TPTP Problem File: SLH0242^1.p
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- Solve Problem
%------------------------------------------------------------------------------
% 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_00446_016786__13249160_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1407 ( 600 unt; 136 typ; 0 def)
% Number of atoms : 3471 (1524 equ; 0 cnn)
% Maximal formula atoms : 12 ( 2 avg)
% Number of connectives : 10275 ( 427 ~; 76 |; 256 &;8112 @)
% ( 0 <=>;1404 =>; 0 <=; 0 <~>)
% Maximal formula depth : 29 ( 6 avg)
% Number of types : 15 ( 14 usr)
% Number of type conns : 404 ( 404 >; 0 *; 0 +; 0 <<)
% Number of symbols : 125 ( 122 usr; 19 con; 0-4 aty)
% Number of variables : 3318 ( 188 ^;2963 !; 167 ?;3318 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 14:33:33.383
%------------------------------------------------------------------------------
% Could-be-implicit typings (14)
thf(ty_n_t__List__Olist_It__List__Olist_It__Set__Oset_Itf__a_J_J_J,type,
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thf(ty_n_t__Set__Oset_It__List__Olist_It__Set__Oset_Itf__a_J_J_J,type,
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thf(ty_n_t__List__Olist_It__Set__Oset_It__Set__Oset_Itf__a_J_J_J,type,
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thf(ty_n_t__Set__Oset_It__Set__Oset_It__Set__Oset_Itf__a_J_J_J,type,
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thf(ty_n_t__List__Olist_It__List__Olist_Itf__a_J_J,type,
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thf(ty_n_t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
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thf(ty_n_t__List__Olist_It__Set__Oset_Itf__a_J_J,type,
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thf(ty_n_t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
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thf(ty_n_t__Extended____Nat__Oenat,type,
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thf(ty_n_t__List__Olist_Itf__a_J,type,
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thf(ty_n_t__Set__Oset_Itf__a_J,type,
set_a: $tType ).
thf(ty_n_t__Num__Onum,type,
num: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
thf(ty_n_tf__a,type,
a: $tType ).
% Explicit typings (122)
thf(sy_c_Finite__Set_Ocard_001t__Set__Oset_Itf__a_J,type,
finite_card_set_a: set_set_a > nat ).
thf(sy_c_Finite__Set_Ocard_001tf__a,type,
finite_card_a: set_a > nat ).
thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_Itf__a_J,type,
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thf(sy_c_Groups_Ominus__class_Ominus_001t__Extended____Nat__Oenat,type,
minus_3235023915231533773d_enat: extended_enat > extended_enat > extended_enat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
minus_minus_nat: nat > nat > nat ).
thf(sy_c_Groups_Oone__class_Oone_001t__Extended____Nat__Oenat,type,
one_on7984719198319812577d_enat: extended_enat ).
thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat,type,
one_one_nat: nat ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Extended____Nat__Oenat,type,
plus_p3455044024723400733d_enat: extended_enat > extended_enat > extended_enat ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat,type,
plus_plus_nat: nat > nat > nat ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Num__Onum,type,
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thf(sy_c_Groups_Ozero__class_Ozero_001t__Extended____Nat__Oenat,type,
zero_z5237406670263579293d_enat: extended_enat ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
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thf(sy_c_If_001t__Nat__Onat,type,
if_nat: $o > nat > nat > nat ).
thf(sy_c_List_Oappend_001t__Set__Oset_Itf__a_J,type,
append_set_a: list_set_a > list_set_a > list_set_a ).
thf(sy_c_List_Oappend_001tf__a,type,
append_a: list_a > list_a > list_a ).
thf(sy_c_List_Odistinct_001t__Set__Oset_Itf__a_J,type,
distinct_set_a: list_set_a > $o ).
thf(sy_c_List_Odistinct_001tf__a,type,
distinct_a: list_a > $o ).
thf(sy_c_List_Odrop_001tf__a,type,
drop_a: nat > list_a > list_a ).
thf(sy_c_List_Olast_001t__Set__Oset_Itf__a_J,type,
last_set_a: list_set_a > set_a ).
thf(sy_c_List_Olast_001tf__a,type,
last_a: list_a > a ).
thf(sy_c_List_Olist_OCons_001t__List__Olist_It__Set__Oset_Itf__a_J_J,type,
cons_list_set_a: list_set_a > list_list_set_a > list_list_set_a ).
thf(sy_c_List_Olist_OCons_001t__List__Olist_Itf__a_J,type,
cons_list_a: list_a > list_list_a > list_list_a ).
thf(sy_c_List_Olist_OCons_001t__Set__Oset_Itf__a_J,type,
cons_set_a: set_a > list_set_a > list_set_a ).
thf(sy_c_List_Olist_OCons_001tf__a,type,
cons_a: a > list_a > list_a ).
thf(sy_c_List_Olist_ONil_001t__List__Olist_It__Set__Oset_Itf__a_J_J,type,
nil_list_set_a: list_list_set_a ).
thf(sy_c_List_Olist_ONil_001t__List__Olist_Itf__a_J,type,
nil_list_a: list_list_a ).
thf(sy_c_List_Olist_ONil_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
nil_set_set_a: list_set_set_a ).
thf(sy_c_List_Olist_ONil_001t__Set__Oset_Itf__a_J,type,
nil_set_a: list_set_a ).
thf(sy_c_List_Olist_ONil_001tf__a,type,
nil_a: list_a ).
thf(sy_c_List_Olist_Ohd_001t__Set__Oset_Itf__a_J,type,
hd_set_a: list_set_a > set_a ).
thf(sy_c_List_Olist_Ohd_001tf__a,type,
hd_a: list_a > a ).
thf(sy_c_List_Olist_Oset_001t__List__Olist_It__Set__Oset_Itf__a_J_J,type,
set_list_set_a2: list_list_set_a > set_list_set_a ).
thf(sy_c_List_Olist_Oset_001t__List__Olist_Itf__a_J,type,
set_list_a2: list_list_a > set_list_a ).
thf(sy_c_List_Olist_Oset_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
set_set_set_a2: list_set_set_a > set_set_set_a ).
thf(sy_c_List_Olist_Oset_001t__Set__Oset_Itf__a_J,type,
set_set_a2: list_set_a > set_set_a ).
thf(sy_c_List_Olist_Oset_001tf__a,type,
set_a2: list_a > set_a ).
thf(sy_c_List_Olist_Otl_001t__Set__Oset_Itf__a_J,type,
tl_set_a: list_set_a > list_set_a ).
thf(sy_c_List_Olist_Otl_001tf__a,type,
tl_a: list_a > list_a ).
thf(sy_c_List_On__lists_001t__Set__Oset_Itf__a_J,type,
n_lists_set_a: nat > list_set_a > list_list_set_a ).
thf(sy_c_List_On__lists_001tf__a,type,
n_lists_a: nat > list_a > list_list_a ).
thf(sy_c_List_Onth_001t__Set__Oset_Itf__a_J,type,
nth_set_a: list_set_a > nat > set_a ).
thf(sy_c_List_Onth_001tf__a,type,
nth_a: list_a > nat > a ).
thf(sy_c_List_Orev_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
rev_set_set_a: list_set_set_a > list_set_set_a ).
thf(sy_c_List_Orev_001t__Set__Oset_Itf__a_J,type,
rev_set_a: list_set_a > list_set_a ).
thf(sy_c_List_Orev_001tf__a,type,
rev_a: list_a > list_a ).
thf(sy_c_List_Otake_001tf__a,type,
take_a: nat > list_a > list_a ).
thf(sy_c_Nat_OSuc,type,
suc: nat > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Set__Oset_Itf__a_J_J,type,
size_size_list_set_a: list_set_a > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_Itf__a_J,type,
size_size_list_a: list_a > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__Num__Onum,type,
size_size_num: num > nat ).
thf(sy_c_Num_Onum_OBit0,type,
bit0: num > num ).
thf(sy_c_Num_Onum_OBit1,type,
bit1: num > num ).
thf(sy_c_Num_Onum_OOne,type,
one: num ).
thf(sy_c_Num_Onumeral__class_Onumeral_001t__Extended____Nat__Oenat,type,
numera1916890842035813515d_enat: num > extended_enat ).
thf(sy_c_Num_Onumeral__class_Onumeral_001t__Nat__Onat,type,
numeral_numeral_nat: num > nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Extended____Nat__Oenat,type,
bot_bo4199563552545308370d_enat: extended_enat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Nat__Onat,type,
bot_bot_nat: nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
bot_bot_set_set_a: set_set_a ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_Itf__a_J,type,
bot_bot_set_a: set_a ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Extended____Nat__Oenat,type,
ord_le72135733267957522d_enat: extended_enat > extended_enat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
ord_less_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Num__Onum,type,
ord_less_num: num > num > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
ord_less_set_set_a: set_set_a > set_set_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_Itf__a_J,type,
ord_less_set_a: set_a > set_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Extended____Nat__Oenat,type,
ord_le2932123472753598470d_enat: extended_enat > extended_enat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
ord_less_eq_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Num__Onum,type,
ord_less_eq_num: num > num > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Set__Oset_Itf__a_J_J_J,type,
ord_le5722252365846178494_set_a: set_set_set_a > set_set_set_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_Itf__a_J,type,
ord_less_eq_set_a: set_a > set_a > $o ).
thf(sy_c_Set_OCollect_001t__Set__Oset_Itf__a_J,type,
collect_set_a: ( set_a > $o ) > set_set_a ).
thf(sy_c_Set_OCollect_001tf__a,type,
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thf(sy_c_Set_Oinsert_001tf__a,type,
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thf(sy_c_Undirected__Graph__Basics_Ograph__system_Oedge__adj_001tf__a,type,
undire4022703626023482010_adj_a: set_set_a > set_a > set_a > $o ).
thf(sy_c_Undirected__Graph__Basics_Ograph__system_Oincident_001t__Set__Oset_Itf__a_J,type,
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thf(sy_c_Undirected__Graph__Basics_Ograph__system_Oincident_001tf__a,type,
undire1521409233611534436dent_a: a > set_a > $o ).
thf(sy_c_Undirected__Graph__Basics_Ograph__system_Oincident__edges_001tf__a,type,
undire3231912044278729248dges_a: set_set_a > a > set_set_a ).
thf(sy_c_Undirected__Graph__Basics_Ograph__system_Oinduced__edges_001tf__a,type,
undire7777452895879145676dges_a: set_set_a > set_a > set_set_a ).
thf(sy_c_Undirected__Graph__Basics_Osubgraph_001t__Set__Oset_Itf__a_J,type,
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thf(sy_c_Undirected__Graph__Basics_Osubgraph_001tf__a,type,
undire7103218114511261257raph_a: set_a > set_set_a > set_a > set_set_a > $o ).
thf(sy_c_Undirected__Graph__Basics_Oulgraph_001t__Set__Oset_Itf__a_J,type,
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thf(sy_c_Undirected__Graph__Basics_Oulgraph_001tf__a,type,
undire7251896706689453996raph_a: set_a > set_set_a > $o ).
thf(sy_c_Undirected__Graph__Basics_Oulgraph_Odegree_001t__Set__Oset_Itf__a_J,type,
undire8939077443744732368_set_a: set_set_set_a > set_a > nat ).
thf(sy_c_Undirected__Graph__Basics_Oulgraph_Odegree_001tf__a,type,
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thf(sy_c_Undirected__Graph__Basics_Oulgraph_Ohas__loop_001t__Set__Oset_Itf__a_J,type,
undire5774735625301615776_set_a: set_set_set_a > set_a > $o ).
thf(sy_c_Undirected__Graph__Basics_Oulgraph_Ohas__loop_001tf__a,type,
undire3617971648856834880loop_a: set_set_a > a > $o ).
thf(sy_c_Undirected__Graph__Basics_Oulgraph_Oincident__loops_001tf__a,type,
undire4753905205749729249oops_a: set_set_a > a > set_set_a ).
thf(sy_c_Undirected__Graph__Basics_Oulgraph_Ois__isolated__vertex_001t__Set__Oset_Itf__a_J,type,
undire6879241558604981877_set_a: set_set_a > set_set_set_a > set_a > $o ).
thf(sy_c_Undirected__Graph__Basics_Oulgraph_Ois__isolated__vertex_001tf__a,type,
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thf(sy_c_Undirected__Graph__Basics_Oulgraph_Ois__loop_001t__Set__Oset_Itf__a_J,type,
undire3618949687197220622_set_a: set_set_a > $o ).
thf(sy_c_Undirected__Graph__Basics_Oulgraph_Ois__loop_001tf__a,type,
undire2905028936066782638loop_a: set_a > $o ).
thf(sy_c_Undirected__Graph__Basics_Oulgraph_Ois__sedge_001t__Set__Oset_Itf__a_J,type,
undire5778062222479071480_set_a: set_set_a > $o ).
thf(sy_c_Undirected__Graph__Basics_Oulgraph_Ois__sedge_001tf__a,type,
undire4917966558017083288edge_a: set_a > $o ).
thf(sy_c_Undirected__Graph__Basics_Oulgraph_Oneighborhood_001tf__a,type,
undire8504279938402040014hood_a: set_a > set_set_a > a > set_a ).
thf(sy_c_Undirected__Graph__Basics_Oulgraph_Overt__adj_001t__Set__Oset_Itf__a_J,type,
undire3510646817838285160_set_a: set_set_set_a > set_a > set_a > $o ).
thf(sy_c_Undirected__Graph__Basics_Oulgraph_Overt__adj_001tf__a,type,
undire397441198561214472_adj_a: set_set_a > a > a > $o ).
thf(sy_c_Undirected__Graph__Walks_Oulgraph_Ois__closed__walk_001t__Set__Oset_Itf__a_J,type,
undire4100213446647512896_set_a: set_set_a > set_set_set_a > list_set_a > $o ).
thf(sy_c_Undirected__Graph__Walks_Oulgraph_Ois__closed__walk_001tf__a,type,
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thf(sy_c_Undirected__Graph__Walks_Oulgraph_Ois__cycle_001t__Set__Oset_Itf__a_J,type,
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thf(sy_c_Undirected__Graph__Walks_Oulgraph_Ois__cycle_001tf__a,type,
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thf(sy_c_Undirected__Graph__Walks_Oulgraph_Ois__gen__path_001t__Set__Oset_Itf__a_J,type,
undire7201326534205417136_set_a: set_set_a > set_set_set_a > list_set_a > $o ).
thf(sy_c_Undirected__Graph__Walks_Oulgraph_Ois__gen__path_001tf__a,type,
undire3562951555376170320path_a: set_a > set_set_a > list_a > $o ).
thf(sy_c_Undirected__Graph__Walks_Oulgraph_Ois__open__walk_001t__Set__Oset_Itf__a_J,type,
undire526879649183275522_set_a: set_set_a > set_set_set_a > list_set_a > $o ).
thf(sy_c_Undirected__Graph__Walks_Oulgraph_Ois__open__walk_001tf__a,type,
undire2427028224930250914walk_a: set_a > set_set_a > list_a > $o ).
thf(sy_c_Undirected__Graph__Walks_Oulgraph_Ois__path_001t__Set__Oset_Itf__a_J,type,
undire8834939040163919632_set_a: set_set_a > set_set_set_a > list_set_a > $o ).
thf(sy_c_Undirected__Graph__Walks_Oulgraph_Ois__path_001tf__a,type,
undire427332500224447920path_a: set_a > set_set_a > list_a > $o ).
thf(sy_c_Undirected__Graph__Walks_Oulgraph_Ois__trail_001t__Set__Oset_Itf__a_J,type,
undire1224551742100448159_set_a: set_set_a > set_set_set_a > list_set_a > $o ).
thf(sy_c_Undirected__Graph__Walks_Oulgraph_Ois__trail_001tf__a,type,
undire7142031287334043199rail_a: set_a > set_set_a > list_a > $o ).
thf(sy_c_Undirected__Graph__Walks_Oulgraph_Ois__walk_001t__Set__Oset_Itf__a_J,type,
undire3014741414213135564_set_a: set_set_a > set_set_set_a > list_set_a > $o ).
thf(sy_c_Undirected__Graph__Walks_Oulgraph_Ois__walk_001tf__a,type,
undire6133010728901294956walk_a: set_a > set_set_a > list_a > $o ).
thf(sy_c_Undirected__Graph__Walks_Oulgraph_Owalk__edges_001t__Set__Oset_Itf__a_J,type,
undire6234387080713648494_set_a: list_set_a > list_set_set_a ).
thf(sy_c_Undirected__Graph__Walks_Oulgraph_Owalk__edges_001tf__a,type,
undire7337870655677353998dges_a: list_a > list_set_a ).
thf(sy_c_Undirected__Graph__Walks_Oulgraph_Owalk__length_001t__Set__Oset_Itf__a_J,type,
undire4424681683220949296_set_a: list_set_a > nat ).
thf(sy_c_Undirected__Graph__Walks_Oulgraph_Owalk__length_001tf__a,type,
undire8849074589633906640ngth_a: list_a > nat ).
thf(sy_c_member_001t__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__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
member_set_set_a: set_set_a > set_set_set_a > $o ).
thf(sy_c_member_001t__Set__Oset_Itf__a_J,type,
member_set_a: set_a > set_set_a > $o ).
thf(sy_c_member_001tf__a,type,
member_a: a > set_a > $o ).
thf(sy_v_edges,type,
edges: set_set_a ).
thf(sy_v_vertices,type,
vertices: set_a ).
thf(sy_v_xs,type,
xs: list_a ).
% Relevant facts (1267)
thf(fact_0_eq,axiom,
( ( hd_a @ xs )
= ( last_a @ xs ) ) ).
% eq
thf(fact_1_dis,axiom,
distinct_a @ ( tl_a @ xs ) ).
% dis
thf(fact_2_w,axiom,
undire6133010728901294956walk_a @ vertices @ edges @ xs ).
% w
thf(fact_3__092_060open_0623_A_092_060le_062_Alength_Axs_A_N_ASuc_A0_092_060close_062,axiom,
ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ ( minus_minus_nat @ ( size_size_list_a @ xs ) @ ( suc @ zero_zero_nat ) ) ).
% \<open>3 \<le> length xs - Suc 0\<close>
thf(fact_4_semiring__norm_I85_J,axiom,
! [M: num] :
( ( bit0 @ M )
!= one ) ).
% semiring_norm(85)
thf(fact_5_semiring__norm_I83_J,axiom,
! [N: num] :
( one
!= ( bit0 @ N ) ) ).
% semiring_norm(83)
thf(fact_6_numeral__le__iff,axiom,
! [M: num,N: num] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
= ( ord_less_eq_num @ M @ N ) ) ).
% numeral_le_iff
thf(fact_7_numeral__le__iff,axiom,
! [M: num,N: num] :
( ( ord_le2932123472753598470d_enat @ ( numera1916890842035813515d_enat @ M ) @ ( numera1916890842035813515d_enat @ N ) )
= ( ord_less_eq_num @ M @ N ) ) ).
% numeral_le_iff
thf(fact_8_verit__eq__simplify_I8_J,axiom,
! [X2: num,Y2: num] :
( ( ( bit0 @ X2 )
= ( bit0 @ Y2 ) )
= ( X2 = Y2 ) ) ).
% verit_eq_simplify(8)
thf(fact_9_semiring__norm_I87_J,axiom,
! [M: num,N: num] :
( ( ( bit0 @ M )
= ( bit0 @ N ) )
= ( M = N ) ) ).
% semiring_norm(87)
thf(fact_10_numeral__eq__iff,axiom,
! [M: num,N: num] :
( ( ( numeral_numeral_nat @ M )
= ( numeral_numeral_nat @ N ) )
= ( M = N ) ) ).
% numeral_eq_iff
thf(fact_11_numeral__eq__iff,axiom,
! [M: num,N: num] :
( ( ( numera1916890842035813515d_enat @ M )
= ( numera1916890842035813515d_enat @ N ) )
= ( M = N ) ) ).
% numeral_eq_iff
thf(fact_12_order__refl,axiom,
! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).
% order_refl
thf(fact_13_order__refl,axiom,
! [X: set_a] : ( ord_less_eq_set_a @ X @ X ) ).
% order_refl
thf(fact_14_order__refl,axiom,
! [X: num] : ( ord_less_eq_num @ X @ X ) ).
% order_refl
thf(fact_15_order__refl,axiom,
! [X: extended_enat] : ( ord_le2932123472753598470d_enat @ X @ X ) ).
% order_refl
thf(fact_16_order__refl,axiom,
! [X: set_set_a] : ( ord_le3724670747650509150_set_a @ X @ X ) ).
% order_refl
thf(fact_17_dual__order_Orefl,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% dual_order.refl
thf(fact_18_dual__order_Orefl,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ A @ A ) ).
% dual_order.refl
thf(fact_19_dual__order_Orefl,axiom,
! [A: num] : ( ord_less_eq_num @ A @ A ) ).
% dual_order.refl
thf(fact_20_dual__order_Orefl,axiom,
! [A: extended_enat] : ( ord_le2932123472753598470d_enat @ A @ A ) ).
% dual_order.refl
thf(fact_21_dual__order_Orefl,axiom,
! [A: set_set_a] : ( ord_le3724670747650509150_set_a @ A @ A ) ).
% dual_order.refl
thf(fact_22_is__walk__tl,axiom,
! [Xs: list_a] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( size_size_list_a @ Xs ) )
=> ( ( undire6133010728901294956walk_a @ vertices @ edges @ Xs )
=> ( undire6133010728901294956walk_a @ vertices @ edges @ ( tl_a @ Xs ) ) ) ) ).
% is_walk_tl
thf(fact_23_verit__eq__simplify_I10_J,axiom,
! [X2: num] :
( one
!= ( bit0 @ X2 ) ) ).
% verit_eq_simplify(10)
thf(fact_24_numeral__2__eq__2,axiom,
( ( numeral_numeral_nat @ ( bit0 @ one ) )
= ( suc @ ( suc @ zero_zero_nat ) ) ) ).
% numeral_2_eq_2
thf(fact_25_wellformed,axiom,
! [E: set_a] :
( ( member_set_a @ E @ edges )
=> ( ord_less_eq_set_a @ E @ vertices ) ) ).
% wellformed
thf(fact_26_verit__eq__simplify_I9_J,axiom,
! [X3: num,Y3: num] :
( ( ( bit1 @ X3 )
= ( bit1 @ Y3 ) )
= ( X3 = Y3 ) ) ).
% verit_eq_simplify(9)
thf(fact_27_semiring__norm_I90_J,axiom,
! [M: num,N: num] :
( ( ( bit1 @ M )
= ( bit1 @ N ) )
= ( M = N ) ) ).
% semiring_norm(90)
thf(fact_28_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_29_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_30_semiring__norm_I88_J,axiom,
! [M: num,N: num] :
( ( bit0 @ M )
!= ( bit1 @ N ) ) ).
% semiring_norm(88)
thf(fact_31_semiring__norm_I89_J,axiom,
! [M: num,N: num] :
( ( bit1 @ M )
!= ( bit0 @ N ) ) ).
% semiring_norm(89)
thf(fact_32_semiring__norm_I84_J,axiom,
! [N: num] :
( one
!= ( bit1 @ N ) ) ).
% semiring_norm(84)
thf(fact_33_semiring__norm_I86_J,axiom,
! [M: num] :
( ( bit1 @ M )
!= one ) ).
% semiring_norm(86)
thf(fact_34_semiring__norm_I71_J,axiom,
! [M: num,N: num] :
( ( ord_less_eq_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
= ( ord_less_eq_num @ M @ N ) ) ).
% semiring_norm(71)
thf(fact_35_semiring__norm_I68_J,axiom,
! [N: num] : ( ord_less_eq_num @ one @ N ) ).
% semiring_norm(68)
thf(fact_36_semiring__norm_I73_J,axiom,
! [M: num,N: num] :
( ( ord_less_eq_num @ ( bit1 @ M ) @ ( bit1 @ N ) )
= ( ord_less_eq_num @ M @ N ) ) ).
% semiring_norm(73)
thf(fact_37_semiring__norm_I69_J,axiom,
! [M: num] :
~ ( ord_less_eq_num @ ( bit0 @ M ) @ one ) ).
% semiring_norm(69)
thf(fact_38_semiring__norm_I72_J,axiom,
! [M: num,N: num] :
( ( ord_less_eq_num @ ( bit0 @ M ) @ ( bit1 @ N ) )
= ( ord_less_eq_num @ M @ N ) ) ).
% semiring_norm(72)
thf(fact_39_semiring__norm_I70_J,axiom,
! [M: num] :
~ ( ord_less_eq_num @ ( bit1 @ M ) @ one ) ).
% semiring_norm(70)
thf(fact_40_ulgraph__axioms,axiom,
undire7251896706689453996raph_a @ vertices @ edges ).
% ulgraph_axioms
thf(fact_41_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_42_edge__adjacent__alt__def,axiom,
! [E1: set_a,E2: set_a] :
( ( member_set_a @ E1 @ edges )
=> ( ( member_set_a @ E2 @ edges )
=> ( ? [X4: a] :
( ( member_a @ X4 @ vertices )
& ( member_a @ X4 @ E1 )
& ( member_a @ X4 @ E2 ) )
=> ( undire4022703626023482010_adj_a @ edges @ E1 @ E2 ) ) ) ) ).
% edge_adjacent_alt_def
thf(fact_43_le__num__One__iff,axiom,
! [X: num] :
( ( ord_less_eq_num @ X @ one )
= ( X = one ) ) ).
% le_num_One_iff
thf(fact_44_ulgraph_Ois__walk_Ocong,axiom,
undire6133010728901294956walk_a = undire6133010728901294956walk_a ).
% ulgraph.is_walk.cong
thf(fact_45_verit__eq__simplify_I14_J,axiom,
! [X2: num,X3: num] :
( ( bit0 @ X2 )
!= ( bit1 @ X3 ) ) ).
% verit_eq_simplify(14)
thf(fact_46_verit__eq__simplify_I12_J,axiom,
! [X3: num] :
( one
!= ( bit1 @ X3 ) ) ).
% verit_eq_simplify(12)
thf(fact_47_le__numeral__extra_I3_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% le_numeral_extra(3)
thf(fact_48_le__numeral__extra_I3_J,axiom,
ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ zero_z5237406670263579293d_enat ).
% le_numeral_extra(3)
thf(fact_49_zero__neq__numeral,axiom,
! [N: num] :
( zero_zero_nat
!= ( numeral_numeral_nat @ N ) ) ).
% zero_neq_numeral
thf(fact_50_zero__neq__numeral,axiom,
! [N: num] :
( zero_z5237406670263579293d_enat
!= ( numera1916890842035813515d_enat @ N ) ) ).
% zero_neq_numeral
thf(fact_51_numeral__3__eq__3,axiom,
( ( numeral_numeral_nat @ ( bit1 @ one ) )
= ( suc @ ( suc @ ( suc @ zero_zero_nat ) ) ) ) ).
% numeral_3_eq_3
thf(fact_52_eval__nat__numeral_I3_J,axiom,
! [N: num] :
( ( numeral_numeral_nat @ ( bit1 @ N ) )
= ( suc @ ( numeral_numeral_nat @ ( bit0 @ N ) ) ) ) ).
% eval_nat_numeral(3)
thf(fact_53_numeral__1__eq__Suc__0,axiom,
( ( numeral_numeral_nat @ one )
= ( suc @ zero_zero_nat ) ) ).
% numeral_1_eq_Suc_0
thf(fact_54_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_55_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_56_order__antisym__conv,axiom,
! [Y: num,X: num] :
( ( ord_less_eq_num @ Y @ X )
=> ( ( ord_less_eq_num @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_57_order__antisym__conv,axiom,
! [Y: extended_enat,X: extended_enat] :
( ( ord_le2932123472753598470d_enat @ Y @ X )
=> ( ( ord_le2932123472753598470d_enat @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_58_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_59_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_60_linorder__le__cases,axiom,
! [X: num,Y: num] :
( ~ ( ord_less_eq_num @ X @ Y )
=> ( ord_less_eq_num @ Y @ X ) ) ).
% linorder_le_cases
thf(fact_61_linorder__le__cases,axiom,
! [X: extended_enat,Y: extended_enat] :
( ~ ( ord_le2932123472753598470d_enat @ X @ Y )
=> ( ord_le2932123472753598470d_enat @ Y @ X ) ) ).
% linorder_le_cases
thf(fact_62_mem__Collect__eq,axiom,
! [A: set_a,P: set_a > $o] :
( ( member_set_a @ A @ ( collect_set_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_63_mem__Collect__eq,axiom,
! [A: a,P: a > $o] :
( ( member_a @ A @ ( collect_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_64_Collect__mem__eq,axiom,
! [A2: set_set_a] :
( ( collect_set_a
@ ^ [X5: set_a] : ( member_set_a @ X5 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_65_Collect__mem__eq,axiom,
! [A2: set_a] :
( ( collect_a
@ ^ [X5: a] : ( member_a @ X5 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_66_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X6: nat,Y4: nat] :
( ( ord_less_eq_nat @ X6 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X6 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_67_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > num,C: num] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X6: nat,Y4: nat] :
( ( ord_less_eq_nat @ X6 @ Y4 )
=> ( ord_less_eq_num @ ( F @ X6 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_68_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > extended_enat,C: extended_enat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X6: nat,Y4: nat] :
( ( ord_less_eq_nat @ X6 @ Y4 )
=> ( ord_le2932123472753598470d_enat @ ( F @ X6 ) @ ( F @ Y4 ) ) )
=> ( ord_le2932123472753598470d_enat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_69_ord__le__eq__subst,axiom,
! [A: num,B: num,F: num > nat,C: nat] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X6: num,Y4: num] :
( ( ord_less_eq_num @ X6 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X6 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_70_ord__le__eq__subst,axiom,
! [A: num,B: num,F: num > num,C: num] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X6: num,Y4: num] :
( ( ord_less_eq_num @ X6 @ Y4 )
=> ( ord_less_eq_num @ ( F @ X6 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_71_ord__le__eq__subst,axiom,
! [A: num,B: num,F: num > extended_enat,C: extended_enat] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X6: num,Y4: num] :
( ( ord_less_eq_num @ X6 @ Y4 )
=> ( ord_le2932123472753598470d_enat @ ( F @ X6 ) @ ( F @ Y4 ) ) )
=> ( ord_le2932123472753598470d_enat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_72_ord__le__eq__subst,axiom,
! [A: extended_enat,B: extended_enat,F: extended_enat > nat,C: nat] :
( ( ord_le2932123472753598470d_enat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X6: extended_enat,Y4: extended_enat] :
( ( ord_le2932123472753598470d_enat @ X6 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X6 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_73_ord__le__eq__subst,axiom,
! [A: extended_enat,B: extended_enat,F: extended_enat > num,C: num] :
( ( ord_le2932123472753598470d_enat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X6: extended_enat,Y4: extended_enat] :
( ( ord_le2932123472753598470d_enat @ X6 @ Y4 )
=> ( ord_less_eq_num @ ( F @ X6 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_74_ord__le__eq__subst,axiom,
! [A: extended_enat,B: extended_enat,F: extended_enat > extended_enat,C: extended_enat] :
( ( ord_le2932123472753598470d_enat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X6: extended_enat,Y4: extended_enat] :
( ( ord_le2932123472753598470d_enat @ X6 @ Y4 )
=> ( ord_le2932123472753598470d_enat @ ( F @ X6 ) @ ( F @ Y4 ) ) )
=> ( ord_le2932123472753598470d_enat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_75_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 )
=> ( ! [X6: nat,Y4: nat] :
( ( ord_less_eq_nat @ X6 @ Y4 )
=> ( ord_less_eq_set_a @ ( F @ X6 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_76_ord__eq__le__subst,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X6: nat,Y4: nat] :
( ( ord_less_eq_nat @ X6 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X6 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_77_ord__eq__le__subst,axiom,
! [A: num,F: nat > num,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X6: nat,Y4: nat] :
( ( ord_less_eq_nat @ X6 @ Y4 )
=> ( ord_less_eq_num @ ( F @ X6 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_78_ord__eq__le__subst,axiom,
! [A: extended_enat,F: nat > extended_enat,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X6: nat,Y4: nat] :
( ( ord_less_eq_nat @ X6 @ Y4 )
=> ( ord_le2932123472753598470d_enat @ ( F @ X6 ) @ ( F @ Y4 ) ) )
=> ( ord_le2932123472753598470d_enat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_79_ord__eq__le__subst,axiom,
! [A: nat,F: num > nat,B: num,C: num] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ! [X6: num,Y4: num] :
( ( ord_less_eq_num @ X6 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X6 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_80_ord__eq__le__subst,axiom,
! [A: num,F: num > num,B: num,C: num] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ! [X6: num,Y4: num] :
( ( ord_less_eq_num @ X6 @ Y4 )
=> ( ord_less_eq_num @ ( F @ X6 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_81_ord__eq__le__subst,axiom,
! [A: extended_enat,F: num > extended_enat,B: num,C: num] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ! [X6: num,Y4: num] :
( ( ord_less_eq_num @ X6 @ Y4 )
=> ( ord_le2932123472753598470d_enat @ ( F @ X6 ) @ ( F @ Y4 ) ) )
=> ( ord_le2932123472753598470d_enat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_82_ord__eq__le__subst,axiom,
! [A: nat,F: extended_enat > nat,B: extended_enat,C: extended_enat] :
( ( A
= ( F @ B ) )
=> ( ( ord_le2932123472753598470d_enat @ B @ C )
=> ( ! [X6: extended_enat,Y4: extended_enat] :
( ( ord_le2932123472753598470d_enat @ X6 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X6 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_83_ord__eq__le__subst,axiom,
! [A: num,F: extended_enat > num,B: extended_enat,C: extended_enat] :
( ( A
= ( F @ B ) )
=> ( ( ord_le2932123472753598470d_enat @ B @ C )
=> ( ! [X6: extended_enat,Y4: extended_enat] :
( ( ord_le2932123472753598470d_enat @ X6 @ Y4 )
=> ( ord_less_eq_num @ ( F @ X6 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_84_ord__eq__le__subst,axiom,
! [A: extended_enat,F: extended_enat > extended_enat,B: extended_enat,C: extended_enat] :
( ( A
= ( F @ B ) )
=> ( ( ord_le2932123472753598470d_enat @ B @ C )
=> ( ! [X6: extended_enat,Y4: extended_enat] :
( ( ord_le2932123472753598470d_enat @ X6 @ Y4 )
=> ( ord_le2932123472753598470d_enat @ ( F @ X6 ) @ ( F @ Y4 ) ) )
=> ( ord_le2932123472753598470d_enat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_85_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 )
=> ( ! [X6: nat,Y4: nat] :
( ( ord_less_eq_nat @ X6 @ Y4 )
=> ( ord_less_eq_set_a @ ( F @ X6 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_86_linorder__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
| ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_linear
thf(fact_87_linorder__linear,axiom,
! [X: num,Y: num] :
( ( ord_less_eq_num @ X @ Y )
| ( ord_less_eq_num @ Y @ X ) ) ).
% linorder_linear
thf(fact_88_linorder__linear,axiom,
! [X: extended_enat,Y: extended_enat] :
( ( ord_le2932123472753598470d_enat @ X @ Y )
| ( ord_le2932123472753598470d_enat @ Y @ X ) ) ).
% linorder_linear
thf(fact_89_verit__la__disequality,axiom,
! [A: nat,B: nat] :
( ( A = B )
| ~ ( ord_less_eq_nat @ A @ B )
| ~ ( ord_less_eq_nat @ B @ A ) ) ).
% verit_la_disequality
thf(fact_90_verit__la__disequality,axiom,
! [A: num,B: num] :
( ( A = B )
| ~ ( ord_less_eq_num @ A @ B )
| ~ ( ord_less_eq_num @ B @ A ) ) ).
% verit_la_disequality
thf(fact_91_verit__la__disequality,axiom,
! [A: extended_enat,B: extended_enat] :
( ( A = B )
| ~ ( ord_le2932123472753598470d_enat @ A @ B )
| ~ ( ord_le2932123472753598470d_enat @ B @ A ) ) ).
% verit_la_disequality
thf(fact_92_order__eq__refl,axiom,
! [X: nat,Y: nat] :
( ( X = Y )
=> ( ord_less_eq_nat @ X @ Y ) ) ).
% order_eq_refl
thf(fact_93_order__eq__refl,axiom,
! [X: set_a,Y: set_a] :
( ( X = Y )
=> ( ord_less_eq_set_a @ X @ Y ) ) ).
% order_eq_refl
thf(fact_94_order__eq__refl,axiom,
! [X: num,Y: num] :
( ( X = Y )
=> ( ord_less_eq_num @ X @ Y ) ) ).
% order_eq_refl
thf(fact_95_order__eq__refl,axiom,
! [X: extended_enat,Y: extended_enat] :
( ( X = Y )
=> ( ord_le2932123472753598470d_enat @ X @ Y ) ) ).
