TPTP Problem File: SLH0459^1.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 : Undirected_Graph_Theory/0016_Undirected_Graph_Walks/prob_00172_006719__13206476_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1486 ( 527 unt; 209 typ; 0 def)
% Number of atoms : 3570 (1520 equ; 0 cnn)
% Maximal formula atoms : 15 ( 2 avg)
% Number of connectives : 11380 ( 457 ~; 49 |; 281 &;9064 @)
% ( 0 <=>;1529 =>; 0 <=; 0 <~>)
% Maximal formula depth : 29 ( 7 avg)
% Number of types : 23 ( 22 usr)
% Number of type conns : 480 ( 480 >; 0 *; 0 +; 0 <<)
% Number of symbols : 190 ( 187 usr; 28 con; 0-4 aty)
% Number of variables : 3644 ( 117 ^;3359 !; 168 ?;3644 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 14:33:15.069
%------------------------------------------------------------------------------
% Could-be-implicit typings (22)
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the_el8589169208993665564od_a_a: set_Product_prod_a_a > product_prod_a_a ).
thf(sy_c_Set_Othe__elem_001t__Set__Oset_Itf__a_J,type,
the_elem_set_a: set_set_a > set_a ).
thf(sy_c_Set_Othe__elem_001tf__a,type,
the_elem_a: set_a > a ).
thf(sy_c_String_Ochar_Osize__char,type,
size_char: char > nat ).
thf(sy_c_Undirected__Graph__Basics_Oall__edges_001t__Nat__Onat,type,
undire463345858124014060es_nat: set_nat > set_set_nat ).
thf(sy_c_Undirected__Graph__Basics_Oall__edges_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
undire6879232364018543115od_a_a: set_Product_prod_a_a > set_se5735800977113168103od_a_a ).
thf(sy_c_Undirected__Graph__Basics_Oall__edges_001t__Set__Oset_Itf__a_J,type,
undire8247866692393712962_set_a: set_set_a > set_set_set_a ).
thf(sy_c_Undirected__Graph__Basics_Oall__edges_001tf__a,type,
undire2918257014606996450dges_a: set_a > set_set_a ).
thf(sy_c_Undirected__Graph__Basics_Ograph__system_001t__Nat__Onat,type,
undire7481384412329822504em_nat: set_nat > set_set_nat > $o ).
thf(sy_c_Undirected__Graph__Basics_Ograph__system_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
undire1860116983885411791od_a_a: set_Product_prod_a_a > set_se5735800977113168103od_a_a > $o ).
thf(sy_c_Undirected__Graph__Basics_Ograph__system_001t__Set__Oset_Itf__a_J,type,
undire7159349782766787846_set_a: set_set_a > set_set_set_a > $o ).
thf(sy_c_Undirected__Graph__Basics_Ograph__system_001tf__a,type,
undire2554140024507503526stem_a: set_a > set_set_a > $o ).
thf(sy_c_Undirected__Graph__Basics_Ograph__system_Oedge__adj_001t__Nat__Onat,type,
undire1664191744716346676dj_nat: set_set_nat > set_nat > set_nat > $o ).
thf(sy_c_Undirected__Graph__Basics_Ograph__system_Oedge__adj_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
undire9186443406341554371od_a_a: set_se5735800977113168103od_a_a > set_Product_prod_a_a > set_Product_prod_a_a > $o ).
thf(sy_c_Undirected__Graph__Basics_Ograph__system_Oedge__adj_001t__Set__Oset_Itf__a_J,type,
undire3485422320110889978_set_a: set_set_set_a > set_set_a > set_set_a > $o ).
thf(sy_c_Undirected__Graph__Basics_Ograph__system_Oedge__adj_001tf__a,type,
undire4022703626023482010_adj_a: set_set_a > set_a > set_a > $o ).
thf(sy_c_Undirected__Graph__Basics_Ograph__system_Oincident_001t__Nat__Onat,type,
undire7858122600432113898nt_nat: nat > set_nat > $o ).
thf(sy_c_Undirected__Graph__Basics_Ograph__system_Oincident_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
undire3369688177417741453od_a_a: product_prod_a_a > set_Product_prod_a_a > $o ).
thf(sy_c_Undirected__Graph__Basics_Ograph__system_Oincident_001t__Set__Oset_Itf__a_J,type,
undire2320338297334612420_set_a: set_a > set_set_a > $o ).
thf(sy_c_Undirected__Graph__Basics_Ograph__system_Oincident_001tf__a,type,
undire1521409233611534436dent_a: a > set_a > $o ).
thf(sy_c_Undirected__Graph__Basics_Ograph__system_Oincident__edges_001tf__a,type,
undire3231912044278729248dges_a: set_set_a > a > set_set_a ).
thf(sy_c_Undirected__Graph__Basics_Ograph__system_Oinduced__edges_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
undire5906991851038061813od_a_a: set_se5735800977113168103od_a_a > set_Product_prod_a_a > set_se5735800977113168103od_a_a ).
thf(sy_c_Undirected__Graph__Basics_Ograph__system_Oinduced__edges_001t__Set__Oset_Itf__a_J,type,
undire7854589003810675244_set_a: set_set_set_a > set_set_a > set_set_set_a ).
thf(sy_c_Undirected__Graph__Basics_Ograph__system_Oinduced__edges_001tf__a,type,
undire7777452895879145676dges_a: set_set_a > set_a > set_set_a ).
thf(sy_c_Undirected__Graph__Basics_Osubgraph_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
undire398746457437328754od_a_a: set_Product_prod_a_a > set_se5735800977113168103od_a_a > set_Product_prod_a_a > set_se5735800977113168103od_a_a > $o ).
thf(sy_c_Undirected__Graph__Basics_Osubgraph_001t__Set__Oset_Itf__a_J,type,
undire1186139521737116585_set_a: set_set_a > set_set_set_a > set_set_a > set_set_set_a > $o ).
thf(sy_c_Undirected__Graph__Basics_Osubgraph_001tf__a,type,
undire7103218114511261257raph_a: set_a > set_set_a > set_a > set_set_a > $o ).
thf(sy_c_Undirected__Graph__Basics_Oulgraph_001t__Nat__Onat,type,
undire3269267262472140706ph_nat: set_nat > set_set_nat > $o ).
thf(sy_c_Undirected__Graph__Basics_Oulgraph_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
undire4585262585102564309od_a_a: set_Product_prod_a_a > set_se5735800977113168103od_a_a > $o ).
thf(sy_c_Undirected__Graph__Basics_Oulgraph_001t__Set__Oset_Itf__a_J,type,
undire6886684016831807756_set_a: set_set_a > set_set_set_a > $o ).
thf(sy_c_Undirected__Graph__Basics_Oulgraph_001tf__a,type,
undire7251896706689453996raph_a: set_a > set_set_a > $o ).
thf(sy_c_Undirected__Graph__Basics_Oulgraph_Oall__edges__between_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
undire4032395788819567636od_a_a: set_se5735800977113168103od_a_a > set_Product_prod_a_a > set_Product_prod_a_a > set_Pr8600417178894128327od_a_a ).
thf(sy_c_Undirected__Graph__Basics_Oulgraph_Oall__edges__between_001t__Set__Oset_Itf__a_J,type,
undire2462398226299384907_set_a: set_set_set_a > set_set_a > set_set_a > set_Pr5845495582615845127_set_a ).
thf(sy_c_Undirected__Graph__Basics_Oulgraph_Oall__edges__between_001tf__a,type,
undire8383842906760478443ween_a: set_set_a > set_a > set_a > set_Product_prod_a_a ).
thf(sy_c_Undirected__Graph__Basics_Oulgraph_Odegree_001t__Nat__Onat,type,
undire6581030323043281630ee_nat: set_set_nat > nat > nat ).
thf(sy_c_Undirected__Graph__Basics_Oulgraph_Odegree_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
undire1436394852029823897od_a_a: set_se5735800977113168103od_a_a > product_prod_a_a > nat ).
thf(sy_c_Undirected__Graph__Basics_Oulgraph_Odegree_001t__Set__Oset_Itf__a_J,type,
undire8939077443744732368_set_a: set_set_set_a > set_a > nat ).
thf(sy_c_Undirected__Graph__Basics_Oulgraph_Odegree_001tf__a,type,
undire8867928226783802224gree_a: set_set_a > a > nat ).
thf(sy_c_Undirected__Graph__Basics_Oulgraph_Oedge__density_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
undire8410861505230878716od_a_a: set_se5735800977113168103od_a_a > set_Product_prod_a_a > set_Product_prod_a_a > real ).
thf(sy_c_Undirected__Graph__Basics_Oulgraph_Oedge__density_001t__Set__Oset_Itf__a_J,type,
undire8927637694342045747_set_a: set_set_set_a > set_set_a > set_set_a > real ).
thf(sy_c_Undirected__Graph__Basics_Oulgraph_Oedge__density_001tf__a,type,
undire297304480579013331sity_a: set_set_a > set_a > set_a > real ).
thf(sy_c_Undirected__Graph__Basics_Oulgraph_Ohas__loop_001t__Nat__Onat,type,
undire5005864372999571214op_nat: set_set_nat > nat > $o ).
thf(sy_c_Undirected__Graph__Basics_Oulgraph_Ohas__loop_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
undire7777398424729533289od_a_a: set_se5735800977113168103od_a_a > product_prod_a_a > $o ).
thf(sy_c_Undirected__Graph__Basics_Oulgraph_Ohas__loop_001t__Set__Oset_Itf__a_J,type,
undire5774735625301615776_set_a: set_set_set_a > set_a > $o ).
thf(sy_c_Undirected__Graph__Basics_Oulgraph_Ohas__loop_001tf__a,type,
undire3617971648856834880loop_a: set_set_a > a > $o ).
thf(sy_c_Undirected__Graph__Basics_Oulgraph_Oincident__loops_001tf__a,type,
undire4753905205749729249oops_a: set_set_a > a > set_set_a ).
thf(sy_c_Undirected__Graph__Basics_Oulgraph_Oincident__sedges_001tf__a,type,
undire1270416042309875431dges_a: set_set_a > a > set_set_a ).
thf(sy_c_Undirected__Graph__Basics_Oulgraph_Ois__edge__between_001tf__a,type,
undire8544646567961481629ween_a: set_a > set_a > set_a > $o ).
thf(sy_c_Undirected__Graph__Basics_Oulgraph_Ois__isolated__vertex_001t__Nat__Onat,type,
undire5609513041723151865ex_nat: set_nat > set_set_nat > nat > $o ).
thf(sy_c_Undirected__Graph__Basics_Oulgraph_Ois__isolated__vertex_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
undire3207556238582723646od_a_a: set_Product_prod_a_a > set_se5735800977113168103od_a_a > product_prod_a_a > $o ).
thf(sy_c_Undirected__Graph__Basics_Oulgraph_Ois__isolated__vertex_001t__Set__Oset_Itf__a_J,type,
undire6879241558604981877_set_a: set_set_a > set_set_set_a > set_a > $o ).
thf(sy_c_Undirected__Graph__Basics_Oulgraph_Ois__isolated__vertex_001tf__a,type,
undire8931668460104145173rtex_a: set_a > set_set_a > a > $o ).
thf(sy_c_Undirected__Graph__Basics_Oulgraph_Ois__loop_001tf__a,type,
undire2905028936066782638loop_a: set_a > $o ).
thf(sy_c_Undirected__Graph__Basics_Oulgraph_Ois__sedge_001tf__a,type,
undire4917966558017083288edge_a: set_a > $o ).
thf(sy_c_Undirected__Graph__Basics_Oulgraph_Oneighborhood_001tf__a,type,
undire8504279938402040014hood_a: set_a > set_set_a > a > set_a ).
thf(sy_c_Undirected__Graph__Basics_Oulgraph_Overt__adj_001t__Nat__Onat,type,
undire1083030068171319366dj_nat: set_set_nat > nat > nat > $o ).
thf(sy_c_Undirected__Graph__Basics_Oulgraph_Overt__adj_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
undire6135774327024169009od_a_a: set_se5735800977113168103od_a_a > product_prod_a_a > product_prod_a_a > $o ).
thf(sy_c_Undirected__Graph__Basics_Oulgraph_Overt__adj_001t__Set__Oset_Itf__a_J,type,
undire3510646817838285160_set_a: set_set_set_a > set_a > set_a > $o ).
thf(sy_c_Undirected__Graph__Basics_Oulgraph_Overt__adj_001tf__a,type,
undire397441198561214472_adj_a: set_set_a > a > a > $o ).
thf(sy_c_Undirected__Graph__Walks_Oulgraph_Ois__walk_001t__Nat__Onat,type,
undire5745680128780950498lk_nat: set_nat > set_set_nat > list_nat > $o ).
thf(sy_c_Undirected__Graph__Walks_Oulgraph_Ois__walk_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
undire3162072421265123221od_a_a: set_Product_prod_a_a > set_se5735800977113168103od_a_a > list_P1396940483166286381od_a_a > $o ).
thf(sy_c_Undirected__Graph__Walks_Oulgraph_Ois__walk_001t__Set__Oset_Itf__a_J,type,
undire3014741414213135564_set_a: set_set_a > set_set_set_a > list_set_a > $o ).
thf(sy_c_Undirected__Graph__Walks_Oulgraph_Ois__walk_001tf__a,type,
undire6133010728901294956walk_a: set_a > set_set_a > list_a > $o ).
thf(sy_c_Undirected__Graph__Walks_Oulgraph_Owalk__edges_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
undire4403264684974754359od_a_a: list_P1396940483166286381od_a_a > list_s9060204159073123853od_a_a ).
thf(sy_c_Undirected__Graph__Walks_Oulgraph_Owalk__edges_001t__Set__Oset_Itf__a_J,type,
undire6234387080713648494_set_a: list_set_a > list_set_set_a ).
thf(sy_c_Undirected__Graph__Walks_Oulgraph_Owalk__edges_001tf__a,type,
undire7337870655677353998dges_a: list_a > list_set_a ).
thf(sy_c_Undirected__Graph__Walks_Oulgraph_Owalk__edges__rel_001tf__a,type,
undire7966302452035489203_rel_a: list_a > list_a > $o ).
thf(sy_c_Undirected__Graph__Walks_Oulgraph_Owalk__length_001t__Set__Oset_Itf__a_J,type,
undire4424681683220949296_set_a: list_set_a > nat ).
thf(sy_c_Undirected__Graph__Walks_Oulgraph_Owalk__length_001tf__a,type,
undire8849074589633906640ngth_a: list_a > nat ).
thf(sy_c_Wellfounded_Oaccp_001t__List__Olist_Itf__a_J,type,
accp_list_a: ( list_a > list_a > $o ) > list_a > $o ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
member1426531477525435216od_a_a: product_prod_a_a > set_Product_prod_a_a > $o ).
thf(sy_c_member_001t__Set__Oset_It__Nat__Onat_J,type,
member_set_nat: set_nat > set_set_nat > $o ).
thf(sy_c_member_001t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
member1816616512716248880od_a_a: set_Product_prod_a_a > set_se5735800977113168103od_a_a > $o ).
thf(sy_c_member_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
member_set_set_a: set_set_a > set_set_set_a > $o ).
thf(sy_c_member_001t__Set__Oset_Itf__a_J,type,
member_set_a: set_a > set_set_a > $o ).
thf(sy_c_member_001tf__a,type,
member_a: a > set_a > $o ).
thf(sy_v_edges,type,
edges: set_set_a ).
thf(sy_v_vertices,type,
vertices: set_a ).
thf(sy_v_xs,type,
xs: list_a ).
thf(sy_v_y,type,
y: a ).
thf(sy_v_ys,type,
ys: list_a ).
thf(sy_v_zs,type,
zs: list_a ).
% Relevant facts (1273)
thf(fact_0_walk__edges_Ocases,axiom,
! [X: list_a] :
( ( X != nil_a )
=> ( ! [X2: a] :
( X
!= ( cons_a @ X2 @ nil_a ) )
=> ~ ! [X2: a,Y: a,Ys: list_a] :
( X
!= ( cons_a @ X2 @ ( cons_a @ Y @ Ys ) ) ) ) ) ).
% walk_edges.cases
thf(fact_1_walk__edges__decomp__ss,axiom,
! [Xs: list_a,Y2: a,Zs: list_a,Ys2: list_a] : ( ord_le3724670747650509150_set_a @ ( set_set_a2 @ ( undire7337870655677353998dges_a @ ( append_a @ Xs @ ( append_a @ ( cons_a @ Y2 @ nil_a ) @ Zs ) ) ) ) @ ( set_set_a2 @ ( undire7337870655677353998dges_a @ ( append_a @ Xs @ ( append_a @ ( cons_a @ Y2 @ nil_a ) @ ( append_a @ Ys2 @ ( append_a @ ( cons_a @ Y2 @ nil_a ) @ Zs ) ) ) ) ) ) ) ).
% walk_edges_decomp_ss
thf(fact_2_walk__edges__append__ss1,axiom,
! [Ys2: list_a,Xs: list_a] : ( ord_le3724670747650509150_set_a @ ( set_set_a2 @ ( undire7337870655677353998dges_a @ Ys2 ) ) @ ( set_set_a2 @ ( undire7337870655677353998dges_a @ ( append_a @ Xs @ Ys2 ) ) ) ) ).
% walk_edges_append_ss1
thf(fact_3_walk__edges__append__ss2,axiom,
! [Xs: list_a,Ys2: list_a] : ( ord_le3724670747650509150_set_a @ ( set_set_a2 @ ( undire7337870655677353998dges_a @ Xs ) ) @ ( set_set_a2 @ ( undire7337870655677353998dges_a @ ( append_a @ Xs @ Ys2 ) ) ) ) ).
% walk_edges_append_ss2
thf(fact_4_is__walk__def,axiom,
! [Xs: list_a] :
( ( undire6133010728901294956walk_a @ vertices @ edges @ Xs )
= ( ( ord_less_eq_set_a @ ( set_a2 @ Xs ) @ vertices )
& ( ord_le3724670747650509150_set_a @ ( set_set_a2 @ ( undire7337870655677353998dges_a @ Xs ) ) @ edges )
& ( Xs != nil_a ) ) ) ).
% is_walk_def
thf(fact_5_assms,axiom,
undire6133010728901294956walk_a @ vertices @ edges @ ( append_a @ xs @ ( append_a @ ( cons_a @ y @ nil_a ) @ ( append_a @ ys @ ( append_a @ ( cons_a @ y @ nil_a ) @ zs ) ) ) ) ).
% assms
thf(fact_6__092_060open_062xs_A_064_A_091y_093_A_064_Azs_A_092_060noteq_062_A_091_093_092_060close_062,axiom,
( ( append_a @ xs @ ( append_a @ ( cons_a @ y @ nil_a ) @ zs ) )
!= nil_a ) ).
% \<open>xs @ [y] @ zs \<noteq> []\<close>
thf(fact_7_edge__adj__inE,axiom,
! [E1: set_a,E2: set_a] :
( ( undire4022703626023482010_adj_a @ edges @ E1 @ E2 )
=> ( ( member_set_a @ E1 @ edges )
& ( member_set_a @ E2 @ edges ) ) ) ).
% edge_adj_inE
thf(fact_8_is__walk__not__empty,axiom,
! [Xs: list_a] :
( ( undire6133010728901294956walk_a @ vertices @ edges @ Xs )
=> ( Xs != nil_a ) ) ).
% is_walk_not_empty
thf(fact_9_is__walk__not__empty2,axiom,
~ ( undire6133010728901294956walk_a @ vertices @ edges @ nil_a ) ).
% is_walk_not_empty2
thf(fact_10_wellformed,axiom,
! [E: set_a] :
( ( member_set_a @ E @ edges )
=> ( ord_less_eq_set_a @ E @ vertices ) ) ).
% wellformed
thf(fact_11_is__walk__singleton,axiom,
! [U: a] :
( ( member_a @ U @ vertices )
=> ( undire6133010728901294956walk_a @ vertices @ edges @ ( cons_a @ U @ nil_a ) ) ) ).
% is_walk_singleton
thf(fact_12_comp__sgraph_Owalk__edges_Osimps_I2_J,axiom,
! [X: set_a] :
( ( undire6234387080713648494_set_a @ ( cons_set_a @ X @ nil_set_a ) )
= nil_set_set_a ) ).
% comp_sgraph.walk_edges.simps(2)
thf(fact_13_comp__sgraph_Owalk__edges_Osimps_I2_J,axiom,
! [X: a] :
( ( undire7337870655677353998dges_a @ ( cons_a @ X @ nil_a ) )
= nil_set_a ) ).
% comp_sgraph.walk_edges.simps(2)
thf(fact_14_comp__sgraph_Owalk__edges_Osimps_I1_J,axiom,
( ( undire6234387080713648494_set_a @ nil_set_a )
= nil_set_set_a ) ).
% comp_sgraph.walk_edges.simps(1)
thf(fact_15_comp__sgraph_Owalk__edges_Osimps_I1_J,axiom,
( ( undire7337870655677353998dges_a @ nil_a )
= nil_set_a ) ).
% comp_sgraph.walk_edges.simps(1)
thf(fact_16_comp__sgraph_Owalk__edges_Ocases,axiom,
! [X: list_a] :
( ( X != nil_a )
=> ( ! [X2: a] :
( X
!= ( cons_a @ X2 @ nil_a ) )
=> ~ ! [X2: a,Y: a,Ys: list_a] :
( X
!= ( cons_a @ X2 @ ( cons_a @ Y @ Ys ) ) ) ) ) ).
% comp_sgraph.walk_edges.cases
thf(fact_17_comp__sgraph_Owalk__edges_Ocases,axiom,
! [X: list_set_a] :
( ( X != nil_set_a )
=> ( ! [X2: set_a] :
( X
!= ( cons_set_a @ X2 @ nil_set_a ) )
=> ~ ! [X2: set_a,Y: set_a,Ys: list_set_a] :
( X
!= ( cons_set_a @ X2 @ ( cons_set_a @ Y @ Ys ) ) ) ) ) ).
% comp_sgraph.walk_edges.cases
thf(fact_18_comp__sgraph_Owalk__edges__decomp__ss,axiom,
! [Xs: list_set_a,Y2: set_a,Zs: list_set_a,Ys2: list_set_a] : ( ord_le5722252365846178494_set_a @ ( set_set_set_a2 @ ( undire6234387080713648494_set_a @ ( append_set_a @ Xs @ ( append_set_a @ ( cons_set_a @ Y2 @ nil_set_a ) @ Zs ) ) ) ) @ ( set_set_set_a2 @ ( undire6234387080713648494_set_a @ ( append_set_a @ Xs @ ( append_set_a @ ( cons_set_a @ Y2 @ nil_set_a ) @ ( append_set_a @ Ys2 @ ( append_set_a @ ( cons_set_a @ Y2 @ nil_set_a ) @ Zs ) ) ) ) ) ) ) ).
% comp_sgraph.walk_edges_decomp_ss
thf(fact_19_comp__sgraph_Owalk__edges__decomp__ss,axiom,
! [Xs: list_a,Y2: a,Zs: list_a,Ys2: list_a] : ( ord_le3724670747650509150_set_a @ ( set_set_a2 @ ( undire7337870655677353998dges_a @ ( append_a @ Xs @ ( append_a @ ( cons_a @ Y2 @ nil_a ) @ Zs ) ) ) ) @ ( set_set_a2 @ ( undire7337870655677353998dges_a @ ( append_a @ Xs @ ( append_a @ ( cons_a @ Y2 @ nil_a ) @ ( append_a @ Ys2 @ ( append_a @ ( cons_a @ Y2 @ nil_a ) @ Zs ) ) ) ) ) ) ) ).
% comp_sgraph.walk_edges_decomp_ss
thf(fact_20_comp__sgraph_Owalk__edges__append__ss1,axiom,
! [Ys2: list_set_a,Xs: list_set_a] : ( ord_le5722252365846178494_set_a @ ( set_set_set_a2 @ ( undire6234387080713648494_set_a @ Ys2 ) ) @ ( set_set_set_a2 @ ( undire6234387080713648494_set_a @ ( append_set_a @ Xs @ Ys2 ) ) ) ) ).
% comp_sgraph.walk_edges_append_ss1
thf(fact_21_comp__sgraph_Owalk__edges__append__ss1,axiom,
! [Ys2: list_a,Xs: list_a] : ( ord_le3724670747650509150_set_a @ ( set_set_a2 @ ( undire7337870655677353998dges_a @ Ys2 ) ) @ ( set_set_a2 @ ( undire7337870655677353998dges_a @ ( append_a @ Xs @ Ys2 ) ) ) ) ).
% comp_sgraph.walk_edges_append_ss1
thf(fact_22_comp__sgraph_Owalk__edges__append__ss2,axiom,
! [Xs: list_set_a,Ys2: list_set_a] : ( ord_le5722252365846178494_set_a @ ( set_set_set_a2 @ ( undire6234387080713648494_set_a @ Xs ) ) @ ( set_set_set_a2 @ ( undire6234387080713648494_set_a @ ( append_set_a @ Xs @ Ys2 ) ) ) ) ).
% comp_sgraph.walk_edges_append_ss2
thf(fact_23_comp__sgraph_Owalk__edges__append__ss2,axiom,
! [Xs: list_a,Ys2: list_a] : ( ord_le3724670747650509150_set_a @ ( set_set_a2 @ ( undire7337870655677353998dges_a @ Xs ) ) @ ( set_set_a2 @ ( undire7337870655677353998dges_a @ ( append_a @ Xs @ Ys2 ) ) ) ) ).
% comp_sgraph.walk_edges_append_ss2
thf(fact_24__092_060open_062set_A_Ixs_A_064_A_091y_093_A_064_Azs_J_A_092_060subseteq_062_AV_092_060close_062,axiom,
ord_less_eq_set_a @ ( set_a2 @ ( append_a @ xs @ ( append_a @ ( cons_a @ y @ nil_a ) @ zs ) ) ) @ vertices ).
% \<open>set (xs @ [y] @ zs) \<subseteq> V\<close>
thf(fact_25_append1__eq__conv,axiom,
! [Xs: list_a,X: a,Ys2: list_a,Y2: a] :
( ( ( append_a @ Xs @ ( cons_a @ X @ nil_a ) )
= ( append_a @ Ys2 @ ( cons_a @ Y2 @ nil_a ) ) )
= ( ( Xs = Ys2 )
& ( X = Y2 ) ) ) ).
% append1_eq_conv
thf(fact_26_append1__eq__conv,axiom,
! [Xs: list_set_a,X: set_a,Ys2: list_set_a,Y2: set_a] :
( ( ( append_set_a @ Xs @ ( cons_set_a @ X @ nil_set_a ) )
= ( append_set_a @ Ys2 @ ( cons_set_a @ Y2 @ nil_set_a ) ) )
= ( ( Xs = Ys2 )
& ( X = Y2 ) ) ) ).
% append1_eq_conv
thf(fact_27_append_Oright__neutral,axiom,
! [A: list_a] :
( ( append_a @ A @ nil_a )
= A ) ).
% append.right_neutral
thf(fact_28_append_Oright__neutral,axiom,
! [A: list_set_a] :
( ( append_set_a @ A @ nil_set_a )
= A ) ).
% append.right_neutral
thf(fact_29_append__Nil2,axiom,
! [Xs: list_a] :
( ( append_a @ Xs @ nil_a )
= Xs ) ).
% append_Nil2
thf(fact_30_append__Nil2,axiom,
! [Xs: list_set_a] :
( ( append_set_a @ Xs @ nil_set_a )
= Xs ) ).
% append_Nil2
thf(fact_31_append__self__conv,axiom,
! [Xs: list_a,Ys2: list_a] :
( ( ( append_a @ Xs @ Ys2 )
= Xs )
= ( Ys2 = nil_a ) ) ).
% append_self_conv
thf(fact_32_append__self__conv,axiom,
! [Xs: list_set_a,Ys2: list_set_a] :
( ( ( append_set_a @ Xs @ Ys2 )
= Xs )
= ( Ys2 = nil_set_a ) ) ).
% append_self_conv
thf(fact_33_self__append__conv,axiom,
! [Y2: list_a,Ys2: list_a] :
( ( Y2
= ( append_a @ Y2 @ Ys2 ) )
= ( Ys2 = nil_a ) ) ).
% self_append_conv
thf(fact_34_self__append__conv,axiom,
! [Y2: list_set_a,Ys2: list_set_a] :
( ( Y2
= ( append_set_a @ Y2 @ Ys2 ) )
= ( Ys2 = nil_set_a ) ) ).
% self_append_conv
thf(fact_35_append__self__conv2,axiom,
! [Xs: list_a,Ys2: list_a] :
( ( ( append_a @ Xs @ Ys2 )
= Ys2 )
= ( Xs = nil_a ) ) ).
% append_self_conv2
thf(fact_36_append__self__conv2,axiom,
! [Xs: list_set_a,Ys2: list_set_a] :
( ( ( append_set_a @ Xs @ Ys2 )
= Ys2 )
= ( Xs = nil_set_a ) ) ).
% append_self_conv2
thf(fact_37_self__append__conv2,axiom,
! [Y2: list_a,Xs: list_a] :
( ( Y2
= ( append_a @ Xs @ Y2 ) )
= ( Xs = nil_a ) ) ).
% self_append_conv2
thf(fact_38_self__append__conv2,axiom,
! [Y2: list_set_a,Xs: list_set_a] :
( ( Y2
= ( append_set_a @ Xs @ Y2 ) )
= ( Xs = nil_set_a ) ) ).
% self_append_conv2
thf(fact_39_Nil__is__append__conv,axiom,
! [Xs: list_a,Ys2: list_a] :
( ( nil_a
= ( append_a @ Xs @ Ys2 ) )
= ( ( Xs = nil_a )
& ( Ys2 = nil_a ) ) ) ).
% Nil_is_append_conv
thf(fact_40_Nil__is__append__conv,axiom,
! [Xs: list_set_a,Ys2: list_set_a] :
( ( nil_set_a
= ( append_set_a @ Xs @ Ys2 ) )
= ( ( Xs = nil_set_a )
& ( Ys2 = nil_set_a ) ) ) ).
% Nil_is_append_conv
thf(fact_41_append__is__Nil__conv,axiom,
! [Xs: list_a,Ys2: list_a] :
( ( ( append_a @ Xs @ Ys2 )
= nil_a )
= ( ( Xs = nil_a )
& ( Ys2 = nil_a ) ) ) ).
% append_is_Nil_conv
thf(fact_42_append__is__Nil__conv,axiom,
! [Xs: list_set_a,Ys2: list_set_a] :
( ( ( append_set_a @ Xs @ Ys2 )
= nil_set_a )
= ( ( Xs = nil_set_a )
& ( Ys2 = nil_set_a ) ) ) ).
% append_is_Nil_conv
thf(fact_43_is__walkI,axiom,
! [Xs: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ Xs ) @ vertices )
=> ( ( ord_le3724670747650509150_set_a @ ( set_set_a2 @ ( undire7337870655677353998dges_a @ Xs ) ) @ edges )
=> ( ( Xs != nil_a )
=> ( undire6133010728901294956walk_a @ vertices @ edges @ Xs ) ) ) ) ).
% is_walkI
thf(fact_44_walk__edges_Osimps_I2_J,axiom,
! [X: a] :
( ( undire7337870655677353998dges_a @ ( cons_a @ X @ nil_a ) )
= nil_set_a ) ).
% walk_edges.simps(2)
thf(fact_45_walk__edges_Osimps_I1_J,axiom,
( ( undire7337870655677353998dges_a @ nil_a )
= nil_set_a ) ).
% walk_edges.simps(1)
thf(fact_46_list_Oinject,axiom,
! [X21: a,X22: list_a,Y21: a,Y22: list_a] :
( ( ( cons_a @ X21 @ X22 )
= ( cons_a @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% list.inject
thf(fact_47_list_Oinject,axiom,
! [X21: set_a,X22: list_set_a,Y21: set_a,Y22: list_set_a] :
( ( ( cons_set_a @ X21 @ X22 )
= ( cons_set_a @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% list.inject
thf(fact_48_same__append__eq,axiom,
! [Xs: list_a,Ys2: list_a,Zs: list_a] :
( ( ( append_a @ Xs @ Ys2 )
= ( append_a @ Xs @ Zs ) )
= ( Ys2 = Zs ) ) ).
% same_append_eq
thf(fact_49_same__append__eq,axiom,
! [Xs: list_set_a,Ys2: list_set_a,Zs: list_set_a] :
( ( ( append_set_a @ Xs @ Ys2 )
= ( append_set_a @ Xs @ Zs ) )
= ( Ys2 = Zs ) ) ).
% same_append_eq
thf(fact_50_append__same__eq,axiom,
! [Ys2: list_a,Xs: list_a,Zs: list_a] :
( ( ( append_a @ Ys2 @ Xs )
= ( append_a @ Zs @ Xs ) )
= ( Ys2 = Zs ) ) ).
% append_same_eq
thf(fact_51_append__same__eq,axiom,
! [Ys2: list_set_a,Xs: list_set_a,Zs: list_set_a] :
( ( ( append_set_a @ Ys2 @ Xs )
= ( append_set_a @ Zs @ Xs ) )
= ( Ys2 = Zs ) ) ).
% append_same_eq
thf(fact_52_append__assoc,axiom,
! [Xs: list_a,Ys2: list_a,Zs: list_a] :
( ( append_a @ ( append_a @ Xs @ Ys2 ) @ Zs )
= ( append_a @ Xs @ ( append_a @ Ys2 @ Zs ) ) ) ).
% append_assoc
thf(fact_53_append__assoc,axiom,
! [Xs: list_set_a,Ys2: list_set_a,Zs: list_set_a] :
( ( append_set_a @ ( append_set_a @ Xs @ Ys2 ) @ Zs )
= ( append_set_a @ Xs @ ( append_set_a @ Ys2 @ Zs ) ) ) ).
% append_assoc
thf(fact_54_append_Oassoc,axiom,
! [A: list_a,B: list_a,C: list_a] :
( ( append_a @ ( append_a @ A @ B ) @ C )
= ( append_a @ A @ ( append_a @ B @ C ) ) ) ).
% append.assoc
thf(fact_55_append_Oassoc,axiom,
! [A: list_set_a,B: list_set_a,C: list_set_a] :
( ( append_set_a @ ( append_set_a @ A @ B ) @ C )
= ( append_set_a @ A @ ( append_set_a @ B @ C ) ) ) ).
% append.assoc
thf(fact_56_edge__adjacent__alt__def,axiom,
! [E1: set_a,E2: set_a] :
( ( member_set_a @ E1 @ edges )
=> ( ( member_set_a @ E2 @ edges )
=> ( ? [X3: a] :
( ( member_a @ X3 @ vertices )
& ( member_a @ X3 @ E1 )
& ( member_a @ X3 @ E2 ) )
=> ( undire4022703626023482010_adj_a @ edges @ E1 @ E2 ) ) ) ) ).
% edge_adjacent_alt_def
thf(fact_57_is__walk__wf,axiom,
! [Xs: list_a] :
( ( undire6133010728901294956walk_a @ vertices @ edges @ Xs )
=> ( ord_less_eq_set_a @ ( set_a2 @ Xs ) @ vertices ) ) ).
% is_walk_wf
thf(fact_58_ulgraph__axioms,axiom,
undire7251896706689453996raph_a @ vertices @ edges ).
% ulgraph_axioms
thf(fact_59_ulgraph_Ois__walk_Ocong,axiom,
undire6133010728901294956walk_a = undire6133010728901294956walk_a ).
% ulgraph.is_walk.cong
thf(fact_60_not__Cons__self2,axiom,
! [X: a,Xs: list_a] :
( ( cons_a @ X @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_61_not__Cons__self2,axiom,
! [X: set_a,Xs: list_set_a] :
( ( cons_set_a @ X @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_62_append__eq__append__conv2,axiom,
! [Xs: list_a,Ys2: list_a,Zs: list_a,Ts: list_a] :
( ( ( append_a @ Xs @ Ys2 )
= ( append_a @ Zs @ Ts ) )
= ( ? [Us: list_a] :
( ( ( Xs
= ( append_a @ Zs @ Us ) )
& ( ( append_a @ Us @ Ys2 )
= Ts ) )
| ( ( ( append_a @ Xs @ Us )
= Zs )
& ( Ys2
= ( append_a @ Us @ Ts ) ) ) ) ) ) ).
% append_eq_append_conv2
thf(fact_63_append__eq__append__conv2,axiom,
! [Xs: list_set_a,Ys2: list_set_a,Zs: list_set_a,Ts: list_set_a] :
( ( ( append_set_a @ Xs @ Ys2 )
= ( append_set_a @ Zs @ Ts ) )
= ( ? [Us: list_set_a] :
( ( ( Xs
= ( append_set_a @ Zs @ Us ) )
& ( ( append_set_a @ Us @ Ys2 )
= Ts ) )
| ( ( ( append_set_a @ Xs @ Us )
= Zs )
& ( Ys2
= ( append_set_a @ Us @ Ts ) ) ) ) ) ) ).
% append_eq_append_conv2
thf(fact_64_mem__Collect__eq,axiom,
! [A: set_a,P: set_a > $o] :
( ( member_set_a @ A @ ( collect_set_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_65_mem__Collect__eq,axiom,
! [A: a,P: a > $o] :
( ( member_a @ A @ ( collect_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_66_mem__Collect__eq,axiom,
! [A: product_prod_a_a,P: product_prod_a_a > $o] :
( ( member1426531477525435216od_a_a @ A @ ( collec3336397797384452498od_a_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_67_mem__Collect__eq,axiom,
! [A: nat,P: nat > $o] :
( ( member_nat @ A @ ( collect_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_68_Collect__mem__eq,axiom,
! [A2: set_set_a] :
( ( collect_set_a
@ ^ [X4: set_a] : ( member_set_a @ X4 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_69_Collect__mem__eq,axiom,
! [A2: set_a] :
( ( collect_a
@ ^ [X4: a] : ( member_a @ X4 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_70_Collect__mem__eq,axiom,
! [A2: set_Product_prod_a_a] :
( ( collec3336397797384452498od_a_a
@ ^ [X4: product_prod_a_a] : ( member1426531477525435216od_a_a @ X4 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_71_Collect__mem__eq,axiom,
! [A2: set_nat] :
( ( collect_nat
@ ^ [X4: nat] : ( member_nat @ X4 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_72_append__eq__appendI,axiom,
! [Xs: list_a,Xs1: list_a,Zs: list_a,Ys2: list_a,Us2: list_a] :
( ( ( append_a @ Xs @ Xs1 )
= Zs )
=> ( ( Ys2
= ( append_a @ Xs1 @ Us2 ) )
=> ( ( append_a @ Xs @ Ys2 )
= ( append_a @ Zs @ Us2 ) ) ) ) ).
% append_eq_appendI
thf(fact_73_append__eq__appendI,axiom,
! [Xs: list_set_a,Xs1: list_set_a,Zs: list_set_a,Ys2: list_set_a,Us2: list_set_a] :
( ( ( append_set_a @ Xs @ Xs1 )
= Zs )
=> ( ( Ys2
= ( append_set_a @ Xs1 @ Us2 ) )
=> ( ( append_set_a @ Xs @ Ys2 )
= ( append_set_a @ Zs @ Us2 ) ) ) ) ).
% append_eq_appendI
thf(fact_74_list__nonempty__induct,axiom,
! [Xs: list_a,P: list_a > $o] :
( ( Xs != nil_a )
=> ( ! [X2: a] : ( P @ ( cons_a @ X2 @ nil_a ) )
=> ( ! [X2: a,Xs2: list_a] :
( ( Xs2 != nil_a )
=> ( ( P @ Xs2 )
=> ( P @ ( cons_a @ X2 @ Xs2 ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_75_list__nonempty__induct,axiom,
! [Xs: list_set_a,P: list_set_a > $o] :
( ( Xs != nil_set_a )
=> ( ! [X2: set_a] : ( P @ ( cons_set_a @ X2 @ nil_set_a ) )
=> ( ! [X2: set_a,Xs2: list_set_a] :
( ( Xs2 != nil_set_a )
=> ( ( P @ Xs2 )
=> ( P @ ( cons_set_a @ X2 @ Xs2 ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_76_list__induct2_H,axiom,
! [P: list_a > list_a > $o,Xs: list_a,Ys2: list_a] :
( ( P @ nil_a @ nil_a )
=> ( ! [X2: a,Xs2: list_a] : ( P @ ( cons_a @ X2 @ Xs2 ) @ nil_a )
=> ( ! [Y: a,Ys: list_a] : ( P @ nil_a @ ( cons_a @ Y @ Ys ) )
=> ( ! [X2: a,Xs2: list_a,Y: a,Ys: list_a] :
( ( P @ Xs2 @ Ys )
=> ( P @ ( cons_a @ X2 @ Xs2 ) @ ( cons_a @ Y @ Ys ) ) )
=> ( P @ Xs @ Ys2 ) ) ) ) ) ).
% list_induct2'
thf(fact_77_list__induct2_H,axiom,
! [P: list_a > list_set_a > $o,Xs: list_a,Ys2: list_set_a] :
( ( P @ nil_a @ nil_set_a )
=> ( ! [X2: a,Xs2: list_a] : ( P @ ( cons_a @ X2 @ Xs2 ) @ nil_set_a )
=> ( ! [Y: set_a,Ys: list_set_a] : ( P @ nil_a @ ( cons_set_a @ Y @ Ys ) )
=> ( ! [X2: a,Xs2: list_a,Y: set_a,Ys: list_set_a] :
( ( P @ Xs2 @ Ys )
=> ( P @ ( cons_a @ X2 @ Xs2 ) @ ( cons_set_a @ Y @ Ys ) ) )
=> ( P @ Xs @ Ys2 ) ) ) ) ) ).
% list_induct2'
thf(fact_78_list__induct2_H,axiom,
! [P: list_set_a > list_a > $o,Xs: list_set_a,Ys2: list_a] :
( ( P @ nil_set_a @ nil_a )
=> ( ! [X2: set_a,Xs2: list_set_a] : ( P @ ( cons_set_a @ X2 @ Xs2 ) @ nil_a )
=> ( ! [Y: a,Ys: list_a] : ( P @ nil_set_a @ ( cons_a @ Y @ Ys ) )
=> ( ! [X2: set_a,Xs2: list_set_a,Y: a,Ys: list_a] :
( ( P @ Xs2 @ Ys )
=> ( P @ ( cons_set_a @ X2 @ Xs2 ) @ ( cons_a @ Y @ Ys ) ) )
=> ( P @ Xs @ Ys2 ) ) ) ) ) ).
% list_induct2'
thf(fact_79_list__induct2_H,axiom,
! [P: list_set_a > list_set_a > $o,Xs: list_set_a,Ys2: list_set_a] :
( ( P @ nil_set_a @ nil_set_a )
=> ( ! [X2: set_a,Xs2: list_set_a] : ( P @ ( cons_set_a @ X2 @ Xs2 ) @ nil_set_a )
=> ( ! [Y: set_a,Ys: list_set_a] : ( P @ nil_set_a @ ( cons_set_a @ Y @ Ys ) )
=> ( ! [X2: set_a,Xs2: list_set_a,Y: set_a,Ys: list_set_a] :
( ( P @ Xs2 @ Ys )
=> ( P @ ( cons_set_a @ X2 @ Xs2 ) @ ( cons_set_a @ Y @ Ys ) ) )
=> ( P @ Xs @ Ys2 ) ) ) ) ) ).
% list_induct2'
thf(fact_80_neq__Nil__conv,axiom,
! [Xs: list_a] :
( ( Xs != nil_a )
= ( ? [Y3: a,Ys3: list_a] :
( Xs
= ( cons_a @ Y3 @ Ys3 ) ) ) ) ).
% neq_Nil_conv
thf(fact_81_neq__Nil__conv,axiom,
! [Xs: list_set_a] :
( ( Xs != nil_set_a )
= ( ? [Y3: set_a,Ys3: list_set_a] :
( Xs
= ( cons_set_a @ Y3 @ Ys3 ) ) ) ) ).
% neq_Nil_conv
thf(fact_82_transpose_Ocases,axiom,
! [X: list_list_a] :
( ( X != nil_list_a )
=> ( ! [Xss: list_list_a] :
( X
!= ( cons_list_a @ nil_a @ Xss ) )
=> ~ ! [X2: a,Xs2: list_a,Xss: list_list_a] :
( X
!= ( cons_list_a @ ( cons_a @ X2 @ Xs2 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_83_transpose_Ocases,axiom,
! [X: list_list_set_a] :
( ( X != nil_list_set_a )
=> ( ! [Xss: list_list_set_a] :
( X
!= ( cons_list_set_a @ nil_set_a @ Xss ) )
=> ~ ! [X2: set_a,Xs2: list_set_a,Xss: list_list_set_a] :
( X
!= ( cons_list_set_a @ ( cons_set_a @ X2 @ Xs2 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_84_min__list_Ocases,axiom,
! [X: list_set_a] :
( ! [X2: set_a,Xs2: list_set_a] :
( X
!= ( cons_set_a @ X2 @ Xs2 ) )
=> ( X = nil_set_a ) ) ).
% min_list.cases
thf(fact_85_list_Oexhaust,axiom,
! [Y2: list_a] :
( ( Y2 != nil_a )
=> ~ ! [X212: a,X222: list_a] :
( Y2
!= ( cons_a @ X212 @ X222 ) ) ) ).
% list.exhaust
thf(fact_86_list_Oexhaust,axiom,
! [Y2: list_set_a] :
( ( Y2 != nil_set_a )
=> ~ ! [X212: set_a,X222: list_set_a] :
( Y2
!= ( cons_set_a @ X212 @ X222 ) ) ) ).
% list.exhaust
thf(fact_87_list_OdiscI,axiom,
! [List: list_a,X21: a,X22: list_a] :
( ( List
= ( cons_a @ X21 @ X22 ) )
=> ( List != nil_a ) ) ).
% list.discI
thf(fact_88_list_OdiscI,axiom,
! [List: list_set_a,X21: set_a,X22: list_set_a] :
( ( List
= ( cons_set_a @ X21 @ X22 ) )
=> ( List != nil_set_a ) ) ).
% list.discI
thf(fact_89_list_Odistinct_I1_J,axiom,
! [X21: a,X22: list_a] :
( nil_a
!= ( cons_a @ X21 @ X22 ) ) ).
% list.distinct(1)
thf(fact_90_list_Odistinct_I1_J,axiom,
! [X21: set_a,X22: list_set_a] :
( nil_set_a
!= ( cons_set_a @ X21 @ X22 ) ) ).
% list.distinct(1)
thf(fact_91_subset__code_I1_J,axiom,
! [Xs: list_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ B2 )
= ( ! [X4: nat] :
( ( member_nat @ X4 @ ( set_nat2 @ Xs ) )
=> ( member_nat @ X4 @ B2 ) ) ) ) ).
% subset_code(1)
thf(fact_92_subset__code_I1_J,axiom,
! [Xs: list_set_a,B2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ ( set_set_a2 @ Xs ) @ B2 )
= ( ! [X4: set_a] :
( ( member_set_a @ X4 @ ( set_set_a2 @ Xs ) )
=> ( member_set_a @ X4 @ B2 ) ) ) ) ).
% subset_code(1)
thf(fact_93_subset__code_I1_J,axiom,
! [Xs: list_a,B2: set_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ Xs ) @ B2 )
= ( ! [X4: a] :
( ( member_a @ X4 @ ( set_a2 @ Xs ) )
=> ( member_a @ X4 @ B2 ) ) ) ) ).
% subset_code(1)
thf(fact_94_subset__code_I1_J,axiom,
! [Xs: list_P1396940483166286381od_a_a,B2: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ ( set_Product_prod_a_a2 @ Xs ) @ B2 )
= ( ! [X4: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X4 @ ( set_Product_prod_a_a2 @ Xs ) )
=> ( member1426531477525435216od_a_a @ X4 @ B2 ) ) ) ) ).
% subset_code(1)
thf(fact_95_set__ConsD,axiom,
! [Y2: product_prod_a_a,X: product_prod_a_a,Xs: list_P1396940483166286381od_a_a] :
( ( member1426531477525435216od_a_a @ Y2 @ ( set_Product_prod_a_a2 @ ( cons_P7316939126706565853od_a_a @ X @ Xs ) ) )
=> ( ( Y2 = X )
| ( member1426531477525435216od_a_a @ Y2 @ ( set_Product_prod_a_a2 @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_96_set__ConsD,axiom,
! [Y2: nat,X: nat,Xs: list_nat] :
( ( member_nat @ Y2 @ ( set_nat2 @ ( cons_nat @ X @ Xs ) ) )
=> ( ( Y2 = X )
| ( member_nat @ Y2 @ ( set_nat2 @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_97_set__ConsD,axiom,
! [Y2: a,X: a,Xs: list_a] :
( ( member_a @ Y2 @ ( set_a2 @ ( cons_a @ X @ Xs ) ) )
=> ( ( Y2 = X )
| ( member_a @ Y2 @ ( set_a2 @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_98_set__ConsD,axiom,
! [Y2: set_a,X: set_a,Xs: list_set_a] :
( ( member_set_a @ Y2 @ ( set_set_a2 @ ( cons_set_a @ X @ Xs ) ) )
=> ( ( Y2 = X )
| ( member_set_a @ Y2 @ ( set_set_a2 @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_99_list_Oset__cases,axiom,
! [E: product_prod_a_a,A: list_P1396940483166286381od_a_a] :
( ( member1426531477525435216od_a_a @ E @ ( set_Product_prod_a_a2 @ A ) )
=> ( ! [Z2: list_P1396940483166286381od_a_a] :
( A
!= ( cons_P7316939126706565853od_a_a @ E @ Z2 ) )
=> ~ ! [Z1: product_prod_a_a,Z2: list_P1396940483166286381od_a_a] :
( ( A
= ( cons_P7316939126706565853od_a_a @ Z1 @ Z2 ) )
=> ~ ( member1426531477525435216od_a_a @ E @ ( set_Product_prod_a_a2 @ Z2 ) ) ) ) ) ).
% list.set_cases
thf(fact_100_list_Oset__cases,axiom,
! [E: nat,A: list_nat] :
( ( member_nat @ E @ ( set_nat2 @ A ) )
=> ( ! [Z2: list_nat] :
( A
!= ( cons_nat @ E @ Z2 ) )
=> ~ ! [Z1: nat,Z2: list_nat] :
( ( A
= ( cons_nat @ Z1 @ Z2 ) )
=> ~ ( member_nat @ E @ ( set_nat2 @ Z2 ) ) ) ) ) ).
% list.set_cases
thf(fact_101_list_Oset__cases,axiom,
! [E: a,A: list_a] :
( ( member_a @ E @ ( set_a2 @ A ) )
=> ( ! [Z2: list_a] :
( A
!= ( cons_a @ E @ Z2 ) )
=> ~ ! [Z1: a,Z2: list_a] :
( ( A
= ( cons_a @ Z1 @ Z2 ) )
=> ~ ( member_a @ E @ ( set_a2 @ Z2 ) ) ) ) ) ).
% list.set_cases
thf(fact_102_list_Oset__cases,axiom,
! [E: set_a,A: list_set_a] :
( ( member_set_a @ E @ ( set_set_a2 @ A ) )
=> ( ! [Z2: list_set_a] :
( A
!= ( cons_set_a @ E @ Z2 ) )
=> ~ ! [Z1: set_a,Z2: list_set_a] :
( ( A
= ( cons_set_a @ Z1 @ Z2 ) )
=> ~ ( member_set_a @ E @ ( set_set_a2 @ Z2 ) ) ) ) ) ).
% list.set_cases
thf(fact_103_list_Oset__intros_I1_J,axiom,
! [X21: product_prod_a_a,X22: list_P1396940483166286381od_a_a] : ( member1426531477525435216od_a_a @ X21 @ ( set_Product_prod_a_a2 @ ( cons_P7316939126706565853od_a_a @ X21 @ X22 ) ) ) ).
% list.set_intros(1)
thf(fact_104_list_Oset__intros_I1_J,axiom,
! [X21: nat,X22: list_nat] : ( member_nat @ X21 @ ( set_nat2 @ ( cons_nat @ X21 @ X22 ) ) ) ).
% list.set_intros(1)
thf(fact_105_list_Oset__intros_I1_J,axiom,
! [X21: a,X22: list_a] : ( member_a @ X21 @ ( set_a2 @ ( cons_a @ X21 @ X22 ) ) ) ).
% list.set_intros(1)
thf(fact_106_list_Oset__intros_I1_J,axiom,
! [X21: set_a,X22: list_set_a] : ( member_set_a @ X21 @ ( set_set_a2 @ ( cons_set_a @ X21 @ X22 ) ) ) ).
% list.set_intros(1)
thf(fact_107_list_Oset__intros_I2_J,axiom,
! [Y2: product_prod_a_a,X22: list_P1396940483166286381od_a_a,X21: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ Y2 @ ( set_Product_prod_a_a2 @ X22 ) )
=> ( member1426531477525435216od_a_a @ Y2 @ ( set_Product_prod_a_a2 @ ( cons_P7316939126706565853od_a_a @ X21 @ X22 ) ) ) ) ).
% list.set_intros(2)
thf(fact_108_list_Oset__intros_I2_J,axiom,
! [Y2: nat,X22: list_nat,X21: nat] :
( ( member_nat @ Y2 @ ( set_nat2 @ X22 ) )
=> ( member_nat @ Y2 @ ( set_nat2 @ ( cons_nat @ X21 @ X22 ) ) ) ) ).
% list.set_intros(2)
thf(fact_109_list_Oset__intros_I2_J,axiom,
! [Y2: a,X22: list_a,X21: a] :
( ( member_a @ Y2 @ ( set_a2 @ X22 ) )
=> ( member_a @ Y2 @ ( set_a2 @ ( cons_a @ X21 @ X22 ) ) ) ) ).
% list.set_intros(2)
thf(fact_110_list_Oset__intros_I2_J,axiom,
! [Y2: set_a,X22: list_set_a,X21: set_a] :
( ( member_set_a @ Y2 @ ( set_set_a2 @ X22 ) )
=> ( member_set_a @ Y2 @ ( set_set_a2 @ ( cons_set_a @ X21 @ X22 ) ) ) ) ).
% list.set_intros(2)
thf(fact_111_Cons__eq__appendI,axiom,
! [X: a,Xs1: list_a,Ys2: list_a,Xs: list_a,Zs: list_a] :
( ( ( cons_a @ X @ Xs1 )
= Ys2 )
=> ( ( Xs
= ( append_a @ Xs1 @ Zs ) )
=> ( ( cons_a @ X @ Xs )
= ( append_a @ Ys2 @ Zs ) ) ) ) ).
% Cons_eq_appendI
thf(fact_112_Cons__eq__appendI,axiom,
! [X: set_a,Xs1: list_set_a,Ys2: list_set_a,Xs: list_set_a,Zs: list_set_a] :
( ( ( cons_set_a @ X @ Xs1 )
= Ys2 )
=> ( ( Xs
= ( append_set_a @ Xs1 @ Zs ) )
=> ( ( cons_set_a @ X @ Xs )
= ( append_set_a @ Ys2 @ Zs ) ) ) ) ).
% Cons_eq_appendI
thf(fact_113_append__Cons,axiom,
! [X: a,Xs: list_a,Ys2: list_a] :
( ( append_a @ ( cons_a @ X @ Xs ) @ Ys2 )
= ( cons_a @ X @ ( append_a @ Xs @ Ys2 ) ) ) ).
% append_Cons
thf(fact_114_append__Cons,axiom,
! [X: set_a,Xs: list_set_a,Ys2: list_set_a] :
( ( append_set_a @ ( cons_set_a @ X @ Xs ) @ Ys2 )
= ( cons_set_a @ X @ ( append_set_a @ Xs @ Ys2 ) ) ) ).
% append_Cons
thf(fact_115_eq__Nil__appendI,axiom,
! [Xs: list_a,Ys2: list_a] :
( ( Xs = Ys2 )
=> ( Xs
= ( append_a @ nil_a @ Ys2 ) ) ) ).
% eq_Nil_appendI
thf(fact_116_eq__Nil__appendI,axiom,
! [Xs: list_set_a,Ys2: list_set_a] :
( ( Xs = Ys2 )
=> ( Xs
= ( append_set_a @ nil_set_a @ Ys2 ) ) ) ).
% eq_Nil_appendI
thf(fact_117_append_Oleft__neutral,axiom,
! [A: list_a] :
( ( append_a @ nil_a @ A )
= A ) ).
% append.left_neutral
thf(fact_118_append_Oleft__neutral,axiom,
! [A: list_set_a] :
( ( append_set_a @ nil_set_a @ A )
= A ) ).
% append.left_neutral
thf(fact_119_append__Nil,axiom,
! [Ys2: list_a] :
( ( append_a @ nil_a @ Ys2 )
= Ys2 ) ).
% append_Nil
thf(fact_120_append__Nil,axiom,
! [Ys2: list_set_a] :
( ( append_set_a @ nil_set_a @ Ys2 )
= Ys2 ) ).
% append_Nil
thf(fact_121_set__subset__Cons,axiom,
! [Xs: list_set_a,X: set_a] : ( ord_le3724670747650509150_set_a @ ( set_set_a2 @ Xs ) @ ( set_set_a2 @ ( cons_set_a @ X @ Xs ) ) ) ).
% set_subset_Cons
thf(fact_122_set__subset__Cons,axiom,
! [Xs: list_a,X: a] : ( ord_less_eq_set_a @ ( set_a2 @ Xs ) @ ( set_a2 @ ( cons_a @ X @ Xs ) ) ) ).
% set_subset_Cons
thf(fact_123_set__subset__Cons,axiom,
! [Xs: list_P1396940483166286381od_a_a,X: product_prod_a_a] : ( ord_le746702958409616551od_a_a @ ( set_Product_prod_a_a2 @ Xs ) @ ( set_Product_prod_a_a2 @ ( cons_P7316939126706565853od_a_a @ X @ Xs ) ) ) ).
