Ograph_Oreachable_001tf__v,type,
sCC_Bl649662514949026229able_v: ( v > set_v ) > v > v > $o ).
thf(sy_c_SCC__Bloemen__Sequential_Ograph_Oreachable__avoiding_001t__Product____Type__Oprod_Itf__v_Mtf__v_J,type,
sCC_Bl5370300055464682748od_v_v: ( product_prod_v_v > set_Product_prod_v_v ) > product_prod_v_v > product_prod_v_v > set_Pr2149350503807050951od_v_v > $o ).
thf(sy_c_SCC__Bloemen__Sequential_Ograph_Oreachable__avoiding_001tf__v,type,
sCC_Bl4291963740693775144ding_v: ( v > set_v ) > v > v > set_Product_prod_v_v > $o ).
thf(sy_c_SCC__Bloemen__Sequential_Ograph_Oreachable__end_001t__Product____Type__Oprod_Itf__v_Mtf__v_J,type,
sCC_Bl4714988717384592488od_v_v: ( product_prod_v_v > set_Product_prod_v_v ) > product_prod_v_v > product_prod_v_v > $o ).
thf(sy_c_SCC__Bloemen__Sequential_Ograph_Oreachable__end_001tf__v,type,
sCC_Bl770211535891879572_end_v: ( v > set_v ) > v > v > $o ).
thf(sy_c_SCC__Bloemen__Sequential_Ograph_Osub__env_001tf__v_001t__Product____Type__Ounit_001t__Product____Type__Ounit,type,
sCC_Bl5768913643336123637t_unit: sCC_Bl1394983891496994913t_unit > sCC_Bl1394983891496994913t_unit > $o ).
thf(sy_c_SCC__Bloemen__Sequential_Ograph_Ounite_001t__Product____Type__Oprod_Itf__v_Mtf__v_J,type,
sCC_Bl4702006153222411093od_v_v: product_prod_v_v > product_prod_v_v > sCC_Bl7326425374436813197t_unit > sCC_Bl7326425374436813197t_unit ).
thf(sy_c_SCC__Bloemen__Sequential_Ograph_Ounite_001tf__v,type,
sCC_Bloemen_unite_v: v > v > sCC_Bl1394983891496994913t_unit > sCC_Bl1394983891496994913t_unit ).
thf(sy_c_SCC__Bloemen__Sequential_Ograph_Ounvisited_001tf__v_001t__Product____Type__Ounit,type,
sCC_Bl3123350270117520219t_unit: ( v > set_v ) > sCC_Bl1394983891496994913t_unit > v > set_Product_prod_v_v ).
thf(sy_c_SCC__Bloemen__Sequential_Ograph_Owf__env_001t__Product____Type__Oprod_Itf__v_Mtf__v_J_001t__Product____Type__Ounit,type,
sCC_Bl7798947040364291444t_unit: ( product_prod_v_v > set_Product_prod_v_v ) > sCC_Bl7326425374436813197t_unit > $o ).
thf(sy_c_SCC__Bloemen__Sequential_Ograph_Owf__env_001tf__v_001t__Product____Type__Ounit,type,
sCC_Bl9196236973127232072t_unit: ( v > set_v ) > sCC_Bl1394983891496994913t_unit > $o ).
thf(sy_c_SCC__Bloemen__Sequential_Oinit__env_001tf__v,type,
sCC_Bl7693227186847904995_env_v: v > sCC_Bl1394983891496994913t_unit ).
thf(sy_c_SCC__Bloemen__Sequential_Oprecedes_001t__Product____Type__Oprod_Itf__v_Mtf__v_J,type,
sCC_Bl2026170059108282219od_v_v: product_prod_v_v > product_prod_v_v > list_P7986770385144383213od_v_v > $o ).
thf(sy_c_SCC__Bloemen__Sequential_Oprecedes_001tf__v,type,
sCC_Bl4022239298816431255edes_v: v > v > list_v > $o ).
thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_Itf__v_Mtf__v_J,type,
collec140062887454715474od_v_v: ( product_prod_v_v > $o ) > set_Product_prod_v_v ).
thf(sy_c_Set_OCollect_001t__Set__Oset_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_Itf__v_J,type,
insert_set_v: set_v > set_set_v > set_set_v ).
thf(sy_c_Set_Oinsert_001tf__v,type,
insert_v: v > set_v > set_v ).
thf(sy_c_Set_Othe__elem_001t__Product____Type__Oprod_Itf__v_Mtf__v_J,type,
the_el5392834299063928540od_v_v: set_Product_prod_v_v > product_prod_v_v ).
thf(sy_c_Set_Othe__elem_001t__Set__Oset_Itf__v_J,type,
the_elem_set_v: set_set_v > set_v ).
thf(sy_c_Set_Othe__elem_001tf__v,type,
the_elem_v: set_v > v ).
thf(sy_c_Sum__Type_OInl_001t__Product____Type__Oprod_Itf__v_Mt__SCC____Bloemen____Sequential__Oenv__Oenv____ext_Itf__v_Mt__Product____Type__Ounit_J_J_001t__Product____Type__Oprod_Itf__v_Mt__SCC____Bloemen____Sequential__Oenv__Oenv____ext_Itf__v_Mt__Product____Type__Ounit_J_J,type,
sum_In526841707622398774t_unit: produc5741669702376414499t_unit > sum_su8181647976486975269t_unit ).
thf(sy_c_Sum__Type_OInr_001t__Product____Type__Oprod_Itf__v_Mt__SCC____Bloemen____Sequential__Oenv__Oenv____ext_Itf__v_Mt__Product____Type__Ounit_J_J_001t__Product____Type__Oprod_Itf__v_Mt__SCC____Bloemen____Sequential__Oenv__Oenv____ext_Itf__v_Mt__Product____Type__Ounit_J_J,type,
sum_In5289330923152326972t_unit: produc5741669702376414499t_unit > sum_su8181647976486975269t_unit ).
thf(sy_c_Wellfounded_Oaccp_001t__Sum____Type__Osum_It__Product____Type__Oprod_Itf__v_Mt__SCC____Bloemen____Sequential__Oenv__Oenv____ext_Itf__v_Mt__Product____Type__Ounit_J_J_Mt__Product____Type__Oprod_Itf__v_Mt__SCC____Bloemen____Sequential__Oenv__Oenv____ext_Itf__v_Mt__Product____Type__Ounit_J_J_J,type,
accp_S2303753412255344476t_unit: ( sum_su8181647976486975269t_unit > sum_su8181647976486975269t_unit > $o ) > sum_su8181647976486975269t_unit > $o ).
thf(sy_c_fChoice_001tf__v,type,
fChoice_v: ( v > $o ) > v ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__v_Mtf__v_J_Mt__Product____Type__Oprod_Itf__v_Mtf__v_J_J,type,
member3038538357316246288od_v_v: produc206430290419586791od_v_v > set_Pr2149350503807050951od_v_v > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_Itf__v_Mtf__v_J,type,
member7453568604450474000od_v_v: product_prod_v_v > set_Product_prod_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_e,type,
e: sCC_Bl1394983891496994913t_unit ).
thf(sy_v_e1,type,
e1: sCC_Bl1394983891496994913t_unit ).
thf(sy_v_e_H,type,
e2: sCC_Bl1394983891496994913t_unit ).
thf(sy_v_e_H_H,type,
e3: sCC_Bl1394983891496994913t_unit ).
thf(sy_v_successors,type,
successors: v > set_v ).
thf(sy_v_u____,type,
u: v ).
thf(sy_v_v,type,
v2: v ).
thf(sy_v_vertices,type,
vertices: set_v ).
thf(sy_v_x____,type,
x: v ).
thf(sy_v_y____,type,
y: v ).
% Relevant facts (1277)
thf(fact_0__092_060open_062x_A_092_060preceq_062_Ay_Ain_Astack_Ae_H_092_060close_062,axiom,
sCC_Bl4022239298816431255edes_v @ x @ y @ ( sCC_Bl8828226123343373779t_unit @ e2 ) ).
% \<open>x \<preceq> y in stack e'\<close>
thf(fact_1_dfs__dfss__rel_Ocong,axiom,
sCC_Bl907557413677168252_rel_v = sCC_Bl907557413677168252_rel_v ).
% dfs_dfss_rel.cong
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_ra__refl,axiom,
! [X: v,E4: set_Product_prod_v_v] : ( sCC_Bl4291963740693775144ding_v @ successors @ X @ X @ E4 ) ).
% ra_refl
thf(fact_4_ra__trans,axiom,
! [X: v,Y: v,E4: set_Product_prod_v_v,Z: v] :
( ( sCC_Bl4291963740693775144ding_v @ successors @ X @ Y @ E4 )
=> ( ( sCC_Bl4291963740693775144ding_v @ successors @ Y @ Z @ E4 )
=> ( sCC_Bl4291963740693775144ding_v @ successors @ X @ Z @ E4 ) ) ) ).
% ra_trans
thf(fact_5_graph_Ounvisited_Ocong,axiom,
sCC_Bl3123350270117520219t_unit = sCC_Bl3123350270117520219t_unit ).
% graph.unvisited.cong
thf(fact_6_graph_Owf__env_Ocong,axiom,
sCC_Bl9196236973127232072t_unit = sCC_Bl9196236973127232072t_unit ).
% graph.wf_env.cong
thf(fact_7_graph_Oreachable__avoiding_Ocong,axiom,
sCC_Bl4291963740693775144ding_v = sCC_Bl4291963740693775144ding_v ).
% graph.reachable_avoiding.cong
thf(fact_8_fold__congs_I6_J,axiom,
! [R: sCC_Bl1394983891496994913t_unit,R2: sCC_Bl1394983891496994913t_unit,V: set_set_v,F: set_set_v > set_set_v,F2: set_set_v > set_set_v] :
( ( R = R2 )
=> ( ( ( sCC_Bl2536197123907397897t_unit @ R2 )
= V )
=> ( ! [V2: set_set_v] :
( ( V = V2 )
=> ( ( F @ V2 )
= ( F2 @ V2 ) ) )
=> ( ( sCC_Bl6816368539212994290t_unit @ F @ R )
= ( sCC_Bl6816368539212994290t_unit @ F2 @ R2 ) ) ) ) ) ).
% fold_congs(6)
thf(fact_9_unfold__congs_I6_J,axiom,
! [R: sCC_Bl1394983891496994913t_unit,R2: sCC_Bl1394983891496994913t_unit,V: set_set_v,F: set_set_v > set_set_v,F2: set_set_v > set_set_v] :
( ( R = R2 )
=> ( ( ( sCC_Bl2536197123907397897t_unit @ R2 )
= V )
=> ( ! [V2: set_set_v] :
( ( V2 = V )
=> ( ( F @ V2 )
= ( F2 @ V2 ) ) )
=> ( ( sCC_Bl6816368539212994290t_unit @ F @ R )
= ( sCC_Bl6816368539212994290t_unit @ F2 @ R2 ) ) ) ) ) ).
% unfold_congs(6)
thf(fact_10_S__reflexive,axiom,
! [E: sCC_Bl1394983891496994913t_unit,N: v] :
( ( sCC_Bl9196236973127232072t_unit @ successors @ E )
=> ( member_v @ N @ ( sCC_Bl1280885523602775798t_unit @ E @ N ) ) ) ).
% S_reflexive
thf(fact_11_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_12_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_13_fold__congs_I3_J,axiom,
! [R: sCC_Bl1394983891496994913t_unit,R2: sCC_Bl1394983891496994913t_unit,V: set_v,F: set_v > set_v,F2: set_v > set_v] :
( ( R = R2 )
=> ( ( ( sCC_Bl157864678168468314t_unit @ R2 )
= V )
=> ( ! [V2: set_v] :
( ( V = V2 )
=> ( ( F @ V2 )
= ( F2 @ V2 ) ) )
=> ( ( sCC_Bl2708505634401380163t_unit @ F @ R )
= ( sCC_Bl2708505634401380163t_unit @ F2 @ R2 ) ) ) ) ) ).
% fold_congs(3)
thf(fact_14_wf_H,axiom,
sCC_Bl9196236973127232072t_unit @ successors @ e2 ).
% wf'
thf(fact_15_xy_I2_J,axiom,
x != y ).
% xy(2)
thf(fact_16_unfold__congs_I3_J,axiom,
! [R: sCC_Bl1394983891496994913t_unit,R2: sCC_Bl1394983891496994913t_unit,V: set_v,F: set_v > set_v,F2: set_v > set_v] :
( ( R = R2 )
=> ( ( ( sCC_Bl157864678168468314t_unit @ R2 )
= V )
=> ( ! [V2: set_v] :
( ( V2 = V )
=> ( ( F @ V2 )
= ( F2 @ V2 ) ) )
=> ( ( sCC_Bl2708505634401380163t_unit @ F @ R )
= ( sCC_Bl2708505634401380163t_unit @ F2 @ R2 ) ) ) ) ) ).
% unfold_congs(3)
thf(fact_17__C3_C,axiom,
sCC_Bl6082031138996704384t_unit @ successors @ v2 @ e1 @ e2 ).
% "3"
thf(fact_18_unfold__congs_I7_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_Bl8828226123343373779t_unit @ R2 )
= V )
=> ( ! [V2: list_v] :
( ( V2 = V )
=> ( ( F @ V2 )
= ( F2 @ V2 ) ) )
=> ( ( sCC_Bl349061681862590396t_unit @ F @ R )
= ( sCC_Bl349061681862590396t_unit @ F2 @ R2 ) ) ) ) ) ).
% unfold_congs(7)
thf(fact_19_fold__congs_I7_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_Bl8828226123343373779t_unit @ R2 )
= V )
=> ( ! [V2: list_v] :
( ( V = V2 )
=> ( ( F @ V2 )
= ( F2 @ V2 ) ) )
=> ( ( sCC_Bl349061681862590396t_unit @ F @ R )
= ( sCC_Bl349061681862590396t_unit @ F2 @ R2 ) ) ) ) ) ).
% fold_congs(7)
thf(fact_20_stack__unexplored,axiom,
! [E: sCC_Bl1394983891496994913t_unit,N: v] :
( ( sCC_Bl9196236973127232072t_unit @ successors @ E )
=> ( ( member_v @ N @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E ) ) )
=> ~ ( member_v @ N @ ( sCC_Bl157864678168468314t_unit @ E ) ) ) ) ).
% stack_unexplored
thf(fact_21_True,axiom,
( v2
= ( hd_v @ ( sCC_Bl8828226123343373779t_unit @ e2 ) ) ) ).
% True
thf(fact_22_e_H__def,axiom,
( e2
= ( sCC_Bloemen_dfss_v @ successors @ v2 @ e1 ) ) ).
% e'_def
thf(fact_23_ra__mono,axiom,
! [X: v,Y: v,E4: set_Product_prod_v_v,E5: set_Product_prod_v_v] :
( ( sCC_Bl4291963740693775144ding_v @ successors @ X @ Y @ E4 )
=> ( ( ord_le7336532860387713383od_v_v @ E5 @ E4 )
=> ( sCC_Bl4291963740693775144ding_v @ successors @ X @ Y @ E5 ) ) ) ).
% ra_mono
thf(fact_24_reachable__avoiding_Ocases,axiom,
! [A1: v,A2: v,A3: set_Product_prod_v_v] :
( ( sCC_Bl4291963740693775144ding_v @ successors @ A1 @ A2 @ A3 )
=> ( ( A2 != A1 )
=> ~ ! [Y2: v] :
( ( sCC_Bl4291963740693775144ding_v @ successors @ A1 @ Y2 @ A3 )
=> ( ( member_v @ A2 @ ( successors @ Y2 ) )
=> ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ Y2 @ A2 ) @ A3 ) ) ) ) ) ).
% reachable_avoiding.cases
thf(fact_25_ra__succ,axiom,
! [X: v,Y: v,E4: set_Product_prod_v_v,Z: v] :
( ( sCC_Bl4291963740693775144ding_v @ successors @ X @ Y @ E4 )
=> ( ( member_v @ Z @ ( successors @ Y ) )
=> ( ~ ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ Y @ Z ) @ E4 )
=> ( sCC_Bl4291963740693775144ding_v @ successors @ X @ Z @ E4 ) ) ) ) ).
% ra_succ
thf(fact_26_reachable__avoiding_Osimps,axiom,
! [A1: v,A2: v,A3: set_Product_prod_v_v] :
( ( sCC_Bl4291963740693775144ding_v @ successors @ A1 @ A2 @ A3 )
= ( ? [X2: v,E6: set_Product_prod_v_v] :
( ( A1 = X2 )
& ( A2 = X2 )
& ( A3 = E6 ) )
| ? [X2: v,Y3: v,E6: set_Product_prod_v_v,Z2: v] :
( ( A1 = X2 )
& ( A2 = Z2 )
& ( A3 = E6 )
& ( sCC_Bl4291963740693775144ding_v @ successors @ X2 @ Y3 @ E6 )
& ( member_v @ Z2 @ ( successors @ Y3 ) )
& ~ ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ Y3 @ Z2 ) @ E6 ) ) ) ) ).
% reachable_avoiding.simps
thf(fact_27_edge__ra,axiom,
! [Y: v,X: v,E4: set_Product_prod_v_v] :
( ( member_v @ Y @ ( successors @ X ) )
=> ( ~ ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ X @ Y ) @ E4 )
=> ( sCC_Bl4291963740693775144ding_v @ successors @ X @ Y @ E4 ) ) ) ).
% edge_ra
thf(fact_28_ra__cases,axiom,
! [X: v,Y: v,E4: set_Product_prod_v_v] :
( ( sCC_Bl4291963740693775144ding_v @ successors @ X @ Y @ E4 )
=> ( ( X = Y )
| ? [Z3: v] :
( ( member_v @ Z3 @ ( successors @ X ) )
& ~ ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ X @ Z3 ) @ E4 )
& ( sCC_Bl4291963740693775144ding_v @ successors @ Z3 @ Y @ E4 ) ) ) ) ).
% ra_cases
thf(fact_29_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_30_precedes__refl,axiom,
! [X: v,Xs: list_v] :
( ( sCC_Bl4022239298816431255edes_v @ X @ X @ Xs )
= ( member_v @ X @ ( set_v2 @ Xs ) ) ) ).
% precedes_refl
thf(fact_31_graph_Odfss_Ocong,axiom,
sCC_Bloemen_dfss_v = sCC_Bloemen_dfss_v ).
% graph.dfss.cong
thf(fact_32_graph_Opost__dfss_Ocong,axiom,
sCC_Bl6082031138996704384t_unit = sCC_Bl6082031138996704384t_unit ).
% graph.post_dfss.cong
thf(fact_33_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_34_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_35_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_36_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_37_visited__unexplored,axiom,
! [E: sCC_Bl1394983891496994913t_unit,M: v] :
( ( sCC_Bl9196236973127232072t_unit @ successors @ E )
=> ( ( member_v @ M @ ( sCC_Bl4645233313691564917t_unit @ E ) )
=> ( ~ ( member_v @ M @ ( sCC_Bl157864678168468314t_unit @ E ) )
=> ~ ! [N2: v] :
( ( member_v @ N2 @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E ) ) )
=> ~ ( member_v @ M @ ( sCC_Bl1280885523602775798t_unit @ E @ N2 ) ) ) ) ) ) ).
% visited_unexplored
thf(fact_38_stack__visited,axiom,
! [E: sCC_Bl1394983891496994913t_unit,N: v] :
( ( sCC_Bl9196236973127232072t_unit @ successors @ E )
=> ( ( member_v @ N @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E ) ) )
=> ( member_v @ N @ ( sCC_Bl4645233313691564917t_unit @ E ) ) ) ) ).
% stack_visited
thf(fact_39_local_Owf,axiom,
sCC_Bl9196236973127232072t_unit @ successors @ e ).
% local.wf
thf(fact_40_subset__antisym,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ B )
=> ( ( ord_le7336532860387713383od_v_v @ B @ A )
=> ( A = B ) ) ) ).
% subset_antisym
thf(fact_41_subset__antisym,axiom,
! [A: set_v,B: set_v] :
( ( ord_less_eq_set_v @ A @ B )
=> ( ( ord_less_eq_set_v @ B @ A )
=> ( A = B ) ) ) ).
% subset_antisym
thf(fact_42_subsetI,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ! [X3: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X3 @ A )
=> ( member7453568604450474000od_v_v @ X3 @ B ) )
=> ( ord_le7336532860387713383od_v_v @ A @ B ) ) ).
% subsetI
thf(fact_43_subsetI,axiom,
! [A: set_v,B: set_v] :
( ! [X3: v] :
( ( member_v @ X3 @ A )
=> ( member_v @ X3 @ B ) )
=> ( ord_less_eq_set_v @ A @ B ) ) ).
% subsetI
thf(fact_44_old_Oprod_Oinject,axiom,
! [A4: v,B2: v,A5: v,B3: v] :
( ( ( product_Pair_v_v @ A4 @ B2 )
= ( product_Pair_v_v @ A5 @ B3 ) )
= ( ( A4 = A5 )
& ( B2 = B3 ) ) ) ).
% old.prod.inject
thf(fact_45_old_Oprod_Oinject,axiom,
! [A4: v,B2: sCC_Bl1394983891496994913t_unit,A5: v,B3: sCC_Bl1394983891496994913t_unit] :
( ( ( produc3862955338007567901t_unit @ A4 @ B2 )
= ( produc3862955338007567901t_unit @ A5 @ B3 ) )
= ( ( A4 = A5 )
& ( B2 = B3 ) ) ) ).
% old.prod.inject
thf(fact_46_prod_Oinject,axiom,
! [X1: v,X22: v,Y1: v,Y22: v] :
( ( ( product_Pair_v_v @ X1 @ X22 )
= ( product_Pair_v_v @ Y1 @ Y22 ) )
= ( ( X1 = Y1 )
& ( X22 = Y22 ) ) ) ).
% prod.inject
thf(fact_47_prod_Oinject,axiom,
! [X1: v,X22: sCC_Bl1394983891496994913t_unit,Y1: v,Y22: sCC_Bl1394983891496994913t_unit] :
( ( ( produc3862955338007567901t_unit @ X1 @ X22 )
= ( produc3862955338007567901t_unit @ Y1 @ Y22 ) )
= ( ( X1 = Y1 )
& ( X22 = Y22 ) ) ) ).
% prod.inject
thf(fact_48__C1_C,axiom,
sCC_Bl36166008131615352t_unit @ successors @ v2 @ e ).
% "1"
thf(fact_49_mem__Collect__eq,axiom,
! [A4: v,P: v > $o] :
( ( member_v @ A4 @ ( collect_v @ P ) )
= ( P @ A4 ) ) ).
% mem_Collect_eq
thf(fact_50_mem__Collect__eq,axiom,
! [A4: product_prod_v_v,P: product_prod_v_v > $o] :
( ( member7453568604450474000od_v_v @ A4 @ ( collec140062887454715474od_v_v @ P ) )
= ( P @ A4 ) ) ).
% mem_Collect_eq
thf(fact_51_mem__Collect__eq,axiom,
! [A4: set_v,P: set_v > $o] :
( ( member_set_v @ A4 @ ( collect_set_v @ P ) )
= ( P @ A4 ) ) ).
% mem_Collect_eq
thf(fact_52_Collect__mem__eq,axiom,
! [A: set_v] :
( ( collect_v
@ ^ [X2: v] : ( member_v @ X2 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_53_Collect__mem__eq,axiom,
! [A: set_Product_prod_v_v] :
( ( collec140062887454715474od_v_v
@ ^ [X2: product_prod_v_v] : ( member7453568604450474000od_v_v @ X2 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_54_Collect__mem__eq,axiom,
! [A: set_set_v] :
( ( collect_set_v
@ ^ [X2: set_v] : ( member_set_v @ X2 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_55_Collect__cong,axiom,
! [P: set_v > $o,Q: set_v > $o] :
( ! [X3: set_v] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collect_set_v @ P )
= ( collect_set_v @ Q ) ) ) ).
% Collect_cong
thf(fact_56_sub,axiom,
sCC_Bl5768913643336123637t_unit @ e @ e1 ).
% sub
thf(fact_57_dual__order_Orefl,axiom,
! [A4: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ A4 @ A4 ) ).
% dual_order.refl
thf(fact_58_dual__order_Orefl,axiom,
! [A4: set_v] : ( ord_less_eq_set_v @ A4 @ A4 ) ).
% dual_order.refl
thf(fact_59_order__refl,axiom,
! [X: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ X @ X ) ).
% order_refl
thf(fact_60_order__refl,axiom,
! [X: set_v] : ( ord_less_eq_set_v @ X @ X ) ).
% order_refl
thf(fact_61_v,axiom,
~ ( member_v @ v2 @ ( sCC_Bl4645233313691564917t_unit @ e ) ) ).
% v
thf(fact_62__092_060open_062sub__env_Ae_Ae_H_092_060close_062,axiom,
sCC_Bl5768913643336123637t_unit @ e @ e2 ).
% \<open>sub_env e e'\<close>
thf(fact_63_init__env__pre__dfs,axiom,
! [V3: v] : ( sCC_Bl36166008131615352t_unit @ successors @ V3 @ ( sCC_Bl7693227186847904995_env_v @ V3 ) ) ).
% init_env_pre_dfs
thf(fact_64_sclosed,axiom,
! [X4: v] :
( ( member_v @ X4 @ vertices )
=> ( ord_less_eq_set_v @ ( successors @ X4 ) @ vertices ) ) ).
% sclosed
thf(fact_65_graph_Opre__dfs_Ocong,axiom,
sCC_Bl36166008131615352t_unit = sCC_Bl36166008131615352t_unit ).
% graph.pre_dfs.cong
thf(fact_66_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_Product_prod_v_v,Z4: set_Product_prod_v_v] : ( Y4 = Z4 ) )
= ( ^ [X2: set_Product_prod_v_v,Y3: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X2 @ Y3 )
& ( ord_le7336532860387713383od_v_v @ Y3 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_67_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_v,Z4: set_v] : ( Y4 = Z4 ) )
= ( ^ [X2: set_v,Y3: set_v] :
( ( ord_less_eq_set_v @ X2 @ Y3 )
& ( ord_less_eq_set_v @ Y3 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_68_ord__eq__le__trans,axiom,
! [A4: set_Product_prod_v_v,B2: set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( A4 = B2 )
=> ( ( ord_le7336532860387713383od_v_v @ B2 @ C )
=> ( ord_le7336532860387713383od_v_v @ A4 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_69_ord__eq__le__trans,axiom,
! [A4: set_v,B2: set_v,C: set_v] :
( ( A4 = B2 )
=> ( ( ord_less_eq_set_v @ B2 @ C )
=> ( ord_less_eq_set_v @ A4 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_70_ord__le__eq__trans,axiom,
! [A4: set_Product_prod_v_v,B2: set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A4 @ B2 )
=> ( ( B2 = C )
=> ( ord_le7336532860387713383od_v_v @ A4 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_71_ord__le__eq__trans,axiom,
! [A4: set_v,B2: set_v,C: set_v] :
( ( ord_less_eq_set_v @ A4 @ B2 )
=> ( ( B2 = C )
=> ( ord_less_eq_set_v @ A4 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_72_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_73_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_74_order_Otrans,axiom,
! [A4: set_Product_prod_v_v,B2: set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A4 @ B2 )
=> ( ( ord_le7336532860387713383od_v_v @ B2 @ C )
=> ( ord_le7336532860387713383od_v_v @ A4 @ C ) ) ) ).
% order.trans
thf(fact_75_order_Otrans,axiom,
! [A4: set_v,B2: set_v,C: set_v] :
( ( ord_less_eq_set_v @ A4 @ B2 )
=> ( ( ord_less_eq_set_v @ B2 @ C )
=> ( ord_less_eq_set_v @ A4 @ C ) ) ) ).
% order.trans
thf(fact_76_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_77_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_78_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: set_Product_prod_v_v,Z4: set_Product_prod_v_v] : ( Y4 = Z4 ) )
= ( ^ [A6: set_Product_prod_v_v,B4: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ B4 @ A6 )
& ( ord_le7336532860387713383od_v_v @ A6 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_79_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: set_v,Z4: set_v] : ( Y4 = Z4 ) )
= ( ^ [A6: set_v,B4: set_v] :
( ( ord_less_eq_set_v @ B4 @ A6 )
& ( ord_less_eq_set_v @ A6 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_80_dual__order_Oantisym,axiom,
! [B2: set_Product_prod_v_v,A4: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ B2 @ A4 )
=> ( ( ord_le7336532860387713383od_v_v @ A4 @ B2 )
=> ( A4 = B2 ) ) ) ).
% dual_order.antisym
thf(fact_81_dual__order_Oantisym,axiom,
! [B2: set_v,A4: set_v] :
( ( ord_less_eq_set_v @ B2 @ A4 )
=> ( ( ord_less_eq_set_v @ A4 @ B2 )
=> ( A4 = B2 ) ) ) ).
% dual_order.antisym
thf(fact_82_dual__order_Otrans,axiom,
! [B2: set_Product_prod_v_v,A4: set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ B2 @ A4 )
=> ( ( ord_le7336532860387713383od_v_v @ C @ B2 )
=> ( ord_le7336532860387713383od_v_v @ C @ A4 ) ) ) ).
% dual_order.trans
thf(fact_83_dual__order_Otrans,axiom,
! [B2: set_v,A4: set_v,C: set_v] :
( ( ord_less_eq_set_v @ B2 @ A4 )
=> ( ( ord_less_eq_set_v @ C @ B2 )
=> ( ord_less_eq_set_v @ C @ A4 ) ) ) ).
% dual_order.trans
thf(fact_84_antisym,axiom,
! [A4: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A4 @ B2 )
=> ( ( ord_le7336532860387713383od_v_v @ B2 @ A4 )
=> ( A4 = B2 ) ) ) ).
% antisym
thf(fact_85_antisym,axiom,
! [A4: set_v,B2: set_v] :
( ( ord_less_eq_set_v @ A4 @ B2 )
=> ( ( ord_less_eq_set_v @ B2 @ A4 )
=> ( A4 = B2 ) ) ) ).
% antisym
thf(fact_86_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_Product_prod_v_v,Z4: set_Product_prod_v_v] : ( Y4 = Z4 ) )
= ( ^ [A6: set_Product_prod_v_v,B4: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A6 @ B4 )
& ( ord_le7336532860387713383od_v_v @ B4 @ A6 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_87_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_v,Z4: set_v] : ( Y4 = Z4 ) )
= ( ^ [A6: set_v,B4: set_v] :
( ( ord_less_eq_set_v @ A6 @ B4 )
& ( ord_less_eq_set_v @ B4 @ A6 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_88_order__subst1,axiom,
! [A4: set_Product_prod_v_v,F: set_Product_prod_v_v > set_Product_prod_v_v,B2: set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A4 @ ( F @ B2 ) )
=> ( ( ord_le7336532860387713383od_v_v @ B2 @ C )
=> ( ! [X3: set_Product_prod_v_v,Y2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X3 @ Y2 )
=> ( ord_le7336532860387713383od_v_v @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le7336532860387713383od_v_v @ A4 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_89_order__subst1,axiom,
! [A4: set_Product_prod_v_v,F: set_v > set_Product_prod_v_v,B2: set_v,C: set_v] :
( ( ord_le7336532860387713383od_v_v @ A4 @ ( F @ B2 ) )
=> ( ( ord_less_eq_set_v @ B2 @ C )
=> ( ! [X3: set_v,Y2: set_v] :
( ( ord_less_eq_set_v @ X3 @ Y2 )
=> ( ord_le7336532860387713383od_v_v @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le7336532860387713383od_v_v @ A4 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_90_order__subst1,axiom,
! [A4: set_v,F: set_Product_prod_v_v > set_v,B2: set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( ord_less_eq_set_v @ A4 @ ( F @ B2 ) )
=> ( ( ord_le7336532860387713383od_v_v @ B2 @ C )
=> ( ! [X3: set_Product_prod_v_v,Y2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X3 @ Y2 )
=> ( ord_less_eq_set_v @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_v @ A4 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_91_order__subst1,axiom,
! [A4: set_v,F: set_v > set_v,B2: set_v,C: set_v] :
( ( ord_less_eq_set_v @ A4 @ ( F @ B2 ) )
=> ( ( ord_less_eq_set_v @ B2 @ C )
=> ( ! [X3: set_v,Y2: set_v] :
( ( ord_less_eq_set_v @ X3 @ Y2 )
=> ( ord_less_eq_set_v @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_v @ A4 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_92_order__subst2,axiom,
! [A4: set_Product_prod_v_v,B2: set_Product_prod_v_v,F: set_Product_prod_v_v > set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A4 @ B2 )
=> ( ( ord_le7336532860387713383od_v_v @ ( F @ B2 ) @ C )
=> ( ! [X3: set_Product_prod_v_v,Y2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X3 @ Y2 )
=> ( ord_le7336532860387713383od_v_v @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le7336532860387713383od_v_v @ ( F @ A4 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_93_order__subst2,axiom,
! [A4: set_Product_prod_v_v,B2: set_Product_prod_v_v,F: set_Product_prod_v_v > set_v,C: set_v] :
( ( ord_le7336532860387713383od_v_v @ A4 @ B2 )
=> ( ( ord_less_eq_set_v @ ( F @ B2 ) @ C )
=> ( ! [X3: set_Product_prod_v_v,Y2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X3 @ Y2 )
=> ( ord_less_eq_set_v @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_v @ ( F @ A4 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_94_order__subst2,axiom,
! [A4: set_v,B2: set_v,F: set_v > set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( ord_less_eq_set_v @ A4 @ B2 )
=> ( ( ord_le7336532860387713383od_v_v @ ( F @ B2 ) @ C )
=> ( ! [X3: set_v,Y2: set_v] :
( ( ord_less_eq_set_v @ X3 @ Y2 )
=> ( ord_le7336532860387713383od_v_v @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le7336532860387713383od_v_v @ ( F @ A4 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_95_order__subst2,axiom,
! [A4: set_v,B2: set_v,F: set_v > set_v,C: set_v] :
( ( ord_less_eq_set_v @ A4 @ B2 )
=> ( ( ord_less_eq_set_v @ ( F @ B2 ) @ C )
=> ( ! [X3: set_v,Y2: set_v] :
( ( ord_less_eq_set_v @ X3 @ Y2 )
=> ( ord_less_eq_set_v @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_v @ ( F @ A4 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_96_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_97_order__eq__refl,axiom,
! [X: set_v,Y: set_v] :
( ( X = Y )
=> ( ord_less_eq_set_v @ X @ Y ) ) ).
% order_eq_refl
thf(fact_98_ord__eq__le__subst,axiom,
! [A4: set_Product_prod_v_v,F: set_Product_prod_v_v > set_Product_prod_v_v,B2: set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( A4
= ( F @ B2 ) )
=> ( ( ord_le7336532860387713383od_v_v @ B2 @ C )
=> ( ! [X3: set_Product_prod_v_v,Y2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X3 @ Y2 )
=> ( ord_le7336532860387713383od_v_v @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le7336532860387713383od_v_v @ A4 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_99_ord__eq__le__subst,axiom,
! [A4: set_v,F: set_Product_prod_v_v > set_v,B2: set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( A4
= ( F @ B2 ) )
=> ( ( ord_le7336532860387713383od_v_v @ B2 @ C )
=> ( ! [X3: set_Product_prod_v_v,Y2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X3 @ Y2 )
=> ( ord_less_eq_set_v @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_v @ A4 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_100_ord__eq__le__subst,axiom,
! [A4: set_Product_prod_v_v,F: set_v > set_Product_prod_v_v,B2: set_v,C: set_v] :
( ( A4
= ( F @ B2 ) )
=> ( ( ord_less_eq_set_v @ B2 @ C )
=> ( ! [X3: set_v,Y2: set_v] :
( ( ord_less_eq_set_v @ X3 @ Y2 )
=> ( ord_le7336532860387713383od_v_v @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le7336532860387713383od_v_v @ A4 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_101_ord__eq__le__subst,axiom,
! [A4: set_v,F: set_v > set_v,B2: set_v,C: set_v] :
( ( A4
= ( F @ B2 ) )
=> ( ( ord_less_eq_set_v @ B2 @ C )
=> ( ! [X3: set_v,Y2: set_v] :
( ( ord_less_eq_set_v @ X3 @ Y2 )
=> ( ord_less_eq_set_v @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_v @ A4 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_102_ord__le__eq__subst,axiom,
! [A4: set_Product_prod_v_v,B2: set_Product_prod_v_v,F: set_Product_prod_v_v > set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A4 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: set_Product_prod_v_v,Y2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X3 @ Y2 )
=> ( ord_le7336532860387713383od_v_v @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le7336532860387713383od_v_v @ ( F @ A4 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_103_ord__le__eq__subst,axiom,
! [A4: set_Product_prod_v_v,B2: set_Product_prod_v_v,F: set_Product_prod_v_v > set_v,C: set_v] :
( ( ord_le7336532860387713383od_v_v @ A4 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: set_Product_prod_v_v,Y2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X3 @ Y2 )
=> ( ord_less_eq_set_v @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_v @ ( F @ A4 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_104_ord__le__eq__subst,axiom,
! [A4: set_v,B2: set_v,F: set_v > set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( ord_less_eq_set_v @ A4 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: set_v,Y2: set_v] :
( ( ord_less_eq_set_v @ X3 @ Y2 )
=> ( ord_le7336532860387713383od_v_v @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le7336532860387713383od_v_v @ ( F @ A4 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_105_ord__le__eq__subst,axiom,
! [A4: set_v,B2: set_v,F: set_v > set_v,C: set_v] :
( ( ord_less_eq_set_v @ A4 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: set_v,Y2: set_v] :
( ( ord_less_eq_set_v @ X3 @ Y2 )
=> ( ord_less_eq_set_v @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_v @ ( F @ A4 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_106_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_107_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_108_old_Oprod_Oexhaust,axiom,
! [Y: product_prod_v_v] :
~ ! [A7: v,B5: v] :
( Y
!= ( product_Pair_v_v @ A7 @ B5 ) ) ).
% old.prod.exhaust
thf(fact_109_old_Oprod_Oexhaust,axiom,
! [Y: produc5741669702376414499t_unit] :
~ ! [A7: v,B5: sCC_Bl1394983891496994913t_unit] :
( Y
!= ( produc3862955338007567901t_unit @ A7 @ B5 ) ) ).
% old.prod.exhaust
thf(fact_110_surj__pair,axiom,
! [P2: product_prod_v_v] :
? [X3: v,Y2: v] :
( P2
= ( product_Pair_v_v @ X3 @ Y2 ) ) ).
% surj_pair
thf(fact_111_surj__pair,axiom,
! [P2: produc5741669702376414499t_unit] :
? [X3: v,Y2: sCC_Bl1394983891496994913t_unit] :
( P2
= ( produc3862955338007567901t_unit @ X3 @ Y2 ) ) ).
% surj_pair
thf(fact_112_prod__cases,axiom,
! [P: product_prod_v_v > $o,P2: product_prod_v_v] :
( ! [A7: v,B5: v] : ( P @ ( product_Pair_v_v @ A7 @ B5 ) )
=> ( P @ P2 ) ) ).
% prod_cases
thf(fact_113_prod__cases,axiom,
! [P: produc5741669702376414499t_unit > $o,P2: produc5741669702376414499t_unit] :
( ! [A7: v,B5: sCC_Bl1394983891496994913t_unit] : ( P @ ( produc3862955338007567901t_unit @ A7 @ B5 ) )
=> ( P @ P2 ) ) ).
% prod_cases
thf(fact_114_Pair__inject,axiom,
! [A4: v,B2: v,A5: v,B3: v] :
( ( ( product_Pair_v_v @ A4 @ B2 )
= ( product_Pair_v_v @ A5 @ B3 ) )
=> ~ ( ( A4 = A5 )
=> ( B2 != B3 ) ) ) ).
% Pair_inject
thf(fact_115_Pair__inject,axiom,
! [A4: v,B2: sCC_Bl1394983891496994913t_unit,A5: v,B3: sCC_Bl1394983891496994913t_unit] :
( ( ( produc3862955338007567901t_unit @ A4 @ B2 )
= ( produc3862955338007567901t_unit @ A5 @ B3 ) )
=> ~ ( ( A4 = A5 )
=> ( B2 != B3 ) ) ) ).
% Pair_inject
thf(fact_116_in__mono,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v,X: product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ B )
=> ( ( member7453568604450474000od_v_v @ X @ A )
=> ( member7453568604450474000od_v_v @ X @ B ) ) ) ).
% in_mono
thf(fact_117_in__mono,axiom,
! [A: set_v,B: set_v,X: v] :
( ( ord_less_eq_set_v @ A @ B )
=> ( ( member_v @ X @ A )
=> ( member_v @ X @ B ) ) ) ).
% in_mono
thf(fact_118_subsetD,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v,C: product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ B )
=> ( ( member7453568604450474000od_v_v @ C @ A )
=> ( member7453568604450474000od_v_v @ C @ B ) ) ) ).
% subsetD
thf(fact_119_subsetD,axiom,
! [A: set_v,B: set_v,C: v] :
( ( ord_less_eq_set_v @ A @ B )
=> ( ( member_v @ C @ A )
=> ( member_v @ C @ B ) ) ) ).
% subsetD
thf(fact_120_equalityE,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( A = B )
=> ~ ( ( ord_le7336532860387713383od_v_v @ A @ B )
=> ~ ( ord_le7336532860387713383od_v_v @ B @ A ) ) ) ).
% equalityE
thf(fact_121_equalityE,axiom,
! [A: set_v,B: set_v] :
( ( A = B )
=> ~ ( ( ord_less_eq_set_v @ A @ B )
=> ~ ( ord_less_eq_set_v @ B @ A ) ) ) ).
% equalityE
thf(fact_122_subset__eq,axiom,
( ord_le7336532860387713383od_v_v
= ( ^ [A8: set_Product_prod_v_v,B6: set_Product_prod_v_v] :
! [X2: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X2 @ A8 )
=> ( member7453568604450474000od_v_v @ X2 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_123_subset__eq,axiom,
( ord_less_eq_set_v
= ( ^ [A8: set_v,B6: set_v] :
! [X2: v] :
( ( member_v @ X2 @ A8 )
=> ( member_v @ X2 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_124_equalityD1,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( A = B )
=> ( ord_le7336532860387713383od_v_v @ A @ B ) ) ).
% equalityD1
thf(fact_125_equalityD1,axiom,
! [A: set_v,B: set_v] :
( ( A = B )
=> ( ord_less_eq_set_v @ A @ B ) ) ).
% equalityD1
thf(fact_126_equalityD2,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( A = B )
=> ( ord_le7336532860387713383od_v_v @ B @ A ) ) ).
% equalityD2
thf(fact_127_equalityD2,axiom,
! [A: set_v,B: set_v] :
( ( A = B )
=> ( ord_less_eq_set_v @ B @ A ) ) ).
% equalityD2
thf(fact_128_subset__iff,axiom,
( ord_le7336532860387713383od_v_v
= ( ^ [A8: set_Product_prod_v_v,B6: set_Product_prod_v_v] :
! [T: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ T @ A8 )
=> ( member7453568604450474000od_v_v @ T @ B6 ) ) ) ) ).
% subset_iff
thf(fact_129_subset__iff,axiom,
( ord_less_eq_set_v
= ( ^ [A8: set_v,B6: set_v] :
! [T: v] :
( ( member_v @ T @ A8 )
=> ( member_v @ T @ B6 ) ) ) ) ).
% subset_iff
thf(fact_130_subset__refl,axiom,
! [A: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ A @ A ) ).
% subset_refl
thf(fact_131_subset__refl,axiom,
! [A: set_v] : ( ord_less_eq_set_v @ A @ A ) ).
% subset_refl
thf(fact_132_Collect__mono,axiom,
! [P: set_v > $o,Q: set_v > $o] :
( ! [X3: set_v] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( ord_le5216385588623774835_set_v @ ( collect_set_v @ P ) @ ( collect_set_v @ Q ) ) ) ).
% Collect_mono
thf(fact_133_Collect__mono,axiom,
! [P: product_prod_v_v > $o,Q: product_prod_v_v > $o] :
( ! [X3: product_prod_v_v] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( ord_le7336532860387713383od_v_v @ ( collec140062887454715474od_v_v @ P ) @ ( collec140062887454715474od_v_v @ Q ) ) ) ).
% Collect_mono
thf(fact_134_Collect__mono,axiom,
! [P: v > $o,Q: v > $o] :
( ! [X3: v] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( ord_less_eq_set_v @ ( collect_v @ P ) @ ( collect_v @ Q ) ) ) ).
% Collect_mono
thf(fact_135_subset__trans,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v,C2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ B )
=> ( ( ord_le7336532860387713383od_v_v @ B @ C2 )
=> ( ord_le7336532860387713383od_v_v @ A @ C2 ) ) ) ).
% subset_trans
thf(fact_136_subset__trans,axiom,
! [A: set_v,B: set_v,C2: set_v] :
( ( ord_less_eq_set_v @ A @ B )
=> ( ( ord_less_eq_set_v @ B @ C2 )
=> ( ord_less_eq_set_v @ A @ C2 ) ) ) ).
% subset_trans
thf(fact_137_set__eq__subset,axiom,
( ( ^ [Y4: set_Product_prod_v_v,Z4: set_Product_prod_v_v] : ( Y4 = Z4 ) )
= ( ^ [A8: set_Product_prod_v_v,B6: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A8 @ B6 )
& ( ord_le7336532860387713383od_v_v @ B6 @ A8 ) ) ) ) ).
% set_eq_subset
thf(fact_138_set__eq__subset,axiom,
( ( ^ [Y4: set_v,Z4: set_v] : ( Y4 = Z4 ) )
= ( ^ [A8: set_v,B6: set_v] :
( ( ord_less_eq_set_v @ A8 @ B6 )
& ( ord_less_eq_set_v @ B6 @ A8 ) ) ) ) ).
% set_eq_subset
thf(fact_139_Collect__mono__iff,axiom,
! [P: set_v > $o,Q: set_v > $o] :
( ( ord_le5216385588623774835_set_v @ ( collect_set_v @ P ) @ ( collect_set_v @ Q ) )
= ( ! [X2: set_v] :
( ( P @ X2 )
=> ( Q @ X2 ) ) ) ) ).
% Collect_mono_iff
thf(fact_140_Collect__mono__iff,axiom,
! [P: product_prod_v_v > $o,Q: product_prod_v_v > $o] :
( ( ord_le7336532860387713383od_v_v @ ( collec140062887454715474od_v_v @ P ) @ ( collec140062887454715474od_v_v @ Q ) )
= ( ! [X2: product_prod_v_v] :
( ( P @ X2 )
=> ( Q @ X2 ) ) ) ) ).
% Collect_mono_iff
thf(fact_141_Collect__mono__iff,axiom,
! [P: v > $o,Q: v > $o] :
( ( ord_less_eq_set_v @ ( collect_v @ P ) @ ( collect_v @ Q ) )
= ( ! [X2: v] :
( ( P @ X2 )
=> ( Q @ X2 ) ) ) ) ).
% Collect_mono_iff
thf(fact_142_pre__dfss__pre__dfs,axiom,
! [V3: v,E: sCC_Bl1394983891496994913t_unit,W: v] :
( ( sCC_Bl1748261141445803503t_unit @ successors @ V3 @ E )
=> ( ~ ( member_v @ W @ ( sCC_Bl4645233313691564917t_unit @ E ) )
=> ( ( member_v @ W @ ( successors @ V3 ) )
=> ( sCC_Bl36166008131615352t_unit @ successors @ W @ E ) ) ) ) ).
% pre_dfss_pre_dfs
thf(fact_143_stack__class,axiom,
! [E: sCC_Bl1394983891496994913t_unit,N: v,M: v] :
( ( sCC_Bl9196236973127232072t_unit @ successors @ E )
=> ( ( member_v @ N @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E ) ) )
=> ( ( member_v @ M @ ( sCC_Bl1280885523602775798t_unit @ E @ N ) )
=> ( member_v @ M @ ( minus_minus_set_v @ ( sCC_Bl4645233313691564917t_unit @ E ) @ ( sCC_Bl157864678168468314t_unit @ E ) ) ) ) ) ) ).
% stack_class
thf(fact_144_subset__empty,axiom,
! [A: set_set_v] :
( ( ord_le5216385588623774835_set_v @ A @ bot_bot_set_set_v )
= ( A = bot_bot_set_set_v ) ) ).
% subset_empty
thf(fact_145_subset__empty,axiom,
! [A: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ bot_bo723834152578015283od_v_v )
= ( A = bot_bo723834152578015283od_v_v ) ) ).
% subset_empty
thf(fact_146_subset__empty,axiom,
! [A: set_v] :
( ( ord_less_eq_set_v @ A @ bot_bot_set_v )
= ( A = bot_bot_set_v ) ) ).
% subset_empty
thf(fact_147_empty__subsetI,axiom,
! [A: set_set_v] : ( ord_le5216385588623774835_set_v @ bot_bot_set_set_v @ A ) ).
% empty_subsetI
thf(fact_148_empty__subsetI,axiom,
! [A: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ bot_bo723834152578015283od_v_v @ A ) ).
% empty_subsetI
thf(fact_149_empty__subsetI,axiom,
! [A: set_v] : ( ord_less_eq_set_v @ bot_bot_set_v @ A ) ).
% empty_subsetI
thf(fact_150_notempty,axiom,
( ( sCC_Bl8828226123343373779t_unit @ e2 )
= ( cons_v @ v2 @ ( sCC_Bl8828226123343373779t_unit @ e ) ) ) ).
% notempty
thf(fact_151_insert__subset,axiom,
! [X: set_v,A: set_set_v,B: set_set_v] :
( ( ord_le5216385588623774835_set_v @ ( insert_set_v @ X @ A ) @ B )
= ( ( member_set_v @ X @ B )
& ( ord_le5216385588623774835_set_v @ A @ B ) ) ) ).
% insert_subset
thf(fact_152_insert__subset,axiom,
! [X: product_prod_v_v,A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ ( insert1338601472111419319od_v_v @ X @ A ) @ B )
= ( ( member7453568604450474000od_v_v @ X @ B )
& ( ord_le7336532860387713383od_v_v @ A @ B ) ) ) ).
% insert_subset
thf(fact_153_insert__subset,axiom,
! [X: v,A: set_v,B: set_v] :
( ( ord_less_eq_set_v @ ( insert_v @ X @ A ) @ B )
= ( ( member_v @ X @ B )
& ( ord_less_eq_set_v @ A @ B ) ) ) ).
% insert_subset
thf(fact_154_Un__subset__iff,axiom,
! [A: set_set_v,B: set_set_v,C2: set_set_v] :
( ( ord_le5216385588623774835_set_v @ ( sup_sup_set_set_v @ A @ B ) @ C2 )
= ( ( ord_le5216385588623774835_set_v @ A @ C2 )
& ( ord_le5216385588623774835_set_v @ B @ C2 ) ) ) ).
% Un_subset_iff
thf(fact_155_Un__subset__iff,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v,C2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ ( sup_su414716646722978715od_v_v @ A @ B ) @ C2 )
= ( ( ord_le7336532860387713383od_v_v @ A @ C2 )
& ( ord_le7336532860387713383od_v_v @ B @ C2 ) ) ) ).
% Un_subset_iff
thf(fact_156_Un__subset__iff,axiom,
! [A: set_v,B: set_v,C2: set_v] :
( ( ord_less_eq_set_v @ ( sup_sup_set_v @ A @ B ) @ C2 )
= ( ( ord_less_eq_set_v @ A @ C2 )
& ( ord_less_eq_set_v @ B @ C2 ) ) ) ).
% Un_subset_iff
thf(fact_157_cst_H,axiom,
( ( sCC_Bl9201514103433284750t_unit @ e2 )
= ( cons_v @ v2 @ ( sCC_Bl9201514103433284750t_unit @ e ) ) ) ).
% cst'
thf(fact_158_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_159_reachable__end_Osimps,axiom,
! [A1: v,A2: v] :
( ( sCC_Bl770211535891879572_end_v @ successors @ A1 @ A2 )
= ( ? [X2: v] :
( ( A1 = X2 )
& ( A2 = X2 ) )
| ? [X2: v,Y3: v,Z2: v] :
( ( A1 = X2 )
& ( A2 = Z2 )
& ( sCC_Bl770211535891879572_end_v @ successors @ X2 @ Y3 )
& ( member_v @ Z2 @ ( successors @ Y3 ) ) ) ) ) ).
% reachable_end.simps
thf(fact_160_reachable__end_Ocases,axiom,
! [A1: v,A2: v] :
( ( sCC_Bl770211535891879572_end_v @ successors @ A1 @ A2 )
=> ( ( A2 != A1 )
=> ~ ! [Y2: v] :
( ( sCC_Bl770211535891879572_end_v @ successors @ A1 @ Y2 )
=> ~ ( member_v @ A2 @ ( successors @ Y2 ) ) ) ) ) ).
% reachable_end.cases
thf(fact_161_re__refl,axiom,
! [X: v] : ( sCC_Bl770211535891879572_end_v @ successors @ X @ X ) ).
% re_refl
thf(fact_162_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_163_ra__add__edge,axiom,
! [X: v,Y: v,E4: set_Product_prod_v_v,V3: v,W: v] :
( ( sCC_Bl4291963740693775144ding_v @ successors @ X @ Y @ E4 )
=> ( ( sCC_Bl4291963740693775144ding_v @ successors @ X @ Y @ ( sup_su414716646722978715od_v_v @ E4 @ ( insert1338601472111419319od_v_v @ ( product_Pair_v_v @ V3 @ W ) @ bot_bo723834152578015283od_v_v ) ) )
| ( ( sCC_Bl4291963740693775144ding_v @ successors @ X @ V3 @ ( sup_su414716646722978715od_v_v @ E4 @ ( insert1338601472111419319od_v_v @ ( product_Pair_v_v @ V3 @ W ) @ bot_bo723834152578015283od_v_v ) ) )
& ( sCC_Bl4291963740693775144ding_v @ successors @ W @ Y @ ( sup_su414716646722978715od_v_v @ E4 @ ( insert1338601472111419319od_v_v @ ( product_Pair_v_v @ V3 @ W ) @ bot_bo723834152578015283od_v_v ) ) ) ) ) ) ).
% ra_add_edge
thf(fact_164_empty__Collect__eq,axiom,
! [P: product_prod_v_v > $o] :
( ( bot_bo723834152578015283od_v_v
= ( collec140062887454715474od_v_v @ P ) )
= ( ! [X2: product_prod_v_v] :
~ ( P @ X2 ) ) ) ).
% empty_Collect_eq
thf(fact_165_empty__Collect__eq,axiom,
! [P: set_v > $o] :
( ( bot_bot_set_set_v
= ( collect_set_v @ P ) )
= ( ! [X2: set_v] :
~ ( P @ X2 ) ) ) ).
% empty_Collect_eq
thf(fact_166_empty__Collect__eq,axiom,
! [P: v > $o] :
( ( bot_bot_set_v
= ( collect_v @ P ) )
= ( ! [X2: v] :
~ ( P @ X2 ) ) ) ).
% empty_Collect_eq
thf(fact_167_Collect__empty__eq,axiom,
! [P: product_prod_v_v > $o] :
( ( ( collec140062887454715474od_v_v @ P )
= bot_bo723834152578015283od_v_v )
= ( ! [X2: product_prod_v_v] :
~ ( P @ X2 ) ) ) ).
% Collect_empty_eq
thf(fact_168_Collect__empty__eq,axiom,
! [P: set_v > $o] :
( ( ( collect_set_v @ P )
= bot_bot_set_set_v )
= ( ! [X2: set_v] :
~ ( P @ X2 ) ) ) ).
% Collect_empty_eq
thf(fact_169_Collect__empty__eq,axiom,
! [P: v > $o] :
( ( ( collect_v @ P )
= bot_bot_set_v )
= ( ! [X2: v] :
~ ( P @ X2 ) ) ) ).
% Collect_empty_eq
thf(fact_170_all__not__in__conv,axiom,
! [A: set_Product_prod_v_v] :
( ( ! [X2: product_prod_v_v] :
~ ( member7453568604450474000od_v_v @ X2 @ A ) )
= ( A = bot_bo723834152578015283od_v_v ) ) ).
% all_not_in_conv
thf(fact_171_all__not__in__conv,axiom,
! [A: set_set_v] :
( ( ! [X2: set_v] :
~ ( member_set_v @ X2 @ A ) )
= ( A = bot_bot_set_set_v ) ) ).
% all_not_in_conv
thf(fact_172_all__not__in__conv,axiom,
! [A: set_v] :
( ( ! [X2: v] :
~ ( member_v @ X2 @ A ) )
= ( A = bot_bot_set_v ) ) ).
% all_not_in_conv
thf(fact_173_empty__iff,axiom,
! [C: product_prod_v_v] :
~ ( member7453568604450474000od_v_v @ C @ bot_bo723834152578015283od_v_v ) ).
% empty_iff
thf(fact_174_empty__iff,axiom,
! [C: set_v] :
~ ( member_set_v @ C @ bot_bot_set_set_v ) ).
% empty_iff
thf(fact_175_empty__iff,axiom,
! [C: v] :
~ ( member_v @ C @ bot_bot_set_v ) ).
% empty_iff
thf(fact_176_insert__absorb2,axiom,
! [X: product_prod_v_v,A: set_Product_prod_v_v] :
( ( insert1338601472111419319od_v_v @ X @ ( insert1338601472111419319od_v_v @ X @ A ) )
= ( insert1338601472111419319od_v_v @ X @ A ) ) ).
% insert_absorb2
thf(fact_177_insert__absorb2,axiom,
! [X: set_v,A: set_set_v] :
( ( insert_set_v @ X @ ( insert_set_v @ X @ A ) )
= ( insert_set_v @ X @ A ) ) ).
