TPTP Problem File: SLH0599^1.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_03025_104369__6587286_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1471 ( 703 unt; 187 typ; 0 def)
% Number of atoms : 3486 (1478 equ; 0 cnn)
% Maximal formula atoms : 13 ( 2 avg)
% Number of connectives : 12214 ( 479 ~; 64 |; 311 &;9941 @)
% ( 0 <=>;1419 =>; 0 <=; 0 <~>)
% Maximal formula depth : 23 ( 6 avg)
% Number of types : 23 ( 22 usr)
% Number of type conns : 553 ( 553 >; 0 *; 0 +; 0 <<)
% Number of symbols : 168 ( 165 usr; 20 con; 0-9 aty)
% Number of variables : 3662 ( 231 ^;3278 !; 153 ?;3662 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 08:54:24.171
%------------------------------------------------------------------------------
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thf(sy_c_SCC__Bloemen__Sequential_Ograph_001tf__v,type,
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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 ).
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thf(sy_c_Set_Oinsert_001t__Set__Oset_It__Set__Oset_Itf__v_J_J,type,
insert_set_set_v: set_set_v > set_set_set_v > set_set_set_v ).
thf(sy_c_Set_Oinsert_001t__Set__Oset_Itf__v_J,type,
insert_set_v: set_v > set_set_v > set_set_v ).
thf(sy_c_Set_Oinsert_001tf__v,type,
insert_v: v > set_v > set_v ).
thf(sy_c_Set_Ois__singleton_001t__Product____Type__Oprod_Itf__v_Mtf__v_J,type,
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thf(sy_c_Set_Ois__singleton_001t__Set__Oset_Itf__v_J,type,
is_singleton_set_v: set_set_v > $o ).
thf(sy_c_Set_Ois__singleton_001tf__v,type,
is_singleton_v: set_v > $o ).
thf(sy_c_Set_Oremove_001t__Product____Type__Oprod_Itf__v_Mtf__v_J,type,
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thf(sy_c_Set_Oremove_001t__Set__Oset_Itf__v_J,type,
remove_set_v: set_v > set_set_v > set_set_v ).
thf(sy_c_Set_Oremove_001tf__v,type,
remove_v: v > set_v > set_v ).
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the_el5392834299063928540od_v_v: set_Product_prod_v_v > product_prod_v_v ).
thf(sy_c_Set_Othe__elem_001t__Set__Oset_Itf__v_J,type,
the_elem_set_v: set_set_v > set_v ).
thf(sy_c_Set_Othe__elem_001tf__v,type,
the_elem_v: set_v > v ).
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thf(sy_c_fChoice_001tf__v,type,
fChoice_v: ( v > $o ) > v ).
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thf(sy_c_member_001t__Set__Oset_It__Set__Oset_Itf__v_J_J,type,
member_set_set_v: set_set_v > set_set_set_v > $o ).
thf(sy_c_member_001t__Set__Oset_Itf__v_J,type,
member_set_v: set_v > set_set_v > $o ).
thf(sy_c_member_001tf__v,type,
member_v: v > set_v > $o ).
thf(sy_v_e_H_H____,type,
e: sCC_Bl1394983891496994913t_unit ).
thf(sy_v_e_H____,type,
e2: sCC_Bl1394983891496994913t_unit ).
thf(sy_v_ea____,type,
ea: sCC_Bl1394983891496994913t_unit ).
thf(sy_v_pfx____,type,
pfx: list_v ).
thf(sy_v_successors,type,
successors: v > set_v ).
thf(sy_v_va____,type,
va: v ).
thf(sy_v_vertices,type,
vertices: set_v ).
thf(sy_v_w____,type,
w: v ).
% Relevant facts (1277)
thf(fact_0_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_1_dfs__dfss__rel_Ocong,axiom,
sCC_Bl907557413677168252_rel_v = sCC_Bl907557413677168252_rel_v ).
% dfs_dfss_rel.cong
thf(fact_2_post_H_H,axiom,
sCC_Bl6082031138996704384t_unit @ successors @ va @ e @ ( sCC_Bloemen_dfss_v @ successors @ va @ e ) ).
% post''
thf(fact_3__092_060open_062sub__env_Ae_A_Idfss_Av_Ae_H_H_J_092_060close_062,axiom,
sCC_Bl5768913643336123637t_unit @ ea @ ( sCC_Bloemen_dfss_v @ successors @ va @ e ) ).
% \<open>sub_env e (dfss v e'')\<close>
thf(fact_4__092_060open_062cstack_A_Idfss_Av_Ae_H_H_J_A_061_Acstack_Ae_092_060close_062,axiom,
( ( sCC_Bl9201514103433284750t_unit @ ( sCC_Bloemen_dfss_v @ successors @ va @ e ) )
= ( sCC_Bl9201514103433284750t_unit @ ea ) ) ).
% \<open>cstack (dfss v e'') = cstack e\<close>
thf(fact_5__092_060open_062_092_060exists_062ns_O_Astack_Ae_A_061_Ans_A_064_Astack_A_Idfss_Av_Ae_H_H_J_092_060close_062,axiom,
? [Ns: list_v] :
( ( sCC_Bl8828226123343373779t_unit @ ea )
= ( append_v @ Ns @ ( sCC_Bl8828226123343373779t_unit @ ( sCC_Bloemen_dfss_v @ successors @ va @ e ) ) ) ) ).
% \<open>\<exists>ns. stack e = ns @ stack (dfss v e'')\<close>
thf(fact_6__092_060open_062v_A_092_060in_062_Avisited_Ae_092_060close_062,axiom,
member_v @ va @ ( sCC_Bl4645233313691564917t_unit @ ea ) ).
% \<open>v \<in> visited e\<close>
thf(fact_7_vs__case,axiom,
( ( minus_minus_set_v @ ( successors @ va ) @ ( sCC_Bl3795065053823578884t_unit @ ea @ va ) )
!= bot_bot_set_v ) ).
% vs_case
thf(fact_8_dfss,axiom,
( ( sCC_Bloemen_dfss_v @ successors @ va @ ea )
= ( sCC_Bloemen_dfss_v @ successors @ va @ e ) ) ).
% dfss
thf(fact_9__092_060open_062_092_060And_062n_O_An_A_092_060in_062_Aset_A_Itl_A_Istack_A_Idfss_Av_Ae_H_H_J_J_J_A_092_060Longrightarrow_062_A_092_060S_062_A_Idfss_Av_Ae_H_H_J_An_A_061_A_092_060S_062_Ae_An_092_060close_062,axiom,
! [N: v] :
( ( member_v @ N @ ( set_v2 @ ( tl_v @ ( sCC_Bl8828226123343373779t_unit @ ( sCC_Bloemen_dfss_v @ successors @ va @ e ) ) ) ) )
=> ( ( sCC_Bl1280885523602775798t_unit @ ( sCC_Bloemen_dfss_v @ successors @ va @ e ) @ N )
= ( sCC_Bl1280885523602775798t_unit @ ea @ N ) ) ) ).
% \<open>\<And>n. n \<in> set (tl (stack (dfss v e''))) \<Longrightarrow> \<S> (dfss v e'') n = \<S> e n\<close>
thf(fact_10_predfss,axiom,
sCC_Bl1748261141445803503t_unit @ successors @ va @ ea ).
% predfss
thf(fact_11__092_060open_062_092_060forall_062u_092_060in_062visited_Ae_A_N_A_123v_125_O_Avsuccs_A_Idfss_Av_Ae_H_H_J_Au_A_061_Avsuccs_Ae_Au_092_060close_062,axiom,
! [X: v] :
( ( member_v @ X @ ( minus_minus_set_v @ ( sCC_Bl4645233313691564917t_unit @ ea ) @ ( insert_v @ va @ bot_bot_set_v ) ) )
=> ( ( sCC_Bl3795065053823578884t_unit @ ( sCC_Bloemen_dfss_v @ successors @ va @ e ) @ X )
= ( sCC_Bl3795065053823578884t_unit @ ea @ X ) ) ) ).
% \<open>\<forall>u\<in>visited e - {v}. vsuccs (dfss v e'') u = vsuccs e u\<close>
thf(fact_12_False,axiom,
member_v @ w @ ( sCC_Bl4645233313691564917t_unit @ ea ) ).
% False
thf(fact_13_graph_Odfss_Ocong,axiom,
sCC_Bloemen_dfss_v = sCC_Bloemen_dfss_v ).
% graph.dfss.cong
thf(fact_14_graph_Opost__dfss_Ocong,axiom,
sCC_Bl6082031138996704384t_unit = sCC_Bl6082031138996704384t_unit ).
% graph.post_dfss.cong
thf(fact_15_local_Owf,axiom,
sCC_Bl9196236973127232072t_unit @ successors @ ea ).
% local.wf
thf(fact_16__092_060open_062v_A_092_060notin_062_Aexplored_Ae_092_060close_062,axiom,
~ ( member_v @ va @ ( sCC_Bl157864678168468314t_unit @ ea ) ) ).
% \<open>v \<notin> explored e\<close>
thf(fact_17_reachable__end_Ocases,axiom,
! [A1: v,A2: v] :
( ( sCC_Bl770211535891879572_end_v @ successors @ A1 @ A2 )
=> ( ( A2 != A1 )
=> ~ ! [Y: v] :
( ( sCC_Bl770211535891879572_end_v @ successors @ A1 @ Y )
=> ~ ( member_v @ A2 @ ( successors @ Y ) ) ) ) ) ).
% reachable_end.cases
thf(fact_18_re__refl,axiom,
! [X2: v] : ( sCC_Bl770211535891879572_end_v @ successors @ X2 @ X2 ) ).
% re_refl
thf(fact_19_re__succ,axiom,
! [X2: v,Y2: v,Z: v] :
( ( sCC_Bl770211535891879572_end_v @ successors @ X2 @ Y2 )
=> ( ( member_v @ Z @ ( successors @ Y2 ) )
=> ( sCC_Bl770211535891879572_end_v @ successors @ X2 @ Z ) ) ) ).
% re_succ
thf(fact_20_reachable__end_Osimps,axiom,
! [A1: v,A2: v] :
( ( sCC_Bl770211535891879572_end_v @ successors @ A1 @ A2 )
= ( ? [X3: v] :
( ( A1 = X3 )
& ( A2 = X3 ) )
| ? [X3: v,Y3: v,Z2: v] :
( ( A1 = X3 )
& ( A2 = Z2 )
& ( sCC_Bl770211535891879572_end_v @ successors @ X3 @ Y3 )
& ( member_v @ Z2 @ ( successors @ Y3 ) ) ) ) ) ).
% reachable_end.simps
thf(fact_21_succ__re,axiom,
! [Y2: v,X2: v,Z: v] :
( ( member_v @ Y2 @ ( successors @ X2 ) )
=> ( ( sCC_Bl770211535891879572_end_v @ successors @ Y2 @ Z )
=> ( sCC_Bl770211535891879572_end_v @ successors @ X2 @ Z ) ) ) ).
% succ_re
thf(fact_22__092_060open_062pre__dfss_Av_Ae_H_H_092_060close_062,axiom,
sCC_Bl1748261141445803503t_unit @ successors @ va @ e ).
% \<open>pre_dfss v e''\<close>
thf(fact_23_wvs,axiom,
member_v @ w @ ( minus_minus_set_v @ ( successors @ va ) @ ( sCC_Bl3795065053823578884t_unit @ ea @ va ) ) ).
% wvs
thf(fact_24_pfx_I1_J,axiom,
( ( sCC_Bl8828226123343373779t_unit @ ea )
= ( append_v @ pfx @ ( sCC_Bl8828226123343373779t_unit @ e2 ) ) ) ).
% pfx(1)
thf(fact_25_init__env__pre__dfs,axiom,
! [V: v] : ( sCC_Bl36166008131615352t_unit @ successors @ V @ ( sCC_Bl7693227186847904995_env_v @ V ) ) ).
% init_env_pre_dfs
thf(fact_26_notexplored,axiom,
~ ( member_v @ w @ ( sCC_Bl157864678168468314t_unit @ ea ) ) ).
% notexplored
thf(fact_27_S__reflexive,axiom,
! [E: sCC_Bl1394983891496994913t_unit,N: v] :
( ( sCC_Bl9196236973127232072t_unit @ successors @ E )
=> ( member_v @ N @ ( sCC_Bl1280885523602775798t_unit @ E @ N ) ) ) ).
% S_reflexive
thf(fact_28__092_060open_062w_A_092_060in_062_Avisited_Ae_092_060close_062,axiom,
member_v @ w @ ( sCC_Bl4645233313691564917t_unit @ ea ) ).
% \<open>w \<in> visited e\<close>
thf(fact_29_pre__dfss__pre__dfs,axiom,
! [V: v,E: sCC_Bl1394983891496994913t_unit,W: v] :
( ( sCC_Bl1748261141445803503t_unit @ successors @ V @ E )
=> ( ~ ( member_v @ W @ ( sCC_Bl4645233313691564917t_unit @ E ) )
=> ( ( member_v @ W @ ( successors @ V ) )
=> ( sCC_Bl36166008131615352t_unit @ successors @ W @ E ) ) ) ) ).
% pre_dfss_pre_dfs
thf(fact_30_stack__visited,axiom,
! [E: sCC_Bl1394983891496994913t_unit,N: v] :
( ( sCC_Bl9196236973127232072t_unit @ successors @ E )
=> ( ( member_v @ N @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E ) ) )
=> ( member_v @ N @ ( sCC_Bl4645233313691564917t_unit @ E ) ) ) ) ).
% stack_visited
thf(fact_31_stack__unexplored,axiom,
! [E: sCC_Bl1394983891496994913t_unit,N: v] :
( ( sCC_Bl9196236973127232072t_unit @ successors @ E )
=> ( ( member_v @ N @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E ) ) )
=> ~ ( member_v @ N @ ( sCC_Bl157864678168468314t_unit @ E ) ) ) ) ).
% stack_unexplored
thf(fact_32_visited__unexplored,axiom,
! [E: sCC_Bl1394983891496994913t_unit,M: v] :
( ( sCC_Bl9196236973127232072t_unit @ successors @ E )
=> ( ( member_v @ M @ ( sCC_Bl4645233313691564917t_unit @ E ) )
=> ( ~ ( member_v @ M @ ( sCC_Bl157864678168468314t_unit @ E ) )
=> ~ ! [N2: v] :
( ( member_v @ N2 @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E ) ) )
=> ~ ( member_v @ M @ ( sCC_Bl1280885523602775798t_unit @ E @ N2 ) ) ) ) ) ) ).
% visited_unexplored
thf(fact_33_stack__class,axiom,
! [E: sCC_Bl1394983891496994913t_unit,N: v,M: v] :
( ( sCC_Bl9196236973127232072t_unit @ successors @ E )
=> ( ( member_v @ N @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E ) ) )
=> ( ( member_v @ M @ ( sCC_Bl1280885523602775798t_unit @ E @ N ) )
=> ( member_v @ M @ ( minus_minus_set_v @ ( sCC_Bl4645233313691564917t_unit @ E ) @ ( sCC_Bl157864678168468314t_unit @ E ) ) ) ) ) ) ).
% stack_class
thf(fact_34_pfx_I2_J,axiom,
( ( sCC_Bl8828226123343373779t_unit @ e2 )
!= nil_v ) ).
% pfx(2)
thf(fact_35_graph_Owf__env_Ocong,axiom,
sCC_Bl9196236973127232072t_unit = sCC_Bl9196236973127232072t_unit ).
% graph.wf_env.cong
thf(fact_36_graph_Oreachable__end_Ocong,axiom,
sCC_Bl770211535891879572_end_v = sCC_Bl770211535891879572_end_v ).
% graph.reachable_end.cong
thf(fact_37_graph_Opre__dfss_Ocong,axiom,
sCC_Bl1748261141445803503t_unit = sCC_Bl1748261141445803503t_unit ).
% graph.pre_dfss.cong
thf(fact_38_graph_Opre__dfs_Ocong,axiom,
sCC_Bl36166008131615352t_unit = sCC_Bl36166008131615352t_unit ).
% graph.pre_dfs.cong
thf(fact_39_unite__S__tl,axiom,
! [E: sCC_Bl1394983891496994913t_unit,W: v,V: v,N: v] :
( ( sCC_Bl9196236973127232072t_unit @ successors @ E )
=> ( ( member_v @ W @ ( successors @ V ) )
=> ( ~ ( member_v @ W @ ( sCC_Bl3795065053823578884t_unit @ E @ V ) )
=> ( ( member_v @ W @ ( sCC_Bl4645233313691564917t_unit @ E ) )
=> ( ~ ( member_v @ W @ ( sCC_Bl157864678168468314t_unit @ E ) )
=> ( ( member_v @ N @ ( set_v2 @ ( tl_v @ ( sCC_Bl8828226123343373779t_unit @ ( sCC_Bloemen_unite_v @ V @ W @ E ) ) ) ) )
=> ( ( sCC_Bl1280885523602775798t_unit @ ( sCC_Bloemen_unite_v @ V @ W @ E ) @ N )
= ( sCC_Bl1280885523602775798t_unit @ E @ N ) ) ) ) ) ) ) ) ).
% unite_S_tl
thf(fact_40_unite__sub__env,axiom,
! [V: v,E: sCC_Bl1394983891496994913t_unit,W: v] :
( ( sCC_Bl1748261141445803503t_unit @ successors @ V @ E )
=> ( ( member_v @ W @ ( successors @ V ) )
=> ( ~ ( member_v @ W @ ( sCC_Bl3795065053823578884t_unit @ E @ V ) )
=> ( ( member_v @ W @ ( sCC_Bl4645233313691564917t_unit @ E ) )
=> ( ~ ( member_v @ W @ ( sCC_Bl157864678168468314t_unit @ E ) )
=> ( sCC_Bl5768913643336123637t_unit @ E @ ( sCC_Bloemen_unite_v @ V @ W @ E ) ) ) ) ) ) ) ).
% unite_sub_env
thf(fact_41_unite__wf__env,axiom,
! [V: v,E: sCC_Bl1394983891496994913t_unit,W: v] :
( ( sCC_Bl1748261141445803503t_unit @ successors @ V @ E )
=> ( ( member_v @ W @ ( successors @ V ) )
=> ( ~ ( member_v @ W @ ( sCC_Bl3795065053823578884t_unit @ E @ V ) )
=> ( ( member_v @ W @ ( sCC_Bl4645233313691564917t_unit @ E ) )
=> ( ~ ( member_v @ W @ ( sCC_Bl157864678168468314t_unit @ E ) )
=> ( sCC_Bl9196236973127232072t_unit @ successors @ ( sCC_Bloemen_unite_v @ V @ W @ E ) ) ) ) ) ) ) ).
% unite_wf_env
thf(fact_42_e_H,axiom,
( e2
= ( sCC_Bloemen_unite_v @ va @ w @ ea ) ) ).
% e'
thf(fact_43_mem__Collect__eq,axiom,
! [A: v,P: v > $o] :
( ( member_v @ A @ ( collect_v @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_44_mem__Collect__eq,axiom,
! [A: product_prod_v_v,P: product_prod_v_v > $o] :
( ( member7453568604450474000od_v_v @ A @ ( collec140062887454715474od_v_v @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_45_mem__Collect__eq,axiom,
! [A: set_v,P: set_v > $o] :
( ( member_set_v @ A @ ( collect_set_v @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_46_Collect__mem__eq,axiom,
! [A3: set_v] :
( ( collect_v
@ ^ [X3: v] : ( member_v @ X3 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_47_Collect__mem__eq,axiom,
! [A3: set_Product_prod_v_v] :
( ( collec140062887454715474od_v_v
@ ^ [X3: product_prod_v_v] : ( member7453568604450474000od_v_v @ X3 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_48_Collect__mem__eq,axiom,
! [A3: set_set_v] :
( ( collect_set_v
@ ^ [X3: set_v] : ( member_set_v @ X3 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_49_Collect__cong,axiom,
! [P: set_v > $o,Q: set_v > $o] :
( ! [X4: set_v] :
( ( P @ X4 )
= ( Q @ X4 ) )
=> ( ( collect_set_v @ P )
= ( collect_set_v @ Q ) ) ) ).
% Collect_cong
thf(fact_50_insert__Diff__single,axiom,
! [A: product_prod_v_v,A3: set_Product_prod_v_v] :
( ( insert1338601472111419319od_v_v @ A @ ( minus_4183494784930505774od_v_v @ A3 @ ( insert1338601472111419319od_v_v @ A @ bot_bo723834152578015283od_v_v ) ) )
= ( insert1338601472111419319od_v_v @ A @ A3 ) ) ).
% insert_Diff_single
thf(fact_51_insert__Diff__single,axiom,
! [A: set_v,A3: set_set_v] :
( ( insert_set_v @ A @ ( minus_7228012346218142266_set_v @ A3 @ ( insert_set_v @ A @ bot_bot_set_set_v ) ) )
= ( insert_set_v @ A @ A3 ) ) ).
% insert_Diff_single
thf(fact_52_insert__Diff__single,axiom,
! [A: v,A3: set_v] :
( ( insert_v @ A @ ( minus_minus_set_v @ A3 @ ( insert_v @ A @ bot_bot_set_v ) ) )
= ( insert_v @ A @ A3 ) ) ).
% insert_Diff_single
thf(fact_53__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_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ( ? [Pfx: list_v] :
( ( sCC_Bl8828226123343373779t_unit @ ea )
= ( append_v @ Pfx @ ( sCC_Bl8828226123343373779t_unit @ e2 ) ) )
=> ( ( sCC_Bl8828226123343373779t_unit @ e2 )
= nil_v ) ) ).
% \<open>\<And>thesis. (\<And>pfx. \<lbrakk>stack e = pfx @ stack e'; stack e' \<noteq> []\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_54_Diff__insert0,axiom,
! [X2: set_v,A3: set_set_v,B: set_set_v] :
( ~ ( member_set_v @ X2 @ A3 )
=> ( ( minus_7228012346218142266_set_v @ A3 @ ( insert_set_v @ X2 @ B ) )
= ( minus_7228012346218142266_set_v @ A3 @ B ) ) ) ).
% Diff_insert0
thf(fact_55_Diff__insert0,axiom,
! [X2: product_prod_v_v,A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ~ ( member7453568604450474000od_v_v @ X2 @ A3 )
=> ( ( minus_4183494784930505774od_v_v @ A3 @ ( insert1338601472111419319od_v_v @ X2 @ B ) )
= ( minus_4183494784930505774od_v_v @ A3 @ B ) ) ) ).
% Diff_insert0
thf(fact_56_Diff__insert0,axiom,
! [X2: v,A3: set_v,B: set_v] :
( ~ ( member_v @ X2 @ A3 )
=> ( ( minus_minus_set_v @ A3 @ ( insert_v @ X2 @ B ) )
= ( minus_minus_set_v @ A3 @ B ) ) ) ).
% Diff_insert0
thf(fact_57_insert__Diff1,axiom,
! [X2: set_v,B: set_set_v,A3: set_set_v] :
( ( member_set_v @ X2 @ B )
=> ( ( minus_7228012346218142266_set_v @ ( insert_set_v @ X2 @ A3 ) @ B )
= ( minus_7228012346218142266_set_v @ A3 @ B ) ) ) ).
% insert_Diff1
thf(fact_58_insert__Diff1,axiom,
! [X2: product_prod_v_v,B: set_Product_prod_v_v,A3: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X2 @ B )
=> ( ( minus_4183494784930505774od_v_v @ ( insert1338601472111419319od_v_v @ X2 @ A3 ) @ B )
= ( minus_4183494784930505774od_v_v @ A3 @ B ) ) ) ).
% insert_Diff1
thf(fact_59_insert__Diff1,axiom,
! [X2: v,B: set_v,A3: set_v] :
( ( member_v @ X2 @ B )
=> ( ( minus_minus_set_v @ ( insert_v @ X2 @ A3 ) @ B )
= ( minus_minus_set_v @ A3 @ B ) ) ) ).
% insert_Diff1
thf(fact_60_Diff__empty,axiom,
! [A3: set_Product_prod_v_v] :
( ( minus_4183494784930505774od_v_v @ A3 @ bot_bo723834152578015283od_v_v )
= A3 ) ).
% Diff_empty
thf(fact_61_Diff__empty,axiom,
! [A3: set_set_v] :
( ( minus_7228012346218142266_set_v @ A3 @ bot_bot_set_set_v )
= A3 ) ).
% Diff_empty
thf(fact_62_Diff__empty,axiom,
! [A3: set_v] :
( ( minus_minus_set_v @ A3 @ bot_bot_set_v )
= A3 ) ).
% Diff_empty
thf(fact_63_empty__Diff,axiom,
! [A3: set_Product_prod_v_v] :
( ( minus_4183494784930505774od_v_v @ bot_bo723834152578015283od_v_v @ A3 )
= bot_bo723834152578015283od_v_v ) ).
% empty_Diff
thf(fact_64_empty__Diff,axiom,
! [A3: set_set_v] :
( ( minus_7228012346218142266_set_v @ bot_bot_set_set_v @ A3 )
= bot_bot_set_set_v ) ).
% empty_Diff
thf(fact_65_empty__Diff,axiom,
! [A3: set_v] :
( ( minus_minus_set_v @ bot_bot_set_v @ A3 )
= bot_bot_set_v ) ).
% empty_Diff
thf(fact_66_Diff__cancel,axiom,
! [A3: set_Product_prod_v_v] :
( ( minus_4183494784930505774od_v_v @ A3 @ A3 )
= bot_bo723834152578015283od_v_v ) ).
% Diff_cancel
thf(fact_67_Diff__cancel,axiom,
! [A3: set_set_v] :
( ( minus_7228012346218142266_set_v @ A3 @ A3 )
= bot_bot_set_set_v ) ).
% Diff_cancel
thf(fact_68_Diff__cancel,axiom,
! [A3: set_v] :
( ( minus_minus_set_v @ A3 @ A3 )
= bot_bot_set_v ) ).
% Diff_cancel
thf(fact_69_empty__Collect__eq,axiom,
! [P: v > $o] :
( ( bot_bot_set_v
= ( collect_v @ P ) )
= ( ! [X3: v] :
~ ( P @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_70_empty__Collect__eq,axiom,
! [P: product_prod_v_v > $o] :
( ( bot_bo723834152578015283od_v_v
= ( collec140062887454715474od_v_v @ P ) )
= ( ! [X3: product_prod_v_v] :
~ ( P @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_71_empty__Collect__eq,axiom,
! [P: set_v > $o] :
( ( bot_bot_set_set_v
= ( collect_set_v @ P ) )
= ( ! [X3: set_v] :
~ ( P @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_72_Collect__empty__eq,axiom,
! [P: v > $o] :
( ( ( collect_v @ P )
= bot_bot_set_v )
= ( ! [X3: v] :
~ ( P @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_73_Collect__empty__eq,axiom,
! [P: product_prod_v_v > $o] :
( ( ( collec140062887454715474od_v_v @ P )
= bot_bo723834152578015283od_v_v )
= ( ! [X3: product_prod_v_v] :
~ ( P @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_74_Collect__empty__eq,axiom,
! [P: set_v > $o] :
( ( ( collect_set_v @ P )
= bot_bot_set_set_v )
= ( ! [X3: set_v] :
~ ( P @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_75_all__not__in__conv,axiom,
! [A3: set_v] :
( ( ! [X3: v] :
~ ( member_v @ X3 @ A3 ) )
= ( A3 = bot_bot_set_v ) ) ).
% all_not_in_conv
thf(fact_76_all__not__in__conv,axiom,
! [A3: set_Product_prod_v_v] :
( ( ! [X3: product_prod_v_v] :
~ ( member7453568604450474000od_v_v @ X3 @ A3 ) )
= ( A3 = bot_bo723834152578015283od_v_v ) ) ).
% all_not_in_conv
thf(fact_77_all__not__in__conv,axiom,
! [A3: set_set_v] :
( ( ! [X3: set_v] :
~ ( member_set_v @ X3 @ A3 ) )
= ( A3 = bot_bot_set_set_v ) ) ).
% all_not_in_conv
thf(fact_78_empty__iff,axiom,
! [C: v] :
~ ( member_v @ C @ bot_bot_set_v ) ).
% empty_iff
thf(fact_79_empty__iff,axiom,
! [C: product_prod_v_v] :
~ ( member7453568604450474000od_v_v @ C @ bot_bo723834152578015283od_v_v ) ).
% empty_iff
thf(fact_80_empty__iff,axiom,
! [C: set_v] :
~ ( member_set_v @ C @ bot_bot_set_set_v ) ).
% empty_iff
thf(fact_81_insert__absorb2,axiom,
! [X2: v,A3: set_v] :
( ( insert_v @ X2 @ ( insert_v @ X2 @ A3 ) )
= ( insert_v @ X2 @ A3 ) ) ).
% insert_absorb2
thf(fact_82_insert__absorb2,axiom,
! [X2: product_prod_v_v,A3: set_Product_prod_v_v] :
( ( insert1338601472111419319od_v_v @ X2 @ ( insert1338601472111419319od_v_v @ X2 @ A3 ) )
= ( insert1338601472111419319od_v_v @ X2 @ A3 ) ) ).
% insert_absorb2
thf(fact_83_insert__absorb2,axiom,
! [X2: set_v,A3: set_set_v] :
( ( insert_set_v @ X2 @ ( insert_set_v @ X2 @ A3 ) )
= ( insert_set_v @ X2 @ A3 ) ) ).
% insert_absorb2
thf(fact_84_insert__iff,axiom,
! [A: set_v,B2: set_v,A3: set_set_v] :
( ( member_set_v @ A @ ( insert_set_v @ B2 @ A3 ) )
= ( ( A = B2 )
| ( member_set_v @ A @ A3 ) ) ) ).
% insert_iff
thf(fact_85_insert__iff,axiom,
! [A: v,B2: v,A3: set_v] :
( ( member_v @ A @ ( insert_v @ B2 @ A3 ) )
= ( ( A = B2 )
| ( member_v @ A @ A3 ) ) ) ).
% insert_iff
thf(fact_86_insert__iff,axiom,
! [A: product_prod_v_v,B2: product_prod_v_v,A3: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ A @ ( insert1338601472111419319od_v_v @ B2 @ A3 ) )
= ( ( A = B2 )
| ( member7453568604450474000od_v_v @ A @ A3 ) ) ) ).
% insert_iff
thf(fact_87_insertCI,axiom,
! [A: set_v,B: set_set_v,B2: set_v] :
( ( ~ ( member_set_v @ A @ B )
=> ( A = B2 ) )
=> ( member_set_v @ A @ ( insert_set_v @ B2 @ B ) ) ) ).
% insertCI
thf(fact_88_insertCI,axiom,
! [A: v,B: set_v,B2: v] :
( ( ~ ( member_v @ A @ B )
=> ( A = B2 ) )
=> ( member_v @ A @ ( insert_v @ B2 @ B ) ) ) ).
% insertCI
thf(fact_89_insertCI,axiom,
! [A: product_prod_v_v,B: set_Product_prod_v_v,B2: product_prod_v_v] :
( ( ~ ( member7453568604450474000od_v_v @ A @ B )
=> ( A = B2 ) )
=> ( member7453568604450474000od_v_v @ A @ ( insert1338601472111419319od_v_v @ B2 @ B ) ) ) ).
% insertCI
thf(fact_90_Diff__idemp,axiom,
! [A3: set_v,B: set_v] :
( ( minus_minus_set_v @ ( minus_minus_set_v @ A3 @ B ) @ B )
= ( minus_minus_set_v @ A3 @ B ) ) ).
% Diff_idemp
thf(fact_91_Diff__iff,axiom,
! [C: product_prod_v_v,A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ C @ ( minus_4183494784930505774od_v_v @ A3 @ B ) )
= ( ( member7453568604450474000od_v_v @ C @ A3 )
& ~ ( member7453568604450474000od_v_v @ C @ B ) ) ) ).
% Diff_iff
thf(fact_92_Diff__iff,axiom,
! [C: v,A3: set_v,B: set_v] :
( ( member_v @ C @ ( minus_minus_set_v @ A3 @ B ) )
= ( ( member_v @ C @ A3 )
& ~ ( member_v @ C @ B ) ) ) ).
% Diff_iff
thf(fact_93_DiffI,axiom,
! [C: product_prod_v_v,A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ C @ A3 )
=> ( ~ ( member7453568604450474000od_v_v @ C @ B )
=> ( member7453568604450474000od_v_v @ C @ ( minus_4183494784930505774od_v_v @ A3 @ B ) ) ) ) ).
% DiffI
thf(fact_94_DiffI,axiom,
! [C: v,A3: set_v,B: set_v] :
( ( member_v @ C @ A3 )
=> ( ~ ( member_v @ C @ B )
=> ( member_v @ C @ ( minus_minus_set_v @ A3 @ B ) ) ) ) ).
% DiffI
thf(fact_95_singletonI,axiom,
! [A: v] : ( member_v @ A @ ( insert_v @ A @ bot_bot_set_v ) ) ).
% singletonI
thf(fact_96_singletonI,axiom,
! [A: product_prod_v_v] : ( member7453568604450474000od_v_v @ A @ ( insert1338601472111419319od_v_v @ A @ bot_bo723834152578015283od_v_v ) ) ).
% singletonI
thf(fact_97_singletonI,axiom,
! [A: set_v] : ( member_set_v @ A @ ( insert_set_v @ A @ bot_bot_set_set_v ) ) ).
% singletonI
thf(fact_98_ex__in__conv,axiom,
! [A3: set_v] :
( ( ? [X3: v] : ( member_v @ X3 @ A3 ) )
= ( A3 != bot_bot_set_v ) ) ).
% ex_in_conv
thf(fact_99_ex__in__conv,axiom,
! [A3: set_Product_prod_v_v] :
( ( ? [X3: product_prod_v_v] : ( member7453568604450474000od_v_v @ X3 @ A3 ) )
= ( A3 != bot_bo723834152578015283od_v_v ) ) ).
% ex_in_conv
thf(fact_100_ex__in__conv,axiom,
! [A3: set_set_v] :
( ( ? [X3: set_v] : ( member_set_v @ X3 @ A3 ) )
= ( A3 != bot_bot_set_set_v ) ) ).
% ex_in_conv
thf(fact_101_equals0I,axiom,
! [A3: set_v] :
( ! [Y: v] :
~ ( member_v @ Y @ A3 )
=> ( A3 = bot_bot_set_v ) ) ).
% equals0I
thf(fact_102_equals0I,axiom,
! [A3: set_Product_prod_v_v] :
( ! [Y: product_prod_v_v] :
~ ( member7453568604450474000od_v_v @ Y @ A3 )
=> ( A3 = bot_bo723834152578015283od_v_v ) ) ).
% equals0I
thf(fact_103_equals0I,axiom,
! [A3: set_set_v] :
( ! [Y: set_v] :
~ ( member_set_v @ Y @ A3 )
=> ( A3 = bot_bot_set_set_v ) ) ).
% equals0I
thf(fact_104_equals0D,axiom,
! [A3: set_v,A: v] :
( ( A3 = bot_bot_set_v )
=> ~ ( member_v @ A @ A3 ) ) ).
% equals0D
thf(fact_105_equals0D,axiom,
! [A3: set_Product_prod_v_v,A: product_prod_v_v] :
( ( A3 = bot_bo723834152578015283od_v_v )
=> ~ ( member7453568604450474000od_v_v @ A @ A3 ) ) ).
% equals0D
thf(fact_106_equals0D,axiom,
! [A3: set_set_v,A: set_v] :
( ( A3 = bot_bot_set_set_v )
=> ~ ( member_set_v @ A @ A3 ) ) ).
% equals0D
thf(fact_107_emptyE,axiom,
! [A: v] :
~ ( member_v @ A @ bot_bot_set_v ) ).
% emptyE
thf(fact_108_emptyE,axiom,
! [A: product_prod_v_v] :
~ ( member7453568604450474000od_v_v @ A @ bot_bo723834152578015283od_v_v ) ).
% emptyE
thf(fact_109_emptyE,axiom,
! [A: set_v] :
~ ( member_set_v @ A @ bot_bot_set_set_v ) ).
% emptyE
thf(fact_110_mk__disjoint__insert,axiom,
! [A: set_v,A3: set_set_v] :
( ( member_set_v @ A @ A3 )
=> ? [B3: set_set_v] :
( ( A3
= ( insert_set_v @ A @ B3 ) )
& ~ ( member_set_v @ A @ B3 ) ) ) ).
% mk_disjoint_insert
thf(fact_111_mk__disjoint__insert,axiom,
! [A: v,A3: set_v] :
( ( member_v @ A @ A3 )
=> ? [B3: set_v] :
( ( A3
= ( insert_v @ A @ B3 ) )
& ~ ( member_v @ A @ B3 ) ) ) ).
% mk_disjoint_insert
thf(fact_112_mk__disjoint__insert,axiom,
! [A: product_prod_v_v,A3: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ A @ A3 )
=> ? [B3: set_Product_prod_v_v] :
( ( A3
= ( insert1338601472111419319od_v_v @ A @ B3 ) )
& ~ ( member7453568604450474000od_v_v @ A @ B3 ) ) ) ).
% mk_disjoint_insert
thf(fact_113_insert__commute,axiom,
! [X2: v,Y2: v,A3: set_v] :
( ( insert_v @ X2 @ ( insert_v @ Y2 @ A3 ) )
= ( insert_v @ Y2 @ ( insert_v @ X2 @ A3 ) ) ) ).
% insert_commute
thf(fact_114_insert__commute,axiom,
! [X2: product_prod_v_v,Y2: product_prod_v_v,A3: set_Product_prod_v_v] :
( ( insert1338601472111419319od_v_v @ X2 @ ( insert1338601472111419319od_v_v @ Y2 @ A3 ) )
= ( insert1338601472111419319od_v_v @ Y2 @ ( insert1338601472111419319od_v_v @ X2 @ A3 ) ) ) ).
% insert_commute
thf(fact_115_insert__commute,axiom,
! [X2: set_v,Y2: set_v,A3: set_set_v] :
( ( insert_set_v @ X2 @ ( insert_set_v @ Y2 @ A3 ) )
= ( insert_set_v @ Y2 @ ( insert_set_v @ X2 @ A3 ) ) ) ).
% insert_commute
thf(fact_116_insert__eq__iff,axiom,
! [A: set_v,A3: set_set_v,B2: set_v,B: set_set_v] :
( ~ ( member_set_v @ A @ A3 )
=> ( ~ ( member_set_v @ B2 @ B )
=> ( ( ( insert_set_v @ A @ A3 )
= ( insert_set_v @ B2 @ B ) )
= ( ( ( A = B2 )
=> ( A3 = B ) )
& ( ( A != B2 )
=> ? [C2: set_set_v] :
( ( A3
= ( insert_set_v @ B2 @ C2 ) )
& ~ ( member_set_v @ B2 @ C2 )
& ( B
= ( insert_set_v @ A @ C2 ) )
& ~ ( member_set_v @ A @ C2 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_117_insert__eq__iff,axiom,
! [A: v,A3: set_v,B2: v,B: set_v] :
( ~ ( member_v @ A @ A3 )
=> ( ~ ( member_v @ B2 @ B )
=> ( ( ( insert_v @ A @ A3 )
= ( insert_v @ B2 @ B ) )
= ( ( ( A = B2 )
=> ( A3 = B ) )
& ( ( A != B2 )
=> ? [C2: set_v] :
( ( A3
= ( insert_v @ B2 @ C2 ) )
& ~ ( member_v @ B2 @ C2 )
& ( B
= ( insert_v @ A @ C2 ) )
& ~ ( member_v @ A @ C2 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_118_insert__eq__iff,axiom,
! [A: product_prod_v_v,A3: set_Product_prod_v_v,B2: product_prod_v_v,B: set_Product_prod_v_v] :
( ~ ( member7453568604450474000od_v_v @ A @ A3 )
=> ( ~ ( member7453568604450474000od_v_v @ B2 @ B )
=> ( ( ( insert1338601472111419319od_v_v @ A @ A3 )
= ( insert1338601472111419319od_v_v @ B2 @ B ) )
= ( ( ( A = B2 )
=> ( A3 = B ) )
& ( ( A != B2 )
=> ? [C2: set_Product_prod_v_v] :
( ( A3
= ( insert1338601472111419319od_v_v @ B2 @ C2 ) )
& ~ ( member7453568604450474000od_v_v @ B2 @ C2 )
& ( B
= ( insert1338601472111419319od_v_v @ A @ C2 ) )
& ~ ( member7453568604450474000od_v_v @ A @ C2 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_119_insert__absorb,axiom,
! [A: set_v,A3: set_set_v] :
( ( member_set_v @ A @ A3 )
=> ( ( insert_set_v @ A @ A3 )
= A3 ) ) ).
% insert_absorb
thf(fact_120_insert__absorb,axiom,
! [A: v,A3: set_v] :
( ( member_v @ A @ A3 )
=> ( ( insert_v @ A @ A3 )
= A3 ) ) ).
% insert_absorb
thf(fact_121_insert__absorb,axiom,
! [A: product_prod_v_v,A3: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ A @ A3 )
=> ( ( insert1338601472111419319od_v_v @ A @ A3 )
= A3 ) ) ).
% insert_absorb
thf(fact_122_insert__ident,axiom,
! [X2: set_v,A3: set_set_v,B: set_set_v] :
( ~ ( member_set_v @ X2 @ A3 )
=> ( ~ ( member_set_v @ X2 @ B )
=> ( ( ( insert_set_v @ X2 @ A3 )
= ( insert_set_v @ X2 @ B ) )
= ( A3 = B ) ) ) ) ).
% insert_ident
thf(fact_123_insert__ident,axiom,
! [X2: v,A3: set_v,B: set_v] :
( ~ ( member_v @ X2 @ A3 )
=> ( ~ ( member_v @ X2 @ B )
=> ( ( ( insert_v @ X2 @ A3 )
= ( insert_v @ X2 @ B ) )
= ( A3 = B ) ) ) ) ).
% insert_ident
thf(fact_124_insert__ident,axiom,
! [X2: product_prod_v_v,A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ~ ( member7453568604450474000od_v_v @ X2 @ A3 )
=> ( ~ ( member7453568604450474000od_v_v @ X2 @ B )
=> ( ( ( insert1338601472111419319od_v_v @ X2 @ A3 )
= ( insert1338601472111419319od_v_v @ X2 @ B ) )
= ( A3 = B ) ) ) ) ).
% insert_ident
thf(fact_125_Set_Oset__insert,axiom,
! [X2: set_v,A3: set_set_v] :
( ( member_set_v @ X2 @ A3 )
=> ~ ! [B3: set_set_v] :
( ( A3
= ( insert_set_v @ X2 @ B3 ) )
=> ( member_set_v @ X2 @ B3 ) ) ) ).
% Set.set_insert
thf(fact_126_Set_Oset__insert,axiom,
! [X2: v,A3: set_v] :
( ( member_v @ X2 @ A3 )
=> ~ ! [B3: set_v] :
( ( A3
= ( insert_v @ X2 @ B3 ) )
=> ( member_v @ X2 @ B3 ) ) ) ).
% Set.set_insert
thf(fact_127_Set_Oset__insert,axiom,
! [X2: product_prod_v_v,A3: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X2 @ A3 )
=> ~ ! [B3: set_Product_prod_v_v] :
( ( A3
= ( insert1338601472111419319od_v_v @ X2 @ B3 ) )
=> ( member7453568604450474000od_v_v @ X2 @ B3 ) ) ) ).
% Set.set_insert
thf(fact_128_insertI2,axiom,
! [A: set_v,B: set_set_v,B2: set_v] :
( ( member_set_v @ A @ B )
=> ( member_set_v @ A @ ( insert_set_v @ B2 @ B ) ) ) ).
% insertI2
thf(fact_129_insertI2,axiom,
! [A: v,B: set_v,B2: v] :
( ( member_v @ A @ B )
=> ( member_v @ A @ ( insert_v @ B2 @ B ) ) ) ).
% insertI2
thf(fact_130_insertI2,axiom,
! [A: product_prod_v_v,B: set_Product_prod_v_v,B2: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ A @ B )
=> ( member7453568604450474000od_v_v @ A @ ( insert1338601472111419319od_v_v @ B2 @ B ) ) ) ).
% insertI2
thf(fact_131_insertI1,axiom,
! [A: set_v,B: set_set_v] : ( member_set_v @ A @ ( insert_set_v @ A @ B ) ) ).
% insertI1
thf(fact_132_insertI1,axiom,
! [A: v,B: set_v] : ( member_v @ A @ ( insert_v @ A @ B ) ) ).
% insertI1
thf(fact_133_insertI1,axiom,
! [A: product_prod_v_v,B: set_Product_prod_v_v] : ( member7453568604450474000od_v_v @ A @ ( insert1338601472111419319od_v_v @ A @ B ) ) ).
% insertI1
thf(fact_134_insertE,axiom,
! [A: set_v,B2: set_v,A3: set_set_v] :
( ( member_set_v @ A @ ( insert_set_v @ B2 @ A3 ) )
=> ( ( A != B2 )
=> ( member_set_v @ A @ A3 ) ) ) ).
% insertE
thf(fact_135_insertE,axiom,
! [A: v,B2: v,A3: set_v] :
( ( member_v @ A @ ( insert_v @ B2 @ A3 ) )
=> ( ( A != B2 )
=> ( member_v @ A @ A3 ) ) ) ).
% insertE
thf(fact_136_insertE,axiom,
! [A: product_prod_v_v,B2: product_prod_v_v,A3: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ A @ ( insert1338601472111419319od_v_v @ B2 @ A3 ) )
=> ( ( A != B2 )
=> ( member7453568604450474000od_v_v @ A @ A3 ) ) ) ).
% insertE
thf(fact_137_DiffD2,axiom,
! [C: product_prod_v_v,A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ C @ ( minus_4183494784930505774od_v_v @ A3 @ B ) )
=> ~ ( member7453568604450474000od_v_v @ C @ B ) ) ).
% DiffD2
thf(fact_138_DiffD2,axiom,
! [C: v,A3: set_v,B: set_v] :
( ( member_v @ C @ ( minus_minus_set_v @ A3 @ B ) )
=> ~ ( member_v @ C @ B ) ) ).
% DiffD2
thf(fact_139_DiffD1,axiom,
! [C: product_prod_v_v,A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ C @ ( minus_4183494784930505774od_v_v @ A3 @ B ) )
=> ( member7453568604450474000od_v_v @ C @ A3 ) ) ).
% DiffD1
thf(fact_140_DiffD1,axiom,
! [C: v,A3: set_v,B: set_v] :
( ( member_v @ C @ ( minus_minus_set_v @ A3 @ B ) )
=> ( member_v @ C @ A3 ) ) ).
% DiffD1
thf(fact_141_DiffE,axiom,
! [C: product_prod_v_v,A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ C @ ( minus_4183494784930505774od_v_v @ A3 @ B ) )
=> ~ ( ( member7453568604450474000od_v_v @ C @ A3 )
=> ( member7453568604450474000od_v_v @ C @ B ) ) ) ).
% DiffE
thf(fact_142_DiffE,axiom,
! [C: v,A3: set_v,B: set_v] :
( ( member_v @ C @ ( minus_minus_set_v @ A3 @ B ) )
=> ~ ( ( member_v @ C @ A3 )
=> ( member_v @ C @ B ) ) ) ).
% DiffE
thf(fact_143_singleton__inject,axiom,
! [A: v,B2: v] :
( ( ( insert_v @ A @ bot_bot_set_v )
= ( insert_v @ B2 @ bot_bot_set_v ) )
=> ( A = B2 ) ) ).
% singleton_inject
thf(fact_144_singleton__inject,axiom,
! [A: product_prod_v_v,B2: product_prod_v_v] :
( ( ( insert1338601472111419319od_v_v @ A @ bot_bo723834152578015283od_v_v )
= ( insert1338601472111419319od_v_v @ B2 @ bot_bo723834152578015283od_v_v ) )
=> ( A = B2 ) ) ).
% singleton_inject
thf(fact_145_singleton__inject,axiom,
! [A: set_v,B2: set_v] :
( ( ( insert_set_v @ A @ bot_bot_set_set_v )
= ( insert_set_v @ B2 @ bot_bot_set_set_v ) )
=> ( A = B2 ) ) ).
% singleton_inject
thf(fact_146_insert__not__empty,axiom,
! [A: v,A3: set_v] :
( ( insert_v @ A @ A3 )
!= bot_bot_set_v ) ).
% insert_not_empty
thf(fact_147_insert__not__empty,axiom,
! [A: product_prod_v_v,A3: set_Product_prod_v_v] :
( ( insert1338601472111419319od_v_v @ A @ A3 )
!= bot_bo723834152578015283od_v_v ) ).
% insert_not_empty
thf(fact_148_insert__not__empty,axiom,
! [A: set_v,A3: set_set_v] :
( ( insert_set_v @ A @ A3 )
!= bot_bot_set_set_v ) ).
% insert_not_empty
thf(fact_149_doubleton__eq__iff,axiom,
! [A: v,B2: v,C: v,D: v] :
( ( ( insert_v @ A @ ( insert_v @ B2 @ bot_bot_set_v ) )
= ( insert_v @ C @ ( insert_v @ D @ bot_bot_set_v ) ) )
= ( ( ( A = C )
& ( B2 = D ) )
| ( ( A = D )
& ( B2 = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_150_doubleton__eq__iff,axiom,
! [A: product_prod_v_v,B2: product_prod_v_v,C: product_prod_v_v,D: product_prod_v_v] :
( ( ( insert1338601472111419319od_v_v @ A @ ( insert1338601472111419319od_v_v @ B2 @ bot_bo723834152578015283od_v_v ) )
= ( insert1338601472111419319od_v_v @ C @ ( insert1338601472111419319od_v_v @ D @ bot_bo723834152578015283od_v_v ) ) )
= ( ( ( A = C )
& ( B2 = D ) )
| ( ( A = D )
& ( B2 = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_151_doubleton__eq__iff,axiom,
! [A: set_v,B2: set_v,C: set_v,D: set_v] :
( ( ( insert_set_v @ A @ ( insert_set_v @ B2 @ bot_bot_set_set_v ) )
= ( insert_set_v @ C @ ( insert_set_v @ D @ bot_bot_set_set_v ) ) )
= ( ( ( A = C )
& ( B2 = D ) )
| ( ( A = D )
& ( B2 = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_152_singleton__iff,axiom,
! [B2: v,A: v] :
( ( member_v @ B2 @ ( insert_v @ A @ bot_bot_set_v ) )
= ( B2 = A ) ) ).
% singleton_iff
thf(fact_153_singleton__iff,axiom,
! [B2: product_prod_v_v,A: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ B2 @ ( insert1338601472111419319od_v_v @ A @ bot_bo723834152578015283od_v_v ) )
= ( B2 = A ) ) ).
% singleton_iff
thf(fact_154_singleton__iff,axiom,
! [B2: set_v,A: set_v] :
( ( member_set_v @ B2 @ ( insert_set_v @ A @ bot_bot_set_set_v ) )
= ( B2 = A ) ) ).
% singleton_iff
thf(fact_155_singletonD,axiom,
! [B2: v,A: v] :
( ( member_v @ B2 @ ( insert_v @ A @ bot_bot_set_v ) )
=> ( B2 = A ) ) ).
% singletonD
thf(fact_156_singletonD,axiom,
! [B2: product_prod_v_v,A: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ B2 @ ( insert1338601472111419319od_v_v @ A @ bot_bo723834152578015283od_v_v ) )
=> ( B2 = A ) ) ).
% singletonD
thf(fact_157_singletonD,axiom,
! [B2: set_v,A: set_v] :
( ( member_set_v @ B2 @ ( insert_set_v @ A @ bot_bot_set_set_v ) )
=> ( B2 = A ) ) ).
% singletonD
thf(fact_158_insert__Diff__if,axiom,
! [X2: set_v,B: set_set_v,A3: set_set_v] :
( ( ( member_set_v @ X2 @ B )
=> ( ( minus_7228012346218142266_set_v @ ( insert_set_v @ X2 @ A3 ) @ B )
= ( minus_7228012346218142266_set_v @ A3 @ B ) ) )
& ( ~ ( member_set_v @ X2 @ B )
=> ( ( minus_7228012346218142266_set_v @ ( insert_set_v @ X2 @ A3 ) @ B )
= ( insert_set_v @ X2 @ ( minus_7228012346218142266_set_v @ A3 @ B ) ) ) ) ) ).
% insert_Diff_if
thf(fact_159_insert__Diff__if,axiom,
! [X2: product_prod_v_v,B: set_Product_prod_v_v,A3: set_Product_prod_v_v] :
( ( ( member7453568604450474000od_v_v @ X2 @ B )
=> ( ( minus_4183494784930505774od_v_v @ ( insert1338601472111419319od_v_v @ X2 @ A3 ) @ B )
= ( minus_4183494784930505774od_v_v @ A3 @ B ) ) )
& ( ~ ( member7453568604450474000od_v_v @ X2 @ B )
=> ( ( minus_4183494784930505774od_v_v @ ( insert1338601472111419319od_v_v @ X2 @ A3 ) @ B )
= ( insert1338601472111419319od_v_v @ X2 @ ( minus_4183494784930505774od_v_v @ A3 @ B ) ) ) ) ) ).
% insert_Diff_if
thf(fact_160_insert__Diff__if,axiom,
! [X2: v,B: set_v,A3: set_v] :
( ( ( member_v @ X2 @ B )
=> ( ( minus_minus_set_v @ ( insert_v @ X2 @ A3 ) @ B )
= ( minus_minus_set_v @ A3 @ B ) ) )
& ( ~ ( member_v @ X2 @ B )
=> ( ( minus_minus_set_v @ ( insert_v @ X2 @ A3 ) @ B )
= ( insert_v @ X2 @ ( minus_minus_set_v @ A3 @ B ) ) ) ) ) ).
% insert_Diff_if
thf(fact_161_Diff__insert__absorb,axiom,
! [X2: product_prod_v_v,A3: set_Product_prod_v_v] :
( ~ ( member7453568604450474000od_v_v @ X2 @ A3 )
=> ( ( minus_4183494784930505774od_v_v @ ( insert1338601472111419319od_v_v @ X2 @ A3 ) @ ( insert1338601472111419319od_v_v @ X2 @ bot_bo723834152578015283od_v_v ) )
= A3 ) ) ).
% Diff_insert_absorb
thf(fact_162_Diff__insert__absorb,axiom,
! [X2: set_v,A3: set_set_v] :
( ~ ( member_set_v @ X2 @ A3 )
=> ( ( minus_7228012346218142266_set_v @ ( insert_set_v @ X2 @ A3 ) @ ( insert_set_v @ X2 @ bot_bot_set_set_v ) )
= A3 ) ) ).
% Diff_insert_absorb
thf(fact_163_Diff__insert__absorb,axiom,
! [X2: v,A3: set_v] :
( ~ ( member_v @ X2 @ A3 )
=> ( ( minus_minus_set_v @ ( insert_v @ X2 @ A3 ) @ ( insert_v @ X2 @ bot_bot_set_v ) )
= A3 ) ) ).
% Diff_insert_absorb
thf(fact_164_Diff__insert2,axiom,
! [A3: set_Product_prod_v_v,A: product_prod_v_v,B: set_Product_prod_v_v] :
( ( minus_4183494784930505774od_v_v @ A3 @ ( insert1338601472111419319od_v_v @ A @ B ) )
= ( minus_4183494784930505774od_v_v @ ( minus_4183494784930505774od_v_v @ A3 @ ( insert1338601472111419319od_v_v @ A @ bot_bo723834152578015283od_v_v ) ) @ B ) ) ).
% Diff_insert2
thf(fact_165_Diff__insert2,axiom,
! [A3: set_set_v,A: set_v,B: set_set_v] :
( ( minus_7228012346218142266_set_v @ A3 @ ( insert_set_v @ A @ B ) )
= ( minus_7228012346218142266_set_v @ ( minus_7228012346218142266_set_v @ A3 @ ( insert_set_v @ A @ bot_bot_set_set_v ) ) @ B ) ) ).
% Diff_insert2
thf(fact_166_Diff__insert2,axiom,
! [A3: set_v,A: v,B: set_v] :
( ( minus_minus_set_v @ A3 @ ( insert_v @ A @ B ) )
= ( minus_minus_set_v @ ( minus_minus_set_v @ A3 @ ( insert_v @ A @ bot_bot_set_v ) ) @ B ) ) ).
% Diff_insert2
thf(fact_167_insert__Diff,axiom,
! [A: product_prod_v_v,A3: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ A @ A3 )
=> ( ( insert1338601472111419319od_v_v @ A @ ( minus_4183494784930505774od_v_v @ A3 @ ( insert1338601472111419319od_v_v @ A @ bot_bo723834152578015283od_v_v ) ) )
= A3 ) ) ).
% insert_Diff
thf(fact_168_insert__Diff,axiom,
! [A: set_v,A3: set_set_v] :
( ( member_set_v @ A @ A3 )
=> ( ( insert_set_v @ A @ ( minus_7228012346218142266_set_v @ A3 @ ( insert_set_v @ A @ bot_bot_set_set_v ) ) )
= A3 ) ) ).
% insert_Diff
thf(fact_169_insert__Diff,axiom,
! [A: v,A3: set_v] :
( ( member_v @ A @ A3 )
=> ( ( insert_v @ A @ ( minus_minus_set_v @ A3 @ ( insert_v @ A @ bot_bot_set_v ) ) )
= A3 ) ) ).
% insert_Diff
thf(fact_170_Diff__insert,axiom,
! [A3: set_Product_prod_v_v,A: product_prod_v_v,B: set_Product_prod_v_v] :
( ( minus_4183494784930505774od_v_v @ A3 @ ( insert1338601472111419319od_v_v @ A @ B ) )
= ( minus_4183494784930505774od_v_v @ ( minus_4183494784930505774od_v_v @ A3 @ B ) @ ( insert1338601472111419319od_v_v @ A @ bot_bo723834152578015283od_v_v ) ) ) ).
% Diff_insert
thf(fact_171_Diff__insert,axiom,
! [A3: set_set_v,A: set_v,B: set_set_v] :
( ( minus_7228012346218142266_set_v @ A3 @ ( insert_set_v @ A @ B ) )
= ( minus_7228012346218142266_set_v @ ( minus_7228012346218142266_set_v @ A3 @ B ) @ ( insert_set_v @ A @ bot_bot_set_set_v ) ) ) ).
% Diff_insert
thf(fact_172_Diff__insert,axiom,
! [A3: set_v,A: v,B: set_v] :
( ( minus_minus_set_v @ A3 @ ( insert_v @ A @ B ) )
= ( minus_minus_set_v @ ( minus_minus_set_v @ A3 @ B ) @ ( insert_v @ A @ bot_bot_set_v ) ) ) ).
% Diff_insert
thf(fact_173_dfs__S__tl__stack_I2_J,axiom,
! [V: v,E: sCC_Bl1394983891496994913t_unit,E2: sCC_Bl1394983891496994913t_unit] :
( ( sCC_Bl8953792750115413617t_unit @ successors @ V @ 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_174_dfs__S__tl__stack_I1_J,axiom,
! [V: v,E: sCC_Bl1394983891496994913t_unit,E2: sCC_Bl1394983891496994913t_unit] :
( ( sCC_Bl8953792750115413617t_unit @ successors @ V @ E @ E2 )
=> ( ( ( sCC_Bl8828226123343373779t_unit @ E )
!= nil_v )
=> ( ( sCC_Bl8828226123343373779t_unit @ E2 )
!= nil_v ) ) ) ).
% dfs_S_tl_stack(1)
thf(fact_175_e_H__def,axiom,
( ( ( member_v @ w @ ( sCC_Bl157864678168468314t_unit @ ea ) )
=> ( e2 = ea ) )
& ( ~ ( member_v @ w @ ( sCC_Bl157864678168468314t_unit @ ea ) )
=> ( ( ~ ( member_v @ w @ ( sCC_Bl4645233313691564917t_unit @ ea ) )
=> ( e2
= ( sCC_Bloemen_dfs_v @ successors @ w @ ea ) ) )
& ( ( member_v @ w @ ( sCC_Bl4645233313691564917t_unit @ ea ) )
=> ( e2
= ( sCC_Bloemen_unite_v @ va @ w @ ea ) ) ) ) ) ) ).
% e'_def
thf(fact_176_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_177_set__empty2,axiom,
! [Xs: list_v] :
( ( bot_bot_set_v
= ( set_v2 @ Xs ) )
= ( Xs = nil_v ) ) ).
% set_empty2
thf(fact_178_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_179_set__empty2,axiom,
! [Xs: list_set_v] :
( ( bot_bot_set_set_v
= ( set_set_v2 @ Xs ) )
= ( Xs = nil_set_v ) ) ).
% set_empty2
thf(fact_180_set__empty,axiom,
! [Xs: list_v] :
( ( ( set_v2 @ Xs )
= bot_bot_set_v )
= ( Xs = nil_v ) ) ).
% set_empty
thf(fact_181_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_182_set__empty,axiom,
! [Xs: list_set_v] :
( ( ( set_set_v2 @ Xs )
= bot_bot_set_set_v )
= ( Xs = nil_set_v ) ) ).
% set_empty
thf(fact_183_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_184_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_185_self__append__conv2,axiom,
! [Y2: list_v,Xs: list_v] :
( ( Y2
= ( append_v @ Xs @ Y2 ) )
= ( Xs = nil_v ) ) ).
% self_append_conv2
thf(fact_186_append__self__conv2,axiom,
! [Xs: list_v,Ys: list_v] :
( ( ( append_v @ Xs @ Ys )
= Ys )
= ( Xs = nil_v ) ) ).
% append_self_conv2
thf(fact_187_self__append__conv,axiom,
! [Y2: list_v,Ys: list_v] :
( ( Y2
= ( append_v @ Y2 @ Ys ) )
= ( Ys = nil_v ) ) ).
% self_append_conv
thf(fact_188_append_Oassoc,axiom,
! [A: list_v,B2: list_v,C: list_v] :
( ( append_v @ ( append_v @ A @ B2 ) @ C )
= ( append_v @ A @ ( append_v @ B2 @ C ) ) ) ).
