TPTP Problem File: SLH0402^1.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/0019_Connectivity/prob_00018_000580__13359516_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1464 ( 516 unt; 189 typ; 0 def)
% Number of atoms : 3791 (1306 equ; 0 cnn)
% Maximal formula atoms : 12 ( 2 avg)
% Number of connectives : 10907 ( 415 ~; 77 |; 310 &;8641 @)
% ( 0 <=>;1464 =>; 0 <=; 0 <~>)
% Maximal formula depth : 21 ( 6 avg)
% Number of types : 20 ( 19 usr)
% Number of type conns : 423 ( 423 >; 0 *; 0 +; 0 <<)
% Number of symbols : 171 ( 170 usr; 21 con; 0-5 aty)
% Number of variables : 3439 ( 199 ^;3083 !; 157 ?;3439 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 14:34:12.548
%------------------------------------------------------------------------------
% Could-be-implicit typings (19)
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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__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__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__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__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__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__closed__walk_001t__Set__Oset_Itf__a_J,type,
undire4100213446647512896_set_a: set_set_a > set_set_set_a > list_set_a > $o ).
thf(sy_c_Undirected__Graph__Walks_Oulgraph_Ois__closed__walk_001tf__a,type,
undire3370724456595283424walk_a: set_a > set_set_a > list_a > $o ).
thf(sy_c_Undirected__Graph__Walks_Oulgraph_Ois__cycle_001t__Set__Oset_Itf__a_J,type,
undire797940137672299967_set_a: set_set_a > set_set_set_a > list_set_a > $o ).
thf(sy_c_Undirected__Graph__Walks_Oulgraph_Ois__cycle_001tf__a,type,
undire2407311113669455967ycle_a: set_a > set_set_a > list_a > $o ).
thf(sy_c_Undirected__Graph__Walks_Oulgraph_Ois__gen__path_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
undire7585867811434966393od_a_a: set_Product_prod_a_a > set_se5735800977113168103od_a_a > list_P1396940483166286381od_a_a > $o ).
thf(sy_c_Undirected__Graph__Walks_Oulgraph_Ois__gen__path_001t__Set__Oset_Itf__a_J,type,
undire7201326534205417136_set_a: set_set_a > set_set_set_a > list_set_a > $o ).
thf(sy_c_Undirected__Graph__Walks_Oulgraph_Ois__gen__path_001tf__a,type,
undire3562951555376170320path_a: set_a > set_set_a > list_a > $o ).
thf(sy_c_Undirected__Graph__Walks_Oulgraph_Ois__open__walk_001t__Set__Oset_Itf__a_J,type,
undire526879649183275522_set_a: set_set_a > set_set_set_a > list_set_a > $o ).
thf(sy_c_Undirected__Graph__Walks_Oulgraph_Ois__open__walk_001tf__a,type,
undire2427028224930250914walk_a: set_a > set_set_a > list_a > $o ).
thf(sy_c_Undirected__Graph__Walks_Oulgraph_Ois__path_001t__Set__Oset_Itf__a_J,type,
undire8834939040163919632_set_a: set_set_a > set_set_set_a > list_set_a > $o ).
thf(sy_c_Undirected__Graph__Walks_Oulgraph_Ois__path_001tf__a,type,
undire427332500224447920path_a: set_a > set_set_a > list_a > $o ).
thf(sy_c_Undirected__Graph__Walks_Oulgraph_Ois__trail_001t__Set__Oset_Itf__a_J,type,
undire1224551742100448159_set_a: set_set_a > set_set_set_a > list_set_a > $o ).
thf(sy_c_Undirected__Graph__Walks_Oulgraph_Ois__trail_001tf__a,type,
undire7142031287334043199rail_a: set_a > set_set_a > list_a > $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__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_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__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_u,type,
u: a ).
thf(sy_v_v,type,
v: a ).
thf(sy_v_vertices,type,
vertices: set_a ).
thf(sy_v_xs,type,
xs: list_a ).
% Relevant facts (1274)
thf(fact_0_ulgraph__axioms,axiom,
undire7251896706689453996raph_a @ vertices @ edges ).
% ulgraph_axioms
thf(fact_1_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_2_edge__adjacent__alt__def,axiom,
! [E1: set_a,E2: set_a] :
( ( member_set_a @ E1 @ edges )
=> ( ( member_set_a @ E2 @ edges )
=> ( ? [X: a] :
( ( member_a @ X @ vertices )
& ( member_a @ X @ E1 )
& ( member_a @ X @ E2 ) )
=> ( undire4022703626023482010_adj_a @ edges @ E1 @ E2 ) ) ) ) ).
% edge_adjacent_alt_def
thf(fact_3_is__trail__rev,axiom,
! [Xs: list_a] :
( ( undire7142031287334043199rail_a @ vertices @ edges @ Xs )
= ( undire7142031287334043199rail_a @ vertices @ edges @ ( rev_a @ Xs ) ) ) ).
% is_trail_rev
thf(fact_4_ulgraph_Oconnecting__walk_Ocong,axiom,
connecting_walk_a = connecting_walk_a ).
% ulgraph.connecting_walk.cong
thf(fact_5_is__closed__walk__rev,axiom,
! [Xs: list_a] :
( ( undire3370724456595283424walk_a @ vertices @ edges @ Xs )
= ( undire3370724456595283424walk_a @ vertices @ edges @ ( rev_a @ Xs ) ) ) ).
% is_closed_walk_rev
thf(fact_6_is__open__walk__rev,axiom,
! [Xs: list_a] :
( ( undire2427028224930250914walk_a @ vertices @ edges @ Xs )
= ( undire2427028224930250914walk_a @ vertices @ edges @ ( rev_a @ Xs ) ) ) ).
% is_open_walk_rev
thf(fact_7_is__path__rev,axiom,
! [Xs: list_a] :
( ( undire427332500224447920path_a @ vertices @ edges @ Xs )
= ( undire427332500224447920path_a @ vertices @ edges @ ( rev_a @ Xs ) ) ) ).
% is_path_rev
thf(fact_8_is__cycle__rev,axiom,
! [Xs: list_a] :
( ( undire2407311113669455967ycle_a @ vertices @ edges @ Xs )
= ( undire2407311113669455967ycle_a @ vertices @ edges @ ( rev_a @ Xs ) ) ) ).
% is_cycle_rev
thf(fact_9_is__gen__path__rev,axiom,
! [P: list_a] :
( ( undire3562951555376170320path_a @ vertices @ edges @ P )
= ( undire3562951555376170320path_a @ vertices @ edges @ ( rev_a @ P ) ) ) ).
% is_gen_path_rev
thf(fact_10_has__loop__in__verts,axiom,
! [V: a] :
( ( undire3617971648856834880loop_a @ edges @ V )
=> ( member_a @ V @ vertices ) ) ).
% has_loop_in_verts
thf(fact_11_incident__edge__in__wf,axiom,
! [E: set_a,V: a] :
( ( member_set_a @ E @ edges )
=> ( ( undire1521409233611534436dent_a @ V @ E )
=> ( member_a @ V @ vertices ) ) ) ).
% incident_edge_in_wf
thf(fact_12_vert__adj__imp__inV,axiom,
! [V1: a,V2: a] :
( ( undire397441198561214472_adj_a @ edges @ V1 @ V2 )
=> ( ( member_a @ V1 @ vertices )
& ( member_a @ V2 @ vertices ) ) ) ).
% vert_adj_imp_inV
thf(fact_13_subgraph__refl,axiom,
undire7103218114511261257raph_a @ vertices @ edges @ vertices @ edges ).
% subgraph_refl
thf(fact_14_is__walk__rev,axiom,
! [Xs: list_a] :
( ( undire6133010728901294956walk_a @ vertices @ edges @ Xs )
= ( undire6133010728901294956walk_a @ vertices @ edges @ ( rev_a @ Xs ) ) ) ).
% is_walk_rev
thf(fact_15_wellformed,axiom,
! [E: set_a] :
( ( member_set_a @ E @ edges )
=> ( ord_less_eq_set_a @ E @ vertices ) ) ).
% wellformed
thf(fact_16_rev__rev__ident,axiom,
! [Xs: list_a] :
( ( rev_a @ ( rev_a @ Xs ) )
= Xs ) ).
% rev_rev_ident
thf(fact_17_rev__rev__ident,axiom,
! [Xs: list_set_a] :
( ( rev_set_a @ ( rev_set_a @ Xs ) )
= Xs ) ).
% rev_rev_ident
thf(fact_18_incident__def,axiom,
undire1521409233611534436dent_a = member_a ).
% incident_def
thf(fact_19_vert__adj__sym,axiom,
! [V1: a,V2: a] :
( ( undire397441198561214472_adj_a @ edges @ V1 @ V2 )
= ( undire397441198561214472_adj_a @ edges @ V2 @ V1 ) ) ).
% vert_adj_sym
thf(fact_20_vert__adj__edge__iff2,axiom,
! [V1: a,V2: a] :
( ( V1 != V2 )
=> ( ( undire397441198561214472_adj_a @ edges @ V1 @ V2 )
= ( ? [X2: set_a] :
( ( member_set_a @ X2 @ edges )
& ( undire1521409233611534436dent_a @ V1 @ X2 )
& ( undire1521409233611534436dent_a @ V2 @ X2 ) ) ) ) ) ).
% vert_adj_edge_iff2
thf(fact_21_rev__is__rev__conv,axiom,
! [Xs: list_a,Ys: list_a] :
( ( ( rev_a @ Xs )
= ( rev_a @ Ys ) )
= ( Xs = Ys ) ) ).
% rev_is_rev_conv
thf(fact_22_rev__is__rev__conv,axiom,
! [Xs: list_set_a,Ys: list_set_a] :
( ( ( rev_set_a @ Xs )
= ( rev_set_a @ Ys ) )
= ( Xs = Ys ) ) ).
% rev_is_rev_conv
thf(fact_23_is__path__walk,axiom,
! [Xs: list_a] :
( ( undire427332500224447920path_a @ vertices @ edges @ Xs )
=> ( undire6133010728901294956walk_a @ vertices @ edges @ Xs ) ) ).
% is_path_walk
thf(fact_24_is__gen__path__cycle,axiom,
! [P: list_a] :
( ( undire2407311113669455967ycle_a @ vertices @ edges @ P )
=> ( undire3562951555376170320path_a @ vertices @ edges @ P ) ) ).
% is_gen_path_cycle
thf(fact_25_is__path__gen__path,axiom,
! [P: list_a] :
( ( undire427332500224447920path_a @ vertices @ edges @ P )
=> ( undire3562951555376170320path_a @ vertices @ edges @ P ) ) ).
% is_path_gen_path
thf(fact_26_rev__swap,axiom,
! [Xs: list_a,Ys: list_a] :
( ( ( rev_a @ Xs )
= Ys )
= ( Xs
= ( rev_a @ Ys ) ) ) ).
% rev_swap
thf(fact_27_rev__swap,axiom,
! [Xs: list_set_a,Ys: list_set_a] :
( ( ( rev_set_a @ Xs )
= Ys )
= ( Xs
= ( rev_set_a @ Ys ) ) ) ).
% rev_swap
thf(fact_28_is__isolated__vertex__no__loop,axiom,
! [V: a] :
( ( undire8931668460104145173rtex_a @ vertices @ edges @ V )
=> ~ ( undire3617971648856834880loop_a @ edges @ V ) ) ).
% is_isolated_vertex_no_loop
thf(fact_29_is__isolated__vertex__def,axiom,
! [V: a] :
( ( undire8931668460104145173rtex_a @ vertices @ edges @ V )
= ( ( member_a @ V @ vertices )
& ! [X2: a] :
( ( member_a @ X2 @ vertices )
=> ~ ( undire397441198561214472_adj_a @ edges @ X2 @ V ) ) ) ) ).
% is_isolated_vertex_def
thf(fact_30_is__isolated__vertex__edge,axiom,
! [V: a,E: set_a] :
( ( undire8931668460104145173rtex_a @ vertices @ edges @ V )
=> ( ( member_set_a @ E @ edges )
=> ~ ( undire1521409233611534436dent_a @ V @ E ) ) ) ).
% is_isolated_vertex_edge
thf(fact_31_induced__is__subgraph,axiom,
! [V3: set_a] :
( ( ord_less_eq_set_a @ V3 @ vertices )
=> ( undire7103218114511261257raph_a @ V3 @ ( undire7777452895879145676dges_a @ edges @ V3 ) @ vertices @ edges ) ) ).
% induced_is_subgraph
thf(fact_32_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_33_is__path__def,axiom,
! [Xs: list_a] :
( ( undire427332500224447920path_a @ vertices @ edges @ Xs )
= ( ( undire2427028224930250914walk_a @ vertices @ edges @ Xs )
& ( distinct_a @ Xs ) ) ) ).
% is_path_def
thf(fact_34_is__walk__not__empty2,axiom,
~ ( undire6133010728901294956walk_a @ vertices @ edges @ nil_a ) ).
% is_walk_not_empty2
thf(fact_35_is__walk__not__empty,axiom,
! [Xs: list_a] :
( ( undire6133010728901294956walk_a @ vertices @ edges @ Xs )
=> ( Xs != nil_a ) ) ).
% is_walk_not_empty
thf(fact_36_ulgraph_Ois__trail__rev,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,Xs: list_set_a] :
( ( undire6886684016831807756_set_a @ Vertices @ Edges )
=> ( ( undire1224551742100448159_set_a @ Vertices @ Edges @ Xs )
= ( undire1224551742100448159_set_a @ Vertices @ Edges @ ( rev_set_a @ Xs ) ) ) ) ).
% ulgraph.is_trail_rev
thf(fact_37_ulgraph_Ois__trail__rev,axiom,
! [Vertices: set_a,Edges: set_set_a,Xs: list_a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( undire7142031287334043199rail_a @ Vertices @ Edges @ Xs )
= ( undire7142031287334043199rail_a @ Vertices @ Edges @ ( rev_a @ Xs ) ) ) ) ).
% ulgraph.is_trail_rev
thf(fact_38_ulgraph_Ois__open__walk__rev,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,Xs: list_set_a] :
( ( undire6886684016831807756_set_a @ Vertices @ Edges )
=> ( ( undire526879649183275522_set_a @ Vertices @ Edges @ Xs )
= ( undire526879649183275522_set_a @ Vertices @ Edges @ ( rev_set_a @ Xs ) ) ) ) ).
% ulgraph.is_open_walk_rev
thf(fact_39_ulgraph_Ois__open__walk__rev,axiom,
! [Vertices: set_a,Edges: set_set_a,Xs: list_a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( undire2427028224930250914walk_a @ Vertices @ Edges @ Xs )
= ( undire2427028224930250914walk_a @ Vertices @ Edges @ ( rev_a @ Xs ) ) ) ) ).
% ulgraph.is_open_walk_rev
thf(fact_40_ulgraph_Ois__closed__walk__rev,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,Xs: list_set_a] :
( ( undire6886684016831807756_set_a @ Vertices @ Edges )
=> ( ( undire4100213446647512896_set_a @ Vertices @ Edges @ Xs )
= ( undire4100213446647512896_set_a @ Vertices @ Edges @ ( rev_set_a @ Xs ) ) ) ) ).
% ulgraph.is_closed_walk_rev
thf(fact_41_ulgraph_Ois__closed__walk__rev,axiom,
! [Vertices: set_a,Edges: set_set_a,Xs: list_a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( undire3370724456595283424walk_a @ Vertices @ Edges @ Xs )
= ( undire3370724456595283424walk_a @ Vertices @ Edges @ ( rev_a @ Xs ) ) ) ) ).
% ulgraph.is_closed_walk_rev
thf(fact_42_ulgraph_Ois__path__gen__path,axiom,
! [Vertices: set_a,Edges: set_set_a,P: list_a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( undire427332500224447920path_a @ Vertices @ Edges @ P )
=> ( undire3562951555376170320path_a @ Vertices @ Edges @ P ) ) ) ).
% ulgraph.is_path_gen_path
thf(fact_43_Nil__is__rev__conv,axiom,
! [Xs: list_a] :
( ( nil_a
= ( rev_a @ Xs ) )
= ( Xs = nil_a ) ) ).
% Nil_is_rev_conv
thf(fact_44_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_45_rev__is__Nil__conv,axiom,
! [Xs: list_a] :
( ( ( rev_a @ Xs )
= nil_a )
= ( Xs = nil_a ) ) ).
% rev_is_Nil_conv
thf(fact_46_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_47_set__rev,axiom,
! [Xs: list_a] :
( ( set_a2 @ ( rev_a @ Xs ) )
= ( set_a2 @ Xs ) ) ).
% set_rev
thf(fact_48_set__rev,axiom,
! [Xs: list_set_a] :
( ( set_set_a2 @ ( rev_set_a @ Xs ) )
= ( set_set_a2 @ Xs ) ) ).
% set_rev
thf(fact_49_distinct__rev,axiom,
! [Xs: list_a] :
( ( distinct_a @ ( rev_a @ Xs ) )
= ( distinct_a @ Xs ) ) ).
% distinct_rev
thf(fact_50_distinct__rev,axiom,
! [Xs: list_set_a] :
( ( distinct_set_a @ ( rev_set_a @ Xs ) )
= ( distinct_set_a @ Xs ) ) ).
% distinct_rev
thf(fact_51_distinct_Osimps_I1_J,axiom,
distinct_a @ nil_a ).
% distinct.simps(1)
thf(fact_52_distinct_Osimps_I1_J,axiom,
distinct_set_a @ nil_set_a ).
% distinct.simps(1)
thf(fact_53_subset__code_I1_J,axiom,
! [Xs: list_a,B: set_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ Xs ) @ B )
= ( ! [X2: a] :
( ( member_a @ X2 @ ( set_a2 @ Xs ) )
=> ( member_a @ X2 @ B ) ) ) ) ).
% subset_code(1)
thf(fact_54_subset__code_I1_J,axiom,
! [Xs: list_set_a,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ ( set_set_a2 @ Xs ) @ B )
= ( ! [X2: set_a] :
( ( member_set_a @ X2 @ ( set_set_a2 @ Xs ) )
=> ( member_set_a @ X2 @ B ) ) ) ) ).
% subset_code(1)
thf(fact_55_subset__code_I1_J,axiom,
! [Xs: list_P1396940483166286381od_a_a,B: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ ( set_Product_prod_a_a2 @ Xs ) @ B )
= ( ! [X2: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X2 @ ( set_Product_prod_a_a2 @ Xs ) )
=> ( member1426531477525435216od_a_a @ X2 @ B ) ) ) ) ).
% subset_code(1)
thf(fact_56_mem__Collect__eq,axiom,
! [A: set_a,P2: set_a > $o] :
( ( member_set_a @ A @ ( collect_set_a @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_57_mem__Collect__eq,axiom,
! [A: a,P2: a > $o] :
( ( member_a @ A @ ( collect_a @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_58_mem__Collect__eq,axiom,
! [A: product_prod_a_a,P2: product_prod_a_a > $o] :
( ( member1426531477525435216od_a_a @ A @ ( collec3336397797384452498od_a_a @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_59_Collect__mem__eq,axiom,
! [A2: set_set_a] :
( ( collect_set_a
@ ^ [X2: set_a] : ( member_set_a @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_60_Collect__mem__eq,axiom,
! [A2: set_a] :
( ( collect_a
@ ^ [X2: a] : ( member_a @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_61_Collect__mem__eq,axiom,
! [A2: set_Product_prod_a_a] :
( ( collec3336397797384452498od_a_a
@ ^ [X2: product_prod_a_a] : ( member1426531477525435216od_a_a @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_62_rev_Osimps_I1_J,axiom,
( ( rev_a @ nil_a )
= nil_a ) ).
% rev.simps(1)
thf(fact_63_rev_Osimps_I1_J,axiom,
( ( rev_set_a @ nil_set_a )
= nil_set_a ) ).
% rev.simps(1)
thf(fact_64_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_65_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_66_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_67_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_68_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_69_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_70_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_71_ulgraph_Ois__path__def,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,Xs: list_set_a] :
( ( undire6886684016831807756_set_a @ Vertices @ Edges )
=> ( ( undire8834939040163919632_set_a @ Vertices @ Edges @ Xs )
= ( ( undire526879649183275522_set_a @ Vertices @ Edges @ Xs )
& ( distinct_set_a @ Xs ) ) ) ) ).
% ulgraph.is_path_def
thf(fact_72_ulgraph_Ois__path__def,axiom,
! [Vertices: set_a,Edges: set_set_a,Xs: list_a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( undire427332500224447920path_a @ Vertices @ Edges @ Xs )
= ( ( undire2427028224930250914walk_a @ Vertices @ Edges @ Xs )
& ( distinct_a @ Xs ) ) ) ) ).
% ulgraph.is_path_def
thf(fact_73_ulgraph_Ois__walk_Ocong,axiom,
undire6133010728901294956walk_a = undire6133010728901294956walk_a ).
% ulgraph.is_walk.cong
thf(fact_74_ulgraph_Ois__gen__path_Ocong,axiom,
undire3562951555376170320path_a = undire3562951555376170320path_a ).
% ulgraph.is_gen_path.cong
thf(fact_75_ulgraph_Ois__cycle_Ocong,axiom,
undire2407311113669455967ycle_a = undire2407311113669455967ycle_a ).
% ulgraph.is_cycle.cong
thf(fact_76_ulgraph_Ois__path_Ocong,axiom,
undire427332500224447920path_a = undire427332500224447920path_a ).
% ulgraph.is_path.cong
thf(fact_77_ulgraph_Ois__closed__walk_Ocong,axiom,
undire3370724456595283424walk_a = undire3370724456595283424walk_a ).
% ulgraph.is_closed_walk.cong
thf(fact_78_ulgraph_Ois__open__walk_Ocong,axiom,
undire2427028224930250914walk_a = undire2427028224930250914walk_a ).
% ulgraph.is_open_walk.cong
thf(fact_79_ulgraph_Ois__trail_Ocong,axiom,
undire7142031287334043199rail_a = undire7142031287334043199rail_a ).
% ulgraph.is_trail.cong
thf(fact_80_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_81_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_82_ulgraph_Ois__gen__path__rev,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,P: list_set_a] :
( ( undire6886684016831807756_set_a @ Vertices @ Edges )
=> ( ( undire7201326534205417136_set_a @ Vertices @ Edges @ P )
= ( undire7201326534205417136_set_a @ Vertices @ Edges @ ( rev_set_a @ P ) ) ) ) ).
% ulgraph.is_gen_path_rev
thf(fact_83_ulgraph_Ois__gen__path__rev,axiom,
! [Vertices: set_a,Edges: set_set_a,P: list_a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( undire3562951555376170320path_a @ Vertices @ Edges @ P )
= ( undire3562951555376170320path_a @ Vertices @ Edges @ ( rev_a @ P ) ) ) ) ).
% ulgraph.is_gen_path_rev
thf(fact_84_ulgraph_Ois__cycle__rev,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,Xs: list_set_a] :
( ( undire6886684016831807756_set_a @ Vertices @ Edges )
=> ( ( undire797940137672299967_set_a @ Vertices @ Edges @ Xs )
= ( undire797940137672299967_set_a @ Vertices @ Edges @ ( rev_set_a @ Xs ) ) ) ) ).
% ulgraph.is_cycle_rev
thf(fact_85_ulgraph_Ois__cycle__rev,axiom,
! [Vertices: set_a,Edges: set_set_a,Xs: list_a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( undire2407311113669455967ycle_a @ Vertices @ Edges @ Xs )
= ( undire2407311113669455967ycle_a @ Vertices @ Edges @ ( rev_a @ Xs ) ) ) ) ).
% ulgraph.is_cycle_rev
thf(fact_86_ulgraph_Ois__path__rev,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,Xs: list_set_a] :
( ( undire6886684016831807756_set_a @ Vertices @ Edges )
=> ( ( undire8834939040163919632_set_a @ Vertices @ Edges @ Xs )
= ( undire8834939040163919632_set_a @ Vertices @ Edges @ ( rev_set_a @ Xs ) ) ) ) ).
% ulgraph.is_path_rev
thf(fact_87_ulgraph_Ois__path__rev,axiom,
! [Vertices: set_a,Edges: set_set_a,Xs: list_a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( undire427332500224447920path_a @ Vertices @ Edges @ Xs )
= ( undire427332500224447920path_a @ Vertices @ Edges @ ( rev_a @ Xs ) ) ) ) ).
% ulgraph.is_path_rev
thf(fact_88_ulgraph_Ois__path__walk,axiom,
! [Vertices: set_a,Edges: set_set_a,Xs: list_a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( undire427332500224447920path_a @ Vertices @ Edges @ Xs )
=> ( undire6133010728901294956walk_a @ Vertices @ Edges @ Xs ) ) ) ).
% ulgraph.is_path_walk
thf(fact_89_ulgraph_Ois__gen__path__cycle,axiom,
! [Vertices: set_a,Edges: set_set_a,P: list_a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( undire2407311113669455967ycle_a @ Vertices @ Edges @ P )
=> ( undire3562951555376170320path_a @ Vertices @ Edges @ P ) ) ) ).
% ulgraph.is_gen_path_cycle
thf(fact_90_is__gen__path__options,axiom,
! [P: list_a] :
( ( undire3562951555376170320path_a @ vertices @ edges @ P )
= ( ( undire2407311113669455967ycle_a @ vertices @ edges @ P )
| ( undire427332500224447920path_a @ vertices @ edges @ P )
| ? [X2: a] :
( ( member_a @ X2 @ vertices )
& ( P
= ( cons_a @ X2 @ nil_a ) ) ) ) ) ).
% is_gen_path_options
thf(fact_91_induced__edges__ss,axiom,
! [V3: set_a] :
( ( ord_less_eq_set_a @ V3 @ vertices )
=> ( ord_le3724670747650509150_set_a @ ( undire7777452895879145676dges_a @ edges @ V3 ) @ edges ) ) ).
% induced_edges_ss
thf(fact_92_is__gen__path__trivial,axiom,
! [X3: a] :
( ( member_a @ X3 @ vertices )
=> ( undire3562951555376170320path_a @ vertices @ edges @ ( cons_a @ X3 @ nil_a ) ) ) ).
% is_gen_path_trivial
thf(fact_93_is__walk__drop__hd,axiom,
! [Ys: list_a,Y: a] :
( ( Ys != nil_a )
=> ( ( undire6133010728901294956walk_a @ vertices @ edges @ ( cons_a @ Y @ Ys ) )
=> ( undire6133010728901294956walk_a @ vertices @ edges @ Ys ) ) ) ).
% is_walk_drop_hd
thf(fact_94_is__walk__singleton,axiom,
! [U: a] :
( ( member_a @ U @ vertices )
=> ( undire6133010728901294956walk_a @ vertices @ edges @ ( cons_a @ U @ nil_a ) ) ) ).
% is_walk_singleton
thf(fact_95_ulgraph_Ois__isolated__vertex__no__loop,axiom,
! [Vertices: set_a,Edges: set_set_a,V: a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( undire8931668460104145173rtex_a @ Vertices @ Edges @ V )
=> ~ ( undire3617971648856834880loop_a @ Edges @ V ) ) ) ).
% ulgraph.is_isolated_vertex_no_loop
thf(fact_96_ulgraph_Ois__isolated__vertex__edge,axiom,
! [Vertices: set_a,Edges: set_set_a,V: a,E: set_a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( undire8931668460104145173rtex_a @ Vertices @ Edges @ V )
=> ( ( member_set_a @ E @ Edges )
=> ~ ( undire1521409233611534436dent_a @ V @ E ) ) ) ) ).
% ulgraph.is_isolated_vertex_edge
thf(fact_97_ulgraph_Ois__isolated__vertex__def,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,V: set_a] :
( ( undire6886684016831807756_set_a @ Vertices @ Edges )
=> ( ( undire6879241558604981877_set_a @ Vertices @ Edges @ V )
= ( ( member_set_a @ V @ Vertices )
& ! [X2: set_a] :
( ( member_set_a @ X2 @ Vertices )
=> ~ ( undire3510646817838285160_set_a @ Edges @ X2 @ V ) ) ) ) ) ).
% ulgraph.is_isolated_vertex_def
thf(fact_98_ulgraph_Ois__isolated__vertex__def,axiom,
! [Vertices: set_Product_prod_a_a,Edges: set_se5735800977113168103od_a_a,V: product_prod_a_a] :
( ( undire4585262585102564309od_a_a @ Vertices @ Edges )
=> ( ( undire3207556238582723646od_a_a @ Vertices @ Edges @ V )
= ( ( member1426531477525435216od_a_a @ V @ Vertices )
& ! [X2: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X2 @ Vertices )
=> ~ ( undire6135774327024169009od_a_a @ Edges @ X2 @ V ) ) ) ) ) ).
% ulgraph.is_isolated_vertex_def
thf(fact_99_ulgraph_Ois__isolated__vertex__def,axiom,
! [Vertices: set_a,Edges: set_set_a,V: a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( undire8931668460104145173rtex_a @ Vertices @ Edges @ V )
= ( ( member_a @ V @ Vertices )
& ! [X2: a] :
( ( member_a @ X2 @ Vertices )
=> ~ ( undire397441198561214472_adj_a @ Edges @ X2 @ V ) ) ) ) ) ).
% ulgraph.is_isolated_vertex_def
thf(fact_100_ulgraph_Overt__adj__edge__iff2,axiom,
! [Vertices: set_a,Edges: set_set_a,V1: a,V2: a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( V1 != V2 )
=> ( ( undire397441198561214472_adj_a @ Edges @ V1 @ V2 )
= ( ? [X2: set_a] :
( ( member_set_a @ X2 @ Edges )
& ( undire1521409233611534436dent_a @ V1 @ X2 )
& ( undire1521409233611534436dent_a @ V2 @ X2 ) ) ) ) ) ) ).
% ulgraph.vert_adj_edge_iff2
thf(fact_101_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_102_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_103_walk__edges_Ocases,axiom,
! [X3: list_a] :
( ( X3 != nil_a )
=> ( ! [X4: a] :
( X3
!= ( cons_a @ X4 @ nil_a ) )
=> ~ ! [X4: a,Y2: a,Ys2: list_a] :
( X3
!= ( cons_a @ X4 @ ( cons_a @ Y2 @ Ys2 ) ) ) ) ) ).
% walk_edges.cases
thf(fact_104_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_105_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_106_is__gen__path__distinct,axiom,
! [P: list_a] :
( ( undire3562951555376170320path_a @ vertices @ edges @ P )
=> ( ( ( hd_a @ P )
!= ( last_a @ P ) )
=> ( distinct_a @ P ) ) ) ).
% is_gen_path_distinct
thf(fact_107_is__open__walk__def,axiom,
! [Xs: list_a] :
( ( undire2427028224930250914walk_a @ vertices @ edges @ Xs )
= ( ( undire6133010728901294956walk_a @ vertices @ edges @ Xs )
& ( ( hd_a @ Xs )
!= ( last_a @ Xs ) ) ) ) ).
% is_open_walk_def
thf(fact_108_is__closed__walk__def,axiom,
! [Xs: list_a] :
( ( undire3370724456595283424walk_a @ vertices @ edges @ Xs )
= ( ( undire6133010728901294956walk_a @ vertices @ edges @ Xs )
& ( ( hd_a @ Xs )
= ( last_a @ Xs ) ) ) ) ).
% is_closed_walk_def
thf(fact_109_connecting__walk__def,axiom,
! [U: a,V: a,Xs: list_a] :
( ( connecting_walk_a @ vertices @ edges @ U @ V @ Xs )
= ( ( undire6133010728901294956walk_a @ vertices @ edges @ Xs )
& ( ( hd_a @ Xs )
= U )
& ( ( last_a @ Xs )
= V ) ) ) ).
% connecting_walk_def
thf(fact_110_rev__singleton__conv,axiom,
! [Xs: list_a,X3: a] :
( ( ( rev_a @ Xs )
= ( cons_a @ X3 @ nil_a ) )
= ( Xs
= ( cons_a @ X3 @ nil_a ) ) ) ).
% rev_singleton_conv
thf(fact_111_rev__singleton__conv,axiom,
! [Xs: list_set_a,X3: set_a] :
( ( ( rev_set_a @ Xs )
= ( cons_set_a @ X3 @ nil_set_a ) )
= ( Xs
= ( cons_set_a @ X3 @ nil_set_a ) ) ) ).
% rev_singleton_conv
thf(fact_112_singleton__rev__conv,axiom,
! [X3: a,Xs: list_a] :
( ( ( cons_a @ X3 @ nil_a )
= ( rev_a @ Xs ) )
= ( ( cons_a @ X3 @ nil_a )
= Xs ) ) ).
% singleton_rev_conv
thf(fact_113_singleton__rev__conv,axiom,
! [X3: set_a,Xs: list_set_a] :
( ( ( cons_set_a @ X3 @ nil_set_a )
= ( rev_set_a @ Xs ) )
= ( ( cons_set_a @ X3 @ nil_set_a )
= Xs ) ) ).
% singleton_rev_conv
thf(fact_114_not__Cons__self2,axiom,
! [X3: a,Xs: list_a] :
( ( cons_a @ X3 @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_115_not__Cons__self2,axiom,
! [X3: set_a,Xs: list_set_a] :
( ( cons_set_a @ X3 @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_116_list_Osel_I1_J,axiom,
! [X21: a,X22: list_a] :
( ( hd_a @ ( cons_a @ X21 @ X22 ) )
= X21 ) ).
% list.sel(1)
thf(fact_117_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_118_hd__Nil__eq__last,axiom,
( ( hd_a @ nil_a )
= ( last_a @ nil_a ) ) ).
% hd_Nil_eq_last
thf(fact_119_hd__Nil__eq__last,axiom,
( ( hd_set_a @ nil_set_a )
= ( last_set_a @ nil_set_a ) ) ).
% hd_Nil_eq_last
thf(fact_120_last__ConsR,axiom,
! [Xs: list_a,X3: a] :
( ( Xs != nil_a )
=> ( ( last_a @ ( cons_a @ X3 @ Xs ) )
= ( last_a @ Xs ) ) ) ).
% last_ConsR
thf(fact_121_last__ConsR,axiom,
! [Xs: list_set_a,X3: set_a] :
( ( Xs != nil_set_a )
=> ( ( last_set_a @ ( cons_set_a @ X3 @ Xs ) )
= ( last_set_a @ Xs ) ) ) ).
% last_ConsR
thf(fact_122_last__ConsL,axiom,
! [Xs: list_a,X3: a] :
( ( Xs = nil_a )
=> ( ( last_a @ ( cons_a @ X3 @ Xs ) )
= X3 ) ) ).
% last_ConsL
thf(fact_123_last__ConsL,axiom,
! [Xs: list_set_a,X3: set_a] :
( ( Xs = nil_set_a )
=> ( ( last_set_a @ ( cons_set_a @ X3 @ Xs ) )
= X3 ) ) ).
% last_ConsL
thf(fact_124_last_Osimps,axiom,
! [Xs: list_a,X3: a] :
( ( ( Xs = nil_a )
=> ( ( last_a @ ( cons_a @ X3 @ Xs ) )
= X3 ) )
& ( ( Xs != nil_a )
=> ( ( last_a @ ( cons_a @ X3 @ Xs ) )
= ( last_a @ Xs ) ) ) ) ).
% last.simps
thf(fact_125_last_Osimps,axiom,
! [Xs: list_set_a,X3: set_a] :
( ( ( Xs = nil_set_a )
=> ( ( last_set_a @ ( cons_set_a @ X3 @ Xs ) )
= X3 ) )
& ( ( Xs != nil_set_a )
=> ( ( last_set_a @ ( cons_set_a @ X3 @ Xs ) )
= ( last_set_a @ Xs ) ) ) ) ).
% last.simps
thf(fact_126_transpose_Ocases,axiom,
! [X3: list_list_a] :
( ( X3 != nil_list_a )
=> ( ! [Xss: list_list_a] :
( X3
!= ( cons_list_a @ nil_a @ Xss ) )
=> ~ ! [X4: a,Xs2: list_a,Xss: list_list_a] :
( X3
!= ( cons_list_a @ ( cons_a @ X4 @ Xs2 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_127_transpose_Ocases,axiom,
! [X3: list_list_set_a] :
( ( X3 != nil_list_set_a )
=> ( ! [Xss: list_list_set_a] :
( X3
!= ( cons_list_set_a @ nil_set_a @ Xss ) )
=> ~ ! [X4: set_a,Xs2: list_set_a,Xss: list_list_set_a] :
( X3
!= ( cons_list_set_a @ ( cons_set_a @ X4 @ Xs2 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_128_hd__rev,axiom,
! [Xs: list_set_a] :
( ( hd_set_a @ ( rev_set_a @ Xs ) )
= ( last_set_a @ Xs ) ) ).
% hd_rev
thf(fact_129_hd__rev,axiom,
! [Xs: list_a] :
( ( hd_a @ ( rev_a @ Xs ) )
= ( last_a @ Xs ) ) ).
% hd_rev
thf(fact_130_last__rev,axiom,
! [Xs: list_set_a] :
( ( last_set_a @ ( rev_set_a @ Xs ) )
= ( hd_set_a @ Xs ) ) ).
% last_rev
thf(fact_131_last__rev,axiom,
! [Xs: list_a] :
( ( last_a @ ( rev_a @ Xs ) )
= ( hd_a @ Xs ) ) ).
% last_rev
thf(fact_132_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_133_comp__sgraph_Owalk__edges_Ocases,axiom,
! [X3: list_a] :
( ( X3 != nil_a )
=> ( ! [X4: a] :
( X3
!= ( cons_a @ X4 @ nil_a ) )
=> ~ ! [X4: a,Y2: a,Ys2: list_a] :
( X3
!= ( cons_a @ X4 @ ( cons_a @ Y2 @ Ys2 ) ) ) ) ) ).
% comp_sgraph.walk_edges.cases
thf(fact_134_comp__sgraph_Owalk__edges_Ocases,axiom,
! [X3: list_set_a] :
( ( X3 != nil_set_a )
=> ( ! [X4: set_a] :
( X3
!= ( cons_set_a @ X4 @ nil_set_a ) )
=> ~ ! [X4: set_a,Y2: set_a,Ys2: list_set_a] :
( X3
!= ( cons_set_a @ X4 @ ( cons_set_a @ Y2 @ Ys2 ) ) ) ) ) ).
% comp_sgraph.walk_edges.cases
thf(fact_135_list__nonempty__induct,axiom,
! [Xs: list_a,P2: list_a > $o] :
( ( Xs != nil_a )
=> ( ! [X4: a] : ( P2 @ ( cons_a @ X4 @ nil_a ) )
=> ( ! [X4: a,Xs2: list_a] :
( ( Xs2 != nil_a )
=> ( ( P2 @ Xs2 )
=> ( P2 @ ( cons_a @ X4 @ Xs2 ) ) ) )
=> ( P2 @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_136_list__nonempty__induct,axiom,
! [Xs: list_set_a,P2: list_set_a > $o] :
( ( Xs != nil_set_a )
=> ( ! [X4: set_a] : ( P2 @ ( cons_set_a @ X4 @ nil_set_a ) )
=> ( ! [X4: set_a,Xs2: list_set_a] :
( ( Xs2 != nil_set_a )
=> ( ( P2 @ Xs2 )
=> ( P2 @ ( cons_set_a @ X4 @ Xs2 ) ) ) )
=> ( P2 @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_137_list__induct2_H,axiom,
! [P2: list_a > list_a > $o,Xs: list_a,Ys: list_a] :
( ( P2 @ nil_a @ nil_a )
=> ( ! [X4: a,Xs2: list_a] : ( P2 @ ( cons_a @ X4 @ Xs2 ) @ nil_a )
=> ( ! [Y2: a,Ys2: list_a] : ( P2 @ nil_a @ ( cons_a @ Y2 @ Ys2 ) )
=> ( ! [X4: a,Xs2: list_a,Y2: a,Ys2: list_a] :
( ( P2 @ Xs2 @ Ys2 )
=> ( P2 @ ( cons_a @ X4 @ Xs2 ) @ ( cons_a @ Y2 @ Ys2 ) ) )
=> ( P2 @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_138_list__induct2_H,axiom,
! [P2: list_a > list_set_a > $o,Xs: list_a,Ys: list_set_a] :
( ( P2 @ nil_a @ nil_set_a )
=> ( ! [X4: a,Xs2: list_a] : ( P2 @ ( cons_a @ X4 @ Xs2 ) @ nil_set_a )
=> ( ! [Y2: set_a,Ys2: list_set_a] : ( P2 @ nil_a @ ( cons_set_a @ Y2 @ Ys2 ) )
=> ( ! [X4: a,Xs2: list_a,Y2: set_a,Ys2: list_set_a] :
( ( P2 @ Xs2 @ Ys2 )
=> ( P2 @ ( cons_a @ X4 @ Xs2 ) @ ( cons_set_a @ Y2 @ Ys2 ) ) )
=> ( P2 @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_139_list__induct2_H,axiom,
! [P2: list_set_a > list_a > $o,Xs: list_set_a,Ys: list_a] :
( ( P2 @ nil_set_a @ nil_a )
=> ( ! [X4: set_a,Xs2: list_set_a] : ( P2 @ ( cons_set_a @ X4 @ Xs2 ) @ nil_a )
=> ( ! [Y2: a,Ys2: list_a] : ( P2 @ nil_set_a @ ( cons_a @ Y2 @ Ys2 ) )
=> ( ! [X4: set_a,Xs2: list_set_a,Y2: a,Ys2: list_a] :
( ( P2 @ Xs2 @ Ys2 )
=> ( P2 @ ( cons_set_a @ X4 @ Xs2 ) @ ( cons_a @ Y2 @ Ys2 ) ) )
=> ( P2 @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_140_list__induct2_H,axiom,
! [P2: list_set_a > list_set_a > $o,Xs: list_set_a,Ys: list_set_a] :
( ( P2 @ nil_set_a @ nil_set_a )
=> ( ! [X4: set_a,Xs2: list_set_a] : ( P2 @ ( cons_set_a @ X4 @ Xs2 ) @ nil_set_a )
=> ( ! [Y2: set_a,Ys2: list_set_a] : ( P2 @ nil_set_a @ ( cons_set_a @ Y2 @ Ys2 ) )
=> ( ! [X4: set_a,Xs2: list_set_a,Y2: set_a,Ys2: list_set_a] :
( ( P2 @ Xs2 @ Ys2 )
=> ( P2 @ ( cons_set_a @ X4 @ Xs2 ) @ ( cons_set_a @ Y2 @ Ys2 ) ) )
=> ( P2 @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_141_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_142_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_143_min__list_Ocases,axiom,
! [X3: list_set_a] :
( ! [X4: set_a,Xs2: list_set_a] :
( X3
!= ( cons_set_a @ X4 @ Xs2 ) )
=> ( X3 = nil_set_a ) ) ).
% min_list.cases
thf(fact_144_list_Oexhaust,axiom,
! [Y: list_a] :
( ( Y != nil_a )
=> ~ ! [X212: a,X222: list_a] :
( Y
!= ( cons_a @ X212 @ X222 ) ) ) ).
