TPTP Problem File: SLH0252^1.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_01269_043626__5919720_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1192 ( 586 unt; 136 typ; 0 def)
% Number of atoms : 2900 (1218 equ; 0 cnn)
% Maximal formula atoms : 10 ( 2 avg)
% Number of connectives : 9529 ( 318 ~; 49 |; 303 &;7807 @)
% ( 0 <=>;1052 =>; 0 <=; 0 <~>)
% Maximal formula depth : 21 ( 6 avg)
% Number of types : 13 ( 12 usr)
% Number of type conns : 470 ( 470 >; 0 *; 0 +; 0 <<)
% Number of symbols : 127 ( 124 usr; 18 con; 0-9 aty)
% Number of variables : 3072 ( 265 ^;2660 !; 147 ?;3072 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 08:53:25.937
%------------------------------------------------------------------------------
% Could-be-implicit typings (12)
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thf(sy_c_SCC__Bloemen__Sequential_Ograph_Oreachable_001tf__v,type,
sCC_Bl649662514949026229able_v: ( v > set_v ) > v > v > $o ).
thf(sy_c_SCC__Bloemen__Sequential_Ograph_Oreachable__avoiding_001tf__v,type,
sCC_Bl4291963740693775144ding_v: ( v > set_v ) > v > v > set_Product_prod_v_v > $o ).
thf(sy_c_SCC__Bloemen__Sequential_Ograph_Oreachable__end_001t__Product____Type__Oprod_Itf__v_Mtf__v_J,type,
sCC_Bl4714988717384592488od_v_v: ( product_prod_v_v > set_Product_prod_v_v ) > product_prod_v_v > product_prod_v_v > $o ).
thf(sy_c_SCC__Bloemen__Sequential_Ograph_Oreachable__end_001tf__v,type,
sCC_Bl770211535891879572_end_v: ( v > set_v ) > v > v > $o ).
thf(sy_c_SCC__Bloemen__Sequential_Ograph_Osub__env_001t__Product____Type__Oprod_Itf__v_Mtf__v_J_001t__Product____Type__Ounit_001t__Product____Type__Ounit,type,
sCC_Bl7963838319573962697t_unit: sCC_Bl7326425374436813197t_unit > sCC_Bl7326425374436813197t_unit > $o ).
thf(sy_c_SCC__Bloemen__Sequential_Ograph_Osub__env_001tf__v_001t__Product____Type__Ounit_001t__Product____Type__Ounit,type,
sCC_Bl5768913643336123637t_unit: sCC_Bl1394983891496994913t_unit > sCC_Bl1394983891496994913t_unit > $o ).
thf(sy_c_SCC__Bloemen__Sequential_Ograph_Ounite_001t__Product____Type__Oprod_Itf__v_Mtf__v_J,type,
sCC_Bl4702006153222411093od_v_v: product_prod_v_v > product_prod_v_v > sCC_Bl7326425374436813197t_unit > sCC_Bl7326425374436813197t_unit ).
thf(sy_c_SCC__Bloemen__Sequential_Ograph_Ounite_001tf__v,type,
sCC_Bloemen_unite_v: v > v > sCC_Bl1394983891496994913t_unit > sCC_Bl1394983891496994913t_unit ).
thf(sy_c_SCC__Bloemen__Sequential_Ograph_Owf__env_001t__Product____Type__Oprod_Itf__v_Mtf__v_J_001t__Product____Type__Ounit,type,
sCC_Bl7798947040364291444t_unit: ( product_prod_v_v > set_Product_prod_v_v ) > sCC_Bl7326425374436813197t_unit > $o ).
thf(sy_c_SCC__Bloemen__Sequential_Ograph_Owf__env_001tf__v_001t__Product____Type__Ounit,type,
sCC_Bl9196236973127232072t_unit: ( v > set_v ) > sCC_Bl1394983891496994913t_unit > $o ).
thf(sy_c_SCC__Bloemen__Sequential_Oinit__env_001tf__v,type,
sCC_Bl7693227186847904995_env_v: v > sCC_Bl1394983891496994913t_unit ).
thf(sy_c_SCC__Bloemen__Sequential_Oprecedes_001t__Product____Type__Oprod_Itf__v_Mtf__v_J,type,
sCC_Bl2026170059108282219od_v_v: product_prod_v_v > product_prod_v_v > list_P7986770385144383213od_v_v > $o ).
thf(sy_c_SCC__Bloemen__Sequential_Oprecedes_001tf__v,type,
sCC_Bl4022239298816431255edes_v: v > v > list_v > $o ).
thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_Itf__v_Mtf__v_J,type,
collec140062887454715474od_v_v: ( product_prod_v_v > $o ) > set_Product_prod_v_v ).
thf(sy_c_Set_OCollect_001t__Set__Oset_It__Product____Type__Oprod_Itf__v_Mtf__v_J_J,type,
collec8263177866097347122od_v_v: ( set_Product_prod_v_v > $o ) > set_se8455005133513928103od_v_v ).
thf(sy_c_Set_OCollect_001t__Set__Oset_Itf__v_J,type,
collect_set_v: ( set_v > $o ) > set_set_v ).
thf(sy_c_Set_OCollect_001tf__v,type,
collect_v: ( v > $o ) > set_v ).
thf(sy_c_Set_Oinsert_001t__Product____Type__Oprod_Itf__v_Mtf__v_J,type,
insert1338601472111419319od_v_v: product_prod_v_v > set_Product_prod_v_v > set_Product_prod_v_v ).
thf(sy_c_Set_Oinsert_001t__Set__Oset_It__Product____Type__Oprod_Itf__v_Mtf__v_J_J,type,
insert7504383016908236695od_v_v: set_Product_prod_v_v > set_se8455005133513928103od_v_v > set_se8455005133513928103od_v_v ).
thf(sy_c_Set_Oinsert_001t__Set__Oset_Itf__v_J,type,
insert_set_v: set_v > set_set_v > set_set_v ).
thf(sy_c_Set_Oinsert_001tf__v,type,
insert_v: v > set_v > set_v ).
thf(sy_c_Set_Othe__elem_001t__Product____Type__Oprod_Itf__v_Mtf__v_J,type,
the_el5392834299063928540od_v_v: set_Product_prod_v_v > product_prod_v_v ).
thf(sy_c_Set_Othe__elem_001tf__v,type,
the_elem_v: set_v > v ).
thf(sy_c_member_001t__Product____Type__Oprod_Itf__v_Mtf__v_J,type,
member7453568604450474000od_v_v: product_prod_v_v > set_Product_prod_v_v > $o ).
thf(sy_c_member_001t__Set__Oset_It__Product____Type__Oprod_Itf__v_Mtf__v_J_J,type,
member8406446414694345712od_v_v: set_Product_prod_v_v > set_se8455005133513928103od_v_v > $o ).
thf(sy_c_member_001t__Set__Oset_Itf__v_J,type,
member_set_v: set_v > set_set_v > $o ).
thf(sy_c_member_001tf__v,type,
member_v: v > set_v > $o ).
thf(sy_v_cc____,type,
cc: set_v ).
thf(sy_v_e,type,
e: sCC_Bl1394983891496994913t_unit ).
thf(sy_v_e_H,type,
e2: sCC_Bl1394983891496994913t_unit ).
thf(sy_v_m____,type,
m: v ).
thf(sy_v_n____,type,
n: v ).
thf(sy_v_pfx____,type,
pfx: list_v ).
thf(sy_v_successors,type,
successors: v > set_v ).
thf(sy_v_v,type,
v2: v ).
thf(sy_v_vertices,type,
vertices: set_v ).
thf(sy_v_w,type,
w: v ).
% Relevant facts (1052)
thf(fact_0__092_060open_062m_A_092_060notin_062_Acc_092_060close_062,axiom,
~ ( member_v @ m @ cc ) ).
% \<open>m \<notin> cc\<close>
thf(fact_1_False,axiom,
~ ( member_v @ n @ cc ) ).
% False
thf(fact_2__092_060open_062_092_060S_062_Ae_An_A_061_A_092_060S_062_Ae_Am_092_060close_062,axiom,
( ( sCC_Bl1280885523602775798t_unit @ e @ n )
= ( sCC_Bl1280885523602775798t_unit @ e @ m ) ) ).
% \<open>\<S> e n = \<S> e m\<close>
thf(fact_3_l,axiom,
member_v @ m @ ( sCC_Bl1280885523602775798t_unit @ e2 @ n ) ).
% l
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_Se_H,axiom,
! [X: v] :
( ( ( member_v @ X @ cc )
=> ( ( sCC_Bl1280885523602775798t_unit @ e2 @ X )
= cc ) )
& ( ~ ( member_v @ X @ cc )
=> ( ( sCC_Bl1280885523602775798t_unit @ e2 @ X )
= ( sCC_Bl1280885523602775798t_unit @ e @ X ) ) ) ) ).
% Se'
thf(fact_6_fold__congs_I2_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_Bl1280885523602775798t_unit @ R2 )
= V )
=> ( ! [V2: v > set_v] :
( ( V = V2 )
=> ( ( F @ V2 )
= ( F2 @ V2 ) ) )
=> ( ( sCC_Bl3155122997657187039t_unit @ F @ R )
= ( sCC_Bl3155122997657187039t_unit @ F2 @ R2 ) ) ) ) ) ).
% fold_congs(2)
thf(fact_7_unfold__congs_I2_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_Bl1280885523602775798t_unit @ R2 )
= V )
=> ( ! [V2: v > set_v] :
( ( V2 = V )
=> ( ( F @ V2 )
= ( F2 @ V2 ) ) )
=> ( ( sCC_Bl3155122997657187039t_unit @ F @ R )
= ( sCC_Bl3155122997657187039t_unit @ F2 @ R2 ) ) ) ) ) ).
% unfold_congs(2)
thf(fact_8_pfx_I2_J,axiom,
( ( sCC_Bl8828226123343373779t_unit @ e2 )
!= nil_v ) ).
% pfx(2)
thf(fact_9_e_H__def,axiom,
( e2
= ( sCC_Bloemen_unite_v @ v2 @ w @ e ) ) ).
% e'_def
thf(fact_10_hd__cc,axiom,
( ( sCC_Bl1280885523602775798t_unit @ e2 @ ( hd_v @ ( sCC_Bl8828226123343373779t_unit @ e2 ) ) )
= cc ) ).
% hd_cc
thf(fact_11__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_12_calculation_I2_J,axiom,
distinct_v @ ( sCC_Bl9201514103433284750t_unit @ e2 ) ).
% calculation(2)
thf(fact_13_select__convs_I2_J,axiom,
! [Root: v,S: v > set_v,Explored: set_v,Visited: set_v,Vsuccs: v > set_v,Sccs: set_set_v,Stack: list_v,Cstack: list_v,More: product_unit] :
( ( sCC_Bl1280885523602775798t_unit @ ( sCC_Bl8064756265740546429t_unit @ Root @ S @ Explored @ Visited @ Vsuccs @ Sccs @ Stack @ Cstack @ More ) )
= S ) ).
% select_convs(2)
thf(fact_14_calculation_I12_J,axiom,
distinct_v @ ( sCC_Bl8828226123343373779t_unit @ e2 ) ).
% calculation(12)
thf(fact_15_pfx_I1_J,axiom,
( ( sCC_Bl8828226123343373779t_unit @ e )
= ( append_v @ pfx @ ( sCC_Bl8828226123343373779t_unit @ e2 ) ) ) ).
% pfx(1)
thf(fact_16_calculation_I11_J,axiom,
( ( comple2307003700295860064_set_v @ ( sCC_Bl2536197123907397897t_unit @ e2 ) )
= ( sCC_Bl157864678168468314t_unit @ e2 ) ) ).
% calculation(11)
thf(fact_17_w_I4_J,axiom,
~ ( member_v @ w @ ( sCC_Bl157864678168468314t_unit @ e ) ) ).
% w(4)
thf(fact_18_w_I3_J,axiom,
member_v @ w @ ( sCC_Bl4645233313691564917t_unit @ e ) ).
% w(3)
thf(fact_19_w_I2_J,axiom,
~ ( member_v @ w @ ( sCC_Bl3795065053823578884t_unit @ e @ v2 ) ) ).
% w(2)
thf(fact_20_select__convs_I8_J,axiom,
! [Root: v,S: v > set_v,Explored: set_v,Visited: set_v,Vsuccs: v > set_v,Sccs: set_set_v,Stack: list_v,Cstack: list_v,More: product_unit] :
( ( sCC_Bl9201514103433284750t_unit @ ( sCC_Bl8064756265740546429t_unit @ Root @ S @ Explored @ Visited @ Vsuccs @ Sccs @ Stack @ Cstack @ More ) )
= Cstack ) ).
% select_convs(8)
thf(fact_21_select__convs_I7_J,axiom,
! [Root: v,S: v > set_v,Explored: set_v,Visited: set_v,Vsuccs: v > set_v,Sccs: set_set_v,Stack: list_v,Cstack: list_v,More: product_unit] :
( ( sCC_Bl8828226123343373779t_unit @ ( sCC_Bl8064756265740546429t_unit @ Root @ S @ Explored @ Visited @ Vsuccs @ Sccs @ Stack @ Cstack @ More ) )
= Stack ) ).
% select_convs(7)
thf(fact_22_select__convs_I6_J,axiom,
! [Root: v,S: v > set_v,Explored: set_v,Visited: set_v,Vsuccs: v > set_v,Sccs: set_set_v,Stack: list_v,Cstack: list_v,More: product_unit] :
( ( sCC_Bl2536197123907397897t_unit @ ( sCC_Bl8064756265740546429t_unit @ Root @ S @ Explored @ Visited @ Vsuccs @ Sccs @ Stack @ Cstack @ More ) )
= Sccs ) ).
% select_convs(6)
thf(fact_23_select__convs_I3_J,axiom,
! [Root: v,S: v > set_v,Explored: set_v,Visited: set_v,Vsuccs: v > set_v,Sccs: set_set_v,Stack: list_v,Cstack: list_v,More: product_unit] :
( ( sCC_Bl157864678168468314t_unit @ ( sCC_Bl8064756265740546429t_unit @ Root @ S @ Explored @ Visited @ Vsuccs @ Sccs @ Stack @ Cstack @ More ) )
= Explored ) ).
% select_convs(3)
thf(fact_24_update__convs_I2_J,axiom,
! [S2: ( v > set_v ) > v > set_v,Root: v,S: v > set_v,Explored: set_v,Visited: set_v,Vsuccs: v > set_v,Sccs: set_set_v,Stack: list_v,Cstack: list_v,More: product_unit] :
( ( sCC_Bl3155122997657187039t_unit @ S2 @ ( sCC_Bl8064756265740546429t_unit @ Root @ S @ Explored @ Visited @ Vsuccs @ Sccs @ Stack @ Cstack @ More ) )
= ( sCC_Bl8064756265740546429t_unit @ Root @ ( S2 @ S ) @ Explored @ Visited @ Vsuccs @ Sccs @ Stack @ Cstack @ More ) ) ).
% update_convs(2)
thf(fact_25_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_26_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_27_append_Oright__neutral,axiom,
! [A: list_v] :
( ( append_v @ A @ nil_v )
= A ) ).
% append.right_neutral
thf(fact_28_append__Nil2,axiom,
! [Xs: list_v] :
( ( append_v @ Xs @ nil_v )
= Xs ) ).
% append_Nil2
thf(fact_29_append__self__conv,axiom,
! [Xs: list_v,Ys: list_v] :
( ( ( append_v @ Xs @ Ys )
= Xs )
= ( Ys = nil_v ) ) ).
% append_self_conv
thf(fact_30_self__append__conv,axiom,
! [Y: list_v,Ys: list_v] :
( ( Y
= ( append_v @ Y @ Ys ) )
= ( Ys = nil_v ) ) ).
% self_append_conv
thf(fact_31_append__self__conv2,axiom,
! [Xs: list_v,Ys: list_v] :
( ( ( append_v @ Xs @ Ys )
= Ys )
= ( Xs = nil_v ) ) ).
% append_self_conv2
thf(fact_32_self__append__conv2,axiom,
! [Y: list_v,Xs: list_v] :
( ( Y
= ( append_v @ Xs @ Y ) )
= ( Xs = nil_v ) ) ).
% self_append_conv2
thf(fact_33_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_34_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_35_w_I1_J,axiom,
member_v @ w @ ( successors @ v2 ) ).
% w(1)
thf(fact_36_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_37_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_38_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_39_mem__Collect__eq,axiom,
! [A: v,P: v > $o] :
( ( member_v @ A @ ( collect_v @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_40_mem__Collect__eq,axiom,
! [A: product_prod_v_v,P: product_prod_v_v > $o] :
( ( member7453568604450474000od_v_v @ A @ ( collec140062887454715474od_v_v @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_41_mem__Collect__eq,axiom,
! [A: set_v,P: set_v > $o] :
( ( member_set_v @ A @ ( collect_set_v @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_42_Collect__mem__eq,axiom,
! [A2: set_v] :
( ( collect_v
@ ^ [X2: v] : ( member_v @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_43_Collect__mem__eq,axiom,
! [A2: set_Product_prod_v_v] :
( ( collec140062887454715474od_v_v
@ ^ [X2: product_prod_v_v] : ( member7453568604450474000od_v_v @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_44_Collect__mem__eq,axiom,
! [A2: set_set_v] :
( ( collect_set_v
@ ^ [X2: set_v] : ( member_set_v @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_45_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_46_append_Oassoc,axiom,
! [A: list_v,B: list_v,C: list_v] :
( ( append_v @ ( append_v @ A @ B ) @ C )
= ( append_v @ A @ ( append_v @ B @ C ) ) ) ).
% append.assoc
thf(fact_47_calculation_I8_J,axiom,
! [X: v] :
( ( member_v @ X @ ( sCC_Bl157864678168468314t_unit @ e2 ) )
=> ( ( sCC_Bl3795065053823578884t_unit @ e2 @ X )
= ( successors @ X ) ) ) ).
% calculation(8)
thf(fact_48_pre,axiom,
sCC_Bl1748261141445803503t_unit @ successors @ v2 @ e ).
% pre
thf(fact_49_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_50_precedes__append__right,axiom,
! [X4: v,Y: v,Xs: list_v,Ys: list_v] :
( ( sCC_Bl4022239298816431255edes_v @ X4 @ Y @ Xs )
=> ( sCC_Bl4022239298816431255edes_v @ X4 @ Y @ ( append_v @ Xs @ Ys ) ) ) ).
% precedes_append_right
thf(fact_51_precedes__append__left,axiom,
! [X4: v,Y: v,Xs: list_v,Ys: list_v] :
( ( sCC_Bl4022239298816431255edes_v @ X4 @ Y @ Xs )
=> ( sCC_Bl4022239298816431255edes_v @ X4 @ Y @ ( append_v @ Ys @ Xs ) ) ) ).
% precedes_append_left
thf(fact_52_precedes__antisym,axiom,
! [X4: v,Y: v,Xs: list_v] :
( ( sCC_Bl4022239298816431255edes_v @ X4 @ Y @ Xs )
=> ( ( sCC_Bl4022239298816431255edes_v @ Y @ X4 @ Xs )
=> ( ( distinct_v @ Xs )
=> ( X4 = Y ) ) ) ) ).
% precedes_antisym
thf(fact_53_precedes__trans,axiom,
! [X4: v,Y: v,Xs: list_v,Z: v] :
( ( sCC_Bl4022239298816431255edes_v @ X4 @ Y @ Xs )
=> ( ( sCC_Bl4022239298816431255edes_v @ Y @ Z @ Xs )
=> ( ( distinct_v @ Xs )
=> ( sCC_Bl4022239298816431255edes_v @ X4 @ Z @ Xs ) ) ) ) ).
% precedes_trans
thf(fact_54_select__convs_I4_J,axiom,
! [Root: v,S: v > set_v,Explored: set_v,Visited: set_v,Vsuccs: v > set_v,Sccs: set_set_v,Stack: list_v,Cstack: list_v,More: product_unit] :
( ( sCC_Bl4645233313691564917t_unit @ ( sCC_Bl8064756265740546429t_unit @ Root @ S @ Explored @ Visited @ Vsuccs @ Sccs @ Stack @ Cstack @ More ) )
= Visited ) ).
% select_convs(4)
thf(fact_55_select__convs_I5_J,axiom,
! [Root: v,S: v > set_v,Explored: set_v,Visited: set_v,Vsuccs: v > set_v,Sccs: set_set_v,Stack: list_v,Cstack: list_v,More: product_unit] :
( ( sCC_Bl3795065053823578884t_unit @ ( sCC_Bl8064756265740546429t_unit @ Root @ S @ Explored @ Visited @ Vsuccs @ Sccs @ Stack @ Cstack @ More ) )
= Vsuccs ) ).
% select_convs(5)
thf(fact_56_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_57_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_58_eq__Nil__appendI,axiom,
! [Xs: list_v,Ys: list_v] :
( ( Xs = Ys )
=> ( Xs
= ( append_v @ nil_v @ Ys ) ) ) ).
% eq_Nil_appendI
thf(fact_59_append_Oleft__neutral,axiom,
! [A: list_v] :
( ( append_v @ nil_v @ A )
= A ) ).
% append.left_neutral
thf(fact_60_append__Nil,axiom,
! [Ys: list_v] :
( ( append_v @ nil_v @ Ys )
= Ys ) ).
% append_Nil
thf(fact_61_distinct_Osimps_I1_J,axiom,
distinct_v @ nil_v ).
% distinct.simps(1)
thf(fact_62_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_63_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_64_calculation_I10_J,axiom,
! [X: set_v] :
( ( member_set_v @ X @ ( sCC_Bl2536197123907397897t_unit @ e2 ) )
=> ( sCC_Bloemen_is_scc_v @ successors @ X ) ) ).
% calculation(10)
thf(fact_65_dfs__S__tl__stack_I1_J,axiom,
! [V3: v,E: sCC_Bl1394983891496994913t_unit,E2: sCC_Bl1394983891496994913t_unit] :
( ( sCC_Bl8953792750115413617t_unit @ successors @ V3 @ E @ E2 )
=> ( ( ( sCC_Bl8828226123343373779t_unit @ E )
!= nil_v )
=> ( ( sCC_Bl8828226123343373779t_unit @ E2 )
!= nil_v ) ) ) ).
% dfs_S_tl_stack(1)
thf(fact_66_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_67_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_68_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_69_local_Owf,axiom,
sCC_Bl9196236973127232072t_unit @ successors @ e ).
% local.wf
thf(fact_70_calculation_I3_J,axiom,
ord_less_eq_set_v @ ( sCC_Bl157864678168468314t_unit @ e2 ) @ ( sCC_Bl4645233313691564917t_unit @ e2 ) ).
% calculation(3)
thf(fact_71_S__reflexive,axiom,
! [E: sCC_Bl1394983891496994913t_unit,N2: v] :
( ( sCC_Bl9196236973127232072t_unit @ successors @ E )
=> ( member_v @ N2 @ ( sCC_Bl1280885523602775798t_unit @ E @ N2 ) ) ) ).
% S_reflexive
thf(fact_72_calculation_I5_J,axiom,
! [X: v] :
( ( member_v @ X @ ( sCC_Bl157864678168468314t_unit @ e2 ) )
=> ! [M: v] :
( ( sCC_Bl649662514949026229able_v @ successors @ X @ M )
=> ( member_v @ M @ ( sCC_Bl157864678168468314t_unit @ e2 ) ) ) ) ).
% calculation(5)
thf(fact_73_Union__iff,axiom,
! [A2: product_prod_v_v,C2: set_se8455005133513928103od_v_v] :
( ( member7453568604450474000od_v_v @ A2 @ ( comple5788137035815166516od_v_v @ C2 ) )
= ( ? [X2: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ X2 @ C2 )
& ( member7453568604450474000od_v_v @ A2 @ X2 ) ) ) ) ).
% Union_iff
thf(fact_74_Union__iff,axiom,
! [A2: v,C2: set_set_v] :
( ( member_v @ A2 @ ( comple2307003700295860064_set_v @ C2 ) )
= ( ? [X2: set_v] :
( ( member_set_v @ X2 @ C2 )
& ( member_v @ A2 @ X2 ) ) ) ) ).
% Union_iff
thf(fact_75_UnionI,axiom,
! [X5: set_Product_prod_v_v,C2: set_se8455005133513928103od_v_v,A2: product_prod_v_v] :
( ( member8406446414694345712od_v_v @ X5 @ C2 )
=> ( ( member7453568604450474000od_v_v @ A2 @ X5 )
=> ( member7453568604450474000od_v_v @ A2 @ ( comple5788137035815166516od_v_v @ C2 ) ) ) ) ).
% UnionI
thf(fact_76_UnionI,axiom,
! [X5: set_v,C2: set_set_v,A2: v] :
( ( member_set_v @ X5 @ C2 )
=> ( ( member_v @ A2 @ X5 )
=> ( member_v @ A2 @ ( comple2307003700295860064_set_v @ C2 ) ) ) ) ).
% UnionI
thf(fact_77_reachable_Ocases,axiom,
! [A1: v,A22: v] :
( ( sCC_Bl649662514949026229able_v @ successors @ A1 @ A22 )
=> ( ( A22 != A1 )
=> ~ ! [Y2: v] :
( ( member_v @ Y2 @ ( successors @ A1 ) )
=> ~ ( sCC_Bl649662514949026229able_v @ successors @ Y2 @ A22 ) ) ) ) ).
% reachable.cases
thf(fact_78_reachable__refl,axiom,
! [X4: v] : ( sCC_Bl649662514949026229able_v @ successors @ X4 @ X4 ) ).
% reachable_refl
thf(fact_79_reachable__succ,axiom,
! [Y: v,X4: v,Z: v] :
( ( member_v @ Y @ ( successors @ X4 ) )
=> ( ( sCC_Bl649662514949026229able_v @ successors @ Y @ Z )
=> ( sCC_Bl649662514949026229able_v @ successors @ X4 @ Z ) ) ) ).
% reachable_succ
thf(fact_80_reachable_Osimps,axiom,
! [A1: v,A22: v] :
( ( sCC_Bl649662514949026229able_v @ successors @ A1 @ A22 )
= ( ? [X2: v] :
( ( A1 = X2 )
& ( A22 = X2 ) )
| ? [X2: v,Y3: v,Z2: v] :
( ( A1 = X2 )
& ( A22 = Z2 )
& ( member_v @ Y3 @ ( successors @ X2 ) )
& ( sCC_Bl649662514949026229able_v @ successors @ Y3 @ Z2 ) ) ) ) ).
% reachable.simps
thf(fact_81_reachable__edge,axiom,
! [Y: v,X4: v] :
( ( member_v @ Y @ ( successors @ X4 ) )
=> ( sCC_Bl649662514949026229able_v @ successors @ X4 @ Y ) ) ).
% reachable_edge
thf(fact_82_reachable__end__induct,axiom,
! [X4: v,Y: v,P: v > v > $o] :
( ( sCC_Bl649662514949026229able_v @ successors @ X4 @ Y )
=> ( ! [X3: v] : ( P @ X3 @ X3 )
=> ( ! [X3: v,Y2: v,Z3: v] :
( ( P @ X3 @ Y2 )
=> ( ( member_v @ Z3 @ ( successors @ Y2 ) )
=> ( P @ X3 @ Z3 ) ) )
=> ( P @ X4 @ Y ) ) ) ) ).
% reachable_end_induct
thf(fact_83_reachable__trans,axiom,
! [X4: v,Y: v,Z: v] :
( ( sCC_Bl649662514949026229able_v @ successors @ X4 @ Y )
=> ( ( sCC_Bl649662514949026229able_v @ successors @ Y @ Z )
=> ( sCC_Bl649662514949026229able_v @ successors @ X4 @ Z ) ) ) ).
% reachable_trans
thf(fact_84_succ__reachable,axiom,
! [X4: v,Y: v,Z: v] :
( ( sCC_Bl649662514949026229able_v @ successors @ X4 @ Y )
=> ( ( member_v @ Z @ ( successors @ Y ) )
=> ( sCC_Bl649662514949026229able_v @ successors @ X4 @ Z ) ) ) ).
% succ_reachable
thf(fact_85_is__subscc__def,axiom,
! [S3: set_v] :
( ( sCC_Bl5398416737448265317bscc_v @ successors @ S3 )
= ( ! [X2: v] :
( ( member_v @ X2 @ S3 )
=> ! [Y3: v] :
( ( member_v @ Y3 @ S3 )
=> ( sCC_Bl649662514949026229able_v @ successors @ X2 @ Y3 ) ) ) ) ) ).
% is_subscc_def
thf(fact_86_sccE,axiom,
! [S3: set_v,X4: v,Y: v] :
( ( sCC_Bloemen_is_scc_v @ successors @ S3 )
=> ( ( member_v @ X4 @ S3 )
=> ( ( sCC_Bl649662514949026229able_v @ successors @ X4 @ Y )
=> ( ( sCC_Bl649662514949026229able_v @ successors @ Y @ X4 )
=> ( member_v @ Y @ S3 ) ) ) ) ) ).
% sccE
thf(fact_87_UN__ball__bex__simps_I3_J,axiom,
! [A2: set_set_v,P: v > $o] :
( ( ? [X2: v] :
( ( member_v @ X2 @ ( comple2307003700295860064_set_v @ A2 ) )
& ( P @ X2 ) ) )
= ( ? [X2: set_v] :
( ( member_set_v @ X2 @ A2 )
& ? [Y3: v] :
( ( member_v @ Y3 @ X2 )
& ( P @ Y3 ) ) ) ) ) ).
% UN_ball_bex_simps(3)
thf(fact_88_UN__ball__bex__simps_I1_J,axiom,
! [A2: set_set_v,P: v > $o] :
( ( ! [X2: v] :
( ( member_v @ X2 @ ( comple2307003700295860064_set_v @ A2 ) )
=> ( P @ X2 ) ) )
= ( ! [X2: set_v] :
( ( member_set_v @ X2 @ A2 )
=> ! [Y3: v] :
( ( member_v @ Y3 @ X2 )
=> ( P @ Y3 ) ) ) ) ) ).
% UN_ball_bex_simps(1)
thf(fact_89_is__scc__def,axiom,
! [S3: set_v] :
( ( sCC_Bloemen_is_scc_v @ successors @ S3 )
= ( ( S3 != bot_bot_set_v )
& ( sCC_Bl5398416737448265317bscc_v @ successors @ S3 )
& ! [S4: set_v] :
( ( ( ord_less_eq_set_v @ S3 @ S4 )
& ( sCC_Bl5398416737448265317bscc_v @ successors @ S4 ) )
=> ( S4 = S3 ) ) ) ) ).
% is_scc_def
thf(fact_90_Sup__bot__conv_I2_J,axiom,
! [A2: set_se8455005133513928103od_v_v] :
( ( bot_bo723834152578015283od_v_v
= ( comple5788137035815166516od_v_v @ A2 ) )
= ( ! [X2: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ X2 @ A2 )
=> ( X2 = bot_bo723834152578015283od_v_v ) ) ) ) ).
% Sup_bot_conv(2)
thf(fact_91_Sup__bot__conv_I2_J,axiom,
! [A2: set_set_v] :
( ( bot_bot_set_v
= ( comple2307003700295860064_set_v @ A2 ) )
= ( ! [X2: set_v] :
( ( member_set_v @ X2 @ A2 )
=> ( X2 = bot_bot_set_v ) ) ) ) ).
% Sup_bot_conv(2)
thf(fact_92_Sup__bot__conv_I1_J,axiom,
! [A2: set_se8455005133513928103od_v_v] :
( ( ( comple5788137035815166516od_v_v @ A2 )
= bot_bo723834152578015283od_v_v )
= ( ! [X2: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ X2 @ A2 )
=> ( X2 = bot_bo723834152578015283od_v_v ) ) ) ) ).
% Sup_bot_conv(1)
thf(fact_93_Sup__bot__conv_I1_J,axiom,
! [A2: set_set_v] :
( ( ( comple2307003700295860064_set_v @ A2 )
= bot_bot_set_v )
= ( ! [X2: set_v] :
( ( member_set_v @ X2 @ A2 )
=> ( X2 = bot_bot_set_v ) ) ) ) ).
% Sup_bot_conv(1)
thf(fact_94_dfs__S__hd__stack_I2_J,axiom,
! [E: sCC_Bl1394983891496994913t_unit,V3: v,E2: sCC_Bl1394983891496994913t_unit,N2: v] :
( ( sCC_Bl9196236973127232072t_unit @ successors @ E )
=> ( ( sCC_Bl8953792750115413617t_unit @ successors @ V3 @ E @ E2 )
=> ( ( ( sCC_Bl8828226123343373779t_unit @ E )
!= nil_v )
=> ( ( member_v @ N2 @ ( sCC_Bl1280885523602775798t_unit @ E @ ( hd_v @ ( sCC_Bl8828226123343373779t_unit @ E ) ) ) )
=> ( member_v @ N2 @ ( sCC_Bl1280885523602775798t_unit @ E2 @ ( hd_v @ ( sCC_Bl8828226123343373779t_unit @ E2 ) ) ) ) ) ) ) ) ).
% dfs_S_hd_stack(2)
thf(fact_95_dfs__S__hd__stack_I1_J,axiom,
! [E: sCC_Bl1394983891496994913t_unit,V3: v,E2: sCC_Bl1394983891496994913t_unit,N2: v] :
( ( sCC_Bl9196236973127232072t_unit @ successors @ E )
=> ( ( sCC_Bl8953792750115413617t_unit @ successors @ V3 @ E @ E2 )
=> ( ( ( sCC_Bl8828226123343373779t_unit @ E )
!= nil_v )
=> ( ( member_v @ N2 @ ( sCC_Bl1280885523602775798t_unit @ E @ ( hd_v @ ( sCC_Bl8828226123343373779t_unit @ E ) ) ) )
=> ( ( sCC_Bl8828226123343373779t_unit @ E2 )
!= nil_v ) ) ) ) ) ).
% dfs_S_hd_stack(1)
thf(fact_96_Sup__empty,axiom,
( ( comple5788137035815166516od_v_v @ bot_bo3497076220358800403od_v_v )
= bot_bo723834152578015283od_v_v ) ).
% Sup_empty
thf(fact_97_Sup__empty,axiom,
( ( comple2307003700295860064_set_v @ bot_bot_set_set_v )
= bot_bot_set_v ) ).
% Sup_empty
thf(fact_98_sclosed,axiom,
! [X: v] :
( ( member_v @ X @ vertices )
=> ( ord_less_eq_set_v @ ( successors @ X ) @ vertices ) ) ).
% sclosed
thf(fact_99_graph_Owf__env_Ocong,axiom,
sCC_Bl9196236973127232072t_unit = sCC_Bl9196236973127232072t_unit ).
% graph.wf_env.cong
thf(fact_100_graph_Oreachable_Ocong,axiom,
sCC_Bl649662514949026229able_v = sCC_Bl649662514949026229able_v ).
% graph.reachable.cong
thf(fact_101_Union__empty,axiom,
( ( comple5788137035815166516od_v_v @ bot_bo3497076220358800403od_v_v )
= bot_bo723834152578015283od_v_v ) ).
% Union_empty
thf(fact_102_Union__empty,axiom,
( ( comple2307003700295860064_set_v @ bot_bot_set_set_v )
= bot_bot_set_v ) ).
% Union_empty
thf(fact_103_Union__mono,axiom,
! [A2: set_se8455005133513928103od_v_v,B2: set_se8455005133513928103od_v_v] :
( ( ord_le4714265922333009223od_v_v @ A2 @ B2 )
=> ( ord_le7336532860387713383od_v_v @ ( comple5788137035815166516od_v_v @ A2 ) @ ( comple5788137035815166516od_v_v @ B2 ) ) ) ).
% Union_mono
thf(fact_104_Union__mono,axiom,
! [A2: set_set_v,B2: set_set_v] :
( ( ord_le5216385588623774835_set_v @ A2 @ B2 )
=> ( ord_less_eq_set_v @ ( comple2307003700295860064_set_v @ A2 ) @ ( comple2307003700295860064_set_v @ B2 ) ) ) ).
% Union_mono
thf(fact_105_Sup__subset__mono,axiom,
! [A2: set_se8455005133513928103od_v_v,B2: set_se8455005133513928103od_v_v] :
( ( ord_le4714265922333009223od_v_v @ A2 @ B2 )
=> ( ord_le7336532860387713383od_v_v @ ( comple5788137035815166516od_v_v @ A2 ) @ ( comple5788137035815166516od_v_v @ B2 ) ) ) ).
% Sup_subset_mono
thf(fact_106_Sup__subset__mono,axiom,
! [A2: set_set_v,B2: set_set_v] :
( ( ord_le5216385588623774835_set_v @ A2 @ B2 )
=> ( ord_less_eq_set_v @ ( comple2307003700295860064_set_v @ A2 ) @ ( comple2307003700295860064_set_v @ B2 ) ) ) ).
% Sup_subset_mono
thf(fact_107_less__eq__Sup,axiom,
! [A2: set_se8455005133513928103od_v_v,U: set_Product_prod_v_v] :
( ! [V2: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ V2 @ A2 )
=> ( ord_le7336532860387713383od_v_v @ U @ V2 ) )
=> ( ( A2 != bot_bo3497076220358800403od_v_v )
=> ( ord_le7336532860387713383od_v_v @ U @ ( comple5788137035815166516od_v_v @ A2 ) ) ) ) ).
% less_eq_Sup
thf(fact_108_less__eq__Sup,axiom,
! [A2: set_set_v,U: set_v] :
( ! [V2: set_v] :
( ( member_set_v @ V2 @ A2 )
=> ( ord_less_eq_set_v @ U @ V2 ) )
=> ( ( A2 != bot_bot_set_set_v )
=> ( ord_less_eq_set_v @ U @ ( comple2307003700295860064_set_v @ A2 ) ) ) ) ).
% less_eq_Sup
thf(fact_109_graph_Ois__subscc_Ocong,axiom,
sCC_Bl5398416737448265317bscc_v = sCC_Bl5398416737448265317bscc_v ).
% graph.is_subscc.cong
thf(fact_110_graph_Ois__scc_Ocong,axiom,
sCC_Bloemen_is_scc_v = sCC_Bloemen_is_scc_v ).
% graph.is_scc.cong
thf(fact_111_Sup__eqI,axiom,
! [A2: set_se8455005133513928103od_v_v,X4: set_Product_prod_v_v] :
( ! [Y2: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ Y2 @ A2 )
=> ( ord_le7336532860387713383od_v_v @ Y2 @ X4 ) )
=> ( ! [Y2: set_Product_prod_v_v] :
( ! [Z4: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ Z4 @ A2 )
=> ( ord_le7336532860387713383od_v_v @ Z4 @ Y2 ) )
=> ( ord_le7336532860387713383od_v_v @ X4 @ Y2 ) )
=> ( ( comple5788137035815166516od_v_v @ A2 )
= X4 ) ) ) ).
% Sup_eqI
thf(fact_112_Sup__eqI,axiom,
! [A2: set_set_v,X4: set_v] :
( ! [Y2: set_v] :
( ( member_set_v @ Y2 @ A2 )
=> ( ord_less_eq_set_v @ Y2 @ X4 ) )
=> ( ! [Y2: set_v] :
( ! [Z4: set_v] :
( ( member_set_v @ Z4 @ A2 )
=> ( ord_less_eq_set_v @ Z4 @ Y2 ) )
=> ( ord_less_eq_set_v @ X4 @ Y2 ) )
=> ( ( comple2307003700295860064_set_v @ A2 )
= X4 ) ) ) ).
% Sup_eqI
thf(fact_113_Sup__mono,axiom,
! [A2: set_se8455005133513928103od_v_v,B2: set_se8455005133513928103od_v_v] :
( ! [A3: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ A3 @ A2 )
=> ? [X: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ X @ B2 )
& ( ord_le7336532860387713383od_v_v @ A3 @ X ) ) )
=> ( ord_le7336532860387713383od_v_v @ ( comple5788137035815166516od_v_v @ A2 ) @ ( comple5788137035815166516od_v_v @ B2 ) ) ) ).
% Sup_mono
thf(fact_114_Sup__mono,axiom,
! [A2: set_set_v,B2: set_set_v] :
( ! [A3: set_v] :
( ( member_set_v @ A3 @ A2 )
=> ? [X: set_v] :
( ( member_set_v @ X @ B2 )
& ( ord_less_eq_set_v @ A3 @ X ) ) )
=> ( ord_less_eq_set_v @ ( comple2307003700295860064_set_v @ A2 ) @ ( comple2307003700295860064_set_v @ B2 ) ) ) ).
% Sup_mono
thf(fact_115_Sup__least,axiom,
! [A2: set_se8455005133513928103od_v_v,Z: set_Product_prod_v_v] :
( ! [X3: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ X3 @ A2 )
=> ( ord_le7336532860387713383od_v_v @ X3 @ Z ) )
=> ( ord_le7336532860387713383od_v_v @ ( comple5788137035815166516od_v_v @ A2 ) @ Z ) ) ).
% Sup_least
thf(fact_116_Sup__least,axiom,
! [A2: set_set_v,Z: set_v] :
( ! [X3: set_v] :
( ( member_set_v @ X3 @ A2 )
=> ( ord_less_eq_set_v @ X3 @ Z ) )
=> ( ord_less_eq_set_v @ ( comple2307003700295860064_set_v @ A2 ) @ Z ) ) ).
% Sup_least
thf(fact_117_Sup__upper,axiom,
! [X4: set_Product_prod_v_v,A2: set_se8455005133513928103od_v_v] :
( ( member8406446414694345712od_v_v @ X4 @ A2 )
=> ( ord_le7336532860387713383od_v_v @ X4 @ ( comple5788137035815166516od_v_v @ A2 ) ) ) ).
% Sup_upper
thf(fact_118_Sup__upper,axiom,
! [X4: set_v,A2: set_set_v] :
( ( member_set_v @ X4 @ A2 )
=> ( ord_less_eq_set_v @ X4 @ ( comple2307003700295860064_set_v @ A2 ) ) ) ).
% Sup_upper
thf(fact_119_Sup__le__iff,axiom,
! [A2: set_se8455005133513928103od_v_v,B: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ ( comple5788137035815166516od_v_v @ A2 ) @ B )
= ( ! [X2: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ X2 @ A2 )
=> ( ord_le7336532860387713383od_v_v @ X2 @ B ) ) ) ) ).
% Sup_le_iff
thf(fact_120_Sup__le__iff,axiom,
! [A2: set_set_v,B: set_v] :
( ( ord_less_eq_set_v @ ( comple2307003700295860064_set_v @ A2 ) @ B )
= ( ! [X2: set_v] :
( ( member_set_v @ X2 @ A2 )
=> ( ord_less_eq_set_v @ X2 @ B ) ) ) ) ).
% Sup_le_iff
thf(fact_121_Sup__upper2,axiom,
! [U: set_Product_prod_v_v,A2: set_se8455005133513928103od_v_v,V3: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ U @ A2 )
=> ( ( ord_le7336532860387713383od_v_v @ V3 @ U )
=> ( ord_le7336532860387713383od_v_v @ V3 @ ( comple5788137035815166516od_v_v @ A2 ) ) ) ) ).
% Sup_upper2
thf(fact_122_Sup__upper2,axiom,
! [U: set_v,A2: set_set_v,V3: set_v] :
( ( member_set_v @ U @ A2 )
=> ( ( ord_less_eq_set_v @ V3 @ U )
=> ( ord_less_eq_set_v @ V3 @ ( comple2307003700295860064_set_v @ A2 ) ) ) ) ).
% Sup_upper2
thf(fact_123_Union__least,axiom,
! [A2: set_se8455005133513928103od_v_v,C2: set_Product_prod_v_v] :
( ! [X6: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ X6 @ A2 )
=> ( ord_le7336532860387713383od_v_v @ X6 @ C2 ) )
=> ( ord_le7336532860387713383od_v_v @ ( comple5788137035815166516od_v_v @ A2 ) @ C2 ) ) ).
% Union_least
thf(fact_124_Union__least,axiom,
! [A2: set_set_v,C2: set_v] :
( ! [X6: set_v] :
( ( member_set_v @ X6 @ A2 )
=> ( ord_less_eq_set_v @ X6 @ C2 ) )
=> ( ord_less_eq_set_v @ ( comple2307003700295860064_set_v @ A2 ) @ C2 ) ) ).
% Union_least
thf(fact_125_Union__upper,axiom,
! [B2: set_Product_prod_v_v,A2: set_se8455005133513928103od_v_v] :
( ( member8406446414694345712od_v_v @ B2 @ A2 )
=> ( ord_le7336532860387713383od_v_v @ B2 @ ( comple5788137035815166516od_v_v @ A2 ) ) ) ).
% Union_upper
thf(fact_126_Union__upper,axiom,
! [B2: set_v,A2: set_set_v] :
( ( member_set_v @ B2 @ A2 )
=> ( ord_less_eq_set_v @ B2 @ ( comple2307003700295860064_set_v @ A2 ) ) ) ).
% Union_upper
thf(fact_127_Union__subsetI,axiom,
! [A2: set_se8455005133513928103od_v_v,B2: set_se8455005133513928103od_v_v] :
( ! [X3: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ X3 @ A2 )
=> ? [Y4: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ Y4 @ B2 )
& ( ord_le7336532860387713383od_v_v @ X3 @ Y4 ) ) )
=> ( ord_le7336532860387713383od_v_v @ ( comple5788137035815166516od_v_v @ A2 ) @ ( comple5788137035815166516od_v_v @ B2 ) ) ) ).
% Union_subsetI
thf(fact_128_Union__subsetI,axiom,
! [A2: set_set_v,B2: set_set_v] :
( ! [X3: set_v] :
( ( member_set_v @ X3 @ A2 )
=> ? [Y4: set_v] :
( ( member_set_v @ Y4 @ B2 )
& ( ord_less_eq_set_v @ X3 @ Y4 ) ) )
=> ( ord_less_eq_set_v @ ( comple2307003700295860064_set_v @ A2 ) @ ( comple2307003700295860064_set_v @ B2 ) ) ) ).
% Union_subsetI
thf(fact_129_Union__empty__conv,axiom,
! [A2: set_se8455005133513928103od_v_v] :
( ( ( comple5788137035815166516od_v_v @ A2 )
= bot_bo723834152578015283od_v_v )
= ( ! [X2: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ X2 @ A2 )
=> ( X2 = bot_bo723834152578015283od_v_v ) ) ) ) ).
% Union_empty_conv
thf(fact_130_Union__empty__conv,axiom,
! [A2: set_set_v] :
( ( ( comple2307003700295860064_set_v @ A2 )
= bot_bot_set_v )
= ( ! [X2: set_v] :
( ( member_set_v @ X2 @ A2 )
=> ( X2 = bot_bot_set_v ) ) ) ) ).
% Union_empty_conv
thf(fact_131_empty__Union__conv,axiom,
! [A2: set_se8455005133513928103od_v_v] :
( ( bot_bo723834152578015283od_v_v
= ( comple5788137035815166516od_v_v @ A2 ) )
= ( ! [X2: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ X2 @ A2 )
=> ( X2 = bot_bo723834152578015283od_v_v ) ) ) ) ).
