TPTP Problem File: SLH0410^1.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_02248_077086__6261568_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1468 ( 666 unt; 184 typ; 0 def)
% Number of atoms : 3600 (1467 equ; 0 cnn)
% Maximal formula atoms : 10 ( 2 avg)
% Number of connectives : 11767 ( 466 ~; 58 |; 337 &;9503 @)
% ( 0 <=>;1403 =>; 0 <=; 0 <~>)
% Maximal formula depth : 23 ( 6 avg)
% Number of types : 22 ( 21 usr)
% Number of type conns : 555 ( 555 >; 0 *; 0 +; 0 <<)
% Number of symbols : 166 ( 163 usr; 20 con; 0-9 aty)
% Number of variables : 3695 ( 223 ^;3296 !; 176 ?;3695 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 08:53:58.007
%------------------------------------------------------------------------------
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thf(sy_c_SCC__Bloemen__Sequential_Ograph_001tf__v,type,
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thf(sy_c_SCC__Bloemen__Sequential_Ograph_Ounite_001tf__v,type,
sCC_Bloemen_unite_v: v > v > sCC_Bl1394983891496994913t_unit > sCC_Bl1394983891496994913t_unit ).
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thf(sy_c_SCC__Bloemen__Sequential_Oinit__env_001tf__v,type,
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thf(sy_c_SCC__Bloemen__Sequential_Oprecedes_001tf__v,type,
sCC_Bl4022239298816431255edes_v: v > v > list_v > $o ).
thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_Itf__v_Mtf__v_J,type,
collec140062887454715474od_v_v: ( product_prod_v_v > $o ) > set_Product_prod_v_v ).
thf(sy_c_Set_OCollect_001t__Set__Oset_Itf__v_J,type,
collect_set_v: ( set_v > $o ) > set_set_v ).
thf(sy_c_Set_OCollect_001tf__v,type,
collect_v: ( v > $o ) > set_v ).
thf(sy_c_Set_Oinsert_001t__Product____Type__Oprod_Itf__v_Mtf__v_J,type,
insert1338601472111419319od_v_v: product_prod_v_v > set_Product_prod_v_v > set_Product_prod_v_v ).
thf(sy_c_Set_Oinsert_001t__Set__Oset_Itf__v_J,type,
insert_set_v2: set_v > set_set_v > set_set_v ).
thf(sy_c_Set_Oinsert_001tf__v,type,
insert_v2: v > set_v > set_v ).
thf(sy_c_Set_Ois__singleton_001t__Product____Type__Oprod_Itf__v_Mtf__v_J,type,
is_sin9198872032823709915od_v_v: set_Product_prod_v_v > $o ).
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_Othe__elem_001t__Product____Type__Oprod_Itf__v_Mtf__v_J,type,
the_el5392834299063928540od_v_v: set_Product_prod_v_v > product_prod_v_v ).
thf(sy_c_Set_Othe__elem_001t__Set__Oset_Itf__v_J,type,
the_elem_set_v: set_set_v > set_v ).
thf(sy_c_Set_Othe__elem_001tf__v,type,
the_elem_v: set_v > v ).
thf(sy_c_Sum__Type_OInl_001t__Product____Type__Oprod_Itf__v_Mt__SCC____Bloemen____Sequential__Oenv__Oenv____ext_Itf__v_Mt__Product____Type__Ounit_J_J_001t__Product____Type__Oprod_Itf__v_Mt__SCC____Bloemen____Sequential__Oenv__Oenv____ext_Itf__v_Mt__Product____Type__Ounit_J_J,type,
sum_In526841707622398774t_unit: produc5741669702376414499t_unit > sum_su8181647976486975269t_unit ).
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sum_In5289330923152326972t_unit: produc5741669702376414499t_unit > sum_su8181647976486975269t_unit ).
thf(sy_c_Wellfounded_Oaccp_001t__Sum____Type__Osum_It__Product____Type__Oprod_Itf__v_Mt__SCC____Bloemen____Sequential__Oenv__Oenv____ext_Itf__v_Mt__Product____Type__Ounit_J_J_Mt__Product____Type__Oprod_Itf__v_Mt__SCC____Bloemen____Sequential__Oenv__Oenv____ext_Itf__v_Mt__Product____Type__Ounit_J_J_J,type,
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thf(sy_c_fChoice_001tf__v,type,
fChoice_v: ( v > $o ) > v ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__v_Mtf__v_J_Mt__Product____Type__Oprod_Itf__v_Mtf__v_J_J,type,
member3038538357316246288od_v_v: produc206430290419586791od_v_v > set_Pr2149350503807050951od_v_v > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_Itf__v_Mt__SCC____Bloemen____Sequential__Oenv__Oenv____ext_Itf__v_Mt__Product____Type__Ounit_J_J,type,
member7924940910754673978t_unit: produc5741669702376414499t_unit > set_Pr6425124735969554649t_unit > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_Itf__v_Mtf__v_J,type,
member7453568604450474000od_v_v: product_prod_v_v > set_Product_prod_v_v > $o ).
thf(sy_c_member_001t__Set__Oset_It__Product____Type__Oprod_Itf__v_Mtf__v_J_J,type,
member8406446414694345712od_v_v: set_Product_prod_v_v > set_se8455005133513928103od_v_v > $o ).
thf(sy_c_member_001t__Set__Oset_Itf__v_J,type,
member_set_v: set_v > set_set_v > $o ).
thf(sy_c_member_001tf__v,type,
member_v: v > set_v > $o ).
thf(sy_v_e,type,
e: sCC_Bl1394983891496994913t_unit ).
thf(sy_v_e1,type,
e1: sCC_Bl1394983891496994913t_unit ).
thf(sy_v_e_H,type,
e2: sCC_Bl1394983891496994913t_unit ).
thf(sy_v_e_H_H,type,
e3: sCC_Bl1394983891496994913t_unit ).
thf(sy_v_ns____,type,
ns: list_v ).
thf(sy_v_successors,type,
successors: v > set_v ).
thf(sy_v_v,type,
v2: v ).
thf(sy_v_vertices,type,
vertices: set_v ).
% Relevant facts (1273)
thf(fact_0_ns_I2_J,axiom,
( ( sCC_Bl8828226123343373779t_unit @ e2 )
!= nil_v ) ).
% ns(2)
thf(fact_1_True,axiom,
( v2
= ( hd_v @ ( sCC_Bl8828226123343373779t_unit @ e2 ) ) ) ).
% True
thf(fact_2_dfs__dfss__rel_Ocong,axiom,
sCC_Bl907557413677168252_rel_v = sCC_Bl907557413677168252_rel_v ).
% dfs_dfss_rel.cong
thf(fact_3_ns_I1_J,axiom,
( ( sCC_Bl8828226123343373779t_unit @ e1 )
= ( append_v @ ns @ ( sCC_Bl8828226123343373779t_unit @ e2 ) ) ) ).
% ns(1)
thf(fact_4_sub__env__trans,axiom,
! [E: sCC_Bl1394983891496994913t_unit,E2: sCC_Bl1394983891496994913t_unit,E3: sCC_Bl1394983891496994913t_unit] :
( ( sCC_Bl5768913643336123637t_unit @ E @ E2 )
=> ( ( sCC_Bl5768913643336123637t_unit @ E2 @ E3 )
=> ( sCC_Bl5768913643336123637t_unit @ E @ E3 ) ) ) ).
% sub_env_trans
thf(fact_5__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062ns_O_A_092_060lbrakk_062stack_Ae1_A_061_Ans_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,
~ ( ? [Ns: list_v] :
( ( sCC_Bl8828226123343373779t_unit @ e1 )
= ( append_v @ Ns @ ( sCC_Bl8828226123343373779t_unit @ e2 ) ) )
=> ( ( sCC_Bl8828226123343373779t_unit @ e2 )
= nil_v ) ) ).
% \<open>\<And>thesis. (\<And>ns. \<lbrakk>stack e1 = ns @ stack e'; stack e' \<noteq> []\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_6_fold__congs_I7_J,axiom,
! [R: sCC_Bl1394983891496994913t_unit,R2: sCC_Bl1394983891496994913t_unit,V: list_v,F: list_v > list_v,F2: list_v > list_v] :
( ( R = R2 )
=> ( ( ( sCC_Bl8828226123343373779t_unit @ R2 )
= V )
=> ( ! [V2: list_v] :
( ( V = V2 )
=> ( ( F @ V2 )
= ( F2 @ V2 ) ) )
=> ( ( sCC_Bl349061681862590396t_unit @ F @ R )
= ( sCC_Bl349061681862590396t_unit @ F2 @ R2 ) ) ) ) ) ).
% fold_congs(7)
thf(fact_7_unfold__congs_I7_J,axiom,
! [R: sCC_Bl1394983891496994913t_unit,R2: sCC_Bl1394983891496994913t_unit,V: list_v,F: list_v > list_v,F2: list_v > list_v] :
( ( R = R2 )
=> ( ( ( sCC_Bl8828226123343373779t_unit @ R2 )
= V )
=> ( ! [V2: list_v] :
( ( V2 = V )
=> ( ( F @ V2 )
= ( F2 @ V2 ) ) )
=> ( ( sCC_Bl349061681862590396t_unit @ F @ R )
= ( sCC_Bl349061681862590396t_unit @ F2 @ R2 ) ) ) ) ) ).
% unfold_congs(7)
thf(fact_8_distinct__union,axiom,
! [Xs: list_v,Ys: list_v] :
( ( distinct_v @ ( union_v @ Xs @ Ys ) )
= ( distinct_v @ Ys ) ) ).
% distinct_union
thf(fact_9__C3_C,axiom,
sCC_Bl6082031138996704384t_unit @ successors @ v2 @ e1 @ e2 ).
% "3"
thf(fact_10_distinct_Osimps_I1_J,axiom,
distinct_v @ nil_v ).
% distinct.simps(1)
thf(fact_11_e_H__def,axiom,
( e2
= ( sCC_Bloemen_dfss_v @ successors @ v2 @ e1 ) ) ).
% e'_def
thf(fact_12_precedes__trans,axiom,
! [X: v,Y: v,Xs: list_v,Z: v] :
( ( sCC_Bl4022239298816431255edes_v @ X @ Y @ Xs )
=> ( ( sCC_Bl4022239298816431255edes_v @ Y @ Z @ Xs )
=> ( ( distinct_v @ Xs )
=> ( sCC_Bl4022239298816431255edes_v @ X @ Z @ Xs ) ) ) ) ).
% precedes_trans
thf(fact_13_precedes__antisym,axiom,
! [X: v,Y: v,Xs: list_v] :
( ( sCC_Bl4022239298816431255edes_v @ X @ Y @ Xs )
=> ( ( sCC_Bl4022239298816431255edes_v @ Y @ X @ Xs )
=> ( ( distinct_v @ Xs )
=> ( X = Y ) ) ) ) ).
% precedes_antisym
thf(fact_14_select__convs_I7_J,axiom,
! [Root: v,S: v > set_v,Explored: set_v,Visited: set_v,Vsuccs: v > set_v,Sccs: set_set_v,Stack: list_v,Cstack: list_v,More: product_unit] :
( ( sCC_Bl8828226123343373779t_unit @ ( sCC_Bl8064756265740546429t_unit @ Root @ S @ Explored @ Visited @ Vsuccs @ Sccs @ Stack @ Cstack @ More ) )
= Stack ) ).
% select_convs(7)
thf(fact_15_distinct__insert,axiom,
! [X: v,Xs: list_v] :
( ( distinct_v @ ( insert_v @ X @ Xs ) )
= ( distinct_v @ Xs ) ) ).
% distinct_insert
thf(fact_16__092_060open_062ns_A_092_060noteq_062_A_091_093_092_060close_062,axiom,
ns != nil_v ).
% \<open>ns \<noteq> []\<close>
thf(fact_17__092_060open_062hd_Ans_A_061_Av_092_060close_062,axiom,
( ( hd_v @ ns )
= v2 ) ).
% \<open>hd ns = v\<close>
thf(fact_18_append_Oassoc,axiom,
! [A: list_v,B: list_v,C: list_v] :
( ( append_v @ ( append_v @ A @ B ) @ C )
= ( append_v @ A @ ( append_v @ B @ C ) ) ) ).
% append.assoc
thf(fact_19_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_20_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_21_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_22_append_Oright__neutral,axiom,
! [A: list_v] :
( ( append_v @ A @ nil_v )
= A ) ).
% append.right_neutral
thf(fact_23_append__Nil2,axiom,
! [Xs: list_v] :
( ( append_v @ Xs @ nil_v )
= Xs ) ).
% append_Nil2
thf(fact_24_append__self__conv,axiom,
! [Xs: list_v,Ys: list_v] :
( ( ( append_v @ Xs @ Ys )
= Xs )
= ( Ys = nil_v ) ) ).
% append_self_conv
thf(fact_25_self__append__conv,axiom,
! [Y: list_v,Ys: list_v] :
( ( Y
= ( append_v @ Y @ Ys ) )
= ( Ys = nil_v ) ) ).
% self_append_conv
thf(fact_26_append__self__conv2,axiom,
! [Xs: list_v,Ys: list_v] :
( ( ( append_v @ Xs @ Ys )
= Ys )
= ( Xs = nil_v ) ) ).
% append_self_conv2
thf(fact_27_self__append__conv2,axiom,
! [Y: list_v,Xs: list_v] :
( ( Y
= ( append_v @ Xs @ Y ) )
= ( Xs = nil_v ) ) ).
% self_append_conv2
thf(fact_28_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_29_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_30_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_31_update__convs_I7_J,axiom,
! [Stack2: list_v > list_v,Root: v,S: v > set_v,Explored: set_v,Visited: set_v,Vsuccs: v > set_v,Sccs: set_set_v,Stack: list_v,Cstack: list_v,More: product_unit] :
( ( sCC_Bl349061681862590396t_unit @ Stack2 @ ( sCC_Bl8064756265740546429t_unit @ Root @ S @ Explored @ Visited @ Vsuccs @ Sccs @ Stack @ Cstack @ More ) )
= ( sCC_Bl8064756265740546429t_unit @ Root @ S @ Explored @ Visited @ Vsuccs @ Sccs @ ( Stack2 @ Stack ) @ Cstack @ More ) ) ).
% update_convs(7)
thf(fact_32_append__Nil,axiom,
! [Ys: list_v] :
( ( append_v @ nil_v @ Ys )
= Ys ) ).
% append_Nil
thf(fact_33_append_Oleft__neutral,axiom,
! [A: list_v] :
( ( append_v @ nil_v @ A )
= A ) ).
% append.left_neutral
thf(fact_34_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_35_eq__Nil__appendI,axiom,
! [Xs: list_v,Ys: list_v] :
( ( Xs = Ys )
=> ( Xs
= ( append_v @ nil_v @ Ys ) ) ) ).
% eq_Nil_appendI
thf(fact_36_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_37_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_38_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_39_mem__Collect__eq,axiom,
! [A: v,P: v > $o] :
( ( member_v @ A @ ( collect_v @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_40_mem__Collect__eq,axiom,
! [A: product_prod_v_v,P: product_prod_v_v > $o] :
( ( member7453568604450474000od_v_v @ A @ ( collec140062887454715474od_v_v @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_41_mem__Collect__eq,axiom,
! [A: set_v,P: set_v > $o] :
( ( member_set_v @ A @ ( collect_set_v @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_42_Collect__mem__eq,axiom,
! [A2: set_v] :
( ( collect_v
@ ^ [X2: v] : ( member_v @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_43_Collect__mem__eq,axiom,
! [A2: set_Product_prod_v_v] :
( ( collec140062887454715474od_v_v
@ ^ [X2: product_prod_v_v] : ( member7453568604450474000od_v_v @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_44_Collect__mem__eq,axiom,
! [A2: set_set_v] :
( ( collect_set_v
@ ^ [X2: set_v] : ( member_set_v @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_45_Collect__cong,axiom,
! [P: set_v > $o,Q: set_v > $o] :
( ! [X3: set_v] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collect_set_v @ P )
= ( collect_set_v @ Q ) ) ) ).
% Collect_cong
thf(fact_46_precedes__append__left,axiom,
! [X: v,Y: v,Xs: list_v,Ys: list_v] :
( ( sCC_Bl4022239298816431255edes_v @ X @ Y @ Xs )
=> ( sCC_Bl4022239298816431255edes_v @ X @ Y @ ( append_v @ Ys @ Xs ) ) ) ).
% precedes_append_left
thf(fact_47_precedes__append__right,axiom,
! [X: v,Y: v,Xs: list_v,Ys: list_v] :
( ( sCC_Bl4022239298816431255edes_v @ X @ Y @ Xs )
=> ( sCC_Bl4022239298816431255edes_v @ X @ Y @ ( append_v @ Xs @ Ys ) ) ) ).
% precedes_append_right
thf(fact_48_graph_Odfss_Ocong,axiom,
sCC_Bloemen_dfss_v = sCC_Bloemen_dfss_v ).
% graph.dfss.cong
thf(fact_49_graph_Opost__dfss_Ocong,axiom,
sCC_Bl6082031138996704384t_unit = sCC_Bl6082031138996704384t_unit ).
% graph.post_dfss.cong
thf(fact_50_dfs__S__tl__stack_I1_J,axiom,
! [V3: v,E: sCC_Bl1394983891496994913t_unit,E2: sCC_Bl1394983891496994913t_unit] :
( ( sCC_Bl8953792750115413617t_unit @ successors @ V3 @ E @ E2 )
=> ( ( ( sCC_Bl8828226123343373779t_unit @ E )
!= nil_v )
=> ( ( sCC_Bl8828226123343373779t_unit @ E2 )
!= nil_v ) ) ) ).
% dfs_S_tl_stack(1)
thf(fact_51__C1_C,axiom,
sCC_Bl36166008131615352t_unit @ successors @ v2 @ e ).
% "1"
thf(fact_52_wf_H,axiom,
sCC_Bl9196236973127232072t_unit @ successors @ e2 ).
% wf'
thf(fact_53_reachable__end_Ocases,axiom,
! [A1: v,A22: v] :
( ( sCC_Bl770211535891879572_end_v @ successors @ A1 @ A22 )
=> ( ( A22 != A1 )
=> ~ ! [Y2: v] :
( ( sCC_Bl770211535891879572_end_v @ successors @ A1 @ Y2 )
=> ~ ( member_v @ A22 @ ( successors @ Y2 ) ) ) ) ) ).
% reachable_end.cases
thf(fact_54_re__refl,axiom,
! [X: v] : ( sCC_Bl770211535891879572_end_v @ successors @ X @ X ) ).
% re_refl
thf(fact_55_re__succ,axiom,
! [X: v,Y: v,Z: v] :
( ( sCC_Bl770211535891879572_end_v @ successors @ X @ Y )
=> ( ( member_v @ Z @ ( successors @ Y ) )
=> ( sCC_Bl770211535891879572_end_v @ successors @ X @ Z ) ) ) ).
% re_succ
thf(fact_56_reachable__end_Osimps,axiom,
! [A1: v,A22: v] :
( ( sCC_Bl770211535891879572_end_v @ successors @ A1 @ A22 )
= ( ? [X2: v] :
( ( A1 = X2 )
& ( A22 = X2 ) )
| ? [X2: v,Y3: v,Z2: v] :
( ( A1 = X2 )
& ( A22 = Z2 )
& ( sCC_Bl770211535891879572_end_v @ successors @ X2 @ Y3 )
& ( member_v @ Z2 @ ( successors @ Y3 ) ) ) ) ) ).
% reachable_end.simps
thf(fact_57_succ__re,axiom,
! [Y: v,X: v,Z: v] :
( ( member_v @ Y @ ( successors @ X ) )
=> ( ( sCC_Bl770211535891879572_end_v @ successors @ Y @ Z )
=> ( sCC_Bl770211535891879572_end_v @ successors @ X @ Z ) ) ) ).
% succ_re
thf(fact_58_init__env__pre__dfs,axiom,
! [V3: v] : ( sCC_Bl36166008131615352t_unit @ successors @ V3 @ ( sCC_Bl7693227186847904995_env_v @ V3 ) ) ).
% init_env_pre_dfs
thf(fact_59_ra__refl,axiom,
! [X: v,E4: set_Product_prod_v_v] : ( sCC_Bl4291963740693775144ding_v @ successors @ X @ X @ E4 ) ).
% ra_refl
thf(fact_60_ra__trans,axiom,
! [X: v,Y: v,E4: set_Product_prod_v_v,Z: v] :
( ( sCC_Bl4291963740693775144ding_v @ successors @ X @ Y @ E4 )
=> ( ( sCC_Bl4291963740693775144ding_v @ successors @ Y @ Z @ E4 )
=> ( sCC_Bl4291963740693775144ding_v @ successors @ X @ Z @ E4 ) ) ) ).
% ra_trans
thf(fact_61_local_Owf,axiom,
sCC_Bl9196236973127232072t_unit @ successors @ e ).
% local.wf
thf(fact_62_graph_Oreachable__avoiding_Ocong,axiom,
sCC_Bl4291963740693775144ding_v = sCC_Bl4291963740693775144ding_v ).
% graph.reachable_avoiding.cong
thf(fact_63_graph_Oreachable__end_Ocong,axiom,
sCC_Bl770211535891879572_end_v = sCC_Bl770211535891879572_end_v ).
% graph.reachable_end.cong
thf(fact_64_graph_Owf__env_Ocong,axiom,
sCC_Bl9196236973127232072t_unit = sCC_Bl9196236973127232072t_unit ).
% graph.wf_env.cong
thf(fact_65_graph_Opost__dfs_Ocong,axiom,
sCC_Bl8953792750115413617t_unit = sCC_Bl8953792750115413617t_unit ).
% graph.post_dfs.cong
thf(fact_66_graph_Opre__dfs_Ocong,axiom,
sCC_Bl36166008131615352t_unit = sCC_Bl36166008131615352t_unit ).
% graph.pre_dfs.cong
thf(fact_67_dfs__S__hd__stack_I1_J,axiom,
! [E: sCC_Bl1394983891496994913t_unit,V3: v,E2: sCC_Bl1394983891496994913t_unit,N: v] :
( ( sCC_Bl9196236973127232072t_unit @ successors @ E )
=> ( ( sCC_Bl8953792750115413617t_unit @ successors @ V3 @ 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_68_dfs__S__hd__stack_I2_J,axiom,
! [E: sCC_Bl1394983891496994913t_unit,V3: v,E2: sCC_Bl1394983891496994913t_unit,N: v] :
( ( sCC_Bl9196236973127232072t_unit @ successors @ E )
=> ( ( sCC_Bl8953792750115413617t_unit @ successors @ V3 @ 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_69_v,axiom,
~ ( member_v @ v2 @ ( sCC_Bl4645233313691564917t_unit @ e ) ) ).
% v
thf(fact_70_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_71_ra__mono,axiom,
! [X: v,Y: v,E4: set_Product_prod_v_v,E5: set_Product_prod_v_v] :
( ( sCC_Bl4291963740693775144ding_v @ successors @ X @ Y @ E4 )
=> ( ( ord_le7336532860387713383od_v_v @ E5 @ E4 )
=> ( sCC_Bl4291963740693775144ding_v @ successors @ X @ Y @ E5 ) ) ) ).
% ra_mono
thf(fact_72_ra__cases,axiom,
! [X: v,Y: v,E4: set_Product_prod_v_v] :
( ( sCC_Bl4291963740693775144ding_v @ successors @ X @ Y @ E4 )
=> ( ( X = Y )
| ? [Z3: v] :
( ( member_v @ Z3 @ ( successors @ X ) )
& ~ ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ X @ Z3 ) @ E4 )
& ( sCC_Bl4291963740693775144ding_v @ successors @ Z3 @ Y @ E4 ) ) ) ) ).
% ra_cases
thf(fact_73_edge__ra,axiom,
! [Y: v,X: v,E4: set_Product_prod_v_v] :
( ( member_v @ Y @ ( successors @ X ) )
=> ( ~ ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ X @ Y ) @ E4 )
=> ( sCC_Bl4291963740693775144ding_v @ successors @ X @ Y @ E4 ) ) ) ).
% edge_ra
thf(fact_74_reachable__avoiding_Osimps,axiom,
! [A1: v,A22: v,A3: set_Product_prod_v_v] :
( ( sCC_Bl4291963740693775144ding_v @ successors @ A1 @ A22 @ A3 )
= ( ? [X2: v,E6: set_Product_prod_v_v] :
( ( A1 = X2 )
& ( A22 = X2 )
& ( A3 = E6 ) )
| ? [X2: v,Y3: v,E6: set_Product_prod_v_v,Z2: v] :
( ( A1 = X2 )
& ( A22 = Z2 )
& ( A3 = E6 )
& ( sCC_Bl4291963740693775144ding_v @ successors @ X2 @ Y3 @ E6 )
& ( member_v @ Z2 @ ( successors @ Y3 ) )
& ~ ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ Y3 @ Z2 ) @ E6 ) ) ) ) ).
% reachable_avoiding.simps
thf(fact_75_ra__succ,axiom,
! [X: v,Y: v,E4: set_Product_prod_v_v,Z: v] :
( ( sCC_Bl4291963740693775144ding_v @ successors @ X @ Y @ E4 )
=> ( ( member_v @ Z @ ( successors @ Y ) )
=> ( ~ ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ Y @ Z ) @ E4 )
=> ( sCC_Bl4291963740693775144ding_v @ successors @ X @ Z @ E4 ) ) ) ) ).
% ra_succ
thf(fact_76_reachable__avoiding_Ocases,axiom,
! [A1: v,A22: v,A3: set_Product_prod_v_v] :
( ( sCC_Bl4291963740693775144ding_v @ successors @ A1 @ A22 @ A3 )
=> ( ( A22 != A1 )
=> ~ ! [Y2: v] :
( ( sCC_Bl4291963740693775144ding_v @ successors @ A1 @ Y2 @ A3 )
=> ( ( member_v @ A22 @ ( successors @ Y2 ) )
=> ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ Y2 @ A22 ) @ A3 ) ) ) ) ) ).
% reachable_avoiding.cases
thf(fact_77_select__convs_I2_J,axiom,
! [Root: v,S: v > set_v,Explored: set_v,Visited: set_v,Vsuccs: v > set_v,Sccs: set_set_v,Stack: list_v,Cstack: list_v,More: product_unit] :
( ( sCC_Bl1280885523602775798t_unit @ ( sCC_Bl8064756265740546429t_unit @ Root @ S @ Explored @ Visited @ Vsuccs @ Sccs @ Stack @ Cstack @ More ) )
= S ) ).
% select_convs(2)
thf(fact_78_select__convs_I4_J,axiom,
! [Root: v,S: v > set_v,Explored: set_v,Visited: set_v,Vsuccs: v > set_v,Sccs: set_set_v,Stack: list_v,Cstack: list_v,More: product_unit] :
( ( sCC_Bl4645233313691564917t_unit @ ( sCC_Bl8064756265740546429t_unit @ Root @ S @ Explored @ Visited @ Vsuccs @ Sccs @ Stack @ Cstack @ More ) )
= Visited ) ).
% select_convs(4)
thf(fact_79_pre__dfss__pre__dfs,axiom,
! [V3: v,E: sCC_Bl1394983891496994913t_unit,W: v] :
( ( sCC_Bl1748261141445803503t_unit @ successors @ V3 @ E )
=> ( ~ ( member_v @ W @ ( sCC_Bl4645233313691564917t_unit @ E ) )
=> ( ( member_v @ W @ ( successors @ V3 ) )
=> ( sCC_Bl36166008131615352t_unit @ successors @ W @ E ) ) ) ) ).
% pre_dfss_pre_dfs
thf(fact_80_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_81_subsetI,axiom,
! [A2: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ! [X3: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X3 @ A2 )
=> ( member7453568604450474000od_v_v @ X3 @ B2 ) )
=> ( ord_le7336532860387713383od_v_v @ A2 @ B2 ) ) ).
% subsetI
thf(fact_82_subsetI,axiom,
! [A2: set_v,B2: set_v] :
( ! [X3: v] :
( ( member_v @ X3 @ A2 )
=> ( member_v @ X3 @ B2 ) )
=> ( ord_less_eq_set_v @ A2 @ B2 ) ) ).
% subsetI
thf(fact_83_subset__antisym,axiom,
! [A2: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A2 @ B2 )
=> ( ( ord_le7336532860387713383od_v_v @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% subset_antisym
thf(fact_84_subset__antisym,axiom,
! [A2: set_v,B2: set_v] :
( ( ord_less_eq_set_v @ A2 @ B2 )
=> ( ( ord_less_eq_set_v @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% subset_antisym
thf(fact_85_prod_Oinject,axiom,
! [X1: v,X22: v,Y1: v,Y22: v] :
( ( ( product_Pair_v_v @ X1 @ X22 )
= ( product_Pair_v_v @ Y1 @ Y22 ) )
= ( ( X1 = Y1 )
& ( X22 = Y22 ) ) ) ).
% prod.inject
thf(fact_86_prod_Oinject,axiom,
! [X1: v,X22: sCC_Bl1394983891496994913t_unit,Y1: v,Y22: sCC_Bl1394983891496994913t_unit] :
( ( ( produc3862955338007567901t_unit @ X1 @ X22 )
= ( produc3862955338007567901t_unit @ Y1 @ Y22 ) )
= ( ( X1 = Y1 )
& ( X22 = Y22 ) ) ) ).
% prod.inject
thf(fact_87_old_Oprod_Oinject,axiom,
! [A: v,B: v,A4: v,B3: v] :
( ( ( product_Pair_v_v @ A @ B )
= ( product_Pair_v_v @ A4 @ B3 ) )
= ( ( A = A4 )
& ( B = B3 ) ) ) ).
% old.prod.inject
thf(fact_88_old_Oprod_Oinject,axiom,
! [A: v,B: sCC_Bl1394983891496994913t_unit,A4: v,B3: sCC_Bl1394983891496994913t_unit] :
( ( ( produc3862955338007567901t_unit @ A @ B )
= ( produc3862955338007567901t_unit @ A4 @ B3 ) )
= ( ( A = A4 )
& ( B = B3 ) ) ) ).
% old.prod.inject
thf(fact_89_order__refl,axiom,
! [X: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ X @ X ) ).
% order_refl
thf(fact_90_order__refl,axiom,
! [X: set_v] : ( ord_less_eq_set_v @ X @ X ) ).
% order_refl
thf(fact_91_dual__order_Orefl,axiom,
! [A: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ A @ A ) ).
% dual_order.refl
thf(fact_92_dual__order_Orefl,axiom,
! [A: set_v] : ( ord_less_eq_set_v @ A @ A ) ).
% dual_order.refl
thf(fact_93_subrelI,axiom,
! [R: set_Pr6425124735969554649t_unit,S2: set_Pr6425124735969554649t_unit] :
( ! [X3: v,Y2: sCC_Bl1394983891496994913t_unit] :
( ( member7924940910754673978t_unit @ ( produc3862955338007567901t_unit @ X3 @ Y2 ) @ R )
=> ( member7924940910754673978t_unit @ ( produc3862955338007567901t_unit @ X3 @ Y2 ) @ S2 ) )
=> ( ord_le7290744839000465721t_unit @ R @ S2 ) ) ).
% subrelI
thf(fact_94_subrelI,axiom,
! [R: set_Product_prod_v_v,S2: set_Product_prod_v_v] :
( ! [X3: v,Y2: v] :
( ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ X3 @ Y2 ) @ R )
=> ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ X3 @ Y2 ) @ S2 ) )
=> ( ord_le7336532860387713383od_v_v @ R @ S2 ) ) ).
% subrelI
thf(fact_95_re__reachable,axiom,
! [X: v,Y: v] :
( ( sCC_Bl770211535891879572_end_v @ successors @ X @ Y )
=> ( sCC_Bl649662514949026229able_v @ successors @ X @ Y ) ) ).
% re_reachable
thf(fact_96_succ__reachable,axiom,
! [X: v,Y: v,Z: v] :
( ( sCC_Bl649662514949026229able_v @ successors @ X @ Y )
=> ( ( member_v @ Z @ ( successors @ Y ) )
=> ( sCC_Bl649662514949026229able_v @ successors @ X @ Z ) ) ) ).
% succ_reachable
thf(fact_97_reachable__trans,axiom,
! [X: v,Y: v,Z: v] :
( ( sCC_Bl649662514949026229able_v @ successors @ X @ Y )
=> ( ( sCC_Bl649662514949026229able_v @ successors @ Y @ Z )
=> ( sCC_Bl649662514949026229able_v @ successors @ X @ Z ) ) ) ).
% reachable_trans
thf(fact_98_reachable__end__induct,axiom,
! [X: v,Y: v,P: v > v > $o] :
( ( sCC_Bl649662514949026229able_v @ successors @ X @ Y )
=> ( ! [X3: v] : ( P @ X3 @ X3 )
=> ( ! [X3: v,Y2: v,Z3: v] :
( ( P @ X3 @ Y2 )
=> ( ( member_v @ Z3 @ ( successors @ Y2 ) )
=> ( P @ X3 @ Z3 ) ) )
=> ( P @ X @ Y ) ) ) ) ).
% reachable_end_induct
thf(fact_99_reachable__edge,axiom,
! [Y: v,X: v] :
( ( member_v @ Y @ ( successors @ X ) )
=> ( sCC_Bl649662514949026229able_v @ successors @ X @ Y ) ) ).
% reachable_edge
thf(fact_100_reachable_Osimps,axiom,
! [A1: v,A22: v] :
( ( sCC_Bl649662514949026229able_v @ successors @ A1 @ A22 )
= ( ? [X2: v] :
( ( A1 = X2 )
& ( A22 = X2 ) )
| ? [X2: v,Y3: v,Z2: v] :
( ( A1 = X2 )
& ( A22 = Z2 )
& ( member_v @ Y3 @ ( successors @ X2 ) )
& ( sCC_Bl649662514949026229able_v @ successors @ Y3 @ Z2 ) ) ) ) ).
% reachable.simps
thf(fact_101_reachable__succ,axiom,
! [Y: v,X: v,Z: v] :
( ( member_v @ Y @ ( successors @ X ) )
=> ( ( sCC_Bl649662514949026229able_v @ successors @ Y @ Z )
=> ( sCC_Bl649662514949026229able_v @ successors @ X @ Z ) ) ) ).
% reachable_succ
thf(fact_102_reachable__refl,axiom,
! [X: v] : ( sCC_Bl649662514949026229able_v @ successors @ X @ X ) ).
% reachable_refl
thf(fact_103_reachable_Ocases,axiom,
! [A1: v,A22: v] :
( ( sCC_Bl649662514949026229able_v @ successors @ A1 @ A22 )
=> ( ( A22 != A1 )
=> ~ ! [Y2: v] :
( ( member_v @ Y2 @ ( successors @ A1 ) )
=> ~ ( sCC_Bl649662514949026229able_v @ successors @ Y2 @ A22 ) ) ) ) ).
% reachable.cases
thf(fact_104_ra__reachable,axiom,
! [X: v,Y: v,E4: set_Product_prod_v_v] :
( ( sCC_Bl4291963740693775144ding_v @ successors @ X @ Y @ E4 )
=> ( sCC_Bl649662514949026229able_v @ successors @ X @ Y ) ) ).
% ra_reachable
thf(fact_105_reachable__re,axiom,
! [X: v,Y: v] :
( ( sCC_Bl649662514949026229able_v @ successors @ X @ Y )
=> ( sCC_Bl770211535891879572_end_v @ successors @ X @ Y ) ) ).
% reachable_re
thf(fact_106_precedes__refl,axiom,
! [X: product_prod_v_v,Xs: list_P7986770385144383213od_v_v] :
( ( sCC_Bl2026170059108282219od_v_v @ X @ X @ Xs )
= ( member7453568604450474000od_v_v @ X @ ( set_Product_prod_v_v2 @ Xs ) ) ) ).
% precedes_refl
thf(fact_107_precedes__refl,axiom,
! [X: v,Xs: list_v] :
( ( sCC_Bl4022239298816431255edes_v @ X @ X @ Xs )
= ( member_v @ X @ ( set_v2 @ Xs ) ) ) ).
% precedes_refl
thf(fact_108_in__set__insert,axiom,
! [X: product_prod_v_v,Xs: list_P7986770385144383213od_v_v] :
( ( member7453568604450474000od_v_v @ X @ ( set_Product_prod_v_v2 @ Xs ) )
=> ( ( insert4539780211034306307od_v_v @ X @ Xs )
= Xs ) ) ).
% in_set_insert
thf(fact_109_in__set__insert,axiom,
! [X: v,Xs: list_v] :
( ( member_v @ X @ ( set_v2 @ Xs ) )
=> ( ( insert_v @ X @ Xs )
= Xs ) ) ).
% in_set_insert
thf(fact_110_sclosed,axiom,
! [X4: v] :
( ( member_v @ X4 @ vertices )
=> ( ord_less_eq_set_v @ ( successors @ X4 ) @ vertices ) ) ).
% sclosed
thf(fact_111_sccE,axiom,
! [S3: set_v,X: v,Y: v] :
( ( sCC_Bloemen_is_scc_v @ successors @ S3 )
=> ( ( member_v @ X @ S3 )
=> ( ( sCC_Bl649662514949026229able_v @ successors @ X @ Y )
=> ( ( sCC_Bl649662514949026229able_v @ successors @ Y @ X )
=> ( member_v @ Y @ S3 ) ) ) ) ) ).
% sccE
thf(fact_112_graph_Oreachable_Ocong,axiom,
sCC_Bl649662514949026229able_v = sCC_Bl649662514949026229able_v ).
% graph.reachable.cong
thf(fact_113_graph_Opre__dfss_Ocong,axiom,
sCC_Bl1748261141445803503t_unit = sCC_Bl1748261141445803503t_unit ).
% graph.pre_dfss.cong
thf(fact_114_subset__code_I1_J,axiom,
! [Xs: list_P7986770385144383213od_v_v,B2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ ( set_Product_prod_v_v2 @ Xs ) @ B2 )
= ( ! [X2: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X2 @ ( set_Product_prod_v_v2 @ Xs ) )
=> ( member7453568604450474000od_v_v @ X2 @ B2 ) ) ) ) ).
% subset_code(1)
thf(fact_115_subset__code_I1_J,axiom,
! [Xs: list_v,B2: set_v] :
( ( ord_less_eq_set_v @ ( set_v2 @ Xs ) @ B2 )
= ( ! [X2: v] :
( ( member_v @ X2 @ ( set_v2 @ Xs ) )
=> ( member_v @ X2 @ B2 ) ) ) ) ).
% subset_code(1)
thf(fact_116_precedes__mem_I1_J,axiom,
! [X: product_prod_v_v,Y: product_prod_v_v,Xs: list_P7986770385144383213od_v_v] :
( ( sCC_Bl2026170059108282219od_v_v @ X @ Y @ Xs )
=> ( member7453568604450474000od_v_v @ X @ ( set_Product_prod_v_v2 @ Xs ) ) ) ).
% precedes_mem(1)
thf(fact_117_precedes__mem_I1_J,axiom,
! [X: v,Y: v,Xs: list_v] :
( ( sCC_Bl4022239298816431255edes_v @ X @ Y @ Xs )
=> ( member_v @ X @ ( set_v2 @ Xs ) ) ) ).
% precedes_mem(1)
thf(fact_118_precedes__mem_I2_J,axiom,
! [X: product_prod_v_v,Y: product_prod_v_v,Xs: list_P7986770385144383213od_v_v] :
( ( sCC_Bl2026170059108282219od_v_v @ X @ Y @ Xs )
=> ( member7453568604450474000od_v_v @ Y @ ( set_Product_prod_v_v2 @ Xs ) ) ) ).
% precedes_mem(2)
thf(fact_119_precedes__mem_I2_J,axiom,
! [X: v,Y: v,Xs: list_v] :
( ( sCC_Bl4022239298816431255edes_v @ X @ Y @ Xs )
=> ( member_v @ Y @ ( set_v2 @ Xs ) ) ) ).
% precedes_mem(2)
thf(fact_120_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_121_hd__in__set,axiom,
! [Xs: list_v] :
( ( Xs != nil_v )
=> ( member_v @ ( hd_v @ Xs ) @ ( set_v2 @ Xs ) ) ) ).
% hd_in_set
thf(fact_122_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_123_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_124_precedes__append__right__iff,axiom,
! [Y: product_prod_v_v,Ys: list_P7986770385144383213od_v_v,X: product_prod_v_v,Xs: list_P7986770385144383213od_v_v] :
( ~ ( member7453568604450474000od_v_v @ Y @ ( set_Product_prod_v_v2 @ Ys ) )
=> ( ( sCC_Bl2026170059108282219od_v_v @ X @ Y @ ( append2138873909117096322od_v_v @ Xs @ Ys ) )
= ( sCC_Bl2026170059108282219od_v_v @ X @ Y @ Xs ) ) ) ).
% precedes_append_right_iff
thf(fact_125_precedes__append__right__iff,axiom,
! [Y: v,Ys: list_v,X: v,Xs: list_v] :
( ~ ( member_v @ Y @ ( set_v2 @ Ys ) )
=> ( ( sCC_Bl4022239298816431255edes_v @ X @ Y @ ( append_v @ Xs @ Ys ) )
= ( sCC_Bl4022239298816431255edes_v @ X @ Y @ Xs ) ) ) ).
% precedes_append_right_iff
thf(fact_126_precedes__append__left__iff,axiom,
! [X: product_prod_v_v,Ys: list_P7986770385144383213od_v_v,Y: product_prod_v_v,Xs: list_P7986770385144383213od_v_v] :
( ~ ( member7453568604450474000od_v_v @ X @ ( set_Product_prod_v_v2 @ Ys ) )
=> ( ( sCC_Bl2026170059108282219od_v_v @ X @ Y @ ( append2138873909117096322od_v_v @ Ys @ Xs ) )
= ( sCC_Bl2026170059108282219od_v_v @ X @ Y @ Xs ) ) ) ).
% precedes_append_left_iff
thf(fact_127_precedes__append__left__iff,axiom,
! [X: v,Ys: list_v,Y: v,Xs: list_v] :
( ~ ( member_v @ X @ ( set_v2 @ Ys ) )
=> ( ( sCC_Bl4022239298816431255edes_v @ X @ Y @ ( append_v @ Ys @ Xs ) )
= ( sCC_Bl4022239298816431255edes_v @ X @ Y @ Xs ) ) ) ).
% precedes_append_left_iff
thf(fact_128_order__antisym__conv,axiom,
! [Y: set_Product_prod_v_v,X: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ Y @ X )
=> ( ( ord_le7336532860387713383od_v_v @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_129_order__antisym__conv,axiom,
! [Y: set_v,X: set_v] :
( ( ord_less_eq_set_v @ Y @ X )
=> ( ( ord_less_eq_set_v @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_130_ord__le__eq__subst,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v,F: set_Product_prod_v_v > set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: set_Product_prod_v_v,Y2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X3 @ Y2 )
=> ( ord_le7336532860387713383od_v_v @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le7336532860387713383od_v_v @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_131_ord__le__eq__subst,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v,F: set_Product_prod_v_v > set_v,C: set_v] :
( ( ord_le7336532860387713383od_v_v @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: set_Product_prod_v_v,Y2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X3 @ Y2 )
=> ( ord_less_eq_set_v @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_v @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_132_ord__le__eq__subst,axiom,
! [A: set_v,B: set_v,F: set_v > set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( ord_less_eq_set_v @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: set_v,Y2: set_v] :
( ( ord_less_eq_set_v @ X3 @ Y2 )
=> ( ord_le7336532860387713383od_v_v @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le7336532860387713383od_v_v @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_133_ord__le__eq__subst,axiom,
! [A: set_v,B: set_v,F: set_v > set_v,C: set_v] :
( ( ord_less_eq_set_v @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: set_v,Y2: set_v] :
( ( ord_less_eq_set_v @ X3 @ Y2 )
=> ( ord_less_eq_set_v @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_v @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_134_ord__eq__le__subst,axiom,
! [A: set_Product_prod_v_v,F: set_Product_prod_v_v > set_Product_prod_v_v,B: set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( A
= ( F @ B ) )
=> ( ( ord_le7336532860387713383od_v_v @ B @ C )
=> ( ! [X3: set_Product_prod_v_v,Y2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X3 @ Y2 )
=> ( ord_le7336532860387713383od_v_v @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le7336532860387713383od_v_v @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_135_ord__eq__le__subst,axiom,
! [A: set_v,F: set_Product_prod_v_v > set_v,B: set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( A
= ( F @ B ) )
=> ( ( ord_le7336532860387713383od_v_v @ B @ C )
=> ( ! [X3: set_Product_prod_v_v,Y2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X3 @ Y2 )
=> ( ord_less_eq_set_v @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_v @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_136_ord__eq__le__subst,axiom,
! [A: set_Product_prod_v_v,F: set_v > set_Product_prod_v_v,B: set_v,C: set_v] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_set_v @ B @ C )
=> ( ! [X3: set_v,Y2: set_v] :
( ( ord_less_eq_set_v @ X3 @ Y2 )
=> ( ord_le7336532860387713383od_v_v @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le7336532860387713383od_v_v @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_137_ord__eq__le__subst,axiom,
! [A: set_v,F: set_v > set_v,B: set_v,C: set_v] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_set_v @ B @ C )
=> ( ! [X3: set_v,Y2: set_v] :
( ( ord_less_eq_set_v @ X3 @ Y2 )
=> ( ord_less_eq_set_v @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_v @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_138_order__eq__refl,axiom,
! [X: set_Product_prod_v_v,Y: set_Product_prod_v_v] :
( ( X = Y )
=> ( ord_le7336532860387713383od_v_v @ X @ Y ) ) ).
% order_eq_refl
thf(fact_139_order__eq__refl,axiom,
! [X: set_v,Y: set_v] :
( ( X = Y )
=> ( ord_less_eq_set_v @ X @ Y ) ) ).
% order_eq_refl
thf(fact_140_order__subst2,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v,F: set_Product_prod_v_v > set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ B )
=> ( ( ord_le7336532860387713383od_v_v @ ( F @ B ) @ C )
=> ( ! [X3: set_Product_prod_v_v,Y2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X3 @ Y2 )
=> ( ord_le7336532860387713383od_v_v @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le7336532860387713383od_v_v @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_141_order__subst2,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v,F: set_Product_prod_v_v > set_v,C: set_v] :
( ( ord_le7336532860387713383od_v_v @ A @ B )
=> ( ( ord_less_eq_set_v @ ( F @ B ) @ C )
=> ( ! [X3: set_Product_prod_v_v,Y2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X3 @ Y2 )
=> ( ord_less_eq_set_v @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_v @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_142_order__subst2,axiom,
! [A: set_v,B: set_v,F: set_v > set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( ord_less_eq_set_v @ A @ B )
=> ( ( ord_le7336532860387713383od_v_v @ ( F @ B ) @ C )
=> ( ! [X3: set_v,Y2: set_v] :
( ( ord_less_eq_set_v @ X3 @ Y2 )
=> ( ord_le7336532860387713383od_v_v @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le7336532860387713383od_v_v @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_143_order__subst2,axiom,
! [A: set_v,B: set_v,F: set_v > set_v,C: set_v] :
( ( ord_less_eq_set_v @ A @ B )
=> ( ( ord_less_eq_set_v @ ( F @ B ) @ C )
=> ( ! [X3: set_v,Y2: set_v] :
( ( ord_less_eq_set_v @ X3 @ Y2 )
=> ( ord_less_eq_set_v @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_v @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_144_order__subst1,axiom,
! [A: set_Product_prod_v_v,F: set_Product_prod_v_v > set_Product_prod_v_v,B: set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ ( F @ B ) )
=> ( ( ord_le7336532860387713383od_v_v @ B @ C )
=> ( ! [X3: set_Product_prod_v_v,Y2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X3 @ Y2 )
=> ( ord_le7336532860387713383od_v_v @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le7336532860387713383od_v_v @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_145_order__subst1,axiom,
! [A: set_Product_prod_v_v,F: set_v > set_Product_prod_v_v,B: set_v,C: set_v] :
( ( ord_le7336532860387713383od_v_v @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_v @ B @ C )
=> ( ! [X3: set_v,Y2: set_v] :
( ( ord_less_eq_set_v @ X3 @ Y2 )
=> ( ord_le7336532860387713383od_v_v @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le7336532860387713383od_v_v @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_146_order__subst1,axiom,
! [A: set_v,F: set_Product_prod_v_v > set_v,B: set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( ord_less_eq_set_v @ A @ ( F @ B ) )
=> ( ( ord_le7336532860387713383od_v_v @ B @ C )
=> ( ! [X3: set_Product_prod_v_v,Y2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X3 @ Y2 )
=> ( ord_less_eq_set_v @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_v @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_147_order__subst1,axiom,
! [A: set_v,F: set_v > set_v,B: set_v,C: set_v] :
( ( ord_less_eq_set_v @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_v @ B @ C )
=> ( ! [X3: set_v,Y2: set_v] :
( ( ord_less_eq_set_v @ X3 @ Y2 )
=> ( ord_less_eq_set_v @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_v @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_148_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_Product_prod_v_v,Z4: set_Product_prod_v_v] : ( Y4 = Z4 ) )
= ( ^ [A5: set_Product_prod_v_v,B4: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A5 @ B4 )
& ( ord_le7336532860387713383od_v_v @ B4 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_149_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_v,Z4: set_v] : ( Y4 = Z4 ) )
= ( ^ [A5: set_v,B4: set_v] :
( ( ord_less_eq_set_v @ A5 @ B4 )
& ( ord_less_eq_set_v @ B4 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_150_antisym,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ B )
=> ( ( ord_le7336532860387713383od_v_v @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_151_antisym,axiom,
! [A: set_v,B: set_v] :
( ( ord_less_eq_set_v @ A @ B )
=> ( ( ord_less_eq_set_v @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_152_dual__order_Otrans,axiom,
! [B: set_Product_prod_v_v,A: set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ B @ A )
=> ( ( ord_le7336532860387713383od_v_v @ C @ B )
=> ( ord_le7336532860387713383od_v_v @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_153_dual__order_Otrans,axiom,
! [B: set_v,A: set_v,C: set_v] :
( ( ord_less_eq_set_v @ B @ A )
=> ( ( ord_less_eq_set_v @ C @ B )
=> ( ord_less_eq_set_v @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_154_dual__order_Oantisym,axiom,
! [B: set_Product_prod_v_v,A: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ B @ A )
=> ( ( ord_le7336532860387713383od_v_v @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_155_dual__order_Oantisym,axiom,
! [B: set_v,A: set_v] :
( ( ord_less_eq_set_v @ B @ A )
=> ( ( ord_less_eq_set_v @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_156_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: set_Product_prod_v_v,Z4: set_Product_prod_v_v] : ( Y4 = Z4 ) )
= ( ^ [A5: set_Product_prod_v_v,B4: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ B4 @ A5 )
& ( ord_le7336532860387713383od_v_v @ A5 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_157_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: set_v,Z4: set_v] : ( Y4 = Z4 ) )
= ( ^ [A5: set_v,B4: set_v] :
( ( ord_less_eq_set_v @ B4 @ A5 )
& ( ord_less_eq_set_v @ A5 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_158_order__trans,axiom,
! [X: set_Product_prod_v_v,Y: set_Product_prod_v_v,Z: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X @ Y )
=> ( ( ord_le7336532860387713383od_v_v @ Y @ Z )
=> ( ord_le7336532860387713383od_v_v @ X @ Z ) ) ) ).
% order_trans
thf(fact_159_order__trans,axiom,
! [X: set_v,Y: set_v,Z: set_v] :
( ( ord_less_eq_set_v @ X @ Y )
=> ( ( ord_less_eq_set_v @ Y @ Z )
=> ( ord_less_eq_set_v @ X @ Z ) ) ) ).
% order_trans
thf(fact_160_order_Otrans,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ B )
=> ( ( ord_le7336532860387713383od_v_v @ B @ C )
=> ( ord_le7336532860387713383od_v_v @ A @ C ) ) ) ).
% order.trans
thf(fact_161_order_Otrans,axiom,
! [A: set_v,B: set_v,C: set_v] :
( ( ord_less_eq_set_v @ A @ B )
=> ( ( ord_less_eq_set_v @ B @ C )
=> ( ord_less_eq_set_v @ A @ C ) ) ) ).
% order.trans
thf(fact_162_order__antisym,axiom,
! [X: set_Product_prod_v_v,Y: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X @ Y )
=> ( ( ord_le7336532860387713383od_v_v @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_163_order__antisym,axiom,
! [X: set_v,Y: set_v] :
( ( ord_less_eq_set_v @ X @ Y )
=> ( ( ord_less_eq_set_v @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_164_ord__le__eq__trans,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ B )
=> ( ( B = C )
=> ( ord_le7336532860387713383od_v_v @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_165_ord__le__eq__trans,axiom,
! [A: set_v,B: set_v,C: set_v] :
( ( ord_less_eq_set_v @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_set_v @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_166_ord__eq__le__trans,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( A = B )
=> ( ( ord_le7336532860387713383od_v_v @ B @ C )
=> ( ord_le7336532860387713383od_v_v @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_167_ord__eq__le__trans,axiom,
! [A: set_v,B: set_v,C: set_v] :
( ( A = B )
=> ( ( ord_less_eq_set_v @ B @ C )
=> ( ord_less_eq_set_v @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_168_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_Product_prod_v_v,Z4: set_Product_prod_v_v] : ( Y4 = Z4 ) )
= ( ^ [X2: set_Product_prod_v_v,Y3: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X2 @ Y3 )
& ( ord_le7336532860387713383od_v_v @ Y3 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_169_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_v,Z4: set_v] : ( Y4 = Z4 ) )
= ( ^ [X2: set_v,Y3: set_v] :
( ( ord_less_eq_set_v @ X2 @ Y3 )
& ( ord_less_eq_set_v @ Y3 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_170_Pair__inject,axiom,
! [A: v,B: v,A4: v,B3: v] :
( ( ( product_Pair_v_v @ A @ B )
= ( product_Pair_v_v @ A4 @ B3 ) )
=> ~ ( ( A = A4 )
=> ( B != B3 ) ) ) ).
% Pair_inject
thf(fact_171_Pair__inject,axiom,
! [A: v,B: sCC_Bl1394983891496994913t_unit,A4: v,B3: sCC_Bl1394983891496994913t_unit] :
( ( ( produc3862955338007567901t_unit @ A @ B )
= ( produc3862955338007567901t_unit @ A4 @ B3 ) )
=> ~ ( ( A = A4 )
=> ( B != B3 ) ) ) ).
% Pair_inject
thf(fact_172_prod__cases,axiom,
! [P: product_prod_v_v > $o,P2: product_prod_v_v] :
( ! [A6: v,B5: v] : ( P @ ( product_Pair_v_v @ A6 @ B5 ) )
=> ( P @ P2 ) ) ).
% prod_cases
thf(fact_173_prod__cases,axiom,
! [P: produc5741669702376414499t_unit > $o,P2: produc5741669702376414499t_unit] :
( ! [A6: v,B5: sCC_Bl1394983891496994913t_unit] : ( P @ ( produc3862955338007567901t_unit @ A6 @ B5 ) )
=> ( P @ P2 ) ) ).
% prod_cases
thf(fact_174_surj__pair,axiom,
! [P2: product_prod_v_v] :
? [X3: v,Y2: v] :
( P2
= ( product_Pair_v_v @ X3 @ Y2 ) ) ).
% surj_pair
thf(fact_175_surj__pair,axiom,
! [P2: produc5741669702376414499t_unit] :
? [X3: v,Y2: sCC_Bl1394983891496994913t_unit] :
( P2
= ( produc3862955338007567901t_unit @ X3 @ Y2 ) ) ).
% surj_pair
thf(fact_176_old_Oprod_Oexhaust,axiom,
! [Y: product_prod_v_v] :
~ ! [A6: v,B5: v] :
( Y
!= ( product_Pair_v_v @ A6 @ B5 ) ) ).
% old.prod.exhaust
thf(fact_177_old_Oprod_Oexhaust,axiom,
! [Y: produc5741669702376414499t_unit] :
~ ! [A6: v,B5: sCC_Bl1394983891496994913t_unit] :
( Y
!= ( produc3862955338007567901t_unit @ A6 @ B5 ) ) ).
% old.prod.exhaust
thf(fact_178_Collect__mono__iff,axiom,
! [P: set_v > $o,Q: set_v > $o] :
( ( ord_le5216385588623774835_set_v @ ( collect_set_v @ P ) @ ( collect_set_v @ Q ) )
= ( ! [X2: set_v] :
( ( P @ X2 )
=> ( Q @ X2 ) ) ) ) ).
% Collect_mono_iff
thf(fact_179_Collect__mono__iff,axiom,
! [P: product_prod_v_v > $o,Q: product_prod_v_v > $o] :
( ( ord_le7336532860387713383od_v_v @ ( collec140062887454715474od_v_v @ P ) @ ( collec140062887454715474od_v_v @ Q ) )
= ( ! [X2: product_prod_v_v] :
( ( P @ X2 )
=> ( Q @ X2 ) ) ) ) ).
% Collect_mono_iff
thf(fact_180_Collect__mono__iff,axiom,
! [P: v > $o,Q: v > $o] :
( ( ord_less_eq_set_v @ ( collect_v @ P ) @ ( collect_v @ Q ) )
= ( ! [X2: v] :
( ( P @ X2 )
=> ( Q @ X2 ) ) ) ) ).
% Collect_mono_iff
thf(fact_181_set__eq__subset,axiom,
( ( ^ [Y4: set_Product_prod_v_v,Z4: set_Product_prod_v_v] : ( Y4 = Z4 ) )
= ( ^ [A7: set_Product_prod_v_v,B6: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A7 @ B6 )
& ( ord_le7336532860387713383od_v_v @ B6 @ A7 ) ) ) ) ).
% set_eq_subset
thf(fact_182_set__eq__subset,axiom,
( ( ^ [Y4: set_v,Z4: set_v] : ( Y4 = Z4 ) )
= ( ^ [A7: set_v,B6: set_v] :
( ( ord_less_eq_set_v @ A7 @ B6 )
& ( ord_less_eq_set_v @ B6 @ A7 ) ) ) ) ).
% set_eq_subset
thf(fact_183_subset__trans,axiom,
! [A2: set_Product_prod_v_v,B2: set_Product_prod_v_v,C2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A2 @ B2 )
=> ( ( ord_le7336532860387713383od_v_v @ B2 @ C2 )
=> ( ord_le7336532860387713383od_v_v @ A2 @ C2 ) ) ) ).
% subset_trans
thf(fact_184_subset__trans,axiom,
! [A2: set_v,B2: set_v,C2: set_v] :
( ( ord_less_eq_set_v @ A2 @ B2 )
=> ( ( ord_less_eq_set_v @ B2 @ C2 )
=> ( ord_less_eq_set_v @ A2 @ C2 ) ) ) ).
% subset_trans
thf(fact_185_Collect__mono,axiom,
! [P: set_v > $o,Q: set_v > $o] :
( ! [X3: set_v] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( ord_le5216385588623774835_set_v @ ( collect_set_v @ P ) @ ( collect_set_v @ Q ) ) ) ).
