TPTP Problem File: SLH0855^1.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain : Archive of Formal Proofs
% Problem :
% Version : Especial.
% English :
% Refs : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source : [Des23]
% Names : SCC_Bloemen_Sequential/0000_SCC_Bloemen_Sequential/prob_01388_047201__5958958_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 339 ( 107 unt; 84 typ; 0 def)
% Number of atoms : 796 ( 234 equ; 0 cnn)
% Maximal formula atoms : 10 ( 3 avg)
% Number of connectives : 2363 ( 58 ~; 9 |; 78 &;1880 @)
% ( 0 <=>; 338 =>; 0 <=; 0 <~>)
% Maximal formula depth : 18 ( 6 avg)
% Number of types : 11 ( 10 usr)
% Number of type conns : 176 ( 176 >; 0 *; 0 +; 0 <<)
% Number of symbols : 77 ( 74 usr; 15 con; 0-4 aty)
% Number of variables : 713 ( 97 ^; 574 !; 42 ?; 713 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 08:53:29.528
%------------------------------------------------------------------------------
% Could-be-implicit typings (10)
thf(ty_n_t__SCC____Bloemen____Sequential__Oenv__Oenv____ext_Itf__v_Mt__Product____Type__Ounit_J,type,
sCC_Bl1394983891496994913t_unit: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Product____Type__Oprod_Itf__v_Mtf__v_J_J_J,type,
set_se8455005133513928103od_v_v: $tType ).
thf(ty_n_t__List__Olist_It__Product____Type__Oprod_Itf__v_Mtf__v_J_J,type,
list_P7986770385144383213od_v_v: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_Itf__v_Mtf__v_J_J,type,
set_Product_prod_v_v: $tType ).
thf(ty_n_t__Product____Type__Oprod_Itf__v_Mtf__v_J,type,
product_prod_v_v: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_Itf__v_J_J,type,
set_set_v: $tType ).
thf(ty_n_t__Product____Type__Ounit,type,
product_unit: $tType ).
thf(ty_n_t__List__Olist_Itf__v_J,type,
list_v: $tType ).
thf(ty_n_t__Set__Oset_Itf__v_J,type,
set_v: $tType ).
thf(ty_n_tf__v,type,
v: $tType ).
% Explicit typings (74)
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_It__Product____Type__Oprod_Itf__v_Mtf__v_J_J,type,
comple5788137035815166516od_v_v: set_se8455005133513928103od_v_v > set_Product_prod_v_v ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_Itf__v_J,type,
comple2307003700295860064_set_v: set_set_v > set_v ).
thf(sy_c_Finite__Set_Ofinite_001tf__v,type,
finite_finite_v: set_v > $o ).
thf(sy_c_Groups_Ominus__class_Ominus_001_062_It__Product____Type__Oprod_Itf__v_Mtf__v_J_M_Eo_J,type,
minus_9095120230875558447_v_v_o: ( product_prod_v_v > $o ) > ( product_prod_v_v > $o ) > product_prod_v_v > $o ).
thf(sy_c_Groups_Ominus__class_Ominus_001_062_It__Set__Oset_Itf__v_J_M_Eo_J,type,
minus_minus_set_v_o: ( set_v > $o ) > ( set_v > $o ) > set_v > $o ).
thf(sy_c_Groups_Ominus__class_Ominus_001_062_Itf__v_M_Eo_J,type,
minus_minus_v_o: ( v > $o ) > ( v > $o ) > v > $o ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Product____Type__Oprod_Itf__v_Mtf__v_J_J,type,
minus_4183494784930505774od_v_v: set_Product_prod_v_v > set_Product_prod_v_v > set_Product_prod_v_v ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Set__Oset_Itf__v_J_J,type,
minus_7228012346218142266_set_v: set_set_v > set_set_v > set_set_v ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_Itf__v_J,type,
minus_minus_set_v: set_v > set_v > set_v ).
thf(sy_c_If_001t__Set__Oset_Itf__v_J,type,
if_set_v: $o > set_v > set_v > set_v ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Product____Type__Ounit,type,
inf_inf_Product_unit: product_unit > product_unit > product_unit ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_Itf__v_J,type,
inf_inf_set_v: set_v > set_v > set_v ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Product____Type__Ounit,type,
sup_sup_Product_unit: product_unit > product_unit > product_unit ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Product____Type__Oprod_Itf__v_Mtf__v_J_J,type,
sup_su414716646722978715od_v_v: set_Product_prod_v_v > set_Product_prod_v_v > set_Product_prod_v_v ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_Itf__v_J,type,
sup_sup_set_v: set_v > set_v > set_v ).
thf(sy_c_List_Oappend_001tf__v,type,
append_v: list_v > list_v > list_v ).
thf(sy_c_List_Odistinct_001tf__v,type,
distinct_v: list_v > $o ).
thf(sy_c_List_Olist_ONil_001t__Product____Type__Oprod_Itf__v_Mtf__v_J,type,
nil_Product_prod_v_v: list_P7986770385144383213od_v_v ).
thf(sy_c_List_Olist_ONil_001tf__v,type,
nil_v: list_v ).
thf(sy_c_List_Olist_Ohd_001t__Product____Type__Oprod_Itf__v_Mtf__v_J,type,
hd_Product_prod_v_v: list_P7986770385144383213od_v_v > product_prod_v_v ).
thf(sy_c_List_Olist_Ohd_001tf__v,type,
hd_v: list_v > v ).
thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_Itf__v_Mtf__v_J,type,
set_Product_prod_v_v2: list_P7986770385144383213od_v_v > set_Product_prod_v_v ).
thf(sy_c_List_Olist_Oset_001tf__v,type,
set_v2: list_v > set_v ).
thf(sy_c_List_Olist_Otl_001tf__v,type,
tl_v: list_v > list_v ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Product____Type__Ounit,type,
bot_bot_Product_unit: product_unit ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Product____Type__Oprod_Itf__v_Mtf__v_J_J,type,
bot_bo723834152578015283od_v_v: set_Product_prod_v_v ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_Itf__v_J,type,
bot_bot_set_v: set_v ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Product____Type__Oprod_Itf__v_Mtf__v_J_M_Eo_J,type,
ord_le5892402249245633078_v_v_o: ( product_prod_v_v > $o ) > ( product_prod_v_v > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_Itf__v_M_Eo_J,type,
ord_less_eq_v_o: ( v > $o ) > ( v > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_Itf__v_Mtf__v_J_J,type,
ord_le7336532860387713383od_v_v: set_Product_prod_v_v > set_Product_prod_v_v > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Product____Type__Oprod_Itf__v_Mtf__v_J_J_J,type,
ord_le4714265922333009223od_v_v: set_se8455005133513928103od_v_v > set_se8455005133513928103od_v_v > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_Itf__v_J_J,type,
ord_le5216385588623774835_set_v: set_set_v > set_set_v > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_Itf__v_J,type,
ord_less_eq_set_v: set_v > set_v > $o ).
thf(sy_c_Product__Type_OPair_001tf__v_001tf__v,type,
product_Pair_v_v: v > v > product_prod_v_v ).
thf(sy_c_Product__Type_OUnity,type,
product_Unity: product_unit ).
thf(sy_c_SCC__Bloemen__Sequential_Oenv_O_092_060S_062_001tf__v_001t__Product____Type__Ounit,type,
sCC_Bl1280885523602775798t_unit: sCC_Bl1394983891496994913t_unit > v > set_v ).
thf(sy_c_SCC__Bloemen__Sequential_Oenv_Ocstack_001tf__v_001t__Product____Type__Ounit,type,
sCC_Bl9201514103433284750t_unit: sCC_Bl1394983891496994913t_unit > list_v ).
thf(sy_c_SCC__Bloemen__Sequential_Oenv_Oexplored_001tf__v_001t__Product____Type__Ounit,type,
sCC_Bl157864678168468314t_unit: sCC_Bl1394983891496994913t_unit > set_v ).
thf(sy_c_SCC__Bloemen__Sequential_Oenv_Omore_001tf__v_001t__Product____Type__Ounit,type,
sCC_Bl3567736435408124606t_unit: sCC_Bl1394983891496994913t_unit > product_unit ).
thf(sy_c_SCC__Bloemen__Sequential_Oenv_Oroot_001tf__v_001t__Product____Type__Ounit,type,
sCC_Bl1090238580953940555t_unit: sCC_Bl1394983891496994913t_unit > v ).
thf(sy_c_SCC__Bloemen__Sequential_Oenv_Osccs_001tf__v_001t__Product____Type__Ounit,type,
sCC_Bl2536197123907397897t_unit: sCC_Bl1394983891496994913t_unit > set_set_v ).
thf(sy_c_SCC__Bloemen__Sequential_Oenv_Ostack_001tf__v_001t__Product____Type__Ounit,type,
sCC_Bl8828226123343373779t_unit: sCC_Bl1394983891496994913t_unit > list_v ).
thf(sy_c_SCC__Bloemen__Sequential_Oenv_Ovisited_001tf__v_001t__Product____Type__Ounit,type,
sCC_Bl4645233313691564917t_unit: sCC_Bl1394983891496994913t_unit > set_v ).
thf(sy_c_SCC__Bloemen__Sequential_Oenv_Ovsuccs_001tf__v_001t__Product____Type__Ounit,type,
sCC_Bl3795065053823578884t_unit: sCC_Bl1394983891496994913t_unit > v > set_v ).
thf(sy_c_SCC__Bloemen__Sequential_Ograph_001tf__v,type,
sCC_Bloemen_graph_v: set_v > ( v > set_v ) > $o ).
thf(sy_c_SCC__Bloemen__Sequential_Ograph_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_001tf__v,type,
sCC_Bl5398416737448265317bscc_v: ( v > set_v ) > set_v > $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_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_001tf__v,type,
sCC_Bl649662514949026229able_v: ( v > set_v ) > v > v > $o ).
thf(sy_c_SCC__Bloemen__Sequential_Ograph_Oreachable__avoiding_001tf__v,type,
sCC_Bl4291963740693775144ding_v: ( v > set_v ) > v > v > set_Product_prod_v_v > $o ).
thf(sy_c_SCC__Bloemen__Sequential_Ograph_Oreachable__end_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_001tf__v,type,
sCC_Bloemen_unite_v: v > v > sCC_Bl1394983891496994913t_unit > sCC_Bl1394983891496994913t_unit ).
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_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_001tf__v,type,
insert_v: v > set_v > set_v ).
thf(sy_c_member_001t__Product____Type__Oprod_Itf__v_Mtf__v_J,type,
member7453568604450474000od_v_v: product_prod_v_v > set_Product_prod_v_v > $o ).
thf(sy_c_member_001t__Set__Oset_It__Product____Type__Oprod_Itf__v_Mtf__v_J_J,type,
member8406446414694345712od_v_v: set_Product_prod_v_v > set_se8455005133513928103od_v_v > $o ).
thf(sy_c_member_001t__Set__Oset_Itf__v_J,type,
member_set_v: set_v > set_set_v > $o ).
thf(sy_c_member_001tf__v,type,
member_v: v > set_v > $o ).
thf(sy_v_cc____,type,
cc: set_v ).
thf(sy_v_e,type,
e: sCC_Bl1394983891496994913t_unit ).
thf(sy_v_e_H,type,
e2: sCC_Bl1394983891496994913t_unit ).
thf(sy_v_pfx____,type,
pfx: list_v ).
thf(sy_v_successors,type,
successors: v > set_v ).
thf(sy_v_v,type,
v2: v ).
thf(sy_v_vertices,type,
vertices: set_v ).
thf(sy_v_w,type,
w: v ).
% Relevant facts (251)
thf(fact_0_calculation_I15_J,axiom,
! [N: v,M: v] :
( ( member_v @ M @ ( sCC_Bl1280885523602775798t_unit @ e2 @ N ) )
= ( ( sCC_Bl1280885523602775798t_unit @ e2 @ N )
= ( sCC_Bl1280885523602775798t_unit @ e2 @ M ) ) ) ).
% calculation(15)
thf(fact_1_w_I3_J,axiom,
member_v @ w @ ( sCC_Bl4645233313691564917t_unit @ e ) ).
% w(3)
thf(fact_2_w_I4_J,axiom,
~ ( member_v @ w @ ( sCC_Bl157864678168468314t_unit @ e ) ) ).
% w(4)
thf(fact_3__092_060open_062_092_060Union_062_A_123_092_060S_062_Ae_H_An_A_124n_O_An_A_092_060in_062_Aset_A_Istack_Ae_H_J_125_A_061_Avisited_Ae_A_N_Aexplored_Ae_092_060close_062,axiom,
( ( comple2307003700295860064_set_v
@ ( collect_set_v
@ ^ [Uu: set_v] :
? [N2: v] :
( ( Uu
= ( sCC_Bl1280885523602775798t_unit @ e2 @ N2 ) )
& ( member_v @ N2 @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ e2 ) ) ) ) ) )
= ( minus_minus_set_v @ ( sCC_Bl4645233313691564917t_unit @ e ) @ ( sCC_Bl157864678168468314t_unit @ e ) ) ) ).