% empty_Union_conv
thf(fact_132_empty__Union__conv,axiom,
! [A2: set_set_v] :
( ( bot_bot_set_v
= ( comple2307003700295860064_set_v @ A2 ) )
= ( ! [X2: set_v] :
( ( member_set_v @ X2 @ A2 )
=> ( X2 = bot_bot_set_v ) ) ) ) ).
% empty_Union_conv
thf(fact_133_graph_Opre__dfss_Ocong,axiom,
sCC_Bl1748261141445803503t_unit = sCC_Bl1748261141445803503t_unit ).
% graph.pre_dfss.cong
thf(fact_134_UnionE,axiom,
! [A2: product_prod_v_v,C2: set_se8455005133513928103od_v_v] :
( ( member7453568604450474000od_v_v @ A2 @ ( comple5788137035815166516od_v_v @ C2 ) )
=> ~ ! [X6: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ A2 @ X6 )
=> ~ ( member8406446414694345712od_v_v @ X6 @ C2 ) ) ) ).
% UnionE
thf(fact_135_UnionE,axiom,
! [A2: v,C2: set_set_v] :
( ( member_v @ A2 @ ( comple2307003700295860064_set_v @ C2 ) )
=> ~ ! [X6: set_v] :
( ( member_v @ A2 @ X6 )
=> ~ ( member_set_v @ X6 @ C2 ) ) ) ).
% UnionE
thf(fact_136_calculation_I1_J,axiom,
! [X: v] :
( ( member_v @ X @ ( sCC_Bl4645233313691564917t_unit @ e2 ) )
=> ( sCC_Bl649662514949026229able_v @ successors @ ( sCC_Bl1090238580953940555t_unit @ e2 ) @ X ) ) ).
% calculation(1)
thf(fact_137_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_138_visited__unexplored,axiom,
! [E: sCC_Bl1394983891496994913t_unit,M2: v] :
( ( sCC_Bl9196236973127232072t_unit @ successors @ E )
=> ( ( member_v @ M2 @ ( sCC_Bl4645233313691564917t_unit @ E ) )
=> ( ~ ( member_v @ M2 @ ( sCC_Bl157864678168468314t_unit @ E ) )
=> ~ ! [N3: v] :
( ( member_v @ N3 @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E ) ) )
=> ~ ( member_v @ M2 @ ( sCC_Bl1280885523602775798t_unit @ E @ N3 ) ) ) ) ) ) ).
% visited_unexplored
thf(fact_139_subset__empty,axiom,
! [A2: set_v] :
( ( ord_less_eq_set_v @ A2 @ bot_bot_set_v )
= ( A2 = bot_bot_set_v ) ) ).
% subset_empty
thf(fact_140_subset__empty,axiom,
! [A2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A2 @ bot_bo723834152578015283od_v_v )
= ( A2 = bot_bo723834152578015283od_v_v ) ) ).
% subset_empty
thf(fact_141_empty__subsetI,axiom,
! [A2: set_v] : ( ord_less_eq_set_v @ bot_bot_set_v @ A2 ) ).
% empty_subsetI
thf(fact_142_empty__subsetI,axiom,
! [A2: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ bot_bo723834152578015283od_v_v @ A2 ) ).
% empty_subsetI
thf(fact_143_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 ) )
=> ! [Xa: v] :
( ( member_v @ Xa @ ( minus_minus_set_v @ ( successors @ X3 ) @ ( sCC_Bl3795065053823578884t_unit @ E @ X3 ) ) )
=> ( ( sCC_Bl649662514949026229able_v @ successors @ V3 @ X3 )
=> ~ ( sCC_Bl649662514949026229able_v @ successors @ Xa @ W ) ) ) )
=> ( member_v @ W @ ( sCC_Bl4645233313691564917t_unit @ E ) ) ) ) ) ) ).
% reachable_visited
thf(fact_144_calculation_I4_J,axiom,
ord_less_eq_set_v @ ( set_v2 @ ( sCC_Bl9201514103433284750t_unit @ e2 ) ) @ ( sCC_Bl4645233313691564917t_unit @ e2 ) ).
% calculation(4)
thf(fact_145_stack__unexplored,axiom,
! [E: sCC_Bl1394983891496994913t_unit,N2: v] :
( ( sCC_Bl9196236973127232072t_unit @ successors @ E )
=> ( ( member_v @ N2 @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E ) ) )
=> ~ ( member_v @ N2 @ ( sCC_Bl157864678168468314t_unit @ E ) ) ) ) ).
% stack_unexplored
thf(fact_146_stack__visited,axiom,
! [E: sCC_Bl1394983891496994913t_unit,N2: v] :
( ( sCC_Bl9196236973127232072t_unit @ successors @ E )
=> ( ( member_v @ N2 @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E ) ) )
=> ( member_v @ N2 @ ( sCC_Bl4645233313691564917t_unit @ E ) ) ) ) ).
% stack_visited
thf(fact_147_post__dfs__def,axiom,
! [V3: v,E: sCC_Bl1394983891496994913t_unit,E2: sCC_Bl1394983891496994913t_unit] :
( ( sCC_Bl8953792750115413617t_unit @ successors @ V3 @ E @ E2 )
= ( ( sCC_Bl9196236973127232072t_unit @ successors @ E2 )
& ( member_v @ V3 @ ( sCC_Bl4645233313691564917t_unit @ E2 ) )
& ( sCC_Bl5768913643336123637t_unit @ E @ E2 )
& ( ( sCC_Bl3795065053823578884t_unit @ E2 @ V3 )
= ( successors @ V3 ) )
& ! [X2: v] :
( ( member_v @ X2 @ ( sCC_Bl4645233313691564917t_unit @ E ) )
=> ( ( sCC_Bl3795065053823578884t_unit @ E2 @ X2 )
= ( sCC_Bl3795065053823578884t_unit @ E @ X2 ) ) )
& ! [X2: v] :
( ( member_v @ X2 @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E2 ) ) )
=> ( sCC_Bl649662514949026229able_v @ successors @ X2 @ V3 ) )
& ? [Ns: list_v] :
( ( sCC_Bl8828226123343373779t_unit @ E )
= ( append_v @ Ns @ ( sCC_Bl8828226123343373779t_unit @ E2 ) ) )
& ( ( ( member_v @ V3 @ ( sCC_Bl157864678168468314t_unit @ E2 ) )
& ( ( sCC_Bl8828226123343373779t_unit @ E2 )
= ( sCC_Bl8828226123343373779t_unit @ E ) )
& ! [X2: v] :
( ( member_v @ X2 @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E2 ) ) )
=> ( ( sCC_Bl1280885523602775798t_unit @ E2 @ X2 )
= ( sCC_Bl1280885523602775798t_unit @ E @ X2 ) ) ) )
| ( ( ( sCC_Bl8828226123343373779t_unit @ E2 )
!= nil_v )
& ( member_v @ V3 @ ( sCC_Bl1280885523602775798t_unit @ E2 @ ( hd_v @ ( sCC_Bl8828226123343373779t_unit @ E2 ) ) ) )
& ! [X2: v] :
( ( member_v @ X2 @ ( set_v2 @ ( tl_v @ ( sCC_Bl8828226123343373779t_unit @ E2 ) ) ) )
=> ( ( sCC_Bl1280885523602775798t_unit @ E2 @ X2 )
= ( sCC_Bl1280885523602775798t_unit @ E @ X2 ) ) ) ) )
& ( ( sCC_Bl9201514103433284750t_unit @ E2 )
= ( sCC_Bl9201514103433284750t_unit @ E ) ) ) ) ).
% post_dfs_def
thf(fact_148_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_149_empty__Collect__eq,axiom,
! [P: v > $o] :
( ( bot_bot_set_v
= ( collect_v @ P ) )
= ( ! [X2: v] :
~ ( P @ X2 ) ) ) ).
% empty_Collect_eq
thf(fact_150_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_151_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_152_Collect__empty__eq,axiom,
! [P: v > $o] :
( ( ( collect_v @ P )
= bot_bot_set_v )
= ( ! [X2: v] :
~ ( P @ X2 ) ) ) ).
% Collect_empty_eq
thf(fact_153_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_154_all__not__in__conv,axiom,
! [A2: set_v] :
( ( ! [X2: v] :
~ ( member_v @ X2 @ A2 ) )
= ( A2 = bot_bot_set_v ) ) ).
% all_not_in_conv
thf(fact_155_all__not__in__conv,axiom,
! [A2: set_Product_prod_v_v] :
( ( ! [X2: product_prod_v_v] :
~ ( member7453568604450474000od_v_v @ X2 @ A2 ) )
= ( A2 = bot_bo723834152578015283od_v_v ) ) ).
% all_not_in_conv
thf(fact_156_empty__iff,axiom,
! [C: v] :
~ ( member_v @ C @ bot_bot_set_v ) ).
% empty_iff
thf(fact_157_empty__iff,axiom,
! [C: product_prod_v_v] :
~ ( member7453568604450474000od_v_v @ C @ bot_bo723834152578015283od_v_v ) ).
% empty_iff
thf(fact_158_subset__antisym,axiom,
! [A2: set_v,B2: set_v] :
( ( ord_less_eq_set_v @ A2 @ B2 )
=> ( ( ord_less_eq_set_v @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% subset_antisym
thf(fact_159_subset__antisym,axiom,
! [A2: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A2 @ B2 )
=> ( ( ord_le7336532860387713383od_v_v @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% subset_antisym
thf(fact_160_subsetI,axiom,
! [A2: set_v,B2: set_v] :
( ! [X3: v] :
( ( member_v @ X3 @ A2 )
=> ( member_v @ X3 @ B2 ) )
=> ( ord_less_eq_set_v @ A2 @ B2 ) ) ).
% subsetI
thf(fact_161_subsetI,axiom,
! [A2: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ! [X3: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X3 @ A2 )
=> ( member7453568604450474000od_v_v @ X3 @ B2 ) )
=> ( ord_le7336532860387713383od_v_v @ A2 @ B2 ) ) ).
% subsetI
thf(fact_162_Int__iff,axiom,
! [C: product_prod_v_v,A2: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ C @ ( inf_in6271465464967711157od_v_v @ A2 @ B2 ) )
= ( ( member7453568604450474000od_v_v @ C @ A2 )
& ( member7453568604450474000od_v_v @ C @ B2 ) ) ) ).
% Int_iff
thf(fact_163_Int__iff,axiom,
! [C: v,A2: set_v,B2: set_v] :
( ( member_v @ C @ ( inf_inf_set_v @ A2 @ B2 ) )
= ( ( member_v @ C @ A2 )
& ( member_v @ C @ B2 ) ) ) ).
% Int_iff
thf(fact_164_IntI,axiom,
! [C: product_prod_v_v,A2: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ C @ A2 )
=> ( ( member7453568604450474000od_v_v @ C @ B2 )
=> ( member7453568604450474000od_v_v @ C @ ( inf_in6271465464967711157od_v_v @ A2 @ B2 ) ) ) ) ).
% IntI
thf(fact_165_IntI,axiom,
! [C: v,A2: set_v,B2: set_v] :
( ( member_v @ C @ A2 )
=> ( ( member_v @ C @ B2 )
=> ( member_v @ C @ ( inf_inf_set_v @ A2 @ B2 ) ) ) ) ).
% IntI
thf(fact_166_Diff__idemp,axiom,
! [A2: set_v,B2: set_v] :
( ( minus_minus_set_v @ ( minus_minus_set_v @ A2 @ B2 ) @ B2 )
= ( minus_minus_set_v @ A2 @ B2 ) ) ).
% Diff_idemp
thf(fact_167_Diff__iff,axiom,
! [C: product_prod_v_v,A2: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ C @ ( minus_4183494784930505774od_v_v @ A2 @ B2 ) )
= ( ( member7453568604450474000od_v_v @ C @ A2 )
& ~ ( member7453568604450474000od_v_v @ C @ B2 ) ) ) ).
% Diff_iff
thf(fact_168_Diff__iff,axiom,
! [C: v,A2: set_v,B2: set_v] :
( ( member_v @ C @ ( minus_minus_set_v @ A2 @ B2 ) )
= ( ( member_v @ C @ A2 )
& ~ ( member_v @ C @ B2 ) ) ) ).
% Diff_iff
thf(fact_169_DiffI,axiom,
! [C: product_prod_v_v,A2: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ C @ A2 )
=> ( ~ ( member7453568604450474000od_v_v @ C @ B2 )
=> ( member7453568604450474000od_v_v @ C @ ( minus_4183494784930505774od_v_v @ A2 @ B2 ) ) ) ) ).
% DiffI
thf(fact_170_DiffI,axiom,
! [C: v,A2: set_v,B2: set_v] :
( ( member_v @ C @ A2 )
=> ( ~ ( member_v @ C @ B2 )
=> ( member_v @ C @ ( minus_minus_set_v @ A2 @ B2 ) ) ) ) ).
% DiffI
thf(fact_171_scc__partition,axiom,
! [S3: set_v,S5: set_v,X4: v] :
( ( sCC_Bloemen_is_scc_v @ successors @ S3 )
=> ( ( sCC_Bloemen_is_scc_v @ successors @ S5 )
=> ( ( member_v @ X4 @ ( inf_inf_set_v @ S3 @ S5 ) )
=> ( S3 = S5 ) ) ) ) ).
% scc_partition
thf(fact_172_Diff__cancel,axiom,
! [A2: set_Product_prod_v_v] :
( ( minus_4183494784930505774od_v_v @ A2 @ A2 )
= bot_bo723834152578015283od_v_v ) ).
% Diff_cancel
thf(fact_173_Diff__cancel,axiom,
! [A2: set_v] :
( ( minus_minus_set_v @ A2 @ A2 )
= bot_bot_set_v ) ).
% Diff_cancel
thf(fact_174_empty__Diff,axiom,
! [A2: set_Product_prod_v_v] :
( ( minus_4183494784930505774od_v_v @ bot_bo723834152578015283od_v_v @ A2 )
= bot_bo723834152578015283od_v_v ) ).
% empty_Diff
thf(fact_175_empty__Diff,axiom,
! [A2: set_v] :
( ( minus_minus_set_v @ bot_bot_set_v @ A2 )
= bot_bot_set_v ) ).
% empty_Diff
thf(fact_176_Diff__empty,axiom,
! [A2: set_Product_prod_v_v] :
( ( minus_4183494784930505774od_v_v @ A2 @ bot_bo723834152578015283od_v_v )
= A2 ) ).
% Diff_empty
thf(fact_177_Diff__empty,axiom,
! [A2: set_v] :
( ( minus_minus_set_v @ A2 @ bot_bot_set_v )
= A2 ) ).
% Diff_empty
thf(fact_178_Int__subset__iff,axiom,
! [C2: set_v,A2: set_v,B2: set_v] :
( ( ord_less_eq_set_v @ C2 @ ( inf_inf_set_v @ A2 @ B2 ) )
= ( ( ord_less_eq_set_v @ C2 @ A2 )
& ( ord_less_eq_set_v @ C2 @ B2 ) ) ) ).
% Int_subset_iff
thf(fact_179_Int__subset__iff,axiom,
! [C2: set_Product_prod_v_v,A2: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ C2 @ ( inf_in6271465464967711157od_v_v @ A2 @ B2 ) )
= ( ( ord_le7336532860387713383od_v_v @ C2 @ A2 )
& ( ord_le7336532860387713383od_v_v @ C2 @ B2 ) ) ) ).
% Int_subset_iff
thf(fact_180_precedes__refl,axiom,
! [X4: product_prod_v_v,Xs: list_P7986770385144383213od_v_v] :
( ( sCC_Bl2026170059108282219od_v_v @ X4 @ X4 @ Xs )
= ( member7453568604450474000od_v_v @ X4 @ ( set_Product_prod_v_v2 @ Xs ) ) ) ).
% precedes_refl
thf(fact_181_precedes__refl,axiom,
! [X4: v,Xs: list_v] :
( ( sCC_Bl4022239298816431255edes_v @ X4 @ X4 @ Xs )
= ( member_v @ X4 @ ( set_v2 @ Xs ) ) ) ).
% precedes_refl
thf(fact_182_dfs__S__tl__stack_I2_J,axiom,
! [V3: v,E: sCC_Bl1394983891496994913t_unit,E2: sCC_Bl1394983891496994913t_unit] :
( ( sCC_Bl8953792750115413617t_unit @ successors @ V3 @ E @ E2 )
=> ( ( ( sCC_Bl8828226123343373779t_unit @ E )
!= nil_v )
=> ! [X: v] :
( ( member_v @ X @ ( set_v2 @ ( tl_v @ ( sCC_Bl8828226123343373779t_unit @ E2 ) ) ) )
=> ( ( sCC_Bl1280885523602775798t_unit @ E2 @ X )
= ( sCC_Bl1280885523602775798t_unit @ E @ X ) ) ) ) ) ).
% dfs_S_tl_stack(2)
thf(fact_183_calculation_I9_J,axiom,
! [X: v] :
( ( member_v @ X @ ( minus_minus_set_v @ ( sCC_Bl4645233313691564917t_unit @ e2 ) @ ( set_v2 @ ( sCC_Bl9201514103433284750t_unit @ e2 ) ) ) )
=> ( ( sCC_Bl3795065053823578884t_unit @ e2 @ X )
= ( successors @ X ) ) ) ).
% calculation(9)
thf(fact_184_stack__class,axiom,
! [E: sCC_Bl1394983891496994913t_unit,N2: v,M2: v] :
( ( sCC_Bl9196236973127232072t_unit @ successors @ E )
=> ( ( member_v @ N2 @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E ) ) )
=> ( ( member_v @ M2 @ ( sCC_Bl1280885523602775798t_unit @ E @ N2 ) )
=> ( member_v @ M2 @ ( minus_minus_set_v @ ( sCC_Bl4645233313691564917t_unit @ E ) @ ( sCC_Bl157864678168468314t_unit @ E ) ) ) ) ) ) ).
% stack_class
thf(fact_185_set__empty,axiom,
! [Xs: list_v] :
( ( ( set_v2 @ Xs )
= bot_bot_set_v )
= ( Xs = nil_v ) ) ).
% set_empty
thf(fact_186_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_187_set__empty2,axiom,
! [Xs: list_v] :
( ( bot_bot_set_v
= ( set_v2 @ Xs ) )
= ( Xs = nil_v ) ) ).
% set_empty2
thf(fact_188_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_189_Diff__eq__empty__iff,axiom,
! [A2: set_v,B2: set_v] :
( ( ( minus_minus_set_v @ A2 @ B2 )
= bot_bot_set_v )
= ( ord_less_eq_set_v @ A2 @ B2 ) ) ).
% Diff_eq_empty_iff
thf(fact_190_Diff__eq__empty__iff,axiom,
! [A2: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( ( minus_4183494784930505774od_v_v @ A2 @ B2 )
= bot_bo723834152578015283od_v_v )
= ( ord_le7336532860387713383od_v_v @ A2 @ B2 ) ) ).
% Diff_eq_empty_iff
thf(fact_191_Diff__disjoint,axiom,
! [A2: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ A2 @ ( minus_4183494784930505774od_v_v @ B2 @ A2 ) )
= bot_bo723834152578015283od_v_v ) ).
% Diff_disjoint
thf(fact_192_Diff__disjoint,axiom,
! [A2: set_v,B2: set_v] :
( ( inf_inf_set_v @ A2 @ ( minus_minus_set_v @ B2 @ A2 ) )
= bot_bot_set_v ) ).
% Diff_disjoint
thf(fact_193_tl__append2,axiom,
! [Xs: list_v,Ys: list_v] :
( ( Xs != nil_v )
=> ( ( tl_v @ ( append_v @ Xs @ Ys ) )
= ( append_v @ ( tl_v @ Xs ) @ Ys ) ) ) ).
% tl_append2
thf(fact_194__092_060open_062_092_060And_062n_Am_O_A_092_060lbrakk_062n_A_092_060in_062_Aset_A_Itl_A_Istack_Ae_H_J_J_059_Am_A_092_060in_062_A_092_060S_062_Ae_An_A_092_060inter_062_Acc_092_060rbrakk_062_A_092_060Longrightarrow_062_AFalse_092_060close_062,axiom,
! [N2: v,M2: v] :
( ( member_v @ N2 @ ( set_v2 @ ( tl_v @ ( sCC_Bl8828226123343373779t_unit @ e2 ) ) ) )
=> ~ ( member_v @ M2 @ ( inf_inf_set_v @ ( sCC_Bl1280885523602775798t_unit @ e @ N2 ) @ cc ) ) ) ).
% \<open>\<And>n m. \<lbrakk>n \<in> set (tl (stack e')); m \<in> \<S> e n \<inter> cc\<rbrakk> \<Longrightarrow> False\<close>
thf(fact_195_tl__cc,axiom,
! [X: v] :
( ( member_v @ X @ ( set_v2 @ ( tl_v @ ( sCC_Bl8828226123343373779t_unit @ e2 ) ) ) )
=> ( ( inf_inf_set_v @ ( sCC_Bl1280885523602775798t_unit @ e @ X ) @ cc )
= bot_bot_set_v ) ) ).
% tl_cc
thf(fact_196_unite__S__tl,axiom,
! [E: sCC_Bl1394983891496994913t_unit,W: v,V3: v,N2: v] :
( ( sCC_Bl9196236973127232072t_unit @ successors @ E )
=> ( ( member_v @ W @ ( successors @ V3 ) )
=> ( ~ ( member_v @ W @ ( sCC_Bl3795065053823578884t_unit @ E @ V3 ) )
=> ( ( member_v @ W @ ( sCC_Bl4645233313691564917t_unit @ E ) )
=> ( ~ ( member_v @ W @ ( sCC_Bl157864678168468314t_unit @ E ) )
=> ( ( member_v @ N2 @ ( set_v2 @ ( tl_v @ ( sCC_Bl8828226123343373779t_unit @ ( sCC_Bloemen_unite_v @ V3 @ W @ E ) ) ) ) )
=> ( ( sCC_Bl1280885523602775798t_unit @ ( sCC_Bloemen_unite_v @ V3 @ W @ E ) @ N2 )
= ( sCC_Bl1280885523602775798t_unit @ E @ N2 ) ) ) ) ) ) ) ) ).
% unite_S_tl
thf(fact_197_distinct__append,axiom,
! [Xs: list_v,Ys: list_v] :
( ( distinct_v @ ( append_v @ Xs @ Ys ) )
= ( ( distinct_v @ Xs )
& ( distinct_v @ Ys )
& ( ( inf_inf_set_v @ ( set_v2 @ Xs ) @ ( set_v2 @ Ys ) )
= bot_bot_set_v ) ) ) ).
% distinct_append
thf(fact_198_distinct__append,axiom,
! [Xs: list_P7986770385144383213od_v_v,Ys: list_P7986770385144383213od_v_v] :
( ( distin6159370996967099744od_v_v @ ( append2138873909117096322od_v_v @ Xs @ Ys ) )
= ( ( distin6159370996967099744od_v_v @ Xs )
& ( distin6159370996967099744od_v_v @ Ys )
& ( ( inf_in6271465464967711157od_v_v @ ( set_Product_prod_v_v2 @ Xs ) @ ( set_Product_prod_v_v2 @ Ys ) )
= bot_bo723834152578015283od_v_v ) ) ) ).
% distinct_append
thf(fact_199_graph__axioms,axiom,
sCC_Bloemen_graph_v @ vertices @ successors ).
% graph_axioms
thf(fact_200_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_201_Diff__triv,axiom,
! [A2: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( ( inf_in6271465464967711157od_v_v @ A2 @ B2 )
= bot_bo723834152578015283od_v_v )
=> ( ( minus_4183494784930505774od_v_v @ A2 @ B2 )
= A2 ) ) ).
% Diff_triv
thf(fact_202_Diff__triv,axiom,
! [A2: set_v,B2: set_v] :
( ( ( inf_inf_set_v @ A2 @ B2 )
= bot_bot_set_v )
=> ( ( minus_minus_set_v @ A2 @ B2 )
= A2 ) ) ).
% Diff_triv
thf(fact_203_Int__Diff__disjoint,axiom,
! [A2: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ ( inf_in6271465464967711157od_v_v @ A2 @ B2 ) @ ( minus_4183494784930505774od_v_v @ A2 @ B2 ) )
= bot_bo723834152578015283od_v_v ) ).
% Int_Diff_disjoint
thf(fact_204_Int__Diff__disjoint,axiom,
! [A2: set_v,B2: set_v] :
( ( inf_inf_set_v @ ( inf_inf_set_v @ A2 @ B2 ) @ ( minus_minus_set_v @ A2 @ B2 ) )
= bot_bot_set_v ) ).
% Int_Diff_disjoint
thf(fact_205_list_Oset__sel_I2_J,axiom,
! [A: list_P7986770385144383213od_v_v,X4: product_prod_v_v] :
( ( A != nil_Product_prod_v_v )
=> ( ( member7453568604450474000od_v_v @ X4 @ ( set_Product_prod_v_v2 @ ( tl_Product_prod_v_v @ A ) ) )
=> ( member7453568604450474000od_v_v @ X4 @ ( set_Product_prod_v_v2 @ A ) ) ) ) ).
% list.set_sel(2)
thf(fact_206_list_Oset__sel_I2_J,axiom,
! [A: list_v,X4: v] :
( ( A != nil_v )
=> ( ( member_v @ X4 @ ( set_v2 @ ( tl_v @ A ) ) )
=> ( member_v @ X4 @ ( set_v2 @ A ) ) ) ) ).
% list.set_sel(2)
thf(fact_207_Diff__Int__distrib2,axiom,
! [A2: set_v,B2: set_v,C2: set_v] :
( ( inf_inf_set_v @ ( minus_minus_set_v @ A2 @ B2 ) @ C2 )
= ( minus_minus_set_v @ ( inf_inf_set_v @ A2 @ C2 ) @ ( inf_inf_set_v @ B2 @ C2 ) ) ) ).
% Diff_Int_distrib2
thf(fact_208_Int__left__commute,axiom,
! [A2: set_v,B2: set_v,C2: set_v] :
( ( inf_inf_set_v @ A2 @ ( inf_inf_set_v @ B2 @ C2 ) )
= ( inf_inf_set_v @ B2 @ ( inf_inf_set_v @ A2 @ C2 ) ) ) ).
% Int_left_commute
thf(fact_209_Diff__Int__distrib,axiom,
! [C2: set_v,A2: set_v,B2: set_v] :
( ( inf_inf_set_v @ C2 @ ( minus_minus_set_v @ A2 @ B2 ) )
= ( minus_minus_set_v @ ( inf_inf_set_v @ C2 @ A2 ) @ ( inf_inf_set_v @ C2 @ B2 ) ) ) ).
% Diff_Int_distrib
thf(fact_210_Int__left__absorb,axiom,
! [A2: set_v,B2: set_v] :
( ( inf_inf_set_v @ A2 @ ( inf_inf_set_v @ A2 @ B2 ) )
= ( inf_inf_set_v @ A2 @ B2 ) ) ).
% Int_left_absorb
thf(fact_211_Diff__Diff__Int,axiom,
! [A2: set_v,B2: set_v] :
( ( minus_minus_set_v @ A2 @ ( minus_minus_set_v @ A2 @ B2 ) )
= ( inf_inf_set_v @ A2 @ B2 ) ) ).
% Diff_Diff_Int
thf(fact_212_Int__commute,axiom,
( inf_inf_set_v
= ( ^ [A4: set_v,B3: set_v] : ( inf_inf_set_v @ B3 @ A4 ) ) ) ).
% Int_commute
thf(fact_213_Int__absorb,axiom,
! [A2: set_v] :
( ( inf_inf_set_v @ A2 @ A2 )
= A2 ) ).
% Int_absorb
thf(fact_214_Int__assoc,axiom,
! [A2: set_v,B2: set_v,C2: set_v] :
( ( inf_inf_set_v @ ( inf_inf_set_v @ A2 @ B2 ) @ C2 )
= ( inf_inf_set_v @ A2 @ ( inf_inf_set_v @ B2 @ C2 ) ) ) ).
% Int_assoc
thf(fact_215_Diff__Int2,axiom,
! [A2: set_v,C2: set_v,B2: set_v] :
( ( minus_minus_set_v @ ( inf_inf_set_v @ A2 @ C2 ) @ ( inf_inf_set_v @ B2 @ C2 ) )
= ( minus_minus_set_v @ ( inf_inf_set_v @ A2 @ C2 ) @ B2 ) ) ).
% Diff_Int2
thf(fact_216_Int__Diff,axiom,
! [A2: set_v,B2: set_v,C2: set_v] :
( ( minus_minus_set_v @ ( inf_inf_set_v @ A2 @ B2 ) @ C2 )
= ( inf_inf_set_v @ A2 @ ( minus_minus_set_v @ B2 @ C2 ) ) ) ).
% Int_Diff
thf(fact_217_DiffD2,axiom,
! [C: product_prod_v_v,A2: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ C @ ( minus_4183494784930505774od_v_v @ A2 @ B2 ) )
=> ~ ( member7453568604450474000od_v_v @ C @ B2 ) ) ).
% DiffD2
thf(fact_218_DiffD2,axiom,
! [C: v,A2: set_v,B2: set_v] :
( ( member_v @ C @ ( minus_minus_set_v @ A2 @ B2 ) )
=> ~ ( member_v @ C @ B2 ) ) ).
% DiffD2
thf(fact_219_DiffD1,axiom,
! [C: product_prod_v_v,A2: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ C @ ( minus_4183494784930505774od_v_v @ A2 @ B2 ) )
=> ( member7453568604450474000od_v_v @ C @ A2 ) ) ).
% DiffD1
thf(fact_220_DiffD1,axiom,
! [C: v,A2: set_v,B2: set_v] :
( ( member_v @ C @ ( minus_minus_set_v @ A2 @ B2 ) )
=> ( member_v @ C @ A2 ) ) ).
% DiffD1
thf(fact_221_IntD2,axiom,
! [C: product_prod_v_v,A2: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ C @ ( inf_in6271465464967711157od_v_v @ A2 @ B2 ) )
=> ( member7453568604450474000od_v_v @ C @ B2 ) ) ).
% IntD2
thf(fact_222_IntD2,axiom,
! [C: v,A2: set_v,B2: set_v] :
( ( member_v @ C @ ( inf_inf_set_v @ A2 @ B2 ) )
=> ( member_v @ C @ B2 ) ) ).
% IntD2
thf(fact_223_IntD1,axiom,
! [C: product_prod_v_v,A2: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ C @ ( inf_in6271465464967711157od_v_v @ A2 @ B2 ) )
=> ( member7453568604450474000od_v_v @ C @ A2 ) ) ).
% IntD1
thf(fact_224_IntD1,axiom,
! [C: v,A2: set_v,B2: set_v] :
( ( member_v @ C @ ( inf_inf_set_v @ A2 @ B2 ) )
=> ( member_v @ C @ A2 ) ) ).
% IntD1
thf(fact_225_DiffE,axiom,
! [C: product_prod_v_v,A2: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ C @ ( minus_4183494784930505774od_v_v @ A2 @ B2 ) )
=> ~ ( ( member7453568604450474000od_v_v @ C @ A2 )
=> ( member7453568604450474000od_v_v @ C @ B2 ) ) ) ).
% DiffE
thf(fact_226_DiffE,axiom,
! [C: v,A2: set_v,B2: set_v] :
( ( member_v @ C @ ( minus_minus_set_v @ A2 @ B2 ) )
=> ~ ( ( member_v @ C @ A2 )
=> ( member_v @ C @ B2 ) ) ) ).
% DiffE
thf(fact_227_IntE,axiom,
! [C: product_prod_v_v,A2: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ C @ ( inf_in6271465464967711157od_v_v @ A2 @ B2 ) )
=> ~ ( ( member7453568604450474000od_v_v @ C @ A2 )
=> ~ ( member7453568604450474000od_v_v @ C @ B2 ) ) ) ).
% IntE
thf(fact_228_IntE,axiom,
! [C: v,A2: set_v,B2: set_v] :
( ( member_v @ C @ ( inf_inf_set_v @ A2 @ B2 ) )
=> ~ ( ( member_v @ C @ A2 )
=> ~ ( member_v @ C @ B2 ) ) ) ).
% IntE
thf(fact_229_bot__set__def,axiom,
( bot_bot_set_set_v
= ( collect_set_v @ bot_bot_set_v_o ) ) ).
% bot_set_def
thf(fact_230_bot__set__def,axiom,
( bot_bot_set_v
= ( collect_v @ bot_bot_v_o ) ) ).
% bot_set_def
thf(fact_231_bot__set__def,axiom,
( bot_bo723834152578015283od_v_v
= ( collec140062887454715474od_v_v @ bot_bo8461541820394803818_v_v_o ) ) ).
% bot_set_def
thf(fact_232_disjoint__iff__not__equal,axiom,
! [A2: set_v,B2: set_v] :
( ( ( inf_inf_set_v @ A2 @ B2 )
= bot_bot_set_v )
= ( ! [X2: v] :
( ( member_v @ X2 @ A2 )
=> ! [Y3: v] :
( ( member_v @ Y3 @ B2 )
=> ( X2 != Y3 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_233_disjoint__iff__not__equal,axiom,
! [A2: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( ( inf_in6271465464967711157od_v_v @ A2 @ B2 )
= bot_bo723834152578015283od_v_v )
= ( ! [X2: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X2 @ A2 )
=> ! [Y3: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ Y3 @ B2 )
=> ( X2 != Y3 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_234_Int__empty__right,axiom,
! [A2: set_v] :
( ( inf_inf_set_v @ A2 @ bot_bot_set_v )
= bot_bot_set_v ) ).
% Int_empty_right
thf(fact_235_Int__empty__right,axiom,
! [A2: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ A2 @ bot_bo723834152578015283od_v_v )
= bot_bo723834152578015283od_v_v ) ).
% Int_empty_right
thf(fact_236_Int__empty__left,axiom,
! [B2: set_v] :
( ( inf_inf_set_v @ bot_bot_set_v @ B2 )
= bot_bot_set_v ) ).
% Int_empty_left
thf(fact_237_Int__empty__left,axiom,
! [B2: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ bot_bo723834152578015283od_v_v @ B2 )
= bot_bo723834152578015283od_v_v ) ).
% Int_empty_left
thf(fact_238_disjoint__iff,axiom,
! [A2: set_v,B2: set_v] :
( ( ( inf_inf_set_v @ A2 @ B2 )
= bot_bot_set_v )
= ( ! [X2: v] :
( ( member_v @ X2 @ A2 )
=> ~ ( member_v @ X2 @ B2 ) ) ) ) ).
% disjoint_iff
thf(fact_239_disjoint__iff,axiom,
! [A2: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( ( inf_in6271465464967711157od_v_v @ A2 @ B2 )
= bot_bo723834152578015283od_v_v )
= ( ! [X2: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X2 @ A2 )
=> ~ ( member7453568604450474000od_v_v @ X2 @ B2 ) ) ) ) ).
% disjoint_iff
thf(fact_240_Int__emptyI,axiom,
! [A2: set_v,B2: set_v] :
( ! [X3: v] :
( ( member_v @ X3 @ A2 )
=> ~ ( member_v @ X3 @ B2 ) )
=> ( ( inf_inf_set_v @ A2 @ B2 )
= bot_bot_set_v ) ) ).
% Int_emptyI
thf(fact_241_Int__emptyI,axiom,
! [A2: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ! [X3: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X3 @ A2 )
=> ~ ( member7453568604450474000od_v_v @ X3 @ B2 ) )
=> ( ( inf_in6271465464967711157od_v_v @ A2 @ B2 )
= bot_bo723834152578015283od_v_v ) ) ).
% Int_emptyI
thf(fact_242_Int__Collect__mono,axiom,
! [A2: set_set_v,B2: set_set_v,P: set_v > $o,Q: set_v > $o] :
( ( ord_le5216385588623774835_set_v @ A2 @ B2 )
=> ( ! [X3: set_v] :
( ( member_set_v @ X3 @ A2 )
=> ( ( P @ X3 )
=> ( Q @ X3 ) ) )
=> ( ord_le5216385588623774835_set_v @ ( inf_inf_set_set_v @ A2 @ ( collect_set_v @ P ) ) @ ( inf_inf_set_set_v @ B2 @ ( collect_set_v @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_243_Int__Collect__mono,axiom,
! [A2: set_v,B2: set_v,P: v > $o,Q: v > $o] :
( ( ord_less_eq_set_v @ A2 @ B2 )
=> ( ! [X3: v] :
( ( member_v @ X3 @ A2 )
=> ( ( P @ X3 )
=> ( Q @ X3 ) ) )
=> ( ord_less_eq_set_v @ ( inf_inf_set_v @ A2 @ ( collect_v @ P ) ) @ ( inf_inf_set_v @ B2 @ ( collect_v @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_244_Int__Collect__mono,axiom,
! [A2: set_Product_prod_v_v,B2: set_Product_prod_v_v,P: product_prod_v_v > $o,Q: product_prod_v_v > $o] :
( ( ord_le7336532860387713383od_v_v @ A2 @ B2 )
=> ( ! [X3: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X3 @ A2 )
=> ( ( P @ X3 )
=> ( Q @ X3 ) ) )
=> ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ A2 @ ( collec140062887454715474od_v_v @ P ) ) @ ( inf_in6271465464967711157od_v_v @ B2 @ ( collec140062887454715474od_v_v @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_245_Int__greatest,axiom,
! [C2: set_v,A2: set_v,B2: set_v] :
( ( ord_less_eq_set_v @ C2 @ A2 )
=> ( ( ord_less_eq_set_v @ C2 @ B2 )
=> ( ord_less_eq_set_v @ C2 @ ( inf_inf_set_v @ A2 @ B2 ) ) ) ) ).
% Int_greatest
thf(fact_246_Int__greatest,axiom,
! [C2: set_Product_prod_v_v,A2: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ C2 @ A2 )
=> ( ( ord_le7336532860387713383od_v_v @ C2 @ B2 )
=> ( ord_le7336532860387713383od_v_v @ C2 @ ( inf_in6271465464967711157od_v_v @ A2 @ B2 ) ) ) ) ).
% Int_greatest
thf(fact_247_Int__absorb2,axiom,
! [A2: set_v,B2: set_v] :
( ( ord_less_eq_set_v @ A2 @ B2 )
=> ( ( inf_inf_set_v @ A2 @ B2 )
= A2 ) ) ).
% Int_absorb2
thf(fact_248_Int__absorb2,axiom,
! [A2: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A2 @ B2 )
=> ( ( inf_in6271465464967711157od_v_v @ A2 @ B2 )
= A2 ) ) ).
% Int_absorb2
thf(fact_249_Int__absorb1,axiom,
! [B2: set_v,A2: set_v] :
( ( ord_less_eq_set_v @ B2 @ A2 )
=> ( ( inf_inf_set_v @ A2 @ B2 )
= B2 ) ) ).
% Int_absorb1
thf(fact_250_Int__absorb1,axiom,
! [B2: set_Product_prod_v_v,A2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ B2 @ A2 )
=> ( ( inf_in6271465464967711157od_v_v @ A2 @ B2 )
= B2 ) ) ).
% Int_absorb1
thf(fact_251_Int__lower2,axiom,
! [A2: set_v,B2: set_v] : ( ord_less_eq_set_v @ ( inf_inf_set_v @ A2 @ B2 ) @ B2 ) ).
% Int_lower2
thf(fact_252_Int__lower2,axiom,
! [A2: set_Product_prod_v_v,B2: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ A2 @ B2 ) @ B2 ) ).
% Int_lower2
thf(fact_253_Int__lower1,axiom,
! [A2: set_v,B2: set_v] : ( ord_less_eq_set_v @ ( inf_inf_set_v @ A2 @ B2 ) @ A2 ) ).
% Int_lower1
thf(fact_254_Int__lower1,axiom,
! [A2: set_Product_prod_v_v,B2: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ A2 @ B2 ) @ A2 ) ).
% Int_lower1
thf(fact_255_Int__mono,axiom,
! [A2: set_v,C2: set_v,B2: set_v,D: set_v] :
( ( ord_less_eq_set_v @ A2 @ C2 )
=> ( ( ord_less_eq_set_v @ B2 @ D )
=> ( ord_less_eq_set_v @ ( inf_inf_set_v @ A2 @ B2 ) @ ( inf_inf_set_v @ C2 @ D ) ) ) ) ).
% Int_mono
thf(fact_256_Int__mono,axiom,
! [A2: set_Product_prod_v_v,C2: set_Product_prod_v_v,B2: set_Product_prod_v_v,D: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A2 @ C2 )
=> ( ( ord_le7336532860387713383od_v_v @ B2 @ D )
=> ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ A2 @ B2 ) @ ( inf_in6271465464967711157od_v_v @ C2 @ D ) ) ) ) ).
% Int_mono
thf(fact_257_double__diff,axiom,
! [A2: set_v,B2: set_v,C2: set_v] :
( ( ord_less_eq_set_v @ A2 @ B2 )
=> ( ( ord_less_eq_set_v @ B2 @ C2 )
=> ( ( minus_minus_set_v @ B2 @ ( minus_minus_set_v @ C2 @ A2 ) )
= A2 ) ) ) ).
% double_diff
thf(fact_258_double__diff,axiom,
! [A2: set_Product_prod_v_v,B2: set_Product_prod_v_v,C2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A2 @ B2 )
=> ( ( ord_le7336532860387713383od_v_v @ B2 @ C2 )
=> ( ( minus_4183494784930505774od_v_v @ B2 @ ( minus_4183494784930505774od_v_v @ C2 @ A2 ) )
= A2 ) ) ) ).
% double_diff
thf(fact_259_Diff__subset,axiom,
! [A2: set_v,B2: set_v] : ( ord_less_eq_set_v @ ( minus_minus_set_v @ A2 @ B2 ) @ A2 ) ).
% Diff_subset
thf(fact_260_Diff__subset,axiom,
! [A2: set_Product_prod_v_v,B2: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ ( minus_4183494784930505774od_v_v @ A2 @ B2 ) @ A2 ) ).
% Diff_subset
thf(fact_261_Diff__mono,axiom,
! [A2: set_v,C2: set_v,D: set_v,B2: set_v] :
( ( ord_less_eq_set_v @ A2 @ C2 )
=> ( ( ord_less_eq_set_v @ D @ B2 )
=> ( ord_less_eq_set_v @ ( minus_minus_set_v @ A2 @ B2 ) @ ( minus_minus_set_v @ C2 @ D ) ) ) ) ).
% Diff_mono
thf(fact_262_Diff__mono,axiom,
! [A2: set_Product_prod_v_v,C2: set_Product_prod_v_v,D: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A2 @ C2 )
=> ( ( ord_le7336532860387713383od_v_v @ D @ B2 )
=> ( ord_le7336532860387713383od_v_v @ ( minus_4183494784930505774od_v_v @ A2 @ B2 ) @ ( minus_4183494784930505774od_v_v @ C2 @ D ) ) ) ) ).
% Diff_mono
thf(fact_263_Sup__inter__less__eq,axiom,
! [A2: set_se8455005133513928103od_v_v,B2: set_se8455005133513928103od_v_v] : ( ord_le7336532860387713383od_v_v @ ( comple5788137035815166516od_v_v @ ( inf_in6058586722421118357od_v_v @ A2 @ B2 ) ) @ ( inf_in6271465464967711157od_v_v @ ( comple5788137035815166516od_v_v @ A2 ) @ ( comple5788137035815166516od_v_v @ B2 ) ) ) ).
% Sup_inter_less_eq
thf(fact_264_Sup__inter__less__eq,axiom,
! [A2: set_set_v,B2: set_set_v] : ( ord_less_eq_set_v @ ( comple2307003700295860064_set_v @ ( inf_inf_set_set_v @ A2 @ B2 ) ) @ ( inf_inf_set_v @ ( comple2307003700295860064_set_v @ A2 ) @ ( comple2307003700295860064_set_v @ B2 ) ) ) ).
% Sup_inter_less_eq
thf(fact_265_Union__Int__subset,axiom,
! [A2: set_se8455005133513928103od_v_v,B2: set_se8455005133513928103od_v_v] : ( ord_le7336532860387713383od_v_v @ ( comple5788137035815166516od_v_v @ ( inf_in6058586722421118357od_v_v @ A2 @ B2 ) ) @ ( inf_in6271465464967711157od_v_v @ ( comple5788137035815166516od_v_v @ A2 ) @ ( comple5788137035815166516od_v_v @ B2 ) ) ) ).
% Union_Int_subset
thf(fact_266_Union__Int__subset,axiom,
! [A2: set_set_v,B2: set_set_v] : ( ord_less_eq_set_v @ ( comple2307003700295860064_set_v @ ( inf_inf_set_set_v @ A2 @ B2 ) ) @ ( inf_inf_set_v @ ( comple2307003700295860064_set_v @ A2 ) @ ( comple2307003700295860064_set_v @ B2 ) ) ) ).
% Union_Int_subset
thf(fact_267_subset__code_I1_J,axiom,
! [Xs: list_v,B2: set_v] :
( ( ord_less_eq_set_v @ ( set_v2 @ Xs ) @ B2 )
= ( ! [X2: v] :
( ( member_v @ X2 @ ( set_v2 @ Xs ) )
=> ( member_v @ X2 @ B2 ) ) ) ) ).
% subset_code(1)
thf(fact_268_subset__code_I1_J,axiom,
! [Xs: list_P7986770385144383213od_v_v,B2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ ( set_Product_prod_v_v2 @ Xs ) @ B2 )
= ( ! [X2: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X2 @ ( set_Product_prod_v_v2 @ Xs ) )
=> ( member7453568604450474000od_v_v @ X2 @ B2 ) ) ) ) ).
% subset_code(1)
thf(fact_269_list_Osel_I2_J,axiom,
( ( tl_v @ nil_v )
= nil_v ) ).
% list.sel(2)
thf(fact_270_distinct__tl,axiom,
! [Xs: list_v] :
( ( distinct_v @ Xs )
=> ( distinct_v @ ( tl_v @ Xs ) ) ) ).
% distinct_tl
thf(fact_271_precedes__mem_I1_J,axiom,
! [X4: product_prod_v_v,Y: product_prod_v_v,Xs: list_P7986770385144383213od_v_v] :
( ( sCC_Bl2026170059108282219od_v_v @ X4 @ Y @ Xs )
=> ( member7453568604450474000od_v_v @ X4 @ ( set_Product_prod_v_v2 @ Xs ) ) ) ).
% precedes_mem(1)
thf(fact_272_precedes__mem_I1_J,axiom,
! [X4: v,Y: v,Xs: list_v] :
( ( sCC_Bl4022239298816431255edes_v @ X4 @ Y @ Xs )
=> ( member_v @ X4 @ ( set_v2 @ Xs ) ) ) ).
% precedes_mem(1)
thf(fact_273_precedes__mem_I2_J,axiom,
! [X4: product_prod_v_v,Y: product_prod_v_v,Xs: list_P7986770385144383213od_v_v] :
( ( sCC_Bl2026170059108282219od_v_v @ X4 @ Y @ Xs )
=> ( member7453568604450474000od_v_v @ Y @ ( set_Product_prod_v_v2 @ Xs ) ) ) ).