% list.exhaust
thf(fact_145_list_Oexhaust,axiom,
! [Y: list_set_a] :
( ( Y != nil_set_a )
=> ~ ! [X212: set_a,X222: list_set_a] :
( Y
!= ( cons_set_a @ X212 @ X222 ) ) ) ).
% list.exhaust
thf(fact_146_list_OdiscI,axiom,
! [List: list_a,X21: a,X22: list_a] :
( ( List
= ( cons_a @ X21 @ X22 ) )
=> ( List != nil_a ) ) ).
% list.discI
thf(fact_147_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_148_list_Odistinct_I1_J,axiom,
! [X21: a,X22: list_a] :
( nil_a
!= ( cons_a @ X21 @ X22 ) ) ).
% list.distinct(1)
thf(fact_149_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_150_set__ConsD,axiom,
! [Y: product_prod_a_a,X3: product_prod_a_a,Xs: list_P1396940483166286381od_a_a] :
( ( member1426531477525435216od_a_a @ Y @ ( set_Product_prod_a_a2 @ ( cons_P7316939126706565853od_a_a @ X3 @ Xs ) ) )
=> ( ( Y = X3 )
| ( member1426531477525435216od_a_a @ Y @ ( set_Product_prod_a_a2 @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_151_set__ConsD,axiom,
! [Y: a,X3: a,Xs: list_a] :
( ( member_a @ Y @ ( set_a2 @ ( cons_a @ X3 @ Xs ) ) )
=> ( ( Y = X3 )
| ( member_a @ Y @ ( set_a2 @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_152_set__ConsD,axiom,
! [Y: set_a,X3: set_a,Xs: list_set_a] :
( ( member_set_a @ Y @ ( set_set_a2 @ ( cons_set_a @ X3 @ Xs ) ) )
=> ( ( Y = X3 )
| ( member_set_a @ Y @ ( set_set_a2 @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_153_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_154_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_155_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_156_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_157_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_158_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_159_list_Oset__intros_I2_J,axiom,
! [Y: product_prod_a_a,X22: list_P1396940483166286381od_a_a,X21: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ Y @ ( set_Product_prod_a_a2 @ X22 ) )
=> ( member1426531477525435216od_a_a @ Y @ ( set_Product_prod_a_a2 @ ( cons_P7316939126706565853od_a_a @ X21 @ X22 ) ) ) ) ).
% list.set_intros(2)
thf(fact_160_list_Oset__intros_I2_J,axiom,
! [Y: a,X22: list_a,X21: a] :
( ( member_a @ Y @ ( set_a2 @ X22 ) )
=> ( member_a @ Y @ ( set_a2 @ ( cons_a @ X21 @ X22 ) ) ) ) ).
% list.set_intros(2)
thf(fact_161_list_Oset__intros_I2_J,axiom,
! [Y: set_a,X22: list_set_a,X21: set_a] :
( ( member_set_a @ Y @ ( set_set_a2 @ X22 ) )
=> ( member_set_a @ Y @ ( set_set_a2 @ ( cons_set_a @ X21 @ X22 ) ) ) ) ).
% list.set_intros(2)
thf(fact_162_distinct__length__2__or__more,axiom,
! [A: a,B2: a,Xs: list_a] :
( ( distinct_a @ ( cons_a @ A @ ( cons_a @ B2 @ Xs ) ) )
= ( ( A != B2 )
& ( distinct_a @ ( cons_a @ A @ Xs ) )
& ( distinct_a @ ( cons_a @ B2 @ Xs ) ) ) ) ).
% distinct_length_2_or_more
thf(fact_163_distinct__length__2__or__more,axiom,
! [A: set_a,B2: set_a,Xs: list_set_a] :
( ( distinct_set_a @ ( cons_set_a @ A @ ( cons_set_a @ B2 @ Xs ) ) )
= ( ( A != B2 )
& ( distinct_set_a @ ( cons_set_a @ A @ Xs ) )
& ( distinct_set_a @ ( cons_set_a @ B2 @ Xs ) ) ) ) ).
% distinct_length_2_or_more
thf(fact_164_ulgraph_Ois__gen__path__distinct,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,P: list_set_a] :
( ( undire6886684016831807756_set_a @ Vertices @ Edges )
=> ( ( undire7201326534205417136_set_a @ Vertices @ Edges @ P )
=> ( ( ( hd_set_a @ P )
!= ( last_set_a @ P ) )
=> ( distinct_set_a @ P ) ) ) ) ).
% ulgraph.is_gen_path_distinct
thf(fact_165_ulgraph_Ois__gen__path__distinct,axiom,
! [Vertices: set_a,Edges: set_set_a,P: list_a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( undire3562951555376170320path_a @ Vertices @ Edges @ P )
=> ( ( ( hd_a @ P )
!= ( last_a @ P ) )
=> ( distinct_a @ P ) ) ) ) ).
% ulgraph.is_gen_path_distinct
thf(fact_166_ulgraph_Ois__open__walk__def,axiom,
! [Vertices: set_a,Edges: set_set_a,Xs: list_a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( undire2427028224930250914walk_a @ Vertices @ Edges @ Xs )
= ( ( undire6133010728901294956walk_a @ Vertices @ Edges @ Xs )
& ( ( hd_a @ Xs )
!= ( last_a @ Xs ) ) ) ) ) ).
% ulgraph.is_open_walk_def
thf(fact_167_ulgraph_Ois__closed__walk__def,axiom,
! [Vertices: set_a,Edges: set_set_a,Xs: list_a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( undire3370724456595283424walk_a @ Vertices @ Edges @ Xs )
= ( ( undire6133010728901294956walk_a @ Vertices @ Edges @ Xs )
& ( ( hd_a @ Xs )
= ( last_a @ Xs ) ) ) ) ) ).
% ulgraph.is_closed_walk_def
thf(fact_168_ulgraph_Oconnecting__walk__def,axiom,
! [Vertices: set_a,Edges: set_set_a,U: a,V: a,Xs: list_a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( connecting_walk_a @ Vertices @ Edges @ U @ V @ Xs )
= ( ( undire6133010728901294956walk_a @ Vertices @ Edges @ Xs )
& ( ( hd_a @ Xs )
= U )
& ( ( last_a @ Xs )
= V ) ) ) ) ).
% ulgraph.connecting_walk_def
thf(fact_169_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_170_hd__in__set,axiom,
! [Xs: list_a] :
( ( Xs != nil_a )
=> ( member_a @ ( hd_a @ Xs ) @ ( set_a2 @ Xs ) ) ) ).
% hd_in_set
thf(fact_171_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_172_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_173_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_174_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_175_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_176_last__in__set,axiom,
! [As: list_a] :
( ( As != nil_a )
=> ( member_a @ ( last_a @ As ) @ ( set_a2 @ As ) ) ) ).
% last_in_set
thf(fact_177_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_178_set__subset__Cons,axiom,
! [Xs: list_a,X3: a] : ( ord_less_eq_set_a @ ( set_a2 @ Xs ) @ ( set_a2 @ ( cons_a @ X3 @ Xs ) ) ) ).
% set_subset_Cons
thf(fact_179_set__subset__Cons,axiom,
! [Xs: list_set_a,X3: set_a] : ( ord_le3724670747650509150_set_a @ ( set_set_a2 @ Xs ) @ ( set_set_a2 @ ( cons_set_a @ X3 @ Xs ) ) ) ).
% set_subset_Cons
thf(fact_180_set__subset__Cons,axiom,
! [Xs: list_P1396940483166286381od_a_a,X3: product_prod_a_a] : ( ord_le746702958409616551od_a_a @ ( set_Product_prod_a_a2 @ Xs ) @ ( set_Product_prod_a_a2 @ ( cons_P7316939126706565853od_a_a @ X3 @ Xs ) ) ) ).
% set_subset_Cons
thf(fact_181_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_182_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_183_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_184_distinct__singleton,axiom,
! [X3: a] : ( distinct_a @ ( cons_a @ X3 @ nil_a ) ) ).
% distinct_singleton
thf(fact_185_distinct__singleton,axiom,
! [X3: set_a] : ( distinct_set_a @ ( cons_set_a @ X3 @ nil_set_a ) ) ).
% distinct_singleton
thf(fact_186_ulgraph_Owalk__edges_Ocases,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,X3: list_set_a] :
( ( undire6886684016831807756_set_a @ Vertices @ Edges )
=> ( ( X3 != nil_set_a )
=> ( ! [X4: set_a] :
( X3
!= ( cons_set_a @ X4 @ nil_set_a ) )
=> ~ ! [X4: set_a,Y2: set_a,Ys2: list_set_a] :
( X3
!= ( cons_set_a @ X4 @ ( cons_set_a @ Y2 @ Ys2 ) ) ) ) ) ) ).
% ulgraph.walk_edges.cases
thf(fact_187_ulgraph_Owalk__edges_Ocases,axiom,
! [Vertices: set_a,Edges: set_set_a,X3: list_a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( X3 != nil_a )
=> ( ! [X4: a] :
( X3
!= ( cons_a @ X4 @ nil_a ) )
=> ~ ! [X4: a,Y2: a,Ys2: list_a] :
( X3
!= ( cons_a @ X4 @ ( cons_a @ Y2 @ Ys2 ) ) ) ) ) ) ).
% ulgraph.walk_edges.cases
thf(fact_188_distinct_Osimps_I2_J,axiom,
! [X3: product_prod_a_a,Xs: list_P1396940483166286381od_a_a] :
( ( distin132333870042060960od_a_a @ ( cons_P7316939126706565853od_a_a @ X3 @ Xs ) )
= ( ~ ( member1426531477525435216od_a_a @ X3 @ ( set_Product_prod_a_a2 @ Xs ) )
& ( distin132333870042060960od_a_a @ Xs ) ) ) ).
% distinct.simps(2)
thf(fact_189_distinct_Osimps_I2_J,axiom,
! [X3: a,Xs: list_a] :
( ( distinct_a @ ( cons_a @ X3 @ Xs ) )
= ( ~ ( member_a @ X3 @ ( set_a2 @ Xs ) )
& ( distinct_a @ Xs ) ) ) ).
% distinct.simps(2)
thf(fact_190_distinct_Osimps_I2_J,axiom,
! [X3: set_a,Xs: list_set_a] :
( ( distinct_set_a @ ( cons_set_a @ X3 @ Xs ) )
= ( ~ ( member_set_a @ X3 @ ( set_set_a2 @ Xs ) )
& ( distinct_set_a @ Xs ) ) ) ).
% distinct.simps(2)
thf(fact_191_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_192_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_193_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_194_ulgraph_Ois__walk__drop__hd,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,Ys: list_set_a,Y: set_a] :
( ( undire6886684016831807756_set_a @ Vertices @ Edges )
=> ( ( Ys != nil_set_a )
=> ( ( undire3014741414213135564_set_a @ Vertices @ Edges @ ( cons_set_a @ Y @ Ys ) )
=> ( undire3014741414213135564_set_a @ Vertices @ Edges @ Ys ) ) ) ) ).
% ulgraph.is_walk_drop_hd
thf(fact_195_ulgraph_Ois__walk__drop__hd,axiom,
! [Vertices: set_a,Edges: set_set_a,Ys: list_a,Y: a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( Ys != nil_a )
=> ( ( undire6133010728901294956walk_a @ Vertices @ Edges @ ( cons_a @ Y @ Ys ) )
=> ( undire6133010728901294956walk_a @ Vertices @ Edges @ Ys ) ) ) ) ).
% ulgraph.is_walk_drop_hd
thf(fact_196_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_197_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_198_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_199_ulgraph_Ois__gen__path__trivial,axiom,
! [Vertices: set_Product_prod_a_a,Edges: set_se5735800977113168103od_a_a,X3: product_prod_a_a] :
( ( undire4585262585102564309od_a_a @ Vertices @ Edges )
=> ( ( member1426531477525435216od_a_a @ X3 @ Vertices )
=> ( undire7585867811434966393od_a_a @ Vertices @ Edges @ ( cons_P7316939126706565853od_a_a @ X3 @ nil_Product_prod_a_a ) ) ) ) ).
% ulgraph.is_gen_path_trivial
thf(fact_200_ulgraph_Ois__gen__path__trivial,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,X3: set_a] :
( ( undire6886684016831807756_set_a @ Vertices @ Edges )
=> ( ( member_set_a @ X3 @ Vertices )
=> ( undire7201326534205417136_set_a @ Vertices @ Edges @ ( cons_set_a @ X3 @ nil_set_a ) ) ) ) ).
% ulgraph.is_gen_path_trivial
thf(fact_201_ulgraph_Ois__gen__path__trivial,axiom,
! [Vertices: set_a,Edges: set_set_a,X3: a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( member_a @ X3 @ Vertices )
=> ( undire3562951555376170320path_a @ Vertices @ Edges @ ( cons_a @ X3 @ nil_a ) ) ) ) ).
% ulgraph.is_gen_path_trivial
thf(fact_202_subgraph_Osubgraph__antisym,axiom,
! [V_H: set_a,E_H: set_set_a,V_G: set_a,E_G: set_set_a,V3: set_a,E3: set_set_a,V4: set_a,E4: set_set_a] :
( ( undire7103218114511261257raph_a @ V_H @ E_H @ V_G @ E_G )
=> ( ( undire7103218114511261257raph_a @ V3 @ E3 @ V4 @ E4 )
=> ( ( undire7103218114511261257raph_a @ V4 @ E4 @ V3 @ E3 )
=> ( ( V4 = V3 )
& ( E4 = E3 ) ) ) ) ) ).
% subgraph.subgraph_antisym
thf(fact_203_graph__system_Oinduced__edges_Ocong,axiom,
undire7777452895879145676dges_a = undire7777452895879145676dges_a ).
% graph_system.induced_edges.cong
thf(fact_204_ulgraph_Overt__adj_Ocong,axiom,
undire397441198561214472_adj_a = undire397441198561214472_adj_a ).
% ulgraph.vert_adj.cong
thf(fact_205_comp__sgraph_Oincident__def,axiom,
undire2320338297334612420_set_a = member_set_a ).
% comp_sgraph.incident_def
thf(fact_206_comp__sgraph_Oincident__def,axiom,
undire3369688177417741453od_a_a = member1426531477525435216od_a_a ).
% comp_sgraph.incident_def
thf(fact_207_comp__sgraph_Oincident__def,axiom,
undire1521409233611534436dent_a = member_a ).
% comp_sgraph.incident_def
thf(fact_208_ulgraph_Ohas__loop_Ocong,axiom,
undire3617971648856834880loop_a = undire3617971648856834880loop_a ).
% ulgraph.has_loop.cong
thf(fact_209_ulgraph_Ois__isolated__vertex_Ocong,axiom,
undire8931668460104145173rtex_a = undire8931668460104145173rtex_a ).
% ulgraph.is_isolated_vertex.cong
thf(fact_210_graph__system_Oedge__adj_Ocong,axiom,
undire4022703626023482010_adj_a = undire4022703626023482010_adj_a ).
% graph_system.edge_adj.cong
thf(fact_211_ulgraph_Ois__gen__path__options,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,P: list_set_a] :
( ( undire6886684016831807756_set_a @ Vertices @ Edges )
=> ( ( undire7201326534205417136_set_a @ Vertices @ Edges @ P )
= ( ( undire797940137672299967_set_a @ Vertices @ Edges @ P )
| ( undire8834939040163919632_set_a @ Vertices @ Edges @ P )
| ? [X2: set_a] :
( ( member_set_a @ X2 @ Vertices )
& ( P
= ( cons_set_a @ X2 @ nil_set_a ) ) ) ) ) ) ).
% ulgraph.is_gen_path_options
thf(fact_212_ulgraph_Ois__gen__path__options,axiom,
! [Vertices: set_a,Edges: set_set_a,P: list_a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( undire3562951555376170320path_a @ Vertices @ Edges @ P )
= ( ( undire2407311113669455967ycle_a @ Vertices @ Edges @ P )
| ( undire427332500224447920path_a @ Vertices @ Edges @ P )
| ? [X2: a] :
( ( member_a @ X2 @ Vertices )
& ( P
= ( cons_a @ X2 @ nil_a ) ) ) ) ) ) ).
% ulgraph.is_gen_path_options
thf(fact_213_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_214_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_215_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_216_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_217_ulgraph_Overt__adj__sym,axiom,
! [Vertices: set_a,Edges: set_set_a,V1: a,V2: a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( undire397441198561214472_adj_a @ Edges @ V1 @ V2 )
= ( undire397441198561214472_adj_a @ Edges @ V2 @ V1 ) ) ) ).
% ulgraph.vert_adj_sym
thf(fact_218_ulgraph_Overt__adj__imp__inV,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,V1: set_a,V2: set_a] :
( ( undire6886684016831807756_set_a @ Vertices @ Edges )
=> ( ( undire3510646817838285160_set_a @ Edges @ V1 @ V2 )
=> ( ( member_set_a @ V1 @ Vertices )
& ( member_set_a @ V2 @ Vertices ) ) ) ) ).
% ulgraph.vert_adj_imp_inV
thf(fact_219_ulgraph_Overt__adj__imp__inV,axiom,
! [Vertices: set_Product_prod_a_a,Edges: set_se5735800977113168103od_a_a,V1: product_prod_a_a,V2: product_prod_a_a] :
( ( undire4585262585102564309od_a_a @ Vertices @ Edges )
=> ( ( undire6135774327024169009od_a_a @ Edges @ V1 @ V2 )
=> ( ( member1426531477525435216od_a_a @ V1 @ Vertices )
& ( member1426531477525435216od_a_a @ V2 @ Vertices ) ) ) ) ).
% ulgraph.vert_adj_imp_inV
thf(fact_220_ulgraph_Overt__adj__imp__inV,axiom,
! [Vertices: set_a,Edges: set_set_a,V1: a,V2: a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( undire397441198561214472_adj_a @ Edges @ V1 @ V2 )
=> ( ( member_a @ V1 @ Vertices )
& ( member_a @ V2 @ Vertices ) ) ) ) ).
% ulgraph.vert_adj_imp_inV
thf(fact_221_ulgraph_Ohas__loop__in__verts,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,V: set_a] :
( ( undire6886684016831807756_set_a @ Vertices @ Edges )
=> ( ( undire5774735625301615776_set_a @ Edges @ V )
=> ( member_set_a @ V @ Vertices ) ) ) ).
% ulgraph.has_loop_in_verts
thf(fact_222_ulgraph_Ohas__loop__in__verts,axiom,
! [Vertices: set_Product_prod_a_a,Edges: set_se5735800977113168103od_a_a,V: product_prod_a_a] :
( ( undire4585262585102564309od_a_a @ Vertices @ Edges )
=> ( ( undire7777398424729533289od_a_a @ Edges @ V )
=> ( member1426531477525435216od_a_a @ V @ Vertices ) ) ) ).
% ulgraph.has_loop_in_verts
thf(fact_223_ulgraph_Ohas__loop__in__verts,axiom,
! [Vertices: set_a,Edges: set_set_a,V: a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( undire3617971648856834880loop_a @ Edges @ V )
=> ( member_a @ V @ Vertices ) ) ) ).
% ulgraph.has_loop_in_verts
thf(fact_224_is__gen__path__def,axiom,
! [P: list_a] :
( ( undire3562951555376170320path_a @ vertices @ edges @ P )
= ( ( undire6133010728901294956walk_a @ vertices @ edges @ P )
& ( ( ( distinct_a @ ( tl_a @ P ) )
& ( ( hd_a @ P )
= ( last_a @ P ) ) )
| ( distinct_a @ P ) ) ) ) ).
% is_gen_path_def
thf(fact_225_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_226_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_227_is__gen__path__distinct__tl,axiom,
! [P: list_a] :
( ( undire3562951555376170320path_a @ vertices @ edges @ P )
=> ( ( ( hd_a @ P )
= ( last_a @ P ) )
=> ( distinct_a @ ( tl_a @ P ) ) ) ) ).
% is_gen_path_distinct_tl
thf(fact_228_is__walk__decomp,axiom,
! [Xs: list_a,Y: a,Ys: list_a,Zs: list_a] :
( ( 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 ) ) ) ) )
=> ( undire6133010728901294956walk_a @ vertices @ edges @ ( append_a @ Xs @ ( append_a @ ( cons_a @ Y @ nil_a ) @ Zs ) ) ) ) ).
% is_walk_decomp
thf(fact_229_is__trail__def,axiom,
! [Xs: list_a] :
( ( undire7142031287334043199rail_a @ vertices @ edges @ Xs )
= ( ( undire6133010728901294956walk_a @ vertices @ edges @ Xs )
& ( distinct_set_a @ ( undire7337870655677353998dges_a @ Xs ) ) ) ) ).
% is_trail_def
thf(fact_230_subsetI,axiom,
! [A2: set_a,B: set_a] :
( ! [X4: a] :
( ( member_a @ X4 @ A2 )
=> ( member_a @ X4 @ B ) )
=> ( ord_less_eq_set_a @ A2 @ B ) ) ).
% subsetI
thf(fact_231_subsetI,axiom,
! [A2: set_set_a,B: set_set_a] :
( ! [X4: set_a] :
( ( member_set_a @ X4 @ A2 )
=> ( member_set_a @ X4 @ B ) )
=> ( ord_le3724670747650509150_set_a @ A2 @ B ) ) ).
% subsetI
thf(fact_232_subsetI,axiom,
! [A2: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ! [X4: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X4 @ A2 )
=> ( member1426531477525435216od_a_a @ X4 @ B ) )
=> ( ord_le746702958409616551od_a_a @ A2 @ B ) ) ).
% subsetI
thf(fact_233_subset__antisym,axiom,
! [A2: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ( ord_less_eq_set_a @ B @ A2 )
=> ( A2 = B ) ) ) ).
% subset_antisym
thf(fact_234_subset__antisym,axiom,
! [A2: set_set_a,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B )
=> ( ( ord_le3724670747650509150_set_a @ B @ A2 )
=> ( A2 = B ) ) ) ).
% subset_antisym
thf(fact_235_subset__antisym,axiom,
! [A2: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A2 @ B )
=> ( ( ord_le746702958409616551od_a_a @ B @ A2 )
=> ( A2 = B ) ) ) ).
% subset_antisym
thf(fact_236_graph__system__axioms,axiom,
undire2554140024507503526stem_a @ vertices @ edges ).
% graph_system_axioms
thf(fact_237_distinct__union,axiom,
! [Xs: list_a,Ys: list_a] :
( ( distinct_a @ ( union_a @ Xs @ Ys ) )
= ( distinct_a @ Ys ) ) ).
% distinct_union
thf(fact_238_distinct__union,axiom,
! [Xs: list_set_a,Ys: list_set_a] :
( ( distinct_set_a @ ( union_set_a @ Xs @ Ys ) )
= ( distinct_set_a @ Ys ) ) ).
% distinct_union
thf(fact_239_walk__edges_Osimps_I1_J,axiom,
( ( undire7337870655677353998dges_a @ nil_a )
= nil_set_a ) ).
% walk_edges.simps(1)
thf(fact_240_walk__edges__rev,axiom,
! [Xs: list_a] :
( ( rev_set_a @ ( undire7337870655677353998dges_a @ Xs ) )
= ( undire7337870655677353998dges_a @ ( rev_a @ Xs ) ) ) ).
% walk_edges_rev
thf(fact_241_walk__edges_Osimps_I2_J,axiom,
! [X3: a] :
( ( undire7337870655677353998dges_a @ ( cons_a @ X3 @ nil_a ) )
= nil_set_a ) ).
% walk_edges.simps(2)
thf(fact_242_induced__is__graph__sys,axiom,
! [V3: set_a] : ( undire2554140024507503526stem_a @ V3 @ ( undire7777452895879145676dges_a @ edges @ V3 ) ) ).
% induced_is_graph_sys
thf(fact_243_distinct__edgesI,axiom,
! [P: list_a] :
( ( distinct_a @ P )
=> ( distinct_set_a @ ( undire7337870655677353998dges_a @ P ) ) ) ).
% distinct_edgesI
thf(fact_244_append_Oassoc,axiom,
! [A: list_a,B2: list_a,C: list_a] :
( ( append_a @ ( append_a @ A @ B2 ) @ C )
= ( append_a @ A @ ( append_a @ B2 @ C ) ) ) ).
% append.assoc
thf(fact_245_append_Oassoc,axiom,
! [A: list_set_a,B2: list_set_a,C: list_set_a] :
( ( append_set_a @ ( append_set_a @ A @ B2 ) @ C )
= ( append_set_a @ A @ ( append_set_a @ B2 @ C ) ) ) ).
% append.assoc
thf(fact_246_append__assoc,axiom,
! [Xs: list_a,Ys: list_a,Zs: list_a] :
( ( append_a @ ( append_a @ Xs @ Ys ) @ Zs )
= ( append_a @ Xs @ ( append_a @ Ys @ Zs ) ) ) ).
% append_assoc
thf(fact_247_append__assoc,axiom,
! [Xs: list_set_a,Ys: list_set_a,Zs: list_set_a] :
( ( append_set_a @ ( append_set_a @ Xs @ Ys ) @ Zs )
= ( append_set_a @ Xs @ ( append_set_a @ Ys @ Zs ) ) ) ).
% append_assoc
thf(fact_248_append__same__eq,axiom,
! [Ys: list_a,Xs: list_a,Zs: list_a] :
( ( ( append_a @ Ys @ Xs )
= ( append_a @ Zs @ Xs ) )
= ( Ys = Zs ) ) ).
% append_same_eq
thf(fact_249_append__same__eq,axiom,
! [Ys: list_set_a,Xs: list_set_a,Zs: list_set_a] :
( ( ( append_set_a @ Ys @ Xs )
= ( append_set_a @ Zs @ Xs ) )
= ( Ys = Zs ) ) ).
% append_same_eq
thf(fact_250_same__append__eq,axiom,
! [Xs: list_a,Ys: list_a,Zs: list_a] :
( ( ( append_a @ Xs @ Ys )
= ( append_a @ Xs @ Zs ) )
= ( Ys = Zs ) ) ).
% same_append_eq
thf(fact_251_same__append__eq,axiom,
! [Xs: list_set_a,Ys: list_set_a,Zs: list_set_a] :
( ( ( append_set_a @ Xs @ Ys )
= ( append_set_a @ Xs @ Zs ) )
= ( Ys = Zs ) ) ).
% same_append_eq
thf(fact_252_walk__edges__append__ss2,axiom,
! [Xs: list_a,Ys: list_a] : ( ord_le3724670747650509150_set_a @ ( set_set_a2 @ ( undire7337870655677353998dges_a @ Xs ) ) @ ( set_set_a2 @ ( undire7337870655677353998dges_a @ ( append_a @ Xs @ Ys ) ) ) ) ).
% walk_edges_append_ss2
thf(fact_253_walk__edges__append__ss1,axiom,
! [Ys: list_a,Xs: list_a] : ( ord_le3724670747650509150_set_a @ ( set_set_a2 @ ( undire7337870655677353998dges_a @ Ys ) ) @ ( set_set_a2 @ ( undire7337870655677353998dges_a @ ( append_a @ Xs @ Ys ) ) ) ) ).
% walk_edges_append_ss1
thf(fact_254_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_255_walk__edges__decomp__ss,axiom,
! [Xs: list_a,Y: a,Zs: list_a,Ys: list_a] : ( ord_le3724670747650509150_set_a @ ( set_set_a2 @ ( undire7337870655677353998dges_a @ ( append_a @ Xs @ ( append_a @ ( cons_a @ Y @ nil_a ) @ Zs ) ) ) ) @ ( set_set_a2 @ ( undire7337870655677353998dges_a @ ( append_a @ Xs @ ( append_a @ ( cons_a @ Y @ nil_a ) @ ( append_a @ Ys @ ( append_a @ ( cons_a @ Y @ nil_a ) @ Zs ) ) ) ) ) ) ) ).
% walk_edges_decomp_ss
thf(fact_256_append_Oright__neutral,axiom,
! [A: list_a] :
( ( append_a @ A @ nil_a )
= A ) ).
% append.right_neutral
thf(fact_257_append_Oright__neutral,axiom,
! [A: list_set_a] :
( ( append_set_a @ A @ nil_set_a )
= A ) ).
% append.right_neutral
thf(fact_258_append__Nil2,axiom,
! [Xs: list_a] :
( ( append_a @ Xs @ nil_a )
= Xs ) ).
% append_Nil2
thf(fact_259_append__Nil2,axiom,
! [Xs: list_set_a] :
( ( append_set_a @ Xs @ nil_set_a )
= Xs ) ).
% append_Nil2
thf(fact_260_append__self__conv,axiom,
! [Xs: list_a,Ys: list_a] :
( ( ( append_a @ Xs @ Ys )
= Xs )
= ( Ys = nil_a ) ) ).
% append_self_conv
thf(fact_261_append__self__conv,axiom,
! [Xs: list_set_a,Ys: list_set_a] :
( ( ( append_set_a @ Xs @ Ys )
= Xs )
= ( Ys = nil_set_a ) ) ).
% append_self_conv
thf(fact_262_self__append__conv,axiom,
! [Y: list_a,Ys: list_a] :
( ( Y
= ( append_a @ Y @ Ys ) )
= ( Ys = nil_a ) ) ).
% self_append_conv
thf(fact_263_self__append__conv,axiom,
! [Y: list_set_a,Ys: list_set_a] :
( ( Y
= ( append_set_a @ Y @ Ys ) )
= ( Ys = nil_set_a ) ) ).
% self_append_conv
thf(fact_264_append__self__conv2,axiom,
! [Xs: list_a,Ys: list_a] :
( ( ( append_a @ Xs @ Ys )
= Ys )
= ( Xs = nil_a ) ) ).
% append_self_conv2
thf(fact_265_append__self__conv2,axiom,
! [Xs: list_set_a,Ys: list_set_a] :
( ( ( append_set_a @ Xs @ Ys )
= Ys )
= ( Xs = nil_set_a ) ) ).
% append_self_conv2
thf(fact_266_self__append__conv2,axiom,
! [Y: list_a,Xs: list_a] :
( ( Y
= ( append_a @ Xs @ Y ) )
= ( Xs = nil_a ) ) ).
% self_append_conv2
thf(fact_267_self__append__conv2,axiom,
! [Y: list_set_a,Xs: list_set_a] :
( ( Y
= ( append_set_a @ Xs @ Y ) )
= ( Xs = nil_set_a ) ) ).
% self_append_conv2
thf(fact_268_Nil__is__append__conv,axiom,
! [Xs: list_a,Ys: list_a] :
( ( nil_a
= ( append_a @ Xs @ Ys ) )
= ( ( Xs = nil_a )
& ( Ys = nil_a ) ) ) ).
% Nil_is_append_conv
thf(fact_269_Nil__is__append__conv,axiom,
! [Xs: list_set_a,Ys: list_set_a] :
( ( nil_set_a
= ( append_set_a @ Xs @ Ys ) )
= ( ( Xs = nil_set_a )
& ( Ys = nil_set_a ) ) ) ).
% Nil_is_append_conv
thf(fact_270_append__is__Nil__conv,axiom,
! [Xs: list_a,Ys: list_a] :
( ( ( append_a @ Xs @ Ys )
= nil_a )
= ( ( Xs = nil_a )
& ( Ys = nil_a ) ) ) ).
% append_is_Nil_conv
thf(fact_271_append__is__Nil__conv,axiom,
! [Xs: list_set_a,Ys: list_set_a] :
( ( ( append_set_a @ Xs @ Ys )
= nil_set_a )
= ( ( Xs = nil_set_a )
& ( Ys = nil_set_a ) ) ) ).
% append_is_Nil_conv
thf(fact_272_rev__append,axiom,
! [Xs: list_a,Ys: list_a] :
( ( rev_a @ ( append_a @ Xs @ Ys ) )
= ( append_a @ ( rev_a @ Ys ) @ ( rev_a @ Xs ) ) ) ).
% rev_append
thf(fact_273_rev__append,axiom,
! [Xs: list_set_a,Ys: list_set_a] :
( ( rev_set_a @ ( append_set_a @ Xs @ Ys ) )
= ( append_set_a @ ( rev_set_a @ Ys ) @ ( rev_set_a @ Xs ) ) ) ).
% rev_append
thf(fact_274_is__walk__append,axiom,
! [Xs: list_a,Ys: list_a] :
( ( undire6133010728901294956walk_a @ vertices @ edges @ Xs )
=> ( ( undire6133010728901294956walk_a @ vertices @ edges @ Ys )
=> ( ( ( last_a @ Xs )
= ( hd_a @ Ys ) )
=> ( undire6133010728901294956walk_a @ vertices @ edges @ ( append_a @ Xs @ ( tl_a @ Ys ) ) ) ) ) ) ).
% is_walk_append
thf(fact_275_append1__eq__conv,axiom,
! [Xs: list_a,X3: a,Ys: list_a,Y: a] :
( ( ( append_a @ Xs @ ( cons_a @ X3 @ nil_a ) )
= ( append_a @ Ys @ ( cons_a @ Y @ nil_a ) ) )
= ( ( Xs = Ys )
& ( X3 = Y ) ) ) ).
% append1_eq_conv
thf(fact_276_append1__eq__conv,axiom,
! [Xs: list_set_a,X3: set_a,Ys: list_set_a,Y: set_a] :
( ( ( append_set_a @ Xs @ ( cons_set_a @ X3 @ nil_set_a ) )
= ( append_set_a @ Ys @ ( cons_set_a @ Y @ nil_set_a ) ) )
= ( ( Xs = Ys )
& ( X3 = Y ) ) ) ).
% append1_eq_conv
thf(fact_277_hd__append2,axiom,
! [Xs: list_a,Ys: list_a] :
( ( Xs != nil_a )
=> ( ( hd_a @ ( append_a @ Xs @ Ys ) )
= ( hd_a @ Xs ) ) ) ).
% hd_append2
thf(fact_278_hd__append2,axiom,
! [Xs: list_set_a,Ys: list_set_a] :
( ( Xs != nil_set_a )
=> ( ( hd_set_a @ ( append_set_a @ Xs @ Ys ) )
= ( hd_set_a @ Xs ) ) ) ).
% hd_append2
thf(fact_279_tl__append2,axiom,
! [Xs: list_a,Ys: list_a] :
( ( Xs != nil_a )
=> ( ( tl_a @ ( append_a @ Xs @ Ys ) )
= ( append_a @ ( tl_a @ Xs ) @ Ys ) ) ) ).
% tl_append2
thf(fact_280_tl__append2,axiom,
! [Xs: list_set_a,Ys: list_set_a] :
( ( Xs != nil_set_a )
=> ( ( tl_set_a @ ( append_set_a @ Xs @ Ys ) )
= ( append_set_a @ ( tl_set_a @ Xs ) @ Ys ) ) ) ).
% tl_append2
thf(fact_281_last__appendL,axiom,
! [Ys: list_a,Xs: list_a] :
( ( Ys = nil_a )
=> ( ( last_a @ ( append_a @ Xs @ Ys ) )
= ( last_a @ Xs ) ) ) ).
% last_appendL
thf(fact_282_last__appendL,axiom,
! [Ys: list_set_a,Xs: list_set_a] :
( ( Ys = nil_set_a )
=> ( ( last_set_a @ ( append_set_a @ Xs @ Ys ) )
= ( last_set_a @ Xs ) ) ) ).
% last_appendL
thf(fact_283_last__appendR,axiom,
! [Ys: list_a,Xs: list_a] :
( ( Ys != nil_a )
=> ( ( last_a @ ( append_a @ Xs @ Ys ) )
= ( last_a @ Ys ) ) ) ).
% last_appendR
thf(fact_284_last__appendR,axiom,
! [Ys: list_set_a,Xs: list_set_a] :
( ( Ys != nil_set_a )
=> ( ( last_set_a @ ( append_set_a @ Xs @ Ys ) )
= ( last_set_a @ Ys ) ) ) ).
% last_appendR
thf(fact_285_rev__eq__Cons__iff,axiom,
! [Xs: list_a,Y: a,Ys: list_a] :
( ( ( rev_a @ Xs )
= ( cons_a @ Y @ Ys ) )
= ( Xs
= ( append_a @ ( rev_a @ Ys ) @ ( cons_a @ Y @ nil_a ) ) ) ) ).
% rev_eq_Cons_iff
thf(fact_286_rev__eq__Cons__iff,axiom,
! [Xs: list_set_a,Y: set_a,Ys: list_set_a] :
( ( ( rev_set_a @ Xs )
= ( cons_set_a @ Y @ Ys ) )
= ( Xs
= ( append_set_a @ ( rev_set_a @ Ys ) @ ( cons_set_a @ Y @ nil_set_a ) ) ) ) ).
% rev_eq_Cons_iff
thf(fact_287_last__snoc,axiom,
! [Xs: list_a,X3: a] :
( ( last_a @ ( append_a @ Xs @ ( cons_a @ X3 @ nil_a ) ) )
= X3 ) ).
% last_snoc
thf(fact_288_last__snoc,axiom,
! [Xs: list_set_a,X3: set_a] :
( ( last_set_a @ ( append_set_a @ Xs @ ( cons_set_a @ X3 @ nil_set_a ) ) )
= X3 ) ).
% last_snoc
thf(fact_289_hd__Cons__tl,axiom,
! [Xs: list_a] :
( ( Xs != nil_a )
=> ( ( cons_a @ ( hd_a @ Xs ) @ ( tl_a @ Xs ) )
= Xs ) ) ).
% hd_Cons_tl
thf(fact_290_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_291_list_Ocollapse,axiom,
! [List: list_a] :
( ( List != nil_a )
=> ( ( cons_a @ ( hd_a @ List ) @ ( tl_a @ List ) )
= List ) ) ).
% list.collapse
thf(fact_292_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_293_is__subgraphI,axiom,
! [V3: set_set_a,V4: set_set_a,E3: set_set_set_a,E4: set_set_set_a] :
( ( ord_le3724670747650509150_set_a @ V3 @ V4 )
=> ( ( ord_le5722252365846178494_set_a @ E3 @ E4 )
=> ( ( undire7159349782766787846_set_a @ V3 @ E3 )
=> ( ( undire7159349782766787846_set_a @ V4 @ E4 )
=> ( undire1186139521737116585_set_a @ V3 @ E3 @ V4 @ E4 ) ) ) ) ) ).
% is_subgraphI
thf(fact_294_is__subgraphI,axiom,
! [V3: set_Product_prod_a_a,V4: set_Product_prod_a_a,E3: set_se5735800977113168103od_a_a,E4: set_se5735800977113168103od_a_a] :
( ( ord_le746702958409616551od_a_a @ V3 @ V4 )
=> ( ( ord_le1995061765932249223od_a_a @ E3 @ E4 )
=> ( ( undire1860116983885411791od_a_a @ V3 @ E3 )
=> ( ( undire1860116983885411791od_a_a @ V4 @ E4 )
=> ( undire398746457437328754od_a_a @ V3 @ E3 @ V4 @ E4 ) ) ) ) ) ).
% is_subgraphI
thf(fact_295_is__subgraphI,axiom,
! [V3: set_a,V4: set_a,E3: set_set_a,E4: set_set_a] :
( ( ord_less_eq_set_a @ V3 @ V4 )
=> ( ( ord_le3724670747650509150_set_a @ E3 @ E4 )
=> ( ( undire2554140024507503526stem_a @ V3 @ E3 )
=> ( ( undire2554140024507503526stem_a @ V4 @ E4 )
=> ( undire7103218114511261257raph_a @ V3 @ E3 @ V4 @ E4 ) ) ) ) ) ).
% is_subgraphI
thf(fact_296_tl__append__if,axiom,
! [Xs: list_a,Ys: list_a] :
( ( ( Xs = nil_a )
=> ( ( tl_a @ ( append_a @ Xs @ Ys ) )
= ( tl_a @ Ys ) ) )
& ( ( Xs != nil_a )
=> ( ( tl_a @ ( append_a @ Xs @ Ys ) )
= ( append_a @ ( tl_a @ Xs ) @ Ys ) ) ) ) ).
% tl_append_if
thf(fact_297_tl__append__if,axiom,
! [Xs: list_set_a,Ys: list_set_a] :
( ( ( Xs = nil_set_a )
=> ( ( tl_set_a @ ( append_set_a @ Xs @ Ys ) )
= ( tl_set_a @ Ys ) ) )
& ( ( Xs != nil_set_a )
=> ( ( tl_set_a @ ( append_set_a @ Xs @ Ys ) )
= ( append_set_a @ ( tl_set_a @ Xs ) @ Ys ) ) ) ) ).
% tl_append_if
thf(fact_298_append__eq__appendI,axiom,
! [Xs: list_a,Xs1: list_a,Zs: list_a,Ys: list_a,Us: list_a] :
( ( ( append_a @ Xs @ Xs1 )
= Zs )
=> ( ( Ys
= ( append_a @ Xs1 @ Us ) )
=> ( ( append_a @ Xs @ Ys )
= ( append_a @ Zs @ Us ) ) ) ) ).
% append_eq_appendI
thf(fact_299_append__eq__appendI,axiom,
! [Xs: list_set_a,Xs1: list_set_a,Zs: list_set_a,Ys: list_set_a,Us: list_set_a] :
( ( ( append_set_a @ Xs @ Xs1 )
= Zs )
=> ( ( Ys
= ( append_set_a @ Xs1 @ Us ) )
=> ( ( append_set_a @ Xs @ Ys )
= ( append_set_a @ Zs @ Us ) ) ) ) ).
% append_eq_appendI
thf(fact_300_append__eq__append__conv2,axiom,
! [Xs: list_a,Ys: list_a,Zs: list_a,Ts: list_a] :
( ( ( append_a @ Xs @ Ys )
= ( append_a @ Zs @ Ts ) )
= ( ? [Us2: list_a] :
( ( ( Xs
= ( append_a @ Zs @ Us2 ) )
& ( ( append_a @ Us2 @ Ys )
= Ts ) )
| ( ( ( append_a @ Xs @ Us2 )
= Zs )
& ( Ys
= ( append_a @ Us2 @ Ts ) ) ) ) ) ) ).
% append_eq_append_conv2
thf(fact_301_append__eq__append__conv2,axiom,
! [Xs: list_set_a,Ys: list_set_a,Zs: list_set_a,Ts: list_set_a] :
( ( ( append_set_a @ Xs @ Ys )
= ( append_set_a @ Zs @ Ts ) )
= ( ? [Us2: list_set_a] :
( ( ( Xs
= ( append_set_a @ Zs @ Us2 ) )
& ( ( append_set_a @ Us2 @ Ys )
= Ts ) )
| ( ( ( append_set_a @ Xs @ Us2 )
= Zs )
& ( Ys
= ( append_set_a @ Us2 @ Ts ) ) ) ) ) ) ).
% append_eq_append_conv2
thf(fact_302_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_303_comp__sgraph_Owalk__edges__append__ss2,axiom,
! [Xs: list_set_a,Ys: 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 @ Ys ) ) ) ) ).