% Collect_mono
thf(fact_186_Collect__mono,axiom,
! [P: product_prod_v_v > $o,Q: product_prod_v_v > $o] :
( ! [X3: product_prod_v_v] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( ord_le7336532860387713383od_v_v @ ( collec140062887454715474od_v_v @ P ) @ ( collec140062887454715474od_v_v @ Q ) ) ) ).
% Collect_mono
thf(fact_187_Collect__mono,axiom,
! [P: v > $o,Q: v > $o] :
( ! [X3: v] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( ord_less_eq_set_v @ ( collect_v @ P ) @ ( collect_v @ Q ) ) ) ).
% Collect_mono
thf(fact_188_subset__refl,axiom,
! [A2: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ A2 @ A2 ) ).
% subset_refl
thf(fact_189_subset__refl,axiom,
! [A2: set_v] : ( ord_less_eq_set_v @ A2 @ A2 ) ).
% subset_refl
thf(fact_190_subset__iff,axiom,
( ord_le7336532860387713383od_v_v
= ( ^ [A7: set_Product_prod_v_v,B6: set_Product_prod_v_v] :
! [T: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ T @ A7 )
=> ( member7453568604450474000od_v_v @ T @ B6 ) ) ) ) ).
% subset_iff
thf(fact_191_subset__iff,axiom,
( ord_less_eq_set_v
= ( ^ [A7: set_v,B6: set_v] :
! [T: v] :
( ( member_v @ T @ A7 )
=> ( member_v @ T @ B6 ) ) ) ) ).
% subset_iff
thf(fact_192_equalityD2,axiom,
! [A2: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( A2 = B2 )
=> ( ord_le7336532860387713383od_v_v @ B2 @ A2 ) ) ).
% equalityD2
thf(fact_193_equalityD2,axiom,
! [A2: set_v,B2: set_v] :
( ( A2 = B2 )
=> ( ord_less_eq_set_v @ B2 @ A2 ) ) ).
% equalityD2
thf(fact_194_equalityD1,axiom,
! [A2: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( A2 = B2 )
=> ( ord_le7336532860387713383od_v_v @ A2 @ B2 ) ) ).
% equalityD1
thf(fact_195_equalityD1,axiom,
! [A2: set_v,B2: set_v] :
( ( A2 = B2 )
=> ( ord_less_eq_set_v @ A2 @ B2 ) ) ).
% equalityD1
thf(fact_196_subset__eq,axiom,
( ord_le7336532860387713383od_v_v
= ( ^ [A7: set_Product_prod_v_v,B6: set_Product_prod_v_v] :
! [X2: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X2 @ A7 )
=> ( member7453568604450474000od_v_v @ X2 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_197_subset__eq,axiom,
( ord_less_eq_set_v
= ( ^ [A7: set_v,B6: set_v] :
! [X2: v] :
( ( member_v @ X2 @ A7 )
=> ( member_v @ X2 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_198_equalityE,axiom,
! [A2: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( A2 = B2 )
=> ~ ( ( ord_le7336532860387713383od_v_v @ A2 @ B2 )
=> ~ ( ord_le7336532860387713383od_v_v @ B2 @ A2 ) ) ) ).
% equalityE
thf(fact_199_equalityE,axiom,
! [A2: set_v,B2: set_v] :
( ( A2 = B2 )
=> ~ ( ( ord_less_eq_set_v @ A2 @ B2 )
=> ~ ( ord_less_eq_set_v @ B2 @ A2 ) ) ) ).
% equalityE
thf(fact_200_subsetD,axiom,
! [A2: set_Product_prod_v_v,B2: set_Product_prod_v_v,C: product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A2 @ B2 )
=> ( ( member7453568604450474000od_v_v @ C @ A2 )
=> ( member7453568604450474000od_v_v @ C @ B2 ) ) ) ).
% subsetD
thf(fact_201_subsetD,axiom,
! [A2: set_v,B2: set_v,C: v] :
( ( ord_less_eq_set_v @ A2 @ B2 )
=> ( ( member_v @ C @ A2 )
=> ( member_v @ C @ B2 ) ) ) ).
% subsetD
thf(fact_202_in__mono,axiom,
! [A2: set_Product_prod_v_v,B2: set_Product_prod_v_v,X: product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A2 @ B2 )
=> ( ( member7453568604450474000od_v_v @ X @ A2 )
=> ( member7453568604450474000od_v_v @ X @ B2 ) ) ) ).
% in_mono
thf(fact_203_in__mono,axiom,
! [A2: set_v,B2: set_v,X: v] :
( ( ord_less_eq_set_v @ A2 @ B2 )
=> ( ( member_v @ X @ A2 )
=> ( member_v @ X @ B2 ) ) ) ).
% in_mono
thf(fact_204_is__subscc__def,axiom,
! [S3: set_v] :
( ( sCC_Bl5398416737448265317bscc_v @ successors @ S3 )
= ( ! [X2: v] :
( ( member_v @ X2 @ S3 )
=> ! [Y3: v] :
( ( member_v @ Y3 @ S3 )
=> ( sCC_Bl649662514949026229able_v @ successors @ X2 @ Y3 ) ) ) ) ) ).
% is_subscc_def
thf(fact_205_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_206_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_207_dfs__S__tl__stack_I2_J,axiom,
! [V3: v,E: sCC_Bl1394983891496994913t_unit,E2: sCC_Bl1394983891496994913t_unit] :
( ( sCC_Bl8953792750115413617t_unit @ successors @ V3 @ E @ E2 )
=> ( ( ( sCC_Bl8828226123343373779t_unit @ E )
!= nil_v )
=> ! [X4: v] :
( ( member_v @ X4 @ ( set_v2 @ ( tl_v @ ( sCC_Bl8828226123343373779t_unit @ E2 ) ) ) )
=> ( ( sCC_Bl1280885523602775798t_unit @ E2 @ X4 )
= ( sCC_Bl1280885523602775798t_unit @ E @ X4 ) ) ) ) ) ).
% dfs_S_tl_stack(2)
thf(fact_208_ra__empty,axiom,
! [X: v,Y: v] :
( ( sCC_Bl4291963740693775144ding_v @ successors @ X @ Y @ bot_bo723834152578015283od_v_v )
= ( sCC_Bl649662514949026229able_v @ successors @ X @ Y ) ) ).
% ra_empty
thf(fact_209__C2_C,axiom,
accp_S2303753412255344476t_unit @ ( sCC_Bl907557413677168252_rel_v @ successors ) @ ( sum_In526841707622398774t_unit @ ( produc3862955338007567901t_unit @ v2 @ e ) ) ).
% "2"
thf(fact_210_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_211_graph_Odfs__S__hd__stack_I1_J,axiom,
! [Vertices: set_v,Successors: v > set_v,E: sCC_Bl1394983891496994913t_unit,V3: v,E2: sCC_Bl1394983891496994913t_unit,N: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl9196236973127232072t_unit @ Successors @ E )
=> ( ( sCC_Bl8953792750115413617t_unit @ Successors @ V3 @ E @ E2 )
=> ( ( ( sCC_Bl8828226123343373779t_unit @ E )
!= nil_v )
=> ( ( member_v @ 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_212_graph_Odfs__S__hd__stack_I2_J,axiom,
! [Vertices: set_v,Successors: v > set_v,E: sCC_Bl1394983891496994913t_unit,V3: v,E2: sCC_Bl1394983891496994913t_unit,N: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl9196236973127232072t_unit @ Successors @ E )
=> ( ( sCC_Bl8953792750115413617t_unit @ Successors @ V3 @ E @ E2 )
=> ( ( ( sCC_Bl8828226123343373779t_unit @ E )
!= nil_v )
=> ( ( member_v @ 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_213_dfss_Ocases,axiom,
! [X: produc5741669702376414499t_unit] :
~ ! [V2: v,E7: sCC_Bl1394983891496994913t_unit] :
( X
!= ( produc3862955338007567901t_unit @ V2 @ E7 ) ) ).
% dfss.cases
thf(fact_214_graph__axioms,axiom,
sCC_Bloemen_graph_v @ vertices @ successors ).
% graph_axioms
thf(fact_215_empty__Collect__eq,axiom,
! [P: product_prod_v_v > $o] :
( ( bot_bo723834152578015283od_v_v
= ( collec140062887454715474od_v_v @ P ) )
= ( ! [X2: product_prod_v_v] :
~ ( P @ X2 ) ) ) ).
% empty_Collect_eq
thf(fact_216_empty__Collect__eq,axiom,
! [P: v > $o] :
( ( bot_bot_set_v
= ( collect_v @ P ) )
= ( ! [X2: v] :
~ ( P @ X2 ) ) ) ).
% empty_Collect_eq
thf(fact_217_empty__Collect__eq,axiom,
! [P: set_v > $o] :
( ( bot_bot_set_set_v
= ( collect_set_v @ P ) )
= ( ! [X2: set_v] :
~ ( P @ X2 ) ) ) ).
% empty_Collect_eq
thf(fact_218_Collect__empty__eq,axiom,
! [P: product_prod_v_v > $o] :
( ( ( collec140062887454715474od_v_v @ P )
= bot_bo723834152578015283od_v_v )
= ( ! [X2: product_prod_v_v] :
~ ( P @ X2 ) ) ) ).
% Collect_empty_eq
thf(fact_219_Collect__empty__eq,axiom,
! [P: v > $o] :
( ( ( collect_v @ P )
= bot_bot_set_v )
= ( ! [X2: v] :
~ ( P @ X2 ) ) ) ).
% Collect_empty_eq
thf(fact_220_Collect__empty__eq,axiom,
! [P: set_v > $o] :
( ( ( collect_set_v @ P )
= bot_bot_set_set_v )
= ( ! [X2: set_v] :
~ ( P @ X2 ) ) ) ).
% Collect_empty_eq
thf(fact_221_all__not__in__conv,axiom,
! [A2: set_Product_prod_v_v] :
( ( ! [X2: product_prod_v_v] :
~ ( member7453568604450474000od_v_v @ X2 @ A2 ) )
= ( A2 = bot_bo723834152578015283od_v_v ) ) ).
% all_not_in_conv
thf(fact_222_all__not__in__conv,axiom,
! [A2: set_v] :
( ( ! [X2: v] :
~ ( member_v @ X2 @ A2 ) )
= ( A2 = bot_bot_set_v ) ) ).
% all_not_in_conv
thf(fact_223_all__not__in__conv,axiom,
! [A2: set_set_v] :
( ( ! [X2: set_v] :
~ ( member_set_v @ X2 @ A2 ) )
= ( A2 = bot_bot_set_set_v ) ) ).
% all_not_in_conv
thf(fact_224_empty__iff,axiom,
! [C: product_prod_v_v] :
~ ( member7453568604450474000od_v_v @ C @ bot_bo723834152578015283od_v_v ) ).
% empty_iff
thf(fact_225_empty__iff,axiom,
! [C: v] :
~ ( member_v @ C @ bot_bot_set_v ) ).
% empty_iff
thf(fact_226_empty__iff,axiom,
! [C: set_v] :
~ ( member_set_v @ C @ bot_bot_set_set_v ) ).
% empty_iff
thf(fact_227_Diff__idemp,axiom,
! [A2: set_v,B2: set_v] :
( ( minus_minus_set_v @ ( minus_minus_set_v @ A2 @ B2 ) @ B2 )
= ( minus_minus_set_v @ A2 @ B2 ) ) ).
% Diff_idemp
thf(fact_228_Diff__iff,axiom,
! [C: product_prod_v_v,A2: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ C @ ( minus_4183494784930505774od_v_v @ A2 @ B2 ) )
= ( ( member7453568604450474000od_v_v @ C @ A2 )
& ~ ( member7453568604450474000od_v_v @ C @ B2 ) ) ) ).
% Diff_iff
thf(fact_229_Diff__iff,axiom,
! [C: v,A2: set_v,B2: set_v] :
( ( member_v @ C @ ( minus_minus_set_v @ A2 @ B2 ) )
= ( ( member_v @ C @ A2 )
& ~ ( member_v @ C @ B2 ) ) ) ).
% Diff_iff
thf(fact_230_DiffI,axiom,
! [C: product_prod_v_v,A2: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ C @ A2 )
=> ( ~ ( member7453568604450474000od_v_v @ C @ B2 )
=> ( member7453568604450474000od_v_v @ C @ ( minus_4183494784930505774od_v_v @ A2 @ B2 ) ) ) ) ).
% DiffI
thf(fact_231_DiffI,axiom,
! [C: v,A2: set_v,B2: set_v] :
( ( member_v @ C @ A2 )
=> ( ~ ( member_v @ C @ B2 )
=> ( member_v @ C @ ( minus_minus_set_v @ A2 @ B2 ) ) ) ) ).
% DiffI
thf(fact_232_subset__empty,axiom,
! [A2: set_set_v] :
( ( ord_le5216385588623774835_set_v @ A2 @ bot_bot_set_set_v )
= ( A2 = bot_bot_set_set_v ) ) ).
% subset_empty
thf(fact_233_subset__empty,axiom,
! [A2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A2 @ bot_bo723834152578015283od_v_v )
= ( A2 = bot_bo723834152578015283od_v_v ) ) ).
% subset_empty
thf(fact_234_subset__empty,axiom,
! [A2: set_v] :
( ( ord_less_eq_set_v @ A2 @ bot_bot_set_v )
= ( A2 = bot_bot_set_v ) ) ).
% subset_empty
thf(fact_235_empty__subsetI,axiom,
! [A2: set_set_v] : ( ord_le5216385588623774835_set_v @ bot_bot_set_set_v @ A2 ) ).
% empty_subsetI
thf(fact_236_empty__subsetI,axiom,
! [A2: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ bot_bo723834152578015283od_v_v @ A2 ) ).
% empty_subsetI
thf(fact_237_empty__subsetI,axiom,
! [A2: set_v] : ( ord_less_eq_set_v @ bot_bot_set_v @ A2 ) ).
% empty_subsetI
thf(fact_238_Diff__cancel,axiom,
! [A2: set_Product_prod_v_v] :
( ( minus_4183494784930505774od_v_v @ A2 @ A2 )
= bot_bo723834152578015283od_v_v ) ).
% Diff_cancel
thf(fact_239_Diff__cancel,axiom,
! [A2: set_set_v] :
( ( minus_7228012346218142266_set_v @ A2 @ A2 )
= bot_bot_set_set_v ) ).
% Diff_cancel
thf(fact_240_Diff__cancel,axiom,
! [A2: set_v] :
( ( minus_minus_set_v @ A2 @ A2 )
= bot_bot_set_v ) ).
% Diff_cancel
thf(fact_241_empty__Diff,axiom,
! [A2: set_Product_prod_v_v] :
( ( minus_4183494784930505774od_v_v @ bot_bo723834152578015283od_v_v @ A2 )
= bot_bo723834152578015283od_v_v ) ).
% empty_Diff
thf(fact_242_empty__Diff,axiom,
! [A2: set_set_v] :
( ( minus_7228012346218142266_set_v @ bot_bot_set_set_v @ A2 )
= bot_bot_set_set_v ) ).
% empty_Diff
thf(fact_243_empty__Diff,axiom,
! [A2: set_v] :
( ( minus_minus_set_v @ bot_bot_set_v @ A2 )
= bot_bot_set_v ) ).
% empty_Diff
thf(fact_244_Diff__empty,axiom,
! [A2: set_Product_prod_v_v] :
( ( minus_4183494784930505774od_v_v @ A2 @ bot_bo723834152578015283od_v_v )
= A2 ) ).
% Diff_empty
thf(fact_245_Diff__empty,axiom,
! [A2: set_set_v] :
( ( minus_7228012346218142266_set_v @ A2 @ bot_bot_set_set_v )
= A2 ) ).
% Diff_empty
thf(fact_246_Diff__empty,axiom,
! [A2: set_v] :
( ( minus_minus_set_v @ A2 @ bot_bot_set_v )
= A2 ) ).
% Diff_empty
thf(fact_247_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_248_set__empty,axiom,
! [Xs: list_v] :
( ( ( set_v2 @ Xs )
= bot_bot_set_v )
= ( Xs = nil_v ) ) ).
% set_empty
thf(fact_249_set__empty,axiom,
! [Xs: list_set_v] :
( ( ( set_set_v2 @ Xs )
= bot_bot_set_set_v )
= ( Xs = nil_set_v ) ) ).
% set_empty
thf(fact_250_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_251_set__empty2,axiom,
! [Xs: list_v] :
( ( bot_bot_set_v
= ( set_v2 @ Xs ) )
= ( Xs = nil_v ) ) ).
% set_empty2
thf(fact_252_set__empty2,axiom,
! [Xs: list_set_v] :
( ( bot_bot_set_set_v
= ( set_set_v2 @ Xs ) )
= ( Xs = nil_set_v ) ) ).
% set_empty2
thf(fact_253_Diff__eq__empty__iff,axiom,
! [A2: set_set_v,B2: set_set_v] :
( ( ( minus_7228012346218142266_set_v @ A2 @ B2 )
= bot_bot_set_set_v )
= ( ord_le5216385588623774835_set_v @ A2 @ B2 ) ) ).
% Diff_eq_empty_iff
thf(fact_254_Diff__eq__empty__iff,axiom,
! [A2: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( ( minus_4183494784930505774od_v_v @ A2 @ B2 )
= bot_bo723834152578015283od_v_v )
= ( ord_le7336532860387713383od_v_v @ A2 @ B2 ) ) ).
% Diff_eq_empty_iff
thf(fact_255_Diff__eq__empty__iff,axiom,
! [A2: set_v,B2: set_v] :
( ( ( minus_minus_set_v @ A2 @ B2 )
= bot_bot_set_v )
= ( ord_less_eq_set_v @ A2 @ B2 ) ) ).
% Diff_eq_empty_iff
thf(fact_256_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_257_graph_Odfss_Ocases,axiom,
! [Vertices: set_v,Successors: v > set_v,X: produc5741669702376414499t_unit] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ~ ! [V2: v,E7: sCC_Bl1394983891496994913t_unit] :
( X
!= ( produc3862955338007567901t_unit @ V2 @ E7 ) ) ) ).
% graph.dfss.cases
thf(fact_258_graph_Ois__scc__def,axiom,
! [Vertices: set_set_v,Successors: set_v > set_set_v,S3: set_set_v] :
( ( sCC_Bl5810666556806954322_set_v @ Vertices @ Successors )
=> ( ( sCC_Bl1515522642333523865_set_v @ Successors @ S3 )
= ( ( S3 != bot_bot_set_set_v )
& ( sCC_Bl7907073126578335045_set_v @ Successors @ S3 )
& ! [S4: set_set_v] :
( ( ( ord_le5216385588623774835_set_v @ S3 @ S4 )
& ( sCC_Bl7907073126578335045_set_v @ Successors @ S4 ) )
=> ( S4 = S3 ) ) ) ) ) ).
% graph.is_scc_def
thf(fact_259_graph_Ois__scc__def,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,S3: set_Product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( sCC_Bl6242042402218619277od_v_v @ Successors @ S3 )
= ( ( S3 != bot_bo723834152578015283od_v_v )
& ( sCC_Bl2301996248249672505od_v_v @ Successors @ S3 )
& ! [S4: set_Product_prod_v_v] :
( ( ( ord_le7336532860387713383od_v_v @ S3 @ S4 )
& ( sCC_Bl2301996248249672505od_v_v @ Successors @ S4 ) )
=> ( S4 = S3 ) ) ) ) ) ).
% graph.is_scc_def
thf(fact_260_graph_Ois__scc__def,axiom,
! [Vertices: set_v,Successors: v > set_v,S3: set_v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bloemen_is_scc_v @ Successors @ S3 )
= ( ( S3 != bot_bot_set_v )
& ( sCC_Bl5398416737448265317bscc_v @ Successors @ S3 )
& ! [S4: set_v] :
( ( ( ord_less_eq_set_v @ S3 @ S4 )
& ( sCC_Bl5398416737448265317bscc_v @ Successors @ S4 ) )
=> ( S4 = S3 ) ) ) ) ) ).
% graph.is_scc_def
thf(fact_261_ex__in__conv,axiom,
! [A2: set_Product_prod_v_v] :
( ( ? [X2: product_prod_v_v] : ( member7453568604450474000od_v_v @ X2 @ A2 ) )
= ( A2 != bot_bo723834152578015283od_v_v ) ) ).
% ex_in_conv
thf(fact_262_ex__in__conv,axiom,
! [A2: set_v] :
( ( ? [X2: v] : ( member_v @ X2 @ A2 ) )
= ( A2 != bot_bot_set_v ) ) ).
% ex_in_conv
thf(fact_263_ex__in__conv,axiom,
! [A2: set_set_v] :
( ( ? [X2: set_v] : ( member_set_v @ X2 @ A2 ) )
= ( A2 != bot_bot_set_set_v ) ) ).
% ex_in_conv
thf(fact_264_equals0I,axiom,
! [A2: set_Product_prod_v_v] :
( ! [Y2: product_prod_v_v] :
~ ( member7453568604450474000od_v_v @ Y2 @ A2 )
=> ( A2 = bot_bo723834152578015283od_v_v ) ) ).
% equals0I
thf(fact_265_equals0I,axiom,
! [A2: set_v] :
( ! [Y2: v] :
~ ( member_v @ Y2 @ A2 )
=> ( A2 = bot_bot_set_v ) ) ).
% equals0I
thf(fact_266_equals0I,axiom,
! [A2: set_set_v] :
( ! [Y2: set_v] :
~ ( member_set_v @ Y2 @ A2 )
=> ( A2 = bot_bot_set_set_v ) ) ).
% equals0I
thf(fact_267_equals0D,axiom,
! [A2: set_Product_prod_v_v,A: product_prod_v_v] :
( ( A2 = bot_bo723834152578015283od_v_v )
=> ~ ( member7453568604450474000od_v_v @ A @ A2 ) ) ).
% equals0D
thf(fact_268_equals0D,axiom,
! [A2: set_v,A: v] :
( ( A2 = bot_bot_set_v )
=> ~ ( member_v @ A @ A2 ) ) ).
% equals0D
thf(fact_269_equals0D,axiom,
! [A2: set_set_v,A: set_v] :
( ( A2 = bot_bot_set_set_v )
=> ~ ( member_set_v @ A @ A2 ) ) ).
% equals0D
thf(fact_270_emptyE,axiom,
! [A: product_prod_v_v] :
~ ( member7453568604450474000od_v_v @ A @ bot_bo723834152578015283od_v_v ) ).
% emptyE
thf(fact_271_emptyE,axiom,
! [A: v] :
~ ( member_v @ A @ bot_bot_set_v ) ).
% emptyE
thf(fact_272_emptyE,axiom,
! [A: set_v] :
~ ( member_set_v @ A @ bot_bot_set_set_v ) ).
% emptyE
thf(fact_273_DiffD2,axiom,
! [C: product_prod_v_v,A2: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ C @ ( minus_4183494784930505774od_v_v @ A2 @ B2 ) )
=> ~ ( member7453568604450474000od_v_v @ C @ B2 ) ) ).
% DiffD2
thf(fact_274_DiffD2,axiom,
! [C: v,A2: set_v,B2: set_v] :
( ( member_v @ C @ ( minus_minus_set_v @ A2 @ B2 ) )
=> ~ ( member_v @ C @ B2 ) ) ).
% DiffD2
thf(fact_275_DiffD1,axiom,
! [C: product_prod_v_v,A2: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ C @ ( minus_4183494784930505774od_v_v @ A2 @ B2 ) )
=> ( member7453568604450474000od_v_v @ C @ A2 ) ) ).
% DiffD1
thf(fact_276_DiffD1,axiom,
! [C: v,A2: set_v,B2: set_v] :
( ( member_v @ C @ ( minus_minus_set_v @ A2 @ B2 ) )
=> ( member_v @ C @ A2 ) ) ).
% DiffD1
thf(fact_277_DiffE,axiom,
! [C: product_prod_v_v,A2: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ C @ ( minus_4183494784930505774od_v_v @ A2 @ B2 ) )
=> ~ ( ( member7453568604450474000od_v_v @ C @ A2 )
=> ( member7453568604450474000od_v_v @ C @ B2 ) ) ) ).
% DiffE
thf(fact_278_DiffE,axiom,
! [C: v,A2: set_v,B2: set_v] :
( ( member_v @ C @ ( minus_minus_set_v @ A2 @ B2 ) )
=> ~ ( ( member_v @ C @ A2 )
=> ( member_v @ C @ B2 ) ) ) ).
% DiffE
thf(fact_279_graph_Ois__subscc_Ocong,axiom,
sCC_Bl5398416737448265317bscc_v = sCC_Bl5398416737448265317bscc_v ).
% graph.is_subscc.cong
thf(fact_280_graph_Ois__scc_Ocong,axiom,
sCC_Bloemen_is_scc_v = sCC_Bloemen_is_scc_v ).
% graph.is_scc.cong
thf(fact_281_graph_Ois__subscc__def,axiom,
! [Vertices: set_v,Successors: v > set_v,S3: set_v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl5398416737448265317bscc_v @ Successors @ S3 )
= ( ! [X2: v] :
( ( member_v @ X2 @ S3 )
=> ! [Y3: v] :
( ( member_v @ Y3 @ S3 )
=> ( sCC_Bl649662514949026229able_v @ Successors @ X2 @ Y3 ) ) ) ) ) ) ).
% graph.is_subscc_def
thf(fact_282_graph_OsccE,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,S3: set_Product_prod_v_v,X: product_prod_v_v,Y: product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( sCC_Bl6242042402218619277od_v_v @ Successors @ S3 )
=> ( ( member7453568604450474000od_v_v @ X @ S3 )
=> ( ( sCC_Bl4981926079593201289od_v_v @ Successors @ X @ Y )
=> ( ( sCC_Bl4981926079593201289od_v_v @ Successors @ Y @ X )
=> ( member7453568604450474000od_v_v @ Y @ S3 ) ) ) ) ) ) ).
% graph.sccE
thf(fact_283_graph_OsccE,axiom,
! [Vertices: set_v,Successors: v > set_v,S3: set_v,X: v,Y: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bloemen_is_scc_v @ Successors @ S3 )
=> ( ( member_v @ X @ S3 )
=> ( ( sCC_Bl649662514949026229able_v @ Successors @ X @ Y )
=> ( ( sCC_Bl649662514949026229able_v @ Successors @ Y @ X )
=> ( member_v @ Y @ S3 ) ) ) ) ) ) ).
% graph.sccE
thf(fact_284_bot_Oextremum,axiom,
! [A: set_set_v] : ( ord_le5216385588623774835_set_v @ bot_bot_set_set_v @ A ) ).
% bot.extremum
thf(fact_285_bot_Oextremum,axiom,
! [A: product_unit] : ( ord_le3221252021190050221t_unit @ bot_bot_Product_unit @ A ) ).
% bot.extremum
thf(fact_286_bot_Oextremum,axiom,
! [A: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ bot_bo723834152578015283od_v_v @ A ) ).
% bot.extremum
thf(fact_287_bot_Oextremum,axiom,
! [A: set_v] : ( ord_less_eq_set_v @ bot_bot_set_v @ A ) ).
% bot.extremum
thf(fact_288_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_289_bot_Oextremum__unique,axiom,
! [A: product_unit] :
( ( ord_le3221252021190050221t_unit @ A @ bot_bot_Product_unit )
= ( A = bot_bot_Product_unit ) ) ).
% bot.extremum_unique
thf(fact_290_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_291_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_292_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_293_bot_Oextremum__uniqueI,axiom,
! [A: product_unit] :
( ( ord_le3221252021190050221t_unit @ A @ bot_bot_Product_unit )
=> ( A = bot_bot_Product_unit ) ) ).
% bot.extremum_uniqueI
thf(fact_294_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_295_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_296_graph_Osclosed,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ! [X4: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X4 @ Vertices )
=> ( ord_le7336532860387713383od_v_v @ ( Successors @ X4 ) @ Vertices ) ) ) ).
% graph.sclosed
thf(fact_297_graph_Osclosed,axiom,
! [Vertices: set_v,Successors: v > set_v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ! [X4: v] :
( ( member_v @ X4 @ Vertices )
=> ( ord_less_eq_set_v @ ( Successors @ X4 ) @ Vertices ) ) ) ).
% graph.sclosed
thf(fact_298_graph_Ora__empty,axiom,
! [Vertices: set_v,Successors: v > set_v,X: v,Y: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl4291963740693775144ding_v @ Successors @ X @ Y @ bot_bo723834152578015283od_v_v )
= ( sCC_Bl649662514949026229able_v @ Successors @ X @ Y ) ) ) ).
% graph.ra_empty
thf(fact_299_list_Osel_I2_J,axiom,
( ( tl_v @ nil_v )
= nil_v ) ).
% list.sel(2)
thf(fact_300_Diff__mono,axiom,
! [A2: set_Product_prod_v_v,C2: set_Product_prod_v_v,D: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A2 @ C2 )
=> ( ( ord_le7336532860387713383od_v_v @ D @ B2 )
=> ( ord_le7336532860387713383od_v_v @ ( minus_4183494784930505774od_v_v @ A2 @ B2 ) @ ( minus_4183494784930505774od_v_v @ C2 @ D ) ) ) ) ).
% Diff_mono
thf(fact_301_Diff__mono,axiom,
! [A2: set_v,C2: set_v,D: set_v,B2: set_v] :
( ( ord_less_eq_set_v @ A2 @ C2 )
=> ( ( ord_less_eq_set_v @ D @ B2 )
=> ( ord_less_eq_set_v @ ( minus_minus_set_v @ A2 @ B2 ) @ ( minus_minus_set_v @ C2 @ D ) ) ) ) ).
% Diff_mono
thf(fact_302_Diff__subset,axiom,
! [A2: set_Product_prod_v_v,B2: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ ( minus_4183494784930505774od_v_v @ A2 @ B2 ) @ A2 ) ).
% Diff_subset
thf(fact_303_Diff__subset,axiom,
! [A2: set_v,B2: set_v] : ( ord_less_eq_set_v @ ( minus_minus_set_v @ A2 @ B2 ) @ A2 ) ).
% Diff_subset
thf(fact_304_double__diff,axiom,
! [A2: set_Product_prod_v_v,B2: set_Product_prod_v_v,C2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A2 @ B2 )
=> ( ( ord_le7336532860387713383od_v_v @ B2 @ C2 )
=> ( ( minus_4183494784930505774od_v_v @ B2 @ ( minus_4183494784930505774od_v_v @ C2 @ A2 ) )
= A2 ) ) ) ).
% double_diff
thf(fact_305_double__diff,axiom,
! [A2: set_v,B2: set_v,C2: set_v] :
( ( ord_less_eq_set_v @ A2 @ B2 )
=> ( ( ord_less_eq_set_v @ B2 @ C2 )
=> ( ( minus_minus_set_v @ B2 @ ( minus_minus_set_v @ C2 @ A2 ) )
= A2 ) ) ) ).
% double_diff
thf(fact_306_graph_Oreachable__edge,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,Y: product_prod_v_v,X: product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( member7453568604450474000od_v_v @ Y @ ( Successors @ X ) )
=> ( sCC_Bl4981926079593201289od_v_v @ Successors @ X @ Y ) ) ) ).
% graph.reachable_edge
thf(fact_307_graph_Oreachable__edge,axiom,
! [Vertices: set_v,Successors: v > set_v,Y: v,X: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( member_v @ Y @ ( Successors @ X ) )
=> ( sCC_Bl649662514949026229able_v @ Successors @ X @ Y ) ) ) ).
% graph.reachable_edge
thf(fact_308_graph_Osucc__reachable,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,X: product_prod_v_v,Y: product_prod_v_v,Z: product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( sCC_Bl4981926079593201289od_v_v @ Successors @ X @ Y )
=> ( ( member7453568604450474000od_v_v @ Z @ ( Successors @ Y ) )
=> ( sCC_Bl4981926079593201289od_v_v @ Successors @ X @ Z ) ) ) ) ).
% graph.succ_reachable
thf(fact_309_graph_Osucc__reachable,axiom,
! [Vertices: set_v,Successors: v > set_v,X: v,Y: v,Z: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl649662514949026229able_v @ Successors @ X @ Y )
=> ( ( member_v @ Z @ ( Successors @ Y ) )
=> ( sCC_Bl649662514949026229able_v @ Successors @ X @ Z ) ) ) ) ).
% graph.succ_reachable
thf(fact_310_graph_Oreachable_Ocases,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,A1: product_prod_v_v,A22: product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( sCC_Bl4981926079593201289od_v_v @ Successors @ A1 @ A22 )
=> ( ( A22 != A1 )
=> ~ ! [Y2: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ Y2 @ ( Successors @ A1 ) )
=> ~ ( sCC_Bl4981926079593201289od_v_v @ Successors @ Y2 @ A22 ) ) ) ) ) ).
% graph.reachable.cases
thf(fact_311_graph_Oreachable_Ocases,axiom,
! [Vertices: set_v,Successors: v > set_v,A1: v,A22: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl649662514949026229able_v @ Successors @ A1 @ A22 )
=> ( ( A22 != A1 )
=> ~ ! [Y2: v] :
( ( member_v @ Y2 @ ( Successors @ A1 ) )
=> ~ ( sCC_Bl649662514949026229able_v @ Successors @ Y2 @ A22 ) ) ) ) ) ).
% graph.reachable.cases
thf(fact_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,A22: product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( sCC_Bl4981926079593201289od_v_v @ Successors @ A1 @ A22 )
= ( ? [X2: product_prod_v_v] :
( ( A1 = X2 )
& ( A22 = X2 ) )
| ? [X2: product_prod_v_v,Y3: product_prod_v_v,Z2: product_prod_v_v] :
( ( A1 = X2 )
& ( A22 = Z2 )
& ( member7453568604450474000od_v_v @ Y3 @ ( Successors @ X2 ) )
& ( sCC_Bl4981926079593201289od_v_v @ Successors @ Y3 @ Z2 ) ) ) ) ) ).
% graph.reachable.simps
thf(fact_313_graph_Oreachable_Osimps,axiom,
! [Vertices: set_v,Successors: v > set_v,A1: v,A22: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl649662514949026229able_v @ Successors @ A1 @ A22 )
= ( ? [X2: v] :
( ( A1 = X2 )
& ( A22 = X2 ) )
| ? [X2: v,Y3: v,Z2: v] :
( ( A1 = X2 )
& ( A22 = Z2 )
& ( member_v @ Y3 @ ( Successors @ X2 ) )
& ( sCC_Bl649662514949026229able_v @ Successors @ Y3 @ Z2 ) ) ) ) ) ).
% graph.reachable.simps
thf(fact_314_graph_Oreachable__trans,axiom,
! [Vertices: set_v,Successors: v > set_v,X: v,Y: v,Z: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl649662514949026229able_v @ Successors @ X @ Y )
=> ( ( sCC_Bl649662514949026229able_v @ Successors @ Y @ Z )
=> ( sCC_Bl649662514949026229able_v @ Successors @ X @ Z ) ) ) ) ).
% graph.reachable_trans
thf(fact_315_graph_Oreachable__end__induct,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,X: product_prod_v_v,Y: product_prod_v_v,P: product_prod_v_v > product_prod_v_v > $o] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( sCC_Bl4981926079593201289od_v_v @ Successors @ X @ Y )
=> ( ! [X3: product_prod_v_v] : ( P @ X3 @ X3 )
=> ( ! [X3: product_prod_v_v,Y2: product_prod_v_v,Z3: product_prod_v_v] :
( ( P @ X3 @ Y2 )
=> ( ( member7453568604450474000od_v_v @ Z3 @ ( Successors @ Y2 ) )
=> ( P @ X3 @ Z3 ) ) )
=> ( P @ X @ Y ) ) ) ) ) ).
% graph.reachable_end_induct
thf(fact_316_graph_Oreachable__end__induct,axiom,
! [Vertices: set_v,Successors: v > set_v,X: v,Y: v,P: v > v > $o] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl649662514949026229able_v @ Successors @ X @ Y )
=> ( ! [X3: v] : ( P @ X3 @ X3 )
=> ( ! [X3: v,Y2: v,Z3: v] :
( ( P @ X3 @ Y2 )
=> ( ( member_v @ Z3 @ ( Successors @ Y2 ) )
=> ( P @ X3 @ Z3 ) ) )
=> ( P @ X @ Y ) ) ) ) ) ).
% graph.reachable_end_induct
thf(fact_317_graph_Oreachable__refl,axiom,
! [Vertices: set_v,Successors: v > set_v,X: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( sCC_Bl649662514949026229able_v @ Successors @ X @ X ) ) ).
% graph.reachable_refl
thf(fact_318_graph_Oreachable__succ,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,Y: product_prod_v_v,X: product_prod_v_v,Z: product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( member7453568604450474000od_v_v @ Y @ ( Successors @ X ) )
=> ( ( sCC_Bl4981926079593201289od_v_v @ Successors @ Y @ Z )
=> ( sCC_Bl4981926079593201289od_v_v @ Successors @ X @ Z ) ) ) ) ).
% graph.reachable_succ
thf(fact_319_graph_Oreachable__succ,axiom,
! [Vertices: set_v,Successors: v > set_v,Y: v,X: v,Z: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( member_v @ Y @ ( Successors @ X ) )
=> ( ( sCC_Bl649662514949026229able_v @ Successors @ Y @ Z )
=> ( sCC_Bl649662514949026229able_v @ Successors @ X @ Z ) ) ) ) ).
% graph.reachable_succ
thf(fact_320_graph_Ora__trans,axiom,
! [Vertices: set_v,Successors: v > set_v,X: v,Y: v,E4: set_Product_prod_v_v,Z: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl4291963740693775144ding_v @ Successors @ X @ Y @ E4 )
=> ( ( sCC_Bl4291963740693775144ding_v @ Successors @ Y @ Z @ E4 )
=> ( sCC_Bl4291963740693775144ding_v @ Successors @ X @ Z @ E4 ) ) ) ) ).
% graph.ra_trans
thf(fact_321_graph_Ora__refl,axiom,
! [Vertices: set_v,Successors: v > set_v,X: v,E4: set_Product_prod_v_v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( sCC_Bl4291963740693775144ding_v @ Successors @ X @ X @ E4 ) ) ).
% graph.ra_refl
thf(fact_322_distinct__tl,axiom,
! [Xs: list_v] :
( ( distinct_v @ Xs )
=> ( distinct_v @ ( tl_v @ Xs ) ) ) ).
% distinct_tl
thf(fact_323_graph_Osucc__re,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,Y: product_prod_v_v,X: product_prod_v_v,Z: product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( member7453568604450474000od_v_v @ Y @ ( Successors @ X ) )
=> ( ( sCC_Bl4714988717384592488od_v_v @ Successors @ Y @ Z )
=> ( sCC_Bl4714988717384592488od_v_v @ Successors @ X @ Z ) ) ) ) ).
% graph.succ_re
thf(fact_324_graph_Osucc__re,axiom,
! [Vertices: set_v,Successors: v > set_v,Y: v,X: v,Z: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( member_v @ Y @ ( Successors @ X ) )
=> ( ( sCC_Bl770211535891879572_end_v @ Successors @ Y @ Z )
=> ( sCC_Bl770211535891879572_end_v @ Successors @ X @ Z ) ) ) ) ).
% graph.succ_re
thf(fact_325_graph_Oreachable__end_Ocases,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,A1: product_prod_v_v,A22: product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( sCC_Bl4714988717384592488od_v_v @ Successors @ A1 @ A22 )
=> ( ( A22 != A1 )
=> ~ ! [Y2: product_prod_v_v] :
( ( sCC_Bl4714988717384592488od_v_v @ Successors @ A1 @ Y2 )
=> ~ ( member7453568604450474000od_v_v @ A22 @ ( Successors @ Y2 ) ) ) ) ) ) ).
% graph.reachable_end.cases
thf(fact_326_graph_Oreachable__end_Ocases,axiom,
! [Vertices: set_v,Successors: v > set_v,A1: v,A22: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl770211535891879572_end_v @ Successors @ A1 @ A22 )
=> ( ( A22 != A1 )
=> ~ ! [Y2: v] :
( ( sCC_Bl770211535891879572_end_v @ Successors @ A1 @ Y2 )
=> ~ ( member_v @ A22 @ ( Successors @ Y2 ) ) ) ) ) ) ).
% graph.reachable_end.cases
thf(fact_327_graph_Oreachable__end_Osimps,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,A1: product_prod_v_v,A22: product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( sCC_Bl4714988717384592488od_v_v @ Successors @ A1 @ A22 )
= ( ? [X2: product_prod_v_v] :
( ( A1 = X2 )
& ( A22 = X2 ) )
| ? [X2: product_prod_v_v,Y3: product_prod_v_v,Z2: product_prod_v_v] :
( ( A1 = X2 )
& ( A22 = Z2 )
& ( sCC_Bl4714988717384592488od_v_v @ Successors @ X2 @ Y3 )
& ( member7453568604450474000od_v_v @ Z2 @ ( Successors @ Y3 ) ) ) ) ) ) ).
% graph.reachable_end.simps
thf(fact_328_graph_Oreachable__end_Osimps,axiom,
! [Vertices: set_v,Successors: v > set_v,A1: v,A22: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl770211535891879572_end_v @ Successors @ A1 @ A22 )
= ( ? [X2: v] :
( ( A1 = X2 )
& ( A22 = X2 ) )
| ? [X2: v,Y3: v,Z2: v] :
( ( A1 = X2 )
& ( A22 = Z2 )
& ( sCC_Bl770211535891879572_end_v @ Successors @ X2 @ Y3 )
& ( member_v @ Z2 @ ( Successors @ Y3 ) ) ) ) ) ) ).
% graph.reachable_end.simps
thf(fact_329_graph_Ore__refl,axiom,
! [Vertices: set_v,Successors: v > set_v,X: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( sCC_Bl770211535891879572_end_v @ Successors @ X @ X ) ) ).
% graph.re_refl
thf(fact_330_graph_Ore__succ,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,X: product_prod_v_v,Y: product_prod_v_v,Z: product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( sCC_Bl4714988717384592488od_v_v @ Successors @ X @ Y )
=> ( ( member7453568604450474000od_v_v @ Z @ ( Successors @ Y ) )
=> ( sCC_Bl4714988717384592488od_v_v @ Successors @ X @ Z ) ) ) ) ).
% graph.re_succ
thf(fact_331_graph_Ore__succ,axiom,
! [Vertices: set_v,Successors: v > set_v,X: v,Y: v,Z: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl770211535891879572_end_v @ Successors @ X @ Y )
=> ( ( member_v @ Z @ ( Successors @ Y ) )
=> ( sCC_Bl770211535891879572_end_v @ Successors @ X @ Z ) ) ) ) ).
% graph.re_succ
thf(fact_332_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_333_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_334_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_335_select__convs_I3_J,axiom,
! [Root: v,S: v > set_v,Explored: set_v,Visited: set_v,Vsuccs: v > set_v,Sccs: set_set_v,Stack: list_v,Cstack: list_v,More: product_unit] :
( ( sCC_Bl157864678168468314t_unit @ ( sCC_Bl8064756265740546429t_unit @ Root @ S @ Explored @ Visited @ Vsuccs @ Sccs @ Stack @ Cstack @ More ) )
= Explored ) ).
% select_convs(3)
thf(fact_336_graph_Oedge__ra,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,Y: product_prod_v_v,X: product_prod_v_v,E4: set_Pr2149350503807050951od_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( member7453568604450474000od_v_v @ Y @ ( Successors @ X ) )
=> ( ~ ( member3038538357316246288od_v_v @ ( produc4031800376763917143od_v_v @ X @ Y ) @ E4 )
=> ( sCC_Bl5370300055464682748od_v_v @ Successors @ X @ Y @ E4 ) ) ) ) ).
% graph.edge_ra
thf(fact_337_graph_Oedge__ra,axiom,
! [Vertices: set_v,Successors: v > set_v,Y: v,X: v,E4: set_Product_prod_v_v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( member_v @ Y @ ( Successors @ X ) )
=> ( ~ ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ X @ Y ) @ E4 )
=> ( sCC_Bl4291963740693775144ding_v @ Successors @ X @ Y @ E4 ) ) ) ) ).
% graph.edge_ra
thf(fact_338_graph_Ora__cases,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,X: product_prod_v_v,Y: product_prod_v_v,E4: set_Pr2149350503807050951od_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( sCC_Bl5370300055464682748od_v_v @ Successors @ X @ Y @ E4 )
=> ( ( X = Y )
| ? [Z3: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ Z3 @ ( Successors @ X ) )
& ~ ( member3038538357316246288od_v_v @ ( produc4031800376763917143od_v_v @ X @ Z3 ) @ E4 )
& ( sCC_Bl5370300055464682748od_v_v @ Successors @ Z3 @ Y @ E4 ) ) ) ) ) ).
% graph.ra_cases
thf(fact_339_graph_Ora__cases,axiom,
! [Vertices: set_v,Successors: v > set_v,X: v,Y: v,E4: set_Product_prod_v_v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl4291963740693775144ding_v @ Successors @ X @ Y @ E4 )
=> ( ( X = Y )
| ? [Z3: v] :
( ( member_v @ Z3 @ ( Successors @ X ) )
& ~ ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ X @ Z3 ) @ E4 )
& ( sCC_Bl4291963740693775144ding_v @ Successors @ Z3 @ Y @ E4 ) ) ) ) ) ).
% graph.ra_cases
thf(fact_340_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,A22: product_prod_v_v,A3: set_Pr2149350503807050951od_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( sCC_Bl5370300055464682748od_v_v @ Successors @ A1 @ A22 @ A3 )
=> ( ( A22 != A1 )
=> ~ ! [Y2: product_prod_v_v] :
( ( sCC_Bl5370300055464682748od_v_v @ Successors @ A1 @ Y2 @ A3 )
=> ( ( member7453568604450474000od_v_v @ A22 @ ( Successors @ Y2 ) )
=> ( member3038538357316246288od_v_v @ ( produc4031800376763917143od_v_v @ Y2 @ A22 ) @ A3 ) ) ) ) ) ) ).
% graph.reachable_avoiding.cases
thf(fact_341_graph_Oreachable__avoiding_Ocases,axiom,
! [Vertices: set_v,Successors: v > set_v,A1: v,A22: v,A3: set_Product_prod_v_v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl4291963740693775144ding_v @ Successors @ A1 @ A22 @ A3 )
=> ( ( A22 != A1 )
=> ~ ! [Y2: v] :
( ( sCC_Bl4291963740693775144ding_v @ Successors @ A1 @ Y2 @ A3 )
=> ( ( member_v @ A22 @ ( Successors @ Y2 ) )
=> ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ Y2 @ A22 ) @ A3 ) ) ) ) ) ) ).
% graph.reachable_avoiding.cases
thf(fact_342_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,A22: product_prod_v_v,A3: set_Pr2149350503807050951od_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( sCC_Bl5370300055464682748od_v_v @ Successors @ A1 @ A22 @ A3 )
= ( ? [X2: product_prod_v_v,E6: set_Pr2149350503807050951od_v_v] :
( ( A1 = X2 )
& ( A22 = X2 )
& ( A3 = E6 ) )
| ? [X2: product_prod_v_v,Y3: product_prod_v_v,E6: set_Pr2149350503807050951od_v_v,Z2: product_prod_v_v] :
( ( A1 = X2 )
& ( A22 = Z2 )
& ( A3 = E6 )
& ( sCC_Bl5370300055464682748od_v_v @ Successors @ X2 @ Y3 @ E6 )
& ( member7453568604450474000od_v_v @ Z2 @ ( Successors @ Y3 ) )
& ~ ( member3038538357316246288od_v_v @ ( produc4031800376763917143od_v_v @ Y3 @ Z2 ) @ E6 ) ) ) ) ) ).
% graph.reachable_avoiding.simps
thf(fact_343_graph_Oreachable__avoiding_Osimps,axiom,
! [Vertices: set_v,Successors: v > set_v,A1: v,A22: v,A3: set_Product_prod_v_v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl4291963740693775144ding_v @ Successors @ A1 @ A22 @ A3 )
= ( ? [X2: v,E6: set_Product_prod_v_v] :
( ( A1 = X2 )
& ( A22 = X2 )
& ( A3 = E6 ) )
| ? [X2: v,Y3: v,E6: set_Product_prod_v_v,Z2: v] :
( ( A1 = X2 )
& ( A22 = Z2 )
& ( A3 = E6 )
& ( sCC_Bl4291963740693775144ding_v @ Successors @ X2 @ Y3 @ E6 )
& ( member_v @ Z2 @ ( Successors @ Y3 ) )
& ~ ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ Y3 @ Z2 ) @ E6 ) ) ) ) ) ).
% graph.reachable_avoiding.simps
thf(fact_344_graph_Ora__succ,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,X: product_prod_v_v,Y: product_prod_v_v,E4: set_Pr2149350503807050951od_v_v,Z: product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( sCC_Bl5370300055464682748od_v_v @ Successors @ X @ Y @ E4 )
=> ( ( member7453568604450474000od_v_v @ Z @ ( Successors @ Y ) )
=> ( ~ ( member3038538357316246288od_v_v @ ( produc4031800376763917143od_v_v @ Y @ Z ) @ E4 )
=> ( sCC_Bl5370300055464682748od_v_v @ Successors @ X @ Z @ E4 ) ) ) ) ) ).
% graph.ra_succ
thf(fact_345_graph_Ora__succ,axiom,
! [Vertices: set_v,Successors: v > set_v,X: v,Y: v,E4: set_Product_prod_v_v,Z: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl4291963740693775144ding_v @ Successors @ X @ Y @ E4 )
=> ( ( member_v @ Z @ ( Successors @ Y ) )
=> ( ~ ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ Y @ Z ) @ E4 )
=> ( sCC_Bl4291963740693775144ding_v @ Successors @ X @ Z @ E4 ) ) ) ) ) ).
% graph.ra_succ
thf(fact_346_list_Oset__sel_I2_J,axiom,
! [A: list_P7986770385144383213od_v_v,X: product_prod_v_v] :
( ( A != nil_Product_prod_v_v )
=> ( ( member7453568604450474000od_v_v @ X @ ( set_Product_prod_v_v2 @ ( tl_Product_prod_v_v @ A ) ) )
=> ( member7453568604450474000od_v_v @ X @ ( set_Product_prod_v_v2 @ A ) ) ) ) ).
% list.set_sel(2)
thf(fact_347_list_Oset__sel_I2_J,axiom,
! [A: list_v,X: v] :
( ( A != nil_v )
=> ( ( member_v @ X @ ( set_v2 @ ( tl_v @ A ) ) )
=> ( member_v @ X @ ( set_v2 @ A ) ) ) ) ).
% list.set_sel(2)
thf(fact_348_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_349_graph_Ora__mono,axiom,
! [Vertices: set_v,Successors: v > set_v,X: v,Y: v,E4: set_Product_prod_v_v,E5: set_Product_prod_v_v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl4291963740693775144ding_v @ Successors @ X @ Y @ E4 )
=> ( ( ord_le7336532860387713383od_v_v @ E5 @ E4 )
=> ( sCC_Bl4291963740693775144ding_v @ Successors @ X @ Y @ E5 ) ) ) ) ).
% graph.ra_mono
thf(fact_350_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_351_graph_Ora__reachable,axiom,
! [Vertices: set_v,Successors: v > set_v,X: v,Y: v,E4: set_Product_prod_v_v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl4291963740693775144ding_v @ Successors @ X @ Y @ E4 )
=> ( sCC_Bl649662514949026229able_v @ Successors @ X @ Y ) ) ) ).
% graph.ra_reachable
thf(fact_352_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_353_empty__set,axiom,
( bot_bo723834152578015283od_v_v
= ( set_Product_prod_v_v2 @ nil_Product_prod_v_v ) ) ).
% empty_set
thf(fact_354_empty__set,axiom,
( bot_bot_set_v
= ( set_v2 @ nil_v ) ) ).
% empty_set
thf(fact_355_empty__set,axiom,
( bot_bot_set_set_v
= ( set_set_v2 @ nil_set_v ) ) ).
% empty_set
thf(fact_356_graph_Ore__reachable,axiom,
! [Vertices: set_v,Successors: v > set_v,X: v,Y: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl770211535891879572_end_v @ Successors @ X @ Y )
=> ( sCC_Bl649662514949026229able_v @ Successors @ X @ Y ) ) ) ).
% graph.re_reachable
thf(fact_357_graph_Oreachable__re,axiom,
! [Vertices: set_v,Successors: v > set_v,X: v,Y: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl649662514949026229able_v @ Successors @ X @ Y )
=> ( sCC_Bl770211535891879572_end_v @ Successors @ X @ Y ) ) ) ).
% graph.reachable_re
thf(fact_358_graph_Odfs__S__tl__stack_I2_J,axiom,
! [Vertices: set_v,Successors: v > set_v,V3: v,E: sCC_Bl1394983891496994913t_unit,E2: sCC_Bl1394983891496994913t_unit] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl8953792750115413617t_unit @ Successors @ V3 @ E @ E2 )
=> ( ( ( sCC_Bl8828226123343373779t_unit @ E )
!= nil_v )
=> ! [X4: v] :
( ( member_v @ X4 @ ( set_v2 @ ( tl_v @ ( sCC_Bl8828226123343373779t_unit @ E2 ) ) ) )
=> ( ( sCC_Bl1280885523602775798t_unit @ E2 @ X4 )
= ( sCC_Bl1280885523602775798t_unit @ E @ X4 ) ) ) ) ) ) ).
% graph.dfs_S_tl_stack(2)
thf(fact_359_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_360_graph_Odfs__S__tl__stack_I1_J,axiom,
! [Vertices: set_v,Successors: v > set_v,V3: v,E: sCC_Bl1394983891496994913t_unit,E2: sCC_Bl1394983891496994913t_unit] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl8953792750115413617t_unit @ Successors @ V3 @ E @ E2 )
=> ( ( ( sCC_Bl8828226123343373779t_unit @ E )
!= nil_v )
=> ( ( sCC_Bl8828226123343373779t_unit @ E2 )
!= nil_v ) ) ) ) ).
% graph.dfs_S_tl_stack(1)
thf(fact_361_graph_Oinit__env__pre__dfs,axiom,
! [Vertices: set_v,Successors: v > set_v,V3: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( sCC_Bl36166008131615352t_unit @ Successors @ V3 @ ( sCC_Bl7693227186847904995_env_v @ V3 ) ) ) ).
% graph.init_env_pre_dfs
thf(fact_362_graph_Opre__dfss__pre__dfs,axiom,
! [Vertices: set_v,Successors: v > set_v,V3: v,E: sCC_Bl1394983891496994913t_unit,W: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl1748261141445803503t_unit @ Successors @ V3 @ E )
=> ( ~ ( member_v @ W @ ( sCC_Bl4645233313691564917t_unit @ E ) )
=> ( ( member_v @ W @ ( Successors @ V3 ) )
=> ( sCC_Bl36166008131615352t_unit @ Successors @ W @ E ) ) ) ) ) ).
% graph.pre_dfss_pre_dfs
thf(fact_363_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_364_reachable__visited,axiom,
! [E: sCC_Bl1394983891496994913t_unit,V3: v,W: v] :
( ( sCC_Bl9196236973127232072t_unit @ successors @ E )
=> ( ( member_v @ V3 @ ( sCC_Bl4645233313691564917t_unit @ E ) )
=> ( ( sCC_Bl649662514949026229able_v @ successors @ V3 @ W )
=> ( ! [X3: v] :
( ( member_v @ X3 @ ( sCC_Bl4645233313691564917t_unit @ E ) )
=> ! [Xa: v] :
( ( member_v @ Xa @ ( minus_minus_set_v @ ( successors @ X3 ) @ ( sCC_Bl3795065053823578884t_unit @ E @ X3 ) ) )
=> ( ( sCC_Bl649662514949026229able_v @ successors @ V3 @ X3 )
=> ~ ( sCC_Bl649662514949026229able_v @ successors @ Xa @ W ) ) ) )
=> ( member_v @ W @ ( sCC_Bl4645233313691564917t_unit @ E ) ) ) ) ) ) ).
% reachable_visited
thf(fact_365_is__scc__def,axiom,
! [S3: set_v] :
( ( sCC_Bloemen_is_scc_v @ successors @ S3 )
= ( ( S3 != bot_bot_set_v )
& ( sCC_Bl5398416737448265317bscc_v @ successors @ S3 )
& ! [S4: set_v] :
( ( ( ord_less_eq_set_v @ S3 @ S4 )
& ( sCC_Bl5398416737448265317bscc_v @ successors @ S4 ) )
=> ( S4 = S3 ) ) ) ) ).
% is_scc_def
thf(fact_366_post__dfs__def,axiom,
! [V3: v,E: sCC_Bl1394983891496994913t_unit,E2: sCC_Bl1394983891496994913t_unit] :
( ( sCC_Bl8953792750115413617t_unit @ successors @ V3 @ E @ E2 )
= ( ( sCC_Bl9196236973127232072t_unit @ successors @ E2 )
& ( member_v @ V3 @ ( sCC_Bl4645233313691564917t_unit @ E2 ) )
& ( sCC_Bl5768913643336123637t_unit @ E @ E2 )
& ( ( sCC_Bl3795065053823578884t_unit @ E2 @ V3 )
= ( successors @ V3 ) )
& ! [X2: v] :
( ( member_v @ X2 @ ( sCC_Bl4645233313691564917t_unit @ E ) )
=> ( ( sCC_Bl3795065053823578884t_unit @ E2 @ X2 )
= ( sCC_Bl3795065053823578884t_unit @ E @ X2 ) ) )
& ! [X2: v] :
( ( member_v @ X2 @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E2 ) ) )
=> ( sCC_Bl649662514949026229able_v @ successors @ X2 @ V3 ) )
& ? [Ns2: list_v] :
( ( sCC_Bl8828226123343373779t_unit @ E )
= ( append_v @ Ns2 @ ( sCC_Bl8828226123343373779t_unit @ E2 ) ) )
& ( ( ( member_v @ V3 @ ( sCC_Bl157864678168468314t_unit @ E2 ) )
& ( ( sCC_Bl8828226123343373779t_unit @ E2 )
= ( sCC_Bl8828226123343373779t_unit @ E ) )
& ! [X2: v] :
( ( member_v @ X2 @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E2 ) ) )
=> ( ( sCC_Bl1280885523602775798t_unit @ E2 @ X2 )
= ( sCC_Bl1280885523602775798t_unit @ E @ X2 ) ) ) )
| ( ( ( sCC_Bl8828226123343373779t_unit @ E2 )
!= nil_v )
& ( member_v @ V3 @ ( sCC_Bl1280885523602775798t_unit @ E2 @ ( hd_v @ ( sCC_Bl8828226123343373779t_unit @ E2 ) ) ) )
& ! [X2: v] :
( ( member_v @ X2 @ ( set_v2 @ ( tl_v @ ( sCC_Bl8828226123343373779t_unit @ E2 ) ) ) )
=> ( ( sCC_Bl1280885523602775798t_unit @ E2 @ X2 )
= ( sCC_Bl1280885523602775798t_unit @ E @ X2 ) ) ) ) )
& ( ( sCC_Bl9201514103433284750t_unit @ E2 )
= ( sCC_Bl9201514103433284750t_unit @ E ) ) ) ) ).
