TPTP Problem File: SLH0241^1.p
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%------------------------------------------------------------------------------
% File : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain : Archive of Formal Proofs
% Problem :
% Version : Especial.
% English :
% Refs : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source : [Des23]
% Names : Undirected_Graph_Theory/0019_Connectivity/prob_00142_007041__13379444_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1420 ( 478 unt; 142 typ; 0 def)
% Number of atoms : 3832 (1347 equ; 0 cnn)
% Maximal formula atoms : 12 ( 2 avg)
% Number of connectives : 12093 ( 415 ~; 71 |; 299 &;9627 @)
% ( 0 <=>;1681 =>; 0 <=; 0 <~>)
% Maximal formula depth : 29 ( 7 avg)
% Number of types : 12 ( 11 usr)
% Number of type conns : 446 ( 446 >; 0 *; 0 +; 0 <<)
% Number of symbols : 132 ( 131 usr; 20 con; 0-5 aty)
% Number of variables : 3723 ( 166 ^;3400 !; 157 ?;3723 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 14:34:18.919
%------------------------------------------------------------------------------
% Could-be-implicit typings (11)
thf(ty_n_t__List__Olist_It__List__Olist_It__Set__Oset_Itf__a_J_J_J,type,
list_list_set_a: $tType ).
thf(ty_n_t__List__Olist_It__Set__Oset_It__Set__Oset_Itf__a_J_J_J,type,
list_set_set_a: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Set__Oset_Itf__a_J_J_J,type,
set_set_set_a: $tType ).
thf(ty_n_t__List__Olist_It__List__Olist_Itf__a_J_J,type,
list_list_a: $tType ).
thf(ty_n_t__List__Olist_It__Set__Oset_Itf__a_J_J,type,
list_set_a: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
set_set_a: $tType ).
thf(ty_n_t__List__Olist_Itf__a_J,type,
list_a: $tType ).
thf(ty_n_t__Set__Oset_Itf__a_J,type,
set_a: $tType ).
thf(ty_n_t__Real__Oreal,type,
real: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
thf(ty_n_tf__a,type,
a: $tType ).
% Explicit typings (131)
thf(sy_c_Connectivity_Oulgraph_Oconnecting__path_001t__Set__Oset_Itf__a_J,type,
connec7350987497872064604_set_a: set_set_a > set_set_set_a > set_a > set_a > list_set_a > $o ).
thf(sy_c_Connectivity_Oulgraph_Oconnecting__path_001tf__a,type,
connecting_path_a: set_a > set_set_a > a > a > list_a > $o ).
thf(sy_c_Connectivity_Oulgraph_Oconnecting__path__str_001tf__a,type,
connec3015921205769380621_str_a: set_a > set_set_a > a > a > list_a > $o ).
thf(sy_c_Connectivity_Oulgraph_Oconnecting__walk_001t__Set__Oset_Itf__a_J,type,
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thf(sy_c_Connectivity_Oulgraph_Oconnecting__walk_001tf__a,type,
connecting_walk_a: set_a > set_set_a > a > a > list_a > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_Itf__a_J,type,
finite_finite_set_a: set_set_a > $o ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
minus_minus_nat: nat > nat > nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Real__Oreal,type,
minus_minus_real: real > real > real ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
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thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_Itf__a_J,type,
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thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat,type,
one_one_nat: nat ).
thf(sy_c_Groups_Oone__class_Oone_001t__Real__Oreal,type,
one_one_real: real ).
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__Real__Oreal,type,
plus_plus_real: real > real > real ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
zero_zero_nat: nat ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Real__Oreal,type,
zero_zero_real: real ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
sup_sup_set_set_a: set_set_a > set_set_a > set_set_a ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_Itf__a_J,type,
sup_sup_set_a: set_a > set_a > set_a ).
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_It__Set__Oset_Itf__a_J_J,type,
distinct_set_set_a: list_set_set_a > $o ).
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__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_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_List_Ounion_001t__Set__Oset_Itf__a_J,type,
union_set_a: list_set_a > list_set_a > list_set_a ).
thf(sy_c_List_Ounion_001tf__a,type,
union_a: list_a > 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_Orderings_Obot__class_Obot_001_062_It__Set__Oset_Itf__a_J_M_Eo_J,type,
bot_bot_set_a_o: set_a > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_Itf__a_M_Eo_J,type,
bot_bot_a_o: a > $o ).
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__Nat__Onat,type,
ord_less_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Real__Oreal,type,
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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__Nat__Onat,type,
ord_less_eq_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Real__Oreal,type,
ord_less_eq_real: real > real > $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,
ord_le3724670747650509150_set_a: set_set_a > set_set_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_Itf__a_J,type,
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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,
collect_a: ( a > $o ) > set_a ).
thf(sy_c_Set_Oinsert_001t__Set__Oset_Itf__a_J,type,
insert_set_a: set_a > set_set_a > set_set_a ).
thf(sy_c_Set_Oinsert_001tf__a,type,
insert_a: a > set_a > set_a ).
thf(sy_c_Set_Othe__elem_001t__Set__Oset_Itf__a_J,type,
the_elem_set_a: set_set_a > set_a ).
thf(sy_c_Set_Othe__elem_001tf__a,type,
the_elem_a: set_a > a ).
thf(sy_c_Undirected__Graph__Basics_Oall__edges_001t__Set__Oset_Itf__a_J,type,
undire8247866692393712962_set_a: set_set_a > set_set_set_a ).
thf(sy_c_Undirected__Graph__Basics_Oall__edges_001tf__a,type,
undire2918257014606996450dges_a: set_a > set_set_a ).
thf(sy_c_Undirected__Graph__Basics_Ograph__system_001t__Set__Oset_Itf__a_J,type,
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thf(sy_c_Undirected__Graph__Basics_Ograph__system_001tf__a,type,
undire2554140024507503526stem_a: set_a > set_set_a > $o ).
thf(sy_c_Undirected__Graph__Basics_Ograph__system_Oedge__adj_001t__Set__Oset_Itf__a_J,type,
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thf(sy_c_Undirected__Graph__Basics_Ograph__system_Oedge__adj_001tf__a,type,
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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,
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thf(sy_c_Undirected__Graph__Basics_Ograph__system_Oincident__edges_001tf__a,type,
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thf(sy_c_Undirected__Graph__Basics_Ograph__system_Oinduced__edges_001t__Set__Oset_Itf__a_J,type,
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thf(sy_c_Undirected__Graph__Basics_Ograph__system_Oinduced__edges_001tf__a,type,
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thf(sy_c_Undirected__Graph__Basics_Osubgraph_001t__Set__Oset_Itf__a_J,type,
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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,
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thf(sy_c_Undirected__Graph__Basics_Oulgraph_Odegree_001t__Set__Oset_Itf__a_J,type,
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thf(sy_c_Undirected__Graph__Basics_Oulgraph_Odegree_001tf__a,type,
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thf(sy_c_Undirected__Graph__Basics_Oulgraph_Oedge__density_001t__Set__Oset_Itf__a_J,type,
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thf(sy_c_Undirected__Graph__Basics_Oulgraph_Oedge__density_001tf__a,type,
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thf(sy_c_Undirected__Graph__Basics_Oulgraph_Ohas__loop_001tf__a,type,
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thf(sy_c_Undirected__Graph__Basics_Oulgraph_Oincident__loops_001tf__a,type,
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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_thesis,type,
thesis: $o ).
thf(sy_v_u,type,
u: a ).
thf(sy_v_v,type,
v: a ).
thf(sy_v_vertices,type,
vertices: set_a ).
thf(sy_v_xs,type,
xs: list_a ).
thf(sy_v_ys,type,
ys: list_a ).
thf(sy_v_z,type,
z: a ).
% Relevant facts (1277)
thf(fact_0_connecting__path__walk,axiom,
! [U: a,V: a,Xs: list_a] :
( ( connecting_path_a @ vertices @ edges @ U @ V @ Xs )
=> ( connecting_walk_a @ vertices @ edges @ U @ V @ Xs ) ) ).
% connecting_path_walk
thf(fact_1_connecting__walk__split,axiom,
! [U: a,V: a,Xs: list_a,Z: a,Ys: list_a] :
( ( connecting_walk_a @ vertices @ edges @ U @ V @ Xs )
=> ( ( connecting_walk_a @ vertices @ edges @ V @ Z @ Ys )
=> ( connecting_walk_a @ vertices @ edges @ U @ Z @ ( append_a @ Xs @ ( tl_a @ Ys ) ) ) ) ) ).
% connecting_walk_split
thf(fact_2_assms_I2_J,axiom,
connecting_path_a @ vertices @ edges @ v @ z @ ys ).
% assms(2)
thf(fact_3_assms_I1_J,axiom,
connecting_path_a @ vertices @ edges @ u @ v @ xs ).
% assms(1)
thf(fact_4_connecting__walk__wf,axiom,
! [U: a,V: a,Xs: list_a] :
( ( connecting_walk_a @ vertices @ edges @ U @ V @ Xs )
=> ( ( member_a @ U @ vertices )
& ( member_a @ V @ vertices ) ) ) ).
% connecting_walk_wf
thf(fact_5_ulgraph__axioms,axiom,
undire7251896706689453996raph_a @ vertices @ edges ).
% ulgraph_axioms
thf(fact_6_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_7_edge__adjacent__alt__def,axiom,
! [E1: set_a,E2: set_a] :
( ( member_set_a @ E1 @ edges )
=> ( ( member_set_a @ E2 @ edges )
=> ( ? [X: a] :
( ( member_a @ X @ vertices )
& ( member_a @ X @ E1 )
& ( member_a @ X @ E2 ) )
=> ( undire4022703626023482010_adj_a @ edges @ E1 @ E2 ) ) ) ) ).
% edge_adjacent_alt_def
thf(fact_8_ulgraph_Oconnecting__walk_Ocong,axiom,
connecting_walk_a = connecting_walk_a ).
% ulgraph.connecting_walk.cong
thf(fact_9_has__loop__in__verts,axiom,
! [V: a] :
( ( undire3617971648856834880loop_a @ edges @ V )
=> ( member_a @ V @ vertices ) ) ).
% has_loop_in_verts
thf(fact_10_connecting__path__str__gen,axiom,
! [U: a,V: a,Xs: list_a] :
( ( connec3015921205769380621_str_a @ vertices @ edges @ U @ V @ Xs )
=> ( connecting_path_a @ vertices @ edges @ U @ V @ Xs ) ) ).
% connecting_path_str_gen
thf(fact_11_incident__edge__in__wf,axiom,
! [E: set_a,V: a] :
( ( member_set_a @ E @ edges )
=> ( ( undire1521409233611534436dent_a @ V @ E )
=> ( member_a @ V @ vertices ) ) ) ).
% incident_edge_in_wf
thf(fact_12_vert__adj__imp__inV,axiom,
! [V1: a,V2: a] :
( ( undire397441198561214472_adj_a @ edges @ V1 @ V2 )
=> ( ( member_a @ V1 @ vertices )
& ( member_a @ V2 @ vertices ) ) ) ).
% vert_adj_imp_inV
thf(fact_13_subgraph__refl,axiom,
undire7103218114511261257raph_a @ vertices @ edges @ vertices @ edges ).
% subgraph_refl
thf(fact_14_connecting__walk__rev,axiom,
! [U: a,V: a,Xs: list_a] :
( ( connecting_walk_a @ vertices @ edges @ U @ V @ Xs )
= ( connecting_walk_a @ vertices @ edges @ V @ U @ ( rev_a @ Xs ) ) ) ).
% connecting_walk_rev
thf(fact_15_wellformed,axiom,
! [E: set_a] :
( ( member_set_a @ E @ edges )
=> ( ord_less_eq_set_a @ E @ vertices ) ) ).
% wellformed
thf(fact_16_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_17_append_Oassoc,axiom,
! [A: list_set_a,B: list_set_a,C: list_set_a] :
( ( append_set_a @ ( append_set_a @ A @ B ) @ C )
= ( append_set_a @ A @ ( append_set_a @ B @ C ) ) ) ).
% append.assoc
thf(fact_18_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_19_append__assoc,axiom,
! [Xs: list_set_a,Ys: list_set_a,Zs: list_set_a] :
( ( append_set_a @ ( append_set_a @ Xs @ Ys ) @ Zs )
= ( append_set_a @ Xs @ ( append_set_a @ Ys @ Zs ) ) ) ).
% append_assoc
thf(fact_20_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_21_append__same__eq,axiom,
! [Ys: list_set_a,Xs: list_set_a,Zs: list_set_a] :
( ( ( append_set_a @ Ys @ Xs )
= ( append_set_a @ Zs @ Xs ) )
= ( Ys = Zs ) ) ).
% append_same_eq
thf(fact_22_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_23_same__append__eq,axiom,
! [Xs: list_set_a,Ys: list_set_a,Zs: list_set_a] :
( ( ( append_set_a @ Xs @ Ys )
= ( append_set_a @ Xs @ Zs ) )
= ( Ys = Zs ) ) ).
% same_append_eq
thf(fact_24_connecting__path__alt__def,axiom,
! [U: a,V: a,Xs: list_a] :
( ( connecting_path_a @ vertices @ edges @ U @ V @ Xs )
= ( ( connecting_walk_a @ vertices @ edges @ U @ V @ Xs )
& ( undire3562951555376170320path_a @ vertices @ edges @ Xs ) ) ) ).
% connecting_path_alt_def
thf(fact_25_incident__def,axiom,
undire1521409233611534436dent_a = member_a ).
% incident_def
thf(fact_26_vert__adj__sym,axiom,
! [V1: a,V2: a] :
( ( undire397441198561214472_adj_a @ edges @ V1 @ V2 )
= ( undire397441198561214472_adj_a @ edges @ V2 @ V1 ) ) ).
% vert_adj_sym
thf(fact_27_vert__adj__edge__iff2,axiom,
! [V1: a,V2: a] :
( ( V1 != V2 )
=> ( ( undire397441198561214472_adj_a @ edges @ V1 @ V2 )
= ( ? [X2: set_a] :
( ( member_set_a @ X2 @ edges )
& ( undire1521409233611534436dent_a @ V1 @ X2 )
& ( undire1521409233611534436dent_a @ V2 @ X2 ) ) ) ) ) ).
% vert_adj_edge_iff2
thf(fact_28_rev__rev__ident,axiom,
! [Xs: list_a] :
( ( rev_a @ ( rev_a @ Xs ) )
= Xs ) ).
% rev_rev_ident
thf(fact_29_rev__rev__ident,axiom,
! [Xs: list_set_a] :
( ( rev_set_a @ ( rev_set_a @ Xs ) )
= Xs ) ).
% rev_rev_ident
thf(fact_30_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_31_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_32_connecting__path__rev,axiom,
! [U: a,V: a,Xs: list_a] :
( ( connecting_path_a @ vertices @ edges @ U @ V @ Xs )
= ( connecting_path_a @ vertices @ edges @ V @ U @ ( rev_a @ Xs ) ) ) ).
% connecting_path_rev
thf(fact_33_is__gen__path__rev,axiom,
! [P: list_a] :
( ( undire3562951555376170320path_a @ vertices @ edges @ P )
= ( undire3562951555376170320path_a @ vertices @ edges @ ( rev_a @ P ) ) ) ).
% is_gen_path_rev
thf(fact_34_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_35_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_36_is__trail__rev,axiom,
! [Xs: list_a] :
( ( undire7142031287334043199rail_a @ vertices @ edges @ Xs )
= ( undire7142031287334043199rail_a @ vertices @ edges @ ( rev_a @ Xs ) ) ) ).
% is_trail_rev
thf(fact_37_ulgraph_Oconnecting__path__rev,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,U: set_a,V: set_a,Xs: list_set_a] :
( ( undire6886684016831807756_set_a @ Vertices @ Edges )
=> ( ( connec7350987497872064604_set_a @ Vertices @ Edges @ U @ V @ Xs )
= ( connec7350987497872064604_set_a @ Vertices @ Edges @ V @ U @ ( rev_set_a @ Xs ) ) ) ) ).
% ulgraph.connecting_path_rev
thf(fact_38_ulgraph_Oconnecting__path__rev,axiom,
! [Vertices: set_a,Edges: set_set_a,U: a,V: a,Xs: list_a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( connecting_path_a @ Vertices @ Edges @ U @ V @ Xs )
= ( connecting_path_a @ Vertices @ Edges @ V @ U @ ( rev_a @ Xs ) ) ) ) ).
% ulgraph.connecting_path_rev
thf(fact_39_ulgraph_Oconnecting__path__str__gen,axiom,
! [Vertices: set_a,Edges: set_set_a,U: a,V: a,Xs: list_a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( connec3015921205769380621_str_a @ Vertices @ Edges @ U @ V @ Xs )
=> ( connecting_path_a @ Vertices @ Edges @ U @ V @ Xs ) ) ) ).
% ulgraph.connecting_path_str_gen
thf(fact_40_rev__swap,axiom,
! [Xs: list_a,Ys: list_a] :
( ( ( rev_a @ Xs )
= Ys )
= ( Xs
= ( rev_a @ Ys ) ) ) ).
% rev_swap
thf(fact_41_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_42_ulgraph_Oconnecting__path__str_Ocong,axiom,
connec3015921205769380621_str_a = connec3015921205769380621_str_a ).
% ulgraph.connecting_path_str.cong
thf(fact_43_ulgraph_Oconnecting__walk__rev,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,U: set_a,V: set_a,Xs: list_set_a] :
( ( undire6886684016831807756_set_a @ Vertices @ Edges )
=> ( ( connec1530789871921280536_set_a @ Vertices @ Edges @ U @ V @ Xs )
= ( connec1530789871921280536_set_a @ Vertices @ Edges @ V @ U @ ( rev_set_a @ Xs ) ) ) ) ).
% ulgraph.connecting_walk_rev
thf(fact_44_ulgraph_Oconnecting__walk__rev,axiom,
! [Vertices: set_a,Edges: set_set_a,U: a,V: a,Xs: list_a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( connecting_walk_a @ Vertices @ Edges @ U @ V @ Xs )
= ( connecting_walk_a @ Vertices @ Edges @ V @ U @ ( rev_a @ Xs ) ) ) ) ).
% ulgraph.connecting_walk_rev
thf(fact_45_ulgraph_Oconnecting__path__alt__def,axiom,
! [Vertices: set_a,Edges: set_set_a,U: a,V: a,Xs: list_a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( connecting_path_a @ Vertices @ Edges @ U @ V @ Xs )
= ( ( connecting_walk_a @ Vertices @ Edges @ U @ V @ Xs )
& ( undire3562951555376170320path_a @ Vertices @ Edges @ Xs ) ) ) ) ).
% ulgraph.connecting_path_alt_def
thf(fact_46_ulgraph_Oconnecting__path__walk,axiom,
! [Vertices: set_a,Edges: set_set_a,U: a,V: a,Xs: list_a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( connecting_path_a @ Vertices @ Edges @ U @ V @ Xs )
=> ( connecting_walk_a @ Vertices @ Edges @ U @ V @ Xs ) ) ) ).
% ulgraph.connecting_path_walk
thf(fact_47_ulgraph_Oconnecting__walk__wf,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,U: set_a,V: set_a,Xs: list_set_a] :
( ( undire6886684016831807756_set_a @ Vertices @ Edges )
=> ( ( connec1530789871921280536_set_a @ Vertices @ Edges @ U @ V @ Xs )
=> ( ( member_set_a @ U @ Vertices )
& ( member_set_a @ V @ Vertices ) ) ) ) ).
% ulgraph.connecting_walk_wf
thf(fact_48_ulgraph_Oconnecting__walk__wf,axiom,
! [Vertices: set_a,Edges: set_set_a,U: a,V: a,Xs: list_a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( connecting_walk_a @ Vertices @ Edges @ U @ V @ Xs )
=> ( ( member_a @ U @ Vertices )
& ( member_a @ V @ Vertices ) ) ) ) ).
% ulgraph.connecting_walk_wf
thf(fact_49_ulgraph_Oconnecting__path_Ocong,axiom,
connecting_path_a = connecting_path_a ).
% ulgraph.connecting_path.cong
thf(fact_50_ulgraph_Oconnecting__walk__split,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,U: set_a,V: set_a,Xs: list_set_a,Z: set_a,Ys: list_set_a] :
( ( undire6886684016831807756_set_a @ Vertices @ Edges )
=> ( ( connec1530789871921280536_set_a @ Vertices @ Edges @ U @ V @ Xs )
=> ( ( connec1530789871921280536_set_a @ Vertices @ Edges @ V @ Z @ Ys )
=> ( connec1530789871921280536_set_a @ Vertices @ Edges @ U @ Z @ ( append_set_a @ Xs @ ( tl_set_a @ Ys ) ) ) ) ) ) ).
% ulgraph.connecting_walk_split
thf(fact_51_ulgraph_Oconnecting__walk__split,axiom,
! [Vertices: set_a,Edges: set_set_a,U: a,V: a,Xs: list_a,Z: a,Ys: list_a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( connecting_walk_a @ Vertices @ Edges @ U @ V @ Xs )
=> ( ( connecting_walk_a @ Vertices @ Edges @ V @ Z @ Ys )
=> ( connecting_walk_a @ Vertices @ Edges @ U @ Z @ ( append_a @ Xs @ ( tl_a @ Ys ) ) ) ) ) ) ).
% ulgraph.connecting_walk_split
thf(fact_52_append__eq__append__conv2,axiom,
! [Xs: list_a,Ys: list_a,Zs: list_a,Ts: list_a] :
( ( ( append_a @ Xs @ Ys )
= ( append_a @ Zs @ Ts ) )
= ( ? [Us: list_a] :
( ( ( Xs
= ( append_a @ Zs @ Us ) )
& ( ( append_a @ Us @ Ys )
= Ts ) )
| ( ( ( append_a @ Xs @ Us )
= Zs )
& ( Ys
= ( append_a @ Us @ Ts ) ) ) ) ) ) ).
% append_eq_append_conv2
thf(fact_53_append__eq__append__conv2,axiom,
! [Xs: list_set_a,Ys: list_set_a,Zs: list_set_a,Ts: list_set_a] :
( ( ( append_set_a @ Xs @ Ys )
= ( append_set_a @ Zs @ Ts ) )
= ( ? [Us: list_set_a] :
( ( ( Xs
= ( append_set_a @ Zs @ Us ) )
& ( ( append_set_a @ Us @ Ys )
= Ts ) )
| ( ( ( append_set_a @ Xs @ Us )
= Zs )
& ( Ys
= ( append_set_a @ Us @ Ts ) ) ) ) ) ) ).
% append_eq_append_conv2
thf(fact_54_append__eq__appendI,axiom,
! [Xs: list_a,Xs1: list_a,Zs: list_a,Ys: list_a,Us2: list_a] :
( ( ( append_a @ Xs @ Xs1 )
= Zs )
=> ( ( Ys
= ( append_a @ Xs1 @ Us2 ) )
=> ( ( append_a @ Xs @ Ys )
= ( append_a @ Zs @ Us2 ) ) ) ) ).
% append_eq_appendI
thf(fact_55_append__eq__appendI,axiom,
! [Xs: list_set_a,Xs1: list_set_a,Zs: list_set_a,Ys: list_set_a,Us2: list_set_a] :
( ( ( append_set_a @ Xs @ Xs1 )
= Zs )
=> ( ( Ys
= ( append_set_a @ Xs1 @ Us2 ) )
=> ( ( append_set_a @ Xs @ Ys )
= ( append_set_a @ Zs @ Us2 ) ) ) ) ).
% append_eq_appendI
thf(fact_56_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_57_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_58_is__isolated__vertex__no__loop,axiom,
! [V: a] :
( ( undire8931668460104145173rtex_a @ vertices @ edges @ V )
=> ~ ( undire3617971648856834880loop_a @ edges @ V ) ) ).
% is_isolated_vertex_no_loop
thf(fact_59_mem__Collect__eq,axiom,
! [A: a,P2: a > $o] :
( ( member_a @ A @ ( collect_a @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_60_mem__Collect__eq,axiom,
! [A: set_a,P2: set_a > $o] :
( ( member_set_a @ A @ ( collect_set_a @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_61_Collect__mem__eq,axiom,
! [A2: set_a] :
( ( collect_a
@ ^ [X2: a] : ( member_a @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_62_Collect__mem__eq,axiom,
! [A2: set_set_a] :
( ( collect_set_a
@ ^ [X2: set_a] : ( member_set_a @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_63_is__isolated__vertex__def,axiom,
! [V: a] :
( ( undire8931668460104145173rtex_a @ vertices @ edges @ V )
= ( ( member_a @ V @ vertices )
& ! [X2: a] :
( ( member_a @ X2 @ vertices )
=> ~ ( undire397441198561214472_adj_a @ edges @ X2 @ V ) ) ) ) ).
% is_isolated_vertex_def
thf(fact_64_is__isolated__vertex__edge,axiom,
! [V: a,E: set_a] :
( ( undire8931668460104145173rtex_a @ vertices @ edges @ V )
=> ( ( member_set_a @ E @ edges )
=> ~ ( undire1521409233611534436dent_a @ V @ E ) ) ) ).
% is_isolated_vertex_edge
thf(fact_65_induced__is__subgraph,axiom,
! [V3: set_a] :
( ( ord_less_eq_set_a @ V3 @ vertices )
=> ( undire7103218114511261257raph_a @ V3 @ ( undire7777452895879145676dges_a @ edges @ V3 ) @ vertices @ edges ) ) ).
% induced_is_subgraph
thf(fact_66_is__path__rev,axiom,
! [Xs: list_a] :
( ( undire427332500224447920path_a @ vertices @ edges @ Xs )
= ( undire427332500224447920path_a @ vertices @ edges @ ( rev_a @ Xs ) ) ) ).
% is_path_rev
thf(fact_67_is__path__gen__path,axiom,
! [P: list_a] :
( ( undire427332500224447920path_a @ vertices @ edges @ P )
=> ( undire3562951555376170320path_a @ vertices @ edges @ P ) ) ).
% is_path_gen_path
thf(fact_68_is__cycle__rev,axiom,
! [Xs: list_a] :
( ( undire2407311113669455967ycle_a @ vertices @ edges @ Xs )
= ( undire2407311113669455967ycle_a @ vertices @ edges @ ( rev_a @ Xs ) ) ) ).
% is_cycle_rev
thf(fact_69_is__gen__path__cycle,axiom,
! [P: list_a] :
( ( undire2407311113669455967ycle_a @ vertices @ edges @ P )
=> ( undire3562951555376170320path_a @ vertices @ edges @ P ) ) ).
% is_gen_path_cycle
thf(fact_70_ulgraph_Overt__adj__edge__iff2,axiom,
! [Vertices: set_a,Edges: set_set_a,V1: a,V2: a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( V1 != V2 )
=> ( ( undire397441198561214472_adj_a @ Edges @ V1 @ V2 )
= ( ? [X2: set_a] :
( ( member_set_a @ X2 @ Edges )
& ( undire1521409233611534436dent_a @ V1 @ X2 )
& ( undire1521409233611534436dent_a @ V2 @ X2 ) ) ) ) ) ) ).
% ulgraph.vert_adj_edge_iff2
thf(fact_71_ulgraph_Ois__isolated__vertex_Ocong,axiom,
undire8931668460104145173rtex_a = undire8931668460104145173rtex_a ).
% ulgraph.is_isolated_vertex.cong
thf(fact_72_graph__system_Oinduced__edges_Ocong,axiom,
undire7777452895879145676dges_a = undire7777452895879145676dges_a ).
% graph_system.induced_edges.cong
thf(fact_73_ulgraph_Ois__isolated__vertex__def,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,V: set_a] :
( ( undire6886684016831807756_set_a @ Vertices @ Edges )
=> ( ( undire6879241558604981877_set_a @ Vertices @ Edges @ V )
= ( ( member_set_a @ V @ Vertices )
& ! [X2: set_a] :
( ( member_set_a @ X2 @ Vertices )
=> ~ ( undire3510646817838285160_set_a @ Edges @ X2 @ V ) ) ) ) ) ).
% ulgraph.is_isolated_vertex_def
thf(fact_74_ulgraph_Ois__isolated__vertex__def,axiom,
! [Vertices: set_a,Edges: set_set_a,V: a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( undire8931668460104145173rtex_a @ Vertices @ Edges @ V )
= ( ( member_a @ V @ Vertices )
& ! [X2: a] :
( ( member_a @ X2 @ Vertices )
=> ~ ( undire397441198561214472_adj_a @ Edges @ X2 @ V ) ) ) ) ) ).
% ulgraph.is_isolated_vertex_def
thf(fact_75_ulgraph_Ois__isolated__vertex__edge,axiom,
! [Vertices: set_a,Edges: set_set_a,V: a,E: set_a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( undire8931668460104145173rtex_a @ Vertices @ Edges @ V )
=> ( ( member_set_a @ E @ Edges )
=> ~ ( undire1521409233611534436dent_a @ V @ E ) ) ) ) ).
% ulgraph.is_isolated_vertex_edge
thf(fact_76_ulgraph_Ois__isolated__vertex__no__loop,axiom,
! [Vertices: set_a,Edges: set_set_a,V: a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( undire8931668460104145173rtex_a @ Vertices @ Edges @ V )
=> ~ ( undire3617971648856834880loop_a @ Edges @ V ) ) ) ).
% ulgraph.is_isolated_vertex_no_loop
thf(fact_77_subgraph_Osubgraph__antisym,axiom,
! [V_H: set_a,E_H: set_set_a,V_G: set_a,E_G: set_set_a,V3: set_a,E3: set_set_a,V4: set_a,E4: set_set_a] :
( ( undire7103218114511261257raph_a @ V_H @ E_H @ V_G @ E_G )
=> ( ( undire7103218114511261257raph_a @ V3 @ E3 @ V4 @ E4 )
=> ( ( undire7103218114511261257raph_a @ V4 @ E4 @ V3 @ E3 )
=> ( ( V4 = V3 )
& ( E4 = E3 ) ) ) ) ) ).
% subgraph.subgraph_antisym
thf(fact_78_ulgraph_Overt__adj_Ocong,axiom,
undire397441198561214472_adj_a = undire397441198561214472_adj_a ).
% ulgraph.vert_adj.cong
thf(fact_79_comp__sgraph_Oincident__def,axiom,
undire2320338297334612420_set_a = member_set_a ).
% comp_sgraph.incident_def
thf(fact_80_comp__sgraph_Oincident__def,axiom,
undire1521409233611534436dent_a = member_a ).
% comp_sgraph.incident_def
thf(fact_81_ulgraph_Ohas__loop_Ocong,axiom,
undire3617971648856834880loop_a = undire3617971648856834880loop_a ).
% ulgraph.has_loop.cong
thf(fact_82_graph__system_Oedge__adj_Ocong,axiom,
undire4022703626023482010_adj_a = undire4022703626023482010_adj_a ).
% graph_system.edge_adj.cong
thf(fact_83_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_84_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_85_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_86_ulgraph_Overt__adj__imp__inV,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,V1: set_a,V2: set_a] :
( ( undire6886684016831807756_set_a @ Vertices @ Edges )
=> ( ( undire3510646817838285160_set_a @ Edges @ V1 @ V2 )
=> ( ( member_set_a @ V1 @ Vertices )
& ( member_set_a @ V2 @ Vertices ) ) ) ) ).
% ulgraph.vert_adj_imp_inV
thf(fact_87_ulgraph_Overt__adj__imp__inV,axiom,
! [Vertices: set_a,Edges: set_set_a,V1: a,V2: a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( undire397441198561214472_adj_a @ Edges @ V1 @ V2 )
=> ( ( member_a @ V1 @ Vertices )
& ( member_a @ V2 @ Vertices ) ) ) ) ).
% ulgraph.vert_adj_imp_inV
thf(fact_88_ulgraph_Overt__adj__sym,axiom,
! [Vertices: set_a,Edges: set_set_a,V1: a,V2: a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( undire397441198561214472_adj_a @ Edges @ V1 @ V2 )
= ( undire397441198561214472_adj_a @ Edges @ V2 @ V1 ) ) ) ).
% ulgraph.vert_adj_sym
thf(fact_89_ulgraph_Ohas__loop__in__verts,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,V: set_a] :
( ( undire6886684016831807756_set_a @ Vertices @ Edges )
=> ( ( undire5774735625301615776_set_a @ Edges @ V )
=> ( member_set_a @ V @ Vertices ) ) ) ).
% ulgraph.has_loop_in_verts
thf(fact_90_ulgraph_Ohas__loop__in__verts,axiom,
! [Vertices: set_a,Edges: set_set_a,V: a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( undire3617971648856834880loop_a @ Edges @ V )
=> ( member_a @ V @ Vertices ) ) ) ).
% ulgraph.has_loop_in_verts
thf(fact_91_induced__edges__ss,axiom,
! [V3: set_a] :
( ( ord_less_eq_set_a @ V3 @ vertices )
=> ( ord_le3724670747650509150_set_a @ ( undire7777452895879145676dges_a @ edges @ V3 ) @ edges ) ) ).
% induced_edges_ss
thf(fact_92_is__path__def,axiom,
! [Xs: list_a] :
( ( undire427332500224447920path_a @ vertices @ edges @ Xs )
= ( ( undire2427028224930250914walk_a @ vertices @ edges @ Xs )
& ( distinct_a @ Xs ) ) ) ).
% is_path_def
thf(fact_93_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_94_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_95_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_96_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_97_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_98_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_99_ulgraph_Ois__path__gen__path,axiom,
! [Vertices: set_a,Edges: set_set_a,P: list_a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( undire427332500224447920path_a @ Vertices @ Edges @ P )
=> ( undire3562951555376170320path_a @ Vertices @ Edges @ P ) ) ) ).
% ulgraph.is_path_gen_path
thf(fact_100_ulgraph_Ois__gen__path__cycle,axiom,
! [Vertices: set_a,Edges: set_set_a,P: list_a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( undire2407311113669455967ycle_a @ Vertices @ Edges @ P )
=> ( undire3562951555376170320path_a @ Vertices @ Edges @ P ) ) ) ).
% ulgraph.is_gen_path_cycle
thf(fact_101_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_102_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_103_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_104_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_105_ulgraph_Ois__gen__path__rev,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,P: list_set_a] :
( ( undire6886684016831807756_set_a @ Vertices @ Edges )
=> ( ( undire7201326534205417136_set_a @ Vertices @ Edges @ P )
= ( undire7201326534205417136_set_a @ Vertices @ Edges @ ( rev_set_a @ P ) ) ) ) ).
% ulgraph.is_gen_path_rev
thf(fact_106_ulgraph_Ois__gen__path__rev,axiom,
! [Vertices: set_a,Edges: set_set_a,P: list_a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( undire3562951555376170320path_a @ Vertices @ Edges @ P )
= ( undire3562951555376170320path_a @ Vertices @ Edges @ ( rev_a @ P ) ) ) ) ).
% ulgraph.is_gen_path_rev
thf(fact_107_is__path__walk,axiom,
! [Xs: list_a] :
( ( undire427332500224447920path_a @ vertices @ edges @ Xs )
=> ( undire6133010728901294956walk_a @ vertices @ edges @ Xs ) ) ).
% is_path_walk
thf(fact_108_is__walk__rev,axiom,
! [Xs: list_a] :
( ( undire6133010728901294956walk_a @ vertices @ edges @ Xs )
= ( undire6133010728901294956walk_a @ vertices @ edges @ ( rev_a @ Xs ) ) ) ).
% is_walk_rev
thf(fact_109_distinct__rev,axiom,
! [Xs: list_a] :
( ( distinct_a @ ( rev_a @ Xs ) )
= ( distinct_a @ Xs ) ) ).
% distinct_rev
thf(fact_110_distinct__rev,axiom,
! [Xs: list_set_a] :
( ( distinct_set_a @ ( rev_set_a @ Xs ) )
= ( distinct_set_a @ Xs ) ) ).
% distinct_rev
thf(fact_111_ulgraph_Ois__walk_Ocong,axiom,
undire6133010728901294956walk_a = undire6133010728901294956walk_a ).
% ulgraph.is_walk.cong
thf(fact_112_distinct__tl,axiom,
! [Xs: list_a] :
( ( distinct_a @ Xs )
=> ( distinct_a @ ( tl_a @ Xs ) ) ) ).
% distinct_tl
thf(fact_113_distinct__tl,axiom,
! [Xs: list_set_a] :
( ( distinct_set_a @ Xs )
=> ( distinct_set_a @ ( tl_set_a @ Xs ) ) ) ).
% distinct_tl
thf(fact_114_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_115_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_116_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_117_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_118_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_119_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_120_ulgraph_Ois__gen__path_Ocong,axiom,
undire3562951555376170320path_a = undire3562951555376170320path_a ).
% ulgraph.is_gen_path.cong
thf(fact_121_ulgraph_Ois__cycle_Ocong,axiom,
undire2407311113669455967ycle_a = undire2407311113669455967ycle_a ).
% ulgraph.is_cycle.cong
thf(fact_122_ulgraph_Ois__path_Ocong,axiom,
undire427332500224447920path_a = undire427332500224447920path_a ).
% ulgraph.is_path.cong
thf(fact_123_ulgraph_Ois__open__walk_Ocong,axiom,
undire2427028224930250914walk_a = undire2427028224930250914walk_a ).
% ulgraph.is_open_walk.cong
thf(fact_124_ulgraph_Ois__closed__walk_Ocong,axiom,
undire3370724456595283424walk_a = undire3370724456595283424walk_a ).
% ulgraph.is_closed_walk.cong
thf(fact_125_ulgraph_Ois__trail_Ocong,axiom,
undire7142031287334043199rail_a = undire7142031287334043199rail_a ).
% ulgraph.is_trail.cong
thf(fact_126_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_127_is__walk__not__empty2,axiom,
~ ( undire6133010728901294956walk_a @ vertices @ edges @ nil_a ) ).
% is_walk_not_empty2
thf(fact_128_is__walk__not__empty,axiom,
! [Xs: list_a] :
( ( undire6133010728901294956walk_a @ vertices @ edges @ Xs )
=> ( Xs != nil_a ) ) ).
% is_walk_not_empty
thf(fact_129_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_130_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_131_subsetI,axiom,
! [A2: set_a,B2: set_a] :
( ! [X3: a] :
( ( member_a @ X3 @ A2 )
=> ( member_a @ X3 @ B2 ) )
=> ( ord_less_eq_set_a @ A2 @ B2 ) ) ).
% subsetI
thf(fact_132_subsetI,axiom,
! [A2: set_set_a,B2: set_set_a] :
( ! [X3: set_a] :
( ( member_set_a @ X3 @ A2 )
=> ( member_set_a @ X3 @ B2 ) )
=> ( ord_le3724670747650509150_set_a @ A2 @ B2 ) ) ).
% subsetI
thf(fact_133_subset__antisym,axiom,
! [A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ord_less_eq_set_a @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% subset_antisym
thf(fact_134_subset__antisym,axiom,
! [A2: set_set_a,B2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B2 )
=> ( ( ord_le3724670747650509150_set_a @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% subset_antisym
thf(fact_135_distinct__union,axiom,
! [Xs: list_a,Ys: list_a] :
( ( distinct_a @ ( union_a @ Xs @ Ys ) )
= ( distinct_a @ Ys ) ) ).
% distinct_union
thf(fact_136_distinct__union,axiom,
! [Xs: list_set_a,Ys: list_set_a] :
( ( distinct_set_a @ ( union_set_a @ Xs @ Ys ) )
= ( distinct_set_a @ Ys ) ) ).
% distinct_union
thf(fact_137_graph__system__axioms,axiom,
undire2554140024507503526stem_a @ vertices @ edges ).
% graph_system_axioms
thf(fact_138_order__refl,axiom,
! [X4: set_a] : ( ord_less_eq_set_a @ X4 @ X4 ) ).
% order_refl
thf(fact_139_order__refl,axiom,
! [X4: set_set_a] : ( ord_le3724670747650509150_set_a @ X4 @ X4 ) ).
% order_refl
thf(fact_140_order__refl,axiom,
! [X4: nat] : ( ord_less_eq_nat @ X4 @ X4 ) ).
% order_refl
thf(fact_141_order__refl,axiom,
! [X4: real] : ( ord_less_eq_real @ X4 @ X4 ) ).
% order_refl
thf(fact_142_induced__is__graph__sys,axiom,
! [V3: set_a] : ( undire2554140024507503526stem_a @ V3 @ ( undire7777452895879145676dges_a @ edges @ V3 ) ) ).
% induced_is_graph_sys
thf(fact_143_dual__order_Orefl,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ A @ A ) ).