% comp_sgraph.walk_edges_append_ss2
thf(fact_304_comp__sgraph_Owalk__edges__append__ss2,axiom,
! [Xs: list_a,Ys: list_a] : ( ord_le3724670747650509150_set_a @ ( set_set_a2 @ ( undire7337870655677353998dges_a @ Xs ) ) @ ( set_set_a2 @ ( undire7337870655677353998dges_a @ ( append_a @ Xs @ Ys ) ) ) ) ).
% comp_sgraph.walk_edges_append_ss2
thf(fact_305_comp__sgraph_Owalk__edges__append__ss1,axiom,
! [Ys: list_set_a,Xs: list_set_a] : ( ord_le5722252365846178494_set_a @ ( set_set_set_a2 @ ( undire6234387080713648494_set_a @ Ys ) ) @ ( set_set_set_a2 @ ( undire6234387080713648494_set_a @ ( append_set_a @ Xs @ Ys ) ) ) ) ).
% comp_sgraph.walk_edges_append_ss1
thf(fact_306_comp__sgraph_Owalk__edges__append__ss1,axiom,
! [Ys: list_a,Xs: list_a] : ( ord_le3724670747650509150_set_a @ ( set_set_a2 @ ( undire7337870655677353998dges_a @ Ys ) ) @ ( set_set_a2 @ ( undire7337870655677353998dges_a @ ( append_a @ Xs @ Ys ) ) ) ) ).
% comp_sgraph.walk_edges_append_ss1
thf(fact_307_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_308_ulgraph_Owalk__edges__append__ss2,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,Xs: list_set_a,Ys: 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 @ Ys ) ) ) ) ) ).
% ulgraph.walk_edges_append_ss2
thf(fact_309_ulgraph_Owalk__edges__append__ss2,axiom,
! [Vertices: set_a,Edges: set_set_a,Xs: list_a,Ys: list_a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ord_le3724670747650509150_set_a @ ( set_set_a2 @ ( undire7337870655677353998dges_a @ Xs ) ) @ ( set_set_a2 @ ( undire7337870655677353998dges_a @ ( append_a @ Xs @ Ys ) ) ) ) ) ).
% ulgraph.walk_edges_append_ss2
thf(fact_310_ulgraph_Owalk__edges__append__ss1,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,Ys: list_set_a,Xs: list_set_a] :
( ( undire6886684016831807756_set_a @ Vertices @ Edges )
=> ( ord_le5722252365846178494_set_a @ ( set_set_set_a2 @ ( undire6234387080713648494_set_a @ Ys ) ) @ ( set_set_set_a2 @ ( undire6234387080713648494_set_a @ ( append_set_a @ Xs @ Ys ) ) ) ) ) ).
% ulgraph.walk_edges_append_ss1
thf(fact_311_ulgraph_Owalk__edges__append__ss1,axiom,
! [Vertices: set_a,Edges: set_set_a,Ys: list_a,Xs: list_a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ord_le3724670747650509150_set_a @ ( set_set_a2 @ ( undire7337870655677353998dges_a @ Ys ) ) @ ( set_set_a2 @ ( undire7337870655677353998dges_a @ ( append_a @ Xs @ Ys ) ) ) ) ) ).
% ulgraph.walk_edges_append_ss1
thf(fact_312_list_Osel_I3_J,axiom,
! [X21: a,X22: list_a] :
( ( tl_a @ ( cons_a @ X21 @ X22 ) )
= X22 ) ).
% list.sel(3)
thf(fact_313_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_314_list_Osel_I2_J,axiom,
( ( tl_a @ nil_a )
= nil_a ) ).
% list.sel(2)
thf(fact_315_list_Osel_I2_J,axiom,
( ( tl_set_a @ nil_set_a )
= nil_set_a ) ).
% list.sel(2)
thf(fact_316_append__Cons,axiom,
! [X3: a,Xs: list_a,Ys: list_a] :
( ( append_a @ ( cons_a @ X3 @ Xs ) @ Ys )
= ( cons_a @ X3 @ ( append_a @ Xs @ Ys ) ) ) ).
% append_Cons
thf(fact_317_append__Cons,axiom,
! [X3: set_a,Xs: list_set_a,Ys: list_set_a] :
( ( append_set_a @ ( cons_set_a @ X3 @ Xs ) @ Ys )
= ( cons_set_a @ X3 @ ( append_set_a @ Xs @ Ys ) ) ) ).
% append_Cons
thf(fact_318_Cons__eq__appendI,axiom,
! [X3: a,Xs1: list_a,Ys: list_a,Xs: list_a,Zs: list_a] :
( ( ( cons_a @ X3 @ Xs1 )
= Ys )
=> ( ( Xs
= ( append_a @ Xs1 @ Zs ) )
=> ( ( cons_a @ X3 @ Xs )
= ( append_a @ Ys @ Zs ) ) ) ) ).
% Cons_eq_appendI
thf(fact_319_Cons__eq__appendI,axiom,
! [X3: set_a,Xs1: list_set_a,Ys: list_set_a,Xs: list_set_a,Zs: list_set_a] :
( ( ( cons_set_a @ X3 @ Xs1 )
= Ys )
=> ( ( Xs
= ( append_set_a @ Xs1 @ Zs ) )
=> ( ( cons_set_a @ X3 @ Xs )
= ( append_set_a @ Ys @ Zs ) ) ) ) ).
% Cons_eq_appendI
thf(fact_320_append__Nil,axiom,
! [Ys: list_a] :
( ( append_a @ nil_a @ Ys )
= Ys ) ).
% append_Nil
thf(fact_321_append__Nil,axiom,
! [Ys: list_set_a] :
( ( append_set_a @ nil_set_a @ Ys )
= Ys ) ).
% append_Nil
thf(fact_322_append_Oleft__neutral,axiom,
! [A: list_a] :
( ( append_a @ nil_a @ A )
= A ) ).
% append.left_neutral
thf(fact_323_append_Oleft__neutral,axiom,
! [A: list_set_a] :
( ( append_set_a @ nil_set_a @ A )
= A ) ).
% append.left_neutral
thf(fact_324_eq__Nil__appendI,axiom,
! [Xs: list_a,Ys: list_a] :
( ( Xs = Ys )
=> ( Xs
= ( append_a @ nil_a @ Ys ) ) ) ).
% eq_Nil_appendI
thf(fact_325_eq__Nil__appendI,axiom,
! [Xs: list_set_a,Ys: list_set_a] :
( ( Xs = Ys )
=> ( Xs
= ( append_set_a @ nil_set_a @ Ys ) ) ) ).
% eq_Nil_appendI
thf(fact_326_distinct__tl,axiom,
! [Xs: list_a] :
( ( distinct_a @ Xs )
=> ( distinct_a @ ( tl_a @ Xs ) ) ) ).
% distinct_tl
thf(fact_327_distinct__tl,axiom,
! [Xs: list_set_a] :
( ( distinct_set_a @ Xs )
=> ( distinct_set_a @ ( tl_set_a @ Xs ) ) ) ).
% distinct_tl
thf(fact_328_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_329_comp__sgraph_Owalk__edges_Osimps_I1_J,axiom,
( ( undire7337870655677353998dges_a @ nil_a )
= nil_set_a ) ).
% comp_sgraph.walk_edges.simps(1)
thf(fact_330_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_331_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_332_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_333_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_334_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_335_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_336_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_337_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_338_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_339_comp__sgraph_Odistinct__edgesI,axiom,
! [P: list_set_a] :
( ( distinct_set_a @ P )
=> ( distinct_set_set_a @ ( undire6234387080713648494_set_a @ P ) ) ) ).
% comp_sgraph.distinct_edgesI
thf(fact_340_comp__sgraph_Odistinct__edgesI,axiom,
! [P: list_a] :
( ( distinct_a @ P )
=> ( distinct_set_a @ ( undire7337870655677353998dges_a @ P ) ) ) ).
% comp_sgraph.distinct_edgesI
thf(fact_341_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_342_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_343_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_344_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_345_subgraph_Osubgraph__trans,axiom,
! [V_H: set_a,E_H: set_set_a,V_G: set_a,E_G: set_set_a,V4: set_a,E4: set_set_a,V3: set_a,E3: set_set_a,V5: set_a,E7: set_set_a] :
( ( undire7103218114511261257raph_a @ V_H @ E_H @ V_G @ E_G )
=> ( ( undire2554140024507503526stem_a @ V4 @ E4 )
=> ( ( undire2554140024507503526stem_a @ V3 @ E3 )
=> ( ( undire2554140024507503526stem_a @ V5 @ E7 )
=> ( ( undire7103218114511261257raph_a @ V5 @ E7 @ V3 @ E3 )
=> ( ( undire7103218114511261257raph_a @ V3 @ E3 @ V4 @ E4 )
=> ( undire7103218114511261257raph_a @ V5 @ E7 @ V4 @ E4 ) ) ) ) ) ) ) ).
% subgraph.subgraph_trans
thf(fact_346_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_347_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_348_graph__system_Oinduced__is__graph__sys,axiom,
! [Vertices: set_a,Edges: set_set_a,V3: set_a] :
( ( undire2554140024507503526stem_a @ Vertices @ Edges )
=> ( undire2554140024507503526stem_a @ V3 @ ( undire7777452895879145676dges_a @ Edges @ V3 ) ) ) ).
% graph_system.induced_is_graph_sys
thf(fact_349_graph__system_Oincident__edge__in__wf,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,E: set_set_a,V: set_a] :
( ( undire7159349782766787846_set_a @ Vertices @ Edges )
=> ( ( member_set_set_a @ E @ Edges )
=> ( ( undire2320338297334612420_set_a @ V @ E )
=> ( member_set_a @ V @ Vertices ) ) ) ) ).
% graph_system.incident_edge_in_wf
thf(fact_350_graph__system_Oincident__edge__in__wf,axiom,
! [Vertices: set_Product_prod_a_a,Edges: set_se5735800977113168103od_a_a,E: set_Product_prod_a_a,V: product_prod_a_a] :
( ( undire1860116983885411791od_a_a @ Vertices @ Edges )
=> ( ( member1816616512716248880od_a_a @ E @ Edges )
=> ( ( undire3369688177417741453od_a_a @ V @ E )
=> ( member1426531477525435216od_a_a @ V @ Vertices ) ) ) ) ).
% graph_system.incident_edge_in_wf
thf(fact_351_graph__system_Oincident__edge__in__wf,axiom,
! [Vertices: set_a,Edges: set_set_a,E: set_a,V: a] :
( ( undire2554140024507503526stem_a @ Vertices @ Edges )
=> ( ( member_set_a @ E @ Edges )
=> ( ( undire1521409233611534436dent_a @ V @ E )
=> ( member_a @ V @ Vertices ) ) ) ) ).
% graph_system.incident_edge_in_wf
thf(fact_352_graph__system_Oincident__def,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,V: set_a,E: set_set_a] :
( ( undire7159349782766787846_set_a @ Vertices @ Edges )
=> ( ( undire2320338297334612420_set_a @ V @ E )
= ( member_set_a @ V @ E ) ) ) ).
% graph_system.incident_def
thf(fact_353_graph__system_Oincident__def,axiom,
! [Vertices: set_Product_prod_a_a,Edges: set_se5735800977113168103od_a_a,V: product_prod_a_a,E: set_Product_prod_a_a] :
( ( undire1860116983885411791od_a_a @ Vertices @ Edges )
=> ( ( undire3369688177417741453od_a_a @ V @ E )
= ( member1426531477525435216od_a_a @ V @ E ) ) ) ).
% graph_system.incident_def
thf(fact_354_graph__system_Oincident__def,axiom,
! [Vertices: set_a,Edges: set_set_a,V: a,E: set_a] :
( ( undire2554140024507503526stem_a @ Vertices @ Edges )
=> ( ( undire1521409233611534436dent_a @ V @ E )
= ( member_a @ V @ E ) ) ) ).
% graph_system.incident_def
thf(fact_355_comp__sgraph_Owalk__edges__decomp__ss,axiom,
! [Xs: list_set_a,Y: set_a,Zs: list_set_a,Ys: list_set_a] : ( ord_le5722252365846178494_set_a @ ( set_set_set_a2 @ ( undire6234387080713648494_set_a @ ( append_set_a @ Xs @ ( append_set_a @ ( cons_set_a @ Y @ nil_set_a ) @ Zs ) ) ) ) @ ( set_set_set_a2 @ ( undire6234387080713648494_set_a @ ( append_set_a @ Xs @ ( append_set_a @ ( cons_set_a @ Y @ nil_set_a ) @ ( append_set_a @ Ys @ ( append_set_a @ ( cons_set_a @ Y @ nil_set_a ) @ Zs ) ) ) ) ) ) ) ).
% comp_sgraph.walk_edges_decomp_ss
thf(fact_356_comp__sgraph_Owalk__edges__decomp__ss,axiom,
! [Xs: list_a,Y: a,Zs: list_a,Ys: list_a] : ( ord_le3724670747650509150_set_a @ ( set_set_a2 @ ( undire7337870655677353998dges_a @ ( append_a @ Xs @ ( append_a @ ( cons_a @ Y @ nil_a ) @ Zs ) ) ) ) @ ( set_set_a2 @ ( undire7337870655677353998dges_a @ ( append_a @ Xs @ ( append_a @ ( cons_a @ Y @ nil_a ) @ ( append_a @ Ys @ ( append_a @ ( cons_a @ Y @ nil_a ) @ Zs ) ) ) ) ) ) ) ).
% comp_sgraph.walk_edges_decomp_ss
thf(fact_357_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 )
=> ( ? [X: set_a] :
( ( member_set_a @ X @ Vertices )
& ( member_set_a @ X @ E1 )
& ( member_set_a @ X @ E2 ) )
=> ( undire3485422320110889978_set_a @ Edges @ E1 @ E2 ) ) ) ) ) ).
% graph_system.edge_adjacent_alt_def
thf(fact_358_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 )
=> ( ? [X: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X @ Vertices )
& ( member1426531477525435216od_a_a @ X @ E1 )
& ( member1426531477525435216od_a_a @ X @ E2 ) )
=> ( undire9186443406341554371od_a_a @ Edges @ E1 @ E2 ) ) ) ) ) ).
% graph_system.edge_adjacent_alt_def
thf(fact_359_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 )
=> ( ? [X: a] :
( ( member_a @ X @ Vertices )
& ( member_a @ X @ E1 )
& ( member_a @ X @ E2 ) )
=> ( undire4022703626023482010_adj_a @ Edges @ E1 @ E2 ) ) ) ) ) ).
% graph_system.edge_adjacent_alt_def
thf(fact_360_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_361_ulgraph_Owalk__edges__decomp__ss,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,Xs: list_set_a,Y: set_a,Zs: list_set_a,Ys: 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 @ Y @ nil_set_a ) @ Zs ) ) ) ) @ ( set_set_set_a2 @ ( undire6234387080713648494_set_a @ ( append_set_a @ Xs @ ( append_set_a @ ( cons_set_a @ Y @ nil_set_a ) @ ( append_set_a @ Ys @ ( append_set_a @ ( cons_set_a @ Y @ nil_set_a ) @ Zs ) ) ) ) ) ) ) ) ).
% ulgraph.walk_edges_decomp_ss
thf(fact_362_ulgraph_Owalk__edges__decomp__ss,axiom,
! [Vertices: set_a,Edges: set_set_a,Xs: list_a,Y: a,Zs: list_a,Ys: list_a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ord_le3724670747650509150_set_a @ ( set_set_a2 @ ( undire7337870655677353998dges_a @ ( append_a @ Xs @ ( append_a @ ( cons_a @ Y @ nil_a ) @ Zs ) ) ) ) @ ( set_set_a2 @ ( undire7337870655677353998dges_a @ ( append_a @ Xs @ ( append_a @ ( cons_a @ Y @ nil_a ) @ ( append_a @ Ys @ ( append_a @ ( cons_a @ Y @ nil_a ) @ Zs ) ) ) ) ) ) ) ) ).
% ulgraph.walk_edges_decomp_ss
thf(fact_363_Nil__tl,axiom,
! [Xs: list_a] :
( ( nil_a
= ( tl_a @ Xs ) )
= ( ( Xs = nil_a )
| ? [X2: a] :
( Xs
= ( cons_a @ X2 @ nil_a ) ) ) ) ).
% Nil_tl
thf(fact_364_Nil__tl,axiom,
! [Xs: list_set_a] :
( ( nil_set_a
= ( tl_set_a @ Xs ) )
= ( ( Xs = nil_set_a )
| ? [X2: set_a] :
( Xs
= ( cons_set_a @ X2 @ nil_set_a ) ) ) ) ).
% Nil_tl
thf(fact_365_tl__Nil,axiom,
! [Xs: list_a] :
( ( ( tl_a @ Xs )
= nil_a )
= ( ( Xs = nil_a )
| ? [X2: a] :
( Xs
= ( cons_a @ X2 @ nil_a ) ) ) ) ).
% tl_Nil
thf(fact_366_tl__Nil,axiom,
! [Xs: list_set_a] :
( ( ( tl_set_a @ Xs )
= nil_set_a )
= ( ( Xs = nil_set_a )
| ? [X2: set_a] :
( Xs
= ( cons_set_a @ X2 @ nil_set_a ) ) ) ) ).
% tl_Nil
thf(fact_367_list_Oset__sel_I2_J,axiom,
! [A: list_P1396940483166286381od_a_a,X3: product_prod_a_a] :
( ( A != nil_Product_prod_a_a )
=> ( ( member1426531477525435216od_a_a @ X3 @ ( set_Product_prod_a_a2 @ ( tl_Product_prod_a_a @ A ) ) )
=> ( member1426531477525435216od_a_a @ X3 @ ( set_Product_prod_a_a2 @ A ) ) ) ) ).
% list.set_sel(2)
thf(fact_368_list_Oset__sel_I2_J,axiom,
! [A: list_a,X3: a] :
( ( A != nil_a )
=> ( ( member_a @ X3 @ ( set_a2 @ ( tl_a @ A ) ) )
=> ( member_a @ X3 @ ( set_a2 @ A ) ) ) ) ).
% list.set_sel(2)
thf(fact_369_list_Oset__sel_I2_J,axiom,
! [A: list_set_a,X3: set_a] :
( ( A != nil_set_a )
=> ( ( member_set_a @ X3 @ ( set_set_a2 @ ( tl_set_a @ A ) ) )
=> ( member_set_a @ X3 @ ( set_set_a2 @ A ) ) ) ) ).
% list.set_sel(2)
thf(fact_370_ulgraph_Ois__walk__append,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,Xs: list_set_a,Ys: list_set_a] :
( ( undire6886684016831807756_set_a @ Vertices @ Edges )
=> ( ( undire3014741414213135564_set_a @ Vertices @ Edges @ Xs )
=> ( ( undire3014741414213135564_set_a @ Vertices @ Edges @ Ys )
=> ( ( ( last_set_a @ Xs )
= ( hd_set_a @ Ys ) )
=> ( undire3014741414213135564_set_a @ Vertices @ Edges @ ( append_set_a @ Xs @ ( tl_set_a @ Ys ) ) ) ) ) ) ) ).
% ulgraph.is_walk_append
thf(fact_371_ulgraph_Ois__walk__append,axiom,
! [Vertices: set_a,Edges: set_set_a,Xs: list_a,Ys: list_a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( undire6133010728901294956walk_a @ Vertices @ Edges @ Xs )
=> ( ( undire6133010728901294956walk_a @ Vertices @ Edges @ Ys )
=> ( ( ( last_a @ Xs )
= ( hd_a @ Ys ) )
=> ( undire6133010728901294956walk_a @ Vertices @ Edges @ ( append_a @ Xs @ ( tl_a @ Ys ) ) ) ) ) ) ) ).
% ulgraph.is_walk_append
thf(fact_372_rev__induct,axiom,
! [P2: list_a > $o,Xs: list_a] :
( ( P2 @ nil_a )
=> ( ! [X4: a,Xs2: list_a] :
( ( P2 @ Xs2 )
=> ( P2 @ ( append_a @ Xs2 @ ( cons_a @ X4 @ nil_a ) ) ) )
=> ( P2 @ Xs ) ) ) ).
% rev_induct
thf(fact_373_rev__induct,axiom,
! [P2: list_set_a > $o,Xs: list_set_a] :
( ( P2 @ nil_set_a )
=> ( ! [X4: set_a,Xs2: list_set_a] :
( ( P2 @ Xs2 )
=> ( P2 @ ( append_set_a @ Xs2 @ ( cons_set_a @ X4 @ nil_set_a ) ) ) )
=> ( P2 @ Xs ) ) ) ).
% rev_induct
thf(fact_374_rev__exhaust,axiom,
! [Xs: list_a] :
( ( Xs != nil_a )
=> ~ ! [Ys2: list_a,Y2: a] :
( Xs
!= ( append_a @ Ys2 @ ( cons_a @ Y2 @ nil_a ) ) ) ) ).
% rev_exhaust
thf(fact_375_rev__exhaust,axiom,
! [Xs: list_set_a] :
( ( Xs != nil_set_a )
=> ~ ! [Ys2: list_set_a,Y2: set_a] :
( Xs
!= ( append_set_a @ Ys2 @ ( cons_set_a @ Y2 @ nil_set_a ) ) ) ) ).
% rev_exhaust
thf(fact_376_Cons__eq__append__conv,axiom,
! [X3: a,Xs: list_a,Ys: list_a,Zs: list_a] :
( ( ( cons_a @ X3 @ Xs )
= ( append_a @ Ys @ Zs ) )
= ( ( ( Ys = nil_a )
& ( ( cons_a @ X3 @ Xs )
= Zs ) )
| ? [Ys4: list_a] :
( ( ( cons_a @ X3 @ Ys4 )
= Ys )
& ( Xs
= ( append_a @ Ys4 @ Zs ) ) ) ) ) ).
% Cons_eq_append_conv
thf(fact_377_Cons__eq__append__conv,axiom,
! [X3: set_a,Xs: list_set_a,Ys: list_set_a,Zs: list_set_a] :
( ( ( cons_set_a @ X3 @ Xs )
= ( append_set_a @ Ys @ Zs ) )
= ( ( ( Ys = nil_set_a )
& ( ( cons_set_a @ X3 @ Xs )
= Zs ) )
| ? [Ys4: list_set_a] :
( ( ( cons_set_a @ X3 @ Ys4 )
= Ys )
& ( Xs
= ( append_set_a @ Ys4 @ Zs ) ) ) ) ) ).
% Cons_eq_append_conv
thf(fact_378_append__eq__Cons__conv,axiom,
! [Ys: list_a,Zs: list_a,X3: a,Xs: list_a] :
( ( ( append_a @ Ys @ Zs )
= ( cons_a @ X3 @ Xs ) )
= ( ( ( Ys = nil_a )
& ( Zs
= ( cons_a @ X3 @ Xs ) ) )
| ? [Ys4: list_a] :
( ( Ys
= ( cons_a @ X3 @ Ys4 ) )
& ( ( append_a @ Ys4 @ Zs )
= Xs ) ) ) ) ).
% append_eq_Cons_conv
thf(fact_379_append__eq__Cons__conv,axiom,
! [Ys: list_set_a,Zs: list_set_a,X3: set_a,Xs: list_set_a] :
( ( ( append_set_a @ Ys @ Zs )
= ( cons_set_a @ X3 @ Xs ) )
= ( ( ( Ys = nil_set_a )
& ( Zs
= ( cons_set_a @ X3 @ Xs ) ) )
| ? [Ys4: list_set_a] :
( ( Ys
= ( cons_set_a @ X3 @ Ys4 ) )
& ( ( append_set_a @ Ys4 @ Zs )
= Xs ) ) ) ) ).
% append_eq_Cons_conv
thf(fact_380_rev__nonempty__induct,axiom,
! [Xs: list_a,P2: list_a > $o] :
( ( Xs != nil_a )
=> ( ! [X4: a] : ( P2 @ ( cons_a @ X4 @ nil_a ) )
=> ( ! [X4: a,Xs2: list_a] :
( ( Xs2 != nil_a )
=> ( ( P2 @ Xs2 )
=> ( P2 @ ( append_a @ Xs2 @ ( cons_a @ X4 @ nil_a ) ) ) ) )
=> ( P2 @ Xs ) ) ) ) ).
% rev_nonempty_induct
thf(fact_381_rev__nonempty__induct,axiom,
! [Xs: list_set_a,P2: list_set_a > $o] :
( ( Xs != nil_set_a )
=> ( ! [X4: set_a] : ( P2 @ ( cons_set_a @ X4 @ nil_set_a ) )
=> ( ! [X4: set_a,Xs2: list_set_a] :
( ( Xs2 != nil_set_a )
=> ( ( P2 @ Xs2 )
=> ( P2 @ ( append_set_a @ Xs2 @ ( cons_set_a @ X4 @ nil_set_a ) ) ) ) )
=> ( P2 @ Xs ) ) ) ) ).
% rev_nonempty_induct
thf(fact_382_split__list,axiom,
! [X3: product_prod_a_a,Xs: list_P1396940483166286381od_a_a] :
( ( member1426531477525435216od_a_a @ X3 @ ( set_Product_prod_a_a2 @ Xs ) )
=> ? [Ys2: list_P1396940483166286381od_a_a,Zs2: list_P1396940483166286381od_a_a] :
( Xs
= ( append5335208819046833346od_a_a @ Ys2 @ ( cons_P7316939126706565853od_a_a @ X3 @ Zs2 ) ) ) ) ).
% split_list
thf(fact_383_split__list,axiom,
! [X3: a,Xs: list_a] :
( ( member_a @ X3 @ ( set_a2 @ Xs ) )
=> ? [Ys2: list_a,Zs2: list_a] :
( Xs
= ( append_a @ Ys2 @ ( cons_a @ X3 @ Zs2 ) ) ) ) ).
% split_list
thf(fact_384_split__list,axiom,
! [X3: set_a,Xs: list_set_a] :
( ( member_set_a @ X3 @ ( set_set_a2 @ Xs ) )
=> ? [Ys2: list_set_a,Zs2: list_set_a] :
( Xs
= ( append_set_a @ Ys2 @ ( cons_set_a @ X3 @ Zs2 ) ) ) ) ).
% split_list
thf(fact_385_split__list__last,axiom,
! [X3: product_prod_a_a,Xs: list_P1396940483166286381od_a_a] :
( ( member1426531477525435216od_a_a @ X3 @ ( set_Product_prod_a_a2 @ Xs ) )
=> ? [Ys2: list_P1396940483166286381od_a_a,Zs2: list_P1396940483166286381od_a_a] :
( ( Xs
= ( append5335208819046833346od_a_a @ Ys2 @ ( cons_P7316939126706565853od_a_a @ X3 @ Zs2 ) ) )
& ~ ( member1426531477525435216od_a_a @ X3 @ ( set_Product_prod_a_a2 @ Zs2 ) ) ) ) ).
% split_list_last
thf(fact_386_split__list__last,axiom,
! [X3: a,Xs: list_a] :
( ( member_a @ X3 @ ( set_a2 @ Xs ) )
=> ? [Ys2: list_a,Zs2: list_a] :
( ( Xs
= ( append_a @ Ys2 @ ( cons_a @ X3 @ Zs2 ) ) )
& ~ ( member_a @ X3 @ ( set_a2 @ Zs2 ) ) ) ) ).
% split_list_last
thf(fact_387_split__list__last,axiom,
! [X3: set_a,Xs: list_set_a] :
( ( member_set_a @ X3 @ ( set_set_a2 @ Xs ) )
=> ? [Ys2: list_set_a,Zs2: list_set_a] :
( ( Xs
= ( append_set_a @ Ys2 @ ( cons_set_a @ X3 @ Zs2 ) ) )
& ~ ( member_set_a @ X3 @ ( set_set_a2 @ Zs2 ) ) ) ) ).
% split_list_last
thf(fact_388_split__list__prop,axiom,
! [Xs: list_a,P2: a > $o] :
( ? [X: a] :
( ( member_a @ X @ ( set_a2 @ Xs ) )
& ( P2 @ X ) )
=> ? [Ys2: list_a,X4: a] :
( ? [Zs2: list_a] :
( Xs
= ( append_a @ Ys2 @ ( cons_a @ X4 @ Zs2 ) ) )
& ( P2 @ X4 ) ) ) ).
% split_list_prop
thf(fact_389_split__list__prop,axiom,
! [Xs: list_set_a,P2: set_a > $o] :
( ? [X: set_a] :
( ( member_set_a @ X @ ( set_set_a2 @ Xs ) )
& ( P2 @ X ) )
=> ? [Ys2: list_set_a,X4: set_a] :
( ? [Zs2: list_set_a] :
( Xs
= ( append_set_a @ Ys2 @ ( cons_set_a @ X4 @ Zs2 ) ) )
& ( P2 @ X4 ) ) ) ).
% split_list_prop
thf(fact_390_split__list__first,axiom,
! [X3: product_prod_a_a,Xs: list_P1396940483166286381od_a_a] :
( ( member1426531477525435216od_a_a @ X3 @ ( set_Product_prod_a_a2 @ Xs ) )
=> ? [Ys2: list_P1396940483166286381od_a_a,Zs2: list_P1396940483166286381od_a_a] :
( ( Xs
= ( append5335208819046833346od_a_a @ Ys2 @ ( cons_P7316939126706565853od_a_a @ X3 @ Zs2 ) ) )
& ~ ( member1426531477525435216od_a_a @ X3 @ ( set_Product_prod_a_a2 @ Ys2 ) ) ) ) ).
% split_list_first
thf(fact_391_split__list__first,axiom,
! [X3: a,Xs: list_a] :
( ( member_a @ X3 @ ( set_a2 @ Xs ) )
=> ? [Ys2: list_a,Zs2: list_a] :
( ( Xs
= ( append_a @ Ys2 @ ( cons_a @ X3 @ Zs2 ) ) )
& ~ ( member_a @ X3 @ ( set_a2 @ Ys2 ) ) ) ) ).
% split_list_first
thf(fact_392_split__list__first,axiom,
! [X3: set_a,Xs: list_set_a] :
( ( member_set_a @ X3 @ ( set_set_a2 @ Xs ) )
=> ? [Ys2: list_set_a,Zs2: list_set_a] :
( ( Xs
= ( append_set_a @ Ys2 @ ( cons_set_a @ X3 @ Zs2 ) ) )
& ~ ( member_set_a @ X3 @ ( set_set_a2 @ Ys2 ) ) ) ) ).
% split_list_first
thf(fact_393_split__list__propE,axiom,
! [Xs: list_a,P2: a > $o] :
( ? [X: a] :
( ( member_a @ X @ ( set_a2 @ Xs ) )
& ( P2 @ X ) )
=> ~ ! [Ys2: list_a,X4: a] :
( ? [Zs2: list_a] :
( Xs
= ( append_a @ Ys2 @ ( cons_a @ X4 @ Zs2 ) ) )
=> ~ ( P2 @ X4 ) ) ) ).
% split_list_propE
thf(fact_394_split__list__propE,axiom,
! [Xs: list_set_a,P2: set_a > $o] :
( ? [X: set_a] :
( ( member_set_a @ X @ ( set_set_a2 @ Xs ) )
& ( P2 @ X ) )
=> ~ ! [Ys2: list_set_a,X4: set_a] :
( ? [Zs2: list_set_a] :
( Xs
= ( append_set_a @ Ys2 @ ( cons_set_a @ X4 @ Zs2 ) ) )
=> ~ ( P2 @ X4 ) ) ) ).
% split_list_propE
thf(fact_395_append__Cons__eq__iff,axiom,
! [X3: product_prod_a_a,Xs: list_P1396940483166286381od_a_a,Ys: list_P1396940483166286381od_a_a,Xs3: list_P1396940483166286381od_a_a,Ys5: list_P1396940483166286381od_a_a] :
( ~ ( member1426531477525435216od_a_a @ X3 @ ( set_Product_prod_a_a2 @ Xs ) )
=> ( ~ ( member1426531477525435216od_a_a @ X3 @ ( set_Product_prod_a_a2 @ Ys ) )
=> ( ( ( append5335208819046833346od_a_a @ Xs @ ( cons_P7316939126706565853od_a_a @ X3 @ Ys ) )
= ( append5335208819046833346od_a_a @ Xs3 @ ( cons_P7316939126706565853od_a_a @ X3 @ Ys5 ) ) )
= ( ( Xs = Xs3 )
& ( Ys = Ys5 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_396_append__Cons__eq__iff,axiom,
! [X3: a,Xs: list_a,Ys: list_a,Xs3: list_a,Ys5: list_a] :
( ~ ( member_a @ X3 @ ( set_a2 @ Xs ) )
=> ( ~ ( member_a @ X3 @ ( set_a2 @ Ys ) )
=> ( ( ( append_a @ Xs @ ( cons_a @ X3 @ Ys ) )
= ( append_a @ Xs3 @ ( cons_a @ X3 @ Ys5 ) ) )
= ( ( Xs = Xs3 )
& ( Ys = Ys5 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_397_append__Cons__eq__iff,axiom,
! [X3: set_a,Xs: list_set_a,Ys: list_set_a,Xs3: list_set_a,Ys5: list_set_a] :
( ~ ( member_set_a @ X3 @ ( set_set_a2 @ Xs ) )
=> ( ~ ( member_set_a @ X3 @ ( set_set_a2 @ Ys ) )
=> ( ( ( append_set_a @ Xs @ ( cons_set_a @ X3 @ Ys ) )
= ( append_set_a @ Xs3 @ ( cons_set_a @ X3 @ Ys5 ) ) )
= ( ( Xs = Xs3 )
& ( Ys = Ys5 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_398_in__set__conv__decomp,axiom,
! [X3: product_prod_a_a,Xs: list_P1396940483166286381od_a_a] :
( ( member1426531477525435216od_a_a @ X3 @ ( set_Product_prod_a_a2 @ Xs ) )
= ( ? [Ys3: list_P1396940483166286381od_a_a,Zs3: list_P1396940483166286381od_a_a] :
( Xs
= ( append5335208819046833346od_a_a @ Ys3 @ ( cons_P7316939126706565853od_a_a @ X3 @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_399_in__set__conv__decomp,axiom,
! [X3: a,Xs: list_a] :
( ( member_a @ X3 @ ( set_a2 @ Xs ) )
= ( ? [Ys3: list_a,Zs3: list_a] :
( Xs
= ( append_a @ Ys3 @ ( cons_a @ X3 @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_400_in__set__conv__decomp,axiom,
! [X3: set_a,Xs: list_set_a] :
( ( member_set_a @ X3 @ ( set_set_a2 @ Xs ) )
= ( ? [Ys3: list_set_a,Zs3: list_set_a] :
( Xs
= ( append_set_a @ Ys3 @ ( cons_set_a @ X3 @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_401_split__list__last__prop,axiom,
! [Xs: list_a,P2: a > $o] :
( ? [X: a] :
( ( member_a @ X @ ( set_a2 @ Xs ) )
& ( P2 @ X ) )
=> ? [Ys2: list_a,X4: a,Zs2: list_a] :
( ( Xs
= ( append_a @ Ys2 @ ( cons_a @ X4 @ Zs2 ) ) )
& ( P2 @ X4 )
& ! [Xa: a] :
( ( member_a @ Xa @ ( set_a2 @ Zs2 ) )
=> ~ ( P2 @ Xa ) ) ) ) ).
% split_list_last_prop
thf(fact_402_split__list__last__prop,axiom,
! [Xs: list_set_a,P2: set_a > $o] :
( ? [X: set_a] :
( ( member_set_a @ X @ ( set_set_a2 @ Xs ) )
& ( P2 @ X ) )
=> ? [Ys2: list_set_a,X4: set_a,Zs2: list_set_a] :
( ( Xs
= ( append_set_a @ Ys2 @ ( cons_set_a @ X4 @ Zs2 ) ) )
& ( P2 @ X4 )
& ! [Xa: set_a] :
( ( member_set_a @ Xa @ ( set_set_a2 @ Zs2 ) )
=> ~ ( P2 @ Xa ) ) ) ) ).
% split_list_last_prop
thf(fact_403_split__list__first__prop,axiom,
! [Xs: list_a,P2: a > $o] :
( ? [X: a] :
( ( member_a @ X @ ( set_a2 @ Xs ) )
& ( P2 @ X ) )
=> ? [Ys2: list_a,X4: a] :
( ? [Zs2: list_a] :
( Xs
= ( append_a @ Ys2 @ ( cons_a @ X4 @ Zs2 ) ) )
& ( P2 @ X4 )
& ! [Xa: a] :
( ( member_a @ Xa @ ( set_a2 @ Ys2 ) )
=> ~ ( P2 @ Xa ) ) ) ) ).
% split_list_first_prop
thf(fact_404_split__list__first__prop,axiom,
! [Xs: list_set_a,P2: set_a > $o] :
( ? [X: set_a] :
( ( member_set_a @ X @ ( set_set_a2 @ Xs ) )
& ( P2 @ X ) )
=> ? [Ys2: list_set_a,X4: set_a] :
( ? [Zs2: list_set_a] :
( Xs
= ( append_set_a @ Ys2 @ ( cons_set_a @ X4 @ Zs2 ) ) )
& ( P2 @ X4 )
& ! [Xa: set_a] :
( ( member_set_a @ Xa @ ( set_set_a2 @ Ys2 ) )
=> ~ ( P2 @ Xa ) ) ) ) ).
% split_list_first_prop
thf(fact_405_split__list__last__propE,axiom,
! [Xs: list_a,P2: a > $o] :
( ? [X: a] :
( ( member_a @ X @ ( set_a2 @ Xs ) )
& ( P2 @ X ) )
=> ~ ! [Ys2: list_a,X4: a,Zs2: list_a] :
( ( Xs
= ( append_a @ Ys2 @ ( cons_a @ X4 @ Zs2 ) ) )
=> ( ( P2 @ X4 )
=> ~ ! [Xa: a] :
( ( member_a @ Xa @ ( set_a2 @ Zs2 ) )
=> ~ ( P2 @ Xa ) ) ) ) ) ).
% split_list_last_propE
thf(fact_406_split__list__last__propE,axiom,
! [Xs: list_set_a,P2: set_a > $o] :
( ? [X: set_a] :
( ( member_set_a @ X @ ( set_set_a2 @ Xs ) )
& ( P2 @ X ) )
=> ~ ! [Ys2: list_set_a,X4: set_a,Zs2: list_set_a] :
( ( Xs
= ( append_set_a @ Ys2 @ ( cons_set_a @ X4 @ Zs2 ) ) )
=> ( ( P2 @ X4 )
=> ~ ! [Xa: set_a] :
( ( member_set_a @ Xa @ ( set_set_a2 @ Zs2 ) )
=> ~ ( P2 @ Xa ) ) ) ) ) ).
% split_list_last_propE
thf(fact_407_split__list__first__propE,axiom,
! [Xs: list_a,P2: a > $o] :
( ? [X: a] :
( ( member_a @ X @ ( set_a2 @ Xs ) )
& ( P2 @ X ) )
=> ~ ! [Ys2: list_a,X4: a] :
( ? [Zs2: list_a] :
( Xs
= ( append_a @ Ys2 @ ( cons_a @ X4 @ Zs2 ) ) )
=> ( ( P2 @ X4 )
=> ~ ! [Xa: a] :
( ( member_a @ Xa @ ( set_a2 @ Ys2 ) )
=> ~ ( P2 @ Xa ) ) ) ) ) ).
% split_list_first_propE
thf(fact_408_split__list__first__propE,axiom,
! [Xs: list_set_a,P2: set_a > $o] :
( ? [X: set_a] :
( ( member_set_a @ X @ ( set_set_a2 @ Xs ) )
& ( P2 @ X ) )
=> ~ ! [Ys2: list_set_a,X4: set_a] :
( ? [Zs2: list_set_a] :
( Xs
= ( append_set_a @ Ys2 @ ( cons_set_a @ X4 @ Zs2 ) ) )
=> ( ( P2 @ X4 )
=> ~ ! [Xa: set_a] :
( ( member_set_a @ Xa @ ( set_set_a2 @ Ys2 ) )
=> ~ ( P2 @ Xa ) ) ) ) ) ).
% split_list_first_propE
thf(fact_409_in__set__conv__decomp__last,axiom,
! [X3: product_prod_a_a,Xs: list_P1396940483166286381od_a_a] :
( ( member1426531477525435216od_a_a @ X3 @ ( set_Product_prod_a_a2 @ Xs ) )
= ( ? [Ys3: list_P1396940483166286381od_a_a,Zs3: list_P1396940483166286381od_a_a] :
( ( Xs
= ( append5335208819046833346od_a_a @ Ys3 @ ( cons_P7316939126706565853od_a_a @ X3 @ Zs3 ) ) )
& ~ ( member1426531477525435216od_a_a @ X3 @ ( set_Product_prod_a_a2 @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_410_in__set__conv__decomp__last,axiom,
! [X3: a,Xs: list_a] :
( ( member_a @ X3 @ ( set_a2 @ Xs ) )
= ( ? [Ys3: list_a,Zs3: list_a] :
( ( Xs
= ( append_a @ Ys3 @ ( cons_a @ X3 @ Zs3 ) ) )
& ~ ( member_a @ X3 @ ( set_a2 @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_411_in__set__conv__decomp__last,axiom,
! [X3: set_a,Xs: list_set_a] :
( ( member_set_a @ X3 @ ( set_set_a2 @ Xs ) )
= ( ? [Ys3: list_set_a,Zs3: list_set_a] :
( ( Xs
= ( append_set_a @ Ys3 @ ( cons_set_a @ X3 @ Zs3 ) ) )
& ~ ( member_set_a @ X3 @ ( set_set_a2 @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_412_in__set__conv__decomp__first,axiom,
! [X3: product_prod_a_a,Xs: list_P1396940483166286381od_a_a] :
( ( member1426531477525435216od_a_a @ X3 @ ( set_Product_prod_a_a2 @ Xs ) )
= ( ? [Ys3: list_P1396940483166286381od_a_a,Zs3: list_P1396940483166286381od_a_a] :
( ( Xs
= ( append5335208819046833346od_a_a @ Ys3 @ ( cons_P7316939126706565853od_a_a @ X3 @ Zs3 ) ) )
& ~ ( member1426531477525435216od_a_a @ X3 @ ( set_Product_prod_a_a2 @ Ys3 ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_413_in__set__conv__decomp__first,axiom,
! [X3: a,Xs: list_a] :
( ( member_a @ X3 @ ( set_a2 @ Xs ) )
= ( ? [Ys3: list_a,Zs3: list_a] :
( ( Xs
= ( append_a @ Ys3 @ ( cons_a @ X3 @ Zs3 ) ) )
& ~ ( member_a @ X3 @ ( set_a2 @ Ys3 ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_414_in__set__conv__decomp__first,axiom,
! [X3: set_a,Xs: list_set_a] :
( ( member_set_a @ X3 @ ( set_set_a2 @ Xs ) )
= ( ? [Ys3: list_set_a,Zs3: list_set_a] :
( ( Xs
= ( append_set_a @ Ys3 @ ( cons_set_a @ X3 @ Zs3 ) ) )
& ~ ( member_set_a @ X3 @ ( set_set_a2 @ Ys3 ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_415_split__list__last__prop__iff,axiom,
! [Xs: list_a,P2: a > $o] :
( ( ? [X2: a] :
( ( member_a @ X2 @ ( set_a2 @ Xs ) )
& ( P2 @ X2 ) ) )
= ( ? [Ys3: list_a,X2: a,Zs3: list_a] :
( ( Xs
= ( append_a @ Ys3 @ ( cons_a @ X2 @ Zs3 ) ) )
& ( P2 @ X2 )
& ! [Y3: a] :
( ( member_a @ Y3 @ ( set_a2 @ Zs3 ) )
=> ~ ( P2 @ Y3 ) ) ) ) ) ).