% append.assoc
thf(fact_189_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_190_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_191_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_192_append_Oright__neutral,axiom,
! [A: list_v] :
( ( append_v @ A @ nil_v )
= A ) ).
% append.right_neutral
thf(fact_193_append__Nil2,axiom,
! [Xs: list_v] :
( ( append_v @ Xs @ nil_v )
= Xs ) ).
% append_Nil2
thf(fact_194_append__self__conv,axiom,
! [Xs: list_v,Ys: list_v] :
( ( ( append_v @ Xs @ Ys )
= Xs )
= ( Ys = nil_v ) ) ).
% append_self_conv
thf(fact_195_graph_Odfs_Ocong,axiom,
sCC_Bloemen_dfs_v = sCC_Bloemen_dfs_v ).
% graph.dfs.cong
thf(fact_196_graph_Opost__dfs_Ocong,axiom,
sCC_Bl8953792750115413617t_unit = sCC_Bl8953792750115413617t_unit ).
% graph.post_dfs.cong
thf(fact_197_append__eq__appendI,axiom,
! [Xs: list_v,Xs1: list_v,Zs: list_v,Ys: list_v,Us: list_v] :
( ( ( append_v @ Xs @ Xs1 )
= Zs )
=> ( ( Ys
= ( append_v @ Xs1 @ Us ) )
=> ( ( append_v @ Xs @ Ys )
= ( append_v @ Zs @ Us ) ) ) ) ).
% append_eq_appendI
thf(fact_198_append__eq__append__conv2,axiom,
! [Xs: list_v,Ys: list_v,Zs: list_v,Ts: list_v] :
( ( ( append_v @ Xs @ Ys )
= ( append_v @ Zs @ Ts ) )
= ( ? [Us2: list_v] :
( ( ( Xs
= ( append_v @ Zs @ Us2 ) )
& ( ( append_v @ Us2 @ Ys )
= Ts ) )
| ( ( ( append_v @ Xs @ Us2 )
= Zs )
& ( Ys
= ( append_v @ Us2 @ Ts ) ) ) ) ) ) ).
% append_eq_append_conv2
thf(fact_199_append__Nil,axiom,
! [Ys: list_v] :
( ( append_v @ nil_v @ Ys )
= Ys ) ).
% append_Nil
thf(fact_200_append_Oleft__neutral,axiom,
! [A: list_v] :
( ( append_v @ nil_v @ A )
= A ) ).
% append.left_neutral
thf(fact_201_eq__Nil__appendI,axiom,
! [Xs: list_v,Ys: list_v] :
( ( Xs = Ys )
=> ( Xs
= ( append_v @ nil_v @ Ys ) ) ) ).
% eq_Nil_appendI
thf(fact_202_list_Osel_I2_J,axiom,
( ( tl_v @ nil_v )
= nil_v ) ).
% list.sel(2)
thf(fact_203_empty__set,axiom,
( bot_bot_set_v
= ( set_v2 @ nil_v ) ) ).
% empty_set
thf(fact_204_empty__set,axiom,
( bot_bo723834152578015283od_v_v
= ( set_Product_prod_v_v2 @ nil_Product_prod_v_v ) ) ).
% empty_set
thf(fact_205_empty__set,axiom,
( bot_bot_set_set_v
= ( set_set_v2 @ nil_set_v ) ) ).
% empty_set
thf(fact_206_list_Oset__sel_I2_J,axiom,
! [A: list_P7986770385144383213od_v_v,X2: product_prod_v_v] :
( ( A != nil_Product_prod_v_v )
=> ( ( member7453568604450474000od_v_v @ X2 @ ( set_Product_prod_v_v2 @ ( tl_Product_prod_v_v @ A ) ) )
=> ( member7453568604450474000od_v_v @ X2 @ ( set_Product_prod_v_v2 @ A ) ) ) ) ).
% list.set_sel(2)
thf(fact_207_list_Oset__sel_I2_J,axiom,
! [A: list_v,X2: v] :
( ( A != nil_v )
=> ( ( member_v @ X2 @ ( set_v2 @ ( tl_v @ A ) ) )
=> ( member_v @ X2 @ ( set_v2 @ A ) ) ) ) ).
% list.set_sel(2)
thf(fact_208_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_209_dfs__S__hd__stack_I1_J,axiom,
! [E: sCC_Bl1394983891496994913t_unit,V: v,E2: sCC_Bl1394983891496994913t_unit,N: v] :
( ( sCC_Bl9196236973127232072t_unit @ successors @ E )
=> ( ( sCC_Bl8953792750115413617t_unit @ successors @ V @ E @ E2 )
=> ( ( ( sCC_Bl8828226123343373779t_unit @ E )
!= nil_v )
=> ( ( member_v @ N @ ( sCC_Bl1280885523602775798t_unit @ E @ ( hd_v @ ( sCC_Bl8828226123343373779t_unit @ E ) ) ) )
=> ( ( sCC_Bl8828226123343373779t_unit @ E2 )
!= nil_v ) ) ) ) ) ).
% dfs_S_hd_stack(1)
thf(fact_210_dfs__S__hd__stack_I2_J,axiom,
! [E: sCC_Bl1394983891496994913t_unit,V: v,E2: sCC_Bl1394983891496994913t_unit,N: v] :
( ( sCC_Bl9196236973127232072t_unit @ successors @ E )
=> ( ( sCC_Bl8953792750115413617t_unit @ successors @ V @ E @ E2 )
=> ( ( ( sCC_Bl8828226123343373779t_unit @ E )
!= nil_v )
=> ( ( member_v @ N @ ( sCC_Bl1280885523602775798t_unit @ E @ ( hd_v @ ( sCC_Bl8828226123343373779t_unit @ E ) ) ) )
=> ( member_v @ N @ ( sCC_Bl1280885523602775798t_unit @ E2 @ ( hd_v @ ( sCC_Bl8828226123343373779t_unit @ E2 ) ) ) ) ) ) ) ) ).
% dfs_S_hd_stack(2)
thf(fact_211_reachable__visited,axiom,
! [E: sCC_Bl1394983891496994913t_unit,V: v,W: v] :
( ( sCC_Bl9196236973127232072t_unit @ successors @ E )
=> ( ( member_v @ V @ ( sCC_Bl4645233313691564917t_unit @ E ) )
=> ( ( sCC_Bl649662514949026229able_v @ successors @ V @ W )
=> ( ! [X4: v] :
( ( member_v @ X4 @ ( sCC_Bl4645233313691564917t_unit @ E ) )
=> ! [Xa: v] :
( ( member_v @ Xa @ ( minus_minus_set_v @ ( successors @ X4 ) @ ( sCC_Bl3795065053823578884t_unit @ E @ X4 ) ) )
=> ( ( sCC_Bl649662514949026229able_v @ successors @ V @ X4 )
=> ~ ( sCC_Bl649662514949026229able_v @ successors @ Xa @ W ) ) ) )
=> ( member_v @ W @ ( sCC_Bl4645233313691564917t_unit @ E ) ) ) ) ) ) ).
% reachable_visited
thf(fact_212_post__dfs__def,axiom,
! [V: v,E: sCC_Bl1394983891496994913t_unit,E2: sCC_Bl1394983891496994913t_unit] :
( ( sCC_Bl8953792750115413617t_unit @ successors @ V @ E @ E2 )
= ( ( sCC_Bl9196236973127232072t_unit @ successors @ E2 )
& ( member_v @ V @ ( sCC_Bl4645233313691564917t_unit @ E2 ) )
& ( sCC_Bl5768913643336123637t_unit @ E @ E2 )
& ( ( sCC_Bl3795065053823578884t_unit @ E2 @ V )
= ( successors @ V ) )
& ! [X3: v] :
( ( member_v @ X3 @ ( sCC_Bl4645233313691564917t_unit @ E ) )
=> ( ( sCC_Bl3795065053823578884t_unit @ E2 @ X3 )
= ( sCC_Bl3795065053823578884t_unit @ E @ X3 ) ) )
& ! [X3: v] :
( ( member_v @ X3 @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E2 ) ) )
=> ( sCC_Bl649662514949026229able_v @ successors @ X3 @ V ) )
& ? [Ns2: list_v] :
( ( sCC_Bl8828226123343373779t_unit @ E )
= ( append_v @ Ns2 @ ( sCC_Bl8828226123343373779t_unit @ E2 ) ) )
& ( ( ( member_v @ V @ ( sCC_Bl157864678168468314t_unit @ E2 ) )
& ( ( sCC_Bl8828226123343373779t_unit @ E2 )
= ( sCC_Bl8828226123343373779t_unit @ E ) )
& ! [X3: v] :
( ( member_v @ X3 @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E2 ) ) )
=> ( ( sCC_Bl1280885523602775798t_unit @ E2 @ X3 )
= ( sCC_Bl1280885523602775798t_unit @ E @ X3 ) ) ) )
| ( ( ( sCC_Bl8828226123343373779t_unit @ E2 )
!= nil_v )
& ( member_v @ V @ ( sCC_Bl1280885523602775798t_unit @ E2 @ ( hd_v @ ( sCC_Bl8828226123343373779t_unit @ E2 ) ) ) )
& ! [X3: v] :
( ( member_v @ X3 @ ( set_v2 @ ( tl_v @ ( sCC_Bl8828226123343373779t_unit @ E2 ) ) ) )
=> ( ( sCC_Bl1280885523602775798t_unit @ E2 @ X3 )
= ( sCC_Bl1280885523602775798t_unit @ E @ X3 ) ) ) ) )
& ( ( sCC_Bl9201514103433284750t_unit @ E2 )
= ( sCC_Bl9201514103433284750t_unit @ E ) ) ) ) ).
% post_dfs_def
thf(fact_213_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,V: product_prod_v_v,N: product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( sCC_Bl7798947040364291444t_unit @ Successors @ E )
=> ( ( member7453568604450474000od_v_v @ W @ ( Successors @ V ) )
=> ( ~ ( member7453568604450474000od_v_v @ W @ ( sCC_Bl3878977043676959280t_unit @ E @ V ) )
=> ( ( member7453568604450474000od_v_v @ W @ ( sCC_Bl5498988629518860705t_unit @ E ) )
=> ( ~ ( member7453568604450474000od_v_v @ W @ ( sCC_Bl5094201334446601350t_unit @ E ) )
=> ( ( member7453568604450474000od_v_v @ N @ ( set_Product_prod_v_v2 @ ( tl_Product_prod_v_v @ ( sCC_Bl2021302119412358655t_unit @ ( sCC_Bl4702006153222411093od_v_v @ V @ W @ E ) ) ) ) )
=> ( ( sCC_Bl8440648026628373538t_unit @ ( sCC_Bl4702006153222411093od_v_v @ V @ W @ E ) @ N )
= ( sCC_Bl8440648026628373538t_unit @ E @ N ) ) ) ) ) ) ) ) ) ).
% graph.unite_S_tl
thf(fact_214_graph_Ounite__S__tl,axiom,
! [Vertices: set_v,Successors: v > set_v,E: sCC_Bl1394983891496994913t_unit,W: v,V: v,N: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl9196236973127232072t_unit @ Successors @ E )
=> ( ( member_v @ W @ ( Successors @ V ) )
=> ( ~ ( member_v @ W @ ( sCC_Bl3795065053823578884t_unit @ E @ V ) )
=> ( ( member_v @ W @ ( sCC_Bl4645233313691564917t_unit @ E ) )
=> ( ~ ( member_v @ W @ ( sCC_Bl157864678168468314t_unit @ E ) )
=> ( ( member_v @ N @ ( set_v2 @ ( tl_v @ ( sCC_Bl8828226123343373779t_unit @ ( sCC_Bloemen_unite_v @ V @ W @ E ) ) ) ) )
=> ( ( sCC_Bl1280885523602775798t_unit @ ( sCC_Bloemen_unite_v @ V @ W @ E ) @ N )
= ( sCC_Bl1280885523602775798t_unit @ E @ N ) ) ) ) ) ) ) ) ) ).
% graph.unite_S_tl
thf(fact_215_pre__dfs__def,axiom,
! [V: v,E: sCC_Bl1394983891496994913t_unit] :
( ( sCC_Bl36166008131615352t_unit @ successors @ V @ E )
= ( ( sCC_Bl9196236973127232072t_unit @ successors @ E )
& ~ ( member_v @ V @ ( sCC_Bl4645233313691564917t_unit @ E ) )
& ( sCC_Bl649662514949026229able_v @ successors @ ( sCC_Bl1090238580953940555t_unit @ E ) @ V )
& ( ( sCC_Bl3795065053823578884t_unit @ E @ V )
= bot_bot_set_v )
& ! [X3: v] :
( ( member_v @ X3 @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E ) ) )
=> ( sCC_Bl649662514949026229able_v @ successors @ X3 @ V ) ) ) ) ).
% pre_dfs_def
thf(fact_216_unite__subscc,axiom,
! [V: v,E: sCC_Bl1394983891496994913t_unit,W: v] :
( ( sCC_Bl1748261141445803503t_unit @ successors @ V @ E )
=> ( ( member_v @ W @ ( successors @ V ) )
=> ( ~ ( member_v @ W @ ( sCC_Bl3795065053823578884t_unit @ E @ V ) )
=> ( ( member_v @ W @ ( sCC_Bl4645233313691564917t_unit @ E ) )
=> ( ~ ( member_v @ W @ ( sCC_Bl157864678168468314t_unit @ E ) )
=> ( sCC_Bl5398416737448265317bscc_v @ successors @ ( sCC_Bl1280885523602775798t_unit @ ( sCC_Bloemen_unite_v @ V @ W @ E ) @ ( hd_v @ ( sCC_Bl8828226123343373779t_unit @ ( sCC_Bloemen_unite_v @ V @ W @ E ) ) ) ) ) ) ) ) ) ) ).
% unite_subscc
thf(fact_217_post__dfss__def,axiom,
! [V: v,E: sCC_Bl1394983891496994913t_unit,E2: sCC_Bl1394983891496994913t_unit] :
( ( sCC_Bl6082031138996704384t_unit @ successors @ V @ E @ E2 )
= ( ( sCC_Bl9196236973127232072t_unit @ successors @ E2 )
& ( ( sCC_Bl3795065053823578884t_unit @ E2 @ V )
= ( successors @ V ) )
& ! [X3: v] :
( ( member_v @ X3 @ ( minus_minus_set_v @ ( sCC_Bl4645233313691564917t_unit @ E ) @ ( insert_v @ V @ bot_bot_set_v ) ) )
=> ( ( sCC_Bl3795065053823578884t_unit @ E2 @ X3 )
= ( sCC_Bl3795065053823578884t_unit @ E @ X3 ) ) )
& ( sCC_Bl5768913643336123637t_unit @ E @ E2 )
& ! [X3: v] :
( ( member_v @ X3 @ ( successors @ V ) )
=> ( member_v @ X3 @ ( sup_sup_set_v @ ( sCC_Bl157864678168468314t_unit @ E2 ) @ ( sCC_Bl1280885523602775798t_unit @ E2 @ ( hd_v @ ( sCC_Bl8828226123343373779t_unit @ E2 ) ) ) ) ) )
& ! [X3: v] :
( ( member_v @ X3 @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E2 ) ) )
=> ( sCC_Bl649662514949026229able_v @ successors @ X3 @ V ) )
& ( ( sCC_Bl8828226123343373779t_unit @ E2 )
!= nil_v )
& ? [Ns2: list_v] :
( ( sCC_Bl8828226123343373779t_unit @ E )
= ( append_v @ Ns2 @ ( sCC_Bl8828226123343373779t_unit @ E2 ) ) )
& ( member_v @ V @ ( sCC_Bl1280885523602775798t_unit @ E2 @ ( hd_v @ ( sCC_Bl8828226123343373779t_unit @ E2 ) ) ) )
& ! [X3: v] :
( ( member_v @ X3 @ ( set_v2 @ ( tl_v @ ( sCC_Bl8828226123343373779t_unit @ E2 ) ) ) )
=> ( ( sCC_Bl1280885523602775798t_unit @ E2 @ X3 )
= ( sCC_Bl1280885523602775798t_unit @ E @ X3 ) ) )
& ( ( ( hd_v @ ( sCC_Bl8828226123343373779t_unit @ E2 ) )
= V )
=> ! [X3: v] :
( ( member_v @ X3 @ ( set_v2 @ ( tl_v @ ( sCC_Bl8828226123343373779t_unit @ E2 ) ) ) )
=> ~ ( sCC_Bl649662514949026229able_v @ successors @ V @ X3 ) ) )
& ( ( sCC_Bl9201514103433284750t_unit @ E2 )
= ( sCC_Bl9201514103433284750t_unit @ E ) ) ) ) ).
% post_dfss_def
thf(fact_218_the__elem__eq,axiom,
! [X2: v] :
( ( the_elem_v @ ( insert_v @ X2 @ bot_bot_set_v ) )
= X2 ) ).
% the_elem_eq
thf(fact_219_the__elem__eq,axiom,
! [X2: product_prod_v_v] :
( ( the_el5392834299063928540od_v_v @ ( insert1338601472111419319od_v_v @ X2 @ bot_bo723834152578015283od_v_v ) )
= X2 ) ).
% the_elem_eq
thf(fact_220_the__elem__eq,axiom,
! [X2: set_v] :
( ( the_elem_set_v @ ( insert_set_v @ X2 @ bot_bot_set_set_v ) )
= X2 ) ).
% the_elem_eq
thf(fact_221_graph__axioms,axiom,
sCC_Bloemen_graph_v @ vertices @ successors ).
% graph_axioms
thf(fact_222_reachable_Ocases,axiom,
! [A1: v,A2: v] :
( ( sCC_Bl649662514949026229able_v @ successors @ A1 @ A2 )
=> ( ( A2 != A1 )
=> ~ ! [Y: v] :
( ( member_v @ Y @ ( successors @ A1 ) )
=> ~ ( sCC_Bl649662514949026229able_v @ successors @ Y @ A2 ) ) ) ) ).
% reachable.cases
thf(fact_223_reachable__refl,axiom,
! [X2: v] : ( sCC_Bl649662514949026229able_v @ successors @ X2 @ X2 ) ).
% reachable_refl
thf(fact_224_reachable__succ,axiom,
! [Y2: v,X2: v,Z: v] :
( ( member_v @ Y2 @ ( successors @ X2 ) )
=> ( ( sCC_Bl649662514949026229able_v @ successors @ Y2 @ Z )
=> ( sCC_Bl649662514949026229able_v @ successors @ X2 @ Z ) ) ) ).
% reachable_succ
thf(fact_225_reachable_Osimps,axiom,
! [A1: v,A2: v] :
( ( sCC_Bl649662514949026229able_v @ successors @ A1 @ A2 )
= ( ? [X3: v] :
( ( A1 = X3 )
& ( A2 = X3 ) )
| ? [X3: v,Y3: v,Z2: v] :
( ( A1 = X3 )
& ( A2 = Z2 )
& ( member_v @ Y3 @ ( successors @ X3 ) )
& ( sCC_Bl649662514949026229able_v @ successors @ Y3 @ Z2 ) ) ) ) ).
% reachable.simps
thf(fact_226_reachable__edge,axiom,
! [Y2: v,X2: v] :
( ( member_v @ Y2 @ ( successors @ X2 ) )
=> ( sCC_Bl649662514949026229able_v @ successors @ X2 @ Y2 ) ) ).
% reachable_edge
thf(fact_227_reachable__end__induct,axiom,
! [X2: v,Y2: v,P: v > v > $o] :
( ( sCC_Bl649662514949026229able_v @ successors @ X2 @ Y2 )
=> ( ! [X4: v] : ( P @ X4 @ X4 )
=> ( ! [X4: v,Y: v,Z3: v] :
( ( P @ X4 @ Y )
=> ( ( member_v @ Z3 @ ( successors @ Y ) )
=> ( P @ X4 @ Z3 ) ) )
=> ( P @ X2 @ Y2 ) ) ) ) ).
% reachable_end_induct
thf(fact_228_reachable__trans,axiom,
! [X2: v,Y2: v,Z: v] :
( ( sCC_Bl649662514949026229able_v @ successors @ X2 @ Y2 )
=> ( ( sCC_Bl649662514949026229able_v @ successors @ Y2 @ Z )
=> ( sCC_Bl649662514949026229able_v @ successors @ X2 @ Z ) ) ) ).
% reachable_trans
thf(fact_229_succ__reachable,axiom,
! [X2: v,Y2: v,Z: v] :
( ( sCC_Bl649662514949026229able_v @ successors @ X2 @ Y2 )
=> ( ( member_v @ Z @ ( successors @ Y2 ) )
=> ( sCC_Bl649662514949026229able_v @ successors @ X2 @ Z ) ) ) ).
% succ_reachable
thf(fact_230_re__reachable,axiom,
! [X2: v,Y2: v] :
( ( sCC_Bl770211535891879572_end_v @ successors @ X2 @ Y2 )
=> ( sCC_Bl649662514949026229able_v @ successors @ X2 @ Y2 ) ) ).
% re_reachable
thf(fact_231_reachable__re,axiom,
! [X2: v,Y2: v] :
( ( sCC_Bl649662514949026229able_v @ successors @ X2 @ Y2 )
=> ( sCC_Bl770211535891879572_end_v @ successors @ X2 @ Y2 ) ) ).
% reachable_re
thf(fact_232_is__subscc__def,axiom,
! [S: set_v] :
( ( sCC_Bl5398416737448265317bscc_v @ successors @ S )
= ( ! [X3: v] :
( ( member_v @ X3 @ S )
=> ! [Y3: v] :
( ( member_v @ Y3 @ S )
=> ( sCC_Bl649662514949026229able_v @ successors @ X3 @ Y3 ) ) ) ) ) ).
% is_subscc_def
thf(fact_233_sup_Oright__idem,axiom,
! [A: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ ( sup_su414716646722978715od_v_v @ A @ B2 ) @ B2 )
= ( sup_su414716646722978715od_v_v @ A @ B2 ) ) ).
% sup.right_idem
thf(fact_234_sup_Oright__idem,axiom,
! [A: set_v,B2: set_v] :
( ( sup_sup_set_v @ ( sup_sup_set_v @ A @ B2 ) @ B2 )
= ( sup_sup_set_v @ A @ B2 ) ) ).
% sup.right_idem
thf(fact_235_sup_Oright__idem,axiom,
! [A: set_set_v,B2: set_set_v] :
( ( sup_sup_set_set_v @ ( sup_sup_set_set_v @ A @ B2 ) @ B2 )
= ( sup_sup_set_set_v @ A @ B2 ) ) ).
% sup.right_idem
thf(fact_236_sup__left__idem,axiom,
! [X2: set_Product_prod_v_v,Y2: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ X2 @ ( sup_su414716646722978715od_v_v @ X2 @ Y2 ) )
= ( sup_su414716646722978715od_v_v @ X2 @ Y2 ) ) ).
% sup_left_idem
thf(fact_237_sup__left__idem,axiom,
! [X2: set_v,Y2: set_v] :
( ( sup_sup_set_v @ X2 @ ( sup_sup_set_v @ X2 @ Y2 ) )
= ( sup_sup_set_v @ X2 @ Y2 ) ) ).
% sup_left_idem
thf(fact_238_sup__left__idem,axiom,
! [X2: set_set_v,Y2: set_set_v] :
( ( sup_sup_set_set_v @ X2 @ ( sup_sup_set_set_v @ X2 @ Y2 ) )
= ( sup_sup_set_set_v @ X2 @ Y2 ) ) ).
% sup_left_idem
thf(fact_239_sup_Oleft__idem,axiom,
! [A: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ A @ ( sup_su414716646722978715od_v_v @ A @ B2 ) )
= ( sup_su414716646722978715od_v_v @ A @ B2 ) ) ).
% sup.left_idem
thf(fact_240_sup_Oleft__idem,axiom,
! [A: set_v,B2: set_v] :
( ( sup_sup_set_v @ A @ ( sup_sup_set_v @ A @ B2 ) )
= ( sup_sup_set_v @ A @ B2 ) ) ).
% sup.left_idem
thf(fact_241_sup_Oleft__idem,axiom,
! [A: set_set_v,B2: set_set_v] :
( ( sup_sup_set_set_v @ A @ ( sup_sup_set_set_v @ A @ B2 ) )
= ( sup_sup_set_set_v @ A @ B2 ) ) ).
% sup.left_idem
thf(fact_242_sup__idem,axiom,
! [X2: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ X2 @ X2 )
= X2 ) ).
% sup_idem
thf(fact_243_sup__idem,axiom,
! [X2: set_v] :
( ( sup_sup_set_v @ X2 @ X2 )
= X2 ) ).
% sup_idem
thf(fact_244_sup__idem,axiom,
! [X2: set_set_v] :
( ( sup_sup_set_set_v @ X2 @ X2 )
= X2 ) ).
% sup_idem
thf(fact_245_sup_Oidem,axiom,
! [A: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ A @ A )
= A ) ).
% sup.idem
thf(fact_246_sup_Oidem,axiom,
! [A: set_v] :
( ( sup_sup_set_v @ A @ A )
= A ) ).
% sup.idem
thf(fact_247_sup_Oidem,axiom,
! [A: set_set_v] :
( ( sup_sup_set_set_v @ A @ A )
= A ) ).
% sup.idem
thf(fact_248_UnCI,axiom,
! [C: product_prod_v_v,B: set_Product_prod_v_v,A3: set_Product_prod_v_v] :
( ( ~ ( member7453568604450474000od_v_v @ C @ B )
=> ( member7453568604450474000od_v_v @ C @ A3 ) )
=> ( member7453568604450474000od_v_v @ C @ ( sup_su414716646722978715od_v_v @ A3 @ B ) ) ) ).
% UnCI
thf(fact_249_UnCI,axiom,
! [C: v,B: set_v,A3: set_v] :
( ( ~ ( member_v @ C @ B )
=> ( member_v @ C @ A3 ) )
=> ( member_v @ C @ ( sup_sup_set_v @ A3 @ B ) ) ) ).
% UnCI
thf(fact_250_UnCI,axiom,
! [C: set_v,B: set_set_v,A3: set_set_v] :
( ( ~ ( member_set_v @ C @ B )
=> ( member_set_v @ C @ A3 ) )
=> ( member_set_v @ C @ ( sup_sup_set_set_v @ A3 @ B ) ) ) ).
% UnCI
thf(fact_251_Un__iff,axiom,
! [C: product_prod_v_v,A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ C @ ( sup_su414716646722978715od_v_v @ A3 @ B ) )
= ( ( member7453568604450474000od_v_v @ C @ A3 )
| ( member7453568604450474000od_v_v @ C @ B ) ) ) ).
% Un_iff
thf(fact_252_Un__iff,axiom,
! [C: v,A3: set_v,B: set_v] :
( ( member_v @ C @ ( sup_sup_set_v @ A3 @ B ) )
= ( ( member_v @ C @ A3 )
| ( member_v @ C @ B ) ) ) ).
% Un_iff
thf(fact_253_Un__iff,axiom,
! [C: set_v,A3: set_set_v,B: set_set_v] :
( ( member_set_v @ C @ ( sup_sup_set_set_v @ A3 @ B ) )
= ( ( member_set_v @ C @ A3 )
| ( member_set_v @ C @ B ) ) ) ).
% Un_iff
thf(fact_254_subscc__add,axiom,
! [S: set_v,X2: v,Y2: v] :
( ( sCC_Bl5398416737448265317bscc_v @ successors @ S )
=> ( ( member_v @ X2 @ S )
=> ( ( sCC_Bl649662514949026229able_v @ successors @ X2 @ Y2 )
=> ( ( sCC_Bl649662514949026229able_v @ successors @ Y2 @ X2 )
=> ( sCC_Bl5398416737448265317bscc_v @ successors @ ( insert_v @ Y2 @ S ) ) ) ) ) ) ).
% subscc_add
thf(fact_255_sup__bot_Oright__neutral,axiom,
! [A: set_v] :
( ( sup_sup_set_v @ A @ bot_bot_set_v )
= A ) ).
% sup_bot.right_neutral
thf(fact_256_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_257_sup__bot_Oright__neutral,axiom,
! [A: set_set_v] :
( ( sup_sup_set_set_v @ A @ bot_bot_set_set_v )
= A ) ).
% sup_bot.right_neutral
thf(fact_258_sup__bot_Oneutr__eq__iff,axiom,
! [A: set_v,B2: set_v] :
( ( bot_bot_set_v
= ( sup_sup_set_v @ A @ B2 ) )
= ( ( A = bot_bot_set_v )
& ( B2 = bot_bot_set_v ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_259_sup__bot_Oneutr__eq__iff,axiom,
! [A: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( bot_bo723834152578015283od_v_v
= ( sup_su414716646722978715od_v_v @ A @ B2 ) )
= ( ( A = bot_bo723834152578015283od_v_v )
& ( B2 = bot_bo723834152578015283od_v_v ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_260_sup__bot_Oneutr__eq__iff,axiom,
! [A: set_set_v,B2: set_set_v] :
( ( bot_bot_set_set_v
= ( sup_sup_set_set_v @ A @ B2 ) )
= ( ( A = bot_bot_set_set_v )
& ( B2 = bot_bot_set_set_v ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_261_sup__bot_Oleft__neutral,axiom,
! [A: set_v] :
( ( sup_sup_set_v @ bot_bot_set_v @ A )
= A ) ).
% sup_bot.left_neutral
thf(fact_262_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_263_sup__bot_Oleft__neutral,axiom,
! [A: set_set_v] :
( ( sup_sup_set_set_v @ bot_bot_set_set_v @ A )
= A ) ).
% sup_bot.left_neutral
thf(fact_264_sup__bot_Oeq__neutr__iff,axiom,
! [A: set_v,B2: set_v] :
( ( ( sup_sup_set_v @ A @ B2 )
= bot_bot_set_v )
= ( ( A = bot_bot_set_v )
& ( B2 = bot_bot_set_v ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_265_sup__bot_Oeq__neutr__iff,axiom,
! [A: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( ( sup_su414716646722978715od_v_v @ A @ B2 )
= bot_bo723834152578015283od_v_v )
= ( ( A = bot_bo723834152578015283od_v_v )
& ( B2 = bot_bo723834152578015283od_v_v ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_266_sup__bot_Oeq__neutr__iff,axiom,
! [A: set_set_v,B2: set_set_v] :
( ( ( sup_sup_set_set_v @ A @ B2 )
= bot_bot_set_set_v )
= ( ( A = bot_bot_set_set_v )
& ( B2 = bot_bot_set_set_v ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_267_sup__eq__bot__iff,axiom,
! [X2: set_v,Y2: set_v] :
( ( ( sup_sup_set_v @ X2 @ Y2 )
= bot_bot_set_v )
= ( ( X2 = bot_bot_set_v )
& ( Y2 = bot_bot_set_v ) ) ) ).
% sup_eq_bot_iff
thf(fact_268_sup__eq__bot__iff,axiom,
! [X2: set_Product_prod_v_v,Y2: set_Product_prod_v_v] :
( ( ( sup_su414716646722978715od_v_v @ X2 @ Y2 )
= bot_bo723834152578015283od_v_v )
= ( ( X2 = bot_bo723834152578015283od_v_v )
& ( Y2 = bot_bo723834152578015283od_v_v ) ) ) ).
% sup_eq_bot_iff
thf(fact_269_sup__eq__bot__iff,axiom,
! [X2: set_set_v,Y2: set_set_v] :
( ( ( sup_sup_set_set_v @ X2 @ Y2 )
= bot_bot_set_set_v )
= ( ( X2 = bot_bot_set_set_v )
& ( Y2 = bot_bot_set_set_v ) ) ) ).
% sup_eq_bot_iff
thf(fact_270_bot__eq__sup__iff,axiom,
! [X2: set_v,Y2: set_v] :
( ( bot_bot_set_v
= ( sup_sup_set_v @ X2 @ Y2 ) )
= ( ( X2 = bot_bot_set_v )
& ( Y2 = bot_bot_set_v ) ) ) ).
% bot_eq_sup_iff
thf(fact_271_bot__eq__sup__iff,axiom,
! [X2: set_Product_prod_v_v,Y2: set_Product_prod_v_v] :
( ( bot_bo723834152578015283od_v_v
= ( sup_su414716646722978715od_v_v @ X2 @ Y2 ) )
= ( ( X2 = bot_bo723834152578015283od_v_v )
& ( Y2 = bot_bo723834152578015283od_v_v ) ) ) ).
% bot_eq_sup_iff
thf(fact_272_bot__eq__sup__iff,axiom,
! [X2: set_set_v,Y2: set_set_v] :
( ( bot_bot_set_set_v
= ( sup_sup_set_set_v @ X2 @ Y2 ) )
= ( ( X2 = bot_bot_set_set_v )
& ( Y2 = bot_bot_set_set_v ) ) ) ).
% bot_eq_sup_iff
thf(fact_273_sup__bot__right,axiom,
! [X2: set_v] :
( ( sup_sup_set_v @ X2 @ bot_bot_set_v )
= X2 ) ).
% sup_bot_right
thf(fact_274_sup__bot__right,axiom,
! [X2: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ X2 @ bot_bo723834152578015283od_v_v )
= X2 ) ).
% sup_bot_right
thf(fact_275_sup__bot__right,axiom,
! [X2: set_set_v] :
( ( sup_sup_set_set_v @ X2 @ bot_bot_set_set_v )
= X2 ) ).
% sup_bot_right
thf(fact_276_sup__bot__left,axiom,
! [X2: set_v] :
( ( sup_sup_set_v @ bot_bot_set_v @ X2 )
= X2 ) ).
% sup_bot_left
thf(fact_277_sup__bot__left,axiom,
! [X2: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ bot_bo723834152578015283od_v_v @ X2 )
= X2 ) ).
% sup_bot_left
thf(fact_278_sup__bot__left,axiom,
! [X2: set_set_v] :
( ( sup_sup_set_set_v @ bot_bot_set_set_v @ X2 )
= X2 ) ).
% sup_bot_left
thf(fact_279_Un__empty,axiom,
! [A3: set_v,B: set_v] :
( ( ( sup_sup_set_v @ A3 @ B )
= bot_bot_set_v )
= ( ( A3 = bot_bot_set_v )
& ( B = bot_bot_set_v ) ) ) ).
% Un_empty
thf(fact_280_Un__empty,axiom,
! [A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ( sup_su414716646722978715od_v_v @ A3 @ B )
= bot_bo723834152578015283od_v_v )
= ( ( A3 = bot_bo723834152578015283od_v_v )
& ( B = bot_bo723834152578015283od_v_v ) ) ) ).
% Un_empty
thf(fact_281_Un__empty,axiom,
! [A3: set_set_v,B: set_set_v] :
( ( ( sup_sup_set_set_v @ A3 @ B )
= bot_bot_set_set_v )
= ( ( A3 = bot_bot_set_set_v )
& ( B = bot_bot_set_set_v ) ) ) ).
% Un_empty
thf(fact_282_Un__insert__right,axiom,
! [A3: set_Product_prod_v_v,A: product_prod_v_v,B: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ A3 @ ( insert1338601472111419319od_v_v @ A @ B ) )
= ( insert1338601472111419319od_v_v @ A @ ( sup_su414716646722978715od_v_v @ A3 @ B ) ) ) ).
% Un_insert_right
thf(fact_283_Un__insert__right,axiom,
! [A3: set_v,A: v,B: set_v] :
( ( sup_sup_set_v @ A3 @ ( insert_v @ A @ B ) )
= ( insert_v @ A @ ( sup_sup_set_v @ A3 @ B ) ) ) ).
% Un_insert_right
thf(fact_284_Un__insert__right,axiom,
! [A3: set_set_v,A: set_v,B: set_set_v] :
( ( sup_sup_set_set_v @ A3 @ ( insert_set_v @ A @ B ) )
= ( insert_set_v @ A @ ( sup_sup_set_set_v @ A3 @ B ) ) ) ).
% Un_insert_right
thf(fact_285_Un__insert__left,axiom,
! [A: product_prod_v_v,B: set_Product_prod_v_v,C3: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ ( insert1338601472111419319od_v_v @ A @ B ) @ C3 )
= ( insert1338601472111419319od_v_v @ A @ ( sup_su414716646722978715od_v_v @ B @ C3 ) ) ) ).
% Un_insert_left
thf(fact_286_Un__insert__left,axiom,
! [A: v,B: set_v,C3: set_v] :
( ( sup_sup_set_v @ ( insert_v @ A @ B ) @ C3 )
= ( insert_v @ A @ ( sup_sup_set_v @ B @ C3 ) ) ) ).
% Un_insert_left
thf(fact_287_Un__insert__left,axiom,
! [A: set_v,B: set_set_v,C3: set_set_v] :
( ( sup_sup_set_set_v @ ( insert_set_v @ A @ B ) @ C3 )
= ( insert_set_v @ A @ ( sup_sup_set_set_v @ B @ C3 ) ) ) ).
% Un_insert_left
thf(fact_288_Un__Diff__cancel2,axiom,
! [B: set_Product_prod_v_v,A3: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ ( minus_4183494784930505774od_v_v @ B @ A3 ) @ A3 )
= ( sup_su414716646722978715od_v_v @ B @ A3 ) ) ).
% Un_Diff_cancel2
thf(fact_289_Un__Diff__cancel2,axiom,
! [B: set_set_v,A3: set_set_v] :
( ( sup_sup_set_set_v @ ( minus_7228012346218142266_set_v @ B @ A3 ) @ A3 )
= ( sup_sup_set_set_v @ B @ A3 ) ) ).
% Un_Diff_cancel2
thf(fact_290_Un__Diff__cancel2,axiom,
! [B: set_v,A3: set_v] :
( ( sup_sup_set_v @ ( minus_minus_set_v @ B @ A3 ) @ A3 )
= ( sup_sup_set_v @ B @ A3 ) ) ).
% Un_Diff_cancel2
thf(fact_291_Un__Diff__cancel,axiom,
! [A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ A3 @ ( minus_4183494784930505774od_v_v @ B @ A3 ) )
= ( sup_su414716646722978715od_v_v @ A3 @ B ) ) ).
% Un_Diff_cancel
thf(fact_292_Un__Diff__cancel,axiom,
! [A3: set_set_v,B: set_set_v] :
( ( sup_sup_set_set_v @ A3 @ ( minus_7228012346218142266_set_v @ B @ A3 ) )
= ( sup_sup_set_set_v @ A3 @ B ) ) ).
% Un_Diff_cancel
thf(fact_293_Un__Diff__cancel,axiom,
! [A3: set_v,B: set_v] :
( ( sup_sup_set_v @ A3 @ ( minus_minus_set_v @ B @ A3 ) )
= ( sup_sup_set_v @ A3 @ B ) ) ).
% Un_Diff_cancel
thf(fact_294_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_295_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_296_set__append,axiom,
! [Xs: list_set_v,Ys: list_set_v] :
( ( set_set_v2 @ ( append_set_v @ Xs @ Ys ) )
= ( sup_sup_set_set_v @ ( set_set_v2 @ Xs ) @ ( set_set_v2 @ Ys ) ) ) ).
% set_append
thf(fact_297_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_298_sccE,axiom,
! [S: set_v,X2: v,Y2: v] :
( ( sCC_Bloemen_is_scc_v @ successors @ S )
=> ( ( member_v @ X2 @ S )
=> ( ( sCC_Bl649662514949026229able_v @ successors @ X2 @ Y2 )
=> ( ( sCC_Bl649662514949026229able_v @ successors @ Y2 @ X2 )
=> ( member_v @ Y2 @ S ) ) ) ) ) ).
% sccE
thf(fact_299_graph_Ois__subscc_Ocong,axiom,
sCC_Bl5398416737448265317bscc_v = sCC_Bl5398416737448265317bscc_v ).
% graph.is_subscc.cong
thf(fact_300_graph_Oreachable__succ,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,Y2: product_prod_v_v,X2: product_prod_v_v,Z: product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( member7453568604450474000od_v_v @ Y2 @ ( Successors @ X2 ) )
=> ( ( sCC_Bl4981926079593201289od_v_v @ Successors @ Y2 @ Z )
=> ( sCC_Bl4981926079593201289od_v_v @ Successors @ X2 @ Z ) ) ) ) ).
% graph.reachable_succ
thf(fact_301_graph_Oreachable__succ,axiom,
! [Vertices: set_v,Successors: v > set_v,Y2: v,X2: v,Z: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( member_v @ Y2 @ ( Successors @ X2 ) )
=> ( ( sCC_Bl649662514949026229able_v @ Successors @ Y2 @ Z )
=> ( sCC_Bl649662514949026229able_v @ Successors @ X2 @ Z ) ) ) ) ).
% graph.reachable_succ
thf(fact_302_graph_Oreachable__refl,axiom,
! [Vertices: set_v,Successors: v > set_v,X2: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( sCC_Bl649662514949026229able_v @ Successors @ X2 @ X2 ) ) ).
% graph.reachable_refl
thf(fact_303_graph_Oreachable__end__induct,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,X2: product_prod_v_v,Y2: product_prod_v_v,P: product_prod_v_v > product_prod_v_v > $o] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( sCC_Bl4981926079593201289od_v_v @ Successors @ X2 @ Y2 )
=> ( ! [X4: product_prod_v_v] : ( P @ X4 @ X4 )
=> ( ! [X4: product_prod_v_v,Y: product_prod_v_v,Z3: product_prod_v_v] :
( ( P @ X4 @ Y )
=> ( ( member7453568604450474000od_v_v @ Z3 @ ( Successors @ Y ) )
=> ( P @ X4 @ Z3 ) ) )
=> ( P @ X2 @ Y2 ) ) ) ) ) ).
% graph.reachable_end_induct
thf(fact_304_graph_Oreachable__end__induct,axiom,
! [Vertices: set_v,Successors: v > set_v,X2: v,Y2: v,P: v > v > $o] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl649662514949026229able_v @ Successors @ X2 @ Y2 )
=> ( ! [X4: v] : ( P @ X4 @ X4 )
=> ( ! [X4: v,Y: v,Z3: v] :
( ( P @ X4 @ Y )
=> ( ( member_v @ Z3 @ ( Successors @ Y ) )
=> ( P @ X4 @ Z3 ) ) )
=> ( P @ X2 @ Y2 ) ) ) ) ) ).
% graph.reachable_end_induct
thf(fact_305_sup__left__commute,axiom,
! [X2: set_Product_prod_v_v,Y2: set_Product_prod_v_v,Z: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ X2 @ ( sup_su414716646722978715od_v_v @ Y2 @ Z ) )
= ( sup_su414716646722978715od_v_v @ Y2 @ ( sup_su414716646722978715od_v_v @ X2 @ Z ) ) ) ).
% sup_left_commute
thf(fact_306_sup__left__commute,axiom,
! [X2: set_v,Y2: set_v,Z: set_v] :
( ( sup_sup_set_v @ X2 @ ( sup_sup_set_v @ Y2 @ Z ) )
= ( sup_sup_set_v @ Y2 @ ( sup_sup_set_v @ X2 @ Z ) ) ) ).
% sup_left_commute
thf(fact_307_sup__left__commute,axiom,
! [X2: set_set_v,Y2: set_set_v,Z: set_set_v] :
( ( sup_sup_set_set_v @ X2 @ ( sup_sup_set_set_v @ Y2 @ Z ) )
= ( sup_sup_set_set_v @ Y2 @ ( sup_sup_set_set_v @ X2 @ Z ) ) ) ).
% sup_left_commute
thf(fact_308_sup_Oleft__commute,axiom,
! [B2: set_Product_prod_v_v,A: set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ B2 @ ( sup_su414716646722978715od_v_v @ A @ C ) )
= ( sup_su414716646722978715od_v_v @ A @ ( sup_su414716646722978715od_v_v @ B2 @ C ) ) ) ).
% sup.left_commute
thf(fact_309_sup_Oleft__commute,axiom,
! [B2: set_v,A: set_v,C: set_v] :
( ( sup_sup_set_v @ B2 @ ( sup_sup_set_v @ A @ C ) )
= ( sup_sup_set_v @ A @ ( sup_sup_set_v @ B2 @ C ) ) ) ).
% sup.left_commute
thf(fact_310_sup_Oleft__commute,axiom,
! [B2: set_set_v,A: set_set_v,C: set_set_v] :
( ( sup_sup_set_set_v @ B2 @ ( sup_sup_set_set_v @ A @ C ) )
= ( sup_sup_set_set_v @ A @ ( sup_sup_set_set_v @ B2 @ C ) ) ) ).
% sup.left_commute
thf(fact_311_graph_Oreachable__trans,axiom,
! [Vertices: set_v,Successors: v > set_v,X2: v,Y2: v,Z: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl649662514949026229able_v @ Successors @ X2 @ Y2 )
=> ( ( sCC_Bl649662514949026229able_v @ Successors @ Y2 @ Z )
=> ( sCC_Bl649662514949026229able_v @ Successors @ X2 @ Z ) ) ) ) ).
% graph.reachable_trans
thf(fact_312_graph_Oreachable_Osimps,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,A1: product_prod_v_v,A2: product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( sCC_Bl4981926079593201289od_v_v @ Successors @ A1 @ A2 )
= ( ? [X3: product_prod_v_v] :
( ( A1 = X3 )
& ( A2 = X3 ) )
| ? [X3: product_prod_v_v,Y3: product_prod_v_v,Z2: product_prod_v_v] :
( ( A1 = X3 )
& ( A2 = Z2 )
& ( member7453568604450474000od_v_v @ Y3 @ ( Successors @ X3 ) )
& ( sCC_Bl4981926079593201289od_v_v @ Successors @ Y3 @ Z2 ) ) ) ) ) ).
% graph.reachable.simps
thf(fact_313_graph_Oreachable_Osimps,axiom,
! [Vertices: set_v,Successors: v > set_v,A1: v,A2: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl649662514949026229able_v @ Successors @ A1 @ A2 )
= ( ? [X3: v] :
( ( A1 = X3 )
& ( A2 = X3 ) )
| ? [X3: v,Y3: v,Z2: v] :
( ( A1 = X3 )
& ( A2 = Z2 )
& ( member_v @ Y3 @ ( Successors @ X3 ) )
& ( sCC_Bl649662514949026229able_v @ Successors @ Y3 @ Z2 ) ) ) ) ) ).
% graph.reachable.simps
thf(fact_314_graph_Oreachable_Ocases,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,A1: product_prod_v_v,A2: product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( sCC_Bl4981926079593201289od_v_v @ Successors @ A1 @ A2 )
=> ( ( A2 != A1 )
=> ~ ! [Y: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ Y @ ( Successors @ A1 ) )
=> ~ ( sCC_Bl4981926079593201289od_v_v @ Successors @ Y @ A2 ) ) ) ) ) ).
% graph.reachable.cases
thf(fact_315_graph_Oreachable_Ocases,axiom,
! [Vertices: set_v,Successors: v > set_v,A1: v,A2: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl649662514949026229able_v @ Successors @ A1 @ A2 )
=> ( ( A2 != A1 )
=> ~ ! [Y: v] :
( ( member_v @ Y @ ( Successors @ A1 ) )
=> ~ ( sCC_Bl649662514949026229able_v @ Successors @ Y @ A2 ) ) ) ) ) ).
% graph.reachable.cases
thf(fact_316_graph_Osucc__reachable,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,X2: product_prod_v_v,Y2: product_prod_v_v,Z: product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( sCC_Bl4981926079593201289od_v_v @ Successors @ X2 @ Y2 )
=> ( ( member7453568604450474000od_v_v @ Z @ ( Successors @ Y2 ) )
=> ( sCC_Bl4981926079593201289od_v_v @ Successors @ X2 @ Z ) ) ) ) ).
% graph.succ_reachable
thf(fact_317_graph_Osucc__reachable,axiom,
! [Vertices: set_v,Successors: v > set_v,X2: v,Y2: v,Z: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl649662514949026229able_v @ Successors @ X2 @ Y2 )
=> ( ( member_v @ Z @ ( Successors @ Y2 ) )
=> ( sCC_Bl649662514949026229able_v @ Successors @ X2 @ Z ) ) ) ) ).
% graph.succ_reachable
thf(fact_318_graph_Oreachable__edge,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,Y2: product_prod_v_v,X2: product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( member7453568604450474000od_v_v @ Y2 @ ( Successors @ X2 ) )
=> ( sCC_Bl4981926079593201289od_v_v @ Successors @ X2 @ Y2 ) ) ) ).
% graph.reachable_edge
thf(fact_319_graph_Oreachable__edge,axiom,
! [Vertices: set_v,Successors: v > set_v,Y2: v,X2: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( member_v @ Y2 @ ( Successors @ X2 ) )
=> ( sCC_Bl649662514949026229able_v @ Successors @ X2 @ Y2 ) ) ) ).
% graph.reachable_edge
thf(fact_320_graph_Oreachable_Ocong,axiom,
sCC_Bl649662514949026229able_v = sCC_Bl649662514949026229able_v ).
% graph.reachable.cong
thf(fact_321_graph_Ois__subscc__def,axiom,
! [Vertices: set_v,Successors: v > set_v,S: set_v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl5398416737448265317bscc_v @ Successors @ S )
= ( ! [X3: v] :
( ( member_v @ X3 @ S )
=> ! [Y3: v] :
( ( member_v @ Y3 @ S )
=> ( sCC_Bl649662514949026229able_v @ Successors @ X3 @ Y3 ) ) ) ) ) ) ).
% graph.is_subscc_def
thf(fact_322_sup__commute,axiom,
( sup_su414716646722978715od_v_v
= ( ^ [X3: set_Product_prod_v_v,Y3: set_Product_prod_v_v] : ( sup_su414716646722978715od_v_v @ Y3 @ X3 ) ) ) ).
% sup_commute
thf(fact_323_sup__commute,axiom,
( sup_sup_set_v
= ( ^ [X3: set_v,Y3: set_v] : ( sup_sup_set_v @ Y3 @ X3 ) ) ) ).
% sup_commute
thf(fact_324_sup__commute,axiom,
( sup_sup_set_set_v
= ( ^ [X3: set_set_v,Y3: set_set_v] : ( sup_sup_set_set_v @ Y3 @ X3 ) ) ) ).
% sup_commute
thf(fact_325_sup_Ocommute,axiom,
( sup_su414716646722978715od_v_v
= ( ^ [A4: set_Product_prod_v_v,B4: set_Product_prod_v_v] : ( sup_su414716646722978715od_v_v @ B4 @ A4 ) ) ) ).
% sup.commute
thf(fact_326_sup_Ocommute,axiom,
( sup_sup_set_v
= ( ^ [A4: set_v,B4: set_v] : ( sup_sup_set_v @ B4 @ A4 ) ) ) ).
% sup.commute
thf(fact_327_sup_Ocommute,axiom,
( sup_sup_set_set_v
= ( ^ [A4: set_set_v,B4: set_set_v] : ( sup_sup_set_set_v @ B4 @ A4 ) ) ) ).
% sup.commute
thf(fact_328_sup__assoc,axiom,
! [X2: set_Product_prod_v_v,Y2: set_Product_prod_v_v,Z: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ ( sup_su414716646722978715od_v_v @ X2 @ Y2 ) @ Z )
= ( sup_su414716646722978715od_v_v @ X2 @ ( sup_su414716646722978715od_v_v @ Y2 @ Z ) ) ) ).
% sup_assoc
thf(fact_329_sup__assoc,axiom,
! [X2: set_v,Y2: set_v,Z: set_v] :
( ( sup_sup_set_v @ ( sup_sup_set_v @ X2 @ Y2 ) @ Z )
= ( sup_sup_set_v @ X2 @ ( sup_sup_set_v @ Y2 @ Z ) ) ) ).
% sup_assoc
thf(fact_330_sup__assoc,axiom,
! [X2: set_set_v,Y2: set_set_v,Z: set_set_v] :
( ( sup_sup_set_set_v @ ( sup_sup_set_set_v @ X2 @ Y2 ) @ Z )
= ( sup_sup_set_set_v @ X2 @ ( sup_sup_set_set_v @ Y2 @ Z ) ) ) ).
% sup_assoc
thf(fact_331_sup_Oassoc,axiom,
! [A: set_Product_prod_v_v,B2: set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ ( sup_su414716646722978715od_v_v @ A @ B2 ) @ C )
= ( sup_su414716646722978715od_v_v @ A @ ( sup_su414716646722978715od_v_v @ B2 @ C ) ) ) ).
% sup.assoc
thf(fact_332_sup_Oassoc,axiom,
! [A: set_v,B2: set_v,C: set_v] :
( ( sup_sup_set_v @ ( sup_sup_set_v @ A @ B2 ) @ C )
= ( sup_sup_set_v @ A @ ( sup_sup_set_v @ B2 @ C ) ) ) ).
% sup.assoc
thf(fact_333_sup_Oassoc,axiom,
! [A: set_set_v,B2: set_set_v,C: set_set_v] :
( ( sup_sup_set_set_v @ ( sup_sup_set_set_v @ A @ B2 ) @ C )
= ( sup_sup_set_set_v @ A @ ( sup_sup_set_set_v @ B2 @ C ) ) ) ).
% sup.assoc
thf(fact_334_inf__sup__aci_I5_J,axiom,
( sup_su414716646722978715od_v_v
= ( ^ [X3: set_Product_prod_v_v,Y3: set_Product_prod_v_v] : ( sup_su414716646722978715od_v_v @ Y3 @ X3 ) ) ) ).
% inf_sup_aci(5)
thf(fact_335_inf__sup__aci_I5_J,axiom,
( sup_sup_set_v
= ( ^ [X3: set_v,Y3: set_v] : ( sup_sup_set_v @ Y3 @ X3 ) ) ) ).
% inf_sup_aci(5)
thf(fact_336_inf__sup__aci_I5_J,axiom,
( sup_sup_set_set_v
= ( ^ [X3: set_set_v,Y3: set_set_v] : ( sup_sup_set_set_v @ Y3 @ X3 ) ) ) ).
% inf_sup_aci(5)
thf(fact_337_inf__sup__aci_I6_J,axiom,
! [X2: set_Product_prod_v_v,Y2: set_Product_prod_v_v,Z: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ ( sup_su414716646722978715od_v_v @ X2 @ Y2 ) @ Z )
= ( sup_su414716646722978715od_v_v @ X2 @ ( sup_su414716646722978715od_v_v @ Y2 @ Z ) ) ) ).
% inf_sup_aci(6)
thf(fact_338_inf__sup__aci_I6_J,axiom,
! [X2: set_v,Y2: set_v,Z: set_v] :
( ( sup_sup_set_v @ ( sup_sup_set_v @ X2 @ Y2 ) @ Z )
= ( sup_sup_set_v @ X2 @ ( sup_sup_set_v @ Y2 @ Z ) ) ) ).
% inf_sup_aci(6)
thf(fact_339_inf__sup__aci_I6_J,axiom,
! [X2: set_set_v,Y2: set_set_v,Z: set_set_v] :
( ( sup_sup_set_set_v @ ( sup_sup_set_set_v @ X2 @ Y2 ) @ Z )
= ( sup_sup_set_set_v @ X2 @ ( sup_sup_set_set_v @ Y2 @ Z ) ) ) ).
% inf_sup_aci(6)
thf(fact_340_inf__sup__aci_I7_J,axiom,
! [X2: set_Product_prod_v_v,Y2: set_Product_prod_v_v,Z: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ X2 @ ( sup_su414716646722978715od_v_v @ Y2 @ Z ) )
= ( sup_su414716646722978715od_v_v @ Y2 @ ( sup_su414716646722978715od_v_v @ X2 @ Z ) ) ) ).
% inf_sup_aci(7)
thf(fact_341_inf__sup__aci_I7_J,axiom,
! [X2: set_v,Y2: set_v,Z: set_v] :
( ( sup_sup_set_v @ X2 @ ( sup_sup_set_v @ Y2 @ Z ) )
= ( sup_sup_set_v @ Y2 @ ( sup_sup_set_v @ X2 @ Z ) ) ) ).
% inf_sup_aci(7)
thf(fact_342_inf__sup__aci_I7_J,axiom,
! [X2: set_set_v,Y2: set_set_v,Z: set_set_v] :
( ( sup_sup_set_set_v @ X2 @ ( sup_sup_set_set_v @ Y2 @ Z ) )
= ( sup_sup_set_set_v @ Y2 @ ( sup_sup_set_set_v @ X2 @ Z ) ) ) ).
% inf_sup_aci(7)
thf(fact_343_inf__sup__aci_I8_J,axiom,
! [X2: set_Product_prod_v_v,Y2: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ X2 @ ( sup_su414716646722978715od_v_v @ X2 @ Y2 ) )
= ( sup_su414716646722978715od_v_v @ X2 @ Y2 ) ) ).
% inf_sup_aci(8)
thf(fact_344_inf__sup__aci_I8_J,axiom,
! [X2: set_v,Y2: set_v] :
( ( sup_sup_set_v @ X2 @ ( sup_sup_set_v @ X2 @ Y2 ) )
= ( sup_sup_set_v @ X2 @ Y2 ) ) ).
% inf_sup_aci(8)
thf(fact_345_inf__sup__aci_I8_J,axiom,
! [X2: set_set_v,Y2: set_set_v] :
( ( sup_sup_set_set_v @ X2 @ ( sup_sup_set_set_v @ X2 @ Y2 ) )
= ( sup_sup_set_set_v @ X2 @ Y2 ) ) ).
% inf_sup_aci(8)
thf(fact_346_UnE,axiom,
! [C: product_prod_v_v,A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ C @ ( sup_su414716646722978715od_v_v @ A3 @ B ) )
=> ( ~ ( member7453568604450474000od_v_v @ C @ A3 )
=> ( member7453568604450474000od_v_v @ C @ B ) ) ) ).
% UnE
thf(fact_347_UnE,axiom,
! [C: v,A3: set_v,B: set_v] :
( ( member_v @ C @ ( sup_sup_set_v @ A3 @ B ) )
=> ( ~ ( member_v @ C @ A3 )
=> ( member_v @ C @ B ) ) ) ).
% UnE
thf(fact_348_UnE,axiom,
! [C: set_v,A3: set_set_v,B: set_set_v] :
( ( member_set_v @ C @ ( sup_sup_set_set_v @ A3 @ B ) )
=> ( ~ ( member_set_v @ C @ A3 )
=> ( member_set_v @ C @ B ) ) ) ).
% UnE
thf(fact_349_UnI1,axiom,
! [C: product_prod_v_v,A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ C @ A3 )
=> ( member7453568604450474000od_v_v @ C @ ( sup_su414716646722978715od_v_v @ A3 @ B ) ) ) ).
% UnI1
thf(fact_350_UnI1,axiom,
! [C: v,A3: set_v,B: set_v] :
( ( member_v @ C @ A3 )
=> ( member_v @ C @ ( sup_sup_set_v @ A3 @ B ) ) ) ).
% UnI1
thf(fact_351_UnI1,axiom,
! [C: set_v,A3: set_set_v,B: set_set_v] :
( ( member_set_v @ C @ A3 )
=> ( member_set_v @ C @ ( sup_sup_set_set_v @ A3 @ B ) ) ) ).
% UnI1
thf(fact_352_UnI2,axiom,
! [C: product_prod_v_v,B: set_Product_prod_v_v,A3: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ C @ B )
=> ( member7453568604450474000od_v_v @ C @ ( sup_su414716646722978715od_v_v @ A3 @ B ) ) ) ).
% UnI2
thf(fact_353_UnI2,axiom,
! [C: v,B: set_v,A3: set_v] :
( ( member_v @ C @ B )
=> ( member_v @ C @ ( sup_sup_set_v @ A3 @ B ) ) ) ).
% UnI2
thf(fact_354_UnI2,axiom,
! [C: set_v,B: set_set_v,A3: set_set_v] :
( ( member_set_v @ C @ B )
=> ( member_set_v @ C @ ( sup_sup_set_set_v @ A3 @ B ) ) ) ).
% UnI2
thf(fact_355_bex__Un,axiom,
! [A3: set_Product_prod_v_v,B: set_Product_prod_v_v,P: product_prod_v_v > $o] :
( ( ? [X3: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X3 @ ( sup_su414716646722978715od_v_v @ A3 @ B ) )
& ( P @ X3 ) ) )
= ( ? [X3: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X3 @ A3 )
& ( P @ X3 ) )
| ? [X3: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X3 @ B )
& ( P @ X3 ) ) ) ) ).
% bex_Un
thf(fact_356_bex__Un,axiom,
! [A3: set_v,B: set_v,P: v > $o] :
( ( ? [X3: v] :
( ( member_v @ X3 @ ( sup_sup_set_v @ A3 @ B ) )
& ( P @ X3 ) ) )
= ( ? [X3: v] :
( ( member_v @ X3 @ A3 )
& ( P @ X3 ) )
| ? [X3: v] :
( ( member_v @ X3 @ B )
& ( P @ X3 ) ) ) ) ).
% bex_Un
thf(fact_357_bex__Un,axiom,
! [A3: set_set_v,B: set_set_v,P: set_v > $o] :
( ( ? [X3: set_v] :
( ( member_set_v @ X3 @ ( sup_sup_set_set_v @ A3 @ B ) )
& ( P @ X3 ) ) )
= ( ? [X3: set_v] :
( ( member_set_v @ X3 @ A3 )
& ( P @ X3 ) )
| ? [X3: set_v] :
( ( member_set_v @ X3 @ B )
& ( P @ X3 ) ) ) ) ).
% bex_Un
thf(fact_358_ball__Un,axiom,
! [A3: set_Product_prod_v_v,B: set_Product_prod_v_v,P: product_prod_v_v > $o] :
( ( ! [X3: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X3 @ ( sup_su414716646722978715od_v_v @ A3 @ B ) )
=> ( P @ X3 ) ) )
= ( ! [X3: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X3 @ A3 )
=> ( P @ X3 ) )
& ! [X3: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X3 @ B )
=> ( P @ X3 ) ) ) ) ).
% ball_Un
thf(fact_359_ball__Un,axiom,
! [A3: set_v,B: set_v,P: v > $o] :
( ( ! [X3: v] :
( ( member_v @ X3 @ ( sup_sup_set_v @ A3 @ B ) )
=> ( P @ X3 ) ) )
= ( ! [X3: v] :
( ( member_v @ X3 @ A3 )
=> ( P @ X3 ) )
& ! [X3: v] :
( ( member_v @ X3 @ B )
=> ( P @ X3 ) ) ) ) ).