% precedes_mem(2)
thf(fact_274_precedes__mem_I2_J,axiom,
! [X4: v,Y: v,Xs: list_v] :
( ( sCC_Bl4022239298816431255edes_v @ X4 @ Y @ Xs )
=> ( member_v @ Y @ ( set_v2 @ Xs ) ) ) ).
% precedes_mem(2)
thf(fact_275_empty__set,axiom,
( bot_bot_set_v
= ( set_v2 @ nil_v ) ) ).
% empty_set
thf(fact_276_empty__set,axiom,
( bot_bo723834152578015283od_v_v
= ( set_Product_prod_v_v2 @ nil_Product_prod_v_v ) ) ).
% empty_set
thf(fact_277_tl__append__if,axiom,
! [Xs: list_v,Ys: list_v] :
( ( ( Xs = nil_v )
=> ( ( tl_v @ ( append_v @ Xs @ Ys ) )
= ( tl_v @ Ys ) ) )
& ( ( Xs != nil_v )
=> ( ( tl_v @ ( append_v @ Xs @ Ys ) )
= ( append_v @ ( tl_v @ Xs ) @ Ys ) ) ) ) ).
% tl_append_if
thf(fact_278_Union__disjoint,axiom,
! [C2: set_se8455005133513928103od_v_v,A2: set_Product_prod_v_v] :
( ( ( inf_in6271465464967711157od_v_v @ ( comple5788137035815166516od_v_v @ C2 ) @ A2 )
= bot_bo723834152578015283od_v_v )
= ( ! [X2: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ X2 @ C2 )
=> ( ( inf_in6271465464967711157od_v_v @ X2 @ A2 )
= bot_bo723834152578015283od_v_v ) ) ) ) ).
% Union_disjoint
thf(fact_279_Union__disjoint,axiom,
! [C2: set_set_v,A2: set_v] :
( ( ( inf_inf_set_v @ ( comple2307003700295860064_set_v @ C2 ) @ A2 )
= bot_bot_set_v )
= ( ! [X2: set_v] :
( ( member_set_v @ X2 @ C2 )
=> ( ( inf_inf_set_v @ X2 @ A2 )
= bot_bot_set_v ) ) ) ) ).
% Union_disjoint
thf(fact_280_list_Oexpand,axiom,
! [List: list_v,List2: list_v] :
( ( ( List = nil_v )
= ( List2 = nil_v ) )
=> ( ( ( List != nil_v )
=> ( ( List2 != nil_v )
=> ( ( ( hd_v @ List )
= ( hd_v @ List2 ) )
& ( ( tl_v @ List )
= ( tl_v @ List2 ) ) ) ) )
=> ( List = List2 ) ) ) ).
% list.expand
thf(fact_281_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_282_hd__in__set,axiom,
! [Xs: list_v] :
( ( Xs != nil_v )
=> ( member_v @ ( hd_v @ Xs ) @ ( set_v2 @ Xs ) ) ) ).
% hd_in_set
thf(fact_283_list_Oset__sel_I1_J,axiom,
! [A: list_P7986770385144383213od_v_v] :
( ( A != nil_Product_prod_v_v )
=> ( member7453568604450474000od_v_v @ ( hd_Product_prod_v_v @ A ) @ ( set_Product_prod_v_v2 @ A ) ) ) ).
% list.set_sel(1)
thf(fact_284_list_Oset__sel_I1_J,axiom,
! [A: list_v] :
( ( A != nil_v )
=> ( member_v @ ( hd_v @ A ) @ ( set_v2 @ A ) ) ) ).
% list.set_sel(1)
thf(fact_285_precedes__append__right__iff,axiom,
! [Y: product_prod_v_v,Ys: list_P7986770385144383213od_v_v,X4: product_prod_v_v,Xs: list_P7986770385144383213od_v_v] :
( ~ ( member7453568604450474000od_v_v @ Y @ ( set_Product_prod_v_v2 @ Ys ) )
=> ( ( sCC_Bl2026170059108282219od_v_v @ X4 @ Y @ ( append2138873909117096322od_v_v @ Xs @ Ys ) )
= ( sCC_Bl2026170059108282219od_v_v @ X4 @ Y @ Xs ) ) ) ).
% precedes_append_right_iff
thf(fact_286_precedes__append__right__iff,axiom,
! [Y: v,Ys: list_v,X4: v,Xs: list_v] :
( ~ ( member_v @ Y @ ( set_v2 @ Ys ) )
=> ( ( sCC_Bl4022239298816431255edes_v @ X4 @ Y @ ( append_v @ Xs @ Ys ) )
= ( sCC_Bl4022239298816431255edes_v @ X4 @ Y @ Xs ) ) ) ).
% precedes_append_right_iff
thf(fact_287_precedes__append__left__iff,axiom,
! [X4: product_prod_v_v,Ys: list_P7986770385144383213od_v_v,Y: product_prod_v_v,Xs: list_P7986770385144383213od_v_v] :
( ~ ( member7453568604450474000od_v_v @ X4 @ ( set_Product_prod_v_v2 @ Ys ) )
=> ( ( sCC_Bl2026170059108282219od_v_v @ X4 @ Y @ ( append2138873909117096322od_v_v @ Ys @ Xs ) )
= ( sCC_Bl2026170059108282219od_v_v @ X4 @ Y @ Xs ) ) ) ).
% precedes_append_left_iff
thf(fact_288_precedes__append__left__iff,axiom,
! [X4: v,Ys: list_v,Y: v,Xs: list_v] :
( ~ ( member_v @ X4 @ ( set_v2 @ Ys ) )
=> ( ( sCC_Bl4022239298816431255edes_v @ X4 @ Y @ ( append_v @ Ys @ Xs ) )
= ( sCC_Bl4022239298816431255edes_v @ X4 @ Y @ Xs ) ) ) ).
% precedes_append_left_iff
thf(fact_289_select__convs_I1_J,axiom,
! [Root: v,S: v > set_v,Explored: set_v,Visited: set_v,Vsuccs: v > set_v,Sccs: set_set_v,Stack: list_v,Cstack: list_v,More: product_unit] :
( ( sCC_Bl1090238580953940555t_unit @ ( sCC_Bl8064756265740546429t_unit @ Root @ S @ Explored @ Visited @ Vsuccs @ Sccs @ Stack @ Cstack @ More ) )
= Root ) ).
% select_convs(1)
thf(fact_290_ex__in__conv,axiom,
! [A2: set_v] :
( ( ? [X2: v] : ( member_v @ X2 @ A2 ) )
= ( A2 != bot_bot_set_v ) ) ).
% ex_in_conv
thf(fact_291_ex__in__conv,axiom,
! [A2: set_Product_prod_v_v] :
( ( ? [X2: product_prod_v_v] : ( member7453568604450474000od_v_v @ X2 @ A2 ) )
= ( A2 != bot_bo723834152578015283od_v_v ) ) ).
% ex_in_conv
thf(fact_292_equals0I,axiom,
! [A2: set_v] :
( ! [Y2: v] :
~ ( member_v @ Y2 @ A2 )
=> ( A2 = bot_bot_set_v ) ) ).
% equals0I
thf(fact_293_equals0I,axiom,
! [A2: set_Product_prod_v_v] :
( ! [Y2: product_prod_v_v] :
~ ( member7453568604450474000od_v_v @ Y2 @ A2 )
=> ( A2 = bot_bo723834152578015283od_v_v ) ) ).
% equals0I
thf(fact_294_equals0D,axiom,
! [A2: set_v,A: v] :
( ( A2 = bot_bot_set_v )
=> ~ ( member_v @ A @ A2 ) ) ).
% equals0D
thf(fact_295_equals0D,axiom,
! [A2: set_Product_prod_v_v,A: product_prod_v_v] :
( ( A2 = bot_bo723834152578015283od_v_v )
=> ~ ( member7453568604450474000od_v_v @ A @ A2 ) ) ).
% equals0D
thf(fact_296_emptyE,axiom,
! [A: v] :
~ ( member_v @ A @ bot_bot_set_v ) ).
% emptyE
thf(fact_297_emptyE,axiom,
! [A: product_prod_v_v] :
~ ( member7453568604450474000od_v_v @ A @ bot_bo723834152578015283od_v_v ) ).
% emptyE
thf(fact_298_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_299_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_300_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_301_set__eq__subset,axiom,
( ( ^ [Y5: set_v,Z5: set_v] : ( Y5 = Z5 ) )
= ( ^ [A4: set_v,B3: set_v] :
( ( ord_less_eq_set_v @ A4 @ B3 )
& ( ord_less_eq_set_v @ B3 @ A4 ) ) ) ) ).
% set_eq_subset
thf(fact_302_set__eq__subset,axiom,
( ( ^ [Y5: set_Product_prod_v_v,Z5: set_Product_prod_v_v] : ( Y5 = Z5 ) )
= ( ^ [A4: set_Product_prod_v_v,B3: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A4 @ B3 )
& ( ord_le7336532860387713383od_v_v @ B3 @ A4 ) ) ) ) ).
% set_eq_subset
thf(fact_303_subset__trans,axiom,
! [A2: set_v,B2: set_v,C2: set_v] :
( ( ord_less_eq_set_v @ A2 @ B2 )
=> ( ( ord_less_eq_set_v @ B2 @ C2 )
=> ( ord_less_eq_set_v @ A2 @ C2 ) ) ) ).
% subset_trans
thf(fact_304_subset__trans,axiom,
! [A2: set_Product_prod_v_v,B2: set_Product_prod_v_v,C2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A2 @ B2 )
=> ( ( ord_le7336532860387713383od_v_v @ B2 @ C2 )
=> ( ord_le7336532860387713383od_v_v @ A2 @ C2 ) ) ) ).
% subset_trans
thf(fact_305_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_306_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_307_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_308_subset__refl,axiom,
! [A2: set_v] : ( ord_less_eq_set_v @ A2 @ A2 ) ).
% subset_refl
thf(fact_309_subset__refl,axiom,
! [A2: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ A2 @ A2 ) ).
% subset_refl
thf(fact_310_subset__iff,axiom,
( ord_less_eq_set_v
= ( ^ [A4: set_v,B3: set_v] :
! [T: v] :
( ( member_v @ T @ A4 )
=> ( member_v @ T @ B3 ) ) ) ) ).
% subset_iff
thf(fact_311_subset__iff,axiom,
( ord_le7336532860387713383od_v_v
= ( ^ [A4: set_Product_prod_v_v,B3: set_Product_prod_v_v] :
! [T: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ T @ A4 )
=> ( member7453568604450474000od_v_v @ T @ B3 ) ) ) ) ).
% subset_iff
thf(fact_312_equalityD2,axiom,
! [A2: set_v,B2: set_v] :
( ( A2 = B2 )
=> ( ord_less_eq_set_v @ B2 @ A2 ) ) ).
% equalityD2
thf(fact_313_equalityD2,axiom,
! [A2: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( A2 = B2 )
=> ( ord_le7336532860387713383od_v_v @ B2 @ A2 ) ) ).
% equalityD2
thf(fact_314_equalityD1,axiom,
! [A2: set_v,B2: set_v] :
( ( A2 = B2 )
=> ( ord_less_eq_set_v @ A2 @ B2 ) ) ).
% equalityD1
thf(fact_315_equalityD1,axiom,
! [A2: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( A2 = B2 )
=> ( ord_le7336532860387713383od_v_v @ A2 @ B2 ) ) ).
% equalityD1
thf(fact_316_subset__eq,axiom,
( ord_less_eq_set_v
= ( ^ [A4: set_v,B3: set_v] :
! [X2: v] :
( ( member_v @ X2 @ A4 )
=> ( member_v @ X2 @ B3 ) ) ) ) ).
% subset_eq
thf(fact_317_subset__eq,axiom,
( ord_le7336532860387713383od_v_v
= ( ^ [A4: set_Product_prod_v_v,B3: set_Product_prod_v_v] :
! [X2: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X2 @ A4 )
=> ( member7453568604450474000od_v_v @ X2 @ B3 ) ) ) ) ).
% subset_eq
thf(fact_318_equalityE,axiom,
! [A2: set_v,B2: set_v] :
( ( A2 = B2 )
=> ~ ( ( ord_less_eq_set_v @ A2 @ B2 )
=> ~ ( ord_less_eq_set_v @ B2 @ A2 ) ) ) ).
% equalityE
thf(fact_319_equalityE,axiom,
! [A2: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( A2 = B2 )
=> ~ ( ( ord_le7336532860387713383od_v_v @ A2 @ B2 )
=> ~ ( ord_le7336532860387713383od_v_v @ B2 @ A2 ) ) ) ).
% equalityE
thf(fact_320_subsetD,axiom,
! [A2: set_v,B2: set_v,C: v] :
( ( ord_less_eq_set_v @ A2 @ B2 )
=> ( ( member_v @ C @ A2 )
=> ( member_v @ C @ B2 ) ) ) ).
% subsetD
thf(fact_321_subsetD,axiom,
! [A2: set_Product_prod_v_v,B2: set_Product_prod_v_v,C: product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A2 @ B2 )
=> ( ( member7453568604450474000od_v_v @ C @ A2 )
=> ( member7453568604450474000od_v_v @ C @ B2 ) ) ) ).
% subsetD
thf(fact_322_in__mono,axiom,
! [A2: set_v,B2: set_v,X4: v] :
( ( ord_less_eq_set_v @ A2 @ B2 )
=> ( ( member_v @ X4 @ A2 )
=> ( member_v @ X4 @ B2 ) ) ) ).
% in_mono
thf(fact_323_in__mono,axiom,
! [A2: set_Product_prod_v_v,B2: set_Product_prod_v_v,X4: product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A2 @ B2 )
=> ( ( member7453568604450474000od_v_v @ X4 @ A2 )
=> ( member7453568604450474000od_v_v @ X4 @ B2 ) ) ) ).
% in_mono
thf(fact_324_surjective,axiom,
! [R: sCC_Bl1394983891496994913t_unit] :
( R
= ( sCC_Bl8064756265740546429t_unit @ ( sCC_Bl1090238580953940555t_unit @ R ) @ ( sCC_Bl1280885523602775798t_unit @ R ) @ ( sCC_Bl157864678168468314t_unit @ R ) @ ( sCC_Bl4645233313691564917t_unit @ R ) @ ( sCC_Bl3795065053823578884t_unit @ R ) @ ( sCC_Bl2536197123907397897t_unit @ R ) @ ( sCC_Bl8828226123343373779t_unit @ R ) @ ( sCC_Bl9201514103433284750t_unit @ R ) @ ( sCC_Bl3567736435408124606t_unit @ R ) ) ) ).
% surjective
thf(fact_325_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_326_boolean__algebra_Oconj__zero__right,axiom,
! [X4: set_v] :
( ( inf_inf_set_v @ X4 @ bot_bot_set_v )
= bot_bot_set_v ) ).
% boolean_algebra.conj_zero_right
thf(fact_327_boolean__algebra_Oconj__zero__right,axiom,
! [X4: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ X4 @ bot_bo723834152578015283od_v_v )
= bot_bo723834152578015283od_v_v ) ).
% boolean_algebra.conj_zero_right
thf(fact_328_boolean__algebra_Oconj__zero__left,axiom,
! [X4: set_v] :
( ( inf_inf_set_v @ bot_bot_set_v @ X4 )
= bot_bot_set_v ) ).
% boolean_algebra.conj_zero_left
thf(fact_329_boolean__algebra_Oconj__zero__left,axiom,
! [X4: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ bot_bo723834152578015283od_v_v @ X4 )
= bot_bo723834152578015283od_v_v ) ).
% boolean_algebra.conj_zero_left
thf(fact_330_inf__bot__right,axiom,
! [X4: set_v] :
( ( inf_inf_set_v @ X4 @ bot_bot_set_v )
= bot_bot_set_v ) ).
% inf_bot_right
thf(fact_331_inf__bot__right,axiom,
! [X4: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ X4 @ bot_bo723834152578015283od_v_v )
= bot_bo723834152578015283od_v_v ) ).
% inf_bot_right
thf(fact_332_inf__bot__left,axiom,
! [X4: set_v] :
( ( inf_inf_set_v @ bot_bot_set_v @ X4 )
= bot_bot_set_v ) ).
% inf_bot_left
thf(fact_333_inf__bot__left,axiom,
! [X4: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ bot_bo723834152578015283od_v_v @ X4 )
= bot_bo723834152578015283od_v_v ) ).
% inf_bot_left
thf(fact_334_inf_Obounded__iff,axiom,
! [A: set_v,B: set_v,C: set_v] :
( ( ord_less_eq_set_v @ A @ ( inf_inf_set_v @ B @ C ) )
= ( ( ord_less_eq_set_v @ A @ B )
& ( ord_less_eq_set_v @ A @ C ) ) ) ).
% inf.bounded_iff
thf(fact_335_inf_Obounded__iff,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ ( inf_in6271465464967711157od_v_v @ B @ C ) )
= ( ( ord_le7336532860387713383od_v_v @ A @ B )
& ( ord_le7336532860387713383od_v_v @ A @ C ) ) ) ).
% inf.bounded_iff
thf(fact_336_le__inf__iff,axiom,
! [X4: set_v,Y: set_v,Z: set_v] :
( ( ord_less_eq_set_v @ X4 @ ( inf_inf_set_v @ Y @ Z ) )
= ( ( ord_less_eq_set_v @ X4 @ Y )
& ( ord_less_eq_set_v @ X4 @ Z ) ) ) ).
% le_inf_iff
thf(fact_337_le__inf__iff,axiom,
! [X4: set_Product_prod_v_v,Y: set_Product_prod_v_v,Z: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X4 @ ( inf_in6271465464967711157od_v_v @ Y @ Z ) )
= ( ( ord_le7336532860387713383od_v_v @ X4 @ Y )
& ( ord_le7336532860387713383od_v_v @ X4 @ Z ) ) ) ).
% le_inf_iff
thf(fact_338_init__env__pre__dfs,axiom,
! [V3: v] : ( sCC_Bl36166008131615352t_unit @ successors @ V3 @ ( sCC_Bl7693227186847904995_env_v @ V3 ) ) ).
% init_env_pre_dfs
thf(fact_339_inf_Oidem,axiom,
! [A: set_v] :
( ( inf_inf_set_v @ A @ A )
= A ) ).
% inf.idem
thf(fact_340_inf__idem,axiom,
! [X4: set_v] :
( ( inf_inf_set_v @ X4 @ X4 )
= X4 ) ).
% inf_idem
thf(fact_341_inf_Oleft__idem,axiom,
! [A: set_v,B: set_v] :
( ( inf_inf_set_v @ A @ ( inf_inf_set_v @ A @ B ) )
= ( inf_inf_set_v @ A @ B ) ) ).
% inf.left_idem
thf(fact_342_inf__left__idem,axiom,
! [X4: set_v,Y: set_v] :
( ( inf_inf_set_v @ X4 @ ( inf_inf_set_v @ X4 @ Y ) )
= ( inf_inf_set_v @ X4 @ Y ) ) ).
% inf_left_idem
thf(fact_343_inf_Oright__idem,axiom,
! [A: set_v,B: set_v] :
( ( inf_inf_set_v @ ( inf_inf_set_v @ A @ B ) @ B )
= ( inf_inf_set_v @ A @ B ) ) ).
% inf.right_idem
thf(fact_344_inf__right__idem,axiom,
! [X4: set_v,Y: set_v] :
( ( inf_inf_set_v @ ( inf_inf_set_v @ X4 @ Y ) @ Y )
= ( inf_inf_set_v @ X4 @ Y ) ) ).
% inf_right_idem
thf(fact_345_graph_Opre__dfs_Ocong,axiom,
sCC_Bl36166008131615352t_unit = sCC_Bl36166008131615352t_unit ).
% graph.pre_dfs.cong
thf(fact_346_graph_Osclosed,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ! [X: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X @ Vertices )
=> ( ord_le7336532860387713383od_v_v @ ( Successors @ X ) @ Vertices ) ) ) ).
% graph.sclosed
thf(fact_347_graph_Osclosed,axiom,
! [Vertices: set_v,Successors: v > set_v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ! [X: v] :
( ( member_v @ X @ Vertices )
=> ( ord_less_eq_set_v @ ( Successors @ X ) @ Vertices ) ) ) ).
% graph.sclosed
thf(fact_348_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,X4: product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( member7453568604450474000od_v_v @ Y @ ( Successors @ X4 ) )
=> ( sCC_Bl4981926079593201289od_v_v @ Successors @ X4 @ Y ) ) ) ).
% graph.reachable_edge
thf(fact_349_graph_Oreachable__edge,axiom,
! [Vertices: set_v,Successors: v > set_v,Y: v,X4: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( member_v @ Y @ ( Successors @ X4 ) )
=> ( sCC_Bl649662514949026229able_v @ Successors @ X4 @ Y ) ) ) ).
% graph.reachable_edge
thf(fact_350_graph_Osucc__reachable,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,X4: 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 @ X4 @ Y )
=> ( ( member7453568604450474000od_v_v @ Z @ ( Successors @ Y ) )
=> ( sCC_Bl4981926079593201289od_v_v @ Successors @ X4 @ Z ) ) ) ) ).
% graph.succ_reachable
thf(fact_351_graph_Osucc__reachable,axiom,
! [Vertices: set_v,Successors: v > set_v,X4: v,Y: v,Z: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl649662514949026229able_v @ Successors @ X4 @ Y )
=> ( ( member_v @ Z @ ( Successors @ Y ) )
=> ( sCC_Bl649662514949026229able_v @ Successors @ X4 @ Z ) ) ) ) ).
% graph.succ_reachable
thf(fact_352_graph_Oreachable_Ocases,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,A1: product_prod_v_v,A22: product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( sCC_Bl4981926079593201289od_v_v @ Successors @ A1 @ A22 )
=> ( ( A22 != A1 )
=> ~ ! [Y2: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ Y2 @ ( Successors @ A1 ) )
=> ~ ( sCC_Bl4981926079593201289od_v_v @ Successors @ Y2 @ A22 ) ) ) ) ) ).
% graph.reachable.cases
thf(fact_353_graph_Oreachable_Ocases,axiom,
! [Vertices: set_v,Successors: v > set_v,A1: v,A22: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl649662514949026229able_v @ Successors @ A1 @ A22 )
=> ( ( A22 != A1 )
=> ~ ! [Y2: v] :
( ( member_v @ Y2 @ ( Successors @ A1 ) )
=> ~ ( sCC_Bl649662514949026229able_v @ Successors @ Y2 @ A22 ) ) ) ) ) ).
% graph.reachable.cases
thf(fact_354_graph_Oreachable_Osimps,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,A1: product_prod_v_v,A22: product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( sCC_Bl4981926079593201289od_v_v @ Successors @ A1 @ A22 )
= ( ? [X2: product_prod_v_v] :
( ( A1 = X2 )
& ( A22 = X2 ) )
| ? [X2: product_prod_v_v,Y3: product_prod_v_v,Z2: product_prod_v_v] :
( ( A1 = X2 )
& ( A22 = Z2 )
& ( member7453568604450474000od_v_v @ Y3 @ ( Successors @ X2 ) )
& ( sCC_Bl4981926079593201289od_v_v @ Successors @ Y3 @ Z2 ) ) ) ) ) ).
% graph.reachable.simps
thf(fact_355_graph_Oreachable_Osimps,axiom,
! [Vertices: set_v,Successors: v > set_v,A1: v,A22: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl649662514949026229able_v @ Successors @ A1 @ A22 )
= ( ? [X2: v] :
( ( A1 = X2 )
& ( A22 = X2 ) )
| ? [X2: v,Y3: v,Z2: v] :
( ( A1 = X2 )
& ( A22 = Z2 )
& ( member_v @ Y3 @ ( Successors @ X2 ) )
& ( sCC_Bl649662514949026229able_v @ Successors @ Y3 @ Z2 ) ) ) ) ) ).
% graph.reachable.simps
thf(fact_356_graph_Oreachable__trans,axiom,
! [Vertices: set_v,Successors: v > set_v,X4: v,Y: v,Z: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl649662514949026229able_v @ Successors @ X4 @ Y )
=> ( ( sCC_Bl649662514949026229able_v @ Successors @ Y @ Z )
=> ( sCC_Bl649662514949026229able_v @ Successors @ X4 @ Z ) ) ) ) ).
% graph.reachable_trans
thf(fact_357_graph_Oreachable__end__induct,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,X4: 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 @ X4 @ 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 @ X4 @ Y ) ) ) ) ) ).
% graph.reachable_end_induct
thf(fact_358_graph_Oreachable__end__induct,axiom,
! [Vertices: set_v,Successors: v > set_v,X4: v,Y: v,P: v > v > $o] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl649662514949026229able_v @ Successors @ X4 @ Y )
=> ( ! [X3: v] : ( P @ X3 @ X3 )
=> ( ! [X3: v,Y2: v,Z3: v] :
( ( P @ X3 @ Y2 )
=> ( ( member_v @ Z3 @ ( Successors @ Y2 ) )
=> ( P @ X3 @ Z3 ) ) )
=> ( P @ X4 @ Y ) ) ) ) ) ).
% graph.reachable_end_induct
thf(fact_359_graph_Oreachable__refl,axiom,
! [Vertices: set_v,Successors: v > set_v,X4: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( sCC_Bl649662514949026229able_v @ Successors @ X4 @ X4 ) ) ).
% graph.reachable_refl
thf(fact_360_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,X4: product_prod_v_v,Z: product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( member7453568604450474000od_v_v @ Y @ ( Successors @ X4 ) )
=> ( ( sCC_Bl4981926079593201289od_v_v @ Successors @ Y @ Z )
=> ( sCC_Bl4981926079593201289od_v_v @ Successors @ X4 @ Z ) ) ) ) ).
% graph.reachable_succ
thf(fact_361_graph_Oreachable__succ,axiom,
! [Vertices: set_v,Successors: v > set_v,Y: v,X4: v,Z: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( member_v @ Y @ ( Successors @ X4 ) )
=> ( ( sCC_Bl649662514949026229able_v @ Successors @ Y @ Z )
=> ( sCC_Bl649662514949026229able_v @ Successors @ X4 @ Z ) ) ) ) ).
% graph.reachable_succ
thf(fact_362_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_363_graph_OS__reflexive,axiom,
! [Vertices: set_v,Successors: v > set_v,E: sCC_Bl1394983891496994913t_unit,N2: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl9196236973127232072t_unit @ Successors @ E )
=> ( member_v @ N2 @ ( sCC_Bl1280885523602775798t_unit @ E @ N2 ) ) ) ) ).
% graph.S_reflexive
thf(fact_364_graph_Oscc__partition,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,S3: set_Product_prod_v_v,S5: set_Product_prod_v_v,X4: product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( sCC_Bl6242042402218619277od_v_v @ Successors @ S3 )
=> ( ( sCC_Bl6242042402218619277od_v_v @ Successors @ S5 )
=> ( ( member7453568604450474000od_v_v @ X4 @ ( inf_in6271465464967711157od_v_v @ S3 @ S5 ) )
=> ( S3 = S5 ) ) ) ) ) ).
% graph.scc_partition
thf(fact_365_graph_Oscc__partition,axiom,
! [Vertices: set_v,Successors: v > set_v,S3: set_v,S5: set_v,X4: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bloemen_is_scc_v @ Successors @ S3 )
=> ( ( sCC_Bloemen_is_scc_v @ Successors @ S5 )
=> ( ( member_v @ X4 @ ( inf_inf_set_v @ S3 @ S5 ) )
=> ( S3 = S5 ) ) ) ) ) ).
% graph.scc_partition
thf(fact_366_graph_Ois__subscc__def,axiom,
! [Vertices: set_v,Successors: v > set_v,S3: set_v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl5398416737448265317bscc_v @ Successors @ S3 )
= ( ! [X2: v] :
( ( member_v @ X2 @ S3 )
=> ! [Y3: v] :
( ( member_v @ Y3 @ S3 )
=> ( sCC_Bl649662514949026229able_v @ Successors @ X2 @ Y3 ) ) ) ) ) ) ).
% graph.is_subscc_def
thf(fact_367_graph_OsccE,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,S3: set_Product_prod_v_v,X4: product_prod_v_v,Y: product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( sCC_Bl6242042402218619277od_v_v @ Successors @ S3 )
=> ( ( member7453568604450474000od_v_v @ X4 @ S3 )
=> ( ( sCC_Bl4981926079593201289od_v_v @ Successors @ X4 @ Y )
=> ( ( sCC_Bl4981926079593201289od_v_v @ Successors @ Y @ X4 )
=> ( member7453568604450474000od_v_v @ Y @ S3 ) ) ) ) ) ) ).
% graph.sccE
thf(fact_368_graph_OsccE,axiom,
! [Vertices: set_v,Successors: v > set_v,S3: set_v,X4: v,Y: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bloemen_is_scc_v @ Successors @ S3 )
=> ( ( member_v @ X4 @ S3 )
=> ( ( sCC_Bl649662514949026229able_v @ Successors @ X4 @ Y )
=> ( ( sCC_Bl649662514949026229able_v @ Successors @ Y @ X4 )
=> ( member_v @ Y @ S3 ) ) ) ) ) ) ).
% graph.sccE
thf(fact_369_inf__sup__aci_I4_J,axiom,
! [X4: set_v,Y: set_v] :
( ( inf_inf_set_v @ X4 @ ( inf_inf_set_v @ X4 @ Y ) )
= ( inf_inf_set_v @ X4 @ Y ) ) ).
% inf_sup_aci(4)
thf(fact_370_inf__sup__aci_I3_J,axiom,
! [X4: set_v,Y: set_v,Z: set_v] :
( ( inf_inf_set_v @ X4 @ ( inf_inf_set_v @ Y @ Z ) )
= ( inf_inf_set_v @ Y @ ( inf_inf_set_v @ X4 @ Z ) ) ) ).
% inf_sup_aci(3)
thf(fact_371_inf__sup__aci_I2_J,axiom,
! [X4: set_v,Y: set_v,Z: set_v] :
( ( inf_inf_set_v @ ( inf_inf_set_v @ X4 @ Y ) @ Z )
= ( inf_inf_set_v @ X4 @ ( inf_inf_set_v @ Y @ Z ) ) ) ).
% inf_sup_aci(2)
thf(fact_372_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_373_inf_Oassoc,axiom,
! [A: set_v,B: set_v,C: set_v] :
( ( inf_inf_set_v @ ( inf_inf_set_v @ A @ B ) @ C )
= ( inf_inf_set_v @ A @ ( inf_inf_set_v @ B @ C ) ) ) ).
% inf.assoc
thf(fact_374_inf__assoc,axiom,
! [X4: set_v,Y: set_v,Z: set_v] :
( ( inf_inf_set_v @ ( inf_inf_set_v @ X4 @ Y ) @ Z )
= ( inf_inf_set_v @ X4 @ ( inf_inf_set_v @ Y @ Z ) ) ) ).
% inf_assoc
thf(fact_375_inf_Ocommute,axiom,
( inf_inf_set_v
= ( ^ [A5: set_v,B4: set_v] : ( inf_inf_set_v @ B4 @ A5 ) ) ) ).
% inf.commute
thf(fact_376_inf__commute,axiom,
( inf_inf_set_v
= ( ^ [X2: set_v,Y3: set_v] : ( inf_inf_set_v @ Y3 @ X2 ) ) ) ).
% inf_commute
thf(fact_377_boolean__algebra__cancel_Oinf1,axiom,
! [A2: set_v,K: set_v,A: set_v,B: set_v] :
( ( A2
= ( inf_inf_set_v @ K @ A ) )
=> ( ( inf_inf_set_v @ A2 @ B )
= ( inf_inf_set_v @ K @ ( inf_inf_set_v @ A @ B ) ) ) ) ).
% boolean_algebra_cancel.inf1
thf(fact_378_boolean__algebra__cancel_Oinf2,axiom,
! [B2: set_v,K: set_v,B: set_v,A: set_v] :
( ( B2
= ( inf_inf_set_v @ K @ B ) )
=> ( ( inf_inf_set_v @ A @ B2 )
= ( inf_inf_set_v @ K @ ( inf_inf_set_v @ A @ B ) ) ) ) ).
% boolean_algebra_cancel.inf2
thf(fact_379_inf_Oleft__commute,axiom,
! [B: set_v,A: set_v,C: set_v] :
( ( inf_inf_set_v @ B @ ( inf_inf_set_v @ A @ C ) )
= ( inf_inf_set_v @ A @ ( inf_inf_set_v @ B @ C ) ) ) ).
% inf.left_commute
thf(fact_380_inf__left__commute,axiom,
! [X4: set_v,Y: set_v,Z: set_v] :
( ( inf_inf_set_v @ X4 @ ( inf_inf_set_v @ Y @ Z ) )
= ( inf_inf_set_v @ Y @ ( inf_inf_set_v @ X4 @ Z ) ) ) ).
% inf_left_commute
thf(fact_381_graph_Odfs__S__tl__stack_I1_J,axiom,
! [Vertices: set_v,Successors: v > set_v,V3: v,E: sCC_Bl1394983891496994913t_unit,E2: sCC_Bl1394983891496994913t_unit] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl8953792750115413617t_unit @ Successors @ V3 @ E @ E2 )
=> ( ( ( sCC_Bl8828226123343373779t_unit @ E )
!= nil_v )
=> ( ( sCC_Bl8828226123343373779t_unit @ E2 )
!= nil_v ) ) ) ) ).
% graph.dfs_S_tl_stack(1)
thf(fact_382_graph_Ostack__visited,axiom,
! [Vertices: set_v,Successors: v > set_v,E: sCC_Bl1394983891496994913t_unit,N2: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl9196236973127232072t_unit @ Successors @ E )
=> ( ( member_v @ N2 @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E ) ) )
=> ( member_v @ N2 @ ( sCC_Bl4645233313691564917t_unit @ E ) ) ) ) ) ).
% graph.stack_visited
thf(fact_383_graph_Ostack__unexplored,axiom,
! [Vertices: set_v,Successors: v > set_v,E: sCC_Bl1394983891496994913t_unit,N2: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl9196236973127232072t_unit @ Successors @ E )
=> ( ( member_v @ N2 @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E ) ) )
=> ~ ( member_v @ N2 @ ( sCC_Bl157864678168468314t_unit @ E ) ) ) ) ) ).
% graph.stack_unexplored
thf(fact_384_graph_Ois__scc__def,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,S3: set_Product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( sCC_Bl6242042402218619277od_v_v @ Successors @ S3 )
= ( ( S3 != bot_bo723834152578015283od_v_v )
& ( sCC_Bl2301996248249672505od_v_v @ Successors @ S3 )
& ! [S4: set_Product_prod_v_v] :
( ( ( ord_le7336532860387713383od_v_v @ S3 @ S4 )
& ( sCC_Bl2301996248249672505od_v_v @ Successors @ S4 ) )
=> ( S4 = S3 ) ) ) ) ) ).
% graph.is_scc_def
thf(fact_385_graph_Ois__scc__def,axiom,
! [Vertices: set_v,Successors: v > set_v,S3: set_v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bloemen_is_scc_v @ Successors @ S3 )
= ( ( S3 != bot_bot_set_v )
& ( sCC_Bl5398416737448265317bscc_v @ Successors @ S3 )
& ! [S4: set_v] :
( ( ( ord_less_eq_set_v @ S3 @ S4 )
& ( sCC_Bl5398416737448265317bscc_v @ Successors @ S4 ) )
=> ( S4 = S3 ) ) ) ) ) ).
% graph.is_scc_def
thf(fact_386_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 ) )
=> ! [Xa: v] :
( ( member_v @ Xa @ ( minus_minus_set_v @ ( Successors @ X3 ) @ ( sCC_Bl3795065053823578884t_unit @ E @ X3 ) ) )
=> ( ( sCC_Bl649662514949026229able_v @ Successors @ V3 @ X3 )
=> ~ ( sCC_Bl649662514949026229able_v @ Successors @ Xa @ W ) ) ) )
=> ( member_v @ W @ ( sCC_Bl4645233313691564917t_unit @ E ) ) ) ) ) ) ) ).
% graph.reachable_visited
thf(fact_387_graph_Odfs__S__tl__stack_I2_J,axiom,
! [Vertices: set_v,Successors: v > set_v,V3: v,E: sCC_Bl1394983891496994913t_unit,E2: sCC_Bl1394983891496994913t_unit] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl8953792750115413617t_unit @ Successors @ V3 @ E @ E2 )
=> ( ( ( sCC_Bl8828226123343373779t_unit @ E )
!= nil_v )
=> ! [X: v] :
( ( member_v @ X @ ( set_v2 @ ( tl_v @ ( sCC_Bl8828226123343373779t_unit @ E2 ) ) ) )
=> ( ( sCC_Bl1280885523602775798t_unit @ E2 @ X )
= ( sCC_Bl1280885523602775798t_unit @ E @ X ) ) ) ) ) ) ).
% graph.dfs_S_tl_stack(2)
thf(fact_388_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_389_graph_Ovisited__unexplored,axiom,
! [Vertices: set_v,Successors: v > set_v,E: sCC_Bl1394983891496994913t_unit,M2: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl9196236973127232072t_unit @ Successors @ E )
=> ( ( member_v @ M2 @ ( sCC_Bl4645233313691564917t_unit @ E ) )
=> ( ~ ( member_v @ M2 @ ( sCC_Bl157864678168468314t_unit @ E ) )
=> ~ ! [N3: v] :
( ( member_v @ N3 @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E ) ) )
=> ~ ( member_v @ M2 @ ( sCC_Bl1280885523602775798t_unit @ E @ N3 ) ) ) ) ) ) ) ).
% graph.visited_unexplored
thf(fact_390_graph_Odfs__S__hd__stack_I2_J,axiom,
! [Vertices: set_v,Successors: v > set_v,E: sCC_Bl1394983891496994913t_unit,V3: v,E2: sCC_Bl1394983891496994913t_unit,N2: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl9196236973127232072t_unit @ Successors @ E )
=> ( ( sCC_Bl8953792750115413617t_unit @ Successors @ V3 @ E @ E2 )
=> ( ( ( sCC_Bl8828226123343373779t_unit @ E )
!= nil_v )
=> ( ( member_v @ N2 @ ( sCC_Bl1280885523602775798t_unit @ E @ ( hd_v @ ( sCC_Bl8828226123343373779t_unit @ E ) ) ) )
=> ( member_v @ N2 @ ( sCC_Bl1280885523602775798t_unit @ E2 @ ( hd_v @ ( sCC_Bl8828226123343373779t_unit @ E2 ) ) ) ) ) ) ) ) ) ).
% graph.dfs_S_hd_stack(2)
thf(fact_391_graph_Odfs__S__hd__stack_I1_J,axiom,
! [Vertices: set_v,Successors: v > set_v,E: sCC_Bl1394983891496994913t_unit,V3: v,E2: sCC_Bl1394983891496994913t_unit,N2: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl9196236973127232072t_unit @ Successors @ E )
=> ( ( sCC_Bl8953792750115413617t_unit @ Successors @ V3 @ E @ E2 )
=> ( ( ( sCC_Bl8828226123343373779t_unit @ E )
!= nil_v )
=> ( ( member_v @ N2 @ ( sCC_Bl1280885523602775798t_unit @ E @ ( hd_v @ ( sCC_Bl8828226123343373779t_unit @ E ) ) ) )
=> ( ( sCC_Bl8828226123343373779t_unit @ E2 )
!= nil_v ) ) ) ) ) ) ).
% graph.dfs_S_hd_stack(1)
thf(fact_392_inf__sup__ord_I2_J,axiom,
! [X4: set_v,Y: set_v] : ( ord_less_eq_set_v @ ( inf_inf_set_v @ X4 @ Y ) @ Y ) ).
% inf_sup_ord(2)
thf(fact_393_inf__sup__ord_I2_J,axiom,
! [X4: set_Product_prod_v_v,Y: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ X4 @ Y ) @ Y ) ).
% inf_sup_ord(2)
thf(fact_394_inf__sup__ord_I1_J,axiom,
! [X4: set_v,Y: set_v] : ( ord_less_eq_set_v @ ( inf_inf_set_v @ X4 @ Y ) @ X4 ) ).
% inf_sup_ord(1)
thf(fact_395_inf__sup__ord_I1_J,axiom,
! [X4: set_Product_prod_v_v,Y: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ X4 @ Y ) @ X4 ) ).
% inf_sup_ord(1)
thf(fact_396_inf__le1,axiom,
! [X4: set_v,Y: set_v] : ( ord_less_eq_set_v @ ( inf_inf_set_v @ X4 @ Y ) @ X4 ) ).
% inf_le1
thf(fact_397_inf__le1,axiom,
! [X4: set_Product_prod_v_v,Y: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ X4 @ Y ) @ X4 ) ).
% inf_le1
thf(fact_398_inf__le2,axiom,
! [X4: set_v,Y: set_v] : ( ord_less_eq_set_v @ ( inf_inf_set_v @ X4 @ Y ) @ Y ) ).
% inf_le2
thf(fact_399_inf__le2,axiom,
! [X4: set_Product_prod_v_v,Y: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ X4 @ Y ) @ Y ) ).
% inf_le2
thf(fact_400_le__infE,axiom,
! [X4: set_v,A: set_v,B: set_v] :
( ( ord_less_eq_set_v @ X4 @ ( inf_inf_set_v @ A @ B ) )
=> ~ ( ( ord_less_eq_set_v @ X4 @ A )
=> ~ ( ord_less_eq_set_v @ X4 @ B ) ) ) ).
% le_infE
thf(fact_401_le__infE,axiom,
! [X4: set_Product_prod_v_v,A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X4 @ ( inf_in6271465464967711157od_v_v @ A @ B ) )
=> ~ ( ( ord_le7336532860387713383od_v_v @ X4 @ A )
=> ~ ( ord_le7336532860387713383od_v_v @ X4 @ B ) ) ) ).
% le_infE
thf(fact_402_le__infI,axiom,
! [X4: set_v,A: set_v,B: set_v] :
( ( ord_less_eq_set_v @ X4 @ A )
=> ( ( ord_less_eq_set_v @ X4 @ B )
=> ( ord_less_eq_set_v @ X4 @ ( inf_inf_set_v @ A @ B ) ) ) ) ).
% le_infI
thf(fact_403_le__infI,axiom,
! [X4: set_Product_prod_v_v,A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X4 @ A )
=> ( ( ord_le7336532860387713383od_v_v @ X4 @ B )
=> ( ord_le7336532860387713383od_v_v @ X4 @ ( inf_in6271465464967711157od_v_v @ A @ B ) ) ) ) ).
% le_infI
thf(fact_404_inf__mono,axiom,
! [A: set_v,C: set_v,B: set_v,D2: set_v] :
( ( ord_less_eq_set_v @ A @ C )
=> ( ( ord_less_eq_set_v @ B @ D2 )
=> ( ord_less_eq_set_v @ ( inf_inf_set_v @ A @ B ) @ ( inf_inf_set_v @ C @ D2 ) ) ) ) ).
% inf_mono
thf(fact_405_inf__mono,axiom,
! [A: set_Product_prod_v_v,C: set_Product_prod_v_v,B: set_Product_prod_v_v,D2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ C )
=> ( ( ord_le7336532860387713383od_v_v @ B @ D2 )
=> ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ A @ B ) @ ( inf_in6271465464967711157od_v_v @ C @ D2 ) ) ) ) ).
% inf_mono
thf(fact_406_le__infI1,axiom,
! [A: set_v,X4: set_v,B: set_v] :
( ( ord_less_eq_set_v @ A @ X4 )
=> ( ord_less_eq_set_v @ ( inf_inf_set_v @ A @ B ) @ X4 ) ) ).
% le_infI1
thf(fact_407_le__infI1,axiom,
! [A: set_Product_prod_v_v,X4: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ X4 )
=> ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ A @ B ) @ X4 ) ) ).
% le_infI1
thf(fact_408_le__infI2,axiom,
! [B: set_v,X4: set_v,A: set_v] :
( ( ord_less_eq_set_v @ B @ X4 )
=> ( ord_less_eq_set_v @ ( inf_inf_set_v @ A @ B ) @ X4 ) ) ).
% le_infI2
thf(fact_409_le__infI2,axiom,
! [B: set_Product_prod_v_v,X4: set_Product_prod_v_v,A: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ B @ X4 )
=> ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ A @ B ) @ X4 ) ) ).
% le_infI2
thf(fact_410_inf_OorderE,axiom,
! [A: set_v,B: set_v] :
( ( ord_less_eq_set_v @ A @ B )
=> ( A
= ( inf_inf_set_v @ A @ B ) ) ) ).
% inf.orderE
thf(fact_411_inf_OorderE,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ B )
=> ( A
= ( inf_in6271465464967711157od_v_v @ A @ B ) ) ) ).
% inf.orderE
thf(fact_412_inf_OorderI,axiom,
! [A: set_v,B: set_v] :
( ( A
= ( inf_inf_set_v @ A @ B ) )
=> ( ord_less_eq_set_v @ A @ B ) ) ).
% inf.orderI
thf(fact_413_inf_OorderI,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( A
= ( inf_in6271465464967711157od_v_v @ A @ B ) )
=> ( ord_le7336532860387713383od_v_v @ A @ B ) ) ).
% inf.orderI
thf(fact_414_inf__unique,axiom,
! [F: set_v > set_v > set_v,X4: 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 @ X4 @ Y )
= ( F @ X4 @ Y ) ) ) ) ) ).
% inf_unique
thf(fact_415_inf__unique,axiom,
! [F: set_Product_prod_v_v > set_Product_prod_v_v > set_Product_prod_v_v,X4: 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 @ X4 @ Y )
= ( F @ X4 @ Y ) ) ) ) ) ).
% inf_unique
thf(fact_416_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_417_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_418_inf_Oabsorb1,axiom,
! [A: set_v,B: set_v] :
( ( ord_less_eq_set_v @ A @ B )
=> ( ( inf_inf_set_v @ A @ B )
= A ) ) ).
% inf.absorb1
thf(fact_419_inf_Oabsorb1,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ B )
=> ( ( inf_in6271465464967711157od_v_v @ A @ B )
= A ) ) ).
% inf.absorb1
thf(fact_420_inf_Oabsorb2,axiom,
! [B: set_v,A: set_v] :
( ( ord_less_eq_set_v @ B @ A )
=> ( ( inf_inf_set_v @ A @ B )
= B ) ) ).
% inf.absorb2
thf(fact_421_inf_Oabsorb2,axiom,
! [B: set_Product_prod_v_v,A: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ B @ A )
=> ( ( inf_in6271465464967711157od_v_v @ A @ B )
= B ) ) ).
% inf.absorb2
thf(fact_422_inf__absorb1,axiom,
! [X4: set_v,Y: set_v] :
( ( ord_less_eq_set_v @ X4 @ Y )
=> ( ( inf_inf_set_v @ X4 @ Y )
= X4 ) ) ).