% post_dfs_def
thf(fact_367_subscc__add,axiom,
! [S3: set_v,X: v,Y: v] :
( ( sCC_Bl5398416737448265317bscc_v @ successors @ S3 )
=> ( ( member_v @ X @ S3 )
=> ( ( sCC_Bl649662514949026229able_v @ successors @ X @ Y )
=> ( ( sCC_Bl649662514949026229able_v @ successors @ Y @ X )
=> ( sCC_Bl5398416737448265317bscc_v @ successors @ ( insert_v2 @ Y @ S3 ) ) ) ) ) ) ).
% subscc_add
thf(fact_368_diff__shunt__var,axiom,
! [X: set_set_v,Y: set_set_v] :
( ( ( minus_7228012346218142266_set_v @ X @ Y )
= bot_bot_set_set_v )
= ( ord_le5216385588623774835_set_v @ X @ Y ) ) ).
% diff_shunt_var
thf(fact_369_diff__shunt__var,axiom,
! [X: product_unit,Y: product_unit] :
( ( ( minus_3524152463667985524t_unit @ X @ Y )
= bot_bot_Product_unit )
= ( ord_le3221252021190050221t_unit @ X @ Y ) ) ).
% diff_shunt_var
thf(fact_370_diff__shunt__var,axiom,
! [X: set_Product_prod_v_v,Y: set_Product_prod_v_v] :
( ( ( minus_4183494784930505774od_v_v @ X @ Y )
= bot_bo723834152578015283od_v_v )
= ( ord_le7336532860387713383od_v_v @ X @ Y ) ) ).
% diff_shunt_var
thf(fact_371_diff__shunt__var,axiom,
! [X: set_v,Y: set_v] :
( ( ( minus_minus_set_v @ X @ Y )
= bot_bot_set_v )
= ( ord_less_eq_set_v @ X @ Y ) ) ).
% diff_shunt_var
thf(fact_372_sum_Oinject_I1_J,axiom,
! [X1: produc5741669702376414499t_unit,Y1: produc5741669702376414499t_unit] :
( ( ( sum_In526841707622398774t_unit @ X1 )
= ( sum_In526841707622398774t_unit @ Y1 ) )
= ( X1 = Y1 ) ) ).
% sum.inject(1)
thf(fact_373_old_Osum_Oinject_I1_J,axiom,
! [A: produc5741669702376414499t_unit,A4: produc5741669702376414499t_unit] :
( ( ( sum_In526841707622398774t_unit @ A )
= ( sum_In526841707622398774t_unit @ A4 ) )
= ( A = A4 ) ) ).
% old.sum.inject(1)
thf(fact_374_insertCI,axiom,
! [A: set_v,B2: set_set_v,B: set_v] :
( ( ~ ( member_set_v @ A @ B2 )
=> ( A = B ) )
=> ( member_set_v @ A @ ( insert_set_v2 @ B @ B2 ) ) ) ).
% insertCI
thf(fact_375_insertCI,axiom,
! [A: v,B2: set_v,B: v] :
( ( ~ ( member_v @ A @ B2 )
=> ( A = B ) )
=> ( member_v @ A @ ( insert_v2 @ B @ B2 ) ) ) ).
% insertCI
thf(fact_376_insertCI,axiom,
! [A: product_prod_v_v,B2: set_Product_prod_v_v,B: product_prod_v_v] :
( ( ~ ( member7453568604450474000od_v_v @ A @ B2 )
=> ( A = B ) )
=> ( member7453568604450474000od_v_v @ A @ ( insert1338601472111419319od_v_v @ B @ B2 ) ) ) ).
% insertCI
thf(fact_377_insert__iff,axiom,
! [A: set_v,B: set_v,A2: set_set_v] :
( ( member_set_v @ A @ ( insert_set_v2 @ B @ A2 ) )
= ( ( A = B )
| ( member_set_v @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_378_insert__iff,axiom,
! [A: v,B: v,A2: set_v] :
( ( member_v @ A @ ( insert_v2 @ B @ A2 ) )
= ( ( A = B )
| ( member_v @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_379_insert__iff,axiom,
! [A: product_prod_v_v,B: product_prod_v_v,A2: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ A @ ( insert1338601472111419319od_v_v @ B @ A2 ) )
= ( ( A = B )
| ( member7453568604450474000od_v_v @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_380_insert__absorb2,axiom,
! [X: v,A2: set_v] :
( ( insert_v2 @ X @ ( insert_v2 @ X @ A2 ) )
= ( insert_v2 @ X @ A2 ) ) ).
% insert_absorb2
thf(fact_381_insert__absorb2,axiom,
! [X: product_prod_v_v,A2: set_Product_prod_v_v] :
( ( insert1338601472111419319od_v_v @ X @ ( insert1338601472111419319od_v_v @ X @ A2 ) )
= ( insert1338601472111419319od_v_v @ X @ A2 ) ) ).
% insert_absorb2
thf(fact_382_insert__absorb2,axiom,
! [X: set_v,A2: set_set_v] :
( ( insert_set_v2 @ X @ ( insert_set_v2 @ X @ A2 ) )
= ( insert_set_v2 @ X @ A2 ) ) ).
% insert_absorb2
thf(fact_383_singletonI,axiom,
! [A: product_prod_v_v] : ( member7453568604450474000od_v_v @ A @ ( insert1338601472111419319od_v_v @ A @ bot_bo723834152578015283od_v_v ) ) ).
% singletonI
thf(fact_384_singletonI,axiom,
! [A: v] : ( member_v @ A @ ( insert_v2 @ A @ bot_bot_set_v ) ) ).
% singletonI
thf(fact_385_singletonI,axiom,
! [A: set_v] : ( member_set_v @ A @ ( insert_set_v2 @ A @ bot_bot_set_set_v ) ) ).
% singletonI
thf(fact_386_insert__subset,axiom,
! [X: set_v,A2: set_set_v,B2: set_set_v] :
( ( ord_le5216385588623774835_set_v @ ( insert_set_v2 @ X @ A2 ) @ B2 )
= ( ( member_set_v @ X @ B2 )
& ( ord_le5216385588623774835_set_v @ A2 @ B2 ) ) ) ).
% insert_subset
thf(fact_387_insert__subset,axiom,
! [X: product_prod_v_v,A2: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ ( insert1338601472111419319od_v_v @ X @ A2 ) @ B2 )
= ( ( member7453568604450474000od_v_v @ X @ B2 )
& ( ord_le7336532860387713383od_v_v @ A2 @ B2 ) ) ) ).
% insert_subset
thf(fact_388_insert__subset,axiom,
! [X: v,A2: set_v,B2: set_v] :
( ( ord_less_eq_set_v @ ( insert_v2 @ X @ A2 ) @ B2 )
= ( ( member_v @ X @ B2 )
& ( ord_less_eq_set_v @ A2 @ B2 ) ) ) ).
% insert_subset
thf(fact_389_Diff__insert0,axiom,
! [X: set_v,A2: set_set_v,B2: set_set_v] :
( ~ ( member_set_v @ X @ A2 )
=> ( ( minus_7228012346218142266_set_v @ A2 @ ( insert_set_v2 @ X @ B2 ) )
= ( minus_7228012346218142266_set_v @ A2 @ B2 ) ) ) ).
% Diff_insert0
thf(fact_390_Diff__insert0,axiom,
! [X: product_prod_v_v,A2: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ~ ( member7453568604450474000od_v_v @ X @ A2 )
=> ( ( minus_4183494784930505774od_v_v @ A2 @ ( insert1338601472111419319od_v_v @ X @ B2 ) )
= ( minus_4183494784930505774od_v_v @ A2 @ B2 ) ) ) ).
% Diff_insert0
thf(fact_391_Diff__insert0,axiom,
! [X: v,A2: set_v,B2: set_v] :
( ~ ( member_v @ X @ A2 )
=> ( ( minus_minus_set_v @ A2 @ ( insert_v2 @ X @ B2 ) )
= ( minus_minus_set_v @ A2 @ B2 ) ) ) ).
% Diff_insert0
thf(fact_392_insert__Diff1,axiom,
! [X: set_v,B2: set_set_v,A2: set_set_v] :
( ( member_set_v @ X @ B2 )
=> ( ( minus_7228012346218142266_set_v @ ( insert_set_v2 @ X @ A2 ) @ B2 )
= ( minus_7228012346218142266_set_v @ A2 @ B2 ) ) ) ).
% insert_Diff1
thf(fact_393_insert__Diff1,axiom,
! [X: product_prod_v_v,B2: set_Product_prod_v_v,A2: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X @ B2 )
=> ( ( minus_4183494784930505774od_v_v @ ( insert1338601472111419319od_v_v @ X @ A2 ) @ B2 )
= ( minus_4183494784930505774od_v_v @ A2 @ B2 ) ) ) ).
% insert_Diff1
thf(fact_394_insert__Diff1,axiom,
! [X: v,B2: set_v,A2: set_v] :
( ( member_v @ X @ B2 )
=> ( ( minus_minus_set_v @ ( insert_v2 @ X @ A2 ) @ B2 )
= ( minus_minus_set_v @ A2 @ B2 ) ) ) ).
% insert_Diff1
thf(fact_395_singleton__insert__inj__eq,axiom,
! [B: set_v,A: set_v,A2: set_set_v] :
( ( ( insert_set_v2 @ B @ bot_bot_set_set_v )
= ( insert_set_v2 @ A @ A2 ) )
= ( ( A = B )
& ( ord_le5216385588623774835_set_v @ A2 @ ( insert_set_v2 @ B @ bot_bot_set_set_v ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_396_singleton__insert__inj__eq,axiom,
! [B: product_prod_v_v,A: product_prod_v_v,A2: set_Product_prod_v_v] :
( ( ( insert1338601472111419319od_v_v @ B @ bot_bo723834152578015283od_v_v )
= ( insert1338601472111419319od_v_v @ A @ A2 ) )
= ( ( A = B )
& ( ord_le7336532860387713383od_v_v @ A2 @ ( insert1338601472111419319od_v_v @ B @ bot_bo723834152578015283od_v_v ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_397_singleton__insert__inj__eq,axiom,
! [B: v,A: v,A2: set_v] :
( ( ( insert_v2 @ B @ bot_bot_set_v )
= ( insert_v2 @ A @ A2 ) )
= ( ( A = B )
& ( ord_less_eq_set_v @ A2 @ ( insert_v2 @ B @ bot_bot_set_v ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_398_singleton__insert__inj__eq_H,axiom,
! [A: set_v,A2: set_set_v,B: set_v] :
( ( ( insert_set_v2 @ A @ A2 )
= ( insert_set_v2 @ B @ bot_bot_set_set_v ) )
= ( ( A = B )
& ( ord_le5216385588623774835_set_v @ A2 @ ( insert_set_v2 @ B @ bot_bot_set_set_v ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_399_singleton__insert__inj__eq_H,axiom,
! [A: product_prod_v_v,A2: set_Product_prod_v_v,B: product_prod_v_v] :
( ( ( insert1338601472111419319od_v_v @ A @ A2 )
= ( insert1338601472111419319od_v_v @ B @ bot_bo723834152578015283od_v_v ) )
= ( ( A = B )
& ( ord_le7336532860387713383od_v_v @ A2 @ ( insert1338601472111419319od_v_v @ B @ bot_bo723834152578015283od_v_v ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_400_singleton__insert__inj__eq_H,axiom,
! [A: v,A2: set_v,B: v] :
( ( ( insert_v2 @ A @ A2 )
= ( insert_v2 @ B @ bot_bot_set_v ) )
= ( ( A = B )
& ( ord_less_eq_set_v @ A2 @ ( insert_v2 @ B @ bot_bot_set_v ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_401_insert__Diff__single,axiom,
! [A: product_prod_v_v,A2: set_Product_prod_v_v] :
( ( insert1338601472111419319od_v_v @ A @ ( minus_4183494784930505774od_v_v @ A2 @ ( insert1338601472111419319od_v_v @ A @ bot_bo723834152578015283od_v_v ) ) )
= ( insert1338601472111419319od_v_v @ A @ A2 ) ) ).
% insert_Diff_single
thf(fact_402_insert__Diff__single,axiom,
! [A: set_v,A2: set_set_v] :
( ( insert_set_v2 @ A @ ( minus_7228012346218142266_set_v @ A2 @ ( insert_set_v2 @ A @ bot_bot_set_set_v ) ) )
= ( insert_set_v2 @ A @ A2 ) ) ).
% insert_Diff_single
thf(fact_403_insert__Diff__single,axiom,
! [A: v,A2: set_v] :
( ( insert_v2 @ A @ ( minus_minus_set_v @ A2 @ ( insert_v2 @ A @ bot_bot_set_v ) ) )
= ( insert_v2 @ A @ A2 ) ) ).
% insert_Diff_single
thf(fact_404_List_Oset__insert,axiom,
! [X: product_prod_v_v,Xs: list_P7986770385144383213od_v_v] :
( ( set_Product_prod_v_v2 @ ( insert4539780211034306307od_v_v @ X @ Xs ) )
= ( insert1338601472111419319od_v_v @ X @ ( set_Product_prod_v_v2 @ Xs ) ) ) ).
% List.set_insert
thf(fact_405_List_Oset__insert,axiom,
! [X: set_v,Xs: list_set_v] :
( ( set_set_v2 @ ( insert_set_v @ X @ Xs ) )
= ( insert_set_v2 @ X @ ( set_set_v2 @ Xs ) ) ) ).
% List.set_insert
thf(fact_406_List_Oset__insert,axiom,
! [X: v,Xs: list_v] :
( ( set_v2 @ ( insert_v @ X @ Xs ) )
= ( insert_v2 @ X @ ( set_v2 @ Xs ) ) ) ).
% List.set_insert
thf(fact_407_insertE,axiom,
! [A: set_v,B: set_v,A2: set_set_v] :
( ( member_set_v @ A @ ( insert_set_v2 @ B @ A2 ) )
=> ( ( A != B )
=> ( member_set_v @ A @ A2 ) ) ) ).
% insertE
thf(fact_408_insertE,axiom,
! [A: v,B: v,A2: set_v] :
( ( member_v @ A @ ( insert_v2 @ B @ A2 ) )
=> ( ( A != B )
=> ( member_v @ A @ A2 ) ) ) ).
% insertE
thf(fact_409_insertE,axiom,
! [A: product_prod_v_v,B: product_prod_v_v,A2: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ A @ ( insert1338601472111419319od_v_v @ B @ A2 ) )
=> ( ( A != B )
=> ( member7453568604450474000od_v_v @ A @ A2 ) ) ) ).
% insertE
thf(fact_410_insertI1,axiom,
! [A: set_v,B2: set_set_v] : ( member_set_v @ A @ ( insert_set_v2 @ A @ B2 ) ) ).
% insertI1
thf(fact_411_insertI1,axiom,
! [A: v,B2: set_v] : ( member_v @ A @ ( insert_v2 @ A @ B2 ) ) ).
% insertI1
thf(fact_412_insertI1,axiom,
! [A: product_prod_v_v,B2: set_Product_prod_v_v] : ( member7453568604450474000od_v_v @ A @ ( insert1338601472111419319od_v_v @ A @ B2 ) ) ).
% insertI1
thf(fact_413_insertI2,axiom,
! [A: set_v,B2: set_set_v,B: set_v] :
( ( member_set_v @ A @ B2 )
=> ( member_set_v @ A @ ( insert_set_v2 @ B @ B2 ) ) ) ).
% insertI2
thf(fact_414_insertI2,axiom,
! [A: v,B2: set_v,B: v] :
( ( member_v @ A @ B2 )
=> ( member_v @ A @ ( insert_v2 @ B @ B2 ) ) ) ).
% insertI2
thf(fact_415_insertI2,axiom,
! [A: product_prod_v_v,B2: set_Product_prod_v_v,B: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ A @ B2 )
=> ( member7453568604450474000od_v_v @ A @ ( insert1338601472111419319od_v_v @ B @ B2 ) ) ) ).
% insertI2
thf(fact_416_Set_Oset__insert,axiom,
! [X: set_v,A2: set_set_v] :
( ( member_set_v @ X @ A2 )
=> ~ ! [B7: set_set_v] :
( ( A2
= ( insert_set_v2 @ X @ B7 ) )
=> ( member_set_v @ X @ B7 ) ) ) ).
% Set.set_insert
thf(fact_417_Set_Oset__insert,axiom,
! [X: v,A2: set_v] :
( ( member_v @ X @ A2 )
=> ~ ! [B7: set_v] :
( ( A2
= ( insert_v2 @ X @ B7 ) )
=> ( member_v @ X @ B7 ) ) ) ).
% Set.set_insert
thf(fact_418_Set_Oset__insert,axiom,
! [X: product_prod_v_v,A2: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X @ A2 )
=> ~ ! [B7: set_Product_prod_v_v] :
( ( A2
= ( insert1338601472111419319od_v_v @ X @ B7 ) )
=> ( member7453568604450474000od_v_v @ X @ B7 ) ) ) ).
% Set.set_insert
thf(fact_419_insert__ident,axiom,
! [X: set_v,A2: set_set_v,B2: set_set_v] :
( ~ ( member_set_v @ X @ A2 )
=> ( ~ ( member_set_v @ X @ B2 )
=> ( ( ( insert_set_v2 @ X @ A2 )
= ( insert_set_v2 @ X @ B2 ) )
= ( A2 = B2 ) ) ) ) ).
% insert_ident
thf(fact_420_insert__ident,axiom,
! [X: v,A2: set_v,B2: set_v] :
( ~ ( member_v @ X @ A2 )
=> ( ~ ( member_v @ X @ B2 )
=> ( ( ( insert_v2 @ X @ A2 )
= ( insert_v2 @ X @ B2 ) )
= ( A2 = B2 ) ) ) ) ).
% insert_ident
thf(fact_421_insert__ident,axiom,
! [X: product_prod_v_v,A2: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ~ ( member7453568604450474000od_v_v @ X @ A2 )
=> ( ~ ( member7453568604450474000od_v_v @ X @ B2 )
=> ( ( ( insert1338601472111419319od_v_v @ X @ A2 )
= ( insert1338601472111419319od_v_v @ X @ B2 ) )
= ( A2 = B2 ) ) ) ) ).
% insert_ident
thf(fact_422_insert__absorb,axiom,
! [A: set_v,A2: set_set_v] :
( ( member_set_v @ A @ A2 )
=> ( ( insert_set_v2 @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_423_insert__absorb,axiom,
! [A: v,A2: set_v] :
( ( member_v @ A @ A2 )
=> ( ( insert_v2 @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_424_insert__absorb,axiom,
! [A: product_prod_v_v,A2: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ A @ A2 )
=> ( ( insert1338601472111419319od_v_v @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_425_insert__eq__iff,axiom,
! [A: set_v,A2: set_set_v,B: set_v,B2: set_set_v] :
( ~ ( member_set_v @ A @ A2 )
=> ( ~ ( member_set_v @ B @ B2 )
=> ( ( ( insert_set_v2 @ A @ A2 )
= ( insert_set_v2 @ B @ B2 ) )
= ( ( ( A = B )
=> ( A2 = B2 ) )
& ( ( A != B )
=> ? [C3: set_set_v] :
( ( A2
= ( insert_set_v2 @ B @ C3 ) )
& ~ ( member_set_v @ B @ C3 )
& ( B2
= ( insert_set_v2 @ A @ C3 ) )
& ~ ( member_set_v @ A @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_426_insert__eq__iff,axiom,
! [A: v,A2: set_v,B: v,B2: set_v] :
( ~ ( member_v @ A @ A2 )
=> ( ~ ( member_v @ B @ B2 )
=> ( ( ( insert_v2 @ A @ A2 )
= ( insert_v2 @ B @ B2 ) )
= ( ( ( A = B )
=> ( A2 = B2 ) )
& ( ( A != B )
=> ? [C3: set_v] :
( ( A2
= ( insert_v2 @ B @ C3 ) )
& ~ ( member_v @ B @ C3 )
& ( B2
= ( insert_v2 @ A @ C3 ) )
& ~ ( member_v @ A @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_427_insert__eq__iff,axiom,
! [A: product_prod_v_v,A2: set_Product_prod_v_v,B: product_prod_v_v,B2: set_Product_prod_v_v] :
( ~ ( member7453568604450474000od_v_v @ A @ A2 )
=> ( ~ ( member7453568604450474000od_v_v @ B @ B2 )
=> ( ( ( insert1338601472111419319od_v_v @ A @ A2 )
= ( insert1338601472111419319od_v_v @ B @ B2 ) )
= ( ( ( A = B )
=> ( A2 = B2 ) )
& ( ( A != B )
=> ? [C3: set_Product_prod_v_v] :
( ( A2
= ( insert1338601472111419319od_v_v @ B @ C3 ) )
& ~ ( member7453568604450474000od_v_v @ B @ C3 )
& ( B2
= ( insert1338601472111419319od_v_v @ A @ C3 ) )
& ~ ( member7453568604450474000od_v_v @ A @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_428_insert__commute,axiom,
! [X: v,Y: v,A2: set_v] :
( ( insert_v2 @ X @ ( insert_v2 @ Y @ A2 ) )
= ( insert_v2 @ Y @ ( insert_v2 @ X @ A2 ) ) ) ).
% insert_commute
thf(fact_429_insert__commute,axiom,
! [X: product_prod_v_v,Y: product_prod_v_v,A2: set_Product_prod_v_v] :
( ( insert1338601472111419319od_v_v @ X @ ( insert1338601472111419319od_v_v @ Y @ A2 ) )
= ( insert1338601472111419319od_v_v @ Y @ ( insert1338601472111419319od_v_v @ X @ A2 ) ) ) ).
% insert_commute
thf(fact_430_insert__commute,axiom,
! [X: set_v,Y: set_v,A2: set_set_v] :
( ( insert_set_v2 @ X @ ( insert_set_v2 @ Y @ A2 ) )
= ( insert_set_v2 @ Y @ ( insert_set_v2 @ X @ A2 ) ) ) ).
% insert_commute
thf(fact_431_mk__disjoint__insert,axiom,
! [A: set_v,A2: set_set_v] :
( ( member_set_v @ A @ A2 )
=> ? [B7: set_set_v] :
( ( A2
= ( insert_set_v2 @ A @ B7 ) )
& ~ ( member_set_v @ A @ B7 ) ) ) ).
% mk_disjoint_insert
thf(fact_432_mk__disjoint__insert,axiom,
! [A: v,A2: set_v] :
( ( member_v @ A @ A2 )
=> ? [B7: set_v] :
( ( A2
= ( insert_v2 @ A @ B7 ) )
& ~ ( member_v @ A @ B7 ) ) ) ).
% mk_disjoint_insert
thf(fact_433_mk__disjoint__insert,axiom,
! [A: product_prod_v_v,A2: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ A @ A2 )
=> ? [B7: set_Product_prod_v_v] :
( ( A2
= ( insert1338601472111419319od_v_v @ A @ B7 ) )
& ~ ( member7453568604450474000od_v_v @ A @ B7 ) ) ) ).
% mk_disjoint_insert
thf(fact_434_bot__set__def,axiom,
( bot_bo723834152578015283od_v_v
= ( collec140062887454715474od_v_v @ bot_bo8461541820394803818_v_v_o ) ) ).
% bot_set_def
thf(fact_435_bot__set__def,axiom,
( bot_bot_set_v
= ( collect_v @ bot_bot_v_o ) ) ).
% bot_set_def
thf(fact_436_bot__set__def,axiom,
( bot_bot_set_set_v
= ( collect_set_v @ bot_bot_set_v_o ) ) ).
% bot_set_def
thf(fact_437_bot__empty__eq,axiom,
( bot_bo8461541820394803818_v_v_o
= ( ^ [X2: product_prod_v_v] : ( member7453568604450474000od_v_v @ X2 @ bot_bo723834152578015283od_v_v ) ) ) ).
% bot_empty_eq
thf(fact_438_bot__empty__eq,axiom,
( bot_bot_v_o
= ( ^ [X2: v] : ( member_v @ X2 @ bot_bot_set_v ) ) ) ).
% bot_empty_eq
thf(fact_439_bot__empty__eq,axiom,
( bot_bot_set_v_o
= ( ^ [X2: set_v] : ( member_set_v @ X2 @ bot_bot_set_set_v ) ) ) ).
% bot_empty_eq
thf(fact_440_singletonD,axiom,
! [B: product_prod_v_v,A: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ B @ ( insert1338601472111419319od_v_v @ A @ bot_bo723834152578015283od_v_v ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_441_singletonD,axiom,
! [B: v,A: v] :
( ( member_v @ B @ ( insert_v2 @ A @ bot_bot_set_v ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_442_singletonD,axiom,
! [B: set_v,A: set_v] :
( ( member_set_v @ B @ ( insert_set_v2 @ A @ bot_bot_set_set_v ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_443_singleton__iff,axiom,
! [B: product_prod_v_v,A: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ B @ ( insert1338601472111419319od_v_v @ A @ bot_bo723834152578015283od_v_v ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_444_singleton__iff,axiom,
! [B: v,A: v] :
( ( member_v @ B @ ( insert_v2 @ A @ bot_bot_set_v ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_445_singleton__iff,axiom,
! [B: set_v,A: set_v] :
( ( member_set_v @ B @ ( insert_set_v2 @ A @ bot_bot_set_set_v ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_446_doubleton__eq__iff,axiom,
! [A: product_prod_v_v,B: product_prod_v_v,C: product_prod_v_v,D2: product_prod_v_v] :
( ( ( insert1338601472111419319od_v_v @ A @ ( insert1338601472111419319od_v_v @ B @ bot_bo723834152578015283od_v_v ) )
= ( insert1338601472111419319od_v_v @ C @ ( insert1338601472111419319od_v_v @ D2 @ bot_bo723834152578015283od_v_v ) ) )
= ( ( ( A = C )
& ( B = D2 ) )
| ( ( A = D2 )
& ( B = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_447_doubleton__eq__iff,axiom,
! [A: v,B: v,C: v,D2: v] :
( ( ( insert_v2 @ A @ ( insert_v2 @ B @ bot_bot_set_v ) )
= ( insert_v2 @ C @ ( insert_v2 @ D2 @ bot_bot_set_v ) ) )
= ( ( ( A = C )
& ( B = D2 ) )
| ( ( A = D2 )
& ( B = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_448_doubleton__eq__iff,axiom,
! [A: set_v,B: set_v,C: set_v,D2: set_v] :
( ( ( insert_set_v2 @ A @ ( insert_set_v2 @ B @ bot_bot_set_set_v ) )
= ( insert_set_v2 @ C @ ( insert_set_v2 @ D2 @ bot_bot_set_set_v ) ) )
= ( ( ( A = C )
& ( B = D2 ) )
| ( ( A = D2 )
& ( B = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_449_insert__not__empty,axiom,
! [A: product_prod_v_v,A2: set_Product_prod_v_v] :
( ( insert1338601472111419319od_v_v @ A @ A2 )
!= bot_bo723834152578015283od_v_v ) ).
% insert_not_empty
thf(fact_450_insert__not__empty,axiom,
! [A: v,A2: set_v] :
( ( insert_v2 @ A @ A2 )
!= bot_bot_set_v ) ).
% insert_not_empty
thf(fact_451_insert__not__empty,axiom,
! [A: set_v,A2: set_set_v] :
( ( insert_set_v2 @ A @ A2 )
!= bot_bot_set_set_v ) ).
% insert_not_empty
thf(fact_452_singleton__inject,axiom,
! [A: product_prod_v_v,B: product_prod_v_v] :
( ( ( insert1338601472111419319od_v_v @ A @ bot_bo723834152578015283od_v_v )
= ( insert1338601472111419319od_v_v @ B @ bot_bo723834152578015283od_v_v ) )
=> ( A = B ) ) ).
% singleton_inject
thf(fact_453_singleton__inject,axiom,
! [A: v,B: v] :
( ( ( insert_v2 @ A @ bot_bot_set_v )
= ( insert_v2 @ B @ bot_bot_set_v ) )
=> ( A = B ) ) ).
% singleton_inject
thf(fact_454_singleton__inject,axiom,
! [A: set_v,B: set_v] :
( ( ( insert_set_v2 @ A @ bot_bot_set_set_v )
= ( insert_set_v2 @ B @ bot_bot_set_set_v ) )
=> ( A = B ) ) ).
% singleton_inject
thf(fact_455_subset__insertI2,axiom,
! [A2: set_set_v,B2: set_set_v,B: set_v] :
( ( ord_le5216385588623774835_set_v @ A2 @ B2 )
=> ( ord_le5216385588623774835_set_v @ A2 @ ( insert_set_v2 @ B @ B2 ) ) ) ).
% subset_insertI2
thf(fact_456_subset__insertI2,axiom,
! [A2: set_Product_prod_v_v,B2: set_Product_prod_v_v,B: product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A2 @ B2 )
=> ( ord_le7336532860387713383od_v_v @ A2 @ ( insert1338601472111419319od_v_v @ B @ B2 ) ) ) ).
% subset_insertI2
thf(fact_457_subset__insertI2,axiom,
! [A2: set_v,B2: set_v,B: v] :
( ( ord_less_eq_set_v @ A2 @ B2 )
=> ( ord_less_eq_set_v @ A2 @ ( insert_v2 @ B @ B2 ) ) ) ).
% subset_insertI2
thf(fact_458_subset__insertI,axiom,
! [B2: set_set_v,A: set_v] : ( ord_le5216385588623774835_set_v @ B2 @ ( insert_set_v2 @ A @ B2 ) ) ).
% subset_insertI
thf(fact_459_subset__insertI,axiom,
! [B2: set_Product_prod_v_v,A: product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ B2 @ ( insert1338601472111419319od_v_v @ A @ B2 ) ) ).
% subset_insertI
thf(fact_460_subset__insertI,axiom,
! [B2: set_v,A: v] : ( ord_less_eq_set_v @ B2 @ ( insert_v2 @ A @ B2 ) ) ).
% subset_insertI
thf(fact_461_subset__insert,axiom,
! [X: set_v,A2: set_set_v,B2: set_set_v] :
( ~ ( member_set_v @ X @ A2 )
=> ( ( ord_le5216385588623774835_set_v @ A2 @ ( insert_set_v2 @ X @ B2 ) )
= ( ord_le5216385588623774835_set_v @ A2 @ B2 ) ) ) ).
% subset_insert
thf(fact_462_subset__insert,axiom,
! [X: product_prod_v_v,A2: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ~ ( member7453568604450474000od_v_v @ X @ A2 )
=> ( ( ord_le7336532860387713383od_v_v @ A2 @ ( insert1338601472111419319od_v_v @ X @ B2 ) )
= ( ord_le7336532860387713383od_v_v @ A2 @ B2 ) ) ) ).
% subset_insert
thf(fact_463_subset__insert,axiom,
! [X: v,A2: set_v,B2: set_v] :
( ~ ( member_v @ X @ A2 )
=> ( ( ord_less_eq_set_v @ A2 @ ( insert_v2 @ X @ B2 ) )
= ( ord_less_eq_set_v @ A2 @ B2 ) ) ) ).
% subset_insert
thf(fact_464_insert__mono,axiom,
! [C2: set_set_v,D: set_set_v,A: set_v] :
( ( ord_le5216385588623774835_set_v @ C2 @ D )
=> ( ord_le5216385588623774835_set_v @ ( insert_set_v2 @ A @ C2 ) @ ( insert_set_v2 @ A @ D ) ) ) ).
% insert_mono
thf(fact_465_insert__mono,axiom,
! [C2: set_Product_prod_v_v,D: set_Product_prod_v_v,A: product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ C2 @ D )
=> ( ord_le7336532860387713383od_v_v @ ( insert1338601472111419319od_v_v @ A @ C2 ) @ ( insert1338601472111419319od_v_v @ A @ D ) ) ) ).
% insert_mono
thf(fact_466_insert__mono,axiom,
! [C2: set_v,D: set_v,A: v] :
( ( ord_less_eq_set_v @ C2 @ D )
=> ( ord_less_eq_set_v @ ( insert_v2 @ A @ C2 ) @ ( insert_v2 @ A @ D ) ) ) ).
% insert_mono
thf(fact_467_insert__Diff__if,axiom,
! [X: set_v,B2: set_set_v,A2: set_set_v] :
( ( ( member_set_v @ X @ B2 )
=> ( ( minus_7228012346218142266_set_v @ ( insert_set_v2 @ X @ A2 ) @ B2 )
= ( minus_7228012346218142266_set_v @ A2 @ B2 ) ) )
& ( ~ ( member_set_v @ X @ B2 )
=> ( ( minus_7228012346218142266_set_v @ ( insert_set_v2 @ X @ A2 ) @ B2 )
= ( insert_set_v2 @ X @ ( minus_7228012346218142266_set_v @ A2 @ B2 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_468_insert__Diff__if,axiom,
! [X: product_prod_v_v,B2: set_Product_prod_v_v,A2: set_Product_prod_v_v] :
( ( ( member7453568604450474000od_v_v @ X @ B2 )
=> ( ( minus_4183494784930505774od_v_v @ ( insert1338601472111419319od_v_v @ X @ A2 ) @ B2 )
= ( minus_4183494784930505774od_v_v @ A2 @ B2 ) ) )
& ( ~ ( member7453568604450474000od_v_v @ X @ B2 )
=> ( ( minus_4183494784930505774od_v_v @ ( insert1338601472111419319od_v_v @ X @ A2 ) @ B2 )
= ( insert1338601472111419319od_v_v @ X @ ( minus_4183494784930505774od_v_v @ A2 @ B2 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_469_insert__Diff__if,axiom,
! [X: v,B2: set_v,A2: set_v] :
( ( ( member_v @ X @ B2 )
=> ( ( minus_minus_set_v @ ( insert_v2 @ X @ A2 ) @ B2 )
= ( minus_minus_set_v @ A2 @ B2 ) ) )
& ( ~ ( member_v @ X @ B2 )
=> ( ( minus_minus_set_v @ ( insert_v2 @ X @ A2 ) @ B2 )
= ( insert_v2 @ X @ ( minus_minus_set_v @ A2 @ B2 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_470_select__convs_I5_J,axiom,
! [Root: v,S: v > set_v,Explored: set_v,Visited: set_v,Vsuccs: v > set_v,Sccs: set_set_v,Stack: list_v,Cstack: list_v,More: product_unit] :
( ( sCC_Bl3795065053823578884t_unit @ ( sCC_Bl8064756265740546429t_unit @ Root @ S @ Explored @ Visited @ Vsuccs @ Sccs @ Stack @ Cstack @ More ) )
= Vsuccs ) ).
% select_convs(5)
thf(fact_471_subset__singleton__iff,axiom,
! [X5: set_set_v,A: set_v] :
( ( ord_le5216385588623774835_set_v @ X5 @ ( insert_set_v2 @ A @ bot_bot_set_set_v ) )
= ( ( X5 = bot_bot_set_set_v )
| ( X5
= ( insert_set_v2 @ A @ bot_bot_set_set_v ) ) ) ) ).
% subset_singleton_iff
thf(fact_472_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_473_subset__singleton__iff,axiom,
! [X5: set_v,A: v] :
( ( ord_less_eq_set_v @ X5 @ ( insert_v2 @ A @ bot_bot_set_v ) )
= ( ( X5 = bot_bot_set_v )
| ( X5
= ( insert_v2 @ A @ bot_bot_set_v ) ) ) ) ).
% subset_singleton_iff
thf(fact_474_subset__singletonD,axiom,
! [A2: set_set_v,X: set_v] :
( ( ord_le5216385588623774835_set_v @ A2 @ ( insert_set_v2 @ X @ bot_bot_set_set_v ) )
=> ( ( A2 = bot_bot_set_set_v )
| ( A2
= ( insert_set_v2 @ X @ bot_bot_set_set_v ) ) ) ) ).
% subset_singletonD
thf(fact_475_subset__singletonD,axiom,
! [A2: set_Product_prod_v_v,X: product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A2 @ ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v ) )
=> ( ( A2 = bot_bo723834152578015283od_v_v )
| ( A2
= ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v ) ) ) ) ).
% subset_singletonD
thf(fact_476_subset__singletonD,axiom,
! [A2: set_v,X: v] :
( ( ord_less_eq_set_v @ A2 @ ( insert_v2 @ X @ bot_bot_set_v ) )
=> ( ( A2 = bot_bot_set_v )
| ( A2
= ( insert_v2 @ X @ bot_bot_set_v ) ) ) ) ).
% subset_singletonD
thf(fact_477_Diff__insert,axiom,
! [A2: set_Product_prod_v_v,A: product_prod_v_v,B2: set_Product_prod_v_v] :
( ( minus_4183494784930505774od_v_v @ A2 @ ( insert1338601472111419319od_v_v @ A @ B2 ) )
= ( minus_4183494784930505774od_v_v @ ( minus_4183494784930505774od_v_v @ A2 @ B2 ) @ ( insert1338601472111419319od_v_v @ A @ bot_bo723834152578015283od_v_v ) ) ) ).
% Diff_insert
thf(fact_478_Diff__insert,axiom,
! [A2: set_set_v,A: set_v,B2: set_set_v] :
( ( minus_7228012346218142266_set_v @ A2 @ ( insert_set_v2 @ A @ B2 ) )
= ( minus_7228012346218142266_set_v @ ( minus_7228012346218142266_set_v @ A2 @ B2 ) @ ( insert_set_v2 @ A @ bot_bot_set_set_v ) ) ) ).
% Diff_insert
thf(fact_479_Diff__insert,axiom,
! [A2: set_v,A: v,B2: set_v] :
( ( minus_minus_set_v @ A2 @ ( insert_v2 @ A @ B2 ) )
= ( minus_minus_set_v @ ( minus_minus_set_v @ A2 @ B2 ) @ ( insert_v2 @ A @ bot_bot_set_v ) ) ) ).
% Diff_insert
thf(fact_480_insert__Diff,axiom,
! [A: product_prod_v_v,A2: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ A @ A2 )
=> ( ( insert1338601472111419319od_v_v @ A @ ( minus_4183494784930505774od_v_v @ A2 @ ( insert1338601472111419319od_v_v @ A @ bot_bo723834152578015283od_v_v ) ) )
= A2 ) ) ).
% insert_Diff
thf(fact_481_insert__Diff,axiom,
! [A: set_v,A2: set_set_v] :
( ( member_set_v @ A @ A2 )
=> ( ( insert_set_v2 @ A @ ( minus_7228012346218142266_set_v @ A2 @ ( insert_set_v2 @ A @ bot_bot_set_set_v ) ) )
= A2 ) ) ).
% insert_Diff
thf(fact_482_insert__Diff,axiom,
! [A: v,A2: set_v] :
( ( member_v @ A @ A2 )
=> ( ( insert_v2 @ A @ ( minus_minus_set_v @ A2 @ ( insert_v2 @ A @ bot_bot_set_v ) ) )
= A2 ) ) ).
% insert_Diff
thf(fact_483_Diff__insert2,axiom,
! [A2: set_Product_prod_v_v,A: product_prod_v_v,B2: set_Product_prod_v_v] :
( ( minus_4183494784930505774od_v_v @ A2 @ ( insert1338601472111419319od_v_v @ A @ B2 ) )
= ( minus_4183494784930505774od_v_v @ ( minus_4183494784930505774od_v_v @ A2 @ ( insert1338601472111419319od_v_v @ A @ bot_bo723834152578015283od_v_v ) ) @ B2 ) ) ).
% Diff_insert2
thf(fact_484_Diff__insert2,axiom,
! [A2: set_set_v,A: set_v,B2: set_set_v] :
( ( minus_7228012346218142266_set_v @ A2 @ ( insert_set_v2 @ A @ B2 ) )
= ( minus_7228012346218142266_set_v @ ( minus_7228012346218142266_set_v @ A2 @ ( insert_set_v2 @ A @ bot_bot_set_set_v ) ) @ B2 ) ) ).
% Diff_insert2
thf(fact_485_Diff__insert2,axiom,
! [A2: set_v,A: v,B2: set_v] :
( ( minus_minus_set_v @ A2 @ ( insert_v2 @ A @ B2 ) )
= ( minus_minus_set_v @ ( minus_minus_set_v @ A2 @ ( insert_v2 @ A @ bot_bot_set_v ) ) @ B2 ) ) ).
% Diff_insert2
thf(fact_486_Diff__insert__absorb,axiom,
! [X: product_prod_v_v,A2: set_Product_prod_v_v] :
( ~ ( member7453568604450474000od_v_v @ X @ A2 )
=> ( ( minus_4183494784930505774od_v_v @ ( insert1338601472111419319od_v_v @ X @ A2 ) @ ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v ) )
= A2 ) ) ).
% Diff_insert_absorb
thf(fact_487_Diff__insert__absorb,axiom,
! [X: set_v,A2: set_set_v] :
( ~ ( member_set_v @ X @ A2 )
=> ( ( minus_7228012346218142266_set_v @ ( insert_set_v2 @ X @ A2 ) @ ( insert_set_v2 @ X @ bot_bot_set_set_v ) )
= A2 ) ) ).
% Diff_insert_absorb
thf(fact_488_Diff__insert__absorb,axiom,
! [X: v,A2: set_v] :
( ~ ( member_v @ X @ A2 )
=> ( ( minus_minus_set_v @ ( insert_v2 @ X @ A2 ) @ ( insert_v2 @ X @ bot_bot_set_v ) )
= A2 ) ) ).
% Diff_insert_absorb
thf(fact_489_subset__Diff__insert,axiom,
! [A2: set_set_v,B2: set_set_v,X: set_v,C2: set_set_v] :
( ( ord_le5216385588623774835_set_v @ A2 @ ( minus_7228012346218142266_set_v @ B2 @ ( insert_set_v2 @ X @ C2 ) ) )
= ( ( ord_le5216385588623774835_set_v @ A2 @ ( minus_7228012346218142266_set_v @ B2 @ C2 ) )
& ~ ( member_set_v @ X @ A2 ) ) ) ).
% subset_Diff_insert
thf(fact_490_subset__Diff__insert,axiom,
! [A2: set_Product_prod_v_v,B2: set_Product_prod_v_v,X: product_prod_v_v,C2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A2 @ ( minus_4183494784930505774od_v_v @ B2 @ ( insert1338601472111419319od_v_v @ X @ C2 ) ) )
= ( ( ord_le7336532860387713383od_v_v @ A2 @ ( minus_4183494784930505774od_v_v @ B2 @ C2 ) )
& ~ ( member7453568604450474000od_v_v @ X @ A2 ) ) ) ).
% subset_Diff_insert
thf(fact_491_subset__Diff__insert,axiom,
! [A2: set_v,B2: set_v,X: v,C2: set_v] :
( ( ord_less_eq_set_v @ A2 @ ( minus_minus_set_v @ B2 @ ( insert_v2 @ X @ C2 ) ) )
= ( ( ord_less_eq_set_v @ A2 @ ( minus_minus_set_v @ B2 @ C2 ) )
& ~ ( member_v @ X @ A2 ) ) ) ).
% subset_Diff_insert
thf(fact_492_select__convs_I8_J,axiom,
! [Root: v,S: v > set_v,Explored: set_v,Visited: set_v,Vsuccs: v > set_v,Sccs: set_set_v,Stack: list_v,Cstack: list_v,More: product_unit] :
( ( sCC_Bl9201514103433284750t_unit @ ( sCC_Bl8064756265740546429t_unit @ Root @ S @ Explored @ Visited @ Vsuccs @ Sccs @ Stack @ Cstack @ More ) )
= Cstack ) ).
% select_convs(8)
thf(fact_493_Diff__single__insert,axiom,
! [A2: set_set_v,X: set_v,B2: set_set_v] :
( ( ord_le5216385588623774835_set_v @ ( minus_7228012346218142266_set_v @ A2 @ ( insert_set_v2 @ X @ bot_bot_set_set_v ) ) @ B2 )
=> ( ord_le5216385588623774835_set_v @ A2 @ ( insert_set_v2 @ X @ B2 ) ) ) ).
% Diff_single_insert
thf(fact_494_Diff__single__insert,axiom,
! [A2: set_Product_prod_v_v,X: product_prod_v_v,B2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ ( minus_4183494784930505774od_v_v @ A2 @ ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v ) ) @ B2 )
=> ( ord_le7336532860387713383od_v_v @ A2 @ ( insert1338601472111419319od_v_v @ X @ B2 ) ) ) ).
% Diff_single_insert
thf(fact_495_Diff__single__insert,axiom,
! [A2: set_v,X: v,B2: set_v] :
( ( ord_less_eq_set_v @ ( minus_minus_set_v @ A2 @ ( insert_v2 @ X @ bot_bot_set_v ) ) @ B2 )
=> ( ord_less_eq_set_v @ A2 @ ( insert_v2 @ X @ B2 ) ) ) ).
% Diff_single_insert
thf(fact_496_subset__insert__iff,axiom,
! [A2: set_set_v,X: set_v,B2: set_set_v] :
( ( ord_le5216385588623774835_set_v @ A2 @ ( insert_set_v2 @ X @ B2 ) )
= ( ( ( member_set_v @ X @ A2 )
=> ( ord_le5216385588623774835_set_v @ ( minus_7228012346218142266_set_v @ A2 @ ( insert_set_v2 @ X @ bot_bot_set_set_v ) ) @ B2 ) )
& ( ~ ( member_set_v @ X @ A2 )
=> ( ord_le5216385588623774835_set_v @ A2 @ B2 ) ) ) ) ).
% subset_insert_iff
thf(fact_497_subset__insert__iff,axiom,
! [A2: set_Product_prod_v_v,X: product_prod_v_v,B2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A2 @ ( insert1338601472111419319od_v_v @ X @ B2 ) )
= ( ( ( member7453568604450474000od_v_v @ X @ A2 )
=> ( ord_le7336532860387713383od_v_v @ ( minus_4183494784930505774od_v_v @ A2 @ ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v ) ) @ B2 ) )
& ( ~ ( member7453568604450474000od_v_v @ X @ A2 )
=> ( ord_le7336532860387713383od_v_v @ A2 @ B2 ) ) ) ) ).
% subset_insert_iff
thf(fact_498_subset__insert__iff,axiom,
! [A2: set_v,X: v,B2: set_v] :
( ( ord_less_eq_set_v @ A2 @ ( insert_v2 @ X @ B2 ) )
= ( ( ( member_v @ X @ A2 )
=> ( ord_less_eq_set_v @ ( minus_minus_set_v @ A2 @ ( insert_v2 @ X @ bot_bot_set_v ) ) @ B2 ) )
& ( ~ ( member_v @ X @ A2 )
=> ( ord_less_eq_set_v @ A2 @ B2 ) ) ) ) ).
% subset_insert_iff
thf(fact_499_graph_Osubscc__add,axiom,
! [Vertices: set_set_v,Successors: set_v > set_set_v,S3: set_set_v,X: set_v,Y: set_v] :
( ( sCC_Bl5810666556806954322_set_v @ Vertices @ Successors )
=> ( ( sCC_Bl7907073126578335045_set_v @ Successors @ S3 )
=> ( ( member_set_v @ X @ S3 )
=> ( ( sCC_Bl7354734129683093653_set_v @ Successors @ X @ Y )
=> ( ( sCC_Bl7354734129683093653_set_v @ Successors @ Y @ X )
=> ( sCC_Bl7907073126578335045_set_v @ Successors @ ( insert_set_v2 @ Y @ S3 ) ) ) ) ) ) ) ).
% graph.subscc_add
thf(fact_500_graph_Osubscc__add,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,S3: set_Product_prod_v_v,X: product_prod_v_v,Y: product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( sCC_Bl2301996248249672505od_v_v @ Successors @ S3 )
=> ( ( member7453568604450474000od_v_v @ X @ S3 )
=> ( ( sCC_Bl4981926079593201289od_v_v @ Successors @ X @ Y )
=> ( ( sCC_Bl4981926079593201289od_v_v @ Successors @ Y @ X )
=> ( sCC_Bl2301996248249672505od_v_v @ Successors @ ( insert1338601472111419319od_v_v @ Y @ S3 ) ) ) ) ) ) ) ).
% graph.subscc_add
thf(fact_501_graph_Osubscc__add,axiom,
! [Vertices: set_v,Successors: v > set_v,S3: set_v,X: v,Y: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl5398416737448265317bscc_v @ Successors @ S3 )
=> ( ( member_v @ X @ S3 )
=> ( ( sCC_Bl649662514949026229able_v @ Successors @ X @ Y )
=> ( ( sCC_Bl649662514949026229able_v @ Successors @ Y @ X )
=> ( sCC_Bl5398416737448265317bscc_v @ Successors @ ( insert_v2 @ Y @ S3 ) ) ) ) ) ) ) ).
% graph.subscc_add
thf(fact_502_Inl__inject,axiom,
! [X: produc5741669702376414499t_unit,Y: produc5741669702376414499t_unit] :
( ( ( sum_In526841707622398774t_unit @ X )
= ( sum_In526841707622398774t_unit @ Y ) )
=> ( X = Y ) ) ).
% Inl_inject
thf(fact_503_graph_Oreachable__visited,axiom,
! [Vertices: set_v,Successors: v > set_v,E: sCC_Bl1394983891496994913t_unit,V3: v,W: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl9196236973127232072t_unit @ Successors @ E )
=> ( ( member_v @ V3 @ ( sCC_Bl4645233313691564917t_unit @ E ) )
=> ( ( sCC_Bl649662514949026229able_v @ Successors @ V3 @ W )
=> ( ! [X3: v] :
( ( member_v @ X3 @ ( sCC_Bl4645233313691564917t_unit @ E ) )
=> ! [Xa: v] :
( ( member_v @ Xa @ ( minus_minus_set_v @ ( Successors @ X3 ) @ ( sCC_Bl3795065053823578884t_unit @ E @ X3 ) ) )
=> ( ( sCC_Bl649662514949026229able_v @ Successors @ V3 @ X3 )
=> ~ ( sCC_Bl649662514949026229able_v @ Successors @ Xa @ W ) ) ) )
=> ( member_v @ W @ ( sCC_Bl4645233313691564917t_unit @ E ) ) ) ) ) ) ) ).
% graph.reachable_visited
thf(fact_504_pre__dfs__def,axiom,
! [V3: v,E: sCC_Bl1394983891496994913t_unit] :
( ( sCC_Bl36166008131615352t_unit @ successors @ V3 @ E )
= ( ( sCC_Bl9196236973127232072t_unit @ successors @ E )
& ~ ( member_v @ V3 @ ( sCC_Bl4645233313691564917t_unit @ E ) )
& ( sCC_Bl649662514949026229able_v @ successors @ ( sCC_Bl1090238580953940555t_unit @ E ) @ V3 )
& ( ( sCC_Bl3795065053823578884t_unit @ E @ V3 )
= bot_bot_set_v )
& ! [X2: v] :
( ( member_v @ X2 @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E ) ) )
=> ( sCC_Bl649662514949026229able_v @ successors @ X2 @ V3 ) ) ) ) ).
% pre_dfs_def
thf(fact_505_post__dfss__def,axiom,
! [V3: v,E: sCC_Bl1394983891496994913t_unit,E2: sCC_Bl1394983891496994913t_unit] :
( ( sCC_Bl6082031138996704384t_unit @ successors @ V3 @ E @ E2 )
= ( ( sCC_Bl9196236973127232072t_unit @ successors @ E2 )
& ( ( sCC_Bl3795065053823578884t_unit @ E2 @ V3 )
= ( successors @ V3 ) )
& ! [X2: v] :
( ( member_v @ X2 @ ( minus_minus_set_v @ ( sCC_Bl4645233313691564917t_unit @ E ) @ ( insert_v2 @ V3 @ bot_bot_set_v ) ) )
=> ( ( sCC_Bl3795065053823578884t_unit @ E2 @ X2 )
= ( sCC_Bl3795065053823578884t_unit @ E @ X2 ) ) )
& ( sCC_Bl5768913643336123637t_unit @ E @ E2 )
& ! [X2: v] :
( ( member_v @ X2 @ ( successors @ V3 ) )
=> ( member_v @ X2 @ ( sup_sup_set_v @ ( sCC_Bl157864678168468314t_unit @ E2 ) @ ( sCC_Bl1280885523602775798t_unit @ E2 @ ( hd_v @ ( sCC_Bl8828226123343373779t_unit @ E2 ) ) ) ) ) )
& ! [X2: v] :
( ( member_v @ X2 @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E2 ) ) )
=> ( sCC_Bl649662514949026229able_v @ successors @ X2 @ V3 ) )
& ( ( sCC_Bl8828226123343373779t_unit @ E2 )
!= nil_v )
& ? [Ns2: list_v] :
( ( sCC_Bl8828226123343373779t_unit @ E )
= ( append_v @ Ns2 @ ( sCC_Bl8828226123343373779t_unit @ E2 ) ) )
& ( member_v @ V3 @ ( sCC_Bl1280885523602775798t_unit @ E2 @ ( hd_v @ ( sCC_Bl8828226123343373779t_unit @ E2 ) ) ) )
& ! [X2: v] :
( ( member_v @ X2 @ ( set_v2 @ ( tl_v @ ( sCC_Bl8828226123343373779t_unit @ E2 ) ) ) )
=> ( ( sCC_Bl1280885523602775798t_unit @ E2 @ X2 )
= ( sCC_Bl1280885523602775798t_unit @ E @ X2 ) ) )
& ( ( ( hd_v @ ( sCC_Bl8828226123343373779t_unit @ E2 ) )
= V3 )
=> ! [X2: v] :
( ( member_v @ X2 @ ( set_v2 @ ( tl_v @ ( sCC_Bl8828226123343373779t_unit @ E2 ) ) ) )
=> ~ ( sCC_Bl649662514949026229able_v @ successors @ V3 @ X2 ) ) )
& ( ( sCC_Bl9201514103433284750t_unit @ E2 )
= ( sCC_Bl9201514103433284750t_unit @ E ) ) ) ) ).
% post_dfss_def
thf(fact_506_unite__S__tl,axiom,
! [E: sCC_Bl1394983891496994913t_unit,W: v,V3: v,N: v] :
( ( sCC_Bl9196236973127232072t_unit @ successors @ E )
=> ( ( member_v @ W @ ( successors @ V3 ) )
=> ( ~ ( member_v @ W @ ( sCC_Bl3795065053823578884t_unit @ E @ V3 ) )
=> ( ( member_v @ W @ ( sCC_Bl4645233313691564917t_unit @ E ) )
=> ( ~ ( member_v @ W @ ( sCC_Bl157864678168468314t_unit @ E ) )
=> ( ( member_v @ N @ ( set_v2 @ ( tl_v @ ( sCC_Bl8828226123343373779t_unit @ ( sCC_Bloemen_unite_v @ V3 @ W @ E ) ) ) ) )
=> ( ( sCC_Bl1280885523602775798t_unit @ ( sCC_Bloemen_unite_v @ V3 @ W @ E ) @ N )
= ( sCC_Bl1280885523602775798t_unit @ E @ N ) ) ) ) ) ) ) ) ).
% unite_S_tl
thf(fact_507_unite__subscc,axiom,
! [V3: v,E: sCC_Bl1394983891496994913t_unit,W: v] :
( ( sCC_Bl1748261141445803503t_unit @ successors @ V3 @ E )
=> ( ( member_v @ W @ ( successors @ V3 ) )
=> ( ~ ( member_v @ W @ ( sCC_Bl3795065053823578884t_unit @ E @ V3 ) )
=> ( ( member_v @ W @ ( sCC_Bl4645233313691564917t_unit @ E ) )
=> ( ~ ( member_v @ W @ ( sCC_Bl157864678168468314t_unit @ E ) )
=> ( sCC_Bl5398416737448265317bscc_v @ successors @ ( sCC_Bl1280885523602775798t_unit @ ( sCC_Bloemen_unite_v @ V3 @ W @ E ) @ ( hd_v @ ( sCC_Bl8828226123343373779t_unit @ ( sCC_Bloemen_unite_v @ V3 @ W @ E ) ) ) ) ) ) ) ) ) ) ).
% unite_subscc
thf(fact_508_unite__wf__env,axiom,
! [V3: v,E: sCC_Bl1394983891496994913t_unit,W: v] :
( ( sCC_Bl1748261141445803503t_unit @ successors @ V3 @ E )
=> ( ( member_v @ W @ ( successors @ V3 ) )
=> ( ~ ( member_v @ W @ ( sCC_Bl3795065053823578884t_unit @ E @ V3 ) )
=> ( ( member_v @ W @ ( sCC_Bl4645233313691564917t_unit @ E ) )
=> ( ~ ( member_v @ W @ ( sCC_Bl157864678168468314t_unit @ E ) )
=> ( sCC_Bl9196236973127232072t_unit @ successors @ ( sCC_Bloemen_unite_v @ V3 @ W @ E ) ) ) ) ) ) ) ).
% unite_wf_env
thf(fact_509_unite__sub__env,axiom,
! [V3: v,E: sCC_Bl1394983891496994913t_unit,W: v] :
( ( sCC_Bl1748261141445803503t_unit @ successors @ V3 @ E )
=> ( ( member_v @ W @ ( successors @ V3 ) )
=> ( ~ ( member_v @ W @ ( sCC_Bl3795065053823578884t_unit @ E @ V3 ) )
=> ( ( member_v @ W @ ( sCC_Bl4645233313691564917t_unit @ E ) )
=> ( ~ ( member_v @ W @ ( sCC_Bl157864678168468314t_unit @ E ) )
=> ( sCC_Bl5768913643336123637t_unit @ E @ ( sCC_Bloemen_unite_v @ V3 @ W @ E ) ) ) ) ) ) ) ).
% unite_sub_env
thf(fact_510_Un__iff,axiom,
! [C: product_prod_v_v,A2: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ C @ ( sup_su414716646722978715od_v_v @ A2 @ B2 ) )
= ( ( member7453568604450474000od_v_v @ C @ A2 )
| ( member7453568604450474000od_v_v @ C @ B2 ) ) ) ).
% Un_iff
thf(fact_511_Un__iff,axiom,
! [C: v,A2: set_v,B2: set_v] :
( ( member_v @ C @ ( sup_sup_set_v @ A2 @ B2 ) )
= ( ( member_v @ C @ A2 )
| ( member_v @ C @ B2 ) ) ) ).
% Un_iff
thf(fact_512_Un__iff,axiom,
! [C: set_v,A2: set_set_v,B2: set_set_v] :
( ( member_set_v @ C @ ( sup_sup_set_set_v @ A2 @ B2 ) )
= ( ( member_set_v @ C @ A2 )
| ( member_set_v @ C @ B2 ) ) ) ).
% Un_iff
thf(fact_513_UnCI,axiom,
! [C: product_prod_v_v,B2: set_Product_prod_v_v,A2: set_Product_prod_v_v] :
( ( ~ ( member7453568604450474000od_v_v @ C @ B2 )
=> ( member7453568604450474000od_v_v @ C @ A2 ) )
=> ( member7453568604450474000od_v_v @ C @ ( sup_su414716646722978715od_v_v @ A2 @ B2 ) ) ) ).
% UnCI
thf(fact_514_UnCI,axiom,
! [C: v,B2: set_v,A2: set_v] :
( ( ~ ( member_v @ C @ B2 )
=> ( member_v @ C @ A2 ) )
=> ( member_v @ C @ ( sup_sup_set_v @ A2 @ B2 ) ) ) ).