% split_list_last_prop_iff
thf(fact_416_split__list__last__prop__iff,axiom,
! [Xs: list_set_a,P2: set_a > $o] :
( ( ? [X2: set_a] :
( ( member_set_a @ X2 @ ( set_set_a2 @ Xs ) )
& ( P2 @ X2 ) ) )
= ( ? [Ys3: list_set_a,X2: set_a,Zs3: list_set_a] :
( ( Xs
= ( append_set_a @ Ys3 @ ( cons_set_a @ X2 @ Zs3 ) ) )
& ( P2 @ X2 )
& ! [Y3: set_a] :
( ( member_set_a @ Y3 @ ( set_set_a2 @ Zs3 ) )
=> ~ ( P2 @ Y3 ) ) ) ) ) ).
% split_list_last_prop_iff
thf(fact_417_split__list__first__prop__iff,axiom,
! [Xs: list_a,P2: a > $o] :
( ( ? [X2: a] :
( ( member_a @ X2 @ ( set_a2 @ Xs ) )
& ( P2 @ X2 ) ) )
= ( ? [Ys3: list_a,X2: a] :
( ? [Zs3: list_a] :
( Xs
= ( append_a @ Ys3 @ ( cons_a @ X2 @ Zs3 ) ) )
& ( P2 @ X2 )
& ! [Y3: a] :
( ( member_a @ Y3 @ ( set_a2 @ Ys3 ) )
=> ~ ( P2 @ Y3 ) ) ) ) ) ).
% split_list_first_prop_iff
thf(fact_418_split__list__first__prop__iff,axiom,
! [Xs: list_set_a,P2: set_a > $o] :
( ( ? [X2: set_a] :
( ( member_set_a @ X2 @ ( set_set_a2 @ Xs ) )
& ( P2 @ X2 ) ) )
= ( ? [Ys3: list_set_a,X2: set_a] :
( ? [Zs3: list_set_a] :
( Xs
= ( append_set_a @ Ys3 @ ( cons_set_a @ X2 @ Zs3 ) ) )
& ( P2 @ X2 )
& ! [Y3: set_a] :
( ( member_set_a @ Y3 @ ( set_set_a2 @ Ys3 ) )
=> ~ ( P2 @ Y3 ) ) ) ) ) ).
% split_list_first_prop_iff
thf(fact_419_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_420_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_421_comp__sgraph_Owalk__edges_Osimps_I2_J,axiom,
! [X3: set_a] :
( ( undire6234387080713648494_set_a @ ( cons_set_a @ X3 @ nil_set_a ) )
= nil_set_set_a ) ).
% comp_sgraph.walk_edges.simps(2)
thf(fact_422_comp__sgraph_Owalk__edges_Osimps_I2_J,axiom,
! [X3: a] :
( ( undire7337870655677353998dges_a @ ( cons_a @ X3 @ nil_a ) )
= nil_set_a ) ).
% comp_sgraph.walk_edges.simps(2)
thf(fact_423_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_424_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_425_hd__append,axiom,
! [Xs: list_a,Ys: list_a] :
( ( ( Xs = nil_a )
=> ( ( hd_a @ ( append_a @ Xs @ Ys ) )
= ( hd_a @ Ys ) ) )
& ( ( Xs != nil_a )
=> ( ( hd_a @ ( append_a @ Xs @ Ys ) )
= ( hd_a @ Xs ) ) ) ) ).
% hd_append
thf(fact_426_hd__append,axiom,
! [Xs: list_set_a,Ys: list_set_a] :
( ( ( Xs = nil_set_a )
=> ( ( hd_set_a @ ( append_set_a @ Xs @ Ys ) )
= ( hd_set_a @ Ys ) ) )
& ( ( Xs != nil_set_a )
=> ( ( hd_set_a @ ( append_set_a @ Xs @ Ys ) )
= ( hd_set_a @ Xs ) ) ) ) ).
% hd_append
thf(fact_427_longest__common__prefix,axiom,
! [Xs: list_a,Ys: list_a] :
? [Ps: list_a,Xs4: list_a,Ys6: list_a] :
( ( Xs
= ( append_a @ Ps @ Xs4 ) )
& ( Ys
= ( append_a @ Ps @ Ys6 ) )
& ( ( Xs4 = nil_a )
| ( Ys6 = nil_a )
| ( ( hd_a @ Xs4 )
!= ( hd_a @ Ys6 ) ) ) ) ).
% longest_common_prefix
thf(fact_428_longest__common__prefix,axiom,
! [Xs: list_set_a,Ys: list_set_a] :
? [Ps: list_set_a,Xs4: list_set_a,Ys6: list_set_a] :
( ( Xs
= ( append_set_a @ Ps @ Xs4 ) )
& ( Ys
= ( 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_429_last__append,axiom,
! [Ys: list_a,Xs: list_a] :
( ( ( Ys = nil_a )
=> ( ( last_a @ ( append_a @ Xs @ Ys ) )
= ( last_a @ Xs ) ) )
& ( ( Ys != nil_a )
=> ( ( last_a @ ( append_a @ Xs @ Ys ) )
= ( last_a @ Ys ) ) ) ) ).
% last_append
thf(fact_430_last__append,axiom,
! [Ys: list_set_a,Xs: list_set_a] :
( ( ( Ys = nil_set_a )
=> ( ( last_set_a @ ( append_set_a @ Xs @ Ys ) )
= ( last_set_a @ Xs ) ) )
& ( ( Ys != nil_set_a )
=> ( ( last_set_a @ ( append_set_a @ Xs @ Ys ) )
= ( last_set_a @ Ys ) ) ) ) ).
% last_append
thf(fact_431_longest__common__suffix,axiom,
! [Xs: list_a,Ys: list_a] :
? [Ss: list_a,Xs4: list_a,Ys6: list_a] :
( ( Xs
= ( append_a @ Xs4 @ Ss ) )
& ( Ys
= ( append_a @ Ys6 @ Ss ) )
& ( ( Xs4 = nil_a )
| ( Ys6 = nil_a )
| ( ( last_a @ Xs4 )
!= ( last_a @ Ys6 ) ) ) ) ).
% longest_common_suffix
thf(fact_432_longest__common__suffix,axiom,
! [Xs: list_set_a,Ys: list_set_a] :
? [Ss: list_set_a,Xs4: list_set_a,Ys6: list_set_a] :
( ( Xs
= ( append_set_a @ Xs4 @ Ss ) )
& ( Ys
= ( 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_433_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_434_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_435_ulgraph_Odistinct__edgesI,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,P: list_set_a] :
( ( undire6886684016831807756_set_a @ Vertices @ Edges )
=> ( ( distinct_set_a @ P )
=> ( distinct_set_set_a @ ( undire6234387080713648494_set_a @ P ) ) ) ) ).
% ulgraph.distinct_edgesI
thf(fact_436_ulgraph_Odistinct__edgesI,axiom,
! [Vertices: set_a,Edges: set_set_a,P: list_a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( distinct_a @ P )
=> ( distinct_set_a @ ( undire7337870655677353998dges_a @ P ) ) ) ) ).
% ulgraph.distinct_edgesI
thf(fact_437_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_438_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_439_list_Oexhaust__sel,axiom,
! [List: list_a] :
( ( List != nil_a )
=> ( List
= ( cons_a @ ( hd_a @ List ) @ ( tl_a @ List ) ) ) ) ).
% list.exhaust_sel
thf(fact_440_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_441_not__distinct__decomp,axiom,
! [Ws: list_a] :
( ~ ( distinct_a @ Ws )
=> ? [Xs2: list_a,Ys2: list_a,Zs2: list_a,Y2: a] :
( Ws
= ( append_a @ Xs2 @ ( append_a @ ( cons_a @ Y2 @ nil_a ) @ ( append_a @ Ys2 @ ( append_a @ ( cons_a @ Y2 @ nil_a ) @ Zs2 ) ) ) ) ) ) ).
% not_distinct_decomp
thf(fact_442_not__distinct__decomp,axiom,
! [Ws: list_set_a] :
( ~ ( distinct_set_a @ Ws )
=> ? [Xs2: list_set_a,Ys2: list_set_a,Zs2: list_set_a,Y2: set_a] :
( Ws
= ( append_set_a @ Xs2 @ ( append_set_a @ ( cons_set_a @ Y2 @ nil_set_a ) @ ( append_set_a @ Ys2 @ ( append_set_a @ ( cons_set_a @ Y2 @ nil_set_a ) @ Zs2 ) ) ) ) ) ) ).
% not_distinct_decomp
thf(fact_443_not__distinct__conv__prefix,axiom,
! [As: list_P1396940483166286381od_a_a] :
( ( ~ ( distin132333870042060960od_a_a @ As ) )
= ( ? [Xs5: list_P1396940483166286381od_a_a,Y3: product_prod_a_a,Ys3: list_P1396940483166286381od_a_a] :
( ( member1426531477525435216od_a_a @ Y3 @ ( set_Product_prod_a_a2 @ Xs5 ) )
& ( distin132333870042060960od_a_a @ Xs5 )
& ( As
= ( append5335208819046833346od_a_a @ Xs5 @ ( cons_P7316939126706565853od_a_a @ Y3 @ Ys3 ) ) ) ) ) ) ).
% not_distinct_conv_prefix
thf(fact_444_not__distinct__conv__prefix,axiom,
! [As: list_a] :
( ( ~ ( distinct_a @ As ) )
= ( ? [Xs5: list_a,Y3: a,Ys3: list_a] :
( ( member_a @ Y3 @ ( set_a2 @ Xs5 ) )
& ( distinct_a @ Xs5 )
& ( As
= ( append_a @ Xs5 @ ( cons_a @ Y3 @ Ys3 ) ) ) ) ) ) ).
% not_distinct_conv_prefix
thf(fact_445_not__distinct__conv__prefix,axiom,
! [As: list_set_a] :
( ( ~ ( distinct_set_a @ As ) )
= ( ? [Xs5: list_set_a,Y3: set_a,Ys3: list_set_a] :
( ( member_set_a @ Y3 @ ( set_set_a2 @ Xs5 ) )
& ( distinct_set_a @ Xs5 )
& ( As
= ( append_set_a @ Xs5 @ ( cons_set_a @ Y3 @ Ys3 ) ) ) ) ) ) ).
% not_distinct_conv_prefix
thf(fact_446_rev_Osimps_I2_J,axiom,
! [X3: a,Xs: list_a] :
( ( rev_a @ ( cons_a @ X3 @ Xs ) )
= ( append_a @ ( rev_a @ Xs ) @ ( cons_a @ X3 @ nil_a ) ) ) ).
% rev.simps(2)
thf(fact_447_rev_Osimps_I2_J,axiom,
! [X3: set_a,Xs: list_set_a] :
( ( rev_set_a @ ( cons_set_a @ X3 @ Xs ) )
= ( append_set_a @ ( rev_set_a @ Xs ) @ ( cons_set_a @ X3 @ nil_set_a ) ) ) ).
% rev.simps(2)
thf(fact_448_ulgraph_Owalk__edges_Osimps_I2_J,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,X3: set_a] :
( ( undire6886684016831807756_set_a @ Vertices @ Edges )
=> ( ( undire6234387080713648494_set_a @ ( cons_set_a @ X3 @ nil_set_a ) )
= nil_set_set_a ) ) ).
% ulgraph.walk_edges.simps(2)
thf(fact_449_ulgraph_Owalk__edges_Osimps_I2_J,axiom,
! [Vertices: set_a,Edges: set_set_a,X3: a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( undire7337870655677353998dges_a @ ( cons_a @ X3 @ nil_a ) )
= nil_set_a ) ) ).
% ulgraph.walk_edges.simps(2)
thf(fact_450_Collect__mono__iff,axiom,
! [P2: a > $o,Q: a > $o] :
( ( ord_less_eq_set_a @ ( collect_a @ P2 ) @ ( collect_a @ Q ) )
= ( ! [X2: a] :
( ( P2 @ X2 )
=> ( Q @ X2 ) ) ) ) ).
% Collect_mono_iff
thf(fact_451_Collect__mono__iff,axiom,
! [P2: set_a > $o,Q: set_a > $o] :
( ( ord_le3724670747650509150_set_a @ ( collect_set_a @ P2 ) @ ( collect_set_a @ Q ) )
= ( ! [X2: set_a] :
( ( P2 @ X2 )
=> ( Q @ X2 ) ) ) ) ).
% Collect_mono_iff
thf(fact_452_Collect__mono__iff,axiom,
! [P2: product_prod_a_a > $o,Q: product_prod_a_a > $o] :
( ( ord_le746702958409616551od_a_a @ ( collec3336397797384452498od_a_a @ P2 ) @ ( collec3336397797384452498od_a_a @ Q ) )
= ( ! [X2: product_prod_a_a] :
( ( P2 @ X2 )
=> ( Q @ X2 ) ) ) ) ).
% Collect_mono_iff
thf(fact_453_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_454_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_455_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_456_subset__trans,axiom,
! [A2: set_a,B: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ( ord_less_eq_set_a @ B @ C2 )
=> ( ord_less_eq_set_a @ A2 @ C2 ) ) ) ).
% subset_trans
thf(fact_457_subset__trans,axiom,
! [A2: set_set_a,B: set_set_a,C2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B )
=> ( ( ord_le3724670747650509150_set_a @ B @ C2 )
=> ( ord_le3724670747650509150_set_a @ A2 @ C2 ) ) ) ).
% subset_trans
thf(fact_458_subset__trans,axiom,
! [A2: set_Product_prod_a_a,B: set_Product_prod_a_a,C2: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A2 @ B )
=> ( ( ord_le746702958409616551od_a_a @ B @ C2 )
=> ( ord_le746702958409616551od_a_a @ A2 @ C2 ) ) ) ).
% subset_trans
thf(fact_459_Collect__mono,axiom,
! [P2: a > $o,Q: a > $o] :
( ! [X4: a] :
( ( P2 @ X4 )
=> ( Q @ X4 ) )
=> ( ord_less_eq_set_a @ ( collect_a @ P2 ) @ ( collect_a @ Q ) ) ) ).
% Collect_mono
thf(fact_460_Collect__mono,axiom,
! [P2: set_a > $o,Q: set_a > $o] :
( ! [X4: set_a] :
( ( P2 @ X4 )
=> ( Q @ X4 ) )
=> ( ord_le3724670747650509150_set_a @ ( collect_set_a @ P2 ) @ ( collect_set_a @ Q ) ) ) ).
% Collect_mono
thf(fact_461_Collect__mono,axiom,
! [P2: product_prod_a_a > $o,Q: product_prod_a_a > $o] :
( ! [X4: product_prod_a_a] :
( ( P2 @ X4 )
=> ( Q @ X4 ) )
=> ( ord_le746702958409616551od_a_a @ ( collec3336397797384452498od_a_a @ P2 ) @ ( collec3336397797384452498od_a_a @ Q ) ) ) ).
% Collect_mono
thf(fact_462_subset__refl,axiom,
! [A2: set_a] : ( ord_less_eq_set_a @ A2 @ A2 ) ).
% subset_refl
thf(fact_463_subset__refl,axiom,
! [A2: set_set_a] : ( ord_le3724670747650509150_set_a @ A2 @ A2 ) ).
% subset_refl
thf(fact_464_subset__refl,axiom,
! [A2: set_Product_prod_a_a] : ( ord_le746702958409616551od_a_a @ A2 @ A2 ) ).
% subset_refl
thf(fact_465_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_466_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_467_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_468_equalityD2,axiom,
! [A2: set_a,B: set_a] :
( ( A2 = B )
=> ( ord_less_eq_set_a @ B @ A2 ) ) ).
% equalityD2
thf(fact_469_equalityD2,axiom,
! [A2: set_set_a,B: set_set_a] :
( ( A2 = B )
=> ( ord_le3724670747650509150_set_a @ B @ A2 ) ) ).
% equalityD2
thf(fact_470_equalityD2,axiom,
! [A2: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ( A2 = B )
=> ( ord_le746702958409616551od_a_a @ B @ A2 ) ) ).
% equalityD2
thf(fact_471_equalityD1,axiom,
! [A2: set_a,B: set_a] :
( ( A2 = B )
=> ( ord_less_eq_set_a @ A2 @ B ) ) ).
% equalityD1
thf(fact_472_equalityD1,axiom,
! [A2: set_set_a,B: set_set_a] :
( ( A2 = B )
=> ( ord_le3724670747650509150_set_a @ A2 @ B ) ) ).
% equalityD1
thf(fact_473_equalityD1,axiom,
! [A2: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ( A2 = B )
=> ( ord_le746702958409616551od_a_a @ A2 @ B ) ) ).
% equalityD1
thf(fact_474_subset__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A3: set_a,B3: set_a] :
! [X2: a] :
( ( member_a @ X2 @ A3 )
=> ( member_a @ X2 @ B3 ) ) ) ) ).
% subset_eq
thf(fact_475_subset__eq,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [A3: set_set_a,B3: set_set_a] :
! [X2: set_a] :
( ( member_set_a @ X2 @ A3 )
=> ( member_set_a @ X2 @ B3 ) ) ) ) ).
% subset_eq
thf(fact_476_subset__eq,axiom,
( ord_le746702958409616551od_a_a
= ( ^ [A3: set_Product_prod_a_a,B3: set_Product_prod_a_a] :
! [X2: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X2 @ A3 )
=> ( member1426531477525435216od_a_a @ X2 @ B3 ) ) ) ) ).
% subset_eq
thf(fact_477_equalityE,axiom,
! [A2: set_a,B: set_a] :
( ( A2 = B )
=> ~ ( ( ord_less_eq_set_a @ A2 @ B )
=> ~ ( ord_less_eq_set_a @ B @ A2 ) ) ) ).
% equalityE
thf(fact_478_equalityE,axiom,
! [A2: set_set_a,B: set_set_a] :
( ( A2 = B )
=> ~ ( ( ord_le3724670747650509150_set_a @ A2 @ B )
=> ~ ( ord_le3724670747650509150_set_a @ B @ A2 ) ) ) ).
% equalityE
thf(fact_479_equalityE,axiom,
! [A2: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ( A2 = B )
=> ~ ( ( ord_le746702958409616551od_a_a @ A2 @ B )
=> ~ ( ord_le746702958409616551od_a_a @ B @ A2 ) ) ) ).
% equalityE
thf(fact_480_subsetD,axiom,
! [A2: set_a,B: set_a,C: a] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ( member_a @ C @ A2 )
=> ( member_a @ C @ B ) ) ) ).
% subsetD
thf(fact_481_subsetD,axiom,
! [A2: set_set_a,B: set_set_a,C: set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B )
=> ( ( member_set_a @ C @ A2 )
=> ( member_set_a @ C @ B ) ) ) ).
% subsetD
thf(fact_482_subsetD,axiom,
! [A2: set_Product_prod_a_a,B: set_Product_prod_a_a,C: product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A2 @ B )
=> ( ( member1426531477525435216od_a_a @ C @ A2 )
=> ( member1426531477525435216od_a_a @ C @ B ) ) ) ).
% subsetD
thf(fact_483_in__mono,axiom,
! [A2: set_a,B: set_a,X3: a] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ( member_a @ X3 @ A2 )
=> ( member_a @ X3 @ B ) ) ) ).
% in_mono
thf(fact_484_in__mono,axiom,
! [A2: set_set_a,B: set_set_a,X3: set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B )
=> ( ( member_set_a @ X3 @ A2 )
=> ( member_set_a @ X3 @ B ) ) ) ).
% in_mono
thf(fact_485_in__mono,axiom,
! [A2: set_Product_prod_a_a,B: set_Product_prod_a_a,X3: product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A2 @ B )
=> ( ( member1426531477525435216od_a_a @ X3 @ A2 )
=> ( member1426531477525435216od_a_a @ X3 @ B ) ) ) ).
% in_mono
thf(fact_486_graph__system_Oinduced__edges__ss,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,V3: set_set_a] :
( ( undire7159349782766787846_set_a @ Vertices @ Edges )
=> ( ( ord_le3724670747650509150_set_a @ V3 @ Vertices )
=> ( ord_le5722252365846178494_set_a @ ( undire7854589003810675244_set_a @ Edges @ V3 ) @ Edges ) ) ) ).
% graph_system.induced_edges_ss
thf(fact_487_graph__system_Oinduced__edges__ss,axiom,
! [Vertices: set_Product_prod_a_a,Edges: set_se5735800977113168103od_a_a,V3: set_Product_prod_a_a] :
( ( undire1860116983885411791od_a_a @ Vertices @ Edges )
=> ( ( ord_le746702958409616551od_a_a @ V3 @ Vertices )
=> ( ord_le1995061765932249223od_a_a @ ( undire5906991851038061813od_a_a @ Edges @ V3 ) @ Edges ) ) ) ).
% graph_system.induced_edges_ss
thf(fact_488_graph__system_Oinduced__edges__ss,axiom,
! [Vertices: set_a,Edges: set_set_a,V3: set_a] :
( ( undire2554140024507503526stem_a @ Vertices @ Edges )
=> ( ( ord_less_eq_set_a @ V3 @ Vertices )
=> ( ord_le3724670747650509150_set_a @ ( undire7777452895879145676dges_a @ Edges @ V3 ) @ Edges ) ) ) ).
% graph_system.induced_edges_ss
thf(fact_489_graph__system_Oinduced__is__subgraph,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,V3: set_set_a] :
( ( undire7159349782766787846_set_a @ Vertices @ Edges )
=> ( ( ord_le3724670747650509150_set_a @ V3 @ Vertices )
=> ( undire1186139521737116585_set_a @ V3 @ ( undire7854589003810675244_set_a @ Edges @ V3 ) @ Vertices @ Edges ) ) ) ).
% graph_system.induced_is_subgraph
thf(fact_490_graph__system_Oinduced__is__subgraph,axiom,
! [Vertices: set_Product_prod_a_a,Edges: set_se5735800977113168103od_a_a,V3: set_Product_prod_a_a] :
( ( undire1860116983885411791od_a_a @ Vertices @ Edges )
=> ( ( ord_le746702958409616551od_a_a @ V3 @ Vertices )
=> ( undire398746457437328754od_a_a @ V3 @ ( undire5906991851038061813od_a_a @ Edges @ V3 ) @ Vertices @ Edges ) ) ) ).
% graph_system.induced_is_subgraph
thf(fact_491_graph__system_Oinduced__is__subgraph,axiom,
! [Vertices: set_a,Edges: set_set_a,V3: set_a] :
( ( undire2554140024507503526stem_a @ Vertices @ Edges )
=> ( ( ord_less_eq_set_a @ V3 @ Vertices )
=> ( undire7103218114511261257raph_a @ V3 @ ( undire7777452895879145676dges_a @ Edges @ V3 ) @ Vertices @ Edges ) ) ) ).
% graph_system.induced_is_subgraph
thf(fact_492_ulgraph_Ois__trail__def,axiom,
! [Vertices: set_a,Edges: set_set_a,Xs: list_a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( undire7142031287334043199rail_a @ Vertices @ Edges @ Xs )
= ( ( undire6133010728901294956walk_a @ Vertices @ Edges @ Xs )
& ( distinct_set_a @ ( undire7337870655677353998dges_a @ Xs ) ) ) ) ) ).
% ulgraph.is_trail_def
thf(fact_493_ulgraph_Ois__walk__decomp,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,Xs: list_set_a,Y: set_a,Ys: list_set_a,Zs: list_set_a] :
( ( undire6886684016831807756_set_a @ Vertices @ Edges )
=> ( ( undire3014741414213135564_set_a @ Vertices @ Edges @ ( append_set_a @ Xs @ ( append_set_a @ ( cons_set_a @ Y @ nil_set_a ) @ ( append_set_a @ Ys @ ( append_set_a @ ( cons_set_a @ Y @ nil_set_a ) @ Zs ) ) ) ) )
=> ( undire3014741414213135564_set_a @ Vertices @ Edges @ ( append_set_a @ Xs @ ( append_set_a @ ( cons_set_a @ Y @ nil_set_a ) @ Zs ) ) ) ) ) ).
% ulgraph.is_walk_decomp
thf(fact_494_ulgraph_Ois__walk__decomp,axiom,
! [Vertices: set_a,Edges: set_set_a,Xs: list_a,Y: a,Ys: list_a,Zs: list_a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( 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 ) ) ) ) )
=> ( undire6133010728901294956walk_a @ Vertices @ Edges @ ( append_a @ Xs @ ( append_a @ ( cons_a @ Y @ nil_a ) @ Zs ) ) ) ) ) ).
% ulgraph.is_walk_decomp
thf(fact_495_distinct__tl__rev,axiom,
! [Xs: list_a] :
( ( ( hd_a @ Xs )
= ( last_a @ Xs ) )
=> ( ( distinct_a @ ( tl_a @ Xs ) )
= ( distinct_a @ ( tl_a @ ( rev_a @ Xs ) ) ) ) ) ).
% distinct_tl_rev
thf(fact_496_distinct__tl__rev,axiom,
! [Xs: list_set_a] :
( ( ( hd_set_a @ Xs )
= ( last_set_a @ Xs ) )
=> ( ( distinct_set_a @ ( tl_set_a @ Xs ) )
= ( distinct_set_a @ ( tl_set_a @ ( rev_set_a @ Xs ) ) ) ) ) ).
% distinct_tl_rev
thf(fact_497_ulgraph_Ois__gen__path__distinct__tl,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,P: list_set_a] :
( ( undire6886684016831807756_set_a @ Vertices @ Edges )
=> ( ( undire7201326534205417136_set_a @ Vertices @ Edges @ P )
=> ( ( ( hd_set_a @ P )
= ( last_set_a @ P ) )
=> ( distinct_set_a @ ( tl_set_a @ P ) ) ) ) ) ).
% ulgraph.is_gen_path_distinct_tl
thf(fact_498_ulgraph_Ois__gen__path__distinct__tl,axiom,
! [Vertices: set_a,Edges: set_set_a,P: list_a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( undire3562951555376170320path_a @ Vertices @ Edges @ P )
=> ( ( ( hd_a @ P )
= ( last_a @ P ) )
=> ( distinct_a @ ( tl_a @ P ) ) ) ) ) ).
% ulgraph.is_gen_path_distinct_tl
thf(fact_499_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_500_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_501_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_502_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_503_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_504_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_505_ulgraph_Ois__gen__path__def,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,P: list_set_a] :
( ( undire6886684016831807756_set_a @ Vertices @ Edges )
=> ( ( undire7201326534205417136_set_a @ Vertices @ Edges @ P )
= ( ( undire3014741414213135564_set_a @ Vertices @ Edges @ P )
& ( ( ( distinct_set_a @ ( tl_set_a @ P ) )
& ( ( hd_set_a @ P )
= ( last_set_a @ P ) ) )
| ( distinct_set_a @ P ) ) ) ) ) ).
% ulgraph.is_gen_path_def
thf(fact_506_ulgraph_Ois__gen__path__def,axiom,
! [Vertices: set_a,Edges: set_set_a,P: list_a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( undire3562951555376170320path_a @ Vertices @ Edges @ P )
= ( ( undire6133010728901294956walk_a @ Vertices @ Edges @ P )
& ( ( ( distinct_a @ ( tl_a @ P ) )
& ( ( hd_a @ P )
= ( last_a @ P ) ) )
| ( distinct_a @ P ) ) ) ) ) ).
% ulgraph.is_gen_path_def
thf(fact_507_dual__order_Orefl,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ A @ A ) ).
% dual_order.refl
thf(fact_508_dual__order_Orefl,axiom,
! [A: set_set_a] : ( ord_le3724670747650509150_set_a @ A @ A ) ).
% dual_order.refl
thf(fact_509_dual__order_Orefl,axiom,
! [A: real] : ( ord_less_eq_real @ A @ A ) ).
% dual_order.refl
thf(fact_510_dual__order_Orefl,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% dual_order.refl
thf(fact_511_dual__order_Orefl,axiom,
! [A: set_Product_prod_a_a] : ( ord_le746702958409616551od_a_a @ A @ A ) ).
% dual_order.refl
thf(fact_512_order__refl,axiom,
! [X3: set_a] : ( ord_less_eq_set_a @ X3 @ X3 ) ).
% order_refl
thf(fact_513_order__refl,axiom,
! [X3: set_set_a] : ( ord_le3724670747650509150_set_a @ X3 @ X3 ) ).
% order_refl
thf(fact_514_order__refl,axiom,
! [X3: real] : ( ord_less_eq_real @ X3 @ X3 ) ).
% order_refl
thf(fact_515_order__refl,axiom,
! [X3: nat] : ( ord_less_eq_nat @ X3 @ X3 ) ).
% order_refl
thf(fact_516_order__refl,axiom,
! [X3: set_Product_prod_a_a] : ( ord_le746702958409616551od_a_a @ X3 @ X3 ) ).
% order_refl
thf(fact_517_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_518_the__elem__set,axiom,
! [X3: a] :
( ( the_elem_a @ ( set_a2 @ ( cons_a @ X3 @ nil_a ) ) )
= X3 ) ).
% the_elem_set
thf(fact_519_the__elem__set,axiom,
! [X3: set_a] :
( ( the_elem_set_a @ ( set_set_a2 @ ( cons_set_a @ X3 @ nil_set_a ) ) )
= X3 ) ).
% the_elem_set
thf(fact_520_list__set__tl,axiom,
! [X3: product_prod_a_a,Xs: list_P1396940483166286381od_a_a] :
( ( member1426531477525435216od_a_a @ X3 @ ( set_Product_prod_a_a2 @ ( tl_Product_prod_a_a @ Xs ) ) )
=> ( member1426531477525435216od_a_a @ X3 @ ( set_Product_prod_a_a2 @ Xs ) ) ) ).
% list_set_tl
thf(fact_521_list__set__tl,axiom,
! [X3: a,Xs: list_a] :
( ( member_a @ X3 @ ( set_a2 @ ( tl_a @ Xs ) ) )
=> ( member_a @ X3 @ ( set_a2 @ Xs ) ) ) ).
% list_set_tl
thf(fact_522_list__set__tl,axiom,
! [X3: set_a,Xs: list_set_a] :
( ( member_set_a @ X3 @ ( set_set_a2 @ ( tl_set_a @ Xs ) ) )
=> ( member_set_a @ X3 @ ( set_set_a2 @ Xs ) ) ) ).
% list_set_tl
thf(fact_523_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_524_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_525_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_526_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_527_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_528_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_529_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_530_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_531_rotate1__is__Nil__conv,axiom,
! [Xs: list_a] :
( ( ( rotate1_a @ Xs )
= nil_a )
= ( Xs = nil_a ) ) ).
% rotate1_is_Nil_conv
thf(fact_532_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_533_set__rotate1,axiom,
! [Xs: list_a] :
( ( set_a2 @ ( rotate1_a @ Xs ) )
= ( set_a2 @ Xs ) ) ).
% set_rotate1
thf(fact_534_set__rotate1,axiom,
! [Xs: list_set_a] :
( ( set_set_a2 @ ( rotate1_set_a @ Xs ) )
= ( set_set_a2 @ Xs ) ) ).
% set_rotate1
thf(fact_535_distinct1__rotate,axiom,
! [Xs: list_a] :
( ( distinct_a @ ( rotate1_a @ Xs ) )
= ( distinct_a @ Xs ) ) ).
% distinct1_rotate
thf(fact_536_distinct1__rotate,axiom,
! [Xs: list_set_a] :
( ( distinct_set_a @ ( rotate1_set_a @ Xs ) )
= ( distinct_set_a @ Xs ) ) ).
% distinct1_rotate
thf(fact_537_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_538_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_539_ulgraph_Oedge__density_Ocong,axiom,
undire297304480579013331sity_a = undire297304480579013331sity_a ).
% ulgraph.edge_density.cong
thf(fact_540_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_541_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_542_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_543_comp__sgraph_Oe__in__all__edges__ss,axiom,
! [E: set_a,S: set_a,V3: set_a] :
( ( member_set_a @ E @ ( undire2918257014606996450dges_a @ S ) )
=> ( ( ord_less_eq_set_a @ E @ V3 )
=> ( ( ord_less_eq_set_a @ V3 @ S )
=> ( member_set_a @ E @ ( undire2918257014606996450dges_a @ V3 ) ) ) ) ) ).
% comp_sgraph.e_in_all_edges_ss
thf(fact_544_comp__sgraph_Oe__in__all__edges__ss,axiom,
! [E: set_set_a,S: set_set_a,V3: set_set_a] :
( ( member_set_set_a @ E @ ( undire8247866692393712962_set_a @ S ) )
=> ( ( ord_le3724670747650509150_set_a @ E @ V3 )
=> ( ( ord_le3724670747650509150_set_a @ V3 @ S )
=> ( member_set_set_a @ E @ ( undire8247866692393712962_set_a @ V3 ) ) ) ) ) ).
% comp_sgraph.e_in_all_edges_ss
thf(fact_545_comp__sgraph_Oe__in__all__edges__ss,axiom,
! [E: set_Product_prod_a_a,S: set_Product_prod_a_a,V3: set_Product_prod_a_a] :
( ( member1816616512716248880od_a_a @ E @ ( undire6879232364018543115od_a_a @ S ) )
=> ( ( ord_le746702958409616551od_a_a @ E @ V3 )
=> ( ( ord_le746702958409616551od_a_a @ V3 @ S )
=> ( member1816616512716248880od_a_a @ E @ ( undire6879232364018543115od_a_a @ V3 ) ) ) ) ) ).
% comp_sgraph.e_in_all_edges_ss
thf(fact_546_comp__sgraph_Oulgraph__axioms,axiom,
! [S: set_a] : ( undire7251896706689453996raph_a @ S @ ( undire2918257014606996450dges_a @ S ) ) ).
% comp_sgraph.ulgraph_axioms
thf(fact_547_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_548_comp__sgraph_Ograph__system__axioms,axiom,
! [S: set_a] : ( undire2554140024507503526stem_a @ S @ ( undire2918257014606996450dges_a @ S ) ) ).
% comp_sgraph.graph_system_axioms
thf(fact_549_comp__sgraph_Osubgraph__complete,axiom,
! [S: set_a] : ( undire7103218114511261257raph_a @ S @ ( undire2918257014606996450dges_a @ S ) @ S @ ( undire2918257014606996450dges_a @ S ) ) ).
% comp_sgraph.subgraph_complete
thf(fact_550_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_551_comp__sgraph_Overt__adj__sym,axiom,
! [S: set_a,V1: a,V2: a] :
( ( undire397441198561214472_adj_a @ ( undire2918257014606996450dges_a @ S ) @ V1 @ V2 )
= ( undire397441198561214472_adj_a @ ( undire2918257014606996450dges_a @ S ) @ V2 @ V1 ) ) ).
% comp_sgraph.vert_adj_sym
thf(fact_552_comp__sgraph_Overt__adj__imp__inV,axiom,
! [S: set_set_a,V1: set_a,V2: set_a] :
( ( undire3510646817838285160_set_a @ ( undire8247866692393712962_set_a @ S ) @ V1 @ V2 )
=> ( ( member_set_a @ V1 @ S )
& ( member_set_a @ V2 @ S ) ) ) ).
% comp_sgraph.vert_adj_imp_inV
thf(fact_553_comp__sgraph_Overt__adj__imp__inV,axiom,
! [S: set_Product_prod_a_a,V1: product_prod_a_a,V2: product_prod_a_a] :
( ( undire6135774327024169009od_a_a @ ( undire6879232364018543115od_a_a @ S ) @ V1 @ V2 )
=> ( ( member1426531477525435216od_a_a @ V1 @ S )
& ( member1426531477525435216od_a_a @ V2 @ S ) ) ) ).
% comp_sgraph.vert_adj_imp_inV
thf(fact_554_comp__sgraph_Overt__adj__imp__inV,axiom,
! [S: set_a,V1: a,V2: a] :
( ( undire397441198561214472_adj_a @ ( undire2918257014606996450dges_a @ S ) @ V1 @ V2 )
=> ( ( member_a @ V1 @ S )
& ( member_a @ V2 @ S ) ) ) ).
% comp_sgraph.vert_adj_imp_inV
thf(fact_555_comp__sgraph_Oincident__edge__in__wf,axiom,
! [E: set_set_a,S: set_set_a,V: set_a] :
( ( member_set_set_a @ E @ ( undire8247866692393712962_set_a @ S ) )
=> ( ( undire2320338297334612420_set_a @ V @ E )
=> ( member_set_a @ V @ S ) ) ) ).
% comp_sgraph.incident_edge_in_wf
thf(fact_556_comp__sgraph_Oincident__edge__in__wf,axiom,
! [E: set_Product_prod_a_a,S: set_Product_prod_a_a,V: product_prod_a_a] :
( ( member1816616512716248880od_a_a @ E @ ( undire6879232364018543115od_a_a @ S ) )
=> ( ( undire3369688177417741453od_a_a @ V @ E )
=> ( member1426531477525435216od_a_a @ V @ S ) ) ) ).
% comp_sgraph.incident_edge_in_wf
thf(fact_557_comp__sgraph_Oincident__edge__in__wf,axiom,
! [E: set_a,S: set_a,V: a] :
( ( member_set_a @ E @ ( undire2918257014606996450dges_a @ S ) )
=> ( ( undire1521409233611534436dent_a @ V @ E )
=> ( member_a @ V @ S ) ) ) ).
% comp_sgraph.incident_edge_in_wf
thf(fact_558_comp__sgraph_Ono__loops,axiom,
! [V: set_a,S: set_set_a] :
( ( member_set_a @ V @ S )
=> ~ ( undire5774735625301615776_set_a @ ( undire8247866692393712962_set_a @ S ) @ V ) ) ).
% comp_sgraph.no_loops
thf(fact_559_comp__sgraph_Ono__loops,axiom,
! [V: product_prod_a_a,S: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ V @ S )
=> ~ ( undire7777398424729533289od_a_a @ ( undire6879232364018543115od_a_a @ S ) @ V ) ) ).
% comp_sgraph.no_loops
thf(fact_560_comp__sgraph_Ono__loops,axiom,
! [V: a,S: set_a] :
( ( member_a @ V @ S )
=> ~ ( undire3617971648856834880loop_a @ ( undire2918257014606996450dges_a @ S ) @ V ) ) ).
% comp_sgraph.no_loops
thf(fact_561_comp__sgraph_Ohas__loop__in__verts,axiom,
! [S: set_set_a,V: set_a] :
( ( undire5774735625301615776_set_a @ ( undire8247866692393712962_set_a @ S ) @ V )
=> ( member_set_a @ V @ S ) ) ).
% comp_sgraph.has_loop_in_verts
thf(fact_562_comp__sgraph_Ohas__loop__in__verts,axiom,
! [S: set_Product_prod_a_a,V: product_prod_a_a] :
( ( undire7777398424729533289od_a_a @ ( undire6879232364018543115od_a_a @ S ) @ V )
=> ( member1426531477525435216od_a_a @ V @ S ) ) ).
% comp_sgraph.has_loop_in_verts
thf(fact_563_comp__sgraph_Ohas__loop__in__verts,axiom,
! [S: set_a,V: a] :
( ( undire3617971648856834880loop_a @ ( undire2918257014606996450dges_a @ S ) @ V )
=> ( member_a @ V @ S ) ) ).
% comp_sgraph.has_loop_in_verts
thf(fact_564_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_565_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 ) )
=> ( ? [X: set_a] :
( ( member_set_a @ X @ S )
& ( member_set_a @ X @ E1 )
& ( member_set_a @ X @ E2 ) )
=> ( undire3485422320110889978_set_a @ ( undire8247866692393712962_set_a @ S ) @ E1 @ E2 ) ) ) ) ).
% comp_sgraph.edge_adjacent_alt_def
thf(fact_566_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 ) )
=> ( ? [X: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X @ S )
& ( member1426531477525435216od_a_a @ X @ E1 )
& ( member1426531477525435216od_a_a @ X @ E2 ) )
=> ( undire9186443406341554371od_a_a @ ( undire6879232364018543115od_a_a @ S ) @ E1 @ E2 ) ) ) ) ).
% comp_sgraph.edge_adjacent_alt_def
thf(fact_567_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 ) )
=> ( ? [X: a] :
( ( member_a @ X @ S )
& ( member_a @ X @ E1 )
& ( member_a @ X @ E2 ) )
=> ( undire4022703626023482010_adj_a @ ( undire2918257014606996450dges_a @ S ) @ E1 @ E2 ) ) ) ) ).
% comp_sgraph.edge_adjacent_alt_def
thf(fact_568_rotate1_Osimps_I1_J,axiom,
( ( rotate1_a @ nil_a )
= nil_a ) ).
% rotate1.simps(1)
thf(fact_569_rotate1_Osimps_I1_J,axiom,
( ( rotate1_set_a @ nil_set_a )
= nil_set_a ) ).
% rotate1.simps(1)
thf(fact_570_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_571_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_572_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_573_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_574_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_575_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_576_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_577_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_578_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_579_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_580_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_581_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_582_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_583_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_584_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_585_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_586_comp__sgraph_Ois__gen__path__rev,axiom,
! [S: set_set_a,P: list_set_a] :
( ( undire7201326534205417136_set_a @ S @ ( undire8247866692393712962_set_a @ S ) @ P )
= ( undire7201326534205417136_set_a @ S @ ( undire8247866692393712962_set_a @ S ) @ ( rev_set_a @ P ) ) ) ).
% comp_sgraph.is_gen_path_rev
thf(fact_587_comp__sgraph_Ois__gen__path__rev,axiom,
! [S: set_a,P: list_a] :
( ( undire3562951555376170320path_a @ S @ ( undire2918257014606996450dges_a @ S ) @ P )
= ( undire3562951555376170320path_a @ S @ ( undire2918257014606996450dges_a @ S ) @ ( rev_a @ P ) ) ) ).
% comp_sgraph.is_gen_path_rev
thf(fact_588_comp__sgraph_Ois__cycle__rev,axiom,
! [S: set_set_a,Xs: list_set_a] :
( ( undire797940137672299967_set_a @ S @ ( undire8247866692393712962_set_a @ S ) @ Xs )
= ( undire797940137672299967_set_a @ S @ ( undire8247866692393712962_set_a @ S ) @ ( rev_set_a @ Xs ) ) ) ).
% comp_sgraph.is_cycle_rev
thf(fact_589_comp__sgraph_Ois__cycle__rev,axiom,
! [S: set_a,Xs: list_a] :
( ( undire2407311113669455967ycle_a @ S @ ( undire2918257014606996450dges_a @ S ) @ Xs )
= ( undire2407311113669455967ycle_a @ S @ ( undire2918257014606996450dges_a @ S ) @ ( rev_a @ Xs ) ) ) ).