% ball_Un
thf(fact_360_ball__Un,axiom,
! [A3: set_set_v,B: set_set_v,P: set_v > $o] :
( ( ! [X3: set_v] :
( ( member_set_v @ X3 @ ( sup_sup_set_set_v @ A3 @ B ) )
=> ( P @ X3 ) ) )
= ( ! [X3: set_v] :
( ( member_set_v @ X3 @ A3 )
=> ( P @ X3 ) )
& ! [X3: set_v] :
( ( member_set_v @ X3 @ B )
=> ( P @ X3 ) ) ) ) ).
% ball_Un
thf(fact_361_Un__assoc,axiom,
! [A3: set_Product_prod_v_v,B: set_Product_prod_v_v,C3: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ ( sup_su414716646722978715od_v_v @ A3 @ B ) @ C3 )
= ( sup_su414716646722978715od_v_v @ A3 @ ( sup_su414716646722978715od_v_v @ B @ C3 ) ) ) ).
% Un_assoc
thf(fact_362_Un__assoc,axiom,
! [A3: set_v,B: set_v,C3: set_v] :
( ( sup_sup_set_v @ ( sup_sup_set_v @ A3 @ B ) @ C3 )
= ( sup_sup_set_v @ A3 @ ( sup_sup_set_v @ B @ C3 ) ) ) ).
% Un_assoc
thf(fact_363_Un__assoc,axiom,
! [A3: set_set_v,B: set_set_v,C3: set_set_v] :
( ( sup_sup_set_set_v @ ( sup_sup_set_set_v @ A3 @ B ) @ C3 )
= ( sup_sup_set_set_v @ A3 @ ( sup_sup_set_set_v @ B @ C3 ) ) ) ).
% Un_assoc
thf(fact_364_Un__absorb,axiom,
! [A3: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ A3 @ A3 )
= A3 ) ).
% Un_absorb
thf(fact_365_Un__absorb,axiom,
! [A3: set_v] :
( ( sup_sup_set_v @ A3 @ A3 )
= A3 ) ).
% Un_absorb
thf(fact_366_Un__absorb,axiom,
! [A3: set_set_v] :
( ( sup_sup_set_set_v @ A3 @ A3 )
= A3 ) ).
% Un_absorb
thf(fact_367_Un__commute,axiom,
( sup_su414716646722978715od_v_v
= ( ^ [A5: set_Product_prod_v_v,B5: set_Product_prod_v_v] : ( sup_su414716646722978715od_v_v @ B5 @ A5 ) ) ) ).
% Un_commute
thf(fact_368_Un__commute,axiom,
( sup_sup_set_v
= ( ^ [A5: set_v,B5: set_v] : ( sup_sup_set_v @ B5 @ A5 ) ) ) ).
% Un_commute
thf(fact_369_Un__commute,axiom,
( sup_sup_set_set_v
= ( ^ [A5: set_set_v,B5: set_set_v] : ( sup_sup_set_set_v @ B5 @ A5 ) ) ) ).
% Un_commute
thf(fact_370_Un__left__absorb,axiom,
! [A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ A3 @ ( sup_su414716646722978715od_v_v @ A3 @ B ) )
= ( sup_su414716646722978715od_v_v @ A3 @ B ) ) ).
% Un_left_absorb
thf(fact_371_Un__left__absorb,axiom,
! [A3: set_v,B: set_v] :
( ( sup_sup_set_v @ A3 @ ( sup_sup_set_v @ A3 @ B ) )
= ( sup_sup_set_v @ A3 @ B ) ) ).
% Un_left_absorb
thf(fact_372_Un__left__absorb,axiom,
! [A3: set_set_v,B: set_set_v] :
( ( sup_sup_set_set_v @ A3 @ ( sup_sup_set_set_v @ A3 @ B ) )
= ( sup_sup_set_set_v @ A3 @ B ) ) ).
% Un_left_absorb
thf(fact_373_Un__left__commute,axiom,
! [A3: set_Product_prod_v_v,B: set_Product_prod_v_v,C3: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ A3 @ ( sup_su414716646722978715od_v_v @ B @ C3 ) )
= ( sup_su414716646722978715od_v_v @ B @ ( sup_su414716646722978715od_v_v @ A3 @ C3 ) ) ) ).
% Un_left_commute
thf(fact_374_Un__left__commute,axiom,
! [A3: set_v,B: set_v,C3: set_v] :
( ( sup_sup_set_v @ A3 @ ( sup_sup_set_v @ B @ C3 ) )
= ( sup_sup_set_v @ B @ ( sup_sup_set_v @ A3 @ C3 ) ) ) ).
% Un_left_commute
thf(fact_375_Un__left__commute,axiom,
! [A3: set_set_v,B: set_set_v,C3: set_set_v] :
( ( sup_sup_set_set_v @ A3 @ ( sup_sup_set_set_v @ B @ C3 ) )
= ( sup_sup_set_set_v @ B @ ( sup_sup_set_set_v @ A3 @ C3 ) ) ) ).
% Un_left_commute
thf(fact_376_graph_Osubscc__add,axiom,
! [Vertices: set_set_v,Successors: set_v > set_set_v,S: set_set_v,X2: set_v,Y2: set_v] :
( ( sCC_Bl5810666556806954322_set_v @ Vertices @ Successors )
=> ( ( sCC_Bl7907073126578335045_set_v @ Successors @ S )
=> ( ( member_set_v @ X2 @ S )
=> ( ( sCC_Bl7354734129683093653_set_v @ Successors @ X2 @ Y2 )
=> ( ( sCC_Bl7354734129683093653_set_v @ Successors @ Y2 @ X2 )
=> ( sCC_Bl7907073126578335045_set_v @ Successors @ ( insert_set_v @ Y2 @ S ) ) ) ) ) ) ) ).
% graph.subscc_add
thf(fact_377_graph_Osubscc__add,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,S: set_Product_prod_v_v,X2: product_prod_v_v,Y2: product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( sCC_Bl2301996248249672505od_v_v @ Successors @ S )
=> ( ( member7453568604450474000od_v_v @ X2 @ S )
=> ( ( sCC_Bl4981926079593201289od_v_v @ Successors @ X2 @ Y2 )
=> ( ( sCC_Bl4981926079593201289od_v_v @ Successors @ Y2 @ X2 )
=> ( sCC_Bl2301996248249672505od_v_v @ Successors @ ( insert1338601472111419319od_v_v @ Y2 @ S ) ) ) ) ) ) ) ).
% graph.subscc_add
thf(fact_378_graph_Osubscc__add,axiom,
! [Vertices: set_v,Successors: v > set_v,S: set_v,X2: v,Y2: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl5398416737448265317bscc_v @ Successors @ S )
=> ( ( member_v @ X2 @ S )
=> ( ( sCC_Bl649662514949026229able_v @ Successors @ X2 @ Y2 )
=> ( ( sCC_Bl649662514949026229able_v @ Successors @ Y2 @ X2 )
=> ( sCC_Bl5398416737448265317bscc_v @ Successors @ ( insert_v @ Y2 @ S ) ) ) ) ) ) ) ).
% graph.subscc_add
thf(fact_379_graph_Oreachable__re,axiom,
! [Vertices: set_v,Successors: v > set_v,X2: v,Y2: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl649662514949026229able_v @ Successors @ X2 @ Y2 )
=> ( sCC_Bl770211535891879572_end_v @ Successors @ X2 @ Y2 ) ) ) ).
% graph.reachable_re
thf(fact_380_graph_Ore__reachable,axiom,
! [Vertices: set_v,Successors: v > set_v,X2: v,Y2: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl770211535891879572_end_v @ Successors @ X2 @ Y2 )
=> ( sCC_Bl649662514949026229able_v @ Successors @ X2 @ Y2 ) ) ) ).
% graph.re_reachable
thf(fact_381_Un__empty__right,axiom,
! [A3: set_v] :
( ( sup_sup_set_v @ A3 @ bot_bot_set_v )
= A3 ) ).
% Un_empty_right
thf(fact_382_Un__empty__right,axiom,
! [A3: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ A3 @ bot_bo723834152578015283od_v_v )
= A3 ) ).
% Un_empty_right
thf(fact_383_Un__empty__right,axiom,
! [A3: set_set_v] :
( ( sup_sup_set_set_v @ A3 @ bot_bot_set_set_v )
= A3 ) ).
% Un_empty_right
thf(fact_384_Un__empty__left,axiom,
! [B: set_v] :
( ( sup_sup_set_v @ bot_bot_set_v @ B )
= B ) ).
% Un_empty_left
thf(fact_385_Un__empty__left,axiom,
! [B: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ bot_bo723834152578015283od_v_v @ B )
= B ) ).
% Un_empty_left
thf(fact_386_Un__empty__left,axiom,
! [B: set_set_v] :
( ( sup_sup_set_set_v @ bot_bot_set_set_v @ B )
= B ) ).
% Un_empty_left
thf(fact_387_Un__Diff,axiom,
! [A3: set_Product_prod_v_v,B: set_Product_prod_v_v,C3: set_Product_prod_v_v] :
( ( minus_4183494784930505774od_v_v @ ( sup_su414716646722978715od_v_v @ A3 @ B ) @ C3 )
= ( sup_su414716646722978715od_v_v @ ( minus_4183494784930505774od_v_v @ A3 @ C3 ) @ ( minus_4183494784930505774od_v_v @ B @ C3 ) ) ) ).
% Un_Diff
thf(fact_388_Un__Diff,axiom,
! [A3: set_set_v,B: set_set_v,C3: set_set_v] :
( ( minus_7228012346218142266_set_v @ ( sup_sup_set_set_v @ A3 @ B ) @ C3 )
= ( sup_sup_set_set_v @ ( minus_7228012346218142266_set_v @ A3 @ C3 ) @ ( minus_7228012346218142266_set_v @ B @ C3 ) ) ) ).
% Un_Diff
thf(fact_389_Un__Diff,axiom,
! [A3: set_v,B: set_v,C3: set_v] :
( ( minus_minus_set_v @ ( sup_sup_set_v @ A3 @ B ) @ C3 )
= ( sup_sup_set_v @ ( minus_minus_set_v @ A3 @ C3 ) @ ( minus_minus_set_v @ B @ C3 ) ) ) ).
% Un_Diff
thf(fact_390_graph_Ore__succ,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,X2: product_prod_v_v,Y2: product_prod_v_v,Z: product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( sCC_Bl4714988717384592488od_v_v @ Successors @ X2 @ Y2 )
=> ( ( member7453568604450474000od_v_v @ Z @ ( Successors @ Y2 ) )
=> ( sCC_Bl4714988717384592488od_v_v @ Successors @ X2 @ Z ) ) ) ) ).
% graph.re_succ
thf(fact_391_graph_Ore__succ,axiom,
! [Vertices: set_v,Successors: v > set_v,X2: v,Y2: v,Z: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl770211535891879572_end_v @ Successors @ X2 @ Y2 )
=> ( ( member_v @ Z @ ( Successors @ Y2 ) )
=> ( sCC_Bl770211535891879572_end_v @ Successors @ X2 @ Z ) ) ) ) ).
% graph.re_succ
thf(fact_392_graph_Ore__refl,axiom,
! [Vertices: set_v,Successors: v > set_v,X2: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( sCC_Bl770211535891879572_end_v @ Successors @ X2 @ X2 ) ) ).
% graph.re_refl
thf(fact_393_graph_Oreachable__end_Osimps,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,A1: product_prod_v_v,A2: product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( sCC_Bl4714988717384592488od_v_v @ Successors @ A1 @ A2 )
= ( ? [X3: product_prod_v_v] :
( ( A1 = X3 )
& ( A2 = X3 ) )
| ? [X3: product_prod_v_v,Y3: product_prod_v_v,Z2: product_prod_v_v] :
( ( A1 = X3 )
& ( A2 = Z2 )
& ( sCC_Bl4714988717384592488od_v_v @ Successors @ X3 @ Y3 )
& ( member7453568604450474000od_v_v @ Z2 @ ( Successors @ Y3 ) ) ) ) ) ) ).
% graph.reachable_end.simps
thf(fact_394_graph_Oreachable__end_Osimps,axiom,
! [Vertices: set_v,Successors: v > set_v,A1: v,A2: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl770211535891879572_end_v @ Successors @ A1 @ A2 )
= ( ? [X3: v] :
( ( A1 = X3 )
& ( A2 = X3 ) )
| ? [X3: v,Y3: v,Z2: v] :
( ( A1 = X3 )
& ( A2 = Z2 )
& ( sCC_Bl770211535891879572_end_v @ Successors @ X3 @ Y3 )
& ( member_v @ Z2 @ ( Successors @ Y3 ) ) ) ) ) ) ).
% graph.reachable_end.simps
thf(fact_395_graph_Oreachable__end_Ocases,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,A1: product_prod_v_v,A2: product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( sCC_Bl4714988717384592488od_v_v @ Successors @ A1 @ A2 )
=> ( ( A2 != A1 )
=> ~ ! [Y: product_prod_v_v] :
( ( sCC_Bl4714988717384592488od_v_v @ Successors @ A1 @ Y )
=> ~ ( member7453568604450474000od_v_v @ A2 @ ( Successors @ Y ) ) ) ) ) ) ).
% graph.reachable_end.cases
thf(fact_396_graph_Oreachable__end_Ocases,axiom,
! [Vertices: set_v,Successors: v > set_v,A1: v,A2: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl770211535891879572_end_v @ Successors @ A1 @ A2 )
=> ( ( A2 != A1 )
=> ~ ! [Y: v] :
( ( sCC_Bl770211535891879572_end_v @ Successors @ A1 @ Y )
=> ~ ( member_v @ A2 @ ( Successors @ Y ) ) ) ) ) ) ).
% graph.reachable_end.cases
thf(fact_397_graph_Osucc__re,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,Y2: product_prod_v_v,X2: product_prod_v_v,Z: product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( member7453568604450474000od_v_v @ Y2 @ ( Successors @ X2 ) )
=> ( ( sCC_Bl4714988717384592488od_v_v @ Successors @ Y2 @ Z )
=> ( sCC_Bl4714988717384592488od_v_v @ Successors @ X2 @ Z ) ) ) ) ).
% graph.succ_re
thf(fact_398_graph_Osucc__re,axiom,
! [Vertices: set_v,Successors: v > set_v,Y2: v,X2: v,Z: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( member_v @ Y2 @ ( Successors @ X2 ) )
=> ( ( sCC_Bl770211535891879572_end_v @ Successors @ Y2 @ Z )
=> ( sCC_Bl770211535891879572_end_v @ Successors @ X2 @ Z ) ) ) ) ).
% graph.succ_re
thf(fact_399_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_400_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_401_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_402_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_403_hd__in__set,axiom,
! [Xs: list_v] :
( ( Xs != nil_v )
=> ( member_v @ ( hd_v @ Xs ) @ ( set_v2 @ Xs ) ) ) ).
% hd_in_set
thf(fact_404_singleton__Un__iff,axiom,
! [X2: v,A3: set_v,B: set_v] :
( ( ( insert_v @ X2 @ bot_bot_set_v )
= ( sup_sup_set_v @ A3 @ B ) )
= ( ( ( A3 = bot_bot_set_v )
& ( B
= ( insert_v @ X2 @ bot_bot_set_v ) ) )
| ( ( A3
= ( insert_v @ X2 @ bot_bot_set_v ) )
& ( B = bot_bot_set_v ) )
| ( ( A3
= ( insert_v @ X2 @ bot_bot_set_v ) )
& ( B
= ( insert_v @ X2 @ bot_bot_set_v ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_405_singleton__Un__iff,axiom,
! [X2: product_prod_v_v,A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ( insert1338601472111419319od_v_v @ X2 @ bot_bo723834152578015283od_v_v )
= ( sup_su414716646722978715od_v_v @ A3 @ B ) )
= ( ( ( A3 = bot_bo723834152578015283od_v_v )
& ( B
= ( insert1338601472111419319od_v_v @ X2 @ bot_bo723834152578015283od_v_v ) ) )
| ( ( A3
= ( insert1338601472111419319od_v_v @ X2 @ bot_bo723834152578015283od_v_v ) )
& ( B = bot_bo723834152578015283od_v_v ) )
| ( ( A3
= ( insert1338601472111419319od_v_v @ X2 @ bot_bo723834152578015283od_v_v ) )
& ( B
= ( insert1338601472111419319od_v_v @ X2 @ bot_bo723834152578015283od_v_v ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_406_singleton__Un__iff,axiom,
! [X2: set_v,A3: set_set_v,B: set_set_v] :
( ( ( insert_set_v @ X2 @ bot_bot_set_set_v )
= ( sup_sup_set_set_v @ A3 @ B ) )
= ( ( ( A3 = bot_bot_set_set_v )
& ( B
= ( insert_set_v @ X2 @ bot_bot_set_set_v ) ) )
| ( ( A3
= ( insert_set_v @ X2 @ bot_bot_set_set_v ) )
& ( B = bot_bot_set_set_v ) )
| ( ( A3
= ( insert_set_v @ X2 @ bot_bot_set_set_v ) )
& ( B
= ( insert_set_v @ X2 @ bot_bot_set_set_v ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_407_Un__singleton__iff,axiom,
! [A3: set_v,B: set_v,X2: v] :
( ( ( sup_sup_set_v @ A3 @ B )
= ( insert_v @ X2 @ bot_bot_set_v ) )
= ( ( ( A3 = bot_bot_set_v )
& ( B
= ( insert_v @ X2 @ bot_bot_set_v ) ) )
| ( ( A3
= ( insert_v @ X2 @ bot_bot_set_v ) )
& ( B = bot_bot_set_v ) )
| ( ( A3
= ( insert_v @ X2 @ bot_bot_set_v ) )
& ( B
= ( insert_v @ X2 @ bot_bot_set_v ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_408_Un__singleton__iff,axiom,
! [A3: set_Product_prod_v_v,B: set_Product_prod_v_v,X2: product_prod_v_v] :
( ( ( sup_su414716646722978715od_v_v @ A3 @ B )
= ( insert1338601472111419319od_v_v @ X2 @ bot_bo723834152578015283od_v_v ) )
= ( ( ( A3 = bot_bo723834152578015283od_v_v )
& ( B
= ( insert1338601472111419319od_v_v @ X2 @ bot_bo723834152578015283od_v_v ) ) )
| ( ( A3
= ( insert1338601472111419319od_v_v @ X2 @ bot_bo723834152578015283od_v_v ) )
& ( B = bot_bo723834152578015283od_v_v ) )
| ( ( A3
= ( insert1338601472111419319od_v_v @ X2 @ bot_bo723834152578015283od_v_v ) )
& ( B
= ( insert1338601472111419319od_v_v @ X2 @ bot_bo723834152578015283od_v_v ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_409_Un__singleton__iff,axiom,
! [A3: set_set_v,B: set_set_v,X2: set_v] :
( ( ( sup_sup_set_set_v @ A3 @ B )
= ( insert_set_v @ X2 @ bot_bot_set_set_v ) )
= ( ( ( A3 = bot_bot_set_set_v )
& ( B
= ( insert_set_v @ X2 @ bot_bot_set_set_v ) ) )
| ( ( A3
= ( insert_set_v @ X2 @ bot_bot_set_set_v ) )
& ( B = bot_bot_set_set_v ) )
| ( ( A3
= ( insert_set_v @ X2 @ bot_bot_set_set_v ) )
& ( B
= ( insert_set_v @ X2 @ bot_bot_set_set_v ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_410_insert__is__Un,axiom,
( insert_v
= ( ^ [A4: v] : ( sup_sup_set_v @ ( insert_v @ A4 @ bot_bot_set_v ) ) ) ) ).
% insert_is_Un
thf(fact_411_insert__is__Un,axiom,
( insert1338601472111419319od_v_v
= ( ^ [A4: product_prod_v_v] : ( sup_su414716646722978715od_v_v @ ( insert1338601472111419319od_v_v @ A4 @ bot_bo723834152578015283od_v_v ) ) ) ) ).
% insert_is_Un
thf(fact_412_insert__is__Un,axiom,
( insert_set_v
= ( ^ [A4: set_v] : ( sup_sup_set_set_v @ ( insert_set_v @ A4 @ bot_bot_set_set_v ) ) ) ) ).
% insert_is_Un
thf(fact_413_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_414_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_415_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_416_graph_Oreachable__visited,axiom,
! [Vertices: set_v,Successors: v > set_v,E: sCC_Bl1394983891496994913t_unit,V: v,W: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl9196236973127232072t_unit @ Successors @ E )
=> ( ( member_v @ V @ ( sCC_Bl4645233313691564917t_unit @ E ) )
=> ( ( sCC_Bl649662514949026229able_v @ Successors @ V @ W )
=> ( ! [X4: v] :
( ( member_v @ X4 @ ( sCC_Bl4645233313691564917t_unit @ E ) )
=> ! [Xa: v] :
( ( member_v @ Xa @ ( minus_minus_set_v @ ( Successors @ X4 ) @ ( sCC_Bl3795065053823578884t_unit @ E @ X4 ) ) )
=> ( ( sCC_Bl649662514949026229able_v @ Successors @ V @ X4 )
=> ~ ( sCC_Bl649662514949026229able_v @ Successors @ Xa @ W ) ) ) )
=> ( member_v @ W @ ( sCC_Bl4645233313691564917t_unit @ E ) ) ) ) ) ) ) ).
% graph.reachable_visited
thf(fact_417_graph_OS__reflexive,axiom,
! [Vertices: set_v,Successors: v > set_v,E: sCC_Bl1394983891496994913t_unit,N: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl9196236973127232072t_unit @ Successors @ E )
=> ( member_v @ N @ ( sCC_Bl1280885523602775798t_unit @ E @ N ) ) ) ) ).
% graph.S_reflexive
thf(fact_418_graph_Odfs__S__hd__stack_I1_J,axiom,
! [Vertices: set_v,Successors: v > set_v,E: sCC_Bl1394983891496994913t_unit,V: v,E2: sCC_Bl1394983891496994913t_unit,N: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl9196236973127232072t_unit @ Successors @ E )
=> ( ( sCC_Bl8953792750115413617t_unit @ Successors @ V @ E @ E2 )
=> ( ( ( sCC_Bl8828226123343373779t_unit @ E )
!= nil_v )
=> ( ( member_v @ N @ ( sCC_Bl1280885523602775798t_unit @ E @ ( hd_v @ ( sCC_Bl8828226123343373779t_unit @ E ) ) ) )
=> ( ( sCC_Bl8828226123343373779t_unit @ E2 )
!= nil_v ) ) ) ) ) ) ).
% graph.dfs_S_hd_stack(1)
thf(fact_419_graph_Odfs__S__hd__stack_I2_J,axiom,
! [Vertices: set_v,Successors: v > set_v,E: sCC_Bl1394983891496994913t_unit,V: v,E2: sCC_Bl1394983891496994913t_unit,N: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl9196236973127232072t_unit @ Successors @ E )
=> ( ( sCC_Bl8953792750115413617t_unit @ Successors @ V @ E @ E2 )
=> ( ( ( sCC_Bl8828226123343373779t_unit @ E )
!= nil_v )
=> ( ( member_v @ N @ ( sCC_Bl1280885523602775798t_unit @ E @ ( hd_v @ ( sCC_Bl8828226123343373779t_unit @ E ) ) ) )
=> ( member_v @ N @ ( sCC_Bl1280885523602775798t_unit @ E2 @ ( hd_v @ ( sCC_Bl8828226123343373779t_unit @ E2 ) ) ) ) ) ) ) ) ) ).
% graph.dfs_S_hd_stack(2)
thf(fact_420_graph_Opre__dfs__def,axiom,
! [Vertices: set_v,Successors: v > set_v,V: v,E: sCC_Bl1394983891496994913t_unit] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl36166008131615352t_unit @ Successors @ V @ E )
= ( ( sCC_Bl9196236973127232072t_unit @ Successors @ E )
& ~ ( member_v @ V @ ( sCC_Bl4645233313691564917t_unit @ E ) )
& ( sCC_Bl649662514949026229able_v @ Successors @ ( sCC_Bl1090238580953940555t_unit @ E ) @ V )
& ( ( sCC_Bl3795065053823578884t_unit @ E @ V )
= bot_bot_set_v )
& ! [X3: v] :
( ( member_v @ X3 @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E ) ) )
=> ( sCC_Bl649662514949026229able_v @ Successors @ X3 @ V ) ) ) ) ) ).
% graph.pre_dfs_def
thf(fact_421_graph_Ounite__subscc,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,V: product_prod_v_v,E: sCC_Bl7326425374436813197t_unit,W: product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( sCC_Bl3607325323686918683t_unit @ Successors @ V @ E )
=> ( ( member7453568604450474000od_v_v @ W @ ( Successors @ V ) )
=> ( ~ ( member7453568604450474000od_v_v @ W @ ( sCC_Bl3878977043676959280t_unit @ E @ V ) )
=> ( ( 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 @ V @ W @ E ) @ ( hd_Product_prod_v_v @ ( sCC_Bl2021302119412358655t_unit @ ( sCC_Bl4702006153222411093od_v_v @ V @ W @ E ) ) ) ) ) ) ) ) ) ) ) ).
% graph.unite_subscc
thf(fact_422_graph_Ounite__subscc,axiom,
! [Vertices: set_v,Successors: v > set_v,V: v,E: sCC_Bl1394983891496994913t_unit,W: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl1748261141445803503t_unit @ Successors @ V @ E )
=> ( ( member_v @ W @ ( Successors @ V ) )
=> ( ~ ( member_v @ W @ ( sCC_Bl3795065053823578884t_unit @ E @ V ) )
=> ( ( member_v @ W @ ( sCC_Bl4645233313691564917t_unit @ E ) )
=> ( ~ ( member_v @ W @ ( sCC_Bl157864678168468314t_unit @ E ) )
=> ( sCC_Bl5398416737448265317bscc_v @ Successors @ ( sCC_Bl1280885523602775798t_unit @ ( sCC_Bloemen_unite_v @ V @ W @ E ) @ ( hd_v @ ( sCC_Bl8828226123343373779t_unit @ ( sCC_Bloemen_unite_v @ V @ W @ E ) ) ) ) ) ) ) ) ) ) ) ).
% graph.unite_subscc
thf(fact_423_graph_Odfs__S__tl__stack_I1_J,axiom,
! [Vertices: set_v,Successors: v > set_v,V: v,E: sCC_Bl1394983891496994913t_unit,E2: sCC_Bl1394983891496994913t_unit] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl8953792750115413617t_unit @ Successors @ V @ E @ E2 )
=> ( ( ( sCC_Bl8828226123343373779t_unit @ E )
!= nil_v )
=> ( ( sCC_Bl8828226123343373779t_unit @ E2 )
!= nil_v ) ) ) ) ).
% graph.dfs_S_tl_stack(1)
thf(fact_424_graph_Oinit__env__pre__dfs,axiom,
! [Vertices: set_v,Successors: v > set_v,V: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( sCC_Bl36166008131615352t_unit @ Successors @ V @ ( sCC_Bl7693227186847904995_env_v @ V ) ) ) ).
% graph.init_env_pre_dfs
thf(fact_425_graph_Opre__dfss__pre__dfs,axiom,
! [Vertices: set_v,Successors: v > set_v,V: v,E: sCC_Bl1394983891496994913t_unit,W: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl1748261141445803503t_unit @ Successors @ V @ E )
=> ( ~ ( member_v @ W @ ( sCC_Bl4645233313691564917t_unit @ E ) )
=> ( ( member_v @ W @ ( Successors @ V ) )
=> ( sCC_Bl36166008131615352t_unit @ Successors @ W @ E ) ) ) ) ) ).
% graph.pre_dfss_pre_dfs
thf(fact_426_graph_Ostack__visited,axiom,
! [Vertices: set_v,Successors: v > set_v,E: sCC_Bl1394983891496994913t_unit,N: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl9196236973127232072t_unit @ Successors @ E )
=> ( ( member_v @ N @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E ) ) )
=> ( member_v @ N @ ( sCC_Bl4645233313691564917t_unit @ E ) ) ) ) ) ).
% graph.stack_visited
thf(fact_427_graph_Ostack__unexplored,axiom,
! [Vertices: set_v,Successors: v > set_v,E: sCC_Bl1394983891496994913t_unit,N: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl9196236973127232072t_unit @ Successors @ E )
=> ( ( member_v @ N @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E ) ) )
=> ~ ( member_v @ N @ ( sCC_Bl157864678168468314t_unit @ E ) ) ) ) ) ).
% graph.stack_unexplored
thf(fact_428_graph_Odfs__S__tl__stack_I2_J,axiom,
! [Vertices: set_v,Successors: v > set_v,V: v,E: sCC_Bl1394983891496994913t_unit,E2: sCC_Bl1394983891496994913t_unit] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl8953792750115413617t_unit @ Successors @ V @ 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_429_graph_Ovisited__unexplored,axiom,
! [Vertices: set_v,Successors: v > set_v,E: sCC_Bl1394983891496994913t_unit,M: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl9196236973127232072t_unit @ Successors @ E )
=> ( ( member_v @ M @ ( sCC_Bl4645233313691564917t_unit @ E ) )
=> ( ~ ( member_v @ M @ ( sCC_Bl157864678168468314t_unit @ E ) )
=> ~ ! [N2: v] :
( ( member_v @ N2 @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E ) ) )
=> ~ ( member_v @ M @ ( sCC_Bl1280885523602775798t_unit @ E @ N2 ) ) ) ) ) ) ) ).
% graph.visited_unexplored
thf(fact_430_graph_Ounite__wf__env,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,V: product_prod_v_v,E: sCC_Bl7326425374436813197t_unit,W: product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( sCC_Bl3607325323686918683t_unit @ Successors @ V @ E )
=> ( ( member7453568604450474000od_v_v @ W @ ( Successors @ V ) )
=> ( ~ ( member7453568604450474000od_v_v @ W @ ( sCC_Bl3878977043676959280t_unit @ E @ V ) )
=> ( ( member7453568604450474000od_v_v @ W @ ( sCC_Bl5498988629518860705t_unit @ E ) )
=> ( ~ ( member7453568604450474000od_v_v @ W @ ( sCC_Bl5094201334446601350t_unit @ E ) )
=> ( sCC_Bl7798947040364291444t_unit @ Successors @ ( sCC_Bl4702006153222411093od_v_v @ V @ W @ E ) ) ) ) ) ) ) ) ).
% graph.unite_wf_env
thf(fact_431_graph_Ounite__wf__env,axiom,
! [Vertices: set_v,Successors: v > set_v,V: v,E: sCC_Bl1394983891496994913t_unit,W: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl1748261141445803503t_unit @ Successors @ V @ E )
=> ( ( member_v @ W @ ( Successors @ V ) )
=> ( ~ ( member_v @ W @ ( sCC_Bl3795065053823578884t_unit @ E @ V ) )
=> ( ( member_v @ W @ ( sCC_Bl4645233313691564917t_unit @ E ) )
=> ( ~ ( member_v @ W @ ( sCC_Bl157864678168468314t_unit @ E ) )
=> ( sCC_Bl9196236973127232072t_unit @ Successors @ ( sCC_Bloemen_unite_v @ V @ W @ E ) ) ) ) ) ) ) ) ).
% graph.unite_wf_env
thf(fact_432_graph_Ostack__class,axiom,
! [Vertices: set_v,Successors: v > set_v,E: sCC_Bl1394983891496994913t_unit,N: v,M: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl9196236973127232072t_unit @ Successors @ E )
=> ( ( member_v @ N @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E ) ) )
=> ( ( member_v @ M @ ( sCC_Bl1280885523602775798t_unit @ E @ N ) )
=> ( member_v @ M @ ( minus_minus_set_v @ ( sCC_Bl4645233313691564917t_unit @ E ) @ ( sCC_Bl157864678168468314t_unit @ E ) ) ) ) ) ) ) ).
% graph.stack_class
thf(fact_433_graph_Ounite__sub__env,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,V: product_prod_v_v,E: sCC_Bl7326425374436813197t_unit,W: product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( sCC_Bl3607325323686918683t_unit @ Successors @ V @ E )
=> ( ( member7453568604450474000od_v_v @ W @ ( Successors @ V ) )
=> ( ~ ( member7453568604450474000od_v_v @ W @ ( sCC_Bl3878977043676959280t_unit @ E @ V ) )
=> ( ( member7453568604450474000od_v_v @ W @ ( sCC_Bl5498988629518860705t_unit @ E ) )
=> ( ~ ( member7453568604450474000od_v_v @ W @ ( sCC_Bl5094201334446601350t_unit @ E ) )
=> ( sCC_Bl7963838319573962697t_unit @ E @ ( sCC_Bl4702006153222411093od_v_v @ V @ W @ E ) ) ) ) ) ) ) ) ).
% graph.unite_sub_env
thf(fact_434_graph_Ounite__sub__env,axiom,
! [Vertices: set_v,Successors: v > set_v,V: v,E: sCC_Bl1394983891496994913t_unit,W: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl1748261141445803503t_unit @ Successors @ V @ E )
=> ( ( member_v @ W @ ( Successors @ V ) )
=> ( ~ ( member_v @ W @ ( sCC_Bl3795065053823578884t_unit @ E @ V ) )
=> ( ( member_v @ W @ ( sCC_Bl4645233313691564917t_unit @ E ) )
=> ( ~ ( member_v @ W @ ( sCC_Bl157864678168468314t_unit @ E ) )
=> ( sCC_Bl5768913643336123637t_unit @ E @ ( sCC_Bloemen_unite_v @ V @ W @ E ) ) ) ) ) ) ) ) ).
% graph.unite_sub_env
thf(fact_435_pre__dfss__def,axiom,
! [V: v,E: sCC_Bl1394983891496994913t_unit] :
( ( sCC_Bl1748261141445803503t_unit @ successors @ V @ E )
= ( ( sCC_Bl9196236973127232072t_unit @ successors @ E )
& ( member_v @ V @ ( sCC_Bl4645233313691564917t_unit @ E ) )
& ( ( sCC_Bl8828226123343373779t_unit @ E )
!= nil_v )
& ( member_v @ V @ ( sCC_Bl1280885523602775798t_unit @ E @ ( hd_v @ ( sCC_Bl8828226123343373779t_unit @ E ) ) ) )
& ! [X3: v] :
( ( member_v @ X3 @ ( sCC_Bl3795065053823578884t_unit @ E @ V ) )
=> ( member_v @ X3 @ ( sup_sup_set_v @ ( sCC_Bl157864678168468314t_unit @ E ) @ ( sCC_Bl1280885523602775798t_unit @ E @ ( hd_v @ ( sCC_Bl8828226123343373779t_unit @ E ) ) ) ) ) )
& ! [X3: v] :
( ( member_v @ X3 @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E ) ) )
=> ( sCC_Bl649662514949026229able_v @ successors @ X3 @ V ) )
& ? [Ns2: list_v] :
( ( sCC_Bl9201514103433284750t_unit @ E )
= ( cons_v @ V @ Ns2 ) ) ) ) ).
% pre_dfss_def
thf(fact_436_graph_Opre__dfss__def,axiom,
! [Vertices: set_v,Successors: v > set_v,V: v,E: sCC_Bl1394983891496994913t_unit] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl1748261141445803503t_unit @ Successors @ V @ E )
= ( ( sCC_Bl9196236973127232072t_unit @ Successors @ E )
& ( member_v @ V @ ( sCC_Bl4645233313691564917t_unit @ E ) )
& ( ( sCC_Bl8828226123343373779t_unit @ E )
!= nil_v )
& ( member_v @ V @ ( sCC_Bl1280885523602775798t_unit @ E @ ( hd_v @ ( sCC_Bl8828226123343373779t_unit @ E ) ) ) )
& ! [X3: v] :
( ( member_v @ X3 @ ( sCC_Bl3795065053823578884t_unit @ E @ V ) )
=> ( member_v @ X3 @ ( sup_sup_set_v @ ( sCC_Bl157864678168468314t_unit @ E ) @ ( sCC_Bl1280885523602775798t_unit @ E @ ( hd_v @ ( sCC_Bl8828226123343373779t_unit @ E ) ) ) ) ) )
& ! [X3: v] :
( ( member_v @ X3 @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E ) ) )
=> ( sCC_Bl649662514949026229able_v @ Successors @ X3 @ V ) )
& ? [Ns2: list_v] :
( ( sCC_Bl9201514103433284750t_unit @ E )
= ( cons_v @ V @ Ns2 ) ) ) ) ) ).
% graph.pre_dfss_def
thf(fact_437_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_438_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_439_set__union,axiom,
! [Xs: list_set_v,Ys: list_set_v] :
( ( set_set_v2 @ ( union_set_v @ Xs @ Ys ) )
= ( sup_sup_set_set_v @ ( set_set_v2 @ Xs ) @ ( set_set_v2 @ Ys ) ) ) ).
% set_union
thf(fact_440_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_441_is__singleton__the__elem,axiom,
( is_singleton_v
= ( ^ [A5: set_v] :
( A5
= ( insert_v @ ( the_elem_v @ A5 ) @ bot_bot_set_v ) ) ) ) ).
% is_singleton_the_elem
thf(fact_442_is__singleton__the__elem,axiom,
( is_sin9198872032823709915od_v_v
= ( ^ [A5: set_Product_prod_v_v] :
( A5
= ( insert1338601472111419319od_v_v @ ( the_el5392834299063928540od_v_v @ A5 ) @ bot_bo723834152578015283od_v_v ) ) ) ) ).
% is_singleton_the_elem
thf(fact_443_is__singleton__the__elem,axiom,
( is_singleton_set_v
= ( ^ [A5: set_set_v] :
( A5
= ( insert_set_v @ ( the_elem_set_v @ A5 ) @ bot_bot_set_set_v ) ) ) ) ).
% is_singleton_the_elem
thf(fact_444_ra__empty,axiom,
! [X2: v,Y2: v] :
( ( sCC_Bl4291963740693775144ding_v @ successors @ X2 @ Y2 @ bot_bo723834152578015283od_v_v )
= ( sCC_Bl649662514949026229able_v @ successors @ X2 @ Y2 ) ) ).
% ra_empty
thf(fact_445_ra__reachable,axiom,
! [X2: v,Y2: v,E4: set_Product_prod_v_v] :
( ( sCC_Bl4291963740693775144ding_v @ successors @ X2 @ Y2 @ E4 )
=> ( sCC_Bl649662514949026229able_v @ successors @ X2 @ Y2 ) ) ).
% ra_reachable
thf(fact_446_is__singletonI,axiom,
! [X2: v] : ( is_singleton_v @ ( insert_v @ X2 @ bot_bot_set_v ) ) ).
% is_singletonI
thf(fact_447_is__singletonI,axiom,
! [X2: product_prod_v_v] : ( is_sin9198872032823709915od_v_v @ ( insert1338601472111419319od_v_v @ X2 @ bot_bo723834152578015283od_v_v ) ) ).
% is_singletonI
thf(fact_448_is__singletonI,axiom,
! [X2: set_v] : ( is_singleton_set_v @ ( insert_set_v @ X2 @ bot_bot_set_set_v ) ) ).
% is_singletonI
thf(fact_449_ra__trans,axiom,
! [X2: v,Y2: v,E4: set_Product_prod_v_v,Z: v] :
( ( sCC_Bl4291963740693775144ding_v @ successors @ X2 @ Y2 @ E4 )
=> ( ( sCC_Bl4291963740693775144ding_v @ successors @ Y2 @ Z @ E4 )
=> ( sCC_Bl4291963740693775144ding_v @ successors @ X2 @ Z @ E4 ) ) ) ).
% ra_trans
thf(fact_450_ra__refl,axiom,
! [X2: v,E4: set_Product_prod_v_v] : ( sCC_Bl4291963740693775144ding_v @ successors @ X2 @ X2 @ E4 ) ).
% ra_refl
thf(fact_451_list_Oinject,axiom,
! [X21: v,X22: list_v,Y21: v,Y22: list_v] :
( ( ( cons_v @ X21 @ X22 )
= ( cons_v @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% list.inject
thf(fact_452_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_453_list_Osimps_I15_J,axiom,
! [X21: set_v,X22: list_set_v] :
( ( set_set_v2 @ ( cons_set_v @ X21 @ X22 ) )
= ( insert_set_v @ X21 @ ( set_set_v2 @ X22 ) ) ) ).
% list.simps(15)
thf(fact_454_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_455_append1__eq__conv,axiom,
! [Xs: list_v,X2: v,Ys: list_v,Y2: v] :
( ( ( append_v @ Xs @ ( cons_v @ X2 @ nil_v ) )
= ( append_v @ Ys @ ( cons_v @ Y2 @ nil_v ) ) )
= ( ( Xs = Ys )
& ( X2 = Y2 ) ) ) ).
% append1_eq_conv
thf(fact_456_hd__Cons__tl,axiom,
! [Xs: list_v] :
( ( Xs != nil_v )
=> ( ( cons_v @ ( hd_v @ Xs ) @ ( tl_v @ Xs ) )
= Xs ) ) ).
% hd_Cons_tl
thf(fact_457_list_Ocollapse,axiom,
! [List: list_v] :
( ( List != nil_v )
=> ( ( cons_v @ ( hd_v @ List ) @ ( tl_v @ List ) )
= List ) ) ).
% list.collapse
thf(fact_458_graph_Ois__scc_Ocong,axiom,
sCC_Bloemen_is_scc_v = sCC_Bloemen_is_scc_v ).
% graph.is_scc.cong
thf(fact_459_graph_Oreachable__avoiding_Ocong,axiom,
sCC_Bl4291963740693775144ding_v = sCC_Bl4291963740693775144ding_v ).
% graph.reachable_avoiding.cong
thf(fact_460_not__Cons__self2,axiom,
! [X2: v,Xs: list_v] :
( ( cons_v @ X2 @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_461_transpose_Ocases,axiom,
! [X2: list_list_v] :
( ( X2 != nil_list_v )
=> ( ! [Xss: list_list_v] :
( X2
!= ( cons_list_v @ nil_v @ Xss ) )
=> ~ ! [X4: v,Xs3: list_v,Xss: list_list_v] :
( X2
!= ( cons_list_v @ ( cons_v @ X4 @ Xs3 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_462_bot__set__def,axiom,
( bot_bot_set_v
= ( collect_v @ bot_bot_v_o ) ) ).
% bot_set_def
thf(fact_463_bot__set__def,axiom,
( bot_bo723834152578015283od_v_v
= ( collec140062887454715474od_v_v @ bot_bo8461541820394803818_v_v_o ) ) ).
% bot_set_def
thf(fact_464_bot__set__def,axiom,
( bot_bot_set_set_v
= ( collect_set_v @ bot_bot_set_v_o ) ) ).
% bot_set_def
thf(fact_465_list_Odistinct_I1_J,axiom,
! [X21: v,X22: list_v] :
( nil_v
!= ( cons_v @ X21 @ X22 ) ) ).
% list.distinct(1)
thf(fact_466_list_OdiscI,axiom,
! [List: list_v,X21: v,X22: list_v] :
( ( List
= ( cons_v @ X21 @ X22 ) )
=> ( List != nil_v ) ) ).
% list.discI
thf(fact_467_list_Oexhaust,axiom,
! [Y2: list_v] :
( ( Y2 != nil_v )
=> ~ ! [X212: v,X222: list_v] :
( Y2
!= ( cons_v @ X212 @ X222 ) ) ) ).
% list.exhaust
thf(fact_468_remdups__adj_Ocases,axiom,
! [X2: list_v] :
( ( X2 != nil_v )
=> ( ! [X4: v] :
( X2
!= ( cons_v @ X4 @ nil_v ) )
=> ~ ! [X4: v,Y: v,Xs3: list_v] :
( X2
!= ( cons_v @ X4 @ ( cons_v @ Y @ Xs3 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_469_neq__Nil__conv,axiom,
! [Xs: list_v] :
( ( Xs != nil_v )
= ( ? [Y3: v,Ys3: list_v] :
( Xs
= ( cons_v @ Y3 @ Ys3 ) ) ) ) ).
% neq_Nil_conv
thf(fact_470_list__induct2_H,axiom,
! [P: list_v > list_v > $o,Xs: list_v,Ys: list_v] :
( ( P @ nil_v @ nil_v )
=> ( ! [X4: v,Xs3: list_v] : ( P @ ( cons_v @ X4 @ Xs3 ) @ nil_v )
=> ( ! [Y: v,Ys4: list_v] : ( P @ nil_v @ ( cons_v @ Y @ Ys4 ) )
=> ( ! [X4: v,Xs3: list_v,Y: v,Ys4: list_v] :
( ( P @ Xs3 @ Ys4 )
=> ( P @ ( cons_v @ X4 @ Xs3 ) @ ( cons_v @ Y @ Ys4 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_471_list__nonempty__induct,axiom,
! [Xs: list_v,P: list_v > $o] :
( ( Xs != nil_v )
=> ( ! [X4: v] : ( P @ ( cons_v @ X4 @ nil_v ) )
=> ( ! [X4: v,Xs3: list_v] :
( ( Xs3 != nil_v )
=> ( ( P @ Xs3 )
=> ( P @ ( cons_v @ X4 @ Xs3 ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_472_list_Oset__intros_I2_J,axiom,
! [Y2: product_prod_v_v,X22: list_P7986770385144383213od_v_v,X21: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ Y2 @ ( set_Product_prod_v_v2 @ X22 ) )
=> ( member7453568604450474000od_v_v @ Y2 @ ( set_Product_prod_v_v2 @ ( cons_P4120604216776828829od_v_v @ X21 @ X22 ) ) ) ) ).
% list.set_intros(2)
thf(fact_473_list_Oset__intros_I2_J,axiom,
! [Y2: v,X22: list_v,X21: v] :
( ( member_v @ Y2 @ ( set_v2 @ X22 ) )
=> ( member_v @ Y2 @ ( set_v2 @ ( cons_v @ X21 @ X22 ) ) ) ) ).
% list.set_intros(2)
thf(fact_474_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_475_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_476_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_477_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_478_set__ConsD,axiom,
! [Y2: product_prod_v_v,X2: product_prod_v_v,Xs: list_P7986770385144383213od_v_v] :
( ( member7453568604450474000od_v_v @ Y2 @ ( set_Product_prod_v_v2 @ ( cons_P4120604216776828829od_v_v @ X2 @ Xs ) ) )
=> ( ( Y2 = X2 )
| ( member7453568604450474000od_v_v @ Y2 @ ( set_Product_prod_v_v2 @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_479_set__ConsD,axiom,
! [Y2: v,X2: v,Xs: list_v] :
( ( member_v @ Y2 @ ( set_v2 @ ( cons_v @ X2 @ Xs ) ) )
=> ( ( Y2 = X2 )
| ( member_v @ Y2 @ ( set_v2 @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_480_append__Cons,axiom,
! [X2: v,Xs: list_v,Ys: list_v] :
( ( append_v @ ( cons_v @ X2 @ Xs ) @ Ys )
= ( cons_v @ X2 @ ( append_v @ Xs @ Ys ) ) ) ).
% append_Cons
thf(fact_481_Cons__eq__appendI,axiom,
! [X2: v,Xs1: list_v,Ys: list_v,Xs: list_v,Zs: list_v] :
( ( ( cons_v @ X2 @ Xs1 )
= Ys )
=> ( ( Xs
= ( append_v @ Xs1 @ Zs ) )
=> ( ( cons_v @ X2 @ Xs )
= ( append_v @ Ys @ Zs ) ) ) ) ).
% Cons_eq_appendI
thf(fact_482_list_Osel_I1_J,axiom,
! [X21: v,X22: list_v] :
( ( hd_v @ ( cons_v @ X21 @ X22 ) )
= X21 ) ).
% list.sel(1)
thf(fact_483_list_Osel_I3_J,axiom,
! [X21: v,X22: list_v] :
( ( tl_v @ ( cons_v @ X21 @ X22 ) )
= X22 ) ).
% list.sel(3)
thf(fact_484_graph_Ora__trans,axiom,
! [Vertices: set_v,Successors: v > set_v,X2: v,Y2: v,E4: set_Product_prod_v_v,Z: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl4291963740693775144ding_v @ Successors @ X2 @ Y2 @ E4 )
=> ( ( sCC_Bl4291963740693775144ding_v @ Successors @ Y2 @ Z @ E4 )
=> ( sCC_Bl4291963740693775144ding_v @ Successors @ X2 @ Z @ E4 ) ) ) ) ).
% graph.ra_trans
thf(fact_485_graph_Ora__refl,axiom,
! [Vertices: set_v,Successors: v > set_v,X2: v,E4: set_Product_prod_v_v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( sCC_Bl4291963740693775144ding_v @ Successors @ X2 @ X2 @ E4 ) ) ).
% graph.ra_refl
thf(fact_486_rev__nonempty__induct,axiom,
! [Xs: list_v,P: list_v > $o] :
( ( Xs != nil_v )
=> ( ! [X4: v] : ( P @ ( cons_v @ X4 @ nil_v ) )
=> ( ! [X4: v,Xs3: list_v] :
( ( Xs3 != nil_v )
=> ( ( P @ Xs3 )
=> ( P @ ( append_v @ Xs3 @ ( cons_v @ X4 @ nil_v ) ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% rev_nonempty_induct
thf(fact_487_append__eq__Cons__conv,axiom,
! [Ys: list_v,Zs: list_v,X2: v,Xs: list_v] :
( ( ( append_v @ Ys @ Zs )
= ( cons_v @ X2 @ Xs ) )
= ( ( ( Ys = nil_v )
& ( Zs
= ( cons_v @ X2 @ Xs ) ) )
| ? [Ys5: list_v] :
( ( Ys
= ( cons_v @ X2 @ Ys5 ) )
& ( ( append_v @ Ys5 @ Zs )
= Xs ) ) ) ) ).
% append_eq_Cons_conv
thf(fact_488_Cons__eq__append__conv,axiom,
! [X2: v,Xs: list_v,Ys: list_v,Zs: list_v] :
( ( ( cons_v @ X2 @ Xs )
= ( append_v @ Ys @ Zs ) )
= ( ( ( Ys = nil_v )
& ( ( cons_v @ X2 @ Xs )
= Zs ) )
| ? [Ys5: list_v] :
( ( ( cons_v @ X2 @ Ys5 )
= Ys )
& ( Xs
= ( append_v @ Ys5 @ Zs ) ) ) ) ) ).
% Cons_eq_append_conv
thf(fact_489_rev__exhaust,axiom,
! [Xs: list_v] :
( ( Xs != nil_v )
=> ~ ! [Ys4: list_v,Y: v] :
( Xs
!= ( append_v @ Ys4 @ ( cons_v @ Y @ nil_v ) ) ) ) ).
% rev_exhaust
thf(fact_490_rev__induct,axiom,
! [P: list_v > $o,Xs: list_v] :
( ( P @ nil_v )
=> ( ! [X4: v,Xs3: list_v] :
( ( P @ Xs3 )
=> ( P @ ( append_v @ Xs3 @ ( cons_v @ X4 @ nil_v ) ) ) )
=> ( P @ Xs ) ) ) ).
% rev_induct
thf(fact_491_split__list,axiom,
! [X2: product_prod_v_v,Xs: list_P7986770385144383213od_v_v] :
( ( member7453568604450474000od_v_v @ X2 @ ( set_Product_prod_v_v2 @ Xs ) )
=> ? [Ys4: list_P7986770385144383213od_v_v,Zs2: list_P7986770385144383213od_v_v] :
( Xs
= ( append2138873909117096322od_v_v @ Ys4 @ ( cons_P4120604216776828829od_v_v @ X2 @ Zs2 ) ) ) ) ).
% split_list
thf(fact_492_split__list,axiom,
! [X2: v,Xs: list_v] :
( ( member_v @ X2 @ ( set_v2 @ Xs ) )
=> ? [Ys4: list_v,Zs2: list_v] :
( Xs
= ( append_v @ Ys4 @ ( cons_v @ X2 @ Zs2 ) ) ) ) ).
% split_list
thf(fact_493_split__list__last,axiom,
! [X2: product_prod_v_v,Xs: list_P7986770385144383213od_v_v] :
( ( member7453568604450474000od_v_v @ X2 @ ( set_Product_prod_v_v2 @ Xs ) )
=> ? [Ys4: list_P7986770385144383213od_v_v,Zs2: list_P7986770385144383213od_v_v] :
( ( Xs
= ( append2138873909117096322od_v_v @ Ys4 @ ( cons_P4120604216776828829od_v_v @ X2 @ Zs2 ) ) )
& ~ ( member7453568604450474000od_v_v @ X2 @ ( set_Product_prod_v_v2 @ Zs2 ) ) ) ) ).
% split_list_last
thf(fact_494_split__list__last,axiom,
! [X2: v,Xs: list_v] :
( ( member_v @ X2 @ ( set_v2 @ Xs ) )
=> ? [Ys4: list_v,Zs2: list_v] :
( ( Xs
= ( append_v @ Ys4 @ ( cons_v @ X2 @ Zs2 ) ) )
& ~ ( member_v @ X2 @ ( set_v2 @ Zs2 ) ) ) ) ).
% split_list_last
thf(fact_495_split__list__prop,axiom,
! [Xs: list_v,P: v > $o] :
( ? [X: v] :
( ( member_v @ X @ ( set_v2 @ Xs ) )
& ( P @ X ) )
=> ? [Ys4: list_v,X4: v] :
( ? [Zs2: list_v] :
( Xs
= ( append_v @ Ys4 @ ( cons_v @ X4 @ Zs2 ) ) )
& ( P @ X4 ) ) ) ).
% split_list_prop
thf(fact_496_split__list__first,axiom,
! [X2: product_prod_v_v,Xs: list_P7986770385144383213od_v_v] :
( ( member7453568604450474000od_v_v @ X2 @ ( set_Product_prod_v_v2 @ Xs ) )
=> ? [Ys4: list_P7986770385144383213od_v_v,Zs2: list_P7986770385144383213od_v_v] :
( ( Xs
= ( append2138873909117096322od_v_v @ Ys4 @ ( cons_P4120604216776828829od_v_v @ X2 @ Zs2 ) ) )
& ~ ( member7453568604450474000od_v_v @ X2 @ ( set_Product_prod_v_v2 @ Ys4 ) ) ) ) ).
% split_list_first
thf(fact_497_split__list__first,axiom,
! [X2: v,Xs: list_v] :
( ( member_v @ X2 @ ( set_v2 @ Xs ) )
=> ? [Ys4: list_v,Zs2: list_v] :
( ( Xs
= ( append_v @ Ys4 @ ( cons_v @ X2 @ Zs2 ) ) )
& ~ ( member_v @ X2 @ ( set_v2 @ Ys4 ) ) ) ) ).
% split_list_first
thf(fact_498_split__list__propE,axiom,
! [Xs: list_v,P: v > $o] :
( ? [X: v] :
( ( member_v @ X @ ( set_v2 @ Xs ) )
& ( P @ X ) )
=> ~ ! [Ys4: list_v,X4: v] :
( ? [Zs2: list_v] :
( Xs
= ( append_v @ Ys4 @ ( cons_v @ X4 @ Zs2 ) ) )
=> ~ ( P @ X4 ) ) ) ).
% split_list_propE
thf(fact_499_append__Cons__eq__iff,axiom,
! [X2: 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 @ X2 @ ( set_Product_prod_v_v2 @ Xs ) )
=> ( ~ ( member7453568604450474000od_v_v @ X2 @ ( set_Product_prod_v_v2 @ Ys ) )
=> ( ( ( append2138873909117096322od_v_v @ Xs @ ( cons_P4120604216776828829od_v_v @ X2 @ Ys ) )
= ( append2138873909117096322od_v_v @ Xs4 @ ( cons_P4120604216776828829od_v_v @ X2 @ Ys6 ) ) )
= ( ( Xs = Xs4 )
& ( Ys = Ys6 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_500_append__Cons__eq__iff,axiom,
! [X2: v,Xs: list_v,Ys: list_v,Xs4: list_v,Ys6: list_v] :
( ~ ( member_v @ X2 @ ( set_v2 @ Xs ) )
=> ( ~ ( member_v @ X2 @ ( set_v2 @ Ys ) )
=> ( ( ( append_v @ Xs @ ( cons_v @ X2 @ Ys ) )
= ( append_v @ Xs4 @ ( cons_v @ X2 @ Ys6 ) ) )
= ( ( Xs = Xs4 )
& ( Ys = Ys6 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_501_in__set__conv__decomp,axiom,
! [X2: product_prod_v_v,Xs: list_P7986770385144383213od_v_v] :
( ( member7453568604450474000od_v_v @ X2 @ ( set_Product_prod_v_v2 @ Xs ) )
= ( ? [Ys3: list_P7986770385144383213od_v_v,Zs3: list_P7986770385144383213od_v_v] :
( Xs
= ( append2138873909117096322od_v_v @ Ys3 @ ( cons_P4120604216776828829od_v_v @ X2 @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_502_in__set__conv__decomp,axiom,
! [X2: v,Xs: list_v] :
( ( member_v @ X2 @ ( set_v2 @ Xs ) )
= ( ? [Ys3: list_v,Zs3: list_v] :
( Xs
= ( append_v @ Ys3 @ ( cons_v @ X2 @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_503_split__list__last__prop,axiom,
! [Xs: list_v,P: v > $o] :
( ? [X: v] :
( ( member_v @ X @ ( set_v2 @ Xs ) )
& ( P @ X ) )
=> ? [Ys4: list_v,X4: v,Zs2: list_v] :
( ( Xs
= ( append_v @ Ys4 @ ( cons_v @ X4 @ Zs2 ) ) )
& ( P @ X4 )
& ! [Xa2: v] :
( ( member_v @ Xa2 @ ( set_v2 @ Zs2 ) )
=> ~ ( P @ Xa2 ) ) ) ) ).
% split_list_last_prop
thf(fact_504_split__list__first__prop,axiom,
! [Xs: list_v,P: v > $o] :
( ? [X: v] :
( ( member_v @ X @ ( set_v2 @ Xs ) )
& ( P @ X ) )
=> ? [Ys4: list_v,X4: v] :
( ? [Zs2: list_v] :
( Xs
= ( append_v @ Ys4 @ ( cons_v @ X4 @ Zs2 ) ) )
& ( P @ X4 )
& ! [Xa2: v] :
( ( member_v @ Xa2 @ ( set_v2 @ Ys4 ) )
=> ~ ( P @ Xa2 ) ) ) ) ).
% split_list_first_prop
thf(fact_505_split__list__last__propE,axiom,
! [Xs: list_v,P: v > $o] :
( ? [X: v] :
( ( member_v @ X @ ( set_v2 @ Xs ) )
& ( P @ X ) )
=> ~ ! [Ys4: list_v,X4: v,Zs2: list_v] :
( ( Xs
= ( append_v @ Ys4 @ ( cons_v @ X4 @ Zs2 ) ) )
=> ( ( P @ X4 )
=> ~ ! [Xa2: v] :
( ( member_v @ Xa2 @ ( set_v2 @ Zs2 ) )
=> ~ ( P @ Xa2 ) ) ) ) ) ).
% split_list_last_propE
thf(fact_506_split__list__first__propE,axiom,
! [Xs: list_v,P: v > $o] :
( ? [X: v] :
( ( member_v @ X @ ( set_v2 @ Xs ) )
& ( P @ X ) )
=> ~ ! [Ys4: list_v,X4: v] :
( ? [Zs2: list_v] :
( Xs
= ( append_v @ Ys4 @ ( cons_v @ X4 @ Zs2 ) ) )
=> ( ( P @ X4 )
=> ~ ! [Xa2: v] :
( ( member_v @ Xa2 @ ( set_v2 @ Ys4 ) )
=> ~ ( P @ Xa2 ) ) ) ) ) ).
% split_list_first_propE
thf(fact_507_in__set__conv__decomp__last,axiom,
! [X2: product_prod_v_v,Xs: list_P7986770385144383213od_v_v] :
( ( member7453568604450474000od_v_v @ X2 @ ( set_Product_prod_v_v2 @ Xs ) )
= ( ? [Ys3: list_P7986770385144383213od_v_v,Zs3: list_P7986770385144383213od_v_v] :
( ( Xs
= ( append2138873909117096322od_v_v @ Ys3 @ ( cons_P4120604216776828829od_v_v @ X2 @ Zs3 ) ) )
& ~ ( member7453568604450474000od_v_v @ X2 @ ( set_Product_prod_v_v2 @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_508_in__set__conv__decomp__last,axiom,
! [X2: v,Xs: list_v] :
( ( member_v @ X2 @ ( set_v2 @ Xs ) )
= ( ? [Ys3: list_v,Zs3: list_v] :
( ( Xs
= ( append_v @ Ys3 @ ( cons_v @ X2 @ Zs3 ) ) )
& ~ ( member_v @ X2 @ ( set_v2 @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_509_in__set__conv__decomp__first,axiom,
! [X2: product_prod_v_v,Xs: list_P7986770385144383213od_v_v] :
( ( member7453568604450474000od_v_v @ X2 @ ( set_Product_prod_v_v2 @ Xs ) )
= ( ? [Ys3: list_P7986770385144383213od_v_v,Zs3: list_P7986770385144383213od_v_v] :
( ( Xs
= ( append2138873909117096322od_v_v @ Ys3 @ ( cons_P4120604216776828829od_v_v @ X2 @ Zs3 ) ) )
& ~ ( member7453568604450474000od_v_v @ X2 @ ( set_Product_prod_v_v2 @ Ys3 ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_510_in__set__conv__decomp__first,axiom,
! [X2: v,Xs: list_v] :
( ( member_v @ X2 @ ( set_v2 @ Xs ) )
= ( ? [Ys3: list_v,Zs3: list_v] :
( ( Xs
= ( append_v @ Ys3 @ ( cons_v @ X2 @ Zs3 ) ) )
& ~ ( member_v @ X2 @ ( set_v2 @ Ys3 ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_511_split__list__last__prop__iff,axiom,
! [Xs: list_v,P: v > $o] :
( ( ? [X3: v] :
( ( member_v @ X3 @ ( set_v2 @ Xs ) )
& ( P @ X3 ) ) )
= ( ? [Ys3: list_v,X3: v,Zs3: list_v] :
( ( Xs
= ( append_v @ Ys3 @ ( cons_v @ X3 @ Zs3 ) ) )
& ( P @ X3 )
& ! [Y3: v] :
( ( member_v @ Y3 @ ( set_v2 @ Zs3 ) )
=> ~ ( P @ Y3 ) ) ) ) ) ).
