TPTP Problem File: SLH0135^1.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_00384_014151__13242966_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1655 ( 571 unt; 372 typ; 0 def)
% Number of atoms : 3649 (1328 equ; 0 cnn)
% Maximal formula atoms : 15 ( 2 avg)
% Number of connectives : 10596 ( 393 ~; 48 |; 318 &;8487 @)
% ( 0 <=>;1350 =>; 0 <=; 0 <~>)
% Maximal formula depth : 21 ( 6 avg)
% Number of types : 48 ( 47 usr)
% Number of type conns : 674 ( 674 >; 0 *; 0 +; 0 <<)
% Number of symbols : 328 ( 325 usr; 32 con; 0-4 aty)
% Number of variables : 3325 ( 128 ^;2997 !; 200 ?;3325 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 14:33:30.754
%------------------------------------------------------------------------------
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thf(sy_c_Undirected__Graph__Walks_Oulgraph_Ois__open__walk_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
undire1203054589613885131od_a_a: set_Product_prod_a_a > set_se5735800977113168103od_a_a > list_P1396940483166286381od_a_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__List__Olist_It__Set__Oset_Itf__a_J_J,type,
undire1052973453303871126_set_a: set_list_set_a > set_set_list_set_a > list_list_set_a > $o ).
thf(sy_c_Undirected__Graph__Walks_Oulgraph_Ois__path_001t__List__Olist_Itf__a_J,type,
undire2586462650415165750list_a: set_list_a > set_set_list_a > list_list_a > $o ).
thf(sy_c_Undirected__Graph__Walks_Oulgraph_Ois__path_001t__Product____Type__Oprod_It__Set__Oset_Itf__a_J_Mtf__a_J,type,
undire8374260845365092473et_a_a: set_Pr2416559167834504103et_a_a > set_se8484413598011947911et_a_a > list_P5740962349794459853et_a_a > $o ).
thf(sy_c_Undirected__Graph__Walks_Oulgraph_Ois__path_001t__Product____Type__Oprod_Itf__a_Mt__Set__Oset_Itf__a_J_J,type,
undire1324510289328068409_set_a: set_Pr6393634178297680487_set_a > set_se4070283622896972359_set_a > list_P494665323402860429_set_a > $o ).
thf(sy_c_Undirected__Graph__Walks_Oulgraph_Ois__path_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
undire9149042980421869017od_a_a: set_Product_prod_a_a > set_se5735800977113168103od_a_a > list_P1396940483166286381od_a_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__List__Olist_It__Set__Oset_Itf__a_J_J,type,
undire2288203741413088850_set_a: set_list_set_a > set_set_list_set_a > list_list_set_a > $o ).
thf(sy_c_Undirected__Graph__Walks_Oulgraph_Ois__walk_001t__List__Olist_Itf__a_J,type,
undire8550186295227992306list_a: set_list_a > set_set_list_a > list_list_a > $o ).
thf(sy_c_Undirected__Graph__Walks_Oulgraph_Ois__walk_001t__Nat__Onat,type,
undire5745680128780950498lk_nat: set_nat > set_set_nat > list_nat > $o ).
thf(sy_c_Undirected__Graph__Walks_Oulgraph_Ois__walk_001t__Product____Type__Oprod_It__Set__Oset_Itf__a_J_Mtf__a_J,type,
undire4537806513611962933et_a_a: set_Pr2416559167834504103et_a_a > set_se8484413598011947911et_a_a > list_P5740962349794459853et_a_a > $o ).
thf(sy_c_Undirected__Graph__Walks_Oulgraph_Ois__walk_001t__Product____Type__Oprod_Itf__a_Mt__Set__Oset_Itf__a_J_J,type,
undire6711427994429714677_set_a: set_Pr6393634178297680487_set_a > set_se4070283622896972359_set_a > list_P494665323402860429_set_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_Opaths_001tf__a,type,
undire1387732426225024653aths_a: set_a > set_set_a > set_list_a ).
thf(sy_c_Undirected__Graph__Walks_Oulgraph_Owalk__edges_001t__List__Olist_Itf__a_J,type,
undire8303882243552421012list_a: list_list_a > list_set_list_a ).
thf(sy_c_Undirected__Graph__Walks_Oulgraph_Owalk__edges_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
undire4403264684974754359od_a_a: list_P1396940483166286381od_a_a > list_s9060204159073123853od_a_a ).
thf(sy_c_Undirected__Graph__Walks_Oulgraph_Owalk__edges_001t__Set__Oset_Itf__a_J,type,
undire6234387080713648494_set_a: list_set_a > list_set_set_a ).
thf(sy_c_Undirected__Graph__Walks_Oulgraph_Owalk__edges_001tf__a,type,
undire7337870655677353998dges_a: list_a > list_set_a ).
thf(sy_c_Undirected__Graph__Walks_Oulgraph_Owalk__edges__rel_001tf__a,type,
undire7966302452035489203_rel_a: list_a > list_a > $o ).
thf(sy_c_Undirected__Graph__Walks_Oulgraph_Owalk__length_001tf__a,type,
undire8849074589633906640ngth_a: list_a > nat ).
thf(sy_c_Undirected__Graph__Walks_Oulgraph_Owalks_001tf__a,type,
undire3736599831911450577alks_a: set_a > set_set_a > set_list_a ).
thf(sy_c_Wellfounded_Oaccp_001t__List__Olist_Itf__a_J,type,
accp_list_a: ( list_a > list_a > $o ) > list_a > $o ).
thf(sy_c_member_001t__List__Olist_It__Set__Oset_Itf__a_J_J,type,
member_list_set_a: list_set_a > set_list_set_a > $o ).
thf(sy_c_member_001t__List__Olist_Itf__a_J,type,
member_list_a: list_a > set_list_a > $o ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Set__Oset_Itf__a_J_Mtf__a_J,type,
member2598349401703774704et_a_a: product_prod_set_a_a > set_Pr2416559167834504103et_a_a > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_Itf__a_Mt__Set__Oset_Itf__a_J_J,type,
member4771970882521526448_set_a: product_prod_a_set_a > set_Pr6393634178297680487_set_a > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
member1426531477525435216od_a_a: product_prod_a_a > set_Product_prod_a_a > $o ).
thf(sy_c_member_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
member_set_list_a: set_list_a > set_set_list_a > $o ).
thf(sy_c_member_001t__Set__Oset_It__Nat__Onat_J,type,
member_set_nat: set_nat > set_set_nat > $o ).
thf(sy_c_member_001t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
member1816616512716248880od_a_a: set_Product_prod_a_a > set_se5735800977113168103od_a_a > $o ).
thf(sy_c_member_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
member_set_set_a: set_set_a > set_set_set_a > $o ).
thf(sy_c_member_001t__Set__Oset_Itf__a_J,type,
member_set_a: set_a > set_set_a > $o ).
thf(sy_c_member_001tf__a,type,
member_a: a > set_a > $o ).
thf(sy_v_edges,type,
edges: set_set_a ).
thf(sy_v_p,type,
p: list_a ).
thf(sy_v_vertices,type,
vertices: set_a ).
% Relevant facts (1273)
thf(fact_0_distinct__tl,axiom,
! [Xs: list_P5740962349794459853et_a_a] :
( ( distin7251654435778379584et_a_a @ Xs )
=> ( distin7251654435778379584et_a_a @ ( tl_Pro2640188747214327222et_a_a @ Xs ) ) ) ).
% distinct_tl
thf(fact_1_distinct__tl,axiom,
! [Xs: list_P494665323402860429_set_a] :
( ( distin201903879741355520_set_a @ Xs )
=> ( distin201903879741355520_set_a @ ( tl_Pro4813810228032078966_set_a @ Xs ) ) ) ).
% distinct_tl
thf(fact_2_distinct__tl,axiom,
! [Xs: list_P1396940483166286381od_a_a] :
( ( distin132333870042060960od_a_a @ Xs )
=> ( distin132333870042060960od_a_a @ ( tl_Product_prod_a_a @ Xs ) ) ) ).
% distinct_tl
thf(fact_3_distinct__tl,axiom,
! [Xs: list_list_set_a] :
( ( distinct_list_set_a @ Xs )
=> ( distinct_list_set_a @ ( tl_list_set_a @ Xs ) ) ) ).
% distinct_tl
thf(fact_4_distinct__tl,axiom,
! [Xs: list_list_a] :
( ( distinct_list_a @ Xs )
=> ( distinct_list_a @ ( tl_list_a @ Xs ) ) ) ).
% distinct_tl
thf(fact_5_distinct__tl,axiom,
! [Xs: list_a] :
( ( distinct_a @ Xs )
=> ( distinct_a @ ( tl_a @ Xs ) ) ) ).
% distinct_tl
thf(fact_6_distinct__tl,axiom,
! [Xs: list_set_a] :
( ( distinct_set_a @ Xs )
=> ( distinct_set_a @ ( tl_set_a @ Xs ) ) ) ).
% distinct_tl
thf(fact_7_assms_I2_J,axiom,
( ( hd_a @ p )
= ( last_a @ p ) ) ).
% assms(2)
thf(fact_8_distinct__union,axiom,
! [Xs: list_P5740962349794459853et_a_a,Ys: list_P5740962349794459853et_a_a] :
( ( distin7251654435778379584et_a_a @ ( union_6946928963749628353et_a_a @ Xs @ Ys ) )
= ( distin7251654435778379584et_a_a @ Ys ) ) ).
% distinct_union
thf(fact_9_distinct__union,axiom,
! [Xs: list_P494665323402860429_set_a,Ys: list_P494665323402860429_set_a] :
( ( distin201903879741355520_set_a @ ( union_9120550444567380097_set_a @ Xs @ Ys ) )
= ( distin201903879741355520_set_a @ Ys ) ) ).
% distinct_union
thf(fact_10_distinct__union,axiom,
! [Xs: list_P1396940483166286381od_a_a,Ys: list_P1396940483166286381od_a_a] :
( ( distin132333870042060960od_a_a @ ( union_7798659288537573153od_a_a @ Xs @ Ys ) )
= ( distin132333870042060960od_a_a @ Ys ) ) ).
% distinct_union
thf(fact_11_distinct__union,axiom,
! [Xs: list_list_set_a,Ys: list_list_set_a] :
( ( distinct_list_set_a @ ( union_list_set_a @ Xs @ Ys ) )
= ( distinct_list_set_a @ Ys ) ) ).
% distinct_union
thf(fact_12_distinct__union,axiom,
! [Xs: list_list_a,Ys: list_list_a] :
( ( distinct_list_a @ ( union_list_a @ Xs @ Ys ) )
= ( distinct_list_a @ Ys ) ) ).
% distinct_union
thf(fact_13_distinct__union,axiom,
! [Xs: list_a,Ys: list_a] :
( ( distinct_a @ ( union_a @ Xs @ Ys ) )
= ( distinct_a @ Ys ) ) ).
% distinct_union
thf(fact_14_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_15_incident__def,axiom,
undire1521409233611534436dent_a = member_a ).
% incident_def
thf(fact_16_distinct__insert,axiom,
! [X: product_prod_set_a_a,Xs: list_P5740962349794459853et_a_a] :
( ( distin7251654435778379584et_a_a @ ( insert2156854329156279523et_a_a @ X @ Xs ) )
= ( distin7251654435778379584et_a_a @ Xs ) ) ).
% distinct_insert
thf(fact_17_distinct__insert,axiom,
! [X: product_prod_a_set_a,Xs: list_P494665323402860429_set_a] :
( ( distin201903879741355520_set_a @ ( insert4330475809974031267_set_a @ X @ Xs ) )
= ( distin201903879741355520_set_a @ Xs ) ) ).
% distinct_insert
thf(fact_18_distinct__insert,axiom,
! [X: product_prod_a_a,Xs: list_P1396940483166286381od_a_a] :
( ( distin132333870042060960od_a_a @ ( insert7736115120964043331od_a_a @ X @ Xs ) )
= ( distin132333870042060960od_a_a @ Xs ) ) ).
% distinct_insert
thf(fact_19_distinct__insert,axiom,
! [X: list_set_a,Xs: list_list_set_a] :
( ( distinct_list_set_a @ ( insert_list_set_a @ X @ Xs ) )
= ( distinct_list_set_a @ Xs ) ) ).
% distinct_insert
thf(fact_20_distinct__insert,axiom,
! [X: list_a,Xs: list_list_a] :
( ( distinct_list_a @ ( insert_list_a @ X @ Xs ) )
= ( distinct_list_a @ Xs ) ) ).
% distinct_insert
thf(fact_21_distinct__insert,axiom,
! [X: a,Xs: list_a] :
( ( distinct_a @ ( insert_a @ X @ Xs ) )
= ( distinct_a @ Xs ) ) ).
% distinct_insert
thf(fact_22_distinct__insert,axiom,
! [X: set_a,Xs: list_set_a] :
( ( distinct_set_a @ ( insert_set_a @ X @ Xs ) )
= ( distinct_set_a @ Xs ) ) ).
% distinct_insert
thf(fact_23_distinct__n__lists,axiom,
! [Xs: list_P5740962349794459853et_a_a,N: nat] :
( ( distin7251654435778379584et_a_a @ Xs )
=> ( distin3418465907611269062et_a_a @ ( n_list7491920702333717973et_a_a @ N @ Xs ) ) ) ).
% distinct_n_lists
thf(fact_24_distinct__n__lists,axiom,
! [Xs: list_P494665323402860429_set_a,N: nat] :
( ( distin201903879741355520_set_a @ Xs )
=> ( distin7395540918074445446_set_a @ ( n_list442170146296693909_set_a @ N @ Xs ) ) ) ).
% distinct_n_lists
thf(fact_25_distinct__n__lists,axiom,
! [Xs: list_P1396940483166286381od_a_a,N: nat] :
( ( distin132333870042060960od_a_a @ Xs )
=> ( distin3546859918470253862od_a_a @ ( n_list8012392499058522933od_a_a @ N @ Xs ) ) ) ).
% distinct_n_lists
thf(fact_26_distinct__n__lists,axiom,
! [Xs: list_list_set_a,N: nat] :
( ( distinct_list_set_a @ Xs )
=> ( distin2165222987885430883_set_a @ ( n_lists_list_set_a @ N @ Xs ) ) ) ).
% distinct_n_lists
thf(fact_27_distinct__n__lists,axiom,
! [Xs: list_list_a,N: nat] :
( ( distinct_list_a @ Xs )
=> ( distinct_list_list_a @ ( n_lists_list_a @ N @ Xs ) ) ) ).
% distinct_n_lists
thf(fact_28_distinct__n__lists,axiom,
! [Xs: list_a,N: nat] :
( ( distinct_a @ Xs )
=> ( distinct_list_a @ ( n_lists_a @ N @ Xs ) ) ) ).
% distinct_n_lists
thf(fact_29_distinct__n__lists,axiom,
! [Xs: list_set_a,N: nat] :
( ( distinct_set_a @ Xs )
=> ( distinct_list_set_a @ ( n_lists_set_a @ N @ Xs ) ) ) ).
% distinct_n_lists
thf(fact_30_distinct__product,axiom,
! [Xs: list_a,Ys: list_a] :
( ( distinct_a @ Xs )
=> ( ( distinct_a @ Ys )
=> ( distin132333870042060960od_a_a @ ( product_a_a @ Xs @ Ys ) ) ) ) ).
% distinct_product
thf(fact_31_distinct__product,axiom,
! [Xs: list_a,Ys: list_set_a] :
( ( distinct_a @ Xs )
=> ( ( distinct_set_a @ Ys )
=> ( distin201903879741355520_set_a @ ( product_a_set_a @ Xs @ Ys ) ) ) ) ).
% distinct_product
thf(fact_32_distinct__product,axiom,
! [Xs: list_set_a,Ys: list_a] :
( ( distinct_set_a @ Xs )
=> ( ( distinct_a @ Ys )
=> ( distin7251654435778379584et_a_a @ ( product_set_a_a @ Xs @ Ys ) ) ) ) ).
% distinct_product
thf(fact_33_distinct__product,axiom,
! [Xs: list_set_a,Ys: list_set_a] :
( ( distinct_set_a @ Xs )
=> ( ( distinct_set_a @ Ys )
=> ( distin4110230307619200160_set_a @ ( product_set_a_set_a @ Xs @ Ys ) ) ) ) ).
% distinct_product
thf(fact_34_distinct__product,axiom,
! [Xs: list_a,Ys: list_list_a] :
( ( distinct_a @ Xs )
=> ( ( distinct_list_a @ Ys )
=> ( distin1612527794249132710list_a @ ( product_a_list_a @ Xs @ Ys ) ) ) ) ).
% distinct_product
thf(fact_35_distinct__product,axiom,
! [Xs: list_list_a,Ys: list_a] :
( ( distinct_list_a @ Xs )
=> ( ( distinct_a @ Ys )
=> ( distin4729310080553060506st_a_a @ ( product_list_a_a @ Xs @ Ys ) ) ) ) ).
% distinct_product
thf(fact_36_distinct__product,axiom,
! [Xs: list_a,Ys: list_P1396940483166286381od_a_a] :
( ( distinct_a @ Xs )
=> ( ( distin132333870042060960od_a_a @ Ys )
=> ( distin4039418177281331017od_a_a @ ( produc6674783723117535214od_a_a @ Xs @ Ys ) ) ) ) ).
% distinct_product
thf(fact_37_distinct__product,axiom,
! [Xs: list_a,Ys: list_list_set_a] :
( ( distinct_a @ Xs )
=> ( ( distinct_list_set_a @ Ys )
=> ( distin8791064984217447430_set_a @ ( product_a_list_set_a @ Xs @ Ys ) ) ) ) ).
% distinct_product
thf(fact_38_distinct__product,axiom,
! [Xs: list_set_a,Ys: list_list_a] :
( ( distinct_set_a @ Xs )
=> ( ( distinct_list_a @ Ys )
=> ( distin1379869053940580678list_a @ ( product_set_a_list_a @ Xs @ Ys ) ) ) ) ).
% distinct_product
thf(fact_39_distinct__product,axiom,
! [Xs: list_P1396940483166286381od_a_a,Ys: list_a] :
( ( distin132333870042060960od_a_a @ Xs )
=> ( ( distinct_a @ Ys )
=> ( distin3798212641234315447_a_a_a @ ( produc2712103193426997468_a_a_a @ Xs @ Ys ) ) ) ) ).
% distinct_product
thf(fact_40_distinct1__rotate,axiom,
! [Xs: list_P5740962349794459853et_a_a] :
( ( distin7251654435778379584et_a_a @ ( rotate2282817068329048218et_a_a @ Xs ) )
= ( distin7251654435778379584et_a_a @ Xs ) ) ).
% distinct1_rotate
thf(fact_41_distinct1__rotate,axiom,
! [Xs: list_P494665323402860429_set_a] :
( ( distin201903879741355520_set_a @ ( rotate4456438549146799962_set_a @ Xs ) )
= ( distin201903879741355520_set_a @ Xs ) ) ).
% distinct1_rotate
thf(fact_42_distinct1__rotate,axiom,
! [Xs: list_P1396940483166286381od_a_a] :
( ( distin132333870042060960od_a_a @ ( rotate5308318543670761978od_a_a @ Xs ) )
= ( distin132333870042060960od_a_a @ Xs ) ) ).
% distinct1_rotate
thf(fact_43_distinct1__rotate,axiom,
! [Xs: list_list_set_a] :
( ( distinct_list_set_a @ ( rotate1_list_set_a @ Xs ) )
= ( distinct_list_set_a @ Xs ) ) ).
% distinct1_rotate
thf(fact_44_distinct1__rotate,axiom,
! [Xs: list_list_a] :
( ( distinct_list_a @ ( rotate1_list_a @ Xs ) )
= ( distinct_list_a @ Xs ) ) ).
% distinct1_rotate
thf(fact_45_distinct1__rotate,axiom,
! [Xs: list_a] :
( ( distinct_a @ ( rotate1_a @ Xs ) )
= ( distinct_a @ Xs ) ) ).
% distinct1_rotate
thf(fact_46_distinct1__rotate,axiom,
! [Xs: list_set_a] :
( ( distinct_set_a @ ( rotate1_set_a @ Xs ) )
= ( distinct_set_a @ Xs ) ) ).
% distinct1_rotate
thf(fact_47_assms_I1_J,axiom,
undire3562951555376170320path_a @ vertices @ edges @ p ).
% assms(1)
thf(fact_48_distinct__rotate,axiom,
! [N: nat,Xs: list_a] :
( ( distinct_a @ ( rotate_a @ N @ Xs ) )
= ( distinct_a @ Xs ) ) ).
% distinct_rotate
thf(fact_49_distinct__rotate,axiom,
! [N: nat,Xs: list_set_a] :
( ( distinct_set_a @ ( rotate_set_a @ N @ Xs ) )
= ( distinct_set_a @ Xs ) ) ).
% distinct_rotate
thf(fact_50_distinct__rotate,axiom,
! [N: nat,Xs: list_P5740962349794459853et_a_a] :
( ( distin7251654435778379584et_a_a @ ( rotate4899871913205339649et_a_a @ N @ Xs ) )
= ( distin7251654435778379584et_a_a @ Xs ) ) ).
% distinct_rotate
thf(fact_51_distinct__rotate,axiom,
! [N: nat,Xs: list_P494665323402860429_set_a] :
( ( distin201903879741355520_set_a @ ( rotate7073493394023091393_set_a @ N @ Xs ) )
= ( distin201903879741355520_set_a @ Xs ) ) ).
% distinct_rotate
thf(fact_52_distinct__rotate,axiom,
! [N: nat,Xs: list_P1396940483166286381od_a_a] :
( ( distin132333870042060960od_a_a @ ( rotate22669329132647265od_a_a @ N @ Xs ) )
= ( distin132333870042060960od_a_a @ Xs ) ) ).
% distinct_rotate
thf(fact_53_distinct__rotate,axiom,
! [N: nat,Xs: list_list_set_a] :
( ( distinct_list_set_a @ ( rotate_list_set_a @ N @ Xs ) )
= ( distinct_list_set_a @ Xs ) ) ).
% distinct_rotate
thf(fact_54_distinct__rotate,axiom,
! [N: nat,Xs: list_list_a] :
( ( distinct_list_a @ ( rotate_list_a @ N @ Xs ) )
= ( distinct_list_a @ Xs ) ) ).
% distinct_rotate
thf(fact_55_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_56_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_57_ulgraph__axioms,axiom,
undire7251896706689453996raph_a @ vertices @ edges ).
% ulgraph_axioms
thf(fact_58_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_59_edge__adjacent__alt__def,axiom,
! [E1: set_a,E2: set_a] :
( ( member_set_a @ E1 @ edges )
=> ( ( member_set_a @ E2 @ edges )
=> ( ? [X2: a] :
( ( member_a @ X2 @ vertices )
& ( member_a @ X2 @ E1 )
& ( member_a @ X2 @ E2 ) )
=> ( undire4022703626023482010_adj_a @ edges @ E1 @ E2 ) ) ) ) ).
% edge_adjacent_alt_def
thf(fact_60_rotate1__rotate__swap,axiom,
! [N: nat,Xs: list_set_a] :
( ( rotate1_set_a @ ( rotate_set_a @ N @ Xs ) )
= ( rotate_set_a @ N @ ( rotate1_set_a @ Xs ) ) ) ).
% rotate1_rotate_swap
thf(fact_61_rotate1__rotate__swap,axiom,
! [N: nat,Xs: list_a] :
( ( rotate1_a @ ( rotate_a @ N @ Xs ) )
= ( rotate_a @ N @ ( rotate1_a @ Xs ) ) ) ).
% rotate1_rotate_swap
thf(fact_62_ulgraph_Ois__gen__path_Ocong,axiom,
undire3562951555376170320path_a = undire3562951555376170320path_a ).
% ulgraph.is_gen_path.cong
thf(fact_63_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_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__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_66_has__loop__in__verts,axiom,
! [V: a] :
( ( undire3617971648856834880loop_a @ edges @ V )
=> ( member_a @ V @ vertices ) ) ).
% has_loop_in_verts
thf(fact_67_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_68_vert__adj__edge__iff2,axiom,
! [V1: a,V2: a] :
( ( V1 != V2 )
=> ( ( undire397441198561214472_adj_a @ edges @ V1 @ V2 )
= ( ? [X3: set_a] :
( ( member_set_a @ X3 @ edges )
& ( undire1521409233611534436dent_a @ V1 @ X3 )
& ( undire1521409233611534436dent_a @ V2 @ X3 ) ) ) ) ) ).
% vert_adj_edge_iff2
thf(fact_69_subgraph__refl,axiom,
undire7103218114511261257raph_a @ vertices @ edges @ vertices @ edges ).
% subgraph_refl
thf(fact_70_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_71_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_72_wellformed,axiom,
! [E: set_a] :
( ( member_set_a @ E @ edges )
=> ( ord_less_eq_set_a @ E @ vertices ) ) ).
% wellformed
thf(fact_73_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_74_graph__system__axioms,axiom,
undire2554140024507503526stem_a @ vertices @ edges ).
% graph_system_axioms
thf(fact_75_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_76_rev__rev__ident,axiom,
! [Xs: list_a] :
( ( rev_a @ ( rev_a @ Xs ) )
= Xs ) ).
% rev_rev_ident
thf(fact_77_rev__rev__ident,axiom,
! [Xs: list_set_a] :
( ( rev_set_a @ ( rev_set_a @ Xs ) )
= Xs ) ).
% rev_rev_ident
thf(fact_78_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_79_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_80_is__walk__rev,axiom,
! [Xs: list_a] :
( ( undire6133010728901294956walk_a @ vertices @ edges @ Xs )
= ( undire6133010728901294956walk_a @ vertices @ edges @ ( rev_a @ Xs ) ) ) ).
% is_walk_rev
thf(fact_81_is__path__walk,axiom,
! [Xs: list_a] :
( ( undire427332500224447920path_a @ vertices @ edges @ Xs )
=> ( undire6133010728901294956walk_a @ vertices @ edges @ Xs ) ) ).
% is_path_walk
thf(fact_82_is__isolated__vertex__def,axiom,
! [V: a] :
( ( undire8931668460104145173rtex_a @ vertices @ edges @ V )
= ( ( member_a @ V @ vertices )
& ! [X3: a] :
( ( member_a @ X3 @ vertices )
=> ~ ( undire397441198561214472_adj_a @ edges @ X3 @ V ) ) ) ) ).
% is_isolated_vertex_def
thf(fact_83_is__path__rev,axiom,
! [Xs: list_a] :
( ( undire427332500224447920path_a @ vertices @ edges @ Xs )
= ( undire427332500224447920path_a @ vertices @ edges @ ( rev_a @ Xs ) ) ) ).
% is_path_rev
thf(fact_84_is__isolated__vertex__no__loop,axiom,
! [V: a] :
( ( undire8931668460104145173rtex_a @ vertices @ edges @ V )
=> ~ ( undire3617971648856834880loop_a @ edges @ V ) ) ).
% is_isolated_vertex_no_loop
thf(fact_85_distinct__rev,axiom,
! [Xs: list_a] :
( ( distinct_a @ ( rev_a @ Xs ) )
= ( distinct_a @ Xs ) ) ).
% distinct_rev
thf(fact_86_distinct__rev,axiom,
! [Xs: list_set_a] :
( ( distinct_set_a @ ( rev_set_a @ Xs ) )
= ( distinct_set_a @ Xs ) ) ).
% distinct_rev
thf(fact_87_distinct__rev,axiom,
! [Xs: list_P5740962349794459853et_a_a] :
( ( distin7251654435778379584et_a_a @ ( rev_Pr4807930434169910317et_a_a @ Xs ) )
= ( distin7251654435778379584et_a_a @ Xs ) ) ).
% distinct_rev
thf(fact_88_distinct__rev,axiom,
! [Xs: list_P494665323402860429_set_a] :
( ( distin201903879741355520_set_a @ ( rev_Pr6981551914987662061_set_a @ Xs ) )
= ( distin201903879741355520_set_a @ Xs ) ) ).
% distinct_rev
thf(fact_89_distinct__rev,axiom,
! [Xs: list_P1396940483166286381od_a_a] :
( ( distin132333870042060960od_a_a @ ( rev_Product_prod_a_a @ Xs ) )
= ( distin132333870042060960od_a_a @ Xs ) ) ).
% distinct_rev
thf(fact_90_distinct__rev,axiom,
! [Xs: list_list_set_a] :
( ( distinct_list_set_a @ ( rev_list_set_a @ Xs ) )
= ( distinct_list_set_a @ Xs ) ) ).
% distinct_rev
thf(fact_91_distinct__rev,axiom,
! [Xs: list_list_a] :
( ( distinct_list_a @ ( rev_list_a @ Xs ) )
= ( distinct_list_a @ Xs ) ) ).
% distinct_rev
thf(fact_92_is__trail__rev,axiom,
! [Xs: list_a] :
( ( undire7142031287334043199rail_a @ vertices @ edges @ Xs )
= ( undire7142031287334043199rail_a @ vertices @ edges @ ( rev_a @ Xs ) ) ) ).
% is_trail_rev
thf(fact_93_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_94_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_95_rev__swap,axiom,
! [Xs: list_a,Ys: list_a] :
( ( ( rev_a @ Xs )
= Ys )
= ( Xs
= ( rev_a @ Ys ) ) ) ).
% rev_swap
thf(fact_96_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_97_ulgraph_Ois__walk_Ocong,axiom,
undire6133010728901294956walk_a = undire6133010728901294956walk_a ).
% ulgraph.is_walk.cong
thf(fact_98_mem__Collect__eq,axiom,
! [A: a,P2: a > $o] :
( ( member_a @ A @ ( collect_a @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_99_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_100_mem__Collect__eq,axiom,
! [A: product_prod_a_a,P2: product_prod_a_a > $o] :
( ( member1426531477525435216od_a_a @ A @ ( collec3336397797384452498od_a_a @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_101_mem__Collect__eq,axiom,
! [A: nat,P2: nat > $o] :
( ( member_nat @ A @ ( collect_nat @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_102_Collect__mem__eq,axiom,
! [A2: set_a] :
( ( collect_a
@ ^ [X3: a] : ( member_a @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_103_Collect__mem__eq,axiom,
! [A2: set_set_a] :
( ( collect_set_a
@ ^ [X3: set_a] : ( member_set_a @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_104_Collect__mem__eq,axiom,
! [A2: set_Product_prod_a_a] :
( ( collec3336397797384452498od_a_a
@ ^ [X3: product_prod_a_a] : ( member1426531477525435216od_a_a @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_105_Collect__mem__eq,axiom,
! [A2: set_nat] :
( ( collect_nat
@ ^ [X3: nat] : ( member_nat @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_106_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_107_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_108_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_109_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_110_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_111_ulgraph_Ois__path_Ocong,axiom,
undire427332500224447920path_a = undire427332500224447920path_a ).
% ulgraph.is_path.cong
thf(fact_112_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_113_ulgraph_Ois__walk__wf__hd,axiom,
! [Vertices: set_Product_prod_a_a,Edges: set_se5735800977113168103od_a_a,Xs: list_P1396940483166286381od_a_a] :
( ( undire4585262585102564309od_a_a @ Vertices @ Edges )
=> ( ( undire3162072421265123221od_a_a @ Vertices @ Edges @ Xs )
=> ( member1426531477525435216od_a_a @ ( hd_Product_prod_a_a @ Xs ) @ Vertices ) ) ) ).
% ulgraph.is_walk_wf_hd
thf(fact_114_ulgraph_Ois__walk__wf__hd,axiom,
! [Vertices: set_nat,Edges: set_set_nat,Xs: list_nat] :
( ( undire3269267262472140706ph_nat @ Vertices @ Edges )
=> ( ( undire5745680128780950498lk_nat @ Vertices @ Edges @ Xs )
=> ( member_nat @ ( hd_nat @ Xs ) @ Vertices ) ) ) ).
% ulgraph.is_walk_wf_hd
thf(fact_115_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_116_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_117_ulgraph_Ois__walk__wf__last,axiom,
! [Vertices: set_Product_prod_a_a,Edges: set_se5735800977113168103od_a_a,Xs: list_P1396940483166286381od_a_a] :
( ( undire4585262585102564309od_a_a @ Vertices @ Edges )
=> ( ( undire3162072421265123221od_a_a @ Vertices @ Edges @ Xs )
=> ( member1426531477525435216od_a_a @ ( last_P8790725268278465478od_a_a @ Xs ) @ Vertices ) ) ) ).
% ulgraph.is_walk_wf_last
thf(fact_118_ulgraph_Ois__walk__wf__last,axiom,
! [Vertices: set_nat,Edges: set_set_nat,Xs: list_nat] :
( ( undire3269267262472140706ph_nat @ Vertices @ Edges )
=> ( ( undire5745680128780950498lk_nat @ Vertices @ Edges @ Xs )
=> ( member_nat @ ( last_nat @ Xs ) @ Vertices ) ) ) ).
% ulgraph.is_walk_wf_last
thf(fact_119_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_120_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_121_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_122_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_123_last__rev,axiom,
! [Xs: list_set_a] :
( ( last_set_a @ ( rev_set_a @ Xs ) )
= ( hd_set_a @ Xs ) ) ).
% last_rev
thf(fact_124_last__rev,axiom,
! [Xs: list_a] :
( ( last_a @ ( rev_a @ Xs ) )
= ( hd_a @ Xs ) ) ).
% last_rev
thf(fact_125_hd__rev,axiom,
! [Xs: list_set_a] :
( ( hd_set_a @ ( rev_set_a @ Xs ) )
= ( last_set_a @ Xs ) ) ).
% hd_rev
thf(fact_126_hd__rev,axiom,
! [Xs: list_a] :
( ( hd_a @ ( rev_a @ Xs ) )
= ( last_a @ Xs ) ) ).
% hd_rev
thf(fact_127_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_128_ulgraph_Ois__gen__path__def,axiom,
! [Vertices: set_Pr2416559167834504103et_a_a,Edges: set_se8484413598011947911et_a_a,P: list_P5740962349794459853et_a_a] :
( ( undire7149961781991362165et_a_a @ Vertices @ Edges )
=> ( ( undire8058049273645199385et_a_a @ Vertices @ Edges @ P )
= ( ( undire4537806513611962933et_a_a @ Vertices @ Edges @ P )
& ( ( ( distin7251654435778379584et_a_a @ ( tl_Pro2640188747214327222et_a_a @ P ) )
& ( ( hd_Pro7221231872133205426et_a_a @ P )
= ( last_P6817006942355120742et_a_a @ P ) ) )
| ( distin7251654435778379584et_a_a @ P ) ) ) ) ) ).
% ulgraph.is_gen_path_def
thf(fact_129_ulgraph_Ois__gen__path__def,axiom,
! [Vertices: set_Pr6393634178297680487_set_a,Edges: set_se4070283622896972359_set_a,P: list_P494665323402860429_set_a] :
( ( undire100211225954338101_set_a @ Vertices @ Edges )
=> ( ( undire1008298717608175321_set_a @ Vertices @ Edges @ P )
= ( ( undire6711427994429714677_set_a @ Vertices @ Edges @ P )
& ( ( ( distin201903879741355520_set_a @ ( tl_Pro4813810228032078966_set_a @ P ) )
& ( ( hd_Pro171481316096181362_set_a @ P )
= ( last_P8990628423172872486_set_a @ P ) ) )
| ( distin201903879741355520_set_a @ P ) ) ) ) ) ).
% ulgraph.is_gen_path_def
thf(fact_130_ulgraph_Ois__gen__path__def,axiom,
! [Vertices: set_Product_prod_a_a,Edges: set_se5735800977113168103od_a_a,P: list_P1396940483166286381od_a_a] :
( ( undire4585262585102564309od_a_a @ Vertices @ Edges )
=> ( ( undire7585867811434966393od_a_a @ Vertices @ Edges @ P )
= ( ( undire3162072421265123221od_a_a @ Vertices @ Edges @ P )
& ( ( ( distin132333870042060960od_a_a @ ( tl_Product_prod_a_a @ P ) )
& ( ( hd_Product_prod_a_a @ P )
= ( last_P8790725268278465478od_a_a @ P ) ) )
| ( distin132333870042060960od_a_a @ P ) ) ) ) ) ).
% ulgraph.is_gen_path_def
thf(fact_131_ulgraph_Ois__gen__path__def,axiom,
! [Vertices: set_list_set_a,Edges: set_set_list_set_a,P: list_list_set_a] :
( ( undire2408673306710905490_set_a @ Vertices @ Edges )
=> ( ( undire5019392814671255094_set_a @ Vertices @ Edges @ P )
= ( ( undire2288203741413088850_set_a @ Vertices @ Edges @ P )
& ( ( ( distinct_list_set_a @ ( tl_list_set_a @ P ) )
& ( ( hd_list_set_a @ P )
= ( last_list_set_a @ P ) ) )
| ( distinct_list_set_a @ P ) ) ) ) ) ).
% ulgraph.is_gen_path_def
thf(fact_132_ulgraph_Ois__gen__path__def,axiom,
! [Vertices: set_list_a,Edges: set_set_list_a,P: list_list_a] :
( ( undire4488935924012268850list_a @ Vertices @ Edges )
=> ( ( undire8568094650444222678list_a @ Vertices @ Edges @ P )
= ( ( undire8550186295227992306list_a @ Vertices @ Edges @ P )
& ( ( ( distinct_list_a @ ( tl_list_a @ P ) )
& ( ( hd_list_a @ P )
= ( last_list_a @ P ) ) )
| ( distinct_list_a @ P ) ) ) ) ) ).
% ulgraph.is_gen_path_def
thf(fact_133_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_134_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_135_ulgraph_Ois__gen__path__distinct,axiom,
! [Vertices: set_Pr2416559167834504103et_a_a,Edges: set_se8484413598011947911et_a_a,P: list_P5740962349794459853et_a_a] :
( ( undire7149961781991362165et_a_a @ Vertices @ Edges )
=> ( ( undire8058049273645199385et_a_a @ Vertices @ Edges @ P )
=> ( ( ( hd_Pro7221231872133205426et_a_a @ P )
!= ( last_P6817006942355120742et_a_a @ P ) )
=> ( distin7251654435778379584et_a_a @ P ) ) ) ) ).
% ulgraph.is_gen_path_distinct
thf(fact_136_ulgraph_Ois__gen__path__distinct,axiom,
! [Vertices: set_Pr6393634178297680487_set_a,Edges: set_se4070283622896972359_set_a,P: list_P494665323402860429_set_a] :
( ( undire100211225954338101_set_a @ Vertices @ Edges )
=> ( ( undire1008298717608175321_set_a @ Vertices @ Edges @ P )
=> ( ( ( hd_Pro171481316096181362_set_a @ P )
!= ( last_P8990628423172872486_set_a @ P ) )
=> ( distin201903879741355520_set_a @ P ) ) ) ) ).
% ulgraph.is_gen_path_distinct
thf(fact_137_ulgraph_Ois__gen__path__distinct,axiom,
! [Vertices: set_Product_prod_a_a,Edges: set_se5735800977113168103od_a_a,P: list_P1396940483166286381od_a_a] :
( ( undire4585262585102564309od_a_a @ Vertices @ Edges )
=> ( ( undire7585867811434966393od_a_a @ Vertices @ Edges @ P )
=> ( ( ( hd_Product_prod_a_a @ P )
!= ( last_P8790725268278465478od_a_a @ P ) )
=> ( distin132333870042060960od_a_a @ P ) ) ) ) ).