% comp_sgraph.is_cycle_rev
thf(fact_590_comp__sgraph_Ois__path__rev,axiom,
! [S: set_set_a,Xs: list_set_a] :
( ( undire8834939040163919632_set_a @ S @ ( undire8247866692393712962_set_a @ S ) @ Xs )
= ( undire8834939040163919632_set_a @ S @ ( undire8247866692393712962_set_a @ S ) @ ( rev_set_a @ Xs ) ) ) ).
% comp_sgraph.is_path_rev
thf(fact_591_comp__sgraph_Ois__path__rev,axiom,
! [S: set_a,Xs: list_a] :
( ( undire427332500224447920path_a @ S @ ( undire2918257014606996450dges_a @ S ) @ Xs )
= ( undire427332500224447920path_a @ S @ ( undire2918257014606996450dges_a @ S ) @ ( rev_a @ Xs ) ) ) ).
% comp_sgraph.is_path_rev
thf(fact_592_comp__sgraph_Oinduced__is__graph__sys,axiom,
! [V3: set_a,S: set_a] : ( undire2554140024507503526stem_a @ V3 @ ( undire7777452895879145676dges_a @ ( undire2918257014606996450dges_a @ S ) @ V3 ) ) ).
% comp_sgraph.induced_is_graph_sys
thf(fact_593_comp__sgraph_Ois__path__walk,axiom,
! [S: set_a,Xs: list_a] :
( ( undire427332500224447920path_a @ S @ ( undire2918257014606996450dges_a @ S ) @ Xs )
=> ( undire6133010728901294956walk_a @ S @ ( undire2918257014606996450dges_a @ S ) @ Xs ) ) ).
% comp_sgraph.is_path_walk
thf(fact_594_comp__sgraph_Ois__gen__path__cycle,axiom,
! [S: set_a,P: list_a] :
( ( undire2407311113669455967ycle_a @ S @ ( undire2918257014606996450dges_a @ S ) @ P )
=> ( undire3562951555376170320path_a @ S @ ( undire2918257014606996450dges_a @ S ) @ P ) ) ).
% comp_sgraph.is_gen_path_cycle
thf(fact_595_comp__sgraph_Overt__adj__edge__iff2,axiom,
! [V1: a,V2: a,S: set_a] :
( ( V1 != V2 )
=> ( ( undire397441198561214472_adj_a @ ( undire2918257014606996450dges_a @ S ) @ V1 @ V2 )
= ( ? [X2: set_a] :
( ( member_set_a @ X2 @ ( undire2918257014606996450dges_a @ S ) )
& ( undire1521409233611534436dent_a @ V1 @ X2 )
& ( undire1521409233611534436dent_a @ V2 @ X2 ) ) ) ) ) ).
% comp_sgraph.vert_adj_edge_iff2
thf(fact_596_comp__sgraph_Ois__path__gen__path,axiom,
! [S: set_a,P: list_a] :
( ( undire427332500224447920path_a @ S @ ( undire2918257014606996450dges_a @ S ) @ P )
=> ( undire3562951555376170320path_a @ S @ ( undire2918257014606996450dges_a @ S ) @ P ) ) ).
% comp_sgraph.is_path_gen_path
thf(fact_597_comp__sgraph_Ois__closed__walk__rev,axiom,
! [S: set_set_a,Xs: list_set_a] :
( ( undire4100213446647512896_set_a @ S @ ( undire8247866692393712962_set_a @ S ) @ Xs )
= ( undire4100213446647512896_set_a @ S @ ( undire8247866692393712962_set_a @ S ) @ ( rev_set_a @ Xs ) ) ) ).
% comp_sgraph.is_closed_walk_rev
thf(fact_598_comp__sgraph_Ois__closed__walk__rev,axiom,
! [S: set_a,Xs: list_a] :
( ( undire3370724456595283424walk_a @ S @ ( undire2918257014606996450dges_a @ S ) @ Xs )
= ( undire3370724456595283424walk_a @ S @ ( undire2918257014606996450dges_a @ S ) @ ( rev_a @ Xs ) ) ) ).
% comp_sgraph.is_closed_walk_rev
thf(fact_599_comp__sgraph_Ois__open__walk__rev,axiom,
! [S: set_set_a,Xs: list_set_a] :
( ( undire526879649183275522_set_a @ S @ ( undire8247866692393712962_set_a @ S ) @ Xs )
= ( undire526879649183275522_set_a @ S @ ( undire8247866692393712962_set_a @ S ) @ ( rev_set_a @ Xs ) ) ) ).
% comp_sgraph.is_open_walk_rev
thf(fact_600_comp__sgraph_Ois__open__walk__rev,axiom,
! [S: set_a,Xs: list_a] :
( ( undire2427028224930250914walk_a @ S @ ( undire2918257014606996450dges_a @ S ) @ Xs )
= ( undire2427028224930250914walk_a @ S @ ( undire2918257014606996450dges_a @ S ) @ ( rev_a @ Xs ) ) ) ).
% comp_sgraph.is_open_walk_rev
thf(fact_601_comp__sgraph_Ois__trail__rev,axiom,
! [S: set_set_a,Xs: list_set_a] :
( ( undire1224551742100448159_set_a @ S @ ( undire8247866692393712962_set_a @ S ) @ Xs )
= ( undire1224551742100448159_set_a @ S @ ( undire8247866692393712962_set_a @ S ) @ ( rev_set_a @ Xs ) ) ) ).
% comp_sgraph.is_trail_rev
thf(fact_602_comp__sgraph_Ois__trail__rev,axiom,
! [S: set_a,Xs: list_a] :
( ( undire7142031287334043199rail_a @ S @ ( undire2918257014606996450dges_a @ S ) @ Xs )
= ( undire7142031287334043199rail_a @ S @ ( undire2918257014606996450dges_a @ S ) @ ( rev_a @ Xs ) ) ) ).
% comp_sgraph.is_trail_rev
thf(fact_603_comp__sgraph_Ois__isolated__vertex__def,axiom,
! [S: set_set_a,V: set_a] :
( ( undire6879241558604981877_set_a @ S @ ( undire8247866692393712962_set_a @ S ) @ V )
= ( ( member_set_a @ V @ S )
& ! [X2: set_a] :
( ( member_set_a @ X2 @ S )
=> ~ ( undire3510646817838285160_set_a @ ( undire8247866692393712962_set_a @ S ) @ X2 @ V ) ) ) ) ).
% comp_sgraph.is_isolated_vertex_def
thf(fact_604_comp__sgraph_Ois__isolated__vertex__def,axiom,
! [S: set_Product_prod_a_a,V: product_prod_a_a] :
( ( undire3207556238582723646od_a_a @ S @ ( undire6879232364018543115od_a_a @ S ) @ V )
= ( ( member1426531477525435216od_a_a @ V @ S )
& ! [X2: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X2 @ S )
=> ~ ( undire6135774327024169009od_a_a @ ( undire6879232364018543115od_a_a @ S ) @ X2 @ V ) ) ) ) ).
% comp_sgraph.is_isolated_vertex_def
thf(fact_605_comp__sgraph_Ois__isolated__vertex__def,axiom,
! [S: set_a,V: a] :
( ( undire8931668460104145173rtex_a @ S @ ( undire2918257014606996450dges_a @ S ) @ V )
= ( ( member_a @ V @ S )
& ! [X2: a] :
( ( member_a @ X2 @ S )
=> ~ ( undire397441198561214472_adj_a @ ( undire2918257014606996450dges_a @ S ) @ X2 @ V ) ) ) ) ).
% comp_sgraph.is_isolated_vertex_def
thf(fact_606_comp__sgraph_Ois__isolated__vertex__edge,axiom,
! [S: set_a,V: a,E: set_a] :
( ( undire8931668460104145173rtex_a @ S @ ( undire2918257014606996450dges_a @ S ) @ V )
=> ( ( member_set_a @ E @ ( undire2918257014606996450dges_a @ S ) )
=> ~ ( undire1521409233611534436dent_a @ V @ E ) ) ) ).
% comp_sgraph.is_isolated_vertex_edge
thf(fact_607_comp__sgraph_Ois__isolated__vertex__no__loop,axiom,
! [S: set_a,V: a] :
( ( undire8931668460104145173rtex_a @ S @ ( undire2918257014606996450dges_a @ S ) @ V )
=> ~ ( undire3617971648856834880loop_a @ ( undire2918257014606996450dges_a @ S ) @ V ) ) ).
% comp_sgraph.is_isolated_vertex_no_loop
thf(fact_608_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_609_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_610_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_611_comp__sgraph_Ois__walk__drop__hd,axiom,
! [Ys: list_set_a,S: set_set_a,Y: set_a] :
( ( Ys != nil_set_a )
=> ( ( undire3014741414213135564_set_a @ S @ ( undire8247866692393712962_set_a @ S ) @ ( cons_set_a @ Y @ Ys ) )
=> ( undire3014741414213135564_set_a @ S @ ( undire8247866692393712962_set_a @ S ) @ Ys ) ) ) ).
% comp_sgraph.is_walk_drop_hd
thf(fact_612_comp__sgraph_Ois__walk__drop__hd,axiom,
! [Ys: list_a,S: set_a,Y: a] :
( ( Ys != nil_a )
=> ( ( undire6133010728901294956walk_a @ S @ ( undire2918257014606996450dges_a @ S ) @ ( cons_a @ Y @ Ys ) )
=> ( undire6133010728901294956walk_a @ S @ ( undire2918257014606996450dges_a @ S ) @ Ys ) ) ) ).
% comp_sgraph.is_walk_drop_hd
thf(fact_613_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_614_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_615_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_616_nle__le,axiom,
! [A: real,B2: real] :
( ( ~ ( ord_less_eq_real @ A @ B2 ) )
= ( ( ord_less_eq_real @ B2 @ A )
& ( B2 != A ) ) ) ).
% nle_le
thf(fact_617_nle__le,axiom,
! [A: nat,B2: nat] :
( ( ~ ( ord_less_eq_nat @ A @ B2 ) )
= ( ( ord_less_eq_nat @ B2 @ A )
& ( B2 != A ) ) ) ).
% nle_le
thf(fact_618_le__cases3,axiom,
! [X3: real,Y: real,Z3: real] :
( ( ( ord_less_eq_real @ X3 @ Y )
=> ~ ( ord_less_eq_real @ Y @ Z3 ) )
=> ( ( ( ord_less_eq_real @ Y @ X3 )
=> ~ ( ord_less_eq_real @ X3 @ Z3 ) )
=> ( ( ( ord_less_eq_real @ X3 @ Z3 )
=> ~ ( ord_less_eq_real @ Z3 @ Y ) )
=> ( ( ( ord_less_eq_real @ Z3 @ Y )
=> ~ ( ord_less_eq_real @ Y @ X3 ) )
=> ( ( ( ord_less_eq_real @ Y @ Z3 )
=> ~ ( ord_less_eq_real @ Z3 @ X3 ) )
=> ~ ( ( ord_less_eq_real @ Z3 @ X3 )
=> ~ ( ord_less_eq_real @ X3 @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_619_le__cases3,axiom,
! [X3: nat,Y: nat,Z3: nat] :
( ( ( ord_less_eq_nat @ X3 @ Y )
=> ~ ( ord_less_eq_nat @ Y @ Z3 ) )
=> ( ( ( ord_less_eq_nat @ Y @ X3 )
=> ~ ( ord_less_eq_nat @ X3 @ Z3 ) )
=> ( ( ( ord_less_eq_nat @ X3 @ Z3 )
=> ~ ( ord_less_eq_nat @ Z3 @ Y ) )
=> ( ( ( ord_less_eq_nat @ Z3 @ Y )
=> ~ ( ord_less_eq_nat @ Y @ X3 ) )
=> ( ( ( ord_less_eq_nat @ Y @ Z3 )
=> ~ ( ord_less_eq_nat @ Z3 @ X3 ) )
=> ~ ( ( ord_less_eq_nat @ Z3 @ X3 )
=> ~ ( ord_less_eq_nat @ X3 @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_620_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_a,Z: set_a] : ( Y4 = Z ) )
= ( ^ [X2: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y3 )
& ( ord_less_eq_set_a @ Y3 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_621_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_set_a,Z: set_set_a] : ( Y4 = Z ) )
= ( ^ [X2: set_set_a,Y3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X2 @ Y3 )
& ( ord_le3724670747650509150_set_a @ Y3 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_622_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: real,Z: real] : ( Y4 = Z ) )
= ( ^ [X2: real,Y3: real] :
( ( ord_less_eq_real @ X2 @ Y3 )
& ( ord_less_eq_real @ Y3 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_623_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: nat,Z: nat] : ( Y4 = Z ) )
= ( ^ [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
& ( ord_less_eq_nat @ Y3 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_624_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_Product_prod_a_a,Z: set_Product_prod_a_a] : ( Y4 = Z ) )
= ( ^ [X2: set_Product_prod_a_a,Y3: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ X2 @ Y3 )
& ( ord_le746702958409616551od_a_a @ Y3 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_625_ord__eq__le__trans,axiom,
! [A: set_a,B2: set_a,C: set_a] :
( ( A = B2 )
=> ( ( ord_less_eq_set_a @ B2 @ C )
=> ( ord_less_eq_set_a @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_626_ord__eq__le__trans,axiom,
! [A: set_set_a,B2: set_set_a,C: set_set_a] :
( ( A = B2 )
=> ( ( ord_le3724670747650509150_set_a @ B2 @ C )
=> ( ord_le3724670747650509150_set_a @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_627_ord__eq__le__trans,axiom,
! [A: real,B2: real,C: real] :
( ( A = B2 )
=> ( ( ord_less_eq_real @ B2 @ C )
=> ( ord_less_eq_real @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_628_ord__eq__le__trans,axiom,
! [A: nat,B2: nat,C: nat] :
( ( A = B2 )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_629_ord__eq__le__trans,axiom,
! [A: set_Product_prod_a_a,B2: set_Product_prod_a_a,C: set_Product_prod_a_a] :
( ( A = B2 )
=> ( ( ord_le746702958409616551od_a_a @ B2 @ C )
=> ( ord_le746702958409616551od_a_a @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_630_ord__le__eq__trans,axiom,
! [A: set_a,B2: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ B2 )
=> ( ( B2 = C )
=> ( ord_less_eq_set_a @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_631_ord__le__eq__trans,axiom,
! [A: set_set_a,B2: set_set_a,C: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ B2 )
=> ( ( B2 = C )
=> ( ord_le3724670747650509150_set_a @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_632_ord__le__eq__trans,axiom,
! [A: real,B2: real,C: real] :
( ( ord_less_eq_real @ A @ B2 )
=> ( ( B2 = C )
=> ( ord_less_eq_real @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_633_ord__le__eq__trans,axiom,
! [A: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( B2 = C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_634_ord__le__eq__trans,axiom,
! [A: set_Product_prod_a_a,B2: set_Product_prod_a_a,C: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A @ B2 )
=> ( ( B2 = C )
=> ( ord_le746702958409616551od_a_a @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_635_order__antisym,axiom,
! [X3: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y )
=> ( ( ord_less_eq_set_a @ Y @ X3 )
=> ( X3 = Y ) ) ) ).
% order_antisym
thf(fact_636_order__antisym,axiom,
! [X3: set_set_a,Y: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X3 @ Y )
=> ( ( ord_le3724670747650509150_set_a @ Y @ X3 )
=> ( X3 = Y ) ) ) ).
% order_antisym
thf(fact_637_order__antisym,axiom,
! [X3: real,Y: real] :
( ( ord_less_eq_real @ X3 @ Y )
=> ( ( ord_less_eq_real @ Y @ X3 )
=> ( X3 = Y ) ) ) ).
% order_antisym
thf(fact_638_order__antisym,axiom,
! [X3: nat,Y: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
=> ( ( ord_less_eq_nat @ Y @ X3 )
=> ( X3 = Y ) ) ) ).
% order_antisym
thf(fact_639_order__antisym,axiom,
! [X3: set_Product_prod_a_a,Y: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ X3 @ Y )
=> ( ( ord_le746702958409616551od_a_a @ Y @ X3 )
=> ( X3 = Y ) ) ) ).
% order_antisym
thf(fact_640_order_Otrans,axiom,
! [A: set_a,B2: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ B2 )
=> ( ( ord_less_eq_set_a @ B2 @ C )
=> ( ord_less_eq_set_a @ A @ C ) ) ) ).
% order.trans
thf(fact_641_order_Otrans,axiom,
! [A: set_set_a,B2: set_set_a,C: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ B2 )
=> ( ( ord_le3724670747650509150_set_a @ B2 @ C )
=> ( ord_le3724670747650509150_set_a @ A @ C ) ) ) ).
% order.trans
thf(fact_642_order_Otrans,axiom,
! [A: real,B2: real,C: real] :
( ( ord_less_eq_real @ A @ B2 )
=> ( ( ord_less_eq_real @ B2 @ C )
=> ( ord_less_eq_real @ A @ C ) ) ) ).
% order.trans
thf(fact_643_order_Otrans,axiom,
! [A: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% order.trans
thf(fact_644_order_Otrans,axiom,
! [A: set_Product_prod_a_a,B2: set_Product_prod_a_a,C: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A @ B2 )
=> ( ( ord_le746702958409616551od_a_a @ B2 @ C )
=> ( ord_le746702958409616551od_a_a @ A @ C ) ) ) ).
% order.trans
thf(fact_645_order__trans,axiom,
! [X3: set_a,Y: set_a,Z3: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y )
=> ( ( ord_less_eq_set_a @ Y @ Z3 )
=> ( ord_less_eq_set_a @ X3 @ Z3 ) ) ) ).
% order_trans
thf(fact_646_order__trans,axiom,
! [X3: set_set_a,Y: set_set_a,Z3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X3 @ Y )
=> ( ( ord_le3724670747650509150_set_a @ Y @ Z3 )
=> ( ord_le3724670747650509150_set_a @ X3 @ Z3 ) ) ) ).
% order_trans
thf(fact_647_order__trans,axiom,
! [X3: real,Y: real,Z3: real] :
( ( ord_less_eq_real @ X3 @ Y )
=> ( ( ord_less_eq_real @ Y @ Z3 )
=> ( ord_less_eq_real @ X3 @ Z3 ) ) ) ).
% order_trans
thf(fact_648_order__trans,axiom,
! [X3: nat,Y: nat,Z3: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z3 )
=> ( ord_less_eq_nat @ X3 @ Z3 ) ) ) ).
% order_trans
thf(fact_649_order__trans,axiom,
! [X3: set_Product_prod_a_a,Y: set_Product_prod_a_a,Z3: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ X3 @ Y )
=> ( ( ord_le746702958409616551od_a_a @ Y @ Z3 )
=> ( ord_le746702958409616551od_a_a @ X3 @ Z3 ) ) ) ).
% order_trans
thf(fact_650_linorder__wlog,axiom,
! [P2: real > real > $o,A: real,B2: real] :
( ! [A4: real,B4: real] :
( ( ord_less_eq_real @ A4 @ B4 )
=> ( P2 @ A4 @ B4 ) )
=> ( ! [A4: real,B4: real] :
( ( P2 @ B4 @ A4 )
=> ( P2 @ A4 @ B4 ) )
=> ( P2 @ A @ B2 ) ) ) ).
% linorder_wlog
thf(fact_651_linorder__wlog,axiom,
! [P2: nat > nat > $o,A: nat,B2: nat] :
( ! [A4: nat,B4: nat] :
( ( ord_less_eq_nat @ A4 @ B4 )
=> ( P2 @ A4 @ B4 ) )
=> ( ! [A4: nat,B4: nat] :
( ( P2 @ B4 @ A4 )
=> ( P2 @ A4 @ B4 ) )
=> ( P2 @ A @ B2 ) ) ) ).
% linorder_wlog
thf(fact_652_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_653_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_654_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_655_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_656_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_657_dual__order_Oantisym,axiom,
! [B2: set_a,A: set_a] :
( ( ord_less_eq_set_a @ B2 @ A )
=> ( ( ord_less_eq_set_a @ A @ B2 )
=> ( A = B2 ) ) ) ).
% dual_order.antisym
thf(fact_658_dual__order_Oantisym,axiom,
! [B2: set_set_a,A: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B2 @ A )
=> ( ( ord_le3724670747650509150_set_a @ A @ B2 )
=> ( A = B2 ) ) ) ).
% dual_order.antisym
thf(fact_659_dual__order_Oantisym,axiom,
! [B2: real,A: real] :
( ( ord_less_eq_real @ B2 @ A )
=> ( ( ord_less_eq_real @ A @ B2 )
=> ( A = B2 ) ) ) ).
% dual_order.antisym
thf(fact_660_dual__order_Oantisym,axiom,
! [B2: nat,A: nat] :
( ( ord_less_eq_nat @ B2 @ A )
=> ( ( ord_less_eq_nat @ A @ B2 )
=> ( A = B2 ) ) ) ).
% dual_order.antisym
thf(fact_661_dual__order_Oantisym,axiom,
! [B2: set_Product_prod_a_a,A: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ B2 @ A )
=> ( ( ord_le746702958409616551od_a_a @ A @ B2 )
=> ( A = B2 ) ) ) ).
% dual_order.antisym
thf(fact_662_dual__order_Otrans,axiom,
! [B2: set_a,A: set_a,C: set_a] :
( ( ord_less_eq_set_a @ B2 @ A )
=> ( ( ord_less_eq_set_a @ C @ B2 )
=> ( ord_less_eq_set_a @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_663_dual__order_Otrans,axiom,
! [B2: set_set_a,A: set_set_a,C: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B2 @ A )
=> ( ( ord_le3724670747650509150_set_a @ C @ B2 )
=> ( ord_le3724670747650509150_set_a @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_664_dual__order_Otrans,axiom,
! [B2: real,A: real,C: real] :
( ( ord_less_eq_real @ B2 @ A )
=> ( ( ord_less_eq_real @ C @ B2 )
=> ( ord_less_eq_real @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_665_dual__order_Otrans,axiom,
! [B2: nat,A: nat,C: nat] :
( ( ord_less_eq_nat @ B2 @ A )
=> ( ( ord_less_eq_nat @ C @ B2 )
=> ( ord_less_eq_nat @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_666_dual__order_Otrans,axiom,
! [B2: set_Product_prod_a_a,A: set_Product_prod_a_a,C: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ B2 @ A )
=> ( ( ord_le746702958409616551od_a_a @ C @ B2 )
=> ( ord_le746702958409616551od_a_a @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_667_antisym,axiom,
! [A: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A @ B2 )
=> ( ( ord_less_eq_set_a @ B2 @ A )
=> ( A = B2 ) ) ) ).
% antisym
thf(fact_668_antisym,axiom,
! [A: set_set_a,B2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ B2 )
=> ( ( ord_le3724670747650509150_set_a @ B2 @ A )
=> ( A = B2 ) ) ) ).
% antisym
thf(fact_669_antisym,axiom,
! [A: real,B2: real] :
( ( ord_less_eq_real @ A @ B2 )
=> ( ( ord_less_eq_real @ B2 @ A )
=> ( A = B2 ) ) ) ).
% antisym
thf(fact_670_antisym,axiom,
! [A: nat,B2: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ord_less_eq_nat @ B2 @ A )
=> ( A = B2 ) ) ) ).
% antisym
thf(fact_671_antisym,axiom,
! [A: set_Product_prod_a_a,B2: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A @ B2 )
=> ( ( ord_le746702958409616551od_a_a @ B2 @ A )
=> ( A = B2 ) ) ) ).
% antisym
thf(fact_672_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_673_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_674_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_675_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_676_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_677_order__subst1,axiom,
! [A: real,F: real > real,B2: real,C: real] :
( ( ord_less_eq_real @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_real @ B2 @ C )
=> ( ! [X4: real,Y2: real] :
( ( ord_less_eq_real @ X4 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_678_order__subst1,axiom,
! [A: real,F: nat > real,B2: nat,C: nat] :
( ( ord_less_eq_real @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X4: nat,Y2: nat] :
( ( ord_less_eq_nat @ X4 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_679_order__subst1,axiom,
! [A: nat,F: real > nat,B2: real,C: real] :
( ( ord_less_eq_nat @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_real @ B2 @ C )
=> ( ! [X4: real,Y2: real] :
( ( ord_less_eq_real @ X4 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_680_order__subst1,axiom,
! [A: nat,F: nat > nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X4: nat,Y2: nat] :
( ( ord_less_eq_nat @ X4 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_681_order__subst1,axiom,
! [A: set_a,F: real > set_a,B2: real,C: real] :
( ( ord_less_eq_set_a @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_real @ B2 @ C )
=> ( ! [X4: real,Y2: real] :
( ( ord_less_eq_real @ X4 @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_682_order__subst1,axiom,
! [A: set_a,F: nat > set_a,B2: nat,C: nat] :
( ( ord_less_eq_set_a @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X4: nat,Y2: nat] :
( ( ord_less_eq_nat @ X4 @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_683_order__subst1,axiom,
! [A: real,F: set_a > real,B2: set_a,C: set_a] :
( ( ord_less_eq_real @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_set_a @ B2 @ C )
=> ( ! [X4: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_684_order__subst1,axiom,
! [A: nat,F: set_a > nat,B2: set_a,C: set_a] :
( ( ord_less_eq_nat @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_set_a @ B2 @ C )
=> ( ! [X4: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_685_order__subst1,axiom,
! [A: set_a,F: set_a > set_a,B2: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_set_a @ B2 @ C )
=> ( ! [X4: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_686_order__subst1,axiom,
! [A: set_set_a,F: real > set_set_a,B2: real,C: real] :
( ( ord_le3724670747650509150_set_a @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_real @ B2 @ C )
=> ( ! [X4: real,Y2: real] :
( ( ord_less_eq_real @ X4 @ Y2 )
=> ( ord_le3724670747650509150_set_a @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_le3724670747650509150_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_687_order__subst2,axiom,
! [A: real,B2: real,F: real > real,C: real] :
( ( ord_less_eq_real @ A @ B2 )
=> ( ( ord_less_eq_real @ ( F @ B2 ) @ C )
=> ( ! [X4: real,Y2: real] :
( ( ord_less_eq_real @ X4 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_688_order__subst2,axiom,
! [A: real,B2: real,F: real > nat,C: nat] :
( ( ord_less_eq_real @ A @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C )
=> ( ! [X4: real,Y2: real] :
( ( ord_less_eq_real @ X4 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_689_order__subst2,axiom,
! [A: nat,B2: nat,F: nat > real,C: real] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ord_less_eq_real @ ( F @ B2 ) @ C )
=> ( ! [X4: nat,Y2: nat] :
( ( ord_less_eq_nat @ X4 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_690_order__subst2,axiom,
! [A: nat,B2: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C )
=> ( ! [X4: nat,Y2: nat] :
( ( ord_less_eq_nat @ X4 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_691_order__subst2,axiom,
! [A: set_a,B2: set_a,F: set_a > real,C: real] :
( ( ord_less_eq_set_a @ A @ B2 )
=> ( ( ord_less_eq_real @ ( F @ B2 ) @ C )
=> ( ! [X4: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_692_order__subst2,axiom,
! [A: set_a,B2: set_a,F: set_a > nat,C: nat] :
( ( ord_less_eq_set_a @ A @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C )
=> ( ! [X4: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_693_order__subst2,axiom,
! [A: real,B2: real,F: real > set_a,C: set_a] :
( ( ord_less_eq_real @ A @ B2 )
=> ( ( ord_less_eq_set_a @ ( F @ B2 ) @ C )
=> ( ! [X4: real,Y2: real] :
( ( ord_less_eq_real @ X4 @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_694_order__subst2,axiom,
! [A: nat,B2: nat,F: nat > set_a,C: set_a] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ord_less_eq_set_a @ ( F @ B2 ) @ C )
=> ( ! [X4: nat,Y2: nat] :
( ( ord_less_eq_nat @ X4 @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_695_order__subst2,axiom,
! [A: set_a,B2: set_a,F: set_a > set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ B2 )
=> ( ( ord_less_eq_set_a @ ( F @ B2 ) @ C )
=> ( ! [X4: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_696_order__subst2,axiom,
! [A: set_set_a,B2: set_set_a,F: set_set_a > real,C: real] :
( ( ord_le3724670747650509150_set_a @ A @ B2 )
=> ( ( ord_less_eq_real @ ( F @ B2 ) @ C )
=> ( ! [X4: set_set_a,Y2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X4 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_697_order__eq__refl,axiom,
! [X3: set_a,Y: set_a] :
( ( X3 = Y )
=> ( ord_less_eq_set_a @ X3 @ Y ) ) ).
% order_eq_refl
thf(fact_698_order__eq__refl,axiom,
! [X3: set_set_a,Y: set_set_a] :
( ( X3 = Y )
=> ( ord_le3724670747650509150_set_a @ X3 @ Y ) ) ).
% order_eq_refl
thf(fact_699_order__eq__refl,axiom,
! [X3: real,Y: real] :
( ( X3 = Y )
=> ( ord_less_eq_real @ X3 @ Y ) ) ).
% order_eq_refl
thf(fact_700_order__eq__refl,axiom,
! [X3: nat,Y: nat] :
( ( X3 = Y )
=> ( ord_less_eq_nat @ X3 @ Y ) ) ).
% order_eq_refl
thf(fact_701_order__eq__refl,axiom,
! [X3: set_Product_prod_a_a,Y: set_Product_prod_a_a] :
( ( X3 = Y )
=> ( ord_le746702958409616551od_a_a @ X3 @ Y ) ) ).
% order_eq_refl
thf(fact_702_linorder__linear,axiom,
! [X3: real,Y: real] :
( ( ord_less_eq_real @ X3 @ Y )
| ( ord_less_eq_real @ Y @ X3 ) ) ).
% linorder_linear
thf(fact_703_linorder__linear,axiom,
! [X3: nat,Y: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
| ( ord_less_eq_nat @ Y @ X3 ) ) ).
% linorder_linear
thf(fact_704_ord__eq__le__subst,axiom,
! [A: real,F: real > real,B2: real,C: real] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_less_eq_real @ B2 @ C )
=> ( ! [X4: real,Y2: real] :
( ( ord_less_eq_real @ X4 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_705_ord__eq__le__subst,axiom,
! [A: nat,F: real > nat,B2: real,C: real] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_less_eq_real @ B2 @ C )
=> ( ! [X4: real,Y2: real] :
( ( ord_less_eq_real @ X4 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_706_ord__eq__le__subst,axiom,
! [A: real,F: nat > real,B2: nat,C: nat] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X4: nat,Y2: nat] :
( ( ord_less_eq_nat @ X4 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_707_ord__eq__le__subst,axiom,
! [A: nat,F: nat > nat,B2: nat,C: nat] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X4: nat,Y2: nat] :
( ( ord_less_eq_nat @ X4 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_708_ord__eq__le__subst,axiom,
! [A: real,F: set_a > real,B2: set_a,C: set_a] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_less_eq_set_a @ B2 @ C )
=> ( ! [X4: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_709_ord__eq__le__subst,axiom,
! [A: nat,F: set_a > nat,B2: set_a,C: set_a] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_less_eq_set_a @ B2 @ C )
=> ( ! [X4: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_710_ord__eq__le__subst,axiom,
! [A: set_a,F: real > set_a,B2: real,C: real] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_less_eq_real @ B2 @ C )
=> ( ! [X4: real,Y2: real] :
( ( ord_less_eq_real @ X4 @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_711_ord__eq__le__subst,axiom,
! [A: set_a,F: nat > set_a,B2: nat,C: nat] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X4: nat,Y2: nat] :
( ( ord_less_eq_nat @ X4 @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_712_ord__eq__le__subst,axiom,
! [A: set_a,F: set_a > set_a,B2: set_a,C: set_a] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_less_eq_set_a @ B2 @ C )
=> ( ! [X4: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_713_ord__eq__le__subst,axiom,
! [A: real,F: set_set_a > real,B2: set_set_a,C: set_set_a] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_le3724670747650509150_set_a @ B2 @ C )
=> ( ! [X4: set_set_a,Y2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X4 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_714_ord__le__eq__subst,axiom,
! [A: real,B2: real,F: real > real,C: real] :
( ( ord_less_eq_real @ A @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X4: real,Y2: real] :
( ( ord_less_eq_real @ X4 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_715_ord__le__eq__subst,axiom,
! [A: real,B2: real,F: real > nat,C: nat] :
( ( ord_less_eq_real @ A @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X4: real,Y2: real] :
( ( ord_less_eq_real @ X4 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_716_ord__le__eq__subst,axiom,
! [A: nat,B2: nat,F: nat > real,C: real] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X4: nat,Y2: nat] :
( ( ord_less_eq_nat @ X4 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_717_ord__le__eq__subst,axiom,
! [A: nat,B2: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X4: nat,Y2: nat] :
( ( ord_less_eq_nat @ X4 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_718_ord__le__eq__subst,axiom,
! [A: set_a,B2: set_a,F: set_a > real,C: real] :
( ( ord_less_eq_set_a @ A @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X4: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_719_ord__le__eq__subst,axiom,
! [A: set_a,B2: set_a,F: set_a > nat,C: nat] :
( ( ord_less_eq_set_a @ A @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X4: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_720_ord__le__eq__subst,axiom,
! [A: real,B2: real,F: real > set_a,C: set_a] :
( ( ord_less_eq_real @ A @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X4: real,Y2: real] :
( ( ord_less_eq_real @ X4 @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_721_ord__le__eq__subst,axiom,
! [A: nat,B2: nat,F: nat > set_a,C: set_a] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X4: nat,Y2: nat] :
( ( ord_less_eq_nat @ X4 @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_722_ord__le__eq__subst,axiom,
! [A: set_a,B2: set_a,F: set_a > set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X4: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_723_ord__le__eq__subst,axiom,
! [A: set_set_a,B2: set_set_a,F: set_set_a > real,C: real] :
( ( ord_le3724670747650509150_set_a @ A @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X4: set_set_a,Y2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X4 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_724_linorder__le__cases,axiom,
! [X3: real,Y: real] :
( ~ ( ord_less_eq_real @ X3 @ Y )
=> ( ord_less_eq_real @ Y @ X3 ) ) ).
% linorder_le_cases
thf(fact_725_linorder__le__cases,axiom,
! [X3: nat,Y: nat] :
( ~ ( ord_less_eq_nat @ X3 @ Y )
=> ( ord_less_eq_nat @ Y @ X3 ) ) ).
% linorder_le_cases
thf(fact_726_order__antisym__conv,axiom,
! [Y: set_a,X3: set_a] :
( ( ord_less_eq_set_a @ Y @ X3 )
=> ( ( ord_less_eq_set_a @ X3 @ Y )
= ( X3 = Y ) ) ) ).
% order_antisym_conv
thf(fact_727_order__antisym__conv,axiom,
! [Y: set_set_a,X3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ Y @ X3 )
=> ( ( ord_le3724670747650509150_set_a @ X3 @ Y )
= ( X3 = Y ) ) ) ).
% order_antisym_conv
thf(fact_728_order__antisym__conv,axiom,
! [Y: real,X3: real] :
( ( ord_less_eq_real @ Y @ X3 )
=> ( ( ord_less_eq_real @ X3 @ Y )
= ( X3 = Y ) ) ) ).
% order_antisym_conv
thf(fact_729_order__antisym__conv,axiom,
! [Y: nat,X3: nat] :
( ( ord_less_eq_nat @ Y @ X3 )
=> ( ( ord_less_eq_nat @ X3 @ Y )
= ( X3 = Y ) ) ) ).
% order_antisym_conv
thf(fact_730_order__antisym__conv,axiom,
! [Y: set_Product_prod_a_a,X3: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ Y @ X3 )
=> ( ( ord_le746702958409616551od_a_a @ X3 @ Y )
= ( X3 = Y ) ) ) ).
% order_antisym_conv
thf(fact_731_comp__sgraph_Ois__gen__path__trivial,axiom,
! [X3: product_prod_a_a,S: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X3 @ S )
=> ( undire7585867811434966393od_a_a @ S @ ( undire6879232364018543115od_a_a @ S ) @ ( cons_P7316939126706565853od_a_a @ X3 @ nil_Product_prod_a_a ) ) ) ).
% comp_sgraph.is_gen_path_trivial
thf(fact_732_comp__sgraph_Ois__gen__path__trivial,axiom,
! [X3: set_a,S: set_set_a] :
( ( member_set_a @ X3 @ S )
=> ( undire7201326534205417136_set_a @ S @ ( undire8247866692393712962_set_a @ S ) @ ( cons_set_a @ X3 @ nil_set_a ) ) ) ).
% comp_sgraph.is_gen_path_trivial
thf(fact_733_comp__sgraph_Ois__gen__path__trivial,axiom,
! [X3: a,S: set_a] :
( ( member_a @ X3 @ S )
=> ( undire3562951555376170320path_a @ S @ ( undire2918257014606996450dges_a @ S ) @ ( cons_a @ X3 @ nil_a ) ) ) ).
% comp_sgraph.is_gen_path_trivial
thf(fact_734_comp__sgraph_Oinduced__edges__ss,axiom,
! [V3: set_set_a,S: set_set_a] :
( ( ord_le3724670747650509150_set_a @ V3 @ S )
=> ( ord_le5722252365846178494_set_a @ ( undire7854589003810675244_set_a @ ( undire8247866692393712962_set_a @ S ) @ V3 ) @ ( undire8247866692393712962_set_a @ S ) ) ) ).
% comp_sgraph.induced_edges_ss
thf(fact_735_comp__sgraph_Oinduced__edges__ss,axiom,
! [V3: set_Product_prod_a_a,S: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ V3 @ S )
=> ( ord_le1995061765932249223od_a_a @ ( undire5906991851038061813od_a_a @ ( undire6879232364018543115od_a_a @ S ) @ V3 ) @ ( undire6879232364018543115od_a_a @ S ) ) ) ).
% comp_sgraph.induced_edges_ss
thf(fact_736_comp__sgraph_Oinduced__edges__ss,axiom,
! [V3: set_a,S: set_a] :
( ( ord_less_eq_set_a @ V3 @ S )
=> ( ord_le3724670747650509150_set_a @ ( undire7777452895879145676dges_a @ ( undire2918257014606996450dges_a @ S ) @ V3 ) @ ( undire2918257014606996450dges_a @ S ) ) ) ).
% comp_sgraph.induced_edges_ss
thf(fact_737_comp__sgraph_Oinduced__is__subgraph,axiom,
! [V3: set_set_a,S: set_set_a] :
( ( ord_le3724670747650509150_set_a @ V3 @ S )
=> ( undire1186139521737116585_set_a @ V3 @ ( undire7854589003810675244_set_a @ ( undire8247866692393712962_set_a @ S ) @ V3 ) @ S @ ( undire8247866692393712962_set_a @ S ) ) ) ).
% comp_sgraph.induced_is_subgraph
thf(fact_738_comp__sgraph_Oinduced__is__subgraph,axiom,
! [V3: set_Product_prod_a_a,S: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ V3 @ S )
=> ( undire398746457437328754od_a_a @ V3 @ ( undire5906991851038061813od_a_a @ ( undire6879232364018543115od_a_a @ S ) @ V3 ) @ S @ ( undire6879232364018543115od_a_a @ S ) ) ) ).
% comp_sgraph.induced_is_subgraph
thf(fact_739_comp__sgraph_Oinduced__is__subgraph,axiom,
! [V3: set_a,S: set_a] :
( ( ord_less_eq_set_a @ V3 @ S )
=> ( undire7103218114511261257raph_a @ V3 @ ( undire7777452895879145676dges_a @ ( undire2918257014606996450dges_a @ S ) @ V3 ) @ S @ ( undire2918257014606996450dges_a @ S ) ) ) ).
% comp_sgraph.induced_is_subgraph
thf(fact_740_comp__sgraph_Ois__path__def,axiom,
! [S: set_set_a,Xs: list_set_a] :
( ( undire8834939040163919632_set_a @ S @ ( undire8247866692393712962_set_a @ S ) @ Xs )
= ( ( undire526879649183275522_set_a @ S @ ( undire8247866692393712962_set_a @ S ) @ Xs )
& ( distinct_set_a @ Xs ) ) ) ).
% comp_sgraph.is_path_def
thf(fact_741_comp__sgraph_Ois__path__def,axiom,
! [S: set_a,Xs: list_a] :
( ( undire427332500224447920path_a @ S @ ( undire2918257014606996450dges_a @ S ) @ Xs )
= ( ( undire2427028224930250914walk_a @ S @ ( undire2918257014606996450dges_a @ S ) @ Xs )
& ( distinct_a @ Xs ) ) ) ).
% comp_sgraph.is_path_def
thf(fact_742_comp__sgraph_Ois__walk__decomp,axiom,
! [S: set_set_a,Xs: list_set_a,Y: set_a,Ys: list_set_a,Zs: list_set_a] :
( ( undire3014741414213135564_set_a @ S @ ( undire8247866692393712962_set_a @ S ) @ ( append_set_a @ Xs @ ( append_set_a @ ( cons_set_a @ Y @ nil_set_a ) @ ( append_set_a @ Ys @ ( append_set_a @ ( cons_set_a @ Y @ nil_set_a ) @ Zs ) ) ) ) )
=> ( undire3014741414213135564_set_a @ S @ ( undire8247866692393712962_set_a @ S ) @ ( append_set_a @ Xs @ ( append_set_a @ ( cons_set_a @ Y @ nil_set_a ) @ Zs ) ) ) ) ).
% comp_sgraph.is_walk_decomp
thf(fact_743_comp__sgraph_Ois__walk__decomp,axiom,
! [S: set_a,Xs: list_a,Y: a,Ys: list_a,Zs: list_a] :
( ( undire6133010728901294956walk_a @ S @ ( undire2918257014606996450dges_a @ S ) @ ( append_a @ Xs @ ( append_a @ ( cons_a @ Y @ nil_a ) @ ( append_a @ Ys @ ( append_a @ ( cons_a @ Y @ nil_a ) @ Zs ) ) ) ) )
=> ( undire6133010728901294956walk_a @ S @ ( undire2918257014606996450dges_a @ S ) @ ( append_a @ Xs @ ( append_a @ ( cons_a @ Y @ nil_a ) @ Zs ) ) ) ) ).
% comp_sgraph.is_walk_decomp
thf(fact_744_rotate1_Osimps_I2_J,axiom,
! [X3: a,Xs: list_a] :
( ( rotate1_a @ ( cons_a @ X3 @ Xs ) )
= ( append_a @ Xs @ ( cons_a @ X3 @ nil_a ) ) ) ).
% rotate1.simps(2)
thf(fact_745_rotate1_Osimps_I2_J,axiom,
! [X3: set_a,Xs: list_set_a] :
( ( rotate1_set_a @ ( cons_set_a @ X3 @ Xs ) )
= ( append_set_a @ Xs @ ( cons_set_a @ X3 @ nil_set_a ) ) ) ).