% \<open>\<Union> {\<S> e' n |n. n \<in> set (stack e')} = visited e - explored e\<close>
thf(fact_4_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_5_pfx_I2_J,axiom,
( ( sCC_Bl8828226123343373779t_unit @ e2 )
!= nil_v ) ).
% pfx(2)
thf(fact_6_calculation_I11_J,axiom,
( ( comple2307003700295860064_set_v @ ( sCC_Bl2536197123907397897t_unit @ e2 ) )
= ( sCC_Bl157864678168468314t_unit @ e2 ) ) ).
% calculation(11)
thf(fact_7_calculation_I3_J,axiom,
ord_less_eq_set_v @ ( sCC_Bl157864678168468314t_unit @ e2 ) @ ( sCC_Bl4645233313691564917t_unit @ e2 ) ).
% calculation(3)
thf(fact_8_cc__Un,axiom,
( cc
= ( comple2307003700295860064_set_v
@ ( collect_set_v
@ ^ [Uu: set_v] :
? [X: v] :
( ( Uu
= ( sCC_Bl1280885523602775798t_unit @ e @ X ) )
& ( member_v @ X @ cc ) ) ) ) ) ).
% cc_Un
thf(fact_9_UN__ball__bex__simps_I3_J,axiom,
! [A: set_set_v,P: v > $o] :
( ( ? [X: v] :
( ( member_v @ X @ ( comple2307003700295860064_set_v @ A ) )
& ( P @ X ) ) )
= ( ? [X: set_v] :
( ( member_set_v @ X @ A )
& ? [Y: v] :
( ( member_v @ Y @ X )
& ( P @ Y ) ) ) ) ) ).
% UN_ball_bex_simps(3)
thf(fact_10_UN__ball__bex__simps_I1_J,axiom,
! [A: set_set_v,P: v > $o] :
( ( ! [X: v] :
( ( member_v @ X @ ( comple2307003700295860064_set_v @ A ) )
=> ( P @ X ) ) )
= ( ! [X: set_v] :
( ( member_set_v @ X @ A )
=> ! [Y: v] :
( ( member_v @ Y @ X )
=> ( P @ Y ) ) ) ) ) ).
% UN_ball_bex_simps(1)
thf(fact_11_UnionI,axiom,
! [X2: set_Product_prod_v_v,C: set_se8455005133513928103od_v_v,A: product_prod_v_v] :
( ( member8406446414694345712od_v_v @ X2 @ C )
=> ( ( member7453568604450474000od_v_v @ A @ X2 )
=> ( member7453568604450474000od_v_v @ A @ ( comple5788137035815166516od_v_v @ C ) ) ) ) ).
% UnionI
thf(fact_12_UnionI,axiom,
! [X2: set_v,C: set_set_v,A: v] :
( ( member_set_v @ X2 @ C )
=> ( ( member_v @ A @ X2 )
=> ( member_v @ A @ ( comple2307003700295860064_set_v @ C ) ) ) ) ).
% UnionI
thf(fact_13_Union__iff,axiom,
! [A: product_prod_v_v,C: set_se8455005133513928103od_v_v] :
( ( member7453568604450474000od_v_v @ A @ ( comple5788137035815166516od_v_v @ C ) )
= ( ? [X: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ X @ C )
& ( member7453568604450474000od_v_v @ A @ X ) ) ) ) ).
% Union_iff
thf(fact_14_Union__iff,axiom,
! [A: v,C: set_set_v] :
( ( member_v @ A @ ( comple2307003700295860064_set_v @ C ) )
= ( ? [X: set_v] :
( ( member_set_v @ X @ C )
& ( member_v @ A @ X ) ) ) ) ).
% Union_iff
thf(fact_15_DiffI,axiom,
! [C2: product_prod_v_v,A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ C2 @ A )
=> ( ~ ( member7453568604450474000od_v_v @ C2 @ B )
=> ( member7453568604450474000od_v_v @ C2 @ ( minus_4183494784930505774od_v_v @ A @ B ) ) ) ) ).
% DiffI
thf(fact_16_DiffI,axiom,
! [C2: v,A: set_v,B: set_v] :
( ( member_v @ C2 @ A )
=> ( ~ ( member_v @ C2 @ B )
=> ( member_v @ C2 @ ( minus_minus_set_v @ A @ B ) ) ) ) ).
% DiffI
thf(fact_17_Diff__iff,axiom,
! [C2: product_prod_v_v,A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ C2 @ ( minus_4183494784930505774od_v_v @ A @ B ) )
= ( ( member7453568604450474000od_v_v @ C2 @ A )
& ~ ( member7453568604450474000od_v_v @ C2 @ B ) ) ) ).
% Diff_iff
thf(fact_18_Diff__iff,axiom,
! [C2: v,A: set_v,B: set_v] :
( ( member_v @ C2 @ ( minus_minus_set_v @ A @ B ) )
= ( ( member_v @ C2 @ A )
& ~ ( member_v @ C2 @ B ) ) ) ).
% Diff_iff
thf(fact_19_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_20_Se_H,axiom,
! [X3: v] :
( ( ( member_v @ X3 @ cc )
=> ( ( sCC_Bl1280885523602775798t_unit @ e2 @ X3 )
= cc ) )
& ( ~ ( member_v @ X3 @ cc )
=> ( ( sCC_Bl1280885523602775798t_unit @ e2 @ X3 )
= ( sCC_Bl1280885523602775798t_unit @ e @ X3 ) ) ) ) ).
% Se'
thf(fact_21_subsetI,axiom,
! [A: set_v,B: set_v] :
( ! [X4: v] :
( ( member_v @ X4 @ A )
=> ( member_v @ X4 @ B ) )
=> ( ord_less_eq_set_v @ A @ B ) ) ).
% subsetI
thf(fact_22_subsetI,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ! [X4: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X4 @ A )
=> ( member7453568604450474000od_v_v @ X4 @ B ) )
=> ( ord_le7336532860387713383od_v_v @ A @ B ) ) ).
% subsetI
thf(fact_23_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_24_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_25_Sup__subset__mono,axiom,
! [A: set_se8455005133513928103od_v_v,B: set_se8455005133513928103od_v_v] :
( ( ord_le4714265922333009223od_v_v @ A @ B )
=> ( ord_le7336532860387713383od_v_v @ ( comple5788137035815166516od_v_v @ A ) @ ( comple5788137035815166516od_v_v @ B ) ) ) ).
% Sup_subset_mono
thf(fact_26_Sup__subset__mono,axiom,
! [A: set_set_v,B: set_set_v] :
( ( ord_le5216385588623774835_set_v @ A @ B )
=> ( ord_less_eq_set_v @ ( comple2307003700295860064_set_v @ A ) @ ( comple2307003700295860064_set_v @ B ) ) ) ).
% Sup_subset_mono
thf(fact_27_in__mono,axiom,
! [A: set_v,B: set_v,X5: v] :
( ( ord_less_eq_set_v @ A @ B )
=> ( ( member_v @ X5 @ A )
=> ( member_v @ X5 @ B ) ) ) ).
% in_mono
thf(fact_28_in__mono,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v,X5: product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ B )
=> ( ( member7453568604450474000od_v_v @ X5 @ A )
=> ( member7453568604450474000od_v_v @ X5 @ B ) ) ) ).
% in_mono
thf(fact_29_subsetD,axiom,
! [A: set_v,B: set_v,C2: v] :
( ( ord_less_eq_set_v @ A @ B )
=> ( ( member_v @ C2 @ A )
=> ( member_v @ C2 @ B ) ) ) ).
% subsetD
thf(fact_30_subsetD,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v,C2: product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ B )
=> ( ( member7453568604450474000od_v_v @ C2 @ A )
=> ( member7453568604450474000od_v_v @ C2 @ B ) ) ) ).
% subsetD
thf(fact_31_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_32_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_33_subset__eq,axiom,
( ord_less_eq_set_v
= ( ^ [A2: set_v,B2: set_v] :
! [X: v] :
( ( member_v @ X @ A2 )
=> ( member_v @ X @ B2 ) ) ) ) ).
% subset_eq
thf(fact_34_subset__eq,axiom,
( ord_le7336532860387713383od_v_v
= ( ^ [A2: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
! [X: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X @ A2 )
=> ( member7453568604450474000od_v_v @ X @ B2 ) ) ) ) ).
% subset_eq
thf(fact_35_equalityD1,axiom,
! [A: set_v,B: set_v] :
( ( A = B )
=> ( ord_less_eq_set_v @ A @ B ) ) ).
% equalityD1
thf(fact_36_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_37_equalityD2,axiom,
! [A: set_v,B: set_v] :
( ( A = B )
=> ( ord_less_eq_set_v @ B @ A ) ) ).
% equalityD2
thf(fact_38_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_39_subset__iff,axiom,
( ord_less_eq_set_v
= ( ^ [A2: set_v,B2: set_v] :
! [T: v] :
( ( member_v @ T @ A2 )
=> ( member_v @ T @ B2 ) ) ) ) ).
% subset_iff
thf(fact_40_subset__iff,axiom,
( ord_le7336532860387713383od_v_v
= ( ^ [A2: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
! [T: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ T @ A2 )
=> ( member7453568604450474000od_v_v @ T @ B2 ) ) ) ) ).
% subset_iff
thf(fact_41_subset__refl,axiom,
! [A: set_v] : ( ord_less_eq_set_v @ A @ A ) ).
% subset_refl
thf(fact_42_subset__refl,axiom,
! [A: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ A @ A ) ).
% subset_refl
thf(fact_43_Collect__mono,axiom,
! [P: set_v > $o,Q: set_v > $o] :
( ! [X4: set_v] :
( ( P @ X4 )
=> ( Q @ X4 ) )
=> ( ord_le5216385588623774835_set_v @ ( collect_set_v @ P ) @ ( collect_set_v @ Q ) ) ) ).
% Collect_mono
thf(fact_44_Collect__mono,axiom,
! [P: v > $o,Q: v > $o] :
( ! [X4: v] :
( ( P @ X4 )
=> ( Q @ X4 ) )
=> ( ord_less_eq_set_v @ ( collect_v @ P ) @ ( collect_v @ Q ) ) ) ).
% Collect_mono
thf(fact_45_Collect__mono,axiom,
! [P: product_prod_v_v > $o,Q: product_prod_v_v > $o] :
( ! [X4: product_prod_v_v] :
( ( P @ X4 )
=> ( Q @ X4 ) )
=> ( ord_le7336532860387713383od_v_v @ ( collec140062887454715474od_v_v @ P ) @ ( collec140062887454715474od_v_v @ Q ) ) ) ).
% Collect_mono
thf(fact_46_subset__trans,axiom,
! [A: set_v,B: set_v,C: set_v] :
( ( ord_less_eq_set_v @ A @ B )
=> ( ( ord_less_eq_set_v @ B @ C )
=> ( ord_less_eq_set_v @ A @ C ) ) ) ).
% subset_trans
thf(fact_47_subset__trans,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ B )
=> ( ( ord_le7336532860387713383od_v_v @ B @ C )
=> ( ord_le7336532860387713383od_v_v @ A @ C ) ) ) ).
% subset_trans
thf(fact_48_set__eq__subset,axiom,
( ( ^ [Y2: set_v,Z: set_v] : ( Y2 = Z ) )
= ( ^ [A2: set_v,B2: set_v] :
( ( ord_less_eq_set_v @ A2 @ B2 )
& ( ord_less_eq_set_v @ B2 @ A2 ) ) ) ) ).
% set_eq_subset
thf(fact_49_set__eq__subset,axiom,
( ( ^ [Y2: set_Product_prod_v_v,Z: set_Product_prod_v_v] : ( Y2 = Z ) )
= ( ^ [A2: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A2 @ B2 )
& ( ord_le7336532860387713383od_v_v @ B2 @ A2 ) ) ) ) ).
% set_eq_subset
thf(fact_50_Collect__subset,axiom,
! [A: set_set_v,P: set_v > $o] :
( ord_le5216385588623774835_set_v
@ ( collect_set_v
@ ^ [X: set_v] :
( ( member_set_v @ X @ A )
& ( P @ X ) ) )
@ A ) ).
% Collect_subset
thf(fact_51_Collect__subset,axiom,
! [A: set_v,P: v > $o] :
( ord_less_eq_set_v
@ ( collect_v
@ ^ [X: v] :
( ( member_v @ X @ A )
& ( P @ X ) ) )
@ A ) ).
% Collect_subset
thf(fact_52_Collect__subset,axiom,
! [A: set_Product_prod_v_v,P: product_prod_v_v > $o] :
( ord_le7336532860387713383od_v_v
@ ( collec140062887454715474od_v_v
@ ^ [X: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X @ A )
& ( P @ X ) ) )
@ A ) ).
% Collect_subset
thf(fact_53_Collect__mono__iff,axiom,
! [P: set_v > $o,Q: set_v > $o] :
( ( ord_le5216385588623774835_set_v @ ( collect_set_v @ P ) @ ( collect_set_v @ Q ) )
= ( ! [X: set_v] :
( ( P @ X )
=> ( Q @ X ) ) ) ) ).
% Collect_mono_iff
thf(fact_54_Collect__mono__iff,axiom,
! [P: v > $o,Q: v > $o] :
( ( ord_less_eq_set_v @ ( collect_v @ P ) @ ( collect_v @ Q ) )
= ( ! [X: v] :
( ( P @ X )
=> ( Q @ X ) ) ) ) ).
