TPTP Problem File: COM177^1.p
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%------------------------------------------------------------------------------
% File : COM177^1 : TPTP v9.0.0. Released v7.0.0.
% Domain : Computing Theory
% Problem : Koenig's lemma 184
% Version : [Bla16] axioms : Especial.
% English :
% Refs : [Loc10] Lochbihler (2010), Coinductive
% : [RB15] Reynolds & Blanchette (2015), A Decision Procedure for
% : [Bla16] Blanchette (2016), Email to Geoff Sutcliffe
% Source : [Bla16]
% Names : koenigslemma__184.p [Bla16]
% Status : Theorem
% Rating : 1.00 v7.5.0, 0.33 v7.2.0, 0.50 v7.1.0
% Syntax : Number of formulae : 351 ( 122 unt; 59 typ; 0 def)
% Number of atoms : 726 ( 225 equ; 0 cnn)
% Maximal formula atoms : 13 ( 2 avg)
% Number of connectives : 3758 ( 102 ~; 9 |; 31 &;3282 @)
% ( 0 <=>; 334 =>; 0 <=; 0 <~>)
% Maximal formula depth : 28 ( 8 avg)
% Number of types : 2 ( 1 usr)
% Number of type conns : 192 ( 192 >; 0 *; 0 +; 0 <<)
% Number of symbols : 60 ( 58 usr; 5 con; 0-5 aty)
% Number of variables : 1011 ( 41 ^; 904 !; 13 ?;1011 :)
% ( 53 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2016-07-13 14:46:59.098
%------------------------------------------------------------------------------
%----Could-be-implicit typings (7)
thf(ty_t_Coinductive__List_Ollist,type,
coinductive_llist: $tType > $tType ).
thf(ty_t_Product__Type_Oprod,type,
product_prod: $tType > $tType > $tType ).
thf(ty_t_Sum__Type_Osum,type,
sum_sum: $tType > $tType > $tType ).
thf(ty_t_Option_Ooption,type,
option: $tType > $tType ).
thf(ty_t_Set_Oset,type,
set: $tType > $tType ).
thf(ty_t_itself,type,
itself: $tType > $tType ).
thf(ty_tf_node,type,
node: $tType ).
%----Explicit typings (52)
thf(sy_cl_HOL_Otype,type,
type:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Ominus,type,
minus:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Ouminus,type,
uminus:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Obot,type,
bot:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Oord,type,
ord:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Otop,type,
top:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Oorder,type,
order:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Ogroup__add,type,
group_add:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Finite__Set_Ofinite,type,
finite_finite:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Olinorder,type,
linorder:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Opreorder,type,
preorder:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Nat_Osemiring__char__0,type,
semiring_char_0:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Oorder__bot,type,
order_bot:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Oorder__top,type,
order_top:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Lattices_Oboolean__algebra,type,
boolean_algebra:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Oordered__ab__group__add,type,
ordered_ab_group_add:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Ocancel__ab__semigroup__add,type,
cancel146912293up_add:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Topological__Spaces_Operfect__space,type,
topolo890362671_space:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Topological__Spaces_Ouniform__space,type,
topolo47006728_space:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_c_Basic__BNFs_Ofsts,type,
basic_fsts:
!>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > ( set @ A ) ) ).
thf(sy_c_Basic__BNFs_Osnds,type,
basic_snds:
!>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > ( set @ B ) ) ).
thf(sy_c_Coinductive__List_Ogen__lset,type,
coinductive_gen_lset:
!>[A: $tType] : ( ( set @ A ) > ( coinductive_llist @ A ) > ( set @ A ) ) ).
thf(sy_c_Coinductive__List_Ollist_Olhd,type,
coinductive_lhd:
!>[A: $tType] : ( ( coinductive_llist @ A ) > A ) ).
thf(sy_c_Coinductive__List_Ollist_Olnull,type,
coinductive_lnull:
!>[A: $tType] : ( ( coinductive_llist @ A ) > $o ) ).
thf(sy_c_Coinductive__List_Ollist_Olset,type,
coinductive_lset:
!>[A: $tType] : ( ( coinductive_llist @ A ) > ( set @ A ) ) ).
thf(sy_c_Finite__Set_Ofinite,type,
finite_finite2:
!>[A: $tType] : ( ( set @ A ) > $o ) ).
thf(sy_c_Groups_Ominus__class_Ominus,type,
minus_minus:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Groups_Ouminus__class_Ouminus,type,
uminus_uminus:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Koenigslemma__Mirabelle__aepjeeakgn_Oconnected,type,
koenig793108494nected:
!>[Node: $tType] : ( ( Node > Node > $o ) > $o ) ).
thf(sy_c_Koenigslemma__Mirabelle__aepjeeakgn_Oreachable__via,type,
koenig317145564le_via:
!>[Node: $tType] : ( ( Node > Node > $o ) > ( set @ Node ) > Node > ( set @ Node ) ) ).