% rotate1.simps(2)
thf(fact_746_comp__sgraph_Ois__gen__path__distinct,axiom,
! [S: set_set_a,P: list_set_a] :
( ( undire7201326534205417136_set_a @ S @ ( undire8247866692393712962_set_a @ S ) @ P )
=> ( ( ( hd_set_a @ P )
!= ( last_set_a @ P ) )
=> ( distinct_set_a @ P ) ) ) ).
% comp_sgraph.is_gen_path_distinct
thf(fact_747_comp__sgraph_Ois__gen__path__distinct,axiom,
! [S: set_a,P: list_a] :
( ( undire3562951555376170320path_a @ S @ ( undire2918257014606996450dges_a @ S ) @ P )
=> ( ( ( hd_a @ P )
!= ( last_a @ P ) )
=> ( distinct_a @ P ) ) ) ).
% comp_sgraph.is_gen_path_distinct
thf(fact_748_comp__sgraph_Ois__open__walk__def,axiom,
! [S: set_a,Xs: list_a] :
( ( undire2427028224930250914walk_a @ S @ ( undire2918257014606996450dges_a @ S ) @ Xs )
= ( ( undire6133010728901294956walk_a @ S @ ( undire2918257014606996450dges_a @ S ) @ Xs )
& ( ( hd_a @ Xs )
!= ( last_a @ Xs ) ) ) ) ).
% comp_sgraph.is_open_walk_def
thf(fact_749_comp__sgraph_Ois__closed__walk__def,axiom,
! [S: set_a,Xs: list_a] :
( ( undire3370724456595283424walk_a @ S @ ( undire2918257014606996450dges_a @ S ) @ Xs )
= ( ( undire6133010728901294956walk_a @ S @ ( undire2918257014606996450dges_a @ S ) @ Xs )
& ( ( hd_a @ Xs )
= ( last_a @ Xs ) ) ) ) ).
% comp_sgraph.is_closed_walk_def
thf(fact_750_comp__sgraph_Oconnecting__walk__def,axiom,
! [S: set_a,U: a,V: a,Xs: list_a] :
( ( connecting_walk_a @ S @ ( undire2918257014606996450dges_a @ S ) @ U @ V @ Xs )
= ( ( undire6133010728901294956walk_a @ S @ ( undire2918257014606996450dges_a @ S ) @ Xs )
& ( ( hd_a @ Xs )
= U )
& ( ( last_a @ Xs )
= V ) ) ) ).
% comp_sgraph.connecting_walk_def
thf(fact_751_comp__sgraph_Ois__trail__def,axiom,
! [S: set_a,Xs: list_a] :
( ( undire7142031287334043199rail_a @ S @ ( undire2918257014606996450dges_a @ S ) @ Xs )
= ( ( undire6133010728901294956walk_a @ S @ ( undire2918257014606996450dges_a @ S ) @ Xs )
& ( distinct_set_a @ ( undire7337870655677353998dges_a @ Xs ) ) ) ) ).
% comp_sgraph.is_trail_def
thf(fact_752_comp__sgraph_Ois__walk__append,axiom,
! [S: set_set_a,Xs: list_set_a,Ys: list_set_a] :
( ( undire3014741414213135564_set_a @ S @ ( undire8247866692393712962_set_a @ S ) @ Xs )
=> ( ( undire3014741414213135564_set_a @ S @ ( undire8247866692393712962_set_a @ S ) @ Ys )
=> ( ( ( last_set_a @ Xs )
= ( hd_set_a @ Ys ) )
=> ( undire3014741414213135564_set_a @ S @ ( undire8247866692393712962_set_a @ S ) @ ( append_set_a @ Xs @ ( tl_set_a @ Ys ) ) ) ) ) ) ).
% comp_sgraph.is_walk_append
thf(fact_753_comp__sgraph_Ois__walk__append,axiom,
! [S: set_a,Xs: list_a,Ys: list_a] :
( ( undire6133010728901294956walk_a @ S @ ( undire2918257014606996450dges_a @ S ) @ Xs )
=> ( ( undire6133010728901294956walk_a @ S @ ( undire2918257014606996450dges_a @ S ) @ Ys )
=> ( ( ( last_a @ Xs )
= ( hd_a @ Ys ) )
=> ( undire6133010728901294956walk_a @ S @ ( undire2918257014606996450dges_a @ S ) @ ( append_a @ Xs @ ( tl_a @ Ys ) ) ) ) ) ) ).
% comp_sgraph.is_walk_append
thf(fact_754_comp__sgraph_Ois__gen__path__distinct__tl,axiom,
! [S: set_set_a,P: list_set_a] :
( ( undire7201326534205417136_set_a @ S @ ( undire8247866692393712962_set_a @ S ) @ P )
=> ( ( ( hd_set_a @ P )
= ( last_set_a @ P ) )
=> ( distinct_set_a @ ( tl_set_a @ P ) ) ) ) ).
% comp_sgraph.is_gen_path_distinct_tl
thf(fact_755_comp__sgraph_Ois__gen__path__distinct__tl,axiom,
! [S: set_a,P: list_a] :
( ( undire3562951555376170320path_a @ S @ ( undire2918257014606996450dges_a @ S ) @ P )
=> ( ( ( hd_a @ P )
= ( last_a @ P ) )
=> ( distinct_a @ ( tl_a @ P ) ) ) ) ).
% comp_sgraph.is_gen_path_distinct_tl
thf(fact_756_comp__sgraph_Ois__gen__path__options,axiom,
! [S: set_set_a,P: list_set_a] :
( ( undire7201326534205417136_set_a @ S @ ( undire8247866692393712962_set_a @ S ) @ P )
= ( ( undire797940137672299967_set_a @ S @ ( undire8247866692393712962_set_a @ S ) @ P )
| ( undire8834939040163919632_set_a @ S @ ( undire8247866692393712962_set_a @ S ) @ P )
| ? [X2: set_a] :
( ( member_set_a @ X2 @ S )
& ( P
= ( cons_set_a @ X2 @ nil_set_a ) ) ) ) ) ).
% comp_sgraph.is_gen_path_options
thf(fact_757_comp__sgraph_Ois__gen__path__options,axiom,
! [S: set_a,P: list_a] :
( ( undire3562951555376170320path_a @ S @ ( undire2918257014606996450dges_a @ S ) @ P )
= ( ( undire2407311113669455967ycle_a @ S @ ( undire2918257014606996450dges_a @ S ) @ P )
| ( undire427332500224447920path_a @ S @ ( undire2918257014606996450dges_a @ S ) @ P )
| ? [X2: a] :
( ( member_a @ X2 @ S )
& ( P
= ( cons_a @ X2 @ nil_a ) ) ) ) ) ).
% comp_sgraph.is_gen_path_options
thf(fact_758_comp__sgraph_Ois__gen__path__def,axiom,
! [S: set_set_a,P: list_set_a] :
( ( undire7201326534205417136_set_a @ S @ ( undire8247866692393712962_set_a @ S ) @ P )
= ( ( undire3014741414213135564_set_a @ S @ ( undire8247866692393712962_set_a @ S ) @ P )
& ( ( ( distinct_set_a @ ( tl_set_a @ P ) )
& ( ( hd_set_a @ P )
= ( last_set_a @ P ) ) )
| ( distinct_set_a @ P ) ) ) ) ).
% comp_sgraph.is_gen_path_def
thf(fact_759_comp__sgraph_Ois__gen__path__def,axiom,
! [S: set_a,P: list_a] :
( ( undire3562951555376170320path_a @ S @ ( undire2918257014606996450dges_a @ S ) @ P )
= ( ( undire6133010728901294956walk_a @ S @ ( undire2918257014606996450dges_a @ S ) @ P )
& ( ( ( distinct_a @ ( tl_a @ P ) )
& ( ( hd_a @ P )
= ( last_a @ P ) ) )
| ( distinct_a @ P ) ) ) ) ).
% comp_sgraph.is_gen_path_def
thf(fact_760_list__exhaust3,axiom,
! [Xs: list_a] :
( ( Xs != nil_a )
=> ( ! [X4: a] :
( Xs
!= ( cons_a @ X4 @ nil_a ) )
=> ~ ! [X4: a,Y2: a,Ys2: list_a] :
( Xs
!= ( cons_a @ X4 @ ( cons_a @ Y2 @ Ys2 ) ) ) ) ) ).
% list_exhaust3
thf(fact_761_list__exhaust3,axiom,
! [Xs: list_set_a] :
( ( Xs != nil_set_a )
=> ( ! [X4: set_a] :
( Xs
!= ( cons_set_a @ X4 @ nil_set_a ) )
=> ~ ! [X4: set_a,Y2: set_a,Ys2: list_set_a] :
( Xs
!= ( cons_set_a @ X4 @ ( cons_set_a @ Y2 @ Ys2 ) ) ) ) ) ).
% list_exhaust3
thf(fact_762_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_763_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_764_is__cycle__alt,axiom,
! [Xs: list_a] :
( ( undire2407311113669455967ycle_a @ vertices @ edges @ Xs )
= ( ( undire6133010728901294956walk_a @ vertices @ edges @ Xs )
& ( distinct_a @ ( tl_a @ Xs ) )
& ( ord_less_eq_nat @ one_one_nat @ ( undire8849074589633906640ngth_a @ Xs ) )
& ( ( hd_a @ Xs )
= ( last_a @ Xs ) ) ) ) ).
% is_cycle_alt
thf(fact_765_is__cycle__def,axiom,
! [Xs: list_a] :
( ( undire2407311113669455967ycle_a @ vertices @ edges @ Xs )
= ( ( undire3370724456595283424walk_a @ vertices @ edges @ Xs )
& ( ord_less_eq_nat @ one_one_nat @ ( undire8849074589633906640ngth_a @ Xs ) )
& ( distinct_a @ ( tl_a @ Xs ) ) ) ) ).
% is_cycle_def
thf(fact_766_is__cycle__alt__gen__path,axiom,
! [Xs: list_a] :
( ( undire2407311113669455967ycle_a @ vertices @ edges @ Xs )
= ( ( undire3562951555376170320path_a @ vertices @ edges @ Xs )
& ( ord_less_eq_nat @ one_one_nat @ ( undire8849074589633906640ngth_a @ Xs ) )
& ( ( hd_a @ Xs )
= ( last_a @ Xs ) ) ) ) ).
% is_cycle_alt_gen_path
thf(fact_767_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_768_walk__length__rev,axiom,
( undire8849074589633906640ngth_a
= ( ^ [P3: list_a] : ( undire8849074589633906640ngth_a @ ( rev_a @ P3 ) ) ) ) ).
% walk_length_rev
thf(fact_769_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_770_ulgraph_Oall__edges__between_Ocong,axiom,
undire8383842906760478443ween_a = undire8383842906760478443ween_a ).
% ulgraph.all_edges_between.cong
thf(fact_771_comp__sgraph_Owalk__length__rev,axiom,
( undire4424681683220949296_set_a
= ( ^ [P3: list_set_a] : ( undire4424681683220949296_set_a @ ( rev_set_a @ P3 ) ) ) ) ).
% comp_sgraph.walk_length_rev
thf(fact_772_comp__sgraph_Owalk__length__rev,axiom,
( undire8849074589633906640ngth_a
= ( ^ [P3: list_a] : ( undire8849074589633906640ngth_a @ ( rev_a @ P3 ) ) ) ) ).
% comp_sgraph.walk_length_rev
thf(fact_773_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_774_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_775_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_776_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_777_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_778_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_779_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_780_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_781_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_782_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_783_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_784_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_785_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_786_ulgraph_Owalk__length__rev,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,P: list_set_a] :
( ( undire6886684016831807756_set_a @ Vertices @ Edges )
=> ( ( undire4424681683220949296_set_a @ P )
= ( undire4424681683220949296_set_a @ ( rev_set_a @ P ) ) ) ) ).
% ulgraph.walk_length_rev
thf(fact_787_ulgraph_Owalk__length__rev,axiom,
! [Vertices: set_a,Edges: set_set_a,P: list_a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( undire8849074589633906640ngth_a @ P )
= ( undire8849074589633906640ngth_a @ ( rev_a @ P ) ) ) ) ).
% ulgraph.walk_length_rev
thf(fact_788_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_789_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_790_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_791_comp__sgraph_Ois__cycle__alt__gen__path,axiom,
! [S: set_a,Xs: list_a] :
( ( undire2407311113669455967ycle_a @ S @ ( undire2918257014606996450dges_a @ S ) @ Xs )
= ( ( undire3562951555376170320path_a @ S @ ( undire2918257014606996450dges_a @ S ) @ Xs )
& ( ord_less_eq_nat @ one_one_nat @ ( undire8849074589633906640ngth_a @ Xs ) )
& ( ( hd_a @ Xs )
= ( last_a @ Xs ) ) ) ) ).
% comp_sgraph.is_cycle_alt_gen_path
thf(fact_792_ulgraph_Ois__cycle__alt__gen__path,axiom,
! [Vertices: set_a,Edges: set_set_a,Xs: list_a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( undire2407311113669455967ycle_a @ Vertices @ Edges @ Xs )
= ( ( undire3562951555376170320path_a @ Vertices @ Edges @ Xs )
& ( ord_less_eq_nat @ one_one_nat @ ( undire8849074589633906640ngth_a @ Xs ) )
& ( ( hd_a @ Xs )
= ( last_a @ Xs ) ) ) ) ) ).
% ulgraph.is_cycle_alt_gen_path
thf(fact_793_comp__sgraph_Ois__cycle__def,axiom,
! [S: set_set_a,Xs: list_set_a] :
( ( undire797940137672299967_set_a @ S @ ( undire8247866692393712962_set_a @ S ) @ Xs )
= ( ( undire4100213446647512896_set_a @ S @ ( undire8247866692393712962_set_a @ S ) @ Xs )
& ( ord_less_eq_nat @ one_one_nat @ ( undire4424681683220949296_set_a @ Xs ) )
& ( distinct_set_a @ ( tl_set_a @ Xs ) ) ) ) ).
% comp_sgraph.is_cycle_def
thf(fact_794_comp__sgraph_Ois__cycle__def,axiom,
! [S: set_a,Xs: list_a] :
( ( undire2407311113669455967ycle_a @ S @ ( undire2918257014606996450dges_a @ S ) @ Xs )
= ( ( undire3370724456595283424walk_a @ S @ ( undire2918257014606996450dges_a @ S ) @ Xs )
& ( ord_less_eq_nat @ one_one_nat @ ( undire8849074589633906640ngth_a @ Xs ) )
& ( distinct_a @ ( tl_a @ Xs ) ) ) ) ).
% comp_sgraph.is_cycle_def
thf(fact_795_ulgraph_Ois__cycle__def,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,Xs: list_set_a] :
( ( undire6886684016831807756_set_a @ Vertices @ Edges )
=> ( ( undire797940137672299967_set_a @ Vertices @ Edges @ Xs )
= ( ( undire4100213446647512896_set_a @ Vertices @ Edges @ Xs )
& ( ord_less_eq_nat @ one_one_nat @ ( undire4424681683220949296_set_a @ Xs ) )
& ( distinct_set_a @ ( tl_set_a @ Xs ) ) ) ) ) ).
% ulgraph.is_cycle_def
thf(fact_796_ulgraph_Ois__cycle__def,axiom,
! [Vertices: set_a,Edges: set_set_a,Xs: list_a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( undire2407311113669455967ycle_a @ Vertices @ Edges @ Xs )
= ( ( undire3370724456595283424walk_a @ Vertices @ Edges @ Xs )
& ( ord_less_eq_nat @ one_one_nat @ ( undire8849074589633906640ngth_a @ Xs ) )
& ( distinct_a @ ( tl_a @ Xs ) ) ) ) ) ).
% ulgraph.is_cycle_def
thf(fact_797_comp__sgraph_Ois__cycle__alt,axiom,
! [S: set_set_a,Xs: list_set_a] :
( ( undire797940137672299967_set_a @ S @ ( undire8247866692393712962_set_a @ S ) @ Xs )
= ( ( undire3014741414213135564_set_a @ S @ ( undire8247866692393712962_set_a @ S ) @ Xs )
& ( distinct_set_a @ ( tl_set_a @ Xs ) )
& ( ord_less_eq_nat @ one_one_nat @ ( undire4424681683220949296_set_a @ Xs ) )
& ( ( hd_set_a @ Xs )
= ( last_set_a @ Xs ) ) ) ) ).
% comp_sgraph.is_cycle_alt
thf(fact_798_comp__sgraph_Ois__cycle__alt,axiom,
! [S: set_a,Xs: list_a] :
( ( undire2407311113669455967ycle_a @ S @ ( undire2918257014606996450dges_a @ S ) @ Xs )
= ( ( undire6133010728901294956walk_a @ S @ ( undire2918257014606996450dges_a @ S ) @ Xs )
& ( distinct_a @ ( tl_a @ Xs ) )
& ( ord_less_eq_nat @ one_one_nat @ ( undire8849074589633906640ngth_a @ Xs ) )
& ( ( hd_a @ Xs )
= ( last_a @ Xs ) ) ) ) ).
% comp_sgraph.is_cycle_alt
thf(fact_799_ulgraph_Ois__cycle__alt,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,Xs: list_set_a] :
( ( undire6886684016831807756_set_a @ Vertices @ Edges )
=> ( ( undire797940137672299967_set_a @ Vertices @ Edges @ Xs )
= ( ( undire3014741414213135564_set_a @ Vertices @ Edges @ Xs )
& ( distinct_set_a @ ( tl_set_a @ Xs ) )
& ( ord_less_eq_nat @ one_one_nat @ ( undire4424681683220949296_set_a @ Xs ) )
& ( ( hd_set_a @ Xs )
= ( last_set_a @ Xs ) ) ) ) ) ).
% ulgraph.is_cycle_alt
thf(fact_800_ulgraph_Ois__cycle__alt,axiom,
! [Vertices: set_a,Edges: set_set_a,Xs: list_a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( undire2407311113669455967ycle_a @ Vertices @ Edges @ Xs )
= ( ( undire6133010728901294956walk_a @ Vertices @ Edges @ Xs )
& ( distinct_a @ ( tl_a @ Xs ) )
& ( ord_less_eq_nat @ one_one_nat @ ( undire8849074589633906640ngth_a @ Xs ) )
& ( ( hd_a @ Xs )
= ( last_a @ Xs ) ) ) ) ) ).
% ulgraph.is_cycle_alt
thf(fact_801_edge__density__eq0,axiom,
! [A2: set_a,B: set_a,X5: set_a,Y5: set_a] :
( ( ( undire8383842906760478443ween_a @ edges @ A2 @ B )
= bot_bo3357376287454694259od_a_a )
=> ( ( ord_less_eq_set_a @ X5 @ A2 )
=> ( ( ord_less_eq_set_a @ Y5 @ B )
=> ( ( undire297304480579013331sity_a @ edges @ X5 @ Y5 )
= zero_zero_real ) ) ) ) ).
% edge_density_eq0
thf(fact_802_le__zero__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_803_walk__length__app,axiom,
! [Xs: list_a,Ys: list_a] :
( ( Xs != nil_a )
=> ( ( Ys != nil_a )
=> ( ( undire8849074589633906640ngth_a @ ( append_a @ Xs @ Ys ) )
= ( plus_plus_nat @ ( plus_plus_nat @ ( undire8849074589633906640ngth_a @ Xs ) @ ( undire8849074589633906640ngth_a @ Ys ) ) @ one_one_nat ) ) ) ) ).
% walk_length_app
thf(fact_804_walk__length__app__ineq,axiom,
! [Xs: list_a,Ys: list_a] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ ( undire8849074589633906640ngth_a @ Xs ) @ ( undire8849074589633906640ngth_a @ Ys ) ) @ ( undire8849074589633906640ngth_a @ ( append_a @ Xs @ Ys ) ) )
& ( ord_less_eq_nat @ ( undire8849074589633906640ngth_a @ ( append_a @ Xs @ Ys ) ) @ ( plus_plus_nat @ ( plus_plus_nat @ ( undire8849074589633906640ngth_a @ Xs ) @ ( undire8849074589633906640ngth_a @ Ys ) ) @ one_one_nat ) ) ) ).
% walk_length_app_ineq
thf(fact_805_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_806_empty__not__edge,axiom,
~ ( member_set_a @ bot_bot_set_a @ edges ) ).
% empty_not_edge
thf(fact_807_empty__Collect__eq,axiom,
! [P2: product_prod_a_a > $o] :
( ( bot_bo3357376287454694259od_a_a
= ( collec3336397797384452498od_a_a @ P2 ) )
= ( ! [X2: product_prod_a_a] :
~ ( P2 @ X2 ) ) ) ).
% empty_Collect_eq
thf(fact_808_empty__Collect__eq,axiom,
! [P2: a > $o] :
( ( bot_bot_set_a
= ( collect_a @ P2 ) )
= ( ! [X2: a] :
~ ( P2 @ X2 ) ) ) ).
% empty_Collect_eq
thf(fact_809_empty__Collect__eq,axiom,
! [P2: set_a > $o] :
( ( bot_bot_set_set_a
= ( collect_set_a @ P2 ) )
= ( ! [X2: set_a] :
~ ( P2 @ X2 ) ) ) ).
% empty_Collect_eq
thf(fact_810_Collect__empty__eq,axiom,
! [P2: product_prod_a_a > $o] :
( ( ( collec3336397797384452498od_a_a @ P2 )
= bot_bo3357376287454694259od_a_a )
= ( ! [X2: product_prod_a_a] :
~ ( P2 @ X2 ) ) ) ).
% Collect_empty_eq
thf(fact_811_Collect__empty__eq,axiom,
! [P2: a > $o] :
( ( ( collect_a @ P2 )
= bot_bot_set_a )
= ( ! [X2: a] :
~ ( P2 @ X2 ) ) ) ).
% Collect_empty_eq
thf(fact_812_Collect__empty__eq,axiom,
! [P2: set_a > $o] :
( ( ( collect_set_a @ P2 )
= bot_bot_set_set_a )
= ( ! [X2: set_a] :
~ ( P2 @ X2 ) ) ) ).
% Collect_empty_eq
thf(fact_813_all__not__in__conv,axiom,
! [A2: set_Product_prod_a_a] :
( ( ! [X2: product_prod_a_a] :
~ ( member1426531477525435216od_a_a @ X2 @ A2 ) )
= ( A2 = bot_bo3357376287454694259od_a_a ) ) ).
% all_not_in_conv
thf(fact_814_all__not__in__conv,axiom,
! [A2: set_a] :
( ( ! [X2: a] :
~ ( member_a @ X2 @ A2 ) )
= ( A2 = bot_bot_set_a ) ) ).
% all_not_in_conv
thf(fact_815_all__not__in__conv,axiom,
! [A2: set_set_a] :
( ( ! [X2: set_a] :
~ ( member_set_a @ X2 @ A2 ) )
= ( A2 = bot_bot_set_set_a ) ) ).
% all_not_in_conv
thf(fact_816_empty__iff,axiom,
! [C: product_prod_a_a] :
~ ( member1426531477525435216od_a_a @ C @ bot_bo3357376287454694259od_a_a ) ).
% empty_iff
thf(fact_817_empty__iff,axiom,
! [C: a] :
~ ( member_a @ C @ bot_bot_set_a ) ).
% empty_iff
thf(fact_818_empty__iff,axiom,
! [C: set_a] :
~ ( member_set_a @ C @ bot_bot_set_set_a ) ).
% empty_iff
thf(fact_819_add__le__cancel__left,axiom,
! [C: real,A: real,B2: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B2 ) )
= ( ord_less_eq_real @ A @ B2 ) ) ).
% add_le_cancel_left
thf(fact_820_add__le__cancel__left,axiom,
! [C: nat,A: nat,B2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B2 ) )
= ( ord_less_eq_nat @ A @ B2 ) ) ).
% add_le_cancel_left
thf(fact_821_add__le__cancel__right,axiom,
! [A: real,C: real,B2: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B2 @ C ) )
= ( ord_less_eq_real @ A @ B2 ) ) ).
% add_le_cancel_right
thf(fact_822_add__le__cancel__right,axiom,
! [A: nat,C: nat,B2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B2 @ C ) )
= ( ord_less_eq_nat @ A @ B2 ) ) ).
% add_le_cancel_right
thf(fact_823_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_824_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_825_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_826_empty__subsetI,axiom,
! [A2: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A2 ) ).
% empty_subsetI
thf(fact_827_empty__subsetI,axiom,
! [A2: set_set_a] : ( ord_le3724670747650509150_set_a @ bot_bot_set_set_a @ A2 ) ).
% empty_subsetI
thf(fact_828_empty__subsetI,axiom,
! [A2: set_Product_prod_a_a] : ( ord_le746702958409616551od_a_a @ bot_bo3357376287454694259od_a_a @ A2 ) ).
% empty_subsetI
thf(fact_829_add__le__same__cancel1,axiom,
! [B2: real,A: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ B2 @ A ) @ B2 )
= ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% add_le_same_cancel1
thf(fact_830_add__le__same__cancel1,axiom,
! [B2: nat,A: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ B2 @ A ) @ B2 )
= ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).
% add_le_same_cancel1
thf(fact_831_add__le__same__cancel2,axiom,
! [A: real,B2: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ A @ B2 ) @ B2 )
= ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% add_le_same_cancel2
thf(fact_832_add__le__same__cancel2,axiom,
! [A: nat,B2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B2 ) @ B2 )
= ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).
% add_le_same_cancel2
thf(fact_833_le__add__same__cancel1,axiom,
! [A: real,B2: real] :
( ( ord_less_eq_real @ A @ ( plus_plus_real @ A @ B2 ) )
= ( ord_less_eq_real @ zero_zero_real @ B2 ) ) ).
% le_add_same_cancel1
thf(fact_834_le__add__same__cancel1,axiom,
! [A: nat,B2: nat] :
( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ A @ B2 ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B2 ) ) ).
% le_add_same_cancel1
thf(fact_835_le__add__same__cancel2,axiom,
! [A: real,B2: real] :
( ( ord_less_eq_real @ A @ ( plus_plus_real @ B2 @ A ) )
= ( ord_less_eq_real @ zero_zero_real @ B2 ) ) ).
% le_add_same_cancel2
thf(fact_836_le__add__same__cancel2,axiom,
! [A: nat,B2: nat] :
( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ B2 @ A ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B2 ) ) ).
% le_add_same_cancel2
thf(fact_837_double__add__le__zero__iff__single__add__le__zero,axiom,
! [A: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ A @ A ) @ zero_zero_real )
= ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% double_add_le_zero_iff_single_add_le_zero
thf(fact_838_zero__le__double__add__iff__zero__le__single__add,axiom,
! [A: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ A @ A ) )
= ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% zero_le_double_add_iff_zero_le_single_add
thf(fact_839_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_840_set__empty2,axiom,
! [Xs: list_a] :
( ( bot_bot_set_a
= ( set_a2 @ Xs ) )
= ( Xs = nil_a ) ) ).
% set_empty2
thf(fact_841_set__empty2,axiom,
! [Xs: list_set_a] :
( ( bot_bot_set_set_a
= ( set_set_a2 @ Xs ) )
= ( Xs = nil_set_a ) ) ).
% set_empty2
thf(fact_842_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_843_set__empty,axiom,
! [Xs: list_a] :
( ( ( set_a2 @ Xs )
= bot_bot_set_a )
= ( Xs = nil_a ) ) ).
% set_empty
thf(fact_844_set__empty,axiom,
! [Xs: list_set_a] :
( ( ( set_set_a2 @ Xs )
= bot_bot_set_set_a )
= ( Xs = nil_set_a ) ) ).
% set_empty
thf(fact_845_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_846_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_847_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_848_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_849_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_850_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_851_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_852_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_853_add__le__imp__le__right,axiom,
! [A: real,C: real,B2: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B2 @ C ) )
=> ( ord_less_eq_real @ A @ B2 ) ) ).
% add_le_imp_le_right
thf(fact_854_add__le__imp__le__right,axiom,
! [A: nat,C: nat,B2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B2 @ C ) )
=> ( ord_less_eq_nat @ A @ B2 ) ) ).
% add_le_imp_le_right
thf(fact_855_add__le__imp__le__left,axiom,
! [C: real,A: real,B2: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B2 ) )
=> ( ord_less_eq_real @ A @ B2 ) ) ).
% add_le_imp_le_left
thf(fact_856_add__le__imp__le__left,axiom,
! [C: nat,A: nat,B2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B2 ) )
=> ( ord_less_eq_nat @ A @ B2 ) ) ).
% add_le_imp_le_left
thf(fact_857_le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [A5: nat,B5: nat] :
? [C3: nat] :
( B5
= ( plus_plus_nat @ A5 @ C3 ) ) ) ) ).
% le_iff_add
thf(fact_858_add__right__mono,axiom,
! [A: real,B2: real,C: real] :
( ( ord_less_eq_real @ A @ B2 )
=> ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B2 @ C ) ) ) ).
% add_right_mono
thf(fact_859_add__right__mono,axiom,
! [A: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B2 @ C ) ) ) ).
% add_right_mono
thf(fact_860_less__eqE,axiom,
! [A: nat,B2: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ~ ! [C4: nat] :
( B2
!= ( plus_plus_nat @ A @ C4 ) ) ) ).
% less_eqE
thf(fact_861_add__left__mono,axiom,
! [A: real,B2: real,C: real] :
( ( ord_less_eq_real @ A @ B2 )
=> ( ord_less_eq_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B2 ) ) ) ).
% add_left_mono
thf(fact_862_add__left__mono,axiom,
! [A: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B2 ) ) ) ).
% add_left_mono
thf(fact_863_add__mono,axiom,
! [A: real,B2: real,C: real,D: real] :
( ( ord_less_eq_real @ A @ B2 )
=> ( ( ord_less_eq_real @ C @ D )
=> ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B2 @ D ) ) ) ) ).
% add_mono
thf(fact_864_add__mono,axiom,
! [A: nat,B2: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B2 @ D ) ) ) ) ).
% add_mono
thf(fact_865_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I: real,J: real,K: real,L: real] :
( ( ( ord_less_eq_real @ I @ J )
& ( ord_less_eq_real @ K @ L ) )
=> ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_866_add__mono__thms__linordered__semiring_I1_J,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_mono_thms_linordered_semiring(1)
thf(fact_867_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I: real,J: real,K: real,L: real] :
( ( ( I = J )
& ( ord_less_eq_real @ K @ L ) )
=> ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_868_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( I = J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_869_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I: real,J: real,K: real,L: real] :
( ( ( ord_less_eq_real @ I @ J )
& ( K = L ) )
=> ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_870_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( K = L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_871_bot_Oextremum,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A ) ).
% bot.extremum
thf(fact_872_bot_Oextremum,axiom,
! [A: set_set_a] : ( ord_le3724670747650509150_set_a @ bot_bot_set_set_a @ A ) ).
% bot.extremum
thf(fact_873_bot_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ bot_bot_nat @ A ) ).
% bot.extremum
thf(fact_874_bot_Oextremum,axiom,
! [A: set_Product_prod_a_a] : ( ord_le746702958409616551od_a_a @ bot_bo3357376287454694259od_a_a @ A ) ).
% bot.extremum
thf(fact_875_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_876_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_877_bot_Oextremum__unique,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ bot_bot_nat )
= ( A = bot_bot_nat ) ) ).
% bot.extremum_unique
thf(fact_878_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_879_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_880_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_881_bot_Oextremum__uniqueI,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ bot_bot_nat )
=> ( A = bot_bot_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_882_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_883_ex__in__conv,axiom,
! [A2: set_Product_prod_a_a] :
( ( ? [X2: product_prod_a_a] : ( member1426531477525435216od_a_a @ X2 @ A2 ) )
= ( A2 != bot_bo3357376287454694259od_a_a ) ) ).
% ex_in_conv
thf(fact_884_ex__in__conv,axiom,
! [A2: set_a] :
( ( ? [X2: a] : ( member_a @ X2 @ A2 ) )
= ( A2 != bot_bot_set_a ) ) ).
% ex_in_conv
thf(fact_885_ex__in__conv,axiom,
! [A2: set_set_a] :
( ( ? [X2: set_a] : ( member_set_a @ X2 @ A2 ) )
= ( A2 != bot_bot_set_set_a ) ) ).
% ex_in_conv
thf(fact_886_equals0I,axiom,
! [A2: set_Product_prod_a_a] :
( ! [Y2: product_prod_a_a] :
~ ( member1426531477525435216od_a_a @ Y2 @ A2 )
=> ( A2 = bot_bo3357376287454694259od_a_a ) ) ).
% equals0I
thf(fact_887_equals0I,axiom,
! [A2: set_a] :
( ! [Y2: a] :
~ ( member_a @ Y2 @ A2 )
=> ( A2 = bot_bot_set_a ) ) ).
% equals0I
thf(fact_888_equals0I,axiom,
! [A2: set_set_a] :
( ! [Y2: set_a] :
~ ( member_set_a @ Y2 @ A2 )
=> ( A2 = bot_bot_set_set_a ) ) ).
% equals0I
thf(fact_889_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_890_equals0D,axiom,
! [A2: set_a,A: a] :
( ( A2 = bot_bot_set_a )
=> ~ ( member_a @ A @ A2 ) ) ).
% equals0D
thf(fact_891_equals0D,axiom,
! [A2: set_set_a,A: set_a] :
( ( A2 = bot_bot_set_set_a )
=> ~ ( member_set_a @ A @ A2 ) ) ).
% equals0D
thf(fact_892_emptyE,axiom,
! [A: product_prod_a_a] :
~ ( member1426531477525435216od_a_a @ A @ bot_bo3357376287454694259od_a_a ) ).
% emptyE
thf(fact_893_emptyE,axiom,
! [A: a] :
~ ( member_a @ A @ bot_bot_set_a ) ).
% emptyE
thf(fact_894_emptyE,axiom,
! [A: set_a] :
~ ( member_set_a @ A @ bot_bot_set_set_a ) ).
% emptyE
thf(fact_895_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_896_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_897_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_898_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_899_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_900_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_901_add__decreasing,axiom,
! [A: real,C: real,B2: real] :
( ( ord_less_eq_real @ A @ zero_zero_real )
=> ( ( ord_less_eq_real @ C @ B2 )
=> ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ B2 ) ) ) ).
% add_decreasing
thf(fact_902_add__decreasing,axiom,
! [A: nat,C: nat,B2: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ C @ B2 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ B2 ) ) ) ).
% add_decreasing
thf(fact_903_add__increasing,axiom,
! [A: real,B2: real,C: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ B2 @ C )
=> ( ord_less_eq_real @ B2 @ ( plus_plus_real @ A @ C ) ) ) ) ).
% add_increasing
thf(fact_904_add__increasing,axiom,
! [A: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ord_less_eq_nat @ B2 @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_increasing
thf(fact_905_add__decreasing2,axiom,
! [C: real,A: real,B2: real] :
( ( ord_less_eq_real @ C @ zero_zero_real )
=> ( ( ord_less_eq_real @ A @ B2 )
=> ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ B2 ) ) ) ).
% add_decreasing2
thf(fact_906_add__decreasing2,axiom,
! [C: nat,A: nat,B2: nat] :
( ( ord_less_eq_nat @ C @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ A @ B2 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ B2 ) ) ) ).
% add_decreasing2
thf(fact_907_add__increasing2,axiom,
! [C: real,B2: real,A: real] :
( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ( ord_less_eq_real @ B2 @ A )
=> ( ord_less_eq_real @ B2 @ ( plus_plus_real @ A @ C ) ) ) ) ).
% add_increasing2
thf(fact_908_add__increasing2,axiom,
! [C: nat,B2: nat,A: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ( ord_less_eq_nat @ B2 @ A )
=> ( ord_less_eq_nat @ B2 @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_increasing2
thf(fact_909_add__nonneg__nonneg,axiom,
! [A: real,B2: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ B2 )
=> ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ A @ B2 ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_910_add__nonneg__nonneg,axiom,
! [A: nat,B2: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B2 ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_911_add__nonpos__nonpos,axiom,
! [A: real,B2: real] :
( ( ord_less_eq_real @ A @ zero_zero_real )
=> ( ( ord_less_eq_real @ B2 @ zero_zero_real )
=> ( ord_less_eq_real @ ( plus_plus_real @ A @ B2 ) @ zero_zero_real ) ) ) ).
% add_nonpos_nonpos
thf(fact_912_add__nonpos__nonpos,axiom,
! [A: nat,B2: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B2 @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B2 ) @ zero_zero_nat ) ) ) ).
% add_nonpos_nonpos
thf(fact_913_add__nonneg__eq__0__iff,axiom,
! [X3: real,Y: real] :
( ( ord_less_eq_real @ zero_zero_real @ X3 )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ( ( plus_plus_real @ X3 @ Y )
= zero_zero_real )
= ( ( X3 = zero_zero_real )
& ( Y = zero_zero_real ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_914_add__nonneg__eq__0__iff,axiom,
! [X3: nat,Y: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ X3 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
=> ( ( ( plus_plus_nat @ X3 @ Y )
= zero_zero_nat )
= ( ( X3 = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_915_add__nonpos__eq__0__iff,axiom,
! [X3: real,Y: real] :
( ( ord_less_eq_real @ X3 @ zero_zero_real )
=> ( ( ord_less_eq_real @ Y @ zero_zero_real )
=> ( ( ( plus_plus_real @ X3 @ Y )
= zero_zero_real )
= ( ( X3 = zero_zero_real )
& ( Y = zero_zero_real ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_916_add__nonpos__eq__0__iff,axiom,
! [X3: nat,Y: nat] :
( ( ord_less_eq_nat @ X3 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ Y @ zero_zero_nat )
=> ( ( ( plus_plus_nat @ X3 @ Y )
= zero_zero_nat )
= ( ( X3 = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_917_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_918_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_919_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_920_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_921_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_922_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_923_empty__set,axiom,
( bot_bo3357376287454694259od_a_a
= ( set_Product_prod_a_a2 @ nil_Product_prod_a_a ) ) ).
% empty_set
thf(fact_924_empty__set,axiom,
( bot_bot_set_a
= ( set_a2 @ nil_a ) ) ).
% empty_set
thf(fact_925_empty__set,axiom,
( bot_bot_set_set_a
= ( set_set_a2 @ nil_set_a ) ) ).
% empty_set
thf(fact_926_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_927_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_928_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_929_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_930_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_931_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_932_comp__sgraph_Owalk__length__app,axiom,
! [Xs: list_set_a,Ys: list_set_a] :
( ( Xs != nil_set_a )
=> ( ( Ys != nil_set_a )
=> ( ( undire4424681683220949296_set_a @ ( append_set_a @ Xs @ Ys ) )
= ( plus_plus_nat @ ( plus_plus_nat @ ( undire4424681683220949296_set_a @ Xs ) @ ( undire4424681683220949296_set_a @ Ys ) ) @ one_one_nat ) ) ) ) ).
% comp_sgraph.walk_length_app
thf(fact_933_comp__sgraph_Owalk__length__app,axiom,
! [Xs: list_a,Ys: list_a] :
( ( Xs != nil_a )
=> ( ( Ys != nil_a )
=> ( ( undire8849074589633906640ngth_a @ ( append_a @ Xs @ Ys ) )
= ( plus_plus_nat @ ( plus_plus_nat @ ( undire8849074589633906640ngth_a @ Xs ) @ ( undire8849074589633906640ngth_a @ Ys ) ) @ one_one_nat ) ) ) ) ).
% comp_sgraph.walk_length_app
thf(fact_934_comp__sgraph_Owalk__length__app__ineq,axiom,
! [Xs: list_set_a,Ys: list_set_a] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ ( undire4424681683220949296_set_a @ Xs ) @ ( undire4424681683220949296_set_a @ Ys ) ) @ ( undire4424681683220949296_set_a @ ( append_set_a @ Xs @ Ys ) ) )
& ( ord_less_eq_nat @ ( undire4424681683220949296_set_a @ ( append_set_a @ Xs @ Ys ) ) @ ( plus_plus_nat @ ( plus_plus_nat @ ( undire4424681683220949296_set_a @ Xs ) @ ( undire4424681683220949296_set_a @ Ys ) ) @ one_one_nat ) ) ) ).
% comp_sgraph.walk_length_app_ineq
thf(fact_935_comp__sgraph_Owalk__length__app__ineq,axiom,
! [Xs: list_a,Ys: list_a] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ ( undire8849074589633906640ngth_a @ Xs ) @ ( undire8849074589633906640ngth_a @ Ys ) ) @ ( undire8849074589633906640ngth_a @ ( append_a @ Xs @ Ys ) ) )
& ( ord_less_eq_nat @ ( undire8849074589633906640ngth_a @ ( append_a @ Xs @ Ys ) ) @ ( plus_plus_nat @ ( plus_plus_nat @ ( undire8849074589633906640ngth_a @ Xs ) @ ( undire8849074589633906640ngth_a @ Ys ) ) @ one_one_nat ) ) ) ).
% comp_sgraph.walk_length_app_ineq
thf(fact_936_ulgraph_Owalk__length__app,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,Xs: list_set_a,Ys: list_set_a] :
( ( undire6886684016831807756_set_a @ Vertices @ Edges )
=> ( ( Xs != nil_set_a )
=> ( ( Ys != nil_set_a )
=> ( ( undire4424681683220949296_set_a @ ( append_set_a @ Xs @ Ys ) )
= ( plus_plus_nat @ ( plus_plus_nat @ ( undire4424681683220949296_set_a @ Xs ) @ ( undire4424681683220949296_set_a @ Ys ) ) @ one_one_nat ) ) ) ) ) ).
% ulgraph.walk_length_app
thf(fact_937_ulgraph_Owalk__length__app,axiom,
! [Vertices: set_a,Edges: set_set_a,Xs: list_a,Ys: list_a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( Xs != nil_a )
=> ( ( Ys != nil_a )
=> ( ( undire8849074589633906640ngth_a @ ( append_a @ Xs @ Ys ) )
= ( plus_plus_nat @ ( plus_plus_nat @ ( undire8849074589633906640ngth_a @ Xs ) @ ( undire8849074589633906640ngth_a @ Ys ) ) @ one_one_nat ) ) ) ) ) ).
% ulgraph.walk_length_app
thf(fact_938_ulgraph_Owalk__length__app__ineq,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,Xs: list_set_a,Ys: list_set_a] :
( ( undire6886684016831807756_set_a @ Vertices @ Edges )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( undire4424681683220949296_set_a @ Xs ) @ ( undire4424681683220949296_set_a @ Ys ) ) @ ( undire4424681683220949296_set_a @ ( append_set_a @ Xs @ Ys ) ) )
& ( ord_less_eq_nat @ ( undire4424681683220949296_set_a @ ( append_set_a @ Xs @ Ys ) ) @ ( plus_plus_nat @ ( plus_plus_nat @ ( undire4424681683220949296_set_a @ Xs ) @ ( undire4424681683220949296_set_a @ Ys ) ) @ one_one_nat ) ) ) ) ).