% order_eq_refl
thf(fact_96_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_97_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 )
=> ( ! [X6: nat,Y4: nat] :
( ( ord_less_eq_nat @ X6 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X6 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_98_order__subst2,axiom,
! [A: nat,B: nat,F: nat > num,C: num] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_num @ ( F @ B ) @ C )
=> ( ! [X6: nat,Y4: nat] :
( ( ord_less_eq_nat @ X6 @ Y4 )
=> ( ord_less_eq_num @ ( F @ X6 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_99_order__subst2,axiom,
! [A: nat,B: nat,F: nat > extended_enat,C: extended_enat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_le2932123472753598470d_enat @ ( F @ B ) @ C )
=> ( ! [X6: nat,Y4: nat] :
( ( ord_less_eq_nat @ X6 @ Y4 )
=> ( ord_le2932123472753598470d_enat @ ( F @ X6 ) @ ( F @ Y4 ) ) )
=> ( ord_le2932123472753598470d_enat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_100_order__subst2,axiom,
! [A: num,B: num,F: num > nat,C: nat] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X6: num,Y4: num] :
( ( ord_less_eq_num @ X6 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X6 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_101_order__subst2,axiom,
! [A: num,B: num,F: num > num,C: num] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ord_less_eq_num @ ( F @ B ) @ C )
=> ( ! [X6: num,Y4: num] :
( ( ord_less_eq_num @ X6 @ Y4 )
=> ( ord_less_eq_num @ ( F @ X6 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_102_order__subst2,axiom,
! [A: num,B: num,F: num > extended_enat,C: extended_enat] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ord_le2932123472753598470d_enat @ ( F @ B ) @ C )
=> ( ! [X6: num,Y4: num] :
( ( ord_less_eq_num @ X6 @ Y4 )
=> ( ord_le2932123472753598470d_enat @ ( F @ X6 ) @ ( F @ Y4 ) ) )
=> ( ord_le2932123472753598470d_enat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_103_order__subst2,axiom,
! [A: extended_enat,B: extended_enat,F: extended_enat > nat,C: nat] :
( ( ord_le2932123472753598470d_enat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X6: extended_enat,Y4: extended_enat] :
( ( ord_le2932123472753598470d_enat @ X6 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X6 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_104_order__subst2,axiom,
! [A: extended_enat,B: extended_enat,F: extended_enat > num,C: num] :
( ( ord_le2932123472753598470d_enat @ A @ B )
=> ( ( ord_less_eq_num @ ( F @ B ) @ C )
=> ( ! [X6: extended_enat,Y4: extended_enat] :
( ( ord_le2932123472753598470d_enat @ X6 @ Y4 )
=> ( ord_less_eq_num @ ( F @ X6 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_105_order__subst2,axiom,
! [A: extended_enat,B: extended_enat,F: extended_enat > extended_enat,C: extended_enat] :
( ( ord_le2932123472753598470d_enat @ A @ B )
=> ( ( ord_le2932123472753598470d_enat @ ( F @ B ) @ C )
=> ( ! [X6: extended_enat,Y4: extended_enat] :
( ( ord_le2932123472753598470d_enat @ X6 @ Y4 )
=> ( ord_le2932123472753598470d_enat @ ( F @ X6 ) @ ( F @ Y4 ) ) )
=> ( ord_le2932123472753598470d_enat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_106_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 )
=> ( ! [X6: nat,Y4: nat] :
( ( ord_less_eq_nat @ X6 @ Y4 )
=> ( ord_less_eq_set_a @ ( F @ X6 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_107_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 )
=> ( ! [X6: nat,Y4: nat] :
( ( ord_less_eq_nat @ X6 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X6 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_108_order__subst1,axiom,
! [A: nat,F: num > nat,B: num,C: num] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ! [X6: num,Y4: num] :
( ( ord_less_eq_num @ X6 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X6 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_109_order__subst1,axiom,
! [A: nat,F: extended_enat > nat,B: extended_enat,C: extended_enat] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_le2932123472753598470d_enat @ B @ C )
=> ( ! [X6: extended_enat,Y4: extended_enat] :
( ( ord_le2932123472753598470d_enat @ X6 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X6 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_110_order__subst1,axiom,
! [A: num,F: nat > num,B: nat,C: nat] :
( ( ord_less_eq_num @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X6: nat,Y4: nat] :
( ( ord_less_eq_nat @ X6 @ Y4 )
=> ( ord_less_eq_num @ ( F @ X6 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_111_order__subst1,axiom,
! [A: num,F: num > num,B: num,C: num] :
( ( ord_less_eq_num @ A @ ( F @ B ) )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ! [X6: num,Y4: num] :
( ( ord_less_eq_num @ X6 @ Y4 )
=> ( ord_less_eq_num @ ( F @ X6 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_112_order__subst1,axiom,
! [A: num,F: extended_enat > num,B: extended_enat,C: extended_enat] :
( ( ord_less_eq_num @ A @ ( F @ B ) )
=> ( ( ord_le2932123472753598470d_enat @ B @ C )
=> ( ! [X6: extended_enat,Y4: extended_enat] :
( ( ord_le2932123472753598470d_enat @ X6 @ Y4 )
=> ( ord_less_eq_num @ ( F @ X6 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_113_order__subst1,axiom,
! [A: extended_enat,F: nat > extended_enat,B: nat,C: nat] :
( ( ord_le2932123472753598470d_enat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X6: nat,Y4: nat] :
( ( ord_less_eq_nat @ X6 @ Y4 )
=> ( ord_le2932123472753598470d_enat @ ( F @ X6 ) @ ( F @ Y4 ) ) )
=> ( ord_le2932123472753598470d_enat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_114_order__subst1,axiom,
! [A: extended_enat,F: num > extended_enat,B: num,C: num] :
( ( ord_le2932123472753598470d_enat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ! [X6: num,Y4: num] :
( ( ord_less_eq_num @ X6 @ Y4 )
=> ( ord_le2932123472753598470d_enat @ ( F @ X6 ) @ ( F @ Y4 ) ) )
=> ( ord_le2932123472753598470d_enat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_115_order__subst1,axiom,
! [A: extended_enat,F: extended_enat > extended_enat,B: extended_enat,C: extended_enat] :
( ( ord_le2932123472753598470d_enat @ A @ ( F @ B ) )
=> ( ( ord_le2932123472753598470d_enat @ B @ C )
=> ( ! [X6: extended_enat,Y4: extended_enat] :
( ( ord_le2932123472753598470d_enat @ X6 @ Y4 )
=> ( ord_le2932123472753598470d_enat @ ( F @ X6 ) @ ( F @ Y4 ) ) )
=> ( ord_le2932123472753598470d_enat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_116_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 )
=> ( ! [X6: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X6 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X6 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_117_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: nat,Z: nat] : ( Y5 = Z ) )
= ( ^ [A3: nat,B2: nat] :
( ( ord_less_eq_nat @ A3 @ B2 )
& ( ord_less_eq_nat @ B2 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_118_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: set_a,Z: set_a] : ( Y5 = Z ) )
= ( ^ [A3: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A3 @ B2 )
& ( ord_less_eq_set_a @ B2 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_119_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: num,Z: num] : ( Y5 = Z ) )
= ( ^ [A3: num,B2: num] :
( ( ord_less_eq_num @ A3 @ B2 )
& ( ord_less_eq_num @ B2 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_120_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: extended_enat,Z: extended_enat] : ( Y5 = Z ) )
= ( ^ [A3: extended_enat,B2: extended_enat] :
( ( ord_le2932123472753598470d_enat @ A3 @ B2 )
& ( ord_le2932123472753598470d_enat @ B2 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_121_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: set_set_a,Z: set_set_a] : ( Y5 = Z ) )
= ( ^ [A3: set_set_a,B2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A3 @ B2 )
& ( ord_le3724670747650509150_set_a @ B2 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_122_antisym,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_123_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_124_antisym,axiom,
! [A: num,B: num] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ord_less_eq_num @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_125_antisym,axiom,
! [A: extended_enat,B: extended_enat] :
( ( ord_le2932123472753598470d_enat @ A @ B )
=> ( ( ord_le2932123472753598470d_enat @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_126_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_127_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_128_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_129_dual__order_Otrans,axiom,
! [B: num,A: num,C: num] :
( ( ord_less_eq_num @ B @ A )
=> ( ( ord_less_eq_num @ C @ B )
=> ( ord_less_eq_num @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_130_dual__order_Otrans,axiom,
! [B: extended_enat,A: extended_enat,C: extended_enat] :
( ( ord_le2932123472753598470d_enat @ B @ A )
=> ( ( ord_le2932123472753598470d_enat @ C @ B )
=> ( ord_le2932123472753598470d_enat @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_131_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_132_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_133_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_134_dual__order_Oantisym,axiom,
! [B: num,A: num] :
( ( ord_less_eq_num @ B @ A )
=> ( ( ord_less_eq_num @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_135_dual__order_Oantisym,axiom,
! [B: extended_enat,A: extended_enat] :
( ( ord_le2932123472753598470d_enat @ B @ A )
=> ( ( ord_le2932123472753598470d_enat @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_136_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_137_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: nat,Z: nat] : ( Y5 = Z ) )
= ( ^ [A3: nat,B2: nat] :
( ( ord_less_eq_nat @ B2 @ A3 )
& ( ord_less_eq_nat @ A3 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_138_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: set_a,Z: set_a] : ( Y5 = Z ) )
= ( ^ [A3: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ B2 @ A3 )
& ( ord_less_eq_set_a @ A3 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_139_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: num,Z: num] : ( Y5 = Z ) )
= ( ^ [A3: num,B2: num] :
( ( ord_less_eq_num @ B2 @ A3 )
& ( ord_less_eq_num @ A3 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_140_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: extended_enat,Z: extended_enat] : ( Y5 = Z ) )
= ( ^ [A3: extended_enat,B2: extended_enat] :
( ( ord_le2932123472753598470d_enat @ B2 @ A3 )
& ( ord_le2932123472753598470d_enat @ A3 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_141_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: set_set_a,Z: set_set_a] : ( Y5 = Z ) )
= ( ^ [A3: set_set_a,B2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B2 @ A3 )
& ( ord_le3724670747650509150_set_a @ A3 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_142_linorder__wlog,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A4: nat,B3: nat] :
( ( ord_less_eq_nat @ A4 @ B3 )
=> ( P @ A4 @ B3 ) )
=> ( ! [A4: nat,B3: nat] :
( ( P @ B3 @ A4 )
=> ( P @ A4 @ B3 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_143_linorder__wlog,axiom,
! [P: num > num > $o,A: num,B: num] :
( ! [A4: num,B3: num] :
( ( ord_less_eq_num @ A4 @ B3 )
=> ( P @ A4 @ B3 ) )
=> ( ! [A4: num,B3: num] :
( ( P @ B3 @ A4 )
=> ( P @ A4 @ B3 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_144_linorder__wlog,axiom,
! [P: extended_enat > extended_enat > $o,A: extended_enat,B: extended_enat] :
( ! [A4: extended_enat,B3: extended_enat] :
( ( ord_le2932123472753598470d_enat @ A4 @ B3 )
=> ( P @ A4 @ B3 ) )
=> ( ! [A4: extended_enat,B3: extended_enat] :
( ( P @ B3 @ A4 )
=> ( P @ A4 @ B3 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_145_order__trans,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z2 )
=> ( ord_less_eq_nat @ X @ Z2 ) ) ) ).
% order_trans
thf(fact_146_order__trans,axiom,
! [X: set_a,Y: set_a,Z2: set_a] :
( ( ord_less_eq_set_a @ X @ Y )
=> ( ( ord_less_eq_set_a @ Y @ Z2 )
=> ( ord_less_eq_set_a @ X @ Z2 ) ) ) ).
% order_trans
thf(fact_147_order__trans,axiom,
! [X: num,Y: num,Z2: num] :
( ( ord_less_eq_num @ X @ Y )
=> ( ( ord_less_eq_num @ Y @ Z2 )
=> ( ord_less_eq_num @ X @ Z2 ) ) ) ).
% order_trans
thf(fact_148_order__trans,axiom,
! [X: extended_enat,Y: extended_enat,Z2: extended_enat] :
( ( ord_le2932123472753598470d_enat @ X @ Y )
=> ( ( ord_le2932123472753598470d_enat @ Y @ Z2 )
=> ( ord_le2932123472753598470d_enat @ X @ Z2 ) ) ) ).
% order_trans
thf(fact_149_order__trans,axiom,
! [X: set_set_a,Y: set_set_a,Z2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X @ Y )
=> ( ( ord_le3724670747650509150_set_a @ Y @ Z2 )
=> ( ord_le3724670747650509150_set_a @ X @ Z2 ) ) ) ).
% order_trans
thf(fact_150_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_151_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_152_order_Otrans,axiom,
! [A: num,B: num,C: num] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ord_less_eq_num @ A @ C ) ) ) ).
% order.trans
thf(fact_153_order_Otrans,axiom,
! [A: extended_enat,B: extended_enat,C: extended_enat] :
( ( ord_le2932123472753598470d_enat @ A @ B )
=> ( ( ord_le2932123472753598470d_enat @ B @ C )
=> ( ord_le2932123472753598470d_enat @ A @ C ) ) ) ).
% order.trans
thf(fact_154_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_155_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_156_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_157_order__antisym,axiom,
! [X: num,Y: num] :
( ( ord_less_eq_num @ X @ Y )
=> ( ( ord_less_eq_num @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_158_order__antisym,axiom,
! [X: extended_enat,Y: extended_enat] :
( ( ord_le2932123472753598470d_enat @ X @ Y )
=> ( ( ord_le2932123472753598470d_enat @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_159_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_160_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_161_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_162_ord__le__eq__trans,axiom,
! [A: num,B: num,C: num] :
( ( ord_less_eq_num @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_num @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_163_ord__le__eq__trans,axiom,
! [A: extended_enat,B: extended_enat,C: extended_enat] :
( ( ord_le2932123472753598470d_enat @ A @ B )
=> ( ( B = C )
=> ( ord_le2932123472753598470d_enat @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_164_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_165_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_166_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_167_ord__eq__le__trans,axiom,
! [A: num,B: num,C: num] :
( ( A = B )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ord_less_eq_num @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_168_ord__eq__le__trans,axiom,
! [A: extended_enat,B: extended_enat,C: extended_enat] :
( ( A = B )
=> ( ( ord_le2932123472753598470d_enat @ B @ C )
=> ( ord_le2932123472753598470d_enat @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_169_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_170_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: nat,Z: nat] : ( Y5 = Z ) )
= ( ^ [X5: nat,Y6: nat] :
( ( ord_less_eq_nat @ X5 @ Y6 )
& ( ord_less_eq_nat @ Y6 @ X5 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_171_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: set_a,Z: set_a] : ( Y5 = Z ) )
= ( ^ [X5: set_a,Y6: set_a] :
( ( ord_less_eq_set_a @ X5 @ Y6 )
& ( ord_less_eq_set_a @ Y6 @ X5 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_172_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: num,Z: num] : ( Y5 = Z ) )
= ( ^ [X5: num,Y6: num] :
( ( ord_less_eq_num @ X5 @ Y6 )
& ( ord_less_eq_num @ Y6 @ X5 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_173_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: extended_enat,Z: extended_enat] : ( Y5 = Z ) )
= ( ^ [X5: extended_enat,Y6: extended_enat] :
( ( ord_le2932123472753598470d_enat @ X5 @ Y6 )
& ( ord_le2932123472753598470d_enat @ Y6 @ X5 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_174_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: set_set_a,Z: set_set_a] : ( Y5 = Z ) )
= ( ^ [X5: set_set_a,Y6: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X5 @ Y6 )
& ( ord_le3724670747650509150_set_a @ Y6 @ X5 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_175_le__cases3,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( ( ord_less_eq_nat @ X @ Y )
=> ~ ( ord_less_eq_nat @ Y @ Z2 ) )
=> ( ( ( ord_less_eq_nat @ Y @ X )
=> ~ ( ord_less_eq_nat @ X @ Z2 ) )
=> ( ( ( ord_less_eq_nat @ X @ Z2 )
=> ~ ( ord_less_eq_nat @ Z2 @ Y ) )
=> ( ( ( ord_less_eq_nat @ Z2 @ Y )
=> ~ ( ord_less_eq_nat @ Y @ X ) )
=> ( ( ( ord_less_eq_nat @ Y @ Z2 )
=> ~ ( ord_less_eq_nat @ Z2 @ X ) )
=> ~ ( ( ord_less_eq_nat @ Z2 @ X )
=> ~ ( ord_less_eq_nat @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_176_le__cases3,axiom,
! [X: num,Y: num,Z2: num] :
( ( ( ord_less_eq_num @ X @ Y )
=> ~ ( ord_less_eq_num @ Y @ Z2 ) )
=> ( ( ( ord_less_eq_num @ Y @ X )
=> ~ ( ord_less_eq_num @ X @ Z2 ) )
=> ( ( ( ord_less_eq_num @ X @ Z2 )
=> ~ ( ord_less_eq_num @ Z2 @ Y ) )
=> ( ( ( ord_less_eq_num @ Z2 @ Y )
=> ~ ( ord_less_eq_num @ Y @ X ) )
=> ( ( ( ord_less_eq_num @ Y @ Z2 )
=> ~ ( ord_less_eq_num @ Z2 @ X ) )
=> ~ ( ( ord_less_eq_num @ Z2 @ X )
=> ~ ( ord_less_eq_num @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_177_le__cases3,axiom,
! [X: extended_enat,Y: extended_enat,Z2: extended_enat] :
( ( ( ord_le2932123472753598470d_enat @ X @ Y )
=> ~ ( ord_le2932123472753598470d_enat @ Y @ Z2 ) )
=> ( ( ( ord_le2932123472753598470d_enat @ Y @ X )
=> ~ ( ord_le2932123472753598470d_enat @ X @ Z2 ) )
=> ( ( ( ord_le2932123472753598470d_enat @ X @ Z2 )
=> ~ ( ord_le2932123472753598470d_enat @ Z2 @ Y ) )
=> ( ( ( ord_le2932123472753598470d_enat @ Z2 @ Y )
=> ~ ( ord_le2932123472753598470d_enat @ Y @ X ) )
=> ( ( ( ord_le2932123472753598470d_enat @ Y @ Z2 )
=> ~ ( ord_le2932123472753598470d_enat @ Z2 @ X ) )
=> ~ ( ( ord_le2932123472753598470d_enat @ Z2 @ X )
=> ~ ( ord_le2932123472753598470d_enat @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_178_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_179_nle__le,axiom,
! [A: num,B: num] :
( ( ~ ( ord_less_eq_num @ A @ B ) )
= ( ( ord_less_eq_num @ B @ A )
& ( B != A ) ) ) ).
% nle_le
thf(fact_180_nle__le,axiom,
! [A: extended_enat,B: extended_enat] :
( ( ~ ( ord_le2932123472753598470d_enat @ A @ B ) )
= ( ( ord_le2932123472753598470d_enat @ B @ A )
& ( B != A ) ) ) ).
% nle_le
thf(fact_181_verit__comp__simplify1_I2_J,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_182_verit__comp__simplify1_I2_J,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_183_verit__comp__simplify1_I2_J,axiom,
! [A: num] : ( ord_less_eq_num @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_184_verit__comp__simplify1_I2_J,axiom,
! [A: extended_enat] : ( ord_le2932123472753598470d_enat @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_185_verit__comp__simplify1_I2_J,axiom,
! [A: set_set_a] : ( ord_le3724670747650509150_set_a @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_186_num_Oexhaust,axiom,
! [Y: num] :
( ( Y != one )
=> ( ! [X22: num] :
( Y
!= ( bit0 @ X22 ) )
=> ~ ! [X32: num] :
( Y
!= ( bit1 @ X32 ) ) ) ) ).
% num.exhaust
thf(fact_187_not__numeral__le__zero,axiom,
! [N: num] :
~ ( ord_less_eq_nat @ ( numeral_numeral_nat @ N ) @ zero_zero_nat ) ).
% not_numeral_le_zero
thf(fact_188_not__numeral__le__zero,axiom,
! [N: num] :
~ ( ord_le2932123472753598470d_enat @ ( numera1916890842035813515d_enat @ N ) @ zero_z5237406670263579293d_enat ) ).
% not_numeral_le_zero
thf(fact_189_zero__le__numeral,axiom,
! [N: num] : ( ord_less_eq_nat @ zero_zero_nat @ ( numeral_numeral_nat @ N ) ) ).
% zero_le_numeral
thf(fact_190_zero__le__numeral,axiom,
! [N: num] : ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ ( numera1916890842035813515d_enat @ N ) ) ).
% zero_le_numeral
thf(fact_191_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_192_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_193_is__gen__path__def,axiom,
! [P2: list_a] :
( ( undire3562951555376170320path_a @ vertices @ edges @ P2 )
= ( ( undire6133010728901294956walk_a @ vertices @ edges @ P2 )
& ( ( ( distinct_a @ ( tl_a @ P2 ) )
& ( ( hd_a @ P2 )
= ( last_a @ P2 ) ) )
| ( distinct_a @ P2 ) ) ) ) ).
% is_gen_path_def
thf(fact_194_is__gen__path__distinct__tl,axiom,
! [P2: list_a] :
( ( undire3562951555376170320path_a @ vertices @ edges @ P2 )
=> ( ( ( hd_a @ P2 )
= ( last_a @ P2 ) )
=> ( distinct_a @ ( tl_a @ P2 ) ) ) ) ).
% is_gen_path_distinct_tl
thf(fact_195_is__path__walk,axiom,
! [Xs: list_a] :
( ( undire427332500224447920path_a @ vertices @ edges @ Xs )
=> ( undire6133010728901294956walk_a @ vertices @ edges @ Xs ) ) ).
% is_path_walk
thf(fact_196_is__gen__path__distinct,axiom,
! [P2: list_a] :
( ( undire3562951555376170320path_a @ vertices @ edges @ P2 )
=> ( ( ( hd_a @ P2 )
!= ( last_a @ P2 ) )
=> ( distinct_a @ P2 ) ) ) ).
% is_gen_path_distinct
thf(fact_197_diff__is__0__eq_H,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( minus_minus_nat @ M @ N )
= zero_zero_nat ) ) ).
% diff_is_0_eq'
thf(fact_198_diff__is__0__eq,axiom,
! [M: nat,N: nat] :
( ( ( minus_minus_nat @ M @ N )
= zero_zero_nat )
= ( ord_less_eq_nat @ M @ N ) ) ).
% diff_is_0_eq
thf(fact_199_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_200_diff__diff__cancel,axiom,
! [I: nat,N: nat] :
( ( ord_less_eq_nat @ I @ N )
=> ( ( minus_minus_nat @ N @ ( minus_minus_nat @ N @ I ) )
= I ) ) ).
% diff_diff_cancel
thf(fact_201_diff__Suc__Suc,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ ( suc @ M ) @ ( suc @ N ) )
= ( minus_minus_nat @ M @ N ) ) ).
% diff_Suc_Suc
thf(fact_202_old_Onat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( suc @ Nat )
= ( suc @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% old.nat.inject
thf(fact_203_nat_Oinject,axiom,
! [X2: nat,Y2: nat] :
( ( ( suc @ X2 )
= ( suc @ Y2 ) )
= ( X2 = Y2 ) ) ).
% nat.inject
thf(fact_204_is__path__gen__path,axiom,
! [P2: list_a] :
( ( undire427332500224447920path_a @ vertices @ edges @ P2 )
=> ( undire3562951555376170320path_a @ vertices @ edges @ P2 ) ) ).
% is_path_gen_path
thf(fact_205_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_206_le__zero__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_207_le__zero__eq,axiom,
! [N: extended_enat] :
( ( ord_le2932123472753598470d_enat @ N @ zero_z5237406670263579293d_enat )
= ( N = zero_z5237406670263579293d_enat ) ) ).
% le_zero_eq
thf(fact_208_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ A )
= zero_zero_nat ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_209_diff__zero,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ zero_zero_nat )
= A ) ).
% diff_zero
thf(fact_210_zero__diff,axiom,
! [A: nat] :
( ( minus_minus_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% zero_diff
thf(fact_211_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_212_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_213_Suc__le__mono,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( suc @ M ) )
= ( ord_less_eq_nat @ N @ M ) ) ).
% Suc_le_mono
thf(fact_214_diff__self__eq__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ M )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_215_diff__0__eq__0,axiom,
! [N: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_216_Suc__diff__diff,axiom,
! [M: nat,N: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ ( suc @ M ) @ N ) @ ( suc @ K ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M @ N ) @ K ) ) ).
% Suc_diff_diff
thf(fact_217_ulgraph_Ois__path_Ocong,axiom,
undire427332500224447920path_a = undire427332500224447920path_a ).
% ulgraph.is_path.cong
thf(fact_218_ulgraph_Ois__path__gen__path,axiom,
! [Vertices: set_a,Edges: set_set_a,P2: list_a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( undire427332500224447920path_a @ Vertices @ Edges @ P2 )
=> ( undire3562951555376170320path_a @ Vertices @ Edges @ P2 ) ) ) ).
% ulgraph.is_path_gen_path
thf(fact_219_ulgraph_Ois__gen__path_Ocong,axiom,
undire3562951555376170320path_a = undire3562951555376170320path_a ).
% ulgraph.is_gen_path.cong
thf(fact_220_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_221_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_222_ulgraph_Ois__open__walk_Ocong,axiom,
undire2427028224930250914walk_a = undire2427028224930250914walk_a ).
% ulgraph.is_open_walk.cong
thf(fact_223_ulgraph_Ois__closed__walk_Ocong,axiom,
undire3370724456595283424walk_a = undire3370724456595283424walk_a ).
% ulgraph.is_closed_walk.cong
thf(fact_224_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_225_ulgraph_Ois__gen__path__distinct,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,P2: list_set_a] :
( ( undire6886684016831807756_set_a @ Vertices @ Edges )
=> ( ( undire7201326534205417136_set_a @ Vertices @ Edges @ P2 )
=> ( ( ( hd_set_a @ P2 )
!= ( last_set_a @ P2 ) )
=> ( distinct_set_a @ P2 ) ) ) ) ).
% ulgraph.is_gen_path_distinct
thf(fact_226_ulgraph_Ois__gen__path__distinct,axiom,
! [Vertices: set_a,Edges: set_set_a,P2: list_a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( undire3562951555376170320path_a @ Vertices @ Edges @ P2 )
=> ( ( ( hd_a @ P2 )
!= ( last_a @ P2 ) )
=> ( distinct_a @ P2 ) ) ) ) ).
% ulgraph.is_gen_path_distinct
thf(fact_227_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_228_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_229_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_230_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_231_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_232_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_233_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_234_ulgraph_Ois__gen__path__distinct__tl,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,P2: list_set_a] :
( ( undire6886684016831807756_set_a @ Vertices @ Edges )
=> ( ( undire7201326534205417136_set_a @ Vertices @ Edges @ P2 )
=> ( ( ( hd_set_a @ P2 )
= ( last_set_a @ P2 ) )
=> ( distinct_set_a @ ( tl_set_a @ P2 ) ) ) ) ) ).
% ulgraph.is_gen_path_distinct_tl
thf(fact_235_ulgraph_Ois__gen__path__distinct__tl,axiom,
! [Vertices: set_a,Edges: set_set_a,P2: list_a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( undire3562951555376170320path_a @ Vertices @ Edges @ P2 )
=> ( ( ( hd_a @ P2 )
= ( last_a @ P2 ) )
=> ( distinct_a @ ( tl_a @ P2 ) ) ) ) ) ).
% ulgraph.is_gen_path_distinct_tl
thf(fact_236_ulgraph_Ois__gen__path__def,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,P2: list_set_a] :
( ( undire6886684016831807756_set_a @ Vertices @ Edges )
=> ( ( undire7201326534205417136_set_a @ Vertices @ Edges @ P2 )
= ( ( undire3014741414213135564_set_a @ Vertices @ Edges @ P2 )
& ( ( ( distinct_set_a @ ( tl_set_a @ P2 ) )
& ( ( hd_set_a @ P2 )
= ( last_set_a @ P2 ) ) )
| ( distinct_set_a @ P2 ) ) ) ) ) ).
% ulgraph.is_gen_path_def
thf(fact_237_ulgraph_Ois__gen__path__def,axiom,
! [Vertices: set_a,Edges: set_set_a,P2: list_a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( undire3562951555376170320path_a @ Vertices @ Edges @ P2 )
= ( ( undire6133010728901294956walk_a @ Vertices @ Edges @ P2 )
& ( ( ( distinct_a @ ( tl_a @ P2 ) )
& ( ( hd_a @ P2 )
= ( last_a @ P2 ) ) )
| ( distinct_a @ P2 ) ) ) ) ) ).
% ulgraph.is_gen_path_def
thf(fact_238_zero__reorient,axiom,
! [X: nat] :
( ( zero_zero_nat = X )
= ( X = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_239_zero__reorient,axiom,
! [X: extended_enat] :
( ( zero_z5237406670263579293d_enat = X )
= ( X = zero_z5237406670263579293d_enat ) ) ).
% zero_reorient
thf(fact_240_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
! [A: nat,C: nat,B: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A @ C ) @ B )
= ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C ) ) ).
% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_241_n__not__Suc__n,axiom,
! [N: nat] :
( N
!= ( suc @ N ) ) ).
% n_not_Suc_n
thf(fact_242_Suc__inject,axiom,
! [X: nat,Y: nat] :
( ( ( suc @ X )
= ( suc @ Y ) )
=> ( X = Y ) ) ).
% Suc_inject
thf(fact_243_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_244_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_245_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% eq_imp_le
thf(fact_246_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_247_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_248_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B: nat] :
( ( P @ K )
=> ( ! [Y4: nat] :
( ( P @ Y4 )
=> ( ord_less_eq_nat @ Y4 @ B ) )
=> ? [X6: nat] :
( ( P @ X6 )
& ! [Y7: nat] :
( ( P @ Y7 )
=> ( ord_less_eq_nat @ Y7 @ X6 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_249_size__neq__size__imp__neq,axiom,
! [X: list_a,Y: list_a] :
( ( ( size_size_list_a @ X )
!= ( size_size_list_a @ Y ) )
=> ( X != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_250_size__neq__size__imp__neq,axiom,
! [X: num,Y: num] :
( ( ( size_size_num @ X )
!= ( size_size_num @ Y ) )
=> ( X != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_251_size__neq__size__imp__neq,axiom,
! [X: list_set_a,Y: list_set_a] :
( ( ( size_size_list_set_a @ X )
!= ( size_size_list_set_a @ Y ) )
=> ( X != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_252_diff__commute,axiom,
! [I: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
= ( minus_minus_nat @ ( minus_minus_nat @ I @ K ) @ J ) ) ).
% diff_commute
thf(fact_253_zero__le,axiom,
! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).
% zero_le
thf(fact_254_zero__le,axiom,
! [X: extended_enat] : ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ X ) ).
% zero_le
thf(fact_255_ulgraph_Ois__walk__tl,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,Xs: list_set_a] :
( ( undire6886684016831807756_set_a @ Vertices @ Edges )
=> ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( size_size_list_set_a @ Xs ) )
=> ( ( undire3014741414213135564_set_a @ Vertices @ Edges @ Xs )
=> ( undire3014741414213135564_set_a @ Vertices @ Edges @ ( tl_set_a @ Xs ) ) ) ) ) ).
% ulgraph.is_walk_tl
thf(fact_256_ulgraph_Ois__walk__tl,axiom,
! [Vertices: set_a,Edges: set_set_a,Xs: list_a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( size_size_list_a @ Xs ) )
=> ( ( undire6133010728901294956walk_a @ Vertices @ Edges @ Xs )
=> ( undire6133010728901294956walk_a @ Vertices @ Edges @ ( tl_a @ Xs ) ) ) ) ) ).
% ulgraph.is_walk_tl
thf(fact_257_not0__implies__Suc,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ? [M2: nat] :
( N
= ( suc @ M2 ) ) ) ).
% not0_implies_Suc
thf(fact_258_Zero__not__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
% Zero_not_Suc
thf(fact_259_Zero__neq__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
% Zero_neq_Suc
thf(fact_260_Suc__neq__Zero,axiom,
! [M: nat] :
( ( suc @ M )
!= zero_zero_nat ) ).
% Suc_neq_Zero
thf(fact_261_zero__induct,axiom,
! [P: nat > $o,K: nat] :
( ( P @ K )
=> ( ! [N2: nat] :
( ( P @ ( suc @ N2 ) )
=> ( P @ N2 ) )
=> ( P @ zero_zero_nat ) ) ) ).
% zero_induct
thf(fact_262_diff__induct,axiom,
! [P: nat > nat > $o,M: nat,N: nat] :
( ! [X6: nat] : ( P @ X6 @ zero_zero_nat )
=> ( ! [Y4: nat] : ( P @ zero_zero_nat @ ( suc @ Y4 ) )
=> ( ! [X6: nat,Y4: nat] :
( ( P @ X6 @ Y4 )
=> ( P @ ( suc @ X6 ) @ ( suc @ Y4 ) ) )
=> ( P @ M @ N ) ) ) ) ).
% diff_induct
thf(fact_263_nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N2: nat] :
( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) )
=> ( P @ N ) ) ) ).
% nat_induct
thf(fact_264_old_Onat_Oexhaust,axiom,
! [Y: nat] :
( ( Y != zero_zero_nat )
=> ~ ! [Nat3: nat] :
( Y
!= ( suc @ Nat3 ) ) ) ).
% old.nat.exhaust
thf(fact_265_nat_OdiscI,axiom,
! [Nat: nat,X2: nat] :
( ( Nat
= ( suc @ X2 ) )
=> ( Nat != zero_zero_nat ) ) ).
% nat.discI
thf(fact_266_old_Onat_Odistinct_I1_J,axiom,
! [Nat2: nat] :
( zero_zero_nat
!= ( suc @ Nat2 ) ) ).
% old.nat.distinct(1)
thf(fact_267_old_Onat_Odistinct_I2_J,axiom,
! [Nat2: nat] :
( ( suc @ Nat2 )
!= zero_zero_nat ) ).
% old.nat.distinct(2)
thf(fact_268_nat_Odistinct_I1_J,axiom,
! [X2: nat] :
( zero_zero_nat
!= ( suc @ X2 ) ) ).
% nat.distinct(1)
thf(fact_269_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_270_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_271_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_272_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_273_Suc__leD,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% Suc_leD
thf(fact_274_le__SucE,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ ( suc @ N ) )
=> ( ~ ( ord_less_eq_nat @ M @ N )
=> ( M
= ( suc @ N ) ) ) ) ).
% le_SucE
thf(fact_275_le__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ M @ ( suc @ N ) ) ) ).
% le_SucI
thf(fact_276_Suc__le__D,axiom,
! [N: nat,M3: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ M3 )
=> ? [M2: nat] :
( M3
= ( suc @ M2 ) ) ) ).
% Suc_le_D
thf(fact_277_le__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ ( suc @ N ) )
= ( ( ord_less_eq_nat @ M @ N )
| ( M
= ( suc @ N ) ) ) ) ).
% le_Suc_eq
thf(fact_278_Suc__n__not__le__n,axiom,
! [N: nat] :
~ ( ord_less_eq_nat @ ( suc @ N ) @ N ) ).
% Suc_n_not_le_n
thf(fact_279_not__less__eq__eq,axiom,
! [M: nat,N: nat] :
( ( ~ ( ord_less_eq_nat @ M @ N ) )
= ( ord_less_eq_nat @ ( suc @ N ) @ M ) ) ).
% not_less_eq_eq
thf(fact_280_full__nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ! [M4: nat] :
( ( ord_less_eq_nat @ ( suc @ M4 ) @ N2 )
=> ( P @ M4 ) )
=> ( P @ N2 ) )
=> ( P @ N ) ) ).
% full_nat_induct
thf(fact_281_nat__induct__at__least,axiom,
! [M: nat,N: nat,P: nat > $o] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( P @ M )
=> ( ! [N2: nat] :
( ( ord_less_eq_nat @ M @ N2 )
=> ( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct_at_least
thf(fact_282_transitive__stepwise__le,axiom,
! [M: nat,N: nat,R: nat > nat > $o] :
( ( ord_less_eq_nat @ M @ N )
=> ( ! [X6: nat] : ( R @ X6 @ X6 )
=> ( ! [X6: nat,Y4: nat,Z3: nat] :
( ( R @ X6 @ Y4 )
=> ( ( R @ Y4 @ Z3 )
=> ( R @ X6 @ Z3 ) ) )
=> ( ! [N2: nat] : ( R @ N2 @ ( suc @ N2 ) )
=> ( R @ M @ N ) ) ) ) ) ).
% transitive_stepwise_le
thf(fact_283_diffs0__imp__equal,axiom,
! [M: nat,N: nat] :
( ( ( minus_minus_nat @ M @ N )
= zero_zero_nat )
=> ( ( ( minus_minus_nat @ N @ M )
= zero_zero_nat )
=> ( M = N ) ) ) ).
% diffs0_imp_equal
thf(fact_284_minus__nat_Odiff__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ zero_zero_nat )
= M ) ).
% minus_nat.diff_0
thf(fact_285_zero__induct__lemma,axiom,
! [P: nat > $o,K: nat,I: nat] :
( ( P @ K )
=> ( ! [N2: nat] :
( ( P @ ( suc @ N2 ) )
=> ( P @ N2 ) )
=> ( P @ ( minus_minus_nat @ K @ I ) ) ) ) ).
% zero_induct_lemma
thf(fact_286_eq__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ( minus_minus_nat @ M @ K )
= ( minus_minus_nat @ N @ K ) )
= ( M = N ) ) ) ) ).
% eq_diff_iff
thf(fact_287_le__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_eq_nat @ M @ N ) ) ) ) ).
% le_diff_iff
thf(fact_288_Nat_Odiff__diff__eq,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( minus_minus_nat @ M @ N ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_289_diff__le__mono,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M @ L ) @ ( minus_minus_nat @ N @ L ) ) ) ).
% diff_le_mono
thf(fact_290_diff__le__self,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ M ) ).
% diff_le_self
thf(fact_291_le__diff__iff_H,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ A @ C )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A ) @ ( minus_minus_nat @ C @ B ) )
= ( ord_less_eq_nat @ B @ A ) ) ) ) ).