% set_subset_Cons
thf(fact_124_rev__nonempty__induct,axiom,
! [Xs: list_a,P: list_a > $o] :
( ( Xs != nil_a )
=> ( ! [X2: a] : ( P @ ( cons_a @ X2 @ nil_a ) )
=> ( ! [X2: a,Xs2: list_a] :
( ( Xs2 != nil_a )
=> ( ( P @ Xs2 )
=> ( P @ ( append_a @ Xs2 @ ( cons_a @ X2 @ nil_a ) ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% rev_nonempty_induct
thf(fact_125_rev__nonempty__induct,axiom,
! [Xs: list_set_a,P: list_set_a > $o] :
( ( Xs != nil_set_a )
=> ( ! [X2: set_a] : ( P @ ( cons_set_a @ X2 @ nil_set_a ) )
=> ( ! [X2: set_a,Xs2: list_set_a] :
( ( Xs2 != nil_set_a )
=> ( ( P @ Xs2 )
=> ( P @ ( append_set_a @ Xs2 @ ( cons_set_a @ X2 @ nil_set_a ) ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% rev_nonempty_induct
thf(fact_126_append__eq__Cons__conv,axiom,
! [Ys2: list_a,Zs: list_a,X: a,Xs: list_a] :
( ( ( append_a @ Ys2 @ Zs )
= ( cons_a @ X @ Xs ) )
= ( ( ( Ys2 = nil_a )
& ( Zs
= ( cons_a @ X @ Xs ) ) )
| ? [Ys4: list_a] :
( ( Ys2
= ( cons_a @ X @ Ys4 ) )
& ( ( append_a @ Ys4 @ Zs )
= Xs ) ) ) ) ).
% append_eq_Cons_conv
thf(fact_127_append__eq__Cons__conv,axiom,
! [Ys2: list_set_a,Zs: list_set_a,X: set_a,Xs: list_set_a] :
( ( ( append_set_a @ Ys2 @ Zs )
= ( cons_set_a @ X @ Xs ) )
= ( ( ( Ys2 = nil_set_a )
& ( Zs
= ( cons_set_a @ X @ Xs ) ) )
| ? [Ys4: list_set_a] :
( ( Ys2
= ( cons_set_a @ X @ Ys4 ) )
& ( ( append_set_a @ Ys4 @ Zs )
= Xs ) ) ) ) ).
% append_eq_Cons_conv
thf(fact_128_Cons__eq__append__conv,axiom,
! [X: a,Xs: list_a,Ys2: list_a,Zs: list_a] :
( ( ( cons_a @ X @ Xs )
= ( append_a @ Ys2 @ Zs ) )
= ( ( ( Ys2 = nil_a )
& ( ( cons_a @ X @ Xs )
= Zs ) )
| ? [Ys4: list_a] :
( ( ( cons_a @ X @ Ys4 )
= Ys2 )
& ( Xs
= ( append_a @ Ys4 @ Zs ) ) ) ) ) ).
% Cons_eq_append_conv
thf(fact_129_Cons__eq__append__conv,axiom,
! [X: set_a,Xs: list_set_a,Ys2: list_set_a,Zs: list_set_a] :
( ( ( cons_set_a @ X @ Xs )
= ( append_set_a @ Ys2 @ Zs ) )
= ( ( ( Ys2 = nil_set_a )
& ( ( cons_set_a @ X @ Xs )
= Zs ) )
| ? [Ys4: list_set_a] :
( ( ( cons_set_a @ X @ Ys4 )
= Ys2 )
& ( Xs
= ( append_set_a @ Ys4 @ Zs ) ) ) ) ) ).
% Cons_eq_append_conv
thf(fact_130_rev__exhaust,axiom,
! [Xs: list_a] :
( ( Xs != nil_a )
=> ~ ! [Ys: list_a,Y: a] :
( Xs
!= ( append_a @ Ys @ ( cons_a @ Y @ nil_a ) ) ) ) ).
% rev_exhaust
thf(fact_131_rev__exhaust,axiom,
! [Xs: list_set_a] :
( ( Xs != nil_set_a )
=> ~ ! [Ys: list_set_a,Y: set_a] :
( Xs
!= ( append_set_a @ Ys @ ( cons_set_a @ Y @ nil_set_a ) ) ) ) ).
% rev_exhaust
thf(fact_132_rev__induct,axiom,
! [P: list_a > $o,Xs: list_a] :
( ( P @ nil_a )
=> ( ! [X2: a,Xs2: list_a] :
( ( P @ Xs2 )
=> ( P @ ( append_a @ Xs2 @ ( cons_a @ X2 @ nil_a ) ) ) )
=> ( P @ Xs ) ) ) ).
% rev_induct
thf(fact_133_rev__induct,axiom,
! [P: list_set_a > $o,Xs: list_set_a] :
( ( P @ nil_set_a )
=> ( ! [X2: set_a,Xs2: list_set_a] :
( ( P @ Xs2 )
=> ( P @ ( append_set_a @ Xs2 @ ( cons_set_a @ X2 @ nil_set_a ) ) ) )
=> ( P @ Xs ) ) ) ).
% rev_induct
thf(fact_134_split__list__first__prop__iff,axiom,
! [Xs: list_a,P: a > $o] :
( ( ? [X4: a] :
( ( member_a @ X4 @ ( set_a2 @ Xs ) )
& ( P @ X4 ) ) )
= ( ? [Ys3: list_a,X4: a] :
( ? [Zs2: list_a] :
( Xs
= ( append_a @ Ys3 @ ( cons_a @ X4 @ Zs2 ) ) )
& ( P @ X4 )
& ! [Y3: a] :
( ( member_a @ Y3 @ ( set_a2 @ Ys3 ) )
=> ~ ( P @ Y3 ) ) ) ) ) ).
% split_list_first_prop_iff
thf(fact_135_split__list__first__prop__iff,axiom,
! [Xs: list_set_a,P: set_a > $o] :
( ( ? [X4: set_a] :
( ( member_set_a @ X4 @ ( set_set_a2 @ Xs ) )
& ( P @ X4 ) ) )
= ( ? [Ys3: list_set_a,X4: set_a] :
( ? [Zs2: list_set_a] :
( Xs
= ( append_set_a @ Ys3 @ ( cons_set_a @ X4 @ Zs2 ) ) )
& ( P @ X4 )
& ! [Y3: set_a] :
( ( member_set_a @ Y3 @ ( set_set_a2 @ Ys3 ) )
=> ~ ( P @ Y3 ) ) ) ) ) ).
% split_list_first_prop_iff
thf(fact_136_split__list__last__prop__iff,axiom,
! [Xs: list_a,P: a > $o] :
( ( ? [X4: a] :
( ( member_a @ X4 @ ( set_a2 @ Xs ) )
& ( P @ X4 ) ) )
= ( ? [Ys3: list_a,X4: a,Zs2: list_a] :
( ( Xs
= ( append_a @ Ys3 @ ( cons_a @ X4 @ Zs2 ) ) )
& ( P @ X4 )
& ! [Y3: a] :
( ( member_a @ Y3 @ ( set_a2 @ Zs2 ) )
=> ~ ( P @ Y3 ) ) ) ) ) ).
% split_list_last_prop_iff
thf(fact_137_split__list__last__prop__iff,axiom,
! [Xs: list_set_a,P: set_a > $o] :
( ( ? [X4: set_a] :
( ( member_set_a @ X4 @ ( set_set_a2 @ Xs ) )
& ( P @ X4 ) ) )
= ( ? [Ys3: list_set_a,X4: set_a,Zs2: list_set_a] :
( ( Xs
= ( append_set_a @ Ys3 @ ( cons_set_a @ X4 @ Zs2 ) ) )
& ( P @ X4 )
& ! [Y3: set_a] :
( ( member_set_a @ Y3 @ ( set_set_a2 @ Zs2 ) )
=> ~ ( P @ Y3 ) ) ) ) ) ).
% split_list_last_prop_iff
thf(fact_138_in__set__conv__decomp__first,axiom,
! [X: product_prod_a_a,Xs: list_P1396940483166286381od_a_a] :
( ( member1426531477525435216od_a_a @ X @ ( set_Product_prod_a_a2 @ Xs ) )
= ( ? [Ys3: list_P1396940483166286381od_a_a,Zs2: list_P1396940483166286381od_a_a] :
( ( Xs
= ( append5335208819046833346od_a_a @ Ys3 @ ( cons_P7316939126706565853od_a_a @ X @ Zs2 ) ) )
& ~ ( member1426531477525435216od_a_a @ X @ ( set_Product_prod_a_a2 @ Ys3 ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_139_in__set__conv__decomp__first,axiom,
! [X: nat,Xs: list_nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs ) )
= ( ? [Ys3: list_nat,Zs2: list_nat] :
( ( Xs
= ( append_nat @ Ys3 @ ( cons_nat @ X @ Zs2 ) ) )
& ~ ( member_nat @ X @ ( set_nat2 @ Ys3 ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_140_in__set__conv__decomp__first,axiom,
! [X: a,Xs: list_a] :
( ( member_a @ X @ ( set_a2 @ Xs ) )
= ( ? [Ys3: list_a,Zs2: list_a] :
( ( Xs
= ( append_a @ Ys3 @ ( cons_a @ X @ Zs2 ) ) )
& ~ ( member_a @ X @ ( set_a2 @ Ys3 ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_141_in__set__conv__decomp__first,axiom,
! [X: set_a,Xs: list_set_a] :
( ( member_set_a @ X @ ( set_set_a2 @ Xs ) )
= ( ? [Ys3: list_set_a,Zs2: list_set_a] :
( ( Xs
= ( append_set_a @ Ys3 @ ( cons_set_a @ X @ Zs2 ) ) )
& ~ ( member_set_a @ X @ ( set_set_a2 @ Ys3 ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_142_in__set__conv__decomp__last,axiom,
! [X: product_prod_a_a,Xs: list_P1396940483166286381od_a_a] :
( ( member1426531477525435216od_a_a @ X @ ( set_Product_prod_a_a2 @ Xs ) )
= ( ? [Ys3: list_P1396940483166286381od_a_a,Zs2: list_P1396940483166286381od_a_a] :
( ( Xs
= ( append5335208819046833346od_a_a @ Ys3 @ ( cons_P7316939126706565853od_a_a @ X @ Zs2 ) ) )
& ~ ( member1426531477525435216od_a_a @ X @ ( set_Product_prod_a_a2 @ Zs2 ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_143_in__set__conv__decomp__last,axiom,
! [X: nat,Xs: list_nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs ) )
= ( ? [Ys3: list_nat,Zs2: list_nat] :
( ( Xs
= ( append_nat @ Ys3 @ ( cons_nat @ X @ Zs2 ) ) )
& ~ ( member_nat @ X @ ( set_nat2 @ Zs2 ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_144_in__set__conv__decomp__last,axiom,
! [X: a,Xs: list_a] :
( ( member_a @ X @ ( set_a2 @ Xs ) )
= ( ? [Ys3: list_a,Zs2: list_a] :
( ( Xs
= ( append_a @ Ys3 @ ( cons_a @ X @ Zs2 ) ) )
& ~ ( member_a @ X @ ( set_a2 @ Zs2 ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_145_in__set__conv__decomp__last,axiom,
! [X: set_a,Xs: list_set_a] :
( ( member_set_a @ X @ ( set_set_a2 @ Xs ) )
= ( ? [Ys3: list_set_a,Zs2: list_set_a] :
( ( Xs
= ( append_set_a @ Ys3 @ ( cons_set_a @ X @ Zs2 ) ) )
& ~ ( member_set_a @ X @ ( set_set_a2 @ Zs2 ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_146_split__list__first__propE,axiom,
! [Xs: list_a,P: a > $o] :
( ? [X3: a] :
( ( member_a @ X3 @ ( set_a2 @ Xs ) )
& ( P @ X3 ) )
=> ~ ! [Ys: list_a,X2: a] :
( ? [Zs3: list_a] :
( Xs
= ( append_a @ Ys @ ( cons_a @ X2 @ Zs3 ) ) )
=> ( ( P @ X2 )
=> ~ ! [Xa: a] :
( ( member_a @ Xa @ ( set_a2 @ Ys ) )
=> ~ ( P @ Xa ) ) ) ) ) ).
% split_list_first_propE
thf(fact_147_split__list__first__propE,axiom,
! [Xs: list_set_a,P: set_a > $o] :
( ? [X3: set_a] :
( ( member_set_a @ X3 @ ( set_set_a2 @ Xs ) )
& ( P @ X3 ) )
=> ~ ! [Ys: list_set_a,X2: set_a] :
( ? [Zs3: list_set_a] :
( Xs
= ( append_set_a @ Ys @ ( cons_set_a @ X2 @ Zs3 ) ) )
=> ( ( P @ X2 )
=> ~ ! [Xa: set_a] :
( ( member_set_a @ Xa @ ( set_set_a2 @ Ys ) )
=> ~ ( P @ Xa ) ) ) ) ) ).
% split_list_first_propE
thf(fact_148_split__list__last__propE,axiom,
! [Xs: list_a,P: a > $o] :
( ? [X3: a] :
( ( member_a @ X3 @ ( set_a2 @ Xs ) )
& ( P @ X3 ) )
=> ~ ! [Ys: list_a,X2: a,Zs3: list_a] :
( ( Xs
= ( append_a @ Ys @ ( cons_a @ X2 @ Zs3 ) ) )
=> ( ( P @ X2 )
=> ~ ! [Xa: a] :
( ( member_a @ Xa @ ( set_a2 @ Zs3 ) )
=> ~ ( P @ Xa ) ) ) ) ) ).
% split_list_last_propE
thf(fact_149_split__list__last__propE,axiom,
! [Xs: list_set_a,P: set_a > $o] :
( ? [X3: set_a] :
( ( member_set_a @ X3 @ ( set_set_a2 @ Xs ) )
& ( P @ X3 ) )
=> ~ ! [Ys: list_set_a,X2: set_a,Zs3: list_set_a] :
( ( Xs
= ( append_set_a @ Ys @ ( cons_set_a @ X2 @ Zs3 ) ) )
=> ( ( P @ X2 )
=> ~ ! [Xa: set_a] :
( ( member_set_a @ Xa @ ( set_set_a2 @ Zs3 ) )
=> ~ ( P @ Xa ) ) ) ) ) ).
% split_list_last_propE
thf(fact_150_split__list__first__prop,axiom,
! [Xs: list_a,P: a > $o] :
( ? [X3: a] :
( ( member_a @ X3 @ ( set_a2 @ Xs ) )
& ( P @ X3 ) )
=> ? [Ys: list_a,X2: a] :
( ? [Zs3: list_a] :
( Xs
= ( append_a @ Ys @ ( cons_a @ X2 @ Zs3 ) ) )
& ( P @ X2 )
& ! [Xa: a] :
( ( member_a @ Xa @ ( set_a2 @ Ys ) )
=> ~ ( P @ Xa ) ) ) ) ).
% split_list_first_prop
thf(fact_151_split__list__first__prop,axiom,
! [Xs: list_set_a,P: set_a > $o] :
( ? [X3: set_a] :
( ( member_set_a @ X3 @ ( set_set_a2 @ Xs ) )
& ( P @ X3 ) )
=> ? [Ys: list_set_a,X2: set_a] :
( ? [Zs3: list_set_a] :
( Xs
= ( append_set_a @ Ys @ ( cons_set_a @ X2 @ Zs3 ) ) )
& ( P @ X2 )
& ! [Xa: set_a] :
( ( member_set_a @ Xa @ ( set_set_a2 @ Ys ) )
=> ~ ( P @ Xa ) ) ) ) ).
% split_list_first_prop
thf(fact_152_split__list__last__prop,axiom,
! [Xs: list_a,P: a > $o] :
( ? [X3: a] :
( ( member_a @ X3 @ ( set_a2 @ Xs ) )
& ( P @ X3 ) )
=> ? [Ys: list_a,X2: a,Zs3: list_a] :
( ( Xs
= ( append_a @ Ys @ ( cons_a @ X2 @ Zs3 ) ) )
& ( P @ X2 )
& ! [Xa: a] :
( ( member_a @ Xa @ ( set_a2 @ Zs3 ) )
=> ~ ( P @ Xa ) ) ) ) ).
% split_list_last_prop
thf(fact_153_split__list__last__prop,axiom,
! [Xs: list_set_a,P: set_a > $o] :
( ? [X3: set_a] :
( ( member_set_a @ X3 @ ( set_set_a2 @ Xs ) )
& ( P @ X3 ) )
=> ? [Ys: list_set_a,X2: set_a,Zs3: list_set_a] :
( ( Xs
= ( append_set_a @ Ys @ ( cons_set_a @ X2 @ Zs3 ) ) )
& ( P @ X2 )
& ! [Xa: set_a] :
( ( member_set_a @ Xa @ ( set_set_a2 @ Zs3 ) )
=> ~ ( P @ Xa ) ) ) ) ).
% split_list_last_prop
thf(fact_154_in__set__conv__decomp,axiom,
! [X: product_prod_a_a,Xs: list_P1396940483166286381od_a_a] :
( ( member1426531477525435216od_a_a @ X @ ( set_Product_prod_a_a2 @ Xs ) )
= ( ? [Ys3: list_P1396940483166286381od_a_a,Zs2: list_P1396940483166286381od_a_a] :
( Xs
= ( append5335208819046833346od_a_a @ Ys3 @ ( cons_P7316939126706565853od_a_a @ X @ Zs2 ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_155_in__set__conv__decomp,axiom,
! [X: nat,Xs: list_nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs ) )
= ( ? [Ys3: list_nat,Zs2: list_nat] :
( Xs
= ( append_nat @ Ys3 @ ( cons_nat @ X @ Zs2 ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_156_in__set__conv__decomp,axiom,
! [X: a,Xs: list_a] :
( ( member_a @ X @ ( set_a2 @ Xs ) )
= ( ? [Ys3: list_a,Zs2: list_a] :
( Xs
= ( append_a @ Ys3 @ ( cons_a @ X @ Zs2 ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_157_in__set__conv__decomp,axiom,
! [X: set_a,Xs: list_set_a] :
( ( member_set_a @ X @ ( set_set_a2 @ Xs ) )
= ( ? [Ys3: list_set_a,Zs2: list_set_a] :
( Xs
= ( append_set_a @ Ys3 @ ( cons_set_a @ X @ Zs2 ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_158_append__Cons__eq__iff,axiom,
! [X: product_prod_a_a,Xs: list_P1396940483166286381od_a_a,Ys2: list_P1396940483166286381od_a_a,Xs3: list_P1396940483166286381od_a_a,Ys5: list_P1396940483166286381od_a_a] :
( ~ ( member1426531477525435216od_a_a @ X @ ( set_Product_prod_a_a2 @ Xs ) )
=> ( ~ ( member1426531477525435216od_a_a @ X @ ( set_Product_prod_a_a2 @ Ys2 ) )
=> ( ( ( append5335208819046833346od_a_a @ Xs @ ( cons_P7316939126706565853od_a_a @ X @ Ys2 ) )
= ( append5335208819046833346od_a_a @ Xs3 @ ( cons_P7316939126706565853od_a_a @ X @ Ys5 ) ) )
= ( ( Xs = Xs3 )
& ( Ys2 = Ys5 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_159_append__Cons__eq__iff,axiom,
! [X: nat,Xs: list_nat,Ys2: list_nat,Xs3: list_nat,Ys5: list_nat] :
( ~ ( member_nat @ X @ ( set_nat2 @ Xs ) )
=> ( ~ ( member_nat @ X @ ( set_nat2 @ Ys2 ) )
=> ( ( ( append_nat @ Xs @ ( cons_nat @ X @ Ys2 ) )
= ( append_nat @ Xs3 @ ( cons_nat @ X @ Ys5 ) ) )
= ( ( Xs = Xs3 )
& ( Ys2 = Ys5 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_160_append__Cons__eq__iff,axiom,
! [X: a,Xs: list_a,Ys2: list_a,Xs3: list_a,Ys5: list_a] :
( ~ ( member_a @ X @ ( set_a2 @ Xs ) )
=> ( ~ ( member_a @ X @ ( set_a2 @ Ys2 ) )
=> ( ( ( append_a @ Xs @ ( cons_a @ X @ Ys2 ) )
= ( append_a @ Xs3 @ ( cons_a @ X @ Ys5 ) ) )
= ( ( Xs = Xs3 )
& ( Ys2 = Ys5 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_161_append__Cons__eq__iff,axiom,
! [X: set_a,Xs: list_set_a,Ys2: list_set_a,Xs3: list_set_a,Ys5: list_set_a] :
( ~ ( member_set_a @ X @ ( set_set_a2 @ Xs ) )
=> ( ~ ( member_set_a @ X @ ( set_set_a2 @ Ys2 ) )
=> ( ( ( append_set_a @ Xs @ ( cons_set_a @ X @ Ys2 ) )
= ( append_set_a @ Xs3 @ ( cons_set_a @ X @ Ys5 ) ) )
= ( ( Xs = Xs3 )
& ( Ys2 = Ys5 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_162_split__list__propE,axiom,
! [Xs: list_a,P: a > $o] :
( ? [X3: a] :
( ( member_a @ X3 @ ( set_a2 @ Xs ) )
& ( P @ X3 ) )
=> ~ ! [Ys: list_a,X2: a] :
( ? [Zs3: list_a] :
( Xs
= ( append_a @ Ys @ ( cons_a @ X2 @ Zs3 ) ) )
=> ~ ( P @ X2 ) ) ) ).
% split_list_propE
thf(fact_163_split__list__propE,axiom,
! [Xs: list_set_a,P: set_a > $o] :
( ? [X3: set_a] :
( ( member_set_a @ X3 @ ( set_set_a2 @ Xs ) )
& ( P @ X3 ) )
=> ~ ! [Ys: list_set_a,X2: set_a] :
( ? [Zs3: list_set_a] :
( Xs
= ( append_set_a @ Ys @ ( cons_set_a @ X2 @ Zs3 ) ) )
=> ~ ( P @ X2 ) ) ) ).
% split_list_propE
thf(fact_164_split__list__first,axiom,
! [X: product_prod_a_a,Xs: list_P1396940483166286381od_a_a] :
( ( member1426531477525435216od_a_a @ X @ ( set_Product_prod_a_a2 @ Xs ) )
=> ? [Ys: list_P1396940483166286381od_a_a,Zs3: list_P1396940483166286381od_a_a] :
( ( Xs
= ( append5335208819046833346od_a_a @ Ys @ ( cons_P7316939126706565853od_a_a @ X @ Zs3 ) ) )
& ~ ( member1426531477525435216od_a_a @ X @ ( set_Product_prod_a_a2 @ Ys ) ) ) ) ).
% split_list_first
thf(fact_165_split__list__first,axiom,
! [X: nat,Xs: list_nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs ) )
=> ? [Ys: list_nat,Zs3: list_nat] :
( ( Xs
= ( append_nat @ Ys @ ( cons_nat @ X @ Zs3 ) ) )
& ~ ( member_nat @ X @ ( set_nat2 @ Ys ) ) ) ) ).
% split_list_first
thf(fact_166_split__list__first,axiom,
! [X: a,Xs: list_a] :
( ( member_a @ X @ ( set_a2 @ Xs ) )
=> ? [Ys: list_a,Zs3: list_a] :
( ( Xs
= ( append_a @ Ys @ ( cons_a @ X @ Zs3 ) ) )
& ~ ( member_a @ X @ ( set_a2 @ Ys ) ) ) ) ).
% split_list_first
thf(fact_167_split__list__first,axiom,
! [X: set_a,Xs: list_set_a] :
( ( member_set_a @ X @ ( set_set_a2 @ Xs ) )
=> ? [Ys: list_set_a,Zs3: list_set_a] :
( ( Xs
= ( append_set_a @ Ys @ ( cons_set_a @ X @ Zs3 ) ) )
& ~ ( member_set_a @ X @ ( set_set_a2 @ Ys ) ) ) ) ).
% split_list_first
thf(fact_168_split__list__prop,axiom,
! [Xs: list_a,P: a > $o] :
( ? [X3: a] :
( ( member_a @ X3 @ ( set_a2 @ Xs ) )
& ( P @ X3 ) )
=> ? [Ys: list_a,X2: a] :
( ? [Zs3: list_a] :
( Xs
= ( append_a @ Ys @ ( cons_a @ X2 @ Zs3 ) ) )
& ( P @ X2 ) ) ) ).
% split_list_prop
thf(fact_169_split__list__prop,axiom,
! [Xs: list_set_a,P: set_a > $o] :
( ? [X3: set_a] :
( ( member_set_a @ X3 @ ( set_set_a2 @ Xs ) )
& ( P @ X3 ) )
=> ? [Ys: list_set_a,X2: set_a] :
( ? [Zs3: list_set_a] :
( Xs
= ( append_set_a @ Ys @ ( cons_set_a @ X2 @ Zs3 ) ) )
& ( P @ X2 ) ) ) ).
% split_list_prop
thf(fact_170_split__list__last,axiom,
! [X: product_prod_a_a,Xs: list_P1396940483166286381od_a_a] :
( ( member1426531477525435216od_a_a @ X @ ( set_Product_prod_a_a2 @ Xs ) )
=> ? [Ys: list_P1396940483166286381od_a_a,Zs3: list_P1396940483166286381od_a_a] :
( ( Xs
= ( append5335208819046833346od_a_a @ Ys @ ( cons_P7316939126706565853od_a_a @ X @ Zs3 ) ) )
& ~ ( member1426531477525435216od_a_a @ X @ ( set_Product_prod_a_a2 @ Zs3 ) ) ) ) ).
% split_list_last
thf(fact_171_split__list__last,axiom,
! [X: nat,Xs: list_nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs ) )
=> ? [Ys: list_nat,Zs3: list_nat] :
( ( Xs
= ( append_nat @ Ys @ ( cons_nat @ X @ Zs3 ) ) )
& ~ ( member_nat @ X @ ( set_nat2 @ Zs3 ) ) ) ) ).
% split_list_last
thf(fact_172_split__list__last,axiom,
! [X: a,Xs: list_a] :
( ( member_a @ X @ ( set_a2 @ Xs ) )
=> ? [Ys: list_a,Zs3: list_a] :
( ( Xs
= ( append_a @ Ys @ ( cons_a @ X @ Zs3 ) ) )
& ~ ( member_a @ X @ ( set_a2 @ Zs3 ) ) ) ) ).
% split_list_last
thf(fact_173_split__list__last,axiom,
! [X: set_a,Xs: list_set_a] :
( ( member_set_a @ X @ ( set_set_a2 @ Xs ) )
=> ? [Ys: list_set_a,Zs3: list_set_a] :
( ( Xs
= ( append_set_a @ Ys @ ( cons_set_a @ X @ Zs3 ) ) )
& ~ ( member_set_a @ X @ ( set_set_a2 @ Zs3 ) ) ) ) ).
% split_list_last
thf(fact_174_split__list,axiom,
! [X: product_prod_a_a,Xs: list_P1396940483166286381od_a_a] :
( ( member1426531477525435216od_a_a @ X @ ( set_Product_prod_a_a2 @ Xs ) )
=> ? [Ys: list_P1396940483166286381od_a_a,Zs3: list_P1396940483166286381od_a_a] :
( Xs
= ( append5335208819046833346od_a_a @ Ys @ ( cons_P7316939126706565853od_a_a @ X @ Zs3 ) ) ) ) ).
% split_list
thf(fact_175_split__list,axiom,
! [X: nat,Xs: list_nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs ) )
=> ? [Ys: list_nat,Zs3: list_nat] :
( Xs
= ( append_nat @ Ys @ ( cons_nat @ X @ Zs3 ) ) ) ) ).
% split_list
thf(fact_176_split__list,axiom,
! [X: a,Xs: list_a] :
( ( member_a @ X @ ( set_a2 @ Xs ) )
=> ? [Ys: list_a,Zs3: list_a] :
( Xs
= ( append_a @ Ys @ ( cons_a @ X @ Zs3 ) ) ) ) ).
% split_list
thf(fact_177_split__list,axiom,
! [X: set_a,Xs: list_set_a] :
( ( member_set_a @ X @ ( set_set_a2 @ Xs ) )
=> ? [Ys: list_set_a,Zs3: list_set_a] :
( Xs
= ( append_set_a @ Ys @ ( cons_set_a @ X @ Zs3 ) ) ) ) ).
% split_list
thf(fact_178_induced__edges__ss,axiom,
! [V: set_a] :
( ( ord_less_eq_set_a @ V @ vertices )
=> ( ord_le3724670747650509150_set_a @ ( undire7777452895879145676dges_a @ edges @ V ) @ edges ) ) ).
% induced_edges_ss
thf(fact_179_has__loop__in__verts,axiom,
! [V2: a] :
( ( undire3617971648856834880loop_a @ edges @ V2 )
=> ( member_a @ V2 @ vertices ) ) ).
% has_loop_in_verts
thf(fact_180_incident__edge__in__wf,axiom,
! [E: set_a,V2: a] :
( ( member_set_a @ E @ edges )
=> ( ( undire1521409233611534436dent_a @ V2 @ E )
=> ( member_a @ V2 @ vertices ) ) ) ).
% incident_edge_in_wf
thf(fact_181_vert__adj__imp__inV,axiom,
! [V1: a,V22: a] :
( ( undire397441198561214472_adj_a @ edges @ V1 @ V22 )
=> ( ( member_a @ V1 @ vertices )
& ( member_a @ V22 @ vertices ) ) ) ).
% vert_adj_imp_inV
thf(fact_182_is__walk__wf__last,axiom,
! [Xs: list_a] :
( ( undire6133010728901294956walk_a @ vertices @ edges @ Xs )
=> ( member_a @ ( last_a @ Xs ) @ vertices ) ) ).
% is_walk_wf_last
thf(fact_183_subgraph__refl,axiom,
undire7103218114511261257raph_a @ vertices @ edges @ vertices @ edges ).
% subgraph_refl
thf(fact_184_is__walk__wf__hd,axiom,
! [Xs: list_a] :
( ( undire6133010728901294956walk_a @ vertices @ edges @ Xs )
=> ( member_a @ ( hd_a @ Xs ) @ vertices ) ) ).
% is_walk_wf_hd
thf(fact_185_is__walk__rev,axiom,
! [Xs: list_a] :
( ( undire6133010728901294956walk_a @ vertices @ edges @ Xs )
= ( undire6133010728901294956walk_a @ vertices @ edges @ ( rev_a @ Xs ) ) ) ).
% is_walk_rev
thf(fact_186_walk__edges__tl__ss,axiom,
! [Xs: list_a] : ( ord_le3724670747650509150_set_a @ ( set_set_a2 @ ( undire7337870655677353998dges_a @ ( tl_a @ Xs ) ) ) @ ( set_set_a2 @ ( undire7337870655677353998dges_a @ Xs ) ) ) ).
% walk_edges_tl_ss
thf(fact_187_graph__system__axioms,axiom,
undire2554140024507503526stem_a @ vertices @ edges ).
% graph_system_axioms
thf(fact_188_subset__antisym,axiom,
! [A2: set_set_a,B2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B2 )
=> ( ( ord_le3724670747650509150_set_a @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% subset_antisym
thf(fact_189_subset__antisym,axiom,
! [A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ord_less_eq_set_a @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% subset_antisym
thf(fact_190_subset__antisym,axiom,
! [A2: set_Product_prod_a_a,B2: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A2 @ B2 )
=> ( ( ord_le746702958409616551od_a_a @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% subset_antisym
thf(fact_191_incident__def,axiom,
undire1521409233611534436dent_a = member_a ).
% incident_def
thf(fact_192_vert__adj__sym,axiom,
! [V1: a,V22: a] :
( ( undire397441198561214472_adj_a @ edges @ V1 @ V22 )
= ( undire397441198561214472_adj_a @ edges @ V22 @ V1 ) ) ).
% vert_adj_sym
thf(fact_193_vert__adj__edge__iff2,axiom,
! [V1: a,V22: a] :
( ( V1 != V22 )
=> ( ( undire397441198561214472_adj_a @ edges @ V1 @ V22 )
= ( ? [X4: set_a] :
( ( member_set_a @ X4 @ edges )
& ( undire1521409233611534436dent_a @ V1 @ X4 )
& ( undire1521409233611534436dent_a @ V22 @ X4 ) ) ) ) ) ).
% vert_adj_edge_iff2
thf(fact_194_induced__is__graph__sys,axiom,
! [V: set_a] : ( undire2554140024507503526stem_a @ V @ ( undire7777452895879145676dges_a @ edges @ V ) ) ).
% induced_is_graph_sys
thf(fact_195_subsetI,axiom,
! [A2: set_nat,B2: set_nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( member_nat @ X2 @ B2 ) )
=> ( ord_less_eq_set_nat @ A2 @ B2 ) ) ).
% subsetI
thf(fact_196_subsetI,axiom,
! [A2: set_set_a,B2: set_set_a] :
( ! [X2: set_a] :
( ( member_set_a @ X2 @ A2 )
=> ( member_set_a @ X2 @ B2 ) )
=> ( ord_le3724670747650509150_set_a @ A2 @ B2 ) ) ).
% subsetI
thf(fact_197_subsetI,axiom,
! [A2: set_a,B2: set_a] :
( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ( member_a @ X2 @ B2 ) )
=> ( ord_less_eq_set_a @ A2 @ B2 ) ) ).
% subsetI
thf(fact_198_subsetI,axiom,
! [A2: set_Product_prod_a_a,B2: set_Product_prod_a_a] :
( ! [X2: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X2 @ A2 )
=> ( member1426531477525435216od_a_a @ X2 @ B2 ) )
=> ( ord_le746702958409616551od_a_a @ A2 @ B2 ) ) ).
% subsetI
thf(fact_199_rev__rev__ident,axiom,
! [Xs: list_a] :
( ( rev_a @ ( rev_a @ Xs ) )
= Xs ) ).
% rev_rev_ident
thf(fact_200_rev__rev__ident,axiom,
! [Xs: list_set_a] :
( ( rev_set_a @ ( rev_set_a @ Xs ) )
= Xs ) ).
% rev_rev_ident
thf(fact_201_rev__is__rev__conv,axiom,
! [Xs: list_a,Ys2: list_a] :
( ( ( rev_a @ Xs )
= ( rev_a @ Ys2 ) )
= ( Xs = Ys2 ) ) ).
% rev_is_rev_conv
thf(fact_202_rev__is__rev__conv,axiom,
! [Xs: list_set_a,Ys2: list_set_a] :
( ( ( rev_set_a @ Xs )
= ( rev_set_a @ Ys2 ) )
= ( Xs = Ys2 ) ) ).
% rev_is_rev_conv
thf(fact_203_induced__is__subgraph,axiom,
! [V: set_a] :
( ( ord_less_eq_set_a @ V @ vertices )
=> ( undire7103218114511261257raph_a @ V @ ( undire7777452895879145676dges_a @ edges @ V ) @ vertices @ edges ) ) ).
% induced_is_subgraph
thf(fact_204_Nil__is__rev__conv,axiom,
! [Xs: list_a] :
( ( nil_a
= ( rev_a @ Xs ) )
= ( Xs = nil_a ) ) ).
% Nil_is_rev_conv
thf(fact_205_Nil__is__rev__conv,axiom,
! [Xs: list_set_a] :
( ( nil_set_a
= ( rev_set_a @ Xs ) )
= ( Xs = nil_set_a ) ) ).
% Nil_is_rev_conv
thf(fact_206_rev__is__Nil__conv,axiom,
! [Xs: list_a] :
( ( ( rev_a @ Xs )
= nil_a )
= ( Xs = nil_a ) ) ).
% rev_is_Nil_conv
thf(fact_207_rev__is__Nil__conv,axiom,
! [Xs: list_set_a] :
( ( ( rev_set_a @ Xs )
= nil_set_a )
= ( Xs = nil_set_a ) ) ).
% rev_is_Nil_conv
thf(fact_208_set__rev,axiom,
! [Xs: list_set_a] :
( ( set_set_a2 @ ( rev_set_a @ Xs ) )
= ( set_set_a2 @ Xs ) ) ).
% set_rev
thf(fact_209_set__rev,axiom,
! [Xs: list_a] :
( ( set_a2 @ ( rev_a @ Xs ) )
= ( set_a2 @ Xs ) ) ).
% set_rev
thf(fact_210_rev__append,axiom,
! [Xs: list_a,Ys2: list_a] :
( ( rev_a @ ( append_a @ Xs @ Ys2 ) )
= ( append_a @ ( rev_a @ Ys2 ) @ ( rev_a @ Xs ) ) ) ).
% rev_append
thf(fact_211_rev__append,axiom,
! [Xs: list_set_a,Ys2: list_set_a] :
( ( rev_set_a @ ( append_set_a @ Xs @ Ys2 ) )
= ( append_set_a @ ( rev_set_a @ Ys2 ) @ ( rev_set_a @ Xs ) ) ) ).
% rev_append
thf(fact_212_is__walk__append,axiom,
! [Xs: list_a,Ys2: list_a] :
( ( undire6133010728901294956walk_a @ vertices @ edges @ Xs )
=> ( ( undire6133010728901294956walk_a @ vertices @ edges @ Ys2 )
=> ( ( ( last_a @ Xs )
= ( hd_a @ Ys2 ) )
=> ( undire6133010728901294956walk_a @ vertices @ edges @ ( append_a @ Xs @ ( tl_a @ Ys2 ) ) ) ) ) ) ).
% is_walk_append
thf(fact_213_rev__singleton__conv,axiom,
! [Xs: list_a,X: a] :
( ( ( rev_a @ Xs )
= ( cons_a @ X @ nil_a ) )
= ( Xs
= ( cons_a @ X @ nil_a ) ) ) ).
% rev_singleton_conv
thf(fact_214_rev__singleton__conv,axiom,
! [Xs: list_set_a,X: set_a] :
( ( ( rev_set_a @ Xs )
= ( cons_set_a @ X @ nil_set_a ) )
= ( Xs
= ( cons_set_a @ X @ nil_set_a ) ) ) ).
% rev_singleton_conv
thf(fact_215_singleton__rev__conv,axiom,
! [X: a,Xs: list_a] :
( ( ( cons_a @ X @ nil_a )
= ( rev_a @ Xs ) )
= ( ( cons_a @ X @ nil_a )
= Xs ) ) ).
% singleton_rev_conv
thf(fact_216_singleton__rev__conv,axiom,
! [X: set_a,Xs: list_set_a] :
( ( ( cons_set_a @ X @ nil_set_a )
= ( rev_set_a @ Xs ) )
= ( ( cons_set_a @ X @ nil_set_a )
= Xs ) ) ).
% singleton_rev_conv
thf(fact_217_hd__append2,axiom,
! [Xs: list_a,Ys2: list_a] :
( ( Xs != nil_a )
=> ( ( hd_a @ ( append_a @ Xs @ Ys2 ) )
= ( hd_a @ Xs ) ) ) ).
% hd_append2
thf(fact_218_hd__append2,axiom,
! [Xs: list_set_a,Ys2: list_set_a] :
( ( Xs != nil_set_a )
=> ( ( hd_set_a @ ( append_set_a @ Xs @ Ys2 ) )
= ( hd_set_a @ Xs ) ) ) ).
% hd_append2
thf(fact_219_tl__append2,axiom,
! [Xs: list_a,Ys2: list_a] :
( ( Xs != nil_a )
=> ( ( tl_a @ ( append_a @ Xs @ Ys2 ) )
= ( append_a @ ( tl_a @ Xs ) @ Ys2 ) ) ) ).
% tl_append2
thf(fact_220_tl__append2,axiom,
! [Xs: list_set_a,Ys2: list_set_a] :
( ( Xs != nil_set_a )
=> ( ( tl_set_a @ ( append_set_a @ Xs @ Ys2 ) )
= ( append_set_a @ ( tl_set_a @ Xs ) @ Ys2 ) ) ) ).
% tl_append2
thf(fact_221_last__appendL,axiom,
! [Ys2: list_a,Xs: list_a] :
( ( Ys2 = nil_a )
=> ( ( last_a @ ( append_a @ Xs @ Ys2 ) )
= ( last_a @ Xs ) ) ) ).
% last_appendL
thf(fact_222_last__appendL,axiom,
! [Ys2: list_set_a,Xs: list_set_a] :
( ( Ys2 = nil_set_a )
=> ( ( last_set_a @ ( append_set_a @ Xs @ Ys2 ) )
= ( last_set_a @ Xs ) ) ) ).
% last_appendL
thf(fact_223_last__appendR,axiom,
! [Ys2: list_a,Xs: list_a] :
( ( Ys2 != nil_a )
=> ( ( last_a @ ( append_a @ Xs @ Ys2 ) )
= ( last_a @ Ys2 ) ) ) ).
% last_appendR
thf(fact_224_last__appendR,axiom,
! [Ys2: list_set_a,Xs: list_set_a] :
( ( Ys2 != nil_set_a )
=> ( ( last_set_a @ ( append_set_a @ Xs @ Ys2 ) )
= ( last_set_a @ Ys2 ) ) ) ).
% last_appendR
thf(fact_225_rev__eq__Cons__iff,axiom,
! [Xs: list_a,Y2: a,Ys2: list_a] :
( ( ( rev_a @ Xs )
= ( cons_a @ Y2 @ Ys2 ) )
= ( Xs
= ( append_a @ ( rev_a @ Ys2 ) @ ( cons_a @ Y2 @ nil_a ) ) ) ) ).
% rev_eq_Cons_iff
thf(fact_226_rev__eq__Cons__iff,axiom,
! [Xs: list_set_a,Y2: set_a,Ys2: list_set_a] :
( ( ( rev_set_a @ Xs )
= ( cons_set_a @ Y2 @ Ys2 ) )
= ( Xs
= ( append_set_a @ ( rev_set_a @ Ys2 ) @ ( cons_set_a @ Y2 @ nil_set_a ) ) ) ) ).
% rev_eq_Cons_iff
thf(fact_227_last__snoc,axiom,
! [Xs: list_a,X: a] :
( ( last_a @ ( append_a @ Xs @ ( cons_a @ X @ nil_a ) ) )
= X ) ).
% last_snoc
thf(fact_228_last__snoc,axiom,
! [Xs: list_set_a,X: set_a] :
( ( last_set_a @ ( append_set_a @ Xs @ ( cons_set_a @ X @ nil_set_a ) ) )
= X ) ).
% last_snoc
thf(fact_229_list_Ocollapse,axiom,
! [List: list_a] :
( ( List != nil_a )
=> ( ( cons_a @ ( hd_a @ List ) @ ( tl_a @ List ) )
= List ) ) ).
% list.collapse
thf(fact_230_list_Ocollapse,axiom,
! [List: list_set_a] :
( ( List != nil_set_a )
=> ( ( cons_set_a @ ( hd_set_a @ List ) @ ( tl_set_a @ List ) )
= List ) ) ).
% list.collapse
thf(fact_231_hd__Cons__tl,axiom,
! [Xs: list_a] :
( ( Xs != nil_a )
=> ( ( cons_a @ ( hd_a @ Xs ) @ ( tl_a @ Xs ) )
= Xs ) ) ).
% hd_Cons_tl
thf(fact_232_hd__Cons__tl,axiom,
! [Xs: list_set_a] :
( ( Xs != nil_set_a )
=> ( ( cons_set_a @ ( hd_set_a @ Xs ) @ ( tl_set_a @ Xs ) )
= Xs ) ) ).
% hd_Cons_tl
thf(fact_233_ulgraph_Owalk__edges__rev,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,Xs: list_set_a] :
( ( undire6886684016831807756_set_a @ Vertices @ Edges )
=> ( ( rev_set_set_a @ ( undire6234387080713648494_set_a @ Xs ) )
= ( undire6234387080713648494_set_a @ ( rev_set_a @ Xs ) ) ) ) ).
% ulgraph.walk_edges_rev
thf(fact_234_ulgraph_Owalk__edges__rev,axiom,
! [Vertices: set_a,Edges: set_set_a,Xs: list_a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( rev_set_a @ ( undire7337870655677353998dges_a @ Xs ) )
= ( undire7337870655677353998dges_a @ ( rev_a @ Xs ) ) ) ) ).
% ulgraph.walk_edges_rev
thf(fact_235_hd__rev,axiom,
! [Xs: list_set_a] :
( ( hd_set_a @ ( rev_set_a @ Xs ) )
= ( last_set_a @ Xs ) ) ).
% hd_rev
thf(fact_236_hd__rev,axiom,
! [Xs: list_a] :
( ( hd_a @ ( rev_a @ Xs ) )
= ( last_a @ Xs ) ) ).
% hd_rev
thf(fact_237_last__rev,axiom,
! [Xs: list_set_a] :
( ( last_set_a @ ( rev_set_a @ Xs ) )
= ( hd_set_a @ Xs ) ) ).
% last_rev
thf(fact_238_last__rev,axiom,
! [Xs: list_a] :
( ( last_a @ ( rev_a @ Xs ) )
= ( hd_a @ Xs ) ) ).
% last_rev
thf(fact_239_rev__swap,axiom,
! [Xs: list_a,Ys2: list_a] :
( ( ( rev_a @ Xs )
= Ys2 )
= ( Xs
= ( rev_a @ Ys2 ) ) ) ).
% rev_swap
thf(fact_240_rev__swap,axiom,
! [Xs: list_set_a,Ys2: list_set_a] :
( ( ( rev_set_a @ Xs )
= Ys2 )
= ( Xs
= ( rev_set_a @ Ys2 ) ) ) ).
% rev_swap
thf(fact_241_ulgraph_Ois__walk__append,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,Xs: list_set_a,Ys2: list_set_a] :
( ( undire6886684016831807756_set_a @ Vertices @ Edges )
=> ( ( undire3014741414213135564_set_a @ Vertices @ Edges @ Xs )
=> ( ( undire3014741414213135564_set_a @ Vertices @ Edges @ Ys2 )
=> ( ( ( last_set_a @ Xs )
= ( hd_set_a @ Ys2 ) )
=> ( undire3014741414213135564_set_a @ Vertices @ Edges @ ( append_set_a @ Xs @ ( tl_set_a @ Ys2 ) ) ) ) ) ) ) ).
% ulgraph.is_walk_append
thf(fact_242_ulgraph_Ois__walk__append,axiom,
! [Vertices: set_a,Edges: set_set_a,Xs: list_a,Ys2: list_a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( undire6133010728901294956walk_a @ Vertices @ Edges @ Xs )
=> ( ( undire6133010728901294956walk_a @ Vertices @ Edges @ Ys2 )
=> ( ( ( last_a @ Xs )
= ( hd_a @ Ys2 ) )
=> ( undire6133010728901294956walk_a @ Vertices @ Edges @ ( append_a @ Xs @ ( tl_a @ Ys2 ) ) ) ) ) ) ) ).
% ulgraph.is_walk_append
thf(fact_243_list_Oexpand,axiom,
! [List: list_a,List2: list_a] :
( ( ( List = nil_a )
= ( List2 = nil_a ) )
=> ( ( ( List != nil_a )
=> ( ( List2 != nil_a )
=> ( ( ( hd_a @ List )
= ( hd_a @ List2 ) )
& ( ( tl_a @ List )
= ( tl_a @ List2 ) ) ) ) )
=> ( List = List2 ) ) ) ).
% list.expand
thf(fact_244_list_Oexpand,axiom,
! [List: list_set_a,List2: list_set_a] :
( ( ( List = nil_set_a )
= ( List2 = nil_set_a ) )
=> ( ( ( List != nil_set_a )
=> ( ( List2 != nil_set_a )
=> ( ( ( hd_set_a @ List )
= ( hd_set_a @ List2 ) )
& ( ( tl_set_a @ List )
= ( tl_set_a @ List2 ) ) ) ) )
=> ( List = List2 ) ) ) ).
% list.expand
thf(fact_245_comp__sgraph_Owalk__edges__rev,axiom,
! [Xs: list_set_a] :
( ( rev_set_set_a @ ( undire6234387080713648494_set_a @ Xs ) )
= ( undire6234387080713648494_set_a @ ( rev_set_a @ Xs ) ) ) ).
% comp_sgraph.walk_edges_rev
thf(fact_246_comp__sgraph_Owalk__edges__rev,axiom,
! [Xs: list_a] :
( ( rev_set_a @ ( undire7337870655677353998dges_a @ Xs ) )
= ( undire7337870655677353998dges_a @ ( rev_a @ Xs ) ) ) ).
% comp_sgraph.walk_edges_rev
thf(fact_247_last__tl,axiom,
! [Xs: list_a] :
( ( ( Xs = nil_a )
| ( ( tl_a @ Xs )
!= nil_a ) )
=> ( ( last_a @ ( tl_a @ Xs ) )
= ( last_a @ Xs ) ) ) ).
% last_tl
thf(fact_248_last__tl,axiom,
! [Xs: list_set_a] :
( ( ( Xs = nil_set_a )
| ( ( tl_set_a @ Xs )
!= nil_set_a ) )
=> ( ( last_set_a @ ( tl_set_a @ Xs ) )
= ( last_set_a @ Xs ) ) ) ).
% last_tl
thf(fact_249_hd__Nil__eq__last,axiom,
( ( hd_a @ nil_a )
= ( last_a @ nil_a ) ) ).
% hd_Nil_eq_last
thf(fact_250_hd__Nil__eq__last,axiom,
( ( hd_set_a @ nil_set_a )
= ( last_set_a @ nil_set_a ) ) ).
% hd_Nil_eq_last
thf(fact_251_ulgraph_Ois__walk__rev,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,Xs: list_set_a] :
( ( undire6886684016831807756_set_a @ Vertices @ Edges )
=> ( ( undire3014741414213135564_set_a @ Vertices @ Edges @ Xs )
= ( undire3014741414213135564_set_a @ Vertices @ Edges @ ( rev_set_a @ Xs ) ) ) ) ).
% ulgraph.is_walk_rev
thf(fact_252_ulgraph_Ois__walk__rev,axiom,
! [Vertices: set_a,Edges: set_set_a,Xs: list_a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( undire6133010728901294956walk_a @ Vertices @ Edges @ Xs )
= ( undire6133010728901294956walk_a @ Vertices @ Edges @ ( rev_a @ Xs ) ) ) ) ).
% ulgraph.is_walk_rev
thf(fact_253_ulgraph_Ois__walk__wf__hd,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,Xs: list_set_a] :
( ( undire6886684016831807756_set_a @ Vertices @ Edges )
=> ( ( undire3014741414213135564_set_a @ Vertices @ Edges @ Xs )
=> ( member_set_a @ ( hd_set_a @ Xs ) @ Vertices ) ) ) ).
% ulgraph.is_walk_wf_hd
thf(fact_254_ulgraph_Ois__walk__wf__hd,axiom,
! [Vertices: set_Product_prod_a_a,Edges: set_se5735800977113168103od_a_a,Xs: list_P1396940483166286381od_a_a] :
( ( undire4585262585102564309od_a_a @ Vertices @ Edges )
=> ( ( undire3162072421265123221od_a_a @ Vertices @ Edges @ Xs )
=> ( member1426531477525435216od_a_a @ ( hd_Product_prod_a_a @ Xs ) @ Vertices ) ) ) ).
% ulgraph.is_walk_wf_hd
thf(fact_255_ulgraph_Ois__walk__wf__hd,axiom,
! [Vertices: set_nat,Edges: set_set_nat,Xs: list_nat] :
( ( undire3269267262472140706ph_nat @ Vertices @ Edges )
=> ( ( undire5745680128780950498lk_nat @ Vertices @ Edges @ Xs )
=> ( member_nat @ ( hd_nat @ Xs ) @ Vertices ) ) ) ).
% ulgraph.is_walk_wf_hd
thf(fact_256_ulgraph_Ois__walk__wf__hd,axiom,
! [Vertices: set_a,Edges: set_set_a,Xs: list_a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( undire6133010728901294956walk_a @ Vertices @ Edges @ Xs )
=> ( member_a @ ( hd_a @ Xs ) @ Vertices ) ) ) ).
% ulgraph.is_walk_wf_hd
thf(fact_257_ulgraph_Ois__walk__wf__last,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,Xs: list_set_a] :
( ( undire6886684016831807756_set_a @ Vertices @ Edges )
=> ( ( undire3014741414213135564_set_a @ Vertices @ Edges @ Xs )
=> ( member_set_a @ ( last_set_a @ Xs ) @ Vertices ) ) ) ).
% ulgraph.is_walk_wf_last
thf(fact_258_ulgraph_Ois__walk__wf__last,axiom,
! [Vertices: set_Product_prod_a_a,Edges: set_se5735800977113168103od_a_a,Xs: list_P1396940483166286381od_a_a] :
( ( undire4585262585102564309od_a_a @ Vertices @ Edges )
=> ( ( undire3162072421265123221od_a_a @ Vertices @ Edges @ Xs )
=> ( member1426531477525435216od_a_a @ ( last_P8790725268278465478od_a_a @ Xs ) @ Vertices ) ) ) ).
% ulgraph.is_walk_wf_last
thf(fact_259_ulgraph_Ois__walk__wf__last,axiom,
! [Vertices: set_nat,Edges: set_set_nat,Xs: list_nat] :
( ( undire3269267262472140706ph_nat @ Vertices @ Edges )
=> ( ( undire5745680128780950498lk_nat @ Vertices @ Edges @ Xs )
=> ( member_nat @ ( last_nat @ Xs ) @ Vertices ) ) ) ).
% ulgraph.is_walk_wf_last
thf(fact_260_ulgraph_Ois__walk__wf__last,axiom,
! [Vertices: set_a,Edges: set_set_a,Xs: list_a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( undire6133010728901294956walk_a @ Vertices @ Edges @ Xs )
=> ( member_a @ ( last_a @ Xs ) @ Vertices ) ) ) ).
% ulgraph.is_walk_wf_last
thf(fact_261_list_Oexhaust__sel,axiom,
! [List: list_a] :
( ( List != nil_a )
=> ( List
= ( cons_a @ ( hd_a @ List ) @ ( tl_a @ List ) ) ) ) ).
% list.exhaust_sel
thf(fact_262_list_Oexhaust__sel,axiom,
! [List: list_set_a] :
( ( List != nil_set_a )
=> ( List
= ( cons_set_a @ ( hd_set_a @ List ) @ ( tl_set_a @ List ) ) ) ) ).
% list.exhaust_sel
thf(fact_263_rev_Osimps_I1_J,axiom,
( ( rev_a @ nil_a )
= nil_a ) ).
% rev.simps(1)
thf(fact_264_rev_Osimps_I1_J,axiom,
( ( rev_set_a @ nil_set_a )
= nil_set_a ) ).
% rev.simps(1)
thf(fact_265_list_Osel_I3_J,axiom,
! [X21: a,X22: list_a] :
( ( tl_a @ ( cons_a @ X21 @ X22 ) )
= X22 ) ).
% list.sel(3)
thf(fact_266_list_Osel_I3_J,axiom,
! [X21: set_a,X22: list_set_a] :
( ( tl_set_a @ ( cons_set_a @ X21 @ X22 ) )
= X22 ) ).
% list.sel(3)
thf(fact_267_list_Osel_I2_J,axiom,
( ( tl_a @ nil_a )
= nil_a ) ).
% list.sel(2)
thf(fact_268_list_Osel_I2_J,axiom,
( ( tl_set_a @ nil_set_a )
= nil_set_a ) ).
% list.sel(2)
thf(fact_269_list_Osel_I1_J,axiom,
! [X21: a,X22: list_a] :
( ( hd_a @ ( cons_a @ X21 @ X22 ) )
= X21 ) ).
% list.sel(1)
thf(fact_270_list_Osel_I1_J,axiom,
! [X21: set_a,X22: list_set_a] :
( ( hd_set_a @ ( cons_set_a @ X21 @ X22 ) )
= X21 ) ).
% list.sel(1)
thf(fact_271_ulgraph_Owalk__edges__tl__ss,axiom,
! [Vertices: set_a,Edges: set_set_a,Xs: list_a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ord_le3724670747650509150_set_a @ ( set_set_a2 @ ( undire7337870655677353998dges_a @ ( tl_a @ Xs ) ) ) @ ( set_set_a2 @ ( undire7337870655677353998dges_a @ Xs ) ) ) ) ).
% ulgraph.walk_edges_tl_ss
thf(fact_272_ulgraph_Owalk__edges_Ocases,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,X: list_set_a] :
( ( undire6886684016831807756_set_a @ Vertices @ Edges )
=> ( ( X != nil_set_a )
=> ( ! [X2: set_a] :
( X
!= ( cons_set_a @ X2 @ nil_set_a ) )
=> ~ ! [X2: set_a,Y: set_a,Ys: list_set_a] :
( X
!= ( cons_set_a @ X2 @ ( cons_set_a @ Y @ Ys ) ) ) ) ) ) ).
% ulgraph.walk_edges.cases
thf(fact_273_ulgraph_Owalk__edges_Ocases,axiom,
! [Vertices: set_a,Edges: set_set_a,X: list_a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( X != nil_a )
=> ( ! [X2: a] :
( X
!= ( cons_a @ X2 @ nil_a ) )
=> ~ ! [X2: a,Y: a,Ys: list_a] :
( X
!= ( cons_a @ X2 @ ( cons_a @ Y @ Ys ) ) ) ) ) ) ).
% ulgraph.walk_edges.cases
thf(fact_274_Nil__tl,axiom,
! [Xs: list_a] :
( ( nil_a
= ( tl_a @ Xs ) )
= ( ( Xs = nil_a )
| ? [X4: a] :
( Xs
= ( cons_a @ X4 @ nil_a ) ) ) ) ).
% Nil_tl
thf(fact_275_Nil__tl,axiom,
! [Xs: list_set_a] :
( ( nil_set_a
= ( tl_set_a @ Xs ) )
= ( ( Xs = nil_set_a )
| ? [X4: set_a] :
( Xs
= ( cons_set_a @ X4 @ nil_set_a ) ) ) ) ).
% Nil_tl
thf(fact_276_tl__Nil,axiom,
! [Xs: list_a] :
( ( ( tl_a @ Xs )
= nil_a )
= ( ( Xs = nil_a )
| ? [X4: a] :
( Xs
= ( cons_a @ X4 @ nil_a ) ) ) ) ).
% tl_Nil
thf(fact_277_tl__Nil,axiom,
! [Xs: list_set_a] :
( ( ( tl_set_a @ Xs )
= nil_set_a )
= ( ( Xs = nil_set_a )
| ? [X4: set_a] :
( Xs
= ( cons_set_a @ X4 @ nil_set_a ) ) ) ) ).