% ulgraph.is_gen_path_distinct
thf(fact_138_ulgraph_Ois__gen__path__distinct,axiom,
! [Vertices: set_list_set_a,Edges: set_set_list_set_a,P: list_list_set_a] :
( ( undire2408673306710905490_set_a @ Vertices @ Edges )
=> ( ( undire5019392814671255094_set_a @ Vertices @ Edges @ P )
=> ( ( ( hd_list_set_a @ P )
!= ( last_list_set_a @ P ) )
=> ( distinct_list_set_a @ P ) ) ) ) ).
% ulgraph.is_gen_path_distinct
thf(fact_139_ulgraph_Ois__gen__path__distinct,axiom,
! [Vertices: set_list_a,Edges: set_set_list_a,P: list_list_a] :
( ( undire4488935924012268850list_a @ Vertices @ Edges )
=> ( ( undire8568094650444222678list_a @ Vertices @ Edges @ P )
=> ( ( ( hd_list_a @ P )
!= ( last_list_a @ P ) )
=> ( distinct_list_a @ P ) ) ) ) ).
% ulgraph.is_gen_path_distinct
thf(fact_140_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_141_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_142_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_143_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_144_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_145_is__subgraphI,axiom,
! [V3: set_list_a,V4: set_list_a,E3: set_set_list_a,E4: set_set_list_a] :
( ( ord_le8861187494160871172list_a @ V3 @ V4 )
=> ( ( ord_le8877086941679407844list_a @ E3 @ E4 )
=> ( ( undire5959234994740280364list_a @ V3 @ E3 )
=> ( ( undire5959234994740280364list_a @ V4 @ E4 )
=> ( undire761398192061991247list_a @ V3 @ E3 @ V4 @ E4 ) ) ) ) ) ).
% is_subgraphI
thf(fact_146_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_147_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_148_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_149_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_150_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_151_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_152_distinct__tl__rev,axiom,
! [Xs: list_P5740962349794459853et_a_a] :
( ( ( hd_Pro7221231872133205426et_a_a @ Xs )
= ( last_P6817006942355120742et_a_a @ Xs ) )
=> ( ( distin7251654435778379584et_a_a @ ( tl_Pro2640188747214327222et_a_a @ Xs ) )
= ( distin7251654435778379584et_a_a @ ( tl_Pro2640188747214327222et_a_a @ ( rev_Pr4807930434169910317et_a_a @ Xs ) ) ) ) ) ).
% distinct_tl_rev
thf(fact_153_distinct__tl__rev,axiom,
! [Xs: list_P494665323402860429_set_a] :
( ( ( hd_Pro171481316096181362_set_a @ Xs )
= ( last_P8990628423172872486_set_a @ Xs ) )
=> ( ( distin201903879741355520_set_a @ ( tl_Pro4813810228032078966_set_a @ Xs ) )
= ( distin201903879741355520_set_a @ ( tl_Pro4813810228032078966_set_a @ ( rev_Pr6981551914987662061_set_a @ Xs ) ) ) ) ) ).
% distinct_tl_rev
thf(fact_154_distinct__tl__rev,axiom,
! [Xs: list_P1396940483166286381od_a_a] :
( ( ( hd_Product_prod_a_a @ Xs )
= ( last_P8790725268278465478od_a_a @ Xs ) )
=> ( ( distin132333870042060960od_a_a @ ( tl_Product_prod_a_a @ Xs ) )
= ( distin132333870042060960od_a_a @ ( tl_Product_prod_a_a @ ( rev_Product_prod_a_a @ Xs ) ) ) ) ) ).
% distinct_tl_rev
thf(fact_155_distinct__tl__rev,axiom,
! [Xs: list_list_set_a] :
( ( ( hd_list_set_a @ Xs )
= ( last_list_set_a @ Xs ) )
=> ( ( distinct_list_set_a @ ( tl_list_set_a @ Xs ) )
= ( distinct_list_set_a @ ( tl_list_set_a @ ( rev_list_set_a @ Xs ) ) ) ) ) ).
% distinct_tl_rev
thf(fact_156_distinct__tl__rev,axiom,
! [Xs: list_list_a] :
( ( ( hd_list_a @ Xs )
= ( last_list_a @ Xs ) )
=> ( ( distinct_list_a @ ( tl_list_a @ Xs ) )
= ( distinct_list_a @ ( tl_list_a @ ( rev_list_a @ Xs ) ) ) ) ) ).
% distinct_tl_rev
thf(fact_157_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_158_is__walk__not__empty2,axiom,
~ ( undire6133010728901294956walk_a @ vertices @ edges @ nil_a ) ).
% is_walk_not_empty2
thf(fact_159_is__walk__not__empty,axiom,
! [Xs: list_a] :
( ( undire6133010728901294956walk_a @ vertices @ edges @ Xs )
=> ( Xs != nil_a ) ) ).
% is_walk_not_empty
thf(fact_160_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_161_induced__is__graph__sys,axiom,
! [V3: set_a] : ( undire2554140024507503526stem_a @ V3 @ ( undire7777452895879145676dges_a @ edges @ V3 ) ) ).
% induced_is_graph_sys
thf(fact_162_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_163_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_164_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_165_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_166_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_167_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_168_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_169_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_170_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_171_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_172_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_173_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_174_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_175_self__append__conv2,axiom,
! [Y: list_a,Xs: list_a] :
( ( Y
= ( append_a @ Xs @ Y ) )
= ( Xs = nil_a ) ) ).
% self_append_conv2
thf(fact_176_self__append__conv2,axiom,
! [Y: list_set_a,Xs: list_set_a] :
( ( Y
= ( append_set_a @ Xs @ Y ) )
= ( Xs = nil_set_a ) ) ).
% self_append_conv2
thf(fact_177_append__self__conv2,axiom,
! [Xs: list_a,Ys: list_a] :
( ( ( append_a @ Xs @ Ys )
= Ys )
= ( Xs = nil_a ) ) ).
% append_self_conv2
thf(fact_178_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_179_self__append__conv,axiom,
! [Y: list_a,Ys: list_a] :
( ( Y
= ( append_a @ Y @ Ys ) )
= ( Ys = nil_a ) ) ).
% self_append_conv
thf(fact_180_self__append__conv,axiom,
! [Y: list_set_a,Ys: list_set_a] :
( ( Y
= ( append_set_a @ Y @ Ys ) )
= ( Ys = nil_set_a ) ) ).
% self_append_conv
thf(fact_181_append__self__conv,axiom,
! [Xs: list_a,Ys: list_a] :
( ( ( append_a @ Xs @ Ys )
= Xs )
= ( Ys = nil_a ) ) ).
% append_self_conv
thf(fact_182_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_183_append__Nil2,axiom,
! [Xs: list_a] :
( ( append_a @ Xs @ nil_a )
= Xs ) ).
% append_Nil2
thf(fact_184_append__Nil2,axiom,
! [Xs: list_set_a] :
( ( append_set_a @ Xs @ nil_set_a )
= Xs ) ).
% append_Nil2
thf(fact_185_append_Oright__neutral,axiom,
! [A: list_a] :
( ( append_a @ A @ nil_a )
= A ) ).
% append.right_neutral
thf(fact_186_append_Oright__neutral,axiom,
! [A: list_set_a] :
( ( append_set_a @ A @ nil_set_a )
= A ) ).
% append.right_neutral
thf(fact_187_rev__is__Nil__conv,axiom,
! [Xs: list_a] :
( ( ( rev_a @ Xs )
= nil_a )
= ( Xs = nil_a ) ) ).
% rev_is_Nil_conv
thf(fact_188_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_189_Nil__is__rev__conv,axiom,
! [Xs: list_a] :
( ( nil_a
= ( rev_a @ Xs ) )
= ( Xs = nil_a ) ) ).
% Nil_is_rev_conv
thf(fact_190_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_191_set__rev,axiom,
! [Xs: list_a] :
( ( set_a2 @ ( rev_a @ Xs ) )
= ( set_a2 @ Xs ) ) ).
% set_rev
thf(fact_192_set__rev,axiom,
! [Xs: list_set_a] :
( ( set_set_a2 @ ( rev_set_a @ Xs ) )
= ( set_set_a2 @ Xs ) ) ).
% set_rev
thf(fact_193_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_194_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_195_rotate__is__Nil__conv,axiom,
! [N: nat,Xs: list_a] :
( ( ( rotate_a @ N @ Xs )
= nil_a )
= ( Xs = nil_a ) ) ).
% rotate_is_Nil_conv
thf(fact_196_rotate__is__Nil__conv,axiom,
! [N: nat,Xs: list_set_a] :
( ( ( rotate_set_a @ N @ Xs )
= nil_set_a )
= ( Xs = nil_set_a ) ) ).
% rotate_is_Nil_conv
thf(fact_197_set__rotate,axiom,
! [N: nat,Xs: list_a] :
( ( set_a2 @ ( rotate_a @ N @ Xs ) )
= ( set_a2 @ Xs ) ) ).
% set_rotate
thf(fact_198_set__rotate,axiom,
! [N: nat,Xs: list_set_a] :
( ( set_set_a2 @ ( rotate_set_a @ N @ Xs ) )
= ( set_set_a2 @ Xs ) ) ).
% set_rotate
thf(fact_199_rotate1__is__Nil__conv,axiom,
! [Xs: list_a] :
( ( ( rotate1_a @ Xs )
= nil_a )
= ( Xs = nil_a ) ) ).
% rotate1_is_Nil_conv
thf(fact_200_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_201_set__rotate1,axiom,
! [Xs: list_a] :
( ( set_a2 @ ( rotate1_a @ Xs ) )
= ( set_a2 @ Xs ) ) ).
% set_rotate1
thf(fact_202_set__rotate1,axiom,
! [Xs: list_set_a] :
( ( set_set_a2 @ ( rotate1_set_a @ Xs ) )
= ( set_set_a2 @ Xs ) ) ).
% set_rotate1
thf(fact_203_in__set__insert,axiom,
! [X: product_prod_a_a,Xs: list_P1396940483166286381od_a_a] :
( ( member1426531477525435216od_a_a @ X @ ( set_Product_prod_a_a2 @ Xs ) )
=> ( ( insert7736115120964043331od_a_a @ X @ Xs )
= Xs ) ) ).
% in_set_insert
thf(fact_204_in__set__insert,axiom,
! [X: nat,Xs: list_nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs ) )
=> ( ( insert_nat @ X @ Xs )
= Xs ) ) ).
% in_set_insert
thf(fact_205_in__set__insert,axiom,
! [X: a,Xs: list_a] :
( ( member_a @ X @ ( set_a2 @ Xs ) )
=> ( ( insert_a @ X @ Xs )
= Xs ) ) ).
% in_set_insert
thf(fact_206_in__set__insert,axiom,
! [X: set_a,Xs: list_set_a] :
( ( member_set_a @ X @ ( set_set_a2 @ Xs ) )
=> ( ( insert_set_a @ X @ Xs )
= Xs ) ) ).
% in_set_insert
thf(fact_207_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_208_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_209_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_210_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_211_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_212_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_213_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_214_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_215_graph__system_Oinduced__edges_Ocong,axiom,
undire7777452895879145676dges_a = undire7777452895879145676dges_a ).
% graph_system.induced_edges.cong
thf(fact_216_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_217_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_218_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_219_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_220_eq__Nil__appendI,axiom,
! [Xs: list_a,Ys: list_a] :
( ( Xs = Ys )
=> ( Xs
= ( append_a @ nil_a @ Ys ) ) ) ).
% eq_Nil_appendI
thf(fact_221_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_222_append_Oleft__neutral,axiom,
! [A: list_a] :
( ( append_a @ nil_a @ A )
= A ) ).
% append.left_neutral
thf(fact_223_append_Oleft__neutral,axiom,
! [A: list_set_a] :
( ( append_set_a @ nil_set_a @ A )
= A ) ).
% append.left_neutral
thf(fact_224_append__Nil,axiom,
! [Ys: list_a] :
( ( append_a @ nil_a @ Ys )
= Ys ) ).
% append_Nil
thf(fact_225_append__Nil,axiom,
! [Ys: list_set_a] :
( ( append_set_a @ nil_set_a @ Ys )
= Ys ) ).
% append_Nil
thf(fact_226_ulgraph_Ois__open__walk_Ocong,axiom,
undire2427028224930250914walk_a = undire2427028224930250914walk_a ).
% ulgraph.is_open_walk.cong
thf(fact_227_ulgraph_Ois__closed__walk_Ocong,axiom,
undire3370724456595283424walk_a = undire3370724456595283424walk_a ).
% ulgraph.is_closed_walk.cong
thf(fact_228_ulgraph_Ois__trail_Ocong,axiom,
undire7142031287334043199rail_a = undire7142031287334043199rail_a ).
% ulgraph.is_trail.cong
thf(fact_229_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_230_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_231_longest__common__prefix,axiom,
! [Xs: list_a,Ys: list_a] :
? [Ps: list_a,Xs2: list_a,Ys2: list_a] :
( ( Xs
= ( append_a @ Ps @ Xs2 ) )
& ( Ys
= ( append_a @ Ps @ Ys2 ) )
& ( ( Xs2 = nil_a )
| ( Ys2 = nil_a )
| ( ( hd_a @ Xs2 )
!= ( hd_a @ Ys2 ) ) ) ) ).
% longest_common_prefix
thf(fact_232_longest__common__prefix,axiom,
! [Xs: list_set_a,Ys: list_set_a] :
? [Ps: list_set_a,Xs2: list_set_a,Ys2: list_set_a] :
( ( Xs
= ( append_set_a @ Ps @ Xs2 ) )
& ( Ys
= ( append_set_a @ Ps @ Ys2 ) )
& ( ( Xs2 = nil_set_a )
| ( Ys2 = nil_set_a )
| ( ( hd_set_a @ Xs2 )
!= ( hd_set_a @ Ys2 ) ) ) ) ).
% longest_common_prefix
thf(fact_233_list_Oset__sel_I1_J,axiom,
! [A: list_P1396940483166286381od_a_a] :
( ( A != nil_Product_prod_a_a )
=> ( member1426531477525435216od_a_a @ ( hd_Product_prod_a_a @ A ) @ ( set_Product_prod_a_a2 @ A ) ) ) ).
% list.set_sel(1)
thf(fact_234_list_Oset__sel_I1_J,axiom,
! [A: list_nat] :
( ( A != nil_nat )
=> ( member_nat @ ( hd_nat @ A ) @ ( set_nat2 @ A ) ) ) ).
% list.set_sel(1)
thf(fact_235_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_236_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_237_hd__in__set,axiom,
! [Xs: list_P1396940483166286381od_a_a] :
( ( Xs != nil_Product_prod_a_a )
=> ( member1426531477525435216od_a_a @ ( hd_Product_prod_a_a @ Xs ) @ ( set_Product_prod_a_a2 @ Xs ) ) ) ).
% hd_in_set
thf(fact_238_hd__in__set,axiom,
! [Xs: list_nat] :
( ( Xs != nil_nat )
=> ( member_nat @ ( hd_nat @ Xs ) @ ( set_nat2 @ Xs ) ) ) ).
% hd_in_set
thf(fact_239_hd__in__set,axiom,
! [Xs: list_a] :
( ( Xs != nil_a )
=> ( member_a @ ( hd_a @ Xs ) @ ( set_a2 @ Xs ) ) ) ).
% hd_in_set
thf(fact_240_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_241_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_242_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_243_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_244_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_245_longest__common__suffix,axiom,
! [Xs: list_a,Ys: list_a] :
? [Ss: list_a,Xs2: list_a,Ys2: list_a] :
( ( Xs
= ( append_a @ Xs2 @ Ss ) )
& ( Ys
= ( append_a @ Ys2 @ Ss ) )
& ( ( Xs2 = nil_a )
| ( Ys2 = nil_a )
| ( ( last_a @ Xs2 )
!= ( last_a @ Ys2 ) ) ) ) ).
% longest_common_suffix
thf(fact_246_longest__common__suffix,axiom,
! [Xs: list_set_a,Ys: list_set_a] :
? [Ss: list_set_a,Xs2: list_set_a,Ys2: list_set_a] :
( ( Xs
= ( append_set_a @ Xs2 @ Ss ) )
& ( Ys
= ( append_set_a @ Ys2 @ Ss ) )
& ( ( Xs2 = nil_set_a )
| ( Ys2 = nil_set_a )
| ( ( last_set_a @ Xs2 )
!= ( last_set_a @ Ys2 ) ) ) ) ).
% longest_common_suffix
thf(fact_247_list_Oset__sel_I2_J,axiom,
! [A: list_P1396940483166286381od_a_a,X: product_prod_a_a] :
( ( A != nil_Product_prod_a_a )
=> ( ( member1426531477525435216od_a_a @ X @ ( set_Product_prod_a_a2 @ ( tl_Product_prod_a_a @ A ) ) )
=> ( member1426531477525435216od_a_a @ X @ ( set_Product_prod_a_a2 @ A ) ) ) ) ).
% list.set_sel(2)
thf(fact_248_list_Oset__sel_I2_J,axiom,
! [A: list_nat,X: nat] :
( ( A != nil_nat )
=> ( ( member_nat @ X @ ( set_nat2 @ ( tl_nat @ A ) ) )
=> ( member_nat @ X @ ( set_nat2 @ A ) ) ) ) ).
% list.set_sel(2)
thf(fact_249_list_Oset__sel_I2_J,axiom,
! [A: list_a,X: a] :
( ( A != nil_a )
=> ( ( member_a @ X @ ( set_a2 @ ( tl_a @ A ) ) )
=> ( member_a @ X @ ( set_a2 @ A ) ) ) ) ).
% list.set_sel(2)
thf(fact_250_list_Oset__sel_I2_J,axiom,
! [A: list_set_a,X: set_a] :
( ( A != nil_set_a )
=> ( ( member_set_a @ X @ ( set_set_a2 @ ( tl_set_a @ A ) ) )
=> ( member_set_a @ X @ ( set_set_a2 @ A ) ) ) ) ).
% list.set_sel(2)
thf(fact_251_last__in__set,axiom,
! [As: list_P1396940483166286381od_a_a] :
( ( As != nil_Product_prod_a_a )
=> ( member1426531477525435216od_a_a @ ( last_P8790725268278465478od_a_a @ As ) @ ( set_Product_prod_a_a2 @ As ) ) ) ).
% last_in_set
thf(fact_252_last__in__set,axiom,
! [As: list_nat] :
( ( As != nil_nat )
=> ( member_nat @ ( last_nat @ As ) @ ( set_nat2 @ As ) ) ) ).
% last_in_set
thf(fact_253_last__in__set,axiom,
! [As: list_a] :
( ( As != nil_a )
=> ( member_a @ ( last_a @ As ) @ ( set_a2 @ As ) ) ) ).
% last_in_set
thf(fact_254_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_255_product_Osimps_I1_J,axiom,
! [Uu: list_set_a] :
( ( product_a_set_a @ nil_a @ Uu )
= nil_Pr883920194014449805_set_a ) ).
% product.simps(1)
thf(fact_256_product_Osimps_I1_J,axiom,
! [Uu: list_a] :
( ( product_a_a @ nil_a @ Uu )
= nil_Product_prod_a_a ) ).
% product.simps(1)
thf(fact_257_product_Osimps_I1_J,axiom,
! [Uu: list_set_a] :
( ( product_set_a_set_a @ nil_set_a @ Uu )
= nil_Pr4665773148637758253_set_a ) ).
% product.simps(1)
thf(fact_258_product_Osimps_I1_J,axiom,
! [Uu: list_a] :
( ( product_set_a_a @ nil_set_a @ Uu )
= nil_Pr7933670750051473869et_a_a ) ).
% product.simps(1)
thf(fact_259_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_260_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_261_subset__code_I1_J,axiom,
! [Xs: list_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ B2 )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
=> ( member_nat @ X3 @ B2 ) ) ) ) ).
% subset_code(1)
thf(fact_262_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 )
= ( ! [X3: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X3 @ ( set_Product_prod_a_a2 @ Xs ) )
=> ( member1426531477525435216od_a_a @ X3 @ B2 ) ) ) ) ).
% subset_code(1)
thf(fact_263_subset__code_I1_J,axiom,
! [Xs: list_list_a,B2: set_list_a] :
( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Xs ) @ B2 )
= ( ! [X3: list_a] :
( ( member_list_a @ X3 @ ( set_list_a2 @ Xs ) )
=> ( member_list_a @ X3 @ B2 ) ) ) ) ).
% subset_code(1)
thf(fact_264_subset__code_I1_J,axiom,
! [Xs: list_set_a,B2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ ( set_set_a2 @ Xs ) @ B2 )
= ( ! [X3: set_a] :
( ( member_set_a @ X3 @ ( set_set_a2 @ Xs ) )
=> ( member_set_a @ X3 @ B2 ) ) ) ) ).
% subset_code(1)
thf(fact_265_subset__code_I1_J,axiom,
! [Xs: list_a,B2: set_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ Xs ) @ B2 )
= ( ! [X3: a] :
( ( member_a @ X3 @ ( set_a2 @ Xs ) )
=> ( member_a @ X3 @ B2 ) ) ) ) ).
% subset_code(1)
thf(fact_266_distinct_Osimps_I1_J,axiom,
distin7251654435778379584et_a_a @ nil_Pr7933670750051473869et_a_a ).
% distinct.simps(1)
thf(fact_267_distinct_Osimps_I1_J,axiom,
distin201903879741355520_set_a @ nil_Pr883920194014449805_set_a ).
% distinct.simps(1)
thf(fact_268_distinct_Osimps_I1_J,axiom,
distin132333870042060960od_a_a @ nil_Product_prod_a_a ).
% distinct.simps(1)
thf(fact_269_distinct_Osimps_I1_J,axiom,
distinct_list_set_a @ nil_list_set_a ).
% distinct.simps(1)
thf(fact_270_distinct_Osimps_I1_J,axiom,
distinct_list_a @ nil_list_a ).
% distinct.simps(1)
thf(fact_271_distinct_Osimps_I1_J,axiom,
distinct_a @ nil_a ).
% distinct.simps(1)
thf(fact_272_distinct_Osimps_I1_J,axiom,
distinct_set_a @ nil_set_a ).
% distinct.simps(1)
thf(fact_273_rev_Osimps_I1_J,axiom,
( ( rev_a @ nil_a )
= nil_a ) ).
% rev.simps(1)
thf(fact_274_rev_Osimps_I1_J,axiom,
( ( rev_set_a @ nil_set_a )
= nil_set_a ) ).
% rev.simps(1)
thf(fact_275_list_Osel_I2_J,axiom,
( ( tl_a @ nil_a )
= nil_a ) ).
% list.sel(2)
thf(fact_276_list_Osel_I2_J,axiom,
( ( tl_set_a @ nil_set_a )
= nil_set_a ) ).
% list.sel(2)
thf(fact_277_rotate1_Osimps_I1_J,axiom,
( ( rotate1_a @ nil_a )
= nil_a ) ).
% rotate1.simps(1)
thf(fact_278_rotate1_Osimps_I1_J,axiom,
( ( rotate1_set_a @ nil_set_a )
= nil_set_a ) ).
% rotate1.simps(1)
thf(fact_279_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_280_graph__system_Oinduced__edges__ss,axiom,
! [Vertices: set_list_a,Edges: set_set_list_a,V3: set_list_a] :
( ( undire5959234994740280364list_a @ Vertices @ Edges )
=> ( ( ord_le8861187494160871172list_a @ V3 @ Vertices )
=> ( ord_le8877086941679407844list_a @ ( undire8521487854958249554list_a @ Edges @ V3 ) @ Edges ) ) ) ).
% graph_system.induced_edges_ss
thf(fact_281_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_282_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_283_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_284_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_285_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_286_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_287_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_288_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_289_hd__Nil__eq__last,axiom,
( ( hd_a @ nil_a )
= ( last_a @ nil_a ) ) ).
% hd_Nil_eq_last
thf(fact_290_hd__Nil__eq__last,axiom,
( ( hd_set_a @ nil_set_a )
= ( last_set_a @ nil_set_a ) ) ).
% hd_Nil_eq_last
thf(fact_291_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_292_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_293_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_294_graph__system_Oinduced__is__subgraph,axiom,
! [Vertices: set_list_a,Edges: set_set_list_a,V3: set_list_a] :
( ( undire5959234994740280364list_a @ Vertices @ Edges )
=> ( ( ord_le8861187494160871172list_a @ V3 @ Vertices )
=> ( undire761398192061991247list_a @ V3 @ ( undire8521487854958249554list_a @ Edges @ V3 ) @ Vertices @ Edges ) ) ) ).
% graph_system.induced_is_subgraph
thf(fact_295_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_296_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_297_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_298_ulgraph_Ois__walk__wf,axiom,
! [Vertices: set_list_a,Edges: set_set_list_a,Xs: list_list_a] :
( ( undire4488935924012268850list_a @ Vertices @ Edges )
=> ( ( undire8550186295227992306list_a @ Vertices @ Edges @ Xs )
=> ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Xs ) @ Vertices ) ) ) ).
% ulgraph.is_walk_wf
thf(fact_299_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_300_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_301_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_302_ulgraph_Overt__adj_Ocong,axiom,
undire397441198561214472_adj_a = undire397441198561214472_adj_a ).
% ulgraph.vert_adj.cong
thf(fact_303_comp__sgraph_Oincident__def,axiom,
undire2320338297334612420_set_a = member_set_a ).
% comp_sgraph.incident_def
thf(fact_304_comp__sgraph_Oincident__def,axiom,
undire3369688177417741453od_a_a = member1426531477525435216od_a_a ).
% comp_sgraph.incident_def
thf(fact_305_comp__sgraph_Oincident__def,axiom,
undire7858122600432113898nt_nat = member_nat ).
% comp_sgraph.incident_def
thf(fact_306_comp__sgraph_Oincident__def,axiom,
undire1521409233611534436dent_a = member_a ).
% comp_sgraph.incident_def
thf(fact_307_ulgraph_Ohas__loop_Ocong,axiom,
undire3617971648856834880loop_a = undire3617971648856834880loop_a ).
% ulgraph.has_loop.cong
thf(fact_308_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_309_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_310_ulgraph_Ois__isolated__vertex_Ocong,axiom,
undire8931668460104145173rtex_a = undire8931668460104145173rtex_a ).
% ulgraph.is_isolated_vertex.cong
thf(fact_311_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_312_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_313_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_314_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_315_graph__system_Oedge__adj_Ocong,axiom,
undire4022703626023482010_adj_a = undire4022703626023482010_adj_a ).
% graph_system.edge_adj.cong
thf(fact_316_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_317_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_318_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_319_ulgraph_Ois__path__def,axiom,
! [Vertices: set_Pr2416559167834504103et_a_a,Edges: set_se8484413598011947911et_a_a,Xs: list_P5740962349794459853et_a_a] :
( ( undire7149961781991362165et_a_a @ Vertices @ Edges )
=> ( ( undire8374260845365092473et_a_a @ Vertices @ Edges @ Xs )
= ( ( undire1907723709775954283et_a_a @ Vertices @ Edges @ Xs )
& ( distin7251654435778379584et_a_a @ Xs ) ) ) ) ).
% ulgraph.is_path_def
thf(fact_320_ulgraph_Ois__path__def,axiom,
! [Vertices: set_Pr6393634178297680487_set_a,Edges: set_se4070283622896972359_set_a,Xs: list_P494665323402860429_set_a] :
( ( undire100211225954338101_set_a @ Vertices @ Edges )
=> ( ( undire1324510289328068409_set_a @ Vertices @ Edges @ Xs )
= ( ( undire4081345190593706027_set_a @ Vertices @ Edges @ Xs )
& ( distin201903879741355520_set_a @ Xs ) ) ) ) ).
% ulgraph.is_path_def
thf(fact_321_ulgraph_Ois__path__def,axiom,
! [Vertices: set_Product_prod_a_a,Edges: set_se5735800977113168103od_a_a,Xs: list_P1396940483166286381od_a_a] :
( ( undire4585262585102564309od_a_a @ Vertices @ Edges )
=> ( ( undire9149042980421869017od_a_a @ Vertices @ Edges @ Xs )
= ( ( undire1203054589613885131od_a_a @ Vertices @ Edges @ Xs )
& ( distin132333870042060960od_a_a @ Xs ) ) ) ) ).
% ulgraph.is_path_def
thf(fact_322_ulgraph_Ois__path__def,axiom,
! [Vertices: set_list_set_a,Edges: set_set_list_set_a,Xs: list_list_set_a] :
( ( undire2408673306710905490_set_a @ Vertices @ Edges )
=> ( ( undire1052973453303871126_set_a @ Vertices @ Edges @ Xs )
= ( ( undire1111087293740939400_set_a @ Vertices @ Edges @ Xs )
& ( distinct_list_set_a @ Xs ) ) ) ) ).
% ulgraph.is_path_def
thf(fact_323_ulgraph_Ois__path__def,axiom,
! [Vertices: set_list_a,Edges: set_set_list_a,Xs: list_list_a] :
( ( undire4488935924012268850list_a @ Vertices @ Edges )
=> ( ( undire2586462650415165750list_a @ Vertices @ Edges @ Xs )
= ( ( undire6929316984140692264list_a @ Vertices @ Edges @ Xs )
& ( distinct_list_a @ Xs ) ) ) ) ).
% ulgraph.is_path_def
thf(fact_324_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_325_graph__system_Ointro,axiom,
! [Edges: set_se5735800977113168103od_a_a,Vertices: set_Product_prod_a_a] :
( ! [E5: set_Product_prod_a_a] :
( ( member1816616512716248880od_a_a @ E5 @ Edges )
=> ( ord_le746702958409616551od_a_a @ E5 @ Vertices ) )
=> ( undire1860116983885411791od_a_a @ Vertices @ Edges ) ) ).
% graph_system.intro
thf(fact_326_graph__system_Ointro,axiom,
! [Edges: set_set_list_a,Vertices: set_list_a] :
( ! [E5: set_list_a] :
( ( member_set_list_a @ E5 @ Edges )
=> ( ord_le8861187494160871172list_a @ E5 @ Vertices ) )
=> ( undire5959234994740280364list_a @ Vertices @ Edges ) ) ).
% graph_system.intro
thf(fact_327_graph__system_Ointro,axiom,
! [Edges: set_set_set_a,Vertices: set_set_a] :
( ! [E5: set_set_a] :
( ( member_set_set_a @ E5 @ Edges )
=> ( ord_le3724670747650509150_set_a @ E5 @ Vertices ) )
=> ( undire7159349782766787846_set_a @ Vertices @ Edges ) ) ).
% graph_system.intro
thf(fact_328_graph__system_Ointro,axiom,
! [Edges: set_set_a,Vertices: set_a] :
( ! [E5: set_a] :
( ( member_set_a @ E5 @ Edges )
=> ( ord_less_eq_set_a @ E5 @ Vertices ) )
=> ( undire2554140024507503526stem_a @ Vertices @ Edges ) ) ).
% graph_system.intro
thf(fact_329_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_330_graph__system_Owellformed,axiom,
! [Vertices: set_list_a,Edges: set_set_list_a,E: set_list_a] :
( ( undire5959234994740280364list_a @ Vertices @ Edges )
=> ( ( member_set_list_a @ E @ Edges )
=> ( ord_le8861187494160871172list_a @ E @ Vertices ) ) ) ).
% graph_system.wellformed
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_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_333_graph__system__def,axiom,
( undire1860116983885411791od_a_a
= ( ^ [Vertices2: set_Product_prod_a_a,Edges2: set_se5735800977113168103od_a_a] :
! [E6: set_Product_prod_a_a] :
( ( member1816616512716248880od_a_a @ E6 @ Edges2 )
=> ( ord_le746702958409616551od_a_a @ E6 @ Vertices2 ) ) ) ) ).
% graph_system_def
thf(fact_334_graph__system__def,axiom,
( undire5959234994740280364list_a
= ( ^ [Vertices2: set_list_a,Edges2: set_set_list_a] :
! [E6: set_list_a] :
( ( member_set_list_a @ E6 @ Edges2 )
=> ( ord_le8861187494160871172list_a @ E6 @ Vertices2 ) ) ) ) ).
% graph_system_def
thf(fact_335_graph__system__def,axiom,
( undire7159349782766787846_set_a
= ( ^ [Vertices2: set_set_a,Edges2: set_set_set_a] :
! [E6: set_set_a] :
( ( member_set_set_a @ E6 @ Edges2 )
=> ( ord_le3724670747650509150_set_a @ E6 @ Vertices2 ) ) ) ) ).
% graph_system_def
thf(fact_336_graph__system__def,axiom,
( undire2554140024507503526stem_a
= ( ^ [Vertices2: set_a,Edges2: set_set_a] :
! [E6: set_a] :
( ( member_set_a @ E6 @ Edges2 )
=> ( ord_less_eq_set_a @ E6 @ Vertices2 ) ) ) ) ).
% graph_system_def
thf(fact_337_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_338_subgraph_Overts__ss,axiom,
! [V_H: set_list_a,E_H: set_set_list_a,V_G: set_list_a,E_G: set_set_list_a] :
( ( undire761398192061991247list_a @ V_H @ E_H @ V_G @ E_G )
=> ( ord_le8861187494160871172list_a @ V_H @ V_G ) ) ).
% subgraph.verts_ss
thf(fact_339_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_340_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_341_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_342_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_343_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_344_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_345_ulgraph_Overt__adj__imp__inV,axiom,
! [Vertices: set_Product_prod_a_a,Edges: set_se5735800977113168103od_a_a,V1: product_prod_a_a,V2: product_prod_a_a] :
( ( undire4585262585102564309od_a_a @ Vertices @ Edges )
=> ( ( undire6135774327024169009od_a_a @ Edges @ V1 @ V2 )
=> ( ( member1426531477525435216od_a_a @ V1 @ Vertices )
& ( member1426531477525435216od_a_a @ V2 @ Vertices ) ) ) ) ).
% ulgraph.vert_adj_imp_inV
thf(fact_346_ulgraph_Overt__adj__imp__inV,axiom,
! [Vertices: set_nat,Edges: set_set_nat,V1: nat,V2: nat] :
( ( undire3269267262472140706ph_nat @ Vertices @ Edges )
=> ( ( undire1083030068171319366dj_nat @ Edges @ V1 @ V2 )
=> ( ( member_nat @ V1 @ Vertices )
& ( member_nat @ V2 @ Vertices ) ) ) ) ).
% ulgraph.vert_adj_imp_inV
thf(fact_347_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_348_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_349_ulgraph_Ohas__loop__in__verts,axiom,
! [Vertices: set_Product_prod_a_a,Edges: set_se5735800977113168103od_a_a,V: product_prod_a_a] :
( ( undire4585262585102564309od_a_a @ Vertices @ Edges )
=> ( ( undire7777398424729533289od_a_a @ Edges @ V )
=> ( member1426531477525435216od_a_a @ V @ Vertices ) ) ) ).
% ulgraph.has_loop_in_verts
thf(fact_350_ulgraph_Ohas__loop__in__verts,axiom,
! [Vertices: set_nat,Edges: set_set_nat,V: nat] :
( ( undire3269267262472140706ph_nat @ Vertices @ Edges )
=> ( ( undire5005864372999571214op_nat @ Edges @ V )
=> ( member_nat @ V @ Vertices ) ) ) ).
% ulgraph.has_loop_in_verts
thf(fact_351_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_352_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_353_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_354_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_355_subgraph_Osubgraph__trans,axiom,
! [V_H: set_a,E_H: set_set_a,V_G: set_a,E_G: set_set_a,V4: set_a,E4: set_set_a,V3: set_a,E3: set_set_a,V5: set_a,E7: set_set_a] :
( ( undire7103218114511261257raph_a @ V_H @ E_H @ V_G @ E_G )
=> ( ( undire2554140024507503526stem_a @ V4 @ E4 )
=> ( ( undire2554140024507503526stem_a @ V3 @ E3 )
=> ( ( undire2554140024507503526stem_a @ V5 @ E7 )
=> ( ( undire7103218114511261257raph_a @ V5 @ E7 @ V3 @ E3 )
=> ( ( undire7103218114511261257raph_a @ V3 @ E3 @ V4 @ E4 )
=> ( undire7103218114511261257raph_a @ V5 @ E7 @ V4 @ E4 ) ) ) ) ) ) ) ).
% subgraph.subgraph_trans
thf(fact_356_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_357_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_358_graph__system_Oincident__def,axiom,
! [Vertices: set_Product_prod_a_a,Edges: set_se5735800977113168103od_a_a,V: product_prod_a_a,E: set_Product_prod_a_a] :
( ( undire1860116983885411791od_a_a @ Vertices @ Edges )
=> ( ( undire3369688177417741453od_a_a @ V @ E )
= ( member1426531477525435216od_a_a @ V @ E ) ) ) ).
% graph_system.incident_def
thf(fact_359_graph__system_Oincident__def,axiom,
! [Vertices: set_nat,Edges: set_set_nat,V: nat,E: set_nat] :
( ( undire7481384412329822504em_nat @ Vertices @ Edges )
=> ( ( undire7858122600432113898nt_nat @ V @ E )
= ( member_nat @ V @ E ) ) ) ).
% graph_system.incident_def
thf(fact_360_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_361_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_362_graph__system_Oincident__edge__in__wf,axiom,
! [Vertices: set_Product_prod_a_a,Edges: set_se5735800977113168103od_a_a,E: set_Product_prod_a_a,V: product_prod_a_a] :
( ( undire1860116983885411791od_a_a @ Vertices @ Edges )
=> ( ( member1816616512716248880od_a_a @ E @ Edges )
=> ( ( undire3369688177417741453od_a_a @ V @ E )
=> ( member1426531477525435216od_a_a @ V @ Vertices ) ) ) ) ).
% graph_system.incident_edge_in_wf
thf(fact_363_graph__system_Oincident__edge__in__wf,axiom,
! [Vertices: set_nat,Edges: set_set_nat,E: set_nat,V: nat] :
( ( undire7481384412329822504em_nat @ Vertices @ Edges )
=> ( ( member_set_nat @ E @ Edges )
=> ( ( undire7858122600432113898nt_nat @ V @ E )
=> ( member_nat @ V @ Vertices ) ) ) ) ).
% graph_system.incident_edge_in_wf
thf(fact_364_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_365_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_366_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_367_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 )
=> ( ? [X2: set_a] :
( ( member_set_a @ X2 @ Vertices )
& ( member_set_a @ X2 @ E1 )
& ( member_set_a @ X2 @ E2 ) )
=> ( undire3485422320110889978_set_a @ Edges @ E1 @ E2 ) ) ) ) ) ).
% graph_system.edge_adjacent_alt_def
thf(fact_368_graph__system_Oedge__adjacent__alt__def,axiom,
! [Vertices: set_Product_prod_a_a,Edges: set_se5735800977113168103od_a_a,E1: set_Product_prod_a_a,E2: set_Product_prod_a_a] :
( ( undire1860116983885411791od_a_a @ Vertices @ Edges )
=> ( ( member1816616512716248880od_a_a @ E1 @ Edges )
=> ( ( member1816616512716248880od_a_a @ E2 @ Edges )
=> ( ? [X2: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X2 @ Vertices )
& ( member1426531477525435216od_a_a @ X2 @ E1 )
& ( member1426531477525435216od_a_a @ X2 @ E2 ) )
=> ( undire9186443406341554371od_a_a @ Edges @ E1 @ E2 ) ) ) ) ) ).