% Collect_mono_iff
thf(fact_55_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 ) )
= ( ! [X: product_prod_v_v] :
( ( P @ X )
=> ( Q @ X ) ) ) ) ).
% Collect_mono_iff
thf(fact_56_Union__mono,axiom,
! [A: set_se8455005133513928103od_v_v,B: set_se8455005133513928103od_v_v] :
( ( ord_le4714265922333009223od_v_v @ A @ B )
=> ( ord_le7336532860387713383od_v_v @ ( comple5788137035815166516od_v_v @ A ) @ ( comple5788137035815166516od_v_v @ B ) ) ) ).
% Union_mono
thf(fact_57_Union__mono,axiom,
! [A: set_set_v,B: set_set_v] :
( ( ord_le5216385588623774835_set_v @ A @ B )
=> ( ord_less_eq_set_v @ ( comple2307003700295860064_set_v @ A ) @ ( comple2307003700295860064_set_v @ B ) ) ) ).
% Union_mono
thf(fact_58_Sup__upper2,axiom,
! [U: set_Product_prod_v_v,A: set_se8455005133513928103od_v_v,V: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ U @ A )
=> ( ( ord_le7336532860387713383od_v_v @ V @ U )
=> ( ord_le7336532860387713383od_v_v @ V @ ( comple5788137035815166516od_v_v @ A ) ) ) ) ).
% Sup_upper2
thf(fact_59_Sup__upper2,axiom,
! [U: set_v,A: set_set_v,V: set_v] :
( ( member_set_v @ U @ A )
=> ( ( ord_less_eq_set_v @ V @ U )
=> ( ord_less_eq_set_v @ V @ ( comple2307003700295860064_set_v @ A ) ) ) ) ).
% Sup_upper2
thf(fact_60_Sup__le__iff,axiom,
! [A: set_se8455005133513928103od_v_v,B3: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ ( comple5788137035815166516od_v_v @ A ) @ B3 )
= ( ! [X: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ X @ A )
=> ( ord_le7336532860387713383od_v_v @ X @ B3 ) ) ) ) ).
% Sup_le_iff
thf(fact_61_Sup__le__iff,axiom,
! [A: set_set_v,B3: set_v] :
( ( ord_less_eq_set_v @ ( comple2307003700295860064_set_v @ A ) @ B3 )
= ( ! [X: set_v] :
( ( member_set_v @ X @ A )
=> ( ord_less_eq_set_v @ X @ B3 ) ) ) ) ).
% Sup_le_iff
thf(fact_62_Sup__upper,axiom,
! [X5: set_Product_prod_v_v,A: set_se8455005133513928103od_v_v] :
( ( member8406446414694345712od_v_v @ X5 @ A )
=> ( ord_le7336532860387713383od_v_v @ X5 @ ( comple5788137035815166516od_v_v @ A ) ) ) ).
% Sup_upper
thf(fact_63_Sup__upper,axiom,
! [X5: set_v,A: set_set_v] :
( ( member_set_v @ X5 @ A )
=> ( ord_less_eq_set_v @ X5 @ ( comple2307003700295860064_set_v @ A ) ) ) ).
% Sup_upper
thf(fact_64_Sup__least,axiom,
! [A: set_se8455005133513928103od_v_v,Z2: set_Product_prod_v_v] :
( ! [X4: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ X4 @ A )
=> ( ord_le7336532860387713383od_v_v @ X4 @ Z2 ) )
=> ( ord_le7336532860387713383od_v_v @ ( comple5788137035815166516od_v_v @ A ) @ Z2 ) ) ).
% Sup_least
thf(fact_65_Sup__least,axiom,
! [A: set_set_v,Z2: set_v] :
( ! [X4: set_v] :
( ( member_set_v @ X4 @ A )
=> ( ord_less_eq_set_v @ X4 @ Z2 ) )
=> ( ord_less_eq_set_v @ ( comple2307003700295860064_set_v @ A ) @ Z2 ) ) ).
% Sup_least
thf(fact_66_Sup__mono,axiom,
! [A: set_se8455005133513928103od_v_v,B: set_se8455005133513928103od_v_v] :
( ! [A3: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ A3 @ A )
=> ? [X3: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ X3 @ B )
& ( ord_le7336532860387713383od_v_v @ A3 @ X3 ) ) )
=> ( ord_le7336532860387713383od_v_v @ ( comple5788137035815166516od_v_v @ A ) @ ( comple5788137035815166516od_v_v @ B ) ) ) ).
% Sup_mono
thf(fact_67_Sup__mono,axiom,
! [A: set_set_v,B: set_set_v] :
( ! [A3: set_v] :
( ( member_set_v @ A3 @ A )
=> ? [X3: set_v] :
( ( member_set_v @ X3 @ B )
& ( ord_less_eq_set_v @ A3 @ X3 ) ) )
=> ( ord_less_eq_set_v @ ( comple2307003700295860064_set_v @ A ) @ ( comple2307003700295860064_set_v @ B ) ) ) ).
% Sup_mono
thf(fact_68_Sup__eqI,axiom,
! [A: set_se8455005133513928103od_v_v,X5: set_Product_prod_v_v] :
( ! [Y3: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ Y3 @ A )
=> ( ord_le7336532860387713383od_v_v @ Y3 @ X5 ) )
=> ( ! [Y3: set_Product_prod_v_v] :
( ! [Z3: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ Z3 @ A )
=> ( ord_le7336532860387713383od_v_v @ Z3 @ Y3 ) )
=> ( ord_le7336532860387713383od_v_v @ X5 @ Y3 ) )
=> ( ( comple5788137035815166516od_v_v @ A )
= X5 ) ) ) ).
% Sup_eqI
thf(fact_69_Sup__eqI,axiom,
! [A: set_set_v,X5: set_v] :
( ! [Y3: set_v] :
( ( member_set_v @ Y3 @ A )
=> ( ord_less_eq_set_v @ Y3 @ X5 ) )
=> ( ! [Y3: set_v] :
( ! [Z3: set_v] :
( ( member_set_v @ Z3 @ A )
=> ( ord_less_eq_set_v @ Z3 @ Y3 ) )
=> ( ord_less_eq_set_v @ X5 @ Y3 ) )
=> ( ( comple2307003700295860064_set_v @ A )
= X5 ) ) ) ).
% Sup_eqI
thf(fact_70_double__diff,axiom,
! [A: set_v,B: set_v,C: set_v] :
( ( ord_less_eq_set_v @ A @ B )
=> ( ( ord_less_eq_set_v @ B @ C )
=> ( ( minus_minus_set_v @ B @ ( minus_minus_set_v @ C @ A ) )
= A ) ) ) ).
% double_diff
thf(fact_71_double__diff,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ B )
=> ( ( ord_le7336532860387713383od_v_v @ B @ C )
=> ( ( minus_4183494784930505774od_v_v @ B @ ( minus_4183494784930505774od_v_v @ C @ A ) )
= A ) ) ) ).
% double_diff
thf(fact_72_mem__Collect__eq,axiom,
! [A4: v,P: v > $o] :
( ( member_v @ A4 @ ( collect_v @ P ) )
= ( P @ A4 ) ) ).
% mem_Collect_eq
thf(fact_73_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_74_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_75_Collect__mem__eq,axiom,
! [A: set_v] :
( ( collect_v
@ ^ [X: v] : ( member_v @ X @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_76_Collect__mem__eq,axiom,
! [A: set_Product_prod_v_v] :
( ( collec140062887454715474od_v_v
@ ^ [X: product_prod_v_v] : ( member7453568604450474000od_v_v @ X @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_77_Collect__mem__eq,axiom,
! [A: set_set_v] :
( ( collect_set_v
@ ^ [X: set_v] : ( member_set_v @ X @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_78_Collect__cong,axiom,
! [P: set_v > $o,Q: set_v > $o] :
( ! [X4: set_v] :
( ( P @ X4 )
= ( Q @ X4 ) )
=> ( ( collect_set_v @ P )
= ( collect_set_v @ Q ) ) ) ).
% Collect_cong
thf(fact_79_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_80_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_81_Diff__mono,axiom,
! [A: set_v,C: set_v,D: set_v,B: set_v] :
( ( ord_less_eq_set_v @ A @ C )
=> ( ( ord_less_eq_set_v @ D @ B )
=> ( ord_less_eq_set_v @ ( minus_minus_set_v @ A @ B ) @ ( minus_minus_set_v @ C @ D ) ) ) ) ).
% Diff_mono
thf(fact_82_Diff__mono,axiom,
! [A: set_Product_prod_v_v,C: set_Product_prod_v_v,D: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ C )
=> ( ( ord_le7336532860387713383od_v_v @ D @ B )
=> ( ord_le7336532860387713383od_v_v @ ( minus_4183494784930505774od_v_v @ A @ B ) @ ( minus_4183494784930505774od_v_v @ C @ D ) ) ) ) ).
% Diff_mono
thf(fact_83_Union__subsetI,axiom,
! [A: set_se8455005133513928103od_v_v,B: set_se8455005133513928103od_v_v] :
( ! [X4: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ X4 @ A )
=> ? [Y4: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ Y4 @ B )
& ( ord_le7336532860387713383od_v_v @ X4 @ Y4 ) ) )
=> ( ord_le7336532860387713383od_v_v @ ( comple5788137035815166516od_v_v @ A ) @ ( comple5788137035815166516od_v_v @ B ) ) ) ).
% Union_subsetI
thf(fact_84_Union__subsetI,axiom,
! [A: set_set_v,B: set_set_v] :
( ! [X4: set_v] :
( ( member_set_v @ X4 @ A )
=> ? [Y4: set_v] :
( ( member_set_v @ Y4 @ B )
& ( ord_less_eq_set_v @ X4 @ Y4 ) ) )
=> ( ord_less_eq_set_v @ ( comple2307003700295860064_set_v @ A ) @ ( comple2307003700295860064_set_v @ B ) ) ) ).
% Union_subsetI
thf(fact_85_Union__upper,axiom,
! [B: set_Product_prod_v_v,A: set_se8455005133513928103od_v_v] :
( ( member8406446414694345712od_v_v @ B @ A )
=> ( ord_le7336532860387713383od_v_v @ B @ ( comple5788137035815166516od_v_v @ A ) ) ) ).
% Union_upper
thf(fact_86_Union__upper,axiom,
! [B: set_v,A: set_set_v] :
( ( member_set_v @ B @ A )
=> ( ord_less_eq_set_v @ B @ ( comple2307003700295860064_set_v @ A ) ) ) ).
% Union_upper
thf(fact_87_Union__least,axiom,
! [A: set_se8455005133513928103od_v_v,C: set_Product_prod_v_v] :
( ! [X6: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ X6 @ A )
=> ( ord_le7336532860387713383od_v_v @ X6 @ C ) )
=> ( ord_le7336532860387713383od_v_v @ ( comple5788137035815166516od_v_v @ A ) @ C ) ) ).
% Union_least
thf(fact_88_Union__least,axiom,
! [A: set_set_v,C: set_v] :
( ! [X6: set_v] :
( ( member_set_v @ X6 @ A )
=> ( ord_less_eq_set_v @ X6 @ C ) )
=> ( ord_less_eq_set_v @ ( comple2307003700295860064_set_v @ A ) @ C ) ) ).
% Union_least
thf(fact_89_minus__set__def,axiom,
( minus_4183494784930505774od_v_v
= ( ^ [A2: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( collec140062887454715474od_v_v
@ ( minus_9095120230875558447_v_v_o
@ ^ [X: product_prod_v_v] : ( member7453568604450474000od_v_v @ X @ A2 )
@ ^ [X: product_prod_v_v] : ( member7453568604450474000od_v_v @ X @ B2 ) ) ) ) ) ).
% minus_set_def
thf(fact_90_minus__set__def,axiom,
( minus_7228012346218142266_set_v
= ( ^ [A2: set_set_v,B2: set_set_v] :
( collect_set_v
@ ( minus_minus_set_v_o
@ ^ [X: set_v] : ( member_set_v @ X @ A2 )
@ ^ [X: set_v] : ( member_set_v @ X @ B2 ) ) ) ) ) ).
% minus_set_def
thf(fact_91_minus__set__def,axiom,
( minus_minus_set_v
= ( ^ [A2: set_v,B2: set_v] :
( collect_v
@ ( minus_minus_v_o
@ ^ [X: v] : ( member_v @ X @ A2 )
@ ^ [X: v] : ( member_v @ X @ B2 ) ) ) ) ) ).
% minus_set_def
thf(fact_92_DiffD2,axiom,
! [C2: product_prod_v_v,A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ C2 @ ( minus_4183494784930505774od_v_v @ A @ B ) )
=> ~ ( member7453568604450474000od_v_v @ C2 @ B ) ) ).
% DiffD2
thf(fact_93_DiffD2,axiom,
! [C2: v,A: set_v,B: set_v] :
( ( member_v @ C2 @ ( minus_minus_set_v @ A @ B ) )
=> ~ ( member_v @ C2 @ B ) ) ).
% DiffD2
thf(fact_94_DiffD1,axiom,
! [C2: product_prod_v_v,A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ C2 @ ( minus_4183494784930505774od_v_v @ A @ B ) )
=> ( member7453568604450474000od_v_v @ C2 @ A ) ) ).