% UnCI
thf(fact_515_UnCI,axiom,
! [C: set_v,B2: set_set_v,A2: set_set_v] :
( ( ~ ( member_set_v @ C @ B2 )
=> ( member_set_v @ C @ A2 ) )
=> ( member_set_v @ C @ ( sup_sup_set_set_v @ A2 @ B2 ) ) ) ).
% UnCI
thf(fact_516_ra__add__edge,axiom,
! [X: v,Y: v,E4: set_Product_prod_v_v,V3: v,W: v] :
( ( sCC_Bl4291963740693775144ding_v @ successors @ X @ Y @ E4 )
=> ( ( sCC_Bl4291963740693775144ding_v @ successors @ X @ Y @ ( sup_su414716646722978715od_v_v @ E4 @ ( insert1338601472111419319od_v_v @ ( product_Pair_v_v @ V3 @ W ) @ bot_bo723834152578015283od_v_v ) ) )
| ( ( sCC_Bl4291963740693775144ding_v @ successors @ X @ V3 @ ( sup_su414716646722978715od_v_v @ E4 @ ( insert1338601472111419319od_v_v @ ( product_Pair_v_v @ V3 @ W ) @ bot_bo723834152578015283od_v_v ) ) )
& ( sCC_Bl4291963740693775144ding_v @ successors @ W @ Y @ ( sup_su414716646722978715od_v_v @ E4 @ ( insert1338601472111419319od_v_v @ ( product_Pair_v_v @ V3 @ W ) @ bot_bo723834152578015283od_v_v ) ) ) ) ) ) ).
% ra_add_edge
thf(fact_517_Un__empty,axiom,
! [A2: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( ( sup_su414716646722978715od_v_v @ A2 @ B2 )
= bot_bo723834152578015283od_v_v )
= ( ( A2 = bot_bo723834152578015283od_v_v )
& ( B2 = bot_bo723834152578015283od_v_v ) ) ) ).
% Un_empty
thf(fact_518_Un__empty,axiom,
! [A2: set_v,B2: set_v] :
( ( ( sup_sup_set_v @ A2 @ B2 )
= bot_bot_set_v )
= ( ( A2 = bot_bot_set_v )
& ( B2 = bot_bot_set_v ) ) ) ).
% Un_empty
thf(fact_519_Un__empty,axiom,
! [A2: set_set_v,B2: set_set_v] :
( ( ( sup_sup_set_set_v @ A2 @ B2 )
= bot_bot_set_set_v )
= ( ( A2 = bot_bot_set_set_v )
& ( B2 = bot_bot_set_set_v ) ) ) ).
% Un_empty
thf(fact_520_Un__subset__iff,axiom,
! [A2: set_set_v,B2: set_set_v,C2: set_set_v] :
( ( ord_le5216385588623774835_set_v @ ( sup_sup_set_set_v @ A2 @ B2 ) @ C2 )
= ( ( ord_le5216385588623774835_set_v @ A2 @ C2 )
& ( ord_le5216385588623774835_set_v @ B2 @ C2 ) ) ) ).
% Un_subset_iff
thf(fact_521_Un__subset__iff,axiom,
! [A2: set_Product_prod_v_v,B2: set_Product_prod_v_v,C2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ ( sup_su414716646722978715od_v_v @ A2 @ B2 ) @ C2 )
= ( ( ord_le7336532860387713383od_v_v @ A2 @ C2 )
& ( ord_le7336532860387713383od_v_v @ B2 @ C2 ) ) ) ).
% Un_subset_iff
thf(fact_522_Un__subset__iff,axiom,
! [A2: set_v,B2: set_v,C2: set_v] :
( ( ord_less_eq_set_v @ ( sup_sup_set_v @ A2 @ B2 ) @ C2 )
= ( ( ord_less_eq_set_v @ A2 @ C2 )
& ( ord_less_eq_set_v @ B2 @ C2 ) ) ) ).
% Un_subset_iff
thf(fact_523_avoiding__explored,axiom,
! [E: sCC_Bl1394983891496994913t_unit,X: v,Y: v,E4: set_Product_prod_v_v,W: v,V3: v] :
( ( sCC_Bl9196236973127232072t_unit @ successors @ E )
=> ( ( sCC_Bl4291963740693775144ding_v @ successors @ X @ Y @ E4 )
=> ( ~ ( member_v @ Y @ ( sCC_Bl157864678168468314t_unit @ E ) )
=> ( ( member_v @ W @ ( sCC_Bl157864678168468314t_unit @ E ) )
=> ( sCC_Bl4291963740693775144ding_v @ successors @ X @ Y @ ( sup_su414716646722978715od_v_v @ E4 @ ( insert1338601472111419319od_v_v @ ( product_Pair_v_v @ V3 @ W ) @ bot_bo723834152578015283od_v_v ) ) ) ) ) ) ) ).
% avoiding_explored
thf(fact_524_Un__insert__right,axiom,
! [A2: set_Product_prod_v_v,A: product_prod_v_v,B2: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ A2 @ ( insert1338601472111419319od_v_v @ A @ B2 ) )
= ( insert1338601472111419319od_v_v @ A @ ( sup_su414716646722978715od_v_v @ A2 @ B2 ) ) ) ).
% Un_insert_right
thf(fact_525_Un__insert__right,axiom,
! [A2: set_v,A: v,B2: set_v] :
( ( sup_sup_set_v @ A2 @ ( insert_v2 @ A @ B2 ) )
= ( insert_v2 @ A @ ( sup_sup_set_v @ A2 @ B2 ) ) ) ).
% Un_insert_right
thf(fact_526_Un__insert__right,axiom,
! [A2: set_set_v,A: set_v,B2: set_set_v] :
( ( sup_sup_set_set_v @ A2 @ ( insert_set_v2 @ A @ B2 ) )
= ( insert_set_v2 @ A @ ( sup_sup_set_set_v @ A2 @ B2 ) ) ) ).
% Un_insert_right
thf(fact_527_Un__insert__left,axiom,
! [A: product_prod_v_v,B2: set_Product_prod_v_v,C2: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ ( insert1338601472111419319od_v_v @ A @ B2 ) @ C2 )
= ( insert1338601472111419319od_v_v @ A @ ( sup_su414716646722978715od_v_v @ B2 @ C2 ) ) ) ).
% Un_insert_left
thf(fact_528_Un__insert__left,axiom,
! [A: v,B2: set_v,C2: set_v] :
( ( sup_sup_set_v @ ( insert_v2 @ A @ B2 ) @ C2 )
= ( insert_v2 @ A @ ( sup_sup_set_v @ B2 @ C2 ) ) ) ).
% Un_insert_left
thf(fact_529_Un__insert__left,axiom,
! [A: set_v,B2: set_set_v,C2: set_set_v] :
( ( sup_sup_set_set_v @ ( insert_set_v2 @ A @ B2 ) @ C2 )
= ( insert_set_v2 @ A @ ( sup_sup_set_set_v @ B2 @ C2 ) ) ) ).
% Un_insert_left
thf(fact_530_Un__Diff__cancel2,axiom,
! [B2: set_Product_prod_v_v,A2: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ ( minus_4183494784930505774od_v_v @ B2 @ A2 ) @ A2 )
= ( sup_su414716646722978715od_v_v @ B2 @ A2 ) ) ).
% Un_Diff_cancel2
thf(fact_531_Un__Diff__cancel2,axiom,
! [B2: set_set_v,A2: set_set_v] :
( ( sup_sup_set_set_v @ ( minus_7228012346218142266_set_v @ B2 @ A2 ) @ A2 )
= ( sup_sup_set_set_v @ B2 @ A2 ) ) ).
% Un_Diff_cancel2
thf(fact_532_Un__Diff__cancel2,axiom,
! [B2: set_v,A2: set_v] :
( ( sup_sup_set_v @ ( minus_minus_set_v @ B2 @ A2 ) @ A2 )
= ( sup_sup_set_v @ B2 @ A2 ) ) ).
% Un_Diff_cancel2
thf(fact_533_Un__Diff__cancel,axiom,
! [A2: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ A2 @ ( minus_4183494784930505774od_v_v @ B2 @ A2 ) )
= ( sup_su414716646722978715od_v_v @ A2 @ B2 ) ) ).
% Un_Diff_cancel
thf(fact_534_Un__Diff__cancel,axiom,
! [A2: set_set_v,B2: set_set_v] :
( ( sup_sup_set_set_v @ A2 @ ( minus_7228012346218142266_set_v @ B2 @ A2 ) )
= ( sup_sup_set_set_v @ A2 @ B2 ) ) ).
% Un_Diff_cancel
thf(fact_535_Un__Diff__cancel,axiom,
! [A2: set_v,B2: set_v] :
( ( sup_sup_set_v @ A2 @ ( minus_minus_set_v @ B2 @ A2 ) )
= ( sup_sup_set_v @ A2 @ B2 ) ) ).
% Un_Diff_cancel
thf(fact_536_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_537_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_538_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_539_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_540_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_541_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_542_Un__left__commute,axiom,
! [A2: set_Product_prod_v_v,B2: set_Product_prod_v_v,C2: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ A2 @ ( sup_su414716646722978715od_v_v @ B2 @ C2 ) )
= ( sup_su414716646722978715od_v_v @ B2 @ ( sup_su414716646722978715od_v_v @ A2 @ C2 ) ) ) ).
% Un_left_commute
thf(fact_543_Un__left__commute,axiom,
! [A2: set_v,B2: set_v,C2: set_v] :
( ( sup_sup_set_v @ A2 @ ( sup_sup_set_v @ B2 @ C2 ) )
= ( sup_sup_set_v @ B2 @ ( sup_sup_set_v @ A2 @ C2 ) ) ) ).
% Un_left_commute
thf(fact_544_Un__left__commute,axiom,
! [A2: set_set_v,B2: set_set_v,C2: set_set_v] :
( ( sup_sup_set_set_v @ A2 @ ( sup_sup_set_set_v @ B2 @ C2 ) )
= ( sup_sup_set_set_v @ B2 @ ( sup_sup_set_set_v @ A2 @ C2 ) ) ) ).
% Un_left_commute
thf(fact_545_Un__left__absorb,axiom,
! [A2: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ A2 @ ( sup_su414716646722978715od_v_v @ A2 @ B2 ) )
= ( sup_su414716646722978715od_v_v @ A2 @ B2 ) ) ).
% Un_left_absorb
thf(fact_546_Un__left__absorb,axiom,
! [A2: set_v,B2: set_v] :
( ( sup_sup_set_v @ A2 @ ( sup_sup_set_v @ A2 @ B2 ) )
= ( sup_sup_set_v @ A2 @ B2 ) ) ).
% Un_left_absorb
thf(fact_547_Un__left__absorb,axiom,
! [A2: set_set_v,B2: set_set_v] :
( ( sup_sup_set_set_v @ A2 @ ( sup_sup_set_set_v @ A2 @ B2 ) )
= ( sup_sup_set_set_v @ A2 @ B2 ) ) ).
% Un_left_absorb
thf(fact_548_Un__commute,axiom,
( sup_su414716646722978715od_v_v
= ( ^ [A7: set_Product_prod_v_v,B6: set_Product_prod_v_v] : ( sup_su414716646722978715od_v_v @ B6 @ A7 ) ) ) ).
% Un_commute
thf(fact_549_Un__commute,axiom,
( sup_sup_set_v
= ( ^ [A7: set_v,B6: set_v] : ( sup_sup_set_v @ B6 @ A7 ) ) ) ).
% Un_commute
thf(fact_550_Un__commute,axiom,
( sup_sup_set_set_v
= ( ^ [A7: set_set_v,B6: set_set_v] : ( sup_sup_set_set_v @ B6 @ A7 ) ) ) ).
% Un_commute
thf(fact_551_Un__absorb,axiom,
! [A2: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ A2 @ A2 )
= A2 ) ).
% Un_absorb
thf(fact_552_Un__absorb,axiom,
! [A2: set_v] :
( ( sup_sup_set_v @ A2 @ A2 )
= A2 ) ).
% Un_absorb
thf(fact_553_Un__absorb,axiom,
! [A2: set_set_v] :
( ( sup_sup_set_set_v @ A2 @ A2 )
= A2 ) ).
% Un_absorb
thf(fact_554_Un__assoc,axiom,
! [A2: set_Product_prod_v_v,B2: set_Product_prod_v_v,C2: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ ( sup_su414716646722978715od_v_v @ A2 @ B2 ) @ C2 )
= ( sup_su414716646722978715od_v_v @ A2 @ ( sup_su414716646722978715od_v_v @ B2 @ C2 ) ) ) ).
% Un_assoc
thf(fact_555_Un__assoc,axiom,
! [A2: set_v,B2: set_v,C2: set_v] :
( ( sup_sup_set_v @ ( sup_sup_set_v @ A2 @ B2 ) @ C2 )
= ( sup_sup_set_v @ A2 @ ( sup_sup_set_v @ B2 @ C2 ) ) ) ).
% Un_assoc
thf(fact_556_Un__assoc,axiom,
! [A2: set_set_v,B2: set_set_v,C2: set_set_v] :
( ( sup_sup_set_set_v @ ( sup_sup_set_set_v @ A2 @ B2 ) @ C2 )
= ( sup_sup_set_set_v @ A2 @ ( sup_sup_set_set_v @ B2 @ C2 ) ) ) ).
% Un_assoc
thf(fact_557_ball__Un,axiom,
! [A2: set_Product_prod_v_v,B2: set_Product_prod_v_v,P: product_prod_v_v > $o] :
( ( ! [X2: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X2 @ ( sup_su414716646722978715od_v_v @ A2 @ B2 ) )
=> ( P @ X2 ) ) )
= ( ! [X2: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X2 @ A2 )
=> ( P @ X2 ) )
& ! [X2: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X2 @ B2 )
=> ( P @ X2 ) ) ) ) ).
% ball_Un
thf(fact_558_ball__Un,axiom,
! [A2: set_v,B2: set_v,P: v > $o] :
( ( ! [X2: v] :
( ( member_v @ X2 @ ( sup_sup_set_v @ A2 @ B2 ) )
=> ( P @ X2 ) ) )
= ( ! [X2: v] :
( ( member_v @ X2 @ A2 )
=> ( P @ X2 ) )
& ! [X2: v] :
( ( member_v @ X2 @ B2 )
=> ( P @ X2 ) ) ) ) ).
% ball_Un
thf(fact_559_ball__Un,axiom,
! [A2: set_set_v,B2: set_set_v,P: set_v > $o] :
( ( ! [X2: set_v] :
( ( member_set_v @ X2 @ ( sup_sup_set_set_v @ A2 @ B2 ) )
=> ( P @ X2 ) ) )
= ( ! [X2: set_v] :
( ( member_set_v @ X2 @ A2 )
=> ( P @ X2 ) )
& ! [X2: set_v] :
( ( member_set_v @ X2 @ B2 )
=> ( P @ X2 ) ) ) ) ).
% ball_Un
thf(fact_560_bex__Un,axiom,
! [A2: set_Product_prod_v_v,B2: set_Product_prod_v_v,P: product_prod_v_v > $o] :
( ( ? [X2: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X2 @ ( sup_su414716646722978715od_v_v @ A2 @ B2 ) )
& ( P @ X2 ) ) )
= ( ? [X2: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X2 @ A2 )
& ( P @ X2 ) )
| ? [X2: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X2 @ B2 )
& ( P @ X2 ) ) ) ) ).
% bex_Un
thf(fact_561_bex__Un,axiom,
! [A2: set_v,B2: set_v,P: v > $o] :
( ( ? [X2: v] :
( ( member_v @ X2 @ ( sup_sup_set_v @ A2 @ B2 ) )
& ( P @ X2 ) ) )
= ( ? [X2: v] :
( ( member_v @ X2 @ A2 )
& ( P @ X2 ) )
| ? [X2: v] :
( ( member_v @ X2 @ B2 )
& ( P @ X2 ) ) ) ) ).
% bex_Un
thf(fact_562_bex__Un,axiom,
! [A2: set_set_v,B2: set_set_v,P: set_v > $o] :
( ( ? [X2: set_v] :
( ( member_set_v @ X2 @ ( sup_sup_set_set_v @ A2 @ B2 ) )
& ( P @ X2 ) ) )
= ( ? [X2: set_v] :
( ( member_set_v @ X2 @ A2 )
& ( P @ X2 ) )
| ? [X2: set_v] :
( ( member_set_v @ X2 @ B2 )
& ( P @ X2 ) ) ) ) ).
% bex_Un
thf(fact_563_UnI2,axiom,
! [C: product_prod_v_v,B2: set_Product_prod_v_v,A2: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ C @ B2 )
=> ( member7453568604450474000od_v_v @ C @ ( sup_su414716646722978715od_v_v @ A2 @ B2 ) ) ) ).
% UnI2
thf(fact_564_UnI2,axiom,
! [C: v,B2: set_v,A2: set_v] :
( ( member_v @ C @ B2 )
=> ( member_v @ C @ ( sup_sup_set_v @ A2 @ B2 ) ) ) ).
% UnI2
thf(fact_565_UnI2,axiom,
! [C: set_v,B2: set_set_v,A2: set_set_v] :
( ( member_set_v @ C @ B2 )
=> ( member_set_v @ C @ ( sup_sup_set_set_v @ A2 @ B2 ) ) ) ).
% UnI2
thf(fact_566_UnI1,axiom,
! [C: product_prod_v_v,A2: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ C @ A2 )
=> ( member7453568604450474000od_v_v @ C @ ( sup_su414716646722978715od_v_v @ A2 @ B2 ) ) ) ).
% UnI1
thf(fact_567_UnI1,axiom,
! [C: v,A2: set_v,B2: set_v] :
( ( member_v @ C @ A2 )
=> ( member_v @ C @ ( sup_sup_set_v @ A2 @ B2 ) ) ) ).
% UnI1
thf(fact_568_UnI1,axiom,
! [C: set_v,A2: set_set_v,B2: set_set_v] :
( ( member_set_v @ C @ A2 )
=> ( member_set_v @ C @ ( sup_sup_set_set_v @ A2 @ B2 ) ) ) ).
% UnI1
thf(fact_569_UnE,axiom,
! [C: product_prod_v_v,A2: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ C @ ( sup_su414716646722978715od_v_v @ A2 @ B2 ) )
=> ( ~ ( member7453568604450474000od_v_v @ C @ A2 )
=> ( member7453568604450474000od_v_v @ C @ B2 ) ) ) ).
% UnE
thf(fact_570_UnE,axiom,
! [C: v,A2: set_v,B2: set_v] :
( ( member_v @ C @ ( sup_sup_set_v @ A2 @ B2 ) )
=> ( ~ ( member_v @ C @ A2 )
=> ( member_v @ C @ B2 ) ) ) ).
% UnE
thf(fact_571_UnE,axiom,
! [C: set_v,A2: set_set_v,B2: set_set_v] :
( ( member_set_v @ C @ ( sup_sup_set_set_v @ A2 @ B2 ) )
=> ( ~ ( member_set_v @ C @ A2 )
=> ( member_set_v @ C @ B2 ) ) ) ).
% UnE
thf(fact_572_boolean__algebra_Odisj__zero__right,axiom,
! [X: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ X @ bot_bo723834152578015283od_v_v )
= X ) ).
% boolean_algebra.disj_zero_right
thf(fact_573_boolean__algebra_Odisj__zero__right,axiom,
! [X: set_v] :
( ( sup_sup_set_v @ X @ bot_bot_set_v )
= X ) ).
% boolean_algebra.disj_zero_right
thf(fact_574_boolean__algebra_Odisj__zero__right,axiom,
! [X: set_set_v] :
( ( sup_sup_set_set_v @ X @ bot_bot_set_set_v )
= X ) ).
% boolean_algebra.disj_zero_right
thf(fact_575_boolean__algebra_Odisj__zero__right,axiom,
! [X: product_unit] :
( ( sup_sup_Product_unit @ X @ bot_bot_Product_unit )
= X ) ).
% boolean_algebra.disj_zero_right
thf(fact_576_Un__empty__right,axiom,
! [A2: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ A2 @ bot_bo723834152578015283od_v_v )
= A2 ) ).
% Un_empty_right
thf(fact_577_Un__empty__right,axiom,
! [A2: set_v] :
( ( sup_sup_set_v @ A2 @ bot_bot_set_v )
= A2 ) ).
% Un_empty_right
thf(fact_578_Un__empty__right,axiom,
! [A2: set_set_v] :
( ( sup_sup_set_set_v @ A2 @ bot_bot_set_set_v )
= A2 ) ).
% Un_empty_right
thf(fact_579_Un__empty__left,axiom,
! [B2: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ bot_bo723834152578015283od_v_v @ B2 )
= B2 ) ).
% Un_empty_left
thf(fact_580_Un__empty__left,axiom,
! [B2: set_v] :
( ( sup_sup_set_v @ bot_bot_set_v @ B2 )
= B2 ) ).
% Un_empty_left
thf(fact_581_Un__empty__left,axiom,
! [B2: set_set_v] :
( ( sup_sup_set_set_v @ bot_bot_set_set_v @ B2 )
= B2 ) ).
% Un_empty_left
thf(fact_582_Un__mono,axiom,
! [A2: set_set_v,C2: set_set_v,B2: set_set_v,D: set_set_v] :
( ( ord_le5216385588623774835_set_v @ A2 @ C2 )
=> ( ( ord_le5216385588623774835_set_v @ B2 @ D )
=> ( ord_le5216385588623774835_set_v @ ( sup_sup_set_set_v @ A2 @ B2 ) @ ( sup_sup_set_set_v @ C2 @ D ) ) ) ) ).
% Un_mono
thf(fact_583_Un__mono,axiom,
! [A2: set_Product_prod_v_v,C2: set_Product_prod_v_v,B2: set_Product_prod_v_v,D: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A2 @ C2 )
=> ( ( ord_le7336532860387713383od_v_v @ B2 @ D )
=> ( ord_le7336532860387713383od_v_v @ ( sup_su414716646722978715od_v_v @ A2 @ B2 ) @ ( sup_su414716646722978715od_v_v @ C2 @ D ) ) ) ) ).
% Un_mono
thf(fact_584_Un__mono,axiom,
! [A2: set_v,C2: set_v,B2: set_v,D: set_v] :
( ( ord_less_eq_set_v @ A2 @ C2 )
=> ( ( ord_less_eq_set_v @ B2 @ D )
=> ( ord_less_eq_set_v @ ( sup_sup_set_v @ A2 @ B2 ) @ ( sup_sup_set_v @ C2 @ D ) ) ) ) ).
% Un_mono
thf(fact_585_Un__least,axiom,
! [A2: set_set_v,C2: set_set_v,B2: set_set_v] :
( ( ord_le5216385588623774835_set_v @ A2 @ C2 )
=> ( ( ord_le5216385588623774835_set_v @ B2 @ C2 )
=> ( ord_le5216385588623774835_set_v @ ( sup_sup_set_set_v @ A2 @ B2 ) @ C2 ) ) ) ).
% Un_least
thf(fact_586_Un__least,axiom,
! [A2: set_Product_prod_v_v,C2: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A2 @ C2 )
=> ( ( ord_le7336532860387713383od_v_v @ B2 @ C2 )
=> ( ord_le7336532860387713383od_v_v @ ( sup_su414716646722978715od_v_v @ A2 @ B2 ) @ C2 ) ) ) ).
% Un_least
thf(fact_587_Un__least,axiom,
! [A2: set_v,C2: set_v,B2: set_v] :
( ( ord_less_eq_set_v @ A2 @ C2 )
=> ( ( ord_less_eq_set_v @ B2 @ C2 )
=> ( ord_less_eq_set_v @ ( sup_sup_set_v @ A2 @ B2 ) @ C2 ) ) ) ).
% Un_least
thf(fact_588_Un__upper1,axiom,
! [A2: set_set_v,B2: set_set_v] : ( ord_le5216385588623774835_set_v @ A2 @ ( sup_sup_set_set_v @ A2 @ B2 ) ) ).
% Un_upper1
thf(fact_589_Un__upper1,axiom,
! [A2: set_Product_prod_v_v,B2: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ A2 @ ( sup_su414716646722978715od_v_v @ A2 @ B2 ) ) ).
% Un_upper1
thf(fact_590_Un__upper1,axiom,
! [A2: set_v,B2: set_v] : ( ord_less_eq_set_v @ A2 @ ( sup_sup_set_v @ A2 @ B2 ) ) ).
% Un_upper1
thf(fact_591_Un__upper2,axiom,
! [B2: set_set_v,A2: set_set_v] : ( ord_le5216385588623774835_set_v @ B2 @ ( sup_sup_set_set_v @ A2 @ B2 ) ) ).
% Un_upper2
thf(fact_592_Un__upper2,axiom,
! [B2: set_Product_prod_v_v,A2: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ B2 @ ( sup_su414716646722978715od_v_v @ A2 @ B2 ) ) ).
% Un_upper2
thf(fact_593_Un__upper2,axiom,
! [B2: set_v,A2: set_v] : ( ord_less_eq_set_v @ B2 @ ( sup_sup_set_v @ A2 @ B2 ) ) ).
% Un_upper2
thf(fact_594_Un__absorb1,axiom,
! [A2: set_set_v,B2: set_set_v] :
( ( ord_le5216385588623774835_set_v @ A2 @ B2 )
=> ( ( sup_sup_set_set_v @ A2 @ B2 )
= B2 ) ) ).
% Un_absorb1
thf(fact_595_Un__absorb1,axiom,
! [A2: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A2 @ B2 )
=> ( ( sup_su414716646722978715od_v_v @ A2 @ B2 )
= B2 ) ) ).
% Un_absorb1
thf(fact_596_Un__absorb1,axiom,
! [A2: set_v,B2: set_v] :
( ( ord_less_eq_set_v @ A2 @ B2 )
=> ( ( sup_sup_set_v @ A2 @ B2 )
= B2 ) ) ).
% Un_absorb1
thf(fact_597_Un__absorb2,axiom,
! [B2: set_set_v,A2: set_set_v] :
( ( ord_le5216385588623774835_set_v @ B2 @ A2 )
=> ( ( sup_sup_set_set_v @ A2 @ B2 )
= A2 ) ) ).
% Un_absorb2
thf(fact_598_Un__absorb2,axiom,
! [B2: set_Product_prod_v_v,A2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ B2 @ A2 )
=> ( ( sup_su414716646722978715od_v_v @ A2 @ B2 )
= A2 ) ) ).
% Un_absorb2
thf(fact_599_Un__absorb2,axiom,
! [B2: set_v,A2: set_v] :
( ( ord_less_eq_set_v @ B2 @ A2 )
=> ( ( sup_sup_set_v @ A2 @ B2 )
= A2 ) ) ).
% Un_absorb2
thf(fact_600_subset__UnE,axiom,
! [C2: set_set_v,A2: set_set_v,B2: set_set_v] :
( ( ord_le5216385588623774835_set_v @ C2 @ ( sup_sup_set_set_v @ A2 @ B2 ) )
=> ~ ! [A8: set_set_v] :
( ( ord_le5216385588623774835_set_v @ A8 @ A2 )
=> ! [B8: set_set_v] :
( ( ord_le5216385588623774835_set_v @ B8 @ B2 )
=> ( C2
!= ( sup_sup_set_set_v @ A8 @ B8 ) ) ) ) ) ).
% subset_UnE
thf(fact_601_subset__UnE,axiom,
! [C2: set_Product_prod_v_v,A2: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ C2 @ ( sup_su414716646722978715od_v_v @ A2 @ B2 ) )
=> ~ ! [A8: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A8 @ A2 )
=> ! [B8: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ B8 @ B2 )
=> ( C2
!= ( sup_su414716646722978715od_v_v @ A8 @ B8 ) ) ) ) ) ).
% subset_UnE
thf(fact_602_subset__UnE,axiom,
! [C2: set_v,A2: set_v,B2: set_v] :
( ( ord_less_eq_set_v @ C2 @ ( sup_sup_set_v @ A2 @ B2 ) )
=> ~ ! [A8: set_v] :
( ( ord_less_eq_set_v @ A8 @ A2 )
=> ! [B8: set_v] :
( ( ord_less_eq_set_v @ B8 @ B2 )
=> ( C2
!= ( sup_sup_set_v @ A8 @ B8 ) ) ) ) ) ).
% subset_UnE
thf(fact_603_subset__Un__eq,axiom,
( ord_le5216385588623774835_set_v
= ( ^ [A7: set_set_v,B6: set_set_v] :
( ( sup_sup_set_set_v @ A7 @ B6 )
= B6 ) ) ) ).
% subset_Un_eq
thf(fact_604_subset__Un__eq,axiom,
( ord_le7336532860387713383od_v_v
= ( ^ [A7: set_Product_prod_v_v,B6: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ A7 @ B6 )
= B6 ) ) ) ).
% subset_Un_eq
thf(fact_605_subset__Un__eq,axiom,
( ord_less_eq_set_v
= ( ^ [A7: set_v,B6: set_v] :
( ( sup_sup_set_v @ A7 @ B6 )
= B6 ) ) ) ).
% subset_Un_eq
thf(fact_606_Un__Diff,axiom,
! [A2: set_Product_prod_v_v,B2: set_Product_prod_v_v,C2: set_Product_prod_v_v] :
( ( minus_4183494784930505774od_v_v @ ( sup_su414716646722978715od_v_v @ A2 @ B2 ) @ C2 )
= ( sup_su414716646722978715od_v_v @ ( minus_4183494784930505774od_v_v @ A2 @ C2 ) @ ( minus_4183494784930505774od_v_v @ B2 @ C2 ) ) ) ).
% Un_Diff
thf(fact_607_Un__Diff,axiom,
! [A2: set_set_v,B2: set_set_v,C2: set_set_v] :
( ( minus_7228012346218142266_set_v @ ( sup_sup_set_set_v @ A2 @ B2 ) @ C2 )
= ( sup_sup_set_set_v @ ( minus_7228012346218142266_set_v @ A2 @ C2 ) @ ( minus_7228012346218142266_set_v @ B2 @ C2 ) ) ) ).
% Un_Diff
thf(fact_608_Un__Diff,axiom,
! [A2: set_v,B2: set_v,C2: set_v] :
( ( minus_minus_set_v @ ( sup_sup_set_v @ A2 @ B2 ) @ C2 )
= ( sup_sup_set_v @ ( minus_minus_set_v @ A2 @ C2 ) @ ( minus_minus_set_v @ B2 @ C2 ) ) ) ).
% Un_Diff
thf(fact_609_insert__is__Un,axiom,
( insert1338601472111419319od_v_v
= ( ^ [A5: product_prod_v_v] : ( sup_su414716646722978715od_v_v @ ( insert1338601472111419319od_v_v @ A5 @ bot_bo723834152578015283od_v_v ) ) ) ) ).
% insert_is_Un
thf(fact_610_insert__is__Un,axiom,
( insert_v2
= ( ^ [A5: v] : ( sup_sup_set_v @ ( insert_v2 @ A5 @ bot_bot_set_v ) ) ) ) ).
% insert_is_Un
thf(fact_611_insert__is__Un,axiom,
( insert_set_v2
= ( ^ [A5: set_v] : ( sup_sup_set_set_v @ ( insert_set_v2 @ A5 @ bot_bot_set_set_v ) ) ) ) ).
% insert_is_Un
thf(fact_612_Un__singleton__iff,axiom,
! [A2: set_Product_prod_v_v,B2: set_Product_prod_v_v,X: product_prod_v_v] :
( ( ( sup_su414716646722978715od_v_v @ A2 @ B2 )
= ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v ) )
= ( ( ( A2 = bot_bo723834152578015283od_v_v )
& ( B2
= ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v ) ) )
| ( ( A2
= ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v ) )
& ( B2 = bot_bo723834152578015283od_v_v ) )
| ( ( A2
= ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v ) )
& ( B2
= ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_613_Un__singleton__iff,axiom,
! [A2: set_v,B2: set_v,X: v] :
( ( ( sup_sup_set_v @ A2 @ B2 )
= ( insert_v2 @ X @ bot_bot_set_v ) )
= ( ( ( A2 = bot_bot_set_v )
& ( B2
= ( insert_v2 @ X @ bot_bot_set_v ) ) )
| ( ( A2
= ( insert_v2 @ X @ bot_bot_set_v ) )
& ( B2 = bot_bot_set_v ) )
| ( ( A2
= ( insert_v2 @ X @ bot_bot_set_v ) )
& ( B2
= ( insert_v2 @ X @ bot_bot_set_v ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_614_Un__singleton__iff,axiom,
! [A2: set_set_v,B2: set_set_v,X: set_v] :
( ( ( sup_sup_set_set_v @ A2 @ B2 )
= ( insert_set_v2 @ X @ bot_bot_set_set_v ) )
= ( ( ( A2 = bot_bot_set_set_v )
& ( B2
= ( insert_set_v2 @ X @ bot_bot_set_set_v ) ) )
| ( ( A2
= ( insert_set_v2 @ X @ bot_bot_set_set_v ) )
& ( B2 = bot_bot_set_set_v ) )
| ( ( A2
= ( insert_set_v2 @ X @ bot_bot_set_set_v ) )
& ( B2
= ( insert_set_v2 @ X @ bot_bot_set_set_v ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_615_singleton__Un__iff,axiom,
! [X: product_prod_v_v,A2: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v )
= ( sup_su414716646722978715od_v_v @ A2 @ B2 ) )
= ( ( ( A2 = bot_bo723834152578015283od_v_v )
& ( B2
= ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v ) ) )
| ( ( A2
= ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v ) )
& ( B2 = bot_bo723834152578015283od_v_v ) )
| ( ( A2
= ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v ) )
& ( B2
= ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_616_singleton__Un__iff,axiom,
! [X: v,A2: set_v,B2: set_v] :
( ( ( insert_v2 @ X @ bot_bot_set_v )
= ( sup_sup_set_v @ A2 @ B2 ) )
= ( ( ( A2 = bot_bot_set_v )
& ( B2
= ( insert_v2 @ X @ bot_bot_set_v ) ) )
| ( ( A2
= ( insert_v2 @ X @ bot_bot_set_v ) )
& ( B2 = bot_bot_set_v ) )
| ( ( A2
= ( insert_v2 @ X @ bot_bot_set_v ) )
& ( B2
= ( insert_v2 @ X @ bot_bot_set_v ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_617_singleton__Un__iff,axiom,
! [X: set_v,A2: set_set_v,B2: set_set_v] :
( ( ( insert_set_v2 @ X @ bot_bot_set_set_v )
= ( sup_sup_set_set_v @ A2 @ B2 ) )
= ( ( ( A2 = bot_bot_set_set_v )
& ( B2
= ( insert_set_v2 @ X @ bot_bot_set_set_v ) ) )
| ( ( A2
= ( insert_set_v2 @ X @ bot_bot_set_set_v ) )
& ( B2 = bot_bot_set_set_v ) )
| ( ( A2
= ( insert_set_v2 @ X @ bot_bot_set_set_v ) )
& ( B2
= ( insert_set_v2 @ X @ bot_bot_set_set_v ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_618_graph_Ora__add__edge,axiom,
! [Vertices: set_v,Successors: v > set_v,X: v,Y: v,E4: set_Product_prod_v_v,V3: v,W: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl4291963740693775144ding_v @ Successors @ X @ Y @ E4 )
=> ( ( sCC_Bl4291963740693775144ding_v @ Successors @ X @ Y @ ( sup_su414716646722978715od_v_v @ E4 @ ( insert1338601472111419319od_v_v @ ( product_Pair_v_v @ V3 @ W ) @ bot_bo723834152578015283od_v_v ) ) )
| ( ( sCC_Bl4291963740693775144ding_v @ Successors @ X @ V3 @ ( sup_su414716646722978715od_v_v @ E4 @ ( insert1338601472111419319od_v_v @ ( product_Pair_v_v @ V3 @ W ) @ bot_bo723834152578015283od_v_v ) ) )
& ( sCC_Bl4291963740693775144ding_v @ Successors @ W @ Y @ ( sup_su414716646722978715od_v_v @ E4 @ ( insert1338601472111419319od_v_v @ ( product_Pair_v_v @ V3 @ W ) @ bot_bo723834152578015283od_v_v ) ) ) ) ) ) ) ).
% graph.ra_add_edge
thf(fact_619_Diff__partition,axiom,
! [A2: set_set_v,B2: set_set_v] :
( ( ord_le5216385588623774835_set_v @ A2 @ B2 )
=> ( ( sup_sup_set_set_v @ A2 @ ( minus_7228012346218142266_set_v @ B2 @ A2 ) )
= B2 ) ) ).
% Diff_partition
thf(fact_620_Diff__partition,axiom,
! [A2: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A2 @ B2 )
=> ( ( sup_su414716646722978715od_v_v @ A2 @ ( minus_4183494784930505774od_v_v @ B2 @ A2 ) )
= B2 ) ) ).
% Diff_partition
thf(fact_621_Diff__partition,axiom,
! [A2: set_v,B2: set_v] :
( ( ord_less_eq_set_v @ A2 @ B2 )
=> ( ( sup_sup_set_v @ A2 @ ( minus_minus_set_v @ B2 @ A2 ) )
= B2 ) ) ).
% Diff_partition
thf(fact_622_Diff__subset__conv,axiom,
! [A2: set_set_v,B2: set_set_v,C2: set_set_v] :
( ( ord_le5216385588623774835_set_v @ ( minus_7228012346218142266_set_v @ A2 @ B2 ) @ C2 )
= ( ord_le5216385588623774835_set_v @ A2 @ ( sup_sup_set_set_v @ B2 @ C2 ) ) ) ).
% Diff_subset_conv
thf(fact_623_Diff__subset__conv,axiom,
! [A2: set_Product_prod_v_v,B2: set_Product_prod_v_v,C2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ ( minus_4183494784930505774od_v_v @ A2 @ B2 ) @ C2 )
= ( ord_le7336532860387713383od_v_v @ A2 @ ( sup_su414716646722978715od_v_v @ B2 @ C2 ) ) ) ).
% Diff_subset_conv
thf(fact_624_Diff__subset__conv,axiom,
! [A2: set_v,B2: set_v,C2: set_v] :
( ( ord_less_eq_set_v @ ( minus_minus_set_v @ A2 @ B2 ) @ C2 )
= ( ord_less_eq_set_v @ A2 @ ( sup_sup_set_v @ B2 @ C2 ) ) ) ).
% Diff_subset_conv
thf(fact_625_graph_Oavoiding__explored,axiom,
! [Vertices: set_v,Successors: v > set_v,E: sCC_Bl1394983891496994913t_unit,X: v,Y: v,E4: set_Product_prod_v_v,W: v,V3: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl9196236973127232072t_unit @ Successors @ E )
=> ( ( sCC_Bl4291963740693775144ding_v @ Successors @ X @ Y @ E4 )
=> ( ~ ( member_v @ Y @ ( sCC_Bl157864678168468314t_unit @ E ) )
=> ( ( member_v @ W @ ( sCC_Bl157864678168468314t_unit @ E ) )
=> ( sCC_Bl4291963740693775144ding_v @ Successors @ X @ Y @ ( sup_su414716646722978715od_v_v @ E4 @ ( insert1338601472111419319od_v_v @ ( product_Pair_v_v @ V3 @ W ) @ bot_bo723834152578015283od_v_v ) ) ) ) ) ) ) ) ).
% graph.avoiding_explored
thf(fact_626_graph_Ounite__sub__env,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,V3: product_prod_v_v,E: sCC_Bl7326425374436813197t_unit,W: product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( sCC_Bl3607325323686918683t_unit @ Successors @ V3 @ E )
=> ( ( member7453568604450474000od_v_v @ W @ ( Successors @ V3 ) )
=> ( ~ ( member7453568604450474000od_v_v @ W @ ( sCC_Bl3878977043676959280t_unit @ E @ V3 ) )
=> ( ( member7453568604450474000od_v_v @ W @ ( sCC_Bl5498988629518860705t_unit @ E ) )
=> ( ~ ( member7453568604450474000od_v_v @ W @ ( sCC_Bl5094201334446601350t_unit @ E ) )
=> ( sCC_Bl7963838319573962697t_unit @ E @ ( sCC_Bl4702006153222411093od_v_v @ V3 @ W @ E ) ) ) ) ) ) ) ) ).
% graph.unite_sub_env
thf(fact_627_graph_Ounite__sub__env,axiom,
! [Vertices: set_v,Successors: v > set_v,V3: v,E: sCC_Bl1394983891496994913t_unit,W: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl1748261141445803503t_unit @ Successors @ V3 @ E )
=> ( ( member_v @ W @ ( Successors @ V3 ) )
=> ( ~ ( member_v @ W @ ( sCC_Bl3795065053823578884t_unit @ E @ V3 ) )
=> ( ( member_v @ W @ ( sCC_Bl4645233313691564917t_unit @ E ) )
=> ( ~ ( member_v @ W @ ( sCC_Bl157864678168468314t_unit @ E ) )
=> ( sCC_Bl5768913643336123637t_unit @ E @ ( sCC_Bloemen_unite_v @ V3 @ W @ E ) ) ) ) ) ) ) ) ).
% graph.unite_sub_env
thf(fact_628_graph_Ounite__wf__env,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,V3: product_prod_v_v,E: sCC_Bl7326425374436813197t_unit,W: product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( sCC_Bl3607325323686918683t_unit @ Successors @ V3 @ E )
=> ( ( member7453568604450474000od_v_v @ W @ ( Successors @ V3 ) )
=> ( ~ ( member7453568604450474000od_v_v @ W @ ( sCC_Bl3878977043676959280t_unit @ E @ V3 ) )
=> ( ( member7453568604450474000od_v_v @ W @ ( sCC_Bl5498988629518860705t_unit @ E ) )
=> ( ~ ( member7453568604450474000od_v_v @ W @ ( sCC_Bl5094201334446601350t_unit @ E ) )
=> ( sCC_Bl7798947040364291444t_unit @ Successors @ ( sCC_Bl4702006153222411093od_v_v @ V3 @ W @ E ) ) ) ) ) ) ) ) ).
% graph.unite_wf_env
thf(fact_629_graph_Ounite__wf__env,axiom,
! [Vertices: set_v,Successors: v > set_v,V3: v,E: sCC_Bl1394983891496994913t_unit,W: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl1748261141445803503t_unit @ Successors @ V3 @ E )
=> ( ( member_v @ W @ ( Successors @ V3 ) )
=> ( ~ ( member_v @ W @ ( sCC_Bl3795065053823578884t_unit @ E @ V3 ) )
=> ( ( member_v @ W @ ( sCC_Bl4645233313691564917t_unit @ E ) )
=> ( ~ ( member_v @ W @ ( sCC_Bl157864678168468314t_unit @ E ) )
=> ( sCC_Bl9196236973127232072t_unit @ Successors @ ( sCC_Bloemen_unite_v @ V3 @ W @ E ) ) ) ) ) ) ) ) ).
% graph.unite_wf_env
thf(fact_630_graph_Ounite__subscc,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,V3: product_prod_v_v,E: sCC_Bl7326425374436813197t_unit,W: product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( sCC_Bl3607325323686918683t_unit @ Successors @ V3 @ E )
=> ( ( member7453568604450474000od_v_v @ W @ ( Successors @ V3 ) )
=> ( ~ ( member7453568604450474000od_v_v @ W @ ( sCC_Bl3878977043676959280t_unit @ E @ V3 ) )
=> ( ( member7453568604450474000od_v_v @ W @ ( sCC_Bl5498988629518860705t_unit @ E ) )
=> ( ~ ( member7453568604450474000od_v_v @ W @ ( sCC_Bl5094201334446601350t_unit @ E ) )
=> ( sCC_Bl2301996248249672505od_v_v @ Successors @ ( sCC_Bl8440648026628373538t_unit @ ( sCC_Bl4702006153222411093od_v_v @ V3 @ W @ E ) @ ( hd_Product_prod_v_v @ ( sCC_Bl2021302119412358655t_unit @ ( sCC_Bl4702006153222411093od_v_v @ V3 @ W @ E ) ) ) ) ) ) ) ) ) ) ) ).
% graph.unite_subscc
thf(fact_631_graph_Ounite__subscc,axiom,
! [Vertices: set_v,Successors: v > set_v,V3: v,E: sCC_Bl1394983891496994913t_unit,W: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl1748261141445803503t_unit @ Successors @ V3 @ E )
=> ( ( member_v @ W @ ( Successors @ V3 ) )
=> ( ~ ( member_v @ W @ ( sCC_Bl3795065053823578884t_unit @ E @ V3 ) )
=> ( ( member_v @ W @ ( sCC_Bl4645233313691564917t_unit @ E ) )
=> ( ~ ( member_v @ W @ ( sCC_Bl157864678168468314t_unit @ E ) )
=> ( sCC_Bl5398416737448265317bscc_v @ Successors @ ( sCC_Bl1280885523602775798t_unit @ ( sCC_Bloemen_unite_v @ V3 @ W @ E ) @ ( hd_v @ ( sCC_Bl8828226123343373779t_unit @ ( sCC_Bloemen_unite_v @ V3 @ W @ E ) ) ) ) ) ) ) ) ) ) ) ).
% graph.unite_subscc
thf(fact_632_graph_Ounite__S__tl,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,E: sCC_Bl7326425374436813197t_unit,W: product_prod_v_v,V3: product_prod_v_v,N: product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( sCC_Bl7798947040364291444t_unit @ Successors @ E )
=> ( ( member7453568604450474000od_v_v @ W @ ( Successors @ V3 ) )
=> ( ~ ( member7453568604450474000od_v_v @ W @ ( sCC_Bl3878977043676959280t_unit @ E @ V3 ) )
=> ( ( member7453568604450474000od_v_v @ W @ ( sCC_Bl5498988629518860705t_unit @ E ) )
=> ( ~ ( member7453568604450474000od_v_v @ W @ ( sCC_Bl5094201334446601350t_unit @ E ) )
=> ( ( member7453568604450474000od_v_v @ N @ ( set_Product_prod_v_v2 @ ( tl_Product_prod_v_v @ ( sCC_Bl2021302119412358655t_unit @ ( sCC_Bl4702006153222411093od_v_v @ V3 @ W @ E ) ) ) ) )
=> ( ( sCC_Bl8440648026628373538t_unit @ ( sCC_Bl4702006153222411093od_v_v @ V3 @ W @ E ) @ N )
= ( sCC_Bl8440648026628373538t_unit @ E @ N ) ) ) ) ) ) ) ) ) ).
% graph.unite_S_tl
thf(fact_633_graph_Ounite__S__tl,axiom,
! [Vertices: set_v,Successors: v > set_v,E: sCC_Bl1394983891496994913t_unit,W: v,V3: v,N: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl9196236973127232072t_unit @ Successors @ E )
=> ( ( member_v @ W @ ( Successors @ V3 ) )
=> ( ~ ( member_v @ W @ ( sCC_Bl3795065053823578884t_unit @ E @ V3 ) )
=> ( ( member_v @ W @ ( sCC_Bl4645233313691564917t_unit @ E ) )
=> ( ~ ( member_v @ W @ ( sCC_Bl157864678168468314t_unit @ E ) )
=> ( ( member_v @ N @ ( set_v2 @ ( tl_v @ ( sCC_Bl8828226123343373779t_unit @ ( sCC_Bloemen_unite_v @ V3 @ W @ E ) ) ) ) )
=> ( ( sCC_Bl1280885523602775798t_unit @ ( sCC_Bloemen_unite_v @ V3 @ W @ E ) @ N )
= ( sCC_Bl1280885523602775798t_unit @ E @ N ) ) ) ) ) ) ) ) ) ).
% graph.unite_S_tl
thf(fact_634_graph_Opre__dfs__def,axiom,
! [Vertices: set_v,Successors: v > set_v,V3: v,E: sCC_Bl1394983891496994913t_unit] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl36166008131615352t_unit @ Successors @ V3 @ E )
= ( ( sCC_Bl9196236973127232072t_unit @ Successors @ E )
& ~ ( member_v @ V3 @ ( sCC_Bl4645233313691564917t_unit @ E ) )
& ( sCC_Bl649662514949026229able_v @ Successors @ ( sCC_Bl1090238580953940555t_unit @ E ) @ V3 )
& ( ( sCC_Bl3795065053823578884t_unit @ E @ V3 )
= bot_bot_set_v )
& ! [X2: v] :
( ( member_v @ X2 @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E ) ) )
=> ( sCC_Bl649662514949026229able_v @ Successors @ X2 @ V3 ) ) ) ) ) ).
% graph.pre_dfs_def
thf(fact_635_pre__dfss__def,axiom,
! [V3: v,E: sCC_Bl1394983891496994913t_unit] :
( ( sCC_Bl1748261141445803503t_unit @ successors @ V3 @ E )
= ( ( sCC_Bl9196236973127232072t_unit @ successors @ E )
& ( member_v @ V3 @ ( sCC_Bl4645233313691564917t_unit @ E ) )
& ( ( sCC_Bl8828226123343373779t_unit @ E )
!= nil_v )
& ( member_v @ V3 @ ( sCC_Bl1280885523602775798t_unit @ E @ ( hd_v @ ( sCC_Bl8828226123343373779t_unit @ E ) ) ) )
& ! [X2: v] :
( ( member_v @ X2 @ ( sCC_Bl3795065053823578884t_unit @ E @ V3 ) )
=> ( member_v @ X2 @ ( sup_sup_set_v @ ( sCC_Bl157864678168468314t_unit @ E ) @ ( sCC_Bl1280885523602775798t_unit @ E @ ( hd_v @ ( sCC_Bl8828226123343373779t_unit @ E ) ) ) ) ) )
& ! [X2: v] :
( ( member_v @ X2 @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E ) ) )
=> ( sCC_Bl649662514949026229able_v @ successors @ X2 @ V3 ) )
& ? [Ns2: list_v] :
( ( sCC_Bl9201514103433284750t_unit @ E )
= ( cons_v @ V3 @ Ns2 ) ) ) ) ).
% pre_dfss_def
thf(fact_636_sup__bot__left,axiom,
! [X: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ bot_bo723834152578015283od_v_v @ X )
= X ) ).
% sup_bot_left
thf(fact_637_sup__bot__left,axiom,
! [X: set_v] :
( ( sup_sup_set_v @ bot_bot_set_v @ X )
= X ) ).
% sup_bot_left
thf(fact_638_sup__bot__left,axiom,
! [X: set_set_v] :
( ( sup_sup_set_set_v @ bot_bot_set_set_v @ X )
= X ) ).
% sup_bot_left
thf(fact_639_sup__bot__left,axiom,
! [X: product_unit] :
( ( sup_sup_Product_unit @ bot_bot_Product_unit @ X )
= X ) ).
% sup_bot_left
thf(fact_640_sup__bot__right,axiom,
! [X: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ X @ bot_bo723834152578015283od_v_v )
= X ) ).
% sup_bot_right
thf(fact_641_sup__bot__right,axiom,
! [X: set_v] :
( ( sup_sup_set_v @ X @ bot_bot_set_v )
= X ) ).
% sup_bot_right
thf(fact_642_sup__bot__right,axiom,
! [X: set_set_v] :
( ( sup_sup_set_set_v @ X @ bot_bot_set_set_v )
= X ) ).
% sup_bot_right
thf(fact_643_sup__bot__right,axiom,
! [X: product_unit] :
( ( sup_sup_Product_unit @ X @ bot_bot_Product_unit )
= X ) ).
% sup_bot_right
thf(fact_644_bot__eq__sup__iff,axiom,
! [X: set_Product_prod_v_v,Y: set_Product_prod_v_v] :
( ( bot_bo723834152578015283od_v_v
= ( sup_su414716646722978715od_v_v @ X @ Y ) )
= ( ( X = bot_bo723834152578015283od_v_v )
& ( Y = bot_bo723834152578015283od_v_v ) ) ) ).
% bot_eq_sup_iff
thf(fact_645_bot__eq__sup__iff,axiom,
! [X: set_v,Y: set_v] :
( ( bot_bot_set_v
= ( sup_sup_set_v @ X @ Y ) )
= ( ( X = bot_bot_set_v )
& ( Y = bot_bot_set_v ) ) ) ).
% bot_eq_sup_iff
thf(fact_646_bot__eq__sup__iff,axiom,
! [X: set_set_v,Y: set_set_v] :
( ( bot_bot_set_set_v
= ( sup_sup_set_set_v @ X @ Y ) )
= ( ( X = bot_bot_set_set_v )
& ( Y = bot_bot_set_set_v ) ) ) ).
% bot_eq_sup_iff
thf(fact_647_bot__eq__sup__iff,axiom,
! [X: product_unit,Y: product_unit] :
( ( bot_bot_Product_unit
= ( sup_sup_Product_unit @ X @ Y ) )
= ( ( X = bot_bot_Product_unit )
& ( Y = bot_bot_Product_unit ) ) ) ).
% bot_eq_sup_iff
thf(fact_648_sup__eq__bot__iff,axiom,
! [X: set_Product_prod_v_v,Y: set_Product_prod_v_v] :
( ( ( sup_su414716646722978715od_v_v @ X @ Y )
= bot_bo723834152578015283od_v_v )
= ( ( X = bot_bo723834152578015283od_v_v )
& ( Y = bot_bo723834152578015283od_v_v ) ) ) ).
% sup_eq_bot_iff
thf(fact_649_sup__eq__bot__iff,axiom,
! [X: set_v,Y: set_v] :
( ( ( sup_sup_set_v @ X @ Y )
= bot_bot_set_v )
= ( ( X = bot_bot_set_v )
& ( Y = bot_bot_set_v ) ) ) ).
% sup_eq_bot_iff
thf(fact_650_sup__eq__bot__iff,axiom,
! [X: set_set_v,Y: set_set_v] :
( ( ( sup_sup_set_set_v @ X @ Y )
= bot_bot_set_set_v )
= ( ( X = bot_bot_set_set_v )
& ( Y = bot_bot_set_set_v ) ) ) ).
% sup_eq_bot_iff
thf(fact_651_sup__eq__bot__iff,axiom,
! [X: product_unit,Y: product_unit] :
( ( ( sup_sup_Product_unit @ X @ Y )
= bot_bot_Product_unit )
= ( ( X = bot_bot_Product_unit )
& ( Y = bot_bot_Product_unit ) ) ) ).
% sup_eq_bot_iff
thf(fact_652_sup__bot_Oeq__neutr__iff,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ( sup_su414716646722978715od_v_v @ A @ B )
= bot_bo723834152578015283od_v_v )
= ( ( A = bot_bo723834152578015283od_v_v )
& ( B = bot_bo723834152578015283od_v_v ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_653_sup__bot_Oeq__neutr__iff,axiom,
! [A: set_v,B: set_v] :
( ( ( sup_sup_set_v @ A @ B )
= bot_bot_set_v )
= ( ( A = bot_bot_set_v )
& ( B = bot_bot_set_v ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_654_sup__bot_Oeq__neutr__iff,axiom,
! [A: set_set_v,B: set_set_v] :
( ( ( sup_sup_set_set_v @ A @ B )
= bot_bot_set_set_v )
= ( ( A = bot_bot_set_set_v )
& ( B = bot_bot_set_set_v ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_655_sup__bot_Oeq__neutr__iff,axiom,
! [A: product_unit,B: product_unit] :
( ( ( sup_sup_Product_unit @ A @ B )
= bot_bot_Product_unit )
= ( ( A = bot_bot_Product_unit )
& ( B = bot_bot_Product_unit ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_656_list_Oinject,axiom,
! [X21: v,X222: list_v,Y21: v,Y222: list_v] :
( ( ( cons_v @ X21 @ X222 )
= ( cons_v @ Y21 @ Y222 ) )
= ( ( X21 = Y21 )
& ( X222 = Y222 ) ) ) ).
% list.inject
thf(fact_657_cst_H,axiom,
( ( sCC_Bl9201514103433284750t_unit @ e2 )
= ( cons_v @ v2 @ ( sCC_Bl9201514103433284750t_unit @ e ) ) ) ).
% cst'
thf(fact_658_le__sup__iff,axiom,
! [X: set_set_v,Y: set_set_v,Z: set_set_v] :
( ( ord_le5216385588623774835_set_v @ ( sup_sup_set_set_v @ X @ Y ) @ Z )
= ( ( ord_le5216385588623774835_set_v @ X @ Z )
& ( ord_le5216385588623774835_set_v @ Y @ Z ) ) ) ).
% le_sup_iff
thf(fact_659_le__sup__iff,axiom,
! [X: product_unit,Y: product_unit,Z: product_unit] :
( ( ord_le3221252021190050221t_unit @ ( sup_sup_Product_unit @ X @ Y ) @ Z )
= ( ( ord_le3221252021190050221t_unit @ X @ Z )
& ( ord_le3221252021190050221t_unit @ Y @ Z ) ) ) ).
% le_sup_iff
thf(fact_660_le__sup__iff,axiom,
! [X: set_Product_prod_v_v,Y: set_Product_prod_v_v,Z: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ ( sup_su414716646722978715od_v_v @ X @ Y ) @ Z )
= ( ( ord_le7336532860387713383od_v_v @ X @ Z )
& ( ord_le7336532860387713383od_v_v @ Y @ Z ) ) ) ).
% le_sup_iff
thf(fact_661_le__sup__iff,axiom,
! [X: set_v,Y: set_v,Z: set_v] :
( ( ord_less_eq_set_v @ ( sup_sup_set_v @ X @ Y ) @ Z )
= ( ( ord_less_eq_set_v @ X @ Z )
& ( ord_less_eq_set_v @ Y @ Z ) ) ) ).
% le_sup_iff
thf(fact_662_sup_Obounded__iff,axiom,
! [B: set_set_v,C: set_set_v,A: set_set_v] :
( ( ord_le5216385588623774835_set_v @ ( sup_sup_set_set_v @ B @ C ) @ A )
= ( ( ord_le5216385588623774835_set_v @ B @ A )
& ( ord_le5216385588623774835_set_v @ C @ A ) ) ) ).
% sup.bounded_iff
thf(fact_663_sup_Obounded__iff,axiom,
! [B: product_unit,C: product_unit,A: product_unit] :
( ( ord_le3221252021190050221t_unit @ ( sup_sup_Product_unit @ B @ C ) @ A )
= ( ( ord_le3221252021190050221t_unit @ B @ A )
& ( ord_le3221252021190050221t_unit @ C @ A ) ) ) ).
% sup.bounded_iff
thf(fact_664_sup_Obounded__iff,axiom,
! [B: set_Product_prod_v_v,C: set_Product_prod_v_v,A: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ ( sup_su414716646722978715od_v_v @ B @ C ) @ A )
= ( ( ord_le7336532860387713383od_v_v @ B @ A )
& ( ord_le7336532860387713383od_v_v @ C @ A ) ) ) ).
% sup.bounded_iff
thf(fact_665_sup_Obounded__iff,axiom,
! [B: set_v,C: set_v,A: set_v] :
( ( ord_less_eq_set_v @ ( sup_sup_set_v @ B @ C ) @ A )
= ( ( ord_less_eq_set_v @ B @ A )
& ( ord_less_eq_set_v @ C @ A ) ) ) ).
% sup.bounded_iff
thf(fact_666_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_667_sup__bot_Oright__neutral,axiom,
! [A: set_v] :
( ( sup_sup_set_v @ A @ bot_bot_set_v )
= A ) ).
