TPTP Problem File: SLH0032^1.p
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%------------------------------------------------------------------------------
% File : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain : Archive of Formal Proofs
% Problem :
% Version : Especial.
% English :
% Refs : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source : [Des23]
% Names : Undirected_Graph_Theory/0016_Undirected_Graph_Walks/prob_00466_017889__13250606_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1433 ( 562 unt; 162 typ; 0 def)
% Number of atoms : 3726 (1356 equ; 0 cnn)
% Maximal formula atoms : 12 ( 2 avg)
% Number of connectives : 10393 ( 411 ~; 70 |; 308 &;8230 @)
% ( 0 <=>;1374 =>; 0 <=; 0 <~>)
% Maximal formula depth : 21 ( 6 avg)
% Number of types : 17 ( 16 usr)
% Number of type conns : 379 ( 379 >; 0 *; 0 +; 0 <<)
% Number of symbols : 147 ( 146 usr; 17 con; 0-4 aty)
% Number of variables : 3288 ( 197 ^;2954 !; 137 ?;3288 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 14:33:33.365
%------------------------------------------------------------------------------
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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__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__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__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__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_p,type,
p: list_a ).
thf(sy_v_vertices,type,
vertices: set_a ).
% Relevant facts (1270)
thf(fact_0_ne,axiom,
( ( hd_a @ p )
!= ( last_a @ p ) ) ).
% ne
thf(fact_1_distinct__tl,axiom,
! [Xs: list_a] :
( ( distinct_a @ Xs )
=> ( distinct_a @ ( tl_a @ Xs ) ) ) ).
% distinct_tl
thf(fact_2_distinct__tl,axiom,
! [Xs: list_set_a] :
( ( distinct_set_a @ Xs )
=> ( distinct_set_a @ ( tl_set_a @ Xs ) ) ) ).
% distinct_tl
thf(fact_3_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_4_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_5_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_6_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_7_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_8_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_9_distinct__union,axiom,
! [Xs: list_a,Ys: list_a] :
( ( distinct_a @ ( union_a @ Xs @ Ys ) )
= ( distinct_a @ Ys ) ) ).
% distinct_union
thf(fact_10_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_11_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_12_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_13_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_14_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_15_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_16_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_17_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_18_hd__Nil__eq__last,axiom,
( ( hd_a @ nil_a )
= ( last_a @ nil_a ) ) ).
% hd_Nil_eq_last
thf(fact_19_hd__Nil__eq__last,axiom,
( ( hd_set_a @ nil_set_a )
= ( last_set_a @ nil_set_a ) ) ).
% hd_Nil_eq_last
thf(fact_20_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_21_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_22_ulgraph__axioms,axiom,
undire7251896706689453996raph_a @ vertices @ edges ).
% ulgraph_axioms
thf(fact_23_assms_I1_J,axiom,
undire3562951555376170320path_a @ vertices @ edges @ p ).
% assms(1)
thf(fact_24_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_25_rev__rev__ident,axiom,
! [Xs: list_a] :
( ( rev_a @ ( rev_a @ Xs ) )
= Xs ) ).
% rev_rev_ident
thf(fact_26_rev__rev__ident,axiom,
! [Xs: list_set_a] :
( ( rev_set_a @ ( rev_set_a @ Xs ) )
= Xs ) ).
% rev_rev_ident
thf(fact_27_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_28_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_29_Nil__is__rev__conv,axiom,
! [Xs: list_a] :
( ( nil_a
= ( rev_a @ Xs ) )
= ( Xs = nil_a ) ) ).
% Nil_is_rev_conv
thf(fact_30_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_31_rev__is__Nil__conv,axiom,
! [Xs: list_a] :
( ( ( rev_a @ Xs )
= nil_a )
= ( Xs = nil_a ) ) ).
% rev_is_Nil_conv
thf(fact_32_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_33_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_34_assms_I3_J,axiom,
~ ( undire2407311113669455967ycle_a @ vertices @ edges @ p ) ).
% assms(3)
thf(fact_35_distinct__rev,axiom,
! [Xs: list_a] :
( ( distinct_a @ ( rev_a @ Xs ) )
= ( distinct_a @ Xs ) ) ).
% distinct_rev
thf(fact_36_distinct__rev,axiom,
! [Xs: list_set_a] :
( ( distinct_set_a @ ( rev_set_a @ Xs ) )
= ( distinct_set_a @ Xs ) ) ).
% distinct_rev
thf(fact_37_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_38_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_39_is__trail__rev,axiom,
! [Xs: list_a] :
( ( undire7142031287334043199rail_a @ vertices @ edges @ Xs )
= ( undire7142031287334043199rail_a @ vertices @ edges @ ( rev_a @ Xs ) ) ) ).
% is_trail_rev
thf(fact_40_rev_Osimps_I1_J,axiom,
( ( rev_a @ nil_a )
= nil_a ) ).
% rev.simps(1)
thf(fact_41_rev_Osimps_I1_J,axiom,
( ( rev_set_a @ nil_set_a )
= nil_set_a ) ).
% rev.simps(1)
thf(fact_42_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_43_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_44_ulgraph_Ois__gen__path_Ocong,axiom,
undire3562951555376170320path_a = undire3562951555376170320path_a ).
% ulgraph.is_gen_path.cong
thf(fact_45_ulgraph_Ois__gen__path_Ocong,axiom,
undire7201326534205417136_set_a = undire7201326534205417136_set_a ).
% ulgraph.is_gen_path.cong
thf(fact_46_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_47_comp__sgraph_Oe__in__all__edges,axiom,
! [E: set_set_a,S: set_set_a] :
( ( member_set_set_a @ E @ ( undire8247866692393712962_set_a @ S ) )
=> ( member_set_set_a @ E @ ( undire8247866692393712962_set_a @ S ) ) ) ).
% comp_sgraph.e_in_all_edges
thf(fact_48_comp__sgraph_Oulgraph__axioms,axiom,
! [S: set_a] : ( undire7251896706689453996raph_a @ S @ ( undire2918257014606996450dges_a @ S ) ) ).
% comp_sgraph.ulgraph_axioms
thf(fact_49_comp__sgraph_Oulgraph__axioms,axiom,
! [S: set_set_a] : ( undire6886684016831807756_set_a @ S @ ( undire8247866692393712962_set_a @ S ) ) ).
% comp_sgraph.ulgraph_axioms
thf(fact_50_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_51_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_52_rev__swap,axiom,
! [Xs: list_a,Ys: list_a] :
( ( ( rev_a @ Xs )
= Ys )
= ( Xs
= ( rev_a @ Ys ) ) ) ).
% rev_swap
thf(fact_53_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_54_distinct_Osimps_I1_J,axiom,
distinct_a @ nil_a ).
% distinct.simps(1)
thf(fact_55_distinct_Osimps_I1_J,axiom,
distinct_set_a @ nil_set_a ).
% distinct.simps(1)
thf(fact_56_list_Osel_I2_J,axiom,
( ( tl_a @ nil_a )
= nil_a ) ).
% list.sel(2)
thf(fact_57_list_Osel_I2_J,axiom,
( ( tl_set_a @ nil_set_a )
= nil_set_a ) ).
% list.sel(2)
thf(fact_58_last__rev,axiom,
! [Xs: list_a] :
( ( last_a @ ( rev_a @ Xs ) )
= ( hd_a @ Xs ) ) ).
% last_rev
thf(fact_59_last__rev,axiom,
! [Xs: list_set_a] :
( ( last_set_a @ ( rev_set_a @ Xs ) )
= ( hd_set_a @ Xs ) ) ).
% last_rev
thf(fact_60_hd__rev,axiom,
! [Xs: list_a] :
( ( hd_a @ ( rev_a @ Xs ) )
= ( last_a @ Xs ) ) ).
% hd_rev
thf(fact_61_hd__rev,axiom,
! [Xs: list_set_a] :
( ( hd_set_a @ ( rev_set_a @ Xs ) )
= ( last_set_a @ Xs ) ) ).
% hd_rev
thf(fact_62_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_63_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_64_is__path__gen__path,axiom,
! [P: list_a] :
( ( undire427332500224447920path_a @ vertices @ edges @ P )
=> ( undire3562951555376170320path_a @ vertices @ edges @ P ) ) ).
% is_path_gen_path
thf(fact_65_is__gen__path__cycle,axiom,
! [P: list_a] :
( ( undire2407311113669455967ycle_a @ vertices @ edges @ P )
=> ( undire3562951555376170320path_a @ vertices @ edges @ P ) ) ).
% is_gen_path_cycle
thf(fact_66_is__path__rev,axiom,
! [Xs: list_a] :
( ( undire427332500224447920path_a @ vertices @ edges @ Xs )
= ( undire427332500224447920path_a @ vertices @ edges @ ( rev_a @ Xs ) ) ) ).
% is_path_rev
thf(fact_67_is__cycle__rev,axiom,
! [Xs: list_a] :
( ( undire2407311113669455967ycle_a @ vertices @ edges @ Xs )
= ( undire2407311113669455967ycle_a @ vertices @ edges @ ( rev_a @ Xs ) ) ) ).
% is_cycle_rev
thf(fact_68_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_69_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_70_mem__Collect__eq,axiom,
! [A: a,P2: a > $o] :
( ( member_a @ A @ ( collect_a @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_71_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_72_Collect__mem__eq,axiom,
! [A2: set_a] :
( ( collect_a
@ ^ [X2: a] : ( member_a @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_73_has__loop__in__verts,axiom,
! [V: a] :
( ( undire3617971648856834880loop_a @ edges @ V )
=> ( member_a @ V @ vertices ) ) ).
% has_loop_in_verts
thf(fact_74_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_75_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_76_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_77__092_060open_062is__walk_Ap_092_060close_062,axiom,
undire6133010728901294956walk_a @ vertices @ edges @ p ).
% \<open>is_walk p\<close>
thf(fact_78_incident__def,axiom,
undire1521409233611534436dent_a = member_a ).
% incident_def
thf(fact_79_walk__edges_Ocases,axiom,
! [X3: list_a] :
( ( X3 != nil_a )
=> ( ! [X4: a] :
( X3
!= ( cons_a @ X4 @ nil_a ) )
=> ~ ! [X4: a,Y: a,Ys2: list_a] :
( X3
!= ( cons_a @ X4 @ ( cons_a @ Y @ Ys2 ) ) ) ) ) ).
% walk_edges.cases
thf(fact_80_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_81_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_82_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_83_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_84_is__walk__not__empty2,axiom,
~ ( undire6133010728901294956walk_a @ vertices @ edges @ nil_a ) ).
% is_walk_not_empty2
thf(fact_85_is__walk__not__empty,axiom,
! [Xs: list_a] :
( ( undire6133010728901294956walk_a @ vertices @ edges @ Xs )
=> ( Xs != nil_a ) ) ).
% is_walk_not_empty
thf(fact_86_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_87_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_88_is__walk__rev,axiom,
! [Xs: list_a] :
( ( undire6133010728901294956walk_a @ vertices @ edges @ Xs )
= ( undire6133010728901294956walk_a @ vertices @ edges @ ( rev_a @ Xs ) ) ) ).
% is_walk_rev
thf(fact_89_is__path__walk,axiom,
! [Xs: list_a] :
( ( undire427332500224447920path_a @ vertices @ edges @ Xs )
=> ( undire6133010728901294956walk_a @ vertices @ edges @ Xs ) ) ).
% is_path_walk
thf(fact_90_is__walk__singleton,axiom,
! [U: a] :
( ( member_a @ U @ vertices )
=> ( undire6133010728901294956walk_a @ vertices @ edges @ ( cons_a @ U @ nil_a ) ) ) ).
% is_walk_singleton
thf(fact_91_is__walk__drop__hd,axiom,
! [Ys: list_a,Y2: a] :
( ( Ys != nil_a )
=> ( ( undire6133010728901294956walk_a @ vertices @ edges @ ( cons_a @ Y2 @ Ys ) )
=> ( undire6133010728901294956walk_a @ vertices @ edges @ Ys ) ) ) ).
% is_walk_drop_hd
thf(fact_92_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_93_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_94_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_95_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_96_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_97_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_98_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_99_list_Ocollapse,axiom,
! [List: list_a] :
( ( List != nil_a )
=> ( ( cons_a @ ( hd_a @ List ) @ ( tl_a @ List ) )
= List ) ) ).
% list.collapse
thf(fact_100_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_101_hd__Cons__tl,axiom,
! [Xs: list_a] :
( ( Xs != nil_a )
=> ( ( cons_a @ ( hd_a @ Xs ) @ ( tl_a @ Xs ) )
= Xs ) ) ).
% hd_Cons_tl
thf(fact_102_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_103_ulgraph_Ohas__loop_Ocong,axiom,
undire3617971648856834880loop_a = undire3617971648856834880loop_a ).
% ulgraph.has_loop.cong
thf(fact_104_ulgraph_Overt__adj_Ocong,axiom,
undire397441198561214472_adj_a = undire397441198561214472_adj_a ).
% ulgraph.vert_adj.cong
thf(fact_105_comp__sgraph_Oincident__def,axiom,
undire2320338297334612420_set_a = member_set_a ).
% comp_sgraph.incident_def
thf(fact_106_comp__sgraph_Oincident__def,axiom,
undire1521409233611534436dent_a = member_a ).
% comp_sgraph.incident_def
thf(fact_107_not__Cons__self2,axiom,
! [X3: a,Xs: list_a] :
( ( cons_a @ X3 @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_108_not__Cons__self2,axiom,
! [X3: set_a,Xs: list_set_a] :
( ( cons_set_a @ X3 @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_109_ulgraph_Ois__cycle_Ocong,axiom,
undire2407311113669455967ycle_a = undire2407311113669455967ycle_a ).
% ulgraph.is_cycle.cong
thf(fact_110_ulgraph_Ois__walk_Ocong,axiom,
undire6133010728901294956walk_a = undire6133010728901294956walk_a ).
% ulgraph.is_walk.cong
thf(fact_111_ulgraph_Ois__path_Ocong,axiom,
undire427332500224447920path_a = undire427332500224447920path_a ).
% ulgraph.is_path.cong
thf(fact_112_ulgraph_Ois__open__walk_Ocong,axiom,
undire2427028224930250914walk_a = undire2427028224930250914walk_a ).
% ulgraph.is_open_walk.cong
thf(fact_113_ulgraph_Ois__closed__walk_Ocong,axiom,
undire3370724456595283424walk_a = undire3370724456595283424walk_a ).
% ulgraph.is_closed_walk.cong
thf(fact_114_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_115_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_116_ulgraph_Ois__trail_Ocong,axiom,
undire7142031287334043199rail_a = undire7142031287334043199rail_a ).
% ulgraph.is_trail.cong
thf(fact_117_graph__system_Oedge__adj_Ocong,axiom,
undire4022703626023482010_adj_a = undire4022703626023482010_adj_a ).
% graph_system.edge_adj.cong
thf(fact_118_comp__sgraph_Ois__path__walk,axiom,
! [S: set_set_a,Xs: list_set_a] :
( ( undire8834939040163919632_set_a @ S @ ( undire8247866692393712962_set_a @ S ) @ Xs )
=> ( undire3014741414213135564_set_a @ S @ ( undire8247866692393712962_set_a @ S ) @ Xs ) ) ).
% comp_sgraph.is_path_walk
thf(fact_119_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_120_ulgraph_Ois__path__walk,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 )
=> ( undire3014741414213135564_set_a @ Vertices @ Edges @ Xs ) ) ) ).
% ulgraph.is_path_walk
thf(fact_121_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_122_comp__sgraph_Overt__adj__edge__iff2,axiom,
! [V1: set_a,V2: set_a,S: set_set_a] :
( ( V1 != V2 )
=> ( ( undire3510646817838285160_set_a @ ( undire8247866692393712962_set_a @ S ) @ V1 @ V2 )
= ( ? [X2: set_set_a] :
( ( member_set_set_a @ X2 @ ( undire8247866692393712962_set_a @ S ) )
& ( undire2320338297334612420_set_a @ V1 @ X2 )
& ( undire2320338297334612420_set_a @ V2 @ X2 ) ) ) ) ) ).
% comp_sgraph.vert_adj_edge_iff2
thf(fact_123_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_124_ulgraph_Overt__adj__edge__iff2,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,V1: set_a,V2: set_a] :
( ( undire6886684016831807756_set_a @ Vertices @ Edges )
=> ( ( V1 != V2 )
=> ( ( undire3510646817838285160_set_a @ Edges @ V1 @ V2 )
= ( ? [X2: set_set_a] :
( ( member_set_set_a @ X2 @ Edges )
& ( undire2320338297334612420_set_a @ V1 @ X2 )
& ( undire2320338297334612420_set_a @ V2 @ X2 ) ) ) ) ) ) ).
% ulgraph.vert_adj_edge_iff2
thf(fact_125_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_126_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_127_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_128_comp__sgraph_Ois__walk__drop__hd,axiom,
! [Ys: list_set_a,S: set_set_a,Y2: set_a] :
( ( Ys != nil_set_a )
=> ( ( undire3014741414213135564_set_a @ S @ ( undire8247866692393712962_set_a @ S ) @ ( cons_set_a @ Y2 @ Ys ) )
=> ( undire3014741414213135564_set_a @ S @ ( undire8247866692393712962_set_a @ S ) @ Ys ) ) ) ).
% comp_sgraph.is_walk_drop_hd
thf(fact_129_comp__sgraph_Ois__walk__drop__hd,axiom,
! [Ys: list_a,S: set_a,Y2: a] :
( ( Ys != nil_a )
=> ( ( undire6133010728901294956walk_a @ S @ ( undire2918257014606996450dges_a @ S ) @ ( cons_a @ Y2 @ Ys ) )
=> ( undire6133010728901294956walk_a @ S @ ( undire2918257014606996450dges_a @ S ) @ Ys ) ) ) ).
% comp_sgraph.is_walk_drop_hd
thf(fact_130_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_131_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_132_ulgraph_Ois__walk__drop__hd,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,Ys: list_set_a,Y2: set_a] :
( ( undire6886684016831807756_set_a @ Vertices @ Edges )
=> ( ( Ys != nil_set_a )
=> ( ( undire3014741414213135564_set_a @ Vertices @ Edges @ ( cons_set_a @ Y2 @ Ys ) )
=> ( undire3014741414213135564_set_a @ Vertices @ Edges @ Ys ) ) ) ) ).
% ulgraph.is_walk_drop_hd
thf(fact_133_ulgraph_Ois__walk__drop__hd,axiom,
! [Vertices: set_a,Edges: set_set_a,Ys: list_a,Y2: a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( Ys != nil_a )
=> ( ( undire6133010728901294956walk_a @ Vertices @ Edges @ ( cons_a @ Y2 @ Ys ) )
=> ( undire6133010728901294956walk_a @ Vertices @ Edges @ Ys ) ) ) ) ).
% ulgraph.is_walk_drop_hd
thf(fact_134_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_135_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_136_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 )
=> ( ! [Y: a,Ys2: list_a] : ( P2 @ nil_a @ ( cons_a @ Y @ Ys2 ) )
=> ( ! [X4: a,Xs2: list_a,Y: a,Ys2: list_a] :
( ( P2 @ Xs2 @ Ys2 )
=> ( P2 @ ( cons_a @ X4 @ Xs2 ) @ ( cons_a @ Y @ Ys2 ) ) )
=> ( P2 @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_137_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 )
=> ( ! [Y: set_a,Ys2: list_set_a] : ( P2 @ nil_a @ ( cons_set_a @ Y @ Ys2 ) )
=> ( ! [X4: a,Xs2: list_a,Y: set_a,Ys2: list_set_a] :
( ( P2 @ Xs2 @ Ys2 )
=> ( P2 @ ( cons_a @ X4 @ Xs2 ) @ ( cons_set_a @ Y @ Ys2 ) ) )
=> ( P2 @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_138_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 )
=> ( ! [Y: a,Ys2: list_a] : ( P2 @ nil_set_a @ ( cons_a @ Y @ Ys2 ) )
=> ( ! [X4: set_a,Xs2: list_set_a,Y: a,Ys2: list_a] :
( ( P2 @ Xs2 @ Ys2 )
=> ( P2 @ ( cons_set_a @ X4 @ Xs2 ) @ ( cons_a @ Y @ Ys2 ) ) )
=> ( P2 @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_139_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 )
=> ( ! [Y: set_a,Ys2: list_set_a] : ( P2 @ nil_set_a @ ( cons_set_a @ Y @ Ys2 ) )
=> ( ! [X4: set_a,Xs2: list_set_a,Y: set_a,Ys2: list_set_a] :
( ( P2 @ Xs2 @ Ys2 )
=> ( P2 @ ( cons_set_a @ X4 @ Xs2 ) @ ( cons_set_a @ Y @ Ys2 ) ) )
=> ( P2 @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_140_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_141_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_142_comp__sgraph_Owalk__edges_Ocases,axiom,
! [X3: list_a] :
( ( X3 != nil_a )
=> ( ! [X4: a] :
( X3
!= ( cons_a @ X4 @ nil_a ) )
=> ~ ! [X4: a,Y: a,Ys2: list_a] :
( X3
!= ( cons_a @ X4 @ ( cons_a @ Y @ Ys2 ) ) ) ) ) ).
% comp_sgraph.walk_edges.cases
thf(fact_143_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,Y: set_a,Ys2: list_set_a] :
( X3
!= ( cons_set_a @ X4 @ ( cons_set_a @ Y @ Ys2 ) ) ) ) ) ).
% comp_sgraph.walk_edges.cases
thf(fact_144_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_145_list_Oexhaust,axiom,
! [Y2: list_a] :
( ( Y2 != nil_a )
=> ~ ! [X212: a,X222: list_a] :
( Y2
!= ( cons_a @ X212 @ X222 ) ) ) ).
% list.exhaust
thf(fact_146_list_Oexhaust,axiom,
! [Y2: list_set_a] :
( ( Y2 != nil_set_a )
=> ~ ! [X212: set_a,X222: list_set_a] :
( Y2
!= ( cons_set_a @ X212 @ X222 ) ) ) ).
% list.exhaust
thf(fact_147_list_OdiscI,axiom,
! [List: list_a,X21: a,X22: list_a] :
( ( List
= ( cons_a @ X21 @ X22 ) )
=> ( List != nil_a ) ) ).
% list.discI
thf(fact_148_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_149_list_Odistinct_I1_J,axiom,
! [X21: a,X22: list_a] :
( nil_a
!= ( cons_a @ X21 @ X22 ) ) ).
% list.distinct(1)
thf(fact_150_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_151_distinct__length__2__or__more,axiom,
! [A: a,B: a,Xs: list_a] :
( ( distinct_a @ ( cons_a @ A @ ( cons_a @ B @ Xs ) ) )
= ( ( A != B )
& ( distinct_a @ ( cons_a @ A @ Xs ) )
& ( distinct_a @ ( cons_a @ B @ Xs ) ) ) ) ).
% distinct_length_2_or_more
thf(fact_152_distinct__length__2__or__more,axiom,
! [A: set_a,B: set_a,Xs: list_set_a] :
( ( distinct_set_a @ ( cons_set_a @ A @ ( cons_set_a @ B @ Xs ) ) )
= ( ( A != B )
& ( distinct_set_a @ ( cons_set_a @ A @ Xs ) )
& ( distinct_set_a @ ( cons_set_a @ B @ Xs ) ) ) ) ).
% distinct_length_2_or_more
thf(fact_153_list_Osel_I1_J,axiom,
! [X21: a,X22: list_a] :
( ( hd_a @ ( cons_a @ X21 @ X22 ) )
= X21 ) ).
% list.sel(1)
thf(fact_154_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_155_list_Osel_I3_J,axiom,
! [X21: a,X22: list_a] :
( ( tl_a @ ( cons_a @ X21 @ X22 ) )
= X22 ) ).