% le_diff_iff'
thf(fact_292_diff__le__mono2,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ).
% diff_le_mono2
thf(fact_293_lift__Suc__mono__le,axiom,
! [F: nat > nat,N: nat,N3: nat] :
( ! [N2: nat] : ( ord_less_eq_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_eq_nat @ N @ N3 )
=> ( ord_less_eq_nat @ ( F @ N ) @ ( F @ N3 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_294_lift__Suc__mono__le,axiom,
! [F: nat > set_a,N: nat,N3: nat] :
( ! [N2: nat] : ( ord_less_eq_set_a @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_eq_nat @ N @ N3 )
=> ( ord_less_eq_set_a @ ( F @ N ) @ ( F @ N3 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_295_lift__Suc__mono__le,axiom,
! [F: nat > num,N: nat,N3: nat] :
( ! [N2: nat] : ( ord_less_eq_num @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_eq_nat @ N @ N3 )
=> ( ord_less_eq_num @ ( F @ N ) @ ( F @ N3 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_296_lift__Suc__mono__le,axiom,
! [F: nat > extended_enat,N: nat,N3: nat] :
( ! [N2: nat] : ( ord_le2932123472753598470d_enat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_eq_nat @ N @ N3 )
=> ( ord_le2932123472753598470d_enat @ ( F @ N ) @ ( F @ N3 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_297_lift__Suc__mono__le,axiom,
! [F: nat > set_set_a,N: nat,N3: nat] :
( ! [N2: nat] : ( ord_le3724670747650509150_set_a @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_eq_nat @ N @ N3 )
=> ( ord_le3724670747650509150_set_a @ ( F @ N ) @ ( F @ N3 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_298_lift__Suc__antimono__le,axiom,
! [F: nat > nat,N: nat,N3: nat] :
( ! [N2: nat] : ( ord_less_eq_nat @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
=> ( ( ord_less_eq_nat @ N @ N3 )
=> ( ord_less_eq_nat @ ( F @ N3 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_299_lift__Suc__antimono__le,axiom,
! [F: nat > set_a,N: nat,N3: nat] :
( ! [N2: nat] : ( ord_less_eq_set_a @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
=> ( ( ord_less_eq_nat @ N @ N3 )
=> ( ord_less_eq_set_a @ ( F @ N3 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_300_lift__Suc__antimono__le,axiom,
! [F: nat > num,N: nat,N3: nat] :
( ! [N2: nat] : ( ord_less_eq_num @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
=> ( ( ord_less_eq_nat @ N @ N3 )
=> ( ord_less_eq_num @ ( F @ N3 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_301_lift__Suc__antimono__le,axiom,
! [F: nat > extended_enat,N: nat,N3: nat] :
( ! [N2: nat] : ( ord_le2932123472753598470d_enat @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
=> ( ( ord_less_eq_nat @ N @ N3 )
=> ( ord_le2932123472753598470d_enat @ ( F @ N3 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_302_lift__Suc__antimono__le,axiom,
! [F: nat > set_set_a,N: nat,N3: nat] :
( ! [N2: nat] : ( ord_le3724670747650509150_set_a @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
=> ( ( ord_less_eq_nat @ N @ N3 )
=> ( ord_le3724670747650509150_set_a @ ( F @ N3 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_303_Suc__diff__le,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ N @ M )
=> ( ( minus_minus_nat @ ( suc @ M ) @ N )
= ( suc @ ( minus_minus_nat @ M @ N ) ) ) ) ).
% Suc_diff_le
thf(fact_304_is__gen__path__cycle,axiom,
! [P2: list_a] :
( ( undire2407311113669455967ycle_a @ vertices @ edges @ P2 )
=> ( undire3562951555376170320path_a @ vertices @ edges @ P2 ) ) ).
% is_gen_path_cycle
thf(fact_305_append__eq__append__conv,axiom,
! [Xs: list_a,Ys: list_a,Us: list_a,Vs: list_a] :
( ( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) )
| ( ( size_size_list_a @ Us )
= ( size_size_list_a @ Vs ) ) )
=> ( ( ( append_a @ Xs @ Us )
= ( append_a @ Ys @ Vs ) )
= ( ( Xs = Ys )
& ( Us = Vs ) ) ) ) ).
% append_eq_append_conv
thf(fact_306_append__eq__append__conv,axiom,
! [Xs: list_set_a,Ys: list_set_a,Us: list_set_a,Vs: list_set_a] :
( ( ( ( size_size_list_set_a @ Xs )
= ( size_size_list_set_a @ Ys ) )
| ( ( size_size_list_set_a @ Us )
= ( size_size_list_set_a @ Vs ) ) )
=> ( ( ( append_set_a @ Xs @ Us )
= ( append_set_a @ Ys @ Vs ) )
= ( ( Xs = Ys )
& ( Us = Vs ) ) ) ) ).
% append_eq_append_conv
thf(fact_307_enat__ord__number_I1_J,axiom,
! [M: num,N: num] :
( ( ord_le2932123472753598470d_enat @ ( numera1916890842035813515d_enat @ M ) @ ( numera1916890842035813515d_enat @ N ) )
= ( ord_less_eq_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) ) ) ).
% enat_ord_number(1)
thf(fact_308_induced__edges__ss,axiom,
! [V: set_a] :
( ( ord_less_eq_set_a @ V @ vertices )
=> ( ord_le3724670747650509150_set_a @ ( undire7777452895879145676dges_a @ edges @ V ) @ edges ) ) ).
% induced_edges_ss
thf(fact_309_has__loop__in__verts,axiom,
! [V2: a] :
( ( undire3617971648856834880loop_a @ edges @ V2 )
=> ( member_a @ V2 @ vertices ) ) ).
% has_loop_in_verts
thf(fact_310_incident__edge__in__wf,axiom,
! [E: set_a,V2: a] :
( ( member_set_a @ E @ edges )
=> ( ( undire1521409233611534436dent_a @ V2 @ E )
=> ( member_a @ V2 @ vertices ) ) ) ).
% incident_edge_in_wf
thf(fact_311_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_312_vert__adj__imp__inV,axiom,
! [V1: a,V22: a] :
( ( undire397441198561214472_adj_a @ edges @ V1 @ V22 )
=> ( ( member_a @ V1 @ vertices )
& ( member_a @ V22 @ vertices ) ) ) ).
% vert_adj_imp_inV
thf(fact_313_is__walk__not__empty2,axiom,
~ ( undire6133010728901294956walk_a @ vertices @ edges @ nil_a ) ).
% is_walk_not_empty2
thf(fact_314_is__walk__not__empty,axiom,
! [Xs: list_a] :
( ( undire6133010728901294956walk_a @ vertices @ edges @ Xs )
=> ( Xs != nil_a ) ) ).
% is_walk_not_empty
thf(fact_315_incident__def,axiom,
undire1521409233611534436dent_a = member_a ).
% incident_def
thf(fact_316_vert__adj__sym,axiom,
! [V1: a,V22: a] :
( ( undire397441198561214472_adj_a @ edges @ V1 @ V22 )
= ( undire397441198561214472_adj_a @ edges @ V22 @ V1 ) ) ).
% vert_adj_sym
thf(fact_317_vert__adj__edge__iff2,axiom,
! [V1: a,V22: a] :
( ( V1 != V22 )
=> ( ( undire397441198561214472_adj_a @ edges @ V1 @ V22 )
= ( ? [X5: set_a] :
( ( member_set_a @ X5 @ edges )
& ( undire1521409233611534436dent_a @ V1 @ X5 )
& ( undire1521409233611534436dent_a @ V22 @ X5 ) ) ) ) ) ).
% vert_adj_edge_iff2
thf(fact_318_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_319_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_320_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_321_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_322_append_Oright__neutral,axiom,
! [A: list_a] :
( ( append_a @ A @ nil_a )
= A ) ).
% append.right_neutral
thf(fact_323_append_Oright__neutral,axiom,
! [A: list_set_a] :
( ( append_set_a @ A @ nil_set_a )
= A ) ).
% append.right_neutral
thf(fact_324_append__Nil2,axiom,
! [Xs: list_a] :
( ( append_a @ Xs @ nil_a )
= Xs ) ).
% append_Nil2
thf(fact_325_append__Nil2,axiom,
! [Xs: list_set_a] :
( ( append_set_a @ Xs @ nil_set_a )
= Xs ) ).
% append_Nil2
thf(fact_326_append__self__conv,axiom,
! [Xs: list_a,Ys: list_a] :
( ( ( append_a @ Xs @ Ys )
= Xs )
= ( Ys = nil_a ) ) ).
% append_self_conv
thf(fact_327_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_328_self__append__conv,axiom,
! [Y: list_a,Ys: list_a] :
( ( Y
= ( append_a @ Y @ Ys ) )
= ( Ys = nil_a ) ) ).
% self_append_conv
thf(fact_329_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_330_append__self__conv2,axiom,
! [Xs: list_a,Ys: list_a] :
( ( ( append_a @ Xs @ Ys )
= Ys )
= ( Xs = nil_a ) ) ).
% append_self_conv2
thf(fact_331_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_332_self__append__conv2,axiom,
! [Y: list_a,Xs: list_a] :
( ( Y
= ( append_a @ Xs @ Y ) )
= ( Xs = nil_a ) ) ).
% self_append_conv2
thf(fact_333_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_334_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_335_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_336_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_337_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_338_length__0__conv,axiom,
! [Xs: list_a] :
( ( ( size_size_list_a @ Xs )
= zero_zero_nat )
= ( Xs = nil_a ) ) ).
% length_0_conv
thf(fact_339_length__0__conv,axiom,
! [Xs: list_set_a] :
( ( ( size_size_list_set_a @ Xs )
= zero_zero_nat )
= ( Xs = nil_set_a ) ) ).
% length_0_conv
thf(fact_340_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_341_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_342_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_343_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_344_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_345_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_346_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_347_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_348_i0__lb,axiom,
! [N: extended_enat] : ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ N ) ).
% i0_lb
thf(fact_349_ile0__eq,axiom,
! [N: extended_enat] :
( ( ord_le2932123472753598470d_enat @ N @ zero_z5237406670263579293d_enat )
= ( N = zero_z5237406670263579293d_enat ) ) ).
% ile0_eq
thf(fact_350_ulgraph_Ois__cycle_Ocong,axiom,
undire2407311113669455967ycle_a = undire2407311113669455967ycle_a ).
% ulgraph.is_cycle.cong
thf(fact_351_last__in__set,axiom,
! [As: list_a] :
( ( As != nil_a )
=> ( member_a @ ( last_a @ As ) @ ( set_a2 @ As ) ) ) ).
% last_in_set
thf(fact_352_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_353_hd__in__set,axiom,
! [Xs: list_a] :
( ( Xs != nil_a )
=> ( member_a @ ( hd_a @ Xs ) @ ( set_a2 @ Xs ) ) ) ).
% hd_in_set
thf(fact_354_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_355_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_356_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_357_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_358_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_359_subset__code_I1_J,axiom,
! [Xs: list_a,B4: set_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ Xs ) @ B4 )
= ( ! [X5: a] :
( ( member_a @ X5 @ ( set_a2 @ Xs ) )
=> ( member_a @ X5 @ B4 ) ) ) ) ).
% subset_code(1)
thf(fact_360_subset__code_I1_J,axiom,
! [Xs: list_set_a,B4: set_set_a] :
( ( ord_le3724670747650509150_set_a @ ( set_set_a2 @ Xs ) @ B4 )
= ( ! [X5: set_a] :
( ( member_set_a @ X5 @ ( set_set_a2 @ Xs ) )
=> ( member_set_a @ X5 @ B4 ) ) ) ) ).
% subset_code(1)
thf(fact_361_append__Nil,axiom,
! [Ys: list_a] :
( ( append_a @ nil_a @ Ys )
= Ys ) ).
% append_Nil
thf(fact_362_append__Nil,axiom,
! [Ys: list_set_a] :
( ( append_set_a @ nil_set_a @ Ys )
= Ys ) ).
% append_Nil
thf(fact_363_append_Oleft__neutral,axiom,
! [A: list_a] :
( ( append_a @ nil_a @ A )
= A ) ).
% append.left_neutral
thf(fact_364_append_Oleft__neutral,axiom,
! [A: list_set_a] :
( ( append_set_a @ nil_set_a @ A )
= A ) ).
% append.left_neutral
thf(fact_365_eq__Nil__appendI,axiom,
! [Xs: list_a,Ys: list_a] :
( ( Xs = Ys )
=> ( Xs
= ( append_a @ nil_a @ Ys ) ) ) ).
% eq_Nil_appendI
thf(fact_366_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_367_distinct_Osimps_I1_J,axiom,
distinct_a @ nil_a ).
% distinct.simps(1)
thf(fact_368_distinct_Osimps_I1_J,axiom,
distinct_set_a @ nil_set_a ).
% distinct.simps(1)
thf(fact_369_list_Osel_I2_J,axiom,
( ( tl_a @ nil_a )
= nil_a ) ).
% list.sel(2)
thf(fact_370_list_Osel_I2_J,axiom,
( ( tl_set_a @ nil_set_a )
= nil_set_a ) ).
% list.sel(2)
thf(fact_371_list_Osize_I3_J,axiom,
( ( size_size_list_a @ nil_a )
= zero_zero_nat ) ).
% list.size(3)
thf(fact_372_list_Osize_I3_J,axiom,
( ( size_size_list_set_a @ nil_set_a )
= zero_zero_nat ) ).
% list.size(3)
thf(fact_373_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_374_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_375_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_376_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_377_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_378_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_379_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_380_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_381_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_382_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_383_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_384_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_385_hd__Nil__eq__last,axiom,
( ( hd_a @ nil_a )
= ( last_a @ nil_a ) ) ).
% hd_Nil_eq_last
thf(fact_386_hd__Nil__eq__last,axiom,
( ( hd_set_a @ nil_set_a )
= ( last_set_a @ nil_set_a ) ) ).
% hd_Nil_eq_last
thf(fact_387_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_388_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_389_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_390_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_391_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_392_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_393_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_394_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_395_neq__if__length__neq,axiom,
! [Xs: list_a,Ys: list_a] :
( ( ( size_size_list_a @ Xs )
!= ( size_size_list_a @ Ys ) )
=> ( Xs != Ys ) ) ).
% neq_if_length_neq
thf(fact_396_neq__if__length__neq,axiom,
! [Xs: list_set_a,Ys: list_set_a] :
( ( ( size_size_list_set_a @ Xs )
!= ( size_size_list_set_a @ Ys ) )
=> ( Xs != Ys ) ) ).
% neq_if_length_neq
thf(fact_397_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs3: list_a] :
( ( size_size_list_a @ Xs3 )
= N ) ).
% Ex_list_of_length
thf(fact_398_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs3: list_set_a] :
( ( size_size_list_set_a @ Xs3 )
= N ) ).
% Ex_list_of_length
thf(fact_399_append__eq__appendI,axiom,
! [Xs: list_a,Xs1: list_a,Zs: list_a,Ys: list_a,Us: list_a] :
( ( ( append_a @ Xs @ Xs1 )
= Zs )
=> ( ( Ys
= ( append_a @ Xs1 @ Us ) )
=> ( ( append_a @ Xs @ Ys )
= ( append_a @ Zs @ Us ) ) ) ) ).
% append_eq_appendI
thf(fact_400_append__eq__append__conv2,axiom,
! [Xs: list_a,Ys: list_a,Zs: list_a,Ts: list_a] :
( ( ( append_a @ Xs @ Ys )
= ( append_a @ Zs @ Ts ) )
= ( ? [Us2: list_a] :
( ( ( Xs
= ( append_a @ Zs @ Us2 ) )
& ( ( append_a @ Us2 @ Ys )
= Ts ) )
| ( ( ( append_a @ Xs @ Us2 )
= Zs )
& ( Ys
= ( append_a @ Us2 @ Ts ) ) ) ) ) ) ).
% append_eq_append_conv2
thf(fact_401_ulgraph_Ois__gen__path__cycle,axiom,
! [Vertices: set_a,Edges: set_set_a,P2: list_a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( undire2407311113669455967ycle_a @ Vertices @ Edges @ P2 )
=> ( undire3562951555376170320path_a @ Vertices @ Edges @ P2 ) ) ) ).
% ulgraph.is_gen_path_cycle
thf(fact_402_num_Osize_I4_J,axiom,
( ( size_size_num @ one )
= zero_zero_nat ) ).
% num.size(4)
thf(fact_403_distinct__tl,axiom,
! [Xs: list_a] :
( ( distinct_a @ Xs )
=> ( distinct_a @ ( tl_a @ Xs ) ) ) ).
% distinct_tl
thf(fact_404_distinct__tl,axiom,
! [Xs: list_set_a] :
( ( distinct_set_a @ Xs )
=> ( distinct_set_a @ ( tl_set_a @ Xs ) ) ) ).
% distinct_tl
thf(fact_405_is__isolated__vertex__no__loop,axiom,
! [V2: a] :
( ( undire8931668460104145173rtex_a @ vertices @ edges @ V2 )
=> ~ ( undire3617971648856834880loop_a @ edges @ V2 ) ) ).
% is_isolated_vertex_no_loop
thf(fact_406_is__isolated__vertex__def,axiom,
! [V2: a] :
( ( undire8931668460104145173rtex_a @ vertices @ edges @ V2 )
= ( ( member_a @ V2 @ vertices )
& ! [X5: a] :
( ( member_a @ X5 @ vertices )
=> ~ ( undire397441198561214472_adj_a @ edges @ X5 @ V2 ) ) ) ) ).
% is_isolated_vertex_def
thf(fact_407_is__isolated__vertex__edge,axiom,
! [V2: a,E: set_a] :
( ( undire8931668460104145173rtex_a @ vertices @ edges @ V2 )
=> ( ( member_set_a @ E @ edges )
=> ~ ( undire1521409233611534436dent_a @ V2 @ E ) ) ) ).
% is_isolated_vertex_edge
thf(fact_408_last__in__list__tl__set,axiom,
! [Xs: list_a] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( size_size_list_a @ Xs ) )
=> ( member_a @ ( last_a @ Xs ) @ ( set_a2 @ ( tl_a @ Xs ) ) ) ) ).
% last_in_list_tl_set
thf(fact_409_last__in__list__tl__set,axiom,
! [Xs: list_set_a] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( size_size_list_set_a @ Xs ) )
=> ( member_set_a @ ( last_set_a @ Xs ) @ ( set_set_a2 @ ( tl_set_a @ Xs ) ) ) ) ).
% last_in_list_tl_set
thf(fact_410_induced__is__subgraph,axiom,
! [V: set_a] :
( ( ord_less_eq_set_a @ V @ vertices )
=> ( undire7103218114511261257raph_a @ V @ ( undire7777452895879145676dges_a @ edges @ V ) @ vertices @ edges ) ) ).
% induced_is_subgraph
thf(fact_411_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_412_Nitpick_Osize__list__simp_I2_J,axiom,
( size_size_list_a
= ( ^ [Xs4: list_a] : ( if_nat @ ( Xs4 = nil_a ) @ zero_zero_nat @ ( suc @ ( size_size_list_a @ ( tl_a @ Xs4 ) ) ) ) ) ) ).
% Nitpick.size_list_simp(2)
thf(fact_413_Nitpick_Osize__list__simp_I2_J,axiom,
( size_size_list_set_a
= ( ^ [Xs4: list_set_a] : ( if_nat @ ( Xs4 = nil_set_a ) @ zero_zero_nat @ ( suc @ ( size_size_list_set_a @ ( tl_set_a @ Xs4 ) ) ) ) ) ) ).
% Nitpick.size_list_simp(2)
thf(fact_414_is__gen__path__trivial,axiom,
! [X: a] :
( ( member_a @ X @ vertices )
=> ( undire3562951555376170320path_a @ vertices @ edges @ ( cons_a @ X @ nil_a ) ) ) ).
% is_gen_path_trivial
thf(fact_415_walk__edges_Ocases,axiom,
! [X: list_a] :
( ( X != nil_a )
=> ( ! [X6: a] :
( X
!= ( cons_a @ X6 @ nil_a ) )
=> ~ ! [X6: a,Y4: a,Ys3: list_a] :
( X
!= ( cons_a @ X6 @ ( cons_a @ Y4 @ Ys3 ) ) ) ) ) ).
% walk_edges.cases
thf(fact_416_list_Oinject,axiom,
! [X21: a,X222: list_a,Y21: a,Y22: list_a] :
( ( ( cons_a @ X21 @ X222 )
= ( cons_a @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X222 = Y22 ) ) ) ).
% list.inject
thf(fact_417_subgraph__refl,axiom,
undire7103218114511261257raph_a @ vertices @ edges @ vertices @ edges ).
% subgraph_refl
thf(fact_418_idiff__0,axiom,
! [N: extended_enat] :
( ( minus_3235023915231533773d_enat @ zero_z5237406670263579293d_enat @ N )
= zero_z5237406670263579293d_enat ) ).
% idiff_0
thf(fact_419_idiff__0__right,axiom,
! [N: extended_enat] :
( ( minus_3235023915231533773d_enat @ N @ zero_z5237406670263579293d_enat )
= N ) ).
% idiff_0_right
thf(fact_420_is__walk__singleton,axiom,
! [U: a] :
( ( member_a @ U @ vertices )
=> ( undire6133010728901294956walk_a @ vertices @ edges @ ( cons_a @ U @ nil_a ) ) ) ).
% is_walk_singleton
thf(fact_421_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_422_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_423_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_424_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_425_last__snoc,axiom,
! [Xs: list_a,X: a] :
( ( last_a @ ( append_a @ Xs @ ( cons_a @ X @ nil_a ) ) )
= X ) ).
% last_snoc
thf(fact_426_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_427_hd__Cons__tl,axiom,
! [Xs: list_a] :
( ( Xs != nil_a )
=> ( ( cons_a @ ( hd_a @ Xs ) @ ( tl_a @ Xs ) )
= Xs ) ) ).
% hd_Cons_tl
thf(fact_428_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_429_list_Ocollapse,axiom,
! [List: list_a] :
( ( List != nil_a )
=> ( ( cons_a @ ( hd_a @ List ) @ ( tl_a @ List ) )
= List ) ) ).
% list.collapse
thf(fact_430_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 ) )
=> ~ ! [X6: set_a,Xs3: list_set_a,Xss: list_list_set_a] :
( X
!= ( cons_list_set_a @ ( cons_set_a @ X6 @ Xs3 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_431_transpose_Ocases,axiom,
! [X: list_list_a] :
( ( X != nil_list_a )
=> ( ! [Xss: list_list_a] :
( X
!= ( cons_list_a @ nil_a @ Xss ) )
=> ~ ! [X6: a,Xs3: list_a,Xss: list_list_a] :
( X
!= ( cons_list_a @ ( cons_a @ X6 @ Xs3 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_432_not__Cons__self2,axiom,
! [X: a,Xs: list_a] :
( ( cons_a @ X @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_433_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_434_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_435_ulgraph_Ois__isolated__vertex_Ocong,axiom,
undire8931668460104145173rtex_a = undire8931668460104145173rtex_a ).
% ulgraph.is_isolated_vertex.cong
thf(fact_436_subgraph_Osubgraph__antisym,axiom,
! [V_H: set_a,E_H: set_set_a,V_G: set_a,E_G: set_set_a,V: set_a,E3: set_set_a,V3: set_a,E4: set_set_a] :
( ( undire7103218114511261257raph_a @ V_H @ E_H @ V_G @ E_G )
=> ( ( undire7103218114511261257raph_a @ V @ E3 @ V3 @ E4 )
=> ( ( undire7103218114511261257raph_a @ V3 @ E4 @ V @ E3 )
=> ( ( V3 = V )
& ( E4 = E3 ) ) ) ) ) ).
% subgraph.subgraph_antisym
thf(fact_437_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_438_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_439_list__nonempty__induct,axiom,
! [Xs: list_set_a,P: list_set_a > $o] :
( ( Xs != nil_set_a )
=> ( ! [X6: set_a] : ( P @ ( cons_set_a @ X6 @ nil_set_a ) )
=> ( ! [X6: set_a,Xs3: list_set_a] :
( ( Xs3 != nil_set_a )
=> ( ( P @ Xs3 )
=> ( P @ ( cons_set_a @ X6 @ Xs3 ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_440_list__nonempty__induct,axiom,
! [Xs: list_a,P: list_a > $o] :
( ( Xs != nil_a )
=> ( ! [X6: a] : ( P @ ( cons_a @ X6 @ nil_a ) )
=> ( ! [X6: a,Xs3: list_a] :
( ( Xs3 != nil_a )
=> ( ( P @ Xs3 )
=> ( P @ ( cons_a @ X6 @ Xs3 ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_441_list__induct2_H,axiom,
! [P: list_set_a > list_set_a > $o,Xs: list_set_a,Ys: list_set_a] :
( ( P @ nil_set_a @ nil_set_a )
=> ( ! [X6: set_a,Xs3: list_set_a] : ( P @ ( cons_set_a @ X6 @ Xs3 ) @ nil_set_a )
=> ( ! [Y4: set_a,Ys3: list_set_a] : ( P @ nil_set_a @ ( cons_set_a @ Y4 @ Ys3 ) )
=> ( ! [X6: set_a,Xs3: list_set_a,Y4: set_a,Ys3: list_set_a] :
( ( P @ Xs3 @ Ys3 )
=> ( P @ ( cons_set_a @ X6 @ Xs3 ) @ ( cons_set_a @ Y4 @ Ys3 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_442_list__induct2_H,axiom,
! [P: list_set_a > list_a > $o,Xs: list_set_a,Ys: list_a] :
( ( P @ nil_set_a @ nil_a )
=> ( ! [X6: set_a,Xs3: list_set_a] : ( P @ ( cons_set_a @ X6 @ Xs3 ) @ nil_a )
=> ( ! [Y4: a,Ys3: list_a] : ( P @ nil_set_a @ ( cons_a @ Y4 @ Ys3 ) )
=> ( ! [X6: set_a,Xs3: list_set_a,Y4: a,Ys3: list_a] :
( ( P @ Xs3 @ Ys3 )
=> ( P @ ( cons_set_a @ X6 @ Xs3 ) @ ( cons_a @ Y4 @ Ys3 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_443_list__induct2_H,axiom,
! [P: list_a > list_set_a > $o,Xs: list_a,Ys: list_set_a] :
( ( P @ nil_a @ nil_set_a )
=> ( ! [X6: a,Xs3: list_a] : ( P @ ( cons_a @ X6 @ Xs3 ) @ nil_set_a )
=> ( ! [Y4: set_a,Ys3: list_set_a] : ( P @ nil_a @ ( cons_set_a @ Y4 @ Ys3 ) )
=> ( ! [X6: a,Xs3: list_a,Y4: set_a,Ys3: list_set_a] :
( ( P @ Xs3 @ Ys3 )
=> ( P @ ( cons_a @ X6 @ Xs3 ) @ ( cons_set_a @ Y4 @ Ys3 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_444_list__induct2_H,axiom,
! [P: list_a > list_a > $o,Xs: list_a,Ys: list_a] :
( ( P @ nil_a @ nil_a )
=> ( ! [X6: a,Xs3: list_a] : ( P @ ( cons_a @ X6 @ Xs3 ) @ nil_a )
=> ( ! [Y4: a,Ys3: list_a] : ( P @ nil_a @ ( cons_a @ Y4 @ Ys3 ) )
=> ( ! [X6: a,Xs3: list_a,Y4: a,Ys3: list_a] :
( ( P @ Xs3 @ Ys3 )
=> ( P @ ( cons_a @ X6 @ Xs3 ) @ ( cons_a @ Y4 @ Ys3 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_445_neq__Nil__conv,axiom,
! [Xs: list_set_a] :
( ( Xs != nil_set_a )
= ( ? [Y6: set_a,Ys4: list_set_a] :
( Xs
= ( cons_set_a @ Y6 @ Ys4 ) ) ) ) ).
% neq_Nil_conv
thf(fact_446_neq__Nil__conv,axiom,
! [Xs: list_a] :
( ( Xs != nil_a )
= ( ? [Y6: a,Ys4: list_a] :
( Xs
= ( cons_a @ Y6 @ Ys4 ) ) ) ) ).
% neq_Nil_conv
thf(fact_447_comp__sgraph_Owalk__edges_Ocases,axiom,
! [X: list_set_a] :
( ( X != nil_set_a )
=> ( ! [X6: set_a] :
( X
!= ( cons_set_a @ X6 @ nil_set_a ) )
=> ~ ! [X6: set_a,Y4: set_a,Ys3: list_set_a] :
( X
!= ( cons_set_a @ X6 @ ( cons_set_a @ Y4 @ Ys3 ) ) ) ) ) ).
% comp_sgraph.walk_edges.cases
thf(fact_448_comp__sgraph_Owalk__edges_Ocases,axiom,
! [X: list_a] :
( ( X != nil_a )
=> ( ! [X6: a] :
( X
!= ( cons_a @ X6 @ nil_a ) )
=> ~ ! [X6: a,Y4: a,Ys3: list_a] :
( X
!= ( cons_a @ X6 @ ( cons_a @ Y4 @ Ys3 ) ) ) ) ) ).
% comp_sgraph.walk_edges.cases
thf(fact_449_min__list_Ocases,axiom,
! [X: list_set_a] :
( ! [X6: set_a,Xs3: list_set_a] :
( X
!= ( cons_set_a @ X6 @ Xs3 ) )
=> ( X = nil_set_a ) ) ).
% min_list.cases
thf(fact_450_list_Oexhaust,axiom,
! [Y: list_set_a] :
( ( Y != nil_set_a )
=> ~ ! [X212: set_a,X223: list_set_a] :
( Y
!= ( cons_set_a @ X212 @ X223 ) ) ) ).
% list.exhaust
thf(fact_451_list_Oexhaust,axiom,
! [Y: list_a] :
( ( Y != nil_a )
=> ~ ! [X212: a,X223: list_a] :
( Y
!= ( cons_a @ X212 @ X223 ) ) ) ).
% list.exhaust
thf(fact_452_list_OdiscI,axiom,
! [List: list_set_a,X21: set_a,X222: list_set_a] :
( ( List
= ( cons_set_a @ X21 @ X222 ) )
=> ( List != nil_set_a ) ) ).
% list.discI
thf(fact_453_list_OdiscI,axiom,
! [List: list_a,X21: a,X222: list_a] :
( ( List
= ( cons_a @ X21 @ X222 ) )
=> ( List != nil_a ) ) ).
% list.discI
thf(fact_454_list_Odistinct_I1_J,axiom,
! [X21: set_a,X222: list_set_a] :
( nil_set_a
!= ( cons_set_a @ X21 @ X222 ) ) ).
% list.distinct(1)
thf(fact_455_list_Odistinct_I1_J,axiom,
! [X21: a,X222: list_a] :
( nil_a
!= ( cons_a @ X21 @ X222 ) ) ).
% list.distinct(1)
thf(fact_456_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_457_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_458_list_Oset__cases,axiom,
! [E: set_a,A: list_set_a] :
( ( member_set_a @ E @ ( set_set_a2 @ A ) )
=> ( ! [Z22: list_set_a] :
( A
!= ( cons_set_a @ E @ Z22 ) )
=> ~ ! [Z1: set_a,Z22: list_set_a] :
( ( A
= ( cons_set_a @ Z1 @ Z22 ) )
=> ~ ( member_set_a @ E @ ( set_set_a2 @ Z22 ) ) ) ) ) ).
% list.set_cases
thf(fact_459_list_Oset__cases,axiom,
! [E: a,A: list_a] :
( ( member_a @ E @ ( set_a2 @ A ) )
=> ( ! [Z22: list_a] :
( A
!= ( cons_a @ E @ Z22 ) )
=> ~ ! [Z1: a,Z22: list_a] :
( ( A
= ( cons_a @ Z1 @ Z22 ) )
=> ~ ( member_a @ E @ ( set_a2 @ Z22 ) ) ) ) ) ).
% list.set_cases
thf(fact_460_list_Oset__intros_I1_J,axiom,
! [X21: set_a,X222: list_set_a] : ( member_set_a @ X21 @ ( set_set_a2 @ ( cons_set_a @ X21 @ X222 ) ) ) ).
% list.set_intros(1)
thf(fact_461_list_Oset__intros_I1_J,axiom,
! [X21: a,X222: list_a] : ( member_a @ X21 @ ( set_a2 @ ( cons_a @ X21 @ X222 ) ) ) ).
% list.set_intros(1)
thf(fact_462_list_Oset__intros_I2_J,axiom,
! [Y: set_a,X222: list_set_a,X21: set_a] :
( ( member_set_a @ Y @ ( set_set_a2 @ X222 ) )
=> ( member_set_a @ Y @ ( set_set_a2 @ ( cons_set_a @ X21 @ X222 ) ) ) ) ).
% list.set_intros(2)
thf(fact_463_list_Oset__intros_I2_J,axiom,
! [Y: a,X222: list_a,X21: a] :
( ( member_a @ Y @ ( set_a2 @ X222 ) )
=> ( member_a @ Y @ ( set_a2 @ ( cons_a @ X21 @ X222 ) ) ) ) ).
% list.set_intros(2)
thf(fact_464_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_465_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_466_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_467_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_468_list_Osel_I1_J,axiom,
! [X21: a,X222: list_a] :
( ( hd_a @ ( cons_a @ X21 @ X222 ) )
= X21 ) ).
% list.sel(1)
thf(fact_469_list_Osel_I3_J,axiom,
! [X21: a,X222: list_a] :
( ( tl_a @ ( cons_a @ X21 @ X222 ) )
= X222 ) ).
% list.sel(3)
thf(fact_470_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_471_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_472_Suc__length__conv,axiom,
! [N: nat,Xs: list_a] :
( ( ( suc @ N )
= ( size_size_list_a @ Xs ) )
= ( ? [Y6: a,Ys4: list_a] :
( ( Xs
= ( cons_a @ Y6 @ Ys4 ) )
& ( ( size_size_list_a @ Ys4 )
= N ) ) ) ) ).
% Suc_length_conv
thf(fact_473_Suc__length__conv,axiom,
! [N: nat,Xs: list_set_a] :
( ( ( suc @ N )
= ( size_size_list_set_a @ Xs ) )
= ( ? [Y6: set_a,Ys4: list_set_a] :
( ( Xs
= ( cons_set_a @ Y6 @ Ys4 ) )
& ( ( size_size_list_set_a @ Ys4 )
= N ) ) ) ) ).
% Suc_length_conv
thf(fact_474_length__Suc__conv,axiom,
! [Xs: list_a,N: nat] :
( ( ( size_size_list_a @ Xs )
= ( suc @ N ) )
= ( ? [Y6: a,Ys4: list_a] :
( ( Xs
= ( cons_a @ Y6 @ Ys4 ) )
& ( ( size_size_list_a @ Ys4 )
= N ) ) ) ) ).
% length_Suc_conv
thf(fact_475_length__Suc__conv,axiom,
! [Xs: list_set_a,N: nat] :
( ( ( size_size_list_set_a @ Xs )
= ( suc @ N ) )
= ( ? [Y6: set_a,Ys4: list_set_a] :
( ( Xs
= ( cons_set_a @ Y6 @ Ys4 ) )
& ( ( size_size_list_set_a @ Ys4 )
= N ) ) ) ) ).