% graph_system.edge_adjacent_alt_def
thf(fact_369_graph__system_Oedge__adjacent__alt__def,axiom,
! [Vertices: set_nat,Edges: set_set_nat,E1: set_nat,E2: set_nat] :
( ( undire7481384412329822504em_nat @ Vertices @ Edges )
=> ( ( member_set_nat @ E1 @ Edges )
=> ( ( member_set_nat @ E2 @ Edges )
=> ( ? [X2: nat] :
( ( member_nat @ X2 @ Vertices )
& ( member_nat @ X2 @ E1 )
& ( member_nat @ X2 @ E2 ) )
=> ( undire1664191744716346676dj_nat @ Edges @ E1 @ E2 ) ) ) ) ) ).
% graph_system.edge_adjacent_alt_def
thf(fact_370_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 )
=> ( ? [X2: a] :
( ( member_a @ X2 @ Vertices )
& ( member_a @ X2 @ E1 )
& ( member_a @ X2 @ E2 ) )
=> ( undire4022703626023482010_adj_a @ Edges @ E1 @ E2 ) ) ) ) ) ).
% graph_system.edge_adjacent_alt_def
thf(fact_371_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 )
= ( ? [X3: set_a] :
( ( member_set_a @ X3 @ Edges )
& ( undire1521409233611534436dent_a @ V1 @ X3 )
& ( undire1521409233611534436dent_a @ V2 @ X3 ) ) ) ) ) ) ).
% ulgraph.vert_adj_edge_iff2
thf(fact_372_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 )
& ! [X3: set_a] :
( ( member_set_a @ X3 @ Vertices )
=> ~ ( undire3510646817838285160_set_a @ Edges @ X3 @ V ) ) ) ) ) ).
% ulgraph.is_isolated_vertex_def
thf(fact_373_ulgraph_Ois__isolated__vertex__def,axiom,
! [Vertices: set_Product_prod_a_a,Edges: set_se5735800977113168103od_a_a,V: product_prod_a_a] :
( ( undire4585262585102564309od_a_a @ Vertices @ Edges )
=> ( ( undire3207556238582723646od_a_a @ Vertices @ Edges @ V )
= ( ( member1426531477525435216od_a_a @ V @ Vertices )
& ! [X3: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X3 @ Vertices )
=> ~ ( undire6135774327024169009od_a_a @ Edges @ X3 @ V ) ) ) ) ) ).
% ulgraph.is_isolated_vertex_def
thf(fact_374_ulgraph_Ois__isolated__vertex__def,axiom,
! [Vertices: set_nat,Edges: set_set_nat,V: nat] :
( ( undire3269267262472140706ph_nat @ Vertices @ Edges )
=> ( ( undire5609513041723151865ex_nat @ Vertices @ Edges @ V )
= ( ( member_nat @ V @ Vertices )
& ! [X3: nat] :
( ( member_nat @ X3 @ Vertices )
=> ~ ( undire1083030068171319366dj_nat @ Edges @ X3 @ V ) ) ) ) ) ).
% ulgraph.is_isolated_vertex_def
thf(fact_375_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 )
& ! [X3: a] :
( ( member_a @ X3 @ Vertices )
=> ~ ( undire397441198561214472_adj_a @ Edges @ X3 @ V ) ) ) ) ) ).
% ulgraph.is_isolated_vertex_def
thf(fact_376_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_377_is__walk__decomp,axiom,
! [Xs: list_a,Y: a,Ys: list_a,Zs: list_a] :
( ( undire6133010728901294956walk_a @ vertices @ edges @ ( append_a @ Xs @ ( append_a @ ( cons_a @ Y @ nil_a ) @ ( append_a @ Ys @ ( append_a @ ( cons_a @ Y @ nil_a ) @ Zs ) ) ) ) )
=> ( undire6133010728901294956walk_a @ vertices @ edges @ ( append_a @ Xs @ ( append_a @ ( cons_a @ Y @ nil_a ) @ Zs ) ) ) ) ).
% is_walk_decomp
thf(fact_378_is__walk__drop__hd,axiom,
! [Ys: list_a,Y: a] :
( ( Ys != nil_a )
=> ( ( undire6133010728901294956walk_a @ vertices @ edges @ ( cons_a @ Y @ Ys ) )
=> ( undire6133010728901294956walk_a @ vertices @ edges @ Ys ) ) ) ).
% is_walk_drop_hd
thf(fact_379_is__walk__singleton,axiom,
! [U: a] :
( ( member_a @ U @ vertices )
=> ( undire6133010728901294956walk_a @ vertices @ edges @ ( cons_a @ U @ nil_a ) ) ) ).
% is_walk_singleton
thf(fact_380_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_381_subsetI,axiom,
! [A2: set_nat,B2: set_nat] :
( ! [X4: nat] :
( ( member_nat @ X4 @ A2 )
=> ( member_nat @ X4 @ B2 ) )
=> ( ord_less_eq_set_nat @ A2 @ B2 ) ) ).
% subsetI
thf(fact_382_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_383_subsetI,axiom,
! [A2: set_list_a,B2: set_list_a] :
( ! [X4: list_a] :
( ( member_list_a @ X4 @ A2 )
=> ( member_list_a @ X4 @ B2 ) )
=> ( ord_le8861187494160871172list_a @ A2 @ B2 ) ) ).
% subsetI
thf(fact_384_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_385_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_386_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_387_subset__antisym,axiom,
! [A2: set_list_a,B2: set_list_a] :
( ( ord_le8861187494160871172list_a @ A2 @ B2 )
=> ( ( ord_le8861187494160871172list_a @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% subset_antisym
thf(fact_388_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_389_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_390_order__refl,axiom,
! [X: set_Product_prod_a_a] : ( ord_le746702958409616551od_a_a @ X @ X ) ).
% order_refl
thf(fact_391_order__refl,axiom,
! [X: set_list_a] : ( ord_le8861187494160871172list_a @ X @ X ) ).
% order_refl
thf(fact_392_order__refl,axiom,
! [X: set_set_a] : ( ord_le3724670747650509150_set_a @ X @ X ) ).
% order_refl
thf(fact_393_order__refl,axiom,
! [X: set_a] : ( ord_less_eq_set_a @ X @ X ) ).
% order_refl
thf(fact_394_order__refl,axiom,
! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).
% order_refl
thf(fact_395_dual__order_Orefl,axiom,
! [A: set_Product_prod_a_a] : ( ord_le746702958409616551od_a_a @ A @ A ) ).
% dual_order.refl
thf(fact_396_dual__order_Orefl,axiom,
! [A: set_list_a] : ( ord_le8861187494160871172list_a @ A @ A ) ).
% dual_order.refl
thf(fact_397_dual__order_Orefl,axiom,
! [A: set_set_a] : ( ord_le3724670747650509150_set_a @ A @ A ) ).
% dual_order.refl
thf(fact_398_dual__order_Orefl,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ A @ A ) ).
% dual_order.refl
thf(fact_399_dual__order_Orefl,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% dual_order.refl
thf(fact_400_edge__density__commute,axiom,
! [X5: set_a,Y2: set_a] :
( ( undire297304480579013331sity_a @ edges @ X5 @ Y2 )
= ( undire297304480579013331sity_a @ edges @ Y2 @ X5 ) ) ).
% edge_density_commute
thf(fact_401_paths__ss__walk,axiom,
ord_le8861187494160871172list_a @ ( undire1387732426225024653aths_a @ vertices @ edges ) @ ( undire3736599831911450577alks_a @ vertices @ edges ) ).
% paths_ss_walk
thf(fact_402_walk__edges_Ocases,axiom,
! [X: list_a] :
( ( X != nil_a )
=> ( ! [X4: a] :
( X
!= ( cons_a @ X4 @ nil_a ) )
=> ~ ! [X4: a,Y3: a,Ys3: list_a] :
( X
!= ( cons_a @ X4 @ ( cons_a @ Y3 @ Ys3 ) ) ) ) ) ).
% walk_edges.cases
thf(fact_403_walk__edges_Osimps_I1_J,axiom,
( ( undire7337870655677353998dges_a @ nil_a )
= nil_set_a ) ).
% walk_edges.simps(1)
thf(fact_404_walk__edges__rev,axiom,
! [Xs: list_a] :
( ( rev_set_a @ ( undire7337870655677353998dges_a @ Xs ) )
= ( undire7337870655677353998dges_a @ ( rev_a @ Xs ) ) ) ).
% walk_edges_rev
thf(fact_405_walk__edges_Osimps_I2_J,axiom,
! [X: a] :
( ( undire7337870655677353998dges_a @ ( cons_a @ X @ nil_a ) )
= nil_set_a ) ).
% walk_edges.simps(2)
thf(fact_406_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_407_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_408_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_409_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_410_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_411_walk__edges__decomp__ss,axiom,
! [Xs: list_a,Y: a,Zs: list_a,Ys: list_a] : ( ord_le3724670747650509150_set_a @ ( set_set_a2 @ ( undire7337870655677353998dges_a @ ( append_a @ Xs @ ( append_a @ ( cons_a @ Y @ nil_a ) @ Zs ) ) ) ) @ ( set_set_a2 @ ( undire7337870655677353998dges_a @ ( append_a @ Xs @ ( append_a @ ( cons_a @ Y @ nil_a ) @ ( append_a @ Ys @ ( append_a @ ( cons_a @ Y @ nil_a ) @ Zs ) ) ) ) ) ) ) ).
% walk_edges_decomp_ss
thf(fact_412_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_413_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_414_append1__eq__conv,axiom,
! [Xs: list_a,X: a,Ys: list_a,Y: a] :
( ( ( append_a @ Xs @ ( cons_a @ X @ nil_a ) )
= ( append_a @ Ys @ ( cons_a @ Y @ nil_a ) ) )
= ( ( Xs = Ys )
& ( X = Y ) ) ) ).
% append1_eq_conv
thf(fact_415_append1__eq__conv,axiom,
! [Xs: list_set_a,X: set_a,Ys: list_set_a,Y: set_a] :
( ( ( append_set_a @ Xs @ ( cons_set_a @ X @ nil_set_a ) )
= ( append_set_a @ Ys @ ( cons_set_a @ Y @ nil_set_a ) ) )
= ( ( Xs = Ys )
& ( X = Y ) ) ) ).
% append1_eq_conv
thf(fact_416_rev__singleton__conv,axiom,
! [Xs: list_a,X: a] :
( ( ( rev_a @ Xs )
= ( cons_a @ X @ nil_a ) )
= ( Xs
= ( cons_a @ X @ nil_a ) ) ) ).
% rev_singleton_conv
thf(fact_417_rev__singleton__conv,axiom,
! [Xs: list_set_a,X: set_a] :
( ( ( rev_set_a @ Xs )
= ( cons_set_a @ X @ nil_set_a ) )
= ( Xs
= ( cons_set_a @ X @ nil_set_a ) ) ) ).
% rev_singleton_conv
thf(fact_418_singleton__rev__conv,axiom,
! [X: a,Xs: list_a] :
( ( ( cons_a @ X @ nil_a )
= ( rev_a @ Xs ) )
= ( ( cons_a @ X @ nil_a )
= Xs ) ) ).
% singleton_rev_conv
thf(fact_419_singleton__rev__conv,axiom,
! [X: set_a,Xs: list_set_a] :
( ( ( cons_set_a @ X @ nil_set_a )
= ( rev_set_a @ Xs ) )
= ( ( cons_set_a @ X @ nil_set_a )
= Xs ) ) ).
% singleton_rev_conv
thf(fact_420_insert__Nil,axiom,
! [X: a] :
( ( insert_a @ X @ nil_a )
= ( cons_a @ X @ nil_a ) ) ).
% insert_Nil
thf(fact_421_insert__Nil,axiom,
! [X: set_a] :
( ( insert_set_a @ X @ nil_set_a )
= ( cons_set_a @ X @ nil_set_a ) ) ).
% insert_Nil
thf(fact_422_not__in__set__insert,axiom,
! [X: product_prod_a_a,Xs: list_P1396940483166286381od_a_a] :
( ~ ( member1426531477525435216od_a_a @ X @ ( set_Product_prod_a_a2 @ Xs ) )
=> ( ( insert7736115120964043331od_a_a @ X @ Xs )
= ( cons_P7316939126706565853od_a_a @ X @ Xs ) ) ) ).
% not_in_set_insert
thf(fact_423_not__in__set__insert,axiom,
! [X: nat,Xs: list_nat] :
( ~ ( member_nat @ X @ ( set_nat2 @ Xs ) )
=> ( ( insert_nat @ X @ Xs )
= ( cons_nat @ X @ Xs ) ) ) ).
% not_in_set_insert
thf(fact_424_not__in__set__insert,axiom,
! [X: a,Xs: list_a] :
( ~ ( member_a @ X @ ( set_a2 @ Xs ) )
=> ( ( insert_a @ X @ Xs )
= ( cons_a @ X @ Xs ) ) ) ).
% not_in_set_insert
thf(fact_425_not__in__set__insert,axiom,
! [X: set_a,Xs: list_set_a] :
( ~ ( member_set_a @ X @ ( set_set_a2 @ Xs ) )
=> ( ( insert_set_a @ X @ Xs )
= ( cons_set_a @ X @ Xs ) ) ) ).
% not_in_set_insert
thf(fact_426_rev__eq__Cons__iff,axiom,
! [Xs: list_a,Y: a,Ys: list_a] :
( ( ( rev_a @ Xs )
= ( cons_a @ Y @ Ys ) )
= ( Xs
= ( append_a @ ( rev_a @ Ys ) @ ( cons_a @ Y @ nil_a ) ) ) ) ).
% rev_eq_Cons_iff
thf(fact_427_rev__eq__Cons__iff,axiom,
! [Xs: list_set_a,Y: set_a,Ys: list_set_a] :
( ( ( rev_set_a @ Xs )
= ( cons_set_a @ Y @ Ys ) )
= ( Xs
= ( append_set_a @ ( rev_set_a @ Ys ) @ ( cons_set_a @ Y @ nil_set_a ) ) ) ) ).
% rev_eq_Cons_iff
thf(fact_428_last__snoc,axiom,
! [Xs: list_a,X: a] :
( ( last_a @ ( append_a @ Xs @ ( cons_a @ X @ nil_a ) ) )
= X ) ).
% last_snoc
thf(fact_429_last__snoc,axiom,
! [Xs: list_set_a,X: set_a] :
( ( last_set_a @ ( append_set_a @ Xs @ ( cons_set_a @ X @ nil_set_a ) ) )
= X ) ).
% last_snoc
thf(fact_430_hd__Cons__tl,axiom,
! [Xs: list_a] :
( ( Xs != nil_a )
=> ( ( cons_a @ ( hd_a @ Xs ) @ ( tl_a @ Xs ) )
= Xs ) ) ).
% hd_Cons_tl
thf(fact_431_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_432_list_Ocollapse,axiom,
! [List: list_a] :
( ( List != nil_a )
=> ( ( cons_a @ ( hd_a @ List ) @ ( tl_a @ List ) )
= List ) ) ).
% list.collapse
thf(fact_433_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_434_ulgraph_Oedge__density_Ocong,axiom,
undire297304480579013331sity_a = undire297304480579013331sity_a ).
% ulgraph.edge_density.cong
thf(fact_435_not__Cons__self2,axiom,
! [X: a,Xs: list_a] :
( ( cons_a @ X @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_436_not__Cons__self2,axiom,
! [X: set_a,Xs: list_set_a] :
( ( cons_set_a @ X @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_437_transpose_Ocases,axiom,
! [X: list_list_a] :
( ( X != nil_list_a )
=> ( ! [Xss: list_list_a] :
( X
!= ( cons_list_a @ nil_a @ Xss ) )
=> ~ ! [X4: a,Xs3: list_a,Xss: list_list_a] :
( X
!= ( cons_list_a @ ( cons_a @ X4 @ Xs3 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_438_transpose_Ocases,axiom,
! [X: list_list_set_a] :
( ( X != nil_list_set_a )
=> ( ! [Xss: list_list_set_a] :
( X
!= ( cons_list_set_a @ nil_set_a @ Xss ) )
=> ~ ! [X4: set_a,Xs3: list_set_a,Xss: list_list_set_a] :
( X
!= ( cons_list_set_a @ ( cons_set_a @ X4 @ Xs3 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_439_comp__sgraph_Owalk__edges_Osimps_I2_J,axiom,
! [X: set_a] :
( ( undire6234387080713648494_set_a @ ( cons_set_a @ X @ nil_set_a ) )
= nil_set_set_a ) ).
% comp_sgraph.walk_edges.simps(2)
thf(fact_440_comp__sgraph_Owalk__edges_Osimps_I2_J,axiom,
! [X: a] :
( ( undire7337870655677353998dges_a @ ( cons_a @ X @ nil_a ) )
= nil_set_a ) ).
% comp_sgraph.walk_edges.simps(2)
thf(fact_441_ulgraph_Opaths_Ocong,axiom,
undire1387732426225024653aths_a = undire1387732426225024653aths_a ).
% ulgraph.paths.cong
thf(fact_442_ulgraph_Owalks_Ocong,axiom,
undire3736599831911450577alks_a = undire3736599831911450577alks_a ).
% ulgraph.walks.cong
thf(fact_443_comp__sgraph_Owalk__edges__decomp__ss,axiom,
! [Xs: list_set_a,Y: set_a,Zs: list_set_a,Ys: list_set_a] : ( ord_le5722252365846178494_set_a @ ( set_set_set_a2 @ ( undire6234387080713648494_set_a @ ( append_set_a @ Xs @ ( append_set_a @ ( cons_set_a @ Y @ nil_set_a ) @ Zs ) ) ) ) @ ( set_set_set_a2 @ ( undire6234387080713648494_set_a @ ( append_set_a @ Xs @ ( append_set_a @ ( cons_set_a @ Y @ nil_set_a ) @ ( append_set_a @ Ys @ ( append_set_a @ ( cons_set_a @ Y @ nil_set_a ) @ Zs ) ) ) ) ) ) ) ).
% comp_sgraph.walk_edges_decomp_ss
thf(fact_444_comp__sgraph_Owalk__edges__decomp__ss,axiom,
! [Xs: list_a,Y: a,Zs: list_a,Ys: list_a] : ( ord_le3724670747650509150_set_a @ ( set_set_a2 @ ( undire7337870655677353998dges_a @ ( append_a @ Xs @ ( append_a @ ( cons_a @ Y @ nil_a ) @ Zs ) ) ) ) @ ( set_set_a2 @ ( undire7337870655677353998dges_a @ ( append_a @ Xs @ ( append_a @ ( cons_a @ Y @ nil_a ) @ ( append_a @ Ys @ ( append_a @ ( cons_a @ Y @ nil_a ) @ Zs ) ) ) ) ) ) ) ).
% comp_sgraph.walk_edges_decomp_ss
thf(fact_445_ulgraph_Owalk__edges_Osimps_I2_J,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,X: set_a] :
( ( undire6886684016831807756_set_a @ Vertices @ Edges )
=> ( ( undire6234387080713648494_set_a @ ( cons_set_a @ X @ nil_set_a ) )
= nil_set_set_a ) ) ).
% ulgraph.walk_edges.simps(2)
thf(fact_446_ulgraph_Owalk__edges_Osimps_I2_J,axiom,
! [Vertices: set_a,Edges: set_set_a,X: a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( undire7337870655677353998dges_a @ ( cons_a @ X @ nil_a ) )
= nil_set_a ) ) ).
% ulgraph.walk_edges.simps(2)
thf(fact_447_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_448_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_449_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_450_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_451_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_452_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_453_list_Odistinct_I1_J,axiom,
! [X21: a,X22: list_a] :
( nil_a
!= ( cons_a @ X21 @ X22 ) ) ).
% list.distinct(1)
thf(fact_454_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_455_list_OdiscI,axiom,
! [List: list_a,X21: a,X22: list_a] :
( ( List
= ( cons_a @ X21 @ X22 ) )
=> ( List != nil_a ) ) ).
% list.discI
thf(fact_456_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_457_list_Oexhaust,axiom,
! [Y: list_a] :
( ( Y != nil_a )
=> ~ ! [X212: a,X222: list_a] :
( Y
!= ( cons_a @ X212 @ X222 ) ) ) ).
% list.exhaust
thf(fact_458_list_Oexhaust,axiom,
! [Y: list_set_a] :
( ( Y != nil_set_a )
=> ~ ! [X212: set_a,X222: list_set_a] :
( Y
!= ( cons_set_a @ X212 @ X222 ) ) ) ).
% list.exhaust
thf(fact_459_min__list_Ocases,axiom,
! [X: list_set_a] :
( ! [X4: set_a,Xs3: list_set_a] :
( X
!= ( cons_set_a @ X4 @ Xs3 ) )
=> ( X = nil_set_a ) ) ).
% min_list.cases
thf(fact_460_neq__Nil__conv,axiom,
! [Xs: list_a] :
( ( Xs != nil_a )
= ( ? [Y4: a,Ys4: list_a] :
( Xs
= ( cons_a @ Y4 @ Ys4 ) ) ) ) ).
% neq_Nil_conv
thf(fact_461_neq__Nil__conv,axiom,
! [Xs: list_set_a] :
( ( Xs != nil_set_a )
= ( ? [Y4: set_a,Ys4: list_set_a] :
( Xs
= ( cons_set_a @ Y4 @ Ys4 ) ) ) ) ).
% neq_Nil_conv
thf(fact_462_list__induct2_H,axiom,
! [P2: list_a > list_a > $o,Xs: list_a,Ys: list_a] :
( ( P2 @ nil_a @ nil_a )
=> ( ! [X4: a,Xs3: list_a] : ( P2 @ ( cons_a @ X4 @ Xs3 ) @ nil_a )
=> ( ! [Y3: a,Ys3: list_a] : ( P2 @ nil_a @ ( cons_a @ Y3 @ Ys3 ) )
=> ( ! [X4: a,Xs3: list_a,Y3: a,Ys3: list_a] :
( ( P2 @ Xs3 @ Ys3 )
=> ( P2 @ ( cons_a @ X4 @ Xs3 ) @ ( cons_a @ Y3 @ Ys3 ) ) )
=> ( P2 @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_463_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,Xs3: list_a] : ( P2 @ ( cons_a @ X4 @ Xs3 ) @ nil_set_a )
=> ( ! [Y3: set_a,Ys3: list_set_a] : ( P2 @ nil_a @ ( cons_set_a @ Y3 @ Ys3 ) )
=> ( ! [X4: a,Xs3: list_a,Y3: set_a,Ys3: list_set_a] :
( ( P2 @ Xs3 @ Ys3 )
=> ( P2 @ ( cons_a @ X4 @ Xs3 ) @ ( cons_set_a @ Y3 @ Ys3 ) ) )
=> ( P2 @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_464_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,Xs3: list_set_a] : ( P2 @ ( cons_set_a @ X4 @ Xs3 ) @ nil_a )
=> ( ! [Y3: a,Ys3: list_a] : ( P2 @ nil_set_a @ ( cons_a @ Y3 @ Ys3 ) )
=> ( ! [X4: set_a,Xs3: list_set_a,Y3: a,Ys3: list_a] :
( ( P2 @ Xs3 @ Ys3 )
=> ( P2 @ ( cons_set_a @ X4 @ Xs3 ) @ ( cons_a @ Y3 @ Ys3 ) ) )
=> ( P2 @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_465_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,Xs3: list_set_a] : ( P2 @ ( cons_set_a @ X4 @ Xs3 ) @ nil_set_a )
=> ( ! [Y3: set_a,Ys3: list_set_a] : ( P2 @ nil_set_a @ ( cons_set_a @ Y3 @ Ys3 ) )
=> ( ! [X4: set_a,Xs3: list_set_a,Y3: set_a,Ys3: list_set_a] :
( ( P2 @ Xs3 @ Ys3 )
=> ( P2 @ ( cons_set_a @ X4 @ Xs3 ) @ ( cons_set_a @ Y3 @ Ys3 ) ) )
=> ( P2 @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_466_list__nonempty__induct,axiom,
! [Xs: list_a,P2: list_a > $o] :
( ( Xs != nil_a )
=> ( ! [X4: a] : ( P2 @ ( cons_a @ X4 @ nil_a ) )
=> ( ! [X4: a,Xs3: list_a] :
( ( Xs3 != nil_a )
=> ( ( P2 @ Xs3 )
=> ( P2 @ ( cons_a @ X4 @ Xs3 ) ) ) )
=> ( P2 @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_467_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,Xs3: list_set_a] :
( ( Xs3 != nil_set_a )
=> ( ( P2 @ Xs3 )
=> ( P2 @ ( cons_set_a @ X4 @ Xs3 ) ) ) )
=> ( P2 @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_468_comp__sgraph_Owalk__edges_Ocases,axiom,
! [X: list_a] :
( ( X != nil_a )
=> ( ! [X4: a] :
( X
!= ( cons_a @ X4 @ nil_a ) )
=> ~ ! [X4: a,Y3: a,Ys3: list_a] :
( X
!= ( cons_a @ X4 @ ( cons_a @ Y3 @ Ys3 ) ) ) ) ) ).
% comp_sgraph.walk_edges.cases
thf(fact_469_comp__sgraph_Owalk__edges_Ocases,axiom,
! [X: list_set_a] :
( ( X != nil_set_a )
=> ( ! [X4: set_a] :
( X
!= ( cons_set_a @ X4 @ nil_set_a ) )
=> ~ ! [X4: set_a,Y3: set_a,Ys3: list_set_a] :
( X
!= ( cons_set_a @ X4 @ ( cons_set_a @ Y3 @ Ys3 ) ) ) ) ) ).
% comp_sgraph.walk_edges.cases
thf(fact_470_list_Oset__intros_I2_J,axiom,
! [Y: product_prod_a_a,X22: list_P1396940483166286381od_a_a,X21: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ Y @ ( set_Product_prod_a_a2 @ X22 ) )
=> ( member1426531477525435216od_a_a @ Y @ ( set_Product_prod_a_a2 @ ( cons_P7316939126706565853od_a_a @ X21 @ X22 ) ) ) ) ).
% list.set_intros(2)
thf(fact_471_list_Oset__intros_I2_J,axiom,
! [Y: nat,X22: list_nat,X21: nat] :
( ( member_nat @ Y @ ( set_nat2 @ X22 ) )
=> ( member_nat @ Y @ ( set_nat2 @ ( cons_nat @ X21 @ X22 ) ) ) ) ).
% list.set_intros(2)
thf(fact_472_list_Oset__intros_I2_J,axiom,
! [Y: a,X22: list_a,X21: a] :
( ( member_a @ Y @ ( set_a2 @ X22 ) )
=> ( member_a @ Y @ ( set_a2 @ ( cons_a @ X21 @ X22 ) ) ) ) ).
% list.set_intros(2)
thf(fact_473_list_Oset__intros_I2_J,axiom,
! [Y: set_a,X22: list_set_a,X21: set_a] :
( ( member_set_a @ Y @ ( set_set_a2 @ X22 ) )
=> ( member_set_a @ Y @ ( set_set_a2 @ ( cons_set_a @ X21 @ X22 ) ) ) ) ).
% list.set_intros(2)
thf(fact_474_list_Oset__intros_I1_J,axiom,
! [X21: product_prod_a_a,X22: list_P1396940483166286381od_a_a] : ( member1426531477525435216od_a_a @ X21 @ ( set_Product_prod_a_a2 @ ( cons_P7316939126706565853od_a_a @ X21 @ X22 ) ) ) ).
% list.set_intros(1)
thf(fact_475_list_Oset__intros_I1_J,axiom,
! [X21: nat,X22: list_nat] : ( member_nat @ X21 @ ( set_nat2 @ ( cons_nat @ X21 @ X22 ) ) ) ).
% list.set_intros(1)
thf(fact_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: product_prod_a_a,A: list_P1396940483166286381od_a_a] :
( ( member1426531477525435216od_a_a @ E @ ( set_Product_prod_a_a2 @ A ) )
=> ( ! [Z2: list_P1396940483166286381od_a_a] :
( A
!= ( cons_P7316939126706565853od_a_a @ E @ Z2 ) )
=> ~ ! [Z1: product_prod_a_a,Z2: list_P1396940483166286381od_a_a] :
( ( A
= ( cons_P7316939126706565853od_a_a @ Z1 @ Z2 ) )
=> ~ ( member1426531477525435216od_a_a @ E @ ( set_Product_prod_a_a2 @ Z2 ) ) ) ) ) ).
% list.set_cases
thf(fact_479_list_Oset__cases,axiom,
! [E: nat,A: list_nat] :
( ( member_nat @ E @ ( set_nat2 @ A ) )
=> ( ! [Z2: list_nat] :
( A
!= ( cons_nat @ E @ Z2 ) )
=> ~ ! [Z1: nat,Z2: list_nat] :
( ( A
= ( cons_nat @ Z1 @ Z2 ) )
=> ~ ( member_nat @ E @ ( set_nat2 @ Z2 ) ) ) ) ) ).
% list.set_cases
thf(fact_480_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_481_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_482_set__ConsD,axiom,
! [Y: product_prod_a_a,X: product_prod_a_a,Xs: list_P1396940483166286381od_a_a] :
( ( member1426531477525435216od_a_a @ Y @ ( set_Product_prod_a_a2 @ ( cons_P7316939126706565853od_a_a @ X @ Xs ) ) )
=> ( ( Y = X )
| ( member1426531477525435216od_a_a @ Y @ ( set_Product_prod_a_a2 @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_483_set__ConsD,axiom,
! [Y: nat,X: nat,Xs: list_nat] :
( ( member_nat @ Y @ ( set_nat2 @ ( cons_nat @ X @ Xs ) ) )
=> ( ( Y = X )
| ( member_nat @ Y @ ( set_nat2 @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_484_set__ConsD,axiom,
! [Y: a,X: a,Xs: list_a] :
( ( member_a @ Y @ ( set_a2 @ ( cons_a @ X @ Xs ) ) )
=> ( ( Y = X )
| ( member_a @ Y @ ( set_a2 @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_485_set__ConsD,axiom,
! [Y: set_a,X: set_a,Xs: list_set_a] :
( ( member_set_a @ Y @ ( set_set_a2 @ ( cons_set_a @ X @ Xs ) ) )
=> ( ( Y = X )
| ( member_set_a @ Y @ ( set_set_a2 @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_486_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_487_comp__sgraph_Owalk__edges_Osimps_I1_J,axiom,
( ( undire7337870655677353998dges_a @ nil_a )
= nil_set_a ) ).
% comp_sgraph.walk_edges.simps(1)
thf(fact_488_append__Cons,axiom,
! [X: a,Xs: list_a,Ys: list_a] :
( ( append_a @ ( cons_a @ X @ Xs ) @ Ys )
= ( cons_a @ X @ ( append_a @ Xs @ Ys ) ) ) ).
% append_Cons
thf(fact_489_append__Cons,axiom,
! [X: set_a,Xs: list_set_a,Ys: list_set_a] :
( ( append_set_a @ ( cons_set_a @ X @ Xs ) @ Ys )
= ( cons_set_a @ X @ ( append_set_a @ Xs @ Ys ) ) ) ).
% append_Cons
thf(fact_490_Cons__eq__appendI,axiom,
! [X: a,Xs1: list_a,Ys: list_a,Xs: list_a,Zs: list_a] :
( ( ( cons_a @ X @ Xs1 )
= Ys )
=> ( ( Xs
= ( append_a @ Xs1 @ Zs ) )
=> ( ( cons_a @ X @ Xs )
= ( append_a @ Ys @ Zs ) ) ) ) ).
% Cons_eq_appendI
thf(fact_491_Cons__eq__appendI,axiom,
! [X: set_a,Xs1: list_set_a,Ys: list_set_a,Xs: list_set_a,Zs: list_set_a] :
( ( ( cons_set_a @ X @ Xs1 )
= Ys )
=> ( ( Xs
= ( append_set_a @ Xs1 @ Zs ) )
=> ( ( cons_set_a @ X @ Xs )
= ( append_set_a @ Ys @ Zs ) ) ) ) ).
% Cons_eq_appendI
thf(fact_492_distinct__length__2__or__more,axiom,
! [A: product_prod_set_a_a,B: product_prod_set_a_a,Xs: list_P5740962349794459853et_a_a] :
( ( distin7251654435778379584et_a_a @ ( cons_P6517022146024809853et_a_a @ A @ ( cons_P6517022146024809853et_a_a @ B @ Xs ) ) )
= ( ( A != B )
& ( distin7251654435778379584et_a_a @ ( cons_P6517022146024809853et_a_a @ A @ Xs ) )
& ( distin7251654435778379584et_a_a @ ( cons_P6517022146024809853et_a_a @ B @ Xs ) ) ) ) ).
% distinct_length_2_or_more
thf(fact_493_distinct__length__2__or__more,axiom,
! [A: product_prod_a_set_a,B: product_prod_a_set_a,Xs: list_P494665323402860429_set_a] :
( ( distin201903879741355520_set_a @ ( cons_P8690643626842561597_set_a @ A @ ( cons_P8690643626842561597_set_a @ B @ Xs ) ) )
= ( ( A != B )
& ( distin201903879741355520_set_a @ ( cons_P8690643626842561597_set_a @ A @ Xs ) )
& ( distin201903879741355520_set_a @ ( cons_P8690643626842561597_set_a @ B @ Xs ) ) ) ) ).
% distinct_length_2_or_more
thf(fact_494_distinct__length__2__or__more,axiom,
! [A: product_prod_a_a,B: product_prod_a_a,Xs: list_P1396940483166286381od_a_a] :
( ( distin132333870042060960od_a_a @ ( cons_P7316939126706565853od_a_a @ A @ ( cons_P7316939126706565853od_a_a @ B @ Xs ) ) )
= ( ( A != B )
& ( distin132333870042060960od_a_a @ ( cons_P7316939126706565853od_a_a @ A @ Xs ) )
& ( distin132333870042060960od_a_a @ ( cons_P7316939126706565853od_a_a @ B @ Xs ) ) ) ) ).
% distinct_length_2_or_more
thf(fact_495_distinct__length__2__or__more,axiom,
! [A: list_set_a,B: list_set_a,Xs: list_list_set_a] :
( ( distinct_list_set_a @ ( cons_list_set_a @ A @ ( cons_list_set_a @ B @ Xs ) ) )
= ( ( A != B )
& ( distinct_list_set_a @ ( cons_list_set_a @ A @ Xs ) )
& ( distinct_list_set_a @ ( cons_list_set_a @ B @ Xs ) ) ) ) ).
% distinct_length_2_or_more
thf(fact_496_distinct__length__2__or__more,axiom,
! [A: list_a,B: list_a,Xs: list_list_a] :
( ( distinct_list_a @ ( cons_list_a @ A @ ( cons_list_a @ B @ Xs ) ) )
= ( ( A != B )
& ( distinct_list_a @ ( cons_list_a @ A @ Xs ) )
& ( distinct_list_a @ ( cons_list_a @ B @ Xs ) ) ) ) ).
% distinct_length_2_or_more
thf(fact_497_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_498_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_499_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_500_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_501_list_Osel_I1_J,axiom,
! [X21: a,X22: list_a] :
( ( hd_a @ ( cons_a @ X21 @ X22 ) )
= X21 ) ).
% list.sel(1)
thf(fact_502_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_503_list_Osel_I3_J,axiom,
! [X21: a,X22: list_a] :
( ( tl_a @ ( cons_a @ X21 @ X22 ) )
= X22 ) ).
% list.sel(3)
thf(fact_504_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_505_ulgraph_Owalk__edges__decomp__ss,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,Xs: list_set_a,Y: set_a,Zs: list_set_a,Ys: list_set_a] :
( ( undire6886684016831807756_set_a @ Vertices @ Edges )
=> ( ord_le5722252365846178494_set_a @ ( set_set_set_a2 @ ( undire6234387080713648494_set_a @ ( append_set_a @ Xs @ ( append_set_a @ ( cons_set_a @ Y @ nil_set_a ) @ Zs ) ) ) ) @ ( set_set_set_a2 @ ( undire6234387080713648494_set_a @ ( append_set_a @ Xs @ ( append_set_a @ ( cons_set_a @ Y @ nil_set_a ) @ ( append_set_a @ Ys @ ( append_set_a @ ( cons_set_a @ Y @ nil_set_a ) @ Zs ) ) ) ) ) ) ) ) ).
% ulgraph.walk_edges_decomp_ss
thf(fact_506_ulgraph_Owalk__edges__decomp__ss,axiom,
! [Vertices: set_a,Edges: set_set_a,Xs: list_a,Y: a,Zs: list_a,Ys: list_a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ord_le3724670747650509150_set_a @ ( set_set_a2 @ ( undire7337870655677353998dges_a @ ( append_a @ Xs @ ( append_a @ ( cons_a @ Y @ nil_a ) @ Zs ) ) ) ) @ ( set_set_a2 @ ( undire7337870655677353998dges_a @ ( append_a @ Xs @ ( append_a @ ( cons_a @ Y @ nil_a ) @ ( append_a @ Ys @ ( append_a @ ( cons_a @ Y @ nil_a ) @ Zs ) ) ) ) ) ) ) ) ).
% ulgraph.walk_edges_decomp_ss
thf(fact_507_ulgraph_Opaths__ss__walk,axiom,
! [Vertices: set_a,Edges: set_set_a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ord_le8861187494160871172list_a @ ( undire1387732426225024653aths_a @ Vertices @ Edges ) @ ( undire3736599831911450577alks_a @ Vertices @ Edges ) ) ) ).
% ulgraph.paths_ss_walk
thf(fact_508_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_509_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_510_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_511_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_512_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_513_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_514_ulgraph_Oedge__density__commute,axiom,
! [Vertices: set_a,Edges: set_set_a,X5: set_a,Y2: set_a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( undire297304480579013331sity_a @ Edges @ X5 @ Y2 )
= ( undire297304480579013331sity_a @ Edges @ Y2 @ X5 ) ) ) ).
% ulgraph.edge_density_commute
thf(fact_515_set__subset__Cons,axiom,
! [Xs: list_P1396940483166286381od_a_a,X: product_prod_a_a] : ( ord_le746702958409616551od_a_a @ ( set_Product_prod_a_a2 @ Xs ) @ ( set_Product_prod_a_a2 @ ( cons_P7316939126706565853od_a_a @ X @ Xs ) ) ) ).
% set_subset_Cons
thf(fact_516_set__subset__Cons,axiom,
! [Xs: list_list_a,X: list_a] : ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Xs ) @ ( set_list_a2 @ ( cons_list_a @ X @ Xs ) ) ) ).
% set_subset_Cons
thf(fact_517_set__subset__Cons,axiom,
! [Xs: list_set_a,X: set_a] : ( ord_le3724670747650509150_set_a @ ( set_set_a2 @ Xs ) @ ( set_set_a2 @ ( cons_set_a @ X @ Xs ) ) ) ).
% set_subset_Cons
thf(fact_518_set__subset__Cons,axiom,
! [Xs: list_a,X: a] : ( ord_less_eq_set_a @ ( set_a2 @ Xs ) @ ( set_a2 @ ( cons_a @ X @ Xs ) ) ) ).
% set_subset_Cons
thf(fact_519_rev__induct,axiom,
! [P2: list_a > $o,Xs: list_a] :
( ( P2 @ nil_a )
=> ( ! [X4: a,Xs3: list_a] :
( ( P2 @ Xs3 )
=> ( P2 @ ( append_a @ Xs3 @ ( cons_a @ X4 @ nil_a ) ) ) )
=> ( P2 @ Xs ) ) ) ).
% rev_induct
thf(fact_520_rev__induct,axiom,
! [P2: list_set_a > $o,Xs: list_set_a] :
( ( P2 @ nil_set_a )
=> ( ! [X4: set_a,Xs3: list_set_a] :
( ( P2 @ Xs3 )
=> ( P2 @ ( append_set_a @ Xs3 @ ( cons_set_a @ X4 @ nil_set_a ) ) ) )
=> ( P2 @ Xs ) ) ) ).