% dual_order.refl
thf(fact_144_dual__order_Orefl,axiom,
! [A: set_set_a] : ( ord_le3724670747650509150_set_a @ A @ A ) ).
% dual_order.refl
thf(fact_145_dual__order_Orefl,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% dual_order.refl
thf(fact_146_dual__order_Orefl,axiom,
! [A: real] : ( ord_less_eq_real @ A @ A ) ).
% dual_order.refl
thf(fact_147_append_Oright__neutral,axiom,
! [A: list_a] :
( ( append_a @ A @ nil_a )
= A ) ).
% append.right_neutral
thf(fact_148_append_Oright__neutral,axiom,
! [A: list_set_a] :
( ( append_set_a @ A @ nil_set_a )
= A ) ).
% append.right_neutral
thf(fact_149_append__Nil2,axiom,
! [Xs: list_a] :
( ( append_a @ Xs @ nil_a )
= Xs ) ).
% append_Nil2
thf(fact_150_append__Nil2,axiom,
! [Xs: list_set_a] :
( ( append_set_a @ Xs @ nil_set_a )
= Xs ) ).
% append_Nil2
thf(fact_151_append__self__conv,axiom,
! [Xs: list_a,Ys: list_a] :
( ( ( append_a @ Xs @ Ys )
= Xs )
= ( Ys = nil_a ) ) ).
% append_self_conv
thf(fact_152_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_153_self__append__conv,axiom,
! [Y: list_a,Ys: list_a] :
( ( Y
= ( append_a @ Y @ Ys ) )
= ( Ys = nil_a ) ) ).
% self_append_conv
thf(fact_154_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_155_append__self__conv2,axiom,
! [Xs: list_a,Ys: list_a] :
( ( ( append_a @ Xs @ Ys )
= Ys )
= ( Xs = nil_a ) ) ).
% append_self_conv2
thf(fact_156_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_157_self__append__conv2,axiom,
! [Y: list_a,Xs: list_a] :
( ( Y
= ( append_a @ Xs @ Y ) )
= ( Xs = nil_a ) ) ).
% self_append_conv2
thf(fact_158_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_159_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_160_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_161_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_162_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_163_connecting__walk__def,axiom,
! [U: a,V: a,Xs: list_a] :
( ( connecting_walk_a @ vertices @ edges @ U @ V @ Xs )
= ( ( undire6133010728901294956walk_a @ vertices @ edges @ Xs )
& ( ( hd_a @ Xs )
= U )
& ( ( last_a @ Xs )
= V ) ) ) ).
% connecting_walk_def
thf(fact_164_connecting__path__def,axiom,
! [U: a,V: a,Xs: list_a] :
( ( connecting_path_a @ vertices @ edges @ U @ V @ Xs )
= ( ( undire3562951555376170320path_a @ vertices @ edges @ Xs )
& ( ( hd_a @ Xs )
= U )
& ( ( last_a @ Xs )
= V ) ) ) ).
% connecting_path_def
thf(fact_165_rev__is__Nil__conv,axiom,
! [Xs: list_a] :
( ( ( rev_a @ Xs )
= nil_a )
= ( Xs = nil_a ) ) ).
% rev_is_Nil_conv
thf(fact_166_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_167_Nil__is__rev__conv,axiom,
! [Xs: list_a] :
( ( nil_a
= ( rev_a @ Xs ) )
= ( Xs = nil_a ) ) ).
% Nil_is_rev_conv
thf(fact_168_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_169_is__gen__path__distinct,axiom,
! [P: list_a] :
( ( undire3562951555376170320path_a @ vertices @ edges @ P )
=> ( ( ( hd_a @ P )
!= ( last_a @ P ) )
=> ( distinct_a @ P ) ) ) ).
% is_gen_path_distinct
thf(fact_170_set__rev,axiom,
! [Xs: list_a] :
( ( set_a2 @ ( rev_a @ Xs ) )
= ( set_a2 @ Xs ) ) ).
% set_rev
thf(fact_171_set__rev,axiom,
! [Xs: list_set_a] :
( ( set_set_a2 @ ( rev_set_a @ Xs ) )
= ( set_set_a2 @ Xs ) ) ).
% set_rev
thf(fact_172_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_173_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_174_connecting__path__str__def,axiom,
! [U: a,V: a,Xs: list_a] :
( ( connec3015921205769380621_str_a @ vertices @ edges @ U @ V @ Xs )
= ( ( undire427332500224447920path_a @ vertices @ edges @ Xs )
& ( ( hd_a @ Xs )
= U )
& ( ( last_a @ Xs )
= V ) ) ) ).
% connecting_path_str_def
thf(fact_175_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_176_is__gen__path__distinct__tl,axiom,
! [P: list_a] :
( ( undire3562951555376170320path_a @ vertices @ edges @ P )
=> ( ( ( hd_a @ P )
= ( last_a @ P ) )
=> ( distinct_a @ ( tl_a @ P ) ) ) ) ).
% is_gen_path_distinct_tl
thf(fact_177_is__gen__path__def,axiom,
! [P: list_a] :
( ( undire3562951555376170320path_a @ vertices @ edges @ P )
= ( ( undire6133010728901294956walk_a @ vertices @ edges @ P )
& ( ( ( distinct_a @ ( tl_a @ P ) )
& ( ( hd_a @ P )
= ( last_a @ P ) ) )
| ( distinct_a @ P ) ) ) ) ).
% is_gen_path_def
thf(fact_178_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_179_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_180_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_181_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_182_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_183_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_184_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_185_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_186_is__subgraphI,axiom,
! [V3: set_set_a,V4: set_set_a,E3: set_set_set_a,E4: set_set_set_a] :
( ( ord_le3724670747650509150_set_a @ V3 @ V4 )
=> ( ( ord_le5722252365846178494_set_a @ E3 @ E4 )
=> ( ( undire7159349782766787846_set_a @ V3 @ E3 )
=> ( ( undire7159349782766787846_set_a @ V4 @ E4 )
=> ( undire1186139521737116585_set_a @ V3 @ E3 @ V4 @ E4 ) ) ) ) ) ).
% is_subgraphI
thf(fact_187_is__subgraphI,axiom,
! [V3: set_a,V4: set_a,E3: set_set_a,E4: set_set_a] :
( ( ord_less_eq_set_a @ V3 @ V4 )
=> ( ( ord_le3724670747650509150_set_a @ E3 @ E4 )
=> ( ( undire2554140024507503526stem_a @ V3 @ E3 )
=> ( ( undire2554140024507503526stem_a @ V4 @ E4 )
=> ( undire7103218114511261257raph_a @ V3 @ E3 @ V4 @ E4 ) ) ) ) ) ).
% is_subgraphI
thf(fact_188_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_189_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_190_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_191_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_192_hd__rev,axiom,
! [Xs: list_set_a] :
( ( hd_set_a @ ( rev_set_a @ Xs ) )
= ( last_set_a @ Xs ) ) ).
% hd_rev
thf(fact_193_hd__rev,axiom,
! [Xs: list_a] :
( ( hd_a @ ( rev_a @ Xs ) )
= ( last_a @ Xs ) ) ).
% hd_rev
thf(fact_194_last__rev,axiom,
! [Xs: list_set_a] :
( ( last_set_a @ ( rev_set_a @ Xs ) )
= ( hd_set_a @ Xs ) ) ).
% last_rev
thf(fact_195_last__rev,axiom,
! [Xs: list_a] :
( ( last_a @ ( rev_a @ Xs ) )
= ( hd_a @ Xs ) ) ).
% last_rev
thf(fact_196_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_197_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_198_hd__in__set,axiom,
! [Xs: list_a] :
( ( Xs != nil_a )
=> ( member_a @ ( hd_a @ Xs ) @ ( set_a2 @ Xs ) ) ) ).
% hd_in_set
thf(fact_199_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_200_last__in__set,axiom,
! [As: list_a] :
( ( As != nil_a )
=> ( member_a @ ( last_a @ As ) @ ( set_a2 @ As ) ) ) ).
% last_in_set
thf(fact_201_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_202_hd__Nil__eq__last,axiom,
( ( hd_a @ nil_a )
= ( last_a @ nil_a ) ) ).
% hd_Nil_eq_last
thf(fact_203_hd__Nil__eq__last,axiom,
( ( hd_set_a @ nil_set_a )
= ( last_set_a @ nil_set_a ) ) ).
% hd_Nil_eq_last
thf(fact_204_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_205_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_206_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_207_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_208_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_209_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_210_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_211_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_212_list_Oset__sel_I2_J,axiom,
! [A: list_a,X4: a] :
( ( A != nil_a )
=> ( ( member_a @ X4 @ ( set_a2 @ ( tl_a @ A ) ) )
=> ( member_a @ X4 @ ( set_a2 @ A ) ) ) ) ).
% list.set_sel(2)
thf(fact_213_list_Oset__sel_I2_J,axiom,
! [A: list_set_a,X4: set_a] :
( ( A != nil_set_a )
=> ( ( member_set_a @ X4 @ ( set_set_a2 @ ( tl_set_a @ A ) ) )
=> ( member_set_a @ X4 @ ( set_set_a2 @ A ) ) ) ) ).
% list.set_sel(2)
thf(fact_214_subset__code_I1_J,axiom,
! [Xs: list_a,B2: set_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ Xs ) @ B2 )
= ( ! [X2: a] :
( ( member_a @ X2 @ ( set_a2 @ Xs ) )
=> ( member_a @ X2 @ B2 ) ) ) ) ).
% subset_code(1)
thf(fact_215_subset__code_I1_J,axiom,
! [Xs: list_set_a,B2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ ( set_set_a2 @ Xs ) @ B2 )
= ( ! [X2: set_a] :
( ( member_set_a @ X2 @ ( set_set_a2 @ Xs ) )
=> ( member_set_a @ X2 @ B2 ) ) ) ) ).
% subset_code(1)
thf(fact_216_graph__system__def,axiom,
( undire7159349782766787846_set_a
= ( ^ [Vertices2: set_set_a,Edges2: set_set_set_a] :
! [E5: set_set_a] :
( ( member_set_set_a @ E5 @ Edges2 )
=> ( ord_le3724670747650509150_set_a @ E5 @ Vertices2 ) ) ) ) ).
% graph_system_def
thf(fact_217_graph__system__def,axiom,
( undire2554140024507503526stem_a
= ( ^ [Vertices2: set_a,Edges2: set_set_a] :
! [E5: set_a] :
( ( member_set_a @ E5 @ Edges2 )
=> ( ord_less_eq_set_a @ E5 @ Vertices2 ) ) ) ) ).
% graph_system_def
thf(fact_218_graph__system_Owellformed,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,E: set_set_a] :
( ( undire7159349782766787846_set_a @ Vertices @ Edges )
=> ( ( member_set_set_a @ E @ Edges )
=> ( ord_le3724670747650509150_set_a @ E @ Vertices ) ) ) ).
% graph_system.wellformed
thf(fact_219_graph__system_Owellformed,axiom,
! [Vertices: set_a,Edges: set_set_a,E: set_a] :
( ( undire2554140024507503526stem_a @ Vertices @ Edges )
=> ( ( member_set_a @ E @ Edges )
=> ( ord_less_eq_set_a @ E @ Vertices ) ) ) ).
% graph_system.wellformed
thf(fact_220_graph__system_Ointro,axiom,
! [Edges: set_set_set_a,Vertices: set_set_a] :
( ! [E6: set_set_a] :
( ( member_set_set_a @ E6 @ Edges )
=> ( ord_le3724670747650509150_set_a @ E6 @ Vertices ) )
=> ( undire7159349782766787846_set_a @ Vertices @ Edges ) ) ).
% graph_system.intro
thf(fact_221_graph__system_Ointro,axiom,
! [Edges: set_set_a,Vertices: set_a] :
( ! [E6: set_a] :
( ( member_set_a @ E6 @ Edges )
=> ( ord_less_eq_set_a @ E6 @ Vertices ) )
=> ( undire2554140024507503526stem_a @ Vertices @ Edges ) ) ).
% graph_system.intro
thf(fact_222_append__Nil,axiom,
! [Ys: list_a] :
( ( append_a @ nil_a @ Ys )
= Ys ) ).
% append_Nil
thf(fact_223_append__Nil,axiom,
! [Ys: list_set_a] :
( ( append_set_a @ nil_set_a @ Ys )
= Ys ) ).
% append_Nil
thf(fact_224_append_Oleft__neutral,axiom,
! [A: list_a] :
( ( append_a @ nil_a @ A )
= A ) ).
% append.left_neutral
thf(fact_225_append_Oleft__neutral,axiom,
! [A: list_set_a] :
( ( append_set_a @ nil_set_a @ A )
= A ) ).
% append.left_neutral
thf(fact_226_eq__Nil__appendI,axiom,
! [Xs: list_a,Ys: list_a] :
( ( Xs = Ys )
=> ( Xs
= ( append_a @ nil_a @ Ys ) ) ) ).
% eq_Nil_appendI
thf(fact_227_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_228_ulgraph_Oaxioms_I1_J,axiom,
! [Vertices: set_a,Edges: set_set_a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( undire2554140024507503526stem_a @ Vertices @ Edges ) ) ).
% ulgraph.axioms(1)
thf(fact_229_distinct_Osimps_I1_J,axiom,
distinct_a @ nil_a ).
% distinct.simps(1)
thf(fact_230_distinct_Osimps_I1_J,axiom,
distinct_set_a @ nil_set_a ).
% distinct.simps(1)
thf(fact_231_rev_Osimps_I1_J,axiom,
( ( rev_a @ nil_a )
= nil_a ) ).
% rev.simps(1)
thf(fact_232_rev_Osimps_I1_J,axiom,
( ( rev_set_a @ nil_set_a )
= nil_set_a ) ).
% rev.simps(1)
thf(fact_233_list_Osel_I2_J,axiom,
( ( tl_a @ nil_a )
= nil_a ) ).
% list.sel(2)
thf(fact_234_list_Osel_I2_J,axiom,
( ( tl_set_a @ nil_set_a )
= nil_set_a ) ).
% list.sel(2)
thf(fact_235_graph__system_Osubgraph__refl,axiom,
! [Vertices: set_a,Edges: set_set_a] :
( ( undire2554140024507503526stem_a @ Vertices @ Edges )
=> ( undire7103218114511261257raph_a @ Vertices @ Edges @ Vertices @ Edges ) ) ).
% graph_system.subgraph_refl
thf(fact_236_subgraph_Osubgraph__trans,axiom,
! [V_H: set_a,E_H: set_set_a,V_G: set_a,E_G: set_set_a,V4: set_a,E4: set_set_a,V3: set_a,E3: set_set_a,V5: set_a,E7: set_set_a] :
( ( undire7103218114511261257raph_a @ V_H @ E_H @ V_G @ E_G )
=> ( ( undire2554140024507503526stem_a @ V4 @ E4 )
=> ( ( undire2554140024507503526stem_a @ V3 @ E3 )
=> ( ( undire2554140024507503526stem_a @ V5 @ E7 )
=> ( ( undire7103218114511261257raph_a @ V5 @ E7 @ V3 @ E3 )
=> ( ( undire7103218114511261257raph_a @ V3 @ E3 @ V4 @ E4 )
=> ( undire7103218114511261257raph_a @ V5 @ E7 @ V4 @ E4 ) ) ) ) ) ) ) ).
% subgraph.subgraph_trans
thf(fact_237_subgraph_Oaxioms_I1_J,axiom,
! [V_H: set_a,E_H: set_set_a,V_G: set_a,E_G: set_set_a] :
( ( undire7103218114511261257raph_a @ V_H @ E_H @ V_G @ E_G )
=> ( undire2554140024507503526stem_a @ V_H @ E_H ) ) ).
% subgraph.axioms(1)
thf(fact_238_subgraph_Oaxioms_I2_J,axiom,
! [V_H: set_a,E_H: set_set_a,V_G: set_a,E_G: set_set_a] :
( ( undire7103218114511261257raph_a @ V_H @ E_H @ V_G @ E_G )
=> ( undire2554140024507503526stem_a @ V_G @ E_G ) ) ).
% subgraph.axioms(2)
thf(fact_239_graph__system_Oinduced__is__graph__sys,axiom,
! [Vertices: set_a,Edges: set_set_a,V3: set_a] :
( ( undire2554140024507503526stem_a @ Vertices @ Edges )
=> ( undire2554140024507503526stem_a @ V3 @ ( undire7777452895879145676dges_a @ Edges @ V3 ) ) ) ).
% graph_system.induced_is_graph_sys
thf(fact_240_graph__system_Oincident__edge__in__wf,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,E: set_set_a,V: set_a] :
( ( undire7159349782766787846_set_a @ Vertices @ Edges )
=> ( ( member_set_set_a @ E @ Edges )
=> ( ( undire2320338297334612420_set_a @ V @ E )
=> ( member_set_a @ V @ Vertices ) ) ) ) ).
% graph_system.incident_edge_in_wf
thf(fact_241_graph__system_Oincident__edge__in__wf,axiom,
! [Vertices: set_a,Edges: set_set_a,E: set_a,V: a] :
( ( undire2554140024507503526stem_a @ Vertices @ Edges )
=> ( ( member_set_a @ E @ Edges )
=> ( ( undire1521409233611534436dent_a @ V @ E )
=> ( member_a @ V @ Vertices ) ) ) ) ).
% graph_system.incident_edge_in_wf
thf(fact_242_graph__system_Oincident__def,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,V: set_a,E: set_set_a] :
( ( undire7159349782766787846_set_a @ Vertices @ Edges )
=> ( ( undire2320338297334612420_set_a @ V @ E )
= ( member_set_a @ V @ E ) ) ) ).
% graph_system.incident_def
thf(fact_243_graph__system_Oincident__def,axiom,
! [Vertices: set_a,Edges: set_set_a,V: a,E: set_a] :
( ( undire2554140024507503526stem_a @ Vertices @ Edges )
=> ( ( undire1521409233611534436dent_a @ V @ E )
= ( member_a @ V @ E ) ) ) ).
% graph_system.incident_def
thf(fact_244_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_245_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_246_ulgraph_Ois__gen__path__distinct,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,P: list_set_a] :
( ( undire6886684016831807756_set_a @ Vertices @ Edges )
=> ( ( undire7201326534205417136_set_a @ Vertices @ Edges @ P )
=> ( ( ( hd_set_a @ P )
!= ( last_set_a @ P ) )
=> ( distinct_set_a @ P ) ) ) ) ).
% ulgraph.is_gen_path_distinct
thf(fact_247_ulgraph_Ois__gen__path__distinct,axiom,
! [Vertices: set_a,Edges: set_set_a,P: list_a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( undire3562951555376170320path_a @ Vertices @ Edges @ P )
=> ( ( ( hd_a @ P )
!= ( last_a @ P ) )
=> ( distinct_a @ P ) ) ) ) ).
% ulgraph.is_gen_path_distinct
thf(fact_248_ulgraph_Oconnecting__walk__def,axiom,
! [Vertices: set_a,Edges: set_set_a,U: a,V: a,Xs: list_a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( connecting_walk_a @ Vertices @ Edges @ U @ V @ Xs )
= ( ( undire6133010728901294956walk_a @ Vertices @ Edges @ Xs )
& ( ( hd_a @ Xs )
= U )
& ( ( last_a @ Xs )
= V ) ) ) ) ).
% ulgraph.connecting_walk_def
thf(fact_249_graph__system_Oedge__adjacent__alt__def,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,E1: set_set_a,E2: set_set_a] :
( ( undire7159349782766787846_set_a @ Vertices @ Edges )
=> ( ( member_set_set_a @ E1 @ Edges )
=> ( ( member_set_set_a @ E2 @ Edges )
=> ( ? [X: set_a] :
( ( member_set_a @ X @ Vertices )
& ( member_set_a @ X @ E1 )
& ( member_set_a @ X @ E2 ) )
=> ( undire3485422320110889978_set_a @ Edges @ E1 @ E2 ) ) ) ) ) ).
% graph_system.edge_adjacent_alt_def
thf(fact_250_graph__system_Oedge__adjacent__alt__def,axiom,
! [Vertices: set_a,Edges: set_set_a,E1: set_a,E2: set_a] :
( ( undire2554140024507503526stem_a @ Vertices @ Edges )
=> ( ( member_set_a @ E1 @ Edges )
=> ( ( member_set_a @ E2 @ Edges )
=> ( ? [X: a] :
( ( member_a @ X @ Vertices )
& ( member_a @ X @ E1 )
& ( member_a @ X @ E2 ) )
=> ( undire4022703626023482010_adj_a @ Edges @ E1 @ E2 ) ) ) ) ) ).
% graph_system.edge_adjacent_alt_def
thf(fact_251_graph__system_Oedge__adj__inE,axiom,
! [Vertices: set_a,Edges: set_set_a,E1: set_a,E2: set_a] :
( ( undire2554140024507503526stem_a @ Vertices @ Edges )
=> ( ( undire4022703626023482010_adj_a @ Edges @ E1 @ E2 )
=> ( ( member_set_a @ E1 @ Edges )
& ( member_set_a @ E2 @ Edges ) ) ) ) ).
% graph_system.edge_adj_inE
thf(fact_252_ulgraph_Oconnecting__path__def,axiom,
! [Vertices: set_a,Edges: set_set_a,U: a,V: a,Xs: list_a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( connecting_path_a @ Vertices @ Edges @ U @ V @ Xs )
= ( ( undire3562951555376170320path_a @ Vertices @ Edges @ Xs )
& ( ( hd_a @ Xs )
= U )
& ( ( last_a @ Xs )
= V ) ) ) ) ).
% ulgraph.connecting_path_def
thf(fact_253_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_254_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_255_ulgraph_Oconnecting__path__str__def,axiom,
! [Vertices: set_a,Edges: set_set_a,U: a,V: a,Xs: list_a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( connec3015921205769380621_str_a @ Vertices @ Edges @ U @ V @ Xs )
= ( ( undire427332500224447920path_a @ Vertices @ Edges @ Xs )
& ( ( hd_a @ Xs )
= U )
& ( ( last_a @ Xs )
= V ) ) ) ) ).
% ulgraph.connecting_path_str_def
thf(fact_256_ulgraph_Ois__walk__append,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,Xs: list_set_a,Ys: list_set_a] :
( ( undire6886684016831807756_set_a @ Vertices @ Edges )
=> ( ( undire3014741414213135564_set_a @ Vertices @ Edges @ Xs )
=> ( ( undire3014741414213135564_set_a @ Vertices @ Edges @ Ys )
=> ( ( ( last_set_a @ Xs )
= ( hd_set_a @ Ys ) )
=> ( undire3014741414213135564_set_a @ Vertices @ Edges @ ( append_set_a @ Xs @ ( tl_set_a @ Ys ) ) ) ) ) ) ) ).
% ulgraph.is_walk_append
thf(fact_257_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_258_ulgraph_Ois__gen__path__distinct__tl,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,P: list_set_a] :
( ( undire6886684016831807756_set_a @ Vertices @ Edges )
=> ( ( undire7201326534205417136_set_a @ Vertices @ Edges @ P )
=> ( ( ( hd_set_a @ P )
= ( last_set_a @ P ) )
=> ( distinct_set_a @ ( tl_set_a @ P ) ) ) ) ) ).
% ulgraph.is_gen_path_distinct_tl
thf(fact_259_ulgraph_Ois__gen__path__distinct__tl,axiom,
! [Vertices: set_a,Edges: set_set_a,P: list_a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( undire3562951555376170320path_a @ Vertices @ Edges @ P )
=> ( ( ( hd_a @ P )
= ( last_a @ P ) )
=> ( distinct_a @ ( tl_a @ P ) ) ) ) ) ).
% ulgraph.is_gen_path_distinct_tl
thf(fact_260_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_261_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_262_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_263_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_264_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_265_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_266_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_267_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_268_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_269_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_270_ulgraph_Ois__gen__path__def,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,P: list_set_a] :
( ( undire6886684016831807756_set_a @ Vertices @ Edges )
=> ( ( undire7201326534205417136_set_a @ Vertices @ Edges @ P )
= ( ( undire3014741414213135564_set_a @ Vertices @ Edges @ P )
& ( ( ( distinct_set_a @ ( tl_set_a @ P ) )
& ( ( hd_set_a @ P )
= ( last_set_a @ P ) ) )
| ( distinct_set_a @ P ) ) ) ) ) ).
% ulgraph.is_gen_path_def
thf(fact_271_ulgraph_Ois__gen__path__def,axiom,
! [Vertices: set_a,Edges: set_set_a,P: list_a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( undire3562951555376170320path_a @ Vertices @ Edges @ P )
= ( ( undire6133010728901294956walk_a @ Vertices @ Edges @ P )
& ( ( ( distinct_a @ ( tl_a @ P ) )
& ( ( hd_a @ P )
= ( last_a @ P ) ) )
| ( distinct_a @ P ) ) ) ) ) ).
% ulgraph.is_gen_path_def
thf(fact_272_order__antisym__conv,axiom,
! [Y: set_a,X4: set_a] :
( ( ord_less_eq_set_a @ Y @ X4 )
=> ( ( ord_less_eq_set_a @ X4 @ Y )
= ( X4 = Y ) ) ) ).
% order_antisym_conv
thf(fact_273_order__antisym__conv,axiom,
! [Y: set_set_a,X4: set_set_a] :
( ( ord_le3724670747650509150_set_a @ Y @ X4 )
=> ( ( ord_le3724670747650509150_set_a @ X4 @ Y )
= ( X4 = Y ) ) ) ).
% order_antisym_conv
thf(fact_274_order__antisym__conv,axiom,
! [Y: nat,X4: nat] :
( ( ord_less_eq_nat @ Y @ X4 )
=> ( ( ord_less_eq_nat @ X4 @ Y )
= ( X4 = Y ) ) ) ).
% order_antisym_conv
thf(fact_275_order__antisym__conv,axiom,
! [Y: real,X4: real] :
( ( ord_less_eq_real @ Y @ X4 )
=> ( ( ord_less_eq_real @ X4 @ Y )
= ( X4 = Y ) ) ) ).
% order_antisym_conv
thf(fact_276_linorder__le__cases,axiom,
! [X4: nat,Y: nat] :
( ~ ( ord_less_eq_nat @ X4 @ Y )
=> ( ord_less_eq_nat @ Y @ X4 ) ) ).
% linorder_le_cases
thf(fact_277_linorder__le__cases,axiom,
! [X4: real,Y: real] :
( ~ ( ord_less_eq_real @ X4 @ Y )
=> ( ord_less_eq_real @ Y @ X4 ) ) ).
% linorder_le_cases
thf(fact_278_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_279_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > real,C: real] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_280_ord__le__eq__subst,axiom,
! [A: real,B: real,F: real > nat,C: nat] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_281_ord__le__eq__subst,axiom,
! [A: real,B: real,F: real > real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_282_ord__le__eq__subst,axiom,
! [A: set_a,B: set_a,F: set_a > nat,C: nat] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_283_ord__le__eq__subst,axiom,
! [A: set_a,B: set_a,F: set_a > real,C: real] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_284_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 )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_285_ord__le__eq__subst,axiom,
! [A: real,B: real,F: real > set_a,C: set_a] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_286_ord__le__eq__subst,axiom,
! [A: set_a,B: set_a,F: set_a > set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_287_ord__le__eq__subst,axiom,
! [A: set_set_a,B: set_set_a,F: set_set_a > nat,C: nat] :
( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: set_set_a,Y2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_288_ord__eq__le__subst,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_289_ord__eq__le__subst,axiom,
! [A: real,F: nat > real,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_290_ord__eq__le__subst,axiom,
! [A: nat,F: real > nat,B: real,C: real] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_291_ord__eq__le__subst,axiom,
! [A: real,F: real > real,B: real,C: real] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_292_ord__eq__le__subst,axiom,
! [A: nat,F: set_a > nat,B: set_a,C: set_a] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ! [X3: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_293_ord__eq__le__subst,axiom,
! [A: real,F: set_a > real,B: set_a,C: set_a] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ! [X3: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_294_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 )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_295_ord__eq__le__subst,axiom,
! [A: set_a,F: real > set_a,B: real,C: real] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_296_ord__eq__le__subst,axiom,
! [A: set_a,F: set_a > set_a,B: set_a,C: set_a] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ! [X3: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_297_ord__eq__le__subst,axiom,
! [A: nat,F: set_set_a > nat,B: set_set_a,C: set_set_a] :
( ( A
= ( F @ B ) )
=> ( ( ord_le3724670747650509150_set_a @ B @ C )
=> ( ! [X3: set_set_a,Y2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_298_linorder__linear,axiom,
! [X4: nat,Y: nat] :
( ( ord_less_eq_nat @ X4 @ Y )
| ( ord_less_eq_nat @ Y @ X4 ) ) ).
% linorder_linear
thf(fact_299_linorder__linear,axiom,
! [X4: real,Y: real] :
( ( ord_less_eq_real @ X4 @ Y )
| ( ord_less_eq_real @ Y @ X4 ) ) ).
% linorder_linear
thf(fact_300_order__eq__refl,axiom,
! [X4: set_a,Y: set_a] :
( ( X4 = Y )
=> ( ord_less_eq_set_a @ X4 @ Y ) ) ).
% order_eq_refl
thf(fact_301_order__eq__refl,axiom,
! [X4: set_set_a,Y: set_set_a] :
( ( X4 = Y )
=> ( ord_le3724670747650509150_set_a @ X4 @ Y ) ) ).
% order_eq_refl
thf(fact_302_order__eq__refl,axiom,
! [X4: nat,Y: nat] :
( ( X4 = Y )
=> ( ord_less_eq_nat @ X4 @ Y ) ) ).
% order_eq_refl
thf(fact_303_order__eq__refl,axiom,
! [X4: real,Y: real] :
( ( X4 = Y )
=> ( ord_less_eq_real @ X4 @ Y ) ) ).
% order_eq_refl
thf(fact_304_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 )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_305_order__subst2,axiom,
! [A: nat,B: nat,F: nat > real,C: real] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_306_order__subst2,axiom,
! [A: real,B: real,F: real > nat,C: nat] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_307_order__subst2,axiom,
! [A: real,B: real,F: real > real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_308_order__subst2,axiom,
! [A: set_a,B: set_a,F: set_a > nat,C: nat] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X3: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_309_order__subst2,axiom,
! [A: set_a,B: set_a,F: set_a > real,C: real] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C )
=> ( ! [X3: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_310_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 )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_311_order__subst2,axiom,
! [A: real,B: real,F: real > set_a,C: set_a] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_set_a @ ( F @ B ) @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_312_order__subst2,axiom,
! [A: set_a,B: set_a,F: set_a > set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_eq_set_a @ ( F @ B ) @ C )
=> ( ! [X3: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_313_order__subst2,axiom,
! [A: set_set_a,B: set_set_a,F: set_set_a > nat,C: nat] :
( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X3: set_set_a,Y2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_314_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 )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_315_order__subst1,axiom,
! [A: nat,F: real > nat,B: real,C: real] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_316_order__subst1,axiom,
! [A: real,F: nat > real,B: nat,C: nat] :
( ( ord_less_eq_real @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_317_order__subst1,axiom,
! [A: real,F: real > real,B: real,C: real] :
( ( ord_less_eq_real @ A @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_318_order__subst1,axiom,
! [A: set_a,F: nat > set_a,B: nat,C: nat] :
( ( ord_less_eq_set_a @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_319_order__subst1,axiom,
! [A: set_a,F: real > set_a,B: real,C: real] :
( ( ord_less_eq_set_a @ A @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_320_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 )
=> ( ! [X3: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_321_order__subst1,axiom,
! [A: real,F: set_a > real,B: set_a,C: set_a] :
( ( ord_less_eq_real @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ! [X3: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_322_order__subst1,axiom,
! [A: set_a,F: set_a > set_a,B: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ! [X3: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_323_order__subst1,axiom,
! [A: set_set_a,F: nat > set_set_a,B: nat,C: nat] :
( ( ord_le3724670747650509150_set_a @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_le3724670747650509150_set_a @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le3724670747650509150_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_324_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: set_a,Z2: set_a] : ( Y3 = Z2 ) )
= ( ^ [A3: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ A3 @ B3 )
& ( ord_less_eq_set_a @ B3 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_325_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: set_set_a,Z2: set_set_a] : ( Y3 = Z2 ) )
= ( ^ [A3: set_set_a,B3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A3 @ B3 )
& ( ord_le3724670747650509150_set_a @ B3 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_326_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: nat,Z2: nat] : ( Y3 = Z2 ) )
= ( ^ [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
& ( ord_less_eq_nat @ B3 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_327_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: real,Z2: real] : ( Y3 = Z2 ) )
= ( ^ [A3: real,B3: real] :
( ( ord_less_eq_real @ A3 @ B3 )
& ( ord_less_eq_real @ B3 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_328_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_329_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_330_antisym,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_331_antisym,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_332_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_333_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_334_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_335_dual__order_Otrans,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( ord_less_eq_real @ C @ B )
=> ( ord_less_eq_real @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_336_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_337_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_338_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_339_dual__order_Oantisym,axiom,
! [B: real,A: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( ord_less_eq_real @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_340_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: set_a,Z2: set_a] : ( Y3 = Z2 ) )
= ( ^ [A3: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ B3 @ A3 )
& ( ord_less_eq_set_a @ A3 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_341_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: set_set_a,Z2: set_set_a] : ( Y3 = Z2 ) )
= ( ^ [A3: set_set_a,B3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B3 @ A3 )
& ( ord_le3724670747650509150_set_a @ A3 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_342_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: nat,Z2: nat] : ( Y3 = Z2 ) )
= ( ^ [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ B3 @ A3 )
& ( ord_less_eq_nat @ A3 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_343_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: real,Z2: real] : ( Y3 = Z2 ) )
= ( ^ [A3: real,B3: real] :
( ( ord_less_eq_real @ B3 @ A3 )
& ( ord_less_eq_real @ A3 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_344_linorder__wlog,axiom,
! [P2: nat > nat > $o,A: nat,B: nat] :
( ! [A4: nat,B4: nat] :
( ( ord_less_eq_nat @ A4 @ B4 )
=> ( P2 @ A4 @ B4 ) )
=> ( ! [A4: nat,B4: nat] :
( ( P2 @ B4 @ A4 )
=> ( P2 @ A4 @ B4 ) )
=> ( P2 @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_345_linorder__wlog,axiom,
! [P2: real > real > $o,A: real,B: real] :
( ! [A4: real,B4: real] :
( ( ord_less_eq_real @ A4 @ B4 )
=> ( P2 @ A4 @ B4 ) )
=> ( ! [A4: real,B4: real] :
( ( P2 @ B4 @ A4 )
=> ( P2 @ A4 @ B4 ) )
=> ( P2 @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_346_order__trans,axiom,
! [X4: set_a,Y: set_a,Z: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y )
=> ( ( ord_less_eq_set_a @ Y @ Z )
=> ( ord_less_eq_set_a @ X4 @ Z ) ) ) ).
% order_trans
thf(fact_347_order__trans,axiom,
! [X4: set_set_a,Y: set_set_a,Z: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X4 @ Y )
=> ( ( ord_le3724670747650509150_set_a @ Y @ Z )
=> ( ord_le3724670747650509150_set_a @ X4 @ Z ) ) ) ).
% order_trans
thf(fact_348_order__trans,axiom,
! [X4: nat,Y: nat,Z: nat] :
( ( ord_less_eq_nat @ X4 @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z )
=> ( ord_less_eq_nat @ X4 @ Z ) ) ) ).
% order_trans
thf(fact_349_order__trans,axiom,
! [X4: real,Y: real,Z: real] :
( ( ord_less_eq_real @ X4 @ Y )
=> ( ( ord_less_eq_real @ Y @ Z )
=> ( ord_less_eq_real @ X4 @ Z ) ) ) ).
% order_trans
thf(fact_350_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_351_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_352_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_353_order_Otrans,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ord_less_eq_real @ A @ C ) ) ) ).
% order.trans
thf(fact_354_order__antisym,axiom,
! [X4: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y )
=> ( ( ord_less_eq_set_a @ Y @ X4 )
=> ( X4 = Y ) ) ) ).
% order_antisym
thf(fact_355_order__antisym,axiom,
! [X4: set_set_a,Y: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X4 @ Y )
=> ( ( ord_le3724670747650509150_set_a @ Y @ X4 )
=> ( X4 = Y ) ) ) ).
% order_antisym
thf(fact_356_order__antisym,axiom,
! [X4: nat,Y: nat] :
( ( ord_less_eq_nat @ X4 @ Y )
=> ( ( ord_less_eq_nat @ Y @ X4 )
=> ( X4 = Y ) ) ) ).
% order_antisym
thf(fact_357_order__antisym,axiom,
! [X4: real,Y: real] :
( ( ord_less_eq_real @ X4 @ Y )
=> ( ( ord_less_eq_real @ Y @ X4 )
=> ( X4 = Y ) ) ) ).
% order_antisym
thf(fact_358_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_359_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_360_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_361_ord__le__eq__trans,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_real @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_362_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_363_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_364_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_365_ord__eq__le__trans,axiom,
! [A: real,B: real,C: real] :
( ( A = B )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ord_less_eq_real @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_366_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: set_a,Z2: set_a] : ( Y3 = Z2 ) )
= ( ^ [X2: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y4 )
& ( ord_less_eq_set_a @ Y4 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_367_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: set_set_a,Z2: set_set_a] : ( Y3 = Z2 ) )
= ( ^ [X2: set_set_a,Y4: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X2 @ Y4 )
& ( ord_le3724670747650509150_set_a @ Y4 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_368_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: nat,Z2: nat] : ( Y3 = Z2 ) )
= ( ^ [X2: nat,Y4: nat] :
( ( ord_less_eq_nat @ X2 @ Y4 )
& ( ord_less_eq_nat @ Y4 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_369_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: real,Z2: real] : ( Y3 = Z2 ) )
= ( ^ [X2: real,Y4: real] :
( ( ord_less_eq_real @ X2 @ Y4 )
& ( ord_less_eq_real @ Y4 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_370_le__cases3,axiom,
! [X4: nat,Y: nat,Z: nat] :
( ( ( ord_less_eq_nat @ X4 @ Y )
=> ~ ( ord_less_eq_nat @ Y @ Z ) )
=> ( ( ( ord_less_eq_nat @ Y @ X4 )
=> ~ ( ord_less_eq_nat @ X4 @ Z ) )
=> ( ( ( ord_less_eq_nat @ X4 @ Z )
=> ~ ( ord_less_eq_nat @ Z @ Y ) )
=> ( ( ( ord_less_eq_nat @ Z @ Y )
=> ~ ( ord_less_eq_nat @ Y @ X4 ) )
=> ( ( ( ord_less_eq_nat @ Y @ Z )
=> ~ ( ord_less_eq_nat @ Z @ X4 ) )
=> ~ ( ( ord_less_eq_nat @ Z @ X4 )
=> ~ ( ord_less_eq_nat @ X4 @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_371_le__cases3,axiom,
! [X4: real,Y: real,Z: real] :
( ( ( ord_less_eq_real @ X4 @ Y )
=> ~ ( ord_less_eq_real @ Y @ Z ) )
=> ( ( ( ord_less_eq_real @ Y @ X4 )
=> ~ ( ord_less_eq_real @ X4 @ Z ) )
=> ( ( ( ord_less_eq_real @ X4 @ Z )
=> ~ ( ord_less_eq_real @ Z @ Y ) )
=> ( ( ( ord_less_eq_real @ Z @ Y )
=> ~ ( ord_less_eq_real @ Y @ X4 ) )
=> ( ( ( ord_less_eq_real @ Y @ Z )
=> ~ ( ord_less_eq_real @ Z @ X4 ) )
=> ~ ( ( ord_less_eq_real @ Z @ X4 )
=> ~ ( ord_less_eq_real @ X4 @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_372_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_373_nle__le,axiom,
! [A: real,B: real] :
( ( ~ ( ord_less_eq_real @ A @ B ) )
= ( ( ord_less_eq_real @ B @ A )
& ( B != A ) ) ) ).
% nle_le
thf(fact_374_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_375_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_376_Collect__mono__iff,axiom,
! [P2: a > $o,Q: a > $o] :
( ( ord_less_eq_set_a @ ( collect_a @ P2 ) @ ( collect_a @ Q ) )
= ( ! [X2: a] :
( ( P2 @ X2 )
=> ( Q @ X2 ) ) ) ) ).
% Collect_mono_iff
thf(fact_377_Collect__mono__iff,axiom,
! [P2: set_a > $o,Q: set_a > $o] :
( ( ord_le3724670747650509150_set_a @ ( collect_set_a @ P2 ) @ ( collect_set_a @ Q ) )
= ( ! [X2: set_a] :
( ( P2 @ X2 )
=> ( Q @ X2 ) ) ) ) ).