% list.sel(3)
thf(fact_156_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_157_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_158_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_159_comp__sgraph_Overt__adj__sym,axiom,
! [S: set_set_a,V1: set_a,V2: set_a] :
( ( undire3510646817838285160_set_a @ ( undire8247866692393712962_set_a @ S ) @ V1 @ V2 )
= ( undire3510646817838285160_set_a @ ( undire8247866692393712962_set_a @ S ) @ V2 @ V1 ) ) ).
% comp_sgraph.vert_adj_sym
thf(fact_160_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_161_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_162_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_163_ulgraph_Overt__adj__sym,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 )
= ( undire3510646817838285160_set_a @ Edges @ V2 @ V1 ) ) ) ).
% ulgraph.vert_adj_sym
thf(fact_164_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_165_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_166_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_167_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_168_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_169_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_170_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_171_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_172_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_173_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_174_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_175_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_176_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_177_comp__sgraph_Ois__open__walk__def,axiom,
! [S: set_set_a,Xs: list_set_a] :
( ( undire526879649183275522_set_a @ S @ ( undire8247866692393712962_set_a @ S ) @ Xs )
= ( ( undire3014741414213135564_set_a @ S @ ( undire8247866692393712962_set_a @ S ) @ Xs )
& ( ( hd_set_a @ Xs )
!= ( last_set_a @ Xs ) ) ) ) ).
% comp_sgraph.is_open_walk_def
thf(fact_178_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_179_comp__sgraph_Ois__closed__walk__def,axiom,
! [S: set_set_a,Xs: list_set_a] :
( ( undire4100213446647512896_set_a @ S @ ( undire8247866692393712962_set_a @ S ) @ Xs )
= ( ( undire3014741414213135564_set_a @ S @ ( undire8247866692393712962_set_a @ S ) @ Xs )
& ( ( hd_set_a @ Xs )
= ( last_set_a @ Xs ) ) ) ) ).
% comp_sgraph.is_closed_walk_def
thf(fact_180_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_181_ulgraph_Ois__open__walk__def,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 )
= ( ( undire3014741414213135564_set_a @ Vertices @ Edges @ Xs )
& ( ( hd_set_a @ Xs )
!= ( last_set_a @ Xs ) ) ) ) ) ).
% ulgraph.is_open_walk_def
thf(fact_182_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_183_ulgraph_Ois__closed__walk__def,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 )
= ( ( undire3014741414213135564_set_a @ Vertices @ Edges @ Xs )
& ( ( hd_set_a @ Xs )
= ( last_set_a @ Xs ) ) ) ) ) ).
% ulgraph.is_closed_walk_def
thf(fact_184_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_185_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_186_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_187_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_188_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_189_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_190_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_191_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_192_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_193_distinct__singleton,axiom,
! [X3: a] : ( distinct_a @ ( cons_a @ X3 @ nil_a ) ) ).
% distinct_singleton
thf(fact_194_distinct__singleton,axiom,
! [X3: set_a] : ( distinct_set_a @ ( cons_set_a @ X3 @ nil_set_a ) ) ).
% distinct_singleton
thf(fact_195_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_196_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_197_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,Y: a,Ys2: list_a] :
( X3
!= ( cons_a @ X4 @ ( cons_a @ Y @ Ys2 ) ) ) ) ) ) ).
% ulgraph.walk_edges.cases
thf(fact_198_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,Y: set_a,Ys2: list_set_a] :
( X3
!= ( cons_set_a @ X4 @ ( cons_set_a @ Y @ Ys2 ) ) ) ) ) ) ).
% ulgraph.walk_edges.cases
thf(fact_199_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_200_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_201_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_202_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_203_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_204_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_205_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_206_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_207_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_208_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_209_last__ConsR,axiom,
! [Xs: list_a,X3: a] :
( ( Xs != nil_a )
=> ( ( last_a @ ( cons_a @ X3 @ Xs ) )
= ( last_a @ Xs ) ) ) ).
% last_ConsR
thf(fact_210_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_211_last__ConsL,axiom,
! [Xs: list_a,X3: a] :
( ( Xs = nil_a )
=> ( ( last_a @ ( cons_a @ X3 @ Xs ) )
= X3 ) ) ).
% last_ConsL
thf(fact_212_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_213_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_214_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_215_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_216_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_217_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_218_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_219_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_220_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_221_comp__sgraph_Oedge__adj__inE,axiom,
! [S: set_set_a,E1: set_set_a,E2: set_set_a] :
( ( undire3485422320110889978_set_a @ ( undire8247866692393712962_set_a @ S ) @ E1 @ E2 )
=> ( ( member_set_set_a @ E1 @ ( undire8247866692393712962_set_a @ S ) )
& ( member_set_set_a @ E2 @ ( undire8247866692393712962_set_a @ S ) ) ) ) ).
% comp_sgraph.edge_adj_inE
thf(fact_222_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_223_list_Oexhaust__sel,axiom,
! [List: list_a] :
( ( List != nil_a )
=> ( List
= ( cons_a @ ( hd_a @ List ) @ ( tl_a @ List ) ) ) ) ).
% list.exhaust_sel
thf(fact_224_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_225_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_226_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_227_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_228_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_229_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_230_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_231_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_232_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_233_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_234_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_235_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_236_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_237_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_238_comp__sgraph_Ois__gen__path__cycle,axiom,
! [S: set_set_a,P: list_set_a] :
( ( undire797940137672299967_set_a @ S @ ( undire8247866692393712962_set_a @ S ) @ P )
=> ( undire7201326534205417136_set_a @ S @ ( undire8247866692393712962_set_a @ S ) @ P ) ) ).
% comp_sgraph.is_gen_path_cycle
thf(fact_239_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_240_comp__sgraph_Ois__path__gen__path,axiom,
! [S: set_set_a,P: list_set_a] :
( ( undire8834939040163919632_set_a @ S @ ( undire8247866692393712962_set_a @ S ) @ P )
=> ( undire7201326534205417136_set_a @ S @ ( undire8247866692393712962_set_a @ S ) @ P ) ) ).
% comp_sgraph.is_path_gen_path
thf(fact_241_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_242_ulgraph_Ois__gen__path__cycle,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,P: list_set_a] :
( ( undire6886684016831807756_set_a @ Vertices @ Edges )
=> ( ( undire797940137672299967_set_a @ Vertices @ Edges @ P )
=> ( undire7201326534205417136_set_a @ Vertices @ Edges @ P ) ) ) ).
% ulgraph.is_gen_path_cycle
thf(fact_243_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_244_ulgraph_Ois__path__gen__path,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,P: list_set_a] :
( ( undire6886684016831807756_set_a @ Vertices @ Edges )
=> ( ( undire8834939040163919632_set_a @ Vertices @ Edges @ P )
=> ( undire7201326534205417136_set_a @ Vertices @ Edges @ P ) ) ) ).
% ulgraph.is_path_gen_path
thf(fact_245_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_246_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_247_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_248_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_249_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_250_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_251_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_252_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_253_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_254_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_255_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_256_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_257_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_258_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_259_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_260_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_261_is__isolated__vertex__no__loop,axiom,
! [V: a] :
( ( undire8931668460104145173rtex_a @ vertices @ edges @ V )
=> ~ ( undire3617971648856834880loop_a @ edges @ V ) ) ).
% is_isolated_vertex_no_loop
thf(fact_262_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_263_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_264_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_265_is__walk__decomp,axiom,
! [Xs: list_a,Y2: a,Ys: list_a,Zs: list_a] :
( ( undire6133010728901294956walk_a @ vertices @ edges @ ( append_a @ Xs @ ( append_a @ ( cons_a @ Y2 @ nil_a ) @ ( append_a @ Ys @ ( append_a @ ( cons_a @ Y2 @ nil_a ) @ Zs ) ) ) ) )
=> ( undire6133010728901294956walk_a @ vertices @ edges @ ( append_a @ Xs @ ( append_a @ ( cons_a @ Y2 @ nil_a ) @ Zs ) ) ) ) ).
% is_walk_decomp
thf(fact_266_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_267_subgraph__refl,axiom,
undire7103218114511261257raph_a @ vertices @ edges @ vertices @ edges ).
% subgraph_refl
thf(fact_268_wellformed,axiom,
! [E: set_a] :
( ( member_set_a @ E @ edges )
=> ( ord_less_eq_set_a @ E @ vertices ) ) ).
% wellformed
thf(fact_269_graph__system__axioms,axiom,
undire2554140024507503526stem_a @ vertices @ edges ).
% graph_system_axioms
thf(fact_270_edge__density__commute,axiom,
! [X5: set_a,Y4: set_a] :
( ( undire297304480579013331sity_a @ edges @ X5 @ Y4 )
= ( undire297304480579013331sity_a @ edges @ Y4 @ X5 ) ) ).
% edge_density_commute
thf(fact_271_list__exhaust3,axiom,
! [Xs: list_a] :
( ( Xs != nil_a )
=> ( ! [X4: a] :
( Xs
!= ( cons_a @ X4 @ nil_a ) )
=> ~ ! [X4: a,Y: a,Ys2: list_a] :
( Xs
!= ( cons_a @ X4 @ ( cons_a @ Y @ Ys2 ) ) ) ) ) ).
% list_exhaust3
thf(fact_272_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,Y: set_a,Ys2: list_set_a] :
( Xs
!= ( cons_set_a @ X4 @ ( cons_set_a @ Y @ Ys2 ) ) ) ) ) ).
% list_exhaust3
thf(fact_273_walk__edges_Osimps_I1_J,axiom,
( ( undire7337870655677353998dges_a @ nil_a )
= nil_set_a ) ).
% walk_edges.simps(1)
thf(fact_274_walk__edges__rev,axiom,
! [Xs: list_a] :
( ( rev_set_a @ ( undire7337870655677353998dges_a @ Xs ) )
= ( undire7337870655677353998dges_a @ ( rev_a @ Xs ) ) ) ).
% walk_edges_rev
thf(fact_275_walk__edges_Osimps_I2_J,axiom,
! [X3: a] :
( ( undire7337870655677353998dges_a @ ( cons_a @ X3 @ nil_a ) )
= nil_set_a ) ).
% walk_edges.simps(2)
thf(fact_276_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_277_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_278_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_279_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_280_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_281_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_282_append_Oassoc,axiom,
! [A: list_a,B: list_a,C: list_a] :
( ( append_a @ ( append_a @ A @ B ) @ C )
= ( append_a @ A @ ( append_a @ B @ C ) ) ) ).
% append.assoc
thf(fact_283_append_Oassoc,axiom,
! [A: list_set_a,B: list_set_a,C: list_set_a] :
( ( append_set_a @ ( append_set_a @ A @ B ) @ C )
= ( append_set_a @ A @ ( append_set_a @ B @ C ) ) ) ).
% append.assoc
thf(fact_284_append_Oright__neutral,axiom,
! [A: list_a] :
( ( append_a @ A @ nil_a )
= A ) ).
% append.right_neutral
thf(fact_285_append_Oright__neutral,axiom,
! [A: list_set_a] :
( ( append_set_a @ A @ nil_set_a )
= A ) ).
% append.right_neutral
thf(fact_286_append__Nil2,axiom,
! [Xs: list_a] :
( ( append_a @ Xs @ nil_a )
= Xs ) ).
% append_Nil2
thf(fact_287_append__Nil2,axiom,
! [Xs: list_set_a] :
( ( append_set_a @ Xs @ nil_set_a )
= Xs ) ).
% append_Nil2
thf(fact_288_append__self__conv,axiom,
! [Xs: list_a,Ys: list_a] :
( ( ( append_a @ Xs @ Ys )
= Xs )
= ( Ys = nil_a ) ) ).
% append_self_conv
thf(fact_289_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_290_self__append__conv,axiom,
! [Y2: list_a,Ys: list_a] :
( ( Y2
= ( append_a @ Y2 @ Ys ) )
= ( Ys = nil_a ) ) ).
% self_append_conv
thf(fact_291_self__append__conv,axiom,
! [Y2: list_set_a,Ys: list_set_a] :
( ( Y2
= ( append_set_a @ Y2 @ Ys ) )
= ( Ys = nil_set_a ) ) ).
% self_append_conv
thf(fact_292_append__self__conv2,axiom,
! [Xs: list_a,Ys: list_a] :
( ( ( append_a @ Xs @ Ys )
= Ys )
= ( Xs = nil_a ) ) ).
% append_self_conv2
thf(fact_293_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_294_self__append__conv2,axiom,
! [Y2: list_a,Xs: list_a] :
( ( Y2
= ( append_a @ Xs @ Y2 ) )
= ( Xs = nil_a ) ) ).
% self_append_conv2
thf(fact_295_self__append__conv2,axiom,
! [Y2: list_set_a,Xs: list_set_a] :
( ( Y2
= ( append_set_a @ Xs @ Y2 ) )
= ( Xs = nil_set_a ) ) ).
% self_append_conv2
thf(fact_296_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_297_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_298_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_299_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_300_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_301_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_302_append1__eq__conv,axiom,
! [Xs: list_a,X3: a,Ys: list_a,Y2: a] :
( ( ( append_a @ Xs @ ( cons_a @ X3 @ nil_a ) )
= ( append_a @ Ys @ ( cons_a @ Y2 @ nil_a ) ) )
= ( ( Xs = Ys )
& ( X3 = Y2 ) ) ) ).
% append1_eq_conv
thf(fact_303_append1__eq__conv,axiom,
! [Xs: list_set_a,X3: set_a,Ys: list_set_a,Y2: set_a] :
( ( ( append_set_a @ Xs @ ( cons_set_a @ X3 @ nil_set_a ) )
= ( append_set_a @ Ys @ ( cons_set_a @ Y2 @ nil_set_a ) ) )
= ( ( Xs = Ys )
& ( X3 = Y2 ) ) ) ).
% append1_eq_conv
thf(fact_304_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_305_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_306_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_307_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_308_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_309_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_310_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_311_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_312_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_313_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_314_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_315_rev__eq__Cons__iff,axiom,
! [Xs: list_a,Y2: a,Ys: list_a] :
( ( ( rev_a @ Xs )
= ( cons_a @ Y2 @ Ys ) )
= ( Xs
= ( append_a @ ( rev_a @ Ys ) @ ( cons_a @ Y2 @ nil_a ) ) ) ) ).
% rev_eq_Cons_iff
thf(fact_316_rev__eq__Cons__iff,axiom,
! [Xs: list_set_a,Y2: set_a,Ys: list_set_a] :
( ( ( rev_set_a @ Xs )
= ( cons_set_a @ Y2 @ Ys ) )
= ( Xs
= ( append_set_a @ ( rev_set_a @ Ys ) @ ( cons_set_a @ Y2 @ nil_set_a ) ) ) ) ).
% rev_eq_Cons_iff
thf(fact_317_last__snoc,axiom,
! [Xs: list_a,X3: a] :
( ( last_a @ ( append_a @ Xs @ ( cons_a @ X3 @ nil_a ) ) )
= X3 ) ).
% last_snoc
thf(fact_318_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_319_ulgraph_Ois__isolated__vertex_Ocong,axiom,
undire8931668460104145173rtex_a = undire8931668460104145173rtex_a ).
% ulgraph.is_isolated_vertex.cong
thf(fact_320_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_321_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_322_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_323_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 ) )
= ( ? [Us: list_a] :
( ( ( Xs
= ( append_a @ Zs @ Us ) )
& ( ( append_a @ Us @ Ys )
= Ts ) )
| ( ( ( append_a @ Xs @ Us )
= Zs )
& ( Ys
= ( append_a @ Us @ Ts ) ) ) ) ) ) ).
% append_eq_append_conv2
thf(fact_324_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 ) )
= ( ? [Us: list_set_a] :
( ( ( Xs
= ( append_set_a @ Zs @ Us ) )
& ( ( append_set_a @ Us @ Ys )
= Ts ) )
| ( ( ( append_set_a @ Xs @ Us )
= Zs )
& ( Ys
= ( append_set_a @ Us @ Ts ) ) ) ) ) ) ).
% append_eq_append_conv2
thf(fact_325_append__eq__appendI,axiom,
! [Xs: list_a,Xs1: list_a,Zs: list_a,Ys: list_a,Us2: list_a] :
( ( ( append_a @ Xs @ Xs1 )
= Zs )
=> ( ( Ys
= ( append_a @ Xs1 @ Us2 ) )
=> ( ( append_a @ Xs @ Ys )
= ( append_a @ Zs @ Us2 ) ) ) ) ).
% append_eq_appendI
thf(fact_326_append__eq__appendI,axiom,
! [Xs: list_set_a,Xs1: list_set_a,Zs: list_set_a,Ys: list_set_a,Us2: list_set_a] :
( ( ( append_set_a @ Xs @ Xs1 )
= Zs )
=> ( ( Ys
= ( append_set_a @ Xs1 @ Us2 ) )
=> ( ( append_set_a @ Xs @ Ys )
= ( append_set_a @ Zs @ Us2 ) ) ) ) ).
% append_eq_appendI
thf(fact_327_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_328_ulgraph_Oedge__density_Ocong,axiom,
undire297304480579013331sity_a = undire297304480579013331sity_a ).
% ulgraph.edge_density.cong
thf(fact_329_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_330_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,E6: set_set_a] :
( ( undire7103218114511261257raph_a @ V_H @ E_H @ V_G @ E_G )
=> ( ( undire2554140024507503526stem_a @ V4 @ E4 )
=> ( ( undire2554140024507503526stem_a @ V3 @ E3 )
=> ( ( undire2554140024507503526stem_a @ V5 @ E6 )
=> ( ( undire7103218114511261257raph_a @ V5 @ E6 @ V3 @ E3 )
=> ( ( undire7103218114511261257raph_a @ V3 @ E3 @ V4 @ E4 )
=> ( undire7103218114511261257raph_a @ V5 @ E6 @ V4 @ E4 ) ) ) ) ) ) ) ).
% subgraph.subgraph_trans
thf(fact_331_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_332_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_333_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_334_graph__system_Ointro,axiom,
! [Edges: set_set_set_a,Vertices: set_set_a] :
( ! [E7: set_set_a] :
( ( member_set_set_a @ E7 @ Edges )
=> ( ord_le3724670747650509150_set_a @ E7 @ Vertices ) )
=> ( undire7159349782766787846_set_a @ Vertices @ Edges ) ) ).
% graph_system.intro
thf(fact_335_graph__system_Ointro,axiom,
! [Edges: set_se5735800977113168103od_a_a,Vertices: set_Product_prod_a_a] :
( ! [E7: set_Product_prod_a_a] :
( ( member1816616512716248880od_a_a @ E7 @ Edges )
=> ( ord_le746702958409616551od_a_a @ E7 @ Vertices ) )
=> ( undire1860116983885411791od_a_a @ Vertices @ Edges ) ) ).
% graph_system.intro
thf(fact_336_graph__system_Ointro,axiom,
! [Edges: set_set_a,Vertices: set_a] :
( ! [E7: set_a] :
( ( member_set_a @ E7 @ Edges )
=> ( ord_less_eq_set_a @ E7 @ Vertices ) )
=> ( undire2554140024507503526stem_a @ Vertices @ Edges ) ) ).
% graph_system.intro
thf(fact_337_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_338_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_339_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_340_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_341_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_342_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_343_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_344_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_345_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_346_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_347_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_348_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_349_append__Nil,axiom,
! [Ys: list_a] :
( ( append_a @ nil_a @ Ys )
= Ys ) ).
% append_Nil
thf(fact_350_append__Nil,axiom,
! [Ys: list_set_a] :
( ( append_set_a @ nil_set_a @ Ys )
= Ys ) ).
% append_Nil
thf(fact_351_append_Oleft__neutral,axiom,
! [A: list_a] :
( ( append_a @ nil_a @ A )
= A ) ).
% append.left_neutral
thf(fact_352_append_Oleft__neutral,axiom,
! [A: list_set_a] :
( ( append_set_a @ nil_set_a @ A )
= A ) ).
% append.left_neutral
thf(fact_353_eq__Nil__appendI,axiom,
! [Xs: list_a,Ys: list_a] :
( ( Xs = Ys )
=> ( Xs
= ( append_a @ nil_a @ Ys ) ) ) ).
% eq_Nil_appendI
thf(fact_354_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_355_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_356_comp__sgraph_Owalk__edges_Osimps_I1_J,axiom,
( ( undire7337870655677353998dges_a @ nil_a )
= nil_set_a ) ).
% comp_sgraph.walk_edges.simps(1)
thf(fact_357_comp__sgraph_Ograph__system__axioms,axiom,
! [S: set_set_a] : ( undire7159349782766787846_set_a @ S @ ( undire8247866692393712962_set_a @ S ) ) ).
% comp_sgraph.graph_system_axioms
thf(fact_358_comp__sgraph_Ograph__system__axioms,axiom,
! [S: set_a] : ( undire2554140024507503526stem_a @ S @ ( undire2918257014606996450dges_a @ S ) ) ).
% comp_sgraph.graph_system_axioms
thf(fact_359_ulgraph_Oaxioms_I1_J,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a] :
( ( undire6886684016831807756_set_a @ Vertices @ Edges )
=> ( undire7159349782766787846_set_a @ Vertices @ Edges ) ) ).
% ulgraph.axioms(1)
thf(fact_360_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_361_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_362_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_363_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_364_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_365_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_366_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_367_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_368_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_369_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_370_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_371_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_372_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_373_comp__sgraph_Osubgraph__complete,axiom,
! [S: set_a] : ( undire7103218114511261257raph_a @ S @ ( undire2918257014606996450dges_a @ S ) @ S @ ( undire2918257014606996450dges_a @ S ) ) ).
% comp_sgraph.subgraph_complete
thf(fact_374_comp__sgraph_Osubgraph__complete,axiom,
! [S: set_set_a] : ( undire1186139521737116585_set_a @ S @ ( undire8247866692393712962_set_a @ S ) @ S @ ( undire8247866692393712962_set_a @ S ) ) ).
% comp_sgraph.subgraph_complete
thf(fact_375_subgraph_Ois__subgraph__ulgraph,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 )
=> ( ( undire6886684016831807756_set_a @ V_G @ E_G )
=> ( undire6886684016831807756_set_a @ V_H @ E_H ) ) ) ).
% subgraph.is_subgraph_ulgraph
thf(fact_376_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_377_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_378_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_379_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_380_comp__sgraph_Oedge__density__commute,axiom,
! [S: set_set_a,X5: set_set_a,Y4: set_set_a] :
( ( undire8927637694342045747_set_a @ ( undire8247866692393712962_set_a @ S ) @ X5 @ Y4 )
= ( undire8927637694342045747_set_a @ ( undire8247866692393712962_set_a @ S ) @ Y4 @ X5 ) ) ).
% comp_sgraph.edge_density_commute
thf(fact_381_comp__sgraph_Oedge__density__commute,axiom,
! [S: set_a,X5: set_a,Y4: set_a] :
( ( undire297304480579013331sity_a @ ( undire2918257014606996450dges_a @ S ) @ X5 @ Y4 )
= ( undire297304480579013331sity_a @ ( undire2918257014606996450dges_a @ S ) @ Y4 @ X5 ) ) ).
% comp_sgraph.edge_density_commute
thf(fact_382_ulgraph_Oedge__density__commute,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,X5: set_set_a,Y4: set_set_a] :
( ( undire6886684016831807756_set_a @ Vertices @ Edges )
=> ( ( undire8927637694342045747_set_a @ Edges @ X5 @ Y4 )
= ( undire8927637694342045747_set_a @ Edges @ Y4 @ X5 ) ) ) ).
% ulgraph.edge_density_commute
thf(fact_383_ulgraph_Oedge__density__commute,axiom,
! [Vertices: set_a,Edges: set_set_a,X5: set_a,Y4: set_a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( undire297304480579013331sity_a @ Edges @ X5 @ Y4 )
= ( undire297304480579013331sity_a @ Edges @ Y4 @ X5 ) ) ) ).
% ulgraph.edge_density_commute
thf(fact_384_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_385_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_386_rev__exhaust,axiom,
! [Xs: list_a] :
( ( Xs != nil_a )
=> ~ ! [Ys2: list_a,Y: a] :
( Xs
!= ( append_a @ Ys2 @ ( cons_a @ Y @ nil_a ) ) ) ) ).