% rev_induct
thf(fact_521_rev__exhaust,axiom,
! [Xs: list_a] :
( ( Xs != nil_a )
=> ~ ! [Ys3: list_a,Y3: a] :
( Xs
!= ( append_a @ Ys3 @ ( cons_a @ Y3 @ nil_a ) ) ) ) ).
% rev_exhaust
thf(fact_522_rev__exhaust,axiom,
! [Xs: list_set_a] :
( ( Xs != nil_set_a )
=> ~ ! [Ys3: list_set_a,Y3: set_a] :
( Xs
!= ( append_set_a @ Ys3 @ ( cons_set_a @ Y3 @ nil_set_a ) ) ) ) ).
% rev_exhaust
thf(fact_523_Cons__eq__append__conv,axiom,
! [X: a,Xs: list_a,Ys: list_a,Zs: list_a] :
( ( ( cons_a @ X @ Xs )
= ( append_a @ Ys @ Zs ) )
= ( ( ( Ys = nil_a )
& ( ( cons_a @ X @ Xs )
= Zs ) )
| ? [Ys5: list_a] :
( ( ( cons_a @ X @ Ys5 )
= Ys )
& ( Xs
= ( append_a @ Ys5 @ Zs ) ) ) ) ) ).
% Cons_eq_append_conv
thf(fact_524_Cons__eq__append__conv,axiom,
! [X: set_a,Xs: list_set_a,Ys: list_set_a,Zs: list_set_a] :
( ( ( cons_set_a @ X @ Xs )
= ( append_set_a @ Ys @ Zs ) )
= ( ( ( Ys = nil_set_a )
& ( ( cons_set_a @ X @ Xs )
= Zs ) )
| ? [Ys5: list_set_a] :
( ( ( cons_set_a @ X @ Ys5 )
= Ys )
& ( Xs
= ( append_set_a @ Ys5 @ Zs ) ) ) ) ) ).
% Cons_eq_append_conv
thf(fact_525_append__eq__Cons__conv,axiom,
! [Ys: list_a,Zs: list_a,X: a,Xs: list_a] :
( ( ( append_a @ Ys @ Zs )
= ( cons_a @ X @ Xs ) )
= ( ( ( Ys = nil_a )
& ( Zs
= ( cons_a @ X @ Xs ) ) )
| ? [Ys5: list_a] :
( ( Ys
= ( cons_a @ X @ Ys5 ) )
& ( ( append_a @ Ys5 @ Zs )
= Xs ) ) ) ) ).
% append_eq_Cons_conv
thf(fact_526_append__eq__Cons__conv,axiom,
! [Ys: list_set_a,Zs: list_set_a,X: set_a,Xs: list_set_a] :
( ( ( append_set_a @ Ys @ Zs )
= ( cons_set_a @ X @ Xs ) )
= ( ( ( Ys = nil_set_a )
& ( Zs
= ( cons_set_a @ X @ Xs ) ) )
| ? [Ys5: list_set_a] :
( ( Ys
= ( cons_set_a @ X @ Ys5 ) )
& ( ( append_set_a @ Ys5 @ Zs )
= Xs ) ) ) ) ).
% append_eq_Cons_conv
thf(fact_527_rev__nonempty__induct,axiom,
! [Xs: list_a,P2: list_a > $o] :
( ( Xs != nil_a )
=> ( ! [X4: a] : ( P2 @ ( cons_a @ X4 @ nil_a ) )
=> ( ! [X4: a,Xs3: list_a] :
( ( Xs3 != nil_a )
=> ( ( P2 @ Xs3 )
=> ( P2 @ ( append_a @ Xs3 @ ( cons_a @ X4 @ nil_a ) ) ) ) )
=> ( P2 @ Xs ) ) ) ) ).
% rev_nonempty_induct
thf(fact_528_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,Xs3: list_set_a] :
( ( Xs3 != nil_set_a )
=> ( ( P2 @ Xs3 )
=> ( P2 @ ( append_set_a @ Xs3 @ ( cons_set_a @ X4 @ nil_set_a ) ) ) ) )
=> ( P2 @ Xs ) ) ) ) ).
% rev_nonempty_induct
thf(fact_529_split__list,axiom,
! [X: product_prod_a_a,Xs: list_P1396940483166286381od_a_a] :
( ( member1426531477525435216od_a_a @ X @ ( set_Product_prod_a_a2 @ Xs ) )
=> ? [Ys3: list_P1396940483166286381od_a_a,Zs2: list_P1396940483166286381od_a_a] :
( Xs
= ( append5335208819046833346od_a_a @ Ys3 @ ( cons_P7316939126706565853od_a_a @ X @ Zs2 ) ) ) ) ).
% split_list
thf(fact_530_split__list,axiom,
! [X: nat,Xs: list_nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs ) )
=> ? [Ys3: list_nat,Zs2: list_nat] :
( Xs
= ( append_nat @ Ys3 @ ( cons_nat @ X @ Zs2 ) ) ) ) ).
% split_list
thf(fact_531_split__list,axiom,
! [X: a,Xs: list_a] :
( ( member_a @ X @ ( set_a2 @ Xs ) )
=> ? [Ys3: list_a,Zs2: list_a] :
( Xs
= ( append_a @ Ys3 @ ( cons_a @ X @ Zs2 ) ) ) ) ).
% split_list
thf(fact_532_split__list,axiom,
! [X: set_a,Xs: list_set_a] :
( ( member_set_a @ X @ ( set_set_a2 @ Xs ) )
=> ? [Ys3: list_set_a,Zs2: list_set_a] :
( Xs
= ( append_set_a @ Ys3 @ ( cons_set_a @ X @ Zs2 ) ) ) ) ).
% split_list
thf(fact_533_split__list__last,axiom,
! [X: product_prod_a_a,Xs: list_P1396940483166286381od_a_a] :
( ( member1426531477525435216od_a_a @ X @ ( set_Product_prod_a_a2 @ Xs ) )
=> ? [Ys3: list_P1396940483166286381od_a_a,Zs2: list_P1396940483166286381od_a_a] :
( ( Xs
= ( append5335208819046833346od_a_a @ Ys3 @ ( cons_P7316939126706565853od_a_a @ X @ Zs2 ) ) )
& ~ ( member1426531477525435216od_a_a @ X @ ( set_Product_prod_a_a2 @ Zs2 ) ) ) ) ).
% split_list_last
thf(fact_534_split__list__last,axiom,
! [X: nat,Xs: list_nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs ) )
=> ? [Ys3: list_nat,Zs2: list_nat] :
( ( Xs
= ( append_nat @ Ys3 @ ( cons_nat @ X @ Zs2 ) ) )
& ~ ( member_nat @ X @ ( set_nat2 @ Zs2 ) ) ) ) ).
% split_list_last
thf(fact_535_split__list__last,axiom,
! [X: a,Xs: list_a] :
( ( member_a @ X @ ( set_a2 @ Xs ) )
=> ? [Ys3: list_a,Zs2: list_a] :
( ( Xs
= ( append_a @ Ys3 @ ( cons_a @ X @ Zs2 ) ) )
& ~ ( member_a @ X @ ( set_a2 @ Zs2 ) ) ) ) ).
% split_list_last
thf(fact_536_split__list__last,axiom,
! [X: set_a,Xs: list_set_a] :
( ( member_set_a @ X @ ( set_set_a2 @ Xs ) )
=> ? [Ys3: list_set_a,Zs2: list_set_a] :
( ( Xs
= ( append_set_a @ Ys3 @ ( cons_set_a @ X @ Zs2 ) ) )
& ~ ( member_set_a @ X @ ( set_set_a2 @ Zs2 ) ) ) ) ).
% split_list_last
thf(fact_537_split__list__prop,axiom,
! [Xs: list_a,P2: a > $o] :
( ? [X2: a] :
( ( member_a @ X2 @ ( set_a2 @ Xs ) )
& ( P2 @ X2 ) )
=> ? [Ys3: list_a,X4: a] :
( ? [Zs2: list_a] :
( Xs
= ( append_a @ Ys3 @ ( cons_a @ X4 @ Zs2 ) ) )
& ( P2 @ X4 ) ) ) ).
% split_list_prop
thf(fact_538_split__list__prop,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,X4: set_a] :
( ? [Zs2: list_set_a] :
( Xs
= ( append_set_a @ Ys3 @ ( cons_set_a @ X4 @ Zs2 ) ) )
& ( P2 @ X4 ) ) ) ).
% split_list_prop
thf(fact_539_split__list__first,axiom,
! [X: product_prod_a_a,Xs: list_P1396940483166286381od_a_a] :
( ( member1426531477525435216od_a_a @ X @ ( set_Product_prod_a_a2 @ Xs ) )
=> ? [Ys3: list_P1396940483166286381od_a_a,Zs2: list_P1396940483166286381od_a_a] :
( ( Xs
= ( append5335208819046833346od_a_a @ Ys3 @ ( cons_P7316939126706565853od_a_a @ X @ Zs2 ) ) )
& ~ ( member1426531477525435216od_a_a @ X @ ( set_Product_prod_a_a2 @ Ys3 ) ) ) ) ).
% split_list_first
thf(fact_540_split__list__first,axiom,
! [X: nat,Xs: list_nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs ) )
=> ? [Ys3: list_nat,Zs2: list_nat] :
( ( Xs
= ( append_nat @ Ys3 @ ( cons_nat @ X @ Zs2 ) ) )
& ~ ( member_nat @ X @ ( set_nat2 @ Ys3 ) ) ) ) ).
% split_list_first
thf(fact_541_split__list__first,axiom,
! [X: a,Xs: list_a] :
( ( member_a @ X @ ( set_a2 @ Xs ) )
=> ? [Ys3: list_a,Zs2: list_a] :
( ( Xs
= ( append_a @ Ys3 @ ( cons_a @ X @ Zs2 ) ) )
& ~ ( member_a @ X @ ( set_a2 @ Ys3 ) ) ) ) ).
% split_list_first
thf(fact_542_split__list__first,axiom,
! [X: set_a,Xs: list_set_a] :
( ( member_set_a @ X @ ( set_set_a2 @ Xs ) )
=> ? [Ys3: list_set_a,Zs2: list_set_a] :
( ( Xs
= ( append_set_a @ Ys3 @ ( cons_set_a @ X @ Zs2 ) ) )
& ~ ( member_set_a @ X @ ( set_set_a2 @ Ys3 ) ) ) ) ).
% split_list_first
thf(fact_543_split__list__propE,axiom,
! [Xs: list_a,P2: a > $o] :
( ? [X2: a] :
( ( member_a @ X2 @ ( set_a2 @ Xs ) )
& ( P2 @ X2 ) )
=> ~ ! [Ys3: list_a,X4: a] :
( ? [Zs2: list_a] :
( Xs
= ( append_a @ Ys3 @ ( cons_a @ X4 @ Zs2 ) ) )
=> ~ ( P2 @ X4 ) ) ) ).
% split_list_propE
thf(fact_544_split__list__propE,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,X4: set_a] :
( ? [Zs2: list_set_a] :
( Xs
= ( append_set_a @ Ys3 @ ( cons_set_a @ X4 @ Zs2 ) ) )
=> ~ ( P2 @ X4 ) ) ) ).
% split_list_propE
thf(fact_545_append__Cons__eq__iff,axiom,
! [X: product_prod_a_a,Xs: list_P1396940483166286381od_a_a,Ys: list_P1396940483166286381od_a_a,Xs4: list_P1396940483166286381od_a_a,Ys6: list_P1396940483166286381od_a_a] :
( ~ ( member1426531477525435216od_a_a @ X @ ( set_Product_prod_a_a2 @ Xs ) )
=> ( ~ ( member1426531477525435216od_a_a @ X @ ( set_Product_prod_a_a2 @ Ys ) )
=> ( ( ( append5335208819046833346od_a_a @ Xs @ ( cons_P7316939126706565853od_a_a @ X @ Ys ) )
= ( append5335208819046833346od_a_a @ Xs4 @ ( cons_P7316939126706565853od_a_a @ X @ Ys6 ) ) )
= ( ( Xs = Xs4 )
& ( Ys = Ys6 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_546_append__Cons__eq__iff,axiom,
! [X: nat,Xs: list_nat,Ys: list_nat,Xs4: list_nat,Ys6: list_nat] :
( ~ ( member_nat @ X @ ( set_nat2 @ Xs ) )
=> ( ~ ( member_nat @ X @ ( set_nat2 @ Ys ) )
=> ( ( ( append_nat @ Xs @ ( cons_nat @ X @ Ys ) )
= ( append_nat @ Xs4 @ ( cons_nat @ X @ Ys6 ) ) )
= ( ( Xs = Xs4 )
& ( Ys = Ys6 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_547_append__Cons__eq__iff,axiom,
! [X: a,Xs: list_a,Ys: list_a,Xs4: list_a,Ys6: list_a] :
( ~ ( member_a @ X @ ( set_a2 @ Xs ) )
=> ( ~ ( member_a @ X @ ( set_a2 @ Ys ) )
=> ( ( ( append_a @ Xs @ ( cons_a @ X @ Ys ) )
= ( append_a @ Xs4 @ ( cons_a @ X @ Ys6 ) ) )
= ( ( Xs = Xs4 )
& ( Ys = Ys6 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_548_append__Cons__eq__iff,axiom,
! [X: set_a,Xs: list_set_a,Ys: list_set_a,Xs4: list_set_a,Ys6: list_set_a] :
( ~ ( member_set_a @ X @ ( set_set_a2 @ Xs ) )
=> ( ~ ( member_set_a @ X @ ( set_set_a2 @ Ys ) )
=> ( ( ( append_set_a @ Xs @ ( cons_set_a @ X @ Ys ) )
= ( append_set_a @ Xs4 @ ( cons_set_a @ X @ Ys6 ) ) )
= ( ( Xs = Xs4 )
& ( Ys = Ys6 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_549_in__set__conv__decomp,axiom,
! [X: product_prod_a_a,Xs: list_P1396940483166286381od_a_a] :
( ( member1426531477525435216od_a_a @ X @ ( set_Product_prod_a_a2 @ Xs ) )
= ( ? [Ys4: list_P1396940483166286381od_a_a,Zs3: list_P1396940483166286381od_a_a] :
( Xs
= ( append5335208819046833346od_a_a @ Ys4 @ ( cons_P7316939126706565853od_a_a @ X @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_550_in__set__conv__decomp,axiom,
! [X: nat,Xs: list_nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs ) )
= ( ? [Ys4: list_nat,Zs3: list_nat] :
( Xs
= ( append_nat @ Ys4 @ ( cons_nat @ X @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_551_in__set__conv__decomp,axiom,
! [X: a,Xs: list_a] :
( ( member_a @ X @ ( set_a2 @ Xs ) )
= ( ? [Ys4: list_a,Zs3: list_a] :
( Xs
= ( append_a @ Ys4 @ ( cons_a @ X @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_552_in__set__conv__decomp,axiom,
! [X: set_a,Xs: list_set_a] :
( ( member_set_a @ X @ ( set_set_a2 @ Xs ) )
= ( ? [Ys4: list_set_a,Zs3: list_set_a] :
( Xs
= ( append_set_a @ Ys4 @ ( cons_set_a @ X @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_553_split__list__last__prop,axiom,
! [Xs: list_a,P2: a > $o] :
( ? [X2: a] :
( ( member_a @ X2 @ ( set_a2 @ Xs ) )
& ( P2 @ X2 ) )
=> ? [Ys3: list_a,X4: a,Zs2: list_a] :
( ( Xs
= ( append_a @ Ys3 @ ( cons_a @ X4 @ Zs2 ) ) )
& ( P2 @ X4 )
& ! [Xa: a] :
( ( member_a @ Xa @ ( set_a2 @ Zs2 ) )
=> ~ ( P2 @ Xa ) ) ) ) ).
% split_list_last_prop
thf(fact_554_split__list__last__prop,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,X4: set_a,Zs2: list_set_a] :
( ( Xs
= ( append_set_a @ Ys3 @ ( 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_555_split__list__first__prop,axiom,
! [Xs: list_a,P2: a > $o] :
( ? [X2: a] :
( ( member_a @ X2 @ ( set_a2 @ Xs ) )
& ( P2 @ X2 ) )
=> ? [Ys3: list_a,X4: a] :
( ? [Zs2: list_a] :
( Xs
= ( append_a @ Ys3 @ ( cons_a @ X4 @ Zs2 ) ) )
& ( P2 @ X4 )
& ! [Xa: a] :
( ( member_a @ Xa @ ( set_a2 @ Ys3 ) )
=> ~ ( P2 @ Xa ) ) ) ) ).
% split_list_first_prop
thf(fact_556_split__list__first__prop,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,X4: set_a] :
( ? [Zs2: list_set_a] :
( Xs
= ( append_set_a @ Ys3 @ ( cons_set_a @ X4 @ Zs2 ) ) )
& ( P2 @ X4 )
& ! [Xa: set_a] :
( ( member_set_a @ Xa @ ( set_set_a2 @ Ys3 ) )
=> ~ ( P2 @ Xa ) ) ) ) ).
% split_list_first_prop
thf(fact_557_split__list__last__propE,axiom,
! [Xs: list_a,P2: a > $o] :
( ? [X2: a] :
( ( member_a @ X2 @ ( set_a2 @ Xs ) )
& ( P2 @ X2 ) )
=> ~ ! [Ys3: list_a,X4: a,Zs2: list_a] :
( ( Xs
= ( append_a @ Ys3 @ ( cons_a @ X4 @ Zs2 ) ) )
=> ( ( P2 @ X4 )
=> ~ ! [Xa: a] :
( ( member_a @ Xa @ ( set_a2 @ Zs2 ) )
=> ~ ( P2 @ Xa ) ) ) ) ) ).
% split_list_last_propE
thf(fact_558_split__list__last__propE,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,X4: set_a,Zs2: list_set_a] :
( ( Xs
= ( append_set_a @ Ys3 @ ( 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_559_split__list__first__propE,axiom,
! [Xs: list_a,P2: a > $o] :
( ? [X2: a] :
( ( member_a @ X2 @ ( set_a2 @ Xs ) )
& ( P2 @ X2 ) )
=> ~ ! [Ys3: list_a,X4: a] :
( ? [Zs2: list_a] :
( Xs
= ( append_a @ Ys3 @ ( cons_a @ X4 @ Zs2 ) ) )
=> ( ( P2 @ X4 )
=> ~ ! [Xa: a] :
( ( member_a @ Xa @ ( set_a2 @ Ys3 ) )
=> ~ ( P2 @ Xa ) ) ) ) ) ).
% split_list_first_propE
thf(fact_560_split__list__first__propE,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,X4: set_a] :
( ? [Zs2: list_set_a] :
( Xs
= ( append_set_a @ Ys3 @ ( cons_set_a @ X4 @ Zs2 ) ) )
=> ( ( P2 @ X4 )
=> ~ ! [Xa: set_a] :
( ( member_set_a @ Xa @ ( set_set_a2 @ Ys3 ) )
=> ~ ( P2 @ Xa ) ) ) ) ) ).
% split_list_first_propE
thf(fact_561_in__set__conv__decomp__last,axiom,
! [X: product_prod_a_a,Xs: list_P1396940483166286381od_a_a] :
( ( member1426531477525435216od_a_a @ X @ ( set_Product_prod_a_a2 @ Xs ) )
= ( ? [Ys4: list_P1396940483166286381od_a_a,Zs3: list_P1396940483166286381od_a_a] :
( ( Xs
= ( append5335208819046833346od_a_a @ Ys4 @ ( cons_P7316939126706565853od_a_a @ X @ Zs3 ) ) )
& ~ ( member1426531477525435216od_a_a @ X @ ( set_Product_prod_a_a2 @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_562_in__set__conv__decomp__last,axiom,
! [X: nat,Xs: list_nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs ) )
= ( ? [Ys4: list_nat,Zs3: list_nat] :
( ( Xs
= ( append_nat @ Ys4 @ ( cons_nat @ X @ Zs3 ) ) )
& ~ ( member_nat @ X @ ( set_nat2 @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_563_in__set__conv__decomp__last,axiom,
! [X: a,Xs: list_a] :
( ( member_a @ X @ ( set_a2 @ Xs ) )
= ( ? [Ys4: list_a,Zs3: list_a] :
( ( Xs
= ( append_a @ Ys4 @ ( cons_a @ X @ Zs3 ) ) )
& ~ ( member_a @ X @ ( set_a2 @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_564_in__set__conv__decomp__last,axiom,
! [X: set_a,Xs: list_set_a] :
( ( member_set_a @ X @ ( set_set_a2 @ Xs ) )
= ( ? [Ys4: list_set_a,Zs3: list_set_a] :
( ( Xs
= ( append_set_a @ Ys4 @ ( cons_set_a @ X @ Zs3 ) ) )
& ~ ( member_set_a @ X @ ( set_set_a2 @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_565_in__set__conv__decomp__first,axiom,
! [X: product_prod_a_a,Xs: list_P1396940483166286381od_a_a] :
( ( member1426531477525435216od_a_a @ X @ ( set_Product_prod_a_a2 @ Xs ) )
= ( ? [Ys4: list_P1396940483166286381od_a_a,Zs3: list_P1396940483166286381od_a_a] :
( ( Xs
= ( append5335208819046833346od_a_a @ Ys4 @ ( cons_P7316939126706565853od_a_a @ X @ Zs3 ) ) )
& ~ ( member1426531477525435216od_a_a @ X @ ( set_Product_prod_a_a2 @ Ys4 ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_566_in__set__conv__decomp__first,axiom,
! [X: nat,Xs: list_nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs ) )
= ( ? [Ys4: list_nat,Zs3: list_nat] :
( ( Xs
= ( append_nat @ Ys4 @ ( cons_nat @ X @ Zs3 ) ) )
& ~ ( member_nat @ X @ ( set_nat2 @ Ys4 ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_567_in__set__conv__decomp__first,axiom,
! [X: a,Xs: list_a] :
( ( member_a @ X @ ( set_a2 @ Xs ) )
= ( ? [Ys4: list_a,Zs3: list_a] :
( ( Xs
= ( append_a @ Ys4 @ ( cons_a @ X @ Zs3 ) ) )
& ~ ( member_a @ X @ ( set_a2 @ Ys4 ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_568_in__set__conv__decomp__first,axiom,
! [X: set_a,Xs: list_set_a] :
( ( member_set_a @ X @ ( set_set_a2 @ Xs ) )
= ( ? [Ys4: list_set_a,Zs3: list_set_a] :
( ( Xs
= ( append_set_a @ Ys4 @ ( cons_set_a @ X @ Zs3 ) ) )
& ~ ( member_set_a @ X @ ( set_set_a2 @ Ys4 ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_569_split__list__last__prop__iff,axiom,
! [Xs: list_a,P2: a > $o] :
( ( ? [X3: a] :
( ( member_a @ X3 @ ( set_a2 @ Xs ) )
& ( P2 @ X3 ) ) )
= ( ? [Ys4: list_a,X3: a,Zs3: list_a] :
( ( Xs
= ( append_a @ Ys4 @ ( cons_a @ X3 @ Zs3 ) ) )
& ( P2 @ X3 )
& ! [Y4: a] :
( ( member_a @ Y4 @ ( set_a2 @ Zs3 ) )
=> ~ ( P2 @ Y4 ) ) ) ) ) ).
% split_list_last_prop_iff
thf(fact_570_split__list__last__prop__iff,axiom,
! [Xs: list_set_a,P2: set_a > $o] :
( ( ? [X3: set_a] :
( ( member_set_a @ X3 @ ( set_set_a2 @ Xs ) )
& ( P2 @ X3 ) ) )
= ( ? [Ys4: list_set_a,X3: set_a,Zs3: list_set_a] :
( ( Xs
= ( append_set_a @ Ys4 @ ( cons_set_a @ X3 @ Zs3 ) ) )
& ( P2 @ X3 )
& ! [Y4: set_a] :
( ( member_set_a @ Y4 @ ( set_set_a2 @ Zs3 ) )
=> ~ ( P2 @ Y4 ) ) ) ) ) ).
% split_list_last_prop_iff
thf(fact_571_split__list__first__prop__iff,axiom,
! [Xs: list_a,P2: a > $o] :
( ( ? [X3: a] :
( ( member_a @ X3 @ ( set_a2 @ Xs ) )
& ( P2 @ X3 ) ) )
= ( ? [Ys4: list_a,X3: a] :
( ? [Zs3: list_a] :
( Xs
= ( append_a @ Ys4 @ ( cons_a @ X3 @ Zs3 ) ) )
& ( P2 @ X3 )
& ! [Y4: a] :
( ( member_a @ Y4 @ ( set_a2 @ Ys4 ) )
=> ~ ( P2 @ Y4 ) ) ) ) ) ).
% split_list_first_prop_iff
thf(fact_572_split__list__first__prop__iff,axiom,
! [Xs: list_set_a,P2: set_a > $o] :
( ( ? [X3: set_a] :
( ( member_set_a @ X3 @ ( set_set_a2 @ Xs ) )
& ( P2 @ X3 ) ) )
= ( ? [Ys4: list_set_a,X3: set_a] :
( ? [Zs3: list_set_a] :
( Xs
= ( append_set_a @ Ys4 @ ( cons_set_a @ X3 @ Zs3 ) ) )
& ( P2 @ X3 )
& ! [Y4: set_a] :
( ( member_set_a @ Y4 @ ( set_set_a2 @ Ys4 ) )
=> ~ ( P2 @ Y4 ) ) ) ) ) ).
% split_list_first_prop_iff
thf(fact_573_distinct__singleton,axiom,
! [X: product_prod_set_a_a] : ( distin7251654435778379584et_a_a @ ( cons_P6517022146024809853et_a_a @ X @ nil_Pr7933670750051473869et_a_a ) ) ).
% distinct_singleton
thf(fact_574_distinct__singleton,axiom,
! [X: product_prod_a_set_a] : ( distin201903879741355520_set_a @ ( cons_P8690643626842561597_set_a @ X @ nil_Pr883920194014449805_set_a ) ) ).
% distinct_singleton
thf(fact_575_distinct__singleton,axiom,
! [X: product_prod_a_a] : ( distin132333870042060960od_a_a @ ( cons_P7316939126706565853od_a_a @ X @ nil_Product_prod_a_a ) ) ).
% distinct_singleton
thf(fact_576_distinct__singleton,axiom,
! [X: list_set_a] : ( distinct_list_set_a @ ( cons_list_set_a @ X @ nil_list_set_a ) ) ).
% distinct_singleton
thf(fact_577_distinct__singleton,axiom,
! [X: list_a] : ( distinct_list_a @ ( cons_list_a @ X @ nil_list_a ) ) ).
% distinct_singleton
thf(fact_578_distinct__singleton,axiom,
! [X: a] : ( distinct_a @ ( cons_a @ X @ nil_a ) ) ).
% distinct_singleton
thf(fact_579_distinct__singleton,axiom,
! [X: set_a] : ( distinct_set_a @ ( cons_set_a @ X @ nil_set_a ) ) ).
% distinct_singleton
thf(fact_580_ulgraph_Owalk__edges_Ocases,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,X: list_set_a] :
( ( undire6886684016831807756_set_a @ Vertices @ Edges )
=> ( ( X != nil_set_a )
=> ( ! [X4: set_a] :
( X
!= ( cons_set_a @ X4 @ nil_set_a ) )
=> ~ ! [X4: set_a,Y3: set_a,Ys3: list_set_a] :
( X
!= ( cons_set_a @ X4 @ ( cons_set_a @ Y3 @ Ys3 ) ) ) ) ) ) ).
% ulgraph.walk_edges.cases
thf(fact_581_ulgraph_Owalk__edges_Ocases,axiom,
! [Vertices: set_a,Edges: set_set_a,X: list_a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( X != nil_a )
=> ( ! [X4: a] :
( X
!= ( cons_a @ X4 @ nil_a ) )
=> ~ ! [X4: a,Y3: a,Ys3: list_a] :
( X
!= ( cons_a @ X4 @ ( cons_a @ Y3 @ Ys3 ) ) ) ) ) ) ).
% ulgraph.walk_edges.cases
thf(fact_582_distinct_Osimps_I2_J,axiom,
! [X: nat,Xs: list_nat] :
( ( distinct_nat @ ( cons_nat @ X @ Xs ) )
= ( ~ ( member_nat @ X @ ( set_nat2 @ Xs ) )
& ( distinct_nat @ Xs ) ) ) ).
% distinct.simps(2)
thf(fact_583_distinct_Osimps_I2_J,axiom,
! [X: product_prod_set_a_a,Xs: list_P5740962349794459853et_a_a] :
( ( distin7251654435778379584et_a_a @ ( cons_P6517022146024809853et_a_a @ X @ Xs ) )
= ( ~ ( member2598349401703774704et_a_a @ X @ ( set_Pr9190486990633797724et_a_a @ Xs ) )
& ( distin7251654435778379584et_a_a @ Xs ) ) ) ).
% distinct.simps(2)
thf(fact_584_distinct_Osimps_I2_J,axiom,
! [X: product_prod_a_set_a,Xs: list_P494665323402860429_set_a] :
( ( distin201903879741355520_set_a @ ( cons_P8690643626842561597_set_a @ X @ Xs ) )
= ( ~ ( member4771970882521526448_set_a @ X @ ( set_Pr2140736434596773660_set_a @ Xs ) )
& ( distin201903879741355520_set_a @ Xs ) ) ) ).
% distinct.simps(2)
thf(fact_585_distinct_Osimps_I2_J,axiom,
! [X: product_prod_a_a,Xs: list_P1396940483166286381od_a_a] :
( ( distin132333870042060960od_a_a @ ( cons_P7316939126706565853od_a_a @ X @ Xs ) )
= ( ~ ( member1426531477525435216od_a_a @ X @ ( set_Product_prod_a_a2 @ Xs ) )
& ( distin132333870042060960od_a_a @ Xs ) ) ) ).
% distinct.simps(2)
thf(fact_586_distinct_Osimps_I2_J,axiom,
! [X: list_set_a,Xs: list_list_set_a] :
( ( distinct_list_set_a @ ( cons_list_set_a @ X @ Xs ) )
= ( ~ ( member_list_set_a @ X @ ( set_list_set_a2 @ Xs ) )
& ( distinct_list_set_a @ Xs ) ) ) ).
% distinct.simps(2)
thf(fact_587_distinct_Osimps_I2_J,axiom,
! [X: list_a,Xs: list_list_a] :
( ( distinct_list_a @ ( cons_list_a @ X @ Xs ) )
= ( ~ ( member_list_a @ X @ ( set_list_a2 @ Xs ) )
& ( distinct_list_a @ Xs ) ) ) ).
% distinct.simps(2)
thf(fact_588_distinct_Osimps_I2_J,axiom,
! [X: a,Xs: list_a] :
( ( distinct_a @ ( cons_a @ X @ Xs ) )
= ( ~ ( member_a @ X @ ( set_a2 @ Xs ) )
& ( distinct_a @ Xs ) ) ) ).
% distinct.simps(2)
thf(fact_589_distinct_Osimps_I2_J,axiom,
! [X: set_a,Xs: list_set_a] :
( ( distinct_set_a @ ( cons_set_a @ X @ Xs ) )
= ( ~ ( member_set_a @ X @ ( set_set_a2 @ Xs ) )
& ( distinct_set_a @ Xs ) ) ) ).
% distinct.simps(2)
thf(fact_590_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_591_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_592_tl__Nil,axiom,
! [Xs: list_a] :
( ( ( tl_a @ Xs )
= nil_a )
= ( ( Xs = nil_a )
| ? [X3: a] :
( Xs
= ( cons_a @ X3 @ nil_a ) ) ) ) ).
% tl_Nil
thf(fact_593_tl__Nil,axiom,
! [Xs: list_set_a] :
( ( ( tl_set_a @ Xs )
= nil_set_a )
= ( ( Xs = nil_set_a )
| ? [X3: set_a] :
( Xs
= ( cons_set_a @ X3 @ nil_set_a ) ) ) ) ).
% tl_Nil
thf(fact_594_Nil__tl,axiom,
! [Xs: list_a] :
( ( nil_a
= ( tl_a @ Xs ) )
= ( ( Xs = nil_a )
| ? [X3: a] :
( Xs
= ( cons_a @ X3 @ nil_a ) ) ) ) ).
% Nil_tl
thf(fact_595_Nil__tl,axiom,
! [Xs: list_set_a] :
( ( nil_set_a
= ( tl_set_a @ Xs ) )
= ( ( Xs = nil_set_a )
| ? [X3: set_a] :
( Xs
= ( cons_set_a @ X3 @ nil_set_a ) ) ) ) ).
% Nil_tl
thf(fact_596_last__ConsR,axiom,
! [Xs: list_a,X: a] :
( ( Xs != nil_a )
=> ( ( last_a @ ( cons_a @ X @ Xs ) )
= ( last_a @ Xs ) ) ) ).
% last_ConsR
thf(fact_597_last__ConsR,axiom,
! [Xs: list_set_a,X: set_a] :
( ( Xs != nil_set_a )
=> ( ( last_set_a @ ( cons_set_a @ X @ Xs ) )
= ( last_set_a @ Xs ) ) ) ).
% last_ConsR
thf(fact_598_last__ConsL,axiom,
! [Xs: list_a,X: a] :
( ( Xs = nil_a )
=> ( ( last_a @ ( cons_a @ X @ Xs ) )
= X ) ) ).
% last_ConsL
thf(fact_599_last__ConsL,axiom,
! [Xs: list_set_a,X: set_a] :
( ( Xs = nil_set_a )
=> ( ( last_set_a @ ( cons_set_a @ X @ Xs ) )
= X ) ) ).
% last_ConsL
thf(fact_600_last_Osimps,axiom,
! [Xs: list_a,X: a] :
( ( ( Xs = nil_a )
=> ( ( last_a @ ( cons_a @ X @ Xs ) )
= X ) )
& ( ( Xs != nil_a )
=> ( ( last_a @ ( cons_a @ X @ Xs ) )
= ( last_a @ Xs ) ) ) ) ).
% last.simps
thf(fact_601_last_Osimps,axiom,
! [Xs: list_set_a,X: set_a] :
( ( ( Xs = nil_set_a )
=> ( ( last_set_a @ ( cons_set_a @ X @ Xs ) )
= X ) )
& ( ( Xs != nil_set_a )
=> ( ( last_set_a @ ( cons_set_a @ X @ Xs ) )
= ( last_set_a @ Xs ) ) ) ) ).
% last.simps
thf(fact_602_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_603_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_604_List_Oinsert__def,axiom,
( insert7736115120964043331od_a_a
= ( ^ [X3: product_prod_a_a,Xs5: list_P1396940483166286381od_a_a] : ( if_lis931442767461590515od_a_a @ ( member1426531477525435216od_a_a @ X3 @ ( set_Product_prod_a_a2 @ Xs5 ) ) @ Xs5 @ ( cons_P7316939126706565853od_a_a @ X3 @ Xs5 ) ) ) ) ).
% List.insert_def
thf(fact_605_List_Oinsert__def,axiom,
( insert_nat
= ( ^ [X3: nat,Xs5: list_nat] : ( if_list_nat @ ( member_nat @ X3 @ ( set_nat2 @ Xs5 ) ) @ Xs5 @ ( cons_nat @ X3 @ Xs5 ) ) ) ) ).
% List.insert_def
thf(fact_606_List_Oinsert__def,axiom,
( insert_a
= ( ^ [X3: a,Xs5: list_a] : ( if_list_a @ ( member_a @ X3 @ ( set_a2 @ Xs5 ) ) @ Xs5 @ ( cons_a @ X3 @ Xs5 ) ) ) ) ).
% List.insert_def
thf(fact_607_List_Oinsert__def,axiom,
( insert_set_a
= ( ^ [X3: set_a,Xs5: list_set_a] : ( if_list_set_a @ ( member_set_a @ X3 @ ( set_set_a2 @ Xs5 ) ) @ Xs5 @ ( cons_set_a @ X3 @ Xs5 ) ) ) ) ).
% List.insert_def
thf(fact_608_order__antisym__conv,axiom,
! [Y: set_Product_prod_a_a,X: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ Y @ X )
=> ( ( ord_le746702958409616551od_a_a @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_609_order__antisym__conv,axiom,
! [Y: set_list_a,X: set_list_a] :
( ( ord_le8861187494160871172list_a @ Y @ X )
=> ( ( ord_le8861187494160871172list_a @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_610_order__antisym__conv,axiom,
! [Y: set_set_a,X: set_set_a] :
( ( ord_le3724670747650509150_set_a @ Y @ X )
=> ( ( ord_le3724670747650509150_set_a @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_611_order__antisym__conv,axiom,
! [Y: set_a,X: set_a] :
( ( ord_less_eq_set_a @ Y @ X )
=> ( ( ord_less_eq_set_a @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_612_order__antisym__conv,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ( ( ord_less_eq_nat @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_613_linorder__le__cases,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_eq_nat @ X @ Y )
=> ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_le_cases
thf(fact_614_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,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_615_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,Y3: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_616_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,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_617_ord__le__eq__subst,axiom,
! [A: set_list_a,B: set_list_a,F: set_list_a > nat,C: nat] :
( ( ord_le8861187494160871172list_a @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: set_list_a,Y3: set_list_a] :
( ( ord_le8861187494160871172list_a @ X4 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_618_ord__le__eq__subst,axiom,
! [A: set_set_a,B: set_set_a,F: set_set_a > nat,C: nat] :
( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: set_set_a,Y3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X4 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_619_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,Y3: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_620_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > set_list_a,C: set_list_a] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ord_le8861187494160871172list_a @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_le8861187494160871172list_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_621_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > set_set_a,C: set_set_a] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ord_le3724670747650509150_set_a @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_le3724670747650509150_set_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_622_ord__le__eq__subst,axiom,
! [A: set_Product_prod_a_a,B: set_Product_prod_a_a,F: set_Product_prod_a_a > nat,C: nat] :
( ( ord_le746702958409616551od_a_a @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: set_Product_prod_a_a,Y3: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ X4 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_623_ord__le__eq__subst,axiom,
! [A: set_list_a,B: set_list_a,F: set_list_a > set_a,C: set_a] :
( ( ord_le8861187494160871172list_a @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: set_list_a,Y3: set_list_a] :
( ( ord_le8861187494160871172list_a @ X4 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_624_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,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_625_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,Y3: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_626_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,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_627_ord__eq__le__subst,axiom,
! [A: nat,F: set_list_a > nat,B: set_list_a,C: set_list_a] :
( ( A
= ( F @ B ) )
=> ( ( ord_le8861187494160871172list_a @ B @ C )
=> ( ! [X4: set_list_a,Y3: set_list_a] :
( ( ord_le8861187494160871172list_a @ X4 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_628_ord__eq__le__subst,axiom,
! [A: nat,F: set_set_a > nat,B: set_set_a,C: set_set_a] :
( ( A
= ( F @ B ) )
=> ( ( ord_le3724670747650509150_set_a @ B @ C )
=> ( ! [X4: set_set_a,Y3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X4 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_629_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,Y3: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_630_ord__eq__le__subst,axiom,
! [A: set_list_a,F: nat > set_list_a,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ord_le8861187494160871172list_a @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_le8861187494160871172list_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_631_ord__eq__le__subst,axiom,
! [A: set_set_a,F: nat > set_set_a,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ord_le3724670747650509150_set_a @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_le3724670747650509150_set_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_632_ord__eq__le__subst,axiom,
! [A: nat,F: set_Product_prod_a_a > nat,B: set_Product_prod_a_a,C: set_Product_prod_a_a] :
( ( A
= ( F @ B ) )
=> ( ( ord_le746702958409616551od_a_a @ B @ C )
=> ( ! [X4: set_Product_prod_a_a,Y3: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ X4 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_633_ord__eq__le__subst,axiom,
! [A: set_a,F: set_list_a > set_a,B: set_list_a,C: set_list_a] :
( ( A
= ( F @ B ) )
=> ( ( ord_le8861187494160871172list_a @ B @ C )
=> ( ! [X4: set_list_a,Y3: set_list_a] :
( ( ord_le8861187494160871172list_a @ X4 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_634_linorder__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
| ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_linear
thf(fact_635_order__eq__refl,axiom,
! [X: set_Product_prod_a_a,Y: set_Product_prod_a_a] :
( ( X = Y )
=> ( ord_le746702958409616551od_a_a @ X @ Y ) ) ).