% tl_Nil
thf(fact_278_list_Oset__sel_I2_J,axiom,
! [A: list_P1396940483166286381od_a_a,X: product_prod_a_a] :
( ( A != nil_Product_prod_a_a )
=> ( ( member1426531477525435216od_a_a @ X @ ( set_Product_prod_a_a2 @ ( tl_Product_prod_a_a @ A ) ) )
=> ( member1426531477525435216od_a_a @ X @ ( set_Product_prod_a_a2 @ A ) ) ) ) ).
% list.set_sel(2)
thf(fact_279_list_Oset__sel_I2_J,axiom,
! [A: list_nat,X: nat] :
( ( A != nil_nat )
=> ( ( member_nat @ X @ ( set_nat2 @ ( tl_nat @ A ) ) )
=> ( member_nat @ X @ ( set_nat2 @ A ) ) ) ) ).
% list.set_sel(2)
thf(fact_280_list_Oset__sel_I2_J,axiom,
! [A: list_set_a,X: set_a] :
( ( A != nil_set_a )
=> ( ( member_set_a @ X @ ( set_set_a2 @ ( tl_set_a @ A ) ) )
=> ( member_set_a @ X @ ( set_set_a2 @ A ) ) ) ) ).
% list.set_sel(2)
thf(fact_281_list_Oset__sel_I2_J,axiom,
! [A: list_a,X: a] :
( ( A != nil_a )
=> ( ( member_a @ X @ ( set_a2 @ ( tl_a @ A ) ) )
=> ( member_a @ X @ ( set_a2 @ A ) ) ) ) ).
% list.set_sel(2)
thf(fact_282_ulgraph_Ois__walk__not__empty,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,Xs: list_set_a] :
( ( undire6886684016831807756_set_a @ Vertices @ Edges )
=> ( ( undire3014741414213135564_set_a @ Vertices @ Edges @ Xs )
=> ( Xs != nil_set_a ) ) ) ).
% ulgraph.is_walk_not_empty
thf(fact_283_ulgraph_Ois__walk__not__empty,axiom,
! [Vertices: set_a,Edges: set_set_a,Xs: list_a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( undire6133010728901294956walk_a @ Vertices @ Edges @ Xs )
=> ( Xs != nil_a ) ) ) ).
% ulgraph.is_walk_not_empty
thf(fact_284_ulgraph_Ois__walk__not__empty2,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a] :
( ( undire6886684016831807756_set_a @ Vertices @ Edges )
=> ~ ( undire3014741414213135564_set_a @ Vertices @ Edges @ nil_set_a ) ) ).
% ulgraph.is_walk_not_empty2
thf(fact_285_ulgraph_Ois__walk__not__empty2,axiom,
! [Vertices: set_a,Edges: set_set_a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ~ ( undire6133010728901294956walk_a @ Vertices @ Edges @ nil_a ) ) ).
% ulgraph.is_walk_not_empty2
thf(fact_286_list_Oset__sel_I1_J,axiom,
! [A: list_P1396940483166286381od_a_a] :
( ( A != nil_Product_prod_a_a )
=> ( member1426531477525435216od_a_a @ ( hd_Product_prod_a_a @ A ) @ ( set_Product_prod_a_a2 @ A ) ) ) ).
% list.set_sel(1)
thf(fact_287_list_Oset__sel_I1_J,axiom,
! [A: list_nat] :
( ( A != nil_nat )
=> ( member_nat @ ( hd_nat @ A ) @ ( set_nat2 @ A ) ) ) ).
% list.set_sel(1)
thf(fact_288_list_Oset__sel_I1_J,axiom,
! [A: list_set_a] :
( ( A != nil_set_a )
=> ( member_set_a @ ( hd_set_a @ A ) @ ( set_set_a2 @ A ) ) ) ).
% list.set_sel(1)
thf(fact_289_list_Oset__sel_I1_J,axiom,
! [A: list_a] :
( ( A != nil_a )
=> ( member_a @ ( hd_a @ A ) @ ( set_a2 @ A ) ) ) ).
% list.set_sel(1)
thf(fact_290_hd__in__set,axiom,
! [Xs: list_P1396940483166286381od_a_a] :
( ( Xs != nil_Product_prod_a_a )
=> ( member1426531477525435216od_a_a @ ( hd_Product_prod_a_a @ Xs ) @ ( set_Product_prod_a_a2 @ Xs ) ) ) ).
% hd_in_set
thf(fact_291_hd__in__set,axiom,
! [Xs: list_nat] :
( ( Xs != nil_nat )
=> ( member_nat @ ( hd_nat @ Xs ) @ ( set_nat2 @ Xs ) ) ) ).
% hd_in_set
thf(fact_292_hd__in__set,axiom,
! [Xs: list_set_a] :
( ( Xs != nil_set_a )
=> ( member_set_a @ ( hd_set_a @ Xs ) @ ( set_set_a2 @ Xs ) ) ) ).
% hd_in_set
thf(fact_293_hd__in__set,axiom,
! [Xs: list_a] :
( ( Xs != nil_a )
=> ( member_a @ ( hd_a @ Xs ) @ ( set_a2 @ Xs ) ) ) ).
% hd_in_set
thf(fact_294_tl__append__if,axiom,
! [Xs: list_a,Ys2: list_a] :
( ( ( Xs = nil_a )
=> ( ( tl_a @ ( append_a @ Xs @ Ys2 ) )
= ( tl_a @ Ys2 ) ) )
& ( ( Xs != nil_a )
=> ( ( tl_a @ ( append_a @ Xs @ Ys2 ) )
= ( append_a @ ( tl_a @ Xs ) @ Ys2 ) ) ) ) ).
% tl_append_if
thf(fact_295_tl__append__if,axiom,
! [Xs: list_set_a,Ys2: list_set_a] :
( ( ( Xs = nil_set_a )
=> ( ( tl_set_a @ ( append_set_a @ Xs @ Ys2 ) )
= ( tl_set_a @ Ys2 ) ) )
& ( ( Xs != nil_set_a )
=> ( ( tl_set_a @ ( append_set_a @ Xs @ Ys2 ) )
= ( append_set_a @ ( tl_set_a @ Xs ) @ Ys2 ) ) ) ) ).
% tl_append_if
thf(fact_296_hd__append,axiom,
! [Xs: list_a,Ys2: list_a] :
( ( ( Xs = nil_a )
=> ( ( hd_a @ ( append_a @ Xs @ Ys2 ) )
= ( hd_a @ Ys2 ) ) )
& ( ( Xs != nil_a )
=> ( ( hd_a @ ( append_a @ Xs @ Ys2 ) )
= ( hd_a @ Xs ) ) ) ) ).
% hd_append
thf(fact_297_hd__append,axiom,
! [Xs: list_set_a,Ys2: list_set_a] :
( ( ( Xs = nil_set_a )
=> ( ( hd_set_a @ ( append_set_a @ Xs @ Ys2 ) )
= ( hd_set_a @ Ys2 ) ) )
& ( ( Xs != nil_set_a )
=> ( ( hd_set_a @ ( append_set_a @ Xs @ Ys2 ) )
= ( hd_set_a @ Xs ) ) ) ) ).
% hd_append
thf(fact_298_longest__common__prefix,axiom,
! [Xs: list_a,Ys2: list_a] :
? [Ps: list_a,Xs4: list_a,Ys6: list_a] :
( ( Xs
= ( append_a @ Ps @ Xs4 ) )
& ( Ys2
= ( append_a @ Ps @ Ys6 ) )
& ( ( Xs4 = nil_a )
| ( Ys6 = nil_a )
| ( ( hd_a @ Xs4 )
!= ( hd_a @ Ys6 ) ) ) ) ).
% longest_common_prefix
thf(fact_299_longest__common__prefix,axiom,
! [Xs: list_set_a,Ys2: list_set_a] :
? [Ps: list_set_a,Xs4: list_set_a,Ys6: list_set_a] :
( ( Xs
= ( append_set_a @ Ps @ Xs4 ) )
& ( Ys2
= ( append_set_a @ Ps @ Ys6 ) )
& ( ( Xs4 = nil_set_a )
| ( Ys6 = nil_set_a )
| ( ( hd_set_a @ Xs4 )
!= ( hd_set_a @ Ys6 ) ) ) ) ).
% longest_common_prefix
thf(fact_300_last_Osimps,axiom,
! [Xs: list_a,X: a] :
( ( ( Xs = nil_a )
=> ( ( last_a @ ( cons_a @ X @ Xs ) )
= X ) )
& ( ( Xs != nil_a )
=> ( ( last_a @ ( cons_a @ X @ Xs ) )
= ( last_a @ Xs ) ) ) ) ).
% last.simps
thf(fact_301_last_Osimps,axiom,
! [Xs: list_set_a,X: set_a] :
( ( ( Xs = nil_set_a )
=> ( ( last_set_a @ ( cons_set_a @ X @ Xs ) )
= X ) )
& ( ( Xs != nil_set_a )
=> ( ( last_set_a @ ( cons_set_a @ X @ Xs ) )
= ( last_set_a @ Xs ) ) ) ) ).
% last.simps
thf(fact_302_last__ConsL,axiom,
! [Xs: list_a,X: a] :
( ( Xs = nil_a )
=> ( ( last_a @ ( cons_a @ X @ Xs ) )
= X ) ) ).
% last_ConsL
thf(fact_303_last__ConsL,axiom,
! [Xs: list_set_a,X: set_a] :
( ( Xs = nil_set_a )
=> ( ( last_set_a @ ( cons_set_a @ X @ Xs ) )
= X ) ) ).
% last_ConsL
thf(fact_304_last__ConsR,axiom,
! [Xs: list_a,X: a] :
( ( Xs != nil_a )
=> ( ( last_a @ ( cons_a @ X @ Xs ) )
= ( last_a @ Xs ) ) ) ).
% last_ConsR
thf(fact_305_last__ConsR,axiom,
! [Xs: list_set_a,X: set_a] :
( ( Xs != nil_set_a )
=> ( ( last_set_a @ ( cons_set_a @ X @ Xs ) )
= ( last_set_a @ Xs ) ) ) ).
% last_ConsR
thf(fact_306_last__in__set,axiom,
! [As: list_P1396940483166286381od_a_a] :
( ( As != nil_Product_prod_a_a )
=> ( member1426531477525435216od_a_a @ ( last_P8790725268278465478od_a_a @ As ) @ ( set_Product_prod_a_a2 @ As ) ) ) ).
% last_in_set
thf(fact_307_last__in__set,axiom,
! [As: list_nat] :
( ( As != nil_nat )
=> ( member_nat @ ( last_nat @ As ) @ ( set_nat2 @ As ) ) ) ).
% last_in_set
thf(fact_308_last__in__set,axiom,
! [As: list_set_a] :
( ( As != nil_set_a )
=> ( member_set_a @ ( last_set_a @ As ) @ ( set_set_a2 @ As ) ) ) ).
% last_in_set
thf(fact_309_last__in__set,axiom,
! [As: list_a] :
( ( As != nil_a )
=> ( member_a @ ( last_a @ As ) @ ( set_a2 @ As ) ) ) ).
% last_in_set
thf(fact_310_last__append,axiom,
! [Ys2: list_a,Xs: list_a] :
( ( ( Ys2 = nil_a )
=> ( ( last_a @ ( append_a @ Xs @ Ys2 ) )
= ( last_a @ Xs ) ) )
& ( ( Ys2 != nil_a )
=> ( ( last_a @ ( append_a @ Xs @ Ys2 ) )
= ( last_a @ Ys2 ) ) ) ) ).
% last_append
thf(fact_311_last__append,axiom,
! [Ys2: list_set_a,Xs: list_set_a] :
( ( ( Ys2 = nil_set_a )
=> ( ( last_set_a @ ( append_set_a @ Xs @ Ys2 ) )
= ( last_set_a @ Xs ) ) )
& ( ( Ys2 != nil_set_a )
=> ( ( last_set_a @ ( append_set_a @ Xs @ Ys2 ) )
= ( last_set_a @ Ys2 ) ) ) ) ).
% last_append
thf(fact_312_longest__common__suffix,axiom,
! [Xs: list_a,Ys2: list_a] :
? [Ss: list_a,Xs4: list_a,Ys6: list_a] :
( ( Xs
= ( append_a @ Xs4 @ Ss ) )
& ( Ys2
= ( append_a @ Ys6 @ Ss ) )
& ( ( Xs4 = nil_a )
| ( Ys6 = nil_a )
| ( ( last_a @ Xs4 )
!= ( last_a @ Ys6 ) ) ) ) ).
% longest_common_suffix
thf(fact_313_longest__common__suffix,axiom,
! [Xs: list_set_a,Ys2: list_set_a] :
? [Ss: list_set_a,Xs4: list_set_a,Ys6: list_set_a] :
( ( Xs
= ( append_set_a @ Xs4 @ Ss ) )
& ( Ys2
= ( append_set_a @ Ys6 @ Ss ) )
& ( ( Xs4 = nil_set_a )
| ( Ys6 = nil_set_a )
| ( ( last_set_a @ Xs4 )
!= ( last_set_a @ Ys6 ) ) ) ) ).
% longest_common_suffix
thf(fact_314_rev_Osimps_I2_J,axiom,
! [X: a,Xs: list_a] :
( ( rev_a @ ( cons_a @ X @ Xs ) )
= ( append_a @ ( rev_a @ Xs ) @ ( cons_a @ X @ nil_a ) ) ) ).
% rev.simps(2)
thf(fact_315_rev_Osimps_I2_J,axiom,
! [X: set_a,Xs: list_set_a] :
( ( rev_set_a @ ( cons_set_a @ X @ Xs ) )
= ( append_set_a @ ( rev_set_a @ Xs ) @ ( cons_set_a @ X @ nil_set_a ) ) ) ).
% rev.simps(2)
thf(fact_316_ulgraph_Ois__walk__singleton,axiom,
! [Vertices: set_Product_prod_a_a,Edges: set_se5735800977113168103od_a_a,U: product_prod_a_a] :
( ( undire4585262585102564309od_a_a @ Vertices @ Edges )
=> ( ( member1426531477525435216od_a_a @ U @ Vertices )
=> ( undire3162072421265123221od_a_a @ Vertices @ Edges @ ( cons_P7316939126706565853od_a_a @ U @ nil_Product_prod_a_a ) ) ) ) ).
% ulgraph.is_walk_singleton
thf(fact_317_ulgraph_Ois__walk__singleton,axiom,
! [Vertices: set_nat,Edges: set_set_nat,U: nat] :
( ( undire3269267262472140706ph_nat @ Vertices @ Edges )
=> ( ( member_nat @ U @ Vertices )
=> ( undire5745680128780950498lk_nat @ Vertices @ Edges @ ( cons_nat @ U @ nil_nat ) ) ) ) ).
% ulgraph.is_walk_singleton
thf(fact_318_ulgraph_Ois__walk__singleton,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,U: set_a] :
( ( undire6886684016831807756_set_a @ Vertices @ Edges )
=> ( ( member_set_a @ U @ Vertices )
=> ( undire3014741414213135564_set_a @ Vertices @ Edges @ ( cons_set_a @ U @ nil_set_a ) ) ) ) ).
% ulgraph.is_walk_singleton
thf(fact_319_ulgraph_Ois__walk__singleton,axiom,
! [Vertices: set_a,Edges: set_set_a,U: a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( member_a @ U @ Vertices )
=> ( undire6133010728901294956walk_a @ Vertices @ Edges @ ( cons_a @ U @ nil_a ) ) ) ) ).
% ulgraph.is_walk_singleton
thf(fact_320_ulgraph_Ois__walk__wf,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,Xs: list_set_a] :
( ( undire6886684016831807756_set_a @ Vertices @ Edges )
=> ( ( undire3014741414213135564_set_a @ Vertices @ Edges @ Xs )
=> ( ord_le3724670747650509150_set_a @ ( set_set_a2 @ Xs ) @ Vertices ) ) ) ).
% ulgraph.is_walk_wf
thf(fact_321_ulgraph_Ois__walk__wf,axiom,
! [Vertices: set_Product_prod_a_a,Edges: set_se5735800977113168103od_a_a,Xs: list_P1396940483166286381od_a_a] :
( ( undire4585262585102564309od_a_a @ Vertices @ Edges )
=> ( ( undire3162072421265123221od_a_a @ Vertices @ Edges @ Xs )
=> ( ord_le746702958409616551od_a_a @ ( set_Product_prod_a_a2 @ Xs ) @ Vertices ) ) ) ).
% ulgraph.is_walk_wf
thf(fact_322_ulgraph_Ois__walk__wf,axiom,
! [Vertices: set_a,Edges: set_set_a,Xs: list_a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( undire6133010728901294956walk_a @ Vertices @ Edges @ Xs )
=> ( ord_less_eq_set_a @ ( set_a2 @ Xs ) @ Vertices ) ) ) ).
% ulgraph.is_walk_wf
thf(fact_323_ulgraph_Owalk__edges_Osimps_I1_J,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a] :
( ( undire6886684016831807756_set_a @ Vertices @ Edges )
=> ( ( undire6234387080713648494_set_a @ nil_set_a )
= nil_set_set_a ) ) ).
% ulgraph.walk_edges.simps(1)
thf(fact_324_ulgraph_Owalk__edges_Osimps_I1_J,axiom,
! [Vertices: set_a,Edges: set_set_a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( undire7337870655677353998dges_a @ nil_a )
= nil_set_a ) ) ).
% ulgraph.walk_edges.simps(1)
thf(fact_325_in__mono,axiom,
! [A2: set_nat,B2: set_nat,X: nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( member_nat @ X @ A2 )
=> ( member_nat @ X @ B2 ) ) ) ).
% in_mono
thf(fact_326_in__mono,axiom,
! [A2: set_set_a,B2: set_set_a,X: set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B2 )
=> ( ( member_set_a @ X @ A2 )
=> ( member_set_a @ X @ B2 ) ) ) ).
% in_mono
thf(fact_327_in__mono,axiom,
! [A2: set_a,B2: set_a,X: a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( member_a @ X @ A2 )
=> ( member_a @ X @ B2 ) ) ) ).
% in_mono
thf(fact_328_in__mono,axiom,
! [A2: set_Product_prod_a_a,B2: set_Product_prod_a_a,X: product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A2 @ B2 )
=> ( ( member1426531477525435216od_a_a @ X @ A2 )
=> ( member1426531477525435216od_a_a @ X @ B2 ) ) ) ).
% in_mono
thf(fact_329_subsetD,axiom,
! [A2: set_nat,B2: set_nat,C: nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( member_nat @ C @ A2 )
=> ( member_nat @ C @ B2 ) ) ) ).
% subsetD
thf(fact_330_subsetD,axiom,
! [A2: set_set_a,B2: set_set_a,C: set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B2 )
=> ( ( member_set_a @ C @ A2 )
=> ( member_set_a @ C @ B2 ) ) ) ).
% subsetD
thf(fact_331_subsetD,axiom,
! [A2: set_a,B2: set_a,C: a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( member_a @ C @ A2 )
=> ( member_a @ C @ B2 ) ) ) ).
% subsetD
thf(fact_332_subsetD,axiom,
! [A2: set_Product_prod_a_a,B2: set_Product_prod_a_a,C: product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A2 @ B2 )
=> ( ( member1426531477525435216od_a_a @ C @ A2 )
=> ( member1426531477525435216od_a_a @ C @ B2 ) ) ) ).
% subsetD
thf(fact_333_equalityE,axiom,
! [A2: set_set_a,B2: set_set_a] :
( ( A2 = B2 )
=> ~ ( ( ord_le3724670747650509150_set_a @ A2 @ B2 )
=> ~ ( ord_le3724670747650509150_set_a @ B2 @ A2 ) ) ) ).
% equalityE
thf(fact_334_equalityE,axiom,
! [A2: set_a,B2: set_a] :
( ( A2 = B2 )
=> ~ ( ( ord_less_eq_set_a @ A2 @ B2 )
=> ~ ( ord_less_eq_set_a @ B2 @ A2 ) ) ) ).
% equalityE
thf(fact_335_equalityE,axiom,
! [A2: set_Product_prod_a_a,B2: set_Product_prod_a_a] :
( ( A2 = B2 )
=> ~ ( ( ord_le746702958409616551od_a_a @ A2 @ B2 )
=> ~ ( ord_le746702958409616551od_a_a @ B2 @ A2 ) ) ) ).
% equalityE
thf(fact_336_subset__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A3: set_nat,B3: set_nat] :
! [X4: nat] :
( ( member_nat @ X4 @ A3 )
=> ( member_nat @ X4 @ B3 ) ) ) ) ).
% subset_eq
thf(fact_337_subset__eq,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [A3: set_set_a,B3: set_set_a] :
! [X4: set_a] :
( ( member_set_a @ X4 @ A3 )
=> ( member_set_a @ X4 @ B3 ) ) ) ) ).
% subset_eq
thf(fact_338_subset__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A3: set_a,B3: set_a] :
! [X4: a] :
( ( member_a @ X4 @ A3 )
=> ( member_a @ X4 @ B3 ) ) ) ) ).
% subset_eq
thf(fact_339_subset__eq,axiom,
( ord_le746702958409616551od_a_a
= ( ^ [A3: set_Product_prod_a_a,B3: set_Product_prod_a_a] :
! [X4: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X4 @ A3 )
=> ( member1426531477525435216od_a_a @ X4 @ B3 ) ) ) ) ).
% subset_eq
thf(fact_340_equalityD1,axiom,
! [A2: set_set_a,B2: set_set_a] :
( ( A2 = B2 )
=> ( ord_le3724670747650509150_set_a @ A2 @ B2 ) ) ).
% equalityD1
thf(fact_341_equalityD1,axiom,
! [A2: set_a,B2: set_a] :
( ( A2 = B2 )
=> ( ord_less_eq_set_a @ A2 @ B2 ) ) ).
% equalityD1
thf(fact_342_equalityD1,axiom,
! [A2: set_Product_prod_a_a,B2: set_Product_prod_a_a] :
( ( A2 = B2 )
=> ( ord_le746702958409616551od_a_a @ A2 @ B2 ) ) ).
% equalityD1
thf(fact_343_equalityD2,axiom,
! [A2: set_set_a,B2: set_set_a] :
( ( A2 = B2 )
=> ( ord_le3724670747650509150_set_a @ B2 @ A2 ) ) ).
% equalityD2
thf(fact_344_equalityD2,axiom,
! [A2: set_a,B2: set_a] :
( ( A2 = B2 )
=> ( ord_less_eq_set_a @ B2 @ A2 ) ) ).
% equalityD2
thf(fact_345_equalityD2,axiom,
! [A2: set_Product_prod_a_a,B2: set_Product_prod_a_a] :
( ( A2 = B2 )
=> ( ord_le746702958409616551od_a_a @ B2 @ A2 ) ) ).
% equalityD2
thf(fact_346_subset__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [A3: set_nat,B3: set_nat] :
! [T: nat] :
( ( member_nat @ T @ A3 )
=> ( member_nat @ T @ B3 ) ) ) ) ).
% subset_iff
thf(fact_347_subset__iff,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [A3: set_set_a,B3: set_set_a] :
! [T: set_a] :
( ( member_set_a @ T @ A3 )
=> ( member_set_a @ T @ B3 ) ) ) ) ).
% subset_iff
thf(fact_348_subset__iff,axiom,
( ord_less_eq_set_a
= ( ^ [A3: set_a,B3: set_a] :
! [T: a] :
( ( member_a @ T @ A3 )
=> ( member_a @ T @ B3 ) ) ) ) ).
% subset_iff
thf(fact_349_subset__iff,axiom,
( ord_le746702958409616551od_a_a
= ( ^ [A3: set_Product_prod_a_a,B3: set_Product_prod_a_a] :
! [T: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ T @ A3 )
=> ( member1426531477525435216od_a_a @ T @ B3 ) ) ) ) ).
% subset_iff
thf(fact_350_subset__refl,axiom,
! [A2: set_set_a] : ( ord_le3724670747650509150_set_a @ A2 @ A2 ) ).
% subset_refl
thf(fact_351_subset__refl,axiom,
! [A2: set_a] : ( ord_less_eq_set_a @ A2 @ A2 ) ).
% subset_refl
thf(fact_352_subset__refl,axiom,
! [A2: set_Product_prod_a_a] : ( ord_le746702958409616551od_a_a @ A2 @ A2 ) ).
% subset_refl
thf(fact_353_Collect__mono,axiom,
! [P: set_a > $o,Q: set_a > $o] :
( ! [X2: set_a] :
( ( P @ X2 )
=> ( Q @ X2 ) )
=> ( ord_le3724670747650509150_set_a @ ( collect_set_a @ P ) @ ( collect_set_a @ Q ) ) ) ).
% Collect_mono
thf(fact_354_Collect__mono,axiom,
! [P: a > $o,Q: a > $o] :
( ! [X2: a] :
( ( P @ X2 )
=> ( Q @ X2 ) )
=> ( ord_less_eq_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) ) ) ).
% Collect_mono
thf(fact_355_Collect__mono,axiom,
! [P: product_prod_a_a > $o,Q: product_prod_a_a > $o] :
( ! [X2: product_prod_a_a] :
( ( P @ X2 )
=> ( Q @ X2 ) )
=> ( ord_le746702958409616551od_a_a @ ( collec3336397797384452498od_a_a @ P ) @ ( collec3336397797384452498od_a_a @ Q ) ) ) ).
% Collect_mono
thf(fact_356_subset__trans,axiom,
! [A2: set_set_a,B2: set_set_a,C2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B2 )
=> ( ( ord_le3724670747650509150_set_a @ B2 @ C2 )
=> ( ord_le3724670747650509150_set_a @ A2 @ C2 ) ) ) ).
% subset_trans
thf(fact_357_subset__trans,axiom,
! [A2: set_a,B2: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ord_less_eq_set_a @ B2 @ C2 )
=> ( ord_less_eq_set_a @ A2 @ C2 ) ) ) ).
% subset_trans
thf(fact_358_subset__trans,axiom,
! [A2: set_Product_prod_a_a,B2: set_Product_prod_a_a,C2: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A2 @ B2 )
=> ( ( ord_le746702958409616551od_a_a @ B2 @ C2 )
=> ( ord_le746702958409616551od_a_a @ A2 @ C2 ) ) ) ).
% subset_trans
thf(fact_359_set__eq__subset,axiom,
( ( ^ [Y4: set_set_a,Z: set_set_a] : ( Y4 = Z ) )
= ( ^ [A3: set_set_a,B3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A3 @ B3 )
& ( ord_le3724670747650509150_set_a @ B3 @ A3 ) ) ) ) ).
% set_eq_subset
thf(fact_360_set__eq__subset,axiom,
( ( ^ [Y4: set_a,Z: set_a] : ( Y4 = Z ) )
= ( ^ [A3: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ A3 @ B3 )
& ( ord_less_eq_set_a @ B3 @ A3 ) ) ) ) ).
% set_eq_subset
thf(fact_361_set__eq__subset,axiom,
( ( ^ [Y4: set_Product_prod_a_a,Z: set_Product_prod_a_a] : ( Y4 = Z ) )
= ( ^ [A3: set_Product_prod_a_a,B3: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A3 @ B3 )
& ( ord_le746702958409616551od_a_a @ B3 @ A3 ) ) ) ) ).
% set_eq_subset
thf(fact_362_Collect__mono__iff,axiom,
! [P: set_a > $o,Q: set_a > $o] :
( ( ord_le3724670747650509150_set_a @ ( collect_set_a @ P ) @ ( collect_set_a @ Q ) )
= ( ! [X4: set_a] :
( ( P @ X4 )
=> ( Q @ X4 ) ) ) ) ).
% Collect_mono_iff
thf(fact_363_Collect__mono__iff,axiom,
! [P: a > $o,Q: a > $o] :
( ( ord_less_eq_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) )
= ( ! [X4: a] :
( ( P @ X4 )
=> ( Q @ X4 ) ) ) ) ).
% Collect_mono_iff
thf(fact_364_Collect__mono__iff,axiom,
! [P: product_prod_a_a > $o,Q: product_prod_a_a > $o] :
( ( ord_le746702958409616551od_a_a @ ( collec3336397797384452498od_a_a @ P ) @ ( collec3336397797384452498od_a_a @ Q ) )
= ( ! [X4: product_prod_a_a] :
( ( P @ X4 )
=> ( Q @ X4 ) ) ) ) ).
% Collect_mono_iff
thf(fact_365_comp__sgraph_Owalk__edges__tl__ss,axiom,
! [Xs: list_a] : ( ord_le3724670747650509150_set_a @ ( set_set_a2 @ ( undire7337870655677353998dges_a @ ( tl_a @ Xs ) ) ) @ ( set_set_a2 @ ( undire7337870655677353998dges_a @ Xs ) ) ) ).
% comp_sgraph.walk_edges_tl_ss
thf(fact_366_ulgraph_Owalk__edges_Osimps_I2_J,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,X: set_a] :
( ( undire6886684016831807756_set_a @ Vertices @ Edges )
=> ( ( undire6234387080713648494_set_a @ ( cons_set_a @ X @ nil_set_a ) )
= nil_set_set_a ) ) ).
% ulgraph.walk_edges.simps(2)
thf(fact_367_ulgraph_Owalk__edges_Osimps_I2_J,axiom,
! [Vertices: set_a,Edges: set_set_a,X: a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( undire7337870655677353998dges_a @ ( cons_a @ X @ nil_a ) )
= nil_set_a ) ) ).
% ulgraph.walk_edges.simps(2)
thf(fact_368_ulgraph_Owalk__edges__append__ss1,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,Ys2: list_set_a,Xs: list_set_a] :
( ( undire6886684016831807756_set_a @ Vertices @ Edges )
=> ( ord_le5722252365846178494_set_a @ ( set_set_set_a2 @ ( undire6234387080713648494_set_a @ Ys2 ) ) @ ( set_set_set_a2 @ ( undire6234387080713648494_set_a @ ( append_set_a @ Xs @ Ys2 ) ) ) ) ) ).
% ulgraph.walk_edges_append_ss1
thf(fact_369_ulgraph_Owalk__edges__append__ss1,axiom,
! [Vertices: set_a,Edges: set_set_a,Ys2: list_a,Xs: list_a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ord_le3724670747650509150_set_a @ ( set_set_a2 @ ( undire7337870655677353998dges_a @ Ys2 ) ) @ ( set_set_a2 @ ( undire7337870655677353998dges_a @ ( append_a @ Xs @ Ys2 ) ) ) ) ) ).
% ulgraph.walk_edges_append_ss1
thf(fact_370_ulgraph_Owalk__edges__append__ss2,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,Xs: list_set_a,Ys2: list_set_a] :
( ( undire6886684016831807756_set_a @ Vertices @ Edges )
=> ( ord_le5722252365846178494_set_a @ ( set_set_set_a2 @ ( undire6234387080713648494_set_a @ Xs ) ) @ ( set_set_set_a2 @ ( undire6234387080713648494_set_a @ ( append_set_a @ Xs @ Ys2 ) ) ) ) ) ).
% ulgraph.walk_edges_append_ss2
thf(fact_371_ulgraph_Owalk__edges__append__ss2,axiom,
! [Vertices: set_a,Edges: set_set_a,Xs: list_a,Ys2: list_a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ord_le3724670747650509150_set_a @ ( set_set_a2 @ ( undire7337870655677353998dges_a @ Xs ) ) @ ( set_set_a2 @ ( undire7337870655677353998dges_a @ ( append_a @ Xs @ Ys2 ) ) ) ) ) ).
% ulgraph.walk_edges_append_ss2
thf(fact_372_ulgraph_Owalk__edges__decomp__ss,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,Xs: list_set_a,Y2: set_a,Zs: list_set_a,Ys2: list_set_a] :
( ( undire6886684016831807756_set_a @ Vertices @ Edges )
=> ( ord_le5722252365846178494_set_a @ ( set_set_set_a2 @ ( undire6234387080713648494_set_a @ ( append_set_a @ Xs @ ( append_set_a @ ( cons_set_a @ Y2 @ nil_set_a ) @ Zs ) ) ) ) @ ( set_set_set_a2 @ ( undire6234387080713648494_set_a @ ( append_set_a @ Xs @ ( append_set_a @ ( cons_set_a @ Y2 @ nil_set_a ) @ ( append_set_a @ Ys2 @ ( append_set_a @ ( cons_set_a @ Y2 @ nil_set_a ) @ Zs ) ) ) ) ) ) ) ) ).
% ulgraph.walk_edges_decomp_ss
thf(fact_373_ulgraph_Owalk__edges__decomp__ss,axiom,
! [Vertices: set_a,Edges: set_set_a,Xs: list_a,Y2: a,Zs: list_a,Ys2: list_a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ord_le3724670747650509150_set_a @ ( set_set_a2 @ ( undire7337870655677353998dges_a @ ( append_a @ Xs @ ( append_a @ ( cons_a @ Y2 @ nil_a ) @ Zs ) ) ) ) @ ( set_set_a2 @ ( undire7337870655677353998dges_a @ ( append_a @ Xs @ ( append_a @ ( cons_a @ Y2 @ nil_a ) @ ( append_a @ Ys2 @ ( append_a @ ( cons_a @ Y2 @ nil_a ) @ Zs ) ) ) ) ) ) ) ) ).
% ulgraph.walk_edges_decomp_ss
thf(fact_374_ulgraph_Ois__walkI,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,Xs: list_set_a] :
( ( undire6886684016831807756_set_a @ Vertices @ Edges )
=> ( ( ord_le3724670747650509150_set_a @ ( set_set_a2 @ Xs ) @ Vertices )
=> ( ( ord_le5722252365846178494_set_a @ ( set_set_set_a2 @ ( undire6234387080713648494_set_a @ Xs ) ) @ Edges )
=> ( ( Xs != nil_set_a )
=> ( undire3014741414213135564_set_a @ Vertices @ Edges @ Xs ) ) ) ) ) ).
% ulgraph.is_walkI
thf(fact_375_ulgraph_Ois__walkI,axiom,
! [Vertices: set_Product_prod_a_a,Edges: set_se5735800977113168103od_a_a,Xs: list_P1396940483166286381od_a_a] :
( ( undire4585262585102564309od_a_a @ Vertices @ Edges )
=> ( ( ord_le746702958409616551od_a_a @ ( set_Product_prod_a_a2 @ Xs ) @ Vertices )
=> ( ( ord_le1995061765932249223od_a_a @ ( set_se8408754101646271900od_a_a @ ( undire4403264684974754359od_a_a @ Xs ) ) @ Edges )
=> ( ( Xs != nil_Product_prod_a_a )
=> ( undire3162072421265123221od_a_a @ Vertices @ Edges @ Xs ) ) ) ) ) ).
% ulgraph.is_walkI
thf(fact_376_ulgraph_Ois__walkI,axiom,
! [Vertices: set_a,Edges: set_set_a,Xs: list_a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Xs ) @ Vertices )
=> ( ( ord_le3724670747650509150_set_a @ ( set_set_a2 @ ( undire7337870655677353998dges_a @ Xs ) ) @ Edges )
=> ( ( Xs != nil_a )
=> ( undire6133010728901294956walk_a @ Vertices @ Edges @ Xs ) ) ) ) ) ).
% ulgraph.is_walkI
thf(fact_377_ulgraph_Ois__walk__def,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,Xs: list_set_a] :
( ( undire6886684016831807756_set_a @ Vertices @ Edges )
=> ( ( undire3014741414213135564_set_a @ Vertices @ Edges @ Xs )
= ( ( ord_le3724670747650509150_set_a @ ( set_set_a2 @ Xs ) @ Vertices )
& ( ord_le5722252365846178494_set_a @ ( set_set_set_a2 @ ( undire6234387080713648494_set_a @ Xs ) ) @ Edges )
& ( Xs != nil_set_a ) ) ) ) ).
% ulgraph.is_walk_def
thf(fact_378_ulgraph_Ois__walk__def,axiom,
! [Vertices: set_Product_prod_a_a,Edges: set_se5735800977113168103od_a_a,Xs: list_P1396940483166286381od_a_a] :
( ( undire4585262585102564309od_a_a @ Vertices @ Edges )
=> ( ( undire3162072421265123221od_a_a @ Vertices @ Edges @ Xs )
= ( ( ord_le746702958409616551od_a_a @ ( set_Product_prod_a_a2 @ Xs ) @ Vertices )
& ( ord_le1995061765932249223od_a_a @ ( set_se8408754101646271900od_a_a @ ( undire4403264684974754359od_a_a @ Xs ) ) @ Edges )
& ( Xs != nil_Product_prod_a_a ) ) ) ) ).
% ulgraph.is_walk_def
thf(fact_379_ulgraph_Ois__walk__def,axiom,
! [Vertices: set_a,Edges: set_set_a,Xs: list_a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( undire6133010728901294956walk_a @ Vertices @ Edges @ Xs )
= ( ( ord_less_eq_set_a @ ( set_a2 @ Xs ) @ Vertices )
& ( ord_le3724670747650509150_set_a @ ( set_set_a2 @ ( undire7337870655677353998dges_a @ Xs ) ) @ Edges )
& ( Xs != nil_a ) ) ) ) ).
% ulgraph.is_walk_def
thf(fact_380_is__isolated__vertex__no__loop,axiom,
! [V2: a] :
( ( undire8931668460104145173rtex_a @ vertices @ edges @ V2 )
=> ~ ( undire3617971648856834880loop_a @ edges @ V2 ) ) ).
% is_isolated_vertex_no_loop
thf(fact_381_is__isolated__vertex__def,axiom,
! [V2: a] :
( ( undire8931668460104145173rtex_a @ vertices @ edges @ V2 )
= ( ( member_a @ V2 @ vertices )
& ! [X4: a] :
( ( member_a @ X4 @ vertices )
=> ~ ( undire397441198561214472_adj_a @ edges @ X4 @ V2 ) ) ) ) ).
% is_isolated_vertex_def
thf(fact_382_is__isolated__vertex__edge,axiom,
! [V2: a,E: set_a] :
( ( undire8931668460104145173rtex_a @ vertices @ edges @ V2 )
=> ( ( member_set_a @ E @ edges )
=> ~ ( undire1521409233611534436dent_a @ V2 @ E ) ) ) ).
% is_isolated_vertex_edge
thf(fact_383_is__subgraphI,axiom,
! [V: set_set_a,V3: set_set_a,E3: set_set_set_a,E4: set_set_set_a] :
( ( ord_le3724670747650509150_set_a @ V @ V3 )
=> ( ( ord_le5722252365846178494_set_a @ E3 @ E4 )
=> ( ( undire7159349782766787846_set_a @ V @ E3 )
=> ( ( undire7159349782766787846_set_a @ V3 @ E4 )
=> ( undire1186139521737116585_set_a @ V @ E3 @ V3 @ E4 ) ) ) ) ) ).
% is_subgraphI
thf(fact_384_is__subgraphI,axiom,
! [V: set_Product_prod_a_a,V3: set_Product_prod_a_a,E3: set_se5735800977113168103od_a_a,E4: set_se5735800977113168103od_a_a] :
( ( ord_le746702958409616551od_a_a @ V @ V3 )
=> ( ( ord_le1995061765932249223od_a_a @ E3 @ E4 )
=> ( ( undire1860116983885411791od_a_a @ V @ E3 )
=> ( ( undire1860116983885411791od_a_a @ V3 @ E4 )
=> ( undire398746457437328754od_a_a @ V @ E3 @ V3 @ E4 ) ) ) ) ) ).
% is_subgraphI
thf(fact_385_is__subgraphI,axiom,
! [V: set_a,V3: set_a,E3: set_set_a,E4: set_set_a] :
( ( ord_less_eq_set_a @ V @ V3 )
=> ( ( ord_le3724670747650509150_set_a @ E3 @ E4 )
=> ( ( undire2554140024507503526stem_a @ V @ E3 )
=> ( ( undire2554140024507503526stem_a @ V3 @ E4 )
=> ( undire7103218114511261257raph_a @ V @ E3 @ V3 @ E4 ) ) ) ) ) ).
% is_subgraphI
thf(fact_386_graph__system_Oinduced__is__subgraph,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,V: set_set_a] :
( ( undire7159349782766787846_set_a @ Vertices @ Edges )
=> ( ( ord_le3724670747650509150_set_a @ V @ Vertices )
=> ( undire1186139521737116585_set_a @ V @ ( undire7854589003810675244_set_a @ Edges @ V ) @ Vertices @ Edges ) ) ) ).
% graph_system.induced_is_subgraph
thf(fact_387_graph__system_Oinduced__is__subgraph,axiom,
! [Vertices: set_Product_prod_a_a,Edges: set_se5735800977113168103od_a_a,V: set_Product_prod_a_a] :
( ( undire1860116983885411791od_a_a @ Vertices @ Edges )
=> ( ( ord_le746702958409616551od_a_a @ V @ Vertices )
=> ( undire398746457437328754od_a_a @ V @ ( undire5906991851038061813od_a_a @ Edges @ V ) @ Vertices @ Edges ) ) ) ).
% graph_system.induced_is_subgraph
thf(fact_388_graph__system_Oinduced__is__subgraph,axiom,
! [Vertices: set_a,Edges: set_set_a,V: set_a] :
( ( undire2554140024507503526stem_a @ Vertices @ Edges )
=> ( ( ord_less_eq_set_a @ V @ Vertices )
=> ( undire7103218114511261257raph_a @ V @ ( undire7777452895879145676dges_a @ Edges @ V ) @ Vertices @ Edges ) ) ) ).
% graph_system.induced_is_subgraph
thf(fact_389_graph__system_Oinduced__edges__ss,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,V: set_set_a] :
( ( undire7159349782766787846_set_a @ Vertices @ Edges )
=> ( ( ord_le3724670747650509150_set_a @ V @ Vertices )
=> ( ord_le5722252365846178494_set_a @ ( undire7854589003810675244_set_a @ Edges @ V ) @ Edges ) ) ) ).
% graph_system.induced_edges_ss
thf(fact_390_graph__system_Oinduced__edges__ss,axiom,
! [Vertices: set_Product_prod_a_a,Edges: set_se5735800977113168103od_a_a,V: set_Product_prod_a_a] :
( ( undire1860116983885411791od_a_a @ Vertices @ Edges )
=> ( ( ord_le746702958409616551od_a_a @ V @ Vertices )
=> ( ord_le1995061765932249223od_a_a @ ( undire5906991851038061813od_a_a @ Edges @ V ) @ Edges ) ) ) ).
% graph_system.induced_edges_ss
thf(fact_391_graph__system_Oinduced__edges__ss,axiom,
! [Vertices: set_a,Edges: set_set_a,V: set_a] :
( ( undire2554140024507503526stem_a @ Vertices @ Edges )
=> ( ( ord_less_eq_set_a @ V @ Vertices )
=> ( ord_le3724670747650509150_set_a @ ( undire7777452895879145676dges_a @ Edges @ V ) @ Edges ) ) ) ).
% graph_system.induced_edges_ss
thf(fact_392_ulgraph_Overt__adj__edge__iff2,axiom,
! [Vertices: set_a,Edges: set_set_a,V1: a,V22: a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( V1 != V22 )
=> ( ( undire397441198561214472_adj_a @ Edges @ V1 @ V22 )
= ( ? [X4: set_a] :
( ( member_set_a @ X4 @ Edges )
& ( undire1521409233611534436dent_a @ V1 @ X4 )
& ( undire1521409233611534436dent_a @ V22 @ X4 ) ) ) ) ) ) ).
% ulgraph.vert_adj_edge_iff2
thf(fact_393_graph__system_Oedge__adjacent__alt__def,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,E1: set_set_a,E2: set_set_a] :
( ( undire7159349782766787846_set_a @ Vertices @ Edges )
=> ( ( member_set_set_a @ E1 @ Edges )
=> ( ( member_set_set_a @ E2 @ Edges )
=> ( ? [X3: set_a] :
( ( member_set_a @ X3 @ Vertices )
& ( member_set_a @ X3 @ E1 )
& ( member_set_a @ X3 @ E2 ) )
=> ( undire3485422320110889978_set_a @ Edges @ E1 @ E2 ) ) ) ) ) ).
% graph_system.edge_adjacent_alt_def
thf(fact_394_graph__system_Oedge__adjacent__alt__def,axiom,
! [Vertices: set_Product_prod_a_a,Edges: set_se5735800977113168103od_a_a,E1: set_Product_prod_a_a,E2: set_Product_prod_a_a] :
( ( undire1860116983885411791od_a_a @ Vertices @ Edges )
=> ( ( member1816616512716248880od_a_a @ E1 @ Edges )
=> ( ( member1816616512716248880od_a_a @ E2 @ Edges )
=> ( ? [X3: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X3 @ Vertices )
& ( member1426531477525435216od_a_a @ X3 @ E1 )
& ( member1426531477525435216od_a_a @ X3 @ E2 ) )
=> ( undire9186443406341554371od_a_a @ Edges @ E1 @ E2 ) ) ) ) ) ).
% graph_system.edge_adjacent_alt_def
thf(fact_395_graph__system_Oedge__adjacent__alt__def,axiom,
! [Vertices: set_nat,Edges: set_set_nat,E1: set_nat,E2: set_nat] :
( ( undire7481384412329822504em_nat @ Vertices @ Edges )
=> ( ( member_set_nat @ E1 @ Edges )
=> ( ( member_set_nat @ E2 @ Edges )
=> ( ? [X3: nat] :
( ( member_nat @ X3 @ Vertices )
& ( member_nat @ X3 @ E1 )
& ( member_nat @ X3 @ E2 ) )
=> ( undire1664191744716346676dj_nat @ Edges @ E1 @ E2 ) ) ) ) ) ).
% graph_system.edge_adjacent_alt_def
thf(fact_396_graph__system_Oedge__adjacent__alt__def,axiom,
! [Vertices: set_a,Edges: set_set_a,E1: set_a,E2: set_a] :
( ( undire2554140024507503526stem_a @ Vertices @ Edges )
=> ( ( member_set_a @ E1 @ Edges )
=> ( ( member_set_a @ E2 @ Edges )
=> ( ? [X3: a] :
( ( member_a @ X3 @ Vertices )
& ( member_a @ X3 @ E1 )
& ( member_a @ X3 @ E2 ) )
=> ( undire4022703626023482010_adj_a @ Edges @ E1 @ E2 ) ) ) ) ) ).
% graph_system.edge_adjacent_alt_def
thf(fact_397_graph__system_Oedge__adj__inE,axiom,
! [Vertices: set_a,Edges: set_set_a,E1: set_a,E2: set_a] :
( ( undire2554140024507503526stem_a @ Vertices @ Edges )
=> ( ( undire4022703626023482010_adj_a @ Edges @ E1 @ E2 )
=> ( ( member_set_a @ E1 @ Edges )
& ( member_set_a @ E2 @ Edges ) ) ) ) ).
% graph_system.edge_adj_inE
thf(fact_398_walk__edges__rev,axiom,
! [Xs: list_a] :
( ( rev_set_a @ ( undire7337870655677353998dges_a @ Xs ) )
= ( undire7337870655677353998dges_a @ ( rev_a @ Xs ) ) ) ).
% walk_edges_rev
thf(fact_399_ulgraph_Ois__isolated__vertex_Ocong,axiom,
undire8931668460104145173rtex_a = undire8931668460104145173rtex_a ).
% ulgraph.is_isolated_vertex.cong
thf(fact_400_ulgraph_Ois__isolated__vertex__def,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,V2: set_a] :
( ( undire6886684016831807756_set_a @ Vertices @ Edges )
=> ( ( undire6879241558604981877_set_a @ Vertices @ Edges @ V2 )
= ( ( member_set_a @ V2 @ Vertices )
& ! [X4: set_a] :
( ( member_set_a @ X4 @ Vertices )
=> ~ ( undire3510646817838285160_set_a @ Edges @ X4 @ V2 ) ) ) ) ) ).
% ulgraph.is_isolated_vertex_def
thf(fact_401_ulgraph_Ois__isolated__vertex__def,axiom,
! [Vertices: set_Product_prod_a_a,Edges: set_se5735800977113168103od_a_a,V2: product_prod_a_a] :
( ( undire4585262585102564309od_a_a @ Vertices @ Edges )
=> ( ( undire3207556238582723646od_a_a @ Vertices @ Edges @ V2 )
= ( ( member1426531477525435216od_a_a @ V2 @ Vertices )
& ! [X4: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X4 @ Vertices )
=> ~ ( undire6135774327024169009od_a_a @ Edges @ X4 @ V2 ) ) ) ) ) ).
% ulgraph.is_isolated_vertex_def
thf(fact_402_ulgraph_Ois__isolated__vertex__def,axiom,
! [Vertices: set_nat,Edges: set_set_nat,V2: nat] :
( ( undire3269267262472140706ph_nat @ Vertices @ Edges )
=> ( ( undire5609513041723151865ex_nat @ Vertices @ Edges @ V2 )
= ( ( member_nat @ V2 @ Vertices )
& ! [X4: nat] :
( ( member_nat @ X4 @ Vertices )
=> ~ ( undire1083030068171319366dj_nat @ Edges @ X4 @ V2 ) ) ) ) ) ).
% ulgraph.is_isolated_vertex_def
thf(fact_403_ulgraph_Ois__isolated__vertex__def,axiom,
! [Vertices: set_a,Edges: set_set_a,V2: a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( undire8931668460104145173rtex_a @ Vertices @ Edges @ V2 )
= ( ( member_a @ V2 @ Vertices )
& ! [X4: a] :
( ( member_a @ X4 @ Vertices )
=> ~ ( undire397441198561214472_adj_a @ Edges @ X4 @ V2 ) ) ) ) ) ).
% ulgraph.is_isolated_vertex_def
thf(fact_404_ulgraph_Ois__isolated__vertex__edge,axiom,
! [Vertices: set_a,Edges: set_set_a,V2: a,E: set_a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( undire8931668460104145173rtex_a @ Vertices @ Edges @ V2 )
=> ( ( member_set_a @ E @ Edges )
=> ~ ( undire1521409233611534436dent_a @ V2 @ E ) ) ) ) ).
% ulgraph.is_isolated_vertex_edge
thf(fact_405_ulgraph_Ois__isolated__vertex__no__loop,axiom,
! [Vertices: set_a,Edges: set_set_a,V2: a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( undire8931668460104145173rtex_a @ Vertices @ Edges @ V2 )
=> ~ ( undire3617971648856834880loop_a @ Edges @ V2 ) ) ) ).
% ulgraph.is_isolated_vertex_no_loop
thf(fact_406_subgraph_Osubgraph__antisym,axiom,
! [V_H: set_a,E_H: set_set_a,V_G: set_a,E_G: set_set_a,V: set_a,E3: set_set_a,V3: set_a,E4: set_set_a] :
( ( undire7103218114511261257raph_a @ V_H @ E_H @ V_G @ E_G )
=> ( ( undire7103218114511261257raph_a @ V @ E3 @ V3 @ E4 )
=> ( ( undire7103218114511261257raph_a @ V3 @ E4 @ V @ E3 )
=> ( ( V3 = V )
& ( E4 = E3 ) ) ) ) ) ).
% subgraph.subgraph_antisym
thf(fact_407_graph__system_Oinduced__edges_Ocong,axiom,
undire7777452895879145676dges_a = undire7777452895879145676dges_a ).
% graph_system.induced_edges.cong
thf(fact_408_ulgraph_Overt__adj_Ocong,axiom,
undire397441198561214472_adj_a = undire397441198561214472_adj_a ).
% ulgraph.vert_adj.cong
thf(fact_409_comp__sgraph_Oincident__def,axiom,
undire2320338297334612420_set_a = member_set_a ).
% comp_sgraph.incident_def
thf(fact_410_comp__sgraph_Oincident__def,axiom,
undire3369688177417741453od_a_a = member1426531477525435216od_a_a ).
% comp_sgraph.incident_def
thf(fact_411_comp__sgraph_Oincident__def,axiom,
undire7858122600432113898nt_nat = member_nat ).
% comp_sgraph.incident_def
thf(fact_412_comp__sgraph_Oincident__def,axiom,
undire1521409233611534436dent_a = member_a ).
% comp_sgraph.incident_def
thf(fact_413_ulgraph_Ohas__loop_Ocong,axiom,
undire3617971648856834880loop_a = undire3617971648856834880loop_a ).
% ulgraph.has_loop.cong
thf(fact_414_graph__system_Oedge__adj_Ocong,axiom,
undire4022703626023482010_adj_a = undire4022703626023482010_adj_a ).
% graph_system.edge_adj.cong
thf(fact_415_graph__system__def,axiom,
( undire7159349782766787846_set_a
= ( ^ [Vertices2: set_set_a,Edges2: set_set_set_a] :
! [E5: set_set_a] :
( ( member_set_set_a @ E5 @ Edges2 )
=> ( ord_le3724670747650509150_set_a @ E5 @ Vertices2 ) ) ) ) ).
% graph_system_def
thf(fact_416_graph__system__def,axiom,
( undire1860116983885411791od_a_a
= ( ^ [Vertices2: set_Product_prod_a_a,Edges2: set_se5735800977113168103od_a_a] :
! [E5: set_Product_prod_a_a] :
( ( member1816616512716248880od_a_a @ E5 @ Edges2 )
=> ( ord_le746702958409616551od_a_a @ E5 @ Vertices2 ) ) ) ) ).
% graph_system_def
thf(fact_417_graph__system__def,axiom,
( undire2554140024507503526stem_a
= ( ^ [Vertices2: set_a,Edges2: set_set_a] :
! [E5: set_a] :
( ( member_set_a @ E5 @ Edges2 )
=> ( ord_less_eq_set_a @ E5 @ Vertices2 ) ) ) ) ).
% graph_system_def
thf(fact_418_graph__system_Owellformed,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,E: set_set_a] :
( ( undire7159349782766787846_set_a @ Vertices @ Edges )
=> ( ( member_set_set_a @ E @ Edges )
=> ( ord_le3724670747650509150_set_a @ E @ Vertices ) ) ) ).
% graph_system.wellformed
thf(fact_419_graph__system_Owellformed,axiom,
! [Vertices: set_Product_prod_a_a,Edges: set_se5735800977113168103od_a_a,E: set_Product_prod_a_a] :
( ( undire1860116983885411791od_a_a @ Vertices @ Edges )
=> ( ( member1816616512716248880od_a_a @ E @ Edges )
=> ( ord_le746702958409616551od_a_a @ E @ Vertices ) ) ) ).
% graph_system.wellformed
thf(fact_420_graph__system_Owellformed,axiom,
! [Vertices: set_a,Edges: set_set_a,E: set_a] :
( ( undire2554140024507503526stem_a @ Vertices @ Edges )
=> ( ( member_set_a @ E @ Edges )
=> ( ord_less_eq_set_a @ E @ Vertices ) ) ) ).
% graph_system.wellformed
thf(fact_421_graph__system_Ointro,axiom,
! [Edges: set_set_set_a,Vertices: set_set_a] :
( ! [E6: set_set_a] :
( ( member_set_set_a @ E6 @ Edges )
=> ( ord_le3724670747650509150_set_a @ E6 @ Vertices ) )
=> ( undire7159349782766787846_set_a @ Vertices @ Edges ) ) ).
% graph_system.intro
thf(fact_422_graph__system_Ointro,axiom,
! [Edges: set_se5735800977113168103od_a_a,Vertices: set_Product_prod_a_a] :
( ! [E6: set_Product_prod_a_a] :
( ( member1816616512716248880od_a_a @ E6 @ Edges )
=> ( ord_le746702958409616551od_a_a @ E6 @ Vertices ) )
=> ( undire1860116983885411791od_a_a @ Vertices @ Edges ) ) ).
% graph_system.intro
thf(fact_423_graph__system_Ointro,axiom,
! [Edges: set_set_a,Vertices: set_a] :
( ! [E6: set_a] :
( ( member_set_a @ E6 @ Edges )
=> ( ord_less_eq_set_a @ E6 @ Vertices ) )
=> ( undire2554140024507503526stem_a @ Vertices @ Edges ) ) ).
% graph_system.intro
thf(fact_424_subgraph_Overts__ss,axiom,
! [V_H: set_set_a,E_H: set_set_set_a,V_G: set_set_a,E_G: set_set_set_a] :
( ( undire1186139521737116585_set_a @ V_H @ E_H @ V_G @ E_G )
=> ( ord_le3724670747650509150_set_a @ V_H @ V_G ) ) ).
% subgraph.verts_ss
thf(fact_425_subgraph_Overts__ss,axiom,
! [V_H: set_Product_prod_a_a,E_H: set_se5735800977113168103od_a_a,V_G: set_Product_prod_a_a,E_G: set_se5735800977113168103od_a_a] :
( ( undire398746457437328754od_a_a @ V_H @ E_H @ V_G @ E_G )
=> ( ord_le746702958409616551od_a_a @ V_H @ V_G ) ) ).
% subgraph.verts_ss
thf(fact_426_subgraph_Overts__ss,axiom,
! [V_H: set_a,E_H: set_set_a,V_G: set_a,E_G: set_set_a] :
( ( undire7103218114511261257raph_a @ V_H @ E_H @ V_G @ E_G )
=> ( ord_less_eq_set_a @ V_H @ V_G ) ) ).
% subgraph.verts_ss
thf(fact_427_ulgraph_Oaxioms_I1_J,axiom,
! [Vertices: set_a,Edges: set_set_a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( undire2554140024507503526stem_a @ Vertices @ Edges ) ) ).
% ulgraph.axioms(1)
thf(fact_428_subgraph_Ois__subgraph__ulgraph,axiom,
! [V_H: set_a,E_H: set_set_a,V_G: set_a,E_G: set_set_a] :
( ( undire7103218114511261257raph_a @ V_H @ E_H @ V_G @ E_G )
=> ( ( undire7251896706689453996raph_a @ V_G @ E_G )
=> ( undire7251896706689453996raph_a @ V_H @ E_H ) ) ) ).
% subgraph.is_subgraph_ulgraph
thf(fact_429_subgraph_Oedges__ss,axiom,
! [V_H: set_a,E_H: set_set_a,V_G: set_a,E_G: set_set_a] :
( ( undire7103218114511261257raph_a @ V_H @ E_H @ V_G @ E_G )
=> ( ord_le3724670747650509150_set_a @ E_H @ E_G ) ) ).