% inf_absorb1
thf(fact_423_inf__absorb1,axiom,
! [X4: set_Product_prod_v_v,Y: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X4 @ Y )
=> ( ( inf_in6271465464967711157od_v_v @ X4 @ Y )
= X4 ) ) ).
% inf_absorb1
thf(fact_424_inf__absorb2,axiom,
! [Y: set_v,X4: set_v] :
( ( ord_less_eq_set_v @ Y @ X4 )
=> ( ( inf_inf_set_v @ X4 @ Y )
= Y ) ) ).
% inf_absorb2
thf(fact_425_inf__absorb2,axiom,
! [Y: set_Product_prod_v_v,X4: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ Y @ X4 )
=> ( ( inf_in6271465464967711157od_v_v @ X4 @ Y )
= Y ) ) ).
% inf_absorb2
thf(fact_426_inf_OboundedE,axiom,
! [A: set_v,B: set_v,C: set_v] :
( ( ord_less_eq_set_v @ A @ ( inf_inf_set_v @ B @ C ) )
=> ~ ( ( ord_less_eq_set_v @ A @ B )
=> ~ ( ord_less_eq_set_v @ A @ C ) ) ) ).
% inf.boundedE
thf(fact_427_inf_OboundedE,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ ( inf_in6271465464967711157od_v_v @ B @ C ) )
=> ~ ( ( ord_le7336532860387713383od_v_v @ A @ B )
=> ~ ( ord_le7336532860387713383od_v_v @ A @ C ) ) ) ).
% inf.boundedE
thf(fact_428_inf_OboundedI,axiom,
! [A: set_v,B: set_v,C: set_v] :
( ( ord_less_eq_set_v @ A @ B )
=> ( ( ord_less_eq_set_v @ A @ C )
=> ( ord_less_eq_set_v @ A @ ( inf_inf_set_v @ B @ C ) ) ) ) ).
% inf.boundedI
thf(fact_429_inf_OboundedI,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ B )
=> ( ( ord_le7336532860387713383od_v_v @ A @ C )
=> ( ord_le7336532860387713383od_v_v @ A @ ( inf_in6271465464967711157od_v_v @ B @ C ) ) ) ) ).
% inf.boundedI
thf(fact_430_inf__greatest,axiom,
! [X4: set_v,Y: set_v,Z: set_v] :
( ( ord_less_eq_set_v @ X4 @ Y )
=> ( ( ord_less_eq_set_v @ X4 @ Z )
=> ( ord_less_eq_set_v @ X4 @ ( inf_inf_set_v @ Y @ Z ) ) ) ) ).
% inf_greatest
thf(fact_431_inf__greatest,axiom,
! [X4: set_Product_prod_v_v,Y: set_Product_prod_v_v,Z: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X4 @ Y )
=> ( ( ord_le7336532860387713383od_v_v @ X4 @ Z )
=> ( ord_le7336532860387713383od_v_v @ X4 @ ( inf_in6271465464967711157od_v_v @ Y @ Z ) ) ) ) ).
% inf_greatest
thf(fact_432_inf_Oorder__iff,axiom,
( ord_less_eq_set_v
= ( ^ [A5: set_v,B4: set_v] :
( A5
= ( inf_inf_set_v @ A5 @ B4 ) ) ) ) ).
% inf.order_iff
thf(fact_433_inf_Oorder__iff,axiom,
( ord_le7336532860387713383od_v_v
= ( ^ [A5: set_Product_prod_v_v,B4: set_Product_prod_v_v] :
( A5
= ( inf_in6271465464967711157od_v_v @ A5 @ B4 ) ) ) ) ).
% inf.order_iff
thf(fact_434_inf_Ocobounded1,axiom,
! [A: set_v,B: set_v] : ( ord_less_eq_set_v @ ( inf_inf_set_v @ A @ B ) @ A ) ).
% inf.cobounded1
thf(fact_435_inf_Ocobounded1,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ A @ B ) @ A ) ).
% inf.cobounded1
thf(fact_436_inf_Ocobounded2,axiom,
! [A: set_v,B: set_v] : ( ord_less_eq_set_v @ ( inf_inf_set_v @ A @ B ) @ B ) ).
% inf.cobounded2
thf(fact_437_inf_Ocobounded2,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ A @ B ) @ B ) ).
% inf.cobounded2
thf(fact_438_inf_Oabsorb__iff1,axiom,
( ord_less_eq_set_v
= ( ^ [A5: set_v,B4: set_v] :
( ( inf_inf_set_v @ A5 @ B4 )
= A5 ) ) ) ).
% inf.absorb_iff1
thf(fact_439_inf_Oabsorb__iff1,axiom,
( ord_le7336532860387713383od_v_v
= ( ^ [A5: set_Product_prod_v_v,B4: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ A5 @ B4 )
= A5 ) ) ) ).
% inf.absorb_iff1
thf(fact_440_inf_Oabsorb__iff2,axiom,
( ord_less_eq_set_v
= ( ^ [B4: set_v,A5: set_v] :
( ( inf_inf_set_v @ A5 @ B4 )
= B4 ) ) ) ).
% inf.absorb_iff2
thf(fact_441_inf_Oabsorb__iff2,axiom,
( ord_le7336532860387713383od_v_v
= ( ^ [B4: set_Product_prod_v_v,A5: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ A5 @ B4 )
= B4 ) ) ) ).
% inf.absorb_iff2
thf(fact_442_inf_OcoboundedI1,axiom,
! [A: set_v,C: set_v,B: set_v] :
( ( ord_less_eq_set_v @ A @ C )
=> ( ord_less_eq_set_v @ ( inf_inf_set_v @ A @ B ) @ C ) ) ).
% inf.coboundedI1
thf(fact_443_inf_OcoboundedI1,axiom,
! [A: set_Product_prod_v_v,C: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ C )
=> ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ A @ B ) @ C ) ) ).
% inf.coboundedI1
thf(fact_444_inf_OcoboundedI2,axiom,
! [B: set_v,C: set_v,A: set_v] :
( ( ord_less_eq_set_v @ B @ C )
=> ( ord_less_eq_set_v @ ( inf_inf_set_v @ A @ B ) @ C ) ) ).
% inf.coboundedI2
thf(fact_445_inf_OcoboundedI2,axiom,
! [B: set_Product_prod_v_v,C: set_Product_prod_v_v,A: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ B @ C )
=> ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ A @ B ) @ C ) ) ).
% inf.coboundedI2
thf(fact_446_graph_Ostack__class,axiom,
! [Vertices: set_v,Successors: v > set_v,E: sCC_Bl1394983891496994913t_unit,N2: v,M2: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl9196236973127232072t_unit @ Successors @ E )
=> ( ( member_v @ N2 @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E ) ) )
=> ( ( member_v @ M2 @ ( sCC_Bl1280885523602775798t_unit @ E @ N2 ) )
=> ( member_v @ M2 @ ( minus_minus_set_v @ ( sCC_Bl4645233313691564917t_unit @ E ) @ ( sCC_Bl157864678168468314t_unit @ E ) ) ) ) ) ) ) ).
% graph.stack_class
thf(fact_447_graph_Ounite__sub__env,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,V3: product_prod_v_v,E: sCC_Bl7326425374436813197t_unit,W: product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( sCC_Bl3607325323686918683t_unit @ Successors @ V3 @ E )
=> ( ( member7453568604450474000od_v_v @ W @ ( Successors @ V3 ) )
=> ( ~ ( member7453568604450474000od_v_v @ W @ ( sCC_Bl3878977043676959280t_unit @ E @ V3 ) )
=> ( ( member7453568604450474000od_v_v @ W @ ( sCC_Bl5498988629518860705t_unit @ E ) )
=> ( ~ ( member7453568604450474000od_v_v @ W @ ( sCC_Bl5094201334446601350t_unit @ E ) )
=> ( sCC_Bl7963838319573962697t_unit @ E @ ( sCC_Bl4702006153222411093od_v_v @ V3 @ W @ E ) ) ) ) ) ) ) ) ).
% graph.unite_sub_env
thf(fact_448_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_449_diff__shunt__var,axiom,
! [X4: set_v,Y: set_v] :
( ( ( minus_minus_set_v @ X4 @ Y )
= bot_bot_set_v )
= ( ord_less_eq_set_v @ X4 @ Y ) ) ).
% diff_shunt_var
thf(fact_450_diff__shunt__var,axiom,
! [X4: set_Product_prod_v_v,Y: set_Product_prod_v_v] :
( ( ( minus_4183494784930505774od_v_v @ X4 @ Y )
= bot_bo723834152578015283od_v_v )
= ( ord_le7336532860387713383od_v_v @ X4 @ Y ) ) ).
% diff_shunt_var
thf(fact_451_graph_Ounite__S__tl,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,E: sCC_Bl7326425374436813197t_unit,W: product_prod_v_v,V3: product_prod_v_v,N2: product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( sCC_Bl7798947040364291444t_unit @ Successors @ E )
=> ( ( member7453568604450474000od_v_v @ W @ ( Successors @ V3 ) )
=> ( ~ ( member7453568604450474000od_v_v @ W @ ( sCC_Bl3878977043676959280t_unit @ E @ V3 ) )
=> ( ( member7453568604450474000od_v_v @ W @ ( sCC_Bl5498988629518860705t_unit @ E ) )
=> ( ~ ( member7453568604450474000od_v_v @ W @ ( sCC_Bl5094201334446601350t_unit @ E ) )
=> ( ( member7453568604450474000od_v_v @ N2 @ ( set_Product_prod_v_v2 @ ( tl_Product_prod_v_v @ ( sCC_Bl2021302119412358655t_unit @ ( sCC_Bl4702006153222411093od_v_v @ V3 @ W @ E ) ) ) ) )
=> ( ( sCC_Bl8440648026628373538t_unit @ ( sCC_Bl4702006153222411093od_v_v @ V3 @ W @ E ) @ N2 )
= ( sCC_Bl8440648026628373538t_unit @ E @ N2 ) ) ) ) ) ) ) ) ) ).
% graph.unite_S_tl
thf(fact_452_graph_Ounite__S__tl,axiom,
! [Vertices: set_v,Successors: v > set_v,E: sCC_Bl1394983891496994913t_unit,W: v,V3: v,N2: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl9196236973127232072t_unit @ Successors @ E )
=> ( ( member_v @ W @ ( Successors @ V3 ) )
=> ( ~ ( member_v @ W @ ( sCC_Bl3795065053823578884t_unit @ E @ V3 ) )
=> ( ( member_v @ W @ ( sCC_Bl4645233313691564917t_unit @ E ) )
=> ( ~ ( member_v @ W @ ( sCC_Bl157864678168468314t_unit @ E ) )
=> ( ( member_v @ N2 @ ( set_v2 @ ( tl_v @ ( sCC_Bl8828226123343373779t_unit @ ( sCC_Bloemen_unite_v @ V3 @ W @ E ) ) ) ) )
=> ( ( sCC_Bl1280885523602775798t_unit @ ( sCC_Bloemen_unite_v @ V3 @ W @ E ) @ N2 )
= ( sCC_Bl1280885523602775798t_unit @ E @ N2 ) ) ) ) ) ) ) ) ) ).
% graph.unite_S_tl
thf(fact_453_graph_Ounite__subscc,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,V3: product_prod_v_v,E: sCC_Bl7326425374436813197t_unit,W: product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( sCC_Bl3607325323686918683t_unit @ Successors @ V3 @ E )
=> ( ( member7453568604450474000od_v_v @ W @ ( Successors @ V3 ) )
=> ( ~ ( member7453568604450474000od_v_v @ W @ ( sCC_Bl3878977043676959280t_unit @ E @ V3 ) )
=> ( ( member7453568604450474000od_v_v @ W @ ( sCC_Bl5498988629518860705t_unit @ E ) )
=> ( ~ ( member7453568604450474000od_v_v @ W @ ( sCC_Bl5094201334446601350t_unit @ E ) )
=> ( sCC_Bl2301996248249672505od_v_v @ Successors @ ( sCC_Bl8440648026628373538t_unit @ ( sCC_Bl4702006153222411093od_v_v @ V3 @ W @ E ) @ ( hd_Product_prod_v_v @ ( sCC_Bl2021302119412358655t_unit @ ( sCC_Bl4702006153222411093od_v_v @ V3 @ W @ E ) ) ) ) ) ) ) ) ) ) ) ).
% graph.unite_subscc
thf(fact_454_graph_Ounite__subscc,axiom,
! [Vertices: set_v,Successors: v > set_v,V3: v,E: sCC_Bl1394983891496994913t_unit,W: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl1748261141445803503t_unit @ Successors @ V3 @ E )
=> ( ( member_v @ W @ ( Successors @ V3 ) )
=> ( ~ ( member_v @ W @ ( sCC_Bl3795065053823578884t_unit @ E @ V3 ) )
=> ( ( member_v @ W @ ( sCC_Bl4645233313691564917t_unit @ E ) )
=> ( ~ ( member_v @ W @ ( sCC_Bl157864678168468314t_unit @ E ) )
=> ( sCC_Bl5398416737448265317bscc_v @ Successors @ ( sCC_Bl1280885523602775798t_unit @ ( sCC_Bloemen_unite_v @ V3 @ W @ E ) @ ( hd_v @ ( sCC_Bl8828226123343373779t_unit @ ( sCC_Bloemen_unite_v @ V3 @ W @ E ) ) ) ) ) ) ) ) ) ) ) ).
% graph.unite_subscc
thf(fact_455_re__reachable,axiom,
! [X4: v,Y: v] :
( ( sCC_Bl770211535891879572_end_v @ successors @ X4 @ Y )
=> ( sCC_Bl649662514949026229able_v @ successors @ X4 @ Y ) ) ).
% re_reachable
thf(fact_456_reachable__re,axiom,
! [X4: v,Y: v] :
( ( sCC_Bl649662514949026229able_v @ successors @ X4 @ Y )
=> ( sCC_Bl770211535891879572_end_v @ successors @ X4 @ Y ) ) ).
% reachable_re
thf(fact_457_post__dfss__def,axiom,
! [V3: v,E: sCC_Bl1394983891496994913t_unit,E2: sCC_Bl1394983891496994913t_unit] :
( ( sCC_Bl6082031138996704384t_unit @ successors @ V3 @ E @ E2 )
= ( ( sCC_Bl9196236973127232072t_unit @ successors @ E2 )
& ( ( sCC_Bl3795065053823578884t_unit @ E2 @ V3 )
= ( successors @ V3 ) )
& ! [X2: v] :
( ( member_v @ X2 @ ( minus_minus_set_v @ ( sCC_Bl4645233313691564917t_unit @ E ) @ ( insert_v @ V3 @ bot_bot_set_v ) ) )
=> ( ( sCC_Bl3795065053823578884t_unit @ E2 @ X2 )
= ( sCC_Bl3795065053823578884t_unit @ E @ X2 ) ) )
& ( sCC_Bl5768913643336123637t_unit @ E @ E2 )
& ! [X2: v] :
( ( member_v @ X2 @ ( successors @ V3 ) )
=> ( member_v @ X2 @ ( sup_sup_set_v @ ( sCC_Bl157864678168468314t_unit @ E2 ) @ ( sCC_Bl1280885523602775798t_unit @ E2 @ ( hd_v @ ( sCC_Bl8828226123343373779t_unit @ E2 ) ) ) ) ) )
& ! [X2: v] :
( ( member_v @ X2 @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E2 ) ) )
=> ( sCC_Bl649662514949026229able_v @ successors @ X2 @ V3 ) )
& ( ( sCC_Bl8828226123343373779t_unit @ E2 )
!= nil_v )
& ? [Ns: list_v] :
( ( sCC_Bl8828226123343373779t_unit @ E )
= ( append_v @ Ns @ ( sCC_Bl8828226123343373779t_unit @ E2 ) ) )
& ( member_v @ V3 @ ( sCC_Bl1280885523602775798t_unit @ E2 @ ( hd_v @ ( sCC_Bl8828226123343373779t_unit @ E2 ) ) ) )
& ! [X2: v] :
( ( member_v @ X2 @ ( set_v2 @ ( tl_v @ ( sCC_Bl8828226123343373779t_unit @ E2 ) ) ) )
=> ( ( sCC_Bl1280885523602775798t_unit @ E2 @ X2 )
= ( sCC_Bl1280885523602775798t_unit @ E @ X2 ) ) )
& ( ( ( hd_v @ ( sCC_Bl8828226123343373779t_unit @ E2 ) )
= V3 )
=> ! [X2: v] :
( ( member_v @ X2 @ ( set_v2 @ ( tl_v @ ( sCC_Bl8828226123343373779t_unit @ E2 ) ) ) )
=> ~ ( sCC_Bl649662514949026229able_v @ successors @ V3 @ X2 ) ) )
& ( ( sCC_Bl9201514103433284750t_unit @ E2 )
= ( sCC_Bl9201514103433284750t_unit @ E ) ) ) ) ).
% post_dfss_def
thf(fact_458_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_459_subscc__add,axiom,
! [S3: set_v,X4: v,Y: v] :
( ( sCC_Bl5398416737448265317bscc_v @ successors @ S3 )
=> ( ( member_v @ X4 @ S3 )
=> ( ( sCC_Bl649662514949026229able_v @ successors @ X4 @ Y )
=> ( ( sCC_Bl649662514949026229able_v @ successors @ Y @ X4 )
=> ( sCC_Bl5398416737448265317bscc_v @ successors @ ( insert_v @ Y @ S3 ) ) ) ) ) ) ).
% subscc_add
thf(fact_460_Sup__inf__eq__bot__iff,axiom,
! [B2: set_se8455005133513928103od_v_v,A: set_Product_prod_v_v] :
( ( ( inf_in6271465464967711157od_v_v @ ( comple5788137035815166516od_v_v @ B2 ) @ A )
= bot_bo723834152578015283od_v_v )
= ( ! [X2: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ X2 @ B2 )
=> ( ( inf_in6271465464967711157od_v_v @ X2 @ A )
= bot_bo723834152578015283od_v_v ) ) ) ) ).
% Sup_inf_eq_bot_iff
thf(fact_461_Sup__inf__eq__bot__iff,axiom,
! [B2: set_set_v,A: set_v] :
( ( ( inf_inf_set_v @ ( comple2307003700295860064_set_v @ B2 ) @ A )
= bot_bot_set_v )
= ( ! [X2: set_v] :
( ( member_set_v @ X2 @ B2 )
=> ( ( inf_inf_set_v @ X2 @ A )
= bot_bot_set_v ) ) ) ) ).
% Sup_inf_eq_bot_iff
thf(fact_462_succ__re,axiom,
! [Y: v,X4: v,Z: v] :
( ( member_v @ Y @ ( successors @ X4 ) )
=> ( ( sCC_Bl770211535891879572_end_v @ successors @ Y @ Z )
=> ( sCC_Bl770211535891879572_end_v @ successors @ X4 @ Z ) ) ) ).
% succ_re
thf(fact_463_reachable__end_Osimps,axiom,
! [A1: v,A22: v] :
( ( sCC_Bl770211535891879572_end_v @ successors @ A1 @ A22 )
= ( ? [X2: v] :
( ( A1 = X2 )
& ( A22 = X2 ) )
| ? [X2: v,Y3: v,Z2: v] :
( ( A1 = X2 )
& ( A22 = Z2 )
& ( sCC_Bl770211535891879572_end_v @ successors @ X2 @ Y3 )
& ( member_v @ Z2 @ ( successors @ Y3 ) ) ) ) ) ).
% reachable_end.simps
thf(fact_464_re__succ,axiom,
! [X4: v,Y: v,Z: v] :
( ( sCC_Bl770211535891879572_end_v @ successors @ X4 @ Y )
=> ( ( member_v @ Z @ ( successors @ Y ) )
=> ( sCC_Bl770211535891879572_end_v @ successors @ X4 @ Z ) ) ) ).
% re_succ
thf(fact_465_re__refl,axiom,
! [X4: v] : ( sCC_Bl770211535891879572_end_v @ successors @ X4 @ X4 ) ).
% re_refl
thf(fact_466_reachable__end_Ocases,axiom,
! [A1: v,A22: v] :
( ( sCC_Bl770211535891879572_end_v @ successors @ A1 @ A22 )
=> ( ( A22 != A1 )
=> ~ ! [Y2: v] :
( ( sCC_Bl770211535891879572_end_v @ successors @ A1 @ Y2 )
=> ~ ( member_v @ A22 @ ( successors @ Y2 ) ) ) ) ) ).
% reachable_end.cases
thf(fact_467_dual__order_Orefl,axiom,
! [A: set_v] : ( ord_less_eq_set_v @ A @ A ) ).
% dual_order.refl
thf(fact_468_dual__order_Orefl,axiom,
! [A: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ A @ A ) ).
% dual_order.refl
thf(fact_469_order__refl,axiom,
! [X4: set_v] : ( ord_less_eq_set_v @ X4 @ X4 ) ).
% order_refl
thf(fact_470_order__refl,axiom,
! [X4: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ X4 @ X4 ) ).
% order_refl
thf(fact_471_insertCI,axiom,
! [A: v,B2: set_v,B: v] :
( ( ~ ( member_v @ A @ B2 )
=> ( A = B ) )
=> ( member_v @ A @ ( insert_v @ B @ B2 ) ) ) ).
% insertCI
thf(fact_472_insertCI,axiom,
! [A: product_prod_v_v,B2: set_Product_prod_v_v,B: product_prod_v_v] :
( ( ~ ( member7453568604450474000od_v_v @ A @ B2 )
=> ( A = B ) )
=> ( member7453568604450474000od_v_v @ A @ ( insert1338601472111419319od_v_v @ B @ B2 ) ) ) ).
% insertCI
thf(fact_473_insert__iff,axiom,
! [A: v,B: v,A2: set_v] :
( ( member_v @ A @ ( insert_v @ B @ A2 ) )
= ( ( A = B )
| ( member_v @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_474_insert__iff,axiom,
! [A: product_prod_v_v,B: product_prod_v_v,A2: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ A @ ( insert1338601472111419319od_v_v @ B @ A2 ) )
= ( ( A = B )
| ( member7453568604450474000od_v_v @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_475_insert__absorb2,axiom,
! [X4: v,A2: set_v] :
( ( insert_v @ X4 @ ( insert_v @ X4 @ A2 ) )
= ( insert_v @ X4 @ A2 ) ) ).
% insert_absorb2
thf(fact_476_insert__absorb2,axiom,
! [X4: product_prod_v_v,A2: set_Product_prod_v_v] :
( ( insert1338601472111419319od_v_v @ X4 @ ( insert1338601472111419319od_v_v @ X4 @ A2 ) )
= ( insert1338601472111419319od_v_v @ X4 @ A2 ) ) ).
% insert_absorb2
thf(fact_477_sup_Oidem,axiom,
! [A: set_v] :
( ( sup_sup_set_v @ A @ A )
= A ) ).
% sup.idem
thf(fact_478_sup_Oidem,axiom,
! [A: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ A @ A )
= A ) ).
% sup.idem
thf(fact_479_sup__idem,axiom,
! [X4: set_v] :
( ( sup_sup_set_v @ X4 @ X4 )
= X4 ) ).
% sup_idem
thf(fact_480_sup__idem,axiom,
! [X4: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ X4 @ X4 )
= X4 ) ).
% sup_idem
thf(fact_481_sup_Oleft__idem,axiom,
! [A: set_v,B: set_v] :
( ( sup_sup_set_v @ A @ ( sup_sup_set_v @ A @ B ) )
= ( sup_sup_set_v @ A @ B ) ) ).
% sup.left_idem
thf(fact_482_sup_Oleft__idem,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ A @ ( sup_su414716646722978715od_v_v @ A @ B ) )
= ( sup_su414716646722978715od_v_v @ A @ B ) ) ).
% sup.left_idem
thf(fact_483_sup__left__idem,axiom,
! [X4: set_v,Y: set_v] :
( ( sup_sup_set_v @ X4 @ ( sup_sup_set_v @ X4 @ Y ) )
= ( sup_sup_set_v @ X4 @ Y ) ) ).
% sup_left_idem
thf(fact_484_sup__left__idem,axiom,
! [X4: set_Product_prod_v_v,Y: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ X4 @ ( sup_su414716646722978715od_v_v @ X4 @ Y ) )
= ( sup_su414716646722978715od_v_v @ X4 @ Y ) ) ).
% sup_left_idem
thf(fact_485_sup_Oright__idem,axiom,
! [A: set_v,B: set_v] :
( ( sup_sup_set_v @ ( sup_sup_set_v @ A @ B ) @ B )
= ( sup_sup_set_v @ A @ B ) ) ).
% sup.right_idem
thf(fact_486_sup_Oright__idem,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ ( sup_su414716646722978715od_v_v @ A @ B ) @ B )
= ( sup_su414716646722978715od_v_v @ A @ B ) ) ).
% sup.right_idem
thf(fact_487_UnCI,axiom,
! [C: v,B2: set_v,A2: set_v] :
( ( ~ ( member_v @ C @ B2 )
=> ( member_v @ C @ A2 ) )
=> ( member_v @ C @ ( sup_sup_set_v @ A2 @ B2 ) ) ) ).
% UnCI
thf(fact_488_UnCI,axiom,
! [C: product_prod_v_v,B2: set_Product_prod_v_v,A2: set_Product_prod_v_v] :
( ( ~ ( member7453568604450474000od_v_v @ C @ B2 )
=> ( member7453568604450474000od_v_v @ C @ A2 ) )
=> ( member7453568604450474000od_v_v @ C @ ( sup_su414716646722978715od_v_v @ A2 @ B2 ) ) ) ).
% UnCI
thf(fact_489_Un__iff,axiom,
! [C: v,A2: set_v,B2: set_v] :
( ( member_v @ C @ ( sup_sup_set_v @ A2 @ B2 ) )
= ( ( member_v @ C @ A2 )
| ( member_v @ C @ B2 ) ) ) ).
% Un_iff
thf(fact_490_Un__iff,axiom,
! [C: product_prod_v_v,A2: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ C @ ( sup_su414716646722978715od_v_v @ A2 @ B2 ) )
= ( ( member7453568604450474000od_v_v @ C @ A2 )
| ( member7453568604450474000od_v_v @ C @ B2 ) ) ) ).
% Un_iff
thf(fact_491_sup_Obounded__iff,axiom,
! [B: set_v,C: set_v,A: set_v] :
( ( ord_less_eq_set_v @ ( sup_sup_set_v @ B @ C ) @ A )
= ( ( ord_less_eq_set_v @ B @ A )
& ( ord_less_eq_set_v @ C @ A ) ) ) ).
% sup.bounded_iff
thf(fact_492_sup_Obounded__iff,axiom,
! [B: set_Product_prod_v_v,C: set_Product_prod_v_v,A: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ ( sup_su414716646722978715od_v_v @ B @ C ) @ A )
= ( ( ord_le7336532860387713383od_v_v @ B @ A )
& ( ord_le7336532860387713383od_v_v @ C @ A ) ) ) ).
% sup.bounded_iff
thf(fact_493_le__sup__iff,axiom,
! [X4: set_v,Y: set_v,Z: set_v] :
( ( ord_less_eq_set_v @ ( sup_sup_set_v @ X4 @ Y ) @ Z )
= ( ( ord_less_eq_set_v @ X4 @ Z )
& ( ord_less_eq_set_v @ Y @ Z ) ) ) ).
% le_sup_iff
thf(fact_494_le__sup__iff,axiom,
! [X4: 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 @ X4 @ Y ) @ Z )
= ( ( ord_le7336532860387713383od_v_v @ X4 @ Z )
& ( ord_le7336532860387713383od_v_v @ Y @ Z ) ) ) ).
% le_sup_iff
thf(fact_495_sup__bot_Oright__neutral,axiom,
! [A: set_v] :
( ( sup_sup_set_v @ A @ bot_bot_set_v )
= A ) ).
% sup_bot.right_neutral
thf(fact_496_sup__bot_Oright__neutral,axiom,
! [A: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ A @ bot_bo723834152578015283od_v_v )
= A ) ).
% sup_bot.right_neutral
thf(fact_497_sup__bot_Oneutr__eq__iff,axiom,
! [A: set_v,B: set_v] :
( ( bot_bot_set_v
= ( sup_sup_set_v @ A @ B ) )
= ( ( A = bot_bot_set_v )
& ( B = bot_bot_set_v ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_498_sup__bot_Oneutr__eq__iff,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( bot_bo723834152578015283od_v_v
= ( sup_su414716646722978715od_v_v @ A @ B ) )
= ( ( A = bot_bo723834152578015283od_v_v )
& ( B = bot_bo723834152578015283od_v_v ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_499_sup__bot_Oleft__neutral,axiom,
! [A: set_v] :
( ( sup_sup_set_v @ bot_bot_set_v @ A )
= A ) ).
% sup_bot.left_neutral
thf(fact_500_sup__bot_Oleft__neutral,axiom,
! [A: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ bot_bo723834152578015283od_v_v @ A )
= A ) ).
% sup_bot.left_neutral
thf(fact_501_sup__bot_Oeq__neutr__iff,axiom,
! [A: set_v,B: set_v] :
( ( ( sup_sup_set_v @ A @ B )
= bot_bot_set_v )
= ( ( A = bot_bot_set_v )
& ( B = bot_bot_set_v ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_502_sup__bot_Oeq__neutr__iff,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ( sup_su414716646722978715od_v_v @ A @ B )
= bot_bo723834152578015283od_v_v )
= ( ( A = bot_bo723834152578015283od_v_v )
& ( B = bot_bo723834152578015283od_v_v ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_503_sup__eq__bot__iff,axiom,
! [X4: set_v,Y: set_v] :
( ( ( sup_sup_set_v @ X4 @ Y )
= bot_bot_set_v )
= ( ( X4 = bot_bot_set_v )
& ( Y = bot_bot_set_v ) ) ) ).
% sup_eq_bot_iff
thf(fact_504_sup__eq__bot__iff,axiom,
! [X4: set_Product_prod_v_v,Y: set_Product_prod_v_v] :
( ( ( sup_su414716646722978715od_v_v @ X4 @ Y )
= bot_bo723834152578015283od_v_v )
= ( ( X4 = bot_bo723834152578015283od_v_v )
& ( Y = bot_bo723834152578015283od_v_v ) ) ) ).
% sup_eq_bot_iff
thf(fact_505_bot__eq__sup__iff,axiom,
! [X4: set_v,Y: set_v] :
( ( bot_bot_set_v
= ( sup_sup_set_v @ X4 @ Y ) )
= ( ( X4 = bot_bot_set_v )
& ( Y = bot_bot_set_v ) ) ) ).
% bot_eq_sup_iff
thf(fact_506_bot__eq__sup__iff,axiom,
! [X4: set_Product_prod_v_v,Y: set_Product_prod_v_v] :
( ( bot_bo723834152578015283od_v_v
= ( sup_su414716646722978715od_v_v @ X4 @ Y ) )
= ( ( X4 = bot_bo723834152578015283od_v_v )
& ( Y = bot_bo723834152578015283od_v_v ) ) ) ).
% bot_eq_sup_iff
thf(fact_507_sup__bot__right,axiom,
! [X4: set_v] :
( ( sup_sup_set_v @ X4 @ bot_bot_set_v )
= X4 ) ).
% sup_bot_right
thf(fact_508_sup__bot__right,axiom,
! [X4: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ X4 @ bot_bo723834152578015283od_v_v )
= X4 ) ).
% sup_bot_right
thf(fact_509_sup__bot__left,axiom,
! [X4: set_v] :
( ( sup_sup_set_v @ bot_bot_set_v @ X4 )
= X4 ) ).
% sup_bot_left
thf(fact_510_sup__bot__left,axiom,
! [X4: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ bot_bo723834152578015283od_v_v @ X4 )
= X4 ) ).
% sup_bot_left
thf(fact_511_singletonI,axiom,
! [A: v] : ( member_v @ A @ ( insert_v @ A @ bot_bot_set_v ) ) ).
% singletonI
thf(fact_512_singletonI,axiom,
! [A: product_prod_v_v] : ( member7453568604450474000od_v_v @ A @ ( insert1338601472111419319od_v_v @ A @ bot_bo723834152578015283od_v_v ) ) ).
% singletonI
thf(fact_513_insert__subset,axiom,
! [X4: v,A2: set_v,B2: set_v] :
( ( ord_less_eq_set_v @ ( insert_v @ X4 @ A2 ) @ B2 )
= ( ( member_v @ X4 @ B2 )
& ( ord_less_eq_set_v @ A2 @ B2 ) ) ) ).
% insert_subset
thf(fact_514_insert__subset,axiom,
! [X4: product_prod_v_v,A2: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ ( insert1338601472111419319od_v_v @ X4 @ A2 ) @ B2 )
= ( ( member7453568604450474000od_v_v @ X4 @ B2 )
& ( ord_le7336532860387713383od_v_v @ A2 @ B2 ) ) ) ).
% insert_subset
thf(fact_515_sup__inf__absorb,axiom,
! [X4: set_v,Y: set_v] :
( ( sup_sup_set_v @ X4 @ ( inf_inf_set_v @ X4 @ Y ) )
= X4 ) ).
% sup_inf_absorb
thf(fact_516_sup__inf__absorb,axiom,
! [X4: set_Product_prod_v_v,Y: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ X4 @ ( inf_in6271465464967711157od_v_v @ X4 @ Y ) )
= X4 ) ).
% sup_inf_absorb
thf(fact_517_inf__sup__absorb,axiom,
! [X4: set_v,Y: set_v] :
( ( inf_inf_set_v @ X4 @ ( sup_sup_set_v @ X4 @ Y ) )
= X4 ) ).
% inf_sup_absorb
thf(fact_518_inf__sup__absorb,axiom,
! [X4: set_Product_prod_v_v,Y: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ X4 @ ( sup_su414716646722978715od_v_v @ X4 @ Y ) )
= X4 ) ).
% inf_sup_absorb
thf(fact_519_Un__empty,axiom,
! [A2: set_v,B2: set_v] :
( ( ( sup_sup_set_v @ A2 @ B2 )
= bot_bot_set_v )
= ( ( A2 = bot_bot_set_v )
& ( B2 = bot_bot_set_v ) ) ) ).
% Un_empty
thf(fact_520_Un__empty,axiom,
! [A2: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( ( sup_su414716646722978715od_v_v @ A2 @ B2 )
= bot_bo723834152578015283od_v_v )
= ( ( A2 = bot_bo723834152578015283od_v_v )
& ( B2 = bot_bo723834152578015283od_v_v ) ) ) ).
% Un_empty
thf(fact_521_Int__insert__right__if1,axiom,
! [A: product_prod_v_v,A2: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ A @ A2 )
=> ( ( inf_in6271465464967711157od_v_v @ A2 @ ( insert1338601472111419319od_v_v @ A @ B2 ) )
= ( insert1338601472111419319od_v_v @ A @ ( inf_in6271465464967711157od_v_v @ A2 @ B2 ) ) ) ) ).
% Int_insert_right_if1
thf(fact_522_Int__insert__right__if1,axiom,
! [A: v,A2: set_v,B2: set_v] :
( ( member_v @ A @ A2 )
=> ( ( inf_inf_set_v @ A2 @ ( insert_v @ A @ B2 ) )
= ( insert_v @ A @ ( inf_inf_set_v @ A2 @ B2 ) ) ) ) ).
% Int_insert_right_if1
thf(fact_523_Int__insert__right__if0,axiom,
! [A: product_prod_v_v,A2: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ~ ( member7453568604450474000od_v_v @ A @ A2 )
=> ( ( inf_in6271465464967711157od_v_v @ A2 @ ( insert1338601472111419319od_v_v @ A @ B2 ) )
= ( inf_in6271465464967711157od_v_v @ A2 @ B2 ) ) ) ).
% Int_insert_right_if0
thf(fact_524_Int__insert__right__if0,axiom,
! [A: v,A2: set_v,B2: set_v] :
( ~ ( member_v @ A @ A2 )
=> ( ( inf_inf_set_v @ A2 @ ( insert_v @ A @ B2 ) )
= ( inf_inf_set_v @ A2 @ B2 ) ) ) ).
% Int_insert_right_if0
thf(fact_525_insert__inter__insert,axiom,
! [A: product_prod_v_v,A2: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ ( insert1338601472111419319od_v_v @ A @ A2 ) @ ( insert1338601472111419319od_v_v @ A @ B2 ) )
= ( insert1338601472111419319od_v_v @ A @ ( inf_in6271465464967711157od_v_v @ A2 @ B2 ) ) ) ).
% insert_inter_insert
thf(fact_526_insert__inter__insert,axiom,
! [A: v,A2: set_v,B2: set_v] :
( ( inf_inf_set_v @ ( insert_v @ A @ A2 ) @ ( insert_v @ A @ B2 ) )
= ( insert_v @ A @ ( inf_inf_set_v @ A2 @ B2 ) ) ) ).
% insert_inter_insert
thf(fact_527_Int__insert__left__if1,axiom,
! [A: product_prod_v_v,C2: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ A @ C2 )
=> ( ( inf_in6271465464967711157od_v_v @ ( insert1338601472111419319od_v_v @ A @ B2 ) @ C2 )
= ( insert1338601472111419319od_v_v @ A @ ( inf_in6271465464967711157od_v_v @ B2 @ C2 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_528_Int__insert__left__if1,axiom,
! [A: v,C2: set_v,B2: set_v] :
( ( member_v @ A @ C2 )
=> ( ( inf_inf_set_v @ ( insert_v @ A @ B2 ) @ C2 )
= ( insert_v @ A @ ( inf_inf_set_v @ B2 @ C2 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_529_Int__insert__left__if0,axiom,
! [A: product_prod_v_v,C2: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ~ ( member7453568604450474000od_v_v @ A @ C2 )
=> ( ( inf_in6271465464967711157od_v_v @ ( insert1338601472111419319od_v_v @ A @ B2 ) @ C2 )
= ( inf_in6271465464967711157od_v_v @ B2 @ C2 ) ) ) ).
% Int_insert_left_if0
thf(fact_530_Int__insert__left__if0,axiom,
! [A: v,C2: set_v,B2: set_v] :
( ~ ( member_v @ A @ C2 )
=> ( ( inf_inf_set_v @ ( insert_v @ A @ B2 ) @ C2 )
= ( inf_inf_set_v @ B2 @ C2 ) ) ) ).
% Int_insert_left_if0
thf(fact_531_Un__subset__iff,axiom,
! [A2: set_v,B2: set_v,C2: set_v] :
( ( ord_less_eq_set_v @ ( sup_sup_set_v @ A2 @ B2 ) @ C2 )
= ( ( ord_less_eq_set_v @ A2 @ C2 )
& ( ord_less_eq_set_v @ B2 @ C2 ) ) ) ).
% Un_subset_iff
thf(fact_532_Un__subset__iff,axiom,
! [A2: set_Product_prod_v_v,B2: set_Product_prod_v_v,C2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ ( sup_su414716646722978715od_v_v @ A2 @ B2 ) @ C2 )
= ( ( ord_le7336532860387713383od_v_v @ A2 @ C2 )
& ( ord_le7336532860387713383od_v_v @ B2 @ C2 ) ) ) ).
% Un_subset_iff
thf(fact_533_Un__insert__left,axiom,
! [A: v,B2: set_v,C2: set_v] :
( ( sup_sup_set_v @ ( insert_v @ A @ B2 ) @ C2 )
= ( insert_v @ A @ ( sup_sup_set_v @ B2 @ C2 ) ) ) ).
% Un_insert_left
thf(fact_534_Un__insert__left,axiom,
! [A: product_prod_v_v,B2: set_Product_prod_v_v,C2: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ ( insert1338601472111419319od_v_v @ A @ B2 ) @ C2 )
= ( insert1338601472111419319od_v_v @ A @ ( sup_su414716646722978715od_v_v @ B2 @ C2 ) ) ) ).
% Un_insert_left
thf(fact_535_Un__insert__right,axiom,
! [A2: set_v,A: v,B2: set_v] :
( ( sup_sup_set_v @ A2 @ ( insert_v @ A @ B2 ) )
= ( insert_v @ A @ ( sup_sup_set_v @ A2 @ B2 ) ) ) ).
% Un_insert_right
thf(fact_536_Un__insert__right,axiom,
! [A2: set_Product_prod_v_v,A: product_prod_v_v,B2: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ A2 @ ( insert1338601472111419319od_v_v @ A @ B2 ) )
= ( insert1338601472111419319od_v_v @ A @ ( sup_su414716646722978715od_v_v @ A2 @ B2 ) ) ) ).
% Un_insert_right
thf(fact_537_insert__Diff1,axiom,
! [X4: product_prod_v_v,B2: set_Product_prod_v_v,A2: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X4 @ B2 )
=> ( ( minus_4183494784930505774od_v_v @ ( insert1338601472111419319od_v_v @ X4 @ A2 ) @ B2 )
= ( minus_4183494784930505774od_v_v @ A2 @ B2 ) ) ) ).
% insert_Diff1
thf(fact_538_insert__Diff1,axiom,
! [X4: v,B2: set_v,A2: set_v] :
( ( member_v @ X4 @ B2 )
=> ( ( minus_minus_set_v @ ( insert_v @ X4 @ A2 ) @ B2 )
= ( minus_minus_set_v @ A2 @ B2 ) ) ) ).
% insert_Diff1
thf(fact_539_Diff__insert0,axiom,
! [X4: product_prod_v_v,A2: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ~ ( member7453568604450474000od_v_v @ X4 @ A2 )
=> ( ( minus_4183494784930505774od_v_v @ A2 @ ( insert1338601472111419319od_v_v @ X4 @ B2 ) )
= ( minus_4183494784930505774od_v_v @ A2 @ B2 ) ) ) ).
% Diff_insert0
thf(fact_540_Diff__insert0,axiom,
! [X4: v,A2: set_v,B2: set_v] :
( ~ ( member_v @ X4 @ A2 )
=> ( ( minus_minus_set_v @ A2 @ ( insert_v @ X4 @ B2 ) )
= ( minus_minus_set_v @ A2 @ B2 ) ) ) ).
% Diff_insert0
thf(fact_541_Un__Int__eq_I1_J,axiom,
! [S3: set_v,T2: set_v] :
( ( inf_inf_set_v @ ( sup_sup_set_v @ S3 @ T2 ) @ S3 )
= S3 ) ).
% Un_Int_eq(1)
thf(fact_542_Un__Int__eq_I1_J,axiom,
! [S3: set_Product_prod_v_v,T2: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ ( sup_su414716646722978715od_v_v @ S3 @ T2 ) @ S3 )
= S3 ) ).
% Un_Int_eq(1)
thf(fact_543_Un__Int__eq_I2_J,axiom,
! [S3: set_v,T2: set_v] :
( ( inf_inf_set_v @ ( sup_sup_set_v @ S3 @ T2 ) @ T2 )
= T2 ) ).
% Un_Int_eq(2)
thf(fact_544_Un__Int__eq_I2_J,axiom,
! [S3: set_Product_prod_v_v,T2: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ ( sup_su414716646722978715od_v_v @ S3 @ T2 ) @ T2 )
= T2 ) ).
% Un_Int_eq(2)
thf(fact_545_Un__Int__eq_I3_J,axiom,
! [S3: set_v,T2: set_v] :
( ( inf_inf_set_v @ S3 @ ( sup_sup_set_v @ S3 @ T2 ) )
= S3 ) ).
% Un_Int_eq(3)
thf(fact_546_Un__Int__eq_I3_J,axiom,
! [S3: set_Product_prod_v_v,T2: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ S3 @ ( sup_su414716646722978715od_v_v @ S3 @ T2 ) )
= S3 ) ).
% Un_Int_eq(3)
thf(fact_547_Un__Int__eq_I4_J,axiom,
! [T2: set_v,S3: set_v] :
( ( inf_inf_set_v @ T2 @ ( sup_sup_set_v @ S3 @ T2 ) )
= T2 ) ).
% Un_Int_eq(4)
thf(fact_548_Un__Int__eq_I4_J,axiom,
! [T2: set_Product_prod_v_v,S3: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ T2 @ ( sup_su414716646722978715od_v_v @ S3 @ T2 ) )
= T2 ) ).
% Un_Int_eq(4)
thf(fact_549_Int__Un__eq_I1_J,axiom,
! [S3: set_v,T2: set_v] :
( ( sup_sup_set_v @ ( inf_inf_set_v @ S3 @ T2 ) @ S3 )
= S3 ) ).
% Int_Un_eq(1)
thf(fact_550_Int__Un__eq_I1_J,axiom,
! [S3: set_Product_prod_v_v,T2: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ S3 @ T2 ) @ S3 )
= S3 ) ).
% Int_Un_eq(1)
thf(fact_551_Int__Un__eq_I2_J,axiom,
! [S3: set_v,T2: set_v] :
( ( sup_sup_set_v @ ( inf_inf_set_v @ S3 @ T2 ) @ T2 )
= T2 ) ).
% Int_Un_eq(2)
thf(fact_552_Int__Un__eq_I2_J,axiom,
! [S3: set_Product_prod_v_v,T2: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ S3 @ T2 ) @ T2 )
= T2 ) ).
% Int_Un_eq(2)
thf(fact_553_Int__Un__eq_I3_J,axiom,
! [S3: set_v,T2: set_v] :
( ( sup_sup_set_v @ S3 @ ( inf_inf_set_v @ S3 @ T2 ) )
= S3 ) ).
% Int_Un_eq(3)
thf(fact_554_Int__Un__eq_I3_J,axiom,
! [S3: set_Product_prod_v_v,T2: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ S3 @ ( inf_in6271465464967711157od_v_v @ S3 @ T2 ) )
= S3 ) ).
% Int_Un_eq(3)
thf(fact_555_Int__Un__eq_I4_J,axiom,
! [T2: set_v,S3: set_v] :
( ( sup_sup_set_v @ T2 @ ( inf_inf_set_v @ S3 @ T2 ) )
= T2 ) ).
% Int_Un_eq(4)
thf(fact_556_Int__Un__eq_I4_J,axiom,
! [T2: set_Product_prod_v_v,S3: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ T2 @ ( inf_in6271465464967711157od_v_v @ S3 @ T2 ) )
= T2 ) ).
% Int_Un_eq(4)
thf(fact_557_Un__Diff__cancel2,axiom,
! [B2: set_Product_prod_v_v,A2: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ ( minus_4183494784930505774od_v_v @ B2 @ A2 ) @ A2 )
= ( sup_su414716646722978715od_v_v @ B2 @ A2 ) ) ).
% Un_Diff_cancel2
thf(fact_558_Un__Diff__cancel2,axiom,
! [B2: set_v,A2: set_v] :
( ( sup_sup_set_v @ ( minus_minus_set_v @ B2 @ A2 ) @ A2 )
= ( sup_sup_set_v @ B2 @ A2 ) ) ).
% Un_Diff_cancel2
thf(fact_559_Un__Diff__cancel,axiom,
! [A2: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ A2 @ ( minus_4183494784930505774od_v_v @ B2 @ A2 ) )
= ( sup_su414716646722978715od_v_v @ A2 @ B2 ) ) ).