% rev_exhaust
thf(fact_387_rev__exhaust,axiom,
! [Xs: list_set_a] :
( ( Xs != nil_set_a )
=> ~ ! [Ys2: list_set_a,Y: set_a] :
( Xs
!= ( append_set_a @ Ys2 @ ( cons_set_a @ Y @ nil_set_a ) ) ) ) ).
% rev_exhaust
thf(fact_388_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_389_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_390_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_391_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_392_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_393_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_394_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_395_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_396_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_397_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_398_longest__common__prefix,axiom,
! [Xs: list_a,Ys: list_a] :
? [Ps: list_a,Xs3: list_a,Ys5: list_a] :
( ( Xs
= ( append_a @ Ps @ Xs3 ) )
& ( Ys
= ( append_a @ Ps @ Ys5 ) )
& ( ( Xs3 = nil_a )
| ( Ys5 = nil_a )
| ( ( hd_a @ Xs3 )
!= ( hd_a @ Ys5 ) ) ) ) ).
% longest_common_prefix
thf(fact_399_longest__common__prefix,axiom,
! [Xs: list_set_a,Ys: list_set_a] :
? [Ps: list_set_a,Xs3: list_set_a,Ys5: list_set_a] :
( ( Xs
= ( append_set_a @ Ps @ Xs3 ) )
& ( Ys
= ( append_set_a @ Ps @ Ys5 ) )
& ( ( Xs3 = nil_set_a )
| ( Ys5 = nil_set_a )
| ( ( hd_set_a @ Xs3 )
!= ( hd_set_a @ Ys5 ) ) ) ) ).
% longest_common_prefix
thf(fact_400_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_401_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_402_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_403_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_404_longest__common__suffix,axiom,
! [Xs: list_a,Ys: list_a] :
? [Ss: list_a,Xs3: list_a,Ys5: list_a] :
( ( Xs
= ( append_a @ Xs3 @ Ss ) )
& ( Ys
= ( append_a @ Ys5 @ Ss ) )
& ( ( Xs3 = nil_a )
| ( Ys5 = nil_a )
| ( ( last_a @ Xs3 )
!= ( last_a @ Ys5 ) ) ) ) ).
% longest_common_suffix
thf(fact_405_longest__common__suffix,axiom,
! [Xs: list_set_a,Ys: list_set_a] :
? [Ss: list_set_a,Xs3: list_set_a,Ys5: list_set_a] :
( ( Xs
= ( append_set_a @ Xs3 @ Ss ) )
& ( Ys
= ( append_set_a @ Ys5 @ Ss ) )
& ( ( Xs3 = nil_set_a )
| ( Ys5 = nil_set_a )
| ( ( last_set_a @ Xs3 )
!= ( last_set_a @ Ys5 ) ) ) ) ).
% longest_common_suffix
thf(fact_406_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_407_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_408_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_409_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_410_not__distinct__decomp,axiom,
! [Ws: list_a] :
( ~ ( distinct_a @ Ws )
=> ? [Xs2: list_a,Ys2: list_a,Zs2: list_a,Y: a] :
( Ws
= ( append_a @ Xs2 @ ( append_a @ ( cons_a @ Y @ nil_a ) @ ( append_a @ Ys2 @ ( append_a @ ( cons_a @ Y @ nil_a ) @ Zs2 ) ) ) ) ) ) ).
% not_distinct_decomp
thf(fact_411_not__distinct__decomp,axiom,
! [Ws: list_set_a] :
( ~ ( distinct_set_a @ Ws )
=> ? [Xs2: list_set_a,Ys2: list_set_a,Zs2: list_set_a,Y: set_a] :
( Ws
= ( append_set_a @ Xs2 @ ( append_set_a @ ( cons_set_a @ Y @ nil_set_a ) @ ( append_set_a @ Ys2 @ ( append_set_a @ ( cons_set_a @ Y @ nil_set_a ) @ Zs2 ) ) ) ) ) ) ).
% not_distinct_decomp
thf(fact_412_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_413_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_414_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_415_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_416_comp__sgraph_Ois__trail__def,axiom,
! [S: set_set_a,Xs: list_set_a] :
( ( undire1224551742100448159_set_a @ S @ ( undire8247866692393712962_set_a @ S ) @ Xs )
= ( ( undire3014741414213135564_set_a @ S @ ( undire8247866692393712962_set_a @ S ) @ Xs )
& ( distinct_set_set_a @ ( undire6234387080713648494_set_a @ Xs ) ) ) ) ).
% comp_sgraph.is_trail_def
thf(fact_417_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_418_ulgraph_Ois__trail__def,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 )
= ( ( undire3014741414213135564_set_a @ Vertices @ Edges @ Xs )
& ( distinct_set_set_a @ ( undire6234387080713648494_set_a @ Xs ) ) ) ) ) ).
% ulgraph.is_trail_def
thf(fact_419_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_420_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_421_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_422_comp__sgraph_Ois__isolated__vertex__edge,axiom,
! [S: set_set_a,V: set_a,E: set_set_a] :
( ( undire6879241558604981877_set_a @ S @ ( undire8247866692393712962_set_a @ S ) @ V )
=> ( ( member_set_set_a @ E @ ( undire8247866692393712962_set_a @ S ) )
=> ~ ( undire2320338297334612420_set_a @ V @ E ) ) ) ).
% comp_sgraph.is_isolated_vertex_edge
thf(fact_423_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_424_comp__sgraph_Ois__isolated__vertex__no__loop,axiom,
! [S: set_set_a,V: set_a] :
( ( undire6879241558604981877_set_a @ S @ ( undire8247866692393712962_set_a @ S ) @ V )
=> ~ ( undire5774735625301615776_set_a @ ( undire8247866692393712962_set_a @ S ) @ V ) ) ).
% comp_sgraph.is_isolated_vertex_no_loop
thf(fact_425_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_426_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_427_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_428_ulgraph_Ois__isolated__vertex__edge,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,V: set_a,E: set_set_a] :
( ( undire6886684016831807756_set_a @ Vertices @ Edges )
=> ( ( undire6879241558604981877_set_a @ Vertices @ Edges @ V )
=> ( ( member_set_set_a @ E @ Edges )
=> ~ ( undire2320338297334612420_set_a @ V @ E ) ) ) ) ).
% ulgraph.is_isolated_vertex_edge
thf(fact_429_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_430_ulgraph_Ois__isolated__vertex__no__loop,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,V: set_a] :
( ( undire6886684016831807756_set_a @ Vertices @ Edges )
=> ( ( undire6879241558604981877_set_a @ Vertices @ Edges @ V )
=> ~ ( undire5774735625301615776_set_a @ Edges @ V ) ) ) ).
% ulgraph.is_isolated_vertex_no_loop
thf(fact_431_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_432_comp__sgraph_Ois__walk__decomp,axiom,
! [S: set_set_a,Xs: list_set_a,Y2: 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 @ Y2 @ nil_set_a ) @ ( append_set_a @ Ys @ ( append_set_a @ ( cons_set_a @ Y2 @ nil_set_a ) @ Zs ) ) ) ) )
=> ( undire3014741414213135564_set_a @ S @ ( undire8247866692393712962_set_a @ S ) @ ( append_set_a @ Xs @ ( append_set_a @ ( cons_set_a @ Y2 @ nil_set_a ) @ Zs ) ) ) ) ).
% comp_sgraph.is_walk_decomp
thf(fact_433_comp__sgraph_Ois__walk__decomp,axiom,
! [S: set_a,Xs: list_a,Y2: a,Ys: list_a,Zs: list_a] :
( ( undire6133010728901294956walk_a @ S @ ( undire2918257014606996450dges_a @ S ) @ ( append_a @ Xs @ ( append_a @ ( cons_a @ Y2 @ nil_a ) @ ( append_a @ Ys @ ( append_a @ ( cons_a @ Y2 @ nil_a ) @ Zs ) ) ) ) )
=> ( undire6133010728901294956walk_a @ S @ ( undire2918257014606996450dges_a @ S ) @ ( append_a @ Xs @ ( append_a @ ( cons_a @ Y2 @ nil_a ) @ Zs ) ) ) ) ).
% comp_sgraph.is_walk_decomp
thf(fact_434_ulgraph_Ois__walk__decomp,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,Xs: list_set_a,Y2: 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 @ Y2 @ nil_set_a ) @ ( append_set_a @ Ys @ ( append_set_a @ ( cons_set_a @ Y2 @ nil_set_a ) @ Zs ) ) ) ) )
=> ( undire3014741414213135564_set_a @ Vertices @ Edges @ ( append_set_a @ Xs @ ( append_set_a @ ( cons_set_a @ Y2 @ nil_set_a ) @ Zs ) ) ) ) ) ).
% ulgraph.is_walk_decomp
thf(fact_435_ulgraph_Ois__walk__decomp,axiom,
! [Vertices: set_a,Edges: set_set_a,Xs: list_a,Y2: a,Ys: list_a,Zs: list_a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( undire6133010728901294956walk_a @ Vertices @ Edges @ ( append_a @ Xs @ ( append_a @ ( cons_a @ Y2 @ nil_a ) @ ( append_a @ Ys @ ( append_a @ ( cons_a @ Y2 @ nil_a ) @ Zs ) ) ) ) )
=> ( undire6133010728901294956walk_a @ Vertices @ Edges @ ( append_a @ Xs @ ( append_a @ ( cons_a @ Y2 @ nil_a ) @ Zs ) ) ) ) ) ).
% ulgraph.is_walk_decomp
thf(fact_436_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_437_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_438_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_439_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_440_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_441_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_442_induced__is__graph__sys,axiom,
! [V3: set_a] : ( undire2554140024507503526stem_a @ V3 @ ( undire7777452895879145676dges_a @ edges @ V3 ) ) ).
% induced_is_graph_sys
thf(fact_443_subsetI,axiom,
! [A2: set_a,B2: set_a] :
( ! [X4: a] :
( ( member_a @ X4 @ A2 )
=> ( member_a @ X4 @ B2 ) )
=> ( ord_less_eq_set_a @ A2 @ B2 ) ) ).
% subsetI
thf(fact_444_subsetI,axiom,
! [A2: set_set_a,B2: set_set_a] :
( ! [X4: set_a] :
( ( member_set_a @ X4 @ A2 )
=> ( member_set_a @ X4 @ B2 ) )
=> ( ord_le3724670747650509150_set_a @ A2 @ B2 ) ) ).
% subsetI
thf(fact_445_subsetI,axiom,
! [A2: set_Product_prod_a_a,B2: set_Product_prod_a_a] :
( ! [X4: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X4 @ A2 )
=> ( member1426531477525435216od_a_a @ X4 @ B2 ) )
=> ( ord_le746702958409616551od_a_a @ A2 @ B2 ) ) ).
% subsetI
thf(fact_446_subset__antisym,axiom,
! [A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ord_less_eq_set_a @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% subset_antisym
thf(fact_447_subset__antisym,axiom,
! [A2: set_set_a,B2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B2 )
=> ( ( ord_le3724670747650509150_set_a @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% subset_antisym
thf(fact_448_subset__antisym,axiom,
! [A2: set_Product_prod_a_a,B2: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A2 @ B2 )
=> ( ( ord_le746702958409616551od_a_a @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% subset_antisym
thf(fact_449_order__refl,axiom,
! [X3: set_a] : ( ord_less_eq_set_a @ X3 @ X3 ) ).
% order_refl
thf(fact_450_order__refl,axiom,
! [X3: set_set_a] : ( ord_le3724670747650509150_set_a @ X3 @ X3 ) ).
% order_refl
thf(fact_451_order__refl,axiom,
! [X3: real] : ( ord_less_eq_real @ X3 @ X3 ) ).
% order_refl
thf(fact_452_order__refl,axiom,
! [X3: nat] : ( ord_less_eq_nat @ X3 @ X3 ) ).
% order_refl
thf(fact_453_order__refl,axiom,
! [X3: set_Product_prod_a_a] : ( ord_le746702958409616551od_a_a @ X3 @ X3 ) ).
% order_refl
thf(fact_454_dual__order_Orefl,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ A @ A ) ).
% dual_order.refl
thf(fact_455_dual__order_Orefl,axiom,
! [A: set_set_a] : ( ord_le3724670747650509150_set_a @ A @ A ) ).
% dual_order.refl
thf(fact_456_dual__order_Orefl,axiom,
! [A: real] : ( ord_less_eq_real @ A @ A ) ).
% dual_order.refl
thf(fact_457_dual__order_Orefl,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% dual_order.refl
thf(fact_458_dual__order_Orefl,axiom,
! [A: set_Product_prod_a_a] : ( ord_le746702958409616551od_a_a @ A @ A ) ).
% dual_order.refl
thf(fact_459_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_460_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_461_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_462_set__rev,axiom,
! [Xs: list_a] :
( ( set_a2 @ ( rev_a @ Xs ) )
= ( set_a2 @ Xs ) ) ).
% set_rev
thf(fact_463_set__rev,axiom,
! [Xs: list_set_a] :
( ( set_set_a2 @ ( rev_set_a @ Xs ) )
= ( set_set_a2 @ Xs ) ) ).
% set_rev
thf(fact_464_rotate1__is__Nil__conv,axiom,
! [Xs: list_a] :
( ( ( rotate1_a @ Xs )
= nil_a )
= ( Xs = nil_a ) ) ).
% rotate1_is_Nil_conv
thf(fact_465_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_466_set__rotate1,axiom,
! [Xs: list_a] :
( ( set_a2 @ ( rotate1_a @ Xs ) )
= ( set_a2 @ Xs ) ) ).
% set_rotate1
thf(fact_467_set__rotate1,axiom,
! [Xs: list_set_a] :
( ( set_set_a2 @ ( rotate1_set_a @ Xs ) )
= ( set_set_a2 @ Xs ) ) ).
% set_rotate1
thf(fact_468_distinct1__rotate,axiom,
! [Xs: list_a] :
( ( distinct_a @ ( rotate1_a @ Xs ) )
= ( distinct_a @ Xs ) ) ).
% distinct1_rotate
thf(fact_469_distinct1__rotate,axiom,
! [Xs: list_set_a] :
( ( distinct_set_a @ ( rotate1_set_a @ Xs ) )
= ( distinct_set_a @ Xs ) ) ).
% distinct1_rotate
thf(fact_470_graph__system_Oinduced__edges_Ocong,axiom,
undire7777452895879145676dges_a = undire7777452895879145676dges_a ).
% graph_system.induced_edges.cong
thf(fact_471_subset__code_I1_J,axiom,
! [Xs: list_a,B2: set_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ Xs ) @ B2 )
= ( ! [X2: a] :
( ( member_a @ X2 @ ( set_a2 @ Xs ) )
=> ( member_a @ X2 @ B2 ) ) ) ) ).
% subset_code(1)
thf(fact_472_subset__code_I1_J,axiom,
! [Xs: list_set_a,B2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ ( set_set_a2 @ Xs ) @ B2 )
= ( ! [X2: set_a] :
( ( member_set_a @ X2 @ ( set_set_a2 @ Xs ) )
=> ( member_set_a @ X2 @ B2 ) ) ) ) ).
% subset_code(1)
thf(fact_473_subset__code_I1_J,axiom,
! [Xs: list_P1396940483166286381od_a_a,B2: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ ( set_Product_prod_a_a2 @ Xs ) @ B2 )
= ( ! [X2: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X2 @ ( set_Product_prod_a_a2 @ Xs ) )
=> ( member1426531477525435216od_a_a @ X2 @ B2 ) ) ) ) ).
% subset_code(1)
thf(fact_474_list_Oset__intros_I2_J,axiom,
! [Y2: a,X22: list_a,X21: a] :
( ( member_a @ Y2 @ ( set_a2 @ X22 ) )
=> ( member_a @ Y2 @ ( set_a2 @ ( cons_a @ X21 @ X22 ) ) ) ) ).
% list.set_intros(2)
thf(fact_475_list_Oset__intros_I2_J,axiom,
! [Y2: set_a,X22: list_set_a,X21: set_a] :
( ( member_set_a @ Y2 @ ( set_set_a2 @ X22 ) )
=> ( member_set_a @ Y2 @ ( set_set_a2 @ ( cons_set_a @ X21 @ X22 ) ) ) ) ).
% list.set_intros(2)
thf(fact_476_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_477_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_478_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_479_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_480_set__ConsD,axiom,
! [Y2: a,X3: a,Xs: list_a] :
( ( member_a @ Y2 @ ( set_a2 @ ( cons_a @ X3 @ Xs ) ) )
=> ( ( Y2 = X3 )
| ( member_a @ Y2 @ ( set_a2 @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_481_set__ConsD,axiom,
! [Y2: set_a,X3: set_a,Xs: list_set_a] :
( ( member_set_a @ Y2 @ ( set_set_a2 @ ( cons_set_a @ X3 @ Xs ) ) )
=> ( ( Y2 = X3 )
| ( member_set_a @ Y2 @ ( set_set_a2 @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_482_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_483_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_484_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_485_comp__sgraph_Owellformed__all__edges,axiom,
! [S: set_set_a] : ( ord_le5722252365846178494_set_a @ ( undire8247866692393712962_set_a @ S ) @ ( undire8247866692393712962_set_a @ S ) ) ).
% comp_sgraph.wellformed_all_edges
thf(fact_486_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_487_comp__sgraph_Oinduced__edges__self,axiom,
! [S: set_set_a] :
( ( undire7854589003810675244_set_a @ ( undire8247866692393712962_set_a @ S ) @ S )
= ( undire8247866692393712962_set_a @ S ) ) ).
% comp_sgraph.induced_edges_self
thf(fact_488_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_489_rotate1_Osimps_I1_J,axiom,
( ( rotate1_a @ nil_a )
= nil_a ) ).
% rotate1.simps(1)
thf(fact_490_rotate1_Osimps_I1_J,axiom,
( ( rotate1_set_a @ nil_set_a )
= nil_set_a ) ).
% rotate1.simps(1)
thf(fact_491_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_492_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_493_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_494_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_495_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_496_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_497_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_498_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_499_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_500_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_501_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_502_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_503_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_504_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_505_append__Cons__eq__iff,axiom,
! [X3: a,Xs: list_a,Ys: list_a,Xs4: list_a,Ys6: list_a] :
( ~ ( member_a @ X3 @ ( set_a2 @ Xs ) )
=> ( ~ ( member_a @ X3 @ ( set_a2 @ Ys ) )
=> ( ( ( append_a @ Xs @ ( cons_a @ X3 @ Ys ) )
= ( append_a @ Xs4 @ ( cons_a @ X3 @ Ys6 ) ) )
= ( ( Xs = Xs4 )
& ( Ys = Ys6 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_506_append__Cons__eq__iff,axiom,
! [X3: set_a,Xs: list_set_a,Ys: list_set_a,Xs4: list_set_a,Ys6: 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 @ Xs4 @ ( cons_set_a @ X3 @ Ys6 ) ) )
= ( ( Xs = Xs4 )
& ( Ys = Ys6 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_507_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_508_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_509_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_510_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_511_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_512_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_513_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_514_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_515_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_516_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_517_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_518_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_519_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_520_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_521_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_522_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_523_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_524_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_525_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_526_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_527_hd__in__set,axiom,
! [Xs: list_a] :
( ( Xs != nil_a )
=> ( member_a @ ( hd_a @ Xs ) @ ( set_a2 @ Xs ) ) ) ).
% hd_in_set
thf(fact_528_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_529_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_530_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_531_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_532_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_533_last__in__set,axiom,
! [As: list_a] :
( ( As != nil_a )
=> ( member_a @ ( last_a @ As ) @ ( set_a2 @ As ) ) ) ).
% last_in_set
thf(fact_534_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_535_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_536_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_537_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_538_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_539_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_540_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_541_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_542_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_543_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_544_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_545_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_546_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_547_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_548_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_549_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_550_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_551_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_552_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_553_comp__sgraph_Oinduced__is__graph__sys,axiom,
! [V3: set_set_a,S: set_set_a] : ( undire7159349782766787846_set_a @ V3 @ ( undire7854589003810675244_set_a @ ( undire8247866692393712962_set_a @ S ) @ V3 ) ) ).
% comp_sgraph.induced_is_graph_sys
thf(fact_554_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_555_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_556_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_557_order__antisym__conv,axiom,
! [Y2: set_a,X3: set_a] :
( ( ord_less_eq_set_a @ Y2 @ X3 )
=> ( ( ord_less_eq_set_a @ X3 @ Y2 )
= ( X3 = Y2 ) ) ) ).
% order_antisym_conv
thf(fact_558_order__antisym__conv,axiom,
! [Y2: set_set_a,X3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ Y2 @ X3 )
=> ( ( ord_le3724670747650509150_set_a @ X3 @ Y2 )
= ( X3 = Y2 ) ) ) ).
% order_antisym_conv
thf(fact_559_order__antisym__conv,axiom,
! [Y2: real,X3: real] :
( ( ord_less_eq_real @ Y2 @ X3 )
=> ( ( ord_less_eq_real @ X3 @ Y2 )
= ( X3 = Y2 ) ) ) ).
% order_antisym_conv
thf(fact_560_order__antisym__conv,axiom,
! [Y2: nat,X3: nat] :
( ( ord_less_eq_nat @ Y2 @ X3 )
=> ( ( ord_less_eq_nat @ X3 @ Y2 )
= ( X3 = Y2 ) ) ) ).
% order_antisym_conv
thf(fact_561_order__antisym__conv,axiom,
! [Y2: set_Product_prod_a_a,X3: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ Y2 @ X3 )
=> ( ( ord_le746702958409616551od_a_a @ X3 @ Y2 )
= ( X3 = Y2 ) ) ) ).
% order_antisym_conv
thf(fact_562_linorder__le__cases,axiom,
! [X3: real,Y2: real] :
( ~ ( ord_less_eq_real @ X3 @ Y2 )
=> ( ord_less_eq_real @ Y2 @ X3 ) ) ).
% linorder_le_cases
thf(fact_563_linorder__le__cases,axiom,
! [X3: nat,Y2: nat] :
( ~ ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ X3 ) ) ).