% subgraph.edges_ss
thf(fact_430_ulgraph_Overt__adj__sym,axiom,
! [Vertices: set_a,Edges: set_set_a,V1: a,V22: a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( undire397441198561214472_adj_a @ Edges @ V1 @ V22 )
= ( undire397441198561214472_adj_a @ Edges @ V22 @ V1 ) ) ) ).
% ulgraph.vert_adj_sym
thf(fact_431_ulgraph_Overt__adj__imp__inV,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,V1: set_a,V22: set_a] :
( ( undire6886684016831807756_set_a @ Vertices @ Edges )
=> ( ( undire3510646817838285160_set_a @ Edges @ V1 @ V22 )
=> ( ( member_set_a @ V1 @ Vertices )
& ( member_set_a @ V22 @ Vertices ) ) ) ) ).
% ulgraph.vert_adj_imp_inV
thf(fact_432_ulgraph_Overt__adj__imp__inV,axiom,
! [Vertices: set_Product_prod_a_a,Edges: set_se5735800977113168103od_a_a,V1: product_prod_a_a,V22: product_prod_a_a] :
( ( undire4585262585102564309od_a_a @ Vertices @ Edges )
=> ( ( undire6135774327024169009od_a_a @ Edges @ V1 @ V22 )
=> ( ( member1426531477525435216od_a_a @ V1 @ Vertices )
& ( member1426531477525435216od_a_a @ V22 @ Vertices ) ) ) ) ).
% ulgraph.vert_adj_imp_inV
thf(fact_433_ulgraph_Overt__adj__imp__inV,axiom,
! [Vertices: set_nat,Edges: set_set_nat,V1: nat,V22: nat] :
( ( undire3269267262472140706ph_nat @ Vertices @ Edges )
=> ( ( undire1083030068171319366dj_nat @ Edges @ V1 @ V22 )
=> ( ( member_nat @ V1 @ Vertices )
& ( member_nat @ V22 @ Vertices ) ) ) ) ).
% ulgraph.vert_adj_imp_inV
thf(fact_434_ulgraph_Overt__adj__imp__inV,axiom,
! [Vertices: set_a,Edges: set_set_a,V1: a,V22: a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( undire397441198561214472_adj_a @ Edges @ V1 @ V22 )
=> ( ( member_a @ V1 @ Vertices )
& ( member_a @ V22 @ Vertices ) ) ) ) ).
% ulgraph.vert_adj_imp_inV
thf(fact_435_ulgraph_Ohas__loop__in__verts,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,V2: set_a] :
( ( undire6886684016831807756_set_a @ Vertices @ Edges )
=> ( ( undire5774735625301615776_set_a @ Edges @ V2 )
=> ( member_set_a @ V2 @ Vertices ) ) ) ).
% ulgraph.has_loop_in_verts
thf(fact_436_ulgraph_Ohas__loop__in__verts,axiom,
! [Vertices: set_Product_prod_a_a,Edges: set_se5735800977113168103od_a_a,V2: product_prod_a_a] :
( ( undire4585262585102564309od_a_a @ Vertices @ Edges )
=> ( ( undire7777398424729533289od_a_a @ Edges @ V2 )
=> ( member1426531477525435216od_a_a @ V2 @ Vertices ) ) ) ).
% ulgraph.has_loop_in_verts
thf(fact_437_ulgraph_Ohas__loop__in__verts,axiom,
! [Vertices: set_nat,Edges: set_set_nat,V2: nat] :
( ( undire3269267262472140706ph_nat @ Vertices @ Edges )
=> ( ( undire5005864372999571214op_nat @ Edges @ V2 )
=> ( member_nat @ V2 @ Vertices ) ) ) ).
% ulgraph.has_loop_in_verts
thf(fact_438_ulgraph_Ohas__loop__in__verts,axiom,
! [Vertices: set_a,Edges: set_set_a,V2: a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( undire3617971648856834880loop_a @ Edges @ V2 )
=> ( member_a @ V2 @ Vertices ) ) ) ).
% ulgraph.has_loop_in_verts
thf(fact_439_subgraph_Oaxioms_I2_J,axiom,
! [V_H: set_a,E_H: set_set_a,V_G: set_a,E_G: set_set_a] :
( ( undire7103218114511261257raph_a @ V_H @ E_H @ V_G @ E_G )
=> ( undire2554140024507503526stem_a @ V_G @ E_G ) ) ).
% subgraph.axioms(2)
thf(fact_440_subgraph_Oaxioms_I1_J,axiom,
! [V_H: set_a,E_H: set_set_a,V_G: set_a,E_G: set_set_a] :
( ( undire7103218114511261257raph_a @ V_H @ E_H @ V_G @ E_G )
=> ( undire2554140024507503526stem_a @ V_H @ E_H ) ) ).
% subgraph.axioms(1)
thf(fact_441_subgraph_Osubgraph__trans,axiom,
! [V_H: set_a,E_H: set_set_a,V_G: set_a,E_G: set_set_a,V3: set_a,E4: set_set_a,V: set_a,E3: set_set_a,V4: set_a,E7: set_set_a] :
( ( undire7103218114511261257raph_a @ V_H @ E_H @ V_G @ E_G )
=> ( ( undire2554140024507503526stem_a @ V3 @ E4 )
=> ( ( undire2554140024507503526stem_a @ V @ E3 )
=> ( ( undire2554140024507503526stem_a @ V4 @ E7 )
=> ( ( undire7103218114511261257raph_a @ V4 @ E7 @ V @ E3 )
=> ( ( undire7103218114511261257raph_a @ V @ E3 @ V3 @ E4 )
=> ( undire7103218114511261257raph_a @ V4 @ E7 @ V3 @ E4 ) ) ) ) ) ) ) ).
% subgraph.subgraph_trans
thf(fact_442_graph__system_Osubgraph__refl,axiom,
! [Vertices: set_a,Edges: set_set_a] :
( ( undire2554140024507503526stem_a @ Vertices @ Edges )
=> ( undire7103218114511261257raph_a @ Vertices @ Edges @ Vertices @ Edges ) ) ).
% graph_system.subgraph_refl
thf(fact_443_graph__system_Oinduced__is__graph__sys,axiom,
! [Vertices: set_a,Edges: set_set_a,V: set_a] :
( ( undire2554140024507503526stem_a @ Vertices @ Edges )
=> ( undire2554140024507503526stem_a @ V @ ( undire7777452895879145676dges_a @ Edges @ V ) ) ) ).
% graph_system.induced_is_graph_sys
thf(fact_444_graph__system_Oincident__def,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,V2: set_a,E: set_set_a] :
( ( undire7159349782766787846_set_a @ Vertices @ Edges )
=> ( ( undire2320338297334612420_set_a @ V2 @ E )
= ( member_set_a @ V2 @ E ) ) ) ).
% graph_system.incident_def
thf(fact_445_graph__system_Oincident__def,axiom,
! [Vertices: set_Product_prod_a_a,Edges: set_se5735800977113168103od_a_a,V2: product_prod_a_a,E: set_Product_prod_a_a] :
( ( undire1860116983885411791od_a_a @ Vertices @ Edges )
=> ( ( undire3369688177417741453od_a_a @ V2 @ E )
= ( member1426531477525435216od_a_a @ V2 @ E ) ) ) ).
% graph_system.incident_def
thf(fact_446_graph__system_Oincident__def,axiom,
! [Vertices: set_nat,Edges: set_set_nat,V2: nat,E: set_nat] :
( ( undire7481384412329822504em_nat @ Vertices @ Edges )
=> ( ( undire7858122600432113898nt_nat @ V2 @ E )
= ( member_nat @ V2 @ E ) ) ) ).
% graph_system.incident_def
thf(fact_447_graph__system_Oincident__def,axiom,
! [Vertices: set_a,Edges: set_set_a,V2: a,E: set_a] :
( ( undire2554140024507503526stem_a @ Vertices @ Edges )
=> ( ( undire1521409233611534436dent_a @ V2 @ E )
= ( member_a @ V2 @ E ) ) ) ).
% graph_system.incident_def
thf(fact_448_graph__system_Oincident__edge__in__wf,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,E: set_set_a,V2: set_a] :
( ( undire7159349782766787846_set_a @ Vertices @ Edges )
=> ( ( member_set_set_a @ E @ Edges )
=> ( ( undire2320338297334612420_set_a @ V2 @ E )
=> ( member_set_a @ V2 @ Vertices ) ) ) ) ).
% graph_system.incident_edge_in_wf
thf(fact_449_graph__system_Oincident__edge__in__wf,axiom,
! [Vertices: set_Product_prod_a_a,Edges: set_se5735800977113168103od_a_a,E: set_Product_prod_a_a,V2: product_prod_a_a] :
( ( undire1860116983885411791od_a_a @ Vertices @ Edges )
=> ( ( member1816616512716248880od_a_a @ E @ Edges )
=> ( ( undire3369688177417741453od_a_a @ V2 @ E )
=> ( member1426531477525435216od_a_a @ V2 @ Vertices ) ) ) ) ).
% graph_system.incident_edge_in_wf
thf(fact_450_graph__system_Oincident__edge__in__wf,axiom,
! [Vertices: set_nat,Edges: set_set_nat,E: set_nat,V2: nat] :
( ( undire7481384412329822504em_nat @ Vertices @ Edges )
=> ( ( member_set_nat @ E @ Edges )
=> ( ( undire7858122600432113898nt_nat @ V2 @ E )
=> ( member_nat @ V2 @ Vertices ) ) ) ) ).
% graph_system.incident_edge_in_wf
thf(fact_451_graph__system_Oincident__edge__in__wf,axiom,
! [Vertices: set_a,Edges: set_set_a,E: set_a,V2: a] :
( ( undire2554140024507503526stem_a @ Vertices @ Edges )
=> ( ( member_set_a @ E @ Edges )
=> ( ( undire1521409233611534436dent_a @ V2 @ E )
=> ( member_a @ V2 @ Vertices ) ) ) ) ).
% graph_system.incident_edge_in_wf
thf(fact_452_dual__order_Orefl,axiom,
! [A: set_set_a] : ( ord_le3724670747650509150_set_a @ A @ A ) ).
% dual_order.refl
thf(fact_453_dual__order_Orefl,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ A @ A ) ).
% dual_order.refl
thf(fact_454_dual__order_Orefl,axiom,
! [A: real] : ( ord_less_eq_real @ A @ A ) ).
% dual_order.refl
thf(fact_455_dual__order_Orefl,axiom,
! [A: set_Product_prod_a_a] : ( ord_le746702958409616551od_a_a @ A @ A ) ).
% dual_order.refl
thf(fact_456_dual__order_Orefl,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% dual_order.refl
thf(fact_457_order__refl,axiom,
! [X: set_set_a] : ( ord_le3724670747650509150_set_a @ X @ X ) ).
% order_refl
thf(fact_458_order__refl,axiom,
! [X: set_a] : ( ord_less_eq_set_a @ X @ X ) ).
% order_refl
thf(fact_459_order__refl,axiom,
! [X: real] : ( ord_less_eq_real @ X @ X ) ).
% order_refl
thf(fact_460_order__refl,axiom,
! [X: set_Product_prod_a_a] : ( ord_le746702958409616551od_a_a @ X @ X ) ).
% order_refl
thf(fact_461_order__refl,axiom,
! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).
% order_refl
thf(fact_462_edge__density__commute,axiom,
! [X5: set_a,Y5: set_a] :
( ( undire297304480579013331sity_a @ edges @ X5 @ Y5 )
= ( undire297304480579013331sity_a @ edges @ Y5 @ X5 ) ) ).
% edge_density_commute
thf(fact_463_walk__length__rev,axiom,
( undire8849074589633906640ngth_a
= ( ^ [P2: list_a] : ( undire8849074589633906640ngth_a @ ( rev_a @ P2 ) ) ) ) ).
% walk_length_rev
thf(fact_464_the__elem__set,axiom,
! [X: a] :
( ( the_elem_a @ ( set_a2 @ ( cons_a @ X @ nil_a ) ) )
= X ) ).
% the_elem_set
thf(fact_465_the__elem__set,axiom,
! [X: set_a] :
( ( the_elem_set_a @ ( set_set_a2 @ ( cons_set_a @ X @ nil_set_a ) ) )
= X ) ).
% the_elem_set
thf(fact_466_list__set__tl,axiom,
! [X: product_prod_a_a,Xs: list_P1396940483166286381od_a_a] :
( ( member1426531477525435216od_a_a @ X @ ( set_Product_prod_a_a2 @ ( tl_Product_prod_a_a @ Xs ) ) )
=> ( member1426531477525435216od_a_a @ X @ ( set_Product_prod_a_a2 @ Xs ) ) ) ).
% list_set_tl
thf(fact_467_list__set__tl,axiom,
! [X: nat,Xs: list_nat] :
( ( member_nat @ X @ ( set_nat2 @ ( tl_nat @ Xs ) ) )
=> ( member_nat @ X @ ( set_nat2 @ Xs ) ) ) ).
% list_set_tl
thf(fact_468_list__set__tl,axiom,
! [X: set_a,Xs: list_set_a] :
( ( member_set_a @ X @ ( set_set_a2 @ ( tl_set_a @ Xs ) ) )
=> ( member_set_a @ X @ ( set_set_a2 @ Xs ) ) ) ).
% list_set_tl
thf(fact_469_list__set__tl,axiom,
! [X: a,Xs: list_a] :
( ( member_a @ X @ ( set_a2 @ ( tl_a @ Xs ) ) )
=> ( member_a @ X @ ( set_a2 @ Xs ) ) ) ).
% list_set_tl
thf(fact_470_comp__sgraph_Ois__walk__def,axiom,
! [S: set_set_a,Xs: list_set_a] :
( ( undire3014741414213135564_set_a @ S @ ( undire8247866692393712962_set_a @ S ) @ Xs )
= ( ( ord_le3724670747650509150_set_a @ ( set_set_a2 @ Xs ) @ S )
& ( ord_le5722252365846178494_set_a @ ( set_set_set_a2 @ ( undire6234387080713648494_set_a @ Xs ) ) @ ( undire8247866692393712962_set_a @ S ) )
& ( Xs != nil_set_a ) ) ) ).
% comp_sgraph.is_walk_def
thf(fact_471_comp__sgraph_Ois__walk__def,axiom,
! [S: set_Product_prod_a_a,Xs: list_P1396940483166286381od_a_a] :
( ( undire3162072421265123221od_a_a @ S @ ( undire6879232364018543115od_a_a @ S ) @ Xs )
= ( ( ord_le746702958409616551od_a_a @ ( set_Product_prod_a_a2 @ Xs ) @ S )
& ( ord_le1995061765932249223od_a_a @ ( set_se8408754101646271900od_a_a @ ( undire4403264684974754359od_a_a @ Xs ) ) @ ( undire6879232364018543115od_a_a @ S ) )
& ( Xs != nil_Product_prod_a_a ) ) ) ).
% comp_sgraph.is_walk_def
thf(fact_472_comp__sgraph_Ois__walk__def,axiom,
! [S: set_a,Xs: list_a] :
( ( undire6133010728901294956walk_a @ S @ ( undire2918257014606996450dges_a @ S ) @ Xs )
= ( ( ord_less_eq_set_a @ ( set_a2 @ Xs ) @ S )
& ( ord_le3724670747650509150_set_a @ ( set_set_a2 @ ( undire7337870655677353998dges_a @ Xs ) ) @ ( undire2918257014606996450dges_a @ S ) )
& ( Xs != nil_a ) ) ) ).
% comp_sgraph.is_walk_def
thf(fact_473_comp__sgraph_Ois__walkI,axiom,
! [Xs: list_set_a,S: set_set_a] :
( ( ord_le3724670747650509150_set_a @ ( set_set_a2 @ Xs ) @ S )
=> ( ( ord_le5722252365846178494_set_a @ ( set_set_set_a2 @ ( undire6234387080713648494_set_a @ Xs ) ) @ ( undire8247866692393712962_set_a @ S ) )
=> ( ( Xs != nil_set_a )
=> ( undire3014741414213135564_set_a @ S @ ( undire8247866692393712962_set_a @ S ) @ Xs ) ) ) ) ).
% comp_sgraph.is_walkI
thf(fact_474_comp__sgraph_Ois__walkI,axiom,
! [Xs: list_P1396940483166286381od_a_a,S: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ ( set_Product_prod_a_a2 @ Xs ) @ S )
=> ( ( ord_le1995061765932249223od_a_a @ ( set_se8408754101646271900od_a_a @ ( undire4403264684974754359od_a_a @ Xs ) ) @ ( undire6879232364018543115od_a_a @ S ) )
=> ( ( Xs != nil_Product_prod_a_a )
=> ( undire3162072421265123221od_a_a @ S @ ( undire6879232364018543115od_a_a @ S ) @ Xs ) ) ) ) ).
% comp_sgraph.is_walkI
thf(fact_475_comp__sgraph_Ois__walkI,axiom,
! [Xs: list_a,S: set_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ Xs ) @ S )
=> ( ( ord_le3724670747650509150_set_a @ ( set_set_a2 @ ( undire7337870655677353998dges_a @ Xs ) ) @ ( undire2918257014606996450dges_a @ S ) )
=> ( ( Xs != nil_a )
=> ( undire6133010728901294956walk_a @ S @ ( undire2918257014606996450dges_a @ S ) @ Xs ) ) ) ) ).
% comp_sgraph.is_walkI
thf(fact_476_ulgraph_Oedge__density_Ocong,axiom,
undire297304480579013331sity_a = undire297304480579013331sity_a ).
% ulgraph.edge_density.cong
thf(fact_477_comp__sgraph_Oe__in__all__edges,axiom,
! [E: set_a,S: set_a] :
( ( member_set_a @ E @ ( undire2918257014606996450dges_a @ S ) )
=> ( member_set_a @ E @ ( undire2918257014606996450dges_a @ S ) ) ) ).
% comp_sgraph.e_in_all_edges
thf(fact_478_comp__sgraph_Oedge__density__commute,axiom,
! [S: set_a,X5: set_a,Y5: set_a] :
( ( undire297304480579013331sity_a @ ( undire2918257014606996450dges_a @ S ) @ X5 @ Y5 )
= ( undire297304480579013331sity_a @ ( undire2918257014606996450dges_a @ S ) @ Y5 @ X5 ) ) ).
% comp_sgraph.edge_density_commute
thf(fact_479_comp__sgraph_Oe__in__all__edges__ss,axiom,
! [E: set_set_a,S: set_set_a,V: set_set_a] :
( ( member_set_set_a @ E @ ( undire8247866692393712962_set_a @ S ) )
=> ( ( ord_le3724670747650509150_set_a @ E @ V )
=> ( ( ord_le3724670747650509150_set_a @ V @ S )
=> ( member_set_set_a @ E @ ( undire8247866692393712962_set_a @ V ) ) ) ) ) ).
% comp_sgraph.e_in_all_edges_ss
thf(fact_480_comp__sgraph_Oe__in__all__edges__ss,axiom,
! [E: set_a,S: set_a,V: set_a] :
( ( member_set_a @ E @ ( undire2918257014606996450dges_a @ S ) )
=> ( ( ord_less_eq_set_a @ E @ V )
=> ( ( ord_less_eq_set_a @ V @ S )
=> ( member_set_a @ E @ ( undire2918257014606996450dges_a @ V ) ) ) ) ) ).
% comp_sgraph.e_in_all_edges_ss
thf(fact_481_comp__sgraph_Oe__in__all__edges__ss,axiom,
! [E: set_Product_prod_a_a,S: set_Product_prod_a_a,V: set_Product_prod_a_a] :
( ( member1816616512716248880od_a_a @ E @ ( undire6879232364018543115od_a_a @ S ) )
=> ( ( ord_le746702958409616551od_a_a @ E @ V )
=> ( ( ord_le746702958409616551od_a_a @ V @ S )
=> ( member1816616512716248880od_a_a @ E @ ( undire6879232364018543115od_a_a @ V ) ) ) ) ) ).
% comp_sgraph.e_in_all_edges_ss
thf(fact_482_comp__sgraph_Owellformed,axiom,
! [E: set_set_a,S: set_set_a] :
( ( member_set_set_a @ E @ ( undire8247866692393712962_set_a @ S ) )
=> ( ord_le3724670747650509150_set_a @ E @ S ) ) ).
% comp_sgraph.wellformed
thf(fact_483_comp__sgraph_Owellformed,axiom,
! [E: set_a,S: set_a] :
( ( member_set_a @ E @ ( undire2918257014606996450dges_a @ S ) )
=> ( ord_less_eq_set_a @ E @ S ) ) ).
% comp_sgraph.wellformed
thf(fact_484_comp__sgraph_Owellformed,axiom,
! [E: set_Product_prod_a_a,S: set_Product_prod_a_a] :
( ( member1816616512716248880od_a_a @ E @ ( undire6879232364018543115od_a_a @ S ) )
=> ( ord_le746702958409616551od_a_a @ E @ S ) ) ).
% comp_sgraph.wellformed
thf(fact_485_comp__sgraph_Oulgraph__axioms,axiom,
! [S: set_a] : ( undire7251896706689453996raph_a @ S @ ( undire2918257014606996450dges_a @ S ) ) ).
% comp_sgraph.ulgraph_axioms
thf(fact_486_comp__sgraph_Owellformed__all__edges,axiom,
! [S: set_a] : ( ord_le3724670747650509150_set_a @ ( undire2918257014606996450dges_a @ S ) @ ( undire2918257014606996450dges_a @ S ) ) ).
% comp_sgraph.wellformed_all_edges
thf(fact_487_comp__sgraph_Ograph__system__axioms,axiom,
! [S: set_a] : ( undire2554140024507503526stem_a @ S @ ( undire2918257014606996450dges_a @ S ) ) ).
% comp_sgraph.graph_system_axioms
thf(fact_488_comp__sgraph_Osubgraph__complete,axiom,
! [S: set_a] : ( undire7103218114511261257raph_a @ S @ ( undire2918257014606996450dges_a @ S ) @ S @ ( undire2918257014606996450dges_a @ S ) ) ).
% comp_sgraph.subgraph_complete
thf(fact_489_comp__sgraph_Oinduced__edges__self,axiom,
! [S: set_a] :
( ( undire7777452895879145676dges_a @ ( undire2918257014606996450dges_a @ S ) @ S )
= ( undire2918257014606996450dges_a @ S ) ) ).
% comp_sgraph.induced_edges_self
thf(fact_490_comp__sgraph_Overt__adj__imp__inV,axiom,
! [S: set_set_a,V1: set_a,V22: set_a] :
( ( undire3510646817838285160_set_a @ ( undire8247866692393712962_set_a @ S ) @ V1 @ V22 )
=> ( ( member_set_a @ V1 @ S )
& ( member_set_a @ V22 @ S ) ) ) ).
% comp_sgraph.vert_adj_imp_inV
thf(fact_491_comp__sgraph_Overt__adj__imp__inV,axiom,
! [S: set_Product_prod_a_a,V1: product_prod_a_a,V22: product_prod_a_a] :
( ( undire6135774327024169009od_a_a @ ( undire6879232364018543115od_a_a @ S ) @ V1 @ V22 )
=> ( ( member1426531477525435216od_a_a @ V1 @ S )
& ( member1426531477525435216od_a_a @ V22 @ S ) ) ) ).
% comp_sgraph.vert_adj_imp_inV
thf(fact_492_comp__sgraph_Overt__adj__imp__inV,axiom,
! [S: set_nat,V1: nat,V22: nat] :
( ( undire1083030068171319366dj_nat @ ( undire463345858124014060es_nat @ S ) @ V1 @ V22 )
=> ( ( member_nat @ V1 @ S )
& ( member_nat @ V22 @ S ) ) ) ).
% comp_sgraph.vert_adj_imp_inV
thf(fact_493_comp__sgraph_Overt__adj__imp__inV,axiom,
! [S: set_a,V1: a,V22: a] :
( ( undire397441198561214472_adj_a @ ( undire2918257014606996450dges_a @ S ) @ V1 @ V22 )
=> ( ( member_a @ V1 @ S )
& ( member_a @ V22 @ S ) ) ) ).
% comp_sgraph.vert_adj_imp_inV
thf(fact_494_comp__sgraph_Overt__adj__sym,axiom,
! [S: set_a,V1: a,V22: a] :
( ( undire397441198561214472_adj_a @ ( undire2918257014606996450dges_a @ S ) @ V1 @ V22 )
= ( undire397441198561214472_adj_a @ ( undire2918257014606996450dges_a @ S ) @ V22 @ V1 ) ) ).
% comp_sgraph.vert_adj_sym
thf(fact_495_comp__sgraph_Oincident__edge__in__wf,axiom,
! [E: set_set_a,S: set_set_a,V2: set_a] :
( ( member_set_set_a @ E @ ( undire8247866692393712962_set_a @ S ) )
=> ( ( undire2320338297334612420_set_a @ V2 @ E )
=> ( member_set_a @ V2 @ S ) ) ) ).
% comp_sgraph.incident_edge_in_wf
thf(fact_496_comp__sgraph_Oincident__edge__in__wf,axiom,
! [E: set_Product_prod_a_a,S: set_Product_prod_a_a,V2: product_prod_a_a] :
( ( member1816616512716248880od_a_a @ E @ ( undire6879232364018543115od_a_a @ S ) )
=> ( ( undire3369688177417741453od_a_a @ V2 @ E )
=> ( member1426531477525435216od_a_a @ V2 @ S ) ) ) ).
% comp_sgraph.incident_edge_in_wf
thf(fact_497_comp__sgraph_Oincident__edge__in__wf,axiom,
! [E: set_nat,S: set_nat,V2: nat] :
( ( member_set_nat @ E @ ( undire463345858124014060es_nat @ S ) )
=> ( ( undire7858122600432113898nt_nat @ V2 @ E )
=> ( member_nat @ V2 @ S ) ) ) ).
% comp_sgraph.incident_edge_in_wf
thf(fact_498_comp__sgraph_Oincident__edge__in__wf,axiom,
! [E: set_a,S: set_a,V2: a] :
( ( member_set_a @ E @ ( undire2918257014606996450dges_a @ S ) )
=> ( ( undire1521409233611534436dent_a @ V2 @ E )
=> ( member_a @ V2 @ S ) ) ) ).
% comp_sgraph.incident_edge_in_wf
thf(fact_499_comp__sgraph_Ohas__loop__in__verts,axiom,
! [S: set_set_a,V2: set_a] :
( ( undire5774735625301615776_set_a @ ( undire8247866692393712962_set_a @ S ) @ V2 )
=> ( member_set_a @ V2 @ S ) ) ).
% comp_sgraph.has_loop_in_verts
thf(fact_500_comp__sgraph_Ohas__loop__in__verts,axiom,
! [S: set_Product_prod_a_a,V2: product_prod_a_a] :
( ( undire7777398424729533289od_a_a @ ( undire6879232364018543115od_a_a @ S ) @ V2 )
=> ( member1426531477525435216od_a_a @ V2 @ S ) ) ).
% comp_sgraph.has_loop_in_verts
thf(fact_501_comp__sgraph_Ohas__loop__in__verts,axiom,
! [S: set_nat,V2: nat] :
( ( undire5005864372999571214op_nat @ ( undire463345858124014060es_nat @ S ) @ V2 )
=> ( member_nat @ V2 @ S ) ) ).
% comp_sgraph.has_loop_in_verts
thf(fact_502_comp__sgraph_Ohas__loop__in__verts,axiom,
! [S: set_a,V2: a] :
( ( undire3617971648856834880loop_a @ ( undire2918257014606996450dges_a @ S ) @ V2 )
=> ( member_a @ V2 @ S ) ) ).
% comp_sgraph.has_loop_in_verts
thf(fact_503_comp__sgraph_Ono__loops,axiom,
! [V2: set_a,S: set_set_a] :
( ( member_set_a @ V2 @ S )
=> ~ ( undire5774735625301615776_set_a @ ( undire8247866692393712962_set_a @ S ) @ V2 ) ) ).
% comp_sgraph.no_loops
thf(fact_504_comp__sgraph_Ono__loops,axiom,
! [V2: product_prod_a_a,S: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ V2 @ S )
=> ~ ( undire7777398424729533289od_a_a @ ( undire6879232364018543115od_a_a @ S ) @ V2 ) ) ).
% comp_sgraph.no_loops
thf(fact_505_comp__sgraph_Ono__loops,axiom,
! [V2: nat,S: set_nat] :
( ( member_nat @ V2 @ S )
=> ~ ( undire5005864372999571214op_nat @ ( undire463345858124014060es_nat @ S ) @ V2 ) ) ).
% comp_sgraph.no_loops
thf(fact_506_comp__sgraph_Ono__loops,axiom,
! [V2: a,S: set_a] :
( ( member_a @ V2 @ S )
=> ~ ( undire3617971648856834880loop_a @ ( undire2918257014606996450dges_a @ S ) @ V2 ) ) ).
% comp_sgraph.no_loops
thf(fact_507_comp__sgraph_Oedge__adjacent__alt__def,axiom,
! [E1: set_set_a,S: set_set_a,E2: set_set_a] :
( ( member_set_set_a @ E1 @ ( undire8247866692393712962_set_a @ S ) )
=> ( ( member_set_set_a @ E2 @ ( undire8247866692393712962_set_a @ S ) )
=> ( ? [X3: set_a] :
( ( member_set_a @ X3 @ S )
& ( member_set_a @ X3 @ E1 )
& ( member_set_a @ X3 @ E2 ) )
=> ( undire3485422320110889978_set_a @ ( undire8247866692393712962_set_a @ S ) @ E1 @ E2 ) ) ) ) ).
% comp_sgraph.edge_adjacent_alt_def
thf(fact_508_comp__sgraph_Oedge__adjacent__alt__def,axiom,
! [E1: set_Product_prod_a_a,S: set_Product_prod_a_a,E2: set_Product_prod_a_a] :
( ( member1816616512716248880od_a_a @ E1 @ ( undire6879232364018543115od_a_a @ S ) )
=> ( ( member1816616512716248880od_a_a @ E2 @ ( undire6879232364018543115od_a_a @ S ) )
=> ( ? [X3: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X3 @ S )
& ( member1426531477525435216od_a_a @ X3 @ E1 )
& ( member1426531477525435216od_a_a @ X3 @ E2 ) )
=> ( undire9186443406341554371od_a_a @ ( undire6879232364018543115od_a_a @ S ) @ E1 @ E2 ) ) ) ) ).
% comp_sgraph.edge_adjacent_alt_def
thf(fact_509_comp__sgraph_Oedge__adjacent__alt__def,axiom,
! [E1: set_nat,S: set_nat,E2: set_nat] :
( ( member_set_nat @ E1 @ ( undire463345858124014060es_nat @ S ) )
=> ( ( member_set_nat @ E2 @ ( undire463345858124014060es_nat @ S ) )
=> ( ? [X3: nat] :
( ( member_nat @ X3 @ S )
& ( member_nat @ X3 @ E1 )
& ( member_nat @ X3 @ E2 ) )
=> ( undire1664191744716346676dj_nat @ ( undire463345858124014060es_nat @ S ) @ E1 @ E2 ) ) ) ) ).
% comp_sgraph.edge_adjacent_alt_def
thf(fact_510_comp__sgraph_Oedge__adjacent__alt__def,axiom,
! [E1: set_a,S: set_a,E2: set_a] :
( ( member_set_a @ E1 @ ( undire2918257014606996450dges_a @ S ) )
=> ( ( member_set_a @ E2 @ ( undire2918257014606996450dges_a @ S ) )
=> ( ? [X3: a] :
( ( member_a @ X3 @ S )
& ( member_a @ X3 @ E1 )
& ( member_a @ X3 @ E2 ) )
=> ( undire4022703626023482010_adj_a @ ( undire2918257014606996450dges_a @ S ) @ E1 @ E2 ) ) ) ) ).
% comp_sgraph.edge_adjacent_alt_def
thf(fact_511_comp__sgraph_Oedge__adj__inE,axiom,
! [S: set_a,E1: set_a,E2: set_a] :
( ( undire4022703626023482010_adj_a @ ( undire2918257014606996450dges_a @ S ) @ E1 @ E2 )
=> ( ( member_set_a @ E1 @ ( undire2918257014606996450dges_a @ S ) )
& ( member_set_a @ E2 @ ( undire2918257014606996450dges_a @ S ) ) ) ) ).
% comp_sgraph.edge_adj_inE
thf(fact_512_comp__sgraph_Owalk__length__rev,axiom,
( undire4424681683220949296_set_a
= ( ^ [P2: list_set_a] : ( undire4424681683220949296_set_a @ ( rev_set_a @ P2 ) ) ) ) ).
% comp_sgraph.walk_length_rev
thf(fact_513_comp__sgraph_Owalk__length__rev,axiom,
( undire8849074589633906640ngth_a
= ( ^ [P2: list_a] : ( undire8849074589633906640ngth_a @ ( rev_a @ P2 ) ) ) ) ).
% comp_sgraph.walk_length_rev
thf(fact_514_ulgraph_Oedge__density__commute,axiom,
! [Vertices: set_a,Edges: set_set_a,X5: set_a,Y5: set_a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( undire297304480579013331sity_a @ Edges @ X5 @ Y5 )
= ( undire297304480579013331sity_a @ Edges @ Y5 @ X5 ) ) ) ).
% ulgraph.edge_density_commute
thf(fact_515_all__edges__mono,axiom,
! [Vs: set_set_a,Ws: set_set_a] :
( ( ord_le3724670747650509150_set_a @ Vs @ Ws )
=> ( ord_le5722252365846178494_set_a @ ( undire8247866692393712962_set_a @ Vs ) @ ( undire8247866692393712962_set_a @ Ws ) ) ) ).
% all_edges_mono
thf(fact_516_all__edges__mono,axiom,
! [Vs: set_a,Ws: set_a] :
( ( ord_less_eq_set_a @ Vs @ Ws )
=> ( ord_le3724670747650509150_set_a @ ( undire2918257014606996450dges_a @ Vs ) @ ( undire2918257014606996450dges_a @ Ws ) ) ) ).
% all_edges_mono
thf(fact_517_all__edges__mono,axiom,
! [Vs: set_Product_prod_a_a,Ws: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ Vs @ Ws )
=> ( ord_le1995061765932249223od_a_a @ ( undire6879232364018543115od_a_a @ Vs ) @ ( undire6879232364018543115od_a_a @ Ws ) ) ) ).
% all_edges_mono
thf(fact_518_comp__sgraph_Ois__walk__not__empty,axiom,
! [S: set_set_a,Xs: list_set_a] :
( ( undire3014741414213135564_set_a @ S @ ( undire8247866692393712962_set_a @ S ) @ Xs )
=> ( Xs != nil_set_a ) ) ).
% comp_sgraph.is_walk_not_empty
thf(fact_519_comp__sgraph_Ois__walk__not__empty,axiom,
! [S: set_a,Xs: list_a] :
( ( undire6133010728901294956walk_a @ S @ ( undire2918257014606996450dges_a @ S ) @ Xs )
=> ( Xs != nil_a ) ) ).
% comp_sgraph.is_walk_not_empty
thf(fact_520_comp__sgraph_Ois__walk__not__empty2,axiom,
! [S: set_set_a] :
~ ( undire3014741414213135564_set_a @ S @ ( undire8247866692393712962_set_a @ S ) @ nil_set_a ) ).
% comp_sgraph.is_walk_not_empty2
thf(fact_521_comp__sgraph_Ois__walk__not__empty2,axiom,
! [S: set_a] :
~ ( undire6133010728901294956walk_a @ S @ ( undire2918257014606996450dges_a @ S ) @ nil_a ) ).
% comp_sgraph.is_walk_not_empty2
thf(fact_522_comp__sgraph_Ois__walk__rev,axiom,
! [S: set_set_a,Xs: list_set_a] :
( ( undire3014741414213135564_set_a @ S @ ( undire8247866692393712962_set_a @ S ) @ Xs )
= ( undire3014741414213135564_set_a @ S @ ( undire8247866692393712962_set_a @ S ) @ ( rev_set_a @ Xs ) ) ) ).
% comp_sgraph.is_walk_rev
thf(fact_523_comp__sgraph_Ois__walk__rev,axiom,
! [S: set_a,Xs: list_a] :
( ( undire6133010728901294956walk_a @ S @ ( undire2918257014606996450dges_a @ S ) @ Xs )
= ( undire6133010728901294956walk_a @ S @ ( undire2918257014606996450dges_a @ S ) @ ( rev_a @ Xs ) ) ) ).
% comp_sgraph.is_walk_rev
thf(fact_524_comp__sgraph_Ois__walk__wf__hd,axiom,
! [S: set_set_a,Xs: list_set_a] :
( ( undire3014741414213135564_set_a @ S @ ( undire8247866692393712962_set_a @ S ) @ Xs )
=> ( member_set_a @ ( hd_set_a @ Xs ) @ S ) ) ).
% comp_sgraph.is_walk_wf_hd
thf(fact_525_comp__sgraph_Ois__walk__wf__hd,axiom,
! [S: set_Product_prod_a_a,Xs: list_P1396940483166286381od_a_a] :
( ( undire3162072421265123221od_a_a @ S @ ( undire6879232364018543115od_a_a @ S ) @ Xs )
=> ( member1426531477525435216od_a_a @ ( hd_Product_prod_a_a @ Xs ) @ S ) ) ).
% comp_sgraph.is_walk_wf_hd
thf(fact_526_comp__sgraph_Ois__walk__wf__hd,axiom,
! [S: set_nat,Xs: list_nat] :
( ( undire5745680128780950498lk_nat @ S @ ( undire463345858124014060es_nat @ S ) @ Xs )
=> ( member_nat @ ( hd_nat @ Xs ) @ S ) ) ).
% comp_sgraph.is_walk_wf_hd
thf(fact_527_comp__sgraph_Ois__walk__wf__hd,axiom,
! [S: set_a,Xs: list_a] :
( ( undire6133010728901294956walk_a @ S @ ( undire2918257014606996450dges_a @ S ) @ Xs )
=> ( member_a @ ( hd_a @ Xs ) @ S ) ) ).
% comp_sgraph.is_walk_wf_hd
thf(fact_528_comp__sgraph_Ois__walk__wf__last,axiom,
! [S: set_set_a,Xs: list_set_a] :
( ( undire3014741414213135564_set_a @ S @ ( undire8247866692393712962_set_a @ S ) @ Xs )
=> ( member_set_a @ ( last_set_a @ Xs ) @ S ) ) ).
% comp_sgraph.is_walk_wf_last
thf(fact_529_comp__sgraph_Ois__walk__wf__last,axiom,
! [S: set_Product_prod_a_a,Xs: list_P1396940483166286381od_a_a] :
( ( undire3162072421265123221od_a_a @ S @ ( undire6879232364018543115od_a_a @ S ) @ Xs )
=> ( member1426531477525435216od_a_a @ ( last_P8790725268278465478od_a_a @ Xs ) @ S ) ) ).
% comp_sgraph.is_walk_wf_last
thf(fact_530_comp__sgraph_Ois__walk__wf__last,axiom,
! [S: set_nat,Xs: list_nat] :
( ( undire5745680128780950498lk_nat @ S @ ( undire463345858124014060es_nat @ S ) @ Xs )
=> ( member_nat @ ( last_nat @ Xs ) @ S ) ) ).
% comp_sgraph.is_walk_wf_last
thf(fact_531_comp__sgraph_Ois__walk__wf__last,axiom,
! [S: set_a,Xs: list_a] :
( ( undire6133010728901294956walk_a @ S @ ( undire2918257014606996450dges_a @ S ) @ Xs )
=> ( member_a @ ( last_a @ Xs ) @ S ) ) ).
% comp_sgraph.is_walk_wf_last
thf(fact_532_comp__sgraph_Oinduced__is__graph__sys,axiom,
! [V: set_a,S: set_a] : ( undire2554140024507503526stem_a @ V @ ( undire7777452895879145676dges_a @ ( undire2918257014606996450dges_a @ S ) @ V ) ) ).
% comp_sgraph.induced_is_graph_sys
thf(fact_533_comp__sgraph_Overt__adj__edge__iff2,axiom,
! [V1: a,V22: a,S: set_a] :
( ( V1 != V22 )
=> ( ( undire397441198561214472_adj_a @ ( undire2918257014606996450dges_a @ S ) @ V1 @ V22 )
= ( ? [X4: set_a] :
( ( member_set_a @ X4 @ ( undire2918257014606996450dges_a @ S ) )
& ( undire1521409233611534436dent_a @ V1 @ X4 )
& ( undire1521409233611534436dent_a @ V22 @ X4 ) ) ) ) ) ).
% comp_sgraph.vert_adj_edge_iff2
thf(fact_534_comp__sgraph_Ois__isolated__vertex__def,axiom,
! [S: set_set_a,V2: set_a] :
( ( undire6879241558604981877_set_a @ S @ ( undire8247866692393712962_set_a @ S ) @ V2 )
= ( ( member_set_a @ V2 @ S )
& ! [X4: set_a] :
( ( member_set_a @ X4 @ S )
=> ~ ( undire3510646817838285160_set_a @ ( undire8247866692393712962_set_a @ S ) @ X4 @ V2 ) ) ) ) ).
% comp_sgraph.is_isolated_vertex_def
thf(fact_535_comp__sgraph_Ois__isolated__vertex__def,axiom,
! [S: set_Product_prod_a_a,V2: product_prod_a_a] :
( ( undire3207556238582723646od_a_a @ S @ ( undire6879232364018543115od_a_a @ S ) @ V2 )
= ( ( member1426531477525435216od_a_a @ V2 @ S )
& ! [X4: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X4 @ S )
=> ~ ( undire6135774327024169009od_a_a @ ( undire6879232364018543115od_a_a @ S ) @ X4 @ V2 ) ) ) ) ).
% comp_sgraph.is_isolated_vertex_def
thf(fact_536_comp__sgraph_Ois__isolated__vertex__def,axiom,
! [S: set_nat,V2: nat] :
( ( undire5609513041723151865ex_nat @ S @ ( undire463345858124014060es_nat @ S ) @ V2 )
= ( ( member_nat @ V2 @ S )
& ! [X4: nat] :
( ( member_nat @ X4 @ S )
=> ~ ( undire1083030068171319366dj_nat @ ( undire463345858124014060es_nat @ S ) @ X4 @ V2 ) ) ) ) ).
% comp_sgraph.is_isolated_vertex_def
thf(fact_537_comp__sgraph_Ois__isolated__vertex__def,axiom,
! [S: set_a,V2: a] :
( ( undire8931668460104145173rtex_a @ S @ ( undire2918257014606996450dges_a @ S ) @ V2 )
= ( ( member_a @ V2 @ S )
& ! [X4: a] :
( ( member_a @ X4 @ S )
=> ~ ( undire397441198561214472_adj_a @ ( undire2918257014606996450dges_a @ S ) @ X4 @ V2 ) ) ) ) ).
% comp_sgraph.is_isolated_vertex_def
thf(fact_538_comp__sgraph_Ois__isolated__vertex__edge,axiom,
! [S: set_a,V2: a,E: set_a] :
( ( undire8931668460104145173rtex_a @ S @ ( undire2918257014606996450dges_a @ S ) @ V2 )
=> ( ( member_set_a @ E @ ( undire2918257014606996450dges_a @ S ) )
=> ~ ( undire1521409233611534436dent_a @ V2 @ E ) ) ) ).
% comp_sgraph.is_isolated_vertex_edge
thf(fact_539_comp__sgraph_Ois__isolated__vertex__no__loop,axiom,
! [S: set_a,V2: a] :
( ( undire8931668460104145173rtex_a @ S @ ( undire2918257014606996450dges_a @ S ) @ V2 )
=> ~ ( undire3617971648856834880loop_a @ ( undire2918257014606996450dges_a @ S ) @ V2 ) ) ).
% comp_sgraph.is_isolated_vertex_no_loop
thf(fact_540_ulgraph_Owalk__length__rev,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,P3: list_set_a] :
( ( undire6886684016831807756_set_a @ Vertices @ Edges )
=> ( ( undire4424681683220949296_set_a @ P3 )
= ( undire4424681683220949296_set_a @ ( rev_set_a @ P3 ) ) ) ) ).
% ulgraph.walk_length_rev
thf(fact_541_ulgraph_Owalk__length__rev,axiom,
! [Vertices: set_a,Edges: set_set_a,P3: list_a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( undire8849074589633906640ngth_a @ P3 )
= ( undire8849074589633906640ngth_a @ ( rev_a @ P3 ) ) ) ) ).
% ulgraph.walk_length_rev
thf(fact_542_comp__sgraph_Ois__walk__singleton,axiom,
! [U: product_prod_a_a,S: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ U @ S )
=> ( undire3162072421265123221od_a_a @ S @ ( undire6879232364018543115od_a_a @ S ) @ ( cons_P7316939126706565853od_a_a @ U @ nil_Product_prod_a_a ) ) ) ).
% comp_sgraph.is_walk_singleton
thf(fact_543_comp__sgraph_Ois__walk__singleton,axiom,
! [U: nat,S: set_nat] :
( ( member_nat @ U @ S )
=> ( undire5745680128780950498lk_nat @ S @ ( undire463345858124014060es_nat @ S ) @ ( cons_nat @ U @ nil_nat ) ) ) ).
% comp_sgraph.is_walk_singleton
thf(fact_544_comp__sgraph_Ois__walk__singleton,axiom,
! [U: set_a,S: set_set_a] :
( ( member_set_a @ U @ S )
=> ( undire3014741414213135564_set_a @ S @ ( undire8247866692393712962_set_a @ S ) @ ( cons_set_a @ U @ nil_set_a ) ) ) ).
% comp_sgraph.is_walk_singleton
thf(fact_545_comp__sgraph_Ois__walk__singleton,axiom,
! [U: a,S: set_a] :
( ( member_a @ U @ S )
=> ( undire6133010728901294956walk_a @ S @ ( undire2918257014606996450dges_a @ S ) @ ( cons_a @ U @ nil_a ) ) ) ).
% comp_sgraph.is_walk_singleton
thf(fact_546_comp__sgraph_Ois__walk__wf,axiom,
! [S: set_set_a,Xs: list_set_a] :
( ( undire3014741414213135564_set_a @ S @ ( undire8247866692393712962_set_a @ S ) @ Xs )
=> ( ord_le3724670747650509150_set_a @ ( set_set_a2 @ Xs ) @ S ) ) ).
% comp_sgraph.is_walk_wf
thf(fact_547_comp__sgraph_Ois__walk__wf,axiom,
! [S: set_Product_prod_a_a,Xs: list_P1396940483166286381od_a_a] :
( ( undire3162072421265123221od_a_a @ S @ ( undire6879232364018543115od_a_a @ S ) @ Xs )
=> ( ord_le746702958409616551od_a_a @ ( set_Product_prod_a_a2 @ Xs ) @ S ) ) ).
% comp_sgraph.is_walk_wf
thf(fact_548_comp__sgraph_Ois__walk__wf,axiom,
! [S: set_a,Xs: list_a] :
( ( undire6133010728901294956walk_a @ S @ ( undire2918257014606996450dges_a @ S ) @ Xs )
=> ( ord_less_eq_set_a @ ( set_a2 @ Xs ) @ S ) ) ).
% comp_sgraph.is_walk_wf
thf(fact_549_nle__le,axiom,
! [A: real,B: real] :
( ( ~ ( ord_less_eq_real @ A @ B ) )
= ( ( ord_less_eq_real @ B @ A )
& ( B != A ) ) ) ).
% nle_le
thf(fact_550_nle__le,axiom,
! [A: nat,B: nat] :
( ( ~ ( ord_less_eq_nat @ A @ B ) )
= ( ( ord_less_eq_nat @ B @ A )
& ( B != A ) ) ) ).
% nle_le
thf(fact_551_le__cases3,axiom,
! [X: real,Y2: real,Z3: real] :
( ( ( ord_less_eq_real @ X @ Y2 )
=> ~ ( ord_less_eq_real @ Y2 @ Z3 ) )
=> ( ( ( ord_less_eq_real @ Y2 @ X )
=> ~ ( ord_less_eq_real @ X @ Z3 ) )
=> ( ( ( ord_less_eq_real @ X @ Z3 )
=> ~ ( ord_less_eq_real @ Z3 @ Y2 ) )
=> ( ( ( ord_less_eq_real @ Z3 @ Y2 )
=> ~ ( ord_less_eq_real @ Y2 @ X ) )
=> ( ( ( ord_less_eq_real @ Y2 @ Z3 )
=> ~ ( ord_less_eq_real @ Z3 @ X ) )
=> ~ ( ( ord_less_eq_real @ Z3 @ X )
=> ~ ( ord_less_eq_real @ X @ Y2 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_552_le__cases3,axiom,
! [X: nat,Y2: nat,Z3: nat] :
( ( ( ord_less_eq_nat @ X @ Y2 )
=> ~ ( ord_less_eq_nat @ Y2 @ Z3 ) )
=> ( ( ( ord_less_eq_nat @ Y2 @ X )
=> ~ ( ord_less_eq_nat @ X @ Z3 ) )
=> ( ( ( ord_less_eq_nat @ X @ Z3 )
=> ~ ( ord_less_eq_nat @ Z3 @ Y2 ) )
=> ( ( ( ord_less_eq_nat @ Z3 @ Y2 )
=> ~ ( ord_less_eq_nat @ Y2 @ X ) )
=> ( ( ( ord_less_eq_nat @ Y2 @ Z3 )
=> ~ ( ord_less_eq_nat @ Z3 @ X ) )
=> ~ ( ( ord_less_eq_nat @ Z3 @ X )
=> ~ ( ord_less_eq_nat @ X @ Y2 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_553_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_set_a,Z: set_set_a] : ( Y4 = Z ) )
= ( ^ [X4: set_set_a,Y3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X4 @ Y3 )
& ( ord_le3724670747650509150_set_a @ Y3 @ X4 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_554_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_a,Z: set_a] : ( Y4 = Z ) )
= ( ^ [X4: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y3 )
& ( ord_less_eq_set_a @ Y3 @ X4 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_555_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: real,Z: real] : ( Y4 = Z ) )
= ( ^ [X4: real,Y3: real] :
( ( ord_less_eq_real @ X4 @ Y3 )
& ( ord_less_eq_real @ Y3 @ X4 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_556_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_Product_prod_a_a,Z: set_Product_prod_a_a] : ( Y4 = Z ) )
= ( ^ [X4: set_Product_prod_a_a,Y3: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ X4 @ Y3 )
& ( ord_le746702958409616551od_a_a @ Y3 @ X4 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_557_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: nat,Z: nat] : ( Y4 = Z ) )
= ( ^ [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
& ( ord_less_eq_nat @ Y3 @ X4 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_558_ord__eq__le__trans,axiom,
! [A: set_set_a,B: set_set_a,C: set_set_a] :
( ( A = B )
=> ( ( ord_le3724670747650509150_set_a @ B @ C )
=> ( ord_le3724670747650509150_set_a @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_559_ord__eq__le__trans,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( A = B )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ord_less_eq_set_a @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_560_ord__eq__le__trans,axiom,
! [A: real,B: real,C: real] :
( ( A = B )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ord_less_eq_real @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_561_ord__eq__le__trans,axiom,
! [A: set_Product_prod_a_a,B: set_Product_prod_a_a,C: set_Product_prod_a_a] :
( ( A = B )
=> ( ( ord_le746702958409616551od_a_a @ B @ C )
=> ( ord_le746702958409616551od_a_a @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_562_ord__eq__le__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( A = B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_563_ord__le__eq__trans,axiom,
! [A: set_set_a,B: set_set_a,C: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( ( B = C )
=> ( ord_le3724670747650509150_set_a @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_564_ord__le__eq__trans,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_set_a @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_565_ord__le__eq__trans,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_real @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_566_ord__le__eq__trans,axiom,
! [A: set_Product_prod_a_a,B: set_Product_prod_a_a,C: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A @ B )
=> ( ( B = C )
=> ( ord_le746702958409616551od_a_a @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_567_ord__le__eq__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_568_order__antisym,axiom,
! [X: set_set_a,Y2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X @ Y2 )
=> ( ( ord_le3724670747650509150_set_a @ Y2 @ X )
=> ( X = Y2 ) ) ) ).
% order_antisym
thf(fact_569_order__antisym,axiom,
! [X: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X @ Y2 )
=> ( ( ord_less_eq_set_a @ Y2 @ X )
=> ( X = Y2 ) ) ) ).
% order_antisym
thf(fact_570_order__antisym,axiom,
! [X: real,Y2: real] :
( ( ord_less_eq_real @ X @ Y2 )
=> ( ( ord_less_eq_real @ Y2 @ X )
=> ( X = Y2 ) ) ) ).
% order_antisym
thf(fact_571_order__antisym,axiom,
! [X: set_Product_prod_a_a,Y2: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ X @ Y2 )
=> ( ( ord_le746702958409616551od_a_a @ Y2 @ X )
=> ( X = Y2 ) ) ) ).
% order_antisym
thf(fact_572_order__antisym,axiom,
! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ( ord_less_eq_nat @ Y2 @ X )
=> ( X = Y2 ) ) ) ).
% order_antisym
thf(fact_573_order_Otrans,axiom,
! [A: set_set_a,B: set_set_a,C: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( ( ord_le3724670747650509150_set_a @ B @ C )
=> ( ord_le3724670747650509150_set_a @ A @ C ) ) ) ).
% order.trans
thf(fact_574_order_Otrans,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ord_less_eq_set_a @ A @ C ) ) ) ).
% order.trans
thf(fact_575_order_Otrans,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ord_less_eq_real @ A @ C ) ) ) ).
% order.trans
thf(fact_576_order_Otrans,axiom,
! [A: set_Product_prod_a_a,B: set_Product_prod_a_a,C: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A @ B )
=> ( ( ord_le746702958409616551od_a_a @ B @ C )
=> ( ord_le746702958409616551od_a_a @ A @ C ) ) ) ).
% order.trans
thf(fact_577_order_Otrans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% order.trans
thf(fact_578_order__trans,axiom,
! [X: set_set_a,Y2: set_set_a,Z3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X @ Y2 )
=> ( ( ord_le3724670747650509150_set_a @ Y2 @ Z3 )
=> ( ord_le3724670747650509150_set_a @ X @ Z3 ) ) ) ).
% order_trans
thf(fact_579_order__trans,axiom,
! [X: set_a,Y2: set_a,Z3: set_a] :
( ( ord_less_eq_set_a @ X @ Y2 )
=> ( ( ord_less_eq_set_a @ Y2 @ Z3 )
=> ( ord_less_eq_set_a @ X @ Z3 ) ) ) ).
% order_trans
thf(fact_580_order__trans,axiom,
! [X: real,Y2: real,Z3: real] :
( ( ord_less_eq_real @ X @ Y2 )
=> ( ( ord_less_eq_real @ Y2 @ Z3 )
=> ( ord_less_eq_real @ X @ Z3 ) ) ) ).
% order_trans
thf(fact_581_order__trans,axiom,
! [X: set_Product_prod_a_a,Y2: set_Product_prod_a_a,Z3: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ X @ Y2 )
=> ( ( ord_le746702958409616551od_a_a @ Y2 @ Z3 )
=> ( ord_le746702958409616551od_a_a @ X @ Z3 ) ) ) ).
% order_trans
thf(fact_582_order__trans,axiom,
! [X: nat,Y2: nat,Z3: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ( ord_less_eq_nat @ Y2 @ Z3 )
=> ( ord_less_eq_nat @ X @ Z3 ) ) ) ).