% insert_absorb2
thf(fact_178_insert__absorb2,axiom,
! [X: v,A: set_v] :
( ( insert_v @ X @ ( insert_v @ X @ A ) )
= ( insert_v @ X @ A ) ) ).
% insert_absorb2
thf(fact_179_insert__iff,axiom,
! [A4: set_v,B2: set_v,A: set_set_v] :
( ( member_set_v @ A4 @ ( insert_set_v @ B2 @ A ) )
= ( ( A4 = B2 )
| ( member_set_v @ A4 @ A ) ) ) ).
% insert_iff
thf(fact_180_insert__iff,axiom,
! [A4: v,B2: v,A: set_v] :
( ( member_v @ A4 @ ( insert_v @ B2 @ A ) )
= ( ( A4 = B2 )
| ( member_v @ A4 @ A ) ) ) ).
% insert_iff
thf(fact_181_insert__iff,axiom,
! [A4: product_prod_v_v,B2: product_prod_v_v,A: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ A4 @ ( insert1338601472111419319od_v_v @ B2 @ A ) )
= ( ( A4 = B2 )
| ( member7453568604450474000od_v_v @ A4 @ A ) ) ) ).
% insert_iff
thf(fact_182_insertCI,axiom,
! [A4: set_v,B: set_set_v,B2: set_v] :
( ( ~ ( member_set_v @ A4 @ B )
=> ( A4 = B2 ) )
=> ( member_set_v @ A4 @ ( insert_set_v @ B2 @ B ) ) ) ).
% insertCI
thf(fact_183_insertCI,axiom,
! [A4: v,B: set_v,B2: v] :
( ( ~ ( member_v @ A4 @ B )
=> ( A4 = B2 ) )
=> ( member_v @ A4 @ ( insert_v @ B2 @ B ) ) ) ).
% insertCI
thf(fact_184_insertCI,axiom,
! [A4: product_prod_v_v,B: set_Product_prod_v_v,B2: product_prod_v_v] :
( ( ~ ( member7453568604450474000od_v_v @ A4 @ B )
=> ( A4 = B2 ) )
=> ( member7453568604450474000od_v_v @ A4 @ ( insert1338601472111419319od_v_v @ B2 @ B ) ) ) ).
% insertCI
thf(fact_185_Un__iff,axiom,
! [C: product_prod_v_v,A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ C @ ( sup_su414716646722978715od_v_v @ A @ B ) )
= ( ( member7453568604450474000od_v_v @ C @ A )
| ( member7453568604450474000od_v_v @ C @ B ) ) ) ).
% Un_iff
thf(fact_186_Un__iff,axiom,
! [C: v,A: set_v,B: set_v] :
( ( member_v @ C @ ( sup_sup_set_v @ A @ B ) )
= ( ( member_v @ C @ A )
| ( member_v @ C @ B ) ) ) ).
% Un_iff
thf(fact_187_Un__iff,axiom,
! [C: set_v,A: set_set_v,B: set_set_v] :
( ( member_set_v @ C @ ( sup_sup_set_set_v @ A @ B ) )
= ( ( member_set_v @ C @ A )
| ( member_set_v @ C @ B ) ) ) ).
% Un_iff
thf(fact_188_UnCI,axiom,
! [C: product_prod_v_v,B: set_Product_prod_v_v,A: set_Product_prod_v_v] :
( ( ~ ( member7453568604450474000od_v_v @ C @ B )
=> ( member7453568604450474000od_v_v @ C @ A ) )
=> ( member7453568604450474000od_v_v @ C @ ( sup_su414716646722978715od_v_v @ A @ B ) ) ) ).
% UnCI
thf(fact_189_UnCI,axiom,
! [C: v,B: set_v,A: set_v] :
( ( ~ ( member_v @ C @ B )
=> ( member_v @ C @ A ) )
=> ( member_v @ C @ ( sup_sup_set_v @ A @ B ) ) ) ).
% UnCI
thf(fact_190_UnCI,axiom,
! [C: set_v,B: set_set_v,A: set_set_v] :
( ( ~ ( member_set_v @ C @ B )
=> ( member_set_v @ C @ A ) )
=> ( member_set_v @ C @ ( sup_sup_set_set_v @ A @ B ) ) ) ).
% UnCI
thf(fact_191_DiffI,axiom,
! [C: product_prod_v_v,A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ C @ A )
=> ( ~ ( member7453568604450474000od_v_v @ C @ B )
=> ( member7453568604450474000od_v_v @ C @ ( minus_4183494784930505774od_v_v @ A @ B ) ) ) ) ).
% DiffI
thf(fact_192_DiffI,axiom,
! [C: v,A: set_v,B: set_v] :
( ( member_v @ C @ A )
=> ( ~ ( member_v @ C @ B )
=> ( member_v @ C @ ( minus_minus_set_v @ A @ B ) ) ) ) ).
% DiffI
thf(fact_193_Diff__iff,axiom,
! [C: product_prod_v_v,A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ C @ ( minus_4183494784930505774od_v_v @ A @ B ) )
= ( ( member7453568604450474000od_v_v @ C @ A )
& ~ ( member7453568604450474000od_v_v @ C @ B ) ) ) ).
% Diff_iff
thf(fact_194_Diff__iff,axiom,
! [C: v,A: set_v,B: set_v] :
( ( member_v @ C @ ( minus_minus_set_v @ A @ B ) )
= ( ( member_v @ C @ A )
& ~ ( member_v @ C @ B ) ) ) ).
% Diff_iff
thf(fact_195_Diff__idemp,axiom,
! [A: set_v,B: set_v] :
( ( minus_minus_set_v @ ( minus_minus_set_v @ A @ B ) @ B )
= ( minus_minus_set_v @ A @ B ) ) ).
% Diff_idemp
thf(fact_196_avoiding__explored,axiom,
! [E: sCC_Bl1394983891496994913t_unit,X: v,Y: v,E4: set_Product_prod_v_v,W: v,V3: v] :
( ( sCC_Bl9196236973127232072t_unit @ successors @ E )
=> ( ( sCC_Bl4291963740693775144ding_v @ successors @ X @ Y @ E4 )
=> ( ~ ( member_v @ Y @ ( sCC_Bl157864678168468314t_unit @ E ) )
=> ( ( member_v @ W @ ( sCC_Bl157864678168468314t_unit @ E ) )
=> ( sCC_Bl4291963740693775144ding_v @ successors @ X @ Y @ ( sup_su414716646722978715od_v_v @ E4 @ ( insert1338601472111419319od_v_v @ ( product_Pair_v_v @ V3 @ W ) @ bot_bo723834152578015283od_v_v ) ) ) ) ) ) ) ).
% avoiding_explored
thf(fact_197_singletonI,axiom,
! [A4: product_prod_v_v] : ( member7453568604450474000od_v_v @ A4 @ ( insert1338601472111419319od_v_v @ A4 @ bot_bo723834152578015283od_v_v ) ) ).
% singletonI
thf(fact_198_singletonI,axiom,
! [A4: set_v] : ( member_set_v @ A4 @ ( insert_set_v @ A4 @ bot_bot_set_set_v ) ) ).
% singletonI
thf(fact_199_singletonI,axiom,
! [A4: v] : ( member_v @ A4 @ ( insert_v @ A4 @ bot_bot_set_v ) ) ).
% singletonI
thf(fact_200_Un__empty,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ( sup_su414716646722978715od_v_v @ A @ B )
= bot_bo723834152578015283od_v_v )
= ( ( A = bot_bo723834152578015283od_v_v )
& ( B = bot_bo723834152578015283od_v_v ) ) ) ).
% Un_empty
thf(fact_201_Un__empty,axiom,
! [A: set_set_v,B: set_set_v] :
( ( ( sup_sup_set_set_v @ A @ B )
= bot_bot_set_set_v )
= ( ( A = bot_bot_set_set_v )
& ( B = bot_bot_set_set_v ) ) ) ).
% Un_empty
thf(fact_202_Un__empty,axiom,
! [A: set_v,B: set_v] :
( ( ( sup_sup_set_v @ A @ B )
= bot_bot_set_v )
= ( ( A = bot_bot_set_v )
& ( B = bot_bot_set_v ) ) ) ).
% Un_empty
thf(fact_203_Diff__empty,axiom,
! [A: set_Product_prod_v_v] :
( ( minus_4183494784930505774od_v_v @ A @ bot_bo723834152578015283od_v_v )
= A ) ).
% Diff_empty
thf(fact_204_Diff__empty,axiom,
! [A: set_set_v] :
( ( minus_7228012346218142266_set_v @ A @ bot_bot_set_set_v )
= A ) ).
% Diff_empty
thf(fact_205_Diff__empty,axiom,
! [A: set_v] :
( ( minus_minus_set_v @ A @ bot_bot_set_v )
= A ) ).
% Diff_empty
thf(fact_206_empty__Diff,axiom,
! [A: set_Product_prod_v_v] :
( ( minus_4183494784930505774od_v_v @ bot_bo723834152578015283od_v_v @ A )
= bot_bo723834152578015283od_v_v ) ).
% empty_Diff
thf(fact_207_empty__Diff,axiom,
! [A: set_set_v] :
( ( minus_7228012346218142266_set_v @ bot_bot_set_set_v @ A )
= bot_bot_set_set_v ) ).
% empty_Diff
thf(fact_208_empty__Diff,axiom,
! [A: set_v] :
( ( minus_minus_set_v @ bot_bot_set_v @ A )
= bot_bot_set_v ) ).
% empty_Diff
thf(fact_209_Diff__cancel,axiom,
! [A: set_Product_prod_v_v] :
( ( minus_4183494784930505774od_v_v @ A @ A )
= bot_bo723834152578015283od_v_v ) ).
% Diff_cancel
thf(fact_210_Diff__cancel,axiom,
! [A: set_set_v] :
( ( minus_7228012346218142266_set_v @ A @ A )
= bot_bot_set_set_v ) ).
% Diff_cancel
thf(fact_211_Diff__cancel,axiom,
! [A: set_v] :
( ( minus_minus_set_v @ A @ A )
= bot_bot_set_v ) ).
% Diff_cancel
thf(fact_212_Un__insert__right,axiom,
! [A: set_Product_prod_v_v,A4: product_prod_v_v,B: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ A @ ( insert1338601472111419319od_v_v @ A4 @ B ) )
= ( insert1338601472111419319od_v_v @ A4 @ ( sup_su414716646722978715od_v_v @ A @ B ) ) ) ).
% Un_insert_right
thf(fact_213_Un__insert__right,axiom,
! [A: set_v,A4: v,B: set_v] :
( ( sup_sup_set_v @ A @ ( insert_v @ A4 @ B ) )
= ( insert_v @ A4 @ ( sup_sup_set_v @ A @ B ) ) ) ).
% Un_insert_right
thf(fact_214_Un__insert__right,axiom,
! [A: set_set_v,A4: set_v,B: set_set_v] :
( ( sup_sup_set_set_v @ A @ ( insert_set_v @ A4 @ B ) )
= ( insert_set_v @ A4 @ ( sup_sup_set_set_v @ A @ B ) ) ) ).
% Un_insert_right
thf(fact_215_Un__insert__left,axiom,
! [A4: product_prod_v_v,B: set_Product_prod_v_v,C2: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ ( insert1338601472111419319od_v_v @ A4 @ B ) @ C2 )
= ( insert1338601472111419319od_v_v @ A4 @ ( sup_su414716646722978715od_v_v @ B @ C2 ) ) ) ).
% Un_insert_left
thf(fact_216_Un__insert__left,axiom,
! [A4: v,B: set_v,C2: set_v] :
( ( sup_sup_set_v @ ( insert_v @ A4 @ B ) @ C2 )
= ( insert_v @ A4 @ ( sup_sup_set_v @ B @ C2 ) ) ) ).
% Un_insert_left
thf(fact_217_Un__insert__left,axiom,
! [A4: set_v,B: set_set_v,C2: set_set_v] :
( ( sup_sup_set_set_v @ ( insert_set_v @ A4 @ B ) @ C2 )
= ( insert_set_v @ A4 @ ( sup_sup_set_set_v @ B @ C2 ) ) ) ).
% Un_insert_left
thf(fact_218_Diff__insert0,axiom,
! [X: set_v,A: set_set_v,B: set_set_v] :
( ~ ( member_set_v @ X @ A )
=> ( ( minus_7228012346218142266_set_v @ A @ ( insert_set_v @ X @ B ) )
= ( minus_7228012346218142266_set_v @ A @ B ) ) ) ).
% Diff_insert0
thf(fact_219_Diff__insert0,axiom,
! [X: product_prod_v_v,A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ~ ( member7453568604450474000od_v_v @ X @ A )
=> ( ( minus_4183494784930505774od_v_v @ A @ ( insert1338601472111419319od_v_v @ X @ B ) )
= ( minus_4183494784930505774od_v_v @ A @ B ) ) ) ).
% Diff_insert0
thf(fact_220_Diff__insert0,axiom,
! [X: v,A: set_v,B: set_v] :
( ~ ( member_v @ X @ A )
=> ( ( minus_minus_set_v @ A @ ( insert_v @ X @ B ) )
= ( minus_minus_set_v @ A @ B ) ) ) ).
% Diff_insert0
thf(fact_221_insert__Diff1,axiom,
! [X: set_v,B: set_set_v,A: set_set_v] :
( ( member_set_v @ X @ B )
=> ( ( minus_7228012346218142266_set_v @ ( insert_set_v @ X @ A ) @ B )
= ( minus_7228012346218142266_set_v @ A @ B ) ) ) ).
% insert_Diff1
thf(fact_222_insert__Diff1,axiom,
! [X: product_prod_v_v,B: set_Product_prod_v_v,A: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X @ B )
=> ( ( minus_4183494784930505774od_v_v @ ( insert1338601472111419319od_v_v @ X @ A ) @ B )
= ( minus_4183494784930505774od_v_v @ A @ B ) ) ) ).
% insert_Diff1
thf(fact_223_insert__Diff1,axiom,
! [X: v,B: set_v,A: set_v] :
( ( member_v @ X @ B )
=> ( ( minus_minus_set_v @ ( insert_v @ X @ A ) @ B )
= ( minus_minus_set_v @ A @ B ) ) ) ).
% insert_Diff1
thf(fact_224_Un__Diff__cancel,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ A @ ( minus_4183494784930505774od_v_v @ B @ A ) )
= ( sup_su414716646722978715od_v_v @ A @ B ) ) ).
% Un_Diff_cancel
thf(fact_225_Un__Diff__cancel,axiom,
! [A: set_set_v,B: set_set_v] :
( ( sup_sup_set_set_v @ A @ ( minus_7228012346218142266_set_v @ B @ A ) )
= ( sup_sup_set_set_v @ A @ B ) ) ).
% Un_Diff_cancel
thf(fact_226_Un__Diff__cancel,axiom,
! [A: set_v,B: set_v] :
( ( sup_sup_set_v @ A @ ( minus_minus_set_v @ B @ A ) )
= ( sup_sup_set_v @ A @ B ) ) ).
% Un_Diff_cancel
thf(fact_227_Un__Diff__cancel2,axiom,
! [B: set_Product_prod_v_v,A: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ ( minus_4183494784930505774od_v_v @ B @ A ) @ A )
= ( sup_su414716646722978715od_v_v @ B @ A ) ) ).
% Un_Diff_cancel2
thf(fact_228_Un__Diff__cancel2,axiom,
! [B: set_set_v,A: set_set_v] :
( ( sup_sup_set_set_v @ ( minus_7228012346218142266_set_v @ B @ A ) @ A )
= ( sup_sup_set_set_v @ B @ A ) ) ).
% Un_Diff_cancel2
thf(fact_229_Un__Diff__cancel2,axiom,
! [B: set_v,A: set_v] :
( ( sup_sup_set_v @ ( minus_minus_set_v @ B @ A ) @ A )
= ( sup_sup_set_v @ B @ A ) ) ).
% Un_Diff_cancel2
thf(fact_230_singleton__insert__inj__eq_H,axiom,
! [A4: set_v,A: set_set_v,B2: set_v] :
( ( ( insert_set_v @ A4 @ A )
= ( insert_set_v @ B2 @ bot_bot_set_set_v ) )
= ( ( A4 = B2 )
& ( ord_le5216385588623774835_set_v @ A @ ( insert_set_v @ B2 @ bot_bot_set_set_v ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_231_singleton__insert__inj__eq_H,axiom,
! [A4: product_prod_v_v,A: set_Product_prod_v_v,B2: product_prod_v_v] :
( ( ( insert1338601472111419319od_v_v @ A4 @ A )
= ( insert1338601472111419319od_v_v @ B2 @ bot_bo723834152578015283od_v_v ) )
= ( ( A4 = B2 )
& ( ord_le7336532860387713383od_v_v @ A @ ( insert1338601472111419319od_v_v @ B2 @ bot_bo723834152578015283od_v_v ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_232_singleton__insert__inj__eq_H,axiom,
! [A4: v,A: set_v,B2: v] :
( ( ( insert_v @ A4 @ A )
= ( insert_v @ B2 @ bot_bot_set_v ) )
= ( ( A4 = B2 )
& ( ord_less_eq_set_v @ A @ ( insert_v @ B2 @ bot_bot_set_v ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_233_singleton__insert__inj__eq,axiom,
! [B2: set_v,A4: set_v,A: set_set_v] :
( ( ( insert_set_v @ B2 @ bot_bot_set_set_v )
= ( insert_set_v @ A4 @ A ) )
= ( ( A4 = B2 )
& ( ord_le5216385588623774835_set_v @ A @ ( insert_set_v @ B2 @ bot_bot_set_set_v ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_234_singleton__insert__inj__eq,axiom,
! [B2: product_prod_v_v,A4: product_prod_v_v,A: set_Product_prod_v_v] :
( ( ( insert1338601472111419319od_v_v @ B2 @ bot_bo723834152578015283od_v_v )
= ( insert1338601472111419319od_v_v @ A4 @ A ) )
= ( ( A4 = B2 )
& ( ord_le7336532860387713383od_v_v @ A @ ( insert1338601472111419319od_v_v @ B2 @ bot_bo723834152578015283od_v_v ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_235_singleton__insert__inj__eq,axiom,
! [B2: v,A4: v,A: set_v] :
( ( ( insert_v @ B2 @ bot_bot_set_v )
= ( insert_v @ A4 @ A ) )
= ( ( A4 = B2 )
& ( ord_less_eq_set_v @ A @ ( insert_v @ B2 @ bot_bot_set_v ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_236_Diff__eq__empty__iff,axiom,
! [A: set_set_v,B: set_set_v] :
( ( ( minus_7228012346218142266_set_v @ A @ B )
= bot_bot_set_set_v )
= ( ord_le5216385588623774835_set_v @ A @ B ) ) ).
% Diff_eq_empty_iff
thf(fact_237_Diff__eq__empty__iff,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ( minus_4183494784930505774od_v_v @ A @ B )
= bot_bo723834152578015283od_v_v )
= ( ord_le7336532860387713383od_v_v @ A @ B ) ) ).
% Diff_eq_empty_iff
thf(fact_238_Diff__eq__empty__iff,axiom,
! [A: set_v,B: set_v] :
( ( ( minus_minus_set_v @ A @ B )
= bot_bot_set_v )
= ( ord_less_eq_set_v @ A @ B ) ) ).
% Diff_eq_empty_iff
thf(fact_239_insert__Diff__single,axiom,
! [A4: product_prod_v_v,A: set_Product_prod_v_v] :
( ( insert1338601472111419319od_v_v @ A4 @ ( minus_4183494784930505774od_v_v @ A @ ( insert1338601472111419319od_v_v @ A4 @ bot_bo723834152578015283od_v_v ) ) )
= ( insert1338601472111419319od_v_v @ A4 @ A ) ) ).
% insert_Diff_single
thf(fact_240_insert__Diff__single,axiom,
! [A4: set_v,A: set_set_v] :
( ( insert_set_v @ A4 @ ( minus_7228012346218142266_set_v @ A @ ( insert_set_v @ A4 @ bot_bot_set_set_v ) ) )
= ( insert_set_v @ A4 @ A ) ) ).
% insert_Diff_single
thf(fact_241_insert__Diff__single,axiom,
! [A4: v,A: set_v] :
( ( insert_v @ A4 @ ( minus_minus_set_v @ A @ ( insert_v @ A4 @ bot_bot_set_v ) ) )
= ( insert_v @ A4 @ A ) ) ).
% insert_Diff_single
thf(fact_242_graph__axioms,axiom,
sCC_Bloemen_graph_v @ vertices @ successors ).
% graph_axioms
thf(fact_243_DiffE,axiom,
! [C: product_prod_v_v,A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ C @ ( minus_4183494784930505774od_v_v @ A @ B ) )
=> ~ ( ( member7453568604450474000od_v_v @ C @ A )
=> ( member7453568604450474000od_v_v @ C @ B ) ) ) ).
% DiffE
thf(fact_244_DiffE,axiom,
! [C: v,A: set_v,B: set_v] :
( ( member_v @ C @ ( minus_minus_set_v @ A @ B ) )
=> ~ ( ( member_v @ C @ A )
=> ( member_v @ C @ B ) ) ) ).
% DiffE
thf(fact_245_DiffD1,axiom,
! [C: product_prod_v_v,A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ C @ ( minus_4183494784930505774od_v_v @ A @ B ) )
=> ( member7453568604450474000od_v_v @ C @ A ) ) ).
% DiffD1
thf(fact_246_DiffD1,axiom,
! [C: v,A: set_v,B: set_v] :
( ( member_v @ C @ ( minus_minus_set_v @ A @ B ) )
=> ( member_v @ C @ A ) ) ).
% DiffD1
thf(fact_247_DiffD2,axiom,
! [C: product_prod_v_v,A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ C @ ( minus_4183494784930505774od_v_v @ A @ B ) )
=> ~ ( member7453568604450474000od_v_v @ C @ B ) ) ).
% DiffD2
thf(fact_248_DiffD2,axiom,
! [C: v,A: set_v,B: set_v] :
( ( member_v @ C @ ( minus_minus_set_v @ A @ B ) )
=> ~ ( member_v @ C @ B ) ) ).
% DiffD2
thf(fact_249_Un__Diff,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v,C2: set_Product_prod_v_v] :
( ( minus_4183494784930505774od_v_v @ ( sup_su414716646722978715od_v_v @ A @ B ) @ C2 )
= ( sup_su414716646722978715od_v_v @ ( minus_4183494784930505774od_v_v @ A @ C2 ) @ ( minus_4183494784930505774od_v_v @ B @ C2 ) ) ) ).
% Un_Diff
thf(fact_250_Un__Diff,axiom,
! [A: set_set_v,B: set_set_v,C2: set_set_v] :
( ( minus_7228012346218142266_set_v @ ( sup_sup_set_set_v @ A @ B ) @ C2 )
= ( sup_sup_set_set_v @ ( minus_7228012346218142266_set_v @ A @ C2 ) @ ( minus_7228012346218142266_set_v @ B @ C2 ) ) ) ).
% Un_Diff
thf(fact_251_Un__Diff,axiom,
! [A: set_v,B: set_v,C2: set_v] :
( ( minus_minus_set_v @ ( sup_sup_set_v @ A @ B ) @ C2 )
= ( sup_sup_set_v @ ( minus_minus_set_v @ A @ C2 ) @ ( minus_minus_set_v @ B @ C2 ) ) ) ).
% Un_Diff
thf(fact_252_Diff__insert,axiom,
! [A: set_Product_prod_v_v,A4: product_prod_v_v,B: set_Product_prod_v_v] :
( ( minus_4183494784930505774od_v_v @ A @ ( insert1338601472111419319od_v_v @ A4 @ B ) )
= ( minus_4183494784930505774od_v_v @ ( minus_4183494784930505774od_v_v @ A @ B ) @ ( insert1338601472111419319od_v_v @ A4 @ bot_bo723834152578015283od_v_v ) ) ) ).
% Diff_insert
thf(fact_253_Diff__insert,axiom,
! [A: set_set_v,A4: set_v,B: set_set_v] :
( ( minus_7228012346218142266_set_v @ A @ ( insert_set_v @ A4 @ B ) )
= ( minus_7228012346218142266_set_v @ ( minus_7228012346218142266_set_v @ A @ B ) @ ( insert_set_v @ A4 @ bot_bot_set_set_v ) ) ) ).
% Diff_insert
thf(fact_254_Diff__insert,axiom,
! [A: set_v,A4: v,B: set_v] :
( ( minus_minus_set_v @ A @ ( insert_v @ A4 @ B ) )
= ( minus_minus_set_v @ ( minus_minus_set_v @ A @ B ) @ ( insert_v @ A4 @ bot_bot_set_v ) ) ) ).
% Diff_insert
thf(fact_255_insert__Diff,axiom,
! [A4: product_prod_v_v,A: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ A4 @ A )
=> ( ( insert1338601472111419319od_v_v @ A4 @ ( minus_4183494784930505774od_v_v @ A @ ( insert1338601472111419319od_v_v @ A4 @ bot_bo723834152578015283od_v_v ) ) )
= A ) ) ).
% insert_Diff
thf(fact_256_insert__Diff,axiom,
! [A4: set_v,A: set_set_v] :
( ( member_set_v @ A4 @ A )
=> ( ( insert_set_v @ A4 @ ( minus_7228012346218142266_set_v @ A @ ( insert_set_v @ A4 @ bot_bot_set_set_v ) ) )
= A ) ) ).
% insert_Diff
thf(fact_257_insert__Diff,axiom,
! [A4: v,A: set_v] :
( ( member_v @ A4 @ A )
=> ( ( insert_v @ A4 @ ( minus_minus_set_v @ A @ ( insert_v @ A4 @ bot_bot_set_v ) ) )
= A ) ) ).
% insert_Diff
thf(fact_258_Diff__insert2,axiom,
! [A: set_Product_prod_v_v,A4: product_prod_v_v,B: set_Product_prod_v_v] :
( ( minus_4183494784930505774od_v_v @ A @ ( insert1338601472111419319od_v_v @ A4 @ B ) )
= ( minus_4183494784930505774od_v_v @ ( minus_4183494784930505774od_v_v @ A @ ( insert1338601472111419319od_v_v @ A4 @ bot_bo723834152578015283od_v_v ) ) @ B ) ) ).
% Diff_insert2
thf(fact_259_Diff__insert2,axiom,
! [A: set_set_v,A4: set_v,B: set_set_v] :
( ( minus_7228012346218142266_set_v @ A @ ( insert_set_v @ A4 @ B ) )
= ( minus_7228012346218142266_set_v @ ( minus_7228012346218142266_set_v @ A @ ( insert_set_v @ A4 @ bot_bot_set_set_v ) ) @ B ) ) ).
% Diff_insert2
thf(fact_260_Diff__insert2,axiom,
! [A: set_v,A4: v,B: set_v] :
( ( minus_minus_set_v @ A @ ( insert_v @ A4 @ B ) )
= ( minus_minus_set_v @ ( minus_minus_set_v @ A @ ( insert_v @ A4 @ bot_bot_set_v ) ) @ B ) ) ).
% Diff_insert2
thf(fact_261_insert__is__Un,axiom,
( insert1338601472111419319od_v_v
= ( ^ [A6: product_prod_v_v] : ( sup_su414716646722978715od_v_v @ ( insert1338601472111419319od_v_v @ A6 @ bot_bo723834152578015283od_v_v ) ) ) ) ).
% insert_is_Un
thf(fact_262_insert__is__Un,axiom,
( insert_set_v
= ( ^ [A6: set_v] : ( sup_sup_set_set_v @ ( insert_set_v @ A6 @ bot_bot_set_set_v ) ) ) ) ).
% insert_is_Un
thf(fact_263_insert__is__Un,axiom,
( insert_v
= ( ^ [A6: v] : ( sup_sup_set_v @ ( insert_v @ A6 @ bot_bot_set_v ) ) ) ) ).
% insert_is_Un
thf(fact_264_insert__Diff__if,axiom,
! [X: set_v,B: set_set_v,A: set_set_v] :
( ( ( member_set_v @ X @ B )
=> ( ( minus_7228012346218142266_set_v @ ( insert_set_v @ X @ A ) @ B )
= ( minus_7228012346218142266_set_v @ A @ B ) ) )
& ( ~ ( member_set_v @ X @ B )
=> ( ( minus_7228012346218142266_set_v @ ( insert_set_v @ X @ A ) @ B )
= ( insert_set_v @ X @ ( minus_7228012346218142266_set_v @ A @ B ) ) ) ) ) ).
% insert_Diff_if
thf(fact_265_insert__Diff__if,axiom,
! [X: product_prod_v_v,B: set_Product_prod_v_v,A: set_Product_prod_v_v] :
( ( ( member7453568604450474000od_v_v @ X @ B )
=> ( ( minus_4183494784930505774od_v_v @ ( insert1338601472111419319od_v_v @ X @ A ) @ B )
= ( minus_4183494784930505774od_v_v @ A @ B ) ) )
& ( ~ ( member7453568604450474000od_v_v @ X @ B )
=> ( ( minus_4183494784930505774od_v_v @ ( insert1338601472111419319od_v_v @ X @ A ) @ B )
= ( insert1338601472111419319od_v_v @ X @ ( minus_4183494784930505774od_v_v @ A @ B ) ) ) ) ) ).
% insert_Diff_if
thf(fact_266_insert__Diff__if,axiom,
! [X: v,B: set_v,A: set_v] :
( ( ( member_v @ X @ B )
=> ( ( minus_minus_set_v @ ( insert_v @ X @ A ) @ B )
= ( minus_minus_set_v @ A @ B ) ) )
& ( ~ ( member_v @ X @ B )
=> ( ( minus_minus_set_v @ ( insert_v @ X @ A ) @ B )
= ( insert_v @ X @ ( minus_minus_set_v @ A @ B ) ) ) ) ) ).
% insert_Diff_if
thf(fact_267_Un__singleton__iff,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v,X: product_prod_v_v] :
( ( ( sup_su414716646722978715od_v_v @ A @ B )
= ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v ) )
= ( ( ( A = bot_bo723834152578015283od_v_v )
& ( B
= ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v ) ) )
| ( ( A
= ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v ) )
& ( B = bot_bo723834152578015283od_v_v ) )
| ( ( A
= ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v ) )
& ( B
= ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_268_Un__singleton__iff,axiom,
! [A: set_set_v,B: set_set_v,X: set_v] :
( ( ( sup_sup_set_set_v @ A @ B )
= ( insert_set_v @ X @ bot_bot_set_set_v ) )
= ( ( ( A = bot_bot_set_set_v )
& ( B
= ( insert_set_v @ X @ bot_bot_set_set_v ) ) )
| ( ( A
= ( insert_set_v @ X @ bot_bot_set_set_v ) )
& ( B = bot_bot_set_set_v ) )
| ( ( A
= ( insert_set_v @ X @ bot_bot_set_set_v ) )
& ( B
= ( insert_set_v @ X @ bot_bot_set_set_v ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_269_Un__singleton__iff,axiom,
! [A: set_v,B: set_v,X: v] :
( ( ( sup_sup_set_v @ A @ B )
= ( insert_v @ X @ bot_bot_set_v ) )
= ( ( ( A = bot_bot_set_v )
& ( B
= ( insert_v @ X @ bot_bot_set_v ) ) )
| ( ( A
= ( insert_v @ X @ bot_bot_set_v ) )
& ( B = bot_bot_set_v ) )
| ( ( A
= ( insert_v @ X @ bot_bot_set_v ) )
& ( B
= ( insert_v @ X @ bot_bot_set_v ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_270_singleton__Un__iff,axiom,
! [X: product_prod_v_v,A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v )
= ( sup_su414716646722978715od_v_v @ A @ B ) )
= ( ( ( A = bot_bo723834152578015283od_v_v )
& ( B
= ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v ) ) )
| ( ( A
= ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v ) )
& ( B = bot_bo723834152578015283od_v_v ) )
| ( ( A
= ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v ) )
& ( B
= ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_271_singleton__Un__iff,axiom,
! [X: set_v,A: set_set_v,B: set_set_v] :
( ( ( insert_set_v @ X @ bot_bot_set_set_v )
= ( sup_sup_set_set_v @ A @ B ) )
= ( ( ( A = bot_bot_set_set_v )
& ( B
= ( insert_set_v @ X @ bot_bot_set_set_v ) ) )
| ( ( A
= ( insert_set_v @ X @ bot_bot_set_set_v ) )
& ( B = bot_bot_set_set_v ) )
| ( ( A
= ( insert_set_v @ X @ bot_bot_set_set_v ) )
& ( B
= ( insert_set_v @ X @ bot_bot_set_set_v ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_272_singleton__Un__iff,axiom,
! [X: v,A: set_v,B: set_v] :
( ( ( insert_v @ X @ bot_bot_set_v )
= ( sup_sup_set_v @ A @ B ) )
= ( ( ( A = bot_bot_set_v )
& ( B
= ( insert_v @ X @ bot_bot_set_v ) ) )
| ( ( A
= ( insert_v @ X @ bot_bot_set_v ) )
& ( B = bot_bot_set_v ) )
| ( ( A
= ( insert_v @ X @ bot_bot_set_v ) )
& ( B
= ( insert_v @ X @ bot_bot_set_v ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_273_Diff__insert__absorb,axiom,
! [X: product_prod_v_v,A: set_Product_prod_v_v] :
( ~ ( member7453568604450474000od_v_v @ X @ A )
=> ( ( minus_4183494784930505774od_v_v @ ( insert1338601472111419319od_v_v @ X @ A ) @ ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v ) )
= A ) ) ).
% Diff_insert_absorb
thf(fact_274_Diff__insert__absorb,axiom,
! [X: set_v,A: set_set_v] :
( ~ ( member_set_v @ X @ A )
=> ( ( minus_7228012346218142266_set_v @ ( insert_set_v @ X @ A ) @ ( insert_set_v @ X @ bot_bot_set_set_v ) )
= A ) ) ).
% Diff_insert_absorb
thf(fact_275_Diff__insert__absorb,axiom,
! [X: v,A: set_v] :
( ~ ( member_v @ X @ A )
=> ( ( minus_minus_set_v @ ( insert_v @ X @ A ) @ ( insert_v @ X @ bot_bot_set_v ) )
= A ) ) ).
% Diff_insert_absorb
thf(fact_276_graph_Oreachable__end_Ocong,axiom,
sCC_Bl770211535891879572_end_v = sCC_Bl770211535891879572_end_v ).
% graph.reachable_end.cong
thf(fact_277_graph_Opre__dfss_Ocong,axiom,
sCC_Bl1748261141445803503t_unit = sCC_Bl1748261141445803503t_unit ).
% graph.pre_dfss.cong
thf(fact_278_Un__empty__right,axiom,
! [A: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ A @ bot_bo723834152578015283od_v_v )
= A ) ).
% Un_empty_right
thf(fact_279_Un__empty__right,axiom,
! [A: set_set_v] :
( ( sup_sup_set_set_v @ A @ bot_bot_set_set_v )
= A ) ).
% Un_empty_right
thf(fact_280_Un__empty__right,axiom,
! [A: set_v] :
( ( sup_sup_set_v @ A @ bot_bot_set_v )
= A ) ).
% Un_empty_right
thf(fact_281_Un__empty__left,axiom,
! [B: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ bot_bo723834152578015283od_v_v @ B )
= B ) ).
% Un_empty_left
thf(fact_282_Un__empty__left,axiom,
! [B: set_set_v] :
( ( sup_sup_set_set_v @ bot_bot_set_set_v @ B )
= B ) ).
% Un_empty_left
thf(fact_283_Un__empty__left,axiom,
! [B: set_v] :
( ( sup_sup_set_v @ bot_bot_set_v @ B )
= B ) ).
% Un_empty_left
thf(fact_284_singleton__inject,axiom,
! [A4: product_prod_v_v,B2: product_prod_v_v] :
( ( ( insert1338601472111419319od_v_v @ A4 @ bot_bo723834152578015283od_v_v )
= ( insert1338601472111419319od_v_v @ B2 @ bot_bo723834152578015283od_v_v ) )
=> ( A4 = B2 ) ) ).
% singleton_inject
thf(fact_285_singleton__inject,axiom,
! [A4: set_v,B2: set_v] :
( ( ( insert_set_v @ A4 @ bot_bot_set_set_v )
= ( insert_set_v @ B2 @ bot_bot_set_set_v ) )
=> ( A4 = B2 ) ) ).
% singleton_inject
thf(fact_286_singleton__inject,axiom,
! [A4: v,B2: v] :
( ( ( insert_v @ A4 @ bot_bot_set_v )
= ( insert_v @ B2 @ bot_bot_set_v ) )
=> ( A4 = B2 ) ) ).
% singleton_inject
thf(fact_287_insert__not__empty,axiom,
! [A4: product_prod_v_v,A: set_Product_prod_v_v] :
( ( insert1338601472111419319od_v_v @ A4 @ A )
!= bot_bo723834152578015283od_v_v ) ).
% insert_not_empty
thf(fact_288_insert__not__empty,axiom,
! [A4: set_v,A: set_set_v] :
( ( insert_set_v @ A4 @ A )
!= bot_bot_set_set_v ) ).
% insert_not_empty
thf(fact_289_insert__not__empty,axiom,
! [A4: v,A: set_v] :
( ( insert_v @ A4 @ A )
!= bot_bot_set_v ) ).
% insert_not_empty
thf(fact_290_doubleton__eq__iff,axiom,
! [A4: product_prod_v_v,B2: product_prod_v_v,C: product_prod_v_v,D: product_prod_v_v] :
( ( ( insert1338601472111419319od_v_v @ A4 @ ( insert1338601472111419319od_v_v @ B2 @ bot_bo723834152578015283od_v_v ) )
= ( insert1338601472111419319od_v_v @ C @ ( insert1338601472111419319od_v_v @ D @ bot_bo723834152578015283od_v_v ) ) )
= ( ( ( A4 = C )
& ( B2 = D ) )
| ( ( A4 = D )
& ( B2 = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_291_doubleton__eq__iff,axiom,
! [A4: set_v,B2: set_v,C: set_v,D: set_v] :
( ( ( insert_set_v @ A4 @ ( insert_set_v @ B2 @ bot_bot_set_set_v ) )
= ( insert_set_v @ C @ ( insert_set_v @ D @ bot_bot_set_set_v ) ) )
= ( ( ( A4 = C )
& ( B2 = D ) )
| ( ( A4 = D )
& ( B2 = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_292_doubleton__eq__iff,axiom,
! [A4: v,B2: v,C: v,D: v] :
( ( ( insert_v @ A4 @ ( insert_v @ B2 @ bot_bot_set_v ) )
= ( insert_v @ C @ ( insert_v @ D @ bot_bot_set_v ) ) )
= ( ( ( A4 = C )
& ( B2 = D ) )
| ( ( A4 = D )
& ( B2 = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_293_singleton__iff,axiom,
! [B2: product_prod_v_v,A4: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ B2 @ ( insert1338601472111419319od_v_v @ A4 @ bot_bo723834152578015283od_v_v ) )
= ( B2 = A4 ) ) ).
% singleton_iff
thf(fact_294_singleton__iff,axiom,
! [B2: set_v,A4: set_v] :
( ( member_set_v @ B2 @ ( insert_set_v @ A4 @ bot_bot_set_set_v ) )
= ( B2 = A4 ) ) ).
% singleton_iff
thf(fact_295_singleton__iff,axiom,
! [B2: v,A4: v] :
( ( member_v @ B2 @ ( insert_v @ A4 @ bot_bot_set_v ) )
= ( B2 = A4 ) ) ).
% singleton_iff
thf(fact_296_singletonD,axiom,
! [B2: product_prod_v_v,A4: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ B2 @ ( insert1338601472111419319od_v_v @ A4 @ bot_bo723834152578015283od_v_v ) )
=> ( B2 = A4 ) ) ).
% singletonD
thf(fact_297_singletonD,axiom,
! [B2: set_v,A4: set_v] :
( ( member_set_v @ B2 @ ( insert_set_v @ A4 @ bot_bot_set_set_v ) )
=> ( B2 = A4 ) ) ).
% singletonD
thf(fact_298_singletonD,axiom,
! [B2: v,A4: v] :
( ( member_v @ B2 @ ( insert_v @ A4 @ bot_bot_set_v ) )
=> ( B2 = A4 ) ) ).
% singletonD
thf(fact_299_subset__insert__iff,axiom,
! [A: set_set_v,X: set_v,B: set_set_v] :
( ( ord_le5216385588623774835_set_v @ A @ ( insert_set_v @ X @ B ) )
= ( ( ( member_set_v @ X @ A )
=> ( ord_le5216385588623774835_set_v @ ( minus_7228012346218142266_set_v @ A @ ( insert_set_v @ X @ bot_bot_set_set_v ) ) @ B ) )
& ( ~ ( member_set_v @ X @ A )
=> ( ord_le5216385588623774835_set_v @ A @ B ) ) ) ) ).
% subset_insert_iff
thf(fact_300_subset__insert__iff,axiom,
! [A: set_Product_prod_v_v,X: product_prod_v_v,B: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ ( insert1338601472111419319od_v_v @ X @ B ) )
= ( ( ( member7453568604450474000od_v_v @ X @ A )
=> ( ord_le7336532860387713383od_v_v @ ( minus_4183494784930505774od_v_v @ A @ ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v ) ) @ B ) )
& ( ~ ( member7453568604450474000od_v_v @ X @ A )
=> ( ord_le7336532860387713383od_v_v @ A @ B ) ) ) ) ).
% subset_insert_iff
thf(fact_301_subset__insert__iff,axiom,
! [A: set_v,X: v,B: set_v] :
( ( ord_less_eq_set_v @ A @ ( insert_v @ X @ B ) )
= ( ( ( member_v @ X @ A )
=> ( ord_less_eq_set_v @ ( minus_minus_set_v @ A @ ( insert_v @ X @ bot_bot_set_v ) ) @ B ) )
& ( ~ ( member_v @ X @ A )
=> ( ord_less_eq_set_v @ A @ B ) ) ) ) ).
% subset_insert_iff
thf(fact_302_Diff__single__insert,axiom,
! [A: set_set_v,X: set_v,B: set_set_v] :
( ( ord_le5216385588623774835_set_v @ ( minus_7228012346218142266_set_v @ A @ ( insert_set_v @ X @ bot_bot_set_set_v ) ) @ B )
=> ( ord_le5216385588623774835_set_v @ A @ ( insert_set_v @ X @ B ) ) ) ).
% Diff_single_insert
thf(fact_303_Diff__single__insert,axiom,
! [A: set_Product_prod_v_v,X: product_prod_v_v,B: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ ( minus_4183494784930505774od_v_v @ A @ ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v ) ) @ B )
=> ( ord_le7336532860387713383od_v_v @ A @ ( insert1338601472111419319od_v_v @ X @ B ) ) ) ).
% Diff_single_insert
thf(fact_304_Diff__single__insert,axiom,
! [A: set_v,X: v,B: set_v] :
( ( ord_less_eq_set_v @ ( minus_minus_set_v @ A @ ( insert_v @ X @ bot_bot_set_v ) ) @ B )
=> ( ord_less_eq_set_v @ A @ ( insert_v @ X @ B ) ) ) ).
% Diff_single_insert
thf(fact_305_Diff__subset__conv,axiom,
! [A: set_set_v,B: set_set_v,C2: set_set_v] :
( ( ord_le5216385588623774835_set_v @ ( minus_7228012346218142266_set_v @ A @ B ) @ C2 )
= ( ord_le5216385588623774835_set_v @ A @ ( sup_sup_set_set_v @ B @ C2 ) ) ) ).
% Diff_subset_conv
thf(fact_306_Diff__subset__conv,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v,C2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ ( minus_4183494784930505774od_v_v @ A @ B ) @ C2 )
= ( ord_le7336532860387713383od_v_v @ A @ ( sup_su414716646722978715od_v_v @ B @ C2 ) ) ) ).
% Diff_subset_conv
thf(fact_307_Diff__subset__conv,axiom,
! [A: set_v,B: set_v,C2: set_v] :
( ( ord_less_eq_set_v @ ( minus_minus_set_v @ A @ B ) @ C2 )
= ( ord_less_eq_set_v @ A @ ( sup_sup_set_v @ B @ C2 ) ) ) ).
% Diff_subset_conv
thf(fact_308_Diff__partition,axiom,
! [A: set_set_v,B: set_set_v] :
( ( ord_le5216385588623774835_set_v @ A @ B )
=> ( ( sup_sup_set_set_v @ A @ ( minus_7228012346218142266_set_v @ B @ A ) )
= B ) ) ).
% Diff_partition
thf(fact_309_Diff__partition,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ B )
=> ( ( sup_su414716646722978715od_v_v @ A @ ( minus_4183494784930505774od_v_v @ B @ A ) )
= B ) ) ).
% Diff_partition
thf(fact_310_Diff__partition,axiom,
! [A: set_v,B: set_v] :
( ( ord_less_eq_set_v @ A @ B )
=> ( ( sup_sup_set_v @ A @ ( minus_minus_set_v @ B @ A ) )
= B ) ) ).
% Diff_partition
thf(fact_311_subset__Diff__insert,axiom,
! [A: set_set_v,B: set_set_v,X: set_v,C2: set_set_v] :
( ( ord_le5216385588623774835_set_v @ A @ ( minus_7228012346218142266_set_v @ B @ ( insert_set_v @ X @ C2 ) ) )
= ( ( ord_le5216385588623774835_set_v @ A @ ( minus_7228012346218142266_set_v @ B @ C2 ) )
& ~ ( member_set_v @ X @ A ) ) ) ).
% subset_Diff_insert
thf(fact_312_subset__Diff__insert,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v,X: product_prod_v_v,C2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ ( minus_4183494784930505774od_v_v @ B @ ( insert1338601472111419319od_v_v @ X @ C2 ) ) )
= ( ( ord_le7336532860387713383od_v_v @ A @ ( minus_4183494784930505774od_v_v @ B @ C2 ) )
& ~ ( member7453568604450474000od_v_v @ X @ A ) ) ) ).
% subset_Diff_insert
thf(fact_313_subset__Diff__insert,axiom,
! [A: set_v,B: set_v,X: v,C2: set_v] :
( ( ord_less_eq_set_v @ A @ ( minus_minus_set_v @ B @ ( insert_v @ X @ C2 ) ) )
= ( ( ord_less_eq_set_v @ A @ ( minus_minus_set_v @ B @ C2 ) )
& ~ ( member_v @ X @ A ) ) ) ).
% subset_Diff_insert
thf(fact_314_double__diff,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v,C2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ B )
=> ( ( ord_le7336532860387713383od_v_v @ B @ C2 )
=> ( ( minus_4183494784930505774od_v_v @ B @ ( minus_4183494784930505774od_v_v @ C2 @ A ) )
= A ) ) ) ).
% double_diff
thf(fact_315_double__diff,axiom,
! [A: set_v,B: set_v,C2: set_v] :
( ( ord_less_eq_set_v @ A @ B )
=> ( ( ord_less_eq_set_v @ B @ C2 )
=> ( ( minus_minus_set_v @ B @ ( minus_minus_set_v @ C2 @ A ) )
= A ) ) ) ).
% double_diff
thf(fact_316_Diff__subset,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ ( minus_4183494784930505774od_v_v @ A @ B ) @ A ) ).
% Diff_subset
thf(fact_317_Diff__subset,axiom,
! [A: set_v,B: set_v] : ( ord_less_eq_set_v @ ( minus_minus_set_v @ A @ B ) @ A ) ).
% Diff_subset
thf(fact_318_Diff__mono,axiom,
! [A: set_Product_prod_v_v,C2: set_Product_prod_v_v,D2: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ C2 )
=> ( ( ord_le7336532860387713383od_v_v @ D2 @ B )
=> ( ord_le7336532860387713383od_v_v @ ( minus_4183494784930505774od_v_v @ A @ B ) @ ( minus_4183494784930505774od_v_v @ C2 @ D2 ) ) ) ) ).
% Diff_mono
thf(fact_319_Diff__mono,axiom,
! [A: set_v,C2: set_v,D2: set_v,B: set_v] :
( ( ord_less_eq_set_v @ A @ C2 )
=> ( ( ord_less_eq_set_v @ D2 @ B )
=> ( ord_less_eq_set_v @ ( minus_minus_set_v @ A @ B ) @ ( minus_minus_set_v @ C2 @ D2 ) ) ) ) ).
% Diff_mono
thf(fact_320_Un__left__commute,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v,C2: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ A @ ( sup_su414716646722978715od_v_v @ B @ C2 ) )
= ( sup_su414716646722978715od_v_v @ B @ ( sup_su414716646722978715od_v_v @ A @ C2 ) ) ) ).
% Un_left_commute
thf(fact_321_Un__left__commute,axiom,
! [A: set_v,B: set_v,C2: set_v] :
( ( sup_sup_set_v @ A @ ( sup_sup_set_v @ B @ C2 ) )
= ( sup_sup_set_v @ B @ ( sup_sup_set_v @ A @ C2 ) ) ) ).
% Un_left_commute
thf(fact_322_Un__left__commute,axiom,
! [A: set_set_v,B: set_set_v,C2: set_set_v] :
( ( sup_sup_set_set_v @ A @ ( sup_sup_set_set_v @ B @ C2 ) )
= ( sup_sup_set_set_v @ B @ ( sup_sup_set_set_v @ A @ C2 ) ) ) ).
% Un_left_commute
thf(fact_323_Un__left__absorb,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ A @ ( sup_su414716646722978715od_v_v @ A @ B ) )
= ( sup_su414716646722978715od_v_v @ A @ B ) ) ).
% Un_left_absorb
thf(fact_324_Un__left__absorb,axiom,
! [A: set_v,B: set_v] :
( ( sup_sup_set_v @ A @ ( sup_sup_set_v @ A @ B ) )
= ( sup_sup_set_v @ A @ B ) ) ).
% Un_left_absorb
thf(fact_325_Un__left__absorb,axiom,
! [A: set_set_v,B: set_set_v] :
( ( sup_sup_set_set_v @ A @ ( sup_sup_set_set_v @ A @ B ) )
= ( sup_sup_set_set_v @ A @ B ) ) ).
% Un_left_absorb
thf(fact_326_Un__commute,axiom,
( sup_su414716646722978715od_v_v
= ( ^ [A8: set_Product_prod_v_v,B6: set_Product_prod_v_v] : ( sup_su414716646722978715od_v_v @ B6 @ A8 ) ) ) ).
% Un_commute
thf(fact_327_Un__commute,axiom,
( sup_sup_set_v
= ( ^ [A8: set_v,B6: set_v] : ( sup_sup_set_v @ B6 @ A8 ) ) ) ).
% Un_commute
thf(fact_328_Un__commute,axiom,
( sup_sup_set_set_v
= ( ^ [A8: set_set_v,B6: set_set_v] : ( sup_sup_set_set_v @ B6 @ A8 ) ) ) ).
% Un_commute
thf(fact_329_Un__absorb,axiom,
! [A: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ A @ A )
= A ) ).
% Un_absorb
thf(fact_330_Un__absorb,axiom,
! [A: set_v] :
( ( sup_sup_set_v @ A @ A )
= A ) ).
% Un_absorb
thf(fact_331_Un__absorb,axiom,
! [A: set_set_v] :
( ( sup_sup_set_set_v @ A @ A )
= A ) ).
% Un_absorb
thf(fact_332_Un__assoc,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v,C2: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ ( sup_su414716646722978715od_v_v @ A @ B ) @ C2 )
= ( sup_su414716646722978715od_v_v @ A @ ( sup_su414716646722978715od_v_v @ B @ C2 ) ) ) ).
% Un_assoc
thf(fact_333_Un__assoc,axiom,
! [A: set_v,B: set_v,C2: set_v] :
( ( sup_sup_set_v @ ( sup_sup_set_v @ A @ B ) @ C2 )
= ( sup_sup_set_v @ A @ ( sup_sup_set_v @ B @ C2 ) ) ) ).
% Un_assoc
thf(fact_334_Un__assoc,axiom,
! [A: set_set_v,B: set_set_v,C2: set_set_v] :
( ( sup_sup_set_set_v @ ( sup_sup_set_set_v @ A @ B ) @ C2 )
= ( sup_sup_set_set_v @ A @ ( sup_sup_set_set_v @ B @ C2 ) ) ) ).
% Un_assoc
thf(fact_335_ball__Un,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v,P: product_prod_v_v > $o] :
( ( ! [X2: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X2 @ ( sup_su414716646722978715od_v_v @ A @ B ) )
=> ( P @ X2 ) ) )
= ( ! [X2: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X2 @ A )
=> ( P @ X2 ) )
& ! [X2: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X2 @ B )
=> ( P @ X2 ) ) ) ) ).
% ball_Un
thf(fact_336_ball__Un,axiom,
! [A: set_v,B: set_v,P: v > $o] :
( ( ! [X2: v] :
( ( member_v @ X2 @ ( sup_sup_set_v @ A @ B ) )
=> ( P @ X2 ) ) )
= ( ! [X2: v] :
( ( member_v @ X2 @ A )
=> ( P @ X2 ) )
& ! [X2: v] :
( ( member_v @ X2 @ B )
=> ( P @ X2 ) ) ) ) ).
% ball_Un
thf(fact_337_ball__Un,axiom,
! [A: set_set_v,B: set_set_v,P: set_v > $o] :
( ( ! [X2: set_v] :
( ( member_set_v @ X2 @ ( sup_sup_set_set_v @ A @ B ) )
=> ( P @ X2 ) ) )
= ( ! [X2: set_v] :
( ( member_set_v @ X2 @ A )
=> ( P @ X2 ) )
& ! [X2: set_v] :
( ( member_set_v @ X2 @ B )
=> ( P @ X2 ) ) ) ) ).
% ball_Un
thf(fact_338_bex__Un,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v,P: product_prod_v_v > $o] :
( ( ? [X2: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X2 @ ( sup_su414716646722978715od_v_v @ A @ B ) )
& ( P @ X2 ) ) )
= ( ? [X2: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X2 @ A )
& ( P @ X2 ) )
| ? [X2: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X2 @ B )
& ( P @ X2 ) ) ) ) ).
% bex_Un
thf(fact_339_bex__Un,axiom,
! [A: set_v,B: set_v,P: v > $o] :
( ( ? [X2: v] :
( ( member_v @ X2 @ ( sup_sup_set_v @ A @ B ) )
& ( P @ X2 ) ) )
= ( ? [X2: v] :
( ( member_v @ X2 @ A )
& ( P @ X2 ) )
| ? [X2: v] :
( ( member_v @ X2 @ B )
& ( P @ X2 ) ) ) ) ).
% bex_Un
thf(fact_340_bex__Un,axiom,
! [A: set_set_v,B: set_set_v,P: set_v > $o] :
( ( ? [X2: set_v] :
( ( member_set_v @ X2 @ ( sup_sup_set_set_v @ A @ B ) )
& ( P @ X2 ) ) )
= ( ? [X2: set_v] :
( ( member_set_v @ X2 @ A )
& ( P @ X2 ) )
| ? [X2: set_v] :
( ( member_set_v @ X2 @ B )
& ( P @ X2 ) ) ) ) ).
% bex_Un
thf(fact_341_UnI2,axiom,
! [C: product_prod_v_v,B: set_Product_prod_v_v,A: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ C @ B )
=> ( member7453568604450474000od_v_v @ C @ ( sup_su414716646722978715od_v_v @ A @ B ) ) ) ).
% UnI2
thf(fact_342_UnI2,axiom,
! [C: v,B: set_v,A: set_v] :
( ( member_v @ C @ B )
=> ( member_v @ C @ ( sup_sup_set_v @ A @ B ) ) ) ).
% UnI2
thf(fact_343_UnI2,axiom,
! [C: set_v,B: set_set_v,A: set_set_v] :
( ( member_set_v @ C @ B )
=> ( member_set_v @ C @ ( sup_sup_set_set_v @ A @ B ) ) ) ).
% UnI2
thf(fact_344_UnI1,axiom,
! [C: product_prod_v_v,A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ C @ A )
=> ( member7453568604450474000od_v_v @ C @ ( sup_su414716646722978715od_v_v @ A @ B ) ) ) ).
% UnI1
thf(fact_345_UnI1,axiom,
! [C: v,A: set_v,B: set_v] :
( ( member_v @ C @ A )
=> ( member_v @ C @ ( sup_sup_set_v @ A @ B ) ) ) ).
% UnI1
thf(fact_346_UnI1,axiom,
! [C: set_v,A: set_set_v,B: set_set_v] :
( ( member_set_v @ C @ A )
=> ( member_set_v @ C @ ( sup_sup_set_set_v @ A @ B ) ) ) ).
% UnI1
thf(fact_347_UnE,axiom,
! [C: product_prod_v_v,A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ C @ ( sup_su414716646722978715od_v_v @ A @ B ) )
=> ( ~ ( member7453568604450474000od_v_v @ C @ A )
=> ( member7453568604450474000od_v_v @ C @ B ) ) ) ).
% UnE
thf(fact_348_UnE,axiom,
! [C: v,A: set_v,B: set_v] :
( ( member_v @ C @ ( sup_sup_set_v @ A @ B ) )
=> ( ~ ( member_v @ C @ A )
=> ( member_v @ C @ B ) ) ) ).
% UnE
thf(fact_349_UnE,axiom,
! [C: set_v,A: set_set_v,B: set_set_v] :
( ( member_set_v @ C @ ( sup_sup_set_set_v @ A @ B ) )
=> ( ~ ( member_set_v @ C @ A )
=> ( member_set_v @ C @ B ) ) ) ).