% linorder_le_cases
thf(fact_564_ord__le__eq__subst,axiom,
! [A: real,B: real,F: real > real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: real,Y: real] :
( ( ord_less_eq_real @ X4 @ Y )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_565_ord__le__eq__subst,axiom,
! [A: real,B: real,F: real > nat,C: nat] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: real,Y: real] :
( ( ord_less_eq_real @ X4 @ Y )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_566_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > real,C: real] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: nat,Y: nat] :
( ( ord_less_eq_nat @ X4 @ Y )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_567_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: nat,Y: nat] :
( ( ord_less_eq_nat @ X4 @ Y )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_568_ord__le__eq__subst,axiom,
! [A: set_a,B: set_a,F: set_a > real,C: real] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_569_ord__le__eq__subst,axiom,
! [A: set_a,B: set_a,F: set_a > nat,C: nat] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_570_ord__le__eq__subst,axiom,
! [A: real,B: real,F: real > set_a,C: set_a] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: real,Y: real] :
( ( ord_less_eq_real @ X4 @ Y )
=> ( ord_less_eq_set_a @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_set_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_571_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > set_a,C: set_a] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: nat,Y: nat] :
( ( ord_less_eq_nat @ X4 @ Y )
=> ( ord_less_eq_set_a @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_set_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_572_ord__le__eq__subst,axiom,
! [A: set_a,B: set_a,F: set_a > set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y )
=> ( ord_less_eq_set_a @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_set_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_573_ord__le__eq__subst,axiom,
! [A: set_set_a,B: set_set_a,F: set_set_a > real,C: real] :
( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: set_set_a,Y: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X4 @ Y )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_574_ord__eq__le__subst,axiom,
! [A: real,F: real > real,B: real,C: real] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X4: real,Y: real] :
( ( ord_less_eq_real @ X4 @ Y )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_575_ord__eq__le__subst,axiom,
! [A: nat,F: real > nat,B: real,C: real] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X4: real,Y: real] :
( ( ord_less_eq_real @ X4 @ Y )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_576_ord__eq__le__subst,axiom,
! [A: real,F: nat > real,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X4: nat,Y: nat] :
( ( ord_less_eq_nat @ X4 @ Y )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_577_ord__eq__le__subst,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X4: nat,Y: nat] :
( ( ord_less_eq_nat @ X4 @ Y )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_578_ord__eq__le__subst,axiom,
! [A: real,F: set_a > real,B: set_a,C: set_a] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ! [X4: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_579_ord__eq__le__subst,axiom,
! [A: nat,F: set_a > nat,B: set_a,C: set_a] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ! [X4: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_580_ord__eq__le__subst,axiom,
! [A: set_a,F: real > set_a,B: real,C: real] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X4: real,Y: real] :
( ( ord_less_eq_real @ X4 @ Y )
=> ( ord_less_eq_set_a @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_set_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_581_ord__eq__le__subst,axiom,
! [A: set_a,F: nat > set_a,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X4: nat,Y: nat] :
( ( ord_less_eq_nat @ X4 @ Y )
=> ( ord_less_eq_set_a @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_set_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_582_ord__eq__le__subst,axiom,
! [A: set_a,F: set_a > set_a,B: set_a,C: set_a] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ! [X4: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y )
=> ( ord_less_eq_set_a @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_set_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_583_ord__eq__le__subst,axiom,
! [A: real,F: set_set_a > real,B: set_set_a,C: set_set_a] :
( ( A
= ( F @ B ) )
=> ( ( ord_le3724670747650509150_set_a @ B @ C )
=> ( ! [X4: set_set_a,Y: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X4 @ Y )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_584_linorder__linear,axiom,
! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
| ( ord_less_eq_real @ Y2 @ X3 ) ) ).
% linorder_linear
thf(fact_585_linorder__linear,axiom,
! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
| ( ord_less_eq_nat @ Y2 @ X3 ) ) ).
% linorder_linear
thf(fact_586_order__eq__refl,axiom,
! [X3: set_a,Y2: set_a] :
( ( X3 = Y2 )
=> ( ord_less_eq_set_a @ X3 @ Y2 ) ) ).
% order_eq_refl
thf(fact_587_order__eq__refl,axiom,
! [X3: set_set_a,Y2: set_set_a] :
( ( X3 = Y2 )
=> ( ord_le3724670747650509150_set_a @ X3 @ Y2 ) ) ).
% order_eq_refl
thf(fact_588_order__eq__refl,axiom,
! [X3: real,Y2: real] :
( ( X3 = Y2 )
=> ( ord_less_eq_real @ X3 @ Y2 ) ) ).
% order_eq_refl
thf(fact_589_order__eq__refl,axiom,
! [X3: nat,Y2: nat] :
( ( X3 = Y2 )
=> ( ord_less_eq_nat @ X3 @ Y2 ) ) ).
% order_eq_refl
thf(fact_590_order__eq__refl,axiom,
! [X3: set_Product_prod_a_a,Y2: set_Product_prod_a_a] :
( ( X3 = Y2 )
=> ( ord_le746702958409616551od_a_a @ X3 @ Y2 ) ) ).
% order_eq_refl
thf(fact_591_order__subst2,axiom,
! [A: real,B: real,F: real > real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C )
=> ( ! [X4: real,Y: real] :
( ( ord_less_eq_real @ X4 @ Y )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_592_order__subst2,axiom,
! [A: real,B: real,F: real > nat,C: nat] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X4: real,Y: real] :
( ( ord_less_eq_real @ X4 @ Y )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_593_order__subst2,axiom,
! [A: nat,B: nat,F: nat > real,C: real] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C )
=> ( ! [X4: nat,Y: nat] :
( ( ord_less_eq_nat @ X4 @ Y )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_594_order__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X4: nat,Y: nat] :
( ( ord_less_eq_nat @ X4 @ Y )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_595_order__subst2,axiom,
! [A: set_a,B: set_a,F: set_a > real,C: real] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C )
=> ( ! [X4: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_596_order__subst2,axiom,
! [A: set_a,B: set_a,F: set_a > nat,C: nat] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X4: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_597_order__subst2,axiom,
! [A: real,B: real,F: real > set_a,C: set_a] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_set_a @ ( F @ B ) @ C )
=> ( ! [X4: real,Y: real] :
( ( ord_less_eq_real @ X4 @ Y )
=> ( ord_less_eq_set_a @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_598_order__subst2,axiom,
! [A: nat,B: nat,F: nat > set_a,C: set_a] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_set_a @ ( F @ B ) @ C )
=> ( ! [X4: nat,Y: nat] :
( ( ord_less_eq_nat @ X4 @ Y )
=> ( ord_less_eq_set_a @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_599_order__subst2,axiom,
! [A: set_a,B: set_a,F: set_a > set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_eq_set_a @ ( F @ B ) @ C )
=> ( ! [X4: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y )
=> ( ord_less_eq_set_a @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_600_order__subst2,axiom,
! [A: set_set_a,B: set_set_a,F: set_set_a > real,C: real] :
( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C )
=> ( ! [X4: set_set_a,Y: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X4 @ Y )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_601_order__subst1,axiom,
! [A: real,F: real > real,B: real,C: real] :
( ( ord_less_eq_real @ A @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X4: real,Y: real] :
( ( ord_less_eq_real @ X4 @ Y )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_602_order__subst1,axiom,
! [A: real,F: nat > real,B: nat,C: nat] :
( ( ord_less_eq_real @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X4: nat,Y: nat] :
( ( ord_less_eq_nat @ X4 @ Y )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_603_order__subst1,axiom,
! [A: nat,F: real > nat,B: real,C: real] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X4: real,Y: real] :
( ( ord_less_eq_real @ X4 @ Y )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_604_order__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X4: nat,Y: nat] :
( ( ord_less_eq_nat @ X4 @ Y )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_605_order__subst1,axiom,
! [A: set_a,F: real > set_a,B: real,C: real] :
( ( ord_less_eq_set_a @ A @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X4: real,Y: real] :
( ( ord_less_eq_real @ X4 @ Y )
=> ( ord_less_eq_set_a @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_606_order__subst1,axiom,
! [A: set_a,F: nat > set_a,B: nat,C: nat] :
( ( ord_less_eq_set_a @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X4: nat,Y: nat] :
( ( ord_less_eq_nat @ X4 @ Y )
=> ( ord_less_eq_set_a @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_607_order__subst1,axiom,
! [A: real,F: set_a > real,B: set_a,C: set_a] :
( ( ord_less_eq_real @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ! [X4: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_608_order__subst1,axiom,
! [A: nat,F: set_a > nat,B: set_a,C: set_a] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ! [X4: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_609_order__subst1,axiom,
! [A: set_a,F: set_a > set_a,B: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ! [X4: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y )
=> ( ord_less_eq_set_a @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_610_order__subst1,axiom,
! [A: set_set_a,F: real > set_set_a,B: real,C: real] :
( ( ord_le3724670747650509150_set_a @ A @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X4: real,Y: real] :
( ( ord_less_eq_real @ X4 @ Y )
=> ( ord_le3724670747650509150_set_a @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_le3724670747650509150_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_611_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: set_a,Z: set_a] : ( Y5 = Z ) )
= ( ^ [A3: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ A3 @ B3 )
& ( ord_less_eq_set_a @ B3 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_612_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: set_set_a,Z: set_set_a] : ( Y5 = Z ) )
= ( ^ [A3: set_set_a,B3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A3 @ B3 )
& ( ord_le3724670747650509150_set_a @ B3 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_613_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: real,Z: real] : ( Y5 = Z ) )
= ( ^ [A3: real,B3: real] :
( ( ord_less_eq_real @ A3 @ B3 )
& ( ord_less_eq_real @ B3 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_614_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: nat,Z: nat] : ( Y5 = Z ) )
= ( ^ [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
& ( ord_less_eq_nat @ B3 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_615_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: set_Product_prod_a_a,Z: set_Product_prod_a_a] : ( Y5 = 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 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_616_antisym,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_eq_set_a @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_617_antisym,axiom,
! [A: set_set_a,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( ( ord_le3724670747650509150_set_a @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_618_antisym,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_619_antisym,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_620_antisym,axiom,
! [A: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A @ B )
=> ( ( ord_le746702958409616551od_a_a @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_621_dual__order_Otrans,axiom,
! [B: set_a,A: set_a,C: set_a] :
( ( ord_less_eq_set_a @ B @ A )
=> ( ( ord_less_eq_set_a @ C @ B )
=> ( ord_less_eq_set_a @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_622_dual__order_Otrans,axiom,
! [B: set_set_a,A: set_set_a,C: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B @ A )
=> ( ( ord_le3724670747650509150_set_a @ C @ B )
=> ( ord_le3724670747650509150_set_a @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_623_dual__order_Otrans,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( ord_less_eq_real @ C @ B )
=> ( ord_less_eq_real @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_624_dual__order_Otrans,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_eq_nat @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_625_dual__order_Otrans,axiom,
! [B: set_Product_prod_a_a,A: set_Product_prod_a_a,C: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ B @ A )
=> ( ( ord_le746702958409616551od_a_a @ C @ B )
=> ( ord_le746702958409616551od_a_a @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_626_dual__order_Oantisym,axiom,
! [B: set_a,A: set_a] :
( ( ord_less_eq_set_a @ B @ A )
=> ( ( ord_less_eq_set_a @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_627_dual__order_Oantisym,axiom,
! [B: set_set_a,A: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B @ A )
=> ( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_628_dual__order_Oantisym,axiom,
! [B: real,A: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( ord_less_eq_real @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_629_dual__order_Oantisym,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_630_dual__order_Oantisym,axiom,
! [B: set_Product_prod_a_a,A: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ B @ A )
=> ( ( ord_le746702958409616551od_a_a @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_631_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: set_a,Z: set_a] : ( Y5 = Z ) )
= ( ^ [A3: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ B3 @ A3 )
& ( ord_less_eq_set_a @ A3 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_632_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: set_set_a,Z: set_set_a] : ( Y5 = Z ) )
= ( ^ [A3: set_set_a,B3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B3 @ A3 )
& ( ord_le3724670747650509150_set_a @ A3 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_633_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: real,Z: real] : ( Y5 = Z ) )
= ( ^ [A3: real,B3: real] :
( ( ord_less_eq_real @ B3 @ A3 )
& ( ord_less_eq_real @ A3 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_634_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: nat,Z: nat] : ( Y5 = Z ) )
= ( ^ [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ B3 @ A3 )
& ( ord_less_eq_nat @ A3 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_635_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: set_Product_prod_a_a,Z: set_Product_prod_a_a] : ( Y5 = Z ) )
= ( ^ [A3: set_Product_prod_a_a,B3: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ B3 @ A3 )
& ( ord_le746702958409616551od_a_a @ A3 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_636_linorder__wlog,axiom,
! [P2: real > real > $o,A: real,B: 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 @ B ) ) ) ).
% linorder_wlog
thf(fact_637_linorder__wlog,axiom,
! [P2: nat > nat > $o,A: nat,B: 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 @ B ) ) ) ).
% linorder_wlog
thf(fact_638_order__trans,axiom,
! [X3: set_a,Y2: set_a,Z3: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y2 )
=> ( ( ord_less_eq_set_a @ Y2 @ Z3 )
=> ( ord_less_eq_set_a @ X3 @ Z3 ) ) ) ).
% order_trans
thf(fact_639_order__trans,axiom,
! [X3: set_set_a,Y2: set_set_a,Z3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X3 @ Y2 )
=> ( ( ord_le3724670747650509150_set_a @ Y2 @ Z3 )
=> ( ord_le3724670747650509150_set_a @ X3 @ Z3 ) ) ) ).
% order_trans
thf(fact_640_order__trans,axiom,
! [X3: real,Y2: real,Z3: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ( ord_less_eq_real @ Y2 @ Z3 )
=> ( ord_less_eq_real @ X3 @ Z3 ) ) ) ).
% order_trans
thf(fact_641_order__trans,axiom,
! [X3: nat,Y2: nat,Z3: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ( ord_less_eq_nat @ Y2 @ Z3 )
=> ( ord_less_eq_nat @ X3 @ Z3 ) ) ) ).
% order_trans
thf(fact_642_order__trans,axiom,
! [X3: set_Product_prod_a_a,Y2: set_Product_prod_a_a,Z3: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ X3 @ Y2 )
=> ( ( ord_le746702958409616551od_a_a @ Y2 @ Z3 )
=> ( ord_le746702958409616551od_a_a @ X3 @ Z3 ) ) ) ).
% order_trans
thf(fact_643_order_Otrans,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ord_less_eq_set_a @ A @ C ) ) ) ).
% order.trans
thf(fact_644_order_Otrans,axiom,
! [A: set_set_a,B: set_set_a,C: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( ( ord_le3724670747650509150_set_a @ B @ C )
=> ( ord_le3724670747650509150_set_a @ A @ C ) ) ) ).
% order.trans
thf(fact_645_order_Otrans,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ord_less_eq_real @ A @ C ) ) ) ).
% order.trans
thf(fact_646_order_Otrans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% order.trans
thf(fact_647_order_Otrans,axiom,
! [A: set_Product_prod_a_a,B: set_Product_prod_a_a,C: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A @ B )
=> ( ( ord_le746702958409616551od_a_a @ B @ C )
=> ( ord_le746702958409616551od_a_a @ A @ C ) ) ) ).
% order.trans
thf(fact_648_order__antisym,axiom,
! [X3: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y2 )
=> ( ( ord_less_eq_set_a @ Y2 @ X3 )
=> ( X3 = Y2 ) ) ) ).
% order_antisym
thf(fact_649_order__antisym,axiom,
! [X3: set_set_a,Y2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X3 @ Y2 )
=> ( ( ord_le3724670747650509150_set_a @ Y2 @ X3 )
=> ( X3 = Y2 ) ) ) ).
% order_antisym
thf(fact_650_order__antisym,axiom,
! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ( ord_less_eq_real @ Y2 @ X3 )
=> ( X3 = Y2 ) ) ) ).
% order_antisym
thf(fact_651_order__antisym,axiom,
! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ( ord_less_eq_nat @ Y2 @ X3 )
=> ( X3 = Y2 ) ) ) ).
% order_antisym
thf(fact_652_order__antisym,axiom,
! [X3: set_Product_prod_a_a,Y2: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ X3 @ Y2 )
=> ( ( ord_le746702958409616551od_a_a @ Y2 @ X3 )
=> ( X3 = Y2 ) ) ) ).
% order_antisym
thf(fact_653_ord__le__eq__trans,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_set_a @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_654_ord__le__eq__trans,axiom,
! [A: set_set_a,B: set_set_a,C: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( ( B = C )
=> ( ord_le3724670747650509150_set_a @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_655_ord__le__eq__trans,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_real @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_656_ord__le__eq__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_657_ord__le__eq__trans,axiom,
! [A: set_Product_prod_a_a,B: set_Product_prod_a_a,C: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A @ B )
=> ( ( B = C )
=> ( ord_le746702958409616551od_a_a @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_658_ord__eq__le__trans,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( A = B )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ord_less_eq_set_a @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_659_ord__eq__le__trans,axiom,
! [A: set_set_a,B: set_set_a,C: set_set_a] :
( ( A = B )
=> ( ( ord_le3724670747650509150_set_a @ B @ C )
=> ( ord_le3724670747650509150_set_a @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_660_ord__eq__le__trans,axiom,
! [A: real,B: real,C: real] :
( ( A = B )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ord_less_eq_real @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_661_ord__eq__le__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( A = B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_662_ord__eq__le__trans,axiom,
! [A: set_Product_prod_a_a,B: set_Product_prod_a_a,C: set_Product_prod_a_a] :
( ( A = B )
=> ( ( ord_le746702958409616551od_a_a @ B @ C )
=> ( ord_le746702958409616551od_a_a @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_663_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: set_a,Z: set_a] : ( Y5 = 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_664_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: set_set_a,Z: set_set_a] : ( Y5 = 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_665_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: real,Z: real] : ( Y5 = Z ) )
= ( ^ [X2: real,Y3: real] :
( ( ord_less_eq_real @ X2 @ Y3 )
& ( ord_less_eq_real @ Y3 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_666_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: nat,Z: nat] : ( Y5 = Z ) )
= ( ^ [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
& ( ord_less_eq_nat @ Y3 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_667_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: set_Product_prod_a_a,Z: set_Product_prod_a_a] : ( Y5 = 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_668_le__cases3,axiom,
! [X3: real,Y2: real,Z3: real] :
( ( ( ord_less_eq_real @ X3 @ Y2 )
=> ~ ( ord_less_eq_real @ Y2 @ Z3 ) )
=> ( ( ( ord_less_eq_real @ Y2 @ X3 )
=> ~ ( ord_less_eq_real @ X3 @ Z3 ) )
=> ( ( ( ord_less_eq_real @ X3 @ Z3 )
=> ~ ( ord_less_eq_real @ Z3 @ Y2 ) )
=> ( ( ( ord_less_eq_real @ Z3 @ Y2 )
=> ~ ( ord_less_eq_real @ Y2 @ X3 ) )
=> ( ( ( ord_less_eq_real @ Y2 @ Z3 )
=> ~ ( ord_less_eq_real @ Z3 @ X3 ) )
=> ~ ( ( ord_less_eq_real @ Z3 @ X3 )
=> ~ ( ord_less_eq_real @ X3 @ Y2 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_669_le__cases3,axiom,
! [X3: nat,Y2: nat,Z3: nat] :
( ( ( ord_less_eq_nat @ X3 @ Y2 )
=> ~ ( ord_less_eq_nat @ Y2 @ Z3 ) )
=> ( ( ( ord_less_eq_nat @ Y2 @ X3 )
=> ~ ( ord_less_eq_nat @ X3 @ Z3 ) )
=> ( ( ( ord_less_eq_nat @ X3 @ Z3 )
=> ~ ( ord_less_eq_nat @ Z3 @ Y2 ) )
=> ( ( ( ord_less_eq_nat @ Z3 @ Y2 )
=> ~ ( ord_less_eq_nat @ Y2 @ X3 ) )
=> ( ( ( ord_less_eq_nat @ Y2 @ Z3 )
=> ~ ( ord_less_eq_nat @ Z3 @ X3 ) )
=> ~ ( ( ord_less_eq_nat @ Z3 @ X3 )
=> ~ ( ord_less_eq_nat @ X3 @ Y2 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_670_nle__le,axiom,
! [A: real,B: real] :
( ( ~ ( ord_less_eq_real @ A @ B ) )
= ( ( ord_less_eq_real @ B @ A )
& ( B != A ) ) ) ).
% nle_le
thf(fact_671_nle__le,axiom,
! [A: nat,B: nat] :
( ( ~ ( ord_less_eq_nat @ A @ B ) )
= ( ( ord_less_eq_nat @ B @ A )
& ( B != A ) ) ) ).
% nle_le
thf(fact_672_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_673_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_674_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_675_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_676_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_677_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_678_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_679_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_680_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_681_set__eq__subset,axiom,
( ( ^ [Y5: set_a,Z: set_a] : ( Y5 = Z ) )
= ( ^ [A5: set_a,B5: set_a] :
( ( ord_less_eq_set_a @ A5 @ B5 )
& ( ord_less_eq_set_a @ B5 @ A5 ) ) ) ) ).
% set_eq_subset
thf(fact_682_set__eq__subset,axiom,
( ( ^ [Y5: set_set_a,Z: set_set_a] : ( Y5 = Z ) )
= ( ^ [A5: set_set_a,B5: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A5 @ B5 )
& ( ord_le3724670747650509150_set_a @ B5 @ A5 ) ) ) ) ).
% set_eq_subset
thf(fact_683_set__eq__subset,axiom,
( ( ^ [Y5: set_Product_prod_a_a,Z: set_Product_prod_a_a] : ( Y5 = 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 ) ) ) ) ).
% set_eq_subset
thf(fact_684_subset__trans,axiom,
! [A2: set_a,B2: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ord_less_eq_set_a @ B2 @ C2 )
=> ( ord_less_eq_set_a @ A2 @ C2 ) ) ) ).
% subset_trans
thf(fact_685_subset__trans,axiom,
! [A2: set_set_a,B2: set_set_a,C2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B2 )
=> ( ( ord_le3724670747650509150_set_a @ B2 @ C2 )
=> ( ord_le3724670747650509150_set_a @ A2 @ C2 ) ) ) ).
% subset_trans
thf(fact_686_subset__trans,axiom,
! [A2: set_Product_prod_a_a,B2: set_Product_prod_a_a,C2: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A2 @ B2 )
=> ( ( ord_le746702958409616551od_a_a @ B2 @ C2 )
=> ( ord_le746702958409616551od_a_a @ A2 @ C2 ) ) ) ).
% subset_trans
thf(fact_687_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_688_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_689_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_690_subset__refl,axiom,
! [A2: set_a] : ( ord_less_eq_set_a @ A2 @ A2 ) ).
% subset_refl
thf(fact_691_subset__refl,axiom,
! [A2: set_set_a] : ( ord_le3724670747650509150_set_a @ A2 @ A2 ) ).
% subset_refl
thf(fact_692_subset__refl,axiom,
! [A2: set_Product_prod_a_a] : ( ord_le746702958409616551od_a_a @ A2 @ A2 ) ).
% subset_refl
thf(fact_693_subset__iff,axiom,
( ord_less_eq_set_a
= ( ^ [A5: set_a,B5: set_a] :
! [T: a] :
( ( member_a @ T @ A5 )
=> ( member_a @ T @ B5 ) ) ) ) ).
% subset_iff
thf(fact_694_subset__iff,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [A5: set_set_a,B5: set_set_a] :
! [T: set_a] :
( ( member_set_a @ T @ A5 )
=> ( member_set_a @ T @ B5 ) ) ) ) ).
% subset_iff
thf(fact_695_subset__iff,axiom,
( ord_le746702958409616551od_a_a
= ( ^ [A5: set_Product_prod_a_a,B5: set_Product_prod_a_a] :
! [T: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ T @ A5 )
=> ( member1426531477525435216od_a_a @ T @ B5 ) ) ) ) ).
% subset_iff
thf(fact_696_equalityD2,axiom,
! [A2: set_a,B2: set_a] :
( ( A2 = B2 )
=> ( ord_less_eq_set_a @ B2 @ A2 ) ) ).
% equalityD2
thf(fact_697_equalityD2,axiom,
! [A2: set_set_a,B2: set_set_a] :
( ( A2 = B2 )
=> ( ord_le3724670747650509150_set_a @ B2 @ A2 ) ) ).
% equalityD2
thf(fact_698_equalityD2,axiom,
! [A2: set_Product_prod_a_a,B2: set_Product_prod_a_a] :
( ( A2 = B2 )
=> ( ord_le746702958409616551od_a_a @ B2 @ A2 ) ) ).
% equalityD2
thf(fact_699_equalityD1,axiom,
! [A2: set_a,B2: set_a] :
( ( A2 = B2 )
=> ( ord_less_eq_set_a @ A2 @ B2 ) ) ).
% equalityD1
thf(fact_700_equalityD1,axiom,
! [A2: set_set_a,B2: set_set_a] :
( ( A2 = B2 )
=> ( ord_le3724670747650509150_set_a @ A2 @ B2 ) ) ).
% equalityD1
thf(fact_701_equalityD1,axiom,
! [A2: set_Product_prod_a_a,B2: set_Product_prod_a_a] :
( ( A2 = B2 )
=> ( ord_le746702958409616551od_a_a @ A2 @ B2 ) ) ).