% order_trans
thf(fact_583_linorder__wlog,axiom,
! [P: real > real > $o,A: real,B: real] :
( ! [A4: real,B4: real] :
( ( ord_less_eq_real @ A4 @ B4 )
=> ( P @ A4 @ B4 ) )
=> ( ! [A4: real,B4: real] :
( ( P @ B4 @ A4 )
=> ( P @ A4 @ B4 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_584_linorder__wlog,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A4: nat,B4: nat] :
( ( ord_less_eq_nat @ A4 @ B4 )
=> ( P @ A4 @ B4 ) )
=> ( ! [A4: nat,B4: nat] :
( ( P @ B4 @ A4 )
=> ( P @ A4 @ B4 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_585_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: set_set_a,Z: set_set_a] : ( Y4 = Z ) )
= ( ^ [A5: set_set_a,B5: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B5 @ A5 )
& ( ord_le3724670747650509150_set_a @ A5 @ B5 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_586_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: set_a,Z: set_a] : ( Y4 = Z ) )
= ( ^ [A5: set_a,B5: set_a] :
( ( ord_less_eq_set_a @ B5 @ A5 )
& ( ord_less_eq_set_a @ A5 @ B5 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_587_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: real,Z: real] : ( Y4 = Z ) )
= ( ^ [A5: real,B5: real] :
( ( ord_less_eq_real @ B5 @ A5 )
& ( ord_less_eq_real @ A5 @ B5 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_588_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: set_Product_prod_a_a,Z: set_Product_prod_a_a] : ( Y4 = Z ) )
= ( ^ [A5: set_Product_prod_a_a,B5: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ B5 @ A5 )
& ( ord_le746702958409616551od_a_a @ A5 @ B5 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_589_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: nat,Z: nat] : ( Y4 = Z ) )
= ( ^ [A5: nat,B5: nat] :
( ( ord_less_eq_nat @ B5 @ A5 )
& ( ord_less_eq_nat @ A5 @ B5 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_590_dual__order_Oantisym,axiom,
! [B: set_set_a,A: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B @ A )
=> ( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_591_dual__order_Oantisym,axiom,
! [B: set_a,A: set_a] :
( ( ord_less_eq_set_a @ B @ A )
=> ( ( ord_less_eq_set_a @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_592_dual__order_Oantisym,axiom,
! [B: real,A: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( ord_less_eq_real @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_593_dual__order_Oantisym,axiom,
! [B: set_Product_prod_a_a,A: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ B @ A )
=> ( ( ord_le746702958409616551od_a_a @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_594_dual__order_Oantisym,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_595_dual__order_Otrans,axiom,
! [B: set_set_a,A: set_set_a,C: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B @ A )
=> ( ( ord_le3724670747650509150_set_a @ C @ B )
=> ( ord_le3724670747650509150_set_a @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_596_dual__order_Otrans,axiom,
! [B: set_a,A: set_a,C: set_a] :
( ( ord_less_eq_set_a @ B @ A )
=> ( ( ord_less_eq_set_a @ C @ B )
=> ( ord_less_eq_set_a @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_597_dual__order_Otrans,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( ord_less_eq_real @ C @ B )
=> ( ord_less_eq_real @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_598_dual__order_Otrans,axiom,
! [B: set_Product_prod_a_a,A: set_Product_prod_a_a,C: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ B @ A )
=> ( ( ord_le746702958409616551od_a_a @ C @ B )
=> ( ord_le746702958409616551od_a_a @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_599_dual__order_Otrans,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_eq_nat @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_600_antisym,axiom,
! [A: set_set_a,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( ( ord_le3724670747650509150_set_a @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_601_antisym,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_eq_set_a @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_602_antisym,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_603_antisym,axiom,
! [A: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A @ B )
=> ( ( ord_le746702958409616551od_a_a @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_604_antisym,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_605_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_set_a,Z: set_set_a] : ( Y4 = Z ) )
= ( ^ [A5: set_set_a,B5: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A5 @ B5 )
& ( ord_le3724670747650509150_set_a @ B5 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_606_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_a,Z: set_a] : ( Y4 = Z ) )
= ( ^ [A5: set_a,B5: set_a] :
( ( ord_less_eq_set_a @ A5 @ B5 )
& ( ord_less_eq_set_a @ B5 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_607_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: real,Z: real] : ( Y4 = Z ) )
= ( ^ [A5: real,B5: real] :
( ( ord_less_eq_real @ A5 @ B5 )
& ( ord_less_eq_real @ B5 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_608_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_Product_prod_a_a,Z: set_Product_prod_a_a] : ( Y4 = Z ) )
= ( ^ [A5: set_Product_prod_a_a,B5: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A5 @ B5 )
& ( ord_le746702958409616551od_a_a @ B5 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_609_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: nat,Z: nat] : ( Y4 = Z ) )
= ( ^ [A5: nat,B5: nat] :
( ( ord_less_eq_nat @ A5 @ B5 )
& ( ord_less_eq_nat @ B5 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_610_order__subst1,axiom,
! [A: real,F: real > real,B: real,C: real] :
( ( ord_less_eq_real @ A @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X2: real,Y: real] :
( ( ord_less_eq_real @ X2 @ Y )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_611_order__subst1,axiom,
! [A: real,F: nat > real,B: nat,C: nat] :
( ( ord_less_eq_real @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_612_order__subst1,axiom,
! [A: nat,F: real > nat,B: real,C: real] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X2: real,Y: real] :
( ( ord_less_eq_real @ X2 @ Y )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_613_order__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_614_order__subst1,axiom,
! [A: set_a,F: real > set_a,B: real,C: real] :
( ( ord_less_eq_set_a @ A @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X2: real,Y: real] :
( ( ord_less_eq_real @ X2 @ Y )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_615_order__subst1,axiom,
! [A: set_a,F: nat > set_a,B: nat,C: nat] :
( ( ord_less_eq_set_a @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_616_order__subst1,axiom,
! [A: real,F: set_a > real,B: set_a,C: set_a] :
( ( ord_less_eq_real @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ! [X2: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_617_order__subst1,axiom,
! [A: nat,F: set_a > nat,B: set_a,C: set_a] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ! [X2: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_618_order__subst1,axiom,
! [A: set_set_a,F: real > set_set_a,B: real,C: real] :
( ( ord_le3724670747650509150_set_a @ A @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X2: real,Y: real] :
( ( ord_less_eq_real @ X2 @ Y )
=> ( ord_le3724670747650509150_set_a @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_le3724670747650509150_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_619_order__subst1,axiom,
! [A: set_set_a,F: nat > set_set_a,B: nat,C: nat] :
( ( ord_le3724670747650509150_set_a @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ord_le3724670747650509150_set_a @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_le3724670747650509150_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_620_order__subst2,axiom,
! [A: real,B: real,F: real > real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C )
=> ( ! [X2: real,Y: real] :
( ( ord_less_eq_real @ X2 @ Y )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_621_order__subst2,axiom,
! [A: real,B: real,F: real > nat,C: nat] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X2: real,Y: real] :
( ( ord_less_eq_real @ X2 @ Y )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_622_order__subst2,axiom,
! [A: nat,B: nat,F: nat > real,C: real] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C )
=> ( ! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_623_order__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_624_order__subst2,axiom,
! [A: set_a,B: set_a,F: set_a > real,C: real] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C )
=> ( ! [X2: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_625_order__subst2,axiom,
! [A: set_a,B: set_a,F: set_a > nat,C: nat] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X2: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_626_order__subst2,axiom,
! [A: real,B: real,F: real > set_a,C: set_a] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_set_a @ ( F @ B ) @ C )
=> ( ! [X2: real,Y: real] :
( ( ord_less_eq_real @ X2 @ Y )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_627_order__subst2,axiom,
! [A: nat,B: nat,F: nat > set_a,C: set_a] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_set_a @ ( F @ B ) @ C )
=> ( ! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_628_order__subst2,axiom,
! [A: set_set_a,B: set_set_a,F: set_set_a > real,C: real] :
( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C )
=> ( ! [X2: set_set_a,Y: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X2 @ Y )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_629_order__subst2,axiom,
! [A: set_set_a,B: set_set_a,F: set_set_a > nat,C: nat] :
( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X2: set_set_a,Y: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X2 @ Y )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_630_order__eq__refl,axiom,
! [X: set_set_a,Y2: set_set_a] :
( ( X = Y2 )
=> ( ord_le3724670747650509150_set_a @ X @ Y2 ) ) ).
% order_eq_refl
thf(fact_631_order__eq__refl,axiom,
! [X: set_a,Y2: set_a] :
( ( X = Y2 )
=> ( ord_less_eq_set_a @ X @ Y2 ) ) ).
% order_eq_refl
thf(fact_632_order__eq__refl,axiom,
! [X: real,Y2: real] :
( ( X = Y2 )
=> ( ord_less_eq_real @ X @ Y2 ) ) ).
% order_eq_refl
thf(fact_633_order__eq__refl,axiom,
! [X: set_Product_prod_a_a,Y2: set_Product_prod_a_a] :
( ( X = Y2 )
=> ( ord_le746702958409616551od_a_a @ X @ Y2 ) ) ).
% order_eq_refl
thf(fact_634_order__eq__refl,axiom,
! [X: nat,Y2: nat] :
( ( X = Y2 )
=> ( ord_less_eq_nat @ X @ Y2 ) ) ).
% order_eq_refl
thf(fact_635_linorder__linear,axiom,
! [X: real,Y2: real] :
( ( ord_less_eq_real @ X @ Y2 )
| ( ord_less_eq_real @ Y2 @ X ) ) ).
% linorder_linear
thf(fact_636_linorder__linear,axiom,
! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
| ( ord_less_eq_nat @ Y2 @ X ) ) ).
% linorder_linear
thf(fact_637_ord__eq__le__subst,axiom,
! [A: real,F: real > real,B: real,C: real] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X2: real,Y: real] :
( ( ord_less_eq_real @ X2 @ Y )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_638_ord__eq__le__subst,axiom,
! [A: nat,F: real > nat,B: real,C: real] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X2: real,Y: real] :
( ( ord_less_eq_real @ X2 @ Y )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_639_ord__eq__le__subst,axiom,
! [A: real,F: nat > real,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_640_ord__eq__le__subst,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_641_ord__eq__le__subst,axiom,
! [A: real,F: set_a > real,B: set_a,C: set_a] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ! [X2: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_642_ord__eq__le__subst,axiom,
! [A: nat,F: set_a > nat,B: set_a,C: set_a] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ! [X2: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_643_ord__eq__le__subst,axiom,
! [A: set_a,F: real > set_a,B: real,C: real] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X2: real,Y: real] :
( ( ord_less_eq_real @ X2 @ Y )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_set_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_644_ord__eq__le__subst,axiom,
! [A: set_a,F: nat > set_a,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_set_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_645_ord__eq__le__subst,axiom,
! [A: real,F: set_set_a > real,B: set_set_a,C: set_set_a] :
( ( A
= ( F @ B ) )
=> ( ( ord_le3724670747650509150_set_a @ B @ C )
=> ( ! [X2: set_set_a,Y: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X2 @ Y )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_646_ord__eq__le__subst,axiom,
! [A: nat,F: set_set_a > nat,B: set_set_a,C: set_set_a] :
( ( A
= ( F @ B ) )
=> ( ( ord_le3724670747650509150_set_a @ B @ C )
=> ( ! [X2: set_set_a,Y: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X2 @ Y )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_647_ord__le__eq__subst,axiom,
! [A: real,B: real,F: real > real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: real,Y: real] :
( ( ord_less_eq_real @ X2 @ Y )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_648_ord__le__eq__subst,axiom,
! [A: real,B: real,F: real > nat,C: nat] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: real,Y: real] :
( ( ord_less_eq_real @ X2 @ Y )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_649_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > real,C: real] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_650_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_651_ord__le__eq__subst,axiom,
! [A: set_a,B: set_a,F: set_a > real,C: real] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_652_ord__le__eq__subst,axiom,
! [A: set_a,B: set_a,F: set_a > nat,C: nat] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_653_ord__le__eq__subst,axiom,
! [A: real,B: real,F: real > set_a,C: set_a] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: real,Y: real] :
( ( ord_less_eq_real @ X2 @ Y )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_set_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_654_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > set_a,C: set_a] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_set_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_655_ord__le__eq__subst,axiom,
! [A: set_set_a,B: set_set_a,F: set_set_a > real,C: real] :
( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: set_set_a,Y: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X2 @ Y )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_656_ord__le__eq__subst,axiom,
! [A: set_set_a,B: set_set_a,F: set_set_a > nat,C: nat] :
( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: set_set_a,Y: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X2 @ Y )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_657_linorder__le__cases,axiom,
! [X: real,Y2: real] :
( ~ ( ord_less_eq_real @ X @ Y2 )
=> ( ord_less_eq_real @ Y2 @ X ) ) ).
% linorder_le_cases
thf(fact_658_linorder__le__cases,axiom,
! [X: nat,Y2: nat] :
( ~ ( ord_less_eq_nat @ X @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ X ) ) ).
% linorder_le_cases
thf(fact_659_order__antisym__conv,axiom,
! [Y2: set_set_a,X: set_set_a] :
( ( ord_le3724670747650509150_set_a @ Y2 @ X )
=> ( ( ord_le3724670747650509150_set_a @ X @ Y2 )
= ( X = Y2 ) ) ) ).
% order_antisym_conv
thf(fact_660_order__antisym__conv,axiom,
! [Y2: set_a,X: set_a] :
( ( ord_less_eq_set_a @ Y2 @ X )
=> ( ( ord_less_eq_set_a @ X @ Y2 )
= ( X = Y2 ) ) ) ).
% order_antisym_conv
thf(fact_661_order__antisym__conv,axiom,
! [Y2: real,X: real] :
( ( ord_less_eq_real @ Y2 @ X )
=> ( ( ord_less_eq_real @ X @ Y2 )
= ( X = Y2 ) ) ) ).
% order_antisym_conv
thf(fact_662_order__antisym__conv,axiom,
! [Y2: set_Product_prod_a_a,X: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ Y2 @ X )
=> ( ( ord_le746702958409616551od_a_a @ X @ Y2 )
= ( X = Y2 ) ) ) ).
% order_antisym_conv
thf(fact_663_order__antisym__conv,axiom,
! [Y2: nat,X: nat] :
( ( ord_less_eq_nat @ Y2 @ X )
=> ( ( ord_less_eq_nat @ X @ Y2 )
= ( X = Y2 ) ) ) ).
% order_antisym_conv
thf(fact_664_comp__sgraph_Oinduced__edges__ss,axiom,
! [V: set_set_a,S: set_set_a] :
( ( ord_le3724670747650509150_set_a @ V @ S )
=> ( ord_le5722252365846178494_set_a @ ( undire7854589003810675244_set_a @ ( undire8247866692393712962_set_a @ S ) @ V ) @ ( undire8247866692393712962_set_a @ S ) ) ) ).
% comp_sgraph.induced_edges_ss
thf(fact_665_comp__sgraph_Oinduced__edges__ss,axiom,
! [V: set_Product_prod_a_a,S: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ V @ S )
=> ( ord_le1995061765932249223od_a_a @ ( undire5906991851038061813od_a_a @ ( undire6879232364018543115od_a_a @ S ) @ V ) @ ( undire6879232364018543115od_a_a @ S ) ) ) ).
% comp_sgraph.induced_edges_ss
thf(fact_666_comp__sgraph_Oinduced__edges__ss,axiom,
! [V: set_a,S: set_a] :
( ( ord_less_eq_set_a @ V @ S )
=> ( ord_le3724670747650509150_set_a @ ( undire7777452895879145676dges_a @ ( undire2918257014606996450dges_a @ S ) @ V ) @ ( undire2918257014606996450dges_a @ S ) ) ) ).
% comp_sgraph.induced_edges_ss
thf(fact_667_comp__sgraph_Oinduced__is__subgraph,axiom,
! [V: set_set_a,S: set_set_a] :
( ( ord_le3724670747650509150_set_a @ V @ S )
=> ( undire1186139521737116585_set_a @ V @ ( undire7854589003810675244_set_a @ ( undire8247866692393712962_set_a @ S ) @ V ) @ S @ ( undire8247866692393712962_set_a @ S ) ) ) ).
% comp_sgraph.induced_is_subgraph
thf(fact_668_comp__sgraph_Oinduced__is__subgraph,axiom,
! [V: set_Product_prod_a_a,S: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ V @ S )
=> ( undire398746457437328754od_a_a @ V @ ( undire5906991851038061813od_a_a @ ( undire6879232364018543115od_a_a @ S ) @ V ) @ S @ ( undire6879232364018543115od_a_a @ S ) ) ) ).
% comp_sgraph.induced_is_subgraph
thf(fact_669_comp__sgraph_Oinduced__is__subgraph,axiom,
! [V: set_a,S: set_a] :
( ( ord_less_eq_set_a @ V @ S )
=> ( undire7103218114511261257raph_a @ V @ ( undire7777452895879145676dges_a @ ( undire2918257014606996450dges_a @ S ) @ V ) @ S @ ( undire2918257014606996450dges_a @ S ) ) ) ).
% comp_sgraph.induced_is_subgraph
thf(fact_670_comp__sgraph_Ois__walk__append,axiom,
! [S: set_set_a,Xs: list_set_a,Ys2: list_set_a] :
( ( undire3014741414213135564_set_a @ S @ ( undire8247866692393712962_set_a @ S ) @ Xs )
=> ( ( undire3014741414213135564_set_a @ S @ ( undire8247866692393712962_set_a @ S ) @ Ys2 )
=> ( ( ( last_set_a @ Xs )
= ( hd_set_a @ Ys2 ) )
=> ( undire3014741414213135564_set_a @ S @ ( undire8247866692393712962_set_a @ S ) @ ( append_set_a @ Xs @ ( tl_set_a @ Ys2 ) ) ) ) ) ) ).
% comp_sgraph.is_walk_append
thf(fact_671_comp__sgraph_Ois__walk__append,axiom,
! [S: set_a,Xs: list_a,Ys2: list_a] :
( ( undire6133010728901294956walk_a @ S @ ( undire2918257014606996450dges_a @ S ) @ Xs )
=> ( ( undire6133010728901294956walk_a @ S @ ( undire2918257014606996450dges_a @ S ) @ Ys2 )
=> ( ( ( last_a @ Xs )
= ( hd_a @ Ys2 ) )
=> ( undire6133010728901294956walk_a @ S @ ( undire2918257014606996450dges_a @ S ) @ ( append_a @ Xs @ ( tl_a @ Ys2 ) ) ) ) ) ) ).
% comp_sgraph.is_walk_append
thf(fact_672_list__exhaust3,axiom,
! [Xs: list_a] :
( ( Xs != nil_a )
=> ( ! [X2: a] :
( Xs
!= ( cons_a @ X2 @ nil_a ) )
=> ~ ! [X2: a,Y: a,Ys: list_a] :
( Xs
!= ( cons_a @ X2 @ ( cons_a @ Y @ Ys ) ) ) ) ) ).
% list_exhaust3
thf(fact_673_list__exhaust3,axiom,
! [Xs: list_set_a] :
( ( Xs != nil_set_a )
=> ( ! [X2: set_a] :
( Xs
!= ( cons_set_a @ X2 @ nil_set_a ) )
=> ~ ! [X2: set_a,Y: set_a,Ys: list_set_a] :
( Xs
!= ( cons_set_a @ X2 @ ( cons_set_a @ Y @ Ys ) ) ) ) ) ).
% list_exhaust3
thf(fact_674_edge__density__ge0,axiom,
! [X5: set_a,Y5: set_a] : ( ord_less_eq_real @ zero_zero_real @ ( undire297304480579013331sity_a @ edges @ X5 @ Y5 ) ) ).
% edge_density_ge0
thf(fact_675_edge__density__le1,axiom,
! [X5: set_a,Y5: set_a] : ( ord_less_eq_real @ ( undire297304480579013331sity_a @ edges @ X5 @ Y5 ) @ one_one_real ) ).
% edge_density_le1
thf(fact_676_rotate1__hd__tl,axiom,
! [Xs: list_a] :
( ( Xs != nil_a )
=> ( ( rotate1_a @ Xs )
= ( append_a @ ( tl_a @ Xs ) @ ( cons_a @ ( hd_a @ Xs ) @ nil_a ) ) ) ) ).
% rotate1_hd_tl
thf(fact_677_rotate1__hd__tl,axiom,
! [Xs: list_set_a] :
( ( Xs != nil_set_a )
=> ( ( rotate1_set_a @ Xs )
= ( append_set_a @ ( tl_set_a @ Xs ) @ ( cons_set_a @ ( hd_set_a @ Xs ) @ nil_set_a ) ) ) ) ).
% rotate1_hd_tl
thf(fact_678_walk__length__def,axiom,
( undire8849074589633906640ngth_a
= ( ^ [P2: list_a] : ( size_size_list_set_a @ ( undire7337870655677353998dges_a @ P2 ) ) ) ) ).
% walk_length_def
thf(fact_679_all__edges__between__mono1,axiom,
! [Y5: set_a,Z4: set_a,X5: set_a] :
( ( ord_less_eq_set_a @ Y5 @ Z4 )
=> ( ord_le746702958409616551od_a_a @ ( undire8383842906760478443ween_a @ edges @ Y5 @ X5 ) @ ( undire8383842906760478443ween_a @ edges @ Z4 @ X5 ) ) ) ).
% all_edges_between_mono1
thf(fact_680_all__edges__between__mono2,axiom,
! [Y5: set_a,Z4: set_a,X5: set_a] :
( ( ord_less_eq_set_a @ Y5 @ Z4 )
=> ( ord_le746702958409616551od_a_a @ ( undire8383842906760478443ween_a @ edges @ X5 @ Y5 ) @ ( undire8383842906760478443ween_a @ edges @ X5 @ Z4 ) ) ) ).
% all_edges_between_mono2
thf(fact_681_append__eq__append__conv,axiom,
! [Xs: list_set_a,Ys2: list_set_a,Us2: list_set_a,Vs: list_set_a] :
( ( ( ( size_size_list_set_a @ Xs )
= ( size_size_list_set_a @ Ys2 ) )
| ( ( size_size_list_set_a @ Us2 )
= ( size_size_list_set_a @ Vs ) ) )
=> ( ( ( append_set_a @ Xs @ Us2 )
= ( append_set_a @ Ys2 @ Vs ) )
= ( ( Xs = Ys2 )
& ( Us2 = Vs ) ) ) ) ).
% append_eq_append_conv
thf(fact_682_append__eq__append__conv,axiom,
! [Xs: list_a,Ys2: list_a,Us2: list_a,Vs: list_a] :
( ( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys2 ) )
| ( ( size_size_list_a @ Us2 )
= ( size_size_list_a @ Vs ) ) )
=> ( ( ( append_a @ Xs @ Us2 )
= ( append_a @ Ys2 @ Vs ) )
= ( ( Xs = Ys2 )
& ( Us2 = Vs ) ) ) ) ).
% append_eq_append_conv
thf(fact_683_length__rev,axiom,
! [Xs: list_set_a] :
( ( size_size_list_set_a @ ( rev_set_a @ Xs ) )
= ( size_size_list_set_a @ Xs ) ) ).
% length_rev
thf(fact_684_length__rev,axiom,
! [Xs: list_a] :
( ( size_size_list_a @ ( rev_a @ Xs ) )
= ( size_size_list_a @ Xs ) ) ).
% length_rev
thf(fact_685_rotate1__is__Nil__conv,axiom,
! [Xs: list_a] :
( ( ( rotate1_a @ Xs )
= nil_a )
= ( Xs = nil_a ) ) ).
% rotate1_is_Nil_conv
thf(fact_686_rotate1__is__Nil__conv,axiom,
! [Xs: list_set_a] :
( ( ( rotate1_set_a @ Xs )
= nil_set_a )
= ( Xs = nil_set_a ) ) ).
% rotate1_is_Nil_conv
thf(fact_687_set__rotate1,axiom,
! [Xs: list_set_a] :
( ( set_set_a2 @ ( rotate1_set_a @ Xs ) )
= ( set_set_a2 @ Xs ) ) ).
% set_rotate1
thf(fact_688_set__rotate1,axiom,
! [Xs: list_a] :
( ( set_a2 @ ( rotate1_a @ Xs ) )
= ( set_a2 @ Xs ) ) ).
% set_rotate1
thf(fact_689_length__rotate1,axiom,
! [Xs: list_set_a] :
( ( size_size_list_set_a @ ( rotate1_set_a @ Xs ) )
= ( size_size_list_set_a @ Xs ) ) ).
% length_rotate1
thf(fact_690_length__rotate1,axiom,
! [Xs: list_a] :
( ( size_size_list_a @ ( rotate1_a @ Xs ) )
= ( size_size_list_a @ Xs ) ) ).
% length_rotate1
thf(fact_691_neq__if__length__neq,axiom,
! [Xs: list_set_a,Ys2: list_set_a] :
( ( ( size_size_list_set_a @ Xs )
!= ( size_size_list_set_a @ Ys2 ) )
=> ( Xs != Ys2 ) ) ).
% neq_if_length_neq
thf(fact_692_neq__if__length__neq,axiom,
! [Xs: list_a,Ys2: list_a] :
( ( ( size_size_list_a @ Xs )
!= ( size_size_list_a @ Ys2 ) )
=> ( Xs != Ys2 ) ) ).
% neq_if_length_neq
thf(fact_693_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs2: list_set_a] :
( ( size_size_list_set_a @ Xs2 )
= N ) ).
% Ex_list_of_length
thf(fact_694_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs2: list_a] :
( ( size_size_list_a @ Xs2 )
= N ) ).
% Ex_list_of_length
thf(fact_695_ulgraph_Oall__edges__between_Ocong,axiom,
undire8383842906760478443ween_a = undire8383842906760478443ween_a ).
% ulgraph.all_edges_between.cong
thf(fact_696_impossible__Cons,axiom,
! [Xs: list_set_a,Ys2: list_set_a,X: set_a] :
( ( ord_less_eq_nat @ ( size_size_list_set_a @ Xs ) @ ( size_size_list_set_a @ Ys2 ) )
=> ( Xs
!= ( cons_set_a @ X @ Ys2 ) ) ) ).
% impossible_Cons
thf(fact_697_impossible__Cons,axiom,
! [Xs: list_a,Ys2: list_a,X: a] :
( ( ord_less_eq_nat @ ( size_size_list_a @ Xs ) @ ( size_size_list_a @ Ys2 ) )
=> ( Xs
!= ( cons_a @ X @ Ys2 ) ) ) ).
% impossible_Cons
thf(fact_698_rotate1_Osimps_I1_J,axiom,
( ( rotate1_a @ nil_a )
= nil_a ) ).
% rotate1.simps(1)
thf(fact_699_rotate1_Osimps_I1_J,axiom,
( ( rotate1_set_a @ nil_set_a )
= nil_set_a ) ).
% rotate1.simps(1)
thf(fact_700_comp__sgraph_Oall__edges__between__mono2,axiom,
! [Y5: set_set_a,Z4: set_set_a,S: set_set_a,X5: set_set_a] :
( ( ord_le3724670747650509150_set_a @ Y5 @ Z4 )
=> ( ord_le8376522849517564071_set_a @ ( undire2462398226299384907_set_a @ ( undire8247866692393712962_set_a @ S ) @ X5 @ Y5 ) @ ( undire2462398226299384907_set_a @ ( undire8247866692393712962_set_a @ S ) @ X5 @ Z4 ) ) ) ).
% comp_sgraph.all_edges_between_mono2
thf(fact_701_comp__sgraph_Oall__edges__between__mono2,axiom,
! [Y5: set_Product_prod_a_a,Z4: set_Product_prod_a_a,S: set_Product_prod_a_a,X5: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ Y5 @ Z4 )
=> ( ord_le3469131294019144807od_a_a @ ( undire4032395788819567636od_a_a @ ( undire6879232364018543115od_a_a @ S ) @ X5 @ Y5 ) @ ( undire4032395788819567636od_a_a @ ( undire6879232364018543115od_a_a @ S ) @ X5 @ Z4 ) ) ) ).
% comp_sgraph.all_edges_between_mono2
thf(fact_702_comp__sgraph_Oall__edges__between__mono2,axiom,
! [Y5: set_a,Z4: set_a,S: set_a,X5: set_a] :
( ( ord_less_eq_set_a @ Y5 @ Z4 )
=> ( ord_le746702958409616551od_a_a @ ( undire8383842906760478443ween_a @ ( undire2918257014606996450dges_a @ S ) @ X5 @ Y5 ) @ ( undire8383842906760478443ween_a @ ( undire2918257014606996450dges_a @ S ) @ X5 @ Z4 ) ) ) ).
% comp_sgraph.all_edges_between_mono2
thf(fact_703_comp__sgraph_Oall__edges__between__mono1,axiom,
! [Y5: set_set_a,Z4: set_set_a,S: set_set_a,X5: set_set_a] :
( ( ord_le3724670747650509150_set_a @ Y5 @ Z4 )
=> ( ord_le8376522849517564071_set_a @ ( undire2462398226299384907_set_a @ ( undire8247866692393712962_set_a @ S ) @ Y5 @ X5 ) @ ( undire2462398226299384907_set_a @ ( undire8247866692393712962_set_a @ S ) @ Z4 @ X5 ) ) ) ).
% comp_sgraph.all_edges_between_mono1
thf(fact_704_comp__sgraph_Oall__edges__between__mono1,axiom,
! [Y5: set_Product_prod_a_a,Z4: set_Product_prod_a_a,S: set_Product_prod_a_a,X5: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ Y5 @ Z4 )
=> ( ord_le3469131294019144807od_a_a @ ( undire4032395788819567636od_a_a @ ( undire6879232364018543115od_a_a @ S ) @ Y5 @ X5 ) @ ( undire4032395788819567636od_a_a @ ( undire6879232364018543115od_a_a @ S ) @ Z4 @ X5 ) ) ) ).
% comp_sgraph.all_edges_between_mono1
thf(fact_705_comp__sgraph_Oall__edges__between__mono1,axiom,
! [Y5: set_a,Z4: set_a,S: set_a,X5: set_a] :
( ( ord_less_eq_set_a @ Y5 @ Z4 )
=> ( ord_le746702958409616551od_a_a @ ( undire8383842906760478443ween_a @ ( undire2918257014606996450dges_a @ S ) @ Y5 @ X5 ) @ ( undire8383842906760478443ween_a @ ( undire2918257014606996450dges_a @ S ) @ Z4 @ X5 ) ) ) ).
% comp_sgraph.all_edges_between_mono1
thf(fact_706_ulgraph_Oall__edges__between__mono2,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,Y5: set_set_a,Z4: set_set_a,X5: set_set_a] :
( ( undire6886684016831807756_set_a @ Vertices @ Edges )
=> ( ( ord_le3724670747650509150_set_a @ Y5 @ Z4 )
=> ( ord_le8376522849517564071_set_a @ ( undire2462398226299384907_set_a @ Edges @ X5 @ Y5 ) @ ( undire2462398226299384907_set_a @ Edges @ X5 @ Z4 ) ) ) ) ).
% ulgraph.all_edges_between_mono2
thf(fact_707_ulgraph_Oall__edges__between__mono2,axiom,
! [Vertices: set_Product_prod_a_a,Edges: set_se5735800977113168103od_a_a,Y5: set_Product_prod_a_a,Z4: set_Product_prod_a_a,X5: set_Product_prod_a_a] :
( ( undire4585262585102564309od_a_a @ Vertices @ Edges )
=> ( ( ord_le746702958409616551od_a_a @ Y5 @ Z4 )
=> ( ord_le3469131294019144807od_a_a @ ( undire4032395788819567636od_a_a @ Edges @ X5 @ Y5 ) @ ( undire4032395788819567636od_a_a @ Edges @ X5 @ Z4 ) ) ) ) ).
% ulgraph.all_edges_between_mono2
thf(fact_708_ulgraph_Oall__edges__between__mono2,axiom,
! [Vertices: set_a,Edges: set_set_a,Y5: set_a,Z4: set_a,X5: set_a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( ord_less_eq_set_a @ Y5 @ Z4 )
=> ( ord_le746702958409616551od_a_a @ ( undire8383842906760478443ween_a @ Edges @ X5 @ Y5 ) @ ( undire8383842906760478443ween_a @ Edges @ X5 @ Z4 ) ) ) ) ).
% ulgraph.all_edges_between_mono2
thf(fact_709_ulgraph_Oall__edges__between__mono1,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,Y5: set_set_a,Z4: set_set_a,X5: set_set_a] :
( ( undire6886684016831807756_set_a @ Vertices @ Edges )
=> ( ( ord_le3724670747650509150_set_a @ Y5 @ Z4 )
=> ( ord_le8376522849517564071_set_a @ ( undire2462398226299384907_set_a @ Edges @ Y5 @ X5 ) @ ( undire2462398226299384907_set_a @ Edges @ Z4 @ X5 ) ) ) ) ).
% ulgraph.all_edges_between_mono1
thf(fact_710_ulgraph_Oall__edges__between__mono1,axiom,
! [Vertices: set_Product_prod_a_a,Edges: set_se5735800977113168103od_a_a,Y5: set_Product_prod_a_a,Z4: set_Product_prod_a_a,X5: set_Product_prod_a_a] :
( ( undire4585262585102564309od_a_a @ Vertices @ Edges )
=> ( ( ord_le746702958409616551od_a_a @ Y5 @ Z4 )
=> ( ord_le3469131294019144807od_a_a @ ( undire4032395788819567636od_a_a @ Edges @ Y5 @ X5 ) @ ( undire4032395788819567636od_a_a @ Edges @ Z4 @ X5 ) ) ) ) ).
% ulgraph.all_edges_between_mono1
thf(fact_711_ulgraph_Oall__edges__between__mono1,axiom,
! [Vertices: set_a,Edges: set_set_a,Y5: set_a,Z4: set_a,X5: set_a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( ord_less_eq_set_a @ Y5 @ Z4 )
=> ( ord_le746702958409616551od_a_a @ ( undire8383842906760478443ween_a @ Edges @ Y5 @ X5 ) @ ( undire8383842906760478443ween_a @ Edges @ Z4 @ X5 ) ) ) ) ).
% ulgraph.all_edges_between_mono1
thf(fact_712_list__induct4,axiom,
! [Xs: list_a,Ys2: list_a,Zs: list_a,Ws: list_a,P: list_a > list_a > list_a > list_a > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys2 ) )
=> ( ( ( size_size_list_a @ Ys2 )
= ( size_size_list_a @ Zs ) )
=> ( ( ( size_size_list_a @ Zs )
= ( size_size_list_a @ Ws ) )
=> ( ( P @ nil_a @ nil_a @ nil_a @ nil_a )
=> ( ! [X2: a,Xs2: list_a,Y: a,Ys: list_a,Z5: a,Zs3: list_a,W: a,Ws2: list_a] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_a @ Ys ) )
=> ( ( ( size_size_list_a @ Ys )
= ( size_size_list_a @ Zs3 ) )
=> ( ( ( size_size_list_a @ Zs3 )
= ( size_size_list_a @ Ws2 ) )
=> ( ( P @ Xs2 @ Ys @ Zs3 @ Ws2 )
=> ( P @ ( cons_a @ X2 @ Xs2 ) @ ( cons_a @ Y @ Ys ) @ ( cons_a @ Z5 @ Zs3 ) @ ( cons_a @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys2 @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_713_list__induct4,axiom,
! [Xs: list_set_a,Ys2: list_a,Zs: list_a,Ws: list_a,P: list_set_a > list_a > list_a > list_a > $o] :
( ( ( size_size_list_set_a @ Xs )
= ( size_size_list_a @ Ys2 ) )
=> ( ( ( size_size_list_a @ Ys2 )
= ( size_size_list_a @ Zs ) )
=> ( ( ( size_size_list_a @ Zs )
= ( size_size_list_a @ Ws ) )
=> ( ( P @ nil_set_a @ nil_a @ nil_a @ nil_a )
=> ( ! [X2: set_a,Xs2: list_set_a,Y: a,Ys: list_a,Z5: a,Zs3: list_a,W: a,Ws2: list_a] :
( ( ( size_size_list_set_a @ Xs2 )
= ( size_size_list_a @ Ys ) )
=> ( ( ( size_size_list_a @ Ys )
= ( size_size_list_a @ Zs3 ) )
=> ( ( ( size_size_list_a @ Zs3 )
= ( size_size_list_a @ Ws2 ) )
=> ( ( P @ Xs2 @ Ys @ Zs3 @ Ws2 )
=> ( P @ ( cons_set_a @ X2 @ Xs2 ) @ ( cons_a @ Y @ Ys ) @ ( cons_a @ Z5 @ Zs3 ) @ ( cons_a @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys2 @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_714_list__induct4,axiom,
! [Xs: list_a,Ys2: list_set_a,Zs: list_a,Ws: list_a,P: list_a > list_set_a > list_a > list_a > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_set_a @ Ys2 ) )
=> ( ( ( size_size_list_set_a @ Ys2 )
= ( size_size_list_a @ Zs ) )
=> ( ( ( size_size_list_a @ Zs )
= ( size_size_list_a @ Ws ) )
=> ( ( P @ nil_a @ nil_set_a @ nil_a @ nil_a )
=> ( ! [X2: a,Xs2: list_a,Y: set_a,Ys: list_set_a,Z5: a,Zs3: list_a,W: a,Ws2: list_a] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_set_a @ Ys ) )
=> ( ( ( size_size_list_set_a @ Ys )
= ( size_size_list_a @ Zs3 ) )
=> ( ( ( size_size_list_a @ Zs3 )
= ( size_size_list_a @ Ws2 ) )
=> ( ( P @ Xs2 @ Ys @ Zs3 @ Ws2 )
=> ( P @ ( cons_a @ X2 @ Xs2 ) @ ( cons_set_a @ Y @ Ys ) @ ( cons_a @ Z5 @ Zs3 ) @ ( cons_a @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys2 @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_715_list__induct4,axiom,
! [Xs: list_a,Ys2: list_a,Zs: list_set_a,Ws: list_a,P: list_a > list_a > list_set_a > list_a > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys2 ) )
=> ( ( ( size_size_list_a @ Ys2 )
= ( size_size_list_set_a @ Zs ) )
=> ( ( ( size_size_list_set_a @ Zs )
= ( size_size_list_a @ Ws ) )
=> ( ( P @ nil_a @ nil_a @ nil_set_a @ nil_a )
=> ( ! [X2: a,Xs2: list_a,Y: a,Ys: list_a,Z5: set_a,Zs3: list_set_a,W: a,Ws2: list_a] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_a @ Ys ) )
=> ( ( ( size_size_list_a @ Ys )
= ( size_size_list_set_a @ Zs3 ) )
=> ( ( ( size_size_list_set_a @ Zs3 )
= ( size_size_list_a @ Ws2 ) )
=> ( ( P @ Xs2 @ Ys @ Zs3 @ Ws2 )
=> ( P @ ( cons_a @ X2 @ Xs2 ) @ ( cons_a @ Y @ Ys ) @ ( cons_set_a @ Z5 @ Zs3 ) @ ( cons_a @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys2 @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_716_list__induct4,axiom,
! [Xs: list_a,Ys2: list_a,Zs: list_a,Ws: list_set_a,P: list_a > list_a > list_a > list_set_a > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys2 ) )
=> ( ( ( size_size_list_a @ Ys2 )
= ( size_size_list_a @ Zs ) )
=> ( ( ( size_size_list_a @ Zs )
= ( size_size_list_set_a @ Ws ) )
=> ( ( P @ nil_a @ nil_a @ nil_a @ nil_set_a )
=> ( ! [X2: a,Xs2: list_a,Y: a,Ys: list_a,Z5: a,Zs3: list_a,W: set_a,Ws2: list_set_a] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_a @ Ys ) )
=> ( ( ( size_size_list_a @ Ys )
= ( size_size_list_a @ Zs3 ) )
=> ( ( ( size_size_list_a @ Zs3 )
= ( size_size_list_set_a @ Ws2 ) )
=> ( ( P @ Xs2 @ Ys @ Zs3 @ Ws2 )
=> ( P @ ( cons_a @ X2 @ Xs2 ) @ ( cons_a @ Y @ Ys ) @ ( cons_a @ Z5 @ Zs3 ) @ ( cons_set_a @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys2 @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_717_list__induct4,axiom,
! [Xs: list_set_a,Ys2: list_set_a,Zs: list_a,Ws: list_a,P: list_set_a > list_set_a > list_a > list_a > $o] :
( ( ( size_size_list_set_a @ Xs )
= ( size_size_list_set_a @ Ys2 ) )
=> ( ( ( size_size_list_set_a @ Ys2 )
= ( size_size_list_a @ Zs ) )
=> ( ( ( size_size_list_a @ Zs )
= ( size_size_list_a @ Ws ) )
=> ( ( P @ nil_set_a @ nil_set_a @ nil_a @ nil_a )
=> ( ! [X2: set_a,Xs2: list_set_a,Y: set_a,Ys: list_set_a,Z5: a,Zs3: list_a,W: a,Ws2: list_a] :
( ( ( size_size_list_set_a @ Xs2 )
= ( size_size_list_set_a @ Ys ) )
=> ( ( ( size_size_list_set_a @ Ys )
= ( size_size_list_a @ Zs3 ) )
=> ( ( ( size_size_list_a @ Zs3 )
= ( size_size_list_a @ Ws2 ) )
=> ( ( P @ Xs2 @ Ys @ Zs3 @ Ws2 )
=> ( P @ ( cons_set_a @ X2 @ Xs2 ) @ ( cons_set_a @ Y @ Ys ) @ ( cons_a @ Z5 @ Zs3 ) @ ( cons_a @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys2 @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_718_list__induct4,axiom,
! [Xs: list_set_a,Ys2: list_a,Zs: list_set_a,Ws: list_a,P: list_set_a > list_a > list_set_a > list_a > $o] :
( ( ( size_size_list_set_a @ Xs )
= ( size_size_list_a @ Ys2 ) )
=> ( ( ( size_size_list_a @ Ys2 )
= ( size_size_list_set_a @ Zs ) )
=> ( ( ( size_size_list_set_a @ Zs )
= ( size_size_list_a @ Ws ) )
=> ( ( P @ nil_set_a @ nil_a @ nil_set_a @ nil_a )
=> ( ! [X2: set_a,Xs2: list_set_a,Y: a,Ys: list_a,Z5: set_a,Zs3: list_set_a,W: a,Ws2: list_a] :
( ( ( size_size_list_set_a @ Xs2 )
= ( size_size_list_a @ Ys ) )
=> ( ( ( size_size_list_a @ Ys )
= ( size_size_list_set_a @ Zs3 ) )
=> ( ( ( size_size_list_set_a @ Zs3 )
= ( size_size_list_a @ Ws2 ) )
=> ( ( P @ Xs2 @ Ys @ Zs3 @ Ws2 )
=> ( P @ ( cons_set_a @ X2 @ Xs2 ) @ ( cons_a @ Y @ Ys ) @ ( cons_set_a @ Z5 @ Zs3 ) @ ( cons_a @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys2 @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_719_list__induct4,axiom,
! [Xs: list_set_a,Ys2: list_a,Zs: list_a,Ws: list_set_a,P: list_set_a > list_a > list_a > list_set_a > $o] :
( ( ( size_size_list_set_a @ Xs )
= ( size_size_list_a @ Ys2 ) )
=> ( ( ( size_size_list_a @ Ys2 )
= ( size_size_list_a @ Zs ) )
=> ( ( ( size_size_list_a @ Zs )
= ( size_size_list_set_a @ Ws ) )
=> ( ( P @ nil_set_a @ nil_a @ nil_a @ nil_set_a )
=> ( ! [X2: set_a,Xs2: list_set_a,Y: a,Ys: list_a,Z5: a,Zs3: list_a,W: set_a,Ws2: list_set_a] :
( ( ( size_size_list_set_a @ Xs2 )
= ( size_size_list_a @ Ys ) )
=> ( ( ( size_size_list_a @ Ys )
= ( size_size_list_a @ Zs3 ) )
=> ( ( ( size_size_list_a @ Zs3 )
= ( size_size_list_set_a @ Ws2 ) )
=> ( ( P @ Xs2 @ Ys @ Zs3 @ Ws2 )
=> ( P @ ( cons_set_a @ X2 @ Xs2 ) @ ( cons_a @ Y @ Ys ) @ ( cons_a @ Z5 @ Zs3 ) @ ( cons_set_a @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys2 @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_720_list__induct4,axiom,
! [Xs: list_a,Ys2: list_set_a,Zs: list_set_a,Ws: list_a,P: list_a > list_set_a > list_set_a > list_a > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_set_a @ Ys2 ) )
=> ( ( ( size_size_list_set_a @ Ys2 )
= ( size_size_list_set_a @ Zs ) )
=> ( ( ( size_size_list_set_a @ Zs )
= ( size_size_list_a @ Ws ) )
=> ( ( P @ nil_a @ nil_set_a @ nil_set_a @ nil_a )
=> ( ! [X2: a,Xs2: list_a,Y: set_a,Ys: list_set_a,Z5: set_a,Zs3: list_set_a,W: a,Ws2: list_a] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_set_a @ Ys ) )
=> ( ( ( size_size_list_set_a @ Ys )
= ( size_size_list_set_a @ Zs3 ) )
=> ( ( ( size_size_list_set_a @ Zs3 )
= ( size_size_list_a @ Ws2 ) )
=> ( ( P @ Xs2 @ Ys @ Zs3 @ Ws2 )
=> ( P @ ( cons_a @ X2 @ Xs2 ) @ ( cons_set_a @ Y @ Ys ) @ ( cons_set_a @ Z5 @ Zs3 ) @ ( cons_a @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys2 @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_721_list__induct4,axiom,
! [Xs: list_a,Ys2: list_set_a,Zs: list_a,Ws: list_set_a,P: list_a > list_set_a > list_a > list_set_a > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_set_a @ Ys2 ) )
=> ( ( ( size_size_list_set_a @ Ys2 )
= ( size_size_list_a @ Zs ) )
=> ( ( ( size_size_list_a @ Zs )
= ( size_size_list_set_a @ Ws ) )
=> ( ( P @ nil_a @ nil_set_a @ nil_a @ nil_set_a )
=> ( ! [X2: a,Xs2: list_a,Y: set_a,Ys: list_set_a,Z5: a,Zs3: list_a,W: set_a,Ws2: list_set_a] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_set_a @ Ys ) )
=> ( ( ( size_size_list_set_a @ Ys )
= ( size_size_list_a @ Zs3 ) )
=> ( ( ( size_size_list_a @ Zs3 )
= ( size_size_list_set_a @ Ws2 ) )
=> ( ( P @ Xs2 @ Ys @ Zs3 @ Ws2 )
=> ( P @ ( cons_a @ X2 @ Xs2 ) @ ( cons_set_a @ Y @ Ys ) @ ( cons_a @ Z5 @ Zs3 ) @ ( cons_set_a @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys2 @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_722_list__induct3,axiom,
! [Xs: list_set_a,Ys2: list_set_a,Zs: list_set_a,P: list_set_a > list_set_a > list_set_a > $o] :
( ( ( size_size_list_set_a @ Xs )
= ( size_size_list_set_a @ Ys2 ) )
=> ( ( ( size_size_list_set_a @ Ys2 )
= ( size_size_list_set_a @ Zs ) )
=> ( ( P @ nil_set_a @ nil_set_a @ nil_set_a )
=> ( ! [X2: set_a,Xs2: list_set_a,Y: set_a,Ys: list_set_a,Z5: set_a,Zs3: list_set_a] :
( ( ( size_size_list_set_a @ Xs2 )
= ( size_size_list_set_a @ Ys ) )
=> ( ( ( size_size_list_set_a @ Ys )
= ( size_size_list_set_a @ Zs3 ) )
=> ( ( P @ Xs2 @ Ys @ Zs3 )
=> ( P @ ( cons_set_a @ X2 @ Xs2 ) @ ( cons_set_a @ Y @ Ys ) @ ( cons_set_a @ Z5 @ Zs3 ) ) ) ) )
=> ( P @ Xs @ Ys2 @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_723_list__induct3,axiom,
! [Xs: list_set_a,Ys2: list_set_a,Zs: list_a,P: list_set_a > list_set_a > list_a > $o] :
( ( ( size_size_list_set_a @ Xs )
= ( size_size_list_set_a @ Ys2 ) )
=> ( ( ( size_size_list_set_a @ Ys2 )
= ( size_size_list_a @ Zs ) )
=> ( ( P @ nil_set_a @ nil_set_a @ nil_a )
=> ( ! [X2: set_a,Xs2: list_set_a,Y: set_a,Ys: list_set_a,Z5: a,Zs3: list_a] :
( ( ( size_size_list_set_a @ Xs2 )
= ( size_size_list_set_a @ Ys ) )
=> ( ( ( size_size_list_set_a @ Ys )
= ( size_size_list_a @ Zs3 ) )
=> ( ( P @ Xs2 @ Ys @ Zs3 )
=> ( P @ ( cons_set_a @ X2 @ Xs2 ) @ ( cons_set_a @ Y @ Ys ) @ ( cons_a @ Z5 @ Zs3 ) ) ) ) )
=> ( P @ Xs @ Ys2 @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_724_list__induct3,axiom,
! [Xs: list_set_a,Ys2: list_a,Zs: list_set_a,P: list_set_a > list_a > list_set_a > $o] :
( ( ( size_size_list_set_a @ Xs )
= ( size_size_list_a @ Ys2 ) )
=> ( ( ( size_size_list_a @ Ys2 )
= ( size_size_list_set_a @ Zs ) )
=> ( ( P @ nil_set_a @ nil_a @ nil_set_a )
=> ( ! [X2: set_a,Xs2: list_set_a,Y: a,Ys: list_a,Z5: set_a,Zs3: list_set_a] :
( ( ( size_size_list_set_a @ Xs2 )
= ( size_size_list_a @ Ys ) )
=> ( ( ( size_size_list_a @ Ys )
= ( size_size_list_set_a @ Zs3 ) )
=> ( ( P @ Xs2 @ Ys @ Zs3 )
=> ( P @ ( cons_set_a @ X2 @ Xs2 ) @ ( cons_a @ Y @ Ys ) @ ( cons_set_a @ Z5 @ Zs3 ) ) ) ) )
=> ( P @ Xs @ Ys2 @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_725_list__induct3,axiom,
! [Xs: list_set_a,Ys2: list_a,Zs: list_a,P: list_set_a > list_a > list_a > $o] :
( ( ( size_size_list_set_a @ Xs )
= ( size_size_list_a @ Ys2 ) )
=> ( ( ( size_size_list_a @ Ys2 )
= ( size_size_list_a @ Zs ) )
=> ( ( P @ nil_set_a @ nil_a @ nil_a )
=> ( ! [X2: set_a,Xs2: list_set_a,Y: a,Ys: list_a,Z5: a,Zs3: list_a] :
( ( ( size_size_list_set_a @ Xs2 )
= ( size_size_list_a @ Ys ) )
=> ( ( ( size_size_list_a @ Ys )
= ( size_size_list_a @ Zs3 ) )
=> ( ( P @ Xs2 @ Ys @ Zs3 )
=> ( P @ ( cons_set_a @ X2 @ Xs2 ) @ ( cons_a @ Y @ Ys ) @ ( cons_a @ Z5 @ Zs3 ) ) ) ) )
=> ( P @ Xs @ Ys2 @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_726_list__induct3,axiom,
! [Xs: list_a,Ys2: list_set_a,Zs: list_set_a,P: list_a > list_set_a > list_set_a > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_set_a @ Ys2 ) )
=> ( ( ( size_size_list_set_a @ Ys2 )
= ( size_size_list_set_a @ Zs ) )
=> ( ( P @ nil_a @ nil_set_a @ nil_set_a )
=> ( ! [X2: a,Xs2: list_a,Y: set_a,Ys: list_set_a,Z5: set_a,Zs3: list_set_a] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_set_a @ Ys ) )
=> ( ( ( size_size_list_set_a @ Ys )
= ( size_size_list_set_a @ Zs3 ) )
=> ( ( P @ Xs2 @ Ys @ Zs3 )
=> ( P @ ( cons_a @ X2 @ Xs2 ) @ ( cons_set_a @ Y @ Ys ) @ ( cons_set_a @ Z5 @ Zs3 ) ) ) ) )
=> ( P @ Xs @ Ys2 @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_727_list__induct3,axiom,
! [Xs: list_a,Ys2: list_set_a,Zs: list_a,P: list_a > list_set_a > list_a > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_set_a @ Ys2 ) )
=> ( ( ( size_size_list_set_a @ Ys2 )
= ( size_size_list_a @ Zs ) )
=> ( ( P @ nil_a @ nil_set_a @ nil_a )
=> ( ! [X2: a,Xs2: list_a,Y: set_a,Ys: list_set_a,Z5: a,Zs3: list_a] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_set_a @ Ys ) )
=> ( ( ( size_size_list_set_a @ Ys )
= ( size_size_list_a @ Zs3 ) )
=> ( ( P @ Xs2 @ Ys @ Zs3 )
=> ( P @ ( cons_a @ X2 @ Xs2 ) @ ( cons_set_a @ Y @ Ys ) @ ( cons_a @ Z5 @ Zs3 ) ) ) ) )
=> ( P @ Xs @ Ys2 @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_728_list__induct3,axiom,
! [Xs: list_a,Ys2: list_a,Zs: list_set_a,P: list_a > list_a > list_set_a > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys2 ) )
=> ( ( ( size_size_list_a @ Ys2 )
= ( size_size_list_set_a @ Zs ) )
=> ( ( P @ nil_a @ nil_a @ nil_set_a )
=> ( ! [X2: a,Xs2: list_a,Y: a,Ys: list_a,Z5: set_a,Zs3: list_set_a] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_a @ Ys ) )
=> ( ( ( size_size_list_a @ Ys )
= ( size_size_list_set_a @ Zs3 ) )
=> ( ( P @ Xs2 @ Ys @ Zs3 )
=> ( P @ ( cons_a @ X2 @ Xs2 ) @ ( cons_a @ Y @ Ys ) @ ( cons_set_a @ Z5 @ Zs3 ) ) ) ) )
=> ( P @ Xs @ Ys2 @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_729_list__induct3,axiom,
! [Xs: list_a,Ys2: list_a,Zs: list_a,P: list_a > list_a > list_a > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys2 ) )
=> ( ( ( size_size_list_a @ Ys2 )
= ( size_size_list_a @ Zs ) )
=> ( ( P @ nil_a @ nil_a @ nil_a )
=> ( ! [X2: a,Xs2: list_a,Y: a,Ys: list_a,Z5: a,Zs3: list_a] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_a @ Ys ) )
=> ( ( ( size_size_list_a @ Ys )
= ( size_size_list_a @ Zs3 ) )
=> ( ( P @ Xs2 @ Ys @ Zs3 )
=> ( P @ ( cons_a @ X2 @ Xs2 ) @ ( cons_a @ Y @ Ys ) @ ( cons_a @ Z5 @ Zs3 ) ) ) ) )
=> ( P @ Xs @ Ys2 @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_730_list__induct2,axiom,
! [Xs: list_set_a,Ys2: list_set_a,P: list_set_a > list_set_a > $o] :
( ( ( size_size_list_set_a @ Xs )
= ( size_size_list_set_a @ Ys2 ) )
=> ( ( P @ nil_set_a @ nil_set_a )
=> ( ! [X2: set_a,Xs2: list_set_a,Y: set_a,Ys: list_set_a] :
( ( ( size_size_list_set_a @ Xs2 )
= ( size_size_list_set_a @ Ys ) )
=> ( ( P @ Xs2 @ Ys )
=> ( P @ ( cons_set_a @ X2 @ Xs2 ) @ ( cons_set_a @ Y @ Ys ) ) ) )
=> ( P @ Xs @ Ys2 ) ) ) ) ).
% list_induct2
thf(fact_731_list__induct2,axiom,
! [Xs: list_set_a,Ys2: list_a,P: list_set_a > list_a > $o] :
( ( ( size_size_list_set_a @ Xs )
= ( size_size_list_a @ Ys2 ) )
=> ( ( P @ nil_set_a @ nil_a )
=> ( ! [X2: set_a,Xs2: list_set_a,Y: a,Ys: list_a] :
( ( ( size_size_list_set_a @ Xs2 )
= ( size_size_list_a @ Ys ) )
=> ( ( P @ Xs2 @ Ys )
=> ( P @ ( cons_set_a @ X2 @ Xs2 ) @ ( cons_a @ Y @ Ys ) ) ) )
=> ( P @ Xs @ Ys2 ) ) ) ) ).
% list_induct2
thf(fact_732_list__induct2,axiom,
! [Xs: list_a,Ys2: list_set_a,P: list_a > list_set_a > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_set_a @ Ys2 ) )
=> ( ( P @ nil_a @ nil_set_a )
=> ( ! [X2: a,Xs2: list_a,Y: set_a,Ys: list_set_a] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_set_a @ Ys ) )
=> ( ( P @ Xs2 @ Ys )
=> ( P @ ( cons_a @ X2 @ Xs2 ) @ ( cons_set_a @ Y @ Ys ) ) ) )
=> ( P @ Xs @ Ys2 ) ) ) ) ).
% list_induct2
thf(fact_733_list__induct2,axiom,
! [Xs: list_a,Ys2: list_a,P: list_a > list_a > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys2 ) )
=> ( ( P @ nil_a @ nil_a )
=> ( ! [X2: a,Xs2: list_a,Y: a,Ys: list_a] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_a @ Ys ) )
=> ( ( P @ Xs2 @ Ys )
=> ( P @ ( cons_a @ X2 @ Xs2 ) @ ( cons_a @ Y @ Ys ) ) ) )
=> ( P @ Xs @ Ys2 ) ) ) ) ).
% list_induct2
thf(fact_734_comp__sgraph_Owalk__length__def,axiom,
( undire8849074589633906640ngth_a
= ( ^ [P2: list_a] : ( size_size_list_set_a @ ( undire7337870655677353998dges_a @ P2 ) ) ) ) ).
% comp_sgraph.walk_length_def
thf(fact_735_comp__sgraph_Oedge__density__le1,axiom,
! [S: set_a,X5: set_a,Y5: set_a] : ( ord_less_eq_real @ ( undire297304480579013331sity_a @ ( undire2918257014606996450dges_a @ S ) @ X5 @ Y5 ) @ one_one_real ) ).
% comp_sgraph.edge_density_le1
thf(fact_736_ulgraph_Oedge__density__le1,axiom,
! [Vertices: set_a,Edges: set_set_a,X5: set_a,Y5: set_a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ord_less_eq_real @ ( undire297304480579013331sity_a @ Edges @ X5 @ Y5 ) @ one_one_real ) ) ).
% ulgraph.edge_density_le1
thf(fact_737_comp__sgraph_Oedge__density__ge0,axiom,
! [S: set_a,X5: set_a,Y5: set_a] : ( ord_less_eq_real @ zero_zero_real @ ( undire297304480579013331sity_a @ ( undire2918257014606996450dges_a @ S ) @ X5 @ Y5 ) ) ).
% comp_sgraph.edge_density_ge0
thf(fact_738_ulgraph_Oedge__density__ge0,axiom,
! [Vertices: set_a,Edges: set_set_a,X5: set_a,Y5: set_a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ord_less_eq_real @ zero_zero_real @ ( undire297304480579013331sity_a @ Edges @ X5 @ Y5 ) ) ) ).
% ulgraph.edge_density_ge0
thf(fact_739_same__length__different,axiom,
! [Xs: list_set_a,Ys2: list_set_a] :
( ( Xs != Ys2 )
=> ( ( ( size_size_list_set_a @ Xs )
= ( size_size_list_set_a @ Ys2 ) )
=> ? [Pre: list_set_a,X2: set_a,Xs4: list_set_a,Y: set_a,Ys6: list_set_a] :
( ( X2 != Y )
& ( Xs
= ( append_set_a @ Pre @ ( append_set_a @ ( cons_set_a @ X2 @ nil_set_a ) @ Xs4 ) ) )
& ( Ys2
= ( append_set_a @ Pre @ ( append_set_a @ ( cons_set_a @ Y @ nil_set_a ) @ Ys6 ) ) ) ) ) ) ).
% same_length_different
thf(fact_740_same__length__different,axiom,
! [Xs: list_a,Ys2: list_a] :
( ( Xs != Ys2 )
=> ( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys2 ) )
=> ? [Pre: list_a,X2: a,Xs4: list_a,Y: a,Ys6: list_a] :
( ( X2 != Y )
& ( Xs
= ( append_a @ Pre @ ( append_a @ ( cons_a @ X2 @ nil_a ) @ Xs4 ) ) )
& ( Ys2
= ( append_a @ Pre @ ( append_a @ ( cons_a @ Y @ nil_a ) @ Ys6 ) ) ) ) ) ) ).
% same_length_different
thf(fact_741_ulgraph_Owalk__length__def,axiom,
! [Vertices: set_a,Edges: set_set_a,P3: list_a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( undire8849074589633906640ngth_a @ P3 )
= ( size_size_list_set_a @ ( undire7337870655677353998dges_a @ P3 ) ) ) ) ).