% split_list_last_prop_iff
thf(fact_512_split__list__first__prop__iff,axiom,
! [Xs: list_v,P: v > $o] :
( ( ? [X3: v] :
( ( member_v @ X3 @ ( set_v2 @ Xs ) )
& ( P @ X3 ) ) )
= ( ? [Ys3: list_v,X3: v] :
( ? [Zs3: list_v] :
( Xs
= ( append_v @ Ys3 @ ( cons_v @ X3 @ Zs3 ) ) )
& ( P @ X3 )
& ! [Y3: v] :
( ( member_v @ Y3 @ ( set_v2 @ Ys3 ) )
=> ~ ( P @ Y3 ) ) ) ) ) ).
% split_list_first_prop_iff
thf(fact_513_tl__Nil,axiom,
! [Xs: list_v] :
( ( ( tl_v @ Xs )
= nil_v )
= ( ( Xs = nil_v )
| ? [X3: v] :
( Xs
= ( cons_v @ X3 @ nil_v ) ) ) ) ).
% tl_Nil
thf(fact_514_Nil__tl,axiom,
! [Xs: list_v] :
( ( nil_v
= ( tl_v @ Xs ) )
= ( ( Xs = nil_v )
| ? [X3: v] :
( Xs
= ( cons_v @ X3 @ nil_v ) ) ) ) ).
% Nil_tl
thf(fact_515_is__singletonI_H,axiom,
! [A3: set_v] :
( ( A3 != bot_bot_set_v )
=> ( ! [X4: v,Y: v] :
( ( member_v @ X4 @ A3 )
=> ( ( member_v @ Y @ A3 )
=> ( X4 = Y ) ) )
=> ( is_singleton_v @ A3 ) ) ) ).
% is_singletonI'
thf(fact_516_is__singletonI_H,axiom,
! [A3: set_Product_prod_v_v] :
( ( A3 != bot_bo723834152578015283od_v_v )
=> ( ! [X4: product_prod_v_v,Y: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X4 @ A3 )
=> ( ( member7453568604450474000od_v_v @ Y @ A3 )
=> ( X4 = Y ) ) )
=> ( is_sin9198872032823709915od_v_v @ A3 ) ) ) ).
% is_singletonI'
thf(fact_517_is__singletonI_H,axiom,
! [A3: set_set_v] :
( ( A3 != bot_bot_set_set_v )
=> ( ! [X4: set_v,Y: set_v] :
( ( member_set_v @ X4 @ A3 )
=> ( ( member_set_v @ Y @ A3 )
=> ( X4 = Y ) ) )
=> ( is_singleton_set_v @ A3 ) ) ) ).
% is_singletonI'
thf(fact_518_graph_Ora__empty,axiom,
! [Vertices: set_v,Successors: v > set_v,X2: v,Y2: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl4291963740693775144ding_v @ Successors @ X2 @ Y2 @ bot_bo723834152578015283od_v_v )
= ( sCC_Bl649662514949026229able_v @ Successors @ X2 @ Y2 ) ) ) ).
% graph.ra_empty
thf(fact_519_graph_Ora__reachable,axiom,
! [Vertices: set_v,Successors: v > set_v,X2: v,Y2: v,E4: set_Product_prod_v_v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl4291963740693775144ding_v @ Successors @ X2 @ Y2 @ E4 )
=> ( sCC_Bl649662514949026229able_v @ Successors @ X2 @ Y2 ) ) ) ).
% graph.ra_reachable
thf(fact_520_list_Oexhaust__sel,axiom,
! [List: list_v] :
( ( List != nil_v )
=> ( List
= ( cons_v @ ( hd_v @ List ) @ ( tl_v @ List ) ) ) ) ).
% list.exhaust_sel
thf(fact_521_the__elem__set,axiom,
! [X2: v] :
( ( the_elem_v @ ( set_v2 @ ( cons_v @ X2 @ nil_v ) ) )
= X2 ) ).
% the_elem_set
thf(fact_522_is__singletonE,axiom,
! [A3: set_v] :
( ( is_singleton_v @ A3 )
=> ~ ! [X4: v] :
( A3
!= ( insert_v @ X4 @ bot_bot_set_v ) ) ) ).
% is_singletonE
thf(fact_523_is__singletonE,axiom,
! [A3: set_Product_prod_v_v] :
( ( is_sin9198872032823709915od_v_v @ A3 )
=> ~ ! [X4: product_prod_v_v] :
( A3
!= ( insert1338601472111419319od_v_v @ X4 @ bot_bo723834152578015283od_v_v ) ) ) ).
% is_singletonE
thf(fact_524_is__singletonE,axiom,
! [A3: set_set_v] :
( ( is_singleton_set_v @ A3 )
=> ~ ! [X4: set_v] :
( A3
!= ( insert_set_v @ X4 @ bot_bot_set_set_v ) ) ) ).
% is_singletonE
thf(fact_525_is__singleton__def,axiom,
( is_singleton_v
= ( ^ [A5: set_v] :
? [X3: v] :
( A5
= ( insert_v @ X3 @ bot_bot_set_v ) ) ) ) ).
% is_singleton_def
thf(fact_526_is__singleton__def,axiom,
( is_sin9198872032823709915od_v_v
= ( ^ [A5: set_Product_prod_v_v] :
? [X3: product_prod_v_v] :
( A5
= ( insert1338601472111419319od_v_v @ X3 @ bot_bo723834152578015283od_v_v ) ) ) ) ).
% is_singleton_def
thf(fact_527_is__singleton__def,axiom,
( is_singleton_set_v
= ( ^ [A5: set_set_v] :
? [X3: set_v] :
( A5
= ( insert_set_v @ X3 @ bot_bot_set_set_v ) ) ) ) ).
% is_singleton_def
thf(fact_528_graph_OsccE,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,S: set_Product_prod_v_v,X2: product_prod_v_v,Y2: product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( sCC_Bl6242042402218619277od_v_v @ Successors @ S )
=> ( ( member7453568604450474000od_v_v @ X2 @ S )
=> ( ( sCC_Bl4981926079593201289od_v_v @ Successors @ X2 @ Y2 )
=> ( ( sCC_Bl4981926079593201289od_v_v @ Successors @ Y2 @ X2 )
=> ( member7453568604450474000od_v_v @ Y2 @ S ) ) ) ) ) ) ).
% graph.sccE
thf(fact_529_graph_OsccE,axiom,
! [Vertices: set_v,Successors: v > set_v,S: set_v,X2: v,Y2: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bloemen_is_scc_v @ Successors @ S )
=> ( ( member_v @ X2 @ S )
=> ( ( sCC_Bl649662514949026229able_v @ Successors @ X2 @ Y2 )
=> ( ( sCC_Bl649662514949026229able_v @ Successors @ Y2 @ X2 )
=> ( member_v @ Y2 @ S ) ) ) ) ) ) ).
% graph.sccE
thf(fact_530_avoiding__explored,axiom,
! [E: sCC_Bl1394983891496994913t_unit,X2: v,Y2: v,E4: set_Product_prod_v_v,W: v,V: v] :
( ( sCC_Bl9196236973127232072t_unit @ successors @ E )
=> ( ( sCC_Bl4291963740693775144ding_v @ successors @ X2 @ Y2 @ E4 )
=> ( ~ ( member_v @ Y2 @ ( sCC_Bl157864678168468314t_unit @ E ) )
=> ( ( member_v @ W @ ( sCC_Bl157864678168468314t_unit @ E ) )
=> ( sCC_Bl4291963740693775144ding_v @ successors @ X2 @ Y2 @ ( sup_su414716646722978715od_v_v @ E4 @ ( insert1338601472111419319od_v_v @ ( product_Pair_v_v @ V @ W ) @ bot_bo723834152578015283od_v_v ) ) ) ) ) ) ) ).
% avoiding_explored
thf(fact_531_is__scc__def,axiom,
! [S: set_v] :
( ( sCC_Bloemen_is_scc_v @ successors @ S )
= ( ( S != bot_bot_set_v )
& ( sCC_Bl5398416737448265317bscc_v @ successors @ S )
& ! [S2: set_v] :
( ( ( ord_less_eq_set_v @ S @ S2 )
& ( sCC_Bl5398416737448265317bscc_v @ successors @ S2 ) )
=> ( S2 = S ) ) ) ) ).
% is_scc_def
thf(fact_532_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_533_sclosed,axiom,
! [X: v] :
( ( member_v @ X @ vertices )
=> ( ord_less_eq_set_v @ ( successors @ X ) @ vertices ) ) ).
% sclosed
thf(fact_534_ra__add__edge,axiom,
! [X2: v,Y2: v,E4: set_Product_prod_v_v,V: v,W: v] :
( ( sCC_Bl4291963740693775144ding_v @ successors @ X2 @ Y2 @ E4 )
=> ( ( sCC_Bl4291963740693775144ding_v @ successors @ X2 @ Y2 @ ( sup_su414716646722978715od_v_v @ E4 @ ( insert1338601472111419319od_v_v @ ( product_Pair_v_v @ V @ W ) @ bot_bo723834152578015283od_v_v ) ) )
| ( ( sCC_Bl4291963740693775144ding_v @ successors @ X2 @ V @ ( sup_su414716646722978715od_v_v @ E4 @ ( insert1338601472111419319od_v_v @ ( product_Pair_v_v @ V @ W ) @ bot_bo723834152578015283od_v_v ) ) )
& ( sCC_Bl4291963740693775144ding_v @ successors @ W @ Y2 @ ( sup_su414716646722978715od_v_v @ E4 @ ( insert1338601472111419319od_v_v @ ( product_Pair_v_v @ V @ W ) @ bot_bo723834152578015283od_v_v ) ) ) ) ) ) ).
% ra_add_edge
thf(fact_535_scc__partition,axiom,
! [S: set_v,S3: set_v,X2: v] :
( ( sCC_Bloemen_is_scc_v @ successors @ S )
=> ( ( sCC_Bloemen_is_scc_v @ successors @ S3 )
=> ( ( member_v @ X2 @ ( inf_inf_set_v @ S @ S3 ) )
=> ( S = S3 ) ) ) ) ).
% scc_partition
thf(fact_536_ra__mono,axiom,
! [X2: v,Y2: v,E4: set_Product_prod_v_v,E5: set_Product_prod_v_v] :
( ( sCC_Bl4291963740693775144ding_v @ successors @ X2 @ Y2 @ E4 )
=> ( ( ord_le7336532860387713383od_v_v @ E5 @ E4 )
=> ( sCC_Bl4291963740693775144ding_v @ successors @ X2 @ Y2 @ E5 ) ) ) ).
% ra_mono
thf(fact_537_reachable__avoiding_Ocases,axiom,
! [A1: v,A2: v,A32: set_Product_prod_v_v] :
( ( sCC_Bl4291963740693775144ding_v @ successors @ A1 @ A2 @ A32 )
=> ( ( A2 != A1 )
=> ~ ! [Y: v] :
( ( sCC_Bl4291963740693775144ding_v @ successors @ A1 @ Y @ A32 )
=> ( ( member_v @ A2 @ ( successors @ Y ) )
=> ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ Y @ A2 ) @ A32 ) ) ) ) ) ).
% reachable_avoiding.cases
thf(fact_538_ra__succ,axiom,
! [X2: v,Y2: v,E4: set_Product_prod_v_v,Z: v] :
( ( sCC_Bl4291963740693775144ding_v @ successors @ X2 @ Y2 @ E4 )
=> ( ( member_v @ Z @ ( successors @ Y2 ) )
=> ( ~ ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ Y2 @ Z ) @ E4 )
=> ( sCC_Bl4291963740693775144ding_v @ successors @ X2 @ Z @ E4 ) ) ) ) ).
% ra_succ
thf(fact_539_reachable__avoiding_Osimps,axiom,
! [A1: v,A2: v,A32: set_Product_prod_v_v] :
( ( sCC_Bl4291963740693775144ding_v @ successors @ A1 @ A2 @ A32 )
= ( ? [X3: v,E6: set_Product_prod_v_v] :
( ( A1 = X3 )
& ( A2 = X3 )
& ( A32 = E6 ) )
| ? [X3: v,Y3: v,E6: set_Product_prod_v_v,Z2: v] :
( ( A1 = X3 )
& ( A2 = Z2 )
& ( A32 = E6 )
& ( sCC_Bl4291963740693775144ding_v @ successors @ X3 @ Y3 @ E6 )
& ( member_v @ Z2 @ ( successors @ Y3 ) )
& ~ ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ Y3 @ Z2 ) @ E6 ) ) ) ) ).
% reachable_avoiding.simps
thf(fact_540_edge__ra,axiom,
! [Y2: v,X2: v,E4: set_Product_prod_v_v] :
( ( member_v @ Y2 @ ( successors @ X2 ) )
=> ( ~ ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ X2 @ Y2 ) @ E4 )
=> ( sCC_Bl4291963740693775144ding_v @ successors @ X2 @ Y2 @ E4 ) ) ) ).
% edge_ra
thf(fact_541_ra__cases,axiom,
! [X2: v,Y2: v,E4: set_Product_prod_v_v] :
( ( sCC_Bl4291963740693775144ding_v @ successors @ X2 @ Y2 @ E4 )
=> ( ( X2 = Y2 )
| ? [Z3: v] :
( ( member_v @ Z3 @ ( successors @ X2 ) )
& ~ ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ X2 @ Z3 ) @ E4 )
& ( sCC_Bl4291963740693775144ding_v @ successors @ Z3 @ Y2 @ E4 ) ) ) ) ).
% ra_cases
thf(fact_542_order__refl,axiom,
! [X2: set_v] : ( ord_less_eq_set_v @ X2 @ X2 ) ).
% order_refl
thf(fact_543_order__refl,axiom,
! [X2: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ X2 @ X2 ) ).
% order_refl
thf(fact_544_dual__order_Orefl,axiom,
! [A: set_v] : ( ord_less_eq_set_v @ A @ A ) ).
% dual_order.refl
thf(fact_545_dual__order_Orefl,axiom,
! [A: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ A @ A ) ).
% dual_order.refl
thf(fact_546_subsetI,axiom,
! [A3: set_v,B: set_v] :
( ! [X4: v] :
( ( member_v @ X4 @ A3 )
=> ( member_v @ X4 @ B ) )
=> ( ord_less_eq_set_v @ A3 @ B ) ) ).
% subsetI
thf(fact_547_subsetI,axiom,
! [A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ! [X4: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X4 @ A3 )
=> ( member7453568604450474000od_v_v @ X4 @ B ) )
=> ( ord_le7336532860387713383od_v_v @ A3 @ B ) ) ).
% subsetI
thf(fact_548_subset__antisym,axiom,
! [A3: set_v,B: set_v] :
( ( ord_less_eq_set_v @ A3 @ B )
=> ( ( ord_less_eq_set_v @ B @ A3 )
=> ( A3 = B ) ) ) ).
% subset_antisym
thf(fact_549_subset__antisym,axiom,
! [A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A3 @ B )
=> ( ( ord_le7336532860387713383od_v_v @ B @ A3 )
=> ( A3 = B ) ) ) ).
% subset_antisym
thf(fact_550_inf_Oidem,axiom,
! [A: set_v] :
( ( inf_inf_set_v @ A @ A )
= A ) ).
% inf.idem
thf(fact_551_inf__idem,axiom,
! [X2: set_v] :
( ( inf_inf_set_v @ X2 @ X2 )
= X2 ) ).
% inf_idem
thf(fact_552_inf_Oleft__idem,axiom,
! [A: set_v,B2: set_v] :
( ( inf_inf_set_v @ A @ ( inf_inf_set_v @ A @ B2 ) )
= ( inf_inf_set_v @ A @ B2 ) ) ).
% inf.left_idem
thf(fact_553_inf__left__idem,axiom,
! [X2: set_v,Y2: set_v] :
( ( inf_inf_set_v @ X2 @ ( inf_inf_set_v @ X2 @ Y2 ) )
= ( inf_inf_set_v @ X2 @ Y2 ) ) ).
% inf_left_idem
thf(fact_554_inf_Oright__idem,axiom,
! [A: set_v,B2: set_v] :
( ( inf_inf_set_v @ ( inf_inf_set_v @ A @ B2 ) @ B2 )
= ( inf_inf_set_v @ A @ B2 ) ) ).
% inf.right_idem
thf(fact_555_inf__right__idem,axiom,
! [X2: set_v,Y2: set_v] :
( ( inf_inf_set_v @ ( inf_inf_set_v @ X2 @ Y2 ) @ Y2 )
= ( inf_inf_set_v @ X2 @ Y2 ) ) ).
% inf_right_idem
thf(fact_556_IntI,axiom,
! [C: product_prod_v_v,A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ C @ A3 )
=> ( ( member7453568604450474000od_v_v @ C @ B )
=> ( member7453568604450474000od_v_v @ C @ ( inf_in6271465464967711157od_v_v @ A3 @ B ) ) ) ) ).
% IntI
thf(fact_557_IntI,axiom,
! [C: v,A3: set_v,B: set_v] :
( ( member_v @ C @ A3 )
=> ( ( member_v @ C @ B )
=> ( member_v @ C @ ( inf_inf_set_v @ A3 @ B ) ) ) ) ).
% IntI
thf(fact_558_Int__iff,axiom,
! [C: product_prod_v_v,A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ C @ ( inf_in6271465464967711157od_v_v @ A3 @ B ) )
= ( ( member7453568604450474000od_v_v @ C @ A3 )
& ( member7453568604450474000od_v_v @ C @ B ) ) ) ).
% Int_iff
thf(fact_559_Int__iff,axiom,
! [C: v,A3: set_v,B: set_v] :
( ( member_v @ C @ ( inf_inf_set_v @ A3 @ B ) )
= ( ( member_v @ C @ A3 )
& ( member_v @ C @ B ) ) ) ).
% Int_iff
thf(fact_560_le__inf__iff,axiom,
! [X2: set_v,Y2: set_v,Z: set_v] :
( ( ord_less_eq_set_v @ X2 @ ( inf_inf_set_v @ Y2 @ Z ) )
= ( ( ord_less_eq_set_v @ X2 @ Y2 )
& ( ord_less_eq_set_v @ X2 @ Z ) ) ) ).
% le_inf_iff
thf(fact_561_le__inf__iff,axiom,
! [X2: set_Product_prod_v_v,Y2: set_Product_prod_v_v,Z: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X2 @ ( inf_in6271465464967711157od_v_v @ Y2 @ Z ) )
= ( ( ord_le7336532860387713383od_v_v @ X2 @ Y2 )
& ( ord_le7336532860387713383od_v_v @ X2 @ Z ) ) ) ).
% le_inf_iff
thf(fact_562_inf_Obounded__iff,axiom,
! [A: set_v,B2: set_v,C: set_v] :
( ( ord_less_eq_set_v @ A @ ( inf_inf_set_v @ B2 @ C ) )
= ( ( ord_less_eq_set_v @ A @ B2 )
& ( ord_less_eq_set_v @ A @ C ) ) ) ).
% inf.bounded_iff
thf(fact_563_inf_Obounded__iff,axiom,
! [A: set_Product_prod_v_v,B2: set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ ( inf_in6271465464967711157od_v_v @ B2 @ C ) )
= ( ( ord_le7336532860387713383od_v_v @ A @ B2 )
& ( ord_le7336532860387713383od_v_v @ A @ C ) ) ) ).
% inf.bounded_iff
thf(fact_564_sup_Obounded__iff,axiom,
! [B2: set_set_v,C: set_set_v,A: set_set_v] :
( ( ord_le5216385588623774835_set_v @ ( sup_sup_set_set_v @ B2 @ C ) @ A )
= ( ( ord_le5216385588623774835_set_v @ B2 @ A )
& ( ord_le5216385588623774835_set_v @ C @ A ) ) ) ).
% sup.bounded_iff
thf(fact_565_sup_Obounded__iff,axiom,
! [B2: set_v,C: set_v,A: set_v] :
( ( ord_less_eq_set_v @ ( sup_sup_set_v @ B2 @ C ) @ A )
= ( ( ord_less_eq_set_v @ B2 @ A )
& ( ord_less_eq_set_v @ C @ A ) ) ) ).
% sup.bounded_iff
thf(fact_566_sup_Obounded__iff,axiom,
! [B2: 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 @ B2 @ C ) @ A )
= ( ( ord_le7336532860387713383od_v_v @ B2 @ A )
& ( ord_le7336532860387713383od_v_v @ C @ A ) ) ) ).
% sup.bounded_iff
thf(fact_567_le__sup__iff,axiom,
! [X2: set_set_v,Y2: set_set_v,Z: set_set_v] :
( ( ord_le5216385588623774835_set_v @ ( sup_sup_set_set_v @ X2 @ Y2 ) @ Z )
= ( ( ord_le5216385588623774835_set_v @ X2 @ Z )
& ( ord_le5216385588623774835_set_v @ Y2 @ Z ) ) ) ).
% le_sup_iff
thf(fact_568_le__sup__iff,axiom,
! [X2: set_v,Y2: set_v,Z: set_v] :
( ( ord_less_eq_set_v @ ( sup_sup_set_v @ X2 @ Y2 ) @ Z )
= ( ( ord_less_eq_set_v @ X2 @ Z )
& ( ord_less_eq_set_v @ Y2 @ Z ) ) ) ).
% le_sup_iff
thf(fact_569_le__sup__iff,axiom,
! [X2: set_Product_prod_v_v,Y2: set_Product_prod_v_v,Z: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ ( sup_su414716646722978715od_v_v @ X2 @ Y2 ) @ Z )
= ( ( ord_le7336532860387713383od_v_v @ X2 @ Z )
& ( ord_le7336532860387713383od_v_v @ Y2 @ Z ) ) ) ).
% le_sup_iff
thf(fact_570_empty__subsetI,axiom,
! [A3: set_set_v] : ( ord_le5216385588623774835_set_v @ bot_bot_set_set_v @ A3 ) ).
% empty_subsetI
thf(fact_571_empty__subsetI,axiom,
! [A3: set_v] : ( ord_less_eq_set_v @ bot_bot_set_v @ A3 ) ).
% empty_subsetI
thf(fact_572_empty__subsetI,axiom,
! [A3: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ bot_bo723834152578015283od_v_v @ A3 ) ).
% empty_subsetI
thf(fact_573_subset__empty,axiom,
! [A3: set_set_v] :
( ( ord_le5216385588623774835_set_v @ A3 @ bot_bot_set_set_v )
= ( A3 = bot_bot_set_set_v ) ) ).
% subset_empty
thf(fact_574_subset__empty,axiom,
! [A3: set_v] :
( ( ord_less_eq_set_v @ A3 @ bot_bot_set_v )
= ( A3 = bot_bot_set_v ) ) ).
% subset_empty
thf(fact_575_subset__empty,axiom,
! [A3: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A3 @ bot_bo723834152578015283od_v_v )
= ( A3 = bot_bo723834152578015283od_v_v ) ) ).
% subset_empty
thf(fact_576_inf__bot__right,axiom,
! [X2: set_v] :
( ( inf_inf_set_v @ X2 @ bot_bot_set_v )
= bot_bot_set_v ) ).
% inf_bot_right
thf(fact_577_inf__bot__right,axiom,
! [X2: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ X2 @ bot_bo723834152578015283od_v_v )
= bot_bo723834152578015283od_v_v ) ).
% inf_bot_right
thf(fact_578_inf__bot__right,axiom,
! [X2: set_set_v] :
( ( inf_inf_set_set_v @ X2 @ bot_bot_set_set_v )
= bot_bot_set_set_v ) ).
% inf_bot_right
thf(fact_579_inf__bot__left,axiom,
! [X2: set_v] :
( ( inf_inf_set_v @ bot_bot_set_v @ X2 )
= bot_bot_set_v ) ).
% inf_bot_left
thf(fact_580_inf__bot__left,axiom,
! [X2: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ bot_bo723834152578015283od_v_v @ X2 )
= bot_bo723834152578015283od_v_v ) ).
% inf_bot_left
thf(fact_581_inf__bot__left,axiom,
! [X2: set_set_v] :
( ( inf_inf_set_set_v @ bot_bot_set_set_v @ X2 )
= bot_bot_set_set_v ) ).
% inf_bot_left
thf(fact_582_insert__subset,axiom,
! [X2: set_v,A3: set_set_v,B: set_set_v] :
( ( ord_le5216385588623774835_set_v @ ( insert_set_v @ X2 @ A3 ) @ B )
= ( ( member_set_v @ X2 @ B )
& ( ord_le5216385588623774835_set_v @ A3 @ B ) ) ) ).
% insert_subset
thf(fact_583_insert__subset,axiom,
! [X2: v,A3: set_v,B: set_v] :
( ( ord_less_eq_set_v @ ( insert_v @ X2 @ A3 ) @ B )
= ( ( member_v @ X2 @ B )
& ( ord_less_eq_set_v @ A3 @ B ) ) ) ).
% insert_subset
thf(fact_584_insert__subset,axiom,
! [X2: product_prod_v_v,A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ ( insert1338601472111419319od_v_v @ X2 @ A3 ) @ B )
= ( ( member7453568604450474000od_v_v @ X2 @ B )
& ( ord_le7336532860387713383od_v_v @ A3 @ B ) ) ) ).
% insert_subset
thf(fact_585_sup__inf__absorb,axiom,
! [X2: set_Product_prod_v_v,Y2: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ X2 @ ( inf_in6271465464967711157od_v_v @ X2 @ Y2 ) )
= X2 ) ).
% sup_inf_absorb
thf(fact_586_sup__inf__absorb,axiom,
! [X2: set_v,Y2: set_v] :
( ( sup_sup_set_v @ X2 @ ( inf_inf_set_v @ X2 @ Y2 ) )
= X2 ) ).
% sup_inf_absorb
thf(fact_587_sup__inf__absorb,axiom,
! [X2: set_set_v,Y2: set_set_v] :
( ( sup_sup_set_set_v @ X2 @ ( inf_inf_set_set_v @ X2 @ Y2 ) )
= X2 ) ).
% sup_inf_absorb
thf(fact_588_inf__sup__absorb,axiom,
! [X2: set_Product_prod_v_v,Y2: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ X2 @ ( sup_su414716646722978715od_v_v @ X2 @ Y2 ) )
= X2 ) ).
% inf_sup_absorb
thf(fact_589_inf__sup__absorb,axiom,
! [X2: set_v,Y2: set_v] :
( ( inf_inf_set_v @ X2 @ ( sup_sup_set_v @ X2 @ Y2 ) )
= X2 ) ).
% inf_sup_absorb
thf(fact_590_inf__sup__absorb,axiom,
! [X2: set_set_v,Y2: set_set_v] :
( ( inf_inf_set_set_v @ X2 @ ( sup_sup_set_set_v @ X2 @ Y2 ) )
= X2 ) ).
% inf_sup_absorb
thf(fact_591_Int__subset__iff,axiom,
! [C3: set_v,A3: set_v,B: set_v] :
( ( ord_less_eq_set_v @ C3 @ ( inf_inf_set_v @ A3 @ B ) )
= ( ( ord_less_eq_set_v @ C3 @ A3 )
& ( ord_less_eq_set_v @ C3 @ B ) ) ) ).
% Int_subset_iff
thf(fact_592_Int__subset__iff,axiom,
! [C3: set_Product_prod_v_v,A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ C3 @ ( inf_in6271465464967711157od_v_v @ A3 @ B ) )
= ( ( ord_le7336532860387713383od_v_v @ C3 @ A3 )
& ( ord_le7336532860387713383od_v_v @ C3 @ B ) ) ) ).
% Int_subset_iff
thf(fact_593_Un__subset__iff,axiom,
! [A3: set_set_v,B: set_set_v,C3: set_set_v] :
( ( ord_le5216385588623774835_set_v @ ( sup_sup_set_set_v @ A3 @ B ) @ C3 )
= ( ( ord_le5216385588623774835_set_v @ A3 @ C3 )
& ( ord_le5216385588623774835_set_v @ B @ C3 ) ) ) ).
% Un_subset_iff
thf(fact_594_Un__subset__iff,axiom,
! [A3: set_v,B: set_v,C3: set_v] :
( ( ord_less_eq_set_v @ ( sup_sup_set_v @ A3 @ B ) @ C3 )
= ( ( ord_less_eq_set_v @ A3 @ C3 )
& ( ord_less_eq_set_v @ B @ C3 ) ) ) ).
% Un_subset_iff
thf(fact_595_Un__subset__iff,axiom,
! [A3: set_Product_prod_v_v,B: set_Product_prod_v_v,C3: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ ( sup_su414716646722978715od_v_v @ A3 @ B ) @ C3 )
= ( ( ord_le7336532860387713383od_v_v @ A3 @ C3 )
& ( ord_le7336532860387713383od_v_v @ B @ C3 ) ) ) ).
% Un_subset_iff
thf(fact_596_Int__insert__right__if1,axiom,
! [A: set_v,A3: set_set_v,B: set_set_v] :
( ( member_set_v @ A @ A3 )
=> ( ( inf_inf_set_set_v @ A3 @ ( insert_set_v @ A @ B ) )
= ( insert_set_v @ A @ ( inf_inf_set_set_v @ A3 @ B ) ) ) ) ).
% Int_insert_right_if1
thf(fact_597_Int__insert__right__if1,axiom,
! [A: product_prod_v_v,A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ A @ A3 )
=> ( ( inf_in6271465464967711157od_v_v @ A3 @ ( insert1338601472111419319od_v_v @ A @ B ) )
= ( insert1338601472111419319od_v_v @ A @ ( inf_in6271465464967711157od_v_v @ A3 @ B ) ) ) ) ).
% Int_insert_right_if1
thf(fact_598_Int__insert__right__if1,axiom,
! [A: v,A3: set_v,B: set_v] :
( ( member_v @ A @ A3 )
=> ( ( inf_inf_set_v @ A3 @ ( insert_v @ A @ B ) )
= ( insert_v @ A @ ( inf_inf_set_v @ A3 @ B ) ) ) ) ).
% Int_insert_right_if1
thf(fact_599_Int__insert__right__if0,axiom,
! [A: set_v,A3: set_set_v,B: set_set_v] :
( ~ ( member_set_v @ A @ A3 )
=> ( ( inf_inf_set_set_v @ A3 @ ( insert_set_v @ A @ B ) )
= ( inf_inf_set_set_v @ A3 @ B ) ) ) ).
% Int_insert_right_if0
thf(fact_600_Int__insert__right__if0,axiom,
! [A: product_prod_v_v,A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ~ ( member7453568604450474000od_v_v @ A @ A3 )
=> ( ( inf_in6271465464967711157od_v_v @ A3 @ ( insert1338601472111419319od_v_v @ A @ B ) )
= ( inf_in6271465464967711157od_v_v @ A3 @ B ) ) ) ).
% Int_insert_right_if0
thf(fact_601_Int__insert__right__if0,axiom,
! [A: v,A3: set_v,B: set_v] :
( ~ ( member_v @ A @ A3 )
=> ( ( inf_inf_set_v @ A3 @ ( insert_v @ A @ B ) )
= ( inf_inf_set_v @ A3 @ B ) ) ) ).
% Int_insert_right_if0
thf(fact_602_insert__inter__insert,axiom,
! [A: product_prod_v_v,A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ ( insert1338601472111419319od_v_v @ A @ A3 ) @ ( insert1338601472111419319od_v_v @ A @ B ) )
= ( insert1338601472111419319od_v_v @ A @ ( inf_in6271465464967711157od_v_v @ A3 @ B ) ) ) ).
% insert_inter_insert
thf(fact_603_insert__inter__insert,axiom,
! [A: set_v,A3: set_set_v,B: set_set_v] :
( ( inf_inf_set_set_v @ ( insert_set_v @ A @ A3 ) @ ( insert_set_v @ A @ B ) )
= ( insert_set_v @ A @ ( inf_inf_set_set_v @ A3 @ B ) ) ) ).
% insert_inter_insert
thf(fact_604_insert__inter__insert,axiom,
! [A: v,A3: set_v,B: set_v] :
( ( inf_inf_set_v @ ( insert_v @ A @ A3 ) @ ( insert_v @ A @ B ) )
= ( insert_v @ A @ ( inf_inf_set_v @ A3 @ B ) ) ) ).
% insert_inter_insert
thf(fact_605_Int__insert__left__if1,axiom,
! [A: set_v,C3: set_set_v,B: set_set_v] :
( ( member_set_v @ A @ C3 )
=> ( ( inf_inf_set_set_v @ ( insert_set_v @ A @ B ) @ C3 )
= ( insert_set_v @ A @ ( inf_inf_set_set_v @ B @ C3 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_606_Int__insert__left__if1,axiom,
! [A: product_prod_v_v,C3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ A @ C3 )
=> ( ( inf_in6271465464967711157od_v_v @ ( insert1338601472111419319od_v_v @ A @ B ) @ C3 )
= ( insert1338601472111419319od_v_v @ A @ ( inf_in6271465464967711157od_v_v @ B @ C3 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_607_Int__insert__left__if1,axiom,
! [A: v,C3: set_v,B: set_v] :
( ( member_v @ A @ C3 )
=> ( ( inf_inf_set_v @ ( insert_v @ A @ B ) @ C3 )
= ( insert_v @ A @ ( inf_inf_set_v @ B @ C3 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_608_Int__insert__left__if0,axiom,
! [A: set_v,C3: set_set_v,B: set_set_v] :
( ~ ( member_set_v @ A @ C3 )
=> ( ( inf_inf_set_set_v @ ( insert_set_v @ A @ B ) @ C3 )
= ( inf_inf_set_set_v @ B @ C3 ) ) ) ).
% Int_insert_left_if0
thf(fact_609_Int__insert__left__if0,axiom,
! [A: product_prod_v_v,C3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ~ ( member7453568604450474000od_v_v @ A @ C3 )
=> ( ( inf_in6271465464967711157od_v_v @ ( insert1338601472111419319od_v_v @ A @ B ) @ C3 )
= ( inf_in6271465464967711157od_v_v @ B @ C3 ) ) ) ).
% Int_insert_left_if0
thf(fact_610_Int__insert__left__if0,axiom,
! [A: v,C3: set_v,B: set_v] :
( ~ ( member_v @ A @ C3 )
=> ( ( inf_inf_set_v @ ( insert_v @ A @ B ) @ C3 )
= ( inf_inf_set_v @ B @ C3 ) ) ) ).
% Int_insert_left_if0
thf(fact_611_Int__Un__eq_I4_J,axiom,
! [T: set_Product_prod_v_v,S: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ T @ ( inf_in6271465464967711157od_v_v @ S @ T ) )
= T ) ).
% Int_Un_eq(4)
thf(fact_612_Int__Un__eq_I4_J,axiom,
! [T: set_v,S: set_v] :
( ( sup_sup_set_v @ T @ ( inf_inf_set_v @ S @ T ) )
= T ) ).
% Int_Un_eq(4)
thf(fact_613_Int__Un__eq_I4_J,axiom,
! [T: set_set_v,S: set_set_v] :
( ( sup_sup_set_set_v @ T @ ( inf_inf_set_set_v @ S @ T ) )
= T ) ).
% Int_Un_eq(4)
thf(fact_614_Int__Un__eq_I3_J,axiom,
! [S: set_Product_prod_v_v,T: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ S @ ( inf_in6271465464967711157od_v_v @ S @ T ) )
= S ) ).
% Int_Un_eq(3)
thf(fact_615_Int__Un__eq_I3_J,axiom,
! [S: set_v,T: set_v] :
( ( sup_sup_set_v @ S @ ( inf_inf_set_v @ S @ T ) )
= S ) ).
% Int_Un_eq(3)
thf(fact_616_Int__Un__eq_I3_J,axiom,
! [S: set_set_v,T: set_set_v] :
( ( sup_sup_set_set_v @ S @ ( inf_inf_set_set_v @ S @ T ) )
= S ) ).
% Int_Un_eq(3)
thf(fact_617_Int__Un__eq_I2_J,axiom,
! [S: set_Product_prod_v_v,T: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ S @ T ) @ T )
= T ) ).
% Int_Un_eq(2)
thf(fact_618_Int__Un__eq_I2_J,axiom,
! [S: set_v,T: set_v] :
( ( sup_sup_set_v @ ( inf_inf_set_v @ S @ T ) @ T )
= T ) ).
% Int_Un_eq(2)
thf(fact_619_Int__Un__eq_I2_J,axiom,
! [S: set_set_v,T: set_set_v] :
( ( sup_sup_set_set_v @ ( inf_inf_set_set_v @ S @ T ) @ T )
= T ) ).
% Int_Un_eq(2)
thf(fact_620_Int__Un__eq_I1_J,axiom,
! [S: set_Product_prod_v_v,T: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ S @ T ) @ S )
= S ) ).
% Int_Un_eq(1)
thf(fact_621_Int__Un__eq_I1_J,axiom,
! [S: set_v,T: set_v] :
( ( sup_sup_set_v @ ( inf_inf_set_v @ S @ T ) @ S )
= S ) ).
% Int_Un_eq(1)
thf(fact_622_Int__Un__eq_I1_J,axiom,
! [S: set_set_v,T: set_set_v] :
( ( sup_sup_set_set_v @ ( inf_inf_set_set_v @ S @ T ) @ S )
= S ) ).
% Int_Un_eq(1)
thf(fact_623_Un__Int__eq_I4_J,axiom,
! [T: set_Product_prod_v_v,S: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ T @ ( sup_su414716646722978715od_v_v @ S @ T ) )
= T ) ).
% Un_Int_eq(4)
thf(fact_624_Un__Int__eq_I4_J,axiom,
! [T: set_v,S: set_v] :
( ( inf_inf_set_v @ T @ ( sup_sup_set_v @ S @ T ) )
= T ) ).
% Un_Int_eq(4)
thf(fact_625_Un__Int__eq_I4_J,axiom,
! [T: set_set_v,S: set_set_v] :
( ( inf_inf_set_set_v @ T @ ( sup_sup_set_set_v @ S @ T ) )
= T ) ).
% Un_Int_eq(4)
thf(fact_626_Un__Int__eq_I3_J,axiom,
! [S: set_Product_prod_v_v,T: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ S @ ( sup_su414716646722978715od_v_v @ S @ T ) )
= S ) ).
% Un_Int_eq(3)
thf(fact_627_Un__Int__eq_I3_J,axiom,
! [S: set_v,T: set_v] :
( ( inf_inf_set_v @ S @ ( sup_sup_set_v @ S @ T ) )
= S ) ).
% Un_Int_eq(3)
thf(fact_628_Un__Int__eq_I3_J,axiom,
! [S: set_set_v,T: set_set_v] :
( ( inf_inf_set_set_v @ S @ ( sup_sup_set_set_v @ S @ T ) )
= S ) ).
% Un_Int_eq(3)
thf(fact_629_Un__Int__eq_I2_J,axiom,
! [S: set_Product_prod_v_v,T: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ ( sup_su414716646722978715od_v_v @ S @ T ) @ T )
= T ) ).
% Un_Int_eq(2)
thf(fact_630_Un__Int__eq_I2_J,axiom,
! [S: set_v,T: set_v] :
( ( inf_inf_set_v @ ( sup_sup_set_v @ S @ T ) @ T )
= T ) ).
% Un_Int_eq(2)
thf(fact_631_Un__Int__eq_I2_J,axiom,
! [S: set_set_v,T: set_set_v] :
( ( inf_inf_set_set_v @ ( sup_sup_set_set_v @ S @ T ) @ T )
= T ) ).
% Un_Int_eq(2)
thf(fact_632_Un__Int__eq_I1_J,axiom,
! [S: set_Product_prod_v_v,T: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ ( sup_su414716646722978715od_v_v @ S @ T ) @ S )
= S ) ).
% Un_Int_eq(1)
thf(fact_633_Un__Int__eq_I1_J,axiom,
! [S: set_v,T: set_v] :
( ( inf_inf_set_v @ ( sup_sup_set_v @ S @ T ) @ S )
= S ) ).
% Un_Int_eq(1)
thf(fact_634_Un__Int__eq_I1_J,axiom,
! [S: set_set_v,T: set_set_v] :
( ( inf_inf_set_set_v @ ( sup_sup_set_set_v @ S @ T ) @ S )
= S ) ).
% Un_Int_eq(1)
thf(fact_635_singleton__insert__inj__eq_H,axiom,
! [A: set_v,A3: set_set_v,B2: set_v] :
( ( ( insert_set_v @ A @ A3 )
= ( insert_set_v @ B2 @ bot_bot_set_set_v ) )
= ( ( A = B2 )
& ( ord_le5216385588623774835_set_v @ A3 @ ( insert_set_v @ B2 @ bot_bot_set_set_v ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_636_singleton__insert__inj__eq_H,axiom,
! [A: v,A3: set_v,B2: v] :
( ( ( insert_v @ A @ A3 )
= ( insert_v @ B2 @ bot_bot_set_v ) )
= ( ( A = B2 )
& ( ord_less_eq_set_v @ A3 @ ( insert_v @ B2 @ bot_bot_set_v ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_637_singleton__insert__inj__eq_H,axiom,
! [A: product_prod_v_v,A3: set_Product_prod_v_v,B2: product_prod_v_v] :
( ( ( insert1338601472111419319od_v_v @ A @ A3 )
= ( insert1338601472111419319od_v_v @ B2 @ bot_bo723834152578015283od_v_v ) )
= ( ( A = B2 )
& ( ord_le7336532860387713383od_v_v @ A3 @ ( insert1338601472111419319od_v_v @ B2 @ bot_bo723834152578015283od_v_v ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_638_singleton__insert__inj__eq,axiom,
! [B2: set_v,A: set_v,A3: set_set_v] :
( ( ( insert_set_v @ B2 @ bot_bot_set_set_v )
= ( insert_set_v @ A @ A3 ) )
= ( ( A = B2 )
& ( ord_le5216385588623774835_set_v @ A3 @ ( insert_set_v @ B2 @ bot_bot_set_set_v ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_639_singleton__insert__inj__eq,axiom,
! [B2: v,A: v,A3: set_v] :
( ( ( insert_v @ B2 @ bot_bot_set_v )
= ( insert_v @ A @ A3 ) )
= ( ( A = B2 )
& ( ord_less_eq_set_v @ A3 @ ( insert_v @ B2 @ bot_bot_set_v ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_640_singleton__insert__inj__eq,axiom,
! [B2: product_prod_v_v,A: product_prod_v_v,A3: set_Product_prod_v_v] :
( ( ( insert1338601472111419319od_v_v @ B2 @ bot_bo723834152578015283od_v_v )
= ( insert1338601472111419319od_v_v @ A @ A3 ) )
= ( ( A = B2 )
& ( ord_le7336532860387713383od_v_v @ A3 @ ( insert1338601472111419319od_v_v @ B2 @ bot_bo723834152578015283od_v_v ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_641_insert__disjoint_I1_J,axiom,
! [A: v,A3: set_v,B: set_v] :
( ( ( inf_inf_set_v @ ( insert_v @ A @ A3 ) @ B )
= bot_bot_set_v )
= ( ~ ( member_v @ A @ B )
& ( ( inf_inf_set_v @ A3 @ B )
= bot_bot_set_v ) ) ) ).
% insert_disjoint(1)
thf(fact_642_insert__disjoint_I1_J,axiom,
! [A: product_prod_v_v,A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ( inf_in6271465464967711157od_v_v @ ( insert1338601472111419319od_v_v @ A @ A3 ) @ B )
= bot_bo723834152578015283od_v_v )
= ( ~ ( member7453568604450474000od_v_v @ A @ B )
& ( ( inf_in6271465464967711157od_v_v @ A3 @ B )
= bot_bo723834152578015283od_v_v ) ) ) ).
% insert_disjoint(1)
thf(fact_643_insert__disjoint_I1_J,axiom,
! [A: set_v,A3: set_set_v,B: set_set_v] :
( ( ( inf_inf_set_set_v @ ( insert_set_v @ A @ A3 ) @ B )
= bot_bot_set_set_v )
= ( ~ ( member_set_v @ A @ B )
& ( ( inf_inf_set_set_v @ A3 @ B )
= bot_bot_set_set_v ) ) ) ).
% insert_disjoint(1)
thf(fact_644_insert__disjoint_I2_J,axiom,
! [A: v,A3: set_v,B: set_v] :
( ( bot_bot_set_v
= ( inf_inf_set_v @ ( insert_v @ A @ A3 ) @ B ) )
= ( ~ ( member_v @ A @ B )
& ( bot_bot_set_v
= ( inf_inf_set_v @ A3 @ B ) ) ) ) ).
% insert_disjoint(2)
thf(fact_645_insert__disjoint_I2_J,axiom,
! [A: product_prod_v_v,A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( bot_bo723834152578015283od_v_v
= ( inf_in6271465464967711157od_v_v @ ( insert1338601472111419319od_v_v @ A @ A3 ) @ B ) )
= ( ~ ( member7453568604450474000od_v_v @ A @ B )
& ( bot_bo723834152578015283od_v_v
= ( inf_in6271465464967711157od_v_v @ A3 @ B ) ) ) ) ).
% insert_disjoint(2)
thf(fact_646_insert__disjoint_I2_J,axiom,
! [A: set_v,A3: set_set_v,B: set_set_v] :
( ( bot_bot_set_set_v
= ( inf_inf_set_set_v @ ( insert_set_v @ A @ A3 ) @ B ) )
= ( ~ ( member_set_v @ A @ B )
& ( bot_bot_set_set_v
= ( inf_inf_set_set_v @ A3 @ B ) ) ) ) ).
% insert_disjoint(2)
thf(fact_647_disjoint__insert_I1_J,axiom,
! [B: set_v,A: v,A3: set_v] :
( ( ( inf_inf_set_v @ B @ ( insert_v @ A @ A3 ) )
= bot_bot_set_v )
= ( ~ ( member_v @ A @ B )
& ( ( inf_inf_set_v @ B @ A3 )
= bot_bot_set_v ) ) ) ).
% disjoint_insert(1)
thf(fact_648_disjoint__insert_I1_J,axiom,
! [B: set_Product_prod_v_v,A: product_prod_v_v,A3: set_Product_prod_v_v] :
( ( ( inf_in6271465464967711157od_v_v @ B @ ( insert1338601472111419319od_v_v @ A @ A3 ) )
= bot_bo723834152578015283od_v_v )
= ( ~ ( member7453568604450474000od_v_v @ A @ B )
& ( ( inf_in6271465464967711157od_v_v @ B @ A3 )
= bot_bo723834152578015283od_v_v ) ) ) ).
% disjoint_insert(1)
thf(fact_649_disjoint__insert_I1_J,axiom,
! [B: set_set_v,A: set_v,A3: set_set_v] :
( ( ( inf_inf_set_set_v @ B @ ( insert_set_v @ A @ A3 ) )
= bot_bot_set_set_v )
= ( ~ ( member_set_v @ A @ B )
& ( ( inf_inf_set_set_v @ B @ A3 )
= bot_bot_set_set_v ) ) ) ).
% disjoint_insert(1)
thf(fact_650_disjoint__insert_I2_J,axiom,
! [A3: set_v,B2: v,B: set_v] :
( ( bot_bot_set_v
= ( inf_inf_set_v @ A3 @ ( insert_v @ B2 @ B ) ) )
= ( ~ ( member_v @ B2 @ A3 )
& ( bot_bot_set_v
= ( inf_inf_set_v @ A3 @ B ) ) ) ) ).
% disjoint_insert(2)
thf(fact_651_disjoint__insert_I2_J,axiom,
! [A3: set_Product_prod_v_v,B2: product_prod_v_v,B: set_Product_prod_v_v] :
( ( bot_bo723834152578015283od_v_v
= ( inf_in6271465464967711157od_v_v @ A3 @ ( insert1338601472111419319od_v_v @ B2 @ B ) ) )
= ( ~ ( member7453568604450474000od_v_v @ B2 @ A3 )
& ( bot_bo723834152578015283od_v_v
= ( inf_in6271465464967711157od_v_v @ A3 @ B ) ) ) ) ).
% disjoint_insert(2)
thf(fact_652_disjoint__insert_I2_J,axiom,
! [A3: set_set_v,B2: set_v,B: set_set_v] :
( ( bot_bot_set_set_v
= ( inf_inf_set_set_v @ A3 @ ( insert_set_v @ B2 @ B ) ) )
= ( ~ ( member_set_v @ B2 @ A3 )
& ( bot_bot_set_set_v
= ( inf_inf_set_set_v @ A3 @ B ) ) ) ) ).
% disjoint_insert(2)
thf(fact_653_Diff__eq__empty__iff,axiom,
! [A3: set_set_v,B: set_set_v] :
( ( ( minus_7228012346218142266_set_v @ A3 @ B )
= bot_bot_set_set_v )
= ( ord_le5216385588623774835_set_v @ A3 @ B ) ) ).
% Diff_eq_empty_iff
thf(fact_654_Diff__eq__empty__iff,axiom,
! [A3: set_v,B: set_v] :
( ( ( minus_minus_set_v @ A3 @ B )
= bot_bot_set_v )
= ( ord_less_eq_set_v @ A3 @ B ) ) ).
% Diff_eq_empty_iff
thf(fact_655_Diff__eq__empty__iff,axiom,
! [A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ( minus_4183494784930505774od_v_v @ A3 @ B )
= bot_bo723834152578015283od_v_v )
= ( ord_le7336532860387713383od_v_v @ A3 @ B ) ) ).
% Diff_eq_empty_iff
thf(fact_656_Diff__disjoint,axiom,
! [A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ A3 @ ( minus_4183494784930505774od_v_v @ B @ A3 ) )
= bot_bo723834152578015283od_v_v ) ).
% Diff_disjoint
thf(fact_657_Diff__disjoint,axiom,
! [A3: set_set_v,B: set_set_v] :
( ( inf_inf_set_set_v @ A3 @ ( minus_7228012346218142266_set_v @ B @ A3 ) )
= bot_bot_set_set_v ) ).
% Diff_disjoint
thf(fact_658_Diff__disjoint,axiom,
! [A3: set_v,B: set_v] :
( ( inf_inf_set_v @ A3 @ ( minus_minus_set_v @ B @ A3 ) )
= bot_bot_set_v ) ).
% Diff_disjoint
thf(fact_659_distrib__sup__le,axiom,
! [X2: set_set_v,Y2: set_set_v,Z: set_set_v] : ( ord_le5216385588623774835_set_v @ ( sup_sup_set_set_v @ X2 @ ( inf_inf_set_set_v @ Y2 @ Z ) ) @ ( inf_inf_set_set_v @ ( sup_sup_set_set_v @ X2 @ Y2 ) @ ( sup_sup_set_set_v @ X2 @ Z ) ) ) ).
% distrib_sup_le
thf(fact_660_distrib__sup__le,axiom,
! [X2: set_v,Y2: set_v,Z: set_v] : ( ord_less_eq_set_v @ ( sup_sup_set_v @ X2 @ ( inf_inf_set_v @ Y2 @ Z ) ) @ ( inf_inf_set_v @ ( sup_sup_set_v @ X2 @ Y2 ) @ ( sup_sup_set_v @ X2 @ Z ) ) ) ).
% distrib_sup_le
thf(fact_661_distrib__sup__le,axiom,
! [X2: set_Product_prod_v_v,Y2: set_Product_prod_v_v,Z: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ ( sup_su414716646722978715od_v_v @ X2 @ ( inf_in6271465464967711157od_v_v @ Y2 @ Z ) ) @ ( inf_in6271465464967711157od_v_v @ ( sup_su414716646722978715od_v_v @ X2 @ Y2 ) @ ( sup_su414716646722978715od_v_v @ X2 @ Z ) ) ) ).
% distrib_sup_le
thf(fact_662_distrib__inf__le,axiom,
! [X2: set_set_v,Y2: set_set_v,Z: set_set_v] : ( ord_le5216385588623774835_set_v @ ( sup_sup_set_set_v @ ( inf_inf_set_set_v @ X2 @ Y2 ) @ ( inf_inf_set_set_v @ X2 @ Z ) ) @ ( inf_inf_set_set_v @ X2 @ ( sup_sup_set_set_v @ Y2 @ Z ) ) ) ).
% distrib_inf_le
thf(fact_663_distrib__inf__le,axiom,
! [X2: set_v,Y2: set_v,Z: set_v] : ( ord_less_eq_set_v @ ( sup_sup_set_v @ ( inf_inf_set_v @ X2 @ Y2 ) @ ( inf_inf_set_v @ X2 @ Z ) ) @ ( inf_inf_set_v @ X2 @ ( sup_sup_set_v @ Y2 @ Z ) ) ) ).
% distrib_inf_le
thf(fact_664_distrib__inf__le,axiom,
! [X2: set_Product_prod_v_v,Y2: set_Product_prod_v_v,Z: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ X2 @ Y2 ) @ ( inf_in6271465464967711157od_v_v @ X2 @ Z ) ) @ ( inf_in6271465464967711157od_v_v @ X2 @ ( sup_su414716646722978715od_v_v @ Y2 @ Z ) ) ) ).
% distrib_inf_le
thf(fact_665_Un__Int__assoc__eq,axiom,
! [A3: set_set_v,B: set_set_v,C3: set_set_v] :
( ( ( sup_sup_set_set_v @ ( inf_inf_set_set_v @ A3 @ B ) @ C3 )
= ( inf_inf_set_set_v @ A3 @ ( sup_sup_set_set_v @ B @ C3 ) ) )
= ( ord_le5216385588623774835_set_v @ C3 @ A3 ) ) ).
% Un_Int_assoc_eq
thf(fact_666_Un__Int__assoc__eq,axiom,
! [A3: set_v,B: set_v,C3: set_v] :
( ( ( sup_sup_set_v @ ( inf_inf_set_v @ A3 @ B ) @ C3 )
= ( inf_inf_set_v @ A3 @ ( sup_sup_set_v @ B @ C3 ) ) )
= ( ord_less_eq_set_v @ C3 @ A3 ) ) ).
% Un_Int_assoc_eq
thf(fact_667_Un__Int__assoc__eq,axiom,
! [A3: set_Product_prod_v_v,B: set_Product_prod_v_v,C3: set_Product_prod_v_v] :
( ( ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ A3 @ B ) @ C3 )
= ( inf_in6271465464967711157od_v_v @ A3 @ ( sup_su414716646722978715od_v_v @ B @ C3 ) ) )
= ( ord_le7336532860387713383od_v_v @ C3 @ A3 ) ) ).
% Un_Int_assoc_eq
thf(fact_668_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_v,Z4: set_v] : ( Y4 = Z4 ) )
= ( ^ [X3: set_v,Y3: set_v] :
( ( ord_less_eq_set_v @ X3 @ Y3 )
& ( ord_less_eq_set_v @ Y3 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_669_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_Product_prod_v_v,Z4: set_Product_prod_v_v] : ( Y4 = Z4 ) )
= ( ^ [X3: set_Product_prod_v_v,Y3: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X3 @ Y3 )
& ( ord_le7336532860387713383od_v_v @ Y3 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_670_ord__eq__le__trans,axiom,
! [A: set_v,B2: set_v,C: set_v] :
( ( A = B2 )
=> ( ( ord_less_eq_set_v @ B2 @ C )
=> ( ord_less_eq_set_v @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_671_ord__eq__le__trans,axiom,
! [A: set_Product_prod_v_v,B2: set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( A = B2 )
=> ( ( ord_le7336532860387713383od_v_v @ B2 @ C )
=> ( ord_le7336532860387713383od_v_v @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_672_ord__le__eq__trans,axiom,
! [A: set_v,B2: set_v,C: set_v] :
( ( ord_less_eq_set_v @ A @ B2 )
=> ( ( B2 = C )
=> ( ord_less_eq_set_v @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_673_ord__le__eq__trans,axiom,
! [A: set_Product_prod_v_v,B2: set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ B2 )
=> ( ( B2 = C )
=> ( ord_le7336532860387713383od_v_v @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_674_order__antisym,axiom,
! [X2: set_v,Y2: set_v] :
( ( ord_less_eq_set_v @ X2 @ Y2 )
=> ( ( ord_less_eq_set_v @ Y2 @ X2 )
=> ( X2 = Y2 ) ) ) ).
% order_antisym
thf(fact_675_order__antisym,axiom,
! [X2: set_Product_prod_v_v,Y2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X2 @ Y2 )
=> ( ( ord_le7336532860387713383od_v_v @ Y2 @ X2 )
=> ( X2 = Y2 ) ) ) ).
% order_antisym
thf(fact_676_order_Otrans,axiom,
! [A: set_v,B2: set_v,C: set_v] :
( ( ord_less_eq_set_v @ A @ B2 )
=> ( ( ord_less_eq_set_v @ B2 @ C )
=> ( ord_less_eq_set_v @ A @ C ) ) ) ).
% order.trans
thf(fact_677_order_Otrans,axiom,
! [A: set_Product_prod_v_v,B2: set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ B2 )
=> ( ( ord_le7336532860387713383od_v_v @ B2 @ C )
=> ( ord_le7336532860387713383od_v_v @ A @ C ) ) ) ).
% order.trans
thf(fact_678_order__trans,axiom,
! [X2: set_v,Y2: set_v,Z: set_v] :
( ( ord_less_eq_set_v @ X2 @ Y2 )
=> ( ( ord_less_eq_set_v @ Y2 @ Z )
=> ( ord_less_eq_set_v @ X2 @ Z ) ) ) ).
% order_trans
thf(fact_679_order__trans,axiom,
! [X2: set_Product_prod_v_v,Y2: set_Product_prod_v_v,Z: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X2 @ Y2 )
=> ( ( ord_le7336532860387713383od_v_v @ Y2 @ Z )
=> ( ord_le7336532860387713383od_v_v @ X2 @ Z ) ) ) ).
% order_trans
thf(fact_680_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: set_v,Z4: set_v] : ( Y4 = Z4 ) )
= ( ^ [A4: set_v,B4: set_v] :
( ( ord_less_eq_set_v @ B4 @ A4 )
& ( ord_less_eq_set_v @ A4 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_681_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: set_Product_prod_v_v,Z4: set_Product_prod_v_v] : ( Y4 = Z4 ) )
= ( ^ [A4: set_Product_prod_v_v,B4: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ B4 @ A4 )
& ( ord_le7336532860387713383od_v_v @ A4 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_682_dual__order_Oantisym,axiom,
! [B2: set_v,A: set_v] :
( ( ord_less_eq_set_v @ B2 @ A )
=> ( ( ord_less_eq_set_v @ A @ B2 )
=> ( A = B2 ) ) ) ).
% dual_order.antisym
thf(fact_683_dual__order_Oantisym,axiom,
! [B2: set_Product_prod_v_v,A: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ B2 @ A )
=> ( ( ord_le7336532860387713383od_v_v @ A @ B2 )
=> ( A = B2 ) ) ) ).
% dual_order.antisym
thf(fact_684_dual__order_Otrans,axiom,
! [B2: set_v,A: set_v,C: set_v] :
( ( ord_less_eq_set_v @ B2 @ A )
=> ( ( ord_less_eq_set_v @ C @ B2 )
=> ( ord_less_eq_set_v @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_685_dual__order_Otrans,axiom,
! [B2: set_Product_prod_v_v,A: set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ B2 @ A )
=> ( ( ord_le7336532860387713383od_v_v @ C @ B2 )
=> ( ord_le7336532860387713383od_v_v @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_686_antisym,axiom,
! [A: set_v,B2: set_v] :
( ( ord_less_eq_set_v @ A @ B2 )
=> ( ( ord_less_eq_set_v @ B2 @ A )
=> ( A = B2 ) ) ) ).
% antisym
thf(fact_687_antisym,axiom,
! [A: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ B2 )
=> ( ( ord_le7336532860387713383od_v_v @ B2 @ A )
=> ( A = B2 ) ) ) ).