% DiffD1
thf(fact_95_DiffD1,axiom,
! [C2: v,A: set_v,B: set_v] :
( ( member_v @ C2 @ ( minus_minus_set_v @ A @ B ) )
=> ( member_v @ C2 @ A ) ) ).
% DiffD1
thf(fact_96_DiffE,axiom,
! [C2: product_prod_v_v,A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ C2 @ ( minus_4183494784930505774od_v_v @ A @ B ) )
=> ~ ( ( member7453568604450474000od_v_v @ C2 @ A )
=> ( member7453568604450474000od_v_v @ C2 @ B ) ) ) ).
% DiffE
thf(fact_97_DiffE,axiom,
! [C2: v,A: set_v,B: set_v] :
( ( member_v @ C2 @ ( minus_minus_set_v @ A @ B ) )
=> ~ ( ( member_v @ C2 @ A )
=> ( member_v @ C2 @ B ) ) ) ).
% DiffE
thf(fact_98_UnionE,axiom,
! [A: product_prod_v_v,C: set_se8455005133513928103od_v_v] :
( ( member7453568604450474000od_v_v @ A @ ( comple5788137035815166516od_v_v @ C ) )
=> ~ ! [X6: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ A @ X6 )
=> ~ ( member8406446414694345712od_v_v @ X6 @ C ) ) ) ).
% UnionE
thf(fact_99_UnionE,axiom,
! [A: v,C: set_set_v] :
( ( member_v @ A @ ( comple2307003700295860064_set_v @ C ) )
=> ~ ! [X6: set_v] :
( ( member_v @ A @ X6 )
=> ~ ( member_set_v @ X6 @ C ) ) ) ).
% UnionE
thf(fact_100_set__diff__eq,axiom,
( minus_4183494784930505774od_v_v
= ( ^ [A2: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( collec140062887454715474od_v_v
@ ^ [X: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X @ A2 )
& ~ ( member7453568604450474000od_v_v @ X @ B2 ) ) ) ) ) ).
% set_diff_eq
thf(fact_101_set__diff__eq,axiom,
( minus_7228012346218142266_set_v
= ( ^ [A2: set_set_v,B2: set_set_v] :
( collect_set_v
@ ^ [X: set_v] :
( ( member_set_v @ X @ A2 )
& ~ ( member_set_v @ X @ B2 ) ) ) ) ) ).
% set_diff_eq
thf(fact_102_set__diff__eq,axiom,
( minus_minus_set_v
= ( ^ [A2: set_v,B2: set_v] :
( collect_v
@ ^ [X: v] :
( ( member_v @ X @ A2 )
& ~ ( member_v @ X @ B2 ) ) ) ) ) ).
% set_diff_eq
thf(fact_103_e_H__def,axiom,
( e2
= ( sCC_Bloemen_unite_v @ v2 @ w @ e ) ) ).
% e'_def
thf(fact_104_calculation_I4_J,axiom,
ord_less_eq_set_v @ ( set_v2 @ ( sCC_Bl9201514103433284750t_unit @ e2 ) ) @ ( sCC_Bl4645233313691564917t_unit @ e2 ) ).
% calculation(4)
thf(fact_105_pfx_I1_J,axiom,
( ( sCC_Bl8828226123343373779t_unit @ e )
= ( append_v @ pfx @ ( sCC_Bl8828226123343373779t_unit @ e2 ) ) ) ).
% pfx(1)
thf(fact_106_sub__env__def,axiom,
( sCC_Bl5768913643336123637t_unit
= ( ^ [E4: sCC_Bl1394983891496994913t_unit,E5: sCC_Bl1394983891496994913t_unit] :
( ( ( sCC_Bl1090238580953940555t_unit @ E5 )
= ( sCC_Bl1090238580953940555t_unit @ E4 ) )
& ( ord_less_eq_set_v @ ( sCC_Bl4645233313691564917t_unit @ E4 ) @ ( sCC_Bl4645233313691564917t_unit @ E5 ) )
& ( ord_less_eq_set_v @ ( sCC_Bl157864678168468314t_unit @ E4 ) @ ( sCC_Bl157864678168468314t_unit @ E5 ) )
& ! [V2: v] : ( ord_less_eq_set_v @ ( sCC_Bl3795065053823578884t_unit @ E4 @ V2 ) @ ( sCC_Bl3795065053823578884t_unit @ E5 @ V2 ) )
& ! [V2: v] : ( ord_less_eq_set_v @ ( sCC_Bl1280885523602775798t_unit @ E4 @ V2 ) @ ( sCC_Bl1280885523602775798t_unit @ E5 @ V2 ) )
& ( ord_less_eq_set_v
@ ( comple2307003700295860064_set_v
@ ( collect_set_v
@ ^ [Uu: set_v] :
? [V2: v] :
( ( Uu
= ( sCC_Bl1280885523602775798t_unit @ E4 @ V2 ) )
& ( member_v @ V2 @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E4 ) ) ) ) ) )
@ ( comple2307003700295860064_set_v
@ ( collect_set_v
@ ^ [Uu: set_v] :
? [V2: v] :
( ( Uu
= ( sCC_Bl1280885523602775798t_unit @ E5 @ V2 ) )
& ( member_v @ V2 @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E5 ) ) ) ) ) ) ) ) ) ) ).
% sub_env_def
thf(fact_107_w_I2_J,axiom,
~ ( member_v @ w @ ( sCC_Bl3795065053823578884t_unit @ e @ v2 ) ) ).
% w(2)
thf(fact_108_hd__cc,axiom,
( ( sCC_Bl1280885523602775798t_unit @ e2 @ ( hd_v @ ( sCC_Bl8828226123343373779t_unit @ e2 ) ) )
= cc ) ).
% hd_cc
thf(fact_109__092_060open_062hd_A_Istack_Ae_H_J_A_092_060in_062_A_092_060S_062_Ae_A_Ihd_A_Istack_Ae_H_J_J_092_060close_062,axiom,
member_v @ ( hd_v @ ( sCC_Bl8828226123343373779t_unit @ e2 ) ) @ ( sCC_Bl1280885523602775798t_unit @ e @ ( hd_v @ ( sCC_Bl8828226123343373779t_unit @ e2 ) ) ) ).
% \<open>hd (stack e') \<in> \<S> e (hd (stack e'))\<close>
thf(fact_110_order__refl,axiom,
! [X5: set_v] : ( ord_less_eq_set_v @ X5 @ X5 ) ).
% order_refl
thf(fact_111_order__refl,axiom,
! [X5: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ X5 @ X5 ) ).
% order_refl
thf(fact_112_dual__order_Orefl,axiom,
! [A4: set_v] : ( ord_less_eq_set_v @ A4 @ A4 ) ).
% dual_order.refl
thf(fact_113_dual__order_Orefl,axiom,
! [A4: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ A4 @ A4 ) ).
% dual_order.refl
thf(fact_114_subset__code_I1_J,axiom,
! [Xs: list_v,B: set_v] :
( ( ord_less_eq_set_v @ ( set_v2 @ Xs ) @ B )
= ( ! [X: v] :
( ( member_v @ X @ ( set_v2 @ Xs ) )
=> ( member_v @ X @ B ) ) ) ) ).
% subset_code(1)
thf(fact_115_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 )
= ( ! [X: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X @ ( set_Product_prod_v_v2 @ Xs ) )
=> ( member7453568604450474000od_v_v @ X @ B ) ) ) ) ).
% subset_code(1)
thf(fact_116_same__append__eq,axiom,
! [Xs: list_v,Ys: list_v,Zs: list_v] :
( ( ( append_v @ Xs @ Ys )
= ( append_v @ Xs @ Zs ) )
= ( Ys = Zs ) ) ).
% same_append_eq
thf(fact_117_append__same__eq,axiom,
! [Ys: list_v,Xs: list_v,Zs: list_v] :
( ( ( append_v @ Ys @ Xs )
= ( append_v @ Zs @ Xs ) )
= ( Ys = Zs ) ) ).
% append_same_eq
thf(fact_118_append__assoc,axiom,
! [Xs: list_v,Ys: list_v,Zs: list_v] :
( ( append_v @ ( append_v @ Xs @ Ys ) @ Zs )
= ( append_v @ Xs @ ( append_v @ Ys @ Zs ) ) ) ).
% append_assoc
thf(fact_119_append_Oassoc,axiom,
! [A4: list_v,B3: list_v,C2: list_v] :
( ( append_v @ ( append_v @ A4 @ B3 ) @ C2 )
= ( append_v @ A4 @ ( append_v @ B3 @ C2 ) ) ) ).
% append.assoc
thf(fact_120_append__is__Nil__conv,axiom,
! [Xs: list_v,Ys: list_v] :
( ( ( append_v @ Xs @ Ys )
= nil_v )
= ( ( Xs = nil_v )
& ( Ys = nil_v ) ) ) ).
% append_is_Nil_conv
thf(fact_121_Nil__is__append__conv,axiom,
! [Xs: list_v,Ys: list_v] :
( ( nil_v
= ( append_v @ Xs @ Ys ) )
= ( ( Xs = nil_v )
& ( Ys = nil_v ) ) ) ).
% Nil_is_append_conv
thf(fact_122_self__append__conv2,axiom,
! [Y5: list_v,Xs: list_v] :
( ( Y5
= ( append_v @ Xs @ Y5 ) )
= ( Xs = nil_v ) ) ).
% self_append_conv2
thf(fact_123_append__self__conv2,axiom,
! [Xs: list_v,Ys: list_v] :
( ( ( append_v @ Xs @ Ys )
= Ys )
= ( Xs = nil_v ) ) ).
% append_self_conv2
thf(fact_124_self__append__conv,axiom,
! [Y5: list_v,Ys: list_v] :
( ( Y5
= ( append_v @ Y5 @ Ys ) )
= ( Ys = nil_v ) ) ).
% self_append_conv
thf(fact_125_append__self__conv,axiom,
! [Xs: list_v,Ys: list_v] :
( ( ( append_v @ Xs @ Ys )
= Xs )
= ( Ys = nil_v ) ) ).
% append_self_conv
thf(fact_126_append__Nil2,axiom,
! [Xs: list_v] :
( ( append_v @ Xs @ nil_v )
= Xs ) ).
% append_Nil2
thf(fact_127_append_Oright__neutral,axiom,
! [A4: list_v] :
( ( append_v @ A4 @ nil_v )
= A4 ) ).
% append.right_neutral
thf(fact_128_hd__append2,axiom,
! [Xs: list_v,Ys: list_v] :
( ( Xs != nil_v )
=> ( ( hd_v @ ( append_v @ Xs @ Ys ) )
= ( hd_v @ Xs ) ) ) ).
% hd_append2
thf(fact_129_hd__append,axiom,
! [Xs: list_v,Ys: list_v] :
( ( ( Xs = nil_v )
=> ( ( hd_v @ ( append_v @ Xs @ Ys ) )
= ( hd_v @ Ys ) ) )
& ( ( Xs != nil_v )
=> ( ( hd_v @ ( append_v @ Xs @ Ys ) )
= ( hd_v @ Xs ) ) ) ) ).
% hd_append
thf(fact_130_longest__common__prefix,axiom,
! [Xs: list_v,Ys: list_v] :
? [Ps: list_v,Xs2: list_v,Ys2: list_v] :
( ( Xs
= ( append_v @ Ps @ Xs2 ) )
& ( Ys
= ( append_v @ Ps @ Ys2 ) )
& ( ( Xs2 = nil_v )
| ( Ys2 = nil_v )
| ( ( hd_v @ Xs2 )
!= ( hd_v @ Ys2 ) ) ) ) ).
% longest_common_prefix
thf(fact_131_append__eq__append__conv2,axiom,
! [Xs: list_v,Ys: list_v,Zs: list_v,Ts: list_v] :
( ( ( append_v @ Xs @ Ys )
= ( append_v @ Zs @ Ts ) )
= ( ? [Us: list_v] :
( ( ( Xs
= ( append_v @ Zs @ Us ) )
& ( ( append_v @ Us @ Ys )
= Ts ) )
| ( ( ( append_v @ Xs @ Us )
= Zs )
& ( Ys
= ( append_v @ Us @ Ts ) ) ) ) ) ) ).
% append_eq_append_conv2
thf(fact_132_append__eq__appendI,axiom,
! [Xs: list_v,Xs1: list_v,Zs: list_v,Ys: list_v,Us2: list_v] :
( ( ( append_v @ Xs @ Xs1 )
= Zs )
=> ( ( Ys
= ( append_v @ Xs1 @ Us2 ) )
=> ( ( append_v @ Xs @ Ys )
= ( append_v @ Zs @ Us2 ) ) ) ) ).
% append_eq_appendI
thf(fact_133_eq__Nil__appendI,axiom,
! [Xs: list_v,Ys: list_v] :
( ( Xs = Ys )
=> ( Xs
= ( append_v @ nil_v @ Ys ) ) ) ).
% eq_Nil_appendI
thf(fact_134_append_Oleft__neutral,axiom,
! [A4: list_v] :
( ( append_v @ nil_v @ A4 )
= A4 ) ).