% ulgraph.walk_length_def
thf(fact_742_rotate1_Osimps_I2_J,axiom,
! [X: a,Xs: list_a] :
( ( rotate1_a @ ( cons_a @ X @ Xs ) )
= ( append_a @ Xs @ ( cons_a @ X @ nil_a ) ) ) ).
% rotate1.simps(2)
thf(fact_743_rotate1_Osimps_I2_J,axiom,
! [X: set_a,Xs: list_set_a] :
( ( rotate1_set_a @ ( cons_set_a @ X @ Xs ) )
= ( append_set_a @ Xs @ ( cons_set_a @ X @ nil_set_a ) ) ) ).
% rotate1.simps(2)
thf(fact_744_edge__density__eq0,axiom,
! [A2: set_a,B2: set_a,X5: set_a,Y5: set_a] :
( ( ( undire8383842906760478443ween_a @ edges @ A2 @ B2 )
= bot_bo3357376287454694259od_a_a )
=> ( ( ord_less_eq_set_a @ X5 @ A2 )
=> ( ( ord_less_eq_set_a @ Y5 @ B2 )
=> ( ( undire297304480579013331sity_a @ edges @ X5 @ Y5 )
= zero_zero_real ) ) ) ) ).
% edge_density_eq0
thf(fact_745_le__zero__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_746_edge__density__zero,axiom,
! [Y5: set_a,X5: set_a] :
( ( Y5 = bot_bot_set_a )
=> ( ( undire297304480579013331sity_a @ edges @ X5 @ Y5 )
= zero_zero_real ) ) ).
% edge_density_zero
thf(fact_747_all__edges__between__rem__wf,axiom,
! [X5: set_a,Y5: set_a] :
( ( undire8383842906760478443ween_a @ edges @ X5 @ Y5 )
= ( undire8383842906760478443ween_a @ edges @ ( inf_inf_set_a @ X5 @ vertices ) @ ( inf_inf_set_a @ Y5 @ vertices ) ) ) ).
% all_edges_between_rem_wf
thf(fact_748_not__one__le__zero,axiom,
~ ( ord_less_eq_real @ one_one_real @ zero_zero_real ) ).
% not_one_le_zero
thf(fact_749_not__one__le__zero,axiom,
~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_le_zero
thf(fact_750_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
ord_less_eq_real @ zero_zero_real @ one_one_real ).
% linordered_nonzero_semiring_class.zero_le_one
thf(fact_751_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% linordered_nonzero_semiring_class.zero_le_one
thf(fact_752_empty__not__edge,axiom,
~ ( member_set_a @ bot_bot_set_a @ edges ) ).
% empty_not_edge
thf(fact_753_empty__iff,axiom,
! [C: nat] :
~ ( member_nat @ C @ bot_bot_set_nat ) ).
% empty_iff
thf(fact_754_empty__iff,axiom,
! [C: product_prod_a_a] :
~ ( member1426531477525435216od_a_a @ C @ bot_bo3357376287454694259od_a_a ) ).
% empty_iff
thf(fact_755_empty__iff,axiom,
! [C: a] :
~ ( member_a @ C @ bot_bot_set_a ) ).
% empty_iff
thf(fact_756_empty__iff,axiom,
! [C: set_a] :
~ ( member_set_a @ C @ bot_bot_set_set_a ) ).
% empty_iff
thf(fact_757_all__not__in__conv,axiom,
! [A2: set_nat] :
( ( ! [X4: nat] :
~ ( member_nat @ X4 @ A2 ) )
= ( A2 = bot_bot_set_nat ) ) ).
% all_not_in_conv
thf(fact_758_all__not__in__conv,axiom,
! [A2: set_Product_prod_a_a] :
( ( ! [X4: product_prod_a_a] :
~ ( member1426531477525435216od_a_a @ X4 @ A2 ) )
= ( A2 = bot_bo3357376287454694259od_a_a ) ) ).
% all_not_in_conv
thf(fact_759_all__not__in__conv,axiom,
! [A2: set_a] :
( ( ! [X4: a] :
~ ( member_a @ X4 @ A2 ) )
= ( A2 = bot_bot_set_a ) ) ).
% all_not_in_conv
thf(fact_760_all__not__in__conv,axiom,
! [A2: set_set_a] :
( ( ! [X4: set_a] :
~ ( member_set_a @ X4 @ A2 ) )
= ( A2 = bot_bot_set_set_a ) ) ).
% all_not_in_conv
thf(fact_761_Collect__empty__eq,axiom,
! [P: product_prod_a_a > $o] :
( ( ( collec3336397797384452498od_a_a @ P )
= bot_bo3357376287454694259od_a_a )
= ( ! [X4: product_prod_a_a] :
~ ( P @ X4 ) ) ) ).
% Collect_empty_eq
thf(fact_762_Collect__empty__eq,axiom,
! [P: a > $o] :
( ( ( collect_a @ P )
= bot_bot_set_a )
= ( ! [X4: a] :
~ ( P @ X4 ) ) ) ).
% Collect_empty_eq
thf(fact_763_Collect__empty__eq,axiom,
! [P: set_a > $o] :
( ( ( collect_set_a @ P )
= bot_bot_set_set_a )
= ( ! [X4: set_a] :
~ ( P @ X4 ) ) ) ).
% Collect_empty_eq
thf(fact_764_empty__Collect__eq,axiom,
! [P: product_prod_a_a > $o] :
( ( bot_bo3357376287454694259od_a_a
= ( collec3336397797384452498od_a_a @ P ) )
= ( ! [X4: product_prod_a_a] :
~ ( P @ X4 ) ) ) ).
% empty_Collect_eq
thf(fact_765_empty__Collect__eq,axiom,
! [P: a > $o] :
( ( bot_bot_set_a
= ( collect_a @ P ) )
= ( ! [X4: a] :
~ ( P @ X4 ) ) ) ).
% empty_Collect_eq
thf(fact_766_empty__Collect__eq,axiom,
! [P: set_a > $o] :
( ( bot_bot_set_set_a
= ( collect_set_a @ P ) )
= ( ! [X4: set_a] :
~ ( P @ X4 ) ) ) ).
% empty_Collect_eq
thf(fact_767_IntI,axiom,
! [C: set_a,A2: set_set_a,B2: set_set_a] :
( ( member_set_a @ C @ A2 )
=> ( ( member_set_a @ C @ B2 )
=> ( member_set_a @ C @ ( inf_inf_set_set_a @ A2 @ B2 ) ) ) ) ).
% IntI
thf(fact_768_IntI,axiom,
! [C: product_prod_a_a,A2: set_Product_prod_a_a,B2: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ C @ A2 )
=> ( ( member1426531477525435216od_a_a @ C @ B2 )
=> ( member1426531477525435216od_a_a @ C @ ( inf_in8905007599844390133od_a_a @ A2 @ B2 ) ) ) ) ).
% IntI
thf(fact_769_IntI,axiom,
! [C: nat,A2: set_nat,B2: set_nat] :
( ( member_nat @ C @ A2 )
=> ( ( member_nat @ C @ B2 )
=> ( member_nat @ C @ ( inf_inf_set_nat @ A2 @ B2 ) ) ) ) ).
% IntI
thf(fact_770_IntI,axiom,
! [C: a,A2: set_a,B2: set_a] :
( ( member_a @ C @ A2 )
=> ( ( member_a @ C @ B2 )
=> ( member_a @ C @ ( inf_inf_set_a @ A2 @ B2 ) ) ) ) ).
% IntI
thf(fact_771_Int__iff,axiom,
! [C: set_a,A2: set_set_a,B2: set_set_a] :
( ( member_set_a @ C @ ( inf_inf_set_set_a @ A2 @ B2 ) )
= ( ( member_set_a @ C @ A2 )
& ( member_set_a @ C @ B2 ) ) ) ).
% Int_iff
thf(fact_772_Int__iff,axiom,
! [C: product_prod_a_a,A2: set_Product_prod_a_a,B2: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ C @ ( inf_in8905007599844390133od_a_a @ A2 @ B2 ) )
= ( ( member1426531477525435216od_a_a @ C @ A2 )
& ( member1426531477525435216od_a_a @ C @ B2 ) ) ) ).
% Int_iff
thf(fact_773_Int__iff,axiom,
! [C: nat,A2: set_nat,B2: set_nat] :
( ( member_nat @ C @ ( inf_inf_set_nat @ A2 @ B2 ) )
= ( ( member_nat @ C @ A2 )
& ( member_nat @ C @ B2 ) ) ) ).
% Int_iff
thf(fact_774_Int__iff,axiom,
! [C: a,A2: set_a,B2: set_a] :
( ( member_a @ C @ ( inf_inf_set_a @ A2 @ B2 ) )
= ( ( member_a @ C @ A2 )
& ( member_a @ C @ B2 ) ) ) ).
% Int_iff
thf(fact_775_edge__adj__def,axiom,
! [E1: set_a,E2: set_a] :
( ( undire4022703626023482010_adj_a @ edges @ E1 @ E2 )
= ( ( ( inf_inf_set_a @ E1 @ E2 )
!= bot_bot_set_a )
& ( member_set_a @ E1 @ edges )
& ( member_set_a @ E2 @ edges ) ) ) ).
% edge_adj_def
thf(fact_776_subset__empty,axiom,
! [A2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ bot_bot_set_set_a )
= ( A2 = bot_bot_set_set_a ) ) ).
% subset_empty
thf(fact_777_subset__empty,axiom,
! [A2: set_a] :
( ( ord_less_eq_set_a @ A2 @ bot_bot_set_a )
= ( A2 = bot_bot_set_a ) ) ).
% subset_empty
thf(fact_778_subset__empty,axiom,
! [A2: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A2 @ bot_bo3357376287454694259od_a_a )
= ( A2 = bot_bo3357376287454694259od_a_a ) ) ).
% subset_empty
thf(fact_779_empty__subsetI,axiom,
! [A2: set_set_a] : ( ord_le3724670747650509150_set_a @ bot_bot_set_set_a @ A2 ) ).
% empty_subsetI
thf(fact_780_empty__subsetI,axiom,
! [A2: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A2 ) ).
% empty_subsetI
thf(fact_781_empty__subsetI,axiom,
! [A2: set_Product_prod_a_a] : ( ord_le746702958409616551od_a_a @ bot_bo3357376287454694259od_a_a @ A2 ) ).
% empty_subsetI
thf(fact_782_Int__subset__iff,axiom,
! [C2: set_set_a,A2: set_set_a,B2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ C2 @ ( inf_inf_set_set_a @ A2 @ B2 ) )
= ( ( ord_le3724670747650509150_set_a @ C2 @ A2 )
& ( ord_le3724670747650509150_set_a @ C2 @ B2 ) ) ) ).
% Int_subset_iff
thf(fact_783_Int__subset__iff,axiom,
! [C2: set_a,A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ C2 @ ( inf_inf_set_a @ A2 @ B2 ) )
= ( ( ord_less_eq_set_a @ C2 @ A2 )
& ( ord_less_eq_set_a @ C2 @ B2 ) ) ) ).
% Int_subset_iff
thf(fact_784_Int__subset__iff,axiom,
! [C2: set_Product_prod_a_a,A2: set_Product_prod_a_a,B2: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ C2 @ ( inf_in8905007599844390133od_a_a @ A2 @ B2 ) )
= ( ( ord_le746702958409616551od_a_a @ C2 @ A2 )
& ( ord_le746702958409616551od_a_a @ C2 @ B2 ) ) ) ).
% Int_subset_iff
thf(fact_785_length__0__conv,axiom,
! [Xs: list_set_a] :
( ( ( size_size_list_set_a @ Xs )
= zero_zero_nat )
= ( Xs = nil_set_a ) ) ).
% length_0_conv
thf(fact_786_length__0__conv,axiom,
! [Xs: list_a] :
( ( ( size_size_list_a @ Xs )
= zero_zero_nat )
= ( Xs = nil_a ) ) ).
% length_0_conv
thf(fact_787_set__empty,axiom,
! [Xs: list_P1396940483166286381od_a_a] :
( ( ( set_Product_prod_a_a2 @ Xs )
= bot_bo3357376287454694259od_a_a )
= ( Xs = nil_Product_prod_a_a ) ) ).
% set_empty
thf(fact_788_set__empty,axiom,
! [Xs: list_a] :
( ( ( set_a2 @ Xs )
= bot_bot_set_a )
= ( Xs = nil_a ) ) ).
% set_empty
thf(fact_789_set__empty,axiom,
! [Xs: list_set_a] :
( ( ( set_set_a2 @ Xs )
= bot_bot_set_set_a )
= ( Xs = nil_set_a ) ) ).
% set_empty
thf(fact_790_set__empty2,axiom,
! [Xs: list_P1396940483166286381od_a_a] :
( ( bot_bo3357376287454694259od_a_a
= ( set_Product_prod_a_a2 @ Xs ) )
= ( Xs = nil_Product_prod_a_a ) ) ).
% set_empty2
thf(fact_791_set__empty2,axiom,
! [Xs: list_a] :
( ( bot_bot_set_a
= ( set_a2 @ Xs ) )
= ( Xs = nil_a ) ) ).
% set_empty2
thf(fact_792_set__empty2,axiom,
! [Xs: list_set_a] :
( ( bot_bot_set_set_a
= ( set_set_a2 @ Xs ) )
= ( Xs = nil_set_a ) ) ).
% set_empty2
thf(fact_793_rotate1__length01,axiom,
! [Xs: list_set_a] :
( ( ord_less_eq_nat @ ( size_size_list_set_a @ Xs ) @ one_one_nat )
=> ( ( rotate1_set_a @ Xs )
= Xs ) ) ).
% rotate1_length01
thf(fact_794_rotate1__length01,axiom,
! [Xs: list_a] :
( ( ord_less_eq_nat @ ( size_size_list_a @ Xs ) @ one_one_nat )
=> ( ( rotate1_a @ Xs )
= Xs ) ) ).
% rotate1_length01
thf(fact_795_comp__sgraph_Oall__edges__between__empty_I1_J,axiom,
! [S: set_Product_prod_a_a,Z4: set_Product_prod_a_a] :
( ( undire4032395788819567636od_a_a @ ( undire6879232364018543115od_a_a @ S ) @ bot_bo3357376287454694259od_a_a @ Z4 )
= bot_bo510284599550014259od_a_a ) ).
% comp_sgraph.all_edges_between_empty(1)
thf(fact_796_comp__sgraph_Oall__edges__between__empty_I1_J,axiom,
! [S: set_set_a,Z4: set_set_a] :
( ( undire2462398226299384907_set_a @ ( undire8247866692393712962_set_a @ S ) @ bot_bot_set_set_a @ Z4 )
= bot_bo5799363139946352499_set_a ) ).
% comp_sgraph.all_edges_between_empty(1)
thf(fact_797_comp__sgraph_Oall__edges__between__empty_I1_J,axiom,
! [S: set_a,Z4: set_a] :
( ( undire8383842906760478443ween_a @ ( undire2918257014606996450dges_a @ S ) @ bot_bot_set_a @ Z4 )
= bot_bo3357376287454694259od_a_a ) ).
% comp_sgraph.all_edges_between_empty(1)
thf(fact_798_comp__sgraph_Oall__edges__between__empty_I2_J,axiom,
! [S: set_Product_prod_a_a,Z4: set_Product_prod_a_a] :
( ( undire4032395788819567636od_a_a @ ( undire6879232364018543115od_a_a @ S ) @ Z4 @ bot_bo3357376287454694259od_a_a )
= bot_bo510284599550014259od_a_a ) ).
% comp_sgraph.all_edges_between_empty(2)
thf(fact_799_comp__sgraph_Oall__edges__between__empty_I2_J,axiom,
! [S: set_set_a,Z4: set_set_a] :
( ( undire2462398226299384907_set_a @ ( undire8247866692393712962_set_a @ S ) @ Z4 @ bot_bot_set_set_a )
= bot_bo5799363139946352499_set_a ) ).
% comp_sgraph.all_edges_between_empty(2)
thf(fact_800_comp__sgraph_Oall__edges__between__empty_I2_J,axiom,
! [S: set_a,Z4: set_a] :
( ( undire8383842906760478443ween_a @ ( undire2918257014606996450dges_a @ S ) @ Z4 @ bot_bot_set_a )
= bot_bo3357376287454694259od_a_a ) ).
% comp_sgraph.all_edges_between_empty(2)
thf(fact_801_all__edges__between__empty_I1_J,axiom,
! [Z4: set_a] :
( ( undire8383842906760478443ween_a @ edges @ bot_bot_set_a @ Z4 )
= bot_bo3357376287454694259od_a_a ) ).
% all_edges_between_empty(1)
thf(fact_802_all__edges__between__empty_I2_J,axiom,
! [Z4: set_a] :
( ( undire8383842906760478443ween_a @ edges @ Z4 @ bot_bot_set_a )
= bot_bo3357376287454694259od_a_a ) ).
% all_edges_between_empty(2)
thf(fact_803_IntE,axiom,
! [C: set_a,A2: set_set_a,B2: set_set_a] :
( ( member_set_a @ C @ ( inf_inf_set_set_a @ A2 @ B2 ) )
=> ~ ( ( member_set_a @ C @ A2 )
=> ~ ( member_set_a @ C @ B2 ) ) ) ).
% IntE
thf(fact_804_IntE,axiom,
! [C: product_prod_a_a,A2: set_Product_prod_a_a,B2: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ C @ ( inf_in8905007599844390133od_a_a @ A2 @ B2 ) )
=> ~ ( ( member1426531477525435216od_a_a @ C @ A2 )
=> ~ ( member1426531477525435216od_a_a @ C @ B2 ) ) ) ).
% IntE
thf(fact_805_IntE,axiom,
! [C: nat,A2: set_nat,B2: set_nat] :
( ( member_nat @ C @ ( inf_inf_set_nat @ A2 @ B2 ) )
=> ~ ( ( member_nat @ C @ A2 )
=> ~ ( member_nat @ C @ B2 ) ) ) ).
% IntE
thf(fact_806_IntE,axiom,
! [C: a,A2: set_a,B2: set_a] :
( ( member_a @ C @ ( inf_inf_set_a @ A2 @ B2 ) )
=> ~ ( ( member_a @ C @ A2 )
=> ~ ( member_a @ C @ B2 ) ) ) ).
% IntE
thf(fact_807_IntD1,axiom,
! [C: set_a,A2: set_set_a,B2: set_set_a] :
( ( member_set_a @ C @ ( inf_inf_set_set_a @ A2 @ B2 ) )
=> ( member_set_a @ C @ A2 ) ) ).
% IntD1
thf(fact_808_IntD1,axiom,
! [C: product_prod_a_a,A2: set_Product_prod_a_a,B2: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ C @ ( inf_in8905007599844390133od_a_a @ A2 @ B2 ) )
=> ( member1426531477525435216od_a_a @ C @ A2 ) ) ).
% IntD1
thf(fact_809_IntD1,axiom,
! [C: nat,A2: set_nat,B2: set_nat] :
( ( member_nat @ C @ ( inf_inf_set_nat @ A2 @ B2 ) )
=> ( member_nat @ C @ A2 ) ) ).
% IntD1
thf(fact_810_IntD1,axiom,
! [C: a,A2: set_a,B2: set_a] :
( ( member_a @ C @ ( inf_inf_set_a @ A2 @ B2 ) )
=> ( member_a @ C @ A2 ) ) ).
% IntD1
thf(fact_811_IntD2,axiom,
! [C: set_a,A2: set_set_a,B2: set_set_a] :
( ( member_set_a @ C @ ( inf_inf_set_set_a @ A2 @ B2 ) )
=> ( member_set_a @ C @ B2 ) ) ).
% IntD2
thf(fact_812_IntD2,axiom,
! [C: product_prod_a_a,A2: set_Product_prod_a_a,B2: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ C @ ( inf_in8905007599844390133od_a_a @ A2 @ B2 ) )
=> ( member1426531477525435216od_a_a @ C @ B2 ) ) ).
% IntD2
thf(fact_813_IntD2,axiom,
! [C: nat,A2: set_nat,B2: set_nat] :
( ( member_nat @ C @ ( inf_inf_set_nat @ A2 @ B2 ) )
=> ( member_nat @ C @ B2 ) ) ).
% IntD2
thf(fact_814_IntD2,axiom,
! [C: a,A2: set_a,B2: set_a] :
( ( member_a @ C @ ( inf_inf_set_a @ A2 @ B2 ) )
=> ( member_a @ C @ B2 ) ) ).
% IntD2
thf(fact_815_emptyE,axiom,
! [A: nat] :
~ ( member_nat @ A @ bot_bot_set_nat ) ).
% emptyE
thf(fact_816_emptyE,axiom,
! [A: product_prod_a_a] :
~ ( member1426531477525435216od_a_a @ A @ bot_bo3357376287454694259od_a_a ) ).
% emptyE
thf(fact_817_emptyE,axiom,
! [A: a] :
~ ( member_a @ A @ bot_bot_set_a ) ).
% emptyE
thf(fact_818_emptyE,axiom,
! [A: set_a] :
~ ( member_set_a @ A @ bot_bot_set_set_a ) ).
% emptyE
thf(fact_819_equals0D,axiom,
! [A2: set_nat,A: nat] :
( ( A2 = bot_bot_set_nat )
=> ~ ( member_nat @ A @ A2 ) ) ).
% equals0D
thf(fact_820_equals0D,axiom,
! [A2: set_Product_prod_a_a,A: product_prod_a_a] :
( ( A2 = bot_bo3357376287454694259od_a_a )
=> ~ ( member1426531477525435216od_a_a @ A @ A2 ) ) ).
% equals0D
thf(fact_821_equals0D,axiom,
! [A2: set_a,A: a] :
( ( A2 = bot_bot_set_a )
=> ~ ( member_a @ A @ A2 ) ) ).
% equals0D
thf(fact_822_equals0D,axiom,
! [A2: set_set_a,A: set_a] :
( ( A2 = bot_bot_set_set_a )
=> ~ ( member_set_a @ A @ A2 ) ) ).
% equals0D
thf(fact_823_equals0I,axiom,
! [A2: set_nat] :
( ! [Y: nat] :
~ ( member_nat @ Y @ A2 )
=> ( A2 = bot_bot_set_nat ) ) ).
% equals0I
thf(fact_824_equals0I,axiom,
! [A2: set_Product_prod_a_a] :
( ! [Y: product_prod_a_a] :
~ ( member1426531477525435216od_a_a @ Y @ A2 )
=> ( A2 = bot_bo3357376287454694259od_a_a ) ) ).
% equals0I
thf(fact_825_equals0I,axiom,
! [A2: set_a] :
( ! [Y: a] :
~ ( member_a @ Y @ A2 )
=> ( A2 = bot_bot_set_a ) ) ).
% equals0I
thf(fact_826_equals0I,axiom,
! [A2: set_set_a] :
( ! [Y: set_a] :
~ ( member_set_a @ Y @ A2 )
=> ( A2 = bot_bot_set_set_a ) ) ).
% equals0I
thf(fact_827_Int__assoc,axiom,
! [A2: set_a,B2: set_a,C2: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ A2 @ B2 ) @ C2 )
= ( inf_inf_set_a @ A2 @ ( inf_inf_set_a @ B2 @ C2 ) ) ) ).
% Int_assoc
thf(fact_828_Int__absorb,axiom,
! [A2: set_a] :
( ( inf_inf_set_a @ A2 @ A2 )
= A2 ) ).
% Int_absorb
thf(fact_829_Int__emptyI,axiom,
! [A2: set_nat,B2: set_nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ~ ( member_nat @ X2 @ B2 ) )
=> ( ( inf_inf_set_nat @ A2 @ B2 )
= bot_bot_set_nat ) ) ).
% Int_emptyI
thf(fact_830_Int__emptyI,axiom,
! [A2: set_Product_prod_a_a,B2: set_Product_prod_a_a] :
( ! [X2: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X2 @ A2 )
=> ~ ( member1426531477525435216od_a_a @ X2 @ B2 ) )
=> ( ( inf_in8905007599844390133od_a_a @ A2 @ B2 )
= bot_bo3357376287454694259od_a_a ) ) ).
% Int_emptyI
thf(fact_831_Int__emptyI,axiom,
! [A2: set_a,B2: set_a] :
( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ~ ( member_a @ X2 @ B2 ) )
=> ( ( inf_inf_set_a @ A2 @ B2 )
= bot_bot_set_a ) ) ).
% Int_emptyI
thf(fact_832_Int__emptyI,axiom,
! [A2: set_set_a,B2: set_set_a] :
( ! [X2: set_a] :
( ( member_set_a @ X2 @ A2 )
=> ~ ( member_set_a @ X2 @ B2 ) )
=> ( ( inf_inf_set_set_a @ A2 @ B2 )
= bot_bot_set_set_a ) ) ).
% Int_emptyI
thf(fact_833_ex__in__conv,axiom,
! [A2: set_nat] :
( ( ? [X4: nat] : ( member_nat @ X4 @ A2 ) )
= ( A2 != bot_bot_set_nat ) ) ).
% ex_in_conv
thf(fact_834_ex__in__conv,axiom,
! [A2: set_Product_prod_a_a] :
( ( ? [X4: product_prod_a_a] : ( member1426531477525435216od_a_a @ X4 @ A2 ) )
= ( A2 != bot_bo3357376287454694259od_a_a ) ) ).
% ex_in_conv
thf(fact_835_ex__in__conv,axiom,
! [A2: set_a] :
( ( ? [X4: a] : ( member_a @ X4 @ A2 ) )
= ( A2 != bot_bot_set_a ) ) ).
% ex_in_conv
thf(fact_836_ex__in__conv,axiom,
! [A2: set_set_a] :
( ( ? [X4: set_a] : ( member_set_a @ X4 @ A2 ) )
= ( A2 != bot_bot_set_set_a ) ) ).
% ex_in_conv
thf(fact_837_Int__commute,axiom,
( inf_inf_set_a
= ( ^ [A3: set_a,B3: set_a] : ( inf_inf_set_a @ B3 @ A3 ) ) ) ).
% Int_commute
thf(fact_838_disjoint__iff,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ( inf_inf_set_nat @ A2 @ B2 )
= bot_bot_set_nat )
= ( ! [X4: nat] :
( ( member_nat @ X4 @ A2 )
=> ~ ( member_nat @ X4 @ B2 ) ) ) ) ).
% disjoint_iff
thf(fact_839_disjoint__iff,axiom,
! [A2: set_Product_prod_a_a,B2: set_Product_prod_a_a] :
( ( ( inf_in8905007599844390133od_a_a @ A2 @ B2 )
= bot_bo3357376287454694259od_a_a )
= ( ! [X4: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X4 @ A2 )
=> ~ ( member1426531477525435216od_a_a @ X4 @ B2 ) ) ) ) ).
% disjoint_iff
thf(fact_840_disjoint__iff,axiom,
! [A2: set_a,B2: set_a] :
( ( ( inf_inf_set_a @ A2 @ B2 )
= bot_bot_set_a )
= ( ! [X4: a] :
( ( member_a @ X4 @ A2 )
=> ~ ( member_a @ X4 @ B2 ) ) ) ) ).
% disjoint_iff
thf(fact_841_disjoint__iff,axiom,
! [A2: set_set_a,B2: set_set_a] :
( ( ( inf_inf_set_set_a @ A2 @ B2 )
= bot_bot_set_set_a )
= ( ! [X4: set_a] :
( ( member_set_a @ X4 @ A2 )
=> ~ ( member_set_a @ X4 @ B2 ) ) ) ) ).
% disjoint_iff
thf(fact_842_Int__empty__left,axiom,
! [B2: set_Product_prod_a_a] :
( ( inf_in8905007599844390133od_a_a @ bot_bo3357376287454694259od_a_a @ B2 )
= bot_bo3357376287454694259od_a_a ) ).
% Int_empty_left
thf(fact_843_Int__empty__left,axiom,
! [B2: set_a] :
( ( inf_inf_set_a @ bot_bot_set_a @ B2 )
= bot_bot_set_a ) ).
% Int_empty_left
thf(fact_844_Int__empty__left,axiom,
! [B2: set_set_a] :
( ( inf_inf_set_set_a @ bot_bot_set_set_a @ B2 )
= bot_bot_set_set_a ) ).
% Int_empty_left
thf(fact_845_Int__empty__right,axiom,
! [A2: set_Product_prod_a_a] :
( ( inf_in8905007599844390133od_a_a @ A2 @ bot_bo3357376287454694259od_a_a )
= bot_bo3357376287454694259od_a_a ) ).
% Int_empty_right
thf(fact_846_Int__empty__right,axiom,
! [A2: set_a] :
( ( inf_inf_set_a @ A2 @ bot_bot_set_a )
= bot_bot_set_a ) ).
% Int_empty_right
thf(fact_847_Int__empty__right,axiom,
! [A2: set_set_a] :
( ( inf_inf_set_set_a @ A2 @ bot_bot_set_set_a )
= bot_bot_set_set_a ) ).
% Int_empty_right
thf(fact_848_Int__left__absorb,axiom,
! [A2: set_a,B2: set_a] :
( ( inf_inf_set_a @ A2 @ ( inf_inf_set_a @ A2 @ B2 ) )
= ( inf_inf_set_a @ A2 @ B2 ) ) ).
% Int_left_absorb
thf(fact_849_Int__left__commute,axiom,
! [A2: set_a,B2: set_a,C2: set_a] :
( ( inf_inf_set_a @ A2 @ ( inf_inf_set_a @ B2 @ C2 ) )
= ( inf_inf_set_a @ B2 @ ( inf_inf_set_a @ A2 @ C2 ) ) ) ).
% Int_left_commute
thf(fact_850_disjoint__iff__not__equal,axiom,
! [A2: set_Product_prod_a_a,B2: set_Product_prod_a_a] :
( ( ( inf_in8905007599844390133od_a_a @ A2 @ B2 )
= bot_bo3357376287454694259od_a_a )
= ( ! [X4: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X4 @ A2 )
=> ! [Y3: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ Y3 @ B2 )
=> ( X4 != Y3 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_851_disjoint__iff__not__equal,axiom,
! [A2: set_a,B2: set_a] :
( ( ( inf_inf_set_a @ A2 @ B2 )
= bot_bot_set_a )
= ( ! [X4: a] :
( ( member_a @ X4 @ A2 )
=> ! [Y3: a] :
( ( member_a @ Y3 @ B2 )
=> ( X4 != Y3 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_852_disjoint__iff__not__equal,axiom,
! [A2: set_set_a,B2: set_set_a] :
( ( ( inf_inf_set_set_a @ A2 @ B2 )
= bot_bot_set_set_a )
= ( ! [X4: set_a] :
( ( member_set_a @ X4 @ A2 )
=> ! [Y3: set_a] :
( ( member_set_a @ Y3 @ B2 )
=> ( X4 != Y3 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_853_all__edges__disjoint,axiom,
! [S: set_Product_prod_a_a,T2: set_Product_prod_a_a] :
( ( ( inf_in8905007599844390133od_a_a @ S @ T2 )
= bot_bo3357376287454694259od_a_a )
=> ( ( inf_in3339382566020358357od_a_a @ ( undire6879232364018543115od_a_a @ S ) @ ( undire6879232364018543115od_a_a @ T2 ) )
= bot_bo777872063958040403od_a_a ) ) ).
% all_edges_disjoint
thf(fact_854_all__edges__disjoint,axiom,
! [S: set_a,T2: set_a] :
( ( ( inf_inf_set_a @ S @ T2 )
= bot_bot_set_a )
=> ( ( inf_inf_set_set_a @ ( undire2918257014606996450dges_a @ S ) @ ( undire2918257014606996450dges_a @ T2 ) )
= bot_bot_set_set_a ) ) ).
% all_edges_disjoint
thf(fact_855_all__edges__disjoint,axiom,
! [S: set_set_a,T2: set_set_a] :
( ( ( inf_inf_set_set_a @ S @ T2 )
= bot_bot_set_set_a )
=> ( ( inf_in1205276777018777868_set_a @ ( undire8247866692393712962_set_a @ S ) @ ( undire8247866692393712962_set_a @ T2 ) )
= bot_bo3380559777022489994_set_a ) ) ).
% all_edges_disjoint
thf(fact_856_bot_Oextremum__uniqueI,axiom,
! [A: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ bot_bot_set_set_a )
=> ( A = bot_bot_set_set_a ) ) ).
% bot.extremum_uniqueI
thf(fact_857_bot_Oextremum__uniqueI,axiom,
! [A: set_a] :
( ( ord_less_eq_set_a @ A @ bot_bot_set_a )
=> ( A = bot_bot_set_a ) ) ).
% bot.extremum_uniqueI
thf(fact_858_bot_Oextremum__uniqueI,axiom,
! [A: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A @ bot_bo3357376287454694259od_a_a )
=> ( A = bot_bo3357376287454694259od_a_a ) ) ).
% bot.extremum_uniqueI
thf(fact_859_bot_Oextremum__uniqueI,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ bot_bot_nat )
=> ( A = bot_bot_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_860_bot_Oextremum__unique,axiom,
! [A: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ bot_bot_set_set_a )
= ( A = bot_bot_set_set_a ) ) ).
% bot.extremum_unique
thf(fact_861_bot_Oextremum__unique,axiom,
! [A: set_a] :
( ( ord_less_eq_set_a @ A @ bot_bot_set_a )
= ( A = bot_bot_set_a ) ) ).
% bot.extremum_unique
thf(fact_862_bot_Oextremum__unique,axiom,
! [A: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A @ bot_bo3357376287454694259od_a_a )
= ( A = bot_bo3357376287454694259od_a_a ) ) ).
% bot.extremum_unique
thf(fact_863_bot_Oextremum__unique,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ bot_bot_nat )
= ( A = bot_bot_nat ) ) ).
% bot.extremum_unique
thf(fact_864_bot_Oextremum,axiom,
! [A: set_set_a] : ( ord_le3724670747650509150_set_a @ bot_bot_set_set_a @ A ) ).
% bot.extremum
thf(fact_865_bot_Oextremum,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A ) ).
% bot.extremum
thf(fact_866_bot_Oextremum,axiom,
! [A: set_Product_prod_a_a] : ( ord_le746702958409616551od_a_a @ bot_bo3357376287454694259od_a_a @ A ) ).
% bot.extremum
thf(fact_867_bot_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ bot_bot_nat @ A ) ).
% bot.extremum
thf(fact_868_Int__mono,axiom,
! [A2: set_set_a,C2: set_set_a,B2: set_set_a,D: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ C2 )
=> ( ( ord_le3724670747650509150_set_a @ B2 @ D )
=> ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ A2 @ B2 ) @ ( inf_inf_set_set_a @ C2 @ D ) ) ) ) ).
% Int_mono
thf(fact_869_Int__mono,axiom,
! [A2: set_a,C2: set_a,B2: set_a,D: set_a] :
( ( ord_less_eq_set_a @ A2 @ C2 )
=> ( ( ord_less_eq_set_a @ B2 @ D )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B2 ) @ ( inf_inf_set_a @ C2 @ D ) ) ) ) ).
% Int_mono
thf(fact_870_Int__mono,axiom,
! [A2: set_Product_prod_a_a,C2: set_Product_prod_a_a,B2: set_Product_prod_a_a,D: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A2 @ C2 )
=> ( ( ord_le746702958409616551od_a_a @ B2 @ D )
=> ( ord_le746702958409616551od_a_a @ ( inf_in8905007599844390133od_a_a @ A2 @ B2 ) @ ( inf_in8905007599844390133od_a_a @ C2 @ D ) ) ) ) ).
% Int_mono
thf(fact_871_Int__lower1,axiom,
! [A2: set_set_a,B2: set_set_a] : ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ A2 @ B2 ) @ A2 ) ).
% Int_lower1
thf(fact_872_Int__lower1,axiom,
! [A2: set_a,B2: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B2 ) @ A2 ) ).
% Int_lower1
thf(fact_873_Int__lower1,axiom,
! [A2: set_Product_prod_a_a,B2: set_Product_prod_a_a] : ( ord_le746702958409616551od_a_a @ ( inf_in8905007599844390133od_a_a @ A2 @ B2 ) @ A2 ) ).
% Int_lower1
thf(fact_874_Int__lower2,axiom,
! [A2: set_set_a,B2: set_set_a] : ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ A2 @ B2 ) @ B2 ) ).
% Int_lower2
thf(fact_875_Int__lower2,axiom,
! [A2: set_a,B2: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B2 ) @ B2 ) ).
% Int_lower2
thf(fact_876_Int__lower2,axiom,
! [A2: set_Product_prod_a_a,B2: set_Product_prod_a_a] : ( ord_le746702958409616551od_a_a @ ( inf_in8905007599844390133od_a_a @ A2 @ B2 ) @ B2 ) ).
% Int_lower2
thf(fact_877_Int__absorb1,axiom,
! [B2: set_set_a,A2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B2 @ A2 )
=> ( ( inf_inf_set_set_a @ A2 @ B2 )
= B2 ) ) ).
% Int_absorb1
thf(fact_878_Int__absorb1,axiom,
! [B2: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ B2 @ A2 )
=> ( ( inf_inf_set_a @ A2 @ B2 )
= B2 ) ) ).
% Int_absorb1
thf(fact_879_Int__absorb1,axiom,
! [B2: set_Product_prod_a_a,A2: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ B2 @ A2 )
=> ( ( inf_in8905007599844390133od_a_a @ A2 @ B2 )
= B2 ) ) ).
% Int_absorb1
thf(fact_880_Int__absorb2,axiom,
! [A2: set_set_a,B2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B2 )
=> ( ( inf_inf_set_set_a @ A2 @ B2 )
= A2 ) ) ).
% Int_absorb2
thf(fact_881_Int__absorb2,axiom,
! [A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( inf_inf_set_a @ A2 @ B2 )
= A2 ) ) ).
% Int_absorb2
thf(fact_882_Int__absorb2,axiom,
! [A2: set_Product_prod_a_a,B2: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A2 @ B2 )
=> ( ( inf_in8905007599844390133od_a_a @ A2 @ B2 )
= A2 ) ) ).
% Int_absorb2
thf(fact_883_Int__greatest,axiom,
! [C2: set_set_a,A2: set_set_a,B2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ C2 @ A2 )
=> ( ( ord_le3724670747650509150_set_a @ C2 @ B2 )
=> ( ord_le3724670747650509150_set_a @ C2 @ ( inf_inf_set_set_a @ A2 @ B2 ) ) ) ) ).
% Int_greatest
thf(fact_884_Int__greatest,axiom,
! [C2: set_a,A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ C2 @ A2 )
=> ( ( ord_less_eq_set_a @ C2 @ B2 )
=> ( ord_less_eq_set_a @ C2 @ ( inf_inf_set_a @ A2 @ B2 ) ) ) ) ).
% Int_greatest
thf(fact_885_Int__greatest,axiom,
! [C2: set_Product_prod_a_a,A2: set_Product_prod_a_a,B2: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ C2 @ A2 )
=> ( ( ord_le746702958409616551od_a_a @ C2 @ B2 )
=> ( ord_le746702958409616551od_a_a @ C2 @ ( inf_in8905007599844390133od_a_a @ A2 @ B2 ) ) ) ) ).
% Int_greatest
thf(fact_886_Int__Collect__mono,axiom,
! [A2: set_nat,B2: set_nat,P: nat > $o,Q: nat > $o] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( ( P @ X2 )
=> ( Q @ X2 ) ) )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A2 @ ( collect_nat @ P ) ) @ ( inf_inf_set_nat @ B2 @ ( collect_nat @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_887_Int__Collect__mono,axiom,
! [A2: set_set_a,B2: set_set_a,P: set_a > $o,Q: set_a > $o] :
( ( ord_le3724670747650509150_set_a @ A2 @ B2 )
=> ( ! [X2: set_a] :
( ( member_set_a @ X2 @ A2 )
=> ( ( P @ X2 )
=> ( Q @ X2 ) ) )
=> ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ A2 @ ( collect_set_a @ P ) ) @ ( inf_inf_set_set_a @ B2 @ ( collect_set_a @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_888_Int__Collect__mono,axiom,
! [A2: set_a,B2: set_a,P: a > $o,Q: a > $o] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ( ( P @ X2 )
=> ( Q @ X2 ) ) )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ ( collect_a @ P ) ) @ ( inf_inf_set_a @ B2 @ ( collect_a @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_889_Int__Collect__mono,axiom,
! [A2: set_Product_prod_a_a,B2: set_Product_prod_a_a,P: product_prod_a_a > $o,Q: product_prod_a_a > $o] :
( ( ord_le746702958409616551od_a_a @ A2 @ B2 )
=> ( ! [X2: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X2 @ A2 )
=> ( ( P @ X2 )
=> ( Q @ X2 ) ) )
=> ( ord_le746702958409616551od_a_a @ ( inf_in8905007599844390133od_a_a @ A2 @ ( collec3336397797384452498od_a_a @ P ) ) @ ( inf_in8905007599844390133od_a_a @ B2 @ ( collec3336397797384452498od_a_a @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_890_ulgraph_Oall__edges__between__empty_I1_J,axiom,
! [Vertices: set_Product_prod_a_a,Edges: set_se5735800977113168103od_a_a,Z4: set_Product_prod_a_a] :
( ( undire4585262585102564309od_a_a @ Vertices @ Edges )
=> ( ( undire4032395788819567636od_a_a @ Edges @ bot_bo3357376287454694259od_a_a @ Z4 )
= bot_bo510284599550014259od_a_a ) ) ).
% ulgraph.all_edges_between_empty(1)
thf(fact_891_ulgraph_Oall__edges__between__empty_I1_J,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,Z4: set_set_a] :
( ( undire6886684016831807756_set_a @ Vertices @ Edges )
=> ( ( undire2462398226299384907_set_a @ Edges @ bot_bot_set_set_a @ Z4 )
= bot_bo5799363139946352499_set_a ) ) ).
% ulgraph.all_edges_between_empty(1)
thf(fact_892_ulgraph_Oall__edges__between__empty_I1_J,axiom,
! [Vertices: set_a,Edges: set_set_a,Z4: set_a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( undire8383842906760478443ween_a @ Edges @ bot_bot_set_a @ Z4 )
= bot_bo3357376287454694259od_a_a ) ) ).
% ulgraph.all_edges_between_empty(1)
thf(fact_893_ulgraph_Oall__edges__between__empty_I2_J,axiom,
! [Vertices: set_Product_prod_a_a,Edges: set_se5735800977113168103od_a_a,Z4: set_Product_prod_a_a] :
( ( undire4585262585102564309od_a_a @ Vertices @ Edges )
=> ( ( undire4032395788819567636od_a_a @ Edges @ Z4 @ bot_bo3357376287454694259od_a_a )
= bot_bo510284599550014259od_a_a ) ) ).
% ulgraph.all_edges_between_empty(2)
thf(fact_894_ulgraph_Oall__edges__between__empty_I2_J,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,Z4: set_set_a] :
( ( undire6886684016831807756_set_a @ Vertices @ Edges )
=> ( ( undire2462398226299384907_set_a @ Edges @ Z4 @ bot_bot_set_set_a )
= bot_bo5799363139946352499_set_a ) ) ).
% ulgraph.all_edges_between_empty(2)
thf(fact_895_ulgraph_Oall__edges__between__empty_I2_J,axiom,
! [Vertices: set_a,Edges: set_set_a,Z4: set_a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( undire8383842906760478443ween_a @ Edges @ Z4 @ bot_bot_set_a )
= bot_bo3357376287454694259od_a_a ) ) ).
% ulgraph.all_edges_between_empty(2)
thf(fact_896_comp__sgraph_Oedge__adj__def,axiom,
! [S: set_Product_prod_a_a,E1: set_Product_prod_a_a,E2: set_Product_prod_a_a] :
( ( undire9186443406341554371od_a_a @ ( undire6879232364018543115od_a_a @ S ) @ E1 @ E2 )
= ( ( ( inf_in8905007599844390133od_a_a @ E1 @ E2 )
!= bot_bo3357376287454694259od_a_a )
& ( member1816616512716248880od_a_a @ E1 @ ( undire6879232364018543115od_a_a @ S ) )
& ( member1816616512716248880od_a_a @ E2 @ ( undire6879232364018543115od_a_a @ S ) ) ) ) ).
% comp_sgraph.edge_adj_def
thf(fact_897_comp__sgraph_Oedge__adj__def,axiom,
! [S: set_set_a,E1: set_set_a,E2: set_set_a] :
( ( undire3485422320110889978_set_a @ ( undire8247866692393712962_set_a @ S ) @ E1 @ E2 )
= ( ( ( inf_inf_set_set_a @ E1 @ E2 )
!= bot_bot_set_set_a )
& ( member_set_set_a @ E1 @ ( undire8247866692393712962_set_a @ S ) )
& ( member_set_set_a @ E2 @ ( undire8247866692393712962_set_a @ S ) ) ) ) ).
% comp_sgraph.edge_adj_def
thf(fact_898_comp__sgraph_Oedge__adj__def,axiom,
! [S: set_a,E1: set_a,E2: set_a] :
( ( undire4022703626023482010_adj_a @ ( undire2918257014606996450dges_a @ S ) @ E1 @ E2 )
= ( ( ( inf_inf_set_a @ E1 @ E2 )
!= bot_bot_set_a )
& ( member_set_a @ E1 @ ( undire2918257014606996450dges_a @ S ) )
& ( member_set_a @ E2 @ ( undire2918257014606996450dges_a @ S ) ) ) ) ).
% comp_sgraph.edge_adj_def
thf(fact_899_graph__system_Oedge__adj__def,axiom,
! [Vertices: set_Product_prod_a_a,Edges: set_se5735800977113168103od_a_a,E1: set_Product_prod_a_a,E2: set_Product_prod_a_a] :
( ( undire1860116983885411791od_a_a @ Vertices @ Edges )
=> ( ( undire9186443406341554371od_a_a @ Edges @ E1 @ E2 )
= ( ( ( inf_in8905007599844390133od_a_a @ E1 @ E2 )
!= bot_bo3357376287454694259od_a_a )
& ( member1816616512716248880od_a_a @ E1 @ Edges )
& ( member1816616512716248880od_a_a @ E2 @ Edges ) ) ) ) ).
% graph_system.edge_adj_def
thf(fact_900_graph__system_Oedge__adj__def,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,E1: set_set_a,E2: set_set_a] :
( ( undire7159349782766787846_set_a @ Vertices @ Edges )
=> ( ( undire3485422320110889978_set_a @ Edges @ E1 @ E2 )
= ( ( ( inf_inf_set_set_a @ E1 @ E2 )
!= bot_bot_set_set_a )
& ( member_set_set_a @ E1 @ Edges )
& ( member_set_set_a @ E2 @ Edges ) ) ) ) ).
% graph_system.edge_adj_def
thf(fact_901_graph__system_Oedge__adj__def,axiom,
! [Vertices: set_a,Edges: set_set_a,E1: set_a,E2: set_a] :
( ( undire2554140024507503526stem_a @ Vertices @ Edges )
=> ( ( undire4022703626023482010_adj_a @ Edges @ E1 @ E2 )
= ( ( ( inf_inf_set_a @ E1 @ E2 )
!= bot_bot_set_a )
& ( member_set_a @ E1 @ Edges )
& ( member_set_a @ E2 @ Edges ) ) ) ) ).
% graph_system.edge_adj_def
thf(fact_902_comp__sgraph_Oempty__not__edge,axiom,
! [S: set_Product_prod_a_a] :
~ ( member1816616512716248880od_a_a @ bot_bo3357376287454694259od_a_a @ ( undire6879232364018543115od_a_a @ S ) ) ).
% comp_sgraph.empty_not_edge
thf(fact_903_comp__sgraph_Oempty__not__edge,axiom,
! [S: set_a] :
~ ( member_set_a @ bot_bot_set_a @ ( undire2918257014606996450dges_a @ S ) ) ).
% comp_sgraph.empty_not_edge
thf(fact_904_comp__sgraph_Oempty__not__edge,axiom,
! [S: set_set_a] :
~ ( member_set_set_a @ bot_bot_set_set_a @ ( undire8247866692393712962_set_a @ S ) ) ).
% comp_sgraph.empty_not_edge
thf(fact_905_ulgraph_Oempty__not__edge,axiom,
! [Vertices: set_Product_prod_a_a,Edges: set_se5735800977113168103od_a_a] :
( ( undire4585262585102564309od_a_a @ Vertices @ Edges )
=> ~ ( member1816616512716248880od_a_a @ bot_bo3357376287454694259od_a_a @ Edges ) ) ).
% ulgraph.empty_not_edge
thf(fact_906_ulgraph_Oempty__not__edge,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a] :
( ( undire6886684016831807756_set_a @ Vertices @ Edges )
=> ~ ( member_set_set_a @ bot_bot_set_set_a @ Edges ) ) ).
% ulgraph.empty_not_edge
thf(fact_907_ulgraph_Oempty__not__edge,axiom,
! [Vertices: set_a,Edges: set_set_a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ~ ( member_set_a @ bot_bot_set_a @ Edges ) ) ).
% ulgraph.empty_not_edge
thf(fact_908_comp__sgraph_Oall__edges__between__rem__wf,axiom,
! [S: set_a,X5: set_a,Y5: set_a] :
( ( undire8383842906760478443ween_a @ ( undire2918257014606996450dges_a @ S ) @ X5 @ Y5 )
= ( undire8383842906760478443ween_a @ ( undire2918257014606996450dges_a @ S ) @ ( inf_inf_set_a @ X5 @ S ) @ ( inf_inf_set_a @ Y5 @ S ) ) ) ).
% comp_sgraph.all_edges_between_rem_wf
thf(fact_909_ulgraph_Oall__edges__between__rem__wf,axiom,
! [Vertices: set_a,Edges: set_set_a,X5: set_a,Y5: set_a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( undire8383842906760478443ween_a @ Edges @ X5 @ Y5 )
= ( undire8383842906760478443ween_a @ Edges @ ( inf_inf_set_a @ X5 @ Vertices ) @ ( inf_inf_set_a @ Y5 @ Vertices ) ) ) ) ).
% ulgraph.all_edges_between_rem_wf
thf(fact_910_empty__set,axiom,
( bot_bo3357376287454694259od_a_a
= ( set_Product_prod_a_a2 @ nil_Product_prod_a_a ) ) ).
% empty_set
thf(fact_911_empty__set,axiom,
( bot_bot_set_a
= ( set_a2 @ nil_a ) ) ).
% empty_set
thf(fact_912_empty__set,axiom,
( bot_bot_set_set_a
= ( set_set_a2 @ nil_set_a ) ) ).
% empty_set
thf(fact_913_list_Osize_I3_J,axiom,
( ( size_size_list_set_a @ nil_set_a )
= zero_zero_nat ) ).
% list.size(3)
thf(fact_914_list_Osize_I3_J,axiom,
( ( size_size_list_a @ nil_a )
= zero_zero_nat ) ).
% list.size(3)
thf(fact_915_zero__reorient,axiom,
! [X: real] :
( ( zero_zero_real = X )
= ( X = zero_zero_real ) ) ).
% zero_reorient
thf(fact_916_zero__reorient,axiom,
! [X: nat] :
( ( zero_zero_nat = X )
= ( X = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_917_one__reorient,axiom,
! [X: real] :
( ( one_one_real = X )
= ( X = one_one_real ) ) ).
% one_reorient
thf(fact_918_one__reorient,axiom,
! [X: nat] :
( ( one_one_nat = X )
= ( X = one_one_nat ) ) ).
% one_reorient
thf(fact_919_last__in__list__set,axiom,
! [Xs: list_P1396940483166286381od_a_a] :
( ( ord_less_eq_nat @ one_one_nat @ ( size_s3885678630836030617od_a_a @ Xs ) )
=> ( member1426531477525435216od_a_a @ ( last_P8790725268278465478od_a_a @ Xs ) @ ( set_Product_prod_a_a2 @ Xs ) ) ) ).
% last_in_list_set
thf(fact_920_last__in__list__set,axiom,
! [Xs: list_nat] :
( ( ord_less_eq_nat @ one_one_nat @ ( size_size_list_nat @ Xs ) )
=> ( member_nat @ ( last_nat @ Xs ) @ ( set_nat2 @ Xs ) ) ) ).
% last_in_list_set
thf(fact_921_last__in__list__set,axiom,
! [Xs: list_set_a] :
( ( ord_less_eq_nat @ one_one_nat @ ( size_size_list_set_a @ Xs ) )
=> ( member_set_a @ ( last_set_a @ Xs ) @ ( set_set_a2 @ Xs ) ) ) ).
% last_in_list_set
thf(fact_922_last__in__list__set,axiom,
! [Xs: list_a] :
( ( ord_less_eq_nat @ one_one_nat @ ( size_size_list_a @ Xs ) )
=> ( member_a @ ( last_a @ Xs ) @ ( set_a2 @ Xs ) ) ) ).
% last_in_list_set
thf(fact_923_comp__sgraph_Oedge__density__zero,axiom,
! [Y5: set_Product_prod_a_a,S: set_Product_prod_a_a,X5: set_Product_prod_a_a] :
( ( Y5 = bot_bo3357376287454694259od_a_a )
=> ( ( undire8410861505230878716od_a_a @ ( undire6879232364018543115od_a_a @ S ) @ X5 @ Y5 )
= zero_zero_real ) ) ).
% comp_sgraph.edge_density_zero
thf(fact_924_comp__sgraph_Oedge__density__zero,axiom,
! [Y5: set_set_a,S: set_set_a,X5: set_set_a] :
( ( Y5 = bot_bot_set_set_a )
=> ( ( undire8927637694342045747_set_a @ ( undire8247866692393712962_set_a @ S ) @ X5 @ Y5 )
= zero_zero_real ) ) ).
% comp_sgraph.edge_density_zero
thf(fact_925_comp__sgraph_Oedge__density__zero,axiom,
! [Y5: set_a,S: set_a,X5: set_a] :
( ( Y5 = bot_bot_set_a )
=> ( ( undire297304480579013331sity_a @ ( undire2918257014606996450dges_a @ S ) @ X5 @ Y5 )
= zero_zero_real ) ) ).
% comp_sgraph.edge_density_zero
thf(fact_926_ulgraph_Oedge__density__zero,axiom,
! [Vertices: set_Product_prod_a_a,Edges: set_se5735800977113168103od_a_a,Y5: set_Product_prod_a_a,X5: set_Product_prod_a_a] :
( ( undire4585262585102564309od_a_a @ Vertices @ Edges )
=> ( ( Y5 = bot_bo3357376287454694259od_a_a )
=> ( ( undire8410861505230878716od_a_a @ Edges @ X5 @ Y5 )
= zero_zero_real ) ) ) ).
% ulgraph.edge_density_zero
thf(fact_927_ulgraph_Oedge__density__zero,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,Y5: set_set_a,X5: set_set_a] :
( ( undire6886684016831807756_set_a @ Vertices @ Edges )
=> ( ( Y5 = bot_bot_set_set_a )
=> ( ( undire8927637694342045747_set_a @ Edges @ X5 @ Y5 )
= zero_zero_real ) ) ) ).
% ulgraph.edge_density_zero
thf(fact_928_ulgraph_Oedge__density__zero,axiom,
! [Vertices: set_a,Edges: set_set_a,Y5: set_a,X5: set_a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( Y5 = bot_bot_set_a )
=> ( ( undire297304480579013331sity_a @ Edges @ X5 @ Y5 )
= zero_zero_real ) ) ) ).
% ulgraph.edge_density_zero
thf(fact_929_comp__sgraph_Oedge__density__eq0,axiom,
! [S: set_set_a,A2: set_set_a,B2: set_set_a,X5: set_set_a,Y5: set_set_a] :
( ( ( undire2462398226299384907_set_a @ ( undire8247866692393712962_set_a @ S ) @ A2 @ B2 )
= bot_bo5799363139946352499_set_a )
=> ( ( ord_le3724670747650509150_set_a @ X5 @ A2 )
=> ( ( ord_le3724670747650509150_set_a @ Y5 @ B2 )
=> ( ( undire8927637694342045747_set_a @ ( undire8247866692393712962_set_a @ S ) @ X5 @ Y5 )
= zero_zero_real ) ) ) ) ).
% comp_sgraph.edge_density_eq0
thf(fact_930_comp__sgraph_Oedge__density__eq0,axiom,
! [S: set_Product_prod_a_a,A2: set_Product_prod_a_a,B2: set_Product_prod_a_a,X5: set_Product_prod_a_a,Y5: set_Product_prod_a_a] :
( ( ( undire4032395788819567636od_a_a @ ( undire6879232364018543115od_a_a @ S ) @ A2 @ B2 )
= bot_bo510284599550014259od_a_a )
=> ( ( ord_le746702958409616551od_a_a @ X5 @ A2 )
=> ( ( ord_le746702958409616551od_a_a @ Y5 @ B2 )
=> ( ( undire8410861505230878716od_a_a @ ( undire6879232364018543115od_a_a @ S ) @ X5 @ Y5 )
= zero_zero_real ) ) ) ) ).
% comp_sgraph.edge_density_eq0
thf(fact_931_comp__sgraph_Oedge__density__eq0,axiom,
! [S: set_a,A2: set_a,B2: set_a,X5: set_a,Y5: set_a] :
( ( ( undire8383842906760478443ween_a @ ( undire2918257014606996450dges_a @ S ) @ A2 @ B2 )
= bot_bo3357376287454694259od_a_a )
=> ( ( ord_less_eq_set_a @ X5 @ A2 )
=> ( ( ord_less_eq_set_a @ Y5 @ B2 )
=> ( ( undire297304480579013331sity_a @ ( undire2918257014606996450dges_a @ S ) @ X5 @ Y5 )
= zero_zero_real ) ) ) ) ).
% comp_sgraph.edge_density_eq0
thf(fact_932_ulgraph_Oedge__density__eq0,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,A2: set_set_a,B2: set_set_a,X5: set_set_a,Y5: set_set_a] :
( ( undire6886684016831807756_set_a @ Vertices @ Edges )
=> ( ( ( undire2462398226299384907_set_a @ Edges @ A2 @ B2 )
= bot_bo5799363139946352499_set_a )
=> ( ( ord_le3724670747650509150_set_a @ X5 @ A2 )
=> ( ( ord_le3724670747650509150_set_a @ Y5 @ B2 )
=> ( ( undire8927637694342045747_set_a @ Edges @ X5 @ Y5 )
= zero_zero_real ) ) ) ) ) ).