% Collect_mono_iff
thf(fact_378_set__eq__subset,axiom,
( ( ^ [Y3: set_a,Z2: set_a] : ( Y3 = Z2 ) )
= ( ^ [A5: set_a,B5: set_a] :
( ( ord_less_eq_set_a @ A5 @ B5 )
& ( ord_less_eq_set_a @ B5 @ A5 ) ) ) ) ).
% set_eq_subset
thf(fact_379_set__eq__subset,axiom,
( ( ^ [Y3: set_set_a,Z2: set_set_a] : ( Y3 = Z2 ) )
= ( ^ [A5: set_set_a,B5: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A5 @ B5 )
& ( ord_le3724670747650509150_set_a @ B5 @ A5 ) ) ) ) ).
% set_eq_subset
thf(fact_380_subset__trans,axiom,
! [A2: set_a,B2: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ord_less_eq_set_a @ B2 @ C2 )
=> ( ord_less_eq_set_a @ A2 @ C2 ) ) ) ).
% subset_trans
thf(fact_381_subset__trans,axiom,
! [A2: set_set_a,B2: set_set_a,C2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B2 )
=> ( ( ord_le3724670747650509150_set_a @ B2 @ C2 )
=> ( ord_le3724670747650509150_set_a @ A2 @ C2 ) ) ) ).
% subset_trans
thf(fact_382_Collect__mono,axiom,
! [P2: a > $o,Q: a > $o] :
( ! [X3: a] :
( ( P2 @ X3 )
=> ( Q @ X3 ) )
=> ( ord_less_eq_set_a @ ( collect_a @ P2 ) @ ( collect_a @ Q ) ) ) ).
% Collect_mono
thf(fact_383_Collect__mono,axiom,
! [P2: set_a > $o,Q: set_a > $o] :
( ! [X3: set_a] :
( ( P2 @ X3 )
=> ( Q @ X3 ) )
=> ( ord_le3724670747650509150_set_a @ ( collect_set_a @ P2 ) @ ( collect_set_a @ Q ) ) ) ).
% Collect_mono
thf(fact_384_subset__refl,axiom,
! [A2: set_a] : ( ord_less_eq_set_a @ A2 @ A2 ) ).
% subset_refl
thf(fact_385_subset__refl,axiom,
! [A2: set_set_a] : ( ord_le3724670747650509150_set_a @ A2 @ A2 ) ).
% subset_refl
thf(fact_386_subset__iff,axiom,
( ord_less_eq_set_a
= ( ^ [A5: set_a,B5: set_a] :
! [T: a] :
( ( member_a @ T @ A5 )
=> ( member_a @ T @ B5 ) ) ) ) ).
% subset_iff
thf(fact_387_subset__iff,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [A5: set_set_a,B5: set_set_a] :
! [T: set_a] :
( ( member_set_a @ T @ A5 )
=> ( member_set_a @ T @ B5 ) ) ) ) ).
% subset_iff
thf(fact_388_equalityD2,axiom,
! [A2: set_a,B2: set_a] :
( ( A2 = B2 )
=> ( ord_less_eq_set_a @ B2 @ A2 ) ) ).
% equalityD2
thf(fact_389_equalityD2,axiom,
! [A2: set_set_a,B2: set_set_a] :
( ( A2 = B2 )
=> ( ord_le3724670747650509150_set_a @ B2 @ A2 ) ) ).
% equalityD2
thf(fact_390_equalityD1,axiom,
! [A2: set_a,B2: set_a] :
( ( A2 = B2 )
=> ( ord_less_eq_set_a @ A2 @ B2 ) ) ).
% equalityD1
thf(fact_391_equalityD1,axiom,
! [A2: set_set_a,B2: set_set_a] :
( ( A2 = B2 )
=> ( ord_le3724670747650509150_set_a @ A2 @ B2 ) ) ).
% equalityD1
thf(fact_392_subset__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A5: set_a,B5: set_a] :
! [X2: a] :
( ( member_a @ X2 @ A5 )
=> ( member_a @ X2 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_393_subset__eq,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [A5: set_set_a,B5: set_set_a] :
! [X2: set_a] :
( ( member_set_a @ X2 @ A5 )
=> ( member_set_a @ X2 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_394_equalityE,axiom,
! [A2: set_a,B2: set_a] :
( ( A2 = B2 )
=> ~ ( ( ord_less_eq_set_a @ A2 @ B2 )
=> ~ ( ord_less_eq_set_a @ B2 @ A2 ) ) ) ).
% equalityE
thf(fact_395_equalityE,axiom,
! [A2: set_set_a,B2: set_set_a] :
( ( A2 = B2 )
=> ~ ( ( ord_le3724670747650509150_set_a @ A2 @ B2 )
=> ~ ( ord_le3724670747650509150_set_a @ B2 @ A2 ) ) ) ).
% equalityE
thf(fact_396_subsetD,axiom,
! [A2: set_a,B2: set_a,C: a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( member_a @ C @ A2 )
=> ( member_a @ C @ B2 ) ) ) ).
% subsetD
thf(fact_397_subsetD,axiom,
! [A2: set_set_a,B2: set_set_a,C: set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B2 )
=> ( ( member_set_a @ C @ A2 )
=> ( member_set_a @ C @ B2 ) ) ) ).
% subsetD
thf(fact_398_in__mono,axiom,
! [A2: set_a,B2: set_a,X4: a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( member_a @ X4 @ A2 )
=> ( member_a @ X4 @ B2 ) ) ) ).
% in_mono
thf(fact_399_in__mono,axiom,
! [A2: set_set_a,B2: set_set_a,X4: set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B2 )
=> ( ( member_set_a @ X4 @ A2 )
=> ( member_set_a @ X4 @ B2 ) ) ) ).
% in_mono
thf(fact_400_graph__system_Oinduced__edges__ss,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,V3: set_set_a] :
( ( undire7159349782766787846_set_a @ Vertices @ Edges )
=> ( ( ord_le3724670747650509150_set_a @ V3 @ Vertices )
=> ( ord_le5722252365846178494_set_a @ ( undire7854589003810675244_set_a @ Edges @ V3 ) @ Edges ) ) ) ).
% graph_system.induced_edges_ss
thf(fact_401_graph__system_Oinduced__edges__ss,axiom,
! [Vertices: set_a,Edges: set_set_a,V3: set_a] :
( ( undire2554140024507503526stem_a @ Vertices @ Edges )
=> ( ( ord_less_eq_set_a @ V3 @ Vertices )
=> ( ord_le3724670747650509150_set_a @ ( undire7777452895879145676dges_a @ Edges @ V3 ) @ Edges ) ) ) ).
% graph_system.induced_edges_ss
thf(fact_402_graph__system_Oinduced__is__subgraph,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,V3: set_set_a] :
( ( undire7159349782766787846_set_a @ Vertices @ Edges )
=> ( ( ord_le3724670747650509150_set_a @ V3 @ Vertices )
=> ( undire1186139521737116585_set_a @ V3 @ ( undire7854589003810675244_set_a @ Edges @ V3 ) @ Vertices @ Edges ) ) ) ).
% graph_system.induced_is_subgraph
thf(fact_403_graph__system_Oinduced__is__subgraph,axiom,
! [Vertices: set_a,Edges: set_set_a,V3: set_a] :
( ( undire2554140024507503526stem_a @ Vertices @ Edges )
=> ( ( ord_less_eq_set_a @ V3 @ Vertices )
=> ( undire7103218114511261257raph_a @ V3 @ ( undire7777452895879145676dges_a @ Edges @ V3 ) @ Vertices @ Edges ) ) ) ).
% graph_system.induced_is_subgraph
thf(fact_404_is__gen__path__options,axiom,
! [P: list_a] :
( ( undire3562951555376170320path_a @ vertices @ edges @ P )
= ( ( undire2407311113669455967ycle_a @ vertices @ edges @ P )
| ( undire427332500224447920path_a @ vertices @ edges @ P )
| ? [X2: a] :
( ( member_a @ X2 @ vertices )
& ( P
= ( cons_a @ X2 @ nil_a ) ) ) ) ) ).
% is_gen_path_options
thf(fact_405_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_406_connecting__path__split,axiom,
! [U: a,V: a,Xs: list_a,Z: a,Ys: list_a] :
( ( connecting_path_a @ vertices @ edges @ U @ V @ Xs )
=> ( ( connecting_path_a @ vertices @ edges @ V @ Z @ Ys )
=> ~ ! [P3: list_a] :
( ( connecting_path_a @ vertices @ edges @ U @ Z @ P3 )
=> ~ ( ord_less_eq_nat @ ( undire8849074589633906640ngth_a @ P3 ) @ ( undire8849074589633906640ngth_a @ ( append_a @ Xs @ ( tl_a @ Ys ) ) ) ) ) ) ) ).
% connecting_path_split
thf(fact_407_connecting__walk__self,axiom,
! [U: a] :
( ( member_a @ U @ vertices )
=> ( connecting_walk_a @ vertices @ edges @ U @ U @ ( cons_a @ U @ nil_a ) ) ) ).
% connecting_walk_self
thf(fact_408_is__gen__path__trivial,axiom,
! [X4: a] :
( ( member_a @ X4 @ vertices )
=> ( undire3562951555376170320path_a @ vertices @ edges @ ( cons_a @ X4 @ nil_a ) ) ) ).
% is_gen_path_trivial
thf(fact_409_connecting__path__self,axiom,
! [U: a] :
( ( member_a @ U @ vertices )
=> ( connecting_path_a @ vertices @ edges @ U @ U @ ( cons_a @ U @ nil_a ) ) ) ).
% connecting_path_self
thf(fact_410_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_411_is__walk__singleton,axiom,
! [U: a] :
( ( member_a @ U @ vertices )
=> ( undire6133010728901294956walk_a @ vertices @ edges @ ( cons_a @ U @ nil_a ) ) ) ).
% is_walk_singleton
thf(fact_412_connecting__walk__path,axiom,
! [U: a,V: a,Xs: list_a] :
( ( connecting_walk_a @ vertices @ edges @ U @ V @ Xs )
=> ? [Ys3: list_a] :
( ( connecting_path_a @ vertices @ edges @ U @ V @ Ys3 )
& ( ord_less_eq_nat @ ( undire8849074589633906640ngth_a @ Ys3 ) @ ( undire8849074589633906640ngth_a @ Xs ) ) ) ) ).
% connecting_walk_path
thf(fact_413_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_414_walk__edges_Ocases,axiom,
! [X4: list_a] :
( ( X4 != nil_a )
=> ( ! [X3: a] :
( X4
!= ( cons_a @ X3 @ nil_a ) )
=> ~ ! [X3: a,Y2: a,Ys3: list_a] :
( X4
!= ( cons_a @ X3 @ ( cons_a @ Y2 @ Ys3 ) ) ) ) ) ).
% walk_edges.cases
thf(fact_415_walk__edges_Osimps_I1_J,axiom,
( ( undire7337870655677353998dges_a @ nil_a )
= nil_set_a ) ).
% walk_edges.simps(1)
thf(fact_416_walk__edges__rev,axiom,
! [Xs: list_a] :
( ( rev_set_a @ ( undire7337870655677353998dges_a @ Xs ) )
= ( undire7337870655677353998dges_a @ ( rev_a @ Xs ) ) ) ).
% walk_edges_rev
thf(fact_417_walk__length__rev,axiom,
( undire8849074589633906640ngth_a
= ( ^ [P4: list_a] : ( undire8849074589633906640ngth_a @ ( rev_a @ P4 ) ) ) ) ).
% walk_length_rev
thf(fact_418_walk__edges_Osimps_I2_J,axiom,
! [X4: a] :
( ( undire7337870655677353998dges_a @ ( cons_a @ X4 @ nil_a ) )
= nil_set_a ) ).
% walk_edges.simps(2)
thf(fact_419_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_420_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_421_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_422_distinct__edgesI,axiom,
! [P: list_a] :
( ( distinct_a @ P )
=> ( distinct_set_a @ ( undire7337870655677353998dges_a @ P ) ) ) ).
% distinct_edgesI
thf(fact_423_list_Oinject,axiom,
! [X21: a,X22: list_a,Y21: a,Y22: list_a] :
( ( ( cons_a @ X21 @ X22 )
= ( cons_a @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% list.inject
thf(fact_424_list_Oinject,axiom,
! [X21: set_a,X22: list_set_a,Y21: set_a,Y22: list_set_a] :
( ( ( cons_set_a @ X21 @ X22 )
= ( cons_set_a @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% list.inject
thf(fact_425_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_426_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_427_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_428_append1__eq__conv,axiom,
! [Xs: list_a,X4: a,Ys: list_a,Y: a] :
( ( ( append_a @ Xs @ ( cons_a @ X4 @ nil_a ) )
= ( append_a @ Ys @ ( cons_a @ Y @ nil_a ) ) )
= ( ( Xs = Ys )
& ( X4 = Y ) ) ) ).
% append1_eq_conv
thf(fact_429_append1__eq__conv,axiom,
! [Xs: list_set_a,X4: set_a,Ys: list_set_a,Y: set_a] :
( ( ( append_set_a @ Xs @ ( cons_set_a @ X4 @ nil_set_a ) )
= ( append_set_a @ Ys @ ( cons_set_a @ Y @ nil_set_a ) ) )
= ( ( Xs = Ys )
& ( X4 = Y ) ) ) ).
% append1_eq_conv
thf(fact_430_rev__singleton__conv,axiom,
! [Xs: list_a,X4: a] :
( ( ( rev_a @ Xs )
= ( cons_a @ X4 @ nil_a ) )
= ( Xs
= ( cons_a @ X4 @ nil_a ) ) ) ).
% rev_singleton_conv
thf(fact_431_rev__singleton__conv,axiom,
! [Xs: list_set_a,X4: set_a] :
( ( ( rev_set_a @ Xs )
= ( cons_set_a @ X4 @ nil_set_a ) )
= ( Xs
= ( cons_set_a @ X4 @ nil_set_a ) ) ) ).
% rev_singleton_conv
thf(fact_432_singleton__rev__conv,axiom,
! [X4: a,Xs: list_a] :
( ( ( cons_a @ X4 @ nil_a )
= ( rev_a @ Xs ) )
= ( ( cons_a @ X4 @ nil_a )
= Xs ) ) ).
% singleton_rev_conv
thf(fact_433_singleton__rev__conv,axiom,
! [X4: set_a,Xs: list_set_a] :
( ( ( cons_set_a @ X4 @ nil_set_a )
= ( rev_set_a @ Xs ) )
= ( ( cons_set_a @ X4 @ nil_set_a )
= Xs ) ) ).
% singleton_rev_conv
thf(fact_434_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_435_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_436_last__snoc,axiom,
! [Xs: list_a,X4: a] :
( ( last_a @ ( append_a @ Xs @ ( cons_a @ X4 @ nil_a ) ) )
= X4 ) ).
% last_snoc
thf(fact_437_last__snoc,axiom,
! [Xs: list_set_a,X4: set_a] :
( ( last_set_a @ ( append_set_a @ Xs @ ( cons_set_a @ X4 @ nil_set_a ) ) )
= X4 ) ).
% last_snoc
thf(fact_438_list_Ocollapse,axiom,
! [List: list_a] :
( ( List != nil_a )
=> ( ( cons_a @ ( hd_a @ List ) @ ( tl_a @ List ) )
= List ) ) ).
% list.collapse
thf(fact_439_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_440_hd__Cons__tl,axiom,
! [Xs: list_a] :
( ( Xs != nil_a )
=> ( ( cons_a @ ( hd_a @ Xs ) @ ( tl_a @ Xs ) )
= Xs ) ) ).
% hd_Cons_tl
thf(fact_441_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_442_comp__sgraph_Owalk__edges_Osimps_I2_J,axiom,
! [X4: set_a] :
( ( undire6234387080713648494_set_a @ ( cons_set_a @ X4 @ nil_set_a ) )
= nil_set_set_a ) ).
% comp_sgraph.walk_edges.simps(2)
thf(fact_443_comp__sgraph_Owalk__edges_Osimps_I2_J,axiom,
! [X4: a] :
( ( undire7337870655677353998dges_a @ ( cons_a @ X4 @ nil_a ) )
= nil_set_a ) ).
% comp_sgraph.walk_edges.simps(2)
thf(fact_444_transpose_Ocases,axiom,
! [X4: list_list_a] :
( ( X4 != nil_list_a )
=> ( ! [Xss: list_list_a] :
( X4
!= ( cons_list_a @ nil_a @ Xss ) )
=> ~ ! [X3: a,Xs3: list_a,Xss: list_list_a] :
( X4
!= ( cons_list_a @ ( cons_a @ X3 @ Xs3 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_445_transpose_Ocases,axiom,
! [X4: list_list_set_a] :
( ( X4 != nil_list_set_a )
=> ( ! [Xss: list_list_set_a] :
( X4
!= ( cons_list_set_a @ nil_set_a @ Xss ) )
=> ~ ! [X3: set_a,Xs3: list_set_a,Xss: list_list_set_a] :
( X4
!= ( cons_list_set_a @ ( cons_set_a @ X3 @ Xs3 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_446_not__Cons__self2,axiom,
! [X4: a,Xs: list_a] :
( ( cons_a @ X4 @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_447_not__Cons__self2,axiom,
! [X4: set_a,Xs: list_set_a] :
( ( cons_set_a @ X4 @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_448_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_449_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_450_ulgraph_Owalk__edges_Osimps_I2_J,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,X4: set_a] :
( ( undire6886684016831807756_set_a @ Vertices @ Edges )
=> ( ( undire6234387080713648494_set_a @ ( cons_set_a @ X4 @ nil_set_a ) )
= nil_set_set_a ) ) ).
% ulgraph.walk_edges.simps(2)
thf(fact_451_ulgraph_Owalk__edges_Osimps_I2_J,axiom,
! [Vertices: set_a,Edges: set_set_a,X4: a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( undire7337870655677353998dges_a @ ( cons_a @ X4 @ nil_a ) )
= nil_set_a ) ) ).
% ulgraph.walk_edges.simps(2)
thf(fact_452_comp__sgraph_Owalk__edges__append__ss2,axiom,
! [Xs: list_set_a,Ys: list_set_a] : ( ord_le5722252365846178494_set_a @ ( set_set_set_a2 @ ( undire6234387080713648494_set_a @ Xs ) ) @ ( set_set_set_a2 @ ( undire6234387080713648494_set_a @ ( append_set_a @ Xs @ Ys ) ) ) ) ).
% comp_sgraph.walk_edges_append_ss2
thf(fact_453_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_454_comp__sgraph_Owalk__edges__append__ss1,axiom,
! [Ys: list_set_a,Xs: list_set_a] : ( ord_le5722252365846178494_set_a @ ( set_set_set_a2 @ ( undire6234387080713648494_set_a @ Ys ) ) @ ( set_set_set_a2 @ ( undire6234387080713648494_set_a @ ( append_set_a @ Xs @ Ys ) ) ) ) ).
% comp_sgraph.walk_edges_append_ss1
thf(fact_455_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_456_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_457_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_458_comp__sgraph_Owalk__edges_Osimps_I1_J,axiom,
( ( undire7337870655677353998dges_a @ nil_a )
= nil_set_a ) ).
% comp_sgraph.walk_edges.simps(1)
thf(fact_459_list__nonempty__induct,axiom,
! [Xs: list_a,P2: list_a > $o] :
( ( Xs != nil_a )
=> ( ! [X3: a] : ( P2 @ ( cons_a @ X3 @ nil_a ) )
=> ( ! [X3: a,Xs3: list_a] :
( ( Xs3 != nil_a )
=> ( ( P2 @ Xs3 )
=> ( P2 @ ( cons_a @ X3 @ Xs3 ) ) ) )
=> ( P2 @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_460_list__nonempty__induct,axiom,
! [Xs: list_set_a,P2: list_set_a > $o] :
( ( Xs != nil_set_a )
=> ( ! [X3: set_a] : ( P2 @ ( cons_set_a @ X3 @ nil_set_a ) )
=> ( ! [X3: set_a,Xs3: list_set_a] :
( ( Xs3 != nil_set_a )
=> ( ( P2 @ Xs3 )
=> ( P2 @ ( cons_set_a @ X3 @ Xs3 ) ) ) )
=> ( P2 @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_461_list__induct2_H,axiom,
! [P2: list_a > list_a > $o,Xs: list_a,Ys: list_a] :
( ( P2 @ nil_a @ nil_a )
=> ( ! [X3: a,Xs3: list_a] : ( P2 @ ( cons_a @ X3 @ Xs3 ) @ nil_a )
=> ( ! [Y2: a,Ys3: list_a] : ( P2 @ nil_a @ ( cons_a @ Y2 @ Ys3 ) )
=> ( ! [X3: a,Xs3: list_a,Y2: a,Ys3: list_a] :
( ( P2 @ Xs3 @ Ys3 )
=> ( P2 @ ( cons_a @ X3 @ Xs3 ) @ ( cons_a @ Y2 @ Ys3 ) ) )
=> ( P2 @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_462_list__induct2_H,axiom,
! [P2: list_a > list_set_a > $o,Xs: list_a,Ys: list_set_a] :
( ( P2 @ nil_a @ nil_set_a )
=> ( ! [X3: a,Xs3: list_a] : ( P2 @ ( cons_a @ X3 @ Xs3 ) @ nil_set_a )
=> ( ! [Y2: set_a,Ys3: list_set_a] : ( P2 @ nil_a @ ( cons_set_a @ Y2 @ Ys3 ) )
=> ( ! [X3: a,Xs3: list_a,Y2: set_a,Ys3: list_set_a] :
( ( P2 @ Xs3 @ Ys3 )
=> ( P2 @ ( cons_a @ X3 @ Xs3 ) @ ( cons_set_a @ Y2 @ Ys3 ) ) )
=> ( P2 @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_463_list__induct2_H,axiom,
! [P2: list_set_a > list_a > $o,Xs: list_set_a,Ys: list_a] :
( ( P2 @ nil_set_a @ nil_a )
=> ( ! [X3: set_a,Xs3: list_set_a] : ( P2 @ ( cons_set_a @ X3 @ Xs3 ) @ nil_a )
=> ( ! [Y2: a,Ys3: list_a] : ( P2 @ nil_set_a @ ( cons_a @ Y2 @ Ys3 ) )
=> ( ! [X3: set_a,Xs3: list_set_a,Y2: a,Ys3: list_a] :
( ( P2 @ Xs3 @ Ys3 )
=> ( P2 @ ( cons_set_a @ X3 @ Xs3 ) @ ( cons_a @ Y2 @ Ys3 ) ) )
=> ( P2 @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_464_list__induct2_H,axiom,
! [P2: list_set_a > list_set_a > $o,Xs: list_set_a,Ys: list_set_a] :
( ( P2 @ nil_set_a @ nil_set_a )
=> ( ! [X3: set_a,Xs3: list_set_a] : ( P2 @ ( cons_set_a @ X3 @ Xs3 ) @ nil_set_a )
=> ( ! [Y2: set_a,Ys3: list_set_a] : ( P2 @ nil_set_a @ ( cons_set_a @ Y2 @ Ys3 ) )
=> ( ! [X3: set_a,Xs3: list_set_a,Y2: set_a,Ys3: list_set_a] :
( ( P2 @ Xs3 @ Ys3 )
=> ( P2 @ ( cons_set_a @ X3 @ Xs3 ) @ ( cons_set_a @ Y2 @ Ys3 ) ) )
=> ( P2 @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_465_neq__Nil__conv,axiom,
! [Xs: list_a] :
( ( Xs != nil_a )
= ( ? [Y4: a,Ys4: list_a] :
( Xs
= ( cons_a @ Y4 @ Ys4 ) ) ) ) ).
% neq_Nil_conv
thf(fact_466_neq__Nil__conv,axiom,
! [Xs: list_set_a] :
( ( Xs != nil_set_a )
= ( ? [Y4: set_a,Ys4: list_set_a] :
( Xs
= ( cons_set_a @ Y4 @ Ys4 ) ) ) ) ).
% neq_Nil_conv
thf(fact_467_min__list_Ocases,axiom,
! [X4: list_set_a] :
( ! [X3: set_a,Xs3: list_set_a] :
( X4
!= ( cons_set_a @ X3 @ Xs3 ) )
=> ( X4 = nil_set_a ) ) ).
% min_list.cases
thf(fact_468_list_Oexhaust,axiom,
! [Y: list_a] :
( ( Y != nil_a )
=> ~ ! [X212: a,X222: list_a] :
( Y
!= ( cons_a @ X212 @ X222 ) ) ) ).
% list.exhaust
thf(fact_469_list_Oexhaust,axiom,
! [Y: list_set_a] :
( ( Y != nil_set_a )
=> ~ ! [X212: set_a,X222: list_set_a] :
( Y
!= ( cons_set_a @ X212 @ X222 ) ) ) ).
% list.exhaust
thf(fact_470_list_OdiscI,axiom,
! [List: list_a,X21: a,X22: list_a] :
( ( List
= ( cons_a @ X21 @ X22 ) )
=> ( List != nil_a ) ) ).
% list.discI
thf(fact_471_list_OdiscI,axiom,
! [List: list_set_a,X21: set_a,X22: list_set_a] :
( ( List
= ( cons_set_a @ X21 @ X22 ) )
=> ( List != nil_set_a ) ) ).
% list.discI
thf(fact_472_list_Odistinct_I1_J,axiom,
! [X21: a,X22: list_a] :
( nil_a
!= ( cons_a @ X21 @ X22 ) ) ).
% list.distinct(1)
thf(fact_473_list_Odistinct_I1_J,axiom,
! [X21: set_a,X22: list_set_a] :
( nil_set_a
!= ( cons_set_a @ X21 @ X22 ) ) ).
% list.distinct(1)
thf(fact_474_comp__sgraph_Owalk__edges_Ocases,axiom,
! [X4: list_a] :
( ( X4 != nil_a )
=> ( ! [X3: a] :
( X4
!= ( cons_a @ X3 @ nil_a ) )
=> ~ ! [X3: a,Y2: a,Ys3: list_a] :
( X4
!= ( cons_a @ X3 @ ( cons_a @ Y2 @ Ys3 ) ) ) ) ) ).
% comp_sgraph.walk_edges.cases
thf(fact_475_comp__sgraph_Owalk__edges_Ocases,axiom,
! [X4: list_set_a] :
( ( X4 != nil_set_a )
=> ( ! [X3: set_a] :
( X4
!= ( cons_set_a @ X3 @ nil_set_a ) )
=> ~ ! [X3: set_a,Y2: set_a,Ys3: list_set_a] :
( X4
!= ( cons_set_a @ X3 @ ( cons_set_a @ Y2 @ Ys3 ) ) ) ) ) ).
% comp_sgraph.walk_edges.cases
thf(fact_476_set__ConsD,axiom,
! [Y: a,X4: a,Xs: list_a] :
( ( member_a @ Y @ ( set_a2 @ ( cons_a @ X4 @ Xs ) ) )
=> ( ( Y = X4 )
| ( member_a @ Y @ ( set_a2 @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_477_set__ConsD,axiom,
! [Y: set_a,X4: set_a,Xs: list_set_a] :
( ( member_set_a @ Y @ ( set_set_a2 @ ( cons_set_a @ X4 @ Xs ) ) )
=> ( ( Y = X4 )
| ( member_set_a @ Y @ ( set_set_a2 @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_478_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_479_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_480_list_Oset__intros_I1_J,axiom,
! [X21: a,X22: list_a] : ( member_a @ X21 @ ( set_a2 @ ( cons_a @ X21 @ X22 ) ) ) ).
% list.set_intros(1)
thf(fact_481_list_Oset__intros_I1_J,axiom,
! [X21: set_a,X22: list_set_a] : ( member_set_a @ X21 @ ( set_set_a2 @ ( cons_set_a @ X21 @ X22 ) ) ) ).
% list.set_intros(1)
thf(fact_482_list_Oset__intros_I2_J,axiom,
! [Y: a,X22: list_a,X21: a] :
( ( member_a @ Y @ ( set_a2 @ X22 ) )
=> ( member_a @ Y @ ( set_a2 @ ( cons_a @ X21 @ X22 ) ) ) ) ).
% list.set_intros(2)
thf(fact_483_list_Oset__intros_I2_J,axiom,
! [Y: set_a,X22: list_set_a,X21: set_a] :
( ( member_set_a @ Y @ ( set_set_a2 @ X22 ) )
=> ( member_set_a @ Y @ ( set_set_a2 @ ( cons_set_a @ X21 @ X22 ) ) ) ) ).
% list.set_intros(2)
thf(fact_484_Cons__eq__appendI,axiom,
! [X4: a,Xs1: list_a,Ys: list_a,Xs: list_a,Zs: list_a] :
( ( ( cons_a @ X4 @ Xs1 )
= Ys )
=> ( ( Xs
= ( append_a @ Xs1 @ Zs ) )
=> ( ( cons_a @ X4 @ Xs )
= ( append_a @ Ys @ Zs ) ) ) ) ).
% Cons_eq_appendI
thf(fact_485_Cons__eq__appendI,axiom,
! [X4: set_a,Xs1: list_set_a,Ys: list_set_a,Xs: list_set_a,Zs: list_set_a] :
( ( ( cons_set_a @ X4 @ Xs1 )
= Ys )
=> ( ( Xs
= ( append_set_a @ Xs1 @ Zs ) )
=> ( ( cons_set_a @ X4 @ Xs )
= ( append_set_a @ Ys @ Zs ) ) ) ) ).
% Cons_eq_appendI
thf(fact_486_append__Cons,axiom,
! [X4: a,Xs: list_a,Ys: list_a] :
( ( append_a @ ( cons_a @ X4 @ Xs ) @ Ys )
= ( cons_a @ X4 @ ( append_a @ Xs @ Ys ) ) ) ).
% append_Cons
thf(fact_487_append__Cons,axiom,
! [X4: set_a,Xs: list_set_a,Ys: list_set_a] :
( ( append_set_a @ ( cons_set_a @ X4 @ Xs ) @ Ys )
= ( cons_set_a @ X4 @ ( append_set_a @ Xs @ Ys ) ) ) ).
% append_Cons
thf(fact_488_comp__sgraph_Odistinct__edgesI,axiom,
! [P: list_set_a] :
( ( distinct_set_a @ P )
=> ( distinct_set_set_a @ ( undire6234387080713648494_set_a @ P ) ) ) ).
% comp_sgraph.distinct_edgesI
thf(fact_489_comp__sgraph_Odistinct__edgesI,axiom,
! [P: list_a] :
( ( distinct_a @ P )
=> ( distinct_set_a @ ( undire7337870655677353998dges_a @ P ) ) ) ).
% comp_sgraph.distinct_edgesI
thf(fact_490_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_491_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_492_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_493_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_494_list_Osel_I1_J,axiom,
! [X21: a,X22: list_a] :
( ( hd_a @ ( cons_a @ X21 @ X22 ) )
= X21 ) ).
% list.sel(1)
thf(fact_495_list_Osel_I1_J,axiom,
! [X21: set_a,X22: list_set_a] :
( ( hd_set_a @ ( cons_set_a @ X21 @ X22 ) )
= X21 ) ).
% list.sel(1)
thf(fact_496_list_Osel_I3_J,axiom,
! [X21: a,X22: list_a] :
( ( tl_a @ ( cons_a @ X21 @ X22 ) )
= X22 ) ).
% list.sel(3)
thf(fact_497_list_Osel_I3_J,axiom,
! [X21: set_a,X22: list_set_a] :
( ( tl_set_a @ ( cons_set_a @ X21 @ X22 ) )
= X22 ) ).
% list.sel(3)
thf(fact_498_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_499_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_500_comp__sgraph_Owalk__length__rev,axiom,
( undire4424681683220949296_set_a
= ( ^ [P4: list_set_a] : ( undire4424681683220949296_set_a @ ( rev_set_a @ P4 ) ) ) ) ).
% comp_sgraph.walk_length_rev
thf(fact_501_comp__sgraph_Owalk__length__rev,axiom,
( undire8849074589633906640ngth_a
= ( ^ [P4: list_a] : ( undire8849074589633906640ngth_a @ ( rev_a @ P4 ) ) ) ) ).
% comp_sgraph.walk_length_rev
thf(fact_502_ulgraph_Owalk__edges__append__ss2,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,Xs: list_set_a,Ys: list_set_a] :
( ( undire6886684016831807756_set_a @ Vertices @ Edges )
=> ( ord_le5722252365846178494_set_a @ ( set_set_set_a2 @ ( undire6234387080713648494_set_a @ Xs ) ) @ ( set_set_set_a2 @ ( undire6234387080713648494_set_a @ ( append_set_a @ Xs @ Ys ) ) ) ) ) ).
% ulgraph.walk_edges_append_ss2
thf(fact_503_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_504_ulgraph_Owalk__edges__append__ss1,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,Ys: list_set_a,Xs: list_set_a] :
( ( undire6886684016831807756_set_a @ Vertices @ Edges )
=> ( ord_le5722252365846178494_set_a @ ( set_set_set_a2 @ ( undire6234387080713648494_set_a @ Ys ) ) @ ( set_set_set_a2 @ ( undire6234387080713648494_set_a @ ( append_set_a @ Xs @ Ys ) ) ) ) ) ).
% ulgraph.walk_edges_append_ss1
thf(fact_505_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_506_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_507_set__subset__Cons,axiom,
! [Xs: list_a,X4: a] : ( ord_less_eq_set_a @ ( set_a2 @ Xs ) @ ( set_a2 @ ( cons_a @ X4 @ Xs ) ) ) ).
% set_subset_Cons
thf(fact_508_set__subset__Cons,axiom,
! [Xs: list_set_a,X4: set_a] : ( ord_le3724670747650509150_set_a @ ( set_set_a2 @ Xs ) @ ( set_set_a2 @ ( cons_set_a @ X4 @ Xs ) ) ) ).
% set_subset_Cons
thf(fact_509_rev__induct,axiom,
! [P2: list_a > $o,Xs: list_a] :
( ( P2 @ nil_a )
=> ( ! [X3: a,Xs3: list_a] :
( ( P2 @ Xs3 )
=> ( P2 @ ( append_a @ Xs3 @ ( cons_a @ X3 @ nil_a ) ) ) )
=> ( P2 @ Xs ) ) ) ).
% rev_induct
thf(fact_510_rev__induct,axiom,
! [P2: list_set_a > $o,Xs: list_set_a] :
( ( P2 @ nil_set_a )
=> ( ! [X3: set_a,Xs3: list_set_a] :
( ( P2 @ Xs3 )
=> ( P2 @ ( append_set_a @ Xs3 @ ( cons_set_a @ X3 @ nil_set_a ) ) ) )
=> ( P2 @ Xs ) ) ) ).
% rev_induct
thf(fact_511_rev__exhaust,axiom,
! [Xs: list_a] :
( ( Xs != nil_a )
=> ~ ! [Ys3: list_a,Y2: a] :
( Xs
!= ( append_a @ Ys3 @ ( cons_a @ Y2 @ nil_a ) ) ) ) ).
% rev_exhaust
thf(fact_512_rev__exhaust,axiom,
! [Xs: list_set_a] :
( ( Xs != nil_set_a )
=> ~ ! [Ys3: list_set_a,Y2: set_a] :
( Xs
!= ( append_set_a @ Ys3 @ ( cons_set_a @ Y2 @ nil_set_a ) ) ) ) ).
% rev_exhaust
thf(fact_513_Cons__eq__append__conv,axiom,
! [X4: a,Xs: list_a,Ys: list_a,Zs: list_a] :
( ( ( cons_a @ X4 @ Xs )
= ( append_a @ Ys @ Zs ) )
= ( ( ( Ys = nil_a )
& ( ( cons_a @ X4 @ Xs )
= Zs ) )
| ? [Ys5: list_a] :
( ( ( cons_a @ X4 @ Ys5 )
= Ys )
& ( Xs
= ( append_a @ Ys5 @ Zs ) ) ) ) ) ).
% Cons_eq_append_conv
thf(fact_514_Cons__eq__append__conv,axiom,
! [X4: set_a,Xs: list_set_a,Ys: list_set_a,Zs: list_set_a] :
( ( ( cons_set_a @ X4 @ Xs )
= ( append_set_a @ Ys @ Zs ) )
= ( ( ( Ys = nil_set_a )
& ( ( cons_set_a @ X4 @ Xs )
= Zs ) )
| ? [Ys5: list_set_a] :
( ( ( cons_set_a @ X4 @ Ys5 )
= Ys )
& ( Xs
= ( append_set_a @ Ys5 @ Zs ) ) ) ) ) ).
% Cons_eq_append_conv
thf(fact_515_append__eq__Cons__conv,axiom,
! [Ys: list_a,Zs: list_a,X4: a,Xs: list_a] :
( ( ( append_a @ Ys @ Zs )
= ( cons_a @ X4 @ Xs ) )
= ( ( ( Ys = nil_a )
& ( Zs
= ( cons_a @ X4 @ Xs ) ) )
| ? [Ys5: list_a] :
( ( Ys
= ( cons_a @ X4 @ Ys5 ) )
& ( ( append_a @ Ys5 @ Zs )
= Xs ) ) ) ) ).
% append_eq_Cons_conv
thf(fact_516_append__eq__Cons__conv,axiom,
! [Ys: list_set_a,Zs: list_set_a,X4: set_a,Xs: list_set_a] :
( ( ( append_set_a @ Ys @ Zs )
= ( cons_set_a @ X4 @ Xs ) )
= ( ( ( Ys = nil_set_a )
& ( Zs
= ( cons_set_a @ X4 @ Xs ) ) )
| ? [Ys5: list_set_a] :
( ( Ys
= ( cons_set_a @ X4 @ Ys5 ) )
& ( ( append_set_a @ Ys5 @ Zs )
= Xs ) ) ) ) ).
% append_eq_Cons_conv
thf(fact_517_rev__nonempty__induct,axiom,
! [Xs: list_a,P2: list_a > $o] :
( ( Xs != nil_a )
=> ( ! [X3: a] : ( P2 @ ( cons_a @ X3 @ nil_a ) )
=> ( ! [X3: a,Xs3: list_a] :
( ( Xs3 != nil_a )
=> ( ( P2 @ Xs3 )
=> ( P2 @ ( append_a @ Xs3 @ ( cons_a @ X3 @ nil_a ) ) ) ) )
=> ( P2 @ Xs ) ) ) ) ).
% rev_nonempty_induct
thf(fact_518_rev__nonempty__induct,axiom,
! [Xs: list_set_a,P2: list_set_a > $o] :
( ( Xs != nil_set_a )
=> ( ! [X3: set_a] : ( P2 @ ( cons_set_a @ X3 @ nil_set_a ) )
=> ( ! [X3: set_a,Xs3: list_set_a] :
( ( Xs3 != nil_set_a )
=> ( ( P2 @ Xs3 )
=> ( P2 @ ( append_set_a @ Xs3 @ ( cons_set_a @ X3 @ nil_set_a ) ) ) ) )
=> ( P2 @ Xs ) ) ) ) ).
% rev_nonempty_induct
thf(fact_519_split__list__first__prop__iff,axiom,
! [Xs: list_a,P2: a > $o] :
( ( ? [X2: a] :
( ( member_a @ X2 @ ( set_a2 @ Xs ) )
& ( P2 @ X2 ) ) )
= ( ? [Ys4: list_a,X2: a] :
( ? [Zs2: list_a] :
( Xs
= ( append_a @ Ys4 @ ( cons_a @ X2 @ Zs2 ) ) )
& ( P2 @ X2 )
& ! [Y4: a] :
( ( member_a @ Y4 @ ( set_a2 @ Ys4 ) )
=> ~ ( P2 @ Y4 ) ) ) ) ) ).
% split_list_first_prop_iff
thf(fact_520_split__list__first__prop__iff,axiom,
! [Xs: list_set_a,P2: set_a > $o] :
( ( ? [X2: set_a] :
( ( member_set_a @ X2 @ ( set_set_a2 @ Xs ) )
& ( P2 @ X2 ) ) )
= ( ? [Ys4: list_set_a,X2: set_a] :
( ? [Zs2: list_set_a] :
( Xs
= ( append_set_a @ Ys4 @ ( cons_set_a @ X2 @ Zs2 ) ) )
& ( P2 @ X2 )
& ! [Y4: set_a] :
( ( member_set_a @ Y4 @ ( set_set_a2 @ Ys4 ) )
=> ~ ( P2 @ Y4 ) ) ) ) ) ).