% antisym
thf(fact_688_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_v,Z4: set_v] : ( Y4 = Z4 ) )
= ( ^ [A4: set_v,B4: set_v] :
( ( ord_less_eq_set_v @ A4 @ B4 )
& ( ord_less_eq_set_v @ B4 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_689_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_Product_prod_v_v,Z4: set_Product_prod_v_v] : ( Y4 = Z4 ) )
= ( ^ [A4: set_Product_prod_v_v,B4: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A4 @ B4 )
& ( ord_le7336532860387713383od_v_v @ B4 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_690_order__subst1,axiom,
! [A: set_v,F: set_v > set_v,B2: set_v,C: set_v] :
( ( ord_less_eq_set_v @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_set_v @ B2 @ C )
=> ( ! [X4: set_v,Y: set_v] :
( ( ord_less_eq_set_v @ X4 @ Y )
=> ( ord_less_eq_set_v @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_set_v @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_691_order__subst1,axiom,
! [A: set_v,F: set_Product_prod_v_v > set_v,B2: set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( ord_less_eq_set_v @ A @ ( F @ B2 ) )
=> ( ( ord_le7336532860387713383od_v_v @ B2 @ C )
=> ( ! [X4: set_Product_prod_v_v,Y: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X4 @ Y )
=> ( ord_less_eq_set_v @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_set_v @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_692_order__subst1,axiom,
! [A: set_Product_prod_v_v,F: set_v > set_Product_prod_v_v,B2: set_v,C: set_v] :
( ( ord_le7336532860387713383od_v_v @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_set_v @ B2 @ C )
=> ( ! [X4: set_v,Y: set_v] :
( ( ord_less_eq_set_v @ X4 @ Y )
=> ( ord_le7336532860387713383od_v_v @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_le7336532860387713383od_v_v @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_693_order__subst1,axiom,
! [A: set_Product_prod_v_v,F: set_Product_prod_v_v > set_Product_prod_v_v,B2: set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ ( F @ B2 ) )
=> ( ( ord_le7336532860387713383od_v_v @ B2 @ C )
=> ( ! [X4: set_Product_prod_v_v,Y: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X4 @ Y )
=> ( ord_le7336532860387713383od_v_v @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_le7336532860387713383od_v_v @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_694_order__subst2,axiom,
! [A: set_v,B2: set_v,F: set_v > set_v,C: set_v] :
( ( ord_less_eq_set_v @ A @ B2 )
=> ( ( ord_less_eq_set_v @ ( F @ B2 ) @ C )
=> ( ! [X4: set_v,Y: set_v] :
( ( ord_less_eq_set_v @ X4 @ Y )
=> ( ord_less_eq_set_v @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_set_v @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_695_order__subst2,axiom,
! [A: set_v,B2: set_v,F: set_v > set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( ord_less_eq_set_v @ A @ B2 )
=> ( ( ord_le7336532860387713383od_v_v @ ( F @ B2 ) @ C )
=> ( ! [X4: set_v,Y: set_v] :
( ( ord_less_eq_set_v @ X4 @ Y )
=> ( ord_le7336532860387713383od_v_v @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_le7336532860387713383od_v_v @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_696_order__subst2,axiom,
! [A: set_Product_prod_v_v,B2: set_Product_prod_v_v,F: set_Product_prod_v_v > set_v,C: set_v] :
( ( ord_le7336532860387713383od_v_v @ A @ B2 )
=> ( ( ord_less_eq_set_v @ ( F @ B2 ) @ C )
=> ( ! [X4: set_Product_prod_v_v,Y: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X4 @ Y )
=> ( ord_less_eq_set_v @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_set_v @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_697_order__subst2,axiom,
! [A: set_Product_prod_v_v,B2: set_Product_prod_v_v,F: set_Product_prod_v_v > set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ B2 )
=> ( ( ord_le7336532860387713383od_v_v @ ( F @ B2 ) @ C )
=> ( ! [X4: set_Product_prod_v_v,Y: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X4 @ Y )
=> ( ord_le7336532860387713383od_v_v @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_le7336532860387713383od_v_v @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_698_order__eq__refl,axiom,
! [X2: set_v,Y2: set_v] :
( ( X2 = Y2 )
=> ( ord_less_eq_set_v @ X2 @ Y2 ) ) ).
% order_eq_refl
thf(fact_699_order__eq__refl,axiom,
! [X2: set_Product_prod_v_v,Y2: set_Product_prod_v_v] :
( ( X2 = Y2 )
=> ( ord_le7336532860387713383od_v_v @ X2 @ Y2 ) ) ).
% order_eq_refl
thf(fact_700_ord__eq__le__subst,axiom,
! [A: set_v,F: set_v > set_v,B2: set_v,C: set_v] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_less_eq_set_v @ B2 @ C )
=> ( ! [X4: set_v,Y: set_v] :
( ( ord_less_eq_set_v @ X4 @ Y )
=> ( ord_less_eq_set_v @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_set_v @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_701_ord__eq__le__subst,axiom,
! [A: set_Product_prod_v_v,F: set_v > set_Product_prod_v_v,B2: set_v,C: set_v] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_less_eq_set_v @ B2 @ C )
=> ( ! [X4: set_v,Y: set_v] :
( ( ord_less_eq_set_v @ X4 @ Y )
=> ( ord_le7336532860387713383od_v_v @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_le7336532860387713383od_v_v @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_702_ord__eq__le__subst,axiom,
! [A: set_v,F: set_Product_prod_v_v > set_v,B2: set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_le7336532860387713383od_v_v @ B2 @ C )
=> ( ! [X4: set_Product_prod_v_v,Y: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X4 @ Y )
=> ( ord_less_eq_set_v @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_set_v @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_703_ord__eq__le__subst,axiom,
! [A: set_Product_prod_v_v,F: set_Product_prod_v_v > set_Product_prod_v_v,B2: set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_le7336532860387713383od_v_v @ B2 @ C )
=> ( ! [X4: set_Product_prod_v_v,Y: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X4 @ Y )
=> ( ord_le7336532860387713383od_v_v @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_le7336532860387713383od_v_v @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_704_ord__le__eq__subst,axiom,
! [A: set_v,B2: set_v,F: set_v > set_v,C: set_v] :
( ( ord_less_eq_set_v @ A @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X4: set_v,Y: set_v] :
( ( ord_less_eq_set_v @ X4 @ Y )
=> ( ord_less_eq_set_v @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_set_v @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_705_ord__le__eq__subst,axiom,
! [A: set_v,B2: set_v,F: set_v > set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( ord_less_eq_set_v @ A @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X4: set_v,Y: set_v] :
( ( ord_less_eq_set_v @ X4 @ Y )
=> ( ord_le7336532860387713383od_v_v @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_le7336532860387713383od_v_v @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_706_ord__le__eq__subst,axiom,
! [A: set_Product_prod_v_v,B2: set_Product_prod_v_v,F: set_Product_prod_v_v > set_v,C: set_v] :
( ( ord_le7336532860387713383od_v_v @ A @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X4: set_Product_prod_v_v,Y: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X4 @ Y )
=> ( ord_less_eq_set_v @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_set_v @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_707_ord__le__eq__subst,axiom,
! [A: set_Product_prod_v_v,B2: set_Product_prod_v_v,F: set_Product_prod_v_v > set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X4: set_Product_prod_v_v,Y: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X4 @ Y )
=> ( ord_le7336532860387713383od_v_v @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_le7336532860387713383od_v_v @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_708_order__antisym__conv,axiom,
! [Y2: set_v,X2: set_v] :
( ( ord_less_eq_set_v @ Y2 @ X2 )
=> ( ( ord_less_eq_set_v @ X2 @ Y2 )
= ( X2 = Y2 ) ) ) ).
% order_antisym_conv
thf(fact_709_order__antisym__conv,axiom,
! [Y2: set_Product_prod_v_v,X2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ Y2 @ X2 )
=> ( ( ord_le7336532860387713383od_v_v @ X2 @ Y2 )
= ( X2 = Y2 ) ) ) ).
% order_antisym_conv
thf(fact_710_IntE,axiom,
! [C: product_prod_v_v,A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ C @ ( inf_in6271465464967711157od_v_v @ A3 @ B ) )
=> ~ ( ( member7453568604450474000od_v_v @ C @ A3 )
=> ~ ( member7453568604450474000od_v_v @ C @ B ) ) ) ).
% IntE
thf(fact_711_IntE,axiom,
! [C: v,A3: set_v,B: set_v] :
( ( member_v @ C @ ( inf_inf_set_v @ A3 @ B ) )
=> ~ ( ( member_v @ C @ A3 )
=> ~ ( member_v @ C @ B ) ) ) ).
% IntE
thf(fact_712_IntD1,axiom,
! [C: product_prod_v_v,A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ C @ ( inf_in6271465464967711157od_v_v @ A3 @ B ) )
=> ( member7453568604450474000od_v_v @ C @ A3 ) ) ).
% IntD1
thf(fact_713_IntD1,axiom,
! [C: v,A3: set_v,B: set_v] :
( ( member_v @ C @ ( inf_inf_set_v @ A3 @ B ) )
=> ( member_v @ C @ A3 ) ) ).
% IntD1
thf(fact_714_IntD2,axiom,
! [C: product_prod_v_v,A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ C @ ( inf_in6271465464967711157od_v_v @ A3 @ B ) )
=> ( member7453568604450474000od_v_v @ C @ B ) ) ).
% IntD2
thf(fact_715_IntD2,axiom,
! [C: v,A3: set_v,B: set_v] :
( ( member_v @ C @ ( inf_inf_set_v @ A3 @ B ) )
=> ( member_v @ C @ B ) ) ).
% IntD2
thf(fact_716_in__mono,axiom,
! [A3: set_v,B: set_v,X2: v] :
( ( ord_less_eq_set_v @ A3 @ B )
=> ( ( member_v @ X2 @ A3 )
=> ( member_v @ X2 @ B ) ) ) ).
% in_mono
thf(fact_717_in__mono,axiom,
! [A3: set_Product_prod_v_v,B: set_Product_prod_v_v,X2: product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A3 @ B )
=> ( ( member7453568604450474000od_v_v @ X2 @ A3 )
=> ( member7453568604450474000od_v_v @ X2 @ B ) ) ) ).
% in_mono
thf(fact_718_subsetD,axiom,
! [A3: set_v,B: set_v,C: v] :
( ( ord_less_eq_set_v @ A3 @ B )
=> ( ( member_v @ C @ A3 )
=> ( member_v @ C @ B ) ) ) ).
% subsetD
thf(fact_719_subsetD,axiom,
! [A3: set_Product_prod_v_v,B: set_Product_prod_v_v,C: product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A3 @ B )
=> ( ( member7453568604450474000od_v_v @ C @ A3 )
=> ( member7453568604450474000od_v_v @ C @ B ) ) ) ).
% subsetD
thf(fact_720_Int__mono,axiom,
! [A3: set_v,C3: set_v,B: set_v,D2: set_v] :
( ( ord_less_eq_set_v @ A3 @ C3 )
=> ( ( ord_less_eq_set_v @ B @ D2 )
=> ( ord_less_eq_set_v @ ( inf_inf_set_v @ A3 @ B ) @ ( inf_inf_set_v @ C3 @ D2 ) ) ) ) ).
% Int_mono
thf(fact_721_Int__mono,axiom,
! [A3: set_Product_prod_v_v,C3: set_Product_prod_v_v,B: set_Product_prod_v_v,D2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A3 @ C3 )
=> ( ( ord_le7336532860387713383od_v_v @ B @ D2 )
=> ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ A3 @ B ) @ ( inf_in6271465464967711157od_v_v @ C3 @ D2 ) ) ) ) ).
% Int_mono
thf(fact_722_Int__assoc,axiom,
! [A3: set_v,B: set_v,C3: set_v] :
( ( inf_inf_set_v @ ( inf_inf_set_v @ A3 @ B ) @ C3 )
= ( inf_inf_set_v @ A3 @ ( inf_inf_set_v @ B @ C3 ) ) ) ).
% Int_assoc
thf(fact_723_equalityE,axiom,
! [A3: set_v,B: set_v] :
( ( A3 = B )
=> ~ ( ( ord_less_eq_set_v @ A3 @ B )
=> ~ ( ord_less_eq_set_v @ B @ A3 ) ) ) ).
% equalityE
thf(fact_724_equalityE,axiom,
! [A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( A3 = B )
=> ~ ( ( ord_le7336532860387713383od_v_v @ A3 @ B )
=> ~ ( ord_le7336532860387713383od_v_v @ B @ A3 ) ) ) ).
% equalityE
thf(fact_725_subset__eq,axiom,
( ord_less_eq_set_v
= ( ^ [A5: set_v,B5: set_v] :
! [X3: v] :
( ( member_v @ X3 @ A5 )
=> ( member_v @ X3 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_726_subset__eq,axiom,
( ord_le7336532860387713383od_v_v
= ( ^ [A5: set_Product_prod_v_v,B5: set_Product_prod_v_v] :
! [X3: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X3 @ A5 )
=> ( member7453568604450474000od_v_v @ X3 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_727_Int__absorb,axiom,
! [A3: set_v] :
( ( inf_inf_set_v @ A3 @ A3 )
= A3 ) ).
% Int_absorb
thf(fact_728_Int__lower1,axiom,
! [A3: set_v,B: set_v] : ( ord_less_eq_set_v @ ( inf_inf_set_v @ A3 @ B ) @ A3 ) ).
% Int_lower1
thf(fact_729_Int__lower1,axiom,
! [A3: set_Product_prod_v_v,B: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ A3 @ B ) @ A3 ) ).
% Int_lower1
thf(fact_730_Int__lower2,axiom,
! [A3: set_v,B: set_v] : ( ord_less_eq_set_v @ ( inf_inf_set_v @ A3 @ B ) @ B ) ).
% Int_lower2
thf(fact_731_Int__lower2,axiom,
! [A3: set_Product_prod_v_v,B: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ A3 @ B ) @ B ) ).
% Int_lower2
thf(fact_732_equalityD1,axiom,
! [A3: set_v,B: set_v] :
( ( A3 = B )
=> ( ord_less_eq_set_v @ A3 @ B ) ) ).
% equalityD1
thf(fact_733_equalityD1,axiom,
! [A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( A3 = B )
=> ( ord_le7336532860387713383od_v_v @ A3 @ B ) ) ).
% equalityD1
thf(fact_734_equalityD2,axiom,
! [A3: set_v,B: set_v] :
( ( A3 = B )
=> ( ord_less_eq_set_v @ B @ A3 ) ) ).
% equalityD2
thf(fact_735_equalityD2,axiom,
! [A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( A3 = B )
=> ( ord_le7336532860387713383od_v_v @ B @ A3 ) ) ).
% equalityD2
thf(fact_736_subset__iff,axiom,
( ord_less_eq_set_v
= ( ^ [A5: set_v,B5: set_v] :
! [T2: v] :
( ( member_v @ T2 @ A5 )
=> ( member_v @ T2 @ B5 ) ) ) ) ).
% subset_iff
thf(fact_737_subset__iff,axiom,
( ord_le7336532860387713383od_v_v
= ( ^ [A5: set_Product_prod_v_v,B5: set_Product_prod_v_v] :
! [T2: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ T2 @ A5 )
=> ( member7453568604450474000od_v_v @ T2 @ B5 ) ) ) ) ).
% subset_iff
thf(fact_738_Int__absorb1,axiom,
! [B: set_v,A3: set_v] :
( ( ord_less_eq_set_v @ B @ A3 )
=> ( ( inf_inf_set_v @ A3 @ B )
= B ) ) ).
% Int_absorb1
thf(fact_739_Int__absorb1,axiom,
! [B: set_Product_prod_v_v,A3: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ B @ A3 )
=> ( ( inf_in6271465464967711157od_v_v @ A3 @ B )
= B ) ) ).
% Int_absorb1
thf(fact_740_Int__absorb2,axiom,
! [A3: set_v,B: set_v] :
( ( ord_less_eq_set_v @ A3 @ B )
=> ( ( inf_inf_set_v @ A3 @ B )
= A3 ) ) ).
% Int_absorb2
thf(fact_741_Int__absorb2,axiom,
! [A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A3 @ B )
=> ( ( inf_in6271465464967711157od_v_v @ A3 @ B )
= A3 ) ) ).
% Int_absorb2
thf(fact_742_Int__commute,axiom,
( inf_inf_set_v
= ( ^ [A5: set_v,B5: set_v] : ( inf_inf_set_v @ B5 @ A5 ) ) ) ).
% Int_commute
thf(fact_743_subset__refl,axiom,
! [A3: set_v] : ( ord_less_eq_set_v @ A3 @ A3 ) ).
% subset_refl
thf(fact_744_subset__refl,axiom,
! [A3: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ A3 @ A3 ) ).
% subset_refl
thf(fact_745_Collect__mono,axiom,
! [P: set_v > $o,Q: set_v > $o] :
( ! [X4: set_v] :
( ( P @ X4 )
=> ( Q @ X4 ) )
=> ( ord_le5216385588623774835_set_v @ ( collect_set_v @ P ) @ ( collect_set_v @ Q ) ) ) ).
% Collect_mono
thf(fact_746_Collect__mono,axiom,
! [P: v > $o,Q: v > $o] :
( ! [X4: v] :
( ( P @ X4 )
=> ( Q @ X4 ) )
=> ( ord_less_eq_set_v @ ( collect_v @ P ) @ ( collect_v @ Q ) ) ) ).
% Collect_mono
thf(fact_747_Collect__mono,axiom,
! [P: product_prod_v_v > $o,Q: product_prod_v_v > $o] :
( ! [X4: product_prod_v_v] :
( ( P @ X4 )
=> ( Q @ X4 ) )
=> ( ord_le7336532860387713383od_v_v @ ( collec140062887454715474od_v_v @ P ) @ ( collec140062887454715474od_v_v @ Q ) ) ) ).
% Collect_mono
thf(fact_748_Int__greatest,axiom,
! [C3: set_v,A3: set_v,B: set_v] :
( ( ord_less_eq_set_v @ C3 @ A3 )
=> ( ( ord_less_eq_set_v @ C3 @ B )
=> ( ord_less_eq_set_v @ C3 @ ( inf_inf_set_v @ A3 @ B ) ) ) ) ).
% Int_greatest
thf(fact_749_Int__greatest,axiom,
! [C3: set_Product_prod_v_v,A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ C3 @ A3 )
=> ( ( ord_le7336532860387713383od_v_v @ C3 @ B )
=> ( ord_le7336532860387713383od_v_v @ C3 @ ( inf_in6271465464967711157od_v_v @ A3 @ B ) ) ) ) ).
% Int_greatest
thf(fact_750_subset__trans,axiom,
! [A3: set_v,B: set_v,C3: set_v] :
( ( ord_less_eq_set_v @ A3 @ B )
=> ( ( ord_less_eq_set_v @ B @ C3 )
=> ( ord_less_eq_set_v @ A3 @ C3 ) ) ) ).
% subset_trans
thf(fact_751_subset__trans,axiom,
! [A3: set_Product_prod_v_v,B: set_Product_prod_v_v,C3: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A3 @ B )
=> ( ( ord_le7336532860387713383od_v_v @ B @ C3 )
=> ( ord_le7336532860387713383od_v_v @ A3 @ C3 ) ) ) ).
% subset_trans
thf(fact_752_set__eq__subset,axiom,
( ( ^ [Y4: set_v,Z4: set_v] : ( Y4 = Z4 ) )
= ( ^ [A5: set_v,B5: set_v] :
( ( ord_less_eq_set_v @ A5 @ B5 )
& ( ord_less_eq_set_v @ B5 @ A5 ) ) ) ) ).
% set_eq_subset
thf(fact_753_set__eq__subset,axiom,
( ( ^ [Y4: set_Product_prod_v_v,Z4: set_Product_prod_v_v] : ( Y4 = Z4 ) )
= ( ^ [A5: set_Product_prod_v_v,B5: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A5 @ B5 )
& ( ord_le7336532860387713383od_v_v @ B5 @ A5 ) ) ) ) ).
% set_eq_subset
thf(fact_754_Int__left__absorb,axiom,
! [A3: set_v,B: set_v] :
( ( inf_inf_set_v @ A3 @ ( inf_inf_set_v @ A3 @ B ) )
= ( inf_inf_set_v @ A3 @ B ) ) ).
% Int_left_absorb
thf(fact_755_Collect__mono__iff,axiom,
! [P: set_v > $o,Q: set_v > $o] :
( ( ord_le5216385588623774835_set_v @ ( collect_set_v @ P ) @ ( collect_set_v @ Q ) )
= ( ! [X3: set_v] :
( ( P @ X3 )
=> ( Q @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_756_Collect__mono__iff,axiom,
! [P: v > $o,Q: v > $o] :
( ( ord_less_eq_set_v @ ( collect_v @ P ) @ ( collect_v @ Q ) )
= ( ! [X3: v] :
( ( P @ X3 )
=> ( Q @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_757_Collect__mono__iff,axiom,
! [P: product_prod_v_v > $o,Q: product_prod_v_v > $o] :
( ( ord_le7336532860387713383od_v_v @ ( collec140062887454715474od_v_v @ P ) @ ( collec140062887454715474od_v_v @ Q ) )
= ( ! [X3: product_prod_v_v] :
( ( P @ X3 )
=> ( Q @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_758_Int__Collect__mono,axiom,
! [A3: set_set_v,B: set_set_v,P: set_v > $o,Q: set_v > $o] :
( ( ord_le5216385588623774835_set_v @ A3 @ B )
=> ( ! [X4: set_v] :
( ( member_set_v @ X4 @ A3 )
=> ( ( P @ X4 )
=> ( Q @ X4 ) ) )
=> ( ord_le5216385588623774835_set_v @ ( inf_inf_set_set_v @ A3 @ ( collect_set_v @ P ) ) @ ( inf_inf_set_set_v @ B @ ( collect_set_v @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_759_Int__Collect__mono,axiom,
! [A3: set_v,B: set_v,P: v > $o,Q: v > $o] :
( ( ord_less_eq_set_v @ A3 @ B )
=> ( ! [X4: v] :
( ( member_v @ X4 @ A3 )
=> ( ( P @ X4 )
=> ( Q @ X4 ) ) )
=> ( ord_less_eq_set_v @ ( inf_inf_set_v @ A3 @ ( collect_v @ P ) ) @ ( inf_inf_set_v @ B @ ( collect_v @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_760_Int__Collect__mono,axiom,
! [A3: set_Product_prod_v_v,B: set_Product_prod_v_v,P: product_prod_v_v > $o,Q: product_prod_v_v > $o] :
( ( ord_le7336532860387713383od_v_v @ A3 @ B )
=> ( ! [X4: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X4 @ A3 )
=> ( ( P @ X4 )
=> ( Q @ X4 ) ) )
=> ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ A3 @ ( collec140062887454715474od_v_v @ P ) ) @ ( inf_in6271465464967711157od_v_v @ B @ ( collec140062887454715474od_v_v @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_761_Int__left__commute,axiom,
! [A3: set_v,B: set_v,C3: set_v] :
( ( inf_inf_set_v @ A3 @ ( inf_inf_set_v @ B @ C3 ) )
= ( inf_inf_set_v @ B @ ( inf_inf_set_v @ A3 @ C3 ) ) ) ).
% Int_left_commute
thf(fact_762_inf__sup__aci_I4_J,axiom,
! [X2: set_v,Y2: set_v] :
( ( inf_inf_set_v @ X2 @ ( inf_inf_set_v @ X2 @ Y2 ) )
= ( inf_inf_set_v @ X2 @ Y2 ) ) ).
% inf_sup_aci(4)
thf(fact_763_inf__sup__aci_I3_J,axiom,
! [X2: set_v,Y2: set_v,Z: set_v] :
( ( inf_inf_set_v @ X2 @ ( inf_inf_set_v @ Y2 @ Z ) )
= ( inf_inf_set_v @ Y2 @ ( inf_inf_set_v @ X2 @ Z ) ) ) ).
% inf_sup_aci(3)
thf(fact_764_inf__sup__aci_I2_J,axiom,
! [X2: set_v,Y2: set_v,Z: set_v] :
( ( inf_inf_set_v @ ( inf_inf_set_v @ X2 @ Y2 ) @ Z )
= ( inf_inf_set_v @ X2 @ ( inf_inf_set_v @ Y2 @ Z ) ) ) ).
% inf_sup_aci(2)
thf(fact_765_inf__sup__aci_I1_J,axiom,
( inf_inf_set_v
= ( ^ [X3: set_v,Y3: set_v] : ( inf_inf_set_v @ Y3 @ X3 ) ) ) ).
% inf_sup_aci(1)
thf(fact_766_inf__sup__ord_I2_J,axiom,
! [X2: set_v,Y2: set_v] : ( ord_less_eq_set_v @ ( inf_inf_set_v @ X2 @ Y2 ) @ Y2 ) ).
% inf_sup_ord(2)
thf(fact_767_inf__sup__ord_I2_J,axiom,
! [X2: set_Product_prod_v_v,Y2: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ X2 @ Y2 ) @ Y2 ) ).
% inf_sup_ord(2)
thf(fact_768_inf__sup__ord_I1_J,axiom,
! [X2: set_v,Y2: set_v] : ( ord_less_eq_set_v @ ( inf_inf_set_v @ X2 @ Y2 ) @ X2 ) ).
% inf_sup_ord(1)
thf(fact_769_inf__sup__ord_I1_J,axiom,
! [X2: set_Product_prod_v_v,Y2: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ X2 @ Y2 ) @ X2 ) ).
% inf_sup_ord(1)
thf(fact_770_inf__le1,axiom,
! [X2: set_v,Y2: set_v] : ( ord_less_eq_set_v @ ( inf_inf_set_v @ X2 @ Y2 ) @ X2 ) ).
% inf_le1
thf(fact_771_inf__le1,axiom,
! [X2: set_Product_prod_v_v,Y2: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ X2 @ Y2 ) @ X2 ) ).
% inf_le1
thf(fact_772_inf__le2,axiom,
! [X2: set_v,Y2: set_v] : ( ord_less_eq_set_v @ ( inf_inf_set_v @ X2 @ Y2 ) @ Y2 ) ).
% inf_le2
thf(fact_773_inf__le2,axiom,
! [X2: set_Product_prod_v_v,Y2: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ X2 @ Y2 ) @ Y2 ) ).
% inf_le2
thf(fact_774_le__infE,axiom,
! [X2: set_v,A: set_v,B2: set_v] :
( ( ord_less_eq_set_v @ X2 @ ( inf_inf_set_v @ A @ B2 ) )
=> ~ ( ( ord_less_eq_set_v @ X2 @ A )
=> ~ ( ord_less_eq_set_v @ X2 @ B2 ) ) ) ).
% le_infE
thf(fact_775_le__infE,axiom,
! [X2: set_Product_prod_v_v,A: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X2 @ ( inf_in6271465464967711157od_v_v @ A @ B2 ) )
=> ~ ( ( ord_le7336532860387713383od_v_v @ X2 @ A )
=> ~ ( ord_le7336532860387713383od_v_v @ X2 @ B2 ) ) ) ).
% le_infE
thf(fact_776_le__infI,axiom,
! [X2: set_v,A: set_v,B2: set_v] :
( ( ord_less_eq_set_v @ X2 @ A )
=> ( ( ord_less_eq_set_v @ X2 @ B2 )
=> ( ord_less_eq_set_v @ X2 @ ( inf_inf_set_v @ A @ B2 ) ) ) ) ).
% le_infI
thf(fact_777_le__infI,axiom,
! [X2: set_Product_prod_v_v,A: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X2 @ A )
=> ( ( ord_le7336532860387713383od_v_v @ X2 @ B2 )
=> ( ord_le7336532860387713383od_v_v @ X2 @ ( inf_in6271465464967711157od_v_v @ A @ B2 ) ) ) ) ).
% le_infI
thf(fact_778_inf__mono,axiom,
! [A: set_v,C: set_v,B2: set_v,D: set_v] :
( ( ord_less_eq_set_v @ A @ C )
=> ( ( ord_less_eq_set_v @ B2 @ D )
=> ( ord_less_eq_set_v @ ( inf_inf_set_v @ A @ B2 ) @ ( inf_inf_set_v @ C @ D ) ) ) ) ).
% inf_mono
thf(fact_779_inf__mono,axiom,
! [A: set_Product_prod_v_v,C: set_Product_prod_v_v,B2: set_Product_prod_v_v,D: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ C )
=> ( ( ord_le7336532860387713383od_v_v @ B2 @ D )
=> ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ A @ B2 ) @ ( inf_in6271465464967711157od_v_v @ C @ D ) ) ) ) ).
% inf_mono
thf(fact_780_le__infI1,axiom,
! [A: set_v,X2: set_v,B2: set_v] :
( ( ord_less_eq_set_v @ A @ X2 )
=> ( ord_less_eq_set_v @ ( inf_inf_set_v @ A @ B2 ) @ X2 ) ) ).
% le_infI1
thf(fact_781_le__infI1,axiom,
! [A: set_Product_prod_v_v,X2: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ X2 )
=> ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ A @ B2 ) @ X2 ) ) ).
% le_infI1
thf(fact_782_le__infI2,axiom,
! [B2: set_v,X2: set_v,A: set_v] :
( ( ord_less_eq_set_v @ B2 @ X2 )
=> ( ord_less_eq_set_v @ ( inf_inf_set_v @ A @ B2 ) @ X2 ) ) ).
% le_infI2
thf(fact_783_le__infI2,axiom,
! [B2: set_Product_prod_v_v,X2: set_Product_prod_v_v,A: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ B2 @ X2 )
=> ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ A @ B2 ) @ X2 ) ) ).
% le_infI2
thf(fact_784_inf_Oassoc,axiom,
! [A: set_v,B2: set_v,C: set_v] :
( ( inf_inf_set_v @ ( inf_inf_set_v @ A @ B2 ) @ C )
= ( inf_inf_set_v @ A @ ( inf_inf_set_v @ B2 @ C ) ) ) ).
% inf.assoc
thf(fact_785_inf__assoc,axiom,
! [X2: set_v,Y2: set_v,Z: set_v] :
( ( inf_inf_set_v @ ( inf_inf_set_v @ X2 @ Y2 ) @ Z )
= ( inf_inf_set_v @ X2 @ ( inf_inf_set_v @ Y2 @ Z ) ) ) ).
% inf_assoc
thf(fact_786_inf_OorderE,axiom,
! [A: set_v,B2: set_v] :
( ( ord_less_eq_set_v @ A @ B2 )
=> ( A
= ( inf_inf_set_v @ A @ B2 ) ) ) ).
% inf.orderE
thf(fact_787_inf_OorderE,axiom,
! [A: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ B2 )
=> ( A
= ( inf_in6271465464967711157od_v_v @ A @ B2 ) ) ) ).
% inf.orderE
thf(fact_788_inf_OorderI,axiom,
! [A: set_v,B2: set_v] :
( ( A
= ( inf_inf_set_v @ A @ B2 ) )
=> ( ord_less_eq_set_v @ A @ B2 ) ) ).
% inf.orderI
thf(fact_789_inf_OorderI,axiom,
! [A: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( A
= ( inf_in6271465464967711157od_v_v @ A @ B2 ) )
=> ( ord_le7336532860387713383od_v_v @ A @ B2 ) ) ).
% inf.orderI
thf(fact_790_inf__unique,axiom,
! [F: set_v > set_v > set_v,X2: set_v,Y2: set_v] :
( ! [X4: set_v,Y: set_v] : ( ord_less_eq_set_v @ ( F @ X4 @ Y ) @ X4 )
=> ( ! [X4: set_v,Y: set_v] : ( ord_less_eq_set_v @ ( F @ X4 @ Y ) @ Y )
=> ( ! [X4: set_v,Y: set_v,Z3: set_v] :
( ( ord_less_eq_set_v @ X4 @ Y )
=> ( ( ord_less_eq_set_v @ X4 @ Z3 )
=> ( ord_less_eq_set_v @ X4 @ ( F @ Y @ Z3 ) ) ) )
=> ( ( inf_inf_set_v @ X2 @ Y2 )
= ( F @ X2 @ Y2 ) ) ) ) ) ).
% inf_unique
thf(fact_791_inf__unique,axiom,
! [F: set_Product_prod_v_v > set_Product_prod_v_v > set_Product_prod_v_v,X2: set_Product_prod_v_v,Y2: set_Product_prod_v_v] :
( ! [X4: set_Product_prod_v_v,Y: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ ( F @ X4 @ Y ) @ X4 )
=> ( ! [X4: set_Product_prod_v_v,Y: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ ( F @ X4 @ Y ) @ Y )
=> ( ! [X4: set_Product_prod_v_v,Y: set_Product_prod_v_v,Z3: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X4 @ Y )
=> ( ( ord_le7336532860387713383od_v_v @ X4 @ Z3 )
=> ( ord_le7336532860387713383od_v_v @ X4 @ ( F @ Y @ Z3 ) ) ) )
=> ( ( inf_in6271465464967711157od_v_v @ X2 @ Y2 )
= ( F @ X2 @ Y2 ) ) ) ) ) ).
% inf_unique
thf(fact_792_le__iff__inf,axiom,
( ord_less_eq_set_v
= ( ^ [X3: set_v,Y3: set_v] :
( ( inf_inf_set_v @ X3 @ Y3 )
= X3 ) ) ) ).
% le_iff_inf
thf(fact_793_le__iff__inf,axiom,
( ord_le7336532860387713383od_v_v
= ( ^ [X3: set_Product_prod_v_v,Y3: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ X3 @ Y3 )
= X3 ) ) ) ).
% le_iff_inf
thf(fact_794_inf_Oabsorb1,axiom,
! [A: set_v,B2: set_v] :
( ( ord_less_eq_set_v @ A @ B2 )
=> ( ( inf_inf_set_v @ A @ B2 )
= A ) ) ).
% inf.absorb1
thf(fact_795_inf_Oabsorb1,axiom,
! [A: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ B2 )
=> ( ( inf_in6271465464967711157od_v_v @ A @ B2 )
= A ) ) ).
% inf.absorb1
thf(fact_796_inf_Oabsorb2,axiom,
! [B2: set_v,A: set_v] :
( ( ord_less_eq_set_v @ B2 @ A )
=> ( ( inf_inf_set_v @ A @ B2 )
= B2 ) ) ).
% inf.absorb2
thf(fact_797_inf_Oabsorb2,axiom,
! [B2: set_Product_prod_v_v,A: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ B2 @ A )
=> ( ( inf_in6271465464967711157od_v_v @ A @ B2 )
= B2 ) ) ).
% inf.absorb2
thf(fact_798_inf_Ocommute,axiom,
( inf_inf_set_v
= ( ^ [A4: set_v,B4: set_v] : ( inf_inf_set_v @ B4 @ A4 ) ) ) ).
% inf.commute
thf(fact_799_inf__absorb1,axiom,
! [X2: set_v,Y2: set_v] :
( ( ord_less_eq_set_v @ X2 @ Y2 )
=> ( ( inf_inf_set_v @ X2 @ Y2 )
= X2 ) ) ).
% inf_absorb1
thf(fact_800_inf__absorb1,axiom,
! [X2: set_Product_prod_v_v,Y2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X2 @ Y2 )
=> ( ( inf_in6271465464967711157od_v_v @ X2 @ Y2 )
= X2 ) ) ).
% inf_absorb1
thf(fact_801_inf__absorb2,axiom,
! [Y2: set_v,X2: set_v] :
( ( ord_less_eq_set_v @ Y2 @ X2 )
=> ( ( inf_inf_set_v @ X2 @ Y2 )
= Y2 ) ) ).
% inf_absorb2
thf(fact_802_inf__absorb2,axiom,
! [Y2: set_Product_prod_v_v,X2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ Y2 @ X2 )
=> ( ( inf_in6271465464967711157od_v_v @ X2 @ Y2 )
= Y2 ) ) ).
% inf_absorb2
thf(fact_803_inf__commute,axiom,
( inf_inf_set_v
= ( ^ [X3: set_v,Y3: set_v] : ( inf_inf_set_v @ Y3 @ X3 ) ) ) ).
% inf_commute
thf(fact_804_inf_OboundedE,axiom,
! [A: set_v,B2: set_v,C: set_v] :
( ( ord_less_eq_set_v @ A @ ( inf_inf_set_v @ B2 @ C ) )
=> ~ ( ( ord_less_eq_set_v @ A @ B2 )
=> ~ ( ord_less_eq_set_v @ A @ C ) ) ) ).
% inf.boundedE
thf(fact_805_inf_OboundedE,axiom,
! [A: set_Product_prod_v_v,B2: set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ ( inf_in6271465464967711157od_v_v @ B2 @ C ) )
=> ~ ( ( ord_le7336532860387713383od_v_v @ A @ B2 )
=> ~ ( ord_le7336532860387713383od_v_v @ A @ C ) ) ) ).
% inf.boundedE
thf(fact_806_inf_OboundedI,axiom,
! [A: set_v,B2: set_v,C: set_v] :
( ( ord_less_eq_set_v @ A @ B2 )
=> ( ( ord_less_eq_set_v @ A @ C )
=> ( ord_less_eq_set_v @ A @ ( inf_inf_set_v @ B2 @ C ) ) ) ) ).
% inf.boundedI
thf(fact_807_inf_OboundedI,axiom,
! [A: set_Product_prod_v_v,B2: set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ B2 )
=> ( ( ord_le7336532860387713383od_v_v @ A @ C )
=> ( ord_le7336532860387713383od_v_v @ A @ ( inf_in6271465464967711157od_v_v @ B2 @ C ) ) ) ) ).
% inf.boundedI
thf(fact_808_inf__greatest,axiom,
! [X2: set_v,Y2: set_v,Z: set_v] :
( ( ord_less_eq_set_v @ X2 @ Y2 )
=> ( ( ord_less_eq_set_v @ X2 @ Z )
=> ( ord_less_eq_set_v @ X2 @ ( inf_inf_set_v @ Y2 @ Z ) ) ) ) ).
% inf_greatest
thf(fact_809_inf__greatest,axiom,
! [X2: set_Product_prod_v_v,Y2: set_Product_prod_v_v,Z: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X2 @ Y2 )
=> ( ( ord_le7336532860387713383od_v_v @ X2 @ Z )
=> ( ord_le7336532860387713383od_v_v @ X2 @ ( inf_in6271465464967711157od_v_v @ Y2 @ Z ) ) ) ) ).
% inf_greatest
thf(fact_810_inf_Oorder__iff,axiom,
( ord_less_eq_set_v
= ( ^ [A4: set_v,B4: set_v] :
( A4
= ( inf_inf_set_v @ A4 @ B4 ) ) ) ) ).
% inf.order_iff
thf(fact_811_inf_Oorder__iff,axiom,
( ord_le7336532860387713383od_v_v
= ( ^ [A4: set_Product_prod_v_v,B4: set_Product_prod_v_v] :
( A4
= ( inf_in6271465464967711157od_v_v @ A4 @ B4 ) ) ) ) ).
% inf.order_iff
thf(fact_812_inf_Ocobounded1,axiom,
! [A: set_v,B2: set_v] : ( ord_less_eq_set_v @ ( inf_inf_set_v @ A @ B2 ) @ A ) ).
% inf.cobounded1
thf(fact_813_inf_Ocobounded1,axiom,
! [A: set_Product_prod_v_v,B2: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ A @ B2 ) @ A ) ).
% inf.cobounded1
thf(fact_814_inf_Ocobounded2,axiom,
! [A: set_v,B2: set_v] : ( ord_less_eq_set_v @ ( inf_inf_set_v @ A @ B2 ) @ B2 ) ).
% inf.cobounded2
thf(fact_815_inf_Ocobounded2,axiom,
! [A: set_Product_prod_v_v,B2: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ A @ B2 ) @ B2 ) ).
% inf.cobounded2
thf(fact_816_inf_Oabsorb__iff1,axiom,
( ord_less_eq_set_v
= ( ^ [A4: set_v,B4: set_v] :
( ( inf_inf_set_v @ A4 @ B4 )
= A4 ) ) ) ).
% inf.absorb_iff1
thf(fact_817_inf_Oabsorb__iff1,axiom,
( ord_le7336532860387713383od_v_v
= ( ^ [A4: set_Product_prod_v_v,B4: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ A4 @ B4 )
= A4 ) ) ) ).
% inf.absorb_iff1
thf(fact_818_inf_Oabsorb__iff2,axiom,
( ord_less_eq_set_v
= ( ^ [B4: set_v,A4: set_v] :
( ( inf_inf_set_v @ A4 @ B4 )
= B4 ) ) ) ).
% inf.absorb_iff2
thf(fact_819_inf_Oabsorb__iff2,axiom,
( ord_le7336532860387713383od_v_v
= ( ^ [B4: set_Product_prod_v_v,A4: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ A4 @ B4 )
= B4 ) ) ) ).
% inf.absorb_iff2
thf(fact_820_inf_OcoboundedI1,axiom,
! [A: set_v,C: set_v,B2: set_v] :
( ( ord_less_eq_set_v @ A @ C )
=> ( ord_less_eq_set_v @ ( inf_inf_set_v @ A @ B2 ) @ C ) ) ).
% inf.coboundedI1
thf(fact_821_inf_OcoboundedI1,axiom,
! [A: set_Product_prod_v_v,C: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ C )
=> ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ A @ B2 ) @ C ) ) ).
% inf.coboundedI1
thf(fact_822_inf_OcoboundedI2,axiom,
! [B2: set_v,C: set_v,A: set_v] :
( ( ord_less_eq_set_v @ B2 @ C )
=> ( ord_less_eq_set_v @ ( inf_inf_set_v @ A @ B2 ) @ C ) ) ).
% inf.coboundedI2
thf(fact_823_inf_OcoboundedI2,axiom,
! [B2: set_Product_prod_v_v,C: set_Product_prod_v_v,A: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ B2 @ C )
=> ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ A @ B2 ) @ C ) ) ).
% inf.coboundedI2
thf(fact_824_inf_Oleft__commute,axiom,
! [B2: set_v,A: set_v,C: set_v] :
( ( inf_inf_set_v @ B2 @ ( inf_inf_set_v @ A @ C ) )
= ( inf_inf_set_v @ A @ ( inf_inf_set_v @ B2 @ C ) ) ) ).
% inf.left_commute
thf(fact_825_inf__left__commute,axiom,
! [X2: set_v,Y2: set_v,Z: set_v] :
( ( inf_inf_set_v @ X2 @ ( inf_inf_set_v @ Y2 @ Z ) )
= ( inf_inf_set_v @ Y2 @ ( inf_inf_set_v @ X2 @ Z ) ) ) ).
% inf_left_commute
thf(fact_826_sup__inf__distrib2,axiom,
! [Y2: set_Product_prod_v_v,Z: set_Product_prod_v_v,X2: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ Y2 @ Z ) @ X2 )
= ( inf_in6271465464967711157od_v_v @ ( sup_su414716646722978715od_v_v @ Y2 @ X2 ) @ ( sup_su414716646722978715od_v_v @ Z @ X2 ) ) ) ).
% sup_inf_distrib2
thf(fact_827_sup__inf__distrib2,axiom,
! [Y2: set_v,Z: set_v,X2: set_v] :
( ( sup_sup_set_v @ ( inf_inf_set_v @ Y2 @ Z ) @ X2 )
= ( inf_inf_set_v @ ( sup_sup_set_v @ Y2 @ X2 ) @ ( sup_sup_set_v @ Z @ X2 ) ) ) ).
% sup_inf_distrib2
thf(fact_828_sup__inf__distrib2,axiom,
! [Y2: set_set_v,Z: set_set_v,X2: set_set_v] :
( ( sup_sup_set_set_v @ ( inf_inf_set_set_v @ Y2 @ Z ) @ X2 )
= ( inf_inf_set_set_v @ ( sup_sup_set_set_v @ Y2 @ X2 ) @ ( sup_sup_set_set_v @ Z @ X2 ) ) ) ).
% sup_inf_distrib2
thf(fact_829_sup__inf__distrib1,axiom,
! [X2: set_Product_prod_v_v,Y2: set_Product_prod_v_v,Z: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ X2 @ ( inf_in6271465464967711157od_v_v @ Y2 @ Z ) )
= ( inf_in6271465464967711157od_v_v @ ( sup_su414716646722978715od_v_v @ X2 @ Y2 ) @ ( sup_su414716646722978715od_v_v @ X2 @ Z ) ) ) ).
% sup_inf_distrib1
thf(fact_830_sup__inf__distrib1,axiom,
! [X2: set_v,Y2: set_v,Z: set_v] :
( ( sup_sup_set_v @ X2 @ ( inf_inf_set_v @ Y2 @ Z ) )
= ( inf_inf_set_v @ ( sup_sup_set_v @ X2 @ Y2 ) @ ( sup_sup_set_v @ X2 @ Z ) ) ) ).
% sup_inf_distrib1
thf(fact_831_sup__inf__distrib1,axiom,
! [X2: set_set_v,Y2: set_set_v,Z: set_set_v] :
( ( sup_sup_set_set_v @ X2 @ ( inf_inf_set_set_v @ Y2 @ Z ) )
= ( inf_inf_set_set_v @ ( sup_sup_set_set_v @ X2 @ Y2 ) @ ( sup_sup_set_set_v @ X2 @ Z ) ) ) ).
% sup_inf_distrib1
thf(fact_832_inf__sup__distrib2,axiom,
! [Y2: set_Product_prod_v_v,Z: set_Product_prod_v_v,X2: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ ( sup_su414716646722978715od_v_v @ Y2 @ Z ) @ X2 )
= ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ Y2 @ X2 ) @ ( inf_in6271465464967711157od_v_v @ Z @ X2 ) ) ) ).
% inf_sup_distrib2
thf(fact_833_inf__sup__distrib2,axiom,
! [Y2: set_v,Z: set_v,X2: set_v] :
( ( inf_inf_set_v @ ( sup_sup_set_v @ Y2 @ Z ) @ X2 )
= ( sup_sup_set_v @ ( inf_inf_set_v @ Y2 @ X2 ) @ ( inf_inf_set_v @ Z @ X2 ) ) ) ).
% inf_sup_distrib2
thf(fact_834_inf__sup__distrib2,axiom,
! [Y2: set_set_v,Z: set_set_v,X2: set_set_v] :
( ( inf_inf_set_set_v @ ( sup_sup_set_set_v @ Y2 @ Z ) @ X2 )
= ( sup_sup_set_set_v @ ( inf_inf_set_set_v @ Y2 @ X2 ) @ ( inf_inf_set_set_v @ Z @ X2 ) ) ) ).
% inf_sup_distrib2
thf(fact_835_inf__sup__distrib1,axiom,
! [X2: set_Product_prod_v_v,Y2: set_Product_prod_v_v,Z: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ X2 @ ( sup_su414716646722978715od_v_v @ Y2 @ Z ) )
= ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ X2 @ Y2 ) @ ( inf_in6271465464967711157od_v_v @ X2 @ Z ) ) ) ).
% inf_sup_distrib1
thf(fact_836_inf__sup__distrib1,axiom,
! [X2: set_v,Y2: set_v,Z: set_v] :
( ( inf_inf_set_v @ X2 @ ( sup_sup_set_v @ Y2 @ Z ) )
= ( sup_sup_set_v @ ( inf_inf_set_v @ X2 @ Y2 ) @ ( inf_inf_set_v @ X2 @ Z ) ) ) ).
% inf_sup_distrib1
thf(fact_837_inf__sup__distrib1,axiom,
! [X2: set_set_v,Y2: set_set_v,Z: set_set_v] :
( ( inf_inf_set_set_v @ X2 @ ( sup_sup_set_set_v @ Y2 @ Z ) )
= ( sup_sup_set_set_v @ ( inf_inf_set_set_v @ X2 @ Y2 ) @ ( inf_inf_set_set_v @ X2 @ Z ) ) ) ).
% inf_sup_distrib1
thf(fact_838_distrib__imp2,axiom,
! [X2: set_Product_prod_v_v,Y2: set_Product_prod_v_v,Z: set_Product_prod_v_v] :
( ! [X4: set_Product_prod_v_v,Y: set_Product_prod_v_v,Z3: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ X4 @ ( inf_in6271465464967711157od_v_v @ Y @ Z3 ) )
= ( inf_in6271465464967711157od_v_v @ ( sup_su414716646722978715od_v_v @ X4 @ Y ) @ ( sup_su414716646722978715od_v_v @ X4 @ Z3 ) ) )
=> ( ( inf_in6271465464967711157od_v_v @ X2 @ ( sup_su414716646722978715od_v_v @ Y2 @ Z ) )
= ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ X2 @ Y2 ) @ ( inf_in6271465464967711157od_v_v @ X2 @ Z ) ) ) ) ).
% distrib_imp2
thf(fact_839_distrib__imp2,axiom,
! [X2: set_v,Y2: set_v,Z: set_v] :
( ! [X4: set_v,Y: set_v,Z3: set_v] :
( ( sup_sup_set_v @ X4 @ ( inf_inf_set_v @ Y @ Z3 ) )
= ( inf_inf_set_v @ ( sup_sup_set_v @ X4 @ Y ) @ ( sup_sup_set_v @ X4 @ Z3 ) ) )
=> ( ( inf_inf_set_v @ X2 @ ( sup_sup_set_v @ Y2 @ Z ) )
= ( sup_sup_set_v @ ( inf_inf_set_v @ X2 @ Y2 ) @ ( inf_inf_set_v @ X2 @ Z ) ) ) ) ).
% distrib_imp2
thf(fact_840_distrib__imp2,axiom,
! [X2: set_set_v,Y2: set_set_v,Z: set_set_v] :
( ! [X4: set_set_v,Y: set_set_v,Z3: set_set_v] :
( ( sup_sup_set_set_v @ X4 @ ( inf_inf_set_set_v @ Y @ Z3 ) )
= ( inf_inf_set_set_v @ ( sup_sup_set_set_v @ X4 @ Y ) @ ( sup_sup_set_set_v @ X4 @ Z3 ) ) )
=> ( ( inf_inf_set_set_v @ X2 @ ( sup_sup_set_set_v @ Y2 @ Z ) )
= ( sup_sup_set_set_v @ ( inf_inf_set_set_v @ X2 @ Y2 ) @ ( inf_inf_set_set_v @ X2 @ Z ) ) ) ) ).
% distrib_imp2
thf(fact_841_distrib__imp1,axiom,
! [X2: set_Product_prod_v_v,Y2: set_Product_prod_v_v,Z: set_Product_prod_v_v] :
( ! [X4: set_Product_prod_v_v,Y: set_Product_prod_v_v,Z3: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ X4 @ ( sup_su414716646722978715od_v_v @ Y @ Z3 ) )
= ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ X4 @ Y ) @ ( inf_in6271465464967711157od_v_v @ X4 @ Z3 ) ) )
=> ( ( sup_su414716646722978715od_v_v @ X2 @ ( inf_in6271465464967711157od_v_v @ Y2 @ Z ) )
= ( inf_in6271465464967711157od_v_v @ ( sup_su414716646722978715od_v_v @ X2 @ Y2 ) @ ( sup_su414716646722978715od_v_v @ X2 @ Z ) ) ) ) ).
% distrib_imp1
thf(fact_842_distrib__imp1,axiom,
! [X2: set_v,Y2: set_v,Z: set_v] :
( ! [X4: set_v,Y: set_v,Z3: set_v] :
( ( inf_inf_set_v @ X4 @ ( sup_sup_set_v @ Y @ Z3 ) )
= ( sup_sup_set_v @ ( inf_inf_set_v @ X4 @ Y ) @ ( inf_inf_set_v @ X4 @ Z3 ) ) )
=> ( ( sup_sup_set_v @ X2 @ ( inf_inf_set_v @ Y2 @ Z ) )
= ( inf_inf_set_v @ ( sup_sup_set_v @ X2 @ Y2 ) @ ( sup_sup_set_v @ X2 @ Z ) ) ) ) ).
% distrib_imp1
thf(fact_843_distrib__imp1,axiom,
! [X2: set_set_v,Y2: set_set_v,Z: set_set_v] :
( ! [X4: set_set_v,Y: set_set_v,Z3: set_set_v] :
( ( inf_inf_set_set_v @ X4 @ ( sup_sup_set_set_v @ Y @ Z3 ) )
= ( sup_sup_set_set_v @ ( inf_inf_set_set_v @ X4 @ Y ) @ ( inf_inf_set_set_v @ X4 @ Z3 ) ) )
=> ( ( sup_sup_set_set_v @ X2 @ ( inf_inf_set_set_v @ Y2 @ Z ) )
= ( inf_inf_set_set_v @ ( sup_sup_set_set_v @ X2 @ Y2 ) @ ( sup_sup_set_set_v @ X2 @ Z ) ) ) ) ).
% distrib_imp1
thf(fact_844_disjoint__iff__not__equal,axiom,
! [A3: set_v,B: set_v] :
( ( ( inf_inf_set_v @ A3 @ B )
= bot_bot_set_v )
= ( ! [X3: v] :
( ( member_v @ X3 @ A3 )
=> ! [Y3: v] :
( ( member_v @ Y3 @ B )
=> ( X3 != Y3 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_845_disjoint__iff__not__equal,axiom,
! [A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ( inf_in6271465464967711157od_v_v @ A3 @ B )
= bot_bo723834152578015283od_v_v )
= ( ! [X3: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X3 @ A3 )
=> ! [Y3: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ Y3 @ B )
=> ( X3 != Y3 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_846_disjoint__iff__not__equal,axiom,
! [A3: set_set_v,B: set_set_v] :
( ( ( inf_inf_set_set_v @ A3 @ B )
= bot_bot_set_set_v )
= ( ! [X3: set_v] :
( ( member_set_v @ X3 @ A3 )
=> ! [Y3: set_v] :
( ( member_set_v @ Y3 @ B )
=> ( X3 != Y3 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_847_Int__empty__right,axiom,
! [A3: set_v] :
( ( inf_inf_set_v @ A3 @ bot_bot_set_v )
= bot_bot_set_v ) ).
% Int_empty_right
thf(fact_848_Int__empty__right,axiom,
! [A3: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ A3 @ bot_bo723834152578015283od_v_v )
= bot_bo723834152578015283od_v_v ) ).
% Int_empty_right
thf(fact_849_Int__empty__right,axiom,
! [A3: set_set_v] :
( ( inf_inf_set_set_v @ A3 @ bot_bot_set_set_v )
= bot_bot_set_set_v ) ).
% Int_empty_right
thf(fact_850_Int__empty__left,axiom,
! [B: set_v] :
( ( inf_inf_set_v @ bot_bot_set_v @ B )
= bot_bot_set_v ) ).
% Int_empty_left
thf(fact_851_Int__empty__left,axiom,
! [B: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ bot_bo723834152578015283od_v_v @ B )
= bot_bo723834152578015283od_v_v ) ).
% Int_empty_left
thf(fact_852_Int__empty__left,axiom,
! [B: set_set_v] :
( ( inf_inf_set_set_v @ bot_bot_set_set_v @ B )
= bot_bot_set_set_v ) ).
% Int_empty_left
thf(fact_853_disjoint__iff,axiom,
! [A3: set_v,B: set_v] :
( ( ( inf_inf_set_v @ A3 @ B )
= bot_bot_set_v )
= ( ! [X3: v] :
( ( member_v @ X3 @ A3 )
=> ~ ( member_v @ X3 @ B ) ) ) ) ).
% disjoint_iff
thf(fact_854_disjoint__iff,axiom,
! [A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ( inf_in6271465464967711157od_v_v @ A3 @ B )
= bot_bo723834152578015283od_v_v )
= ( ! [X3: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X3 @ A3 )
=> ~ ( member7453568604450474000od_v_v @ X3 @ B ) ) ) ) ).
% disjoint_iff
thf(fact_855_disjoint__iff,axiom,
! [A3: set_set_v,B: set_set_v] :
( ( ( inf_inf_set_set_v @ A3 @ B )
= bot_bot_set_set_v )
= ( ! [X3: set_v] :
( ( member_set_v @ X3 @ A3 )
=> ~ ( member_set_v @ X3 @ B ) ) ) ) ).
% disjoint_iff
thf(fact_856_Int__emptyI,axiom,
! [A3: set_v,B: set_v] :
( ! [X4: v] :
( ( member_v @ X4 @ A3 )
=> ~ ( member_v @ X4 @ B ) )
=> ( ( inf_inf_set_v @ A3 @ B )
= bot_bot_set_v ) ) ).
% Int_emptyI
thf(fact_857_Int__emptyI,axiom,
! [A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ! [X4: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X4 @ A3 )
=> ~ ( member7453568604450474000od_v_v @ X4 @ B ) )
=> ( ( inf_in6271465464967711157od_v_v @ A3 @ B )
= bot_bo723834152578015283od_v_v ) ) ).
% Int_emptyI
thf(fact_858_Int__emptyI,axiom,
! [A3: set_set_v,B: set_set_v] :
( ! [X4: set_v] :
( ( member_set_v @ X4 @ A3 )
=> ~ ( member_set_v @ X4 @ B ) )
=> ( ( inf_inf_set_set_v @ A3 @ B )
= bot_bot_set_set_v ) ) ).
% Int_emptyI
thf(fact_859_Int__insert__right,axiom,
! [A: set_v,A3: set_set_v,B: set_set_v] :
( ( ( member_set_v @ A @ A3 )
=> ( ( inf_inf_set_set_v @ A3 @ ( insert_set_v @ A @ B ) )
= ( insert_set_v @ A @ ( inf_inf_set_set_v @ A3 @ B ) ) ) )
& ( ~ ( member_set_v @ A @ A3 )
=> ( ( inf_inf_set_set_v @ A3 @ ( insert_set_v @ A @ B ) )
= ( inf_inf_set_set_v @ A3 @ B ) ) ) ) ).
% Int_insert_right
thf(fact_860_Int__insert__right,axiom,
! [A: product_prod_v_v,A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ( member7453568604450474000od_v_v @ A @ A3 )
=> ( ( inf_in6271465464967711157od_v_v @ A3 @ ( insert1338601472111419319od_v_v @ A @ B ) )
= ( insert1338601472111419319od_v_v @ A @ ( inf_in6271465464967711157od_v_v @ A3 @ B ) ) ) )
& ( ~ ( member7453568604450474000od_v_v @ A @ A3 )
=> ( ( inf_in6271465464967711157od_v_v @ A3 @ ( insert1338601472111419319od_v_v @ A @ B ) )
= ( inf_in6271465464967711157od_v_v @ A3 @ B ) ) ) ) ).
% Int_insert_right
thf(fact_861_Int__insert__right,axiom,
! [A: v,A3: set_v,B: set_v] :
( ( ( member_v @ A @ A3 )
=> ( ( inf_inf_set_v @ A3 @ ( insert_v @ A @ B ) )
= ( insert_v @ A @ ( inf_inf_set_v @ A3 @ B ) ) ) )
& ( ~ ( member_v @ A @ A3 )
=> ( ( inf_inf_set_v @ A3 @ ( insert_v @ A @ B ) )
= ( inf_inf_set_v @ A3 @ B ) ) ) ) ).
% Int_insert_right
thf(fact_862_Int__insert__left,axiom,
! [A: set_v,C3: set_set_v,B: set_set_v] :
( ( ( member_set_v @ A @ C3 )
=> ( ( inf_inf_set_set_v @ ( insert_set_v @ A @ B ) @ C3 )
= ( insert_set_v @ A @ ( inf_inf_set_set_v @ B @ C3 ) ) ) )
& ( ~ ( member_set_v @ A @ C3 )
=> ( ( inf_inf_set_set_v @ ( insert_set_v @ A @ B ) @ C3 )
= ( inf_inf_set_set_v @ B @ C3 ) ) ) ) ).
% Int_insert_left
thf(fact_863_Int__insert__left,axiom,
! [A: product_prod_v_v,C3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ( member7453568604450474000od_v_v @ A @ C3 )
=> ( ( inf_in6271465464967711157od_v_v @ ( insert1338601472111419319od_v_v @ A @ B ) @ C3 )
= ( insert1338601472111419319od_v_v @ A @ ( inf_in6271465464967711157od_v_v @ B @ C3 ) ) ) )
& ( ~ ( member7453568604450474000od_v_v @ A @ C3 )
=> ( ( inf_in6271465464967711157od_v_v @ ( insert1338601472111419319od_v_v @ A @ B ) @ C3 )
= ( inf_in6271465464967711157od_v_v @ B @ C3 ) ) ) ) ).
% Int_insert_left
thf(fact_864_Int__insert__left,axiom,
! [A: v,C3: set_v,B: set_v] :
( ( ( member_v @ A @ C3 )
=> ( ( inf_inf_set_v @ ( insert_v @ A @ B ) @ C3 )
= ( insert_v @ A @ ( inf_inf_set_v @ B @ C3 ) ) ) )
& ( ~ ( member_v @ A @ C3 )
=> ( ( inf_inf_set_v @ ( insert_v @ A @ B ) @ C3 )
= ( inf_inf_set_v @ B @ C3 ) ) ) ) ).
% Int_insert_left
thf(fact_865_Un__Int__crazy,axiom,
! [A3: set_Product_prod_v_v,B: set_Product_prod_v_v,C3: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ A3 @ B ) @ ( inf_in6271465464967711157od_v_v @ B @ C3 ) ) @ ( inf_in6271465464967711157od_v_v @ C3 @ A3 ) )
= ( inf_in6271465464967711157od_v_v @ ( inf_in6271465464967711157od_v_v @ ( sup_su414716646722978715od_v_v @ A3 @ B ) @ ( sup_su414716646722978715od_v_v @ B @ C3 ) ) @ ( sup_su414716646722978715od_v_v @ C3 @ A3 ) ) ) ).
% Un_Int_crazy
thf(fact_866_Un__Int__crazy,axiom,
! [A3: set_v,B: set_v,C3: set_v] :
( ( sup_sup_set_v @ ( sup_sup_set_v @ ( inf_inf_set_v @ A3 @ B ) @ ( inf_inf_set_v @ B @ C3 ) ) @ ( inf_inf_set_v @ C3 @ A3 ) )
= ( inf_inf_set_v @ ( inf_inf_set_v @ ( sup_sup_set_v @ A3 @ B ) @ ( sup_sup_set_v @ B @ C3 ) ) @ ( sup_sup_set_v @ C3 @ A3 ) ) ) ).
% Un_Int_crazy
thf(fact_867_Un__Int__crazy,axiom,
! [A3: set_set_v,B: set_set_v,C3: set_set_v] :
( ( sup_sup_set_set_v @ ( sup_sup_set_set_v @ ( inf_inf_set_set_v @ A3 @ B ) @ ( inf_inf_set_set_v @ B @ C3 ) ) @ ( inf_inf_set_set_v @ C3 @ A3 ) )
= ( inf_inf_set_set_v @ ( inf_inf_set_set_v @ ( sup_sup_set_set_v @ A3 @ B ) @ ( sup_sup_set_set_v @ B @ C3 ) ) @ ( sup_sup_set_set_v @ C3 @ A3 ) ) ) ).
% Un_Int_crazy
thf(fact_868_Int__Un__distrib,axiom,
! [A3: set_Product_prod_v_v,B: set_Product_prod_v_v,C3: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ A3 @ ( sup_su414716646722978715od_v_v @ B @ C3 ) )
= ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ A3 @ B ) @ ( inf_in6271465464967711157od_v_v @ A3 @ C3 ) ) ) ).
% Int_Un_distrib
thf(fact_869_Int__Un__distrib,axiom,
! [A3: set_v,B: set_v,C3: set_v] :
( ( inf_inf_set_v @ A3 @ ( sup_sup_set_v @ B @ C3 ) )
= ( sup_sup_set_v @ ( inf_inf_set_v @ A3 @ B ) @ ( inf_inf_set_v @ A3 @ C3 ) ) ) ).
% Int_Un_distrib
thf(fact_870_Int__Un__distrib,axiom,
! [A3: set_set_v,B: set_set_v,C3: set_set_v] :
( ( inf_inf_set_set_v @ A3 @ ( sup_sup_set_set_v @ B @ C3 ) )
= ( sup_sup_set_set_v @ ( inf_inf_set_set_v @ A3 @ B ) @ ( inf_inf_set_set_v @ A3 @ C3 ) ) ) ).