% ulgraph.walk_length_app_ineq
thf(fact_939_ulgraph_Owalk__length__app__ineq,axiom,
! [Vertices: set_a,Edges: set_set_a,Xs: list_a,Ys: list_a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( undire8849074589633906640ngth_a @ Xs ) @ ( undire8849074589633906640ngth_a @ Ys ) ) @ ( undire8849074589633906640ngth_a @ ( append_a @ Xs @ Ys ) ) )
& ( ord_less_eq_nat @ ( undire8849074589633906640ngth_a @ ( append_a @ Xs @ Ys ) ) @ ( plus_plus_nat @ ( plus_plus_nat @ ( undire8849074589633906640ngth_a @ Xs ) @ ( undire8849074589633906640ngth_a @ Ys ) ) @ one_one_nat ) ) ) ) ).
% ulgraph.walk_length_app_ineq
thf(fact_940_comp__sgraph_Oedge__density__eq0,axiom,
! [S: set_set_a,A2: set_set_a,B: set_set_a,X5: set_set_a,Y5: set_set_a] :
( ( ( undire2462398226299384907_set_a @ ( undire8247866692393712962_set_a @ S ) @ A2 @ B )
= bot_bo5799363139946352499_set_a )
=> ( ( ord_le3724670747650509150_set_a @ X5 @ A2 )
=> ( ( ord_le3724670747650509150_set_a @ Y5 @ B )
=> ( ( undire8927637694342045747_set_a @ ( undire8247866692393712962_set_a @ S ) @ X5 @ Y5 )
= zero_zero_real ) ) ) ) ).
% comp_sgraph.edge_density_eq0
thf(fact_941_comp__sgraph_Oedge__density__eq0,axiom,
! [S: set_Product_prod_a_a,A2: set_Product_prod_a_a,B: 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 @ B )
= bot_bo510284599550014259od_a_a )
=> ( ( ord_le746702958409616551od_a_a @ X5 @ A2 )
=> ( ( ord_le746702958409616551od_a_a @ Y5 @ B )
=> ( ( undire8410861505230878716od_a_a @ ( undire6879232364018543115od_a_a @ S ) @ X5 @ Y5 )
= zero_zero_real ) ) ) ) ).
% comp_sgraph.edge_density_eq0
thf(fact_942_comp__sgraph_Oedge__density__eq0,axiom,
! [S: set_a,A2: set_a,B: set_a,X5: set_a,Y5: set_a] :
( ( ( undire8383842906760478443ween_a @ ( undire2918257014606996450dges_a @ S ) @ A2 @ B )
= bot_bo3357376287454694259od_a_a )
=> ( ( ord_less_eq_set_a @ X5 @ A2 )
=> ( ( ord_less_eq_set_a @ Y5 @ B )
=> ( ( undire297304480579013331sity_a @ ( undire2918257014606996450dges_a @ S ) @ X5 @ Y5 )
= zero_zero_real ) ) ) ) ).
% comp_sgraph.edge_density_eq0
thf(fact_943_ulgraph_Oedge__density__eq0,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,A2: set_set_a,B: set_set_a,X5: set_set_a,Y5: set_set_a] :
( ( undire6886684016831807756_set_a @ Vertices @ Edges )
=> ( ( ( undire2462398226299384907_set_a @ Edges @ A2 @ B )
= bot_bo5799363139946352499_set_a )
=> ( ( ord_le3724670747650509150_set_a @ X5 @ A2 )
=> ( ( ord_le3724670747650509150_set_a @ Y5 @ B )
=> ( ( undire8927637694342045747_set_a @ Edges @ X5 @ Y5 )
= zero_zero_real ) ) ) ) ) ).
% ulgraph.edge_density_eq0
thf(fact_944_ulgraph_Oedge__density__eq0,axiom,
! [Vertices: set_Product_prod_a_a,Edges: set_se5735800977113168103od_a_a,A2: set_Product_prod_a_a,B: 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 @ B )
= bot_bo510284599550014259od_a_a )
=> ( ( ord_le746702958409616551od_a_a @ X5 @ A2 )
=> ( ( ord_le746702958409616551od_a_a @ Y5 @ B )
=> ( ( undire8410861505230878716od_a_a @ Edges @ X5 @ Y5 )
= zero_zero_real ) ) ) ) ) ).
% ulgraph.edge_density_eq0
thf(fact_945_ulgraph_Oedge__density__eq0,axiom,
! [Vertices: set_a,Edges: set_set_a,A2: set_a,B: set_a,X5: set_a,Y5: set_a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( ( undire8383842906760478443ween_a @ Edges @ A2 @ B )
= bot_bo3357376287454694259od_a_a )
=> ( ( ord_less_eq_set_a @ X5 @ A2 )
=> ( ( ord_less_eq_set_a @ Y5 @ B )
=> ( ( undire297304480579013331sity_a @ Edges @ X5 @ Y5 )
= zero_zero_real ) ) ) ) ) ).
% ulgraph.edge_density_eq0
thf(fact_946_zero__le,axiom,
! [X3: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X3 ) ).
% zero_le
thf(fact_947_iso__vertex__empty__neighborhood,axiom,
! [V: a] :
( ( undire8931668460104145173rtex_a @ vertices @ edges @ V )
=> ( ( undire8504279938402040014hood_a @ vertices @ edges @ V )
= bot_bot_set_a ) ) ).
% iso_vertex_empty_neighborhood
thf(fact_948_is__gen__path__path,axiom,
! [P: list_a] :
( ( undire3562951555376170320path_a @ vertices @ edges @ P )
=> ( ( ord_less_nat @ zero_zero_nat @ ( undire8849074589633906640ngth_a @ P ) )
=> ( ~ ( undire2407311113669455967ycle_a @ vertices @ edges @ P )
=> ( undire427332500224447920path_a @ vertices @ edges @ P ) ) ) ) ).
% is_gen_path_path
thf(fact_949_is__isolated__vertex__degree0,axiom,
! [V: a] :
( ( undire8931668460104145173rtex_a @ vertices @ edges @ V )
=> ( ( undire8867928226783802224gree_a @ edges @ V )
= zero_zero_nat ) ) ).
% is_isolated_vertex_degree0
thf(fact_950_walk__edges__app,axiom,
! [Xs: list_a,Y: a,X3: a] :
( ( undire7337870655677353998dges_a @ ( append_a @ Xs @ ( cons_a @ Y @ ( cons_a @ X3 @ nil_a ) ) ) )
= ( append_set_a @ ( undire7337870655677353998dges_a @ ( append_a @ Xs @ ( cons_a @ Y @ nil_a ) ) ) @ ( cons_set_a @ ( insert_a @ Y @ ( insert_a @ X3 @ bot_bot_set_a ) ) @ nil_set_a ) ) ) ).
% walk_edges_app
thf(fact_951_walk__edges__singleton__app,axiom,
! [Ys: list_a,X3: a] :
( ( Ys != nil_a )
=> ( ( undire7337870655677353998dges_a @ ( append_a @ ( cons_a @ X3 @ nil_a ) @ Ys ) )
= ( cons_set_a @ ( insert_a @ X3 @ ( insert_a @ ( hd_a @ Ys ) @ bot_bot_set_a ) ) @ ( undire7337870655677353998dges_a @ Ys ) ) ) ) ).
% walk_edges_singleton_app
thf(fact_952_insertCI,axiom,
! [A: set_a,B: set_set_a,B2: set_a] :
( ( ~ ( member_set_a @ A @ B )
=> ( A = B2 ) )
=> ( member_set_a @ A @ ( insert_set_a @ B2 @ B ) ) ) ).
% insertCI
thf(fact_953_insertCI,axiom,
! [A: a,B: set_a,B2: a] :
( ( ~ ( member_a @ A @ B )
=> ( A = B2 ) )
=> ( member_a @ A @ ( insert_a @ B2 @ B ) ) ) ).
% insertCI
thf(fact_954_insertCI,axiom,
! [A: product_prod_a_a,B: set_Product_prod_a_a,B2: product_prod_a_a] :
( ( ~ ( member1426531477525435216od_a_a @ A @ B )
=> ( A = B2 ) )
=> ( member1426531477525435216od_a_a @ A @ ( insert4534936382041156343od_a_a @ B2 @ B ) ) ) ).
% insertCI
thf(fact_955_insert__iff,axiom,
! [A: set_a,B2: set_a,A2: set_set_a] :
( ( member_set_a @ A @ ( insert_set_a @ B2 @ A2 ) )
= ( ( A = B2 )
| ( member_set_a @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_956_insert__iff,axiom,
! [A: a,B2: a,A2: set_a] :
( ( member_a @ A @ ( insert_a @ B2 @ A2 ) )
= ( ( A = B2 )
| ( member_a @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_957_insert__iff,axiom,
! [A: product_prod_a_a,B2: product_prod_a_a,A2: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ A @ ( insert4534936382041156343od_a_a @ B2 @ A2 ) )
= ( ( A = B2 )
| ( member1426531477525435216od_a_a @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_958_insert__absorb2,axiom,
! [X3: a,A2: set_a] :
( ( insert_a @ X3 @ ( insert_a @ X3 @ A2 ) )
= ( insert_a @ X3 @ A2 ) ) ).
% insert_absorb2
thf(fact_959_insert__absorb2,axiom,
! [X3: set_a,A2: set_set_a] :
( ( insert_set_a @ X3 @ ( insert_set_a @ X3 @ A2 ) )
= ( insert_set_a @ X3 @ A2 ) ) ).
% insert_absorb2
thf(fact_960_not__vert__adj,axiom,
! [V: a,U: a] :
( ~ ( undire397441198561214472_adj_a @ edges @ V @ U )
=> ~ ( member_set_a @ ( insert_a @ V @ ( insert_a @ U @ bot_bot_set_a ) ) @ edges ) ) ).
% not_vert_adj
thf(fact_961_vert__adj__def,axiom,
! [V1: a,V2: a] :
( ( undire397441198561214472_adj_a @ edges @ V1 @ V2 )
= ( member_set_a @ ( insert_a @ V1 @ ( insert_a @ V2 @ bot_bot_set_a ) ) @ edges ) ) ).
% vert_adj_def
thf(fact_962_has__loop__def,axiom,
! [V: a] :
( ( undire3617971648856834880loop_a @ edges @ V )
= ( member_set_a @ ( insert_a @ V @ bot_bot_set_a ) @ edges ) ) ).
% has_loop_def
thf(fact_963_wellformed__alt__fst,axiom,
! [X3: a,Y: a] :
( ( member_set_a @ ( insert_a @ X3 @ ( insert_a @ Y @ bot_bot_set_a ) ) @ edges )
=> ( member_a @ X3 @ vertices ) ) ).
% wellformed_alt_fst
thf(fact_964_wellformed__alt__snd,axiom,
! [X3: a,Y: a] :
( ( member_set_a @ ( insert_a @ X3 @ ( insert_a @ Y @ bot_bot_set_a ) ) @ edges )
=> ( member_a @ Y @ vertices ) ) ).
% wellformed_alt_snd
thf(fact_965_is__edge__between__def,axiom,
( undire8544646567961481629ween_a
= ( ^ [X6: set_a,Y6: set_a,E5: set_a] :
? [X2: a,Y3: a] :
( ( E5
= ( insert_a @ X2 @ ( insert_a @ Y3 @ bot_bot_set_a ) ) )
& ( member_a @ X2 @ X6 )
& ( member_a @ Y3 @ Y6 ) ) ) ) ).
% is_edge_between_def
thf(fact_966_walk__edges_Osimps_I3_J,axiom,
! [X3: a,Y: a,Ys: list_a] :
( ( undire7337870655677353998dges_a @ ( cons_a @ X3 @ ( cons_a @ Y @ Ys ) ) )
= ( cons_set_a @ ( insert_a @ X3 @ ( insert_a @ Y @ bot_bot_set_a ) ) @ ( undire7337870655677353998dges_a @ ( cons_a @ Y @ Ys ) ) ) ) ).
% walk_edges.simps(3)
thf(fact_967_vert__adj__inc__edge__iff,axiom,
! [V1: a,V2: a] :
( ( undire397441198561214472_adj_a @ edges @ V1 @ V2 )
= ( ( undire1521409233611534436dent_a @ V1 @ ( insert_a @ V1 @ ( insert_a @ V2 @ bot_bot_set_a ) ) )
& ( undire1521409233611534436dent_a @ V2 @ ( insert_a @ V1 @ ( insert_a @ V2 @ bot_bot_set_a ) ) )
& ( member_set_a @ ( insert_a @ V1 @ ( insert_a @ V2 @ bot_bot_set_a ) ) @ edges ) ) ) ).
% vert_adj_inc_edge_iff
thf(fact_968_is__walk__hd__tl,axiom,
! [Y: a,Ys: list_a,X3: a] :
( ( undire6133010728901294956walk_a @ vertices @ edges @ ( cons_a @ Y @ Ys ) )
=> ( ( member_set_a @ ( insert_a @ X3 @ ( insert_a @ Y @ bot_bot_set_a ) ) @ edges )
=> ( undire6133010728901294956walk_a @ vertices @ edges @ ( cons_a @ X3 @ ( cons_a @ Y @ Ys ) ) ) ) ) ).
% is_walk_hd_tl
thf(fact_969_singletonI,axiom,
! [A: product_prod_a_a] : ( member1426531477525435216od_a_a @ A @ ( insert4534936382041156343od_a_a @ A @ bot_bo3357376287454694259od_a_a ) ) ).
% singletonI
thf(fact_970_singletonI,axiom,
! [A: a] : ( member_a @ A @ ( insert_a @ A @ bot_bot_set_a ) ) ).
% singletonI
thf(fact_971_singletonI,axiom,
! [A: set_a] : ( member_set_a @ A @ ( insert_set_a @ A @ bot_bot_set_set_a ) ) ).
% singletonI
thf(fact_972_insert__subset,axiom,
! [X3: a,A2: set_a,B: set_a] :
( ( ord_less_eq_set_a @ ( insert_a @ X3 @ A2 ) @ B )
= ( ( member_a @ X3 @ B )
& ( ord_less_eq_set_a @ A2 @ B ) ) ) ).
% insert_subset
thf(fact_973_insert__subset,axiom,
! [X3: set_a,A2: set_set_a,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ ( insert_set_a @ X3 @ A2 ) @ B )
= ( ( member_set_a @ X3 @ B )
& ( ord_le3724670747650509150_set_a @ A2 @ B ) ) ) ).
% insert_subset
thf(fact_974_insert__subset,axiom,
! [X3: product_prod_a_a,A2: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ ( insert4534936382041156343od_a_a @ X3 @ A2 ) @ B )
= ( ( member1426531477525435216od_a_a @ X3 @ B )
& ( ord_le746702958409616551od_a_a @ A2 @ B ) ) ) ).
% insert_subset
thf(fact_975_walk__edges_Oelims,axiom,
! [X3: list_a,Y: list_set_a] :
( ( ( undire7337870655677353998dges_a @ X3 )
= Y )
=> ( ( ( X3 = nil_a )
=> ( Y != nil_set_a ) )
=> ( ( ? [X4: a] :
( X3
= ( cons_a @ X4 @ nil_a ) )
=> ( Y != nil_set_a ) )
=> ~ ! [X4: a,Y2: a,Ys2: list_a] :
( ( X3
= ( cons_a @ X4 @ ( cons_a @ Y2 @ Ys2 ) ) )
=> ( Y
!= ( cons_set_a @ ( insert_a @ X4 @ ( insert_a @ Y2 @ bot_bot_set_a ) ) @ ( undire7337870655677353998dges_a @ ( cons_a @ Y2 @ Ys2 ) ) ) ) ) ) ) ) ).
% walk_edges.elims
thf(fact_976_singleton__insert__inj__eq,axiom,
! [B2: a,A: a,A2: set_a] :
( ( ( insert_a @ B2 @ bot_bot_set_a )
= ( insert_a @ A @ A2 ) )
= ( ( A = B2 )
& ( ord_less_eq_set_a @ A2 @ ( insert_a @ B2 @ bot_bot_set_a ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_977_singleton__insert__inj__eq,axiom,
! [B2: set_a,A: set_a,A2: set_set_a] :
( ( ( insert_set_a @ B2 @ bot_bot_set_set_a )
= ( insert_set_a @ A @ A2 ) )
= ( ( A = B2 )
& ( ord_le3724670747650509150_set_a @ A2 @ ( insert_set_a @ B2 @ bot_bot_set_set_a ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_978_singleton__insert__inj__eq,axiom,
! [B2: product_prod_a_a,A: product_prod_a_a,A2: set_Product_prod_a_a] :
( ( ( insert4534936382041156343od_a_a @ B2 @ bot_bo3357376287454694259od_a_a )
= ( insert4534936382041156343od_a_a @ A @ A2 ) )
= ( ( A = B2 )
& ( ord_le746702958409616551od_a_a @ A2 @ ( insert4534936382041156343od_a_a @ B2 @ bot_bo3357376287454694259od_a_a ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_979_singleton__insert__inj__eq_H,axiom,
! [A: a,A2: set_a,B2: a] :
( ( ( insert_a @ A @ A2 )
= ( insert_a @ B2 @ bot_bot_set_a ) )
= ( ( A = B2 )
& ( ord_less_eq_set_a @ A2 @ ( insert_a @ B2 @ bot_bot_set_a ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_980_singleton__insert__inj__eq_H,axiom,
! [A: set_a,A2: set_set_a,B2: set_a] :
( ( ( insert_set_a @ A @ A2 )
= ( insert_set_a @ B2 @ bot_bot_set_set_a ) )
= ( ( A = B2 )
& ( ord_le3724670747650509150_set_a @ A2 @ ( insert_set_a @ B2 @ bot_bot_set_set_a ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_981_singleton__insert__inj__eq_H,axiom,
! [A: product_prod_a_a,A2: set_Product_prod_a_a,B2: product_prod_a_a] :
( ( ( insert4534936382041156343od_a_a @ A @ A2 )
= ( insert4534936382041156343od_a_a @ B2 @ bot_bo3357376287454694259od_a_a ) )
= ( ( A = B2 )
& ( ord_le746702958409616551od_a_a @ A2 @ ( insert4534936382041156343od_a_a @ B2 @ bot_bo3357376287454694259od_a_a ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_982_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_983_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_984_comp__sgraph_Odegree__none,axiom,
! [V: set_a,S: set_set_a] :
( ~ ( member_set_a @ V @ S )
=> ( ( undire8939077443744732368_set_a @ ( undire8247866692393712962_set_a @ S ) @ V )
= zero_zero_nat ) ) ).
% comp_sgraph.degree_none
thf(fact_985_comp__sgraph_Odegree__none,axiom,
! [V: product_prod_a_a,S: set_Product_prod_a_a] :
( ~ ( member1426531477525435216od_a_a @ V @ S )
=> ( ( undire1436394852029823897od_a_a @ ( undire6879232364018543115od_a_a @ S ) @ V )
= zero_zero_nat ) ) ).
% comp_sgraph.degree_none
thf(fact_986_comp__sgraph_Odegree__none,axiom,
! [V: a,S: set_a] :
( ~ ( member_a @ V @ S )
=> ( ( undire8867928226783802224gree_a @ ( undire2918257014606996450dges_a @ S ) @ V )
= zero_zero_nat ) ) ).
% comp_sgraph.degree_none
thf(fact_987_the__elem__eq,axiom,
! [X3: product_prod_a_a] :
( ( the_el8589169208993665564od_a_a @ ( insert4534936382041156343od_a_a @ X3 @ bot_bo3357376287454694259od_a_a ) )
= X3 ) ).
% the_elem_eq
thf(fact_988_the__elem__eq,axiom,
! [X3: a] :
( ( the_elem_a @ ( insert_a @ X3 @ bot_bot_set_a ) )
= X3 ) ).
% the_elem_eq
thf(fact_989_the__elem__eq,axiom,
! [X3: set_a] :
( ( the_elem_set_a @ ( insert_set_a @ X3 @ bot_bot_set_set_a ) )
= X3 ) ).
% the_elem_eq
thf(fact_990_degree__none,axiom,
! [V: a] :
( ~ ( member_a @ V @ vertices )
=> ( ( undire8867928226783802224gree_a @ edges @ V )
= zero_zero_nat ) ) ).
% degree_none
thf(fact_991_singleton__inject,axiom,
! [A: product_prod_a_a,B2: product_prod_a_a] :
( ( ( insert4534936382041156343od_a_a @ A @ bot_bo3357376287454694259od_a_a )
= ( insert4534936382041156343od_a_a @ B2 @ bot_bo3357376287454694259od_a_a ) )
=> ( A = B2 ) ) ).
% singleton_inject
thf(fact_992_singleton__inject,axiom,
! [A: a,B2: a] :
( ( ( insert_a @ A @ bot_bot_set_a )
= ( insert_a @ B2 @ bot_bot_set_a ) )
=> ( A = B2 ) ) ).
% singleton_inject
thf(fact_993_singleton__inject,axiom,
! [A: set_a,B2: set_a] :
( ( ( insert_set_a @ A @ bot_bot_set_set_a )
= ( insert_set_a @ B2 @ bot_bot_set_set_a ) )
=> ( A = B2 ) ) ).
% singleton_inject
thf(fact_994_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_995_insert__not__empty,axiom,
! [A: a,A2: set_a] :
( ( insert_a @ A @ A2 )
!= bot_bot_set_a ) ).
% insert_not_empty
thf(fact_996_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_997_doubleton__eq__iff,axiom,
! [A: product_prod_a_a,B2: product_prod_a_a,C: product_prod_a_a,D: product_prod_a_a] :
( ( ( insert4534936382041156343od_a_a @ A @ ( insert4534936382041156343od_a_a @ B2 @ bot_bo3357376287454694259od_a_a ) )
= ( insert4534936382041156343od_a_a @ C @ ( insert4534936382041156343od_a_a @ D @ bot_bo3357376287454694259od_a_a ) ) )
= ( ( ( A = C )
& ( B2 = D ) )
| ( ( A = D )
& ( B2 = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_998_doubleton__eq__iff,axiom,
! [A: a,B2: a,C: a,D: a] :
( ( ( insert_a @ A @ ( insert_a @ B2 @ bot_bot_set_a ) )
= ( insert_a @ C @ ( insert_a @ D @ bot_bot_set_a ) ) )
= ( ( ( A = C )
& ( B2 = D ) )
| ( ( A = D )
& ( B2 = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_999_doubleton__eq__iff,axiom,
! [A: set_a,B2: set_a,C: set_a,D: set_a] :
( ( ( insert_set_a @ A @ ( insert_set_a @ B2 @ bot_bot_set_set_a ) )
= ( insert_set_a @ C @ ( insert_set_a @ D @ bot_bot_set_set_a ) ) )
= ( ( ( A = C )
& ( B2 = D ) )
| ( ( A = D )
& ( B2 = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_1000_singleton__iff,axiom,
! [B2: product_prod_a_a,A: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ B2 @ ( insert4534936382041156343od_a_a @ A @ bot_bo3357376287454694259od_a_a ) )
= ( B2 = A ) ) ).
% singleton_iff
thf(fact_1001_singleton__iff,axiom,
! [B2: a,A: a] :
( ( member_a @ B2 @ ( insert_a @ A @ bot_bot_set_a ) )
= ( B2 = A ) ) ).
% singleton_iff
thf(fact_1002_singleton__iff,axiom,
! [B2: set_a,A: set_a] :
( ( member_set_a @ B2 @ ( insert_set_a @ A @ bot_bot_set_set_a ) )
= ( B2 = A ) ) ).
% singleton_iff
thf(fact_1003_bot__set__def,axiom,
( bot_bo3357376287454694259od_a_a
= ( collec3336397797384452498od_a_a @ bot_bo4160289986317612842_a_a_o ) ) ).
% bot_set_def
thf(fact_1004_bot__set__def,axiom,
( bot_bot_set_a
= ( collect_a @ bot_bot_a_o ) ) ).
% bot_set_def
thf(fact_1005_bot__set__def,axiom,
( bot_bot_set_set_a
= ( collect_set_a @ bot_bot_set_a_o ) ) ).
% bot_set_def
thf(fact_1006_singletonD,axiom,
! [B2: product_prod_a_a,A: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ B2 @ ( insert4534936382041156343od_a_a @ A @ bot_bo3357376287454694259od_a_a ) )
=> ( B2 = A ) ) ).
% singletonD
thf(fact_1007_singletonD,axiom,
! [B2: a,A: a] :
( ( member_a @ B2 @ ( insert_a @ A @ bot_bot_set_a ) )
=> ( B2 = A ) ) ).
% singletonD
thf(fact_1008_singletonD,axiom,
! [B2: set_a,A: set_a] :
( ( member_set_a @ B2 @ ( insert_set_a @ A @ bot_bot_set_set_a ) )
=> ( B2 = A ) ) ).
% singletonD
thf(fact_1009_bot_Onot__eq__extremum,axiom,
! [A: set_Product_prod_a_a] :
( ( A != bot_bo3357376287454694259od_a_a )
= ( ord_le6819997720685908915od_a_a @ bot_bo3357376287454694259od_a_a @ A ) ) ).
% bot.not_eq_extremum
thf(fact_1010_bot_Onot__eq__extremum,axiom,
! [A: set_a] :
( ( A != bot_bot_set_a )
= ( ord_less_set_a @ bot_bot_set_a @ A ) ) ).
% bot.not_eq_extremum
thf(fact_1011_bot_Onot__eq__extremum,axiom,
! [A: set_set_a] :
( ( A != bot_bot_set_set_a )
= ( ord_less_set_set_a @ bot_bot_set_set_a @ A ) ) ).
% bot.not_eq_extremum
thf(fact_1012_bot_Onot__eq__extremum,axiom,
! [A: nat] :
( ( A != bot_bot_nat )
= ( ord_less_nat @ bot_bot_nat @ A ) ) ).
% bot.not_eq_extremum
thf(fact_1013_bot_Oextremum__strict,axiom,
! [A: set_Product_prod_a_a] :
~ ( ord_le6819997720685908915od_a_a @ A @ bot_bo3357376287454694259od_a_a ) ).
% bot.extremum_strict
thf(fact_1014_bot_Oextremum__strict,axiom,
! [A: set_a] :
~ ( ord_less_set_a @ A @ bot_bot_set_a ) ).
% bot.extremum_strict
thf(fact_1015_bot_Oextremum__strict,axiom,
! [A: set_set_a] :
~ ( ord_less_set_set_a @ A @ bot_bot_set_set_a ) ).
% bot.extremum_strict
thf(fact_1016_bot_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ bot_bot_nat ) ).
% bot.extremum_strict
thf(fact_1017_order__le__imp__less__or__eq,axiom,
! [X3: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y )
=> ( ( ord_less_set_a @ X3 @ Y )
| ( X3 = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_1018_order__le__imp__less__or__eq,axiom,
! [X3: set_set_a,Y: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X3 @ Y )
=> ( ( ord_less_set_set_a @ X3 @ Y )
| ( X3 = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_1019_order__le__imp__less__or__eq,axiom,
! [X3: real,Y: real] :
( ( ord_less_eq_real @ X3 @ Y )
=> ( ( ord_less_real @ X3 @ Y )
| ( X3 = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_1020_order__le__imp__less__or__eq,axiom,
! [X3: nat,Y: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
=> ( ( ord_less_nat @ X3 @ Y )
| ( X3 = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_1021_order__le__imp__less__or__eq,axiom,
! [X3: set_Product_prod_a_a,Y: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ X3 @ Y )
=> ( ( ord_le6819997720685908915od_a_a @ X3 @ Y )
| ( X3 = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_1022_linorder__le__less__linear,axiom,
! [X3: real,Y: real] :
( ( ord_less_eq_real @ X3 @ Y )
| ( ord_less_real @ Y @ X3 ) ) ).
% linorder_le_less_linear
thf(fact_1023_linorder__le__less__linear,axiom,
! [X3: nat,Y: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
| ( ord_less_nat @ Y @ X3 ) ) ).
% linorder_le_less_linear
thf(fact_1024_order__less__le__subst2,axiom,
! [A: nat,B2: nat,F: nat > set_a,C: set_a] :
( ( ord_less_nat @ A @ B2 )
=> ( ( ord_less_eq_set_a @ ( F @ B2 ) @ C )
=> ( ! [X4: nat,Y2: nat] :
( ( ord_less_nat @ X4 @ Y2 )
=> ( ord_less_set_a @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_1025_order__less__le__subst2,axiom,
! [A: nat,B2: nat,F: nat > set_set_a,C: set_set_a] :
( ( ord_less_nat @ A @ B2 )
=> ( ( ord_le3724670747650509150_set_a @ ( F @ B2 ) @ C )
=> ( ! [X4: nat,Y2: nat] :
( ( ord_less_nat @ X4 @ Y2 )
=> ( ord_less_set_set_a @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_1026_order__less__le__subst2,axiom,
! [A: nat,B2: nat,F: nat > real,C: real] :
( ( ord_less_nat @ A @ B2 )
=> ( ( ord_less_eq_real @ ( F @ B2 ) @ C )
=> ( ! [X4: nat,Y2: nat] :
( ( ord_less_nat @ X4 @ Y2 )
=> ( ord_less_real @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_1027_order__less__le__subst2,axiom,
! [A: nat,B2: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C )
=> ( ! [X4: nat,Y2: nat] :
( ( ord_less_nat @ X4 @ Y2 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_1028_order__less__le__subst2,axiom,
! [A: nat,B2: nat,F: nat > set_Product_prod_a_a,C: set_Product_prod_a_a] :
( ( ord_less_nat @ A @ B2 )
=> ( ( ord_le746702958409616551od_a_a @ ( F @ B2 ) @ C )
=> ( ! [X4: nat,Y2: nat] :
( ( ord_less_nat @ X4 @ Y2 )
=> ( ord_le6819997720685908915od_a_a @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_le6819997720685908915od_a_a @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_1029_order__less__le__subst1,axiom,
! [A: real,F: real > real,B2: real,C: real] :
( ( ord_less_real @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_real @ B2 @ C )
=> ( ! [X4: real,Y2: real] :
( ( ord_less_eq_real @ X4 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_1030_order__less__le__subst1,axiom,
! [A: nat,F: real > nat,B2: real,C: real] :
( ( ord_less_nat @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_real @ B2 @ C )
=> ( ! [X4: real,Y2: real] :
( ( ord_less_eq_real @ X4 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_1031_order__less__le__subst1,axiom,
! [A: real,F: nat > real,B2: nat,C: nat] :
( ( ord_less_real @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X4: nat,Y2: nat] :
( ( ord_less_eq_nat @ X4 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_1032_order__less__le__subst1,axiom,
! [A: nat,F: nat > nat,B2: nat,C: nat] :
( ( ord_less_nat @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X4: nat,Y2: nat] :
( ( ord_less_eq_nat @ X4 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_1033_order__less__le__subst1,axiom,
! [A: real,F: set_a > real,B2: set_a,C: set_a] :
( ( ord_less_real @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_set_a @ B2 @ C )
=> ( ! [X4: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_1034_order__less__le__subst1,axiom,
! [A: nat,F: set_a > nat,B2: set_a,C: set_a] :
( ( ord_less_nat @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_set_a @ B2 @ C )
=> ( ! [X4: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_1035_order__less__le__subst1,axiom,
! [A: set_a,F: real > set_a,B2: real,C: real] :
( ( ord_less_set_a @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_real @ B2 @ C )
=> ( ! [X4: real,Y2: real] :
( ( ord_less_eq_real @ X4 @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_1036_order__less__le__subst1,axiom,
! [A: set_a,F: nat > set_a,B2: nat,C: nat] :
( ( ord_less_set_a @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X4: nat,Y2: nat] :
( ( ord_less_eq_nat @ X4 @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_1037_order__less__le__subst1,axiom,
! [A: set_a,F: set_a > set_a,B2: set_a,C: set_a] :
( ( ord_less_set_a @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_set_a @ B2 @ C )
=> ( ! [X4: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_1038_order__less__le__subst1,axiom,
! [A: real,F: set_set_a > real,B2: set_set_a,C: set_set_a] :
( ( ord_less_real @ A @ ( F @ B2 ) )
=> ( ( ord_le3724670747650509150_set_a @ B2 @ C )
=> ( ! [X4: set_set_a,Y2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X4 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_1039_order__le__less__subst2,axiom,
! [A: real,B2: real,F: real > real,C: real] :
( ( ord_less_eq_real @ A @ B2 )
=> ( ( ord_less_real @ ( F @ B2 ) @ C )
=> ( ! [X4: real,Y2: real] :
( ( ord_less_eq_real @ X4 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_1040_order__le__less__subst2,axiom,
! [A: real,B2: real,F: real > nat,C: nat] :
( ( ord_less_eq_real @ A @ B2 )
=> ( ( ord_less_nat @ ( F @ B2 ) @ C )
=> ( ! [X4: real,Y2: real] :
( ( ord_less_eq_real @ X4 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_1041_order__le__less__subst2,axiom,
! [A: nat,B2: nat,F: nat > real,C: real] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ord_less_real @ ( F @ B2 ) @ C )
=> ( ! [X4: nat,Y2: nat] :
( ( ord_less_eq_nat @ X4 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_1042_order__le__less__subst2,axiom,
! [A: nat,B2: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ord_less_nat @ ( F @ B2 ) @ C )
=> ( ! [X4: nat,Y2: nat] :
( ( ord_less_eq_nat @ X4 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_1043_order__le__less__subst2,axiom,
! [A: set_a,B2: set_a,F: set_a > real,C: real] :
( ( ord_less_eq_set_a @ A @ B2 )
=> ( ( ord_less_real @ ( F @ B2 ) @ C )
=> ( ! [X4: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_1044_order__le__less__subst2,axiom,
! [A: set_a,B2: set_a,F: set_a > nat,C: nat] :
( ( ord_less_eq_set_a @ A @ B2 )
=> ( ( ord_less_nat @ ( F @ B2 ) @ C )
=> ( ! [X4: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_1045_order__le__less__subst2,axiom,
! [A: real,B2: real,F: real > set_a,C: set_a] :
( ( ord_less_eq_real @ A @ B2 )
=> ( ( ord_less_set_a @ ( F @ B2 ) @ C )
=> ( ! [X4: real,Y2: real] :
( ( ord_less_eq_real @ X4 @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_1046_order__le__less__subst2,axiom,
! [A: nat,B2: nat,F: nat > set_a,C: set_a] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ord_less_set_a @ ( F @ B2 ) @ C )
=> ( ! [X4: nat,Y2: nat] :
( ( ord_less_eq_nat @ X4 @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_1047_order__le__less__subst2,axiom,
! [A: set_a,B2: set_a,F: set_a > set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ B2 )
=> ( ( ord_less_set_a @ ( F @ B2 ) @ C )
=> ( ! [X4: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_1048_order__le__less__subst2,axiom,
! [A: set_set_a,B2: set_set_a,F: set_set_a > real,C: real] :
( ( ord_le3724670747650509150_set_a @ A @ B2 )
=> ( ( ord_less_real @ ( F @ B2 ) @ C )
=> ( ! [X4: set_set_a,Y2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X4 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_1049_order__le__less__subst1,axiom,
! [A: set_a,F: nat > set_a,B2: nat,C: nat] :
( ( ord_less_eq_set_a @ A @ ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ! [X4: nat,Y2: nat] :
( ( ord_less_nat @ X4 @ Y2 )
=> ( ord_less_set_a @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_1050_order__le__less__subst1,axiom,
! [A: set_set_a,F: nat > set_set_a,B2: nat,C: nat] :
( ( ord_le3724670747650509150_set_a @ A @ ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ! [X4: nat,Y2: nat] :
( ( ord_less_nat @ X4 @ Y2 )
=> ( ord_less_set_set_a @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_1051_order__le__less__subst1,axiom,
! [A: real,F: nat > real,B2: nat,C: nat] :
( ( ord_less_eq_real @ A @ ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ! [X4: nat,Y2: nat] :
( ( ord_less_nat @ X4 @ Y2 )
=> ( ord_less_real @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_1052_order__le__less__subst1,axiom,
! [A: nat,F: nat > nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ! [X4: nat,Y2: nat] :
( ( ord_less_nat @ X4 @ Y2 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_1053_order__le__less__subst1,axiom,
! [A: set_Product_prod_a_a,F: nat > set_Product_prod_a_a,B2: nat,C: nat] :
( ( ord_le746702958409616551od_a_a @ A @ ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ! [X4: nat,Y2: nat] :
( ( ord_less_nat @ X4 @ Y2 )
=> ( ord_le6819997720685908915od_a_a @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_le6819997720685908915od_a_a @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_1054_order__less__le__trans,axiom,
! [X3: set_a,Y: set_a,Z3: set_a] :
( ( ord_less_set_a @ X3 @ Y )
=> ( ( ord_less_eq_set_a @ Y @ Z3 )
=> ( ord_less_set_a @ X3 @ Z3 ) ) ) ).
% order_less_le_trans
thf(fact_1055_order__less__le__trans,axiom,
! [X3: set_set_a,Y: set_set_a,Z3: set_set_a] :
( ( ord_less_set_set_a @ X3 @ Y )
=> ( ( ord_le3724670747650509150_set_a @ Y @ Z3 )
=> ( ord_less_set_set_a @ X3 @ Z3 ) ) ) ).
% order_less_le_trans
thf(fact_1056_order__less__le__trans,axiom,
! [X3: real,Y: real,Z3: real] :
( ( ord_less_real @ X3 @ Y )
=> ( ( ord_less_eq_real @ Y @ Z3 )
=> ( ord_less_real @ X3 @ Z3 ) ) ) ).
% order_less_le_trans
thf(fact_1057_order__less__le__trans,axiom,
! [X3: nat,Y: nat,Z3: nat] :
( ( ord_less_nat @ X3 @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z3 )
=> ( ord_less_nat @ X3 @ Z3 ) ) ) ).
% order_less_le_trans
thf(fact_1058_order__less__le__trans,axiom,
! [X3: set_Product_prod_a_a,Y: set_Product_prod_a_a,Z3: set_Product_prod_a_a] :
( ( ord_le6819997720685908915od_a_a @ X3 @ Y )
=> ( ( ord_le746702958409616551od_a_a @ Y @ Z3 )
=> ( ord_le6819997720685908915od_a_a @ X3 @ Z3 ) ) ) ).
% order_less_le_trans
thf(fact_1059_order__le__less__trans,axiom,
! [X3: set_a,Y: set_a,Z3: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y )
=> ( ( ord_less_set_a @ Y @ Z3 )
=> ( ord_less_set_a @ X3 @ Z3 ) ) ) ).
% order_le_less_trans
thf(fact_1060_order__le__less__trans,axiom,
! [X3: set_set_a,Y: set_set_a,Z3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X3 @ Y )
=> ( ( ord_less_set_set_a @ Y @ Z3 )
=> ( ord_less_set_set_a @ X3 @ Z3 ) ) ) ).
% order_le_less_trans
thf(fact_1061_order__le__less__trans,axiom,
! [X3: real,Y: real,Z3: real] :
( ( ord_less_eq_real @ X3 @ Y )
=> ( ( ord_less_real @ Y @ Z3 )
=> ( ord_less_real @ X3 @ Z3 ) ) ) ).
% order_le_less_trans
thf(fact_1062_order__le__less__trans,axiom,
! [X3: nat,Y: nat,Z3: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
=> ( ( ord_less_nat @ Y @ Z3 )
=> ( ord_less_nat @ X3 @ Z3 ) ) ) ).
% order_le_less_trans
thf(fact_1063_order__le__less__trans,axiom,
! [X3: set_Product_prod_a_a,Y: set_Product_prod_a_a,Z3: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ X3 @ Y )
=> ( ( ord_le6819997720685908915od_a_a @ Y @ Z3 )
=> ( ord_le6819997720685908915od_a_a @ X3 @ Z3 ) ) ) ).
% order_le_less_trans
thf(fact_1064_order__neq__le__trans,axiom,
! [A: set_a,B2: set_a] :
( ( A != B2 )
=> ( ( ord_less_eq_set_a @ A @ B2 )
=> ( ord_less_set_a @ A @ B2 ) ) ) ).
% order_neq_le_trans
thf(fact_1065_order__neq__le__trans,axiom,
! [A: set_set_a,B2: set_set_a] :
( ( A != B2 )
=> ( ( ord_le3724670747650509150_set_a @ A @ B2 )
=> ( ord_less_set_set_a @ A @ B2 ) ) ) ).
% order_neq_le_trans
thf(fact_1066_order__neq__le__trans,axiom,
! [A: real,B2: real] :
( ( A != B2 )
=> ( ( ord_less_eq_real @ A @ B2 )
=> ( ord_less_real @ A @ B2 ) ) ) ).
% order_neq_le_trans
thf(fact_1067_order__neq__le__trans,axiom,
! [A: nat,B2: nat] :
( ( A != B2 )
=> ( ( ord_less_eq_nat @ A @ B2 )
=> ( ord_less_nat @ A @ B2 ) ) ) ).
% order_neq_le_trans
thf(fact_1068_order__neq__le__trans,axiom,
! [A: set_Product_prod_a_a,B2: set_Product_prod_a_a] :
( ( A != B2 )
=> ( ( ord_le746702958409616551od_a_a @ A @ B2 )
=> ( ord_le6819997720685908915od_a_a @ A @ B2 ) ) ) ).
% order_neq_le_trans
thf(fact_1069_order__le__neq__trans,axiom,
! [A: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A @ B2 )
=> ( ( A != B2 )
=> ( ord_less_set_a @ A @ B2 ) ) ) ).
% order_le_neq_trans
thf(fact_1070_order__le__neq__trans,axiom,
! [A: set_set_a,B2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ B2 )
=> ( ( A != B2 )
=> ( ord_less_set_set_a @ A @ B2 ) ) ) ).
% order_le_neq_trans
thf(fact_1071_order__le__neq__trans,axiom,
! [A: real,B2: real] :
( ( ord_less_eq_real @ A @ B2 )
=> ( ( A != B2 )
=> ( ord_less_real @ A @ B2 ) ) ) ).
% order_le_neq_trans
thf(fact_1072_order__le__neq__trans,axiom,
! [A: nat,B2: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( A != B2 )
=> ( ord_less_nat @ A @ B2 ) ) ) ).
% order_le_neq_trans
thf(fact_1073_order__le__neq__trans,axiom,
! [A: set_Product_prod_a_a,B2: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A @ B2 )
=> ( ( A != B2 )
=> ( ord_le6819997720685908915od_a_a @ A @ B2 ) ) ) ).
% order_le_neq_trans
thf(fact_1074_order__less__imp__le,axiom,
! [X3: set_a,Y: set_a] :
( ( ord_less_set_a @ X3 @ Y )
=> ( ord_less_eq_set_a @ X3 @ Y ) ) ).
% order_less_imp_le
thf(fact_1075_order__less__imp__le,axiom,
! [X3: set_set_a,Y: set_set_a] :
( ( ord_less_set_set_a @ X3 @ Y )
=> ( ord_le3724670747650509150_set_a @ X3 @ Y ) ) ).
% order_less_imp_le
thf(fact_1076_order__less__imp__le,axiom,
! [X3: real,Y: real] :
( ( ord_less_real @ X3 @ Y )
=> ( ord_less_eq_real @ X3 @ Y ) ) ).