% equalityD1
thf(fact_702_subset__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A5: set_a,B5: set_a] :
! [X2: a] :
( ( member_a @ X2 @ A5 )
=> ( member_a @ X2 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_703_subset__eq,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [A5: set_set_a,B5: set_set_a] :
! [X2: set_a] :
( ( member_set_a @ X2 @ A5 )
=> ( member_set_a @ X2 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_704_subset__eq,axiom,
( ord_le746702958409616551od_a_a
= ( ^ [A5: set_Product_prod_a_a,B5: set_Product_prod_a_a] :
! [X2: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X2 @ A5 )
=> ( member1426531477525435216od_a_a @ X2 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_705_equalityE,axiom,
! [A2: set_a,B2: set_a] :
( ( A2 = B2 )
=> ~ ( ( ord_less_eq_set_a @ A2 @ B2 )
=> ~ ( ord_less_eq_set_a @ B2 @ A2 ) ) ) ).
% equalityE
thf(fact_706_equalityE,axiom,
! [A2: set_set_a,B2: set_set_a] :
( ( A2 = B2 )
=> ~ ( ( ord_le3724670747650509150_set_a @ A2 @ B2 )
=> ~ ( ord_le3724670747650509150_set_a @ B2 @ A2 ) ) ) ).
% equalityE
thf(fact_707_equalityE,axiom,
! [A2: set_Product_prod_a_a,B2: set_Product_prod_a_a] :
( ( A2 = B2 )
=> ~ ( ( ord_le746702958409616551od_a_a @ A2 @ B2 )
=> ~ ( ord_le746702958409616551od_a_a @ B2 @ A2 ) ) ) ).
% equalityE
thf(fact_708_subsetD,axiom,
! [A2: set_a,B2: set_a,C: a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( member_a @ C @ A2 )
=> ( member_a @ C @ B2 ) ) ) ).
% subsetD
thf(fact_709_subsetD,axiom,
! [A2: set_set_a,B2: set_set_a,C: set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B2 )
=> ( ( member_set_a @ C @ A2 )
=> ( member_set_a @ C @ B2 ) ) ) ).
% subsetD
thf(fact_710_subsetD,axiom,
! [A2: set_Product_prod_a_a,B2: set_Product_prod_a_a,C: product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A2 @ B2 )
=> ( ( member1426531477525435216od_a_a @ C @ A2 )
=> ( member1426531477525435216od_a_a @ C @ B2 ) ) ) ).
% subsetD
thf(fact_711_in__mono,axiom,
! [A2: set_a,B2: set_a,X3: a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( member_a @ X3 @ A2 )
=> ( member_a @ X3 @ B2 ) ) ) ).
% in_mono
thf(fact_712_in__mono,axiom,
! [A2: set_set_a,B2: set_set_a,X3: set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B2 )
=> ( ( member_set_a @ X3 @ A2 )
=> ( member_set_a @ X3 @ B2 ) ) ) ).
% in_mono
thf(fact_713_in__mono,axiom,
! [A2: set_Product_prod_a_a,B2: set_Product_prod_a_a,X3: product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A2 @ B2 )
=> ( ( member1426531477525435216od_a_a @ X3 @ A2 )
=> ( member1426531477525435216od_a_a @ X3 @ B2 ) ) ) ).
% in_mono
thf(fact_714_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_715_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_716_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_717_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_718_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_719_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_720_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_721_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_722_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_723_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_724_walk__edges__decomp__ss,axiom,
! [Xs: list_a,Y2: a,Zs: list_a,Ys: list_a] : ( ord_le3724670747650509150_set_a @ ( set_set_a2 @ ( undire7337870655677353998dges_a @ ( append_a @ Xs @ ( append_a @ ( cons_a @ Y2 @ nil_a ) @ Zs ) ) ) ) @ ( set_set_a2 @ ( undire7337870655677353998dges_a @ ( append_a @ Xs @ ( append_a @ ( cons_a @ Y2 @ nil_a ) @ ( append_a @ Ys @ ( append_a @ ( cons_a @ Y2 @ nil_a ) @ Zs ) ) ) ) ) ) ) ).
% walk_edges_decomp_ss
thf(fact_725_edge__density__ge0,axiom,
! [X5: set_a,Y4: set_a] : ( ord_less_eq_real @ zero_zero_real @ ( undire297304480579013331sity_a @ edges @ X5 @ Y4 ) ) ).
% edge_density_ge0
thf(fact_726_edge__density__le1,axiom,
! [X5: set_a,Y4: set_a] : ( ord_less_eq_real @ ( undire297304480579013331sity_a @ edges @ X5 @ Y4 ) @ one_one_real ) ).
% edge_density_le1
thf(fact_727_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_728_the__elem__set,axiom,
! [X3: a] :
( ( the_elem_a @ ( set_a2 @ ( cons_a @ X3 @ nil_a ) ) )
= X3 ) ).
% the_elem_set
thf(fact_729_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_730_walk__length__rev,axiom,
( undire8849074589633906640ngth_a
= ( ^ [P3: list_a] : ( undire8849074589633906640ngth_a @ ( rev_a @ P3 ) ) ) ) ).
% walk_length_rev
thf(fact_731_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_732_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_733_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_734_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_735_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_736_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_737_comp__sgraph_Owalk__length__rev,axiom,
( undire8849074589633906640ngth_a
= ( ^ [P3: list_a] : ( undire8849074589633906640ngth_a @ ( rev_a @ P3 ) ) ) ) ).
% comp_sgraph.walk_length_rev
thf(fact_738_comp__sgraph_Oedge__density__le1,axiom,
! [S: set_set_a,X5: set_set_a,Y4: set_set_a] : ( ord_less_eq_real @ ( undire8927637694342045747_set_a @ ( undire8247866692393712962_set_a @ S ) @ X5 @ Y4 ) @ one_one_real ) ).
% comp_sgraph.edge_density_le1
thf(fact_739_comp__sgraph_Oedge__density__le1,axiom,
! [S: set_a,X5: set_a,Y4: set_a] : ( ord_less_eq_real @ ( undire297304480579013331sity_a @ ( undire2918257014606996450dges_a @ S ) @ X5 @ Y4 ) @ one_one_real ) ).
% comp_sgraph.edge_density_le1
thf(fact_740_ulgraph_Oedge__density__le1,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,X5: set_set_a,Y4: set_set_a] :
( ( undire6886684016831807756_set_a @ Vertices @ Edges )
=> ( ord_less_eq_real @ ( undire8927637694342045747_set_a @ Edges @ X5 @ Y4 ) @ one_one_real ) ) ).
% ulgraph.edge_density_le1
thf(fact_741_ulgraph_Oedge__density__le1,axiom,
! [Vertices: set_a,Edges: set_set_a,X5: set_a,Y4: set_a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ord_less_eq_real @ ( undire297304480579013331sity_a @ Edges @ X5 @ Y4 ) @ one_one_real ) ) ).
% ulgraph.edge_density_le1
thf(fact_742_comp__sgraph_Oedge__density__ge0,axiom,
! [S: set_set_a,X5: set_set_a,Y4: set_set_a] : ( ord_less_eq_real @ zero_zero_real @ ( undire8927637694342045747_set_a @ ( undire8247866692393712962_set_a @ S ) @ X5 @ Y4 ) ) ).
% comp_sgraph.edge_density_ge0
thf(fact_743_comp__sgraph_Oedge__density__ge0,axiom,
! [S: set_a,X5: set_a,Y4: set_a] : ( ord_less_eq_real @ zero_zero_real @ ( undire297304480579013331sity_a @ ( undire2918257014606996450dges_a @ S ) @ X5 @ Y4 ) ) ).
% comp_sgraph.edge_density_ge0
thf(fact_744_ulgraph_Oedge__density__ge0,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,X5: set_set_a,Y4: set_set_a] :
( ( undire6886684016831807756_set_a @ Vertices @ Edges )
=> ( ord_less_eq_real @ zero_zero_real @ ( undire8927637694342045747_set_a @ Edges @ X5 @ Y4 ) ) ) ).
% ulgraph.edge_density_ge0
thf(fact_745_ulgraph_Oedge__density__ge0,axiom,
! [Vertices: set_a,Edges: set_set_a,X5: set_a,Y4: set_a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ord_less_eq_real @ zero_zero_real @ ( undire297304480579013331sity_a @ Edges @ X5 @ Y4 ) ) ) ).
% ulgraph.edge_density_ge0
thf(fact_746_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_747_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_748_comp__sgraph_Ois__cycle__alt__gen__path,axiom,
! [S: set_set_a,Xs: list_set_a] :
( ( undire797940137672299967_set_a @ S @ ( undire8247866692393712962_set_a @ S ) @ Xs )
= ( ( undire7201326534205417136_set_a @ S @ ( undire8247866692393712962_set_a @ S ) @ 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_gen_path
thf(fact_749_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_750_ulgraph_Ois__cycle__alt__gen__path,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 )
= ( ( undire7201326534205417136_set_a @ Vertices @ Edges @ Xs )
& ( ord_less_eq_nat @ one_one_nat @ ( undire4424681683220949296_set_a @ Xs ) )
& ( ( hd_set_a @ Xs )
= ( last_set_a @ Xs ) ) ) ) ) ).
% ulgraph.is_cycle_alt_gen_path
thf(fact_751_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_752_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_753_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_754_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_755_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_756_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_757_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_758_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_759_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_760_comp__sgraph_Owalk__edges__tl__ss,axiom,
! [Xs: list_set_a] : ( ord_le5722252365846178494_set_a @ ( set_set_set_a2 @ ( undire6234387080713648494_set_a @ ( tl_set_a @ Xs ) ) ) @ ( set_set_set_a2 @ ( undire6234387080713648494_set_a @ Xs ) ) ) ).
% comp_sgraph.walk_edges_tl_ss
thf(fact_761_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_762_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_763_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_764_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_765_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_766_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_767_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_768_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_769_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_770_ulgraph_Owalk__edges__tl__ss,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,Xs: list_set_a] :
( ( undire6886684016831807756_set_a @ Vertices @ Edges )
=> ( ord_le5722252365846178494_set_a @ ( set_set_set_a2 @ ( undire6234387080713648494_set_a @ ( tl_set_a @ Xs ) ) ) @ ( set_set_set_a2 @ ( undire6234387080713648494_set_a @ Xs ) ) ) ) ).
% ulgraph.walk_edges_tl_ss
thf(fact_771_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_772_comp__sgraph_Owalk__edges__decomp__ss,axiom,
! [Xs: list_set_a,Y2: 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 @ Y2 @ nil_set_a ) @ Zs ) ) ) ) @ ( set_set_set_a2 @ ( undire6234387080713648494_set_a @ ( append_set_a @ Xs @ ( append_set_a @ ( cons_set_a @ Y2 @ nil_set_a ) @ ( append_set_a @ Ys @ ( append_set_a @ ( cons_set_a @ Y2 @ nil_set_a ) @ Zs ) ) ) ) ) ) ) ).
% comp_sgraph.walk_edges_decomp_ss
thf(fact_773_comp__sgraph_Owalk__edges__decomp__ss,axiom,
! [Xs: list_a,Y2: a,Zs: list_a,Ys: list_a] : ( ord_le3724670747650509150_set_a @ ( set_set_a2 @ ( undire7337870655677353998dges_a @ ( append_a @ Xs @ ( append_a @ ( cons_a @ Y2 @ nil_a ) @ Zs ) ) ) ) @ ( set_set_a2 @ ( undire7337870655677353998dges_a @ ( append_a @ Xs @ ( append_a @ ( cons_a @ Y2 @ nil_a ) @ ( append_a @ Ys @ ( append_a @ ( cons_a @ Y2 @ nil_a ) @ Zs ) ) ) ) ) ) ) ).
% comp_sgraph.walk_edges_decomp_ss
thf(fact_774_ulgraph_Owalk__edges__decomp__ss,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,Xs: list_set_a,Y2: set_a,Zs: list_set_a,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 @ Y2 @ nil_set_a ) @ Zs ) ) ) ) @ ( set_set_set_a2 @ ( undire6234387080713648494_set_a @ ( append_set_a @ Xs @ ( append_set_a @ ( cons_set_a @ Y2 @ nil_set_a ) @ ( append_set_a @ Ys @ ( append_set_a @ ( cons_set_a @ Y2 @ nil_set_a ) @ Zs ) ) ) ) ) ) ) ) ).
% ulgraph.walk_edges_decomp_ss
thf(fact_775_ulgraph_Owalk__edges__decomp__ss,axiom,
! [Vertices: set_a,Edges: set_set_a,Xs: list_a,Y2: 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 @ Y2 @ nil_a ) @ Zs ) ) ) ) @ ( set_set_a2 @ ( undire7337870655677353998dges_a @ ( append_a @ Xs @ ( append_a @ ( cons_a @ Y2 @ nil_a ) @ ( append_a @ Ys @ ( append_a @ ( cons_a @ Y2 @ nil_a ) @ Zs ) ) ) ) ) ) ) ) ).
% ulgraph.walk_edges_decomp_ss
thf(fact_776_le__zero__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_777_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_778_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_779_edge__density__zero,axiom,
! [Y4: set_a,X5: set_a] :
( ( Y4 = bot_bot_set_a )
=> ( ( undire297304480579013331sity_a @ edges @ X5 @ Y4 )
= zero_zero_real ) ) ).
% edge_density_zero
thf(fact_780_zero__less__one__class_Ozero__le__one,axiom,
ord_less_eq_real @ zero_zero_real @ one_one_real ).
% zero_less_one_class.zero_le_one
thf(fact_781_zero__less__one__class_Ozero__le__one,axiom,
ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one_class.zero_le_one
thf(fact_782_empty__not__edge,axiom,
~ ( member_set_a @ bot_bot_set_a @ edges ) ).
% empty_not_edge
thf(fact_783_empty__iff,axiom,
! [C: a] :
~ ( member_a @ C @ bot_bot_set_a ) ).
% empty_iff
thf(fact_784_empty__iff,axiom,
! [C: set_a] :
~ ( member_set_a @ C @ bot_bot_set_set_a ) ).
% empty_iff
thf(fact_785_empty__iff,axiom,
! [C: product_prod_a_a] :
~ ( member1426531477525435216od_a_a @ C @ bot_bo3357376287454694259od_a_a ) ).
% empty_iff
thf(fact_786_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_787_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_788_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_789_Collect__empty__eq,axiom,
! [P2: a > $o] :
( ( ( collect_a @ P2 )
= bot_bot_set_a )
= ( ! [X2: a] :
~ ( P2 @ X2 ) ) ) ).
% Collect_empty_eq
thf(fact_790_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_791_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_792_empty__Collect__eq,axiom,
! [P2: a > $o] :
( ( bot_bot_set_a
= ( collect_a @ P2 ) )
= ( ! [X2: a] :
~ ( P2 @ X2 ) ) ) ).
% empty_Collect_eq
thf(fact_793_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_794_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_795_add__le__cancel__left,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
= ( ord_less_eq_real @ A @ B ) ) ).
% add_le_cancel_left
thf(fact_796_add__le__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
= ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_cancel_left
thf(fact_797_add__le__cancel__right,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
= ( ord_less_eq_real @ A @ B ) ) ).
% add_le_cancel_right
thf(fact_798_add__le__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_cancel_right
thf(fact_799_add_Oright__neutral,axiom,
! [A: real] :
( ( plus_plus_real @ A @ zero_zero_real )
= A ) ).
% add.right_neutral
thf(fact_800_add_Oright__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.right_neutral
thf(fact_801_double__zero__sym,axiom,
! [A: real] :
( ( zero_zero_real
= ( plus_plus_real @ A @ A ) )
= ( A = zero_zero_real ) ) ).
% double_zero_sym
thf(fact_802_add__cancel__left__left,axiom,
! [B: real,A: real] :
( ( ( plus_plus_real @ B @ A )
= A )
= ( B = zero_zero_real ) ) ).
% add_cancel_left_left
thf(fact_803_add__cancel__left__left,axiom,
! [B: nat,A: nat] :
( ( ( plus_plus_nat @ B @ A )
= A )
= ( B = zero_zero_nat ) ) ).
% add_cancel_left_left
thf(fact_804_add__cancel__left__right,axiom,
! [A: real,B: real] :
( ( ( plus_plus_real @ A @ B )
= A )
= ( B = zero_zero_real ) ) ).
% add_cancel_left_right
thf(fact_805_add__cancel__left__right,axiom,
! [A: nat,B: nat] :
( ( ( plus_plus_nat @ A @ B )
= A )
= ( B = zero_zero_nat ) ) ).
% add_cancel_left_right
thf(fact_806_add__cancel__right__left,axiom,
! [A: real,B: real] :
( ( A
= ( plus_plus_real @ B @ A ) )
= ( B = zero_zero_real ) ) ).
% add_cancel_right_left
thf(fact_807_add__cancel__right__left,axiom,
! [A: nat,B: nat] :
( ( A
= ( plus_plus_nat @ B @ A ) )
= ( B = zero_zero_nat ) ) ).
% add_cancel_right_left
thf(fact_808_add__cancel__right__right,axiom,
! [A: real,B: real] :
( ( A
= ( plus_plus_real @ A @ B ) )
= ( B = zero_zero_real ) ) ).
% add_cancel_right_right
thf(fact_809_add__cancel__right__right,axiom,
! [A: nat,B: nat] :
( ( A
= ( plus_plus_nat @ A @ B ) )
= ( B = zero_zero_nat ) ) ).
% add_cancel_right_right
thf(fact_810_add__eq__0__iff__both__eq__0,axiom,
! [X3: nat,Y2: nat] :
( ( ( plus_plus_nat @ X3 @ Y2 )
= zero_zero_nat )
= ( ( X3 = zero_zero_nat )
& ( Y2 = zero_zero_nat ) ) ) ).
% add_eq_0_iff_both_eq_0
thf(fact_811_zero__eq__add__iff__both__eq__0,axiom,
! [X3: nat,Y2: nat] :
( ( zero_zero_nat
= ( plus_plus_nat @ X3 @ Y2 ) )
= ( ( X3 = zero_zero_nat )
& ( Y2 = zero_zero_nat ) ) ) ).
% zero_eq_add_iff_both_eq_0
thf(fact_812_add__0,axiom,
! [A: real] :
( ( plus_plus_real @ zero_zero_real @ A )
= A ) ).
% add_0
thf(fact_813_add__0,axiom,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ) ).
% add_0
thf(fact_814_empty__subsetI,axiom,
! [A2: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A2 ) ).
% empty_subsetI
thf(fact_815_empty__subsetI,axiom,
! [A2: set_set_a] : ( ord_le3724670747650509150_set_a @ bot_bot_set_set_a @ A2 ) ).
% empty_subsetI
thf(fact_816_empty__subsetI,axiom,
! [A2: set_Product_prod_a_a] : ( ord_le746702958409616551od_a_a @ bot_bo3357376287454694259od_a_a @ A2 ) ).
% empty_subsetI
thf(fact_817_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_818_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_819_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_820_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_821_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_822_le__add__same__cancel2,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ ( plus_plus_real @ B @ A ) )
= ( ord_less_eq_real @ zero_zero_real @ B ) ) ).
% le_add_same_cancel2
thf(fact_823_le__add__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ B @ A ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% le_add_same_cancel2
thf(fact_824_le__add__same__cancel1,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ ( plus_plus_real @ A @ B ) )
= ( ord_less_eq_real @ zero_zero_real @ B ) ) ).
% le_add_same_cancel1
thf(fact_825_le__add__same__cancel1,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ A @ B ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% le_add_same_cancel1
thf(fact_826_add__le__same__cancel2,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ A @ B ) @ B )
= ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% add_le_same_cancel2
thf(fact_827_add__le__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ B )
= ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).
% add_le_same_cancel2
thf(fact_828_add__le__same__cancel1,axiom,
! [B: real,A: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ B @ A ) @ B )
= ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% add_le_same_cancel1
thf(fact_829_add__le__same__cancel1,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ B @ A ) @ B )
= ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).
% add_le_same_cancel1
thf(fact_830_set__empty,axiom,
! [Xs: list_a] :
( ( ( set_a2 @ Xs )
= bot_bot_set_a )
= ( Xs = nil_a ) ) ).
% set_empty
thf(fact_831_set__empty,axiom,
! [Xs: list_set_a] :
( ( ( set_set_a2 @ Xs )
= bot_bot_set_set_a )
= ( Xs = nil_set_a ) ) ).
% set_empty
thf(fact_832_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_833_set__empty2,axiom,
! [Xs: list_a] :
( ( bot_bot_set_a
= ( set_a2 @ Xs ) )
= ( Xs = nil_a ) ) ).
% set_empty2
thf(fact_834_set__empty2,axiom,
! [Xs: list_set_a] :
( ( bot_bot_set_set_a
= ( set_set_a2 @ Xs ) )
= ( Xs = nil_set_a ) ) ).
% set_empty2
thf(fact_835_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_836_assms_I2_J,axiom,
ord_less_nat @ zero_zero_nat @ ( undire8849074589633906640ngth_a @ p ) ).
% assms(2)
thf(fact_837_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_838_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_839_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_840_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_841_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_842_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_843_add__mono,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ C @ D )
=> ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) ) ) ) ).
% add_mono
thf(fact_844_add__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_mono
thf(fact_845_add__left__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ord_less_eq_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) ) ) ).
% add_left_mono
thf(fact_846_add__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) ) ) ).
% add_left_mono
thf(fact_847_less__eqE,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ~ ! [C3: nat] :
( B
!= ( plus_plus_nat @ A @ C3 ) ) ) ).
% less_eqE
thf(fact_848_add__right__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) ) ) ).
% add_right_mono
thf(fact_849_add__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).
% add_right_mono
thf(fact_850_le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [A3: nat,B3: nat] :
? [C4: nat] :
( B3
= ( plus_plus_nat @ A3 @ C4 ) ) ) ) ).
% le_iff_add
thf(fact_851_add__le__imp__le__left,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
=> ( ord_less_eq_real @ A @ B ) ) ).
% add_le_imp_le_left
thf(fact_852_add__le__imp__le__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_imp_le_left
thf(fact_853_add__le__imp__le__right,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
=> ( ord_less_eq_real @ A @ B ) ) ).
% add_le_imp_le_right
thf(fact_854_add__le__imp__le__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_imp_le_right
thf(fact_855_comm__monoid__add__class_Oadd__0,axiom,
! [A: real] :
( ( plus_plus_real @ zero_zero_real @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_856_comm__monoid__add__class_Oadd__0,axiom,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_857_add_Ocomm__neutral,axiom,
! [A: real] :
( ( plus_plus_real @ A @ zero_zero_real )
= A ) ).
% add.comm_neutral
thf(fact_858_add_Ocomm__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.comm_neutral
thf(fact_859_add_Ogroup__left__neutral,axiom,
! [A: real] :
( ( plus_plus_real @ zero_zero_real @ A )
= A ) ).
% add.group_left_neutral
thf(fact_860_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_861_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_862_bot_Oextremum__uniqueI,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ bot_bot_nat )
=> ( A = bot_bot_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_863_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_864_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_865_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_866_bot_Oextremum__unique,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ bot_bot_nat )
= ( A = bot_bot_nat ) ) ).
% bot.extremum_unique
thf(fact_867_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_868_bot_Oextremum,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A ) ).
% bot.extremum
thf(fact_869_bot_Oextremum,axiom,
! [A: set_set_a] : ( ord_le3724670747650509150_set_a @ bot_bot_set_set_a @ A ) ).
% bot.extremum
thf(fact_870_bot_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ bot_bot_nat @ A ) ).
% bot.extremum
thf(fact_871_bot_Oextremum,axiom,
! [A: set_Product_prod_a_a] : ( ord_le746702958409616551od_a_a @ bot_bo3357376287454694259od_a_a @ A ) ).
% bot.extremum
thf(fact_872_emptyE,axiom,
! [A: a] :
~ ( member_a @ A @ bot_bot_set_a ) ).
% emptyE
thf(fact_873_emptyE,axiom,
! [A: set_a] :
~ ( member_set_a @ A @ bot_bot_set_set_a ) ).
% emptyE
thf(fact_874_emptyE,axiom,
! [A: product_prod_a_a] :
~ ( member1426531477525435216od_a_a @ A @ bot_bo3357376287454694259od_a_a ) ).