% Int_Un_distrib
thf(fact_871_Un__Int__distrib,axiom,
! [A3: set_Product_prod_v_v,B: set_Product_prod_v_v,C3: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ A3 @ ( inf_in6271465464967711157od_v_v @ B @ C3 ) )
= ( inf_in6271465464967711157od_v_v @ ( sup_su414716646722978715od_v_v @ A3 @ B ) @ ( sup_su414716646722978715od_v_v @ A3 @ C3 ) ) ) ).
% Un_Int_distrib
thf(fact_872_Un__Int__distrib,axiom,
! [A3: set_v,B: set_v,C3: set_v] :
( ( sup_sup_set_v @ A3 @ ( inf_inf_set_v @ B @ C3 ) )
= ( inf_inf_set_v @ ( sup_sup_set_v @ A3 @ B ) @ ( sup_sup_set_v @ A3 @ C3 ) ) ) ).
% Un_Int_distrib
thf(fact_873_Un__Int__distrib,axiom,
! [A3: set_set_v,B: set_set_v,C3: set_set_v] :
( ( sup_sup_set_set_v @ A3 @ ( inf_inf_set_set_v @ B @ C3 ) )
= ( inf_inf_set_set_v @ ( sup_sup_set_set_v @ A3 @ B ) @ ( sup_sup_set_set_v @ A3 @ C3 ) ) ) ).
% Un_Int_distrib
thf(fact_874_Int__Un__distrib2,axiom,
! [B: set_Product_prod_v_v,C3: set_Product_prod_v_v,A3: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ ( sup_su414716646722978715od_v_v @ B @ C3 ) @ A3 )
= ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ B @ A3 ) @ ( inf_in6271465464967711157od_v_v @ C3 @ A3 ) ) ) ).
% Int_Un_distrib2
thf(fact_875_Int__Un__distrib2,axiom,
! [B: set_v,C3: set_v,A3: set_v] :
( ( inf_inf_set_v @ ( sup_sup_set_v @ B @ C3 ) @ A3 )
= ( sup_sup_set_v @ ( inf_inf_set_v @ B @ A3 ) @ ( inf_inf_set_v @ C3 @ A3 ) ) ) ).
% Int_Un_distrib2
thf(fact_876_Int__Un__distrib2,axiom,
! [B: set_set_v,C3: set_set_v,A3: set_set_v] :
( ( inf_inf_set_set_v @ ( sup_sup_set_set_v @ B @ C3 ) @ A3 )
= ( sup_sup_set_set_v @ ( inf_inf_set_set_v @ B @ A3 ) @ ( inf_inf_set_set_v @ C3 @ A3 ) ) ) ).
% Int_Un_distrib2
thf(fact_877_Un__Int__distrib2,axiom,
! [B: set_Product_prod_v_v,C3: set_Product_prod_v_v,A3: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ B @ C3 ) @ A3 )
= ( inf_in6271465464967711157od_v_v @ ( sup_su414716646722978715od_v_v @ B @ A3 ) @ ( sup_su414716646722978715od_v_v @ C3 @ A3 ) ) ) ).
% Un_Int_distrib2
thf(fact_878_Un__Int__distrib2,axiom,
! [B: set_v,C3: set_v,A3: set_v] :
( ( sup_sup_set_v @ ( inf_inf_set_v @ B @ C3 ) @ A3 )
= ( inf_inf_set_v @ ( sup_sup_set_v @ B @ A3 ) @ ( sup_sup_set_v @ C3 @ A3 ) ) ) ).
% Un_Int_distrib2
thf(fact_879_Un__Int__distrib2,axiom,
! [B: set_set_v,C3: set_set_v,A3: set_set_v] :
( ( sup_sup_set_set_v @ ( inf_inf_set_set_v @ B @ C3 ) @ A3 )
= ( inf_inf_set_set_v @ ( sup_sup_set_set_v @ B @ A3 ) @ ( sup_sup_set_set_v @ C3 @ A3 ) ) ) ).
% Un_Int_distrib2
thf(fact_880_Diff__Int__distrib2,axiom,
! [A3: set_v,B: set_v,C3: set_v] :
( ( inf_inf_set_v @ ( minus_minus_set_v @ A3 @ B ) @ C3 )
= ( minus_minus_set_v @ ( inf_inf_set_v @ A3 @ C3 ) @ ( inf_inf_set_v @ B @ C3 ) ) ) ).
% Diff_Int_distrib2
thf(fact_881_Diff__Int__distrib,axiom,
! [C3: set_v,A3: set_v,B: set_v] :
( ( inf_inf_set_v @ C3 @ ( minus_minus_set_v @ A3 @ B ) )
= ( minus_minus_set_v @ ( inf_inf_set_v @ C3 @ A3 ) @ ( inf_inf_set_v @ C3 @ B ) ) ) ).
% Diff_Int_distrib
thf(fact_882_Diff__Diff__Int,axiom,
! [A3: set_v,B: set_v] :
( ( minus_minus_set_v @ A3 @ ( minus_minus_set_v @ A3 @ B ) )
= ( inf_inf_set_v @ A3 @ B ) ) ).
% Diff_Diff_Int
thf(fact_883_Diff__Int2,axiom,
! [A3: set_v,C3: set_v,B: set_v] :
( ( minus_minus_set_v @ ( inf_inf_set_v @ A3 @ C3 ) @ ( inf_inf_set_v @ B @ C3 ) )
= ( minus_minus_set_v @ ( inf_inf_set_v @ A3 @ C3 ) @ B ) ) ).
% Diff_Int2
thf(fact_884_Int__Diff,axiom,
! [A3: set_v,B: set_v,C3: set_v] :
( ( minus_minus_set_v @ ( inf_inf_set_v @ A3 @ B ) @ C3 )
= ( inf_inf_set_v @ A3 @ ( minus_minus_set_v @ B @ C3 ) ) ) ).
% Int_Diff
thf(fact_885_bot_Oextremum,axiom,
! [A: set_set_v] : ( ord_le5216385588623774835_set_v @ bot_bot_set_set_v @ A ) ).
% bot.extremum
thf(fact_886_bot_Oextremum,axiom,
! [A: set_v] : ( ord_less_eq_set_v @ bot_bot_set_v @ A ) ).
% bot.extremum
thf(fact_887_bot_Oextremum,axiom,
! [A: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ bot_bo723834152578015283od_v_v @ A ) ).
% bot.extremum
thf(fact_888_bot_Oextremum__unique,axiom,
! [A: set_set_v] :
( ( ord_le5216385588623774835_set_v @ A @ bot_bot_set_set_v )
= ( A = bot_bot_set_set_v ) ) ).
% bot.extremum_unique
thf(fact_889_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_890_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_891_bot_Oextremum__uniqueI,axiom,
! [A: set_set_v] :
( ( ord_le5216385588623774835_set_v @ A @ bot_bot_set_set_v )
=> ( A = bot_bot_set_set_v ) ) ).
% bot.extremum_uniqueI
thf(fact_892_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_893_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_894_sup_OcoboundedI2,axiom,
! [C: set_set_v,B2: set_set_v,A: set_set_v] :
( ( ord_le5216385588623774835_set_v @ C @ B2 )
=> ( ord_le5216385588623774835_set_v @ C @ ( sup_sup_set_set_v @ A @ B2 ) ) ) ).
% sup.coboundedI2
thf(fact_895_sup_OcoboundedI2,axiom,
! [C: set_v,B2: set_v,A: set_v] :
( ( ord_less_eq_set_v @ C @ B2 )
=> ( ord_less_eq_set_v @ C @ ( sup_sup_set_v @ A @ B2 ) ) ) ).
% sup.coboundedI2
thf(fact_896_sup_OcoboundedI2,axiom,
! [C: set_Product_prod_v_v,B2: set_Product_prod_v_v,A: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ C @ B2 )
=> ( ord_le7336532860387713383od_v_v @ C @ ( sup_su414716646722978715od_v_v @ A @ B2 ) ) ) ).
% sup.coboundedI2
thf(fact_897_sup_OcoboundedI1,axiom,
! [C: set_set_v,A: set_set_v,B2: set_set_v] :
( ( ord_le5216385588623774835_set_v @ C @ A )
=> ( ord_le5216385588623774835_set_v @ C @ ( sup_sup_set_set_v @ A @ B2 ) ) ) ).
% sup.coboundedI1
thf(fact_898_sup_OcoboundedI1,axiom,
! [C: set_v,A: set_v,B2: set_v] :
( ( ord_less_eq_set_v @ C @ A )
=> ( ord_less_eq_set_v @ C @ ( sup_sup_set_v @ A @ B2 ) ) ) ).
% sup.coboundedI1
thf(fact_899_sup_OcoboundedI1,axiom,
! [C: set_Product_prod_v_v,A: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ C @ A )
=> ( ord_le7336532860387713383od_v_v @ C @ ( sup_su414716646722978715od_v_v @ A @ B2 ) ) ) ).
% sup.coboundedI1
thf(fact_900_sup_Oabsorb__iff2,axiom,
( ord_le5216385588623774835_set_v
= ( ^ [A4: set_set_v,B4: set_set_v] :
( ( sup_sup_set_set_v @ A4 @ B4 )
= B4 ) ) ) ).
% sup.absorb_iff2
thf(fact_901_sup_Oabsorb__iff2,axiom,
( ord_less_eq_set_v
= ( ^ [A4: set_v,B4: set_v] :
( ( sup_sup_set_v @ A4 @ B4 )
= B4 ) ) ) ).
% sup.absorb_iff2
thf(fact_902_sup_Oabsorb__iff2,axiom,
( ord_le7336532860387713383od_v_v
= ( ^ [A4: set_Product_prod_v_v,B4: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ A4 @ B4 )
= B4 ) ) ) ).
% sup.absorb_iff2
thf(fact_903_sup_Oabsorb__iff1,axiom,
( ord_le5216385588623774835_set_v
= ( ^ [B4: set_set_v,A4: set_set_v] :
( ( sup_sup_set_set_v @ A4 @ B4 )
= A4 ) ) ) ).
% sup.absorb_iff1
thf(fact_904_sup_Oabsorb__iff1,axiom,
( ord_less_eq_set_v
= ( ^ [B4: set_v,A4: set_v] :
( ( sup_sup_set_v @ A4 @ B4 )
= A4 ) ) ) ).
% sup.absorb_iff1
thf(fact_905_sup_Oabsorb__iff1,axiom,
( ord_le7336532860387713383od_v_v
= ( ^ [B4: set_Product_prod_v_v,A4: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ A4 @ B4 )
= A4 ) ) ) ).
% sup.absorb_iff1
thf(fact_906_sup_Ocobounded2,axiom,
! [B2: set_set_v,A: set_set_v] : ( ord_le5216385588623774835_set_v @ B2 @ ( sup_sup_set_set_v @ A @ B2 ) ) ).
% sup.cobounded2
thf(fact_907_sup_Ocobounded2,axiom,
! [B2: set_v,A: set_v] : ( ord_less_eq_set_v @ B2 @ ( sup_sup_set_v @ A @ B2 ) ) ).
% sup.cobounded2
thf(fact_908_sup_Ocobounded2,axiom,
! [B2: set_Product_prod_v_v,A: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ B2 @ ( sup_su414716646722978715od_v_v @ A @ B2 ) ) ).
% sup.cobounded2
thf(fact_909_sup_Ocobounded1,axiom,
! [A: set_set_v,B2: set_set_v] : ( ord_le5216385588623774835_set_v @ A @ ( sup_sup_set_set_v @ A @ B2 ) ) ).
% sup.cobounded1
thf(fact_910_sup_Ocobounded1,axiom,
! [A: set_v,B2: set_v] : ( ord_less_eq_set_v @ A @ ( sup_sup_set_v @ A @ B2 ) ) ).
% sup.cobounded1
thf(fact_911_sup_Ocobounded1,axiom,
! [A: set_Product_prod_v_v,B2: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ A @ ( sup_su414716646722978715od_v_v @ A @ B2 ) ) ).
% sup.cobounded1
thf(fact_912_sup_Oorder__iff,axiom,
( ord_le5216385588623774835_set_v
= ( ^ [B4: set_set_v,A4: set_set_v] :
( A4
= ( sup_sup_set_set_v @ A4 @ B4 ) ) ) ) ).
% sup.order_iff
thf(fact_913_sup_Oorder__iff,axiom,
( ord_less_eq_set_v
= ( ^ [B4: set_v,A4: set_v] :
( A4
= ( sup_sup_set_v @ A4 @ B4 ) ) ) ) ).
% sup.order_iff
thf(fact_914_sup_Oorder__iff,axiom,
( ord_le7336532860387713383od_v_v
= ( ^ [B4: set_Product_prod_v_v,A4: set_Product_prod_v_v] :
( A4
= ( sup_su414716646722978715od_v_v @ A4 @ B4 ) ) ) ) ).
% sup.order_iff
thf(fact_915_sup_OboundedI,axiom,
! [B2: set_set_v,A: set_set_v,C: set_set_v] :
( ( ord_le5216385588623774835_set_v @ B2 @ A )
=> ( ( ord_le5216385588623774835_set_v @ C @ A )
=> ( ord_le5216385588623774835_set_v @ ( sup_sup_set_set_v @ B2 @ C ) @ A ) ) ) ).
% sup.boundedI
thf(fact_916_sup_OboundedI,axiom,
! [B2: set_v,A: set_v,C: set_v] :
( ( ord_less_eq_set_v @ B2 @ A )
=> ( ( ord_less_eq_set_v @ C @ A )
=> ( ord_less_eq_set_v @ ( sup_sup_set_v @ B2 @ C ) @ A ) ) ) ).
% sup.boundedI
thf(fact_917_sup_OboundedI,axiom,
! [B2: set_Product_prod_v_v,A: set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ B2 @ A )
=> ( ( ord_le7336532860387713383od_v_v @ C @ A )
=> ( ord_le7336532860387713383od_v_v @ ( sup_su414716646722978715od_v_v @ B2 @ C ) @ A ) ) ) ).
% sup.boundedI
thf(fact_918_sup_OboundedE,axiom,
! [B2: set_set_v,C: set_set_v,A: set_set_v] :
( ( ord_le5216385588623774835_set_v @ ( sup_sup_set_set_v @ B2 @ C ) @ A )
=> ~ ( ( ord_le5216385588623774835_set_v @ B2 @ A )
=> ~ ( ord_le5216385588623774835_set_v @ C @ A ) ) ) ).
% sup.boundedE
thf(fact_919_sup_OboundedE,axiom,
! [B2: set_v,C: set_v,A: set_v] :
( ( ord_less_eq_set_v @ ( sup_sup_set_v @ B2 @ C ) @ A )
=> ~ ( ( ord_less_eq_set_v @ B2 @ A )
=> ~ ( ord_less_eq_set_v @ C @ A ) ) ) ).
% sup.boundedE
thf(fact_920_sup_OboundedE,axiom,
! [B2: 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 @ B2 @ C ) @ A )
=> ~ ( ( ord_le7336532860387713383od_v_v @ B2 @ A )
=> ~ ( ord_le7336532860387713383od_v_v @ C @ A ) ) ) ).
% sup.boundedE
thf(fact_921_sup__absorb2,axiom,
! [X2: set_set_v,Y2: set_set_v] :
( ( ord_le5216385588623774835_set_v @ X2 @ Y2 )
=> ( ( sup_sup_set_set_v @ X2 @ Y2 )
= Y2 ) ) ).
% sup_absorb2
thf(fact_922_sup__absorb2,axiom,
! [X2: set_v,Y2: set_v] :
( ( ord_less_eq_set_v @ X2 @ Y2 )
=> ( ( sup_sup_set_v @ X2 @ Y2 )
= Y2 ) ) ).
% sup_absorb2
thf(fact_923_sup__absorb2,axiom,
! [X2: set_Product_prod_v_v,Y2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X2 @ Y2 )
=> ( ( sup_su414716646722978715od_v_v @ X2 @ Y2 )
= Y2 ) ) ).
% sup_absorb2
thf(fact_924_sup__absorb1,axiom,
! [Y2: set_set_v,X2: set_set_v] :
( ( ord_le5216385588623774835_set_v @ Y2 @ X2 )
=> ( ( sup_sup_set_set_v @ X2 @ Y2 )
= X2 ) ) ).
% sup_absorb1
thf(fact_925_sup__absorb1,axiom,
! [Y2: set_v,X2: set_v] :
( ( ord_less_eq_set_v @ Y2 @ X2 )
=> ( ( sup_sup_set_v @ X2 @ Y2 )
= X2 ) ) ).
% sup_absorb1
thf(fact_926_sup__absorb1,axiom,
! [Y2: set_Product_prod_v_v,X2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ Y2 @ X2 )
=> ( ( sup_su414716646722978715od_v_v @ X2 @ Y2 )
= X2 ) ) ).
% sup_absorb1
thf(fact_927_sup_Oabsorb2,axiom,
! [A: set_set_v,B2: set_set_v] :
( ( ord_le5216385588623774835_set_v @ A @ B2 )
=> ( ( sup_sup_set_set_v @ A @ B2 )
= B2 ) ) ).
% sup.absorb2
thf(fact_928_sup_Oabsorb2,axiom,
! [A: set_v,B2: set_v] :
( ( ord_less_eq_set_v @ A @ B2 )
=> ( ( sup_sup_set_v @ A @ B2 )
= B2 ) ) ).
% sup.absorb2
thf(fact_929_sup_Oabsorb2,axiom,
! [A: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ B2 )
=> ( ( sup_su414716646722978715od_v_v @ A @ B2 )
= B2 ) ) ).
% sup.absorb2
thf(fact_930_sup_Oabsorb1,axiom,
! [B2: set_set_v,A: set_set_v] :
( ( ord_le5216385588623774835_set_v @ B2 @ A )
=> ( ( sup_sup_set_set_v @ A @ B2 )
= A ) ) ).
% sup.absorb1
thf(fact_931_sup_Oabsorb1,axiom,
! [B2: set_v,A: set_v] :
( ( ord_less_eq_set_v @ B2 @ A )
=> ( ( sup_sup_set_v @ A @ B2 )
= A ) ) ).
% sup.absorb1
thf(fact_932_sup_Oabsorb1,axiom,
! [B2: set_Product_prod_v_v,A: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ B2 @ A )
=> ( ( sup_su414716646722978715od_v_v @ A @ B2 )
= A ) ) ).
% sup.absorb1
thf(fact_933_sup__unique,axiom,
! [F: set_set_v > set_set_v > set_set_v,X2: set_set_v,Y2: set_set_v] :
( ! [X4: set_set_v,Y: set_set_v] : ( ord_le5216385588623774835_set_v @ X4 @ ( F @ X4 @ Y ) )
=> ( ! [X4: set_set_v,Y: set_set_v] : ( ord_le5216385588623774835_set_v @ Y @ ( F @ X4 @ Y ) )
=> ( ! [X4: set_set_v,Y: set_set_v,Z3: set_set_v] :
( ( ord_le5216385588623774835_set_v @ Y @ X4 )
=> ( ( ord_le5216385588623774835_set_v @ Z3 @ X4 )
=> ( ord_le5216385588623774835_set_v @ ( F @ Y @ Z3 ) @ X4 ) ) )
=> ( ( sup_sup_set_set_v @ X2 @ Y2 )
= ( F @ X2 @ Y2 ) ) ) ) ) ).
% sup_unique
thf(fact_934_sup__unique,axiom,
! [F: set_v > set_v > set_v,X2: set_v,Y2: set_v] :
( ! [X4: set_v,Y: set_v] : ( ord_less_eq_set_v @ X4 @ ( F @ X4 @ Y ) )
=> ( ! [X4: set_v,Y: set_v] : ( ord_less_eq_set_v @ Y @ ( F @ X4 @ Y ) )
=> ( ! [X4: set_v,Y: set_v,Z3: set_v] :
( ( ord_less_eq_set_v @ Y @ X4 )
=> ( ( ord_less_eq_set_v @ Z3 @ X4 )
=> ( ord_less_eq_set_v @ ( F @ Y @ Z3 ) @ X4 ) ) )
=> ( ( sup_sup_set_v @ X2 @ Y2 )
= ( F @ X2 @ Y2 ) ) ) ) ) ).
% sup_unique
thf(fact_935_sup__unique,axiom,
! [F: set_Product_prod_v_v > set_Product_prod_v_v > set_Product_prod_v_v,X2: set_Product_prod_v_v,Y2: set_Product_prod_v_v] :
( ! [X4: set_Product_prod_v_v,Y: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ X4 @ ( F @ X4 @ Y ) )
=> ( ! [X4: set_Product_prod_v_v,Y: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ Y @ ( F @ X4 @ Y ) )
=> ( ! [X4: set_Product_prod_v_v,Y: set_Product_prod_v_v,Z3: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ Y @ X4 )
=> ( ( ord_le7336532860387713383od_v_v @ Z3 @ X4 )
=> ( ord_le7336532860387713383od_v_v @ ( F @ Y @ Z3 ) @ X4 ) ) )
=> ( ( sup_su414716646722978715od_v_v @ X2 @ Y2 )
= ( F @ X2 @ Y2 ) ) ) ) ) ).
% sup_unique
thf(fact_936_sup_OorderI,axiom,
! [A: set_set_v,B2: set_set_v] :
( ( A
= ( sup_sup_set_set_v @ A @ B2 ) )
=> ( ord_le5216385588623774835_set_v @ B2 @ A ) ) ).
% sup.orderI
thf(fact_937_sup_OorderI,axiom,
! [A: set_v,B2: set_v] :
( ( A
= ( sup_sup_set_v @ A @ B2 ) )
=> ( ord_less_eq_set_v @ B2 @ A ) ) ).
% sup.orderI
thf(fact_938_sup_OorderI,axiom,
! [A: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( A
= ( sup_su414716646722978715od_v_v @ A @ B2 ) )
=> ( ord_le7336532860387713383od_v_v @ B2 @ A ) ) ).
% sup.orderI
thf(fact_939_sup_OorderE,axiom,
! [B2: set_set_v,A: set_set_v] :
( ( ord_le5216385588623774835_set_v @ B2 @ A )
=> ( A
= ( sup_sup_set_set_v @ A @ B2 ) ) ) ).
% sup.orderE
thf(fact_940_sup_OorderE,axiom,
! [B2: set_v,A: set_v] :
( ( ord_less_eq_set_v @ B2 @ A )
=> ( A
= ( sup_sup_set_v @ A @ B2 ) ) ) ).
% sup.orderE
thf(fact_941_sup_OorderE,axiom,
! [B2: set_Product_prod_v_v,A: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ B2 @ A )
=> ( A
= ( sup_su414716646722978715od_v_v @ A @ B2 ) ) ) ).
% sup.orderE
thf(fact_942_le__iff__sup,axiom,
( ord_le5216385588623774835_set_v
= ( ^ [X3: set_set_v,Y3: set_set_v] :
( ( sup_sup_set_set_v @ X3 @ Y3 )
= Y3 ) ) ) ).
% le_iff_sup
thf(fact_943_le__iff__sup,axiom,
( ord_less_eq_set_v
= ( ^ [X3: set_v,Y3: set_v] :
( ( sup_sup_set_v @ X3 @ Y3 )
= Y3 ) ) ) ).
% le_iff_sup
thf(fact_944_le__iff__sup,axiom,
( ord_le7336532860387713383od_v_v
= ( ^ [X3: set_Product_prod_v_v,Y3: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ X3 @ Y3 )
= Y3 ) ) ) ).
% le_iff_sup
thf(fact_945_sup__least,axiom,
! [Y2: set_set_v,X2: set_set_v,Z: set_set_v] :
( ( ord_le5216385588623774835_set_v @ Y2 @ X2 )
=> ( ( ord_le5216385588623774835_set_v @ Z @ X2 )
=> ( ord_le5216385588623774835_set_v @ ( sup_sup_set_set_v @ Y2 @ Z ) @ X2 ) ) ) ).
% sup_least
thf(fact_946_sup__least,axiom,
! [Y2: set_v,X2: set_v,Z: set_v] :
( ( ord_less_eq_set_v @ Y2 @ X2 )
=> ( ( ord_less_eq_set_v @ Z @ X2 )
=> ( ord_less_eq_set_v @ ( sup_sup_set_v @ Y2 @ Z ) @ X2 ) ) ) ).
% sup_least
thf(fact_947_sup__least,axiom,
! [Y2: set_Product_prod_v_v,X2: set_Product_prod_v_v,Z: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ Y2 @ X2 )
=> ( ( ord_le7336532860387713383od_v_v @ Z @ X2 )
=> ( ord_le7336532860387713383od_v_v @ ( sup_su414716646722978715od_v_v @ Y2 @ Z ) @ X2 ) ) ) ).
% sup_least
thf(fact_948_sup__mono,axiom,
! [A: set_set_v,C: set_set_v,B2: set_set_v,D: set_set_v] :
( ( ord_le5216385588623774835_set_v @ A @ C )
=> ( ( ord_le5216385588623774835_set_v @ B2 @ D )
=> ( ord_le5216385588623774835_set_v @ ( sup_sup_set_set_v @ A @ B2 ) @ ( sup_sup_set_set_v @ C @ D ) ) ) ) ).
% sup_mono
thf(fact_949_sup__mono,axiom,
! [A: set_v,C: set_v,B2: set_v,D: set_v] :
( ( ord_less_eq_set_v @ A @ C )
=> ( ( ord_less_eq_set_v @ B2 @ D )
=> ( ord_less_eq_set_v @ ( sup_sup_set_v @ A @ B2 ) @ ( sup_sup_set_v @ C @ D ) ) ) ) ).
% sup_mono
thf(fact_950_sup__mono,axiom,
! [A: set_Product_prod_v_v,C: set_Product_prod_v_v,B2: set_Product_prod_v_v,D: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ C )
=> ( ( ord_le7336532860387713383od_v_v @ B2 @ D )
=> ( ord_le7336532860387713383od_v_v @ ( sup_su414716646722978715od_v_v @ A @ B2 ) @ ( sup_su414716646722978715od_v_v @ C @ D ) ) ) ) ).
% sup_mono
thf(fact_951_sup_Omono,axiom,
! [C: set_set_v,A: set_set_v,D: set_set_v,B2: set_set_v] :
( ( ord_le5216385588623774835_set_v @ C @ A )
=> ( ( ord_le5216385588623774835_set_v @ D @ B2 )
=> ( ord_le5216385588623774835_set_v @ ( sup_sup_set_set_v @ C @ D ) @ ( sup_sup_set_set_v @ A @ B2 ) ) ) ) ).
% sup.mono
thf(fact_952_sup_Omono,axiom,
! [C: set_v,A: set_v,D: set_v,B2: set_v] :
( ( ord_less_eq_set_v @ C @ A )
=> ( ( ord_less_eq_set_v @ D @ B2 )
=> ( ord_less_eq_set_v @ ( sup_sup_set_v @ C @ D ) @ ( sup_sup_set_v @ A @ B2 ) ) ) ) ).
% sup.mono
thf(fact_953_sup_Omono,axiom,
! [C: set_Product_prod_v_v,A: set_Product_prod_v_v,D: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ C @ A )
=> ( ( ord_le7336532860387713383od_v_v @ D @ B2 )
=> ( ord_le7336532860387713383od_v_v @ ( sup_su414716646722978715od_v_v @ C @ D ) @ ( sup_su414716646722978715od_v_v @ A @ B2 ) ) ) ) ).
% sup.mono
thf(fact_954_le__supI2,axiom,
! [X2: set_set_v,B2: set_set_v,A: set_set_v] :
( ( ord_le5216385588623774835_set_v @ X2 @ B2 )
=> ( ord_le5216385588623774835_set_v @ X2 @ ( sup_sup_set_set_v @ A @ B2 ) ) ) ).
% le_supI2
thf(fact_955_le__supI2,axiom,
! [X2: set_v,B2: set_v,A: set_v] :
( ( ord_less_eq_set_v @ X2 @ B2 )
=> ( ord_less_eq_set_v @ X2 @ ( sup_sup_set_v @ A @ B2 ) ) ) ).
% le_supI2
thf(fact_956_le__supI2,axiom,
! [X2: set_Product_prod_v_v,B2: set_Product_prod_v_v,A: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X2 @ B2 )
=> ( ord_le7336532860387713383od_v_v @ X2 @ ( sup_su414716646722978715od_v_v @ A @ B2 ) ) ) ).
% le_supI2
thf(fact_957_le__supI1,axiom,
! [X2: set_set_v,A: set_set_v,B2: set_set_v] :
( ( ord_le5216385588623774835_set_v @ X2 @ A )
=> ( ord_le5216385588623774835_set_v @ X2 @ ( sup_sup_set_set_v @ A @ B2 ) ) ) ).
% le_supI1
thf(fact_958_le__supI1,axiom,
! [X2: set_v,A: set_v,B2: set_v] :
( ( ord_less_eq_set_v @ X2 @ A )
=> ( ord_less_eq_set_v @ X2 @ ( sup_sup_set_v @ A @ B2 ) ) ) ).
% le_supI1
thf(fact_959_le__supI1,axiom,
! [X2: set_Product_prod_v_v,A: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X2 @ A )
=> ( ord_le7336532860387713383od_v_v @ X2 @ ( sup_su414716646722978715od_v_v @ A @ B2 ) ) ) ).
% le_supI1
thf(fact_960_sup__ge2,axiom,
! [Y2: set_set_v,X2: set_set_v] : ( ord_le5216385588623774835_set_v @ Y2 @ ( sup_sup_set_set_v @ X2 @ Y2 ) ) ).
% sup_ge2
thf(fact_961_sup__ge2,axiom,
! [Y2: set_v,X2: set_v] : ( ord_less_eq_set_v @ Y2 @ ( sup_sup_set_v @ X2 @ Y2 ) ) ).
% sup_ge2
thf(fact_962_sup__ge2,axiom,
! [Y2: set_Product_prod_v_v,X2: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ Y2 @ ( sup_su414716646722978715od_v_v @ X2 @ Y2 ) ) ).
% sup_ge2
thf(fact_963_sup__ge1,axiom,
! [X2: set_set_v,Y2: set_set_v] : ( ord_le5216385588623774835_set_v @ X2 @ ( sup_sup_set_set_v @ X2 @ Y2 ) ) ).
% sup_ge1
thf(fact_964_sup__ge1,axiom,
! [X2: set_v,Y2: set_v] : ( ord_less_eq_set_v @ X2 @ ( sup_sup_set_v @ X2 @ Y2 ) ) ).
% sup_ge1
thf(fact_965_sup__ge1,axiom,
! [X2: set_Product_prod_v_v,Y2: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ X2 @ ( sup_su414716646722978715od_v_v @ X2 @ Y2 ) ) ).
% sup_ge1
thf(fact_966_le__supI,axiom,
! [A: set_set_v,X2: set_set_v,B2: set_set_v] :
( ( ord_le5216385588623774835_set_v @ A @ X2 )
=> ( ( ord_le5216385588623774835_set_v @ B2 @ X2 )
=> ( ord_le5216385588623774835_set_v @ ( sup_sup_set_set_v @ A @ B2 ) @ X2 ) ) ) ).
% le_supI
thf(fact_967_le__supI,axiom,
! [A: set_v,X2: set_v,B2: set_v] :
( ( ord_less_eq_set_v @ A @ X2 )
=> ( ( ord_less_eq_set_v @ B2 @ X2 )
=> ( ord_less_eq_set_v @ ( sup_sup_set_v @ A @ B2 ) @ X2 ) ) ) ).
% le_supI
thf(fact_968_le__supI,axiom,
! [A: set_Product_prod_v_v,X2: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ X2 )
=> ( ( ord_le7336532860387713383od_v_v @ B2 @ X2 )
=> ( ord_le7336532860387713383od_v_v @ ( sup_su414716646722978715od_v_v @ A @ B2 ) @ X2 ) ) ) ).
% le_supI
thf(fact_969_le__supE,axiom,
! [A: set_set_v,B2: set_set_v,X2: set_set_v] :
( ( ord_le5216385588623774835_set_v @ ( sup_sup_set_set_v @ A @ B2 ) @ X2 )
=> ~ ( ( ord_le5216385588623774835_set_v @ A @ X2 )
=> ~ ( ord_le5216385588623774835_set_v @ B2 @ X2 ) ) ) ).
% le_supE
thf(fact_970_le__supE,axiom,
! [A: set_v,B2: set_v,X2: set_v] :
( ( ord_less_eq_set_v @ ( sup_sup_set_v @ A @ B2 ) @ X2 )
=> ~ ( ( ord_less_eq_set_v @ A @ X2 )
=> ~ ( ord_less_eq_set_v @ B2 @ X2 ) ) ) ).
% le_supE
thf(fact_971_le__supE,axiom,
! [A: set_Product_prod_v_v,B2: set_Product_prod_v_v,X2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ ( sup_su414716646722978715od_v_v @ A @ B2 ) @ X2 )
=> ~ ( ( ord_le7336532860387713383od_v_v @ A @ X2 )
=> ~ ( ord_le7336532860387713383od_v_v @ B2 @ X2 ) ) ) ).
% le_supE
thf(fact_972_inf__sup__ord_I3_J,axiom,
! [X2: set_set_v,Y2: set_set_v] : ( ord_le5216385588623774835_set_v @ X2 @ ( sup_sup_set_set_v @ X2 @ Y2 ) ) ).
% inf_sup_ord(3)
thf(fact_973_inf__sup__ord_I3_J,axiom,
! [X2: set_v,Y2: set_v] : ( ord_less_eq_set_v @ X2 @ ( sup_sup_set_v @ X2 @ Y2 ) ) ).
% inf_sup_ord(3)
thf(fact_974_inf__sup__ord_I3_J,axiom,
! [X2: set_Product_prod_v_v,Y2: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ X2 @ ( sup_su414716646722978715od_v_v @ X2 @ Y2 ) ) ).
% inf_sup_ord(3)
thf(fact_975_inf__sup__ord_I4_J,axiom,
! [Y2: set_set_v,X2: set_set_v] : ( ord_le5216385588623774835_set_v @ Y2 @ ( sup_sup_set_set_v @ X2 @ Y2 ) ) ).
% inf_sup_ord(4)
thf(fact_976_inf__sup__ord_I4_J,axiom,
! [Y2: set_v,X2: set_v] : ( ord_less_eq_set_v @ Y2 @ ( sup_sup_set_v @ X2 @ Y2 ) ) ).
% inf_sup_ord(4)
thf(fact_977_inf__sup__ord_I4_J,axiom,
! [Y2: set_Product_prod_v_v,X2: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ Y2 @ ( sup_su414716646722978715od_v_v @ X2 @ Y2 ) ) ).
% inf_sup_ord(4)
thf(fact_978_subset__insertI2,axiom,
! [A3: set_set_v,B: set_set_v,B2: set_v] :
( ( ord_le5216385588623774835_set_v @ A3 @ B )
=> ( ord_le5216385588623774835_set_v @ A3 @ ( insert_set_v @ B2 @ B ) ) ) ).
% subset_insertI2
thf(fact_979_subset__insertI2,axiom,
! [A3: set_v,B: set_v,B2: v] :
( ( ord_less_eq_set_v @ A3 @ B )
=> ( ord_less_eq_set_v @ A3 @ ( insert_v @ B2 @ B ) ) ) ).
% subset_insertI2
thf(fact_980_subset__insertI2,axiom,
! [A3: set_Product_prod_v_v,B: set_Product_prod_v_v,B2: product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A3 @ B )
=> ( ord_le7336532860387713383od_v_v @ A3 @ ( insert1338601472111419319od_v_v @ B2 @ B ) ) ) ).
% subset_insertI2
thf(fact_981_subset__insertI,axiom,
! [B: set_set_v,A: set_v] : ( ord_le5216385588623774835_set_v @ B @ ( insert_set_v @ A @ B ) ) ).
% subset_insertI
thf(fact_982_subset__insertI,axiom,
! [B: set_v,A: v] : ( ord_less_eq_set_v @ B @ ( insert_v @ A @ B ) ) ).
% subset_insertI
thf(fact_983_subset__insertI,axiom,
! [B: set_Product_prod_v_v,A: product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ B @ ( insert1338601472111419319od_v_v @ A @ B ) ) ).
% subset_insertI
thf(fact_984_subset__insert,axiom,
! [X2: set_v,A3: set_set_v,B: set_set_v] :
( ~ ( member_set_v @ X2 @ A3 )
=> ( ( ord_le5216385588623774835_set_v @ A3 @ ( insert_set_v @ X2 @ B ) )
= ( ord_le5216385588623774835_set_v @ A3 @ B ) ) ) ).
% subset_insert
thf(fact_985_subset__insert,axiom,
! [X2: v,A3: set_v,B: set_v] :
( ~ ( member_v @ X2 @ A3 )
=> ( ( ord_less_eq_set_v @ A3 @ ( insert_v @ X2 @ B ) )
= ( ord_less_eq_set_v @ A3 @ B ) ) ) ).
% subset_insert
thf(fact_986_subset__insert,axiom,
! [X2: product_prod_v_v,A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ~ ( member7453568604450474000od_v_v @ X2 @ A3 )
=> ( ( ord_le7336532860387713383od_v_v @ A3 @ ( insert1338601472111419319od_v_v @ X2 @ B ) )
= ( ord_le7336532860387713383od_v_v @ A3 @ B ) ) ) ).
% subset_insert
thf(fact_987_insert__mono,axiom,
! [C3: set_set_v,D2: set_set_v,A: set_v] :
( ( ord_le5216385588623774835_set_v @ C3 @ D2 )
=> ( ord_le5216385588623774835_set_v @ ( insert_set_v @ A @ C3 ) @ ( insert_set_v @ A @ D2 ) ) ) ).
% insert_mono
thf(fact_988_insert__mono,axiom,
! [C3: set_v,D2: set_v,A: v] :
( ( ord_less_eq_set_v @ C3 @ D2 )
=> ( ord_less_eq_set_v @ ( insert_v @ A @ C3 ) @ ( insert_v @ A @ D2 ) ) ) ).
% insert_mono
thf(fact_989_insert__mono,axiom,
! [C3: set_Product_prod_v_v,D2: set_Product_prod_v_v,A: product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ C3 @ D2 )
=> ( ord_le7336532860387713383od_v_v @ ( insert1338601472111419319od_v_v @ A @ C3 ) @ ( insert1338601472111419319od_v_v @ A @ D2 ) ) ) ).
% insert_mono
thf(fact_990_subset__code_I1_J,axiom,
! [Xs: list_v,B: set_v] :
( ( ord_less_eq_set_v @ ( set_v2 @ Xs ) @ B )
= ( ! [X3: v] :
( ( member_v @ X3 @ ( set_v2 @ Xs ) )
=> ( member_v @ X3 @ B ) ) ) ) ).
% subset_code(1)
thf(fact_991_subset__code_I1_J,axiom,
! [Xs: list_P7986770385144383213od_v_v,B: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ ( set_Product_prod_v_v2 @ Xs ) @ B )
= ( ! [X3: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X3 @ ( set_Product_prod_v_v2 @ Xs ) )
=> ( member7453568604450474000od_v_v @ X3 @ B ) ) ) ) ).
% subset_code(1)
thf(fact_992_Un__mono,axiom,
! [A3: set_set_v,C3: set_set_v,B: set_set_v,D2: set_set_v] :
( ( ord_le5216385588623774835_set_v @ A3 @ C3 )
=> ( ( ord_le5216385588623774835_set_v @ B @ D2 )
=> ( ord_le5216385588623774835_set_v @ ( sup_sup_set_set_v @ A3 @ B ) @ ( sup_sup_set_set_v @ C3 @ D2 ) ) ) ) ).
% Un_mono
thf(fact_993_Un__mono,axiom,
! [A3: set_v,C3: set_v,B: set_v,D2: set_v] :
( ( ord_less_eq_set_v @ A3 @ C3 )
=> ( ( ord_less_eq_set_v @ B @ D2 )
=> ( ord_less_eq_set_v @ ( sup_sup_set_v @ A3 @ B ) @ ( sup_sup_set_v @ C3 @ D2 ) ) ) ) ).
% Un_mono
thf(fact_994_Un__mono,axiom,
! [A3: set_Product_prod_v_v,C3: set_Product_prod_v_v,B: set_Product_prod_v_v,D2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A3 @ C3 )
=> ( ( ord_le7336532860387713383od_v_v @ B @ D2 )
=> ( ord_le7336532860387713383od_v_v @ ( sup_su414716646722978715od_v_v @ A3 @ B ) @ ( sup_su414716646722978715od_v_v @ C3 @ D2 ) ) ) ) ).
% Un_mono
thf(fact_995_Un__least,axiom,
! [A3: set_set_v,C3: set_set_v,B: set_set_v] :
( ( ord_le5216385588623774835_set_v @ A3 @ C3 )
=> ( ( ord_le5216385588623774835_set_v @ B @ C3 )
=> ( ord_le5216385588623774835_set_v @ ( sup_sup_set_set_v @ A3 @ B ) @ C3 ) ) ) ).
% Un_least
thf(fact_996_Un__least,axiom,
! [A3: set_v,C3: set_v,B: set_v] :
( ( ord_less_eq_set_v @ A3 @ C3 )
=> ( ( ord_less_eq_set_v @ B @ C3 )
=> ( ord_less_eq_set_v @ ( sup_sup_set_v @ A3 @ B ) @ C3 ) ) ) ).
% Un_least
thf(fact_997_Un__least,axiom,
! [A3: set_Product_prod_v_v,C3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A3 @ C3 )
=> ( ( ord_le7336532860387713383od_v_v @ B @ C3 )
=> ( ord_le7336532860387713383od_v_v @ ( sup_su414716646722978715od_v_v @ A3 @ B ) @ C3 ) ) ) ).
% Un_least
thf(fact_998_Un__upper1,axiom,
! [A3: set_set_v,B: set_set_v] : ( ord_le5216385588623774835_set_v @ A3 @ ( sup_sup_set_set_v @ A3 @ B ) ) ).
% Un_upper1
thf(fact_999_Un__upper1,axiom,
! [A3: set_v,B: set_v] : ( ord_less_eq_set_v @ A3 @ ( sup_sup_set_v @ A3 @ B ) ) ).
% Un_upper1
thf(fact_1000_Un__upper1,axiom,
! [A3: set_Product_prod_v_v,B: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ A3 @ ( sup_su414716646722978715od_v_v @ A3 @ B ) ) ).
% Un_upper1
thf(fact_1001_Un__upper2,axiom,
! [B: set_set_v,A3: set_set_v] : ( ord_le5216385588623774835_set_v @ B @ ( sup_sup_set_set_v @ A3 @ B ) ) ).
% Un_upper2
thf(fact_1002_Un__upper2,axiom,
! [B: set_v,A3: set_v] : ( ord_less_eq_set_v @ B @ ( sup_sup_set_v @ A3 @ B ) ) ).
% Un_upper2
thf(fact_1003_Un__upper2,axiom,
! [B: set_Product_prod_v_v,A3: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ B @ ( sup_su414716646722978715od_v_v @ A3 @ B ) ) ).
% Un_upper2
thf(fact_1004_Un__absorb1,axiom,
! [A3: set_set_v,B: set_set_v] :
( ( ord_le5216385588623774835_set_v @ A3 @ B )
=> ( ( sup_sup_set_set_v @ A3 @ B )
= B ) ) ).
% Un_absorb1
thf(fact_1005_Un__absorb1,axiom,
! [A3: set_v,B: set_v] :
( ( ord_less_eq_set_v @ A3 @ B )
=> ( ( sup_sup_set_v @ A3 @ B )
= B ) ) ).
% Un_absorb1
thf(fact_1006_Un__absorb1,axiom,
! [A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A3 @ B )
=> ( ( sup_su414716646722978715od_v_v @ A3 @ B )
= B ) ) ).
% Un_absorb1
thf(fact_1007_Un__absorb2,axiom,
! [B: set_set_v,A3: set_set_v] :
( ( ord_le5216385588623774835_set_v @ B @ A3 )
=> ( ( sup_sup_set_set_v @ A3 @ B )
= A3 ) ) ).
% Un_absorb2
thf(fact_1008_Un__absorb2,axiom,
! [B: set_v,A3: set_v] :
( ( ord_less_eq_set_v @ B @ A3 )
=> ( ( sup_sup_set_v @ A3 @ B )
= A3 ) ) ).
% Un_absorb2
thf(fact_1009_Un__absorb2,axiom,
! [B: set_Product_prod_v_v,A3: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ B @ A3 )
=> ( ( sup_su414716646722978715od_v_v @ A3 @ B )
= A3 ) ) ).
% Un_absorb2
thf(fact_1010_subset__UnE,axiom,
! [C3: set_set_v,A3: set_set_v,B: set_set_v] :
( ( ord_le5216385588623774835_set_v @ C3 @ ( sup_sup_set_set_v @ A3 @ B ) )
=> ~ ! [A6: set_set_v] :
( ( ord_le5216385588623774835_set_v @ A6 @ A3 )
=> ! [B6: set_set_v] :
( ( ord_le5216385588623774835_set_v @ B6 @ B )
=> ( C3
!= ( sup_sup_set_set_v @ A6 @ B6 ) ) ) ) ) ).
% subset_UnE
thf(fact_1011_subset__UnE,axiom,
! [C3: set_v,A3: set_v,B: set_v] :
( ( ord_less_eq_set_v @ C3 @ ( sup_sup_set_v @ A3 @ B ) )
=> ~ ! [A6: set_v] :
( ( ord_less_eq_set_v @ A6 @ A3 )
=> ! [B6: set_v] :
( ( ord_less_eq_set_v @ B6 @ B )
=> ( C3
!= ( sup_sup_set_v @ A6 @ B6 ) ) ) ) ) ).
% subset_UnE
thf(fact_1012_subset__UnE,axiom,
! [C3: set_Product_prod_v_v,A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ C3 @ ( sup_su414716646722978715od_v_v @ A3 @ B ) )
=> ~ ! [A6: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A6 @ A3 )
=> ! [B6: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ B6 @ B )
=> ( C3
!= ( sup_su414716646722978715od_v_v @ A6 @ B6 ) ) ) ) ) ).
% subset_UnE
thf(fact_1013_subset__Un__eq,axiom,
( ord_le5216385588623774835_set_v
= ( ^ [A5: set_set_v,B5: set_set_v] :
( ( sup_sup_set_set_v @ A5 @ B5 )
= B5 ) ) ) ).
% subset_Un_eq
thf(fact_1014_subset__Un__eq,axiom,
( ord_less_eq_set_v
= ( ^ [A5: set_v,B5: set_v] :
( ( sup_sup_set_v @ A5 @ B5 )
= B5 ) ) ) ).
% subset_Un_eq
thf(fact_1015_subset__Un__eq,axiom,
( ord_le7336532860387713383od_v_v
= ( ^ [A5: set_Product_prod_v_v,B5: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ A5 @ B5 )
= B5 ) ) ) ).
% subset_Un_eq
thf(fact_1016_double__diff,axiom,
! [A3: set_v,B: set_v,C3: set_v] :
( ( ord_less_eq_set_v @ A3 @ B )
=> ( ( ord_less_eq_set_v @ B @ C3 )
=> ( ( minus_minus_set_v @ B @ ( minus_minus_set_v @ C3 @ A3 ) )
= A3 ) ) ) ).
% double_diff
thf(fact_1017_double__diff,axiom,
! [A3: set_Product_prod_v_v,B: set_Product_prod_v_v,C3: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A3 @ B )
=> ( ( ord_le7336532860387713383od_v_v @ B @ C3 )
=> ( ( minus_4183494784930505774od_v_v @ B @ ( minus_4183494784930505774od_v_v @ C3 @ A3 ) )
= A3 ) ) ) ).
% double_diff
thf(fact_1018_Diff__subset,axiom,
! [A3: set_v,B: set_v] : ( ord_less_eq_set_v @ ( minus_minus_set_v @ A3 @ B ) @ A3 ) ).
% Diff_subset
thf(fact_1019_Diff__subset,axiom,
! [A3: set_Product_prod_v_v,B: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ ( minus_4183494784930505774od_v_v @ A3 @ B ) @ A3 ) ).
% Diff_subset
thf(fact_1020_Diff__mono,axiom,
! [A3: set_v,C3: set_v,D2: set_v,B: set_v] :
( ( ord_less_eq_set_v @ A3 @ C3 )
=> ( ( ord_less_eq_set_v @ D2 @ B )
=> ( ord_less_eq_set_v @ ( minus_minus_set_v @ A3 @ B ) @ ( minus_minus_set_v @ C3 @ D2 ) ) ) ) ).
% Diff_mono
thf(fact_1021_Diff__mono,axiom,
! [A3: set_Product_prod_v_v,C3: set_Product_prod_v_v,D2: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A3 @ C3 )
=> ( ( ord_le7336532860387713383od_v_v @ D2 @ B )
=> ( ord_le7336532860387713383od_v_v @ ( minus_4183494784930505774od_v_v @ A3 @ B ) @ ( minus_4183494784930505774od_v_v @ C3 @ D2 ) ) ) ) ).
% Diff_mono
thf(fact_1022_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_1023_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_1024_select__convs_I7_J,axiom,
! [Root: v,S4: v > set_v,Explored: set_v,Visited: set_v,Vsuccs: v > set_v,Sccs: set_set_v,Stack: list_v,Cstack: list_v,More: product_unit] :
( ( sCC_Bl8828226123343373779t_unit @ ( sCC_Bl8064756265740546429t_unit @ Root @ S4 @ Explored @ Visited @ Vsuccs @ Sccs @ Stack @ Cstack @ More ) )
= Stack ) ).
% select_convs(7)
thf(fact_1025_select__convs_I2_J,axiom,
! [Root: v,S4: v > set_v,Explored: set_v,Visited: set_v,Vsuccs: v > set_v,Sccs: set_set_v,Stack: list_v,Cstack: list_v,More: product_unit] :
( ( sCC_Bl1280885523602775798t_unit @ ( sCC_Bl8064756265740546429t_unit @ Root @ S4 @ Explored @ Visited @ Vsuccs @ Sccs @ Stack @ Cstack @ More ) )
= S4 ) ).
% select_convs(2)
thf(fact_1026_select__convs_I5_J,axiom,
! [Root: v,S4: v > set_v,Explored: set_v,Visited: set_v,Vsuccs: v > set_v,Sccs: set_set_v,Stack: list_v,Cstack: list_v,More: product_unit] :
( ( sCC_Bl3795065053823578884t_unit @ ( sCC_Bl8064756265740546429t_unit @ Root @ S4 @ Explored @ Visited @ Vsuccs @ Sccs @ Stack @ Cstack @ More ) )
= Vsuccs ) ).
% select_convs(5)
thf(fact_1027_select__convs_I4_J,axiom,
! [Root: v,S4: v > set_v,Explored: set_v,Visited: set_v,Vsuccs: v > set_v,Sccs: set_set_v,Stack: list_v,Cstack: list_v,More: product_unit] :
( ( sCC_Bl4645233313691564917t_unit @ ( sCC_Bl8064756265740546429t_unit @ Root @ S4 @ Explored @ Visited @ Vsuccs @ Sccs @ Stack @ Cstack @ More ) )
= Visited ) ).
% select_convs(4)
thf(fact_1028_select__convs_I3_J,axiom,
! [Root: v,S4: v > set_v,Explored: set_v,Visited: set_v,Vsuccs: v > set_v,Sccs: set_set_v,Stack: list_v,Cstack: list_v,More: product_unit] :
( ( sCC_Bl157864678168468314t_unit @ ( sCC_Bl8064756265740546429t_unit @ Root @ S4 @ Explored @ Visited @ Vsuccs @ Sccs @ Stack @ Cstack @ More ) )
= Explored ) ).
% select_convs(3)
thf(fact_1029_select__convs_I8_J,axiom,
! [Root: v,S4: v > set_v,Explored: set_v,Visited: set_v,Vsuccs: v > set_v,Sccs: set_set_v,Stack: list_v,Cstack: list_v,More: product_unit] :
( ( sCC_Bl9201514103433284750t_unit @ ( sCC_Bl8064756265740546429t_unit @ Root @ S4 @ Explored @ Visited @ Vsuccs @ Sccs @ Stack @ Cstack @ More ) )
= Cstack ) ).
% select_convs(8)
thf(fact_1030_select__convs_I6_J,axiom,
! [Root: v,S4: v > set_v,Explored: set_v,Visited: set_v,Vsuccs: v > set_v,Sccs: set_set_v,Stack: list_v,Cstack: list_v,More: product_unit] :
( ( sCC_Bl2536197123907397897t_unit @ ( sCC_Bl8064756265740546429t_unit @ Root @ S4 @ Explored @ Visited @ Vsuccs @ Sccs @ Stack @ Cstack @ More ) )
= Sccs ) ).
% select_convs(6)
thf(fact_1031_Int__Diff__disjoint,axiom,
! [A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ ( inf_in6271465464967711157od_v_v @ A3 @ B ) @ ( minus_4183494784930505774od_v_v @ A3 @ B ) )
= bot_bo723834152578015283od_v_v ) ).
% Int_Diff_disjoint
thf(fact_1032_Int__Diff__disjoint,axiom,
! [A3: set_set_v,B: set_set_v] :
( ( inf_inf_set_set_v @ ( inf_inf_set_set_v @ A3 @ B ) @ ( minus_7228012346218142266_set_v @ A3 @ B ) )
= bot_bot_set_set_v ) ).
% Int_Diff_disjoint
thf(fact_1033_Int__Diff__disjoint,axiom,
! [A3: set_v,B: set_v] :
( ( inf_inf_set_v @ ( inf_inf_set_v @ A3 @ B ) @ ( minus_minus_set_v @ A3 @ B ) )
= bot_bot_set_v ) ).
% Int_Diff_disjoint
thf(fact_1034_Diff__triv,axiom,
! [A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ( inf_in6271465464967711157od_v_v @ A3 @ B )
= bot_bo723834152578015283od_v_v )
=> ( ( minus_4183494784930505774od_v_v @ A3 @ B )
= A3 ) ) ).
% Diff_triv
thf(fact_1035_Diff__triv,axiom,
! [A3: set_set_v,B: set_set_v] :
( ( ( inf_inf_set_set_v @ A3 @ B )
= bot_bot_set_set_v )
=> ( ( minus_7228012346218142266_set_v @ A3 @ B )
= A3 ) ) ).
% Diff_triv
thf(fact_1036_Diff__triv,axiom,
! [A3: set_v,B: set_v] :
( ( ( inf_inf_set_v @ A3 @ B )
= bot_bot_set_v )
=> ( ( minus_minus_set_v @ A3 @ B )
= A3 ) ) ).
% Diff_triv
thf(fact_1037_Un__Diff__Int,axiom,
! [A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ ( minus_4183494784930505774od_v_v @ A3 @ B ) @ ( inf_in6271465464967711157od_v_v @ A3 @ B ) )
= A3 ) ).
% Un_Diff_Int
thf(fact_1038_Un__Diff__Int,axiom,
! [A3: set_set_v,B: set_set_v] :
( ( sup_sup_set_set_v @ ( minus_7228012346218142266_set_v @ A3 @ B ) @ ( inf_inf_set_set_v @ A3 @ B ) )
= A3 ) ).
% Un_Diff_Int
thf(fact_1039_Un__Diff__Int,axiom,
! [A3: set_v,B: set_v] :
( ( sup_sup_set_v @ ( minus_minus_set_v @ A3 @ B ) @ ( inf_inf_set_v @ A3 @ B ) )
= A3 ) ).
% Un_Diff_Int
thf(fact_1040_Int__Diff__Un,axiom,
! [A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ A3 @ B ) @ ( minus_4183494784930505774od_v_v @ A3 @ B ) )
= A3 ) ).
% Int_Diff_Un
thf(fact_1041_Int__Diff__Un,axiom,
! [A3: set_set_v,B: set_set_v] :
( ( sup_sup_set_set_v @ ( inf_inf_set_set_v @ A3 @ B ) @ ( minus_7228012346218142266_set_v @ A3 @ B ) )
= A3 ) ).
% Int_Diff_Un
thf(fact_1042_Int__Diff__Un,axiom,
! [A3: set_v,B: set_v] :
( ( sup_sup_set_v @ ( inf_inf_set_v @ A3 @ B ) @ ( minus_minus_set_v @ A3 @ B ) )
= A3 ) ).
% Int_Diff_Un
thf(fact_1043_Diff__Int,axiom,
! [A3: set_Product_prod_v_v,B: set_Product_prod_v_v,C3: set_Product_prod_v_v] :
( ( minus_4183494784930505774od_v_v @ A3 @ ( inf_in6271465464967711157od_v_v @ B @ C3 ) )
= ( sup_su414716646722978715od_v_v @ ( minus_4183494784930505774od_v_v @ A3 @ B ) @ ( minus_4183494784930505774od_v_v @ A3 @ C3 ) ) ) ).
% Diff_Int
thf(fact_1044_Diff__Int,axiom,
! [A3: set_set_v,B: set_set_v,C3: set_set_v] :
( ( minus_7228012346218142266_set_v @ A3 @ ( inf_inf_set_set_v @ B @ C3 ) )
= ( sup_sup_set_set_v @ ( minus_7228012346218142266_set_v @ A3 @ B ) @ ( minus_7228012346218142266_set_v @ A3 @ C3 ) ) ) ).
% Diff_Int
thf(fact_1045_Diff__Int,axiom,
! [A3: set_v,B: set_v,C3: set_v] :
( ( minus_minus_set_v @ A3 @ ( inf_inf_set_v @ B @ C3 ) )
= ( sup_sup_set_v @ ( minus_minus_set_v @ A3 @ B ) @ ( minus_minus_set_v @ A3 @ C3 ) ) ) ).
% Diff_Int
thf(fact_1046_Diff__Un,axiom,
! [A3: set_Product_prod_v_v,B: set_Product_prod_v_v,C3: set_Product_prod_v_v] :
( ( minus_4183494784930505774od_v_v @ A3 @ ( sup_su414716646722978715od_v_v @ B @ C3 ) )
= ( inf_in6271465464967711157od_v_v @ ( minus_4183494784930505774od_v_v @ A3 @ B ) @ ( minus_4183494784930505774od_v_v @ A3 @ C3 ) ) ) ).
% Diff_Un
thf(fact_1047_Diff__Un,axiom,
! [A3: set_set_v,B: set_set_v,C3: set_set_v] :
( ( minus_7228012346218142266_set_v @ A3 @ ( sup_sup_set_set_v @ B @ C3 ) )
= ( inf_inf_set_set_v @ ( minus_7228012346218142266_set_v @ A3 @ B ) @ ( minus_7228012346218142266_set_v @ A3 @ C3 ) ) ) ).
% Diff_Un
thf(fact_1048_Diff__Un,axiom,
! [A3: set_v,B: set_v,C3: set_v] :
( ( minus_minus_set_v @ A3 @ ( sup_sup_set_v @ B @ C3 ) )
= ( inf_inf_set_v @ ( minus_minus_set_v @ A3 @ B ) @ ( minus_minus_set_v @ A3 @ C3 ) ) ) ).
% Diff_Un
thf(fact_1049_subset__singleton__iff,axiom,
! [X5: set_set_v,A: set_v] :
( ( ord_le5216385588623774835_set_v @ X5 @ ( insert_set_v @ A @ bot_bot_set_set_v ) )
= ( ( X5 = bot_bot_set_set_v )
| ( X5
= ( insert_set_v @ A @ bot_bot_set_set_v ) ) ) ) ).
% subset_singleton_iff
thf(fact_1050_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_1051_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_1052_subset__singletonD,axiom,
! [A3: set_set_v,X2: set_v] :
( ( ord_le5216385588623774835_set_v @ A3 @ ( insert_set_v @ X2 @ bot_bot_set_set_v ) )
=> ( ( A3 = bot_bot_set_set_v )
| ( A3
= ( insert_set_v @ X2 @ bot_bot_set_set_v ) ) ) ) ).
% subset_singletonD
thf(fact_1053_subset__singletonD,axiom,
! [A3: set_v,X2: v] :
( ( ord_less_eq_set_v @ A3 @ ( insert_v @ X2 @ bot_bot_set_v ) )
=> ( ( A3 = bot_bot_set_v )
| ( A3
= ( insert_v @ X2 @ bot_bot_set_v ) ) ) ) ).
% subset_singletonD
thf(fact_1054_subset__singletonD,axiom,
! [A3: set_Product_prod_v_v,X2: product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A3 @ ( insert1338601472111419319od_v_v @ X2 @ bot_bo723834152578015283od_v_v ) )
=> ( ( A3 = bot_bo723834152578015283od_v_v )
| ( A3
= ( insert1338601472111419319od_v_v @ X2 @ bot_bo723834152578015283od_v_v ) ) ) ) ).
% subset_singletonD
thf(fact_1055_set__subset__Cons,axiom,
! [Xs: list_v,X2: v] : ( ord_less_eq_set_v @ ( set_v2 @ Xs ) @ ( set_v2 @ ( cons_v @ X2 @ Xs ) ) ) ).
% set_subset_Cons
thf(fact_1056_set__subset__Cons,axiom,
! [Xs: list_P7986770385144383213od_v_v,X2: product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ ( set_Product_prod_v_v2 @ Xs ) @ ( set_Product_prod_v_v2 @ ( cons_P4120604216776828829od_v_v @ X2 @ Xs ) ) ) ).
% set_subset_Cons
thf(fact_1057_subset__Diff__insert,axiom,
! [A3: set_set_v,B: set_set_v,X2: set_v,C3: set_set_v] :
( ( ord_le5216385588623774835_set_v @ A3 @ ( minus_7228012346218142266_set_v @ B @ ( insert_set_v @ X2 @ C3 ) ) )
= ( ( ord_le5216385588623774835_set_v @ A3 @ ( minus_7228012346218142266_set_v @ B @ C3 ) )
& ~ ( member_set_v @ X2 @ A3 ) ) ) ).
% subset_Diff_insert
thf(fact_1058_subset__Diff__insert,axiom,
! [A3: set_v,B: set_v,X2: v,C3: set_v] :
( ( ord_less_eq_set_v @ A3 @ ( minus_minus_set_v @ B @ ( insert_v @ X2 @ C3 ) ) )
= ( ( ord_less_eq_set_v @ A3 @ ( minus_minus_set_v @ B @ C3 ) )
& ~ ( member_v @ X2 @ A3 ) ) ) ).