% order_less_imp_le
thf(fact_1077_order__less__imp__le,axiom,
! [X3: nat,Y: nat] :
( ( ord_less_nat @ X3 @ Y )
=> ( ord_less_eq_nat @ X3 @ Y ) ) ).
% order_less_imp_le
thf(fact_1078_order__less__imp__le,axiom,
! [X3: set_Product_prod_a_a,Y: set_Product_prod_a_a] :
( ( ord_le6819997720685908915od_a_a @ X3 @ Y )
=> ( ord_le746702958409616551od_a_a @ X3 @ Y ) ) ).
% order_less_imp_le
thf(fact_1079_linorder__not__less,axiom,
! [X3: real,Y: real] :
( ( ~ ( ord_less_real @ X3 @ Y ) )
= ( ord_less_eq_real @ Y @ X3 ) ) ).
% linorder_not_less
thf(fact_1080_linorder__not__less,axiom,
! [X3: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X3 @ Y ) )
= ( ord_less_eq_nat @ Y @ X3 ) ) ).
% linorder_not_less
thf(fact_1081_linorder__not__le,axiom,
! [X3: real,Y: real] :
( ( ~ ( ord_less_eq_real @ X3 @ Y ) )
= ( ord_less_real @ Y @ X3 ) ) ).
% linorder_not_le
thf(fact_1082_linorder__not__le,axiom,
! [X3: nat,Y: nat] :
( ( ~ ( ord_less_eq_nat @ X3 @ Y ) )
= ( ord_less_nat @ Y @ X3 ) ) ).
% linorder_not_le
thf(fact_1083_order__less__le,axiom,
( ord_less_set_a
= ( ^ [X2: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y3 )
& ( X2 != Y3 ) ) ) ) ).
% order_less_le
thf(fact_1084_order__less__le,axiom,
( ord_less_set_set_a
= ( ^ [X2: set_set_a,Y3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X2 @ Y3 )
& ( X2 != Y3 ) ) ) ) ).
% order_less_le
thf(fact_1085_order__less__le,axiom,
( ord_less_real
= ( ^ [X2: real,Y3: real] :
( ( ord_less_eq_real @ X2 @ Y3 )
& ( X2 != Y3 ) ) ) ) ).
% order_less_le
thf(fact_1086_order__less__le,axiom,
( ord_less_nat
= ( ^ [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
& ( X2 != Y3 ) ) ) ) ).
% order_less_le
thf(fact_1087_order__less__le,axiom,
( ord_le6819997720685908915od_a_a
= ( ^ [X2: set_Product_prod_a_a,Y3: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ X2 @ Y3 )
& ( X2 != Y3 ) ) ) ) ).
% order_less_le
thf(fact_1088_order__le__less,axiom,
( ord_less_eq_set_a
= ( ^ [X2: set_a,Y3: set_a] :
( ( ord_less_set_a @ X2 @ Y3 )
| ( X2 = Y3 ) ) ) ) ).
% order_le_less
thf(fact_1089_order__le__less,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [X2: set_set_a,Y3: set_set_a] :
( ( ord_less_set_set_a @ X2 @ Y3 )
| ( X2 = Y3 ) ) ) ) ).
% order_le_less
thf(fact_1090_order__le__less,axiom,
( ord_less_eq_real
= ( ^ [X2: real,Y3: real] :
( ( ord_less_real @ X2 @ Y3 )
| ( X2 = Y3 ) ) ) ) ).
% order_le_less
thf(fact_1091_order__le__less,axiom,
( ord_less_eq_nat
= ( ^ [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
| ( X2 = Y3 ) ) ) ) ).
% order_le_less
thf(fact_1092_order__le__less,axiom,
( ord_le746702958409616551od_a_a
= ( ^ [X2: set_Product_prod_a_a,Y3: set_Product_prod_a_a] :
( ( ord_le6819997720685908915od_a_a @ X2 @ Y3 )
| ( X2 = Y3 ) ) ) ) ).
% order_le_less
thf(fact_1093_dual__order_Ostrict__implies__order,axiom,
! [B2: set_a,A: set_a] :
( ( ord_less_set_a @ B2 @ A )
=> ( ord_less_eq_set_a @ B2 @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_1094_dual__order_Ostrict__implies__order,axiom,
! [B2: set_set_a,A: set_set_a] :
( ( ord_less_set_set_a @ B2 @ A )
=> ( ord_le3724670747650509150_set_a @ B2 @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_1095_dual__order_Ostrict__implies__order,axiom,
! [B2: real,A: real] :
( ( ord_less_real @ B2 @ A )
=> ( ord_less_eq_real @ B2 @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_1096_dual__order_Ostrict__implies__order,axiom,
! [B2: nat,A: nat] :
( ( ord_less_nat @ B2 @ A )
=> ( ord_less_eq_nat @ B2 @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_1097_dual__order_Ostrict__implies__order,axiom,
! [B2: set_Product_prod_a_a,A: set_Product_prod_a_a] :
( ( ord_le6819997720685908915od_a_a @ B2 @ A )
=> ( ord_le746702958409616551od_a_a @ B2 @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_1098_order_Ostrict__implies__order,axiom,
! [A: set_a,B2: set_a] :
( ( ord_less_set_a @ A @ B2 )
=> ( ord_less_eq_set_a @ A @ B2 ) ) ).
% order.strict_implies_order
thf(fact_1099_order_Ostrict__implies__order,axiom,
! [A: set_set_a,B2: set_set_a] :
( ( ord_less_set_set_a @ A @ B2 )
=> ( ord_le3724670747650509150_set_a @ A @ B2 ) ) ).
% order.strict_implies_order
thf(fact_1100_order_Ostrict__implies__order,axiom,
! [A: real,B2: real] :
( ( ord_less_real @ A @ B2 )
=> ( ord_less_eq_real @ A @ B2 ) ) ).
% order.strict_implies_order
thf(fact_1101_order_Ostrict__implies__order,axiom,
! [A: nat,B2: nat] :
( ( ord_less_nat @ A @ B2 )
=> ( ord_less_eq_nat @ A @ B2 ) ) ).
% order.strict_implies_order
thf(fact_1102_order_Ostrict__implies__order,axiom,
! [A: set_Product_prod_a_a,B2: set_Product_prod_a_a] :
( ( ord_le6819997720685908915od_a_a @ A @ B2 )
=> ( ord_le746702958409616551od_a_a @ A @ B2 ) ) ).
% order.strict_implies_order
thf(fact_1103_dual__order_Ostrict__iff__not,axiom,
( ord_less_set_a
= ( ^ [B5: set_a,A5: set_a] :
( ( ord_less_eq_set_a @ B5 @ A5 )
& ~ ( ord_less_eq_set_a @ A5 @ B5 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_1104_dual__order_Ostrict__iff__not,axiom,
( ord_less_set_set_a
= ( ^ [B5: set_set_a,A5: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B5 @ A5 )
& ~ ( ord_le3724670747650509150_set_a @ A5 @ B5 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_1105_dual__order_Ostrict__iff__not,axiom,
( ord_less_real
= ( ^ [B5: real,A5: real] :
( ( ord_less_eq_real @ B5 @ A5 )
& ~ ( ord_less_eq_real @ A5 @ B5 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_1106_dual__order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [B5: nat,A5: nat] :
( ( ord_less_eq_nat @ B5 @ A5 )
& ~ ( ord_less_eq_nat @ A5 @ B5 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_1107_dual__order_Ostrict__iff__not,axiom,
( ord_le6819997720685908915od_a_a
= ( ^ [B5: set_Product_prod_a_a,A5: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ B5 @ A5 )
& ~ ( ord_le746702958409616551od_a_a @ A5 @ B5 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_1108_dual__order_Ostrict__trans2,axiom,
! [B2: set_a,A: set_a,C: set_a] :
( ( ord_less_set_a @ B2 @ A )
=> ( ( ord_less_eq_set_a @ C @ B2 )
=> ( ord_less_set_a @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_1109_dual__order_Ostrict__trans2,axiom,
! [B2: set_set_a,A: set_set_a,C: set_set_a] :
( ( ord_less_set_set_a @ B2 @ A )
=> ( ( ord_le3724670747650509150_set_a @ C @ B2 )
=> ( ord_less_set_set_a @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_1110_dual__order_Ostrict__trans2,axiom,
! [B2: real,A: real,C: real] :
( ( ord_less_real @ B2 @ A )
=> ( ( ord_less_eq_real @ C @ B2 )
=> ( ord_less_real @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_1111_dual__order_Ostrict__trans2,axiom,
! [B2: nat,A: nat,C: nat] :
( ( ord_less_nat @ B2 @ A )
=> ( ( ord_less_eq_nat @ C @ B2 )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_1112_dual__order_Ostrict__trans2,axiom,
! [B2: set_Product_prod_a_a,A: set_Product_prod_a_a,C: set_Product_prod_a_a] :
( ( ord_le6819997720685908915od_a_a @ B2 @ A )
=> ( ( ord_le746702958409616551od_a_a @ C @ B2 )
=> ( ord_le6819997720685908915od_a_a @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_1113_dual__order_Ostrict__trans1,axiom,
! [B2: set_a,A: set_a,C: set_a] :
( ( ord_less_eq_set_a @ B2 @ A )
=> ( ( ord_less_set_a @ C @ B2 )
=> ( ord_less_set_a @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_1114_dual__order_Ostrict__trans1,axiom,
! [B2: set_set_a,A: set_set_a,C: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B2 @ A )
=> ( ( ord_less_set_set_a @ C @ B2 )
=> ( ord_less_set_set_a @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_1115_dual__order_Ostrict__trans1,axiom,
! [B2: real,A: real,C: real] :
( ( ord_less_eq_real @ B2 @ A )
=> ( ( ord_less_real @ C @ B2 )
=> ( ord_less_real @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_1116_dual__order_Ostrict__trans1,axiom,
! [B2: nat,A: nat,C: nat] :
( ( ord_less_eq_nat @ B2 @ A )
=> ( ( ord_less_nat @ C @ B2 )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_1117_dual__order_Ostrict__trans1,axiom,
! [B2: set_Product_prod_a_a,A: set_Product_prod_a_a,C: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ B2 @ A )
=> ( ( ord_le6819997720685908915od_a_a @ C @ B2 )
=> ( ord_le6819997720685908915od_a_a @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_1118_dual__order_Ostrict__iff__order,axiom,
( ord_less_set_a
= ( ^ [B5: set_a,A5: set_a] :
( ( ord_less_eq_set_a @ B5 @ A5 )
& ( A5 != B5 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_1119_dual__order_Ostrict__iff__order,axiom,
( ord_less_set_set_a
= ( ^ [B5: set_set_a,A5: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B5 @ A5 )
& ( A5 != B5 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_1120_dual__order_Ostrict__iff__order,axiom,
( ord_less_real
= ( ^ [B5: real,A5: real] :
( ( ord_less_eq_real @ B5 @ A5 )
& ( A5 != B5 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_1121_dual__order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [B5: nat,A5: nat] :
( ( ord_less_eq_nat @ B5 @ A5 )
& ( A5 != B5 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_1122_dual__order_Ostrict__iff__order,axiom,
( ord_le6819997720685908915od_a_a
= ( ^ [B5: set_Product_prod_a_a,A5: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ B5 @ A5 )
& ( A5 != B5 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_1123_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_set_a
= ( ^ [B5: set_a,A5: set_a] :
( ( ord_less_set_a @ B5 @ A5 )
| ( A5 = B5 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_1124_dual__order_Oorder__iff__strict,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [B5: set_set_a,A5: set_set_a] :
( ( ord_less_set_set_a @ B5 @ A5 )
| ( A5 = B5 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_1125_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_real
= ( ^ [B5: real,A5: real] :
( ( ord_less_real @ B5 @ A5 )
| ( A5 = B5 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_1126_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [B5: nat,A5: nat] :
( ( ord_less_nat @ B5 @ A5 )
| ( A5 = B5 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_1127_dual__order_Oorder__iff__strict,axiom,
( ord_le746702958409616551od_a_a
= ( ^ [B5: set_Product_prod_a_a,A5: set_Product_prod_a_a] :
( ( ord_le6819997720685908915od_a_a @ B5 @ A5 )
| ( A5 = B5 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_1128_dense__le__bounded,axiom,
! [X3: real,Y: real,Z3: real] :
( ( ord_less_real @ X3 @ Y )
=> ( ! [W: real] :
( ( ord_less_real @ X3 @ W )
=> ( ( ord_less_real @ W @ Y )
=> ( ord_less_eq_real @ W @ Z3 ) ) )
=> ( ord_less_eq_real @ Y @ Z3 ) ) ) ).
% dense_le_bounded
thf(fact_1129_dense__ge__bounded,axiom,
! [Z3: real,X3: real,Y: real] :
( ( ord_less_real @ Z3 @ X3 )
=> ( ! [W: real] :
( ( ord_less_real @ Z3 @ W )
=> ( ( ord_less_real @ W @ X3 )
=> ( ord_less_eq_real @ Y @ W ) ) )
=> ( ord_less_eq_real @ Y @ Z3 ) ) ) ).
% dense_ge_bounded
thf(fact_1130_order_Ostrict__iff__not,axiom,
( ord_less_set_a
= ( ^ [A5: set_a,B5: set_a] :
( ( ord_less_eq_set_a @ A5 @ B5 )
& ~ ( ord_less_eq_set_a @ B5 @ A5 ) ) ) ) ).
% order.strict_iff_not
thf(fact_1131_order_Ostrict__iff__not,axiom,
( ord_less_set_set_a
= ( ^ [A5: set_set_a,B5: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A5 @ B5 )
& ~ ( ord_le3724670747650509150_set_a @ B5 @ A5 ) ) ) ) ).
% order.strict_iff_not
thf(fact_1132_order_Ostrict__iff__not,axiom,
( ord_less_real
= ( ^ [A5: real,B5: real] :
( ( ord_less_eq_real @ A5 @ B5 )
& ~ ( ord_less_eq_real @ B5 @ A5 ) ) ) ) ).
% order.strict_iff_not
thf(fact_1133_order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [A5: nat,B5: nat] :
( ( ord_less_eq_nat @ A5 @ B5 )
& ~ ( ord_less_eq_nat @ B5 @ A5 ) ) ) ) ).
% order.strict_iff_not
thf(fact_1134_order_Ostrict__iff__not,axiom,
( ord_le6819997720685908915od_a_a
= ( ^ [A5: set_Product_prod_a_a,B5: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A5 @ B5 )
& ~ ( ord_le746702958409616551od_a_a @ B5 @ A5 ) ) ) ) ).
% order.strict_iff_not
thf(fact_1135_order_Ostrict__trans2,axiom,
! [A: set_a,B2: set_a,C: set_a] :
( ( ord_less_set_a @ A @ B2 )
=> ( ( ord_less_eq_set_a @ B2 @ C )
=> ( ord_less_set_a @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_1136_order_Ostrict__trans2,axiom,
! [A: set_set_a,B2: set_set_a,C: set_set_a] :
( ( ord_less_set_set_a @ A @ B2 )
=> ( ( ord_le3724670747650509150_set_a @ B2 @ C )
=> ( ord_less_set_set_a @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_1137_order_Ostrict__trans2,axiom,
! [A: real,B2: real,C: real] :
( ( ord_less_real @ A @ B2 )
=> ( ( ord_less_eq_real @ B2 @ C )
=> ( ord_less_real @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_1138_order_Ostrict__trans2,axiom,
! [A: nat,B2: nat,C: nat] :
( ( ord_less_nat @ A @ B2 )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_1139_order_Ostrict__trans2,axiom,
! [A: set_Product_prod_a_a,B2: set_Product_prod_a_a,C: set_Product_prod_a_a] :
( ( ord_le6819997720685908915od_a_a @ A @ B2 )
=> ( ( ord_le746702958409616551od_a_a @ B2 @ C )
=> ( ord_le6819997720685908915od_a_a @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_1140_order_Ostrict__trans1,axiom,
! [A: set_a,B2: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ B2 )
=> ( ( ord_less_set_a @ B2 @ C )
=> ( ord_less_set_a @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_1141_order_Ostrict__trans1,axiom,
! [A: set_set_a,B2: set_set_a,C: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ B2 )
=> ( ( ord_less_set_set_a @ B2 @ C )
=> ( ord_less_set_set_a @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_1142_order_Ostrict__trans1,axiom,
! [A: real,B2: real,C: real] :
( ( ord_less_eq_real @ A @ B2 )
=> ( ( ord_less_real @ B2 @ C )
=> ( ord_less_real @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_1143_order_Ostrict__trans1,axiom,
! [A: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_1144_order_Ostrict__trans1,axiom,
! [A: set_Product_prod_a_a,B2: set_Product_prod_a_a,C: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A @ B2 )
=> ( ( ord_le6819997720685908915od_a_a @ B2 @ C )
=> ( ord_le6819997720685908915od_a_a @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_1145_order_Ostrict__iff__order,axiom,
( ord_less_set_a
= ( ^ [A5: set_a,B5: set_a] :
( ( ord_less_eq_set_a @ A5 @ B5 )
& ( A5 != B5 ) ) ) ) ).
% order.strict_iff_order
thf(fact_1146_order_Ostrict__iff__order,axiom,
( ord_less_set_set_a
= ( ^ [A5: set_set_a,B5: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A5 @ B5 )
& ( A5 != B5 ) ) ) ) ).
% order.strict_iff_order
thf(fact_1147_order_Ostrict__iff__order,axiom,
( ord_less_real
= ( ^ [A5: real,B5: real] :
( ( ord_less_eq_real @ A5 @ B5 )
& ( A5 != B5 ) ) ) ) ).
% order.strict_iff_order
thf(fact_1148_order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [A5: nat,B5: nat] :
( ( ord_less_eq_nat @ A5 @ B5 )
& ( A5 != B5 ) ) ) ) ).
% order.strict_iff_order
thf(fact_1149_order_Ostrict__iff__order,axiom,
( ord_le6819997720685908915od_a_a
= ( ^ [A5: set_Product_prod_a_a,B5: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A5 @ B5 )
& ( A5 != B5 ) ) ) ) ).
% order.strict_iff_order
thf(fact_1150_order_Oorder__iff__strict,axiom,
( ord_less_eq_set_a
= ( ^ [A5: set_a,B5: set_a] :
( ( ord_less_set_a @ A5 @ B5 )
| ( A5 = B5 ) ) ) ) ).
% order.order_iff_strict
thf(fact_1151_order_Oorder__iff__strict,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [A5: set_set_a,B5: set_set_a] :
( ( ord_less_set_set_a @ A5 @ B5 )
| ( A5 = B5 ) ) ) ) ).
% order.order_iff_strict
thf(fact_1152_order_Oorder__iff__strict,axiom,
( ord_less_eq_real
= ( ^ [A5: real,B5: real] :
( ( ord_less_real @ A5 @ B5 )
| ( A5 = B5 ) ) ) ) ).
% order.order_iff_strict
thf(fact_1153_order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [A5: nat,B5: nat] :
( ( ord_less_nat @ A5 @ B5 )
| ( A5 = B5 ) ) ) ) ).
% order.order_iff_strict
thf(fact_1154_order_Oorder__iff__strict,axiom,
( ord_le746702958409616551od_a_a
= ( ^ [A5: set_Product_prod_a_a,B5: set_Product_prod_a_a] :
( ( ord_le6819997720685908915od_a_a @ A5 @ B5 )
| ( A5 = B5 ) ) ) ) ).
% order.order_iff_strict
thf(fact_1155_not__le__imp__less,axiom,
! [Y: real,X3: real] :
( ~ ( ord_less_eq_real @ Y @ X3 )
=> ( ord_less_real @ X3 @ Y ) ) ).
% not_le_imp_less
thf(fact_1156_not__le__imp__less,axiom,
! [Y: nat,X3: nat] :
( ~ ( ord_less_eq_nat @ Y @ X3 )
=> ( ord_less_nat @ X3 @ Y ) ) ).
% not_le_imp_less
thf(fact_1157_less__le__not__le,axiom,
( ord_less_set_a
= ( ^ [X2: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y3 )
& ~ ( ord_less_eq_set_a @ Y3 @ X2 ) ) ) ) ).
% less_le_not_le
thf(fact_1158_less__le__not__le,axiom,
( ord_less_set_set_a
= ( ^ [X2: set_set_a,Y3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X2 @ Y3 )
& ~ ( ord_le3724670747650509150_set_a @ Y3 @ X2 ) ) ) ) ).
% less_le_not_le
thf(fact_1159_less__le__not__le,axiom,
( ord_less_real
= ( ^ [X2: real,Y3: real] :
( ( ord_less_eq_real @ X2 @ Y3 )
& ~ ( ord_less_eq_real @ Y3 @ X2 ) ) ) ) ).
% less_le_not_le
thf(fact_1160_less__le__not__le,axiom,
( ord_less_nat
= ( ^ [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
& ~ ( ord_less_eq_nat @ Y3 @ X2 ) ) ) ) ).
% less_le_not_le
thf(fact_1161_less__le__not__le,axiom,
( ord_le6819997720685908915od_a_a
= ( ^ [X2: set_Product_prod_a_a,Y3: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ X2 @ Y3 )
& ~ ( ord_le746702958409616551od_a_a @ Y3 @ X2 ) ) ) ) ).
% less_le_not_le
thf(fact_1162_dense__le,axiom,
! [Y: real,Z3: real] :
( ! [X4: real] :
( ( ord_less_real @ X4 @ Y )
=> ( ord_less_eq_real @ X4 @ Z3 ) )
=> ( ord_less_eq_real @ Y @ Z3 ) ) ).
% dense_le
thf(fact_1163_dense__ge,axiom,
! [Z3: real,Y: real] :
( ! [X4: real] :
( ( ord_less_real @ Z3 @ X4 )
=> ( ord_less_eq_real @ Y @ X4 ) )
=> ( ord_less_eq_real @ Y @ Z3 ) ) ).
% dense_ge
thf(fact_1164_antisym__conv2,axiom,
! [X3: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y )
=> ( ( ~ ( ord_less_set_a @ X3 @ Y ) )
= ( X3 = Y ) ) ) ).
% antisym_conv2
thf(fact_1165_antisym__conv2,axiom,
! [X3: set_set_a,Y: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X3 @ Y )
=> ( ( ~ ( ord_less_set_set_a @ X3 @ Y ) )
= ( X3 = Y ) ) ) ).
% antisym_conv2
thf(fact_1166_antisym__conv2,axiom,
! [X3: real,Y: real] :
( ( ord_less_eq_real @ X3 @ Y )
=> ( ( ~ ( ord_less_real @ X3 @ Y ) )
= ( X3 = Y ) ) ) ).
% antisym_conv2
thf(fact_1167_antisym__conv2,axiom,
! [X3: nat,Y: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
=> ( ( ~ ( ord_less_nat @ X3 @ Y ) )
= ( X3 = Y ) ) ) ).
% antisym_conv2
thf(fact_1168_antisym__conv2,axiom,
! [X3: set_Product_prod_a_a,Y: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ X3 @ Y )
=> ( ( ~ ( ord_le6819997720685908915od_a_a @ X3 @ Y ) )
= ( X3 = Y ) ) ) ).
% antisym_conv2
thf(fact_1169_antisym__conv1,axiom,
! [X3: set_a,Y: set_a] :
( ~ ( ord_less_set_a @ X3 @ Y )
=> ( ( ord_less_eq_set_a @ X3 @ Y )
= ( X3 = Y ) ) ) ).
% antisym_conv1
thf(fact_1170_antisym__conv1,axiom,
! [X3: set_set_a,Y: set_set_a] :
( ~ ( ord_less_set_set_a @ X3 @ Y )
=> ( ( ord_le3724670747650509150_set_a @ X3 @ Y )
= ( X3 = Y ) ) ) ).
% antisym_conv1
thf(fact_1171_antisym__conv1,axiom,
! [X3: real,Y: real] :
( ~ ( ord_less_real @ X3 @ Y )
=> ( ( ord_less_eq_real @ X3 @ Y )
= ( X3 = Y ) ) ) ).
% antisym_conv1
thf(fact_1172_antisym__conv1,axiom,
! [X3: nat,Y: nat] :
( ~ ( ord_less_nat @ X3 @ Y )
=> ( ( ord_less_eq_nat @ X3 @ Y )
= ( X3 = Y ) ) ) ).
% antisym_conv1
thf(fact_1173_antisym__conv1,axiom,
! [X3: set_Product_prod_a_a,Y: set_Product_prod_a_a] :
( ~ ( ord_le6819997720685908915od_a_a @ X3 @ Y )
=> ( ( ord_le746702958409616551od_a_a @ X3 @ Y )
= ( X3 = Y ) ) ) ).
% antisym_conv1
thf(fact_1174_nless__le,axiom,
! [A: set_a,B2: set_a] :
( ( ~ ( ord_less_set_a @ A @ B2 ) )
= ( ~ ( ord_less_eq_set_a @ A @ B2 )
| ( A = B2 ) ) ) ).
% nless_le
thf(fact_1175_nless__le,axiom,
! [A: set_set_a,B2: set_set_a] :
( ( ~ ( ord_less_set_set_a @ A @ B2 ) )
= ( ~ ( ord_le3724670747650509150_set_a @ A @ B2 )
| ( A = B2 ) ) ) ).
% nless_le
thf(fact_1176_nless__le,axiom,
! [A: real,B2: real] :
( ( ~ ( ord_less_real @ A @ B2 ) )
= ( ~ ( ord_less_eq_real @ A @ B2 )
| ( A = B2 ) ) ) ).
% nless_le
thf(fact_1177_nless__le,axiom,
! [A: nat,B2: nat] :
( ( ~ ( ord_less_nat @ A @ B2 ) )
= ( ~ ( ord_less_eq_nat @ A @ B2 )
| ( A = B2 ) ) ) ).
% nless_le
thf(fact_1178_nless__le,axiom,
! [A: set_Product_prod_a_a,B2: set_Product_prod_a_a] :
( ( ~ ( ord_le6819997720685908915od_a_a @ A @ B2 ) )
= ( ~ ( ord_le746702958409616551od_a_a @ A @ B2 )
| ( A = B2 ) ) ) ).
% nless_le
thf(fact_1179_leI,axiom,
! [X3: real,Y: real] :
( ~ ( ord_less_real @ X3 @ Y )
=> ( ord_less_eq_real @ Y @ X3 ) ) ).
% leI
thf(fact_1180_leI,axiom,
! [X3: nat,Y: nat] :
( ~ ( ord_less_nat @ X3 @ Y )
=> ( ord_less_eq_nat @ Y @ X3 ) ) ).
% leI
thf(fact_1181_leD,axiom,
! [Y: set_a,X3: set_a] :
( ( ord_less_eq_set_a @ Y @ X3 )
=> ~ ( ord_less_set_a @ X3 @ Y ) ) ).
% leD
thf(fact_1182_leD,axiom,
! [Y: set_set_a,X3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ Y @ X3 )
=> ~ ( ord_less_set_set_a @ X3 @ Y ) ) ).
% leD
thf(fact_1183_leD,axiom,
! [Y: real,X3: real] :
( ( ord_less_eq_real @ Y @ X3 )
=> ~ ( ord_less_real @ X3 @ Y ) ) ).
% leD
thf(fact_1184_leD,axiom,
! [Y: nat,X3: nat] :
( ( ord_less_eq_nat @ Y @ X3 )
=> ~ ( ord_less_nat @ X3 @ Y ) ) ).
% leD
thf(fact_1185_leD,axiom,
! [Y: set_Product_prod_a_a,X3: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ Y @ X3 )
=> ~ ( ord_le6819997720685908915od_a_a @ X3 @ Y ) ) ).
% leD
thf(fact_1186_insert__mono,axiom,
! [C2: set_a,D2: set_a,A: a] :
( ( ord_less_eq_set_a @ C2 @ D2 )
=> ( ord_less_eq_set_a @ ( insert_a @ A @ C2 ) @ ( insert_a @ A @ D2 ) ) ) ).
% insert_mono
thf(fact_1187_insert__mono,axiom,
! [C2: set_set_a,D2: set_set_a,A: set_a] :
( ( ord_le3724670747650509150_set_a @ C2 @ D2 )
=> ( ord_le3724670747650509150_set_a @ ( insert_set_a @ A @ C2 ) @ ( insert_set_a @ A @ D2 ) ) ) ).
% insert_mono
thf(fact_1188_insert__mono,axiom,
! [C2: set_Product_prod_a_a,D2: set_Product_prod_a_a,A: product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ C2 @ D2 )
=> ( ord_le746702958409616551od_a_a @ ( insert4534936382041156343od_a_a @ A @ C2 ) @ ( insert4534936382041156343od_a_a @ A @ D2 ) ) ) ).
% insert_mono
thf(fact_1189_subset__insert,axiom,
! [X3: a,A2: set_a,B: set_a] :
( ~ ( member_a @ X3 @ A2 )
=> ( ( ord_less_eq_set_a @ A2 @ ( insert_a @ X3 @ B ) )
= ( ord_less_eq_set_a @ A2 @ B ) ) ) ).
% subset_insert
thf(fact_1190_subset__insert,axiom,
! [X3: set_a,A2: set_set_a,B: set_set_a] :
( ~ ( member_set_a @ X3 @ A2 )
=> ( ( ord_le3724670747650509150_set_a @ A2 @ ( insert_set_a @ X3 @ B ) )
= ( ord_le3724670747650509150_set_a @ A2 @ B ) ) ) ).
% subset_insert
thf(fact_1191_subset__insert,axiom,
! [X3: product_prod_a_a,A2: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ~ ( member1426531477525435216od_a_a @ X3 @ A2 )
=> ( ( ord_le746702958409616551od_a_a @ A2 @ ( insert4534936382041156343od_a_a @ X3 @ B ) )
= ( ord_le746702958409616551od_a_a @ A2 @ B ) ) ) ).
% subset_insert
thf(fact_1192_subset__insertI,axiom,
! [B: set_a,A: a] : ( ord_less_eq_set_a @ B @ ( insert_a @ A @ B ) ) ).
% subset_insertI
thf(fact_1193_subset__insertI,axiom,
! [B: set_set_a,A: set_a] : ( ord_le3724670747650509150_set_a @ B @ ( insert_set_a @ A @ B ) ) ).
% subset_insertI
thf(fact_1194_subset__insertI,axiom,
! [B: set_Product_prod_a_a,A: product_prod_a_a] : ( ord_le746702958409616551od_a_a @ B @ ( insert4534936382041156343od_a_a @ A @ B ) ) ).
% subset_insertI
thf(fact_1195_subset__insertI2,axiom,
! [A2: set_a,B: set_a,B2: a] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ord_less_eq_set_a @ A2 @ ( insert_a @ B2 @ B ) ) ) ).
% subset_insertI2
thf(fact_1196_subset__insertI2,axiom,
! [A2: set_set_a,B: set_set_a,B2: set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B )
=> ( ord_le3724670747650509150_set_a @ A2 @ ( insert_set_a @ B2 @ B ) ) ) ).
% subset_insertI2
thf(fact_1197_subset__insertI2,axiom,
! [A2: set_Product_prod_a_a,B: set_Product_prod_a_a,B2: product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A2 @ B )
=> ( ord_le746702958409616551od_a_a @ A2 @ ( insert4534936382041156343od_a_a @ B2 @ B ) ) ) ).
% subset_insertI2
thf(fact_1198_gt__ex,axiom,
! [X3: nat] :
? [X_1: nat] : ( ord_less_nat @ X3 @ X_1 ) ).
% gt_ex
thf(fact_1199_less__imp__neq,axiom,
! [X3: nat,Y: nat] :
( ( ord_less_nat @ X3 @ Y )
=> ( X3 != Y ) ) ).
% less_imp_neq
thf(fact_1200_order_Oasym,axiom,
! [A: nat,B2: nat] :
( ( ord_less_nat @ A @ B2 )
=> ~ ( ord_less_nat @ B2 @ A ) ) ).
% order.asym
thf(fact_1201_neighborhood__incident,axiom,
! [U: a,V: a] :
( ( member_a @ U @ ( undire8504279938402040014hood_a @ vertices @ edges @ V ) )
= ( member_set_a @ ( insert_a @ U @ ( insert_a @ V @ bot_bot_set_a ) ) @ ( undire3231912044278729248dges_a @ edges @ V ) ) ) ).
% neighborhood_incident
thf(fact_1202_degree0__neighborhood__empt__iff,axiom,
! [V: a] :
( ( finite_finite_set_a @ edges )
=> ( ( ( undire8867928226783802224gree_a @ edges @ V )
= zero_zero_nat )
= ( ( undire8504279938402040014hood_a @ vertices @ edges @ V )
= bot_bot_set_a ) ) ) ).
% degree0_neighborhood_empt_iff
thf(fact_1203_walk__edges__index,axiom,
! [I: nat,W2: list_a] :
( ( ord_less_eq_nat @ zero_zero_nat @ I )
=> ( ( ord_less_nat @ I @ ( undire8849074589633906640ngth_a @ W2 ) )
=> ( ( undire6133010728901294956walk_a @ vertices @ edges @ W2 )
=> ( member_set_a @ ( nth_set_a @ ( undire7337870655677353998dges_a @ W2 ) @ I ) @ edges ) ) ) ) ).
% walk_edges_index
thf(fact_1204_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_1205_finite__incident__edges,axiom,
! [V: a] :
( ( finite_finite_set_a @ edges )
=> ( finite_finite_set_a @ ( undire3231912044278729248dges_a @ edges @ V ) ) ) ).
% finite_incident_edges
thf(fact_1206_incident__edges__empty,axiom,
! [V: a] :
( ~ ( member_a @ V @ vertices )
=> ( ( undire3231912044278729248dges_a @ edges @ V )
= bot_bot_set_set_a ) ) ).
% incident_edges_empty
thf(fact_1207_degree0__inc__edges__empt__iff,axiom,
! [V: a] :
( ( finite_finite_set_a @ edges )
=> ( ( ( undire8867928226783802224gree_a @ edges @ V )
= zero_zero_nat )
= ( ( undire3231912044278729248dges_a @ edges @ V )
= bot_bot_set_set_a ) ) ) ).
% degree0_inc_edges_empt_iff
thf(fact_1208_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_1209_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_1210_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_1211_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_1212_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% eq_imp_le
thf(fact_1213_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_1214_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_1215_Nat_Oex__has__greatest__nat,axiom,
! [P2: nat > $o,K: nat,B2: nat] :
( ( P2 @ K )
=> ( ! [Y2: nat] :
( ( P2 @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ B2 ) )
=> ? [X4: nat] :
( ( P2 @ X4 )
& ! [Y7: nat] :
( ( P2 @ Y7 )
=> ( ord_less_eq_nat @ Y7 @ X4 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_1216_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_1217_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_1218_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_1219_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_1220_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
& ( M2 != N2 ) ) ) ) ).
% nat_less_le
thf(fact_1221_less__imp__le__nat,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% less_imp_le_nat
thf(fact_1222_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ N2 )
| ( M2 = N2 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_1223_less__or__eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( ( ord_less_nat @ M @ N )
| ( M = N ) )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% less_or_eq_imp_le
thf(fact_1224_le__neq__implies__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( M != N )
=> ( ord_less_nat @ M @ N ) ) ) ).
% le_neq_implies_less
thf(fact_1225_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I: nat,J: nat] :
( ! [I2: nat,J2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ord_less_nat @ ( F @ I2 ) @ ( F @ J2 ) ) )
=> ( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_1226_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_1227_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_1228_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_1229_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_1230_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_1231_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_1232_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_1233_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_1234_le__add2,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M @ N ) ) ).
% le_add2
thf(fact_1235_le__add1,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M ) ) ).
% le_add1
thf(fact_1236_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_1237_ex__least__nat__le,axiom,
! [P2: nat > $o,N: nat] :
( ( P2 @ N )
=> ( ~ ( P2 @ zero_zero_nat )
=> ? [K3: nat] :
( ( ord_less_eq_nat @ K3 @ N )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ K3 )
=> ~ ( P2 @ I3 ) )
& ( P2 @ K3 ) ) ) ) ).
% ex_least_nat_le
thf(fact_1238_mono__nat__linear__lb,axiom,
! [F: nat > nat,M: nat,K: nat] :
( ! [M3: nat,N3: nat] :
( ( ord_less_nat @ M3 @ N3 )
=> ( ord_less_nat @ ( F @ M3 ) @ ( F @ N3 ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M ) @ K ) @ ( F @ ( plus_plus_nat @ M @ K ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_1239_incident__loops__simp_I1_J,axiom,
! [V: a] :
( ( undire3617971648856834880loop_a @ edges @ V )
=> ( ( undire4753905205749729249oops_a @ edges @ V )
= ( insert_set_a @ ( insert_a @ V @ bot_bot_set_a ) @ bot_bot_set_set_a ) ) ) ).
% incident_loops_simp(1)
thf(fact_1240_walk__edges__append__union,axiom,
! [Xs: list_a,Ys: list_a] :
( ( Xs != nil_a )
=> ( ( Ys != nil_a )
=> ( ( set_set_a2 @ ( undire7337870655677353998dges_a @ ( append_a @ Xs @ Ys ) ) )
= ( sup_sup_set_set_a @ ( sup_sup_set_set_a @ ( set_set_a2 @ ( undire7337870655677353998dges_a @ Xs ) ) @ ( set_set_a2 @ ( undire7337870655677353998dges_a @ Ys ) ) ) @ ( insert_set_a @ ( insert_a @ ( last_a @ Xs ) @ ( insert_a @ ( hd_a @ Ys ) @ bot_bot_set_a ) ) @ bot_bot_set_set_a ) ) ) ) ) ).
% walk_edges_append_union
thf(fact_1241_finite__inc__sedges,axiom,
! [V: a] :
( ( finite_finite_set_a @ edges )
=> ( finite_finite_set_a @ ( undire1270416042309875431dges_a @ edges @ V ) ) ) ).
% finite_inc_sedges
thf(fact_1242_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_1243_finite__incident__loops,axiom,
! [V: a] : ( finite_finite_set_a @ ( undire4753905205749729249oops_a @ edges @ V ) ) ).
% finite_incident_loops
thf(fact_1244_incident__loops__simp_I2_J,axiom,
! [V: a] :
( ~ ( undire3617971648856834880loop_a @ edges @ V )
=> ( ( undire4753905205749729249oops_a @ edges @ V )
= bot_bot_set_set_a ) ) ).
% incident_loops_simp(2)
thf(fact_1245_incident__edges__union,axiom,
! [V: a] :
( ( undire3231912044278729248dges_a @ edges @ V )
= ( sup_sup_set_set_a @ ( undire1270416042309875431dges_a @ edges @ V ) @ ( undire4753905205749729249oops_a @ edges @ V ) ) ) ).
% incident_edges_union
thf(fact_1246_incident__edges__sedges,axiom,
! [V: a] :
( ~ ( undire3617971648856834880loop_a @ edges @ V )
=> ( ( undire3231912044278729248dges_a @ edges @ V )
= ( undire1270416042309875431dges_a @ edges @ V ) ) ) ).
% incident_edges_sedges
thf(fact_1247_incident__sedges__empty,axiom,
! [V: a] :
( ~ ( member_a @ V @ vertices )
=> ( ( undire1270416042309875431dges_a @ edges @ V )
= bot_bot_set_set_a ) ) ).
% incident_sedges_empty
thf(fact_1248_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_1249_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_1250_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_1251_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_1252_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_1253_all__edges__betw__I,axiom,
! [X3: a,X5: set_a,Y: a,Y5: set_a] :
( ( member_a @ X3 @ X5 )
=> ( ( member_a @ Y @ Y5 )
=> ( ( member_set_a @ ( insert_a @ X3 @ ( insert_a @ Y @ bot_bot_set_a ) ) @ edges )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Y ) @ ( undire8383842906760478443ween_a @ edges @ X5 @ Y5 ) ) ) ) ) ).
% all_edges_betw_I
thf(fact_1254_all__edges__betw__D3,axiom,
! [X3: a,Y: a,X5: set_a,Y5: set_a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Y ) @ ( undire8383842906760478443ween_a @ edges @ X5 @ Y5 ) )
=> ( member_set_a @ ( insert_a @ X3 @ ( insert_a @ Y @ bot_bot_set_a ) ) @ edges ) ) ).
% all_edges_betw_D3
thf(fact_1255_incident__loops__card,axiom,
! [V: a] : ( ord_less_eq_nat @ ( finite_card_set_a @ ( undire4753905205749729249oops_a @ edges @ V ) ) @ one_one_nat ) ).
% incident_loops_card
thf(fact_1256_degree__no__loops,axiom,
! [V: a] :
( ~ ( undire3617971648856834880loop_a @ edges @ V )
=> ( ( undire8867928226783802224gree_a @ edges @ V )
= ( finite_card_set_a @ ( undire3231912044278729248dges_a @ edges @ V ) ) ) ) ).
% degree_no_loops
thf(fact_1257_card__incident__sedges__neighborhood,axiom,
! [V: a] :
( ( finite_card_set_a @ ( undire3231912044278729248dges_a @ edges @ V ) )
= ( finite_card_a @ ( undire8504279938402040014hood_a @ vertices @ edges @ V ) ) ) ).
% card_incident_sedges_neighborhood
thf(fact_1258_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_1259_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_1260_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_1261_card1__incident__imp__vert,axiom,
! [V: a,E: set_a] :
( ( ( undire1521409233611534436dent_a @ V @ E )
& ( ( finite_card_a @ E )
= one_one_nat ) )
=> ( E
= ( insert_a @ V @ bot_bot_set_a ) ) ) ).
% card1_incident_imp_vert
thf(fact_1262_is__loop__def,axiom,
( undire2905028936066782638loop_a
= ( ^ [E5: set_a] :
( ( finite_card_a @ E5 )
= one_one_nat ) ) ) ).
% is_loop_def
thf(fact_1263_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_1264_mult__le__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_eq_nat @ M @ N ) ) ) ).
% mult_le_cancel2
thf(fact_1265_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_1266_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_1267_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_1268_le__square,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ M ) ) ).
% le_square
thf(fact_1269_le__cube,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ ( times_times_nat @ M @ M ) ) ) ).
% le_cube
thf(fact_1270_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_1271_nat__mult__le__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_eq_nat @ M @ N ) ) ) ).
% nat_mult_le_cancel_disj
thf(fact_1272_nat__mult__le__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ord_less_eq_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ) ).
% nat_mult_le_cancel1
thf(fact_1273_finite__nat__set__iff__bounded__le,axiom,
( finite_finite_nat
= ( ^ [N4: set_nat] :
? [M2: nat] :
! [X2: nat] :
( ( member_nat @ X2 @ N4 )
=> ( ord_less_eq_nat @ X2 @ M2 ) ) ) ) ).
% finite_nat_set_iff_bounded_le
% Conjectures (1)
thf(conj_0,conjecture,
( ( connecting_walk_a @ vertices @ edges @ u @ v @ xs )
= ( connecting_walk_a @ vertices @ edges @ v @ u @ ( rev_a @ xs ) ) ) ).
%------------------------------------------------------------------------------