% UnE
thf(fact_350_mk__disjoint__insert,axiom,
! [A4: set_v,A: set_set_v] :
( ( member_set_v @ A4 @ A )
=> ? [B7: set_set_v] :
( ( A
= ( insert_set_v @ A4 @ B7 ) )
& ~ ( member_set_v @ A4 @ B7 ) ) ) ).
% mk_disjoint_insert
thf(fact_351_mk__disjoint__insert,axiom,
! [A4: v,A: set_v] :
( ( member_v @ A4 @ A )
=> ? [B7: set_v] :
( ( A
= ( insert_v @ A4 @ B7 ) )
& ~ ( member_v @ A4 @ B7 ) ) ) ).
% mk_disjoint_insert
thf(fact_352_mk__disjoint__insert,axiom,
! [A4: product_prod_v_v,A: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ A4 @ A )
=> ? [B7: set_Product_prod_v_v] :
( ( A
= ( insert1338601472111419319od_v_v @ A4 @ B7 ) )
& ~ ( member7453568604450474000od_v_v @ A4 @ B7 ) ) ) ).
% mk_disjoint_insert
thf(fact_353_insert__commute,axiom,
! [X: product_prod_v_v,Y: product_prod_v_v,A: set_Product_prod_v_v] :
( ( insert1338601472111419319od_v_v @ X @ ( insert1338601472111419319od_v_v @ Y @ A ) )
= ( insert1338601472111419319od_v_v @ Y @ ( insert1338601472111419319od_v_v @ X @ A ) ) ) ).
% insert_commute
thf(fact_354_insert__commute,axiom,
! [X: set_v,Y: set_v,A: set_set_v] :
( ( insert_set_v @ X @ ( insert_set_v @ Y @ A ) )
= ( insert_set_v @ Y @ ( insert_set_v @ X @ A ) ) ) ).
% insert_commute
thf(fact_355_insert__commute,axiom,
! [X: v,Y: v,A: set_v] :
( ( insert_v @ X @ ( insert_v @ Y @ A ) )
= ( insert_v @ Y @ ( insert_v @ X @ A ) ) ) ).
% insert_commute
thf(fact_356_insert__eq__iff,axiom,
! [A4: set_v,A: set_set_v,B2: set_v,B: set_set_v] :
( ~ ( member_set_v @ A4 @ A )
=> ( ~ ( member_set_v @ B2 @ B )
=> ( ( ( insert_set_v @ A4 @ A )
= ( insert_set_v @ B2 @ B ) )
= ( ( ( A4 = B2 )
=> ( A = B ) )
& ( ( A4 != B2 )
=> ? [C3: set_set_v] :
( ( A
= ( insert_set_v @ B2 @ C3 ) )
& ~ ( member_set_v @ B2 @ C3 )
& ( B
= ( insert_set_v @ A4 @ C3 ) )
& ~ ( member_set_v @ A4 @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_357_insert__eq__iff,axiom,
! [A4: v,A: set_v,B2: v,B: set_v] :
( ~ ( member_v @ A4 @ A )
=> ( ~ ( member_v @ B2 @ B )
=> ( ( ( insert_v @ A4 @ A )
= ( insert_v @ B2 @ B ) )
= ( ( ( A4 = B2 )
=> ( A = B ) )
& ( ( A4 != B2 )
=> ? [C3: set_v] :
( ( A
= ( insert_v @ B2 @ C3 ) )
& ~ ( member_v @ B2 @ C3 )
& ( B
= ( insert_v @ A4 @ C3 ) )
& ~ ( member_v @ A4 @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_358_insert__eq__iff,axiom,
! [A4: product_prod_v_v,A: set_Product_prod_v_v,B2: product_prod_v_v,B: set_Product_prod_v_v] :
( ~ ( member7453568604450474000od_v_v @ A4 @ A )
=> ( ~ ( member7453568604450474000od_v_v @ B2 @ B )
=> ( ( ( insert1338601472111419319od_v_v @ A4 @ A )
= ( insert1338601472111419319od_v_v @ B2 @ B ) )
= ( ( ( A4 = B2 )
=> ( A = B ) )
& ( ( A4 != B2 )
=> ? [C3: set_Product_prod_v_v] :
( ( A
= ( insert1338601472111419319od_v_v @ B2 @ C3 ) )
& ~ ( member7453568604450474000od_v_v @ B2 @ C3 )
& ( B
= ( insert1338601472111419319od_v_v @ A4 @ C3 ) )
& ~ ( member7453568604450474000od_v_v @ A4 @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_359_insert__absorb,axiom,
! [A4: set_v,A: set_set_v] :
( ( member_set_v @ A4 @ A )
=> ( ( insert_set_v @ A4 @ A )
= A ) ) ).
% insert_absorb
thf(fact_360_insert__absorb,axiom,
! [A4: v,A: set_v] :
( ( member_v @ A4 @ A )
=> ( ( insert_v @ A4 @ A )
= A ) ) ).
% insert_absorb
thf(fact_361_insert__absorb,axiom,
! [A4: product_prod_v_v,A: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ A4 @ A )
=> ( ( insert1338601472111419319od_v_v @ A4 @ A )
= A ) ) ).
% insert_absorb
thf(fact_362_insert__ident,axiom,
! [X: set_v,A: set_set_v,B: set_set_v] :
( ~ ( member_set_v @ X @ A )
=> ( ~ ( member_set_v @ X @ B )
=> ( ( ( insert_set_v @ X @ A )
= ( insert_set_v @ X @ B ) )
= ( A = B ) ) ) ) ).
% insert_ident
thf(fact_363_insert__ident,axiom,
! [X: v,A: set_v,B: set_v] :
( ~ ( member_v @ X @ A )
=> ( ~ ( member_v @ X @ B )
=> ( ( ( insert_v @ X @ A )
= ( insert_v @ X @ B ) )
= ( A = B ) ) ) ) ).
% insert_ident
thf(fact_364_insert__ident,axiom,
! [X: product_prod_v_v,A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ~ ( member7453568604450474000od_v_v @ X @ A )
=> ( ~ ( member7453568604450474000od_v_v @ X @ B )
=> ( ( ( insert1338601472111419319od_v_v @ X @ A )
= ( insert1338601472111419319od_v_v @ X @ B ) )
= ( A = B ) ) ) ) ).
% insert_ident
thf(fact_365_Set_Oset__insert,axiom,
! [X: set_v,A: set_set_v] :
( ( member_set_v @ X @ A )
=> ~ ! [B7: set_set_v] :
( ( A
= ( insert_set_v @ X @ B7 ) )
=> ( member_set_v @ X @ B7 ) ) ) ).
% Set.set_insert
thf(fact_366_Set_Oset__insert,axiom,
! [X: v,A: set_v] :
( ( member_v @ X @ A )
=> ~ ! [B7: set_v] :
( ( A
= ( insert_v @ X @ B7 ) )
=> ( member_v @ X @ B7 ) ) ) ).
% Set.set_insert
thf(fact_367_Set_Oset__insert,axiom,
! [X: product_prod_v_v,A: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X @ A )
=> ~ ! [B7: set_Product_prod_v_v] :
( ( A
= ( insert1338601472111419319od_v_v @ X @ B7 ) )
=> ( member7453568604450474000od_v_v @ X @ B7 ) ) ) ).
% Set.set_insert
thf(fact_368_insertI2,axiom,
! [A4: set_v,B: set_set_v,B2: set_v] :
( ( member_set_v @ A4 @ B )
=> ( member_set_v @ A4 @ ( insert_set_v @ B2 @ B ) ) ) ).
% insertI2
thf(fact_369_insertI2,axiom,
! [A4: v,B: set_v,B2: v] :
( ( member_v @ A4 @ B )
=> ( member_v @ A4 @ ( insert_v @ B2 @ B ) ) ) ).
% insertI2
thf(fact_370_insertI2,axiom,
! [A4: product_prod_v_v,B: set_Product_prod_v_v,B2: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ A4 @ B )
=> ( member7453568604450474000od_v_v @ A4 @ ( insert1338601472111419319od_v_v @ B2 @ B ) ) ) ).
% insertI2
thf(fact_371_insertI1,axiom,
! [A4: set_v,B: set_set_v] : ( member_set_v @ A4 @ ( insert_set_v @ A4 @ B ) ) ).
% insertI1
thf(fact_372_insertI1,axiom,
! [A4: v,B: set_v] : ( member_v @ A4 @ ( insert_v @ A4 @ B ) ) ).
% insertI1
thf(fact_373_insertI1,axiom,
! [A4: product_prod_v_v,B: set_Product_prod_v_v] : ( member7453568604450474000od_v_v @ A4 @ ( insert1338601472111419319od_v_v @ A4 @ B ) ) ).
% insertI1
thf(fact_374_insertE,axiom,
! [A4: set_v,B2: set_v,A: set_set_v] :
( ( member_set_v @ A4 @ ( insert_set_v @ B2 @ A ) )
=> ( ( A4 != B2 )
=> ( member_set_v @ A4 @ A ) ) ) ).
% insertE
thf(fact_375_insertE,axiom,
! [A4: v,B2: v,A: set_v] :
( ( member_v @ A4 @ ( insert_v @ B2 @ A ) )
=> ( ( A4 != B2 )
=> ( member_v @ A4 @ A ) ) ) ).
% insertE
thf(fact_376_insertE,axiom,
! [A4: product_prod_v_v,B2: product_prod_v_v,A: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ A4 @ ( insert1338601472111419319od_v_v @ B2 @ A ) )
=> ( ( A4 != B2 )
=> ( member7453568604450474000od_v_v @ A4 @ A ) ) ) ).
% insertE
thf(fact_377_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_378_ex__in__conv,axiom,
! [A: set_Product_prod_v_v] :
( ( ? [X2: product_prod_v_v] : ( member7453568604450474000od_v_v @ X2 @ A ) )
= ( A != bot_bo723834152578015283od_v_v ) ) ).
% ex_in_conv
thf(fact_379_ex__in__conv,axiom,
! [A: set_set_v] :
( ( ? [X2: set_v] : ( member_set_v @ X2 @ A ) )
= ( A != bot_bot_set_set_v ) ) ).
% ex_in_conv
thf(fact_380_ex__in__conv,axiom,
! [A: set_v] :
( ( ? [X2: v] : ( member_v @ X2 @ A ) )
= ( A != bot_bot_set_v ) ) ).
% ex_in_conv
thf(fact_381_equals0I,axiom,
! [A: set_Product_prod_v_v] :
( ! [Y2: product_prod_v_v] :
~ ( member7453568604450474000od_v_v @ Y2 @ A )
=> ( A = bot_bo723834152578015283od_v_v ) ) ).
% equals0I
thf(fact_382_equals0I,axiom,
! [A: set_set_v] :
( ! [Y2: set_v] :
~ ( member_set_v @ Y2 @ A )
=> ( A = bot_bot_set_set_v ) ) ).
% equals0I
thf(fact_383_equals0I,axiom,
! [A: set_v] :
( ! [Y2: v] :
~ ( member_v @ Y2 @ A )
=> ( A = bot_bot_set_v ) ) ).
% equals0I
thf(fact_384_equals0D,axiom,
! [A: set_Product_prod_v_v,A4: product_prod_v_v] :
( ( A = bot_bo723834152578015283od_v_v )
=> ~ ( member7453568604450474000od_v_v @ A4 @ A ) ) ).
% equals0D
thf(fact_385_equals0D,axiom,
! [A: set_set_v,A4: set_v] :
( ( A = bot_bot_set_set_v )
=> ~ ( member_set_v @ A4 @ A ) ) ).
% equals0D
thf(fact_386_equals0D,axiom,
! [A: set_v,A4: v] :
( ( A = bot_bot_set_v )
=> ~ ( member_v @ A4 @ A ) ) ).
% equals0D
thf(fact_387_emptyE,axiom,
! [A4: product_prod_v_v] :
~ ( member7453568604450474000od_v_v @ A4 @ bot_bo723834152578015283od_v_v ) ).
% emptyE
thf(fact_388_emptyE,axiom,
! [A4: set_v] :
~ ( member_set_v @ A4 @ bot_bot_set_set_v ) ).
% emptyE
thf(fact_389_emptyE,axiom,
! [A4: v] :
~ ( member_v @ A4 @ bot_bot_set_v ) ).
% emptyE
thf(fact_390_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_391_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_392_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_393_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_394_subset__singleton__iff,axiom,
! [X5: set_set_v,A4: set_v] :
( ( ord_le5216385588623774835_set_v @ X5 @ ( insert_set_v @ A4 @ bot_bot_set_set_v ) )
= ( ( X5 = bot_bot_set_set_v )
| ( X5
= ( insert_set_v @ A4 @ bot_bot_set_set_v ) ) ) ) ).
% subset_singleton_iff
thf(fact_395_subset__singleton__iff,axiom,
! [X5: set_Product_prod_v_v,A4: product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X5 @ ( insert1338601472111419319od_v_v @ A4 @ bot_bo723834152578015283od_v_v ) )
= ( ( X5 = bot_bo723834152578015283od_v_v )
| ( X5
= ( insert1338601472111419319od_v_v @ A4 @ bot_bo723834152578015283od_v_v ) ) ) ) ).
% subset_singleton_iff
thf(fact_396_subset__singleton__iff,axiom,
! [X5: set_v,A4: v] :
( ( ord_less_eq_set_v @ X5 @ ( insert_v @ A4 @ bot_bot_set_v ) )
= ( ( X5 = bot_bot_set_v )
| ( X5
= ( insert_v @ A4 @ bot_bot_set_v ) ) ) ) ).
% subset_singleton_iff
thf(fact_397_subset__singletonD,axiom,
! [A: set_set_v,X: set_v] :
( ( ord_le5216385588623774835_set_v @ A @ ( insert_set_v @ X @ bot_bot_set_set_v ) )
=> ( ( A = bot_bot_set_set_v )
| ( A
= ( insert_set_v @ X @ bot_bot_set_set_v ) ) ) ) ).
% subset_singletonD
thf(fact_398_subset__singletonD,axiom,
! [A: set_Product_prod_v_v,X: product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v ) )
=> ( ( A = bot_bo723834152578015283od_v_v )
| ( A
= ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v ) ) ) ) ).
% subset_singletonD
thf(fact_399_subset__singletonD,axiom,
! [A: set_v,X: v] :
( ( ord_less_eq_set_v @ A @ ( insert_v @ X @ bot_bot_set_v ) )
=> ( ( A = bot_bot_set_v )
| ( A
= ( insert_v @ X @ bot_bot_set_v ) ) ) ) ).
% subset_singletonD
thf(fact_400_bot_Oextremum__uniqueI,axiom,
! [A4: set_set_v] :
( ( ord_le5216385588623774835_set_v @ A4 @ bot_bot_set_set_v )
=> ( A4 = bot_bot_set_set_v ) ) ).
% bot.extremum_uniqueI
thf(fact_401_bot_Oextremum__uniqueI,axiom,
! [A4: product_unit] :
( ( ord_le3221252021190050221t_unit @ A4 @ bot_bot_Product_unit )
=> ( A4 = bot_bot_Product_unit ) ) ).
% bot.extremum_uniqueI
thf(fact_402_bot_Oextremum__uniqueI,axiom,
! [A4: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A4 @ bot_bo723834152578015283od_v_v )
=> ( A4 = bot_bo723834152578015283od_v_v ) ) ).
% bot.extremum_uniqueI
thf(fact_403_bot_Oextremum__uniqueI,axiom,
! [A4: set_v] :
( ( ord_less_eq_set_v @ A4 @ bot_bot_set_v )
=> ( A4 = bot_bot_set_v ) ) ).
% bot.extremum_uniqueI
thf(fact_404_bot_Oextremum__unique,axiom,
! [A4: set_set_v] :
( ( ord_le5216385588623774835_set_v @ A4 @ bot_bot_set_set_v )
= ( A4 = bot_bot_set_set_v ) ) ).
% bot.extremum_unique
thf(fact_405_bot_Oextremum__unique,axiom,
! [A4: product_unit] :
( ( ord_le3221252021190050221t_unit @ A4 @ bot_bot_Product_unit )
= ( A4 = bot_bot_Product_unit ) ) ).
% bot.extremum_unique
thf(fact_406_bot_Oextremum__unique,axiom,
! [A4: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A4 @ bot_bo723834152578015283od_v_v )
= ( A4 = bot_bo723834152578015283od_v_v ) ) ).
% bot.extremum_unique
thf(fact_407_bot_Oextremum__unique,axiom,
! [A4: set_v] :
( ( ord_less_eq_set_v @ A4 @ bot_bot_set_v )
= ( A4 = bot_bot_set_v ) ) ).
% bot.extremum_unique
thf(fact_408_bot_Oextremum,axiom,
! [A4: set_set_v] : ( ord_le5216385588623774835_set_v @ bot_bot_set_set_v @ A4 ) ).
% bot.extremum
thf(fact_409_bot_Oextremum,axiom,
! [A4: product_unit] : ( ord_le3221252021190050221t_unit @ bot_bot_Product_unit @ A4 ) ).
% bot.extremum
thf(fact_410_bot_Oextremum,axiom,
! [A4: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ bot_bo723834152578015283od_v_v @ A4 ) ).
% bot.extremum
thf(fact_411_bot_Oextremum,axiom,
! [A4: set_v] : ( ord_less_eq_set_v @ bot_bot_set_v @ A4 ) ).
% bot.extremum
thf(fact_412_subset__Un__eq,axiom,
( ord_le5216385588623774835_set_v
= ( ^ [A8: set_set_v,B6: set_set_v] :
( ( sup_sup_set_set_v @ A8 @ B6 )
= B6 ) ) ) ).
% subset_Un_eq
thf(fact_413_subset__Un__eq,axiom,
( ord_le7336532860387713383od_v_v
= ( ^ [A8: set_Product_prod_v_v,B6: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ A8 @ B6 )
= B6 ) ) ) ).
% subset_Un_eq
thf(fact_414_subset__Un__eq,axiom,
( ord_less_eq_set_v
= ( ^ [A8: set_v,B6: set_v] :
( ( sup_sup_set_v @ A8 @ B6 )
= B6 ) ) ) ).
% subset_Un_eq
thf(fact_415_subset__UnE,axiom,
! [C2: set_set_v,A: set_set_v,B: set_set_v] :
( ( ord_le5216385588623774835_set_v @ C2 @ ( sup_sup_set_set_v @ A @ B ) )
=> ~ ! [A9: set_set_v] :
( ( ord_le5216385588623774835_set_v @ A9 @ A )
=> ! [B8: set_set_v] :
( ( ord_le5216385588623774835_set_v @ B8 @ B )
=> ( C2
!= ( sup_sup_set_set_v @ A9 @ B8 ) ) ) ) ) ).
% subset_UnE
thf(fact_416_subset__UnE,axiom,
! [C2: set_Product_prod_v_v,A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ C2 @ ( sup_su414716646722978715od_v_v @ A @ B ) )
=> ~ ! [A9: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A9 @ A )
=> ! [B8: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ B8 @ B )
=> ( C2
!= ( sup_su414716646722978715od_v_v @ A9 @ B8 ) ) ) ) ) ).
% subset_UnE
thf(fact_417_subset__UnE,axiom,
! [C2: set_v,A: set_v,B: set_v] :
( ( ord_less_eq_set_v @ C2 @ ( sup_sup_set_v @ A @ B ) )
=> ~ ! [A9: set_v] :
( ( ord_less_eq_set_v @ A9 @ A )
=> ! [B8: set_v] :
( ( ord_less_eq_set_v @ B8 @ B )
=> ( C2
!= ( sup_sup_set_v @ A9 @ B8 ) ) ) ) ) ).
% subset_UnE
thf(fact_418_Un__absorb2,axiom,
! [B: set_set_v,A: set_set_v] :
( ( ord_le5216385588623774835_set_v @ B @ A )
=> ( ( sup_sup_set_set_v @ A @ B )
= A ) ) ).
% Un_absorb2
thf(fact_419_Un__absorb2,axiom,
! [B: set_Product_prod_v_v,A: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ B @ A )
=> ( ( sup_su414716646722978715od_v_v @ A @ B )
= A ) ) ).
% Un_absorb2
thf(fact_420_Un__absorb2,axiom,
! [B: set_v,A: set_v] :
( ( ord_less_eq_set_v @ B @ A )
=> ( ( sup_sup_set_v @ A @ B )
= A ) ) ).
% Un_absorb2
thf(fact_421_Un__absorb1,axiom,
! [A: set_set_v,B: set_set_v] :
( ( ord_le5216385588623774835_set_v @ A @ B )
=> ( ( sup_sup_set_set_v @ A @ B )
= B ) ) ).
% Un_absorb1
thf(fact_422_Un__absorb1,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ B )
=> ( ( sup_su414716646722978715od_v_v @ A @ B )
= B ) ) ).
% Un_absorb1
thf(fact_423_Un__absorb1,axiom,
! [A: set_v,B: set_v] :
( ( ord_less_eq_set_v @ A @ B )
=> ( ( sup_sup_set_v @ A @ B )
= B ) ) ).
% Un_absorb1
thf(fact_424_Un__upper2,axiom,
! [B: set_set_v,A: set_set_v] : ( ord_le5216385588623774835_set_v @ B @ ( sup_sup_set_set_v @ A @ B ) ) ).
% Un_upper2
thf(fact_425_Un__upper2,axiom,
! [B: set_Product_prod_v_v,A: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ B @ ( sup_su414716646722978715od_v_v @ A @ B ) ) ).
% Un_upper2
thf(fact_426_Un__upper2,axiom,
! [B: set_v,A: set_v] : ( ord_less_eq_set_v @ B @ ( sup_sup_set_v @ A @ B ) ) ).
% Un_upper2
thf(fact_427_Un__upper1,axiom,
! [A: set_set_v,B: set_set_v] : ( ord_le5216385588623774835_set_v @ A @ ( sup_sup_set_set_v @ A @ B ) ) ).
% Un_upper1
thf(fact_428_Un__upper1,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ A @ ( sup_su414716646722978715od_v_v @ A @ B ) ) ).
% Un_upper1
thf(fact_429_Un__upper1,axiom,
! [A: set_v,B: set_v] : ( ord_less_eq_set_v @ A @ ( sup_sup_set_v @ A @ B ) ) ).
% Un_upper1
thf(fact_430_Un__least,axiom,
! [A: set_set_v,C2: set_set_v,B: set_set_v] :
( ( ord_le5216385588623774835_set_v @ A @ C2 )
=> ( ( ord_le5216385588623774835_set_v @ B @ C2 )
=> ( ord_le5216385588623774835_set_v @ ( sup_sup_set_set_v @ A @ B ) @ C2 ) ) ) ).
% Un_least
thf(fact_431_Un__least,axiom,
! [A: set_Product_prod_v_v,C2: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ C2 )
=> ( ( ord_le7336532860387713383od_v_v @ B @ C2 )
=> ( ord_le7336532860387713383od_v_v @ ( sup_su414716646722978715od_v_v @ A @ B ) @ C2 ) ) ) ).
% Un_least
thf(fact_432_Un__least,axiom,
! [A: set_v,C2: set_v,B: set_v] :
( ( ord_less_eq_set_v @ A @ C2 )
=> ( ( ord_less_eq_set_v @ B @ C2 )
=> ( ord_less_eq_set_v @ ( sup_sup_set_v @ A @ B ) @ C2 ) ) ) ).
% Un_least
thf(fact_433_Un__mono,axiom,
! [A: set_set_v,C2: set_set_v,B: set_set_v,D2: set_set_v] :
( ( ord_le5216385588623774835_set_v @ A @ C2 )
=> ( ( ord_le5216385588623774835_set_v @ B @ D2 )
=> ( ord_le5216385588623774835_set_v @ ( sup_sup_set_set_v @ A @ B ) @ ( sup_sup_set_set_v @ C2 @ D2 ) ) ) ) ).
% Un_mono
thf(fact_434_Un__mono,axiom,
! [A: set_Product_prod_v_v,C2: set_Product_prod_v_v,B: set_Product_prod_v_v,D2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ C2 )
=> ( ( ord_le7336532860387713383od_v_v @ B @ D2 )
=> ( ord_le7336532860387713383od_v_v @ ( sup_su414716646722978715od_v_v @ A @ B ) @ ( sup_su414716646722978715od_v_v @ C2 @ D2 ) ) ) ) ).
% Un_mono
thf(fact_435_Un__mono,axiom,
! [A: set_v,C2: set_v,B: set_v,D2: set_v] :
( ( ord_less_eq_set_v @ A @ C2 )
=> ( ( ord_less_eq_set_v @ B @ D2 )
=> ( ord_less_eq_set_v @ ( sup_sup_set_v @ A @ B ) @ ( sup_sup_set_v @ C2 @ D2 ) ) ) ) ).
% Un_mono
thf(fact_436_subset__insertI2,axiom,
! [A: set_set_v,B: set_set_v,B2: set_v] :
( ( ord_le5216385588623774835_set_v @ A @ B )
=> ( ord_le5216385588623774835_set_v @ A @ ( insert_set_v @ B2 @ B ) ) ) ).
% subset_insertI2
thf(fact_437_subset__insertI2,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v,B2: product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ B )
=> ( ord_le7336532860387713383od_v_v @ A @ ( insert1338601472111419319od_v_v @ B2 @ B ) ) ) ).
% subset_insertI2
thf(fact_438_subset__insertI2,axiom,
! [A: set_v,B: set_v,B2: v] :
( ( ord_less_eq_set_v @ A @ B )
=> ( ord_less_eq_set_v @ A @ ( insert_v @ B2 @ B ) ) ) ).
% subset_insertI2
thf(fact_439_subset__insertI,axiom,
! [B: set_set_v,A4: set_v] : ( ord_le5216385588623774835_set_v @ B @ ( insert_set_v @ A4 @ B ) ) ).
% subset_insertI
thf(fact_440_subset__insertI,axiom,
! [B: set_Product_prod_v_v,A4: product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ B @ ( insert1338601472111419319od_v_v @ A4 @ B ) ) ).
% subset_insertI
thf(fact_441_subset__insertI,axiom,
! [B: set_v,A4: v] : ( ord_less_eq_set_v @ B @ ( insert_v @ A4 @ B ) ) ).
% subset_insertI
thf(fact_442_subset__insert,axiom,
! [X: set_v,A: set_set_v,B: set_set_v] :
( ~ ( member_set_v @ X @ A )
=> ( ( ord_le5216385588623774835_set_v @ A @ ( insert_set_v @ X @ B ) )
= ( ord_le5216385588623774835_set_v @ A @ B ) ) ) ).
% subset_insert
thf(fact_443_subset__insert,axiom,
! [X: product_prod_v_v,A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ~ ( member7453568604450474000od_v_v @ X @ A )
=> ( ( ord_le7336532860387713383od_v_v @ A @ ( insert1338601472111419319od_v_v @ X @ B ) )
= ( ord_le7336532860387713383od_v_v @ A @ B ) ) ) ).
% subset_insert
thf(fact_444_subset__insert,axiom,
! [X: v,A: set_v,B: set_v] :
( ~ ( member_v @ X @ A )
=> ( ( ord_less_eq_set_v @ A @ ( insert_v @ X @ B ) )
= ( ord_less_eq_set_v @ A @ B ) ) ) ).
% subset_insert
thf(fact_445_insert__mono,axiom,
! [C2: set_set_v,D2: set_set_v,A4: set_v] :
( ( ord_le5216385588623774835_set_v @ C2 @ D2 )
=> ( ord_le5216385588623774835_set_v @ ( insert_set_v @ A4 @ C2 ) @ ( insert_set_v @ A4 @ D2 ) ) ) ).
% insert_mono
thf(fact_446_insert__mono,axiom,
! [C2: set_Product_prod_v_v,D2: set_Product_prod_v_v,A4: product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ C2 @ D2 )
=> ( ord_le7336532860387713383od_v_v @ ( insert1338601472111419319od_v_v @ A4 @ C2 ) @ ( insert1338601472111419319od_v_v @ A4 @ D2 ) ) ) ).
% insert_mono
thf(fact_447_insert__mono,axiom,
! [C2: set_v,D2: set_v,A4: v] :
( ( ord_less_eq_set_v @ C2 @ D2 )
=> ( ord_less_eq_set_v @ ( insert_v @ A4 @ C2 ) @ ( insert_v @ A4 @ D2 ) ) ) ).
% insert_mono
thf(fact_448_list_Osimps_I15_J,axiom,
! [X21: product_prod_v_v,X222: list_P7986770385144383213od_v_v] :
( ( set_Product_prod_v_v2 @ ( cons_P4120604216776828829od_v_v @ X21 @ X222 ) )
= ( insert1338601472111419319od_v_v @ X21 @ ( set_Product_prod_v_v2 @ X222 ) ) ) ).
% list.simps(15)
thf(fact_449_list_Osimps_I15_J,axiom,
! [X21: set_v,X222: list_set_v] :
( ( set_set_v2 @ ( cons_set_v @ X21 @ X222 ) )
= ( insert_set_v @ X21 @ ( set_set_v2 @ X222 ) ) ) ).
% list.simps(15)
thf(fact_450_list_Osimps_I15_J,axiom,
! [X21: v,X222: list_v] :
( ( set_v2 @ ( cons_v @ X21 @ X222 ) )
= ( insert_v @ X21 @ ( set_v2 @ X222 ) ) ) ).
% list.simps(15)
thf(fact_451_sup__bot_Oright__neutral,axiom,
! [A4: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ A4 @ bot_bo723834152578015283od_v_v )
= A4 ) ).
% sup_bot.right_neutral
thf(fact_452_sup__bot_Oright__neutral,axiom,
! [A4: set_set_v] :
( ( sup_sup_set_set_v @ A4 @ bot_bot_set_set_v )
= A4 ) ).
% sup_bot.right_neutral
thf(fact_453_sup__bot_Oright__neutral,axiom,
! [A4: set_v] :
( ( sup_sup_set_v @ A4 @ bot_bot_set_v )
= A4 ) ).
% sup_bot.right_neutral
thf(fact_454_sup__bot_Oright__neutral,axiom,
! [A4: product_unit] :
( ( sup_sup_Product_unit @ A4 @ bot_bot_Product_unit )
= A4 ) ).
% sup_bot.right_neutral
thf(fact_455_sup__bot_Oneutr__eq__iff,axiom,
! [A4: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( bot_bo723834152578015283od_v_v
= ( sup_su414716646722978715od_v_v @ A4 @ B2 ) )
= ( ( A4 = bot_bo723834152578015283od_v_v )
& ( B2 = bot_bo723834152578015283od_v_v ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_456_sup__bot_Oneutr__eq__iff,axiom,
! [A4: set_set_v,B2: set_set_v] :
( ( bot_bot_set_set_v
= ( sup_sup_set_set_v @ A4 @ B2 ) )
= ( ( A4 = bot_bot_set_set_v )
& ( B2 = bot_bot_set_set_v ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_457_sup__bot_Oneutr__eq__iff,axiom,
! [A4: set_v,B2: set_v] :
( ( bot_bot_set_v
= ( sup_sup_set_v @ A4 @ B2 ) )
= ( ( A4 = bot_bot_set_v )
& ( B2 = bot_bot_set_v ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_458_sup__bot_Oneutr__eq__iff,axiom,
! [A4: product_unit,B2: product_unit] :
( ( bot_bot_Product_unit
= ( sup_sup_Product_unit @ A4 @ B2 ) )
= ( ( A4 = bot_bot_Product_unit )
& ( B2 = bot_bot_Product_unit ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_459_sup__bot_Oleft__neutral,axiom,
! [A4: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ bot_bo723834152578015283od_v_v @ A4 )
= A4 ) ).
% sup_bot.left_neutral
thf(fact_460_sup__bot_Oleft__neutral,axiom,
! [A4: set_set_v] :
( ( sup_sup_set_set_v @ bot_bot_set_set_v @ A4 )
= A4 ) ).
% sup_bot.left_neutral
thf(fact_461_sup__bot_Oleft__neutral,axiom,
! [A4: set_v] :
( ( sup_sup_set_v @ bot_bot_set_v @ A4 )
= A4 ) ).
% sup_bot.left_neutral
thf(fact_462_sup__bot_Oleft__neutral,axiom,
! [A4: product_unit] :
( ( sup_sup_Product_unit @ bot_bot_Product_unit @ A4 )
= A4 ) ).
% sup_bot.left_neutral
thf(fact_463_sup__bot_Oeq__neutr__iff,axiom,
! [A4: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( ( sup_su414716646722978715od_v_v @ A4 @ B2 )
= bot_bo723834152578015283od_v_v )
= ( ( A4 = bot_bo723834152578015283od_v_v )
& ( B2 = bot_bo723834152578015283od_v_v ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_464_sup__bot_Oeq__neutr__iff,axiom,
! [A4: set_set_v,B2: set_set_v] :
( ( ( sup_sup_set_set_v @ A4 @ B2 )
= bot_bot_set_set_v )
= ( ( A4 = bot_bot_set_set_v )
& ( B2 = bot_bot_set_set_v ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_465_sup__bot_Oeq__neutr__iff,axiom,
! [A4: set_v,B2: set_v] :
( ( ( sup_sup_set_v @ A4 @ B2 )
= bot_bot_set_v )
= ( ( A4 = bot_bot_set_v )
& ( B2 = bot_bot_set_v ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_466_sup__bot_Oeq__neutr__iff,axiom,
! [A4: product_unit,B2: product_unit] :
( ( ( sup_sup_Product_unit @ A4 @ B2 )
= bot_bot_Product_unit )
= ( ( A4 = bot_bot_Product_unit )
& ( B2 = bot_bot_Product_unit ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_467_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_468_sup__eq__bot__iff,axiom,
! [X: set_set_v,Y: set_set_v] :
( ( ( sup_sup_set_set_v @ X @ Y )
= bot_bot_set_set_v )
= ( ( X = bot_bot_set_set_v )
& ( Y = bot_bot_set_set_v ) ) ) ).
% sup_eq_bot_iff
thf(fact_469_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_470_sup__eq__bot__iff,axiom,
! [X: product_unit,Y: product_unit] :
( ( ( sup_sup_Product_unit @ X @ Y )
= bot_bot_Product_unit )
= ( ( X = bot_bot_Product_unit )
& ( Y = bot_bot_Product_unit ) ) ) ).
% sup_eq_bot_iff
thf(fact_471_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_472_bot__eq__sup__iff,axiom,
! [X: set_set_v,Y: set_set_v] :
( ( bot_bot_set_set_v
= ( sup_sup_set_set_v @ X @ Y ) )
= ( ( X = bot_bot_set_set_v )
& ( Y = bot_bot_set_set_v ) ) ) ).
% bot_eq_sup_iff
thf(fact_473_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_474_bot__eq__sup__iff,axiom,
! [X: product_unit,Y: product_unit] :
( ( bot_bot_Product_unit
= ( sup_sup_Product_unit @ X @ Y ) )
= ( ( X = bot_bot_Product_unit )
& ( Y = bot_bot_Product_unit ) ) ) ).
% bot_eq_sup_iff
thf(fact_475_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_476_sup__bot__right,axiom,
! [X: set_set_v] :
( ( sup_sup_set_set_v @ X @ bot_bot_set_set_v )
= X ) ).
% sup_bot_right
thf(fact_477_sup__bot__right,axiom,
! [X: set_v] :
( ( sup_sup_set_v @ X @ bot_bot_set_v )
= X ) ).
% sup_bot_right
thf(fact_478_sup__bot__right,axiom,
! [X: product_unit] :
( ( sup_sup_Product_unit @ X @ bot_bot_Product_unit )
= X ) ).
% sup_bot_right
thf(fact_479_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_480_sup__bot__left,axiom,
! [X: set_set_v] :
( ( sup_sup_set_set_v @ bot_bot_set_set_v @ X )
= X ) ).
% sup_bot_left
thf(fact_481_sup__bot__left,axiom,
! [X: set_v] :
( ( sup_sup_set_v @ bot_bot_set_v @ X )
= X ) ).
% sup_bot_left
thf(fact_482_sup__bot__left,axiom,
! [X: product_unit] :
( ( sup_sup_Product_unit @ bot_bot_Product_unit @ X )
= X ) ).
% sup_bot_left
thf(fact_483_list_Oinject,axiom,
! [X21: v,X222: list_v,Y21: v,Y222: list_v] :
( ( ( cons_v @ X21 @ X222 )
= ( cons_v @ Y21 @ Y222 ) )
= ( ( X21 = Y21 )
& ( X222 = Y222 ) ) ) ).
% list.inject
thf(fact_484_sup_Oidem,axiom,
! [A4: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ A4 @ A4 )
= A4 ) ).
% sup.idem
thf(fact_485_sup_Oidem,axiom,
! [A4: set_v] :
( ( sup_sup_set_v @ A4 @ A4 )
= A4 ) ).
% sup.idem
thf(fact_486_sup_Oidem,axiom,
! [A4: set_set_v] :
( ( sup_sup_set_set_v @ A4 @ A4 )
= A4 ) ).
% sup.idem
thf(fact_487_sup_Oidem,axiom,
! [A4: product_unit] :
( ( sup_sup_Product_unit @ A4 @ A4 )
= A4 ) ).
% sup.idem
thf(fact_488_sup__idem,axiom,
! [X: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ X @ X )
= X ) ).
% sup_idem
thf(fact_489_sup__idem,axiom,
! [X: set_v] :
( ( sup_sup_set_v @ X @ X )
= X ) ).
% sup_idem
thf(fact_490_sup__idem,axiom,
! [X: set_set_v] :
( ( sup_sup_set_set_v @ X @ X )
= X ) ).
% sup_idem
thf(fact_491_sup__idem,axiom,
! [X: product_unit] :
( ( sup_sup_Product_unit @ X @ X )
= X ) ).
% sup_idem
thf(fact_492_sup_Oleft__idem,axiom,
! [A4: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ A4 @ ( sup_su414716646722978715od_v_v @ A4 @ B2 ) )
= ( sup_su414716646722978715od_v_v @ A4 @ B2 ) ) ).
% sup.left_idem
thf(fact_493_sup_Oleft__idem,axiom,
! [A4: set_v,B2: set_v] :
( ( sup_sup_set_v @ A4 @ ( sup_sup_set_v @ A4 @ B2 ) )
= ( sup_sup_set_v @ A4 @ B2 ) ) ).
% sup.left_idem
thf(fact_494_sup_Oleft__idem,axiom,
! [A4: set_set_v,B2: set_set_v] :
( ( sup_sup_set_set_v @ A4 @ ( sup_sup_set_set_v @ A4 @ B2 ) )
= ( sup_sup_set_set_v @ A4 @ B2 ) ) ).
% sup.left_idem
thf(fact_495_sup_Oleft__idem,axiom,
! [A4: product_unit,B2: product_unit] :
( ( sup_sup_Product_unit @ A4 @ ( sup_sup_Product_unit @ A4 @ B2 ) )
= ( sup_sup_Product_unit @ A4 @ B2 ) ) ).
% sup.left_idem
thf(fact_496_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_497_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_498_sup__left__idem,axiom,
! [X: set_set_v,Y: set_set_v] :
( ( sup_sup_set_set_v @ X @ ( sup_sup_set_set_v @ X @ Y ) )
= ( sup_sup_set_set_v @ X @ Y ) ) ).
% sup_left_idem
thf(fact_499_sup__left__idem,axiom,
! [X: product_unit,Y: product_unit] :
( ( sup_sup_Product_unit @ X @ ( sup_sup_Product_unit @ X @ Y ) )
= ( sup_sup_Product_unit @ X @ Y ) ) ).
% sup_left_idem
thf(fact_500_sup_Oright__idem,axiom,
! [A4: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ ( sup_su414716646722978715od_v_v @ A4 @ B2 ) @ B2 )
= ( sup_su414716646722978715od_v_v @ A4 @ B2 ) ) ).
% sup.right_idem
thf(fact_501_sup_Oright__idem,axiom,
! [A4: set_v,B2: set_v] :
( ( sup_sup_set_v @ ( sup_sup_set_v @ A4 @ B2 ) @ B2 )
= ( sup_sup_set_v @ A4 @ B2 ) ) ).
% sup.right_idem
thf(fact_502_sup_Oright__idem,axiom,
! [A4: set_set_v,B2: set_set_v] :
( ( sup_sup_set_set_v @ ( sup_sup_set_set_v @ A4 @ B2 ) @ B2 )
= ( sup_sup_set_set_v @ A4 @ B2 ) ) ).
% sup.right_idem
thf(fact_503_sup_Oright__idem,axiom,
! [A4: product_unit,B2: product_unit] :
( ( sup_sup_Product_unit @ ( sup_sup_Product_unit @ A4 @ B2 ) @ B2 )
= ( sup_sup_Product_unit @ A4 @ B2 ) ) ).
% sup.right_idem
thf(fact_504_le__sup__iff,axiom,
! [X: set_set_v,Y: set_set_v,Z: set_set_v] :
( ( ord_le5216385588623774835_set_v @ ( sup_sup_set_set_v @ X @ Y ) @ Z )
= ( ( ord_le5216385588623774835_set_v @ X @ Z )
& ( ord_le5216385588623774835_set_v @ Y @ Z ) ) ) ).
% le_sup_iff
thf(fact_505_le__sup__iff,axiom,
! [X: product_unit,Y: product_unit,Z: product_unit] :
( ( ord_le3221252021190050221t_unit @ ( sup_sup_Product_unit @ X @ Y ) @ Z )
= ( ( ord_le3221252021190050221t_unit @ X @ Z )
& ( ord_le3221252021190050221t_unit @ Y @ Z ) ) ) ).
% le_sup_iff
thf(fact_506_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_507_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_508_sup_Obounded__iff,axiom,
! [B2: set_set_v,C: set_set_v,A4: set_set_v] :
( ( ord_le5216385588623774835_set_v @ ( sup_sup_set_set_v @ B2 @ C ) @ A4 )
= ( ( ord_le5216385588623774835_set_v @ B2 @ A4 )
& ( ord_le5216385588623774835_set_v @ C @ A4 ) ) ) ).
% sup.bounded_iff
thf(fact_509_sup_Obounded__iff,axiom,
! [B2: product_unit,C: product_unit,A4: product_unit] :
( ( ord_le3221252021190050221t_unit @ ( sup_sup_Product_unit @ B2 @ C ) @ A4 )
= ( ( ord_le3221252021190050221t_unit @ B2 @ A4 )
& ( ord_le3221252021190050221t_unit @ C @ A4 ) ) ) ).
% sup.bounded_iff
thf(fact_510_sup_Obounded__iff,axiom,
! [B2: set_Product_prod_v_v,C: set_Product_prod_v_v,A4: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ ( sup_su414716646722978715od_v_v @ B2 @ C ) @ A4 )
= ( ( ord_le7336532860387713383od_v_v @ B2 @ A4 )
& ( ord_le7336532860387713383od_v_v @ C @ A4 ) ) ) ).
% sup.bounded_iff
thf(fact_511_sup_Obounded__iff,axiom,
! [B2: set_v,C: set_v,A4: set_v] :
( ( ord_less_eq_set_v @ ( sup_sup_set_v @ B2 @ C ) @ A4 )
= ( ( ord_less_eq_set_v @ B2 @ A4 )
& ( ord_less_eq_set_v @ C @ A4 ) ) ) ).
% sup.bounded_iff
thf(fact_512_graph_Odfss_Ocases,axiom,
! [Vertices: set_v,Successors: v > set_v,X: produc5741669702376414499t_unit] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ~ ! [V2: v,E7: sCC_Bl1394983891496994913t_unit] :
( X
!= ( produc3862955338007567901t_unit @ V2 @ E7 ) ) ) ).
% graph.dfss.cases
thf(fact_513_bot__set__def,axiom,
( bot_bo723834152578015283od_v_v
= ( collec140062887454715474od_v_v @ bot_bo8461541820394803818_v_v_o ) ) ).
% bot_set_def
thf(fact_514_bot__set__def,axiom,
( bot_bot_set_set_v
= ( collect_set_v @ bot_bot_set_v_o ) ) ).
% bot_set_def
thf(fact_515_bot__set__def,axiom,
( bot_bot_set_v
= ( collect_v @ bot_bot_v_o ) ) ).
% bot_set_def
thf(fact_516_graph_Osclosed,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ! [X4: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X4 @ Vertices )
=> ( ord_le7336532860387713383od_v_v @ ( Successors @ X4 ) @ Vertices ) ) ) ).
% graph.sclosed
thf(fact_517_graph_Osclosed,axiom,
! [Vertices: set_v,Successors: v > set_v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ! [X4: v] :
( ( member_v @ X4 @ Vertices )
=> ( ord_less_eq_set_v @ ( Successors @ X4 ) @ Vertices ) ) ) ).
% graph.sclosed
thf(fact_518_graph_Ora__trans,axiom,
! [Vertices: set_v,Successors: v > set_v,X: v,Y: v,E4: set_Product_prod_v_v,Z: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl4291963740693775144ding_v @ Successors @ X @ Y @ E4 )
=> ( ( sCC_Bl4291963740693775144ding_v @ Successors @ Y @ Z @ E4 )
=> ( sCC_Bl4291963740693775144ding_v @ Successors @ X @ Z @ E4 ) ) ) ) ).
% graph.ra_trans
thf(fact_519_graph_Ora__refl,axiom,
! [Vertices: set_v,Successors: v > set_v,X: v,E4: set_Product_prod_v_v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( sCC_Bl4291963740693775144ding_v @ Successors @ X @ X @ E4 ) ) ).
% graph.ra_refl
thf(fact_520_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_521_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_522_graph_Oreachable__end_Ocases,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,A1: product_prod_v_v,A2: product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( sCC_Bl4714988717384592488od_v_v @ Successors @ A1 @ A2 )
=> ( ( A2 != A1 )
=> ~ ! [Y2: product_prod_v_v] :
( ( sCC_Bl4714988717384592488od_v_v @ Successors @ A1 @ Y2 )
=> ~ ( member7453568604450474000od_v_v @ A2 @ ( Successors @ Y2 ) ) ) ) ) ) ).
% graph.reachable_end.cases
thf(fact_523_graph_Oreachable__end_Ocases,axiom,
! [Vertices: set_v,Successors: v > set_v,A1: v,A2: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl770211535891879572_end_v @ Successors @ A1 @ A2 )
=> ( ( A2 != A1 )
=> ~ ! [Y2: v] :
( ( sCC_Bl770211535891879572_end_v @ Successors @ A1 @ Y2 )
=> ~ ( member_v @ A2 @ ( Successors @ Y2 ) ) ) ) ) ) ).
% graph.reachable_end.cases
thf(fact_524_graph_Oreachable__end_Osimps,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,A1: product_prod_v_v,A2: product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( sCC_Bl4714988717384592488od_v_v @ Successors @ A1 @ A2 )
= ( ? [X2: product_prod_v_v] :
( ( A1 = X2 )
& ( A2 = X2 ) )
| ? [X2: product_prod_v_v,Y3: product_prod_v_v,Z2: product_prod_v_v] :
( ( A1 = X2 )
& ( A2 = Z2 )
& ( sCC_Bl4714988717384592488od_v_v @ Successors @ X2 @ Y3 )
& ( member7453568604450474000od_v_v @ Z2 @ ( Successors @ Y3 ) ) ) ) ) ) ).
% graph.reachable_end.simps
thf(fact_525_graph_Oreachable__end_Osimps,axiom,
! [Vertices: set_v,Successors: v > set_v,A1: v,A2: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl770211535891879572_end_v @ Successors @ A1 @ A2 )
= ( ? [X2: v] :
( ( A1 = X2 )
& ( A2 = X2 ) )
| ? [X2: v,Y3: v,Z2: v] :
( ( A1 = X2 )
& ( A2 = Z2 )
& ( sCC_Bl770211535891879572_end_v @ Successors @ X2 @ Y3 )
& ( member_v @ Z2 @ ( Successors @ Y3 ) ) ) ) ) ) ).
% graph.reachable_end.simps
thf(fact_526_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_527_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_528_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_529_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_530_graph_Oedge__ra,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,Y: product_prod_v_v,X: product_prod_v_v,E4: set_Pr2149350503807050951od_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( member7453568604450474000od_v_v @ Y @ ( Successors @ X ) )
=> ( ~ ( member3038538357316246288od_v_v @ ( produc4031800376763917143od_v_v @ X @ Y ) @ E4 )
=> ( sCC_Bl5370300055464682748od_v_v @ Successors @ X @ Y @ E4 ) ) ) ) ).
% graph.edge_ra
thf(fact_531_graph_Oedge__ra,axiom,
! [Vertices: set_v,Successors: v > set_v,Y: v,X: v,E4: set_Product_prod_v_v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( member_v @ Y @ ( Successors @ X ) )
=> ( ~ ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ X @ Y ) @ E4 )
=> ( sCC_Bl4291963740693775144ding_v @ Successors @ X @ Y @ E4 ) ) ) ) ).
% graph.edge_ra
thf(fact_532_graph_Ora__cases,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,X: product_prod_v_v,Y: product_prod_v_v,E4: set_Pr2149350503807050951od_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( sCC_Bl5370300055464682748od_v_v @ Successors @ X @ Y @ E4 )
=> ( ( X = Y )
| ? [Z3: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ Z3 @ ( Successors @ X ) )
& ~ ( member3038538357316246288od_v_v @ ( produc4031800376763917143od_v_v @ X @ Z3 ) @ E4 )
& ( sCC_Bl5370300055464682748od_v_v @ Successors @ Z3 @ Y @ E4 ) ) ) ) ) ).
% graph.ra_cases
thf(fact_533_graph_Ora__cases,axiom,
! [Vertices: set_v,Successors: v > set_v,X: v,Y: v,E4: set_Product_prod_v_v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl4291963740693775144ding_v @ Successors @ X @ Y @ E4 )
=> ( ( X = Y )
| ? [Z3: v] :
( ( member_v @ Z3 @ ( Successors @ X ) )
& ~ ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ X @ Z3 ) @ E4 )
& ( sCC_Bl4291963740693775144ding_v @ Successors @ Z3 @ Y @ E4 ) ) ) ) ) ).
% graph.ra_cases
thf(fact_534_graph_Oreachable__avoiding_Ocases,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,A1: product_prod_v_v,A2: product_prod_v_v,A3: set_Pr2149350503807050951od_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( sCC_Bl5370300055464682748od_v_v @ Successors @ A1 @ A2 @ A3 )
=> ( ( A2 != A1 )
=> ~ ! [Y2: product_prod_v_v] :
( ( sCC_Bl5370300055464682748od_v_v @ Successors @ A1 @ Y2 @ A3 )
=> ( ( member7453568604450474000od_v_v @ A2 @ ( Successors @ Y2 ) )
=> ( member3038538357316246288od_v_v @ ( produc4031800376763917143od_v_v @ Y2 @ A2 ) @ A3 ) ) ) ) ) ) ).
% graph.reachable_avoiding.cases
thf(fact_535_graph_Oreachable__avoiding_Ocases,axiom,
! [Vertices: set_v,Successors: v > set_v,A1: v,A2: v,A3: set_Product_prod_v_v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl4291963740693775144ding_v @ Successors @ A1 @ A2 @ A3 )
=> ( ( A2 != A1 )
=> ~ ! [Y2: v] :
( ( sCC_Bl4291963740693775144ding_v @ Successors @ A1 @ Y2 @ A3 )
=> ( ( member_v @ A2 @ ( Successors @ Y2 ) )
=> ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ Y2 @ A2 ) @ A3 ) ) ) ) ) ) ).
% graph.reachable_avoiding.cases
thf(fact_536_graph_Oreachable__avoiding_Osimps,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,A1: product_prod_v_v,A2: product_prod_v_v,A3: set_Pr2149350503807050951od_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( sCC_Bl5370300055464682748od_v_v @ Successors @ A1 @ A2 @ A3 )
= ( ? [X2: product_prod_v_v,E6: set_Pr2149350503807050951od_v_v] :
( ( A1 = X2 )
& ( A2 = X2 )
& ( A3 = E6 ) )
| ? [X2: product_prod_v_v,Y3: product_prod_v_v,E6: set_Pr2149350503807050951od_v_v,Z2: product_prod_v_v] :
( ( A1 = X2 )
& ( A2 = Z2 )
& ( A3 = E6 )
& ( sCC_Bl5370300055464682748od_v_v @ Successors @ X2 @ Y3 @ E6 )
& ( member7453568604450474000od_v_v @ Z2 @ ( Successors @ Y3 ) )
& ~ ( member3038538357316246288od_v_v @ ( produc4031800376763917143od_v_v @ Y3 @ Z2 ) @ E6 ) ) ) ) ) ).
% graph.reachable_avoiding.simps
thf(fact_537_graph_Oreachable__avoiding_Osimps,axiom,
! [Vertices: set_v,Successors: v > set_v,A1: v,A2: v,A3: set_Product_prod_v_v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl4291963740693775144ding_v @ Successors @ A1 @ A2 @ A3 )
= ( ? [X2: v,E6: set_Product_prod_v_v] :
( ( A1 = X2 )
& ( A2 = X2 )
& ( A3 = E6 ) )
| ? [X2: v,Y3: v,E6: set_Product_prod_v_v,Z2: v] :
( ( A1 = X2 )
& ( A2 = Z2 )
& ( A3 = E6 )
& ( sCC_Bl4291963740693775144ding_v @ Successors @ X2 @ Y3 @ E6 )
& ( member_v @ Z2 @ ( Successors @ Y3 ) )
& ~ ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ Y3 @ Z2 ) @ E6 ) ) ) ) ) ).
% graph.reachable_avoiding.simps
thf(fact_538_graph_Ora__succ,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,X: product_prod_v_v,Y: product_prod_v_v,E4: set_Pr2149350503807050951od_v_v,Z: product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( sCC_Bl5370300055464682748od_v_v @ Successors @ X @ Y @ E4 )
=> ( ( member7453568604450474000od_v_v @ Z @ ( Successors @ Y ) )
=> ( ~ ( member3038538357316246288od_v_v @ ( produc4031800376763917143od_v_v @ Y @ Z ) @ E4 )
=> ( sCC_Bl5370300055464682748od_v_v @ Successors @ X @ Z @ E4 ) ) ) ) ) ).
% graph.ra_succ
thf(fact_539_graph_Ora__succ,axiom,
! [Vertices: set_v,Successors: v > set_v,X: v,Y: v,E4: set_Product_prod_v_v,Z: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl4291963740693775144ding_v @ Successors @ X @ Y @ E4 )
=> ( ( member_v @ Z @ ( Successors @ Y ) )
=> ( ~ ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ Y @ Z ) @ E4 )
=> ( sCC_Bl4291963740693775144ding_v @ Successors @ X @ Z @ E4 ) ) ) ) ) ).
% graph.ra_succ
thf(fact_540_graph_Ora__add__edge,axiom,
! [Vertices: set_v,Successors: v > set_v,X: v,Y: v,E4: set_Product_prod_v_v,V3: v,W: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl4291963740693775144ding_v @ Successors @ X @ Y @ E4 )
=> ( ( sCC_Bl4291963740693775144ding_v @ Successors @ X @ Y @ ( sup_su414716646722978715od_v_v @ E4 @ ( insert1338601472111419319od_v_v @ ( product_Pair_v_v @ V3 @ W ) @ bot_bo723834152578015283od_v_v ) ) )
| ( ( sCC_Bl4291963740693775144ding_v @ Successors @ X @ V3 @ ( sup_su414716646722978715od_v_v @ E4 @ ( insert1338601472111419319od_v_v @ ( product_Pair_v_v @ V3 @ W ) @ bot_bo723834152578015283od_v_v ) ) )
& ( sCC_Bl4291963740693775144ding_v @ Successors @ W @ Y @ ( sup_su414716646722978715od_v_v @ E4 @ ( insert1338601472111419319od_v_v @ ( product_Pair_v_v @ V3 @ W ) @ bot_bo723834152578015283od_v_v ) ) ) ) ) ) ) ).
% graph.ra_add_edge
thf(fact_541_graph_Ora__mono,axiom,
! [Vertices: set_v,Successors: v > set_v,X: v,Y: v,E4: set_Product_prod_v_v,E5: set_Product_prod_v_v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl4291963740693775144ding_v @ Successors @ X @ Y @ E4 )
=> ( ( ord_le7336532860387713383od_v_v @ E5 @ E4 )
=> ( sCC_Bl4291963740693775144ding_v @ Successors @ X @ Y @ E5 ) ) ) ) ).
% graph.ra_mono
thf(fact_542_graph_OS__reflexive,axiom,
! [Vertices: set_v,Successors: v > set_v,E: sCC_Bl1394983891496994913t_unit,N: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl9196236973127232072t_unit @ Successors @ E )
=> ( member_v @ N @ ( sCC_Bl1280885523602775798t_unit @ E @ N ) ) ) ) ).
% graph.S_reflexive
thf(fact_543_not__Cons__self2,axiom,
! [X: v,Xs: list_v] :
( ( cons_v @ X @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_544_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_545_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_546_inf__sup__aci_I8_J,axiom,
! [X: set_set_v,Y: set_set_v] :
( ( sup_sup_set_set_v @ X @ ( sup_sup_set_set_v @ X @ Y ) )
= ( sup_sup_set_set_v @ X @ Y ) ) ).
% inf_sup_aci(8)
thf(fact_547_inf__sup__aci_I8_J,axiom,
! [X: product_unit,Y: product_unit] :
( ( sup_sup_Product_unit @ X @ ( sup_sup_Product_unit @ X @ Y ) )
= ( sup_sup_Product_unit @ X @ Y ) ) ).
% inf_sup_aci(8)
thf(fact_548_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_549_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_550_inf__sup__aci_I7_J,axiom,
! [X: set_set_v,Y: set_set_v,Z: set_set_v] :
( ( sup_sup_set_set_v @ X @ ( sup_sup_set_set_v @ Y @ Z ) )
= ( sup_sup_set_set_v @ Y @ ( sup_sup_set_set_v @ X @ Z ) ) ) ).