% Un_Diff_cancel
thf(fact_560_Un__Diff__cancel,axiom,
! [A2: set_v,B2: set_v] :
( ( sup_sup_set_v @ A2 @ ( minus_minus_set_v @ B2 @ A2 ) )
= ( sup_sup_set_v @ A2 @ B2 ) ) ).
% Un_Diff_cancel
thf(fact_561_Union__Un__distrib,axiom,
! [A2: set_se8455005133513928103od_v_v,B2: set_se8455005133513928103od_v_v] :
( ( comple5788137035815166516od_v_v @ ( sup_su335656005089752955od_v_v @ A2 @ B2 ) )
= ( sup_su414716646722978715od_v_v @ ( comple5788137035815166516od_v_v @ A2 ) @ ( comple5788137035815166516od_v_v @ B2 ) ) ) ).
% Union_Un_distrib
thf(fact_562_Union__Un__distrib,axiom,
! [A2: set_set_v,B2: set_set_v] :
( ( comple2307003700295860064_set_v @ ( sup_sup_set_set_v @ A2 @ B2 ) )
= ( sup_sup_set_v @ ( comple2307003700295860064_set_v @ A2 ) @ ( comple2307003700295860064_set_v @ B2 ) ) ) ).
% Union_Un_distrib
thf(fact_563_singleton__insert__inj__eq_H,axiom,
! [A: v,A2: set_v,B: v] :
( ( ( insert_v @ A @ A2 )
= ( insert_v @ B @ bot_bot_set_v ) )
= ( ( A = B )
& ( ord_less_eq_set_v @ A2 @ ( insert_v @ B @ bot_bot_set_v ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_564_singleton__insert__inj__eq_H,axiom,
! [A: product_prod_v_v,A2: set_Product_prod_v_v,B: product_prod_v_v] :
( ( ( insert1338601472111419319od_v_v @ A @ A2 )
= ( insert1338601472111419319od_v_v @ B @ bot_bo723834152578015283od_v_v ) )
= ( ( A = B )
& ( ord_le7336532860387713383od_v_v @ A2 @ ( insert1338601472111419319od_v_v @ B @ bot_bo723834152578015283od_v_v ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_565_singleton__insert__inj__eq,axiom,
! [B: v,A: v,A2: set_v] :
( ( ( insert_v @ B @ bot_bot_set_v )
= ( insert_v @ A @ A2 ) )
= ( ( A = B )
& ( ord_less_eq_set_v @ A2 @ ( insert_v @ B @ bot_bot_set_v ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_566_singleton__insert__inj__eq,axiom,
! [B: product_prod_v_v,A: product_prod_v_v,A2: set_Product_prod_v_v] :
( ( ( insert1338601472111419319od_v_v @ B @ bot_bo723834152578015283od_v_v )
= ( insert1338601472111419319od_v_v @ A @ A2 ) )
= ( ( A = B )
& ( ord_le7336532860387713383od_v_v @ A2 @ ( insert1338601472111419319od_v_v @ B @ bot_bo723834152578015283od_v_v ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_567_list_Osimps_I15_J,axiom,
! [X21: product_prod_v_v,X22: list_P7986770385144383213od_v_v] :
( ( set_Product_prod_v_v2 @ ( cons_P4120604216776828829od_v_v @ X21 @ X22 ) )
= ( insert1338601472111419319od_v_v @ X21 @ ( set_Product_prod_v_v2 @ X22 ) ) ) ).
% list.simps(15)
thf(fact_568_list_Osimps_I15_J,axiom,
! [X21: v,X22: list_v] :
( ( set_v2 @ ( cons_v @ X21 @ X22 ) )
= ( insert_v @ X21 @ ( set_v2 @ X22 ) ) ) ).
% list.simps(15)
thf(fact_569_disjoint__insert_I2_J,axiom,
! [A2: set_v,B: v,B2: set_v] :
( ( bot_bot_set_v
= ( inf_inf_set_v @ A2 @ ( insert_v @ B @ B2 ) ) )
= ( ~ ( member_v @ B @ A2 )
& ( bot_bot_set_v
= ( inf_inf_set_v @ A2 @ B2 ) ) ) ) ).
% disjoint_insert(2)
thf(fact_570_disjoint__insert_I2_J,axiom,
! [A2: set_Product_prod_v_v,B: product_prod_v_v,B2: set_Product_prod_v_v] :
( ( bot_bo723834152578015283od_v_v
= ( inf_in6271465464967711157od_v_v @ A2 @ ( insert1338601472111419319od_v_v @ B @ B2 ) ) )
= ( ~ ( member7453568604450474000od_v_v @ B @ A2 )
& ( bot_bo723834152578015283od_v_v
= ( inf_in6271465464967711157od_v_v @ A2 @ B2 ) ) ) ) ).
% disjoint_insert(2)
thf(fact_571_disjoint__insert_I1_J,axiom,
! [B2: set_v,A: v,A2: set_v] :
( ( ( inf_inf_set_v @ B2 @ ( insert_v @ A @ A2 ) )
= bot_bot_set_v )
= ( ~ ( member_v @ A @ B2 )
& ( ( inf_inf_set_v @ B2 @ A2 )
= bot_bot_set_v ) ) ) ).
% disjoint_insert(1)
thf(fact_572_disjoint__insert_I1_J,axiom,
! [B2: set_Product_prod_v_v,A: product_prod_v_v,A2: set_Product_prod_v_v] :
( ( ( inf_in6271465464967711157od_v_v @ B2 @ ( insert1338601472111419319od_v_v @ A @ A2 ) )
= bot_bo723834152578015283od_v_v )
= ( ~ ( member7453568604450474000od_v_v @ A @ B2 )
& ( ( inf_in6271465464967711157od_v_v @ B2 @ A2 )
= bot_bo723834152578015283od_v_v ) ) ) ).
% disjoint_insert(1)
thf(fact_573_insert__disjoint_I2_J,axiom,
! [A: v,A2: set_v,B2: set_v] :
( ( bot_bot_set_v
= ( inf_inf_set_v @ ( insert_v @ A @ A2 ) @ B2 ) )
= ( ~ ( member_v @ A @ B2 )
& ( bot_bot_set_v
= ( inf_inf_set_v @ A2 @ B2 ) ) ) ) ).
% insert_disjoint(2)
thf(fact_574_insert__disjoint_I2_J,axiom,
! [A: product_prod_v_v,A2: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( bot_bo723834152578015283od_v_v
= ( inf_in6271465464967711157od_v_v @ ( insert1338601472111419319od_v_v @ A @ A2 ) @ B2 ) )
= ( ~ ( member7453568604450474000od_v_v @ A @ B2 )
& ( bot_bo723834152578015283od_v_v
= ( inf_in6271465464967711157od_v_v @ A2 @ B2 ) ) ) ) ).
% insert_disjoint(2)
thf(fact_575_insert__disjoint_I1_J,axiom,
! [A: v,A2: set_v,B2: set_v] :
( ( ( inf_inf_set_v @ ( insert_v @ A @ A2 ) @ B2 )
= bot_bot_set_v )
= ( ~ ( member_v @ A @ B2 )
& ( ( inf_inf_set_v @ A2 @ B2 )
= bot_bot_set_v ) ) ) ).
% insert_disjoint(1)
thf(fact_576_insert__disjoint_I1_J,axiom,
! [A: product_prod_v_v,A2: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( ( inf_in6271465464967711157od_v_v @ ( insert1338601472111419319od_v_v @ A @ A2 ) @ B2 )
= bot_bo723834152578015283od_v_v )
= ( ~ ( member7453568604450474000od_v_v @ A @ B2 )
& ( ( inf_in6271465464967711157od_v_v @ A2 @ B2 )
= bot_bo723834152578015283od_v_v ) ) ) ).
% insert_disjoint(1)
thf(fact_577_Sup__insert,axiom,
! [A: set_Product_prod_v_v,A2: set_se8455005133513928103od_v_v] :
( ( comple5788137035815166516od_v_v @ ( insert7504383016908236695od_v_v @ A @ A2 ) )
= ( sup_su414716646722978715od_v_v @ A @ ( comple5788137035815166516od_v_v @ A2 ) ) ) ).
% Sup_insert
thf(fact_578_Sup__insert,axiom,
! [A: set_v,A2: set_set_v] :
( ( comple2307003700295860064_set_v @ ( insert_set_v @ A @ A2 ) )
= ( sup_sup_set_v @ A @ ( comple2307003700295860064_set_v @ A2 ) ) ) ).
% Sup_insert
thf(fact_579_insert__Diff__single,axiom,
! [A: product_prod_v_v,A2: set_Product_prod_v_v] :
( ( insert1338601472111419319od_v_v @ A @ ( minus_4183494784930505774od_v_v @ A2 @ ( insert1338601472111419319od_v_v @ A @ bot_bo723834152578015283od_v_v ) ) )
= ( insert1338601472111419319od_v_v @ A @ A2 ) ) ).
% insert_Diff_single
thf(fact_580_insert__Diff__single,axiom,
! [A: v,A2: set_v] :
( ( insert_v @ A @ ( minus_minus_set_v @ A2 @ ( insert_v @ A @ bot_bot_set_v ) ) )
= ( insert_v @ A @ A2 ) ) ).
% insert_Diff_single
thf(fact_581_append1__eq__conv,axiom,
! [Xs: list_v,X4: v,Ys: list_v,Y: v] :
( ( ( append_v @ Xs @ ( cons_v @ X4 @ nil_v ) )
= ( append_v @ Ys @ ( cons_v @ Y @ nil_v ) ) )
= ( ( Xs = Ys )
& ( X4 = Y ) ) ) ).
% append1_eq_conv
thf(fact_582_set__append,axiom,
! [Xs: list_v,Ys: list_v] :
( ( set_v2 @ ( append_v @ Xs @ Ys ) )
= ( sup_sup_set_v @ ( set_v2 @ Xs ) @ ( set_v2 @ Ys ) ) ) ).
% set_append
thf(fact_583_set__append,axiom,
! [Xs: list_P7986770385144383213od_v_v,Ys: list_P7986770385144383213od_v_v] :
( ( set_Product_prod_v_v2 @ ( append2138873909117096322od_v_v @ Xs @ Ys ) )
= ( sup_su414716646722978715od_v_v @ ( set_Product_prod_v_v2 @ Xs ) @ ( set_Product_prod_v_v2 @ Ys ) ) ) ).
% set_append
thf(fact_584_hd__Cons__tl,axiom,
! [Xs: list_v] :
( ( Xs != nil_v )
=> ( ( cons_v @ ( hd_v @ Xs ) @ ( tl_v @ Xs ) )
= Xs ) ) ).
% hd_Cons_tl
thf(fact_585_list_Ocollapse,axiom,
! [List: list_v] :
( ( List != nil_v )
=> ( ( cons_v @ ( hd_v @ List ) @ ( tl_v @ List ) )
= List ) ) ).
% list.collapse
thf(fact_586_insert__is__Un,axiom,
( insert_v
= ( ^ [A5: v] : ( sup_sup_set_v @ ( insert_v @ A5 @ bot_bot_set_v ) ) ) ) ).
% insert_is_Un
thf(fact_587_insert__is__Un,axiom,
( insert1338601472111419319od_v_v
= ( ^ [A5: product_prod_v_v] : ( sup_su414716646722978715od_v_v @ ( insert1338601472111419319od_v_v @ A5 @ bot_bo723834152578015283od_v_v ) ) ) ) ).
% insert_is_Un
thf(fact_588_Un__singleton__iff,axiom,
! [A2: set_v,B2: set_v,X4: v] :
( ( ( sup_sup_set_v @ A2 @ B2 )
= ( insert_v @ X4 @ bot_bot_set_v ) )
= ( ( ( A2 = bot_bot_set_v )
& ( B2
= ( insert_v @ X4 @ bot_bot_set_v ) ) )
| ( ( A2
= ( insert_v @ X4 @ bot_bot_set_v ) )
& ( B2 = bot_bot_set_v ) )
| ( ( A2
= ( insert_v @ X4 @ bot_bot_set_v ) )
& ( B2
= ( insert_v @ X4 @ bot_bot_set_v ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_589_Un__singleton__iff,axiom,
! [A2: set_Product_prod_v_v,B2: set_Product_prod_v_v,X4: product_prod_v_v] :
( ( ( sup_su414716646722978715od_v_v @ A2 @ B2 )
= ( insert1338601472111419319od_v_v @ X4 @ bot_bo723834152578015283od_v_v ) )
= ( ( ( A2 = bot_bo723834152578015283od_v_v )
& ( B2
= ( insert1338601472111419319od_v_v @ X4 @ bot_bo723834152578015283od_v_v ) ) )
| ( ( A2
= ( insert1338601472111419319od_v_v @ X4 @ bot_bo723834152578015283od_v_v ) )
& ( B2 = bot_bo723834152578015283od_v_v ) )
| ( ( A2
= ( insert1338601472111419319od_v_v @ X4 @ bot_bo723834152578015283od_v_v ) )
& ( B2
= ( insert1338601472111419319od_v_v @ X4 @ bot_bo723834152578015283od_v_v ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_590_singleton__Un__iff,axiom,
! [X4: v,A2: set_v,B2: set_v] :
( ( ( insert_v @ X4 @ bot_bot_set_v )
= ( sup_sup_set_v @ A2 @ B2 ) )
= ( ( ( A2 = bot_bot_set_v )
& ( B2
= ( insert_v @ X4 @ bot_bot_set_v ) ) )
| ( ( A2
= ( insert_v @ X4 @ bot_bot_set_v ) )
& ( B2 = bot_bot_set_v ) )
| ( ( A2
= ( insert_v @ X4 @ bot_bot_set_v ) )
& ( B2
= ( insert_v @ X4 @ bot_bot_set_v ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_591_singleton__Un__iff,axiom,
! [X4: product_prod_v_v,A2: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( ( insert1338601472111419319od_v_v @ X4 @ bot_bo723834152578015283od_v_v )
= ( sup_su414716646722978715od_v_v @ A2 @ B2 ) )
= ( ( ( A2 = bot_bo723834152578015283od_v_v )
& ( B2
= ( insert1338601472111419319od_v_v @ X4 @ bot_bo723834152578015283od_v_v ) ) )
| ( ( A2
= ( insert1338601472111419319od_v_v @ X4 @ bot_bo723834152578015283od_v_v ) )
& ( B2 = bot_bo723834152578015283od_v_v ) )
| ( ( A2
= ( insert1338601472111419319od_v_v @ X4 @ bot_bo723834152578015283od_v_v ) )
& ( B2
= ( insert1338601472111419319od_v_v @ X4 @ bot_bo723834152578015283od_v_v ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_592_inf__sup__aci_I8_J,axiom,
! [X4: set_v,Y: set_v] :
( ( sup_sup_set_v @ X4 @ ( sup_sup_set_v @ X4 @ Y ) )
= ( sup_sup_set_v @ X4 @ Y ) ) ).
% inf_sup_aci(8)
thf(fact_593_inf__sup__aci_I8_J,axiom,
! [X4: set_Product_prod_v_v,Y: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ X4 @ ( sup_su414716646722978715od_v_v @ X4 @ Y ) )
= ( sup_su414716646722978715od_v_v @ X4 @ Y ) ) ).
% inf_sup_aci(8)
thf(fact_594_inf__sup__aci_I7_J,axiom,
! [X4: set_v,Y: set_v,Z: set_v] :
( ( sup_sup_set_v @ X4 @ ( sup_sup_set_v @ Y @ Z ) )
= ( sup_sup_set_v @ Y @ ( sup_sup_set_v @ X4 @ Z ) ) ) ).
% inf_sup_aci(7)
thf(fact_595_inf__sup__aci_I7_J,axiom,
! [X4: set_Product_prod_v_v,Y: set_Product_prod_v_v,Z: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ X4 @ ( sup_su414716646722978715od_v_v @ Y @ Z ) )
= ( sup_su414716646722978715od_v_v @ Y @ ( sup_su414716646722978715od_v_v @ X4 @ Z ) ) ) ).
% inf_sup_aci(7)
thf(fact_596_inf__sup__aci_I6_J,axiom,
! [X4: set_v,Y: set_v,Z: set_v] :
( ( sup_sup_set_v @ ( sup_sup_set_v @ X4 @ Y ) @ Z )
= ( sup_sup_set_v @ X4 @ ( sup_sup_set_v @ Y @ Z ) ) ) ).
% inf_sup_aci(6)
thf(fact_597_inf__sup__aci_I6_J,axiom,
! [X4: 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 @ X4 @ Y ) @ Z )
= ( sup_su414716646722978715od_v_v @ X4 @ ( sup_su414716646722978715od_v_v @ Y @ Z ) ) ) ).
% inf_sup_aci(6)
thf(fact_598_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_599_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_600_sup_Oassoc,axiom,
! [A: set_v,B: set_v,C: set_v] :
( ( sup_sup_set_v @ ( sup_sup_set_v @ A @ B ) @ C )
= ( sup_sup_set_v @ A @ ( sup_sup_set_v @ B @ C ) ) ) ).
% sup.assoc
thf(fact_601_sup_Oassoc,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ ( sup_su414716646722978715od_v_v @ A @ B ) @ C )
= ( sup_su414716646722978715od_v_v @ A @ ( sup_su414716646722978715od_v_v @ B @ C ) ) ) ).
% sup.assoc
thf(fact_602_sup__assoc,axiom,
! [X4: set_v,Y: set_v,Z: set_v] :
( ( sup_sup_set_v @ ( sup_sup_set_v @ X4 @ Y ) @ Z )
= ( sup_sup_set_v @ X4 @ ( sup_sup_set_v @ Y @ Z ) ) ) ).
% sup_assoc
thf(fact_603_sup__assoc,axiom,
! [X4: 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 @ X4 @ Y ) @ Z )
= ( sup_su414716646722978715od_v_v @ X4 @ ( sup_su414716646722978715od_v_v @ Y @ Z ) ) ) ).
% sup_assoc
thf(fact_604_sup_Ocommute,axiom,
( sup_sup_set_v
= ( ^ [A5: set_v,B4: set_v] : ( sup_sup_set_v @ B4 @ A5 ) ) ) ).
% sup.commute
thf(fact_605_sup_Ocommute,axiom,
( sup_su414716646722978715od_v_v
= ( ^ [A5: set_Product_prod_v_v,B4: set_Product_prod_v_v] : ( sup_su414716646722978715od_v_v @ B4 @ A5 ) ) ) ).
% sup.commute
thf(fact_606_sup__commute,axiom,
( sup_sup_set_v
= ( ^ [X2: set_v,Y3: set_v] : ( sup_sup_set_v @ Y3 @ X2 ) ) ) ).
% sup_commute
thf(fact_607_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_608_boolean__algebra__cancel_Osup1,axiom,
! [A2: set_v,K: set_v,A: set_v,B: set_v] :
( ( A2
= ( sup_sup_set_v @ K @ A ) )
=> ( ( sup_sup_set_v @ A2 @ B )
= ( sup_sup_set_v @ K @ ( sup_sup_set_v @ A @ B ) ) ) ) ).
% boolean_algebra_cancel.sup1
thf(fact_609_boolean__algebra__cancel_Osup1,axiom,
! [A2: set_Product_prod_v_v,K: set_Product_prod_v_v,A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( A2
= ( sup_su414716646722978715od_v_v @ K @ A ) )
=> ( ( sup_su414716646722978715od_v_v @ A2 @ B )
= ( sup_su414716646722978715od_v_v @ K @ ( sup_su414716646722978715od_v_v @ A @ B ) ) ) ) ).
% boolean_algebra_cancel.sup1
thf(fact_610_boolean__algebra__cancel_Osup2,axiom,
! [B2: set_v,K: set_v,B: set_v,A: set_v] :
( ( B2
= ( sup_sup_set_v @ K @ B ) )
=> ( ( sup_sup_set_v @ A @ B2 )
= ( sup_sup_set_v @ K @ ( sup_sup_set_v @ A @ B ) ) ) ) ).
% boolean_algebra_cancel.sup2
thf(fact_611_boolean__algebra__cancel_Osup2,axiom,
! [B2: set_Product_prod_v_v,K: set_Product_prod_v_v,B: set_Product_prod_v_v,A: set_Product_prod_v_v] :
( ( B2
= ( sup_su414716646722978715od_v_v @ K @ B ) )
=> ( ( sup_su414716646722978715od_v_v @ A @ B2 )
= ( sup_su414716646722978715od_v_v @ K @ ( sup_su414716646722978715od_v_v @ A @ B ) ) ) ) ).
% boolean_algebra_cancel.sup2
thf(fact_612_sup_Oleft__commute,axiom,
! [B: set_v,A: set_v,C: set_v] :
( ( sup_sup_set_v @ B @ ( sup_sup_set_v @ A @ C ) )
= ( sup_sup_set_v @ A @ ( sup_sup_set_v @ B @ C ) ) ) ).
% sup.left_commute
thf(fact_613_sup_Oleft__commute,axiom,
! [B: set_Product_prod_v_v,A: set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ B @ ( sup_su414716646722978715od_v_v @ A @ C ) )
= ( sup_su414716646722978715od_v_v @ A @ ( sup_su414716646722978715od_v_v @ B @ C ) ) ) ).
% sup.left_commute
thf(fact_614_sup__left__commute,axiom,
! [X4: set_v,Y: set_v,Z: set_v] :
( ( sup_sup_set_v @ X4 @ ( sup_sup_set_v @ Y @ Z ) )
= ( sup_sup_set_v @ Y @ ( sup_sup_set_v @ X4 @ Z ) ) ) ).
% sup_left_commute
thf(fact_615_sup__left__commute,axiom,
! [X4: set_Product_prod_v_v,Y: set_Product_prod_v_v,Z: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ X4 @ ( sup_su414716646722978715od_v_v @ Y @ Z ) )
= ( sup_su414716646722978715od_v_v @ Y @ ( sup_su414716646722978715od_v_v @ X4 @ Z ) ) ) ).
% sup_left_commute
thf(fact_616_transpose_Ocases,axiom,
! [X4: list_list_v] :
( ( X4 != nil_list_v )
=> ( ! [Xss: list_list_v] :
( X4
!= ( cons_list_v @ nil_v @ Xss ) )
=> ~ ! [X3: v,Xs3: list_v,Xss: list_list_v] :
( X4
!= ( cons_list_v @ ( cons_v @ X3 @ Xs3 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_617_graph_Oreachable__end_Ocong,axiom,
sCC_Bl770211535891879572_end_v = sCC_Bl770211535891879572_end_v ).
% graph.reachable_end.cong
thf(fact_618_UnE,axiom,
! [C: v,A2: set_v,B2: set_v] :
( ( member_v @ C @ ( sup_sup_set_v @ A2 @ B2 ) )
=> ( ~ ( member_v @ C @ A2 )
=> ( member_v @ C @ B2 ) ) ) ).
% UnE
thf(fact_619_UnE,axiom,
! [C: product_prod_v_v,A2: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ C @ ( sup_su414716646722978715od_v_v @ A2 @ B2 ) )
=> ( ~ ( member7453568604450474000od_v_v @ C @ A2 )
=> ( member7453568604450474000od_v_v @ C @ B2 ) ) ) ).
% UnE
thf(fact_620_UnI1,axiom,
! [C: v,A2: set_v,B2: set_v] :
( ( member_v @ C @ A2 )
=> ( member_v @ C @ ( sup_sup_set_v @ A2 @ B2 ) ) ) ).
% UnI1
thf(fact_621_UnI1,axiom,
! [C: product_prod_v_v,A2: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ C @ A2 )
=> ( member7453568604450474000od_v_v @ C @ ( sup_su414716646722978715od_v_v @ A2 @ B2 ) ) ) ).
% UnI1
thf(fact_622_UnI2,axiom,
! [C: v,B2: set_v,A2: set_v] :
( ( member_v @ C @ B2 )
=> ( member_v @ C @ ( sup_sup_set_v @ A2 @ B2 ) ) ) ).
% UnI2
thf(fact_623_UnI2,axiom,
! [C: product_prod_v_v,B2: set_Product_prod_v_v,A2: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ C @ B2 )
=> ( member7453568604450474000od_v_v @ C @ ( sup_su414716646722978715od_v_v @ A2 @ B2 ) ) ) ).
% UnI2
thf(fact_624_bex__Un,axiom,
! [A2: set_v,B2: set_v,P: v > $o] :
( ( ? [X2: v] :
( ( member_v @ X2 @ ( sup_sup_set_v @ A2 @ B2 ) )
& ( P @ X2 ) ) )
= ( ? [X2: v] :
( ( member_v @ X2 @ A2 )
& ( P @ X2 ) )
| ? [X2: v] :
( ( member_v @ X2 @ B2 )
& ( P @ X2 ) ) ) ) ).
% bex_Un
thf(fact_625_bex__Un,axiom,
! [A2: set_Product_prod_v_v,B2: set_Product_prod_v_v,P: product_prod_v_v > $o] :
( ( ? [X2: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X2 @ ( sup_su414716646722978715od_v_v @ A2 @ B2 ) )
& ( P @ X2 ) ) )
= ( ? [X2: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X2 @ A2 )
& ( P @ X2 ) )
| ? [X2: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X2 @ B2 )
& ( P @ X2 ) ) ) ) ).
% bex_Un
thf(fact_626_ball__Un,axiom,
! [A2: set_v,B2: set_v,P: v > $o] :
( ( ! [X2: v] :
( ( member_v @ X2 @ ( sup_sup_set_v @ A2 @ B2 ) )
=> ( P @ X2 ) ) )
= ( ! [X2: v] :
( ( member_v @ X2 @ A2 )
=> ( P @ X2 ) )
& ! [X2: v] :
( ( member_v @ X2 @ B2 )
=> ( P @ X2 ) ) ) ) ).
% ball_Un
thf(fact_627_ball__Un,axiom,
! [A2: set_Product_prod_v_v,B2: set_Product_prod_v_v,P: product_prod_v_v > $o] :
( ( ! [X2: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X2 @ ( sup_su414716646722978715od_v_v @ A2 @ B2 ) )
=> ( P @ X2 ) ) )
= ( ! [X2: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X2 @ A2 )
=> ( P @ X2 ) )
& ! [X2: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X2 @ B2 )
=> ( P @ X2 ) ) ) ) ).
% ball_Un
thf(fact_628_insertE,axiom,
! [A: v,B: v,A2: set_v] :
( ( member_v @ A @ ( insert_v @ B @ A2 ) )
=> ( ( A != B )
=> ( member_v @ A @ A2 ) ) ) ).
% insertE
thf(fact_629_insertE,axiom,
! [A: product_prod_v_v,B: product_prod_v_v,A2: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ A @ ( insert1338601472111419319od_v_v @ B @ A2 ) )
=> ( ( A != B )
=> ( member7453568604450474000od_v_v @ A @ A2 ) ) ) ).
% insertE
thf(fact_630_Un__assoc,axiom,
! [A2: set_v,B2: set_v,C2: set_v] :
( ( sup_sup_set_v @ ( sup_sup_set_v @ A2 @ B2 ) @ C2 )
= ( sup_sup_set_v @ A2 @ ( sup_sup_set_v @ B2 @ C2 ) ) ) ).
% Un_assoc
thf(fact_631_Un__assoc,axiom,
! [A2: set_Product_prod_v_v,B2: set_Product_prod_v_v,C2: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ ( sup_su414716646722978715od_v_v @ A2 @ B2 ) @ C2 )
= ( sup_su414716646722978715od_v_v @ A2 @ ( sup_su414716646722978715od_v_v @ B2 @ C2 ) ) ) ).
% Un_assoc
thf(fact_632_insertI1,axiom,
! [A: v,B2: set_v] : ( member_v @ A @ ( insert_v @ A @ B2 ) ) ).
% insertI1
thf(fact_633_insertI1,axiom,
! [A: product_prod_v_v,B2: set_Product_prod_v_v] : ( member7453568604450474000od_v_v @ A @ ( insert1338601472111419319od_v_v @ A @ B2 ) ) ).
% insertI1
thf(fact_634_insertI2,axiom,
! [A: v,B2: set_v,B: v] :
( ( member_v @ A @ B2 )
=> ( member_v @ A @ ( insert_v @ B @ B2 ) ) ) ).
% insertI2
thf(fact_635_insertI2,axiom,
! [A: product_prod_v_v,B2: set_Product_prod_v_v,B: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ A @ B2 )
=> ( member7453568604450474000od_v_v @ A @ ( insert1338601472111419319od_v_v @ B @ B2 ) ) ) ).
% insertI2
thf(fact_636_Un__absorb,axiom,
! [A2: set_v] :
( ( sup_sup_set_v @ A2 @ A2 )
= A2 ) ).
% Un_absorb
thf(fact_637_Un__absorb,axiom,
! [A2: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ A2 @ A2 )
= A2 ) ).
% Un_absorb
thf(fact_638_Un__commute,axiom,
( sup_sup_set_v
= ( ^ [A4: set_v,B3: set_v] : ( sup_sup_set_v @ B3 @ A4 ) ) ) ).
% Un_commute
thf(fact_639_Un__commute,axiom,
( sup_su414716646722978715od_v_v
= ( ^ [A4: set_Product_prod_v_v,B3: set_Product_prod_v_v] : ( sup_su414716646722978715od_v_v @ B3 @ A4 ) ) ) ).
% Un_commute
thf(fact_640_Set_Oset__insert,axiom,
! [X4: v,A2: set_v] :
( ( member_v @ X4 @ A2 )
=> ~ ! [B5: set_v] :
( ( A2
= ( insert_v @ X4 @ B5 ) )
=> ( member_v @ X4 @ B5 ) ) ) ).
% Set.set_insert
thf(fact_641_Set_Oset__insert,axiom,
! [X4: product_prod_v_v,A2: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X4 @ A2 )
=> ~ ! [B5: set_Product_prod_v_v] :
( ( A2
= ( insert1338601472111419319od_v_v @ X4 @ B5 ) )
=> ( member7453568604450474000od_v_v @ X4 @ B5 ) ) ) ).
% Set.set_insert
thf(fact_642_insert__ident,axiom,
! [X4: v,A2: set_v,B2: set_v] :
( ~ ( member_v @ X4 @ A2 )
=> ( ~ ( member_v @ X4 @ B2 )
=> ( ( ( insert_v @ X4 @ A2 )
= ( insert_v @ X4 @ B2 ) )
= ( A2 = B2 ) ) ) ) ).
% insert_ident
thf(fact_643_insert__ident,axiom,
! [X4: product_prod_v_v,A2: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ~ ( member7453568604450474000od_v_v @ X4 @ A2 )
=> ( ~ ( member7453568604450474000od_v_v @ X4 @ B2 )
=> ( ( ( insert1338601472111419319od_v_v @ X4 @ A2 )
= ( insert1338601472111419319od_v_v @ X4 @ B2 ) )
= ( A2 = B2 ) ) ) ) ).
% insert_ident
thf(fact_644_insert__absorb,axiom,
! [A: v,A2: set_v] :
( ( member_v @ A @ A2 )
=> ( ( insert_v @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_645_insert__absorb,axiom,
! [A: product_prod_v_v,A2: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ A @ A2 )
=> ( ( insert1338601472111419319od_v_v @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_646_insert__eq__iff,axiom,
! [A: v,A2: set_v,B: v,B2: set_v] :
( ~ ( member_v @ A @ A2 )
=> ( ~ ( member_v @ B @ B2 )
=> ( ( ( insert_v @ A @ A2 )
= ( insert_v @ B @ B2 ) )
= ( ( ( A = B )
=> ( A2 = B2 ) )
& ( ( A != B )
=> ? [C3: set_v] :
( ( A2
= ( insert_v @ B @ C3 ) )
& ~ ( member_v @ B @ C3 )
& ( B2
= ( insert_v @ A @ C3 ) )
& ~ ( member_v @ A @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_647_insert__eq__iff,axiom,
! [A: product_prod_v_v,A2: set_Product_prod_v_v,B: product_prod_v_v,B2: set_Product_prod_v_v] :
( ~ ( member7453568604450474000od_v_v @ A @ A2 )
=> ( ~ ( member7453568604450474000od_v_v @ B @ B2 )
=> ( ( ( insert1338601472111419319od_v_v @ A @ A2 )
= ( insert1338601472111419319od_v_v @ B @ B2 ) )
= ( ( ( A = B )
=> ( A2 = B2 ) )
& ( ( A != B )
=> ? [C3: set_Product_prod_v_v] :
( ( A2
= ( insert1338601472111419319od_v_v @ B @ C3 ) )
& ~ ( member7453568604450474000od_v_v @ B @ C3 )
& ( B2
= ( insert1338601472111419319od_v_v @ A @ C3 ) )
& ~ ( member7453568604450474000od_v_v @ A @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_648_Un__left__absorb,axiom,
! [A2: set_v,B2: set_v] :
( ( sup_sup_set_v @ A2 @ ( sup_sup_set_v @ A2 @ B2 ) )
= ( sup_sup_set_v @ A2 @ B2 ) ) ).
% Un_left_absorb
thf(fact_649_Un__left__absorb,axiom,
! [A2: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ A2 @ ( sup_su414716646722978715od_v_v @ A2 @ B2 ) )
= ( sup_su414716646722978715od_v_v @ A2 @ B2 ) ) ).
% Un_left_absorb
thf(fact_650_insert__commute,axiom,
! [X4: v,Y: v,A2: set_v] :
( ( insert_v @ X4 @ ( insert_v @ Y @ A2 ) )
= ( insert_v @ Y @ ( insert_v @ X4 @ A2 ) ) ) ).
% insert_commute
thf(fact_651_insert__commute,axiom,
! [X4: product_prod_v_v,Y: product_prod_v_v,A2: set_Product_prod_v_v] :
( ( insert1338601472111419319od_v_v @ X4 @ ( insert1338601472111419319od_v_v @ Y @ A2 ) )
= ( insert1338601472111419319od_v_v @ Y @ ( insert1338601472111419319od_v_v @ X4 @ A2 ) ) ) ).
% insert_commute
thf(fact_652_Un__left__commute,axiom,
! [A2: set_v,B2: set_v,C2: set_v] :
( ( sup_sup_set_v @ A2 @ ( sup_sup_set_v @ B2 @ C2 ) )
= ( sup_sup_set_v @ B2 @ ( sup_sup_set_v @ A2 @ C2 ) ) ) ).
% Un_left_commute
thf(fact_653_Un__left__commute,axiom,
! [A2: set_Product_prod_v_v,B2: set_Product_prod_v_v,C2: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ A2 @ ( sup_su414716646722978715od_v_v @ B2 @ C2 ) )
= ( sup_su414716646722978715od_v_v @ B2 @ ( sup_su414716646722978715od_v_v @ A2 @ C2 ) ) ) ).
% Un_left_commute
thf(fact_654_mk__disjoint__insert,axiom,
! [A: v,A2: set_v] :
( ( member_v @ A @ A2 )
=> ? [B5: set_v] :
( ( A2
= ( insert_v @ A @ B5 ) )
& ~ ( member_v @ A @ B5 ) ) ) ).
% mk_disjoint_insert
thf(fact_655_mk__disjoint__insert,axiom,
! [A: product_prod_v_v,A2: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ A @ A2 )
=> ? [B5: set_Product_prod_v_v] :
( ( A2
= ( insert1338601472111419319od_v_v @ A @ B5 ) )
& ~ ( member7453568604450474000od_v_v @ A @ B5 ) ) ) ).
% mk_disjoint_insert
thf(fact_656_Sup__union__distrib,axiom,
! [A2: set_se8455005133513928103od_v_v,B2: set_se8455005133513928103od_v_v] :
( ( comple5788137035815166516od_v_v @ ( sup_su335656005089752955od_v_v @ A2 @ B2 ) )
= ( sup_su414716646722978715od_v_v @ ( comple5788137035815166516od_v_v @ A2 ) @ ( comple5788137035815166516od_v_v @ B2 ) ) ) ).
% Sup_union_distrib
thf(fact_657_Sup__union__distrib,axiom,
! [A2: set_set_v,B2: set_set_v] :
( ( comple2307003700295860064_set_v @ ( sup_sup_set_set_v @ A2 @ B2 ) )
= ( sup_sup_set_v @ ( comple2307003700295860064_set_v @ A2 ) @ ( comple2307003700295860064_set_v @ B2 ) ) ) ).
% Sup_union_distrib
thf(fact_658_inf__sup__ord_I4_J,axiom,
! [Y: set_v,X4: set_v] : ( ord_less_eq_set_v @ Y @ ( sup_sup_set_v @ X4 @ Y ) ) ).
% inf_sup_ord(4)
thf(fact_659_inf__sup__ord_I4_J,axiom,
! [Y: set_Product_prod_v_v,X4: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ Y @ ( sup_su414716646722978715od_v_v @ X4 @ Y ) ) ).
% inf_sup_ord(4)
thf(fact_660_inf__sup__ord_I3_J,axiom,
! [X4: set_v,Y: set_v] : ( ord_less_eq_set_v @ X4 @ ( sup_sup_set_v @ X4 @ Y ) ) ).
% inf_sup_ord(3)
thf(fact_661_inf__sup__ord_I3_J,axiom,
! [X4: set_Product_prod_v_v,Y: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ X4 @ ( sup_su414716646722978715od_v_v @ X4 @ Y ) ) ).
% inf_sup_ord(3)
thf(fact_662_le__supE,axiom,
! [A: set_v,B: set_v,X4: set_v] :
( ( ord_less_eq_set_v @ ( sup_sup_set_v @ A @ B ) @ X4 )
=> ~ ( ( ord_less_eq_set_v @ A @ X4 )
=> ~ ( ord_less_eq_set_v @ B @ X4 ) ) ) ).
% le_supE
thf(fact_663_le__supE,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v,X4: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ ( sup_su414716646722978715od_v_v @ A @ B ) @ X4 )
=> ~ ( ( ord_le7336532860387713383od_v_v @ A @ X4 )
=> ~ ( ord_le7336532860387713383od_v_v @ B @ X4 ) ) ) ).
% le_supE
thf(fact_664_le__supI,axiom,
! [A: set_v,X4: set_v,B: set_v] :
( ( ord_less_eq_set_v @ A @ X4 )
=> ( ( ord_less_eq_set_v @ B @ X4 )
=> ( ord_less_eq_set_v @ ( sup_sup_set_v @ A @ B ) @ X4 ) ) ) ).
% le_supI
thf(fact_665_le__supI,axiom,
! [A: set_Product_prod_v_v,X4: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ X4 )
=> ( ( ord_le7336532860387713383od_v_v @ B @ X4 )
=> ( ord_le7336532860387713383od_v_v @ ( sup_su414716646722978715od_v_v @ A @ B ) @ X4 ) ) ) ).
% le_supI
thf(fact_666_sup__ge1,axiom,
! [X4: set_v,Y: set_v] : ( ord_less_eq_set_v @ X4 @ ( sup_sup_set_v @ X4 @ Y ) ) ).
% sup_ge1
thf(fact_667_sup__ge1,axiom,
! [X4: set_Product_prod_v_v,Y: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ X4 @ ( sup_su414716646722978715od_v_v @ X4 @ Y ) ) ).
% sup_ge1
thf(fact_668_sup__ge2,axiom,
! [Y: set_v,X4: set_v] : ( ord_less_eq_set_v @ Y @ ( sup_sup_set_v @ X4 @ Y ) ) ).
% sup_ge2
thf(fact_669_sup__ge2,axiom,
! [Y: set_Product_prod_v_v,X4: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ Y @ ( sup_su414716646722978715od_v_v @ X4 @ Y ) ) ).
% sup_ge2
thf(fact_670_le__supI1,axiom,
! [X4: set_v,A: set_v,B: set_v] :
( ( ord_less_eq_set_v @ X4 @ A )
=> ( ord_less_eq_set_v @ X4 @ ( sup_sup_set_v @ A @ B ) ) ) ).
% le_supI1
thf(fact_671_le__supI1,axiom,
! [X4: set_Product_prod_v_v,A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X4 @ A )
=> ( ord_le7336532860387713383od_v_v @ X4 @ ( sup_su414716646722978715od_v_v @ A @ B ) ) ) ).
% le_supI1
thf(fact_672_le__supI2,axiom,
! [X4: set_v,B: set_v,A: set_v] :
( ( ord_less_eq_set_v @ X4 @ B )
=> ( ord_less_eq_set_v @ X4 @ ( sup_sup_set_v @ A @ B ) ) ) ).
% le_supI2
thf(fact_673_le__supI2,axiom,
! [X4: set_Product_prod_v_v,B: set_Product_prod_v_v,A: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X4 @ B )
=> ( ord_le7336532860387713383od_v_v @ X4 @ ( sup_su414716646722978715od_v_v @ A @ B ) ) ) ).
% le_supI2
thf(fact_674_sup_Omono,axiom,
! [C: set_v,A: set_v,D2: set_v,B: set_v] :
( ( ord_less_eq_set_v @ C @ A )
=> ( ( ord_less_eq_set_v @ D2 @ B )
=> ( ord_less_eq_set_v @ ( sup_sup_set_v @ C @ D2 ) @ ( sup_sup_set_v @ A @ B ) ) ) ) ).
% sup.mono
thf(fact_675_sup_Omono,axiom,
! [C: set_Product_prod_v_v,A: set_Product_prod_v_v,D2: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ C @ A )
=> ( ( ord_le7336532860387713383od_v_v @ D2 @ B )
=> ( ord_le7336532860387713383od_v_v @ ( sup_su414716646722978715od_v_v @ C @ D2 ) @ ( sup_su414716646722978715od_v_v @ A @ B ) ) ) ) ).
% sup.mono
thf(fact_676_sup__mono,axiom,
! [A: set_v,C: set_v,B: set_v,D2: set_v] :
( ( ord_less_eq_set_v @ A @ C )
=> ( ( ord_less_eq_set_v @ B @ D2 )
=> ( ord_less_eq_set_v @ ( sup_sup_set_v @ A @ B ) @ ( sup_sup_set_v @ C @ D2 ) ) ) ) ).
% sup_mono
thf(fact_677_sup__mono,axiom,
! [A: set_Product_prod_v_v,C: set_Product_prod_v_v,B: set_Product_prod_v_v,D2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ C )
=> ( ( ord_le7336532860387713383od_v_v @ B @ D2 )
=> ( ord_le7336532860387713383od_v_v @ ( sup_su414716646722978715od_v_v @ A @ B ) @ ( sup_su414716646722978715od_v_v @ C @ D2 ) ) ) ) ).
% sup_mono
thf(fact_678_sup__least,axiom,
! [Y: set_v,X4: set_v,Z: set_v] :
( ( ord_less_eq_set_v @ Y @ X4 )
=> ( ( ord_less_eq_set_v @ Z @ X4 )
=> ( ord_less_eq_set_v @ ( sup_sup_set_v @ Y @ Z ) @ X4 ) ) ) ).
% sup_least
thf(fact_679_sup__least,axiom,
! [Y: set_Product_prod_v_v,X4: set_Product_prod_v_v,Z: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ Y @ X4 )
=> ( ( ord_le7336532860387713383od_v_v @ Z @ X4 )
=> ( ord_le7336532860387713383od_v_v @ ( sup_su414716646722978715od_v_v @ Y @ Z ) @ X4 ) ) ) ).
% sup_least
thf(fact_680_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_681_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_682_sup_OorderE,axiom,
! [B: set_v,A: set_v] :
( ( ord_less_eq_set_v @ B @ A )
=> ( A
= ( sup_sup_set_v @ A @ B ) ) ) ).
% sup.orderE
thf(fact_683_sup_OorderE,axiom,
! [B: set_Product_prod_v_v,A: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ B @ A )
=> ( A
= ( sup_su414716646722978715od_v_v @ A @ B ) ) ) ).
% sup.orderE
thf(fact_684_sup_OorderI,axiom,
! [A: set_v,B: set_v] :
( ( A
= ( sup_sup_set_v @ A @ B ) )
=> ( ord_less_eq_set_v @ B @ A ) ) ).
% sup.orderI
thf(fact_685_sup_OorderI,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( A
= ( sup_su414716646722978715od_v_v @ A @ B ) )
=> ( ord_le7336532860387713383od_v_v @ B @ A ) ) ).
% sup.orderI
thf(fact_686_sup__unique,axiom,
! [F: set_v > set_v > set_v,X4: 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 @ X4 @ Y )
= ( F @ X4 @ Y ) ) ) ) ) ).
% sup_unique
thf(fact_687_sup__unique,axiom,
! [F: set_Product_prod_v_v > set_Product_prod_v_v > set_Product_prod_v_v,X4: 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 @ X4 @ Y )
= ( F @ X4 @ Y ) ) ) ) ) ).
% sup_unique
thf(fact_688_sup_Oabsorb1,axiom,
! [B: set_v,A: set_v] :
( ( ord_less_eq_set_v @ B @ A )
=> ( ( sup_sup_set_v @ A @ B )
= A ) ) ).
% sup.absorb1
thf(fact_689_sup_Oabsorb1,axiom,
! [B: set_Product_prod_v_v,A: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ B @ A )
=> ( ( sup_su414716646722978715od_v_v @ A @ B )
= A ) ) ).
% sup.absorb1
thf(fact_690_sup_Oabsorb2,axiom,
! [A: set_v,B: set_v] :
( ( ord_less_eq_set_v @ A @ B )
=> ( ( sup_sup_set_v @ A @ B )
= B ) ) ).
% sup.absorb2
thf(fact_691_sup_Oabsorb2,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ B )
=> ( ( sup_su414716646722978715od_v_v @ A @ B )
= B ) ) ).
% sup.absorb2
thf(fact_692_sup__absorb1,axiom,
! [Y: set_v,X4: set_v] :
( ( ord_less_eq_set_v @ Y @ X4 )
=> ( ( sup_sup_set_v @ X4 @ Y )
= X4 ) ) ).
% sup_absorb1
thf(fact_693_sup__absorb1,axiom,
! [Y: set_Product_prod_v_v,X4: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ Y @ X4 )
=> ( ( sup_su414716646722978715od_v_v @ X4 @ Y )
= X4 ) ) ).
% sup_absorb1
thf(fact_694_sup__absorb2,axiom,
! [X4: set_v,Y: set_v] :
( ( ord_less_eq_set_v @ X4 @ Y )
=> ( ( sup_sup_set_v @ X4 @ Y )
= Y ) ) ).
% sup_absorb2
thf(fact_695_sup__absorb2,axiom,
! [X4: set_Product_prod_v_v,Y: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X4 @ Y )
=> ( ( sup_su414716646722978715od_v_v @ X4 @ Y )
= Y ) ) ).
% sup_absorb2
thf(fact_696_sup_OboundedE,axiom,
! [B: set_v,C: set_v,A: set_v] :
( ( ord_less_eq_set_v @ ( sup_sup_set_v @ B @ C ) @ A )
=> ~ ( ( ord_less_eq_set_v @ B @ A )
=> ~ ( ord_less_eq_set_v @ C @ A ) ) ) ).
% sup.boundedE
thf(fact_697_sup_OboundedE,axiom,
! [B: set_Product_prod_v_v,C: set_Product_prod_v_v,A: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ ( sup_su414716646722978715od_v_v @ B @ C ) @ A )
=> ~ ( ( ord_le7336532860387713383od_v_v @ B @ A )
=> ~ ( ord_le7336532860387713383od_v_v @ C @ A ) ) ) ).