% length_Suc_conv
thf(fact_476_list__induct4,axiom,
! [Xs: list_a,Ys: list_a,Zs: list_a,Ws: list_a,P: list_a > list_a > list_a > list_a > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( ( size_size_list_a @ Ys )
= ( size_size_list_a @ Zs ) )
=> ( ( ( size_size_list_a @ Zs )
= ( size_size_list_a @ Ws ) )
=> ( ( P @ nil_a @ nil_a @ nil_a @ nil_a )
=> ( ! [X6: a,Xs3: list_a,Y4: a,Ys3: list_a,Z3: a,Zs2: list_a,W: a,Ws2: list_a] :
( ( ( size_size_list_a @ Xs3 )
= ( size_size_list_a @ Ys3 ) )
=> ( ( ( size_size_list_a @ Ys3 )
= ( size_size_list_a @ Zs2 ) )
=> ( ( ( size_size_list_a @ Zs2 )
= ( size_size_list_a @ Ws2 ) )
=> ( ( P @ Xs3 @ Ys3 @ Zs2 @ Ws2 )
=> ( P @ ( cons_a @ X6 @ Xs3 ) @ ( cons_a @ Y4 @ Ys3 ) @ ( cons_a @ Z3 @ Zs2 ) @ ( cons_a @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_477_list__induct4,axiom,
! [Xs: list_a,Ys: list_a,Zs: list_a,Ws: list_set_a,P: list_a > list_a > list_a > list_set_a > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( ( size_size_list_a @ Ys )
= ( size_size_list_a @ Zs ) )
=> ( ( ( size_size_list_a @ Zs )
= ( size_size_list_set_a @ Ws ) )
=> ( ( P @ nil_a @ nil_a @ nil_a @ nil_set_a )
=> ( ! [X6: a,Xs3: list_a,Y4: a,Ys3: list_a,Z3: a,Zs2: list_a,W: set_a,Ws2: list_set_a] :
( ( ( size_size_list_a @ Xs3 )
= ( size_size_list_a @ Ys3 ) )
=> ( ( ( size_size_list_a @ Ys3 )
= ( size_size_list_a @ Zs2 ) )
=> ( ( ( size_size_list_a @ Zs2 )
= ( size_size_list_set_a @ Ws2 ) )
=> ( ( P @ Xs3 @ Ys3 @ Zs2 @ Ws2 )
=> ( P @ ( cons_a @ X6 @ Xs3 ) @ ( cons_a @ Y4 @ Ys3 ) @ ( cons_a @ Z3 @ Zs2 ) @ ( cons_set_a @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_478_list__induct4,axiom,
! [Xs: list_a,Ys: list_a,Zs: list_set_a,Ws: list_a,P: list_a > list_a > list_set_a > list_a > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( ( size_size_list_a @ Ys )
= ( size_size_list_set_a @ Zs ) )
=> ( ( ( size_size_list_set_a @ Zs )
= ( size_size_list_a @ Ws ) )
=> ( ( P @ nil_a @ nil_a @ nil_set_a @ nil_a )
=> ( ! [X6: a,Xs3: list_a,Y4: a,Ys3: list_a,Z3: set_a,Zs2: list_set_a,W: a,Ws2: list_a] :
( ( ( size_size_list_a @ Xs3 )
= ( size_size_list_a @ Ys3 ) )
=> ( ( ( size_size_list_a @ Ys3 )
= ( size_size_list_set_a @ Zs2 ) )
=> ( ( ( size_size_list_set_a @ Zs2 )
= ( size_size_list_a @ Ws2 ) )
=> ( ( P @ Xs3 @ Ys3 @ Zs2 @ Ws2 )
=> ( P @ ( cons_a @ X6 @ Xs3 ) @ ( cons_a @ Y4 @ Ys3 ) @ ( cons_set_a @ Z3 @ Zs2 ) @ ( cons_a @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_479_list__induct4,axiom,
! [Xs: list_a,Ys: list_set_a,Zs: list_a,Ws: list_a,P: list_a > list_set_a > list_a > list_a > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_set_a @ Ys ) )
=> ( ( ( size_size_list_set_a @ Ys )
= ( size_size_list_a @ Zs ) )
=> ( ( ( size_size_list_a @ Zs )
= ( size_size_list_a @ Ws ) )
=> ( ( P @ nil_a @ nil_set_a @ nil_a @ nil_a )
=> ( ! [X6: a,Xs3: list_a,Y4: set_a,Ys3: list_set_a,Z3: a,Zs2: list_a,W: a,Ws2: list_a] :
( ( ( size_size_list_a @ Xs3 )
= ( size_size_list_set_a @ Ys3 ) )
=> ( ( ( size_size_list_set_a @ Ys3 )
= ( size_size_list_a @ Zs2 ) )
=> ( ( ( size_size_list_a @ Zs2 )
= ( size_size_list_a @ Ws2 ) )
=> ( ( P @ Xs3 @ Ys3 @ Zs2 @ Ws2 )
=> ( P @ ( cons_a @ X6 @ Xs3 ) @ ( cons_set_a @ Y4 @ Ys3 ) @ ( cons_a @ Z3 @ Zs2 ) @ ( cons_a @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_480_list__induct4,axiom,
! [Xs: list_set_a,Ys: list_a,Zs: list_a,Ws: list_a,P: list_set_a > list_a > list_a > list_a > $o] :
( ( ( size_size_list_set_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( ( size_size_list_a @ Ys )
= ( size_size_list_a @ Zs ) )
=> ( ( ( size_size_list_a @ Zs )
= ( size_size_list_a @ Ws ) )
=> ( ( P @ nil_set_a @ nil_a @ nil_a @ nil_a )
=> ( ! [X6: set_a,Xs3: list_set_a,Y4: a,Ys3: list_a,Z3: a,Zs2: list_a,W: a,Ws2: list_a] :
( ( ( size_size_list_set_a @ Xs3 )
= ( size_size_list_a @ Ys3 ) )
=> ( ( ( size_size_list_a @ Ys3 )
= ( size_size_list_a @ Zs2 ) )
=> ( ( ( size_size_list_a @ Zs2 )
= ( size_size_list_a @ Ws2 ) )
=> ( ( P @ Xs3 @ Ys3 @ Zs2 @ Ws2 )
=> ( P @ ( cons_set_a @ X6 @ Xs3 ) @ ( cons_a @ Y4 @ Ys3 ) @ ( cons_a @ Z3 @ Zs2 ) @ ( cons_a @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_481_list__induct4,axiom,
! [Xs: list_a,Ys: list_a,Zs: list_set_a,Ws: list_set_a,P: list_a > list_a > list_set_a > list_set_a > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( ( size_size_list_a @ Ys )
= ( size_size_list_set_a @ Zs ) )
=> ( ( ( size_size_list_set_a @ Zs )
= ( size_size_list_set_a @ Ws ) )
=> ( ( P @ nil_a @ nil_a @ nil_set_a @ nil_set_a )
=> ( ! [X6: a,Xs3: list_a,Y4: a,Ys3: list_a,Z3: set_a,Zs2: list_set_a,W: set_a,Ws2: list_set_a] :
( ( ( size_size_list_a @ Xs3 )
= ( size_size_list_a @ Ys3 ) )
=> ( ( ( size_size_list_a @ Ys3 )
= ( size_size_list_set_a @ Zs2 ) )
=> ( ( ( size_size_list_set_a @ Zs2 )
= ( size_size_list_set_a @ Ws2 ) )
=> ( ( P @ Xs3 @ Ys3 @ Zs2 @ Ws2 )
=> ( P @ ( cons_a @ X6 @ Xs3 ) @ ( cons_a @ Y4 @ Ys3 ) @ ( cons_set_a @ Z3 @ Zs2 ) @ ( cons_set_a @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_482_list__induct4,axiom,
! [Xs: list_a,Ys: list_set_a,Zs: list_a,Ws: list_set_a,P: list_a > list_set_a > list_a > list_set_a > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_set_a @ Ys ) )
=> ( ( ( size_size_list_set_a @ Ys )
= ( size_size_list_a @ Zs ) )
=> ( ( ( size_size_list_a @ Zs )
= ( size_size_list_set_a @ Ws ) )
=> ( ( P @ nil_a @ nil_set_a @ nil_a @ nil_set_a )
=> ( ! [X6: a,Xs3: list_a,Y4: set_a,Ys3: list_set_a,Z3: a,Zs2: list_a,W: set_a,Ws2: list_set_a] :
( ( ( size_size_list_a @ Xs3 )
= ( size_size_list_set_a @ Ys3 ) )
=> ( ( ( size_size_list_set_a @ Ys3 )
= ( size_size_list_a @ Zs2 ) )
=> ( ( ( size_size_list_a @ Zs2 )
= ( size_size_list_set_a @ Ws2 ) )
=> ( ( P @ Xs3 @ Ys3 @ Zs2 @ Ws2 )
=> ( P @ ( cons_a @ X6 @ Xs3 ) @ ( cons_set_a @ Y4 @ Ys3 ) @ ( cons_a @ Z3 @ Zs2 ) @ ( cons_set_a @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_483_list__induct4,axiom,
! [Xs: list_a,Ys: list_set_a,Zs: list_set_a,Ws: list_a,P: list_a > list_set_a > list_set_a > list_a > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_set_a @ Ys ) )
=> ( ( ( size_size_list_set_a @ Ys )
= ( size_size_list_set_a @ Zs ) )
=> ( ( ( size_size_list_set_a @ Zs )
= ( size_size_list_a @ Ws ) )
=> ( ( P @ nil_a @ nil_set_a @ nil_set_a @ nil_a )
=> ( ! [X6: a,Xs3: list_a,Y4: set_a,Ys3: list_set_a,Z3: set_a,Zs2: list_set_a,W: a,Ws2: list_a] :
( ( ( size_size_list_a @ Xs3 )
= ( size_size_list_set_a @ Ys3 ) )
=> ( ( ( size_size_list_set_a @ Ys3 )
= ( size_size_list_set_a @ Zs2 ) )
=> ( ( ( size_size_list_set_a @ Zs2 )
= ( size_size_list_a @ Ws2 ) )
=> ( ( P @ Xs3 @ Ys3 @ Zs2 @ Ws2 )
=> ( P @ ( cons_a @ X6 @ Xs3 ) @ ( cons_set_a @ Y4 @ Ys3 ) @ ( cons_set_a @ Z3 @ Zs2 ) @ ( cons_a @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_484_list__induct4,axiom,
! [Xs: list_set_a,Ys: list_a,Zs: list_a,Ws: list_set_a,P: list_set_a > list_a > list_a > list_set_a > $o] :
( ( ( size_size_list_set_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( ( size_size_list_a @ Ys )
= ( size_size_list_a @ Zs ) )
=> ( ( ( size_size_list_a @ Zs )
= ( size_size_list_set_a @ Ws ) )
=> ( ( P @ nil_set_a @ nil_a @ nil_a @ nil_set_a )
=> ( ! [X6: set_a,Xs3: list_set_a,Y4: a,Ys3: list_a,Z3: a,Zs2: list_a,W: set_a,Ws2: list_set_a] :
( ( ( size_size_list_set_a @ Xs3 )
= ( size_size_list_a @ Ys3 ) )
=> ( ( ( size_size_list_a @ Ys3 )
= ( size_size_list_a @ Zs2 ) )
=> ( ( ( size_size_list_a @ Zs2 )
= ( size_size_list_set_a @ Ws2 ) )
=> ( ( P @ Xs3 @ Ys3 @ Zs2 @ Ws2 )
=> ( P @ ( cons_set_a @ X6 @ Xs3 ) @ ( cons_a @ Y4 @ Ys3 ) @ ( cons_a @ Z3 @ Zs2 ) @ ( cons_set_a @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_485_list__induct4,axiom,
! [Xs: list_set_a,Ys: list_a,Zs: list_set_a,Ws: list_a,P: list_set_a > list_a > list_set_a > list_a > $o] :
( ( ( size_size_list_set_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( ( size_size_list_a @ Ys )
= ( size_size_list_set_a @ Zs ) )
=> ( ( ( size_size_list_set_a @ Zs )
= ( size_size_list_a @ Ws ) )
=> ( ( P @ nil_set_a @ nil_a @ nil_set_a @ nil_a )
=> ( ! [X6: set_a,Xs3: list_set_a,Y4: a,Ys3: list_a,Z3: set_a,Zs2: list_set_a,W: a,Ws2: list_a] :
( ( ( size_size_list_set_a @ Xs3 )
= ( size_size_list_a @ Ys3 ) )
=> ( ( ( size_size_list_a @ Ys3 )
= ( size_size_list_set_a @ Zs2 ) )
=> ( ( ( size_size_list_set_a @ Zs2 )
= ( size_size_list_a @ Ws2 ) )
=> ( ( P @ Xs3 @ Ys3 @ Zs2 @ Ws2 )
=> ( P @ ( cons_set_a @ X6 @ Xs3 ) @ ( cons_a @ Y4 @ Ys3 ) @ ( cons_set_a @ Z3 @ Zs2 ) @ ( cons_a @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_486_list__induct3,axiom,
! [Xs: list_a,Ys: list_a,Zs: list_a,P: list_a > list_a > list_a > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( ( size_size_list_a @ Ys )
= ( size_size_list_a @ Zs ) )
=> ( ( P @ nil_a @ nil_a @ nil_a )
=> ( ! [X6: a,Xs3: list_a,Y4: a,Ys3: list_a,Z3: a,Zs2: list_a] :
( ( ( size_size_list_a @ Xs3 )
= ( size_size_list_a @ Ys3 ) )
=> ( ( ( size_size_list_a @ Ys3 )
= ( size_size_list_a @ Zs2 ) )
=> ( ( P @ Xs3 @ Ys3 @ Zs2 )
=> ( P @ ( cons_a @ X6 @ Xs3 ) @ ( cons_a @ Y4 @ Ys3 ) @ ( cons_a @ Z3 @ Zs2 ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_487_list__induct3,axiom,
! [Xs: list_a,Ys: list_a,Zs: list_set_a,P: list_a > list_a > list_set_a > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( ( size_size_list_a @ Ys )
= ( size_size_list_set_a @ Zs ) )
=> ( ( P @ nil_a @ nil_a @ nil_set_a )
=> ( ! [X6: a,Xs3: list_a,Y4: a,Ys3: list_a,Z3: set_a,Zs2: list_set_a] :
( ( ( size_size_list_a @ Xs3 )
= ( size_size_list_a @ Ys3 ) )
=> ( ( ( size_size_list_a @ Ys3 )
= ( size_size_list_set_a @ Zs2 ) )
=> ( ( P @ Xs3 @ Ys3 @ Zs2 )
=> ( P @ ( cons_a @ X6 @ Xs3 ) @ ( cons_a @ Y4 @ Ys3 ) @ ( cons_set_a @ Z3 @ Zs2 ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_488_list__induct3,axiom,
! [Xs: list_a,Ys: list_set_a,Zs: list_a,P: list_a > list_set_a > list_a > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_set_a @ Ys ) )
=> ( ( ( size_size_list_set_a @ Ys )
= ( size_size_list_a @ Zs ) )
=> ( ( P @ nil_a @ nil_set_a @ nil_a )
=> ( ! [X6: a,Xs3: list_a,Y4: set_a,Ys3: list_set_a,Z3: a,Zs2: list_a] :
( ( ( size_size_list_a @ Xs3 )
= ( size_size_list_set_a @ Ys3 ) )
=> ( ( ( size_size_list_set_a @ Ys3 )
= ( size_size_list_a @ Zs2 ) )
=> ( ( P @ Xs3 @ Ys3 @ Zs2 )
=> ( P @ ( cons_a @ X6 @ Xs3 ) @ ( cons_set_a @ Y4 @ Ys3 ) @ ( cons_a @ Z3 @ Zs2 ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_489_list__induct3,axiom,
! [Xs: list_a,Ys: list_set_a,Zs: list_set_a,P: list_a > list_set_a > list_set_a > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_set_a @ Ys ) )
=> ( ( ( size_size_list_set_a @ Ys )
= ( size_size_list_set_a @ Zs ) )
=> ( ( P @ nil_a @ nil_set_a @ nil_set_a )
=> ( ! [X6: a,Xs3: list_a,Y4: set_a,Ys3: list_set_a,Z3: set_a,Zs2: list_set_a] :
( ( ( size_size_list_a @ Xs3 )
= ( size_size_list_set_a @ Ys3 ) )
=> ( ( ( size_size_list_set_a @ Ys3 )
= ( size_size_list_set_a @ Zs2 ) )
=> ( ( P @ Xs3 @ Ys3 @ Zs2 )
=> ( P @ ( cons_a @ X6 @ Xs3 ) @ ( cons_set_a @ Y4 @ Ys3 ) @ ( cons_set_a @ Z3 @ Zs2 ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_490_list__induct3,axiom,
! [Xs: list_set_a,Ys: list_a,Zs: list_a,P: list_set_a > list_a > list_a > $o] :
( ( ( size_size_list_set_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( ( size_size_list_a @ Ys )
= ( size_size_list_a @ Zs ) )
=> ( ( P @ nil_set_a @ nil_a @ nil_a )
=> ( ! [X6: set_a,Xs3: list_set_a,Y4: a,Ys3: list_a,Z3: a,Zs2: list_a] :
( ( ( size_size_list_set_a @ Xs3 )
= ( size_size_list_a @ Ys3 ) )
=> ( ( ( size_size_list_a @ Ys3 )
= ( size_size_list_a @ Zs2 ) )
=> ( ( P @ Xs3 @ Ys3 @ Zs2 )
=> ( P @ ( cons_set_a @ X6 @ Xs3 ) @ ( cons_a @ Y4 @ Ys3 ) @ ( cons_a @ Z3 @ Zs2 ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_491_list__induct3,axiom,
! [Xs: list_set_a,Ys: list_a,Zs: list_set_a,P: list_set_a > list_a > list_set_a > $o] :
( ( ( size_size_list_set_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( ( size_size_list_a @ Ys )
= ( size_size_list_set_a @ Zs ) )
=> ( ( P @ nil_set_a @ nil_a @ nil_set_a )
=> ( ! [X6: set_a,Xs3: list_set_a,Y4: a,Ys3: list_a,Z3: set_a,Zs2: list_set_a] :
( ( ( size_size_list_set_a @ Xs3 )
= ( size_size_list_a @ Ys3 ) )
=> ( ( ( size_size_list_a @ Ys3 )
= ( size_size_list_set_a @ Zs2 ) )
=> ( ( P @ Xs3 @ Ys3 @ Zs2 )
=> ( P @ ( cons_set_a @ X6 @ Xs3 ) @ ( cons_a @ Y4 @ Ys3 ) @ ( cons_set_a @ Z3 @ Zs2 ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_492_list__induct3,axiom,
! [Xs: list_set_a,Ys: list_set_a,Zs: list_a,P: list_set_a > list_set_a > list_a > $o] :
( ( ( size_size_list_set_a @ Xs )
= ( size_size_list_set_a @ Ys ) )
=> ( ( ( size_size_list_set_a @ Ys )
= ( size_size_list_a @ Zs ) )
=> ( ( P @ nil_set_a @ nil_set_a @ nil_a )
=> ( ! [X6: set_a,Xs3: list_set_a,Y4: set_a,Ys3: list_set_a,Z3: a,Zs2: list_a] :
( ( ( size_size_list_set_a @ Xs3 )
= ( size_size_list_set_a @ Ys3 ) )
=> ( ( ( size_size_list_set_a @ Ys3 )
= ( size_size_list_a @ Zs2 ) )
=> ( ( P @ Xs3 @ Ys3 @ Zs2 )
=> ( P @ ( cons_set_a @ X6 @ Xs3 ) @ ( cons_set_a @ Y4 @ Ys3 ) @ ( cons_a @ Z3 @ Zs2 ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_493_list__induct3,axiom,
! [Xs: list_set_a,Ys: list_set_a,Zs: list_set_a,P: list_set_a > list_set_a > list_set_a > $o] :
( ( ( size_size_list_set_a @ Xs )
= ( size_size_list_set_a @ Ys ) )
=> ( ( ( size_size_list_set_a @ Ys )
= ( size_size_list_set_a @ Zs ) )
=> ( ( P @ nil_set_a @ nil_set_a @ nil_set_a )
=> ( ! [X6: set_a,Xs3: list_set_a,Y4: set_a,Ys3: list_set_a,Z3: set_a,Zs2: list_set_a] :
( ( ( size_size_list_set_a @ Xs3 )
= ( size_size_list_set_a @ Ys3 ) )
=> ( ( ( size_size_list_set_a @ Ys3 )
= ( size_size_list_set_a @ Zs2 ) )
=> ( ( P @ Xs3 @ Ys3 @ Zs2 )
=> ( P @ ( cons_set_a @ X6 @ Xs3 ) @ ( cons_set_a @ Y4 @ Ys3 ) @ ( cons_set_a @ Z3 @ Zs2 ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_494_list__induct2,axiom,
! [Xs: list_a,Ys: list_a,P: list_a > list_a > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( P @ nil_a @ nil_a )
=> ( ! [X6: a,Xs3: list_a,Y4: a,Ys3: list_a] :
( ( ( size_size_list_a @ Xs3 )
= ( size_size_list_a @ Ys3 ) )
=> ( ( P @ Xs3 @ Ys3 )
=> ( P @ ( cons_a @ X6 @ Xs3 ) @ ( cons_a @ Y4 @ Ys3 ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_495_list__induct2,axiom,
! [Xs: list_a,Ys: list_set_a,P: list_a > list_set_a > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_set_a @ Ys ) )
=> ( ( P @ nil_a @ nil_set_a )
=> ( ! [X6: a,Xs3: list_a,Y4: set_a,Ys3: list_set_a] :
( ( ( size_size_list_a @ Xs3 )
= ( size_size_list_set_a @ Ys3 ) )
=> ( ( P @ Xs3 @ Ys3 )
=> ( P @ ( cons_a @ X6 @ Xs3 ) @ ( cons_set_a @ Y4 @ Ys3 ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_496_list__induct2,axiom,
! [Xs: list_set_a,Ys: list_a,P: list_set_a > list_a > $o] :
( ( ( size_size_list_set_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( P @ nil_set_a @ nil_a )
=> ( ! [X6: set_a,Xs3: list_set_a,Y4: a,Ys3: list_a] :
( ( ( size_size_list_set_a @ Xs3 )
= ( size_size_list_a @ Ys3 ) )
=> ( ( P @ Xs3 @ Ys3 )
=> ( P @ ( cons_set_a @ X6 @ Xs3 ) @ ( cons_a @ Y4 @ Ys3 ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_497_list__induct2,axiom,
! [Xs: list_set_a,Ys: list_set_a,P: list_set_a > list_set_a > $o] :
( ( ( size_size_list_set_a @ Xs )
= ( size_size_list_set_a @ Ys ) )
=> ( ( P @ nil_set_a @ nil_set_a )
=> ( ! [X6: set_a,Xs3: list_set_a,Y4: set_a,Ys3: list_set_a] :
( ( ( size_size_list_set_a @ Xs3 )
= ( size_size_list_set_a @ Ys3 ) )
=> ( ( P @ Xs3 @ Ys3 )
=> ( P @ ( cons_set_a @ X6 @ Xs3 ) @ ( cons_set_a @ Y4 @ Ys3 ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_498_impossible__Cons,axiom,
! [Xs: list_a,Ys: list_a,X: a] :
( ( ord_less_eq_nat @ ( size_size_list_a @ Xs ) @ ( size_size_list_a @ Ys ) )
=> ( Xs
!= ( cons_a @ X @ Ys ) ) ) ).
% impossible_Cons
thf(fact_499_impossible__Cons,axiom,
! [Xs: list_set_a,Ys: list_set_a,X: set_a] :
( ( ord_less_eq_nat @ ( size_size_list_set_a @ Xs ) @ ( size_size_list_set_a @ Ys ) )
=> ( Xs
!= ( cons_set_a @ X @ Ys ) ) ) ).
% impossible_Cons
thf(fact_500_rev__induct,axiom,
! [P: list_set_a > $o,Xs: list_set_a] :
( ( P @ nil_set_a )
=> ( ! [X6: set_a,Xs3: list_set_a] :
( ( P @ Xs3 )
=> ( P @ ( append_set_a @ Xs3 @ ( cons_set_a @ X6 @ nil_set_a ) ) ) )
=> ( P @ Xs ) ) ) ).
% rev_induct
thf(fact_501_rev__induct,axiom,
! [P: list_a > $o,Xs: list_a] :
( ( P @ nil_a )
=> ( ! [X6: a,Xs3: list_a] :
( ( P @ Xs3 )
=> ( P @ ( append_a @ Xs3 @ ( cons_a @ X6 @ nil_a ) ) ) )
=> ( P @ Xs ) ) ) ).
% rev_induct
thf(fact_502_rev__exhaust,axiom,
! [Xs: list_set_a] :
( ( Xs != nil_set_a )
=> ~ ! [Ys3: list_set_a,Y4: set_a] :
( Xs
!= ( append_set_a @ Ys3 @ ( cons_set_a @ Y4 @ nil_set_a ) ) ) ) ).
% rev_exhaust
thf(fact_503_rev__exhaust,axiom,
! [Xs: list_a] :
( ( Xs != nil_a )
=> ~ ! [Ys3: list_a,Y4: a] :
( Xs
!= ( append_a @ Ys3 @ ( cons_a @ Y4 @ nil_a ) ) ) ) ).
% rev_exhaust
thf(fact_504_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_505_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_506_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_507_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_508_rev__nonempty__induct,axiom,
! [Xs: list_set_a,P: list_set_a > $o] :
( ( Xs != nil_set_a )
=> ( ! [X6: set_a] : ( P @ ( cons_set_a @ X6 @ nil_set_a ) )
=> ( ! [X6: set_a,Xs3: list_set_a] :
( ( Xs3 != nil_set_a )
=> ( ( P @ Xs3 )
=> ( P @ ( append_set_a @ Xs3 @ ( cons_set_a @ X6 @ nil_set_a ) ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% rev_nonempty_induct
thf(fact_509_rev__nonempty__induct,axiom,
! [Xs: list_a,P: list_a > $o] :
( ( Xs != nil_a )
=> ( ! [X6: a] : ( P @ ( cons_a @ X6 @ nil_a ) )
=> ( ! [X6: a,Xs3: list_a] :
( ( Xs3 != nil_a )
=> ( ( P @ Xs3 )
=> ( P @ ( append_a @ Xs3 @ ( cons_a @ X6 @ nil_a ) ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% rev_nonempty_induct
thf(fact_510_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_511_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_512_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_513_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_514_split__list__prop,axiom,
! [Xs: list_set_a,P: set_a > $o] :
( ? [X4: set_a] :
( ( member_set_a @ X4 @ ( set_set_a2 @ Xs ) )
& ( P @ X4 ) )
=> ? [Ys3: list_set_a,X6: set_a] :
( ? [Zs2: list_set_a] :
( Xs
= ( append_set_a @ Ys3 @ ( cons_set_a @ X6 @ Zs2 ) ) )
& ( P @ X6 ) ) ) ).
% split_list_prop
thf(fact_515_split__list__prop,axiom,
! [Xs: list_a,P: a > $o] :
( ? [X4: a] :
( ( member_a @ X4 @ ( set_a2 @ Xs ) )
& ( P @ X4 ) )
=> ? [Ys3: list_a,X6: a] :
( ? [Zs2: list_a] :
( Xs
= ( append_a @ Ys3 @ ( cons_a @ X6 @ Zs2 ) ) )
& ( P @ X6 ) ) ) ).
% split_list_prop
thf(fact_516_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_517_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_518_split__list__propE,axiom,
! [Xs: list_set_a,P: set_a > $o] :
( ? [X4: set_a] :
( ( member_set_a @ X4 @ ( set_set_a2 @ Xs ) )
& ( P @ X4 ) )
=> ~ ! [Ys3: list_set_a,X6: set_a] :
( ? [Zs2: list_set_a] :
( Xs
= ( append_set_a @ Ys3 @ ( cons_set_a @ X6 @ Zs2 ) ) )
=> ~ ( P @ X6 ) ) ) ).
% split_list_propE
thf(fact_519_split__list__propE,axiom,
! [Xs: list_a,P: a > $o] :
( ? [X4: a] :
( ( member_a @ X4 @ ( set_a2 @ Xs ) )
& ( P @ X4 ) )
=> ~ ! [Ys3: list_a,X6: a] :
( ? [Zs2: list_a] :
( Xs
= ( append_a @ Ys3 @ ( cons_a @ X6 @ Zs2 ) ) )
=> ~ ( P @ X6 ) ) ) ).
% split_list_propE
thf(fact_520_append__Cons__eq__iff,axiom,
! [X: set_a,Xs: list_set_a,Ys: list_set_a,Xs5: 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 @ Xs5 @ ( cons_set_a @ X @ Ys6 ) ) )
= ( ( Xs = Xs5 )
& ( Ys = Ys6 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_521_append__Cons__eq__iff,axiom,
! [X: a,Xs: list_a,Ys: list_a,Xs5: 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 @ Xs5 @ ( cons_a @ X @ Ys6 ) ) )
= ( ( Xs = Xs5 )
& ( Ys = Ys6 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_522_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_523_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_524_split__list__last__prop,axiom,
! [Xs: list_set_a,P: set_a > $o] :
( ? [X4: set_a] :
( ( member_set_a @ X4 @ ( set_set_a2 @ Xs ) )
& ( P @ X4 ) )
=> ? [Ys3: list_set_a,X6: set_a,Zs2: list_set_a] :
( ( Xs
= ( append_set_a @ Ys3 @ ( cons_set_a @ X6 @ Zs2 ) ) )
& ( P @ X6 )
& ! [Xa: set_a] :
( ( member_set_a @ Xa @ ( set_set_a2 @ Zs2 ) )
=> ~ ( P @ Xa ) ) ) ) ).
% split_list_last_prop
thf(fact_525_split__list__last__prop,axiom,
! [Xs: list_a,P: a > $o] :
( ? [X4: a] :
( ( member_a @ X4 @ ( set_a2 @ Xs ) )
& ( P @ X4 ) )
=> ? [Ys3: list_a,X6: a,Zs2: list_a] :
( ( Xs
= ( append_a @ Ys3 @ ( cons_a @ X6 @ Zs2 ) ) )
& ( P @ X6 )
& ! [Xa: a] :
( ( member_a @ Xa @ ( set_a2 @ Zs2 ) )
=> ~ ( P @ Xa ) ) ) ) ).
% split_list_last_prop
thf(fact_526_split__list__first__prop,axiom,
! [Xs: list_set_a,P: set_a > $o] :
( ? [X4: set_a] :
( ( member_set_a @ X4 @ ( set_set_a2 @ Xs ) )
& ( P @ X4 ) )
=> ? [Ys3: list_set_a,X6: set_a] :
( ? [Zs2: list_set_a] :
( Xs
= ( append_set_a @ Ys3 @ ( cons_set_a @ X6 @ Zs2 ) ) )
& ( P @ X6 )
& ! [Xa: set_a] :
( ( member_set_a @ Xa @ ( set_set_a2 @ Ys3 ) )
=> ~ ( P @ Xa ) ) ) ) ).
% split_list_first_prop
thf(fact_527_split__list__first__prop,axiom,
! [Xs: list_a,P: a > $o] :
( ? [X4: a] :
( ( member_a @ X4 @ ( set_a2 @ Xs ) )
& ( P @ X4 ) )
=> ? [Ys3: list_a,X6: a] :
( ? [Zs2: list_a] :
( Xs
= ( append_a @ Ys3 @ ( cons_a @ X6 @ Zs2 ) ) )
& ( P @ X6 )
& ! [Xa: a] :
( ( member_a @ Xa @ ( set_a2 @ Ys3 ) )
=> ~ ( P @ Xa ) ) ) ) ).
% split_list_first_prop
thf(fact_528_split__list__last__propE,axiom,
! [Xs: list_set_a,P: set_a > $o] :
( ? [X4: set_a] :
( ( member_set_a @ X4 @ ( set_set_a2 @ Xs ) )
& ( P @ X4 ) )
=> ~ ! [Ys3: list_set_a,X6: set_a,Zs2: list_set_a] :
( ( Xs
= ( append_set_a @ Ys3 @ ( cons_set_a @ X6 @ Zs2 ) ) )
=> ( ( P @ X6 )
=> ~ ! [Xa: set_a] :
( ( member_set_a @ Xa @ ( set_set_a2 @ Zs2 ) )
=> ~ ( P @ Xa ) ) ) ) ) ).
% split_list_last_propE
thf(fact_529_split__list__last__propE,axiom,
! [Xs: list_a,P: a > $o] :
( ? [X4: a] :
( ( member_a @ X4 @ ( set_a2 @ Xs ) )
& ( P @ X4 ) )
=> ~ ! [Ys3: list_a,X6: a,Zs2: list_a] :
( ( Xs
= ( append_a @ Ys3 @ ( cons_a @ X6 @ Zs2 ) ) )
=> ( ( P @ X6 )
=> ~ ! [Xa: a] :
( ( member_a @ Xa @ ( set_a2 @ Zs2 ) )
=> ~ ( P @ Xa ) ) ) ) ) ).
% split_list_last_propE
thf(fact_530_split__list__first__propE,axiom,
! [Xs: list_set_a,P: set_a > $o] :
( ? [X4: set_a] :
( ( member_set_a @ X4 @ ( set_set_a2 @ Xs ) )
& ( P @ X4 ) )
=> ~ ! [Ys3: list_set_a,X6: set_a] :
( ? [Zs2: list_set_a] :
( Xs
= ( append_set_a @ Ys3 @ ( cons_set_a @ X6 @ Zs2 ) ) )
=> ( ( P @ X6 )
=> ~ ! [Xa: set_a] :
( ( member_set_a @ Xa @ ( set_set_a2 @ Ys3 ) )
=> ~ ( P @ Xa ) ) ) ) ) ).
% split_list_first_propE
thf(fact_531_split__list__first__propE,axiom,
! [Xs: list_a,P: a > $o] :
( ? [X4: a] :
( ( member_a @ X4 @ ( set_a2 @ Xs ) )
& ( P @ X4 ) )
=> ~ ! [Ys3: list_a,X6: a] :
( ? [Zs2: list_a] :
( Xs
= ( append_a @ Ys3 @ ( cons_a @ X6 @ Zs2 ) ) )
=> ( ( P @ X6 )
=> ~ ! [Xa: a] :
( ( member_a @ Xa @ ( set_a2 @ Ys3 ) )
=> ~ ( P @ Xa ) ) ) ) ) ).
% split_list_first_propE
thf(fact_532_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_533_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_534_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_535_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_536_split__list__last__prop__iff,axiom,
! [Xs: list_set_a,P: set_a > $o] :
( ( ? [X5: set_a] :
( ( member_set_a @ X5 @ ( set_set_a2 @ Xs ) )
& ( P @ X5 ) ) )
= ( ? [Ys4: list_set_a,X5: set_a,Zs3: list_set_a] :
( ( Xs
= ( append_set_a @ Ys4 @ ( cons_set_a @ X5 @ Zs3 ) ) )
& ( P @ X5 )
& ! [Y6: set_a] :
( ( member_set_a @ Y6 @ ( set_set_a2 @ Zs3 ) )
=> ~ ( P @ Y6 ) ) ) ) ) ).
% split_list_last_prop_iff
thf(fact_537_split__list__last__prop__iff,axiom,
! [Xs: list_a,P: a > $o] :
( ( ? [X5: a] :
( ( member_a @ X5 @ ( set_a2 @ Xs ) )
& ( P @ X5 ) ) )
= ( ? [Ys4: list_a,X5: a,Zs3: list_a] :
( ( Xs
= ( append_a @ Ys4 @ ( cons_a @ X5 @ Zs3 ) ) )
& ( P @ X5 )
& ! [Y6: a] :
( ( member_a @ Y6 @ ( set_a2 @ Zs3 ) )
=> ~ ( P @ Y6 ) ) ) ) ) ).
% split_list_last_prop_iff
thf(fact_538_split__list__first__prop__iff,axiom,
! [Xs: list_set_a,P: set_a > $o] :
( ( ? [X5: set_a] :
( ( member_set_a @ X5 @ ( set_set_a2 @ Xs ) )
& ( P @ X5 ) ) )
= ( ? [Ys4: list_set_a,X5: set_a] :
( ? [Zs3: list_set_a] :
( Xs
= ( append_set_a @ Ys4 @ ( cons_set_a @ X5 @ Zs3 ) ) )
& ( P @ X5 )
& ! [Y6: set_a] :
( ( member_set_a @ Y6 @ ( set_set_a2 @ Ys4 ) )
=> ~ ( P @ Y6 ) ) ) ) ) ).
% split_list_first_prop_iff
thf(fact_539_split__list__first__prop__iff,axiom,
! [Xs: list_a,P: a > $o] :
( ( ? [X5: a] :
( ( member_a @ X5 @ ( set_a2 @ Xs ) )
& ( P @ X5 ) ) )
= ( ? [Ys4: list_a,X5: a] :
( ? [Zs3: list_a] :
( Xs
= ( append_a @ Ys4 @ ( cons_a @ X5 @ Zs3 ) ) )
& ( P @ X5 )
& ! [Y6: a] :
( ( member_a @ Y6 @ ( set_a2 @ Ys4 ) )
=> ~ ( P @ Y6 ) ) ) ) ) ).
% split_list_first_prop_iff
thf(fact_540_distinct__singleton,axiom,
! [X: set_a] : ( distinct_set_a @ ( cons_set_a @ X @ nil_set_a ) ) ).
% distinct_singleton
thf(fact_541_distinct__singleton,axiom,
! [X: a] : ( distinct_a @ ( cons_a @ X @ nil_a ) ) ).
% distinct_singleton
thf(fact_542_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 )
=> ( ! [X6: set_a] :
( X
!= ( cons_set_a @ X6 @ nil_set_a ) )
=> ~ ! [X6: set_a,Y4: set_a,Ys3: list_set_a] :
( X
!= ( cons_set_a @ X6 @ ( cons_set_a @ Y4 @ Ys3 ) ) ) ) ) ) ).
% ulgraph.walk_edges.cases
thf(fact_543_ulgraph_Owalk__edges_Ocases,axiom,
! [Vertices: set_a,Edges: set_set_a,X: list_a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( X != nil_a )
=> ( ! [X6: a] :
( X
!= ( cons_a @ X6 @ nil_a ) )
=> ~ ! [X6: a,Y4: a,Ys3: list_a] :
( X
!= ( cons_a @ X6 @ ( cons_a @ Y4 @ Ys3 ) ) ) ) ) ) ).
% ulgraph.walk_edges.cases
thf(fact_544_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_545_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_546_Nil__tl,axiom,
! [Xs: list_set_a] :
( ( nil_set_a
= ( tl_set_a @ Xs ) )
= ( ( Xs = nil_set_a )
| ? [X5: set_a] :
( Xs
= ( cons_set_a @ X5 @ nil_set_a ) ) ) ) ).
% Nil_tl
thf(fact_547_Nil__tl,axiom,
! [Xs: list_a] :
( ( nil_a
= ( tl_a @ Xs ) )
= ( ( Xs = nil_a )
| ? [X5: a] :
( Xs
= ( cons_a @ X5 @ nil_a ) ) ) ) ).
% Nil_tl
thf(fact_548_tl__Nil,axiom,
! [Xs: list_set_a] :
( ( ( tl_set_a @ Xs )
= nil_set_a )
= ( ( Xs = nil_set_a )
| ? [X5: set_a] :
( Xs
= ( cons_set_a @ X5 @ nil_set_a ) ) ) ) ).
% tl_Nil
thf(fact_549_tl__Nil,axiom,
! [Xs: list_a] :
( ( ( tl_a @ Xs )
= nil_a )
= ( ( Xs = nil_a )
| ? [X5: a] :
( Xs
= ( cons_a @ X5 @ nil_a ) ) ) ) ).