% ulgraph.edge_density_eq0
thf(fact_933_ulgraph_Oedge__density__eq0,axiom,
! [Vertices: set_Product_prod_a_a,Edges: set_se5735800977113168103od_a_a,A2: set_Product_prod_a_a,B2: set_Product_prod_a_a,X5: set_Product_prod_a_a,Y5: set_Product_prod_a_a] :
( ( undire4585262585102564309od_a_a @ Vertices @ Edges )
=> ( ( ( undire4032395788819567636od_a_a @ Edges @ A2 @ B2 )
= bot_bo510284599550014259od_a_a )
=> ( ( ord_le746702958409616551od_a_a @ X5 @ A2 )
=> ( ( ord_le746702958409616551od_a_a @ Y5 @ B2 )
=> ( ( undire8410861505230878716od_a_a @ Edges @ X5 @ Y5 )
= zero_zero_real ) ) ) ) ) ).
% ulgraph.edge_density_eq0
thf(fact_934_ulgraph_Oedge__density__eq0,axiom,
! [Vertices: set_a,Edges: set_set_a,A2: set_a,B2: set_a,X5: set_a,Y5: set_a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( ( undire8383842906760478443ween_a @ Edges @ A2 @ B2 )
= bot_bo3357376287454694259od_a_a )
=> ( ( ord_less_eq_set_a @ X5 @ A2 )
=> ( ( ord_less_eq_set_a @ Y5 @ B2 )
=> ( ( undire297304480579013331sity_a @ Edges @ X5 @ Y5 )
= zero_zero_real ) ) ) ) ) ).
% ulgraph.edge_density_eq0
thf(fact_935_zero__le,axiom,
! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).
% zero_le
thf(fact_936_zero__neq__one,axiom,
zero_zero_real != one_one_real ).
% zero_neq_one
thf(fact_937_zero__neq__one,axiom,
zero_zero_nat != one_one_nat ).
% zero_neq_one
thf(fact_938_zero__less__one__class_Ozero__le__one,axiom,
ord_less_eq_real @ zero_zero_real @ one_one_real ).
% zero_less_one_class.zero_le_one
thf(fact_939_zero__less__one__class_Ozero__le__one,axiom,
ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one_class.zero_le_one
thf(fact_940_iso__vertex__empty__neighborhood,axiom,
! [V2: a] :
( ( undire8931668460104145173rtex_a @ vertices @ edges @ V2 )
=> ( ( undire8504279938402040014hood_a @ vertices @ edges @ V2 )
= bot_bot_set_a ) ) ).
% iso_vertex_empty_neighborhood
thf(fact_941_is__isolated__vertex__degree0,axiom,
! [V2: a] :
( ( undire8931668460104145173rtex_a @ vertices @ edges @ V2 )
=> ( ( undire8867928226783802224gree_a @ edges @ V2 )
= zero_zero_nat ) ) ).
% is_isolated_vertex_degree0
thf(fact_942_walk__edges__singleton__app,axiom,
! [Ys2: list_a,X: a] :
( ( Ys2 != nil_a )
=> ( ( undire7337870655677353998dges_a @ ( append_a @ ( cons_a @ X @ nil_a ) @ Ys2 ) )
= ( cons_set_a @ ( insert_a @ X @ ( insert_a @ ( hd_a @ Ys2 ) @ bot_bot_set_a ) ) @ ( undire7337870655677353998dges_a @ Ys2 ) ) ) ) ).
% walk_edges_singleton_app
thf(fact_943_walk__edges__app,axiom,
! [Xs: list_a,Y2: a,X: a] :
( ( undire7337870655677353998dges_a @ ( append_a @ Xs @ ( cons_a @ Y2 @ ( cons_a @ X @ nil_a ) ) ) )
= ( append_set_a @ ( undire7337870655677353998dges_a @ ( append_a @ Xs @ ( cons_a @ Y2 @ nil_a ) ) ) @ ( cons_set_a @ ( insert_a @ Y2 @ ( insert_a @ X @ bot_bot_set_a ) ) @ nil_set_a ) ) ) ).
% walk_edges_app
thf(fact_944_inf_Obounded__iff,axiom,
! [A: set_set_a,B: set_set_a,C: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ ( inf_inf_set_set_a @ B @ C ) )
= ( ( ord_le3724670747650509150_set_a @ A @ B )
& ( ord_le3724670747650509150_set_a @ A @ C ) ) ) ).
% inf.bounded_iff
thf(fact_945_inf_Obounded__iff,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ ( inf_inf_set_a @ B @ C ) )
= ( ( ord_less_eq_set_a @ A @ B )
& ( ord_less_eq_set_a @ A @ C ) ) ) ).
% inf.bounded_iff
thf(fact_946_inf_Obounded__iff,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ ( inf_inf_real @ B @ C ) )
= ( ( ord_less_eq_real @ A @ B )
& ( ord_less_eq_real @ A @ C ) ) ) ).
% inf.bounded_iff
thf(fact_947_inf_Obounded__iff,axiom,
! [A: set_Product_prod_a_a,B: set_Product_prod_a_a,C: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A @ ( inf_in8905007599844390133od_a_a @ B @ C ) )
= ( ( ord_le746702958409616551od_a_a @ A @ B )
& ( ord_le746702958409616551od_a_a @ A @ C ) ) ) ).
% inf.bounded_iff
thf(fact_948_inf_Obounded__iff,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( inf_inf_nat @ B @ C ) )
= ( ( ord_less_eq_nat @ A @ B )
& ( ord_less_eq_nat @ A @ C ) ) ) ).
% inf.bounded_iff
thf(fact_949_le__inf__iff,axiom,
! [X: set_set_a,Y2: set_set_a,Z3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X @ ( inf_inf_set_set_a @ Y2 @ Z3 ) )
= ( ( ord_le3724670747650509150_set_a @ X @ Y2 )
& ( ord_le3724670747650509150_set_a @ X @ Z3 ) ) ) ).
% le_inf_iff
thf(fact_950_le__inf__iff,axiom,
! [X: set_a,Y2: set_a,Z3: set_a] :
( ( ord_less_eq_set_a @ X @ ( inf_inf_set_a @ Y2 @ Z3 ) )
= ( ( ord_less_eq_set_a @ X @ Y2 )
& ( ord_less_eq_set_a @ X @ Z3 ) ) ) ).
% le_inf_iff
thf(fact_951_le__inf__iff,axiom,
! [X: real,Y2: real,Z3: real] :
( ( ord_less_eq_real @ X @ ( inf_inf_real @ Y2 @ Z3 ) )
= ( ( ord_less_eq_real @ X @ Y2 )
& ( ord_less_eq_real @ X @ Z3 ) ) ) ).
% le_inf_iff
thf(fact_952_le__inf__iff,axiom,
! [X: set_Product_prod_a_a,Y2: set_Product_prod_a_a,Z3: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ X @ ( inf_in8905007599844390133od_a_a @ Y2 @ Z3 ) )
= ( ( ord_le746702958409616551od_a_a @ X @ Y2 )
& ( ord_le746702958409616551od_a_a @ X @ Z3 ) ) ) ).
% le_inf_iff
thf(fact_953_le__inf__iff,axiom,
! [X: nat,Y2: nat,Z3: nat] :
( ( ord_less_eq_nat @ X @ ( inf_inf_nat @ Y2 @ Z3 ) )
= ( ( ord_less_eq_nat @ X @ Y2 )
& ( ord_less_eq_nat @ X @ Z3 ) ) ) ).
% le_inf_iff
thf(fact_954_insert__absorb2,axiom,
! [X: a,A2: set_a] :
( ( insert_a @ X @ ( insert_a @ X @ A2 ) )
= ( insert_a @ X @ A2 ) ) ).
% insert_absorb2
thf(fact_955_insert__absorb2,axiom,
! [X: set_a,A2: set_set_a] :
( ( insert_set_a @ X @ ( insert_set_a @ X @ A2 ) )
= ( insert_set_a @ X @ A2 ) ) ).
% insert_absorb2
thf(fact_956_insert__iff,axiom,
! [A: set_a,B: set_a,A2: set_set_a] :
( ( member_set_a @ A @ ( insert_set_a @ B @ A2 ) )
= ( ( A = B )
| ( member_set_a @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_957_insert__iff,axiom,
! [A: a,B: a,A2: set_a] :
( ( member_a @ A @ ( insert_a @ B @ A2 ) )
= ( ( A = B )
| ( member_a @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_958_insert__iff,axiom,
! [A: product_prod_a_a,B: product_prod_a_a,A2: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ A @ ( insert4534936382041156343od_a_a @ B @ A2 ) )
= ( ( A = B )
| ( member1426531477525435216od_a_a @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_959_insert__iff,axiom,
! [A: nat,B: nat,A2: set_nat] :
( ( member_nat @ A @ ( insert_nat @ B @ A2 ) )
= ( ( A = B )
| ( member_nat @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_960_insertCI,axiom,
! [A: set_a,B2: set_set_a,B: set_a] :
( ( ~ ( member_set_a @ A @ B2 )
=> ( A = B ) )
=> ( member_set_a @ A @ ( insert_set_a @ B @ B2 ) ) ) ).
% insertCI
thf(fact_961_insertCI,axiom,
! [A: a,B2: set_a,B: a] :
( ( ~ ( member_a @ A @ B2 )
=> ( A = B ) )
=> ( member_a @ A @ ( insert_a @ B @ B2 ) ) ) ).
% insertCI
thf(fact_962_insertCI,axiom,
! [A: product_prod_a_a,B2: set_Product_prod_a_a,B: product_prod_a_a] :
( ( ~ ( member1426531477525435216od_a_a @ A @ B2 )
=> ( A = B ) )
=> ( member1426531477525435216od_a_a @ A @ ( insert4534936382041156343od_a_a @ B @ B2 ) ) ) ).
% insertCI
thf(fact_963_insertCI,axiom,
! [A: nat,B2: set_nat,B: nat] :
( ( ~ ( member_nat @ A @ B2 )
=> ( A = B ) )
=> ( member_nat @ A @ ( insert_nat @ B @ B2 ) ) ) ).
% insertCI
thf(fact_964_vert__adj__def,axiom,
! [V1: a,V22: a] :
( ( undire397441198561214472_adj_a @ edges @ V1 @ V22 )
= ( member_set_a @ ( insert_a @ V1 @ ( insert_a @ V22 @ bot_bot_set_a ) ) @ edges ) ) ).
% vert_adj_def
thf(fact_965_not__vert__adj,axiom,
! [V2: a,U: a] :
( ~ ( undire397441198561214472_adj_a @ edges @ V2 @ U )
=> ~ ( member_set_a @ ( insert_a @ V2 @ ( insert_a @ U @ bot_bot_set_a ) ) @ edges ) ) ).
% not_vert_adj
thf(fact_966_has__loop__def,axiom,
! [V2: a] :
( ( undire3617971648856834880loop_a @ edges @ V2 )
= ( member_set_a @ ( insert_a @ V2 @ bot_bot_set_a ) @ edges ) ) ).
% has_loop_def
thf(fact_967_wellformed__alt__snd,axiom,
! [X: a,Y2: a] :
( ( member_set_a @ ( insert_a @ X @ ( insert_a @ Y2 @ bot_bot_set_a ) ) @ edges )
=> ( member_a @ Y2 @ vertices ) ) ).
% wellformed_alt_snd
thf(fact_968_wellformed__alt__fst,axiom,
! [X: a,Y2: a] :
( ( member_set_a @ ( insert_a @ X @ ( insert_a @ Y2 @ bot_bot_set_a ) ) @ edges )
=> ( member_a @ X @ vertices ) ) ).
% wellformed_alt_fst
thf(fact_969_is__edge__between__def,axiom,
( undire8544646567961481629ween_a
= ( ^ [X6: set_a,Y6: set_a,E5: set_a] :
? [X4: a,Y3: a] :
( ( E5
= ( insert_a @ X4 @ ( insert_a @ Y3 @ bot_bot_set_a ) ) )
& ( member_a @ X4 @ X6 )
& ( member_a @ Y3 @ Y6 ) ) ) ) ).
% is_edge_between_def
thf(fact_970_walk__edges_Osimps_I3_J,axiom,
! [X: a,Y2: a,Ys2: list_a] :
( ( undire7337870655677353998dges_a @ ( cons_a @ X @ ( cons_a @ Y2 @ Ys2 ) ) )
= ( cons_set_a @ ( insert_a @ X @ ( insert_a @ Y2 @ bot_bot_set_a ) ) @ ( undire7337870655677353998dges_a @ ( cons_a @ Y2 @ Ys2 ) ) ) ) ).
% walk_edges.simps(3)
thf(fact_971_vert__adj__inc__edge__iff,axiom,
! [V1: a,V22: a] :
( ( undire397441198561214472_adj_a @ edges @ V1 @ V22 )
= ( ( undire1521409233611534436dent_a @ V1 @ ( insert_a @ V1 @ ( insert_a @ V22 @ bot_bot_set_a ) ) )
& ( undire1521409233611534436dent_a @ V22 @ ( insert_a @ V1 @ ( insert_a @ V22 @ bot_bot_set_a ) ) )
& ( member_set_a @ ( insert_a @ V1 @ ( insert_a @ V22 @ bot_bot_set_a ) ) @ edges ) ) ) ).
% vert_adj_inc_edge_iff
thf(fact_972_singletonI,axiom,
! [A: nat] : ( member_nat @ A @ ( insert_nat @ A @ bot_bot_set_nat ) ) ).
% singletonI
thf(fact_973_singletonI,axiom,
! [A: product_prod_a_a] : ( member1426531477525435216od_a_a @ A @ ( insert4534936382041156343od_a_a @ A @ bot_bo3357376287454694259od_a_a ) ) ).
% singletonI
thf(fact_974_singletonI,axiom,
! [A: a] : ( member_a @ A @ ( insert_a @ A @ bot_bot_set_a ) ) ).
% singletonI
thf(fact_975_singletonI,axiom,
! [A: set_a] : ( member_set_a @ A @ ( insert_set_a @ A @ bot_bot_set_set_a ) ) ).
% singletonI
thf(fact_976_insert__subset,axiom,
! [X: nat,A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ ( insert_nat @ X @ A2 ) @ B2 )
= ( ( member_nat @ X @ B2 )
& ( ord_less_eq_set_nat @ A2 @ B2 ) ) ) ).
% insert_subset
thf(fact_977_insert__subset,axiom,
! [X: set_a,A2: set_set_a,B2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ ( insert_set_a @ X @ A2 ) @ B2 )
= ( ( member_set_a @ X @ B2 )
& ( ord_le3724670747650509150_set_a @ A2 @ B2 ) ) ) ).
% insert_subset
thf(fact_978_insert__subset,axiom,
! [X: a,A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ ( insert_a @ X @ A2 ) @ B2 )
= ( ( member_a @ X @ B2 )
& ( ord_less_eq_set_a @ A2 @ B2 ) ) ) ).
% insert_subset
thf(fact_979_insert__subset,axiom,
! [X: product_prod_a_a,A2: set_Product_prod_a_a,B2: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ ( insert4534936382041156343od_a_a @ X @ A2 ) @ B2 )
= ( ( member1426531477525435216od_a_a @ X @ B2 )
& ( ord_le746702958409616551od_a_a @ A2 @ B2 ) ) ) ).
% insert_subset
thf(fact_980_Int__insert__right__if1,axiom,
! [A: set_a,A2: set_set_a,B2: set_set_a] :
( ( member_set_a @ A @ A2 )
=> ( ( inf_inf_set_set_a @ A2 @ ( insert_set_a @ A @ B2 ) )
= ( insert_set_a @ A @ ( inf_inf_set_set_a @ A2 @ B2 ) ) ) ) ).
% Int_insert_right_if1
thf(fact_981_Int__insert__right__if1,axiom,
! [A: product_prod_a_a,A2: set_Product_prod_a_a,B2: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ A @ A2 )
=> ( ( inf_in8905007599844390133od_a_a @ A2 @ ( insert4534936382041156343od_a_a @ A @ B2 ) )
= ( insert4534936382041156343od_a_a @ A @ ( inf_in8905007599844390133od_a_a @ A2 @ B2 ) ) ) ) ).
% Int_insert_right_if1
thf(fact_982_Int__insert__right__if1,axiom,
! [A: nat,A2: set_nat,B2: set_nat] :
( ( member_nat @ A @ A2 )
=> ( ( inf_inf_set_nat @ A2 @ ( insert_nat @ A @ B2 ) )
= ( insert_nat @ A @ ( inf_inf_set_nat @ A2 @ B2 ) ) ) ) ).
% Int_insert_right_if1
thf(fact_983_Int__insert__right__if1,axiom,
! [A: a,A2: set_a,B2: set_a] :
( ( member_a @ A @ A2 )
=> ( ( inf_inf_set_a @ A2 @ ( insert_a @ A @ B2 ) )
= ( insert_a @ A @ ( inf_inf_set_a @ A2 @ B2 ) ) ) ) ).
% Int_insert_right_if1
thf(fact_984_Int__insert__right__if0,axiom,
! [A: set_a,A2: set_set_a,B2: set_set_a] :
( ~ ( member_set_a @ A @ A2 )
=> ( ( inf_inf_set_set_a @ A2 @ ( insert_set_a @ A @ B2 ) )
= ( inf_inf_set_set_a @ A2 @ B2 ) ) ) ).
% Int_insert_right_if0
thf(fact_985_Int__insert__right__if0,axiom,
! [A: product_prod_a_a,A2: set_Product_prod_a_a,B2: set_Product_prod_a_a] :
( ~ ( member1426531477525435216od_a_a @ A @ A2 )
=> ( ( inf_in8905007599844390133od_a_a @ A2 @ ( insert4534936382041156343od_a_a @ A @ B2 ) )
= ( inf_in8905007599844390133od_a_a @ A2 @ B2 ) ) ) ).
% Int_insert_right_if0
thf(fact_986_Int__insert__right__if0,axiom,
! [A: nat,A2: set_nat,B2: set_nat] :
( ~ ( member_nat @ A @ A2 )
=> ( ( inf_inf_set_nat @ A2 @ ( insert_nat @ A @ B2 ) )
= ( inf_inf_set_nat @ A2 @ B2 ) ) ) ).
% Int_insert_right_if0
thf(fact_987_Int__insert__right__if0,axiom,
! [A: a,A2: set_a,B2: set_a] :
( ~ ( member_a @ A @ A2 )
=> ( ( inf_inf_set_a @ A2 @ ( insert_a @ A @ B2 ) )
= ( inf_inf_set_a @ A2 @ B2 ) ) ) ).
% Int_insert_right_if0
thf(fact_988_insert__inter__insert,axiom,
! [A: set_a,A2: set_set_a,B2: set_set_a] :
( ( inf_inf_set_set_a @ ( insert_set_a @ A @ A2 ) @ ( insert_set_a @ A @ B2 ) )
= ( insert_set_a @ A @ ( inf_inf_set_set_a @ A2 @ B2 ) ) ) ).
% insert_inter_insert
thf(fact_989_insert__inter__insert,axiom,
! [A: a,A2: set_a,B2: set_a] :
( ( inf_inf_set_a @ ( insert_a @ A @ A2 ) @ ( insert_a @ A @ B2 ) )
= ( insert_a @ A @ ( inf_inf_set_a @ A2 @ B2 ) ) ) ).
% insert_inter_insert
thf(fact_990_Int__insert__left__if1,axiom,
! [A: set_a,C2: set_set_a,B2: set_set_a] :
( ( member_set_a @ A @ C2 )
=> ( ( inf_inf_set_set_a @ ( insert_set_a @ A @ B2 ) @ C2 )
= ( insert_set_a @ A @ ( inf_inf_set_set_a @ B2 @ C2 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_991_Int__insert__left__if1,axiom,
! [A: product_prod_a_a,C2: set_Product_prod_a_a,B2: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ A @ C2 )
=> ( ( inf_in8905007599844390133od_a_a @ ( insert4534936382041156343od_a_a @ A @ B2 ) @ C2 )
= ( insert4534936382041156343od_a_a @ A @ ( inf_in8905007599844390133od_a_a @ B2 @ C2 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_992_Int__insert__left__if1,axiom,
! [A: nat,C2: set_nat,B2: set_nat] :
( ( member_nat @ A @ C2 )
=> ( ( inf_inf_set_nat @ ( insert_nat @ A @ B2 ) @ C2 )
= ( insert_nat @ A @ ( inf_inf_set_nat @ B2 @ C2 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_993_Int__insert__left__if1,axiom,
! [A: a,C2: set_a,B2: set_a] :
( ( member_a @ A @ C2 )
=> ( ( inf_inf_set_a @ ( insert_a @ A @ B2 ) @ C2 )
= ( insert_a @ A @ ( inf_inf_set_a @ B2 @ C2 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_994_Int__insert__left__if0,axiom,
! [A: set_a,C2: set_set_a,B2: set_set_a] :
( ~ ( member_set_a @ A @ C2 )
=> ( ( inf_inf_set_set_a @ ( insert_set_a @ A @ B2 ) @ C2 )
= ( inf_inf_set_set_a @ B2 @ C2 ) ) ) ).
% Int_insert_left_if0
thf(fact_995_Int__insert__left__if0,axiom,
! [A: product_prod_a_a,C2: set_Product_prod_a_a,B2: set_Product_prod_a_a] :
( ~ ( member1426531477525435216od_a_a @ A @ C2 )
=> ( ( inf_in8905007599844390133od_a_a @ ( insert4534936382041156343od_a_a @ A @ B2 ) @ C2 )
= ( inf_in8905007599844390133od_a_a @ B2 @ C2 ) ) ) ).
% Int_insert_left_if0
thf(fact_996_Int__insert__left__if0,axiom,
! [A: nat,C2: set_nat,B2: set_nat] :
( ~ ( member_nat @ A @ C2 )
=> ( ( inf_inf_set_nat @ ( insert_nat @ A @ B2 ) @ C2 )
= ( inf_inf_set_nat @ B2 @ C2 ) ) ) ).
% Int_insert_left_if0
thf(fact_997_Int__insert__left__if0,axiom,
! [A: a,C2: set_a,B2: set_a] :
( ~ ( member_a @ A @ C2 )
=> ( ( inf_inf_set_a @ ( insert_a @ A @ B2 ) @ C2 )
= ( inf_inf_set_a @ B2 @ C2 ) ) ) ).
% Int_insert_left_if0
thf(fact_998_walk__edges_Oelims,axiom,
! [X: list_a,Y2: list_set_a] :
( ( ( undire7337870655677353998dges_a @ X )
= Y2 )
=> ( ( ( X = nil_a )
=> ( Y2 != nil_set_a ) )
=> ( ( ? [X2: a] :
( X
= ( cons_a @ X2 @ nil_a ) )
=> ( Y2 != nil_set_a ) )
=> ~ ! [X2: a,Y: a,Ys: list_a] :
( ( X
= ( cons_a @ X2 @ ( cons_a @ Y @ Ys ) ) )
=> ( Y2
!= ( cons_set_a @ ( insert_a @ X2 @ ( insert_a @ Y @ bot_bot_set_a ) ) @ ( undire7337870655677353998dges_a @ ( cons_a @ Y @ Ys ) ) ) ) ) ) ) ) ).
% walk_edges.elims
thf(fact_999_singleton__insert__inj__eq,axiom,
! [B: set_a,A: set_a,A2: set_set_a] :
( ( ( insert_set_a @ B @ bot_bot_set_set_a )
= ( insert_set_a @ A @ A2 ) )
= ( ( A = B )
& ( ord_le3724670747650509150_set_a @ A2 @ ( insert_set_a @ B @ bot_bot_set_set_a ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_1000_singleton__insert__inj__eq,axiom,
! [B: a,A: a,A2: set_a] :
( ( ( insert_a @ B @ bot_bot_set_a )
= ( insert_a @ A @ A2 ) )
= ( ( A = B )
& ( ord_less_eq_set_a @ A2 @ ( insert_a @ B @ bot_bot_set_a ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_1001_singleton__insert__inj__eq,axiom,
! [B: product_prod_a_a,A: product_prod_a_a,A2: set_Product_prod_a_a] :
( ( ( insert4534936382041156343od_a_a @ B @ bot_bo3357376287454694259od_a_a )
= ( insert4534936382041156343od_a_a @ A @ A2 ) )
= ( ( A = B )
& ( ord_le746702958409616551od_a_a @ A2 @ ( insert4534936382041156343od_a_a @ B @ bot_bo3357376287454694259od_a_a ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_1002_singleton__insert__inj__eq_H,axiom,
! [A: set_a,A2: set_set_a,B: set_a] :
( ( ( insert_set_a @ A @ A2 )
= ( insert_set_a @ B @ bot_bot_set_set_a ) )
= ( ( A = B )
& ( ord_le3724670747650509150_set_a @ A2 @ ( insert_set_a @ B @ bot_bot_set_set_a ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_1003_singleton__insert__inj__eq_H,axiom,
! [A: a,A2: set_a,B: a] :
( ( ( insert_a @ A @ A2 )
= ( insert_a @ B @ bot_bot_set_a ) )
= ( ( A = B )
& ( ord_less_eq_set_a @ A2 @ ( insert_a @ B @ bot_bot_set_a ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_1004_singleton__insert__inj__eq_H,axiom,
! [A: product_prod_a_a,A2: set_Product_prod_a_a,B: product_prod_a_a] :
( ( ( insert4534936382041156343od_a_a @ A @ A2 )
= ( insert4534936382041156343od_a_a @ B @ bot_bo3357376287454694259od_a_a ) )
= ( ( A = B )
& ( ord_le746702958409616551od_a_a @ A2 @ ( insert4534936382041156343od_a_a @ B @ bot_bo3357376287454694259od_a_a ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_1005_insert__disjoint_I1_J,axiom,
! [A: nat,A2: set_nat,B2: set_nat] :
( ( ( inf_inf_set_nat @ ( insert_nat @ A @ A2 ) @ B2 )
= bot_bot_set_nat )
= ( ~ ( member_nat @ A @ B2 )
& ( ( inf_inf_set_nat @ A2 @ B2 )
= bot_bot_set_nat ) ) ) ).
% insert_disjoint(1)
thf(fact_1006_insert__disjoint_I1_J,axiom,
! [A: product_prod_a_a,A2: set_Product_prod_a_a,B2: set_Product_prod_a_a] :
( ( ( inf_in8905007599844390133od_a_a @ ( insert4534936382041156343od_a_a @ A @ A2 ) @ B2 )
= bot_bo3357376287454694259od_a_a )
= ( ~ ( member1426531477525435216od_a_a @ A @ B2 )
& ( ( inf_in8905007599844390133od_a_a @ A2 @ B2 )
= bot_bo3357376287454694259od_a_a ) ) ) ).
% insert_disjoint(1)
thf(fact_1007_insert__disjoint_I1_J,axiom,
! [A: a,A2: set_a,B2: set_a] :
( ( ( inf_inf_set_a @ ( insert_a @ A @ A2 ) @ B2 )
= bot_bot_set_a )
= ( ~ ( member_a @ A @ B2 )
& ( ( inf_inf_set_a @ A2 @ B2 )
= bot_bot_set_a ) ) ) ).
% insert_disjoint(1)
thf(fact_1008_insert__disjoint_I1_J,axiom,
! [A: set_a,A2: set_set_a,B2: set_set_a] :
( ( ( inf_inf_set_set_a @ ( insert_set_a @ A @ A2 ) @ B2 )
= bot_bot_set_set_a )
= ( ~ ( member_set_a @ A @ B2 )
& ( ( inf_inf_set_set_a @ A2 @ B2 )
= bot_bot_set_set_a ) ) ) ).
% insert_disjoint(1)
thf(fact_1009_insert__disjoint_I2_J,axiom,
! [A: nat,A2: set_nat,B2: set_nat] :
( ( bot_bot_set_nat
= ( inf_inf_set_nat @ ( insert_nat @ A @ A2 ) @ B2 ) )
= ( ~ ( member_nat @ A @ B2 )
& ( bot_bot_set_nat
= ( inf_inf_set_nat @ A2 @ B2 ) ) ) ) ).
% insert_disjoint(2)
thf(fact_1010_insert__disjoint_I2_J,axiom,
! [A: product_prod_a_a,A2: set_Product_prod_a_a,B2: set_Product_prod_a_a] :
( ( bot_bo3357376287454694259od_a_a
= ( inf_in8905007599844390133od_a_a @ ( insert4534936382041156343od_a_a @ A @ A2 ) @ B2 ) )
= ( ~ ( member1426531477525435216od_a_a @ A @ B2 )
& ( bot_bo3357376287454694259od_a_a
= ( inf_in8905007599844390133od_a_a @ A2 @ B2 ) ) ) ) ).
% insert_disjoint(2)
thf(fact_1011_insert__disjoint_I2_J,axiom,
! [A: a,A2: set_a,B2: set_a] :
( ( bot_bot_set_a
= ( inf_inf_set_a @ ( insert_a @ A @ A2 ) @ B2 ) )
= ( ~ ( member_a @ A @ B2 )
& ( bot_bot_set_a
= ( inf_inf_set_a @ A2 @ B2 ) ) ) ) ).
% insert_disjoint(2)
thf(fact_1012_insert__disjoint_I2_J,axiom,
! [A: set_a,A2: set_set_a,B2: set_set_a] :
( ( bot_bot_set_set_a
= ( inf_inf_set_set_a @ ( insert_set_a @ A @ A2 ) @ B2 ) )
= ( ~ ( member_set_a @ A @ B2 )
& ( bot_bot_set_set_a
= ( inf_inf_set_set_a @ A2 @ B2 ) ) ) ) ).
% insert_disjoint(2)
thf(fact_1013_disjoint__insert_I1_J,axiom,
! [B2: set_nat,A: nat,A2: set_nat] :
( ( ( inf_inf_set_nat @ B2 @ ( insert_nat @ A @ A2 ) )
= bot_bot_set_nat )
= ( ~ ( member_nat @ A @ B2 )
& ( ( inf_inf_set_nat @ B2 @ A2 )
= bot_bot_set_nat ) ) ) ).
% disjoint_insert(1)
thf(fact_1014_disjoint__insert_I1_J,axiom,
! [B2: set_Product_prod_a_a,A: product_prod_a_a,A2: set_Product_prod_a_a] :
( ( ( inf_in8905007599844390133od_a_a @ B2 @ ( insert4534936382041156343od_a_a @ A @ A2 ) )
= bot_bo3357376287454694259od_a_a )
= ( ~ ( member1426531477525435216od_a_a @ A @ B2 )
& ( ( inf_in8905007599844390133od_a_a @ B2 @ A2 )
= bot_bo3357376287454694259od_a_a ) ) ) ).
% disjoint_insert(1)
thf(fact_1015_disjoint__insert_I1_J,axiom,
! [B2: set_a,A: a,A2: set_a] :
( ( ( inf_inf_set_a @ B2 @ ( insert_a @ A @ A2 ) )
= bot_bot_set_a )
= ( ~ ( member_a @ A @ B2 )
& ( ( inf_inf_set_a @ B2 @ A2 )
= bot_bot_set_a ) ) ) ).
% disjoint_insert(1)
thf(fact_1016_disjoint__insert_I1_J,axiom,
! [B2: set_set_a,A: set_a,A2: set_set_a] :
( ( ( inf_inf_set_set_a @ B2 @ ( insert_set_a @ A @ A2 ) )
= bot_bot_set_set_a )
= ( ~ ( member_set_a @ A @ B2 )
& ( ( inf_inf_set_set_a @ B2 @ A2 )
= bot_bot_set_set_a ) ) ) ).
% disjoint_insert(1)
thf(fact_1017_disjoint__insert_I2_J,axiom,
! [A2: set_nat,B: nat,B2: set_nat] :
( ( bot_bot_set_nat
= ( inf_inf_set_nat @ A2 @ ( insert_nat @ B @ B2 ) ) )
= ( ~ ( member_nat @ B @ A2 )
& ( bot_bot_set_nat
= ( inf_inf_set_nat @ A2 @ B2 ) ) ) ) ).
% disjoint_insert(2)
thf(fact_1018_disjoint__insert_I2_J,axiom,
! [A2: set_Product_prod_a_a,B: product_prod_a_a,B2: set_Product_prod_a_a] :
( ( bot_bo3357376287454694259od_a_a
= ( inf_in8905007599844390133od_a_a @ A2 @ ( insert4534936382041156343od_a_a @ B @ B2 ) ) )
= ( ~ ( member1426531477525435216od_a_a @ B @ A2 )
& ( bot_bo3357376287454694259od_a_a
= ( inf_in8905007599844390133od_a_a @ A2 @ B2 ) ) ) ) ).
% disjoint_insert(2)
thf(fact_1019_disjoint__insert_I2_J,axiom,
! [A2: set_a,B: a,B2: set_a] :
( ( bot_bot_set_a
= ( inf_inf_set_a @ A2 @ ( insert_a @ B @ B2 ) ) )
= ( ~ ( member_a @ B @ A2 )
& ( bot_bot_set_a
= ( inf_inf_set_a @ A2 @ B2 ) ) ) ) ).
% disjoint_insert(2)
thf(fact_1020_disjoint__insert_I2_J,axiom,
! [A2: set_set_a,B: set_a,B2: set_set_a] :
( ( bot_bot_set_set_a
= ( inf_inf_set_set_a @ A2 @ ( insert_set_a @ B @ B2 ) ) )
= ( ~ ( member_set_a @ B @ A2 )
& ( bot_bot_set_set_a
= ( inf_inf_set_set_a @ A2 @ B2 ) ) ) ) ).
% disjoint_insert(2)
thf(fact_1021_list_Osimps_I15_J,axiom,
! [X21: a,X22: list_a] :
( ( set_a2 @ ( cons_a @ X21 @ X22 ) )
= ( insert_a @ X21 @ ( set_a2 @ X22 ) ) ) ).
% list.simps(15)
thf(fact_1022_list_Osimps_I15_J,axiom,
! [X21: set_a,X22: list_set_a] :
( ( set_set_a2 @ ( cons_set_a @ X21 @ X22 ) )
= ( insert_set_a @ X21 @ ( set_set_a2 @ X22 ) ) ) ).
% list.simps(15)
thf(fact_1023_comp__sgraph_Odegree__none,axiom,
! [V2: set_a,S: set_set_a] :
( ~ ( member_set_a @ V2 @ S )
=> ( ( undire8939077443744732368_set_a @ ( undire8247866692393712962_set_a @ S ) @ V2 )
= zero_zero_nat ) ) ).
% comp_sgraph.degree_none
thf(fact_1024_comp__sgraph_Odegree__none,axiom,
! [V2: product_prod_a_a,S: set_Product_prod_a_a] :
( ~ ( member1426531477525435216od_a_a @ V2 @ S )
=> ( ( undire1436394852029823897od_a_a @ ( undire6879232364018543115od_a_a @ S ) @ V2 )
= zero_zero_nat ) ) ).
% comp_sgraph.degree_none
thf(fact_1025_comp__sgraph_Odegree__none,axiom,
! [V2: nat,S: set_nat] :
( ~ ( member_nat @ V2 @ S )
=> ( ( undire6581030323043281630ee_nat @ ( undire463345858124014060es_nat @ S ) @ V2 )
= zero_zero_nat ) ) ).
% comp_sgraph.degree_none
thf(fact_1026_comp__sgraph_Odegree__none,axiom,
! [V2: a,S: set_a] :
( ~ ( member_a @ V2 @ S )
=> ( ( undire8867928226783802224gree_a @ ( undire2918257014606996450dges_a @ S ) @ V2 )
= zero_zero_nat ) ) ).
% comp_sgraph.degree_none
thf(fact_1027_the__elem__eq,axiom,
! [X: product_prod_a_a] :
( ( the_el8589169208993665564od_a_a @ ( insert4534936382041156343od_a_a @ X @ bot_bo3357376287454694259od_a_a ) )
= X ) ).
% the_elem_eq
thf(fact_1028_the__elem__eq,axiom,
! [X: a] :
( ( the_elem_a @ ( insert_a @ X @ bot_bot_set_a ) )
= X ) ).
% the_elem_eq
thf(fact_1029_the__elem__eq,axiom,
! [X: set_a] :
( ( the_elem_set_a @ ( insert_set_a @ X @ bot_bot_set_set_a ) )
= X ) ).
% the_elem_eq
thf(fact_1030_degree__none,axiom,
! [V2: a] :
( ~ ( member_a @ V2 @ vertices )
=> ( ( undire8867928226783802224gree_a @ edges @ V2 )
= zero_zero_nat ) ) ).
% degree_none
thf(fact_1031_mk__disjoint__insert,axiom,
! [A: set_a,A2: set_set_a] :
( ( member_set_a @ A @ A2 )
=> ? [B6: set_set_a] :
( ( A2
= ( insert_set_a @ A @ B6 ) )
& ~ ( member_set_a @ A @ B6 ) ) ) ).
% mk_disjoint_insert
thf(fact_1032_mk__disjoint__insert,axiom,
! [A: a,A2: set_a] :
( ( member_a @ A @ A2 )
=> ? [B6: set_a] :
( ( A2
= ( insert_a @ A @ B6 ) )
& ~ ( member_a @ A @ B6 ) ) ) ).
% mk_disjoint_insert
thf(fact_1033_mk__disjoint__insert,axiom,
! [A: product_prod_a_a,A2: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ A @ A2 )
=> ? [B6: set_Product_prod_a_a] :
( ( A2
= ( insert4534936382041156343od_a_a @ A @ B6 ) )
& ~ ( member1426531477525435216od_a_a @ A @ B6 ) ) ) ).
% mk_disjoint_insert
thf(fact_1034_mk__disjoint__insert,axiom,
! [A: nat,A2: set_nat] :
( ( member_nat @ A @ A2 )
=> ? [B6: set_nat] :
( ( A2
= ( insert_nat @ A @ B6 ) )
& ~ ( member_nat @ A @ B6 ) ) ) ).
% mk_disjoint_insert
thf(fact_1035_insert__commute,axiom,
! [X: a,Y2: a,A2: set_a] :
( ( insert_a @ X @ ( insert_a @ Y2 @ A2 ) )
= ( insert_a @ Y2 @ ( insert_a @ X @ A2 ) ) ) ).
% insert_commute
thf(fact_1036_insert__commute,axiom,
! [X: set_a,Y2: set_a,A2: set_set_a] :
( ( insert_set_a @ X @ ( insert_set_a @ Y2 @ A2 ) )
= ( insert_set_a @ Y2 @ ( insert_set_a @ X @ A2 ) ) ) ).
% insert_commute
thf(fact_1037_insert__eq__iff,axiom,
! [A: set_a,A2: set_set_a,B: set_a,B2: set_set_a] :
( ~ ( member_set_a @ A @ A2 )
=> ( ~ ( member_set_a @ B @ B2 )
=> ( ( ( insert_set_a @ A @ A2 )
= ( insert_set_a @ B @ B2 ) )
= ( ( ( A = B )
=> ( A2 = B2 ) )
& ( ( A != B )
=> ? [C3: set_set_a] :
( ( A2
= ( insert_set_a @ B @ C3 ) )
& ~ ( member_set_a @ B @ C3 )
& ( B2
= ( insert_set_a @ A @ C3 ) )
& ~ ( member_set_a @ A @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_1038_insert__eq__iff,axiom,
! [A: a,A2: set_a,B: a,B2: set_a] :
( ~ ( member_a @ A @ A2 )
=> ( ~ ( member_a @ B @ B2 )
=> ( ( ( insert_a @ A @ A2 )
= ( insert_a @ B @ B2 ) )
= ( ( ( A = B )
=> ( A2 = B2 ) )
& ( ( A != B )
=> ? [C3: set_a] :
( ( A2
= ( insert_a @ B @ C3 ) )
& ~ ( member_a @ B @ C3 )
& ( B2
= ( insert_a @ A @ C3 ) )
& ~ ( member_a @ A @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_1039_insert__eq__iff,axiom,
! [A: product_prod_a_a,A2: set_Product_prod_a_a,B: product_prod_a_a,B2: set_Product_prod_a_a] :
( ~ ( member1426531477525435216od_a_a @ A @ A2 )
=> ( ~ ( member1426531477525435216od_a_a @ B @ B2 )
=> ( ( ( insert4534936382041156343od_a_a @ A @ A2 )
= ( insert4534936382041156343od_a_a @ B @ B2 ) )
= ( ( ( A = B )
=> ( A2 = B2 ) )
& ( ( A != B )
=> ? [C3: set_Product_prod_a_a] :
( ( A2
= ( insert4534936382041156343od_a_a @ B @ C3 ) )
& ~ ( member1426531477525435216od_a_a @ B @ C3 )
& ( B2
= ( insert4534936382041156343od_a_a @ A @ C3 ) )
& ~ ( member1426531477525435216od_a_a @ A @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_1040_insert__eq__iff,axiom,
! [A: nat,A2: set_nat,B: nat,B2: set_nat] :
( ~ ( member_nat @ A @ A2 )
=> ( ~ ( member_nat @ B @ B2 )
=> ( ( ( insert_nat @ A @ A2 )
= ( insert_nat @ B @ B2 ) )
= ( ( ( A = B )
=> ( A2 = B2 ) )
& ( ( A != B )
=> ? [C3: set_nat] :
( ( A2
= ( insert_nat @ B @ C3 ) )
& ~ ( member_nat @ B @ C3 )
& ( B2
= ( insert_nat @ A @ C3 ) )
& ~ ( member_nat @ A @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_1041_insert__absorb,axiom,
! [A: set_a,A2: set_set_a] :
( ( member_set_a @ A @ A2 )
=> ( ( insert_set_a @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_1042_insert__absorb,axiom,
! [A: a,A2: set_a] :
( ( member_a @ A @ A2 )
=> ( ( insert_a @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_1043_insert__absorb,axiom,
! [A: product_prod_a_a,A2: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ A @ A2 )
=> ( ( insert4534936382041156343od_a_a @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_1044_insert__absorb,axiom,
! [A: nat,A2: set_nat] :
( ( member_nat @ A @ A2 )
=> ( ( insert_nat @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_1045_insert__ident,axiom,
! [X: set_a,A2: set_set_a,B2: set_set_a] :
( ~ ( member_set_a @ X @ A2 )
=> ( ~ ( member_set_a @ X @ B2 )
=> ( ( ( insert_set_a @ X @ A2 )
= ( insert_set_a @ X @ B2 ) )
= ( A2 = B2 ) ) ) ) ).
% insert_ident
thf(fact_1046_insert__ident,axiom,
! [X: a,A2: set_a,B2: set_a] :
( ~ ( member_a @ X @ A2 )
=> ( ~ ( member_a @ X @ B2 )
=> ( ( ( insert_a @ X @ A2 )
= ( insert_a @ X @ B2 ) )
= ( A2 = B2 ) ) ) ) ).
% insert_ident
thf(fact_1047_insert__ident,axiom,
! [X: product_prod_a_a,A2: set_Product_prod_a_a,B2: set_Product_prod_a_a] :
( ~ ( member1426531477525435216od_a_a @ X @ A2 )
=> ( ~ ( member1426531477525435216od_a_a @ X @ B2 )
=> ( ( ( insert4534936382041156343od_a_a @ X @ A2 )
= ( insert4534936382041156343od_a_a @ X @ B2 ) )
= ( A2 = B2 ) ) ) ) ).
% insert_ident
thf(fact_1048_insert__ident,axiom,
! [X: nat,A2: set_nat,B2: set_nat] :
( ~ ( member_nat @ X @ A2 )
=> ( ~ ( member_nat @ X @ B2 )
=> ( ( ( insert_nat @ X @ A2 )
= ( insert_nat @ X @ B2 ) )
= ( A2 = B2 ) ) ) ) ).
% insert_ident
thf(fact_1049_Set_Oset__insert,axiom,
! [X: set_a,A2: set_set_a] :
( ( member_set_a @ X @ A2 )
=> ~ ! [B6: set_set_a] :
( ( A2
= ( insert_set_a @ X @ B6 ) )
=> ( member_set_a @ X @ B6 ) ) ) ).
% Set.set_insert
thf(fact_1050_Set_Oset__insert,axiom,
! [X: a,A2: set_a] :
( ( member_a @ X @ A2 )
=> ~ ! [B6: set_a] :
( ( A2
= ( insert_a @ X @ B6 ) )
=> ( member_a @ X @ B6 ) ) ) ).
% Set.set_insert
thf(fact_1051_Set_Oset__insert,axiom,
! [X: product_prod_a_a,A2: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X @ A2 )
=> ~ ! [B6: set_Product_prod_a_a] :
( ( A2
= ( insert4534936382041156343od_a_a @ X @ B6 ) )
=> ( member1426531477525435216od_a_a @ X @ B6 ) ) ) ).
% Set.set_insert
thf(fact_1052_Set_Oset__insert,axiom,
! [X: nat,A2: set_nat] :
( ( member_nat @ X @ A2 )
=> ~ ! [B6: set_nat] :
( ( A2
= ( insert_nat @ X @ B6 ) )
=> ( member_nat @ X @ B6 ) ) ) ).
% Set.set_insert
thf(fact_1053_insertI2,axiom,
! [A: set_a,B2: set_set_a,B: set_a] :
( ( member_set_a @ A @ B2 )
=> ( member_set_a @ A @ ( insert_set_a @ B @ B2 ) ) ) ).
% insertI2
thf(fact_1054_insertI2,axiom,
! [A: a,B2: set_a,B: a] :
( ( member_a @ A @ B2 )
=> ( member_a @ A @ ( insert_a @ B @ B2 ) ) ) ).
% insertI2
thf(fact_1055_insertI2,axiom,
! [A: product_prod_a_a,B2: set_Product_prod_a_a,B: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ A @ B2 )
=> ( member1426531477525435216od_a_a @ A @ ( insert4534936382041156343od_a_a @ B @ B2 ) ) ) ).
% insertI2
thf(fact_1056_insertI2,axiom,
! [A: nat,B2: set_nat,B: nat] :
( ( member_nat @ A @ B2 )
=> ( member_nat @ A @ ( insert_nat @ B @ B2 ) ) ) ).
% insertI2
thf(fact_1057_insertI1,axiom,
! [A: set_a,B2: set_set_a] : ( member_set_a @ A @ ( insert_set_a @ A @ B2 ) ) ).
% insertI1
thf(fact_1058_insertI1,axiom,
! [A: a,B2: set_a] : ( member_a @ A @ ( insert_a @ A @ B2 ) ) ).
% insertI1
thf(fact_1059_insertI1,axiom,
! [A: product_prod_a_a,B2: set_Product_prod_a_a] : ( member1426531477525435216od_a_a @ A @ ( insert4534936382041156343od_a_a @ A @ B2 ) ) ).
% insertI1
thf(fact_1060_insertI1,axiom,
! [A: nat,B2: set_nat] : ( member_nat @ A @ ( insert_nat @ A @ B2 ) ) ).
% insertI1
thf(fact_1061_insertE,axiom,
! [A: set_a,B: set_a,A2: set_set_a] :
( ( member_set_a @ A @ ( insert_set_a @ B @ A2 ) )
=> ( ( A != B )
=> ( member_set_a @ A @ A2 ) ) ) ).
% insertE
thf(fact_1062_insertE,axiom,
! [A: a,B: a,A2: set_a] :
( ( member_a @ A @ ( insert_a @ B @ A2 ) )
=> ( ( A != B )
=> ( member_a @ A @ A2 ) ) ) ).
% insertE
thf(fact_1063_insertE,axiom,
! [A: product_prod_a_a,B: product_prod_a_a,A2: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ A @ ( insert4534936382041156343od_a_a @ B @ A2 ) )
=> ( ( A != B )
=> ( member1426531477525435216od_a_a @ A @ A2 ) ) ) ).
% insertE
thf(fact_1064_insertE,axiom,
! [A: nat,B: nat,A2: set_nat] :
( ( member_nat @ A @ ( insert_nat @ B @ A2 ) )
=> ( ( A != B )
=> ( member_nat @ A @ A2 ) ) ) ).
% insertE
thf(fact_1065_ulgraph_Oneighborhood_Ocong,axiom,
undire8504279938402040014hood_a = undire8504279938402040014hood_a ).
% ulgraph.neighborhood.cong
thf(fact_1066_ulgraph_Odegree_Ocong,axiom,
undire8867928226783802224gree_a = undire8867928226783802224gree_a ).
% ulgraph.degree.cong
thf(fact_1067_bot__set__def,axiom,
( bot_bo3357376287454694259od_a_a
= ( collec3336397797384452498od_a_a @ bot_bo4160289986317612842_a_a_o ) ) ).
% bot_set_def
thf(fact_1068_bot__set__def,axiom,
( bot_bot_set_a
= ( collect_a @ bot_bot_a_o ) ) ).
% bot_set_def
thf(fact_1069_bot__set__def,axiom,
( bot_bot_set_set_a
= ( collect_set_a @ bot_bot_set_a_o ) ) ).
% bot_set_def
thf(fact_1070_singleton__inject,axiom,
! [A: product_prod_a_a,B: product_prod_a_a] :
( ( ( insert4534936382041156343od_a_a @ A @ bot_bo3357376287454694259od_a_a )
= ( insert4534936382041156343od_a_a @ B @ bot_bo3357376287454694259od_a_a ) )
=> ( A = B ) ) ).
% singleton_inject
thf(fact_1071_singleton__inject,axiom,
! [A: a,B: a] :
( ( ( insert_a @ A @ bot_bot_set_a )
= ( insert_a @ B @ bot_bot_set_a ) )
=> ( A = B ) ) ).
% singleton_inject
thf(fact_1072_singleton__inject,axiom,
! [A: set_a,B: set_a] :
( ( ( insert_set_a @ A @ bot_bot_set_set_a )
= ( insert_set_a @ B @ bot_bot_set_set_a ) )
=> ( A = B ) ) ).
% singleton_inject
thf(fact_1073_insert__not__empty,axiom,
! [A: product_prod_a_a,A2: set_Product_prod_a_a] :
( ( insert4534936382041156343od_a_a @ A @ A2 )
!= bot_bo3357376287454694259od_a_a ) ).
% insert_not_empty
thf(fact_1074_insert__not__empty,axiom,
! [A: a,A2: set_a] :
( ( insert_a @ A @ A2 )
!= bot_bot_set_a ) ).
% insert_not_empty
thf(fact_1075_insert__not__empty,axiom,
! [A: set_a,A2: set_set_a] :
( ( insert_set_a @ A @ A2 )
!= bot_bot_set_set_a ) ).
% insert_not_empty
thf(fact_1076_doubleton__eq__iff,axiom,
! [A: product_prod_a_a,B: product_prod_a_a,C: product_prod_a_a,D2: product_prod_a_a] :
( ( ( insert4534936382041156343od_a_a @ A @ ( insert4534936382041156343od_a_a @ B @ bot_bo3357376287454694259od_a_a ) )
= ( insert4534936382041156343od_a_a @ C @ ( insert4534936382041156343od_a_a @ D2 @ bot_bo3357376287454694259od_a_a ) ) )
= ( ( ( A = C )
& ( B = D2 ) )
| ( ( A = D2 )
& ( B = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_1077_doubleton__eq__iff,axiom,
! [A: a,B: a,C: a,D2: a] :
( ( ( insert_a @ A @ ( insert_a @ B @ bot_bot_set_a ) )
= ( insert_a @ C @ ( insert_a @ D2 @ bot_bot_set_a ) ) )
= ( ( ( A = C )
& ( B = D2 ) )
| ( ( A = D2 )
& ( B = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_1078_doubleton__eq__iff,axiom,
! [A: set_a,B: set_a,C: set_a,D2: set_a] :
( ( ( insert_set_a @ A @ ( insert_set_a @ B @ bot_bot_set_set_a ) )
= ( insert_set_a @ C @ ( insert_set_a @ D2 @ bot_bot_set_set_a ) ) )
= ( ( ( A = C )
& ( B = D2 ) )
| ( ( A = D2 )
& ( B = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_1079_singleton__iff,axiom,
! [B: nat,A: nat] :
( ( member_nat @ B @ ( insert_nat @ A @ bot_bot_set_nat ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_1080_singleton__iff,axiom,
! [B: product_prod_a_a,A: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ B @ ( insert4534936382041156343od_a_a @ A @ bot_bo3357376287454694259od_a_a ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_1081_singleton__iff,axiom,
! [B: a,A: a] :
( ( member_a @ B @ ( insert_a @ A @ bot_bot_set_a ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_1082_singleton__iff,axiom,
! [B: set_a,A: set_a] :
( ( member_set_a @ B @ ( insert_set_a @ A @ bot_bot_set_set_a ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_1083_singletonD,axiom,
! [B: nat,A: nat] :
( ( member_nat @ B @ ( insert_nat @ A @ bot_bot_set_nat ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_1084_singletonD,axiom,
! [B: product_prod_a_a,A: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ B @ ( insert4534936382041156343od_a_a @ A @ bot_bo3357376287454694259od_a_a ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_1085_singletonD,axiom,
! [B: a,A: a] :
( ( member_a @ B @ ( insert_a @ A @ bot_bot_set_a ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_1086_singletonD,axiom,
! [B: set_a,A: set_a] :
( ( member_set_a @ B @ ( insert_set_a @ A @ bot_bot_set_set_a ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_1087_subset__insertI2,axiom,
! [A2: set_set_a,B2: set_set_a,B: set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B2 )
=> ( ord_le3724670747650509150_set_a @ A2 @ ( insert_set_a @ B @ B2 ) ) ) ).
% subset_insertI2
thf(fact_1088_subset__insertI2,axiom,
! [A2: set_a,B2: set_a,B: a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ord_less_eq_set_a @ A2 @ ( insert_a @ B @ B2 ) ) ) ).
% subset_insertI2
thf(fact_1089_subset__insertI2,axiom,
! [A2: set_Product_prod_a_a,B2: set_Product_prod_a_a,B: product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A2 @ B2 )
=> ( ord_le746702958409616551od_a_a @ A2 @ ( insert4534936382041156343od_a_a @ B @ B2 ) ) ) ).
% subset_insertI2
thf(fact_1090_subset__insertI,axiom,
! [B2: set_set_a,A: set_a] : ( ord_le3724670747650509150_set_a @ B2 @ ( insert_set_a @ A @ B2 ) ) ).
% subset_insertI
thf(fact_1091_subset__insertI,axiom,
! [B2: set_a,A: a] : ( ord_less_eq_set_a @ B2 @ ( insert_a @ A @ B2 ) ) ).
% subset_insertI
thf(fact_1092_subset__insertI,axiom,
! [B2: set_Product_prod_a_a,A: product_prod_a_a] : ( ord_le746702958409616551od_a_a @ B2 @ ( insert4534936382041156343od_a_a @ A @ B2 ) ) ).
% subset_insertI
thf(fact_1093_subset__insert,axiom,
! [X: nat,A2: set_nat,B2: set_nat] :
( ~ ( member_nat @ X @ A2 )
=> ( ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ X @ B2 ) )
= ( ord_less_eq_set_nat @ A2 @ B2 ) ) ) ).
% subset_insert
thf(fact_1094_subset__insert,axiom,
! [X: set_a,A2: set_set_a,B2: set_set_a] :
( ~ ( member_set_a @ X @ A2 )
=> ( ( ord_le3724670747650509150_set_a @ A2 @ ( insert_set_a @ X @ B2 ) )
= ( ord_le3724670747650509150_set_a @ A2 @ B2 ) ) ) ).
% subset_insert
thf(fact_1095_subset__insert,axiom,
! [X: a,A2: set_a,B2: set_a] :
( ~ ( member_a @ X @ A2 )
=> ( ( ord_less_eq_set_a @ A2 @ ( insert_a @ X @ B2 ) )
= ( ord_less_eq_set_a @ A2 @ B2 ) ) ) ).
% subset_insert
thf(fact_1096_subset__insert,axiom,
! [X: product_prod_a_a,A2: set_Product_prod_a_a,B2: set_Product_prod_a_a] :
( ~ ( member1426531477525435216od_a_a @ X @ A2 )
=> ( ( ord_le746702958409616551od_a_a @ A2 @ ( insert4534936382041156343od_a_a @ X @ B2 ) )
= ( ord_le746702958409616551od_a_a @ A2 @ B2 ) ) ) ).
% subset_insert
thf(fact_1097_insert__mono,axiom,
! [C2: set_set_a,D: set_set_a,A: set_a] :
( ( ord_le3724670747650509150_set_a @ C2 @ D )
=> ( ord_le3724670747650509150_set_a @ ( insert_set_a @ A @ C2 ) @ ( insert_set_a @ A @ D ) ) ) ).
% insert_mono
thf(fact_1098_insert__mono,axiom,
! [C2: set_a,D: set_a,A: a] :
( ( ord_less_eq_set_a @ C2 @ D )
=> ( ord_less_eq_set_a @ ( insert_a @ A @ C2 ) @ ( insert_a @ A @ D ) ) ) ).
% insert_mono
thf(fact_1099_insert__mono,axiom,
! [C2: set_Product_prod_a_a,D: set_Product_prod_a_a,A: product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ C2 @ D )
=> ( ord_le746702958409616551od_a_a @ ( insert4534936382041156343od_a_a @ A @ C2 ) @ ( insert4534936382041156343od_a_a @ A @ D ) ) ) ).
% insert_mono
thf(fact_1100_Int__insert__right,axiom,
! [A: set_a,A2: set_set_a,B2: set_set_a] :
( ( ( member_set_a @ A @ A2 )
=> ( ( inf_inf_set_set_a @ A2 @ ( insert_set_a @ A @ B2 ) )
= ( insert_set_a @ A @ ( inf_inf_set_set_a @ A2 @ B2 ) ) ) )
& ( ~ ( member_set_a @ A @ A2 )
=> ( ( inf_inf_set_set_a @ A2 @ ( insert_set_a @ A @ B2 ) )
= ( inf_inf_set_set_a @ A2 @ B2 ) ) ) ) ).