% inf_sup_aci(7)
thf(fact_551_inf__sup__aci_I7_J,axiom,
! [X: product_unit,Y: product_unit,Z: product_unit] :
( ( sup_sup_Product_unit @ X @ ( sup_sup_Product_unit @ Y @ Z ) )
= ( sup_sup_Product_unit @ Y @ ( sup_sup_Product_unit @ X @ Z ) ) ) ).
% inf_sup_aci(7)
thf(fact_552_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_553_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_554_inf__sup__aci_I6_J,axiom,
! [X: set_set_v,Y: set_set_v,Z: set_set_v] :
( ( sup_sup_set_set_v @ ( sup_sup_set_set_v @ X @ Y ) @ Z )
= ( sup_sup_set_set_v @ X @ ( sup_sup_set_set_v @ Y @ Z ) ) ) ).
% inf_sup_aci(6)
thf(fact_555_inf__sup__aci_I6_J,axiom,
! [X: product_unit,Y: product_unit,Z: product_unit] :
( ( sup_sup_Product_unit @ ( sup_sup_Product_unit @ X @ Y ) @ Z )
= ( sup_sup_Product_unit @ X @ ( sup_sup_Product_unit @ Y @ Z ) ) ) ).
% inf_sup_aci(6)
thf(fact_556_inf__sup__aci_I5_J,axiom,
( sup_su414716646722978715od_v_v
= ( ^ [X2: set_Product_prod_v_v,Y3: set_Product_prod_v_v] : ( sup_su414716646722978715od_v_v @ Y3 @ X2 ) ) ) ).
% inf_sup_aci(5)
thf(fact_557_inf__sup__aci_I5_J,axiom,
( sup_sup_set_v
= ( ^ [X2: set_v,Y3: set_v] : ( sup_sup_set_v @ Y3 @ X2 ) ) ) ).
% inf_sup_aci(5)
thf(fact_558_inf__sup__aci_I5_J,axiom,
( sup_sup_set_set_v
= ( ^ [X2: set_set_v,Y3: set_set_v] : ( sup_sup_set_set_v @ Y3 @ X2 ) ) ) ).
% inf_sup_aci(5)
thf(fact_559_inf__sup__aci_I5_J,axiom,
( sup_sup_Product_unit
= ( ^ [X2: product_unit,Y3: product_unit] : ( sup_sup_Product_unit @ Y3 @ X2 ) ) ) ).
% inf_sup_aci(5)
thf(fact_560_sup_Oassoc,axiom,
! [A4: set_Product_prod_v_v,B2: set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ ( sup_su414716646722978715od_v_v @ A4 @ B2 ) @ C )
= ( sup_su414716646722978715od_v_v @ A4 @ ( sup_su414716646722978715od_v_v @ B2 @ C ) ) ) ).
% sup.assoc
thf(fact_561_sup_Oassoc,axiom,
! [A4: set_v,B2: set_v,C: set_v] :
( ( sup_sup_set_v @ ( sup_sup_set_v @ A4 @ B2 ) @ C )
= ( sup_sup_set_v @ A4 @ ( sup_sup_set_v @ B2 @ C ) ) ) ).
% sup.assoc
thf(fact_562_sup_Oassoc,axiom,
! [A4: set_set_v,B2: set_set_v,C: set_set_v] :
( ( sup_sup_set_set_v @ ( sup_sup_set_set_v @ A4 @ B2 ) @ C )
= ( sup_sup_set_set_v @ A4 @ ( sup_sup_set_set_v @ B2 @ C ) ) ) ).
% sup.assoc
thf(fact_563_sup_Oassoc,axiom,
! [A4: product_unit,B2: product_unit,C: product_unit] :
( ( sup_sup_Product_unit @ ( sup_sup_Product_unit @ A4 @ B2 ) @ C )
= ( sup_sup_Product_unit @ A4 @ ( sup_sup_Product_unit @ B2 @ C ) ) ) ).
% sup.assoc
thf(fact_564_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_565_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_566_sup__assoc,axiom,
! [X: set_set_v,Y: set_set_v,Z: set_set_v] :
( ( sup_sup_set_set_v @ ( sup_sup_set_set_v @ X @ Y ) @ Z )
= ( sup_sup_set_set_v @ X @ ( sup_sup_set_set_v @ Y @ Z ) ) ) ).
% sup_assoc
thf(fact_567_sup__assoc,axiom,
! [X: product_unit,Y: product_unit,Z: product_unit] :
( ( sup_sup_Product_unit @ ( sup_sup_Product_unit @ X @ Y ) @ Z )
= ( sup_sup_Product_unit @ X @ ( sup_sup_Product_unit @ Y @ Z ) ) ) ).
% sup_assoc
thf(fact_568_sup_Ocommute,axiom,
( sup_su414716646722978715od_v_v
= ( ^ [A6: set_Product_prod_v_v,B4: set_Product_prod_v_v] : ( sup_su414716646722978715od_v_v @ B4 @ A6 ) ) ) ).
% sup.commute
thf(fact_569_sup_Ocommute,axiom,
( sup_sup_set_v
= ( ^ [A6: set_v,B4: set_v] : ( sup_sup_set_v @ B4 @ A6 ) ) ) ).
% sup.commute
thf(fact_570_sup_Ocommute,axiom,
( sup_sup_set_set_v
= ( ^ [A6: set_set_v,B4: set_set_v] : ( sup_sup_set_set_v @ B4 @ A6 ) ) ) ).
% sup.commute
thf(fact_571_sup_Ocommute,axiom,
( sup_sup_Product_unit
= ( ^ [A6: product_unit,B4: product_unit] : ( sup_sup_Product_unit @ B4 @ A6 ) ) ) ).
% sup.commute
thf(fact_572_sup__commute,axiom,
( sup_su414716646722978715od_v_v
= ( ^ [X2: set_Product_prod_v_v,Y3: set_Product_prod_v_v] : ( sup_su414716646722978715od_v_v @ Y3 @ X2 ) ) ) ).
% sup_commute
thf(fact_573_sup__commute,axiom,
( sup_sup_set_v
= ( ^ [X2: set_v,Y3: set_v] : ( sup_sup_set_v @ Y3 @ X2 ) ) ) ).
% sup_commute
thf(fact_574_sup__commute,axiom,
( sup_sup_set_set_v
= ( ^ [X2: set_set_v,Y3: set_set_v] : ( sup_sup_set_set_v @ Y3 @ X2 ) ) ) ).
% sup_commute
thf(fact_575_sup__commute,axiom,
( sup_sup_Product_unit
= ( ^ [X2: product_unit,Y3: product_unit] : ( sup_sup_Product_unit @ Y3 @ X2 ) ) ) ).
% sup_commute
thf(fact_576_sup_Oleft__commute,axiom,
! [B2: set_Product_prod_v_v,A4: set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ B2 @ ( sup_su414716646722978715od_v_v @ A4 @ C ) )
= ( sup_su414716646722978715od_v_v @ A4 @ ( sup_su414716646722978715od_v_v @ B2 @ C ) ) ) ).
% sup.left_commute
thf(fact_577_sup_Oleft__commute,axiom,
! [B2: set_v,A4: set_v,C: set_v] :
( ( sup_sup_set_v @ B2 @ ( sup_sup_set_v @ A4 @ C ) )
= ( sup_sup_set_v @ A4 @ ( sup_sup_set_v @ B2 @ C ) ) ) ).
% sup.left_commute
thf(fact_578_sup_Oleft__commute,axiom,
! [B2: set_set_v,A4: set_set_v,C: set_set_v] :
( ( sup_sup_set_set_v @ B2 @ ( sup_sup_set_set_v @ A4 @ C ) )
= ( sup_sup_set_set_v @ A4 @ ( sup_sup_set_set_v @ B2 @ C ) ) ) ).
% sup.left_commute
thf(fact_579_sup_Oleft__commute,axiom,
! [B2: product_unit,A4: product_unit,C: product_unit] :
( ( sup_sup_Product_unit @ B2 @ ( sup_sup_Product_unit @ A4 @ C ) )
= ( sup_sup_Product_unit @ A4 @ ( sup_sup_Product_unit @ B2 @ C ) ) ) ).
% sup.left_commute
thf(fact_580_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_581_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_582_sup__left__commute,axiom,
! [X: set_set_v,Y: set_set_v,Z: set_set_v] :
( ( sup_sup_set_set_v @ X @ ( sup_sup_set_set_v @ Y @ Z ) )
= ( sup_sup_set_set_v @ Y @ ( sup_sup_set_set_v @ X @ Z ) ) ) ).
% sup_left_commute
thf(fact_583_sup__left__commute,axiom,
! [X: product_unit,Y: product_unit,Z: product_unit] :
( ( sup_sup_Product_unit @ X @ ( sup_sup_Product_unit @ Y @ Z ) )
= ( sup_sup_Product_unit @ Y @ ( sup_sup_Product_unit @ X @ Z ) ) ) ).
% sup_left_commute
thf(fact_584_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_585_graph_Opre__dfss__pre__dfs,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 @ ( sCC_Bl4645233313691564917t_unit @ E ) )
=> ( ( member_v @ W @ ( Successors @ V3 ) )
=> ( sCC_Bl36166008131615352t_unit @ Successors @ W @ E ) ) ) ) ) ).
% graph.pre_dfss_pre_dfs
thf(fact_586_graph_Ostack__unexplored,axiom,
! [Vertices: set_v,Successors: v > set_v,E: sCC_Bl1394983891496994913t_unit,N: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl9196236973127232072t_unit @ Successors @ E )
=> ( ( member_v @ N @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E ) ) )
=> ~ ( member_v @ N @ ( sCC_Bl157864678168468314t_unit @ E ) ) ) ) ) ).
% graph.stack_unexplored
thf(fact_587_graph_Ostack__visited,axiom,
! [Vertices: set_v,Successors: v > set_v,E: sCC_Bl1394983891496994913t_unit,N: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl9196236973127232072t_unit @ Successors @ E )
=> ( ( member_v @ N @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E ) ) )
=> ( member_v @ N @ ( sCC_Bl4645233313691564917t_unit @ E ) ) ) ) ) ).
% graph.stack_visited
thf(fact_588_graph_Oavoiding__explored,axiom,
! [Vertices: set_v,Successors: v > set_v,E: sCC_Bl1394983891496994913t_unit,X: v,Y: v,E4: set_Product_prod_v_v,W: v,V3: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl9196236973127232072t_unit @ Successors @ E )
=> ( ( sCC_Bl4291963740693775144ding_v @ Successors @ X @ Y @ E4 )
=> ( ~ ( member_v @ Y @ ( sCC_Bl157864678168468314t_unit @ E ) )
=> ( ( member_v @ W @ ( sCC_Bl157864678168468314t_unit @ E ) )
=> ( sCC_Bl4291963740693775144ding_v @ Successors @ X @ Y @ ( sup_su414716646722978715od_v_v @ E4 @ ( insert1338601472111419319od_v_v @ ( product_Pair_v_v @ V3 @ W ) @ bot_bo723834152578015283od_v_v ) ) ) ) ) ) ) ) ).
% graph.avoiding_explored
thf(fact_589_graph_Ovisited__unexplored,axiom,
! [Vertices: set_v,Successors: v > set_v,E: sCC_Bl1394983891496994913t_unit,M: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl9196236973127232072t_unit @ Successors @ E )
=> ( ( member_v @ M @ ( sCC_Bl4645233313691564917t_unit @ E ) )
=> ( ~ ( member_v @ M @ ( sCC_Bl157864678168468314t_unit @ E ) )
=> ~ ! [N2: v] :
( ( member_v @ N2 @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E ) ) )
=> ~ ( member_v @ M @ ( sCC_Bl1280885523602775798t_unit @ E @ N2 ) ) ) ) ) ) ) ).
% graph.visited_unexplored
thf(fact_590_inf__sup__ord_I4_J,axiom,
! [Y: set_set_v,X: set_set_v] : ( ord_le5216385588623774835_set_v @ Y @ ( sup_sup_set_set_v @ X @ Y ) ) ).
% inf_sup_ord(4)
thf(fact_591_inf__sup__ord_I4_J,axiom,
! [Y: product_unit,X: product_unit] : ( ord_le3221252021190050221t_unit @ Y @ ( sup_sup_Product_unit @ X @ Y ) ) ).
% inf_sup_ord(4)
thf(fact_592_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_593_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_594_inf__sup__ord_I3_J,axiom,
! [X: set_set_v,Y: set_set_v] : ( ord_le5216385588623774835_set_v @ X @ ( sup_sup_set_set_v @ X @ Y ) ) ).
% inf_sup_ord(3)
thf(fact_595_inf__sup__ord_I3_J,axiom,
! [X: product_unit,Y: product_unit] : ( ord_le3221252021190050221t_unit @ X @ ( sup_sup_Product_unit @ X @ Y ) ) ).
% inf_sup_ord(3)
thf(fact_596_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_597_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_598_le__supE,axiom,
! [A4: set_set_v,B2: set_set_v,X: set_set_v] :
( ( ord_le5216385588623774835_set_v @ ( sup_sup_set_set_v @ A4 @ B2 ) @ X )
=> ~ ( ( ord_le5216385588623774835_set_v @ A4 @ X )
=> ~ ( ord_le5216385588623774835_set_v @ B2 @ X ) ) ) ).
% le_supE
thf(fact_599_le__supE,axiom,
! [A4: product_unit,B2: product_unit,X: product_unit] :
( ( ord_le3221252021190050221t_unit @ ( sup_sup_Product_unit @ A4 @ B2 ) @ X )
=> ~ ( ( ord_le3221252021190050221t_unit @ A4 @ X )
=> ~ ( ord_le3221252021190050221t_unit @ B2 @ X ) ) ) ).
% le_supE
thf(fact_600_le__supE,axiom,
! [A4: set_Product_prod_v_v,B2: set_Product_prod_v_v,X: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ ( sup_su414716646722978715od_v_v @ A4 @ B2 ) @ X )
=> ~ ( ( ord_le7336532860387713383od_v_v @ A4 @ X )
=> ~ ( ord_le7336532860387713383od_v_v @ B2 @ X ) ) ) ).
% le_supE
thf(fact_601_le__supE,axiom,
! [A4: set_v,B2: set_v,X: set_v] :
( ( ord_less_eq_set_v @ ( sup_sup_set_v @ A4 @ B2 ) @ X )
=> ~ ( ( ord_less_eq_set_v @ A4 @ X )
=> ~ ( ord_less_eq_set_v @ B2 @ X ) ) ) ).
% le_supE
thf(fact_602_le__supI,axiom,
! [A4: set_set_v,X: set_set_v,B2: set_set_v] :
( ( ord_le5216385588623774835_set_v @ A4 @ X )
=> ( ( ord_le5216385588623774835_set_v @ B2 @ X )
=> ( ord_le5216385588623774835_set_v @ ( sup_sup_set_set_v @ A4 @ B2 ) @ X ) ) ) ).
% le_supI
thf(fact_603_le__supI,axiom,
! [A4: product_unit,X: product_unit,B2: product_unit] :
( ( ord_le3221252021190050221t_unit @ A4 @ X )
=> ( ( ord_le3221252021190050221t_unit @ B2 @ X )
=> ( ord_le3221252021190050221t_unit @ ( sup_sup_Product_unit @ A4 @ B2 ) @ X ) ) ) ).
% le_supI
thf(fact_604_le__supI,axiom,
! [A4: set_Product_prod_v_v,X: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A4 @ X )
=> ( ( ord_le7336532860387713383od_v_v @ B2 @ X )
=> ( ord_le7336532860387713383od_v_v @ ( sup_su414716646722978715od_v_v @ A4 @ B2 ) @ X ) ) ) ).
% le_supI
thf(fact_605_le__supI,axiom,
! [A4: set_v,X: set_v,B2: set_v] :
( ( ord_less_eq_set_v @ A4 @ X )
=> ( ( ord_less_eq_set_v @ B2 @ X )
=> ( ord_less_eq_set_v @ ( sup_sup_set_v @ A4 @ B2 ) @ X ) ) ) ).
% le_supI
thf(fact_606_sup__ge1,axiom,
! [X: set_set_v,Y: set_set_v] : ( ord_le5216385588623774835_set_v @ X @ ( sup_sup_set_set_v @ X @ Y ) ) ).
% sup_ge1
thf(fact_607_sup__ge1,axiom,
! [X: product_unit,Y: product_unit] : ( ord_le3221252021190050221t_unit @ X @ ( sup_sup_Product_unit @ 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__ge1,axiom,
! [X: set_v,Y: set_v] : ( ord_less_eq_set_v @ X @ ( sup_sup_set_v @ X @ Y ) ) ).
% sup_ge1
thf(fact_610_sup__ge2,axiom,
! [Y: set_set_v,X: set_set_v] : ( ord_le5216385588623774835_set_v @ Y @ ( sup_sup_set_set_v @ X @ Y ) ) ).
% sup_ge2
thf(fact_611_sup__ge2,axiom,
! [Y: product_unit,X: product_unit] : ( ord_le3221252021190050221t_unit @ Y @ ( sup_sup_Product_unit @ X @ Y ) ) ).
% sup_ge2
thf(fact_612_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_613_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_614_le__supI1,axiom,
! [X: set_set_v,A4: set_set_v,B2: set_set_v] :
( ( ord_le5216385588623774835_set_v @ X @ A4 )
=> ( ord_le5216385588623774835_set_v @ X @ ( sup_sup_set_set_v @ A4 @ B2 ) ) ) ).
% le_supI1
thf(fact_615_le__supI1,axiom,
! [X: product_unit,A4: product_unit,B2: product_unit] :
( ( ord_le3221252021190050221t_unit @ X @ A4 )
=> ( ord_le3221252021190050221t_unit @ X @ ( sup_sup_Product_unit @ A4 @ B2 ) ) ) ).
% le_supI1
thf(fact_616_le__supI1,axiom,
! [X: set_Product_prod_v_v,A4: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X @ A4 )
=> ( ord_le7336532860387713383od_v_v @ X @ ( sup_su414716646722978715od_v_v @ A4 @ B2 ) ) ) ).
% le_supI1
thf(fact_617_le__supI1,axiom,
! [X: set_v,A4: set_v,B2: set_v] :
( ( ord_less_eq_set_v @ X @ A4 )
=> ( ord_less_eq_set_v @ X @ ( sup_sup_set_v @ A4 @ B2 ) ) ) ).
% le_supI1
thf(fact_618_le__supI2,axiom,
! [X: set_set_v,B2: set_set_v,A4: set_set_v] :
( ( ord_le5216385588623774835_set_v @ X @ B2 )
=> ( ord_le5216385588623774835_set_v @ X @ ( sup_sup_set_set_v @ A4 @ B2 ) ) ) ).
% le_supI2
thf(fact_619_le__supI2,axiom,
! [X: product_unit,B2: product_unit,A4: product_unit] :
( ( ord_le3221252021190050221t_unit @ X @ B2 )
=> ( ord_le3221252021190050221t_unit @ X @ ( sup_sup_Product_unit @ A4 @ B2 ) ) ) ).
% le_supI2
thf(fact_620_le__supI2,axiom,
! [X: set_Product_prod_v_v,B2: set_Product_prod_v_v,A4: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X @ B2 )
=> ( ord_le7336532860387713383od_v_v @ X @ ( sup_su414716646722978715od_v_v @ A4 @ B2 ) ) ) ).
% le_supI2
thf(fact_621_le__supI2,axiom,
! [X: set_v,B2: set_v,A4: set_v] :
( ( ord_less_eq_set_v @ X @ B2 )
=> ( ord_less_eq_set_v @ X @ ( sup_sup_set_v @ A4 @ B2 ) ) ) ).
% le_supI2
thf(fact_622_sup_Omono,axiom,
! [C: set_set_v,A4: set_set_v,D: set_set_v,B2: set_set_v] :
( ( ord_le5216385588623774835_set_v @ C @ A4 )
=> ( ( ord_le5216385588623774835_set_v @ D @ B2 )
=> ( ord_le5216385588623774835_set_v @ ( sup_sup_set_set_v @ C @ D ) @ ( sup_sup_set_set_v @ A4 @ B2 ) ) ) ) ).
% sup.mono
thf(fact_623_sup_Omono,axiom,
! [C: product_unit,A4: product_unit,D: product_unit,B2: product_unit] :
( ( ord_le3221252021190050221t_unit @ C @ A4 )
=> ( ( ord_le3221252021190050221t_unit @ D @ B2 )
=> ( ord_le3221252021190050221t_unit @ ( sup_sup_Product_unit @ C @ D ) @ ( sup_sup_Product_unit @ A4 @ B2 ) ) ) ) ).
% sup.mono
thf(fact_624_sup_Omono,axiom,
! [C: set_Product_prod_v_v,A4: set_Product_prod_v_v,D: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ C @ A4 )
=> ( ( ord_le7336532860387713383od_v_v @ D @ B2 )
=> ( ord_le7336532860387713383od_v_v @ ( sup_su414716646722978715od_v_v @ C @ D ) @ ( sup_su414716646722978715od_v_v @ A4 @ B2 ) ) ) ) ).
% sup.mono
thf(fact_625_sup_Omono,axiom,
! [C: set_v,A4: set_v,D: set_v,B2: set_v] :
( ( ord_less_eq_set_v @ C @ A4 )
=> ( ( ord_less_eq_set_v @ D @ B2 )
=> ( ord_less_eq_set_v @ ( sup_sup_set_v @ C @ D ) @ ( sup_sup_set_v @ A4 @ B2 ) ) ) ) ).
% sup.mono
thf(fact_626_sup__mono,axiom,
! [A4: set_set_v,C: set_set_v,B2: set_set_v,D: set_set_v] :
( ( ord_le5216385588623774835_set_v @ A4 @ C )
=> ( ( ord_le5216385588623774835_set_v @ B2 @ D )
=> ( ord_le5216385588623774835_set_v @ ( sup_sup_set_set_v @ A4 @ B2 ) @ ( sup_sup_set_set_v @ C @ D ) ) ) ) ).
% sup_mono
thf(fact_627_sup__mono,axiom,
! [A4: product_unit,C: product_unit,B2: product_unit,D: product_unit] :
( ( ord_le3221252021190050221t_unit @ A4 @ C )
=> ( ( ord_le3221252021190050221t_unit @ B2 @ D )
=> ( ord_le3221252021190050221t_unit @ ( sup_sup_Product_unit @ A4 @ B2 ) @ ( sup_sup_Product_unit @ C @ D ) ) ) ) ).
% sup_mono
thf(fact_628_sup__mono,axiom,
! [A4: set_Product_prod_v_v,C: set_Product_prod_v_v,B2: set_Product_prod_v_v,D: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A4 @ C )
=> ( ( ord_le7336532860387713383od_v_v @ B2 @ D )
=> ( ord_le7336532860387713383od_v_v @ ( sup_su414716646722978715od_v_v @ A4 @ B2 ) @ ( sup_su414716646722978715od_v_v @ C @ D ) ) ) ) ).
% sup_mono
thf(fact_629_sup__mono,axiom,
! [A4: set_v,C: set_v,B2: set_v,D: set_v] :
( ( ord_less_eq_set_v @ A4 @ C )
=> ( ( ord_less_eq_set_v @ B2 @ D )
=> ( ord_less_eq_set_v @ ( sup_sup_set_v @ A4 @ B2 ) @ ( sup_sup_set_v @ C @ D ) ) ) ) ).
% sup_mono
thf(fact_630_sup__least,axiom,
! [Y: set_set_v,X: set_set_v,Z: set_set_v] :
( ( ord_le5216385588623774835_set_v @ Y @ X )
=> ( ( ord_le5216385588623774835_set_v @ Z @ X )
=> ( ord_le5216385588623774835_set_v @ ( sup_sup_set_set_v @ Y @ Z ) @ X ) ) ) ).
% sup_least
thf(fact_631_sup__least,axiom,
! [Y: product_unit,X: product_unit,Z: product_unit] :
( ( ord_le3221252021190050221t_unit @ Y @ X )
=> ( ( ord_le3221252021190050221t_unit @ Z @ X )
=> ( ord_le3221252021190050221t_unit @ ( sup_sup_Product_unit @ Y @ Z ) @ X ) ) ) ).
% sup_least
thf(fact_632_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_633_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_634_le__iff__sup,axiom,
( ord_le5216385588623774835_set_v
= ( ^ [X2: set_set_v,Y3: set_set_v] :
( ( sup_sup_set_set_v @ X2 @ Y3 )
= Y3 ) ) ) ).
% le_iff_sup
thf(fact_635_le__iff__sup,axiom,
( ord_le3221252021190050221t_unit
= ( ^ [X2: product_unit,Y3: product_unit] :
( ( sup_sup_Product_unit @ X2 @ Y3 )
= Y3 ) ) ) ).
% le_iff_sup
thf(fact_636_le__iff__sup,axiom,
( ord_le7336532860387713383od_v_v
= ( ^ [X2: set_Product_prod_v_v,Y3: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ X2 @ Y3 )
= Y3 ) ) ) ).
% le_iff_sup
thf(fact_637_le__iff__sup,axiom,
( ord_less_eq_set_v
= ( ^ [X2: set_v,Y3: set_v] :
( ( sup_sup_set_v @ X2 @ Y3 )
= Y3 ) ) ) ).
% le_iff_sup
thf(fact_638_sup_OorderE,axiom,
! [B2: set_set_v,A4: set_set_v] :
( ( ord_le5216385588623774835_set_v @ B2 @ A4 )
=> ( A4
= ( sup_sup_set_set_v @ A4 @ B2 ) ) ) ).
% sup.orderE
thf(fact_639_sup_OorderE,axiom,
! [B2: product_unit,A4: product_unit] :
( ( ord_le3221252021190050221t_unit @ B2 @ A4 )
=> ( A4
= ( sup_sup_Product_unit @ A4 @ B2 ) ) ) ).
% sup.orderE
thf(fact_640_sup_OorderE,axiom,
! [B2: set_Product_prod_v_v,A4: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ B2 @ A4 )
=> ( A4
= ( sup_su414716646722978715od_v_v @ A4 @ B2 ) ) ) ).
% sup.orderE
thf(fact_641_sup_OorderE,axiom,
! [B2: set_v,A4: set_v] :
( ( ord_less_eq_set_v @ B2 @ A4 )
=> ( A4
= ( sup_sup_set_v @ A4 @ B2 ) ) ) ).
% sup.orderE
thf(fact_642_sup_OorderI,axiom,
! [A4: set_set_v,B2: set_set_v] :
( ( A4
= ( sup_sup_set_set_v @ A4 @ B2 ) )
=> ( ord_le5216385588623774835_set_v @ B2 @ A4 ) ) ).
% sup.orderI
thf(fact_643_sup_OorderI,axiom,
! [A4: product_unit,B2: product_unit] :
( ( A4
= ( sup_sup_Product_unit @ A4 @ B2 ) )
=> ( ord_le3221252021190050221t_unit @ B2 @ A4 ) ) ).
% sup.orderI
thf(fact_644_sup_OorderI,axiom,
! [A4: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( A4
= ( sup_su414716646722978715od_v_v @ A4 @ B2 ) )
=> ( ord_le7336532860387713383od_v_v @ B2 @ A4 ) ) ).
% sup.orderI
thf(fact_645_sup_OorderI,axiom,
! [A4: set_v,B2: set_v] :
( ( A4
= ( sup_sup_set_v @ A4 @ B2 ) )
=> ( ord_less_eq_set_v @ B2 @ A4 ) ) ).
% sup.orderI
thf(fact_646_sup__unique,axiom,
! [F: set_set_v > set_set_v > set_set_v,X: set_set_v,Y: set_set_v] :
( ! [X3: set_set_v,Y2: set_set_v] : ( ord_le5216385588623774835_set_v @ X3 @ ( F @ X3 @ Y2 ) )
=> ( ! [X3: set_set_v,Y2: set_set_v] : ( ord_le5216385588623774835_set_v @ Y2 @ ( F @ X3 @ Y2 ) )
=> ( ! [X3: set_set_v,Y2: set_set_v,Z3: set_set_v] :
( ( ord_le5216385588623774835_set_v @ Y2 @ X3 )
=> ( ( ord_le5216385588623774835_set_v @ Z3 @ X3 )
=> ( ord_le5216385588623774835_set_v @ ( F @ Y2 @ Z3 ) @ X3 ) ) )
=> ( ( sup_sup_set_set_v @ X @ Y )
= ( F @ X @ Y ) ) ) ) ) ).
% sup_unique
thf(fact_647_sup__unique,axiom,
! [F: product_unit > product_unit > product_unit,X: product_unit,Y: product_unit] :
( ! [X3: product_unit,Y2: product_unit] : ( ord_le3221252021190050221t_unit @ X3 @ ( F @ X3 @ Y2 ) )
=> ( ! [X3: product_unit,Y2: product_unit] : ( ord_le3221252021190050221t_unit @ Y2 @ ( F @ X3 @ Y2 ) )
=> ( ! [X3: product_unit,Y2: product_unit,Z3: product_unit] :
( ( ord_le3221252021190050221t_unit @ Y2 @ X3 )
=> ( ( ord_le3221252021190050221t_unit @ Z3 @ X3 )
=> ( ord_le3221252021190050221t_unit @ ( F @ Y2 @ Z3 ) @ X3 ) ) )
=> ( ( sup_sup_Product_unit @ X @ Y )
= ( F @ X @ Y ) ) ) ) ) ).
% sup_unique
thf(fact_648_sup__unique,axiom,
! [F: set_Product_prod_v_v > set_Product_prod_v_v > set_Product_prod_v_v,X: set_Product_prod_v_v,Y: set_Product_prod_v_v] :
( ! [X3: set_Product_prod_v_v,Y2: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ X3 @ ( F @ X3 @ Y2 ) )
=> ( ! [X3: set_Product_prod_v_v,Y2: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ Y2 @ ( F @ X3 @ Y2 ) )
=> ( ! [X3: set_Product_prod_v_v,Y2: set_Product_prod_v_v,Z3: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ Y2 @ X3 )
=> ( ( ord_le7336532860387713383od_v_v @ Z3 @ X3 )
=> ( ord_le7336532860387713383od_v_v @ ( F @ Y2 @ Z3 ) @ X3 ) ) )
=> ( ( sup_su414716646722978715od_v_v @ X @ Y )
= ( F @ X @ Y ) ) ) ) ) ).
% sup_unique
thf(fact_649_sup__unique,axiom,
! [F: set_v > set_v > set_v,X: set_v,Y: set_v] :
( ! [X3: set_v,Y2: set_v] : ( ord_less_eq_set_v @ X3 @ ( F @ X3 @ Y2 ) )
=> ( ! [X3: set_v,Y2: set_v] : ( ord_less_eq_set_v @ Y2 @ ( F @ X3 @ Y2 ) )
=> ( ! [X3: set_v,Y2: set_v,Z3: set_v] :
( ( ord_less_eq_set_v @ Y2 @ X3 )
=> ( ( ord_less_eq_set_v @ Z3 @ X3 )
=> ( ord_less_eq_set_v @ ( F @ Y2 @ Z3 ) @ X3 ) ) )
=> ( ( sup_sup_set_v @ X @ Y )
= ( F @ X @ Y ) ) ) ) ) ).
% sup_unique
thf(fact_650_sup_Oabsorb1,axiom,
! [B2: set_set_v,A4: set_set_v] :
( ( ord_le5216385588623774835_set_v @ B2 @ A4 )
=> ( ( sup_sup_set_set_v @ A4 @ B2 )
= A4 ) ) ).
% sup.absorb1
thf(fact_651_sup_Oabsorb1,axiom,
! [B2: product_unit,A4: product_unit] :
( ( ord_le3221252021190050221t_unit @ B2 @ A4 )
=> ( ( sup_sup_Product_unit @ A4 @ B2 )
= A4 ) ) ).
% sup.absorb1
thf(fact_652_sup_Oabsorb1,axiom,
! [B2: set_Product_prod_v_v,A4: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ B2 @ A4 )
=> ( ( sup_su414716646722978715od_v_v @ A4 @ B2 )
= A4 ) ) ).
% sup.absorb1
thf(fact_653_sup_Oabsorb1,axiom,
! [B2: set_v,A4: set_v] :
( ( ord_less_eq_set_v @ B2 @ A4 )
=> ( ( sup_sup_set_v @ A4 @ B2 )
= A4 ) ) ).
% sup.absorb1
thf(fact_654_sup_Oabsorb2,axiom,
! [A4: set_set_v,B2: set_set_v] :
( ( ord_le5216385588623774835_set_v @ A4 @ B2 )
=> ( ( sup_sup_set_set_v @ A4 @ B2 )
= B2 ) ) ).
% sup.absorb2
thf(fact_655_sup_Oabsorb2,axiom,
! [A4: product_unit,B2: product_unit] :
( ( ord_le3221252021190050221t_unit @ A4 @ B2 )
=> ( ( sup_sup_Product_unit @ A4 @ B2 )
= B2 ) ) ).
% sup.absorb2
thf(fact_656_sup_Oabsorb2,axiom,
! [A4: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A4 @ B2 )
=> ( ( sup_su414716646722978715od_v_v @ A4 @ B2 )
= B2 ) ) ).
% sup.absorb2
thf(fact_657_sup_Oabsorb2,axiom,
! [A4: set_v,B2: set_v] :
( ( ord_less_eq_set_v @ A4 @ B2 )
=> ( ( sup_sup_set_v @ A4 @ B2 )
= B2 ) ) ).
% sup.absorb2
thf(fact_658_sup__absorb1,axiom,
! [Y: set_set_v,X: set_set_v] :
( ( ord_le5216385588623774835_set_v @ Y @ X )
=> ( ( sup_sup_set_set_v @ X @ Y )
= X ) ) ).
% sup_absorb1
thf(fact_659_sup__absorb1,axiom,
! [Y: product_unit,X: product_unit] :
( ( ord_le3221252021190050221t_unit @ Y @ X )
=> ( ( sup_sup_Product_unit @ X @ Y )
= X ) ) ).
% sup_absorb1
thf(fact_660_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_661_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_662_sup__absorb2,axiom,
! [X: set_set_v,Y: set_set_v] :
( ( ord_le5216385588623774835_set_v @ X @ Y )
=> ( ( sup_sup_set_set_v @ X @ Y )
= Y ) ) ).
% sup_absorb2
thf(fact_663_sup__absorb2,axiom,
! [X: product_unit,Y: product_unit] :
( ( ord_le3221252021190050221t_unit @ X @ Y )
=> ( ( sup_sup_Product_unit @ X @ Y )
= Y ) ) ).
% sup_absorb2
thf(fact_664_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_665_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_666_sup_OboundedE,axiom,
! [B2: set_set_v,C: set_set_v,A4: set_set_v] :
( ( ord_le5216385588623774835_set_v @ ( sup_sup_set_set_v @ B2 @ C ) @ A4 )
=> ~ ( ( ord_le5216385588623774835_set_v @ B2 @ A4 )
=> ~ ( ord_le5216385588623774835_set_v @ C @ A4 ) ) ) ).
% sup.boundedE
thf(fact_667_sup_OboundedE,axiom,
! [B2: product_unit,C: product_unit,A4: product_unit] :
( ( ord_le3221252021190050221t_unit @ ( sup_sup_Product_unit @ B2 @ C ) @ A4 )
=> ~ ( ( ord_le3221252021190050221t_unit @ B2 @ A4 )
=> ~ ( ord_le3221252021190050221t_unit @ C @ A4 ) ) ) ).
% sup.boundedE
thf(fact_668_sup_OboundedE,axiom,
! [B2: set_Product_prod_v_v,C: set_Product_prod_v_v,A4: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ ( sup_su414716646722978715od_v_v @ B2 @ C ) @ A4 )
=> ~ ( ( ord_le7336532860387713383od_v_v @ B2 @ A4 )
=> ~ ( ord_le7336532860387713383od_v_v @ C @ A4 ) ) ) ).
% sup.boundedE
thf(fact_669_sup_OboundedE,axiom,
! [B2: set_v,C: set_v,A4: set_v] :
( ( ord_less_eq_set_v @ ( sup_sup_set_v @ B2 @ C ) @ A4 )
=> ~ ( ( ord_less_eq_set_v @ B2 @ A4 )
=> ~ ( ord_less_eq_set_v @ C @ A4 ) ) ) ).
% sup.boundedE
thf(fact_670_sup_OboundedI,axiom,
! [B2: set_set_v,A4: set_set_v,C: set_set_v] :
( ( ord_le5216385588623774835_set_v @ B2 @ A4 )
=> ( ( ord_le5216385588623774835_set_v @ C @ A4 )
=> ( ord_le5216385588623774835_set_v @ ( sup_sup_set_set_v @ B2 @ C ) @ A4 ) ) ) ).
% sup.boundedI
thf(fact_671_sup_OboundedI,axiom,
! [B2: product_unit,A4: product_unit,C: product_unit] :
( ( ord_le3221252021190050221t_unit @ B2 @ A4 )
=> ( ( ord_le3221252021190050221t_unit @ C @ A4 )
=> ( ord_le3221252021190050221t_unit @ ( sup_sup_Product_unit @ B2 @ C ) @ A4 ) ) ) ).
% sup.boundedI
thf(fact_672_sup_OboundedI,axiom,
! [B2: set_Product_prod_v_v,A4: set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ B2 @ A4 )
=> ( ( ord_le7336532860387713383od_v_v @ C @ A4 )
=> ( ord_le7336532860387713383od_v_v @ ( sup_su414716646722978715od_v_v @ B2 @ C ) @ A4 ) ) ) ).
% sup.boundedI
thf(fact_673_sup_OboundedI,axiom,
! [B2: set_v,A4: set_v,C: set_v] :
( ( ord_less_eq_set_v @ B2 @ A4 )
=> ( ( ord_less_eq_set_v @ C @ A4 )
=> ( ord_less_eq_set_v @ ( sup_sup_set_v @ B2 @ C ) @ A4 ) ) ) ).
% sup.boundedI
thf(fact_674_sup_Oorder__iff,axiom,
( ord_le5216385588623774835_set_v
= ( ^ [B4: set_set_v,A6: set_set_v] :
( A6
= ( sup_sup_set_set_v @ A6 @ B4 ) ) ) ) ).
% sup.order_iff
thf(fact_675_sup_Oorder__iff,axiom,
( ord_le3221252021190050221t_unit
= ( ^ [B4: product_unit,A6: product_unit] :
( A6
= ( sup_sup_Product_unit @ A6 @ B4 ) ) ) ) ).
% sup.order_iff
thf(fact_676_sup_Oorder__iff,axiom,
( ord_le7336532860387713383od_v_v
= ( ^ [B4: set_Product_prod_v_v,A6: set_Product_prod_v_v] :
( A6
= ( sup_su414716646722978715od_v_v @ A6 @ B4 ) ) ) ) ).
% sup.order_iff
thf(fact_677_sup_Oorder__iff,axiom,
( ord_less_eq_set_v
= ( ^ [B4: set_v,A6: set_v] :
( A6
= ( sup_sup_set_v @ A6 @ B4 ) ) ) ) ).
% sup.order_iff
thf(fact_678_sup_Ocobounded1,axiom,
! [A4: set_set_v,B2: set_set_v] : ( ord_le5216385588623774835_set_v @ A4 @ ( sup_sup_set_set_v @ A4 @ B2 ) ) ).
% sup.cobounded1
thf(fact_679_sup_Ocobounded1,axiom,
! [A4: product_unit,B2: product_unit] : ( ord_le3221252021190050221t_unit @ A4 @ ( sup_sup_Product_unit @ A4 @ B2 ) ) ).
% sup.cobounded1
thf(fact_680_sup_Ocobounded1,axiom,
! [A4: set_Product_prod_v_v,B2: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ A4 @ ( sup_su414716646722978715od_v_v @ A4 @ B2 ) ) ).
% sup.cobounded1
thf(fact_681_sup_Ocobounded1,axiom,
! [A4: set_v,B2: set_v] : ( ord_less_eq_set_v @ A4 @ ( sup_sup_set_v @ A4 @ B2 ) ) ).
% sup.cobounded1
thf(fact_682_sup_Ocobounded2,axiom,
! [B2: set_set_v,A4: set_set_v] : ( ord_le5216385588623774835_set_v @ B2 @ ( sup_sup_set_set_v @ A4 @ B2 ) ) ).
% sup.cobounded2
thf(fact_683_sup_Ocobounded2,axiom,
! [B2: product_unit,A4: product_unit] : ( ord_le3221252021190050221t_unit @ B2 @ ( sup_sup_Product_unit @ A4 @ B2 ) ) ).
% sup.cobounded2
thf(fact_684_sup_Ocobounded2,axiom,
! [B2: set_Product_prod_v_v,A4: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ B2 @ ( sup_su414716646722978715od_v_v @ A4 @ B2 ) ) ).
% sup.cobounded2
thf(fact_685_sup_Ocobounded2,axiom,
! [B2: set_v,A4: set_v] : ( ord_less_eq_set_v @ B2 @ ( sup_sup_set_v @ A4 @ B2 ) ) ).
% sup.cobounded2
thf(fact_686_sup_Oabsorb__iff1,axiom,
( ord_le5216385588623774835_set_v
= ( ^ [B4: set_set_v,A6: set_set_v] :
( ( sup_sup_set_set_v @ A6 @ B4 )
= A6 ) ) ) ).
% sup.absorb_iff1
thf(fact_687_sup_Oabsorb__iff1,axiom,
( ord_le3221252021190050221t_unit
= ( ^ [B4: product_unit,A6: product_unit] :
( ( sup_sup_Product_unit @ A6 @ B4 )
= A6 ) ) ) ).
% sup.absorb_iff1
thf(fact_688_sup_Oabsorb__iff1,axiom,
( ord_le7336532860387713383od_v_v
= ( ^ [B4: set_Product_prod_v_v,A6: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ A6 @ B4 )
= A6 ) ) ) ).
% sup.absorb_iff1
thf(fact_689_sup_Oabsorb__iff1,axiom,
( ord_less_eq_set_v
= ( ^ [B4: set_v,A6: set_v] :
( ( sup_sup_set_v @ A6 @ B4 )
= A6 ) ) ) ).
% sup.absorb_iff1
thf(fact_690_sup_Oabsorb__iff2,axiom,
( ord_le5216385588623774835_set_v
= ( ^ [A6: set_set_v,B4: set_set_v] :
( ( sup_sup_set_set_v @ A6 @ B4 )
= B4 ) ) ) ).
% sup.absorb_iff2
thf(fact_691_sup_Oabsorb__iff2,axiom,
( ord_le3221252021190050221t_unit
= ( ^ [A6: product_unit,B4: product_unit] :
( ( sup_sup_Product_unit @ A6 @ B4 )
= B4 ) ) ) ).
% sup.absorb_iff2
thf(fact_692_sup_Oabsorb__iff2,axiom,
( ord_le7336532860387713383od_v_v
= ( ^ [A6: set_Product_prod_v_v,B4: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ A6 @ B4 )
= B4 ) ) ) ).
% sup.absorb_iff2
thf(fact_693_sup_Oabsorb__iff2,axiom,
( ord_less_eq_set_v
= ( ^ [A6: set_v,B4: set_v] :
( ( sup_sup_set_v @ A6 @ B4 )
= B4 ) ) ) ).
% sup.absorb_iff2
thf(fact_694_sup_OcoboundedI1,axiom,
! [C: set_set_v,A4: set_set_v,B2: set_set_v] :
( ( ord_le5216385588623774835_set_v @ C @ A4 )
=> ( ord_le5216385588623774835_set_v @ C @ ( sup_sup_set_set_v @ A4 @ B2 ) ) ) ).
% sup.coboundedI1
thf(fact_695_sup_OcoboundedI1,axiom,
! [C: product_unit,A4: product_unit,B2: product_unit] :
( ( ord_le3221252021190050221t_unit @ C @ A4 )
=> ( ord_le3221252021190050221t_unit @ C @ ( sup_sup_Product_unit @ A4 @ B2 ) ) ) ).
% sup.coboundedI1
thf(fact_696_sup_OcoboundedI1,axiom,
! [C: set_Product_prod_v_v,A4: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ C @ A4 )
=> ( ord_le7336532860387713383od_v_v @ C @ ( sup_su414716646722978715od_v_v @ A4 @ B2 ) ) ) ).
% sup.coboundedI1
thf(fact_697_sup_OcoboundedI1,axiom,
! [C: set_v,A4: set_v,B2: set_v] :
( ( ord_less_eq_set_v @ C @ A4 )
=> ( ord_less_eq_set_v @ C @ ( sup_sup_set_v @ A4 @ B2 ) ) ) ).
% sup.coboundedI1
thf(fact_698_sup_OcoboundedI2,axiom,
! [C: set_set_v,B2: set_set_v,A4: set_set_v] :
( ( ord_le5216385588623774835_set_v @ C @ B2 )
=> ( ord_le5216385588623774835_set_v @ C @ ( sup_sup_set_set_v @ A4 @ B2 ) ) ) ).
% sup.coboundedI2
thf(fact_699_sup_OcoboundedI2,axiom,
! [C: product_unit,B2: product_unit,A4: product_unit] :
( ( ord_le3221252021190050221t_unit @ C @ B2 )
=> ( ord_le3221252021190050221t_unit @ C @ ( sup_sup_Product_unit @ A4 @ B2 ) ) ) ).
% sup.coboundedI2
thf(fact_700_sup_OcoboundedI2,axiom,
! [C: set_Product_prod_v_v,B2: set_Product_prod_v_v,A4: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ C @ B2 )
=> ( ord_le7336532860387713383od_v_v @ C @ ( sup_su414716646722978715od_v_v @ A4 @ B2 ) ) ) ).
% sup.coboundedI2
thf(fact_701_sup_OcoboundedI2,axiom,
! [C: set_v,B2: set_v,A4: set_v] :
( ( ord_less_eq_set_v @ C @ B2 )
=> ( ord_less_eq_set_v @ C @ ( sup_sup_set_v @ A4 @ B2 ) ) ) ).
% sup.coboundedI2
thf(fact_702_list_Oset__intros_I2_J,axiom,
! [Y: product_prod_v_v,X222: list_P7986770385144383213od_v_v,X21: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ Y @ ( set_Product_prod_v_v2 @ X222 ) )
=> ( member7453568604450474000od_v_v @ Y @ ( set_Product_prod_v_v2 @ ( cons_P4120604216776828829od_v_v @ X21 @ X222 ) ) ) ) ).
% list.set_intros(2)
thf(fact_703_list_Oset__intros_I2_J,axiom,
! [Y: v,X222: list_v,X21: v] :
( ( member_v @ Y @ ( set_v2 @ X222 ) )
=> ( member_v @ Y @ ( set_v2 @ ( cons_v @ X21 @ X222 ) ) ) ) ).
% list.set_intros(2)
thf(fact_704_list_Oset__intros_I1_J,axiom,
! [X21: product_prod_v_v,X222: list_P7986770385144383213od_v_v] : ( member7453568604450474000od_v_v @ X21 @ ( set_Product_prod_v_v2 @ ( cons_P4120604216776828829od_v_v @ X21 @ X222 ) ) ) ).
% list.set_intros(1)
thf(fact_705_list_Oset__intros_I1_J,axiom,
! [X21: v,X222: list_v] : ( member_v @ X21 @ ( set_v2 @ ( cons_v @ X21 @ X222 ) ) ) ).
% list.set_intros(1)
thf(fact_706_list_Oset__cases,axiom,
! [E: product_prod_v_v,A4: list_P7986770385144383213od_v_v] :
( ( member7453568604450474000od_v_v @ E @ ( set_Product_prod_v_v2 @ A4 ) )
=> ( ! [Z22: list_P7986770385144383213od_v_v] :
( A4
!= ( cons_P4120604216776828829od_v_v @ E @ Z22 ) )
=> ~ ! [Z1: product_prod_v_v,Z22: list_P7986770385144383213od_v_v] :
( ( A4
= ( cons_P4120604216776828829od_v_v @ Z1 @ Z22 ) )
=> ~ ( member7453568604450474000od_v_v @ E @ ( set_Product_prod_v_v2 @ Z22 ) ) ) ) ) ).
% list.set_cases
thf(fact_707_list_Oset__cases,axiom,
! [E: v,A4: list_v] :
( ( member_v @ E @ ( set_v2 @ A4 ) )
=> ( ! [Z22: list_v] :
( A4
!= ( cons_v @ E @ Z22 ) )
=> ~ ! [Z1: v,Z22: list_v] :
( ( A4
= ( cons_v @ Z1 @ Z22 ) )
=> ~ ( member_v @ E @ ( set_v2 @ Z22 ) ) ) ) ) ).
% list.set_cases
thf(fact_708_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_709_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_710_subset__code_I1_J,axiom,
! [Xs: list_P7986770385144383213od_v_v,B: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ ( set_Product_prod_v_v2 @ Xs ) @ B )
= ( ! [X2: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X2 @ ( set_Product_prod_v_v2 @ Xs ) )
=> ( member7453568604450474000od_v_v @ X2 @ B ) ) ) ) ).
% subset_code(1)
thf(fact_711_subset__code_I1_J,axiom,
! [Xs: list_v,B: set_v] :
( ( ord_less_eq_set_v @ ( set_v2 @ Xs ) @ B )
= ( ! [X2: v] :
( ( member_v @ X2 @ ( set_v2 @ Xs ) )
=> ( member_v @ X2 @ B ) ) ) ) ).
% subset_code(1)
thf(fact_712_graph_Ostack__class,axiom,
! [Vertices: set_v,Successors: v > set_v,E: sCC_Bl1394983891496994913t_unit,N: v,M: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl9196236973127232072t_unit @ Successors @ E )
=> ( ( member_v @ N @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E ) ) )
=> ( ( member_v @ M @ ( sCC_Bl1280885523602775798t_unit @ E @ N ) )
=> ( member_v @ M @ ( minus_minus_set_v @ ( sCC_Bl4645233313691564917t_unit @ E ) @ ( sCC_Bl157864678168468314t_unit @ E ) ) ) ) ) ) ) ).
% graph.stack_class
thf(fact_713_list_Osel_I1_J,axiom,
! [X21: v,X222: list_v] :
( ( hd_v @ ( cons_v @ X21 @ X222 ) )
= X21 ) ).
% list.sel(1)
thf(fact_714_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_715_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_716_u_I2_J,axiom,
( sCC_Bl4291963740693775144ding_v @ successors @ u @ y
@ ( sCC_Bl3123350270117520219t_unit @ successors
@ ( sCC_Bl7876664385711583351t_unit
@ ^ [Uu: list_v] : ( tl_v @ ( sCC_Bl9201514103433284750t_unit @ e2 ) )
@ ( sCC_Bl349061681862590396t_unit
@ ^ [Uu: list_v] : ( tl_v @ ( sCC_Bl8828226123343373779t_unit @ e2 ) )
@ ( sCC_Bl2708505634401380163t_unit
@ ^ [Uu: set_v] : ( sup_sup_set_v @ ( sCC_Bl157864678168468314t_unit @ e2 ) @ ( sCC_Bl1280885523602775798t_unit @ e2 @ v2 ) )
@ ( sCC_Bl6816368539212994290t_unit
@ ^ [Uu: set_set_v] : ( sup_sup_set_set_v @ ( sCC_Bl2536197123907397897t_unit @ e2 ) @ ( insert_set_v @ ( sCC_Bl1280885523602775798t_unit @ e2 @ v2 ) @ bot_bot_set_set_v ) )
@ e2 ) ) ) )
@ x ) ) ).
% u(2)
thf(fact_717_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_718_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_719_set__union,axiom,
! [Xs: list_set_v,Ys: list_set_v] :
( ( set_set_v2 @ ( union_set_v @ Xs @ Ys ) )
= ( sup_sup_set_set_v @ ( set_set_v2 @ Xs ) @ ( set_set_v2 @ Ys ) ) ) ).
% set_union
thf(fact_720_diff__shunt__var,axiom,
! [X: set_set_v,Y: set_set_v] :
( ( ( minus_7228012346218142266_set_v @ X @ Y )
= bot_bot_set_set_v )
= ( ord_le5216385588623774835_set_v @ X @ Y ) ) ).
% diff_shunt_var
thf(fact_721_diff__shunt__var,axiom,
! [X: product_unit,Y: product_unit] :
( ( ( minus_3524152463667985524t_unit @ X @ Y )
= bot_bot_Product_unit )
= ( ord_le3221252021190050221t_unit @ X @ Y ) ) ).
% diff_shunt_var
thf(fact_722_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_723_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_724_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_725_xy_I1_J,axiom,
( sCC_Bl4022239298816431255edes_v @ x @ y
@ ( sCC_Bl8828226123343373779t_unit
@ ( sCC_Bl7876664385711583351t_unit
@ ^ [Uu: list_v] : ( tl_v @ ( sCC_Bl9201514103433284750t_unit @ e2 ) )
@ ( sCC_Bl349061681862590396t_unit
@ ^ [Uu: list_v] : ( tl_v @ ( sCC_Bl8828226123343373779t_unit @ e2 ) )
@ ( sCC_Bl2708505634401380163t_unit
@ ^ [Uu: set_v] : ( sup_sup_set_v @ ( sCC_Bl157864678168468314t_unit @ e2 ) @ ( sCC_Bl1280885523602775798t_unit @ e2 @ v2 ) )
@ ( sCC_Bl6816368539212994290t_unit
@ ^ [Uu: set_set_v] : ( sup_sup_set_set_v @ ( sCC_Bl2536197123907397897t_unit @ e2 ) @ ( insert_set_v @ ( sCC_Bl1280885523602775798t_unit @ e2 @ v2 ) @ bot_bot_set_set_v ) )
@ e2 ) ) ) ) ) ) ).
% xy(1)
thf(fact_726_u_I1_J,axiom,
( member_v @ u
@ ( sCC_Bl1280885523602775798t_unit
@ ( sCC_Bl7876664385711583351t_unit
@ ^ [Uu: list_v] : ( tl_v @ ( sCC_Bl9201514103433284750t_unit @ e2 ) )
@ ( sCC_Bl349061681862590396t_unit
@ ^ [Uu: list_v] : ( tl_v @ ( sCC_Bl8828226123343373779t_unit @ e2 ) )
@ ( sCC_Bl2708505634401380163t_unit
@ ^ [Uu: set_v] : ( sup_sup_set_v @ ( sCC_Bl157864678168468314t_unit @ e2 ) @ ( sCC_Bl1280885523602775798t_unit @ e2 @ v2 ) )
@ ( sCC_Bl6816368539212994290t_unit
@ ^ [Uu: set_set_v] : ( sup_sup_set_set_v @ ( sCC_Bl2536197123907397897t_unit @ e2 ) @ ( insert_set_v @ ( sCC_Bl1280885523602775798t_unit @ e2 @ v2 ) @ bot_bot_set_set_v ) )
@ e2 ) ) ) )
@ x ) ) ).
% u(1)
thf(fact_727_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_728_the__elem__eq,axiom,
! [X: set_v] :
( ( the_elem_set_v @ ( insert_set_v @ X @ bot_bot_set_set_v ) )
= X ) ).
% the_elem_eq
thf(fact_729_the__elem__eq,axiom,
! [X: v] :
( ( the_elem_v @ ( insert_v @ X @ bot_bot_set_v ) )
= X ) ).
% the_elem_eq
thf(fact_730_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_731_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_732_reachable__end__induct,axiom,
! [X: v,Y: v,P: v > v > $o] :
( ( sCC_Bl649662514949026229able_v @ successors @ X @ Y )
=> ( ! [X3: v] : ( P @ X3 @ X3 )
=> ( ! [X3: v,Y2: v,Z3: v] :
( ( P @ X3 @ Y2 )
=> ( ( member_v @ Z3 @ ( successors @ Y2 ) )
=> ( P @ X3 @ Z3 ) ) )
=> ( P @ X @ Y ) ) ) ) ).
% reachable_end_induct
thf(fact_733_reachable__edge,axiom,
! [Y: v,X: v] :
( ( member_v @ Y @ ( successors @ X ) )
=> ( sCC_Bl649662514949026229able_v @ successors @ X @ Y ) ) ).
% reachable_edge
thf(fact_734_reachable_Osimps,axiom,
! [A1: v,A2: v] :
( ( sCC_Bl649662514949026229able_v @ successors @ A1 @ A2 )
= ( ? [X2: v] :
( ( A1 = X2 )
& ( A2 = X2 ) )
| ? [X2: v,Y3: v,Z2: v] :
( ( A1 = X2 )
& ( A2 = Z2 )
& ( member_v @ Y3 @ ( successors @ X2 ) )
& ( sCC_Bl649662514949026229able_v @ successors @ Y3 @ Z2 ) ) ) ) ).
% reachable.simps
thf(fact_735_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_736_reachable__refl,axiom,
! [X: v] : ( sCC_Bl649662514949026229able_v @ successors @ X @ X ) ).
% reachable_refl
thf(fact_737_reachable_Ocases,axiom,
! [A1: v,A2: v] :
( ( sCC_Bl649662514949026229able_v @ successors @ A1 @ A2 )
=> ( ( A2 != A1 )
=> ~ ! [Y2: v] :
( ( member_v @ Y2 @ ( successors @ A1 ) )
=> ~ ( sCC_Bl649662514949026229able_v @ successors @ Y2 @ A2 ) ) ) ) ).
% reachable.cases
thf(fact_738_e_H_H__def,axiom,
( e3
= ( sCC_Bl7876664385711583351t_unit
@ ^ [Uu: list_v] : ( tl_v @ ( sCC_Bl9201514103433284750t_unit @ e2 ) )
@ e2 ) ) ).