% tl_Nil
thf(fact_550_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_551_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_552_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_553_last__ConsL,axiom,
! [Xs: list_a,X: a] :
( ( Xs = nil_a )
=> ( ( last_a @ ( cons_a @ X @ Xs ) )
= X ) ) ).
% last_ConsL
thf(fact_554_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_555_last__ConsR,axiom,
! [Xs: list_a,X: a] :
( ( Xs != nil_a )
=> ( ( last_a @ ( cons_a @ X @ Xs ) )
= ( last_a @ Xs ) ) ) ).
% last_ConsR
thf(fact_556_ulgraph_Ois__isolated__vertex__def,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,V2: set_a] :
( ( undire6886684016831807756_set_a @ Vertices @ Edges )
=> ( ( undire6879241558604981877_set_a @ Vertices @ Edges @ V2 )
= ( ( member_set_a @ V2 @ Vertices )
& ! [X5: set_a] :
( ( member_set_a @ X5 @ Vertices )
=> ~ ( undire3510646817838285160_set_a @ Edges @ X5 @ V2 ) ) ) ) ) ).
% ulgraph.is_isolated_vertex_def
thf(fact_557_ulgraph_Ois__isolated__vertex__def,axiom,
! [Vertices: set_a,Edges: set_set_a,V2: a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( undire8931668460104145173rtex_a @ Vertices @ Edges @ V2 )
= ( ( member_a @ V2 @ Vertices )
& ! [X5: a] :
( ( member_a @ X5 @ Vertices )
=> ~ ( undire397441198561214472_adj_a @ Edges @ X5 @ V2 ) ) ) ) ) ).
% ulgraph.is_isolated_vertex_def
thf(fact_558_ulgraph_Ois__isolated__vertex__edge,axiom,
! [Vertices: set_a,Edges: set_set_a,V2: a,E: set_a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( undire8931668460104145173rtex_a @ Vertices @ Edges @ V2 )
=> ( ( member_set_a @ E @ Edges )
=> ~ ( undire1521409233611534436dent_a @ V2 @ E ) ) ) ) ).
% ulgraph.is_isolated_vertex_edge
thf(fact_559_ulgraph_Ois__isolated__vertex__no__loop,axiom,
! [Vertices: set_a,Edges: set_set_a,V2: a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( undire8931668460104145173rtex_a @ Vertices @ Edges @ V2 )
=> ~ ( undire3617971648856834880loop_a @ Edges @ V2 ) ) ) ).
% ulgraph.is_isolated_vertex_no_loop
thf(fact_560_Suc__le__length__iff,axiom,
! [N: nat,Xs: list_a] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( size_size_list_a @ Xs ) )
= ( ? [X5: a,Ys4: list_a] :
( ( Xs
= ( cons_a @ X5 @ Ys4 ) )
& ( ord_less_eq_nat @ N @ ( size_size_list_a @ Ys4 ) ) ) ) ) ).
% Suc_le_length_iff
thf(fact_561_Suc__le__length__iff,axiom,
! [N: nat,Xs: list_set_a] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( size_size_list_set_a @ Xs ) )
= ( ? [X5: set_a,Ys4: list_set_a] :
( ( Xs
= ( cons_set_a @ X5 @ Ys4 ) )
& ( ord_less_eq_nat @ N @ ( size_size_list_set_a @ Ys4 ) ) ) ) ) ).
% Suc_le_length_iff
thf(fact_562_same__length__different,axiom,
! [Xs: list_a,Ys: list_a] :
( ( Xs != Ys )
=> ( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ? [Pre: list_a,X6: a,Xs2: list_a,Y4: a,Ys2: list_a] :
( ( X6 != Y4 )
& ( Xs
= ( append_a @ Pre @ ( append_a @ ( cons_a @ X6 @ nil_a ) @ Xs2 ) ) )
& ( Ys
= ( append_a @ Pre @ ( append_a @ ( cons_a @ Y4 @ nil_a ) @ Ys2 ) ) ) ) ) ) ).
% same_length_different
thf(fact_563_same__length__different,axiom,
! [Xs: list_set_a,Ys: list_set_a] :
( ( Xs != Ys )
=> ( ( ( size_size_list_set_a @ Xs )
= ( size_size_list_set_a @ Ys ) )
=> ? [Pre: list_set_a,X6: set_a,Xs2: list_set_a,Y4: set_a,Ys2: list_set_a] :
( ( X6 != Y4 )
& ( Xs
= ( append_set_a @ Pre @ ( append_set_a @ ( cons_set_a @ X6 @ nil_set_a ) @ Xs2 ) ) )
& ( Ys
= ( append_set_a @ Pre @ ( append_set_a @ ( cons_set_a @ Y4 @ nil_set_a ) @ Ys2 ) ) ) ) ) ) ).
% same_length_different
thf(fact_564_not__distinct__decomp,axiom,
! [Ws: list_set_a] :
( ~ ( distinct_set_a @ Ws )
=> ? [Xs3: list_set_a,Ys3: list_set_a,Zs2: list_set_a,Y4: set_a] :
( Ws
= ( append_set_a @ Xs3 @ ( append_set_a @ ( cons_set_a @ Y4 @ nil_set_a ) @ ( append_set_a @ Ys3 @ ( append_set_a @ ( cons_set_a @ Y4 @ nil_set_a ) @ Zs2 ) ) ) ) ) ) ).
% not_distinct_decomp
thf(fact_565_not__distinct__decomp,axiom,
! [Ws: list_a] :
( ~ ( distinct_a @ Ws )
=> ? [Xs3: list_a,Ys3: list_a,Zs2: list_a,Y4: a] :
( Ws
= ( append_a @ Xs3 @ ( append_a @ ( cons_a @ Y4 @ nil_a ) @ ( append_a @ Ys3 @ ( append_a @ ( cons_a @ Y4 @ nil_a ) @ Zs2 ) ) ) ) ) ) ).
% not_distinct_decomp
thf(fact_566_not__distinct__conv__prefix,axiom,
! [As: list_set_a] :
( ( ~ ( distinct_set_a @ As ) )
= ( ? [Xs4: list_set_a,Y6: set_a,Ys4: list_set_a] :
( ( member_set_a @ Y6 @ ( set_set_a2 @ Xs4 ) )
& ( distinct_set_a @ Xs4 )
& ( As
= ( append_set_a @ Xs4 @ ( cons_set_a @ Y6 @ Ys4 ) ) ) ) ) ) ).
% not_distinct_conv_prefix
thf(fact_567_not__distinct__conv__prefix,axiom,
! [As: list_a] :
( ( ~ ( distinct_a @ As ) )
= ( ? [Xs4: list_a,Y6: a,Ys4: list_a] :
( ( member_a @ Y6 @ ( set_a2 @ Xs4 ) )
& ( distinct_a @ Xs4 )
& ( As
= ( append_a @ Xs4 @ ( cons_a @ Y6 @ Ys4 ) ) ) ) ) ) ).
% not_distinct_conv_prefix
thf(fact_568_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_569_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_570_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_571_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_572_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_573_list_Oexhaust__sel,axiom,
! [List: list_a] :
( ( List != nil_a )
=> ( List
= ( cons_a @ ( hd_a @ List ) @ ( tl_a @ List ) ) ) ) ).
% list.exhaust_sel
thf(fact_574_ulgraph_Ois__gen__path__trivial,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,X: set_a] :
( ( undire6886684016831807756_set_a @ Vertices @ Edges )
=> ( ( member_set_a @ X @ Vertices )
=> ( undire7201326534205417136_set_a @ Vertices @ Edges @ ( cons_set_a @ X @ nil_set_a ) ) ) ) ).
% ulgraph.is_gen_path_trivial
thf(fact_575_ulgraph_Ois__gen__path__trivial,axiom,
! [Vertices: set_a,Edges: set_set_a,X: a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( member_a @ X @ Vertices )
=> ( undire3562951555376170320path_a @ Vertices @ Edges @ ( cons_a @ X @ nil_a ) ) ) ) ).
% ulgraph.is_gen_path_trivial
thf(fact_576_graph__system_Oinduced__edges_Ocong,axiom,
undire7777452895879145676dges_a = undire7777452895879145676dges_a ).
% graph_system.induced_edges.cong
thf(fact_577_ulgraph_Overt__adj_Ocong,axiom,
undire397441198561214472_adj_a = undire397441198561214472_adj_a ).
% ulgraph.vert_adj.cong
thf(fact_578_comp__sgraph_Oincident__def,axiom,
undire2320338297334612420_set_a = member_set_a ).
% comp_sgraph.incident_def
thf(fact_579_comp__sgraph_Oincident__def,axiom,
undire1521409233611534436dent_a = member_a ).
% comp_sgraph.incident_def
thf(fact_580_length__Suc__conv__rev,axiom,
! [Xs: list_a,N: nat] :
( ( ( size_size_list_a @ Xs )
= ( suc @ N ) )
= ( ? [Y6: a,Ys4: list_a] :
( ( Xs
= ( append_a @ Ys4 @ ( cons_a @ Y6 @ nil_a ) ) )
& ( ( size_size_list_a @ Ys4 )
= N ) ) ) ) ).
% length_Suc_conv_rev
thf(fact_581_length__Suc__conv__rev,axiom,
! [Xs: list_set_a,N: nat] :
( ( ( size_size_list_set_a @ Xs )
= ( suc @ N ) )
= ( ? [Y6: set_a,Ys4: list_set_a] :
( ( Xs
= ( append_set_a @ Ys4 @ ( cons_set_a @ Y6 @ nil_set_a ) ) )
& ( ( size_size_list_set_a @ Ys4 )
= N ) ) ) ) ).
% length_Suc_conv_rev
thf(fact_582_ulgraph_Ohas__loop_Ocong,axiom,
undire3617971648856834880loop_a = undire3617971648856834880loop_a ).
% ulgraph.has_loop.cong
thf(fact_583_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_584_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_585_graph__system_Oedge__adj_Ocong,axiom,
undire4022703626023482010_adj_a = undire4022703626023482010_adj_a ).
% graph_system.edge_adj.cong
thf(fact_586_ulgraph_Overt__adj__imp__inV,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,V1: set_a,V22: set_a] :
( ( undire6886684016831807756_set_a @ Vertices @ Edges )
=> ( ( undire3510646817838285160_set_a @ Edges @ V1 @ V22 )
=> ( ( member_set_a @ V1 @ Vertices )
& ( member_set_a @ V22 @ Vertices ) ) ) ) ).
% ulgraph.vert_adj_imp_inV
thf(fact_587_ulgraph_Overt__adj__imp__inV,axiom,
! [Vertices: set_a,Edges: set_set_a,V1: a,V22: a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( undire397441198561214472_adj_a @ Edges @ V1 @ V22 )
=> ( ( member_a @ V1 @ Vertices )
& ( member_a @ V22 @ Vertices ) ) ) ) ).
% ulgraph.vert_adj_imp_inV
thf(fact_588_ulgraph_Overt__adj__sym,axiom,
! [Vertices: set_a,Edges: set_set_a,V1: a,V22: a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( undire397441198561214472_adj_a @ Edges @ V1 @ V22 )
= ( undire397441198561214472_adj_a @ Edges @ V22 @ V1 ) ) ) ).
% ulgraph.vert_adj_sym
thf(fact_589_ulgraph_Ohas__loop__in__verts,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,V2: set_a] :
( ( undire6886684016831807756_set_a @ Vertices @ Edges )
=> ( ( undire5774735625301615776_set_a @ Edges @ V2 )
=> ( member_set_a @ V2 @ Vertices ) ) ) ).
% ulgraph.has_loop_in_verts
thf(fact_590_ulgraph_Ohas__loop__in__verts,axiom,
! [Vertices: set_a,Edges: set_set_a,V2: a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( undire3617971648856834880loop_a @ Edges @ V2 )
=> ( member_a @ V2 @ Vertices ) ) ) ).
% ulgraph.has_loop_in_verts
thf(fact_591_ulgraph_Overt__adj__edge__iff2,axiom,
! [Vertices: set_a,Edges: set_set_a,V1: a,V22: a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( V1 != V22 )
=> ( ( undire397441198561214472_adj_a @ Edges @ V1 @ V22 )
= ( ? [X5: set_a] :
( ( member_set_a @ X5 @ Edges )
& ( undire1521409233611534436dent_a @ V1 @ X5 )
& ( undire1521409233611534436dent_a @ V22 @ X5 ) ) ) ) ) ) ).
% ulgraph.vert_adj_edge_iff2
thf(fact_592_is__isolated__vertex__degree0,axiom,
! [V2: a] :
( ( undire8931668460104145173rtex_a @ vertices @ edges @ V2 )
=> ( ( undire8867928226783802224gree_a @ edges @ V2 )
= zero_zero_nat ) ) ).
% is_isolated_vertex_degree0
thf(fact_593_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_594_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_595_length__append__singleton,axiom,
! [Xs: list_a,X: a] :
( ( size_size_list_a @ ( append_a @ Xs @ ( cons_a @ X @ nil_a ) ) )
= ( suc @ ( size_size_list_a @ Xs ) ) ) ).
% length_append_singleton
thf(fact_596_length__append__singleton,axiom,
! [Xs: list_set_a,X: set_a] :
( ( size_size_list_set_a @ ( append_set_a @ Xs @ ( cons_set_a @ X @ nil_set_a ) ) )
= ( suc @ ( size_size_list_set_a @ Xs ) ) ) ).
% length_append_singleton
thf(fact_597_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_598_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_599_walk__edges_Osimps_I1_J,axiom,
( ( undire7337870655677353998dges_a @ nil_a )
= nil_set_a ) ).
% walk_edges.simps(1)
thf(fact_600_walk__edges_Osimps_I2_J,axiom,
! [X: a] :
( ( undire7337870655677353998dges_a @ ( cons_a @ X @ nil_a ) )
= nil_set_a ) ).
% walk_edges.simps(2)
thf(fact_601_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_602_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_603_rev__rev__ident,axiom,
! [Xs: list_a] :
( ( rev_a @ ( rev_a @ Xs ) )
= Xs ) ).
% rev_rev_ident
thf(fact_604_rev__rev__ident,axiom,
! [Xs: list_set_a] :
( ( rev_set_a @ ( rev_set_a @ Xs ) )
= Xs ) ).
% rev_rev_ident
thf(fact_605_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_606_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_607_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_608_is__walk__rev,axiom,
! [Xs: list_a] :
( ( undire6133010728901294956walk_a @ vertices @ edges @ Xs )
= ( undire6133010728901294956walk_a @ vertices @ edges @ ( rev_a @ Xs ) ) ) ).
% is_walk_rev
thf(fact_609_is__gen__path__rev,axiom,
! [P2: list_a] :
( ( undire3562951555376170320path_a @ vertices @ edges @ P2 )
= ( undire3562951555376170320path_a @ vertices @ edges @ ( rev_a @ P2 ) ) ) ).
% is_gen_path_rev
thf(fact_610_is__cycle__rev,axiom,
! [Xs: list_a] :
( ( undire2407311113669455967ycle_a @ vertices @ edges @ Xs )
= ( undire2407311113669455967ycle_a @ vertices @ edges @ ( rev_a @ Xs ) ) ) ).
% is_cycle_rev
thf(fact_611_is__path__rev,axiom,
! [Xs: list_a] :
( ( undire427332500224447920path_a @ vertices @ edges @ Xs )
= ( undire427332500224447920path_a @ vertices @ edges @ ( rev_a @ Xs ) ) ) ).
% is_path_rev
thf(fact_612_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_613_Nil__is__rev__conv,axiom,
! [Xs: list_a] :
( ( nil_a
= ( rev_a @ Xs ) )
= ( Xs = nil_a ) ) ).
% Nil_is_rev_conv
thf(fact_614_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_615_rev__is__Nil__conv,axiom,
! [Xs: list_a] :
( ( ( rev_a @ Xs )
= nil_a )
= ( Xs = nil_a ) ) ).
% rev_is_Nil_conv
thf(fact_616_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_617_set__rev,axiom,
! [Xs: list_a] :
( ( set_a2 @ ( rev_a @ Xs ) )
= ( set_a2 @ Xs ) ) ).
% set_rev
thf(fact_618_set__rev,axiom,
! [Xs: list_set_a] :
( ( set_set_a2 @ ( rev_set_a @ Xs ) )
= ( set_set_a2 @ Xs ) ) ).
% set_rev
thf(fact_619_length__rev,axiom,
! [Xs: list_a] :
( ( size_size_list_a @ ( rev_a @ Xs ) )
= ( size_size_list_a @ Xs ) ) ).
% length_rev
thf(fact_620_length__rev,axiom,
! [Xs: list_set_a] :
( ( size_size_list_set_a @ ( rev_set_a @ Xs ) )
= ( size_size_list_set_a @ Xs ) ) ).
% length_rev
thf(fact_621_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_622_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_623_distinct__rev,axiom,
! [Xs: list_a] :
( ( distinct_a @ ( rev_a @ Xs ) )
= ( distinct_a @ Xs ) ) ).
% distinct_rev
thf(fact_624_distinct__rev,axiom,
! [Xs: list_set_a] :
( ( distinct_set_a @ ( rev_set_a @ Xs ) )
= ( distinct_set_a @ Xs ) ) ).
% distinct_rev
thf(fact_625_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_626_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_627_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_628_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_629_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_630_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_631_degree__none,axiom,
! [V2: a] :
( ~ ( member_a @ V2 @ vertices )
=> ( ( undire8867928226783802224gree_a @ edges @ V2 )
= zero_zero_nat ) ) ).
% degree_none
thf(fact_632_is__trail__rev,axiom,
! [Xs: list_a] :
( ( undire7142031287334043199rail_a @ vertices @ edges @ Xs )
= ( undire7142031287334043199rail_a @ vertices @ edges @ ( rev_a @ Xs ) ) ) ).
% is_trail_rev
thf(fact_633_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_634_ulgraph_Odegree_Ocong,axiom,
undire8867928226783802224gree_a = undire8867928226783802224gree_a ).
% ulgraph.degree.cong
thf(fact_635_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_636_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_637_rev__swap,axiom,
! [Xs: list_a,Ys: list_a] :
( ( ( rev_a @ Xs )
= Ys )
= ( Xs
= ( rev_a @ Ys ) ) ) ).
% rev_swap
thf(fact_638_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_639_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_640_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_641_rev_Osimps_I1_J,axiom,
( ( rev_a @ nil_a )
= nil_a ) ).
% rev.simps(1)
thf(fact_642_rev_Osimps_I1_J,axiom,
( ( rev_set_a @ nil_set_a )
= nil_set_a ) ).
% rev.simps(1)
thf(fact_643_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_644_comp__sgraph_Owalk__edges_Osimps_I1_J,axiom,
( ( undire7337870655677353998dges_a @ nil_a )
= nil_set_a ) ).
% comp_sgraph.walk_edges.simps(1)
thf(fact_645_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_646_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_647_hd__rev,axiom,
! [Xs: list_set_a] :
( ( hd_set_a @ ( rev_set_a @ Xs ) )
= ( last_set_a @ Xs ) ) ).
% hd_rev
thf(fact_648_hd__rev,axiom,
! [Xs: list_a] :
( ( hd_a @ ( rev_a @ Xs ) )
= ( last_a @ Xs ) ) ).
% hd_rev
thf(fact_649_last__rev,axiom,
! [Xs: list_set_a] :
( ( last_set_a @ ( rev_set_a @ Xs ) )
= ( hd_set_a @ Xs ) ) ).
% last_rev
thf(fact_650_last__rev,axiom,
! [Xs: list_a] :
( ( last_a @ ( rev_a @ Xs ) )
= ( hd_a @ Xs ) ) ).
% last_rev
thf(fact_651_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_652_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_653_ulgraph_Ois__gen__path__rev,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,P2: list_set_a] :
( ( undire6886684016831807756_set_a @ Vertices @ Edges )
=> ( ( undire7201326534205417136_set_a @ Vertices @ Edges @ P2 )
= ( undire7201326534205417136_set_a @ Vertices @ Edges @ ( rev_set_a @ P2 ) ) ) ) ).
% ulgraph.is_gen_path_rev
thf(fact_654_ulgraph_Ois__gen__path__rev,axiom,
! [Vertices: set_a,Edges: set_set_a,P2: list_a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( undire3562951555376170320path_a @ Vertices @ Edges @ P2 )
= ( undire3562951555376170320path_a @ Vertices @ Edges @ ( rev_a @ P2 ) ) ) ) ).
% ulgraph.is_gen_path_rev
thf(fact_655_ulgraph_Ois__cycle__rev,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,Xs: list_set_a] :
( ( undire6886684016831807756_set_a @ Vertices @ Edges )
=> ( ( undire797940137672299967_set_a @ Vertices @ Edges @ Xs )
= ( undire797940137672299967_set_a @ Vertices @ Edges @ ( rev_set_a @ Xs ) ) ) ) ).
% ulgraph.is_cycle_rev
thf(fact_656_ulgraph_Ois__cycle__rev,axiom,
! [Vertices: set_a,Edges: set_set_a,Xs: list_a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( undire2407311113669455967ycle_a @ Vertices @ Edges @ Xs )
= ( undire2407311113669455967ycle_a @ Vertices @ Edges @ ( rev_a @ Xs ) ) ) ) ).
% ulgraph.is_cycle_rev
thf(fact_657_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_658_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_659_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_660_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_661_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_662_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_663_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_664_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_665_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_666_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_667_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_668_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_669_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_670_ulgraph_Odegree__none,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,V2: set_a] :
( ( undire6886684016831807756_set_a @ Vertices @ Edges )
=> ( ~ ( member_set_a @ V2 @ Vertices )
=> ( ( undire8939077443744732368_set_a @ Edges @ V2 )
= zero_zero_nat ) ) ) ).
% ulgraph.degree_none
thf(fact_671_ulgraph_Odegree__none,axiom,
! [Vertices: set_a,Edges: set_set_a,V2: a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ~ ( member_a @ V2 @ Vertices )
=> ( ( undire8867928226783802224gree_a @ Edges @ V2 )
= zero_zero_nat ) ) ) ).
% ulgraph.degree_none
thf(fact_672_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_673_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_674_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_675_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_676_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_677_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_678_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_679_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_680_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_681_ulgraph_Ois__isolated__vertex__degree0,axiom,
! [Vertices: set_a,Edges: set_set_a,V2: a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( undire8931668460104145173rtex_a @ Vertices @ Edges @ V2 )
=> ( ( undire8867928226783802224gree_a @ Edges @ V2 )
= zero_zero_nat ) ) ) ).
% ulgraph.is_isolated_vertex_degree0
thf(fact_682_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_683_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_684_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_685_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_686_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_687_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_688_list__decode_Ocases,axiom,
! [X: nat] :
( ( X != zero_zero_nat )
=> ~ ! [N2: nat] :
( X
!= ( suc @ N2 ) ) ) ).
% list_decode.cases
thf(fact_689_length__Cons,axiom,
! [X: a,Xs: list_a] :
( ( size_size_list_a @ ( cons_a @ X @ Xs ) )
= ( suc @ ( size_size_list_a @ Xs ) ) ) ).
% length_Cons
thf(fact_690_length__Cons,axiom,
! [X: set_a,Xs: list_set_a] :
( ( size_size_list_set_a @ ( cons_set_a @ X @ Xs ) )
= ( suc @ ( size_size_list_set_a @ Xs ) ) ) ).
% length_Cons
thf(fact_691_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_692_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_693_is__cycle__alt,axiom,
! [Xs: list_a] :
( ( undire2407311113669455967ycle_a @ vertices @ edges @ Xs )
= ( ( undire6133010728901294956walk_a @ vertices @ edges @ Xs )
& ( distinct_a @ ( tl_a @ Xs ) )
& ( ord_less_eq_nat @ one_one_nat @ ( undire8849074589633906640ngth_a @ Xs ) )
& ( ( hd_a @ Xs )
= ( last_a @ Xs ) ) ) ) ).
% is_cycle_alt
thf(fact_694_ediff__le__self,axiom,
! [X: extended_enat,Y: extended_enat] : ( ord_le2932123472753598470d_enat @ ( minus_3235023915231533773d_enat @ X @ Y ) @ X ) ).
% ediff_le_self
thf(fact_695_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_696_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_697_walk__length__def,axiom,
( undire8849074589633906640ngth_a
= ( ^ [P3: list_a] : ( size_size_list_set_a @ ( undire7337870655677353998dges_a @ P3 ) ) ) ) ).
% walk_length_def
thf(fact_698_walk__edges__rev,axiom,
! [Xs: list_a] :
( ( rev_set_a @ ( undire7337870655677353998dges_a @ Xs ) )
= ( undire7337870655677353998dges_a @ ( rev_a @ Xs ) ) ) ).
% walk_edges_rev
thf(fact_699_walk__length__rev,axiom,
( undire8849074589633906640ngth_a
= ( ^ [P3: list_a] : ( undire8849074589633906640ngth_a @ ( rev_a @ P3 ) ) ) ) ).
% walk_length_rev
thf(fact_700_walk__length__conv,axiom,
( undire8849074589633906640ngth_a
= ( ^ [P3: list_a] : ( minus_minus_nat @ ( size_size_list_a @ P3 ) @ one_one_nat ) ) ) ).
% walk_length_conv
thf(fact_701_numeral__eq__one__iff,axiom,
! [N: num] :
( ( ( numeral_numeral_nat @ N )
= one_one_nat )
= ( N = one ) ) ).
% numeral_eq_one_iff
thf(fact_702_numeral__eq__one__iff,axiom,
! [N: num] :
( ( ( numera1916890842035813515d_enat @ N )
= one_on7984719198319812577d_enat )
= ( N = one ) ) ).
% numeral_eq_one_iff
thf(fact_703_one__eq__numeral__iff,axiom,
! [N: num] :
( ( one_one_nat
= ( numeral_numeral_nat @ N ) )
= ( one = N ) ) ).
% one_eq_numeral_iff
thf(fact_704_one__eq__numeral__iff,axiom,
! [N: num] :
( ( one_on7984719198319812577d_enat
= ( numera1916890842035813515d_enat @ N ) )
= ( one = N ) ) ).
% one_eq_numeral_iff
thf(fact_705_diff__Suc__1,axiom,
! [N: nat] :
( ( minus_minus_nat @ ( suc @ N ) @ one_one_nat )
= N ) ).
% diff_Suc_1
thf(fact_706_is__cycle__alt__gen__path,axiom,
! [Xs: list_a] :
( ( undire2407311113669455967ycle_a @ vertices @ edges @ Xs )
= ( ( undire3562951555376170320path_a @ vertices @ edges @ Xs )
& ( ord_less_eq_nat @ one_one_nat @ ( undire8849074589633906640ngth_a @ Xs ) )
& ( ( hd_a @ Xs )
= ( last_a @ Xs ) ) ) ) ).
% is_cycle_alt_gen_path
thf(fact_707_is__cycle__def,axiom,
! [Xs: list_a] :
( ( undire2407311113669455967ycle_a @ vertices @ edges @ Xs )
= ( ( undire3370724456595283424walk_a @ vertices @ edges @ Xs )
& ( ord_less_eq_nat @ one_one_nat @ ( undire8849074589633906640ngth_a @ Xs ) )
& ( distinct_a @ ( tl_a @ Xs ) ) ) ) ).
% is_cycle_def
thf(fact_708_length__tl,axiom,
! [Xs: list_a] :
( ( size_size_list_a @ ( tl_a @ Xs ) )
= ( minus_minus_nat @ ( size_size_list_a @ Xs ) @ one_one_nat ) ) ).
% length_tl
thf(fact_709_length__tl,axiom,
! [Xs: list_set_a] :
( ( size_size_list_set_a @ ( tl_set_a @ Xs ) )
= ( minus_minus_nat @ ( size_size_list_set_a @ Xs ) @ one_one_nat ) ) ).
% length_tl
thf(fact_710_Suc__1,axiom,
( ( suc @ one_one_nat )
= ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).
% Suc_1
thf(fact_711_numeral__le__one__iff,axiom,
! [N: num] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ N ) @ one_one_nat )
= ( ord_less_eq_num @ N @ one ) ) ).
% numeral_le_one_iff
thf(fact_712_numeral__le__one__iff,axiom,
! [N: num] :
( ( ord_le2932123472753598470d_enat @ ( numera1916890842035813515d_enat @ N ) @ one_on7984719198319812577d_enat )
= ( ord_less_eq_num @ N @ one ) ) ).
% numeral_le_one_iff
thf(fact_713_ulgraph_Ois__trail_Ocong,axiom,
undire7142031287334043199rail_a = undire7142031287334043199rail_a ).
% ulgraph.is_trail.cong
thf(fact_714_one__reorient,axiom,
! [X: nat] :
( ( one_one_nat = X )
= ( X = one_one_nat ) ) ).
% one_reorient
thf(fact_715_one__reorient,axiom,
! [X: extended_enat] :
( ( one_on7984719198319812577d_enat = X )
= ( X = one_on7984719198319812577d_enat ) ) ).
% one_reorient
thf(fact_716_comp__sgraph_Owalk__length__conv,axiom,
( undire4424681683220949296_set_a
= ( ^ [P3: list_set_a] : ( minus_minus_nat @ ( size_size_list_set_a @ P3 ) @ one_one_nat ) ) ) ).
% comp_sgraph.walk_length_conv
thf(fact_717_comp__sgraph_Owalk__length__conv,axiom,
( undire8849074589633906640ngth_a
= ( ^ [P3: list_a] : ( minus_minus_nat @ ( size_size_list_a @ P3 ) @ one_one_nat ) ) ) ).
% comp_sgraph.walk_length_conv
thf(fact_718_le__numeral__extra_I4_J,axiom,
ord_less_eq_nat @ one_one_nat @ one_one_nat ).
% le_numeral_extra(4)
thf(fact_719_le__numeral__extra_I4_J,axiom,
ord_le2932123472753598470d_enat @ one_on7984719198319812577d_enat @ one_on7984719198319812577d_enat ).
% le_numeral_extra(4)
thf(fact_720_comp__sgraph_Owalk__length__rev,axiom,
( undire4424681683220949296_set_a
= ( ^ [P3: list_set_a] : ( undire4424681683220949296_set_a @ ( rev_set_a @ P3 ) ) ) ) ).
% comp_sgraph.walk_length_rev
thf(fact_721_comp__sgraph_Owalk__length__rev,axiom,
( undire8849074589633906640ngth_a
= ( ^ [P3: list_a] : ( undire8849074589633906640ngth_a @ ( rev_a @ P3 ) ) ) ) ).
% comp_sgraph.walk_length_rev
thf(fact_722_comp__sgraph_Owalk__length__def,axiom,
( undire8849074589633906640ngth_a
= ( ^ [P3: list_a] : ( size_size_list_set_a @ ( undire7337870655677353998dges_a @ P3 ) ) ) ) ).
% comp_sgraph.walk_length_def
thf(fact_723_ulgraph_Owalk__length__conv,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,P2: list_set_a] :
( ( undire6886684016831807756_set_a @ Vertices @ Edges )
=> ( ( undire4424681683220949296_set_a @ P2 )
= ( minus_minus_nat @ ( size_size_list_set_a @ P2 ) @ one_one_nat ) ) ) ).
% ulgraph.walk_length_conv
thf(fact_724_ulgraph_Owalk__length__conv,axiom,
! [Vertices: set_a,Edges: set_set_a,P2: list_a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( undire8849074589633906640ngth_a @ P2 )
= ( minus_minus_nat @ ( size_size_list_a @ P2 ) @ one_one_nat ) ) ) ).
% ulgraph.walk_length_conv
thf(fact_725_one__le__numeral,axiom,
! [N: num] : ( ord_less_eq_nat @ one_one_nat @ ( numeral_numeral_nat @ N ) ) ).
% one_le_numeral
thf(fact_726_one__le__numeral,axiom,
! [N: num] : ( ord_le2932123472753598470d_enat @ one_on7984719198319812577d_enat @ ( numera1916890842035813515d_enat @ N ) ) ).
% one_le_numeral
thf(fact_727_numeral__One,axiom,
( ( numeral_numeral_nat @ one )
= one_one_nat ) ).
% numeral_One
thf(fact_728_numeral__One,axiom,
( ( numera1916890842035813515d_enat @ one )
= one_on7984719198319812577d_enat ) ).
% numeral_One
thf(fact_729_numerals_I1_J,axiom,
( ( numeral_numeral_nat @ one )
= one_one_nat ) ).
% numerals(1)
thf(fact_730_One__nat__def,axiom,
( one_one_nat
= ( suc @ zero_zero_nat ) ) ).
% One_nat_def
thf(fact_731_diff__Suc__eq__diff__pred,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ M @ ( suc @ N ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M @ one_one_nat ) @ N ) ) ).
% diff_Suc_eq_diff_pred
thf(fact_732_length__n__lists__elem,axiom,
! [Ys: list_a,N: nat,Xs: list_a] :
( ( member_list_a @ Ys @ ( set_list_a2 @ ( n_lists_a @ N @ Xs ) ) )
=> ( ( size_size_list_a @ Ys )
= N ) ) ).
% length_n_lists_elem
thf(fact_733_length__n__lists__elem,axiom,
! [Ys: list_set_a,N: nat,Xs: list_set_a] :
( ( member_list_set_a @ Ys @ ( set_list_set_a2 @ ( n_lists_set_a @ N @ Xs ) ) )
=> ( ( size_size_list_set_a @ Ys )
= N ) ) ).
% length_n_lists_elem
thf(fact_734_ulgraph_Owalk__length__rev,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,P2: list_set_a] :
( ( undire6886684016831807756_set_a @ Vertices @ Edges )
=> ( ( undire4424681683220949296_set_a @ P2 )
= ( undire4424681683220949296_set_a @ ( rev_set_a @ P2 ) ) ) ) ).
% ulgraph.walk_length_rev
thf(fact_735_ulgraph_Owalk__length__rev,axiom,
! [Vertices: set_a,Edges: set_set_a,P2: list_a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( undire8849074589633906640ngth_a @ P2 )
= ( undire8849074589633906640ngth_a @ ( rev_a @ P2 ) ) ) ) ).
% ulgraph.walk_length_rev
thf(fact_736_ulgraph_Owalk__length__def,axiom,
! [Vertices: set_a,Edges: set_set_a,P2: list_a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( undire8849074589633906640ngth_a @ P2 )
= ( size_size_list_set_a @ ( undire7337870655677353998dges_a @ P2 ) ) ) ) ).
% ulgraph.walk_length_def
thf(fact_737_ulgraph_Ois__cycle__alt__gen__path,axiom,
! [Vertices: set_a,Edges: set_set_a,Xs: list_a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( undire2407311113669455967ycle_a @ Vertices @ Edges @ Xs )
= ( ( undire3562951555376170320path_a @ Vertices @ Edges @ Xs )
& ( ord_less_eq_nat @ one_one_nat @ ( undire8849074589633906640ngth_a @ Xs ) )
& ( ( hd_a @ Xs )
= ( last_a @ Xs ) ) ) ) ) ).
% ulgraph.is_cycle_alt_gen_path
thf(fact_738_ulgraph_Ois__cycle__def,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,Xs: list_set_a] :
( ( undire6886684016831807756_set_a @ Vertices @ Edges )
=> ( ( undire797940137672299967_set_a @ Vertices @ Edges @ Xs )
= ( ( undire4100213446647512896_set_a @ Vertices @ Edges @ Xs )
& ( ord_less_eq_nat @ one_one_nat @ ( undire4424681683220949296_set_a @ Xs ) )
& ( distinct_set_a @ ( tl_set_a @ Xs ) ) ) ) ) ).
% ulgraph.is_cycle_def
thf(fact_739_ulgraph_Ois__cycle__def,axiom,
! [Vertices: set_a,Edges: set_set_a,Xs: list_a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( undire2407311113669455967ycle_a @ Vertices @ Edges @ Xs )
= ( ( undire3370724456595283424walk_a @ Vertices @ Edges @ Xs )
& ( ord_less_eq_nat @ one_one_nat @ ( undire8849074589633906640ngth_a @ Xs ) )
& ( distinct_a @ ( tl_a @ Xs ) ) ) ) ) ).
% ulgraph.is_cycle_def
thf(fact_740_last__in__list__set,axiom,
! [Xs: list_a] :
( ( ord_less_eq_nat @ one_one_nat @ ( size_size_list_a @ Xs ) )
=> ( member_a @ ( last_a @ Xs ) @ ( set_a2 @ Xs ) ) ) ).
% last_in_list_set
thf(fact_741_last__in__list__set,axiom,
! [Xs: list_set_a] :
( ( ord_less_eq_nat @ one_one_nat @ ( size_size_list_set_a @ Xs ) )
=> ( member_set_a @ ( last_set_a @ Xs ) @ ( set_set_a2 @ Xs ) ) ) ).
% last_in_list_set
thf(fact_742_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_743_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_744_ulgraph_Ois__cycle__alt,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,Xs: list_set_a] :
( ( undire6886684016831807756_set_a @ Vertices @ Edges )
=> ( ( undire797940137672299967_set_a @ Vertices @ Edges @ Xs )
= ( ( undire3014741414213135564_set_a @ Vertices @ Edges @ Xs )
& ( distinct_set_a @ ( tl_set_a @ Xs ) )
& ( ord_less_eq_nat @ one_one_nat @ ( undire4424681683220949296_set_a @ Xs ) )
& ( ( hd_set_a @ Xs )
= ( last_set_a @ Xs ) ) ) ) ) ).