% sup.boundedE
thf(fact_698_sup_OboundedI,axiom,
! [B: set_v,A: set_v,C: set_v] :
( ( ord_less_eq_set_v @ B @ A )
=> ( ( ord_less_eq_set_v @ C @ A )
=> ( ord_less_eq_set_v @ ( sup_sup_set_v @ B @ C ) @ A ) ) ) ).
% sup.boundedI
thf(fact_699_sup_OboundedI,axiom,
! [B: set_Product_prod_v_v,A: set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ B @ A )
=> ( ( ord_le7336532860387713383od_v_v @ C @ A )
=> ( ord_le7336532860387713383od_v_v @ ( sup_su414716646722978715od_v_v @ B @ C ) @ A ) ) ) ).
% sup.boundedI
thf(fact_700_sup_Oorder__iff,axiom,
( ord_less_eq_set_v
= ( ^ [B4: set_v,A5: set_v] :
( A5
= ( sup_sup_set_v @ A5 @ B4 ) ) ) ) ).
% sup.order_iff
thf(fact_701_sup_Oorder__iff,axiom,
( ord_le7336532860387713383od_v_v
= ( ^ [B4: set_Product_prod_v_v,A5: set_Product_prod_v_v] :
( A5
= ( sup_su414716646722978715od_v_v @ A5 @ B4 ) ) ) ) ).
% sup.order_iff
thf(fact_702_sup_Ocobounded1,axiom,
! [A: set_v,B: set_v] : ( ord_less_eq_set_v @ A @ ( sup_sup_set_v @ A @ B ) ) ).
% sup.cobounded1
thf(fact_703_sup_Ocobounded1,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ A @ ( sup_su414716646722978715od_v_v @ A @ B ) ) ).
% sup.cobounded1
thf(fact_704_sup_Ocobounded2,axiom,
! [B: set_v,A: set_v] : ( ord_less_eq_set_v @ B @ ( sup_sup_set_v @ A @ B ) ) ).
% sup.cobounded2
thf(fact_705_sup_Ocobounded2,axiom,
! [B: set_Product_prod_v_v,A: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ B @ ( sup_su414716646722978715od_v_v @ A @ B ) ) ).
% sup.cobounded2
thf(fact_706_sup_Oabsorb__iff1,axiom,
( ord_less_eq_set_v
= ( ^ [B4: set_v,A5: set_v] :
( ( sup_sup_set_v @ A5 @ B4 )
= A5 ) ) ) ).
% sup.absorb_iff1
thf(fact_707_sup_Oabsorb__iff1,axiom,
( ord_le7336532860387713383od_v_v
= ( ^ [B4: set_Product_prod_v_v,A5: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ A5 @ B4 )
= A5 ) ) ) ).
% sup.absorb_iff1
thf(fact_708_sup_Oabsorb__iff2,axiom,
( ord_less_eq_set_v
= ( ^ [A5: set_v,B4: set_v] :
( ( sup_sup_set_v @ A5 @ B4 )
= B4 ) ) ) ).
% sup.absorb_iff2
thf(fact_709_sup_Oabsorb__iff2,axiom,
( ord_le7336532860387713383od_v_v
= ( ^ [A5: set_Product_prod_v_v,B4: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ A5 @ B4 )
= B4 ) ) ) ).
% sup.absorb_iff2
thf(fact_710_sup_OcoboundedI1,axiom,
! [C: set_v,A: set_v,B: set_v] :
( ( ord_less_eq_set_v @ C @ A )
=> ( ord_less_eq_set_v @ C @ ( sup_sup_set_v @ A @ B ) ) ) ).
% sup.coboundedI1
thf(fact_711_sup_OcoboundedI1,axiom,
! [C: set_Product_prod_v_v,A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ C @ A )
=> ( ord_le7336532860387713383od_v_v @ C @ ( sup_su414716646722978715od_v_v @ A @ B ) ) ) ).
% sup.coboundedI1
thf(fact_712_sup_OcoboundedI2,axiom,
! [C: set_v,B: set_v,A: set_v] :
( ( ord_less_eq_set_v @ C @ B )
=> ( ord_less_eq_set_v @ C @ ( sup_sup_set_v @ A @ B ) ) ) ).
% sup.coboundedI2
thf(fact_713_sup_OcoboundedI2,axiom,
! [C: set_Product_prod_v_v,B: set_Product_prod_v_v,A: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ C @ B )
=> ( ord_le7336532860387713383od_v_v @ C @ ( sup_su414716646722978715od_v_v @ A @ B ) ) ) ).
% sup.coboundedI2
thf(fact_714_boolean__algebra_Odisj__zero__right,axiom,
! [X4: set_v] :
( ( sup_sup_set_v @ X4 @ bot_bot_set_v )
= X4 ) ).
% boolean_algebra.disj_zero_right
thf(fact_715_boolean__algebra_Odisj__zero__right,axiom,
! [X4: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ X4 @ bot_bo723834152578015283od_v_v )
= X4 ) ).
% boolean_algebra.disj_zero_right
thf(fact_716_distrib__imp1,axiom,
! [X4: 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 @ X4 @ ( inf_inf_set_v @ Y @ Z ) )
= ( inf_inf_set_v @ ( sup_sup_set_v @ X4 @ Y ) @ ( sup_sup_set_v @ X4 @ Z ) ) ) ) ).
% distrib_imp1
thf(fact_717_distrib__imp1,axiom,
! [X4: 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 @ X4 @ ( inf_in6271465464967711157od_v_v @ Y @ Z ) )
= ( inf_in6271465464967711157od_v_v @ ( sup_su414716646722978715od_v_v @ X4 @ Y ) @ ( sup_su414716646722978715od_v_v @ X4 @ Z ) ) ) ) ).
% distrib_imp1
thf(fact_718_distrib__imp2,axiom,
! [X4: 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 @ X4 @ ( sup_sup_set_v @ Y @ Z ) )
= ( sup_sup_set_v @ ( inf_inf_set_v @ X4 @ Y ) @ ( inf_inf_set_v @ X4 @ Z ) ) ) ) ).
% distrib_imp2
thf(fact_719_distrib__imp2,axiom,
! [X4: 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 @ X4 @ ( sup_su414716646722978715od_v_v @ Y @ Z ) )
= ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ X4 @ Y ) @ ( inf_in6271465464967711157od_v_v @ X4 @ Z ) ) ) ) ).
% distrib_imp2
thf(fact_720_inf__sup__distrib1,axiom,
! [X4: set_v,Y: set_v,Z: set_v] :
( ( inf_inf_set_v @ X4 @ ( sup_sup_set_v @ Y @ Z ) )
= ( sup_sup_set_v @ ( inf_inf_set_v @ X4 @ Y ) @ ( inf_inf_set_v @ X4 @ Z ) ) ) ).
% inf_sup_distrib1
thf(fact_721_inf__sup__distrib1,axiom,
! [X4: set_Product_prod_v_v,Y: set_Product_prod_v_v,Z: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ X4 @ ( sup_su414716646722978715od_v_v @ Y @ Z ) )
= ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ X4 @ Y ) @ ( inf_in6271465464967711157od_v_v @ X4 @ Z ) ) ) ).
% inf_sup_distrib1
thf(fact_722_inf__sup__distrib2,axiom,
! [Y: set_v,Z: set_v,X4: set_v] :
( ( inf_inf_set_v @ ( sup_sup_set_v @ Y @ Z ) @ X4 )
= ( sup_sup_set_v @ ( inf_inf_set_v @ Y @ X4 ) @ ( inf_inf_set_v @ Z @ X4 ) ) ) ).
% inf_sup_distrib2
thf(fact_723_inf__sup__distrib2,axiom,
! [Y: set_Product_prod_v_v,Z: set_Product_prod_v_v,X4: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ ( sup_su414716646722978715od_v_v @ Y @ Z ) @ X4 )
= ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ Y @ X4 ) @ ( inf_in6271465464967711157od_v_v @ Z @ X4 ) ) ) ).
% inf_sup_distrib2
thf(fact_724_sup__inf__distrib1,axiom,
! [X4: set_v,Y: set_v,Z: set_v] :
( ( sup_sup_set_v @ X4 @ ( inf_inf_set_v @ Y @ Z ) )
= ( inf_inf_set_v @ ( sup_sup_set_v @ X4 @ Y ) @ ( sup_sup_set_v @ X4 @ Z ) ) ) ).
% sup_inf_distrib1
thf(fact_725_sup__inf__distrib1,axiom,
! [X4: set_Product_prod_v_v,Y: set_Product_prod_v_v,Z: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ X4 @ ( inf_in6271465464967711157od_v_v @ Y @ Z ) )
= ( inf_in6271465464967711157od_v_v @ ( sup_su414716646722978715od_v_v @ X4 @ Y ) @ ( sup_su414716646722978715od_v_v @ X4 @ Z ) ) ) ).
% sup_inf_distrib1
thf(fact_726_sup__inf__distrib2,axiom,
! [Y: set_v,Z: set_v,X4: set_v] :
( ( sup_sup_set_v @ ( inf_inf_set_v @ Y @ Z ) @ X4 )
= ( inf_inf_set_v @ ( sup_sup_set_v @ Y @ X4 ) @ ( sup_sup_set_v @ Z @ X4 ) ) ) ).
% sup_inf_distrib2
thf(fact_727_sup__inf__distrib2,axiom,
! [Y: set_Product_prod_v_v,Z: set_Product_prod_v_v,X4: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ Y @ Z ) @ X4 )
= ( inf_in6271465464967711157od_v_v @ ( sup_su414716646722978715od_v_v @ Y @ X4 ) @ ( sup_su414716646722978715od_v_v @ Z @ X4 ) ) ) ).
% sup_inf_distrib2
thf(fact_728_boolean__algebra_Oconj__disj__distrib,axiom,
! [X4: set_v,Y: set_v,Z: set_v] :
( ( inf_inf_set_v @ X4 @ ( sup_sup_set_v @ Y @ Z ) )
= ( sup_sup_set_v @ ( inf_inf_set_v @ X4 @ Y ) @ ( inf_inf_set_v @ X4 @ Z ) ) ) ).
% boolean_algebra.conj_disj_distrib
thf(fact_729_boolean__algebra_Oconj__disj__distrib,axiom,
! [X4: set_Product_prod_v_v,Y: set_Product_prod_v_v,Z: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ X4 @ ( sup_su414716646722978715od_v_v @ Y @ Z ) )
= ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ X4 @ Y ) @ ( inf_in6271465464967711157od_v_v @ X4 @ Z ) ) ) ).
% boolean_algebra.conj_disj_distrib
thf(fact_730_boolean__algebra_Odisj__conj__distrib,axiom,
! [X4: set_v,Y: set_v,Z: set_v] :
( ( sup_sup_set_v @ X4 @ ( inf_inf_set_v @ Y @ Z ) )
= ( inf_inf_set_v @ ( sup_sup_set_v @ X4 @ Y ) @ ( sup_sup_set_v @ X4 @ Z ) ) ) ).
% boolean_algebra.disj_conj_distrib
thf(fact_731_boolean__algebra_Odisj__conj__distrib,axiom,
! [X4: set_Product_prod_v_v,Y: set_Product_prod_v_v,Z: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ X4 @ ( inf_in6271465464967711157od_v_v @ Y @ Z ) )
= ( inf_in6271465464967711157od_v_v @ ( sup_su414716646722978715od_v_v @ X4 @ Y ) @ ( sup_su414716646722978715od_v_v @ X4 @ Z ) ) ) ).
% boolean_algebra.disj_conj_distrib
thf(fact_732_boolean__algebra_Oconj__disj__distrib2,axiom,
! [Y: set_v,Z: set_v,X4: set_v] :
( ( inf_inf_set_v @ ( sup_sup_set_v @ Y @ Z ) @ X4 )
= ( sup_sup_set_v @ ( inf_inf_set_v @ Y @ X4 ) @ ( inf_inf_set_v @ Z @ X4 ) ) ) ).
% boolean_algebra.conj_disj_distrib2
thf(fact_733_boolean__algebra_Oconj__disj__distrib2,axiom,
! [Y: set_Product_prod_v_v,Z: set_Product_prod_v_v,X4: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ ( sup_su414716646722978715od_v_v @ Y @ Z ) @ X4 )
= ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ Y @ X4 ) @ ( inf_in6271465464967711157od_v_v @ Z @ X4 ) ) ) ).
% boolean_algebra.conj_disj_distrib2
thf(fact_734_boolean__algebra_Odisj__conj__distrib2,axiom,
! [Y: set_v,Z: set_v,X4: set_v] :
( ( sup_sup_set_v @ ( inf_inf_set_v @ Y @ Z ) @ X4 )
= ( inf_inf_set_v @ ( sup_sup_set_v @ Y @ X4 ) @ ( sup_sup_set_v @ Z @ X4 ) ) ) ).
% boolean_algebra.disj_conj_distrib2
thf(fact_735_boolean__algebra_Odisj__conj__distrib2,axiom,
! [Y: set_Product_prod_v_v,Z: set_Product_prod_v_v,X4: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ Y @ Z ) @ X4 )
= ( inf_in6271465464967711157od_v_v @ ( sup_su414716646722978715od_v_v @ Y @ X4 ) @ ( sup_su414716646722978715od_v_v @ Z @ X4 ) ) ) ).
% boolean_algebra.disj_conj_distrib2
thf(fact_736_Un__empty__right,axiom,
! [A2: set_v] :
( ( sup_sup_set_v @ A2 @ bot_bot_set_v )
= A2 ) ).
% Un_empty_right
thf(fact_737_Un__empty__right,axiom,
! [A2: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ A2 @ bot_bo723834152578015283od_v_v )
= A2 ) ).
% Un_empty_right
thf(fact_738_Un__empty__left,axiom,
! [B2: set_v] :
( ( sup_sup_set_v @ bot_bot_set_v @ B2 )
= B2 ) ).
% Un_empty_left
thf(fact_739_Un__empty__left,axiom,
! [B2: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ bot_bo723834152578015283od_v_v @ B2 )
= B2 ) ).
% Un_empty_left
thf(fact_740_singleton__inject,axiom,
! [A: v,B: v] :
( ( ( insert_v @ A @ bot_bot_set_v )
= ( insert_v @ B @ bot_bot_set_v ) )
=> ( A = B ) ) ).
% singleton_inject
thf(fact_741_singleton__inject,axiom,
! [A: product_prod_v_v,B: product_prod_v_v] :
( ( ( insert1338601472111419319od_v_v @ A @ bot_bo723834152578015283od_v_v )
= ( insert1338601472111419319od_v_v @ B @ bot_bo723834152578015283od_v_v ) )
=> ( A = B ) ) ).
% singleton_inject
thf(fact_742_insert__not__empty,axiom,
! [A: v,A2: set_v] :
( ( insert_v @ A @ A2 )
!= bot_bot_set_v ) ).
% insert_not_empty
thf(fact_743_insert__not__empty,axiom,
! [A: product_prod_v_v,A2: set_Product_prod_v_v] :
( ( insert1338601472111419319od_v_v @ A @ A2 )
!= bot_bo723834152578015283od_v_v ) ).
% insert_not_empty
thf(fact_744_doubleton__eq__iff,axiom,
! [A: v,B: v,C: v,D2: v] :
( ( ( insert_v @ A @ ( insert_v @ B @ bot_bot_set_v ) )
= ( insert_v @ C @ ( insert_v @ D2 @ bot_bot_set_v ) ) )
= ( ( ( A = C )
& ( B = D2 ) )
| ( ( A = D2 )
& ( B = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_745_doubleton__eq__iff,axiom,
! [A: product_prod_v_v,B: product_prod_v_v,C: product_prod_v_v,D2: product_prod_v_v] :
( ( ( insert1338601472111419319od_v_v @ A @ ( insert1338601472111419319od_v_v @ B @ bot_bo723834152578015283od_v_v ) )
= ( insert1338601472111419319od_v_v @ C @ ( insert1338601472111419319od_v_v @ D2 @ bot_bo723834152578015283od_v_v ) ) )
= ( ( ( A = C )
& ( B = D2 ) )
| ( ( A = D2 )
& ( B = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_746_singleton__iff,axiom,
! [B: v,A: v] :
( ( member_v @ B @ ( insert_v @ A @ bot_bot_set_v ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_747_singleton__iff,axiom,
! [B: product_prod_v_v,A: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ B @ ( insert1338601472111419319od_v_v @ A @ bot_bo723834152578015283od_v_v ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_748_singletonD,axiom,
! [B: v,A: v] :
( ( member_v @ B @ ( insert_v @ A @ bot_bot_set_v ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_749_singletonD,axiom,
! [B: product_prod_v_v,A: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ B @ ( insert1338601472111419319od_v_v @ A @ bot_bo723834152578015283od_v_v ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_750_subset__Un__eq,axiom,
( ord_less_eq_set_v
= ( ^ [A4: set_v,B3: set_v] :
( ( sup_sup_set_v @ A4 @ B3 )
= B3 ) ) ) ).
% subset_Un_eq
thf(fact_751_subset__Un__eq,axiom,
( ord_le7336532860387713383od_v_v
= ( ^ [A4: set_Product_prod_v_v,B3: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ A4 @ B3 )
= B3 ) ) ) ).
% subset_Un_eq
thf(fact_752_subset__UnE,axiom,
! [C2: set_v,A2: set_v,B2: set_v] :
( ( ord_less_eq_set_v @ C2 @ ( sup_sup_set_v @ A2 @ B2 ) )
=> ~ ! [A6: set_v] :
( ( ord_less_eq_set_v @ A6 @ A2 )
=> ! [B6: set_v] :
( ( ord_less_eq_set_v @ B6 @ B2 )
=> ( C2
!= ( sup_sup_set_v @ A6 @ B6 ) ) ) ) ) ).
% subset_UnE
thf(fact_753_subset__UnE,axiom,
! [C2: set_Product_prod_v_v,A2: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ C2 @ ( sup_su414716646722978715od_v_v @ A2 @ B2 ) )
=> ~ ! [A6: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A6 @ A2 )
=> ! [B6: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ B6 @ B2 )
=> ( C2
!= ( sup_su414716646722978715od_v_v @ A6 @ B6 ) ) ) ) ) ).
% subset_UnE
thf(fact_754_Un__absorb2,axiom,
! [B2: set_v,A2: set_v] :
( ( ord_less_eq_set_v @ B2 @ A2 )
=> ( ( sup_sup_set_v @ A2 @ B2 )
= A2 ) ) ).
% Un_absorb2
thf(fact_755_Un__absorb2,axiom,
! [B2: set_Product_prod_v_v,A2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ B2 @ A2 )
=> ( ( sup_su414716646722978715od_v_v @ A2 @ B2 )
= A2 ) ) ).
% Un_absorb2
thf(fact_756_Un__absorb1,axiom,
! [A2: set_v,B2: set_v] :
( ( ord_less_eq_set_v @ A2 @ B2 )
=> ( ( sup_sup_set_v @ A2 @ B2 )
= B2 ) ) ).
% Un_absorb1
thf(fact_757_Un__absorb1,axiom,
! [A2: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A2 @ B2 )
=> ( ( sup_su414716646722978715od_v_v @ A2 @ B2 )
= B2 ) ) ).
% Un_absorb1
thf(fact_758_Un__upper2,axiom,
! [B2: set_v,A2: set_v] : ( ord_less_eq_set_v @ B2 @ ( sup_sup_set_v @ A2 @ B2 ) ) ).
% Un_upper2
thf(fact_759_Un__upper2,axiom,
! [B2: set_Product_prod_v_v,A2: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ B2 @ ( sup_su414716646722978715od_v_v @ A2 @ B2 ) ) ).
% Un_upper2
thf(fact_760_Un__upper1,axiom,
! [A2: set_v,B2: set_v] : ( ord_less_eq_set_v @ A2 @ ( sup_sup_set_v @ A2 @ B2 ) ) ).
% Un_upper1
thf(fact_761_Un__upper1,axiom,
! [A2: set_Product_prod_v_v,B2: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ A2 @ ( sup_su414716646722978715od_v_v @ A2 @ B2 ) ) ).
% Un_upper1
thf(fact_762_Un__least,axiom,
! [A2: set_v,C2: set_v,B2: set_v] :
( ( ord_less_eq_set_v @ A2 @ C2 )
=> ( ( ord_less_eq_set_v @ B2 @ C2 )
=> ( ord_less_eq_set_v @ ( sup_sup_set_v @ A2 @ B2 ) @ C2 ) ) ) ).
% Un_least
thf(fact_763_Un__least,axiom,
! [A2: set_Product_prod_v_v,C2: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A2 @ C2 )
=> ( ( ord_le7336532860387713383od_v_v @ B2 @ C2 )
=> ( ord_le7336532860387713383od_v_v @ ( sup_su414716646722978715od_v_v @ A2 @ B2 ) @ C2 ) ) ) ).
% Un_least
thf(fact_764_Un__mono,axiom,
! [A2: set_v,C2: set_v,B2: set_v,D: set_v] :
( ( ord_less_eq_set_v @ A2 @ C2 )
=> ( ( ord_less_eq_set_v @ B2 @ D )
=> ( ord_less_eq_set_v @ ( sup_sup_set_v @ A2 @ B2 ) @ ( sup_sup_set_v @ C2 @ D ) ) ) ) ).
% Un_mono
thf(fact_765_Un__mono,axiom,
! [A2: set_Product_prod_v_v,C2: set_Product_prod_v_v,B2: set_Product_prod_v_v,D: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A2 @ C2 )
=> ( ( ord_le7336532860387713383od_v_v @ B2 @ D )
=> ( ord_le7336532860387713383od_v_v @ ( sup_su414716646722978715od_v_v @ A2 @ B2 ) @ ( sup_su414716646722978715od_v_v @ C2 @ D ) ) ) ) ).
% Un_mono
thf(fact_766_subset__insertI2,axiom,
! [A2: set_v,B2: set_v,B: v] :
( ( ord_less_eq_set_v @ A2 @ B2 )
=> ( ord_less_eq_set_v @ A2 @ ( insert_v @ B @ B2 ) ) ) ).
% subset_insertI2
thf(fact_767_subset__insertI2,axiom,
! [A2: set_Product_prod_v_v,B2: set_Product_prod_v_v,B: product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A2 @ B2 )
=> ( ord_le7336532860387713383od_v_v @ A2 @ ( insert1338601472111419319od_v_v @ B @ B2 ) ) ) ).
% subset_insertI2
thf(fact_768_subset__insertI,axiom,
! [B2: set_v,A: v] : ( ord_less_eq_set_v @ B2 @ ( insert_v @ A @ B2 ) ) ).
% subset_insertI
thf(fact_769_subset__insertI,axiom,
! [B2: set_Product_prod_v_v,A: product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ B2 @ ( insert1338601472111419319od_v_v @ A @ B2 ) ) ).
% subset_insertI
thf(fact_770_subset__insert,axiom,
! [X4: v,A2: set_v,B2: set_v] :
( ~ ( member_v @ X4 @ A2 )
=> ( ( ord_less_eq_set_v @ A2 @ ( insert_v @ X4 @ B2 ) )
= ( ord_less_eq_set_v @ A2 @ B2 ) ) ) ).
% subset_insert
thf(fact_771_subset__insert,axiom,
! [X4: product_prod_v_v,A2: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ~ ( member7453568604450474000od_v_v @ X4 @ A2 )
=> ( ( ord_le7336532860387713383od_v_v @ A2 @ ( insert1338601472111419319od_v_v @ X4 @ B2 ) )
= ( ord_le7336532860387713383od_v_v @ A2 @ B2 ) ) ) ).
% subset_insert
thf(fact_772_insert__mono,axiom,
! [C2: set_v,D: set_v,A: v] :
( ( ord_less_eq_set_v @ C2 @ D )
=> ( ord_less_eq_set_v @ ( insert_v @ A @ C2 ) @ ( insert_v @ A @ D ) ) ) ).
% insert_mono
thf(fact_773_insert__mono,axiom,
! [C2: set_Product_prod_v_v,D: set_Product_prod_v_v,A: product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ C2 @ D )
=> ( ord_le7336532860387713383od_v_v @ ( insert1338601472111419319od_v_v @ A @ C2 ) @ ( insert1338601472111419319od_v_v @ A @ D ) ) ) ).
% insert_mono
thf(fact_774_list__nonempty__induct,axiom,
! [Xs: list_v,P: list_v > $o] :
( ( Xs != nil_v )
=> ( ! [X3: v] : ( P @ ( cons_v @ X3 @ nil_v ) )
=> ( ! [X3: v,Xs3: list_v] :
( ( Xs3 != nil_v )
=> ( ( P @ Xs3 )
=> ( P @ ( cons_v @ X3 @ Xs3 ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_775_list__induct2_H,axiom,
! [P: list_v > list_v > $o,Xs: list_v,Ys: list_v] :
( ( P @ nil_v @ nil_v )
=> ( ! [X3: v,Xs3: list_v] : ( P @ ( cons_v @ X3 @ Xs3 ) @ nil_v )
=> ( ! [Y2: v,Ys3: list_v] : ( P @ nil_v @ ( cons_v @ Y2 @ Ys3 ) )
=> ( ! [X3: v,Xs3: list_v,Y2: v,Ys3: list_v] :
( ( P @ Xs3 @ Ys3 )
=> ( P @ ( cons_v @ X3 @ Xs3 ) @ ( cons_v @ Y2 @ Ys3 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_776_neq__Nil__conv,axiom,
! [Xs: list_v] :
( ( Xs != nil_v )
= ( ? [Y3: v,Ys4: list_v] :
( Xs
= ( cons_v @ Y3 @ Ys4 ) ) ) ) ).
% neq_Nil_conv
thf(fact_777_remdups__adj_Ocases,axiom,
! [X4: list_v] :
( ( X4 != nil_v )
=> ( ! [X3: v] :
( X4
!= ( cons_v @ X3 @ nil_v ) )
=> ~ ! [X3: v,Y2: v,Xs3: list_v] :
( X4
!= ( cons_v @ X3 @ ( cons_v @ Y2 @ Xs3 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_778_list_Oexhaust,axiom,
! [Y: list_v] :
( ( Y != nil_v )
=> ~ ! [X212: v,X222: list_v] :
( Y
!= ( cons_v @ X212 @ X222 ) ) ) ).
% list.exhaust
thf(fact_779_list_OdiscI,axiom,
! [List: list_v,X21: v,X22: list_v] :
( ( List
= ( cons_v @ X21 @ X22 ) )
=> ( List != nil_v ) ) ).
% list.discI
thf(fact_780_list_Odistinct_I1_J,axiom,
! [X21: v,X22: list_v] :
( nil_v
!= ( cons_v @ X21 @ X22 ) ) ).
% list.distinct(1)
thf(fact_781_Un__Int__distrib2,axiom,
! [B2: set_v,C2: set_v,A2: set_v] :
( ( sup_sup_set_v @ ( inf_inf_set_v @ B2 @ C2 ) @ A2 )
= ( inf_inf_set_v @ ( sup_sup_set_v @ B2 @ A2 ) @ ( sup_sup_set_v @ C2 @ A2 ) ) ) ).
% Un_Int_distrib2
thf(fact_782_Un__Int__distrib2,axiom,
! [B2: set_Product_prod_v_v,C2: set_Product_prod_v_v,A2: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ B2 @ C2 ) @ A2 )
= ( inf_in6271465464967711157od_v_v @ ( sup_su414716646722978715od_v_v @ B2 @ A2 ) @ ( sup_su414716646722978715od_v_v @ C2 @ A2 ) ) ) ).
% Un_Int_distrib2
thf(fact_783_Int__Un__distrib2,axiom,
! [B2: set_v,C2: set_v,A2: set_v] :
( ( inf_inf_set_v @ ( sup_sup_set_v @ B2 @ C2 ) @ A2 )
= ( sup_sup_set_v @ ( inf_inf_set_v @ B2 @ A2 ) @ ( inf_inf_set_v @ C2 @ A2 ) ) ) ).
% Int_Un_distrib2
thf(fact_784_Int__Un__distrib2,axiom,
! [B2: set_Product_prod_v_v,C2: set_Product_prod_v_v,A2: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ ( sup_su414716646722978715od_v_v @ B2 @ C2 ) @ A2 )
= ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ B2 @ A2 ) @ ( inf_in6271465464967711157od_v_v @ C2 @ A2 ) ) ) ).
% Int_Un_distrib2
thf(fact_785_Un__Int__distrib,axiom,
! [A2: set_v,B2: set_v,C2: set_v] :
( ( sup_sup_set_v @ A2 @ ( inf_inf_set_v @ B2 @ C2 ) )
= ( inf_inf_set_v @ ( sup_sup_set_v @ A2 @ B2 ) @ ( sup_sup_set_v @ A2 @ C2 ) ) ) ).
% Un_Int_distrib
thf(fact_786_Un__Int__distrib,axiom,
! [A2: set_Product_prod_v_v,B2: set_Product_prod_v_v,C2: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ A2 @ ( inf_in6271465464967711157od_v_v @ B2 @ C2 ) )
= ( inf_in6271465464967711157od_v_v @ ( sup_su414716646722978715od_v_v @ A2 @ B2 ) @ ( sup_su414716646722978715od_v_v @ A2 @ C2 ) ) ) ).
% Un_Int_distrib
thf(fact_787_Int__Un__distrib,axiom,
! [A2: set_v,B2: set_v,C2: set_v] :
( ( inf_inf_set_v @ A2 @ ( sup_sup_set_v @ B2 @ C2 ) )
= ( sup_sup_set_v @ ( inf_inf_set_v @ A2 @ B2 ) @ ( inf_inf_set_v @ A2 @ C2 ) ) ) ).
% Int_Un_distrib
thf(fact_788_Int__Un__distrib,axiom,
! [A2: set_Product_prod_v_v,B2: set_Product_prod_v_v,C2: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ A2 @ ( sup_su414716646722978715od_v_v @ B2 @ C2 ) )
= ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ A2 @ B2 ) @ ( inf_in6271465464967711157od_v_v @ A2 @ C2 ) ) ) ).
% Int_Un_distrib
thf(fact_789_Un__Int__crazy,axiom,
! [A2: set_v,B2: set_v,C2: set_v] :
( ( sup_sup_set_v @ ( sup_sup_set_v @ ( inf_inf_set_v @ A2 @ B2 ) @ ( inf_inf_set_v @ B2 @ C2 ) ) @ ( inf_inf_set_v @ C2 @ A2 ) )
= ( inf_inf_set_v @ ( inf_inf_set_v @ ( sup_sup_set_v @ A2 @ B2 ) @ ( sup_sup_set_v @ B2 @ C2 ) ) @ ( sup_sup_set_v @ C2 @ A2 ) ) ) ).
% Un_Int_crazy
thf(fact_790_Un__Int__crazy,axiom,
! [A2: set_Product_prod_v_v,B2: set_Product_prod_v_v,C2: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ A2 @ B2 ) @ ( inf_in6271465464967711157od_v_v @ B2 @ C2 ) ) @ ( inf_in6271465464967711157od_v_v @ C2 @ A2 ) )
= ( inf_in6271465464967711157od_v_v @ ( inf_in6271465464967711157od_v_v @ ( sup_su414716646722978715od_v_v @ A2 @ B2 ) @ ( sup_su414716646722978715od_v_v @ B2 @ C2 ) ) @ ( sup_su414716646722978715od_v_v @ C2 @ A2 ) ) ) ).
% Un_Int_crazy
thf(fact_791_list_Oset__intros_I2_J,axiom,
! [Y: product_prod_v_v,X22: list_P7986770385144383213od_v_v,X21: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ Y @ ( set_Product_prod_v_v2 @ X22 ) )
=> ( member7453568604450474000od_v_v @ Y @ ( set_Product_prod_v_v2 @ ( cons_P4120604216776828829od_v_v @ X21 @ X22 ) ) ) ) ).
% list.set_intros(2)
thf(fact_792_list_Oset__intros_I2_J,axiom,
! [Y: v,X22: list_v,X21: v] :
( ( member_v @ Y @ ( set_v2 @ X22 ) )
=> ( member_v @ Y @ ( set_v2 @ ( cons_v @ X21 @ X22 ) ) ) ) ).
% list.set_intros(2)
thf(fact_793_list_Oset__intros_I1_J,axiom,
! [X21: product_prod_v_v,X22: list_P7986770385144383213od_v_v] : ( member7453568604450474000od_v_v @ X21 @ ( set_Product_prod_v_v2 @ ( cons_P4120604216776828829od_v_v @ X21 @ X22 ) ) ) ).
% list.set_intros(1)
thf(fact_794_list_Oset__intros_I1_J,axiom,
! [X21: v,X22: list_v] : ( member_v @ X21 @ ( set_v2 @ ( cons_v @ X21 @ X22 ) ) ) ).
% list.set_intros(1)
thf(fact_795_list_Oset__cases,axiom,
! [E: product_prod_v_v,A: list_P7986770385144383213od_v_v] :
( ( member7453568604450474000od_v_v @ E @ ( set_Product_prod_v_v2 @ A ) )
=> ( ! [Z22: list_P7986770385144383213od_v_v] :
( A
!= ( cons_P4120604216776828829od_v_v @ E @ Z22 ) )
=> ~ ! [Z1: product_prod_v_v,Z22: list_P7986770385144383213od_v_v] :
( ( A
= ( cons_P4120604216776828829od_v_v @ Z1 @ Z22 ) )
=> ~ ( member7453568604450474000od_v_v @ E @ ( set_Product_prod_v_v2 @ Z22 ) ) ) ) ) ).
% list.set_cases
thf(fact_796_list_Oset__cases,axiom,
! [E: v,A: list_v] :
( ( member_v @ E @ ( set_v2 @ A ) )
=> ( ! [Z22: list_v] :
( A
!= ( cons_v @ E @ Z22 ) )
=> ~ ! [Z1: v,Z22: list_v] :
( ( A
= ( cons_v @ Z1 @ Z22 ) )
=> ~ ( member_v @ E @ ( set_v2 @ Z22 ) ) ) ) ) ).
% list.set_cases
thf(fact_797_set__ConsD,axiom,
! [Y: product_prod_v_v,X4: product_prod_v_v,Xs: list_P7986770385144383213od_v_v] :
( ( member7453568604450474000od_v_v @ Y @ ( set_Product_prod_v_v2 @ ( cons_P4120604216776828829od_v_v @ X4 @ Xs ) ) )
=> ( ( Y = X4 )
| ( member7453568604450474000od_v_v @ Y @ ( set_Product_prod_v_v2 @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_798_set__ConsD,axiom,
! [Y: v,X4: v,Xs: list_v] :
( ( member_v @ Y @ ( set_v2 @ ( cons_v @ X4 @ Xs ) ) )
=> ( ( Y = X4 )
| ( member_v @ Y @ ( set_v2 @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_799_Int__insert__right,axiom,
! [A: product_prod_v_v,A2: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( ( member7453568604450474000od_v_v @ A @ A2 )
=> ( ( inf_in6271465464967711157od_v_v @ A2 @ ( insert1338601472111419319od_v_v @ A @ B2 ) )
= ( insert1338601472111419319od_v_v @ A @ ( inf_in6271465464967711157od_v_v @ A2 @ B2 ) ) ) )
& ( ~ ( member7453568604450474000od_v_v @ A @ A2 )
=> ( ( inf_in6271465464967711157od_v_v @ A2 @ ( insert1338601472111419319od_v_v @ A @ B2 ) )
= ( inf_in6271465464967711157od_v_v @ A2 @ B2 ) ) ) ) ).
% Int_insert_right
thf(fact_800_Int__insert__right,axiom,
! [A: v,A2: set_v,B2: set_v] :
( ( ( member_v @ A @ A2 )
=> ( ( inf_inf_set_v @ A2 @ ( insert_v @ A @ B2 ) )
= ( insert_v @ A @ ( inf_inf_set_v @ A2 @ B2 ) ) ) )
& ( ~ ( member_v @ A @ A2 )
=> ( ( inf_inf_set_v @ A2 @ ( insert_v @ A @ B2 ) )
= ( inf_inf_set_v @ A2 @ B2 ) ) ) ) ).
% Int_insert_right
thf(fact_801_Int__insert__left,axiom,
! [A: product_prod_v_v,C2: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( ( member7453568604450474000od_v_v @ A @ C2 )
=> ( ( inf_in6271465464967711157od_v_v @ ( insert1338601472111419319od_v_v @ A @ B2 ) @ C2 )
= ( insert1338601472111419319od_v_v @ A @ ( inf_in6271465464967711157od_v_v @ B2 @ C2 ) ) ) )
& ( ~ ( member7453568604450474000od_v_v @ A @ C2 )
=> ( ( inf_in6271465464967711157od_v_v @ ( insert1338601472111419319od_v_v @ A @ B2 ) @ C2 )
= ( inf_in6271465464967711157od_v_v @ B2 @ C2 ) ) ) ) ).
% Int_insert_left
thf(fact_802_Int__insert__left,axiom,
! [A: v,C2: set_v,B2: set_v] :
( ( ( member_v @ A @ C2 )
=> ( ( inf_inf_set_v @ ( insert_v @ A @ B2 ) @ C2 )
= ( insert_v @ A @ ( inf_inf_set_v @ B2 @ C2 ) ) ) )
& ( ~ ( member_v @ A @ C2 )
=> ( ( inf_inf_set_v @ ( insert_v @ A @ B2 ) @ C2 )
= ( inf_inf_set_v @ B2 @ C2 ) ) ) ) ).
% Int_insert_left
thf(fact_803_Un__Diff,axiom,
! [A2: set_Product_prod_v_v,B2: set_Product_prod_v_v,C2: set_Product_prod_v_v] :
( ( minus_4183494784930505774od_v_v @ ( sup_su414716646722978715od_v_v @ A2 @ B2 ) @ C2 )
= ( sup_su414716646722978715od_v_v @ ( minus_4183494784930505774od_v_v @ A2 @ C2 ) @ ( minus_4183494784930505774od_v_v @ B2 @ C2 ) ) ) ).
% Un_Diff
thf(fact_804_Un__Diff,axiom,
! [A2: set_v,B2: set_v,C2: set_v] :
( ( minus_minus_set_v @ ( sup_sup_set_v @ A2 @ B2 ) @ C2 )
= ( sup_sup_set_v @ ( minus_minus_set_v @ A2 @ C2 ) @ ( minus_minus_set_v @ B2 @ C2 ) ) ) ).
% Un_Diff
thf(fact_805_insert__Diff__if,axiom,
! [X4: product_prod_v_v,B2: set_Product_prod_v_v,A2: set_Product_prod_v_v] :
( ( ( member7453568604450474000od_v_v @ X4 @ B2 )
=> ( ( minus_4183494784930505774od_v_v @ ( insert1338601472111419319od_v_v @ X4 @ A2 ) @ B2 )
= ( minus_4183494784930505774od_v_v @ A2 @ B2 ) ) )
& ( ~ ( member7453568604450474000od_v_v @ X4 @ B2 )
=> ( ( minus_4183494784930505774od_v_v @ ( insert1338601472111419319od_v_v @ X4 @ A2 ) @ B2 )
= ( insert1338601472111419319od_v_v @ X4 @ ( minus_4183494784930505774od_v_v @ A2 @ B2 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_806_insert__Diff__if,axiom,
! [X4: v,B2: set_v,A2: set_v] :
( ( ( member_v @ X4 @ B2 )
=> ( ( minus_minus_set_v @ ( insert_v @ X4 @ A2 ) @ B2 )
= ( minus_minus_set_v @ A2 @ B2 ) ) )
& ( ~ ( member_v @ X4 @ B2 )
=> ( ( minus_minus_set_v @ ( insert_v @ X4 @ A2 ) @ B2 )
= ( insert_v @ X4 @ ( minus_minus_set_v @ A2 @ B2 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_807_Cons__eq__appendI,axiom,
! [X4: v,Xs1: list_v,Ys: list_v,Xs: list_v,Zs: list_v] :
( ( ( cons_v @ X4 @ Xs1 )
= Ys )
=> ( ( Xs
= ( append_v @ Xs1 @ Zs ) )
=> ( ( cons_v @ X4 @ Xs )
= ( append_v @ Ys @ Zs ) ) ) ) ).
% Cons_eq_appendI
thf(fact_808_append__Cons,axiom,
! [X4: v,Xs: list_v,Ys: list_v] :
( ( append_v @ ( cons_v @ X4 @ Xs ) @ Ys )
= ( cons_v @ X4 @ ( append_v @ Xs @ Ys ) ) ) ).
% append_Cons
thf(fact_809_distinct__length__2__or__more,axiom,
! [A: v,B: v,Xs: list_v] :
( ( distinct_v @ ( cons_v @ A @ ( cons_v @ B @ Xs ) ) )
= ( ( A != B )
& ( distinct_v @ ( cons_v @ A @ Xs ) )
& ( distinct_v @ ( cons_v @ B @ Xs ) ) ) ) ).
% distinct_length_2_or_more
thf(fact_810_list_Osel_I1_J,axiom,
! [X21: v,X22: list_v] :
( ( hd_v @ ( cons_v @ X21 @ X22 ) )
= X21 ) ).
% list.sel(1)
thf(fact_811_list_Osel_I3_J,axiom,
! [X21: v,X22: list_v] :
( ( tl_v @ ( cons_v @ X21 @ X22 ) )
= X22 ) ).
% list.sel(3)
thf(fact_812_precedes__in__tail,axiom,
! [X4: v,Z: v,Y: v,Zs: list_v] :
( ( X4 != Z )
=> ( ( sCC_Bl4022239298816431255edes_v @ X4 @ Y @ ( cons_v @ Z @ Zs ) )
= ( sCC_Bl4022239298816431255edes_v @ X4 @ Y @ Zs ) ) ) ).
% precedes_in_tail
thf(fact_813_graph_Ore__succ,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,X4: 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 @ X4 @ Y )
=> ( ( member7453568604450474000od_v_v @ Z @ ( Successors @ Y ) )
=> ( sCC_Bl4714988717384592488od_v_v @ Successors @ X4 @ Z ) ) ) ) ).
% graph.re_succ
thf(fact_814_graph_Ore__succ,axiom,
! [Vertices: set_v,Successors: v > set_v,X4: v,Y: v,Z: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl770211535891879572_end_v @ Successors @ X4 @ Y )
=> ( ( member_v @ Z @ ( Successors @ Y ) )
=> ( sCC_Bl770211535891879572_end_v @ Successors @ X4 @ Z ) ) ) ) ).
% graph.re_succ
thf(fact_815_graph_Ore__refl,axiom,
! [Vertices: set_v,Successors: v > set_v,X4: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( sCC_Bl770211535891879572_end_v @ Successors @ X4 @ X4 ) ) ).
% graph.re_refl
thf(fact_816_graph_Oreachable__end_Osimps,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,A1: product_prod_v_v,A22: product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( sCC_Bl4714988717384592488od_v_v @ Successors @ A1 @ A22 )
= ( ? [X2: product_prod_v_v] :
( ( A1 = X2 )
& ( A22 = X2 ) )
| ? [X2: product_prod_v_v,Y3: product_prod_v_v,Z2: product_prod_v_v] :
( ( A1 = X2 )
& ( A22 = Z2 )
& ( sCC_Bl4714988717384592488od_v_v @ Successors @ X2 @ Y3 )
& ( member7453568604450474000od_v_v @ Z2 @ ( Successors @ Y3 ) ) ) ) ) ) ).
% graph.reachable_end.simps
thf(fact_817_graph_Oreachable__end_Osimps,axiom,
! [Vertices: set_v,Successors: v > set_v,A1: v,A22: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl770211535891879572_end_v @ Successors @ A1 @ A22 )
= ( ? [X2: v] :
( ( A1 = X2 )
& ( A22 = X2 ) )
| ? [X2: v,Y3: v,Z2: v] :
( ( A1 = X2 )
& ( A22 = Z2 )
& ( sCC_Bl770211535891879572_end_v @ Successors @ X2 @ Y3 )
& ( member_v @ Z2 @ ( Successors @ Y3 ) ) ) ) ) ) ).
% graph.reachable_end.simps
thf(fact_818_graph_Oreachable__end_Ocases,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,A1: product_prod_v_v,A22: product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( sCC_Bl4714988717384592488od_v_v @ Successors @ A1 @ A22 )
=> ( ( A22 != A1 )
=> ~ ! [Y2: product_prod_v_v] :
( ( sCC_Bl4714988717384592488od_v_v @ Successors @ A1 @ Y2 )
=> ~ ( member7453568604450474000od_v_v @ A22 @ ( Successors @ Y2 ) ) ) ) ) ) ).
% graph.reachable_end.cases
thf(fact_819_graph_Oreachable__end_Ocases,axiom,
! [Vertices: set_v,Successors: v > set_v,A1: v,A22: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl770211535891879572_end_v @ Successors @ A1 @ A22 )
=> ( ( A22 != A1 )
=> ~ ! [Y2: v] :
( ( sCC_Bl770211535891879572_end_v @ Successors @ A1 @ Y2 )
=> ~ ( member_v @ A22 @ ( Successors @ Y2 ) ) ) ) ) ) ).
% graph.reachable_end.cases
thf(fact_820_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,X4: product_prod_v_v,Z: product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( member7453568604450474000od_v_v @ Y @ ( Successors @ X4 ) )
=> ( ( sCC_Bl4714988717384592488od_v_v @ Successors @ Y @ Z )
=> ( sCC_Bl4714988717384592488od_v_v @ Successors @ X4 @ Z ) ) ) ) ).
% graph.succ_re
thf(fact_821_graph_Osucc__re,axiom,
! [Vertices: set_v,Successors: v > set_v,Y: v,X4: v,Z: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( member_v @ Y @ ( Successors @ X4 ) )
=> ( ( sCC_Bl770211535891879572_end_v @ Successors @ Y @ Z )
=> ( sCC_Bl770211535891879572_end_v @ Successors @ X4 @ Z ) ) ) ) ).
% graph.succ_re
thf(fact_822_distrib__inf__le,axiom,
! [X4: set_v,Y: set_v,Z: set_v] : ( ord_less_eq_set_v @ ( sup_sup_set_v @ ( inf_inf_set_v @ X4 @ Y ) @ ( inf_inf_set_v @ X4 @ Z ) ) @ ( inf_inf_set_v @ X4 @ ( sup_sup_set_v @ Y @ Z ) ) ) ).
% distrib_inf_le
thf(fact_823_distrib__inf__le,axiom,
! [X4: 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 @ X4 @ Y ) @ ( inf_in6271465464967711157od_v_v @ X4 @ Z ) ) @ ( inf_in6271465464967711157od_v_v @ X4 @ ( sup_su414716646722978715od_v_v @ Y @ Z ) ) ) ).
% distrib_inf_le
thf(fact_824_distrib__sup__le,axiom,
! [X4: set_v,Y: set_v,Z: set_v] : ( ord_less_eq_set_v @ ( sup_sup_set_v @ X4 @ ( inf_inf_set_v @ Y @ Z ) ) @ ( inf_inf_set_v @ ( sup_sup_set_v @ X4 @ Y ) @ ( sup_sup_set_v @ X4 @ Z ) ) ) ).