thf(sy_c_Koenigslemma__Mirabelle__aepjeeakgn_Oreachable__viap,type,
koenig1757754772e_viap:
!>[Node: $tType] : ( ( Node > Node > $o ) > ( set @ Node ) > Node > Node > $o ) ).
thf(sy_c_Orderings_Obot__class_Obot,type,
bot_bot:
!>[A: $tType] : A ).
thf(sy_c_Orderings_Oord__class_Oless__eq,type,
ord_less_eq:
!>[A: $tType] : ( A > A > $o ) ).
thf(sy_c_Orderings_Otop__class_Otop,type,
top_top:
!>[A: $tType] : A ).
thf(sy_c_Product__Type_OPair,type,
product_Pair:
!>[A: $tType,B: $tType] : ( A > B > ( product_prod @ A @ B ) ) ).
thf(sy_c_Product__Type_Ointernal__case__prod,type,
produc2004651681e_prod:
!>[A: $tType,B: $tType,C: $tType] : ( ( A > B > C ) > ( product_prod @ A @ B ) > C ) ).
thf(sy_c_Product__Type_Oold_Oprod_Orec__prod,type,
product_rec_prod:
!>[A: $tType,B: $tType,T: $tType] : ( ( A > B > T ) > ( product_prod @ A @ B ) > T ) ).
thf(sy_c_Pure_Otype,type,
type2:
!>[A: $tType] : ( itself @ A ) ).
thf(sy_c_Set_OCollect,type,
collect:
!>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
thf(sy_c_Set_Oinsert,type,
insert:
!>[A: $tType] : ( A > ( set @ A ) > ( set @ A ) ) ).
thf(sy_c_Set_Ois__empty,type,
is_empty:
!>[A: $tType] : ( ( set @ A ) > $o ) ).
thf(sy_c_Set_Ois__singleton,type,
is_singleton:
!>[A: $tType] : ( ( set @ A ) > $o ) ).
thf(sy_c_Set_Opairwise,type,
pairwise:
!>[A: $tType] : ( ( A > A > $o ) > ( set @ A ) > $o ) ).
thf(sy_c_Set_Othe__elem,type,
the_elem:
!>[A: $tType] : ( ( set @ A ) > A ) ).
thf(sy_c_Topological__Spaces_Ouniform__space__class_Ototally__bounded,type,
topolo406746546ounded:
!>[A: $tType] : ( ( set @ A ) > $o ) ).
thf(sy_c_member,type,
member:
!>[A: $tType] : ( A > ( set @ A ) > $o ) ).
thf(sy_v_f____,type,
f: ( product_prod @ node @ ( set @ node ) ) > ( coinductive_llist @ node ) ).
thf(sy_v_graph,type,
graph: node > node > $o ).
thf(sy_v_na____,type,
na: node ).
thf(sy_v_ns____,type,
ns: set @ node ).
thf(sy_v_nsa____,type,
nsa: set @ node ).
thf(sy_v_x____,type,
x: node ).
%----Relevant facts (254)
thf(fact_0__092_060open_062n_A_092_060notin_062_Ans_092_060close_062,axiom,
~ ( member @ node @ na @ nsa ) ).
% \<open>n \<notin> ns\<close>
thf(fact_1__092_060open_062finite_Ans_092_060close_062,axiom,
finite_finite2 @ node @ nsa ).
% \<open>finite ns\<close>
thf(fact_2__092_060open_062infinite_A_Ireachable__via_Agraph_A_I_N_Ainsert_An_Ans_J_An_J_092_060close_062,axiom,
~ ( finite_finite2 @ node @ ( koenig317145564le_via @ node @ graph @ ( uminus_uminus @ ( set @ node ) @ ( insert @ node @ na @ nsa ) ) @ na ) ) ).
% \<open>infinite (reachable_via graph (- insert n ns) n)\<close>
thf(fact_3__092_060open_062x_A_092_060in_062_Alset_A_If_A_In_M_Ans_J_J_092_060close_062,axiom,
member @ node @ x @ ( coinductive_lset @ node @ ( f @ ( product_Pair @ node @ ( set @ node ) @ na @ nsa ) ) ) ).
% \<open>x \<in> lset (f (n, ns))\<close>
thf(fact_4_ns__def,axiom,
( ns
= ( bot_bot @ ( set @ node ) ) ) ).
% ns_def
thf(fact_5_infinite,axiom,
~ ( finite_finite2 @ node @ ( top_top @ ( set @ node ) ) ) ).
% infinite
thf(fact_6_reachable__viap__reachable__via__eq,axiom,
! [Node: $tType] :
( ( koenig1757754772e_viap @ Node )
= ( ^ [Graph: Node > Node > $o,Ns: set @ Node,N: Node,X: Node] : ( member @ Node @ X @ ( koenig317145564le_via @ Node @ Graph @ Ns @ N ) ) ) ) ).
% reachable_viap_reachable_via_eq
thf(fact_7_connected,axiom,
koenig793108494nected @ node @ graph ).