% emptyE
thf(fact_875_equals0D,axiom,
! [A2: set_a,A: a] :
( ( A2 = bot_bot_set_a )
=> ~ ( member_a @ A @ A2 ) ) ).
% equals0D
thf(fact_876_equals0D,axiom,
! [A2: set_set_a,A: set_a] :
( ( A2 = bot_bot_set_set_a )
=> ~ ( member_set_a @ A @ A2 ) ) ).
% equals0D
thf(fact_877_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_878_equals0I,axiom,
! [A2: set_a] :
( ! [Y: a] :
~ ( member_a @ Y @ A2 )
=> ( A2 = bot_bot_set_a ) ) ).
% equals0I
thf(fact_879_equals0I,axiom,
! [A2: set_set_a] :
( ! [Y: set_a] :
~ ( member_set_a @ Y @ A2 )
=> ( A2 = bot_bot_set_set_a ) ) ).
% equals0I
thf(fact_880_equals0I,axiom,
! [A2: set_Product_prod_a_a] :
( ! [Y: product_prod_a_a] :
~ ( member1426531477525435216od_a_a @ Y @ A2 )
=> ( A2 = bot_bo3357376287454694259od_a_a ) ) ).
% equals0I
thf(fact_881_ex__in__conv,axiom,
! [A2: set_a] :
( ( ? [X2: a] : ( member_a @ X2 @ A2 ) )
= ( A2 != bot_bot_set_a ) ) ).
% ex_in_conv
thf(fact_882_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_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_add__nonpos__eq__0__iff,axiom,
! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ zero_zero_real )
=> ( ( ord_less_eq_real @ Y2 @ zero_zero_real )
=> ( ( ( plus_plus_real @ X3 @ Y2 )
= zero_zero_real )
= ( ( X3 = zero_zero_real )
& ( Y2 = zero_zero_real ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_885_add__nonpos__eq__0__iff,axiom,
! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ Y2 @ zero_zero_nat )
=> ( ( ( plus_plus_nat @ X3 @ Y2 )
= zero_zero_nat )
= ( ( X3 = zero_zero_nat )
& ( Y2 = zero_zero_nat ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_886_add__nonneg__eq__0__iff,axiom,
! [X3: real,Y2: real] :
( ( ord_less_eq_real @ zero_zero_real @ X3 )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y2 )
=> ( ( ( plus_plus_real @ X3 @ Y2 )
= zero_zero_real )
= ( ( X3 = zero_zero_real )
& ( Y2 = zero_zero_real ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_887_add__nonneg__eq__0__iff,axiom,
! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ X3 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ Y2 )
=> ( ( ( plus_plus_nat @ X3 @ Y2 )
= zero_zero_nat )
= ( ( X3 = zero_zero_nat )
& ( Y2 = zero_zero_nat ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_888_add__nonpos__nonpos,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ zero_zero_real )
=> ( ( ord_less_eq_real @ B @ zero_zero_real )
=> ( ord_less_eq_real @ ( plus_plus_real @ A @ B ) @ zero_zero_real ) ) ) ).
% add_nonpos_nonpos
thf(fact_889_add__nonpos__nonpos,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_nonpos_nonpos
thf(fact_890_add__nonneg__nonneg,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ A @ B ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_891_add__nonneg__nonneg,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_892_add__increasing2,axiom,
! [C: real,B: real,A: real] :
( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ( ord_less_eq_real @ B @ A )
=> ( ord_less_eq_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).
% add_increasing2
thf(fact_893_add__increasing2,axiom,
! [C: nat,B: nat,A: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ( ord_less_eq_nat @ B @ A )
=> ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_increasing2
thf(fact_894_add__decreasing2,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_eq_real @ C @ zero_zero_real )
=> ( ( ord_less_eq_real @ A @ B )
=> ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ B ) ) ) ).
% add_decreasing2
thf(fact_895_add__decreasing2,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ C @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ B ) ) ) ).
% add_decreasing2
thf(fact_896_add__increasing,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ord_less_eq_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).
% add_increasing
thf(fact_897_add__increasing,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_increasing
thf(fact_898_add__decreasing,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_eq_real @ A @ zero_zero_real )
=> ( ( ord_less_eq_real @ C @ B )
=> ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ B ) ) ) ).
% add_decreasing
thf(fact_899_add__decreasing,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ B ) ) ) ).
% add_decreasing
thf(fact_900_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_901_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_902_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_903_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_904_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_905_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_906_empty__set,axiom,
( bot_bot_set_a
= ( set_a2 @ nil_a ) ) ).
% empty_set
thf(fact_907_empty__set,axiom,
( bot_bot_set_set_a
= ( set_set_a2 @ nil_set_a ) ) ).
% empty_set
thf(fact_908_empty__set,axiom,
( bot_bo3357376287454694259od_a_a
= ( set_Product_prod_a_a2 @ nil_Product_prod_a_a ) ) ).
% empty_set
thf(fact_909_zero__reorient,axiom,
! [X3: real] :
( ( zero_zero_real = X3 )
= ( X3 = zero_zero_real ) ) ).
% zero_reorient
thf(fact_910_zero__reorient,axiom,
! [X3: nat] :
( ( zero_zero_nat = X3 )
= ( X3 = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_911_one__reorient,axiom,
! [X3: real] :
( ( one_one_real = X3 )
= ( X3 = one_one_real ) ) ).
% one_reorient
thf(fact_912_one__reorient,axiom,
! [X3: nat] :
( ( one_one_nat = X3 )
= ( X3 = one_one_nat ) ) ).
% one_reorient
thf(fact_913_comp__sgraph_Oedge__density__zero,axiom,
! [Y4: set_Product_prod_a_a,S: set_Product_prod_a_a,X5: set_Product_prod_a_a] :
( ( Y4 = bot_bo3357376287454694259od_a_a )
=> ( ( undire8410861505230878716od_a_a @ ( undire6879232364018543115od_a_a @ S ) @ X5 @ Y4 )
= zero_zero_real ) ) ).
% comp_sgraph.edge_density_zero
thf(fact_914_comp__sgraph_Oedge__density__zero,axiom,
! [Y4: set_set_a,S: set_set_a,X5: set_set_a] :
( ( Y4 = bot_bot_set_set_a )
=> ( ( undire8927637694342045747_set_a @ ( undire8247866692393712962_set_a @ S ) @ X5 @ Y4 )
= zero_zero_real ) ) ).
% comp_sgraph.edge_density_zero
thf(fact_915_comp__sgraph_Oedge__density__zero,axiom,
! [Y4: set_a,S: set_a,X5: set_a] :
( ( Y4 = bot_bot_set_a )
=> ( ( undire297304480579013331sity_a @ ( undire2918257014606996450dges_a @ S ) @ X5 @ Y4 )
= zero_zero_real ) ) ).
% comp_sgraph.edge_density_zero
thf(fact_916_ulgraph_Oedge__density__zero,axiom,
! [Vertices: set_Product_prod_a_a,Edges: set_se5735800977113168103od_a_a,Y4: set_Product_prod_a_a,X5: set_Product_prod_a_a] :
( ( undire4585262585102564309od_a_a @ Vertices @ Edges )
=> ( ( Y4 = bot_bo3357376287454694259od_a_a )
=> ( ( undire8410861505230878716od_a_a @ Edges @ X5 @ Y4 )
= zero_zero_real ) ) ) ).
% ulgraph.edge_density_zero
thf(fact_917_ulgraph_Oedge__density__zero,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,Y4: set_set_a,X5: set_set_a] :
( ( undire6886684016831807756_set_a @ Vertices @ Edges )
=> ( ( Y4 = bot_bot_set_set_a )
=> ( ( undire8927637694342045747_set_a @ Edges @ X5 @ Y4 )
= zero_zero_real ) ) ) ).
% ulgraph.edge_density_zero
thf(fact_918_ulgraph_Oedge__density__zero,axiom,
! [Vertices: set_a,Edges: set_set_a,Y4: set_a,X5: set_a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( Y4 = bot_bot_set_a )
=> ( ( undire297304480579013331sity_a @ Edges @ X5 @ Y4 )
= zero_zero_real ) ) ) ).
% ulgraph.edge_density_zero
thf(fact_919_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_920_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_921_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_922_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_923_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_924_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_925_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_926_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_927_zero__le,axiom,
! [X3: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X3 ) ).
% zero_le
thf(fact_928_zero__neq__one,axiom,
zero_zero_real != one_one_real ).
% zero_neq_one
thf(fact_929_zero__neq__one,axiom,
zero_zero_nat != one_one_nat ).
% zero_neq_one
thf(fact_930_not__one__le__zero,axiom,
~ ( ord_less_eq_real @ one_one_real @ zero_zero_real ) ).
% not_one_le_zero
thf(fact_931_not__one__le__zero,axiom,
~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_le_zero
thf(fact_932_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
ord_less_eq_real @ zero_zero_real @ one_one_real ).
% linordered_nonzero_semiring_class.zero_le_one
thf(fact_933_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% linordered_nonzero_semiring_class.zero_le_one
thf(fact_934_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_935_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_936_walk__edges__app,axiom,
! [Xs: list_a,Y2: a,X3: a] :
( ( undire7337870655677353998dges_a @ ( append_a @ Xs @ ( cons_a @ Y2 @ ( cons_a @ X3 @ nil_a ) ) ) )
= ( append_set_a @ ( undire7337870655677353998dges_a @ ( append_a @ Xs @ ( cons_a @ Y2 @ nil_a ) ) ) @ ( cons_set_a @ ( insert_a @ Y2 @ ( insert_a @ X3 @ bot_bot_set_a ) ) @ nil_set_a ) ) ) ).
% walk_edges_app
thf(fact_937_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_938_is__walk__hd__tl,axiom,
! [Y2: a,Ys: list_a,X3: a] :
( ( undire6133010728901294956walk_a @ vertices @ edges @ ( cons_a @ Y2 @ Ys ) )
=> ( ( member_set_a @ ( insert_a @ X3 @ ( insert_a @ Y2 @ bot_bot_set_a ) ) @ edges )
=> ( undire6133010728901294956walk_a @ vertices @ edges @ ( cons_a @ X3 @ ( cons_a @ Y2 @ Ys ) ) ) ) ) ).
% is_walk_hd_tl
thf(fact_939_insert__absorb2,axiom,
! [X3: a,A2: set_a] :
( ( insert_a @ X3 @ ( insert_a @ X3 @ A2 ) )
= ( insert_a @ X3 @ A2 ) ) ).
% insert_absorb2
thf(fact_940_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_941_insert__iff,axiom,
! [A: set_a,B: set_a,A2: set_set_a] :
( ( member_set_a @ A @ ( insert_set_a @ B @ A2 ) )
= ( ( A = B )
| ( member_set_a @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_942_insert__iff,axiom,
! [A: a,B: a,A2: set_a] :
( ( member_a @ A @ ( insert_a @ B @ A2 ) )
= ( ( A = B )
| ( member_a @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_943_insertCI,axiom,
! [A: set_a,B2: set_set_a,B: set_a] :
( ( ~ ( member_set_a @ A @ B2 )
=> ( A = B ) )
=> ( member_set_a @ A @ ( insert_set_a @ B @ B2 ) ) ) ).
% insertCI
thf(fact_944_insertCI,axiom,
! [A: a,B2: set_a,B: a] :
( ( ~ ( member_a @ A @ B2 )
=> ( A = B ) )
=> ( member_a @ A @ ( insert_a @ B @ B2 ) ) ) ).
% insertCI
thf(fact_945_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_946_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_947_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_948_wellformed__alt__fst,axiom,
! [X3: a,Y2: a] :
( ( member_set_a @ ( insert_a @ X3 @ ( insert_a @ Y2 @ bot_bot_set_a ) ) @ edges )
=> ( member_a @ X3 @ vertices ) ) ).
% wellformed_alt_fst
thf(fact_949_wellformed__alt__snd,axiom,
! [X3: a,Y2: a] :
( ( member_set_a @ ( insert_a @ X3 @ ( insert_a @ Y2 @ bot_bot_set_a ) ) @ edges )
=> ( member_a @ Y2 @ vertices ) ) ).
% wellformed_alt_snd
thf(fact_950_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_951_walk__edges_Osimps_I3_J,axiom,
! [X3: a,Y2: a,Ys: list_a] :
( ( undire7337870655677353998dges_a @ ( cons_a @ X3 @ ( cons_a @ Y2 @ Ys ) ) )
= ( cons_set_a @ ( insert_a @ X3 @ ( insert_a @ Y2 @ bot_bot_set_a ) ) @ ( undire7337870655677353998dges_a @ ( cons_a @ Y2 @ Ys ) ) ) ) ).
% walk_edges.simps(3)
thf(fact_952_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_953_not__gr__zero,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_954_singletonI,axiom,
! [A: a] : ( member_a @ A @ ( insert_a @ A @ bot_bot_set_a ) ) ).
% singletonI
thf(fact_955_singletonI,axiom,
! [A: set_a] : ( member_set_a @ A @ ( insert_set_a @ A @ bot_bot_set_set_a ) ) ).
% singletonI
thf(fact_956_singletonI,axiom,
! [A: product_prod_a_a] : ( member1426531477525435216od_a_a @ A @ ( insert4534936382041156343od_a_a @ A @ bot_bo3357376287454694259od_a_a ) ) ).
% singletonI
thf(fact_957_insert__subset,axiom,
! [X3: a,A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ ( insert_a @ X3 @ A2 ) @ B2 )
= ( ( member_a @ X3 @ B2 )
& ( ord_less_eq_set_a @ A2 @ B2 ) ) ) ).
% insert_subset
thf(fact_958_insert__subset,axiom,
! [X3: set_a,A2: set_set_a,B2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ ( insert_set_a @ X3 @ A2 ) @ B2 )
= ( ( member_set_a @ X3 @ B2 )
& ( ord_le3724670747650509150_set_a @ A2 @ B2 ) ) ) ).
% insert_subset
thf(fact_959_insert__subset,axiom,
! [X3: product_prod_a_a,A2: set_Product_prod_a_a,B2: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ ( insert4534936382041156343od_a_a @ X3 @ A2 ) @ B2 )
= ( ( member1426531477525435216od_a_a @ X3 @ B2 )
& ( ord_le746702958409616551od_a_a @ A2 @ B2 ) ) ) ).
% insert_subset
thf(fact_960_walk__edges_Oelims,axiom,
! [X3: list_a,Y2: list_set_a] :
( ( ( undire7337870655677353998dges_a @ X3 )
= Y2 )
=> ( ( ( X3 = nil_a )
=> ( Y2 != nil_set_a ) )
=> ( ( ? [X4: a] :
( X3
= ( cons_a @ X4 @ nil_a ) )
=> ( Y2 != nil_set_a ) )
=> ~ ! [X4: a,Y: a,Ys2: list_a] :
( ( X3
= ( cons_a @ X4 @ ( cons_a @ Y @ Ys2 ) ) )
=> ( Y2
!= ( cons_set_a @ ( insert_a @ X4 @ ( insert_a @ Y @ bot_bot_set_a ) ) @ ( undire7337870655677353998dges_a @ ( cons_a @ Y @ Ys2 ) ) ) ) ) ) ) ) ).
% walk_edges.elims
thf(fact_961_zero__less__double__add__iff__zero__less__single__add,axiom,
! [A: real] :
( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ A ) )
= ( ord_less_real @ zero_zero_real @ A ) ) ).
% zero_less_double_add_iff_zero_less_single_add
thf(fact_962_double__add__less__zero__iff__single__add__less__zero,axiom,
! [A: real] :
( ( ord_less_real @ ( plus_plus_real @ A @ A ) @ zero_zero_real )
= ( ord_less_real @ A @ zero_zero_real ) ) ).
% double_add_less_zero_iff_single_add_less_zero
thf(fact_963_less__add__same__cancel2,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ ( plus_plus_real @ B @ A ) )
= ( ord_less_real @ zero_zero_real @ B ) ) ).
% less_add_same_cancel2
thf(fact_964_less__add__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ ( plus_plus_nat @ B @ A ) )
= ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% less_add_same_cancel2
thf(fact_965_less__add__same__cancel1,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ ( plus_plus_real @ A @ B ) )
= ( ord_less_real @ zero_zero_real @ B ) ) ).
% less_add_same_cancel1
thf(fact_966_less__add__same__cancel1,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ ( plus_plus_nat @ A @ B ) )
= ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% less_add_same_cancel1
thf(fact_967_add__less__same__cancel2,axiom,
! [A: real,B: real] :
( ( ord_less_real @ ( plus_plus_real @ A @ B ) @ B )
= ( ord_less_real @ A @ zero_zero_real ) ) ).
% add_less_same_cancel2
thf(fact_968_add__less__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ B )
= ( ord_less_nat @ A @ zero_zero_nat ) ) ).
% add_less_same_cancel2
thf(fact_969_add__less__same__cancel1,axiom,
! [B: real,A: real] :
( ( ord_less_real @ ( plus_plus_real @ B @ A ) @ B )
= ( ord_less_real @ A @ zero_zero_real ) ) ).
% add_less_same_cancel1
thf(fact_970_add__less__same__cancel1,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ B @ A ) @ B )
= ( ord_less_nat @ A @ zero_zero_nat ) ) ).
% add_less_same_cancel1
thf(fact_971_singleton__insert__inj__eq_H,axiom,
! [A: a,A2: set_a,B: a] :
( ( ( insert_a @ A @ A2 )
= ( insert_a @ B @ bot_bot_set_a ) )
= ( ( A = B )
& ( ord_less_eq_set_a @ A2 @ ( insert_a @ B @ bot_bot_set_a ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_972_singleton__insert__inj__eq_H,axiom,
! [A: set_a,A2: set_set_a,B: set_a] :
( ( ( insert_set_a @ A @ A2 )
= ( insert_set_a @ B @ bot_bot_set_set_a ) )
= ( ( A = B )
& ( ord_le3724670747650509150_set_a @ A2 @ ( insert_set_a @ B @ bot_bot_set_set_a ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_973_singleton__insert__inj__eq_H,axiom,
! [A: product_prod_a_a,A2: set_Product_prod_a_a,B: product_prod_a_a] :
( ( ( insert4534936382041156343od_a_a @ A @ A2 )
= ( insert4534936382041156343od_a_a @ B @ bot_bo3357376287454694259od_a_a ) )
= ( ( A = B )
& ( ord_le746702958409616551od_a_a @ A2 @ ( insert4534936382041156343od_a_a @ B @ bot_bo3357376287454694259od_a_a ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_974_singleton__insert__inj__eq,axiom,
! [B: a,A: a,A2: set_a] :
( ( ( insert_a @ B @ bot_bot_set_a )
= ( insert_a @ A @ A2 ) )
= ( ( A = B )
& ( ord_less_eq_set_a @ A2 @ ( insert_a @ B @ bot_bot_set_a ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_975_singleton__insert__inj__eq,axiom,
! [B: set_a,A: set_a,A2: set_set_a] :
( ( ( insert_set_a @ B @ bot_bot_set_set_a )
= ( insert_set_a @ A @ A2 ) )
= ( ( A = B )
& ( ord_le3724670747650509150_set_a @ A2 @ ( insert_set_a @ B @ bot_bot_set_set_a ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_976_singleton__insert__inj__eq,axiom,
! [B: product_prod_a_a,A: product_prod_a_a,A2: set_Product_prod_a_a] :
( ( ( insert4534936382041156343od_a_a @ B @ bot_bo3357376287454694259od_a_a )
= ( insert4534936382041156343od_a_a @ A @ A2 ) )
= ( ( A = B )
& ( ord_le746702958409616551od_a_a @ A2 @ ( insert4534936382041156343od_a_a @ B @ bot_bo3357376287454694259od_a_a ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_977_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_978_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_979_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_980_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_981_the__elem__eq,axiom,
! [X3: a] :
( ( the_elem_a @ ( insert_a @ X3 @ bot_bot_set_a ) )
= X3 ) ).
% the_elem_eq
thf(fact_982_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_983_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_984_degree__none,axiom,
! [V: a] :
( ~ ( member_a @ V @ vertices )
=> ( ( undire8867928226783802224gree_a @ edges @ V )
= zero_zero_nat ) ) ).
% degree_none
thf(fact_985_gr__zeroI,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr_zeroI
thf(fact_986_not__less__zero,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less_zero
thf(fact_987_gr__implies__not__zero,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_988_zero__less__iff__neq__zero,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( N != zero_zero_nat ) ) ).
% zero_less_iff_neq_zero
thf(fact_989_singletonD,axiom,
! [B: a,A: a] :
( ( member_a @ B @ ( insert_a @ A @ bot_bot_set_a ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_990_singletonD,axiom,
! [B: set_a,A: set_a] :
( ( member_set_a @ B @ ( insert_set_a @ A @ bot_bot_set_set_a ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_991_singletonD,axiom,
! [B: product_prod_a_a,A: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ B @ ( insert4534936382041156343od_a_a @ A @ bot_bo3357376287454694259od_a_a ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_992_bot__set__def,axiom,
( bot_bot_set_a
= ( collect_a @ bot_bot_a_o ) ) ).
% bot_set_def
thf(fact_993_bot__set__def,axiom,
( bot_bot_set_set_a
= ( collect_set_a @ bot_bot_set_a_o ) ) ).
% bot_set_def
thf(fact_994_bot__set__def,axiom,
( bot_bo3357376287454694259od_a_a
= ( collec3336397797384452498od_a_a @ bot_bo4160289986317612842_a_a_o ) ) ).
% bot_set_def
thf(fact_995_singleton__iff,axiom,
! [B: a,A: a] :
( ( member_a @ B @ ( insert_a @ A @ bot_bot_set_a ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_996_singleton__iff,axiom,
! [B: set_a,A: set_a] :
( ( member_set_a @ B @ ( insert_set_a @ A @ bot_bot_set_set_a ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_997_singleton__iff,axiom,
! [B: product_prod_a_a,A: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ B @ ( insert4534936382041156343od_a_a @ A @ bot_bo3357376287454694259od_a_a ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_998_doubleton__eq__iff,axiom,
! [A: a,B: a,C: a,D: a] :
( ( ( insert_a @ A @ ( insert_a @ B @ bot_bot_set_a ) )
= ( insert_a @ C @ ( insert_a @ D @ bot_bot_set_a ) ) )
= ( ( ( A = C )
& ( B = D ) )
| ( ( A = D )
& ( B = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_999_doubleton__eq__iff,axiom,
! [A: set_a,B: set_a,C: set_a,D: set_a] :
( ( ( insert_set_a @ A @ ( insert_set_a @ B @ bot_bot_set_set_a ) )
= ( insert_set_a @ C @ ( insert_set_a @ D @ bot_bot_set_set_a ) ) )
= ( ( ( A = C )
& ( B = D ) )
| ( ( A = D )
& ( B = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_1000_doubleton__eq__iff,axiom,
! [A: product_prod_a_a,B: product_prod_a_a,C: product_prod_a_a,D: product_prod_a_a] :
( ( ( insert4534936382041156343od_a_a @ A @ ( insert4534936382041156343od_a_a @ B @ bot_bo3357376287454694259od_a_a ) )
= ( insert4534936382041156343od_a_a @ C @ ( insert4534936382041156343od_a_a @ D @ bot_bo3357376287454694259od_a_a ) ) )
= ( ( ( A = C )
& ( B = D ) )
| ( ( A = D )
& ( B = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_1001_insert__not__empty,axiom,
! [A: a,A2: set_a] :
( ( insert_a @ A @ A2 )
!= bot_bot_set_a ) ).
% insert_not_empty
thf(fact_1002_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_1003_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_1004_singleton__inject,axiom,
! [A: a,B: a] :
( ( ( insert_a @ A @ bot_bot_set_a )
= ( insert_a @ B @ bot_bot_set_a ) )
=> ( A = B ) ) ).