% e''_def
thf(fact_739_ra__reachable,axiom,
! [X: v,Y: v,E4: set_Product_prod_v_v] :
( ( sCC_Bl4291963740693775144ding_v @ successors @ X @ Y @ E4 )
=> ( sCC_Bl649662514949026229able_v @ successors @ X @ Y ) ) ).
% ra_reachable
thf(fact_740_reachable__re,axiom,
! [X: v,Y: v] :
( ( sCC_Bl649662514949026229able_v @ successors @ X @ Y )
=> ( sCC_Bl770211535891879572_end_v @ successors @ X @ Y ) ) ).
% reachable_re
thf(fact_741_re__reachable,axiom,
! [X: v,Y: v] :
( ( sCC_Bl770211535891879572_end_v @ successors @ X @ Y )
=> ( sCC_Bl649662514949026229able_v @ successors @ X @ Y ) ) ).
% re_reachable
thf(fact_742_singleton__conv,axiom,
! [A4: product_prod_v_v] :
( ( collec140062887454715474od_v_v
@ ^ [X2: product_prod_v_v] : ( X2 = A4 ) )
= ( insert1338601472111419319od_v_v @ A4 @ bot_bo723834152578015283od_v_v ) ) ).
% singleton_conv
thf(fact_743_singleton__conv,axiom,
! [A4: set_v] :
( ( collect_set_v
@ ^ [X2: set_v] : ( X2 = A4 ) )
= ( insert_set_v @ A4 @ bot_bot_set_set_v ) ) ).
% singleton_conv
thf(fact_744_singleton__conv,axiom,
! [A4: v] :
( ( collect_v
@ ^ [X2: v] : ( X2 = A4 ) )
= ( insert_v @ A4 @ bot_bot_set_v ) ) ).
% singleton_conv
thf(fact_745_singleton__conv2,axiom,
! [A4: product_prod_v_v] :
( ( collec140062887454715474od_v_v
@ ( ^ [Y4: product_prod_v_v,Z4: product_prod_v_v] : ( Y4 = Z4 )
@ A4 ) )
= ( insert1338601472111419319od_v_v @ A4 @ bot_bo723834152578015283od_v_v ) ) ).
% singleton_conv2
thf(fact_746_singleton__conv2,axiom,
! [A4: set_v] :
( ( collect_set_v
@ ( ^ [Y4: set_v,Z4: set_v] : ( Y4 = Z4 )
@ A4 ) )
= ( insert_set_v @ A4 @ bot_bot_set_set_v ) ) ).
% singleton_conv2
thf(fact_747_singleton__conv2,axiom,
! [A4: v] :
( ( collect_v
@ ( ^ [Y4: v,Z4: v] : ( Y4 = Z4 )
@ A4 ) )
= ( insert_v @ A4 @ bot_bot_set_v ) ) ).
% singleton_conv2
thf(fact_748_e1__def,axiom,
( e1
= ( sCC_Bl7876664385711583351t_unit
@ ^ [Uu: list_v] : ( cons_v @ v2 @ ( sCC_Bl9201514103433284750t_unit @ e ) )
@ ( sCC_Bl349061681862590396t_unit
@ ^ [Uu: list_v] : ( cons_v @ v2 @ ( sCC_Bl8828226123343373779t_unit @ e ) )
@ ( sCC_Bl7870604408699998558t_unit
@ ^ [Uu: set_v] : ( sup_sup_set_v @ ( sCC_Bl4645233313691564917t_unit @ e ) @ ( insert_v @ v2 @ bot_bot_set_v ) )
@ e ) ) ) ) ).
% e1_def
thf(fact_749_calculation_I5_J,axiom,
! [N3: v,M2: v] :
( ( member_v @ M2
@ ( sCC_Bl1280885523602775798t_unit
@ ( sCC_Bl7876664385711583351t_unit
@ ^ [Uu: list_v] : ( tl_v @ ( sCC_Bl9201514103433284750t_unit @ e2 ) )
@ ( sCC_Bl349061681862590396t_unit
@ ^ [Uu: list_v] : ( tl_v @ ( sCC_Bl8828226123343373779t_unit @ e2 ) )
@ ( sCC_Bl2708505634401380163t_unit
@ ^ [Uu: set_v] : ( sup_sup_set_v @ ( sCC_Bl157864678168468314t_unit @ e2 ) @ ( sCC_Bl1280885523602775798t_unit @ e2 @ v2 ) )
@ ( sCC_Bl6816368539212994290t_unit
@ ^ [Uu: set_set_v] : ( sup_sup_set_set_v @ ( sCC_Bl2536197123907397897t_unit @ e2 ) @ ( insert_set_v @ ( sCC_Bl1280885523602775798t_unit @ e2 @ v2 ) @ bot_bot_set_set_v ) )
@ e2 ) ) ) )
@ N3 ) )
= ( ( sCC_Bl1280885523602775798t_unit
@ ( sCC_Bl7876664385711583351t_unit
@ ^ [Uu: list_v] : ( tl_v @ ( sCC_Bl9201514103433284750t_unit @ e2 ) )
@ ( sCC_Bl349061681862590396t_unit
@ ^ [Uu: list_v] : ( tl_v @ ( sCC_Bl8828226123343373779t_unit @ e2 ) )
@ ( sCC_Bl2708505634401380163t_unit
@ ^ [Uu: set_v] : ( sup_sup_set_v @ ( sCC_Bl157864678168468314t_unit @ e2 ) @ ( sCC_Bl1280885523602775798t_unit @ e2 @ v2 ) )
@ ( sCC_Bl6816368539212994290t_unit
@ ^ [Uu: set_set_v] : ( sup_sup_set_set_v @ ( sCC_Bl2536197123907397897t_unit @ e2 ) @ ( insert_set_v @ ( sCC_Bl1280885523602775798t_unit @ e2 @ v2 ) @ bot_bot_set_set_v ) )
@ e2 ) ) ) )
@ N3 )
= ( sCC_Bl1280885523602775798t_unit
@ ( sCC_Bl7876664385711583351t_unit
@ ^ [Uu: list_v] : ( tl_v @ ( sCC_Bl9201514103433284750t_unit @ e2 ) )
@ ( sCC_Bl349061681862590396t_unit
@ ^ [Uu: list_v] : ( tl_v @ ( sCC_Bl8828226123343373779t_unit @ e2 ) )
@ ( sCC_Bl2708505634401380163t_unit
@ ^ [Uu: set_v] : ( sup_sup_set_v @ ( sCC_Bl157864678168468314t_unit @ e2 ) @ ( sCC_Bl1280885523602775798t_unit @ e2 @ v2 ) )
@ ( sCC_Bl6816368539212994290t_unit
@ ^ [Uu: set_set_v] : ( sup_sup_set_set_v @ ( sCC_Bl2536197123907397897t_unit @ e2 ) @ ( insert_set_v @ ( sCC_Bl1280885523602775798t_unit @ e2 @ v2 ) @ bot_bot_set_set_v ) )
@ e2 ) ) ) )
@ M2 ) ) ) ).
% calculation(5)
thf(fact_750_calculation_I6_J,axiom,
! [N3: v] :
( ~ ( member_v @ N3
@ ( sCC_Bl4645233313691564917t_unit
@ ( sCC_Bl7876664385711583351t_unit
@ ^ [Uu: list_v] : ( tl_v @ ( sCC_Bl9201514103433284750t_unit @ e2 ) )
@ ( sCC_Bl349061681862590396t_unit
@ ^ [Uu: list_v] : ( tl_v @ ( sCC_Bl8828226123343373779t_unit @ e2 ) )
@ ( sCC_Bl2708505634401380163t_unit
@ ^ [Uu: set_v] : ( sup_sup_set_v @ ( sCC_Bl157864678168468314t_unit @ e2 ) @ ( sCC_Bl1280885523602775798t_unit @ e2 @ v2 ) )
@ ( sCC_Bl6816368539212994290t_unit
@ ^ [Uu: set_set_v] : ( sup_sup_set_set_v @ ( sCC_Bl2536197123907397897t_unit @ e2 ) @ ( insert_set_v @ ( sCC_Bl1280885523602775798t_unit @ e2 @ v2 ) @ bot_bot_set_set_v ) )
@ e2 ) ) ) ) ) )
=> ( ( sCC_Bl1280885523602775798t_unit
@ ( sCC_Bl7876664385711583351t_unit
@ ^ [Uu: list_v] : ( tl_v @ ( sCC_Bl9201514103433284750t_unit @ e2 ) )
@ ( sCC_Bl349061681862590396t_unit
@ ^ [Uu: list_v] : ( tl_v @ ( sCC_Bl8828226123343373779t_unit @ e2 ) )
@ ( sCC_Bl2708505634401380163t_unit
@ ^ [Uu: set_v] : ( sup_sup_set_v @ ( sCC_Bl157864678168468314t_unit @ e2 ) @ ( sCC_Bl1280885523602775798t_unit @ e2 @ v2 ) )
@ ( sCC_Bl6816368539212994290t_unit
@ ^ [Uu: set_set_v] : ( sup_sup_set_set_v @ ( sCC_Bl2536197123907397897t_unit @ e2 ) @ ( insert_set_v @ ( sCC_Bl1280885523602775798t_unit @ e2 @ v2 ) @ bot_bot_set_set_v ) )
@ e2 ) ) ) )
@ N3 )
= ( insert_v @ N3 @ bot_bot_set_v ) ) ) ).
% calculation(6)
thf(fact_751__092_060open_062v_A_092_060in_062_Avisited_A_Ie_H_092_060lparr_062sccs_A_058_061_Asccs_Ae_H_A_092_060union_062_A_123_092_060S_062_Ae_H_Av_125_M_Aexplored_A_058_061_Aexplored_Ae_H_A_092_060union_062_A_092_060S_062_Ae_H_Av_M_Astack_A_058_061_Atl_A_Istack_Ae_H_J_M_Acstack_A_058_061_Atl_A_Icstack_Ae_H_J_092_060rparr_062_J_092_060close_062,axiom,
( member_v @ v2
@ ( sCC_Bl4645233313691564917t_unit
@ ( sCC_Bl7876664385711583351t_unit
@ ^ [Uu: list_v] : ( tl_v @ ( sCC_Bl9201514103433284750t_unit @ e2 ) )
@ ( sCC_Bl349061681862590396t_unit
@ ^ [Uu: list_v] : ( tl_v @ ( sCC_Bl8828226123343373779t_unit @ e2 ) )
@ ( sCC_Bl2708505634401380163t_unit
@ ^ [Uu: set_v] : ( sup_sup_set_v @ ( sCC_Bl157864678168468314t_unit @ e2 ) @ ( sCC_Bl1280885523602775798t_unit @ e2 @ v2 ) )
@ ( sCC_Bl6816368539212994290t_unit
@ ^ [Uu: set_set_v] : ( sup_sup_set_set_v @ ( sCC_Bl2536197123907397897t_unit @ e2 ) @ ( insert_set_v @ ( sCC_Bl1280885523602775798t_unit @ e2 @ v2 ) @ bot_bot_set_set_v ) )
@ e2 ) ) ) ) ) ) ).
% \<open>v \<in> visited (e'\<lparr>sccs := sccs e' \<union> {\<S> e' v}, explored := explored e' \<union> \<S> e' v, stack := tl (stack e'), cstack := tl (cstack e')\<rparr>)\<close>
thf(fact_752_calculation_I11_J,axiom,
! [N3: v,M2: v] :
( ( sCC_Bl4022239298816431255edes_v @ N3 @ M2
@ ( sCC_Bl8828226123343373779t_unit
@ ( sCC_Bl7876664385711583351t_unit
@ ^ [Uu: list_v] : ( tl_v @ ( sCC_Bl9201514103433284750t_unit @ e2 ) )
@ ( sCC_Bl349061681862590396t_unit
@ ^ [Uu: list_v] : ( tl_v @ ( sCC_Bl8828226123343373779t_unit @ e2 ) )
@ ( sCC_Bl2708505634401380163t_unit
@ ^ [Uu: set_v] : ( sup_sup_set_v @ ( sCC_Bl157864678168468314t_unit @ e2 ) @ ( sCC_Bl1280885523602775798t_unit @ e2 @ v2 ) )
@ ( sCC_Bl6816368539212994290t_unit
@ ^ [Uu: set_set_v] : ( sup_sup_set_set_v @ ( sCC_Bl2536197123907397897t_unit @ e2 ) @ ( insert_set_v @ ( sCC_Bl1280885523602775798t_unit @ e2 @ v2 ) @ bot_bot_set_set_v ) )
@ e2 ) ) ) ) ) )
=> ( sCC_Bl4022239298816431255edes_v @ N3 @ M2
@ ( sCC_Bl9201514103433284750t_unit
@ ( sCC_Bl7876664385711583351t_unit
@ ^ [Uu: list_v] : ( tl_v @ ( sCC_Bl9201514103433284750t_unit @ e2 ) )
@ ( sCC_Bl349061681862590396t_unit
@ ^ [Uu: list_v] : ( tl_v @ ( sCC_Bl8828226123343373779t_unit @ e2 ) )
@ ( sCC_Bl2708505634401380163t_unit
@ ^ [Uu: set_v] : ( sup_sup_set_v @ ( sCC_Bl157864678168468314t_unit @ e2 ) @ ( sCC_Bl1280885523602775798t_unit @ e2 @ v2 ) )
@ ( sCC_Bl6816368539212994290t_unit
@ ^ [Uu: set_set_v] : ( sup_sup_set_set_v @ ( sCC_Bl2536197123907397897t_unit @ e2 ) @ ( insert_set_v @ ( sCC_Bl1280885523602775798t_unit @ e2 @ v2 ) @ bot_bot_set_set_v ) )
@ e2 ) ) ) ) ) ) ) ).
% calculation(11)
thf(fact_753_stack2,axiom,
( ( sCC_Bl8828226123343373779t_unit
@ ( sCC_Bl7876664385711583351t_unit
@ ^ [Uu: list_v] : ( tl_v @ ( sCC_Bl9201514103433284750t_unit @ e2 ) )
@ ( sCC_Bl349061681862590396t_unit
@ ^ [Uu: list_v] : ( tl_v @ ( sCC_Bl8828226123343373779t_unit @ e2 ) )
@ ( sCC_Bl2708505634401380163t_unit
@ ^ [Uu: set_v] : ( sup_sup_set_v @ ( sCC_Bl157864678168468314t_unit @ e2 ) @ ( sCC_Bl1280885523602775798t_unit @ e2 @ v2 ) )
@ ( sCC_Bl6816368539212994290t_unit
@ ^ [Uu: set_set_v] : ( sup_sup_set_set_v @ ( sCC_Bl2536197123907397897t_unit @ e2 ) @ ( insert_set_v @ ( sCC_Bl1280885523602775798t_unit @ e2 @ v2 ) @ bot_bot_set_set_v ) )
@ e2 ) ) ) ) )
= ( sCC_Bl8828226123343373779t_unit @ e ) ) ).
% stack2
thf(fact_754_calculation_I15_J,axiom,
! [X4: v] :
( ( member_v @ X4
@ ( sCC_Bl157864678168468314t_unit
@ ( sCC_Bl7876664385711583351t_unit
@ ^ [Uu: list_v] : ( tl_v @ ( sCC_Bl9201514103433284750t_unit @ e2 ) )
@ ( sCC_Bl349061681862590396t_unit
@ ^ [Uu: list_v] : ( tl_v @ ( sCC_Bl8828226123343373779t_unit @ e2 ) )
@ ( sCC_Bl2708505634401380163t_unit
@ ^ [Uu: set_v] : ( sup_sup_set_v @ ( sCC_Bl157864678168468314t_unit @ e2 ) @ ( sCC_Bl1280885523602775798t_unit @ e2 @ v2 ) )
@ ( sCC_Bl6816368539212994290t_unit
@ ^ [Uu: set_set_v] : ( sup_sup_set_set_v @ ( sCC_Bl2536197123907397897t_unit @ e2 ) @ ( insert_set_v @ ( sCC_Bl1280885523602775798t_unit @ e2 @ v2 ) @ bot_bot_set_set_v ) )
@ e2 ) ) ) ) ) )
=> ! [M2: v] :
( ( sCC_Bl649662514949026229able_v @ successors @ X4 @ M2 )
=> ( member_v @ M2
@ ( sCC_Bl157864678168468314t_unit
@ ( sCC_Bl7876664385711583351t_unit
@ ^ [Uu: list_v] : ( tl_v @ ( sCC_Bl9201514103433284750t_unit @ e2 ) )
@ ( sCC_Bl349061681862590396t_unit
@ ^ [Uu: list_v] : ( tl_v @ ( sCC_Bl8828226123343373779t_unit @ e2 ) )
@ ( sCC_Bl2708505634401380163t_unit
@ ^ [Uu: set_v] : ( sup_sup_set_v @ ( sCC_Bl157864678168468314t_unit @ e2 ) @ ( sCC_Bl1280885523602775798t_unit @ e2 @ v2 ) )
@ ( sCC_Bl6816368539212994290t_unit
@ ^ [Uu: set_set_v] : ( sup_sup_set_set_v @ ( sCC_Bl2536197123907397897t_unit @ e2 ) @ ( insert_set_v @ ( sCC_Bl1280885523602775798t_unit @ e2 @ v2 ) @ bot_bot_set_set_v ) )
@ e2 ) ) ) ) ) ) ) ) ).
% calculation(15)
thf(fact_755_calculation_I19_J,axiom,
! [X4: v] :
( ( member_v @ X4
@ ( set_v2
@ ( sCC_Bl8828226123343373779t_unit
@ ( sCC_Bl7876664385711583351t_unit
@ ^ [Uu: list_v] : ( tl_v @ ( sCC_Bl9201514103433284750t_unit @ e2 ) )
@ ( sCC_Bl349061681862590396t_unit
@ ^ [Uu: list_v] : ( tl_v @ ( sCC_Bl8828226123343373779t_unit @ e2 ) )
@ ( sCC_Bl2708505634401380163t_unit
@ ^ [Uu: set_v] : ( sup_sup_set_v @ ( sCC_Bl157864678168468314t_unit @ e2 ) @ ( sCC_Bl1280885523602775798t_unit @ e2 @ v2 ) )
@ ( sCC_Bl6816368539212994290t_unit
@ ^ [Uu: set_set_v] : ( sup_sup_set_set_v @ ( sCC_Bl2536197123907397897t_unit @ e2 ) @ ( insert_set_v @ ( sCC_Bl1280885523602775798t_unit @ e2 @ v2 ) @ bot_bot_set_set_v ) )
@ e2 ) ) ) ) ) ) )
=> ! [Xa: v] :
( ( member_v @ Xa
@ ( sCC_Bl1280885523602775798t_unit
@ ( sCC_Bl7876664385711583351t_unit
@ ^ [Uu: list_v] : ( tl_v @ ( sCC_Bl9201514103433284750t_unit @ e2 ) )
@ ( sCC_Bl349061681862590396t_unit
@ ^ [Uu: list_v] : ( tl_v @ ( sCC_Bl8828226123343373779t_unit @ e2 ) )
@ ( sCC_Bl2708505634401380163t_unit
@ ^ [Uu: set_v] : ( sup_sup_set_v @ ( sCC_Bl157864678168468314t_unit @ e2 ) @ ( sCC_Bl1280885523602775798t_unit @ e2 @ v2 ) )
@ ( sCC_Bl6816368539212994290t_unit
@ ^ [Uu: set_set_v] : ( sup_sup_set_set_v @ ( sCC_Bl2536197123907397897t_unit @ e2 ) @ ( insert_set_v @ ( sCC_Bl1280885523602775798t_unit @ e2 @ v2 ) @ bot_bot_set_set_v ) )
@ e2 ) ) ) )
@ X4 ) )
=> ( ( member_v @ Xa
@ ( set_v2
@ ( sCC_Bl9201514103433284750t_unit
@ ( sCC_Bl7876664385711583351t_unit
@ ^ [Uu: list_v] : ( tl_v @ ( sCC_Bl9201514103433284750t_unit @ e2 ) )
@ ( sCC_Bl349061681862590396t_unit
@ ^ [Uu: list_v] : ( tl_v @ ( sCC_Bl8828226123343373779t_unit @ e2 ) )
@ ( sCC_Bl2708505634401380163t_unit
@ ^ [Uu: set_v] : ( sup_sup_set_v @ ( sCC_Bl157864678168468314t_unit @ e2 ) @ ( sCC_Bl1280885523602775798t_unit @ e2 @ v2 ) )
@ ( sCC_Bl6816368539212994290t_unit
@ ^ [Uu: set_set_v] : ( sup_sup_set_set_v @ ( sCC_Bl2536197123907397897t_unit @ e2 ) @ ( insert_set_v @ ( sCC_Bl1280885523602775798t_unit @ e2 @ v2 ) @ bot_bot_set_set_v ) )
@ e2 ) ) ) ) ) ) )
=> ( sCC_Bl4022239298816431255edes_v @ Xa @ X4
@ ( sCC_Bl9201514103433284750t_unit
@ ( sCC_Bl7876664385711583351t_unit
@ ^ [Uu: list_v] : ( tl_v @ ( sCC_Bl9201514103433284750t_unit @ e2 ) )
@ ( sCC_Bl349061681862590396t_unit
@ ^ [Uu: list_v] : ( tl_v @ ( sCC_Bl8828226123343373779t_unit @ e2 ) )
@ ( sCC_Bl2708505634401380163t_unit
@ ^ [Uu: set_v] : ( sup_sup_set_v @ ( sCC_Bl157864678168468314t_unit @ e2 ) @ ( sCC_Bl1280885523602775798t_unit @ e2 @ v2 ) )
@ ( sCC_Bl6816368539212994290t_unit
@ ^ [Uu: set_set_v] : ( sup_sup_set_set_v @ ( sCC_Bl2536197123907397897t_unit @ e2 ) @ ( insert_set_v @ ( sCC_Bl1280885523602775798t_unit @ e2 @ v2 ) @ bot_bot_set_set_v ) )
@ e2 ) ) ) ) ) ) ) ) ) ).
% calculation(19)
thf(fact_756_calculation_I12_J,axiom,
( ord_less_eq_set_v
@ ( sCC_Bl157864678168468314t_unit
@ ( sCC_Bl7876664385711583351t_unit
@ ^ [Uu: list_v] : ( tl_v @ ( sCC_Bl9201514103433284750t_unit @ e2 ) )
@ ( sCC_Bl349061681862590396t_unit
@ ^ [Uu: list_v] : ( tl_v @ ( sCC_Bl8828226123343373779t_unit @ e2 ) )
@ ( sCC_Bl2708505634401380163t_unit
@ ^ [Uu: set_v] : ( sup_sup_set_v @ ( sCC_Bl157864678168468314t_unit @ e2 ) @ ( sCC_Bl1280885523602775798t_unit @ e2 @ v2 ) )
@ ( sCC_Bl6816368539212994290t_unit
@ ^ [Uu: set_set_v] : ( sup_sup_set_set_v @ ( sCC_Bl2536197123907397897t_unit @ e2 ) @ ( insert_set_v @ ( sCC_Bl1280885523602775798t_unit @ e2 @ v2 ) @ bot_bot_set_set_v ) )
@ e2 ) ) ) ) )
@ ( sCC_Bl4645233313691564917t_unit
@ ( sCC_Bl7876664385711583351t_unit
@ ^ [Uu: list_v] : ( tl_v @ ( sCC_Bl9201514103433284750t_unit @ e2 ) )
@ ( sCC_Bl349061681862590396t_unit
@ ^ [Uu: list_v] : ( tl_v @ ( sCC_Bl8828226123343373779t_unit @ e2 ) )
@ ( sCC_Bl2708505634401380163t_unit
@ ^ [Uu: set_v] : ( sup_sup_set_v @ ( sCC_Bl157864678168468314t_unit @ e2 ) @ ( sCC_Bl1280885523602775798t_unit @ e2 @ v2 ) )
@ ( sCC_Bl6816368539212994290t_unit
@ ^ [Uu: set_set_v] : ( sup_sup_set_set_v @ ( sCC_Bl2536197123907397897t_unit @ e2 ) @ ( insert_set_v @ ( sCC_Bl1280885523602775798t_unit @ e2 @ v2 ) @ bot_bot_set_set_v ) )
@ e2 ) ) ) ) ) ) ).
% calculation(12)
thf(fact_757_subenv,axiom,
( sCC_Bl5768913643336123637t_unit @ e
@ ( sCC_Bl7876664385711583351t_unit
@ ^ [Uu: list_v] : ( tl_v @ ( sCC_Bl9201514103433284750t_unit @ e2 ) )
@ ( sCC_Bl349061681862590396t_unit
@ ^ [Uu: list_v] : ( tl_v @ ( sCC_Bl8828226123343373779t_unit @ e2 ) )
@ ( sCC_Bl2708505634401380163t_unit
@ ^ [Uu: set_v] : ( sup_sup_set_v @ ( sCC_Bl157864678168468314t_unit @ e2 ) @ ( sCC_Bl1280885523602775798t_unit @ e2 @ v2 ) )
@ ( sCC_Bl6816368539212994290t_unit
@ ^ [Uu: set_set_v] : ( sup_sup_set_set_v @ ( sCC_Bl2536197123907397897t_unit @ e2 ) @ ( insert_set_v @ ( sCC_Bl1280885523602775798t_unit @ e2 @ v2 ) @ bot_bot_set_set_v ) )
@ e2 ) ) ) ) ) ).
% subenv
thf(fact_758_calculation_I13_J,axiom,
( ord_less_eq_set_v
@ ( set_v2
@ ( sCC_Bl9201514103433284750t_unit
@ ( sCC_Bl7876664385711583351t_unit
@ ^ [Uu: list_v] : ( tl_v @ ( sCC_Bl9201514103433284750t_unit @ e2 ) )
@ ( sCC_Bl349061681862590396t_unit
@ ^ [Uu: list_v] : ( tl_v @ ( sCC_Bl8828226123343373779t_unit @ e2 ) )
@ ( sCC_Bl2708505634401380163t_unit
@ ^ [Uu: set_v] : ( sup_sup_set_v @ ( sCC_Bl157864678168468314t_unit @ e2 ) @ ( sCC_Bl1280885523602775798t_unit @ e2 @ v2 ) )
@ ( sCC_Bl6816368539212994290t_unit
@ ^ [Uu: set_set_v] : ( sup_sup_set_set_v @ ( sCC_Bl2536197123907397897t_unit @ e2 ) @ ( insert_set_v @ ( sCC_Bl1280885523602775798t_unit @ e2 @ v2 ) @ bot_bot_set_set_v ) )
@ e2 ) ) ) ) ) )
@ ( sCC_Bl4645233313691564917t_unit
@ ( sCC_Bl7876664385711583351t_unit
@ ^ [Uu: list_v] : ( tl_v @ ( sCC_Bl9201514103433284750t_unit @ e2 ) )
@ ( sCC_Bl349061681862590396t_unit
@ ^ [Uu: list_v] : ( tl_v @ ( sCC_Bl8828226123343373779t_unit @ e2 ) )
@ ( sCC_Bl2708505634401380163t_unit
@ ^ [Uu: set_v] : ( sup_sup_set_v @ ( sCC_Bl157864678168468314t_unit @ e2 ) @ ( sCC_Bl1280885523602775798t_unit @ e2 @ v2 ) )
@ ( sCC_Bl6816368539212994290t_unit
@ ^ [Uu: set_set_v] : ( sup_sup_set_set_v @ ( sCC_Bl2536197123907397897t_unit @ e2 ) @ ( insert_set_v @ ( sCC_Bl1280885523602775798t_unit @ e2 @ v2 ) @ bot_bot_set_set_v ) )
@ e2 ) ) ) ) ) ) ).
% calculation(13)
thf(fact_759_calculation_I10_J,axiom,
! [N3: v,M2: v] :
( ( sCC_Bl4022239298816431255edes_v @ N3 @ M2
@ ( sCC_Bl8828226123343373779t_unit
@ ( sCC_Bl7876664385711583351t_unit
@ ^ [Uu: list_v] : ( tl_v @ ( sCC_Bl9201514103433284750t_unit @ e2 ) )
@ ( sCC_Bl349061681862590396t_unit
@ ^ [Uu: list_v] : ( tl_v @ ( sCC_Bl8828226123343373779t_unit @ e2 ) )
@ ( sCC_Bl2708505634401380163t_unit
@ ^ [Uu: set_v] : ( sup_sup_set_v @ ( sCC_Bl157864678168468314t_unit @ e2 ) @ ( sCC_Bl1280885523602775798t_unit @ e2 @ v2 ) )
@ ( sCC_Bl6816368539212994290t_unit
@ ^ [Uu: set_set_v] : ( sup_sup_set_set_v @ ( sCC_Bl2536197123907397897t_unit @ e2 ) @ ( insert_set_v @ ( sCC_Bl1280885523602775798t_unit @ e2 @ v2 ) @ bot_bot_set_set_v ) )
@ e2 ) ) ) ) ) )
=> ( sCC_Bl649662514949026229able_v @ successors @ M2 @ N3 ) ) ).
% calculation(10)
thf(fact_760_pre__dfs__pre__dfss,axiom,
! [V3: v,E: sCC_Bl1394983891496994913t_unit] :
( ( sCC_Bl36166008131615352t_unit @ successors @ V3 @ E )
=> ( sCC_Bl1748261141445803503t_unit @ successors @ V3
@ ( sCC_Bl7876664385711583351t_unit
@ ^ [Uu: list_v] : ( cons_v @ V3 @ ( sCC_Bl9201514103433284750t_unit @ E ) )
@ ( sCC_Bl349061681862590396t_unit
@ ^ [Uu: list_v] : ( cons_v @ V3 @ ( sCC_Bl8828226123343373779t_unit @ E ) )
@ ( sCC_Bl7870604408699998558t_unit
@ ^ [Uu: set_v] : ( sup_sup_set_v @ ( sCC_Bl4645233313691564917t_unit @ E ) @ ( insert_v @ V3 @ bot_bot_set_v ) )
@ E ) ) ) ) ) ).
% pre_dfs_pre_dfss
thf(fact_761_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_762_Un__def,axiom,
( sup_su414716646722978715od_v_v
= ( ^ [A8: set_Product_prod_v_v,B6: set_Product_prod_v_v] :
( collec140062887454715474od_v_v
@ ^ [X2: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X2 @ A8 )
| ( member7453568604450474000od_v_v @ X2 @ B6 ) ) ) ) ) ).
% Un_def
thf(fact_763_Un__def,axiom,
( sup_sup_set_v
= ( ^ [A8: set_v,B6: set_v] :
( collect_v
@ ^ [X2: v] :
( ( member_v @ X2 @ A8 )
| ( member_v @ X2 @ B6 ) ) ) ) ) ).
% Un_def
thf(fact_764_Un__def,axiom,
( sup_sup_set_set_v
= ( ^ [A8: set_set_v,B6: set_set_v] :
( collect_set_v
@ ^ [X2: set_v] :
( ( member_set_v @ X2 @ A8 )
| ( member_set_v @ X2 @ B6 ) ) ) ) ) ).
% Un_def
thf(fact_765_empty__def,axiom,
( bot_bo723834152578015283od_v_v
= ( collec140062887454715474od_v_v
@ ^ [X2: product_prod_v_v] : $false ) ) ).
% empty_def
thf(fact_766_empty__def,axiom,
( bot_bot_set_set_v
= ( collect_set_v
@ ^ [X2: set_v] : $false ) ) ).
% empty_def
thf(fact_767_empty__def,axiom,
( bot_bot_set_v
= ( collect_v
@ ^ [X2: v] : $false ) ) ).
% empty_def
thf(fact_768_set__diff__eq,axiom,
( minus_4183494784930505774od_v_v
= ( ^ [A8: set_Product_prod_v_v,B6: set_Product_prod_v_v] :
( collec140062887454715474od_v_v
@ ^ [X2: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X2 @ A8 )
& ~ ( member7453568604450474000od_v_v @ X2 @ B6 ) ) ) ) ) ).
% set_diff_eq
thf(fact_769_set__diff__eq,axiom,
( minus_7228012346218142266_set_v
= ( ^ [A8: set_set_v,B6: set_set_v] :
( collect_set_v
@ ^ [X2: set_v] :
( ( member_set_v @ X2 @ A8 )
& ~ ( member_set_v @ X2 @ B6 ) ) ) ) ) ).
% set_diff_eq
thf(fact_770_set__diff__eq,axiom,
( minus_minus_set_v
= ( ^ [A8: set_v,B6: set_v] :
( collect_v
@ ^ [X2: v] :
( ( member_v @ X2 @ A8 )
& ~ ( member_v @ X2 @ B6 ) ) ) ) ) ).
% set_diff_eq
thf(fact_771_sup__set__def,axiom,
( sup_su414716646722978715od_v_v
= ( ^ [A8: set_Product_prod_v_v,B6: set_Product_prod_v_v] :
( collec140062887454715474od_v_v
@ ( sup_su5941406310530359554_v_v_o
@ ^ [X2: product_prod_v_v] : ( member7453568604450474000od_v_v @ X2 @ A8 )
@ ^ [X2: product_prod_v_v] : ( member7453568604450474000od_v_v @ X2 @ B6 ) ) ) ) ) ).
% sup_set_def
thf(fact_772_sup__set__def,axiom,
( sup_sup_set_v
= ( ^ [A8: set_v,B6: set_v] :
( collect_v
@ ( sup_sup_v_o
@ ^ [X2: v] : ( member_v @ X2 @ A8 )
@ ^ [X2: v] : ( member_v @ X2 @ B6 ) ) ) ) ) ).
% sup_set_def
thf(fact_773_sup__set__def,axiom,
( sup_sup_set_set_v
= ( ^ [A8: set_set_v,B6: set_set_v] :
( collect_set_v
@ ( sup_sup_set_v_o
@ ^ [X2: set_v] : ( member_set_v @ X2 @ A8 )
@ ^ [X2: set_v] : ( member_set_v @ X2 @ B6 ) ) ) ) ) ).
% sup_set_def
thf(fact_774_insert__compr,axiom,
( insert_v
= ( ^ [A6: v,B6: set_v] :
( collect_v
@ ^ [X2: v] :
( ( X2 = A6 )
| ( member_v @ X2 @ B6 ) ) ) ) ) ).
% insert_compr
thf(fact_775_insert__compr,axiom,
( insert1338601472111419319od_v_v
= ( ^ [A6: product_prod_v_v,B6: set_Product_prod_v_v] :
( collec140062887454715474od_v_v
@ ^ [X2: product_prod_v_v] :
( ( X2 = A6 )
| ( member7453568604450474000od_v_v @ X2 @ B6 ) ) ) ) ) ).
% insert_compr
thf(fact_776_insert__compr,axiom,
( insert_set_v
= ( ^ [A6: set_v,B6: set_set_v] :
( collect_set_v
@ ^ [X2: set_v] :
( ( X2 = A6 )
| ( member_set_v @ X2 @ B6 ) ) ) ) ) ).
% insert_compr
thf(fact_777_minus__set__def,axiom,
( minus_4183494784930505774od_v_v
= ( ^ [A8: set_Product_prod_v_v,B6: set_Product_prod_v_v] :
( collec140062887454715474od_v_v
@ ( minus_9095120230875558447_v_v_o
@ ^ [X2: product_prod_v_v] : ( member7453568604450474000od_v_v @ X2 @ A8 )
@ ^ [X2: product_prod_v_v] : ( member7453568604450474000od_v_v @ X2 @ B6 ) ) ) ) ) ).
% minus_set_def
thf(fact_778_minus__set__def,axiom,
( minus_7228012346218142266_set_v
= ( ^ [A8: set_set_v,B6: set_set_v] :
( collect_set_v
@ ( minus_minus_set_v_o
@ ^ [X2: set_v] : ( member_set_v @ X2 @ A8 )
@ ^ [X2: set_v] : ( member_set_v @ X2 @ B6 ) ) ) ) ) ).
% minus_set_def
thf(fact_779_minus__set__def,axiom,
( minus_minus_set_v
= ( ^ [A8: set_v,B6: set_v] :
( collect_v
@ ( minus_minus_v_o
@ ^ [X2: v] : ( member_v @ X2 @ A8 )
@ ^ [X2: v] : ( member_v @ X2 @ B6 ) ) ) ) ) ).
% minus_set_def
thf(fact_780_insert__Collect,axiom,
! [A4: product_prod_v_v,P: product_prod_v_v > $o] :
( ( insert1338601472111419319od_v_v @ A4 @ ( collec140062887454715474od_v_v @ P ) )
= ( collec140062887454715474od_v_v
@ ^ [U: product_prod_v_v] :
( ( U != A4 )
=> ( P @ U ) ) ) ) ).
% insert_Collect
thf(fact_781_insert__Collect,axiom,
! [A4: v,P: v > $o] :
( ( insert_v @ A4 @ ( collect_v @ P ) )
= ( collect_v
@ ^ [U: v] :
( ( U != A4 )
=> ( P @ U ) ) ) ) ).
% insert_Collect
thf(fact_782_insert__Collect,axiom,
! [A4: set_v,P: set_v > $o] :
( ( insert_set_v @ A4 @ ( collect_set_v @ P ) )
= ( collect_set_v
@ ^ [U: set_v] :
( ( U != A4 )
=> ( P @ U ) ) ) ) ).
% insert_Collect
thf(fact_783_Collect__disj__eq,axiom,
! [P: product_prod_v_v > $o,Q: product_prod_v_v > $o] :
( ( collec140062887454715474od_v_v
@ ^ [X2: product_prod_v_v] :
( ( P @ X2 )
| ( Q @ X2 ) ) )
= ( sup_su414716646722978715od_v_v @ ( collec140062887454715474od_v_v @ P ) @ ( collec140062887454715474od_v_v @ Q ) ) ) ).
% Collect_disj_eq
thf(fact_784_Collect__disj__eq,axiom,
! [P: v > $o,Q: v > $o] :
( ( collect_v
@ ^ [X2: v] :
( ( P @ X2 )
| ( Q @ X2 ) ) )
= ( sup_sup_set_v @ ( collect_v @ P ) @ ( collect_v @ Q ) ) ) ).
% Collect_disj_eq
thf(fact_785_Collect__disj__eq,axiom,
! [P: set_v > $o,Q: set_v > $o] :
( ( collect_set_v
@ ^ [X2: set_v] :
( ( P @ X2 )
| ( Q @ X2 ) ) )
= ( sup_sup_set_set_v @ ( collect_set_v @ P ) @ ( collect_set_v @ Q ) ) ) ).
% Collect_disj_eq
thf(fact_786_graph_Oreachable_Ocong,axiom,
sCC_Bl649662514949026229able_v = sCC_Bl649662514949026229able_v ).
% graph.reachable.cong
thf(fact_787_Collect__subset,axiom,
! [A: set_set_v,P: set_v > $o] :
( ord_le5216385588623774835_set_v
@ ( collect_set_v
@ ^ [X2: set_v] :
( ( member_set_v @ X2 @ A )
& ( P @ X2 ) ) )
@ A ) ).
% Collect_subset
thf(fact_788_Collect__subset,axiom,
! [A: set_Product_prod_v_v,P: product_prod_v_v > $o] :
( ord_le7336532860387713383od_v_v
@ ( collec140062887454715474od_v_v
@ ^ [X2: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X2 @ A )
& ( P @ X2 ) ) )
@ A ) ).
% Collect_subset
thf(fact_789_Collect__subset,axiom,
! [A: set_v,P: v > $o] :
( ord_less_eq_set_v
@ ( collect_v
@ ^ [X2: v] :
( ( member_v @ X2 @ A )
& ( P @ X2 ) ) )
@ A ) ).
% Collect_subset
thf(fact_790_less__eq__set__def,axiom,
( ord_le7336532860387713383od_v_v
= ( ^ [A8: set_Product_prod_v_v,B6: set_Product_prod_v_v] :
( ord_le5892402249245633078_v_v_o
@ ^ [X2: product_prod_v_v] : ( member7453568604450474000od_v_v @ X2 @ A8 )
@ ^ [X2: product_prod_v_v] : ( member7453568604450474000od_v_v @ X2 @ B6 ) ) ) ) ).
% less_eq_set_def
thf(fact_791_less__eq__set__def,axiom,
( ord_less_eq_set_v
= ( ^ [A8: set_v,B6: set_v] :
( ord_less_eq_v_o
@ ^ [X2: v] : ( member_v @ X2 @ A8 )
@ ^ [X2: v] : ( member_v @ X2 @ B6 ) ) ) ) ).
% less_eq_set_def
thf(fact_792_Collect__conv__if,axiom,
! [P: product_prod_v_v > $o,A4: product_prod_v_v] :
( ( ( P @ A4 )
=> ( ( collec140062887454715474od_v_v
@ ^ [X2: product_prod_v_v] :
( ( X2 = A4 )
& ( P @ X2 ) ) )
= ( insert1338601472111419319od_v_v @ A4 @ bot_bo723834152578015283od_v_v ) ) )
& ( ~ ( P @ A4 )
=> ( ( collec140062887454715474od_v_v
@ ^ [X2: product_prod_v_v] :
( ( X2 = A4 )
& ( P @ X2 ) ) )
= bot_bo723834152578015283od_v_v ) ) ) ).
% Collect_conv_if
thf(fact_793_Collect__conv__if,axiom,
! [P: set_v > $o,A4: set_v] :
( ( ( P @ A4 )
=> ( ( collect_set_v
@ ^ [X2: set_v] :
( ( X2 = A4 )
& ( P @ X2 ) ) )
= ( insert_set_v @ A4 @ bot_bot_set_set_v ) ) )
& ( ~ ( P @ A4 )
=> ( ( collect_set_v
@ ^ [X2: set_v] :
( ( X2 = A4 )
& ( P @ X2 ) ) )
= bot_bot_set_set_v ) ) ) ).
% Collect_conv_if
thf(fact_794_Collect__conv__if,axiom,
! [P: v > $o,A4: v] :
( ( ( P @ A4 )
=> ( ( collect_v
@ ^ [X2: v] :
( ( X2 = A4 )
& ( P @ X2 ) ) )
= ( insert_v @ A4 @ bot_bot_set_v ) ) )
& ( ~ ( P @ A4 )
=> ( ( collect_v
@ ^ [X2: v] :
( ( X2 = A4 )
& ( P @ X2 ) ) )
= bot_bot_set_v ) ) ) ).
% Collect_conv_if
thf(fact_795_Collect__conv__if2,axiom,
! [P: product_prod_v_v > $o,A4: product_prod_v_v] :
( ( ( P @ A4 )
=> ( ( collec140062887454715474od_v_v
@ ^ [X2: product_prod_v_v] :
( ( A4 = X2 )
& ( P @ X2 ) ) )
= ( insert1338601472111419319od_v_v @ A4 @ bot_bo723834152578015283od_v_v ) ) )
& ( ~ ( P @ A4 )
=> ( ( collec140062887454715474od_v_v
@ ^ [X2: product_prod_v_v] :
( ( A4 = X2 )
& ( P @ X2 ) ) )
= bot_bo723834152578015283od_v_v ) ) ) ).
% Collect_conv_if2
thf(fact_796_Collect__conv__if2,axiom,
! [P: set_v > $o,A4: set_v] :
( ( ( P @ A4 )
=> ( ( collect_set_v
@ ^ [X2: set_v] :
( ( A4 = X2 )
& ( P @ X2 ) ) )
= ( insert_set_v @ A4 @ bot_bot_set_set_v ) ) )
& ( ~ ( P @ A4 )
=> ( ( collect_set_v
@ ^ [X2: set_v] :
( ( A4 = X2 )
& ( P @ X2 ) ) )
= bot_bot_set_set_v ) ) ) ).
% Collect_conv_if2
thf(fact_797_Collect__conv__if2,axiom,
! [P: v > $o,A4: v] :
( ( ( P @ A4 )
=> ( ( collect_v
@ ^ [X2: v] :
( ( A4 = X2 )
& ( P @ X2 ) ) )
= ( insert_v @ A4 @ bot_bot_set_v ) ) )
& ( ~ ( P @ A4 )
=> ( ( collect_v
@ ^ [X2: v] :
( ( A4 = X2 )
& ( P @ X2 ) ) )
= bot_bot_set_v ) ) ) ).
% Collect_conv_if2
thf(fact_798_insert__def,axiom,
( insert1338601472111419319od_v_v
= ( ^ [A6: product_prod_v_v] :
( sup_su414716646722978715od_v_v
@ ( collec140062887454715474od_v_v
@ ^ [X2: product_prod_v_v] : ( X2 = A6 ) ) ) ) ) ).
% insert_def
thf(fact_799_insert__def,axiom,
( insert_v
= ( ^ [A6: v] :
( sup_sup_set_v
@ ( collect_v
@ ^ [X2: v] : ( X2 = A6 ) ) ) ) ) ).
% insert_def
thf(fact_800_insert__def,axiom,
( insert_set_v
= ( ^ [A6: set_v] :
( sup_sup_set_set_v
@ ( collect_set_v
@ ^ [X2: set_v] : ( X2 = A6 ) ) ) ) ) ).
% insert_def
thf(fact_801_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_802_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_803_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_804_graph_Oreachable__end__induct,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,X: product_prod_v_v,Y: product_prod_v_v,P: product_prod_v_v > product_prod_v_v > $o] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( sCC_Bl4981926079593201289od_v_v @ Successors @ X @ Y )
=> ( ! [X3: product_prod_v_v] : ( P @ X3 @ X3 )
=> ( ! [X3: product_prod_v_v,Y2: product_prod_v_v,Z3: product_prod_v_v] :
( ( P @ X3 @ Y2 )
=> ( ( member7453568604450474000od_v_v @ Z3 @ ( Successors @ Y2 ) )
=> ( P @ X3 @ Z3 ) ) )
=> ( P @ X @ Y ) ) ) ) ) ).
% graph.reachable_end_induct
thf(fact_805_graph_Oreachable__end__induct,axiom,
! [Vertices: set_v,Successors: v > set_v,X: v,Y: v,P: v > v > $o] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl649662514949026229able_v @ Successors @ X @ Y )
=> ( ! [X3: v] : ( P @ X3 @ X3 )
=> ( ! [X3: v,Y2: v,Z3: v] :
( ( P @ X3 @ Y2 )
=> ( ( member_v @ Z3 @ ( Successors @ Y2 ) )
=> ( P @ X3 @ Z3 ) ) )
=> ( P @ X @ Y ) ) ) ) ) ).
% graph.reachable_end_induct
thf(fact_806_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_807_graph_Oreachable_Osimps,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,A1: product_prod_v_v,A2: product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( sCC_Bl4981926079593201289od_v_v @ Successors @ A1 @ A2 )
= ( ? [X2: product_prod_v_v] :
( ( A1 = X2 )
& ( A2 = X2 ) )
| ? [X2: product_prod_v_v,Y3: product_prod_v_v,Z2: product_prod_v_v] :
( ( A1 = X2 )
& ( A2 = Z2 )
& ( member7453568604450474000od_v_v @ Y3 @ ( Successors @ X2 ) )
& ( sCC_Bl4981926079593201289od_v_v @ Successors @ Y3 @ Z2 ) ) ) ) ) ).
% graph.reachable.simps
thf(fact_808_graph_Oreachable_Osimps,axiom,
! [Vertices: set_v,Successors: v > set_v,A1: v,A2: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl649662514949026229able_v @ Successors @ A1 @ A2 )
= ( ? [X2: v] :
( ( A1 = X2 )
& ( A2 = X2 ) )
| ? [X2: v,Y3: v,Z2: v] :
( ( A1 = X2 )
& ( A2 = Z2 )
& ( member_v @ Y3 @ ( Successors @ X2 ) )
& ( sCC_Bl649662514949026229able_v @ Successors @ Y3 @ Z2 ) ) ) ) ) ).
% graph.reachable.simps
thf(fact_809_graph_Oreachable_Ocases,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,A1: product_prod_v_v,A2: product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( sCC_Bl4981926079593201289od_v_v @ Successors @ A1 @ A2 )
=> ( ( A2 != A1 )
=> ~ ! [Y2: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ Y2 @ ( Successors @ A1 ) )
=> ~ ( sCC_Bl4981926079593201289od_v_v @ Successors @ Y2 @ A2 ) ) ) ) ) ).
% graph.reachable.cases
thf(fact_810_graph_Oreachable_Ocases,axiom,
! [Vertices: set_v,Successors: v > set_v,A1: v,A2: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl649662514949026229able_v @ Successors @ A1 @ A2 )
=> ( ( A2 != A1 )
=> ~ ! [Y2: v] :
( ( member_v @ Y2 @ ( Successors @ A1 ) )
=> ~ ( sCC_Bl649662514949026229able_v @ Successors @ Y2 @ A2 ) ) ) ) ) ).
% graph.reachable.cases
thf(fact_811_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_812_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_813_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_814_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_815_graph_Ora__reachable,axiom,
! [Vertices: set_v,Successors: v > set_v,X: v,Y: v,E4: set_Product_prod_v_v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl4291963740693775144ding_v @ Successors @ X @ Y @ E4 )
=> ( sCC_Bl649662514949026229able_v @ Successors @ X @ Y ) ) ) ).
% graph.ra_reachable
thf(fact_816_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_817_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_818_boolean__algebra__cancel_Osup2,axiom,
! [B: set_Product_prod_v_v,K: set_Product_prod_v_v,B2: set_Product_prod_v_v,A4: set_Product_prod_v_v] :
( ( B
= ( sup_su414716646722978715od_v_v @ K @ B2 ) )
=> ( ( sup_su414716646722978715od_v_v @ A4 @ B )
= ( sup_su414716646722978715od_v_v @ K @ ( sup_su414716646722978715od_v_v @ A4 @ B2 ) ) ) ) ).
% boolean_algebra_cancel.sup2
thf(fact_819_boolean__algebra__cancel_Osup2,axiom,
! [B: set_v,K: set_v,B2: set_v,A4: set_v] :
( ( B
= ( sup_sup_set_v @ K @ B2 ) )
=> ( ( sup_sup_set_v @ A4 @ B )
= ( sup_sup_set_v @ K @ ( sup_sup_set_v @ A4 @ B2 ) ) ) ) ).
% boolean_algebra_cancel.sup2
thf(fact_820_boolean__algebra__cancel_Osup2,axiom,
! [B: set_set_v,K: set_set_v,B2: set_set_v,A4: set_set_v] :
( ( B
= ( sup_sup_set_set_v @ K @ B2 ) )
=> ( ( sup_sup_set_set_v @ A4 @ B )
= ( sup_sup_set_set_v @ K @ ( sup_sup_set_set_v @ A4 @ B2 ) ) ) ) ).
% boolean_algebra_cancel.sup2
thf(fact_821_boolean__algebra__cancel_Osup2,axiom,
! [B: product_unit,K: product_unit,B2: product_unit,A4: product_unit] :
( ( B
= ( sup_sup_Product_unit @ K @ B2 ) )
=> ( ( sup_sup_Product_unit @ A4 @ B )
= ( sup_sup_Product_unit @ K @ ( sup_sup_Product_unit @ A4 @ B2 ) ) ) ) ).
% boolean_algebra_cancel.sup2
thf(fact_822_boolean__algebra__cancel_Osup1,axiom,
! [A: set_Product_prod_v_v,K: set_Product_prod_v_v,A4: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( A
= ( sup_su414716646722978715od_v_v @ K @ A4 ) )
=> ( ( sup_su414716646722978715od_v_v @ A @ B2 )
= ( sup_su414716646722978715od_v_v @ K @ ( sup_su414716646722978715od_v_v @ A4 @ B2 ) ) ) ) ).
% boolean_algebra_cancel.sup1
thf(fact_823_boolean__algebra__cancel_Osup1,axiom,
! [A: set_v,K: set_v,A4: set_v,B2: set_v] :
( ( A
= ( sup_sup_set_v @ K @ A4 ) )
=> ( ( sup_sup_set_v @ A @ B2 )
= ( sup_sup_set_v @ K @ ( sup_sup_set_v @ A4 @ B2 ) ) ) ) ).
% boolean_algebra_cancel.sup1
thf(fact_824_boolean__algebra__cancel_Osup1,axiom,
! [A: set_set_v,K: set_set_v,A4: set_set_v,B2: set_set_v] :
( ( A
= ( sup_sup_set_set_v @ K @ A4 ) )
=> ( ( sup_sup_set_set_v @ A @ B2 )
= ( sup_sup_set_set_v @ K @ ( sup_sup_set_set_v @ A4 @ B2 ) ) ) ) ).
% boolean_algebra_cancel.sup1
thf(fact_825_boolean__algebra__cancel_Osup1,axiom,
! [A: product_unit,K: product_unit,A4: product_unit,B2: product_unit] :
( ( A
= ( sup_sup_Product_unit @ K @ A4 ) )
=> ( ( sup_sup_Product_unit @ A @ B2 )
= ( sup_sup_Product_unit @ K @ ( sup_sup_Product_unit @ A4 @ B2 ) ) ) ) ).
% boolean_algebra_cancel.sup1
thf(fact_826_graph_Ora__empty,axiom,
! [Vertices: set_v,Successors: v > set_v,X: v,Y: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl4291963740693775144ding_v @ Successors @ X @ Y @ bot_bo723834152578015283od_v_v )
= ( sCC_Bl649662514949026229able_v @ Successors @ X @ Y ) ) ) ).
% graph.ra_empty
thf(fact_827_list_Osel_I3_J,axiom,
! [X21: v,X222: list_v] :
( ( tl_v @ ( cons_v @ X21 @ X222 ) )
= X222 ) ).
% list.sel(3)
thf(fact_828_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_829_boolean__algebra_Odisj__zero__right,axiom,
! [X: set_set_v] :
( ( sup_sup_set_set_v @ X @ bot_bot_set_set_v )
= X ) ).
% boolean_algebra.disj_zero_right
thf(fact_830_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_831_boolean__algebra_Odisj__zero__right,axiom,
! [X: product_unit] :
( ( sup_sup_Product_unit @ X @ bot_bot_Product_unit )
= X ) ).
% boolean_algebra.disj_zero_right
thf(fact_832_calculation_I1_J,axiom,
! [X4: v] :
( ( member_v @ X4
@ ( sCC_Bl4645233313691564917t_unit
@ ( sCC_Bl7876664385711583351t_unit
@ ^ [Uu: list_v] : ( tl_v @ ( sCC_Bl9201514103433284750t_unit @ e2 ) )
@ ( sCC_Bl349061681862590396t_unit
@ ^ [Uu: list_v] : ( tl_v @ ( sCC_Bl8828226123343373779t_unit @ e2 ) )
@ ( sCC_Bl2708505634401380163t_unit
@ ^ [Uu: set_v] : ( sup_sup_set_v @ ( sCC_Bl157864678168468314t_unit @ e2 ) @ ( sCC_Bl1280885523602775798t_unit @ e2 @ v2 ) )
@ ( sCC_Bl6816368539212994290t_unit
@ ^ [Uu: set_set_v] : ( sup_sup_set_set_v @ ( sCC_Bl2536197123907397897t_unit @ e2 ) @ ( insert_set_v @ ( sCC_Bl1280885523602775798t_unit @ e2 @ v2 ) @ bot_bot_set_set_v ) )
@ e2 ) ) ) ) ) )
=> ( sCC_Bl649662514949026229able_v @ successors
@ ( sCC_Bl1090238580953940555t_unit
@ ( sCC_Bl7876664385711583351t_unit
@ ^ [Uu: list_v] : ( tl_v @ ( sCC_Bl9201514103433284750t_unit @ e2 ) )
@ ( sCC_Bl349061681862590396t_unit
@ ^ [Uu: list_v] : ( tl_v @ ( sCC_Bl8828226123343373779t_unit @ e2 ) )
@ ( sCC_Bl2708505634401380163t_unit
@ ^ [Uu: set_v] : ( sup_sup_set_v @ ( sCC_Bl157864678168468314t_unit @ e2 ) @ ( sCC_Bl1280885523602775798t_unit @ e2 @ v2 ) )
@ ( sCC_Bl6816368539212994290t_unit
@ ^ [Uu: set_set_v] : ( sup_sup_set_set_v @ ( sCC_Bl2536197123907397897t_unit @ e2 ) @ ( insert_set_v @ ( sCC_Bl1280885523602775798t_unit @ e2 @ v2 ) @ bot_bot_set_set_v ) )
@ e2 ) ) ) ) )
@ X4 ) ) ).
% calculation(1)
thf(fact_833_e2,axiom,
( ( sCC_Bloemen_dfs_v @ successors @ v2 @ e )
= ( sCC_Bl7876664385711583351t_unit
@ ^ [Uu: list_v] : ( tl_v @ ( sCC_Bl9201514103433284750t_unit @ e2 ) )
@ ( sCC_Bl349061681862590396t_unit
@ ^ [Uu: list_v] : ( tl_v @ ( sCC_Bl8828226123343373779t_unit @ e2 ) )
@ ( sCC_Bl2708505634401380163t_unit
@ ^ [Uu: set_v] : ( sup_sup_set_v @ ( sCC_Bl157864678168468314t_unit @ e2 ) @ ( sCC_Bl1280885523602775798t_unit @ e2 @ v2 ) )
@ ( sCC_Bl6816368539212994290t_unit
@ ^ [Uu: set_set_v] : ( sup_sup_set_set_v @ ( sCC_Bl2536197123907397897t_unit @ e2 ) @ ( insert_set_v @ ( sCC_Bl1280885523602775798t_unit @ e2 @ v2 ) @ bot_bot_set_set_v ) )
@ e2 ) ) ) ) ) ).