% ulgraph.is_cycle_alt
thf(fact_745_ulgraph_Ois__cycle__alt,axiom,
! [Vertices: set_a,Edges: set_set_a,Xs: list_a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( undire2407311113669455967ycle_a @ Vertices @ Edges @ Xs )
= ( ( undire6133010728901294956walk_a @ Vertices @ Edges @ Xs )
& ( distinct_a @ ( tl_a @ Xs ) )
& ( ord_less_eq_nat @ one_one_nat @ ( undire8849074589633906640ngth_a @ Xs ) )
& ( ( hd_a @ Xs )
= ( last_a @ Xs ) ) ) ) ) ).
% ulgraph.is_cycle_alt
thf(fact_746_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_747_alt__edge__size,axiom,
! [E: set_a] :
( ( member_set_a @ E @ edges )
=> ( ( ( finite_card_a @ E )
= one_one_nat )
| ( ( finite_card_a @ E )
= ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% alt_edge_size
thf(fact_748_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_749_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_750_add__right__cancel,axiom,
! [B: nat,A: nat,C: nat] :
( ( ( plus_plus_nat @ B @ A )
= ( plus_plus_nat @ C @ A ) )
= ( B = C ) ) ).
% add_right_cancel
thf(fact_751_add__left__cancel,axiom,
! [A: nat,B: nat,C: nat] :
( ( ( plus_plus_nat @ A @ B )
= ( plus_plus_nat @ A @ C ) )
= ( B = C ) ) ).
% add_left_cancel
thf(fact_752_add__le__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_cancel_right
thf(fact_753_add__le__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
= ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_cancel_left
thf(fact_754_add_Oright__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.right_neutral
thf(fact_755_add_Oright__neutral,axiom,
! [A: extended_enat] :
( ( plus_p3455044024723400733d_enat @ A @ zero_z5237406670263579293d_enat )
= A ) ).
% add.right_neutral
thf(fact_756_add__cancel__left__left,axiom,
! [B: nat,A: nat] :
( ( ( plus_plus_nat @ B @ A )
= A )
= ( B = zero_zero_nat ) ) ).
% add_cancel_left_left
thf(fact_757_add__cancel__left__right,axiom,
! [A: nat,B: nat] :
( ( ( plus_plus_nat @ A @ B )
= A )
= ( B = zero_zero_nat ) ) ).
% add_cancel_left_right
thf(fact_758_add__cancel__right__left,axiom,
! [A: nat,B: nat] :
( ( A
= ( plus_plus_nat @ B @ A ) )
= ( B = zero_zero_nat ) ) ).
% add_cancel_right_left
thf(fact_759_add__cancel__right__right,axiom,
! [A: nat,B: nat] :
( ( A
= ( plus_plus_nat @ A @ B ) )
= ( B = zero_zero_nat ) ) ).
% add_cancel_right_right
thf(fact_760_add__eq__0__iff__both__eq__0,axiom,
! [X: nat,Y: nat] :
( ( ( plus_plus_nat @ X @ Y )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ).
% add_eq_0_iff_both_eq_0
thf(fact_761_add__eq__0__iff__both__eq__0,axiom,
! [X: extended_enat,Y: extended_enat] :
( ( ( plus_p3455044024723400733d_enat @ X @ Y )
= zero_z5237406670263579293d_enat )
= ( ( X = zero_z5237406670263579293d_enat )
& ( Y = zero_z5237406670263579293d_enat ) ) ) ).
% add_eq_0_iff_both_eq_0
thf(fact_762_zero__eq__add__iff__both__eq__0,axiom,
! [X: nat,Y: nat] :
( ( zero_zero_nat
= ( plus_plus_nat @ X @ Y ) )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ).
% zero_eq_add_iff_both_eq_0
thf(fact_763_zero__eq__add__iff__both__eq__0,axiom,
! [X: extended_enat,Y: extended_enat] :
( ( zero_z5237406670263579293d_enat
= ( plus_p3455044024723400733d_enat @ X @ Y ) )
= ( ( X = zero_z5237406670263579293d_enat )
& ( Y = zero_z5237406670263579293d_enat ) ) ) ).
% zero_eq_add_iff_both_eq_0
thf(fact_764_add__0,axiom,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ) ).
% add_0
thf(fact_765_add__0,axiom,
! [A: extended_enat] :
( ( plus_p3455044024723400733d_enat @ zero_z5237406670263579293d_enat @ A )
= A ) ).
% add_0
thf(fact_766_add__numeral__left,axiom,
! [V2: num,W2: num,Z2: nat] :
( ( plus_plus_nat @ ( numeral_numeral_nat @ V2 ) @ ( plus_plus_nat @ ( numeral_numeral_nat @ W2 ) @ Z2 ) )
= ( plus_plus_nat @ ( numeral_numeral_nat @ ( plus_plus_num @ V2 @ W2 ) ) @ Z2 ) ) ).
% add_numeral_left
thf(fact_767_add__numeral__left,axiom,
! [V2: num,W2: num,Z2: extended_enat] :
( ( plus_p3455044024723400733d_enat @ ( numera1916890842035813515d_enat @ V2 ) @ ( plus_p3455044024723400733d_enat @ ( numera1916890842035813515d_enat @ W2 ) @ Z2 ) )
= ( plus_p3455044024723400733d_enat @ ( numera1916890842035813515d_enat @ ( plus_plus_num @ V2 @ W2 ) ) @ Z2 ) ) ).
% add_numeral_left
thf(fact_768_numeral__plus__numeral,axiom,
! [M: num,N: num] :
( ( plus_plus_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
= ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) ).
% numeral_plus_numeral
thf(fact_769_numeral__plus__numeral,axiom,
! [M: num,N: num] :
( ( plus_p3455044024723400733d_enat @ ( numera1916890842035813515d_enat @ M ) @ ( numera1916890842035813515d_enat @ N ) )
= ( numera1916890842035813515d_enat @ ( plus_plus_num @ M @ N ) ) ) ).
% numeral_plus_numeral
thf(fact_770_add__diff__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
= ( minus_minus_nat @ A @ B ) ) ).
% add_diff_cancel_left
thf(fact_771_add__diff__cancel__left_H,axiom,
! [A: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ A )
= B ) ).
% add_diff_cancel_left'
thf(fact_772_add__diff__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( minus_minus_nat @ A @ B ) ) ).
% add_diff_cancel_right
thf(fact_773_add__diff__cancel__right_H,axiom,
! [A: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ B )
= A ) ).
% add_diff_cancel_right'
thf(fact_774_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_775_Nat_Oadd__0__right,axiom,
! [M: nat] :
( ( plus_plus_nat @ M @ zero_zero_nat )
= M ) ).
% Nat.add_0_right
thf(fact_776_add__Suc__right,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ M @ ( suc @ N ) )
= ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).
% add_Suc_right
thf(fact_777_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_778_diff__diff__left,axiom,
! [I: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
= ( minus_minus_nat @ I @ ( plus_plus_nat @ J @ K ) ) ) ).
% diff_diff_left
thf(fact_779_le__add__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ B @ A ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% le_add_same_cancel2
thf(fact_780_le__add__same__cancel1,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ A @ B ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% le_add_same_cancel1
thf(fact_781_add__le__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ B )
= ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).
% add_le_same_cancel2
thf(fact_782_add__le__same__cancel1,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ B @ A ) @ B )
= ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).
% add_le_same_cancel1
thf(fact_783_diff__add__zero,axiom,
! [A: nat,B: nat] :
( ( minus_minus_nat @ A @ ( plus_plus_nat @ A @ B ) )
= zero_zero_nat ) ).
% diff_add_zero
thf(fact_784_Nat_Odiff__diff__right,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ) ).
% Nat.diff_diff_right
thf(fact_785_Nat_Oadd__diff__assoc2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I )
= ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K ) ) ) ).
% Nat.add_diff_assoc2
thf(fact_786_Nat_Oadd__diff__assoc,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K ) ) ) ).
% Nat.add_diff_assoc
thf(fact_787_length__append,axiom,
! [Xs: list_a,Ys: list_a] :
( ( size_size_list_a @ ( append_a @ Xs @ Ys ) )
= ( plus_plus_nat @ ( size_size_list_a @ Xs ) @ ( size_size_list_a @ Ys ) ) ) ).
% length_append
thf(fact_788_length__append,axiom,
! [Xs: list_set_a,Ys: list_set_a] :
( ( size_size_list_set_a @ ( append_set_a @ Xs @ Ys ) )
= ( plus_plus_nat @ ( size_size_list_set_a @ Xs ) @ ( size_size_list_set_a @ Ys ) ) ) ).
% length_append
thf(fact_789_numeral__plus__one,axiom,
! [N: num] :
( ( plus_plus_nat @ ( numeral_numeral_nat @ N ) @ one_one_nat )
= ( numeral_numeral_nat @ ( plus_plus_num @ N @ one ) ) ) ).
% numeral_plus_one
thf(fact_790_numeral__plus__one,axiom,
! [N: num] :
( ( plus_p3455044024723400733d_enat @ ( numera1916890842035813515d_enat @ N ) @ one_on7984719198319812577d_enat )
= ( numera1916890842035813515d_enat @ ( plus_plus_num @ N @ one ) ) ) ).
% numeral_plus_one
thf(fact_791_one__plus__numeral,axiom,
! [N: num] :
( ( plus_plus_nat @ one_one_nat @ ( numeral_numeral_nat @ N ) )
= ( numeral_numeral_nat @ ( plus_plus_num @ one @ N ) ) ) ).
% one_plus_numeral
thf(fact_792_one__plus__numeral,axiom,
! [N: num] :
( ( plus_p3455044024723400733d_enat @ one_on7984719198319812577d_enat @ ( numera1916890842035813515d_enat @ N ) )
= ( numera1916890842035813515d_enat @ ( plus_plus_num @ one @ N ) ) ) ).
% one_plus_numeral
thf(fact_793_diff__Suc__diff__eq2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( suc @ ( minus_minus_nat @ J @ K ) ) @ I )
= ( minus_minus_nat @ ( suc @ J ) @ ( plus_plus_nat @ K @ I ) ) ) ) ).
% diff_Suc_diff_eq2
thf(fact_794_diff__Suc__diff__eq1,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ I @ ( suc @ ( minus_minus_nat @ J @ K ) ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ K ) @ ( suc @ J ) ) ) ) ).
% diff_Suc_diff_eq1
thf(fact_795_one__add__one,axiom,
( ( plus_plus_nat @ one_one_nat @ one_one_nat )
= ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).
% one_add_one
thf(fact_796_one__add__one,axiom,
( ( plus_p3455044024723400733d_enat @ one_on7984719198319812577d_enat @ one_on7984719198319812577d_enat )
= ( numera1916890842035813515d_enat @ ( bit0 @ one ) ) ) ).
% one_add_one
thf(fact_797_add__2__eq__Suc_H,axiom,
! [N: nat] :
( ( plus_plus_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( suc @ ( suc @ N ) ) ) ).
% add_2_eq_Suc'
thf(fact_798_add__2__eq__Suc,axiom,
! [N: nat] :
( ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
= ( suc @ ( suc @ N ) ) ) ).
% add_2_eq_Suc
thf(fact_799_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( K = L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_800_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I: extended_enat,J: extended_enat,K: extended_enat,L: extended_enat] :
( ( ( ord_le2932123472753598470d_enat @ I @ J )
& ( K = L ) )
=> ( ord_le2932123472753598470d_enat @ ( plus_p3455044024723400733d_enat @ I @ K ) @ ( plus_p3455044024723400733d_enat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_801_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( I = J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_802_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I: extended_enat,J: extended_enat,K: extended_enat,L: extended_enat] :
( ( ( I = J )
& ( ord_le2932123472753598470d_enat @ K @ L ) )
=> ( ord_le2932123472753598470d_enat @ ( plus_p3455044024723400733d_enat @ I @ K ) @ ( plus_p3455044024723400733d_enat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_803_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_804_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I: extended_enat,J: extended_enat,K: extended_enat,L: extended_enat] :
( ( ( ord_le2932123472753598470d_enat @ I @ J )
& ( ord_le2932123472753598470d_enat @ K @ L ) )
=> ( ord_le2932123472753598470d_enat @ ( plus_p3455044024723400733d_enat @ I @ K ) @ ( plus_p3455044024723400733d_enat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_805_add__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_mono
thf(fact_806_add__mono,axiom,
! [A: extended_enat,B: extended_enat,C: extended_enat,D: extended_enat] :
( ( ord_le2932123472753598470d_enat @ A @ B )
=> ( ( ord_le2932123472753598470d_enat @ C @ D )
=> ( ord_le2932123472753598470d_enat @ ( plus_p3455044024723400733d_enat @ A @ C ) @ ( plus_p3455044024723400733d_enat @ B @ D ) ) ) ) ).
% add_mono
thf(fact_807_add__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) ) ) ).
% add_left_mono
thf(fact_808_add__left__mono,axiom,
! [A: extended_enat,B: extended_enat,C: extended_enat] :
( ( ord_le2932123472753598470d_enat @ A @ B )
=> ( ord_le2932123472753598470d_enat @ ( plus_p3455044024723400733d_enat @ C @ A ) @ ( plus_p3455044024723400733d_enat @ C @ B ) ) ) ).
% add_left_mono
thf(fact_809_less__eqE,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ~ ! [C2: nat] :
( B
!= ( plus_plus_nat @ A @ C2 ) ) ) ).
% less_eqE
thf(fact_810_less__eqE,axiom,
! [A: extended_enat,B: extended_enat] :
( ( ord_le2932123472753598470d_enat @ A @ B )
=> ~ ! [C2: extended_enat] :
( B
!= ( plus_p3455044024723400733d_enat @ A @ C2 ) ) ) ).
% less_eqE
thf(fact_811_add__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).
% add_right_mono
thf(fact_812_add__right__mono,axiom,
! [A: extended_enat,B: extended_enat,C: extended_enat] :
( ( ord_le2932123472753598470d_enat @ A @ B )
=> ( ord_le2932123472753598470d_enat @ ( plus_p3455044024723400733d_enat @ A @ C ) @ ( plus_p3455044024723400733d_enat @ B @ C ) ) ) ).
% add_right_mono
thf(fact_813_le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [A3: nat,B2: nat] :
? [C3: nat] :
( B2
= ( plus_plus_nat @ A3 @ C3 ) ) ) ) ).
% le_iff_add
thf(fact_814_le__iff__add,axiom,
( ord_le2932123472753598470d_enat
= ( ^ [A3: extended_enat,B2: extended_enat] :
? [C3: extended_enat] :
( B2
= ( plus_p3455044024723400733d_enat @ A3 @ C3 ) ) ) ) ).
% le_iff_add
thf(fact_815_add__le__imp__le__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_imp_le_left
thf(fact_816_add__le__imp__le__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_imp_le_right
thf(fact_817_add_Ocomm__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.comm_neutral
thf(fact_818_add_Ocomm__neutral,axiom,
! [A: extended_enat] :
( ( plus_p3455044024723400733d_enat @ A @ zero_z5237406670263579293d_enat )
= A ) ).
% add.comm_neutral
thf(fact_819_comm__monoid__add__class_Oadd__0,axiom,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_820_comm__monoid__add__class_Oadd__0,axiom,
! [A: extended_enat] :
( ( plus_p3455044024723400733d_enat @ zero_z5237406670263579293d_enat @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_821_verit__sum__simplify,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% verit_sum_simplify
thf(fact_822_diff__diff__eq,axiom,
! [A: nat,B: nat,C: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C )
= ( minus_minus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% diff_diff_eq
thf(fact_823_add__implies__diff,axiom,
! [C: nat,B: nat,A: nat] :
( ( ( plus_plus_nat @ C @ B )
= A )
=> ( C
= ( minus_minus_nat @ A @ B ) ) ) ).
% add_implies_diff
thf(fact_824_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_825_plus__nat_Oadd__0,axiom,
! [N: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N )
= N ) ).
% plus_nat.add_0
thf(fact_826_add__Suc__shift,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ ( suc @ M ) @ N )
= ( plus_plus_nat @ M @ ( suc @ N ) ) ) ).
% add_Suc_shift
thf(fact_827_add__Suc,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ ( suc @ M ) @ N )
= ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).
% add_Suc
thf(fact_828_nat__arith_Osuc1,axiom,
! [A2: nat,K: nat,A: nat] :
( ( A2
= ( plus_plus_nat @ K @ A ) )
=> ( ( suc @ A2 )
= ( plus_plus_nat @ K @ ( suc @ A ) ) ) ) ).
% nat_arith.suc1
thf(fact_829_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_830_le__add1,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M ) ) ).
% le_add1
thf(fact_831_le__add2,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M @ N ) ) ).
% le_add2
thf(fact_832_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_833_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_834_le__Suc__ex,axiom,
! [K: nat,L: nat] :
( ( ord_less_eq_nat @ K @ L )
=> ? [N2: nat] :
( L
= ( plus_plus_nat @ K @ N2 ) ) ) ).
% le_Suc_ex
thf(fact_835_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_836_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_837_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_838_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_839_nat__le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [M5: nat,N4: nat] :
? [K2: nat] :
( N4
= ( plus_plus_nat @ M5 @ K2 ) ) ) ) ).
% nat_le_iff_add
thf(fact_840_diff__add__inverse2,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ N )
= M ) ).
% diff_add_inverse2
thf(fact_841_diff__add__inverse,axiom,
! [N: nat,M: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ N @ M ) @ N )
= M ) ).
% diff_add_inverse
thf(fact_842_diff__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N @ K ) )
= ( minus_minus_nat @ M @ N ) ) ).
% diff_cancel2
thf(fact_843_Nat_Odiff__cancel,axiom,
! [K: nat,M: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
= ( minus_minus_nat @ M @ N ) ) ).
% Nat.diff_cancel
thf(fact_844_add__right__imp__eq,axiom,
! [B: nat,A: nat,C: nat] :
( ( ( plus_plus_nat @ B @ A )
= ( plus_plus_nat @ C @ A ) )
=> ( B = C ) ) ).
% add_right_imp_eq
thf(fact_845_add__left__imp__eq,axiom,
! [A: nat,B: nat,C: nat] :
( ( ( plus_plus_nat @ A @ B )
= ( plus_plus_nat @ A @ C ) )
=> ( B = C ) ) ).
% add_left_imp_eq
thf(fact_846_add_Oleft__commute,axiom,
! [B: nat,A: nat,C: nat] :
( ( plus_plus_nat @ B @ ( plus_plus_nat @ A @ C ) )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% add.left_commute
thf(fact_847_add_Oleft__commute,axiom,
! [B: extended_enat,A: extended_enat,C: extended_enat] :
( ( plus_p3455044024723400733d_enat @ B @ ( plus_p3455044024723400733d_enat @ A @ C ) )
= ( plus_p3455044024723400733d_enat @ A @ ( plus_p3455044024723400733d_enat @ B @ C ) ) ) ).
% add.left_commute
thf(fact_848_add_Ocommute,axiom,
( plus_plus_nat
= ( ^ [A3: nat,B2: nat] : ( plus_plus_nat @ B2 @ A3 ) ) ) ).
% add.commute
thf(fact_849_add_Ocommute,axiom,
( plus_p3455044024723400733d_enat
= ( ^ [A3: extended_enat,B2: extended_enat] : ( plus_p3455044024723400733d_enat @ B2 @ A3 ) ) ) ).
% add.commute
thf(fact_850_add_Oassoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% add.assoc
thf(fact_851_add_Oassoc,axiom,
! [A: extended_enat,B: extended_enat,C: extended_enat] :
( ( plus_p3455044024723400733d_enat @ ( plus_p3455044024723400733d_enat @ A @ B ) @ C )
= ( plus_p3455044024723400733d_enat @ A @ ( plus_p3455044024723400733d_enat @ B @ C ) ) ) ).
% add.assoc
thf(fact_852_group__cancel_Oadd2,axiom,
! [B4: nat,K: nat,B: nat,A: nat] :
( ( B4
= ( plus_plus_nat @ K @ B ) )
=> ( ( plus_plus_nat @ A @ B4 )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_853_group__cancel_Oadd2,axiom,
! [B4: extended_enat,K: extended_enat,B: extended_enat,A: extended_enat] :
( ( B4
= ( plus_p3455044024723400733d_enat @ K @ B ) )
=> ( ( plus_p3455044024723400733d_enat @ A @ B4 )
= ( plus_p3455044024723400733d_enat @ K @ ( plus_p3455044024723400733d_enat @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_854_group__cancel_Oadd1,axiom,
! [A2: nat,K: nat,A: nat,B: nat] :
( ( A2
= ( plus_plus_nat @ K @ A ) )
=> ( ( plus_plus_nat @ A2 @ B )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_855_group__cancel_Oadd1,axiom,
! [A2: extended_enat,K: extended_enat,A: extended_enat,B: extended_enat] :
( ( A2
= ( plus_p3455044024723400733d_enat @ K @ A ) )
=> ( ( plus_p3455044024723400733d_enat @ A2 @ B )
= ( plus_p3455044024723400733d_enat @ K @ ( plus_p3455044024723400733d_enat @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_856_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( I = J )
& ( K = L ) )
=> ( ( plus_plus_nat @ I @ K )
= ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_857_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I: extended_enat,J: extended_enat,K: extended_enat,L: extended_enat] :
( ( ( I = J )
& ( K = L ) )
=> ( ( plus_p3455044024723400733d_enat @ I @ K )
= ( plus_p3455044024723400733d_enat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_858_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_859_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: extended_enat,B: extended_enat,C: extended_enat] :
( ( plus_p3455044024723400733d_enat @ ( plus_p3455044024723400733d_enat @ A @ B ) @ C )
= ( plus_p3455044024723400733d_enat @ A @ ( plus_p3455044024723400733d_enat @ B @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_860_zero__one__enat__neq_I1_J,axiom,
zero_z5237406670263579293d_enat != one_on7984719198319812577d_enat ).
% zero_one_enat_neq(1)
thf(fact_861_add__nonpos__eq__0__iff,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ Y @ zero_zero_nat )
=> ( ( ( plus_plus_nat @ X @ Y )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_862_add__nonpos__eq__0__iff,axiom,
! [X: extended_enat,Y: extended_enat] :
( ( ord_le2932123472753598470d_enat @ X @ zero_z5237406670263579293d_enat )
=> ( ( ord_le2932123472753598470d_enat @ Y @ zero_z5237406670263579293d_enat )
=> ( ( ( plus_p3455044024723400733d_enat @ X @ Y )
= zero_z5237406670263579293d_enat )
= ( ( X = zero_z5237406670263579293d_enat )
& ( Y = zero_z5237406670263579293d_enat ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_863_add__nonneg__eq__0__iff,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ X )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
=> ( ( ( plus_plus_nat @ X @ Y )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_864_add__nonneg__eq__0__iff,axiom,
! [X: extended_enat,Y: extended_enat] :
( ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ X )
=> ( ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ Y )
=> ( ( ( plus_p3455044024723400733d_enat @ X @ Y )
= zero_z5237406670263579293d_enat )
= ( ( X = zero_z5237406670263579293d_enat )
& ( Y = zero_z5237406670263579293d_enat ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_865_add__nonpos__nonpos,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_nonpos_nonpos
thf(fact_866_add__nonpos__nonpos,axiom,
! [A: extended_enat,B: extended_enat] :
( ( ord_le2932123472753598470d_enat @ A @ zero_z5237406670263579293d_enat )
=> ( ( ord_le2932123472753598470d_enat @ B @ zero_z5237406670263579293d_enat )
=> ( ord_le2932123472753598470d_enat @ ( plus_p3455044024723400733d_enat @ A @ B ) @ zero_z5237406670263579293d_enat ) ) ) ).
% add_nonpos_nonpos
thf(fact_867_add__nonneg__nonneg,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_868_add__nonneg__nonneg,axiom,
! [A: extended_enat,B: extended_enat] :
( ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ A )
=> ( ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ B )
=> ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ ( plus_p3455044024723400733d_enat @ A @ B ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_869_add__increasing2,axiom,
! [C: nat,B: nat,A: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ( ord_less_eq_nat @ B @ A )
=> ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_increasing2
thf(fact_870_add__increasing2,axiom,
! [C: extended_enat,B: extended_enat,A: extended_enat] :
( ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ C )
=> ( ( ord_le2932123472753598470d_enat @ B @ A )
=> ( ord_le2932123472753598470d_enat @ B @ ( plus_p3455044024723400733d_enat @ A @ C ) ) ) ) ).
% add_increasing2
thf(fact_871_add__decreasing2,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ C @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ B ) ) ) ).
% add_decreasing2
thf(fact_872_add__decreasing2,axiom,
! [C: extended_enat,A: extended_enat,B: extended_enat] :
( ( ord_le2932123472753598470d_enat @ C @ zero_z5237406670263579293d_enat )
=> ( ( ord_le2932123472753598470d_enat @ A @ B )
=> ( ord_le2932123472753598470d_enat @ ( plus_p3455044024723400733d_enat @ A @ C ) @ B ) ) ) ).
% add_decreasing2
thf(fact_873_add__increasing,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_increasing
thf(fact_874_add__increasing,axiom,
! [A: extended_enat,B: extended_enat,C: extended_enat] :
( ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ A )
=> ( ( ord_le2932123472753598470d_enat @ B @ C )
=> ( ord_le2932123472753598470d_enat @ B @ ( plus_p3455044024723400733d_enat @ A @ C ) ) ) ) ).
% add_increasing
thf(fact_875_add__decreasing,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ B ) ) ) ).
% add_decreasing
thf(fact_876_add__decreasing,axiom,
! [A: extended_enat,C: extended_enat,B: extended_enat] :
( ( ord_le2932123472753598470d_enat @ A @ zero_z5237406670263579293d_enat )
=> ( ( ord_le2932123472753598470d_enat @ C @ B )
=> ( ord_le2932123472753598470d_enat @ ( plus_p3455044024723400733d_enat @ A @ C ) @ B ) ) ) ).
% add_decreasing
thf(fact_877_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( ( ( minus_minus_nat @ B @ A )
= C )
= ( B
= ( plus_plus_nat @ C @ A ) ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
thf(fact_878_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ A @ ( minus_minus_nat @ B @ A ) )
= B ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_inverse
thf(fact_879_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( minus_minus_nat @ C @ ( minus_minus_nat @ B @ A ) )
= ( minus_minus_nat @ ( plus_plus_nat @ C @ A ) @ B ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_diff_right
thf(fact_880_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A )
= ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ C ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
thf(fact_881_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ C )
= ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
thf(fact_882_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ C @ B ) @ A )
= ( plus_plus_nat @ C @ ( minus_minus_nat @ B @ A ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc
thf(fact_883_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ C @ ( minus_minus_nat @ B @ A ) )
= ( minus_minus_nat @ ( plus_plus_nat @ C @ B ) @ A ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc
thf(fact_884_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ ( minus_minus_nat @ B @ A ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ B ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_diff_conv2
thf(fact_885_le__add__diff,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ C @ ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A ) ) ) ).
% le_add_diff
thf(fact_886_ordered__cancel__comm__monoid__diff__class_Odiff__add,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ A )
= B ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add
thf(fact_887_one__plus__numeral__commute,axiom,
! [X: num] :
( ( plus_plus_nat @ one_one_nat @ ( numeral_numeral_nat @ X ) )
= ( plus_plus_nat @ ( numeral_numeral_nat @ X ) @ one_one_nat ) ) ).
% one_plus_numeral_commute
thf(fact_888_one__plus__numeral__commute,axiom,
! [X: num] :
( ( plus_p3455044024723400733d_enat @ one_on7984719198319812577d_enat @ ( numera1916890842035813515d_enat @ X ) )
= ( plus_p3455044024723400733d_enat @ ( numera1916890842035813515d_enat @ X ) @ one_on7984719198319812577d_enat ) ) ).
% one_plus_numeral_commute
thf(fact_889_numeral__Bit0,axiom,
! [N: num] :
( ( numeral_numeral_nat @ ( bit0 @ N ) )
= ( plus_plus_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ N ) ) ) ).
% numeral_Bit0
thf(fact_890_numeral__Bit0,axiom,
! [N: num] :
( ( numera1916890842035813515d_enat @ ( bit0 @ N ) )
= ( plus_p3455044024723400733d_enat @ ( numera1916890842035813515d_enat @ N ) @ ( numera1916890842035813515d_enat @ N ) ) ) ).
% numeral_Bit0
thf(fact_891_add__is__1,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= ( suc @ zero_zero_nat ) )
= ( ( ( M
= ( suc @ zero_zero_nat ) )
& ( N = zero_zero_nat ) )
| ( ( M = zero_zero_nat )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ) ).
% add_is_1
thf(fact_892_one__is__add,axiom,
! [M: nat,N: nat] :
( ( ( suc @ zero_zero_nat )
= ( plus_plus_nat @ M @ N ) )
= ( ( ( M
= ( suc @ zero_zero_nat ) )
& ( N = zero_zero_nat ) )
| ( ( M = zero_zero_nat )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ) ).
% one_is_add
thf(fact_893_diff__add__0,axiom,
! [N: nat,M: nat] :
( ( minus_minus_nat @ N @ ( plus_plus_nat @ N @ M ) )
= zero_zero_nat ) ).
% diff_add_0
thf(fact_894_Nat_Ole__imp__diff__is__add,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ( minus_minus_nat @ J @ I )
= K )
= ( J
= ( plus_plus_nat @ K @ I ) ) ) ) ).
% Nat.le_imp_diff_is_add
thf(fact_895_Nat_Odiff__add__assoc2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K )
= ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I ) ) ) ).
% Nat.diff_add_assoc2
thf(fact_896_Nat_Odiff__add__assoc,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K )
= ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K ) ) ) ) ).
% Nat.diff_add_assoc
thf(fact_897_Nat_Ole__diff__conv2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( ord_less_eq_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ) ).
% Nat.le_diff_conv2
thf(fact_898_le__diff__conv,axiom,
! [J: nat,K: nat,I: nat] :
( ( ord_less_eq_nat @ ( minus_minus_nat @ J @ K ) @ I )
= ( ord_less_eq_nat @ J @ ( plus_plus_nat @ I @ K ) ) ) ).
% le_diff_conv
thf(fact_899_Suc__eq__plus1,axiom,
( suc
= ( ^ [N4: nat] : ( plus_plus_nat @ N4 @ one_one_nat ) ) ) ).
% Suc_eq_plus1
thf(fact_900_plus__1__eq__Suc,axiom,
( ( plus_plus_nat @ one_one_nat )
= suc ) ).
% plus_1_eq_Suc
thf(fact_901_Suc__eq__plus1__left,axiom,
( suc
= ( plus_plus_nat @ one_one_nat ) ) ).
% Suc_eq_plus1_left
thf(fact_902_card__length,axiom,
! [Xs: list_a] : ( ord_less_eq_nat @ ( finite_card_a @ ( set_a2 @ Xs ) ) @ ( size_size_list_a @ Xs ) ) ).
% card_length
thf(fact_903_card__length,axiom,
! [Xs: list_set_a] : ( ord_less_eq_nat @ ( finite_card_set_a @ ( set_set_a2 @ Xs ) ) @ ( size_size_list_set_a @ Xs ) ) ).
% card_length
thf(fact_904_numeral__Bit1,axiom,
! [N: num] :
( ( numeral_numeral_nat @ ( bit1 @ N ) )
= ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ N ) ) @ one_one_nat ) ) ).
% numeral_Bit1
thf(fact_905_numeral__Bit1,axiom,
! [N: num] :
( ( numera1916890842035813515d_enat @ ( bit1 @ N ) )
= ( plus_p3455044024723400733d_enat @ ( plus_p3455044024723400733d_enat @ ( numera1916890842035813515d_enat @ N ) @ ( numera1916890842035813515d_enat @ N ) ) @ one_on7984719198319812577d_enat ) ) ).
% numeral_Bit1
thf(fact_906_card__distinct,axiom,
! [Xs: list_a] :
( ( ( finite_card_a @ ( set_a2 @ Xs ) )
= ( size_size_list_a @ Xs ) )
=> ( distinct_a @ Xs ) ) ).
% card_distinct
thf(fact_907_card__distinct,axiom,
! [Xs: list_set_a] :
( ( ( finite_card_set_a @ ( set_set_a2 @ Xs ) )
= ( size_size_list_set_a @ Xs ) )
=> ( distinct_set_a @ Xs ) ) ).
% card_distinct
thf(fact_908_distinct__card,axiom,
! [Xs: list_a] :
( ( distinct_a @ Xs )
=> ( ( finite_card_a @ ( set_a2 @ Xs ) )
= ( size_size_list_a @ Xs ) ) ) ).
% distinct_card
thf(fact_909_distinct__card,axiom,
! [Xs: list_set_a] :
( ( distinct_set_a @ Xs )
=> ( ( finite_card_set_a @ ( set_set_a2 @ Xs ) )
= ( size_size_list_set_a @ Xs ) ) ) ).
% distinct_card
thf(fact_910_Suc3__eq__add__3,axiom,
! [N: nat] :
( ( suc @ ( suc @ ( suc @ N ) ) )
= ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ N ) ) ).
% Suc3_eq_add_3
thf(fact_911_list_Osize_I4_J,axiom,
! [X21: a,X222: list_a] :
( ( size_size_list_a @ ( cons_a @ X21 @ X222 ) )
= ( plus_plus_nat @ ( size_size_list_a @ X222 ) @ ( suc @ zero_zero_nat ) ) ) ).
% list.size(4)
thf(fact_912_list_Osize_I4_J,axiom,
! [X21: set_a,X222: list_set_a] :
( ( size_size_list_set_a @ ( cons_set_a @ X21 @ X222 ) )
= ( plus_plus_nat @ ( size_size_list_set_a @ X222 ) @ ( suc @ zero_zero_nat ) ) ) ).
% list.size(4)
thf(fact_913_nat__1__add__1,axiom,
( ( plus_plus_nat @ one_one_nat @ one_one_nat )
= ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).
% nat_1_add_1
thf(fact_914_add__eq__if,axiom,
( plus_plus_nat
= ( ^ [M5: nat,N4: nat] : ( if_nat @ ( M5 = zero_zero_nat ) @ N4 @ ( suc @ ( plus_plus_nat @ ( minus_minus_nat @ M5 @ one_one_nat ) @ N4 ) ) ) ) ) ).
% add_eq_if
thf(fact_915_comp__sgraph_Owalk__length__app,axiom,
! [Xs: list_set_a,Ys: list_set_a] :
( ( Xs != nil_set_a )
=> ( ( Ys != nil_set_a )
=> ( ( undire4424681683220949296_set_a @ ( append_set_a @ Xs @ Ys ) )
= ( plus_plus_nat @ ( plus_plus_nat @ ( undire4424681683220949296_set_a @ Xs ) @ ( undire4424681683220949296_set_a @ Ys ) ) @ one_one_nat ) ) ) ) ).
% comp_sgraph.walk_length_app
thf(fact_916_comp__sgraph_Owalk__length__app,axiom,
! [Xs: list_a,Ys: list_a] :
( ( Xs != nil_a )
=> ( ( Ys != nil_a )
=> ( ( undire8849074589633906640ngth_a @ ( append_a @ Xs @ Ys ) )
= ( plus_plus_nat @ ( plus_plus_nat @ ( undire8849074589633906640ngth_a @ Xs ) @ ( undire8849074589633906640ngth_a @ Ys ) ) @ one_one_nat ) ) ) ) ).
% comp_sgraph.walk_length_app
thf(fact_917_comp__sgraph_Owalk__length__app__ineq,axiom,
! [Xs: list_a,Ys: list_a] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ ( undire8849074589633906640ngth_a @ Xs ) @ ( undire8849074589633906640ngth_a @ Ys ) ) @ ( undire8849074589633906640ngth_a @ ( append_a @ Xs @ Ys ) ) )
& ( ord_less_eq_nat @ ( undire8849074589633906640ngth_a @ ( append_a @ Xs @ Ys ) ) @ ( plus_plus_nat @ ( plus_plus_nat @ ( undire8849074589633906640ngth_a @ Xs ) @ ( undire8849074589633906640ngth_a @ Ys ) ) @ one_one_nat ) ) ) ).
% comp_sgraph.walk_length_app_ineq
thf(fact_918_num_Osize_I5_J,axiom,
! [X2: num] :
( ( size_size_num @ ( bit0 @ X2 ) )
= ( plus_plus_nat @ ( size_size_num @ X2 ) @ ( suc @ zero_zero_nat ) ) ) ).
% num.size(5)
thf(fact_919_num_Osize_I6_J,axiom,
! [X3: num] :
( ( size_size_num @ ( bit1 @ X3 ) )
= ( plus_plus_nat @ ( size_size_num @ X3 ) @ ( suc @ zero_zero_nat ) ) ) ).
% num.size(6)
thf(fact_920_ulgraph_Oalt__edge__size,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,E: set_set_a] :
( ( undire6886684016831807756_set_a @ Vertices @ Edges )
=> ( ( member_set_set_a @ E @ Edges )
=> ( ( ( finite_card_set_a @ E )
= one_one_nat )
| ( ( finite_card_set_a @ E )
= ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% ulgraph.alt_edge_size
thf(fact_921_ulgraph_Oalt__edge__size,axiom,
! [Vertices: set_a,Edges: set_set_a,E: set_a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( member_set_a @ E @ Edges )
=> ( ( ( finite_card_a @ E )
= one_one_nat )
| ( ( finite_card_a @ E )
= ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% ulgraph.alt_edge_size
thf(fact_922_ulgraph_Owalk__length__app,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,Xs: list_set_a,Ys: list_set_a] :
( ( undire6886684016831807756_set_a @ Vertices @ Edges )
=> ( ( Xs != nil_set_a )
=> ( ( Ys != nil_set_a )
=> ( ( undire4424681683220949296_set_a @ ( append_set_a @ Xs @ Ys ) )
= ( plus_plus_nat @ ( plus_plus_nat @ ( undire4424681683220949296_set_a @ Xs ) @ ( undire4424681683220949296_set_a @ Ys ) ) @ one_one_nat ) ) ) ) ) ).