% split_list_first_prop_iff
thf(fact_521_split__list__last__prop__iff,axiom,
! [Xs: list_a,P2: a > $o] :
( ( ? [X2: a] :
( ( member_a @ X2 @ ( set_a2 @ Xs ) )
& ( P2 @ X2 ) ) )
= ( ? [Ys4: list_a,X2: a,Zs2: list_a] :
( ( Xs
= ( append_a @ Ys4 @ ( cons_a @ X2 @ Zs2 ) ) )
& ( P2 @ X2 )
& ! [Y4: a] :
( ( member_a @ Y4 @ ( set_a2 @ Zs2 ) )
=> ~ ( P2 @ Y4 ) ) ) ) ) ).
% split_list_last_prop_iff
thf(fact_522_split__list__last__prop__iff,axiom,
! [Xs: list_set_a,P2: set_a > $o] :
( ( ? [X2: set_a] :
( ( member_set_a @ X2 @ ( set_set_a2 @ Xs ) )
& ( P2 @ X2 ) ) )
= ( ? [Ys4: list_set_a,X2: set_a,Zs2: list_set_a] :
( ( Xs
= ( append_set_a @ Ys4 @ ( cons_set_a @ X2 @ Zs2 ) ) )
& ( P2 @ X2 )
& ! [Y4: set_a] :
( ( member_set_a @ Y4 @ ( set_set_a2 @ Zs2 ) )
=> ~ ( P2 @ Y4 ) ) ) ) ) ).
% split_list_last_prop_iff
thf(fact_523_in__set__conv__decomp__first,axiom,
! [X4: a,Xs: list_a] :
( ( member_a @ X4 @ ( set_a2 @ Xs ) )
= ( ? [Ys4: list_a,Zs2: list_a] :
( ( Xs
= ( append_a @ Ys4 @ ( cons_a @ X4 @ Zs2 ) ) )
& ~ ( member_a @ X4 @ ( set_a2 @ Ys4 ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_524_in__set__conv__decomp__first,axiom,
! [X4: set_a,Xs: list_set_a] :
( ( member_set_a @ X4 @ ( set_set_a2 @ Xs ) )
= ( ? [Ys4: list_set_a,Zs2: list_set_a] :
( ( Xs
= ( append_set_a @ Ys4 @ ( cons_set_a @ X4 @ Zs2 ) ) )
& ~ ( member_set_a @ X4 @ ( set_set_a2 @ Ys4 ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_525_in__set__conv__decomp__last,axiom,
! [X4: a,Xs: list_a] :
( ( member_a @ X4 @ ( set_a2 @ Xs ) )
= ( ? [Ys4: list_a,Zs2: list_a] :
( ( Xs
= ( append_a @ Ys4 @ ( cons_a @ X4 @ Zs2 ) ) )
& ~ ( member_a @ X4 @ ( set_a2 @ Zs2 ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_526_in__set__conv__decomp__last,axiom,
! [X4: set_a,Xs: list_set_a] :
( ( member_set_a @ X4 @ ( set_set_a2 @ Xs ) )
= ( ? [Ys4: list_set_a,Zs2: list_set_a] :
( ( Xs
= ( append_set_a @ Ys4 @ ( cons_set_a @ X4 @ Zs2 ) ) )
& ~ ( member_set_a @ X4 @ ( set_set_a2 @ Zs2 ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_527_split__list__first__propE,axiom,
! [Xs: list_a,P2: a > $o] :
( ? [X: a] :
( ( member_a @ X @ ( set_a2 @ Xs ) )
& ( P2 @ X ) )
=> ~ ! [Ys3: list_a,X3: a] :
( ? [Zs3: list_a] :
( Xs
= ( append_a @ Ys3 @ ( cons_a @ X3 @ Zs3 ) ) )
=> ( ( P2 @ X3 )
=> ~ ! [Xa: a] :
( ( member_a @ Xa @ ( set_a2 @ Ys3 ) )
=> ~ ( P2 @ Xa ) ) ) ) ) ).
% split_list_first_propE
thf(fact_528_split__list__first__propE,axiom,
! [Xs: list_set_a,P2: set_a > $o] :
( ? [X: set_a] :
( ( member_set_a @ X @ ( set_set_a2 @ Xs ) )
& ( P2 @ X ) )
=> ~ ! [Ys3: list_set_a,X3: set_a] :
( ? [Zs3: list_set_a] :
( Xs
= ( append_set_a @ Ys3 @ ( cons_set_a @ X3 @ Zs3 ) ) )
=> ( ( P2 @ X3 )
=> ~ ! [Xa: set_a] :
( ( member_set_a @ Xa @ ( set_set_a2 @ Ys3 ) )
=> ~ ( P2 @ Xa ) ) ) ) ) ).
% split_list_first_propE
thf(fact_529_split__list__last__propE,axiom,
! [Xs: list_a,P2: a > $o] :
( ? [X: a] :
( ( member_a @ X @ ( set_a2 @ Xs ) )
& ( P2 @ X ) )
=> ~ ! [Ys3: list_a,X3: a,Zs3: list_a] :
( ( Xs
= ( append_a @ Ys3 @ ( cons_a @ X3 @ Zs3 ) ) )
=> ( ( P2 @ X3 )
=> ~ ! [Xa: a] :
( ( member_a @ Xa @ ( set_a2 @ Zs3 ) )
=> ~ ( P2 @ Xa ) ) ) ) ) ).
% split_list_last_propE
thf(fact_530_split__list__last__propE,axiom,
! [Xs: list_set_a,P2: set_a > $o] :
( ? [X: set_a] :
( ( member_set_a @ X @ ( set_set_a2 @ Xs ) )
& ( P2 @ X ) )
=> ~ ! [Ys3: list_set_a,X3: set_a,Zs3: list_set_a] :
( ( Xs
= ( append_set_a @ Ys3 @ ( cons_set_a @ X3 @ Zs3 ) ) )
=> ( ( P2 @ X3 )
=> ~ ! [Xa: set_a] :
( ( member_set_a @ Xa @ ( set_set_a2 @ Zs3 ) )
=> ~ ( P2 @ Xa ) ) ) ) ) ).
% split_list_last_propE
thf(fact_531_split__list__first__prop,axiom,
! [Xs: list_a,P2: a > $o] :
( ? [X: a] :
( ( member_a @ X @ ( set_a2 @ Xs ) )
& ( P2 @ X ) )
=> ? [Ys3: list_a,X3: a] :
( ? [Zs3: list_a] :
( Xs
= ( append_a @ Ys3 @ ( cons_a @ X3 @ Zs3 ) ) )
& ( P2 @ X3 )
& ! [Xa: a] :
( ( member_a @ Xa @ ( set_a2 @ Ys3 ) )
=> ~ ( P2 @ Xa ) ) ) ) ).
% split_list_first_prop
thf(fact_532_split__list__first__prop,axiom,
! [Xs: list_set_a,P2: set_a > $o] :
( ? [X: set_a] :
( ( member_set_a @ X @ ( set_set_a2 @ Xs ) )
& ( P2 @ X ) )
=> ? [Ys3: list_set_a,X3: set_a] :
( ? [Zs3: list_set_a] :
( Xs
= ( append_set_a @ Ys3 @ ( cons_set_a @ X3 @ Zs3 ) ) )
& ( P2 @ X3 )
& ! [Xa: set_a] :
( ( member_set_a @ Xa @ ( set_set_a2 @ Ys3 ) )
=> ~ ( P2 @ Xa ) ) ) ) ).
% split_list_first_prop
thf(fact_533_split__list__last__prop,axiom,
! [Xs: list_a,P2: a > $o] :
( ? [X: a] :
( ( member_a @ X @ ( set_a2 @ Xs ) )
& ( P2 @ X ) )
=> ? [Ys3: list_a,X3: a,Zs3: list_a] :
( ( Xs
= ( append_a @ Ys3 @ ( cons_a @ X3 @ Zs3 ) ) )
& ( P2 @ X3 )
& ! [Xa: a] :
( ( member_a @ Xa @ ( set_a2 @ Zs3 ) )
=> ~ ( P2 @ Xa ) ) ) ) ).
% split_list_last_prop
thf(fact_534_split__list__last__prop,axiom,
! [Xs: list_set_a,P2: set_a > $o] :
( ? [X: set_a] :
( ( member_set_a @ X @ ( set_set_a2 @ Xs ) )
& ( P2 @ X ) )
=> ? [Ys3: list_set_a,X3: set_a,Zs3: list_set_a] :
( ( Xs
= ( append_set_a @ Ys3 @ ( cons_set_a @ X3 @ Zs3 ) ) )
& ( P2 @ X3 )
& ! [Xa: set_a] :
( ( member_set_a @ Xa @ ( set_set_a2 @ Zs3 ) )
=> ~ ( P2 @ Xa ) ) ) ) ).
% split_list_last_prop
thf(fact_535_in__set__conv__decomp,axiom,
! [X4: a,Xs: list_a] :
( ( member_a @ X4 @ ( set_a2 @ Xs ) )
= ( ? [Ys4: list_a,Zs2: list_a] :
( Xs
= ( append_a @ Ys4 @ ( cons_a @ X4 @ Zs2 ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_536_in__set__conv__decomp,axiom,
! [X4: set_a,Xs: list_set_a] :
( ( member_set_a @ X4 @ ( set_set_a2 @ Xs ) )
= ( ? [Ys4: list_set_a,Zs2: list_set_a] :
( Xs
= ( append_set_a @ Ys4 @ ( cons_set_a @ X4 @ Zs2 ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_537_append__Cons__eq__iff,axiom,
! [X4: a,Xs: list_a,Ys: list_a,Xs4: list_a,Ys6: list_a] :
( ~ ( member_a @ X4 @ ( set_a2 @ Xs ) )
=> ( ~ ( member_a @ X4 @ ( set_a2 @ Ys ) )
=> ( ( ( append_a @ Xs @ ( cons_a @ X4 @ Ys ) )
= ( append_a @ Xs4 @ ( cons_a @ X4 @ Ys6 ) ) )
= ( ( Xs = Xs4 )
& ( Ys = Ys6 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_538_append__Cons__eq__iff,axiom,
! [X4: set_a,Xs: list_set_a,Ys: list_set_a,Xs4: list_set_a,Ys6: list_set_a] :
( ~ ( member_set_a @ X4 @ ( set_set_a2 @ Xs ) )
=> ( ~ ( member_set_a @ X4 @ ( set_set_a2 @ Ys ) )
=> ( ( ( append_set_a @ Xs @ ( cons_set_a @ X4 @ Ys ) )
= ( append_set_a @ Xs4 @ ( cons_set_a @ X4 @ Ys6 ) ) )
= ( ( Xs = Xs4 )
& ( Ys = Ys6 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_539_split__list__propE,axiom,
! [Xs: list_a,P2: a > $o] :
( ? [X: a] :
( ( member_a @ X @ ( set_a2 @ Xs ) )
& ( P2 @ X ) )
=> ~ ! [Ys3: list_a,X3: a] :
( ? [Zs3: list_a] :
( Xs
= ( append_a @ Ys3 @ ( cons_a @ X3 @ Zs3 ) ) )
=> ~ ( P2 @ X3 ) ) ) ).
% split_list_propE
thf(fact_540_split__list__propE,axiom,
! [Xs: list_set_a,P2: set_a > $o] :
( ? [X: set_a] :
( ( member_set_a @ X @ ( set_set_a2 @ Xs ) )
& ( P2 @ X ) )
=> ~ ! [Ys3: list_set_a,X3: set_a] :
( ? [Zs3: list_set_a] :
( Xs
= ( append_set_a @ Ys3 @ ( cons_set_a @ X3 @ Zs3 ) ) )
=> ~ ( P2 @ X3 ) ) ) ).
% split_list_propE
thf(fact_541_split__list__first,axiom,
! [X4: a,Xs: list_a] :
( ( member_a @ X4 @ ( set_a2 @ Xs ) )
=> ? [Ys3: list_a,Zs3: list_a] :
( ( Xs
= ( append_a @ Ys3 @ ( cons_a @ X4 @ Zs3 ) ) )
& ~ ( member_a @ X4 @ ( set_a2 @ Ys3 ) ) ) ) ).
% split_list_first
thf(fact_542_split__list__first,axiom,
! [X4: set_a,Xs: list_set_a] :
( ( member_set_a @ X4 @ ( set_set_a2 @ Xs ) )
=> ? [Ys3: list_set_a,Zs3: list_set_a] :
( ( Xs
= ( append_set_a @ Ys3 @ ( cons_set_a @ X4 @ Zs3 ) ) )
& ~ ( member_set_a @ X4 @ ( set_set_a2 @ Ys3 ) ) ) ) ).
% split_list_first
thf(fact_543_split__list__prop,axiom,
! [Xs: list_a,P2: a > $o] :
( ? [X: a] :
( ( member_a @ X @ ( set_a2 @ Xs ) )
& ( P2 @ X ) )
=> ? [Ys3: list_a,X3: a] :
( ? [Zs3: list_a] :
( Xs
= ( append_a @ Ys3 @ ( cons_a @ X3 @ Zs3 ) ) )
& ( P2 @ X3 ) ) ) ).
% split_list_prop
thf(fact_544_split__list__prop,axiom,
! [Xs: list_set_a,P2: set_a > $o] :
( ? [X: set_a] :
( ( member_set_a @ X @ ( set_set_a2 @ Xs ) )
& ( P2 @ X ) )
=> ? [Ys3: list_set_a,X3: set_a] :
( ? [Zs3: list_set_a] :
( Xs
= ( append_set_a @ Ys3 @ ( cons_set_a @ X3 @ Zs3 ) ) )
& ( P2 @ X3 ) ) ) ).
% split_list_prop
thf(fact_545_split__list__last,axiom,
! [X4: a,Xs: list_a] :
( ( member_a @ X4 @ ( set_a2 @ Xs ) )
=> ? [Ys3: list_a,Zs3: list_a] :
( ( Xs
= ( append_a @ Ys3 @ ( cons_a @ X4 @ Zs3 ) ) )
& ~ ( member_a @ X4 @ ( set_a2 @ Zs3 ) ) ) ) ).
% split_list_last
thf(fact_546_split__list__last,axiom,
! [X4: set_a,Xs: list_set_a] :
( ( member_set_a @ X4 @ ( set_set_a2 @ Xs ) )
=> ? [Ys3: list_set_a,Zs3: list_set_a] :
( ( Xs
= ( append_set_a @ Ys3 @ ( cons_set_a @ X4 @ Zs3 ) ) )
& ~ ( member_set_a @ X4 @ ( set_set_a2 @ Zs3 ) ) ) ) ).
% split_list_last
thf(fact_547_split__list,axiom,
! [X4: a,Xs: list_a] :
( ( member_a @ X4 @ ( set_a2 @ Xs ) )
=> ? [Ys3: list_a,Zs3: list_a] :
( Xs
= ( append_a @ Ys3 @ ( cons_a @ X4 @ Zs3 ) ) ) ) ).
% split_list
thf(fact_548_split__list,axiom,
! [X4: set_a,Xs: list_set_a] :
( ( member_set_a @ X4 @ ( set_set_a2 @ Xs ) )
=> ? [Ys3: list_set_a,Zs3: list_set_a] :
( Xs
= ( append_set_a @ Ys3 @ ( cons_set_a @ X4 @ Zs3 ) ) ) ) ).
% split_list
thf(fact_549_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_550_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_551_distinct__singleton,axiom,
! [X4: a] : ( distinct_a @ ( cons_a @ X4 @ nil_a ) ) ).
% distinct_singleton
thf(fact_552_distinct__singleton,axiom,
! [X4: set_a] : ( distinct_set_a @ ( cons_set_a @ X4 @ nil_set_a ) ) ).
% distinct_singleton
thf(fact_553_ulgraph_Owalk__edges_Ocases,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,X4: list_set_a] :
( ( undire6886684016831807756_set_a @ Vertices @ Edges )
=> ( ( X4 != nil_set_a )
=> ( ! [X3: set_a] :
( X4
!= ( cons_set_a @ X3 @ nil_set_a ) )
=> ~ ! [X3: set_a,Y2: set_a,Ys3: list_set_a] :
( X4
!= ( cons_set_a @ X3 @ ( cons_set_a @ Y2 @ Ys3 ) ) ) ) ) ) ).
% ulgraph.walk_edges.cases
thf(fact_554_ulgraph_Owalk__edges_Ocases,axiom,
! [Vertices: set_a,Edges: set_set_a,X4: list_a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( X4 != nil_a )
=> ( ! [X3: a] :
( X4
!= ( cons_a @ X3 @ nil_a ) )
=> ~ ! [X3: a,Y2: a,Ys3: list_a] :
( X4
!= ( cons_a @ X3 @ ( cons_a @ Y2 @ Ys3 ) ) ) ) ) ) ).
% ulgraph.walk_edges.cases
thf(fact_555_distinct_Osimps_I2_J,axiom,
! [X4: a,Xs: list_a] :
( ( distinct_a @ ( cons_a @ X4 @ Xs ) )
= ( ~ ( member_a @ X4 @ ( set_a2 @ Xs ) )
& ( distinct_a @ Xs ) ) ) ).
% distinct.simps(2)
thf(fact_556_distinct_Osimps_I2_J,axiom,
! [X4: set_a,Xs: list_set_a] :
( ( distinct_set_a @ ( cons_set_a @ X4 @ Xs ) )
= ( ~ ( member_set_a @ X4 @ ( set_set_a2 @ Xs ) )
& ( distinct_set_a @ Xs ) ) ) ).
% distinct.simps(2)
thf(fact_557_Nil__tl,axiom,
! [Xs: list_a] :
( ( nil_a
= ( tl_a @ Xs ) )
= ( ( Xs = nil_a )
| ? [X2: a] :
( Xs
= ( cons_a @ X2 @ nil_a ) ) ) ) ).
% Nil_tl
thf(fact_558_Nil__tl,axiom,
! [Xs: list_set_a] :
( ( nil_set_a
= ( tl_set_a @ Xs ) )
= ( ( Xs = nil_set_a )
| ? [X2: set_a] :
( Xs
= ( cons_set_a @ X2 @ nil_set_a ) ) ) ) ).
% Nil_tl
thf(fact_559_tl__Nil,axiom,
! [Xs: list_a] :
( ( ( tl_a @ Xs )
= nil_a )
= ( ( Xs = nil_a )
| ? [X2: a] :
( Xs
= ( cons_a @ X2 @ nil_a ) ) ) ) ).
% tl_Nil
thf(fact_560_tl__Nil,axiom,
! [Xs: list_set_a] :
( ( ( tl_set_a @ Xs )
= nil_set_a )
= ( ( Xs = nil_set_a )
| ? [X2: set_a] :
( Xs
= ( cons_set_a @ X2 @ nil_set_a ) ) ) ) ).
% tl_Nil
thf(fact_561_last__ConsR,axiom,
! [Xs: list_a,X4: a] :
( ( Xs != nil_a )
=> ( ( last_a @ ( cons_a @ X4 @ Xs ) )
= ( last_a @ Xs ) ) ) ).
% last_ConsR
thf(fact_562_last__ConsR,axiom,
! [Xs: list_set_a,X4: set_a] :
( ( Xs != nil_set_a )
=> ( ( last_set_a @ ( cons_set_a @ X4 @ Xs ) )
= ( last_set_a @ Xs ) ) ) ).
% last_ConsR
thf(fact_563_last__ConsL,axiom,
! [Xs: list_a,X4: a] :
( ( Xs = nil_a )
=> ( ( last_a @ ( cons_a @ X4 @ Xs ) )
= X4 ) ) ).
% last_ConsL
thf(fact_564_last__ConsL,axiom,
! [Xs: list_set_a,X4: set_a] :
( ( Xs = nil_set_a )
=> ( ( last_set_a @ ( cons_set_a @ X4 @ Xs ) )
= X4 ) ) ).
% last_ConsL
thf(fact_565_last_Osimps,axiom,
! [Xs: list_a,X4: a] :
( ( ( Xs = nil_a )
=> ( ( last_a @ ( cons_a @ X4 @ Xs ) )
= X4 ) )
& ( ( Xs != nil_a )
=> ( ( last_a @ ( cons_a @ X4 @ Xs ) )
= ( last_a @ Xs ) ) ) ) ).
% last.simps
thf(fact_566_last_Osimps,axiom,
! [Xs: list_set_a,X4: set_a] :
( ( ( Xs = nil_set_a )
=> ( ( last_set_a @ ( cons_set_a @ X4 @ Xs ) )
= X4 ) )
& ( ( Xs != nil_set_a )
=> ( ( last_set_a @ ( cons_set_a @ X4 @ Xs ) )
= ( last_set_a @ Xs ) ) ) ) ).
% last.simps
thf(fact_567_ulgraph_Odistinct__edgesI,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,P: list_set_a] :
( ( undire6886684016831807756_set_a @ Vertices @ Edges )
=> ( ( distinct_set_a @ P )
=> ( distinct_set_set_a @ ( undire6234387080713648494_set_a @ P ) ) ) ) ).
% ulgraph.distinct_edgesI
thf(fact_568_ulgraph_Odistinct__edgesI,axiom,
! [Vertices: set_a,Edges: set_set_a,P: list_a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( distinct_a @ P )
=> ( distinct_set_a @ ( undire7337870655677353998dges_a @ P ) ) ) ) ).
% ulgraph.distinct_edgesI
thf(fact_569_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_570_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_571_ulgraph_Owalk__length__rev,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,P: list_set_a] :
( ( undire6886684016831807756_set_a @ Vertices @ Edges )
=> ( ( undire4424681683220949296_set_a @ P )
= ( undire4424681683220949296_set_a @ ( rev_set_a @ P ) ) ) ) ).
% ulgraph.walk_length_rev
thf(fact_572_ulgraph_Owalk__length__rev,axiom,
! [Vertices: set_a,Edges: set_set_a,P: list_a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( undire8849074589633906640ngth_a @ P )
= ( undire8849074589633906640ngth_a @ ( rev_a @ P ) ) ) ) ).
% ulgraph.walk_length_rev
thf(fact_573_not__distinct__decomp,axiom,
! [Ws: list_a] :
( ~ ( distinct_a @ Ws )
=> ? [Xs3: list_a,Ys3: list_a,Zs3: list_a,Y2: a] :
( Ws
= ( append_a @ Xs3 @ ( append_a @ ( cons_a @ Y2 @ nil_a ) @ ( append_a @ Ys3 @ ( append_a @ ( cons_a @ Y2 @ nil_a ) @ Zs3 ) ) ) ) ) ) ).
% not_distinct_decomp
thf(fact_574_not__distinct__decomp,axiom,
! [Ws: list_set_a] :
( ~ ( distinct_set_a @ Ws )
=> ? [Xs3: list_set_a,Ys3: list_set_a,Zs3: list_set_a,Y2: set_a] :
( Ws
= ( append_set_a @ Xs3 @ ( append_set_a @ ( cons_set_a @ Y2 @ nil_set_a ) @ ( append_set_a @ Ys3 @ ( append_set_a @ ( cons_set_a @ Y2 @ nil_set_a ) @ Zs3 ) ) ) ) ) ) ).
% not_distinct_decomp
thf(fact_575_not__distinct__conv__prefix,axiom,
! [As: list_a] :
( ( ~ ( distinct_a @ As ) )
= ( ? [Xs5: list_a,Y4: a,Ys4: list_a] :
( ( member_a @ Y4 @ ( set_a2 @ Xs5 ) )
& ( distinct_a @ Xs5 )
& ( As
= ( append_a @ Xs5 @ ( cons_a @ Y4 @ Ys4 ) ) ) ) ) ) ).
% not_distinct_conv_prefix
thf(fact_576_not__distinct__conv__prefix,axiom,
! [As: list_set_a] :
( ( ~ ( distinct_set_a @ As ) )
= ( ? [Xs5: list_set_a,Y4: set_a,Ys4: list_set_a] :
( ( member_set_a @ Y4 @ ( set_set_a2 @ Xs5 ) )
& ( distinct_set_a @ Xs5 )
& ( As
= ( append_set_a @ Xs5 @ ( cons_set_a @ Y4 @ Ys4 ) ) ) ) ) ) ).
% not_distinct_conv_prefix
thf(fact_577_rev_Osimps_I2_J,axiom,
! [X4: a,Xs: list_a] :
( ( rev_a @ ( cons_a @ X4 @ Xs ) )
= ( append_a @ ( rev_a @ Xs ) @ ( cons_a @ X4 @ nil_a ) ) ) ).
% rev.simps(2)
thf(fact_578_rev_Osimps_I2_J,axiom,
! [X4: set_a,Xs: list_set_a] :
( ( rev_set_a @ ( cons_set_a @ X4 @ Xs ) )
= ( append_set_a @ ( rev_set_a @ Xs ) @ ( cons_set_a @ X4 @ nil_set_a ) ) ) ).
% rev.simps(2)
thf(fact_579_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_580_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_581_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_582_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_583_list_Oexhaust__sel,axiom,
! [List: list_a] :
( ( List != nil_a )
=> ( List
= ( cons_a @ ( hd_a @ List ) @ ( tl_a @ List ) ) ) ) ).
% list.exhaust_sel
thf(fact_584_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_585_ulgraph_Oconnecting__path__self,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,U: set_a] :
( ( undire6886684016831807756_set_a @ Vertices @ Edges )
=> ( ( member_set_a @ U @ Vertices )
=> ( connec7350987497872064604_set_a @ Vertices @ Edges @ U @ U @ ( cons_set_a @ U @ nil_set_a ) ) ) ) ).
% ulgraph.connecting_path_self
thf(fact_586_ulgraph_Oconnecting__path__self,axiom,
! [Vertices: set_a,Edges: set_set_a,U: a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( member_a @ U @ Vertices )
=> ( connecting_path_a @ Vertices @ Edges @ U @ U @ ( cons_a @ U @ nil_a ) ) ) ) ).
% ulgraph.connecting_path_self
thf(fact_587_ulgraph_Ois__gen__path__trivial,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,X4: set_a] :
( ( undire6886684016831807756_set_a @ Vertices @ Edges )
=> ( ( member_set_a @ X4 @ Vertices )
=> ( undire7201326534205417136_set_a @ Vertices @ Edges @ ( cons_set_a @ X4 @ nil_set_a ) ) ) ) ).
% ulgraph.is_gen_path_trivial
thf(fact_588_ulgraph_Ois__gen__path__trivial,axiom,
! [Vertices: set_a,Edges: set_set_a,X4: a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( member_a @ X4 @ Vertices )
=> ( undire3562951555376170320path_a @ Vertices @ Edges @ ( cons_a @ X4 @ nil_a ) ) ) ) ).
% ulgraph.is_gen_path_trivial
thf(fact_589_ulgraph_Oconnecting__walk__self,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,U: set_a] :
( ( undire6886684016831807756_set_a @ Vertices @ Edges )
=> ( ( member_set_a @ U @ Vertices )
=> ( connec1530789871921280536_set_a @ Vertices @ Edges @ U @ U @ ( cons_set_a @ U @ nil_set_a ) ) ) ) ).
% ulgraph.connecting_walk_self
thf(fact_590_ulgraph_Oconnecting__walk__self,axiom,
! [Vertices: set_a,Edges: set_set_a,U: a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( member_a @ U @ Vertices )
=> ( connecting_walk_a @ Vertices @ Edges @ U @ U @ ( cons_a @ U @ nil_a ) ) ) ) ).
% ulgraph.connecting_walk_self
thf(fact_591_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_592_ulgraph_Oconnecting__walk__path,axiom,
! [Vertices: set_a,Edges: set_set_a,U: a,V: a,Xs: list_a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( connecting_walk_a @ Vertices @ Edges @ U @ V @ Xs )
=> ? [Ys3: list_a] :
( ( connecting_path_a @ Vertices @ Edges @ U @ V @ Ys3 )
& ( ord_less_eq_nat @ ( undire8849074589633906640ngth_a @ Ys3 ) @ ( undire8849074589633906640ngth_a @ Xs ) ) ) ) ) ).
% ulgraph.connecting_walk_path
thf(fact_593_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_594_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_595_ulgraph_Oconnecting__path__split,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,U: set_a,V: set_a,Xs: list_set_a,Z: set_a,Ys: list_set_a] :
( ( undire6886684016831807756_set_a @ Vertices @ Edges )
=> ( ( connec7350987497872064604_set_a @ Vertices @ Edges @ U @ V @ Xs )
=> ( ( connec7350987497872064604_set_a @ Vertices @ Edges @ V @ Z @ Ys )
=> ~ ! [P3: list_set_a] :
( ( connec7350987497872064604_set_a @ Vertices @ Edges @ U @ Z @ P3 )
=> ~ ( ord_less_eq_nat @ ( undire4424681683220949296_set_a @ P3 ) @ ( undire4424681683220949296_set_a @ ( append_set_a @ Xs @ ( tl_set_a @ Ys ) ) ) ) ) ) ) ) ).
% ulgraph.connecting_path_split
thf(fact_596_ulgraph_Oconnecting__path__split,axiom,
! [Vertices: set_a,Edges: set_set_a,U: a,V: a,Xs: list_a,Z: a,Ys: list_a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( connecting_path_a @ Vertices @ Edges @ U @ V @ Xs )
=> ( ( connecting_path_a @ Vertices @ Edges @ V @ Z @ Ys )
=> ~ ! [P3: list_a] :
( ( connecting_path_a @ Vertices @ Edges @ U @ Z @ P3 )
=> ~ ( ord_less_eq_nat @ ( undire8849074589633906640ngth_a @ P3 ) @ ( undire8849074589633906640ngth_a @ ( append_a @ Xs @ ( tl_a @ Ys ) ) ) ) ) ) ) ) ).
% ulgraph.connecting_path_split
thf(fact_597_ulgraph_Ois__gen__path__options,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,P: list_set_a] :
( ( undire6886684016831807756_set_a @ Vertices @ Edges )
=> ( ( undire7201326534205417136_set_a @ Vertices @ Edges @ P )
= ( ( undire797940137672299967_set_a @ Vertices @ Edges @ P )
| ( undire8834939040163919632_set_a @ Vertices @ Edges @ P )
| ? [X2: set_a] :
( ( member_set_a @ X2 @ Vertices )
& ( P
= ( cons_set_a @ X2 @ nil_set_a ) ) ) ) ) ) ).
% ulgraph.is_gen_path_options
thf(fact_598_ulgraph_Ois__gen__path__options,axiom,
! [Vertices: set_a,Edges: set_set_a,P: list_a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( undire3562951555376170320path_a @ Vertices @ Edges @ P )
= ( ( undire2407311113669455967ycle_a @ Vertices @ Edges @ P )
| ( undire427332500224447920path_a @ Vertices @ Edges @ P )
| ? [X2: a] :
( ( member_a @ X2 @ Vertices )
& ( P
= ( cons_a @ X2 @ nil_a ) ) ) ) ) ) ).
% ulgraph.is_gen_path_options
thf(fact_599_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_600_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_601_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_602_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_603_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_604_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_605_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_606_that,axiom,
! [P: list_a] :
( ( connecting_path_a @ vertices @ edges @ u @ z @ P )
=> ( ( ord_less_eq_nat @ ( undire8849074589633906640ngth_a @ P ) @ ( plus_plus_nat @ ( undire8849074589633906640ngth_a @ xs ) @ ( undire8849074589633906640ngth_a @ ys ) ) )
=> thesis ) ) ).
% that
thf(fact_607_connecting__path__length__bound,axiom,
! [U: a,V: a,P: list_a] :
( ( U != V )
=> ( ( connecting_path_a @ vertices @ edges @ U @ V @ P )
=> ( ord_less_eq_nat @ one_one_nat @ ( undire8849074589633906640ngth_a @ P ) ) ) ) ).
% connecting_path_length_bound
thf(fact_608_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_609_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_610_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_611_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_612_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_613_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_614_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_615_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_616_comp__sgraph_Owalk__length__app__ineq,axiom,
! [Xs: list_set_a,Ys: list_set_a] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ ( undire4424681683220949296_set_a @ Xs ) @ ( undire4424681683220949296_set_a @ Ys ) ) @ ( undire4424681683220949296_set_a @ ( append_set_a @ Xs @ Ys ) ) )
& ( ord_less_eq_nat @ ( undire4424681683220949296_set_a @ ( append_set_a @ Xs @ Ys ) ) @ ( plus_plus_nat @ ( plus_plus_nat @ ( undire4424681683220949296_set_a @ Xs ) @ ( undire4424681683220949296_set_a @ Ys ) ) @ one_one_nat ) ) ) ).
% comp_sgraph.walk_length_app_ineq
thf(fact_617_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_618_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_619_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_620_ulgraph_Owalk__length__app__ineq,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,Xs: list_set_a,Ys: list_set_a] :
( ( undire6886684016831807756_set_a @ Vertices @ Edges )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( undire4424681683220949296_set_a @ Xs ) @ ( undire4424681683220949296_set_a @ Ys ) ) @ ( undire4424681683220949296_set_a @ ( append_set_a @ Xs @ Ys ) ) )
& ( ord_less_eq_nat @ ( undire4424681683220949296_set_a @ ( append_set_a @ Xs @ Ys ) ) @ ( plus_plus_nat @ ( plus_plus_nat @ ( undire4424681683220949296_set_a @ Xs ) @ ( undire4424681683220949296_set_a @ Ys ) ) @ one_one_nat ) ) ) ) ).
% ulgraph.walk_length_app_ineq
thf(fact_621_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_622_ulgraph_Oconnecting__path__length__bound,axiom,
! [Vertices: set_a,Edges: set_set_a,U: a,V: a,P: list_a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( U != V )
=> ( ( connecting_path_a @ Vertices @ Edges @ U @ V @ P )
=> ( ord_less_eq_nat @ one_one_nat @ ( undire8849074589633906640ngth_a @ P ) ) ) ) ) ).
% ulgraph.connecting_path_length_bound
thf(fact_623_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_624_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_625_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_626_add__le__cancel__left,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
= ( ord_less_eq_real @ A @ B ) ) ).
% add_le_cancel_left
thf(fact_627_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_628_add__le__cancel__right,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
= ( ord_less_eq_real @ A @ B ) ) ).
% add_le_cancel_right
thf(fact_629_connecting__path__singleton,axiom,
! [U: a,V: a,Xs: list_a] :
( ( connecting_path_a @ vertices @ edges @ U @ V @ Xs )
=> ( ( ( size_size_list_a @ Xs )
= one_one_nat )
=> ( U = V ) ) ) ).
% connecting_path_singleton
thf(fact_630_comp__sgraph_Ois__cycle__alt,axiom,
! [S: set_set_a,Xs: list_set_a] :
( ( undire797940137672299967_set_a @ S @ ( undire8247866692393712962_set_a @ S ) @ Xs )
= ( ( undire3014741414213135564_set_a @ S @ ( undire8247866692393712962_set_a @ S ) @ Xs )
& ( distinct_set_a @ ( tl_set_a @ Xs ) )
& ( ord_less_eq_nat @ one_one_nat @ ( undire4424681683220949296_set_a @ Xs ) )
& ( ( hd_set_a @ Xs )
= ( last_set_a @ Xs ) ) ) ) ).
% comp_sgraph.is_cycle_alt
thf(fact_631_comp__sgraph_Ois__cycle__alt,axiom,
! [S: set_a,Xs: list_a] :
( ( undire2407311113669455967ycle_a @ S @ ( undire2918257014606996450dges_a @ S ) @ Xs )
= ( ( undire6133010728901294956walk_a @ S @ ( undire2918257014606996450dges_a @ S ) @ Xs )
& ( distinct_a @ ( tl_a @ Xs ) )
& ( ord_less_eq_nat @ one_one_nat @ ( undire8849074589633906640ngth_a @ Xs ) )
& ( ( hd_a @ Xs )
= ( last_a @ Xs ) ) ) ) ).
% comp_sgraph.is_cycle_alt
thf(fact_632_edge__density__commute,axiom,
! [X5: set_a,Y5: set_a] :
( ( undire297304480579013331sity_a @ edges @ X5 @ Y5 )
= ( undire297304480579013331sity_a @ edges @ Y5 @ X5 ) ) ).
% edge_density_commute
thf(fact_633_append__eq__append__conv,axiom,
! [Xs: list_a,Ys: list_a,Us2: list_a,Vs: list_a] :
( ( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) )
| ( ( size_size_list_a @ Us2 )
= ( size_size_list_a @ Vs ) ) )
=> ( ( ( append_a @ Xs @ Us2 )
= ( append_a @ Ys @ Vs ) )
= ( ( Xs = Ys )
& ( Us2 = Vs ) ) ) ) ).
% append_eq_append_conv
thf(fact_634_append__eq__append__conv,axiom,
! [Xs: list_set_a,Ys: list_set_a,Us2: list_set_a,Vs: list_set_a] :
( ( ( ( size_size_list_set_a @ Xs )
= ( size_size_list_set_a @ Ys ) )
| ( ( size_size_list_set_a @ Us2 )
= ( size_size_list_set_a @ Vs ) ) )
=> ( ( ( append_set_a @ Xs @ Us2 )
= ( append_set_a @ Ys @ Vs ) )
= ( ( Xs = Ys )
& ( Us2 = Vs ) ) ) ) ).
% append_eq_append_conv
thf(fact_635_length__rev,axiom,
! [Xs: list_a] :
( ( size_size_list_a @ ( rev_a @ Xs ) )
= ( size_size_list_a @ Xs ) ) ).
% length_rev
thf(fact_636_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_637_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_638_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_639_comp__sgraph_Oconnecting__path__singleton,axiom,
! [S: set_set_a,U: set_a,V: set_a,Xs: list_set_a] :
( ( connec7350987497872064604_set_a @ S @ ( undire8247866692393712962_set_a @ S ) @ U @ V @ Xs )
=> ( ( ( size_size_list_set_a @ Xs )
= one_one_nat )
=> ( U = V ) ) ) ).
% comp_sgraph.connecting_path_singleton
thf(fact_640_comp__sgraph_Oconnecting__path__singleton,axiom,
! [S: set_a,U: a,V: a,Xs: list_a] :
( ( connecting_path_a @ S @ ( undire2918257014606996450dges_a @ S ) @ U @ V @ Xs )
=> ( ( ( size_size_list_a @ Xs )
= one_one_nat )
=> ( U = V ) ) ) ).
% comp_sgraph.connecting_path_singleton
thf(fact_641_comp__sgraph_Oedge__density__commute,axiom,
! [S: set_a,X5: set_a,Y5: set_a] :
( ( undire297304480579013331sity_a @ ( undire2918257014606996450dges_a @ S ) @ X5 @ Y5 )
= ( undire297304480579013331sity_a @ ( undire2918257014606996450dges_a @ S ) @ Y5 @ X5 ) ) ).
% comp_sgraph.edge_density_commute
thf(fact_642_comp__sgraph_Oe__in__all__edges,axiom,
! [E: set_a,S: set_a] :
( ( member_set_a @ E @ ( undire2918257014606996450dges_a @ S ) )
=> ( member_set_a @ E @ ( undire2918257014606996450dges_a @ S ) ) ) ).
% comp_sgraph.e_in_all_edges
thf(fact_643_ulgraph_Oedge__density_Ocong,axiom,
undire297304480579013331sity_a = undire297304480579013331sity_a ).
% ulgraph.edge_density.cong
thf(fact_644_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_645_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_646_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs3: list_a] :
( ( size_size_list_a @ Xs3 )
= N ) ).
% Ex_list_of_length
thf(fact_647_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs3: list_set_a] :
( ( size_size_list_set_a @ Xs3 )
= N ) ).
% Ex_list_of_length
thf(fact_648_comp__sgraph_Owellformed,axiom,
! [E: set_a,S: set_a] :
( ( member_set_a @ E @ ( undire2918257014606996450dges_a @ S ) )
=> ( ord_less_eq_set_a @ E @ S ) ) ).
% comp_sgraph.wellformed
thf(fact_649_comp__sgraph_Owellformed,axiom,
! [E: set_set_a,S: set_set_a] :
( ( member_set_set_a @ E @ ( undire8247866692393712962_set_a @ S ) )
=> ( ord_le3724670747650509150_set_a @ E @ S ) ) ).
% comp_sgraph.wellformed
thf(fact_650_comp__sgraph_Oe__in__all__edges__ss,axiom,
! [E: set_a,S: set_a,V3: set_a] :
( ( member_set_a @ E @ ( undire2918257014606996450dges_a @ S ) )
=> ( ( ord_less_eq_set_a @ E @ V3 )
=> ( ( ord_less_eq_set_a @ V3 @ S )
=> ( member_set_a @ E @ ( undire2918257014606996450dges_a @ V3 ) ) ) ) ) ).