% e2
thf(fact_834_calculation_I7_J,axiom,
! [N3: v] :
( sCC_Bl5398416737448265317bscc_v @ successors
@ ( sCC_Bl1280885523602775798t_unit
@ ( sCC_Bl7876664385711583351t_unit
@ ^ [Uu: list_v] : ( tl_v @ ( sCC_Bl9201514103433284750t_unit @ e2 ) )
@ ( sCC_Bl349061681862590396t_unit
@ ^ [Uu: list_v] : ( tl_v @ ( sCC_Bl8828226123343373779t_unit @ e2 ) )
@ ( sCC_Bl2708505634401380163t_unit
@ ^ [Uu: set_v] : ( sup_sup_set_v @ ( sCC_Bl157864678168468314t_unit @ e2 ) @ ( sCC_Bl1280885523602775798t_unit @ e2 @ v2 ) )
@ ( sCC_Bl6816368539212994290t_unit
@ ^ [Uu: set_set_v] : ( sup_sup_set_set_v @ ( sCC_Bl2536197123907397897t_unit @ e2 ) @ ( insert_set_v @ ( sCC_Bl1280885523602775798t_unit @ e2 @ v2 ) @ bot_bot_set_set_v ) )
@ e2 ) ) ) )
@ N3 ) ) ).
% calculation(7)
thf(fact_835_calculation_I16_J,axiom,
! [X4: v] :
( ( member_v @ X4
@ ( minus_minus_set_v
@ ( sCC_Bl4645233313691564917t_unit
@ ( sCC_Bl7876664385711583351t_unit
@ ^ [Uu: list_v] : ( tl_v @ ( sCC_Bl9201514103433284750t_unit @ e2 ) )
@ ( sCC_Bl349061681862590396t_unit
@ ^ [Uu: list_v] : ( tl_v @ ( sCC_Bl8828226123343373779t_unit @ e2 ) )
@ ( sCC_Bl2708505634401380163t_unit
@ ^ [Uu: set_v] : ( sup_sup_set_v @ ( sCC_Bl157864678168468314t_unit @ e2 ) @ ( sCC_Bl1280885523602775798t_unit @ e2 @ v2 ) )
@ ( sCC_Bl6816368539212994290t_unit
@ ^ [Uu: set_set_v] : ( sup_sup_set_set_v @ ( sCC_Bl2536197123907397897t_unit @ e2 ) @ ( insert_set_v @ ( sCC_Bl1280885523602775798t_unit @ e2 @ v2 ) @ bot_bot_set_set_v ) )
@ e2 ) ) ) ) )
@ ( set_v2
@ ( sCC_Bl9201514103433284750t_unit
@ ( sCC_Bl7876664385711583351t_unit
@ ^ [Uu: list_v] : ( tl_v @ ( sCC_Bl9201514103433284750t_unit @ e2 ) )
@ ( sCC_Bl349061681862590396t_unit
@ ^ [Uu: list_v] : ( tl_v @ ( sCC_Bl8828226123343373779t_unit @ e2 ) )
@ ( sCC_Bl2708505634401380163t_unit
@ ^ [Uu: set_v] : ( sup_sup_set_v @ ( sCC_Bl157864678168468314t_unit @ e2 ) @ ( sCC_Bl1280885523602775798t_unit @ e2 @ v2 ) )
@ ( sCC_Bl6816368539212994290t_unit
@ ^ [Uu: set_set_v] : ( sup_sup_set_set_v @ ( sCC_Bl2536197123907397897t_unit @ e2 ) @ ( insert_set_v @ ( sCC_Bl1280885523602775798t_unit @ e2 @ v2 ) @ bot_bot_set_set_v ) )
@ e2 ) ) ) ) ) ) ) )
=> ( ( sCC_Bl3795065053823578884t_unit
@ ( sCC_Bl7876664385711583351t_unit
@ ^ [Uu: list_v] : ( tl_v @ ( sCC_Bl9201514103433284750t_unit @ e2 ) )
@ ( sCC_Bl349061681862590396t_unit
@ ^ [Uu: list_v] : ( tl_v @ ( sCC_Bl8828226123343373779t_unit @ e2 ) )
@ ( sCC_Bl2708505634401380163t_unit
@ ^ [Uu: set_v] : ( sup_sup_set_v @ ( sCC_Bl157864678168468314t_unit @ e2 ) @ ( sCC_Bl1280885523602775798t_unit @ e2 @ v2 ) )
@ ( sCC_Bl6816368539212994290t_unit
@ ^ [Uu: set_set_v] : ( sup_sup_set_set_v @ ( sCC_Bl2536197123907397897t_unit @ e2 ) @ ( insert_set_v @ ( sCC_Bl1280885523602775798t_unit @ e2 @ v2 ) @ bot_bot_set_set_v ) )
@ e2 ) ) ) )
@ X4 )
= ( successors @ X4 ) ) ) ).
% calculation(16)
thf(fact_836_explored__vsuccs,axiom,
! [U2: v] :
( ( member_v @ U2
@ ( sCC_Bl157864678168468314t_unit
@ ( sCC_Bl7876664385711583351t_unit
@ ^ [Uu: list_v] : ( tl_v @ ( sCC_Bl9201514103433284750t_unit @ e2 ) )
@ ( sCC_Bl349061681862590396t_unit
@ ^ [Uu: list_v] : ( tl_v @ ( sCC_Bl8828226123343373779t_unit @ e2 ) )
@ ( sCC_Bl2708505634401380163t_unit
@ ^ [Uu: set_v] : ( sup_sup_set_v @ ( sCC_Bl157864678168468314t_unit @ e2 ) @ ( sCC_Bl1280885523602775798t_unit @ e2 @ v2 ) )
@ ( sCC_Bl6816368539212994290t_unit
@ ^ [Uu: set_set_v] : ( sup_sup_set_set_v @ ( sCC_Bl2536197123907397897t_unit @ e2 ) @ ( insert_set_v @ ( sCC_Bl1280885523602775798t_unit @ e2 @ v2 ) @ bot_bot_set_set_v ) )
@ e2 ) ) ) ) ) )
=> ( ( sCC_Bl3795065053823578884t_unit
@ ( sCC_Bl7876664385711583351t_unit
@ ^ [Uu: list_v] : ( tl_v @ ( sCC_Bl9201514103433284750t_unit @ e2 ) )
@ ( sCC_Bl349061681862590396t_unit
@ ^ [Uu: list_v] : ( tl_v @ ( sCC_Bl8828226123343373779t_unit @ e2 ) )
@ ( sCC_Bl2708505634401380163t_unit
@ ^ [Uu: set_v] : ( sup_sup_set_v @ ( sCC_Bl157864678168468314t_unit @ e2 ) @ ( sCC_Bl1280885523602775798t_unit @ e2 @ v2 ) )
@ ( sCC_Bl6816368539212994290t_unit
@ ^ [Uu: set_set_v] : ( sup_sup_set_set_v @ ( sCC_Bl2536197123907397897t_unit @ e2 ) @ ( insert_set_v @ ( sCC_Bl1280885523602775798t_unit @ e2 @ v2 ) @ bot_bot_set_set_v ) )
@ e2 ) ) ) )
@ U2 )
= ( successors @ U2 ) ) ) ).
% explored_vsuccs
thf(fact_837_calculation_I4_J,axiom,
! [N3: v] :
( ~ ( member_v @ N3
@ ( sCC_Bl4645233313691564917t_unit
@ ( sCC_Bl7876664385711583351t_unit
@ ^ [Uu: list_v] : ( tl_v @ ( sCC_Bl9201514103433284750t_unit @ e2 ) )
@ ( sCC_Bl349061681862590396t_unit
@ ^ [Uu: list_v] : ( tl_v @ ( sCC_Bl8828226123343373779t_unit @ e2 ) )
@ ( sCC_Bl2708505634401380163t_unit
@ ^ [Uu: set_v] : ( sup_sup_set_v @ ( sCC_Bl157864678168468314t_unit @ e2 ) @ ( sCC_Bl1280885523602775798t_unit @ e2 @ v2 ) )
@ ( sCC_Bl6816368539212994290t_unit
@ ^ [Uu: set_set_v] : ( sup_sup_set_set_v @ ( sCC_Bl2536197123907397897t_unit @ e2 ) @ ( insert_set_v @ ( sCC_Bl1280885523602775798t_unit @ e2 @ v2 ) @ bot_bot_set_set_v ) )
@ e2 ) ) ) ) ) )
=> ( ( sCC_Bl3795065053823578884t_unit
@ ( sCC_Bl7876664385711583351t_unit
@ ^ [Uu: list_v] : ( tl_v @ ( sCC_Bl9201514103433284750t_unit @ e2 ) )
@ ( sCC_Bl349061681862590396t_unit
@ ^ [Uu: list_v] : ( tl_v @ ( sCC_Bl8828226123343373779t_unit @ e2 ) )
@ ( sCC_Bl2708505634401380163t_unit
@ ^ [Uu: set_v] : ( sup_sup_set_v @ ( sCC_Bl157864678168468314t_unit @ e2 ) @ ( sCC_Bl1280885523602775798t_unit @ e2 @ v2 ) )
@ ( sCC_Bl6816368539212994290t_unit
@ ^ [Uu: set_set_v] : ( sup_sup_set_set_v @ ( sCC_Bl2536197123907397897t_unit @ e2 ) @ ( insert_set_v @ ( sCC_Bl1280885523602775798t_unit @ e2 @ v2 ) @ bot_bot_set_set_v ) )
@ e2 ) ) ) )
@ N3 )
= bot_bot_set_v ) ) ).
% calculation(4)
thf(fact_838_is__subscc__def,axiom,
! [S: set_v] :
( ( sCC_Bl5398416737448265317bscc_v @ successors @ S )
= ( ! [X2: v] :
( ( member_v @ X2 @ S )
=> ! [Y3: v] :
( ( member_v @ Y3 @ S )
=> ( sCC_Bl649662514949026229able_v @ successors @ X2 @ Y3 ) ) ) ) ) ).
% is_subscc_def
thf(fact_839_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_840_is__scc__def,axiom,
! [S: set_v] :
( ( sCC_Bloemen_is_scc_v @ successors @ S )
= ( ( S != bot_bot_set_v )
& ( sCC_Bl5398416737448265317bscc_v @ successors @ S )
& ! [S2: set_v] :
( ( ( ord_less_eq_set_v @ S @ S2 )
& ( sCC_Bl5398416737448265317bscc_v @ successors @ S2 ) )
=> ( S2 = S ) ) ) ) ).
% is_scc_def
thf(fact_841_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 )
=> ( ! [X3: v] :
( ( member_v @ X3 @ ( sCC_Bl4645233313691564917t_unit @ E ) )
=> ! [Xa2: v] :
( ( member_v @ Xa2 @ ( minus_minus_set_v @ ( successors @ X3 ) @ ( sCC_Bl3795065053823578884t_unit @ E @ X3 ) ) )
=> ( ( sCC_Bl649662514949026229able_v @ successors @ V3 @ X3 )
=> ~ ( sCC_Bl649662514949026229able_v @ successors @ Xa2 @ W ) ) ) )
=> ( member_v @ W @ ( sCC_Bl4645233313691564917t_unit @ E ) ) ) ) ) ) ).
% reachable_visited
thf(fact_842_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 )
& ! [X2: v] :
( ( member_v @ X2 @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E ) ) )
=> ( sCC_Bl649662514949026229able_v @ successors @ X2 @ V3 ) ) ) ) ).
% pre_dfs_def
thf(fact_843_pre__dfss__explored__pre__dfss,axiom,
! [V3: v,E: sCC_Bl1394983891496994913t_unit,W: v] :
( ( sCC_Bl1748261141445803503t_unit @ successors @ V3 @ E )
=> ( ( member_v @ W @ ( successors @ V3 ) )
=> ( ( member_v @ W @ ( sCC_Bl157864678168468314t_unit @ E ) )
=> ( sCC_Bl1748261141445803503t_unit @ successors @ V3
@ ( sCC_Bl48393358579903213t_unit
@ ^ [Uu: v > set_v,X2: v] : ( if_set_v @ ( X2 = V3 ) @ ( sup_sup_set_v @ ( sCC_Bl3795065053823578884t_unit @ E @ V3 ) @ ( insert_v @ W @ bot_bot_set_v ) ) @ ( sCC_Bl3795065053823578884t_unit @ E @ X2 ) )
@ E ) ) ) ) ) ).
% pre_dfss_explored_pre_dfss
thf(fact_844_graph_Odfs_Ocong,axiom,
sCC_Bloemen_dfs_v = sCC_Bloemen_dfs_v ).
% graph.dfs.cong
thf(fact_845_graph_Ois__subscc_Ocong,axiom,
sCC_Bl5398416737448265317bscc_v = sCC_Bl5398416737448265317bscc_v ).
% graph.is_subscc.cong
thf(fact_846_graph_Ois__scc_Ocong,axiom,
sCC_Bloemen_is_scc_v = sCC_Bloemen_is_scc_v ).
% graph.is_scc.cong
thf(fact_847_fold__congs_I4_J,axiom,
! [R: sCC_Bl1394983891496994913t_unit,R2: sCC_Bl1394983891496994913t_unit,V: set_v,F: set_v > set_v,F2: set_v > set_v] :
( ( R = R2 )
=> ( ( ( sCC_Bl4645233313691564917t_unit @ R2 )
= V )
=> ( ! [V2: set_v] :
( ( V = V2 )
=> ( ( F @ V2 )
= ( F2 @ V2 ) ) )
=> ( ( sCC_Bl7870604408699998558t_unit @ F @ R )
= ( sCC_Bl7870604408699998558t_unit @ F2 @ R2 ) ) ) ) ) ).
% fold_congs(4)
thf(fact_848_unfold__congs_I4_J,axiom,
! [R: sCC_Bl1394983891496994913t_unit,R2: sCC_Bl1394983891496994913t_unit,V: set_v,F: set_v > set_v,F2: set_v > set_v] :
( ( R = R2 )
=> ( ( ( sCC_Bl4645233313691564917t_unit @ R2 )
= V )
=> ( ! [V2: set_v] :
( ( V2 = V )
=> ( ( F @ V2 )
= ( F2 @ V2 ) ) )
=> ( ( sCC_Bl7870604408699998558t_unit @ F @ R )
= ( sCC_Bl7870604408699998558t_unit @ F2 @ R2 ) ) ) ) ) ).
% unfold_congs(4)
thf(fact_849_graph_Ois__scc__def,axiom,
! [Vertices: set_set_v,Successors: set_v > set_set_v,S: set_set_v] :
( ( sCC_Bl5810666556806954322_set_v @ Vertices @ Successors )
=> ( ( sCC_Bl1515522642333523865_set_v @ Successors @ S )
= ( ( S != bot_bot_set_set_v )
& ( sCC_Bl7907073126578335045_set_v @ Successors @ S )
& ! [S2: set_set_v] :
( ( ( ord_le5216385588623774835_set_v @ S @ S2 )
& ( sCC_Bl7907073126578335045_set_v @ Successors @ S2 ) )
=> ( S2 = S ) ) ) ) ) ).
% graph.is_scc_def
thf(fact_850_graph_Ois__scc__def,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,S: set_Product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( sCC_Bl6242042402218619277od_v_v @ Successors @ S )
= ( ( S != bot_bo723834152578015283od_v_v )
& ( sCC_Bl2301996248249672505od_v_v @ Successors @ S )
& ! [S2: set_Product_prod_v_v] :
( ( ( ord_le7336532860387713383od_v_v @ S @ S2 )
& ( sCC_Bl2301996248249672505od_v_v @ Successors @ S2 ) )
=> ( S2 = S ) ) ) ) ) ).
% graph.is_scc_def
thf(fact_851_graph_Ois__scc__def,axiom,
! [Vertices: set_v,Successors: v > set_v,S: set_v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bloemen_is_scc_v @ Successors @ S )
= ( ( S != bot_bot_set_v )
& ( sCC_Bl5398416737448265317bscc_v @ Successors @ S )
& ! [S2: set_v] :
( ( ( ord_less_eq_set_v @ S @ S2 )
& ( sCC_Bl5398416737448265317bscc_v @ Successors @ S2 ) )
=> ( S2 = S ) ) ) ) ) ).
% graph.is_scc_def
thf(fact_852_graph_Ois__subscc__def,axiom,
! [Vertices: set_v,Successors: v > set_v,S: set_v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl5398416737448265317bscc_v @ Successors @ S )
= ( ! [X2: v] :
( ( member_v @ X2 @ S )
=> ! [Y3: v] :
( ( member_v @ Y3 @ S )
=> ( sCC_Bl649662514949026229able_v @ Successors @ X2 @ Y3 ) ) ) ) ) ) ).
% graph.is_subscc_def
thf(fact_853_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_854_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_855_graph_Osubscc__add,axiom,
! [Vertices: set_set_v,Successors: set_v > set_set_v,S: set_set_v,X: set_v,Y: set_v] :
( ( sCC_Bl5810666556806954322_set_v @ Vertices @ Successors )
=> ( ( sCC_Bl7907073126578335045_set_v @ Successors @ S )
=> ( ( member_set_v @ X @ S )
=> ( ( sCC_Bl7354734129683093653_set_v @ Successors @ X @ Y )
=> ( ( sCC_Bl7354734129683093653_set_v @ Successors @ Y @ X )
=> ( sCC_Bl7907073126578335045_set_v @ Successors @ ( insert_set_v @ Y @ S ) ) ) ) ) ) ) ).
% graph.subscc_add
thf(fact_856_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_857_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_858_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 )
& ! [X2: v] :
( ( member_v @ X2 @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E ) ) )
=> ( sCC_Bl649662514949026229able_v @ Successors @ X2 @ V3 ) ) ) ) ) ).
% graph.pre_dfs_def
thf(fact_859_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 )
=> ( ! [X3: v] :
( ( member_v @ X3 @ ( sCC_Bl4645233313691564917t_unit @ E ) )
=> ! [Xa2: v] :
( ( member_v @ Xa2 @ ( minus_minus_set_v @ ( Successors @ X3 ) @ ( sCC_Bl3795065053823578884t_unit @ E @ X3 ) ) )
=> ( ( sCC_Bl649662514949026229able_v @ Successors @ V3 @ X3 )
=> ~ ( sCC_Bl649662514949026229able_v @ Successors @ Xa2 @ W ) ) ) )
=> ( member_v @ W @ ( sCC_Bl4645233313691564917t_unit @ E ) ) ) ) ) ) ) ).
% graph.reachable_visited
thf(fact_860_graph_Opre__dfs__pre__dfss,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_Bl1748261141445803503t_unit @ Successors @ V3
@ ( sCC_Bl7876664385711583351t_unit
@ ^ [Uu: list_v] : ( cons_v @ V3 @ ( sCC_Bl9201514103433284750t_unit @ E ) )
@ ( sCC_Bl349061681862590396t_unit
@ ^ [Uu: list_v] : ( cons_v @ V3 @ ( sCC_Bl8828226123343373779t_unit @ E ) )
@ ( sCC_Bl7870604408699998558t_unit
@ ^ [Uu: set_v] : ( sup_sup_set_v @ ( sCC_Bl4645233313691564917t_unit @ E ) @ ( insert_v @ V3 @ bot_bot_set_v ) )
@ E ) ) ) ) ) ) ).
% graph.pre_dfs_pre_dfss
thf(fact_861_unite__S__tl,axiom,
! [E: sCC_Bl1394983891496994913t_unit,W: v,V3: v,N: 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 @ N @ ( set_v2 @ ( tl_v @ ( sCC_Bl8828226123343373779t_unit @ ( sCC_Bloemen_unite_v @ V3 @ W @ E ) ) ) ) )
=> ( ( sCC_Bl1280885523602775798t_unit @ ( sCC_Bloemen_unite_v @ V3 @ W @ E ) @ N )
= ( sCC_Bl1280885523602775798t_unit @ E @ N ) ) ) ) ) ) ) ) ).
% unite_S_tl
thf(fact_862_calculation_I3_J,axiom,
! [N3: v] :
( ord_less_eq_set_v
@ ( sCC_Bl3795065053823578884t_unit
@ ( sCC_Bl7876664385711583351t_unit
@ ^ [Uu: list_v] : ( tl_v @ ( sCC_Bl9201514103433284750t_unit @ e2 ) )
@ ( sCC_Bl349061681862590396t_unit
@ ^ [Uu: list_v] : ( tl_v @ ( sCC_Bl8828226123343373779t_unit @ e2 ) )
@ ( sCC_Bl2708505634401380163t_unit
@ ^ [Uu: set_v] : ( sup_sup_set_v @ ( sCC_Bl157864678168468314t_unit @ e2 ) @ ( sCC_Bl1280885523602775798t_unit @ e2 @ v2 ) )
@ ( sCC_Bl6816368539212994290t_unit
@ ^ [Uu: set_set_v] : ( sup_sup_set_set_v @ ( sCC_Bl2536197123907397897t_unit @ e2 ) @ ( insert_set_v @ ( sCC_Bl1280885523602775798t_unit @ e2 @ v2 ) @ bot_bot_set_set_v ) )
@ e2 ) ) ) )
@ N3 )
@ ( inf_inf_set_v @ ( successors @ N3 )
@ ( sCC_Bl4645233313691564917t_unit
@ ( sCC_Bl7876664385711583351t_unit
@ ^ [Uu: list_v] : ( tl_v @ ( sCC_Bl9201514103433284750t_unit @ e2 ) )
@ ( sCC_Bl349061681862590396t_unit
@ ^ [Uu: list_v] : ( tl_v @ ( sCC_Bl8828226123343373779t_unit @ e2 ) )
@ ( sCC_Bl2708505634401380163t_unit
@ ^ [Uu: set_v] : ( sup_sup_set_v @ ( sCC_Bl157864678168468314t_unit @ e2 ) @ ( sCC_Bl1280885523602775798t_unit @ e2 @ v2 ) )
@ ( sCC_Bl6816368539212994290t_unit
@ ^ [Uu: set_set_v] : ( sup_sup_set_set_v @ ( sCC_Bl2536197123907397897t_unit @ e2 ) @ ( insert_set_v @ ( sCC_Bl1280885523602775798t_unit @ e2 @ v2 ) @ bot_bot_set_set_v ) )
@ e2 ) ) ) ) ) ) ) ).
% calculation(3)
thf(fact_863_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_864_calculation_I17_J,axiom,
! [N3: v,M2: v] :
( ( ( member_v @ N3
@ ( set_v2
@ ( sCC_Bl8828226123343373779t_unit
@ ( sCC_Bl7876664385711583351t_unit
@ ^ [Uu: list_v] : ( tl_v @ ( sCC_Bl9201514103433284750t_unit @ e2 ) )
@ ( sCC_Bl349061681862590396t_unit
@ ^ [Uu: list_v] : ( tl_v @ ( sCC_Bl8828226123343373779t_unit @ e2 ) )
@ ( sCC_Bl2708505634401380163t_unit
@ ^ [Uu: set_v] : ( sup_sup_set_v @ ( sCC_Bl157864678168468314t_unit @ e2 ) @ ( sCC_Bl1280885523602775798t_unit @ e2 @ v2 ) )
@ ( sCC_Bl6816368539212994290t_unit
@ ^ [Uu: set_set_v] : ( sup_sup_set_set_v @ ( sCC_Bl2536197123907397897t_unit @ e2 ) @ ( insert_set_v @ ( sCC_Bl1280885523602775798t_unit @ e2 @ v2 ) @ bot_bot_set_set_v ) )
@ e2 ) ) ) ) ) ) )
& ( member_v @ M2
@ ( set_v2
@ ( sCC_Bl8828226123343373779t_unit
@ ( sCC_Bl7876664385711583351t_unit
@ ^ [Uu: list_v] : ( tl_v @ ( sCC_Bl9201514103433284750t_unit @ e2 ) )
@ ( sCC_Bl349061681862590396t_unit
@ ^ [Uu: list_v] : ( tl_v @ ( sCC_Bl8828226123343373779t_unit @ e2 ) )
@ ( sCC_Bl2708505634401380163t_unit
@ ^ [Uu: set_v] : ( sup_sup_set_v @ ( sCC_Bl157864678168468314t_unit @ e2 ) @ ( sCC_Bl1280885523602775798t_unit @ e2 @ v2 ) )
@ ( sCC_Bl6816368539212994290t_unit
@ ^ [Uu: set_set_v] : ( sup_sup_set_set_v @ ( sCC_Bl2536197123907397897t_unit @ e2 ) @ ( insert_set_v @ ( sCC_Bl1280885523602775798t_unit @ e2 @ v2 ) @ bot_bot_set_set_v ) )
@ e2 ) ) ) ) ) ) )
& ( N3 != M2 ) )
=> ( ( inf_inf_set_v
@ ( sCC_Bl1280885523602775798t_unit
@ ( sCC_Bl7876664385711583351t_unit
@ ^ [Uu: list_v] : ( tl_v @ ( sCC_Bl9201514103433284750t_unit @ e2 ) )
@ ( sCC_Bl349061681862590396t_unit
@ ^ [Uu: list_v] : ( tl_v @ ( sCC_Bl8828226123343373779t_unit @ e2 ) )
@ ( sCC_Bl2708505634401380163t_unit
@ ^ [Uu: set_v] : ( sup_sup_set_v @ ( sCC_Bl157864678168468314t_unit @ e2 ) @ ( sCC_Bl1280885523602775798t_unit @ e2 @ v2 ) )
@ ( sCC_Bl6816368539212994290t_unit
@ ^ [Uu: set_set_v] : ( sup_sup_set_set_v @ ( sCC_Bl2536197123907397897t_unit @ e2 ) @ ( insert_set_v @ ( sCC_Bl1280885523602775798t_unit @ e2 @ v2 ) @ bot_bot_set_set_v ) )
@ e2 ) ) ) )
@ N3 )
@ ( sCC_Bl1280885523602775798t_unit
@ ( sCC_Bl7876664385711583351t_unit
@ ^ [Uu: list_v] : ( tl_v @ ( sCC_Bl9201514103433284750t_unit @ e2 ) )
@ ( sCC_Bl349061681862590396t_unit
@ ^ [Uu: list_v] : ( tl_v @ ( sCC_Bl8828226123343373779t_unit @ e2 ) )
@ ( sCC_Bl2708505634401380163t_unit
@ ^ [Uu: set_v] : ( sup_sup_set_v @ ( sCC_Bl157864678168468314t_unit @ e2 ) @ ( sCC_Bl1280885523602775798t_unit @ e2 @ v2 ) )
@ ( sCC_Bl6816368539212994290t_unit
@ ^ [Uu: set_set_v] : ( sup_sup_set_set_v @ ( sCC_Bl2536197123907397897t_unit @ e2 ) @ ( insert_set_v @ ( sCC_Bl1280885523602775798t_unit @ e2 @ v2 ) @ bot_bot_set_set_v ) )
@ e2 ) ) ) )
@ M2 ) )
= bot_bot_set_v ) ) ).
% calculation(17)
thf(fact_865_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 ) ) ) )
& ! [X2: v] :
( ( member_v @ X2 @ ( sCC_Bl3795065053823578884t_unit @ E @ V3 ) )
=> ( member_v @ X2 @ ( sup_sup_set_v @ ( sCC_Bl157864678168468314t_unit @ E ) @ ( sCC_Bl1280885523602775798t_unit @ E @ ( hd_v @ ( sCC_Bl8828226123343373779t_unit @ E ) ) ) ) ) )
& ! [X2: v] :
( ( member_v @ X2 @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E ) ) )
=> ( sCC_Bl649662514949026229able_v @ successors @ X2 @ V3 ) )
& ? [Ns: list_v] :
( ( sCC_Bl9201514103433284750t_unit @ E )
= ( cons_v @ V3 @ Ns ) ) ) ) ).
% pre_dfss_def
thf(fact_866_inf_Oidem,axiom,
! [A4: set_v] :
( ( inf_inf_set_v @ A4 @ A4 )
= A4 ) ).
% inf.idem
thf(fact_867_inf_Oidem,axiom,
! [A4: product_unit] :
( ( inf_inf_Product_unit @ A4 @ A4 )
= A4 ) ).
% inf.idem
thf(fact_868_inf__idem,axiom,
! [X: set_v] :
( ( inf_inf_set_v @ X @ X )
= X ) ).
% inf_idem
thf(fact_869_inf__idem,axiom,
! [X: product_unit] :
( ( inf_inf_Product_unit @ X @ X )
= X ) ).
% inf_idem
thf(fact_870_inf_Oleft__idem,axiom,
! [A4: set_v,B2: set_v] :
( ( inf_inf_set_v @ A4 @ ( inf_inf_set_v @ A4 @ B2 ) )
= ( inf_inf_set_v @ A4 @ B2 ) ) ).
% inf.left_idem
thf(fact_871_inf_Oleft__idem,axiom,
! [A4: product_unit,B2: product_unit] :
( ( inf_inf_Product_unit @ A4 @ ( inf_inf_Product_unit @ A4 @ B2 ) )
= ( inf_inf_Product_unit @ A4 @ B2 ) ) ).
% inf.left_idem
thf(fact_872_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_873_inf__left__idem,axiom,
! [X: product_unit,Y: product_unit] :
( ( inf_inf_Product_unit @ X @ ( inf_inf_Product_unit @ X @ Y ) )
= ( inf_inf_Product_unit @ X @ Y ) ) ).
% inf_left_idem
thf(fact_874_inf_Oright__idem,axiom,
! [A4: set_v,B2: set_v] :
( ( inf_inf_set_v @ ( inf_inf_set_v @ A4 @ B2 ) @ B2 )
= ( inf_inf_set_v @ A4 @ B2 ) ) ).
% inf.right_idem
thf(fact_875_inf_Oright__idem,axiom,
! [A4: product_unit,B2: product_unit] :
( ( inf_inf_Product_unit @ ( inf_inf_Product_unit @ A4 @ B2 ) @ B2 )
= ( inf_inf_Product_unit @ A4 @ B2 ) ) ).
% inf.right_idem
thf(fact_876_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_877_inf__right__idem,axiom,
! [X: product_unit,Y: product_unit] :
( ( inf_inf_Product_unit @ ( inf_inf_Product_unit @ X @ Y ) @ Y )
= ( inf_inf_Product_unit @ X @ Y ) ) ).
% inf_right_idem
thf(fact_878_IntI,axiom,
! [C: product_prod_v_v,A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ C @ A )
=> ( ( member7453568604450474000od_v_v @ C @ B )
=> ( member7453568604450474000od_v_v @ C @ ( inf_in6271465464967711157od_v_v @ A @ B ) ) ) ) ).
% IntI
thf(fact_879_IntI,axiom,
! [C: v,A: set_v,B: set_v] :
( ( member_v @ C @ A )
=> ( ( member_v @ C @ B )
=> ( member_v @ C @ ( inf_inf_set_v @ A @ B ) ) ) ) ).
% IntI
thf(fact_880_Int__iff,axiom,
! [C: product_prod_v_v,A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ C @ ( inf_in6271465464967711157od_v_v @ A @ B ) )
= ( ( member7453568604450474000od_v_v @ C @ A )
& ( member7453568604450474000od_v_v @ C @ B ) ) ) ).
% Int_iff
thf(fact_881_Int__iff,axiom,
! [C: v,A: set_v,B: set_v] :
( ( member_v @ C @ ( inf_inf_set_v @ A @ B ) )
= ( ( member_v @ C @ A )
& ( member_v @ C @ B ) ) ) ).
% Int_iff
thf(fact_882_scc__partition,axiom,
! [S: set_v,S3: set_v,X: v] :
( ( sCC_Bloemen_is_scc_v @ successors @ S )
=> ( ( sCC_Bloemen_is_scc_v @ successors @ S3 )
=> ( ( member_v @ X @ ( inf_inf_set_v @ S @ S3 ) )
=> ( S = S3 ) ) ) ) ).
% scc_partition
thf(fact_883_inf_Obounded__iff,axiom,
! [A4: product_unit,B2: product_unit,C: product_unit] :
( ( ord_le3221252021190050221t_unit @ A4 @ ( inf_inf_Product_unit @ B2 @ C ) )
= ( ( ord_le3221252021190050221t_unit @ A4 @ B2 )
& ( ord_le3221252021190050221t_unit @ A4 @ C ) ) ) ).
% inf.bounded_iff
thf(fact_884_inf_Obounded__iff,axiom,
! [A4: set_Product_prod_v_v,B2: set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A4 @ ( inf_in6271465464967711157od_v_v @ B2 @ C ) )
= ( ( ord_le7336532860387713383od_v_v @ A4 @ B2 )
& ( ord_le7336532860387713383od_v_v @ A4 @ C ) ) ) ).
% inf.bounded_iff
thf(fact_885_inf_Obounded__iff,axiom,
! [A4: set_v,B2: set_v,C: set_v] :
( ( ord_less_eq_set_v @ A4 @ ( inf_inf_set_v @ B2 @ C ) )
= ( ( ord_less_eq_set_v @ A4 @ B2 )
& ( ord_less_eq_set_v @ A4 @ C ) ) ) ).
% inf.bounded_iff
thf(fact_886_le__inf__iff,axiom,
! [X: product_unit,Y: product_unit,Z: product_unit] :
( ( ord_le3221252021190050221t_unit @ X @ ( inf_inf_Product_unit @ Y @ Z ) )
= ( ( ord_le3221252021190050221t_unit @ X @ Y )
& ( ord_le3221252021190050221t_unit @ X @ Z ) ) ) ).
% le_inf_iff
thf(fact_887_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_888_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_889_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_890_boolean__algebra_Oconj__zero__left,axiom,
! [X: set_set_v] :
( ( inf_inf_set_set_v @ bot_bot_set_set_v @ X )
= bot_bot_set_set_v ) ).
% boolean_algebra.conj_zero_left
thf(fact_891_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_892_boolean__algebra_Oconj__zero__left,axiom,
! [X: product_unit] :
( ( inf_inf_Product_unit @ bot_bot_Product_unit @ X )
= bot_bot_Product_unit ) ).
% boolean_algebra.conj_zero_left
thf(fact_893_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_894_boolean__algebra_Oconj__zero__right,axiom,
! [X: set_set_v] :
( ( inf_inf_set_set_v @ X @ bot_bot_set_set_v )
= bot_bot_set_set_v ) ).
% boolean_algebra.conj_zero_right
thf(fact_895_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_896_boolean__algebra_Oconj__zero__right,axiom,
! [X: product_unit] :
( ( inf_inf_Product_unit @ X @ bot_bot_Product_unit )
= bot_bot_Product_unit ) ).
% boolean_algebra.conj_zero_right
thf(fact_897_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_898_inf__bot__right,axiom,
! [X: set_set_v] :
( ( inf_inf_set_set_v @ X @ bot_bot_set_set_v )
= bot_bot_set_set_v ) ).
% inf_bot_right
thf(fact_899_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_900_inf__bot__right,axiom,
! [X: product_unit] :
( ( inf_inf_Product_unit @ X @ bot_bot_Product_unit )
= bot_bot_Product_unit ) ).
% inf_bot_right
thf(fact_901_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_902_inf__bot__left,axiom,
! [X: set_set_v] :
( ( inf_inf_set_set_v @ bot_bot_set_set_v @ X )
= bot_bot_set_set_v ) ).
% inf_bot_left
thf(fact_903_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_904_inf__bot__left,axiom,
! [X: product_unit] :
( ( inf_inf_Product_unit @ bot_bot_Product_unit @ X )
= bot_bot_Product_unit ) ).
% inf_bot_left
thf(fact_905_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_906_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_907_sup__inf__absorb,axiom,
! [X: set_set_v,Y: set_set_v] :
( ( sup_sup_set_set_v @ X @ ( inf_inf_set_set_v @ X @ Y ) )
= X ) ).
% sup_inf_absorb
thf(fact_908_sup__inf__absorb,axiom,
! [X: product_unit,Y: product_unit] :
( ( sup_sup_Product_unit @ X @ ( inf_inf_Product_unit @ X @ Y ) )
= X ) ).
% sup_inf_absorb
thf(fact_909_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_910_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_911_inf__sup__absorb,axiom,
! [X: set_set_v,Y: set_set_v] :
( ( inf_inf_set_set_v @ X @ ( sup_sup_set_set_v @ X @ Y ) )
= X ) ).
% inf_sup_absorb
thf(fact_912_inf__sup__absorb,axiom,
! [X: product_unit,Y: product_unit] :
( ( inf_inf_Product_unit @ X @ ( sup_sup_Product_unit @ X @ Y ) )
= X ) ).
% inf_sup_absorb
thf(fact_913_Int__subset__iff,axiom,
! [C2: set_Product_prod_v_v,A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ C2 @ ( inf_in6271465464967711157od_v_v @ A @ B ) )
= ( ( ord_le7336532860387713383od_v_v @ C2 @ A )
& ( ord_le7336532860387713383od_v_v @ C2 @ B ) ) ) ).
% Int_subset_iff
thf(fact_914_Int__subset__iff,axiom,
! [C2: set_v,A: set_v,B: set_v] :
( ( ord_less_eq_set_v @ C2 @ ( inf_inf_set_v @ A @ B ) )
= ( ( ord_less_eq_set_v @ C2 @ A )
& ( ord_less_eq_set_v @ C2 @ B ) ) ) ).
% Int_subset_iff
thf(fact_915_Int__insert__left__if0,axiom,
! [A4: set_v,C2: set_set_v,B: set_set_v] :
( ~ ( member_set_v @ A4 @ C2 )
=> ( ( inf_inf_set_set_v @ ( insert_set_v @ A4 @ B ) @ C2 )
= ( inf_inf_set_set_v @ B @ C2 ) ) ) ).
% Int_insert_left_if0
thf(fact_916_Int__insert__left__if0,axiom,
! [A4: product_prod_v_v,C2: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ~ ( member7453568604450474000od_v_v @ A4 @ C2 )
=> ( ( inf_in6271465464967711157od_v_v @ ( insert1338601472111419319od_v_v @ A4 @ B ) @ C2 )
= ( inf_in6271465464967711157od_v_v @ B @ C2 ) ) ) ).
% Int_insert_left_if0
thf(fact_917_Int__insert__left__if0,axiom,
! [A4: v,C2: set_v,B: set_v] :
( ~ ( member_v @ A4 @ C2 )
=> ( ( inf_inf_set_v @ ( insert_v @ A4 @ B ) @ C2 )
= ( inf_inf_set_v @ B @ C2 ) ) ) ).
% Int_insert_left_if0
thf(fact_918_Int__insert__left__if1,axiom,
! [A4: set_v,C2: set_set_v,B: set_set_v] :
( ( member_set_v @ A4 @ C2 )
=> ( ( inf_inf_set_set_v @ ( insert_set_v @ A4 @ B ) @ C2 )
= ( insert_set_v @ A4 @ ( inf_inf_set_set_v @ B @ C2 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_919_Int__insert__left__if1,axiom,
! [A4: product_prod_v_v,C2: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ A4 @ C2 )
=> ( ( inf_in6271465464967711157od_v_v @ ( insert1338601472111419319od_v_v @ A4 @ B ) @ C2 )
= ( insert1338601472111419319od_v_v @ A4 @ ( inf_in6271465464967711157od_v_v @ B @ C2 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_920_Int__insert__left__if1,axiom,
! [A4: v,C2: set_v,B: set_v] :
( ( member_v @ A4 @ C2 )
=> ( ( inf_inf_set_v @ ( insert_v @ A4 @ B ) @ C2 )
= ( insert_v @ A4 @ ( inf_inf_set_v @ B @ C2 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_921_insert__inter__insert,axiom,
! [A4: product_prod_v_v,A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ ( insert1338601472111419319od_v_v @ A4 @ A ) @ ( insert1338601472111419319od_v_v @ A4 @ B ) )
= ( insert1338601472111419319od_v_v @ A4 @ ( inf_in6271465464967711157od_v_v @ A @ B ) ) ) ).
% insert_inter_insert
thf(fact_922_insert__inter__insert,axiom,
! [A4: set_v,A: set_set_v,B: set_set_v] :
( ( inf_inf_set_set_v @ ( insert_set_v @ A4 @ A ) @ ( insert_set_v @ A4 @ B ) )
= ( insert_set_v @ A4 @ ( inf_inf_set_set_v @ A @ B ) ) ) ).
% insert_inter_insert
thf(fact_923_insert__inter__insert,axiom,
! [A4: v,A: set_v,B: set_v] :
( ( inf_inf_set_v @ ( insert_v @ A4 @ A ) @ ( insert_v @ A4 @ B ) )
= ( insert_v @ A4 @ ( inf_inf_set_v @ A @ B ) ) ) ).
% insert_inter_insert
thf(fact_924_Int__insert__right__if0,axiom,
! [A4: set_v,A: set_set_v,B: set_set_v] :
( ~ ( member_set_v @ A4 @ A )
=> ( ( inf_inf_set_set_v @ A @ ( insert_set_v @ A4 @ B ) )
= ( inf_inf_set_set_v @ A @ B ) ) ) ).
% Int_insert_right_if0
thf(fact_925_Int__insert__right__if0,axiom,
! [A4: product_prod_v_v,A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ~ ( member7453568604450474000od_v_v @ A4 @ A )
=> ( ( inf_in6271465464967711157od_v_v @ A @ ( insert1338601472111419319od_v_v @ A4 @ B ) )
= ( inf_in6271465464967711157od_v_v @ A @ B ) ) ) ).
% Int_insert_right_if0
thf(fact_926_Int__insert__right__if0,axiom,
! [A4: v,A: set_v,B: set_v] :
( ~ ( member_v @ A4 @ A )
=> ( ( inf_inf_set_v @ A @ ( insert_v @ A4 @ B ) )
= ( inf_inf_set_v @ A @ B ) ) ) ).
% Int_insert_right_if0
thf(fact_927_Int__insert__right__if1,axiom,
! [A4: set_v,A: set_set_v,B: set_set_v] :
( ( member_set_v @ A4 @ A )
=> ( ( inf_inf_set_set_v @ A @ ( insert_set_v @ A4 @ B ) )
= ( insert_set_v @ A4 @ ( inf_inf_set_set_v @ A @ B ) ) ) ) ).
% Int_insert_right_if1
thf(fact_928_Int__insert__right__if1,axiom,
! [A4: product_prod_v_v,A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ A4 @ A )
=> ( ( inf_in6271465464967711157od_v_v @ A @ ( insert1338601472111419319od_v_v @ A4 @ B ) )
= ( insert1338601472111419319od_v_v @ A4 @ ( inf_in6271465464967711157od_v_v @ A @ B ) ) ) ) ).
% Int_insert_right_if1
thf(fact_929_Int__insert__right__if1,axiom,
! [A4: v,A: set_v,B: set_v] :
( ( member_v @ A4 @ A )
=> ( ( inf_inf_set_v @ A @ ( insert_v @ A4 @ B ) )
= ( insert_v @ A4 @ ( inf_inf_set_v @ A @ B ) ) ) ) ).
% Int_insert_right_if1
thf(fact_930_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_931_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_932_Int__Un__eq_I4_J,axiom,
! [T2: set_set_v,S: set_set_v] :
( ( sup_sup_set_set_v @ T2 @ ( inf_inf_set_set_v @ S @ T2 ) )
= T2 ) ).
% Int_Un_eq(4)
thf(fact_933_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_934_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_935_Int__Un__eq_I3_J,axiom,
! [S: set_set_v,T2: set_set_v] :
( ( sup_sup_set_set_v @ S @ ( inf_inf_set_set_v @ S @ T2 ) )
= S ) ).
% Int_Un_eq(3)
thf(fact_936_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_937_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_938_Int__Un__eq_I2_J,axiom,
! [S: set_set_v,T2: set_set_v] :
( ( sup_sup_set_set_v @ ( inf_inf_set_set_v @ S @ T2 ) @ T2 )
= T2 ) ).
% Int_Un_eq(2)
thf(fact_939_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_940_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_941_Int__Un__eq_I1_J,axiom,
! [S: set_set_v,T2: set_set_v] :
( ( sup_sup_set_set_v @ ( inf_inf_set_set_v @ S @ T2 ) @ S )
= S ) ).
% Int_Un_eq(1)
thf(fact_942_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_943_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_944_Un__Int__eq_I4_J,axiom,
! [T2: set_set_v,S: set_set_v] :
( ( inf_inf_set_set_v @ T2 @ ( sup_sup_set_set_v @ S @ T2 ) )
= T2 ) ).
% Un_Int_eq(4)
thf(fact_945_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_946_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_947_Un__Int__eq_I3_J,axiom,
! [S: set_set_v,T2: set_set_v] :
( ( inf_inf_set_set_v @ S @ ( sup_sup_set_set_v @ S @ T2 ) )
= S ) ).
% Un_Int_eq(3)
thf(fact_948_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_949_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_950_Un__Int__eq_I2_J,axiom,
! [S: set_set_v,T2: set_set_v] :
( ( inf_inf_set_set_v @ ( sup_sup_set_set_v @ S @ T2 ) @ T2 )
= T2 ) ).
% Un_Int_eq(2)
thf(fact_951_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_952_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_953_Un__Int__eq_I1_J,axiom,
! [S: set_set_v,T2: set_set_v] :
( ( inf_inf_set_set_v @ ( sup_sup_set_set_v @ S @ T2 ) @ S )
= S ) ).
% Un_Int_eq(1)
thf(fact_954_unite__wf__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_Bl9196236973127232072t_unit @ successors @ ( sCC_Bloemen_unite_v @ V3 @ W @ E ) ) ) ) ) ) ) ).
% unite_wf_env
thf(fact_955_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_956_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_957_set__empty2,axiom,
! [Xs: list_set_v] :
( ( bot_bot_set_set_v
= ( set_set_v2 @ Xs ) )
= ( Xs = nil_set_v ) ) ).
% set_empty2
thf(fact_958_set__empty2,axiom,
! [Xs: list_v] :
( ( bot_bot_set_v
= ( set_v2 @ Xs ) )
= ( Xs = nil_v ) ) ).
% set_empty2
thf(fact_959_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_960_set__empty,axiom,
! [Xs: list_set_v] :
( ( ( set_set_v2 @ Xs )
= bot_bot_set_set_v )
= ( Xs = nil_set_v ) ) ).
% set_empty
thf(fact_961_set__empty,axiom,
! [Xs: list_v] :
( ( ( set_v2 @ Xs )
= bot_bot_set_v )
= ( Xs = nil_v ) ) ).
% set_empty
thf(fact_962_disjoint__insert_I2_J,axiom,
! [A: set_Product_prod_v_v,B2: product_prod_v_v,B: set_Product_prod_v_v] :
( ( bot_bo723834152578015283od_v_v
= ( inf_in6271465464967711157od_v_v @ A @ ( insert1338601472111419319od_v_v @ B2 @ B ) ) )
= ( ~ ( member7453568604450474000od_v_v @ B2 @ A )
& ( bot_bo723834152578015283od_v_v
= ( inf_in6271465464967711157od_v_v @ A @ B ) ) ) ) ).
% disjoint_insert(2)
thf(fact_963_disjoint__insert_I2_J,axiom,
! [A: set_set_v,B2: set_v,B: set_set_v] :
( ( bot_bot_set_set_v
= ( inf_inf_set_set_v @ A @ ( insert_set_v @ B2 @ B ) ) )
= ( ~ ( member_set_v @ B2 @ A )
& ( bot_bot_set_set_v
= ( inf_inf_set_set_v @ A @ B ) ) ) ) ).
% disjoint_insert(2)
thf(fact_964_disjoint__insert_I2_J,axiom,
! [A: set_v,B2: v,B: set_v] :
( ( bot_bot_set_v
= ( inf_inf_set_v @ A @ ( insert_v @ B2 @ B ) ) )
= ( ~ ( member_v @ B2 @ A )
& ( bot_bot_set_v
= ( inf_inf_set_v @ A @ B ) ) ) ) ).
% disjoint_insert(2)
thf(fact_965_disjoint__insert_I1_J,axiom,
! [B: set_Product_prod_v_v,A4: product_prod_v_v,A: set_Product_prod_v_v] :
( ( ( inf_in6271465464967711157od_v_v @ B @ ( insert1338601472111419319od_v_v @ A4 @ A ) )
= bot_bo723834152578015283od_v_v )
= ( ~ ( member7453568604450474000od_v_v @ A4 @ B )
& ( ( inf_in6271465464967711157od_v_v @ B @ A )
= bot_bo723834152578015283od_v_v ) ) ) ).
% disjoint_insert(1)
thf(fact_966_disjoint__insert_I1_J,axiom,
! [B: set_set_v,A4: set_v,A: set_set_v] :
( ( ( inf_inf_set_set_v @ B @ ( insert_set_v @ A4 @ A ) )
= bot_bot_set_set_v )
= ( ~ ( member_set_v @ A4 @ B )
& ( ( inf_inf_set_set_v @ B @ A )
= bot_bot_set_set_v ) ) ) ).
% disjoint_insert(1)
thf(fact_967_disjoint__insert_I1_J,axiom,
! [B: set_v,A4: v,A: set_v] :
( ( ( inf_inf_set_v @ B @ ( insert_v @ A4 @ A ) )
= bot_bot_set_v )
= ( ~ ( member_v @ A4 @ B )
& ( ( inf_inf_set_v @ B @ A )
= bot_bot_set_v ) ) ) ).
% disjoint_insert(1)
thf(fact_968_insert__disjoint_I2_J,axiom,
! [A4: product_prod_v_v,A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( bot_bo723834152578015283od_v_v
= ( inf_in6271465464967711157od_v_v @ ( insert1338601472111419319od_v_v @ A4 @ A ) @ B ) )
= ( ~ ( member7453568604450474000od_v_v @ A4 @ B )
& ( bot_bo723834152578015283od_v_v
= ( inf_in6271465464967711157od_v_v @ A @ B ) ) ) ) ).
% insert_disjoint(2)
thf(fact_969_insert__disjoint_I2_J,axiom,
! [A4: set_v,A: set_set_v,B: set_set_v] :
( ( bot_bot_set_set_v
= ( inf_inf_set_set_v @ ( insert_set_v @ A4 @ A ) @ B ) )
= ( ~ ( member_set_v @ A4 @ B )
& ( bot_bot_set_set_v
= ( inf_inf_set_set_v @ A @ B ) ) ) ) ).
% insert_disjoint(2)
thf(fact_970_insert__disjoint_I2_J,axiom,
! [A4: v,A: set_v,B: set_v] :
( ( bot_bot_set_v
= ( inf_inf_set_v @ ( insert_v @ A4 @ A ) @ B ) )
= ( ~ ( member_v @ A4 @ B )
& ( bot_bot_set_v
= ( inf_inf_set_v @ A @ B ) ) ) ) ).
% insert_disjoint(2)
thf(fact_971_insert__disjoint_I1_J,axiom,
! [A4: product_prod_v_v,A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ( inf_in6271465464967711157od_v_v @ ( insert1338601472111419319od_v_v @ A4 @ A ) @ B )
= bot_bo723834152578015283od_v_v )
= ( ~ ( member7453568604450474000od_v_v @ A4 @ B )
& ( ( inf_in6271465464967711157od_v_v @ A @ B )
= bot_bo723834152578015283od_v_v ) ) ) ).
% insert_disjoint(1)
thf(fact_972_insert__disjoint_I1_J,axiom,
! [A4: set_v,A: set_set_v,B: set_set_v] :
( ( ( inf_inf_set_set_v @ ( insert_set_v @ A4 @ A ) @ B )
= bot_bot_set_set_v )
= ( ~ ( member_set_v @ A4 @ B )
& ( ( inf_inf_set_set_v @ A @ B )
= bot_bot_set_set_v ) ) ) ).
% insert_disjoint(1)
thf(fact_973_insert__disjoint_I1_J,axiom,
! [A4: v,A: set_v,B: set_v] :
( ( ( inf_inf_set_v @ ( insert_v @ A4 @ A ) @ B )
= bot_bot_set_v )
= ( ~ ( member_v @ A4 @ B )
& ( ( inf_inf_set_v @ A @ B )
= bot_bot_set_v ) ) ) ).
% insert_disjoint(1)
thf(fact_974_Diff__disjoint,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ A @ ( minus_4183494784930505774od_v_v @ B @ A ) )
= bot_bo723834152578015283od_v_v ) ).
% Diff_disjoint
thf(fact_975_Diff__disjoint,axiom,
! [A: set_set_v,B: set_set_v] :
( ( inf_inf_set_set_v @ A @ ( minus_7228012346218142266_set_v @ B @ A ) )
= bot_bot_set_set_v ) ).
% Diff_disjoint
thf(fact_976_Diff__disjoint,axiom,
! [A: set_v,B: set_v] :
( ( inf_inf_set_v @ A @ ( minus_minus_set_v @ B @ A ) )
= bot_bot_set_v ) ).
% Diff_disjoint
thf(fact_977_pre__dfss__unite__pre__dfss,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_Bl1748261141445803503t_unit @ successors @ V3
@ ( sCC_Bl48393358579903213t_unit
@ ^ [Uu: v > set_v,X2: v] : ( if_set_v @ ( X2 = V3 ) @ ( sup_sup_set_v @ ( sCC_Bl3795065053823578884t_unit @ ( sCC_Bloemen_unite_v @ V3 @ W @ E ) @ V3 ) @ ( insert_v @ W @ bot_bot_set_v ) ) @ ( sCC_Bl3795065053823578884t_unit @ ( sCC_Bloemen_unite_v @ V3 @ W @ E ) @ X2 ) )
@ ( sCC_Bloemen_unite_v @ V3 @ W @ E ) ) ) ) ) ) ) ) ).
% pre_dfss_unite_pre_dfss
thf(fact_978_list_Ocollapse,axiom,
! [List: list_v] :
( ( List != nil_v )
=> ( ( cons_v @ ( hd_v @ List ) @ ( tl_v @ List ) )
= List ) ) ).
% list.collapse
thf(fact_979_hd__Cons__tl,axiom,
! [Xs: list_v] :
( ( Xs != nil_v )
=> ( ( cons_v @ ( hd_v @ Xs ) @ ( tl_v @ Xs ) )
= Xs ) ) ).
% hd_Cons_tl
thf(fact_980_inf_OcoboundedI2,axiom,
! [B2: product_unit,C: product_unit,A4: product_unit] :
( ( ord_le3221252021190050221t_unit @ B2 @ C )
=> ( ord_le3221252021190050221t_unit @ ( inf_inf_Product_unit @ A4 @ B2 ) @ C ) ) ).
% inf.coboundedI2
thf(fact_981_inf_OcoboundedI2,axiom,
! [B2: set_Product_prod_v_v,C: set_Product_prod_v_v,A4: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ B2 @ C )
=> ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ A4 @ B2 ) @ C ) ) ).
% inf.coboundedI2
thf(fact_982_inf_OcoboundedI2,axiom,
! [B2: set_v,C: set_v,A4: set_v] :
( ( ord_less_eq_set_v @ B2 @ C )
=> ( ord_less_eq_set_v @ ( inf_inf_set_v @ A4 @ B2 ) @ C ) ) ).
% inf.coboundedI2
thf(fact_983_inf_OcoboundedI1,axiom,
! [A4: product_unit,C: product_unit,B2: product_unit] :
( ( ord_le3221252021190050221t_unit @ A4 @ C )
=> ( ord_le3221252021190050221t_unit @ ( inf_inf_Product_unit @ A4 @ B2 ) @ C ) ) ).
% inf.coboundedI1
thf(fact_984_inf_OcoboundedI1,axiom,
! [A4: set_Product_prod_v_v,C: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A4 @ C )
=> ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ A4 @ B2 ) @ C ) ) ).
% inf.coboundedI1
thf(fact_985_inf_OcoboundedI1,axiom,
! [A4: set_v,C: set_v,B2: set_v] :
( ( ord_less_eq_set_v @ A4 @ C )
=> ( ord_less_eq_set_v @ ( inf_inf_set_v @ A4 @ B2 ) @ C ) ) ).
% inf.coboundedI1
thf(fact_986_inf_Oabsorb__iff2,axiom,
( ord_le3221252021190050221t_unit
= ( ^ [B4: product_unit,A6: product_unit] :
( ( inf_inf_Product_unit @ A6 @ B4 )
= B4 ) ) ) ).
% inf.absorb_iff2
thf(fact_987_inf_Oabsorb__iff2,axiom,
( ord_le7336532860387713383od_v_v
= ( ^ [B4: set_Product_prod_v_v,A6: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ A6 @ B4 )
= B4 ) ) ) ).
% inf.absorb_iff2
thf(fact_988_inf_Oabsorb__iff2,axiom,
( ord_less_eq_set_v
= ( ^ [B4: set_v,A6: set_v] :
( ( inf_inf_set_v @ A6 @ B4 )
= B4 ) ) ) ).
% inf.absorb_iff2
thf(fact_989_inf_Oabsorb__iff1,axiom,
( ord_le3221252021190050221t_unit
= ( ^ [A6: product_unit,B4: product_unit] :
( ( inf_inf_Product_unit @ A6 @ B4 )
= A6 ) ) ) ).
% inf.absorb_iff1
thf(fact_990_inf_Oabsorb__iff1,axiom,
( ord_le7336532860387713383od_v_v
= ( ^ [A6: set_Product_prod_v_v,B4: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ A6 @ B4 )
= A6 ) ) ) ).
% inf.absorb_iff1
thf(fact_991_inf_Oabsorb__iff1,axiom,
( ord_less_eq_set_v
= ( ^ [A6: set_v,B4: set_v] :
( ( inf_inf_set_v @ A6 @ B4 )
= A6 ) ) ) ).
% inf.absorb_iff1
thf(fact_992_inf_Ocobounded2,axiom,
! [A4: product_unit,B2: product_unit] : ( ord_le3221252021190050221t_unit @ ( inf_inf_Product_unit @ A4 @ B2 ) @ B2 ) ).
% inf.cobounded2
thf(fact_993_inf_Ocobounded2,axiom,
! [A4: set_Product_prod_v_v,B2: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ A4 @ B2 ) @ B2 ) ).