% ulgraph.walk_length_app
thf(fact_923_ulgraph_Owalk__length__app,axiom,
! [Vertices: set_a,Edges: set_set_a,Xs: list_a,Ys: list_a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( Xs != nil_a )
=> ( ( Ys != nil_a )
=> ( ( undire8849074589633906640ngth_a @ ( append_a @ Xs @ Ys ) )
= ( plus_plus_nat @ ( plus_plus_nat @ ( undire8849074589633906640ngth_a @ Xs ) @ ( undire8849074589633906640ngth_a @ Ys ) ) @ one_one_nat ) ) ) ) ) ).
% ulgraph.walk_length_app
thf(fact_924_ulgraph_Owalk__length__app__ineq,axiom,
! [Vertices: set_a,Edges: set_set_a,Xs: list_a,Ys: list_a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( undire8849074589633906640ngth_a @ Xs ) @ ( undire8849074589633906640ngth_a @ Ys ) ) @ ( undire8849074589633906640ngth_a @ ( append_a @ Xs @ Ys ) ) )
& ( ord_less_eq_nat @ ( undire8849074589633906640ngth_a @ ( append_a @ Xs @ Ys ) ) @ ( plus_plus_nat @ ( plus_plus_nat @ ( undire8849074589633906640ngth_a @ Xs ) @ ( undire8849074589633906640ngth_a @ Ys ) ) @ one_one_nat ) ) ) ) ).
% ulgraph.walk_length_app_ineq
thf(fact_925_is__sedge__def,axiom,
( undire4917966558017083288edge_a
= ( ^ [E5: set_a] :
( ( finite_card_a @ E5 )
= ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% is_sedge_def
thf(fact_926_le__add__diff__inverse2,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ A @ B ) @ B )
= A ) ) ).
% le_add_diff_inverse2
thf(fact_927_le__add__diff__inverse,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( plus_plus_nat @ B @ ( minus_minus_nat @ A @ B ) )
= A ) ) ).
% le_add_diff_inverse
thf(fact_928_is__loop__def,axiom,
( undire2905028936066782638loop_a
= ( ^ [E5: set_a] :
( ( finite_card_a @ E5 )
= one_one_nat ) ) ) ).
% is_loop_def
thf(fact_929_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_930_semiring__norm_I6_J,axiom,
! [M: num,N: num] :
( ( plus_plus_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
= ( bit0 @ ( plus_plus_num @ M @ N ) ) ) ).
% semiring_norm(6)
thf(fact_931_semiring__norm_I2_J,axiom,
( ( plus_plus_num @ one @ one )
= ( bit0 @ one ) ) ).
% semiring_norm(2)
thf(fact_932_semiring__norm_I9_J,axiom,
! [M: num,N: num] :
( ( plus_plus_num @ ( bit1 @ M ) @ ( bit0 @ N ) )
= ( bit1 @ ( plus_plus_num @ M @ N ) ) ) ).
% semiring_norm(9)
thf(fact_933_semiring__norm_I7_J,axiom,
! [M: num,N: num] :
( ( plus_plus_num @ ( bit0 @ M ) @ ( bit1 @ N ) )
= ( bit1 @ ( plus_plus_num @ M @ N ) ) ) ).
% semiring_norm(7)
thf(fact_934_semiring__norm_I10_J,axiom,
! [M: num,N: num] :
( ( plus_plus_num @ ( bit1 @ M ) @ ( bit1 @ N ) )
= ( bit0 @ ( plus_plus_num @ ( plus_plus_num @ M @ N ) @ one ) ) ) ).
% semiring_norm(10)
thf(fact_935_semiring__norm_I8_J,axiom,
! [M: num] :
( ( plus_plus_num @ ( bit1 @ M ) @ one )
= ( bit0 @ ( plus_plus_num @ M @ one ) ) ) ).
% semiring_norm(8)
thf(fact_936_semiring__norm_I5_J,axiom,
! [M: num] :
( ( plus_plus_num @ ( bit0 @ M ) @ one )
= ( bit1 @ M ) ) ).
% semiring_norm(5)
thf(fact_937_semiring__norm_I4_J,axiom,
! [N: num] :
( ( plus_plus_num @ one @ ( bit1 @ N ) )
= ( bit0 @ ( plus_plus_num @ N @ one ) ) ) ).
% semiring_norm(4)
thf(fact_938_semiring__norm_I3_J,axiom,
! [N: num] :
( ( plus_plus_num @ one @ ( bit0 @ N ) )
= ( bit1 @ N ) ) ).
% semiring_norm(3)
thf(fact_939_Suc__numeral,axiom,
! [N: num] :
( ( suc @ ( numeral_numeral_nat @ N ) )
= ( numeral_numeral_nat @ ( plus_plus_num @ N @ one ) ) ) ).
% Suc_numeral
thf(fact_940_add__One__commute,axiom,
! [N: num] :
( ( plus_plus_num @ one @ N )
= ( plus_plus_num @ N @ one ) ) ).
% add_One_commute
thf(fact_941_iadd__is__0,axiom,
! [M: extended_enat,N: extended_enat] :
( ( ( plus_p3455044024723400733d_enat @ M @ N )
= zero_z5237406670263579293d_enat )
= ( ( M = zero_z5237406670263579293d_enat )
& ( N = zero_z5237406670263579293d_enat ) ) ) ).
% iadd_is_0
thf(fact_942_ulgraph_Ois__edge__or__loop,axiom,
! [Vertices: set_a,Edges: set_set_a,E: set_a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( member_set_a @ E @ Edges )
=> ( ( undire2905028936066782638loop_a @ E )
| ( undire4917966558017083288edge_a @ E ) ) ) ) ).
% ulgraph.is_edge_or_loop
thf(fact_943_add__diff__assoc__enat,axiom,
! [Z2: extended_enat,Y: extended_enat,X: extended_enat] :
( ( ord_le2932123472753598470d_enat @ Z2 @ Y )
=> ( ( plus_p3455044024723400733d_enat @ X @ ( minus_3235023915231533773d_enat @ Y @ Z2 ) )
= ( minus_3235023915231533773d_enat @ ( plus_p3455044024723400733d_enat @ X @ Y ) @ Z2 ) ) ) ).
% add_diff_assoc_enat
thf(fact_944_comp__sgraph_Ois__loop__def,axiom,
( undire3618949687197220622_set_a
= ( ^ [E5: set_set_a] :
( ( finite_card_set_a @ E5 )
= one_one_nat ) ) ) ).
% comp_sgraph.is_loop_def
thf(fact_945_comp__sgraph_Ois__loop__def,axiom,
( undire2905028936066782638loop_a
= ( ^ [E5: set_a] :
( ( finite_card_a @ E5 )
= one_one_nat ) ) ) ).
% comp_sgraph.is_loop_def
thf(fact_946_Suc__nat__number__of__add,axiom,
! [V2: num,N: nat] :
( ( suc @ ( plus_plus_nat @ ( numeral_numeral_nat @ V2 ) @ N ) )
= ( plus_plus_nat @ ( numeral_numeral_nat @ ( plus_plus_num @ V2 @ one ) ) @ N ) ) ).
% Suc_nat_number_of_add
thf(fact_947_ulgraph_Ois__loop__def,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,E: set_set_a] :
( ( undire6886684016831807756_set_a @ Vertices @ Edges )
=> ( ( undire3618949687197220622_set_a @ E )
= ( ( finite_card_set_a @ E )
= one_one_nat ) ) ) ).
% ulgraph.is_loop_def
thf(fact_948_ulgraph_Ois__loop__def,axiom,
! [Vertices: set_a,Edges: set_set_a,E: set_a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( undire2905028936066782638loop_a @ E )
= ( ( finite_card_a @ E )
= one_one_nat ) ) ) ).
% ulgraph.is_loop_def
thf(fact_949_zero__neq__one,axiom,
zero_zero_nat != one_one_nat ).
% zero_neq_one
thf(fact_950_zero__neq__one,axiom,
zero_z5237406670263579293d_enat != one_on7984719198319812577d_enat ).
% zero_neq_one
thf(fact_951_comp__sgraph_Ois__sedge__def,axiom,
( undire5778062222479071480_set_a
= ( ^ [E5: set_set_a] :
( ( finite_card_set_a @ E5 )
= ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% comp_sgraph.is_sedge_def
thf(fact_952_comp__sgraph_Ois__sedge__def,axiom,
( undire4917966558017083288edge_a
= ( ^ [E5: set_a] :
( ( finite_card_a @ E5 )
= ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% comp_sgraph.is_sedge_def
thf(fact_953_ulgraph_Ois__sedge__def,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,E: set_set_a] :
( ( undire6886684016831807756_set_a @ Vertices @ Edges )
=> ( ( undire5778062222479071480_set_a @ E )
= ( ( finite_card_set_a @ E )
= ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% ulgraph.is_sedge_def
thf(fact_954_ulgraph_Ois__sedge__def,axiom,
! [Vertices: set_a,Edges: set_set_a,E: set_a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( undire4917966558017083288edge_a @ E )
= ( ( finite_card_a @ E )
= ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% ulgraph.is_sedge_def
thf(fact_955_zero__less__one__class_Ozero__le__one,axiom,
ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one_class.zero_le_one
thf(fact_956_zero__less__one__class_Ozero__le__one,axiom,
ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ one_on7984719198319812577d_enat ).
% zero_less_one_class.zero_le_one
thf(fact_957_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_958_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ one_on7984719198319812577d_enat ).
% linordered_nonzero_semiring_class.zero_le_one
thf(fact_959_not__one__le__zero,axiom,
~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_le_zero
thf(fact_960_not__one__le__zero,axiom,
~ ( ord_le2932123472753598470d_enat @ one_on7984719198319812577d_enat @ zero_z5237406670263579293d_enat ) ).
% not_one_le_zero
thf(fact_961_add__le__imp__le__diff,axiom,
! [I: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N )
=> ( ord_less_eq_nat @ I @ ( minus_minus_nat @ N @ K ) ) ) ).
% add_le_imp_le_diff
thf(fact_962_add__le__add__imp__diff__le,axiom,
! [I: nat,K: nat,N: nat,J: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J @ K ) )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J @ K ) )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ N @ K ) @ J ) ) ) ) ) ).
% add_le_add_imp_diff_le
thf(fact_963_edge__size,axiom,
! [E: set_a] :
( ( member_set_a @ E @ edges )
=> ( ( ord_less_nat @ zero_zero_nat @ ( finite_card_a @ E ) )
& ( ord_less_eq_nat @ ( finite_card_a @ E ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% edge_size
thf(fact_964_nat__induct2,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ( P @ one_one_nat )
=> ( ! [N2: nat] :
( ( P @ N2 )
=> ( P @ ( plus_plus_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct2
thf(fact_965_incident__loops__card,axiom,
! [V2: a] : ( ord_less_eq_nat @ ( finite_card_set_a @ ( undire4753905205749729249oops_a @ edges @ V2 ) ) @ one_one_nat ) ).
% incident_loops_card
thf(fact_966_not__gr__zero,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_967_not__gr__zero,axiom,
! [N: extended_enat] :
( ( ~ ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ N ) )
= ( N = zero_z5237406670263579293d_enat ) ) ).
% not_gr_zero
thf(fact_968_numeral__less__iff,axiom,
! [M: num,N: num] :
( ( ord_less_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
= ( ord_less_num @ M @ N ) ) ).
% numeral_less_iff
thf(fact_969_numeral__less__iff,axiom,
! [M: num,N: num] :
( ( ord_le72135733267957522d_enat @ ( numera1916890842035813515d_enat @ M ) @ ( numera1916890842035813515d_enat @ N ) )
= ( ord_less_num @ M @ N ) ) ).
% numeral_less_iff
thf(fact_970_add__less__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
= ( ord_less_nat @ A @ B ) ) ).
% add_less_cancel_left
thf(fact_971_add__less__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( ord_less_nat @ A @ B ) ) ).
% add_less_cancel_right
thf(fact_972_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_973_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_974_bot__nat__0_Onot__eq__extremum,axiom,
! [A: nat] :
( ( A != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ A ) ) ).
% bot_nat_0.not_eq_extremum
thf(fact_975_Suc__less__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% Suc_less_eq
thf(fact_976_Suc__mono,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) ) ) ).
% Suc_mono
thf(fact_977_lessI,axiom,
! [N: nat] : ( ord_less_nat @ N @ ( suc @ N ) ) ).
% lessI
thf(fact_978_nat__add__left__cancel__less,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% nat_add_left_cancel_less
thf(fact_979_less__add__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ ( plus_plus_nat @ B @ A ) )
= ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% less_add_same_cancel2
thf(fact_980_less__add__same__cancel1,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ ( plus_plus_nat @ A @ B ) )
= ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% less_add_same_cancel1
thf(fact_981_add__less__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ B )
= ( ord_less_nat @ A @ zero_zero_nat ) ) ).
% add_less_same_cancel2
thf(fact_982_add__less__same__cancel1,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ B @ A ) @ B )
= ( ord_less_nat @ A @ zero_zero_nat ) ) ).
% add_less_same_cancel1
thf(fact_983_less__Suc0,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ ( suc @ zero_zero_nat ) )
= ( N = zero_zero_nat ) ) ).
% less_Suc0
thf(fact_984_zero__less__Suc,axiom,
! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N ) ) ).
% zero_less_Suc
thf(fact_985_add__gr__0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ M )
| ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% add_gr_0
thf(fact_986_zero__less__diff,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N @ M ) )
= ( ord_less_nat @ M @ N ) ) ).
% zero_less_diff
thf(fact_987_less__one,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ one_one_nat )
= ( N = zero_zero_nat ) ) ).
% less_one
thf(fact_988_one__less__numeral__iff,axiom,
! [N: num] :
( ( ord_less_nat @ one_one_nat @ ( numeral_numeral_nat @ N ) )
= ( ord_less_num @ one @ N ) ) ).
% one_less_numeral_iff
thf(fact_989_one__less__numeral__iff,axiom,
! [N: num] :
( ( ord_le72135733267957522d_enat @ one_on7984719198319812577d_enat @ ( numera1916890842035813515d_enat @ N ) )
= ( ord_less_num @ one @ N ) ) ).
% one_less_numeral_iff
thf(fact_990_length__greater__0__conv,axiom,
! [Xs: list_a] :
( ( ord_less_nat @ zero_zero_nat @ ( size_size_list_a @ Xs ) )
= ( Xs != nil_a ) ) ).
% length_greater_0_conv
thf(fact_991_length__greater__0__conv,axiom,
! [Xs: list_set_a] :
( ( ord_less_nat @ zero_zero_nat @ ( size_size_list_set_a @ Xs ) )
= ( Xs != nil_set_a ) ) ).
% length_greater_0_conv
thf(fact_992_Suc__pred,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( suc @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) )
= N ) ) ).
% Suc_pred
thf(fact_993_Suc__diff__1,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( suc @ ( minus_minus_nat @ N @ one_one_nat ) )
= N ) ) ).
% Suc_diff_1
thf(fact_994_less__numeral__extra_I4_J,axiom,
~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).
% less_numeral_extra(4)
thf(fact_995_less__numeral__extra_I4_J,axiom,
~ ( ord_le72135733267957522d_enat @ one_on7984719198319812577d_enat @ one_on7984719198319812577d_enat ) ).
% less_numeral_extra(4)
thf(fact_996_less__add__eq__less,axiom,
! [K: nat,L: nat,M: nat,N: nat] :
( ( ord_less_nat @ K @ L )
=> ( ( ( plus_plus_nat @ M @ L )
= ( plus_plus_nat @ K @ N ) )
=> ( ord_less_nat @ M @ N ) ) ) ).
% less_add_eq_less
thf(fact_997_trans__less__add2,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ M @ J ) ) ) ).
% trans_less_add2
thf(fact_998_trans__less__add1,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ J @ M ) ) ) ).
% trans_less_add1
thf(fact_999_add__less__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% add_less_mono1
thf(fact_1000_not__add__less2,axiom,
! [J: nat,I: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J @ I ) @ I ) ).
% not_add_less2
thf(fact_1001_not__add__less1,axiom,
! [I: nat,J: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ I ) ).
% not_add_less1
thf(fact_1002_add__less__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ K @ L )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_less_mono
thf(fact_1003_add__lessD1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ K )
=> ( ord_less_nat @ I @ K ) ) ).
% add_lessD1
thf(fact_1004_add__less__imp__less__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
=> ( ord_less_nat @ A @ B ) ) ).
% add_less_imp_less_right
thf(fact_1005_add__less__imp__less__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
=> ( ord_less_nat @ A @ B ) ) ).
% add_less_imp_less_left
thf(fact_1006_add__strict__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).
% add_strict_right_mono
thf(fact_1007_add__strict__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) ) ) ).
% add_strict_left_mono
thf(fact_1008_add__strict__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_strict_mono
thf(fact_1009_add__strict__mono,axiom,
! [A: extended_enat,B: extended_enat,C: extended_enat,D: extended_enat] :
( ( ord_le72135733267957522d_enat @ A @ B )
=> ( ( ord_le72135733267957522d_enat @ C @ D )
=> ( ord_le72135733267957522d_enat @ ( plus_p3455044024723400733d_enat @ A @ C ) @ ( plus_p3455044024723400733d_enat @ B @ D ) ) ) ) ).
% add_strict_mono
thf(fact_1010_add__mono__thms__linordered__field_I1_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I @ J )
& ( K = L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_1011_add__mono__thms__linordered__field_I2_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( I = J )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_1012_add__mono__thms__linordered__field_I5_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I @ J )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_1013_verit__comp__simplify1_I3_J,axiom,
! [B5: nat,A5: nat] :
( ( ~ ( ord_less_eq_nat @ B5 @ A5 ) )
= ( ord_less_nat @ A5 @ B5 ) ) ).
% verit_comp_simplify1(3)
thf(fact_1014_verit__comp__simplify1_I3_J,axiom,
! [B5: num,A5: num] :
( ( ~ ( ord_less_eq_num @ B5 @ A5 ) )
= ( ord_less_num @ A5 @ B5 ) ) ).
% verit_comp_simplify1(3)
thf(fact_1015_verit__comp__simplify1_I3_J,axiom,
! [B5: extended_enat,A5: extended_enat] :
( ( ~ ( ord_le2932123472753598470d_enat @ B5 @ A5 ) )
= ( ord_le72135733267957522d_enat @ A5 @ B5 ) ) ).
% verit_comp_simplify1(3)
thf(fact_1016_leD,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ~ ( ord_less_nat @ X @ Y ) ) ).
% leD
thf(fact_1017_leD,axiom,
! [Y: set_a,X: set_a] :
( ( ord_less_eq_set_a @ Y @ X )
=> ~ ( ord_less_set_a @ X @ Y ) ) ).
% leD
thf(fact_1018_leD,axiom,
! [Y: num,X: num] :
( ( ord_less_eq_num @ Y @ X )
=> ~ ( ord_less_num @ X @ Y ) ) ).
% leD
thf(fact_1019_leD,axiom,
! [Y: extended_enat,X: extended_enat] :
( ( ord_le2932123472753598470d_enat @ Y @ X )
=> ~ ( ord_le72135733267957522d_enat @ X @ Y ) ) ).
% leD
thf(fact_1020_leD,axiom,
! [Y: set_set_a,X: set_set_a] :
( ( ord_le3724670747650509150_set_a @ Y @ X )
=> ~ ( ord_less_set_set_a @ X @ Y ) ) ).
% leD
thf(fact_1021_leI,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_eq_nat @ Y @ X ) ) ).
% leI
thf(fact_1022_leI,axiom,
! [X: num,Y: num] :
( ~ ( ord_less_num @ X @ Y )
=> ( ord_less_eq_num @ Y @ X ) ) ).
% leI
thf(fact_1023_leI,axiom,
! [X: extended_enat,Y: extended_enat] :
( ~ ( ord_le72135733267957522d_enat @ X @ Y )
=> ( ord_le2932123472753598470d_enat @ Y @ X ) ) ).
% leI
thf(fact_1024_nless__le,axiom,
! [A: nat,B: nat] :
( ( ~ ( ord_less_nat @ A @ B ) )
= ( ~ ( ord_less_eq_nat @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_1025_nless__le,axiom,
! [A: set_a,B: set_a] :
( ( ~ ( ord_less_set_a @ A @ B ) )
= ( ~ ( ord_less_eq_set_a @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_1026_nless__le,axiom,
! [A: num,B: num] :
( ( ~ ( ord_less_num @ A @ B ) )
= ( ~ ( ord_less_eq_num @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_1027_nless__le,axiom,
! [A: extended_enat,B: extended_enat] :
( ( ~ ( ord_le72135733267957522d_enat @ A @ B ) )
= ( ~ ( ord_le2932123472753598470d_enat @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_1028_nless__le,axiom,
! [A: set_set_a,B: set_set_a] :
( ( ~ ( ord_less_set_set_a @ A @ B ) )
= ( ~ ( ord_le3724670747650509150_set_a @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_1029_antisym__conv1,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_1030_antisym__conv1,axiom,
! [X: set_a,Y: set_a] :
( ~ ( ord_less_set_a @ X @ Y )
=> ( ( ord_less_eq_set_a @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_1031_antisym__conv1,axiom,
! [X: num,Y: num] :
( ~ ( ord_less_num @ X @ Y )
=> ( ( ord_less_eq_num @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_1032_antisym__conv1,axiom,
! [X: extended_enat,Y: extended_enat] :
( ~ ( ord_le72135733267957522d_enat @ X @ Y )
=> ( ( ord_le2932123472753598470d_enat @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_1033_antisym__conv1,axiom,
! [X: set_set_a,Y: set_set_a] :
( ~ ( ord_less_set_set_a @ X @ Y )
=> ( ( ord_le3724670747650509150_set_a @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_1034_antisym__conv2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ~ ( ord_less_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_1035_antisym__conv2,axiom,
! [X: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X @ Y )
=> ( ( ~ ( ord_less_set_a @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_1036_antisym__conv2,axiom,
! [X: num,Y: num] :
( ( ord_less_eq_num @ X @ Y )
=> ( ( ~ ( ord_less_num @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_1037_antisym__conv2,axiom,
! [X: extended_enat,Y: extended_enat] :
( ( ord_le2932123472753598470d_enat @ X @ Y )
=> ( ( ~ ( ord_le72135733267957522d_enat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_1038_antisym__conv2,axiom,
! [X: set_set_a,Y: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X @ Y )
=> ( ( ~ ( ord_less_set_set_a @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_1039_less__le__not__le,axiom,
( ord_less_nat
= ( ^ [X5: nat,Y6: nat] :
( ( ord_less_eq_nat @ X5 @ Y6 )
& ~ ( ord_less_eq_nat @ Y6 @ X5 ) ) ) ) ).
% less_le_not_le
thf(fact_1040_less__le__not__le,axiom,
( ord_less_set_a
= ( ^ [X5: set_a,Y6: set_a] :
( ( ord_less_eq_set_a @ X5 @ Y6 )
& ~ ( ord_less_eq_set_a @ Y6 @ X5 ) ) ) ) ).
% less_le_not_le
thf(fact_1041_less__le__not__le,axiom,
( ord_less_num
= ( ^ [X5: num,Y6: num] :
( ( ord_less_eq_num @ X5 @ Y6 )
& ~ ( ord_less_eq_num @ Y6 @ X5 ) ) ) ) ).
% less_le_not_le
thf(fact_1042_less__le__not__le,axiom,
( ord_le72135733267957522d_enat
= ( ^ [X5: extended_enat,Y6: extended_enat] :
( ( ord_le2932123472753598470d_enat @ X5 @ Y6 )
& ~ ( ord_le2932123472753598470d_enat @ Y6 @ X5 ) ) ) ) ).
% less_le_not_le
thf(fact_1043_less__le__not__le,axiom,
( ord_less_set_set_a
= ( ^ [X5: set_set_a,Y6: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X5 @ Y6 )
& ~ ( ord_le3724670747650509150_set_a @ Y6 @ X5 ) ) ) ) ).
% less_le_not_le
thf(fact_1044_not__le__imp__less,axiom,
! [Y: nat,X: nat] :
( ~ ( ord_less_eq_nat @ Y @ X )
=> ( ord_less_nat @ X @ Y ) ) ).
% not_le_imp_less
thf(fact_1045_not__le__imp__less,axiom,
! [Y: num,X: num] :
( ~ ( ord_less_eq_num @ Y @ X )
=> ( ord_less_num @ X @ Y ) ) ).
% not_le_imp_less
thf(fact_1046_not__le__imp__less,axiom,
! [Y: extended_enat,X: extended_enat] :
( ~ ( ord_le2932123472753598470d_enat @ Y @ X )
=> ( ord_le72135733267957522d_enat @ X @ Y ) ) ).
% not_le_imp_less
thf(fact_1047_order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [A3: nat,B2: nat] :
( ( ord_less_nat @ A3 @ B2 )
| ( A3 = B2 ) ) ) ) ).
% order.order_iff_strict
thf(fact_1048_order_Oorder__iff__strict,axiom,
( ord_less_eq_set_a
= ( ^ [A3: set_a,B2: set_a] :
( ( ord_less_set_a @ A3 @ B2 )
| ( A3 = B2 ) ) ) ) ).
% order.order_iff_strict
thf(fact_1049_order_Oorder__iff__strict,axiom,
( ord_less_eq_num
= ( ^ [A3: num,B2: num] :
( ( ord_less_num @ A3 @ B2 )
| ( A3 = B2 ) ) ) ) ).
% order.order_iff_strict
thf(fact_1050_order_Oorder__iff__strict,axiom,
( ord_le2932123472753598470d_enat
= ( ^ [A3: extended_enat,B2: extended_enat] :
( ( ord_le72135733267957522d_enat @ A3 @ B2 )
| ( A3 = B2 ) ) ) ) ).
% order.order_iff_strict
thf(fact_1051_order_Oorder__iff__strict,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [A3: set_set_a,B2: set_set_a] :
( ( ord_less_set_set_a @ A3 @ B2 )
| ( A3 = B2 ) ) ) ) ).
% order.order_iff_strict
thf(fact_1052_order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [A3: nat,B2: nat] :
( ( ord_less_eq_nat @ A3 @ B2 )
& ( A3 != B2 ) ) ) ) ).
% order.strict_iff_order
thf(fact_1053_order_Ostrict__iff__order,axiom,
( ord_less_set_a
= ( ^ [A3: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A3 @ B2 )
& ( A3 != B2 ) ) ) ) ).
% order.strict_iff_order
thf(fact_1054_order_Ostrict__iff__order,axiom,
( ord_less_num
= ( ^ [A3: num,B2: num] :
( ( ord_less_eq_num @ A3 @ B2 )
& ( A3 != B2 ) ) ) ) ).
% order.strict_iff_order
thf(fact_1055_order_Ostrict__iff__order,axiom,
( ord_le72135733267957522d_enat
= ( ^ [A3: extended_enat,B2: extended_enat] :
( ( ord_le2932123472753598470d_enat @ A3 @ B2 )
& ( A3 != B2 ) ) ) ) ).
% order.strict_iff_order
thf(fact_1056_order_Ostrict__iff__order,axiom,
( ord_less_set_set_a
= ( ^ [A3: set_set_a,B2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A3 @ B2 )
& ( A3 != B2 ) ) ) ) ).
% order.strict_iff_order
thf(fact_1057_order_Ostrict__trans1,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_1058_order_Ostrict__trans1,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_set_a @ B @ C )
=> ( ord_less_set_a @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_1059_order_Ostrict__trans1,axiom,
! [A: num,B: num,C: num] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ord_less_num @ B @ C )
=> ( ord_less_num @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_1060_order_Ostrict__trans1,axiom,
! [A: extended_enat,B: extended_enat,C: extended_enat] :
( ( ord_le2932123472753598470d_enat @ A @ B )
=> ( ( ord_le72135733267957522d_enat @ B @ C )
=> ( ord_le72135733267957522d_enat @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_1061_order_Ostrict__trans1,axiom,
! [A: set_set_a,B: set_set_a,C: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( ( ord_less_set_set_a @ B @ C )
=> ( ord_less_set_set_a @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_1062_order_Ostrict__trans2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_1063_order_Ostrict__trans2,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( ord_less_set_a @ A @ B )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ord_less_set_a @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_1064_order_Ostrict__trans2,axiom,
! [A: num,B: num,C: num] :
( ( ord_less_num @ A @ B )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ord_less_num @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_1065_order_Ostrict__trans2,axiom,
! [A: extended_enat,B: extended_enat,C: extended_enat] :
( ( ord_le72135733267957522d_enat @ A @ B )
=> ( ( ord_le2932123472753598470d_enat @ B @ C )
=> ( ord_le72135733267957522d_enat @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_1066_order_Ostrict__trans2,axiom,
! [A: set_set_a,B: set_set_a,C: set_set_a] :
( ( ord_less_set_set_a @ A @ B )
=> ( ( ord_le3724670747650509150_set_a @ B @ C )
=> ( ord_less_set_set_a @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_1067_order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [A3: nat,B2: nat] :
( ( ord_less_eq_nat @ A3 @ B2 )
& ~ ( ord_less_eq_nat @ B2 @ A3 ) ) ) ) ).
% order.strict_iff_not
thf(fact_1068_order_Ostrict__iff__not,axiom,
( ord_less_set_a
= ( ^ [A3: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A3 @ B2 )
& ~ ( ord_less_eq_set_a @ B2 @ A3 ) ) ) ) ).
% order.strict_iff_not
thf(fact_1069_order_Ostrict__iff__not,axiom,
( ord_less_num
= ( ^ [A3: num,B2: num] :
( ( ord_less_eq_num @ A3 @ B2 )
& ~ ( ord_less_eq_num @ B2 @ A3 ) ) ) ) ).
% order.strict_iff_not
thf(fact_1070_order_Ostrict__iff__not,axiom,
( ord_le72135733267957522d_enat
= ( ^ [A3: extended_enat,B2: extended_enat] :
( ( ord_le2932123472753598470d_enat @ A3 @ B2 )
& ~ ( ord_le2932123472753598470d_enat @ B2 @ A3 ) ) ) ) ).
% order.strict_iff_not
thf(fact_1071_order_Ostrict__iff__not,axiom,
( ord_less_set_set_a
= ( ^ [A3: set_set_a,B2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A3 @ B2 )
& ~ ( ord_le3724670747650509150_set_a @ B2 @ A3 ) ) ) ) ).
% order.strict_iff_not
thf(fact_1072_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [B2: nat,A3: nat] :
( ( ord_less_nat @ B2 @ A3 )
| ( A3 = B2 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_1073_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_set_a
= ( ^ [B2: set_a,A3: set_a] :
( ( ord_less_set_a @ B2 @ A3 )
| ( A3 = B2 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_1074_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_num
= ( ^ [B2: num,A3: num] :
( ( ord_less_num @ B2 @ A3 )
| ( A3 = B2 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_1075_dual__order_Oorder__iff__strict,axiom,
( ord_le2932123472753598470d_enat
= ( ^ [B2: extended_enat,A3: extended_enat] :
( ( ord_le72135733267957522d_enat @ B2 @ A3 )
| ( A3 = B2 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_1076_dual__order_Oorder__iff__strict,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [B2: set_set_a,A3: set_set_a] :
( ( ord_less_set_set_a @ B2 @ A3 )
| ( A3 = B2 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_1077_dual__order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [B2: nat,A3: nat] :
( ( ord_less_eq_nat @ B2 @ A3 )
& ( A3 != B2 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_1078_dual__order_Ostrict__iff__order,axiom,
( ord_less_set_a
= ( ^ [B2: set_a,A3: set_a] :
( ( ord_less_eq_set_a @ B2 @ A3 )
& ( A3 != B2 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_1079_dual__order_Ostrict__iff__order,axiom,
( ord_less_num
= ( ^ [B2: num,A3: num] :
( ( ord_less_eq_num @ B2 @ A3 )
& ( A3 != B2 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_1080_dual__order_Ostrict__iff__order,axiom,
( ord_le72135733267957522d_enat
= ( ^ [B2: extended_enat,A3: extended_enat] :
( ( ord_le2932123472753598470d_enat @ B2 @ A3 )
& ( A3 != B2 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_1081_dual__order_Ostrict__iff__order,axiom,
( ord_less_set_set_a
= ( ^ [B2: set_set_a,A3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B2 @ A3 )
& ( A3 != B2 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_1082_dual__order_Ostrict__trans1,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_nat @ C @ B )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_1083_dual__order_Ostrict__trans1,axiom,
! [B: set_a,A: set_a,C: set_a] :
( ( ord_less_eq_set_a @ B @ A )
=> ( ( ord_less_set_a @ C @ B )
=> ( ord_less_set_a @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_1084_dual__order_Ostrict__trans1,axiom,
! [B: num,A: num,C: num] :
( ( ord_less_eq_num @ B @ A )
=> ( ( ord_less_num @ C @ B )
=> ( ord_less_num @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_1085_dual__order_Ostrict__trans1,axiom,
! [B: extended_enat,A: extended_enat,C: extended_enat] :
( ( ord_le2932123472753598470d_enat @ B @ A )
=> ( ( ord_le72135733267957522d_enat @ C @ B )
=> ( ord_le72135733267957522d_enat @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_1086_dual__order_Ostrict__trans1,axiom,
! [B: set_set_a,A: set_set_a,C: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B @ A )
=> ( ( ord_less_set_set_a @ C @ B )
=> ( ord_less_set_set_a @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_1087_dual__order_Ostrict__trans2,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_1088_dual__order_Ostrict__trans2,axiom,
! [B: set_a,A: set_a,C: set_a] :
( ( ord_less_set_a @ B @ A )
=> ( ( ord_less_eq_set_a @ C @ B )
=> ( ord_less_set_a @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_1089_dual__order_Ostrict__trans2,axiom,
! [B: num,A: num,C: num] :
( ( ord_less_num @ B @ A )
=> ( ( ord_less_eq_num @ C @ B )
=> ( ord_less_num @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_1090_dual__order_Ostrict__trans2,axiom,
! [B: extended_enat,A: extended_enat,C: extended_enat] :
( ( ord_le72135733267957522d_enat @ B @ A )
=> ( ( ord_le2932123472753598470d_enat @ C @ B )
=> ( ord_le72135733267957522d_enat @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_1091_dual__order_Ostrict__trans2,axiom,
! [B: set_set_a,A: set_set_a,C: set_set_a] :
( ( ord_less_set_set_a @ B @ A )
=> ( ( ord_le3724670747650509150_set_a @ C @ B )
=> ( ord_less_set_set_a @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_1092_dual__order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [B2: nat,A3: nat] :
( ( ord_less_eq_nat @ B2 @ A3 )
& ~ ( ord_less_eq_nat @ A3 @ B2 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_1093_dual__order_Ostrict__iff__not,axiom,
( ord_less_set_a
= ( ^ [B2: set_a,A3: set_a] :
( ( ord_less_eq_set_a @ B2 @ A3 )
& ~ ( ord_less_eq_set_a @ A3 @ B2 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_1094_dual__order_Ostrict__iff__not,axiom,
( ord_less_num
= ( ^ [B2: num,A3: num] :
( ( ord_less_eq_num @ B2 @ A3 )
& ~ ( ord_less_eq_num @ A3 @ B2 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_1095_dual__order_Ostrict__iff__not,axiom,
( ord_le72135733267957522d_enat
= ( ^ [B2: extended_enat,A3: extended_enat] :
( ( ord_le2932123472753598470d_enat @ B2 @ A3 )
& ~ ( ord_le2932123472753598470d_enat @ A3 @ B2 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_1096_dual__order_Ostrict__iff__not,axiom,
( ord_less_set_set_a
= ( ^ [B2: set_set_a,A3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B2 @ A3 )
& ~ ( ord_le3724670747650509150_set_a @ A3 @ B2 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_1097_order_Ostrict__implies__order,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_1098_order_Ostrict__implies__order,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_set_a @ A @ B )
=> ( ord_less_eq_set_a @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_1099_order_Ostrict__implies__order,axiom,
! [A: num,B: num] :
( ( ord_less_num @ A @ B )
=> ( ord_less_eq_num @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_1100_order_Ostrict__implies__order,axiom,
! [A: extended_enat,B: extended_enat] :
( ( ord_le72135733267957522d_enat @ A @ B )
=> ( ord_le2932123472753598470d_enat @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_1101_order_Ostrict__implies__order,axiom,
! [A: set_set_a,B: set_set_a] :
( ( ord_less_set_set_a @ A @ B )
=> ( ord_le3724670747650509150_set_a @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_1102_dual__order_Ostrict__implies__order,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( ord_less_eq_nat @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_1103_dual__order_Ostrict__implies__order,axiom,
! [B: set_a,A: set_a] :
( ( ord_less_set_a @ B @ A )
=> ( ord_less_eq_set_a @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_1104_dual__order_Ostrict__implies__order,axiom,
! [B: num,A: num] :
( ( ord_less_num @ B @ A )
=> ( ord_less_eq_num @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_1105_dual__order_Ostrict__implies__order,axiom,
! [B: extended_enat,A: extended_enat] :
( ( ord_le72135733267957522d_enat @ B @ A )
=> ( ord_le2932123472753598470d_enat @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_1106_dual__order_Ostrict__implies__order,axiom,
! [B: set_set_a,A: set_set_a] :
( ( ord_less_set_set_a @ B @ A )
=> ( ord_le3724670747650509150_set_a @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_1107_order__le__less,axiom,
( ord_less_eq_nat
= ( ^ [X5: nat,Y6: nat] :
( ( ord_less_nat @ X5 @ Y6 )
| ( X5 = Y6 ) ) ) ) ).