% order_eq_refl
thf(fact_636_order__eq__refl,axiom,
! [X: set_list_a,Y: set_list_a] :
( ( X = Y )
=> ( ord_le8861187494160871172list_a @ X @ Y ) ) ).
% order_eq_refl
thf(fact_637_order__eq__refl,axiom,
! [X: set_set_a,Y: set_set_a] :
( ( X = Y )
=> ( ord_le3724670747650509150_set_a @ X @ Y ) ) ).
% order_eq_refl
thf(fact_638_order__eq__refl,axiom,
! [X: set_a,Y: set_a] :
( ( X = Y )
=> ( ord_less_eq_set_a @ X @ Y ) ) ).
% order_eq_refl
thf(fact_639_order__eq__refl,axiom,
! [X: nat,Y: nat] :
( ( X = Y )
=> ( ord_less_eq_nat @ X @ Y ) ) ).
% order_eq_refl
thf(fact_640_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,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_641_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,Y3: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_642_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,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_643_order__subst2,axiom,
! [A: set_list_a,B: set_list_a,F: set_list_a > nat,C: nat] :
( ( ord_le8861187494160871172list_a @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X4: set_list_a,Y3: set_list_a] :
( ( ord_le8861187494160871172list_a @ X4 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_644_order__subst2,axiom,
! [A: set_set_a,B: set_set_a,F: set_set_a > nat,C: nat] :
( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X4: set_set_a,Y3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X4 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_645_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,Y3: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_646_order__subst2,axiom,
! [A: nat,B: nat,F: nat > set_list_a,C: set_list_a] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_le8861187494160871172list_a @ ( F @ B ) @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ord_le8861187494160871172list_a @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_le8861187494160871172list_a @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_647_order__subst2,axiom,
! [A: nat,B: nat,F: nat > set_set_a,C: set_set_a] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_le3724670747650509150_set_a @ ( F @ B ) @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ord_le3724670747650509150_set_a @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_le3724670747650509150_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_648_order__subst2,axiom,
! [A: set_Product_prod_a_a,B: set_Product_prod_a_a,F: set_Product_prod_a_a > nat,C: nat] :
( ( ord_le746702958409616551od_a_a @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X4: set_Product_prod_a_a,Y3: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ X4 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_649_order__subst2,axiom,
! [A: set_list_a,B: set_list_a,F: set_list_a > set_a,C: set_a] :
( ( ord_le8861187494160871172list_a @ A @ B )
=> ( ( ord_less_eq_set_a @ ( F @ B ) @ C )
=> ( ! [X4: set_list_a,Y3: set_list_a] :
( ( ord_le8861187494160871172list_a @ X4 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_650_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,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_651_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,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_652_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,Y3: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_653_order__subst1,axiom,
! [A: set_list_a,F: nat > set_list_a,B: nat,C: nat] :
( ( ord_le8861187494160871172list_a @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ord_le8861187494160871172list_a @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_le8861187494160871172list_a @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_654_order__subst1,axiom,
! [A: set_set_a,F: nat > set_set_a,B: nat,C: nat] :
( ( ord_le3724670747650509150_set_a @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ord_le3724670747650509150_set_a @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_le3724670747650509150_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_655_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,Y3: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_656_order__subst1,axiom,
! [A: nat,F: set_list_a > nat,B: set_list_a,C: set_list_a] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_le8861187494160871172list_a @ B @ C )
=> ( ! [X4: set_list_a,Y3: set_list_a] :
( ( ord_le8861187494160871172list_a @ X4 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_657_order__subst1,axiom,
! [A: nat,F: set_set_a > nat,B: set_set_a,C: set_set_a] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_le3724670747650509150_set_a @ B @ C )
=> ( ! [X4: set_set_a,Y3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X4 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_658_order__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_eq_nat @ B @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ord_le746702958409616551od_a_a @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_le746702958409616551od_a_a @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_659_order__subst1,axiom,
! [A: set_list_a,F: set_a > set_list_a,B: set_a,C: set_a] :
( ( ord_le8861187494160871172list_a @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ! [X4: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y3 )
=> ( ord_le8861187494160871172list_a @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_le8861187494160871172list_a @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_660_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_661_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: set_list_a,Z: set_list_a] : ( Y5 = Z ) )
= ( ^ [A3: set_list_a,B3: set_list_a] :
( ( ord_le8861187494160871172list_a @ A3 @ B3 )
& ( ord_le8861187494160871172list_a @ B3 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_662_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_663_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_664_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_665_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_666_antisym,axiom,
! [A: set_list_a,B: set_list_a] :
( ( ord_le8861187494160871172list_a @ A @ B )
=> ( ( ord_le8861187494160871172list_a @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_667_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_668_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_669_antisym,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_670_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_671_dual__order_Otrans,axiom,
! [B: set_list_a,A: set_list_a,C: set_list_a] :
( ( ord_le8861187494160871172list_a @ B @ A )
=> ( ( ord_le8861187494160871172list_a @ C @ B )
=> ( ord_le8861187494160871172list_a @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_672_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_673_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_674_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_675_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_676_dual__order_Oantisym,axiom,
! [B: set_list_a,A: set_list_a] :
( ( ord_le8861187494160871172list_a @ B @ A )
=> ( ( ord_le8861187494160871172list_a @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_677_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_678_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_679_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_680_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_681_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: set_list_a,Z: set_list_a] : ( Y5 = Z ) )
= ( ^ [A3: set_list_a,B3: set_list_a] :
( ( ord_le8861187494160871172list_a @ B3 @ A3 )
& ( ord_le8861187494160871172list_a @ A3 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_682_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_683_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_684_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_685_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_686_order__trans,axiom,
! [X: set_Product_prod_a_a,Y: set_Product_prod_a_a,Z3: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ X @ Y )
=> ( ( ord_le746702958409616551od_a_a @ Y @ Z3 )
=> ( ord_le746702958409616551od_a_a @ X @ Z3 ) ) ) ).
% order_trans
thf(fact_687_order__trans,axiom,
! [X: set_list_a,Y: set_list_a,Z3: set_list_a] :
( ( ord_le8861187494160871172list_a @ X @ Y )
=> ( ( ord_le8861187494160871172list_a @ Y @ Z3 )
=> ( ord_le8861187494160871172list_a @ X @ Z3 ) ) ) ).
% order_trans
thf(fact_688_order__trans,axiom,
! [X: set_set_a,Y: set_set_a,Z3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X @ Y )
=> ( ( ord_le3724670747650509150_set_a @ Y @ Z3 )
=> ( ord_le3724670747650509150_set_a @ X @ Z3 ) ) ) ).
% order_trans
thf(fact_689_order__trans,axiom,
! [X: set_a,Y: set_a,Z3: set_a] :
( ( ord_less_eq_set_a @ X @ Y )
=> ( ( ord_less_eq_set_a @ Y @ Z3 )
=> ( ord_less_eq_set_a @ X @ Z3 ) ) ) ).
% order_trans
thf(fact_690_order__trans,axiom,
! [X: nat,Y: nat,Z3: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z3 )
=> ( ord_less_eq_nat @ X @ Z3 ) ) ) ).
% order_trans
thf(fact_691_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_692_order_Otrans,axiom,
! [A: set_list_a,B: set_list_a,C: set_list_a] :
( ( ord_le8861187494160871172list_a @ A @ B )
=> ( ( ord_le8861187494160871172list_a @ B @ C )
=> ( ord_le8861187494160871172list_a @ A @ C ) ) ) ).
% order.trans
thf(fact_693_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_694_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_695_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_696_order__antisym,axiom,
! [X: set_Product_prod_a_a,Y: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ X @ Y )
=> ( ( ord_le746702958409616551od_a_a @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_697_order__antisym,axiom,
! [X: set_list_a,Y: set_list_a] :
( ( ord_le8861187494160871172list_a @ X @ Y )
=> ( ( ord_le8861187494160871172list_a @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_698_order__antisym,axiom,
! [X: set_set_a,Y: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X @ Y )
=> ( ( ord_le3724670747650509150_set_a @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_699_order__antisym,axiom,
! [X: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X @ Y )
=> ( ( ord_less_eq_set_a @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_700_order__antisym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_701_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_702_ord__le__eq__trans,axiom,
! [A: set_list_a,B: set_list_a,C: set_list_a] :
( ( ord_le8861187494160871172list_a @ A @ B )
=> ( ( B = C )
=> ( ord_le8861187494160871172list_a @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_703_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_704_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_705_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_706_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_707_ord__eq__le__trans,axiom,
! [A: set_list_a,B: set_list_a,C: set_list_a] :
( ( A = B )
=> ( ( ord_le8861187494160871172list_a @ B @ C )
=> ( ord_le8861187494160871172list_a @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_708_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_709_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_710_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_711_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: set_Product_prod_a_a,Z: set_Product_prod_a_a] : ( Y5 = Z ) )
= ( ^ [X3: set_Product_prod_a_a,Y4: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ X3 @ Y4 )
& ( ord_le746702958409616551od_a_a @ Y4 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_712_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: set_list_a,Z: set_list_a] : ( Y5 = Z ) )
= ( ^ [X3: set_list_a,Y4: set_list_a] :
( ( ord_le8861187494160871172list_a @ X3 @ Y4 )
& ( ord_le8861187494160871172list_a @ Y4 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_713_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: set_set_a,Z: set_set_a] : ( Y5 = Z ) )
= ( ^ [X3: set_set_a,Y4: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X3 @ Y4 )
& ( ord_le3724670747650509150_set_a @ Y4 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_714_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: set_a,Z: set_a] : ( Y5 = Z ) )
= ( ^ [X3: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y4 )
& ( ord_less_eq_set_a @ Y4 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_715_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: nat,Z: nat] : ( Y5 = Z ) )
= ( ^ [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
& ( ord_less_eq_nat @ Y4 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_716_le__cases3,axiom,
! [X: nat,Y: nat,Z3: nat] :
( ( ( ord_less_eq_nat @ X @ Y )
=> ~ ( ord_less_eq_nat @ Y @ Z3 ) )
=> ( ( ( ord_less_eq_nat @ Y @ X )
=> ~ ( ord_less_eq_nat @ X @ Z3 ) )
=> ( ( ( ord_less_eq_nat @ X @ Z3 )
=> ~ ( ord_less_eq_nat @ Z3 @ Y ) )
=> ( ( ( ord_less_eq_nat @ Z3 @ Y )
=> ~ ( ord_less_eq_nat @ Y @ X ) )
=> ( ( ( ord_less_eq_nat @ Y @ Z3 )
=> ~ ( ord_less_eq_nat @ Z3 @ X ) )
=> ~ ( ( ord_less_eq_nat @ Z3 @ X )
=> ~ ( ord_less_eq_nat @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_717_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_718_not__distinct__decomp,axiom,
! [Ws: list_P5740962349794459853et_a_a] :
( ~ ( distin7251654435778379584et_a_a @ Ws )
=> ? [Xs3: list_P5740962349794459853et_a_a,Ys3: list_P5740962349794459853et_a_a,Zs2: list_P5740962349794459853et_a_a,Y3: product_prod_set_a_a] :
( Ws
= ( append8719822696595618658et_a_a @ Xs3 @ ( append8719822696595618658et_a_a @ ( cons_P6517022146024809853et_a_a @ Y3 @ nil_Pr7933670750051473869et_a_a ) @ ( append8719822696595618658et_a_a @ Ys3 @ ( append8719822696595618658et_a_a @ ( cons_P6517022146024809853et_a_a @ Y3 @ nil_Pr7933670750051473869et_a_a ) @ Zs2 ) ) ) ) ) ) ).
% not_distinct_decomp
thf(fact_719_not__distinct__decomp,axiom,
! [Ws: list_P494665323402860429_set_a] :
( ~ ( distin201903879741355520_set_a @ Ws )
=> ? [Xs3: list_P494665323402860429_set_a,Ys3: list_P494665323402860429_set_a,Zs2: list_P494665323402860429_set_a,Y3: product_prod_a_set_a] :
( Ws
= ( append1670072140558594594_set_a @ Xs3 @ ( append1670072140558594594_set_a @ ( cons_P8690643626842561597_set_a @ Y3 @ nil_Pr883920194014449805_set_a ) @ ( append1670072140558594594_set_a @ Ys3 @ ( append1670072140558594594_set_a @ ( cons_P8690643626842561597_set_a @ Y3 @ nil_Pr883920194014449805_set_a ) @ Zs2 ) ) ) ) ) ) ).
% not_distinct_decomp
thf(fact_720_not__distinct__decomp,axiom,
! [Ws: list_P1396940483166286381od_a_a] :
( ~ ( distin132333870042060960od_a_a @ Ws )
=> ? [Xs3: list_P1396940483166286381od_a_a,Ys3: list_P1396940483166286381od_a_a,Zs2: list_P1396940483166286381od_a_a,Y3: product_prod_a_a] :
( Ws
= ( append5335208819046833346od_a_a @ Xs3 @ ( append5335208819046833346od_a_a @ ( cons_P7316939126706565853od_a_a @ Y3 @ nil_Product_prod_a_a ) @ ( append5335208819046833346od_a_a @ Ys3 @ ( append5335208819046833346od_a_a @ ( cons_P7316939126706565853od_a_a @ Y3 @ nil_Product_prod_a_a ) @ Zs2 ) ) ) ) ) ) ).
% not_distinct_decomp
thf(fact_721_not__distinct__decomp,axiom,
! [Ws: list_list_set_a] :
( ~ ( distinct_list_set_a @ Ws )
=> ? [Xs3: list_list_set_a,Ys3: list_list_set_a,Zs2: list_list_set_a,Y3: list_set_a] :
( Ws
= ( append_list_set_a @ Xs3 @ ( append_list_set_a @ ( cons_list_set_a @ Y3 @ nil_list_set_a ) @ ( append_list_set_a @ Ys3 @ ( append_list_set_a @ ( cons_list_set_a @ Y3 @ nil_list_set_a ) @ Zs2 ) ) ) ) ) ) ).
% not_distinct_decomp
thf(fact_722_not__distinct__decomp,axiom,
! [Ws: list_list_a] :
( ~ ( distinct_list_a @ Ws )
=> ? [Xs3: list_list_a,Ys3: list_list_a,Zs2: list_list_a,Y3: list_a] :
( Ws
= ( append_list_a @ Xs3 @ ( append_list_a @ ( cons_list_a @ Y3 @ nil_list_a ) @ ( append_list_a @ Ys3 @ ( append_list_a @ ( cons_list_a @ Y3 @ nil_list_a ) @ Zs2 ) ) ) ) ) ) ).
% not_distinct_decomp
thf(fact_723_not__distinct__decomp,axiom,
! [Ws: list_a] :
( ~ ( distinct_a @ Ws )
=> ? [Xs3: list_a,Ys3: list_a,Zs2: list_a,Y3: a] :
( Ws
= ( append_a @ Xs3 @ ( append_a @ ( cons_a @ Y3 @ nil_a ) @ ( append_a @ Ys3 @ ( append_a @ ( cons_a @ Y3 @ nil_a ) @ Zs2 ) ) ) ) ) ) ).
% not_distinct_decomp
thf(fact_724_not__distinct__decomp,axiom,
! [Ws: list_set_a] :
( ~ ( distinct_set_a @ Ws )
=> ? [Xs3: list_set_a,Ys3: list_set_a,Zs2: list_set_a,Y3: set_a] :
( Ws
= ( append_set_a @ Xs3 @ ( append_set_a @ ( cons_set_a @ Y3 @ nil_set_a ) @ ( append_set_a @ Ys3 @ ( append_set_a @ ( cons_set_a @ Y3 @ nil_set_a ) @ Zs2 ) ) ) ) ) ) ).
% not_distinct_decomp
thf(fact_725_not__distinct__conv__prefix,axiom,
! [As: list_nat] :
( ( ~ ( distinct_nat @ As ) )
= ( ? [Xs5: list_nat,Y4: nat,Ys4: list_nat] :
( ( member_nat @ Y4 @ ( set_nat2 @ Xs5 ) )
& ( distinct_nat @ Xs5 )
& ( As
= ( append_nat @ Xs5 @ ( cons_nat @ Y4 @ Ys4 ) ) ) ) ) ) ).
% not_distinct_conv_prefix
thf(fact_726_not__distinct__conv__prefix,axiom,
! [As: list_P5740962349794459853et_a_a] :
( ( ~ ( distin7251654435778379584et_a_a @ As ) )
= ( ? [Xs5: list_P5740962349794459853et_a_a,Y4: product_prod_set_a_a,Ys4: list_P5740962349794459853et_a_a] :
( ( member2598349401703774704et_a_a @ Y4 @ ( set_Pr9190486990633797724et_a_a @ Xs5 ) )
& ( distin7251654435778379584et_a_a @ Xs5 )
& ( As
= ( append8719822696595618658et_a_a @ Xs5 @ ( cons_P6517022146024809853et_a_a @ Y4 @ Ys4 ) ) ) ) ) ) ).
% not_distinct_conv_prefix
thf(fact_727_not__distinct__conv__prefix,axiom,
! [As: list_P494665323402860429_set_a] :
( ( ~ ( distin201903879741355520_set_a @ As ) )
= ( ? [Xs5: list_P494665323402860429_set_a,Y4: product_prod_a_set_a,Ys4: list_P494665323402860429_set_a] :
( ( member4771970882521526448_set_a @ Y4 @ ( set_Pr2140736434596773660_set_a @ Xs5 ) )
& ( distin201903879741355520_set_a @ Xs5 )
& ( As
= ( append1670072140558594594_set_a @ Xs5 @ ( cons_P8690643626842561597_set_a @ Y4 @ Ys4 ) ) ) ) ) ) ).
% not_distinct_conv_prefix
thf(fact_728_not__distinct__conv__prefix,axiom,
! [As: list_P1396940483166286381od_a_a] :
( ( ~ ( distin132333870042060960od_a_a @ As ) )
= ( ? [Xs5: list_P1396940483166286381od_a_a,Y4: product_prod_a_a,Ys4: list_P1396940483166286381od_a_a] :
( ( member1426531477525435216od_a_a @ Y4 @ ( set_Product_prod_a_a2 @ Xs5 ) )
& ( distin132333870042060960od_a_a @ Xs5 )
& ( As
= ( append5335208819046833346od_a_a @ Xs5 @ ( cons_P7316939126706565853od_a_a @ Y4 @ Ys4 ) ) ) ) ) ) ).
% not_distinct_conv_prefix
thf(fact_729_not__distinct__conv__prefix,axiom,
! [As: list_list_set_a] :
( ( ~ ( distinct_list_set_a @ As ) )
= ( ? [Xs5: list_list_set_a,Y4: list_set_a,Ys4: list_list_set_a] :
( ( member_list_set_a @ Y4 @ ( set_list_set_a2 @ Xs5 ) )
& ( distinct_list_set_a @ Xs5 )
& ( As
= ( append_list_set_a @ Xs5 @ ( cons_list_set_a @ Y4 @ Ys4 ) ) ) ) ) ) ).
% not_distinct_conv_prefix
thf(fact_730_not__distinct__conv__prefix,axiom,
! [As: list_list_a] :
( ( ~ ( distinct_list_a @ As ) )
= ( ? [Xs5: list_list_a,Y4: list_a,Ys4: list_list_a] :
( ( member_list_a @ Y4 @ ( set_list_a2 @ Xs5 ) )
& ( distinct_list_a @ Xs5 )
& ( As
= ( append_list_a @ Xs5 @ ( cons_list_a @ Y4 @ Ys4 ) ) ) ) ) ) ).
% not_distinct_conv_prefix
thf(fact_731_not__distinct__conv__prefix,axiom,
! [As: list_a] :
( ( ~ ( distinct_a @ As ) )
= ( ? [Xs5: list_a,Y4: a,Ys4: list_a] :
( ( member_a @ Y4 @ ( set_a2 @ Xs5 ) )
& ( distinct_a @ Xs5 )
& ( As
= ( append_a @ Xs5 @ ( cons_a @ Y4 @ Ys4 ) ) ) ) ) ) ).
% not_distinct_conv_prefix
thf(fact_732_not__distinct__conv__prefix,axiom,
! [As: list_set_a] :
( ( ~ ( distinct_set_a @ As ) )
= ( ? [Xs5: list_set_a,Y4: set_a,Ys4: list_set_a] :
( ( member_set_a @ Y4 @ ( set_set_a2 @ Xs5 ) )
& ( distinct_set_a @ Xs5 )
& ( As
= ( append_set_a @ Xs5 @ ( cons_set_a @ Y4 @ Ys4 ) ) ) ) ) ) ).
% not_distinct_conv_prefix
thf(fact_733_rev_Osimps_I2_J,axiom,
! [X: a,Xs: list_a] :
( ( rev_a @ ( cons_a @ X @ Xs ) )
= ( append_a @ ( rev_a @ Xs ) @ ( cons_a @ X @ nil_a ) ) ) ).
% rev.simps(2)
thf(fact_734_rev_Osimps_I2_J,axiom,
! [X: set_a,Xs: list_set_a] :
( ( rev_set_a @ ( cons_set_a @ X @ Xs ) )
= ( append_set_a @ ( rev_set_a @ Xs ) @ ( cons_set_a @ X @ nil_set_a ) ) ) ).
% rev.simps(2)
thf(fact_735_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 ) )
= ( ! [X3: product_prod_a_a] :
( ( P2 @ X3 )
=> ( Q @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_736_Collect__mono__iff,axiom,
! [P2: list_a > $o,Q: list_a > $o] :
( ( ord_le8861187494160871172list_a @ ( collect_list_a @ P2 ) @ ( collect_list_a @ Q ) )
= ( ! [X3: list_a] :
( ( P2 @ X3 )
=> ( Q @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_737_Collect__mono__iff,axiom,
! [P2: set_a > $o,Q: set_a > $o] :
( ( ord_le3724670747650509150_set_a @ ( collect_set_a @ P2 ) @ ( collect_set_a @ Q ) )
= ( ! [X3: set_a] :
( ( P2 @ X3 )
=> ( Q @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_738_Collect__mono__iff,axiom,
! [P2: a > $o,Q: a > $o] :
( ( ord_less_eq_set_a @ ( collect_a @ P2 ) @ ( collect_a @ Q ) )
= ( ! [X3: a] :
( ( P2 @ X3 )
=> ( Q @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_739_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_740_set__eq__subset,axiom,
( ( ^ [Y5: set_list_a,Z: set_list_a] : ( Y5 = Z ) )
= ( ^ [A5: set_list_a,B5: set_list_a] :
( ( ord_le8861187494160871172list_a @ A5 @ B5 )
& ( ord_le8861187494160871172list_a @ B5 @ A5 ) ) ) ) ).
% set_eq_subset
thf(fact_741_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_742_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_743_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_744_subset__trans,axiom,
! [A2: set_list_a,B2: set_list_a,C2: set_list_a] :
( ( ord_le8861187494160871172list_a @ A2 @ B2 )
=> ( ( ord_le8861187494160871172list_a @ B2 @ C2 )
=> ( ord_le8861187494160871172list_a @ A2 @ C2 ) ) ) ).
% subset_trans
thf(fact_745_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_746_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_747_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_748_Collect__mono,axiom,
! [P2: list_a > $o,Q: list_a > $o] :
( ! [X4: list_a] :
( ( P2 @ X4 )
=> ( Q @ X4 ) )
=> ( ord_le8861187494160871172list_a @ ( collect_list_a @ P2 ) @ ( collect_list_a @ Q ) ) ) ).
% Collect_mono
thf(fact_749_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_750_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_751_subset__refl,axiom,
! [A2: set_Product_prod_a_a] : ( ord_le746702958409616551od_a_a @ A2 @ A2 ) ).
% subset_refl
thf(fact_752_subset__refl,axiom,
! [A2: set_list_a] : ( ord_le8861187494160871172list_a @ A2 @ A2 ) ).
% subset_refl
thf(fact_753_subset__refl,axiom,
! [A2: set_set_a] : ( ord_le3724670747650509150_set_a @ A2 @ A2 ) ).
% subset_refl
thf(fact_754_subset__refl,axiom,
! [A2: set_a] : ( ord_less_eq_set_a @ A2 @ A2 ) ).
% subset_refl
thf(fact_755_subset__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [A5: set_nat,B5: set_nat] :
! [T: nat] :
( ( member_nat @ T @ A5 )
=> ( member_nat @ T @ B5 ) ) ) ) ).
% subset_iff
thf(fact_756_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_757_subset__iff,axiom,
( ord_le8861187494160871172list_a
= ( ^ [A5: set_list_a,B5: set_list_a] :
! [T: list_a] :
( ( member_list_a @ T @ A5 )
=> ( member_list_a @ T @ B5 ) ) ) ) ).
% subset_iff
thf(fact_758_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_759_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_760_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_761_equalityD2,axiom,
! [A2: set_list_a,B2: set_list_a] :
( ( A2 = B2 )
=> ( ord_le8861187494160871172list_a @ B2 @ A2 ) ) ).
% equalityD2
thf(fact_762_equalityD2,axiom,
! [A2: set_set_a,B2: set_set_a] :
( ( A2 = B2 )
=> ( ord_le3724670747650509150_set_a @ B2 @ A2 ) ) ).
% equalityD2
thf(fact_763_equalityD2,axiom,
! [A2: set_a,B2: set_a] :
( ( A2 = B2 )
=> ( ord_less_eq_set_a @ B2 @ A2 ) ) ).
% equalityD2
thf(fact_764_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_765_equalityD1,axiom,
! [A2: set_list_a,B2: set_list_a] :
( ( A2 = B2 )
=> ( ord_le8861187494160871172list_a @ A2 @ B2 ) ) ).
% equalityD1
thf(fact_766_equalityD1,axiom,
! [A2: set_set_a,B2: set_set_a] :
( ( A2 = B2 )
=> ( ord_le3724670747650509150_set_a @ A2 @ B2 ) ) ).
% equalityD1
thf(fact_767_equalityD1,axiom,
! [A2: set_a,B2: set_a] :
( ( A2 = B2 )
=> ( ord_less_eq_set_a @ A2 @ B2 ) ) ).
% equalityD1
thf(fact_768_subset__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A5: set_nat,B5: set_nat] :
! [X3: nat] :
( ( member_nat @ X3 @ A5 )
=> ( member_nat @ X3 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_769_subset__eq,axiom,
( ord_le746702958409616551od_a_a
= ( ^ [A5: set_Product_prod_a_a,B5: set_Product_prod_a_a] :
! [X3: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X3 @ A5 )
=> ( member1426531477525435216od_a_a @ X3 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_770_subset__eq,axiom,
( ord_le8861187494160871172list_a
= ( ^ [A5: set_list_a,B5: set_list_a] :
! [X3: list_a] :
( ( member_list_a @ X3 @ A5 )
=> ( member_list_a @ X3 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_771_subset__eq,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [A5: set_set_a,B5: set_set_a] :
! [X3: set_a] :
( ( member_set_a @ X3 @ A5 )
=> ( member_set_a @ X3 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_772_subset__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A5: set_a,B5: set_a] :
! [X3: a] :
( ( member_a @ X3 @ A5 )
=> ( member_a @ X3 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_773_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_774_equalityE,axiom,
! [A2: set_list_a,B2: set_list_a] :
( ( A2 = B2 )
=> ~ ( ( ord_le8861187494160871172list_a @ A2 @ B2 )
=> ~ ( ord_le8861187494160871172list_a @ B2 @ A2 ) ) ) ).
% equalityE
thf(fact_775_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_776_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_777_subsetD,axiom,
! [A2: set_nat,B2: set_nat,C: nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( member_nat @ C @ A2 )
=> ( member_nat @ C @ B2 ) ) ) ).
% subsetD
thf(fact_778_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_779_subsetD,axiom,
! [A2: set_list_a,B2: set_list_a,C: list_a] :
( ( ord_le8861187494160871172list_a @ A2 @ B2 )
=> ( ( member_list_a @ C @ A2 )
=> ( member_list_a @ C @ B2 ) ) ) ).
% subsetD
thf(fact_780_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_781_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_782_in__mono,axiom,
! [A2: set_nat,B2: set_nat,X: nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( member_nat @ X @ A2 )
=> ( member_nat @ X @ B2 ) ) ) ).
% in_mono
thf(fact_783_in__mono,axiom,
! [A2: set_Product_prod_a_a,B2: set_Product_prod_a_a,X: product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A2 @ B2 )
=> ( ( member1426531477525435216od_a_a @ X @ A2 )
=> ( member1426531477525435216od_a_a @ X @ B2 ) ) ) ).
% in_mono
thf(fact_784_in__mono,axiom,
! [A2: set_list_a,B2: set_list_a,X: list_a] :
( ( ord_le8861187494160871172list_a @ A2 @ B2 )
=> ( ( member_list_a @ X @ A2 )
=> ( member_list_a @ X @ B2 ) ) ) ).
% in_mono
thf(fact_785_in__mono,axiom,
! [A2: set_set_a,B2: set_set_a,X: set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B2 )
=> ( ( member_set_a @ X @ A2 )
=> ( member_set_a @ X @ B2 ) ) ) ).
% in_mono
thf(fact_786_in__mono,axiom,
! [A2: set_a,B2: set_a,X: a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( member_a @ X @ A2 )
=> ( member_a @ X @ B2 ) ) ) ).
% in_mono
thf(fact_787_ulgraph_Ois__walk__singleton,axiom,
! [Vertices: set_Product_prod_a_a,Edges: set_se5735800977113168103od_a_a,U: product_prod_a_a] :
( ( undire4585262585102564309od_a_a @ Vertices @ Edges )
=> ( ( member1426531477525435216od_a_a @ U @ Vertices )
=> ( undire3162072421265123221od_a_a @ Vertices @ Edges @ ( cons_P7316939126706565853od_a_a @ U @ nil_Product_prod_a_a ) ) ) ) ).
% ulgraph.is_walk_singleton
thf(fact_788_ulgraph_Ois__walk__singleton,axiom,
! [Vertices: set_nat,Edges: set_set_nat,U: nat] :
( ( undire3269267262472140706ph_nat @ Vertices @ Edges )
=> ( ( member_nat @ U @ Vertices )
=> ( undire5745680128780950498lk_nat @ Vertices @ Edges @ ( cons_nat @ U @ nil_nat ) ) ) ) ).
% ulgraph.is_walk_singleton
thf(fact_789_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_790_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_791_ulgraph_Ois__walk__drop__hd,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,Ys: list_set_a,Y: set_a] :
( ( undire6886684016831807756_set_a @ Vertices @ Edges )
=> ( ( Ys != nil_set_a )
=> ( ( undire3014741414213135564_set_a @ Vertices @ Edges @ ( cons_set_a @ Y @ Ys ) )
=> ( undire3014741414213135564_set_a @ Vertices @ Edges @ Ys ) ) ) ) ).
% ulgraph.is_walk_drop_hd
thf(fact_792_ulgraph_Ois__walk__drop__hd,axiom,
! [Vertices: set_a,Edges: set_set_a,Ys: list_a,Y: a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( Ys != nil_a )
=> ( ( undire6133010728901294956walk_a @ Vertices @ Edges @ ( cons_a @ Y @ Ys ) )
=> ( undire6133010728901294956walk_a @ Vertices @ Edges @ Ys ) ) ) ) ).
% ulgraph.is_walk_drop_hd
thf(fact_793_list_Oexhaust__sel,axiom,
! [List: list_a] :
( ( List != nil_a )
=> ( List
= ( cons_a @ ( hd_a @ List ) @ ( tl_a @ List ) ) ) ) ).
% list.exhaust_sel
thf(fact_794_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_795_rotate1_Osimps_I2_J,axiom,
! [X: a,Xs: list_a] :
( ( rotate1_a @ ( cons_a @ X @ Xs ) )
= ( append_a @ Xs @ ( cons_a @ X @ nil_a ) ) ) ).
% rotate1.simps(2)
thf(fact_796_rotate1_Osimps_I2_J,axiom,
! [X: set_a,Xs: list_set_a] :
( ( rotate1_set_a @ ( cons_set_a @ X @ Xs ) )
= ( append_set_a @ Xs @ ( cons_set_a @ X @ nil_set_a ) ) ) ).
% rotate1.simps(2)
thf(fact_797_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_798_ulgraph_Ois__walk__decomp,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,Xs: list_set_a,Y: set_a,Ys: list_set_a,Zs: list_set_a] :
( ( undire6886684016831807756_set_a @ Vertices @ Edges )
=> ( ( undire3014741414213135564_set_a @ Vertices @ Edges @ ( append_set_a @ Xs @ ( append_set_a @ ( cons_set_a @ Y @ nil_set_a ) @ ( append_set_a @ Ys @ ( append_set_a @ ( cons_set_a @ Y @ nil_set_a ) @ Zs ) ) ) ) )
=> ( undire3014741414213135564_set_a @ Vertices @ Edges @ ( append_set_a @ Xs @ ( append_set_a @ ( cons_set_a @ Y @ nil_set_a ) @ Zs ) ) ) ) ) ).
% ulgraph.is_walk_decomp
thf(fact_799_ulgraph_Ois__walk__decomp,axiom,
! [Vertices: set_a,Edges: set_set_a,Xs: list_a,Y: a,Ys: list_a,Zs: list_a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( undire6133010728901294956walk_a @ Vertices @ Edges @ ( append_a @ Xs @ ( append_a @ ( cons_a @ Y @ nil_a ) @ ( append_a @ Ys @ ( append_a @ ( cons_a @ Y @ nil_a ) @ Zs ) ) ) ) )
=> ( undire6133010728901294956walk_a @ Vertices @ Edges @ ( append_a @ Xs @ ( append_a @ ( cons_a @ Y @ nil_a ) @ Zs ) ) ) ) ) ).
% ulgraph.is_walk_decomp
thf(fact_800_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_801_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_802_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_803_ulgraph_Ois__walk__def,axiom,
! [Vertices: set_list_a,Edges: set_set_list_a,Xs: list_list_a] :
( ( undire4488935924012268850list_a @ Vertices @ Edges )
=> ( ( undire8550186295227992306list_a @ Vertices @ Edges @ Xs )
= ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Xs ) @ Vertices )
& ( ord_le8877086941679407844list_a @ ( set_set_list_a2 @ ( undire8303882243552421012list_a @ Xs ) ) @ Edges )
& ( Xs != nil_list_a ) ) ) ) ).
% ulgraph.is_walk_def
thf(fact_804_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_805_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_806_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_807_ulgraph_Ois__walkI,axiom,
! [Vertices: set_list_a,Edges: set_set_list_a,Xs: list_list_a] :
( ( undire4488935924012268850list_a @ Vertices @ Edges )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Xs ) @ Vertices )
=> ( ( ord_le8877086941679407844list_a @ ( set_set_list_a2 @ ( undire8303882243552421012list_a @ Xs ) ) @ Edges )
=> ( ( Xs != nil_list_a )
=> ( undire8550186295227992306list_a @ Vertices @ Edges @ Xs ) ) ) ) ) ).
% ulgraph.is_walkI
thf(fact_808_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_809_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_810_edge__density__ge0,axiom,
! [X5: set_a,Y2: set_a] : ( ord_less_eq_real @ zero_zero_real @ ( undire297304480579013331sity_a @ edges @ X5 @ Y2 ) ) ).
% edge_density_ge0
thf(fact_811_edge__density__le1,axiom,
! [X5: set_a,Y2: set_a] : ( ord_less_eq_real @ ( undire297304480579013331sity_a @ edges @ X5 @ Y2 ) @ one_one_real ) ).
% edge_density_le1
thf(fact_812_the__elem__set,axiom,
! [X: a] :
( ( the_elem_a @ ( set_a2 @ ( cons_a @ X @ nil_a ) ) )
= X ) ).
% the_elem_set
thf(fact_813_the__elem__set,axiom,
! [X: set_a] :
( ( the_elem_set_a @ ( set_set_a2 @ ( cons_set_a @ X @ nil_set_a ) ) )
= X ) ).
% the_elem_set
thf(fact_814_list__set__tl,axiom,
! [X: product_prod_a_a,Xs: list_P1396940483166286381od_a_a] :
( ( member1426531477525435216od_a_a @ X @ ( set_Product_prod_a_a2 @ ( tl_Product_prod_a_a @ Xs ) ) )
=> ( member1426531477525435216od_a_a @ X @ ( set_Product_prod_a_a2 @ Xs ) ) ) ).
% list_set_tl
thf(fact_815_list__set__tl,axiom,
! [X: nat,Xs: list_nat] :
( ( member_nat @ X @ ( set_nat2 @ ( tl_nat @ Xs ) ) )
=> ( member_nat @ X @ ( set_nat2 @ Xs ) ) ) ).
% list_set_tl
thf(fact_816_list__set__tl,axiom,
! [X: a,Xs: list_a] :
( ( member_a @ X @ ( set_a2 @ ( tl_a @ Xs ) ) )
=> ( member_a @ X @ ( set_a2 @ Xs ) ) ) ).
% list_set_tl
thf(fact_817_list__set__tl,axiom,
! [X: set_a,Xs: list_set_a] :
( ( member_set_a @ X @ ( set_set_a2 @ ( tl_set_a @ Xs ) ) )
=> ( member_set_a @ X @ ( set_set_a2 @ Xs ) ) ) ).
% list_set_tl
thf(fact_818_list__exhaust3,axiom,
! [Xs: list_a] :
( ( Xs != nil_a )
=> ( ! [X4: a] :
( Xs
!= ( cons_a @ X4 @ nil_a ) )
=> ~ ! [X4: a,Y3: a,Ys3: list_a] :
( Xs
!= ( cons_a @ X4 @ ( cons_a @ Y3 @ Ys3 ) ) ) ) ) ).
% list_exhaust3
thf(fact_819_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,Y3: set_a,Ys3: list_set_a] :
( Xs
!= ( cons_set_a @ X4 @ ( cons_set_a @ Y3 @ Ys3 ) ) ) ) ) ).
% list_exhaust3
thf(fact_820_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_821_comp__sgraph_Ois__gen__path__def,axiom,
! [S: set_Pr2416559167834504103et_a_a,P: list_P5740962349794459853et_a_a] :
( ( undire8058049273645199385et_a_a @ S @ ( undire4745662417834845355et_a_a @ S ) @ P )
= ( ( undire4537806513611962933et_a_a @ S @ ( undire4745662417834845355et_a_a @ S ) @ P )
& ( ( ( distin7251654435778379584et_a_a @ ( tl_Pro2640188747214327222et_a_a @ P ) )
& ( ( hd_Pro7221231872133205426et_a_a @ P )
= ( last_P6817006942355120742et_a_a @ P ) ) )
| ( distin7251654435778379584et_a_a @ P ) ) ) ) ).