% inf.cobounded2
thf(fact_994_inf_Ocobounded2,axiom,
! [A4: set_v,B2: set_v] : ( ord_less_eq_set_v @ ( inf_inf_set_v @ A4 @ B2 ) @ B2 ) ).
% inf.cobounded2
thf(fact_995_inf_Ocobounded1,axiom,
! [A4: product_unit,B2: product_unit] : ( ord_le3221252021190050221t_unit @ ( inf_inf_Product_unit @ A4 @ B2 ) @ A4 ) ).
% inf.cobounded1
thf(fact_996_inf_Ocobounded1,axiom,
! [A4: set_Product_prod_v_v,B2: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ A4 @ B2 ) @ A4 ) ).
% inf.cobounded1
thf(fact_997_inf_Ocobounded1,axiom,
! [A4: set_v,B2: set_v] : ( ord_less_eq_set_v @ ( inf_inf_set_v @ A4 @ B2 ) @ A4 ) ).
% inf.cobounded1
thf(fact_998_inf_Oorder__iff,axiom,
( ord_le3221252021190050221t_unit
= ( ^ [A6: product_unit,B4: product_unit] :
( A6
= ( inf_inf_Product_unit @ A6 @ B4 ) ) ) ) ).
% inf.order_iff
thf(fact_999_inf_Oorder__iff,axiom,
( ord_le7336532860387713383od_v_v
= ( ^ [A6: set_Product_prod_v_v,B4: set_Product_prod_v_v] :
( A6
= ( inf_in6271465464967711157od_v_v @ A6 @ B4 ) ) ) ) ).
% inf.order_iff
thf(fact_1000_inf_Oorder__iff,axiom,
( ord_less_eq_set_v
= ( ^ [A6: set_v,B4: set_v] :
( A6
= ( inf_inf_set_v @ A6 @ B4 ) ) ) ) ).
% inf.order_iff
thf(fact_1001_inf__greatest,axiom,
! [X: product_unit,Y: product_unit,Z: product_unit] :
( ( ord_le3221252021190050221t_unit @ X @ Y )
=> ( ( ord_le3221252021190050221t_unit @ X @ Z )
=> ( ord_le3221252021190050221t_unit @ X @ ( inf_inf_Product_unit @ Y @ Z ) ) ) ) ).
% inf_greatest
thf(fact_1002_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_1003_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_1004_inf_OboundedI,axiom,
! [A4: product_unit,B2: product_unit,C: product_unit] :
( ( ord_le3221252021190050221t_unit @ A4 @ B2 )
=> ( ( ord_le3221252021190050221t_unit @ A4 @ C )
=> ( ord_le3221252021190050221t_unit @ A4 @ ( inf_inf_Product_unit @ B2 @ C ) ) ) ) ).
% inf.boundedI
thf(fact_1005_inf_OboundedI,axiom,
! [A4: set_Product_prod_v_v,B2: set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A4 @ B2 )
=> ( ( ord_le7336532860387713383od_v_v @ A4 @ C )
=> ( ord_le7336532860387713383od_v_v @ A4 @ ( inf_in6271465464967711157od_v_v @ B2 @ C ) ) ) ) ).
% inf.boundedI
thf(fact_1006_inf_OboundedI,axiom,
! [A4: set_v,B2: set_v,C: set_v] :
( ( ord_less_eq_set_v @ A4 @ B2 )
=> ( ( ord_less_eq_set_v @ A4 @ C )
=> ( ord_less_eq_set_v @ A4 @ ( inf_inf_set_v @ B2 @ C ) ) ) ) ).
% inf.boundedI
thf(fact_1007_inf_OboundedE,axiom,
! [A4: product_unit,B2: product_unit,C: product_unit] :
( ( ord_le3221252021190050221t_unit @ A4 @ ( inf_inf_Product_unit @ B2 @ C ) )
=> ~ ( ( ord_le3221252021190050221t_unit @ A4 @ B2 )
=> ~ ( ord_le3221252021190050221t_unit @ A4 @ C ) ) ) ).
% inf.boundedE
thf(fact_1008_inf_OboundedE,axiom,
! [A4: set_Product_prod_v_v,B2: set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A4 @ ( inf_in6271465464967711157od_v_v @ B2 @ C ) )
=> ~ ( ( ord_le7336532860387713383od_v_v @ A4 @ B2 )
=> ~ ( ord_le7336532860387713383od_v_v @ A4 @ C ) ) ) ).
% inf.boundedE
thf(fact_1009_inf_OboundedE,axiom,
! [A4: set_v,B2: set_v,C: set_v] :
( ( ord_less_eq_set_v @ A4 @ ( inf_inf_set_v @ B2 @ C ) )
=> ~ ( ( ord_less_eq_set_v @ A4 @ B2 )
=> ~ ( ord_less_eq_set_v @ A4 @ C ) ) ) ).
% inf.boundedE
thf(fact_1010_inf__absorb2,axiom,
! [Y: product_unit,X: product_unit] :
( ( ord_le3221252021190050221t_unit @ Y @ X )
=> ( ( inf_inf_Product_unit @ X @ Y )
= Y ) ) ).
% inf_absorb2
thf(fact_1011_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_1012_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_1013_inf__absorb1,axiom,
! [X: product_unit,Y: product_unit] :
( ( ord_le3221252021190050221t_unit @ X @ Y )
=> ( ( inf_inf_Product_unit @ X @ Y )
= X ) ) ).
% inf_absorb1
thf(fact_1014_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_1015_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_1016_inf_Oabsorb2,axiom,
! [B2: product_unit,A4: product_unit] :
( ( ord_le3221252021190050221t_unit @ B2 @ A4 )
=> ( ( inf_inf_Product_unit @ A4 @ B2 )
= B2 ) ) ).
% inf.absorb2
thf(fact_1017_inf_Oabsorb2,axiom,
! [B2: set_Product_prod_v_v,A4: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ B2 @ A4 )
=> ( ( inf_in6271465464967711157od_v_v @ A4 @ B2 )
= B2 ) ) ).
% inf.absorb2
thf(fact_1018_inf_Oabsorb2,axiom,
! [B2: set_v,A4: set_v] :
( ( ord_less_eq_set_v @ B2 @ A4 )
=> ( ( inf_inf_set_v @ A4 @ B2 )
= B2 ) ) ).
% inf.absorb2
thf(fact_1019_inf_Oabsorb1,axiom,
! [A4: product_unit,B2: product_unit] :
( ( ord_le3221252021190050221t_unit @ A4 @ B2 )
=> ( ( inf_inf_Product_unit @ A4 @ B2 )
= A4 ) ) ).
% inf.absorb1
thf(fact_1020_inf_Oabsorb1,axiom,
! [A4: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A4 @ B2 )
=> ( ( inf_in6271465464967711157od_v_v @ A4 @ B2 )
= A4 ) ) ).
% inf.absorb1
thf(fact_1021_inf_Oabsorb1,axiom,
! [A4: set_v,B2: set_v] :
( ( ord_less_eq_set_v @ A4 @ B2 )
=> ( ( inf_inf_set_v @ A4 @ B2 )
= A4 ) ) ).
% inf.absorb1
thf(fact_1022_le__iff__inf,axiom,
( ord_le3221252021190050221t_unit
= ( ^ [X2: product_unit,Y3: product_unit] :
( ( inf_inf_Product_unit @ X2 @ Y3 )
= X2 ) ) ) ).
% le_iff_inf
thf(fact_1023_le__iff__inf,axiom,
( ord_le7336532860387713383od_v_v
= ( ^ [X2: set_Product_prod_v_v,Y3: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ X2 @ Y3 )
= X2 ) ) ) ).
% le_iff_inf
thf(fact_1024_le__iff__inf,axiom,
( ord_less_eq_set_v
= ( ^ [X2: set_v,Y3: set_v] :
( ( inf_inf_set_v @ X2 @ Y3 )
= X2 ) ) ) ).
% le_iff_inf
thf(fact_1025_inf__unique,axiom,
! [F: product_unit > product_unit > product_unit,X: product_unit,Y: product_unit] :
( ! [X3: product_unit,Y2: product_unit] : ( ord_le3221252021190050221t_unit @ ( F @ X3 @ Y2 ) @ X3 )
=> ( ! [X3: product_unit,Y2: product_unit] : ( ord_le3221252021190050221t_unit @ ( F @ X3 @ Y2 ) @ Y2 )
=> ( ! [X3: product_unit,Y2: product_unit,Z3: product_unit] :
( ( ord_le3221252021190050221t_unit @ X3 @ Y2 )
=> ( ( ord_le3221252021190050221t_unit @ X3 @ Z3 )
=> ( ord_le3221252021190050221t_unit @ X3 @ ( F @ Y2 @ Z3 ) ) ) )
=> ( ( inf_inf_Product_unit @ X @ Y )
= ( F @ X @ Y ) ) ) ) ) ).
% inf_unique
thf(fact_1026_inf__unique,axiom,
! [F: set_Product_prod_v_v > set_Product_prod_v_v > set_Product_prod_v_v,X: set_Product_prod_v_v,Y: set_Product_prod_v_v] :
( ! [X3: set_Product_prod_v_v,Y2: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ ( F @ X3 @ Y2 ) @ X3 )
=> ( ! [X3: set_Product_prod_v_v,Y2: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ ( F @ X3 @ Y2 ) @ Y2 )
=> ( ! [X3: set_Product_prod_v_v,Y2: set_Product_prod_v_v,Z3: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X3 @ Y2 )
=> ( ( ord_le7336532860387713383od_v_v @ X3 @ Z3 )
=> ( ord_le7336532860387713383od_v_v @ X3 @ ( F @ Y2 @ Z3 ) ) ) )
=> ( ( inf_in6271465464967711157od_v_v @ X @ Y )
= ( F @ X @ Y ) ) ) ) ) ).
% inf_unique
thf(fact_1027_inf__unique,axiom,
! [F: set_v > set_v > set_v,X: set_v,Y: set_v] :
( ! [X3: set_v,Y2: set_v] : ( ord_less_eq_set_v @ ( F @ X3 @ Y2 ) @ X3 )
=> ( ! [X3: set_v,Y2: set_v] : ( ord_less_eq_set_v @ ( F @ X3 @ Y2 ) @ Y2 )
=> ( ! [X3: set_v,Y2: set_v,Z3: set_v] :
( ( ord_less_eq_set_v @ X3 @ Y2 )
=> ( ( ord_less_eq_set_v @ X3 @ Z3 )
=> ( ord_less_eq_set_v @ X3 @ ( F @ Y2 @ Z3 ) ) ) )
=> ( ( inf_inf_set_v @ X @ Y )
= ( F @ X @ Y ) ) ) ) ) ).
% inf_unique
thf(fact_1028_inf_OorderI,axiom,
! [A4: product_unit,B2: product_unit] :
( ( A4
= ( inf_inf_Product_unit @ A4 @ B2 ) )
=> ( ord_le3221252021190050221t_unit @ A4 @ B2 ) ) ).
% inf.orderI
thf(fact_1029_inf_OorderI,axiom,
! [A4: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( A4
= ( inf_in6271465464967711157od_v_v @ A4 @ B2 ) )
=> ( ord_le7336532860387713383od_v_v @ A4 @ B2 ) ) ).
% inf.orderI
thf(fact_1030_inf_OorderI,axiom,
! [A4: set_v,B2: set_v] :
( ( A4
= ( inf_inf_set_v @ A4 @ B2 ) )
=> ( ord_less_eq_set_v @ A4 @ B2 ) ) ).
% inf.orderI
thf(fact_1031_inf_OorderE,axiom,
! [A4: product_unit,B2: product_unit] :
( ( ord_le3221252021190050221t_unit @ A4 @ B2 )
=> ( A4
= ( inf_inf_Product_unit @ A4 @ B2 ) ) ) ).
% inf.orderE
thf(fact_1032_inf_OorderE,axiom,
! [A4: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A4 @ B2 )
=> ( A4
= ( inf_in6271465464967711157od_v_v @ A4 @ B2 ) ) ) ).
% inf.orderE
thf(fact_1033_inf_OorderE,axiom,
! [A4: set_v,B2: set_v] :
( ( ord_less_eq_set_v @ A4 @ B2 )
=> ( A4
= ( inf_inf_set_v @ A4 @ B2 ) ) ) ).
% inf.orderE
thf(fact_1034_le__infI2,axiom,
! [B2: product_unit,X: product_unit,A4: product_unit] :
( ( ord_le3221252021190050221t_unit @ B2 @ X )
=> ( ord_le3221252021190050221t_unit @ ( inf_inf_Product_unit @ A4 @ B2 ) @ X ) ) ).
% le_infI2
thf(fact_1035_le__infI2,axiom,
! [B2: set_Product_prod_v_v,X: set_Product_prod_v_v,A4: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ B2 @ X )
=> ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ A4 @ B2 ) @ X ) ) ).
% le_infI2
thf(fact_1036_le__infI2,axiom,
! [B2: set_v,X: set_v,A4: set_v] :
( ( ord_less_eq_set_v @ B2 @ X )
=> ( ord_less_eq_set_v @ ( inf_inf_set_v @ A4 @ B2 ) @ X ) ) ).
% le_infI2
thf(fact_1037_le__infI1,axiom,
! [A4: product_unit,X: product_unit,B2: product_unit] :
( ( ord_le3221252021190050221t_unit @ A4 @ X )
=> ( ord_le3221252021190050221t_unit @ ( inf_inf_Product_unit @ A4 @ B2 ) @ X ) ) ).
% le_infI1
thf(fact_1038_le__infI1,axiom,
! [A4: set_Product_prod_v_v,X: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A4 @ X )
=> ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ A4 @ B2 ) @ X ) ) ).
% le_infI1
thf(fact_1039_le__infI1,axiom,
! [A4: set_v,X: set_v,B2: set_v] :
( ( ord_less_eq_set_v @ A4 @ X )
=> ( ord_less_eq_set_v @ ( inf_inf_set_v @ A4 @ B2 ) @ X ) ) ).
% le_infI1
thf(fact_1040_inf__mono,axiom,
! [A4: product_unit,C: product_unit,B2: product_unit,D: product_unit] :
( ( ord_le3221252021190050221t_unit @ A4 @ C )
=> ( ( ord_le3221252021190050221t_unit @ B2 @ D )
=> ( ord_le3221252021190050221t_unit @ ( inf_inf_Product_unit @ A4 @ B2 ) @ ( inf_inf_Product_unit @ C @ D ) ) ) ) ).
% inf_mono
thf(fact_1041_inf__mono,axiom,
! [A4: set_Product_prod_v_v,C: set_Product_prod_v_v,B2: set_Product_prod_v_v,D: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A4 @ C )
=> ( ( ord_le7336532860387713383od_v_v @ B2 @ D )
=> ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ A4 @ B2 ) @ ( inf_in6271465464967711157od_v_v @ C @ D ) ) ) ) ).
% inf_mono
thf(fact_1042_inf__mono,axiom,
! [A4: set_v,C: set_v,B2: set_v,D: set_v] :
( ( ord_less_eq_set_v @ A4 @ C )
=> ( ( ord_less_eq_set_v @ B2 @ D )
=> ( ord_less_eq_set_v @ ( inf_inf_set_v @ A4 @ B2 ) @ ( inf_inf_set_v @ C @ D ) ) ) ) ).
% inf_mono
thf(fact_1043_le__infI,axiom,
! [X: product_unit,A4: product_unit,B2: product_unit] :
( ( ord_le3221252021190050221t_unit @ X @ A4 )
=> ( ( ord_le3221252021190050221t_unit @ X @ B2 )
=> ( ord_le3221252021190050221t_unit @ X @ ( inf_inf_Product_unit @ A4 @ B2 ) ) ) ) ).
% le_infI
thf(fact_1044_le__infI,axiom,
! [X: set_Product_prod_v_v,A4: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X @ A4 )
=> ( ( ord_le7336532860387713383od_v_v @ X @ B2 )
=> ( ord_le7336532860387713383od_v_v @ X @ ( inf_in6271465464967711157od_v_v @ A4 @ B2 ) ) ) ) ).
% le_infI
thf(fact_1045_le__infI,axiom,
! [X: set_v,A4: set_v,B2: set_v] :
( ( ord_less_eq_set_v @ X @ A4 )
=> ( ( ord_less_eq_set_v @ X @ B2 )
=> ( ord_less_eq_set_v @ X @ ( inf_inf_set_v @ A4 @ B2 ) ) ) ) ).
% le_infI
thf(fact_1046_le__infE,axiom,
! [X: product_unit,A4: product_unit,B2: product_unit] :
( ( ord_le3221252021190050221t_unit @ X @ ( inf_inf_Product_unit @ A4 @ B2 ) )
=> ~ ( ( ord_le3221252021190050221t_unit @ X @ A4 )
=> ~ ( ord_le3221252021190050221t_unit @ X @ B2 ) ) ) ).
% le_infE
thf(fact_1047_le__infE,axiom,
! [X: set_Product_prod_v_v,A4: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X @ ( inf_in6271465464967711157od_v_v @ A4 @ B2 ) )
=> ~ ( ( ord_le7336532860387713383od_v_v @ X @ A4 )
=> ~ ( ord_le7336532860387713383od_v_v @ X @ B2 ) ) ) ).
% le_infE
thf(fact_1048_le__infE,axiom,
! [X: set_v,A4: set_v,B2: set_v] :
( ( ord_less_eq_set_v @ X @ ( inf_inf_set_v @ A4 @ B2 ) )
=> ~ ( ( ord_less_eq_set_v @ X @ A4 )
=> ~ ( ord_less_eq_set_v @ X @ B2 ) ) ) ).
% le_infE
thf(fact_1049_inf__le2,axiom,
! [X: product_unit,Y: product_unit] : ( ord_le3221252021190050221t_unit @ ( inf_inf_Product_unit @ X @ Y ) @ Y ) ).
% inf_le2
thf(fact_1050_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_1051_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_1052_inf__le1,axiom,
! [X: product_unit,Y: product_unit] : ( ord_le3221252021190050221t_unit @ ( inf_inf_Product_unit @ X @ Y ) @ X ) ).
% inf_le1
thf(fact_1053_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_1054_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_1055_inf__sup__ord_I1_J,axiom,
! [X: product_unit,Y: product_unit] : ( ord_le3221252021190050221t_unit @ ( inf_inf_Product_unit @ X @ Y ) @ X ) ).
% inf_sup_ord(1)
thf(fact_1056_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_1057_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_1058_inf__sup__ord_I2_J,axiom,
! [X: product_unit,Y: product_unit] : ( ord_le3221252021190050221t_unit @ ( inf_inf_Product_unit @ X @ Y ) @ Y ) ).
% inf_sup_ord(2)
thf(fact_1059_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_1060_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_1061_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_1062_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_1063_sup__inf__distrib2,axiom,
! [Y: set_set_v,Z: set_set_v,X: set_set_v] :
( ( sup_sup_set_set_v @ ( inf_inf_set_set_v @ Y @ Z ) @ X )
= ( inf_inf_set_set_v @ ( sup_sup_set_set_v @ Y @ X ) @ ( sup_sup_set_set_v @ Z @ X ) ) ) ).
% sup_inf_distrib2
thf(fact_1064_sup__inf__distrib2,axiom,
! [Y: product_unit,Z: product_unit,X: product_unit] :
( ( sup_sup_Product_unit @ ( inf_inf_Product_unit @ Y @ Z ) @ X )
= ( inf_inf_Product_unit @ ( sup_sup_Product_unit @ Y @ X ) @ ( sup_sup_Product_unit @ Z @ X ) ) ) ).
% sup_inf_distrib2
thf(fact_1065_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_1066_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_1067_sup__inf__distrib1,axiom,
! [X: set_set_v,Y: set_set_v,Z: set_set_v] :
( ( sup_sup_set_set_v @ X @ ( inf_inf_set_set_v @ Y @ Z ) )
= ( inf_inf_set_set_v @ ( sup_sup_set_set_v @ X @ Y ) @ ( sup_sup_set_set_v @ X @ Z ) ) ) ).
% sup_inf_distrib1
thf(fact_1068_sup__inf__distrib1,axiom,
! [X: product_unit,Y: product_unit,Z: product_unit] :
( ( sup_sup_Product_unit @ X @ ( inf_inf_Product_unit @ Y @ Z ) )
= ( inf_inf_Product_unit @ ( sup_sup_Product_unit @ X @ Y ) @ ( sup_sup_Product_unit @ X @ Z ) ) ) ).
% sup_inf_distrib1
thf(fact_1069_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_1070_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_1071_inf__sup__distrib2,axiom,
! [Y: set_set_v,Z: set_set_v,X: set_set_v] :
( ( inf_inf_set_set_v @ ( sup_sup_set_set_v @ Y @ Z ) @ X )
= ( sup_sup_set_set_v @ ( inf_inf_set_set_v @ Y @ X ) @ ( inf_inf_set_set_v @ Z @ X ) ) ) ).
% inf_sup_distrib2
thf(fact_1072_inf__sup__distrib2,axiom,
! [Y: product_unit,Z: product_unit,X: product_unit] :
( ( inf_inf_Product_unit @ ( sup_sup_Product_unit @ Y @ Z ) @ X )
= ( sup_sup_Product_unit @ ( inf_inf_Product_unit @ Y @ X ) @ ( inf_inf_Product_unit @ Z @ X ) ) ) ).
% inf_sup_distrib2
thf(fact_1073_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_1074_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_1075_inf__sup__distrib1,axiom,
! [X: set_set_v,Y: set_set_v,Z: set_set_v] :
( ( inf_inf_set_set_v @ X @ ( sup_sup_set_set_v @ Y @ Z ) )
= ( sup_sup_set_set_v @ ( inf_inf_set_set_v @ X @ Y ) @ ( inf_inf_set_set_v @ X @ Z ) ) ) ).
% inf_sup_distrib1
thf(fact_1076_inf__sup__distrib1,axiom,
! [X: product_unit,Y: product_unit,Z: product_unit] :
( ( inf_inf_Product_unit @ X @ ( sup_sup_Product_unit @ Y @ Z ) )
= ( sup_sup_Product_unit @ ( inf_inf_Product_unit @ X @ Y ) @ ( inf_inf_Product_unit @ X @ Z ) ) ) ).
% inf_sup_distrib1
thf(fact_1077_distrib__imp2,axiom,
! [X: set_Product_prod_v_v,Y: set_Product_prod_v_v,Z: set_Product_prod_v_v] :
( ! [X3: set_Product_prod_v_v,Y2: set_Product_prod_v_v,Z3: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ X3 @ ( inf_in6271465464967711157od_v_v @ Y2 @ Z3 ) )
= ( inf_in6271465464967711157od_v_v @ ( sup_su414716646722978715od_v_v @ X3 @ Y2 ) @ ( sup_su414716646722978715od_v_v @ X3 @ Z3 ) ) )
=> ( ( 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_1078_distrib__imp2,axiom,
! [X: set_v,Y: set_v,Z: set_v] :
( ! [X3: set_v,Y2: set_v,Z3: set_v] :
( ( sup_sup_set_v @ X3 @ ( inf_inf_set_v @ Y2 @ Z3 ) )
= ( inf_inf_set_v @ ( sup_sup_set_v @ X3 @ Y2 ) @ ( sup_sup_set_v @ X3 @ Z3 ) ) )
=> ( ( 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_1079_distrib__imp2,axiom,
! [X: set_set_v,Y: set_set_v,Z: set_set_v] :
( ! [X3: set_set_v,Y2: set_set_v,Z3: set_set_v] :
( ( sup_sup_set_set_v @ X3 @ ( inf_inf_set_set_v @ Y2 @ Z3 ) )
= ( inf_inf_set_set_v @ ( sup_sup_set_set_v @ X3 @ Y2 ) @ ( sup_sup_set_set_v @ X3 @ Z3 ) ) )
=> ( ( inf_inf_set_set_v @ X @ ( sup_sup_set_set_v @ Y @ Z ) )
= ( sup_sup_set_set_v @ ( inf_inf_set_set_v @ X @ Y ) @ ( inf_inf_set_set_v @ X @ Z ) ) ) ) ).
% distrib_imp2
thf(fact_1080_distrib__imp2,axiom,
! [X: product_unit,Y: product_unit,Z: product_unit] :
( ! [X3: product_unit,Y2: product_unit,Z3: product_unit] :
( ( sup_sup_Product_unit @ X3 @ ( inf_inf_Product_unit @ Y2 @ Z3 ) )
= ( inf_inf_Product_unit @ ( sup_sup_Product_unit @ X3 @ Y2 ) @ ( sup_sup_Product_unit @ X3 @ Z3 ) ) )
=> ( ( inf_inf_Product_unit @ X @ ( sup_sup_Product_unit @ Y @ Z ) )
= ( sup_sup_Product_unit @ ( inf_inf_Product_unit @ X @ Y ) @ ( inf_inf_Product_unit @ X @ Z ) ) ) ) ).
% distrib_imp2
thf(fact_1081_distrib__imp1,axiom,
! [X: set_Product_prod_v_v,Y: set_Product_prod_v_v,Z: set_Product_prod_v_v] :
( ! [X3: set_Product_prod_v_v,Y2: set_Product_prod_v_v,Z3: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ X3 @ ( sup_su414716646722978715od_v_v @ Y2 @ Z3 ) )
= ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ X3 @ Y2 ) @ ( inf_in6271465464967711157od_v_v @ X3 @ Z3 ) ) )
=> ( ( 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_1082_distrib__imp1,axiom,
! [X: set_v,Y: set_v,Z: set_v] :
( ! [X3: set_v,Y2: set_v,Z3: set_v] :
( ( inf_inf_set_v @ X3 @ ( sup_sup_set_v @ Y2 @ Z3 ) )
= ( sup_sup_set_v @ ( inf_inf_set_v @ X3 @ Y2 ) @ ( inf_inf_set_v @ X3 @ Z3 ) ) )
=> ( ( 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_1083_distrib__imp1,axiom,
! [X: set_set_v,Y: set_set_v,Z: set_set_v] :
( ! [X3: set_set_v,Y2: set_set_v,Z3: set_set_v] :
( ( inf_inf_set_set_v @ X3 @ ( sup_sup_set_set_v @ Y2 @ Z3 ) )
= ( sup_sup_set_set_v @ ( inf_inf_set_set_v @ X3 @ Y2 ) @ ( inf_inf_set_set_v @ X3 @ Z3 ) ) )
=> ( ( sup_sup_set_set_v @ X @ ( inf_inf_set_set_v @ Y @ Z ) )
= ( inf_inf_set_set_v @ ( sup_sup_set_set_v @ X @ Y ) @ ( sup_sup_set_set_v @ X @ Z ) ) ) ) ).
% distrib_imp1
thf(fact_1084_distrib__imp1,axiom,
! [X: product_unit,Y: product_unit,Z: product_unit] :
( ! [X3: product_unit,Y2: product_unit,Z3: product_unit] :
( ( inf_inf_Product_unit @ X3 @ ( sup_sup_Product_unit @ Y2 @ Z3 ) )
= ( sup_sup_Product_unit @ ( inf_inf_Product_unit @ X3 @ Y2 ) @ ( inf_inf_Product_unit @ X3 @ Z3 ) ) )
=> ( ( sup_sup_Product_unit @ X @ ( inf_inf_Product_unit @ Y @ Z ) )
= ( inf_inf_Product_unit @ ( sup_sup_Product_unit @ X @ Y ) @ ( sup_sup_Product_unit @ X @ Z ) ) ) ) ).
% distrib_imp1
thf(fact_1085_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_1086_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_1087_boolean__algebra_Oconj__disj__distrib,axiom,
! [X: set_set_v,Y: set_set_v,Z: set_set_v] :
( ( inf_inf_set_set_v @ X @ ( sup_sup_set_set_v @ Y @ Z ) )
= ( sup_sup_set_set_v @ ( inf_inf_set_set_v @ X @ Y ) @ ( inf_inf_set_set_v @ X @ Z ) ) ) ).
% boolean_algebra.conj_disj_distrib
thf(fact_1088_boolean__algebra_Oconj__disj__distrib,axiom,
! [X: product_unit,Y: product_unit,Z: product_unit] :
( ( inf_inf_Product_unit @ X @ ( sup_sup_Product_unit @ Y @ Z ) )
= ( sup_sup_Product_unit @ ( inf_inf_Product_unit @ X @ Y ) @ ( inf_inf_Product_unit @ X @ Z ) ) ) ).
% boolean_algebra.conj_disj_distrib
thf(fact_1089_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_1090_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_1091_boolean__algebra_Odisj__conj__distrib,axiom,
! [X: set_set_v,Y: set_set_v,Z: set_set_v] :
( ( sup_sup_set_set_v @ X @ ( inf_inf_set_set_v @ Y @ Z ) )
= ( inf_inf_set_set_v @ ( sup_sup_set_set_v @ X @ Y ) @ ( sup_sup_set_set_v @ X @ Z ) ) ) ).
% boolean_algebra.disj_conj_distrib
thf(fact_1092_boolean__algebra_Odisj__conj__distrib,axiom,
! [X: product_unit,Y: product_unit,Z: product_unit] :
( ( sup_sup_Product_unit @ X @ ( inf_inf_Product_unit @ Y @ Z ) )
= ( inf_inf_Product_unit @ ( sup_sup_Product_unit @ X @ Y ) @ ( sup_sup_Product_unit @ X @ Z ) ) ) ).
% boolean_algebra.disj_conj_distrib
thf(fact_1093_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_1094_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_1095_boolean__algebra_Oconj__disj__distrib2,axiom,
! [Y: set_set_v,Z: set_set_v,X: set_set_v] :
( ( inf_inf_set_set_v @ ( sup_sup_set_set_v @ Y @ Z ) @ X )
= ( sup_sup_set_set_v @ ( inf_inf_set_set_v @ Y @ X ) @ ( inf_inf_set_set_v @ Z @ X ) ) ) ).
% boolean_algebra.conj_disj_distrib2
thf(fact_1096_boolean__algebra_Oconj__disj__distrib2,axiom,
! [Y: product_unit,Z: product_unit,X: product_unit] :
( ( inf_inf_Product_unit @ ( sup_sup_Product_unit @ Y @ Z ) @ X )
= ( sup_sup_Product_unit @ ( inf_inf_Product_unit @ Y @ X ) @ ( inf_inf_Product_unit @ Z @ X ) ) ) ).
% boolean_algebra.conj_disj_distrib2
thf(fact_1097_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_1098_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_1099_boolean__algebra_Odisj__conj__distrib2,axiom,
! [Y: set_set_v,Z: set_set_v,X: set_set_v] :
( ( sup_sup_set_set_v @ ( inf_inf_set_set_v @ Y @ Z ) @ X )
= ( inf_inf_set_set_v @ ( sup_sup_set_set_v @ Y @ X ) @ ( sup_sup_set_set_v @ Z @ X ) ) ) ).
% boolean_algebra.disj_conj_distrib2
thf(fact_1100_boolean__algebra_Odisj__conj__distrib2,axiom,
! [Y: product_unit,Z: product_unit,X: product_unit] :
( ( sup_sup_Product_unit @ ( inf_inf_Product_unit @ Y @ Z ) @ X )
= ( inf_inf_Product_unit @ ( sup_sup_Product_unit @ Y @ X ) @ ( sup_sup_Product_unit @ Z @ X ) ) ) ).
% boolean_algebra.disj_conj_distrib2
thf(fact_1101_disjoint__iff__not__equal,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ( inf_in6271465464967711157od_v_v @ A @ B )
= bot_bo723834152578015283od_v_v )
= ( ! [X2: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X2 @ A )
=> ! [Y3: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ Y3 @ B )
=> ( X2 != Y3 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_1102_disjoint__iff__not__equal,axiom,
! [A: set_set_v,B: set_set_v] :
( ( ( inf_inf_set_set_v @ A @ B )
= bot_bot_set_set_v )
= ( ! [X2: set_v] :
( ( member_set_v @ X2 @ A )
=> ! [Y3: set_v] :
( ( member_set_v @ Y3 @ B )
=> ( X2 != Y3 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_1103_disjoint__iff__not__equal,axiom,
! [A: set_v,B: set_v] :
( ( ( inf_inf_set_v @ A @ B )
= bot_bot_set_v )
= ( ! [X2: v] :
( ( member_v @ X2 @ A )
=> ! [Y3: v] :
( ( member_v @ Y3 @ B )
=> ( X2 != Y3 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_1104_Int__empty__right,axiom,
! [A: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ A @ bot_bo723834152578015283od_v_v )
= bot_bo723834152578015283od_v_v ) ).
% Int_empty_right
thf(fact_1105_Int__empty__right,axiom,
! [A: set_set_v] :
( ( inf_inf_set_set_v @ A @ bot_bot_set_set_v )
= bot_bot_set_set_v ) ).
% Int_empty_right
thf(fact_1106_Int__empty__right,axiom,
! [A: set_v] :
( ( inf_inf_set_v @ A @ bot_bot_set_v )
= bot_bot_set_v ) ).
% Int_empty_right
thf(fact_1107_Int__empty__left,axiom,
! [B: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ bot_bo723834152578015283od_v_v @ B )
= bot_bo723834152578015283od_v_v ) ).
% Int_empty_left
thf(fact_1108_Int__empty__left,axiom,
! [B: set_set_v] :
( ( inf_inf_set_set_v @ bot_bot_set_set_v @ B )
= bot_bot_set_set_v ) ).
% Int_empty_left
thf(fact_1109_Int__empty__left,axiom,
! [B: set_v] :
( ( inf_inf_set_v @ bot_bot_set_v @ B )
= bot_bot_set_v ) ).
% Int_empty_left
thf(fact_1110_disjoint__iff,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ( inf_in6271465464967711157od_v_v @ A @ B )
= bot_bo723834152578015283od_v_v )
= ( ! [X2: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X2 @ A )
=> ~ ( member7453568604450474000od_v_v @ X2 @ B ) ) ) ) ).
% disjoint_iff
thf(fact_1111_disjoint__iff,axiom,
! [A: set_set_v,B: set_set_v] :
( ( ( inf_inf_set_set_v @ A @ B )
= bot_bot_set_set_v )
= ( ! [X2: set_v] :
( ( member_set_v @ X2 @ A )
=> ~ ( member_set_v @ X2 @ B ) ) ) ) ).
% disjoint_iff
thf(fact_1112_disjoint__iff,axiom,
! [A: set_v,B: set_v] :
( ( ( inf_inf_set_v @ A @ B )
= bot_bot_set_v )
= ( ! [X2: v] :
( ( member_v @ X2 @ A )
=> ~ ( member_v @ X2 @ B ) ) ) ) ).
% disjoint_iff
thf(fact_1113_Int__emptyI,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ! [X3: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X3 @ A )
=> ~ ( member7453568604450474000od_v_v @ X3 @ B ) )
=> ( ( inf_in6271465464967711157od_v_v @ A @ B )
= bot_bo723834152578015283od_v_v ) ) ).
% Int_emptyI
thf(fact_1114_Int__emptyI,axiom,
! [A: set_set_v,B: set_set_v] :
( ! [X3: set_v] :
( ( member_set_v @ X3 @ A )
=> ~ ( member_set_v @ X3 @ B ) )
=> ( ( inf_inf_set_set_v @ A @ B )
= bot_bot_set_set_v ) ) ).
% Int_emptyI
thf(fact_1115_Int__emptyI,axiom,
! [A: set_v,B: set_v] :
( ! [X3: v] :
( ( member_v @ X3 @ A )
=> ~ ( member_v @ X3 @ B ) )
=> ( ( inf_inf_set_v @ A @ B )
= bot_bot_set_v ) ) ).
% Int_emptyI
thf(fact_1116_Int__mono,axiom,
! [A: set_Product_prod_v_v,C2: set_Product_prod_v_v,B: set_Product_prod_v_v,D2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ C2 )
=> ( ( ord_le7336532860387713383od_v_v @ B @ D2 )
=> ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ A @ B ) @ ( inf_in6271465464967711157od_v_v @ C2 @ D2 ) ) ) ) ).
% Int_mono
thf(fact_1117_Int__mono,axiom,
! [A: set_v,C2: set_v,B: set_v,D2: set_v] :
( ( ord_less_eq_set_v @ A @ C2 )
=> ( ( ord_less_eq_set_v @ B @ D2 )
=> ( ord_less_eq_set_v @ ( inf_inf_set_v @ A @ B ) @ ( inf_inf_set_v @ C2 @ D2 ) ) ) ) ).
% Int_mono
thf(fact_1118_Int__lower1,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ A @ B ) @ A ) ).
% Int_lower1
thf(fact_1119_Int__lower1,axiom,
! [A: set_v,B: set_v] : ( ord_less_eq_set_v @ ( inf_inf_set_v @ A @ B ) @ A ) ).
% Int_lower1
thf(fact_1120_Int__lower2,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ A @ B ) @ B ) ).
% Int_lower2
thf(fact_1121_Int__lower2,axiom,
! [A: set_v,B: set_v] : ( ord_less_eq_set_v @ ( inf_inf_set_v @ A @ B ) @ B ) ).
% Int_lower2
thf(fact_1122_Int__absorb1,axiom,
! [B: set_Product_prod_v_v,A: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ B @ A )
=> ( ( inf_in6271465464967711157od_v_v @ A @ B )
= B ) ) ).
% Int_absorb1
thf(fact_1123_Int__absorb1,axiom,
! [B: set_v,A: set_v] :
( ( ord_less_eq_set_v @ B @ A )
=> ( ( inf_inf_set_v @ A @ B )
= B ) ) ).
% Int_absorb1
thf(fact_1124_Int__absorb2,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ B )
=> ( ( inf_in6271465464967711157od_v_v @ A @ B )
= A ) ) ).
% Int_absorb2
thf(fact_1125_Int__absorb2,axiom,
! [A: set_v,B: set_v] :
( ( ord_less_eq_set_v @ A @ B )
=> ( ( inf_inf_set_v @ A @ B )
= A ) ) ).
% Int_absorb2
thf(fact_1126_Int__greatest,axiom,
! [C2: set_Product_prod_v_v,A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ C2 @ A )
=> ( ( ord_le7336532860387713383od_v_v @ C2 @ B )
=> ( ord_le7336532860387713383od_v_v @ C2 @ ( inf_in6271465464967711157od_v_v @ A @ B ) ) ) ) ).
% Int_greatest
thf(fact_1127_Int__greatest,axiom,
! [C2: set_v,A: set_v,B: set_v] :
( ( ord_less_eq_set_v @ C2 @ A )
=> ( ( ord_less_eq_set_v @ C2 @ B )
=> ( ord_less_eq_set_v @ C2 @ ( inf_inf_set_v @ A @ B ) ) ) ) ).
% Int_greatest
thf(fact_1128_Int__Collect__mono,axiom,
! [A: set_set_v,B: set_set_v,P: set_v > $o,Q: set_v > $o] :
( ( ord_le5216385588623774835_set_v @ A @ B )
=> ( ! [X3: set_v] :
( ( member_set_v @ X3 @ A )
=> ( ( P @ X3 )
=> ( Q @ X3 ) ) )
=> ( ord_le5216385588623774835_set_v @ ( inf_inf_set_set_v @ A @ ( collect_set_v @ P ) ) @ ( inf_inf_set_set_v @ B @ ( collect_set_v @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_1129_Int__Collect__mono,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v,P: product_prod_v_v > $o,Q: product_prod_v_v > $o] :
( ( ord_le7336532860387713383od_v_v @ A @ B )
=> ( ! [X3: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X3 @ A )
=> ( ( P @ X3 )
=> ( Q @ X3 ) ) )
=> ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ A @ ( collec140062887454715474od_v_v @ P ) ) @ ( inf_in6271465464967711157od_v_v @ B @ ( collec140062887454715474od_v_v @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_1130_Int__Collect__mono,axiom,
! [A: set_v,B: set_v,P: v > $o,Q: v > $o] :
( ( ord_less_eq_set_v @ A @ B )
=> ( ! [X3: v] :
( ( member_v @ X3 @ A )
=> ( ( P @ X3 )
=> ( Q @ X3 ) ) )
=> ( ord_less_eq_set_v @ ( inf_inf_set_v @ A @ ( collect_v @ P ) ) @ ( inf_inf_set_v @ B @ ( collect_v @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_1131_transpose_Ocases,axiom,
! [X: list_list_v] :
( ( X != nil_list_v )
=> ( ! [Xss: list_list_v] :
( X
!= ( cons_list_v @ nil_v @ Xss ) )
=> ~ ! [X3: v,Xs2: list_v,Xss: list_list_v] :
( X
!= ( cons_list_v @ ( cons_v @ X3 @ Xs2 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_1132_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_1133_inf__sup__aci_I4_J,axiom,
! [X: product_unit,Y: product_unit] :
( ( inf_inf_Product_unit @ X @ ( inf_inf_Product_unit @ X @ Y ) )
= ( inf_inf_Product_unit @ X @ Y ) ) ).
% inf_sup_aci(4)
thf(fact_1134_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_1135_inf__sup__aci_I3_J,axiom,
! [X: product_unit,Y: product_unit,Z: product_unit] :
( ( inf_inf_Product_unit @ X @ ( inf_inf_Product_unit @ Y @ Z ) )
= ( inf_inf_Product_unit @ Y @ ( inf_inf_Product_unit @ X @ Z ) ) ) ).
% inf_sup_aci(3)
thf(fact_1136_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_1137_inf__sup__aci_I2_J,axiom,
! [X: product_unit,Y: product_unit,Z: product_unit] :
( ( inf_inf_Product_unit @ ( inf_inf_Product_unit @ X @ Y ) @ Z )
= ( inf_inf_Product_unit @ X @ ( inf_inf_Product_unit @ Y @ Z ) ) ) ).
% inf_sup_aci(2)
thf(fact_1138_inf__sup__aci_I1_J,axiom,
( inf_inf_set_v
= ( ^ [X2: set_v,Y3: set_v] : ( inf_inf_set_v @ Y3 @ X2 ) ) ) ).
% inf_sup_aci(1)
thf(fact_1139_inf__sup__aci_I1_J,axiom,
( inf_inf_Product_unit
= ( ^ [X2: product_unit,Y3: product_unit] : ( inf_inf_Product_unit @ Y3 @ X2 ) ) ) ).
% inf_sup_aci(1)
thf(fact_1140_inf_Oassoc,axiom,
! [A4: set_v,B2: set_v,C: set_v] :
( ( inf_inf_set_v @ ( inf_inf_set_v @ A4 @ B2 ) @ C )
= ( inf_inf_set_v @ A4 @ ( inf_inf_set_v @ B2 @ C ) ) ) ).
% inf.assoc
thf(fact_1141_inf_Oassoc,axiom,
! [A4: product_unit,B2: product_unit,C: product_unit] :
( ( inf_inf_Product_unit @ ( inf_inf_Product_unit @ A4 @ B2 ) @ C )
= ( inf_inf_Product_unit @ A4 @ ( inf_inf_Product_unit @ B2 @ C ) ) ) ).
% inf.assoc
thf(fact_1142_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_1143_inf__assoc,axiom,
! [X: product_unit,Y: product_unit,Z: product_unit] :
( ( inf_inf_Product_unit @ ( inf_inf_Product_unit @ X @ Y ) @ Z )
= ( inf_inf_Product_unit @ X @ ( inf_inf_Product_unit @ Y @ Z ) ) ) ).
% inf_assoc
thf(fact_1144_inf_Ocommute,axiom,
( inf_inf_set_v
= ( ^ [A6: set_v,B4: set_v] : ( inf_inf_set_v @ B4 @ A6 ) ) ) ).
% inf.commute
thf(fact_1145_inf_Ocommute,axiom,
( inf_inf_Product_unit
= ( ^ [A6: product_unit,B4: product_unit] : ( inf_inf_Product_unit @ B4 @ A6 ) ) ) ).
% inf.commute
thf(fact_1146_inf__commute,axiom,
( inf_inf_set_v
= ( ^ [X2: set_v,Y3: set_v] : ( inf_inf_set_v @ Y3 @ X2 ) ) ) ).
% inf_commute
thf(fact_1147_inf__commute,axiom,
( inf_inf_Product_unit
= ( ^ [X2: product_unit,Y3: product_unit] : ( inf_inf_Product_unit @ Y3 @ X2 ) ) ) ).
% inf_commute
thf(fact_1148_inf_Oleft__commute,axiom,
! [B2: set_v,A4: set_v,C: set_v] :
( ( inf_inf_set_v @ B2 @ ( inf_inf_set_v @ A4 @ C ) )
= ( inf_inf_set_v @ A4 @ ( inf_inf_set_v @ B2 @ C ) ) ) ).
% inf.left_commute
thf(fact_1149_inf_Oleft__commute,axiom,
! [B2: product_unit,A4: product_unit,C: product_unit] :
( ( inf_inf_Product_unit @ B2 @ ( inf_inf_Product_unit @ A4 @ C ) )
= ( inf_inf_Product_unit @ A4 @ ( inf_inf_Product_unit @ B2 @ C ) ) ) ).
% inf.left_commute
thf(fact_1150_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_1151_inf__left__commute,axiom,
! [X: product_unit,Y: product_unit,Z: product_unit] :
( ( inf_inf_Product_unit @ X @ ( inf_inf_Product_unit @ Y @ Z ) )
= ( inf_inf_Product_unit @ Y @ ( inf_inf_Product_unit @ X @ Z ) ) ) ).
% inf_left_commute
thf(fact_1152_IntE,axiom,
! [C: product_prod_v_v,A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ C @ ( inf_in6271465464967711157od_v_v @ A @ B ) )
=> ~ ( ( member7453568604450474000od_v_v @ C @ A )
=> ~ ( member7453568604450474000od_v_v @ C @ B ) ) ) ).
% IntE
thf(fact_1153_IntE,axiom,
! [C: v,A: set_v,B: set_v] :
( ( member_v @ C @ ( inf_inf_set_v @ A @ B ) )
=> ~ ( ( member_v @ C @ A )
=> ~ ( member_v @ C @ B ) ) ) ).
% IntE
thf(fact_1154_IntD1,axiom,
! [C: product_prod_v_v,A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ C @ ( inf_in6271465464967711157od_v_v @ A @ B ) )
=> ( member7453568604450474000od_v_v @ C @ A ) ) ).
% IntD1
thf(fact_1155_IntD1,axiom,
! [C: v,A: set_v,B: set_v] :
( ( member_v @ C @ ( inf_inf_set_v @ A @ B ) )
=> ( member_v @ C @ A ) ) ).
% IntD1
thf(fact_1156_IntD2,axiom,
! [C: product_prod_v_v,A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ C @ ( inf_in6271465464967711157od_v_v @ A @ B ) )
=> ( member7453568604450474000od_v_v @ C @ B ) ) ).
% IntD2
thf(fact_1157_IntD2,axiom,
! [C: v,A: set_v,B: set_v] :
( ( member_v @ C @ ( inf_inf_set_v @ A @ B ) )
=> ( member_v @ C @ B ) ) ).
% IntD2
thf(fact_1158_Int__assoc,axiom,
! [A: set_v,B: set_v,C2: set_v] :
( ( inf_inf_set_v @ ( inf_inf_set_v @ A @ B ) @ C2 )
= ( inf_inf_set_v @ A @ ( inf_inf_set_v @ B @ C2 ) ) ) ).
% Int_assoc
thf(fact_1159_Int__absorb,axiom,
! [A: set_v] :
( ( inf_inf_set_v @ A @ A )
= A ) ).
% Int_absorb
thf(fact_1160_Int__commute,axiom,
( inf_inf_set_v
= ( ^ [A8: set_v,B6: set_v] : ( inf_inf_set_v @ B6 @ A8 ) ) ) ).
% Int_commute
thf(fact_1161_Int__left__absorb,axiom,
! [A: set_v,B: set_v] :
( ( inf_inf_set_v @ A @ ( inf_inf_set_v @ A @ B ) )
= ( inf_inf_set_v @ A @ B ) ) ).
% Int_left_absorb
thf(fact_1162_Int__left__commute,axiom,
! [A: set_v,B: set_v,C2: set_v] :
( ( inf_inf_set_v @ A @ ( inf_inf_set_v @ B @ C2 ) )
= ( inf_inf_set_v @ B @ ( inf_inf_set_v @ A @ C2 ) ) ) ).
% Int_left_commute
thf(fact_1163_Collect__conj__eq,axiom,
! [P: set_v > $o,Q: set_v > $o] :
( ( collect_set_v
@ ^ [X2: set_v] :
( ( P @ X2 )
& ( Q @ X2 ) ) )
= ( inf_inf_set_set_v @ ( collect_set_v @ P ) @ ( collect_set_v @ Q ) ) ) ).
% Collect_conj_eq
thf(fact_1164_Collect__conj__eq,axiom,
! [P: v > $o,Q: v > $o] :
( ( collect_v
@ ^ [X2: v] :
( ( P @ X2 )
& ( Q @ X2 ) ) )
= ( inf_inf_set_v @ ( collect_v @ P ) @ ( collect_v @ Q ) ) ) ).
% Collect_conj_eq
thf(fact_1165_Int__Collect,axiom,
! [X: product_prod_v_v,A: set_Product_prod_v_v,P: product_prod_v_v > $o] :
( ( member7453568604450474000od_v_v @ X @ ( inf_in6271465464967711157od_v_v @ A @ ( collec140062887454715474od_v_v @ P ) ) )
= ( ( member7453568604450474000od_v_v @ X @ A )
& ( P @ X ) ) ) ).
% Int_Collect
thf(fact_1166_Int__Collect,axiom,
! [X: set_v,A: set_set_v,P: set_v > $o] :
( ( member_set_v @ X @ ( inf_inf_set_set_v @ A @ ( collect_set_v @ P ) ) )
= ( ( member_set_v @ X @ A )
& ( P @ X ) ) ) ).
% Int_Collect
thf(fact_1167_Int__Collect,axiom,
! [X: v,A: set_v,P: v > $o] :
( ( member_v @ X @ ( inf_inf_set_v @ A @ ( collect_v @ P ) ) )
= ( ( member_v @ X @ A )
& ( P @ X ) ) ) ).
% Int_Collect
thf(fact_1168_Int__def,axiom,
( inf_in6271465464967711157od_v_v
= ( ^ [A8: set_Product_prod_v_v,B6: set_Product_prod_v_v] :
( collec140062887454715474od_v_v
@ ^ [X2: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X2 @ A8 )
& ( member7453568604450474000od_v_v @ X2 @ B6 ) ) ) ) ) ).
% Int_def
thf(fact_1169_Int__def,axiom,
( inf_inf_set_set_v
= ( ^ [A8: set_set_v,B6: set_set_v] :
( collect_set_v
@ ^ [X2: set_v] :
( ( member_set_v @ X2 @ A8 )
& ( member_set_v @ X2 @ B6 ) ) ) ) ) ).
% Int_def
thf(fact_1170_Int__def,axiom,
( inf_inf_set_v
= ( ^ [A8: set_v,B6: set_v] :
( collect_v
@ ^ [X2: v] :
( ( member_v @ X2 @ A8 )
& ( member_v @ X2 @ B6 ) ) ) ) ) ).
% Int_def
thf(fact_1171_Int__insert__left,axiom,
! [A4: set_v,C2: set_set_v,B: set_set_v] :
( ( ( member_set_v @ A4 @ C2 )
=> ( ( inf_inf_set_set_v @ ( insert_set_v @ A4 @ B ) @ C2 )
= ( insert_set_v @ A4 @ ( inf_inf_set_set_v @ B @ C2 ) ) ) )
& ( ~ ( member_set_v @ A4 @ C2 )
=> ( ( inf_inf_set_set_v @ ( insert_set_v @ A4 @ B ) @ C2 )
= ( inf_inf_set_set_v @ B @ C2 ) ) ) ) ).
% Int_insert_left
thf(fact_1172_Int__insert__left,axiom,
! [A4: product_prod_v_v,C2: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ( member7453568604450474000od_v_v @ A4 @ C2 )
=> ( ( inf_in6271465464967711157od_v_v @ ( insert1338601472111419319od_v_v @ A4 @ B ) @ C2 )
= ( insert1338601472111419319od_v_v @ A4 @ ( inf_in6271465464967711157od_v_v @ B @ C2 ) ) ) )
& ( ~ ( member7453568604450474000od_v_v @ A4 @ C2 )
=> ( ( inf_in6271465464967711157od_v_v @ ( insert1338601472111419319od_v_v @ A4 @ B ) @ C2 )
= ( inf_in6271465464967711157od_v_v @ B @ C2 ) ) ) ) ).
% Int_insert_left
thf(fact_1173_Int__insert__left,axiom,
! [A4: v,C2: set_v,B: set_v] :
( ( ( member_v @ A4 @ C2 )
=> ( ( inf_inf_set_v @ ( insert_v @ A4 @ B ) @ C2 )
= ( insert_v @ A4 @ ( inf_inf_set_v @ B @ C2 ) ) ) )
& ( ~ ( member_v @ A4 @ C2 )
=> ( ( inf_inf_set_v @ ( insert_v @ A4 @ B ) @ C2 )
= ( inf_inf_set_v @ B @ C2 ) ) ) ) ).
% Int_insert_left
thf(fact_1174_Int__insert__right,axiom,
! [A4: set_v,A: set_set_v,B: set_set_v] :
( ( ( member_set_v @ A4 @ A )
=> ( ( inf_inf_set_set_v @ A @ ( insert_set_v @ A4 @ B ) )
= ( insert_set_v @ A4 @ ( inf_inf_set_set_v @ A @ B ) ) ) )
& ( ~ ( member_set_v @ A4 @ A )
=> ( ( inf_inf_set_set_v @ A @ ( insert_set_v @ A4 @ B ) )
= ( inf_inf_set_set_v @ A @ B ) ) ) ) ).
% Int_insert_right
thf(fact_1175_Int__insert__right,axiom,
! [A4: product_prod_v_v,A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ( member7453568604450474000od_v_v @ A4 @ A )
=> ( ( inf_in6271465464967711157od_v_v @ A @ ( insert1338601472111419319od_v_v @ A4 @ B ) )
= ( insert1338601472111419319od_v_v @ A4 @ ( inf_in6271465464967711157od_v_v @ A @ B ) ) ) )
& ( ~ ( member7453568604450474000od_v_v @ A4 @ A )
=> ( ( inf_in6271465464967711157od_v_v @ A @ ( insert1338601472111419319od_v_v @ A4 @ B ) )
= ( inf_in6271465464967711157od_v_v @ A @ B ) ) ) ) ).
% Int_insert_right
thf(fact_1176_Int__insert__right,axiom,
! [A4: v,A: set_v,B: set_v] :
( ( ( member_v @ A4 @ A )
=> ( ( inf_inf_set_v @ A @ ( insert_v @ A4 @ B ) )
= ( insert_v @ A4 @ ( inf_inf_set_v @ A @ B ) ) ) )
& ( ~ ( member_v @ A4 @ A )
=> ( ( inf_inf_set_v @ A @ ( insert_v @ A4 @ B ) )
= ( inf_inf_set_v @ A @ B ) ) ) ) ).
% Int_insert_right
thf(fact_1177_Un__Int__crazy,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v,C2: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ A @ B ) @ ( inf_in6271465464967711157od_v_v @ B @ C2 ) ) @ ( inf_in6271465464967711157od_v_v @ C2 @ A ) )
= ( inf_in6271465464967711157od_v_v @ ( inf_in6271465464967711157od_v_v @ ( sup_su414716646722978715od_v_v @ A @ B ) @ ( sup_su414716646722978715od_v_v @ B @ C2 ) ) @ ( sup_su414716646722978715od_v_v @ C2 @ A ) ) ) ).
% Un_Int_crazy
thf(fact_1178_Un__Int__crazy,axiom,
! [A: set_v,B: set_v,C2: set_v] :
( ( sup_sup_set_v @ ( sup_sup_set_v @ ( inf_inf_set_v @ A @ B ) @ ( inf_inf_set_v @ B @ C2 ) ) @ ( inf_inf_set_v @ C2 @ A ) )
= ( inf_inf_set_v @ ( inf_inf_set_v @ ( sup_sup_set_v @ A @ B ) @ ( sup_sup_set_v @ B @ C2 ) ) @ ( sup_sup_set_v @ C2 @ A ) ) ) ).
% Un_Int_crazy
thf(fact_1179_Un__Int__crazy,axiom,
! [A: set_set_v,B: set_set_v,C2: set_set_v] :
( ( sup_sup_set_set_v @ ( sup_sup_set_set_v @ ( inf_inf_set_set_v @ A @ B ) @ ( inf_inf_set_set_v @ B @ C2 ) ) @ ( inf_inf_set_set_v @ C2 @ A ) )
= ( inf_inf_set_set_v @ ( inf_inf_set_set_v @ ( sup_sup_set_set_v @ A @ B ) @ ( sup_sup_set_set_v @ B @ C2 ) ) @ ( sup_sup_set_set_v @ C2 @ A ) ) ) ).
% Un_Int_crazy
thf(fact_1180_Int__Un__distrib,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v,C2: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ A @ ( sup_su414716646722978715od_v_v @ B @ C2 ) )
= ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ A @ B ) @ ( inf_in6271465464967711157od_v_v @ A @ C2 ) ) ) ).
% Int_Un_distrib
thf(fact_1181_Int__Un__distrib,axiom,
! [A: set_v,B: set_v,C2: set_v] :
( ( inf_inf_set_v @ A @ ( sup_sup_set_v @ B @ C2 ) )
= ( sup_sup_set_v @ ( inf_inf_set_v @ A @ B ) @ ( inf_inf_set_v @ A @ C2 ) ) ) ).