% append.left_neutral
thf(fact_135_append__Nil,axiom,
! [Ys: list_v] :
( ( append_v @ nil_v @ Ys )
= Ys ) ).
% append_Nil
thf(fact_136_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_137_hd__in__set,axiom,
! [Xs: list_v] :
( ( Xs != nil_v )
=> ( member_v @ ( hd_v @ Xs ) @ ( set_v2 @ Xs ) ) ) ).
% hd_in_set
thf(fact_138_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_139_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_140_less__eq__set__def,axiom,
( ord_less_eq_set_v
= ( ^ [A2: set_v,B2: set_v] :
( ord_less_eq_v_o
@ ^ [X: v] : ( member_v @ X @ A2 )
@ ^ [X: v] : ( member_v @ X @ B2 ) ) ) ) ).
% less_eq_set_def
thf(fact_141_less__eq__set__def,axiom,
( ord_le7336532860387713383od_v_v
= ( ^ [A2: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ord_le5892402249245633078_v_v_o
@ ^ [X: product_prod_v_v] : ( member7453568604450474000od_v_v @ X @ A2 )
@ ^ [X: product_prod_v_v] : ( member7453568604450474000od_v_v @ X @ B2 ) ) ) ) ).
% less_eq_set_def
thf(fact_142_order__antisym__conv,axiom,
! [Y5: set_v,X5: set_v] :
( ( ord_less_eq_set_v @ Y5 @ X5 )
=> ( ( ord_less_eq_set_v @ X5 @ Y5 )
= ( X5 = Y5 ) ) ) ).
% order_antisym_conv
thf(fact_143_order__antisym__conv,axiom,
! [Y5: set_Product_prod_v_v,X5: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ Y5 @ X5 )
=> ( ( ord_le7336532860387713383od_v_v @ X5 @ Y5 )
= ( X5 = Y5 ) ) ) ).
% order_antisym_conv
thf(fact_144_ord__le__eq__subst,axiom,
! [A4: set_v,B3: set_v,F: set_v > set_v,C2: set_v] :
( ( ord_less_eq_set_v @ A4 @ B3 )
=> ( ( ( F @ B3 )
= C2 )
=> ( ! [X4: set_v,Y3: set_v] :
( ( ord_less_eq_set_v @ X4 @ Y3 )
=> ( ord_less_eq_set_v @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_v @ ( F @ A4 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_145_ord__le__eq__subst,axiom,
! [A4: set_v,B3: set_v,F: set_v > set_Product_prod_v_v,C2: set_Product_prod_v_v] :
( ( ord_less_eq_set_v @ A4 @ B3 )
=> ( ( ( F @ B3 )
= C2 )
=> ( ! [X4: set_v,Y3: set_v] :
( ( ord_less_eq_set_v @ X4 @ Y3 )
=> ( ord_le7336532860387713383od_v_v @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_le7336532860387713383od_v_v @ ( F @ A4 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_146_ord__le__eq__subst,axiom,
! [A4: set_Product_prod_v_v,B3: set_Product_prod_v_v,F: set_Product_prod_v_v > set_v,C2: set_v] :
( ( ord_le7336532860387713383od_v_v @ A4 @ B3 )
=> ( ( ( F @ B3 )
= C2 )
=> ( ! [X4: set_Product_prod_v_v,Y3: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X4 @ Y3 )
=> ( ord_less_eq_set_v @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_v @ ( F @ A4 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_147_ord__le__eq__subst,axiom,
! [A4: set_Product_prod_v_v,B3: set_Product_prod_v_v,F: set_Product_prod_v_v > set_Product_prod_v_v,C2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A4 @ B3 )
=> ( ( ( F @ B3 )
= C2 )
=> ( ! [X4: set_Product_prod_v_v,Y3: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X4 @ Y3 )
=> ( ord_le7336532860387713383od_v_v @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_le7336532860387713383od_v_v @ ( F @ A4 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_148_ord__eq__le__subst,axiom,
! [A4: set_v,F: set_v > set_v,B3: set_v,C2: set_v] :
( ( A4
= ( F @ B3 ) )
=> ( ( ord_less_eq_set_v @ B3 @ C2 )
=> ( ! [X4: set_v,Y3: set_v] :
( ( ord_less_eq_set_v @ X4 @ Y3 )
=> ( ord_less_eq_set_v @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_v @ A4 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_149_ord__eq__le__subst,axiom,
! [A4: set_Product_prod_v_v,F: set_v > set_Product_prod_v_v,B3: set_v,C2: set_v] :
( ( A4
= ( F @ B3 ) )
=> ( ( ord_less_eq_set_v @ B3 @ C2 )
=> ( ! [X4: set_v,Y3: set_v] :
( ( ord_less_eq_set_v @ X4 @ Y3 )
=> ( ord_le7336532860387713383od_v_v @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_le7336532860387713383od_v_v @ A4 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_150_ord__eq__le__subst,axiom,
! [A4: set_v,F: set_Product_prod_v_v > set_v,B3: set_Product_prod_v_v,C2: set_Product_prod_v_v] :
( ( A4
= ( F @ B3 ) )
=> ( ( ord_le7336532860387713383od_v_v @ B3 @ C2 )
=> ( ! [X4: set_Product_prod_v_v,Y3: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X4 @ Y3 )
=> ( ord_less_eq_set_v @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_v @ A4 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_151_ord__eq__le__subst,axiom,
! [A4: set_Product_prod_v_v,F: set_Product_prod_v_v > set_Product_prod_v_v,B3: set_Product_prod_v_v,C2: set_Product_prod_v_v] :
( ( A4
= ( F @ B3 ) )
=> ( ( ord_le7336532860387713383od_v_v @ B3 @ C2 )
=> ( ! [X4: set_Product_prod_v_v,Y3: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X4 @ Y3 )
=> ( ord_le7336532860387713383od_v_v @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_le7336532860387713383od_v_v @ A4 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_152_order__eq__refl,axiom,
! [X5: set_v,Y5: set_v] :
( ( X5 = Y5 )
=> ( ord_less_eq_set_v @ X5 @ Y5 ) ) ).
% order_eq_refl
thf(fact_153_order__eq__refl,axiom,
! [X5: set_Product_prod_v_v,Y5: set_Product_prod_v_v] :
( ( X5 = Y5 )
=> ( ord_le7336532860387713383od_v_v @ X5 @ Y5 ) ) ).
% order_eq_refl
thf(fact_154_order__subst2,axiom,
! [A4: set_v,B3: set_v,F: set_v > set_v,C2: set_v] :
( ( ord_less_eq_set_v @ A4 @ B3 )
=> ( ( ord_less_eq_set_v @ ( F @ B3 ) @ C2 )
=> ( ! [X4: set_v,Y3: set_v] :
( ( ord_less_eq_set_v @ X4 @ Y3 )
=> ( ord_less_eq_set_v @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_v @ ( F @ A4 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_155_order__subst2,axiom,
! [A4: set_v,B3: set_v,F: set_v > set_Product_prod_v_v,C2: set_Product_prod_v_v] :
( ( ord_less_eq_set_v @ A4 @ B3 )
=> ( ( ord_le7336532860387713383od_v_v @ ( F @ B3 ) @ C2 )
=> ( ! [X4: set_v,Y3: set_v] :
( ( ord_less_eq_set_v @ X4 @ Y3 )
=> ( ord_le7336532860387713383od_v_v @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_le7336532860387713383od_v_v @ ( F @ A4 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_156_order__subst2,axiom,
! [A4: set_Product_prod_v_v,B3: set_Product_prod_v_v,F: set_Product_prod_v_v > set_v,C2: set_v] :
( ( ord_le7336532860387713383od_v_v @ A4 @ B3 )
=> ( ( ord_less_eq_set_v @ ( F @ B3 ) @ C2 )
=> ( ! [X4: set_Product_prod_v_v,Y3: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X4 @ Y3 )
=> ( ord_less_eq_set_v @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_v @ ( F @ A4 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_157_order__subst2,axiom,
! [A4: set_Product_prod_v_v,B3: set_Product_prod_v_v,F: set_Product_prod_v_v > set_Product_prod_v_v,C2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A4 @ B3 )
=> ( ( ord_le7336532860387713383od_v_v @ ( F @ B3 ) @ C2 )
=> ( ! [X4: set_Product_prod_v_v,Y3: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X4 @ Y3 )
=> ( ord_le7336532860387713383od_v_v @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_le7336532860387713383od_v_v @ ( F @ A4 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_158_order__subst1,axiom,
! [A4: set_v,F: set_v > set_v,B3: set_v,C2: set_v] :
( ( ord_less_eq_set_v @ A4 @ ( F @ B3 ) )
=> ( ( ord_less_eq_set_v @ B3 @ C2 )
=> ( ! [X4: set_v,Y3: set_v] :
( ( ord_less_eq_set_v @ X4 @ Y3 )
=> ( ord_less_eq_set_v @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_v @ A4 @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_159_order__subst1,axiom,
! [A4: set_v,F: set_Product_prod_v_v > set_v,B3: set_Product_prod_v_v,C2: set_Product_prod_v_v] :
( ( ord_less_eq_set_v @ A4 @ ( F @ B3 ) )
=> ( ( ord_le7336532860387713383od_v_v @ B3 @ C2 )
=> ( ! [X4: set_Product_prod_v_v,Y3: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X4 @ Y3 )
=> ( ord_less_eq_set_v @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_v @ A4 @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_160_order__subst1,axiom,
! [A4: set_Product_prod_v_v,F: set_v > set_Product_prod_v_v,B3: set_v,C2: set_v] :
( ( ord_le7336532860387713383od_v_v @ A4 @ ( F @ B3 ) )
=> ( ( ord_less_eq_set_v @ B3 @ C2 )
=> ( ! [X4: set_v,Y3: set_v] :
( ( ord_less_eq_set_v @ X4 @ Y3 )
=> ( ord_le7336532860387713383od_v_v @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_le7336532860387713383od_v_v @ A4 @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_161_order__subst1,axiom,
! [A4: set_Product_prod_v_v,F: set_Product_prod_v_v > set_Product_prod_v_v,B3: set_Product_prod_v_v,C2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A4 @ ( F @ B3 ) )
=> ( ( ord_le7336532860387713383od_v_v @ B3 @ C2 )
=> ( ! [X4: set_Product_prod_v_v,Y3: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X4 @ Y3 )
=> ( ord_le7336532860387713383od_v_v @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_le7336532860387713383od_v_v @ A4 @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_162_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y2: set_v,Z: set_v] : ( Y2 = Z ) )
= ( ^ [A5: set_v,B4: set_v] :
( ( ord_less_eq_set_v @ A5 @ B4 )
& ( ord_less_eq_set_v @ B4 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_163_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y2: set_Product_prod_v_v,Z: set_Product_prod_v_v] : ( Y2 = Z ) )
= ( ^ [A5: set_Product_prod_v_v,B4: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A5 @ B4 )
& ( ord_le7336532860387713383od_v_v @ B4 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_164_antisym,axiom,
! [A4: set_v,B3: set_v] :
( ( ord_less_eq_set_v @ A4 @ B3 )
=> ( ( ord_less_eq_set_v @ B3 @ A4 )
=> ( A4 = B3 ) ) ) ).
% antisym
thf(fact_165_antisym,axiom,
! [A4: set_Product_prod_v_v,B3: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A4 @ B3 )
=> ( ( ord_le7336532860387713383od_v_v @ B3 @ A4 )
=> ( A4 = B3 ) ) ) ).
% antisym
thf(fact_166_dual__order_Otrans,axiom,
! [B3: set_v,A4: set_v,C2: set_v] :
( ( ord_less_eq_set_v @ B3 @ A4 )
=> ( ( ord_less_eq_set_v @ C2 @ B3 )
=> ( ord_less_eq_set_v @ C2 @ A4 ) ) ) ).
% dual_order.trans
thf(fact_167_dual__order_Otrans,axiom,
! [B3: set_Product_prod_v_v,A4: set_Product_prod_v_v,C2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ B3 @ A4 )
=> ( ( ord_le7336532860387713383od_v_v @ C2 @ B3 )
=> ( ord_le7336532860387713383od_v_v @ C2 @ A4 ) ) ) ).
% dual_order.trans
thf(fact_168_dual__order_Oantisym,axiom,
! [B3: set_v,A4: set_v] :
( ( ord_less_eq_set_v @ B3 @ A4 )
=> ( ( ord_less_eq_set_v @ A4 @ B3 )
=> ( A4 = B3 ) ) ) ).
% dual_order.antisym
thf(fact_169_dual__order_Oantisym,axiom,
! [B3: set_Product_prod_v_v,A4: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ B3 @ A4 )
=> ( ( ord_le7336532860387713383od_v_v @ A4 @ B3 )
=> ( A4 = B3 ) ) ) ).