% comp_sgraph.is_gen_path_def
thf(fact_822_comp__sgraph_Ois__gen__path__def,axiom,
! [S: set_Pr6393634178297680487_set_a,P: list_P494665323402860429_set_a] :
( ( undire1008298717608175321_set_a @ S @ ( undire6919283898652597099_set_a @ S ) @ P )
= ( ( undire6711427994429714677_set_a @ S @ ( undire6919283898652597099_set_a @ S ) @ P )
& ( ( ( distin201903879741355520_set_a @ ( tl_Pro4813810228032078966_set_a @ P ) )
& ( ( hd_Pro171481316096181362_set_a @ P )
= ( last_P8990628423172872486_set_a @ P ) ) )
| ( distin201903879741355520_set_a @ P ) ) ) ) ).
% comp_sgraph.is_gen_path_def
thf(fact_823_comp__sgraph_Ois__gen__path__def,axiom,
! [S: set_Product_prod_a_a,P: list_P1396940483166286381od_a_a] :
( ( undire7585867811434966393od_a_a @ S @ ( undire6879232364018543115od_a_a @ S ) @ P )
= ( ( undire3162072421265123221od_a_a @ S @ ( undire6879232364018543115od_a_a @ S ) @ P )
& ( ( ( distin132333870042060960od_a_a @ ( tl_Product_prod_a_a @ P ) )
& ( ( hd_Product_prod_a_a @ P )
= ( last_P8790725268278465478od_a_a @ P ) ) )
| ( distin132333870042060960od_a_a @ P ) ) ) ) ).
% comp_sgraph.is_gen_path_def
thf(fact_824_comp__sgraph_Ois__gen__path__def,axiom,
! [S: set_list_set_a,P: list_list_set_a] :
( ( undire5019392814671255094_set_a @ S @ ( undire5484946175218547656_set_a @ S ) @ P )
= ( ( undire2288203741413088850_set_a @ S @ ( undire5484946175218547656_set_a @ S ) @ P )
& ( ( ( distinct_list_set_a @ ( tl_list_set_a @ P ) )
& ( ( hd_list_set_a @ P )
= ( last_list_set_a @ P ) ) )
| ( distinct_list_set_a @ P ) ) ) ) ).
% comp_sgraph.is_gen_path_def
thf(fact_825_comp__sgraph_Ois__gen__path__def,axiom,
! [S: set_list_a,P: list_list_a] :
( ( undire8568094650444222678list_a @ S @ ( undire517252296021605992list_a @ S ) @ P )
= ( ( undire8550186295227992306list_a @ S @ ( undire517252296021605992list_a @ S ) @ P )
& ( ( ( distinct_list_a @ ( tl_list_a @ P ) )
& ( ( hd_list_a @ P )
= ( last_list_a @ P ) ) )
| ( distinct_list_a @ P ) ) ) ) ).
% comp_sgraph.is_gen_path_def
thf(fact_826_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_827_all__edges__between__mono2,axiom,
! [Y2: set_a,Z4: set_a,X5: set_a] :
( ( ord_less_eq_set_a @ Y2 @ Z4 )
=> ( ord_le746702958409616551od_a_a @ ( undire8383842906760478443ween_a @ edges @ X5 @ Y2 ) @ ( undire8383842906760478443ween_a @ edges @ X5 @ Z4 ) ) ) ).
% all_edges_between_mono2
thf(fact_828_all__edges__between__mono1,axiom,
! [Y2: set_a,Z4: set_a,X5: set_a] :
( ( ord_less_eq_set_a @ Y2 @ Z4 )
=> ( ord_le746702958409616551od_a_a @ ( undire8383842906760478443ween_a @ edges @ Y2 @ X5 ) @ ( undire8383842906760478443ween_a @ edges @ Z4 @ X5 ) ) ) ).
% all_edges_between_mono1
thf(fact_829_comp__sgraph_Oall__edges__between__mono2,axiom,
! [Y2: set_Product_prod_a_a,Z4: set_Product_prod_a_a,S: set_Product_prod_a_a,X5: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ Y2 @ Z4 )
=> ( ord_le3469131294019144807od_a_a @ ( undire4032395788819567636od_a_a @ ( undire6879232364018543115od_a_a @ S ) @ X5 @ Y2 ) @ ( undire4032395788819567636od_a_a @ ( undire6879232364018543115od_a_a @ S ) @ X5 @ Z4 ) ) ) ).
% comp_sgraph.all_edges_between_mono2
thf(fact_830_comp__sgraph_Oall__edges__between__mono2,axiom,
! [Y2: set_list_a,Z4: set_list_a,S: set_list_a,X5: set_list_a] :
( ( ord_le8861187494160871172list_a @ Y2 @ Z4 )
=> ( ord_le7857023143581076903list_a @ ( undire3588171647663456497list_a @ ( undire517252296021605992list_a @ S ) @ X5 @ Y2 ) @ ( undire3588171647663456497list_a @ ( undire517252296021605992list_a @ S ) @ X5 @ Z4 ) ) ) ).
% comp_sgraph.all_edges_between_mono2
thf(fact_831_comp__sgraph_Oall__edges__between__mono2,axiom,
! [Y2: set_set_a,Z4: set_set_a,S: set_set_a,X5: set_set_a] :
( ( ord_le3724670747650509150_set_a @ Y2 @ Z4 )
=> ( ord_le8376522849517564071_set_a @ ( undire2462398226299384907_set_a @ ( undire8247866692393712962_set_a @ S ) @ X5 @ Y2 ) @ ( undire2462398226299384907_set_a @ ( undire8247866692393712962_set_a @ S ) @ X5 @ Z4 ) ) ) ).
% comp_sgraph.all_edges_between_mono2
thf(fact_832_comp__sgraph_Oall__edges__between__mono2,axiom,
! [Y2: set_a,Z4: set_a,S: set_a,X5: set_a] :
( ( ord_less_eq_set_a @ Y2 @ Z4 )
=> ( ord_le746702958409616551od_a_a @ ( undire8383842906760478443ween_a @ ( undire2918257014606996450dges_a @ S ) @ X5 @ Y2 ) @ ( undire8383842906760478443ween_a @ ( undire2918257014606996450dges_a @ S ) @ X5 @ Z4 ) ) ) ).
% comp_sgraph.all_edges_between_mono2
thf(fact_833_comp__sgraph_Oall__edges__between__mono1,axiom,
! [Y2: set_Product_prod_a_a,Z4: set_Product_prod_a_a,S: set_Product_prod_a_a,X5: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ Y2 @ Z4 )
=> ( ord_le3469131294019144807od_a_a @ ( undire4032395788819567636od_a_a @ ( undire6879232364018543115od_a_a @ S ) @ Y2 @ X5 ) @ ( undire4032395788819567636od_a_a @ ( undire6879232364018543115od_a_a @ S ) @ Z4 @ X5 ) ) ) ).
% comp_sgraph.all_edges_between_mono1
thf(fact_834_comp__sgraph_Oall__edges__between__mono1,axiom,
! [Y2: set_list_a,Z4: set_list_a,S: set_list_a,X5: set_list_a] :
( ( ord_le8861187494160871172list_a @ Y2 @ Z4 )
=> ( ord_le7857023143581076903list_a @ ( undire3588171647663456497list_a @ ( undire517252296021605992list_a @ S ) @ Y2 @ X5 ) @ ( undire3588171647663456497list_a @ ( undire517252296021605992list_a @ S ) @ Z4 @ X5 ) ) ) ).
% comp_sgraph.all_edges_between_mono1
thf(fact_835_comp__sgraph_Oall__edges__between__mono1,axiom,
! [Y2: set_set_a,Z4: set_set_a,S: set_set_a,X5: set_set_a] :
( ( ord_le3724670747650509150_set_a @ Y2 @ Z4 )
=> ( ord_le8376522849517564071_set_a @ ( undire2462398226299384907_set_a @ ( undire8247866692393712962_set_a @ S ) @ Y2 @ X5 ) @ ( undire2462398226299384907_set_a @ ( undire8247866692393712962_set_a @ S ) @ Z4 @ X5 ) ) ) ).
% comp_sgraph.all_edges_between_mono1
thf(fact_836_comp__sgraph_Oall__edges__between__mono1,axiom,
! [Y2: set_a,Z4: set_a,S: set_a,X5: set_a] :
( ( ord_less_eq_set_a @ Y2 @ Z4 )
=> ( ord_le746702958409616551od_a_a @ ( undire8383842906760478443ween_a @ ( undire2918257014606996450dges_a @ S ) @ Y2 @ X5 ) @ ( undire8383842906760478443ween_a @ ( undire2918257014606996450dges_a @ S ) @ Z4 @ X5 ) ) ) ).
% comp_sgraph.all_edges_between_mono1
thf(fact_837_comp__sgraph_Oedge__density__le1,axiom,
! [S: set_a,X5: set_a,Y2: set_a] : ( ord_less_eq_real @ ( undire297304480579013331sity_a @ ( undire2918257014606996450dges_a @ S ) @ X5 @ Y2 ) @ one_one_real ) ).
% comp_sgraph.edge_density_le1
thf(fact_838_comp__sgraph_Oedge__density__ge0,axiom,
! [S: set_a,X5: set_a,Y2: set_a] : ( ord_less_eq_real @ zero_zero_real @ ( undire297304480579013331sity_a @ ( undire2918257014606996450dges_a @ S ) @ X5 @ Y2 ) ) ).
% comp_sgraph.edge_density_ge0
thf(fact_839_ulgraph_Oall__edges__between_Ocong,axiom,
undire8383842906760478443ween_a = undire8383842906760478443ween_a ).
% ulgraph.all_edges_between.cong
thf(fact_840_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_841_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_842_comp__sgraph_Owellformed,axiom,
! [E: set_list_a,S: set_list_a] :
( ( member_set_list_a @ E @ ( undire517252296021605992list_a @ S ) )
=> ( ord_le8861187494160871172list_a @ E @ S ) ) ).
% comp_sgraph.wellformed
thf(fact_843_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_844_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_845_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_846_comp__sgraph_Oe__in__all__edges__ss,axiom,
! [E: set_list_a,S: set_list_a,V3: set_list_a] :
( ( member_set_list_a @ E @ ( undire517252296021605992list_a @ S ) )
=> ( ( ord_le8861187494160871172list_a @ E @ V3 )
=> ( ( ord_le8861187494160871172list_a @ V3 @ S )
=> ( member_set_list_a @ E @ ( undire517252296021605992list_a @ V3 ) ) ) ) ) ).
% comp_sgraph.e_in_all_edges_ss
thf(fact_847_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_848_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_849_comp__sgraph_Oulgraph__axioms,axiom,
! [S: set_a] : ( undire7251896706689453996raph_a @ S @ ( undire2918257014606996450dges_a @ S ) ) ).
% comp_sgraph.ulgraph_axioms
thf(fact_850_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_851_comp__sgraph_Ograph__system__axioms,axiom,
! [S: set_a] : ( undire2554140024507503526stem_a @ S @ ( undire2918257014606996450dges_a @ S ) ) ).
% comp_sgraph.graph_system_axioms
thf(fact_852_comp__sgraph_Osubgraph__complete,axiom,
! [S: set_a] : ( undire7103218114511261257raph_a @ S @ ( undire2918257014606996450dges_a @ S ) @ S @ ( undire2918257014606996450dges_a @ S ) ) ).
% comp_sgraph.subgraph_complete
thf(fact_853_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_854_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_855_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_856_comp__sgraph_Overt__adj__imp__inV,axiom,
! [S: set_Product_prod_a_a,V1: product_prod_a_a,V2: product_prod_a_a] :
( ( undire6135774327024169009od_a_a @ ( undire6879232364018543115od_a_a @ S ) @ V1 @ V2 )
=> ( ( member1426531477525435216od_a_a @ V1 @ S )
& ( member1426531477525435216od_a_a @ V2 @ S ) ) ) ).
% comp_sgraph.vert_adj_imp_inV
thf(fact_857_comp__sgraph_Overt__adj__imp__inV,axiom,
! [S: set_nat,V1: nat,V2: nat] :
( ( undire1083030068171319366dj_nat @ ( undire463345858124014060es_nat @ S ) @ V1 @ V2 )
=> ( ( member_nat @ V1 @ S )
& ( member_nat @ V2 @ S ) ) ) ).
% comp_sgraph.vert_adj_imp_inV
thf(fact_858_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_859_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_860_comp__sgraph_Oincident__edge__in__wf,axiom,
! [E: set_Product_prod_a_a,S: set_Product_prod_a_a,V: product_prod_a_a] :
( ( member1816616512716248880od_a_a @ E @ ( undire6879232364018543115od_a_a @ S ) )
=> ( ( undire3369688177417741453od_a_a @ V @ E )
=> ( member1426531477525435216od_a_a @ V @ S ) ) ) ).
% comp_sgraph.incident_edge_in_wf
thf(fact_861_comp__sgraph_Oincident__edge__in__wf,axiom,
! [E: set_nat,S: set_nat,V: nat] :
( ( member_set_nat @ E @ ( undire463345858124014060es_nat @ S ) )
=> ( ( undire7858122600432113898nt_nat @ V @ E )
=> ( member_nat @ V @ S ) ) ) ).
% comp_sgraph.incident_edge_in_wf
thf(fact_862_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_863_comp__sgraph_Oedge__density__commute,axiom,
! [S: set_a,X5: set_a,Y2: set_a] :
( ( undire297304480579013331sity_a @ ( undire2918257014606996450dges_a @ S ) @ X5 @ Y2 )
= ( undire297304480579013331sity_a @ ( undire2918257014606996450dges_a @ S ) @ Y2 @ X5 ) ) ).
% comp_sgraph.edge_density_commute
thf(fact_864_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_865_comp__sgraph_Ono__loops,axiom,
! [V: product_prod_a_a,S: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ V @ S )
=> ~ ( undire7777398424729533289od_a_a @ ( undire6879232364018543115od_a_a @ S ) @ V ) ) ).
% comp_sgraph.no_loops
thf(fact_866_comp__sgraph_Ono__loops,axiom,
! [V: nat,S: set_nat] :
( ( member_nat @ V @ S )
=> ~ ( undire5005864372999571214op_nat @ ( undire463345858124014060es_nat @ S ) @ V ) ) ).
% comp_sgraph.no_loops
thf(fact_867_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_868_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_869_comp__sgraph_Ohas__loop__in__verts,axiom,
! [S: set_Product_prod_a_a,V: product_prod_a_a] :
( ( undire7777398424729533289od_a_a @ ( undire6879232364018543115od_a_a @ S ) @ V )
=> ( member1426531477525435216od_a_a @ V @ S ) ) ).
% comp_sgraph.has_loop_in_verts
thf(fact_870_comp__sgraph_Ohas__loop__in__verts,axiom,
! [S: set_nat,V: nat] :
( ( undire5005864372999571214op_nat @ ( undire463345858124014060es_nat @ S ) @ V )
=> ( member_nat @ V @ S ) ) ).
% comp_sgraph.has_loop_in_verts
thf(fact_871_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_872_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_873_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 ) )
=> ( ? [X2: set_a] :
( ( member_set_a @ X2 @ S )
& ( member_set_a @ X2 @ E1 )
& ( member_set_a @ X2 @ E2 ) )
=> ( undire3485422320110889978_set_a @ ( undire8247866692393712962_set_a @ S ) @ E1 @ E2 ) ) ) ) ).
% comp_sgraph.edge_adjacent_alt_def
thf(fact_874_comp__sgraph_Oedge__adjacent__alt__def,axiom,
! [E1: set_Product_prod_a_a,S: set_Product_prod_a_a,E2: set_Product_prod_a_a] :
( ( member1816616512716248880od_a_a @ E1 @ ( undire6879232364018543115od_a_a @ S ) )
=> ( ( member1816616512716248880od_a_a @ E2 @ ( undire6879232364018543115od_a_a @ S ) )
=> ( ? [X2: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X2 @ S )
& ( member1426531477525435216od_a_a @ X2 @ E1 )
& ( member1426531477525435216od_a_a @ X2 @ E2 ) )
=> ( undire9186443406341554371od_a_a @ ( undire6879232364018543115od_a_a @ S ) @ E1 @ E2 ) ) ) ) ).
% comp_sgraph.edge_adjacent_alt_def
thf(fact_875_comp__sgraph_Oedge__adjacent__alt__def,axiom,
! [E1: set_nat,S: set_nat,E2: set_nat] :
( ( member_set_nat @ E1 @ ( undire463345858124014060es_nat @ S ) )
=> ( ( member_set_nat @ E2 @ ( undire463345858124014060es_nat @ S ) )
=> ( ? [X2: nat] :
( ( member_nat @ X2 @ S )
& ( member_nat @ X2 @ E1 )
& ( member_nat @ X2 @ E2 ) )
=> ( undire1664191744716346676dj_nat @ ( undire463345858124014060es_nat @ S ) @ E1 @ E2 ) ) ) ) ).
% comp_sgraph.edge_adjacent_alt_def
thf(fact_876_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 ) )
=> ( ? [X2: a] :
( ( member_a @ X2 @ S )
& ( member_a @ X2 @ E1 )
& ( member_a @ X2 @ E2 ) )
=> ( undire4022703626023482010_adj_a @ ( undire2918257014606996450dges_a @ S ) @ E1 @ E2 ) ) ) ) ).
% comp_sgraph.edge_adjacent_alt_def
thf(fact_877_ulgraph_Oall__edges__between__mono1,axiom,
! [Vertices: set_Product_prod_a_a,Edges: set_se5735800977113168103od_a_a,Y2: set_Product_prod_a_a,Z4: set_Product_prod_a_a,X5: set_Product_prod_a_a] :
( ( undire4585262585102564309od_a_a @ Vertices @ Edges )
=> ( ( ord_le746702958409616551od_a_a @ Y2 @ Z4 )
=> ( ord_le3469131294019144807od_a_a @ ( undire4032395788819567636od_a_a @ Edges @ Y2 @ X5 ) @ ( undire4032395788819567636od_a_a @ Edges @ Z4 @ X5 ) ) ) ) ).
% ulgraph.all_edges_between_mono1
thf(fact_878_ulgraph_Oall__edges__between__mono1,axiom,
! [Vertices: set_list_a,Edges: set_set_list_a,Y2: set_list_a,Z4: set_list_a,X5: set_list_a] :
( ( undire4488935924012268850list_a @ Vertices @ Edges )
=> ( ( ord_le8861187494160871172list_a @ Y2 @ Z4 )
=> ( ord_le7857023143581076903list_a @ ( undire3588171647663456497list_a @ Edges @ Y2 @ X5 ) @ ( undire3588171647663456497list_a @ Edges @ Z4 @ X5 ) ) ) ) ).
% ulgraph.all_edges_between_mono1
thf(fact_879_ulgraph_Oall__edges__between__mono1,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,Y2: set_set_a,Z4: set_set_a,X5: set_set_a] :
( ( undire6886684016831807756_set_a @ Vertices @ Edges )
=> ( ( ord_le3724670747650509150_set_a @ Y2 @ Z4 )
=> ( ord_le8376522849517564071_set_a @ ( undire2462398226299384907_set_a @ Edges @ Y2 @ X5 ) @ ( undire2462398226299384907_set_a @ Edges @ Z4 @ X5 ) ) ) ) ).
% ulgraph.all_edges_between_mono1
thf(fact_880_ulgraph_Oall__edges__between__mono1,axiom,
! [Vertices: set_a,Edges: set_set_a,Y2: set_a,Z4: set_a,X5: set_a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( ord_less_eq_set_a @ Y2 @ Z4 )
=> ( ord_le746702958409616551od_a_a @ ( undire8383842906760478443ween_a @ Edges @ Y2 @ X5 ) @ ( undire8383842906760478443ween_a @ Edges @ Z4 @ X5 ) ) ) ) ).
% ulgraph.all_edges_between_mono1
thf(fact_881_ulgraph_Oall__edges__between__mono2,axiom,
! [Vertices: set_Product_prod_a_a,Edges: set_se5735800977113168103od_a_a,Y2: set_Product_prod_a_a,Z4: set_Product_prod_a_a,X5: set_Product_prod_a_a] :
( ( undire4585262585102564309od_a_a @ Vertices @ Edges )
=> ( ( ord_le746702958409616551od_a_a @ Y2 @ Z4 )
=> ( ord_le3469131294019144807od_a_a @ ( undire4032395788819567636od_a_a @ Edges @ X5 @ Y2 ) @ ( undire4032395788819567636od_a_a @ Edges @ X5 @ Z4 ) ) ) ) ).
% ulgraph.all_edges_between_mono2
thf(fact_882_ulgraph_Oall__edges__between__mono2,axiom,
! [Vertices: set_list_a,Edges: set_set_list_a,Y2: set_list_a,Z4: set_list_a,X5: set_list_a] :
( ( undire4488935924012268850list_a @ Vertices @ Edges )
=> ( ( ord_le8861187494160871172list_a @ Y2 @ Z4 )
=> ( ord_le7857023143581076903list_a @ ( undire3588171647663456497list_a @ Edges @ X5 @ Y2 ) @ ( undire3588171647663456497list_a @ Edges @ X5 @ Z4 ) ) ) ) ).
% ulgraph.all_edges_between_mono2
thf(fact_883_ulgraph_Oall__edges__between__mono2,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,Y2: set_set_a,Z4: set_set_a,X5: set_set_a] :
( ( undire6886684016831807756_set_a @ Vertices @ Edges )
=> ( ( ord_le3724670747650509150_set_a @ Y2 @ Z4 )
=> ( ord_le8376522849517564071_set_a @ ( undire2462398226299384907_set_a @ Edges @ X5 @ Y2 ) @ ( undire2462398226299384907_set_a @ Edges @ X5 @ Z4 ) ) ) ) ).
% ulgraph.all_edges_between_mono2
thf(fact_884_ulgraph_Oall__edges__between__mono2,axiom,
! [Vertices: set_a,Edges: set_set_a,Y2: set_a,Z4: set_a,X5: set_a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( ord_less_eq_set_a @ Y2 @ Z4 )
=> ( ord_le746702958409616551od_a_a @ ( undire8383842906760478443ween_a @ Edges @ X5 @ Y2 ) @ ( undire8383842906760478443ween_a @ Edges @ X5 @ Z4 ) ) ) ) ).
% ulgraph.all_edges_between_mono2
thf(fact_885_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_886_all__edges__mono,axiom,
! [Vs: set_list_a,Ws: set_list_a] :
( ( ord_le8861187494160871172list_a @ Vs @ Ws )
=> ( ord_le8877086941679407844list_a @ ( undire517252296021605992list_a @ Vs ) @ ( undire517252296021605992list_a @ Ws ) ) ) ).
% all_edges_mono
thf(fact_887_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_888_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_889_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_890_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_891_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_892_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_893_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_894_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_895_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_896_comp__sgraph_Ois__walk__wf__hd,axiom,
! [S: set_Product_prod_a_a,Xs: list_P1396940483166286381od_a_a] :
( ( undire3162072421265123221od_a_a @ S @ ( undire6879232364018543115od_a_a @ S ) @ Xs )
=> ( member1426531477525435216od_a_a @ ( hd_Product_prod_a_a @ Xs ) @ S ) ) ).
% comp_sgraph.is_walk_wf_hd
thf(fact_897_comp__sgraph_Ois__walk__wf__hd,axiom,
! [S: set_nat,Xs: list_nat] :
( ( undire5745680128780950498lk_nat @ S @ ( undire463345858124014060es_nat @ S ) @ Xs )
=> ( member_nat @ ( hd_nat @ Xs ) @ S ) ) ).
% comp_sgraph.is_walk_wf_hd
thf(fact_898_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_899_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_900_comp__sgraph_Ois__walk__wf__last,axiom,
! [S: set_Product_prod_a_a,Xs: list_P1396940483166286381od_a_a] :
( ( undire3162072421265123221od_a_a @ S @ ( undire6879232364018543115od_a_a @ S ) @ Xs )
=> ( member1426531477525435216od_a_a @ ( last_P8790725268278465478od_a_a @ Xs ) @ S ) ) ).
% comp_sgraph.is_walk_wf_last
thf(fact_901_comp__sgraph_Ois__walk__wf__last,axiom,
! [S: set_nat,Xs: list_nat] :
( ( undire5745680128780950498lk_nat @ S @ ( undire463345858124014060es_nat @ S ) @ Xs )
=> ( member_nat @ ( last_nat @ Xs ) @ S ) ) ).
% comp_sgraph.is_walk_wf_last
thf(fact_902_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_903_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_904_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_905_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_906_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_907_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_908_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_909_comp__sgraph_Overt__adj__edge__iff2,axiom,
! [V1: a,V2: a,S: set_a] :
( ( V1 != V2 )
=> ( ( undire397441198561214472_adj_a @ ( undire2918257014606996450dges_a @ S ) @ V1 @ V2 )
= ( ? [X3: set_a] :
( ( member_set_a @ X3 @ ( undire2918257014606996450dges_a @ S ) )
& ( undire1521409233611534436dent_a @ V1 @ X3 )
& ( undire1521409233611534436dent_a @ V2 @ X3 ) ) ) ) ) ).
% comp_sgraph.vert_adj_edge_iff2
thf(fact_910_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_911_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_912_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_913_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_914_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_915_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_916_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 )
& ! [X3: set_a] :
( ( member_set_a @ X3 @ S )
=> ~ ( undire3510646817838285160_set_a @ ( undire8247866692393712962_set_a @ S ) @ X3 @ V ) ) ) ) ).
% comp_sgraph.is_isolated_vertex_def
thf(fact_917_comp__sgraph_Ois__isolated__vertex__def,axiom,
! [S: set_Product_prod_a_a,V: product_prod_a_a] :
( ( undire3207556238582723646od_a_a @ S @ ( undire6879232364018543115od_a_a @ S ) @ V )
= ( ( member1426531477525435216od_a_a @ V @ S )
& ! [X3: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X3 @ S )
=> ~ ( undire6135774327024169009od_a_a @ ( undire6879232364018543115od_a_a @ S ) @ X3 @ V ) ) ) ) ).
% comp_sgraph.is_isolated_vertex_def
thf(fact_918_comp__sgraph_Ois__isolated__vertex__def,axiom,
! [S: set_nat,V: nat] :
( ( undire5609513041723151865ex_nat @ S @ ( undire463345858124014060es_nat @ S ) @ V )
= ( ( member_nat @ V @ S )
& ! [X3: nat] :
( ( member_nat @ X3 @ S )
=> ~ ( undire1083030068171319366dj_nat @ ( undire463345858124014060es_nat @ S ) @ X3 @ V ) ) ) ) ).
% comp_sgraph.is_isolated_vertex_def
thf(fact_919_comp__sgraph_Ois__isolated__vertex__def,axiom,
! [S: set_a,V: a] :
( ( undire8931668460104145173rtex_a @ S @ ( undire2918257014606996450dges_a @ S ) @ V )
= ( ( member_a @ V @ S )
& ! [X3: a] :
( ( member_a @ X3 @ S )
=> ~ ( undire397441198561214472_adj_a @ ( undire2918257014606996450dges_a @ S ) @ X3 @ V ) ) ) ) ).
% comp_sgraph.is_isolated_vertex_def
thf(fact_920_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_921_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_922_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_923_ulgraph_Oedge__density__le1,axiom,
! [Vertices: set_a,Edges: set_set_a,X5: set_a,Y2: set_a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ord_less_eq_real @ ( undire297304480579013331sity_a @ Edges @ X5 @ Y2 ) @ one_one_real ) ) ).
% ulgraph.edge_density_le1
thf(fact_924_ulgraph_Oedge__density__ge0,axiom,
! [Vertices: set_a,Edges: set_set_a,X5: set_a,Y2: set_a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ord_less_eq_real @ zero_zero_real @ ( undire297304480579013331sity_a @ Edges @ X5 @ Y2 ) ) ) ).
% ulgraph.edge_density_ge0
thf(fact_925_comp__sgraph_Ois__walk__singleton,axiom,
! [U: product_prod_a_a,S: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ U @ S )
=> ( undire3162072421265123221od_a_a @ S @ ( undire6879232364018543115od_a_a @ S ) @ ( cons_P7316939126706565853od_a_a @ U @ nil_Product_prod_a_a ) ) ) ).
% comp_sgraph.is_walk_singleton
thf(fact_926_comp__sgraph_Ois__walk__singleton,axiom,
! [U: nat,S: set_nat] :
( ( member_nat @ U @ S )
=> ( undire5745680128780950498lk_nat @ S @ ( undire463345858124014060es_nat @ S ) @ ( cons_nat @ U @ nil_nat ) ) ) ).
% comp_sgraph.is_walk_singleton
thf(fact_927_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_928_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_929_comp__sgraph_Ois__walk__drop__hd,axiom,
! [Ys: list_set_a,S: set_set_a,Y: set_a] :
( ( Ys != nil_set_a )
=> ( ( undire3014741414213135564_set_a @ S @ ( undire8247866692393712962_set_a @ S ) @ ( cons_set_a @ Y @ Ys ) )
=> ( undire3014741414213135564_set_a @ S @ ( undire8247866692393712962_set_a @ S ) @ Ys ) ) ) ).
% comp_sgraph.is_walk_drop_hd
thf(fact_930_comp__sgraph_Ois__walk__drop__hd,axiom,
! [Ys: list_a,S: set_a,Y: a] :
( ( Ys != nil_a )
=> ( ( undire6133010728901294956walk_a @ S @ ( undire2918257014606996450dges_a @ S ) @ ( cons_a @ Y @ Ys ) )
=> ( undire6133010728901294956walk_a @ S @ ( undire2918257014606996450dges_a @ S ) @ Ys ) ) ) ).
% comp_sgraph.is_walk_drop_hd
thf(fact_931_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_932_comp__sgraph_Ois__walk__wf,axiom,
! [S: set_list_a,Xs: list_list_a] :
( ( undire8550186295227992306list_a @ S @ ( undire517252296021605992list_a @ S ) @ Xs )
=> ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Xs ) @ S ) ) ).
% comp_sgraph.is_walk_wf
thf(fact_933_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_934_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_935_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_936_comp__sgraph_Oinduced__edges__ss,axiom,
! [V3: set_list_a,S: set_list_a] :
( ( ord_le8861187494160871172list_a @ V3 @ S )
=> ( ord_le8877086941679407844list_a @ ( undire8521487854958249554list_a @ ( undire517252296021605992list_a @ S ) @ V3 ) @ ( undire517252296021605992list_a @ S ) ) ) ).
% comp_sgraph.induced_edges_ss
thf(fact_937_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_938_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_939_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_940_comp__sgraph_Oinduced__is__subgraph,axiom,
! [V3: set_list_a,S: set_list_a] :
( ( ord_le8861187494160871172list_a @ V3 @ S )
=> ( undire761398192061991247list_a @ V3 @ ( undire8521487854958249554list_a @ ( undire517252296021605992list_a @ S ) @ V3 ) @ S @ ( undire517252296021605992list_a @ S ) ) ) ).
% comp_sgraph.induced_is_subgraph
thf(fact_941_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_942_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_943_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_944_comp__sgraph_Ois__path__def,axiom,
! [S: set_Pr2416559167834504103et_a_a,Xs: list_P5740962349794459853et_a_a] :
( ( undire8374260845365092473et_a_a @ S @ ( undire4745662417834845355et_a_a @ S ) @ Xs )
= ( ( undire1907723709775954283et_a_a @ S @ ( undire4745662417834845355et_a_a @ S ) @ Xs )
& ( distin7251654435778379584et_a_a @ Xs ) ) ) ).
% comp_sgraph.is_path_def
thf(fact_945_comp__sgraph_Ois__path__def,axiom,
! [S: set_Pr6393634178297680487_set_a,Xs: list_P494665323402860429_set_a] :
( ( undire1324510289328068409_set_a @ S @ ( undire6919283898652597099_set_a @ S ) @ Xs )
= ( ( undire4081345190593706027_set_a @ S @ ( undire6919283898652597099_set_a @ S ) @ Xs )
& ( distin201903879741355520_set_a @ Xs ) ) ) ).
% comp_sgraph.is_path_def
thf(fact_946_comp__sgraph_Ois__path__def,axiom,
! [S: set_Product_prod_a_a,Xs: list_P1396940483166286381od_a_a] :
( ( undire9149042980421869017od_a_a @ S @ ( undire6879232364018543115od_a_a @ S ) @ Xs )
= ( ( undire1203054589613885131od_a_a @ S @ ( undire6879232364018543115od_a_a @ S ) @ Xs )
& ( distin132333870042060960od_a_a @ Xs ) ) ) ).
% comp_sgraph.is_path_def
thf(fact_947_comp__sgraph_Ois__path__def,axiom,
! [S: set_list_set_a,Xs: list_list_set_a] :
( ( undire1052973453303871126_set_a @ S @ ( undire5484946175218547656_set_a @ S ) @ Xs )
= ( ( undire1111087293740939400_set_a @ S @ ( undire5484946175218547656_set_a @ S ) @ Xs )
& ( distinct_list_set_a @ Xs ) ) ) ).
% comp_sgraph.is_path_def
thf(fact_948_comp__sgraph_Ois__path__def,axiom,
! [S: set_list_a,Xs: list_list_a] :
( ( undire2586462650415165750list_a @ S @ ( undire517252296021605992list_a @ S ) @ Xs )
= ( ( undire6929316984140692264list_a @ S @ ( undire517252296021605992list_a @ S ) @ Xs )
& ( distinct_list_a @ Xs ) ) ) ).
% comp_sgraph.is_path_def
thf(fact_949_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_950_comp__sgraph_Ois__walk__decomp,axiom,
! [S: set_set_a,Xs: list_set_a,Y: set_a,Ys: list_set_a,Zs: list_set_a] :
( ( undire3014741414213135564_set_a @ S @ ( undire8247866692393712962_set_a @ S ) @ ( append_set_a @ Xs @ ( append_set_a @ ( cons_set_a @ Y @ nil_set_a ) @ ( append_set_a @ Ys @ ( append_set_a @ ( cons_set_a @ Y @ nil_set_a ) @ Zs ) ) ) ) )
=> ( undire3014741414213135564_set_a @ S @ ( undire8247866692393712962_set_a @ S ) @ ( append_set_a @ Xs @ ( append_set_a @ ( cons_set_a @ Y @ nil_set_a ) @ Zs ) ) ) ) ).
% comp_sgraph.is_walk_decomp
thf(fact_951_comp__sgraph_Ois__walk__decomp,axiom,
! [S: set_a,Xs: list_a,Y: a,Ys: list_a,Zs: list_a] :
( ( undire6133010728901294956walk_a @ S @ ( undire2918257014606996450dges_a @ S ) @ ( append_a @ Xs @ ( append_a @ ( cons_a @ Y @ nil_a ) @ ( append_a @ Ys @ ( append_a @ ( cons_a @ Y @ nil_a ) @ Zs ) ) ) ) )
=> ( undire6133010728901294956walk_a @ S @ ( undire2918257014606996450dges_a @ S ) @ ( append_a @ Xs @ ( append_a @ ( cons_a @ Y @ nil_a ) @ Zs ) ) ) ) ).
% comp_sgraph.is_walk_decomp
thf(fact_952_comp__sgraph_Opaths__ss__walk,axiom,
! [S: set_a] : ( ord_le8861187494160871172list_a @ ( undire1387732426225024653aths_a @ S @ ( undire2918257014606996450dges_a @ S ) ) @ ( undire3736599831911450577alks_a @ S @ ( undire2918257014606996450dges_a @ S ) ) ) ).
% comp_sgraph.paths_ss_walk
thf(fact_953_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_954_comp__sgraph_Ois__gen__path__distinct,axiom,
! [S: set_Pr2416559167834504103et_a_a,P: list_P5740962349794459853et_a_a] :
( ( undire8058049273645199385et_a_a @ S @ ( undire4745662417834845355et_a_a @ S ) @ P )
=> ( ( ( hd_Pro7221231872133205426et_a_a @ P )
!= ( last_P6817006942355120742et_a_a @ P ) )
=> ( distin7251654435778379584et_a_a @ P ) ) ) ).
% comp_sgraph.is_gen_path_distinct
thf(fact_955_comp__sgraph_Ois__gen__path__distinct,axiom,
! [S: set_Pr6393634178297680487_set_a,P: list_P494665323402860429_set_a] :
( ( undire1008298717608175321_set_a @ S @ ( undire6919283898652597099_set_a @ S ) @ P )
=> ( ( ( hd_Pro171481316096181362_set_a @ P )
!= ( last_P8990628423172872486_set_a @ P ) )
=> ( distin201903879741355520_set_a @ P ) ) ) ).
% comp_sgraph.is_gen_path_distinct
thf(fact_956_comp__sgraph_Ois__gen__path__distinct,axiom,
! [S: set_Product_prod_a_a,P: list_P1396940483166286381od_a_a] :
( ( undire7585867811434966393od_a_a @ S @ ( undire6879232364018543115od_a_a @ S ) @ P )
=> ( ( ( hd_Product_prod_a_a @ P )
!= ( last_P8790725268278465478od_a_a @ P ) )
=> ( distin132333870042060960od_a_a @ P ) ) ) ).
% comp_sgraph.is_gen_path_distinct
thf(fact_957_comp__sgraph_Ois__gen__path__distinct,axiom,
! [S: set_list_set_a,P: list_list_set_a] :
( ( undire5019392814671255094_set_a @ S @ ( undire5484946175218547656_set_a @ S ) @ P )
=> ( ( ( hd_list_set_a @ P )
!= ( last_list_set_a @ P ) )
=> ( distinct_list_set_a @ P ) ) ) ).
% comp_sgraph.is_gen_path_distinct
thf(fact_958_comp__sgraph_Ois__gen__path__distinct,axiom,
! [S: set_list_a,P: list_list_a] :
( ( undire8568094650444222678list_a @ S @ ( undire517252296021605992list_a @ S ) @ P )
=> ( ( ( hd_list_a @ P )
!= ( last_list_a @ P ) )
=> ( distinct_list_a @ P ) ) ) ).
% comp_sgraph.is_gen_path_distinct
thf(fact_959_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_960_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_961_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_962_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_963_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_964_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_965_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_966_comp__sgraph_Ois__walk__def,axiom,
! [S: set_list_a,Xs: list_list_a] :
( ( undire8550186295227992306list_a @ S @ ( undire517252296021605992list_a @ S ) @ Xs )
= ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Xs ) @ S )
& ( ord_le8877086941679407844list_a @ ( set_set_list_a2 @ ( undire8303882243552421012list_a @ Xs ) ) @ ( undire517252296021605992list_a @ S ) )
& ( Xs != nil_list_a ) ) ) ).
% comp_sgraph.is_walk_def
thf(fact_967_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_968_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_969_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_970_comp__sgraph_Ois__walkI,axiom,
! [Xs: list_list_a,S: set_list_a] :
( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Xs ) @ S )
=> ( ( ord_le8877086941679407844list_a @ ( set_set_list_a2 @ ( undire8303882243552421012list_a @ Xs ) ) @ ( undire517252296021605992list_a @ S ) )
=> ( ( Xs != nil_list_a )
=> ( undire8550186295227992306list_a @ S @ ( undire517252296021605992list_a @ S ) @ Xs ) ) ) ) ).
% comp_sgraph.is_walkI
thf(fact_971_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_972_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_973_edge__density__eq0,axiom,
! [A2: set_a,B2: set_a,X5: set_a,Y2: set_a] :
( ( ( undire8383842906760478443ween_a @ edges @ A2 @ B2 )
= bot_bo3357376287454694259od_a_a )
=> ( ( ord_less_eq_set_a @ X5 @ A2 )
=> ( ( ord_less_eq_set_a @ Y2 @ B2 )
=> ( ( undire297304480579013331sity_a @ edges @ X5 @ Y2 )
= zero_zero_real ) ) ) ) ).