% sup_bot.right_neutral
thf(fact_668_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_669_sup__bot_Oright__neutral,axiom,
! [A: product_unit] :
( ( sup_sup_Product_unit @ A @ bot_bot_Product_unit )
= A ) ).
% sup_bot.right_neutral
thf(fact_670_sup__bot_Oneutr__eq__iff,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( bot_bo723834152578015283od_v_v
= ( sup_su414716646722978715od_v_v @ A @ B ) )
= ( ( A = bot_bo723834152578015283od_v_v )
& ( B = bot_bo723834152578015283od_v_v ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_671_sup__bot_Oneutr__eq__iff,axiom,
! [A: set_v,B: set_v] :
( ( bot_bot_set_v
= ( sup_sup_set_v @ A @ B ) )
= ( ( A = bot_bot_set_v )
& ( B = bot_bot_set_v ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_672_sup__bot_Oneutr__eq__iff,axiom,
! [A: set_set_v,B: set_set_v] :
( ( bot_bot_set_set_v
= ( sup_sup_set_set_v @ A @ B ) )
= ( ( A = bot_bot_set_set_v )
& ( B = bot_bot_set_set_v ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_673_sup__bot_Oneutr__eq__iff,axiom,
! [A: product_unit,B: product_unit] :
( ( bot_bot_Product_unit
= ( sup_sup_Product_unit @ A @ B ) )
= ( ( A = bot_bot_Product_unit )
& ( B = bot_bot_Product_unit ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_674_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_675_sup__bot_Oleft__neutral,axiom,
! [A: set_v] :
( ( sup_sup_set_v @ bot_bot_set_v @ A )
= A ) ).
% sup_bot.left_neutral
thf(fact_676_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_677_sup__bot_Oleft__neutral,axiom,
! [A: product_unit] :
( ( sup_sup_Product_unit @ bot_bot_Product_unit @ A )
= A ) ).
% sup_bot.left_neutral
thf(fact_678_list_Osimps_I15_J,axiom,
! [X21: product_prod_v_v,X222: list_P7986770385144383213od_v_v] :
( ( set_Product_prod_v_v2 @ ( cons_P4120604216776828829od_v_v @ X21 @ X222 ) )
= ( insert1338601472111419319od_v_v @ X21 @ ( set_Product_prod_v_v2 @ X222 ) ) ) ).
% list.simps(15)
thf(fact_679_list_Osimps_I15_J,axiom,
! [X21: set_v,X222: list_set_v] :
( ( set_set_v2 @ ( cons_set_v @ X21 @ X222 ) )
= ( insert_set_v2 @ X21 @ ( set_set_v2 @ X222 ) ) ) ).
% list.simps(15)
thf(fact_680_list_Osimps_I15_J,axiom,
! [X21: v,X222: list_v] :
( ( set_v2 @ ( cons_v @ X21 @ X222 ) )
= ( insert_v2 @ X21 @ ( set_v2 @ X222 ) ) ) ).
% list.simps(15)
thf(fact_681_append1__eq__conv,axiom,
! [Xs: list_v,X: v,Ys: list_v,Y: v] :
( ( ( append_v @ Xs @ ( cons_v @ X @ nil_v ) )
= ( append_v @ Ys @ ( cons_v @ Y @ nil_v ) ) )
= ( ( Xs = Ys )
& ( X = Y ) ) ) ).
% append1_eq_conv
thf(fact_682_insert__Nil,axiom,
! [X: v] :
( ( insert_v @ X @ nil_v )
= ( cons_v @ X @ nil_v ) ) ).
% insert_Nil
thf(fact_683_not__in__set__insert,axiom,
! [X: product_prod_v_v,Xs: list_P7986770385144383213od_v_v] :
( ~ ( member7453568604450474000od_v_v @ X @ ( set_Product_prod_v_v2 @ Xs ) )
=> ( ( insert4539780211034306307od_v_v @ X @ Xs )
= ( cons_P4120604216776828829od_v_v @ X @ Xs ) ) ) ).
% not_in_set_insert
thf(fact_684_not__in__set__insert,axiom,
! [X: v,Xs: list_v] :
( ~ ( member_v @ X @ ( set_v2 @ Xs ) )
=> ( ( insert_v @ X @ Xs )
= ( cons_v @ X @ Xs ) ) ) ).
% not_in_set_insert
thf(fact_685_hd__Cons__tl,axiom,
! [Xs: list_v] :
( ( Xs != nil_v )
=> ( ( cons_v @ ( hd_v @ Xs ) @ ( tl_v @ Xs ) )
= Xs ) ) ).
% hd_Cons_tl
thf(fact_686_list_Ocollapse,axiom,
! [List: list_v] :
( ( List != nil_v )
=> ( ( cons_v @ ( hd_v @ List ) @ ( tl_v @ List ) )
= List ) ) ).
% list.collapse
thf(fact_687_transpose_Ocases,axiom,
! [X: list_list_v] :
( ( X != nil_list_v )
=> ( ! [Xss: list_list_v] :
( X
!= ( cons_list_v @ nil_v @ Xss ) )
=> ~ ! [X3: v,Xs3: list_v,Xss: list_list_v] :
( X
!= ( cons_list_v @ ( cons_v @ X3 @ Xs3 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_688_not__Cons__self2,axiom,
! [X: v,Xs: list_v] :
( ( cons_v @ X @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_689_sorted__wrt_Ocases,axiom,
! [X: produc8237170675765753490list_v] :
( ! [P3: v > v > $o] :
( X
!= ( produc601102195597853570list_v @ P3 @ nil_v ) )
=> ~ ! [P3: v > v > $o,X3: v,Ys3: list_v] :
( X
!= ( produc601102195597853570list_v @ P3 @ ( cons_v @ X3 @ Ys3 ) ) ) ) ).
% sorted_wrt.cases
thf(fact_690_successively_Ocases,axiom,
! [X: produc8237170675765753490list_v] :
( ! [P3: v > v > $o] :
( X
!= ( produc601102195597853570list_v @ P3 @ nil_v ) )
=> ( ! [P3: v > v > $o,X3: v] :
( X
!= ( produc601102195597853570list_v @ P3 @ ( cons_v @ X3 @ nil_v ) ) )
=> ~ ! [P3: v > v > $o,X3: v,Y2: v,Xs3: list_v] :
( X
!= ( produc601102195597853570list_v @ P3 @ ( cons_v @ X3 @ ( cons_v @ Y2 @ Xs3 ) ) ) ) ) ) ).
% successively.cases
thf(fact_691_splice_Ocases,axiom,
! [X: produc1391462591744249447list_v] :
( ! [Ys3: list_v] :
( X
!= ( produc6795410681906604247list_v @ nil_v @ Ys3 ) )
=> ~ ! [X3: v,Xs3: list_v,Ys3: list_v] :
( X
!= ( produc6795410681906604247list_v @ ( cons_v @ X3 @ Xs3 ) @ Ys3 ) ) ) ).
% splice.cases
thf(fact_692_shuffles_Ocases,axiom,
! [X: produc1391462591744249447list_v] :
( ! [Ys3: list_v] :
( X
!= ( produc6795410681906604247list_v @ nil_v @ Ys3 ) )
=> ( ! [Xs3: list_v] :
( X
!= ( produc6795410681906604247list_v @ Xs3 @ nil_v ) )
=> ~ ! [X3: v,Xs3: list_v,Y2: v,Ys3: list_v] :
( X
!= ( produc6795410681906604247list_v @ ( cons_v @ X3 @ Xs3 ) @ ( cons_v @ Y2 @ Ys3 ) ) ) ) ) ).
% shuffles.cases
thf(fact_693_list__nonempty__induct,axiom,
! [Xs: list_v,P: list_v > $o] :
( ( Xs != nil_v )
=> ( ! [X3: v] : ( P @ ( cons_v @ X3 @ nil_v ) )
=> ( ! [X3: v,Xs3: list_v] :
( ( Xs3 != nil_v )
=> ( ( P @ Xs3 )
=> ( P @ ( cons_v @ X3 @ Xs3 ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_694_list__induct2_H,axiom,
! [P: list_v > list_v > $o,Xs: list_v,Ys: list_v] :
( ( P @ nil_v @ nil_v )
=> ( ! [X3: v,Xs3: list_v] : ( P @ ( cons_v @ X3 @ Xs3 ) @ nil_v )
=> ( ! [Y2: v,Ys3: list_v] : ( P @ nil_v @ ( cons_v @ Y2 @ Ys3 ) )
=> ( ! [X3: v,Xs3: list_v,Y2: v,Ys3: list_v] :
( ( P @ Xs3 @ Ys3 )
=> ( P @ ( cons_v @ X3 @ Xs3 ) @ ( cons_v @ Y2 @ Ys3 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_695_neq__Nil__conv,axiom,
! [Xs: list_v] :
( ( Xs != nil_v )
= ( ? [Y3: v,Ys4: list_v] :
( Xs
= ( cons_v @ Y3 @ Ys4 ) ) ) ) ).
% neq_Nil_conv
thf(fact_696_remdups__adj_Ocases,axiom,
! [X: list_v] :
( ( X != nil_v )
=> ( ! [X3: v] :
( X
!= ( cons_v @ X3 @ nil_v ) )
=> ~ ! [X3: v,Y2: v,Xs3: list_v] :
( X
!= ( cons_v @ X3 @ ( cons_v @ Y2 @ Xs3 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_697_list_Oexhaust,axiom,
! [Y: list_v] :
( ( Y != nil_v )
=> ~ ! [X212: v,X223: list_v] :
( Y
!= ( cons_v @ X212 @ X223 ) ) ) ).
% list.exhaust
thf(fact_698_list_OdiscI,axiom,
! [List: list_v,X21: v,X222: list_v] :
( ( List
= ( cons_v @ X21 @ X222 ) )
=> ( List != nil_v ) ) ).
% list.discI
thf(fact_699_list_Odistinct_I1_J,axiom,
! [X21: v,X222: list_v] :
( nil_v
!= ( cons_v @ X21 @ X222 ) ) ).
% list.distinct(1)
thf(fact_700_list_Oset__intros_I2_J,axiom,
! [Y: product_prod_v_v,X222: list_P7986770385144383213od_v_v,X21: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ Y @ ( set_Product_prod_v_v2 @ X222 ) )
=> ( member7453568604450474000od_v_v @ Y @ ( set_Product_prod_v_v2 @ ( cons_P4120604216776828829od_v_v @ X21 @ X222 ) ) ) ) ).
% list.set_intros(2)
thf(fact_701_list_Oset__intros_I2_J,axiom,
! [Y: v,X222: list_v,X21: v] :
( ( member_v @ Y @ ( set_v2 @ X222 ) )
=> ( member_v @ Y @ ( set_v2 @ ( cons_v @ X21 @ X222 ) ) ) ) ).
% list.set_intros(2)
thf(fact_702_list_Oset__intros_I1_J,axiom,
! [X21: product_prod_v_v,X222: list_P7986770385144383213od_v_v] : ( member7453568604450474000od_v_v @ X21 @ ( set_Product_prod_v_v2 @ ( cons_P4120604216776828829od_v_v @ X21 @ X222 ) ) ) ).
% list.set_intros(1)
thf(fact_703_list_Oset__intros_I1_J,axiom,
! [X21: v,X222: list_v] : ( member_v @ X21 @ ( set_v2 @ ( cons_v @ X21 @ X222 ) ) ) ).
% list.set_intros(1)
thf(fact_704_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_705_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_706_set__ConsD,axiom,
! [Y: product_prod_v_v,X: product_prod_v_v,Xs: list_P7986770385144383213od_v_v] :
( ( member7453568604450474000od_v_v @ Y @ ( set_Product_prod_v_v2 @ ( cons_P4120604216776828829od_v_v @ X @ Xs ) ) )
=> ( ( Y = X )
| ( member7453568604450474000od_v_v @ Y @ ( set_Product_prod_v_v2 @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_707_set__ConsD,axiom,
! [Y: v,X: v,Xs: list_v] :
( ( member_v @ Y @ ( set_v2 @ ( cons_v @ X @ Xs ) ) )
=> ( ( Y = X )
| ( member_v @ Y @ ( set_v2 @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_708_Cons__eq__appendI,axiom,
! [X: v,Xs1: list_v,Ys: list_v,Xs: list_v,Zs: list_v] :
( ( ( cons_v @ X @ Xs1 )
= Ys )
=> ( ( Xs
= ( append_v @ Xs1 @ Zs ) )
=> ( ( cons_v @ X @ Xs )
= ( append_v @ Ys @ Zs ) ) ) ) ).
% Cons_eq_appendI
thf(fact_709_append__Cons,axiom,
! [X: v,Xs: list_v,Ys: list_v] :
( ( append_v @ ( cons_v @ X @ Xs ) @ Ys )
= ( cons_v @ X @ ( append_v @ Xs @ Ys ) ) ) ).
% append_Cons
thf(fact_710_distinct__length__2__or__more,axiom,
! [A: v,B: v,Xs: list_v] :
( ( distinct_v @ ( cons_v @ A @ ( cons_v @ B @ Xs ) ) )
= ( ( A != B )
& ( distinct_v @ ( cons_v @ A @ Xs ) )
& ( distinct_v @ ( cons_v @ B @ Xs ) ) ) ) ).
% distinct_length_2_or_more
thf(fact_711_list_Osel_I1_J,axiom,
! [X21: v,X222: list_v] :
( ( hd_v @ ( cons_v @ X21 @ X222 ) )
= X21 ) ).
% list.sel(1)
thf(fact_712_list_Osel_I3_J,axiom,
! [X21: v,X222: list_v] :
( ( tl_v @ ( cons_v @ X21 @ X222 ) )
= X222 ) ).
% list.sel(3)
thf(fact_713_precedes__in__tail,axiom,
! [X: v,Z: v,Y: v,Zs: list_v] :
( ( X != Z )
=> ( ( sCC_Bl4022239298816431255edes_v @ X @ Y @ ( cons_v @ Z @ Zs ) )
= ( sCC_Bl4022239298816431255edes_v @ X @ Y @ Zs ) ) ) ).
% precedes_in_tail
thf(fact_714_set__subset__Cons,axiom,
! [Xs: list_P7986770385144383213od_v_v,X: product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ ( set_Product_prod_v_v2 @ Xs ) @ ( set_Product_prod_v_v2 @ ( cons_P4120604216776828829od_v_v @ X @ Xs ) ) ) ).
% set_subset_Cons
thf(fact_715_set__subset__Cons,axiom,
! [Xs: list_v,X: v] : ( ord_less_eq_set_v @ ( set_v2 @ Xs ) @ ( set_v2 @ ( cons_v @ X @ Xs ) ) ) ).
% set_subset_Cons
thf(fact_716_rev__nonempty__induct,axiom,
! [Xs: list_v,P: list_v > $o] :
( ( Xs != nil_v )
=> ( ! [X3: v] : ( P @ ( cons_v @ X3 @ nil_v ) )
=> ( ! [X3: v,Xs3: list_v] :
( ( Xs3 != nil_v )
=> ( ( P @ Xs3 )
=> ( P @ ( append_v @ Xs3 @ ( cons_v @ X3 @ nil_v ) ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% rev_nonempty_induct
thf(fact_717_append__eq__Cons__conv,axiom,
! [Ys: list_v,Zs: list_v,X: v,Xs: list_v] :
( ( ( append_v @ Ys @ Zs )
= ( cons_v @ X @ Xs ) )
= ( ( ( Ys = nil_v )
& ( Zs
= ( cons_v @ X @ Xs ) ) )
| ? [Ys5: list_v] :
( ( Ys
= ( cons_v @ X @ Ys5 ) )
& ( ( append_v @ Ys5 @ Zs )
= Xs ) ) ) ) ).
% append_eq_Cons_conv
thf(fact_718_Cons__eq__append__conv,axiom,
! [X: v,Xs: list_v,Ys: list_v,Zs: list_v] :
( ( ( cons_v @ X @ Xs )
= ( append_v @ Ys @ Zs ) )
= ( ( ( Ys = nil_v )
& ( ( cons_v @ X @ Xs )
= Zs ) )
| ? [Ys5: list_v] :
( ( ( cons_v @ X @ Ys5 )
= Ys )
& ( Xs
= ( append_v @ Ys5 @ Zs ) ) ) ) ) ).
% Cons_eq_append_conv
thf(fact_719_rev__exhaust,axiom,
! [Xs: list_v] :
( ( Xs != nil_v )
=> ~ ! [Ys3: list_v,Y2: v] :
( Xs
!= ( append_v @ Ys3 @ ( cons_v @ Y2 @ nil_v ) ) ) ) ).
% rev_exhaust
thf(fact_720_rev__induct,axiom,
! [P: list_v > $o,Xs: list_v] :
( ( P @ nil_v )
=> ( ! [X3: v,Xs3: list_v] :
( ( P @ Xs3 )
=> ( P @ ( append_v @ Xs3 @ ( cons_v @ X3 @ nil_v ) ) ) )
=> ( P @ Xs ) ) ) ).
% rev_induct
thf(fact_721_split__list__first__prop__iff,axiom,
! [Xs: list_v,P: v > $o] :
( ( ? [X2: v] :
( ( member_v @ X2 @ ( set_v2 @ Xs ) )
& ( P @ X2 ) ) )
= ( ? [Ys4: list_v,X2: v] :
( ? [Zs2: list_v] :
( Xs
= ( append_v @ Ys4 @ ( cons_v @ X2 @ Zs2 ) ) )
& ( P @ X2 )
& ! [Y3: v] :
( ( member_v @ Y3 @ ( set_v2 @ Ys4 ) )
=> ~ ( P @ Y3 ) ) ) ) ) ).
% split_list_first_prop_iff
thf(fact_722_split__list__last__prop__iff,axiom,
! [Xs: list_v,P: v > $o] :
( ( ? [X2: v] :
( ( member_v @ X2 @ ( set_v2 @ Xs ) )
& ( P @ X2 ) ) )
= ( ? [Ys4: list_v,X2: v,Zs2: list_v] :
( ( Xs
= ( append_v @ Ys4 @ ( cons_v @ X2 @ Zs2 ) ) )
& ( P @ X2 )
& ! [Y3: v] :
( ( member_v @ Y3 @ ( set_v2 @ Zs2 ) )
=> ~ ( P @ Y3 ) ) ) ) ) ).
% split_list_last_prop_iff
thf(fact_723_in__set__conv__decomp__first,axiom,
! [X: product_prod_v_v,Xs: list_P7986770385144383213od_v_v] :
( ( member7453568604450474000od_v_v @ X @ ( set_Product_prod_v_v2 @ Xs ) )
= ( ? [Ys4: list_P7986770385144383213od_v_v,Zs2: list_P7986770385144383213od_v_v] :
( ( Xs
= ( append2138873909117096322od_v_v @ Ys4 @ ( cons_P4120604216776828829od_v_v @ X @ Zs2 ) ) )
& ~ ( member7453568604450474000od_v_v @ X @ ( set_Product_prod_v_v2 @ Ys4 ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_724_in__set__conv__decomp__first,axiom,
! [X: v,Xs: list_v] :
( ( member_v @ X @ ( set_v2 @ Xs ) )
= ( ? [Ys4: list_v,Zs2: list_v] :
( ( Xs
= ( append_v @ Ys4 @ ( cons_v @ X @ Zs2 ) ) )
& ~ ( member_v @ X @ ( set_v2 @ Ys4 ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_725_in__set__conv__decomp__last,axiom,
! [X: product_prod_v_v,Xs: list_P7986770385144383213od_v_v] :
( ( member7453568604450474000od_v_v @ X @ ( set_Product_prod_v_v2 @ Xs ) )
= ( ? [Ys4: list_P7986770385144383213od_v_v,Zs2: list_P7986770385144383213od_v_v] :
( ( Xs
= ( append2138873909117096322od_v_v @ Ys4 @ ( cons_P4120604216776828829od_v_v @ X @ Zs2 ) ) )
& ~ ( member7453568604450474000od_v_v @ X @ ( set_Product_prod_v_v2 @ Zs2 ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_726_in__set__conv__decomp__last,axiom,
! [X: v,Xs: list_v] :
( ( member_v @ X @ ( set_v2 @ Xs ) )
= ( ? [Ys4: list_v,Zs2: list_v] :
( ( Xs
= ( append_v @ Ys4 @ ( cons_v @ X @ Zs2 ) ) )
& ~ ( member_v @ X @ ( set_v2 @ Zs2 ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_727_split__list__first__propE,axiom,
! [Xs: list_v,P: v > $o] :
( ? [X4: v] :
( ( member_v @ X4 @ ( set_v2 @ Xs ) )
& ( P @ X4 ) )
=> ~ ! [Ys3: list_v,X3: v] :
( ? [Zs3: list_v] :
( Xs
= ( append_v @ Ys3 @ ( cons_v @ X3 @ Zs3 ) ) )
=> ( ( P @ X3 )
=> ~ ! [Xa2: v] :
( ( member_v @ Xa2 @ ( set_v2 @ Ys3 ) )
=> ~ ( P @ Xa2 ) ) ) ) ) ).
% split_list_first_propE
thf(fact_728_split__list__last__propE,axiom,
! [Xs: list_v,P: v > $o] :
( ? [X4: v] :
( ( member_v @ X4 @ ( set_v2 @ Xs ) )
& ( P @ X4 ) )
=> ~ ! [Ys3: list_v,X3: v,Zs3: list_v] :
( ( Xs
= ( append_v @ Ys3 @ ( cons_v @ X3 @ Zs3 ) ) )
=> ( ( P @ X3 )
=> ~ ! [Xa2: v] :
( ( member_v @ Xa2 @ ( set_v2 @ Zs3 ) )
=> ~ ( P @ Xa2 ) ) ) ) ) ).
% split_list_last_propE
thf(fact_729_split__list__first__prop,axiom,
! [Xs: list_v,P: v > $o] :
( ? [X4: v] :
( ( member_v @ X4 @ ( set_v2 @ Xs ) )
& ( P @ X4 ) )
=> ? [Ys3: list_v,X3: v] :
( ? [Zs3: list_v] :
( Xs
= ( append_v @ Ys3 @ ( cons_v @ X3 @ Zs3 ) ) )
& ( P @ X3 )
& ! [Xa2: v] :
( ( member_v @ Xa2 @ ( set_v2 @ Ys3 ) )
=> ~ ( P @ Xa2 ) ) ) ) ).
% split_list_first_prop
thf(fact_730_split__list__last__prop,axiom,
! [Xs: list_v,P: v > $o] :
( ? [X4: v] :
( ( member_v @ X4 @ ( set_v2 @ Xs ) )
& ( P @ X4 ) )
=> ? [Ys3: list_v,X3: v,Zs3: list_v] :
( ( Xs
= ( append_v @ Ys3 @ ( cons_v @ X3 @ Zs3 ) ) )
& ( P @ X3 )
& ! [Xa2: v] :
( ( member_v @ Xa2 @ ( set_v2 @ Zs3 ) )
=> ~ ( P @ Xa2 ) ) ) ) ).
% split_list_last_prop
thf(fact_731_in__set__conv__decomp,axiom,
! [X: product_prod_v_v,Xs: list_P7986770385144383213od_v_v] :
( ( member7453568604450474000od_v_v @ X @ ( set_Product_prod_v_v2 @ Xs ) )
= ( ? [Ys4: list_P7986770385144383213od_v_v,Zs2: list_P7986770385144383213od_v_v] :
( Xs
= ( append2138873909117096322od_v_v @ Ys4 @ ( cons_P4120604216776828829od_v_v @ X @ Zs2 ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_732_in__set__conv__decomp,axiom,
! [X: v,Xs: list_v] :
( ( member_v @ X @ ( set_v2 @ Xs ) )
= ( ? [Ys4: list_v,Zs2: list_v] :
( Xs
= ( append_v @ Ys4 @ ( cons_v @ X @ Zs2 ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_733_append__Cons__eq__iff,axiom,
! [X: product_prod_v_v,Xs: list_P7986770385144383213od_v_v,Ys: list_P7986770385144383213od_v_v,Xs4: list_P7986770385144383213od_v_v,Ys6: list_P7986770385144383213od_v_v] :
( ~ ( member7453568604450474000od_v_v @ X @ ( set_Product_prod_v_v2 @ Xs ) )
=> ( ~ ( member7453568604450474000od_v_v @ X @ ( set_Product_prod_v_v2 @ Ys ) )
=> ( ( ( append2138873909117096322od_v_v @ Xs @ ( cons_P4120604216776828829od_v_v @ X @ Ys ) )
= ( append2138873909117096322od_v_v @ Xs4 @ ( cons_P4120604216776828829od_v_v @ X @ Ys6 ) ) )
= ( ( Xs = Xs4 )
& ( Ys = Ys6 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_734_append__Cons__eq__iff,axiom,
! [X: v,Xs: list_v,Ys: list_v,Xs4: list_v,Ys6: list_v] :
( ~ ( member_v @ X @ ( set_v2 @ Xs ) )
=> ( ~ ( member_v @ X @ ( set_v2 @ Ys ) )
=> ( ( ( append_v @ Xs @ ( cons_v @ X @ Ys ) )
= ( append_v @ Xs4 @ ( cons_v @ X @ Ys6 ) ) )
= ( ( Xs = Xs4 )
& ( Ys = Ys6 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_735_split__list__propE,axiom,
! [Xs: list_v,P: v > $o] :
( ? [X4: v] :
( ( member_v @ X4 @ ( set_v2 @ Xs ) )
& ( P @ X4 ) )
=> ~ ! [Ys3: list_v,X3: v] :
( ? [Zs3: list_v] :
( Xs
= ( append_v @ Ys3 @ ( cons_v @ X3 @ Zs3 ) ) )
=> ~ ( P @ X3 ) ) ) ).
% split_list_propE
thf(fact_736_split__list__first,axiom,
! [X: product_prod_v_v,Xs: list_P7986770385144383213od_v_v] :
( ( member7453568604450474000od_v_v @ X @ ( set_Product_prod_v_v2 @ Xs ) )
=> ? [Ys3: list_P7986770385144383213od_v_v,Zs3: list_P7986770385144383213od_v_v] :
( ( Xs
= ( append2138873909117096322od_v_v @ Ys3 @ ( cons_P4120604216776828829od_v_v @ X @ Zs3 ) ) )
& ~ ( member7453568604450474000od_v_v @ X @ ( set_Product_prod_v_v2 @ Ys3 ) ) ) ) ).
% split_list_first
thf(fact_737_split__list__first,axiom,
! [X: v,Xs: list_v] :
( ( member_v @ X @ ( set_v2 @ Xs ) )
=> ? [Ys3: list_v,Zs3: list_v] :
( ( Xs
= ( append_v @ Ys3 @ ( cons_v @ X @ Zs3 ) ) )
& ~ ( member_v @ X @ ( set_v2 @ Ys3 ) ) ) ) ).
% split_list_first
thf(fact_738_split__list__prop,axiom,
! [Xs: list_v,P: v > $o] :
( ? [X4: v] :
( ( member_v @ X4 @ ( set_v2 @ Xs ) )
& ( P @ X4 ) )
=> ? [Ys3: list_v,X3: v] :
( ? [Zs3: list_v] :
( Xs
= ( append_v @ Ys3 @ ( cons_v @ X3 @ Zs3 ) ) )
& ( P @ X3 ) ) ) ).
% split_list_prop
thf(fact_739_split__list__last,axiom,
! [X: product_prod_v_v,Xs: list_P7986770385144383213od_v_v] :
( ( member7453568604450474000od_v_v @ X @ ( set_Product_prod_v_v2 @ Xs ) )
=> ? [Ys3: list_P7986770385144383213od_v_v,Zs3: list_P7986770385144383213od_v_v] :
( ( Xs
= ( append2138873909117096322od_v_v @ Ys3 @ ( cons_P4120604216776828829od_v_v @ X @ Zs3 ) ) )
& ~ ( member7453568604450474000od_v_v @ X @ ( set_Product_prod_v_v2 @ Zs3 ) ) ) ) ).
% split_list_last
thf(fact_740_split__list__last,axiom,
! [X: v,Xs: list_v] :
( ( member_v @ X @ ( set_v2 @ Xs ) )
=> ? [Ys3: list_v,Zs3: list_v] :
( ( Xs
= ( append_v @ Ys3 @ ( cons_v @ X @ Zs3 ) ) )
& ~ ( member_v @ X @ ( set_v2 @ Zs3 ) ) ) ) ).
% split_list_last
thf(fact_741_split__list,axiom,
! [X: product_prod_v_v,Xs: list_P7986770385144383213od_v_v] :
( ( member7453568604450474000od_v_v @ X @ ( set_Product_prod_v_v2 @ Xs ) )
=> ? [Ys3: list_P7986770385144383213od_v_v,Zs3: list_P7986770385144383213od_v_v] :
( Xs
= ( append2138873909117096322od_v_v @ Ys3 @ ( cons_P4120604216776828829od_v_v @ X @ Zs3 ) ) ) ) ).
% split_list
thf(fact_742_split__list,axiom,
! [X: v,Xs: list_v] :
( ( member_v @ X @ ( set_v2 @ Xs ) )
=> ? [Ys3: list_v,Zs3: list_v] :
( Xs
= ( append_v @ Ys3 @ ( cons_v @ X @ Zs3 ) ) ) ) ).
% split_list
thf(fact_743_distinct__singleton,axiom,
! [X: v] : ( distinct_v @ ( cons_v @ X @ nil_v ) ) ).
% distinct_singleton
thf(fact_744_distinct_Osimps_I2_J,axiom,
! [X: product_prod_v_v,Xs: list_P7986770385144383213od_v_v] :
( ( distin6159370996967099744od_v_v @ ( cons_P4120604216776828829od_v_v @ X @ Xs ) )
= ( ~ ( member7453568604450474000od_v_v @ X @ ( set_Product_prod_v_v2 @ Xs ) )
& ( distin6159370996967099744od_v_v @ Xs ) ) ) ).
% distinct.simps(2)
thf(fact_745_distinct_Osimps_I2_J,axiom,
! [X: v,Xs: list_v] :
( ( distinct_v @ ( cons_v @ X @ Xs ) )
= ( ~ ( member_v @ X @ ( set_v2 @ Xs ) )
& ( distinct_v @ Xs ) ) ) ).
% distinct.simps(2)
thf(fact_746_Nil__tl,axiom,
! [Xs: list_v] :
( ( nil_v
= ( tl_v @ Xs ) )
= ( ( Xs = nil_v )
| ? [X2: v] :
( Xs
= ( cons_v @ X2 @ nil_v ) ) ) ) ).
% Nil_tl
thf(fact_747_tl__Nil,axiom,
! [Xs: list_v] :
( ( ( tl_v @ Xs )
= nil_v )
= ( ( Xs = nil_v )
| ? [X2: v] :
( Xs
= ( cons_v @ X2 @ nil_v ) ) ) ) ).
% tl_Nil
thf(fact_748_tail__not__precedes,axiom,
! [Y: product_prod_v_v,X: product_prod_v_v,Xs: list_P7986770385144383213od_v_v] :
( ( sCC_Bl2026170059108282219od_v_v @ Y @ X @ ( cons_P4120604216776828829od_v_v @ X @ Xs ) )
=> ( ~ ( member7453568604450474000od_v_v @ X @ ( set_Product_prod_v_v2 @ Xs ) )
=> ( X = Y ) ) ) ).
% tail_not_precedes
thf(fact_749_tail__not__precedes,axiom,
! [Y: v,X: v,Xs: list_v] :
( ( sCC_Bl4022239298816431255edes_v @ Y @ X @ ( cons_v @ X @ Xs ) )
=> ( ~ ( member_v @ X @ ( set_v2 @ Xs ) )
=> ( X = Y ) ) ) ).
% tail_not_precedes
thf(fact_750_head__precedes,axiom,
! [Y: product_prod_v_v,X: product_prod_v_v,Xs: list_P7986770385144383213od_v_v] :
( ( member7453568604450474000od_v_v @ Y @ ( set_Product_prod_v_v2 @ ( cons_P4120604216776828829od_v_v @ X @ Xs ) ) )
=> ( sCC_Bl2026170059108282219od_v_v @ X @ Y @ ( cons_P4120604216776828829od_v_v @ X @ Xs ) ) ) ).
% head_precedes
thf(fact_751_head__precedes,axiom,
! [Y: v,X: v,Xs: list_v] :
( ( member_v @ Y @ ( set_v2 @ ( cons_v @ X @ Xs ) ) )
=> ( sCC_Bl4022239298816431255edes_v @ X @ Y @ ( cons_v @ X @ Xs ) ) ) ).
% head_precedes
thf(fact_752_List_Oinsert__def,axiom,
( insert4539780211034306307od_v_v
= ( ^ [X2: product_prod_v_v,Xs5: list_P7986770385144383213od_v_v] : ( if_lis7521272669439687347od_v_v @ ( member7453568604450474000od_v_v @ X2 @ ( set_Product_prod_v_v2 @ Xs5 ) ) @ Xs5 @ ( cons_P4120604216776828829od_v_v @ X2 @ Xs5 ) ) ) ) ).
% List.insert_def
thf(fact_753_List_Oinsert__def,axiom,
( insert_v
= ( ^ [X2: v,Xs5: list_v] : ( if_list_v @ ( member_v @ X2 @ ( set_v2 @ Xs5 ) ) @ Xs5 @ ( cons_v @ X2 @ Xs5 ) ) ) ) ).
% List.insert_def
thf(fact_754_not__distinct__decomp,axiom,
! [Ws: list_v] :
( ~ ( distinct_v @ Ws )
=> ? [Xs3: list_v,Ys3: list_v,Zs3: list_v,Y2: v] :
( Ws
= ( append_v @ Xs3 @ ( append_v @ ( cons_v @ Y2 @ nil_v ) @ ( append_v @ Ys3 @ ( append_v @ ( cons_v @ Y2 @ nil_v ) @ Zs3 ) ) ) ) ) ) ).
% not_distinct_decomp
thf(fact_755_not__distinct__conv__prefix,axiom,
! [As: list_P7986770385144383213od_v_v] :
( ( ~ ( distin6159370996967099744od_v_v @ As ) )
= ( ? [Xs5: list_P7986770385144383213od_v_v,Y3: product_prod_v_v,Ys4: list_P7986770385144383213od_v_v] :
( ( member7453568604450474000od_v_v @ Y3 @ ( set_Product_prod_v_v2 @ Xs5 ) )
& ( distin6159370996967099744od_v_v @ Xs5 )
& ( As
= ( append2138873909117096322od_v_v @ Xs5 @ ( cons_P4120604216776828829od_v_v @ Y3 @ Ys4 ) ) ) ) ) ) ).
% not_distinct_conv_prefix
thf(fact_756_not__distinct__conv__prefix,axiom,
! [As: list_v] :
( ( ~ ( distinct_v @ As ) )
= ( ? [Xs5: list_v,Y3: v,Ys4: list_v] :
( ( member_v @ Y3 @ ( set_v2 @ Xs5 ) )
& ( distinct_v @ Xs5 )
& ( As
= ( append_v @ Xs5 @ ( cons_v @ Y3 @ Ys4 ) ) ) ) ) ) ).
% not_distinct_conv_prefix
thf(fact_757_list_Oexhaust__sel,axiom,
! [List: list_v] :
( ( List != nil_v )
=> ( List
= ( cons_v @ ( hd_v @ List ) @ ( tl_v @ List ) ) ) ) ).
% list.exhaust_sel
thf(fact_758_precedes__def,axiom,
( sCC_Bl2026170059108282219od_v_v
= ( ^ [X2: product_prod_v_v,Y3: product_prod_v_v,Xs5: list_P7986770385144383213od_v_v] :
? [L: list_P7986770385144383213od_v_v,R3: list_P7986770385144383213od_v_v] :
( ( Xs5
= ( append2138873909117096322od_v_v @ L @ ( cons_P4120604216776828829od_v_v @ X2 @ R3 ) ) )
& ( member7453568604450474000od_v_v @ Y3 @ ( set_Product_prod_v_v2 @ ( cons_P4120604216776828829od_v_v @ X2 @ R3 ) ) ) ) ) ) ).
% precedes_def
thf(fact_759_precedes__def,axiom,
( sCC_Bl4022239298816431255edes_v
= ( ^ [X2: v,Y3: v,Xs5: list_v] :
? [L: list_v,R3: list_v] :
( ( Xs5
= ( append_v @ L @ ( cons_v @ X2 @ R3 ) ) )
& ( member_v @ Y3 @ ( set_v2 @ ( cons_v @ X2 @ R3 ) ) ) ) ) ) ).
% precedes_def
thf(fact_760_split__list__precedes,axiom,
! [Y: product_prod_v_v,Ys: list_P7986770385144383213od_v_v,X: product_prod_v_v,Xs: list_P7986770385144383213od_v_v] :
( ( member7453568604450474000od_v_v @ Y @ ( set_Product_prod_v_v2 @ ( append2138873909117096322od_v_v @ Ys @ ( cons_P4120604216776828829od_v_v @ X @ nil_Product_prod_v_v ) ) ) )
=> ( sCC_Bl2026170059108282219od_v_v @ Y @ X @ ( append2138873909117096322od_v_v @ Ys @ ( cons_P4120604216776828829od_v_v @ X @ Xs ) ) ) ) ).
% split_list_precedes
thf(fact_761_split__list__precedes,axiom,
! [Y: v,Ys: list_v,X: v,Xs: list_v] :
( ( member_v @ Y @ ( set_v2 @ ( append_v @ Ys @ ( cons_v @ X @ nil_v ) ) ) )
=> ( sCC_Bl4022239298816431255edes_v @ Y @ X @ ( append_v @ Ys @ ( cons_v @ X @ Xs ) ) ) ) ).
% split_list_precedes
thf(fact_762_inf__sup__ord_I4_J,axiom,
! [Y: set_set_v,X: set_set_v] : ( ord_le5216385588623774835_set_v @ Y @ ( sup_sup_set_set_v @ X @ Y ) ) ).
% inf_sup_ord(4)
thf(fact_763_inf__sup__ord_I4_J,axiom,
! [Y: product_unit,X: product_unit] : ( ord_le3221252021190050221t_unit @ Y @ ( sup_sup_Product_unit @ X @ Y ) ) ).
% inf_sup_ord(4)
thf(fact_764_inf__sup__ord_I4_J,axiom,
! [Y: set_Product_prod_v_v,X: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ Y @ ( sup_su414716646722978715od_v_v @ X @ Y ) ) ).
% inf_sup_ord(4)
thf(fact_765_inf__sup__ord_I4_J,axiom,
! [Y: set_v,X: set_v] : ( ord_less_eq_set_v @ Y @ ( sup_sup_set_v @ X @ Y ) ) ).
% inf_sup_ord(4)
thf(fact_766_inf__sup__ord_I3_J,axiom,
! [X: set_set_v,Y: set_set_v] : ( ord_le5216385588623774835_set_v @ X @ ( sup_sup_set_set_v @ X @ Y ) ) ).
% inf_sup_ord(3)
thf(fact_767_inf__sup__ord_I3_J,axiom,
! [X: product_unit,Y: product_unit] : ( ord_le3221252021190050221t_unit @ X @ ( sup_sup_Product_unit @ X @ Y ) ) ).
% inf_sup_ord(3)
thf(fact_768_inf__sup__ord_I3_J,axiom,
! [X: set_Product_prod_v_v,Y: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ X @ ( sup_su414716646722978715od_v_v @ X @ Y ) ) ).
% inf_sup_ord(3)
thf(fact_769_inf__sup__ord_I3_J,axiom,
! [X: set_v,Y: set_v] : ( ord_less_eq_set_v @ X @ ( sup_sup_set_v @ X @ Y ) ) ).
% inf_sup_ord(3)
thf(fact_770_le__supE,axiom,
! [A: set_set_v,B: set_set_v,X: set_set_v] :
( ( ord_le5216385588623774835_set_v @ ( sup_sup_set_set_v @ A @ B ) @ X )
=> ~ ( ( ord_le5216385588623774835_set_v @ A @ X )
=> ~ ( ord_le5216385588623774835_set_v @ B @ X ) ) ) ).
% le_supE
thf(fact_771_le__supE,axiom,
! [A: product_unit,B: product_unit,X: product_unit] :
( ( ord_le3221252021190050221t_unit @ ( sup_sup_Product_unit @ A @ B ) @ X )
=> ~ ( ( ord_le3221252021190050221t_unit @ A @ X )
=> ~ ( ord_le3221252021190050221t_unit @ B @ X ) ) ) ).
% le_supE
thf(fact_772_le__supE,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v,X: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ ( sup_su414716646722978715od_v_v @ A @ B ) @ X )
=> ~ ( ( ord_le7336532860387713383od_v_v @ A @ X )
=> ~ ( ord_le7336532860387713383od_v_v @ B @ X ) ) ) ).
% le_supE
thf(fact_773_le__supE,axiom,
! [A: set_v,B: set_v,X: set_v] :
( ( ord_less_eq_set_v @ ( sup_sup_set_v @ A @ B ) @ X )
=> ~ ( ( ord_less_eq_set_v @ A @ X )
=> ~ ( ord_less_eq_set_v @ B @ X ) ) ) ).
% le_supE
thf(fact_774_le__supI,axiom,
! [A: set_set_v,X: set_set_v,B: set_set_v] :
( ( ord_le5216385588623774835_set_v @ A @ X )
=> ( ( ord_le5216385588623774835_set_v @ B @ X )
=> ( ord_le5216385588623774835_set_v @ ( sup_sup_set_set_v @ A @ B ) @ X ) ) ) ).
% le_supI
thf(fact_775_le__supI,axiom,
! [A: product_unit,X: product_unit,B: product_unit] :
( ( ord_le3221252021190050221t_unit @ A @ X )
=> ( ( ord_le3221252021190050221t_unit @ B @ X )
=> ( ord_le3221252021190050221t_unit @ ( sup_sup_Product_unit @ A @ B ) @ X ) ) ) ).
% le_supI
thf(fact_776_le__supI,axiom,
! [A: set_Product_prod_v_v,X: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ X )
=> ( ( ord_le7336532860387713383od_v_v @ B @ X )
=> ( ord_le7336532860387713383od_v_v @ ( sup_su414716646722978715od_v_v @ A @ B ) @ X ) ) ) ).
% le_supI
thf(fact_777_le__supI,axiom,
! [A: set_v,X: set_v,B: set_v] :
( ( ord_less_eq_set_v @ A @ X )
=> ( ( ord_less_eq_set_v @ B @ X )
=> ( ord_less_eq_set_v @ ( sup_sup_set_v @ A @ B ) @ X ) ) ) ).
% le_supI
thf(fact_778_sup__ge1,axiom,
! [X: set_set_v,Y: set_set_v] : ( ord_le5216385588623774835_set_v @ X @ ( sup_sup_set_set_v @ X @ Y ) ) ).
% sup_ge1
thf(fact_779_sup__ge1,axiom,
! [X: product_unit,Y: product_unit] : ( ord_le3221252021190050221t_unit @ X @ ( sup_sup_Product_unit @ X @ Y ) ) ).
% sup_ge1
thf(fact_780_sup__ge1,axiom,
! [X: set_Product_prod_v_v,Y: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ X @ ( sup_su414716646722978715od_v_v @ X @ Y ) ) ).
% sup_ge1
thf(fact_781_sup__ge1,axiom,
! [X: set_v,Y: set_v] : ( ord_less_eq_set_v @ X @ ( sup_sup_set_v @ X @ Y ) ) ).
% sup_ge1
thf(fact_782_sup__ge2,axiom,
! [Y: set_set_v,X: set_set_v] : ( ord_le5216385588623774835_set_v @ Y @ ( sup_sup_set_set_v @ X @ Y ) ) ).
% sup_ge2
thf(fact_783_sup__ge2,axiom,
! [Y: product_unit,X: product_unit] : ( ord_le3221252021190050221t_unit @ Y @ ( sup_sup_Product_unit @ X @ Y ) ) ).
% sup_ge2
thf(fact_784_sup__ge2,axiom,
! [Y: set_Product_prod_v_v,X: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ Y @ ( sup_su414716646722978715od_v_v @ X @ Y ) ) ).
% sup_ge2
thf(fact_785_sup__ge2,axiom,
! [Y: set_v,X: set_v] : ( ord_less_eq_set_v @ Y @ ( sup_sup_set_v @ X @ Y ) ) ).
% sup_ge2
thf(fact_786_le__supI1,axiom,
! [X: set_set_v,A: set_set_v,B: set_set_v] :
( ( ord_le5216385588623774835_set_v @ X @ A )
=> ( ord_le5216385588623774835_set_v @ X @ ( sup_sup_set_set_v @ A @ B ) ) ) ).
% le_supI1
thf(fact_787_le__supI1,axiom,
! [X: product_unit,A: product_unit,B: product_unit] :
( ( ord_le3221252021190050221t_unit @ X @ A )
=> ( ord_le3221252021190050221t_unit @ X @ ( sup_sup_Product_unit @ A @ B ) ) ) ).
% le_supI1
thf(fact_788_le__supI1,axiom,
! [X: set_Product_prod_v_v,A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X @ A )
=> ( ord_le7336532860387713383od_v_v @ X @ ( sup_su414716646722978715od_v_v @ A @ B ) ) ) ).
% le_supI1
thf(fact_789_le__supI1,axiom,
! [X: set_v,A: set_v,B: set_v] :
( ( ord_less_eq_set_v @ X @ A )
=> ( ord_less_eq_set_v @ X @ ( sup_sup_set_v @ A @ B ) ) ) ).
% le_supI1
thf(fact_790_le__supI2,axiom,
! [X: set_set_v,B: set_set_v,A: set_set_v] :
( ( ord_le5216385588623774835_set_v @ X @ B )
=> ( ord_le5216385588623774835_set_v @ X @ ( sup_sup_set_set_v @ A @ B ) ) ) ).
% le_supI2
thf(fact_791_le__supI2,axiom,
! [X: product_unit,B: product_unit,A: product_unit] :
( ( ord_le3221252021190050221t_unit @ X @ B )
=> ( ord_le3221252021190050221t_unit @ X @ ( sup_sup_Product_unit @ A @ B ) ) ) ).
% le_supI2
thf(fact_792_le__supI2,axiom,
! [X: set_Product_prod_v_v,B: set_Product_prod_v_v,A: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X @ B )
=> ( ord_le7336532860387713383od_v_v @ X @ ( sup_su414716646722978715od_v_v @ A @ B ) ) ) ).
% le_supI2
thf(fact_793_le__supI2,axiom,
! [X: set_v,B: set_v,A: set_v] :
( ( ord_less_eq_set_v @ X @ B )
=> ( ord_less_eq_set_v @ X @ ( sup_sup_set_v @ A @ B ) ) ) ).
% le_supI2
thf(fact_794_sup_Omono,axiom,
! [C: set_set_v,A: set_set_v,D2: set_set_v,B: set_set_v] :
( ( ord_le5216385588623774835_set_v @ C @ A )
=> ( ( ord_le5216385588623774835_set_v @ D2 @ B )
=> ( ord_le5216385588623774835_set_v @ ( sup_sup_set_set_v @ C @ D2 ) @ ( sup_sup_set_set_v @ A @ B ) ) ) ) ).
% sup.mono
thf(fact_795_sup_Omono,axiom,
! [C: product_unit,A: product_unit,D2: product_unit,B: product_unit] :
( ( ord_le3221252021190050221t_unit @ C @ A )
=> ( ( ord_le3221252021190050221t_unit @ D2 @ B )
=> ( ord_le3221252021190050221t_unit @ ( sup_sup_Product_unit @ C @ D2 ) @ ( sup_sup_Product_unit @ A @ B ) ) ) ) ).
% sup.mono
thf(fact_796_sup_Omono,axiom,
! [C: set_Product_prod_v_v,A: set_Product_prod_v_v,D2: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ C @ A )
=> ( ( ord_le7336532860387713383od_v_v @ D2 @ B )
=> ( ord_le7336532860387713383od_v_v @ ( sup_su414716646722978715od_v_v @ C @ D2 ) @ ( sup_su414716646722978715od_v_v @ A @ B ) ) ) ) ).
% sup.mono
thf(fact_797_sup_Omono,axiom,
! [C: set_v,A: set_v,D2: set_v,B: set_v] :
( ( ord_less_eq_set_v @ C @ A )
=> ( ( ord_less_eq_set_v @ D2 @ B )
=> ( ord_less_eq_set_v @ ( sup_sup_set_v @ C @ D2 ) @ ( sup_sup_set_v @ A @ B ) ) ) ) ).
% sup.mono
thf(fact_798_sup__mono,axiom,
! [A: set_set_v,C: set_set_v,B: set_set_v,D2: set_set_v] :
( ( ord_le5216385588623774835_set_v @ A @ C )
=> ( ( ord_le5216385588623774835_set_v @ B @ D2 )
=> ( ord_le5216385588623774835_set_v @ ( sup_sup_set_set_v @ A @ B ) @ ( sup_sup_set_set_v @ C @ D2 ) ) ) ) ).
% sup_mono
thf(fact_799_sup__mono,axiom,
! [A: product_unit,C: product_unit,B: product_unit,D2: product_unit] :
( ( ord_le3221252021190050221t_unit @ A @ C )
=> ( ( ord_le3221252021190050221t_unit @ B @ D2 )
=> ( ord_le3221252021190050221t_unit @ ( sup_sup_Product_unit @ A @ B ) @ ( sup_sup_Product_unit @ C @ D2 ) ) ) ) ).
% sup_mono
thf(fact_800_sup__mono,axiom,
! [A: set_Product_prod_v_v,C: set_Product_prod_v_v,B: set_Product_prod_v_v,D2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ C )
=> ( ( ord_le7336532860387713383od_v_v @ B @ D2 )
=> ( ord_le7336532860387713383od_v_v @ ( sup_su414716646722978715od_v_v @ A @ B ) @ ( sup_su414716646722978715od_v_v @ C @ D2 ) ) ) ) ).
% sup_mono
thf(fact_801_sup__mono,axiom,
! [A: set_v,C: set_v,B: set_v,D2: set_v] :
( ( ord_less_eq_set_v @ A @ C )
=> ( ( ord_less_eq_set_v @ B @ D2 )
=> ( ord_less_eq_set_v @ ( sup_sup_set_v @ A @ B ) @ ( sup_sup_set_v @ C @ D2 ) ) ) ) ).
% sup_mono
thf(fact_802_sup__least,axiom,
! [Y: set_set_v,X: set_set_v,Z: set_set_v] :
( ( ord_le5216385588623774835_set_v @ Y @ X )
=> ( ( ord_le5216385588623774835_set_v @ Z @ X )
=> ( ord_le5216385588623774835_set_v @ ( sup_sup_set_set_v @ Y @ Z ) @ X ) ) ) ).
% sup_least
thf(fact_803_sup__least,axiom,
! [Y: product_unit,X: product_unit,Z: product_unit] :
( ( ord_le3221252021190050221t_unit @ Y @ X )
=> ( ( ord_le3221252021190050221t_unit @ Z @ X )
=> ( ord_le3221252021190050221t_unit @ ( sup_sup_Product_unit @ Y @ Z ) @ X ) ) ) ).
% sup_least
thf(fact_804_sup__least,axiom,
! [Y: set_Product_prod_v_v,X: set_Product_prod_v_v,Z: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ Y @ X )
=> ( ( ord_le7336532860387713383od_v_v @ Z @ X )
=> ( ord_le7336532860387713383od_v_v @ ( sup_su414716646722978715od_v_v @ Y @ Z ) @ X ) ) ) ).
% sup_least
thf(fact_805_sup__least,axiom,
! [Y: set_v,X: set_v,Z: set_v] :
( ( ord_less_eq_set_v @ Y @ X )
=> ( ( ord_less_eq_set_v @ Z @ X )
=> ( ord_less_eq_set_v @ ( sup_sup_set_v @ Y @ Z ) @ X ) ) ) ).
% sup_least
thf(fact_806_le__iff__sup,axiom,
( ord_le5216385588623774835_set_v
= ( ^ [X2: set_set_v,Y3: set_set_v] :
( ( sup_sup_set_set_v @ X2 @ Y3 )
= Y3 ) ) ) ).
% le_iff_sup
thf(fact_807_le__iff__sup,axiom,
( ord_le3221252021190050221t_unit
= ( ^ [X2: product_unit,Y3: product_unit] :
( ( sup_sup_Product_unit @ X2 @ Y3 )
= Y3 ) ) ) ).
% le_iff_sup
thf(fact_808_le__iff__sup,axiom,
( ord_le7336532860387713383od_v_v
= ( ^ [X2: set_Product_prod_v_v,Y3: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ X2 @ Y3 )
= Y3 ) ) ) ).
% le_iff_sup
thf(fact_809_le__iff__sup,axiom,
( ord_less_eq_set_v
= ( ^ [X2: set_v,Y3: set_v] :
( ( sup_sup_set_v @ X2 @ Y3 )
= Y3 ) ) ) ).
% le_iff_sup
thf(fact_810_sup_OorderE,axiom,
! [B: set_set_v,A: set_set_v] :
( ( ord_le5216385588623774835_set_v @ B @ A )
=> ( A
= ( sup_sup_set_set_v @ A @ B ) ) ) ).
% sup.orderE
thf(fact_811_sup_OorderE,axiom,
! [B: product_unit,A: product_unit] :
( ( ord_le3221252021190050221t_unit @ B @ A )
=> ( A
= ( sup_sup_Product_unit @ A @ B ) ) ) ).
% sup.orderE
thf(fact_812_sup_OorderE,axiom,
! [B: set_Product_prod_v_v,A: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ B @ A )
=> ( A
= ( sup_su414716646722978715od_v_v @ A @ B ) ) ) ).
% sup.orderE
thf(fact_813_sup_OorderE,axiom,
! [B: set_v,A: set_v] :
( ( ord_less_eq_set_v @ B @ A )
=> ( A
= ( sup_sup_set_v @ A @ B ) ) ) ).
% sup.orderE
thf(fact_814_sup_OorderI,axiom,
! [A: set_set_v,B: set_set_v] :
( ( A
= ( sup_sup_set_set_v @ A @ B ) )
=> ( ord_le5216385588623774835_set_v @ B @ A ) ) ).
% sup.orderI
thf(fact_815_sup_OorderI,axiom,
! [A: product_unit,B: product_unit] :
( ( A
= ( sup_sup_Product_unit @ A @ B ) )
=> ( ord_le3221252021190050221t_unit @ B @ A ) ) ).
% sup.orderI
thf(fact_816_sup_OorderI,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( A
= ( sup_su414716646722978715od_v_v @ A @ B ) )
=> ( ord_le7336532860387713383od_v_v @ B @ A ) ) ).
% sup.orderI
thf(fact_817_sup_OorderI,axiom,
! [A: set_v,B: set_v] :
( ( A
= ( sup_sup_set_v @ A @ B ) )
=> ( ord_less_eq_set_v @ B @ A ) ) ).
% sup.orderI
thf(fact_818_sup__unique,axiom,
! [F: set_set_v > set_set_v > set_set_v,X: set_set_v,Y: set_set_v] :
( ! [X3: set_set_v,Y2: set_set_v] : ( ord_le5216385588623774835_set_v @ X3 @ ( F @ X3 @ Y2 ) )
=> ( ! [X3: set_set_v,Y2: set_set_v] : ( ord_le5216385588623774835_set_v @ Y2 @ ( F @ X3 @ Y2 ) )
=> ( ! [X3: set_set_v,Y2: set_set_v,Z3: set_set_v] :
( ( ord_le5216385588623774835_set_v @ Y2 @ X3 )
=> ( ( ord_le5216385588623774835_set_v @ Z3 @ X3 )
=> ( ord_le5216385588623774835_set_v @ ( F @ Y2 @ Z3 ) @ X3 ) ) )
=> ( ( sup_sup_set_set_v @ X @ Y )
= ( F @ X @ Y ) ) ) ) ) ).
% sup_unique
thf(fact_819_sup__unique,axiom,
! [F: product_unit > product_unit > product_unit,X: product_unit,Y: product_unit] :
( ! [X3: product_unit,Y2: product_unit] : ( ord_le3221252021190050221t_unit @ X3 @ ( F @ X3 @ Y2 ) )
=> ( ! [X3: product_unit,Y2: product_unit] : ( ord_le3221252021190050221t_unit @ Y2 @ ( F @ X3 @ Y2 ) )
=> ( ! [X3: product_unit,Y2: product_unit,Z3: product_unit] :
( ( ord_le3221252021190050221t_unit @ Y2 @ X3 )
=> ( ( ord_le3221252021190050221t_unit @ Z3 @ X3 )
=> ( ord_le3221252021190050221t_unit @ ( F @ Y2 @ Z3 ) @ X3 ) ) )
=> ( ( sup_sup_Product_unit @ X @ Y )
= ( F @ X @ Y ) ) ) ) ) ).
% sup_unique
thf(fact_820_sup__unique,axiom,
! [F: set_Product_prod_v_v > set_Product_prod_v_v > set_Product_prod_v_v,X: set_Product_prod_v_v,Y: set_Product_prod_v_v] :
( ! [X3: set_Product_prod_v_v,Y2: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ X3 @ ( F @ X3 @ Y2 ) )
=> ( ! [X3: set_Product_prod_v_v,Y2: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ Y2 @ ( F @ X3 @ Y2 ) )
=> ( ! [X3: set_Product_prod_v_v,Y2: set_Product_prod_v_v,Z3: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ Y2 @ X3 )
=> ( ( ord_le7336532860387713383od_v_v @ Z3 @ X3 )
=> ( ord_le7336532860387713383od_v_v @ ( F @ Y2 @ Z3 ) @ X3 ) ) )
=> ( ( sup_su414716646722978715od_v_v @ X @ Y )
= ( F @ X @ Y ) ) ) ) ) ).
% sup_unique
thf(fact_821_sup__unique,axiom,
! [F: set_v > set_v > set_v,X: set_v,Y: set_v] :
( ! [X3: set_v,Y2: set_v] : ( ord_less_eq_set_v @ X3 @ ( F @ X3 @ Y2 ) )
=> ( ! [X3: set_v,Y2: set_v] : ( ord_less_eq_set_v @ Y2 @ ( F @ X3 @ Y2 ) )
=> ( ! [X3: set_v,Y2: set_v,Z3: set_v] :
( ( ord_less_eq_set_v @ Y2 @ X3 )
=> ( ( ord_less_eq_set_v @ Z3 @ X3 )
=> ( ord_less_eq_set_v @ ( F @ Y2 @ Z3 ) @ X3 ) ) )
=> ( ( sup_sup_set_v @ X @ Y )
= ( F @ X @ Y ) ) ) ) ) ).
% sup_unique
thf(fact_822_sup_Oabsorb1,axiom,
! [B: set_set_v,A: set_set_v] :
( ( ord_le5216385588623774835_set_v @ B @ A )
=> ( ( sup_sup_set_set_v @ A @ B )
= A ) ) ).
% sup.absorb1
thf(fact_823_sup_Oabsorb1,axiom,
! [B: product_unit,A: product_unit] :
( ( ord_le3221252021190050221t_unit @ B @ A )
=> ( ( sup_sup_Product_unit @ A @ B )
= A ) ) ).
% sup.absorb1
thf(fact_824_sup_Oabsorb1,axiom,
! [B: set_Product_prod_v_v,A: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ B @ A )
=> ( ( sup_su414716646722978715od_v_v @ A @ B )
= A ) ) ).