% distrib_sup_le
thf(fact_825_distrib__sup__le,axiom,
! [X4: 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 @ X4 @ ( inf_in6271465464967711157od_v_v @ Y @ Z ) ) @ ( inf_in6271465464967711157od_v_v @ ( sup_su414716646722978715od_v_v @ X4 @ Y ) @ ( sup_su414716646722978715od_v_v @ X4 @ Z ) ) ) ).
% distrib_sup_le
thf(fact_826_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_827_subset__singleton__iff,axiom,
! [X5: set_v,A: v] :
( ( ord_less_eq_set_v @ X5 @ ( insert_v @ A @ bot_bot_set_v ) )
= ( ( X5 = bot_bot_set_v )
| ( X5
= ( insert_v @ A @ bot_bot_set_v ) ) ) ) ).
% subset_singleton_iff
thf(fact_828_subset__singleton__iff,axiom,
! [X5: set_Product_prod_v_v,A: product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X5 @ ( insert1338601472111419319od_v_v @ A @ bot_bo723834152578015283od_v_v ) )
= ( ( X5 = bot_bo723834152578015283od_v_v )
| ( X5
= ( insert1338601472111419319od_v_v @ A @ bot_bo723834152578015283od_v_v ) ) ) ) ).
% subset_singleton_iff
thf(fact_829_subset__singletonD,axiom,
! [A2: set_v,X4: v] :
( ( ord_less_eq_set_v @ A2 @ ( insert_v @ X4 @ bot_bot_set_v ) )
=> ( ( A2 = bot_bot_set_v )
| ( A2
= ( insert_v @ X4 @ bot_bot_set_v ) ) ) ) ).
% subset_singletonD
thf(fact_830_subset__singletonD,axiom,
! [A2: set_Product_prod_v_v,X4: product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A2 @ ( insert1338601472111419319od_v_v @ X4 @ bot_bo723834152578015283od_v_v ) )
=> ( ( A2 = bot_bo723834152578015283od_v_v )
| ( A2
= ( insert1338601472111419319od_v_v @ X4 @ bot_bo723834152578015283od_v_v ) ) ) ) ).
% subset_singletonD
thf(fact_831_Un__Int__assoc__eq,axiom,
! [A2: set_v,B2: set_v,C2: set_v] :
( ( ( sup_sup_set_v @ ( inf_inf_set_v @ A2 @ B2 ) @ C2 )
= ( inf_inf_set_v @ A2 @ ( sup_sup_set_v @ B2 @ C2 ) ) )
= ( ord_less_eq_set_v @ C2 @ A2 ) ) ).
% Un_Int_assoc_eq
thf(fact_832_Un__Int__assoc__eq,axiom,
! [A2: set_Product_prod_v_v,B2: set_Product_prod_v_v,C2: set_Product_prod_v_v] :
( ( ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ A2 @ B2 ) @ C2 )
= ( inf_in6271465464967711157od_v_v @ A2 @ ( sup_su414716646722978715od_v_v @ B2 @ C2 ) ) )
= ( ord_le7336532860387713383od_v_v @ C2 @ A2 ) ) ).
% Un_Int_assoc_eq
thf(fact_833_Diff__partition,axiom,
! [A2: set_v,B2: set_v] :
( ( ord_less_eq_set_v @ A2 @ B2 )
=> ( ( sup_sup_set_v @ A2 @ ( minus_minus_set_v @ B2 @ A2 ) )
= B2 ) ) ).
% Diff_partition
thf(fact_834_Diff__partition,axiom,
! [A2: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A2 @ B2 )
=> ( ( sup_su414716646722978715od_v_v @ A2 @ ( minus_4183494784930505774od_v_v @ B2 @ A2 ) )
= B2 ) ) ).
% Diff_partition
thf(fact_835_Diff__subset__conv,axiom,
! [A2: set_v,B2: set_v,C2: set_v] :
( ( ord_less_eq_set_v @ ( minus_minus_set_v @ A2 @ B2 ) @ C2 )
= ( ord_less_eq_set_v @ A2 @ ( sup_sup_set_v @ B2 @ C2 ) ) ) ).
% Diff_subset_conv
thf(fact_836_Diff__subset__conv,axiom,
! [A2: set_Product_prod_v_v,B2: set_Product_prod_v_v,C2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ ( minus_4183494784930505774od_v_v @ A2 @ B2 ) @ C2 )
= ( ord_le7336532860387713383od_v_v @ A2 @ ( sup_su414716646722978715od_v_v @ B2 @ C2 ) ) ) ).
% Diff_subset_conv
thf(fact_837_Diff__insert,axiom,
! [A2: set_Product_prod_v_v,A: product_prod_v_v,B2: set_Product_prod_v_v] :
( ( minus_4183494784930505774od_v_v @ A2 @ ( insert1338601472111419319od_v_v @ A @ B2 ) )
= ( minus_4183494784930505774od_v_v @ ( minus_4183494784930505774od_v_v @ A2 @ B2 ) @ ( insert1338601472111419319od_v_v @ A @ bot_bo723834152578015283od_v_v ) ) ) ).
% Diff_insert
thf(fact_838_Diff__insert,axiom,
! [A2: set_v,A: v,B2: set_v] :
( ( minus_minus_set_v @ A2 @ ( insert_v @ A @ B2 ) )
= ( minus_minus_set_v @ ( minus_minus_set_v @ A2 @ B2 ) @ ( insert_v @ A @ bot_bot_set_v ) ) ) ).
% Diff_insert
thf(fact_839_insert__Diff,axiom,
! [A: product_prod_v_v,A2: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ A @ A2 )
=> ( ( insert1338601472111419319od_v_v @ A @ ( minus_4183494784930505774od_v_v @ A2 @ ( insert1338601472111419319od_v_v @ A @ bot_bo723834152578015283od_v_v ) ) )
= A2 ) ) ).
% insert_Diff
thf(fact_840_insert__Diff,axiom,
! [A: v,A2: set_v] :
( ( member_v @ A @ A2 )
=> ( ( insert_v @ A @ ( minus_minus_set_v @ A2 @ ( insert_v @ A @ bot_bot_set_v ) ) )
= A2 ) ) ).
% insert_Diff
thf(fact_841_Diff__insert2,axiom,
! [A2: set_Product_prod_v_v,A: product_prod_v_v,B2: set_Product_prod_v_v] :
( ( minus_4183494784930505774od_v_v @ A2 @ ( insert1338601472111419319od_v_v @ A @ B2 ) )
= ( minus_4183494784930505774od_v_v @ ( minus_4183494784930505774od_v_v @ A2 @ ( insert1338601472111419319od_v_v @ A @ bot_bo723834152578015283od_v_v ) ) @ B2 ) ) ).
% Diff_insert2
thf(fact_842_Diff__insert2,axiom,
! [A2: set_v,A: v,B2: set_v] :
( ( minus_minus_set_v @ A2 @ ( insert_v @ A @ B2 ) )
= ( minus_minus_set_v @ ( minus_minus_set_v @ A2 @ ( insert_v @ A @ bot_bot_set_v ) ) @ B2 ) ) ).
% Diff_insert2
thf(fact_843_Diff__insert__absorb,axiom,
! [X4: product_prod_v_v,A2: set_Product_prod_v_v] :
( ~ ( member7453568604450474000od_v_v @ X4 @ A2 )
=> ( ( minus_4183494784930505774od_v_v @ ( insert1338601472111419319od_v_v @ X4 @ A2 ) @ ( insert1338601472111419319od_v_v @ X4 @ bot_bo723834152578015283od_v_v ) )
= A2 ) ) ).
% Diff_insert_absorb
thf(fact_844_Diff__insert__absorb,axiom,
! [X4: v,A2: set_v] :
( ~ ( member_v @ X4 @ A2 )
=> ( ( minus_minus_set_v @ ( insert_v @ X4 @ A2 ) @ ( insert_v @ X4 @ bot_bot_set_v ) )
= A2 ) ) ).
% Diff_insert_absorb
thf(fact_845_set__subset__Cons,axiom,
! [Xs: list_v,X4: v] : ( ord_less_eq_set_v @ ( set_v2 @ Xs ) @ ( set_v2 @ ( cons_v @ X4 @ Xs ) ) ) ).
% set_subset_Cons
thf(fact_846_set__subset__Cons,axiom,
! [Xs: list_P7986770385144383213od_v_v,X4: product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ ( set_Product_prod_v_v2 @ Xs ) @ ( set_Product_prod_v_v2 @ ( cons_P4120604216776828829od_v_v @ X4 @ Xs ) ) ) ).
% set_subset_Cons
thf(fact_847_subset__Diff__insert,axiom,
! [A2: set_v,B2: set_v,X4: v,C2: set_v] :
( ( ord_less_eq_set_v @ A2 @ ( minus_minus_set_v @ B2 @ ( insert_v @ X4 @ C2 ) ) )
= ( ( ord_less_eq_set_v @ A2 @ ( minus_minus_set_v @ B2 @ C2 ) )
& ~ ( member_v @ X4 @ A2 ) ) ) ).
% subset_Diff_insert
thf(fact_848_subset__Diff__insert,axiom,
! [A2: set_Product_prod_v_v,B2: set_Product_prod_v_v,X4: product_prod_v_v,C2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A2 @ ( minus_4183494784930505774od_v_v @ B2 @ ( insert1338601472111419319od_v_v @ X4 @ C2 ) ) )
= ( ( ord_le7336532860387713383od_v_v @ A2 @ ( minus_4183494784930505774od_v_v @ B2 @ C2 ) )
& ~ ( member7453568604450474000od_v_v @ X4 @ A2 ) ) ) ).
% subset_Diff_insert
thf(fact_849_Un__Diff__Int,axiom,
! [A2: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ ( minus_4183494784930505774od_v_v @ A2 @ B2 ) @ ( inf_in6271465464967711157od_v_v @ A2 @ B2 ) )
= A2 ) ).
% Un_Diff_Int
thf(fact_850_Un__Diff__Int,axiom,
! [A2: set_v,B2: set_v] :
( ( sup_sup_set_v @ ( minus_minus_set_v @ A2 @ B2 ) @ ( inf_inf_set_v @ A2 @ B2 ) )
= A2 ) ).
% Un_Diff_Int
thf(fact_851_Int__Diff__Un,axiom,
! [A2: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ A2 @ B2 ) @ ( minus_4183494784930505774od_v_v @ A2 @ B2 ) )
= A2 ) ).
% Int_Diff_Un
thf(fact_852_Int__Diff__Un,axiom,
! [A2: set_v,B2: set_v] :
( ( sup_sup_set_v @ ( inf_inf_set_v @ A2 @ B2 ) @ ( minus_minus_set_v @ A2 @ B2 ) )
= A2 ) ).
% Int_Diff_Un
thf(fact_853_Diff__Int,axiom,
! [A2: set_Product_prod_v_v,B2: set_Product_prod_v_v,C2: set_Product_prod_v_v] :
( ( minus_4183494784930505774od_v_v @ A2 @ ( inf_in6271465464967711157od_v_v @ B2 @ C2 ) )
= ( sup_su414716646722978715od_v_v @ ( minus_4183494784930505774od_v_v @ A2 @ B2 ) @ ( minus_4183494784930505774od_v_v @ A2 @ C2 ) ) ) ).
% Diff_Int
thf(fact_854_Diff__Int,axiom,
! [A2: set_v,B2: set_v,C2: set_v] :
( ( minus_minus_set_v @ A2 @ ( inf_inf_set_v @ B2 @ C2 ) )
= ( sup_sup_set_v @ ( minus_minus_set_v @ A2 @ B2 ) @ ( minus_minus_set_v @ A2 @ C2 ) ) ) ).
% Diff_Int
thf(fact_855_Diff__Un,axiom,
! [A2: set_Product_prod_v_v,B2: set_Product_prod_v_v,C2: set_Product_prod_v_v] :
( ( minus_4183494784930505774od_v_v @ A2 @ ( sup_su414716646722978715od_v_v @ B2 @ C2 ) )
= ( inf_in6271465464967711157od_v_v @ ( minus_4183494784930505774od_v_v @ A2 @ B2 ) @ ( minus_4183494784930505774od_v_v @ A2 @ C2 ) ) ) ).
% Diff_Un
thf(fact_856_Diff__Un,axiom,
! [A2: set_v,B2: set_v,C2: set_v] :
( ( minus_minus_set_v @ A2 @ ( sup_sup_set_v @ B2 @ C2 ) )
= ( inf_inf_set_v @ ( minus_minus_set_v @ A2 @ B2 ) @ ( minus_minus_set_v @ A2 @ C2 ) ) ) ).
% Diff_Un
thf(fact_857_rev__nonempty__induct,axiom,
! [Xs: list_v,P: list_v > $o] :
( ( Xs != nil_v )
=> ( ! [X3: v] : ( P @ ( cons_v @ X3 @ nil_v ) )
=> ( ! [X3: v,Xs3: list_v] :
( ( Xs3 != nil_v )
=> ( ( P @ Xs3 )
=> ( P @ ( append_v @ Xs3 @ ( cons_v @ X3 @ nil_v ) ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% rev_nonempty_induct
thf(fact_858_append__eq__Cons__conv,axiom,
! [Ys: list_v,Zs: list_v,X4: v,Xs: list_v] :
( ( ( append_v @ Ys @ Zs )
= ( cons_v @ X4 @ Xs ) )
= ( ( ( Ys = nil_v )
& ( Zs
= ( cons_v @ X4 @ Xs ) ) )
| ? [Ys5: list_v] :
( ( Ys
= ( cons_v @ X4 @ Ys5 ) )
& ( ( append_v @ Ys5 @ Zs )
= Xs ) ) ) ) ).
% append_eq_Cons_conv
thf(fact_859_Cons__eq__append__conv,axiom,
! [X4: v,Xs: list_v,Ys: list_v,Zs: list_v] :
( ( ( cons_v @ X4 @ Xs )
= ( append_v @ Ys @ Zs ) )
= ( ( ( Ys = nil_v )
& ( ( cons_v @ X4 @ Xs )
= Zs ) )
| ? [Ys5: list_v] :
( ( ( cons_v @ X4 @ Ys5 )
= Ys )
& ( Xs
= ( append_v @ Ys5 @ Zs ) ) ) ) ) ).
% Cons_eq_append_conv
thf(fact_860_rev__exhaust,axiom,
! [Xs: list_v] :
( ( Xs != nil_v )
=> ~ ! [Ys3: list_v,Y2: v] :
( Xs
!= ( append_v @ Ys3 @ ( cons_v @ Y2 @ nil_v ) ) ) ) ).
% rev_exhaust
thf(fact_861_rev__induct,axiom,
! [P: list_v > $o,Xs: list_v] :
( ( P @ nil_v )
=> ( ! [X3: v,Xs3: list_v] :
( ( P @ Xs3 )
=> ( P @ ( append_v @ Xs3 @ ( cons_v @ X3 @ nil_v ) ) ) )
=> ( P @ Xs ) ) ) ).
% rev_induct
thf(fact_862_split__list,axiom,
! [X4: product_prod_v_v,Xs: list_P7986770385144383213od_v_v] :
( ( member7453568604450474000od_v_v @ X4 @ ( set_Product_prod_v_v2 @ Xs ) )
=> ? [Ys3: list_P7986770385144383213od_v_v,Zs2: list_P7986770385144383213od_v_v] :
( Xs
= ( append2138873909117096322od_v_v @ Ys3 @ ( cons_P4120604216776828829od_v_v @ X4 @ Zs2 ) ) ) ) ).
% split_list
thf(fact_863_split__list,axiom,
! [X4: v,Xs: list_v] :
( ( member_v @ X4 @ ( set_v2 @ Xs ) )
=> ? [Ys3: list_v,Zs2: list_v] :
( Xs
= ( append_v @ Ys3 @ ( cons_v @ X4 @ Zs2 ) ) ) ) ).
% split_list
thf(fact_864_split__list__last,axiom,
! [X4: product_prod_v_v,Xs: list_P7986770385144383213od_v_v] :
( ( member7453568604450474000od_v_v @ X4 @ ( set_Product_prod_v_v2 @ Xs ) )
=> ? [Ys3: list_P7986770385144383213od_v_v,Zs2: list_P7986770385144383213od_v_v] :
( ( Xs
= ( append2138873909117096322od_v_v @ Ys3 @ ( cons_P4120604216776828829od_v_v @ X4 @ Zs2 ) ) )
& ~ ( member7453568604450474000od_v_v @ X4 @ ( set_Product_prod_v_v2 @ Zs2 ) ) ) ) ).
% split_list_last
thf(fact_865_split__list__last,axiom,
! [X4: v,Xs: list_v] :
( ( member_v @ X4 @ ( set_v2 @ Xs ) )
=> ? [Ys3: list_v,Zs2: list_v] :
( ( Xs
= ( append_v @ Ys3 @ ( cons_v @ X4 @ Zs2 ) ) )
& ~ ( member_v @ X4 @ ( set_v2 @ Zs2 ) ) ) ) ).
% split_list_last
thf(fact_866_split__list__prop,axiom,
! [Xs: list_v,P: v > $o] :
( ? [X: v] :
( ( member_v @ X @ ( set_v2 @ Xs ) )
& ( P @ X ) )
=> ? [Ys3: list_v,X3: v] :
( ? [Zs2: list_v] :
( Xs
= ( append_v @ Ys3 @ ( cons_v @ X3 @ Zs2 ) ) )
& ( P @ X3 ) ) ) ).
% split_list_prop
thf(fact_867_split__list__first,axiom,
! [X4: product_prod_v_v,Xs: list_P7986770385144383213od_v_v] :
( ( member7453568604450474000od_v_v @ X4 @ ( set_Product_prod_v_v2 @ Xs ) )
=> ? [Ys3: list_P7986770385144383213od_v_v,Zs2: list_P7986770385144383213od_v_v] :
( ( Xs
= ( append2138873909117096322od_v_v @ Ys3 @ ( cons_P4120604216776828829od_v_v @ X4 @ Zs2 ) ) )
& ~ ( member7453568604450474000od_v_v @ X4 @ ( set_Product_prod_v_v2 @ Ys3 ) ) ) ) ).
% split_list_first
thf(fact_868_split__list__first,axiom,
! [X4: v,Xs: list_v] :
( ( member_v @ X4 @ ( set_v2 @ Xs ) )
=> ? [Ys3: list_v,Zs2: list_v] :
( ( Xs
= ( append_v @ Ys3 @ ( cons_v @ X4 @ Zs2 ) ) )
& ~ ( member_v @ X4 @ ( set_v2 @ Ys3 ) ) ) ) ).
% split_list_first
thf(fact_869_split__list__propE,axiom,
! [Xs: list_v,P: v > $o] :
( ? [X: v] :
( ( member_v @ X @ ( set_v2 @ Xs ) )
& ( P @ X ) )
=> ~ ! [Ys3: list_v,X3: v] :
( ? [Zs2: list_v] :
( Xs
= ( append_v @ Ys3 @ ( cons_v @ X3 @ Zs2 ) ) )
=> ~ ( P @ X3 ) ) ) ).
% split_list_propE
thf(fact_870_append__Cons__eq__iff,axiom,
! [X4: product_prod_v_v,Xs: list_P7986770385144383213od_v_v,Ys: list_P7986770385144383213od_v_v,Xs4: list_P7986770385144383213od_v_v,Ys6: list_P7986770385144383213od_v_v] :
( ~ ( member7453568604450474000od_v_v @ X4 @ ( set_Product_prod_v_v2 @ Xs ) )
=> ( ~ ( member7453568604450474000od_v_v @ X4 @ ( set_Product_prod_v_v2 @ Ys ) )
=> ( ( ( append2138873909117096322od_v_v @ Xs @ ( cons_P4120604216776828829od_v_v @ X4 @ Ys ) )
= ( append2138873909117096322od_v_v @ Xs4 @ ( cons_P4120604216776828829od_v_v @ X4 @ Ys6 ) ) )
= ( ( Xs = Xs4 )
& ( Ys = Ys6 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_871_append__Cons__eq__iff,axiom,
! [X4: v,Xs: list_v,Ys: list_v,Xs4: list_v,Ys6: list_v] :
( ~ ( member_v @ X4 @ ( set_v2 @ Xs ) )
=> ( ~ ( member_v @ X4 @ ( set_v2 @ Ys ) )
=> ( ( ( append_v @ Xs @ ( cons_v @ X4 @ Ys ) )
= ( append_v @ Xs4 @ ( cons_v @ X4 @ Ys6 ) ) )
= ( ( Xs = Xs4 )
& ( Ys = Ys6 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_872_in__set__conv__decomp,axiom,
! [X4: product_prod_v_v,Xs: list_P7986770385144383213od_v_v] :
( ( member7453568604450474000od_v_v @ X4 @ ( set_Product_prod_v_v2 @ Xs ) )
= ( ? [Ys4: list_P7986770385144383213od_v_v,Zs3: list_P7986770385144383213od_v_v] :
( Xs
= ( append2138873909117096322od_v_v @ Ys4 @ ( cons_P4120604216776828829od_v_v @ X4 @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_873_in__set__conv__decomp,axiom,
! [X4: v,Xs: list_v] :
( ( member_v @ X4 @ ( set_v2 @ Xs ) )
= ( ? [Ys4: list_v,Zs3: list_v] :
( Xs
= ( append_v @ Ys4 @ ( cons_v @ X4 @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_874_split__list__last__prop,axiom,
! [Xs: list_v,P: v > $o] :
( ? [X: v] :
( ( member_v @ X @ ( set_v2 @ Xs ) )
& ( P @ X ) )
=> ? [Ys3: list_v,X3: v,Zs2: list_v] :
( ( Xs
= ( append_v @ Ys3 @ ( cons_v @ X3 @ Zs2 ) ) )
& ( P @ X3 )
& ! [Xa2: v] :
( ( member_v @ Xa2 @ ( set_v2 @ Zs2 ) )
=> ~ ( P @ Xa2 ) ) ) ) ).
% split_list_last_prop
thf(fact_875_split__list__first__prop,axiom,
! [Xs: list_v,P: v > $o] :
( ? [X: v] :
( ( member_v @ X @ ( set_v2 @ Xs ) )
& ( P @ X ) )
=> ? [Ys3: list_v,X3: v] :
( ? [Zs2: list_v] :
( Xs
= ( append_v @ Ys3 @ ( cons_v @ X3 @ Zs2 ) ) )
& ( P @ X3 )
& ! [Xa2: v] :
( ( member_v @ Xa2 @ ( set_v2 @ Ys3 ) )
=> ~ ( P @ Xa2 ) ) ) ) ).
% split_list_first_prop
thf(fact_876_split__list__last__propE,axiom,
! [Xs: list_v,P: v > $o] :
( ? [X: v] :
( ( member_v @ X @ ( set_v2 @ Xs ) )
& ( P @ X ) )
=> ~ ! [Ys3: list_v,X3: v,Zs2: list_v] :
( ( Xs
= ( append_v @ Ys3 @ ( cons_v @ X3 @ Zs2 ) ) )
=> ( ( P @ X3 )
=> ~ ! [Xa2: v] :
( ( member_v @ Xa2 @ ( set_v2 @ Zs2 ) )
=> ~ ( P @ Xa2 ) ) ) ) ) ).
% split_list_last_propE
thf(fact_877_split__list__first__propE,axiom,
! [Xs: list_v,P: v > $o] :
( ? [X: v] :
( ( member_v @ X @ ( set_v2 @ Xs ) )
& ( P @ X ) )
=> ~ ! [Ys3: list_v,X3: v] :
( ? [Zs2: list_v] :
( Xs
= ( append_v @ Ys3 @ ( cons_v @ X3 @ Zs2 ) ) )
=> ( ( P @ X3 )
=> ~ ! [Xa2: v] :
( ( member_v @ Xa2 @ ( set_v2 @ Ys3 ) )
=> ~ ( P @ Xa2 ) ) ) ) ) ).
% split_list_first_propE
thf(fact_878_in__set__conv__decomp__last,axiom,
! [X4: product_prod_v_v,Xs: list_P7986770385144383213od_v_v] :
( ( member7453568604450474000od_v_v @ X4 @ ( set_Product_prod_v_v2 @ Xs ) )
= ( ? [Ys4: list_P7986770385144383213od_v_v,Zs3: list_P7986770385144383213od_v_v] :
( ( Xs
= ( append2138873909117096322od_v_v @ Ys4 @ ( cons_P4120604216776828829od_v_v @ X4 @ Zs3 ) ) )
& ~ ( member7453568604450474000od_v_v @ X4 @ ( set_Product_prod_v_v2 @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_879_in__set__conv__decomp__last,axiom,
! [X4: v,Xs: list_v] :
( ( member_v @ X4 @ ( set_v2 @ Xs ) )
= ( ? [Ys4: list_v,Zs3: list_v] :
( ( Xs
= ( append_v @ Ys4 @ ( cons_v @ X4 @ Zs3 ) ) )
& ~ ( member_v @ X4 @ ( set_v2 @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_880_in__set__conv__decomp__first,axiom,
! [X4: product_prod_v_v,Xs: list_P7986770385144383213od_v_v] :
( ( member7453568604450474000od_v_v @ X4 @ ( set_Product_prod_v_v2 @ Xs ) )
= ( ? [Ys4: list_P7986770385144383213od_v_v,Zs3: list_P7986770385144383213od_v_v] :
( ( Xs
= ( append2138873909117096322od_v_v @ Ys4 @ ( cons_P4120604216776828829od_v_v @ X4 @ Zs3 ) ) )
& ~ ( member7453568604450474000od_v_v @ X4 @ ( set_Product_prod_v_v2 @ Ys4 ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_881_in__set__conv__decomp__first,axiom,
! [X4: v,Xs: list_v] :
( ( member_v @ X4 @ ( set_v2 @ Xs ) )
= ( ? [Ys4: list_v,Zs3: list_v] :
( ( Xs
= ( append_v @ Ys4 @ ( cons_v @ X4 @ Zs3 ) ) )
& ~ ( member_v @ X4 @ ( set_v2 @ Ys4 ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_882_split__list__last__prop__iff,axiom,
! [Xs: list_v,P: v > $o] :
( ( ? [X2: v] :
( ( member_v @ X2 @ ( set_v2 @ Xs ) )
& ( P @ X2 ) ) )
= ( ? [Ys4: list_v,X2: v,Zs3: list_v] :
( ( Xs
= ( append_v @ Ys4 @ ( cons_v @ X2 @ Zs3 ) ) )
& ( P @ X2 )
& ! [Y3: v] :
( ( member_v @ Y3 @ ( set_v2 @ Zs3 ) )
=> ~ ( P @ Y3 ) ) ) ) ) ).
% split_list_last_prop_iff
thf(fact_883_split__list__first__prop__iff,axiom,
! [Xs: list_v,P: v > $o] :
( ( ? [X2: v] :
( ( member_v @ X2 @ ( set_v2 @ Xs ) )
& ( P @ X2 ) ) )
= ( ? [Ys4: list_v,X2: v] :
( ? [Zs3: list_v] :
( Xs
= ( append_v @ Ys4 @ ( cons_v @ X2 @ Zs3 ) ) )
& ( P @ X2 )
& ! [Y3: v] :
( ( member_v @ Y3 @ ( set_v2 @ Ys4 ) )
=> ~ ( P @ Y3 ) ) ) ) ) ).
% split_list_first_prop_iff
thf(fact_884_distinct__singleton,axiom,
! [X4: v] : ( distinct_v @ ( cons_v @ X4 @ nil_v ) ) ).
% distinct_singleton
thf(fact_885_distinct_Osimps_I2_J,axiom,
! [X4: product_prod_v_v,Xs: list_P7986770385144383213od_v_v] :
( ( distin6159370996967099744od_v_v @ ( cons_P4120604216776828829od_v_v @ X4 @ Xs ) )
= ( ~ ( member7453568604450474000od_v_v @ X4 @ ( set_Product_prod_v_v2 @ Xs ) )
& ( distin6159370996967099744od_v_v @ Xs ) ) ) ).
% distinct.simps(2)
thf(fact_886_distinct_Osimps_I2_J,axiom,
! [X4: v,Xs: list_v] :
( ( distinct_v @ ( cons_v @ X4 @ Xs ) )
= ( ~ ( member_v @ X4 @ ( set_v2 @ Xs ) )
& ( distinct_v @ Xs ) ) ) ).
% distinct.simps(2)
thf(fact_887_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_888_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_889_tail__not__precedes,axiom,
! [Y: product_prod_v_v,X4: product_prod_v_v,Xs: list_P7986770385144383213od_v_v] :
( ( sCC_Bl2026170059108282219od_v_v @ Y @ X4 @ ( cons_P4120604216776828829od_v_v @ X4 @ Xs ) )
=> ( ~ ( member7453568604450474000od_v_v @ X4 @ ( set_Product_prod_v_v2 @ Xs ) )
=> ( X4 = Y ) ) ) ).
% tail_not_precedes
thf(fact_890_tail__not__precedes,axiom,
! [Y: v,X4: v,Xs: list_v] :
( ( sCC_Bl4022239298816431255edes_v @ Y @ X4 @ ( cons_v @ X4 @ Xs ) )
=> ( ~ ( member_v @ X4 @ ( set_v2 @ Xs ) )
=> ( X4 = Y ) ) ) ).
% tail_not_precedes
thf(fact_891_head__precedes,axiom,
! [Y: product_prod_v_v,X4: product_prod_v_v,Xs: list_P7986770385144383213od_v_v] :
( ( member7453568604450474000od_v_v @ Y @ ( set_Product_prod_v_v2 @ ( cons_P4120604216776828829od_v_v @ X4 @ Xs ) ) )
=> ( sCC_Bl2026170059108282219od_v_v @ X4 @ Y @ ( cons_P4120604216776828829od_v_v @ X4 @ Xs ) ) ) ).
% head_precedes
thf(fact_892_head__precedes,axiom,
! [Y: v,X4: v,Xs: list_v] :
( ( member_v @ Y @ ( set_v2 @ ( cons_v @ X4 @ Xs ) ) )
=> ( sCC_Bl4022239298816431255edes_v @ X4 @ Y @ ( cons_v @ X4 @ Xs ) ) ) ).
% head_precedes
thf(fact_893_Diff__single__insert,axiom,
! [A2: set_v,X4: v,B2: set_v] :
( ( ord_less_eq_set_v @ ( minus_minus_set_v @ A2 @ ( insert_v @ X4 @ bot_bot_set_v ) ) @ B2 )
=> ( ord_less_eq_set_v @ A2 @ ( insert_v @ X4 @ B2 ) ) ) ).
% Diff_single_insert
thf(fact_894_Diff__single__insert,axiom,
! [A2: set_Product_prod_v_v,X4: product_prod_v_v,B2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ ( minus_4183494784930505774od_v_v @ A2 @ ( insert1338601472111419319od_v_v @ X4 @ bot_bo723834152578015283od_v_v ) ) @ B2 )
=> ( ord_le7336532860387713383od_v_v @ A2 @ ( insert1338601472111419319od_v_v @ X4 @ B2 ) ) ) ).
% Diff_single_insert
thf(fact_895_subset__insert__iff,axiom,
! [A2: set_v,X4: v,B2: set_v] :
( ( ord_less_eq_set_v @ A2 @ ( insert_v @ X4 @ B2 ) )
= ( ( ( member_v @ X4 @ A2 )
=> ( ord_less_eq_set_v @ ( minus_minus_set_v @ A2 @ ( insert_v @ X4 @ bot_bot_set_v ) ) @ B2 ) )
& ( ~ ( member_v @ X4 @ A2 )
=> ( ord_less_eq_set_v @ A2 @ B2 ) ) ) ) ).
% subset_insert_iff
thf(fact_896_subset__insert__iff,axiom,
! [A2: set_Product_prod_v_v,X4: product_prod_v_v,B2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A2 @ ( insert1338601472111419319od_v_v @ X4 @ B2 ) )
= ( ( ( member7453568604450474000od_v_v @ X4 @ A2 )
=> ( ord_le7336532860387713383od_v_v @ ( minus_4183494784930505774od_v_v @ A2 @ ( insert1338601472111419319od_v_v @ X4 @ bot_bo723834152578015283od_v_v ) ) @ B2 ) )
& ( ~ ( member7453568604450474000od_v_v @ X4 @ A2 )
=> ( ord_le7336532860387713383od_v_v @ A2 @ B2 ) ) ) ) ).
% subset_insert_iff
thf(fact_897_order__antisym__conv,axiom,
! [Y: set_v,X4: set_v] :
( ( ord_less_eq_set_v @ Y @ X4 )
=> ( ( ord_less_eq_set_v @ X4 @ Y )
= ( X4 = Y ) ) ) ).
% order_antisym_conv
thf(fact_898_order__antisym__conv,axiom,
! [Y: set_Product_prod_v_v,X4: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ Y @ X4 )
=> ( ( ord_le7336532860387713383od_v_v @ X4 @ Y )
= ( X4 = Y ) ) ) ).
% order_antisym_conv
thf(fact_899_ord__le__eq__subst,axiom,
! [A: set_v,B: set_v,F: set_v > set_v,C: set_v] :
( ( ord_less_eq_set_v @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [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 @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_900_ord__le__eq__subst,axiom,
! [A: set_v,B: set_v,F: set_v > set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( ord_less_eq_set_v @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [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 @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_901_ord__le__eq__subst,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v,F: set_Product_prod_v_v > set_v,C: set_v] :
( ( ord_le7336532860387713383od_v_v @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [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 @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_902_ord__le__eq__subst,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v,F: set_Product_prod_v_v > set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [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 @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_903_ord__eq__le__subst,axiom,
! [A: set_v,F: set_v > set_v,B: set_v,C: set_v] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_set_v @ B @ C )
=> ( ! [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 @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_904_ord__eq__le__subst,axiom,
! [A: set_Product_prod_v_v,F: set_v > set_Product_prod_v_v,B: set_v,C: set_v] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_set_v @ B @ C )
=> ( ! [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 @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_905_ord__eq__le__subst,axiom,
! [A: set_v,F: set_Product_prod_v_v > set_v,B: set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( A
= ( F @ B ) )
=> ( ( ord_le7336532860387713383od_v_v @ B @ C )
=> ( ! [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 @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_906_ord__eq__le__subst,axiom,
! [A: set_Product_prod_v_v,F: set_Product_prod_v_v > set_Product_prod_v_v,B: set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( A
= ( F @ B ) )
=> ( ( ord_le7336532860387713383od_v_v @ B @ C )
=> ( ! [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 @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_907_order__eq__refl,axiom,
! [X4: set_v,Y: set_v] :
( ( X4 = Y )
=> ( ord_less_eq_set_v @ X4 @ Y ) ) ).
% order_eq_refl
thf(fact_908_order__eq__refl,axiom,
! [X4: set_Product_prod_v_v,Y: set_Product_prod_v_v] :
( ( X4 = Y )
=> ( ord_le7336532860387713383od_v_v @ X4 @ Y ) ) ).
% order_eq_refl
thf(fact_909_order__subst2,axiom,
! [A: set_v,B: set_v,F: set_v > set_v,C: set_v] :
( ( ord_less_eq_set_v @ A @ B )
=> ( ( ord_less_eq_set_v @ ( F @ B ) @ C )
=> ( ! [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 @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_910_order__subst2,axiom,
! [A: set_v,B: set_v,F: set_v > set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( ord_less_eq_set_v @ A @ B )
=> ( ( ord_le7336532860387713383od_v_v @ ( F @ B ) @ C )
=> ( ! [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 @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_911_order__subst2,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v,F: set_Product_prod_v_v > set_v,C: set_v] :
( ( ord_le7336532860387713383od_v_v @ A @ B )
=> ( ( ord_less_eq_set_v @ ( F @ B ) @ C )
=> ( ! [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 @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_912_order__subst2,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v,F: set_Product_prod_v_v > set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ B )
=> ( ( ord_le7336532860387713383od_v_v @ ( F @ B ) @ C )
=> ( ! [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 @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_913_order__subst1,axiom,
! [A: set_v,F: set_v > set_v,B: set_v,C: set_v] :
( ( ord_less_eq_set_v @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_v @ B @ C )
=> ( ! [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 @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_914_order__subst1,axiom,
! [A: set_v,F: set_Product_prod_v_v > set_v,B: set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( ord_less_eq_set_v @ A @ ( F @ B ) )
=> ( ( ord_le7336532860387713383od_v_v @ B @ C )
=> ( ! [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 @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_915_order__subst1,axiom,
! [A: set_Product_prod_v_v,F: set_v > set_Product_prod_v_v,B: set_v,C: set_v] :
( ( ord_le7336532860387713383od_v_v @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_v @ B @ C )
=> ( ! [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 @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_916_order__subst1,axiom,
! [A: set_Product_prod_v_v,F: set_Product_prod_v_v > set_Product_prod_v_v,B: set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ ( F @ B ) )
=> ( ( ord_le7336532860387713383od_v_v @ B @ C )
=> ( ! [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 @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_917_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: set_v,Z5: set_v] : ( Y5 = Z5 ) )
= ( ^ [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_918_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: set_Product_prod_v_v,Z5: set_Product_prod_v_v] : ( Y5 = Z5 ) )
= ( ^ [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_919_antisym,axiom,
! [A: set_v,B: set_v] :
( ( ord_less_eq_set_v @ A @ B )
=> ( ( ord_less_eq_set_v @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_920_antisym,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ B )
=> ( ( ord_le7336532860387713383od_v_v @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_921_dual__order_Otrans,axiom,
! [B: set_v,A: set_v,C: set_v] :
( ( ord_less_eq_set_v @ B @ A )
=> ( ( ord_less_eq_set_v @ C @ B )
=> ( ord_less_eq_set_v @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_922_dual__order_Otrans,axiom,
! [B: set_Product_prod_v_v,A: set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ B @ A )
=> ( ( ord_le7336532860387713383od_v_v @ C @ B )
=> ( ord_le7336532860387713383od_v_v @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_923_dual__order_Oantisym,axiom,
! [B: set_v,A: set_v] :
( ( ord_less_eq_set_v @ B @ A )
=> ( ( ord_less_eq_set_v @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_924_dual__order_Oantisym,axiom,
! [B: set_Product_prod_v_v,A: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ B @ A )
=> ( ( ord_le7336532860387713383od_v_v @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_925_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: set_v,Z5: set_v] : ( Y5 = Z5 ) )
= ( ^ [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_926_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: set_Product_prod_v_v,Z5: set_Product_prod_v_v] : ( Y5 = Z5 ) )
= ( ^ [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_927_order__trans,axiom,
! [X4: set_v,Y: set_v,Z: set_v] :
( ( ord_less_eq_set_v @ X4 @ Y )
=> ( ( ord_less_eq_set_v @ Y @ Z )
=> ( ord_less_eq_set_v @ X4 @ Z ) ) ) ).
% order_trans
thf(fact_928_order__trans,axiom,
! [X4: set_Product_prod_v_v,Y: set_Product_prod_v_v,Z: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X4 @ Y )
=> ( ( ord_le7336532860387713383od_v_v @ Y @ Z )
=> ( ord_le7336532860387713383od_v_v @ X4 @ Z ) ) ) ).
% order_trans
thf(fact_929_order_Otrans,axiom,
! [A: set_v,B: set_v,C: set_v] :
( ( ord_less_eq_set_v @ A @ B )
=> ( ( ord_less_eq_set_v @ B @ C )
=> ( ord_less_eq_set_v @ A @ C ) ) ) ).
% order.trans
thf(fact_930_order_Otrans,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ B )
=> ( ( ord_le7336532860387713383od_v_v @ B @ C )
=> ( ord_le7336532860387713383od_v_v @ A @ C ) ) ) ).
% order.trans
thf(fact_931_order__antisym,axiom,
! [X4: set_v,Y: set_v] :
( ( ord_less_eq_set_v @ X4 @ Y )
=> ( ( ord_less_eq_set_v @ Y @ X4 )
=> ( X4 = Y ) ) ) ).
% order_antisym
thf(fact_932_order__antisym,axiom,
! [X4: set_Product_prod_v_v,Y: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X4 @ Y )
=> ( ( ord_le7336532860387713383od_v_v @ Y @ X4 )
=> ( X4 = Y ) ) ) ).
% order_antisym
thf(fact_933_ord__le__eq__trans,axiom,
! [A: set_v,B: set_v,C: set_v] :
( ( ord_less_eq_set_v @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_set_v @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_934_ord__le__eq__trans,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ B )
=> ( ( B = C )
=> ( ord_le7336532860387713383od_v_v @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_935_ord__eq__le__trans,axiom,
! [A: set_v,B: set_v,C: set_v] :
( ( A = B )
=> ( ( ord_less_eq_set_v @ B @ C )
=> ( ord_less_eq_set_v @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_936_ord__eq__le__trans,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( A = B )
=> ( ( ord_le7336532860387713383od_v_v @ B @ C )
=> ( ord_le7336532860387713383od_v_v @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_937_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: set_v,Z5: set_v] : ( Y5 = Z5 ) )
= ( ^ [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_938_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: set_Product_prod_v_v,Z5: set_Product_prod_v_v] : ( Y5 = Z5 ) )
= ( ^ [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_939_not__distinct__decomp,axiom,
! [Ws: list_v] :
( ~ ( distinct_v @ Ws )
=> ? [Xs3: list_v,Ys3: list_v,Zs2: list_v,Y2: v] :
( Ws
= ( append_v @ Xs3 @ ( append_v @ ( cons_v @ Y2 @ nil_v ) @ ( append_v @ Ys3 @ ( append_v @ ( cons_v @ Y2 @ nil_v ) @ Zs2 ) ) ) ) ) ) ).
% not_distinct_decomp
thf(fact_940_not__distinct__conv__prefix,axiom,
! [As: list_P7986770385144383213od_v_v] :
( ( ~ ( distin6159370996967099744od_v_v @ As ) )
= ( ? [Xs5: list_P7986770385144383213od_v_v,Y3: product_prod_v_v,Ys4: list_P7986770385144383213od_v_v] :
( ( member7453568604450474000od_v_v @ Y3 @ ( set_Product_prod_v_v2 @ Xs5 ) )
& ( distin6159370996967099744od_v_v @ Xs5 )
& ( As
= ( append2138873909117096322od_v_v @ Xs5 @ ( cons_P4120604216776828829od_v_v @ Y3 @ Ys4 ) ) ) ) ) ) ).
% not_distinct_conv_prefix
thf(fact_941_not__distinct__conv__prefix,axiom,
! [As: list_v] :
( ( ~ ( distinct_v @ As ) )
= ( ? [Xs5: list_v,Y3: v,Ys4: list_v] :
( ( member_v @ Y3 @ ( set_v2 @ Xs5 ) )
& ( distinct_v @ Xs5 )
& ( As
= ( append_v @ Xs5 @ ( cons_v @ Y3 @ Ys4 ) ) ) ) ) ) ).
% not_distinct_conv_prefix
thf(fact_942_list_Oexhaust__sel,axiom,
! [List: list_v] :
( ( List != nil_v )
=> ( List
= ( cons_v @ ( hd_v @ List ) @ ( tl_v @ List ) ) ) ) ).
% list.exhaust_sel
thf(fact_943_precedes__def,axiom,
( sCC_Bl2026170059108282219od_v_v
= ( ^ [X2: product_prod_v_v,Y3: product_prod_v_v,Xs5: list_P7986770385144383213od_v_v] :
? [L: list_P7986770385144383213od_v_v,R3: list_P7986770385144383213od_v_v] :
( ( Xs5
= ( append2138873909117096322od_v_v @ L @ ( cons_P4120604216776828829od_v_v @ X2 @ R3 ) ) )
& ( member7453568604450474000od_v_v @ Y3 @ ( set_Product_prod_v_v2 @ ( cons_P4120604216776828829od_v_v @ X2 @ R3 ) ) ) ) ) ) ).
% precedes_def
thf(fact_944_precedes__def,axiom,
( sCC_Bl4022239298816431255edes_v
= ( ^ [X2: v,Y3: v,Xs5: list_v] :
? [L: list_v,R3: list_v] :
( ( Xs5
= ( append_v @ L @ ( cons_v @ X2 @ R3 ) ) )
& ( member_v @ Y3 @ ( set_v2 @ ( cons_v @ X2 @ R3 ) ) ) ) ) ) ).
% precedes_def
thf(fact_945_graph_Oreachable__re,axiom,
! [Vertices: set_v,Successors: v > set_v,X4: v,Y: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl649662514949026229able_v @ Successors @ X4 @ Y )
=> ( sCC_Bl770211535891879572_end_v @ Successors @ X4 @ Y ) ) ) ).
% graph.reachable_re
thf(fact_946_graph_Ore__reachable,axiom,
! [Vertices: set_v,Successors: v > set_v,X4: v,Y: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl770211535891879572_end_v @ Successors @ X4 @ Y )
=> ( sCC_Bl649662514949026229able_v @ Successors @ X4 @ Y ) ) ) ).
% graph.re_reachable
thf(fact_947_graph_Osubscc__add,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,S3: set_Product_prod_v_v,X4: product_prod_v_v,Y: product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( sCC_Bl2301996248249672505od_v_v @ Successors @ S3 )
=> ( ( member7453568604450474000od_v_v @ X4 @ S3 )
=> ( ( sCC_Bl4981926079593201289od_v_v @ Successors @ X4 @ Y )
=> ( ( sCC_Bl4981926079593201289od_v_v @ Successors @ Y @ X4 )
=> ( sCC_Bl2301996248249672505od_v_v @ Successors @ ( insert1338601472111419319od_v_v @ Y @ S3 ) ) ) ) ) ) ) ).
% graph.subscc_add
thf(fact_948_graph_Osubscc__add,axiom,
! [Vertices: set_v,Successors: v > set_v,S3: set_v,X4: v,Y: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl5398416737448265317bscc_v @ Successors @ S3 )
=> ( ( member_v @ X4 @ S3 )
=> ( ( sCC_Bl649662514949026229able_v @ Successors @ X4 @ Y )
=> ( ( sCC_Bl649662514949026229able_v @ Successors @ Y @ X4 )
=> ( sCC_Bl5398416737448265317bscc_v @ Successors @ ( insert_v @ Y @ S3 ) ) ) ) ) ) ) ).
% graph.subscc_add
thf(fact_949_split__list__precedes,axiom,
! [Y: product_prod_v_v,Ys: list_P7986770385144383213od_v_v,X4: product_prod_v_v,Xs: list_P7986770385144383213od_v_v] :
( ( member7453568604450474000od_v_v @ Y @ ( set_Product_prod_v_v2 @ ( append2138873909117096322od_v_v @ Ys @ ( cons_P4120604216776828829od_v_v @ X4 @ nil_Product_prod_v_v ) ) ) )
=> ( sCC_Bl2026170059108282219od_v_v @ Y @ X4 @ ( append2138873909117096322od_v_v @ Ys @ ( cons_P4120604216776828829od_v_v @ X4 @ Xs ) ) ) ) ).