% edge_density_eq0
thf(fact_974_le__zero__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_975_edge__density__zero,axiom,
! [Y2: set_a,X5: set_a] :
( ( Y2 = bot_bot_set_a )
=> ( ( undire297304480579013331sity_a @ edges @ X5 @ Y2 )
= zero_zero_real ) ) ).
% edge_density_zero
thf(fact_976_all__edges__between__rem__wf,axiom,
! [X5: set_a,Y2: set_a] :
( ( undire8383842906760478443ween_a @ edges @ X5 @ Y2 )
= ( undire8383842906760478443ween_a @ edges @ ( inf_inf_set_a @ X5 @ vertices ) @ ( inf_inf_set_a @ Y2 @ vertices ) ) ) ).
% all_edges_between_rem_wf
thf(fact_977_not__one__le__zero,axiom,
~ ( ord_less_eq_real @ one_one_real @ zero_zero_real ) ).
% not_one_le_zero
thf(fact_978_not__one__le__zero,axiom,
~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_le_zero
thf(fact_979_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_980_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_981_empty__not__edge,axiom,
~ ( member_set_a @ bot_bot_set_a @ edges ) ).
% empty_not_edge
thf(fact_982_empty__iff,axiom,
! [C: nat] :
~ ( member_nat @ C @ bot_bot_set_nat ) ).
% empty_iff
thf(fact_983_empty__iff,axiom,
! [C: product_prod_a_a] :
~ ( member1426531477525435216od_a_a @ C @ bot_bo3357376287454694259od_a_a ) ).
% empty_iff
thf(fact_984_empty__iff,axiom,
! [C: a] :
~ ( member_a @ C @ bot_bot_set_a ) ).
% empty_iff
thf(fact_985_empty__iff,axiom,
! [C: set_a] :
~ ( member_set_a @ C @ bot_bot_set_set_a ) ).
% empty_iff
thf(fact_986_all__not__in__conv,axiom,
! [A2: set_nat] :
( ( ! [X3: nat] :
~ ( member_nat @ X3 @ A2 ) )
= ( A2 = bot_bot_set_nat ) ) ).
% all_not_in_conv
thf(fact_987_all__not__in__conv,axiom,
! [A2: set_Product_prod_a_a] :
( ( ! [X3: product_prod_a_a] :
~ ( member1426531477525435216od_a_a @ X3 @ A2 ) )
= ( A2 = bot_bo3357376287454694259od_a_a ) ) ).
% all_not_in_conv
thf(fact_988_all__not__in__conv,axiom,
! [A2: set_a] :
( ( ! [X3: a] :
~ ( member_a @ X3 @ A2 ) )
= ( A2 = bot_bot_set_a ) ) ).
% all_not_in_conv
thf(fact_989_all__not__in__conv,axiom,
! [A2: set_set_a] :
( ( ! [X3: set_a] :
~ ( member_set_a @ X3 @ A2 ) )
= ( A2 = bot_bot_set_set_a ) ) ).
% all_not_in_conv
thf(fact_990_Collect__empty__eq,axiom,
! [P2: product_prod_a_a > $o] :
( ( ( collec3336397797384452498od_a_a @ P2 )
= bot_bo3357376287454694259od_a_a )
= ( ! [X3: product_prod_a_a] :
~ ( P2 @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_991_Collect__empty__eq,axiom,
! [P2: a > $o] :
( ( ( collect_a @ P2 )
= bot_bot_set_a )
= ( ! [X3: a] :
~ ( P2 @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_992_Collect__empty__eq,axiom,
! [P2: set_a > $o] :
( ( ( collect_set_a @ P2 )
= bot_bot_set_set_a )
= ( ! [X3: set_a] :
~ ( P2 @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_993_empty__Collect__eq,axiom,
! [P2: product_prod_a_a > $o] :
( ( bot_bo3357376287454694259od_a_a
= ( collec3336397797384452498od_a_a @ P2 ) )
= ( ! [X3: product_prod_a_a] :
~ ( P2 @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_994_empty__Collect__eq,axiom,
! [P2: a > $o] :
( ( bot_bot_set_a
= ( collect_a @ P2 ) )
= ( ! [X3: a] :
~ ( P2 @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_995_empty__Collect__eq,axiom,
! [P2: set_a > $o] :
( ( bot_bot_set_set_a
= ( collect_set_a @ P2 ) )
= ( ! [X3: set_a] :
~ ( P2 @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_996_IntI,axiom,
! [C: set_a,A2: set_set_a,B2: set_set_a] :
( ( member_set_a @ C @ A2 )
=> ( ( member_set_a @ C @ B2 )
=> ( member_set_a @ C @ ( inf_inf_set_set_a @ A2 @ B2 ) ) ) ) ).
% IntI
thf(fact_997_IntI,axiom,
! [C: product_prod_a_a,A2: set_Product_prod_a_a,B2: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ C @ A2 )
=> ( ( member1426531477525435216od_a_a @ C @ B2 )
=> ( member1426531477525435216od_a_a @ C @ ( inf_in8905007599844390133od_a_a @ A2 @ B2 ) ) ) ) ).
% IntI
thf(fact_998_IntI,axiom,
! [C: nat,A2: set_nat,B2: set_nat] :
( ( member_nat @ C @ A2 )
=> ( ( member_nat @ C @ B2 )
=> ( member_nat @ C @ ( inf_inf_set_nat @ A2 @ B2 ) ) ) ) ).
% IntI
thf(fact_999_IntI,axiom,
! [C: a,A2: set_a,B2: set_a] :
( ( member_a @ C @ A2 )
=> ( ( member_a @ C @ B2 )
=> ( member_a @ C @ ( inf_inf_set_a @ A2 @ B2 ) ) ) ) ).
% IntI
thf(fact_1000_Int__iff,axiom,
! [C: set_a,A2: set_set_a,B2: set_set_a] :
( ( member_set_a @ C @ ( inf_inf_set_set_a @ A2 @ B2 ) )
= ( ( member_set_a @ C @ A2 )
& ( member_set_a @ C @ B2 ) ) ) ).
% Int_iff
thf(fact_1001_Int__iff,axiom,
! [C: product_prod_a_a,A2: set_Product_prod_a_a,B2: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ C @ ( inf_in8905007599844390133od_a_a @ A2 @ B2 ) )
= ( ( member1426531477525435216od_a_a @ C @ A2 )
& ( member1426531477525435216od_a_a @ C @ B2 ) ) ) ).
% Int_iff
thf(fact_1002_Int__iff,axiom,
! [C: nat,A2: set_nat,B2: set_nat] :
( ( member_nat @ C @ ( inf_inf_set_nat @ A2 @ B2 ) )
= ( ( member_nat @ C @ A2 )
& ( member_nat @ C @ B2 ) ) ) ).
% Int_iff
thf(fact_1003_Int__iff,axiom,
! [C: a,A2: set_a,B2: set_a] :
( ( member_a @ C @ ( inf_inf_set_a @ A2 @ B2 ) )
= ( ( member_a @ C @ A2 )
& ( member_a @ C @ B2 ) ) ) ).
% Int_iff
thf(fact_1004_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_1005_empty__subsetI,axiom,
! [A2: set_Product_prod_a_a] : ( ord_le746702958409616551od_a_a @ bot_bo3357376287454694259od_a_a @ A2 ) ).
% empty_subsetI
thf(fact_1006_empty__subsetI,axiom,
! [A2: set_list_a] : ( ord_le8861187494160871172list_a @ bot_bot_set_list_a @ A2 ) ).
% empty_subsetI
thf(fact_1007_empty__subsetI,axiom,
! [A2: set_set_a] : ( ord_le3724670747650509150_set_a @ bot_bot_set_set_a @ A2 ) ).
% empty_subsetI
thf(fact_1008_empty__subsetI,axiom,
! [A2: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A2 ) ).
% empty_subsetI
thf(fact_1009_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_1010_subset__empty,axiom,
! [A2: set_list_a] :
( ( ord_le8861187494160871172list_a @ A2 @ bot_bot_set_list_a )
= ( A2 = bot_bot_set_list_a ) ) ).
% subset_empty
thf(fact_1011_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_1012_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_1013_Int__subset__iff,axiom,
! [C2: set_Product_prod_a_a,A2: set_Product_prod_a_a,B2: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ C2 @ ( inf_in8905007599844390133od_a_a @ A2 @ B2 ) )
= ( ( ord_le746702958409616551od_a_a @ C2 @ A2 )
& ( ord_le746702958409616551od_a_a @ C2 @ B2 ) ) ) ).
% Int_subset_iff
thf(fact_1014_Int__subset__iff,axiom,
! [C2: set_list_a,A2: set_list_a,B2: set_list_a] :
( ( ord_le8861187494160871172list_a @ C2 @ ( inf_inf_set_list_a @ A2 @ B2 ) )
= ( ( ord_le8861187494160871172list_a @ C2 @ A2 )
& ( ord_le8861187494160871172list_a @ C2 @ B2 ) ) ) ).
% Int_subset_iff
thf(fact_1015_Int__subset__iff,axiom,
! [C2: set_set_a,A2: set_set_a,B2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ C2 @ ( inf_inf_set_set_a @ A2 @ B2 ) )
= ( ( ord_le3724670747650509150_set_a @ C2 @ A2 )
& ( ord_le3724670747650509150_set_a @ C2 @ B2 ) ) ) ).
% Int_subset_iff
thf(fact_1016_Int__subset__iff,axiom,
! [C2: set_a,A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ C2 @ ( inf_inf_set_a @ A2 @ B2 ) )
= ( ( ord_less_eq_set_a @ C2 @ A2 )
& ( ord_less_eq_set_a @ C2 @ B2 ) ) ) ).
% Int_subset_iff
thf(fact_1017_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_1018_set__empty,axiom,
! [Xs: list_a] :
( ( ( set_a2 @ Xs )
= bot_bot_set_a )
= ( Xs = nil_a ) ) ).
% set_empty
thf(fact_1019_set__empty,axiom,
! [Xs: list_set_a] :
( ( ( set_set_a2 @ Xs )
= bot_bot_set_set_a )
= ( Xs = nil_set_a ) ) ).
% set_empty
thf(fact_1020_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_1021_set__empty2,axiom,
! [Xs: list_a] :
( ( bot_bot_set_a
= ( set_a2 @ Xs ) )
= ( Xs = nil_a ) ) ).
% set_empty2
thf(fact_1022_set__empty2,axiom,
! [Xs: list_set_a] :
( ( bot_bot_set_set_a
= ( set_set_a2 @ Xs ) )
= ( Xs = nil_set_a ) ) ).
% set_empty2
thf(fact_1023_comp__sgraph_Oall__edges__between__empty_I2_J,axiom,
! [S: set_Product_prod_a_a,Z4: set_Product_prod_a_a] :
( ( undire4032395788819567636od_a_a @ ( undire6879232364018543115od_a_a @ S ) @ Z4 @ bot_bo3357376287454694259od_a_a )
= bot_bo510284599550014259od_a_a ) ).
% comp_sgraph.all_edges_between_empty(2)
thf(fact_1024_comp__sgraph_Oall__edges__between__empty_I2_J,axiom,
! [S: set_set_a,Z4: set_set_a] :
( ( undire2462398226299384907_set_a @ ( undire8247866692393712962_set_a @ S ) @ Z4 @ bot_bot_set_set_a )
= bot_bo5799363139946352499_set_a ) ).
% comp_sgraph.all_edges_between_empty(2)
thf(fact_1025_comp__sgraph_Oall__edges__between__empty_I2_J,axiom,
! [S: set_a,Z4: set_a] :
( ( undire8383842906760478443ween_a @ ( undire2918257014606996450dges_a @ S ) @ Z4 @ bot_bot_set_a )
= bot_bo3357376287454694259od_a_a ) ).
% comp_sgraph.all_edges_between_empty(2)
thf(fact_1026_comp__sgraph_Oall__edges__between__empty_I1_J,axiom,
! [S: set_Product_prod_a_a,Z4: set_Product_prod_a_a] :
( ( undire4032395788819567636od_a_a @ ( undire6879232364018543115od_a_a @ S ) @ bot_bo3357376287454694259od_a_a @ Z4 )
= bot_bo510284599550014259od_a_a ) ).
% comp_sgraph.all_edges_between_empty(1)
thf(fact_1027_comp__sgraph_Oall__edges__between__empty_I1_J,axiom,
! [S: set_set_a,Z4: set_set_a] :
( ( undire2462398226299384907_set_a @ ( undire8247866692393712962_set_a @ S ) @ bot_bot_set_set_a @ Z4 )
= bot_bo5799363139946352499_set_a ) ).
% comp_sgraph.all_edges_between_empty(1)
thf(fact_1028_comp__sgraph_Oall__edges__between__empty_I1_J,axiom,
! [S: set_a,Z4: set_a] :
( ( undire8383842906760478443ween_a @ ( undire2918257014606996450dges_a @ S ) @ bot_bot_set_a @ Z4 )
= bot_bo3357376287454694259od_a_a ) ).
% comp_sgraph.all_edges_between_empty(1)
thf(fact_1029_n__lists__Nil,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( n_lists_a @ N @ nil_a )
= ( cons_list_a @ nil_a @ nil_list_a ) ) )
& ( ( N != zero_zero_nat )
=> ( ( n_lists_a @ N @ nil_a )
= nil_list_a ) ) ) ).
% n_lists_Nil
thf(fact_1030_n__lists__Nil,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( n_lists_set_a @ N @ nil_set_a )
= ( cons_list_set_a @ nil_set_a @ nil_list_set_a ) ) )
& ( ( N != zero_zero_nat )
=> ( ( n_lists_set_a @ N @ nil_set_a )
= nil_list_set_a ) ) ) ).
% n_lists_Nil
thf(fact_1031_distinct__append,axiom,
! [Xs: list_P5740962349794459853et_a_a,Ys: list_P5740962349794459853et_a_a] :
( ( distin7251654435778379584et_a_a @ ( append8719822696595618658et_a_a @ Xs @ Ys ) )
= ( ( distin7251654435778379584et_a_a @ Xs )
& ( distin7251654435778379584et_a_a @ Ys )
& ( ( inf_in20140756741694357et_a_a @ ( set_Pr9190486990633797724et_a_a @ Xs ) @ ( set_Pr9190486990633797724et_a_a @ Ys ) )
= bot_bo6682002291534152211et_a_a ) ) ) ).
% distinct_append
thf(fact_1032_distinct__append,axiom,
! [Xs: list_P494665323402860429_set_a,Ys: list_P494665323402860429_set_a] :
( ( distin201903879741355520_set_a @ ( append1670072140558594594_set_a @ Xs @ Ys ) )
= ( ( distin201903879741355520_set_a @ Xs )
& ( distin201903879741355520_set_a @ Ys )
& ( ( inf_in3997215767204870741_set_a @ ( set_Pr2140736434596773660_set_a @ Xs ) @ ( set_Pr2140736434596773660_set_a @ Ys ) )
= bot_bo1435705265142552787_set_a ) ) ) ).
% distinct_append
thf(fact_1033_distinct__append,axiom,
! [Xs: list_list_set_a,Ys: list_list_set_a] :
( ( distinct_list_set_a @ ( append_list_set_a @ Xs @ Ys ) )
= ( ( distinct_list_set_a @ Xs )
& ( distinct_list_set_a @ Ys )
& ( ( inf_in5868711818016843698_set_a @ ( set_list_set_a2 @ Xs ) @ ( set_list_set_a2 @ Ys ) )
= bot_bo4397488018069675312_set_a ) ) ) ).
% distinct_append
thf(fact_1034_distinct__append,axiom,
! [Xs: list_list_a,Ys: list_list_a] :
( ( distinct_list_a @ ( append_list_a @ Xs @ Ys ) )
= ( ( distinct_list_a @ Xs )
& ( distinct_list_a @ Ys )
& ( ( inf_inf_set_list_a @ ( set_list_a2 @ Xs ) @ ( set_list_a2 @ Ys ) )
= bot_bot_set_list_a ) ) ) ).
% distinct_append
thf(fact_1035_distinct__append,axiom,
! [Xs: list_P1396940483166286381od_a_a,Ys: list_P1396940483166286381od_a_a] :
( ( distin132333870042060960od_a_a @ ( append5335208819046833346od_a_a @ Xs @ Ys ) )
= ( ( distin132333870042060960od_a_a @ Xs )
& ( distin132333870042060960od_a_a @ Ys )
& ( ( inf_in8905007599844390133od_a_a @ ( set_Product_prod_a_a2 @ Xs ) @ ( set_Product_prod_a_a2 @ Ys ) )
= bot_bo3357376287454694259od_a_a ) ) ) ).
% distinct_append
thf(fact_1036_distinct__append,axiom,
! [Xs: list_a,Ys: list_a] :
( ( distinct_a @ ( append_a @ Xs @ Ys ) )
= ( ( distinct_a @ Xs )
& ( distinct_a @ Ys )
& ( ( inf_inf_set_a @ ( set_a2 @ Xs ) @ ( set_a2 @ Ys ) )
= bot_bot_set_a ) ) ) ).
% distinct_append
thf(fact_1037_distinct__append,axiom,
! [Xs: list_set_a,Ys: list_set_a] :
( ( distinct_set_a @ ( append_set_a @ Xs @ Ys ) )
= ( ( distinct_set_a @ Xs )
& ( distinct_set_a @ Ys )
& ( ( inf_inf_set_set_a @ ( set_set_a2 @ Xs ) @ ( set_set_a2 @ Ys ) )
= bot_bot_set_set_a ) ) ) ).
% distinct_append
thf(fact_1038_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_1039_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_1040_all__edges__disjoint,axiom,
! [S: set_Product_prod_a_a,T2: set_Product_prod_a_a] :
( ( ( inf_in8905007599844390133od_a_a @ S @ T2 )
= bot_bo3357376287454694259od_a_a )
=> ( ( inf_in3339382566020358357od_a_a @ ( undire6879232364018543115od_a_a @ S ) @ ( undire6879232364018543115od_a_a @ T2 ) )
= bot_bo777872063958040403od_a_a ) ) ).
% all_edges_disjoint
thf(fact_1041_all__edges__disjoint,axiom,
! [S: set_a,T2: set_a] :
( ( ( inf_inf_set_a @ S @ T2 )
= bot_bot_set_a )
=> ( ( inf_inf_set_set_a @ ( undire2918257014606996450dges_a @ S ) @ ( undire2918257014606996450dges_a @ T2 ) )
= bot_bot_set_set_a ) ) ).
% all_edges_disjoint
thf(fact_1042_all__edges__disjoint,axiom,
! [S: set_set_a,T2: set_set_a] :
( ( ( inf_inf_set_set_a @ S @ T2 )
= bot_bot_set_set_a )
=> ( ( inf_in1205276777018777868_set_a @ ( undire8247866692393712962_set_a @ S ) @ ( undire8247866692393712962_set_a @ T2 ) )
= bot_bo3380559777022489994_set_a ) ) ).
% all_edges_disjoint
thf(fact_1043_IntE,axiom,
! [C: set_a,A2: set_set_a,B2: set_set_a] :
( ( member_set_a @ C @ ( inf_inf_set_set_a @ A2 @ B2 ) )
=> ~ ( ( member_set_a @ C @ A2 )
=> ~ ( member_set_a @ C @ B2 ) ) ) ).
% IntE
thf(fact_1044_IntE,axiom,
! [C: product_prod_a_a,A2: set_Product_prod_a_a,B2: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ C @ ( inf_in8905007599844390133od_a_a @ A2 @ B2 ) )
=> ~ ( ( member1426531477525435216od_a_a @ C @ A2 )
=> ~ ( member1426531477525435216od_a_a @ C @ B2 ) ) ) ).
% IntE
thf(fact_1045_IntE,axiom,
! [C: nat,A2: set_nat,B2: set_nat] :
( ( member_nat @ C @ ( inf_inf_set_nat @ A2 @ B2 ) )
=> ~ ( ( member_nat @ C @ A2 )
=> ~ ( member_nat @ C @ B2 ) ) ) ).
% IntE
thf(fact_1046_IntE,axiom,
! [C: a,A2: set_a,B2: set_a] :
( ( member_a @ C @ ( inf_inf_set_a @ A2 @ B2 ) )
=> ~ ( ( member_a @ C @ A2 )
=> ~ ( member_a @ C @ B2 ) ) ) ).
% IntE
thf(fact_1047_IntD1,axiom,
! [C: set_a,A2: set_set_a,B2: set_set_a] :
( ( member_set_a @ C @ ( inf_inf_set_set_a @ A2 @ B2 ) )
=> ( member_set_a @ C @ A2 ) ) ).
% IntD1
thf(fact_1048_IntD1,axiom,
! [C: product_prod_a_a,A2: set_Product_prod_a_a,B2: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ C @ ( inf_in8905007599844390133od_a_a @ A2 @ B2 ) )
=> ( member1426531477525435216od_a_a @ C @ A2 ) ) ).
% IntD1
thf(fact_1049_IntD1,axiom,
! [C: nat,A2: set_nat,B2: set_nat] :
( ( member_nat @ C @ ( inf_inf_set_nat @ A2 @ B2 ) )
=> ( member_nat @ C @ A2 ) ) ).
% IntD1
thf(fact_1050_IntD1,axiom,
! [C: a,A2: set_a,B2: set_a] :
( ( member_a @ C @ ( inf_inf_set_a @ A2 @ B2 ) )
=> ( member_a @ C @ A2 ) ) ).
% IntD1
thf(fact_1051_IntD2,axiom,
! [C: set_a,A2: set_set_a,B2: set_set_a] :
( ( member_set_a @ C @ ( inf_inf_set_set_a @ A2 @ B2 ) )
=> ( member_set_a @ C @ B2 ) ) ).
% IntD2
thf(fact_1052_IntD2,axiom,
! [C: product_prod_a_a,A2: set_Product_prod_a_a,B2: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ C @ ( inf_in8905007599844390133od_a_a @ A2 @ B2 ) )
=> ( member1426531477525435216od_a_a @ C @ B2 ) ) ).
% IntD2
thf(fact_1053_IntD2,axiom,
! [C: nat,A2: set_nat,B2: set_nat] :
( ( member_nat @ C @ ( inf_inf_set_nat @ A2 @ B2 ) )
=> ( member_nat @ C @ B2 ) ) ).
% IntD2
thf(fact_1054_IntD2,axiom,
! [C: a,A2: set_a,B2: set_a] :
( ( member_a @ C @ ( inf_inf_set_a @ A2 @ B2 ) )
=> ( member_a @ C @ B2 ) ) ).
% IntD2
thf(fact_1055_emptyE,axiom,
! [A: nat] :
~ ( member_nat @ A @ bot_bot_set_nat ) ).
% emptyE
thf(fact_1056_emptyE,axiom,
! [A: product_prod_a_a] :
~ ( member1426531477525435216od_a_a @ A @ bot_bo3357376287454694259od_a_a ) ).
% emptyE
thf(fact_1057_emptyE,axiom,
! [A: a] :
~ ( member_a @ A @ bot_bot_set_a ) ).
% emptyE
thf(fact_1058_emptyE,axiom,
! [A: set_a] :
~ ( member_set_a @ A @ bot_bot_set_set_a ) ).
% emptyE
thf(fact_1059_equals0D,axiom,
! [A2: set_nat,A: nat] :
( ( A2 = bot_bot_set_nat )
=> ~ ( member_nat @ A @ A2 ) ) ).
% equals0D
thf(fact_1060_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_1061_equals0D,axiom,
! [A2: set_a,A: a] :
( ( A2 = bot_bot_set_a )
=> ~ ( member_a @ A @ A2 ) ) ).
% equals0D
thf(fact_1062_equals0D,axiom,
! [A2: set_set_a,A: set_a] :
( ( A2 = bot_bot_set_set_a )
=> ~ ( member_set_a @ A @ A2 ) ) ).
% equals0D
thf(fact_1063_equals0I,axiom,
! [A2: set_nat] :
( ! [Y3: nat] :
~ ( member_nat @ Y3 @ A2 )
=> ( A2 = bot_bot_set_nat ) ) ).
% equals0I
thf(fact_1064_equals0I,axiom,
! [A2: set_Product_prod_a_a] :
( ! [Y3: product_prod_a_a] :
~ ( member1426531477525435216od_a_a @ Y3 @ A2 )
=> ( A2 = bot_bo3357376287454694259od_a_a ) ) ).
% equals0I
thf(fact_1065_equals0I,axiom,
! [A2: set_a] :
( ! [Y3: a] :
~ ( member_a @ Y3 @ A2 )
=> ( A2 = bot_bot_set_a ) ) ).
% equals0I
thf(fact_1066_equals0I,axiom,
! [A2: set_set_a] :
( ! [Y3: set_a] :
~ ( member_set_a @ Y3 @ A2 )
=> ( A2 = bot_bot_set_set_a ) ) ).
% equals0I
thf(fact_1067_Int__assoc,axiom,
! [A2: set_a,B2: set_a,C2: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ A2 @ B2 ) @ C2 )
= ( inf_inf_set_a @ A2 @ ( inf_inf_set_a @ B2 @ C2 ) ) ) ).
% Int_assoc
thf(fact_1068_Int__absorb,axiom,
! [A2: set_a] :
( ( inf_inf_set_a @ A2 @ A2 )
= A2 ) ).
% Int_absorb
thf(fact_1069_Int__emptyI,axiom,
! [A2: set_nat,B2: set_nat] :
( ! [X4: nat] :
( ( member_nat @ X4 @ A2 )
=> ~ ( member_nat @ X4 @ B2 ) )
=> ( ( inf_inf_set_nat @ A2 @ B2 )
= bot_bot_set_nat ) ) ).
% Int_emptyI
thf(fact_1070_Int__emptyI,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 ) )
=> ( ( inf_in8905007599844390133od_a_a @ A2 @ B2 )
= bot_bo3357376287454694259od_a_a ) ) ).
% Int_emptyI
thf(fact_1071_Int__emptyI,axiom,
! [A2: set_a,B2: set_a] :
( ! [X4: a] :
( ( member_a @ X4 @ A2 )
=> ~ ( member_a @ X4 @ B2 ) )
=> ( ( inf_inf_set_a @ A2 @ B2 )
= bot_bot_set_a ) ) ).
% Int_emptyI
thf(fact_1072_Int__emptyI,axiom,
! [A2: set_set_a,B2: set_set_a] :
( ! [X4: set_a] :
( ( member_set_a @ X4 @ A2 )
=> ~ ( member_set_a @ X4 @ B2 ) )
=> ( ( inf_inf_set_set_a @ A2 @ B2 )
= bot_bot_set_set_a ) ) ).
% Int_emptyI
thf(fact_1073_ex__in__conv,axiom,
! [A2: set_nat] :
( ( ? [X3: nat] : ( member_nat @ X3 @ A2 ) )
= ( A2 != bot_bot_set_nat ) ) ).
% ex_in_conv
thf(fact_1074_ex__in__conv,axiom,
! [A2: set_Product_prod_a_a] :
( ( ? [X3: product_prod_a_a] : ( member1426531477525435216od_a_a @ X3 @ A2 ) )
= ( A2 != bot_bo3357376287454694259od_a_a ) ) ).
% ex_in_conv
thf(fact_1075_ex__in__conv,axiom,
! [A2: set_a] :
( ( ? [X3: a] : ( member_a @ X3 @ A2 ) )
= ( A2 != bot_bot_set_a ) ) ).
% ex_in_conv
thf(fact_1076_ex__in__conv,axiom,
! [A2: set_set_a] :
( ( ? [X3: set_a] : ( member_set_a @ X3 @ A2 ) )
= ( A2 != bot_bot_set_set_a ) ) ).
% ex_in_conv
thf(fact_1077_Int__commute,axiom,
( inf_inf_set_a
= ( ^ [A5: set_a,B5: set_a] : ( inf_inf_set_a @ B5 @ A5 ) ) ) ).
% Int_commute
thf(fact_1078_disjoint__iff,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ( inf_inf_set_nat @ A2 @ B2 )
= bot_bot_set_nat )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ~ ( member_nat @ X3 @ B2 ) ) ) ) ).
% disjoint_iff
thf(fact_1079_disjoint__iff,axiom,
! [A2: set_Product_prod_a_a,B2: set_Product_prod_a_a] :
( ( ( inf_in8905007599844390133od_a_a @ A2 @ B2 )
= bot_bo3357376287454694259od_a_a )
= ( ! [X3: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X3 @ A2 )
=> ~ ( member1426531477525435216od_a_a @ X3 @ B2 ) ) ) ) ).
% disjoint_iff
thf(fact_1080_disjoint__iff,axiom,
! [A2: set_a,B2: set_a] :
( ( ( inf_inf_set_a @ A2 @ B2 )
= bot_bot_set_a )
= ( ! [X3: a] :
( ( member_a @ X3 @ A2 )
=> ~ ( member_a @ X3 @ B2 ) ) ) ) ).
% disjoint_iff
thf(fact_1081_disjoint__iff,axiom,
! [A2: set_set_a,B2: set_set_a] :
( ( ( inf_inf_set_set_a @ A2 @ B2 )
= bot_bot_set_set_a )
= ( ! [X3: set_a] :
( ( member_set_a @ X3 @ A2 )
=> ~ ( member_set_a @ X3 @ B2 ) ) ) ) ).
% disjoint_iff
thf(fact_1082_Int__empty__left,axiom,
! [B2: set_Product_prod_a_a] :
( ( inf_in8905007599844390133od_a_a @ bot_bo3357376287454694259od_a_a @ B2 )
= bot_bo3357376287454694259od_a_a ) ).
% Int_empty_left
thf(fact_1083_Int__empty__left,axiom,
! [B2: set_a] :
( ( inf_inf_set_a @ bot_bot_set_a @ B2 )
= bot_bot_set_a ) ).
% Int_empty_left
thf(fact_1084_Int__empty__left,axiom,
! [B2: set_set_a] :
( ( inf_inf_set_set_a @ bot_bot_set_set_a @ B2 )
= bot_bot_set_set_a ) ).
% Int_empty_left
thf(fact_1085_Int__empty__right,axiom,
! [A2: set_Product_prod_a_a] :
( ( inf_in8905007599844390133od_a_a @ A2 @ bot_bo3357376287454694259od_a_a )
= bot_bo3357376287454694259od_a_a ) ).
% Int_empty_right
thf(fact_1086_Int__empty__right,axiom,
! [A2: set_a] :
( ( inf_inf_set_a @ A2 @ bot_bot_set_a )
= bot_bot_set_a ) ).
% Int_empty_right
thf(fact_1087_Int__empty__right,axiom,
! [A2: set_set_a] :
( ( inf_inf_set_set_a @ A2 @ bot_bot_set_set_a )
= bot_bot_set_set_a ) ).
% Int_empty_right
thf(fact_1088_Int__left__absorb,axiom,
! [A2: set_a,B2: set_a] :
( ( inf_inf_set_a @ A2 @ ( inf_inf_set_a @ A2 @ B2 ) )
= ( inf_inf_set_a @ A2 @ B2 ) ) ).
% Int_left_absorb
thf(fact_1089_Int__left__commute,axiom,
! [A2: set_a,B2: set_a,C2: set_a] :
( ( inf_inf_set_a @ A2 @ ( inf_inf_set_a @ B2 @ C2 ) )
= ( inf_inf_set_a @ B2 @ ( inf_inf_set_a @ A2 @ C2 ) ) ) ).
% Int_left_commute
thf(fact_1090_disjoint__iff__not__equal,axiom,
! [A2: set_Product_prod_a_a,B2: set_Product_prod_a_a] :
( ( ( inf_in8905007599844390133od_a_a @ A2 @ B2 )
= bot_bo3357376287454694259od_a_a )
= ( ! [X3: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X3 @ A2 )
=> ! [Y4: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ Y4 @ B2 )
=> ( X3 != Y4 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_1091_disjoint__iff__not__equal,axiom,
! [A2: set_a,B2: set_a] :
( ( ( inf_inf_set_a @ A2 @ B2 )
= bot_bot_set_a )
= ( ! [X3: a] :
( ( member_a @ X3 @ A2 )
=> ! [Y4: a] :
( ( member_a @ Y4 @ B2 )
=> ( X3 != Y4 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_1092_disjoint__iff__not__equal,axiom,
! [A2: set_set_a,B2: set_set_a] :
( ( ( inf_inf_set_set_a @ A2 @ B2 )
= bot_bot_set_set_a )
= ( ! [X3: set_a] :
( ( member_set_a @ X3 @ A2 )
=> ! [Y4: set_a] :
( ( member_set_a @ Y4 @ B2 )
=> ( X3 != Y4 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_1093_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_1094_bot_Oextremum__uniqueI,axiom,
! [A: set_list_a] :
( ( ord_le8861187494160871172list_a @ A @ bot_bot_set_list_a )
=> ( A = bot_bot_set_list_a ) ) ).
% bot.extremum_uniqueI
thf(fact_1095_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_1096_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_1097_bot_Oextremum__uniqueI,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ bot_bot_nat )
=> ( A = bot_bot_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_1098_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_1099_bot_Oextremum__unique,axiom,
! [A: set_list_a] :
( ( ord_le8861187494160871172list_a @ A @ bot_bot_set_list_a )
= ( A = bot_bot_set_list_a ) ) ).
% bot.extremum_unique
thf(fact_1100_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_1101_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_1102_bot_Oextremum__unique,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ bot_bot_nat )
= ( A = bot_bot_nat ) ) ).
% bot.extremum_unique
thf(fact_1103_bot_Oextremum,axiom,
! [A: set_Product_prod_a_a] : ( ord_le746702958409616551od_a_a @ bot_bo3357376287454694259od_a_a @ A ) ).
% bot.extremum
thf(fact_1104_bot_Oextremum,axiom,
! [A: set_list_a] : ( ord_le8861187494160871172list_a @ bot_bot_set_list_a @ A ) ).
% bot.extremum
thf(fact_1105_bot_Oextremum,axiom,
! [A: set_set_a] : ( ord_le3724670747650509150_set_a @ bot_bot_set_set_a @ A ) ).
% bot.extremum
thf(fact_1106_bot_Oextremum,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A ) ).
% bot.extremum
thf(fact_1107_bot_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ bot_bot_nat @ A ) ).
% bot.extremum
thf(fact_1108_ulgraph_Oall__edges__between__empty_I1_J,axiom,
! [Vertices: set_Product_prod_a_a,Edges: set_se5735800977113168103od_a_a,Z4: set_Product_prod_a_a] :
( ( undire4585262585102564309od_a_a @ Vertices @ Edges )
=> ( ( undire4032395788819567636od_a_a @ Edges @ bot_bo3357376287454694259od_a_a @ Z4 )
= bot_bo510284599550014259od_a_a ) ) ).
% ulgraph.all_edges_between_empty(1)
thf(fact_1109_ulgraph_Oall__edges__between__empty_I1_J,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,Z4: set_set_a] :
( ( undire6886684016831807756_set_a @ Vertices @ Edges )
=> ( ( undire2462398226299384907_set_a @ Edges @ bot_bot_set_set_a @ Z4 )
= bot_bo5799363139946352499_set_a ) ) ).
% ulgraph.all_edges_between_empty(1)
thf(fact_1110_ulgraph_Oall__edges__between__empty_I1_J,axiom,
! [Vertices: set_a,Edges: set_set_a,Z4: set_a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( undire8383842906760478443ween_a @ Edges @ bot_bot_set_a @ Z4 )
= bot_bo3357376287454694259od_a_a ) ) ).
% ulgraph.all_edges_between_empty(1)
thf(fact_1111_ulgraph_Oall__edges__between__empty_I2_J,axiom,
! [Vertices: set_Product_prod_a_a,Edges: set_se5735800977113168103od_a_a,Z4: set_Product_prod_a_a] :
( ( undire4585262585102564309od_a_a @ Vertices @ Edges )
=> ( ( undire4032395788819567636od_a_a @ Edges @ Z4 @ bot_bo3357376287454694259od_a_a )
= bot_bo510284599550014259od_a_a ) ) ).
% ulgraph.all_edges_between_empty(2)
thf(fact_1112_ulgraph_Oall__edges__between__empty_I2_J,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,Z4: set_set_a] :
( ( undire6886684016831807756_set_a @ Vertices @ Edges )
=> ( ( undire2462398226299384907_set_a @ Edges @ Z4 @ bot_bot_set_set_a )
= bot_bo5799363139946352499_set_a ) ) ).
% ulgraph.all_edges_between_empty(2)
thf(fact_1113_ulgraph_Oall__edges__between__empty_I2_J,axiom,
! [Vertices: set_a,Edges: set_set_a,Z4: set_a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( undire8383842906760478443ween_a @ Edges @ Z4 @ bot_bot_set_a )
= bot_bo3357376287454694259od_a_a ) ) ).
% ulgraph.all_edges_between_empty(2)
thf(fact_1114_Int__Collect__mono,axiom,
! [A2: set_nat,B2: set_nat,P2: nat > $o,Q: nat > $o] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ A2 )
=> ( ( P2 @ X4 )
=> ( Q @ X4 ) ) )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A2 @ ( collect_nat @ P2 ) ) @ ( inf_inf_set_nat @ B2 @ ( collect_nat @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_1115_Int__Collect__mono,axiom,
! [A2: set_Product_prod_a_a,B2: set_Product_prod_a_a,P2: product_prod_a_a > $o,Q: product_prod_a_a > $o] :
( ( ord_le746702958409616551od_a_a @ A2 @ B2 )
=> ( ! [X4: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X4 @ A2 )
=> ( ( P2 @ X4 )
=> ( Q @ X4 ) ) )
=> ( ord_le746702958409616551od_a_a @ ( inf_in8905007599844390133od_a_a @ A2 @ ( collec3336397797384452498od_a_a @ P2 ) ) @ ( inf_in8905007599844390133od_a_a @ B2 @ ( collec3336397797384452498od_a_a @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_1116_Int__Collect__mono,axiom,
! [A2: set_list_a,B2: set_list_a,P2: list_a > $o,Q: list_a > $o] :
( ( ord_le8861187494160871172list_a @ A2 @ B2 )
=> ( ! [X4: list_a] :
( ( member_list_a @ X4 @ A2 )
=> ( ( P2 @ X4 )
=> ( Q @ X4 ) ) )
=> ( ord_le8861187494160871172list_a @ ( inf_inf_set_list_a @ A2 @ ( collect_list_a @ P2 ) ) @ ( inf_inf_set_list_a @ B2 @ ( collect_list_a @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_1117_Int__Collect__mono,axiom,
! [A2: set_set_a,B2: set_set_a,P2: set_a > $o,Q: set_a > $o] :
( ( ord_le3724670747650509150_set_a @ A2 @ B2 )
=> ( ! [X4: set_a] :
( ( member_set_a @ X4 @ A2 )
=> ( ( P2 @ X4 )
=> ( Q @ X4 ) ) )
=> ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ A2 @ ( collect_set_a @ P2 ) ) @ ( inf_inf_set_set_a @ B2 @ ( collect_set_a @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_1118_Int__Collect__mono,axiom,
! [A2: set_a,B2: set_a,P2: a > $o,Q: a > $o] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ! [X4: a] :
( ( member_a @ X4 @ A2 )
=> ( ( P2 @ X4 )
=> ( Q @ X4 ) ) )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ ( collect_a @ P2 ) ) @ ( inf_inf_set_a @ B2 @ ( collect_a @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_1119_Int__greatest,axiom,
! [C2: set_Product_prod_a_a,A2: set_Product_prod_a_a,B2: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ C2 @ A2 )
=> ( ( ord_le746702958409616551od_a_a @ C2 @ B2 )
=> ( ord_le746702958409616551od_a_a @ C2 @ ( inf_in8905007599844390133od_a_a @ A2 @ B2 ) ) ) ) ).