% order_le_less
thf(fact_1108_order__le__less,axiom,
( ord_less_eq_set_a
= ( ^ [X5: set_a,Y6: set_a] :
( ( ord_less_set_a @ X5 @ Y6 )
| ( X5 = Y6 ) ) ) ) ).
% order_le_less
thf(fact_1109_order__le__less,axiom,
( ord_less_eq_num
= ( ^ [X5: num,Y6: num] :
( ( ord_less_num @ X5 @ Y6 )
| ( X5 = Y6 ) ) ) ) ).
% order_le_less
thf(fact_1110_order__le__less,axiom,
( ord_le2932123472753598470d_enat
= ( ^ [X5: extended_enat,Y6: extended_enat] :
( ( ord_le72135733267957522d_enat @ X5 @ Y6 )
| ( X5 = Y6 ) ) ) ) ).
% order_le_less
thf(fact_1111_order__le__less,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [X5: set_set_a,Y6: set_set_a] :
( ( ord_less_set_set_a @ X5 @ Y6 )
| ( X5 = Y6 ) ) ) ) ).
% order_le_less
thf(fact_1112_order__less__le,axiom,
( ord_less_nat
= ( ^ [X5: nat,Y6: nat] :
( ( ord_less_eq_nat @ X5 @ Y6 )
& ( X5 != Y6 ) ) ) ) ).
% order_less_le
thf(fact_1113_order__less__le,axiom,
( ord_less_set_a
= ( ^ [X5: set_a,Y6: set_a] :
( ( ord_less_eq_set_a @ X5 @ Y6 )
& ( X5 != Y6 ) ) ) ) ).
% order_less_le
thf(fact_1114_order__less__le,axiom,
( ord_less_num
= ( ^ [X5: num,Y6: num] :
( ( ord_less_eq_num @ X5 @ Y6 )
& ( X5 != Y6 ) ) ) ) ).
% order_less_le
thf(fact_1115_order__less__le,axiom,
( ord_le72135733267957522d_enat
= ( ^ [X5: extended_enat,Y6: extended_enat] :
( ( ord_le2932123472753598470d_enat @ X5 @ Y6 )
& ( X5 != Y6 ) ) ) ) ).
% order_less_le
thf(fact_1116_order__less__le,axiom,
( ord_less_set_set_a
= ( ^ [X5: set_set_a,Y6: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X5 @ Y6 )
& ( X5 != Y6 ) ) ) ) ).
% order_less_le
thf(fact_1117_linorder__not__le,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_eq_nat @ X @ Y ) )
= ( ord_less_nat @ Y @ X ) ) ).
% linorder_not_le
thf(fact_1118_linorder__not__le,axiom,
! [X: num,Y: num] :
( ( ~ ( ord_less_eq_num @ X @ Y ) )
= ( ord_less_num @ Y @ X ) ) ).
% linorder_not_le
thf(fact_1119_linorder__not__le,axiom,
! [X: extended_enat,Y: extended_enat] :
( ( ~ ( ord_le2932123472753598470d_enat @ X @ Y ) )
= ( ord_le72135733267957522d_enat @ Y @ X ) ) ).
% linorder_not_le
thf(fact_1120_linorder__not__less,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X @ Y ) )
= ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_not_less
thf(fact_1121_linorder__not__less,axiom,
! [X: num,Y: num] :
( ( ~ ( ord_less_num @ X @ Y ) )
= ( ord_less_eq_num @ Y @ X ) ) ).
% linorder_not_less
thf(fact_1122_linorder__not__less,axiom,
! [X: extended_enat,Y: extended_enat] :
( ( ~ ( ord_le72135733267957522d_enat @ X @ Y ) )
= ( ord_le2932123472753598470d_enat @ Y @ X ) ) ).
% linorder_not_less
thf(fact_1123_order__less__imp__le,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ord_less_eq_nat @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_1124_order__less__imp__le,axiom,
! [X: set_a,Y: set_a] :
( ( ord_less_set_a @ X @ Y )
=> ( ord_less_eq_set_a @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_1125_order__less__imp__le,axiom,
! [X: num,Y: num] :
( ( ord_less_num @ X @ Y )
=> ( ord_less_eq_num @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_1126_order__less__imp__le,axiom,
! [X: extended_enat,Y: extended_enat] :
( ( ord_le72135733267957522d_enat @ X @ Y )
=> ( ord_le2932123472753598470d_enat @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_1127_order__less__imp__le,axiom,
! [X: set_set_a,Y: set_set_a] :
( ( ord_less_set_set_a @ X @ Y )
=> ( ord_le3724670747650509150_set_a @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_1128_order__le__neq__trans,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( A != B )
=> ( ord_less_nat @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_1129_order__le__neq__trans,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( A != B )
=> ( ord_less_set_a @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_1130_order__le__neq__trans,axiom,
! [A: num,B: num] :
( ( ord_less_eq_num @ A @ B )
=> ( ( A != B )
=> ( ord_less_num @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_1131_order__le__neq__trans,axiom,
! [A: extended_enat,B: extended_enat] :
( ( ord_le2932123472753598470d_enat @ A @ B )
=> ( ( A != B )
=> ( ord_le72135733267957522d_enat @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_1132_order__le__neq__trans,axiom,
! [A: set_set_a,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( ( A != B )
=> ( ord_less_set_set_a @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_1133_order__neq__le__trans,axiom,
! [A: nat,B: nat] :
( ( A != B )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_nat @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_1134_order__neq__le__trans,axiom,
! [A: set_a,B: set_a] :
( ( A != B )
=> ( ( ord_less_eq_set_a @ A @ B )
=> ( ord_less_set_a @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_1135_order__neq__le__trans,axiom,
! [A: num,B: num] :
( ( A != B )
=> ( ( ord_less_eq_num @ A @ B )
=> ( ord_less_num @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_1136_order__neq__le__trans,axiom,
! [A: extended_enat,B: extended_enat] :
( ( A != B )
=> ( ( ord_le2932123472753598470d_enat @ A @ B )
=> ( ord_le72135733267957522d_enat @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_1137_order__neq__le__trans,axiom,
! [A: set_set_a,B: set_set_a] :
( ( A != B )
=> ( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( ord_less_set_set_a @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_1138_order__le__less__trans,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ Z2 )
=> ( ord_less_nat @ X @ Z2 ) ) ) ).
% order_le_less_trans
thf(fact_1139_order__le__less__trans,axiom,
! [X: set_a,Y: set_a,Z2: set_a] :
( ( ord_less_eq_set_a @ X @ Y )
=> ( ( ord_less_set_a @ Y @ Z2 )
=> ( ord_less_set_a @ X @ Z2 ) ) ) ).
% order_le_less_trans
thf(fact_1140_order__le__less__trans,axiom,
! [X: num,Y: num,Z2: num] :
( ( ord_less_eq_num @ X @ Y )
=> ( ( ord_less_num @ Y @ Z2 )
=> ( ord_less_num @ X @ Z2 ) ) ) ).
% order_le_less_trans
thf(fact_1141_order__le__less__trans,axiom,
! [X: extended_enat,Y: extended_enat,Z2: extended_enat] :
( ( ord_le2932123472753598470d_enat @ X @ Y )
=> ( ( ord_le72135733267957522d_enat @ Y @ Z2 )
=> ( ord_le72135733267957522d_enat @ X @ Z2 ) ) ) ).
% order_le_less_trans
thf(fact_1142_order__le__less__trans,axiom,
! [X: set_set_a,Y: set_set_a,Z2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X @ Y )
=> ( ( ord_less_set_set_a @ Y @ Z2 )
=> ( ord_less_set_set_a @ X @ Z2 ) ) ) ).
% order_le_less_trans
thf(fact_1143_order__less__le__trans,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z2 )
=> ( ord_less_nat @ X @ Z2 ) ) ) ).
% order_less_le_trans
thf(fact_1144_order__less__le__trans,axiom,
! [X: set_a,Y: set_a,Z2: set_a] :
( ( ord_less_set_a @ X @ Y )
=> ( ( ord_less_eq_set_a @ Y @ Z2 )
=> ( ord_less_set_a @ X @ Z2 ) ) ) ).
% order_less_le_trans
thf(fact_1145_order__less__le__trans,axiom,
! [X: num,Y: num,Z2: num] :
( ( ord_less_num @ X @ Y )
=> ( ( ord_less_eq_num @ Y @ Z2 )
=> ( ord_less_num @ X @ Z2 ) ) ) ).
% order_less_le_trans
thf(fact_1146_order__less__le__trans,axiom,
! [X: extended_enat,Y: extended_enat,Z2: extended_enat] :
( ( ord_le72135733267957522d_enat @ X @ Y )
=> ( ( ord_le2932123472753598470d_enat @ Y @ Z2 )
=> ( ord_le72135733267957522d_enat @ X @ Z2 ) ) ) ).
% order_less_le_trans
thf(fact_1147_order__less__le__trans,axiom,
! [X: set_set_a,Y: set_set_a,Z2: set_set_a] :
( ( ord_less_set_set_a @ X @ Y )
=> ( ( ord_le3724670747650509150_set_a @ Y @ Z2 )
=> ( ord_less_set_set_a @ X @ Z2 ) ) ) ).
% order_less_le_trans
thf(fact_1148_order__le__less__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X6: nat,Y4: nat] :
( ( ord_less_nat @ X6 @ Y4 )
=> ( ord_less_nat @ ( F @ X6 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_1149_order__le__less__subst1,axiom,
! [A: nat,F: extended_enat > nat,B: extended_enat,C: extended_enat] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_le72135733267957522d_enat @ B @ C )
=> ( ! [X6: extended_enat,Y4: extended_enat] :
( ( ord_le72135733267957522d_enat @ X6 @ Y4 )
=> ( ord_less_nat @ ( F @ X6 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_1150_order__le__less__subst1,axiom,
! [A: nat,F: num > nat,B: num,C: num] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_num @ B @ C )
=> ( ! [X6: num,Y4: num] :
( ( ord_less_num @ X6 @ Y4 )
=> ( ord_less_nat @ ( F @ X6 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_1151_order__le__less__subst1,axiom,
! [A: num,F: nat > num,B: nat,C: nat] :
( ( ord_less_eq_num @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X6: nat,Y4: nat] :
( ( ord_less_nat @ X6 @ Y4 )
=> ( ord_less_num @ ( F @ X6 ) @ ( F @ Y4 ) ) )
=> ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_1152_order__le__less__subst1,axiom,
! [A: num,F: extended_enat > num,B: extended_enat,C: extended_enat] :
( ( ord_less_eq_num @ A @ ( F @ B ) )
=> ( ( ord_le72135733267957522d_enat @ B @ C )
=> ( ! [X6: extended_enat,Y4: extended_enat] :
( ( ord_le72135733267957522d_enat @ X6 @ Y4 )
=> ( ord_less_num @ ( F @ X6 ) @ ( F @ Y4 ) ) )
=> ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_1153_order__le__less__subst1,axiom,
! [A: num,F: num > num,B: num,C: num] :
( ( ord_less_eq_num @ A @ ( F @ B ) )
=> ( ( ord_less_num @ B @ C )
=> ( ! [X6: num,Y4: num] :
( ( ord_less_num @ X6 @ Y4 )
=> ( ord_less_num @ ( F @ X6 ) @ ( F @ Y4 ) ) )
=> ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_1154_order__le__less__subst1,axiom,
! [A: extended_enat,F: nat > extended_enat,B: nat,C: nat] :
( ( ord_le2932123472753598470d_enat @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X6: nat,Y4: nat] :
( ( ord_less_nat @ X6 @ Y4 )
=> ( ord_le72135733267957522d_enat @ ( F @ X6 ) @ ( F @ Y4 ) ) )
=> ( ord_le72135733267957522d_enat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_1155_order__le__less__subst1,axiom,
! [A: extended_enat,F: extended_enat > extended_enat,B: extended_enat,C: extended_enat] :
( ( ord_le2932123472753598470d_enat @ A @ ( F @ B ) )
=> ( ( ord_le72135733267957522d_enat @ B @ C )
=> ( ! [X6: extended_enat,Y4: extended_enat] :
( ( ord_le72135733267957522d_enat @ X6 @ Y4 )
=> ( ord_le72135733267957522d_enat @ ( F @ X6 ) @ ( F @ Y4 ) ) )
=> ( ord_le72135733267957522d_enat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_1156_order__le__less__subst1,axiom,
! [A: extended_enat,F: num > extended_enat,B: num,C: num] :
( ( ord_le2932123472753598470d_enat @ A @ ( F @ B ) )
=> ( ( ord_less_num @ B @ C )
=> ( ! [X6: num,Y4: num] :
( ( ord_less_num @ X6 @ Y4 )
=> ( ord_le72135733267957522d_enat @ ( F @ X6 ) @ ( F @ Y4 ) ) )
=> ( ord_le72135733267957522d_enat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_1157_order__le__less__subst1,axiom,
! [A: set_a,F: nat > set_a,B: nat,C: nat] :
( ( ord_less_eq_set_a @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X6: nat,Y4: nat] :
( ( ord_less_nat @ X6 @ Y4 )
=> ( ord_less_set_a @ ( F @ X6 ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_1158_order__le__less__subst2,axiom,
! [A: set_set_a,B: set_set_a,F: set_set_a > num,C: num] :
( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( ( ord_less_num @ ( F @ B ) @ C )
=> ( ! [X6: set_set_a,Y4: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X6 @ Y4 )
=> ( ord_less_eq_num @ ( F @ X6 ) @ ( F @ Y4 ) ) )
=> ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_1159_order__le__less__subst2,axiom,
! [A: set_set_a,B: set_set_a,F: set_set_a > extended_enat,C: extended_enat] :
( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( ( ord_le72135733267957522d_enat @ ( F @ B ) @ C )
=> ( ! [X6: set_set_a,Y4: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X6 @ Y4 )
=> ( ord_le2932123472753598470d_enat @ ( F @ X6 ) @ ( F @ Y4 ) ) )
=> ( ord_le72135733267957522d_enat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_1160_order__le__less__subst2,axiom,
! [A: set_set_a,B: set_set_a,F: set_set_a > set_set_a,C: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( ( ord_less_set_set_a @ ( F @ B ) @ C )
=> ( ! [X6: set_set_a,Y4: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X6 @ Y4 )
=> ( ord_le3724670747650509150_set_a @ ( F @ X6 ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_1161_infinite__descent0,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ~ ( P @ N2 )
=> ? [M4: nat] :
( ( ord_less_nat @ M4 @ N2 )
& ~ ( P @ M4 ) ) ) )
=> ( P @ N ) ) ) ).
% infinite_descent0
thf(fact_1162_gr__implies__not0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_1163_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_zeroE
thf(fact_1164_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less0
thf(fact_1165_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr0
thf(fact_1166_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr0I
thf(fact_1167_bot__nat__0_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_1168_not__less__less__Suc__eq,axiom,
! [N: nat,M: nat] :
( ~ ( ord_less_nat @ N @ M )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
= ( N = M ) ) ) ).
% not_less_less_Suc_eq
thf(fact_1169_strict__inc__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_nat @ I @ J )
=> ( ! [I2: nat] :
( ( J
= ( suc @ I2 ) )
=> ( P @ I2 ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ( P @ ( suc @ I2 ) )
=> ( P @ I2 ) ) )
=> ( P @ I ) ) ) ) ).
% strict_inc_induct
thf(fact_1170_less__Suc__induct,axiom,
! [I: nat,J: nat,P: nat > nat > $o] :
( ( ord_less_nat @ I @ J )
=> ( ! [I2: nat] : ( P @ I2 @ ( suc @ I2 ) )
=> ( ! [I2: nat,J2: nat,K3: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ( ord_less_nat @ J2 @ K3 )
=> ( ( P @ I2 @ J2 )
=> ( ( P @ J2 @ K3 )
=> ( P @ I2 @ K3 ) ) ) ) )
=> ( P @ I @ J ) ) ) ) ).
% less_Suc_induct
thf(fact_1171_less__trans__Suc,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ J @ K )
=> ( ord_less_nat @ ( suc @ I ) @ K ) ) ) ).
% less_trans_Suc
thf(fact_1172_Suc__less__SucD,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_less_SucD
thf(fact_1173_less__antisym,axiom,
! [N: nat,M: nat] :
( ~ ( ord_less_nat @ N @ M )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
=> ( M = N ) ) ) ).
% less_antisym
thf(fact_1174_Suc__less__eq2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ ( suc @ N ) @ M )
= ( ? [M6: nat] :
( ( M
= ( suc @ M6 ) )
& ( ord_less_nat @ N @ M6 ) ) ) ) ).
% Suc_less_eq2
thf(fact_1175_All__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( suc @ N ) )
=> ( P @ I3 ) ) )
= ( ( P @ N )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ N )
=> ( P @ I3 ) ) ) ) ).
% All_less_Suc
thf(fact_1176_not__less__eq,axiom,
! [M: nat,N: nat] :
( ( ~ ( ord_less_nat @ M @ N ) )
= ( ord_less_nat @ N @ ( suc @ M ) ) ) ).
% not_less_eq
thf(fact_1177_less__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
= ( ( ord_less_nat @ M @ N )
| ( M = N ) ) ) ).
% less_Suc_eq
thf(fact_1178_Ex__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I3: nat] :
( ( ord_less_nat @ I3 @ ( suc @ N ) )
& ( P @ I3 ) ) )
= ( ( P @ N )
| ? [I3: nat] :
( ( ord_less_nat @ I3 @ N )
& ( P @ I3 ) ) ) ) ).
% Ex_less_Suc
thf(fact_1179_less__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
% less_SucI
thf(fact_1180_less__SucE,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
=> ( ~ ( ord_less_nat @ M @ N )
=> ( M = N ) ) ) ).
% less_SucE
thf(fact_1181_Suc__lessI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( ( suc @ M )
!= N )
=> ( ord_less_nat @ ( suc @ M ) @ N ) ) ) ).
% Suc_lessI
thf(fact_1182_Suc__lessE,axiom,
! [I: nat,K: nat] :
( ( ord_less_nat @ ( suc @ I ) @ K )
=> ~ ! [J2: nat] :
( ( ord_less_nat @ I @ J2 )
=> ( K
!= ( suc @ J2 ) ) ) ) ).
% Suc_lessE
thf(fact_1183_Suc__lessD,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ N )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_lessD
thf(fact_1184_Nat_OlessE,axiom,
! [I: nat,K: nat] :
( ( ord_less_nat @ I @ K )
=> ( ( K
!= ( suc @ I ) )
=> ~ ! [J2: nat] :
( ( ord_less_nat @ I @ J2 )
=> ( K
!= ( suc @ J2 ) ) ) ) ) ).
% Nat.lessE
thf(fact_1185_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M5: nat,N4: nat] :
( ( ord_less_eq_nat @ M5 @ N4 )
& ( M5 != N4 ) ) ) ) ).
% nat_less_le
thf(fact_1186_less__imp__le__nat,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% less_imp_le_nat
thf(fact_1187_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M5: nat,N4: nat] :
( ( ord_less_nat @ M5 @ N4 )
| ( M5 = N4 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_1188_less__or__eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( ( ord_less_nat @ M @ N )
| ( M = N ) )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% less_or_eq_imp_le
thf(fact_1189_le__neq__implies__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( M != N )
=> ( ord_less_nat @ M @ N ) ) ) ).
% le_neq_implies_less
thf(fact_1190_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I: nat,J: nat] :
( ! [I2: nat,J2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ord_less_nat @ ( F @ I2 ) @ ( F @ J2 ) ) )
=> ( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_1191_less__imp__diff__less,axiom,
! [J: nat,K: nat,N: nat] :
( ( ord_less_nat @ J @ K )
=> ( ord_less_nat @ ( minus_minus_nat @ J @ N ) @ K ) ) ).
% less_imp_diff_less
thf(fact_1192_diff__less__mono2,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( ord_less_nat @ M @ L )
=> ( ord_less_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ) ).
% diff_less_mono2
thf(fact_1193_linorder__neqE__nat,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neqE_nat
thf(fact_1194_infinite__descent,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ~ ( P @ N2 )
=> ? [M4: nat] :
( ( ord_less_nat @ M4 @ N2 )
& ~ ( P @ M4 ) ) )
=> ( P @ N ) ) ).
% infinite_descent
thf(fact_1195_nat__less__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ! [M4: nat] :
( ( ord_less_nat @ M4 @ N2 )
=> ( P @ M4 ) )
=> ( P @ N2 ) )
=> ( P @ N ) ) ).
% nat_less_induct
thf(fact_1196_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_1197_less__not__refl3,axiom,
! [S: nat,T: nat] :
( ( ord_less_nat @ S @ T )
=> ( S != T ) ) ).
% less_not_refl3
thf(fact_1198_less__not__refl2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( M != N ) ) ).
% less_not_refl2
thf(fact_1199_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_not_refl
thf(fact_1200_nat__neq__iff,axiom,
! [M: nat,N: nat] :
( ( M != N )
= ( ( ord_less_nat @ M @ N )
| ( ord_less_nat @ N @ M ) ) ) ).
% nat_neq_iff
thf(fact_1201_Ex__less__Suc2,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I3: nat] :
( ( ord_less_nat @ I3 @ ( suc @ N ) )
& ( P @ I3 ) ) )
= ( ( P @ zero_zero_nat )
| ? [I3: nat] :
( ( ord_less_nat @ I3 @ N )
& ( P @ ( suc @ I3 ) ) ) ) ) ).
% Ex_less_Suc2
thf(fact_1202_gr0__conv__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( ? [M5: nat] :
( N
= ( suc @ M5 ) ) ) ) ).
% gr0_conv_Suc
thf(fact_1203_All__less__Suc2,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( suc @ N ) )
=> ( P @ I3 ) ) )
= ( ( P @ zero_zero_nat )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ N )
=> ( P @ ( suc @ I3 ) ) ) ) ) ).
% All_less_Suc2
thf(fact_1204_gr0__implies__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ? [M2: nat] :
( N
= ( suc @ M2 ) ) ) ).
% gr0_implies_Suc
thf(fact_1205_less__Suc__eq__0__disj,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
= ( ( M = zero_zero_nat )
| ? [J3: nat] :
( ( M
= ( suc @ J3 ) )
& ( ord_less_nat @ J3 @ N ) ) ) ) ).
% less_Suc_eq_0_disj
thf(fact_1206_ex__least__nat__le,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K3: nat] :
( ( ord_less_eq_nat @ K3 @ N )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ K3 )
=> ~ ( P @ I4 ) )
& ( P @ K3 ) ) ) ) ).
% ex_least_nat_le
thf(fact_1207_le__imp__less__Suc,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
% le_imp_less_Suc
thf(fact_1208_less__eq__Suc__le,axiom,
( ord_less_nat
= ( ^ [N4: nat] : ( ord_less_eq_nat @ ( suc @ N4 ) ) ) ) ).
% less_eq_Suc_le
thf(fact_1209_less__Suc__eq__le,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% less_Suc_eq_le
thf(fact_1210_le__less__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
= ( N = M ) ) ) ).
% le_less_Suc_eq
thf(fact_1211_Suc__le__lessD,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_le_lessD
thf(fact_1212_inc__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( P @ J )
=> ( ! [N2: nat] :
( ( ord_less_eq_nat @ I @ N2 )
=> ( ( ord_less_nat @ N2 @ J )
=> ( ( P @ ( suc @ N2 ) )
=> ( P @ N2 ) ) ) )
=> ( P @ I ) ) ) ) ).
% inc_induct
thf(fact_1213_dec__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( P @ I )
=> ( ! [N2: nat] :
( ( ord_less_eq_nat @ I @ N2 )
=> ( ( ord_less_nat @ N2 @ J )
=> ( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) ) ) )
=> ( P @ J ) ) ) ) ).
% dec_induct
thf(fact_1214_Suc__le__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
= ( ord_less_nat @ M @ N ) ) ).
% Suc_le_eq
thf(fact_1215_Suc__leI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_eq_nat @ ( suc @ M ) @ N ) ) ).
% Suc_leI
thf(fact_1216_less__imp__add__positive,axiom,
! [I: nat,J: nat] :
( ( ord_less_nat @ I @ J )
=> ? [K3: nat] :
( ( ord_less_nat @ zero_zero_nat @ K3 )
& ( ( plus_plus_nat @ I @ K3 )
= J ) ) ) ).
% less_imp_add_positive
thf(fact_1217_less__natE,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ~ ! [Q: nat] :
( N
!= ( suc @ ( plus_plus_nat @ M @ Q ) ) ) ) ).
% less_natE
thf(fact_1218_less__add__Suc1,axiom,
! [I: nat,M: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ I @ M ) ) ) ).
% less_add_Suc1
thf(fact_1219_less__add__Suc2,axiom,
! [I: nat,M: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ M @ I ) ) ) ).
% less_add_Suc2
thf(fact_1220_less__iff__Suc__add,axiom,
( ord_less_nat
= ( ^ [M5: nat,N4: nat] :
? [K2: nat] :
( N4
= ( suc @ ( plus_plus_nat @ M5 @ K2 ) ) ) ) ) ).
% less_iff_Suc_add
thf(fact_1221_less__imp__Suc__add,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ? [K3: nat] :
( N
= ( suc @ ( plus_plus_nat @ M @ K3 ) ) ) ) ).
% less_imp_Suc_add
thf(fact_1222_diff__less,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ M ) ) ) ).
% diff_less
thf(fact_1223_mono__nat__linear__lb,axiom,
! [F: nat > nat,M: nat,K: nat] :
( ! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( ord_less_nat @ ( F @ M2 ) @ ( F @ N2 ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M ) @ K ) @ ( F @ ( plus_plus_nat @ M @ K ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_1224_Suc__diff__Suc,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( ( suc @ ( minus_minus_nat @ M @ ( suc @ N ) ) )
= ( minus_minus_nat @ M @ N ) ) ) ).
% Suc_diff_Suc
thf(fact_1225_diff__less__Suc,axiom,
! [M: nat,N: nat] : ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ ( suc @ M ) ) ).
% diff_less_Suc
thf(fact_1226_diff__less__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ A )
=> ( ord_less_nat @ ( minus_minus_nat @ A @ C ) @ ( minus_minus_nat @ B @ C ) ) ) ) ).
% diff_less_mono
thf(fact_1227_less__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_nat @ M @ N ) ) ) ) ).
% less_diff_iff
thf(fact_1228_add__diff__inverse__nat,axiom,
! [M: nat,N: nat] :
( ~ ( ord_less_nat @ M @ N )
=> ( ( plus_plus_nat @ N @ ( minus_minus_nat @ M @ N ) )
= M ) ) ).
% add_diff_inverse_nat
thf(fact_1229_less__diff__conv,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ).
% less_diff_conv
thf(fact_1230_ex__least__nat__less,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K3: nat] :
( ( ord_less_nat @ K3 @ N )
& ! [I4: nat] :
( ( ord_less_eq_nat @ I4 @ K3 )
=> ~ ( P @ I4 ) )
& ( P @ ( suc @ K3 ) ) ) ) ) ).
% ex_least_nat_less
thf(fact_1231_diff__Suc__less,axiom,
! [N: nat,I: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_nat @ ( minus_minus_nat @ N @ ( suc @ I ) ) @ N ) ) ).
% diff_Suc_less
thf(fact_1232_nat__induct__non__zero,axiom,
! [N: nat,P: nat > $o] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( P @ one_one_nat )
=> ( ! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct_non_zero
thf(fact_1233_nat__diff__split__asm,axiom,
! [P: nat > $o,A: nat,B: nat] :
( ( P @ ( minus_minus_nat @ A @ B ) )
= ( ~ ( ( ( ord_less_nat @ A @ B )
& ~ ( P @ zero_zero_nat ) )
| ? [D2: nat] :
( ( A
= ( plus_plus_nat @ B @ D2 ) )
& ~ ( P @ D2 ) ) ) ) ) ).
% nat_diff_split_asm
thf(fact_1234_nat__diff__split,axiom,
! [P: nat > $o,A: nat,B: nat] :
( ( P @ ( minus_minus_nat @ A @ B ) )
= ( ( ( ord_less_nat @ A @ B )
=> ( P @ zero_zero_nat ) )
& ! [D2: nat] :
( ( A
= ( plus_plus_nat @ B @ D2 ) )
=> ( P @ D2 ) ) ) ) ).
% nat_diff_split
thf(fact_1235_less__diff__conv2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( ord_less_nat @ ( minus_minus_nat @ J @ K ) @ I )
= ( ord_less_nat @ J @ ( plus_plus_nat @ I @ K ) ) ) ) ).
% less_diff_conv2
thf(fact_1236_Suc__pred_H,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( N
= ( suc @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% Suc_pred'
thf(fact_1237_Suc__diff__eq__diff__pred,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( minus_minus_nat @ ( suc @ M ) @ N )
= ( minus_minus_nat @ M @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% Suc_diff_eq_diff_pred
thf(fact_1238_less__2__cases__iff,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( ( N = zero_zero_nat )
| ( N
= ( suc @ zero_zero_nat ) ) ) ) ).
% less_2_cases_iff
thf(fact_1239_less__2__cases,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
=> ( ( N = zero_zero_nat )
| ( N
= ( suc @ zero_zero_nat ) ) ) ) ).
% less_2_cases
thf(fact_1240_walk__edges__index,axiom,
! [I: nat,W2: list_a] :
( ( ord_less_eq_nat @ zero_zero_nat @ I )
=> ( ( ord_less_nat @ I @ ( undire8849074589633906640ngth_a @ W2 ) )
=> ( ( undire6133010728901294956walk_a @ vertices @ edges @ W2 )
=> ( member_set_a @ ( nth_set_a @ ( undire7337870655677353998dges_a @ W2 ) @ I ) @ edges ) ) ) ) ).
% walk_edges_index
thf(fact_1241_is__walk__take,axiom,
! [W2: list_a,N: nat] :
( ( undire6133010728901294956walk_a @ vertices @ edges @ W2 )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_nat @ N @ ( size_size_list_a @ W2 ) )
=> ( undire6133010728901294956walk_a @ vertices @ edges @ ( take_a @ N @ W2 ) ) ) ) ) ).
% is_walk_take
thf(fact_1242_pos2,axiom,
ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ).
% pos2
thf(fact_1243_i0__less,axiom,
! [N: extended_enat] :
( ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ N )
= ( N != zero_z5237406670263579293d_enat ) ) ).
% i0_less
thf(fact_1244_semiring__norm_I78_J,axiom,
! [M: num,N: num] :
( ( ord_less_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
= ( ord_less_num @ M @ N ) ) ).
% semiring_norm(78)
thf(fact_1245_semiring__norm_I75_J,axiom,
! [M: num] :
~ ( ord_less_num @ M @ one ) ).
% semiring_norm(75)
thf(fact_1246_semiring__norm_I80_J,axiom,
! [M: num,N: num] :
( ( ord_less_num @ ( bit1 @ M ) @ ( bit1 @ N ) )
= ( ord_less_num @ M @ N ) ) ).
% semiring_norm(80)
thf(fact_1247_semiring__norm_I76_J,axiom,
! [N: num] : ( ord_less_num @ one @ ( bit0 @ N ) ) ).
% semiring_norm(76)
thf(fact_1248_semiring__norm_I81_J,axiom,
! [M: num,N: num] :
( ( ord_less_num @ ( bit1 @ M ) @ ( bit0 @ N ) )
= ( ord_less_num @ M @ N ) ) ).
% semiring_norm(81)
thf(fact_1249_semiring__norm_I77_J,axiom,
! [N: num] : ( ord_less_num @ one @ ( bit1 @ N ) ) ).
% semiring_norm(77)
thf(fact_1250_enat__ord__number_I2_J,axiom,
! [M: num,N: num] :
( ( ord_le72135733267957522d_enat @ ( numera1916890842035813515d_enat @ M ) @ ( numera1916890842035813515d_enat @ N ) )
= ( ord_less_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) ) ) ).
% enat_ord_number(2)
thf(fact_1251_semiring__norm_I74_J,axiom,
! [M: num,N: num] :
( ( ord_less_eq_num @ ( bit1 @ M ) @ ( bit0 @ N ) )
= ( ord_less_num @ M @ N ) ) ).
% semiring_norm(74)
thf(fact_1252_semiring__norm_I79_J,axiom,
! [M: num,N: num] :
( ( ord_less_num @ ( bit0 @ M ) @ ( bit1 @ N ) )
= ( ord_less_eq_num @ M @ N ) ) ).
% semiring_norm(79)
thf(fact_1253_not__iless0,axiom,
! [N: extended_enat] :
~ ( ord_le72135733267957522d_enat @ N @ zero_z5237406670263579293d_enat ) ).
% not_iless0
thf(fact_1254_enat__less__induct,axiom,
! [P: extended_enat > $o,N: extended_enat] :
( ! [N2: extended_enat] :
( ! [M4: extended_enat] :
( ( ord_le72135733267957522d_enat @ M4 @ N2 )
=> ( P @ M4 ) )
=> ( P @ N2 ) )
=> ( P @ N ) ) ).
% enat_less_induct
thf(fact_1255_is__walk__drop,axiom,
! [W2: list_a,N: nat] :
( ( undire6133010728901294956walk_a @ vertices @ edges @ W2 )
=> ( ( ord_less_nat @ N @ ( size_size_list_a @ W2 ) )
=> ( undire6133010728901294956walk_a @ vertices @ edges @ ( drop_a @ N @ W2 ) ) ) ) ).
% is_walk_drop
thf(fact_1256_incident__loops__simp_I2_J,axiom,
! [V2: a] :
( ~ ( undire3617971648856834880loop_a @ edges @ V2 )
=> ( ( undire4753905205749729249oops_a @ edges @ V2 )
= bot_bot_set_set_a ) ) ).
% incident_loops_simp(2)
thf(fact_1257_incident__edges__empty,axiom,
! [V2: a] :
( ~ ( member_a @ V2 @ vertices )
=> ( ( undire3231912044278729248dges_a @ edges @ V2 )
= bot_bot_set_set_a ) ) ).
% incident_edges_empty
thf(fact_1258_empty__not__edge,axiom,
~ ( member_set_a @ bot_bot_set_a @ edges ) ).
% empty_not_edge
thf(fact_1259_degree__no__loops,axiom,
! [V2: a] :
( ~ ( undire3617971648856834880loop_a @ edges @ V2 )
=> ( ( undire8867928226783802224gree_a @ edges @ V2 )
= ( finite_card_set_a @ ( undire3231912044278729248dges_a @ edges @ V2 ) ) ) ) ).
% degree_no_loops
thf(fact_1260_bot__enat__def,axiom,
bot_bo4199563552545308370d_enat = zero_z5237406670263579293d_enat ).
% bot_enat_def
thf(fact_1261_bot__nat__def,axiom,
bot_bot_nat = zero_zero_nat ).
% bot_nat_def
thf(fact_1262_card__incident__sedges__neighborhood,axiom,
! [V2: a] :
( ( finite_card_set_a @ ( undire3231912044278729248dges_a @ edges @ V2 ) )
= ( finite_card_a @ ( undire8504279938402040014hood_a @ vertices @ edges @ V2 ) ) ) ).
% card_incident_sedges_neighborhood
thf(fact_1263_iso__vertex__empty__neighborhood,axiom,
! [V2: a] :
( ( undire8931668460104145173rtex_a @ vertices @ edges @ V2 )
=> ( ( undire8504279938402040014hood_a @ vertices @ edges @ V2 )
= bot_bot_set_a ) ) ).
% iso_vertex_empty_neighborhood
thf(fact_1264_is__walk__index,axiom,
! [I: nat,W2: list_a] :
( ( ord_less_eq_nat @ zero_zero_nat @ I )
=> ( ( ord_less_nat @ ( suc @ I ) @ ( size_size_list_a @ W2 ) )
=> ( ( undire6133010728901294956walk_a @ vertices @ edges @ W2 )
=> ( member_set_a @ ( insert_a @ ( nth_a @ W2 @ I ) @ ( insert_a @ ( nth_a @ W2 @ ( plus_plus_nat @ I @ one_one_nat ) ) @ bot_bot_set_a ) ) @ edges ) ) ) ) ).
% is_walk_index
thf(fact_1265_degree0__neighborhood__empt__iff,axiom,
! [V2: a] :
( ( finite_finite_set_a @ edges )
=> ( ( ( undire8867928226783802224gree_a @ edges @ V2 )
= zero_zero_nat )
= ( ( undire8504279938402040014hood_a @ vertices @ edges @ V2 )
= bot_bot_set_a ) ) ) ).
% degree0_neighborhood_empt_iff
thf(fact_1266_finite__incident__edges,axiom,
! [V2: a] :
( ( finite_finite_set_a @ edges )
=> ( finite_finite_set_a @ ( undire3231912044278729248dges_a @ edges @ V2 ) ) ) ).
% finite_incident_edges
% Helper facts (3)
thf(help_If_3_1_If_001t__Nat__Onat_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ( if_nat @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ( if_nat @ $true @ X @ Y )
= X ) ).
% Conjectures (1)
thf(conj_0,conjecture,
ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) @ ( size_size_list_a @ xs ) ).
%------------------------------------------------------------------------------