% singleton_inject
thf(fact_1005_singleton__inject,axiom,
! [A: set_a,B: set_a] :
( ( ( insert_set_a @ A @ bot_bot_set_set_a )
= ( insert_set_a @ B @ bot_bot_set_set_a ) )
=> ( A = B ) ) ).
% singleton_inject
thf(fact_1006_singleton__inject,axiom,
! [A: product_prod_a_a,B: product_prod_a_a] :
( ( ( insert4534936382041156343od_a_a @ A @ bot_bo3357376287454694259od_a_a )
= ( insert4534936382041156343od_a_a @ B @ bot_bo3357376287454694259od_a_a ) )
=> ( A = B ) ) ).
% singleton_inject
thf(fact_1007_leD,axiom,
! [Y2: set_a,X3: set_a] :
( ( ord_less_eq_set_a @ Y2 @ X3 )
=> ~ ( ord_less_set_a @ X3 @ Y2 ) ) ).
% leD
thf(fact_1008_leD,axiom,
! [Y2: set_set_a,X3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ Y2 @ X3 )
=> ~ ( ord_less_set_set_a @ X3 @ Y2 ) ) ).
% leD
thf(fact_1009_leD,axiom,
! [Y2: real,X3: real] :
( ( ord_less_eq_real @ Y2 @ X3 )
=> ~ ( ord_less_real @ X3 @ Y2 ) ) ).
% leD
thf(fact_1010_leD,axiom,
! [Y2: nat,X3: nat] :
( ( ord_less_eq_nat @ Y2 @ X3 )
=> ~ ( ord_less_nat @ X3 @ Y2 ) ) ).
% leD
thf(fact_1011_leD,axiom,
! [Y2: set_Product_prod_a_a,X3: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ Y2 @ X3 )
=> ~ ( ord_le6819997720685908915od_a_a @ X3 @ Y2 ) ) ).
% leD
thf(fact_1012_leI,axiom,
! [X3: real,Y2: real] :
( ~ ( ord_less_real @ X3 @ Y2 )
=> ( ord_less_eq_real @ Y2 @ X3 ) ) ).
% leI
thf(fact_1013_leI,axiom,
! [X3: nat,Y2: nat] :
( ~ ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ X3 ) ) ).
% leI
thf(fact_1014_nless__le,axiom,
! [A: set_a,B: set_a] :
( ( ~ ( ord_less_set_a @ A @ B ) )
= ( ~ ( ord_less_eq_set_a @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_1015_nless__le,axiom,
! [A: set_set_a,B: set_set_a] :
( ( ~ ( ord_less_set_set_a @ A @ B ) )
= ( ~ ( ord_le3724670747650509150_set_a @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_1016_nless__le,axiom,
! [A: real,B: real] :
( ( ~ ( ord_less_real @ A @ B ) )
= ( ~ ( ord_less_eq_real @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_1017_nless__le,axiom,
! [A: nat,B: nat] :
( ( ~ ( ord_less_nat @ A @ B ) )
= ( ~ ( ord_less_eq_nat @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_1018_nless__le,axiom,
! [A: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ( ~ ( ord_le6819997720685908915od_a_a @ A @ B ) )
= ( ~ ( ord_le746702958409616551od_a_a @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_1019_antisym__conv1,axiom,
! [X3: set_a,Y2: set_a] :
( ~ ( ord_less_set_a @ X3 @ Y2 )
=> ( ( ord_less_eq_set_a @ X3 @ Y2 )
= ( X3 = Y2 ) ) ) ).
% antisym_conv1
thf(fact_1020_antisym__conv1,axiom,
! [X3: set_set_a,Y2: set_set_a] :
( ~ ( ord_less_set_set_a @ X3 @ Y2 )
=> ( ( ord_le3724670747650509150_set_a @ X3 @ Y2 )
= ( X3 = Y2 ) ) ) ).
% antisym_conv1
thf(fact_1021_antisym__conv1,axiom,
! [X3: real,Y2: real] :
( ~ ( ord_less_real @ X3 @ Y2 )
=> ( ( ord_less_eq_real @ X3 @ Y2 )
= ( X3 = Y2 ) ) ) ).
% antisym_conv1
thf(fact_1022_antisym__conv1,axiom,
! [X3: nat,Y2: nat] :
( ~ ( ord_less_nat @ X3 @ Y2 )
=> ( ( ord_less_eq_nat @ X3 @ Y2 )
= ( X3 = Y2 ) ) ) ).
% antisym_conv1
thf(fact_1023_antisym__conv1,axiom,
! [X3: set_Product_prod_a_a,Y2: set_Product_prod_a_a] :
( ~ ( ord_le6819997720685908915od_a_a @ X3 @ Y2 )
=> ( ( ord_le746702958409616551od_a_a @ X3 @ Y2 )
= ( X3 = Y2 ) ) ) ).
% antisym_conv1
thf(fact_1024_antisym__conv2,axiom,
! [X3: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y2 )
=> ( ( ~ ( ord_less_set_a @ X3 @ Y2 ) )
= ( X3 = Y2 ) ) ) ).
% antisym_conv2
thf(fact_1025_antisym__conv2,axiom,
! [X3: set_set_a,Y2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X3 @ Y2 )
=> ( ( ~ ( ord_less_set_set_a @ X3 @ Y2 ) )
= ( X3 = Y2 ) ) ) ).
% antisym_conv2
thf(fact_1026_antisym__conv2,axiom,
! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ( ~ ( ord_less_real @ X3 @ Y2 ) )
= ( X3 = Y2 ) ) ) ).
% antisym_conv2
thf(fact_1027_antisym__conv2,axiom,
! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ( ~ ( ord_less_nat @ X3 @ Y2 ) )
= ( X3 = Y2 ) ) ) ).
% antisym_conv2
thf(fact_1028_antisym__conv2,axiom,
! [X3: set_Product_prod_a_a,Y2: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ X3 @ Y2 )
=> ( ( ~ ( ord_le6819997720685908915od_a_a @ X3 @ Y2 ) )
= ( X3 = Y2 ) ) ) ).
% antisym_conv2
thf(fact_1029_dense__ge,axiom,
! [Z3: real,Y2: real] :
( ! [X4: real] :
( ( ord_less_real @ Z3 @ X4 )
=> ( ord_less_eq_real @ Y2 @ X4 ) )
=> ( ord_less_eq_real @ Y2 @ Z3 ) ) ).
% dense_ge
thf(fact_1030_dense__le,axiom,
! [Y2: real,Z3: real] :
( ! [X4: real] :
( ( ord_less_real @ X4 @ Y2 )
=> ( ord_less_eq_real @ X4 @ Z3 ) )
=> ( ord_less_eq_real @ Y2 @ Z3 ) ) ).
% dense_le
thf(fact_1031_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_1032_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_1033_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_1034_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_1035_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_1036_not__le__imp__less,axiom,
! [Y2: real,X3: real] :
( ~ ( ord_less_eq_real @ Y2 @ X3 )
=> ( ord_less_real @ X3 @ Y2 ) ) ).
% not_le_imp_less
thf(fact_1037_not__le__imp__less,axiom,
! [Y2: nat,X3: nat] :
( ~ ( ord_less_eq_nat @ Y2 @ X3 )
=> ( ord_less_nat @ X3 @ Y2 ) ) ).
% not_le_imp_less
thf(fact_1038_order_Oorder__iff__strict,axiom,
( ord_less_eq_set_a
= ( ^ [A3: set_a,B3: set_a] :
( ( ord_less_set_a @ A3 @ B3 )
| ( A3 = B3 ) ) ) ) ).
% order.order_iff_strict
thf(fact_1039_order_Oorder__iff__strict,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [A3: set_set_a,B3: set_set_a] :
( ( ord_less_set_set_a @ A3 @ B3 )
| ( A3 = B3 ) ) ) ) ).
% order.order_iff_strict
thf(fact_1040_order_Oorder__iff__strict,axiom,
( ord_less_eq_real
= ( ^ [A3: real,B3: real] :
( ( ord_less_real @ A3 @ B3 )
| ( A3 = B3 ) ) ) ) ).
% order.order_iff_strict
thf(fact_1041_order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [A3: nat,B3: nat] :
( ( ord_less_nat @ A3 @ B3 )
| ( A3 = B3 ) ) ) ) ).
% order.order_iff_strict
thf(fact_1042_order_Oorder__iff__strict,axiom,
( ord_le746702958409616551od_a_a
= ( ^ [A3: set_Product_prod_a_a,B3: set_Product_prod_a_a] :
( ( ord_le6819997720685908915od_a_a @ A3 @ B3 )
| ( A3 = B3 ) ) ) ) ).
% order.order_iff_strict
thf(fact_1043_order_Ostrict__iff__order,axiom,
( ord_less_set_a
= ( ^ [A3: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ A3 @ B3 )
& ( A3 != B3 ) ) ) ) ).
% order.strict_iff_order
thf(fact_1044_order_Ostrict__iff__order,axiom,
( ord_less_set_set_a
= ( ^ [A3: set_set_a,B3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A3 @ B3 )
& ( A3 != B3 ) ) ) ) ).
% order.strict_iff_order
thf(fact_1045_order_Ostrict__iff__order,axiom,
( ord_less_real
= ( ^ [A3: real,B3: real] :
( ( ord_less_eq_real @ A3 @ B3 )
& ( A3 != B3 ) ) ) ) ).
% order.strict_iff_order
thf(fact_1046_order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
& ( A3 != B3 ) ) ) ) ).
% order.strict_iff_order
thf(fact_1047_order_Ostrict__iff__order,axiom,
( ord_le6819997720685908915od_a_a
= ( ^ [A3: set_Product_prod_a_a,B3: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A3 @ B3 )
& ( A3 != B3 ) ) ) ) ).
% order.strict_iff_order
thf(fact_1048_order_Ostrict__trans1,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_set_a @ B @ C )
=> ( ord_less_set_a @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_1049_order_Ostrict__trans1,axiom,
! [A: set_set_a,B: set_set_a,C: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( ( ord_less_set_set_a @ B @ C )
=> ( ord_less_set_set_a @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_1050_order_Ostrict__trans1,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_real @ B @ C )
=> ( ord_less_real @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_1051_order_Ostrict__trans1,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_1052_order_Ostrict__trans1,axiom,
! [A: set_Product_prod_a_a,B: set_Product_prod_a_a,C: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A @ B )
=> ( ( ord_le6819997720685908915od_a_a @ B @ C )
=> ( ord_le6819997720685908915od_a_a @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_1053_order_Ostrict__trans2,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( ord_less_set_a @ A @ B )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ord_less_set_a @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_1054_order_Ostrict__trans2,axiom,
! [A: set_set_a,B: set_set_a,C: set_set_a] :
( ( ord_less_set_set_a @ A @ B )
=> ( ( ord_le3724670747650509150_set_a @ B @ C )
=> ( ord_less_set_set_a @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_1055_order_Ostrict__trans2,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ord_less_real @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_1056_order_Ostrict__trans2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_1057_order_Ostrict__trans2,axiom,
! [A: set_Product_prod_a_a,B: set_Product_prod_a_a,C: set_Product_prod_a_a] :
( ( ord_le6819997720685908915od_a_a @ A @ B )
=> ( ( ord_le746702958409616551od_a_a @ B @ C )
=> ( ord_le6819997720685908915od_a_a @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_1058_order_Ostrict__iff__not,axiom,
( ord_less_set_a
= ( ^ [A3: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ A3 @ B3 )
& ~ ( ord_less_eq_set_a @ B3 @ A3 ) ) ) ) ).
% order.strict_iff_not
thf(fact_1059_order_Ostrict__iff__not,axiom,
( ord_less_set_set_a
= ( ^ [A3: set_set_a,B3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A3 @ B3 )
& ~ ( ord_le3724670747650509150_set_a @ B3 @ A3 ) ) ) ) ).
% order.strict_iff_not
thf(fact_1060_order_Ostrict__iff__not,axiom,
( ord_less_real
= ( ^ [A3: real,B3: real] :
( ( ord_less_eq_real @ A3 @ B3 )
& ~ ( ord_less_eq_real @ B3 @ A3 ) ) ) ) ).
% order.strict_iff_not
thf(fact_1061_order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
& ~ ( ord_less_eq_nat @ B3 @ A3 ) ) ) ) ).
% order.strict_iff_not
thf(fact_1062_order_Ostrict__iff__not,axiom,
( ord_le6819997720685908915od_a_a
= ( ^ [A3: set_Product_prod_a_a,B3: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A3 @ B3 )
& ~ ( ord_le746702958409616551od_a_a @ B3 @ A3 ) ) ) ) ).
% order.strict_iff_not
thf(fact_1063_dense__ge__bounded,axiom,
! [Z3: real,X3: real,Y2: real] :
( ( ord_less_real @ Z3 @ X3 )
=> ( ! [W: real] :
( ( ord_less_real @ Z3 @ W )
=> ( ( ord_less_real @ W @ X3 )
=> ( ord_less_eq_real @ Y2 @ W ) ) )
=> ( ord_less_eq_real @ Y2 @ Z3 ) ) ) ).
% dense_ge_bounded
thf(fact_1064_dense__le__bounded,axiom,
! [X3: real,Y2: real,Z3: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ( ! [W: real] :
( ( ord_less_real @ X3 @ W )
=> ( ( ord_less_real @ W @ Y2 )
=> ( ord_less_eq_real @ W @ Z3 ) ) )
=> ( ord_less_eq_real @ Y2 @ Z3 ) ) ) ).
% dense_le_bounded
thf(fact_1065_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_set_a
= ( ^ [B3: set_a,A3: set_a] :
( ( ord_less_set_a @ B3 @ A3 )
| ( A3 = B3 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_1066_dual__order_Oorder__iff__strict,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [B3: set_set_a,A3: set_set_a] :
( ( ord_less_set_set_a @ B3 @ A3 )
| ( A3 = B3 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_1067_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_real
= ( ^ [B3: real,A3: real] :
( ( ord_less_real @ B3 @ A3 )
| ( A3 = B3 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_1068_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [B3: nat,A3: nat] :
( ( ord_less_nat @ B3 @ A3 )
| ( A3 = B3 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_1069_dual__order_Oorder__iff__strict,axiom,
( ord_le746702958409616551od_a_a
= ( ^ [B3: set_Product_prod_a_a,A3: set_Product_prod_a_a] :
( ( ord_le6819997720685908915od_a_a @ B3 @ A3 )
| ( A3 = B3 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_1070_dual__order_Ostrict__iff__order,axiom,
( ord_less_set_a
= ( ^ [B3: set_a,A3: set_a] :
( ( ord_less_eq_set_a @ B3 @ A3 )
& ( A3 != B3 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_1071_dual__order_Ostrict__iff__order,axiom,
( ord_less_set_set_a
= ( ^ [B3: set_set_a,A3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B3 @ A3 )
& ( A3 != B3 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_1072_dual__order_Ostrict__iff__order,axiom,
( ord_less_real
= ( ^ [B3: real,A3: real] :
( ( ord_less_eq_real @ B3 @ A3 )
& ( A3 != B3 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_1073_dual__order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [B3: nat,A3: nat] :
( ( ord_less_eq_nat @ B3 @ A3 )
& ( A3 != B3 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_1074_dual__order_Ostrict__iff__order,axiom,
( ord_le6819997720685908915od_a_a
= ( ^ [B3: set_Product_prod_a_a,A3: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ B3 @ A3 )
& ( A3 != B3 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_1075_dual__order_Ostrict__trans1,axiom,
! [B: set_a,A: set_a,C: set_a] :
( ( ord_less_eq_set_a @ B @ A )
=> ( ( ord_less_set_a @ C @ B )
=> ( ord_less_set_a @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_1076_dual__order_Ostrict__trans1,axiom,
! [B: set_set_a,A: set_set_a,C: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B @ A )
=> ( ( ord_less_set_set_a @ C @ B )
=> ( ord_less_set_set_a @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_1077_dual__order_Ostrict__trans1,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( ord_less_real @ C @ B )
=> ( ord_less_real @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_1078_dual__order_Ostrict__trans1,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_nat @ C @ B )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_1079_dual__order_Ostrict__trans1,axiom,
! [B: set_Product_prod_a_a,A: set_Product_prod_a_a,C: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ B @ A )
=> ( ( ord_le6819997720685908915od_a_a @ C @ B )
=> ( ord_le6819997720685908915od_a_a @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_1080_dual__order_Ostrict__trans2,axiom,
! [B: set_a,A: set_a,C: set_a] :
( ( ord_less_set_a @ B @ A )
=> ( ( ord_less_eq_set_a @ C @ B )
=> ( ord_less_set_a @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_1081_dual__order_Ostrict__trans2,axiom,
! [B: set_set_a,A: set_set_a,C: set_set_a] :
( ( ord_less_set_set_a @ B @ A )
=> ( ( ord_le3724670747650509150_set_a @ C @ B )
=> ( ord_less_set_set_a @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_1082_dual__order_Ostrict__trans2,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_real @ B @ A )
=> ( ( ord_less_eq_real @ C @ B )
=> ( ord_less_real @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_1083_dual__order_Ostrict__trans2,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_1084_dual__order_Ostrict__trans2,axiom,
! [B: set_Product_prod_a_a,A: set_Product_prod_a_a,C: set_Product_prod_a_a] :
( ( ord_le6819997720685908915od_a_a @ B @ A )
=> ( ( ord_le746702958409616551od_a_a @ C @ B )
=> ( ord_le6819997720685908915od_a_a @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_1085_dual__order_Ostrict__iff__not,axiom,
( ord_less_set_a
= ( ^ [B3: set_a,A3: set_a] :
( ( ord_less_eq_set_a @ B3 @ A3 )
& ~ ( ord_less_eq_set_a @ A3 @ B3 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_1086_dual__order_Ostrict__iff__not,axiom,
( ord_less_set_set_a
= ( ^ [B3: set_set_a,A3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B3 @ A3 )
& ~ ( ord_le3724670747650509150_set_a @ A3 @ B3 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_1087_dual__order_Ostrict__iff__not,axiom,
( ord_less_real
= ( ^ [B3: real,A3: real] :
( ( ord_less_eq_real @ B3 @ A3 )
& ~ ( ord_less_eq_real @ A3 @ B3 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_1088_dual__order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [B3: nat,A3: nat] :
( ( ord_less_eq_nat @ B3 @ A3 )
& ~ ( ord_less_eq_nat @ A3 @ B3 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_1089_dual__order_Ostrict__iff__not,axiom,
( ord_le6819997720685908915od_a_a
= ( ^ [B3: set_Product_prod_a_a,A3: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ B3 @ A3 )
& ~ ( ord_le746702958409616551od_a_a @ A3 @ B3 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_1090_order_Ostrict__implies__order,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_set_a @ A @ B )
=> ( ord_less_eq_set_a @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_1091_order_Ostrict__implies__order,axiom,
! [A: set_set_a,B: set_set_a] :
( ( ord_less_set_set_a @ A @ B )
=> ( ord_le3724670747650509150_set_a @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_1092_order_Ostrict__implies__order,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ( ord_less_eq_real @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_1093_order_Ostrict__implies__order,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_1094_order_Ostrict__implies__order,axiom,
! [A: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ( ord_le6819997720685908915od_a_a @ A @ B )
=> ( ord_le746702958409616551od_a_a @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_1095_dual__order_Ostrict__implies__order,axiom,
! [B: set_a,A: set_a] :
( ( ord_less_set_a @ B @ A )
=> ( ord_less_eq_set_a @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_1096_dual__order_Ostrict__implies__order,axiom,
! [B: set_set_a,A: set_set_a] :
( ( ord_less_set_set_a @ B @ A )
=> ( ord_le3724670747650509150_set_a @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_1097_dual__order_Ostrict__implies__order,axiom,
! [B: real,A: real] :
( ( ord_less_real @ B @ A )
=> ( ord_less_eq_real @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_1098_dual__order_Ostrict__implies__order,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( ord_less_eq_nat @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_1099_dual__order_Ostrict__implies__order,axiom,
! [B: set_Product_prod_a_a,A: set_Product_prod_a_a] :
( ( ord_le6819997720685908915od_a_a @ B @ A )
=> ( ord_le746702958409616551od_a_a @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_1100_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_1101_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_1102_order__le__less,axiom,
( ord_less_eq_real
= ( ^ [X2: real,Y3: real] :
( ( ord_less_real @ X2 @ Y3 )
| ( X2 = Y3 ) ) ) ) ).
% order_le_less
thf(fact_1103_order__le__less,axiom,
( ord_less_eq_nat
= ( ^ [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
| ( X2 = Y3 ) ) ) ) ).
% order_le_less
thf(fact_1104_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_1105_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_1106_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_1107_order__less__le,axiom,
( ord_less_real
= ( ^ [X2: real,Y3: real] :
( ( ord_less_eq_real @ X2 @ Y3 )
& ( X2 != Y3 ) ) ) ) ).
% order_less_le
thf(fact_1108_order__less__le,axiom,
( ord_less_nat
= ( ^ [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
& ( X2 != Y3 ) ) ) ) ).
% order_less_le
thf(fact_1109_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_1110_linorder__not__le,axiom,
! [X3: real,Y2: real] :
( ( ~ ( ord_less_eq_real @ X3 @ Y2 ) )
= ( ord_less_real @ Y2 @ X3 ) ) ).
% linorder_not_le
thf(fact_1111_linorder__not__le,axiom,
! [X3: nat,Y2: nat] :
( ( ~ ( ord_less_eq_nat @ X3 @ Y2 ) )
= ( ord_less_nat @ Y2 @ X3 ) ) ).
% linorder_not_le
thf(fact_1112_linorder__not__less,axiom,
! [X3: real,Y2: real] :
( ( ~ ( ord_less_real @ X3 @ Y2 ) )
= ( ord_less_eq_real @ Y2 @ X3 ) ) ).
% linorder_not_less
thf(fact_1113_linorder__not__less,axiom,
! [X3: nat,Y2: nat] :
( ( ~ ( ord_less_nat @ X3 @ Y2 ) )
= ( ord_less_eq_nat @ Y2 @ X3 ) ) ).
% linorder_not_less
thf(fact_1114_order__less__imp__le,axiom,
! [X3: set_a,Y2: set_a] :
( ( ord_less_set_a @ X3 @ Y2 )
=> ( ord_less_eq_set_a @ X3 @ Y2 ) ) ).
% order_less_imp_le
thf(fact_1115_order__less__imp__le,axiom,
! [X3: set_set_a,Y2: set_set_a] :
( ( ord_less_set_set_a @ X3 @ Y2 )
=> ( ord_le3724670747650509150_set_a @ X3 @ Y2 ) ) ).
% order_less_imp_le
thf(fact_1116_order__less__imp__le,axiom,
! [X3: real,Y2: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ( ord_less_eq_real @ X3 @ Y2 ) ) ).
% order_less_imp_le
thf(fact_1117_order__less__imp__le,axiom,
! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ X3 @ Y2 ) ) ).
% order_less_imp_le
thf(fact_1118_order__less__imp__le,axiom,
! [X3: set_Product_prod_a_a,Y2: set_Product_prod_a_a] :
( ( ord_le6819997720685908915od_a_a @ X3 @ Y2 )
=> ( ord_le746702958409616551od_a_a @ X3 @ Y2 ) ) ).
% order_less_imp_le
thf(fact_1119_order__le__neq__trans,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( A != B )
=> ( ord_less_set_a @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_1120_order__le__neq__trans,axiom,
! [A: set_set_a,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( ( A != B )
=> ( ord_less_set_set_a @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_1121_order__le__neq__trans,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( A != B )
=> ( ord_less_real @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_1122_order__le__neq__trans,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( A != B )
=> ( ord_less_nat @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_1123_order__le__neq__trans,axiom,
! [A: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A @ B )
=> ( ( A != B )
=> ( ord_le6819997720685908915od_a_a @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_1124_order__neq__le__trans,axiom,
! [A: set_a,B: set_a] :
( ( A != B )
=> ( ( ord_less_eq_set_a @ A @ B )
=> ( ord_less_set_a @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_1125_order__neq__le__trans,axiom,
! [A: set_set_a,B: set_set_a] :
( ( A != B )
=> ( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( ord_less_set_set_a @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_1126_order__neq__le__trans,axiom,
! [A: real,B: real] :
( ( A != B )
=> ( ( ord_less_eq_real @ A @ B )
=> ( ord_less_real @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_1127_order__neq__le__trans,axiom,
! [A: nat,B: nat] :
( ( A != B )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_nat @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_1128_order__neq__le__trans,axiom,
! [A: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ( A != B )
=> ( ( ord_le746702958409616551od_a_a @ A @ B )
=> ( ord_le6819997720685908915od_a_a @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_1129_order__le__less__trans,axiom,
! [X3: set_a,Y2: set_a,Z3: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y2 )
=> ( ( ord_less_set_a @ Y2 @ Z3 )
=> ( ord_less_set_a @ X3 @ Z3 ) ) ) ).