% Int_Un_distrib
thf(fact_1182_Int__Un__distrib,axiom,
! [A: set_set_v,B: set_set_v,C2: set_set_v] :
( ( inf_inf_set_set_v @ A @ ( sup_sup_set_set_v @ B @ C2 ) )
= ( sup_sup_set_set_v @ ( inf_inf_set_set_v @ A @ B ) @ ( inf_inf_set_set_v @ A @ C2 ) ) ) ).
% Int_Un_distrib
thf(fact_1183_Un__Int__distrib,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v,C2: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ A @ ( inf_in6271465464967711157od_v_v @ B @ C2 ) )
= ( inf_in6271465464967711157od_v_v @ ( sup_su414716646722978715od_v_v @ A @ B ) @ ( sup_su414716646722978715od_v_v @ A @ C2 ) ) ) ).
% Un_Int_distrib
thf(fact_1184_Un__Int__distrib,axiom,
! [A: set_v,B: set_v,C2: set_v] :
( ( sup_sup_set_v @ A @ ( inf_inf_set_v @ B @ C2 ) )
= ( inf_inf_set_v @ ( sup_sup_set_v @ A @ B ) @ ( sup_sup_set_v @ A @ C2 ) ) ) ).
% Un_Int_distrib
thf(fact_1185_Un__Int__distrib,axiom,
! [A: set_set_v,B: set_set_v,C2: set_set_v] :
( ( sup_sup_set_set_v @ A @ ( inf_inf_set_set_v @ B @ C2 ) )
= ( inf_inf_set_set_v @ ( sup_sup_set_set_v @ A @ B ) @ ( sup_sup_set_set_v @ A @ C2 ) ) ) ).
% Un_Int_distrib
thf(fact_1186_Int__Un__distrib2,axiom,
! [B: set_Product_prod_v_v,C2: set_Product_prod_v_v,A: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ ( sup_su414716646722978715od_v_v @ B @ C2 ) @ A )
= ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ B @ A ) @ ( inf_in6271465464967711157od_v_v @ C2 @ A ) ) ) ).
% Int_Un_distrib2
thf(fact_1187_Int__Un__distrib2,axiom,
! [B: set_v,C2: set_v,A: set_v] :
( ( inf_inf_set_v @ ( sup_sup_set_v @ B @ C2 ) @ A )
= ( sup_sup_set_v @ ( inf_inf_set_v @ B @ A ) @ ( inf_inf_set_v @ C2 @ A ) ) ) ).
% Int_Un_distrib2
thf(fact_1188_Int__Un__distrib2,axiom,
! [B: set_set_v,C2: set_set_v,A: set_set_v] :
( ( inf_inf_set_set_v @ ( sup_sup_set_set_v @ B @ C2 ) @ A )
= ( sup_sup_set_set_v @ ( inf_inf_set_set_v @ B @ A ) @ ( inf_inf_set_set_v @ C2 @ A ) ) ) ).
% Int_Un_distrib2
thf(fact_1189_Un__Int__distrib2,axiom,
! [B: set_Product_prod_v_v,C2: set_Product_prod_v_v,A: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ B @ C2 ) @ A )
= ( inf_in6271465464967711157od_v_v @ ( sup_su414716646722978715od_v_v @ B @ A ) @ ( sup_su414716646722978715od_v_v @ C2 @ A ) ) ) ).
% Un_Int_distrib2
thf(fact_1190_Un__Int__distrib2,axiom,
! [B: set_v,C2: set_v,A: set_v] :
( ( sup_sup_set_v @ ( inf_inf_set_v @ B @ C2 ) @ A )
= ( inf_inf_set_v @ ( sup_sup_set_v @ B @ A ) @ ( sup_sup_set_v @ C2 @ A ) ) ) ).
% Un_Int_distrib2
thf(fact_1191_Un__Int__distrib2,axiom,
! [B: set_set_v,C2: set_set_v,A: set_set_v] :
( ( sup_sup_set_set_v @ ( inf_inf_set_set_v @ B @ C2 ) @ A )
= ( inf_inf_set_set_v @ ( sup_sup_set_set_v @ B @ A ) @ ( sup_sup_set_set_v @ C2 @ A ) ) ) ).
% Un_Int_distrib2
thf(fact_1192_Int__Diff,axiom,
! [A: set_v,B: set_v,C2: set_v] :
( ( minus_minus_set_v @ ( inf_inf_set_v @ A @ B ) @ C2 )
= ( inf_inf_set_v @ A @ ( minus_minus_set_v @ B @ C2 ) ) ) ).
% Int_Diff
thf(fact_1193_Diff__Int2,axiom,
! [A: set_v,C2: set_v,B: set_v] :
( ( minus_minus_set_v @ ( inf_inf_set_v @ A @ C2 ) @ ( inf_inf_set_v @ B @ C2 ) )
= ( minus_minus_set_v @ ( inf_inf_set_v @ A @ C2 ) @ B ) ) ).
% Diff_Int2
thf(fact_1194_Diff__Diff__Int,axiom,
! [A: set_v,B: set_v] :
( ( minus_minus_set_v @ A @ ( minus_minus_set_v @ A @ B ) )
= ( inf_inf_set_v @ A @ B ) ) ).
% Diff_Diff_Int
thf(fact_1195_Diff__Int__distrib,axiom,
! [C2: set_v,A: set_v,B: set_v] :
( ( inf_inf_set_v @ C2 @ ( minus_minus_set_v @ A @ B ) )
= ( minus_minus_set_v @ ( inf_inf_set_v @ C2 @ A ) @ ( inf_inf_set_v @ C2 @ B ) ) ) ).
% Diff_Int_distrib
thf(fact_1196_Diff__Int__distrib2,axiom,
! [A: set_v,B: set_v,C2: set_v] :
( ( inf_inf_set_v @ ( minus_minus_set_v @ A @ B ) @ C2 )
= ( minus_minus_set_v @ ( inf_inf_set_v @ A @ C2 ) @ ( inf_inf_set_v @ B @ C2 ) ) ) ).
% Diff_Int_distrib2
thf(fact_1197_successively_Ocases,axiom,
! [X: produc8237170675765753490list_v] :
( ! [P3: v > v > $o] :
( X
!= ( produc601102195597853570list_v @ P3 @ nil_v ) )
=> ( ! [P3: v > v > $o,X3: v] :
( X
!= ( produc601102195597853570list_v @ P3 @ ( cons_v @ X3 @ nil_v ) ) )
=> ~ ! [P3: v > v > $o,X3: v,Y2: v,Xs2: list_v] :
( X
!= ( produc601102195597853570list_v @ P3 @ ( cons_v @ X3 @ ( cons_v @ Y2 @ Xs2 ) ) ) ) ) ) ).
% successively.cases
thf(fact_1198_sorted__wrt_Ocases,axiom,
! [X: produc8237170675765753490list_v] :
( ! [P3: v > v > $o] :
( X
!= ( produc601102195597853570list_v @ P3 @ nil_v ) )
=> ~ ! [P3: v > v > $o,X3: v,Ys2: list_v] :
( X
!= ( produc601102195597853570list_v @ P3 @ ( cons_v @ X3 @ Ys2 ) ) ) ) ).
% sorted_wrt.cases
thf(fact_1199_shuffles_Ocases,axiom,
! [X: produc1391462591744249447list_v] :
( ! [Ys2: list_v] :
( X
!= ( produc6795410681906604247list_v @ nil_v @ Ys2 ) )
=> ( ! [Xs2: list_v] :
( X
!= ( produc6795410681906604247list_v @ Xs2 @ nil_v ) )
=> ~ ! [X3: v,Xs2: list_v,Y2: v,Ys2: list_v] :
( X
!= ( produc6795410681906604247list_v @ ( cons_v @ X3 @ Xs2 ) @ ( cons_v @ Y2 @ Ys2 ) ) ) ) ) ).
% shuffles.cases
thf(fact_1200_splice_Ocases,axiom,
! [X: produc1391462591744249447list_v] :
( ! [Ys2: list_v] :
( X
!= ( produc6795410681906604247list_v @ nil_v @ Ys2 ) )
=> ~ ! [X3: v,Xs2: list_v,Ys2: list_v] :
( X
!= ( produc6795410681906604247list_v @ ( cons_v @ X3 @ Xs2 ) @ Ys2 ) ) ) ).
% splice.cases
thf(fact_1201_list__nonempty__induct,axiom,
! [Xs: list_v,P: list_v > $o] :
( ( Xs != nil_v )
=> ( ! [X3: v] : ( P @ ( cons_v @ X3 @ nil_v ) )
=> ( ! [X3: v,Xs2: list_v] :
( ( Xs2 != nil_v )
=> ( ( P @ Xs2 )
=> ( P @ ( cons_v @ X3 @ Xs2 ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_1202_list__induct2_H,axiom,
! [P: list_v > list_v > $o,Xs: list_v,Ys: list_v] :
( ( P @ nil_v @ nil_v )
=> ( ! [X3: v,Xs2: list_v] : ( P @ ( cons_v @ X3 @ Xs2 ) @ nil_v )
=> ( ! [Y2: v,Ys2: list_v] : ( P @ nil_v @ ( cons_v @ Y2 @ Ys2 ) )
=> ( ! [X3: v,Xs2: list_v,Y2: v,Ys2: list_v] :
( ( P @ Xs2 @ Ys2 )
=> ( P @ ( cons_v @ X3 @ Xs2 ) @ ( cons_v @ Y2 @ Ys2 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_1203_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_1204_remdups__adj_Ocases,axiom,
! [X: list_v] :
( ( X != nil_v )
=> ( ! [X3: v] :
( X
!= ( cons_v @ X3 @ nil_v ) )
=> ~ ! [X3: v,Y2: v,Xs2: list_v] :
( X
!= ( cons_v @ X3 @ ( cons_v @ Y2 @ Xs2 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_1205_list_Oexhaust,axiom,
! [Y: list_v] :
( ( Y != nil_v )
=> ~ ! [X212: v,X223: list_v] :
( Y
!= ( cons_v @ X212 @ X223 ) ) ) ).
% list.exhaust
thf(fact_1206_list_OdiscI,axiom,
! [List: list_v,X21: v,X222: list_v] :
( ( List
= ( cons_v @ X21 @ X222 ) )
=> ( List != nil_v ) ) ).
% list.discI
thf(fact_1207_list_Odistinct_I1_J,axiom,
! [X21: v,X222: list_v] :
( nil_v
!= ( cons_v @ X21 @ X222 ) ) ).
% list.distinct(1)
thf(fact_1208_list_Osel_I2_J,axiom,
( ( tl_v @ nil_v )
= nil_v ) ).
% list.sel(2)
thf(fact_1209_distrib__sup__le,axiom,
! [X: set_set_v,Y: set_set_v,Z: set_set_v] : ( ord_le5216385588623774835_set_v @ ( sup_sup_set_set_v @ X @ ( inf_inf_set_set_v @ Y @ Z ) ) @ ( inf_inf_set_set_v @ ( sup_sup_set_set_v @ X @ Y ) @ ( sup_sup_set_set_v @ X @ Z ) ) ) ).
% distrib_sup_le
thf(fact_1210_distrib__sup__le,axiom,
! [X: product_unit,Y: product_unit,Z: product_unit] : ( ord_le3221252021190050221t_unit @ ( sup_sup_Product_unit @ X @ ( inf_inf_Product_unit @ Y @ Z ) ) @ ( inf_inf_Product_unit @ ( sup_sup_Product_unit @ X @ Y ) @ ( sup_sup_Product_unit @ X @ Z ) ) ) ).
% distrib_sup_le
thf(fact_1211_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_1212_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_1213_distrib__inf__le,axiom,
! [X: set_set_v,Y: set_set_v,Z: set_set_v] : ( ord_le5216385588623774835_set_v @ ( sup_sup_set_set_v @ ( inf_inf_set_set_v @ X @ Y ) @ ( inf_inf_set_set_v @ X @ Z ) ) @ ( inf_inf_set_set_v @ X @ ( sup_sup_set_set_v @ Y @ Z ) ) ) ).
% distrib_inf_le
thf(fact_1214_distrib__inf__le,axiom,
! [X: product_unit,Y: product_unit,Z: product_unit] : ( ord_le3221252021190050221t_unit @ ( sup_sup_Product_unit @ ( inf_inf_Product_unit @ X @ Y ) @ ( inf_inf_Product_unit @ X @ Z ) ) @ ( inf_inf_Product_unit @ X @ ( sup_sup_Product_unit @ Y @ Z ) ) ) ).
% distrib_inf_le
thf(fact_1215_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_1216_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_1217_Un__Int__assoc__eq,axiom,
! [A: set_set_v,B: set_set_v,C2: set_set_v] :
( ( ( sup_sup_set_set_v @ ( inf_inf_set_set_v @ A @ B ) @ C2 )
= ( inf_inf_set_set_v @ A @ ( sup_sup_set_set_v @ B @ C2 ) ) )
= ( ord_le5216385588623774835_set_v @ C2 @ A ) ) ).
% Un_Int_assoc_eq
thf(fact_1218_Un__Int__assoc__eq,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v,C2: set_Product_prod_v_v] :
( ( ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ A @ B ) @ C2 )
= ( inf_in6271465464967711157od_v_v @ A @ ( sup_su414716646722978715od_v_v @ B @ C2 ) ) )
= ( ord_le7336532860387713383od_v_v @ C2 @ A ) ) ).
% Un_Int_assoc_eq
thf(fact_1219_Un__Int__assoc__eq,axiom,
! [A: set_v,B: set_v,C2: set_v] :
( ( ( sup_sup_set_v @ ( inf_inf_set_v @ A @ B ) @ C2 )
= ( inf_inf_set_v @ A @ ( sup_sup_set_v @ B @ C2 ) ) )
= ( ord_less_eq_set_v @ C2 @ A ) ) ).
% Un_Int_assoc_eq
thf(fact_1220_Diff__triv,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ( inf_in6271465464967711157od_v_v @ A @ B )
= bot_bo723834152578015283od_v_v )
=> ( ( minus_4183494784930505774od_v_v @ A @ B )
= A ) ) ).
% Diff_triv
thf(fact_1221_Diff__triv,axiom,
! [A: set_set_v,B: set_set_v] :
( ( ( inf_inf_set_set_v @ A @ B )
= bot_bot_set_set_v )
=> ( ( minus_7228012346218142266_set_v @ A @ B )
= A ) ) ).
% Diff_triv
thf(fact_1222_Diff__triv,axiom,
! [A: set_v,B: set_v] :
( ( ( inf_inf_set_v @ A @ B )
= bot_bot_set_v )
=> ( ( minus_minus_set_v @ A @ B )
= A ) ) ).
% Diff_triv
thf(fact_1223_Int__Diff__disjoint,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ ( inf_in6271465464967711157od_v_v @ A @ B ) @ ( minus_4183494784930505774od_v_v @ A @ B ) )
= bot_bo723834152578015283od_v_v ) ).
% Int_Diff_disjoint
thf(fact_1224_Int__Diff__disjoint,axiom,
! [A: set_set_v,B: set_set_v] :
( ( inf_inf_set_set_v @ ( inf_inf_set_set_v @ A @ B ) @ ( minus_7228012346218142266_set_v @ A @ B ) )
= bot_bot_set_set_v ) ).
% Int_Diff_disjoint
thf(fact_1225_Int__Diff__disjoint,axiom,
! [A: set_v,B: set_v] :
( ( inf_inf_set_v @ ( inf_inf_set_v @ A @ B ) @ ( minus_minus_set_v @ A @ B ) )
= bot_bot_set_v ) ).
% Int_Diff_disjoint
thf(fact_1226_Diff__Un,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v,C2: set_Product_prod_v_v] :
( ( minus_4183494784930505774od_v_v @ A @ ( sup_su414716646722978715od_v_v @ B @ C2 ) )
= ( inf_in6271465464967711157od_v_v @ ( minus_4183494784930505774od_v_v @ A @ B ) @ ( minus_4183494784930505774od_v_v @ A @ C2 ) ) ) ).
% Diff_Un
thf(fact_1227_Diff__Un,axiom,
! [A: set_set_v,B: set_set_v,C2: set_set_v] :
( ( minus_7228012346218142266_set_v @ A @ ( sup_sup_set_set_v @ B @ C2 ) )
= ( inf_inf_set_set_v @ ( minus_7228012346218142266_set_v @ A @ B ) @ ( minus_7228012346218142266_set_v @ A @ C2 ) ) ) ).
% Diff_Un
thf(fact_1228_Diff__Un,axiom,
! [A: set_v,B: set_v,C2: set_v] :
( ( minus_minus_set_v @ A @ ( sup_sup_set_v @ B @ C2 ) )
= ( inf_inf_set_v @ ( minus_minus_set_v @ A @ B ) @ ( minus_minus_set_v @ A @ C2 ) ) ) ).
% Diff_Un
thf(fact_1229_Diff__Int,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v,C2: set_Product_prod_v_v] :
( ( minus_4183494784930505774od_v_v @ A @ ( inf_in6271465464967711157od_v_v @ B @ C2 ) )
= ( sup_su414716646722978715od_v_v @ ( minus_4183494784930505774od_v_v @ A @ B ) @ ( minus_4183494784930505774od_v_v @ A @ C2 ) ) ) ).
% Diff_Int
thf(fact_1230_Diff__Int,axiom,
! [A: set_set_v,B: set_set_v,C2: set_set_v] :
( ( minus_7228012346218142266_set_v @ A @ ( inf_inf_set_set_v @ B @ C2 ) )
= ( sup_sup_set_set_v @ ( minus_7228012346218142266_set_v @ A @ B ) @ ( minus_7228012346218142266_set_v @ A @ C2 ) ) ) ).
% Diff_Int
thf(fact_1231_Diff__Int,axiom,
! [A: set_v,B: set_v,C2: set_v] :
( ( minus_minus_set_v @ A @ ( inf_inf_set_v @ B @ C2 ) )
= ( sup_sup_set_v @ ( minus_minus_set_v @ A @ B ) @ ( minus_minus_set_v @ A @ C2 ) ) ) ).
% Diff_Int
thf(fact_1232_Int__Diff__Un,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ A @ B ) @ ( minus_4183494784930505774od_v_v @ A @ B ) )
= A ) ).
% Int_Diff_Un
thf(fact_1233_Int__Diff__Un,axiom,
! [A: set_set_v,B: set_set_v] :
( ( sup_sup_set_set_v @ ( inf_inf_set_set_v @ A @ B ) @ ( minus_7228012346218142266_set_v @ A @ B ) )
= A ) ).
% Int_Diff_Un
thf(fact_1234_Int__Diff__Un,axiom,
! [A: set_v,B: set_v] :
( ( sup_sup_set_v @ ( inf_inf_set_v @ A @ B ) @ ( minus_minus_set_v @ A @ B ) )
= A ) ).
% Int_Diff_Un
thf(fact_1235_Un__Diff__Int,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ ( minus_4183494784930505774od_v_v @ A @ B ) @ ( inf_in6271465464967711157od_v_v @ A @ B ) )
= A ) ).
% Un_Diff_Int
thf(fact_1236_Un__Diff__Int,axiom,
! [A: set_set_v,B: set_set_v] :
( ( sup_sup_set_set_v @ ( minus_7228012346218142266_set_v @ A @ B ) @ ( inf_inf_set_set_v @ A @ B ) )
= A ) ).
% Un_Diff_Int
thf(fact_1237_Un__Diff__Int,axiom,
! [A: set_v,B: set_v] :
( ( sup_sup_set_v @ ( minus_minus_set_v @ A @ B ) @ ( inf_inf_set_v @ A @ B ) )
= A ) ).
% Un_Diff_Int
thf(fact_1238_empty__set,axiom,
( bot_bo723834152578015283od_v_v
= ( set_Product_prod_v_v2 @ nil_Product_prod_v_v ) ) ).
% empty_set
thf(fact_1239_empty__set,axiom,
( bot_bot_set_set_v
= ( set_set_v2 @ nil_set_v ) ) ).
% empty_set
thf(fact_1240_empty__set,axiom,
( bot_bot_set_v
= ( set_v2 @ nil_v ) ) ).
% empty_set
thf(fact_1241_Nil__tl,axiom,
! [Xs: list_v] :
( ( nil_v
= ( tl_v @ Xs ) )
= ( ( Xs = nil_v )
| ? [X2: v] :
( Xs
= ( cons_v @ X2 @ nil_v ) ) ) ) ).
% Nil_tl
thf(fact_1242_tl__Nil,axiom,
! [Xs: list_v] :
( ( ( tl_v @ Xs )
= nil_v )
= ( ( Xs = nil_v )
| ? [X2: v] :
( Xs
= ( cons_v @ X2 @ nil_v ) ) ) ) ).
% tl_Nil
thf(fact_1243_list_Oset__sel_I2_J,axiom,
! [A4: list_P7986770385144383213od_v_v,X: product_prod_v_v] :
( ( A4 != nil_Product_prod_v_v )
=> ( ( member7453568604450474000od_v_v @ X @ ( set_Product_prod_v_v2 @ ( tl_Product_prod_v_v @ A4 ) ) )
=> ( member7453568604450474000od_v_v @ X @ ( set_Product_prod_v_v2 @ A4 ) ) ) ) ).
% list.set_sel(2)
thf(fact_1244_list_Oset__sel_I2_J,axiom,
! [A4: list_v,X: v] :
( ( A4 != nil_v )
=> ( ( member_v @ X @ ( set_v2 @ ( tl_v @ A4 ) ) )
=> ( member_v @ X @ ( set_v2 @ A4 ) ) ) ) ).
% list.set_sel(2)
thf(fact_1245_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_1246_hd__in__set,axiom,
! [Xs: list_v] :
( ( Xs != nil_v )
=> ( member_v @ ( hd_v @ Xs ) @ ( set_v2 @ Xs ) ) ) ).
% hd_in_set
thf(fact_1247_list_Oset__sel_I1_J,axiom,
! [A4: list_P7986770385144383213od_v_v] :
( ( A4 != nil_Product_prod_v_v )
=> ( member7453568604450474000od_v_v @ ( hd_Product_prod_v_v @ A4 ) @ ( set_Product_prod_v_v2 @ A4 ) ) ) ).
% list.set_sel(1)
thf(fact_1248_list_Oset__sel_I1_J,axiom,
! [A4: list_v] :
( ( A4 != nil_v )
=> ( member_v @ ( hd_v @ A4 ) @ ( set_v2 @ A4 ) ) ) ).
% list.set_sel(1)
thf(fact_1249_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_1250_graph_Oscc__partition,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,S: set_Product_prod_v_v,S3: set_Product_prod_v_v,X: product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( sCC_Bl6242042402218619277od_v_v @ Successors @ S )
=> ( ( sCC_Bl6242042402218619277od_v_v @ Successors @ S3 )
=> ( ( member7453568604450474000od_v_v @ X @ ( inf_in6271465464967711157od_v_v @ S @ S3 ) )
=> ( S = S3 ) ) ) ) ) ).
% graph.scc_partition
thf(fact_1251_graph_Oscc__partition,axiom,
! [Vertices: set_v,Successors: v > set_v,S: set_v,S3: set_v,X: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bloemen_is_scc_v @ Successors @ S )
=> ( ( sCC_Bloemen_is_scc_v @ Successors @ S3 )
=> ( ( member_v @ X @ ( inf_inf_set_v @ S @ S3 ) )
=> ( S = S3 ) ) ) ) ) ).
% graph.scc_partition
thf(fact_1252_unfold__congs_I5_J,axiom,
! [R: sCC_Bl1394983891496994913t_unit,R2: sCC_Bl1394983891496994913t_unit,V: v > set_v,F: ( v > set_v ) > v > set_v,F2: ( v > set_v ) > v > set_v] :
( ( R = R2 )
=> ( ( ( sCC_Bl3795065053823578884t_unit @ R2 )
= V )
=> ( ! [V2: v > set_v] :
( ( V2 = V )
=> ( ( F @ V2 )
= ( F2 @ V2 ) ) )
=> ( ( sCC_Bl48393358579903213t_unit @ F @ R )
= ( sCC_Bl48393358579903213t_unit @ F2 @ R2 ) ) ) ) ) ).
% unfold_congs(5)
thf(fact_1253_fold__congs_I5_J,axiom,
! [R: sCC_Bl1394983891496994913t_unit,R2: sCC_Bl1394983891496994913t_unit,V: v > set_v,F: ( v > set_v ) > v > set_v,F2: ( v > set_v ) > v > set_v] :
( ( R = R2 )
=> ( ( ( sCC_Bl3795065053823578884t_unit @ R2 )
= V )
=> ( ! [V2: v > set_v] :
( ( V = V2 )
=> ( ( F @ V2 )
= ( F2 @ V2 ) ) )
=> ( ( sCC_Bl48393358579903213t_unit @ F @ R )
= ( sCC_Bl48393358579903213t_unit @ F2 @ R2 ) ) ) ) ) ).
% fold_congs(5)
thf(fact_1254_list_Oexhaust__sel,axiom,
! [List: list_v] :
( ( List != nil_v )
=> ( List
= ( cons_v @ ( hd_v @ List ) @ ( tl_v @ List ) ) ) ) ).
% list.exhaust_sel
thf(fact_1255_the__elem__set,axiom,
! [X: v] :
( ( the_elem_v @ ( set_v2 @ ( cons_v @ X @ nil_v ) ) )
= X ) ).
% the_elem_set
thf(fact_1256_graph_Ounite__wf__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_Bl7798947040364291444t_unit @ Successors @ ( sCC_Bl4702006153222411093od_v_v @ V3 @ W @ E ) ) ) ) ) ) ) ) ).
% graph.unite_wf_env
thf(fact_1257_graph_Ounite__wf__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_Bl9196236973127232072t_unit @ Successors @ ( sCC_Bloemen_unite_v @ V3 @ W @ E ) ) ) ) ) ) ) ) ).
% graph.unite_wf_env
thf(fact_1258_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_1259_pre__dfss__post__dfs__pre__dfss,axiom,
! [V3: v,E: sCC_Bl1394983891496994913t_unit,W: v] :
( ( sCC_Bl1748261141445803503t_unit @ successors @ V3 @ E )
=> ( ( member_v @ W @ ( successors @ V3 ) )
=> ( ~ ( member_v @ W @ ( sCC_Bl4645233313691564917t_unit @ E ) )
=> ( ( sCC_Bl8953792750115413617t_unit @ successors @ W @ E @ ( sCC_Bloemen_dfs_v @ successors @ W @ E ) )
=> ( sCC_Bl1748261141445803503t_unit @ successors @ V3
@ ( sCC_Bl48393358579903213t_unit
@ ^ [Uu: v > set_v,X2: v] : ( if_set_v @ ( X2 = V3 ) @ ( sup_sup_set_v @ ( sCC_Bl3795065053823578884t_unit @ ( sCC_Bloemen_dfs_v @ successors @ W @ E ) @ V3 ) @ ( insert_v @ W @ bot_bot_set_v ) ) @ ( sCC_Bl3795065053823578884t_unit @ ( sCC_Bloemen_dfs_v @ successors @ W @ E ) @ X2 ) )
@ ( sCC_Bloemen_dfs_v @ successors @ W @ E ) ) ) ) ) ) ) ).
% pre_dfss_post_dfs_pre_dfss
thf(fact_1260_calculation_I8_J,axiom,
( ( comple2307003700295860064_set_v
@ ( sCC_Bl2536197123907397897t_unit
@ ( sCC_Bl7876664385711583351t_unit
@ ^ [Uu: list_v] : ( tl_v @ ( sCC_Bl9201514103433284750t_unit @ e2 ) )
@ ( sCC_Bl349061681862590396t_unit
@ ^ [Uu: list_v] : ( tl_v @ ( sCC_Bl8828226123343373779t_unit @ e2 ) )
@ ( sCC_Bl2708505634401380163t_unit
@ ^ [Uu: set_v] : ( sup_sup_set_v @ ( sCC_Bl157864678168468314t_unit @ e2 ) @ ( sCC_Bl1280885523602775798t_unit @ e2 @ v2 ) )
@ ( sCC_Bl6816368539212994290t_unit
@ ^ [Uu: set_set_v] : ( sup_sup_set_set_v @ ( sCC_Bl2536197123907397897t_unit @ e2 ) @ ( insert_set_v @ ( sCC_Bl1280885523602775798t_unit @ e2 @ v2 ) @ bot_bot_set_set_v ) )
@ e2 ) ) ) ) ) )
= ( sCC_Bl157864678168468314t_unit
@ ( sCC_Bl7876664385711583351t_unit
@ ^ [Uu: list_v] : ( tl_v @ ( sCC_Bl9201514103433284750t_unit @ e2 ) )
@ ( sCC_Bl349061681862590396t_unit
@ ^ [Uu: list_v] : ( tl_v @ ( sCC_Bl8828226123343373779t_unit @ e2 ) )
@ ( sCC_Bl2708505634401380163t_unit
@ ^ [Uu: set_v] : ( sup_sup_set_v @ ( sCC_Bl157864678168468314t_unit @ e2 ) @ ( sCC_Bl1280885523602775798t_unit @ e2 @ v2 ) )
@ ( sCC_Bl6816368539212994290t_unit
@ ^ [Uu: set_set_v] : ( sup_sup_set_set_v @ ( sCC_Bl2536197123907397897t_unit @ e2 ) @ ( insert_set_v @ ( sCC_Bl1280885523602775798t_unit @ e2 @ v2 ) @ bot_bot_set_set_v ) )
@ e2 ) ) ) ) ) ) ).
% calculation(8)
thf(fact_1261_calculation_I18_J,axiom,
( ( comple2307003700295860064_set_v
@ ( collect_set_v
@ ^ [Uu: set_v] :
? [N4: v] :
( ( Uu
= ( sCC_Bl1280885523602775798t_unit
@ ( sCC_Bl7876664385711583351t_unit
@ ^ [Uv: list_v] : ( tl_v @ ( sCC_Bl9201514103433284750t_unit @ e2 ) )
@ ( sCC_Bl349061681862590396t_unit
@ ^ [Uv: list_v] : ( tl_v @ ( sCC_Bl8828226123343373779t_unit @ e2 ) )
@ ( sCC_Bl2708505634401380163t_unit
@ ^ [Uv: set_v] : ( sup_sup_set_v @ ( sCC_Bl157864678168468314t_unit @ e2 ) @ ( sCC_Bl1280885523602775798t_unit @ e2 @ v2 ) )
@ ( sCC_Bl6816368539212994290t_unit
@ ^ [Uv: set_set_v] : ( sup_sup_set_set_v @ ( sCC_Bl2536197123907397897t_unit @ e2 ) @ ( insert_set_v @ ( sCC_Bl1280885523602775798t_unit @ e2 @ v2 ) @ bot_bot_set_set_v ) )
@ e2 ) ) ) )
@ N4 ) )
& ( member_v @ N4
@ ( set_v2
@ ( sCC_Bl8828226123343373779t_unit
@ ( sCC_Bl7876664385711583351t_unit
@ ^ [Uv: list_v] : ( tl_v @ ( sCC_Bl9201514103433284750t_unit @ e2 ) )
@ ( sCC_Bl349061681862590396t_unit
@ ^ [Uv: list_v] : ( tl_v @ ( sCC_Bl8828226123343373779t_unit @ e2 ) )
@ ( sCC_Bl2708505634401380163t_unit
@ ^ [Uv: set_v] : ( sup_sup_set_v @ ( sCC_Bl157864678168468314t_unit @ e2 ) @ ( sCC_Bl1280885523602775798t_unit @ e2 @ v2 ) )
@ ( sCC_Bl6816368539212994290t_unit
@ ^ [Uv: set_set_v] : ( sup_sup_set_set_v @ ( sCC_Bl2536197123907397897t_unit @ e2 ) @ ( insert_set_v @ ( sCC_Bl1280885523602775798t_unit @ e2 @ v2 ) @ bot_bot_set_set_v ) )
@ e2 ) ) ) ) ) ) ) ) ) )
= ( minus_minus_set_v
@ ( sCC_Bl4645233313691564917t_unit
@ ( sCC_Bl7876664385711583351t_unit
@ ^ [Uu: list_v] : ( tl_v @ ( sCC_Bl9201514103433284750t_unit @ e2 ) )
@ ( sCC_Bl349061681862590396t_unit
@ ^ [Uu: list_v] : ( tl_v @ ( sCC_Bl8828226123343373779t_unit @ e2 ) )
@ ( sCC_Bl2708505634401380163t_unit
@ ^ [Uu: set_v] : ( sup_sup_set_v @ ( sCC_Bl157864678168468314t_unit @ e2 ) @ ( sCC_Bl1280885523602775798t_unit @ e2 @ v2 ) )
@ ( sCC_Bl6816368539212994290t_unit
@ ^ [Uu: set_set_v] : ( sup_sup_set_set_v @ ( sCC_Bl2536197123907397897t_unit @ e2 ) @ ( insert_set_v @ ( sCC_Bl1280885523602775798t_unit @ e2 @ v2 ) @ bot_bot_set_set_v ) )
@ e2 ) ) ) ) )
@ ( sCC_Bl157864678168468314t_unit
@ ( sCC_Bl7876664385711583351t_unit
@ ^ [Uu: list_v] : ( tl_v @ ( sCC_Bl9201514103433284750t_unit @ e2 ) )
@ ( sCC_Bl349061681862590396t_unit
@ ^ [Uu: list_v] : ( tl_v @ ( sCC_Bl8828226123343373779t_unit @ e2 ) )
@ ( sCC_Bl2708505634401380163t_unit
@ ^ [Uu: set_v] : ( sup_sup_set_v @ ( sCC_Bl157864678168468314t_unit @ e2 ) @ ( sCC_Bl1280885523602775798t_unit @ e2 @ v2 ) )
@ ( sCC_Bl6816368539212994290t_unit
@ ^ [Uu: set_set_v] : ( sup_sup_set_set_v @ ( sCC_Bl2536197123907397897t_unit @ e2 ) @ ( insert_set_v @ ( sCC_Bl1280885523602775798t_unit @ e2 @ v2 ) @ bot_bot_set_set_v ) )
@ e2 ) ) ) ) ) ) ) ).
% calculation(18)
thf(fact_1262_calculation_I2_J,axiom,
( distinct_v
@ ( sCC_Bl8828226123343373779t_unit
@ ( sCC_Bl7876664385711583351t_unit
@ ^ [Uu: list_v] : ( tl_v @ ( sCC_Bl9201514103433284750t_unit @ e2 ) )
@ ( sCC_Bl349061681862590396t_unit
@ ^ [Uu: list_v] : ( tl_v @ ( sCC_Bl8828226123343373779t_unit @ e2 ) )
@ ( sCC_Bl2708505634401380163t_unit
@ ^ [Uu: set_v] : ( sup_sup_set_v @ ( sCC_Bl157864678168468314t_unit @ e2 ) @ ( sCC_Bl1280885523602775798t_unit @ e2 @ v2 ) )
@ ( sCC_Bl6816368539212994290t_unit
@ ^ [Uu: set_set_v] : ( sup_sup_set_set_v @ ( sCC_Bl2536197123907397897t_unit @ e2 ) @ ( insert_set_v @ ( sCC_Bl1280885523602775798t_unit @ e2 @ v2 ) @ bot_bot_set_set_v ) )
@ e2 ) ) ) ) ) ) ).
% calculation(2)
thf(fact_1263_calculation_I9_J,axiom,
( distinct_v
@ ( sCC_Bl9201514103433284750t_unit
@ ( sCC_Bl7876664385711583351t_unit
@ ^ [Uu: list_v] : ( tl_v @ ( sCC_Bl9201514103433284750t_unit @ e2 ) )
@ ( sCC_Bl349061681862590396t_unit
@ ^ [Uu: list_v] : ( tl_v @ ( sCC_Bl8828226123343373779t_unit @ e2 ) )
@ ( sCC_Bl2708505634401380163t_unit
@ ^ [Uu: set_v] : ( sup_sup_set_v @ ( sCC_Bl157864678168468314t_unit @ e2 ) @ ( sCC_Bl1280885523602775798t_unit @ e2 @ v2 ) )
@ ( sCC_Bl6816368539212994290t_unit
@ ^ [Uu: set_set_v] : ( sup_sup_set_set_v @ ( sCC_Bl2536197123907397897t_unit @ e2 ) @ ( insert_set_v @ ( sCC_Bl1280885523602775798t_unit @ e2 @ v2 ) @ bot_bot_set_set_v ) )
@ e2 ) ) ) ) ) ) ).
% calculation(9)
thf(fact_1264_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_1265_dfs_Opsimps,axiom,
! [V3: v,E: sCC_Bl1394983891496994913t_unit] :
( ( accp_S2303753412255344476t_unit @ ( sCC_Bl907557413677168252_rel_v @ successors ) @ ( sum_In526841707622398774t_unit @ ( produc3862955338007567901t_unit @ V3 @ E ) ) )
=> ( ( sCC_Bloemen_dfs_v @ successors @ V3 @ E )
= ( if_SCC4926449794303880475t_unit
@ ( V3
= ( hd_v
@ ( sCC_Bl8828226123343373779t_unit
@ ( sCC_Bloemen_dfss_v @ successors @ V3
@ ( sCC_Bl7876664385711583351t_unit
@ ^ [Uu: list_v] : ( cons_v @ V3 @ ( sCC_Bl9201514103433284750t_unit @ E ) )
@ ( sCC_Bl349061681862590396t_unit
@ ^ [Uu: list_v] : ( cons_v @ V3 @ ( sCC_Bl8828226123343373779t_unit @ E ) )
@ ( sCC_Bl7870604408699998558t_unit
@ ^ [Uu: set_v] : ( sup_sup_set_v @ ( sCC_Bl4645233313691564917t_unit @ E ) @ ( insert_v @ V3 @ bot_bot_set_v ) )
@ E ) ) ) ) ) ) )
@ ( sCC_Bl7876664385711583351t_unit
@ ^ [Uu: list_v] :
( tl_v
@ ( sCC_Bl9201514103433284750t_unit
@ ( sCC_Bloemen_dfss_v @ successors @ V3
@ ( sCC_Bl7876664385711583351t_unit
@ ^ [Uv: list_v] : ( cons_v @ V3 @ ( sCC_Bl9201514103433284750t_unit @ E ) )
@ ( sCC_Bl349061681862590396t_unit
@ ^ [Uv: list_v] : ( cons_v @ V3 @ ( sCC_Bl8828226123343373779t_unit @ E ) )
@ ( sCC_Bl7870604408699998558t_unit
@ ^ [Uv: set_v] : ( sup_sup_set_v @ ( sCC_Bl4645233313691564917t_unit @ E ) @ ( insert_v @ V3 @ bot_bot_set_v ) )
@ E ) ) ) ) ) )
@ ( sCC_Bl349061681862590396t_unit
@ ^ [Uu: list_v] :
( tl_v
@ ( sCC_Bl8828226123343373779t_unit
@ ( sCC_Bloemen_dfss_v @ successors @ V3
@ ( sCC_Bl7876664385711583351t_unit
@ ^ [Uv: list_v] : ( cons_v @ V3 @ ( sCC_Bl9201514103433284750t_unit @ E ) )
@ ( sCC_Bl349061681862590396t_unit
@ ^ [Uv: list_v] : ( cons_v @ V3 @ ( sCC_Bl8828226123343373779t_unit @ E ) )
@ ( sCC_Bl7870604408699998558t_unit
@ ^ [Uv: set_v] : ( sup_sup_set_v @ ( sCC_Bl4645233313691564917t_unit @ E ) @ ( insert_v @ V3 @ bot_bot_set_v ) )
@ E ) ) ) ) ) )
@ ( sCC_Bl2708505634401380163t_unit
@ ^ [Uu: set_v] :
( sup_sup_set_v
@ ( sCC_Bl157864678168468314t_unit
@ ( sCC_Bloemen_dfss_v @ successors @ V3
@ ( sCC_Bl7876664385711583351t_unit
@ ^ [Uv: list_v] : ( cons_v @ V3 @ ( sCC_Bl9201514103433284750t_unit @ E ) )
@ ( sCC_Bl349061681862590396t_unit
@ ^ [Uv: list_v] : ( cons_v @ V3 @ ( sCC_Bl8828226123343373779t_unit @ E ) )
@ ( sCC_Bl7870604408699998558t_unit
@ ^ [Uv: set_v] : ( sup_sup_set_v @ ( sCC_Bl4645233313691564917t_unit @ E ) @ ( insert_v @ V3 @ bot_bot_set_v ) )
@ E ) ) ) ) )
@ ( sCC_Bl1280885523602775798t_unit
@ ( sCC_Bloemen_dfss_v @ successors @ V3
@ ( sCC_Bl7876664385711583351t_unit
@ ^ [Uv: list_v] : ( cons_v @ V3 @ ( sCC_Bl9201514103433284750t_unit @ E ) )
@ ( sCC_Bl349061681862590396t_unit
@ ^ [Uv: list_v] : ( cons_v @ V3 @ ( sCC_Bl8828226123343373779t_unit @ E ) )
@ ( sCC_Bl7870604408699998558t_unit
@ ^ [Uv: set_v] : ( sup_sup_set_v @ ( sCC_Bl4645233313691564917t_unit @ E ) @ ( insert_v @ V3 @ bot_bot_set_v ) )
@ E ) ) ) )
@ V3 ) )
@ ( sCC_Bl6816368539212994290t_unit
@ ^ [Uu: set_set_v] :
( sup_sup_set_set_v
@ ( sCC_Bl2536197123907397897t_unit
@ ( sCC_Bloemen_dfss_v @ successors @ V3
@ ( sCC_Bl7876664385711583351t_unit
@ ^ [Uv: list_v] : ( cons_v @ V3 @ ( sCC_Bl9201514103433284750t_unit @ E ) )
@ ( sCC_Bl349061681862590396t_unit
@ ^ [Uv: list_v] : ( cons_v @ V3 @ ( sCC_Bl8828226123343373779t_unit @ E ) )
@ ( sCC_Bl7870604408699998558t_unit
@ ^ [Uv: set_v] : ( sup_sup_set_v @ ( sCC_Bl4645233313691564917t_unit @ E ) @ ( insert_v @ V3 @ bot_bot_set_v ) )
@ E ) ) ) ) )
@ ( insert_set_v
@ ( sCC_Bl1280885523602775798t_unit
@ ( sCC_Bloemen_dfss_v @ successors @ V3
@ ( sCC_Bl7876664385711583351t_unit
@ ^ [Uv: list_v] : ( cons_v @ V3 @ ( sCC_Bl9201514103433284750t_unit @ E ) )
@ ( sCC_Bl349061681862590396t_unit
@ ^ [Uv: list_v] : ( cons_v @ V3 @ ( sCC_Bl8828226123343373779t_unit @ E ) )
@ ( sCC_Bl7870604408699998558t_unit
@ ^ [Uv: set_v] : ( sup_sup_set_v @ ( sCC_Bl4645233313691564917t_unit @ E ) @ ( insert_v @ V3 @ bot_bot_set_v ) )
@ E ) ) ) )
@ V3 )
@ bot_bot_set_set_v ) )
@ ( sCC_Bloemen_dfss_v @ successors @ V3
@ ( sCC_Bl7876664385711583351t_unit
@ ^ [Uu: list_v] : ( cons_v @ V3 @ ( sCC_Bl9201514103433284750t_unit @ E ) )
@ ( sCC_Bl349061681862590396t_unit
@ ^ [Uu: list_v] : ( cons_v @ V3 @ ( sCC_Bl8828226123343373779t_unit @ E ) )
@ ( sCC_Bl7870604408699998558t_unit
@ ^ [Uu: set_v] : ( sup_sup_set_v @ ( sCC_Bl4645233313691564917t_unit @ E ) @ ( insert_v @ V3 @ bot_bot_set_v ) )
@ E ) ) ) ) ) ) ) )
@ ( sCC_Bl7876664385711583351t_unit
@ ^ [Uu: list_v] :
( tl_v
@ ( sCC_Bl9201514103433284750t_unit
@ ( sCC_Bloemen_dfss_v @ successors @ V3
@ ( sCC_Bl7876664385711583351t_unit
@ ^ [Uv: list_v] : ( cons_v @ V3 @ ( sCC_Bl9201514103433284750t_unit @ E ) )
@ ( sCC_Bl349061681862590396t_unit
@ ^ [Uv: list_v] : ( cons_v @ V3 @ ( sCC_Bl8828226123343373779t_unit @ E ) )
@ ( sCC_Bl7870604408699998558t_unit
@ ^ [Uv: set_v] : ( sup_sup_set_v @ ( sCC_Bl4645233313691564917t_unit @ E ) @ ( insert_v @ V3 @ bot_bot_set_v ) )
@ E ) ) ) ) ) )
@ ( sCC_Bloemen_dfss_v @ successors @ V3
@ ( sCC_Bl7876664385711583351t_unit
@ ^ [Uu: list_v] : ( cons_v @ V3 @ ( sCC_Bl9201514103433284750t_unit @ E ) )
@ ( sCC_Bl349061681862590396t_unit
@ ^ [Uu: list_v] : ( cons_v @ V3 @ ( sCC_Bl8828226123343373779t_unit @ E ) )
@ ( sCC_Bl7870604408699998558t_unit
@ ^ [Uu: set_v] : ( sup_sup_set_v @ ( sCC_Bl4645233313691564917t_unit @ E ) @ ( insert_v @ V3 @ bot_bot_set_v ) )
@ E ) ) ) ) ) ) ) ) ).
% dfs.psimps
thf(fact_1266__C2_C,axiom,
accp_S2303753412255344476t_unit @ ( sCC_Bl907557413677168252_rel_v @ successors ) @ ( sum_In526841707622398774t_unit @ ( produc3862955338007567901t_unit @ v2 @ e ) ) ).
% "2"
thf(fact_1267_dfs__dfss_Odomintros_I1_J,axiom,
! [V3: v,E: sCC_Bl1394983891496994913t_unit] :
( ( accp_S2303753412255344476t_unit @ ( sCC_Bl907557413677168252_rel_v @ successors )
@ ( sum_In5289330923152326972t_unit
@ ( produc3862955338007567901t_unit @ V3
@ ( sCC_Bl7876664385711583351t_unit
@ ^ [Uu: list_v] : ( cons_v @ V3 @ ( sCC_Bl9201514103433284750t_unit @ E ) )
@ ( sCC_Bl349061681862590396t_unit
@ ^ [Uu: list_v] : ( cons_v @ V3 @ ( sCC_Bl8828226123343373779t_unit @ E ) )
@ ( sCC_Bl7870604408699998558t_unit
@ ^ [Uu: set_v] : ( insert_v @ V3 @ ( sCC_Bl4645233313691564917t_unit @ E ) )
@ E ) ) ) ) ) )
=> ( accp_S2303753412255344476t_unit @ ( sCC_Bl907557413677168252_rel_v @ successors ) @ ( sum_In526841707622398774t_unit @ ( produc3862955338007567901t_unit @ V3 @ E ) ) ) ) ).
% dfs_dfss.domintros(1)
thf(fact_1268_dfss_Opsimps,axiom,
! [V3: v,E: sCC_Bl1394983891496994913t_unit] :
( ( accp_S2303753412255344476t_unit @ ( sCC_Bl907557413677168252_rel_v @ successors ) @ ( sum_In5289330923152326972t_unit @ ( produc3862955338007567901t_unit @ V3 @ E ) ) )
=> ( ( sCC_Bloemen_dfss_v @ successors @ V3 @ E )
= ( if_SCC4926449794303880475t_unit
@ ( ( minus_minus_set_v @ ( successors @ V3 ) @ ( sCC_Bl3795065053823578884t_unit @ E @ V3 ) )
= bot_bot_set_v )
@ E
@ ( sCC_Bloemen_dfss_v @ successors @ V3
@ ( sCC_Bl48393358579903213t_unit
@ ^ [Uu: v > set_v,X2: v] :
( if_set_v @ ( X2 = V3 )
@ ( sup_sup_set_v
@ ( sCC_Bl3795065053823578884t_unit
@ ( if_SCC4926449794303880475t_unit
@ ( member_v
@ ( fChoice_v
@ ^ [Y3: v] : ( member_v @ Y3 @ ( minus_minus_set_v @ ( successors @ V3 ) @ ( sCC_Bl3795065053823578884t_unit @ E @ V3 ) ) ) )
@ ( sCC_Bl157864678168468314t_unit @ E ) )
@ E
@ ( if_SCC4926449794303880475t_unit
@ ~ ( member_v
@ ( fChoice_v
@ ^ [Y3: v] : ( member_v @ Y3 @ ( minus_minus_set_v @ ( successors @ V3 ) @ ( sCC_Bl3795065053823578884t_unit @ E @ V3 ) ) ) )
@ ( sCC_Bl4645233313691564917t_unit @ E ) )
@ ( sCC_Bloemen_dfs_v @ successors
@ ( fChoice_v
@ ^ [Y3: v] : ( member_v @ Y3 @ ( minus_minus_set_v @ ( successors @ V3 ) @ ( sCC_Bl3795065053823578884t_unit @ E @ V3 ) ) ) )
@ E )
@ ( sCC_Bloemen_unite_v @ V3
@ ( fChoice_v
@ ^ [Y3: v] : ( member_v @ Y3 @ ( minus_minus_set_v @ ( successors @ V3 ) @ ( sCC_Bl3795065053823578884t_unit @ E @ V3 ) ) ) )
@ E ) ) )
@ V3 )
@ ( insert_v
@ ( fChoice_v
@ ^ [Y3: v] : ( member_v @ Y3 @ ( minus_minus_set_v @ ( successors @ V3 ) @ ( sCC_Bl3795065053823578884t_unit @ E @ V3 ) ) ) )
@ bot_bot_set_v ) )
@ ( sCC_Bl3795065053823578884t_unit
@ ( if_SCC4926449794303880475t_unit
@ ( member_v
@ ( fChoice_v
@ ^ [Y3: v] : ( member_v @ Y3 @ ( minus_minus_set_v @ ( successors @ V3 ) @ ( sCC_Bl3795065053823578884t_unit @ E @ V3 ) ) ) )
@ ( sCC_Bl157864678168468314t_unit @ E ) )
@ E
@ ( if_SCC4926449794303880475t_unit
@ ~ ( member_v
@ ( fChoice_v
@ ^ [Y3: v] : ( member_v @ Y3 @ ( minus_minus_set_v @ ( successors @ V3 ) @ ( sCC_Bl3795065053823578884t_unit @ E @ V3 ) ) ) )
@ ( sCC_Bl4645233313691564917t_unit @ E ) )
@ ( sCC_Bloemen_dfs_v @ successors
@ ( fChoice_v
@ ^ [Y3: v] : ( member_v @ Y3 @ ( minus_minus_set_v @ ( successors @ V3 ) @ ( sCC_Bl3795065053823578884t_unit @ E @ V3 ) ) ) )
@ E )
@ ( sCC_Bloemen_unite_v @ V3
@ ( fChoice_v
@ ^ [Y3: v] : ( member_v @ Y3 @ ( minus_minus_set_v @ ( successors @ V3 ) @ ( sCC_Bl3795065053823578884t_unit @ E @ V3 ) ) ) )
@ E ) ) )
@ X2 ) )
@ ( if_SCC4926449794303880475t_unit
@ ( member_v
@ ( fChoice_v
@ ^ [X2: v] : ( member_v @ X2 @ ( minus_minus_set_v @ ( successors @ V3 ) @ ( sCC_Bl3795065053823578884t_unit @ E @ V3 ) ) ) )
@ ( sCC_Bl157864678168468314t_unit @ E ) )
@ E
@ ( if_SCC4926449794303880475t_unit
@ ~ ( member_v
@ ( fChoice_v
@ ^ [X2: v] : ( member_v @ X2 @ ( minus_minus_set_v @ ( successors @ V3 ) @ ( sCC_Bl3795065053823578884t_unit @ E @ V3 ) ) ) )
@ ( sCC_Bl4645233313691564917t_unit @ E ) )
@ ( sCC_Bloemen_dfs_v @ successors
@ ( fChoice_v
@ ^ [X2: v] : ( member_v @ X2 @ ( minus_minus_set_v @ ( successors @ V3 ) @ ( sCC_Bl3795065053823578884t_unit @ E @ V3 ) ) ) )
@ E )
@ ( sCC_Bloemen_unite_v @ V3
@ ( fChoice_v
@ ^ [X2: v] : ( member_v @ X2 @ ( minus_minus_set_v @ ( successors @ V3 ) @ ( sCC_Bl3795065053823578884t_unit @ E @ V3 ) ) ) )
@ E ) ) ) ) ) ) ) ) ).
% dfss.psimps
thf(fact_1269_unite__def,axiom,
( sCC_Bloemen_unite_v
= ( ^ [V4: v,W2: v,E8: sCC_Bl1394983891496994913t_unit] :
( sCC_Bl349061681862590396t_unit
@ ^ [Uu: list_v] :
( dropWhile_v
@ ^ [X2: v] :
~ ( member_v @ W2 @ ( sCC_Bl1280885523602775798t_unit @ E8 @ X2 ) )
@ ( sCC_Bl8828226123343373779t_unit @ E8 ) )
@ ( sCC_Bl3155122997657187039t_unit
@ ^ [Uu: v > set_v,X2: v] :
( if_set_v
@ ( member_v @ X2
@ ( comple2307003700295860064_set_v
@ ( collect_set_v
@ ^ [Uv: set_v] :
? [Y3: v] :
( ( Uv
= ( sCC_Bl1280885523602775798t_unit @ E8 @ Y3 ) )
& ( member_v @ Y3
@ ( sup_sup_set_v
@ ( set_v2
@ ( takeWhile_v
@ ^ [Z2: v] :
~ ( member_v @ W2 @ ( sCC_Bl1280885523602775798t_unit @ E8 @ Z2 ) )
@ ( sCC_Bl8828226123343373779t_unit @ E8 ) ) )
@ ( insert_v
@ ( hd_v
@ ( dropWhile_v
@ ^ [Z2: v] :
~ ( member_v @ W2 @ ( sCC_Bl1280885523602775798t_unit @ E8 @ Z2 ) )
@ ( sCC_Bl8828226123343373779t_unit @ E8 ) ) )
@ bot_bot_set_v ) ) ) ) ) ) )
@ ( comple2307003700295860064_set_v
@ ( collect_set_v
@ ^ [Uv: set_v] :
? [Y3: v] :
( ( Uv
= ( sCC_Bl1280885523602775798t_unit @ E8 @ Y3 ) )
& ( member_v @ Y3
@ ( sup_sup_set_v
@ ( set_v2
@ ( takeWhile_v
@ ^ [Z2: v] :
~ ( member_v @ W2 @ ( sCC_Bl1280885523602775798t_unit @ E8 @ Z2 ) )
@ ( sCC_Bl8828226123343373779t_unit @ E8 ) ) )
@ ( insert_v
@ ( hd_v
@ ( dropWhile_v
@ ^ [Z2: v] :
~ ( member_v @ W2 @ ( sCC_Bl1280885523602775798t_unit @ E8 @ Z2 ) )
@ ( sCC_Bl8828226123343373779t_unit @ E8 ) ) )
@ bot_bot_set_v ) ) ) ) ) )
@ ( sCC_Bl1280885523602775798t_unit @ E8 @ X2 ) )
@ E8 ) ) ) ) ).
% unite_def
thf(fact_1270_vfin,axiom,
finite_finite_v @ vertices ).
% vfin
thf(fact_1271_Sup__unit__def,axiom,
( comple4687483117567863418t_unit
= ( ^ [Uu2: set_Product_unit] : product_Unity ) ) ).
% Sup_unit_def
thf(fact_1272_old_Ounit_Oexhaust,axiom,
! [Y: product_unit] : ( Y = product_Unity ) ).
% old.unit.exhaust
thf(fact_1273_inf__unit__def,axiom,
( inf_inf_Product_unit
= ( ^ [Uu2: product_unit,Uv2: product_unit] : product_Unity ) ) ).
% inf_unit_def
thf(fact_1274_bot__unit__def,axiom,
bot_bot_Product_unit = product_Unity ).
% bot_unit_def
thf(fact_1275_sup__unit__def,axiom,
( sup_sup_Product_unit
= ( ^ [Uu2: product_unit,Uv2: product_unit] : product_Unity ) ) ).
% sup_unit_def
thf(fact_1276_default__unit__def,axiom,
defaul566961228789861419t_unit = product_Unity ).
% default_unit_def
% Helper facts (6)
thf(help_fChoice_1_1_fChoice_001tf__v_T,axiom,
! [P: v > $o] :
( ( P @ ( fChoice_v @ P ) )
= ( ? [X6: v] : ( P @ X6 ) ) ) ).
thf(help_If_2_1_If_001t__Set__Oset_Itf__v_J_T,axiom,
! [X: set_v,Y: set_v] :
( ( if_set_v @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Set__Oset_Itf__v_J_T,axiom,
! [X: set_v,Y: set_v] :
( ( if_set_v @ $true @ X @ Y )
= X ) ).
thf(help_If_3_1_If_001t__SCC____Bloemen____Sequential__Oenv__Oenv____ext_Itf__v_Mt__Product____Type__Ounit_J_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__SCC____Bloemen____Sequential__Oenv__Oenv____ext_Itf__v_Mt__Product____Type__Ounit_J_T,axiom,
! [X: sCC_Bl1394983891496994913t_unit,Y: sCC_Bl1394983891496994913t_unit] :
( ( if_SCC4926449794303880475t_unit @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__SCC____Bloemen____Sequential__Oenv__Oenv____ext_Itf__v_Mt__Product____Type__Ounit_J_T,axiom,
! [X: sCC_Bl1394983891496994913t_unit,Y: sCC_Bl1394983891496994913t_unit] :
( ( if_SCC4926449794303880475t_unit @ $true @ X @ Y )
= X ) ).
% Conjectures (1)
thf(conj_0,conjecture,
$false ).
%------------------------------------------------------------------------------