% comp_sgraph.e_in_all_edges_ss
thf(fact_651_comp__sgraph_Oe__in__all__edges__ss,axiom,
! [E: set_set_a,S: set_set_a,V3: set_set_a] :
( ( member_set_set_a @ E @ ( undire8247866692393712962_set_a @ S ) )
=> ( ( ord_le3724670747650509150_set_a @ E @ V3 )
=> ( ( ord_le3724670747650509150_set_a @ V3 @ S )
=> ( member_set_set_a @ E @ ( undire8247866692393712962_set_a @ V3 ) ) ) ) ) ).
% comp_sgraph.e_in_all_edges_ss
thf(fact_652_comp__sgraph_Oulgraph__axioms,axiom,
! [S: set_a] : ( undire7251896706689453996raph_a @ S @ ( undire2918257014606996450dges_a @ S ) ) ).
% comp_sgraph.ulgraph_axioms
thf(fact_653_comp__sgraph_Owellformed__all__edges,axiom,
! [S: set_a] : ( ord_le3724670747650509150_set_a @ ( undire2918257014606996450dges_a @ S ) @ ( undire2918257014606996450dges_a @ S ) ) ).
% comp_sgraph.wellformed_all_edges
thf(fact_654_comp__sgraph_Ograph__system__axioms,axiom,
! [S: set_a] : ( undire2554140024507503526stem_a @ S @ ( undire2918257014606996450dges_a @ S ) ) ).
% comp_sgraph.graph_system_axioms
thf(fact_655_comp__sgraph_Osubgraph__complete,axiom,
! [S: set_a] : ( undire7103218114511261257raph_a @ S @ ( undire2918257014606996450dges_a @ S ) @ S @ ( undire2918257014606996450dges_a @ S ) ) ).
% comp_sgraph.subgraph_complete
thf(fact_656_comp__sgraph_Oinduced__edges__self,axiom,
! [S: set_a] :
( ( undire7777452895879145676dges_a @ ( undire2918257014606996450dges_a @ S ) @ S )
= ( undire2918257014606996450dges_a @ S ) ) ).
% comp_sgraph.induced_edges_self
thf(fact_657_comp__sgraph_Oconnecting__walk__wf,axiom,
! [S: set_set_a,U: set_a,V: set_a,Xs: list_set_a] :
( ( connec1530789871921280536_set_a @ S @ ( undire8247866692393712962_set_a @ S ) @ U @ V @ Xs )
=> ( ( member_set_a @ U @ S )
& ( member_set_a @ V @ S ) ) ) ).
% comp_sgraph.connecting_walk_wf
thf(fact_658_comp__sgraph_Oconnecting__walk__wf,axiom,
! [S: set_a,U: a,V: a,Xs: list_a] :
( ( connecting_walk_a @ S @ ( undire2918257014606996450dges_a @ S ) @ U @ V @ Xs )
=> ( ( member_a @ U @ S )
& ( member_a @ V @ S ) ) ) ).
% comp_sgraph.connecting_walk_wf
thf(fact_659_comp__sgraph_Overt__adj__sym,axiom,
! [S: set_a,V1: a,V2: a] :
( ( undire397441198561214472_adj_a @ ( undire2918257014606996450dges_a @ S ) @ V1 @ V2 )
= ( undire397441198561214472_adj_a @ ( undire2918257014606996450dges_a @ S ) @ V2 @ V1 ) ) ).
% comp_sgraph.vert_adj_sym
thf(fact_660_comp__sgraph_Overt__adj__imp__inV,axiom,
! [S: set_set_a,V1: set_a,V2: set_a] :
( ( undire3510646817838285160_set_a @ ( undire8247866692393712962_set_a @ S ) @ V1 @ V2 )
=> ( ( member_set_a @ V1 @ S )
& ( member_set_a @ V2 @ S ) ) ) ).
% comp_sgraph.vert_adj_imp_inV
thf(fact_661_comp__sgraph_Overt__adj__imp__inV,axiom,
! [S: set_a,V1: a,V2: a] :
( ( undire397441198561214472_adj_a @ ( undire2918257014606996450dges_a @ S ) @ V1 @ V2 )
=> ( ( member_a @ V1 @ S )
& ( member_a @ V2 @ S ) ) ) ).
% comp_sgraph.vert_adj_imp_inV
thf(fact_662_comp__sgraph_Oincident__edge__in__wf,axiom,
! [E: set_set_a,S: set_set_a,V: set_a] :
( ( member_set_set_a @ E @ ( undire8247866692393712962_set_a @ S ) )
=> ( ( undire2320338297334612420_set_a @ V @ E )
=> ( member_set_a @ V @ S ) ) ) ).
% comp_sgraph.incident_edge_in_wf
thf(fact_663_comp__sgraph_Oincident__edge__in__wf,axiom,
! [E: set_a,S: set_a,V: a] :
( ( member_set_a @ E @ ( undire2918257014606996450dges_a @ S ) )
=> ( ( undire1521409233611534436dent_a @ V @ E )
=> ( member_a @ V @ S ) ) ) ).
% comp_sgraph.incident_edge_in_wf
thf(fact_664_comp__sgraph_Ono__loops,axiom,
! [V: set_a,S: set_set_a] :
( ( member_set_a @ V @ S )
=> ~ ( undire5774735625301615776_set_a @ ( undire8247866692393712962_set_a @ S ) @ V ) ) ).
% comp_sgraph.no_loops
thf(fact_665_comp__sgraph_Ono__loops,axiom,
! [V: a,S: set_a] :
( ( member_a @ V @ S )
=> ~ ( undire3617971648856834880loop_a @ ( undire2918257014606996450dges_a @ S ) @ V ) ) ).
% comp_sgraph.no_loops
thf(fact_666_comp__sgraph_Ohas__loop__in__verts,axiom,
! [S: set_set_a,V: set_a] :
( ( undire5774735625301615776_set_a @ ( undire8247866692393712962_set_a @ S ) @ V )
=> ( member_set_a @ V @ S ) ) ).
% comp_sgraph.has_loop_in_verts
thf(fact_667_comp__sgraph_Ohas__loop__in__verts,axiom,
! [S: set_a,V: a] :
( ( undire3617971648856834880loop_a @ ( undire2918257014606996450dges_a @ S ) @ V )
=> ( member_a @ V @ S ) ) ).
% comp_sgraph.has_loop_in_verts
thf(fact_668_comp__sgraph_Oedge__adj__inE,axiom,
! [S: set_a,E1: set_a,E2: set_a] :
( ( undire4022703626023482010_adj_a @ ( undire2918257014606996450dges_a @ S ) @ E1 @ E2 )
=> ( ( member_set_a @ E1 @ ( undire2918257014606996450dges_a @ S ) )
& ( member_set_a @ E2 @ ( undire2918257014606996450dges_a @ S ) ) ) ) ).
% comp_sgraph.edge_adj_inE
thf(fact_669_comp__sgraph_Oedge__adjacent__alt__def,axiom,
! [E1: set_set_a,S: set_set_a,E2: set_set_a] :
( ( member_set_set_a @ E1 @ ( undire8247866692393712962_set_a @ S ) )
=> ( ( member_set_set_a @ E2 @ ( undire8247866692393712962_set_a @ S ) )
=> ( ? [X: set_a] :
( ( member_set_a @ X @ S )
& ( member_set_a @ X @ E1 )
& ( member_set_a @ X @ E2 ) )
=> ( undire3485422320110889978_set_a @ ( undire8247866692393712962_set_a @ S ) @ E1 @ E2 ) ) ) ) ).
% comp_sgraph.edge_adjacent_alt_def
thf(fact_670_comp__sgraph_Oedge__adjacent__alt__def,axiom,
! [E1: set_a,S: set_a,E2: set_a] :
( ( member_set_a @ E1 @ ( undire2918257014606996450dges_a @ S ) )
=> ( ( member_set_a @ E2 @ ( undire2918257014606996450dges_a @ S ) )
=> ( ? [X: a] :
( ( member_a @ X @ S )
& ( member_a @ X @ E1 )
& ( member_a @ X @ E2 ) )
=> ( undire4022703626023482010_adj_a @ ( undire2918257014606996450dges_a @ S ) @ E1 @ E2 ) ) ) ) ).
% comp_sgraph.edge_adjacent_alt_def
thf(fact_671_ulgraph_Oedge__density__commute,axiom,
! [Vertices: set_a,Edges: set_set_a,X5: set_a,Y5: set_a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( undire297304480579013331sity_a @ Edges @ X5 @ Y5 )
= ( undire297304480579013331sity_a @ Edges @ Y5 @ X5 ) ) ) ).
% ulgraph.edge_density_commute
thf(fact_672_list__induct2,axiom,
! [Xs: list_a,Ys: list_a,P2: list_a > list_a > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( P2 @ nil_a @ nil_a )
=> ( ! [X3: a,Xs3: list_a,Y2: a,Ys3: list_a] :
( ( ( size_size_list_a @ Xs3 )
= ( size_size_list_a @ Ys3 ) )
=> ( ( P2 @ Xs3 @ Ys3 )
=> ( P2 @ ( cons_a @ X3 @ Xs3 ) @ ( cons_a @ Y2 @ Ys3 ) ) ) )
=> ( P2 @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_673_list__induct2,axiom,
! [Xs: list_a,Ys: list_set_a,P2: list_a > list_set_a > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_set_a @ Ys ) )
=> ( ( P2 @ nil_a @ nil_set_a )
=> ( ! [X3: a,Xs3: list_a,Y2: set_a,Ys3: list_set_a] :
( ( ( size_size_list_a @ Xs3 )
= ( size_size_list_set_a @ Ys3 ) )
=> ( ( P2 @ Xs3 @ Ys3 )
=> ( P2 @ ( cons_a @ X3 @ Xs3 ) @ ( cons_set_a @ Y2 @ Ys3 ) ) ) )
=> ( P2 @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_674_list__induct2,axiom,
! [Xs: list_set_a,Ys: list_a,P2: list_set_a > list_a > $o] :
( ( ( size_size_list_set_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( P2 @ nil_set_a @ nil_a )
=> ( ! [X3: set_a,Xs3: list_set_a,Y2: a,Ys3: list_a] :
( ( ( size_size_list_set_a @ Xs3 )
= ( size_size_list_a @ Ys3 ) )
=> ( ( P2 @ Xs3 @ Ys3 )
=> ( P2 @ ( cons_set_a @ X3 @ Xs3 ) @ ( cons_a @ Y2 @ Ys3 ) ) ) )
=> ( P2 @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_675_list__induct2,axiom,
! [Xs: list_set_a,Ys: list_set_a,P2: list_set_a > list_set_a > $o] :
( ( ( size_size_list_set_a @ Xs )
= ( size_size_list_set_a @ Ys ) )
=> ( ( P2 @ nil_set_a @ nil_set_a )
=> ( ! [X3: set_a,Xs3: list_set_a,Y2: set_a,Ys3: list_set_a] :
( ( ( size_size_list_set_a @ Xs3 )
= ( size_size_list_set_a @ Ys3 ) )
=> ( ( P2 @ Xs3 @ Ys3 )
=> ( P2 @ ( cons_set_a @ X3 @ Xs3 ) @ ( cons_set_a @ Y2 @ Ys3 ) ) ) )
=> ( P2 @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_676_list__induct3,axiom,
! [Xs: list_a,Ys: list_a,Zs: list_a,P2: 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 ) )
=> ( ( P2 @ nil_a @ nil_a @ nil_a )
=> ( ! [X3: a,Xs3: list_a,Y2: a,Ys3: list_a,Z3: a,Zs3: list_a] :
( ( ( size_size_list_a @ Xs3 )
= ( size_size_list_a @ Ys3 ) )
=> ( ( ( size_size_list_a @ Ys3 )
= ( size_size_list_a @ Zs3 ) )
=> ( ( P2 @ Xs3 @ Ys3 @ Zs3 )
=> ( P2 @ ( cons_a @ X3 @ Xs3 ) @ ( cons_a @ Y2 @ Ys3 ) @ ( cons_a @ Z3 @ Zs3 ) ) ) ) )
=> ( P2 @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_677_list__induct3,axiom,
! [Xs: list_a,Ys: list_a,Zs: list_set_a,P2: 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 ) )
=> ( ( P2 @ nil_a @ nil_a @ nil_set_a )
=> ( ! [X3: a,Xs3: list_a,Y2: a,Ys3: list_a,Z3: set_a,Zs3: list_set_a] :
( ( ( size_size_list_a @ Xs3 )
= ( size_size_list_a @ Ys3 ) )
=> ( ( ( size_size_list_a @ Ys3 )
= ( size_size_list_set_a @ Zs3 ) )
=> ( ( P2 @ Xs3 @ Ys3 @ Zs3 )
=> ( P2 @ ( cons_a @ X3 @ Xs3 ) @ ( cons_a @ Y2 @ Ys3 ) @ ( cons_set_a @ Z3 @ Zs3 ) ) ) ) )
=> ( P2 @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_678_list__induct3,axiom,
! [Xs: list_a,Ys: list_set_a,Zs: list_a,P2: 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 ) )
=> ( ( P2 @ nil_a @ nil_set_a @ nil_a )
=> ( ! [X3: a,Xs3: list_a,Y2: set_a,Ys3: list_set_a,Z3: a,Zs3: list_a] :
( ( ( size_size_list_a @ Xs3 )
= ( size_size_list_set_a @ Ys3 ) )
=> ( ( ( size_size_list_set_a @ Ys3 )
= ( size_size_list_a @ Zs3 ) )
=> ( ( P2 @ Xs3 @ Ys3 @ Zs3 )
=> ( P2 @ ( cons_a @ X3 @ Xs3 ) @ ( cons_set_a @ Y2 @ Ys3 ) @ ( cons_a @ Z3 @ Zs3 ) ) ) ) )
=> ( P2 @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_679_list__induct3,axiom,
! [Xs: list_a,Ys: list_set_a,Zs: list_set_a,P2: 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 ) )
=> ( ( P2 @ nil_a @ nil_set_a @ nil_set_a )
=> ( ! [X3: a,Xs3: list_a,Y2: set_a,Ys3: list_set_a,Z3: set_a,Zs3: 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 @ Zs3 ) )
=> ( ( P2 @ Xs3 @ Ys3 @ Zs3 )
=> ( P2 @ ( cons_a @ X3 @ Xs3 ) @ ( cons_set_a @ Y2 @ Ys3 ) @ ( cons_set_a @ Z3 @ Zs3 ) ) ) ) )
=> ( P2 @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_680_list__induct3,axiom,
! [Xs: list_set_a,Ys: list_a,Zs: list_a,P2: 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 ) )
=> ( ( P2 @ nil_set_a @ nil_a @ nil_a )
=> ( ! [X3: set_a,Xs3: list_set_a,Y2: a,Ys3: list_a,Z3: a,Zs3: list_a] :
( ( ( size_size_list_set_a @ Xs3 )
= ( size_size_list_a @ Ys3 ) )
=> ( ( ( size_size_list_a @ Ys3 )
= ( size_size_list_a @ Zs3 ) )
=> ( ( P2 @ Xs3 @ Ys3 @ Zs3 )
=> ( P2 @ ( cons_set_a @ X3 @ Xs3 ) @ ( cons_a @ Y2 @ Ys3 ) @ ( cons_a @ Z3 @ Zs3 ) ) ) ) )
=> ( P2 @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_681_list__induct3,axiom,
! [Xs: list_set_a,Ys: list_a,Zs: list_set_a,P2: 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 ) )
=> ( ( P2 @ nil_set_a @ nil_a @ nil_set_a )
=> ( ! [X3: set_a,Xs3: list_set_a,Y2: a,Ys3: list_a,Z3: set_a,Zs3: list_set_a] :
( ( ( size_size_list_set_a @ Xs3 )
= ( size_size_list_a @ Ys3 ) )
=> ( ( ( size_size_list_a @ Ys3 )
= ( size_size_list_set_a @ Zs3 ) )
=> ( ( P2 @ Xs3 @ Ys3 @ Zs3 )
=> ( P2 @ ( cons_set_a @ X3 @ Xs3 ) @ ( cons_a @ Y2 @ Ys3 ) @ ( cons_set_a @ Z3 @ Zs3 ) ) ) ) )
=> ( P2 @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_682_list__induct3,axiom,
! [Xs: list_set_a,Ys: list_set_a,Zs: list_a,P2: 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 ) )
=> ( ( P2 @ nil_set_a @ nil_set_a @ nil_a )
=> ( ! [X3: set_a,Xs3: list_set_a,Y2: set_a,Ys3: list_set_a,Z3: a,Zs3: list_a] :
( ( ( size_size_list_set_a @ Xs3 )
= ( size_size_list_set_a @ Ys3 ) )
=> ( ( ( size_size_list_set_a @ Ys3 )
= ( size_size_list_a @ Zs3 ) )
=> ( ( P2 @ Xs3 @ Ys3 @ Zs3 )
=> ( P2 @ ( cons_set_a @ X3 @ Xs3 ) @ ( cons_set_a @ Y2 @ Ys3 ) @ ( cons_a @ Z3 @ Zs3 ) ) ) ) )
=> ( P2 @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_683_list__induct3,axiom,
! [Xs: list_set_a,Ys: list_set_a,Zs: list_set_a,P2: 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 ) )
=> ( ( P2 @ nil_set_a @ nil_set_a @ nil_set_a )
=> ( ! [X3: set_a,Xs3: list_set_a,Y2: set_a,Ys3: list_set_a,Z3: set_a,Zs3: 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 @ Zs3 ) )
=> ( ( P2 @ Xs3 @ Ys3 @ Zs3 )
=> ( P2 @ ( cons_set_a @ X3 @ Xs3 ) @ ( cons_set_a @ Y2 @ Ys3 ) @ ( cons_set_a @ Z3 @ Zs3 ) ) ) ) )
=> ( P2 @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_684_list__induct4,axiom,
! [Xs: list_a,Ys: list_a,Zs: list_a,Ws: list_a,P2: 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 ) )
=> ( ( P2 @ nil_a @ nil_a @ nil_a @ nil_a )
=> ( ! [X3: a,Xs3: list_a,Y2: a,Ys3: list_a,Z3: a,Zs3: 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 @ Zs3 ) )
=> ( ( ( size_size_list_a @ Zs3 )
= ( size_size_list_a @ Ws2 ) )
=> ( ( P2 @ Xs3 @ Ys3 @ Zs3 @ Ws2 )
=> ( P2 @ ( cons_a @ X3 @ Xs3 ) @ ( cons_a @ Y2 @ Ys3 ) @ ( cons_a @ Z3 @ Zs3 ) @ ( cons_a @ W @ Ws2 ) ) ) ) ) )
=> ( P2 @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_685_list__induct4,axiom,
! [Xs: list_a,Ys: list_a,Zs: list_a,Ws: list_set_a,P2: 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 ) )
=> ( ( P2 @ nil_a @ nil_a @ nil_a @ nil_set_a )
=> ( ! [X3: a,Xs3: list_a,Y2: a,Ys3: list_a,Z3: a,Zs3: 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 @ Zs3 ) )
=> ( ( ( size_size_list_a @ Zs3 )
= ( size_size_list_set_a @ Ws2 ) )
=> ( ( P2 @ Xs3 @ Ys3 @ Zs3 @ Ws2 )
=> ( P2 @ ( cons_a @ X3 @ Xs3 ) @ ( cons_a @ Y2 @ Ys3 ) @ ( cons_a @ Z3 @ Zs3 ) @ ( cons_set_a @ W @ Ws2 ) ) ) ) ) )
=> ( P2 @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_686_list__induct4,axiom,
! [Xs: list_a,Ys: list_a,Zs: list_set_a,Ws: list_a,P2: 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 ) )
=> ( ( P2 @ nil_a @ nil_a @ nil_set_a @ nil_a )
=> ( ! [X3: a,Xs3: list_a,Y2: a,Ys3: list_a,Z3: set_a,Zs3: 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 @ Zs3 ) )
=> ( ( ( size_size_list_set_a @ Zs3 )
= ( size_size_list_a @ Ws2 ) )
=> ( ( P2 @ Xs3 @ Ys3 @ Zs3 @ Ws2 )
=> ( P2 @ ( cons_a @ X3 @ Xs3 ) @ ( cons_a @ Y2 @ Ys3 ) @ ( cons_set_a @ Z3 @ Zs3 ) @ ( cons_a @ W @ Ws2 ) ) ) ) ) )
=> ( P2 @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_687_list__induct4,axiom,
! [Xs: list_a,Ys: list_set_a,Zs: list_a,Ws: list_a,P2: 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 ) )
=> ( ( P2 @ nil_a @ nil_set_a @ nil_a @ nil_a )
=> ( ! [X3: a,Xs3: list_a,Y2: set_a,Ys3: list_set_a,Z3: a,Zs3: 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 @ Zs3 ) )
=> ( ( ( size_size_list_a @ Zs3 )
= ( size_size_list_a @ Ws2 ) )
=> ( ( P2 @ Xs3 @ Ys3 @ Zs3 @ Ws2 )
=> ( P2 @ ( cons_a @ X3 @ Xs3 ) @ ( cons_set_a @ Y2 @ Ys3 ) @ ( cons_a @ Z3 @ Zs3 ) @ ( cons_a @ W @ Ws2 ) ) ) ) ) )
=> ( P2 @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_688_list__induct4,axiom,
! [Xs: list_set_a,Ys: list_a,Zs: list_a,Ws: list_a,P2: 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 ) )
=> ( ( P2 @ nil_set_a @ nil_a @ nil_a @ nil_a )
=> ( ! [X3: set_a,Xs3: list_set_a,Y2: a,Ys3: list_a,Z3: a,Zs3: 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 @ Zs3 ) )
=> ( ( ( size_size_list_a @ Zs3 )
= ( size_size_list_a @ Ws2 ) )
=> ( ( P2 @ Xs3 @ Ys3 @ Zs3 @ Ws2 )
=> ( P2 @ ( cons_set_a @ X3 @ Xs3 ) @ ( cons_a @ Y2 @ Ys3 ) @ ( cons_a @ Z3 @ Zs3 ) @ ( cons_a @ W @ Ws2 ) ) ) ) ) )
=> ( P2 @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_689_list__induct4,axiom,
! [Xs: list_a,Ys: list_a,Zs: list_set_a,Ws: list_set_a,P2: 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 ) )
=> ( ( P2 @ nil_a @ nil_a @ nil_set_a @ nil_set_a )
=> ( ! [X3: a,Xs3: list_a,Y2: a,Ys3: list_a,Z3: set_a,Zs3: 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 @ Zs3 ) )
=> ( ( ( size_size_list_set_a @ Zs3 )
= ( size_size_list_set_a @ Ws2 ) )
=> ( ( P2 @ Xs3 @ Ys3 @ Zs3 @ Ws2 )
=> ( P2 @ ( cons_a @ X3 @ Xs3 ) @ ( cons_a @ Y2 @ Ys3 ) @ ( cons_set_a @ Z3 @ Zs3 ) @ ( cons_set_a @ W @ Ws2 ) ) ) ) ) )
=> ( P2 @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_690_list__induct4,axiom,
! [Xs: list_a,Ys: list_set_a,Zs: list_a,Ws: list_set_a,P2: 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 ) )
=> ( ( P2 @ nil_a @ nil_set_a @ nil_a @ nil_set_a )
=> ( ! [X3: a,Xs3: list_a,Y2: set_a,Ys3: list_set_a,Z3: a,Zs3: 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 @ Zs3 ) )
=> ( ( ( size_size_list_a @ Zs3 )
= ( size_size_list_set_a @ Ws2 ) )
=> ( ( P2 @ Xs3 @ Ys3 @ Zs3 @ Ws2 )
=> ( P2 @ ( cons_a @ X3 @ Xs3 ) @ ( cons_set_a @ Y2 @ Ys3 ) @ ( cons_a @ Z3 @ Zs3 ) @ ( cons_set_a @ W @ Ws2 ) ) ) ) ) )
=> ( P2 @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_691_list__induct4,axiom,
! [Xs: list_a,Ys: list_set_a,Zs: list_set_a,Ws: list_a,P2: 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 ) )
=> ( ( P2 @ nil_a @ nil_set_a @ nil_set_a @ nil_a )
=> ( ! [X3: a,Xs3: list_a,Y2: set_a,Ys3: list_set_a,Z3: set_a,Zs3: 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 @ Zs3 ) )
=> ( ( ( size_size_list_set_a @ Zs3 )
= ( size_size_list_a @ Ws2 ) )
=> ( ( P2 @ Xs3 @ Ys3 @ Zs3 @ Ws2 )
=> ( P2 @ ( cons_a @ X3 @ Xs3 ) @ ( cons_set_a @ Y2 @ Ys3 ) @ ( cons_set_a @ Z3 @ Zs3 ) @ ( cons_a @ W @ Ws2 ) ) ) ) ) )
=> ( P2 @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_692_list__induct4,axiom,
! [Xs: list_set_a,Ys: list_a,Zs: list_a,Ws: list_set_a,P2: 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 ) )
=> ( ( P2 @ nil_set_a @ nil_a @ nil_a @ nil_set_a )
=> ( ! [X3: set_a,Xs3: list_set_a,Y2: a,Ys3: list_a,Z3: a,Zs3: 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 @ Zs3 ) )
=> ( ( ( size_size_list_a @ Zs3 )
= ( size_size_list_set_a @ Ws2 ) )
=> ( ( P2 @ Xs3 @ Ys3 @ Zs3 @ Ws2 )
=> ( P2 @ ( cons_set_a @ X3 @ Xs3 ) @ ( cons_a @ Y2 @ Ys3 ) @ ( cons_a @ Z3 @ Zs3 ) @ ( cons_set_a @ W @ Ws2 ) ) ) ) ) )
=> ( P2 @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_693_list__induct4,axiom,
! [Xs: list_set_a,Ys: list_a,Zs: list_set_a,Ws: list_a,P2: 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 ) )
=> ( ( P2 @ nil_set_a @ nil_a @ nil_set_a @ nil_a )
=> ( ! [X3: set_a,Xs3: list_set_a,Y2: a,Ys3: list_a,Z3: set_a,Zs3: 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 @ Zs3 ) )
=> ( ( ( size_size_list_set_a @ Zs3 )
= ( size_size_list_a @ Ws2 ) )
=> ( ( P2 @ Xs3 @ Ys3 @ Zs3 @ Ws2 )
=> ( P2 @ ( cons_set_a @ X3 @ Xs3 ) @ ( cons_a @ Y2 @ Ys3 ) @ ( cons_set_a @ Z3 @ Zs3 ) @ ( cons_a @ W @ Ws2 ) ) ) ) ) )
=> ( P2 @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_694_impossible__Cons,axiom,
! [Xs: list_a,Ys: list_a,X4: a] :
( ( ord_less_eq_nat @ ( size_size_list_a @ Xs ) @ ( size_size_list_a @ Ys ) )
=> ( Xs
!= ( cons_a @ X4 @ Ys ) ) ) ).
% impossible_Cons
thf(fact_695_impossible__Cons,axiom,
! [Xs: list_set_a,Ys: list_set_a,X4: set_a] :
( ( ord_less_eq_nat @ ( size_size_list_set_a @ Xs ) @ ( size_size_list_set_a @ Ys ) )
=> ( Xs
!= ( cons_set_a @ X4 @ Ys ) ) ) ).
% impossible_Cons
thf(fact_696_all__edges__mono,axiom,
! [Vs: set_a,Ws: set_a] :
( ( ord_less_eq_set_a @ Vs @ Ws )
=> ( ord_le3724670747650509150_set_a @ ( undire2918257014606996450dges_a @ Vs ) @ ( undire2918257014606996450dges_a @ Ws ) ) ) ).
% all_edges_mono
thf(fact_697_all__edges__mono,axiom,
! [Vs: set_set_a,Ws: set_set_a] :
( ( ord_le3724670747650509150_set_a @ Vs @ Ws )
=> ( ord_le5722252365846178494_set_a @ ( undire8247866692393712962_set_a @ Vs ) @ ( undire8247866692393712962_set_a @ Ws ) ) ) ).
% all_edges_mono
thf(fact_698_comp__sgraph_Ois__walk__not__empty,axiom,
! [S: set_set_a,Xs: list_set_a] :
( ( undire3014741414213135564_set_a @ S @ ( undire8247866692393712962_set_a @ S ) @ Xs )
=> ( Xs != nil_set_a ) ) ).
% comp_sgraph.is_walk_not_empty
thf(fact_699_comp__sgraph_Ois__walk__not__empty,axiom,
! [S: set_a,Xs: list_a] :
( ( undire6133010728901294956walk_a @ S @ ( undire2918257014606996450dges_a @ S ) @ Xs )
=> ( Xs != nil_a ) ) ).
% comp_sgraph.is_walk_not_empty
thf(fact_700_comp__sgraph_Ois__walk__not__empty2,axiom,
! [S: set_set_a] :
~ ( undire3014741414213135564_set_a @ S @ ( undire8247866692393712962_set_a @ S ) @ nil_set_a ) ).
% comp_sgraph.is_walk_not_empty2
thf(fact_701_comp__sgraph_Ois__walk__not__empty2,axiom,
! [S: set_a] :
~ ( undire6133010728901294956walk_a @ S @ ( undire2918257014606996450dges_a @ S ) @ nil_a ) ).
% comp_sgraph.is_walk_not_empty2
thf(fact_702_comp__sgraph_Ois__walk__rev,axiom,
! [S: set_set_a,Xs: list_set_a] :
( ( undire3014741414213135564_set_a @ S @ ( undire8247866692393712962_set_a @ S ) @ Xs )
= ( undire3014741414213135564_set_a @ S @ ( undire8247866692393712962_set_a @ S ) @ ( rev_set_a @ Xs ) ) ) ).
% comp_sgraph.is_walk_rev
thf(fact_703_comp__sgraph_Ois__walk__rev,axiom,
! [S: set_a,Xs: list_a] :
( ( undire6133010728901294956walk_a @ S @ ( undire2918257014606996450dges_a @ S ) @ Xs )
= ( undire6133010728901294956walk_a @ S @ ( undire2918257014606996450dges_a @ S ) @ ( rev_a @ Xs ) ) ) ).
% comp_sgraph.is_walk_rev
thf(fact_704_comp__sgraph_Ois__walk__wf__hd,axiom,
! [S: set_set_a,Xs: list_set_a] :
( ( undire3014741414213135564_set_a @ S @ ( undire8247866692393712962_set_a @ S ) @ Xs )
=> ( member_set_a @ ( hd_set_a @ Xs ) @ S ) ) ).
% comp_sgraph.is_walk_wf_hd
thf(fact_705_comp__sgraph_Ois__walk__wf__hd,axiom,
! [S: set_a,Xs: list_a] :
( ( undire6133010728901294956walk_a @ S @ ( undire2918257014606996450dges_a @ S ) @ Xs )
=> ( member_a @ ( hd_a @ Xs ) @ S ) ) ).
% comp_sgraph.is_walk_wf_hd
thf(fact_706_comp__sgraph_Ois__walk__wf__last,axiom,
! [S: set_set_a,Xs: list_set_a] :
( ( undire3014741414213135564_set_a @ S @ ( undire8247866692393712962_set_a @ S ) @ Xs )
=> ( member_set_a @ ( last_set_a @ Xs ) @ S ) ) ).
% comp_sgraph.is_walk_wf_last
thf(fact_707_comp__sgraph_Ois__walk__wf__last,axiom,
! [S: set_a,Xs: list_a] :
( ( undire6133010728901294956walk_a @ S @ ( undire2918257014606996450dges_a @ S ) @ Xs )
=> ( member_a @ ( last_a @ Xs ) @ S ) ) ).
% comp_sgraph.is_walk_wf_last
thf(fact_708_comp__sgraph_Oconnecting__path__rev,axiom,
! [S: set_set_a,U: set_a,V: set_a,Xs: list_set_a] :
( ( connec7350987497872064604_set_a @ S @ ( undire8247866692393712962_set_a @ S ) @ U @ V @ Xs )
= ( connec7350987497872064604_set_a @ S @ ( undire8247866692393712962_set_a @ S ) @ V @ U @ ( rev_set_a @ Xs ) ) ) ).
% comp_sgraph.connecting_path_rev
thf(fact_709_comp__sgraph_Oconnecting__path__rev,axiom,
! [S: set_a,U: a,V: a,Xs: list_a] :
( ( connecting_path_a @ S @ ( undire2918257014606996450dges_a @ S ) @ U @ V @ Xs )
= ( connecting_path_a @ S @ ( undire2918257014606996450dges_a @ S ) @ V @ U @ ( rev_a @ Xs ) ) ) ).
% comp_sgraph.connecting_path_rev
thf(fact_710_comp__sgraph_Ois__gen__path__rev,axiom,
! [S: set_set_a,P: list_set_a] :
( ( undire7201326534205417136_set_a @ S @ ( undire8247866692393712962_set_a @ S ) @ P )
= ( undire7201326534205417136_set_a @ S @ ( undire8247866692393712962_set_a @ S ) @ ( rev_set_a @ P ) ) ) ).
% comp_sgraph.is_gen_path_rev
thf(fact_711_comp__sgraph_Ois__gen__path__rev,axiom,
! [S: set_a,P: list_a] :
( ( undire3562951555376170320path_a @ S @ ( undire2918257014606996450dges_a @ S ) @ P )
= ( undire3562951555376170320path_a @ S @ ( undire2918257014606996450dges_a @ S ) @ ( rev_a @ P ) ) ) ).
% comp_sgraph.is_gen_path_rev
thf(fact_712_comp__sgraph_Ois__cycle__rev,axiom,
! [S: set_set_a,Xs: list_set_a] :
( ( undire797940137672299967_set_a @ S @ ( undire8247866692393712962_set_a @ S ) @ Xs )
= ( undire797940137672299967_set_a @ S @ ( undire8247866692393712962_set_a @ S ) @ ( rev_set_a @ Xs ) ) ) ).
% comp_sgraph.is_cycle_rev
thf(fact_713_comp__sgraph_Ois__cycle__rev,axiom,
! [S: set_a,Xs: list_a] :
( ( undire2407311113669455967ycle_a @ S @ ( undire2918257014606996450dges_a @ S ) @ Xs )
= ( undire2407311113669455967ycle_a @ S @ ( undire2918257014606996450dges_a @ S ) @ ( rev_a @ Xs ) ) ) ).
% comp_sgraph.is_cycle_rev
thf(fact_714_comp__sgraph_Oconnecting__walk__rev,axiom,
! [S: set_set_a,U: set_a,V: set_a,Xs: list_set_a] :
( ( connec1530789871921280536_set_a @ S @ ( undire8247866692393712962_set_a @ S ) @ U @ V @ Xs )
= ( connec1530789871921280536_set_a @ S @ ( undire8247866692393712962_set_a @ S ) @ V @ U @ ( rev_set_a @ Xs ) ) ) ).
% comp_sgraph.connecting_walk_rev
thf(fact_715_comp__sgraph_Oconnecting__walk__rev,axiom,
! [S: set_a,U: a,V: a,Xs: list_a] :
( ( connecting_walk_a @ S @ ( undire2918257014606996450dges_a @ S ) @ U @ V @ Xs )
= ( connecting_walk_a @ S @ ( undire2918257014606996450dges_a @ S ) @ V @ U @ ( rev_a @ Xs ) ) ) ).
% comp_sgraph.connecting_walk_rev
thf(fact_716_comp__sgraph_Ois__path__rev,axiom,
! [S: set_set_a,Xs: list_set_a] :
( ( undire8834939040163919632_set_a @ S @ ( undire8247866692393712962_set_a @ S ) @ Xs )
= ( undire8834939040163919632_set_a @ S @ ( undire8247866692393712962_set_a @ S ) @ ( rev_set_a @ Xs ) ) ) ).
% comp_sgraph.is_path_rev
thf(fact_717_comp__sgraph_Ois__path__rev,axiom,
! [S: set_a,Xs: list_a] :
( ( undire427332500224447920path_a @ S @ ( undire2918257014606996450dges_a @ S ) @ Xs )
= ( undire427332500224447920path_a @ S @ ( undire2918257014606996450dges_a @ S ) @ ( rev_a @ Xs ) ) ) ).
% comp_sgraph.is_path_rev
thf(fact_718_comp__sgraph_Ois__path__walk,axiom,
! [S: set_a,Xs: list_a] :
( ( undire427332500224447920path_a @ S @ ( undire2918257014606996450dges_a @ S ) @ Xs )
=> ( undire6133010728901294956walk_a @ S @ ( undire2918257014606996450dges_a @ S ) @ Xs ) ) ).
% comp_sgraph.is_path_walk
thf(fact_719_comp__sgraph_Oinduced__is__graph__sys,axiom,
! [V3: set_a,S: set_a] : ( undire2554140024507503526stem_a @ V3 @ ( undire7777452895879145676dges_a @ ( undire2918257014606996450dges_a @ S ) @ V3 ) ) ).
% comp_sgraph.induced_is_graph_sys
thf(fact_720_comp__sgraph_Oconnecting__path__walk,axiom,
! [S: set_a,U: a,V: a,Xs: list_a] :
( ( connecting_path_a @ S @ ( undire2918257014606996450dges_a @ S ) @ U @ V @ Xs )
=> ( connecting_walk_a @ S @ ( undire2918257014606996450dges_a @ S ) @ U @ V @ Xs ) ) ).
% comp_sgraph.connecting_path_walk
thf(fact_721_comp__sgraph_Ois__gen__path__cycle,axiom,
! [S: set_a,P: list_a] :
( ( undire2407311113669455967ycle_a @ S @ ( undire2918257014606996450dges_a @ S ) @ P )
=> ( undire3562951555376170320path_a @ S @ ( undire2918257014606996450dges_a @ S ) @ P ) ) ).
% comp_sgraph.is_gen_path_cycle
thf(fact_722_comp__sgraph_Ois__path__gen__path,axiom,
! [S: set_a,P: list_a] :
( ( undire427332500224447920path_a @ S @ ( undire2918257014606996450dges_a @ S ) @ P )
=> ( undire3562951555376170320path_a @ S @ ( undire2918257014606996450dges_a @ S ) @ P ) ) ).
% comp_sgraph.is_path_gen_path
thf(fact_723_comp__sgraph_Overt__adj__edge__iff2,axiom,
! [V1: a,V2: a,S: set_a] :
( ( V1 != V2 )
=> ( ( undire397441198561214472_adj_a @ ( undire2918257014606996450dges_a @ S ) @ V1 @ V2 )
= ( ? [X2: set_a] :
( ( member_set_a @ X2 @ ( undire2918257014606996450dges_a @ S ) )
& ( undire1521409233611534436dent_a @ V1 @ X2 )
& ( undire1521409233611534436dent_a @ V2 @ X2 ) ) ) ) ) ).
% comp_sgraph.vert_adj_edge_iff2
thf(fact_724_comp__sgraph_Ois__closed__walk__rev,axiom,
! [S: set_set_a,Xs: list_set_a] :
( ( undire4100213446647512896_set_a @ S @ ( undire8247866692393712962_set_a @ S ) @ Xs )
= ( undire4100213446647512896_set_a @ S @ ( undire8247866692393712962_set_a @ S ) @ ( rev_set_a @ Xs ) ) ) ).
% comp_sgraph.is_closed_walk_rev
thf(fact_725_comp__sgraph_Ois__closed__walk__rev,axiom,
! [S: set_a,Xs: list_a] :
( ( undire3370724456595283424walk_a @ S @ ( undire2918257014606996450dges_a @ S ) @ Xs )
= ( undire3370724456595283424walk_a @ S @ ( undire2918257014606996450dges_a @ S ) @ ( rev_a @ Xs ) ) ) ).
% comp_sgraph.is_closed_walk_rev
thf(fact_726_comp__sgraph_Ois__open__walk__rev,axiom,
! [S: set_set_a,Xs: list_set_a] :
( ( undire526879649183275522_set_a @ S @ ( undire8247866692393712962_set_a @ S ) @ Xs )
= ( undire526879649183275522_set_a @ S @ ( undire8247866692393712962_set_a @ S ) @ ( rev_set_a @ Xs ) ) ) ).
% comp_sgraph.is_open_walk_rev
thf(fact_727_comp__sgraph_Ois__open__walk__rev,axiom,
! [S: set_a,Xs: list_a] :
( ( undire2427028224930250914walk_a @ S @ ( undire2918257014606996450dges_a @ S ) @ Xs )
= ( undire2427028224930250914walk_a @ S @ ( undire2918257014606996450dges_a @ S ) @ ( rev_a @ Xs ) ) ) ).