% order_le_less_trans
thf(fact_1130_order__le__less__trans,axiom,
! [X3: set_set_a,Y2: set_set_a,Z3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X3 @ Y2 )
=> ( ( ord_less_set_set_a @ Y2 @ Z3 )
=> ( ord_less_set_set_a @ X3 @ Z3 ) ) ) ).
% order_le_less_trans
thf(fact_1131_order__le__less__trans,axiom,
! [X3: real,Y2: real,Z3: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ( ord_less_real @ Y2 @ Z3 )
=> ( ord_less_real @ X3 @ Z3 ) ) ) ).
% order_le_less_trans
thf(fact_1132_order__le__less__trans,axiom,
! [X3: nat,Y2: nat,Z3: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ( ord_less_nat @ Y2 @ Z3 )
=> ( ord_less_nat @ X3 @ Z3 ) ) ) ).
% order_le_less_trans
thf(fact_1133_order__le__less__trans,axiom,
! [X3: set_Product_prod_a_a,Y2: set_Product_prod_a_a,Z3: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ X3 @ Y2 )
=> ( ( ord_le6819997720685908915od_a_a @ Y2 @ Z3 )
=> ( ord_le6819997720685908915od_a_a @ X3 @ Z3 ) ) ) ).
% order_le_less_trans
thf(fact_1134_order__less__le__trans,axiom,
! [X3: set_a,Y2: set_a,Z3: set_a] :
( ( ord_less_set_a @ X3 @ Y2 )
=> ( ( ord_less_eq_set_a @ Y2 @ Z3 )
=> ( ord_less_set_a @ X3 @ Z3 ) ) ) ).
% order_less_le_trans
thf(fact_1135_order__less__le__trans,axiom,
! [X3: set_set_a,Y2: set_set_a,Z3: set_set_a] :
( ( ord_less_set_set_a @ X3 @ Y2 )
=> ( ( ord_le3724670747650509150_set_a @ Y2 @ Z3 )
=> ( ord_less_set_set_a @ X3 @ Z3 ) ) ) ).
% order_less_le_trans
thf(fact_1136_order__less__le__trans,axiom,
! [X3: real,Y2: real,Z3: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ( ( ord_less_eq_real @ Y2 @ Z3 )
=> ( ord_less_real @ X3 @ Z3 ) ) ) ).
% order_less_le_trans
thf(fact_1137_order__less__le__trans,axiom,
! [X3: nat,Y2: nat,Z3: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ( ord_less_eq_nat @ Y2 @ Z3 )
=> ( ord_less_nat @ X3 @ Z3 ) ) ) ).
% order_less_le_trans
thf(fact_1138_order__less__le__trans,axiom,
! [X3: set_Product_prod_a_a,Y2: set_Product_prod_a_a,Z3: set_Product_prod_a_a] :
( ( ord_le6819997720685908915od_a_a @ X3 @ Y2 )
=> ( ( ord_le746702958409616551od_a_a @ Y2 @ Z3 )
=> ( ord_le6819997720685908915od_a_a @ X3 @ Z3 ) ) ) ).
% order_less_le_trans
thf(fact_1139_order__le__less__subst1,axiom,
! [A: set_a,F: nat > set_a,B: nat,C: nat] :
( ( ord_less_eq_set_a @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X4: nat,Y: nat] :
( ( ord_less_nat @ X4 @ Y )
=> ( ord_less_set_a @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_1140_order__le__less__subst1,axiom,
! [A: set_set_a,F: nat > set_set_a,B: nat,C: nat] :
( ( ord_le3724670747650509150_set_a @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X4: nat,Y: nat] :
( ( ord_less_nat @ X4 @ Y )
=> ( ord_less_set_set_a @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_set_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_1141_order__le__less__subst1,axiom,
! [A: real,F: nat > real,B: nat,C: nat] :
( ( ord_less_eq_real @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X4: nat,Y: nat] :
( ( ord_less_nat @ X4 @ Y )
=> ( ord_less_real @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_1142_order__le__less__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X4: nat,Y: nat] :
( ( ord_less_nat @ X4 @ Y )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_1143_order__le__less__subst1,axiom,
! [A: set_Product_prod_a_a,F: nat > set_Product_prod_a_a,B: nat,C: nat] :
( ( ord_le746702958409616551od_a_a @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X4: nat,Y: nat] :
( ( ord_less_nat @ X4 @ Y )
=> ( ord_le6819997720685908915od_a_a @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_le6819997720685908915od_a_a @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_1144_order__le__less__subst2,axiom,
! [A: real,B: real,F: real > real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C )
=> ( ! [X4: real,Y: real] :
( ( ord_less_eq_real @ X4 @ Y )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_1145_order__le__less__subst2,axiom,
! [A: real,B: real,F: real > nat,C: nat] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X4: real,Y: real] :
( ( ord_less_eq_real @ X4 @ Y )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_1146_order__le__less__subst2,axiom,
! [A: nat,B: nat,F: nat > real,C: real] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C )
=> ( ! [X4: nat,Y: nat] :
( ( ord_less_eq_nat @ X4 @ Y )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_1147_order__le__less__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X4: nat,Y: nat] :
( ( ord_less_eq_nat @ X4 @ Y )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_1148_order__le__less__subst2,axiom,
! [A: set_a,B: set_a,F: set_a > real,C: real] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C )
=> ( ! [X4: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_1149_order__le__less__subst2,axiom,
! [A: set_a,B: set_a,F: set_a > nat,C: nat] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X4: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_1150_order__le__less__subst2,axiom,
! [A: real,B: real,F: real > set_a,C: set_a] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_set_a @ ( F @ B ) @ C )
=> ( ! [X4: real,Y: real] :
( ( ord_less_eq_real @ X4 @ Y )
=> ( ord_less_eq_set_a @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_1151_order__le__less__subst2,axiom,
! [A: nat,B: nat,F: nat > set_a,C: set_a] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_set_a @ ( F @ B ) @ C )
=> ( ! [X4: nat,Y: nat] :
( ( ord_less_eq_nat @ X4 @ Y )
=> ( ord_less_eq_set_a @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_1152_order__le__less__subst2,axiom,
! [A: set_a,B: set_a,F: set_a > set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_set_a @ ( F @ B ) @ C )
=> ( ! [X4: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y )
=> ( ord_less_eq_set_a @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_1153_order__le__less__subst2,axiom,
! [A: set_set_a,B: set_set_a,F: set_set_a > real,C: real] :
( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C )
=> ( ! [X4: set_set_a,Y: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X4 @ Y )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_1154_order__less__le__subst1,axiom,
! [A: real,F: real > real,B: real,C: real] :
( ( ord_less_real @ A @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X4: real,Y: real] :
( ( ord_less_eq_real @ X4 @ Y )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_1155_order__less__le__subst1,axiom,
! [A: nat,F: real > nat,B: real,C: real] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X4: real,Y: real] :
( ( ord_less_eq_real @ X4 @ Y )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_1156_order__less__le__subst1,axiom,
! [A: real,F: nat > real,B: nat,C: nat] :
( ( ord_less_real @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X4: nat,Y: nat] :
( ( ord_less_eq_nat @ X4 @ Y )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_1157_order__less__le__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X4: nat,Y: nat] :
( ( ord_less_eq_nat @ X4 @ Y )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_1158_order__less__le__subst1,axiom,
! [A: real,F: set_a > real,B: set_a,C: set_a] :
( ( ord_less_real @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ! [X4: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_1159_order__less__le__subst1,axiom,
! [A: nat,F: set_a > nat,B: set_a,C: set_a] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ! [X4: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_1160_order__less__le__subst1,axiom,
! [A: set_a,F: real > set_a,B: real,C: real] :
( ( ord_less_set_a @ A @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X4: real,Y: real] :
( ( ord_less_eq_real @ X4 @ Y )
=> ( ord_less_eq_set_a @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_1161_order__less__le__subst1,axiom,
! [A: set_a,F: nat > set_a,B: nat,C: nat] :
( ( ord_less_set_a @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X4: nat,Y: nat] :
( ( ord_less_eq_nat @ X4 @ Y )
=> ( ord_less_eq_set_a @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_1162_order__less__le__subst1,axiom,
! [A: set_a,F: set_a > set_a,B: set_a,C: set_a] :
( ( ord_less_set_a @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ! [X4: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y )
=> ( ord_less_eq_set_a @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_1163_order__less__le__subst1,axiom,
! [A: real,F: set_set_a > real,B: set_set_a,C: set_set_a] :
( ( ord_less_real @ A @ ( F @ B ) )
=> ( ( ord_le3724670747650509150_set_a @ B @ C )
=> ( ! [X4: set_set_a,Y: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X4 @ Y )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_1164_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > set_a,C: set_a] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_set_a @ ( F @ B ) @ C )
=> ( ! [X4: nat,Y: nat] :
( ( ord_less_nat @ X4 @ Y )
=> ( ord_less_set_a @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_1165_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > set_set_a,C: set_set_a] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_le3724670747650509150_set_a @ ( F @ B ) @ C )
=> ( ! [X4: nat,Y: nat] :
( ( ord_less_nat @ X4 @ Y )
=> ( ord_less_set_set_a @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_set_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_1166_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > real,C: real] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C )
=> ( ! [X4: nat,Y: nat] :
( ( ord_less_nat @ X4 @ Y )
=> ( ord_less_real @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_1167_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X4: nat,Y: nat] :
( ( ord_less_nat @ X4 @ Y )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_1168_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > set_Product_prod_a_a,C: set_Product_prod_a_a] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_le746702958409616551od_a_a @ ( F @ B ) @ C )
=> ( ! [X4: nat,Y: nat] :
( ( ord_less_nat @ X4 @ Y )
=> ( ord_le6819997720685908915od_a_a @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_le6819997720685908915od_a_a @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_1169_linorder__le__less__linear,axiom,
! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
| ( ord_less_real @ Y2 @ X3 ) ) ).
% linorder_le_less_linear
thf(fact_1170_linorder__le__less__linear,axiom,
! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
| ( ord_less_nat @ Y2 @ X3 ) ) ).
% linorder_le_less_linear
thf(fact_1171_order__le__imp__less__or__eq,axiom,
! [X3: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y2 )
=> ( ( ord_less_set_a @ X3 @ Y2 )
| ( X3 = Y2 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_1172_order__le__imp__less__or__eq,axiom,
! [X3: set_set_a,Y2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X3 @ Y2 )
=> ( ( ord_less_set_set_a @ X3 @ Y2 )
| ( X3 = Y2 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_1173_order__le__imp__less__or__eq,axiom,
! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ( ord_less_real @ X3 @ Y2 )
| ( X3 = Y2 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_1174_order__le__imp__less__or__eq,axiom,
! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ( ord_less_nat @ X3 @ Y2 )
| ( X3 = Y2 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_1175_order__le__imp__less__or__eq,axiom,
! [X3: set_Product_prod_a_a,Y2: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ X3 @ Y2 )
=> ( ( ord_le6819997720685908915od_a_a @ X3 @ Y2 )
| ( X3 = Y2 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_1176_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_1177_less__one,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ one_one_nat )
= ( N = zero_zero_nat ) ) ).
% less_one
thf(fact_1178_add__gr__0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ M )
| ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% add_gr_0
thf(fact_1179_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_1180_finite__incident__edges,axiom,
! [V: a] :
( ( finite_finite_set_a @ edges )
=> ( finite_finite_set_a @ ( undire3231912044278729248dges_a @ edges @ V ) ) ) ).
% finite_incident_edges
thf(fact_1181_incident__edges__empty,axiom,
! [V: a] :
( ~ ( member_a @ V @ vertices )
=> ( ( undire3231912044278729248dges_a @ edges @ V )
= bot_bot_set_set_a ) ) ).
% incident_edges_empty
thf(fact_1182_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_1183_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_1184_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_1185_bot__nat__0_Onot__eq__extremum,axiom,
! [A: nat] :
( ( A != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ A ) ) ).
% bot_nat_0.not_eq_extremum
thf(fact_1186_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_1187_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_1188_add__is__0,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= zero_zero_nat )
= ( ( M = zero_zero_nat )
& ( N = zero_zero_nat ) ) ) ).
% add_is_0
thf(fact_1189_Nat_Oadd__0__right,axiom,
! [M: nat] :
( ( plus_plus_nat @ M @ zero_zero_nat )
= M ) ).
% Nat.add_0_right
thf(fact_1190_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_1191_bot__nat__def,axiom,
bot_bot_nat = zero_zero_nat ).
% bot_nat_def
thf(fact_1192_Nat_Oex__has__greatest__nat,axiom,
! [P2: nat > $o,K: nat,B: nat] :
( ( P2 @ K )
=> ( ! [Y: nat] :
( ( P2 @ Y )
=> ( ord_less_eq_nat @ Y @ B ) )
=> ? [X4: nat] :
( ( P2 @ X4 )
& ! [Y7: nat] :
( ( P2 @ Y7 )
=> ( ord_less_eq_nat @ Y7 @ X4 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_1193_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_1194_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_1195_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% eq_imp_le
thf(fact_1196_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_1197_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_1198_infinite__descent0,axiom,
! [P2: nat > $o,N: nat] :
( ( P2 @ zero_zero_nat )
=> ( ! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ~ ( P2 @ N2 )
=> ? [M2: nat] :
( ( ord_less_nat @ M2 @ N2 )
& ~ ( P2 @ M2 ) ) ) )
=> ( P2 @ N ) ) ) ).
% infinite_descent0
thf(fact_1199_gr__implies__not0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_1200_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_zeroE
thf(fact_1201_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less0
thf(fact_1202_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr0
thf(fact_1203_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr0I
thf(fact_1204_bot__nat__0_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_1205_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_1206_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_1207_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_1208_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_1209_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M3: nat,N3: nat] :
( ( ord_less_eq_nat @ M3 @ N3 )
& ( M3 != N3 ) ) ) ) ).
% nat_less_le
thf(fact_1210_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_1211_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M3: nat,N3: nat] :
( ( ord_less_nat @ M3 @ N3 )
| ( M3 = N3 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_1212_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_1213_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_1214_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_1215_plus__nat_Oadd__0,axiom,
! [N: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N )
= N ) ).
% plus_nat.add_0
thf(fact_1216_add__eq__self__zero,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= M )
=> ( N = zero_zero_nat ) ) ).
% add_eq_self_zero
thf(fact_1217_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_1218_le__add1,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M ) ) ).
% le_add1
thf(fact_1219_le__add2,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M @ N ) ) ).
% le_add2
thf(fact_1220_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_1221_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_1222_le__Suc__ex,axiom,
! [K: nat,L: nat] :
( ( ord_less_eq_nat @ K @ L )
=> ? [N2: nat] :
( L
= ( plus_plus_nat @ K @ N2 ) ) ) ).
% le_Suc_ex
thf(fact_1223_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_1224_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_1225_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_1226_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_1227_nat__le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [M3: nat,N3: nat] :
? [K2: nat] :
( N3
= ( plus_plus_nat @ M3 @ K2 ) ) ) ) ).
% nat_le_iff_add
thf(fact_1228_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_1229_less__imp__add__positive,axiom,
! [I: nat,J: nat] :
( ( ord_less_nat @ I @ J )
=> ? [K3: nat] :
( ( ord_less_nat @ zero_zero_nat @ K3 )
& ( ( plus_plus_nat @ I @ K3 )
= J ) ) ) ).
% less_imp_add_positive
thf(fact_1230_mono__nat__linear__lb,axiom,
! [F: nat > nat,M: nat,K: nat] :
( ! [M4: nat,N2: nat] :
( ( ord_less_nat @ M4 @ N2 )
=> ( ord_less_nat @ ( F @ M4 ) @ ( F @ N2 ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M ) @ K ) @ ( F @ ( plus_plus_nat @ M @ K ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_1231_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_1232_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_1233_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_1234_finite__incident__loops,axiom,
! [V: a] : ( finite_finite_set_a @ ( undire4753905205749729249oops_a @ edges @ V ) ) ).
% finite_incident_loops
thf(fact_1235_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_1236_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_1237_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_1238_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_1239_finite__inc__sedges,axiom,
! [V: a] :
( ( finite_finite_set_a @ edges )
=> ( finite_finite_set_a @ ( undire1270416042309875431dges_a @ edges @ V ) ) ) ).
% finite_inc_sedges
thf(fact_1240_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_1241_incident__edges__sedges,axiom,
! [V: a] :
( ~ ( undire3617971648856834880loop_a @ edges @ V )
=> ( ( undire3231912044278729248dges_a @ edges @ V )
= ( undire1270416042309875431dges_a @ edges @ V ) ) ) ).
% incident_edges_sedges
thf(fact_1242_incident__sedges__empty,axiom,
! [V: a] :
( ~ ( member_a @ V @ vertices )
=> ( ( undire1270416042309875431dges_a @ edges @ V )
= bot_bot_set_set_a ) ) ).
% incident_sedges_empty
thf(fact_1243_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_1244_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_1245_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_1246_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_1247_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_1248_is__loop__def,axiom,
( undire2905028936066782638loop_a
= ( ^ [E5: set_a] :
( ( finite_card_a @ E5 )
= one_one_nat ) ) ) ).
% is_loop_def
thf(fact_1249_edge__density__eq0,axiom,
! [A2: set_a,B2: set_a,X5: set_a,Y4: set_a] :
( ( ( undire8383842906760478443ween_a @ edges @ A2 @ B2 )
= bot_bo3357376287454694259od_a_a )
=> ( ( ord_less_eq_set_a @ X5 @ A2 )
=> ( ( ord_less_eq_set_a @ Y4 @ B2 )
=> ( ( undire297304480579013331sity_a @ edges @ X5 @ Y4 )
= zero_zero_real ) ) ) ) ).
% edge_density_eq0
thf(fact_1250_finite__all__edges__between,axiom,
! [X5: set_a,Y4: set_a] :
( ( finite_finite_a @ X5 )
=> ( ( finite_finite_a @ Y4 )
=> ( finite6544458595007987280od_a_a @ ( undire8383842906760478443ween_a @ edges @ X5 @ Y4 ) ) ) ) ).
% finite_all_edges_between
thf(fact_1251_card__all__edges__between__commute,axiom,
! [X5: set_a,Y4: set_a] :
( ( finite4795055649997197647od_a_a @ ( undire8383842906760478443ween_a @ edges @ X5 @ Y4 ) )
= ( finite4795055649997197647od_a_a @ ( undire8383842906760478443ween_a @ edges @ Y4 @ X5 ) ) ) ).
% card_all_edges_between_commute
thf(fact_1252_all__edges__between__mono2,axiom,
! [Y4: set_a,Z4: set_a,X5: set_a] :
( ( ord_less_eq_set_a @ Y4 @ Z4 )
=> ( ord_le746702958409616551od_a_a @ ( undire8383842906760478443ween_a @ edges @ X5 @ Y4 ) @ ( undire8383842906760478443ween_a @ edges @ X5 @ Z4 ) ) ) ).
% all_edges_between_mono2
thf(fact_1253_all__edges__between__mono1,axiom,
! [Y4: set_a,Z4: set_a,X5: set_a] :
( ( ord_less_eq_set_a @ Y4 @ Z4 )
=> ( ord_le746702958409616551od_a_a @ ( undire8383842906760478443ween_a @ edges @ Y4 @ X5 ) @ ( undire8383842906760478443ween_a @ edges @ Z4 @ X5 ) ) ) ).
% all_edges_between_mono1
thf(fact_1254_all__edges__between__Un2,axiom,
! [X5: set_a,Y4: set_a,Z4: set_a] :
( ( undire8383842906760478443ween_a @ edges @ X5 @ ( sup_sup_set_a @ Y4 @ Z4 ) )
= ( sup_su3048258781599657691od_a_a @ ( undire8383842906760478443ween_a @ edges @ X5 @ Y4 ) @ ( undire8383842906760478443ween_a @ edges @ X5 @ Z4 ) ) ) ).
% all_edges_between_Un2
thf(fact_1255_all__edges__between__Un1,axiom,
! [X5: set_a,Y4: set_a,Z4: set_a] :
( ( undire8383842906760478443ween_a @ edges @ ( sup_sup_set_a @ X5 @ Y4 ) @ Z4 )
= ( sup_su3048258781599657691od_a_a @ ( undire8383842906760478443ween_a @ edges @ X5 @ Z4 ) @ ( undire8383842906760478443ween_a @ edges @ Y4 @ Z4 ) ) ) ).
% all_edges_between_Un1
thf(fact_1256_all__edges__between__rem__wf,axiom,
! [X5: set_a,Y4: set_a] :
( ( undire8383842906760478443ween_a @ edges @ X5 @ Y4 )
= ( undire8383842906760478443ween_a @ edges @ ( inf_inf_set_a @ X5 @ vertices ) @ ( inf_inf_set_a @ Y4 @ vertices ) ) ) ).
% all_edges_between_rem_wf
thf(fact_1257_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_1258_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_1259_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_1260_max__all__edges__between,axiom,
! [X5: set_a,Y4: set_a] :
( ( finite_finite_a @ X5 )
=> ( ( finite_finite_a @ Y4 )
=> ( ord_less_eq_nat @ ( finite4795055649997197647od_a_a @ ( undire8383842906760478443ween_a @ edges @ X5 @ Y4 ) ) @ ( times_times_nat @ ( finite_card_a @ X5 ) @ ( finite_card_a @ Y4 ) ) ) ) ) ).
% max_all_edges_between
thf(fact_1261_mult__is__0,axiom,
! [M: nat,N: nat] :
( ( ( times_times_nat @ M @ N )
= zero_zero_nat )
= ( ( M = zero_zero_nat )
| ( N = zero_zero_nat ) ) ) ).
% mult_is_0
thf(fact_1262_mult__0__right,axiom,
! [M: nat] :
( ( times_times_nat @ M @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_0_right
thf(fact_1263_mult__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( times_times_nat @ K @ M )
= ( times_times_nat @ K @ N ) )
= ( ( M = N )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel1
thf(fact_1264_mult__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ( times_times_nat @ M @ K )
= ( times_times_nat @ N @ K ) )
= ( ( M = N )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel2
thf(fact_1265_nat__mult__eq__1__iff,axiom,
! [M: nat,N: nat] :
( ( ( times_times_nat @ M @ N )
= one_one_nat )
= ( ( M = one_one_nat )
& ( N = one_one_nat ) ) ) ).
% nat_mult_eq_1_iff
thf(fact_1266_nat__1__eq__mult__iff,axiom,
! [M: nat,N: nat] :
( ( one_one_nat
= ( times_times_nat @ M @ N ) )
= ( ( M = one_one_nat )
& ( N = one_one_nat ) ) ) ).
% nat_1_eq_mult_iff
thf(fact_1267_nat__0__less__mult__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ M @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ M )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% nat_0_less_mult_iff
thf(fact_1268_mult__less__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
& ( ord_less_nat @ M @ N ) ) ) ).
% mult_less_cancel2
thf(fact_1269_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
% Conjectures (1)
thf(conj_0,conjecture,
( ( ( distinct_a @ ( tl_a @ p ) )
& ( ( hd_a @ p )
= ( last_a @ p ) ) )
| ( distinct_a @ p ) ) ).
%------------------------------------------------------------------------------