% dual_order.antisym
thf(fact_170_dual__order_Oeq__iff,axiom,
( ( ^ [Y2: set_v,Z: set_v] : ( Y2 = Z ) )
= ( ^ [A5: set_v,B4: set_v] :
( ( ord_less_eq_set_v @ B4 @ A5 )
& ( ord_less_eq_set_v @ A5 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_171_dual__order_Oeq__iff,axiom,
( ( ^ [Y2: set_Product_prod_v_v,Z: set_Product_prod_v_v] : ( Y2 = Z ) )
= ( ^ [A5: set_Product_prod_v_v,B4: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ B4 @ A5 )
& ( ord_le7336532860387713383od_v_v @ A5 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_172_order__trans,axiom,
! [X5: set_v,Y5: set_v,Z2: set_v] :
( ( ord_less_eq_set_v @ X5 @ Y5 )
=> ( ( ord_less_eq_set_v @ Y5 @ Z2 )
=> ( ord_less_eq_set_v @ X5 @ Z2 ) ) ) ).
% order_trans
thf(fact_173_order__trans,axiom,
! [X5: set_Product_prod_v_v,Y5: set_Product_prod_v_v,Z2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X5 @ Y5 )
=> ( ( ord_le7336532860387713383od_v_v @ Y5 @ Z2 )
=> ( ord_le7336532860387713383od_v_v @ X5 @ Z2 ) ) ) ).
% order_trans
thf(fact_174_order_Otrans,axiom,
! [A4: set_v,B3: set_v,C2: set_v] :
( ( ord_less_eq_set_v @ A4 @ B3 )
=> ( ( ord_less_eq_set_v @ B3 @ C2 )
=> ( ord_less_eq_set_v @ A4 @ C2 ) ) ) ).
% order.trans
thf(fact_175_order_Otrans,axiom,
! [A4: set_Product_prod_v_v,B3: set_Product_prod_v_v,C2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A4 @ B3 )
=> ( ( ord_le7336532860387713383od_v_v @ B3 @ C2 )
=> ( ord_le7336532860387713383od_v_v @ A4 @ C2 ) ) ) ).
% order.trans
thf(fact_176_order__antisym,axiom,
! [X5: set_v,Y5: set_v] :
( ( ord_less_eq_set_v @ X5 @ Y5 )
=> ( ( ord_less_eq_set_v @ Y5 @ X5 )
=> ( X5 = Y5 ) ) ) ).
% order_antisym
thf(fact_177_order__antisym,axiom,
! [X5: set_Product_prod_v_v,Y5: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X5 @ Y5 )
=> ( ( ord_le7336532860387713383od_v_v @ Y5 @ X5 )
=> ( X5 = Y5 ) ) ) ).
% order_antisym
thf(fact_178_ord__le__eq__trans,axiom,
! [A4: set_v,B3: set_v,C2: set_v] :
( ( ord_less_eq_set_v @ A4 @ B3 )
=> ( ( B3 = C2 )
=> ( ord_less_eq_set_v @ A4 @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_179_ord__le__eq__trans,axiom,
! [A4: set_Product_prod_v_v,B3: set_Product_prod_v_v,C2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A4 @ B3 )
=> ( ( B3 = C2 )
=> ( ord_le7336532860387713383od_v_v @ A4 @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_180_ord__eq__le__trans,axiom,
! [A4: set_v,B3: set_v,C2: set_v] :
( ( A4 = B3 )
=> ( ( ord_less_eq_set_v @ B3 @ C2 )
=> ( ord_less_eq_set_v @ A4 @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_181_ord__eq__le__trans,axiom,
! [A4: set_Product_prod_v_v,B3: set_Product_prod_v_v,C2: set_Product_prod_v_v] :
( ( A4 = B3 )
=> ( ( ord_le7336532860387713383od_v_v @ B3 @ C2 )
=> ( ord_le7336532860387713383od_v_v @ A4 @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_182_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y2: set_v,Z: set_v] : ( Y2 = Z ) )
= ( ^ [X: set_v,Y: set_v] :
( ( ord_less_eq_set_v @ X @ Y )
& ( ord_less_eq_set_v @ Y @ X ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_183_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y2: set_Product_prod_v_v,Z: set_Product_prod_v_v] : ( Y2 = Z ) )
= ( ^ [X: set_Product_prod_v_v,Y: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X @ Y )
& ( ord_le7336532860387713383od_v_v @ Y @ X ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_184_cSup__eq__maximum,axiom,
! [Z2: set_Product_prod_v_v,X2: set_se8455005133513928103od_v_v] :
( ( member8406446414694345712od_v_v @ Z2 @ X2 )
=> ( ! [X4: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ X4 @ X2 )
=> ( ord_le7336532860387713383od_v_v @ X4 @ Z2 ) )
=> ( ( comple5788137035815166516od_v_v @ X2 )
= Z2 ) ) ) ).
% cSup_eq_maximum
thf(fact_185_cSup__eq__maximum,axiom,
! [Z2: set_v,X2: set_set_v] :
( ( member_set_v @ Z2 @ X2 )
=> ( ! [X4: set_v] :
( ( member_set_v @ X4 @ X2 )
=> ( ord_less_eq_set_v @ X4 @ Z2 ) )
=> ( ( comple2307003700295860064_set_v @ X2 )
= Z2 ) ) ) ).
% cSup_eq_maximum
thf(fact_186_calculation_I13_J,axiom,
! [N: v,M: v] :
( ( sCC_Bl4022239298816431255edes_v @ N @ M @ ( sCC_Bl8828226123343373779t_unit @ e2 ) )
=> ( sCC_Bl4022239298816431255edes_v @ N @ M @ ( sCC_Bl9201514103433284750t_unit @ e2 ) ) ) ).
% calculation(13)
thf(fact_187_equality,axiom,
! [R: sCC_Bl1394983891496994913t_unit,R2: sCC_Bl1394983891496994913t_unit] :
( ( ( sCC_Bl1090238580953940555t_unit @ R )
= ( sCC_Bl1090238580953940555t_unit @ R2 ) )
=> ( ( ( sCC_Bl1280885523602775798t_unit @ R )
= ( sCC_Bl1280885523602775798t_unit @ R2 ) )
=> ( ( ( sCC_Bl157864678168468314t_unit @ R )
= ( sCC_Bl157864678168468314t_unit @ R2 ) )
=> ( ( ( sCC_Bl4645233313691564917t_unit @ R )
= ( sCC_Bl4645233313691564917t_unit @ R2 ) )
=> ( ( ( sCC_Bl3795065053823578884t_unit @ R )
= ( sCC_Bl3795065053823578884t_unit @ R2 ) )
=> ( ( ( sCC_Bl2536197123907397897t_unit @ R )
= ( sCC_Bl2536197123907397897t_unit @ R2 ) )
=> ( ( ( sCC_Bl8828226123343373779t_unit @ R )
= ( sCC_Bl8828226123343373779t_unit @ R2 ) )
=> ( ( ( sCC_Bl9201514103433284750t_unit @ R )
= ( sCC_Bl9201514103433284750t_unit @ R2 ) )
=> ( ( ( sCC_Bl3567736435408124606t_unit @ R )
= ( sCC_Bl3567736435408124606t_unit @ R2 ) )
=> ( R = R2 ) ) ) ) ) ) ) ) ) ) ).
% equality
thf(fact_188_calculation_I7_J,axiom,
! [N: v] :
( ~ ( member_v @ N @ ( sCC_Bl4645233313691564917t_unit @ e2 ) )
=> ( ( sCC_Bl3795065053823578884t_unit @ e2 @ N )
= bot_bot_set_v ) ) ).
% calculation(7)
thf(fact_189_graph_Osub__env__def,axiom,
! [Vertices: set_v,Successors: v > set_v,E: sCC_Bl1394983891496994913t_unit,E2: sCC_Bl1394983891496994913t_unit] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl5768913643336123637t_unit @ E @ E2 )
= ( ( ( sCC_Bl1090238580953940555t_unit @ E2 )
= ( sCC_Bl1090238580953940555t_unit @ E ) )
& ( ord_less_eq_set_v @ ( sCC_Bl4645233313691564917t_unit @ E ) @ ( sCC_Bl4645233313691564917t_unit @ E2 ) )
& ( ord_less_eq_set_v @ ( sCC_Bl157864678168468314t_unit @ E ) @ ( sCC_Bl157864678168468314t_unit @ E2 ) )
& ! [V2: v] : ( ord_less_eq_set_v @ ( sCC_Bl3795065053823578884t_unit @ E @ V2 ) @ ( sCC_Bl3795065053823578884t_unit @ E2 @ V2 ) )
& ! [V2: v] : ( ord_less_eq_set_v @ ( sCC_Bl1280885523602775798t_unit @ E @ V2 ) @ ( sCC_Bl1280885523602775798t_unit @ E2 @ V2 ) )
& ( ord_less_eq_set_v
@ ( comple2307003700295860064_set_v
@ ( collect_set_v
@ ^ [Uu: set_v] :
? [V2: v] :
( ( Uu
= ( sCC_Bl1280885523602775798t_unit @ E @ V2 ) )
& ( member_v @ V2 @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E ) ) ) ) ) )
@ ( comple2307003700295860064_set_v
@ ( collect_set_v
@ ^ [Uu: set_v] :
? [V2: v] :
( ( Uu
= ( sCC_Bl1280885523602775798t_unit @ E2 @ V2 ) )
& ( member_v @ V2 @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E2 ) ) ) ) ) ) ) ) ) ) ).
% graph.sub_env_def
thf(fact_190_w_I1_J,axiom,
member_v @ w @ ( successors @ v2 ) ).
% w(1)
thf(fact_191_cc__def,axiom,
( cc
= ( comple2307003700295860064_set_v
@ ( collect_set_v
@ ^ [Uu: set_v] :
? [N2: v] :
( ( Uu
= ( sCC_Bl1280885523602775798t_unit @ e @ N2 ) )
& ( member_v @ N2 @ ( sup_sup_set_v @ ( set_v2 @ pfx ) @ ( insert_v @ ( hd_v @ ( sCC_Bl8828226123343373779t_unit @ e2 ) ) @ bot_bot_set_v ) ) ) ) ) ) ) ).
% cc_def
thf(fact_192_calculation_I9_J,axiom,
! [X3: v] :
( ( member_v @ X3 @ ( minus_minus_set_v @ ( sCC_Bl4645233313691564917t_unit @ e2 ) @ ( set_v2 @ ( sCC_Bl9201514103433284750t_unit @ e2 ) ) ) )
=> ( ( sCC_Bl3795065053823578884t_unit @ e2 @ X3 )
= ( successors @ X3 ) ) ) ).
% calculation(9)
thf(fact_193_pfx_I3_J,axiom,
( ( sCC_Bl1280885523602775798t_unit @ e2 )
= ( ^ [X: v] :
( if_set_v
@ ( member_v @ X
@ ( comple2307003700295860064_set_v
@ ( collect_set_v
@ ^ [Uu: set_v] :
? [N2: v] :
( ( Uu
= ( sCC_Bl1280885523602775798t_unit @ e @ N2 ) )
& ( member_v @ N2 @ ( sup_sup_set_v @ ( set_v2 @ pfx ) @ ( insert_v @ ( hd_v @ ( sCC_Bl8828226123343373779t_unit @ e2 ) ) @ bot_bot_set_v ) ) ) ) ) ) )
@ ( comple2307003700295860064_set_v
@ ( collect_set_v
@ ^ [Uu: set_v] :
? [N2: v] :
( ( Uu
= ( sCC_Bl1280885523602775798t_unit @ e @ N2 ) )
& ( member_v @ N2 @ ( sup_sup_set_v @ ( set_v2 @ pfx ) @ ( insert_v @ ( hd_v @ ( sCC_Bl8828226123343373779t_unit @ e2 ) ) @ bot_bot_set_v ) ) ) ) ) )
@ ( sCC_Bl1280885523602775798t_unit @ e @ X ) ) ) ) ).
% pfx(3)
thf(fact_194_calculation_I2_J,axiom,
distinct_v @ ( sCC_Bl9201514103433284750t_unit @ e2 ) ).
% calculation(2)
thf(fact_195_graph__axioms,axiom,
sCC_Bloemen_graph_v @ vertices @ successors ).
% graph_axioms
thf(fact_196_calculation_I12_J,axiom,
distinct_v @ ( sCC_Bl8828226123343373779t_unit @ e2 ) ).
% calculation(12)
thf(fact_197_calculation_I8_J,axiom,
! [X3: v] :
( ( member_v @ X3 @ ( sCC_Bl157864678168468314t_unit @ e2 ) )
=> ( ( sCC_Bl3795065053823578884t_unit @ e2 @ X3 )
= ( successors @ X3 ) ) ) ).
% calculation(8)
thf(fact_198_pre,axiom,
sCC_Bl1748261141445803503t_unit @ successors @ v2 @ e ).
% pre
thf(fact_199_calculation_I16_J,axiom,
! [N: v] :
( ~ ( member_v @ N @ ( sCC_Bl4645233313691564917t_unit @ e2 ) )
=> ( ( sCC_Bl1280885523602775798t_unit @ e2 @ N )
= ( insert_v @ N @ bot_bot_set_v ) ) ) ).