% split_list_precedes
thf(fact_950_split__list__precedes,axiom,
! [Y: v,Ys: list_v,X4: v,Xs: list_v] :
( ( member_v @ Y @ ( set_v2 @ ( append_v @ Ys @ ( cons_v @ X4 @ nil_v ) ) ) )
=> ( sCC_Bl4022239298816431255edes_v @ Y @ X4 @ ( append_v @ Ys @ ( cons_v @ X4 @ Xs ) ) ) ) ).
% split_list_precedes
thf(fact_951_bot_Oextremum__uniqueI,axiom,
! [A: set_v] :
( ( ord_less_eq_set_v @ A @ bot_bot_set_v )
=> ( A = bot_bot_set_v ) ) ).
% bot.extremum_uniqueI
thf(fact_952_bot_Oextremum__uniqueI,axiom,
! [A: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ bot_bo723834152578015283od_v_v )
=> ( A = bot_bo723834152578015283od_v_v ) ) ).
% bot.extremum_uniqueI
thf(fact_953_bot_Oextremum__unique,axiom,
! [A: set_v] :
( ( ord_less_eq_set_v @ A @ bot_bot_set_v )
= ( A = bot_bot_set_v ) ) ).
% bot.extremum_unique
thf(fact_954_bot_Oextremum__unique,axiom,
! [A: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ bot_bo723834152578015283od_v_v )
= ( A = bot_bo723834152578015283od_v_v ) ) ).
% bot.extremum_unique
thf(fact_955_bot_Oextremum,axiom,
! [A: set_v] : ( ord_less_eq_set_v @ bot_bot_set_v @ A ) ).
% bot.extremum
thf(fact_956_bot_Oextremum,axiom,
! [A: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ bot_bo723834152578015283od_v_v @ A ) ).
% bot.extremum
thf(fact_957_graph_Opre__dfss__def,axiom,
! [Vertices: set_v,Successors: v > set_v,V3: v,E: sCC_Bl1394983891496994913t_unit] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl1748261141445803503t_unit @ Successors @ V3 @ E )
= ( ( sCC_Bl9196236973127232072t_unit @ Successors @ E )
& ( member_v @ V3 @ ( sCC_Bl4645233313691564917t_unit @ E ) )
& ( ( sCC_Bl8828226123343373779t_unit @ E )
!= nil_v )
& ( member_v @ V3 @ ( sCC_Bl1280885523602775798t_unit @ E @ ( hd_v @ ( sCC_Bl8828226123343373779t_unit @ E ) ) ) )
& ! [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 ) ) ) ) ) ).
% graph.pre_dfss_def
thf(fact_958_ccpo__Sup__singleton,axiom,
! [X4: set_v] :
( ( comple2307003700295860064_set_v @ ( insert_set_v @ X4 @ bot_bot_set_set_v ) )
= X4 ) ).
% ccpo_Sup_singleton
thf(fact_959_cSup__singleton,axiom,
! [X4: set_v] :
( ( comple2307003700295860064_set_v @ ( insert_set_v @ X4 @ bot_bot_set_set_v ) )
= X4 ) ).
% cSup_singleton
thf(fact_960_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_961_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_962_cSup__least,axiom,
! [X5: set_se8455005133513928103od_v_v,Z: set_Product_prod_v_v] :
( ( X5 != bot_bo3497076220358800403od_v_v )
=> ( ! [X3: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ X3 @ X5 )
=> ( ord_le7336532860387713383od_v_v @ X3 @ Z ) )
=> ( ord_le7336532860387713383od_v_v @ ( comple5788137035815166516od_v_v @ X5 ) @ Z ) ) ) ).
% cSup_least
thf(fact_963_cSup__least,axiom,
! [X5: set_set_v,Z: set_v] :
( ( X5 != bot_bot_set_set_v )
=> ( ! [X3: set_v] :
( ( member_set_v @ X3 @ X5 )
=> ( ord_less_eq_set_v @ X3 @ Z ) )
=> ( ord_less_eq_set_v @ ( comple2307003700295860064_set_v @ X5 ) @ Z ) ) ) ).
% cSup_least
thf(fact_964_cSup__eq__non__empty,axiom,
! [X5: set_se8455005133513928103od_v_v,A: set_Product_prod_v_v] :
( ( X5 != bot_bo3497076220358800403od_v_v )
=> ( ! [X3: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ X3 @ X5 )
=> ( ord_le7336532860387713383od_v_v @ X3 @ A ) )
=> ( ! [Y2: set_Product_prod_v_v] :
( ! [X: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ X @ X5 )
=> ( ord_le7336532860387713383od_v_v @ X @ Y2 ) )
=> ( ord_le7336532860387713383od_v_v @ A @ Y2 ) )
=> ( ( comple5788137035815166516od_v_v @ X5 )
= A ) ) ) ) ).
% cSup_eq_non_empty
thf(fact_965_cSup__eq__non__empty,axiom,
! [X5: set_set_v,A: set_v] :
( ( X5 != bot_bot_set_set_v )
=> ( ! [X3: set_v] :
( ( member_set_v @ X3 @ X5 )
=> ( ord_less_eq_set_v @ X3 @ A ) )
=> ( ! [Y2: set_v] :
( ! [X: set_v] :
( ( member_set_v @ X @ X5 )
=> ( ord_less_eq_set_v @ X @ Y2 ) )
=> ( ord_less_eq_set_v @ A @ Y2 ) )
=> ( ( comple2307003700295860064_set_v @ X5 )
= A ) ) ) ) ).
% cSup_eq_non_empty
thf(fact_966_distinct__union,axiom,
! [Xs: list_v,Ys: list_v] :
( ( distinct_v @ ( union_v @ Xs @ Ys ) )
= ( distinct_v @ Ys ) ) ).
% distinct_union
thf(fact_967_Union__insert,axiom,
! [A: set_Product_prod_v_v,B2: set_se8455005133513928103od_v_v] :
( ( comple5788137035815166516od_v_v @ ( insert7504383016908236695od_v_v @ A @ B2 ) )
= ( sup_su414716646722978715od_v_v @ A @ ( comple5788137035815166516od_v_v @ B2 ) ) ) ).
% Union_insert
thf(fact_968_Union__insert,axiom,
! [A: set_v,B2: set_set_v] :
( ( comple2307003700295860064_set_v @ ( insert_set_v @ A @ B2 ) )
= ( sup_sup_set_v @ A @ ( comple2307003700295860064_set_v @ B2 ) ) ) ).
% Union_insert
thf(fact_969_cSup__eq__maximum,axiom,
! [Z: set_Product_prod_v_v,X5: set_se8455005133513928103od_v_v] :
( ( member8406446414694345712od_v_v @ Z @ X5 )
=> ( ! [X3: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ X3 @ X5 )
=> ( ord_le7336532860387713383od_v_v @ X3 @ Z ) )
=> ( ( comple5788137035815166516od_v_v @ X5 )
= Z ) ) ) ).
% cSup_eq_maximum
thf(fact_970_cSup__eq__maximum,axiom,
! [Z: set_v,X5: set_set_v] :
( ( member_set_v @ Z @ X5 )
=> ( ! [X3: set_v] :
( ( member_set_v @ X3 @ X5 )
=> ( ord_less_eq_set_v @ X3 @ Z ) )
=> ( ( comple2307003700295860064_set_v @ X5 )
= Z ) ) ) ).
% cSup_eq_maximum
thf(fact_971_insert__partition,axiom,
! [X4: set_Product_prod_v_v,F3: set_se8455005133513928103od_v_v] :
( ~ ( member8406446414694345712od_v_v @ X4 @ F3 )
=> ( ! [X3: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ X3 @ ( insert7504383016908236695od_v_v @ X4 @ F3 ) )
=> ! [Xa: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ Xa @ ( insert7504383016908236695od_v_v @ X4 @ F3 ) )
=> ( ( X3 != Xa )
=> ( ( inf_in6271465464967711157od_v_v @ X3 @ Xa )
= bot_bo723834152578015283od_v_v ) ) ) )
=> ( ( inf_in6271465464967711157od_v_v @ X4 @ ( comple5788137035815166516od_v_v @ F3 ) )
= bot_bo723834152578015283od_v_v ) ) ) ).
% insert_partition
thf(fact_972_insert__partition,axiom,
! [X4: set_v,F3: set_set_v] :
( ~ ( member_set_v @ X4 @ F3 )
=> ( ! [X3: set_v] :
( ( member_set_v @ X3 @ ( insert_set_v @ X4 @ F3 ) )
=> ! [Xa: set_v] :
( ( member_set_v @ Xa @ ( insert_set_v @ X4 @ F3 ) )
=> ( ( X3 != Xa )
=> ( ( inf_inf_set_v @ X3 @ Xa )
= bot_bot_set_v ) ) ) )
=> ( ( inf_inf_set_v @ X4 @ ( comple2307003700295860064_set_v @ F3 ) )
= bot_bot_set_v ) ) ) ).
% insert_partition
thf(fact_973_cc__def,axiom,
( cc
= ( comple2307003700295860064_set_v
@ ( collect_set_v
@ ^ [Uu: set_v] :
? [N4: v] :
( ( Uu
= ( sCC_Bl1280885523602775798t_unit @ e @ N4 ) )
& ( member_v @ N4 @ ( sup_sup_set_v @ ( set_v2 @ pfx ) @ ( insert_v @ ( hd_v @ ( sCC_Bl8828226123343373779t_unit @ e2 ) ) @ bot_bot_set_v ) ) ) ) ) ) ) ).
% cc_def
thf(fact_974_the__elem__eq,axiom,
! [X4: v] :
( ( the_elem_v @ ( insert_v @ X4 @ bot_bot_set_v ) )
= X4 ) ).
% the_elem_eq
thf(fact_975_the__elem__eq,axiom,
! [X4: product_prod_v_v] :
( ( the_el5392834299063928540od_v_v @ ( insert1338601472111419319od_v_v @ X4 @ bot_bo723834152578015283od_v_v ) )
= X4 ) ).
% the_elem_eq
thf(fact_976_cc__Un,axiom,
( cc
= ( comple2307003700295860064_set_v
@ ( collect_set_v
@ ^ [Uu: set_v] :
? [X2: v] :
( ( Uu
= ( sCC_Bl1280885523602775798t_unit @ e @ X2 ) )
& ( member_v @ X2 @ cc ) ) ) ) ) ).
% cc_Un
thf(fact_977_pfx_I3_J,axiom,
( ( sCC_Bl1280885523602775798t_unit @ e2 )
= ( ^ [X2: v] :
( if_set_v
@ ( member_v @ X2
@ ( comple2307003700295860064_set_v
@ ( collect_set_v
@ ^ [Uu: set_v] :
? [N4: v] :
( ( Uu
= ( sCC_Bl1280885523602775798t_unit @ e @ N4 ) )
& ( member_v @ N4 @ ( sup_sup_set_v @ ( set_v2 @ pfx ) @ ( insert_v @ ( hd_v @ ( sCC_Bl8828226123343373779t_unit @ e2 ) ) @ bot_bot_set_v ) ) ) ) ) ) )
@ ( comple2307003700295860064_set_v
@ ( collect_set_v
@ ^ [Uu: set_v] :
? [N4: v] :
( ( Uu
= ( sCC_Bl1280885523602775798t_unit @ e @ N4 ) )
& ( member_v @ N4 @ ( sup_sup_set_v @ ( set_v2 @ pfx ) @ ( insert_v @ ( hd_v @ ( sCC_Bl8828226123343373779t_unit @ e2 ) ) @ bot_bot_set_v ) ) ) ) ) )
@ ( sCC_Bl1280885523602775798t_unit @ e @ X2 ) ) ) ) ).
% pfx(3)
thf(fact_978_singleton__conv,axiom,
! [A: set_v] :
( ( collect_set_v
@ ^ [X2: set_v] : ( X2 = A ) )
= ( insert_set_v @ A @ bot_bot_set_set_v ) ) ).
% singleton_conv
thf(fact_979_singleton__conv,axiom,
! [A: v] :
( ( collect_v
@ ^ [X2: v] : ( X2 = A ) )
= ( insert_v @ A @ bot_bot_set_v ) ) ).
% singleton_conv
thf(fact_980_singleton__conv,axiom,
! [A: product_prod_v_v] :
( ( collec140062887454715474od_v_v
@ ^ [X2: product_prod_v_v] : ( X2 = A ) )
= ( insert1338601472111419319od_v_v @ A @ bot_bo723834152578015283od_v_v ) ) ).
% singleton_conv
thf(fact_981_singleton__conv2,axiom,
! [A: set_v] :
( ( collect_set_v
@ ( ^ [Y5: set_v,Z5: set_v] : ( Y5 = Z5 )
@ A ) )
= ( insert_set_v @ A @ bot_bot_set_set_v ) ) ).
% singleton_conv2
thf(fact_982_singleton__conv2,axiom,
! [A: v] :
( ( collect_v
@ ( ^ [Y5: v,Z5: v] : ( Y5 = Z5 )
@ A ) )
= ( insert_v @ A @ bot_bot_set_v ) ) ).
% singleton_conv2
thf(fact_983_singleton__conv2,axiom,
! [A: product_prod_v_v] :
( ( collec140062887454715474od_v_v
@ ( ^ [Y5: product_prod_v_v,Z5: product_prod_v_v] : ( Y5 = Z5 )
@ A ) )
= ( insert1338601472111419319od_v_v @ A @ bot_bo723834152578015283od_v_v ) ) ).
% singleton_conv2
thf(fact_984_sub__env__def,axiom,
( sCC_Bl5768913643336123637t_unit
= ( ^ [E4: sCC_Bl1394983891496994913t_unit,E5: sCC_Bl1394983891496994913t_unit] :
( ( ( sCC_Bl1090238580953940555t_unit @ E5 )
= ( sCC_Bl1090238580953940555t_unit @ E4 ) )
& ( ord_less_eq_set_v @ ( sCC_Bl4645233313691564917t_unit @ E4 ) @ ( sCC_Bl4645233313691564917t_unit @ E5 ) )
& ( ord_less_eq_set_v @ ( sCC_Bl157864678168468314t_unit @ E4 ) @ ( sCC_Bl157864678168468314t_unit @ E5 ) )
& ! [V4: v] : ( ord_less_eq_set_v @ ( sCC_Bl3795065053823578884t_unit @ E4 @ V4 ) @ ( sCC_Bl3795065053823578884t_unit @ E5 @ V4 ) )
& ! [V4: v] : ( ord_less_eq_set_v @ ( sCC_Bl1280885523602775798t_unit @ E4 @ V4 ) @ ( sCC_Bl1280885523602775798t_unit @ E5 @ V4 ) )
& ( ord_less_eq_set_v
@ ( comple2307003700295860064_set_v
@ ( collect_set_v
@ ^ [Uu: set_v] :
? [V4: v] :
( ( Uu
= ( sCC_Bl1280885523602775798t_unit @ E4 @ V4 ) )
& ( member_v @ V4 @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E4 ) ) ) ) ) )
@ ( comple2307003700295860064_set_v
@ ( collect_set_v
@ ^ [Uu: set_v] :
? [V4: v] :
( ( Uu
= ( sCC_Bl1280885523602775798t_unit @ E5 @ V4 ) )
& ( member_v @ V4 @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E5 ) ) ) ) ) ) ) ) ) ) ).
% sub_env_def
thf(fact_985_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_986__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 )
!= ( ^ [X2: v] :
( if_set_v
@ ( member_v @ X2
@ ( comple2307003700295860064_set_v
@ ( collect_set_v
@ ^ [Uu: set_v] :
? [N4: v] :
( ( Uu
= ( sCC_Bl1280885523602775798t_unit @ e @ N4 ) )
& ( member_v @ N4 @ ( sup_sup_set_v @ ( set_v2 @ Pfx ) @ ( insert_v @ ( hd_v @ ( sCC_Bl8828226123343373779t_unit @ e2 ) ) @ bot_bot_set_v ) ) ) ) ) ) )
@ ( comple2307003700295860064_set_v
@ ( collect_set_v
@ ^ [Uu: set_v] :
? [N4: v] :
( ( Uu
= ( sCC_Bl1280885523602775798t_unit @ e @ N4 ) )
& ( member_v @ N4 @ ( sup_sup_set_v @ ( set_v2 @ Pfx ) @ ( insert_v @ ( hd_v @ ( sCC_Bl8828226123343373779t_unit @ e2 ) ) @ bot_bot_set_v ) ) ) ) ) )
@ ( sCC_Bl1280885523602775798t_unit @ e @ X2 ) ) ) ) ) ) ).
% \<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_987_Collect__conv__if,axiom,
! [P: set_v > $o,A: set_v] :
( ( ( P @ A )
=> ( ( collect_set_v
@ ^ [X2: set_v] :
( ( X2 = A )
& ( P @ X2 ) ) )
= ( insert_set_v @ A @ bot_bot_set_set_v ) ) )
& ( ~ ( P @ A )
=> ( ( collect_set_v
@ ^ [X2: set_v] :
( ( X2 = A )
& ( P @ X2 ) ) )
= bot_bot_set_set_v ) ) ) ).
% Collect_conv_if
thf(fact_988_Collect__conv__if,axiom,
! [P: v > $o,A: v] :
( ( ( P @ A )
=> ( ( collect_v
@ ^ [X2: v] :
( ( X2 = A )
& ( P @ X2 ) ) )
= ( insert_v @ A @ bot_bot_set_v ) ) )
& ( ~ ( P @ A )
=> ( ( collect_v
@ ^ [X2: v] :
( ( X2 = A )
& ( P @ X2 ) ) )
= bot_bot_set_v ) ) ) ).
% Collect_conv_if
thf(fact_989_Collect__conv__if,axiom,
! [P: product_prod_v_v > $o,A: product_prod_v_v] :
( ( ( P @ A )
=> ( ( collec140062887454715474od_v_v
@ ^ [X2: product_prod_v_v] :
( ( X2 = A )
& ( P @ X2 ) ) )
= ( insert1338601472111419319od_v_v @ A @ bot_bo723834152578015283od_v_v ) ) )
& ( ~ ( P @ A )
=> ( ( collec140062887454715474od_v_v
@ ^ [X2: product_prod_v_v] :
( ( X2 = A )
& ( P @ X2 ) ) )
= bot_bo723834152578015283od_v_v ) ) ) ).
% Collect_conv_if
thf(fact_990_Collect__conv__if2,axiom,
! [P: set_v > $o,A: set_v] :
( ( ( P @ A )
=> ( ( collect_set_v
@ ^ [X2: set_v] :
( ( A = X2 )
& ( P @ X2 ) ) )
= ( insert_set_v @ A @ bot_bot_set_set_v ) ) )
& ( ~ ( P @ A )
=> ( ( collect_set_v
@ ^ [X2: set_v] :
( ( A = X2 )
& ( P @ X2 ) ) )
= bot_bot_set_set_v ) ) ) ).
% Collect_conv_if2
thf(fact_991_Collect__conv__if2,axiom,
! [P: v > $o,A: v] :
( ( ( P @ A )
=> ( ( collect_v
@ ^ [X2: v] :
( ( A = X2 )
& ( P @ X2 ) ) )
= ( insert_v @ A @ bot_bot_set_v ) ) )
& ( ~ ( P @ A )
=> ( ( collect_v
@ ^ [X2: v] :
( ( A = X2 )
& ( P @ X2 ) ) )
= bot_bot_set_v ) ) ) ).
% Collect_conv_if2
thf(fact_992_Collect__conv__if2,axiom,
! [P: product_prod_v_v > $o,A: product_prod_v_v] :
( ( ( P @ A )
=> ( ( collec140062887454715474od_v_v
@ ^ [X2: product_prod_v_v] :
( ( A = X2 )
& ( P @ X2 ) ) )
= ( insert1338601472111419319od_v_v @ A @ bot_bo723834152578015283od_v_v ) ) )
& ( ~ ( P @ A )
=> ( ( collec140062887454715474od_v_v
@ ^ [X2: product_prod_v_v] :
( ( A = X2 )
& ( P @ X2 ) ) )
= bot_bo723834152578015283od_v_v ) ) ) ).
% Collect_conv_if2
thf(fact_993_Un__def,axiom,
( sup_sup_set_set_v
= ( ^ [A4: set_set_v,B3: set_set_v] :
( collect_set_v
@ ^ [X2: set_v] :
( ( member_set_v @ X2 @ A4 )
| ( member_set_v @ X2 @ B3 ) ) ) ) ) ).
% Un_def
thf(fact_994_Un__def,axiom,
( sup_sup_set_v
= ( ^ [A4: set_v,B3: set_v] :
( collect_v
@ ^ [X2: v] :
( ( member_v @ X2 @ A4 )
| ( member_v @ X2 @ B3 ) ) ) ) ) ).
% Un_def
thf(fact_995_Un__def,axiom,
( sup_su414716646722978715od_v_v
= ( ^ [A4: set_Product_prod_v_v,B3: set_Product_prod_v_v] :
( collec140062887454715474od_v_v
@ ^ [X2: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X2 @ A4 )
| ( member7453568604450474000od_v_v @ X2 @ B3 ) ) ) ) ) ).
% Un_def
thf(fact_996_insert__def,axiom,
( insert_set_v
= ( ^ [A5: set_v] :
( sup_sup_set_set_v
@ ( collect_set_v
@ ^ [X2: set_v] : ( X2 = A5 ) ) ) ) ) ).
% insert_def
thf(fact_997_insert__def,axiom,
( insert_v
= ( ^ [A5: v] :
( sup_sup_set_v
@ ( collect_v
@ ^ [X2: v] : ( X2 = A5 ) ) ) ) ) ).
% insert_def
thf(fact_998_insert__def,axiom,
( insert1338601472111419319od_v_v
= ( ^ [A5: product_prod_v_v] :
( sup_su414716646722978715od_v_v
@ ( collec140062887454715474od_v_v
@ ^ [X2: product_prod_v_v] : ( X2 = A5 ) ) ) ) ) ).
% insert_def
thf(fact_999_sup__set__def,axiom,
( sup_sup_set_set_v
= ( ^ [A4: set_set_v,B3: set_set_v] :
( collect_set_v
@ ( sup_sup_set_v_o
@ ^ [X2: set_v] : ( member_set_v @ X2 @ A4 )
@ ^ [X2: set_v] : ( member_set_v @ X2 @ B3 ) ) ) ) ) ).
% sup_set_def
thf(fact_1000_sup__set__def,axiom,
( sup_sup_set_v
= ( ^ [A4: set_v,B3: set_v] :
( collect_v
@ ( sup_sup_v_o
@ ^ [X2: v] : ( member_v @ X2 @ A4 )
@ ^ [X2: v] : ( member_v @ X2 @ B3 ) ) ) ) ) ).
% sup_set_def
thf(fact_1001_sup__set__def,axiom,
( sup_su414716646722978715od_v_v
= ( ^ [A4: set_Product_prod_v_v,B3: set_Product_prod_v_v] :
( collec140062887454715474od_v_v
@ ( sup_su5941406310530359554_v_v_o
@ ^ [X2: product_prod_v_v] : ( member7453568604450474000od_v_v @ X2 @ A4 )
@ ^ [X2: product_prod_v_v] : ( member7453568604450474000od_v_v @ X2 @ B3 ) ) ) ) ) ).
% sup_set_def
thf(fact_1002_insert__compr,axiom,
( insert_v
= ( ^ [A5: v,B3: set_v] :
( collect_v
@ ^ [X2: v] :
( ( X2 = A5 )
| ( member_v @ X2 @ B3 ) ) ) ) ) ).
% insert_compr
thf(fact_1003_insert__compr,axiom,
( insert1338601472111419319od_v_v
= ( ^ [A5: product_prod_v_v,B3: set_Product_prod_v_v] :
( collec140062887454715474od_v_v
@ ^ [X2: product_prod_v_v] :
( ( X2 = A5 )
| ( member7453568604450474000od_v_v @ X2 @ B3 ) ) ) ) ) ).
% insert_compr
thf(fact_1004_insert__compr,axiom,
( insert_set_v
= ( ^ [A5: set_v,B3: set_set_v] :
( collect_set_v
@ ^ [X2: set_v] :
( ( X2 = A5 )
| ( member_set_v @ X2 @ B3 ) ) ) ) ) ).
% insert_compr
thf(fact_1005_insert__Collect,axiom,
! [A: v,P: v > $o] :
( ( insert_v @ A @ ( collect_v @ P ) )
= ( collect_v
@ ^ [U2: v] :
( ( U2 != A )
=> ( P @ U2 ) ) ) ) ).
% insert_Collect
thf(fact_1006_insert__Collect,axiom,
! [A: product_prod_v_v,P: product_prod_v_v > $o] :
( ( insert1338601472111419319od_v_v @ A @ ( collec140062887454715474od_v_v @ P ) )
= ( collec140062887454715474od_v_v
@ ^ [U2: product_prod_v_v] :
( ( U2 != A )
=> ( P @ U2 ) ) ) ) ).
% insert_Collect
thf(fact_1007_insert__Collect,axiom,
! [A: set_v,P: set_v > $o] :
( ( insert_set_v @ A @ ( collect_set_v @ P ) )
= ( collect_set_v
@ ^ [U2: set_v] :
( ( U2 != A )
=> ( P @ U2 ) ) ) ) ).
% insert_Collect
thf(fact_1008_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_1009_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_1010_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_1011_Collect__subset,axiom,
! [A2: set_set_v,P: set_v > $o] :
( ord_le5216385588623774835_set_v
@ ( collect_set_v
@ ^ [X2: set_v] :
( ( member_set_v @ X2 @ A2 )
& ( P @ X2 ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_1012_Collect__subset,axiom,
! [A2: set_v,P: v > $o] :
( ord_less_eq_set_v
@ ( collect_v
@ ^ [X2: v] :
( ( member_v @ X2 @ A2 )
& ( P @ X2 ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_1013_Collect__subset,axiom,
! [A2: 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 @ A2 )
& ( P @ X2 ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_1014_less__eq__set__def,axiom,
( ord_less_eq_set_v
= ( ^ [A4: set_v,B3: set_v] :
( ord_less_eq_v_o
@ ^ [X2: v] : ( member_v @ X2 @ A4 )
@ ^ [X2: v] : ( member_v @ X2 @ B3 ) ) ) ) ).
% less_eq_set_def
thf(fact_1015_less__eq__set__def,axiom,
( ord_le7336532860387713383od_v_v
= ( ^ [A4: set_Product_prod_v_v,B3: set_Product_prod_v_v] :
( ord_le5892402249245633078_v_v_o
@ ^ [X2: product_prod_v_v] : ( member7453568604450474000od_v_v @ X2 @ A4 )
@ ^ [X2: product_prod_v_v] : ( member7453568604450474000od_v_v @ X2 @ B3 ) ) ) ) ).
% less_eq_set_def
thf(fact_1016_Int__def,axiom,
( inf_in6271465464967711157od_v_v
= ( ^ [A4: set_Product_prod_v_v,B3: set_Product_prod_v_v] :
( collec140062887454715474od_v_v
@ ^ [X2: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X2 @ A4 )
& ( member7453568604450474000od_v_v @ X2 @ B3 ) ) ) ) ) ).
% Int_def
thf(fact_1017_Int__def,axiom,
( inf_inf_set_set_v
= ( ^ [A4: set_set_v,B3: set_set_v] :
( collect_set_v
@ ^ [X2: set_v] :
( ( member_set_v @ X2 @ A4 )
& ( member_set_v @ X2 @ B3 ) ) ) ) ) ).
% Int_def
thf(fact_1018_Int__def,axiom,
( inf_inf_set_v
= ( ^ [A4: set_v,B3: set_v] :
( collect_v
@ ^ [X2: v] :
( ( member_v @ X2 @ A4 )
& ( member_v @ X2 @ B3 ) ) ) ) ) ).
% Int_def
thf(fact_1019_Int__Collect,axiom,
! [X4: product_prod_v_v,A2: set_Product_prod_v_v,P: product_prod_v_v > $o] :
( ( member7453568604450474000od_v_v @ X4 @ ( inf_in6271465464967711157od_v_v @ A2 @ ( collec140062887454715474od_v_v @ P ) ) )
= ( ( member7453568604450474000od_v_v @ X4 @ A2 )
& ( P @ X4 ) ) ) ).
% Int_Collect
thf(fact_1020_Int__Collect,axiom,
! [X4: set_v,A2: set_set_v,P: set_v > $o] :
( ( member_set_v @ X4 @ ( inf_inf_set_set_v @ A2 @ ( collect_set_v @ P ) ) )
= ( ( member_set_v @ X4 @ A2 )
& ( P @ X4 ) ) ) ).
% Int_Collect
thf(fact_1021_Int__Collect,axiom,
! [X4: v,A2: set_v,P: v > $o] :
( ( member_v @ X4 @ ( inf_inf_set_v @ A2 @ ( collect_v @ P ) ) )
= ( ( member_v @ X4 @ A2 )
& ( P @ X4 ) ) ) ).
% Int_Collect
thf(fact_1022_inf__set__def,axiom,
( inf_in6271465464967711157od_v_v
= ( ^ [A4: set_Product_prod_v_v,B3: set_Product_prod_v_v] :
( collec140062887454715474od_v_v
@ ( inf_in6860806757119575912_v_v_o
@ ^ [X2: product_prod_v_v] : ( member7453568604450474000od_v_v @ X2 @ A4 )
@ ^ [X2: product_prod_v_v] : ( member7453568604450474000od_v_v @ X2 @ B3 ) ) ) ) ) ).
% inf_set_def
thf(fact_1023_inf__set__def,axiom,
( inf_inf_set_set_v
= ( ^ [A4: set_set_v,B3: set_set_v] :
( collect_set_v
@ ( inf_inf_set_v_o
@ ^ [X2: set_v] : ( member_set_v @ X2 @ A4 )
@ ^ [X2: set_v] : ( member_set_v @ X2 @ B3 ) ) ) ) ) ).
% inf_set_def
thf(fact_1024_inf__set__def,axiom,
( inf_inf_set_v
= ( ^ [A4: set_v,B3: set_v] :
( collect_v
@ ( inf_inf_v_o
@ ^ [X2: v] : ( member_v @ X2 @ A4 )
@ ^ [X2: v] : ( member_v @ X2 @ B3 ) ) ) ) ) ).
% inf_set_def
thf(fact_1025_set__diff__eq,axiom,
( minus_4183494784930505774od_v_v
= ( ^ [A4: set_Product_prod_v_v,B3: set_Product_prod_v_v] :
( collec140062887454715474od_v_v
@ ^ [X2: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X2 @ A4 )
& ~ ( member7453568604450474000od_v_v @ X2 @ B3 ) ) ) ) ) ).
% set_diff_eq
thf(fact_1026_set__diff__eq,axiom,
( minus_7228012346218142266_set_v
= ( ^ [A4: set_set_v,B3: set_set_v] :
( collect_set_v
@ ^ [X2: set_v] :
( ( member_set_v @ X2 @ A4 )
& ~ ( member_set_v @ X2 @ B3 ) ) ) ) ) ).
% set_diff_eq
thf(fact_1027_set__diff__eq,axiom,
( minus_minus_set_v
= ( ^ [A4: set_v,B3: set_v] :
( collect_v
@ ^ [X2: v] :
( ( member_v @ X2 @ A4 )
& ~ ( member_v @ X2 @ B3 ) ) ) ) ) ).
% set_diff_eq
thf(fact_1028_minus__set__def,axiom,
( minus_4183494784930505774od_v_v
= ( ^ [A4: set_Product_prod_v_v,B3: set_Product_prod_v_v] :
( collec140062887454715474od_v_v
@ ( minus_9095120230875558447_v_v_o
@ ^ [X2: product_prod_v_v] : ( member7453568604450474000od_v_v @ X2 @ A4 )
@ ^ [X2: product_prod_v_v] : ( member7453568604450474000od_v_v @ X2 @ B3 ) ) ) ) ) ).
% minus_set_def
thf(fact_1029_minus__set__def,axiom,
( minus_7228012346218142266_set_v
= ( ^ [A4: set_set_v,B3: set_set_v] :
( collect_set_v
@ ( minus_minus_set_v_o
@ ^ [X2: set_v] : ( member_set_v @ X2 @ A4 )
@ ^ [X2: set_v] : ( member_set_v @ X2 @ B3 ) ) ) ) ) ).
% minus_set_def
thf(fact_1030_minus__set__def,axiom,
( minus_minus_set_v
= ( ^ [A4: set_v,B3: set_v] :
( collect_v
@ ( minus_minus_v_o
@ ^ [X2: v] : ( member_v @ X2 @ A4 )
@ ^ [X2: v] : ( member_v @ X2 @ B3 ) ) ) ) ) ).
% minus_set_def
thf(fact_1031_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_1032_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_1033_empty__def,axiom,
( bot_bot_set_set_v
= ( collect_set_v
@ ^ [X2: set_v] : $false ) ) ).
% empty_def
thf(fact_1034_empty__def,axiom,
( bot_bot_set_v
= ( collect_v
@ ^ [X2: v] : $false ) ) ).
% empty_def
thf(fact_1035_empty__def,axiom,
( bot_bo723834152578015283od_v_v
= ( collec140062887454715474od_v_v
@ ^ [X2: product_prod_v_v] : $false ) ) ).
% empty_def
thf(fact_1036_graph_Ounite__S__equal,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,E: sCC_Bl7326425374436813197t_unit,W: product_prod_v_v,V3: product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( sCC_Bl7798947040364291444t_unit @ Successors @ E )
=> ( ( member7453568604450474000od_v_v @ W @ ( Successors @ V3 ) )
=> ( ~ ( member7453568604450474000od_v_v @ W @ ( sCC_Bl3878977043676959280t_unit @ E @ V3 ) )
=> ( ( member7453568604450474000od_v_v @ W @ ( sCC_Bl5498988629518860705t_unit @ E ) )
=> ( ~ ( member7453568604450474000od_v_v @ W @ ( sCC_Bl5094201334446601350t_unit @ E ) )
=> ( ( comple5788137035815166516od_v_v
@ ( collec8263177866097347122od_v_v
@ ^ [Uu: set_Product_prod_v_v] :
? [N4: product_prod_v_v] :
( ( Uu
= ( sCC_Bl8440648026628373538t_unit @ ( sCC_Bl4702006153222411093od_v_v @ V3 @ W @ E ) @ N4 ) )
& ( member7453568604450474000od_v_v @ N4 @ ( set_Product_prod_v_v2 @ ( sCC_Bl2021302119412358655t_unit @ ( sCC_Bl4702006153222411093od_v_v @ V3 @ W @ E ) ) ) ) ) ) )
= ( comple5788137035815166516od_v_v
@ ( collec8263177866097347122od_v_v
@ ^ [Uu: set_Product_prod_v_v] :
? [N4: product_prod_v_v] :
( ( Uu
= ( sCC_Bl8440648026628373538t_unit @ E @ N4 ) )
& ( member7453568604450474000od_v_v @ N4 @ ( set_Product_prod_v_v2 @ ( sCC_Bl2021302119412358655t_unit @ E ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% graph.unite_S_equal
thf(fact_1037_graph_Ounite__S__equal,axiom,
! [Vertices: set_v,Successors: v > set_v,E: sCC_Bl1394983891496994913t_unit,W: v,V3: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl9196236973127232072t_unit @ Successors @ E )
=> ( ( member_v @ W @ ( Successors @ V3 ) )
=> ( ~ ( member_v @ W @ ( sCC_Bl3795065053823578884t_unit @ E @ V3 ) )
=> ( ( member_v @ W @ ( sCC_Bl4645233313691564917t_unit @ E ) )
=> ( ~ ( member_v @ W @ ( sCC_Bl157864678168468314t_unit @ E ) )
=> ( ( comple2307003700295860064_set_v
@ ( collect_set_v
@ ^ [Uu: set_v] :
? [N4: v] :
( ( Uu
= ( sCC_Bl1280885523602775798t_unit @ ( sCC_Bloemen_unite_v @ V3 @ W @ E ) @ N4 ) )
& ( member_v @ N4 @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ ( sCC_Bloemen_unite_v @ V3 @ W @ E ) ) ) ) ) ) )
= ( comple2307003700295860064_set_v
@ ( collect_set_v
@ ^ [Uu: set_v] :
? [N4: v] :
( ( Uu
= ( sCC_Bl1280885523602775798t_unit @ E @ N4 ) )
& ( member_v @ N4 @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% graph.unite_S_equal
thf(fact_1038_graph_Osub__env__def,axiom,
! [Vertices: set_v,Successors: v > set_v,E: sCC_Bl1394983891496994913t_unit,E2: sCC_Bl1394983891496994913t_unit] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl5768913643336123637t_unit @ E @ E2 )
= ( ( ( sCC_Bl1090238580953940555t_unit @ E2 )
= ( sCC_Bl1090238580953940555t_unit @ E ) )
& ( ord_less_eq_set_v @ ( sCC_Bl4645233313691564917t_unit @ E ) @ ( sCC_Bl4645233313691564917t_unit @ E2 ) )
& ( ord_less_eq_set_v @ ( sCC_Bl157864678168468314t_unit @ E ) @ ( sCC_Bl157864678168468314t_unit @ E2 ) )
& ! [V4: v] : ( ord_less_eq_set_v @ ( sCC_Bl3795065053823578884t_unit @ E @ V4 ) @ ( sCC_Bl3795065053823578884t_unit @ E2 @ V4 ) )
& ! [V4: v] : ( ord_less_eq_set_v @ ( sCC_Bl1280885523602775798t_unit @ E @ V4 ) @ ( sCC_Bl1280885523602775798t_unit @ E2 @ V4 ) )
& ( ord_less_eq_set_v
@ ( comple2307003700295860064_set_v
@ ( collect_set_v
@ ^ [Uu: set_v] :
? [V4: v] :
( ( Uu
= ( sCC_Bl1280885523602775798t_unit @ E @ V4 ) )
& ( member_v @ V4 @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E ) ) ) ) ) )
@ ( comple2307003700295860064_set_v
@ ( collect_set_v
@ ^ [Uu: set_v] :
? [V4: v] :
( ( Uu
= ( sCC_Bl1280885523602775798t_unit @ E2 @ V4 ) )
& ( member_v @ V4 @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E2 ) ) ) ) ) ) ) ) ) ) ).
% graph.sub_env_def
thf(fact_1039_ra__add__edge,axiom,
! [X4: v,Y: v,E6: set_Product_prod_v_v,V3: v,W: v] :
( ( sCC_Bl4291963740693775144ding_v @ successors @ X4 @ Y @ E6 )
=> ( ( sCC_Bl4291963740693775144ding_v @ successors @ X4 @ Y @ ( sup_su414716646722978715od_v_v @ E6 @ ( insert1338601472111419319od_v_v @ ( product_Pair_v_v @ V3 @ W ) @ bot_bo723834152578015283od_v_v ) ) )
| ( ( sCC_Bl4291963740693775144ding_v @ successors @ X4 @ V3 @ ( sup_su414716646722978715od_v_v @ E6 @ ( insert1338601472111419319od_v_v @ ( product_Pair_v_v @ V3 @ W ) @ bot_bo723834152578015283od_v_v ) ) )
& ( sCC_Bl4291963740693775144ding_v @ successors @ W @ Y @ ( sup_su414716646722978715od_v_v @ E6 @ ( insert1338601472111419319od_v_v @ ( product_Pair_v_v @ V3 @ W ) @ bot_bo723834152578015283od_v_v ) ) ) ) ) ) ).
% ra_add_edge
thf(fact_1040_ra__cases,axiom,
! [X4: v,Y: v,E6: set_Product_prod_v_v] :
( ( sCC_Bl4291963740693775144ding_v @ successors @ X4 @ Y @ E6 )
=> ( ( X4 = Y )
| ? [Z3: v] :
( ( member_v @ Z3 @ ( successors @ X4 ) )
& ~ ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ X4 @ Z3 ) @ E6 )
& ( sCC_Bl4291963740693775144ding_v @ successors @ Z3 @ Y @ E6 ) ) ) ) ).
% ra_cases
thf(fact_1041_ra__mono,axiom,
! [X4: v,Y: v,E6: set_Product_prod_v_v,E7: set_Product_prod_v_v] :
( ( sCC_Bl4291963740693775144ding_v @ successors @ X4 @ Y @ E6 )
=> ( ( ord_le7336532860387713383od_v_v @ E7 @ E6 )
=> ( sCC_Bl4291963740693775144ding_v @ successors @ X4 @ Y @ E7 ) ) ) ).
% ra_mono
thf(fact_1042_ra__trans,axiom,
! [X4: v,Y: v,E6: set_Product_prod_v_v,Z: v] :
( ( sCC_Bl4291963740693775144ding_v @ successors @ X4 @ Y @ E6 )
=> ( ( sCC_Bl4291963740693775144ding_v @ successors @ Y @ Z @ E6 )
=> ( sCC_Bl4291963740693775144ding_v @ successors @ X4 @ Z @ E6 ) ) ) ).
% ra_trans
thf(fact_1043_ra__refl,axiom,
! [X4: v,E6: set_Product_prod_v_v] : ( sCC_Bl4291963740693775144ding_v @ successors @ X4 @ X4 @ E6 ) ).
% ra_refl
thf(fact_1044_ra__reachable,axiom,
! [X4: v,Y: v,E6: set_Product_prod_v_v] :
( ( sCC_Bl4291963740693775144ding_v @ successors @ X4 @ Y @ E6 )
=> ( sCC_Bl649662514949026229able_v @ successors @ X4 @ Y ) ) ).
% ra_reachable
thf(fact_1045_reachable__avoiding_Ocases,axiom,
! [A1: v,A22: v,A32: set_Product_prod_v_v] :
( ( sCC_Bl4291963740693775144ding_v @ successors @ A1 @ A22 @ A32 )
=> ( ( A22 != A1 )
=> ~ ! [Y2: v] :
( ( sCC_Bl4291963740693775144ding_v @ successors @ A1 @ Y2 @ A32 )
=> ( ( member_v @ A22 @ ( successors @ Y2 ) )
=> ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ Y2 @ A22 ) @ A32 ) ) ) ) ) ).
% reachable_avoiding.cases
thf(fact_1046_ra__succ,axiom,
! [X4: v,Y: v,E6: set_Product_prod_v_v,Z: v] :
( ( sCC_Bl4291963740693775144ding_v @ successors @ X4 @ Y @ E6 )
=> ( ( member_v @ Z @ ( successors @ Y ) )
=> ( ~ ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ Y @ Z ) @ E6 )
=> ( sCC_Bl4291963740693775144ding_v @ successors @ X4 @ Z @ E6 ) ) ) ) ).
% ra_succ
thf(fact_1047_reachable__avoiding_Osimps,axiom,
! [A1: v,A22: v,A32: set_Product_prod_v_v] :
( ( sCC_Bl4291963740693775144ding_v @ successors @ A1 @ A22 @ A32 )
= ( ? [X2: v,E8: set_Product_prod_v_v] :
( ( A1 = X2 )
& ( A22 = X2 )
& ( A32 = E8 ) )
| ? [X2: v,Y3: v,E8: set_Product_prod_v_v,Z2: v] :
( ( A1 = X2 )
& ( A22 = Z2 )
& ( A32 = E8 )
& ( sCC_Bl4291963740693775144ding_v @ successors @ X2 @ Y3 @ E8 )
& ( member_v @ Z2 @ ( successors @ Y3 ) )
& ~ ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ Y3 @ Z2 ) @ E8 ) ) ) ) ).
% reachable_avoiding.simps
thf(fact_1048_edge__ra,axiom,
! [Y: v,X4: v,E6: set_Product_prod_v_v] :
( ( member_v @ Y @ ( successors @ X4 ) )
=> ( ~ ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ X4 @ Y ) @ E6 )
=> ( sCC_Bl4291963740693775144ding_v @ successors @ X4 @ Y @ E6 ) ) ) ).
% edge_ra
thf(fact_1049_ra__empty,axiom,
! [X4: v,Y: v] :
( ( sCC_Bl4291963740693775144ding_v @ successors @ X4 @ Y @ bot_bo723834152578015283od_v_v )
= ( sCC_Bl649662514949026229able_v @ successors @ X4 @ Y ) ) ).
% ra_empty
thf(fact_1050_vfin,axiom,
finite_finite_v @ vertices ).
% vfin
thf(fact_1051_unite__def,axiom,
( sCC_Bloemen_unite_v
= ( ^ [V4: v,W2: v,E4: sCC_Bl1394983891496994913t_unit] :
( sCC_Bl349061681862590396t_unit
@ ^ [Uu: list_v] :
( dropWhile_v
@ ^ [X2: v] :
~ ( member_v @ W2 @ ( sCC_Bl1280885523602775798t_unit @ E4 @ X2 ) )
@ ( sCC_Bl8828226123343373779t_unit @ E4 ) )
@ ( 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 @ E4 @ Y3 ) )
& ( member_v @ Y3
@ ( sup_sup_set_v
@ ( set_v2
@ ( takeWhile_v
@ ^ [Z2: v] :
~ ( member_v @ W2 @ ( sCC_Bl1280885523602775798t_unit @ E4 @ Z2 ) )
@ ( sCC_Bl8828226123343373779t_unit @ E4 ) ) )
@ ( insert_v
@ ( hd_v
@ ( dropWhile_v
@ ^ [Z2: v] :
~ ( member_v @ W2 @ ( sCC_Bl1280885523602775798t_unit @ E4 @ Z2 ) )
@ ( sCC_Bl8828226123343373779t_unit @ E4 ) ) )
@ bot_bot_set_v ) ) ) ) ) ) )
@ ( comple2307003700295860064_set_v
@ ( collect_set_v
@ ^ [Uv: set_v] :
? [Y3: v] :
( ( Uv
= ( sCC_Bl1280885523602775798t_unit @ E4 @ Y3 ) )
& ( member_v @ Y3
@ ( sup_sup_set_v
@ ( set_v2
@ ( takeWhile_v
@ ^ [Z2: v] :
~ ( member_v @ W2 @ ( sCC_Bl1280885523602775798t_unit @ E4 @ Z2 ) )
@ ( sCC_Bl8828226123343373779t_unit @ E4 ) ) )
@ ( insert_v
@ ( hd_v
@ ( dropWhile_v
@ ^ [Z2: v] :
~ ( member_v @ W2 @ ( sCC_Bl1280885523602775798t_unit @ E4 @ Z2 ) )
@ ( sCC_Bl8828226123343373779t_unit @ E4 ) ) )
@ bot_bot_set_v ) ) ) ) ) )
@ ( sCC_Bl1280885523602775798t_unit @ E4 @ X2 ) )
@ E4 ) ) ) ) ).
% unite_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,
! [X4: set_v,Y: set_v] :
( ( if_set_v @ $false @ X4 @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Set__Oset_Itf__v_J_T,axiom,
! [X4: set_v,Y: set_v] :
( ( if_set_v @ $true @ X4 @ Y )
= X4 ) ).
% Conjectures (1)
thf(conj_0,conjecture,
( ( sCC_Bl1280885523602775798t_unit @ e2 @ n )
= ( sCC_Bl1280885523602775798t_unit @ e2 @ m ) ) ).
%------------------------------------------------------------------------------