% subset_Diff_insert
thf(fact_1059_subset__Diff__insert,axiom,
! [A3: set_Product_prod_v_v,B: set_Product_prod_v_v,X2: product_prod_v_v,C3: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A3 @ ( minus_4183494784930505774od_v_v @ B @ ( insert1338601472111419319od_v_v @ X2 @ C3 ) ) )
= ( ( ord_le7336532860387713383od_v_v @ A3 @ ( minus_4183494784930505774od_v_v @ B @ C3 ) )
& ~ ( member7453568604450474000od_v_v @ X2 @ A3 ) ) ) ).
% subset_Diff_insert
thf(fact_1060_Diff__subset__conv,axiom,
! [A3: set_set_v,B: set_set_v,C3: set_set_v] :
( ( ord_le5216385588623774835_set_v @ ( minus_7228012346218142266_set_v @ A3 @ B ) @ C3 )
= ( ord_le5216385588623774835_set_v @ A3 @ ( sup_sup_set_set_v @ B @ C3 ) ) ) ).
% Diff_subset_conv
thf(fact_1061_Diff__subset__conv,axiom,
! [A3: set_v,B: set_v,C3: set_v] :
( ( ord_less_eq_set_v @ ( minus_minus_set_v @ A3 @ B ) @ C3 )
= ( ord_less_eq_set_v @ A3 @ ( sup_sup_set_v @ B @ C3 ) ) ) ).
% Diff_subset_conv
thf(fact_1062_Diff__subset__conv,axiom,
! [A3: set_Product_prod_v_v,B: set_Product_prod_v_v,C3: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ ( minus_4183494784930505774od_v_v @ A3 @ B ) @ C3 )
= ( ord_le7336532860387713383od_v_v @ A3 @ ( sup_su414716646722978715od_v_v @ B @ C3 ) ) ) ).
% Diff_subset_conv
thf(fact_1063_Diff__partition,axiom,
! [A3: set_set_v,B: set_set_v] :
( ( ord_le5216385588623774835_set_v @ A3 @ B )
=> ( ( sup_sup_set_set_v @ A3 @ ( minus_7228012346218142266_set_v @ B @ A3 ) )
= B ) ) ).
% Diff_partition
thf(fact_1064_Diff__partition,axiom,
! [A3: set_v,B: set_v] :
( ( ord_less_eq_set_v @ A3 @ B )
=> ( ( sup_sup_set_v @ A3 @ ( minus_minus_set_v @ B @ A3 ) )
= B ) ) ).
% Diff_partition
thf(fact_1065_Diff__partition,axiom,
! [A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A3 @ B )
=> ( ( sup_su414716646722978715od_v_v @ A3 @ ( minus_4183494784930505774od_v_v @ B @ A3 ) )
= B ) ) ).
% Diff_partition
thf(fact_1066_graph_Ora__succ,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,X2: product_prod_v_v,Y2: product_prod_v_v,E4: set_Pr2149350503807050951od_v_v,Z: product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( sCC_Bl5370300055464682748od_v_v @ Successors @ X2 @ Y2 @ E4 )
=> ( ( member7453568604450474000od_v_v @ Z @ ( Successors @ Y2 ) )
=> ( ~ ( member3038538357316246288od_v_v @ ( produc4031800376763917143od_v_v @ Y2 @ Z ) @ E4 )
=> ( sCC_Bl5370300055464682748od_v_v @ Successors @ X2 @ Z @ E4 ) ) ) ) ) ).
% graph.ra_succ
thf(fact_1067_graph_Ora__succ,axiom,
! [Vertices: set_v,Successors: v > set_v,X2: v,Y2: v,E4: set_Product_prod_v_v,Z: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl4291963740693775144ding_v @ Successors @ X2 @ Y2 @ E4 )
=> ( ( member_v @ Z @ ( Successors @ Y2 ) )
=> ( ~ ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ Y2 @ Z ) @ E4 )
=> ( sCC_Bl4291963740693775144ding_v @ Successors @ X2 @ Z @ E4 ) ) ) ) ) ).
% graph.ra_succ
thf(fact_1068_graph_Oreachable__avoiding_Osimps,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,A1: product_prod_v_v,A2: product_prod_v_v,A32: set_Pr2149350503807050951od_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( sCC_Bl5370300055464682748od_v_v @ Successors @ A1 @ A2 @ A32 )
= ( ? [X3: product_prod_v_v,E6: set_Pr2149350503807050951od_v_v] :
( ( A1 = X3 )
& ( A2 = X3 )
& ( A32 = E6 ) )
| ? [X3: product_prod_v_v,Y3: product_prod_v_v,E6: set_Pr2149350503807050951od_v_v,Z2: product_prod_v_v] :
( ( A1 = X3 )
& ( A2 = Z2 )
& ( A32 = E6 )
& ( sCC_Bl5370300055464682748od_v_v @ Successors @ X3 @ Y3 @ E6 )
& ( member7453568604450474000od_v_v @ Z2 @ ( Successors @ Y3 ) )
& ~ ( member3038538357316246288od_v_v @ ( produc4031800376763917143od_v_v @ Y3 @ Z2 ) @ E6 ) ) ) ) ) ).
% graph.reachable_avoiding.simps
thf(fact_1069_graph_Oreachable__avoiding_Osimps,axiom,
! [Vertices: set_v,Successors: v > set_v,A1: v,A2: v,A32: set_Product_prod_v_v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl4291963740693775144ding_v @ Successors @ A1 @ A2 @ A32 )
= ( ? [X3: v,E6: set_Product_prod_v_v] :
( ( A1 = X3 )
& ( A2 = X3 )
& ( A32 = E6 ) )
| ? [X3: v,Y3: v,E6: set_Product_prod_v_v,Z2: v] :
( ( A1 = X3 )
& ( A2 = Z2 )
& ( A32 = E6 )
& ( sCC_Bl4291963740693775144ding_v @ Successors @ X3 @ Y3 @ E6 )
& ( member_v @ Z2 @ ( Successors @ Y3 ) )
& ~ ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ Y3 @ Z2 ) @ E6 ) ) ) ) ) ).
% graph.reachable_avoiding.simps
thf(fact_1070_graph_Oreachable__avoiding_Ocases,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,A1: product_prod_v_v,A2: product_prod_v_v,A32: set_Pr2149350503807050951od_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( sCC_Bl5370300055464682748od_v_v @ Successors @ A1 @ A2 @ A32 )
=> ( ( A2 != A1 )
=> ~ ! [Y: product_prod_v_v] :
( ( sCC_Bl5370300055464682748od_v_v @ Successors @ A1 @ Y @ A32 )
=> ( ( member7453568604450474000od_v_v @ A2 @ ( Successors @ Y ) )
=> ( member3038538357316246288od_v_v @ ( produc4031800376763917143od_v_v @ Y @ A2 ) @ A32 ) ) ) ) ) ) ).
% graph.reachable_avoiding.cases
thf(fact_1071_graph_Oreachable__avoiding_Ocases,axiom,
! [Vertices: set_v,Successors: v > set_v,A1: v,A2: v,A32: set_Product_prod_v_v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl4291963740693775144ding_v @ Successors @ A1 @ A2 @ A32 )
=> ( ( A2 != A1 )
=> ~ ! [Y: v] :
( ( sCC_Bl4291963740693775144ding_v @ Successors @ A1 @ Y @ A32 )
=> ( ( member_v @ A2 @ ( Successors @ Y ) )
=> ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ Y @ A2 ) @ A32 ) ) ) ) ) ) ).
% graph.reachable_avoiding.cases
thf(fact_1072_graph_Ora__cases,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,X2: product_prod_v_v,Y2: product_prod_v_v,E4: set_Pr2149350503807050951od_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( sCC_Bl5370300055464682748od_v_v @ Successors @ X2 @ Y2 @ E4 )
=> ( ( X2 = Y2 )
| ? [Z3: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ Z3 @ ( Successors @ X2 ) )
& ~ ( member3038538357316246288od_v_v @ ( produc4031800376763917143od_v_v @ X2 @ Z3 ) @ E4 )
& ( sCC_Bl5370300055464682748od_v_v @ Successors @ Z3 @ Y2 @ E4 ) ) ) ) ) ).
% graph.ra_cases
thf(fact_1073_graph_Ora__cases,axiom,
! [Vertices: set_v,Successors: v > set_v,X2: v,Y2: v,E4: set_Product_prod_v_v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl4291963740693775144ding_v @ Successors @ X2 @ Y2 @ E4 )
=> ( ( X2 = Y2 )
| ? [Z3: v] :
( ( member_v @ Z3 @ ( Successors @ X2 ) )
& ~ ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ X2 @ Z3 ) @ E4 )
& ( sCC_Bl4291963740693775144ding_v @ Successors @ Z3 @ Y2 @ E4 ) ) ) ) ) ).
% graph.ra_cases
thf(fact_1074_graph_Oedge__ra,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,Y2: product_prod_v_v,X2: product_prod_v_v,E4: set_Pr2149350503807050951od_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( member7453568604450474000od_v_v @ Y2 @ ( Successors @ X2 ) )
=> ( ~ ( member3038538357316246288od_v_v @ ( produc4031800376763917143od_v_v @ X2 @ Y2 ) @ E4 )
=> ( sCC_Bl5370300055464682748od_v_v @ Successors @ X2 @ Y2 @ E4 ) ) ) ) ).
% graph.edge_ra
thf(fact_1075_graph_Oedge__ra,axiom,
! [Vertices: set_v,Successors: v > set_v,Y2: v,X2: v,E4: set_Product_prod_v_v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( member_v @ Y2 @ ( Successors @ X2 ) )
=> ( ~ ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ X2 @ Y2 ) @ E4 )
=> ( sCC_Bl4291963740693775144ding_v @ Successors @ X2 @ Y2 @ E4 ) ) ) ) ).
% graph.edge_ra
thf(fact_1076_graph_Ora__mono,axiom,
! [Vertices: set_v,Successors: v > set_v,X2: v,Y2: v,E4: set_Product_prod_v_v,E5: set_Product_prod_v_v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl4291963740693775144ding_v @ Successors @ X2 @ Y2 @ E4 )
=> ( ( ord_le7336532860387713383od_v_v @ E5 @ E4 )
=> ( sCC_Bl4291963740693775144ding_v @ Successors @ X2 @ Y2 @ E5 ) ) ) ) ).
% graph.ra_mono
thf(fact_1077_graph_Oscc__partition,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,S: set_Product_prod_v_v,S3: set_Product_prod_v_v,X2: product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( sCC_Bl6242042402218619277od_v_v @ Successors @ S )
=> ( ( sCC_Bl6242042402218619277od_v_v @ Successors @ S3 )
=> ( ( member7453568604450474000od_v_v @ X2 @ ( inf_in6271465464967711157od_v_v @ S @ S3 ) )
=> ( S = S3 ) ) ) ) ) ).
% graph.scc_partition
thf(fact_1078_graph_Oscc__partition,axiom,
! [Vertices: set_v,Successors: v > set_v,S: set_v,S3: set_v,X2: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bloemen_is_scc_v @ Successors @ S )
=> ( ( sCC_Bloemen_is_scc_v @ Successors @ S3 )
=> ( ( member_v @ X2 @ ( inf_inf_set_v @ S @ S3 ) )
=> ( S = S3 ) ) ) ) ) ).
% graph.scc_partition
thf(fact_1079_Diff__single__insert,axiom,
! [A3: set_set_v,X2: set_v,B: set_set_v] :
( ( ord_le5216385588623774835_set_v @ ( minus_7228012346218142266_set_v @ A3 @ ( insert_set_v @ X2 @ bot_bot_set_set_v ) ) @ B )
=> ( ord_le5216385588623774835_set_v @ A3 @ ( insert_set_v @ X2 @ B ) ) ) ).
% Diff_single_insert
thf(fact_1080_Diff__single__insert,axiom,
! [A3: set_v,X2: v,B: set_v] :
( ( ord_less_eq_set_v @ ( minus_minus_set_v @ A3 @ ( insert_v @ X2 @ bot_bot_set_v ) ) @ B )
=> ( ord_less_eq_set_v @ A3 @ ( insert_v @ X2 @ B ) ) ) ).
% Diff_single_insert
thf(fact_1081_Diff__single__insert,axiom,
! [A3: set_Product_prod_v_v,X2: product_prod_v_v,B: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ ( minus_4183494784930505774od_v_v @ A3 @ ( insert1338601472111419319od_v_v @ X2 @ bot_bo723834152578015283od_v_v ) ) @ B )
=> ( ord_le7336532860387713383od_v_v @ A3 @ ( insert1338601472111419319od_v_v @ X2 @ B ) ) ) ).
% Diff_single_insert
thf(fact_1082_subset__insert__iff,axiom,
! [A3: set_set_v,X2: set_v,B: set_set_v] :
( ( ord_le5216385588623774835_set_v @ A3 @ ( insert_set_v @ X2 @ B ) )
= ( ( ( member_set_v @ X2 @ A3 )
=> ( ord_le5216385588623774835_set_v @ ( minus_7228012346218142266_set_v @ A3 @ ( insert_set_v @ X2 @ bot_bot_set_set_v ) ) @ B ) )
& ( ~ ( member_set_v @ X2 @ A3 )
=> ( ord_le5216385588623774835_set_v @ A3 @ B ) ) ) ) ).
% subset_insert_iff
thf(fact_1083_subset__insert__iff,axiom,
! [A3: set_v,X2: v,B: set_v] :
( ( ord_less_eq_set_v @ A3 @ ( insert_v @ X2 @ B ) )
= ( ( ( member_v @ X2 @ A3 )
=> ( ord_less_eq_set_v @ ( minus_minus_set_v @ A3 @ ( insert_v @ X2 @ bot_bot_set_v ) ) @ B ) )
& ( ~ ( member_v @ X2 @ A3 )
=> ( ord_less_eq_set_v @ A3 @ B ) ) ) ) ).
% subset_insert_iff
thf(fact_1084_subset__insert__iff,axiom,
! [A3: set_Product_prod_v_v,X2: product_prod_v_v,B: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A3 @ ( insert1338601472111419319od_v_v @ X2 @ B ) )
= ( ( ( member7453568604450474000od_v_v @ X2 @ A3 )
=> ( ord_le7336532860387713383od_v_v @ ( minus_4183494784930505774od_v_v @ A3 @ ( insert1338601472111419319od_v_v @ X2 @ bot_bo723834152578015283od_v_v ) ) @ B ) )
& ( ~ ( member7453568604450474000od_v_v @ X2 @ A3 )
=> ( ord_le7336532860387713383od_v_v @ A3 @ B ) ) ) ) ).
% subset_insert_iff
thf(fact_1085_graph_Ora__add__edge,axiom,
! [Vertices: set_v,Successors: v > set_v,X2: v,Y2: v,E4: set_Product_prod_v_v,V: v,W: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl4291963740693775144ding_v @ Successors @ X2 @ Y2 @ E4 )
=> ( ( sCC_Bl4291963740693775144ding_v @ Successors @ X2 @ Y2 @ ( sup_su414716646722978715od_v_v @ E4 @ ( insert1338601472111419319od_v_v @ ( product_Pair_v_v @ V @ W ) @ bot_bo723834152578015283od_v_v ) ) )
| ( ( sCC_Bl4291963740693775144ding_v @ Successors @ X2 @ V @ ( sup_su414716646722978715od_v_v @ E4 @ ( insert1338601472111419319od_v_v @ ( product_Pair_v_v @ V @ W ) @ bot_bo723834152578015283od_v_v ) ) )
& ( sCC_Bl4291963740693775144ding_v @ Successors @ W @ Y2 @ ( sup_su414716646722978715od_v_v @ E4 @ ( insert1338601472111419319od_v_v @ ( product_Pair_v_v @ V @ W ) @ bot_bo723834152578015283od_v_v ) ) ) ) ) ) ) ).
% graph.ra_add_edge
thf(fact_1086_graph_Ois__scc__def,axiom,
! [Vertices: set_set_v,Successors: set_v > set_set_v,S: set_set_v] :
( ( sCC_Bl5810666556806954322_set_v @ Vertices @ Successors )
=> ( ( sCC_Bl1515522642333523865_set_v @ Successors @ S )
= ( ( S != bot_bot_set_set_v )
& ( sCC_Bl7907073126578335045_set_v @ Successors @ S )
& ! [S2: set_set_v] :
( ( ( ord_le5216385588623774835_set_v @ S @ S2 )
& ( sCC_Bl7907073126578335045_set_v @ Successors @ S2 ) )
=> ( S2 = S ) ) ) ) ) ).
% graph.is_scc_def
thf(fact_1087_graph_Ois__scc__def,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,S: set_Product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( sCC_Bl6242042402218619277od_v_v @ Successors @ S )
= ( ( S != bot_bo723834152578015283od_v_v )
& ( sCC_Bl2301996248249672505od_v_v @ Successors @ S )
& ! [S2: set_Product_prod_v_v] :
( ( ( ord_le7336532860387713383od_v_v @ S @ S2 )
& ( sCC_Bl2301996248249672505od_v_v @ Successors @ S2 ) )
=> ( S2 = S ) ) ) ) ) ).
% graph.is_scc_def
thf(fact_1088_graph_Ois__scc__def,axiom,
! [Vertices: set_v,Successors: v > set_v,S: set_v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bloemen_is_scc_v @ Successors @ S )
= ( ( S != bot_bot_set_v )
& ( sCC_Bl5398416737448265317bscc_v @ Successors @ S )
& ! [S2: set_v] :
( ( ( ord_less_eq_set_v @ S @ S2 )
& ( sCC_Bl5398416737448265317bscc_v @ Successors @ S2 ) )
=> ( S2 = S ) ) ) ) ) ).
% graph.is_scc_def
thf(fact_1089_graph_Oavoiding__explored,axiom,
! [Vertices: set_v,Successors: v > set_v,E: sCC_Bl1394983891496994913t_unit,X2: v,Y2: v,E4: set_Product_prod_v_v,W: v,V: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl9196236973127232072t_unit @ Successors @ E )
=> ( ( sCC_Bl4291963740693775144ding_v @ Successors @ X2 @ Y2 @ E4 )
=> ( ~ ( member_v @ Y2 @ ( sCC_Bl157864678168468314t_unit @ E ) )
=> ( ( member_v @ W @ ( sCC_Bl157864678168468314t_unit @ E ) )
=> ( sCC_Bl4291963740693775144ding_v @ Successors @ X2 @ Y2 @ ( sup_su414716646722978715od_v_v @ E4 @ ( insert1338601472111419319od_v_v @ ( product_Pair_v_v @ V @ W ) @ bot_bo723834152578015283od_v_v ) ) ) ) ) ) ) ) ).
% graph.avoiding_explored
thf(fact_1090_boolean__algebra_Oconj__zero__left,axiom,
! [X2: set_v] :
( ( inf_inf_set_v @ bot_bot_set_v @ X2 )
= bot_bot_set_v ) ).
% boolean_algebra.conj_zero_left
thf(fact_1091_boolean__algebra_Oconj__zero__left,axiom,
! [X2: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ bot_bo723834152578015283od_v_v @ X2 )
= bot_bo723834152578015283od_v_v ) ).
% boolean_algebra.conj_zero_left
thf(fact_1092_boolean__algebra_Oconj__zero__left,axiom,
! [X2: set_set_v] :
( ( inf_inf_set_set_v @ bot_bot_set_set_v @ X2 )
= bot_bot_set_set_v ) ).
% boolean_algebra.conj_zero_left
thf(fact_1093_boolean__algebra_Oconj__zero__right,axiom,
! [X2: set_v] :
( ( inf_inf_set_v @ X2 @ bot_bot_set_v )
= bot_bot_set_v ) ).
% boolean_algebra.conj_zero_right
thf(fact_1094_boolean__algebra_Oconj__zero__right,axiom,
! [X2: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ X2 @ bot_bo723834152578015283od_v_v )
= bot_bo723834152578015283od_v_v ) ).
% boolean_algebra.conj_zero_right
thf(fact_1095_boolean__algebra_Oconj__zero__right,axiom,
! [X2: set_set_v] :
( ( inf_inf_set_set_v @ X2 @ bot_bot_set_set_v )
= bot_bot_set_set_v ) ).
% boolean_algebra.conj_zero_right
thf(fact_1096_diff__shunt__var,axiom,
! [X2: set_set_v,Y2: set_set_v] :
( ( ( minus_7228012346218142266_set_v @ X2 @ Y2 )
= bot_bot_set_set_v )
= ( ord_le5216385588623774835_set_v @ X2 @ Y2 ) ) ).
% diff_shunt_var
thf(fact_1097_diff__shunt__var,axiom,
! [X2: set_v,Y2: set_v] :
( ( ( minus_minus_set_v @ X2 @ Y2 )
= bot_bot_set_v )
= ( ord_less_eq_set_v @ X2 @ Y2 ) ) ).
% diff_shunt_var
thf(fact_1098_diff__shunt__var,axiom,
! [X2: set_Product_prod_v_v,Y2: set_Product_prod_v_v] :
( ( ( minus_4183494784930505774od_v_v @ X2 @ Y2 )
= bot_bo723834152578015283od_v_v )
= ( ord_le7336532860387713383od_v_v @ X2 @ Y2 ) ) ).
% diff_shunt_var
thf(fact_1099_bot__empty__eq,axiom,
( bot_bot_v_o
= ( ^ [X3: v] : ( member_v @ X3 @ bot_bot_set_v ) ) ) ).
% bot_empty_eq
thf(fact_1100_bot__empty__eq,axiom,
( bot_bo8461541820394803818_v_v_o
= ( ^ [X3: product_prod_v_v] : ( member7453568604450474000od_v_v @ X3 @ bot_bo723834152578015283od_v_v ) ) ) ).
% bot_empty_eq
thf(fact_1101_bot__empty__eq,axiom,
( bot_bot_set_v_o
= ( ^ [X3: set_v] : ( member_set_v @ X3 @ bot_bot_set_set_v ) ) ) ).
% bot_empty_eq
thf(fact_1102_Collect__empty__eq__bot,axiom,
! [P: v > $o] :
( ( ( collect_v @ P )
= bot_bot_set_v )
= ( P = bot_bot_v_o ) ) ).
% Collect_empty_eq_bot
thf(fact_1103_Collect__empty__eq__bot,axiom,
! [P: product_prod_v_v > $o] :
( ( ( collec140062887454715474od_v_v @ P )
= bot_bo723834152578015283od_v_v )
= ( P = bot_bo8461541820394803818_v_v_o ) ) ).
% Collect_empty_eq_bot
thf(fact_1104_Collect__empty__eq__bot,axiom,
! [P: set_v > $o] :
( ( ( collect_set_v @ P )
= bot_bot_set_set_v )
= ( P = bot_bot_set_v_o ) ) ).
% Collect_empty_eq_bot
thf(fact_1105_less__by__empty,axiom,
! [A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( A3 = bot_bo723834152578015283od_v_v )
=> ( ord_le7336532860387713383od_v_v @ A3 @ B ) ) ).
% less_by_empty
thf(fact_1106_graph_Odfss_Ocases,axiom,
! [Vertices: set_v,Successors: v > set_v,X2: produc5741669702376414499t_unit] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ~ ! [V2: v,E7: sCC_Bl1394983891496994913t_unit] :
( X2
!= ( produc3862955338007567901t_unit @ V2 @ E7 ) ) ) ).
% graph.dfss.cases
thf(fact_1107_sorted__wrt_Ocases,axiom,
! [X2: produc8237170675765753490list_v] :
( ! [P2: v > v > $o] :
( X2
!= ( produc601102195597853570list_v @ P2 @ nil_v ) )
=> ~ ! [P2: v > v > $o,X4: v,Ys4: list_v] :
( X2
!= ( produc601102195597853570list_v @ P2 @ ( cons_v @ X4 @ Ys4 ) ) ) ) ).
% sorted_wrt.cases
thf(fact_1108_successively_Ocases,axiom,
! [X2: produc8237170675765753490list_v] :
( ! [P2: v > v > $o] :
( X2
!= ( produc601102195597853570list_v @ P2 @ nil_v ) )
=> ( ! [P2: v > v > $o,X4: v] :
( X2
!= ( produc601102195597853570list_v @ P2 @ ( cons_v @ X4 @ nil_v ) ) )
=> ~ ! [P2: v > v > $o,X4: v,Y: v,Xs3: list_v] :
( X2
!= ( produc601102195597853570list_v @ P2 @ ( cons_v @ X4 @ ( cons_v @ Y @ Xs3 ) ) ) ) ) ) ).
% successively.cases
thf(fact_1109_splice_Ocases,axiom,
! [X2: produc1391462591744249447list_v] :
( ! [Ys4: list_v] :
( X2
!= ( produc6795410681906604247list_v @ nil_v @ Ys4 ) )
=> ~ ! [X4: v,Xs3: list_v,Ys4: list_v] :
( X2
!= ( produc6795410681906604247list_v @ ( cons_v @ X4 @ Xs3 ) @ Ys4 ) ) ) ).
% splice.cases
thf(fact_1110_shuffles_Ocases,axiom,
! [X2: produc1391462591744249447list_v] :
( ! [Ys4: list_v] :
( X2
!= ( produc6795410681906604247list_v @ nil_v @ Ys4 ) )
=> ( ! [Xs3: list_v] :
( X2
!= ( produc6795410681906604247list_v @ Xs3 @ nil_v ) )
=> ~ ! [X4: v,Xs3: list_v,Y: v,Ys4: list_v] :
( X2
!= ( produc6795410681906604247list_v @ ( cons_v @ X4 @ Xs3 ) @ ( cons_v @ Y @ Ys4 ) ) ) ) ) ).
% shuffles.cases
thf(fact_1111_boolean__algebra__cancel_Osup1,axiom,
! [A3: set_Product_prod_v_v,K: set_Product_prod_v_v,A: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( A3
= ( sup_su414716646722978715od_v_v @ K @ A ) )
=> ( ( sup_su414716646722978715od_v_v @ A3 @ B2 )
= ( sup_su414716646722978715od_v_v @ K @ ( sup_su414716646722978715od_v_v @ A @ B2 ) ) ) ) ).
% boolean_algebra_cancel.sup1
thf(fact_1112_boolean__algebra__cancel_Osup1,axiom,
! [A3: set_v,K: set_v,A: set_v,B2: set_v] :
( ( A3
= ( sup_sup_set_v @ K @ A ) )
=> ( ( sup_sup_set_v @ A3 @ B2 )
= ( sup_sup_set_v @ K @ ( sup_sup_set_v @ A @ B2 ) ) ) ) ).
% boolean_algebra_cancel.sup1
thf(fact_1113_boolean__algebra__cancel_Osup1,axiom,
! [A3: set_set_v,K: set_set_v,A: set_set_v,B2: set_set_v] :
( ( A3
= ( sup_sup_set_set_v @ K @ A ) )
=> ( ( sup_sup_set_set_v @ A3 @ B2 )
= ( sup_sup_set_set_v @ K @ ( sup_sup_set_set_v @ A @ B2 ) ) ) ) ).
% boolean_algebra_cancel.sup1
thf(fact_1114_boolean__algebra__cancel_Osup2,axiom,
! [B: set_Product_prod_v_v,K: set_Product_prod_v_v,B2: set_Product_prod_v_v,A: set_Product_prod_v_v] :
( ( B
= ( sup_su414716646722978715od_v_v @ K @ B2 ) )
=> ( ( sup_su414716646722978715od_v_v @ A @ B )
= ( sup_su414716646722978715od_v_v @ K @ ( sup_su414716646722978715od_v_v @ A @ B2 ) ) ) ) ).
% boolean_algebra_cancel.sup2
thf(fact_1115_boolean__algebra__cancel_Osup2,axiom,
! [B: set_v,K: set_v,B2: set_v,A: set_v] :
( ( B
= ( sup_sup_set_v @ K @ B2 ) )
=> ( ( sup_sup_set_v @ A @ B )
= ( sup_sup_set_v @ K @ ( sup_sup_set_v @ A @ B2 ) ) ) ) ).
% boolean_algebra_cancel.sup2
thf(fact_1116_boolean__algebra__cancel_Osup2,axiom,
! [B: set_set_v,K: set_set_v,B2: set_set_v,A: set_set_v] :
( ( B
= ( sup_sup_set_set_v @ K @ B2 ) )
=> ( ( sup_sup_set_set_v @ A @ B )
= ( sup_sup_set_set_v @ K @ ( sup_sup_set_set_v @ A @ B2 ) ) ) ) ).
% boolean_algebra_cancel.sup2
thf(fact_1117_boolean__algebra_Odisj__zero__right,axiom,
! [X2: set_v] :
( ( sup_sup_set_v @ X2 @ bot_bot_set_v )
= X2 ) ).
% boolean_algebra.disj_zero_right
thf(fact_1118_boolean__algebra_Odisj__zero__right,axiom,
! [X2: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ X2 @ bot_bo723834152578015283od_v_v )
= X2 ) ).
% boolean_algebra.disj_zero_right
thf(fact_1119_boolean__algebra_Odisj__zero__right,axiom,
! [X2: set_set_v] :
( ( sup_sup_set_set_v @ X2 @ bot_bot_set_set_v )
= X2 ) ).
% boolean_algebra.disj_zero_right
thf(fact_1120_boolean__algebra_Oconj__disj__distrib,axiom,
! [X2: set_Product_prod_v_v,Y2: set_Product_prod_v_v,Z: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ X2 @ ( sup_su414716646722978715od_v_v @ Y2 @ Z ) )
= ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ X2 @ Y2 ) @ ( inf_in6271465464967711157od_v_v @ X2 @ Z ) ) ) ).
% boolean_algebra.conj_disj_distrib
thf(fact_1121_boolean__algebra_Oconj__disj__distrib,axiom,
! [X2: set_v,Y2: set_v,Z: set_v] :
( ( inf_inf_set_v @ X2 @ ( sup_sup_set_v @ Y2 @ Z ) )
= ( sup_sup_set_v @ ( inf_inf_set_v @ X2 @ Y2 ) @ ( inf_inf_set_v @ X2 @ Z ) ) ) ).
% boolean_algebra.conj_disj_distrib
thf(fact_1122_boolean__algebra_Oconj__disj__distrib,axiom,
! [X2: set_set_v,Y2: set_set_v,Z: set_set_v] :
( ( inf_inf_set_set_v @ X2 @ ( sup_sup_set_set_v @ Y2 @ Z ) )
= ( sup_sup_set_set_v @ ( inf_inf_set_set_v @ X2 @ Y2 ) @ ( inf_inf_set_set_v @ X2 @ Z ) ) ) ).
% boolean_algebra.conj_disj_distrib
thf(fact_1123_boolean__algebra_Odisj__conj__distrib,axiom,
! [X2: set_Product_prod_v_v,Y2: set_Product_prod_v_v,Z: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ X2 @ ( inf_in6271465464967711157od_v_v @ Y2 @ Z ) )
= ( inf_in6271465464967711157od_v_v @ ( sup_su414716646722978715od_v_v @ X2 @ Y2 ) @ ( sup_su414716646722978715od_v_v @ X2 @ Z ) ) ) ).
% boolean_algebra.disj_conj_distrib
thf(fact_1124_boolean__algebra_Odisj__conj__distrib,axiom,
! [X2: set_v,Y2: set_v,Z: set_v] :
( ( sup_sup_set_v @ X2 @ ( inf_inf_set_v @ Y2 @ Z ) )
= ( inf_inf_set_v @ ( sup_sup_set_v @ X2 @ Y2 ) @ ( sup_sup_set_v @ X2 @ Z ) ) ) ).
% boolean_algebra.disj_conj_distrib
thf(fact_1125_boolean__algebra_Odisj__conj__distrib,axiom,
! [X2: set_set_v,Y2: set_set_v,Z: set_set_v] :
( ( sup_sup_set_set_v @ X2 @ ( inf_inf_set_set_v @ Y2 @ Z ) )
= ( inf_inf_set_set_v @ ( sup_sup_set_set_v @ X2 @ Y2 ) @ ( sup_sup_set_set_v @ X2 @ Z ) ) ) ).
% boolean_algebra.disj_conj_distrib
thf(fact_1126_boolean__algebra_Oconj__disj__distrib2,axiom,
! [Y2: set_Product_prod_v_v,Z: set_Product_prod_v_v,X2: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ ( sup_su414716646722978715od_v_v @ Y2 @ Z ) @ X2 )
= ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ Y2 @ X2 ) @ ( inf_in6271465464967711157od_v_v @ Z @ X2 ) ) ) ).
% boolean_algebra.conj_disj_distrib2
thf(fact_1127_boolean__algebra_Oconj__disj__distrib2,axiom,
! [Y2: set_v,Z: set_v,X2: set_v] :
( ( inf_inf_set_v @ ( sup_sup_set_v @ Y2 @ Z ) @ X2 )
= ( sup_sup_set_v @ ( inf_inf_set_v @ Y2 @ X2 ) @ ( inf_inf_set_v @ Z @ X2 ) ) ) ).
% boolean_algebra.conj_disj_distrib2
thf(fact_1128_boolean__algebra_Oconj__disj__distrib2,axiom,
! [Y2: set_set_v,Z: set_set_v,X2: set_set_v] :
( ( inf_inf_set_set_v @ ( sup_sup_set_set_v @ Y2 @ Z ) @ X2 )
= ( sup_sup_set_set_v @ ( inf_inf_set_set_v @ Y2 @ X2 ) @ ( inf_inf_set_set_v @ Z @ X2 ) ) ) ).
% boolean_algebra.conj_disj_distrib2
thf(fact_1129_boolean__algebra_Odisj__conj__distrib2,axiom,
! [Y2: set_Product_prod_v_v,Z: set_Product_prod_v_v,X2: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ Y2 @ Z ) @ X2 )
= ( inf_in6271465464967711157od_v_v @ ( sup_su414716646722978715od_v_v @ Y2 @ X2 ) @ ( sup_su414716646722978715od_v_v @ Z @ X2 ) ) ) ).
% boolean_algebra.disj_conj_distrib2
thf(fact_1130_boolean__algebra_Odisj__conj__distrib2,axiom,
! [Y2: set_v,Z: set_v,X2: set_v] :
( ( sup_sup_set_v @ ( inf_inf_set_v @ Y2 @ Z ) @ X2 )
= ( inf_inf_set_v @ ( sup_sup_set_v @ Y2 @ X2 ) @ ( sup_sup_set_v @ Z @ X2 ) ) ) ).
% boolean_algebra.disj_conj_distrib2
thf(fact_1131_boolean__algebra_Odisj__conj__distrib2,axiom,
! [Y2: set_set_v,Z: set_set_v,X2: set_set_v] :
( ( sup_sup_set_set_v @ ( inf_inf_set_set_v @ Y2 @ Z ) @ X2 )
= ( inf_inf_set_set_v @ ( sup_sup_set_set_v @ Y2 @ X2 ) @ ( sup_sup_set_set_v @ Z @ X2 ) ) ) ).
% boolean_algebra.disj_conj_distrib2
thf(fact_1132_vfin,axiom,
finite_finite_v @ vertices ).
% vfin
thf(fact_1133_rotate1__hd__tl,axiom,
! [Xs: list_v] :
( ( Xs != nil_v )
=> ( ( rotate1_v @ Xs )
= ( append_v @ ( tl_v @ Xs ) @ ( cons_v @ ( hd_v @ Xs ) @ nil_v ) ) ) ) ).
% rotate1_hd_tl
thf(fact_1134_set__removeAll,axiom,
! [X2: product_prod_v_v,Xs: list_P7986770385144383213od_v_v] :
( ( set_Product_prod_v_v2 @ ( remove481895986417801203od_v_v @ X2 @ Xs ) )
= ( minus_4183494784930505774od_v_v @ ( set_Product_prod_v_v2 @ Xs ) @ ( insert1338601472111419319od_v_v @ X2 @ bot_bo723834152578015283od_v_v ) ) ) ).
% set_removeAll
thf(fact_1135_set__removeAll,axiom,
! [X2: set_v,Xs: list_set_v] :
( ( set_set_v2 @ ( removeAll_set_v @ X2 @ Xs ) )
= ( minus_7228012346218142266_set_v @ ( set_set_v2 @ Xs ) @ ( insert_set_v @ X2 @ bot_bot_set_set_v ) ) ) ).
% set_removeAll
thf(fact_1136_set__removeAll,axiom,
! [X2: v,Xs: list_v] :
( ( set_v2 @ ( removeAll_v @ X2 @ Xs ) )
= ( minus_minus_set_v @ ( set_v2 @ Xs ) @ ( insert_v @ X2 @ bot_bot_set_v ) ) ) ).
% set_removeAll
thf(fact_1137_List_Ofinite__set,axiom,
! [Xs: list_v] : ( finite_finite_v @ ( set_v2 @ Xs ) ) ).
% List.finite_set
thf(fact_1138_removeAll__id,axiom,
! [X2: product_prod_v_v,Xs: list_P7986770385144383213od_v_v] :
( ~ ( member7453568604450474000od_v_v @ X2 @ ( set_Product_prod_v_v2 @ Xs ) )
=> ( ( remove481895986417801203od_v_v @ X2 @ Xs )
= Xs ) ) ).
% removeAll_id
thf(fact_1139_removeAll__id,axiom,
! [X2: v,Xs: list_v] :
( ~ ( member_v @ X2 @ ( set_v2 @ Xs ) )
=> ( ( removeAll_v @ X2 @ Xs )
= Xs ) ) ).
% removeAll_id
thf(fact_1140_rotate1__is__Nil__conv,axiom,
! [Xs: list_v] :
( ( ( rotate1_v @ Xs )
= nil_v )
= ( Xs = nil_v ) ) ).
% rotate1_is_Nil_conv
thf(fact_1141_set__rotate1,axiom,
! [Xs: list_v] :
( ( set_v2 @ ( rotate1_v @ Xs ) )
= ( set_v2 @ Xs ) ) ).
% set_rotate1
thf(fact_1142_removeAll__append,axiom,
! [X2: v,Xs: list_v,Ys: list_v] :
( ( removeAll_v @ X2 @ ( append_v @ Xs @ Ys ) )
= ( append_v @ ( removeAll_v @ X2 @ Xs ) @ ( removeAll_v @ X2 @ Ys ) ) ) ).
% removeAll_append
thf(fact_1143_finite__list,axiom,
! [A3: set_v] :
( ( finite_finite_v @ A3 )
=> ? [Xs3: list_v] :
( ( set_v2 @ Xs3 )
= A3 ) ) ).
% finite_list
thf(fact_1144_graph_Ovfin,axiom,
! [Vertices: set_v,Successors: v > set_v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( finite_finite_v @ Vertices ) ) ).
% graph.vfin
thf(fact_1145_removeAll_Osimps_I2_J,axiom,
! [X2: v,Y2: v,Xs: list_v] :
( ( ( X2 = Y2 )
=> ( ( removeAll_v @ X2 @ ( cons_v @ Y2 @ Xs ) )
= ( removeAll_v @ X2 @ Xs ) ) )
& ( ( X2 != Y2 )
=> ( ( removeAll_v @ X2 @ ( cons_v @ Y2 @ Xs ) )
= ( cons_v @ Y2 @ ( removeAll_v @ X2 @ Xs ) ) ) ) ) ).
% removeAll.simps(2)
thf(fact_1146_removeAll_Osimps_I1_J,axiom,
! [X2: v] :
( ( removeAll_v @ X2 @ nil_v )
= nil_v ) ).
% removeAll.simps(1)
thf(fact_1147_rotate1_Osimps_I1_J,axiom,
( ( rotate1_v @ nil_v )
= nil_v ) ).
% rotate1.simps(1)
thf(fact_1148_SCC__Bloemen__Sequential_Ograph__def,axiom,
( sCC_Bl8307124943676871238od_v_v
= ( ^ [Vertices2: set_Product_prod_v_v,Successors2: product_prod_v_v > set_Product_prod_v_v] :
( ( finite3348123685078250256od_v_v @ Vertices2 )
& ! [X3: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X3 @ Vertices2 )
=> ( ord_le7336532860387713383od_v_v @ ( Successors2 @ X3 ) @ Vertices2 ) ) ) ) ) ).
% SCC_Bloemen_Sequential.graph_def
thf(fact_1149_SCC__Bloemen__Sequential_Ograph__def,axiom,
( sCC_Bloemen_graph_v
= ( ^ [Vertices2: set_v,Successors2: v > set_v] :
( ( finite_finite_v @ Vertices2 )
& ! [X3: v] :
( ( member_v @ X3 @ Vertices2 )
=> ( ord_less_eq_set_v @ ( Successors2 @ X3 ) @ Vertices2 ) ) ) ) ) ).
% SCC_Bloemen_Sequential.graph_def
thf(fact_1150_graph_Ointro,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v] :
( ( finite3348123685078250256od_v_v @ Vertices )
=> ( ! [X4: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X4 @ Vertices )
=> ( ord_le7336532860387713383od_v_v @ ( Successors @ X4 ) @ Vertices ) )
=> ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors ) ) ) ).
% graph.intro
thf(fact_1151_graph_Ointro,axiom,
! [Vertices: set_v,Successors: v > set_v] :
( ( finite_finite_v @ Vertices )
=> ( ! [X4: v] :
( ( member_v @ X4 @ Vertices )
=> ( ord_less_eq_set_v @ ( Successors @ X4 ) @ Vertices ) )
=> ( sCC_Bloemen_graph_v @ Vertices @ Successors ) ) ) ).
% graph.intro
thf(fact_1152_rotate1_Osimps_I2_J,axiom,
! [X2: v,Xs: list_v] :
( ( rotate1_v @ ( cons_v @ X2 @ Xs ) )
= ( append_v @ Xs @ ( cons_v @ X2 @ nil_v ) ) ) ).
% rotate1.simps(2)
thf(fact_1153_finite__Diff__insert,axiom,
! [A3: set_Product_prod_v_v,A: product_prod_v_v,B: set_Product_prod_v_v] :
( ( finite3348123685078250256od_v_v @ ( minus_4183494784930505774od_v_v @ A3 @ ( insert1338601472111419319od_v_v @ A @ B ) ) )
= ( finite3348123685078250256od_v_v @ ( minus_4183494784930505774od_v_v @ A3 @ B ) ) ) ).
% finite_Diff_insert
thf(fact_1154_finite__Diff__insert,axiom,
! [A3: set_set_v,A: set_v,B: set_set_v] :
( ( finite_finite_set_v @ ( minus_7228012346218142266_set_v @ A3 @ ( insert_set_v @ A @ B ) ) )
= ( finite_finite_set_v @ ( minus_7228012346218142266_set_v @ A3 @ B ) ) ) ).
% finite_Diff_insert
thf(fact_1155_finite__Diff__insert,axiom,
! [A3: set_v,A: v,B: set_v] :
( ( finite_finite_v @ ( minus_minus_set_v @ A3 @ ( insert_v @ A @ B ) ) )
= ( finite_finite_v @ ( minus_minus_set_v @ A3 @ B ) ) ) ).
% finite_Diff_insert
thf(fact_1156_finite__Diff,axiom,
! [A3: set_v,B: set_v] :
( ( finite_finite_v @ A3 )
=> ( finite_finite_v @ ( minus_minus_set_v @ A3 @ B ) ) ) ).
% finite_Diff
thf(fact_1157_finite__Diff2,axiom,
! [B: set_v,A3: set_v] :
( ( finite_finite_v @ B )
=> ( ( finite_finite_v @ ( minus_minus_set_v @ A3 @ B ) )
= ( finite_finite_v @ A3 ) ) ) ).
% finite_Diff2
thf(fact_1158_finite__insert,axiom,
! [A: product_prod_v_v,A3: set_Product_prod_v_v] :
( ( finite3348123685078250256od_v_v @ ( insert1338601472111419319od_v_v @ A @ A3 ) )
= ( finite3348123685078250256od_v_v @ A3 ) ) ).
% finite_insert
thf(fact_1159_finite__insert,axiom,
! [A: set_v,A3: set_set_v] :
( ( finite_finite_set_v @ ( insert_set_v @ A @ A3 ) )
= ( finite_finite_set_v @ A3 ) ) ).
% finite_insert
thf(fact_1160_finite__insert,axiom,
! [A: v,A3: set_v] :
( ( finite_finite_v @ ( insert_v @ A @ A3 ) )
= ( finite_finite_v @ A3 ) ) ).
% finite_insert
thf(fact_1161_finite__Un,axiom,
! [F2: set_Product_prod_v_v,G: set_Product_prod_v_v] :
( ( finite3348123685078250256od_v_v @ ( sup_su414716646722978715od_v_v @ F2 @ G ) )
= ( ( finite3348123685078250256od_v_v @ F2 )
& ( finite3348123685078250256od_v_v @ G ) ) ) ).
% finite_Un
thf(fact_1162_finite__Un,axiom,
! [F2: set_v,G: set_v] :
( ( finite_finite_v @ ( sup_sup_set_v @ F2 @ G ) )
= ( ( finite_finite_v @ F2 )
& ( finite_finite_v @ G ) ) ) ).
% finite_Un
thf(fact_1163_finite__Un,axiom,
! [F2: set_set_v,G: set_set_v] :
( ( finite_finite_set_v @ ( sup_sup_set_set_v @ F2 @ G ) )
= ( ( finite_finite_set_v @ F2 )
& ( finite_finite_set_v @ G ) ) ) ).
% finite_Un
thf(fact_1164_infinite__imp__nonempty,axiom,
! [S: set_v] :
( ~ ( finite_finite_v @ S )
=> ( S != bot_bot_set_v ) ) ).
% infinite_imp_nonempty
thf(fact_1165_infinite__imp__nonempty,axiom,
! [S: set_Product_prod_v_v] :
( ~ ( finite3348123685078250256od_v_v @ S )
=> ( S != bot_bo723834152578015283od_v_v ) ) ).
% infinite_imp_nonempty
thf(fact_1166_infinite__imp__nonempty,axiom,
! [S: set_set_v] :
( ~ ( finite_finite_set_v @ S )
=> ( S != bot_bot_set_set_v ) ) ).
% infinite_imp_nonempty
thf(fact_1167_finite_OemptyI,axiom,
finite_finite_v @ bot_bot_set_v ).
% finite.emptyI
thf(fact_1168_finite_OemptyI,axiom,
finite3348123685078250256od_v_v @ bot_bo723834152578015283od_v_v ).
% finite.emptyI
thf(fact_1169_finite_OemptyI,axiom,
finite_finite_set_v @ bot_bot_set_set_v ).
% finite.emptyI
thf(fact_1170_finite_OinsertI,axiom,
! [A3: set_Product_prod_v_v,A: product_prod_v_v] :
( ( finite3348123685078250256od_v_v @ A3 )
=> ( finite3348123685078250256od_v_v @ ( insert1338601472111419319od_v_v @ A @ A3 ) ) ) ).
% finite.insertI
thf(fact_1171_finite_OinsertI,axiom,
! [A3: set_set_v,A: set_v] :
( ( finite_finite_set_v @ A3 )
=> ( finite_finite_set_v @ ( insert_set_v @ A @ A3 ) ) ) ).
% finite.insertI
thf(fact_1172_finite_OinsertI,axiom,
! [A3: set_v,A: v] :
( ( finite_finite_v @ A3 )
=> ( finite_finite_v @ ( insert_v @ A @ A3 ) ) ) ).
% finite.insertI
thf(fact_1173_infinite__Un,axiom,
! [S: set_Product_prod_v_v,T: set_Product_prod_v_v] :
( ( ~ ( finite3348123685078250256od_v_v @ ( sup_su414716646722978715od_v_v @ S @ T ) ) )
= ( ~ ( finite3348123685078250256od_v_v @ S )
| ~ ( finite3348123685078250256od_v_v @ T ) ) ) ).
% infinite_Un
thf(fact_1174_infinite__Un,axiom,
! [S: set_v,T: set_v] :
( ( ~ ( finite_finite_v @ ( sup_sup_set_v @ S @ T ) ) )
= ( ~ ( finite_finite_v @ S )
| ~ ( finite_finite_v @ T ) ) ) ).
% infinite_Un
thf(fact_1175_infinite__Un,axiom,
! [S: set_set_v,T: set_set_v] :
( ( ~ ( finite_finite_set_v @ ( sup_sup_set_set_v @ S @ T ) ) )
= ( ~ ( finite_finite_set_v @ S )
| ~ ( finite_finite_set_v @ T ) ) ) ).
% infinite_Un
thf(fact_1176_Un__infinite,axiom,
! [S: set_Product_prod_v_v,T: set_Product_prod_v_v] :
( ~ ( finite3348123685078250256od_v_v @ S )
=> ~ ( finite3348123685078250256od_v_v @ ( sup_su414716646722978715od_v_v @ S @ T ) ) ) ).
% Un_infinite
thf(fact_1177_Un__infinite,axiom,
! [S: set_v,T: set_v] :
( ~ ( finite_finite_v @ S )
=> ~ ( finite_finite_v @ ( sup_sup_set_v @ S @ T ) ) ) ).
% Un_infinite
thf(fact_1178_Un__infinite,axiom,
! [S: set_set_v,T: set_set_v] :
( ~ ( finite_finite_set_v @ S )
=> ~ ( finite_finite_set_v @ ( sup_sup_set_set_v @ S @ T ) ) ) ).
% Un_infinite
thf(fact_1179_finite__UnI,axiom,
! [F2: set_Product_prod_v_v,G: set_Product_prod_v_v] :
( ( finite3348123685078250256od_v_v @ F2 )
=> ( ( finite3348123685078250256od_v_v @ G )
=> ( finite3348123685078250256od_v_v @ ( sup_su414716646722978715od_v_v @ F2 @ G ) ) ) ) ).
% finite_UnI
thf(fact_1180_finite__UnI,axiom,
! [F2: set_v,G: set_v] :
( ( finite_finite_v @ F2 )
=> ( ( finite_finite_v @ G )
=> ( finite_finite_v @ ( sup_sup_set_v @ F2 @ G ) ) ) ) ).
% finite_UnI
thf(fact_1181_finite__UnI,axiom,
! [F2: set_set_v,G: set_set_v] :
( ( finite_finite_set_v @ F2 )
=> ( ( finite_finite_set_v @ G )
=> ( finite_finite_set_v @ ( sup_sup_set_set_v @ F2 @ G ) ) ) ) ).
% finite_UnI
thf(fact_1182_Diff__infinite__finite,axiom,
! [T: set_v,S: set_v] :
( ( finite_finite_v @ T )
=> ( ~ ( finite_finite_v @ S )
=> ~ ( finite_finite_v @ ( minus_minus_set_v @ S @ T ) ) ) ) ).
% Diff_infinite_finite
thf(fact_1183_finite__has__maximal,axiom,
! [A3: set_set_v] :
( ( finite_finite_set_v @ A3 )
=> ( ( A3 != bot_bot_set_set_v )
=> ? [X4: set_v] :
( ( member_set_v @ X4 @ A3 )
& ! [Xa2: set_v] :
( ( member_set_v @ Xa2 @ A3 )
=> ( ( ord_less_eq_set_v @ X4 @ Xa2 )
=> ( X4 = Xa2 ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_1184_finite__has__maximal,axiom,
! [A3: set_se8455005133513928103od_v_v] :
( ( finite6084192165098772208od_v_v @ A3 )
=> ( ( A3 != bot_bo3497076220358800403od_v_v )
=> ? [X4: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ X4 @ A3 )
& ! [Xa2: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ Xa2 @ A3 )
=> ( ( ord_le7336532860387713383od_v_v @ X4 @ Xa2 )
=> ( X4 = Xa2 ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_1185_finite__has__minimal,axiom,
! [A3: set_set_v] :
( ( finite_finite_set_v @ A3 )
=> ( ( A3 != bot_bot_set_set_v )
=> ? [X4: set_v] :
( ( member_set_v @ X4 @ A3 )
& ! [Xa2: set_v] :
( ( member_set_v @ Xa2 @ A3 )
=> ( ( ord_less_eq_set_v @ Xa2 @ X4 )
=> ( X4 = Xa2 ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_1186_finite__has__minimal,axiom,
! [A3: set_se8455005133513928103od_v_v] :
( ( finite6084192165098772208od_v_v @ A3 )
=> ( ( A3 != bot_bo3497076220358800403od_v_v )
=> ? [X4: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ X4 @ A3 )
& ! [Xa2: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ Xa2 @ A3 )
=> ( ( ord_le7336532860387713383od_v_v @ Xa2 @ X4 )
=> ( X4 = Xa2 ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_1187_infinite__finite__induct,axiom,
! [P: set_v > $o,A3: set_v] :
( ! [A7: set_v] :
( ~ ( finite_finite_v @ A7 )
=> ( P @ A7 ) )
=> ( ( P @ bot_bot_set_v )
=> ( ! [X4: v,F3: set_v] :
( ( finite_finite_v @ F3 )
=> ( ~ ( member_v @ X4 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_v @ X4 @ F3 ) ) ) ) )
=> ( P @ A3 ) ) ) ) ).
% infinite_finite_induct
thf(fact_1188_infinite__finite__induct,axiom,
! [P: set_Product_prod_v_v > $o,A3: set_Product_prod_v_v] :
( ! [A7: set_Product_prod_v_v] :
( ~ ( finite3348123685078250256od_v_v @ A7 )
=> ( P @ A7 ) )
=> ( ( P @ bot_bo723834152578015283od_v_v )
=> ( ! [X4: product_prod_v_v,F3: set_Product_prod_v_v] :
( ( finite3348123685078250256od_v_v @ F3 )
=> ( ~ ( member7453568604450474000od_v_v @ X4 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert1338601472111419319od_v_v @ X4 @ F3 ) ) ) ) )
=> ( P @ A3 ) ) ) ) ).
% infinite_finite_induct
thf(fact_1189_infinite__finite__induct,axiom,
! [P: set_set_v > $o,A3: set_set_v] :
( ! [A7: set_set_v] :
( ~ ( finite_finite_set_v @ A7 )
=> ( P @ A7 ) )
=> ( ( P @ bot_bot_set_set_v )
=> ( ! [X4: set_v,F3: set_set_v] :
( ( finite_finite_set_v @ F3 )
=> ( ~ ( member_set_v @ X4 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_set_v @ X4 @ F3 ) ) ) ) )
=> ( P @ A3 ) ) ) ) ).
% infinite_finite_induct
thf(fact_1190_finite__ne__induct,axiom,
! [F2: set_v,P: set_v > $o] :
( ( finite_finite_v @ F2 )
=> ( ( F2 != bot_bot_set_v )
=> ( ! [X4: v] : ( P @ ( insert_v @ X4 @ bot_bot_set_v ) )
=> ( ! [X4: v,F3: set_v] :
( ( finite_finite_v @ F3 )
=> ( ( F3 != bot_bot_set_v )
=> ( ~ ( member_v @ X4 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_v @ X4 @ F3 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_1191_finite__ne__induct,axiom,
! [F2: set_Product_prod_v_v,P: set_Product_prod_v_v > $o] :
( ( finite3348123685078250256od_v_v @ F2 )
=> ( ( F2 != bot_bo723834152578015283od_v_v )
=> ( ! [X4: product_prod_v_v] : ( P @ ( insert1338601472111419319od_v_v @ X4 @ bot_bo723834152578015283od_v_v ) )
=> ( ! [X4: product_prod_v_v,F3: set_Product_prod_v_v] :
( ( finite3348123685078250256od_v_v @ F3 )
=> ( ( F3 != bot_bo723834152578015283od_v_v )
=> ( ~ ( member7453568604450474000od_v_v @ X4 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert1338601472111419319od_v_v @ X4 @ F3 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_1192_finite__ne__induct,axiom,
! [F2: set_set_v,P: set_set_v > $o] :
( ( finite_finite_set_v @ F2 )
=> ( ( F2 != bot_bot_set_set_v )
=> ( ! [X4: set_v] : ( P @ ( insert_set_v @ X4 @ bot_bot_set_set_v ) )
=> ( ! [X4: set_v,F3: set_set_v] :
( ( finite_finite_set_v @ F3 )
=> ( ( F3 != bot_bot_set_set_v )
=> ( ~ ( member_set_v @ X4 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_set_v @ X4 @ F3 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_1193_finite__induct,axiom,
! [F2: set_v,P: set_v > $o] :
( ( finite_finite_v @ F2 )
=> ( ( P @ bot_bot_set_v )
=> ( ! [X4: v,F3: set_v] :
( ( finite_finite_v @ F3 )
=> ( ~ ( member_v @ X4 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_v @ X4 @ F3 ) ) ) ) )
=> ( P @ F2 ) ) ) ) ).
% finite_induct
thf(fact_1194_finite__induct,axiom,
! [F2: set_Product_prod_v_v,P: set_Product_prod_v_v > $o] :
( ( finite3348123685078250256od_v_v @ F2 )
=> ( ( P @ bot_bo723834152578015283od_v_v )
=> ( ! [X4: product_prod_v_v,F3: set_Product_prod_v_v] :
( ( finite3348123685078250256od_v_v @ F3 )
=> ( ~ ( member7453568604450474000od_v_v @ X4 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert1338601472111419319od_v_v @ X4 @ F3 ) ) ) ) )
=> ( P @ F2 ) ) ) ) ).
% finite_induct
thf(fact_1195_finite__induct,axiom,
! [F2: set_set_v,P: set_set_v > $o] :
( ( finite_finite_set_v @ F2 )
=> ( ( P @ bot_bot_set_set_v )
=> ( ! [X4: set_v,F3: set_set_v] :
( ( finite_finite_set_v @ F3 )
=> ( ~ ( member_set_v @ X4 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_set_v @ X4 @ F3 ) ) ) ) )
=> ( P @ F2 ) ) ) ) ).
% finite_induct
thf(fact_1196_finite_Osimps,axiom,
( finite_finite_v
= ( ^ [A4: set_v] :
( ( A4 = bot_bot_set_v )
| ? [A5: set_v,B4: v] :
( ( A4
= ( insert_v @ B4 @ A5 ) )
& ( finite_finite_v @ A5 ) ) ) ) ) ).
% finite.simps
thf(fact_1197_finite_Osimps,axiom,
( finite3348123685078250256od_v_v
= ( ^ [A4: set_Product_prod_v_v] :
( ( A4 = bot_bo723834152578015283od_v_v )
| ? [A5: set_Product_prod_v_v,B4: product_prod_v_v] :
( ( A4
= ( insert1338601472111419319od_v_v @ B4 @ A5 ) )
& ( finite3348123685078250256od_v_v @ A5 ) ) ) ) ) ).
% finite.simps
thf(fact_1198_finite_Osimps,axiom,
( finite_finite_set_v
= ( ^ [A4: set_set_v] :
( ( A4 = bot_bot_set_set_v )
| ? [A5: set_set_v,B4: set_v] :
( ( A4
= ( insert_set_v @ B4 @ A5 ) )
& ( finite_finite_set_v @ A5 ) ) ) ) ) ).
% finite.simps
thf(fact_1199_finite_Ocases,axiom,
! [A: set_v] :
( ( finite_finite_v @ A )
=> ( ( A != bot_bot_set_v )
=> ~ ! [A7: set_v] :
( ? [A8: v] :
( A
= ( insert_v @ A8 @ A7 ) )
=> ~ ( finite_finite_v @ A7 ) ) ) ) ).
% finite.cases
thf(fact_1200_finite_Ocases,axiom,
! [A: set_Product_prod_v_v] :
( ( finite3348123685078250256od_v_v @ A )
=> ( ( A != bot_bo723834152578015283od_v_v )
=> ~ ! [A7: set_Product_prod_v_v] :
( ? [A8: product_prod_v_v] :
( A
= ( insert1338601472111419319od_v_v @ A8 @ A7 ) )
=> ~ ( finite3348123685078250256od_v_v @ A7 ) ) ) ) ).
% finite.cases
thf(fact_1201_finite_Ocases,axiom,
! [A: set_set_v] :
( ( finite_finite_set_v @ A )
=> ( ( A != bot_bot_set_set_v )
=> ~ ! [A7: set_set_v] :
( ? [A8: set_v] :
( A
= ( insert_set_v @ A8 @ A7 ) )
=> ~ ( finite_finite_set_v @ A7 ) ) ) ) ).