% calculation(16)
thf(fact_200__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062pfx_O_A_092_060lbrakk_062stack_Ae_A_061_Apfx_A_064_Astack_Ae_H_059_Astack_Ae_H_A_092_060noteq_062_A_091_093_059_Alet_Acc_A_061_A_092_060Union_062_A_123_092_060S_062_Ae_An_A_124n_O_An_A_092_060in_062_Aset_Apfx_A_092_060union_062_A_123hd_A_Istack_Ae_H_J_125_125_Ain_A_092_060S_062_Ae_H_A_061_A_I_092_060lambda_062x_O_Aif_Ax_A_092_060in_062_Acc_Athen_Acc_Aelse_A_092_060S_062_Ae_Ax_J_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [Pfx: list_v] :
( ( ( sCC_Bl8828226123343373779t_unit @ e )
= ( append_v @ Pfx @ ( sCC_Bl8828226123343373779t_unit @ e2 ) ) )
=> ( ( ( sCC_Bl8828226123343373779t_unit @ e2 )
!= nil_v )
=> ( ( sCC_Bl1280885523602775798t_unit @ e2 )
!= ( ^ [X: v] :
( if_set_v
@ ( member_v @ X
@ ( comple2307003700295860064_set_v
@ ( collect_set_v
@ ^ [Uu: set_v] :
? [N2: v] :
( ( Uu
= ( sCC_Bl1280885523602775798t_unit @ e @ N2 ) )
& ( member_v @ N2 @ ( sup_sup_set_v @ ( set_v2 @ Pfx ) @ ( insert_v @ ( hd_v @ ( sCC_Bl8828226123343373779t_unit @ e2 ) ) @ bot_bot_set_v ) ) ) ) ) ) )
@ ( comple2307003700295860064_set_v
@ ( collect_set_v
@ ^ [Uu: set_v] :
? [N2: v] :
( ( Uu
= ( sCC_Bl1280885523602775798t_unit @ e @ N2 ) )
& ( member_v @ N2 @ ( sup_sup_set_v @ ( set_v2 @ Pfx ) @ ( insert_v @ ( hd_v @ ( sCC_Bl8828226123343373779t_unit @ e2 ) ) @ bot_bot_set_v ) ) ) ) ) )
@ ( sCC_Bl1280885523602775798t_unit @ e @ X ) ) ) ) ) ) ).
% \<open>\<And>thesis. (\<And>pfx. \<lbrakk>stack e = pfx @ stack e'; stack e' \<noteq> []; let cc = \<Union> {\<S> e n |n. n \<in> set pfx \<union> {hd (stack e')}} in \<S> e' = (\<lambda>x. if x \<in> cc then cc else \<S> e x)\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_201_sclosed,axiom,
! [X3: v] :
( ( member_v @ X3 @ vertices )
=> ( ord_less_eq_set_v @ ( successors @ X3 ) @ vertices ) ) ).
% sclosed
thf(fact_202_unite__sub__env,axiom,
! [V: v,E: sCC_Bl1394983891496994913t_unit,W: v] :
( ( sCC_Bl1748261141445803503t_unit @ successors @ V @ E )
=> ( ( member_v @ W @ ( successors @ V ) )
=> ( ~ ( member_v @ W @ ( sCC_Bl3795065053823578884t_unit @ E @ V ) )
=> ( ( member_v @ W @ ( sCC_Bl4645233313691564917t_unit @ E ) )
=> ( ~ ( member_v @ W @ ( sCC_Bl157864678168468314t_unit @ E ) )
=> ( sCC_Bl5768913643336123637t_unit @ E @ ( sCC_Bloemen_unite_v @ V @ W @ E ) ) ) ) ) ) ) ).
% unite_sub_env
thf(fact_203_calculation_I10_J,axiom,
! [X3: set_v] :
( ( member_set_v @ X3 @ ( sCC_Bl2536197123907397897t_unit @ e2 ) )
=> ( sCC_Bloemen_is_scc_v @ successors @ X3 ) ) ).
% calculation(10)
thf(fact_204_unite__S__equal,axiom,
! [E: sCC_Bl1394983891496994913t_unit,W: v,V: v] :
( ( sCC_Bl9196236973127232072t_unit @ successors @ E )
=> ( ( member_v @ W @ ( successors @ V ) )
=> ( ~ ( member_v @ W @ ( sCC_Bl3795065053823578884t_unit @ E @ V ) )
=> ( ( member_v @ W @ ( sCC_Bl4645233313691564917t_unit @ E ) )
=> ( ~ ( member_v @ W @ ( sCC_Bl157864678168468314t_unit @ E ) )
=> ( ( comple2307003700295860064_set_v
@ ( collect_set_v
@ ^ [Uu: set_v] :
? [N2: v] :
( ( Uu
= ( sCC_Bl1280885523602775798t_unit @ ( sCC_Bloemen_unite_v @ V @ W @ E ) @ N2 ) )
& ( member_v @ N2 @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ ( sCC_Bloemen_unite_v @ V @ W @ E ) ) ) ) ) ) )
= ( comple2307003700295860064_set_v
@ ( collect_set_v
@ ^ [Uu: set_v] :
? [N2: v] :
( ( Uu
= ( sCC_Bl1280885523602775798t_unit @ E @ N2 ) )
& ( member_v @ N2 @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E ) ) ) ) ) ) ) ) ) ) ) ) ).
% unite_S_equal
thf(fact_205_calculation_I6_J,axiom,
! [N: v] : ( ord_less_eq_set_v @ ( sCC_Bl3795065053823578884t_unit @ e2 @ N ) @ ( inf_inf_set_v @ ( successors @ N ) @ ( sCC_Bl4645233313691564917t_unit @ e2 ) ) ) ).
% calculation(6)
thf(fact_206_scc__partition,axiom,
! [S: set_v,S2: set_v,X5: v] :
( ( sCC_Bloemen_is_scc_v @ successors @ S )
=> ( ( sCC_Bloemen_is_scc_v @ successors @ S2 )
=> ( ( member_v @ X5 @ ( inf_inf_set_v @ S @ S2 ) )
=> ( S = S2 ) ) ) ) ).
% scc_partition
thf(fact_207_local_Owf,axiom,
sCC_Bl9196236973127232072t_unit @ successors @ e ).
% local.wf
thf(fact_208_calculation_I17_J,axiom,
! [X3: v] :
( ( member_v @ X3 @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ e2 ) ) )
=> ! [Xa: v] :
( ( member_v @ Xa @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ e2 ) ) )
=> ( ( X3 != Xa )
=> ( ( inf_inf_set_v @ ( sCC_Bl1280885523602775798t_unit @ e2 @ X3 ) @ ( sCC_Bl1280885523602775798t_unit @ e2 @ Xa ) )
= bot_bot_set_v ) ) ) ) ).
% calculation(17)
thf(fact_209_unite__subscc,axiom,
! [V: v,E: sCC_Bl1394983891496994913t_unit,W: v] :
( ( sCC_Bl1748261141445803503t_unit @ successors @ V @ E )
=> ( ( member_v @ W @ ( successors @ V ) )
=> ( ~ ( member_v @ W @ ( sCC_Bl3795065053823578884t_unit @ E @ V ) )
=> ( ( 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 @ V @ W @ E ) @ ( hd_v @ ( sCC_Bl8828226123343373779t_unit @ ( sCC_Bloemen_unite_v @ V @ W @ E ) ) ) ) ) ) ) ) ) ) ).
% unite_subscc
thf(fact_210_is__scc__def,axiom,
! [S: set_v] :
( ( sCC_Bloemen_is_scc_v @ successors @ S )
= ( ( S != bot_bot_set_v )
& ( sCC_Bl5398416737448265317bscc_v @ successors @ S )
& ! [S3: set_v] :
( ( ( ord_less_eq_set_v @ S @ S3 )
& ( sCC_Bl5398416737448265317bscc_v @ successors @ S3 ) )
=> ( S3 = S ) ) ) ) ).
% is_scc_def
thf(fact_211_unite__S__tl,axiom,
! [E: sCC_Bl1394983891496994913t_unit,W: v,V: v,N3: v] :
( ( sCC_Bl9196236973127232072t_unit @ successors @ E )
=> ( ( member_v @ W @ ( successors @ V ) )
=> ( ~ ( member_v @ W @ ( sCC_Bl3795065053823578884t_unit @ E @ V ) )
=> ( ( member_v @ W @ ( sCC_Bl4645233313691564917t_unit @ E ) )
=> ( ~ ( member_v @ W @ ( sCC_Bl157864678168468314t_unit @ E ) )
=> ( ( member_v @ N3 @ ( set_v2 @ ( tl_v @ ( sCC_Bl8828226123343373779t_unit @ ( sCC_Bloemen_unite_v @ V @ W @ E ) ) ) ) )
=> ( ( sCC_Bl1280885523602775798t_unit @ ( sCC_Bloemen_unite_v @ V @ W @ E ) @ N3 )
= ( sCC_Bl1280885523602775798t_unit @ E @ N3 ) ) ) ) ) ) ) ) ).
% unite_S_tl
thf(fact_212_tl__cc,axiom,
! [X3: v] :
( ( member_v @ X3 @ ( set_v2 @ ( tl_v @ ( sCC_Bl8828226123343373779t_unit @ e2 ) ) ) )
=> ( ( inf_inf_set_v @ ( sCC_Bl1280885523602775798t_unit @ e @ X3 ) @ cc )
= bot_bot_set_v ) ) ).
% tl_cc
thf(fact_213__092_060open_062_092_060And_062n_Am_O_A_092_060lbrakk_062n_A_092_060in_062_Aset_A_Itl_A_Istack_Ae_H_J_J_059_Am_A_092_060in_062_A_092_060S_062_Ae_An_A_092_060inter_062_Acc_092_060rbrakk_062_A_092_060Longrightarrow_062_AFalse_092_060close_062,axiom,
! [N3: v,M2: v] :
( ( member_v @ N3 @ ( set_v2 @ ( tl_v @ ( sCC_Bl8828226123343373779t_unit @ e2 ) ) ) )
=> ~ ( member_v @ M2 @ ( inf_inf_set_v @ ( sCC_Bl1280885523602775798t_unit @ e @ N3 ) @ cc ) ) ) ).
% \<open>\<And>n m. \<lbrakk>n \<in> set (tl (stack e')); m \<in> \<S> e n \<inter> cc\<rbrakk> \<Longrightarrow> False\<close>
thf(fact_214_succ__reachable,axiom,
! [X5: v,Y5: v,Z2: v] :
( ( sCC_Bl649662514949026229able_v @ successors @ X5 @ Y5 )
=> ( ( member_v @ Z2 @ ( successors @ Y5 ) )
=> ( sCC_Bl649662514949026229able_v @ successors @ X5 @ Z2 ) ) ) ).
% succ_reachable
thf(fact_215_reachable__trans,axiom,
! [X5: v,Y5: v,Z2: v] :
( ( sCC_Bl649662514949026229able_v @ successors @ X5 @ Y5 )
=> ( ( sCC_Bl649662514949026229able_v @ successors @ Y5 @ Z2 )
=> ( sCC_Bl649662514949026229able_v @ successors @ X5 @ Z2 ) ) ) ).
% reachable_trans
thf(fact_216_reachable__end__induct,axiom,
! [X5: v,Y5: v,P: v > v > $o] :
( ( sCC_Bl649662514949026229able_v @ successors @ X5 @ Y5 )
=> ( ! [X4: v] : ( P @ X4 @ X4 )
=> ( ! [X4: v,Y3: v,Z4: v] :
( ( P @ X4 @ Y3 )
=> ( ( member_v @ Z4 @ ( successors @ Y3 ) )
=> ( P @ X4 @ Z4 ) ) )
=> ( P @ X5 @ Y5 ) ) ) ) ).
% reachable_end_induct
thf(fact_217_reachable__edge,axiom,
! [Y5: v,X5: v] :
( ( member_v @ Y5 @ ( successors @ X5 ) )
=> ( sCC_Bl649662514949026229able_v @ successors @ X5 @ Y5 ) ) ).
% reachable_edge
thf(fact_218_reachable_Osimps,axiom,
! [A1: v,A22: v] :
( ( sCC_Bl649662514949026229able_v @ successors @ A1 @ A22 )
= ( ? [X: v] :
( ( A1 = X )
& ( A22 = X ) )
| ? [X: v,Y: v,Z5: v] :
( ( A1 = X )
& ( A22 = Z5 )
& ( member_v @ Y @ ( successors @ X ) )
& ( sCC_Bl649662514949026229able_v @ successors @ Y @ Z5 ) ) ) ) ).
% reachable.simps
thf(fact_219_reachable__succ,axiom,
! [Y5: v,X5: v,Z2: v] :
( ( member_v @ Y5 @ ( successors @ X5 ) )
=> ( ( sCC_Bl649662514949026229able_v @ successors @ Y5 @ Z2 )
=> ( sCC_Bl649662514949026229able_v @ successors @ X5 @ Z2 ) ) ) ).
% reachable_succ
thf(fact_220_reachable__refl,axiom,
! [X5: v] : ( sCC_Bl649662514949026229able_v @ successors @ X5 @ X5 ) ).
% reachable_refl
thf(fact_221_reachable_Ocases,axiom,
! [A1: v,A22: v] :
( ( sCC_Bl649662514949026229able_v @ successors @ A1 @ A22 )
=> ( ( A22 != A1 )
=> ~ ! [Y3: v] :
( ( member_v @ Y3 @ ( successors @ A1 ) )
=> ~ ( sCC_Bl649662514949026229able_v @ successors @ Y3 @ A22 ) ) ) ) ).