% comp_sgraph.is_open_walk_rev
thf(fact_728_comp__sgraph_Ois__trail__rev,axiom,
! [S: set_set_a,Xs: list_set_a] :
( ( undire1224551742100448159_set_a @ S @ ( undire8247866692393712962_set_a @ S ) @ Xs )
= ( undire1224551742100448159_set_a @ S @ ( undire8247866692393712962_set_a @ S ) @ ( rev_set_a @ Xs ) ) ) ).
% comp_sgraph.is_trail_rev
thf(fact_729_comp__sgraph_Ois__trail__rev,axiom,
! [S: set_a,Xs: list_a] :
( ( undire7142031287334043199rail_a @ S @ ( undire2918257014606996450dges_a @ S ) @ Xs )
= ( undire7142031287334043199rail_a @ S @ ( undire2918257014606996450dges_a @ S ) @ ( rev_a @ Xs ) ) ) ).
% comp_sgraph.is_trail_rev
thf(fact_730_comp__sgraph_Ois__isolated__vertex__def,axiom,
! [S: set_set_a,V: set_a] :
( ( undire6879241558604981877_set_a @ S @ ( undire8247866692393712962_set_a @ S ) @ V )
= ( ( member_set_a @ V @ S )
& ! [X2: set_a] :
( ( member_set_a @ X2 @ S )
=> ~ ( undire3510646817838285160_set_a @ ( undire8247866692393712962_set_a @ S ) @ X2 @ V ) ) ) ) ).
% comp_sgraph.is_isolated_vertex_def
thf(fact_731_comp__sgraph_Ois__isolated__vertex__def,axiom,
! [S: set_a,V: a] :
( ( undire8931668460104145173rtex_a @ S @ ( undire2918257014606996450dges_a @ S ) @ V )
= ( ( member_a @ V @ S )
& ! [X2: a] :
( ( member_a @ X2 @ S )
=> ~ ( undire397441198561214472_adj_a @ ( undire2918257014606996450dges_a @ S ) @ X2 @ V ) ) ) ) ).
% comp_sgraph.is_isolated_vertex_def
thf(fact_732_comp__sgraph_Ois__isolated__vertex__edge,axiom,
! [S: set_a,V: a,E: set_a] :
( ( undire8931668460104145173rtex_a @ S @ ( undire2918257014606996450dges_a @ S ) @ V )
=> ( ( member_set_a @ E @ ( undire2918257014606996450dges_a @ S ) )
=> ~ ( undire1521409233611534436dent_a @ V @ E ) ) ) ).
% comp_sgraph.is_isolated_vertex_edge
thf(fact_733_comp__sgraph_Ois__isolated__vertex__no__loop,axiom,
! [S: set_a,V: a] :
( ( undire8931668460104145173rtex_a @ S @ ( undire2918257014606996450dges_a @ S ) @ V )
=> ~ ( undire3617971648856834880loop_a @ ( undire2918257014606996450dges_a @ S ) @ V ) ) ).
% comp_sgraph.is_isolated_vertex_no_loop
thf(fact_734_comp__sgraph_Oconnecting__path__str__gen,axiom,
! [S: set_a,U: a,V: a,Xs: list_a] :
( ( connec3015921205769380621_str_a @ S @ ( undire2918257014606996450dges_a @ S ) @ U @ V @ Xs )
=> ( connecting_path_a @ S @ ( undire2918257014606996450dges_a @ S ) @ U @ V @ Xs ) ) ).
% comp_sgraph.connecting_path_str_gen
thf(fact_735_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,X3: a,Xs2: list_a,Y2: a,Ys2: list_a] :
( ( X3 != Y2 )
& ( Xs
= ( append_a @ Pre @ ( append_a @ ( cons_a @ X3 @ nil_a ) @ Xs2 ) ) )
& ( Ys
= ( append_a @ Pre @ ( append_a @ ( cons_a @ Y2 @ nil_a ) @ Ys2 ) ) ) ) ) ) ).
% same_length_different
thf(fact_736_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,X3: set_a,Xs2: list_set_a,Y2: set_a,Ys2: list_set_a] :
( ( X3 != Y2 )
& ( Xs
= ( append_set_a @ Pre @ ( append_set_a @ ( cons_set_a @ X3 @ nil_set_a ) @ Xs2 ) ) )
& ( Ys
= ( append_set_a @ Pre @ ( append_set_a @ ( cons_set_a @ Y2 @ nil_set_a ) @ Ys2 ) ) ) ) ) ) ).
% same_length_different
thf(fact_737_comp__sgraph_Ois__walk__singleton,axiom,
! [U: set_a,S: set_set_a] :
( ( member_set_a @ U @ S )
=> ( undire3014741414213135564_set_a @ S @ ( undire8247866692393712962_set_a @ S ) @ ( cons_set_a @ U @ nil_set_a ) ) ) ).
% comp_sgraph.is_walk_singleton
thf(fact_738_comp__sgraph_Ois__walk__singleton,axiom,
! [U: a,S: set_a] :
( ( member_a @ U @ S )
=> ( undire6133010728901294956walk_a @ S @ ( undire2918257014606996450dges_a @ S ) @ ( cons_a @ U @ nil_a ) ) ) ).
% comp_sgraph.is_walk_singleton
thf(fact_739_comp__sgraph_Ois__walk__drop__hd,axiom,
! [Ys: list_set_a,S: set_set_a,Y: set_a] :
( ( Ys != nil_set_a )
=> ( ( undire3014741414213135564_set_a @ S @ ( undire8247866692393712962_set_a @ S ) @ ( cons_set_a @ Y @ Ys ) )
=> ( undire3014741414213135564_set_a @ S @ ( undire8247866692393712962_set_a @ S ) @ Ys ) ) ) ).
% comp_sgraph.is_walk_drop_hd
thf(fact_740_comp__sgraph_Ois__walk__drop__hd,axiom,
! [Ys: list_a,S: set_a,Y: a] :
( ( Ys != nil_a )
=> ( ( undire6133010728901294956walk_a @ S @ ( undire2918257014606996450dges_a @ S ) @ ( cons_a @ Y @ Ys ) )
=> ( undire6133010728901294956walk_a @ S @ ( undire2918257014606996450dges_a @ S ) @ Ys ) ) ) ).
% comp_sgraph.is_walk_drop_hd
thf(fact_741_comp__sgraph_Ois__walk__wf,axiom,
! [S: set_set_a,Xs: list_set_a] :
( ( undire3014741414213135564_set_a @ S @ ( undire8247866692393712962_set_a @ S ) @ Xs )
=> ( ord_le3724670747650509150_set_a @ ( set_set_a2 @ Xs ) @ S ) ) ).
% comp_sgraph.is_walk_wf
thf(fact_742_comp__sgraph_Ois__walk__wf,axiom,
! [S: set_a,Xs: list_a] :
( ( undire6133010728901294956walk_a @ S @ ( undire2918257014606996450dges_a @ S ) @ Xs )
=> ( ord_less_eq_set_a @ ( set_a2 @ Xs ) @ S ) ) ).
% comp_sgraph.is_walk_wf
thf(fact_743_comp__sgraph_Oconnecting__path__self,axiom,
! [U: set_a,S: set_set_a] :
( ( member_set_a @ U @ S )
=> ( connec7350987497872064604_set_a @ S @ ( undire8247866692393712962_set_a @ S ) @ U @ U @ ( cons_set_a @ U @ nil_set_a ) ) ) ).
% comp_sgraph.connecting_path_self
thf(fact_744_comp__sgraph_Oconnecting__path__self,axiom,
! [U: a,S: set_a] :
( ( member_a @ U @ S )
=> ( connecting_path_a @ S @ ( undire2918257014606996450dges_a @ S ) @ U @ U @ ( cons_a @ U @ nil_a ) ) ) ).
% comp_sgraph.connecting_path_self
thf(fact_745_comp__sgraph_Ois__gen__path__trivial,axiom,
! [X4: set_a,S: set_set_a] :
( ( member_set_a @ X4 @ S )
=> ( undire7201326534205417136_set_a @ S @ ( undire8247866692393712962_set_a @ S ) @ ( cons_set_a @ X4 @ nil_set_a ) ) ) ).
% comp_sgraph.is_gen_path_trivial
thf(fact_746_comp__sgraph_Ois__gen__path__trivial,axiom,
! [X4: a,S: set_a] :
( ( member_a @ X4 @ S )
=> ( undire3562951555376170320path_a @ S @ ( undire2918257014606996450dges_a @ S ) @ ( cons_a @ X4 @ nil_a ) ) ) ).
% comp_sgraph.is_gen_path_trivial
thf(fact_747_comp__sgraph_Oconnecting__walk__self,axiom,
! [U: set_a,S: set_set_a] :
( ( member_set_a @ U @ S )
=> ( connec1530789871921280536_set_a @ S @ ( undire8247866692393712962_set_a @ S ) @ U @ U @ ( cons_set_a @ U @ nil_set_a ) ) ) ).
% comp_sgraph.connecting_walk_self
thf(fact_748_comp__sgraph_Oconnecting__walk__self,axiom,
! [U: a,S: set_a] :
( ( member_a @ U @ S )
=> ( connecting_walk_a @ S @ ( undire2918257014606996450dges_a @ S ) @ U @ U @ ( cons_a @ U @ nil_a ) ) ) ).
% comp_sgraph.connecting_walk_self
thf(fact_749_comp__sgraph_Oinduced__edges__ss,axiom,
! [V3: set_set_a,S: set_set_a] :
( ( ord_le3724670747650509150_set_a @ V3 @ S )
=> ( ord_le5722252365846178494_set_a @ ( undire7854589003810675244_set_a @ ( undire8247866692393712962_set_a @ S ) @ V3 ) @ ( undire8247866692393712962_set_a @ S ) ) ) ).
% comp_sgraph.induced_edges_ss
thf(fact_750_comp__sgraph_Oinduced__edges__ss,axiom,
! [V3: set_a,S: set_a] :
( ( ord_less_eq_set_a @ V3 @ S )
=> ( ord_le3724670747650509150_set_a @ ( undire7777452895879145676dges_a @ ( undire2918257014606996450dges_a @ S ) @ V3 ) @ ( undire2918257014606996450dges_a @ S ) ) ) ).
% comp_sgraph.induced_edges_ss
thf(fact_751_ulgraph_Oconnecting__path__singleton,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,U: set_a,V: set_a,Xs: list_set_a] :
( ( undire6886684016831807756_set_a @ Vertices @ Edges )
=> ( ( connec7350987497872064604_set_a @ Vertices @ Edges @ U @ V @ Xs )
=> ( ( ( size_size_list_set_a @ Xs )
= one_one_nat )
=> ( U = V ) ) ) ) ).
% ulgraph.connecting_path_singleton
thf(fact_752_ulgraph_Oconnecting__path__singleton,axiom,
! [Vertices: set_a,Edges: set_set_a,U: a,V: a,Xs: list_a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( connecting_path_a @ Vertices @ Edges @ U @ V @ Xs )
=> ( ( ( size_size_list_a @ Xs )
= one_one_nat )
=> ( U = V ) ) ) ) ).
% ulgraph.connecting_path_singleton
thf(fact_753_comp__sgraph_Oconnecting__walk__split,axiom,
! [S: set_set_a,U: set_a,V: set_a,Xs: list_set_a,Z: set_a,Ys: list_set_a] :
( ( connec1530789871921280536_set_a @ S @ ( undire8247866692393712962_set_a @ S ) @ U @ V @ Xs )
=> ( ( connec1530789871921280536_set_a @ S @ ( undire8247866692393712962_set_a @ S ) @ V @ Z @ Ys )
=> ( connec1530789871921280536_set_a @ S @ ( undire8247866692393712962_set_a @ S ) @ U @ Z @ ( append_set_a @ Xs @ ( tl_set_a @ Ys ) ) ) ) ) ).
% comp_sgraph.connecting_walk_split
thf(fact_754_comp__sgraph_Oconnecting__walk__split,axiom,
! [S: set_a,U: a,V: a,Xs: list_a,Z: a,Ys: list_a] :
( ( connecting_walk_a @ S @ ( undire2918257014606996450dges_a @ S ) @ U @ V @ Xs )
=> ( ( connecting_walk_a @ S @ ( undire2918257014606996450dges_a @ S ) @ V @ Z @ Ys )
=> ( connecting_walk_a @ S @ ( undire2918257014606996450dges_a @ S ) @ U @ Z @ ( append_a @ Xs @ ( tl_a @ Ys ) ) ) ) ) ).
% comp_sgraph.connecting_walk_split
thf(fact_755_Nat_Oex__has__greatest__nat,axiom,
! [P2: nat > $o,K: nat,B: nat] :
( ( P2 @ K )
=> ( ! [Y2: nat] :
( ( P2 @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ B ) )
=> ? [X3: nat] :
( ( P2 @ X3 )
& ! [Y6: nat] :
( ( P2 @ Y6 )
=> ( ord_less_eq_nat @ Y6 @ X3 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_756_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_757_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_758_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% eq_imp_le
thf(fact_759_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_760_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_761_comp__sgraph_Oinduced__is__subgraph,axiom,
! [V3: set_set_a,S: set_set_a] :
( ( ord_le3724670747650509150_set_a @ V3 @ S )
=> ( undire1186139521737116585_set_a @ V3 @ ( undire7854589003810675244_set_a @ ( undire8247866692393712962_set_a @ S ) @ V3 ) @ S @ ( undire8247866692393712962_set_a @ S ) ) ) ).
% comp_sgraph.induced_is_subgraph
thf(fact_762_comp__sgraph_Oinduced__is__subgraph,axiom,
! [V3: set_a,S: set_a] :
( ( ord_less_eq_set_a @ V3 @ S )
=> ( undire7103218114511261257raph_a @ V3 @ ( undire7777452895879145676dges_a @ ( undire2918257014606996450dges_a @ S ) @ V3 ) @ S @ ( undire2918257014606996450dges_a @ S ) ) ) ).
% comp_sgraph.induced_is_subgraph
thf(fact_763_comp__sgraph_Oconnecting__path__alt__def,axiom,
! [S: set_a,U: a,V: a,Xs: list_a] :
( ( connecting_path_a @ S @ ( undire2918257014606996450dges_a @ S ) @ U @ V @ Xs )
= ( ( connecting_walk_a @ S @ ( undire2918257014606996450dges_a @ S ) @ U @ V @ Xs )
& ( undire3562951555376170320path_a @ S @ ( undire2918257014606996450dges_a @ S ) @ Xs ) ) ) ).
% comp_sgraph.connecting_path_alt_def
thf(fact_764_comp__sgraph_Ois__path__def,axiom,
! [S: set_set_a,Xs: list_set_a] :
( ( undire8834939040163919632_set_a @ S @ ( undire8247866692393712962_set_a @ S ) @ Xs )
= ( ( undire526879649183275522_set_a @ S @ ( undire8247866692393712962_set_a @ S ) @ Xs )
& ( distinct_set_a @ Xs ) ) ) ).
% comp_sgraph.is_path_def
thf(fact_765_comp__sgraph_Ois__path__def,axiom,
! [S: set_a,Xs: list_a] :
( ( undire427332500224447920path_a @ S @ ( undire2918257014606996450dges_a @ S ) @ Xs )
= ( ( undire2427028224930250914walk_a @ S @ ( undire2918257014606996450dges_a @ S ) @ Xs )
& ( distinct_a @ Xs ) ) ) ).
% comp_sgraph.is_path_def
thf(fact_766_comp__sgraph_Ois__walk__decomp,axiom,
! [S: set_set_a,Xs: list_set_a,Y: set_a,Ys: list_set_a,Zs: list_set_a] :
( ( undire3014741414213135564_set_a @ S @ ( undire8247866692393712962_set_a @ S ) @ ( append_set_a @ Xs @ ( append_set_a @ ( cons_set_a @ Y @ nil_set_a ) @ ( append_set_a @ Ys @ ( append_set_a @ ( cons_set_a @ Y @ nil_set_a ) @ Zs ) ) ) ) )
=> ( undire3014741414213135564_set_a @ S @ ( undire8247866692393712962_set_a @ S ) @ ( append_set_a @ Xs @ ( append_set_a @ ( cons_set_a @ Y @ nil_set_a ) @ Zs ) ) ) ) ).
% comp_sgraph.is_walk_decomp
thf(fact_767_comp__sgraph_Ois__walk__decomp,axiom,
! [S: set_a,Xs: list_a,Y: a,Ys: list_a,Zs: list_a] :
( ( undire6133010728901294956walk_a @ S @ ( undire2918257014606996450dges_a @ S ) @ ( append_a @ Xs @ ( append_a @ ( cons_a @ Y @ nil_a ) @ ( append_a @ Ys @ ( append_a @ ( cons_a @ Y @ nil_a ) @ Zs ) ) ) ) )
=> ( undire6133010728901294956walk_a @ S @ ( undire2918257014606996450dges_a @ S ) @ ( append_a @ Xs @ ( append_a @ ( cons_a @ Y @ nil_a ) @ Zs ) ) ) ) ).
% comp_sgraph.is_walk_decomp
thf(fact_768_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_769_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_770_comp__sgraph_Oconnecting__path__length__bound,axiom,
! [U: a,V: a,S: set_a,P: list_a] :
( ( U != V )
=> ( ( connecting_path_a @ S @ ( undire2918257014606996450dges_a @ S ) @ U @ V @ P )
=> ( ord_less_eq_nat @ one_one_nat @ ( undire8849074589633906640ngth_a @ P ) ) ) ) ).
% comp_sgraph.connecting_path_length_bound
thf(fact_771_comp__sgraph_Ois__gen__path__distinct,axiom,
! [S: set_set_a,P: list_set_a] :
( ( undire7201326534205417136_set_a @ S @ ( undire8247866692393712962_set_a @ S ) @ P )
=> ( ( ( hd_set_a @ P )
!= ( last_set_a @ P ) )
=> ( distinct_set_a @ P ) ) ) ).
% comp_sgraph.is_gen_path_distinct
thf(fact_772_comp__sgraph_Ois__gen__path__distinct,axiom,
! [S: set_a,P: list_a] :
( ( undire3562951555376170320path_a @ S @ ( undire2918257014606996450dges_a @ S ) @ P )
=> ( ( ( hd_a @ P )
!= ( last_a @ P ) )
=> ( distinct_a @ P ) ) ) ).
% comp_sgraph.is_gen_path_distinct
thf(fact_773_comp__sgraph_Oconnecting__walk__path,axiom,
! [S: set_a,U: a,V: a,Xs: list_a] :
( ( connecting_walk_a @ S @ ( undire2918257014606996450dges_a @ S ) @ U @ V @ Xs )
=> ? [Ys3: list_a] :
( ( connecting_path_a @ S @ ( undire2918257014606996450dges_a @ S ) @ U @ V @ Ys3 )
& ( ord_less_eq_nat @ ( undire8849074589633906640ngth_a @ Ys3 ) @ ( undire8849074589633906640ngth_a @ Xs ) ) ) ) ).
% comp_sgraph.connecting_walk_path
thf(fact_774_comp__sgraph_Oconnecting__walk__def,axiom,
! [S: set_a,U: a,V: a,Xs: list_a] :
( ( connecting_walk_a @ S @ ( undire2918257014606996450dges_a @ S ) @ U @ V @ Xs )
= ( ( undire6133010728901294956walk_a @ S @ ( undire2918257014606996450dges_a @ S ) @ Xs )
& ( ( hd_a @ Xs )
= U )
& ( ( last_a @ Xs )
= V ) ) ) ).
% comp_sgraph.connecting_walk_def
thf(fact_775_comp__sgraph_Oconnecting__path__def,axiom,
! [S: set_a,U: a,V: a,Xs: list_a] :
( ( connecting_path_a @ S @ ( undire2918257014606996450dges_a @ S ) @ U @ V @ Xs )
= ( ( undire3562951555376170320path_a @ S @ ( undire2918257014606996450dges_a @ S ) @ Xs )
& ( ( hd_a @ Xs )
= U )
& ( ( last_a @ Xs )
= V ) ) ) ).
% comp_sgraph.connecting_path_def
thf(fact_776_comp__sgraph_Ois__closed__walk__def,axiom,
! [S: set_a,Xs: list_a] :
( ( undire3370724456595283424walk_a @ S @ ( undire2918257014606996450dges_a @ S ) @ Xs )
= ( ( undire6133010728901294956walk_a @ S @ ( undire2918257014606996450dges_a @ S ) @ Xs )
& ( ( hd_a @ Xs )
= ( last_a @ Xs ) ) ) ) ).
% comp_sgraph.is_closed_walk_def
thf(fact_777_comp__sgraph_Ois__open__walk__def,axiom,
! [S: set_a,Xs: list_a] :
( ( undire2427028224930250914walk_a @ S @ ( undire2918257014606996450dges_a @ S ) @ Xs )
= ( ( undire6133010728901294956walk_a @ S @ ( undire2918257014606996450dges_a @ S ) @ Xs )
& ( ( hd_a @ Xs )
!= ( last_a @ Xs ) ) ) ) ).
% comp_sgraph.is_open_walk_def
thf(fact_778_comp__sgraph_Oconnecting__path__str__def,axiom,
! [S: set_a,U: a,V: a,Xs: list_a] :
( ( connec3015921205769380621_str_a @ S @ ( undire2918257014606996450dges_a @ S ) @ U @ V @ Xs )
= ( ( undire427332500224447920path_a @ S @ ( undire2918257014606996450dges_a @ S ) @ Xs )
& ( ( hd_a @ Xs )
= U )
& ( ( last_a @ Xs )
= V ) ) ) ).
% comp_sgraph.connecting_path_str_def
thf(fact_779_comp__sgraph_Ois__trail__def,axiom,
! [S: set_a,Xs: list_a] :
( ( undire7142031287334043199rail_a @ S @ ( undire2918257014606996450dges_a @ S ) @ Xs )
= ( ( undire6133010728901294956walk_a @ S @ ( undire2918257014606996450dges_a @ S ) @ Xs )
& ( distinct_set_a @ ( undire7337870655677353998dges_a @ Xs ) ) ) ) ).
% comp_sgraph.is_trail_def
thf(fact_780_comp__sgraph_Oconnecting__path__split,axiom,
! [S: set_set_a,U: set_a,V: set_a,Xs: list_set_a,Z: set_a,Ys: list_set_a] :
( ( connec7350987497872064604_set_a @ S @ ( undire8247866692393712962_set_a @ S ) @ U @ V @ Xs )
=> ( ( connec7350987497872064604_set_a @ S @ ( undire8247866692393712962_set_a @ S ) @ V @ Z @ Ys )
=> ~ ! [P3: list_set_a] :
( ( connec7350987497872064604_set_a @ S @ ( undire8247866692393712962_set_a @ S ) @ U @ Z @ P3 )
=> ~ ( ord_less_eq_nat @ ( undire4424681683220949296_set_a @ P3 ) @ ( undire4424681683220949296_set_a @ ( append_set_a @ Xs @ ( tl_set_a @ Ys ) ) ) ) ) ) ) ).
% comp_sgraph.connecting_path_split
thf(fact_781_comp__sgraph_Oconnecting__path__split,axiom,
! [S: set_a,U: a,V: a,Xs: list_a,Z: a,Ys: list_a] :
( ( connecting_path_a @ S @ ( undire2918257014606996450dges_a @ S ) @ U @ V @ Xs )
=> ( ( connecting_path_a @ S @ ( undire2918257014606996450dges_a @ S ) @ V @ Z @ Ys )
=> ~ ! [P3: list_a] :
( ( connecting_path_a @ S @ ( undire2918257014606996450dges_a @ S ) @ U @ Z @ P3 )
=> ~ ( ord_less_eq_nat @ ( undire8849074589633906640ngth_a @ P3 ) @ ( undire8849074589633906640ngth_a @ ( append_a @ Xs @ ( tl_a @ Ys ) ) ) ) ) ) ) ).
% comp_sgraph.connecting_path_split
thf(fact_782_comp__sgraph_Ois__walk__append,axiom,
! [S: set_set_a,Xs: list_set_a,Ys: list_set_a] :
( ( undire3014741414213135564_set_a @ S @ ( undire8247866692393712962_set_a @ S ) @ Xs )
=> ( ( undire3014741414213135564_set_a @ S @ ( undire8247866692393712962_set_a @ S ) @ Ys )
=> ( ( ( last_set_a @ Xs )
= ( hd_set_a @ Ys ) )
=> ( undire3014741414213135564_set_a @ S @ ( undire8247866692393712962_set_a @ S ) @ ( append_set_a @ Xs @ ( tl_set_a @ Ys ) ) ) ) ) ) ).
% comp_sgraph.is_walk_append
thf(fact_783_comp__sgraph_Ois__walk__append,axiom,
! [S: set_a,Xs: list_a,Ys: list_a] :
( ( undire6133010728901294956walk_a @ S @ ( undire2918257014606996450dges_a @ S ) @ Xs )
=> ( ( undire6133010728901294956walk_a @ S @ ( undire2918257014606996450dges_a @ S ) @ Ys )
=> ( ( ( last_a @ Xs )
= ( hd_a @ Ys ) )
=> ( undire6133010728901294956walk_a @ S @ ( undire2918257014606996450dges_a @ S ) @ ( append_a @ Xs @ ( tl_a @ Ys ) ) ) ) ) ) ).
% comp_sgraph.is_walk_append
thf(fact_784_comp__sgraph_Ois__gen__path__distinct__tl,axiom,
! [S: set_set_a,P: list_set_a] :
( ( undire7201326534205417136_set_a @ S @ ( undire8247866692393712962_set_a @ S ) @ P )
=> ( ( ( hd_set_a @ P )
= ( last_set_a @ P ) )
=> ( distinct_set_a @ ( tl_set_a @ P ) ) ) ) ).
% comp_sgraph.is_gen_path_distinct_tl
thf(fact_785_comp__sgraph_Ois__gen__path__distinct__tl,axiom,
! [S: set_a,P: list_a] :
( ( undire3562951555376170320path_a @ S @ ( undire2918257014606996450dges_a @ S ) @ P )
=> ( ( ( hd_a @ P )
= ( last_a @ P ) )
=> ( distinct_a @ ( tl_a @ P ) ) ) ) ).
% comp_sgraph.is_gen_path_distinct_tl
thf(fact_786_comp__sgraph_Ois__gen__path__options,axiom,
! [S: set_set_a,P: list_set_a] :
( ( undire7201326534205417136_set_a @ S @ ( undire8247866692393712962_set_a @ S ) @ P )
= ( ( undire797940137672299967_set_a @ S @ ( undire8247866692393712962_set_a @ S ) @ P )
| ( undire8834939040163919632_set_a @ S @ ( undire8247866692393712962_set_a @ S ) @ P )
| ? [X2: set_a] :
( ( member_set_a @ X2 @ S )
& ( P
= ( cons_set_a @ X2 @ nil_set_a ) ) ) ) ) ).
% comp_sgraph.is_gen_path_options
thf(fact_787_comp__sgraph_Ois__gen__path__options,axiom,
! [S: set_a,P: list_a] :
( ( undire3562951555376170320path_a @ S @ ( undire2918257014606996450dges_a @ S ) @ P )
= ( ( undire2407311113669455967ycle_a @ S @ ( undire2918257014606996450dges_a @ S ) @ P )
| ( undire427332500224447920path_a @ S @ ( undire2918257014606996450dges_a @ S ) @ P )
| ? [X2: a] :
( ( member_a @ X2 @ S )
& ( P
= ( cons_a @ X2 @ nil_a ) ) ) ) ) ).
% comp_sgraph.is_gen_path_options
thf(fact_788_comp__sgraph_Ois__gen__path__def,axiom,
! [S: set_set_a,P: list_set_a] :
( ( undire7201326534205417136_set_a @ S @ ( undire8247866692393712962_set_a @ S ) @ P )
= ( ( undire3014741414213135564_set_a @ S @ ( undire8247866692393712962_set_a @ S ) @ P )
& ( ( ( distinct_set_a @ ( tl_set_a @ P ) )
& ( ( hd_set_a @ P )
= ( last_set_a @ P ) ) )
| ( distinct_set_a @ P ) ) ) ) ).
% comp_sgraph.is_gen_path_def
thf(fact_789_comp__sgraph_Ois__gen__path__def,axiom,
! [S: set_a,P: list_a] :
( ( undire3562951555376170320path_a @ S @ ( undire2918257014606996450dges_a @ S ) @ P )
= ( ( undire6133010728901294956walk_a @ S @ ( undire2918257014606996450dges_a @ S ) @ P )
& ( ( ( distinct_a @ ( tl_a @ P ) )
& ( ( hd_a @ P )
= ( last_a @ P ) ) )
| ( distinct_a @ P ) ) ) ) ).
% comp_sgraph.is_gen_path_def
thf(fact_790_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_791_add__le__imp__le__right,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
=> ( ord_less_eq_real @ A @ B ) ) ).
% add_le_imp_le_right
thf(fact_792_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_793_add__le__imp__le__left,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
=> ( ord_less_eq_real @ A @ B ) ) ).
% add_le_imp_le_left
thf(fact_794_le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [A3: nat,B3: nat] :
? [C3: nat] :
( B3
= ( plus_plus_nat @ A3 @ C3 ) ) ) ) ).
% le_iff_add
thf(fact_795_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_796_add__right__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) ) ) ).
% add_right_mono
thf(fact_797_less__eqE,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ~ ! [C4: nat] :
( B
!= ( plus_plus_nat @ A @ C4 ) ) ) ).
% less_eqE
thf(fact_798_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_799_add__left__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ord_less_eq_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) ) ) ).
% add_left_mono
thf(fact_800_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_801_add__mono,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ C @ D )
=> ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) ) ) ) ).
% add_mono
thf(fact_802_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_803_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I: real,J: real,K: real,L: real] :
( ( ( ord_less_eq_real @ I @ J )
& ( ord_less_eq_real @ K @ L ) )
=> ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_804_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_805_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I: real,J: real,K: real,L: real] :
( ( ( I = J )
& ( ord_less_eq_real @ K @ L ) )
=> ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_806_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_807_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I: real,J: real,K: real,L: real] :
( ( ( ord_less_eq_real @ I @ J )
& ( K = L ) )
=> ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_808_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_809_le__add1,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M ) ) ).
% le_add1
thf(fact_810_le__add2,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M @ N ) ) ).
% le_add2
thf(fact_811_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_812_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_813_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_814_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_815_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_816_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_817_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_818_nat__le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [M2: nat,N3: nat] :
? [K2: nat] :
( N3
= ( plus_plus_nat @ M2 @ K2 ) ) ) ) ).
% nat_le_iff_add
thf(fact_819_comp__sgraph_Ois__walk__def,axiom,
! [S: set_set_a,Xs: list_set_a] :
( ( undire3014741414213135564_set_a @ S @ ( undire8247866692393712962_set_a @ S ) @ Xs )
= ( ( ord_le3724670747650509150_set_a @ ( set_set_a2 @ Xs ) @ S )
& ( ord_le5722252365846178494_set_a @ ( set_set_set_a2 @ ( undire6234387080713648494_set_a @ Xs ) ) @ ( undire8247866692393712962_set_a @ S ) )
& ( Xs != nil_set_a ) ) ) ).
% comp_sgraph.is_walk_def
thf(fact_820_comp__sgraph_Ois__walk__def,axiom,
! [S: set_a,Xs: list_a] :
( ( undire6133010728901294956walk_a @ S @ ( undire2918257014606996450dges_a @ S ) @ Xs )
= ( ( ord_less_eq_set_a @ ( set_a2 @ Xs ) @ S )
& ( ord_le3724670747650509150_set_a @ ( set_set_a2 @ ( undire7337870655677353998dges_a @ Xs ) ) @ ( undire2918257014606996450dges_a @ S ) )
& ( Xs != nil_a ) ) ) ).
% comp_sgraph.is_walk_def
thf(fact_821_comp__sgraph_Ois__walkI,axiom,
! [Xs: list_set_a,S: set_set_a] :
( ( ord_le3724670747650509150_set_a @ ( set_set_a2 @ Xs ) @ S )
=> ( ( ord_le5722252365846178494_set_a @ ( set_set_set_a2 @ ( undire6234387080713648494_set_a @ Xs ) ) @ ( undire8247866692393712962_set_a @ S ) )
=> ( ( Xs != nil_set_a )
=> ( undire3014741414213135564_set_a @ S @ ( undire8247866692393712962_set_a @ S ) @ Xs ) ) ) ) ).
% comp_sgraph.is_walkI
thf(fact_822_comp__sgraph_Ois__walkI,axiom,
! [Xs: list_a,S: set_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ Xs ) @ S )
=> ( ( ord_le3724670747650509150_set_a @ ( set_set_a2 @ ( undire7337870655677353998dges_a @ Xs ) ) @ ( undire2918257014606996450dges_a @ S ) )
=> ( ( Xs != nil_a )
=> ( undire6133010728901294956walk_a @ S @ ( undire2918257014606996450dges_a @ S ) @ Xs ) ) ) ) ).
% comp_sgraph.is_walkI
thf(fact_823_comp__sgraph_Ois__cycle__alt__gen__path,axiom,
! [S: set_a,Xs: list_a] :
( ( undire2407311113669455967ycle_a @ S @ ( undire2918257014606996450dges_a @ S ) @ Xs )
= ( ( undire3562951555376170320path_a @ S @ ( undire2918257014606996450dges_a @ S ) @ Xs )
& ( ord_less_eq_nat @ one_one_nat @ ( undire8849074589633906640ngth_a @ Xs ) )
& ( ( hd_a @ Xs )
= ( last_a @ Xs ) ) ) ) ).
% comp_sgraph.is_cycle_alt_gen_path
thf(fact_824_comp__sgraph_Ois__cycle__def,axiom,
! [S: set_set_a,Xs: list_set_a] :
( ( undire797940137672299967_set_a @ S @ ( undire8247866692393712962_set_a @ S ) @ Xs )
= ( ( undire4100213446647512896_set_a @ S @ ( undire8247866692393712962_set_a @ S ) @ Xs )
& ( ord_less_eq_nat @ one_one_nat @ ( undire4424681683220949296_set_a @ Xs ) )
& ( distinct_set_a @ ( tl_set_a @ Xs ) ) ) ) ).
% comp_sgraph.is_cycle_def
thf(fact_825_comp__sgraph_Ois__cycle__def,axiom,
! [S: set_a,Xs: list_a] :
( ( undire2407311113669455967ycle_a @ S @ ( undire2918257014606996450dges_a @ S ) @ Xs )
= ( ( undire3370724456595283424walk_a @ S @ ( undire2918257014606996450dges_a @ S ) @ Xs )
& ( ord_less_eq_nat @ one_one_nat @ ( undire8849074589633906640ngth_a @ Xs ) )
& ( distinct_a @ ( tl_a @ Xs ) ) ) ) ).
% comp_sgraph.is_cycle_def
thf(fact_826_walk__length__conv,axiom,
( undire8849074589633906640ngth_a
= ( ^ [P4: list_a] : ( minus_minus_nat @ ( size_size_list_a @ P4 ) @ one_one_nat ) ) ) ).
% walk_length_conv
thf(fact_827_edge__density__ge0,axiom,
! [X5: set_a,Y5: set_a] : ( ord_less_eq_real @ zero_zero_real @ ( undire297304480579013331sity_a @ edges @ X5 @ Y5 ) ) ).
% edge_density_ge0
thf(fact_828_edge__density__le1,axiom,
! [X5: set_a,Y5: set_a] : ( ord_less_eq_real @ ( undire297304480579013331sity_a @ edges @ X5 @ Y5 ) @ one_one_real ) ).
% edge_density_le1
thf(fact_829_the__elem__set,axiom,
! [X4: a] :
( ( the_elem_a @ ( set_a2 @ ( cons_a @ X4 @ nil_a ) ) )
= X4 ) ).
% the_elem_set
thf(fact_830_the__elem__set,axiom,
! [X4: set_a] :
( ( the_elem_set_a @ ( set_set_a2 @ ( cons_set_a @ X4 @ nil_set_a ) ) )
= X4 ) ).
% the_elem_set
thf(fact_831_list__set__tl,axiom,
! [X4: a,Xs: list_a] :
( ( member_a @ X4 @ ( set_a2 @ ( tl_a @ Xs ) ) )
=> ( member_a @ X4 @ ( set_a2 @ Xs ) ) ) ).
% list_set_tl
thf(fact_832_list__set__tl,axiom,
! [X4: set_a,Xs: list_set_a] :
( ( member_set_a @ X4 @ ( set_set_a2 @ ( tl_set_a @ Xs ) ) )
=> ( member_set_a @ X4 @ ( set_set_a2 @ Xs ) ) ) ).
% list_set_tl
thf(fact_833_walk__length__def,axiom,
( undire8849074589633906640ngth_a
= ( ^ [P4: list_a] : ( size_size_list_set_a @ ( undire7337870655677353998dges_a @ P4 ) ) ) ) ).
% walk_length_def
thf(fact_834_le__zero__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_835_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_836_zero__le__double__add__iff__zero__le__single__add,axiom,
! [A: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ A @ A ) )
= ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% zero_le_double_add_iff_zero_le_single_add
thf(fact_837_double__add__le__zero__iff__single__add__le__zero,axiom,
! [A: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ A @ A ) @ zero_zero_real )
= ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% double_add_le_zero_iff_single_add_le_zero
thf(fact_838_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_839_le__add__same__cancel2,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ ( plus_plus_real @ B @ A ) )
= ( ord_less_eq_real @ zero_zero_real @ B ) ) ).
% le_add_same_cancel2
thf(fact_840_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_841_le__add__same__cancel1,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ ( plus_plus_real @ A @ B ) )
= ( ord_less_eq_real @ zero_zero_real @ B ) ) ).
% le_add_same_cancel1
thf(fact_842_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_843_add__le__same__cancel2,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ A @ B ) @ B )
= ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% add_le_same_cancel2
thf(fact_844_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_845_add__le__same__cancel1,axiom,
! [B: real,A: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ B @ A ) @ B )
= ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% add_le_same_cancel1
thf(fact_846_diff__ge__0__iff__ge,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( minus_minus_real @ A @ B ) )
= ( ord_less_eq_real @ B @ A ) ) ).
% diff_ge_0_iff_ge
thf(fact_847_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_848_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_849_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_850_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_851_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_852_le__iff__diff__le__0,axiom,
( ord_less_eq_real
= ( ^ [A3: real,B3: real] : ( ord_less_eq_real @ ( minus_minus_real @ A3 @ B3 ) @ zero_zero_real ) ) ) ).
% le_iff_diff_le_0
thf(fact_853_diff__eq__diff__less__eq,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ( minus_minus_real @ A @ B )
= ( minus_minus_real @ C @ D ) )
=> ( ( ord_less_eq_real @ A @ B )
= ( ord_less_eq_real @ C @ D ) ) ) ).
% diff_eq_diff_less_eq
thf(fact_854_diff__right__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ord_less_eq_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ C ) ) ) ).
% diff_right_mono
thf(fact_855_diff__left__mono,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ord_less_eq_real @ ( minus_minus_real @ C @ A ) @ ( minus_minus_real @ C @ B ) ) ) ).
% diff_left_mono
thf(fact_856_diff__mono,axiom,
! [A: real,B: real,D: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ D @ C )
=> ( ord_less_eq_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ D ) ) ) ) ).
% diff_mono
thf(fact_857_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_858_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_859_diff__le__self,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ M ) ).
% diff_le_self
thf(fact_860_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_861_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_862_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_863_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_864_zero__le,axiom,
! [X4: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X4 ) ).
% zero_le
thf(fact_865_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_866_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_867_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_868_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_869_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_870_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_871_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_872_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_873_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_874_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_875_le__diff__eq,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_eq_real @ A @ ( minus_minus_real @ C @ B ) )
= ( ord_less_eq_real @ ( plus_plus_real @ A @ B ) @ C ) ) ).
% le_diff_eq
thf(fact_876_diff__le__eq,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ ( minus_minus_real @ A @ B ) @ C )
= ( ord_less_eq_real @ A @ ( plus_plus_real @ C @ B ) ) ) ).