% Int_greatest
thf(fact_1120_Int__greatest,axiom,
! [C2: set_list_a,A2: set_list_a,B2: set_list_a] :
( ( ord_le8861187494160871172list_a @ C2 @ A2 )
=> ( ( ord_le8861187494160871172list_a @ C2 @ B2 )
=> ( ord_le8861187494160871172list_a @ C2 @ ( inf_inf_set_list_a @ A2 @ B2 ) ) ) ) ).
% Int_greatest
thf(fact_1121_Int__greatest,axiom,
! [C2: set_set_a,A2: set_set_a,B2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ C2 @ A2 )
=> ( ( ord_le3724670747650509150_set_a @ C2 @ B2 )
=> ( ord_le3724670747650509150_set_a @ C2 @ ( inf_inf_set_set_a @ A2 @ B2 ) ) ) ) ).
% Int_greatest
thf(fact_1122_Int__greatest,axiom,
! [C2: set_a,A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ C2 @ A2 )
=> ( ( ord_less_eq_set_a @ C2 @ B2 )
=> ( ord_less_eq_set_a @ C2 @ ( inf_inf_set_a @ A2 @ B2 ) ) ) ) ).
% Int_greatest
thf(fact_1123_Int__absorb2,axiom,
! [A2: set_Product_prod_a_a,B2: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A2 @ B2 )
=> ( ( inf_in8905007599844390133od_a_a @ A2 @ B2 )
= A2 ) ) ).
% Int_absorb2
thf(fact_1124_Int__absorb2,axiom,
! [A2: set_list_a,B2: set_list_a] :
( ( ord_le8861187494160871172list_a @ A2 @ B2 )
=> ( ( inf_inf_set_list_a @ A2 @ B2 )
= A2 ) ) ).
% Int_absorb2
thf(fact_1125_Int__absorb2,axiom,
! [A2: set_set_a,B2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B2 )
=> ( ( inf_inf_set_set_a @ A2 @ B2 )
= A2 ) ) ).
% Int_absorb2
thf(fact_1126_Int__absorb2,axiom,
! [A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( inf_inf_set_a @ A2 @ B2 )
= A2 ) ) ).
% Int_absorb2
thf(fact_1127_Int__absorb1,axiom,
! [B2: set_Product_prod_a_a,A2: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ B2 @ A2 )
=> ( ( inf_in8905007599844390133od_a_a @ A2 @ B2 )
= B2 ) ) ).
% Int_absorb1
thf(fact_1128_Int__absorb1,axiom,
! [B2: set_list_a,A2: set_list_a] :
( ( ord_le8861187494160871172list_a @ B2 @ A2 )
=> ( ( inf_inf_set_list_a @ A2 @ B2 )
= B2 ) ) ).
% Int_absorb1
thf(fact_1129_Int__absorb1,axiom,
! [B2: set_set_a,A2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B2 @ A2 )
=> ( ( inf_inf_set_set_a @ A2 @ B2 )
= B2 ) ) ).
% Int_absorb1
thf(fact_1130_Int__absorb1,axiom,
! [B2: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ B2 @ A2 )
=> ( ( inf_inf_set_a @ A2 @ B2 )
= B2 ) ) ).
% Int_absorb1
thf(fact_1131_Int__lower2,axiom,
! [A2: set_Product_prod_a_a,B2: set_Product_prod_a_a] : ( ord_le746702958409616551od_a_a @ ( inf_in8905007599844390133od_a_a @ A2 @ B2 ) @ B2 ) ).
% Int_lower2
thf(fact_1132_Int__lower2,axiom,
! [A2: set_list_a,B2: set_list_a] : ( ord_le8861187494160871172list_a @ ( inf_inf_set_list_a @ A2 @ B2 ) @ B2 ) ).
% Int_lower2
thf(fact_1133_Int__lower2,axiom,
! [A2: set_set_a,B2: set_set_a] : ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ A2 @ B2 ) @ B2 ) ).
% Int_lower2
thf(fact_1134_Int__lower2,axiom,
! [A2: set_a,B2: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B2 ) @ B2 ) ).
% Int_lower2
thf(fact_1135_Int__lower1,axiom,
! [A2: set_Product_prod_a_a,B2: set_Product_prod_a_a] : ( ord_le746702958409616551od_a_a @ ( inf_in8905007599844390133od_a_a @ A2 @ B2 ) @ A2 ) ).
% Int_lower1
thf(fact_1136_Int__lower1,axiom,
! [A2: set_list_a,B2: set_list_a] : ( ord_le8861187494160871172list_a @ ( inf_inf_set_list_a @ A2 @ B2 ) @ A2 ) ).
% Int_lower1
thf(fact_1137_Int__lower1,axiom,
! [A2: set_set_a,B2: set_set_a] : ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ A2 @ B2 ) @ A2 ) ).
% Int_lower1
thf(fact_1138_Int__lower1,axiom,
! [A2: set_a,B2: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B2 ) @ A2 ) ).
% Int_lower1
thf(fact_1139_Int__mono,axiom,
! [A2: set_Product_prod_a_a,C2: set_Product_prod_a_a,B2: set_Product_prod_a_a,D: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A2 @ C2 )
=> ( ( ord_le746702958409616551od_a_a @ B2 @ D )
=> ( ord_le746702958409616551od_a_a @ ( inf_in8905007599844390133od_a_a @ A2 @ B2 ) @ ( inf_in8905007599844390133od_a_a @ C2 @ D ) ) ) ) ).
% Int_mono
thf(fact_1140_Int__mono,axiom,
! [A2: set_list_a,C2: set_list_a,B2: set_list_a,D: set_list_a] :
( ( ord_le8861187494160871172list_a @ A2 @ C2 )
=> ( ( ord_le8861187494160871172list_a @ B2 @ D )
=> ( ord_le8861187494160871172list_a @ ( inf_inf_set_list_a @ A2 @ B2 ) @ ( inf_inf_set_list_a @ C2 @ D ) ) ) ) ).
% Int_mono
thf(fact_1141_Int__mono,axiom,
! [A2: set_set_a,C2: set_set_a,B2: set_set_a,D: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ C2 )
=> ( ( ord_le3724670747650509150_set_a @ B2 @ D )
=> ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ A2 @ B2 ) @ ( inf_inf_set_set_a @ C2 @ D ) ) ) ) ).
% Int_mono
thf(fact_1142_Int__mono,axiom,
! [A2: set_a,C2: set_a,B2: set_a,D: set_a] :
( ( ord_less_eq_set_a @ A2 @ C2 )
=> ( ( ord_less_eq_set_a @ B2 @ D )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B2 ) @ ( inf_inf_set_a @ C2 @ D ) ) ) ) ).
% Int_mono
thf(fact_1143_comp__sgraph_Oedge__adj__def,axiom,
! [S: set_Product_prod_a_a,E1: set_Product_prod_a_a,E2: set_Product_prod_a_a] :
( ( undire9186443406341554371od_a_a @ ( undire6879232364018543115od_a_a @ S ) @ E1 @ E2 )
= ( ( ( inf_in8905007599844390133od_a_a @ E1 @ E2 )
!= bot_bo3357376287454694259od_a_a )
& ( member1816616512716248880od_a_a @ E1 @ ( undire6879232364018543115od_a_a @ S ) )
& ( member1816616512716248880od_a_a @ E2 @ ( undire6879232364018543115od_a_a @ S ) ) ) ) ).
% comp_sgraph.edge_adj_def
thf(fact_1144_comp__sgraph_Oedge__adj__def,axiom,
! [S: set_set_a,E1: set_set_a,E2: set_set_a] :
( ( undire3485422320110889978_set_a @ ( undire8247866692393712962_set_a @ S ) @ E1 @ E2 )
= ( ( ( inf_inf_set_set_a @ E1 @ E2 )
!= bot_bot_set_set_a )
& ( member_set_set_a @ E1 @ ( undire8247866692393712962_set_a @ S ) )
& ( member_set_set_a @ E2 @ ( undire8247866692393712962_set_a @ S ) ) ) ) ).
% comp_sgraph.edge_adj_def
thf(fact_1145_comp__sgraph_Oedge__adj__def,axiom,
! [S: set_a,E1: set_a,E2: set_a] :
( ( undire4022703626023482010_adj_a @ ( undire2918257014606996450dges_a @ S ) @ E1 @ E2 )
= ( ( ( inf_inf_set_a @ E1 @ E2 )
!= bot_bot_set_a )
& ( member_set_a @ E1 @ ( undire2918257014606996450dges_a @ S ) )
& ( member_set_a @ E2 @ ( undire2918257014606996450dges_a @ S ) ) ) ) ).
% comp_sgraph.edge_adj_def
thf(fact_1146_graph__system_Oedge__adj__def,axiom,
! [Vertices: set_Product_prod_a_a,Edges: set_se5735800977113168103od_a_a,E1: set_Product_prod_a_a,E2: set_Product_prod_a_a] :
( ( undire1860116983885411791od_a_a @ Vertices @ Edges )
=> ( ( undire9186443406341554371od_a_a @ Edges @ E1 @ E2 )
= ( ( ( inf_in8905007599844390133od_a_a @ E1 @ E2 )
!= bot_bo3357376287454694259od_a_a )
& ( member1816616512716248880od_a_a @ E1 @ Edges )
& ( member1816616512716248880od_a_a @ E2 @ Edges ) ) ) ) ).
% graph_system.edge_adj_def
thf(fact_1147_graph__system_Oedge__adj__def,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,E1: set_set_a,E2: set_set_a] :
( ( undire7159349782766787846_set_a @ Vertices @ Edges )
=> ( ( undire3485422320110889978_set_a @ Edges @ E1 @ E2 )
= ( ( ( inf_inf_set_set_a @ E1 @ E2 )
!= bot_bot_set_set_a )
& ( member_set_set_a @ E1 @ Edges )
& ( member_set_set_a @ E2 @ Edges ) ) ) ) ).
% graph_system.edge_adj_def
thf(fact_1148_graph__system_Oedge__adj__def,axiom,
! [Vertices: set_a,Edges: set_set_a,E1: set_a,E2: set_a] :
( ( undire2554140024507503526stem_a @ Vertices @ Edges )
=> ( ( undire4022703626023482010_adj_a @ Edges @ E1 @ E2 )
= ( ( ( inf_inf_set_a @ E1 @ E2 )
!= bot_bot_set_a )
& ( member_set_a @ E1 @ Edges )
& ( member_set_a @ E2 @ Edges ) ) ) ) ).
% graph_system.edge_adj_def
thf(fact_1149_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_1150_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_1151_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_1152_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_1153_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_1154_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_1155_comp__sgraph_Oall__edges__between__rem__wf,axiom,
! [S: set_a,X5: set_a,Y2: set_a] :
( ( undire8383842906760478443ween_a @ ( undire2918257014606996450dges_a @ S ) @ X5 @ Y2 )
= ( undire8383842906760478443ween_a @ ( undire2918257014606996450dges_a @ S ) @ ( inf_inf_set_a @ X5 @ S ) @ ( inf_inf_set_a @ Y2 @ S ) ) ) ).
% comp_sgraph.all_edges_between_rem_wf
thf(fact_1156_ulgraph_Oall__edges__between__rem__wf,axiom,
! [Vertices: set_a,Edges: set_set_a,X5: set_a,Y2: set_a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( undire8383842906760478443ween_a @ Edges @ X5 @ Y2 )
= ( undire8383842906760478443ween_a @ Edges @ ( inf_inf_set_a @ X5 @ Vertices ) @ ( inf_inf_set_a @ Y2 @ Vertices ) ) ) ) ).
% ulgraph.all_edges_between_rem_wf
thf(fact_1157_empty__set,axiom,
( bot_bo3357376287454694259od_a_a
= ( set_Product_prod_a_a2 @ nil_Product_prod_a_a ) ) ).
% empty_set
thf(fact_1158_empty__set,axiom,
( bot_bot_set_a
= ( set_a2 @ nil_a ) ) ).
% empty_set
thf(fact_1159_empty__set,axiom,
( bot_bot_set_set_a
= ( set_set_a2 @ nil_set_a ) ) ).
% empty_set
thf(fact_1160_zero__reorient,axiom,
! [X: real] :
( ( zero_zero_real = X )
= ( X = zero_zero_real ) ) ).
% zero_reorient
thf(fact_1161_zero__reorient,axiom,
! [X: nat] :
( ( zero_zero_nat = X )
= ( X = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_1162_one__reorient,axiom,
! [X: real] :
( ( one_one_real = X )
= ( X = one_one_real ) ) ).
% one_reorient
thf(fact_1163_one__reorient,axiom,
! [X: nat] :
( ( one_one_nat = X )
= ( X = one_one_nat ) ) ).
% one_reorient
thf(fact_1164_comp__sgraph_Oedge__density__zero,axiom,
! [Y2: set_Product_prod_a_a,S: set_Product_prod_a_a,X5: set_Product_prod_a_a] :
( ( Y2 = bot_bo3357376287454694259od_a_a )
=> ( ( undire8410861505230878716od_a_a @ ( undire6879232364018543115od_a_a @ S ) @ X5 @ Y2 )
= zero_zero_real ) ) ).
% comp_sgraph.edge_density_zero
thf(fact_1165_comp__sgraph_Oedge__density__zero,axiom,
! [Y2: set_set_a,S: set_set_a,X5: set_set_a] :
( ( Y2 = bot_bot_set_set_a )
=> ( ( undire8927637694342045747_set_a @ ( undire8247866692393712962_set_a @ S ) @ X5 @ Y2 )
= zero_zero_real ) ) ).
% comp_sgraph.edge_density_zero
thf(fact_1166_comp__sgraph_Oedge__density__zero,axiom,
! [Y2: set_a,S: set_a,X5: set_a] :
( ( Y2 = bot_bot_set_a )
=> ( ( undire297304480579013331sity_a @ ( undire2918257014606996450dges_a @ S ) @ X5 @ Y2 )
= zero_zero_real ) ) ).
% comp_sgraph.edge_density_zero
thf(fact_1167_ulgraph_Oedge__density__zero,axiom,
! [Vertices: set_Product_prod_a_a,Edges: set_se5735800977113168103od_a_a,Y2: set_Product_prod_a_a,X5: set_Product_prod_a_a] :
( ( undire4585262585102564309od_a_a @ Vertices @ Edges )
=> ( ( Y2 = bot_bo3357376287454694259od_a_a )
=> ( ( undire8410861505230878716od_a_a @ Edges @ X5 @ Y2 )
= zero_zero_real ) ) ) ).
% ulgraph.edge_density_zero
thf(fact_1168_ulgraph_Oedge__density__zero,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,Y2: set_set_a,X5: set_set_a] :
( ( undire6886684016831807756_set_a @ Vertices @ Edges )
=> ( ( Y2 = bot_bot_set_set_a )
=> ( ( undire8927637694342045747_set_a @ Edges @ X5 @ Y2 )
= zero_zero_real ) ) ) ).
% ulgraph.edge_density_zero
thf(fact_1169_ulgraph_Oedge__density__zero,axiom,
! [Vertices: set_a,Edges: set_set_a,Y2: set_a,X5: set_a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( Y2 = bot_bot_set_a )
=> ( ( undire297304480579013331sity_a @ Edges @ X5 @ Y2 )
= zero_zero_real ) ) ) ).
% ulgraph.edge_density_zero
thf(fact_1170_n__lists_Osimps_I1_J,axiom,
! [Xs: list_a] :
( ( n_lists_a @ zero_zero_nat @ Xs )
= ( cons_list_a @ nil_a @ nil_list_a ) ) ).
% n_lists.simps(1)
thf(fact_1171_n__lists_Osimps_I1_J,axiom,
! [Xs: list_set_a] :
( ( n_lists_set_a @ zero_zero_nat @ Xs )
= ( cons_list_set_a @ nil_set_a @ nil_list_set_a ) ) ).
% n_lists.simps(1)
thf(fact_1172_comp__sgraph_Oedge__density__eq0,axiom,
! [S: set_set_a,A2: set_set_a,B2: set_set_a,X5: set_set_a,Y2: set_set_a] :
( ( ( undire2462398226299384907_set_a @ ( undire8247866692393712962_set_a @ S ) @ A2 @ B2 )
= bot_bo5799363139946352499_set_a )
=> ( ( ord_le3724670747650509150_set_a @ X5 @ A2 )
=> ( ( ord_le3724670747650509150_set_a @ Y2 @ B2 )
=> ( ( undire8927637694342045747_set_a @ ( undire8247866692393712962_set_a @ S ) @ X5 @ Y2 )
= zero_zero_real ) ) ) ) ).
% comp_sgraph.edge_density_eq0
thf(fact_1173_comp__sgraph_Oedge__density__eq0,axiom,
! [S: set_a,A2: set_a,B2: set_a,X5: set_a,Y2: set_a] :
( ( ( undire8383842906760478443ween_a @ ( undire2918257014606996450dges_a @ S ) @ A2 @ B2 )
= bot_bo3357376287454694259od_a_a )
=> ( ( ord_less_eq_set_a @ X5 @ A2 )
=> ( ( ord_less_eq_set_a @ Y2 @ B2 )
=> ( ( undire297304480579013331sity_a @ ( undire2918257014606996450dges_a @ S ) @ X5 @ Y2 )
= zero_zero_real ) ) ) ) ).
% comp_sgraph.edge_density_eq0
thf(fact_1174_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_1175_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_1176_walk__edges__app,axiom,
! [Xs: list_a,Y: a,X: a] :
( ( undire7337870655677353998dges_a @ ( append_a @ Xs @ ( cons_a @ Y @ ( cons_a @ X @ nil_a ) ) ) )
= ( append_set_a @ ( undire7337870655677353998dges_a @ ( append_a @ Xs @ ( cons_a @ Y @ nil_a ) ) ) @ ( cons_set_a @ ( insert_a2 @ Y @ ( insert_a2 @ X @ bot_bot_set_a ) ) @ nil_set_a ) ) ) ).
% walk_edges_app
thf(fact_1177_walk__edges__singleton__app,axiom,
! [Ys: list_a,X: a] :
( ( Ys != nil_a )
=> ( ( undire7337870655677353998dges_a @ ( append_a @ ( cons_a @ X @ nil_a ) @ Ys ) )
= ( cons_set_a @ ( insert_a2 @ X @ ( insert_a2 @ ( hd_a @ Ys ) @ bot_bot_set_a ) ) @ ( undire7337870655677353998dges_a @ Ys ) ) ) ) ).
% walk_edges_singleton_app
thf(fact_1178_is__walk__hd__tl,axiom,
! [Y: a,Ys: list_a,X: a] :
( ( undire6133010728901294956walk_a @ vertices @ edges @ ( cons_a @ Y @ Ys ) )
=> ( ( member_set_a @ ( insert_a2 @ X @ ( insert_a2 @ Y @ bot_bot_set_a ) ) @ edges )
=> ( undire6133010728901294956walk_a @ vertices @ edges @ ( cons_a @ X @ ( cons_a @ Y @ Ys ) ) ) ) ) ).
% is_walk_hd_tl
thf(fact_1179_not__vert__adj,axiom,
! [V: a,U: a] :
( ~ ( undire397441198561214472_adj_a @ edges @ V @ U )
=> ~ ( member_set_a @ ( insert_a2 @ V @ ( insert_a2 @ U @ bot_bot_set_a ) ) @ edges ) ) ).
% not_vert_adj
thf(fact_1180_vert__adj__def,axiom,
! [V1: a,V2: a] :
( ( undire397441198561214472_adj_a @ edges @ V1 @ V2 )
= ( member_set_a @ ( insert_a2 @ V1 @ ( insert_a2 @ V2 @ bot_bot_set_a ) ) @ edges ) ) ).
% vert_adj_def
thf(fact_1181_has__loop__def,axiom,
! [V: a] :
( ( undire3617971648856834880loop_a @ edges @ V )
= ( member_set_a @ ( insert_a2 @ V @ bot_bot_set_a ) @ edges ) ) ).
% has_loop_def
thf(fact_1182_wellformed__alt__fst,axiom,
! [X: a,Y: a] :
( ( member_set_a @ ( insert_a2 @ X @ ( insert_a2 @ Y @ bot_bot_set_a ) ) @ edges )
=> ( member_a @ X @ vertices ) ) ).
% wellformed_alt_fst
thf(fact_1183_wellformed__alt__snd,axiom,
! [X: a,Y: a] :
( ( member_set_a @ ( insert_a2 @ X @ ( insert_a2 @ Y @ bot_bot_set_a ) ) @ edges )
=> ( member_a @ Y @ vertices ) ) ).
% wellformed_alt_snd
thf(fact_1184_is__edge__between__def,axiom,
( undire8544646567961481629ween_a
= ( ^ [X6: set_a,Y6: set_a,E6: set_a] :
? [X3: a,Y4: a] :
( ( E6
= ( insert_a2 @ X3 @ ( insert_a2 @ Y4 @ bot_bot_set_a ) ) )
& ( member_a @ X3 @ X6 )
& ( member_a @ Y4 @ Y6 ) ) ) ) ).
% is_edge_between_def
thf(fact_1185_walk__edges_Osimps_I3_J,axiom,
! [X: a,Y: a,Ys: list_a] :
( ( undire7337870655677353998dges_a @ ( cons_a @ X @ ( cons_a @ Y @ Ys ) ) )
= ( cons_set_a @ ( insert_a2 @ X @ ( insert_a2 @ Y @ bot_bot_set_a ) ) @ ( undire7337870655677353998dges_a @ ( cons_a @ Y @ Ys ) ) ) ) ).
% walk_edges.simps(3)
thf(fact_1186_vert__adj__inc__edge__iff,axiom,
! [V1: a,V2: a] :
( ( undire397441198561214472_adj_a @ edges @ V1 @ V2 )
= ( ( undire1521409233611534436dent_a @ V1 @ ( insert_a2 @ V1 @ ( insert_a2 @ V2 @ bot_bot_set_a ) ) )
& ( undire1521409233611534436dent_a @ V2 @ ( insert_a2 @ V1 @ ( insert_a2 @ V2 @ bot_bot_set_a ) ) )
& ( member_set_a @ ( insert_a2 @ V1 @ ( insert_a2 @ V2 @ bot_bot_set_a ) ) @ edges ) ) ) ).
% vert_adj_inc_edge_iff
thf(fact_1187_walk__edges_Oelims,axiom,
! [X: list_a,Y: list_set_a] :
( ( ( undire7337870655677353998dges_a @ X )
= Y )
=> ( ( ( X = nil_a )
=> ( Y != nil_set_a ) )
=> ( ( ? [X4: a] :
( X
= ( cons_a @ X4 @ nil_a ) )
=> ( Y != nil_set_a ) )
=> ~ ! [X4: a,Y3: a,Ys3: list_a] :
( ( X
= ( cons_a @ X4 @ ( cons_a @ Y3 @ Ys3 ) ) )
=> ( Y
!= ( cons_set_a @ ( insert_a2 @ X4 @ ( insert_a2 @ Y3 @ bot_bot_set_a ) ) @ ( undire7337870655677353998dges_a @ ( cons_a @ Y3 @ Ys3 ) ) ) ) ) ) ) ) ).
% walk_edges.elims
thf(fact_1188_degree__none,axiom,
! [V: a] :
( ~ ( member_a @ V @ vertices )
=> ( ( undire8867928226783802224gree_a @ edges @ V )
= zero_zero_nat ) ) ).
% degree_none
thf(fact_1189_neighborhood__incident,axiom,
! [U: a,V: a] :
( ( member_a @ U @ ( undire8504279938402040014hood_a @ vertices @ edges @ V ) )
= ( member_set_a @ ( insert_a2 @ U @ ( insert_a2 @ V @ bot_bot_set_a ) ) @ ( undire3231912044278729248dges_a @ edges @ V ) ) ) ).
% neighborhood_incident
thf(fact_1190_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_1191_finite__incident__edges,axiom,
! [V: a] :
( ( finite_finite_set_a @ edges )
=> ( finite_finite_set_a @ ( undire3231912044278729248dges_a @ edges @ V ) ) ) ).
% finite_incident_edges
thf(fact_1192_incident__edges__empty,axiom,
! [V: a] :
( ~ ( member_a @ V @ vertices )
=> ( ( undire3231912044278729248dges_a @ edges @ V )
= bot_bot_set_set_a ) ) ).
% incident_edges_empty
thf(fact_1193_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_1194_incident__loops__simp_I1_J,axiom,
! [V: a] :
( ( undire3617971648856834880loop_a @ edges @ V )
=> ( ( undire4753905205749729249oops_a @ edges @ V )
= ( insert_set_a2 @ ( insert_a2 @ V @ bot_bot_set_a ) @ bot_bot_set_set_a ) ) ) ).
% incident_loops_simp(1)
thf(fact_1195_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_a2 @ ( insert_a2 @ ( last_a @ Xs ) @ ( insert_a2 @ ( hd_a @ Ys ) @ bot_bot_set_a ) ) @ bot_bot_set_set_a ) ) ) ) ) ).
% walk_edges_append_union
thf(fact_1196_finite__inc__sedges,axiom,
! [V: a] :
( ( finite_finite_set_a @ edges )
=> ( finite_finite_set_a @ ( undire1270416042309875431dges_a @ edges @ V ) ) ) ).
% finite_inc_sedges
thf(fact_1197_finite__incident__loops,axiom,
! [V: a] : ( finite_finite_set_a @ ( undire4753905205749729249oops_a @ edges @ V ) ) ).
% finite_incident_loops
thf(fact_1198_finite__all__edges__between,axiom,
! [X5: set_a,Y2: set_a] :
( ( finite_finite_a @ X5 )
=> ( ( finite_finite_a @ Y2 )
=> ( finite6544458595007987280od_a_a @ ( undire8383842906760478443ween_a @ edges @ X5 @ Y2 ) ) ) ) ).
% finite_all_edges_between
thf(fact_1199_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_1200_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_1201_incident__edges__sedges,axiom,
! [V: a] :
( ~ ( undire3617971648856834880loop_a @ edges @ V )
=> ( ( undire3231912044278729248dges_a @ edges @ V )
= ( undire1270416042309875431dges_a @ edges @ V ) ) ) ).
% incident_edges_sedges
thf(fact_1202_incident__sedges__empty,axiom,
! [V: a] :
( ~ ( member_a @ V @ vertices )
=> ( ( undire1270416042309875431dges_a @ edges @ V )
= bot_bot_set_set_a ) ) ).
% incident_sedges_empty
thf(fact_1203_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_1204_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_1205_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_1206_all__edges__between__Un1,axiom,
! [X5: set_a,Y2: set_a,Z4: set_a] :
( ( undire8383842906760478443ween_a @ edges @ ( sup_sup_set_a @ X5 @ Y2 ) @ Z4 )
= ( sup_su3048258781599657691od_a_a @ ( undire8383842906760478443ween_a @ edges @ X5 @ Z4 ) @ ( undire8383842906760478443ween_a @ edges @ Y2 @ Z4 ) ) ) ).
% all_edges_between_Un1
thf(fact_1207_all__edges__between__Un2,axiom,
! [X5: set_a,Y2: set_a,Z4: set_a] :
( ( undire8383842906760478443ween_a @ edges @ X5 @ ( sup_sup_set_a @ Y2 @ Z4 ) )
= ( sup_su3048258781599657691od_a_a @ ( undire8383842906760478443ween_a @ edges @ X5 @ Y2 ) @ ( undire8383842906760478443ween_a @ edges @ X5 @ Z4 ) ) ) ).
% all_edges_between_Un2
thf(fact_1208_all__edges__betw__D3,axiom,
! [X: a,Y: a,X5: set_a,Y2: set_a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y ) @ ( undire8383842906760478443ween_a @ edges @ X5 @ Y2 ) )
=> ( member_set_a @ ( insert_a2 @ X @ ( insert_a2 @ Y @ bot_bot_set_a ) ) @ edges ) ) ).
% all_edges_betw_D3
thf(fact_1209_all__edges__betw__I,axiom,
! [X: a,X5: set_a,Y: a,Y2: set_a] :
( ( member_a @ X @ X5 )
=> ( ( member_a @ Y @ Y2 )
=> ( ( member_set_a @ ( insert_a2 @ X @ ( insert_a2 @ Y @ bot_bot_set_a ) ) @ edges )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y ) @ ( undire8383842906760478443ween_a @ edges @ X5 @ Y2 ) ) ) ) ) ).
% all_edges_betw_I
thf(fact_1210_card1__incident__imp__vert,axiom,
! [V: a,E: set_a] :
( ( ( undire1521409233611534436dent_a @ V @ E )
& ( ( finite_card_a @ E )
= one_one_nat ) )
=> ( E
= ( insert_a2 @ V @ bot_bot_set_a ) ) ) ).
% card1_incident_imp_vert
thf(fact_1211_walk__edges_Opelims,axiom,
! [X: list_a,Y: list_set_a] :
( ( ( undire7337870655677353998dges_a @ X )
= Y )
=> ( ( accp_list_a @ undire7966302452035489203_rel_a @ X )
=> ( ( ( X = nil_a )
=> ( ( Y = nil_set_a )
=> ~ ( accp_list_a @ undire7966302452035489203_rel_a @ nil_a ) ) )
=> ( ! [X4: a] :
( ( X
= ( cons_a @ X4 @ nil_a ) )
=> ( ( Y = nil_set_a )
=> ~ ( accp_list_a @ undire7966302452035489203_rel_a @ ( cons_a @ X4 @ nil_a ) ) ) )
=> ~ ! [X4: a,Y3: a,Ys3: list_a] :
( ( X
= ( cons_a @ X4 @ ( cons_a @ Y3 @ Ys3 ) ) )
=> ( ( Y
= ( cons_set_a @ ( insert_a2 @ X4 @ ( insert_a2 @ Y3 @ bot_bot_set_a ) ) @ ( undire7337870655677353998dges_a @ ( cons_a @ Y3 @ Ys3 ) ) ) )
=> ~ ( accp_list_a @ undire7966302452035489203_rel_a @ ( cons_a @ X4 @ ( cons_a @ Y3 @ Ys3 ) ) ) ) ) ) ) ) ) ).
% walk_edges.pelims
thf(fact_1212_is__loop__def,axiom,
( undire2905028936066782638loop_a
= ( ^ [E6: set_a] :
( ( finite_card_a @ E6 )
= one_one_nat ) ) ) ).
% is_loop_def
thf(fact_1213_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_1214_card__all__edges__between__commute,axiom,
! [X5: set_a,Y2: set_a] :
( ( finite4795055649997197647od_a_a @ ( undire8383842906760478443ween_a @ edges @ X5 @ Y2 ) )
= ( finite4795055649997197647od_a_a @ ( undire8383842906760478443ween_a @ edges @ Y2 @ X5 ) ) ) ).
% card_all_edges_between_commute
thf(fact_1215_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_1216_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_1217_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_1218_max__all__edges__between,axiom,
! [X5: set_a,Y2: set_a] :
( ( finite_finite_a @ X5 )
=> ( ( finite_finite_a @ Y2 )
=> ( ord_less_eq_nat @ ( finite4795055649997197647od_a_a @ ( undire8383842906760478443ween_a @ edges @ X5 @ Y2 ) ) @ ( times_times_nat @ ( finite_card_a @ X5 ) @ ( finite_card_a @ Y2 ) ) ) ) ) ).
% max_all_edges_between
thf(fact_1219_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_1220_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_1221_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_1222_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_1223_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_1224_mult__0__right,axiom,
! [M: nat] :
( ( times_times_nat @ M @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_0_right
thf(fact_1225_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_1226_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_1227_bot__nat__def,axiom,
bot_bot_nat = zero_zero_nat ).
% bot_nat_def
thf(fact_1228_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_1229_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_1230_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% eq_imp_le
thf(fact_1231_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_1232_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_1233_Nat_Oex__has__greatest__nat,axiom,
! [P2: nat > $o,K: nat,B: nat] :
( ( P2 @ K )
=> ( ! [Y3: nat] :
( ( P2 @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ B ) )
=> ? [X4: nat] :
( ( P2 @ X4 )
& ! [Y7: nat] :
( ( P2 @ Y7 )
=> ( ord_less_eq_nat @ Y7 @ X4 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_1234_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_1235_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_1236_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_1237_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_1238_mult__0,axiom,
! [N: nat] :
( ( times_times_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% mult_0
thf(fact_1239_mult__le__mono2,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( times_times_nat @ K @ I ) @ ( times_times_nat @ K @ J ) ) ) ).
% mult_le_mono2
thf(fact_1240_mult__le__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ K ) ) ) ).
% mult_le_mono1
thf(fact_1241_mult__le__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ L ) ) ) ) ).
% mult_le_mono
thf(fact_1242_le__square,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ M ) ) ).
% le_square
thf(fact_1243_le__cube,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ ( times_times_nat @ M @ M ) ) ) ).
% le_cube
thf(fact_1244_nat__mult__1,axiom,
! [N: nat] :
( ( times_times_nat @ one_one_nat @ N )
= N ) ).
% nat_mult_1
thf(fact_1245_nat__mult__1__right,axiom,
! [N: nat] :
( ( times_times_nat @ N @ one_one_nat )
= N ) ).
% nat_mult_1_right
thf(fact_1246_mult__eq__self__implies__10,axiom,
! [M: nat,N: nat] :
( ( M
= ( times_times_nat @ M @ N ) )
=> ( ( N = one_one_nat )
| ( M = zero_zero_nat ) ) ) ).
% mult_eq_self_implies_10
thf(fact_1247_finite__nat__set__iff__bounded__le,axiom,
( finite_finite_nat
= ( ^ [N2: set_nat] :
? [M2: nat] :
! [X3: nat] :
( ( member_nat @ X3 @ N2 )
=> ( ord_less_eq_nat @ X3 @ M2 ) ) ) ) ).
% finite_nat_set_iff_bounded_le
thf(fact_1248_bounded__Max__nat,axiom,
! [P2: nat > $o,X: nat,M3: nat] :
( ( P2 @ X )
=> ( ! [X4: nat] :
( ( P2 @ X4 )
=> ( ord_less_eq_nat @ X4 @ M3 ) )
=> ~ ! [M4: nat] :
( ( P2 @ M4 )
=> ~ ! [X2: nat] :
( ( P2 @ X2 )
=> ( ord_less_eq_nat @ X2 @ M4 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_1249_infinite__nat__iff__unbounded__le,axiom,
! [S: set_nat] :
( ( ~ ( finite_finite_nat @ S ) )
= ( ! [M2: nat] :
? [N3: nat] :
( ( ord_less_eq_nat @ M2 @ N3 )
& ( member_nat @ N3 @ S ) ) ) ) ).
% infinite_nat_iff_unbounded_le
thf(fact_1250_nat__mult__eq__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( times_times_nat @ K @ M )
= ( times_times_nat @ K @ N ) )
= ( ( K = zero_zero_nat )
| ( M = N ) ) ) ).
% nat_mult_eq_cancel_disj
thf(fact_1251_walk__length__rev,axiom,
( undire8849074589633906640ngth_a
= ( ^ [P3: list_a] : ( undire8849074589633906640ngth_a @ ( rev_a @ P3 ) ) ) ) ).
% walk_length_rev
thf(fact_1252_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_1253_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_1254_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_1255_Nat_Oadd__0__right,axiom,
! [M: nat] :
( ( plus_plus_nat @ M @ zero_zero_nat )
= M ) ).
% Nat.add_0_right
thf(fact_1256_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_1257_plus__nat_Oadd__0,axiom,
! [N: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N )
= N ) ).
% plus_nat.add_0
thf(fact_1258_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_1259_nat__le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [M2: nat,N3: nat] :
? [K2: nat] :
( N3
= ( plus_plus_nat @ M2 @ K2 ) ) ) ) ).
% nat_le_iff_add
thf(fact_1260_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_1261_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_1262_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_1263_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_1264_le__Suc__ex,axiom,
! [K: nat,L: nat] :
( ( ord_less_eq_nat @ K @ L )
=> ? [N4: nat] :
( L
= ( plus_plus_nat @ K @ N4 ) ) ) ).
% le_Suc_ex
thf(fact_1265_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_1266_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_1267_le__add2,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M @ N ) ) ).
% le_add2
thf(fact_1268_le__add1,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M ) ) ).
% le_add1
thf(fact_1269_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_1270_walk__length__def,axiom,
( undire8849074589633906640ngth_a
= ( ^ [P3: list_a] : ( size_size_list_set_a @ ( undire7337870655677353998dges_a @ P3 ) ) ) ) ).
% walk_length_def
thf(fact_1271_Euclid__induct,axiom,
! [P2: nat > nat > $o,A: nat,B: nat] :
( ! [A4: nat,B4: nat] :
( ( P2 @ A4 @ B4 )
= ( P2 @ B4 @ A4 ) )
=> ( ! [A4: nat] : ( P2 @ A4 @ zero_zero_nat )
=> ( ! [A4: nat,B4: nat] :
( ( P2 @ A4 @ B4 )
=> ( P2 @ A4 @ ( plus_plus_nat @ A4 @ B4 ) ) )
=> ( P2 @ A @ B ) ) ) ) ).
% Euclid_induct
thf(fact_1272_walk__length__conv,axiom,
( undire8849074589633906640ngth_a
= ( ^ [P3: list_a] : ( minus_minus_nat @ ( size_size_list_a @ P3 ) @ one_one_nat ) ) ) ).
% walk_length_conv
% Helper facts (9)
thf(help_If_2_1_If_001t__List__Olist_Itf__a_J_T,axiom,
! [X: list_a,Y: list_a] :
( ( if_list_a @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__List__Olist_Itf__a_J_T,axiom,
! [X: list_a,Y: list_a] :
( ( if_list_a @ $true @ X @ Y )
= X ) ).
thf(help_If_2_1_If_001t__List__Olist_It__Nat__Onat_J_T,axiom,
! [X: list_nat,Y: list_nat] :
( ( if_list_nat @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__List__Olist_It__Nat__Onat_J_T,axiom,
! [X: list_nat,Y: list_nat] :
( ( if_list_nat @ $true @ X @ Y )
= X ) ).
thf(help_If_2_1_If_001t__List__Olist_It__Set__Oset_Itf__a_J_J_T,axiom,
! [X: list_set_a,Y: list_set_a] :
( ( if_list_set_a @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__List__Olist_It__Set__Oset_Itf__a_J_J_T,axiom,
! [X: list_set_a,Y: list_set_a] :
( ( if_list_set_a @ $true @ X @ Y )
= X ) ).
thf(help_If_3_1_If_001t__List__Olist_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J_T,axiom,
! [P2: $o] :
( ( P2 = $true )
| ( P2 = $false ) ) ).
thf(help_If_2_1_If_001t__List__Olist_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J_T,axiom,
! [X: list_P1396940483166286381od_a_a,Y: list_P1396940483166286381od_a_a] :
( ( if_lis931442767461590515od_a_a @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__List__Olist_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J_T,axiom,
! [X: list_P1396940483166286381od_a_a,Y: list_P1396940483166286381od_a_a] :
( ( if_lis931442767461590515od_a_a @ $true @ X @ Y )
= X ) ).
% Conjectures (1)
thf(conj_0,conjecture,
distinct_a @ ( tl_a @ p ) ).
%------------------------------------------------------------------------------