% reachable.cases
thf(fact_222_is__subscc__def,axiom,
! [S: set_v] :
( ( sCC_Bl5398416737448265317bscc_v @ successors @ S )
= ( ! [X: v] :
( ( member_v @ X @ S )
=> ! [Y: v] :
( ( member_v @ Y @ S )
=> ( sCC_Bl649662514949026229able_v @ successors @ X @ Y ) ) ) ) ) ).
% is_subscc_def
thf(fact_223_sccE,axiom,
! [S: set_v,X5: v,Y5: v] :
( ( sCC_Bloemen_is_scc_v @ successors @ S )
=> ( ( member_v @ X5 @ S )
=> ( ( sCC_Bl649662514949026229able_v @ successors @ X5 @ Y5 )
=> ( ( sCC_Bl649662514949026229able_v @ successors @ Y5 @ X5 )
=> ( member_v @ Y5 @ S ) ) ) ) ) ).
% sccE
thf(fact_224_subscc__add,axiom,
! [S: set_v,X5: v,Y5: v] :
( ( sCC_Bl5398416737448265317bscc_v @ successors @ S )
=> ( ( member_v @ X5 @ S )
=> ( ( sCC_Bl649662514949026229able_v @ successors @ X5 @ Y5 )
=> ( ( sCC_Bl649662514949026229able_v @ successors @ Y5 @ X5 )
=> ( sCC_Bl5398416737448265317bscc_v @ successors @ ( insert_v @ Y5 @ S ) ) ) ) ) ) ).
% subscc_add
thf(fact_225_calculation_I1_J,axiom,
! [X3: v] :
( ( member_v @ X3 @ ( sCC_Bl4645233313691564917t_unit @ e2 ) )
=> ( sCC_Bl649662514949026229able_v @ successors @ ( sCC_Bl1090238580953940555t_unit @ e2 ) @ X3 ) ) ).
% calculation(1)
thf(fact_226_calculation_I5_J,axiom,
! [X3: v] :
( ( member_v @ X3 @ ( sCC_Bl157864678168468314t_unit @ e2 ) )
=> ! [M: v] :
( ( sCC_Bl649662514949026229able_v @ successors @ X3 @ M )
=> ( member_v @ M @ ( sCC_Bl157864678168468314t_unit @ e2 ) ) ) ) ).
% calculation(5)
thf(fact_227_calculation_I14_J,axiom,
! [N: v,M: v] :
( ( sCC_Bl4022239298816431255edes_v @ N @ M @ ( sCC_Bl8828226123343373779t_unit @ e2 ) )
=> ( sCC_Bl649662514949026229able_v @ successors @ M @ N ) ) ).
% calculation(14)
thf(fact_228_init__env__pre__dfs,axiom,
! [V: v] : ( sCC_Bl36166008131615352t_unit @ successors @ V @ ( sCC_Bl7693227186847904995_env_v @ V ) ) ).
% init_env_pre_dfs
thf(fact_229_reachable__re,axiom,
! [X5: v,Y5: v] :
( ( sCC_Bl649662514949026229able_v @ successors @ X5 @ Y5 )
=> ( sCC_Bl770211535891879572_end_v @ successors @ X5 @ Y5 ) ) ).
% reachable_re
thf(fact_230_re__reachable,axiom,
! [X5: v,Y5: v] :
( ( sCC_Bl770211535891879572_end_v @ successors @ X5 @ Y5 )
=> ( sCC_Bl649662514949026229able_v @ successors @ X5 @ Y5 ) ) ).
% re_reachable
thf(fact_231_succ__re,axiom,
! [Y5: v,X5: v,Z2: v] :
( ( member_v @ Y5 @ ( successors @ X5 ) )
=> ( ( sCC_Bl770211535891879572_end_v @ successors @ Y5 @ Z2 )
=> ( sCC_Bl770211535891879572_end_v @ successors @ X5 @ Z2 ) ) ) ).
% succ_re
thf(fact_232_reachable__end_Osimps,axiom,
! [A1: v,A22: v] :
( ( sCC_Bl770211535891879572_end_v @ successors @ A1 @ A22 )
= ( ? [X: v] :
( ( A1 = X )
& ( A22 = X ) )
| ? [X: v,Y: v,Z5: v] :
( ( A1 = X )
& ( A22 = Z5 )
& ( sCC_Bl770211535891879572_end_v @ successors @ X @ Y )
& ( member_v @ Z5 @ ( successors @ Y ) ) ) ) ) ).
% reachable_end.simps
thf(fact_233_re__succ,axiom,
! [X5: v,Y5: v,Z2: v] :
( ( sCC_Bl770211535891879572_end_v @ successors @ X5 @ Y5 )
=> ( ( member_v @ Z2 @ ( successors @ Y5 ) )
=> ( sCC_Bl770211535891879572_end_v @ successors @ X5 @ Z2 ) ) ) ).
% re_succ
thf(fact_234_re__refl,axiom,
! [X5: v] : ( sCC_Bl770211535891879572_end_v @ successors @ X5 @ X5 ) ).
% re_refl
thf(fact_235_reachable__end_Ocases,axiom,
! [A1: v,A22: v] :
( ( sCC_Bl770211535891879572_end_v @ successors @ A1 @ A22 )
=> ( ( A22 != A1 )
=> ~ ! [Y3: v] :
( ( sCC_Bl770211535891879572_end_v @ successors @ A1 @ Y3 )
=> ~ ( member_v @ A22 @ ( successors @ Y3 ) ) ) ) ) ).
% reachable_end.cases
thf(fact_236_reachable__avoiding_Ocases,axiom,
! [A1: v,A22: v,A32: set_Product_prod_v_v] :
( ( sCC_Bl4291963740693775144ding_v @ successors @ A1 @ A22 @ A32 )
=> ( ( A22 != A1 )
=> ~ ! [Y3: v] :
( ( sCC_Bl4291963740693775144ding_v @ successors @ A1 @ Y3 @ A32 )
=> ( ( member_v @ A22 @ ( successors @ Y3 ) )
=> ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ Y3 @ A22 ) @ A32 ) ) ) ) ) ).
% reachable_avoiding.cases
thf(fact_237_ra__mono,axiom,
! [X5: v,Y5: v,E6: set_Product_prod_v_v,E7: set_Product_prod_v_v] :
( ( sCC_Bl4291963740693775144ding_v @ successors @ X5 @ Y5 @ E6 )
=> ( ( ord_le7336532860387713383od_v_v @ E7 @ E6 )
=> ( sCC_Bl4291963740693775144ding_v @ successors @ X5 @ Y5 @ E7 ) ) ) ).
% ra_mono
thf(fact_238_ra__refl,axiom,
! [X5: v,E6: set_Product_prod_v_v] : ( sCC_Bl4291963740693775144ding_v @ successors @ X5 @ X5 @ E6 ) ).
% ra_refl
thf(fact_239_ra__trans,axiom,
! [X5: v,Y5: v,E6: set_Product_prod_v_v,Z2: v] :
( ( sCC_Bl4291963740693775144ding_v @ successors @ X5 @ Y5 @ E6 )
=> ( ( sCC_Bl4291963740693775144ding_v @ successors @ Y5 @ Z2 @ E6 )
=> ( sCC_Bl4291963740693775144ding_v @ successors @ X5 @ Z2 @ E6 ) ) ) ).
% ra_trans
thf(fact_240_ra__reachable,axiom,
! [X5: v,Y5: v,E6: set_Product_prod_v_v] :
( ( sCC_Bl4291963740693775144ding_v @ successors @ X5 @ Y5 @ E6 )
=> ( sCC_Bl649662514949026229able_v @ successors @ X5 @ Y5 ) ) ).
% ra_reachable
thf(fact_241_ra__cases,axiom,
! [X5: v,Y5: v,E6: set_Product_prod_v_v] :
( ( sCC_Bl4291963740693775144ding_v @ successors @ X5 @ Y5 @ E6 )
=> ( ( X5 = Y5 )
| ? [Z4: v] :
( ( member_v @ Z4 @ ( successors @ X5 ) )
& ~ ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ X5 @ Z4 ) @ E6 )
& ( sCC_Bl4291963740693775144ding_v @ successors @ Z4 @ Y5 @ E6 ) ) ) ) ).
% ra_cases
thf(fact_242_edge__ra,axiom,
! [Y5: v,X5: v,E6: set_Product_prod_v_v] :
( ( member_v @ Y5 @ ( successors @ X5 ) )
=> ( ~ ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ X5 @ Y5 ) @ E6 )
=> ( sCC_Bl4291963740693775144ding_v @ successors @ X5 @ Y5 @ E6 ) ) ) ).
% edge_ra
thf(fact_243_reachable__avoiding_Osimps,axiom,
! [A1: v,A22: v,A32: set_Product_prod_v_v] :
( ( sCC_Bl4291963740693775144ding_v @ successors @ A1 @ A22 @ A32 )
= ( ? [X: v,E8: set_Product_prod_v_v] :
( ( A1 = X )
& ( A22 = X )
& ( A32 = E8 ) )
| ? [X: v,Y: v,E8: set_Product_prod_v_v,Z5: v] :
( ( A1 = X )
& ( A22 = Z5 )
& ( A32 = E8 )
& ( sCC_Bl4291963740693775144ding_v @ successors @ X @ Y @ E8 )
& ( member_v @ Z5 @ ( successors @ Y ) )
& ~ ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ Y @ Z5 ) @ E8 ) ) ) ) ).
% reachable_avoiding.simps
thf(fact_244_ra__succ,axiom,
! [X5: v,Y5: v,E6: set_Product_prod_v_v,Z2: v] :
( ( sCC_Bl4291963740693775144ding_v @ successors @ X5 @ Y5 @ E6 )
=> ( ( member_v @ Z2 @ ( successors @ Y5 ) )
=> ( ~ ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ Y5 @ Z2 ) @ E6 )
=> ( sCC_Bl4291963740693775144ding_v @ successors @ X5 @ Z2 @ E6 ) ) ) ) ).
% ra_succ
thf(fact_245_ra__empty,axiom,
! [X5: v,Y5: v] :
( ( sCC_Bl4291963740693775144ding_v @ successors @ X5 @ Y5 @ bot_bo723834152578015283od_v_v )
= ( sCC_Bl649662514949026229able_v @ successors @ X5 @ Y5 ) ) ).
% ra_empty
thf(fact_246_ra__add__edge,axiom,
! [X5: v,Y5: v,E6: set_Product_prod_v_v,V: v,W: v] :
( ( sCC_Bl4291963740693775144ding_v @ successors @ X5 @ Y5 @ E6 )
=> ( ( sCC_Bl4291963740693775144ding_v @ successors @ X5 @ Y5 @ ( sup_su414716646722978715od_v_v @ E6 @ ( insert1338601472111419319od_v_v @ ( product_Pair_v_v @ V @ W ) @ bot_bo723834152578015283od_v_v ) ) )
| ( ( sCC_Bl4291963740693775144ding_v @ successors @ X5 @ V @ ( sup_su414716646722978715od_v_v @ E6 @ ( insert1338601472111419319od_v_v @ ( product_Pair_v_v @ V @ W ) @ bot_bo723834152578015283od_v_v ) ) )
& ( sCC_Bl4291963740693775144ding_v @ successors @ W @ Y5 @ ( sup_su414716646722978715od_v_v @ E6 @ ( insert1338601472111419319od_v_v @ ( product_Pair_v_v @ V @ W ) @ bot_bo723834152578015283od_v_v ) ) ) ) ) ) ).
% ra_add_edge
thf(fact_247_vfin,axiom,
finite_finite_v @ vertices ).
% vfin
thf(fact_248_bot__unit__def,axiom,
bot_bot_Product_unit = product_Unity ).
% bot_unit_def
thf(fact_249_sup__unit__def,axiom,
( sup_sup_Product_unit
= ( ^ [Uu2: product_unit,Uv: product_unit] : product_Unity ) ) ).
% sup_unit_def
thf(fact_250_inf__unit__def,axiom,
( inf_inf_Product_unit
= ( ^ [Uu2: product_unit,Uv: product_unit] : product_Unity ) ) ).
% inf_unit_def
% Helper facts (3)
thf(help_If_3_1_If_001t__Set__Oset_Itf__v_J_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__Set__Oset_Itf__v_J_T,axiom,
! [X5: set_v,Y5: set_v] :
( ( if_set_v @ $false @ X5 @ Y5 )
= Y5 ) ).
thf(help_If_1_1_If_001t__Set__Oset_Itf__v_J_T,axiom,
! [X5: set_v,Y5: set_v] :
( ( if_set_v @ $true @ X5 @ Y5 )
= X5 ) ).
% Conjectures (1)
thf(conj_0,conjecture,
( ( comple2307003700295860064_set_v
@ ( collect_set_v
@ ^ [Uu: set_v] :
? [N2: v] :
( ( Uu
= ( sCC_Bl1280885523602775798t_unit @ e2 @ N2 ) )
& ( member_v @ N2 @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ e2 ) ) ) ) ) )
= ( minus_minus_set_v @ ( sCC_Bl4645233313691564917t_unit @ e2 ) @ ( sCC_Bl157864678168468314t_unit @ e2 ) ) ) ).
%------------------------------------------------------------------------------