% diff_le_eq
thf(fact_877_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_878_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_879_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_880_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_881_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_882_add__nonpos__eq__0__iff,axiom,
! [X4: nat,Y: nat] :
( ( ord_less_eq_nat @ X4 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ Y @ zero_zero_nat )
=> ( ( ( plus_plus_nat @ X4 @ Y )
= zero_zero_nat )
= ( ( X4 = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_883_add__nonpos__eq__0__iff,axiom,
! [X4: real,Y: real] :
( ( ord_less_eq_real @ X4 @ zero_zero_real )
=> ( ( ord_less_eq_real @ Y @ zero_zero_real )
=> ( ( ( plus_plus_real @ X4 @ Y )
= zero_zero_real )
= ( ( X4 = zero_zero_real )
& ( Y = zero_zero_real ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_884_add__nonneg__eq__0__iff,axiom,
! [X4: nat,Y: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ X4 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
=> ( ( ( plus_plus_nat @ X4 @ Y )
= zero_zero_nat )
= ( ( X4 = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_885_add__nonneg__eq__0__iff,axiom,
! [X4: real,Y: real] :
( ( ord_less_eq_real @ zero_zero_real @ X4 )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ( ( plus_plus_real @ X4 @ Y )
= zero_zero_real )
= ( ( X4 = zero_zero_real )
& ( Y = zero_zero_real ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_886_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_887_add__nonpos__nonpos,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ zero_zero_real )
=> ( ( ord_less_eq_real @ B @ zero_zero_real )
=> ( ord_less_eq_real @ ( plus_plus_real @ A @ B ) @ zero_zero_real ) ) ) ).
% add_nonpos_nonpos
thf(fact_888_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_889_add__nonneg__nonneg,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ A @ B ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_890_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_891_add__increasing2,axiom,
! [C: real,B: real,A: real] :
( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ( ord_less_eq_real @ B @ A )
=> ( ord_less_eq_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).
% add_increasing2
thf(fact_892_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_893_add__decreasing2,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_eq_real @ C @ zero_zero_real )
=> ( ( ord_less_eq_real @ A @ B )
=> ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ B ) ) ) ).
% add_decreasing2
thf(fact_894_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_895_add__increasing,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ord_less_eq_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).
% add_increasing
thf(fact_896_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_897_add__decreasing,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_eq_real @ A @ zero_zero_real )
=> ( ( ord_less_eq_real @ C @ B )
=> ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ B ) ) ) ).
% add_decreasing
thf(fact_898_comp__sgraph_Owalk__length__def,axiom,
( undire8849074589633906640ngth_a
= ( ^ [P4: list_a] : ( size_size_list_set_a @ ( undire7337870655677353998dges_a @ P4 ) ) ) ) ).
% comp_sgraph.walk_length_def
thf(fact_899_comp__sgraph_Oedge__density__le1,axiom,
! [S: set_a,X5: set_a,Y5: set_a] : ( ord_less_eq_real @ ( undire297304480579013331sity_a @ ( undire2918257014606996450dges_a @ S ) @ X5 @ Y5 ) @ one_one_real ) ).
% comp_sgraph.edge_density_le1
thf(fact_900_ulgraph_Oedge__density__le1,axiom,
! [Vertices: set_a,Edges: set_set_a,X5: set_a,Y5: set_a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ord_less_eq_real @ ( undire297304480579013331sity_a @ Edges @ X5 @ Y5 ) @ one_one_real ) ) ).
% ulgraph.edge_density_le1
thf(fact_901_comp__sgraph_Oedge__density__ge0,axiom,
! [S: set_a,X5: set_a,Y5: set_a] : ( ord_less_eq_real @ zero_zero_real @ ( undire297304480579013331sity_a @ ( undire2918257014606996450dges_a @ S ) @ X5 @ Y5 ) ) ).
% comp_sgraph.edge_density_ge0
thf(fact_902_ulgraph_Oedge__density__ge0,axiom,
! [Vertices: set_a,Edges: set_set_a,X5: set_a,Y5: set_a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ord_less_eq_real @ zero_zero_real @ ( undire297304480579013331sity_a @ Edges @ X5 @ Y5 ) ) ) ).
% ulgraph.edge_density_ge0
thf(fact_903_comp__sgraph_Owalk__length__conv,axiom,
( undire4424681683220949296_set_a
= ( ^ [P4: list_set_a] : ( minus_minus_nat @ ( size_size_list_set_a @ P4 ) @ one_one_nat ) ) ) ).
% comp_sgraph.walk_length_conv
thf(fact_904_comp__sgraph_Owalk__length__conv,axiom,
( undire8849074589633906640ngth_a
= ( ^ [P4: list_a] : ( minus_minus_nat @ ( size_size_list_a @ P4 ) @ one_one_nat ) ) ) ).
% comp_sgraph.walk_length_conv
thf(fact_905_ulgraph_Owalk__length__def,axiom,
! [Vertices: set_a,Edges: set_set_a,P: list_a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( undire8849074589633906640ngth_a @ P )
= ( size_size_list_set_a @ ( undire7337870655677353998dges_a @ P ) ) ) ) ).
% ulgraph.walk_length_def
thf(fact_906_ulgraph_Owalk__length__conv,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,P: list_set_a] :
( ( undire6886684016831807756_set_a @ Vertices @ Edges )
=> ( ( undire4424681683220949296_set_a @ P )
= ( minus_minus_nat @ ( size_size_list_set_a @ P ) @ one_one_nat ) ) ) ).
% ulgraph.walk_length_conv
thf(fact_907_ulgraph_Owalk__length__conv,axiom,
! [Vertices: set_a,Edges: set_set_a,P: list_a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( undire8849074589633906640ngth_a @ P )
= ( minus_minus_nat @ ( size_size_list_a @ P ) @ one_one_nat ) ) ) ).
% ulgraph.walk_length_conv
thf(fact_908_list__exhaust3,axiom,
! [Xs: list_a] :
( ( Xs != nil_a )
=> ( ! [X3: a] :
( Xs
!= ( cons_a @ X3 @ nil_a ) )
=> ~ ! [X3: a,Y2: a,Ys3: list_a] :
( Xs
!= ( cons_a @ X3 @ ( cons_a @ Y2 @ Ys3 ) ) ) ) ) ).
% list_exhaust3
thf(fact_909_list__exhaust3,axiom,
! [Xs: list_set_a] :
( ( Xs != nil_set_a )
=> ( ! [X3: set_a] :
( Xs
!= ( cons_set_a @ X3 @ nil_set_a ) )
=> ~ ! [X3: set_a,Y2: set_a,Ys3: list_set_a] :
( Xs
!= ( cons_set_a @ X3 @ ( cons_set_a @ Y2 @ Ys3 ) ) ) ) ) ).
% list_exhaust3
thf(fact_910_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_911_le__add__diff__inverse2,axiom,
! [B: real,A: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( plus_plus_real @ ( minus_minus_real @ A @ B ) @ B )
= A ) ) ).
% le_add_diff_inverse2
thf(fact_912_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_913_le__add__diff__inverse,axiom,
! [B: real,A: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( plus_plus_real @ B @ ( minus_minus_real @ A @ B ) )
= A ) ) ).
% le_add_diff_inverse
thf(fact_914_edge__density__zero,axiom,
! [Y5: set_a,X5: set_a] :
( ( Y5 = bot_bot_set_a )
=> ( ( undire297304480579013331sity_a @ edges @ X5 @ Y5 )
= zero_zero_real ) ) ).
% edge_density_zero
thf(fact_915_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_916_add__le__add__imp__diff__le,axiom,
! [I: real,K: real,N: real,J: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ N )
=> ( ( ord_less_eq_real @ N @ ( plus_plus_real @ J @ K ) )
=> ( ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ N )
=> ( ( ord_less_eq_real @ N @ ( plus_plus_real @ J @ K ) )
=> ( ord_less_eq_real @ ( minus_minus_real @ N @ K ) @ J ) ) ) ) ) ).
% add_le_add_imp_diff_le
thf(fact_917_empty__not__edge,axiom,
~ ( member_set_a @ bot_bot_set_a @ edges ) ).
% empty_not_edge
thf(fact_918_Diff__empty,axiom,
! [A2: set_a] :
( ( minus_minus_set_a @ A2 @ bot_bot_set_a )
= A2 ) ).
% Diff_empty
thf(fact_919_Diff__empty,axiom,
! [A2: set_set_a] :
( ( minus_5736297505244876581_set_a @ A2 @ bot_bot_set_set_a )
= A2 ) ).
% Diff_empty
thf(fact_920_empty__Diff,axiom,
! [A2: set_a] :
( ( minus_minus_set_a @ bot_bot_set_a @ A2 )
= bot_bot_set_a ) ).
% empty_Diff
thf(fact_921_empty__Diff,axiom,
! [A2: set_set_a] :
( ( minus_5736297505244876581_set_a @ bot_bot_set_set_a @ A2 )
= bot_bot_set_set_a ) ).
% empty_Diff
thf(fact_922_Diff__cancel,axiom,
! [A2: set_a] :
( ( minus_minus_set_a @ A2 @ A2 )
= bot_bot_set_a ) ).
% Diff_cancel
thf(fact_923_Diff__cancel,axiom,
! [A2: set_set_a] :
( ( minus_5736297505244876581_set_a @ A2 @ A2 )
= bot_bot_set_set_a ) ).
% Diff_cancel
thf(fact_924_empty__Collect__eq,axiom,
! [P2: a > $o] :
( ( bot_bot_set_a
= ( collect_a @ P2 ) )
= ( ! [X2: a] :
~ ( P2 @ X2 ) ) ) ).
% empty_Collect_eq
thf(fact_925_empty__Collect__eq,axiom,
! [P2: set_a > $o] :
( ( bot_bot_set_set_a
= ( collect_set_a @ P2 ) )
= ( ! [X2: set_a] :
~ ( P2 @ X2 ) ) ) ).
% empty_Collect_eq
thf(fact_926_Collect__empty__eq,axiom,
! [P2: a > $o] :
( ( ( collect_a @ P2 )
= bot_bot_set_a )
= ( ! [X2: a] :
~ ( P2 @ X2 ) ) ) ).
% Collect_empty_eq
thf(fact_927_Collect__empty__eq,axiom,
! [P2: set_a > $o] :
( ( ( collect_set_a @ P2 )
= bot_bot_set_set_a )
= ( ! [X2: set_a] :
~ ( P2 @ X2 ) ) ) ).
% Collect_empty_eq
thf(fact_928_all__not__in__conv,axiom,
! [A2: set_a] :
( ( ! [X2: a] :
~ ( member_a @ X2 @ A2 ) )
= ( A2 = bot_bot_set_a ) ) ).
% all_not_in_conv
thf(fact_929_all__not__in__conv,axiom,
! [A2: set_set_a] :
( ( ! [X2: set_a] :
~ ( member_set_a @ X2 @ A2 ) )
= ( A2 = bot_bot_set_set_a ) ) ).
% all_not_in_conv
thf(fact_930_empty__iff,axiom,
! [C: a] :
~ ( member_a @ C @ bot_bot_set_a ) ).
% empty_iff
thf(fact_931_empty__iff,axiom,
! [C: set_a] :
~ ( member_set_a @ C @ bot_bot_set_set_a ) ).
% empty_iff
thf(fact_932_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_933_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_934_Diff__eq__empty__iff,axiom,
! [A2: set_a,B2: set_a] :
( ( ( minus_minus_set_a @ A2 @ B2 )
= bot_bot_set_a )
= ( ord_less_eq_set_a @ A2 @ B2 ) ) ).
% Diff_eq_empty_iff
thf(fact_935_Diff__eq__empty__iff,axiom,
! [A2: set_set_a,B2: set_set_a] :
( ( ( minus_5736297505244876581_set_a @ A2 @ B2 )
= bot_bot_set_set_a )
= ( ord_le3724670747650509150_set_a @ A2 @ B2 ) ) ).
% Diff_eq_empty_iff
thf(fact_936_subset__empty,axiom,
! [A2: set_a] :
( ( ord_less_eq_set_a @ A2 @ bot_bot_set_a )
= ( A2 = bot_bot_set_a ) ) ).
% subset_empty
thf(fact_937_subset__empty,axiom,
! [A2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ bot_bot_set_set_a )
= ( A2 = bot_bot_set_set_a ) ) ).
% subset_empty
thf(fact_938_empty__subsetI,axiom,
! [A2: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A2 ) ).
% empty_subsetI
thf(fact_939_empty__subsetI,axiom,
! [A2: set_set_a] : ( ord_le3724670747650509150_set_a @ bot_bot_set_set_a @ A2 ) ).
% empty_subsetI
thf(fact_940_length__0__conv,axiom,
! [Xs: list_a] :
( ( ( size_size_list_a @ Xs )
= zero_zero_nat )
= ( Xs = nil_a ) ) ).
% length_0_conv
thf(fact_941_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_942_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_943_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_944_set__empty2,axiom,
! [Xs: list_a] :
( ( bot_bot_set_a
= ( set_a2 @ Xs ) )
= ( Xs = nil_a ) ) ).
% set_empty2
thf(fact_945_set__empty2,axiom,
! [Xs: list_set_a] :
( ( bot_bot_set_set_a
= ( set_set_a2 @ Xs ) )
= ( Xs = nil_set_a ) ) ).
% set_empty2
thf(fact_946_set__empty,axiom,
! [Xs: list_a] :
( ( ( set_a2 @ Xs )
= bot_bot_set_a )
= ( Xs = nil_a ) ) ).
% set_empty
thf(fact_947_set__empty,axiom,
! [Xs: list_set_a] :
( ( ( set_set_a2 @ Xs )
= bot_bot_set_set_a )
= ( Xs = nil_set_a ) ) ).
% set_empty
thf(fact_948_bot_Oextremum,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A ) ).
% bot.extremum
thf(fact_949_bot_Oextremum,axiom,
! [A: set_set_a] : ( ord_le3724670747650509150_set_a @ bot_bot_set_set_a @ A ) ).
% bot.extremum
thf(fact_950_bot_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ bot_bot_nat @ A ) ).
% bot.extremum
thf(fact_951_bot_Oextremum__unique,axiom,
! [A: set_a] :
( ( ord_less_eq_set_a @ A @ bot_bot_set_a )
= ( A = bot_bot_set_a ) ) ).
% bot.extremum_unique
thf(fact_952_bot_Oextremum__unique,axiom,
! [A: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ bot_bot_set_set_a )
= ( A = bot_bot_set_set_a ) ) ).
% bot.extremum_unique
thf(fact_953_bot_Oextremum__unique,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ bot_bot_nat )
= ( A = bot_bot_nat ) ) ).
% bot.extremum_unique
thf(fact_954_bot_Oextremum__uniqueI,axiom,
! [A: set_a] :
( ( ord_less_eq_set_a @ A @ bot_bot_set_a )
=> ( A = bot_bot_set_a ) ) ).
% bot.extremum_uniqueI
thf(fact_955_bot_Oextremum__uniqueI,axiom,
! [A: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ bot_bot_set_set_a )
=> ( A = bot_bot_set_set_a ) ) ).
% bot.extremum_uniqueI
thf(fact_956_bot_Oextremum__uniqueI,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ bot_bot_nat )
=> ( A = bot_bot_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_957_double__diff,axiom,
! [A2: set_a,B2: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ord_less_eq_set_a @ B2 @ C2 )
=> ( ( minus_minus_set_a @ B2 @ ( minus_minus_set_a @ C2 @ A2 ) )
= A2 ) ) ) ).
% double_diff
thf(fact_958_double__diff,axiom,
! [A2: set_set_a,B2: set_set_a,C2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B2 )
=> ( ( ord_le3724670747650509150_set_a @ B2 @ C2 )
=> ( ( minus_5736297505244876581_set_a @ B2 @ ( minus_5736297505244876581_set_a @ C2 @ A2 ) )
= A2 ) ) ) ).
% double_diff
thf(fact_959_Diff__subset,axiom,
! [A2: set_a,B2: set_a] : ( ord_less_eq_set_a @ ( minus_minus_set_a @ A2 @ B2 ) @ A2 ) ).
% Diff_subset
thf(fact_960_Diff__subset,axiom,
! [A2: set_set_a,B2: set_set_a] : ( ord_le3724670747650509150_set_a @ ( minus_5736297505244876581_set_a @ A2 @ B2 ) @ A2 ) ).
% Diff_subset
thf(fact_961_Diff__mono,axiom,
! [A2: set_a,C2: set_a,D2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ C2 )
=> ( ( ord_less_eq_set_a @ D2 @ B2 )
=> ( ord_less_eq_set_a @ ( minus_minus_set_a @ A2 @ B2 ) @ ( minus_minus_set_a @ C2 @ D2 ) ) ) ) ).
% Diff_mono
thf(fact_962_Diff__mono,axiom,
! [A2: set_set_a,C2: set_set_a,D2: set_set_a,B2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ C2 )
=> ( ( ord_le3724670747650509150_set_a @ D2 @ B2 )
=> ( ord_le3724670747650509150_set_a @ ( minus_5736297505244876581_set_a @ A2 @ B2 ) @ ( minus_5736297505244876581_set_a @ C2 @ D2 ) ) ) ) ).
% Diff_mono
thf(fact_963_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_964_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_965_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_966_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_967_comp__sgraph_Oempty__not__edge,axiom,
! [S: set_a] :
~ ( member_set_a @ bot_bot_set_a @ ( undire2918257014606996450dges_a @ S ) ) ).
% comp_sgraph.empty_not_edge
thf(fact_968_comp__sgraph_Oempty__not__edge,axiom,
! [S: set_set_a] :
~ ( member_set_set_a @ bot_bot_set_set_a @ ( undire8247866692393712962_set_a @ S ) ) ).
% comp_sgraph.empty_not_edge
thf(fact_969_ulgraph_Oempty__not__edge,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a] :
( ( undire6886684016831807756_set_a @ Vertices @ Edges )
=> ~ ( member_set_set_a @ bot_bot_set_set_a @ Edges ) ) ).
% ulgraph.empty_not_edge
thf(fact_970_ulgraph_Oempty__not__edge,axiom,
! [Vertices: set_a,Edges: set_set_a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ~ ( member_set_a @ bot_bot_set_a @ Edges ) ) ).
% ulgraph.empty_not_edge
thf(fact_971_ex__in__conv,axiom,
! [A2: set_a] :
( ( ? [X2: a] : ( member_a @ X2 @ A2 ) )
= ( A2 != bot_bot_set_a ) ) ).
% ex_in_conv
thf(fact_972_ex__in__conv,axiom,
! [A2: set_set_a] :
( ( ? [X2: set_a] : ( member_set_a @ X2 @ A2 ) )
= ( A2 != bot_bot_set_set_a ) ) ).
% ex_in_conv
thf(fact_973_equals0I,axiom,
! [A2: set_a] :
( ! [Y2: a] :
~ ( member_a @ Y2 @ A2 )
=> ( A2 = bot_bot_set_a ) ) ).
% equals0I
thf(fact_974_equals0I,axiom,
! [A2: set_set_a] :
( ! [Y2: set_a] :
~ ( member_set_a @ Y2 @ A2 )
=> ( A2 = bot_bot_set_set_a ) ) ).
% equals0I
thf(fact_975_equals0D,axiom,
! [A2: set_a,A: a] :
( ( A2 = bot_bot_set_a )
=> ~ ( member_a @ A @ A2 ) ) ).
% equals0D
thf(fact_976_equals0D,axiom,
! [A2: set_set_a,A: set_a] :
( ( A2 = bot_bot_set_set_a )
=> ~ ( member_set_a @ A @ A2 ) ) ).
% equals0D
thf(fact_977_emptyE,axiom,
! [A: a] :
~ ( member_a @ A @ bot_bot_set_a ) ).
% emptyE
thf(fact_978_emptyE,axiom,
! [A: set_a] :
~ ( member_set_a @ A @ bot_bot_set_set_a ) ).
% emptyE
thf(fact_979_empty__set,axiom,
( bot_bot_set_a
= ( set_a2 @ nil_a ) ) ).
% empty_set
thf(fact_980_empty__set,axiom,
( bot_bot_set_set_a
= ( set_set_a2 @ nil_set_a ) ) ).
% empty_set
thf(fact_981_list_Osize_I3_J,axiom,
( ( size_size_list_a @ nil_a )
= zero_zero_nat ) ).
% list.size(3)
thf(fact_982_list_Osize_I3_J,axiom,
( ( size_size_list_set_a @ nil_set_a )
= zero_zero_nat ) ).
% list.size(3)
thf(fact_983_comp__sgraph_Oedge__density__zero,axiom,
! [Y5: set_set_a,S: set_set_a,X5: set_set_a] :
( ( Y5 = bot_bot_set_set_a )
=> ( ( undire8927637694342045747_set_a @ ( undire8247866692393712962_set_a @ S ) @ X5 @ Y5 )
= zero_zero_real ) ) ).
% comp_sgraph.edge_density_zero
thf(fact_984_comp__sgraph_Oedge__density__zero,axiom,
! [Y5: set_a,S: set_a,X5: set_a] :
( ( Y5 = bot_bot_set_a )
=> ( ( undire297304480579013331sity_a @ ( undire2918257014606996450dges_a @ S ) @ X5 @ Y5 )
= zero_zero_real ) ) ).
% comp_sgraph.edge_density_zero
thf(fact_985_ulgraph_Oedge__density__zero,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,Y5: set_set_a,X5: set_set_a] :
( ( undire6886684016831807756_set_a @ Vertices @ Edges )
=> ( ( Y5 = bot_bot_set_set_a )
=> ( ( undire8927637694342045747_set_a @ Edges @ X5 @ Y5 )
= zero_zero_real ) ) ) ).
% ulgraph.edge_density_zero
thf(fact_986_ulgraph_Oedge__density__zero,axiom,
! [Vertices: set_a,Edges: set_set_a,Y5: set_a,X5: set_a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( Y5 = bot_bot_set_a )
=> ( ( undire297304480579013331sity_a @ Edges @ X5 @ Y5 )
= zero_zero_real ) ) ) ).
% ulgraph.edge_density_zero
thf(fact_987_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_988_zero__less__one__class_Ozero__le__one,axiom,
ord_less_eq_real @ zero_zero_real @ one_one_real ).
% zero_less_one_class.zero_le_one
thf(fact_989_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_990_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
ord_less_eq_real @ zero_zero_real @ one_one_real ).
% linordered_nonzero_semiring_class.zero_le_one
thf(fact_991_not__one__le__zero,axiom,
~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_le_zero
thf(fact_992_not__one__le__zero,axiom,
~ ( ord_less_eq_real @ one_one_real @ zero_zero_real ) ).
% not_one_le_zero
thf(fact_993_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_994_add__le__imp__le__diff,axiom,
! [I: real,K: real,N: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ N )
=> ( ord_less_eq_real @ I @ ( minus_minus_real @ N @ K ) ) ) ).
% add_le_imp_le_diff
thf(fact_995_iso__vertex__empty__neighborhood,axiom,
! [V: a] :
( ( undire8931668460104145173rtex_a @ vertices @ edges @ V )
=> ( ( undire8504279938402040014hood_a @ vertices @ edges @ V )
= bot_bot_set_a ) ) ).
% iso_vertex_empty_neighborhood
thf(fact_996_connecting__path__gen__str,axiom,
! [U: a,V: a,Xs: list_a] :
( ( connecting_path_a @ vertices @ edges @ U @ V @ Xs )
=> ( ~ ( undire2407311113669455967ycle_a @ vertices @ edges @ Xs )
=> ( ( ord_less_nat @ zero_zero_nat @ ( undire8849074589633906640ngth_a @ Xs ) )
=> ( connec3015921205769380621_str_a @ vertices @ edges @ U @ V @ Xs ) ) ) ) ).
% connecting_path_gen_str
thf(fact_997_is__gen__path__path,axiom,
! [P: list_a] :
( ( undire3562951555376170320path_a @ vertices @ edges @ P )
=> ( ( ord_less_nat @ zero_zero_nat @ ( undire8849074589633906640ngth_a @ P ) )
=> ( ~ ( undire2407311113669455967ycle_a @ vertices @ edges @ P )
=> ( undire427332500224447920path_a @ vertices @ edges @ P ) ) ) ) ).
% is_gen_path_path
thf(fact_998_is__isolated__vertex__degree0,axiom,
! [V: a] :
( ( undire8931668460104145173rtex_a @ vertices @ edges @ V )
=> ( ( undire8867928226783802224gree_a @ edges @ V )
= zero_zero_nat ) ) ).
% is_isolated_vertex_degree0
thf(fact_999_DiffI,axiom,
! [C: a,A2: set_a,B2: set_a] :
( ( member_a @ C @ A2 )
=> ( ~ ( member_a @ C @ B2 )
=> ( member_a @ C @ ( minus_minus_set_a @ A2 @ B2 ) ) ) ) ).
% DiffI
thf(fact_1000_DiffI,axiom,
! [C: set_a,A2: set_set_a,B2: set_set_a] :
( ( member_set_a @ C @ A2 )
=> ( ~ ( member_set_a @ C @ B2 )
=> ( member_set_a @ C @ ( minus_5736297505244876581_set_a @ A2 @ B2 ) ) ) ) ).
% DiffI
thf(fact_1001_Diff__iff,axiom,
! [C: a,A2: set_a,B2: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A2 @ B2 ) )
= ( ( member_a @ C @ A2 )
& ~ ( member_a @ C @ B2 ) ) ) ).
% Diff_iff
thf(fact_1002_Diff__iff,axiom,
! [C: set_a,A2: set_set_a,B2: set_set_a] :
( ( member_set_a @ C @ ( minus_5736297505244876581_set_a @ A2 @ B2 ) )
= ( ( member_set_a @ C @ A2 )
& ~ ( member_set_a @ C @ B2 ) ) ) ).
% Diff_iff
thf(fact_1003_comp__sgraph_Odegree__none,axiom,
! [V: set_a,S: set_set_a] :
( ~ ( member_set_a @ V @ S )
=> ( ( undire8939077443744732368_set_a @ ( undire8247866692393712962_set_a @ S ) @ V )
= zero_zero_nat ) ) ).
% comp_sgraph.degree_none
thf(fact_1004_comp__sgraph_Odegree__none,axiom,
! [V: a,S: set_a] :
( ~ ( member_a @ V @ S )
=> ( ( undire8867928226783802224gree_a @ ( undire2918257014606996450dges_a @ S ) @ V )
= zero_zero_nat ) ) ).
% comp_sgraph.degree_none
thf(fact_1005_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_1006_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_1007_degree__none,axiom,
! [V: a] :
( ~ ( member_a @ V @ vertices )
=> ( ( undire8867928226783802224gree_a @ edges @ V )
= zero_zero_nat ) ) ).
% degree_none
thf(fact_1008_ex__least__nat__le,axiom,
! [P2: nat > $o,N: nat] :
( ( P2 @ N )
=> ( ~ ( P2 @ zero_zero_nat )
=> ? [K3: nat] :
( ( ord_less_eq_nat @ K3 @ N )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ K3 )
=> ~ ( P2 @ I2 ) )
& ( P2 @ K3 ) ) ) ) ).
% ex_least_nat_le
thf(fact_1009_DiffE,axiom,
! [C: a,A2: set_a,B2: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A2 @ B2 ) )
=> ~ ( ( member_a @ C @ A2 )
=> ( member_a @ C @ B2 ) ) ) ).
% DiffE
thf(fact_1010_DiffE,axiom,
! [C: set_a,A2: set_set_a,B2: set_set_a] :
( ( member_set_a @ C @ ( minus_5736297505244876581_set_a @ A2 @ B2 ) )
=> ~ ( ( member_set_a @ C @ A2 )
=> ( member_set_a @ C @ B2 ) ) ) ).
% DiffE
thf(fact_1011_DiffD1,axiom,
! [C: a,A2: set_a,B2: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A2 @ B2 ) )
=> ( member_a @ C @ A2 ) ) ).
% DiffD1
thf(fact_1012_DiffD1,axiom,
! [C: set_a,A2: set_set_a,B2: set_set_a] :
( ( member_set_a @ C @ ( minus_5736297505244876581_set_a @ A2 @ B2 ) )
=> ( member_set_a @ C @ A2 ) ) ).
% DiffD1
thf(fact_1013_DiffD2,axiom,
! [C: a,A2: set_a,B2: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A2 @ B2 ) )
=> ~ ( member_a @ C @ B2 ) ) ).
% DiffD2
thf(fact_1014_DiffD2,axiom,
! [C: set_a,A2: set_set_a,B2: set_set_a] :
( ( member_set_a @ C @ ( minus_5736297505244876581_set_a @ A2 @ B2 ) )
=> ~ ( member_set_a @ C @ B2 ) ) ).
% DiffD2
thf(fact_1015_length__induct,axiom,
! [P2: list_a > $o,Xs: list_a] :
( ! [Xs3: list_a] :
( ! [Ys7: list_a] :
( ( ord_less_nat @ ( size_size_list_a @ Ys7 ) @ ( size_size_list_a @ Xs3 ) )
=> ( P2 @ Ys7 ) )
=> ( P2 @ Xs3 ) )
=> ( P2 @ Xs ) ) ).
% length_induct
thf(fact_1016_length__induct,axiom,
! [P2: list_set_a > $o,Xs: list_set_a] :
( ! [Xs3: list_set_a] :
( ! [Ys7: list_set_a] :
( ( ord_less_nat @ ( size_size_list_set_a @ Ys7 ) @ ( size_size_list_set_a @ Xs3 ) )
=> ( P2 @ Ys7 ) )
=> ( P2 @ Xs3 ) )
=> ( P2 @ Xs ) ) ).
% length_induct
thf(fact_1017_bot__set__def,axiom,
( bot_bot_set_a
= ( collect_a @ bot_bot_a_o ) ) ).
% bot_set_def
thf(fact_1018_bot__set__def,axiom,
( bot_bot_set_set_a
= ( collect_set_a @ bot_bot_set_a_o ) ) ).
% bot_set_def
thf(fact_1019_bot_Onot__eq__extremum,axiom,
! [A: set_a] :
( ( A != bot_bot_set_a )
= ( ord_less_set_a @ bot_bot_set_a @ A ) ) ).
% bot.not_eq_extremum
thf(fact_1020_bot_Onot__eq__extremum,axiom,
! [A: set_set_a] :
( ( A != bot_bot_set_set_a )
= ( ord_less_set_set_a @ bot_bot_set_set_a @ A ) ) ).
% bot.not_eq_extremum
thf(fact_1021_bot_Onot__eq__extremum,axiom,
! [A: nat] :
( ( A != bot_bot_nat )
= ( ord_less_nat @ bot_bot_nat @ A ) ) ).
% bot.not_eq_extremum
thf(fact_1022_bot_Oextremum__strict,axiom,
! [A: set_a] :
~ ( ord_less_set_a @ A @ bot_bot_set_a ) ).
% bot.extremum_strict
thf(fact_1023_bot_Oextremum__strict,axiom,
! [A: set_set_a] :
~ ( ord_less_set_set_a @ A @ bot_bot_set_set_a ) ).
% bot.extremum_strict
thf(fact_1024_bot_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ bot_bot_nat ) ).
% bot.extremum_strict
thf(fact_1025_ulgraph_Odegree_Ocong,axiom,
undire8867928226783802224gree_a = undire8867928226783802224gree_a ).
% ulgraph.degree.cong
thf(fact_1026_ulgraph_Oneighborhood_Ocong,axiom,
undire8504279938402040014hood_a = undire8504279938402040014hood_a ).
% ulgraph.neighborhood.cong
thf(fact_1027_gt__ex,axiom,
! [X4: nat] :
? [X_1: nat] : ( ord_less_nat @ X4 @ X_1 ) ).
% gt_ex
thf(fact_1028_less__imp__neq,axiom,
! [X4: nat,Y: nat] :
( ( ord_less_nat @ X4 @ Y )
=> ( X4 != Y ) ) ).
% less_imp_neq
thf(fact_1029_order_Oasym,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order.asym
thf(fact_1030_ord__eq__less__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( A = B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_1031_ord__less__eq__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( B = C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_1032_less__induct,axiom,
! [P2: nat > $o,A: nat] :
( ! [X3: nat] :
( ! [Y6: nat] :
( ( ord_less_nat @ Y6 @ X3 )
=> ( P2 @ Y6 ) )
=> ( P2 @ X3 ) )
=> ( P2 @ A ) ) ).
% less_induct
thf(fact_1033_antisym__conv3,axiom,
! [Y: nat,X4: nat] :
( ~ ( ord_less_nat @ Y @ X4 )
=> ( ( ~ ( ord_less_nat @ X4 @ Y ) )
= ( X4 = Y ) ) ) ).
% antisym_conv3
thf(fact_1034_linorder__cases,axiom,
! [X4: nat,Y: nat] :
( ~ ( ord_less_nat @ X4 @ Y )
=> ( ( X4 != Y )
=> ( ord_less_nat @ Y @ X4 ) ) ) ).
% linorder_cases
thf(fact_1035_dual__order_Oasym,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ~ ( ord_less_nat @ A @ B ) ) ).
% dual_order.asym
thf(fact_1036_dual__order_Oirrefl,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% dual_order.irrefl
thf(fact_1037_exists__least__iff,axiom,
( ( ^ [P5: nat > $o] :
? [X6: nat] : ( P5 @ X6 ) )
= ( ^ [P6: nat > $o] :
? [N3: nat] :
( ( P6 @ N3 )
& ! [M2: nat] :
( ( ord_less_nat @ M2 @ N3 )
=> ~ ( P6 @ M2 ) ) ) ) ) ).
% exists_least_iff
thf(fact_1038_linorder__less__wlog,axiom,
! [P2: nat > nat > $o,A: nat,B: nat] :
( ! [A4: nat,B4: nat] :
( ( ord_less_nat @ A4 @ B4 )
=> ( P2 @ A4 @ B4 ) )
=> ( ! [A4: nat] : ( P2 @ A4 @ A4 )
=> ( ! [A4: nat,B4: nat] :
( ( P2 @ B4 @ A4 )
=> ( P2 @ A4 @ B4 ) )
=> ( P2 @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_1039_order_Ostrict__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans
thf(fact_1040_not__less__iff__gr__or__eq,axiom,
! [X4: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X4 @ Y ) )
= ( ( ord_less_nat @ Y @ X4 )
| ( X4 = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_1041_dual__order_Ostrict__trans,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( ord_less_nat @ C @ B )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_1042_order_Ostrict__implies__not__eq,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_1043_dual__order_Ostrict__implies__not__eq,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_1044_linorder__neqE,axiom,
! [X4: nat,Y: nat] :
( ( X4 != Y )
=> ( ~ ( ord_less_nat @ X4 @ Y )
=> ( ord_less_nat @ Y @ X4 ) ) ) ).
% linorder_neqE
thf(fact_1045_order__less__asym,axiom,
! [X4: nat,Y: nat] :
( ( ord_less_nat @ X4 @ Y )
=> ~ ( ord_less_nat @ Y @ X4 ) ) ).
% order_less_asym
thf(fact_1046_linorder__neq__iff,axiom,
! [X4: nat,Y: nat] :
( ( X4 != Y )
= ( ( ord_less_nat @ X4 @ Y )
| ( ord_less_nat @ Y @ X4 ) ) ) ).
% linorder_neq_iff
thf(fact_1047_order__less__asym_H,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order_less_asym'
thf(fact_1048_order__less__trans,axiom,
! [X4: nat,Y: nat,Z: nat] :
( ( ord_less_nat @ X4 @ Y )
=> ( ( ord_less_nat @ Y @ Z )
=> ( ord_less_nat @ X4 @ Z ) ) ) ).
% order_less_trans
thf(fact_1049_ord__eq__less__subst,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_1050_ord__less__eq__subst,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_1051_order__less__irrefl,axiom,
! [X4: nat] :
~ ( ord_less_nat @ X4 @ X4 ) ).
% order_less_irrefl
thf(fact_1052_order__less__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_1053_order__less__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_1054_order__less__not__sym,axiom,
! [X4: nat,Y: nat] :
( ( ord_less_nat @ X4 @ Y )
=> ~ ( ord_less_nat @ Y @ X4 ) ) ).
% order_less_not_sym
thf(fact_1055_order__less__imp__triv,axiom,
! [X4: nat,Y: nat,P2: $o] :
( ( ord_less_nat @ X4 @ Y )
=> ( ( ord_less_nat @ Y @ X4 )
=> P2 ) ) ).
% order_less_imp_triv
thf(fact_1056_linorder__less__linear,axiom,
! [X4: nat,Y: nat] :
( ( ord_less_nat @ X4 @ Y )
| ( X4 = Y )
| ( ord_less_nat @ Y @ X4 ) ) ).
% linorder_less_linear
thf(fact_1057_order__less__imp__not__eq,axiom,
! [X4: nat,Y: nat] :
( ( ord_less_nat @ X4 @ Y )
=> ( X4 != Y ) ) ).
% order_less_imp_not_eq
thf(fact_1058_order__less__imp__not__eq2,axiom,
! [X4: nat,Y: nat] :
( ( ord_less_nat @ X4 @ Y )
=> ( Y != X4 ) ) ).
% order_less_imp_not_eq2
thf(fact_1059_order__less__imp__not__less,axiom,
! [X4: nat,Y: nat] :
( ( ord_less_nat @ X4 @ Y )
=> ~ ( ord_less_nat @ Y @ X4 ) ) ).
% order_less_imp_not_less
thf(fact_1060_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M2: nat,N3: nat] :
( ( ord_less_eq_nat @ M2 @ N3 )
& ( M2 != N3 ) ) ) ) ).
% nat_less_le
thf(fact_1061_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_1062_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M2: nat,N3: nat] :
( ( ord_less_nat @ M2 @ N3 )
| ( M2 = N3 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_1063_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_1064_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_1065_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I: nat,J: nat] :
( ! [I3: nat,J2: nat] :
( ( ord_less_nat @ I3 @ J2 )
=> ( ord_less_nat @ ( F @ I3 ) @ ( F @ J2 ) ) )
=> ( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_1066_leD,axiom,
! [Y: set_a,X4: set_a] :
( ( ord_less_eq_set_a @ Y @ X4 )
=> ~ ( ord_less_set_a @ X4 @ Y ) ) ).
% leD
thf(fact_1067_leD,axiom,
! [Y: set_set_a,X4: set_set_a] :
( ( ord_le3724670747650509150_set_a @ Y @ X4 )
=> ~ ( ord_less_set_set_a @ X4 @ Y ) ) ).
% leD
thf(fact_1068_leD,axiom,
! [Y: nat,X4: nat] :
( ( ord_less_eq_nat @ Y @ X4 )
=> ~ ( ord_less_nat @ X4 @ Y ) ) ).
% leD
thf(fact_1069_leD,axiom,
! [Y: real,X4: real] :
( ( ord_less_eq_real @ Y @ X4 )
=> ~ ( ord_less_real @ X4 @ Y ) ) ).
% leD
thf(fact_1070_leI,axiom,
! [X4: nat,Y: nat] :
( ~ ( ord_less_nat @ X4 @ Y )
=> ( ord_less_eq_nat @ Y @ X4 ) ) ).
% leI
thf(fact_1071_leI,axiom,
! [X4: real,Y: real] :
( ~ ( ord_less_real @ X4 @ Y )
=> ( ord_less_eq_real @ Y @ X4 ) ) ).
% leI
thf(fact_1072_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_1073_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_1074_nless__le,axiom,
! [A: nat,B: nat] :
( ( ~ ( ord_less_nat @ A @ B ) )
= ( ~ ( ord_less_eq_nat @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_1075_nless__le,axiom,
! [A: real,B: real] :
( ( ~ ( ord_less_real @ A @ B ) )
= ( ~ ( ord_less_eq_real @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_1076_antisym__conv1,axiom,
! [X4: set_a,Y: set_a] :
( ~ ( ord_less_set_a @ X4 @ Y )
=> ( ( ord_less_eq_set_a @ X4 @ Y )
= ( X4 = Y ) ) ) ).
% antisym_conv1
thf(fact_1077_antisym__conv1,axiom,
! [X4: set_set_a,Y: set_set_a] :
( ~ ( ord_less_set_set_a @ X4 @ Y )
=> ( ( ord_le3724670747650509150_set_a @ X4 @ Y )
= ( X4 = Y ) ) ) ).
% antisym_conv1
thf(fact_1078_antisym__conv1,axiom,
! [X4: nat,Y: nat] :
( ~ ( ord_less_nat @ X4 @ Y )
=> ( ( ord_less_eq_nat @ X4 @ Y )
= ( X4 = Y ) ) ) ).
% antisym_conv1
thf(fact_1079_antisym__conv1,axiom,
! [X4: real,Y: real] :
( ~ ( ord_less_real @ X4 @ Y )
=> ( ( ord_less_eq_real @ X4 @ Y )
= ( X4 = Y ) ) ) ).
% antisym_conv1
thf(fact_1080_antisym__conv2,axiom,
! [X4: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y )
=> ( ( ~ ( ord_less_set_a @ X4 @ Y ) )
= ( X4 = Y ) ) ) ).
% antisym_conv2
thf(fact_1081_antisym__conv2,axiom,
! [X4: set_set_a,Y: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X4 @ Y )
=> ( ( ~ ( ord_less_set_set_a @ X4 @ Y ) )
= ( X4 = Y ) ) ) ).