% finite.cases
thf(fact_1202_finite__subset__induct_H,axiom,
! [F2: set_set_v,A3: set_set_v,P: set_set_v > $o] :
( ( finite_finite_set_v @ F2 )
=> ( ( ord_le5216385588623774835_set_v @ F2 @ A3 )
=> ( ( P @ bot_bot_set_set_v )
=> ( ! [A8: set_v,F3: set_set_v] :
( ( finite_finite_set_v @ F3 )
=> ( ( member_set_v @ A8 @ A3 )
=> ( ( ord_le5216385588623774835_set_v @ F3 @ A3 )
=> ( ~ ( member_set_v @ A8 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_set_v @ A8 @ F3 ) ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_1203_finite__subset__induct_H,axiom,
! [F2: set_v,A3: set_v,P: set_v > $o] :
( ( finite_finite_v @ F2 )
=> ( ( ord_less_eq_set_v @ F2 @ A3 )
=> ( ( P @ bot_bot_set_v )
=> ( ! [A8: v,F3: set_v] :
( ( finite_finite_v @ F3 )
=> ( ( member_v @ A8 @ A3 )
=> ( ( ord_less_eq_set_v @ F3 @ A3 )
=> ( ~ ( member_v @ A8 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_v @ A8 @ F3 ) ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_1204_finite__subset__induct_H,axiom,
! [F2: set_Product_prod_v_v,A3: set_Product_prod_v_v,P: set_Product_prod_v_v > $o] :
( ( finite3348123685078250256od_v_v @ F2 )
=> ( ( ord_le7336532860387713383od_v_v @ F2 @ A3 )
=> ( ( P @ bot_bo723834152578015283od_v_v )
=> ( ! [A8: product_prod_v_v,F3: set_Product_prod_v_v] :
( ( finite3348123685078250256od_v_v @ F3 )
=> ( ( member7453568604450474000od_v_v @ A8 @ A3 )
=> ( ( ord_le7336532860387713383od_v_v @ F3 @ A3 )
=> ( ~ ( member7453568604450474000od_v_v @ A8 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert1338601472111419319od_v_v @ A8 @ F3 ) ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_1205_finite__subset__induct,axiom,
! [F2: set_set_v,A3: set_set_v,P: set_set_v > $o] :
( ( finite_finite_set_v @ F2 )
=> ( ( ord_le5216385588623774835_set_v @ F2 @ A3 )
=> ( ( P @ bot_bot_set_set_v )
=> ( ! [A8: set_v,F3: set_set_v] :
( ( finite_finite_set_v @ F3 )
=> ( ( member_set_v @ A8 @ A3 )
=> ( ~ ( member_set_v @ A8 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_set_v @ A8 @ F3 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_1206_finite__subset__induct,axiom,
! [F2: set_v,A3: set_v,P: set_v > $o] :
( ( finite_finite_v @ F2 )
=> ( ( ord_less_eq_set_v @ F2 @ A3 )
=> ( ( P @ bot_bot_set_v )
=> ( ! [A8: v,F3: set_v] :
( ( finite_finite_v @ F3 )
=> ( ( member_v @ A8 @ A3 )
=> ( ~ ( member_v @ A8 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_v @ A8 @ F3 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_1207_finite__subset__induct,axiom,
! [F2: set_Product_prod_v_v,A3: set_Product_prod_v_v,P: set_Product_prod_v_v > $o] :
( ( finite3348123685078250256od_v_v @ F2 )
=> ( ( ord_le7336532860387713383od_v_v @ F2 @ A3 )
=> ( ( P @ bot_bo723834152578015283od_v_v )
=> ( ! [A8: product_prod_v_v,F3: set_Product_prod_v_v] :
( ( finite3348123685078250256od_v_v @ F3 )
=> ( ( member7453568604450474000od_v_v @ A8 @ A3 )
=> ( ~ ( member7453568604450474000od_v_v @ A8 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert1338601472111419319od_v_v @ A8 @ F3 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_1208_infinite__remove,axiom,
! [S: set_Product_prod_v_v,A: product_prod_v_v] :
( ~ ( finite3348123685078250256od_v_v @ S )
=> ~ ( finite3348123685078250256od_v_v @ ( minus_4183494784930505774od_v_v @ S @ ( insert1338601472111419319od_v_v @ A @ bot_bo723834152578015283od_v_v ) ) ) ) ).
% infinite_remove
thf(fact_1209_infinite__remove,axiom,
! [S: set_set_v,A: set_v] :
( ~ ( finite_finite_set_v @ S )
=> ~ ( finite_finite_set_v @ ( minus_7228012346218142266_set_v @ S @ ( insert_set_v @ A @ bot_bot_set_set_v ) ) ) ) ).
% infinite_remove
thf(fact_1210_infinite__remove,axiom,
! [S: set_v,A: v] :
( ~ ( finite_finite_v @ S )
=> ~ ( finite_finite_v @ ( minus_minus_set_v @ S @ ( insert_v @ A @ bot_bot_set_v ) ) ) ) ).
% infinite_remove
thf(fact_1211_infinite__coinduct,axiom,
! [X5: set_Product_prod_v_v > $o,A3: set_Product_prod_v_v] :
( ( X5 @ A3 )
=> ( ! [A7: set_Product_prod_v_v] :
( ( X5 @ A7 )
=> ? [X: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X @ A7 )
& ( ( X5 @ ( minus_4183494784930505774od_v_v @ A7 @ ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v ) ) )
| ~ ( finite3348123685078250256od_v_v @ ( minus_4183494784930505774od_v_v @ A7 @ ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v ) ) ) ) ) )
=> ~ ( finite3348123685078250256od_v_v @ A3 ) ) ) ).
% infinite_coinduct
thf(fact_1212_infinite__coinduct,axiom,
! [X5: set_set_v > $o,A3: set_set_v] :
( ( X5 @ A3 )
=> ( ! [A7: set_set_v] :
( ( X5 @ A7 )
=> ? [X: set_v] :
( ( member_set_v @ X @ A7 )
& ( ( X5 @ ( minus_7228012346218142266_set_v @ A7 @ ( insert_set_v @ X @ bot_bot_set_set_v ) ) )
| ~ ( finite_finite_set_v @ ( minus_7228012346218142266_set_v @ A7 @ ( insert_set_v @ X @ bot_bot_set_set_v ) ) ) ) ) )
=> ~ ( finite_finite_set_v @ A3 ) ) ) ).
% infinite_coinduct
thf(fact_1213_infinite__coinduct,axiom,
! [X5: set_v > $o,A3: set_v] :
( ( X5 @ A3 )
=> ( ! [A7: set_v] :
( ( X5 @ A7 )
=> ? [X: v] :
( ( member_v @ X @ A7 )
& ( ( X5 @ ( minus_minus_set_v @ A7 @ ( insert_v @ X @ bot_bot_set_v ) ) )
| ~ ( finite_finite_v @ ( minus_minus_set_v @ A7 @ ( insert_v @ X @ bot_bot_set_v ) ) ) ) ) )
=> ~ ( finite_finite_v @ A3 ) ) ) ).
% infinite_coinduct
thf(fact_1214_finite__empty__induct,axiom,
! [A3: set_Product_prod_v_v,P: set_Product_prod_v_v > $o] :
( ( finite3348123685078250256od_v_v @ A3 )
=> ( ( P @ A3 )
=> ( ! [A8: product_prod_v_v,A7: set_Product_prod_v_v] :
( ( finite3348123685078250256od_v_v @ A7 )
=> ( ( member7453568604450474000od_v_v @ A8 @ A7 )
=> ( ( P @ A7 )
=> ( P @ ( minus_4183494784930505774od_v_v @ A7 @ ( insert1338601472111419319od_v_v @ A8 @ bot_bo723834152578015283od_v_v ) ) ) ) ) )
=> ( P @ bot_bo723834152578015283od_v_v ) ) ) ) ).
% finite_empty_induct
thf(fact_1215_finite__empty__induct,axiom,
! [A3: set_set_v,P: set_set_v > $o] :
( ( finite_finite_set_v @ A3 )
=> ( ( P @ A3 )
=> ( ! [A8: set_v,A7: set_set_v] :
( ( finite_finite_set_v @ A7 )
=> ( ( member_set_v @ A8 @ A7 )
=> ( ( P @ A7 )
=> ( P @ ( minus_7228012346218142266_set_v @ A7 @ ( insert_set_v @ A8 @ bot_bot_set_set_v ) ) ) ) ) )
=> ( P @ bot_bot_set_set_v ) ) ) ) ).
% finite_empty_induct
thf(fact_1216_finite__empty__induct,axiom,
! [A3: set_v,P: set_v > $o] :
( ( finite_finite_v @ A3 )
=> ( ( P @ A3 )
=> ( ! [A8: v,A7: set_v] :
( ( finite_finite_v @ A7 )
=> ( ( member_v @ A8 @ A7 )
=> ( ( P @ A7 )
=> ( P @ ( minus_minus_set_v @ A7 @ ( insert_v @ A8 @ bot_bot_set_v ) ) ) ) ) )
=> ( P @ bot_bot_set_v ) ) ) ) ).
% finite_empty_induct
thf(fact_1217_remove__induct,axiom,
! [P: set_set_v > $o,B: set_set_v] :
( ( P @ bot_bot_set_set_v )
=> ( ( ~ ( finite_finite_set_v @ B )
=> ( P @ B ) )
=> ( ! [A7: set_set_v] :
( ( finite_finite_set_v @ A7 )
=> ( ( A7 != bot_bot_set_set_v )
=> ( ( ord_le5216385588623774835_set_v @ A7 @ B )
=> ( ! [X: set_v] :
( ( member_set_v @ X @ A7 )
=> ( P @ ( minus_7228012346218142266_set_v @ A7 @ ( insert_set_v @ X @ bot_bot_set_set_v ) ) ) )
=> ( P @ A7 ) ) ) ) )
=> ( P @ B ) ) ) ) ).
% remove_induct
thf(fact_1218_remove__induct,axiom,
! [P: set_v > $o,B: set_v] :
( ( P @ bot_bot_set_v )
=> ( ( ~ ( finite_finite_v @ B )
=> ( P @ B ) )
=> ( ! [A7: set_v] :
( ( finite_finite_v @ A7 )
=> ( ( A7 != bot_bot_set_v )
=> ( ( ord_less_eq_set_v @ A7 @ B )
=> ( ! [X: v] :
( ( member_v @ X @ A7 )
=> ( P @ ( minus_minus_set_v @ A7 @ ( insert_v @ X @ bot_bot_set_v ) ) ) )
=> ( P @ A7 ) ) ) ) )
=> ( P @ B ) ) ) ) ).
% remove_induct
thf(fact_1219_remove__induct,axiom,
! [P: set_Product_prod_v_v > $o,B: set_Product_prod_v_v] :
( ( P @ bot_bo723834152578015283od_v_v )
=> ( ( ~ ( finite3348123685078250256od_v_v @ B )
=> ( P @ B ) )
=> ( ! [A7: set_Product_prod_v_v] :
( ( finite3348123685078250256od_v_v @ A7 )
=> ( ( A7 != bot_bo723834152578015283od_v_v )
=> ( ( ord_le7336532860387713383od_v_v @ A7 @ B )
=> ( ! [X: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X @ A7 )
=> ( P @ ( minus_4183494784930505774od_v_v @ A7 @ ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v ) ) ) )
=> ( P @ A7 ) ) ) ) )
=> ( P @ B ) ) ) ) ).
% remove_induct
thf(fact_1220_finite__remove__induct,axiom,
! [B: set_set_v,P: set_set_v > $o] :
( ( finite_finite_set_v @ B )
=> ( ( P @ bot_bot_set_set_v )
=> ( ! [A7: set_set_v] :
( ( finite_finite_set_v @ A7 )
=> ( ( A7 != bot_bot_set_set_v )
=> ( ( ord_le5216385588623774835_set_v @ A7 @ B )
=> ( ! [X: set_v] :
( ( member_set_v @ X @ A7 )
=> ( P @ ( minus_7228012346218142266_set_v @ A7 @ ( insert_set_v @ X @ bot_bot_set_set_v ) ) ) )
=> ( P @ A7 ) ) ) ) )
=> ( P @ B ) ) ) ) ).
% finite_remove_induct
thf(fact_1221_finite__remove__induct,axiom,
! [B: set_v,P: set_v > $o] :
( ( finite_finite_v @ B )
=> ( ( P @ bot_bot_set_v )
=> ( ! [A7: set_v] :
( ( finite_finite_v @ A7 )
=> ( ( A7 != bot_bot_set_v )
=> ( ( ord_less_eq_set_v @ A7 @ B )
=> ( ! [X: v] :
( ( member_v @ X @ A7 )
=> ( P @ ( minus_minus_set_v @ A7 @ ( insert_v @ X @ bot_bot_set_v ) ) ) )
=> ( P @ A7 ) ) ) ) )
=> ( P @ B ) ) ) ) ).
% finite_remove_induct
thf(fact_1222_finite__remove__induct,axiom,
! [B: set_Product_prod_v_v,P: set_Product_prod_v_v > $o] :
( ( finite3348123685078250256od_v_v @ B )
=> ( ( P @ bot_bo723834152578015283od_v_v )
=> ( ! [A7: set_Product_prod_v_v] :
( ( finite3348123685078250256od_v_v @ A7 )
=> ( ( A7 != bot_bo723834152578015283od_v_v )
=> ( ( ord_le7336532860387713383od_v_v @ A7 @ B )
=> ( ! [X: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X @ A7 )
=> ( P @ ( minus_4183494784930505774od_v_v @ A7 @ ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v ) ) ) )
=> ( P @ A7 ) ) ) ) )
=> ( P @ B ) ) ) ) ).
% finite_remove_induct
thf(fact_1223_insert__subsetI,axiom,
! [X2: set_v,A3: set_set_v,X5: set_set_v] :
( ( member_set_v @ X2 @ A3 )
=> ( ( ord_le5216385588623774835_set_v @ X5 @ A3 )
=> ( ord_le5216385588623774835_set_v @ ( insert_set_v @ X2 @ X5 ) @ A3 ) ) ) ).
% insert_subsetI
thf(fact_1224_insert__subsetI,axiom,
! [X2: v,A3: set_v,X5: set_v] :
( ( member_v @ X2 @ A3 )
=> ( ( ord_less_eq_set_v @ X5 @ A3 )
=> ( ord_less_eq_set_v @ ( insert_v @ X2 @ X5 ) @ A3 ) ) ) ).
% insert_subsetI
thf(fact_1225_insert__subsetI,axiom,
! [X2: product_prod_v_v,A3: set_Product_prod_v_v,X5: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X2 @ A3 )
=> ( ( ord_le7336532860387713383od_v_v @ X5 @ A3 )
=> ( ord_le7336532860387713383od_v_v @ ( insert1338601472111419319od_v_v @ X2 @ X5 ) @ A3 ) ) ) ).
% insert_subsetI
thf(fact_1226_subset__emptyI,axiom,
! [A3: set_set_v] :
( ! [X4: set_v] :
~ ( member_set_v @ X4 @ A3 )
=> ( ord_le5216385588623774835_set_v @ A3 @ bot_bot_set_set_v ) ) ).
% subset_emptyI
thf(fact_1227_subset__emptyI,axiom,
! [A3: set_v] :
( ! [X4: v] :
~ ( member_v @ X4 @ A3 )
=> ( ord_less_eq_set_v @ A3 @ bot_bot_set_v ) ) ).
% subset_emptyI
thf(fact_1228_subset__emptyI,axiom,
! [A3: set_Product_prod_v_v] :
( ! [X4: product_prod_v_v] :
~ ( member7453568604450474000od_v_v @ X4 @ A3 )
=> ( ord_le7336532860387713383od_v_v @ A3 @ bot_bo723834152578015283od_v_v ) ) ).
% subset_emptyI
thf(fact_1229_product__lists_Osimps_I1_J,axiom,
( ( product_lists_v @ nil_list_v )
= ( cons_list_v @ nil_v @ nil_list_v ) ) ).
% product_lists.simps(1)
thf(fact_1230_bind__simps_I2_J,axiom,
! [X2: v,Xs: list_v,F: v > list_v] :
( ( bind_v_v @ ( cons_v @ X2 @ Xs ) @ F )
= ( append_v @ ( F @ X2 ) @ ( bind_v_v @ Xs @ F ) ) ) ).
% bind_simps(2)
thf(fact_1231_Field__insert,axiom,
! [A: v,B2: v,R: set_Product_prod_v_v] :
( ( field_v @ ( insert1338601472111419319od_v_v @ ( product_Pair_v_v @ A @ B2 ) @ R ) )
= ( sup_sup_set_v @ ( insert_v @ A @ ( insert_v @ B2 @ bot_bot_set_v ) ) @ ( field_v @ R ) ) ) ).
% Field_insert
thf(fact_1232_Field__insert,axiom,
! [A: product_prod_v_v,B2: product_prod_v_v,R: set_Pr2149350503807050951od_v_v] :
( ( field_7153129647634986036od_v_v @ ( insert5641704497130386615od_v_v @ ( produc4031800376763917143od_v_v @ A @ B2 ) @ R ) )
= ( sup_su414716646722978715od_v_v @ ( insert1338601472111419319od_v_v @ A @ ( insert1338601472111419319od_v_v @ B2 @ bot_bo723834152578015283od_v_v ) ) @ ( field_7153129647634986036od_v_v @ R ) ) ) ).
% Field_insert
thf(fact_1233_Field__insert,axiom,
! [A: set_v,B2: set_v,R: set_Pr8199228935972127175_set_v] :
( ( field_set_v @ ( insert1457770702614273975_set_v @ ( produc3441907479644599895_set_v @ A @ B2 ) @ R ) )
= ( sup_sup_set_set_v @ ( insert_set_v @ A @ ( insert_set_v @ B2 @ bot_bot_set_set_v ) ) @ ( field_set_v @ R ) ) ) ).
% Field_insert
thf(fact_1234_remove__def,axiom,
( remove5001965847480235980od_v_v
= ( ^ [X3: product_prod_v_v,A5: set_Product_prod_v_v] : ( minus_4183494784930505774od_v_v @ A5 @ ( insert1338601472111419319od_v_v @ X3 @ bot_bo723834152578015283od_v_v ) ) ) ) ).
% remove_def
thf(fact_1235_remove__def,axiom,
( remove_set_v
= ( ^ [X3: set_v,A5: set_set_v] : ( minus_7228012346218142266_set_v @ A5 @ ( insert_set_v @ X3 @ bot_bot_set_set_v ) ) ) ) ).
% remove_def
thf(fact_1236_remove__def,axiom,
( remove_v
= ( ^ [X3: v,A5: set_v] : ( minus_minus_set_v @ A5 @ ( insert_v @ X3 @ bot_bot_set_v ) ) ) ) ).
% remove_def
thf(fact_1237_member__remove,axiom,
! [X2: v,Y2: v,A3: set_v] :
( ( member_v @ X2 @ ( remove_v @ Y2 @ A3 ) )
= ( ( member_v @ X2 @ A3 )
& ( X2 != Y2 ) ) ) ).
% member_remove
thf(fact_1238_member__remove,axiom,
! [X2: product_prod_v_v,Y2: product_prod_v_v,A3: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X2 @ ( remove5001965847480235980od_v_v @ Y2 @ A3 ) )
= ( ( member7453568604450474000od_v_v @ X2 @ A3 )
& ( X2 != Y2 ) ) ) ).
% member_remove
thf(fact_1239_bind__simps_I1_J,axiom,
! [F: v > list_v] :
( ( bind_v_v @ nil_v @ F )
= nil_v ) ).
% bind_simps(1)
thf(fact_1240_Field__empty,axiom,
( ( field_7153129647634986036od_v_v @ bot_bo3282589961317712691od_v_v )
= bot_bo723834152578015283od_v_v ) ).
% Field_empty
thf(fact_1241_Field__empty,axiom,
( ( field_set_v @ bot_bo8153096493302634547_set_v )
= bot_bot_set_set_v ) ).
% Field_empty
thf(fact_1242_Field__empty,axiom,
( ( field_v @ bot_bo723834152578015283od_v_v )
= bot_bot_set_v ) ).
% Field_empty
thf(fact_1243_Field__Un,axiom,
! [R: set_Pr2149350503807050951od_v_v,S5: set_Pr2149350503807050951od_v_v] :
( ( field_7153129647634986036od_v_v @ ( sup_su1742609618068805275od_v_v @ R @ S5 ) )
= ( sup_su414716646722978715od_v_v @ ( field_7153129647634986036od_v_v @ R ) @ ( field_7153129647634986036od_v_v @ S5 ) ) ) ).
% Field_Un
thf(fact_1244_Field__Un,axiom,
! [R: set_Pr8199228935972127175_set_v,S5: set_Pr8199228935972127175_set_v] :
( ( field_set_v @ ( sup_su7977902838240902043_set_v @ R @ S5 ) )
= ( sup_sup_set_set_v @ ( field_set_v @ R ) @ ( field_set_v @ S5 ) ) ) ).
% Field_Un
thf(fact_1245_Field__Un,axiom,
! [R: set_Product_prod_v_v,S5: set_Product_prod_v_v] :
( ( field_v @ ( sup_su414716646722978715od_v_v @ R @ S5 ) )
= ( sup_sup_set_v @ ( field_v @ R ) @ ( field_v @ S5 ) ) ) ).
% Field_Un
thf(fact_1246_remove__code_I1_J,axiom,
! [X2: v,Xs: list_v] :
( ( remove_v @ X2 @ ( set_v2 @ Xs ) )
= ( set_v2 @ ( removeAll_v @ X2 @ Xs ) ) ) ).
% remove_code(1)
thf(fact_1247_dom,axiom,
accp_S2303753412255344476t_unit @ ( sCC_Bl907557413677168252_rel_v @ successors ) @ ( sum_In5289330923152326972t_unit @ ( produc3862955338007567901t_unit @ va @ ea ) ) ).
% dom
thf(fact_1248_Sup__fin_Oinsert__remove,axiom,
! [A3: set_se8455005133513928103od_v_v,X2: set_Product_prod_v_v] :
( ( finite6084192165098772208od_v_v @ A3 )
=> ( ( ( ( minus_7679383599658060814od_v_v @ A3 @ ( insert7504383016908236695od_v_v @ X2 @ bot_bo3497076220358800403od_v_v ) )
= bot_bo3497076220358800403od_v_v )
=> ( ( lattic5151207300795964030od_v_v @ ( insert7504383016908236695od_v_v @ X2 @ A3 ) )
= X2 ) )
& ( ( ( minus_7679383599658060814od_v_v @ A3 @ ( insert7504383016908236695od_v_v @ X2 @ bot_bo3497076220358800403od_v_v ) )
!= bot_bo3497076220358800403od_v_v )
=> ( ( lattic5151207300795964030od_v_v @ ( insert7504383016908236695od_v_v @ X2 @ A3 ) )
= ( sup_su414716646722978715od_v_v @ X2 @ ( lattic5151207300795964030od_v_v @ ( minus_7679383599658060814od_v_v @ A3 @ ( insert7504383016908236695od_v_v @ X2 @ bot_bo3497076220358800403od_v_v ) ) ) ) ) ) ) ) ).
% Sup_fin.insert_remove
thf(fact_1249_Sup__fin_Oinsert__remove,axiom,
! [A3: set_set_set_v,X2: set_set_v] :
( ( finite8701002811114149628_set_v @ A3 )
=> ( ( ( ( minus_5754555218759951642_set_v @ A3 @ ( insert_set_set_v @ X2 @ bot_bo5775917114081396255_set_v ) )
= bot_bo5775917114081396255_set_v )
=> ( ( lattic1829858174534819978_set_v @ ( insert_set_set_v @ X2 @ A3 ) )
= X2 ) )
& ( ( ( minus_5754555218759951642_set_v @ A3 @ ( insert_set_set_v @ X2 @ bot_bo5775917114081396255_set_v ) )
!= bot_bo5775917114081396255_set_v )
=> ( ( lattic1829858174534819978_set_v @ ( insert_set_set_v @ X2 @ A3 ) )
= ( sup_sup_set_set_v @ X2 @ ( lattic1829858174534819978_set_v @ ( minus_5754555218759951642_set_v @ A3 @ ( insert_set_set_v @ X2 @ bot_bo5775917114081396255_set_v ) ) ) ) ) ) ) ) ).
% Sup_fin.insert_remove
thf(fact_1250_Sup__fin_Oinsert__remove,axiom,
! [A3: set_set_v,X2: set_v] :
( ( finite_finite_set_v @ A3 )
=> ( ( ( ( minus_7228012346218142266_set_v @ A3 @ ( insert_set_v @ X2 @ bot_bot_set_set_v ) )
= bot_bot_set_set_v )
=> ( ( lattic2918178447194608042_set_v @ ( insert_set_v @ X2 @ A3 ) )
= X2 ) )
& ( ( ( minus_7228012346218142266_set_v @ A3 @ ( insert_set_v @ X2 @ bot_bot_set_set_v ) )
!= bot_bot_set_set_v )
=> ( ( lattic2918178447194608042_set_v @ ( insert_set_v @ X2 @ A3 ) )
= ( sup_sup_set_v @ X2 @ ( lattic2918178447194608042_set_v @ ( minus_7228012346218142266_set_v @ A3 @ ( insert_set_v @ X2 @ bot_bot_set_set_v ) ) ) ) ) ) ) ) ).
% Sup_fin.insert_remove
thf(fact_1251_Sup__fin_Oremove,axiom,
! [A3: set_se8455005133513928103od_v_v,X2: set_Product_prod_v_v] :
( ( finite6084192165098772208od_v_v @ A3 )
=> ( ( member8406446414694345712od_v_v @ X2 @ A3 )
=> ( ( ( ( minus_7679383599658060814od_v_v @ A3 @ ( insert7504383016908236695od_v_v @ X2 @ bot_bo3497076220358800403od_v_v ) )
= bot_bo3497076220358800403od_v_v )
=> ( ( lattic5151207300795964030od_v_v @ A3 )
= X2 ) )
& ( ( ( minus_7679383599658060814od_v_v @ A3 @ ( insert7504383016908236695od_v_v @ X2 @ bot_bo3497076220358800403od_v_v ) )
!= bot_bo3497076220358800403od_v_v )
=> ( ( lattic5151207300795964030od_v_v @ A3 )
= ( sup_su414716646722978715od_v_v @ X2 @ ( lattic5151207300795964030od_v_v @ ( minus_7679383599658060814od_v_v @ A3 @ ( insert7504383016908236695od_v_v @ X2 @ bot_bo3497076220358800403od_v_v ) ) ) ) ) ) ) ) ) ).
% Sup_fin.remove
thf(fact_1252_Sup__fin_Oremove,axiom,
! [A3: set_set_set_v,X2: set_set_v] :
( ( finite8701002811114149628_set_v @ A3 )
=> ( ( member_set_set_v @ X2 @ A3 )
=> ( ( ( ( minus_5754555218759951642_set_v @ A3 @ ( insert_set_set_v @ X2 @ bot_bo5775917114081396255_set_v ) )
= bot_bo5775917114081396255_set_v )
=> ( ( lattic1829858174534819978_set_v @ A3 )
= X2 ) )
& ( ( ( minus_5754555218759951642_set_v @ A3 @ ( insert_set_set_v @ X2 @ bot_bo5775917114081396255_set_v ) )
!= bot_bo5775917114081396255_set_v )
=> ( ( lattic1829858174534819978_set_v @ A3 )
= ( sup_sup_set_set_v @ X2 @ ( lattic1829858174534819978_set_v @ ( minus_5754555218759951642_set_v @ A3 @ ( insert_set_set_v @ X2 @ bot_bo5775917114081396255_set_v ) ) ) ) ) ) ) ) ) ).
% Sup_fin.remove
thf(fact_1253_Sup__fin_Oremove,axiom,
! [A3: set_set_v,X2: set_v] :
( ( finite_finite_set_v @ A3 )
=> ( ( member_set_v @ X2 @ A3 )
=> ( ( ( ( minus_7228012346218142266_set_v @ A3 @ ( insert_set_v @ X2 @ bot_bot_set_set_v ) )
= bot_bot_set_set_v )
=> ( ( lattic2918178447194608042_set_v @ A3 )
= X2 ) )
& ( ( ( minus_7228012346218142266_set_v @ A3 @ ( insert_set_v @ X2 @ bot_bot_set_set_v ) )
!= bot_bot_set_set_v )
=> ( ( lattic2918178447194608042_set_v @ A3 )
= ( sup_sup_set_v @ X2 @ ( lattic2918178447194608042_set_v @ ( minus_7228012346218142266_set_v @ A3 @ ( insert_set_v @ X2 @ bot_bot_set_set_v ) ) ) ) ) ) ) ) ) ).
% Sup_fin.remove
thf(fact_1254_dfss_Ocases,axiom,
! [X2: produc5741669702376414499t_unit] :
~ ! [V2: v,E7: sCC_Bl1394983891496994913t_unit] :
( X2
!= ( produc3862955338007567901t_unit @ V2 @ E7 ) ) ).
% dfss.cases
thf(fact_1255_Sup__fin_Osingleton,axiom,
! [X2: set_v] :
( ( lattic2918178447194608042_set_v @ ( insert_set_v @ X2 @ bot_bot_set_set_v ) )
= X2 ) ).
% Sup_fin.singleton
thf(fact_1256_Sup__fin_Oinsert,axiom,
! [A3: set_se8455005133513928103od_v_v,X2: set_Product_prod_v_v] :
( ( finite6084192165098772208od_v_v @ A3 )
=> ( ( A3 != bot_bo3497076220358800403od_v_v )
=> ( ( lattic5151207300795964030od_v_v @ ( insert7504383016908236695od_v_v @ X2 @ A3 ) )
= ( sup_su414716646722978715od_v_v @ X2 @ ( lattic5151207300795964030od_v_v @ A3 ) ) ) ) ) ).
% Sup_fin.insert
thf(fact_1257_Sup__fin_Oinsert,axiom,
! [A3: set_set_set_v,X2: set_set_v] :
( ( finite8701002811114149628_set_v @ A3 )
=> ( ( A3 != bot_bo5775917114081396255_set_v )
=> ( ( lattic1829858174534819978_set_v @ ( insert_set_set_v @ X2 @ A3 ) )
= ( sup_sup_set_set_v @ X2 @ ( lattic1829858174534819978_set_v @ A3 ) ) ) ) ) ).
% Sup_fin.insert
thf(fact_1258_Sup__fin_Oinsert,axiom,
! [A3: set_set_v,X2: set_v] :
( ( finite_finite_set_v @ A3 )
=> ( ( A3 != bot_bot_set_set_v )
=> ( ( lattic2918178447194608042_set_v @ ( insert_set_v @ X2 @ A3 ) )
= ( sup_sup_set_v @ X2 @ ( lattic2918178447194608042_set_v @ A3 ) ) ) ) ) ).
% Sup_fin.insert
thf(fact_1259_Sup__fin_Oin__idem,axiom,
! [A3: set_se8455005133513928103od_v_v,X2: set_Product_prod_v_v] :
( ( finite6084192165098772208od_v_v @ A3 )
=> ( ( member8406446414694345712od_v_v @ X2 @ A3 )
=> ( ( sup_su414716646722978715od_v_v @ X2 @ ( lattic5151207300795964030od_v_v @ A3 ) )
= ( lattic5151207300795964030od_v_v @ A3 ) ) ) ) ).
% Sup_fin.in_idem
thf(fact_1260_Sup__fin_Oin__idem,axiom,
! [A3: set_set_v,X2: set_v] :
( ( finite_finite_set_v @ A3 )
=> ( ( member_set_v @ X2 @ A3 )
=> ( ( sup_sup_set_v @ X2 @ ( lattic2918178447194608042_set_v @ A3 ) )
= ( lattic2918178447194608042_set_v @ A3 ) ) ) ) ).
% Sup_fin.in_idem
thf(fact_1261_Sup__fin_Oin__idem,axiom,
! [A3: set_set_set_v,X2: set_set_v] :
( ( finite8701002811114149628_set_v @ A3 )
=> ( ( member_set_set_v @ X2 @ A3 )
=> ( ( sup_sup_set_set_v @ X2 @ ( lattic1829858174534819978_set_v @ A3 ) )
= ( lattic1829858174534819978_set_v @ A3 ) ) ) ) ).
% Sup_fin.in_idem
thf(fact_1262_Sup__fin_OboundedE,axiom,
! [A3: set_set_v,X2: set_v] :
( ( finite_finite_set_v @ A3 )
=> ( ( A3 != bot_bot_set_set_v )
=> ( ( ord_less_eq_set_v @ ( lattic2918178447194608042_set_v @ A3 ) @ X2 )
=> ! [A9: set_v] :
( ( member_set_v @ A9 @ A3 )
=> ( ord_less_eq_set_v @ A9 @ X2 ) ) ) ) ) ).
% Sup_fin.boundedE
thf(fact_1263_Sup__fin_OboundedE,axiom,
! [A3: set_se8455005133513928103od_v_v,X2: set_Product_prod_v_v] :
( ( finite6084192165098772208od_v_v @ A3 )
=> ( ( A3 != bot_bo3497076220358800403od_v_v )
=> ( ( ord_le7336532860387713383od_v_v @ ( lattic5151207300795964030od_v_v @ A3 ) @ X2 )
=> ! [A9: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ A9 @ A3 )
=> ( ord_le7336532860387713383od_v_v @ A9 @ X2 ) ) ) ) ) ).
% Sup_fin.boundedE
thf(fact_1264_Sup__fin_OboundedI,axiom,
! [A3: set_se8455005133513928103od_v_v,X2: set_Product_prod_v_v] :
( ( finite6084192165098772208od_v_v @ A3 )
=> ( ( A3 != bot_bo3497076220358800403od_v_v )
=> ( ! [A8: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ A8 @ A3 )
=> ( ord_le7336532860387713383od_v_v @ A8 @ X2 ) )
=> ( ord_le7336532860387713383od_v_v @ ( lattic5151207300795964030od_v_v @ A3 ) @ X2 ) ) ) ) ).
% Sup_fin.boundedI
thf(fact_1265_prepostdfs,axiom,
! [Vs: set_v,W: v] :
( ( Vs
= ( minus_minus_set_v @ ( successors @ va ) @ ( sCC_Bl3795065053823578884t_unit @ ea @ va ) ) )
=> ( ( Vs != bot_bot_set_v )
=> ( ( W
= ( fChoice_v
@ ^ [X3: v] : ( member_v @ X3 @ Vs ) ) )
=> ( ~ ( member_v @ W @ ( sCC_Bl157864678168468314t_unit @ ea ) )
=> ( ~ ( member_v @ W @ ( sCC_Bl4645233313691564917t_unit @ ea ) )
=> ( ( sCC_Bl36166008131615352t_unit @ successors @ W @ ea )
=> ( sCC_Bl8953792750115413617t_unit @ successors @ W @ ea @ ( sCC_Bloemen_dfs_v @ successors @ W @ ea ) ) ) ) ) ) ) ) ).
% prepostdfs
thf(fact_1266_w__def,axiom,
( w
= ( fChoice_v
@ ^ [X3: v] : ( member_v @ X3 @ ( minus_minus_set_v @ ( successors @ va ) @ ( sCC_Bl3795065053823578884t_unit @ ea @ va ) ) ) ) ) ).
% w_def
thf(fact_1267_prepostdfss,axiom,
! [Vs: set_v,W: v,E2: sCC_Bl1394983891496994913t_unit,E3: sCC_Bl1394983891496994913t_unit] :
( ( Vs
= ( minus_minus_set_v @ ( successors @ va ) @ ( sCC_Bl3795065053823578884t_unit @ ea @ va ) ) )
=> ( ( Vs != bot_bot_set_v )
=> ( ( W
= ( fChoice_v
@ ^ [X3: v] : ( member_v @ X3 @ Vs ) ) )
=> ( ( ( ( member_v @ W @ ( sCC_Bl157864678168468314t_unit @ ea ) )
=> ( E2 = ea ) )
& ( ~ ( member_v @ W @ ( sCC_Bl157864678168468314t_unit @ ea ) )
=> ( ( ~ ( member_v @ W @ ( sCC_Bl4645233313691564917t_unit @ ea ) )
=> ( E2
= ( sCC_Bloemen_dfs_v @ successors @ W @ ea ) ) )
& ( ( member_v @ W @ ( sCC_Bl4645233313691564917t_unit @ ea ) )
=> ( E2
= ( sCC_Bloemen_unite_v @ va @ W @ ea ) ) ) ) ) )
=> ( ( E3
= ( sCC_Bl48393358579903213t_unit
@ ^ [Uu: v > set_v,X3: v] : ( if_set_v @ ( X3 = va ) @ ( sup_sup_set_v @ ( sCC_Bl3795065053823578884t_unit @ E2 @ va ) @ ( insert_v @ W @ bot_bot_set_v ) ) @ ( sCC_Bl3795065053823578884t_unit @ E2 @ X3 ) )
@ E2 ) )
=> ( ( sCC_Bl1748261141445803503t_unit @ successors @ va @ E3 )
=> ( sCC_Bl6082031138996704384t_unit @ successors @ va @ E3 @ ( sCC_Bloemen_dfss_v @ successors @ va @ E3 ) ) ) ) ) ) ) ) ).
% prepostdfss
thf(fact_1268_dfss_Opsimps,axiom,
! [V: v,E: sCC_Bl1394983891496994913t_unit] :
( ( accp_S2303753412255344476t_unit @ ( sCC_Bl907557413677168252_rel_v @ successors ) @ ( sum_In5289330923152326972t_unit @ ( produc3862955338007567901t_unit @ V @ E ) ) )
=> ( ( sCC_Bloemen_dfss_v @ successors @ V @ E )
= ( if_SCC4926449794303880475t_unit
@ ( ( minus_minus_set_v @ ( successors @ V ) @ ( sCC_Bl3795065053823578884t_unit @ E @ V ) )
= bot_bot_set_v )
@ E
@ ( sCC_Bloemen_dfss_v @ successors @ V
@ ( sCC_Bl48393358579903213t_unit
@ ^ [Uu: v > set_v,X3: v] :
( if_set_v @ ( X3 = V )
@ ( sup_sup_set_v
@ ( sCC_Bl3795065053823578884t_unit
@ ( if_SCC4926449794303880475t_unit
@ ( member_v
@ ( fChoice_v
@ ^ [Y3: v] : ( member_v @ Y3 @ ( minus_minus_set_v @ ( successors @ V ) @ ( sCC_Bl3795065053823578884t_unit @ E @ V ) ) ) )
@ ( sCC_Bl157864678168468314t_unit @ E ) )
@ E
@ ( if_SCC4926449794303880475t_unit
@ ~ ( member_v
@ ( fChoice_v
@ ^ [Y3: v] : ( member_v @ Y3 @ ( minus_minus_set_v @ ( successors @ V ) @ ( sCC_Bl3795065053823578884t_unit @ E @ V ) ) ) )
@ ( sCC_Bl4645233313691564917t_unit @ E ) )
@ ( sCC_Bloemen_dfs_v @ successors
@ ( fChoice_v
@ ^ [Y3: v] : ( member_v @ Y3 @ ( minus_minus_set_v @ ( successors @ V ) @ ( sCC_Bl3795065053823578884t_unit @ E @ V ) ) ) )
@ E )
@ ( sCC_Bloemen_unite_v @ V
@ ( fChoice_v
@ ^ [Y3: v] : ( member_v @ Y3 @ ( minus_minus_set_v @ ( successors @ V ) @ ( sCC_Bl3795065053823578884t_unit @ E @ V ) ) ) )
@ E ) ) )
@ V )
@ ( insert_v
@ ( fChoice_v
@ ^ [Y3: v] : ( member_v @ Y3 @ ( minus_minus_set_v @ ( successors @ V ) @ ( sCC_Bl3795065053823578884t_unit @ E @ V ) ) ) )
@ bot_bot_set_v ) )
@ ( sCC_Bl3795065053823578884t_unit
@ ( if_SCC4926449794303880475t_unit
@ ( member_v
@ ( fChoice_v
@ ^ [Y3: v] : ( member_v @ Y3 @ ( minus_minus_set_v @ ( successors @ V ) @ ( sCC_Bl3795065053823578884t_unit @ E @ V ) ) ) )
@ ( sCC_Bl157864678168468314t_unit @ E ) )
@ E
@ ( if_SCC4926449794303880475t_unit
@ ~ ( member_v
@ ( fChoice_v
@ ^ [Y3: v] : ( member_v @ Y3 @ ( minus_minus_set_v @ ( successors @ V ) @ ( sCC_Bl3795065053823578884t_unit @ E @ V ) ) ) )
@ ( sCC_Bl4645233313691564917t_unit @ E ) )
@ ( sCC_Bloemen_dfs_v @ successors
@ ( fChoice_v
@ ^ [Y3: v] : ( member_v @ Y3 @ ( minus_minus_set_v @ ( successors @ V ) @ ( sCC_Bl3795065053823578884t_unit @ E @ V ) ) ) )
@ E )
@ ( sCC_Bloemen_unite_v @ V
@ ( fChoice_v
@ ^ [Y3: v] : ( member_v @ Y3 @ ( minus_minus_set_v @ ( successors @ V ) @ ( sCC_Bl3795065053823578884t_unit @ E @ V ) ) ) )
@ E ) ) )
@ X3 ) )
@ ( if_SCC4926449794303880475t_unit
@ ( member_v
@ ( fChoice_v
@ ^ [X3: v] : ( member_v @ X3 @ ( minus_minus_set_v @ ( successors @ V ) @ ( sCC_Bl3795065053823578884t_unit @ E @ V ) ) ) )
@ ( sCC_Bl157864678168468314t_unit @ E ) )
@ E
@ ( if_SCC4926449794303880475t_unit
@ ~ ( member_v
@ ( fChoice_v
@ ^ [X3: v] : ( member_v @ X3 @ ( minus_minus_set_v @ ( successors @ V ) @ ( sCC_Bl3795065053823578884t_unit @ E @ V ) ) ) )
@ ( sCC_Bl4645233313691564917t_unit @ E ) )
@ ( sCC_Bloemen_dfs_v @ successors
@ ( fChoice_v
@ ^ [X3: v] : ( member_v @ X3 @ ( minus_minus_set_v @ ( successors @ V ) @ ( sCC_Bl3795065053823578884t_unit @ E @ V ) ) ) )
@ E )
@ ( sCC_Bloemen_unite_v @ V
@ ( fChoice_v
@ ^ [X3: v] : ( member_v @ X3 @ ( minus_minus_set_v @ ( successors @ V ) @ ( sCC_Bl3795065053823578884t_unit @ E @ V ) ) ) )
@ E ) ) ) ) ) ) ) ) ).
% dfss.psimps
thf(fact_1269_e_H_H__def,axiom,
( e
= ( sCC_Bl48393358579903213t_unit
@ ^ [Uu: v > set_v,X3: v] : ( if_set_v @ ( X3 = va ) @ ( sup_sup_set_v @ ( sCC_Bl3795065053823578884t_unit @ e2 @ va ) @ ( insert_v @ w @ bot_bot_set_v ) ) @ ( sCC_Bl3795065053823578884t_unit @ e2 @ X3 ) )
@ e2 ) ) ).
% e''_def
thf(fact_1270_pre__dfss__unite__pre__dfss,axiom,
! [V: v,E: sCC_Bl1394983891496994913t_unit,W: v] :
( ( sCC_Bl1748261141445803503t_unit @ successors @ V @ E )
=> ( ( member_v @ W @ ( successors @ V ) )
=> ( ~ ( member_v @ W @ ( sCC_Bl3795065053823578884t_unit @ E @ V ) )
=> ( ( member_v @ W @ ( sCC_Bl4645233313691564917t_unit @ E ) )
=> ( ~ ( member_v @ W @ ( sCC_Bl157864678168468314t_unit @ E ) )
=> ( sCC_Bl1748261141445803503t_unit @ successors @ V
@ ( sCC_Bl48393358579903213t_unit
@ ^ [Uu: v > set_v,X3: v] : ( if_set_v @ ( X3 = V ) @ ( sup_sup_set_v @ ( sCC_Bl3795065053823578884t_unit @ ( sCC_Bloemen_unite_v @ V @ W @ E ) @ V ) @ ( insert_v @ W @ bot_bot_set_v ) ) @ ( sCC_Bl3795065053823578884t_unit @ ( sCC_Bloemen_unite_v @ V @ W @ E ) @ X3 ) )
@ ( sCC_Bloemen_unite_v @ V @ W @ E ) ) ) ) ) ) ) ) ).
% pre_dfss_unite_pre_dfss
thf(fact_1271_pre__dfss__post__dfs__pre__dfss,axiom,
! [V: v,E: sCC_Bl1394983891496994913t_unit,W: v] :
( ( sCC_Bl1748261141445803503t_unit @ successors @ V @ E )
=> ( ( member_v @ W @ ( successors @ V ) )
=> ( ~ ( member_v @ W @ ( sCC_Bl4645233313691564917t_unit @ E ) )
=> ( ( sCC_Bl8953792750115413617t_unit @ successors @ W @ E @ ( sCC_Bloemen_dfs_v @ successors @ W @ E ) )
=> ( sCC_Bl1748261141445803503t_unit @ successors @ V
@ ( sCC_Bl48393358579903213t_unit
@ ^ [Uu: v > set_v,X3: v] : ( if_set_v @ ( X3 = V ) @ ( sup_sup_set_v @ ( sCC_Bl3795065053823578884t_unit @ ( sCC_Bloemen_dfs_v @ successors @ W @ E ) @ V ) @ ( insert_v @ W @ bot_bot_set_v ) ) @ ( sCC_Bl3795065053823578884t_unit @ ( sCC_Bloemen_dfs_v @ successors @ W @ E ) @ X3 ) )
@ ( sCC_Bloemen_dfs_v @ successors @ W @ E ) ) ) ) ) ) ) ).
% pre_dfss_post_dfs_pre_dfss
thf(fact_1272_pre__dfs__implies__post__dfs,axiom,
! [V: v,E: sCC_Bl1394983891496994913t_unit] :
( ( sCC_Bl36166008131615352t_unit @ successors @ V @ E )
=> ( ( accp_S2303753412255344476t_unit @ ( sCC_Bl907557413677168252_rel_v @ successors ) @ ( sum_In526841707622398774t_unit @ ( produc3862955338007567901t_unit @ V @ E ) ) )
=> ( ( sCC_Bl6082031138996704384t_unit @ successors @ V
@ ( sCC_Bl7876664385711583351t_unit
@ ^ [Uu: list_v] : ( cons_v @ V @ ( sCC_Bl9201514103433284750t_unit @ E ) )
@ ( sCC_Bl349061681862590396t_unit
@ ^ [Uu: list_v] : ( cons_v @ V @ ( sCC_Bl8828226123343373779t_unit @ E ) )
@ ( sCC_Bl7870604408699998558t_unit
@ ^ [Uu: set_v] : ( sup_sup_set_v @ ( sCC_Bl4645233313691564917t_unit @ E ) @ ( insert_v @ V @ bot_bot_set_v ) )
@ E ) ) )
@ ( sCC_Bloemen_dfss_v @ successors @ V
@ ( sCC_Bl7876664385711583351t_unit
@ ^ [Uu: list_v] : ( cons_v @ V @ ( sCC_Bl9201514103433284750t_unit @ E ) )
@ ( sCC_Bl349061681862590396t_unit
@ ^ [Uu: list_v] : ( cons_v @ V @ ( sCC_Bl8828226123343373779t_unit @ E ) )
@ ( sCC_Bl7870604408699998558t_unit
@ ^ [Uu: set_v] : ( sup_sup_set_v @ ( sCC_Bl4645233313691564917t_unit @ E ) @ ( insert_v @ V @ bot_bot_set_v ) )
@ E ) ) ) ) )
=> ( sCC_Bl8953792750115413617t_unit @ successors @ V @ E @ ( sCC_Bloemen_dfs_v @ successors @ V @ E ) ) ) ) ) ).
% pre_dfs_implies_post_dfs
thf(fact_1273_dfs__dfss_Odomintros_I1_J,axiom,
! [V: v,E: sCC_Bl1394983891496994913t_unit] :
( ( accp_S2303753412255344476t_unit @ ( sCC_Bl907557413677168252_rel_v @ successors )
@ ( sum_In5289330923152326972t_unit
@ ( produc3862955338007567901t_unit @ V
@ ( sCC_Bl7876664385711583351t_unit
@ ^ [Uu: list_v] : ( cons_v @ V @ ( sCC_Bl9201514103433284750t_unit @ E ) )
@ ( sCC_Bl349061681862590396t_unit
@ ^ [Uu: list_v] : ( cons_v @ V @ ( sCC_Bl8828226123343373779t_unit @ E ) )
@ ( sCC_Bl7870604408699998558t_unit
@ ^ [Uu: set_v] : ( insert_v @ V @ ( sCC_Bl4645233313691564917t_unit @ E ) )
@ E ) ) ) ) ) )
=> ( accp_S2303753412255344476t_unit @ ( sCC_Bl907557413677168252_rel_v @ successors ) @ ( sum_In526841707622398774t_unit @ ( produc3862955338007567901t_unit @ V @ E ) ) ) ) ).
% dfs_dfss.domintros(1)
thf(fact_1274_dfs_Opsimps,axiom,
! [V: v,E: sCC_Bl1394983891496994913t_unit] :
( ( accp_S2303753412255344476t_unit @ ( sCC_Bl907557413677168252_rel_v @ successors ) @ ( sum_In526841707622398774t_unit @ ( produc3862955338007567901t_unit @ V @ E ) ) )
=> ( ( sCC_Bloemen_dfs_v @ successors @ V @ E )
= ( if_SCC4926449794303880475t_unit
@ ( V
= ( hd_v
@ ( sCC_Bl8828226123343373779t_unit
@ ( sCC_Bloemen_dfss_v @ successors @ V
@ ( sCC_Bl7876664385711583351t_unit
@ ^ [Uu: list_v] : ( cons_v @ V @ ( sCC_Bl9201514103433284750t_unit @ E ) )
@ ( sCC_Bl349061681862590396t_unit
@ ^ [Uu: list_v] : ( cons_v @ V @ ( sCC_Bl8828226123343373779t_unit @ E ) )
@ ( sCC_Bl7870604408699998558t_unit
@ ^ [Uu: set_v] : ( sup_sup_set_v @ ( sCC_Bl4645233313691564917t_unit @ E ) @ ( insert_v @ V @ bot_bot_set_v ) )
@ E ) ) ) ) ) ) )
@ ( sCC_Bl7876664385711583351t_unit
@ ^ [Uu: list_v] :
( tl_v
@ ( sCC_Bl9201514103433284750t_unit
@ ( sCC_Bloemen_dfss_v @ successors @ V
@ ( sCC_Bl7876664385711583351t_unit
@ ^ [Uv: list_v] : ( cons_v @ V @ ( sCC_Bl9201514103433284750t_unit @ E ) )
@ ( sCC_Bl349061681862590396t_unit
@ ^ [Uv: list_v] : ( cons_v @ V @ ( sCC_Bl8828226123343373779t_unit @ E ) )
@ ( sCC_Bl7870604408699998558t_unit
@ ^ [Uv: set_v] : ( sup_sup_set_v @ ( sCC_Bl4645233313691564917t_unit @ E ) @ ( insert_v @ V @ bot_bot_set_v ) )
@ E ) ) ) ) ) )
@ ( sCC_Bl349061681862590396t_unit
@ ^ [Uu: list_v] :
( tl_v
@ ( sCC_Bl8828226123343373779t_unit
@ ( sCC_Bloemen_dfss_v @ successors @ V
@ ( sCC_Bl7876664385711583351t_unit
@ ^ [Uv: list_v] : ( cons_v @ V @ ( sCC_Bl9201514103433284750t_unit @ E ) )
@ ( sCC_Bl349061681862590396t_unit
@ ^ [Uv: list_v] : ( cons_v @ V @ ( sCC_Bl8828226123343373779t_unit @ E ) )
@ ( sCC_Bl7870604408699998558t_unit
@ ^ [Uv: set_v] : ( sup_sup_set_v @ ( sCC_Bl4645233313691564917t_unit @ E ) @ ( insert_v @ V @ bot_bot_set_v ) )
@ E ) ) ) ) ) )
@ ( sCC_Bl2708505634401380163t_unit
@ ^ [Uu: set_v] :
( sup_sup_set_v
@ ( sCC_Bl157864678168468314t_unit
@ ( sCC_Bloemen_dfss_v @ successors @ V
@ ( sCC_Bl7876664385711583351t_unit
@ ^ [Uv: list_v] : ( cons_v @ V @ ( sCC_Bl9201514103433284750t_unit @ E ) )
@ ( sCC_Bl349061681862590396t_unit
@ ^ [Uv: list_v] : ( cons_v @ V @ ( sCC_Bl8828226123343373779t_unit @ E ) )
@ ( sCC_Bl7870604408699998558t_unit
@ ^ [Uv: set_v] : ( sup_sup_set_v @ ( sCC_Bl4645233313691564917t_unit @ E ) @ ( insert_v @ V @ bot_bot_set_v ) )
@ E ) ) ) ) )
@ ( sCC_Bl1280885523602775798t_unit
@ ( sCC_Bloemen_dfss_v @ successors @ V
@ ( sCC_Bl7876664385711583351t_unit
@ ^ [Uv: list_v] : ( cons_v @ V @ ( sCC_Bl9201514103433284750t_unit @ E ) )
@ ( sCC_Bl349061681862590396t_unit
@ ^ [Uv: list_v] : ( cons_v @ V @ ( sCC_Bl8828226123343373779t_unit @ E ) )
@ ( sCC_Bl7870604408699998558t_unit
@ ^ [Uv: set_v] : ( sup_sup_set_v @ ( sCC_Bl4645233313691564917t_unit @ E ) @ ( insert_v @ V @ bot_bot_set_v ) )
@ E ) ) ) )
@ V ) )
@ ( sCC_Bl6816368539212994290t_unit
@ ^ [Uu: set_set_v] :
( sup_sup_set_set_v
@ ( sCC_Bl2536197123907397897t_unit
@ ( sCC_Bloemen_dfss_v @ successors @ V
@ ( sCC_Bl7876664385711583351t_unit
@ ^ [Uv: list_v] : ( cons_v @ V @ ( sCC_Bl9201514103433284750t_unit @ E ) )
@ ( sCC_Bl349061681862590396t_unit
@ ^ [Uv: list_v] : ( cons_v @ V @ ( sCC_Bl8828226123343373779t_unit @ E ) )
@ ( sCC_Bl7870604408699998558t_unit
@ ^ [Uv: set_v] : ( sup_sup_set_v @ ( sCC_Bl4645233313691564917t_unit @ E ) @ ( insert_v @ V @ bot_bot_set_v ) )
@ E ) ) ) ) )
@ ( insert_set_v
@ ( sCC_Bl1280885523602775798t_unit
@ ( sCC_Bloemen_dfss_v @ successors @ V
@ ( sCC_Bl7876664385711583351t_unit
@ ^ [Uv: list_v] : ( cons_v @ V @ ( sCC_Bl9201514103433284750t_unit @ E ) )
@ ( sCC_Bl349061681862590396t_unit
@ ^ [Uv: list_v] : ( cons_v @ V @ ( sCC_Bl8828226123343373779t_unit @ E ) )
@ ( sCC_Bl7870604408699998558t_unit
@ ^ [Uv: set_v] : ( sup_sup_set_v @ ( sCC_Bl4645233313691564917t_unit @ E ) @ ( insert_v @ V @ bot_bot_set_v ) )
@ E ) ) ) )
@ V )
@ bot_bot_set_set_v ) )
@ ( sCC_Bloemen_dfss_v @ successors @ V
@ ( sCC_Bl7876664385711583351t_unit
@ ^ [Uu: list_v] : ( cons_v @ V @ ( sCC_Bl9201514103433284750t_unit @ E ) )
@ ( sCC_Bl349061681862590396t_unit
@ ^ [Uu: list_v] : ( cons_v @ V @ ( sCC_Bl8828226123343373779t_unit @ E ) )
@ ( sCC_Bl7870604408699998558t_unit
@ ^ [Uu: set_v] : ( sup_sup_set_v @ ( sCC_Bl4645233313691564917t_unit @ E ) @ ( insert_v @ V @ bot_bot_set_v ) )
@ E ) ) ) ) ) ) ) )
@ ( sCC_Bl7876664385711583351t_unit
@ ^ [Uu: list_v] :
( tl_v
@ ( sCC_Bl9201514103433284750t_unit
@ ( sCC_Bloemen_dfss_v @ successors @ V
@ ( sCC_Bl7876664385711583351t_unit
@ ^ [Uv: list_v] : ( cons_v @ V @ ( sCC_Bl9201514103433284750t_unit @ E ) )
@ ( sCC_Bl349061681862590396t_unit
@ ^ [Uv: list_v] : ( cons_v @ V @ ( sCC_Bl8828226123343373779t_unit @ E ) )
@ ( sCC_Bl7870604408699998558t_unit
@ ^ [Uv: set_v] : ( sup_sup_set_v @ ( sCC_Bl4645233313691564917t_unit @ E ) @ ( insert_v @ V @ bot_bot_set_v ) )
@ E ) ) ) ) ) )
@ ( sCC_Bloemen_dfss_v @ successors @ V
@ ( sCC_Bl7876664385711583351t_unit
@ ^ [Uu: list_v] : ( cons_v @ V @ ( sCC_Bl9201514103433284750t_unit @ E ) )
@ ( sCC_Bl349061681862590396t_unit
@ ^ [Uu: list_v] : ( cons_v @ V @ ( sCC_Bl8828226123343373779t_unit @ E ) )
@ ( sCC_Bl7870604408699998558t_unit
@ ^ [Uu: set_v] : ( sup_sup_set_v @ ( sCC_Bl4645233313691564917t_unit @ E ) @ ( insert_v @ V @ bot_bot_set_v ) )
@ E ) ) ) ) ) ) ) ) ).
% dfs.psimps
thf(fact_1275_unite__S__equal,axiom,
! [E: sCC_Bl1394983891496994913t_unit,W: v,V: v] :
( ( sCC_Bl9196236973127232072t_unit @ successors @ E )
=> ( ( member_v @ W @ ( successors @ V ) )
=> ( ~ ( member_v @ W @ ( sCC_Bl3795065053823578884t_unit @ E @ V ) )
=> ( ( member_v @ W @ ( sCC_Bl4645233313691564917t_unit @ E ) )
=> ( ~ ( member_v @ W @ ( sCC_Bl157864678168468314t_unit @ E ) )
=> ( ( comple2307003700295860064_set_v
@ ( collect_set_v
@ ^ [Uu: set_v] :
? [N3: v] :
( ( Uu
= ( sCC_Bl1280885523602775798t_unit @ ( sCC_Bloemen_unite_v @ V @ W @ E ) @ N3 ) )
& ( member_v @ N3 @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ ( sCC_Bloemen_unite_v @ V @ W @ E ) ) ) ) ) ) )
= ( comple2307003700295860064_set_v
@ ( collect_set_v
@ ^ [Uu: set_v] :
? [N3: v] :
( ( Uu
= ( sCC_Bl1280885523602775798t_unit @ E @ N3 ) )
& ( member_v @ N3 @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E ) ) ) ) ) ) ) ) ) ) ) ) ).
% unite_S_equal
thf(fact_1276_unite__def,axiom,
( sCC_Bloemen_unite_v
= ( ^ [V3: v,W2: v,E8: sCC_Bl1394983891496994913t_unit] :
( sCC_Bl349061681862590396t_unit
@ ^ [Uu: list_v] :
( dropWhile_v
@ ^ [X3: v] :
~ ( member_v @ W2 @ ( sCC_Bl1280885523602775798t_unit @ E8 @ X3 ) )
@ ( sCC_Bl8828226123343373779t_unit @ E8 ) )
@ ( sCC_Bl3155122997657187039t_unit
@ ^ [Uu: v > set_v,X3: v] :
( if_set_v
@ ( member_v @ X3
@ ( comple2307003700295860064_set_v
@ ( collect_set_v
@ ^ [Uv: set_v] :
? [Y3: v] :
( ( Uv
= ( sCC_Bl1280885523602775798t_unit @ E8 @ Y3 ) )
& ( member_v @ Y3
@ ( sup_sup_set_v
@ ( set_v2
@ ( takeWhile_v
@ ^ [Z2: v] :
~ ( member_v @ W2 @ ( sCC_Bl1280885523602775798t_unit @ E8 @ Z2 ) )
@ ( sCC_Bl8828226123343373779t_unit @ E8 ) ) )
@ ( insert_v
@ ( hd_v
@ ( dropWhile_v
@ ^ [Z2: v] :
~ ( member_v @ W2 @ ( sCC_Bl1280885523602775798t_unit @ E8 @ Z2 ) )
@ ( sCC_Bl8828226123343373779t_unit @ E8 ) ) )
@ bot_bot_set_v ) ) ) ) ) ) )
@ ( comple2307003700295860064_set_v
@ ( collect_set_v
@ ^ [Uv: set_v] :
? [Y3: v] :
( ( Uv
= ( sCC_Bl1280885523602775798t_unit @ E8 @ Y3 ) )
& ( member_v @ Y3
@ ( sup_sup_set_v
@ ( set_v2
@ ( takeWhile_v
@ ^ [Z2: v] :
~ ( member_v @ W2 @ ( sCC_Bl1280885523602775798t_unit @ E8 @ Z2 ) )
@ ( sCC_Bl8828226123343373779t_unit @ E8 ) ) )
@ ( insert_v
@ ( hd_v
@ ( dropWhile_v
@ ^ [Z2: v] :
~ ( member_v @ W2 @ ( sCC_Bl1280885523602775798t_unit @ E8 @ Z2 ) )
@ ( sCC_Bl8828226123343373779t_unit @ E8 ) ) )
@ bot_bot_set_v ) ) ) ) ) )
@ ( sCC_Bl1280885523602775798t_unit @ E8 @ X3 ) )
@ E8 ) ) ) ) ).
% unite_def
% Helper facts (6)
thf(help_fChoice_1_1_fChoice_001tf__v_T,axiom,
! [P: v > $o] :
( ( P @ ( fChoice_v @ P ) )
= ( ? [X6: v] : ( P @ X6 ) ) ) ).
thf(help_If_2_1_If_001t__Set__Oset_Itf__v_J_T,axiom,
! [X2: set_v,Y2: set_v] :
( ( if_set_v @ $false @ X2 @ Y2 )
= Y2 ) ).
thf(help_If_1_1_If_001t__Set__Oset_Itf__v_J_T,axiom,
! [X2: set_v,Y2: set_v] :
( ( if_set_v @ $true @ X2 @ Y2 )
= X2 ) ).
thf(help_If_3_1_If_001t__SCC____Bloemen____Sequential__Oenv__Oenv____ext_Itf__v_Mt__Product____Type__Ounit_J_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__SCC____Bloemen____Sequential__Oenv__Oenv____ext_Itf__v_Mt__Product____Type__Ounit_J_T,axiom,
! [X2: sCC_Bl1394983891496994913t_unit,Y2: sCC_Bl1394983891496994913t_unit] :
( ( if_SCC4926449794303880475t_unit @ $false @ X2 @ Y2 )
= Y2 ) ).
thf(help_If_1_1_If_001t__SCC____Bloemen____Sequential__Oenv__Oenv____ext_Itf__v_Mt__Product____Type__Ounit_J_T,axiom,
! [X2: sCC_Bl1394983891496994913t_unit,Y2: sCC_Bl1394983891496994913t_unit] :
( ( if_SCC4926449794303880475t_unit @ $true @ X2 @ Y2 )
= X2 ) ).
% Conjectures (1)
thf(conj_0,conjecture,
sCC_Bl6082031138996704384t_unit @ successors @ va @ ea @ ( sCC_Bloemen_dfss_v @ successors @ va @ ea ) ).
%------------------------------------------------------------------------------