% sup.absorb1
thf(fact_825_sup_Oabsorb1,axiom,
! [B: set_v,A: set_v] :
( ( ord_less_eq_set_v @ B @ A )
=> ( ( sup_sup_set_v @ A @ B )
= A ) ) ).
% sup.absorb1
thf(fact_826_sup_Oabsorb2,axiom,
! [A: set_set_v,B: set_set_v] :
( ( ord_le5216385588623774835_set_v @ A @ B )
=> ( ( sup_sup_set_set_v @ A @ B )
= B ) ) ).
% sup.absorb2
thf(fact_827_sup_Oabsorb2,axiom,
! [A: product_unit,B: product_unit] :
( ( ord_le3221252021190050221t_unit @ A @ B )
=> ( ( sup_sup_Product_unit @ A @ B )
= B ) ) ).
% sup.absorb2
thf(fact_828_sup_Oabsorb2,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ B )
=> ( ( sup_su414716646722978715od_v_v @ A @ B )
= B ) ) ).
% sup.absorb2
thf(fact_829_sup_Oabsorb2,axiom,
! [A: set_v,B: set_v] :
( ( ord_less_eq_set_v @ A @ B )
=> ( ( sup_sup_set_v @ A @ B )
= B ) ) ).
% sup.absorb2
thf(fact_830_sup__absorb1,axiom,
! [Y: set_set_v,X: set_set_v] :
( ( ord_le5216385588623774835_set_v @ Y @ X )
=> ( ( sup_sup_set_set_v @ X @ Y )
= X ) ) ).
% sup_absorb1
thf(fact_831_sup__absorb1,axiom,
! [Y: product_unit,X: product_unit] :
( ( ord_le3221252021190050221t_unit @ Y @ X )
=> ( ( sup_sup_Product_unit @ X @ Y )
= X ) ) ).
% sup_absorb1
thf(fact_832_sup__absorb1,axiom,
! [Y: set_Product_prod_v_v,X: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ Y @ X )
=> ( ( sup_su414716646722978715od_v_v @ X @ Y )
= X ) ) ).
% sup_absorb1
thf(fact_833_sup__absorb1,axiom,
! [Y: set_v,X: set_v] :
( ( ord_less_eq_set_v @ Y @ X )
=> ( ( sup_sup_set_v @ X @ Y )
= X ) ) ).
% sup_absorb1
thf(fact_834_sup__absorb2,axiom,
! [X: set_set_v,Y: set_set_v] :
( ( ord_le5216385588623774835_set_v @ X @ Y )
=> ( ( sup_sup_set_set_v @ X @ Y )
= Y ) ) ).
% sup_absorb2
thf(fact_835_sup__absorb2,axiom,
! [X: product_unit,Y: product_unit] :
( ( ord_le3221252021190050221t_unit @ X @ Y )
=> ( ( sup_sup_Product_unit @ X @ Y )
= Y ) ) ).
% sup_absorb2
thf(fact_836_sup__absorb2,axiom,
! [X: set_Product_prod_v_v,Y: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X @ Y )
=> ( ( sup_su414716646722978715od_v_v @ X @ Y )
= Y ) ) ).
% sup_absorb2
thf(fact_837_sup__absorb2,axiom,
! [X: set_v,Y: set_v] :
( ( ord_less_eq_set_v @ X @ Y )
=> ( ( sup_sup_set_v @ X @ Y )
= Y ) ) ).
% sup_absorb2
thf(fact_838_sup_OboundedE,axiom,
! [B: set_set_v,C: set_set_v,A: set_set_v] :
( ( ord_le5216385588623774835_set_v @ ( sup_sup_set_set_v @ B @ C ) @ A )
=> ~ ( ( ord_le5216385588623774835_set_v @ B @ A )
=> ~ ( ord_le5216385588623774835_set_v @ C @ A ) ) ) ).
% sup.boundedE
thf(fact_839_sup_OboundedE,axiom,
! [B: product_unit,C: product_unit,A: product_unit] :
( ( ord_le3221252021190050221t_unit @ ( sup_sup_Product_unit @ B @ C ) @ A )
=> ~ ( ( ord_le3221252021190050221t_unit @ B @ A )
=> ~ ( ord_le3221252021190050221t_unit @ C @ A ) ) ) ).
% sup.boundedE
thf(fact_840_sup_OboundedE,axiom,
! [B: set_Product_prod_v_v,C: set_Product_prod_v_v,A: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ ( sup_su414716646722978715od_v_v @ B @ C ) @ A )
=> ~ ( ( ord_le7336532860387713383od_v_v @ B @ A )
=> ~ ( ord_le7336532860387713383od_v_v @ C @ A ) ) ) ).
% sup.boundedE
thf(fact_841_sup_OboundedE,axiom,
! [B: set_v,C: set_v,A: set_v] :
( ( ord_less_eq_set_v @ ( sup_sup_set_v @ B @ C ) @ A )
=> ~ ( ( ord_less_eq_set_v @ B @ A )
=> ~ ( ord_less_eq_set_v @ C @ A ) ) ) ).
% sup.boundedE
thf(fact_842_sup_OboundedI,axiom,
! [B: set_set_v,A: set_set_v,C: set_set_v] :
( ( ord_le5216385588623774835_set_v @ B @ A )
=> ( ( ord_le5216385588623774835_set_v @ C @ A )
=> ( ord_le5216385588623774835_set_v @ ( sup_sup_set_set_v @ B @ C ) @ A ) ) ) ).
% sup.boundedI
thf(fact_843_sup_OboundedI,axiom,
! [B: product_unit,A: product_unit,C: product_unit] :
( ( ord_le3221252021190050221t_unit @ B @ A )
=> ( ( ord_le3221252021190050221t_unit @ C @ A )
=> ( ord_le3221252021190050221t_unit @ ( sup_sup_Product_unit @ B @ C ) @ A ) ) ) ).
% sup.boundedI
thf(fact_844_sup_OboundedI,axiom,
! [B: set_Product_prod_v_v,A: set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ B @ A )
=> ( ( ord_le7336532860387713383od_v_v @ C @ A )
=> ( ord_le7336532860387713383od_v_v @ ( sup_su414716646722978715od_v_v @ B @ C ) @ A ) ) ) ).
% sup.boundedI
thf(fact_845_sup_OboundedI,axiom,
! [B: set_v,A: set_v,C: set_v] :
( ( ord_less_eq_set_v @ B @ A )
=> ( ( ord_less_eq_set_v @ C @ A )
=> ( ord_less_eq_set_v @ ( sup_sup_set_v @ B @ C ) @ A ) ) ) ).
% sup.boundedI
thf(fact_846_sup_Oorder__iff,axiom,
( ord_le5216385588623774835_set_v
= ( ^ [B4: set_set_v,A5: set_set_v] :
( A5
= ( sup_sup_set_set_v @ A5 @ B4 ) ) ) ) ).
% sup.order_iff
thf(fact_847_sup_Oorder__iff,axiom,
( ord_le3221252021190050221t_unit
= ( ^ [B4: product_unit,A5: product_unit] :
( A5
= ( sup_sup_Product_unit @ A5 @ B4 ) ) ) ) ).
% sup.order_iff
thf(fact_848_sup_Oorder__iff,axiom,
( ord_le7336532860387713383od_v_v
= ( ^ [B4: set_Product_prod_v_v,A5: set_Product_prod_v_v] :
( A5
= ( sup_su414716646722978715od_v_v @ A5 @ B4 ) ) ) ) ).
% sup.order_iff
thf(fact_849_sup_Oorder__iff,axiom,
( ord_less_eq_set_v
= ( ^ [B4: set_v,A5: set_v] :
( A5
= ( sup_sup_set_v @ A5 @ B4 ) ) ) ) ).
% sup.order_iff
thf(fact_850_sup_Ocobounded1,axiom,
! [A: set_set_v,B: set_set_v] : ( ord_le5216385588623774835_set_v @ A @ ( sup_sup_set_set_v @ A @ B ) ) ).
% sup.cobounded1
thf(fact_851_sup_Ocobounded1,axiom,
! [A: product_unit,B: product_unit] : ( ord_le3221252021190050221t_unit @ A @ ( sup_sup_Product_unit @ A @ B ) ) ).
% sup.cobounded1
thf(fact_852_sup_Ocobounded1,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ A @ ( sup_su414716646722978715od_v_v @ A @ B ) ) ).
% sup.cobounded1
thf(fact_853_sup_Ocobounded1,axiom,
! [A: set_v,B: set_v] : ( ord_less_eq_set_v @ A @ ( sup_sup_set_v @ A @ B ) ) ).
% sup.cobounded1
thf(fact_854_sup_Ocobounded2,axiom,
! [B: set_set_v,A: set_set_v] : ( ord_le5216385588623774835_set_v @ B @ ( sup_sup_set_set_v @ A @ B ) ) ).
% sup.cobounded2
thf(fact_855_sup_Ocobounded2,axiom,
! [B: product_unit,A: product_unit] : ( ord_le3221252021190050221t_unit @ B @ ( sup_sup_Product_unit @ A @ B ) ) ).
% sup.cobounded2
thf(fact_856_sup_Ocobounded2,axiom,
! [B: set_Product_prod_v_v,A: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ B @ ( sup_su414716646722978715od_v_v @ A @ B ) ) ).
% sup.cobounded2
thf(fact_857_sup_Ocobounded2,axiom,
! [B: set_v,A: set_v] : ( ord_less_eq_set_v @ B @ ( sup_sup_set_v @ A @ B ) ) ).
% sup.cobounded2
thf(fact_858_sup_Oabsorb__iff1,axiom,
( ord_le5216385588623774835_set_v
= ( ^ [B4: set_set_v,A5: set_set_v] :
( ( sup_sup_set_set_v @ A5 @ B4 )
= A5 ) ) ) ).
% sup.absorb_iff1
thf(fact_859_sup_Oabsorb__iff1,axiom,
( ord_le3221252021190050221t_unit
= ( ^ [B4: product_unit,A5: product_unit] :
( ( sup_sup_Product_unit @ A5 @ B4 )
= A5 ) ) ) ).
% sup.absorb_iff1
thf(fact_860_sup_Oabsorb__iff1,axiom,
( ord_le7336532860387713383od_v_v
= ( ^ [B4: set_Product_prod_v_v,A5: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ A5 @ B4 )
= A5 ) ) ) ).
% sup.absorb_iff1
thf(fact_861_sup_Oabsorb__iff1,axiom,
( ord_less_eq_set_v
= ( ^ [B4: set_v,A5: set_v] :
( ( sup_sup_set_v @ A5 @ B4 )
= A5 ) ) ) ).
% sup.absorb_iff1
thf(fact_862_sup_Oabsorb__iff2,axiom,
( ord_le5216385588623774835_set_v
= ( ^ [A5: set_set_v,B4: set_set_v] :
( ( sup_sup_set_set_v @ A5 @ B4 )
= B4 ) ) ) ).
% sup.absorb_iff2
thf(fact_863_sup_Oabsorb__iff2,axiom,
( ord_le3221252021190050221t_unit
= ( ^ [A5: product_unit,B4: product_unit] :
( ( sup_sup_Product_unit @ A5 @ B4 )
= B4 ) ) ) ).
% sup.absorb_iff2
thf(fact_864_sup_Oabsorb__iff2,axiom,
( ord_le7336532860387713383od_v_v
= ( ^ [A5: set_Product_prod_v_v,B4: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ A5 @ B4 )
= B4 ) ) ) ).
% sup.absorb_iff2
thf(fact_865_sup_Oabsorb__iff2,axiom,
( ord_less_eq_set_v
= ( ^ [A5: set_v,B4: set_v] :
( ( sup_sup_set_v @ A5 @ B4 )
= B4 ) ) ) ).
% sup.absorb_iff2
thf(fact_866_sup_OcoboundedI1,axiom,
! [C: set_set_v,A: set_set_v,B: set_set_v] :
( ( ord_le5216385588623774835_set_v @ C @ A )
=> ( ord_le5216385588623774835_set_v @ C @ ( sup_sup_set_set_v @ A @ B ) ) ) ).
% sup.coboundedI1
thf(fact_867_sup_OcoboundedI1,axiom,
! [C: product_unit,A: product_unit,B: product_unit] :
( ( ord_le3221252021190050221t_unit @ C @ A )
=> ( ord_le3221252021190050221t_unit @ C @ ( sup_sup_Product_unit @ A @ B ) ) ) ).
% sup.coboundedI1
thf(fact_868_sup_OcoboundedI1,axiom,
! [C: set_Product_prod_v_v,A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ C @ A )
=> ( ord_le7336532860387713383od_v_v @ C @ ( sup_su414716646722978715od_v_v @ A @ B ) ) ) ).
% sup.coboundedI1
thf(fact_869_sup_OcoboundedI1,axiom,
! [C: set_v,A: set_v,B: set_v] :
( ( ord_less_eq_set_v @ C @ A )
=> ( ord_less_eq_set_v @ C @ ( sup_sup_set_v @ A @ B ) ) ) ).
% sup.coboundedI1
thf(fact_870_sup_OcoboundedI2,axiom,
! [C: set_set_v,B: set_set_v,A: set_set_v] :
( ( ord_le5216385588623774835_set_v @ C @ B )
=> ( ord_le5216385588623774835_set_v @ C @ ( sup_sup_set_set_v @ A @ B ) ) ) ).
% sup.coboundedI2
thf(fact_871_sup_OcoboundedI2,axiom,
! [C: product_unit,B: product_unit,A: product_unit] :
( ( ord_le3221252021190050221t_unit @ C @ B )
=> ( ord_le3221252021190050221t_unit @ C @ ( sup_sup_Product_unit @ A @ B ) ) ) ).
% sup.coboundedI2
thf(fact_872_sup_OcoboundedI2,axiom,
! [C: set_Product_prod_v_v,B: set_Product_prod_v_v,A: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ C @ B )
=> ( ord_le7336532860387713383od_v_v @ C @ ( sup_su414716646722978715od_v_v @ A @ B ) ) ) ).
% sup.coboundedI2
thf(fact_873_sup_OcoboundedI2,axiom,
! [C: set_v,B: set_v,A: set_v] :
( ( ord_less_eq_set_v @ C @ B )
=> ( ord_less_eq_set_v @ C @ ( sup_sup_set_v @ A @ B ) ) ) ).
% sup.coboundedI2
thf(fact_874_graph_Opre__dfss__def,axiom,
! [Vertices: set_v,Successors: v > set_v,V3: v,E: sCC_Bl1394983891496994913t_unit] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl1748261141445803503t_unit @ Successors @ V3 @ E )
= ( ( sCC_Bl9196236973127232072t_unit @ Successors @ E )
& ( member_v @ V3 @ ( sCC_Bl4645233313691564917t_unit @ E ) )
& ( ( sCC_Bl8828226123343373779t_unit @ E )
!= nil_v )
& ( member_v @ V3 @ ( sCC_Bl1280885523602775798t_unit @ E @ ( hd_v @ ( sCC_Bl8828226123343373779t_unit @ E ) ) ) )
& ! [X2: v] :
( ( member_v @ X2 @ ( sCC_Bl3795065053823578884t_unit @ E @ V3 ) )
=> ( member_v @ X2 @ ( sup_sup_set_v @ ( sCC_Bl157864678168468314t_unit @ E ) @ ( sCC_Bl1280885523602775798t_unit @ E @ ( hd_v @ ( sCC_Bl8828226123343373779t_unit @ E ) ) ) ) ) )
& ! [X2: v] :
( ( member_v @ X2 @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E ) ) )
=> ( sCC_Bl649662514949026229able_v @ Successors @ X2 @ V3 ) )
& ? [Ns2: list_v] :
( ( sCC_Bl9201514103433284750t_unit @ E )
= ( cons_v @ V3 @ Ns2 ) ) ) ) ) ).
% graph.pre_dfss_def
thf(fact_875_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_876_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_877_scc__partition,axiom,
! [S3: set_v,S5: set_v,X: v] :
( ( sCC_Bloemen_is_scc_v @ successors @ S3 )
=> ( ( sCC_Bloemen_is_scc_v @ successors @ S5 )
=> ( ( member_v @ X @ ( inf_inf_set_v @ S3 @ S5 ) )
=> ( S3 = S5 ) ) ) ) ).
% scc_partition
thf(fact_878_the__elem__eq,axiom,
! [X: product_prod_v_v] :
( ( the_el5392834299063928540od_v_v @ ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v ) )
= X ) ).
% the_elem_eq
thf(fact_879_the__elem__eq,axiom,
! [X: v] :
( ( the_elem_v @ ( insert_v2 @ X @ bot_bot_set_v ) )
= X ) ).
% the_elem_eq
thf(fact_880_the__elem__eq,axiom,
! [X: set_v] :
( ( the_elem_set_v @ ( insert_set_v2 @ X @ bot_bot_set_set_v ) )
= X ) ).
% the_elem_eq
thf(fact_881_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_882_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_883_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_884_Int__iff,axiom,
! [C: product_prod_v_v,A2: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ C @ ( inf_in6271465464967711157od_v_v @ A2 @ B2 ) )
= ( ( member7453568604450474000od_v_v @ C @ A2 )
& ( member7453568604450474000od_v_v @ C @ B2 ) ) ) ).
% Int_iff
thf(fact_885_Int__iff,axiom,
! [C: v,A2: set_v,B2: set_v] :
( ( member_v @ C @ ( inf_inf_set_v @ A2 @ B2 ) )
= ( ( member_v @ C @ A2 )
& ( member_v @ C @ B2 ) ) ) ).
% Int_iff
thf(fact_886_IntI,axiom,
! [C: product_prod_v_v,A2: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ C @ A2 )
=> ( ( member7453568604450474000od_v_v @ C @ B2 )
=> ( member7453568604450474000od_v_v @ C @ ( inf_in6271465464967711157od_v_v @ A2 @ B2 ) ) ) ) ).
% IntI
thf(fact_887_IntI,axiom,
! [C: v,A2: set_v,B2: set_v] :
( ( member_v @ C @ A2 )
=> ( ( member_v @ C @ B2 )
=> ( member_v @ C @ ( inf_inf_set_v @ A2 @ B2 ) ) ) ) ).
% IntI
thf(fact_888_inf_Obounded__iff,axiom,
! [A: product_unit,B: product_unit,C: product_unit] :
( ( ord_le3221252021190050221t_unit @ A @ ( inf_inf_Product_unit @ B @ C ) )
= ( ( ord_le3221252021190050221t_unit @ A @ B )
& ( ord_le3221252021190050221t_unit @ A @ C ) ) ) ).
% inf.bounded_iff
thf(fact_889_inf_Obounded__iff,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ ( inf_in6271465464967711157od_v_v @ B @ C ) )
= ( ( ord_le7336532860387713383od_v_v @ A @ B )
& ( ord_le7336532860387713383od_v_v @ A @ C ) ) ) ).
% inf.bounded_iff
thf(fact_890_inf_Obounded__iff,axiom,
! [A: set_v,B: set_v,C: set_v] :
( ( ord_less_eq_set_v @ A @ ( inf_inf_set_v @ B @ C ) )
= ( ( ord_less_eq_set_v @ A @ B )
& ( ord_less_eq_set_v @ A @ C ) ) ) ).
% inf.bounded_iff
thf(fact_891_le__inf__iff,axiom,
! [X: product_unit,Y: product_unit,Z: product_unit] :
( ( ord_le3221252021190050221t_unit @ X @ ( inf_inf_Product_unit @ Y @ Z ) )
= ( ( ord_le3221252021190050221t_unit @ X @ Y )
& ( ord_le3221252021190050221t_unit @ X @ Z ) ) ) ).
% le_inf_iff
thf(fact_892_le__inf__iff,axiom,
! [X: set_Product_prod_v_v,Y: set_Product_prod_v_v,Z: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X @ ( inf_in6271465464967711157od_v_v @ Y @ Z ) )
= ( ( ord_le7336532860387713383od_v_v @ X @ Y )
& ( ord_le7336532860387713383od_v_v @ X @ Z ) ) ) ).
% le_inf_iff
thf(fact_893_le__inf__iff,axiom,
! [X: set_v,Y: set_v,Z: set_v] :
( ( ord_less_eq_set_v @ X @ ( inf_inf_set_v @ Y @ Z ) )
= ( ( ord_less_eq_set_v @ X @ Y )
& ( ord_less_eq_set_v @ X @ Z ) ) ) ).
% le_inf_iff
thf(fact_894_boolean__algebra_Oconj__zero__right,axiom,
! [X: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ X @ bot_bo723834152578015283od_v_v )
= bot_bo723834152578015283od_v_v ) ).
% boolean_algebra.conj_zero_right
thf(fact_895_boolean__algebra_Oconj__zero__right,axiom,
! [X: set_v] :
( ( inf_inf_set_v @ X @ bot_bot_set_v )
= bot_bot_set_v ) ).
% boolean_algebra.conj_zero_right
thf(fact_896_boolean__algebra_Oconj__zero__right,axiom,
! [X: set_set_v] :
( ( inf_inf_set_set_v @ X @ bot_bot_set_set_v )
= bot_bot_set_set_v ) ).
% boolean_algebra.conj_zero_right
thf(fact_897_boolean__algebra_Oconj__zero__right,axiom,
! [X: product_unit] :
( ( inf_inf_Product_unit @ X @ bot_bot_Product_unit )
= bot_bot_Product_unit ) ).
% boolean_algebra.conj_zero_right
thf(fact_898_boolean__algebra_Oconj__zero__left,axiom,
! [X: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ bot_bo723834152578015283od_v_v @ X )
= bot_bo723834152578015283od_v_v ) ).
% boolean_algebra.conj_zero_left
thf(fact_899_boolean__algebra_Oconj__zero__left,axiom,
! [X: set_v] :
( ( inf_inf_set_v @ bot_bot_set_v @ X )
= bot_bot_set_v ) ).
% boolean_algebra.conj_zero_left
thf(fact_900_boolean__algebra_Oconj__zero__left,axiom,
! [X: set_set_v] :
( ( inf_inf_set_set_v @ bot_bot_set_set_v @ X )
= bot_bot_set_set_v ) ).
% boolean_algebra.conj_zero_left
thf(fact_901_boolean__algebra_Oconj__zero__left,axiom,
! [X: product_unit] :
( ( inf_inf_Product_unit @ bot_bot_Product_unit @ X )
= bot_bot_Product_unit ) ).
% boolean_algebra.conj_zero_left
thf(fact_902_inf__bot__right,axiom,
! [X: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ X @ bot_bo723834152578015283od_v_v )
= bot_bo723834152578015283od_v_v ) ).
% inf_bot_right
thf(fact_903_inf__bot__right,axiom,
! [X: set_v] :
( ( inf_inf_set_v @ X @ bot_bot_set_v )
= bot_bot_set_v ) ).
% inf_bot_right
thf(fact_904_inf__bot__right,axiom,
! [X: set_set_v] :
( ( inf_inf_set_set_v @ X @ bot_bot_set_set_v )
= bot_bot_set_set_v ) ).
% inf_bot_right
thf(fact_905_inf__bot__right,axiom,
! [X: product_unit] :
( ( inf_inf_Product_unit @ X @ bot_bot_Product_unit )
= bot_bot_Product_unit ) ).
% inf_bot_right
thf(fact_906_inf__bot__left,axiom,
! [X: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ bot_bo723834152578015283od_v_v @ X )
= bot_bo723834152578015283od_v_v ) ).
% inf_bot_left
thf(fact_907_inf__bot__left,axiom,
! [X: set_v] :
( ( inf_inf_set_v @ bot_bot_set_v @ X )
= bot_bot_set_v ) ).
% inf_bot_left
thf(fact_908_inf__bot__left,axiom,
! [X: set_set_v] :
( ( inf_inf_set_set_v @ bot_bot_set_set_v @ X )
= bot_bot_set_set_v ) ).
% inf_bot_left
thf(fact_909_inf__bot__left,axiom,
! [X: product_unit] :
( ( inf_inf_Product_unit @ bot_bot_Product_unit @ X )
= bot_bot_Product_unit ) ).
% inf_bot_left
thf(fact_910_Int__subset__iff,axiom,
! [C2: set_Product_prod_v_v,A2: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ C2 @ ( inf_in6271465464967711157od_v_v @ A2 @ B2 ) )
= ( ( ord_le7336532860387713383od_v_v @ C2 @ A2 )
& ( ord_le7336532860387713383od_v_v @ C2 @ B2 ) ) ) ).
% Int_subset_iff
thf(fact_911_Int__subset__iff,axiom,
! [C2: set_v,A2: set_v,B2: set_v] :
( ( ord_less_eq_set_v @ C2 @ ( inf_inf_set_v @ A2 @ B2 ) )
= ( ( ord_less_eq_set_v @ C2 @ A2 )
& ( ord_less_eq_set_v @ C2 @ B2 ) ) ) ).
% Int_subset_iff
thf(fact_912_Int__insert__right__if1,axiom,
! [A: set_v,A2: set_set_v,B2: set_set_v] :
( ( member_set_v @ A @ A2 )
=> ( ( inf_inf_set_set_v @ A2 @ ( insert_set_v2 @ A @ B2 ) )
= ( insert_set_v2 @ A @ ( inf_inf_set_set_v @ A2 @ B2 ) ) ) ) ).
% Int_insert_right_if1
thf(fact_913_Int__insert__right__if1,axiom,
! [A: product_prod_v_v,A2: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ A @ A2 )
=> ( ( inf_in6271465464967711157od_v_v @ A2 @ ( insert1338601472111419319od_v_v @ A @ B2 ) )
= ( insert1338601472111419319od_v_v @ A @ ( inf_in6271465464967711157od_v_v @ A2 @ B2 ) ) ) ) ).
% Int_insert_right_if1
thf(fact_914_Int__insert__right__if1,axiom,
! [A: v,A2: set_v,B2: set_v] :
( ( member_v @ A @ A2 )
=> ( ( inf_inf_set_v @ A2 @ ( insert_v2 @ A @ B2 ) )
= ( insert_v2 @ A @ ( inf_inf_set_v @ A2 @ B2 ) ) ) ) ).
% Int_insert_right_if1
thf(fact_915_Int__insert__right__if0,axiom,
! [A: set_v,A2: set_set_v,B2: set_set_v] :
( ~ ( member_set_v @ A @ A2 )
=> ( ( inf_inf_set_set_v @ A2 @ ( insert_set_v2 @ A @ B2 ) )
= ( inf_inf_set_set_v @ A2 @ B2 ) ) ) ).
% Int_insert_right_if0
thf(fact_916_Int__insert__right__if0,axiom,
! [A: product_prod_v_v,A2: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ~ ( member7453568604450474000od_v_v @ A @ A2 )
=> ( ( inf_in6271465464967711157od_v_v @ A2 @ ( insert1338601472111419319od_v_v @ A @ B2 ) )
= ( inf_in6271465464967711157od_v_v @ A2 @ B2 ) ) ) ).
% Int_insert_right_if0
thf(fact_917_Int__insert__right__if0,axiom,
! [A: v,A2: set_v,B2: set_v] :
( ~ ( member_v @ A @ A2 )
=> ( ( inf_inf_set_v @ A2 @ ( insert_v2 @ A @ B2 ) )
= ( inf_inf_set_v @ A2 @ B2 ) ) ) ).
% Int_insert_right_if0
thf(fact_918_insert__inter__insert,axiom,
! [A: product_prod_v_v,A2: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ ( insert1338601472111419319od_v_v @ A @ A2 ) @ ( insert1338601472111419319od_v_v @ A @ B2 ) )
= ( insert1338601472111419319od_v_v @ A @ ( inf_in6271465464967711157od_v_v @ A2 @ B2 ) ) ) ).
% insert_inter_insert
thf(fact_919_insert__inter__insert,axiom,
! [A: set_v,A2: set_set_v,B2: set_set_v] :
( ( inf_inf_set_set_v @ ( insert_set_v2 @ A @ A2 ) @ ( insert_set_v2 @ A @ B2 ) )
= ( insert_set_v2 @ A @ ( inf_inf_set_set_v @ A2 @ B2 ) ) ) ).
% insert_inter_insert
thf(fact_920_insert__inter__insert,axiom,
! [A: v,A2: set_v,B2: set_v] :
( ( inf_inf_set_v @ ( insert_v2 @ A @ A2 ) @ ( insert_v2 @ A @ B2 ) )
= ( insert_v2 @ A @ ( inf_inf_set_v @ A2 @ B2 ) ) ) ).
% insert_inter_insert
thf(fact_921_Int__insert__left__if1,axiom,
! [A: set_v,C2: set_set_v,B2: set_set_v] :
( ( member_set_v @ A @ C2 )
=> ( ( inf_inf_set_set_v @ ( insert_set_v2 @ A @ B2 ) @ C2 )
= ( insert_set_v2 @ A @ ( inf_inf_set_set_v @ B2 @ C2 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_922_Int__insert__left__if1,axiom,
! [A: product_prod_v_v,C2: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ A @ C2 )
=> ( ( inf_in6271465464967711157od_v_v @ ( insert1338601472111419319od_v_v @ A @ B2 ) @ C2 )
= ( insert1338601472111419319od_v_v @ A @ ( inf_in6271465464967711157od_v_v @ B2 @ C2 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_923_Int__insert__left__if1,axiom,
! [A: v,C2: set_v,B2: set_v] :
( ( member_v @ A @ C2 )
=> ( ( inf_inf_set_v @ ( insert_v2 @ A @ B2 ) @ C2 )
= ( insert_v2 @ A @ ( inf_inf_set_v @ B2 @ C2 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_924_Int__insert__left__if0,axiom,
! [A: set_v,C2: set_set_v,B2: set_set_v] :
( ~ ( member_set_v @ A @ C2 )
=> ( ( inf_inf_set_set_v @ ( insert_set_v2 @ A @ B2 ) @ C2 )
= ( inf_inf_set_set_v @ B2 @ C2 ) ) ) ).
% Int_insert_left_if0
thf(fact_925_Int__insert__left__if0,axiom,
! [A: product_prod_v_v,C2: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ~ ( member7453568604450474000od_v_v @ A @ C2 )
=> ( ( inf_in6271465464967711157od_v_v @ ( insert1338601472111419319od_v_v @ A @ B2 ) @ C2 )
= ( inf_in6271465464967711157od_v_v @ B2 @ C2 ) ) ) ).
% Int_insert_left_if0
thf(fact_926_Int__insert__left__if0,axiom,
! [A: v,C2: set_v,B2: set_v] :
( ~ ( member_v @ A @ C2 )
=> ( ( inf_inf_set_v @ ( insert_v2 @ A @ B2 ) @ C2 )
= ( inf_inf_set_v @ B2 @ C2 ) ) ) ).
% Int_insert_left_if0
thf(fact_927_Un__Int__eq_I1_J,axiom,
! [S3: set_Product_prod_v_v,T2: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ ( sup_su414716646722978715od_v_v @ S3 @ T2 ) @ S3 )
= S3 ) ).
% Un_Int_eq(1)
thf(fact_928_Un__Int__eq_I1_J,axiom,
! [S3: set_v,T2: set_v] :
( ( inf_inf_set_v @ ( sup_sup_set_v @ S3 @ T2 ) @ S3 )
= S3 ) ).
% Un_Int_eq(1)
thf(fact_929_Un__Int__eq_I1_J,axiom,
! [S3: set_set_v,T2: set_set_v] :
( ( inf_inf_set_set_v @ ( sup_sup_set_set_v @ S3 @ T2 ) @ S3 )
= S3 ) ).
% Un_Int_eq(1)
thf(fact_930_Un__Int__eq_I2_J,axiom,
! [S3: set_Product_prod_v_v,T2: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ ( sup_su414716646722978715od_v_v @ S3 @ T2 ) @ T2 )
= T2 ) ).
% Un_Int_eq(2)
thf(fact_931_Un__Int__eq_I2_J,axiom,
! [S3: set_v,T2: set_v] :
( ( inf_inf_set_v @ ( sup_sup_set_v @ S3 @ T2 ) @ T2 )
= T2 ) ).
% Un_Int_eq(2)
thf(fact_932_Un__Int__eq_I2_J,axiom,
! [S3: set_set_v,T2: set_set_v] :
( ( inf_inf_set_set_v @ ( sup_sup_set_set_v @ S3 @ T2 ) @ T2 )
= T2 ) ).
% Un_Int_eq(2)
thf(fact_933_Un__Int__eq_I3_J,axiom,
! [S3: set_Product_prod_v_v,T2: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ S3 @ ( sup_su414716646722978715od_v_v @ S3 @ T2 ) )
= S3 ) ).
% Un_Int_eq(3)
thf(fact_934_Un__Int__eq_I3_J,axiom,
! [S3: set_v,T2: set_v] :
( ( inf_inf_set_v @ S3 @ ( sup_sup_set_v @ S3 @ T2 ) )
= S3 ) ).
% Un_Int_eq(3)
thf(fact_935_Un__Int__eq_I3_J,axiom,
! [S3: set_set_v,T2: set_set_v] :
( ( inf_inf_set_set_v @ S3 @ ( sup_sup_set_set_v @ S3 @ T2 ) )
= S3 ) ).
% Un_Int_eq(3)
thf(fact_936_Un__Int__eq_I4_J,axiom,
! [T2: set_Product_prod_v_v,S3: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ T2 @ ( sup_su414716646722978715od_v_v @ S3 @ T2 ) )
= T2 ) ).
% Un_Int_eq(4)
thf(fact_937_Un__Int__eq_I4_J,axiom,
! [T2: set_v,S3: set_v] :
( ( inf_inf_set_v @ T2 @ ( sup_sup_set_v @ S3 @ T2 ) )
= T2 ) ).
% Un_Int_eq(4)
thf(fact_938_Un__Int__eq_I4_J,axiom,
! [T2: set_set_v,S3: set_set_v] :
( ( inf_inf_set_set_v @ T2 @ ( sup_sup_set_set_v @ S3 @ T2 ) )
= T2 ) ).
% Un_Int_eq(4)
thf(fact_939_Int__Un__eq_I1_J,axiom,
! [S3: set_Product_prod_v_v,T2: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ S3 @ T2 ) @ S3 )
= S3 ) ).
% Int_Un_eq(1)
thf(fact_940_Int__Un__eq_I1_J,axiom,
! [S3: set_v,T2: set_v] :
( ( sup_sup_set_v @ ( inf_inf_set_v @ S3 @ T2 ) @ S3 )
= S3 ) ).
% Int_Un_eq(1)
thf(fact_941_Int__Un__eq_I1_J,axiom,
! [S3: set_set_v,T2: set_set_v] :
( ( sup_sup_set_set_v @ ( inf_inf_set_set_v @ S3 @ T2 ) @ S3 )
= S3 ) ).
% Int_Un_eq(1)
thf(fact_942_Int__Un__eq_I2_J,axiom,
! [S3: set_Product_prod_v_v,T2: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ S3 @ T2 ) @ T2 )
= T2 ) ).
% Int_Un_eq(2)
thf(fact_943_Int__Un__eq_I2_J,axiom,
! [S3: set_v,T2: set_v] :
( ( sup_sup_set_v @ ( inf_inf_set_v @ S3 @ T2 ) @ T2 )
= T2 ) ).
% Int_Un_eq(2)
thf(fact_944_Int__Un__eq_I2_J,axiom,
! [S3: set_set_v,T2: set_set_v] :
( ( sup_sup_set_set_v @ ( inf_inf_set_set_v @ S3 @ T2 ) @ T2 )
= T2 ) ).
% Int_Un_eq(2)
thf(fact_945_Int__Un__eq_I3_J,axiom,
! [S3: set_Product_prod_v_v,T2: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ S3 @ ( inf_in6271465464967711157od_v_v @ S3 @ T2 ) )
= S3 ) ).
% Int_Un_eq(3)
thf(fact_946_Int__Un__eq_I3_J,axiom,
! [S3: set_v,T2: set_v] :
( ( sup_sup_set_v @ S3 @ ( inf_inf_set_v @ S3 @ T2 ) )
= S3 ) ).
% Int_Un_eq(3)
thf(fact_947_Int__Un__eq_I3_J,axiom,
! [S3: set_set_v,T2: set_set_v] :
( ( sup_sup_set_set_v @ S3 @ ( inf_inf_set_set_v @ S3 @ T2 ) )
= S3 ) ).
% Int_Un_eq(3)
thf(fact_948_Int__Un__eq_I4_J,axiom,
! [T2: set_Product_prod_v_v,S3: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ T2 @ ( inf_in6271465464967711157od_v_v @ S3 @ T2 ) )
= T2 ) ).
% Int_Un_eq(4)
thf(fact_949_Int__Un__eq_I4_J,axiom,
! [T2: set_v,S3: set_v] :
( ( sup_sup_set_v @ T2 @ ( inf_inf_set_v @ S3 @ T2 ) )
= T2 ) ).
% Int_Un_eq(4)
thf(fact_950_Int__Un__eq_I4_J,axiom,
! [T2: set_set_v,S3: set_set_v] :
( ( sup_sup_set_set_v @ T2 @ ( inf_inf_set_set_v @ S3 @ T2 ) )
= T2 ) ).
% Int_Un_eq(4)
thf(fact_951_disjoint__insert_I2_J,axiom,
! [A2: set_Product_prod_v_v,B: product_prod_v_v,B2: set_Product_prod_v_v] :
( ( bot_bo723834152578015283od_v_v
= ( inf_in6271465464967711157od_v_v @ A2 @ ( insert1338601472111419319od_v_v @ B @ B2 ) ) )
= ( ~ ( member7453568604450474000od_v_v @ B @ A2 )
& ( bot_bo723834152578015283od_v_v
= ( inf_in6271465464967711157od_v_v @ A2 @ B2 ) ) ) ) ).
% disjoint_insert(2)
thf(fact_952_disjoint__insert_I2_J,axiom,
! [A2: set_v,B: v,B2: set_v] :
( ( bot_bot_set_v
= ( inf_inf_set_v @ A2 @ ( insert_v2 @ B @ B2 ) ) )
= ( ~ ( member_v @ B @ A2 )
& ( bot_bot_set_v
= ( inf_inf_set_v @ A2 @ B2 ) ) ) ) ).
% disjoint_insert(2)
thf(fact_953_disjoint__insert_I2_J,axiom,
! [A2: set_set_v,B: set_v,B2: set_set_v] :
( ( bot_bot_set_set_v
= ( inf_inf_set_set_v @ A2 @ ( insert_set_v2 @ B @ B2 ) ) )
= ( ~ ( member_set_v @ B @ A2 )
& ( bot_bot_set_set_v
= ( inf_inf_set_set_v @ A2 @ B2 ) ) ) ) ).
% disjoint_insert(2)
thf(fact_954_disjoint__insert_I1_J,axiom,
! [B2: set_Product_prod_v_v,A: product_prod_v_v,A2: set_Product_prod_v_v] :
( ( ( inf_in6271465464967711157od_v_v @ B2 @ ( insert1338601472111419319od_v_v @ A @ A2 ) )
= bot_bo723834152578015283od_v_v )
= ( ~ ( member7453568604450474000od_v_v @ A @ B2 )
& ( ( inf_in6271465464967711157od_v_v @ B2 @ A2 )
= bot_bo723834152578015283od_v_v ) ) ) ).
% disjoint_insert(1)
thf(fact_955_disjoint__insert_I1_J,axiom,
! [B2: set_v,A: v,A2: set_v] :
( ( ( inf_inf_set_v @ B2 @ ( insert_v2 @ A @ A2 ) )
= bot_bot_set_v )
= ( ~ ( member_v @ A @ B2 )
& ( ( inf_inf_set_v @ B2 @ A2 )
= bot_bot_set_v ) ) ) ).
% disjoint_insert(1)
thf(fact_956_disjoint__insert_I1_J,axiom,
! [B2: set_set_v,A: set_v,A2: set_set_v] :
( ( ( inf_inf_set_set_v @ B2 @ ( insert_set_v2 @ A @ A2 ) )
= bot_bot_set_set_v )
= ( ~ ( member_set_v @ A @ B2 )
& ( ( inf_inf_set_set_v @ B2 @ A2 )
= bot_bot_set_set_v ) ) ) ).
% disjoint_insert(1)
thf(fact_957_insert__disjoint_I2_J,axiom,
! [A: product_prod_v_v,A2: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( bot_bo723834152578015283od_v_v
= ( inf_in6271465464967711157od_v_v @ ( insert1338601472111419319od_v_v @ A @ A2 ) @ B2 ) )
= ( ~ ( member7453568604450474000od_v_v @ A @ B2 )
& ( bot_bo723834152578015283od_v_v
= ( inf_in6271465464967711157od_v_v @ A2 @ B2 ) ) ) ) ).
% insert_disjoint(2)
thf(fact_958_insert__disjoint_I2_J,axiom,
! [A: v,A2: set_v,B2: set_v] :
( ( bot_bot_set_v
= ( inf_inf_set_v @ ( insert_v2 @ A @ A2 ) @ B2 ) )
= ( ~ ( member_v @ A @ B2 )
& ( bot_bot_set_v
= ( inf_inf_set_v @ A2 @ B2 ) ) ) ) ).
% insert_disjoint(2)
thf(fact_959_insert__disjoint_I2_J,axiom,
! [A: set_v,A2: set_set_v,B2: set_set_v] :
( ( bot_bot_set_set_v
= ( inf_inf_set_set_v @ ( insert_set_v2 @ A @ A2 ) @ B2 ) )
= ( ~ ( member_set_v @ A @ B2 )
& ( bot_bot_set_set_v
= ( inf_inf_set_set_v @ A2 @ B2 ) ) ) ) ).
% insert_disjoint(2)
thf(fact_960_insert__disjoint_I1_J,axiom,
! [A: product_prod_v_v,A2: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( ( inf_in6271465464967711157od_v_v @ ( insert1338601472111419319od_v_v @ A @ A2 ) @ B2 )
= bot_bo723834152578015283od_v_v )
= ( ~ ( member7453568604450474000od_v_v @ A @ B2 )
& ( ( inf_in6271465464967711157od_v_v @ A2 @ B2 )
= bot_bo723834152578015283od_v_v ) ) ) ).
% insert_disjoint(1)
thf(fact_961_insert__disjoint_I1_J,axiom,
! [A: v,A2: set_v,B2: set_v] :
( ( ( inf_inf_set_v @ ( insert_v2 @ A @ A2 ) @ B2 )
= bot_bot_set_v )
= ( ~ ( member_v @ A @ B2 )
& ( ( inf_inf_set_v @ A2 @ B2 )
= bot_bot_set_v ) ) ) ).
% insert_disjoint(1)
thf(fact_962_insert__disjoint_I1_J,axiom,
! [A: set_v,A2: set_set_v,B2: set_set_v] :
( ( ( inf_inf_set_set_v @ ( insert_set_v2 @ A @ A2 ) @ B2 )
= bot_bot_set_set_v )
= ( ~ ( member_set_v @ A @ B2 )
& ( ( inf_inf_set_set_v @ A2 @ B2 )
= bot_bot_set_set_v ) ) ) ).
% insert_disjoint(1)
thf(fact_963_Diff__disjoint,axiom,
! [A2: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ A2 @ ( minus_4183494784930505774od_v_v @ B2 @ A2 ) )
= bot_bo723834152578015283od_v_v ) ).
% Diff_disjoint
thf(fact_964_Diff__disjoint,axiom,
! [A2: set_set_v,B2: set_set_v] :
( ( inf_inf_set_set_v @ A2 @ ( minus_7228012346218142266_set_v @ B2 @ A2 ) )
= bot_bot_set_set_v ) ).
% Diff_disjoint
thf(fact_965_Diff__disjoint,axiom,
! [A2: set_v,B2: set_v] :
( ( inf_inf_set_v @ A2 @ ( minus_minus_set_v @ B2 @ A2 ) )
= bot_bot_set_v ) ).
% Diff_disjoint
thf(fact_966_distinct__append,axiom,
! [Xs: list_P7986770385144383213od_v_v,Ys: list_P7986770385144383213od_v_v] :
( ( distin6159370996967099744od_v_v @ ( append2138873909117096322od_v_v @ Xs @ Ys ) )
= ( ( distin6159370996967099744od_v_v @ Xs )
& ( distin6159370996967099744od_v_v @ Ys )
& ( ( inf_in6271465464967711157od_v_v @ ( set_Product_prod_v_v2 @ Xs ) @ ( set_Product_prod_v_v2 @ Ys ) )
= bot_bo723834152578015283od_v_v ) ) ) ).
% distinct_append
thf(fact_967_distinct__append,axiom,
! [Xs: list_v,Ys: list_v] :
( ( distinct_v @ ( append_v @ Xs @ Ys ) )
= ( ( distinct_v @ Xs )
& ( distinct_v @ Ys )
& ( ( inf_inf_set_v @ ( set_v2 @ Xs ) @ ( set_v2 @ Ys ) )
= bot_bot_set_v ) ) ) ).
% distinct_append
thf(fact_968_distinct__append,axiom,
! [Xs: list_set_v,Ys: list_set_v] :
( ( distinct_set_v @ ( append_set_v @ Xs @ Ys ) )
= ( ( distinct_set_v @ Xs )
& ( distinct_set_v @ Ys )
& ( ( inf_inf_set_set_v @ ( set_set_v2 @ Xs ) @ ( set_set_v2 @ Ys ) )
= bot_bot_set_set_v ) ) ) ).
% distinct_append
thf(fact_969_inf__sup__ord_I2_J,axiom,
! [X: product_unit,Y: product_unit] : ( ord_le3221252021190050221t_unit @ ( inf_inf_Product_unit @ X @ Y ) @ Y ) ).
% inf_sup_ord(2)
thf(fact_970_inf__sup__ord_I2_J,axiom,
! [X: set_Product_prod_v_v,Y: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ X @ Y ) @ Y ) ).
% inf_sup_ord(2)
thf(fact_971_inf__sup__ord_I2_J,axiom,
! [X: set_v,Y: set_v] : ( ord_less_eq_set_v @ ( inf_inf_set_v @ X @ Y ) @ Y ) ).
% inf_sup_ord(2)
thf(fact_972_inf__sup__ord_I1_J,axiom,
! [X: product_unit,Y: product_unit] : ( ord_le3221252021190050221t_unit @ ( inf_inf_Product_unit @ X @ Y ) @ X ) ).
% inf_sup_ord(1)
thf(fact_973_inf__sup__ord_I1_J,axiom,
! [X: set_Product_prod_v_v,Y: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ X @ Y ) @ X ) ).
% inf_sup_ord(1)
thf(fact_974_inf__sup__ord_I1_J,axiom,
! [X: set_v,Y: set_v] : ( ord_less_eq_set_v @ ( inf_inf_set_v @ X @ Y ) @ X ) ).
% inf_sup_ord(1)
thf(fact_975_inf__le1,axiom,
! [X: product_unit,Y: product_unit] : ( ord_le3221252021190050221t_unit @ ( inf_inf_Product_unit @ X @ Y ) @ X ) ).
% inf_le1
thf(fact_976_inf__le1,axiom,
! [X: set_Product_prod_v_v,Y: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ X @ Y ) @ X ) ).
% inf_le1
thf(fact_977_inf__le1,axiom,
! [X: set_v,Y: set_v] : ( ord_less_eq_set_v @ ( inf_inf_set_v @ X @ Y ) @ X ) ).
% inf_le1
thf(fact_978_inf__le2,axiom,
! [X: product_unit,Y: product_unit] : ( ord_le3221252021190050221t_unit @ ( inf_inf_Product_unit @ X @ Y ) @ Y ) ).
% inf_le2
thf(fact_979_inf__le2,axiom,
! [X: set_Product_prod_v_v,Y: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ X @ Y ) @ Y ) ).
% inf_le2
thf(fact_980_inf__le2,axiom,
! [X: set_v,Y: set_v] : ( ord_less_eq_set_v @ ( inf_inf_set_v @ X @ Y ) @ Y ) ).
% inf_le2
thf(fact_981_le__infE,axiom,
! [X: product_unit,A: product_unit,B: product_unit] :
( ( ord_le3221252021190050221t_unit @ X @ ( inf_inf_Product_unit @ A @ B ) )
=> ~ ( ( ord_le3221252021190050221t_unit @ X @ A )
=> ~ ( ord_le3221252021190050221t_unit @ X @ B ) ) ) ).
% le_infE
thf(fact_982_le__infE,axiom,
! [X: set_Product_prod_v_v,A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X @ ( inf_in6271465464967711157od_v_v @ A @ B ) )
=> ~ ( ( ord_le7336532860387713383od_v_v @ X @ A )
=> ~ ( ord_le7336532860387713383od_v_v @ X @ B ) ) ) ).
% le_infE
thf(fact_983_le__infE,axiom,
! [X: set_v,A: set_v,B: set_v] :
( ( ord_less_eq_set_v @ X @ ( inf_inf_set_v @ A @ B ) )
=> ~ ( ( ord_less_eq_set_v @ X @ A )
=> ~ ( ord_less_eq_set_v @ X @ B ) ) ) ).
% le_infE
thf(fact_984_le__infI,axiom,
! [X: product_unit,A: product_unit,B: product_unit] :
( ( ord_le3221252021190050221t_unit @ X @ A )
=> ( ( ord_le3221252021190050221t_unit @ X @ B )
=> ( ord_le3221252021190050221t_unit @ X @ ( inf_inf_Product_unit @ A @ B ) ) ) ) ).
% le_infI
thf(fact_985_le__infI,axiom,
! [X: set_Product_prod_v_v,A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X @ A )
=> ( ( ord_le7336532860387713383od_v_v @ X @ B )
=> ( ord_le7336532860387713383od_v_v @ X @ ( inf_in6271465464967711157od_v_v @ A @ B ) ) ) ) ).
% le_infI
thf(fact_986_le__infI,axiom,
! [X: set_v,A: set_v,B: set_v] :
( ( ord_less_eq_set_v @ X @ A )
=> ( ( ord_less_eq_set_v @ X @ B )
=> ( ord_less_eq_set_v @ X @ ( inf_inf_set_v @ A @ B ) ) ) ) ).
% le_infI
thf(fact_987_inf__mono,axiom,
! [A: product_unit,C: product_unit,B: product_unit,D2: product_unit] :
( ( ord_le3221252021190050221t_unit @ A @ C )
=> ( ( ord_le3221252021190050221t_unit @ B @ D2 )
=> ( ord_le3221252021190050221t_unit @ ( inf_inf_Product_unit @ A @ B ) @ ( inf_inf_Product_unit @ C @ D2 ) ) ) ) ).
% inf_mono
thf(fact_988_inf__mono,axiom,
! [A: set_Product_prod_v_v,C: set_Product_prod_v_v,B: set_Product_prod_v_v,D2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ C )
=> ( ( ord_le7336532860387713383od_v_v @ B @ D2 )
=> ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ A @ B ) @ ( inf_in6271465464967711157od_v_v @ C @ D2 ) ) ) ) ).
% inf_mono
thf(fact_989_inf__mono,axiom,
! [A: set_v,C: set_v,B: set_v,D2: set_v] :
( ( ord_less_eq_set_v @ A @ C )
=> ( ( ord_less_eq_set_v @ B @ D2 )
=> ( ord_less_eq_set_v @ ( inf_inf_set_v @ A @ B ) @ ( inf_inf_set_v @ C @ D2 ) ) ) ) ).
% inf_mono
thf(fact_990_le__infI1,axiom,
! [A: product_unit,X: product_unit,B: product_unit] :
( ( ord_le3221252021190050221t_unit @ A @ X )
=> ( ord_le3221252021190050221t_unit @ ( inf_inf_Product_unit @ A @ B ) @ X ) ) ).
% le_infI1
thf(fact_991_le__infI1,axiom,
! [A: set_Product_prod_v_v,X: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ X )
=> ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ A @ B ) @ X ) ) ).
% le_infI1
thf(fact_992_le__infI1,axiom,
! [A: set_v,X: set_v,B: set_v] :
( ( ord_less_eq_set_v @ A @ X )
=> ( ord_less_eq_set_v @ ( inf_inf_set_v @ A @ B ) @ X ) ) ).
% le_infI1
thf(fact_993_le__infI2,axiom,
! [B: product_unit,X: product_unit,A: product_unit] :
( ( ord_le3221252021190050221t_unit @ B @ X )
=> ( ord_le3221252021190050221t_unit @ ( inf_inf_Product_unit @ A @ B ) @ X ) ) ).
% le_infI2
thf(fact_994_le__infI2,axiom,
! [B: set_Product_prod_v_v,X: set_Product_prod_v_v,A: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ B @ X )
=> ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ A @ B ) @ X ) ) ).
% le_infI2
thf(fact_995_le__infI2,axiom,
! [B: set_v,X: set_v,A: set_v] :
( ( ord_less_eq_set_v @ B @ X )
=> ( ord_less_eq_set_v @ ( inf_inf_set_v @ A @ B ) @ X ) ) ).
% le_infI2
thf(fact_996_inf_OorderE,axiom,
! [A: product_unit,B: product_unit] :
( ( ord_le3221252021190050221t_unit @ A @ B )
=> ( A
= ( inf_inf_Product_unit @ A @ B ) ) ) ).
% inf.orderE
thf(fact_997_inf_OorderE,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ B )
=> ( A
= ( inf_in6271465464967711157od_v_v @ A @ B ) ) ) ).
% inf.orderE
thf(fact_998_inf_OorderE,axiom,
! [A: set_v,B: set_v] :
( ( ord_less_eq_set_v @ A @ B )
=> ( A
= ( inf_inf_set_v @ A @ B ) ) ) ).
% inf.orderE
thf(fact_999_inf_OorderI,axiom,
! [A: product_unit,B: product_unit] :
( ( A
= ( inf_inf_Product_unit @ A @ B ) )
=> ( ord_le3221252021190050221t_unit @ A @ B ) ) ).
% inf.orderI
thf(fact_1000_inf_OorderI,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( A
= ( inf_in6271465464967711157od_v_v @ A @ B ) )
=> ( ord_le7336532860387713383od_v_v @ A @ B ) ) ).
% inf.orderI
thf(fact_1001_inf_OorderI,axiom,
! [A: set_v,B: set_v] :
( ( A
= ( inf_inf_set_v @ A @ B ) )
=> ( ord_less_eq_set_v @ A @ B ) ) ).
% inf.orderI
thf(fact_1002_inf__unique,axiom,
! [F: product_unit > product_unit > product_unit,X: product_unit,Y: product_unit] :
( ! [X3: product_unit,Y2: product_unit] : ( ord_le3221252021190050221t_unit @ ( F @ X3 @ Y2 ) @ X3 )
=> ( ! [X3: product_unit,Y2: product_unit] : ( ord_le3221252021190050221t_unit @ ( F @ X3 @ Y2 ) @ Y2 )
=> ( ! [X3: product_unit,Y2: product_unit,Z3: product_unit] :
( ( ord_le3221252021190050221t_unit @ X3 @ Y2 )
=> ( ( ord_le3221252021190050221t_unit @ X3 @ Z3 )
=> ( ord_le3221252021190050221t_unit @ X3 @ ( F @ Y2 @ Z3 ) ) ) )
=> ( ( inf_inf_Product_unit @ X @ Y )
= ( F @ X @ Y ) ) ) ) ) ).
% inf_unique
thf(fact_1003_inf__unique,axiom,
! [F: set_Product_prod_v_v > set_Product_prod_v_v > set_Product_prod_v_v,X: set_Product_prod_v_v,Y: set_Product_prod_v_v] :
( ! [X3: set_Product_prod_v_v,Y2: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ ( F @ X3 @ Y2 ) @ X3 )
=> ( ! [X3: set_Product_prod_v_v,Y2: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ ( F @ X3 @ Y2 ) @ Y2 )
=> ( ! [X3: set_Product_prod_v_v,Y2: set_Product_prod_v_v,Z3: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X3 @ Y2 )
=> ( ( ord_le7336532860387713383od_v_v @ X3 @ Z3 )
=> ( ord_le7336532860387713383od_v_v @ X3 @ ( F @ Y2 @ Z3 ) ) ) )
=> ( ( inf_in6271465464967711157od_v_v @ X @ Y )
= ( F @ X @ Y ) ) ) ) ) ).
% inf_unique
thf(fact_1004_inf__unique,axiom,
! [F: set_v > set_v > set_v,X: set_v,Y: set_v] :
( ! [X3: set_v,Y2: set_v] : ( ord_less_eq_set_v @ ( F @ X3 @ Y2 ) @ X3 )
=> ( ! [X3: set_v,Y2: set_v] : ( ord_less_eq_set_v @ ( F @ X3 @ Y2 ) @ Y2 )
=> ( ! [X3: set_v,Y2: set_v,Z3: set_v] :
( ( ord_less_eq_set_v @ X3 @ Y2 )
=> ( ( ord_less_eq_set_v @ X3 @ Z3 )
=> ( ord_less_eq_set_v @ X3 @ ( F @ Y2 @ Z3 ) ) ) )
=> ( ( inf_inf_set_v @ X @ Y )
= ( F @ X @ Y ) ) ) ) ) ).
% inf_unique
thf(fact_1005_le__iff__inf,axiom,
( ord_le3221252021190050221t_unit
= ( ^ [X2: product_unit,Y3: product_unit] :
( ( inf_inf_Product_unit @ X2 @ Y3 )
= X2 ) ) ) ).
% le_iff_inf
thf(fact_1006_le__iff__inf,axiom,
( ord_le7336532860387713383od_v_v
= ( ^ [X2: set_Product_prod_v_v,Y3: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ X2 @ Y3 )
= X2 ) ) ) ).
% le_iff_inf
thf(fact_1007_le__iff__inf,axiom,
( ord_less_eq_set_v
= ( ^ [X2: set_v,Y3: set_v] :
( ( inf_inf_set_v @ X2 @ Y3 )
= X2 ) ) ) ).
% le_iff_inf
thf(fact_1008_inf_Oabsorb1,axiom,
! [A: product_unit,B: product_unit] :
( ( ord_le3221252021190050221t_unit @ A @ B )
=> ( ( inf_inf_Product_unit @ A @ B )
= A ) ) ).
% inf.absorb1
thf(fact_1009_inf_Oabsorb1,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ B )
=> ( ( inf_in6271465464967711157od_v_v @ A @ B )
= A ) ) ).
% inf.absorb1
thf(fact_1010_inf_Oabsorb1,axiom,
! [A: set_v,B: set_v] :
( ( ord_less_eq_set_v @ A @ B )
=> ( ( inf_inf_set_v @ A @ B )
= A ) ) ).
% inf.absorb1
thf(fact_1011_inf_Oabsorb2,axiom,
! [B: product_unit,A: product_unit] :
( ( ord_le3221252021190050221t_unit @ B @ A )
=> ( ( inf_inf_Product_unit @ A @ B )
= B ) ) ).
% inf.absorb2
thf(fact_1012_inf_Oabsorb2,axiom,
! [B: set_Product_prod_v_v,A: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ B @ A )
=> ( ( inf_in6271465464967711157od_v_v @ A @ B )
= B ) ) ).
% inf.absorb2
thf(fact_1013_inf_Oabsorb2,axiom,
! [B: set_v,A: set_v] :
( ( ord_less_eq_set_v @ B @ A )
=> ( ( inf_inf_set_v @ A @ B )
= B ) ) ).