% Int_insert_right
thf(fact_1101_Int__insert__right,axiom,
! [A: product_prod_a_a,A2: set_Product_prod_a_a,B2: set_Product_prod_a_a] :
( ( ( member1426531477525435216od_a_a @ A @ A2 )
=> ( ( inf_in8905007599844390133od_a_a @ A2 @ ( insert4534936382041156343od_a_a @ A @ B2 ) )
= ( insert4534936382041156343od_a_a @ A @ ( inf_in8905007599844390133od_a_a @ A2 @ B2 ) ) ) )
& ( ~ ( member1426531477525435216od_a_a @ A @ A2 )
=> ( ( inf_in8905007599844390133od_a_a @ A2 @ ( insert4534936382041156343od_a_a @ A @ B2 ) )
= ( inf_in8905007599844390133od_a_a @ A2 @ B2 ) ) ) ) ).
% Int_insert_right
thf(fact_1102_Int__insert__right,axiom,
! [A: nat,A2: set_nat,B2: set_nat] :
( ( ( member_nat @ A @ A2 )
=> ( ( inf_inf_set_nat @ A2 @ ( insert_nat @ A @ B2 ) )
= ( insert_nat @ A @ ( inf_inf_set_nat @ A2 @ B2 ) ) ) )
& ( ~ ( member_nat @ A @ A2 )
=> ( ( inf_inf_set_nat @ A2 @ ( insert_nat @ A @ B2 ) )
= ( inf_inf_set_nat @ A2 @ B2 ) ) ) ) ).
% Int_insert_right
thf(fact_1103_Int__insert__right,axiom,
! [A: a,A2: set_a,B2: set_a] :
( ( ( member_a @ A @ A2 )
=> ( ( inf_inf_set_a @ A2 @ ( insert_a @ A @ B2 ) )
= ( insert_a @ A @ ( inf_inf_set_a @ A2 @ B2 ) ) ) )
& ( ~ ( member_a @ A @ A2 )
=> ( ( inf_inf_set_a @ A2 @ ( insert_a @ A @ B2 ) )
= ( inf_inf_set_a @ A2 @ B2 ) ) ) ) ).
% Int_insert_right
thf(fact_1104_Int__insert__left,axiom,
! [A: set_a,C2: set_set_a,B2: set_set_a] :
( ( ( member_set_a @ A @ C2 )
=> ( ( inf_inf_set_set_a @ ( insert_set_a @ A @ B2 ) @ C2 )
= ( insert_set_a @ A @ ( inf_inf_set_set_a @ B2 @ C2 ) ) ) )
& ( ~ ( member_set_a @ A @ C2 )
=> ( ( inf_inf_set_set_a @ ( insert_set_a @ A @ B2 ) @ C2 )
= ( inf_inf_set_set_a @ B2 @ C2 ) ) ) ) ).
% Int_insert_left
thf(fact_1105_Int__insert__left,axiom,
! [A: product_prod_a_a,C2: set_Product_prod_a_a,B2: set_Product_prod_a_a] :
( ( ( member1426531477525435216od_a_a @ A @ C2 )
=> ( ( inf_in8905007599844390133od_a_a @ ( insert4534936382041156343od_a_a @ A @ B2 ) @ C2 )
= ( insert4534936382041156343od_a_a @ A @ ( inf_in8905007599844390133od_a_a @ B2 @ C2 ) ) ) )
& ( ~ ( member1426531477525435216od_a_a @ A @ C2 )
=> ( ( inf_in8905007599844390133od_a_a @ ( insert4534936382041156343od_a_a @ A @ B2 ) @ C2 )
= ( inf_in8905007599844390133od_a_a @ B2 @ C2 ) ) ) ) ).
% Int_insert_left
thf(fact_1106_Int__insert__left,axiom,
! [A: nat,C2: set_nat,B2: set_nat] :
( ( ( member_nat @ A @ C2 )
=> ( ( inf_inf_set_nat @ ( insert_nat @ A @ B2 ) @ C2 )
= ( insert_nat @ A @ ( inf_inf_set_nat @ B2 @ C2 ) ) ) )
& ( ~ ( member_nat @ A @ C2 )
=> ( ( inf_inf_set_nat @ ( insert_nat @ A @ B2 ) @ C2 )
= ( inf_inf_set_nat @ B2 @ C2 ) ) ) ) ).
% Int_insert_left
thf(fact_1107_Int__insert__left,axiom,
! [A: a,C2: set_a,B2: set_a] :
( ( ( member_a @ A @ C2 )
=> ( ( inf_inf_set_a @ ( insert_a @ A @ B2 ) @ C2 )
= ( insert_a @ A @ ( inf_inf_set_a @ B2 @ C2 ) ) ) )
& ( ~ ( member_a @ A @ C2 )
=> ( ( inf_inf_set_a @ ( insert_a @ A @ B2 ) @ C2 )
= ( inf_inf_set_a @ B2 @ C2 ) ) ) ) ).
% Int_insert_left
thf(fact_1108_ulgraph_Odegree__none,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,V2: set_a] :
( ( undire6886684016831807756_set_a @ Vertices @ Edges )
=> ( ~ ( member_set_a @ V2 @ Vertices )
=> ( ( undire8939077443744732368_set_a @ Edges @ V2 )
= zero_zero_nat ) ) ) ).
% ulgraph.degree_none
thf(fact_1109_ulgraph_Odegree__none,axiom,
! [Vertices: set_Product_prod_a_a,Edges: set_se5735800977113168103od_a_a,V2: product_prod_a_a] :
( ( undire4585262585102564309od_a_a @ Vertices @ Edges )
=> ( ~ ( member1426531477525435216od_a_a @ V2 @ Vertices )
=> ( ( undire1436394852029823897od_a_a @ Edges @ V2 )
= zero_zero_nat ) ) ) ).
% ulgraph.degree_none
thf(fact_1110_ulgraph_Odegree__none,axiom,
! [Vertices: set_nat,Edges: set_set_nat,V2: nat] :
( ( undire3269267262472140706ph_nat @ Vertices @ Edges )
=> ( ~ ( member_nat @ V2 @ Vertices )
=> ( ( undire6581030323043281630ee_nat @ Edges @ V2 )
= zero_zero_nat ) ) ) ).
% ulgraph.degree_none
thf(fact_1111_ulgraph_Odegree__none,axiom,
! [Vertices: set_a,Edges: set_set_a,V2: a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ~ ( member_a @ V2 @ Vertices )
=> ( ( undire8867928226783802224gree_a @ Edges @ V2 )
= zero_zero_nat ) ) ) ).
% ulgraph.degree_none
thf(fact_1112_subset__singletonD,axiom,
! [A2: set_set_a,X: set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ ( insert_set_a @ X @ bot_bot_set_set_a ) )
=> ( ( A2 = bot_bot_set_set_a )
| ( A2
= ( insert_set_a @ X @ bot_bot_set_set_a ) ) ) ) ).
% subset_singletonD
thf(fact_1113_subset__singletonD,axiom,
! [A2: set_a,X: a] :
( ( ord_less_eq_set_a @ A2 @ ( insert_a @ X @ bot_bot_set_a ) )
=> ( ( A2 = bot_bot_set_a )
| ( A2
= ( insert_a @ X @ bot_bot_set_a ) ) ) ) ).
% subset_singletonD
thf(fact_1114_subset__singletonD,axiom,
! [A2: set_Product_prod_a_a,X: product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A2 @ ( insert4534936382041156343od_a_a @ X @ bot_bo3357376287454694259od_a_a ) )
=> ( ( A2 = bot_bo3357376287454694259od_a_a )
| ( A2
= ( insert4534936382041156343od_a_a @ X @ bot_bo3357376287454694259od_a_a ) ) ) ) ).
% subset_singletonD
thf(fact_1115_subset__singleton__iff,axiom,
! [X5: set_set_a,A: set_a] :
( ( ord_le3724670747650509150_set_a @ X5 @ ( insert_set_a @ A @ bot_bot_set_set_a ) )
= ( ( X5 = bot_bot_set_set_a )
| ( X5
= ( insert_set_a @ A @ bot_bot_set_set_a ) ) ) ) ).
% subset_singleton_iff
thf(fact_1116_subset__singleton__iff,axiom,
! [X5: set_a,A: a] :
( ( ord_less_eq_set_a @ X5 @ ( insert_a @ A @ bot_bot_set_a ) )
= ( ( X5 = bot_bot_set_a )
| ( X5
= ( insert_a @ A @ bot_bot_set_a ) ) ) ) ).
% subset_singleton_iff
thf(fact_1117_subset__singleton__iff,axiom,
! [X5: set_Product_prod_a_a,A: product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ X5 @ ( insert4534936382041156343od_a_a @ A @ bot_bo3357376287454694259od_a_a ) )
= ( ( X5 = bot_bo3357376287454694259od_a_a )
| ( X5
= ( insert4534936382041156343od_a_a @ A @ bot_bo3357376287454694259od_a_a ) ) ) ) ).
% subset_singleton_iff
thf(fact_1118_comp__sgraph_Osingleton__not__edge,axiom,
! [X: product_prod_a_a,S: set_Product_prod_a_a] :
~ ( member1816616512716248880od_a_a @ ( insert4534936382041156343od_a_a @ X @ bot_bo3357376287454694259od_a_a ) @ ( undire6879232364018543115od_a_a @ S ) ) ).
% comp_sgraph.singleton_not_edge
thf(fact_1119_comp__sgraph_Osingleton__not__edge,axiom,
! [X: a,S: set_a] :
~ ( member_set_a @ ( insert_a @ X @ bot_bot_set_a ) @ ( undire2918257014606996450dges_a @ S ) ) ).
% comp_sgraph.singleton_not_edge
thf(fact_1120_comp__sgraph_Osingleton__not__edge,axiom,
! [X: set_a,S: set_set_a] :
~ ( member_set_set_a @ ( insert_set_a @ X @ bot_bot_set_set_a ) @ ( undire8247866692393712962_set_a @ S ) ) ).
% comp_sgraph.singleton_not_edge
thf(fact_1121_comp__sgraph_Owellformed__alt__fst,axiom,
! [X: nat,Y2: nat,S: set_nat] :
( ( member_set_nat @ ( insert_nat @ X @ ( insert_nat @ Y2 @ bot_bot_set_nat ) ) @ ( undire463345858124014060es_nat @ S ) )
=> ( member_nat @ X @ S ) ) ).
% comp_sgraph.wellformed_alt_fst
thf(fact_1122_comp__sgraph_Owellformed__alt__fst,axiom,
! [X: product_prod_a_a,Y2: product_prod_a_a,S: set_Product_prod_a_a] :
( ( member1816616512716248880od_a_a @ ( insert4534936382041156343od_a_a @ X @ ( insert4534936382041156343od_a_a @ Y2 @ bot_bo3357376287454694259od_a_a ) ) @ ( undire6879232364018543115od_a_a @ S ) )
=> ( member1426531477525435216od_a_a @ X @ S ) ) ).
% comp_sgraph.wellformed_alt_fst
thf(fact_1123_comp__sgraph_Owellformed__alt__fst,axiom,
! [X: a,Y2: a,S: set_a] :
( ( member_set_a @ ( insert_a @ X @ ( insert_a @ Y2 @ bot_bot_set_a ) ) @ ( undire2918257014606996450dges_a @ S ) )
=> ( member_a @ X @ S ) ) ).
% comp_sgraph.wellformed_alt_fst
thf(fact_1124_comp__sgraph_Owellformed__alt__fst,axiom,
! [X: set_a,Y2: set_a,S: set_set_a] :
( ( member_set_set_a @ ( insert_set_a @ X @ ( insert_set_a @ Y2 @ bot_bot_set_set_a ) ) @ ( undire8247866692393712962_set_a @ S ) )
=> ( member_set_a @ X @ S ) ) ).
% comp_sgraph.wellformed_alt_fst
thf(fact_1125_comp__sgraph_Owellformed__alt__snd,axiom,
! [X: nat,Y2: nat,S: set_nat] :
( ( member_set_nat @ ( insert_nat @ X @ ( insert_nat @ Y2 @ bot_bot_set_nat ) ) @ ( undire463345858124014060es_nat @ S ) )
=> ( member_nat @ Y2 @ S ) ) ).
% comp_sgraph.wellformed_alt_snd
thf(fact_1126_comp__sgraph_Owellformed__alt__snd,axiom,
! [X: product_prod_a_a,Y2: product_prod_a_a,S: set_Product_prod_a_a] :
( ( member1816616512716248880od_a_a @ ( insert4534936382041156343od_a_a @ X @ ( insert4534936382041156343od_a_a @ Y2 @ bot_bo3357376287454694259od_a_a ) ) @ ( undire6879232364018543115od_a_a @ S ) )
=> ( member1426531477525435216od_a_a @ Y2 @ S ) ) ).
% comp_sgraph.wellformed_alt_snd
thf(fact_1127_comp__sgraph_Owellformed__alt__snd,axiom,
! [X: a,Y2: a,S: set_a] :
( ( member_set_a @ ( insert_a @ X @ ( insert_a @ Y2 @ bot_bot_set_a ) ) @ ( undire2918257014606996450dges_a @ S ) )
=> ( member_a @ Y2 @ S ) ) ).
% comp_sgraph.wellformed_alt_snd
thf(fact_1128_comp__sgraph_Owellformed__alt__snd,axiom,
! [X: set_a,Y2: set_a,S: set_set_a] :
( ( member_set_set_a @ ( insert_set_a @ X @ ( insert_set_a @ Y2 @ bot_bot_set_set_a ) ) @ ( undire8247866692393712962_set_a @ S ) )
=> ( member_set_a @ Y2 @ S ) ) ).
% comp_sgraph.wellformed_alt_snd
thf(fact_1129_graph__system_Owellformed__alt__fst,axiom,
! [Vertices: set_nat,Edges: set_set_nat,X: nat,Y2: nat] :
( ( undire7481384412329822504em_nat @ Vertices @ Edges )
=> ( ( member_set_nat @ ( insert_nat @ X @ ( insert_nat @ Y2 @ bot_bot_set_nat ) ) @ Edges )
=> ( member_nat @ X @ Vertices ) ) ) ).
% graph_system.wellformed_alt_fst
thf(fact_1130_graph__system_Owellformed__alt__fst,axiom,
! [Vertices: set_Product_prod_a_a,Edges: set_se5735800977113168103od_a_a,X: product_prod_a_a,Y2: product_prod_a_a] :
( ( undire1860116983885411791od_a_a @ Vertices @ Edges )
=> ( ( member1816616512716248880od_a_a @ ( insert4534936382041156343od_a_a @ X @ ( insert4534936382041156343od_a_a @ Y2 @ bot_bo3357376287454694259od_a_a ) ) @ Edges )
=> ( member1426531477525435216od_a_a @ X @ Vertices ) ) ) ).
% graph_system.wellformed_alt_fst
thf(fact_1131_graph__system_Owellformed__alt__fst,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,X: set_a,Y2: set_a] :
( ( undire7159349782766787846_set_a @ Vertices @ Edges )
=> ( ( member_set_set_a @ ( insert_set_a @ X @ ( insert_set_a @ Y2 @ bot_bot_set_set_a ) ) @ Edges )
=> ( member_set_a @ X @ Vertices ) ) ) ).
% graph_system.wellformed_alt_fst
thf(fact_1132_graph__system_Owellformed__alt__fst,axiom,
! [Vertices: set_a,Edges: set_set_a,X: a,Y2: a] :
( ( undire2554140024507503526stem_a @ Vertices @ Edges )
=> ( ( member_set_a @ ( insert_a @ X @ ( insert_a @ Y2 @ bot_bot_set_a ) ) @ Edges )
=> ( member_a @ X @ Vertices ) ) ) ).
% graph_system.wellformed_alt_fst
thf(fact_1133_graph__system_Owellformed__alt__snd,axiom,
! [Vertices: set_nat,Edges: set_set_nat,X: nat,Y2: nat] :
( ( undire7481384412329822504em_nat @ Vertices @ Edges )
=> ( ( member_set_nat @ ( insert_nat @ X @ ( insert_nat @ Y2 @ bot_bot_set_nat ) ) @ Edges )
=> ( member_nat @ Y2 @ Vertices ) ) ) ).
% graph_system.wellformed_alt_snd
thf(fact_1134_graph__system_Owellformed__alt__snd,axiom,
! [Vertices: set_Product_prod_a_a,Edges: set_se5735800977113168103od_a_a,X: product_prod_a_a,Y2: product_prod_a_a] :
( ( undire1860116983885411791od_a_a @ Vertices @ Edges )
=> ( ( member1816616512716248880od_a_a @ ( insert4534936382041156343od_a_a @ X @ ( insert4534936382041156343od_a_a @ Y2 @ bot_bo3357376287454694259od_a_a ) ) @ Edges )
=> ( member1426531477525435216od_a_a @ Y2 @ Vertices ) ) ) ).
% graph_system.wellformed_alt_snd
thf(fact_1135_graph__system_Owellformed__alt__snd,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,X: set_a,Y2: set_a] :
( ( undire7159349782766787846_set_a @ Vertices @ Edges )
=> ( ( member_set_set_a @ ( insert_set_a @ X @ ( insert_set_a @ Y2 @ bot_bot_set_set_a ) ) @ Edges )
=> ( member_set_a @ Y2 @ Vertices ) ) ) ).
% graph_system.wellformed_alt_snd
thf(fact_1136_graph__system_Owellformed__alt__snd,axiom,
! [Vertices: set_a,Edges: set_set_a,X: a,Y2: a] :
( ( undire2554140024507503526stem_a @ Vertices @ Edges )
=> ( ( member_set_a @ ( insert_a @ X @ ( insert_a @ Y2 @ bot_bot_set_a ) ) @ Edges )
=> ( member_a @ Y2 @ Vertices ) ) ) ).
% graph_system.wellformed_alt_snd
thf(fact_1137_comp__sgraph_Oinduced__edges__alt,axiom,
! [S: set_a,V: set_a] :
( ( undire7777452895879145676dges_a @ ( undire2918257014606996450dges_a @ S ) @ V )
= ( inf_inf_set_set_a @ ( undire2918257014606996450dges_a @ S ) @ ( undire2918257014606996450dges_a @ V ) ) ) ).
% comp_sgraph.induced_edges_alt
thf(fact_1138_comp__sgraph_Owalk__edges_Osimps_I3_J,axiom,
! [X: product_prod_a_a,Y2: product_prod_a_a,Ys2: list_P1396940483166286381od_a_a] :
( ( undire4403264684974754359od_a_a @ ( cons_P7316939126706565853od_a_a @ X @ ( cons_P7316939126706565853od_a_a @ Y2 @ Ys2 ) ) )
= ( cons_s5735289257037284029od_a_a @ ( insert4534936382041156343od_a_a @ X @ ( insert4534936382041156343od_a_a @ Y2 @ bot_bo3357376287454694259od_a_a ) ) @ ( undire4403264684974754359od_a_a @ ( cons_P7316939126706565853od_a_a @ Y2 @ Ys2 ) ) ) ) ).
% comp_sgraph.walk_edges.simps(3)
thf(fact_1139_comp__sgraph_Owalk__edges_Osimps_I3_J,axiom,
! [X: set_a,Y2: set_a,Ys2: list_set_a] :
( ( undire6234387080713648494_set_a @ ( cons_set_a @ X @ ( cons_set_a @ Y2 @ Ys2 ) ) )
= ( cons_set_set_a @ ( insert_set_a @ X @ ( insert_set_a @ Y2 @ bot_bot_set_set_a ) ) @ ( undire6234387080713648494_set_a @ ( cons_set_a @ Y2 @ Ys2 ) ) ) ) ).
% comp_sgraph.walk_edges.simps(3)
thf(fact_1140_comp__sgraph_Owalk__edges_Osimps_I3_J,axiom,
! [X: a,Y2: a,Ys2: list_a] :
( ( undire7337870655677353998dges_a @ ( cons_a @ X @ ( cons_a @ Y2 @ Ys2 ) ) )
= ( cons_set_a @ ( insert_a @ X @ ( insert_a @ Y2 @ bot_bot_set_a ) ) @ ( undire7337870655677353998dges_a @ ( cons_a @ Y2 @ Ys2 ) ) ) ) ).
% comp_sgraph.walk_edges.simps(3)
thf(fact_1141_comp__sgraph_Onot__vert__adj,axiom,
! [S: set_Product_prod_a_a,V2: product_prod_a_a,U: product_prod_a_a] :
( ~ ( undire6135774327024169009od_a_a @ ( undire6879232364018543115od_a_a @ S ) @ V2 @ U )
=> ~ ( member1816616512716248880od_a_a @ ( insert4534936382041156343od_a_a @ V2 @ ( insert4534936382041156343od_a_a @ U @ bot_bo3357376287454694259od_a_a ) ) @ ( undire6879232364018543115od_a_a @ S ) ) ) ).
% comp_sgraph.not_vert_adj
thf(fact_1142_comp__sgraph_Onot__vert__adj,axiom,
! [S: set_set_a,V2: set_a,U: set_a] :
( ~ ( undire3510646817838285160_set_a @ ( undire8247866692393712962_set_a @ S ) @ V2 @ U )
=> ~ ( member_set_set_a @ ( insert_set_a @ V2 @ ( insert_set_a @ U @ bot_bot_set_set_a ) ) @ ( undire8247866692393712962_set_a @ S ) ) ) ).
% comp_sgraph.not_vert_adj
thf(fact_1143_comp__sgraph_Onot__vert__adj,axiom,
! [S: set_a,V2: a,U: a] :
( ~ ( undire397441198561214472_adj_a @ ( undire2918257014606996450dges_a @ S ) @ V2 @ U )
=> ~ ( member_set_a @ ( insert_a @ V2 @ ( insert_a @ U @ bot_bot_set_a ) ) @ ( undire2918257014606996450dges_a @ S ) ) ) ).
% comp_sgraph.not_vert_adj
thf(fact_1144_comp__sgraph_Overt__adj__def,axiom,
! [S: set_Product_prod_a_a,V1: product_prod_a_a,V22: product_prod_a_a] :
( ( undire6135774327024169009od_a_a @ ( undire6879232364018543115od_a_a @ S ) @ V1 @ V22 )
= ( member1816616512716248880od_a_a @ ( insert4534936382041156343od_a_a @ V1 @ ( insert4534936382041156343od_a_a @ V22 @ bot_bo3357376287454694259od_a_a ) ) @ ( undire6879232364018543115od_a_a @ S ) ) ) ).
% comp_sgraph.vert_adj_def
thf(fact_1145_comp__sgraph_Overt__adj__def,axiom,
! [S: set_set_a,V1: set_a,V22: set_a] :
( ( undire3510646817838285160_set_a @ ( undire8247866692393712962_set_a @ S ) @ V1 @ V22 )
= ( member_set_set_a @ ( insert_set_a @ V1 @ ( insert_set_a @ V22 @ bot_bot_set_set_a ) ) @ ( undire8247866692393712962_set_a @ S ) ) ) ).
% comp_sgraph.vert_adj_def
thf(fact_1146_comp__sgraph_Overt__adj__def,axiom,
! [S: set_a,V1: a,V22: a] :
( ( undire397441198561214472_adj_a @ ( undire2918257014606996450dges_a @ S ) @ V1 @ V22 )
= ( member_set_a @ ( insert_a @ V1 @ ( insert_a @ V22 @ bot_bot_set_a ) ) @ ( undire2918257014606996450dges_a @ S ) ) ) ).
% comp_sgraph.vert_adj_def
thf(fact_1147_comp__sgraph_Ohas__loop__def,axiom,
! [S: set_Product_prod_a_a,V2: product_prod_a_a] :
( ( undire7777398424729533289od_a_a @ ( undire6879232364018543115od_a_a @ S ) @ V2 )
= ( member1816616512716248880od_a_a @ ( insert4534936382041156343od_a_a @ V2 @ bot_bo3357376287454694259od_a_a ) @ ( undire6879232364018543115od_a_a @ S ) ) ) ).
% comp_sgraph.has_loop_def
thf(fact_1148_comp__sgraph_Ohas__loop__def,axiom,
! [S: set_set_a,V2: set_a] :
( ( undire5774735625301615776_set_a @ ( undire8247866692393712962_set_a @ S ) @ V2 )
= ( member_set_set_a @ ( insert_set_a @ V2 @ bot_bot_set_set_a ) @ ( undire8247866692393712962_set_a @ S ) ) ) ).
% comp_sgraph.has_loop_def
thf(fact_1149_comp__sgraph_Ohas__loop__def,axiom,
! [S: set_a,V2: a] :
( ( undire3617971648856834880loop_a @ ( undire2918257014606996450dges_a @ S ) @ V2 )
= ( member_set_a @ ( insert_a @ V2 @ bot_bot_set_a ) @ ( undire2918257014606996450dges_a @ S ) ) ) ).
% comp_sgraph.has_loop_def
thf(fact_1150_comp__sgraph_Ois__isolated__vertex__degree0,axiom,
! [S: set_a,V2: a] :
( ( undire8931668460104145173rtex_a @ S @ ( undire2918257014606996450dges_a @ S ) @ V2 )
=> ( ( undire8867928226783802224gree_a @ ( undire2918257014606996450dges_a @ S ) @ V2 )
= zero_zero_nat ) ) ).
% comp_sgraph.is_isolated_vertex_degree0
thf(fact_1151_ulgraph_Onot__vert__adj,axiom,
! [Vertices: set_Product_prod_a_a,Edges: set_se5735800977113168103od_a_a,V2: product_prod_a_a,U: product_prod_a_a] :
( ( undire4585262585102564309od_a_a @ Vertices @ Edges )
=> ( ~ ( undire6135774327024169009od_a_a @ Edges @ V2 @ U )
=> ~ ( member1816616512716248880od_a_a @ ( insert4534936382041156343od_a_a @ V2 @ ( insert4534936382041156343od_a_a @ U @ bot_bo3357376287454694259od_a_a ) ) @ Edges ) ) ) ).
% ulgraph.not_vert_adj
thf(fact_1152_ulgraph_Onot__vert__adj,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,V2: set_a,U: set_a] :
( ( undire6886684016831807756_set_a @ Vertices @ Edges )
=> ( ~ ( undire3510646817838285160_set_a @ Edges @ V2 @ U )
=> ~ ( member_set_set_a @ ( insert_set_a @ V2 @ ( insert_set_a @ U @ bot_bot_set_set_a ) ) @ Edges ) ) ) ).
% ulgraph.not_vert_adj
thf(fact_1153_ulgraph_Onot__vert__adj,axiom,
! [Vertices: set_a,Edges: set_set_a,V2: a,U: a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ~ ( undire397441198561214472_adj_a @ Edges @ V2 @ U )
=> ~ ( member_set_a @ ( insert_a @ V2 @ ( insert_a @ U @ bot_bot_set_a ) ) @ Edges ) ) ) ).
% ulgraph.not_vert_adj
thf(fact_1154_ulgraph_Overt__adj__def,axiom,
! [Vertices: set_Product_prod_a_a,Edges: set_se5735800977113168103od_a_a,V1: product_prod_a_a,V22: product_prod_a_a] :
( ( undire4585262585102564309od_a_a @ Vertices @ Edges )
=> ( ( undire6135774327024169009od_a_a @ Edges @ V1 @ V22 )
= ( member1816616512716248880od_a_a @ ( insert4534936382041156343od_a_a @ V1 @ ( insert4534936382041156343od_a_a @ V22 @ bot_bo3357376287454694259od_a_a ) ) @ Edges ) ) ) ).
% ulgraph.vert_adj_def
thf(fact_1155_ulgraph_Overt__adj__def,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,V1: set_a,V22: set_a] :
( ( undire6886684016831807756_set_a @ Vertices @ Edges )
=> ( ( undire3510646817838285160_set_a @ Edges @ V1 @ V22 )
= ( member_set_set_a @ ( insert_set_a @ V1 @ ( insert_set_a @ V22 @ bot_bot_set_set_a ) ) @ Edges ) ) ) ).
% ulgraph.vert_adj_def
thf(fact_1156_ulgraph_Overt__adj__def,axiom,
! [Vertices: set_a,Edges: set_set_a,V1: a,V22: a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( undire397441198561214472_adj_a @ Edges @ V1 @ V22 )
= ( member_set_a @ ( insert_a @ V1 @ ( insert_a @ V22 @ bot_bot_set_a ) ) @ Edges ) ) ) ).
% ulgraph.vert_adj_def
thf(fact_1157_ulgraph_Ohas__loop__def,axiom,
! [Vertices: set_Product_prod_a_a,Edges: set_se5735800977113168103od_a_a,V2: product_prod_a_a] :
( ( undire4585262585102564309od_a_a @ Vertices @ Edges )
=> ( ( undire7777398424729533289od_a_a @ Edges @ V2 )
= ( member1816616512716248880od_a_a @ ( insert4534936382041156343od_a_a @ V2 @ bot_bo3357376287454694259od_a_a ) @ Edges ) ) ) ).
% ulgraph.has_loop_def
thf(fact_1158_ulgraph_Ohas__loop__def,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,V2: set_a] :
( ( undire6886684016831807756_set_a @ Vertices @ Edges )
=> ( ( undire5774735625301615776_set_a @ Edges @ V2 )
= ( member_set_set_a @ ( insert_set_a @ V2 @ bot_bot_set_set_a ) @ Edges ) ) ) ).
% ulgraph.has_loop_def
thf(fact_1159_ulgraph_Ohas__loop__def,axiom,
! [Vertices: set_a,Edges: set_set_a,V2: a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( undire3617971648856834880loop_a @ Edges @ V2 )
= ( member_set_a @ ( insert_a @ V2 @ bot_bot_set_a ) @ Edges ) ) ) ).
% ulgraph.has_loop_def
thf(fact_1160_neighborhood__incident,axiom,
! [U: a,V2: a] :
( ( member_a @ U @ ( undire8504279938402040014hood_a @ vertices @ edges @ V2 ) )
= ( member_set_a @ ( insert_a @ U @ ( insert_a @ V2 @ bot_bot_set_a ) ) @ ( undire3231912044278729248dges_a @ edges @ V2 ) ) ) ).
% neighborhood_incident
thf(fact_1161_degree0__neighborhood__empt__iff,axiom,
! [V2: a] :
( ( finite_finite_set_a @ edges )
=> ( ( ( undire8867928226783802224gree_a @ edges @ V2 )
= zero_zero_nat )
= ( ( undire8504279938402040014hood_a @ vertices @ edges @ V2 )
= bot_bot_set_a ) ) ) ).
% degree0_neighborhood_empt_iff
thf(fact_1162_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_1163_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_1164_incident__loops__simp_I2_J,axiom,
! [V2: a] :
( ~ ( undire3617971648856834880loop_a @ edges @ V2 )
=> ( ( undire4753905205749729249oops_a @ edges @ V2 )
= bot_bot_set_set_a ) ) ).
% incident_loops_simp(2)
thf(fact_1165_finite__incident__edges,axiom,
! [V2: a] :
( ( finite_finite_set_a @ edges )
=> ( finite_finite_set_a @ ( undire3231912044278729248dges_a @ edges @ V2 ) ) ) ).
% finite_incident_edges
thf(fact_1166_finite__incident__loops,axiom,
! [V2: a] : ( finite_finite_set_a @ ( undire4753905205749729249oops_a @ edges @ V2 ) ) ).
% finite_incident_loops
thf(fact_1167_incident__edges__empty,axiom,
! [V2: a] :
( ~ ( member_a @ V2 @ vertices )
=> ( ( undire3231912044278729248dges_a @ edges @ V2 )
= bot_bot_set_set_a ) ) ).
% incident_edges_empty
thf(fact_1168_incident__loops__simp_I1_J,axiom,
! [V2: a] :
( ( undire3617971648856834880loop_a @ edges @ V2 )
=> ( ( undire4753905205749729249oops_a @ edges @ V2 )
= ( insert_set_a @ ( insert_a @ V2 @ bot_bot_set_a ) @ bot_bot_set_set_a ) ) ) ).
% incident_loops_simp(1)
thf(fact_1169_degree0__inc__edges__empt__iff,axiom,
! [V2: a] :
( ( finite_finite_set_a @ edges )
=> ( ( ( undire8867928226783802224gree_a @ edges @ V2 )
= zero_zero_nat )
= ( ( undire3231912044278729248dges_a @ edges @ V2 )
= bot_bot_set_set_a ) ) ) ).
% degree0_inc_edges_empt_iff
thf(fact_1170_bot__nat__def,axiom,
bot_bot_nat = zero_zero_nat ).
% bot_nat_def
thf(fact_1171_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B: nat] :
( ( P @ K )
=> ( ! [Y: nat] :
( ( P @ Y )
=> ( ord_less_eq_nat @ Y @ B ) )
=> ? [X2: nat] :
( ( P @ X2 )
& ! [Y7: nat] :
( ( P @ Y7 )
=> ( ord_less_eq_nat @ Y7 @ X2 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_1172_nat__le__linear,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
| ( ord_less_eq_nat @ N @ M ) ) ).
% nat_le_linear
thf(fact_1173_le__antisym,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( M = N ) ) ) ).
% le_antisym
thf(fact_1174_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% eq_imp_le
thf(fact_1175_le__trans,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ J @ K )
=> ( ord_less_eq_nat @ I @ K ) ) ) ).
% le_trans
thf(fact_1176_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_1177_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_1178_bot__nat__0_Oextremum__uniqueI,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_uniqueI
thf(fact_1179_bot__nat__0_Oextremum__unique,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
= ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_unique
thf(fact_1180_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_1181_walk__edges__append__union,axiom,
! [Xs: list_a,Ys2: list_a] :
( ( Xs != nil_a )
=> ( ( Ys2 != nil_a )
=> ( ( set_set_a2 @ ( undire7337870655677353998dges_a @ ( append_a @ Xs @ Ys2 ) ) )
= ( sup_sup_set_set_a @ ( sup_sup_set_set_a @ ( set_set_a2 @ ( undire7337870655677353998dges_a @ Xs ) ) @ ( set_set_a2 @ ( undire7337870655677353998dges_a @ Ys2 ) ) ) @ ( insert_set_a @ ( insert_a @ ( last_a @ Xs ) @ ( insert_a @ ( hd_a @ Ys2 ) @ bot_bot_set_a ) ) @ bot_bot_set_set_a ) ) ) ) ) ).
% walk_edges_append_union
thf(fact_1182_finite__inc__sedges,axiom,
! [V2: a] :
( ( finite_finite_set_a @ edges )
=> ( finite_finite_set_a @ ( undire1270416042309875431dges_a @ edges @ V2 ) ) ) ).
% finite_inc_sedges
thf(fact_1183_all__edges__betw__I,axiom,
! [X: a,X5: set_a,Y2: a,Y5: set_a] :
( ( member_a @ X @ X5 )
=> ( ( member_a @ Y2 @ Y5 )
=> ( ( member_set_a @ ( insert_a @ X @ ( insert_a @ Y2 @ bot_bot_set_a ) ) @ edges )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y2 ) @ ( undire8383842906760478443ween_a @ edges @ X5 @ Y5 ) ) ) ) ) ).
% all_edges_betw_I
thf(fact_1184_all__edges__betw__D3,axiom,
! [X: a,Y2: a,X5: set_a,Y5: set_a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y2 ) @ ( undire8383842906760478443ween_a @ edges @ X5 @ Y5 ) )
=> ( member_set_a @ ( insert_a @ X @ ( insert_a @ Y2 @ bot_bot_set_a ) ) @ edges ) ) ).
% all_edges_betw_D3
thf(fact_1185_finite__all__edges__between,axiom,
! [X5: set_a,Y5: set_a] :
( ( finite_finite_a @ X5 )
=> ( ( finite_finite_a @ Y5 )
=> ( finite6544458595007987280od_a_a @ ( undire8383842906760478443ween_a @ edges @ X5 @ Y5 ) ) ) ) ).
% finite_all_edges_between
thf(fact_1186_incident__edges__union,axiom,
! [V2: a] :
( ( undire3231912044278729248dges_a @ edges @ V2 )
= ( sup_sup_set_set_a @ ( undire1270416042309875431dges_a @ edges @ V2 ) @ ( undire4753905205749729249oops_a @ edges @ V2 ) ) ) ).
% incident_edges_union
thf(fact_1187_incident__edges__sedges,axiom,
! [V2: a] :
( ~ ( undire3617971648856834880loop_a @ edges @ V2 )
=> ( ( undire3231912044278729248dges_a @ edges @ V2 )
= ( undire1270416042309875431dges_a @ edges @ V2 ) ) ) ).
% incident_edges_sedges
thf(fact_1188_incident__sedges__empty,axiom,
! [V2: a] :
( ~ ( member_a @ V2 @ vertices )
=> ( ( undire1270416042309875431dges_a @ edges @ V2 )
= bot_bot_set_set_a ) ) ).
% incident_sedges_empty
thf(fact_1189_induced__edges__union__subgraph__single,axiom,
! [VH1: set_a,S: set_a,VH2: set_a,T2: set_a,EH1: set_set_a,EH2: set_set_a] :
( ( ord_less_eq_set_a @ VH1 @ S )
=> ( ( ord_less_eq_set_a @ VH2 @ T2 )
=> ( ( undire2554140024507503526stem_a @ VH1 @ EH1 )
=> ( ( undire2554140024507503526stem_a @ VH2 @ EH2 )
=> ( ( undire7103218114511261257raph_a @ ( sup_sup_set_a @ VH1 @ VH2 ) @ ( sup_sup_set_set_a @ EH1 @ EH2 ) @ ( sup_sup_set_a @ S @ T2 ) @ ( undire7777452895879145676dges_a @ edges @ ( sup_sup_set_a @ S @ T2 ) ) )
=> ( undire7103218114511261257raph_a @ VH1 @ EH1 @ S @ ( undire7777452895879145676dges_a @ edges @ S ) ) ) ) ) ) ) ).
% induced_edges_union_subgraph_single
thf(fact_1190_induced__union__subgraph,axiom,
! [VH1: set_a,S: set_a,VH2: set_a,T2: set_a,EH1: set_set_a,EH2: set_set_a] :
( ( ord_less_eq_set_a @ VH1 @ S )
=> ( ( ord_less_eq_set_a @ VH2 @ T2 )
=> ( ( undire2554140024507503526stem_a @ VH1 @ EH1 )
=> ( ( undire2554140024507503526stem_a @ VH2 @ EH2 )
=> ( ( ( undire7103218114511261257raph_a @ VH1 @ EH1 @ S @ ( undire7777452895879145676dges_a @ edges @ S ) )
& ( undire7103218114511261257raph_a @ VH2 @ EH2 @ T2 @ ( undire7777452895879145676dges_a @ edges @ T2 ) ) )
= ( undire7103218114511261257raph_a @ ( sup_sup_set_a @ VH1 @ VH2 ) @ ( sup_sup_set_set_a @ EH1 @ EH2 ) @ ( sup_sup_set_a @ S @ T2 ) @ ( undire7777452895879145676dges_a @ edges @ ( sup_sup_set_a @ S @ T2 ) ) ) ) ) ) ) ) ).
% induced_union_subgraph
thf(fact_1191_induced__edges__union,axiom,
! [VH1: set_a,S: set_a,VH2: set_a,T2: set_a,EH1: set_set_a,EH2: set_set_a] :
( ( ord_less_eq_set_a @ VH1 @ S )
=> ( ( ord_less_eq_set_a @ VH2 @ T2 )
=> ( ( undire2554140024507503526stem_a @ VH1 @ EH1 )
=> ( ( undire2554140024507503526stem_a @ VH2 @ EH2 )
=> ( ( ord_le3724670747650509150_set_a @ ( sup_sup_set_set_a @ EH1 @ EH2 ) @ ( undire7777452895879145676dges_a @ edges @ ( sup_sup_set_a @ S @ T2 ) ) )
=> ( ord_le3724670747650509150_set_a @ EH1 @ ( undire7777452895879145676dges_a @ edges @ S ) ) ) ) ) ) ) ).
% induced_edges_union
thf(fact_1192_all__edges__between__Un2,axiom,
! [X5: set_a,Y5: set_a,Z4: set_a] :
( ( undire8383842906760478443ween_a @ edges @ X5 @ ( sup_sup_set_a @ Y5 @ Z4 ) )
= ( sup_su3048258781599657691od_a_a @ ( undire8383842906760478443ween_a @ edges @ X5 @ Y5 ) @ ( undire8383842906760478443ween_a @ edges @ X5 @ Z4 ) ) ) ).
% all_edges_between_Un2
thf(fact_1193_all__edges__between__Un1,axiom,
! [X5: set_a,Y5: set_a,Z4: set_a] :
( ( undire8383842906760478443ween_a @ edges @ ( sup_sup_set_a @ X5 @ Y5 ) @ Z4 )
= ( sup_su3048258781599657691od_a_a @ ( undire8383842906760478443ween_a @ edges @ X5 @ Z4 ) @ ( undire8383842906760478443ween_a @ edges @ Y5 @ Z4 ) ) ) ).
% all_edges_between_Un1
thf(fact_1194_walk__length__app,axiom,
! [Xs: list_a,Ys2: list_a] :
( ( Xs != nil_a )
=> ( ( Ys2 != nil_a )
=> ( ( undire8849074589633906640ngth_a @ ( append_a @ Xs @ Ys2 ) )
= ( plus_plus_nat @ ( plus_plus_nat @ ( undire8849074589633906640ngth_a @ Xs ) @ ( undire8849074589633906640ngth_a @ Ys2 ) ) @ one_one_nat ) ) ) ) ).
% walk_length_app
thf(fact_1195_walk__length__app__ineq,axiom,
! [Xs: list_a,Ys2: list_a] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ ( undire8849074589633906640ngth_a @ Xs ) @ ( undire8849074589633906640ngth_a @ Ys2 ) ) @ ( undire8849074589633906640ngth_a @ ( append_a @ Xs @ Ys2 ) ) )
& ( ord_less_eq_nat @ ( undire8849074589633906640ngth_a @ ( append_a @ Xs @ Ys2 ) ) @ ( plus_plus_nat @ ( plus_plus_nat @ ( undire8849074589633906640ngth_a @ Xs ) @ ( undire8849074589633906640ngth_a @ Ys2 ) ) @ one_one_nat ) ) ) ).
% walk_length_app_ineq
thf(fact_1196_Nat_Oadd__0__right,axiom,
! [M: nat] :
( ( plus_plus_nat @ M @ zero_zero_nat )
= M ) ).
% Nat.add_0_right
thf(fact_1197_add__is__0,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= zero_zero_nat )
= ( ( M = zero_zero_nat )
& ( N = zero_zero_nat ) ) ) ).
% add_is_0
thf(fact_1198_nat__add__left__cancel__le,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% nat_add_left_cancel_le
thf(fact_1199_nat__le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [M2: nat,N2: nat] :
? [K2: nat] :
( N2
= ( plus_plus_nat @ M2 @ K2 ) ) ) ) ).
% nat_le_iff_add
thf(fact_1200_trans__le__add2,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ M @ J ) ) ) ).
% trans_le_add2
thf(fact_1201_trans__le__add1,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ J @ M ) ) ) ).
% trans_le_add1
thf(fact_1202_add__le__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% add_le_mono1
thf(fact_1203_add__le__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_le_mono
thf(fact_1204_le__Suc__ex,axiom,
! [K: nat,L: nat] :
( ( ord_less_eq_nat @ K @ L )
=> ? [N3: nat] :
( L
= ( plus_plus_nat @ K @ N3 ) ) ) ).
% le_Suc_ex
thf(fact_1205_add__leD2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ( ord_less_eq_nat @ K @ N ) ) ).
% add_leD2
thf(fact_1206_add__leD1,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% add_leD1
thf(fact_1207_le__add2,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M @ N ) ) ).
% le_add2
thf(fact_1208_le__add1,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M ) ) ).
% le_add1
thf(fact_1209_add__leE,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ~ ( ( ord_less_eq_nat @ M @ N )
=> ~ ( ord_less_eq_nat @ K @ N ) ) ) ).
% add_leE
thf(fact_1210_add__eq__self__zero,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= M )
=> ( N = zero_zero_nat ) ) ).
% add_eq_self_zero
thf(fact_1211_plus__nat_Oadd__0,axiom,
! [N: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N )
= N ) ).
% plus_nat.add_0
thf(fact_1212_incident__loops__card,axiom,
! [V2: a] : ( ord_less_eq_nat @ ( finite_card_set_a @ ( undire4753905205749729249oops_a @ edges @ V2 ) ) @ one_one_nat ) ).
% incident_loops_card
thf(fact_1213_degree__no__loops,axiom,
! [V2: a] :
( ~ ( undire3617971648856834880loop_a @ edges @ V2 )
=> ( ( undire8867928226783802224gree_a @ edges @ V2 )
= ( finite_card_set_a @ ( undire3231912044278729248dges_a @ edges @ V2 ) ) ) ) ).
% degree_no_loops
thf(fact_1214_card__incident__sedges__neighborhood,axiom,
! [V2: a] :
( ( finite_card_set_a @ ( undire3231912044278729248dges_a @ edges @ V2 ) )
= ( finite_card_a @ ( undire8504279938402040014hood_a @ vertices @ edges @ V2 ) ) ) ).
% card_incident_sedges_neighborhood
thf(fact_1215_card1__incident__imp__vert,axiom,
! [V2: a,E: set_a] :
( ( ( undire1521409233611534436dent_a @ V2 @ E )
& ( ( finite_card_a @ E )
= one_one_nat ) )
=> ( E
= ( insert_a @ V2 @ bot_bot_set_a ) ) ) ).
% card1_incident_imp_vert
thf(fact_1216_card__all__edges__between__commute,axiom,
! [X5: set_a,Y5: set_a] :
( ( finite4795055649997197647od_a_a @ ( undire8383842906760478443ween_a @ edges @ X5 @ Y5 ) )
= ( finite4795055649997197647od_a_a @ ( undire8383842906760478443ween_a @ edges @ Y5 @ X5 ) ) ) ).
% card_all_edges_between_commute
thf(fact_1217_is__loop__def,axiom,
( undire2905028936066782638loop_a
= ( ^ [E5: set_a] :
( ( finite_card_a @ E5 )
= one_one_nat ) ) ) ).
% is_loop_def
thf(fact_1218_max__all__edges__between,axiom,
! [X5: set_a,Y5: set_a] :
( ( finite_finite_a @ X5 )
=> ( ( finite_finite_a @ Y5 )
=> ( ord_less_eq_nat @ ( finite4795055649997197647od_a_a @ ( undire8383842906760478443ween_a @ edges @ X5 @ Y5 ) ) @ ( times_times_nat @ ( finite_card_a @ X5 ) @ ( finite_card_a @ Y5 ) ) ) ) ) ).
% max_all_edges_between
thf(fact_1219_mult__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ( times_times_nat @ M @ K )
= ( times_times_nat @ N @ K ) )
= ( ( M = N )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel2
thf(fact_1220_mult__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( times_times_nat @ K @ M )
= ( times_times_nat @ K @ N ) )
= ( ( M = N )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel1
thf(fact_1221_mult__0__right,axiom,
! [M: nat] :
( ( times_times_nat @ M @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_0_right
thf(fact_1222_mult__is__0,axiom,
! [M: nat,N: nat] :
( ( ( times_times_nat @ M @ N )
= zero_zero_nat )
= ( ( M = zero_zero_nat )
| ( N = zero_zero_nat ) ) ) ).
% mult_is_0
thf(fact_1223_nat__mult__eq__1__iff,axiom,
! [M: nat,N: nat] :
( ( ( times_times_nat @ M @ N )
= one_one_nat )
= ( ( M = one_one_nat )
& ( N = one_one_nat ) ) ) ).
% nat_mult_eq_1_iff
thf(fact_1224_nat__1__eq__mult__iff,axiom,
! [M: nat,N: nat] :
( ( one_one_nat
= ( times_times_nat @ M @ N ) )
= ( ( M = one_one_nat )
& ( N = one_one_nat ) ) ) ).
% nat_1_eq_mult_iff
thf(fact_1225_mult__eq__self__implies__10,axiom,
! [M: nat,N: nat] :
( ( M
= ( times_times_nat @ M @ N ) )
=> ( ( N = one_one_nat )
| ( M = zero_zero_nat ) ) ) ).
% mult_eq_self_implies_10
thf(fact_1226_mult__le__mono2,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( times_times_nat @ K @ I ) @ ( times_times_nat @ K @ J ) ) ) ).
% mult_le_mono2
thf(fact_1227_mult__le__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ K ) ) ) ).
% mult_le_mono1
thf(fact_1228_mult__le__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ L ) ) ) ) ).
% mult_le_mono
thf(fact_1229_le__square,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ M ) ) ).
% le_square
thf(fact_1230_le__cube,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ ( times_times_nat @ M @ M ) ) ) ).
% le_cube
thf(fact_1231_nat__mult__1__right,axiom,
! [N: nat] :
( ( times_times_nat @ N @ one_one_nat )
= N ) ).
% nat_mult_1_right
thf(fact_1232_nat__mult__1,axiom,
! [N: nat] :
( ( times_times_nat @ one_one_nat @ N )
= N ) ).
% nat_mult_1
thf(fact_1233_mult__0,axiom,
! [N: nat] :
( ( times_times_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% mult_0
thf(fact_1234_is__edge__or__loop,axiom,
! [E: set_a] :
( ( member_set_a @ E @ edges )
=> ( ( undire2905028936066782638loop_a @ E )
| ( undire4917966558017083288edge_a @ E ) ) ) ).
% is_edge_or_loop
thf(fact_1235_bounded__Max__nat,axiom,
! [P: nat > $o,X: nat,M3: nat] :
( ( P @ X )
=> ( ! [X2: nat] :
( ( P @ X2 )
=> ( ord_less_eq_nat @ X2 @ M3 ) )
=> ~ ! [M4: nat] :
( ( P @ M4 )
=> ~ ! [X3: nat] :
( ( P @ X3 )
=> ( ord_less_eq_nat @ X3 @ M4 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_1236_finite__nat__set__iff__bounded__le,axiom,
( finite_finite_nat
= ( ^ [N4: set_nat] :
? [M2: nat] :
! [X4: nat] :
( ( member_nat @ X4 @ N4 )
=> ( ord_less_eq_nat @ X4 @ M2 ) ) ) ) ).
% finite_nat_set_iff_bounded_le
thf(fact_1237_walk__edges_Opelims,axiom,
! [X: list_a,Y2: list_set_a] :
( ( ( undire7337870655677353998dges_a @ X )
= Y2 )
=> ( ( accp_list_a @ undire7966302452035489203_rel_a @ X )
=> ( ( ( X = nil_a )
=> ( ( Y2 = nil_set_a )
=> ~ ( accp_list_a @ undire7966302452035489203_rel_a @ nil_a ) ) )
=> ( ! [X2: a] :
( ( X
= ( cons_a @ X2 @ nil_a ) )
=> ( ( Y2 = nil_set_a )
=> ~ ( accp_list_a @ undire7966302452035489203_rel_a @ ( cons_a @ X2 @ nil_a ) ) ) )
=> ~ ! [X2: a,Y: a,Ys: list_a] :
( ( X
= ( cons_a @ X2 @ ( cons_a @ Y @ Ys ) ) )
=> ( ( Y2
= ( cons_set_a @ ( insert_a @ X2 @ ( insert_a @ Y @ bot_bot_set_a ) ) @ ( undire7337870655677353998dges_a @ ( cons_a @ Y @ Ys ) ) ) )
=> ~ ( accp_list_a @ undire7966302452035489203_rel_a @ ( cons_a @ X2 @ ( cons_a @ Y @ Ys ) ) ) ) ) ) ) ) ) ).
% walk_edges.pelims
thf(fact_1238_size__char__eq__0,axiom,
( size_size_char
= ( ^ [C4: char] : zero_zero_nat ) ) ).
% size_char_eq_0
thf(fact_1239_size_H__char__eq__0,axiom,
( size_char
= ( ^ [C4: char] : zero_zero_nat ) ) ).
% size'_char_eq_0
thf(fact_1240_infinite__nat__iff__unbounded__le,axiom,
! [S: set_nat] :
( ( ~ ( finite_finite_nat @ S ) )
= ( ! [M2: nat] :
? [N2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
& ( member_nat @ N2 @ S ) ) ) ) ).
% infinite_nat_iff_unbounded_le
thf(fact_1241_walk__length__conv,axiom,
( undire8849074589633906640ngth_a
= ( ^ [P2: list_a] : ( minus_minus_nat @ ( size_size_list_a @ P2 ) @ one_one_nat ) ) ) ).
% walk_length_conv
thf(fact_1242_diff__0__eq__0,axiom,
! [N: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_1243_diff__self__eq__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ M )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_1244_diff__diff__cancel,axiom,
! [I: nat,N: nat] :
( ( ord_less_eq_nat @ I @ N )
=> ( ( minus_minus_nat @ N @ ( minus_minus_nat @ N @ I ) )
= I ) ) ).
% diff_diff_cancel
thf(fact_1245_diff__is__0__eq,axiom,
! [M: nat,N: nat] :
( ( ( minus_minus_nat @ M @ N )
= zero_zero_nat )
= ( ord_less_eq_nat @ M @ N ) ) ).
% diff_is_0_eq
thf(fact_1246_diff__is__0__eq_H,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( minus_minus_nat @ M @ N )
= zero_zero_nat ) ) ).
% diff_is_0_eq'
thf(fact_1247_Nat_Odiff__diff__right,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ) ).
% Nat.diff_diff_right
thf(fact_1248_Nat_Oadd__diff__assoc2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I )
= ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K ) ) ) ).
% Nat.add_diff_assoc2
thf(fact_1249_Nat_Oadd__diff__assoc,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K ) ) ) ).
% Nat.add_diff_assoc
thf(fact_1250_minus__nat_Odiff__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ zero_zero_nat )
= M ) ).
% minus_nat.diff_0
thf(fact_1251_diffs0__imp__equal,axiom,
! [M: nat,N: nat] :
( ( ( minus_minus_nat @ M @ N )
= zero_zero_nat )
=> ( ( ( minus_minus_nat @ N @ M )
= zero_zero_nat )
=> ( M = N ) ) ) ).
% diffs0_imp_equal
thf(fact_1252_Nat_Ole__imp__diff__is__add,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ( minus_minus_nat @ J @ I )
= K )
= ( J
= ( plus_plus_nat @ K @ I ) ) ) ) ).
% Nat.le_imp_diff_is_add
thf(fact_1253_Nat_Odiff__add__assoc2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K )
= ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I ) ) ) ).
% Nat.diff_add_assoc2
thf(fact_1254_Nat_Odiff__add__assoc,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K )
= ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K ) ) ) ) ).
% Nat.diff_add_assoc
thf(fact_1255_Nat_Ole__diff__conv2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( ord_less_eq_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ) ).
% Nat.le_diff_conv2
thf(fact_1256_le__diff__conv,axiom,
! [J: nat,K: nat,I: nat] :
( ( ord_less_eq_nat @ ( minus_minus_nat @ J @ K ) @ I )
= ( ord_less_eq_nat @ J @ ( plus_plus_nat @ I @ K ) ) ) ).
% le_diff_conv
thf(fact_1257_diff__add__0,axiom,
! [N: nat,M: nat] :
( ( minus_minus_nat @ N @ ( plus_plus_nat @ N @ M ) )
= zero_zero_nat ) ).
% diff_add_0
thf(fact_1258_diff__le__mono2,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ).
% diff_le_mono2
thf(fact_1259_le__diff__iff_H,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ A @ C )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A ) @ ( minus_minus_nat @ C @ B ) )
= ( ord_less_eq_nat @ B @ A ) ) ) ) ).
% le_diff_iff'
thf(fact_1260_diff__le__self,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ M ) ).
% diff_le_self
thf(fact_1261_diff__le__mono,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M @ L ) @ ( minus_minus_nat @ N @ L ) ) ) ).
% diff_le_mono
thf(fact_1262_Nat_Odiff__diff__eq,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( minus_minus_nat @ M @ N ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_1263_le__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_eq_nat @ M @ N ) ) ) ) ).
% le_diff_iff
thf(fact_1264_eq__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ( minus_minus_nat @ M @ K )
= ( minus_minus_nat @ N @ K ) )
= ( M = N ) ) ) ) ).
% eq_diff_iff
thf(fact_1265_mult__eq__if,axiom,
( times_times_nat
= ( ^ [M2: nat,N2: nat] : ( if_nat @ ( M2 = zero_zero_nat ) @ zero_zero_nat @ ( plus_plus_nat @ N2 @ ( times_times_nat @ ( minus_minus_nat @ M2 @ one_one_nat ) @ N2 ) ) ) ) ) ).
% mult_eq_if
thf(fact_1266_nat__diff__add__eq2,axiom,
! [I: nat,J: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( minus_minus_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N ) ) ) ) ).
% nat_diff_add_eq2
thf(fact_1267_nat__mult__eq__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( times_times_nat @ K @ M )
= ( times_times_nat @ K @ N ) )
= ( ( K = zero_zero_nat )
| ( M = N ) ) ) ).
% nat_mult_eq_cancel_disj
thf(fact_1268_nat__eq__add__iff1,axiom,
! [J: nat,I: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ J @ I )
=> ( ( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M )
= ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M )
= N ) ) ) ).
% nat_eq_add_iff1
thf(fact_1269_nat__eq__add__iff2,axiom,
! [I: nat,J: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M )
= ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( M
= ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N ) ) ) ) ).
% nat_eq_add_iff2
thf(fact_1270_nat__le__add__iff1,axiom,
! [J: nat,I: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ J @ I )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M ) @ N ) ) ) ).
% nat_le_add_iff1
thf(fact_1271_nat__le__add__iff2,axiom,
! [I: nat,J: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( ord_less_eq_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N ) ) ) ) ).
% nat_le_add_iff2
thf(fact_1272_nat__diff__add__eq1,axiom,
! [J: nat,I: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ J @ I )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M ) @ N ) ) ) ).
% nat_diff_add_eq1
% Helper facts (3)
thf(help_If_3_1_If_001t__Nat__Onat_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y2: nat] :
( ( if_nat @ $false @ X @ Y2 )
= Y2 ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y2: nat] :
( ( if_nat @ $true @ X @ Y2 )
= X ) ).
% Conjectures (1)
thf(conj_0,conjecture,
ord_le3724670747650509150_set_a @ ( set_set_a2 @ ( undire7337870655677353998dges_a @ ( append_a @ xs @ ( append_a @ ( cons_a @ y @ nil_a ) @ zs ) ) ) ) @ edges ).
%------------------------------------------------------------------------------