% antisym_conv2
thf(fact_1082_antisym__conv2,axiom,
! [X4: nat,Y: nat] :
( ( ord_less_eq_nat @ X4 @ Y )
=> ( ( ~ ( ord_less_nat @ X4 @ Y ) )
= ( X4 = Y ) ) ) ).
% antisym_conv2
thf(fact_1083_antisym__conv2,axiom,
! [X4: real,Y: real] :
( ( ord_less_eq_real @ X4 @ Y )
=> ( ( ~ ( ord_less_real @ X4 @ Y ) )
= ( X4 = Y ) ) ) ).
% antisym_conv2
thf(fact_1084_dense__ge,axiom,
! [Z: real,Y: real] :
( ! [X3: real] :
( ( ord_less_real @ Z @ X3 )
=> ( ord_less_eq_real @ Y @ X3 ) )
=> ( ord_less_eq_real @ Y @ Z ) ) ).
% dense_ge
thf(fact_1085_dense__le,axiom,
! [Y: real,Z: real] :
( ! [X3: real] :
( ( ord_less_real @ X3 @ Y )
=> ( ord_less_eq_real @ X3 @ Z ) )
=> ( ord_less_eq_real @ Y @ Z ) ) ).
% dense_le
thf(fact_1086_less__le__not__le,axiom,
( ord_less_set_a
= ( ^ [X2: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y4 )
& ~ ( ord_less_eq_set_a @ Y4 @ X2 ) ) ) ) ).
% less_le_not_le
thf(fact_1087_less__le__not__le,axiom,
( ord_less_set_set_a
= ( ^ [X2: set_set_a,Y4: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X2 @ Y4 )
& ~ ( ord_le3724670747650509150_set_a @ Y4 @ X2 ) ) ) ) ).
% less_le_not_le
thf(fact_1088_less__le__not__le,axiom,
( ord_less_nat
= ( ^ [X2: nat,Y4: nat] :
( ( ord_less_eq_nat @ X2 @ Y4 )
& ~ ( ord_less_eq_nat @ Y4 @ X2 ) ) ) ) ).
% less_le_not_le
thf(fact_1089_less__le__not__le,axiom,
( ord_less_real
= ( ^ [X2: real,Y4: real] :
( ( ord_less_eq_real @ X2 @ Y4 )
& ~ ( ord_less_eq_real @ Y4 @ X2 ) ) ) ) ).
% less_le_not_le
thf(fact_1090_not__le__imp__less,axiom,
! [Y: nat,X4: nat] :
( ~ ( ord_less_eq_nat @ Y @ X4 )
=> ( ord_less_nat @ X4 @ Y ) ) ).
% not_le_imp_less
thf(fact_1091_not__le__imp__less,axiom,
! [Y: real,X4: real] :
( ~ ( ord_less_eq_real @ Y @ X4 )
=> ( ord_less_real @ X4 @ Y ) ) ).
% not_le_imp_less
thf(fact_1092_order_Oorder__iff__strict,axiom,
( ord_less_eq_set_a
= ( ^ [A3: set_a,B3: set_a] :
( ( ord_less_set_a @ A3 @ B3 )
| ( A3 = B3 ) ) ) ) ).
% order.order_iff_strict
thf(fact_1093_order_Oorder__iff__strict,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [A3: set_set_a,B3: set_set_a] :
( ( ord_less_set_set_a @ A3 @ B3 )
| ( A3 = B3 ) ) ) ) ).
% order.order_iff_strict
thf(fact_1094_order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [A3: nat,B3: nat] :
( ( ord_less_nat @ A3 @ B3 )
| ( A3 = B3 ) ) ) ) ).
% order.order_iff_strict
thf(fact_1095_order_Oorder__iff__strict,axiom,
( ord_less_eq_real
= ( ^ [A3: real,B3: real] :
( ( ord_less_real @ A3 @ B3 )
| ( A3 = B3 ) ) ) ) ).
% order.order_iff_strict
thf(fact_1096_order_Ostrict__iff__order,axiom,
( ord_less_set_a
= ( ^ [A3: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ A3 @ B3 )
& ( A3 != B3 ) ) ) ) ).
% order.strict_iff_order
thf(fact_1097_order_Ostrict__iff__order,axiom,
( ord_less_set_set_a
= ( ^ [A3: set_set_a,B3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A3 @ B3 )
& ( A3 != B3 ) ) ) ) ).
% order.strict_iff_order
thf(fact_1098_order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
& ( A3 != B3 ) ) ) ) ).
% order.strict_iff_order
thf(fact_1099_order_Ostrict__iff__order,axiom,
( ord_less_real
= ( ^ [A3: real,B3: real] :
( ( ord_less_eq_real @ A3 @ B3 )
& ( A3 != B3 ) ) ) ) ).
% order.strict_iff_order
thf(fact_1100_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_1101_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_1102_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_1103_order_Ostrict__trans1,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_real @ B @ C )
=> ( ord_less_real @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_1104_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_1105_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_1106_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_1107_order_Ostrict__trans2,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ord_less_real @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_1108_order_Ostrict__iff__not,axiom,
( ord_less_set_a
= ( ^ [A3: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ A3 @ B3 )
& ~ ( ord_less_eq_set_a @ B3 @ A3 ) ) ) ) ).
% order.strict_iff_not
thf(fact_1109_order_Ostrict__iff__not,axiom,
( ord_less_set_set_a
= ( ^ [A3: set_set_a,B3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A3 @ B3 )
& ~ ( ord_le3724670747650509150_set_a @ B3 @ A3 ) ) ) ) ).
% order.strict_iff_not
thf(fact_1110_order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
& ~ ( ord_less_eq_nat @ B3 @ A3 ) ) ) ) ).
% order.strict_iff_not
thf(fact_1111_order_Ostrict__iff__not,axiom,
( ord_less_real
= ( ^ [A3: real,B3: real] :
( ( ord_less_eq_real @ A3 @ B3 )
& ~ ( ord_less_eq_real @ B3 @ A3 ) ) ) ) ).
% order.strict_iff_not
thf(fact_1112_dense__ge__bounded,axiom,
! [Z: real,X4: real,Y: real] :
( ( ord_less_real @ Z @ X4 )
=> ( ! [W: real] :
( ( ord_less_real @ Z @ W )
=> ( ( ord_less_real @ W @ X4 )
=> ( ord_less_eq_real @ Y @ W ) ) )
=> ( ord_less_eq_real @ Y @ Z ) ) ) ).
% dense_ge_bounded
thf(fact_1113_dense__le__bounded,axiom,
! [X4: real,Y: real,Z: real] :
( ( ord_less_real @ X4 @ Y )
=> ( ! [W: real] :
( ( ord_less_real @ X4 @ W )
=> ( ( ord_less_real @ W @ Y )
=> ( ord_less_eq_real @ W @ Z ) ) )
=> ( ord_less_eq_real @ Y @ Z ) ) ) ).
% dense_le_bounded
thf(fact_1114_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_set_a
= ( ^ [B3: set_a,A3: set_a] :
( ( ord_less_set_a @ B3 @ A3 )
| ( A3 = B3 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_1115_dual__order_Oorder__iff__strict,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [B3: set_set_a,A3: set_set_a] :
( ( ord_less_set_set_a @ B3 @ A3 )
| ( A3 = B3 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_1116_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [B3: nat,A3: nat] :
( ( ord_less_nat @ B3 @ A3 )
| ( A3 = B3 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_1117_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_real
= ( ^ [B3: real,A3: real] :
( ( ord_less_real @ B3 @ A3 )
| ( A3 = B3 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_1118_dual__order_Ostrict__iff__order,axiom,
( ord_less_set_a
= ( ^ [B3: set_a,A3: set_a] :
( ( ord_less_eq_set_a @ B3 @ A3 )
& ( A3 != B3 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_1119_dual__order_Ostrict__iff__order,axiom,
( ord_less_set_set_a
= ( ^ [B3: set_set_a,A3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B3 @ A3 )
& ( A3 != B3 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_1120_dual__order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [B3: nat,A3: nat] :
( ( ord_less_eq_nat @ B3 @ A3 )
& ( A3 != B3 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_1121_dual__order_Ostrict__iff__order,axiom,
( ord_less_real
= ( ^ [B3: real,A3: real] :
( ( ord_less_eq_real @ B3 @ A3 )
& ( A3 != B3 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_1122_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_1123_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_1124_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_1125_dual__order_Ostrict__trans1,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( ord_less_real @ C @ B )
=> ( ord_less_real @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_1126_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_1127_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_1128_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_1129_dual__order_Ostrict__trans2,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_real @ B @ A )
=> ( ( ord_less_eq_real @ C @ B )
=> ( ord_less_real @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_1130_dual__order_Ostrict__iff__not,axiom,
( ord_less_set_a
= ( ^ [B3: set_a,A3: set_a] :
( ( ord_less_eq_set_a @ B3 @ A3 )
& ~ ( ord_less_eq_set_a @ A3 @ B3 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_1131_dual__order_Ostrict__iff__not,axiom,
( ord_less_set_set_a
= ( ^ [B3: set_set_a,A3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B3 @ A3 )
& ~ ( ord_le3724670747650509150_set_a @ A3 @ B3 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_1132_dual__order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [B3: nat,A3: nat] :
( ( ord_less_eq_nat @ B3 @ A3 )
& ~ ( ord_less_eq_nat @ A3 @ B3 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_1133_dual__order_Ostrict__iff__not,axiom,
( ord_less_real
= ( ^ [B3: real,A3: real] :
( ( ord_less_eq_real @ B3 @ A3 )
& ~ ( ord_less_eq_real @ A3 @ B3 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_1134_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_1135_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_1136_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_1137_order_Ostrict__implies__order,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ( ord_less_eq_real @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_1138_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_1139_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_1140_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_1141_dual__order_Ostrict__implies__order,axiom,
! [B: real,A: real] :
( ( ord_less_real @ B @ A )
=> ( ord_less_eq_real @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_1142_order__le__less,axiom,
( ord_less_eq_set_a
= ( ^ [X2: set_a,Y4: set_a] :
( ( ord_less_set_a @ X2 @ Y4 )
| ( X2 = Y4 ) ) ) ) ).
% order_le_less
thf(fact_1143_order__le__less,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [X2: set_set_a,Y4: set_set_a] :
( ( ord_less_set_set_a @ X2 @ Y4 )
| ( X2 = Y4 ) ) ) ) ).
% order_le_less
thf(fact_1144_order__le__less,axiom,
( ord_less_eq_nat
= ( ^ [X2: nat,Y4: nat] :
( ( ord_less_nat @ X2 @ Y4 )
| ( X2 = Y4 ) ) ) ) ).
% order_le_less
thf(fact_1145_order__le__less,axiom,
( ord_less_eq_real
= ( ^ [X2: real,Y4: real] :
( ( ord_less_real @ X2 @ Y4 )
| ( X2 = Y4 ) ) ) ) ).
% order_le_less
thf(fact_1146_order__less__le,axiom,
( ord_less_set_a
= ( ^ [X2: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y4 )
& ( X2 != Y4 ) ) ) ) ).
% order_less_le
thf(fact_1147_order__less__le,axiom,
( ord_less_set_set_a
= ( ^ [X2: set_set_a,Y4: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X2 @ Y4 )
& ( X2 != Y4 ) ) ) ) ).
% order_less_le
thf(fact_1148_order__less__le,axiom,
( ord_less_nat
= ( ^ [X2: nat,Y4: nat] :
( ( ord_less_eq_nat @ X2 @ Y4 )
& ( X2 != Y4 ) ) ) ) ).
% order_less_le
thf(fact_1149_order__less__le,axiom,
( ord_less_real
= ( ^ [X2: real,Y4: real] :
( ( ord_less_eq_real @ X2 @ Y4 )
& ( X2 != Y4 ) ) ) ) ).
% order_less_le
thf(fact_1150_linorder__not__le,axiom,
! [X4: nat,Y: nat] :
( ( ~ ( ord_less_eq_nat @ X4 @ Y ) )
= ( ord_less_nat @ Y @ X4 ) ) ).
% linorder_not_le
thf(fact_1151_linorder__not__le,axiom,
! [X4: real,Y: real] :
( ( ~ ( ord_less_eq_real @ X4 @ Y ) )
= ( ord_less_real @ Y @ X4 ) ) ).
% linorder_not_le
thf(fact_1152_linorder__not__less,axiom,
! [X4: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X4 @ Y ) )
= ( ord_less_eq_nat @ Y @ X4 ) ) ).
% linorder_not_less
thf(fact_1153_linorder__not__less,axiom,
! [X4: real,Y: real] :
( ( ~ ( ord_less_real @ X4 @ Y ) )
= ( ord_less_eq_real @ Y @ X4 ) ) ).
% linorder_not_less
thf(fact_1154_order__less__imp__le,axiom,
! [X4: set_a,Y: set_a] :
( ( ord_less_set_a @ X4 @ Y )
=> ( ord_less_eq_set_a @ X4 @ Y ) ) ).
% order_less_imp_le
thf(fact_1155_order__less__imp__le,axiom,
! [X4: set_set_a,Y: set_set_a] :
( ( ord_less_set_set_a @ X4 @ Y )
=> ( ord_le3724670747650509150_set_a @ X4 @ Y ) ) ).
% order_less_imp_le
thf(fact_1156_order__less__imp__le,axiom,
! [X4: nat,Y: nat] :
( ( ord_less_nat @ X4 @ Y )
=> ( ord_less_eq_nat @ X4 @ Y ) ) ).
% order_less_imp_le
thf(fact_1157_order__less__imp__le,axiom,
! [X4: real,Y: real] :
( ( ord_less_real @ X4 @ Y )
=> ( ord_less_eq_real @ X4 @ Y ) ) ).
% order_less_imp_le
thf(fact_1158_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_1159_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_1160_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_1161_order__le__neq__trans,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( A != B )
=> ( ord_less_real @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_1162_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_1163_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_1164_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_1165_order__neq__le__trans,axiom,
! [A: real,B: real] :
( ( A != B )
=> ( ( ord_less_eq_real @ A @ B )
=> ( ord_less_real @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_1166_order__le__less__trans,axiom,
! [X4: set_a,Y: set_a,Z: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y )
=> ( ( ord_less_set_a @ Y @ Z )
=> ( ord_less_set_a @ X4 @ Z ) ) ) ).
% order_le_less_trans
thf(fact_1167_order__le__less__trans,axiom,
! [X4: set_set_a,Y: set_set_a,Z: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X4 @ Y )
=> ( ( ord_less_set_set_a @ Y @ Z )
=> ( ord_less_set_set_a @ X4 @ Z ) ) ) ).
% order_le_less_trans
thf(fact_1168_order__le__less__trans,axiom,
! [X4: nat,Y: nat,Z: nat] :
( ( ord_less_eq_nat @ X4 @ Y )
=> ( ( ord_less_nat @ Y @ Z )
=> ( ord_less_nat @ X4 @ Z ) ) ) ).
% order_le_less_trans
thf(fact_1169_order__le__less__trans,axiom,
! [X4: real,Y: real,Z: real] :
( ( ord_less_eq_real @ X4 @ Y )
=> ( ( ord_less_real @ Y @ Z )
=> ( ord_less_real @ X4 @ Z ) ) ) ).
% order_le_less_trans
thf(fact_1170_order__less__le__trans,axiom,
! [X4: set_a,Y: set_a,Z: set_a] :
( ( ord_less_set_a @ X4 @ Y )
=> ( ( ord_less_eq_set_a @ Y @ Z )
=> ( ord_less_set_a @ X4 @ Z ) ) ) ).
% order_less_le_trans
thf(fact_1171_order__less__le__trans,axiom,
! [X4: set_set_a,Y: set_set_a,Z: set_set_a] :
( ( ord_less_set_set_a @ X4 @ Y )
=> ( ( ord_le3724670747650509150_set_a @ Y @ Z )
=> ( ord_less_set_set_a @ X4 @ Z ) ) ) ).
% order_less_le_trans
thf(fact_1172_order__less__le__trans,axiom,
! [X4: nat,Y: nat,Z: nat] :
( ( ord_less_nat @ X4 @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z )
=> ( ord_less_nat @ X4 @ Z ) ) ) ).
% order_less_le_trans
thf(fact_1173_order__less__le__trans,axiom,
! [X4: real,Y: real,Z: real] :
( ( ord_less_real @ X4 @ Y )
=> ( ( ord_less_eq_real @ Y @ Z )
=> ( ord_less_real @ X4 @ Z ) ) ) ).
% order_less_le_trans
thf(fact_1174_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 )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_set_a @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_1175_order__le__less__subst1,axiom,
! [A: set_set_a,F: nat > set_set_a,B: nat,C: nat] :
( ( ord_le3724670747650509150_set_a @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_set_set_a @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_1176_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 )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_1177_order__le__less__subst1,axiom,
! [A: real,F: nat > real,B: nat,C: nat] :
( ( ord_less_eq_real @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_1178_order__le__less__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_1179_order__le__less__subst2,axiom,
! [A: nat,B: nat,F: nat > real,C: real] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_1180_order__le__less__subst2,axiom,
! [A: real,B: real,F: real > nat,C: nat] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_1181_order__le__less__subst2,axiom,
! [A: real,B: real,F: real > real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_1182_order__le__less__subst2,axiom,
! [A: set_a,B: set_a,F: set_a > nat,C: nat] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X3: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_1183_order__le__less__subst2,axiom,
! [A: set_a,B: set_a,F: set_a > real,C: real] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C )
=> ( ! [X3: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_1184_order__le__less__subst2,axiom,
! [A: nat,B: nat,F: nat > set_a,C: set_a] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_set_a @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_1185_order__le__less__subst2,axiom,
! [A: real,B: real,F: real > set_a,C: set_a] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_set_a @ ( F @ B ) @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_1186_order__le__less__subst2,axiom,
! [A: set_a,B: set_a,F: set_a > set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_set_a @ ( F @ B ) @ C )
=> ( ! [X3: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_1187_order__le__less__subst2,axiom,
! [A: set_set_a,B: set_set_a,F: set_set_a > nat,C: nat] :
( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X3: set_set_a,Y2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_1188_order__less__le__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_1189_order__less__le__subst1,axiom,
! [A: real,F: nat > real,B: nat,C: nat] :
( ( ord_less_real @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_1190_order__less__le__subst1,axiom,
! [A: nat,F: real > nat,B: real,C: real] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_1191_order__less__le__subst1,axiom,
! [A: real,F: real > real,B: real,C: real] :
( ( ord_less_real @ A @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_1192_order__less__le__subst1,axiom,
! [A: nat,F: set_a > nat,B: set_a,C: set_a] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ! [X3: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_1193_order__less__le__subst1,axiom,
! [A: real,F: set_a > real,B: set_a,C: set_a] :
( ( ord_less_real @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ! [X3: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_1194_order__less__le__subst1,axiom,
! [A: set_a,F: nat > set_a,B: nat,C: nat] :
( ( ord_less_set_a @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_1195_order__less__le__subst1,axiom,
! [A: set_a,F: real > set_a,B: real,C: real] :
( ( ord_less_set_a @ A @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_1196_order__less__le__subst1,axiom,
! [A: set_a,F: set_a > set_a,B: set_a,C: set_a] :
( ( ord_less_set_a @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ! [X3: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_1197_order__less__le__subst1,axiom,
! [A: nat,F: set_set_a > nat,B: set_set_a,C: set_set_a] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_le3724670747650509150_set_a @ B @ C )
=> ( ! [X3: set_set_a,Y2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_1198_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > set_a,C: set_a] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_set_a @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_set_a @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_1199_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > set_set_a,C: set_set_a] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_le3724670747650509150_set_a @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_set_set_a @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_1200_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_1201_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > real,C: real] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_1202_linorder__le__less__linear,axiom,
! [X4: nat,Y: nat] :
( ( ord_less_eq_nat @ X4 @ Y )
| ( ord_less_nat @ Y @ X4 ) ) ).
% linorder_le_less_linear
thf(fact_1203_linorder__le__less__linear,axiom,
! [X4: real,Y: real] :
( ( ord_less_eq_real @ X4 @ Y )
| ( ord_less_real @ Y @ X4 ) ) ).
% linorder_le_less_linear
thf(fact_1204_order__le__imp__less__or__eq,axiom,
! [X4: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y )
=> ( ( ord_less_set_a @ X4 @ Y )
| ( X4 = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_1205_order__le__imp__less__or__eq,axiom,
! [X4: set_set_a,Y: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X4 @ Y )
=> ( ( ord_less_set_set_a @ X4 @ Y )
| ( X4 = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_1206_order__le__imp__less__or__eq,axiom,
! [X4: nat,Y: nat] :
( ( ord_less_eq_nat @ X4 @ Y )
=> ( ( ord_less_nat @ X4 @ Y )
| ( X4 = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_1207_order__le__imp__less__or__eq,axiom,
! [X4: real,Y: real] :
( ( ord_less_eq_real @ X4 @ Y )
=> ( ( ord_less_real @ X4 @ Y )
| ( X4 = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_1208_ulgraph_Odegree__none,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,V: set_a] :
( ( undire6886684016831807756_set_a @ Vertices @ Edges )
=> ( ~ ( member_set_a @ V @ Vertices )
=> ( ( undire8939077443744732368_set_a @ Edges @ V )
= zero_zero_nat ) ) ) ).
% ulgraph.degree_none
thf(fact_1209_ulgraph_Odegree__none,axiom,
! [Vertices: set_a,Edges: set_set_a,V: a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ~ ( member_a @ V @ Vertices )
=> ( ( undire8867928226783802224gree_a @ Edges @ V )
= zero_zero_nat ) ) ) ).
% ulgraph.degree_none
thf(fact_1210_add__less__le__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_less_le_mono
thf(fact_1211_add__less__le__mono,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_eq_real @ C @ D )
=> ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) ) ) ) ).
% add_less_le_mono
thf(fact_1212_add__le__less__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_le_less_mono
thf(fact_1213_add__le__less__mono,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_real @ C @ D )
=> ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) ) ) ) ).
% add_le_less_mono
thf(fact_1214_add__mono__thms__linordered__field_I3_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I @ J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_1215_add__mono__thms__linordered__field_I3_J,axiom,
! [I: real,J: real,K: real,L: real] :
( ( ( ord_less_real @ I @ J )
& ( ord_less_eq_real @ K @ L ) )
=> ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_1216_add__mono__thms__linordered__field_I4_J,axiom,
! [I: real,J: real,K: real,L: real] :
( ( ( ord_less_eq_real @ I @ J )
& ( ord_less_real @ K @ L ) )
=> ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_1217_mono__nat__linear__lb,axiom,
! [F: nat > nat,M: nat,K: nat] :
( ! [M3: nat,N2: nat] :
( ( ord_less_nat @ M3 @ N2 )
=> ( ord_less_nat @ ( F @ M3 ) @ ( F @ N2 ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M ) @ K ) @ ( F @ ( plus_plus_nat @ M @ K ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_1218_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_1219_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_1220_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_1221_degree0__neighborhood__empt__iff,axiom,
! [V: a] :
( ( finite_finite_set_a @ edges )
=> ( ( ( undire8867928226783802224gree_a @ edges @ V )
= zero_zero_nat )
= ( ( undire8504279938402040014hood_a @ vertices @ edges @ V )
= bot_bot_set_a ) ) ) ).
% degree0_neighborhood_empt_iff
thf(fact_1222_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_1223_walk__edges__app,axiom,
! [Xs: list_a,Y: a,X4: a] :
( ( undire7337870655677353998dges_a @ ( append_a @ Xs @ ( cons_a @ Y @ ( cons_a @ X4 @ nil_a ) ) ) )
= ( append_set_a @ ( undire7337870655677353998dges_a @ ( append_a @ Xs @ ( cons_a @ Y @ nil_a ) ) ) @ ( cons_set_a @ ( insert_a @ Y @ ( insert_a @ X4 @ bot_bot_set_a ) ) @ nil_set_a ) ) ) ).
% walk_edges_app
thf(fact_1224_vert__adj__def,axiom,
! [V1: a,V2: a] :
( ( undire397441198561214472_adj_a @ edges @ V1 @ V2 )
= ( member_set_a @ ( insert_a @ V1 @ ( insert_a @ V2 @ bot_bot_set_a ) ) @ edges ) ) ).
% vert_adj_def
thf(fact_1225_not__vert__adj,axiom,
! [V: a,U: a] :
( ~ ( undire397441198561214472_adj_a @ edges @ V @ U )
=> ~ ( member_set_a @ ( insert_a @ V @ ( insert_a @ U @ bot_bot_set_a ) ) @ edges ) ) ).
% not_vert_adj
thf(fact_1226_has__loop__def,axiom,
! [V: a] :
( ( undire3617971648856834880loop_a @ edges @ V )
= ( member_set_a @ ( insert_a @ V @ bot_bot_set_a ) @ edges ) ) ).
% has_loop_def
thf(fact_1227_wellformed__alt__snd,axiom,
! [X4: a,Y: a] :
( ( member_set_a @ ( insert_a @ X4 @ ( insert_a @ Y @ bot_bot_set_a ) ) @ edges )
=> ( member_a @ Y @ vertices ) ) ).
% wellformed_alt_snd
thf(fact_1228_wellformed__alt__fst,axiom,
! [X4: a,Y: a] :
( ( member_set_a @ ( insert_a @ X4 @ ( insert_a @ Y @ bot_bot_set_a ) ) @ edges )
=> ( member_a @ X4 @ vertices ) ) ).
% wellformed_alt_fst
thf(fact_1229_is__edge__between__def,axiom,
( undire8544646567961481629ween_a
= ( ^ [X7: set_a,Y7: set_a,E5: set_a] :
? [X2: a,Y4: a] :
( ( E5
= ( insert_a @ X2 @ ( insert_a @ Y4 @ bot_bot_set_a ) ) )
& ( member_a @ X2 @ X7 )
& ( member_a @ Y4 @ Y7 ) ) ) ) ).
% is_edge_between_def
thf(fact_1230_walk__edges_Osimps_I3_J,axiom,
! [X4: a,Y: a,Ys: list_a] :
( ( undire7337870655677353998dges_a @ ( cons_a @ X4 @ ( cons_a @ Y @ Ys ) ) )
= ( cons_set_a @ ( insert_a @ X4 @ ( insert_a @ Y @ bot_bot_set_a ) ) @ ( undire7337870655677353998dges_a @ ( cons_a @ Y @ Ys ) ) ) ) ).
% walk_edges.simps(3)
thf(fact_1231_vert__adj__inc__edge__iff,axiom,
! [V1: a,V2: a] :
( ( undire397441198561214472_adj_a @ edges @ V1 @ V2 )
= ( ( undire1521409233611534436dent_a @ V1 @ ( insert_a @ V1 @ ( insert_a @ V2 @ bot_bot_set_a ) ) )
& ( undire1521409233611534436dent_a @ V2 @ ( insert_a @ V1 @ ( insert_a @ V2 @ bot_bot_set_a ) ) )
& ( member_set_a @ ( insert_a @ V1 @ ( insert_a @ V2 @ bot_bot_set_a ) ) @ edges ) ) ) ).
% vert_adj_inc_edge_iff
thf(fact_1232_is__walk__hd__tl,axiom,
! [Y: a,Ys: list_a,X4: a] :
( ( undire6133010728901294956walk_a @ vertices @ edges @ ( cons_a @ Y @ Ys ) )
=> ( ( member_set_a @ ( insert_a @ X4 @ ( insert_a @ Y @ bot_bot_set_a ) ) @ edges )
=> ( undire6133010728901294956walk_a @ vertices @ edges @ ( cons_a @ X4 @ ( cons_a @ Y @ Ys ) ) ) ) ) ).
% is_walk_hd_tl
thf(fact_1233_walk__edges_Oelims,axiom,
! [X4: list_a,Y: list_set_a] :
( ( ( undire7337870655677353998dges_a @ X4 )
= Y )
=> ( ( ( X4 = nil_a )
=> ( Y != nil_set_a ) )
=> ( ( ? [X3: a] :
( X4
= ( cons_a @ X3 @ nil_a ) )
=> ( Y != nil_set_a ) )
=> ~ ! [X3: a,Y2: a,Ys3: list_a] :
( ( X4
= ( cons_a @ X3 @ ( cons_a @ Y2 @ Ys3 ) ) )
=> ( Y
!= ( cons_set_a @ ( insert_a @ X3 @ ( insert_a @ Y2 @ bot_bot_set_a ) ) @ ( undire7337870655677353998dges_a @ ( cons_a @ Y2 @ Ys3 ) ) ) ) ) ) ) ) ).
% walk_edges.elims
thf(fact_1234_walk__edges__singleton__app,axiom,
! [Ys: list_a,X4: a] :
( ( Ys != nil_a )
=> ( ( undire7337870655677353998dges_a @ ( append_a @ ( cons_a @ X4 @ nil_a ) @ Ys ) )
= ( cons_set_a @ ( insert_a @ X4 @ ( insert_a @ ( hd_a @ Ys ) @ bot_bot_set_a ) ) @ ( undire7337870655677353998dges_a @ Ys ) ) ) ) ).
% walk_edges_singleton_app
thf(fact_1235_neighborhood__incident,axiom,
! [U: a,V: a] :
( ( member_a @ U @ ( undire8504279938402040014hood_a @ vertices @ edges @ V ) )
= ( member_set_a @ ( insert_a @ U @ ( insert_a @ V @ bot_bot_set_a ) ) @ ( undire3231912044278729248dges_a @ edges @ V ) ) ) ).
% neighborhood_incident
thf(fact_1236_degree0__inc__edges__empt__iff,axiom,
! [V: a] :
( ( finite_finite_set_a @ edges )
=> ( ( ( undire8867928226783802224gree_a @ edges @ V )
= zero_zero_nat )
= ( ( undire3231912044278729248dges_a @ edges @ V )
= bot_bot_set_set_a ) ) ) ).
% degree0_inc_edges_empt_iff
thf(fact_1237_finite__inc__sedges,axiom,
! [V: a] :
( ( finite_finite_set_a @ edges )
=> ( finite_finite_set_a @ ( undire1270416042309875431dges_a @ edges @ V ) ) ) ).
% finite_inc_sedges
thf(fact_1238_finite__incident__edges,axiom,
! [V: a] :
( ( finite_finite_set_a @ edges )
=> ( finite_finite_set_a @ ( undire3231912044278729248dges_a @ edges @ V ) ) ) ).
% finite_incident_edges
thf(fact_1239_incident__edges__empty,axiom,
! [V: a] :
( ~ ( member_a @ V @ vertices )
=> ( ( undire3231912044278729248dges_a @ edges @ V )
= bot_bot_set_set_a ) ) ).
% incident_edges_empty
thf(fact_1240_incident__edges__sedges,axiom,
! [V: a] :
( ~ ( undire3617971648856834880loop_a @ edges @ V )
=> ( ( undire3231912044278729248dges_a @ edges @ V )
= ( undire1270416042309875431dges_a @ edges @ V ) ) ) ).
% incident_edges_sedges
thf(fact_1241_incident__sedges__empty,axiom,
! [V: a] :
( ~ ( member_a @ V @ vertices )
=> ( ( undire1270416042309875431dges_a @ edges @ V )
= bot_bot_set_set_a ) ) ).
% incident_sedges_empty
thf(fact_1242_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_1243_incident__loops__simp_I1_J,axiom,
! [V: a] :
( ( undire3617971648856834880loop_a @ edges @ V )
=> ( ( undire4753905205749729249oops_a @ edges @ V )
= ( insert_set_a @ ( insert_a @ V @ bot_bot_set_a ) @ bot_bot_set_set_a ) ) ) ).
% incident_loops_simp(1)
thf(fact_1244_finite__incident__loops,axiom,
! [V: a] : ( finite_finite_set_a @ ( undire4753905205749729249oops_a @ edges @ V ) ) ).
% finite_incident_loops
thf(fact_1245_incident__loops__simp_I2_J,axiom,
! [V: a] :
( ~ ( undire3617971648856834880loop_a @ edges @ V )
=> ( ( undire4753905205749729249oops_a @ edges @ V )
= bot_bot_set_set_a ) ) ).
% incident_loops_simp(2)
thf(fact_1246_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_1247_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_1248_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_1249_Suc__leI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_eq_nat @ ( suc @ M ) @ N ) ) ).
% Suc_leI
thf(fact_1250_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_1251_dec__induct,axiom,
! [I: nat,J: nat,P2: nat > $o] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( P2 @ I )
=> ( ! [N2: nat] :
( ( ord_less_eq_nat @ I @ N2 )
=> ( ( ord_less_nat @ N2 @ J )
=> ( ( P2 @ N2 )
=> ( P2 @ ( suc @ N2 ) ) ) ) )
=> ( P2 @ J ) ) ) ) ).
% dec_induct
thf(fact_1252_inc__induct,axiom,
! [I: nat,J: nat,P2: nat > $o] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( P2 @ J )
=> ( ! [N2: nat] :
( ( ord_less_eq_nat @ I @ N2 )
=> ( ( ord_less_nat @ N2 @ J )
=> ( ( P2 @ ( suc @ N2 ) )
=> ( P2 @ N2 ) ) ) )
=> ( P2 @ I ) ) ) ) ).
% inc_induct
thf(fact_1253_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_1254_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_1255_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_1256_less__eq__Suc__le,axiom,
( ord_less_nat
= ( ^ [N3: nat] : ( ord_less_eq_nat @ ( suc @ N3 ) ) ) ) ).
% less_eq_Suc_le
thf(fact_1257_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_1258_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_1259_transitive__stepwise__le,axiom,
! [M: nat,N: nat,R: nat > nat > $o] :
( ( ord_less_eq_nat @ M @ N )
=> ( ! [X3: nat] : ( R @ X3 @ X3 )
=> ( ! [X3: nat,Y2: nat,Z3: nat] :
( ( R @ X3 @ Y2 )
=> ( ( R @ Y2 @ Z3 )
=> ( R @ X3 @ Z3 ) ) )
=> ( ! [N2: nat] : ( R @ N2 @ ( suc @ N2 ) )
=> ( R @ M @ N ) ) ) ) ) ).
% transitive_stepwise_le
thf(fact_1260_nat__induct__at__least,axiom,
! [M: nat,N: nat,P2: nat > $o] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( P2 @ M )
=> ( ! [N2: nat] :
( ( ord_less_eq_nat @ M @ N2 )
=> ( ( P2 @ N2 )
=> ( P2 @ ( suc @ N2 ) ) ) )
=> ( P2 @ N ) ) ) ) ).
% nat_induct_at_least
thf(fact_1261_full__nat__induct,axiom,
! [P2: nat > $o,N: nat] :
( ! [N2: nat] :
( ! [M4: nat] :
( ( ord_less_eq_nat @ ( suc @ M4 ) @ N2 )
=> ( P2 @ M4 ) )
=> ( P2 @ N2 ) )
=> ( P2 @ N ) ) ).
% full_nat_induct
thf(fact_1262_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_1263_Suc__n__not__le__n,axiom,
! [N: nat] :
~ ( ord_less_eq_nat @ ( suc @ N ) @ N ) ).
% Suc_n_not_le_n
thf(fact_1264_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_1265_Suc__le__D,axiom,
! [N: nat,M5: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ M5 )
=> ? [M3: nat] :
( M5
= ( suc @ M3 ) ) ) ).
% Suc_le_D
thf(fact_1266_le__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ M @ ( suc @ N ) ) ) ).
% le_SucI
thf(fact_1267_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_1268_Suc__leD,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% Suc_leD
thf(fact_1269_ex__least__nat__less,axiom,
! [P2: nat > $o,N: nat] :
( ( P2 @ N )
=> ( ~ ( P2 @ zero_zero_nat )
=> ? [K3: nat] :
( ( ord_less_nat @ K3 @ N )
& ! [I2: nat] :
( ( ord_less_eq_nat @ I2 @ K3 )
=> ~ ( P2 @ I2 ) )
& ( P2 @ ( suc @ K3 ) ) ) ) ) ).
% ex_least_nat_less
thf(fact_1270_walk__edges__append__union,axiom,
! [Xs: list_a,Ys: list_a] :
( ( Xs != nil_a )
=> ( ( Ys != nil_a )
=> ( ( set_set_a2 @ ( undire7337870655677353998dges_a @ ( append_a @ Xs @ Ys ) ) )
= ( sup_sup_set_set_a @ ( sup_sup_set_set_a @ ( set_set_a2 @ ( undire7337870655677353998dges_a @ Xs ) ) @ ( set_set_a2 @ ( undire7337870655677353998dges_a @ Ys ) ) ) @ ( insert_set_a @ ( insert_a @ ( last_a @ Xs ) @ ( insert_a @ ( hd_a @ Ys ) @ bot_bot_set_a ) ) @ bot_bot_set_set_a ) ) ) ) ) ).
% walk_edges_append_union
thf(fact_1271_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_1272_incident__edges__union,axiom,
! [V: a] :
( ( undire3231912044278729248dges_a @ edges @ V )
= ( sup_sup_set_set_a @ ( undire1270416042309875431dges_a @ edges @ V ) @ ( undire4753905205749729249oops_a @ edges @ V ) ) ) ).
% incident_edges_union
thf(fact_1273_induced__union__subgraph,axiom,
! [VH1: set_a,S: set_a,VH2: set_a,T2: set_a,EH1: set_set_a,EH2: set_set_a] :
( ( ord_less_eq_set_a @ VH1 @ S )
=> ( ( ord_less_eq_set_a @ VH2 @ T2 )
=> ( ( undire2554140024507503526stem_a @ VH1 @ EH1 )
=> ( ( undire2554140024507503526stem_a @ VH2 @ EH2 )
=> ( ( ( undire7103218114511261257raph_a @ VH1 @ EH1 @ S @ ( undire7777452895879145676dges_a @ edges @ S ) )
& ( undire7103218114511261257raph_a @ VH2 @ EH2 @ T2 @ ( undire7777452895879145676dges_a @ edges @ T2 ) ) )
= ( undire7103218114511261257raph_a @ ( sup_sup_set_a @ VH1 @ VH2 ) @ ( sup_sup_set_set_a @ EH1 @ EH2 ) @ ( sup_sup_set_a @ S @ T2 ) @ ( undire7777452895879145676dges_a @ edges @ ( sup_sup_set_a @ S @ T2 ) ) ) ) ) ) ) ) ).
% induced_union_subgraph
thf(fact_1274_induced__edges__union__subgraph__single,axiom,
! [VH1: set_a,S: set_a,VH2: set_a,T2: set_a,EH1: set_set_a,EH2: set_set_a] :
( ( ord_less_eq_set_a @ VH1 @ S )
=> ( ( ord_less_eq_set_a @ VH2 @ T2 )
=> ( ( undire2554140024507503526stem_a @ VH1 @ EH1 )
=> ( ( undire2554140024507503526stem_a @ VH2 @ EH2 )
=> ( ( undire7103218114511261257raph_a @ ( sup_sup_set_a @ VH1 @ VH2 ) @ ( sup_sup_set_set_a @ EH1 @ EH2 ) @ ( sup_sup_set_a @ S @ T2 ) @ ( undire7777452895879145676dges_a @ edges @ ( sup_sup_set_a @ S @ T2 ) ) )
=> ( undire7103218114511261257raph_a @ VH1 @ EH1 @ S @ ( undire7777452895879145676dges_a @ edges @ S ) ) ) ) ) ) ) ).
% induced_edges_union_subgraph_single
thf(fact_1275_induced__edges__union,axiom,
! [VH1: set_a,S: set_a,VH2: set_a,T2: set_a,EH1: set_set_a,EH2: set_set_a] :
( ( ord_less_eq_set_a @ VH1 @ S )
=> ( ( ord_less_eq_set_a @ VH2 @ T2 )
=> ( ( undire2554140024507503526stem_a @ VH1 @ EH1 )
=> ( ( undire2554140024507503526stem_a @ VH2 @ EH2 )
=> ( ( ord_le3724670747650509150_set_a @ ( sup_sup_set_set_a @ EH1 @ EH2 ) @ ( undire7777452895879145676dges_a @ edges @ ( sup_sup_set_a @ S @ T2 ) ) )
=> ( ord_le3724670747650509150_set_a @ EH1 @ ( undire7777452895879145676dges_a @ edges @ S ) ) ) ) ) ) ) ).
% induced_edges_union
thf(fact_1276_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
% Conjectures (1)
thf(conj_0,conjecture,
connecting_walk_a @ vertices @ edges @ u @ z @ ( append_a @ xs @ ( tl_a @ ys ) ) ).
%------------------------------------------------------------------------------