% inf.absorb2
thf(fact_1014_inf__absorb1,axiom,
! [X: product_unit,Y: product_unit] :
( ( ord_le3221252021190050221t_unit @ X @ Y )
=> ( ( inf_inf_Product_unit @ X @ Y )
= X ) ) ).
% inf_absorb1
thf(fact_1015_inf__absorb1,axiom,
! [X: set_Product_prod_v_v,Y: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X @ Y )
=> ( ( inf_in6271465464967711157od_v_v @ X @ Y )
= X ) ) ).
% inf_absorb1
thf(fact_1016_inf__absorb1,axiom,
! [X: set_v,Y: set_v] :
( ( ord_less_eq_set_v @ X @ Y )
=> ( ( inf_inf_set_v @ X @ Y )
= X ) ) ).
% inf_absorb1
thf(fact_1017_inf__absorb2,axiom,
! [Y: product_unit,X: product_unit] :
( ( ord_le3221252021190050221t_unit @ Y @ X )
=> ( ( inf_inf_Product_unit @ X @ Y )
= Y ) ) ).
% inf_absorb2
thf(fact_1018_inf__absorb2,axiom,
! [Y: set_Product_prod_v_v,X: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ Y @ X )
=> ( ( inf_in6271465464967711157od_v_v @ X @ Y )
= Y ) ) ).
% inf_absorb2
thf(fact_1019_inf__absorb2,axiom,
! [Y: set_v,X: set_v] :
( ( ord_less_eq_set_v @ Y @ X )
=> ( ( inf_inf_set_v @ X @ Y )
= Y ) ) ).
% inf_absorb2
thf(fact_1020_inf_OboundedE,axiom,
! [A: product_unit,B: product_unit,C: product_unit] :
( ( ord_le3221252021190050221t_unit @ A @ ( inf_inf_Product_unit @ B @ C ) )
=> ~ ( ( ord_le3221252021190050221t_unit @ A @ B )
=> ~ ( ord_le3221252021190050221t_unit @ A @ C ) ) ) ).
% inf.boundedE
thf(fact_1021_inf_OboundedE,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ ( inf_in6271465464967711157od_v_v @ B @ C ) )
=> ~ ( ( ord_le7336532860387713383od_v_v @ A @ B )
=> ~ ( ord_le7336532860387713383od_v_v @ A @ C ) ) ) ).
% inf.boundedE
thf(fact_1022_inf_OboundedE,axiom,
! [A: set_v,B: set_v,C: set_v] :
( ( ord_less_eq_set_v @ A @ ( inf_inf_set_v @ B @ C ) )
=> ~ ( ( ord_less_eq_set_v @ A @ B )
=> ~ ( ord_less_eq_set_v @ A @ C ) ) ) ).
% inf.boundedE
thf(fact_1023_inf_OboundedI,axiom,
! [A: product_unit,B: product_unit,C: product_unit] :
( ( ord_le3221252021190050221t_unit @ A @ B )
=> ( ( ord_le3221252021190050221t_unit @ A @ C )
=> ( ord_le3221252021190050221t_unit @ A @ ( inf_inf_Product_unit @ B @ C ) ) ) ) ).
% inf.boundedI
thf(fact_1024_inf_OboundedI,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ B )
=> ( ( ord_le7336532860387713383od_v_v @ A @ C )
=> ( ord_le7336532860387713383od_v_v @ A @ ( inf_in6271465464967711157od_v_v @ B @ C ) ) ) ) ).
% inf.boundedI
thf(fact_1025_inf_OboundedI,axiom,
! [A: set_v,B: set_v,C: set_v] :
( ( ord_less_eq_set_v @ A @ B )
=> ( ( ord_less_eq_set_v @ A @ C )
=> ( ord_less_eq_set_v @ A @ ( inf_inf_set_v @ B @ C ) ) ) ) ).
% inf.boundedI
thf(fact_1026_inf__greatest,axiom,
! [X: product_unit,Y: product_unit,Z: product_unit] :
( ( ord_le3221252021190050221t_unit @ X @ Y )
=> ( ( ord_le3221252021190050221t_unit @ X @ Z )
=> ( ord_le3221252021190050221t_unit @ X @ ( inf_inf_Product_unit @ Y @ Z ) ) ) ) ).
% inf_greatest
thf(fact_1027_inf__greatest,axiom,
! [X: set_Product_prod_v_v,Y: set_Product_prod_v_v,Z: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X @ Y )
=> ( ( ord_le7336532860387713383od_v_v @ X @ Z )
=> ( ord_le7336532860387713383od_v_v @ X @ ( inf_in6271465464967711157od_v_v @ Y @ Z ) ) ) ) ).
% inf_greatest
thf(fact_1028_inf__greatest,axiom,
! [X: set_v,Y: set_v,Z: set_v] :
( ( ord_less_eq_set_v @ X @ Y )
=> ( ( ord_less_eq_set_v @ X @ Z )
=> ( ord_less_eq_set_v @ X @ ( inf_inf_set_v @ Y @ Z ) ) ) ) ).
% inf_greatest
thf(fact_1029_inf_Oorder__iff,axiom,
( ord_le3221252021190050221t_unit
= ( ^ [A5: product_unit,B4: product_unit] :
( A5
= ( inf_inf_Product_unit @ A5 @ B4 ) ) ) ) ).
% inf.order_iff
thf(fact_1030_inf_Oorder__iff,axiom,
( ord_le7336532860387713383od_v_v
= ( ^ [A5: set_Product_prod_v_v,B4: set_Product_prod_v_v] :
( A5
= ( inf_in6271465464967711157od_v_v @ A5 @ B4 ) ) ) ) ).
% inf.order_iff
thf(fact_1031_inf_Oorder__iff,axiom,
( ord_less_eq_set_v
= ( ^ [A5: set_v,B4: set_v] :
( A5
= ( inf_inf_set_v @ A5 @ B4 ) ) ) ) ).
% inf.order_iff
thf(fact_1032_inf_Ocobounded1,axiom,
! [A: product_unit,B: product_unit] : ( ord_le3221252021190050221t_unit @ ( inf_inf_Product_unit @ A @ B ) @ A ) ).
% inf.cobounded1
thf(fact_1033_inf_Ocobounded1,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ A @ B ) @ A ) ).
% inf.cobounded1
thf(fact_1034_inf_Ocobounded1,axiom,
! [A: set_v,B: set_v] : ( ord_less_eq_set_v @ ( inf_inf_set_v @ A @ B ) @ A ) ).
% inf.cobounded1
thf(fact_1035_inf_Ocobounded2,axiom,
! [A: product_unit,B: product_unit] : ( ord_le3221252021190050221t_unit @ ( inf_inf_Product_unit @ A @ B ) @ B ) ).
% inf.cobounded2
thf(fact_1036_inf_Ocobounded2,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ A @ B ) @ B ) ).
% inf.cobounded2
thf(fact_1037_inf_Ocobounded2,axiom,
! [A: set_v,B: set_v] : ( ord_less_eq_set_v @ ( inf_inf_set_v @ A @ B ) @ B ) ).
% inf.cobounded2
thf(fact_1038_inf_Oabsorb__iff1,axiom,
( ord_le3221252021190050221t_unit
= ( ^ [A5: product_unit,B4: product_unit] :
( ( inf_inf_Product_unit @ A5 @ B4 )
= A5 ) ) ) ).
% inf.absorb_iff1
thf(fact_1039_inf_Oabsorb__iff1,axiom,
( ord_le7336532860387713383od_v_v
= ( ^ [A5: set_Product_prod_v_v,B4: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ A5 @ B4 )
= A5 ) ) ) ).
% inf.absorb_iff1
thf(fact_1040_inf_Oabsorb__iff1,axiom,
( ord_less_eq_set_v
= ( ^ [A5: set_v,B4: set_v] :
( ( inf_inf_set_v @ A5 @ B4 )
= A5 ) ) ) ).
% inf.absorb_iff1
thf(fact_1041_inf_Oabsorb__iff2,axiom,
( ord_le3221252021190050221t_unit
= ( ^ [B4: product_unit,A5: product_unit] :
( ( inf_inf_Product_unit @ A5 @ B4 )
= B4 ) ) ) ).
% inf.absorb_iff2
thf(fact_1042_inf_Oabsorb__iff2,axiom,
( ord_le7336532860387713383od_v_v
= ( ^ [B4: set_Product_prod_v_v,A5: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ A5 @ B4 )
= B4 ) ) ) ).
% inf.absorb_iff2
thf(fact_1043_inf_Oabsorb__iff2,axiom,
( ord_less_eq_set_v
= ( ^ [B4: set_v,A5: set_v] :
( ( inf_inf_set_v @ A5 @ B4 )
= B4 ) ) ) ).
% inf.absorb_iff2
thf(fact_1044_inf_OcoboundedI1,axiom,
! [A: product_unit,C: product_unit,B: product_unit] :
( ( ord_le3221252021190050221t_unit @ A @ C )
=> ( ord_le3221252021190050221t_unit @ ( inf_inf_Product_unit @ A @ B ) @ C ) ) ).
% inf.coboundedI1
thf(fact_1045_inf_OcoboundedI1,axiom,
! [A: set_Product_prod_v_v,C: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ C )
=> ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ A @ B ) @ C ) ) ).
% inf.coboundedI1
thf(fact_1046_inf_OcoboundedI1,axiom,
! [A: set_v,C: set_v,B: set_v] :
( ( ord_less_eq_set_v @ A @ C )
=> ( ord_less_eq_set_v @ ( inf_inf_set_v @ A @ B ) @ C ) ) ).
% inf.coboundedI1
thf(fact_1047_inf_OcoboundedI2,axiom,
! [B: product_unit,C: product_unit,A: product_unit] :
( ( ord_le3221252021190050221t_unit @ B @ C )
=> ( ord_le3221252021190050221t_unit @ ( inf_inf_Product_unit @ A @ B ) @ C ) ) ).
% inf.coboundedI2
thf(fact_1048_inf_OcoboundedI2,axiom,
! [B: set_Product_prod_v_v,C: set_Product_prod_v_v,A: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ B @ C )
=> ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ A @ B ) @ C ) ) ).
% inf.coboundedI2
thf(fact_1049_inf_OcoboundedI2,axiom,
! [B: set_v,C: set_v,A: set_v] :
( ( ord_less_eq_set_v @ B @ C )
=> ( ord_less_eq_set_v @ ( inf_inf_set_v @ A @ B ) @ C ) ) ).
% inf.coboundedI2
thf(fact_1050_Int__left__commute,axiom,
! [A2: set_v,B2: set_v,C2: set_v] :
( ( inf_inf_set_v @ A2 @ ( inf_inf_set_v @ B2 @ C2 ) )
= ( inf_inf_set_v @ B2 @ ( inf_inf_set_v @ A2 @ C2 ) ) ) ).
% Int_left_commute
thf(fact_1051_Int__left__absorb,axiom,
! [A2: set_v,B2: set_v] :
( ( inf_inf_set_v @ A2 @ ( inf_inf_set_v @ A2 @ B2 ) )
= ( inf_inf_set_v @ A2 @ B2 ) ) ).
% Int_left_absorb
thf(fact_1052_Int__commute,axiom,
( inf_inf_set_v
= ( ^ [A7: set_v,B6: set_v] : ( inf_inf_set_v @ B6 @ A7 ) ) ) ).
% Int_commute
thf(fact_1053_Int__absorb,axiom,
! [A2: set_v] :
( ( inf_inf_set_v @ A2 @ A2 )
= A2 ) ).
% Int_absorb
thf(fact_1054_Int__assoc,axiom,
! [A2: set_v,B2: set_v,C2: set_v] :
( ( inf_inf_set_v @ ( inf_inf_set_v @ A2 @ B2 ) @ C2 )
= ( inf_inf_set_v @ A2 @ ( inf_inf_set_v @ B2 @ C2 ) ) ) ).
% Int_assoc
thf(fact_1055_IntD2,axiom,
! [C: product_prod_v_v,A2: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ C @ ( inf_in6271465464967711157od_v_v @ A2 @ B2 ) )
=> ( member7453568604450474000od_v_v @ C @ B2 ) ) ).
% IntD2
thf(fact_1056_IntD2,axiom,
! [C: v,A2: set_v,B2: set_v] :
( ( member_v @ C @ ( inf_inf_set_v @ A2 @ B2 ) )
=> ( member_v @ C @ B2 ) ) ).
% IntD2
thf(fact_1057_IntD1,axiom,
! [C: product_prod_v_v,A2: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ C @ ( inf_in6271465464967711157od_v_v @ A2 @ B2 ) )
=> ( member7453568604450474000od_v_v @ C @ A2 ) ) ).
% IntD1
thf(fact_1058_IntD1,axiom,
! [C: v,A2: set_v,B2: set_v] :
( ( member_v @ C @ ( inf_inf_set_v @ A2 @ B2 ) )
=> ( member_v @ C @ A2 ) ) ).
% IntD1
thf(fact_1059_IntE,axiom,
! [C: product_prod_v_v,A2: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ C @ ( inf_in6271465464967711157od_v_v @ A2 @ B2 ) )
=> ~ ( ( member7453568604450474000od_v_v @ C @ A2 )
=> ~ ( member7453568604450474000od_v_v @ C @ B2 ) ) ) ).
% IntE
thf(fact_1060_IntE,axiom,
! [C: v,A2: set_v,B2: set_v] :
( ( member_v @ C @ ( inf_inf_set_v @ A2 @ B2 ) )
=> ~ ( ( member_v @ C @ A2 )
=> ~ ( member_v @ C @ B2 ) ) ) ).
% IntE
thf(fact_1061_disjoint__iff__not__equal,axiom,
! [A2: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( ( inf_in6271465464967711157od_v_v @ A2 @ B2 )
= bot_bo723834152578015283od_v_v )
= ( ! [X2: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X2 @ A2 )
=> ! [Y3: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ Y3 @ B2 )
=> ( X2 != Y3 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_1062_disjoint__iff__not__equal,axiom,
! [A2: set_v,B2: set_v] :
( ( ( inf_inf_set_v @ A2 @ B2 )
= bot_bot_set_v )
= ( ! [X2: v] :
( ( member_v @ X2 @ A2 )
=> ! [Y3: v] :
( ( member_v @ Y3 @ B2 )
=> ( X2 != Y3 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_1063_disjoint__iff__not__equal,axiom,
! [A2: set_set_v,B2: set_set_v] :
( ( ( inf_inf_set_set_v @ A2 @ B2 )
= bot_bot_set_set_v )
= ( ! [X2: set_v] :
( ( member_set_v @ X2 @ A2 )
=> ! [Y3: set_v] :
( ( member_set_v @ Y3 @ B2 )
=> ( X2 != Y3 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_1064_Int__empty__right,axiom,
! [A2: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ A2 @ bot_bo723834152578015283od_v_v )
= bot_bo723834152578015283od_v_v ) ).
% Int_empty_right
thf(fact_1065_Int__empty__right,axiom,
! [A2: set_v] :
( ( inf_inf_set_v @ A2 @ bot_bot_set_v )
= bot_bot_set_v ) ).
% Int_empty_right
thf(fact_1066_Int__empty__right,axiom,
! [A2: set_set_v] :
( ( inf_inf_set_set_v @ A2 @ bot_bot_set_set_v )
= bot_bot_set_set_v ) ).
% Int_empty_right
thf(fact_1067_Int__empty__left,axiom,
! [B2: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ bot_bo723834152578015283od_v_v @ B2 )
= bot_bo723834152578015283od_v_v ) ).
% Int_empty_left
thf(fact_1068_Int__empty__left,axiom,
! [B2: set_v] :
( ( inf_inf_set_v @ bot_bot_set_v @ B2 )
= bot_bot_set_v ) ).
% Int_empty_left
thf(fact_1069_Int__empty__left,axiom,
! [B2: set_set_v] :
( ( inf_inf_set_set_v @ bot_bot_set_set_v @ B2 )
= bot_bot_set_set_v ) ).
% Int_empty_left
thf(fact_1070_disjoint__iff,axiom,
! [A2: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( ( inf_in6271465464967711157od_v_v @ A2 @ B2 )
= bot_bo723834152578015283od_v_v )
= ( ! [X2: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X2 @ A2 )
=> ~ ( member7453568604450474000od_v_v @ X2 @ B2 ) ) ) ) ).
% disjoint_iff
thf(fact_1071_disjoint__iff,axiom,
! [A2: set_v,B2: set_v] :
( ( ( inf_inf_set_v @ A2 @ B2 )
= bot_bot_set_v )
= ( ! [X2: v] :
( ( member_v @ X2 @ A2 )
=> ~ ( member_v @ X2 @ B2 ) ) ) ) ).
% disjoint_iff
thf(fact_1072_disjoint__iff,axiom,
! [A2: set_set_v,B2: set_set_v] :
( ( ( inf_inf_set_set_v @ A2 @ B2 )
= bot_bot_set_set_v )
= ( ! [X2: set_v] :
( ( member_set_v @ X2 @ A2 )
=> ~ ( member_set_v @ X2 @ B2 ) ) ) ) ).
% disjoint_iff
thf(fact_1073_Int__emptyI,axiom,
! [A2: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ! [X3: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X3 @ A2 )
=> ~ ( member7453568604450474000od_v_v @ X3 @ B2 ) )
=> ( ( inf_in6271465464967711157od_v_v @ A2 @ B2 )
= bot_bo723834152578015283od_v_v ) ) ).
% Int_emptyI
thf(fact_1074_Int__emptyI,axiom,
! [A2: set_v,B2: set_v] :
( ! [X3: v] :
( ( member_v @ X3 @ A2 )
=> ~ ( member_v @ X3 @ B2 ) )
=> ( ( inf_inf_set_v @ A2 @ B2 )
= bot_bot_set_v ) ) ).
% Int_emptyI
thf(fact_1075_Int__emptyI,axiom,
! [A2: set_set_v,B2: set_set_v] :
( ! [X3: set_v] :
( ( member_set_v @ X3 @ A2 )
=> ~ ( member_set_v @ X3 @ B2 ) )
=> ( ( inf_inf_set_set_v @ A2 @ B2 )
= bot_bot_set_set_v ) ) ).
% Int_emptyI
thf(fact_1076_Int__mono,axiom,
! [A2: set_Product_prod_v_v,C2: set_Product_prod_v_v,B2: set_Product_prod_v_v,D: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A2 @ C2 )
=> ( ( ord_le7336532860387713383od_v_v @ B2 @ D )
=> ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ A2 @ B2 ) @ ( inf_in6271465464967711157od_v_v @ C2 @ D ) ) ) ) ).
% Int_mono
thf(fact_1077_Int__mono,axiom,
! [A2: set_v,C2: set_v,B2: set_v,D: set_v] :
( ( ord_less_eq_set_v @ A2 @ C2 )
=> ( ( ord_less_eq_set_v @ B2 @ D )
=> ( ord_less_eq_set_v @ ( inf_inf_set_v @ A2 @ B2 ) @ ( inf_inf_set_v @ C2 @ D ) ) ) ) ).
% Int_mono
thf(fact_1078_Int__lower1,axiom,
! [A2: set_Product_prod_v_v,B2: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ A2 @ B2 ) @ A2 ) ).
% Int_lower1
thf(fact_1079_Int__lower1,axiom,
! [A2: set_v,B2: set_v] : ( ord_less_eq_set_v @ ( inf_inf_set_v @ A2 @ B2 ) @ A2 ) ).
% Int_lower1
thf(fact_1080_Int__lower2,axiom,
! [A2: set_Product_prod_v_v,B2: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ A2 @ B2 ) @ B2 ) ).
% Int_lower2
thf(fact_1081_Int__lower2,axiom,
! [A2: set_v,B2: set_v] : ( ord_less_eq_set_v @ ( inf_inf_set_v @ A2 @ B2 ) @ B2 ) ).
% Int_lower2
thf(fact_1082_Int__absorb1,axiom,
! [B2: set_Product_prod_v_v,A2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ B2 @ A2 )
=> ( ( inf_in6271465464967711157od_v_v @ A2 @ B2 )
= B2 ) ) ).
% Int_absorb1
thf(fact_1083_Int__absorb1,axiom,
! [B2: set_v,A2: set_v] :
( ( ord_less_eq_set_v @ B2 @ A2 )
=> ( ( inf_inf_set_v @ A2 @ B2 )
= B2 ) ) ).
% Int_absorb1
thf(fact_1084_Int__absorb2,axiom,
! [A2: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A2 @ B2 )
=> ( ( inf_in6271465464967711157od_v_v @ A2 @ B2 )
= A2 ) ) ).
% Int_absorb2
thf(fact_1085_Int__absorb2,axiom,
! [A2: set_v,B2: set_v] :
( ( ord_less_eq_set_v @ A2 @ B2 )
=> ( ( inf_inf_set_v @ A2 @ B2 )
= A2 ) ) ).
% Int_absorb2
thf(fact_1086_Int__greatest,axiom,
! [C2: set_Product_prod_v_v,A2: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ C2 @ A2 )
=> ( ( ord_le7336532860387713383od_v_v @ C2 @ B2 )
=> ( ord_le7336532860387713383od_v_v @ C2 @ ( inf_in6271465464967711157od_v_v @ A2 @ B2 ) ) ) ) ).
% Int_greatest
thf(fact_1087_Int__greatest,axiom,
! [C2: set_v,A2: set_v,B2: set_v] :
( ( ord_less_eq_set_v @ C2 @ A2 )
=> ( ( ord_less_eq_set_v @ C2 @ B2 )
=> ( ord_less_eq_set_v @ C2 @ ( inf_inf_set_v @ A2 @ B2 ) ) ) ) ).
% Int_greatest
thf(fact_1088_Int__Collect__mono,axiom,
! [A2: set_set_v,B2: set_set_v,P: set_v > $o,Q: set_v > $o] :
( ( ord_le5216385588623774835_set_v @ A2 @ B2 )
=> ( ! [X3: set_v] :
( ( member_set_v @ X3 @ A2 )
=> ( ( P @ X3 )
=> ( Q @ X3 ) ) )
=> ( ord_le5216385588623774835_set_v @ ( inf_inf_set_set_v @ A2 @ ( collect_set_v @ P ) ) @ ( inf_inf_set_set_v @ B2 @ ( collect_set_v @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_1089_Int__Collect__mono,axiom,
! [A2: set_Product_prod_v_v,B2: set_Product_prod_v_v,P: product_prod_v_v > $o,Q: product_prod_v_v > $o] :
( ( ord_le7336532860387713383od_v_v @ A2 @ B2 )
=> ( ! [X3: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X3 @ A2 )
=> ( ( P @ X3 )
=> ( Q @ X3 ) ) )
=> ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ A2 @ ( collec140062887454715474od_v_v @ P ) ) @ ( inf_in6271465464967711157od_v_v @ B2 @ ( collec140062887454715474od_v_v @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_1090_Int__Collect__mono,axiom,
! [A2: set_v,B2: set_v,P: v > $o,Q: v > $o] :
( ( ord_less_eq_set_v @ A2 @ B2 )
=> ( ! [X3: v] :
( ( member_v @ X3 @ A2 )
=> ( ( P @ X3 )
=> ( Q @ X3 ) ) )
=> ( ord_less_eq_set_v @ ( inf_inf_set_v @ A2 @ ( collect_v @ P ) ) @ ( inf_inf_set_v @ B2 @ ( collect_v @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_1091_Int__insert__right,axiom,
! [A: set_v,A2: set_set_v,B2: set_set_v] :
( ( ( member_set_v @ A @ A2 )
=> ( ( inf_inf_set_set_v @ A2 @ ( insert_set_v2 @ A @ B2 ) )
= ( insert_set_v2 @ A @ ( inf_inf_set_set_v @ A2 @ B2 ) ) ) )
& ( ~ ( member_set_v @ A @ A2 )
=> ( ( inf_inf_set_set_v @ A2 @ ( insert_set_v2 @ A @ B2 ) )
= ( inf_inf_set_set_v @ A2 @ B2 ) ) ) ) ).
% Int_insert_right
thf(fact_1092_Int__insert__right,axiom,
! [A: product_prod_v_v,A2: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( ( member7453568604450474000od_v_v @ A @ A2 )
=> ( ( inf_in6271465464967711157od_v_v @ A2 @ ( insert1338601472111419319od_v_v @ A @ B2 ) )
= ( insert1338601472111419319od_v_v @ A @ ( inf_in6271465464967711157od_v_v @ A2 @ B2 ) ) ) )
& ( ~ ( member7453568604450474000od_v_v @ A @ A2 )
=> ( ( inf_in6271465464967711157od_v_v @ A2 @ ( insert1338601472111419319od_v_v @ A @ B2 ) )
= ( inf_in6271465464967711157od_v_v @ A2 @ B2 ) ) ) ) ).
% Int_insert_right
thf(fact_1093_Int__insert__right,axiom,
! [A: v,A2: set_v,B2: set_v] :
( ( ( member_v @ A @ A2 )
=> ( ( inf_inf_set_v @ A2 @ ( insert_v2 @ A @ B2 ) )
= ( insert_v2 @ A @ ( inf_inf_set_v @ A2 @ B2 ) ) ) )
& ( ~ ( member_v @ A @ A2 )
=> ( ( inf_inf_set_v @ A2 @ ( insert_v2 @ A @ B2 ) )
= ( inf_inf_set_v @ A2 @ B2 ) ) ) ) ).
% Int_insert_right
thf(fact_1094_Int__insert__left,axiom,
! [A: set_v,C2: set_set_v,B2: set_set_v] :
( ( ( member_set_v @ A @ C2 )
=> ( ( inf_inf_set_set_v @ ( insert_set_v2 @ A @ B2 ) @ C2 )
= ( insert_set_v2 @ A @ ( inf_inf_set_set_v @ B2 @ C2 ) ) ) )
& ( ~ ( member_set_v @ A @ C2 )
=> ( ( inf_inf_set_set_v @ ( insert_set_v2 @ A @ B2 ) @ C2 )
= ( inf_inf_set_set_v @ B2 @ C2 ) ) ) ) ).
% Int_insert_left
thf(fact_1095_Int__insert__left,axiom,
! [A: product_prod_v_v,C2: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( ( member7453568604450474000od_v_v @ A @ C2 )
=> ( ( inf_in6271465464967711157od_v_v @ ( insert1338601472111419319od_v_v @ A @ B2 ) @ C2 )
= ( insert1338601472111419319od_v_v @ A @ ( inf_in6271465464967711157od_v_v @ B2 @ C2 ) ) ) )
& ( ~ ( member7453568604450474000od_v_v @ A @ C2 )
=> ( ( inf_in6271465464967711157od_v_v @ ( insert1338601472111419319od_v_v @ A @ B2 ) @ C2 )
= ( inf_in6271465464967711157od_v_v @ B2 @ C2 ) ) ) ) ).
% Int_insert_left
thf(fact_1096_Int__insert__left,axiom,
! [A: v,C2: set_v,B2: set_v] :
( ( ( member_v @ A @ C2 )
=> ( ( inf_inf_set_v @ ( insert_v2 @ A @ B2 ) @ C2 )
= ( insert_v2 @ A @ ( inf_inf_set_v @ B2 @ C2 ) ) ) )
& ( ~ ( member_v @ A @ C2 )
=> ( ( inf_inf_set_v @ ( insert_v2 @ A @ B2 ) @ C2 )
= ( inf_inf_set_v @ B2 @ C2 ) ) ) ) ).
% Int_insert_left
thf(fact_1097_Un__Int__distrib2,axiom,
! [B2: set_Product_prod_v_v,C2: set_Product_prod_v_v,A2: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ B2 @ C2 ) @ A2 )
= ( inf_in6271465464967711157od_v_v @ ( sup_su414716646722978715od_v_v @ B2 @ A2 ) @ ( sup_su414716646722978715od_v_v @ C2 @ A2 ) ) ) ).
% Un_Int_distrib2
thf(fact_1098_Un__Int__distrib2,axiom,
! [B2: set_v,C2: set_v,A2: set_v] :
( ( sup_sup_set_v @ ( inf_inf_set_v @ B2 @ C2 ) @ A2 )
= ( inf_inf_set_v @ ( sup_sup_set_v @ B2 @ A2 ) @ ( sup_sup_set_v @ C2 @ A2 ) ) ) ).
% Un_Int_distrib2
thf(fact_1099_Un__Int__distrib2,axiom,
! [B2: set_set_v,C2: set_set_v,A2: set_set_v] :
( ( sup_sup_set_set_v @ ( inf_inf_set_set_v @ B2 @ C2 ) @ A2 )
= ( inf_inf_set_set_v @ ( sup_sup_set_set_v @ B2 @ A2 ) @ ( sup_sup_set_set_v @ C2 @ A2 ) ) ) ).
% Un_Int_distrib2
thf(fact_1100_Int__Un__distrib2,axiom,
! [B2: set_Product_prod_v_v,C2: set_Product_prod_v_v,A2: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ ( sup_su414716646722978715od_v_v @ B2 @ C2 ) @ A2 )
= ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ B2 @ A2 ) @ ( inf_in6271465464967711157od_v_v @ C2 @ A2 ) ) ) ).
% Int_Un_distrib2
thf(fact_1101_Int__Un__distrib2,axiom,
! [B2: set_v,C2: set_v,A2: set_v] :
( ( inf_inf_set_v @ ( sup_sup_set_v @ B2 @ C2 ) @ A2 )
= ( sup_sup_set_v @ ( inf_inf_set_v @ B2 @ A2 ) @ ( inf_inf_set_v @ C2 @ A2 ) ) ) ).
% Int_Un_distrib2
thf(fact_1102_Int__Un__distrib2,axiom,
! [B2: set_set_v,C2: set_set_v,A2: set_set_v] :
( ( inf_inf_set_set_v @ ( sup_sup_set_set_v @ B2 @ C2 ) @ A2 )
= ( sup_sup_set_set_v @ ( inf_inf_set_set_v @ B2 @ A2 ) @ ( inf_inf_set_set_v @ C2 @ A2 ) ) ) ).
% Int_Un_distrib2
thf(fact_1103_Un__Int__distrib,axiom,
! [A2: set_Product_prod_v_v,B2: set_Product_prod_v_v,C2: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ A2 @ ( inf_in6271465464967711157od_v_v @ B2 @ C2 ) )
= ( inf_in6271465464967711157od_v_v @ ( sup_su414716646722978715od_v_v @ A2 @ B2 ) @ ( sup_su414716646722978715od_v_v @ A2 @ C2 ) ) ) ).
% Un_Int_distrib
thf(fact_1104_Un__Int__distrib,axiom,
! [A2: set_v,B2: set_v,C2: set_v] :
( ( sup_sup_set_v @ A2 @ ( inf_inf_set_v @ B2 @ C2 ) )
= ( inf_inf_set_v @ ( sup_sup_set_v @ A2 @ B2 ) @ ( sup_sup_set_v @ A2 @ C2 ) ) ) ).
% Un_Int_distrib
thf(fact_1105_Un__Int__distrib,axiom,
! [A2: set_set_v,B2: set_set_v,C2: set_set_v] :
( ( sup_sup_set_set_v @ A2 @ ( inf_inf_set_set_v @ B2 @ C2 ) )
= ( inf_inf_set_set_v @ ( sup_sup_set_set_v @ A2 @ B2 ) @ ( sup_sup_set_set_v @ A2 @ C2 ) ) ) ).
% Un_Int_distrib
thf(fact_1106_Int__Un__distrib,axiom,
! [A2: set_Product_prod_v_v,B2: set_Product_prod_v_v,C2: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ A2 @ ( sup_su414716646722978715od_v_v @ B2 @ C2 ) )
= ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ A2 @ B2 ) @ ( inf_in6271465464967711157od_v_v @ A2 @ C2 ) ) ) ).
% Int_Un_distrib
thf(fact_1107_Int__Un__distrib,axiom,
! [A2: set_v,B2: set_v,C2: set_v] :
( ( inf_inf_set_v @ A2 @ ( sup_sup_set_v @ B2 @ C2 ) )
= ( sup_sup_set_v @ ( inf_inf_set_v @ A2 @ B2 ) @ ( inf_inf_set_v @ A2 @ C2 ) ) ) ).
% Int_Un_distrib
thf(fact_1108_Int__Un__distrib,axiom,
! [A2: set_set_v,B2: set_set_v,C2: set_set_v] :
( ( inf_inf_set_set_v @ A2 @ ( sup_sup_set_set_v @ B2 @ C2 ) )
= ( sup_sup_set_set_v @ ( inf_inf_set_set_v @ A2 @ B2 ) @ ( inf_inf_set_set_v @ A2 @ C2 ) ) ) ).
% Int_Un_distrib
thf(fact_1109_Un__Int__crazy,axiom,
! [A2: set_Product_prod_v_v,B2: set_Product_prod_v_v,C2: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ A2 @ B2 ) @ ( inf_in6271465464967711157od_v_v @ B2 @ C2 ) ) @ ( inf_in6271465464967711157od_v_v @ C2 @ A2 ) )
= ( inf_in6271465464967711157od_v_v @ ( inf_in6271465464967711157od_v_v @ ( sup_su414716646722978715od_v_v @ A2 @ B2 ) @ ( sup_su414716646722978715od_v_v @ B2 @ C2 ) ) @ ( sup_su414716646722978715od_v_v @ C2 @ A2 ) ) ) ).
% Un_Int_crazy
thf(fact_1110_Un__Int__crazy,axiom,
! [A2: set_v,B2: set_v,C2: set_v] :
( ( sup_sup_set_v @ ( sup_sup_set_v @ ( inf_inf_set_v @ A2 @ B2 ) @ ( inf_inf_set_v @ B2 @ C2 ) ) @ ( inf_inf_set_v @ C2 @ A2 ) )
= ( inf_inf_set_v @ ( inf_inf_set_v @ ( sup_sup_set_v @ A2 @ B2 ) @ ( sup_sup_set_v @ B2 @ C2 ) ) @ ( sup_sup_set_v @ C2 @ A2 ) ) ) ).
% Un_Int_crazy
thf(fact_1111_Un__Int__crazy,axiom,
! [A2: set_set_v,B2: set_set_v,C2: set_set_v] :
( ( sup_sup_set_set_v @ ( sup_sup_set_set_v @ ( inf_inf_set_set_v @ A2 @ B2 ) @ ( inf_inf_set_set_v @ B2 @ C2 ) ) @ ( inf_inf_set_set_v @ C2 @ A2 ) )
= ( inf_inf_set_set_v @ ( inf_inf_set_set_v @ ( sup_sup_set_set_v @ A2 @ B2 ) @ ( sup_sup_set_set_v @ B2 @ C2 ) ) @ ( sup_sup_set_set_v @ C2 @ A2 ) ) ) ).
% Un_Int_crazy
thf(fact_1112_Diff__Int__distrib2,axiom,
! [A2: set_v,B2: set_v,C2: set_v] :
( ( inf_inf_set_v @ ( minus_minus_set_v @ A2 @ B2 ) @ C2 )
= ( minus_minus_set_v @ ( inf_inf_set_v @ A2 @ C2 ) @ ( inf_inf_set_v @ B2 @ C2 ) ) ) ).
% Diff_Int_distrib2
thf(fact_1113_Diff__Int__distrib,axiom,
! [C2: set_v,A2: set_v,B2: set_v] :
( ( inf_inf_set_v @ C2 @ ( minus_minus_set_v @ A2 @ B2 ) )
= ( minus_minus_set_v @ ( inf_inf_set_v @ C2 @ A2 ) @ ( inf_inf_set_v @ C2 @ B2 ) ) ) ).
% Diff_Int_distrib
thf(fact_1114_Diff__Diff__Int,axiom,
! [A2: set_v,B2: set_v] :
( ( minus_minus_set_v @ A2 @ ( minus_minus_set_v @ A2 @ B2 ) )
= ( inf_inf_set_v @ A2 @ B2 ) ) ).
% Diff_Diff_Int
thf(fact_1115_Diff__Int2,axiom,
! [A2: set_v,C2: set_v,B2: set_v] :
( ( minus_minus_set_v @ ( inf_inf_set_v @ A2 @ C2 ) @ ( inf_inf_set_v @ B2 @ C2 ) )
= ( minus_minus_set_v @ ( inf_inf_set_v @ A2 @ C2 ) @ B2 ) ) ).
% Diff_Int2
thf(fact_1116_Int__Diff,axiom,
! [A2: set_v,B2: set_v,C2: set_v] :
( ( minus_minus_set_v @ ( inf_inf_set_v @ A2 @ B2 ) @ C2 )
= ( inf_inf_set_v @ A2 @ ( minus_minus_set_v @ B2 @ C2 ) ) ) ).
% Int_Diff
thf(fact_1117_distrib__sup__le,axiom,
! [X: set_set_v,Y: set_set_v,Z: set_set_v] : ( ord_le5216385588623774835_set_v @ ( sup_sup_set_set_v @ X @ ( inf_inf_set_set_v @ Y @ Z ) ) @ ( inf_inf_set_set_v @ ( sup_sup_set_set_v @ X @ Y ) @ ( sup_sup_set_set_v @ X @ Z ) ) ) ).
% distrib_sup_le
thf(fact_1118_distrib__sup__le,axiom,
! [X: product_unit,Y: product_unit,Z: product_unit] : ( ord_le3221252021190050221t_unit @ ( sup_sup_Product_unit @ X @ ( inf_inf_Product_unit @ Y @ Z ) ) @ ( inf_inf_Product_unit @ ( sup_sup_Product_unit @ X @ Y ) @ ( sup_sup_Product_unit @ X @ Z ) ) ) ).
% distrib_sup_le
thf(fact_1119_distrib__sup__le,axiom,
! [X: set_Product_prod_v_v,Y: set_Product_prod_v_v,Z: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ ( sup_su414716646722978715od_v_v @ X @ ( inf_in6271465464967711157od_v_v @ Y @ Z ) ) @ ( inf_in6271465464967711157od_v_v @ ( sup_su414716646722978715od_v_v @ X @ Y ) @ ( sup_su414716646722978715od_v_v @ X @ Z ) ) ) ).
% distrib_sup_le
thf(fact_1120_distrib__sup__le,axiom,
! [X: set_v,Y: set_v,Z: set_v] : ( ord_less_eq_set_v @ ( sup_sup_set_v @ X @ ( inf_inf_set_v @ Y @ Z ) ) @ ( inf_inf_set_v @ ( sup_sup_set_v @ X @ Y ) @ ( sup_sup_set_v @ X @ Z ) ) ) ).
% distrib_sup_le
thf(fact_1121_distrib__inf__le,axiom,
! [X: set_set_v,Y: set_set_v,Z: set_set_v] : ( ord_le5216385588623774835_set_v @ ( sup_sup_set_set_v @ ( inf_inf_set_set_v @ X @ Y ) @ ( inf_inf_set_set_v @ X @ Z ) ) @ ( inf_inf_set_set_v @ X @ ( sup_sup_set_set_v @ Y @ Z ) ) ) ).
% distrib_inf_le
thf(fact_1122_distrib__inf__le,axiom,
! [X: product_unit,Y: product_unit,Z: product_unit] : ( ord_le3221252021190050221t_unit @ ( sup_sup_Product_unit @ ( inf_inf_Product_unit @ X @ Y ) @ ( inf_inf_Product_unit @ X @ Z ) ) @ ( inf_inf_Product_unit @ X @ ( sup_sup_Product_unit @ Y @ Z ) ) ) ).
% distrib_inf_le
thf(fact_1123_distrib__inf__le,axiom,
! [X: set_Product_prod_v_v,Y: set_Product_prod_v_v,Z: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ X @ Y ) @ ( inf_in6271465464967711157od_v_v @ X @ Z ) ) @ ( inf_in6271465464967711157od_v_v @ X @ ( sup_su414716646722978715od_v_v @ Y @ Z ) ) ) ).
% distrib_inf_le
thf(fact_1124_distrib__inf__le,axiom,
! [X: set_v,Y: set_v,Z: set_v] : ( ord_less_eq_set_v @ ( sup_sup_set_v @ ( inf_inf_set_v @ X @ Y ) @ ( inf_inf_set_v @ X @ Z ) ) @ ( inf_inf_set_v @ X @ ( sup_sup_set_v @ Y @ Z ) ) ) ).
% distrib_inf_le
thf(fact_1125_Int__Diff__disjoint,axiom,
! [A2: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ ( inf_in6271465464967711157od_v_v @ A2 @ B2 ) @ ( minus_4183494784930505774od_v_v @ A2 @ B2 ) )
= bot_bo723834152578015283od_v_v ) ).
% Int_Diff_disjoint
thf(fact_1126_Int__Diff__disjoint,axiom,
! [A2: set_set_v,B2: set_set_v] :
( ( inf_inf_set_set_v @ ( inf_inf_set_set_v @ A2 @ B2 ) @ ( minus_7228012346218142266_set_v @ A2 @ B2 ) )
= bot_bot_set_set_v ) ).
% Int_Diff_disjoint
thf(fact_1127_Int__Diff__disjoint,axiom,
! [A2: set_v,B2: set_v] :
( ( inf_inf_set_v @ ( inf_inf_set_v @ A2 @ B2 ) @ ( minus_minus_set_v @ A2 @ B2 ) )
= bot_bot_set_v ) ).
% Int_Diff_disjoint
thf(fact_1128_Diff__triv,axiom,
! [A2: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( ( inf_in6271465464967711157od_v_v @ A2 @ B2 )
= bot_bo723834152578015283od_v_v )
=> ( ( minus_4183494784930505774od_v_v @ A2 @ B2 )
= A2 ) ) ).
% Diff_triv
thf(fact_1129_Diff__triv,axiom,
! [A2: set_set_v,B2: set_set_v] :
( ( ( inf_inf_set_set_v @ A2 @ B2 )
= bot_bot_set_set_v )
=> ( ( minus_7228012346218142266_set_v @ A2 @ B2 )
= A2 ) ) ).
% Diff_triv
thf(fact_1130_Diff__triv,axiom,
! [A2: set_v,B2: set_v] :
( ( ( inf_inf_set_v @ A2 @ B2 )
= bot_bot_set_v )
=> ( ( minus_minus_set_v @ A2 @ B2 )
= A2 ) ) ).
% Diff_triv
thf(fact_1131_Un__Int__assoc__eq,axiom,
! [A2: set_set_v,B2: set_set_v,C2: set_set_v] :
( ( ( sup_sup_set_set_v @ ( inf_inf_set_set_v @ A2 @ B2 ) @ C2 )
= ( inf_inf_set_set_v @ A2 @ ( sup_sup_set_set_v @ B2 @ C2 ) ) )
= ( ord_le5216385588623774835_set_v @ C2 @ A2 ) ) ).
% Un_Int_assoc_eq
thf(fact_1132_Un__Int__assoc__eq,axiom,
! [A2: set_Product_prod_v_v,B2: set_Product_prod_v_v,C2: set_Product_prod_v_v] :
( ( ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ A2 @ B2 ) @ C2 )
= ( inf_in6271465464967711157od_v_v @ A2 @ ( sup_su414716646722978715od_v_v @ B2 @ C2 ) ) )
= ( ord_le7336532860387713383od_v_v @ C2 @ A2 ) ) ).
% Un_Int_assoc_eq
thf(fact_1133_Un__Int__assoc__eq,axiom,
! [A2: set_v,B2: set_v,C2: set_v] :
( ( ( sup_sup_set_v @ ( inf_inf_set_v @ A2 @ B2 ) @ C2 )
= ( inf_inf_set_v @ A2 @ ( sup_sup_set_v @ B2 @ C2 ) ) )
= ( ord_less_eq_set_v @ C2 @ A2 ) ) ).
% Un_Int_assoc_eq
thf(fact_1134_Un__Diff__Int,axiom,
! [A2: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ ( minus_4183494784930505774od_v_v @ A2 @ B2 ) @ ( inf_in6271465464967711157od_v_v @ A2 @ B2 ) )
= A2 ) ).
% Un_Diff_Int
thf(fact_1135_Un__Diff__Int,axiom,
! [A2: set_set_v,B2: set_set_v] :
( ( sup_sup_set_set_v @ ( minus_7228012346218142266_set_v @ A2 @ B2 ) @ ( inf_inf_set_set_v @ A2 @ B2 ) )
= A2 ) ).
% Un_Diff_Int
thf(fact_1136_Un__Diff__Int,axiom,
! [A2: set_v,B2: set_v] :
( ( sup_sup_set_v @ ( minus_minus_set_v @ A2 @ B2 ) @ ( inf_inf_set_v @ A2 @ B2 ) )
= A2 ) ).
% Un_Diff_Int
thf(fact_1137_Int__Diff__Un,axiom,
! [A2: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ A2 @ B2 ) @ ( minus_4183494784930505774od_v_v @ A2 @ B2 ) )
= A2 ) ).
% Int_Diff_Un
thf(fact_1138_Int__Diff__Un,axiom,
! [A2: set_set_v,B2: set_set_v] :
( ( sup_sup_set_set_v @ ( inf_inf_set_set_v @ A2 @ B2 ) @ ( minus_7228012346218142266_set_v @ A2 @ B2 ) )
= A2 ) ).
% Int_Diff_Un
thf(fact_1139_Int__Diff__Un,axiom,
! [A2: set_v,B2: set_v] :
( ( sup_sup_set_v @ ( inf_inf_set_v @ A2 @ B2 ) @ ( minus_minus_set_v @ A2 @ B2 ) )
= A2 ) ).
% Int_Diff_Un
thf(fact_1140_Diff__Int,axiom,
! [A2: set_Product_prod_v_v,B2: set_Product_prod_v_v,C2: set_Product_prod_v_v] :
( ( minus_4183494784930505774od_v_v @ A2 @ ( inf_in6271465464967711157od_v_v @ B2 @ C2 ) )
= ( sup_su414716646722978715od_v_v @ ( minus_4183494784930505774od_v_v @ A2 @ B2 ) @ ( minus_4183494784930505774od_v_v @ A2 @ C2 ) ) ) ).
% Diff_Int
thf(fact_1141_Diff__Int,axiom,
! [A2: set_set_v,B2: set_set_v,C2: set_set_v] :
( ( minus_7228012346218142266_set_v @ A2 @ ( inf_inf_set_set_v @ B2 @ C2 ) )
= ( sup_sup_set_set_v @ ( minus_7228012346218142266_set_v @ A2 @ B2 ) @ ( minus_7228012346218142266_set_v @ A2 @ C2 ) ) ) ).
% Diff_Int
thf(fact_1142_Diff__Int,axiom,
! [A2: set_v,B2: set_v,C2: set_v] :
( ( minus_minus_set_v @ A2 @ ( inf_inf_set_v @ B2 @ C2 ) )
= ( sup_sup_set_v @ ( minus_minus_set_v @ A2 @ B2 ) @ ( minus_minus_set_v @ A2 @ C2 ) ) ) ).
% Diff_Int
thf(fact_1143_Diff__Un,axiom,
! [A2: set_Product_prod_v_v,B2: set_Product_prod_v_v,C2: set_Product_prod_v_v] :
( ( minus_4183494784930505774od_v_v @ A2 @ ( sup_su414716646722978715od_v_v @ B2 @ C2 ) )
= ( inf_in6271465464967711157od_v_v @ ( minus_4183494784930505774od_v_v @ A2 @ B2 ) @ ( minus_4183494784930505774od_v_v @ A2 @ C2 ) ) ) ).
% Diff_Un
thf(fact_1144_Diff__Un,axiom,
! [A2: set_set_v,B2: set_set_v,C2: set_set_v] :
( ( minus_7228012346218142266_set_v @ A2 @ ( sup_sup_set_set_v @ B2 @ C2 ) )
= ( inf_inf_set_set_v @ ( minus_7228012346218142266_set_v @ A2 @ B2 ) @ ( minus_7228012346218142266_set_v @ A2 @ C2 ) ) ) ).
% Diff_Un
thf(fact_1145_Diff__Un,axiom,
! [A2: set_v,B2: set_v,C2: set_v] :
( ( minus_minus_set_v @ A2 @ ( sup_sup_set_v @ B2 @ C2 ) )
= ( inf_inf_set_v @ ( minus_minus_set_v @ A2 @ B2 ) @ ( minus_minus_set_v @ A2 @ C2 ) ) ) ).
% Diff_Un
thf(fact_1146_graph_Oscc__partition,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,S3: set_Product_prod_v_v,S5: set_Product_prod_v_v,X: product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( sCC_Bl6242042402218619277od_v_v @ Successors @ S3 )
=> ( ( sCC_Bl6242042402218619277od_v_v @ Successors @ S5 )
=> ( ( member7453568604450474000od_v_v @ X @ ( inf_in6271465464967711157od_v_v @ S3 @ S5 ) )
=> ( S3 = S5 ) ) ) ) ) ).
% graph.scc_partition
thf(fact_1147_graph_Oscc__partition,axiom,
! [Vertices: set_v,Successors: v > set_v,S3: set_v,S5: set_v,X: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bloemen_is_scc_v @ Successors @ S3 )
=> ( ( sCC_Bloemen_is_scc_v @ Successors @ S5 )
=> ( ( member_v @ X @ ( inf_inf_set_v @ S3 @ S5 ) )
=> ( S3 = S5 ) ) ) ) ) ).
% graph.scc_partition
thf(fact_1148_select__convs_I6_J,axiom,
! [Root: v,S: v > set_v,Explored: set_v,Visited: set_v,Vsuccs: v > set_v,Sccs: set_set_v,Stack: list_v,Cstack: list_v,More: product_unit] :
( ( sCC_Bl2536197123907397897t_unit @ ( sCC_Bl8064756265740546429t_unit @ Root @ S @ Explored @ Visited @ Vsuccs @ Sccs @ Stack @ Cstack @ More ) )
= Sccs ) ).
% select_convs(6)
thf(fact_1149_the__elem__set,axiom,
! [X: v] :
( ( the_elem_v @ ( set_v2 @ ( cons_v @ X @ nil_v ) ) )
= X ) ).
% the_elem_set
thf(fact_1150_is__singleton__the__elem,axiom,
( is_sin9198872032823709915od_v_v
= ( ^ [A7: set_Product_prod_v_v] :
( A7
= ( insert1338601472111419319od_v_v @ ( the_el5392834299063928540od_v_v @ A7 ) @ bot_bo723834152578015283od_v_v ) ) ) ) ).
% is_singleton_the_elem
thf(fact_1151_is__singleton__the__elem,axiom,
( is_singleton_v
= ( ^ [A7: set_v] :
( A7
= ( insert_v2 @ ( the_elem_v @ A7 ) @ bot_bot_set_v ) ) ) ) ).
% is_singleton_the_elem
thf(fact_1152_is__singleton__the__elem,axiom,
( is_singleton_set_v
= ( ^ [A7: set_set_v] :
( A7
= ( insert_set_v2 @ ( the_elem_set_v @ A7 ) @ bot_bot_set_set_v ) ) ) ) ).
% is_singleton_the_elem
thf(fact_1153_less__by__empty,axiom,
! [A2: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( A2 = bot_bo723834152578015283od_v_v )
=> ( ord_le7336532860387713383od_v_v @ A2 @ B2 ) ) ).
% less_by_empty
thf(fact_1154_set__remove1__eq,axiom,
! [Xs: list_P7986770385144383213od_v_v,X: product_prod_v_v] :
( ( distin6159370996967099744od_v_v @ Xs )
=> ( ( set_Product_prod_v_v2 @ ( remove333779696311199107od_v_v @ X @ Xs ) )
= ( minus_4183494784930505774od_v_v @ ( set_Product_prod_v_v2 @ Xs ) @ ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v ) ) ) ) ).
% set_remove1_eq
thf(fact_1155_set__remove1__eq,axiom,
! [Xs: list_set_v,X: set_v] :
( ( distinct_set_v @ Xs )
=> ( ( set_set_v2 @ ( remove1_set_v @ X @ Xs ) )
= ( minus_7228012346218142266_set_v @ ( set_set_v2 @ Xs ) @ ( insert_set_v2 @ X @ bot_bot_set_set_v ) ) ) ) ).
% set_remove1_eq
thf(fact_1156_set__remove1__eq,axiom,
! [Xs: list_v,X: v] :
( ( distinct_v @ Xs )
=> ( ( set_v2 @ ( remove1_v @ X @ Xs ) )
= ( minus_minus_set_v @ ( set_v2 @ Xs ) @ ( insert_v2 @ X @ bot_bot_set_v ) ) ) ) ).
% set_remove1_eq
thf(fact_1157_in__set__remove1,axiom,
! [A: product_prod_v_v,B: product_prod_v_v,Xs: list_P7986770385144383213od_v_v] :
( ( A != B )
=> ( ( member7453568604450474000od_v_v @ A @ ( set_Product_prod_v_v2 @ ( remove333779696311199107od_v_v @ B @ Xs ) ) )
= ( member7453568604450474000od_v_v @ A @ ( set_Product_prod_v_v2 @ Xs ) ) ) ) ).
% in_set_remove1
thf(fact_1158_in__set__remove1,axiom,
! [A: v,B: v,Xs: list_v] :
( ( A != B )
=> ( ( member_v @ A @ ( set_v2 @ ( remove1_v @ B @ Xs ) ) )
= ( member_v @ A @ ( set_v2 @ Xs ) ) ) ) ).
% in_set_remove1
thf(fact_1159_is__singletonI,axiom,
! [X: product_prod_v_v] : ( is_sin9198872032823709915od_v_v @ ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v ) ) ).
% is_singletonI
thf(fact_1160_is__singletonI,axiom,
! [X: v] : ( is_singleton_v @ ( insert_v2 @ X @ bot_bot_set_v ) ) ).
% is_singletonI
thf(fact_1161_is__singletonI,axiom,
! [X: set_v] : ( is_singleton_set_v @ ( insert_set_v2 @ X @ bot_bot_set_set_v ) ) ).
% is_singletonI
thf(fact_1162_remove1_Osimps_I2_J,axiom,
! [X: v,Y: v,Xs: list_v] :
( ( ( X = Y )
=> ( ( remove1_v @ X @ ( cons_v @ Y @ Xs ) )
= Xs ) )
& ( ( X != Y )
=> ( ( remove1_v @ X @ ( cons_v @ Y @ Xs ) )
= ( cons_v @ Y @ ( remove1_v @ X @ Xs ) ) ) ) ) ).
% remove1.simps(2)
thf(fact_1163_remove1_Osimps_I1_J,axiom,
! [X: v] :
( ( remove1_v @ X @ nil_v )
= nil_v ) ).
% remove1.simps(1)
thf(fact_1164_notin__set__remove1,axiom,
! [X: product_prod_v_v,Xs: list_P7986770385144383213od_v_v,Y: product_prod_v_v] :
( ~ ( member7453568604450474000od_v_v @ X @ ( set_Product_prod_v_v2 @ Xs ) )
=> ~ ( member7453568604450474000od_v_v @ X @ ( set_Product_prod_v_v2 @ ( remove333779696311199107od_v_v @ Y @ Xs ) ) ) ) ).
% notin_set_remove1
thf(fact_1165_notin__set__remove1,axiom,
! [X: v,Xs: list_v,Y: v] :
( ~ ( member_v @ X @ ( set_v2 @ Xs ) )
=> ~ ( member_v @ X @ ( set_v2 @ ( remove1_v @ Y @ Xs ) ) ) ) ).
% notin_set_remove1
thf(fact_1166_remove1__idem,axiom,
! [X: product_prod_v_v,Xs: list_P7986770385144383213od_v_v] :
( ~ ( member7453568604450474000od_v_v @ X @ ( set_Product_prod_v_v2 @ Xs ) )
=> ( ( remove333779696311199107od_v_v @ X @ Xs )
= Xs ) ) ).
% remove1_idem
thf(fact_1167_remove1__idem,axiom,
! [X: v,Xs: list_v] :
( ~ ( member_v @ X @ ( set_v2 @ Xs ) )
=> ( ( remove1_v @ X @ Xs )
= Xs ) ) ).
% remove1_idem
thf(fact_1168_distinct__remove1,axiom,
! [Xs: list_v,X: v] :
( ( distinct_v @ Xs )
=> ( distinct_v @ ( remove1_v @ X @ Xs ) ) ) ).
% distinct_remove1
thf(fact_1169_set__remove1__subset,axiom,
! [X: product_prod_v_v,Xs: list_P7986770385144383213od_v_v] : ( ord_le7336532860387713383od_v_v @ ( set_Product_prod_v_v2 @ ( remove333779696311199107od_v_v @ X @ Xs ) ) @ ( set_Product_prod_v_v2 @ Xs ) ) ).
% set_remove1_subset
thf(fact_1170_set__remove1__subset,axiom,
! [X: v,Xs: list_v] : ( ord_less_eq_set_v @ ( set_v2 @ ( remove1_v @ X @ Xs ) ) @ ( set_v2 @ Xs ) ) ).
% set_remove1_subset
thf(fact_1171_remove1__append,axiom,
! [X: product_prod_v_v,Xs: list_P7986770385144383213od_v_v,Ys: list_P7986770385144383213od_v_v] :
( ( ( member7453568604450474000od_v_v @ X @ ( set_Product_prod_v_v2 @ Xs ) )
=> ( ( remove333779696311199107od_v_v @ X @ ( append2138873909117096322od_v_v @ Xs @ Ys ) )
= ( append2138873909117096322od_v_v @ ( remove333779696311199107od_v_v @ X @ Xs ) @ Ys ) ) )
& ( ~ ( member7453568604450474000od_v_v @ X @ ( set_Product_prod_v_v2 @ Xs ) )
=> ( ( remove333779696311199107od_v_v @ X @ ( append2138873909117096322od_v_v @ Xs @ Ys ) )
= ( append2138873909117096322od_v_v @ Xs @ ( remove333779696311199107od_v_v @ X @ Ys ) ) ) ) ) ).
% remove1_append
thf(fact_1172_remove1__append,axiom,
! [X: v,Xs: list_v,Ys: list_v] :
( ( ( member_v @ X @ ( set_v2 @ Xs ) )
=> ( ( remove1_v @ X @ ( append_v @ Xs @ Ys ) )
= ( append_v @ ( remove1_v @ X @ Xs ) @ Ys ) ) )
& ( ~ ( member_v @ X @ ( set_v2 @ Xs ) )
=> ( ( remove1_v @ X @ ( append_v @ Xs @ Ys ) )
= ( append_v @ Xs @ ( remove1_v @ X @ Ys ) ) ) ) ) ).
% remove1_append
thf(fact_1173_is__singletonI_H,axiom,
! [A2: set_Product_prod_v_v] :
( ( A2 != bot_bo723834152578015283od_v_v )
=> ( ! [X3: product_prod_v_v,Y2: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X3 @ A2 )
=> ( ( member7453568604450474000od_v_v @ Y2 @ A2 )
=> ( X3 = Y2 ) ) )
=> ( is_sin9198872032823709915od_v_v @ A2 ) ) ) ).
% is_singletonI'
thf(fact_1174_is__singletonI_H,axiom,
! [A2: set_v] :
( ( A2 != bot_bot_set_v )
=> ( ! [X3: v,Y2: v] :
( ( member_v @ X3 @ A2 )
=> ( ( member_v @ Y2 @ A2 )
=> ( X3 = Y2 ) ) )
=> ( is_singleton_v @ A2 ) ) ) ).
% is_singletonI'
thf(fact_1175_is__singletonI_H,axiom,
! [A2: set_set_v] :
( ( A2 != bot_bot_set_set_v )
=> ( ! [X3: set_v,Y2: set_v] :
( ( member_set_v @ X3 @ A2 )
=> ( ( member_set_v @ Y2 @ A2 )
=> ( X3 = Y2 ) ) )
=> ( is_singleton_set_v @ A2 ) ) ) ).
% is_singletonI'
thf(fact_1176_remove1__split,axiom,
! [A: product_prod_v_v,Xs: list_P7986770385144383213od_v_v,Ys: list_P7986770385144383213od_v_v] :
( ( member7453568604450474000od_v_v @ A @ ( set_Product_prod_v_v2 @ Xs ) )
=> ( ( ( remove333779696311199107od_v_v @ A @ Xs )
= Ys )
= ( ? [Ls: list_P7986770385144383213od_v_v,Rs: list_P7986770385144383213od_v_v] :
( ( Xs
= ( append2138873909117096322od_v_v @ Ls @ ( cons_P4120604216776828829od_v_v @ A @ Rs ) ) )
& ~ ( member7453568604450474000od_v_v @ A @ ( set_Product_prod_v_v2 @ Ls ) )
& ( Ys
= ( append2138873909117096322od_v_v @ Ls @ Rs ) ) ) ) ) ) ).
% remove1_split
thf(fact_1177_remove1__split,axiom,
! [A: v,Xs: list_v,Ys: list_v] :
( ( member_v @ A @ ( set_v2 @ Xs ) )
=> ( ( ( remove1_v @ A @ Xs )
= Ys )
= ( ? [Ls: list_v,Rs: list_v] :
( ( Xs
= ( append_v @ Ls @ ( cons_v @ A @ Rs ) ) )
& ~ ( member_v @ A @ ( set_v2 @ Ls ) )
& ( Ys
= ( append_v @ Ls @ Rs ) ) ) ) ) ) ).
% remove1_split
thf(fact_1178_is__singletonE,axiom,
! [A2: set_Product_prod_v_v] :
( ( is_sin9198872032823709915od_v_v @ A2 )
=> ~ ! [X3: product_prod_v_v] :
( A2
!= ( insert1338601472111419319od_v_v @ X3 @ bot_bo723834152578015283od_v_v ) ) ) ).
% is_singletonE
thf(fact_1179_is__singletonE,axiom,
! [A2: set_v] :
( ( is_singleton_v @ A2 )
=> ~ ! [X3: v] :
( A2
!= ( insert_v2 @ X3 @ bot_bot_set_v ) ) ) ).
% is_singletonE
thf(fact_1180_is__singletonE,axiom,
! [A2: set_set_v] :
( ( is_singleton_set_v @ A2 )
=> ~ ! [X3: set_v] :
( A2
!= ( insert_set_v2 @ X3 @ bot_bot_set_set_v ) ) ) ).
% is_singletonE
thf(fact_1181_is__singleton__def,axiom,
( is_sin9198872032823709915od_v_v
= ( ^ [A7: set_Product_prod_v_v] :
? [X2: product_prod_v_v] :
( A7
= ( insert1338601472111419319od_v_v @ X2 @ bot_bo723834152578015283od_v_v ) ) ) ) ).
% is_singleton_def
thf(fact_1182_is__singleton__def,axiom,
( is_singleton_v
= ( ^ [A7: set_v] :
? [X2: v] :
( A7
= ( insert_v2 @ X2 @ bot_bot_set_v ) ) ) ) ).
% is_singleton_def
thf(fact_1183_is__singleton__def,axiom,
( is_singleton_set_v
= ( ^ [A7: set_set_v] :
? [X2: set_v] :
( A7
= ( insert_set_v2 @ X2 @ bot_bot_set_set_v ) ) ) ) ).
% is_singleton_def
thf(fact_1184_vfin,axiom,
finite_finite_v @ vertices ).
% vfin
thf(fact_1185_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_1186_set__removeAll,axiom,
! [X: product_prod_v_v,Xs: list_P7986770385144383213od_v_v] :
( ( set_Product_prod_v_v2 @ ( remove481895986417801203od_v_v @ X @ Xs ) )
= ( minus_4183494784930505774od_v_v @ ( set_Product_prod_v_v2 @ Xs ) @ ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v ) ) ) ).
% set_removeAll
thf(fact_1187_set__removeAll,axiom,
! [X: set_v,Xs: list_set_v] :
( ( set_set_v2 @ ( removeAll_set_v @ X @ Xs ) )
= ( minus_7228012346218142266_set_v @ ( set_set_v2 @ Xs ) @ ( insert_set_v2 @ X @ bot_bot_set_set_v ) ) ) ).
% set_removeAll
thf(fact_1188_set__removeAll,axiom,
! [X: v,Xs: list_v] :
( ( set_v2 @ ( removeAll_v @ X @ Xs ) )
= ( minus_minus_set_v @ ( set_v2 @ Xs ) @ ( insert_v2 @ X @ bot_bot_set_v ) ) ) ).
% set_removeAll
thf(fact_1189_List_Ofinite__set,axiom,
! [Xs: list_v] : ( finite_finite_v @ ( set_v2 @ Xs ) ) ).
% List.finite_set
thf(fact_1190_removeAll__id,axiom,
! [X: product_prod_v_v,Xs: list_P7986770385144383213od_v_v] :
( ~ ( member7453568604450474000od_v_v @ X @ ( set_Product_prod_v_v2 @ Xs ) )
=> ( ( remove481895986417801203od_v_v @ X @ Xs )
= Xs ) ) ).
% removeAll_id
thf(fact_1191_removeAll__id,axiom,
! [X: v,Xs: list_v] :
( ~ ( member_v @ X @ ( set_v2 @ Xs ) )
=> ( ( removeAll_v @ X @ Xs )
= Xs ) ) ).
% removeAll_id
thf(fact_1192_rotate1__is__Nil__conv,axiom,
! [Xs: list_v] :
( ( ( rotate1_v @ Xs )
= nil_v )
= ( Xs = nil_v ) ) ).
% rotate1_is_Nil_conv
thf(fact_1193_set__rotate1,axiom,
! [Xs: list_v] :
( ( set_v2 @ ( rotate1_v @ Xs ) )
= ( set_v2 @ Xs ) ) ).
% set_rotate1
thf(fact_1194_removeAll__append,axiom,
! [X: v,Xs: list_v,Ys: list_v] :
( ( removeAll_v @ X @ ( append_v @ Xs @ Ys ) )
= ( append_v @ ( removeAll_v @ X @ Xs ) @ ( removeAll_v @ X @ Ys ) ) ) ).
% removeAll_append
thf(fact_1195_distinct1__rotate,axiom,
! [Xs: list_v] :
( ( distinct_v @ ( rotate1_v @ Xs ) )
= ( distinct_v @ Xs ) ) ).
% distinct1_rotate
thf(fact_1196_removeAll_Osimps_I2_J,axiom,
! [X: v,Y: v,Xs: list_v] :
( ( ( X = Y )
=> ( ( removeAll_v @ X @ ( cons_v @ Y @ Xs ) )
= ( removeAll_v @ X @ Xs ) ) )
& ( ( X != Y )
=> ( ( removeAll_v @ X @ ( cons_v @ Y @ Xs ) )
= ( cons_v @ Y @ ( removeAll_v @ X @ Xs ) ) ) ) ) ).
% removeAll.simps(2)
thf(fact_1197_removeAll_Osimps_I1_J,axiom,
! [X: v] :
( ( removeAll_v @ X @ nil_v )
= nil_v ) ).
% removeAll.simps(1)
thf(fact_1198_rotate1_Osimps_I1_J,axiom,
( ( rotate1_v @ nil_v )
= nil_v ) ).
% rotate1.simps(1)
thf(fact_1199_distinct__removeAll,axiom,
! [Xs: list_v,X: v] :
( ( distinct_v @ Xs )
=> ( distinct_v @ ( removeAll_v @ X @ Xs ) ) ) ).
% distinct_removeAll
thf(fact_1200_graph_Ovfin,axiom,
! [Vertices: set_v,Successors: v > set_v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( finite_finite_v @ Vertices ) ) ).
% graph.vfin
thf(fact_1201_finite__list,axiom,
! [A2: set_v] :
( ( finite_finite_v @ A2 )
=> ? [Xs3: list_v] :
( ( set_v2 @ Xs3 )
= A2 ) ) ).
% finite_list
thf(fact_1202_finite__distinct__list,axiom,
! [A2: set_v] :
( ( finite_finite_v @ A2 )
=> ? [Xs3: list_v] :
( ( ( set_v2 @ Xs3 )
= A2 )
& ( distinct_v @ Xs3 ) ) ) ).
% finite_distinct_list
thf(fact_1203_graph_Ointro,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v] :
( ( finite3348123685078250256od_v_v @ Vertices )
=> ( ! [X3: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X3 @ Vertices )
=> ( ord_le7336532860387713383od_v_v @ ( Successors @ X3 ) @ Vertices ) )
=> ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors ) ) ) ).
% graph.intro
thf(fact_1204_graph_Ointro,axiom,
! [Vertices: set_v,Successors: v > set_v] :
( ( finite_finite_v @ Vertices )
=> ( ! [X3: v] :
( ( member_v @ X3 @ Vertices )
=> ( ord_less_eq_set_v @ ( Successors @ X3 ) @ Vertices ) )
=> ( sCC_Bloemen_graph_v @ Vertices @ Successors ) ) ) ).
% graph.intro
thf(fact_1205_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 )
& ! [X2: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X2 @ Vertices2 )
=> ( ord_le7336532860387713383od_v_v @ ( Successors2 @ X2 ) @ Vertices2 ) ) ) ) ) ).
% SCC_Bloemen_Sequential.graph_def
thf(fact_1206_SCC__Bloemen__Sequential_Ograph__def,axiom,
( sCC_Bloemen_graph_v
= ( ^ [Vertices2: set_v,Successors2: v > set_v] :
( ( finite_finite_v @ Vertices2 )
& ! [X2: v] :
( ( member_v @ X2 @ Vertices2 )
=> ( ord_less_eq_set_v @ ( Successors2 @ X2 ) @ Vertices2 ) ) ) ) ) ).
% SCC_Bloemen_Sequential.graph_def
thf(fact_1207_distinct__remove1__removeAll,axiom,
! [Xs: list_v,X: v] :
( ( distinct_v @ Xs )
=> ( ( remove1_v @ X @ Xs )
= ( removeAll_v @ X @ Xs ) ) ) ).
% distinct_remove1_removeAll
thf(fact_1208_rotate1_Osimps_I2_J,axiom,
! [X: v,Xs: list_v] :
( ( rotate1_v @ ( cons_v @ X @ Xs ) )
= ( append_v @ Xs @ ( cons_v @ X @ nil_v ) ) ) ).
% rotate1.simps(2)
thf(fact_1209_finite__Diff__insert,axiom,
! [A2: set_Product_prod_v_v,A: product_prod_v_v,B2: set_Product_prod_v_v] :
( ( finite3348123685078250256od_v_v @ ( minus_4183494784930505774od_v_v @ A2 @ ( insert1338601472111419319od_v_v @ A @ B2 ) ) )
= ( finite3348123685078250256od_v_v @ ( minus_4183494784930505774od_v_v @ A2 @ B2 ) ) ) ).
% finite_Diff_insert
thf(fact_1210_finite__Diff__insert,axiom,
! [A2: set_set_v,A: set_v,B2: set_set_v] :
( ( finite_finite_set_v @ ( minus_7228012346218142266_set_v @ A2 @ ( insert_set_v2 @ A @ B2 ) ) )
= ( finite_finite_set_v @ ( minus_7228012346218142266_set_v @ A2 @ B2 ) ) ) ).
% finite_Diff_insert
thf(fact_1211_finite__Diff__insert,axiom,
! [A2: set_v,A: v,B2: set_v] :
( ( finite_finite_v @ ( minus_minus_set_v @ A2 @ ( insert_v2 @ A @ B2 ) ) )
= ( finite_finite_v @ ( minus_minus_set_v @ A2 @ B2 ) ) ) ).
% finite_Diff_insert
thf(fact_1212_finite__Diff,axiom,
! [A2: set_v,B2: set_v] :
( ( finite_finite_v @ A2 )
=> ( finite_finite_v @ ( minus_minus_set_v @ A2 @ B2 ) ) ) ).
% finite_Diff
thf(fact_1213_finite__Diff2,axiom,
! [B2: set_v,A2: set_v] :
( ( finite_finite_v @ B2 )
=> ( ( finite_finite_v @ ( minus_minus_set_v @ A2 @ B2 ) )
= ( finite_finite_v @ A2 ) ) ) ).
% finite_Diff2
thf(fact_1214_finite__has__maximal2,axiom,
! [A2: set_se8455005133513928103od_v_v,A: set_Product_prod_v_v] :
( ( finite6084192165098772208od_v_v @ A2 )
=> ( ( member8406446414694345712od_v_v @ A @ A2 )
=> ? [X3: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ X3 @ A2 )
& ( ord_le7336532860387713383od_v_v @ A @ X3 )
& ! [Xa2: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ Xa2 @ A2 )
=> ( ( ord_le7336532860387713383od_v_v @ X3 @ Xa2 )
=> ( X3 = Xa2 ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_1215_finite__has__maximal2,axiom,
! [A2: set_set_v,A: set_v] :
( ( finite_finite_set_v @ A2 )
=> ( ( member_set_v @ A @ A2 )
=> ? [X3: set_v] :
( ( member_set_v @ X3 @ A2 )
& ( ord_less_eq_set_v @ A @ X3 )
& ! [Xa2: set_v] :
( ( member_set_v @ Xa2 @ A2 )
=> ( ( ord_less_eq_set_v @ X3 @ Xa2 )
=> ( X3 = Xa2 ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_1216_finite__has__minimal2,axiom,
! [A2: set_se8455005133513928103od_v_v,A: set_Product_prod_v_v] :
( ( finite6084192165098772208od_v_v @ A2 )
=> ( ( member8406446414694345712od_v_v @ A @ A2 )
=> ? [X3: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ X3 @ A2 )
& ( ord_le7336532860387713383od_v_v @ X3 @ A )
& ! [Xa2: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ Xa2 @ A2 )
=> ( ( ord_le7336532860387713383od_v_v @ Xa2 @ X3 )
=> ( X3 = Xa2 ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_1217_finite__has__minimal2,axiom,
! [A2: set_set_v,A: set_v] :
( ( finite_finite_set_v @ A2 )
=> ( ( member_set_v @ A @ A2 )
=> ? [X3: set_v] :
( ( member_set_v @ X3 @ A2 )
& ( ord_less_eq_set_v @ X3 @ A )
& ! [Xa2: set_v] :
( ( member_set_v @ Xa2 @ A2 )
=> ( ( ord_less_eq_set_v @ Xa2 @ X3 )
=> ( X3 = Xa2 ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_1218_finite_OemptyI,axiom,
finite3348123685078250256od_v_v @ bot_bo723834152578015283od_v_v ).
% finite.emptyI
thf(fact_1219_finite_OemptyI,axiom,
finite_finite_v @ bot_bot_set_v ).
% finite.emptyI
thf(fact_1220_finite_OemptyI,axiom,
finite_finite_set_v @ bot_bot_set_set_v ).
% finite.emptyI
thf(fact_1221_infinite__imp__nonempty,axiom,
! [S3: set_Product_prod_v_v] :
( ~ ( finite3348123685078250256od_v_v @ S3 )
=> ( S3 != bot_bo723834152578015283od_v_v ) ) ).
% infinite_imp_nonempty
thf(fact_1222_infinite__imp__nonempty,axiom,
! [S3: set_v] :
( ~ ( finite_finite_v @ S3 )
=> ( S3 != bot_bot_set_v ) ) ).
% infinite_imp_nonempty
thf(fact_1223_infinite__imp__nonempty,axiom,
! [S3: set_set_v] :
( ~ ( finite_finite_set_v @ S3 )
=> ( S3 != bot_bot_set_set_v ) ) ).
% infinite_imp_nonempty
thf(fact_1224_finite__subset,axiom,
! [A2: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A2 @ B2 )
=> ( ( finite3348123685078250256od_v_v @ B2 )
=> ( finite3348123685078250256od_v_v @ A2 ) ) ) ).
% finite_subset
thf(fact_1225_finite__subset,axiom,
! [A2: set_v,B2: set_v] :
( ( ord_less_eq_set_v @ A2 @ B2 )
=> ( ( finite_finite_v @ B2 )
=> ( finite_finite_v @ A2 ) ) ) ).
% finite_subset
thf(fact_1226_infinite__super,axiom,
! [S3: set_Product_prod_v_v,T2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ S3 @ T2 )
=> ( ~ ( finite3348123685078250256od_v_v @ S3 )
=> ~ ( finite3348123685078250256od_v_v @ T2 ) ) ) ).
% infinite_super
thf(fact_1227_infinite__super,axiom,
! [S3: set_v,T2: set_v] :
( ( ord_less_eq_set_v @ S3 @ T2 )
=> ( ~ ( finite_finite_v @ S3 )
=> ~ ( finite_finite_v @ T2 ) ) ) ).
% infinite_super
thf(fact_1228_rev__finite__subset,axiom,
! [B2: set_Product_prod_v_v,A2: set_Product_prod_v_v] :
( ( finite3348123685078250256od_v_v @ B2 )
=> ( ( ord_le7336532860387713383od_v_v @ A2 @ B2 )
=> ( finite3348123685078250256od_v_v @ A2 ) ) ) ).
% rev_finite_subset
thf(fact_1229_rev__finite__subset,axiom,
! [B2: set_v,A2: set_v] :
( ( finite_finite_v @ B2 )
=> ( ( ord_less_eq_set_v @ A2 @ B2 )
=> ( finite_finite_v @ A2 ) ) ) ).
% rev_finite_subset
thf(fact_1230_Diff__infinite__finite,axiom,
! [T2: set_v,S3: set_v] :
( ( finite_finite_v @ T2 )
=> ( ~ ( finite_finite_v @ S3 )
=> ~ ( finite_finite_v @ ( minus_minus_set_v @ S3 @ T2 ) ) ) ) ).
% Diff_infinite_finite
thf(fact_1231_finite__has__maximal,axiom,
! [A2: set_se8455005133513928103od_v_v] :
( ( finite6084192165098772208od_v_v @ A2 )
=> ( ( A2 != bot_bo3497076220358800403od_v_v )
=> ? [X3: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ X3 @ A2 )
& ! [Xa2: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ Xa2 @ A2 )
=> ( ( ord_le7336532860387713383od_v_v @ X3 @ Xa2 )
=> ( X3 = Xa2 ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_1232_finite__has__maximal,axiom,
! [A2: set_set_v] :
( ( finite_finite_set_v @ A2 )
=> ( ( A2 != bot_bot_set_set_v )
=> ? [X3: set_v] :
( ( member_set_v @ X3 @ A2 )
& ! [Xa2: set_v] :
( ( member_set_v @ Xa2 @ A2 )
=> ( ( ord_less_eq_set_v @ X3 @ Xa2 )
=> ( X3 = Xa2 ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_1233_finite__has__minimal,axiom,
! [A2: set_se8455005133513928103od_v_v] :
( ( finite6084192165098772208od_v_v @ A2 )
=> ( ( A2 != bot_bo3497076220358800403od_v_v )
=> ? [X3: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ X3 @ A2 )
& ! [Xa2: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ Xa2 @ A2 )
=> ( ( ord_le7336532860387713383od_v_v @ Xa2 @ X3 )
=> ( X3 = Xa2 ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_1234_finite__has__minimal,axiom,
! [A2: set_set_v] :
( ( finite_finite_set_v @ A2 )
=> ( ( A2 != bot_bot_set_set_v )
=> ? [X3: set_v] :
( ( member_set_v @ X3 @ A2 )
& ! [Xa2: set_v] :
( ( member_set_v @ Xa2 @ A2 )
=> ( ( ord_less_eq_set_v @ Xa2 @ X3 )
=> ( X3 = Xa2 ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_1235_infinite__finite__induct,axiom,
! [P: set_Product_prod_v_v > $o,A2: set_Product_prod_v_v] :
( ! [A9: set_Product_prod_v_v] :
( ~ ( finite3348123685078250256od_v_v @ A9 )
=> ( P @ A9 ) )
=> ( ( P @ bot_bo723834152578015283od_v_v )
=> ( ! [X3: product_prod_v_v,F3: set_Product_prod_v_v] :
( ( finite3348123685078250256od_v_v @ F3 )
=> ( ~ ( member7453568604450474000od_v_v @ X3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert1338601472111419319od_v_v @ X3 @ F3 ) ) ) ) )
=> ( P @ A2 ) ) ) ) ).
% infinite_finite_induct
thf(fact_1236_infinite__finite__induct,axiom,
! [P: set_v > $o,A2: set_v] :
( ! [A9: set_v] :
( ~ ( finite_finite_v @ A9 )
=> ( P @ A9 ) )
=> ( ( P @ bot_bot_set_v )
=> ( ! [X3: v,F3: set_v] :
( ( finite_finite_v @ F3 )
=> ( ~ ( member_v @ X3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_v2 @ X3 @ F3 ) ) ) ) )
=> ( P @ A2 ) ) ) ) ).
% infinite_finite_induct
thf(fact_1237_infinite__finite__induct,axiom,
! [P: set_set_v > $o,A2: set_set_v] :
( ! [A9: set_set_v] :
( ~ ( finite_finite_set_v @ A9 )
=> ( P @ A9 ) )
=> ( ( P @ bot_bot_set_set_v )
=> ( ! [X3: set_v,F3: set_set_v] :
( ( finite_finite_set_v @ F3 )
=> ( ~ ( member_set_v @ X3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_set_v2 @ X3 @ F3 ) ) ) ) )
=> ( P @ A2 ) ) ) ) ).
% infinite_finite_induct
thf(fact_1238_finite__ne__induct,axiom,
! [F4: set_Product_prod_v_v,P: set_Product_prod_v_v > $o] :
( ( finite3348123685078250256od_v_v @ F4 )
=> ( ( F4 != bot_bo723834152578015283od_v_v )
=> ( ! [X3: product_prod_v_v] : ( P @ ( insert1338601472111419319od_v_v @ X3 @ bot_bo723834152578015283od_v_v ) )
=> ( ! [X3: product_prod_v_v,F3: set_Product_prod_v_v] :
( ( finite3348123685078250256od_v_v @ F3 )
=> ( ( F3 != bot_bo723834152578015283od_v_v )
=> ( ~ ( member7453568604450474000od_v_v @ X3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert1338601472111419319od_v_v @ X3 @ F3 ) ) ) ) ) )
=> ( P @ F4 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_1239_finite__ne__induct,axiom,
! [F4: set_v,P: set_v > $o] :
( ( finite_finite_v @ F4 )
=> ( ( F4 != bot_bot_set_v )
=> ( ! [X3: v] : ( P @ ( insert_v2 @ X3 @ bot_bot_set_v ) )
=> ( ! [X3: v,F3: set_v] :
( ( finite_finite_v @ F3 )
=> ( ( F3 != bot_bot_set_v )
=> ( ~ ( member_v @ X3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_v2 @ X3 @ F3 ) ) ) ) ) )
=> ( P @ F4 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_1240_finite__ne__induct,axiom,
! [F4: set_set_v,P: set_set_v > $o] :
( ( finite_finite_set_v @ F4 )
=> ( ( F4 != bot_bot_set_set_v )
=> ( ! [X3: set_v] : ( P @ ( insert_set_v2 @ X3 @ bot_bot_set_set_v ) )
=> ( ! [X3: set_v,F3: set_set_v] :
( ( finite_finite_set_v @ F3 )
=> ( ( F3 != bot_bot_set_set_v )
=> ( ~ ( member_set_v @ X3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_set_v2 @ X3 @ F3 ) ) ) ) ) )
=> ( P @ F4 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_1241_finite__induct,axiom,
! [F4: set_Product_prod_v_v,P: set_Product_prod_v_v > $o] :
( ( finite3348123685078250256od_v_v @ F4 )
=> ( ( P @ bot_bo723834152578015283od_v_v )
=> ( ! [X3: product_prod_v_v,F3: set_Product_prod_v_v] :
( ( finite3348123685078250256od_v_v @ F3 )
=> ( ~ ( member7453568604450474000od_v_v @ X3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert1338601472111419319od_v_v @ X3 @ F3 ) ) ) ) )
=> ( P @ F4 ) ) ) ) ).
% finite_induct
thf(fact_1242_finite__induct,axiom,
! [F4: set_v,P: set_v > $o] :
( ( finite_finite_v @ F4 )
=> ( ( P @ bot_bot_set_v )
=> ( ! [X3: v,F3: set_v] :
( ( finite_finite_v @ F3 )
=> ( ~ ( member_v @ X3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_v2 @ X3 @ F3 ) ) ) ) )
=> ( P @ F4 ) ) ) ) ).
% finite_induct
thf(fact_1243_finite__induct,axiom,
! [F4: set_set_v,P: set_set_v > $o] :
( ( finite_finite_set_v @ F4 )
=> ( ( P @ bot_bot_set_set_v )
=> ( ! [X3: set_v,F3: set_set_v] :
( ( finite_finite_set_v @ F3 )
=> ( ~ ( member_set_v @ X3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_set_v2 @ X3 @ F3 ) ) ) ) )
=> ( P @ F4 ) ) ) ) ).
% finite_induct
thf(fact_1244_finite_Osimps,axiom,
( finite3348123685078250256od_v_v
= ( ^ [A5: set_Product_prod_v_v] :
( ( A5 = bot_bo723834152578015283od_v_v )
| ? [A7: set_Product_prod_v_v,B4: product_prod_v_v] :
( ( A5
= ( insert1338601472111419319od_v_v @ B4 @ A7 ) )
& ( finite3348123685078250256od_v_v @ A7 ) ) ) ) ) ).
% finite.simps
thf(fact_1245_finite_Osimps,axiom,
( finite_finite_v
= ( ^ [A5: set_v] :
( ( A5 = bot_bot_set_v )
| ? [A7: set_v,B4: v] :
( ( A5
= ( insert_v2 @ B4 @ A7 ) )
& ( finite_finite_v @ A7 ) ) ) ) ) ).
% finite.simps
thf(fact_1246_finite_Osimps,axiom,
( finite_finite_set_v
= ( ^ [A5: set_set_v] :
( ( A5 = bot_bot_set_set_v )
| ? [A7: set_set_v,B4: set_v] :
( ( A5
= ( insert_set_v2 @ B4 @ A7 ) )
& ( finite_finite_set_v @ A7 ) ) ) ) ) ).
% finite.simps
thf(fact_1247_finite_Ocases,axiom,
! [A: set_Product_prod_v_v] :
( ( finite3348123685078250256od_v_v @ A )
=> ( ( A != bot_bo723834152578015283od_v_v )
=> ~ ! [A9: set_Product_prod_v_v] :
( ? [A6: product_prod_v_v] :
( A
= ( insert1338601472111419319od_v_v @ A6 @ A9 ) )
=> ~ ( finite3348123685078250256od_v_v @ A9 ) ) ) ) ).
% finite.cases
thf(fact_1248_finite_Ocases,axiom,
! [A: set_v] :
( ( finite_finite_v @ A )
=> ( ( A != bot_bot_set_v )
=> ~ ! [A9: set_v] :
( ? [A6: v] :
( A
= ( insert_v2 @ A6 @ A9 ) )
=> ~ ( finite_finite_v @ A9 ) ) ) ) ).
% finite.cases
thf(fact_1249_finite_Ocases,axiom,
! [A: set_set_v] :
( ( finite_finite_set_v @ A )
=> ( ( A != bot_bot_set_set_v )
=> ~ ! [A9: set_set_v] :
( ? [A6: set_v] :
( A
= ( insert_set_v2 @ A6 @ A9 ) )
=> ~ ( finite_finite_set_v @ A9 ) ) ) ) ).
% finite.cases
thf(fact_1250_finite__subset__induct,axiom,
! [F4: set_set_v,A2: set_set_v,P: set_set_v > $o] :
( ( finite_finite_set_v @ F4 )
=> ( ( ord_le5216385588623774835_set_v @ F4 @ A2 )
=> ( ( P @ bot_bot_set_set_v )
=> ( ! [A6: set_v,F3: set_set_v] :
( ( finite_finite_set_v @ F3 )
=> ( ( member_set_v @ A6 @ A2 )
=> ( ~ ( member_set_v @ A6 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_set_v2 @ A6 @ F3 ) ) ) ) ) )
=> ( P @ F4 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_1251_finite__subset__induct,axiom,
! [F4: set_Product_prod_v_v,A2: set_Product_prod_v_v,P: set_Product_prod_v_v > $o] :
( ( finite3348123685078250256od_v_v @ F4 )
=> ( ( ord_le7336532860387713383od_v_v @ F4 @ A2 )
=> ( ( P @ bot_bo723834152578015283od_v_v )
=> ( ! [A6: product_prod_v_v,F3: set_Product_prod_v_v] :
( ( finite3348123685078250256od_v_v @ F3 )
=> ( ( member7453568604450474000od_v_v @ A6 @ A2 )
=> ( ~ ( member7453568604450474000od_v_v @ A6 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert1338601472111419319od_v_v @ A6 @ F3 ) ) ) ) ) )
=> ( P @ F4 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_1252_finite__subset__induct,axiom,
! [F4: set_v,A2: set_v,P: set_v > $o] :
( ( finite_finite_v @ F4 )
=> ( ( ord_less_eq_set_v @ F4 @ A2 )
=> ( ( P @ bot_bot_set_v )
=> ( ! [A6: v,F3: set_v] :
( ( finite_finite_v @ F3 )
=> ( ( member_v @ A6 @ A2 )
=> ( ~ ( member_v @ A6 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_v2 @ A6 @ F3 ) ) ) ) ) )
=> ( P @ F4 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_1253_finite__subset__induct_H,axiom,
! [F4: set_set_v,A2: set_set_v,P: set_set_v > $o] :
( ( finite_finite_set_v @ F4 )
=> ( ( ord_le5216385588623774835_set_v @ F4 @ A2 )
=> ( ( P @ bot_bot_set_set_v )
=> ( ! [A6: set_v,F3: set_set_v] :
( ( finite_finite_set_v @ F3 )
=> ( ( member_set_v @ A6 @ A2 )
=> ( ( ord_le5216385588623774835_set_v @ F3 @ A2 )
=> ( ~ ( member_set_v @ A6 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_set_v2 @ A6 @ F3 ) ) ) ) ) ) )
=> ( P @ F4 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_1254_finite__subset__induct_H,axiom,
! [F4: set_Product_prod_v_v,A2: set_Product_prod_v_v,P: set_Product_prod_v_v > $o] :
( ( finite3348123685078250256od_v_v @ F4 )
=> ( ( ord_le7336532860387713383od_v_v @ F4 @ A2 )
=> ( ( P @ bot_bo723834152578015283od_v_v )
=> ( ! [A6: product_prod_v_v,F3: set_Product_prod_v_v] :
( ( finite3348123685078250256od_v_v @ F3 )
=> ( ( member7453568604450474000od_v_v @ A6 @ A2 )
=> ( ( ord_le7336532860387713383od_v_v @ F3 @ A2 )
=> ( ~ ( member7453568604450474000od_v_v @ A6 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert1338601472111419319od_v_v @ A6 @ F3 ) ) ) ) ) ) )
=> ( P @ F4 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_1255_finite__subset__induct_H,axiom,
! [F4: set_v,A2: set_v,P: set_v > $o] :
( ( finite_finite_v @ F4 )
=> ( ( ord_less_eq_set_v @ F4 @ A2 )
=> ( ( P @ bot_bot_set_v )
=> ( ! [A6: v,F3: set_v] :
( ( finite_finite_v @ F3 )
=> ( ( member_v @ A6 @ A2 )
=> ( ( ord_less_eq_set_v @ F3 @ A2 )
=> ( ~ ( member_v @ A6 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_v2 @ A6 @ F3 ) ) ) ) ) ) )
=> ( P @ F4 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_1256_finite__empty__induct,axiom,
! [A2: set_set_v,P: set_set_v > $o] :
( ( finite_finite_set_v @ A2 )
=> ( ( P @ A2 )
=> ( ! [A6: set_v,A9: set_set_v] :
( ( finite_finite_set_v @ A9 )
=> ( ( member_set_v @ A6 @ A9 )
=> ( ( P @ A9 )
=> ( P @ ( minus_7228012346218142266_set_v @ A9 @ ( insert_set_v2 @ A6 @ bot_bot_set_set_v ) ) ) ) ) )
=> ( P @ bot_bot_set_set_v ) ) ) ) ).
% finite_empty_induct
thf(fact_1257_finite__empty__induct,axiom,
! [A2: set_v,P: set_v > $o] :
( ( finite_finite_v @ A2 )
=> ( ( P @ A2 )
=> ( ! [A6: v,A9: set_v] :
( ( finite_finite_v @ A9 )
=> ( ( member_v @ A6 @ A9 )
=> ( ( P @ A9 )
=> ( P @ ( minus_minus_set_v @ A9 @ ( insert_v2 @ A6 @ bot_bot_set_v ) ) ) ) ) )
=> ( P @ bot_bot_set_v ) ) ) ) ).
% finite_empty_induct
thf(fact_1258_e1__def,axiom,
( e1
= ( sCC_Bl7876664385711583351t_unit
@ ^ [Uu: list_v] : ( cons_v @ v2 @ ( sCC_Bl9201514103433284750t_unit @ e ) )
@ ( sCC_Bl349061681862590396t_unit
@ ^ [Uu: list_v] : ( cons_v @ v2 @ ( sCC_Bl8828226123343373779t_unit @ e ) )
@ ( sCC_Bl7870604408699998558t_unit
@ ^ [Uu: set_v] : ( sup_sup_set_v @ ( sCC_Bl4645233313691564917t_unit @ e ) @ ( insert_v2 @ v2 @ bot_bot_set_v ) )
@ e ) ) ) ) ).
% e1_def
thf(fact_1259_e_H_H__def,axiom,
( e3
= ( sCC_Bl7876664385711583351t_unit
@ ^ [Uu: list_v] : ( tl_v @ ( sCC_Bl9201514103433284750t_unit @ e2 ) )
@ e2 ) ) ).
% e''_def
thf(fact_1260_pre__dfss__unite__pre__dfss,axiom,
! [V3: v,E: sCC_Bl1394983891496994913t_unit,W: v] :
( ( sCC_Bl1748261141445803503t_unit @ successors @ V3 @ E )
=> ( ( member_v @ W @ ( successors @ V3 ) )
=> ( ~ ( member_v @ W @ ( sCC_Bl3795065053823578884t_unit @ E @ V3 ) )
=> ( ( member_v @ W @ ( sCC_Bl4645233313691564917t_unit @ E ) )
=> ( ~ ( member_v @ W @ ( sCC_Bl157864678168468314t_unit @ E ) )
=> ( sCC_Bl1748261141445803503t_unit @ successors @ V3
@ ( sCC_Bl48393358579903213t_unit
@ ^ [Uu: v > set_v,X2: v] : ( if_set_v @ ( X2 = V3 ) @ ( sup_sup_set_v @ ( sCC_Bl3795065053823578884t_unit @ ( sCC_Bloemen_unite_v @ V3 @ W @ E ) @ V3 ) @ ( insert_v2 @ W @ bot_bot_set_v ) ) @ ( sCC_Bl3795065053823578884t_unit @ ( sCC_Bloemen_unite_v @ V3 @ W @ E ) @ X2 ) )
@ ( sCC_Bloemen_unite_v @ V3 @ W @ E ) ) ) ) ) ) ) ) ).
% pre_dfss_unite_pre_dfss
thf(fact_1261_dfs__dfss_Odomintros_I1_J,axiom,
! [V3: v,E: sCC_Bl1394983891496994913t_unit] :
( ( accp_S2303753412255344476t_unit @ ( sCC_Bl907557413677168252_rel_v @ successors )
@ ( sum_In5289330923152326972t_unit
@ ( produc3862955338007567901t_unit @ V3
@ ( sCC_Bl7876664385711583351t_unit
@ ^ [Uu: list_v] : ( cons_v @ V3 @ ( sCC_Bl9201514103433284750t_unit @ E ) )
@ ( sCC_Bl349061681862590396t_unit
@ ^ [Uu: list_v] : ( cons_v @ V3 @ ( sCC_Bl8828226123343373779t_unit @ E ) )
@ ( sCC_Bl7870604408699998558t_unit
@ ^ [Uu: set_v] : ( insert_v2 @ V3 @ ( sCC_Bl4645233313691564917t_unit @ E ) )
@ E ) ) ) ) ) )
=> ( accp_S2303753412255344476t_unit @ ( sCC_Bl907557413677168252_rel_v @ successors ) @ ( sum_In526841707622398774t_unit @ ( produc3862955338007567901t_unit @ V3 @ E ) ) ) ) ).
% dfs_dfss.domintros(1)
thf(fact_1262_dfs_Opsimps,axiom,
! [V3: v,E: sCC_Bl1394983891496994913t_unit] :
( ( accp_S2303753412255344476t_unit @ ( sCC_Bl907557413677168252_rel_v @ successors ) @ ( sum_In526841707622398774t_unit @ ( produc3862955338007567901t_unit @ V3 @ E ) ) )
=> ( ( sCC_Bloemen_dfs_v @ successors @ V3 @ E )
= ( if_SCC4926449794303880475t_unit
@ ( V3
= ( hd_v
@ ( sCC_Bl8828226123343373779t_unit
@ ( sCC_Bloemen_dfss_v @ successors @ V3
@ ( sCC_Bl7876664385711583351t_unit
@ ^ [Uu: list_v] : ( cons_v @ V3 @ ( sCC_Bl9201514103433284750t_unit @ E ) )
@ ( sCC_Bl349061681862590396t_unit
@ ^ [Uu: list_v] : ( cons_v @ V3 @ ( sCC_Bl8828226123343373779t_unit @ E ) )
@ ( sCC_Bl7870604408699998558t_unit
@ ^ [Uu: set_v] : ( sup_sup_set_v @ ( sCC_Bl4645233313691564917t_unit @ E ) @ ( insert_v2 @ V3 @ bot_bot_set_v ) )
@ E ) ) ) ) ) ) )
@ ( sCC_Bl7876664385711583351t_unit
@ ^ [Uu: list_v] :
( tl_v
@ ( sCC_Bl9201514103433284750t_unit
@ ( sCC_Bloemen_dfss_v @ successors @ V3
@ ( sCC_Bl7876664385711583351t_unit
@ ^ [Uv: list_v] : ( cons_v @ V3 @ ( sCC_Bl9201514103433284750t_unit @ E ) )
@ ( sCC_Bl349061681862590396t_unit
@ ^ [Uv: list_v] : ( cons_v @ V3 @ ( sCC_Bl8828226123343373779t_unit @ E ) )
@ ( sCC_Bl7870604408699998558t_unit
@ ^ [Uv: set_v] : ( sup_sup_set_v @ ( sCC_Bl4645233313691564917t_unit @ E ) @ ( insert_v2 @ V3 @ bot_bot_set_v ) )
@ E ) ) ) ) ) )
@ ( sCC_Bl349061681862590396t_unit
@ ^ [Uu: list_v] :
( tl_v
@ ( sCC_Bl8828226123343373779t_unit
@ ( sCC_Bloemen_dfss_v @ successors @ V3
@ ( sCC_Bl7876664385711583351t_unit
@ ^ [Uv: list_v] : ( cons_v @ V3 @ ( sCC_Bl9201514103433284750t_unit @ E ) )
@ ( sCC_Bl349061681862590396t_unit
@ ^ [Uv: list_v] : ( cons_v @ V3 @ ( sCC_Bl8828226123343373779t_unit @ E ) )
@ ( sCC_Bl7870604408699998558t_unit
@ ^ [Uv: set_v] : ( sup_sup_set_v @ ( sCC_Bl4645233313691564917t_unit @ E ) @ ( insert_v2 @ V3 @ bot_bot_set_v ) )
@ E ) ) ) ) ) )
@ ( sCC_Bl2708505634401380163t_unit
@ ^ [Uu: set_v] :
( sup_sup_set_v
@ ( sCC_Bl157864678168468314t_unit
@ ( sCC_Bloemen_dfss_v @ successors @ V3
@ ( sCC_Bl7876664385711583351t_unit
@ ^ [Uv: list_v] : ( cons_v @ V3 @ ( sCC_Bl9201514103433284750t_unit @ E ) )
@ ( sCC_Bl349061681862590396t_unit
@ ^ [Uv: list_v] : ( cons_v @ V3 @ ( sCC_Bl8828226123343373779t_unit @ E ) )
@ ( sCC_Bl7870604408699998558t_unit
@ ^ [Uv: set_v] : ( sup_sup_set_v @ ( sCC_Bl4645233313691564917t_unit @ E ) @ ( insert_v2 @ V3 @ bot_bot_set_v ) )
@ E ) ) ) ) )
@ ( sCC_Bl1280885523602775798t_unit
@ ( sCC_Bloemen_dfss_v @ successors @ V3
@ ( sCC_Bl7876664385711583351t_unit
@ ^ [Uv: list_v] : ( cons_v @ V3 @ ( sCC_Bl9201514103433284750t_unit @ E ) )
@ ( sCC_Bl349061681862590396t_unit
@ ^ [Uv: list_v] : ( cons_v @ V3 @ ( sCC_Bl8828226123343373779t_unit @ E ) )
@ ( sCC_Bl7870604408699998558t_unit
@ ^ [Uv: set_v] : ( sup_sup_set_v @ ( sCC_Bl4645233313691564917t_unit @ E ) @ ( insert_v2 @ V3 @ bot_bot_set_v ) )
@ E ) ) ) )
@ V3 ) )
@ ( sCC_Bl6816368539212994290t_unit
@ ^ [Uu: set_set_v] :
( sup_sup_set_set_v
@ ( sCC_Bl2536197123907397897t_unit
@ ( sCC_Bloemen_dfss_v @ successors @ V3
@ ( sCC_Bl7876664385711583351t_unit
@ ^ [Uv: list_v] : ( cons_v @ V3 @ ( sCC_Bl9201514103433284750t_unit @ E ) )
@ ( sCC_Bl349061681862590396t_unit
@ ^ [Uv: list_v] : ( cons_v @ V3 @ ( sCC_Bl8828226123343373779t_unit @ E ) )
@ ( sCC_Bl7870604408699998558t_unit
@ ^ [Uv: set_v] : ( sup_sup_set_v @ ( sCC_Bl4645233313691564917t_unit @ E ) @ ( insert_v2 @ V3 @ bot_bot_set_v ) )
@ E ) ) ) ) )
@ ( insert_set_v2
@ ( sCC_Bl1280885523602775798t_unit
@ ( sCC_Bloemen_dfss_v @ successors @ V3
@ ( sCC_Bl7876664385711583351t_unit
@ ^ [Uv: list_v] : ( cons_v @ V3 @ ( sCC_Bl9201514103433284750t_unit @ E ) )
@ ( sCC_Bl349061681862590396t_unit
@ ^ [Uv: list_v] : ( cons_v @ V3 @ ( sCC_Bl8828226123343373779t_unit @ E ) )
@ ( sCC_Bl7870604408699998558t_unit
@ ^ [Uv: set_v] : ( sup_sup_set_v @ ( sCC_Bl4645233313691564917t_unit @ E ) @ ( insert_v2 @ V3 @ bot_bot_set_v ) )
@ E ) ) ) )
@ V3 )
@ bot_bot_set_set_v ) )
@ ( sCC_Bloemen_dfss_v @ successors @ V3
@ ( sCC_Bl7876664385711583351t_unit
@ ^ [Uu: list_v] : ( cons_v @ V3 @ ( sCC_Bl9201514103433284750t_unit @ E ) )
@ ( sCC_Bl349061681862590396t_unit
@ ^ [Uu: list_v] : ( cons_v @ V3 @ ( sCC_Bl8828226123343373779t_unit @ E ) )
@ ( sCC_Bl7870604408699998558t_unit
@ ^ [Uu: set_v] : ( sup_sup_set_v @ ( sCC_Bl4645233313691564917t_unit @ E ) @ ( insert_v2 @ V3 @ bot_bot_set_v ) )
@ E ) ) ) ) ) ) ) )
@ ( sCC_Bl7876664385711583351t_unit
@ ^ [Uu: list_v] :
( tl_v
@ ( sCC_Bl9201514103433284750t_unit
@ ( sCC_Bloemen_dfss_v @ successors @ V3
@ ( sCC_Bl7876664385711583351t_unit
@ ^ [Uv: list_v] : ( cons_v @ V3 @ ( sCC_Bl9201514103433284750t_unit @ E ) )
@ ( sCC_Bl349061681862590396t_unit
@ ^ [Uv: list_v] : ( cons_v @ V3 @ ( sCC_Bl8828226123343373779t_unit @ E ) )
@ ( sCC_Bl7870604408699998558t_unit
@ ^ [Uv: set_v] : ( sup_sup_set_v @ ( sCC_Bl4645233313691564917t_unit @ E ) @ ( insert_v2 @ V3 @ bot_bot_set_v ) )
@ E ) ) ) ) ) )
@ ( sCC_Bloemen_dfss_v @ successors @ V3
@ ( sCC_Bl7876664385711583351t_unit
@ ^ [Uu: list_v] : ( cons_v @ V3 @ ( sCC_Bl9201514103433284750t_unit @ E ) )
@ ( sCC_Bl349061681862590396t_unit
@ ^ [Uu: list_v] : ( cons_v @ V3 @ ( sCC_Bl8828226123343373779t_unit @ E ) )
@ ( sCC_Bl7870604408699998558t_unit
@ ^ [Uu: set_v] : ( sup_sup_set_v @ ( sCC_Bl4645233313691564917t_unit @ E ) @ ( insert_v2 @ V3 @ bot_bot_set_v ) )
@ E ) ) ) ) ) ) ) ) ).
% dfs.psimps
thf(fact_1263_pre__dfss__post__dfs__pre__dfss,axiom,
! [V3: v,E: sCC_Bl1394983891496994913t_unit,W: v] :
( ( sCC_Bl1748261141445803503t_unit @ successors @ V3 @ E )
=> ( ( member_v @ W @ ( successors @ V3 ) )
=> ( ~ ( member_v @ W @ ( sCC_Bl4645233313691564917t_unit @ E ) )
=> ( ( sCC_Bl8953792750115413617t_unit @ successors @ W @ E @ ( sCC_Bloemen_dfs_v @ successors @ W @ E ) )
=> ( sCC_Bl1748261141445803503t_unit @ successors @ V3
@ ( sCC_Bl48393358579903213t_unit
@ ^ [Uu: v > set_v,X2: v] : ( if_set_v @ ( X2 = V3 ) @ ( sup_sup_set_v @ ( sCC_Bl3795065053823578884t_unit @ ( sCC_Bloemen_dfs_v @ successors @ W @ E ) @ V3 ) @ ( insert_v2 @ W @ bot_bot_set_v ) ) @ ( sCC_Bl3795065053823578884t_unit @ ( sCC_Bloemen_dfs_v @ successors @ W @ E ) @ X2 ) )
@ ( sCC_Bloemen_dfs_v @ successors @ W @ E ) ) ) ) ) ) ) ).
% pre_dfss_post_dfs_pre_dfss
thf(fact_1264_dfss_Opsimps,axiom,
! [V3: v,E: sCC_Bl1394983891496994913t_unit] :
( ( accp_S2303753412255344476t_unit @ ( sCC_Bl907557413677168252_rel_v @ successors ) @ ( sum_In5289330923152326972t_unit @ ( produc3862955338007567901t_unit @ V3 @ E ) ) )
=> ( ( sCC_Bloemen_dfss_v @ successors @ V3 @ E )
= ( if_SCC4926449794303880475t_unit
@ ( ( minus_minus_set_v @ ( successors @ V3 ) @ ( sCC_Bl3795065053823578884t_unit @ E @ V3 ) )
= bot_bot_set_v )
@ E
@ ( sCC_Bloemen_dfss_v @ successors @ V3
@ ( sCC_Bl48393358579903213t_unit
@ ^ [Uu: v > set_v,X2: v] :
( if_set_v @ ( X2 = V3 )
@ ( sup_sup_set_v
@ ( sCC_Bl3795065053823578884t_unit
@ ( if_SCC4926449794303880475t_unit
@ ( member_v
@ ( fChoice_v
@ ^ [Y3: v] : ( member_v @ Y3 @ ( minus_minus_set_v @ ( successors @ V3 ) @ ( sCC_Bl3795065053823578884t_unit @ E @ V3 ) ) ) )
@ ( sCC_Bl157864678168468314t_unit @ E ) )
@ E
@ ( if_SCC4926449794303880475t_unit
@ ~ ( member_v
@ ( fChoice_v
@ ^ [Y3: v] : ( member_v @ Y3 @ ( minus_minus_set_v @ ( successors @ V3 ) @ ( sCC_Bl3795065053823578884t_unit @ E @ V3 ) ) ) )
@ ( sCC_Bl4645233313691564917t_unit @ E ) )
@ ( sCC_Bloemen_dfs_v @ successors
@ ( fChoice_v
@ ^ [Y3: v] : ( member_v @ Y3 @ ( minus_minus_set_v @ ( successors @ V3 ) @ ( sCC_Bl3795065053823578884t_unit @ E @ V3 ) ) ) )
@ E )
@ ( sCC_Bloemen_unite_v @ V3
@ ( fChoice_v
@ ^ [Y3: v] : ( member_v @ Y3 @ ( minus_minus_set_v @ ( successors @ V3 ) @ ( sCC_Bl3795065053823578884t_unit @ E @ V3 ) ) ) )
@ E ) ) )
@ V3 )
@ ( insert_v2
@ ( fChoice_v
@ ^ [Y3: v] : ( member_v @ Y3 @ ( minus_minus_set_v @ ( successors @ V3 ) @ ( sCC_Bl3795065053823578884t_unit @ E @ V3 ) ) ) )
@ bot_bot_set_v ) )
@ ( sCC_Bl3795065053823578884t_unit
@ ( if_SCC4926449794303880475t_unit
@ ( member_v
@ ( fChoice_v
@ ^ [Y3: v] : ( member_v @ Y3 @ ( minus_minus_set_v @ ( successors @ V3 ) @ ( sCC_Bl3795065053823578884t_unit @ E @ V3 ) ) ) )
@ ( sCC_Bl157864678168468314t_unit @ E ) )
@ E
@ ( if_SCC4926449794303880475t_unit
@ ~ ( member_v
@ ( fChoice_v
@ ^ [Y3: v] : ( member_v @ Y3 @ ( minus_minus_set_v @ ( successors @ V3 ) @ ( sCC_Bl3795065053823578884t_unit @ E @ V3 ) ) ) )
@ ( sCC_Bl4645233313691564917t_unit @ E ) )
@ ( sCC_Bloemen_dfs_v @ successors
@ ( fChoice_v
@ ^ [Y3: v] : ( member_v @ Y3 @ ( minus_minus_set_v @ ( successors @ V3 ) @ ( sCC_Bl3795065053823578884t_unit @ E @ V3 ) ) ) )
@ E )
@ ( sCC_Bloemen_unite_v @ V3
@ ( fChoice_v
@ ^ [Y3: v] : ( member_v @ Y3 @ ( minus_minus_set_v @ ( successors @ V3 ) @ ( sCC_Bl3795065053823578884t_unit @ E @ V3 ) ) ) )
@ E ) ) )
@ X2 ) )
@ ( if_SCC4926449794303880475t_unit
@ ( member_v
@ ( fChoice_v
@ ^ [X2: v] : ( member_v @ X2 @ ( minus_minus_set_v @ ( successors @ V3 ) @ ( sCC_Bl3795065053823578884t_unit @ E @ V3 ) ) ) )
@ ( sCC_Bl157864678168468314t_unit @ E ) )
@ E
@ ( if_SCC4926449794303880475t_unit
@ ~ ( member_v
@ ( fChoice_v
@ ^ [X2: v] : ( member_v @ X2 @ ( minus_minus_set_v @ ( successors @ V3 ) @ ( sCC_Bl3795065053823578884t_unit @ E @ V3 ) ) ) )
@ ( sCC_Bl4645233313691564917t_unit @ E ) )
@ ( sCC_Bloemen_dfs_v @ successors
@ ( fChoice_v
@ ^ [X2: v] : ( member_v @ X2 @ ( minus_minus_set_v @ ( successors @ V3 ) @ ( sCC_Bl3795065053823578884t_unit @ E @ V3 ) ) ) )
@ E )
@ ( sCC_Bloemen_unite_v @ V3
@ ( fChoice_v
@ ^ [X2: v] : ( member_v @ X2 @ ( minus_minus_set_v @ ( successors @ V3 ) @ ( sCC_Bl3795065053823578884t_unit @ E @ V3 ) ) ) )
@ E ) ) ) ) ) ) ) ) ).
% dfss.psimps
thf(fact_1265_unite__S__equal,axiom,
! [E: sCC_Bl1394983891496994913t_unit,W: v,V3: v] :
( ( sCC_Bl9196236973127232072t_unit @ successors @ E )
=> ( ( member_v @ W @ ( successors @ V3 ) )
=> ( ~ ( member_v @ W @ ( sCC_Bl3795065053823578884t_unit @ E @ V3 ) )
=> ( ( member_v @ W @ ( sCC_Bl4645233313691564917t_unit @ E ) )
=> ( ~ ( member_v @ W @ ( sCC_Bl157864678168468314t_unit @ E ) )
=> ( ( comple2307003700295860064_set_v
@ ( collect_set_v
@ ^ [Uu: set_v] :
? [N3: v] :
( ( Uu
= ( sCC_Bl1280885523602775798t_unit @ ( sCC_Bloemen_unite_v @ V3 @ W @ E ) @ N3 ) )
& ( member_v @ N3 @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ ( sCC_Bloemen_unite_v @ V3 @ 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_1266_unite__def,axiom,
( sCC_Bloemen_unite_v
= ( ^ [V4: v,W2: v,E8: sCC_Bl1394983891496994913t_unit] :
( sCC_Bl349061681862590396t_unit
@ ^ [Uu: list_v] :
( dropWhile_v
@ ^ [X2: v] :
~ ( member_v @ W2 @ ( sCC_Bl1280885523602775798t_unit @ E8 @ X2 ) )
@ ( sCC_Bl8828226123343373779t_unit @ E8 ) )
@ ( sCC_Bl3155122997657187039t_unit
@ ^ [Uu: v > set_v,X2: v] :
( if_set_v
@ ( member_v @ X2
@ ( comple2307003700295860064_set_v
@ ( collect_set_v
@ ^ [Uv: set_v] :
? [Y3: v] :
( ( Uv
= ( sCC_Bl1280885523602775798t_unit @ E8 @ Y3 ) )
& ( member_v @ Y3
@ ( sup_sup_set_v
@ ( set_v2
@ ( takeWhile_v
@ ^ [Z2: v] :
~ ( member_v @ W2 @ ( sCC_Bl1280885523602775798t_unit @ E8 @ Z2 ) )
@ ( sCC_Bl8828226123343373779t_unit @ E8 ) ) )
@ ( insert_v2
@ ( 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_v2
@ ( hd_v
@ ( dropWhile_v
@ ^ [Z2: v] :
~ ( member_v @ W2 @ ( sCC_Bl1280885523602775798t_unit @ E8 @ Z2 ) )
@ ( sCC_Bl8828226123343373779t_unit @ E8 ) ) )
@ bot_bot_set_v ) ) ) ) ) )
@ ( sCC_Bl1280885523602775798t_unit @ E8 @ X2 ) )
@ E8 ) ) ) ) ).
% unite_def
thf(fact_1267_old_Ounit_Oexhaust,axiom,
! [Y: product_unit] : ( Y = product_Unity ) ).
% old.unit.exhaust
thf(fact_1268_sup__unit__def,axiom,
( sup_sup_Product_unit
= ( ^ [Uu2: product_unit,Uv2: product_unit] : product_Unity ) ) ).
% sup_unit_def
thf(fact_1269_bot__unit__def,axiom,
bot_bot_Product_unit = product_Unity ).
% bot_unit_def
thf(fact_1270_Sup__unit__def,axiom,
( comple4687483117567863418t_unit
= ( ^ [Uu2: set_Product_unit] : product_Unity ) ) ).
% Sup_unit_def
thf(fact_1271_inf__unit__def,axiom,
( inf_inf_Product_unit
= ( ^ [Uu2: product_unit,Uv2: product_unit] : product_Unity ) ) ).
% inf_unit_def
thf(fact_1272_default__unit__def,axiom,
defaul566961228789861419t_unit = product_Unity ).
% default_unit_def
% Helper facts (10)
thf(help_fChoice_1_1_fChoice_001tf__v_T,axiom,
! [P: v > $o] :
( ( P @ ( fChoice_v @ P ) )
= ( ? [X6: v] : ( P @ X6 ) ) ) ).
thf(help_If_2_1_If_001t__Set__Oset_Itf__v_J_T,axiom,
! [X: set_v,Y: set_v] :
( ( if_set_v @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Set__Oset_Itf__v_J_T,axiom,
! [X: set_v,Y: set_v] :
( ( if_set_v @ $true @ X @ Y )
= X ) ).
thf(help_If_2_1_If_001t__List__Olist_Itf__v_J_T,axiom,
! [X: list_v,Y: list_v] :
( ( if_list_v @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__List__Olist_Itf__v_J_T,axiom,
! [X: list_v,Y: list_v] :
( ( if_list_v @ $true @ X @ Y )
= X ) ).
thf(help_If_2_1_If_001t__List__Olist_It__Product____Type__Oprod_Itf__v_Mtf__v_J_J_T,axiom,
! [X: list_P7986770385144383213od_v_v,Y: list_P7986770385144383213od_v_v] :
( ( if_lis7521272669439687347od_v_v @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__List__Olist_It__Product____Type__Oprod_Itf__v_Mtf__v_J_J_T,axiom,
! [X: list_P7986770385144383213od_v_v,Y: list_P7986770385144383213od_v_v] :
( ( if_lis7521272669439687347od_v_v @ $true @ X @ Y )
= X ) ).
thf(help_If_3_1_If_001t__SCC____Bloemen____Sequential__Oenv__Oenv____ext_Itf__v_Mt__Product____Type__Ounit_J_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__SCC____Bloemen____Sequential__Oenv__Oenv____ext_Itf__v_Mt__Product____Type__Ounit_J_T,axiom,
! [X: sCC_Bl1394983891496994913t_unit,Y: sCC_Bl1394983891496994913t_unit] :
( ( if_SCC4926449794303880475t_unit @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__SCC____Bloemen____Sequential__Oenv__Oenv____ext_Itf__v_Mt__Product____Type__Ounit_J_T,axiom,
! [X: sCC_Bl1394983891496994913t_unit,Y: sCC_Bl1394983891496994913t_unit] :
( ( if_SCC4926449794303880475t_unit @ $true @ X @ Y )
= X ) ).
% Conjectures (1)
thf(conj_0,conjecture,
~ ( distinct_v @ ( sCC_Bl8828226123343373779t_unit @ e1 ) ) ).
%------------------------------------------------------------------------------