% connected
thf(fact_8_finite__compl,axiom,
! [A: $tType,A2: set @ A] :
( ( finite_finite2 @ A @ A2 )
=> ( ( finite_finite2 @ A @ ( uminus_uminus @ ( set @ A ) @ A2 ) )
= ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) ) ) ) ).
% finite_compl
thf(fact_9_compl__bot__eq,axiom,
! [A: $tType] :
( ( boolean_algebra @ A @ ( type2 @ A ) )
=> ( ( uminus_uminus @ A @ ( bot_bot @ A ) )
= ( top_top @ A ) ) ) ).
% compl_bot_eq
thf(fact_10_compl__top__eq,axiom,
! [A: $tType] :
( ( boolean_algebra @ A @ ( type2 @ A ) )
=> ( ( uminus_uminus @ A @ ( top_top @ A ) )
= ( bot_bot @ A ) ) ) ).
% compl_top_eq
thf(fact_11_finite__insert,axiom,
! [A: $tType,A3: A,A2: set @ A] :
( ( finite_finite2 @ A @ ( insert @ A @ A3 @ A2 ) )
= ( finite_finite2 @ A @ A2 ) ) ).
% finite_insert
thf(fact_12_singletonI,axiom,
! [A: $tType,A3: A] : ( member @ A @ A3 @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% singletonI
thf(fact_13_finite__option__UNIV,axiom,
! [A: $tType] :
( ( finite_finite2 @ ( option @ A ) @ ( top_top @ ( set @ ( option @ A ) ) ) )
= ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) ) ) ).
% finite_option_UNIV
thf(fact_14_finite__Plus__UNIV__iff,axiom,
! [A: $tType,B: $tType] :
( ( finite_finite2 @ ( sum_sum @ A @ B ) @ ( top_top @ ( set @ ( sum_sum @ A @ B ) ) ) )
= ( ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) )
& ( finite_finite2 @ B @ ( top_top @ ( set @ B ) ) ) ) ) ).
% finite_Plus_UNIV_iff
thf(fact_15_Compl__UNIV__eq,axiom,
! [A: $tType] :
( ( uminus_uminus @ ( set @ A ) @ ( top_top @ ( set @ A ) ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% Compl_UNIV_eq
thf(fact_16_Compl__empty__eq,axiom,
! [A: $tType] :
( ( uminus_uminus @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) )
= ( top_top @ ( set @ A ) ) ) ).
% Compl_empty_eq
thf(fact_17_finite_Ocases,axiom,
! [A: $tType,A3: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ~ ! [A4: set @ A] :
( ? [A5: A] :
( A3
= ( insert @ A @ A5 @ A4 ) )
=> ~ ( finite_finite2 @ A @ A4 ) ) ) ) ).
% finite.cases
thf(fact_18_finite_Osimps,axiom,
! [A: $tType] :
( ( finite_finite2 @ A )
= ( ^ [A6: set @ A] :
( ( A6
= ( bot_bot @ ( set @ A ) ) )
| ? [A7: set @ A,B2: A] :
( ( A6
= ( insert @ A @ B2 @ A7 ) )
& ( finite_finite2 @ A @ A7 ) ) ) ) ) ).
% finite.simps
thf(fact_19_compl__eq__compl__iff,axiom,
! [A: $tType] :
( ( boolean_algebra @ A @ ( type2 @ A ) )
=> ! [X2: A,Y: A] :
( ( ( uminus_uminus @ A @ X2 )
= ( uminus_uminus @ A @ Y ) )
= ( X2 = Y ) ) ) ).
% compl_eq_compl_iff
thf(fact_20_double__compl,axiom,
! [A: $tType] :
( ( boolean_algebra @ A @ ( type2 @ A ) )
=> ! [X2: A] :
( ( uminus_uminus @ A @ ( uminus_uminus @ A @ X2 ) )
= X2 ) ) ).
% double_compl
thf(fact_21_uminus__apply,axiom,
! [B: $tType,A: $tType] :
( ( uminus @ B @ ( type2 @ B ) )
=> ( ( uminus_uminus @ ( A > B ) )
= ( ^ [A7: A > B,X: A] : ( uminus_uminus @ B @ ( A7 @ X ) ) ) ) ) ).
% uminus_apply
thf(fact_22_empty__Collect__eq,axiom,
! [A: $tType,P: A > $o] :
( ( ( bot_bot @ ( set @ A ) )
= ( collect @ A @ P ) )
= ( ! [X: A] :
~ ( P @ X ) ) ) ).
% empty_Collect_eq
thf(fact_23_Collect__empty__eq,axiom,
! [A: $tType,P: A > $o] :
( ( ( collect @ A @ P )
= ( bot_bot @ ( set @ A ) ) )
= ( ! [X: A] :
~ ( P @ X ) ) ) ).
% Collect_empty_eq
thf(fact_24_all__not__in__conv,axiom,
! [A: $tType,A2: set @ A] :
( ( ! [X: A] :
~ ( member @ A @ X @ A2 ) )
= ( A2
= ( bot_bot @ ( set @ A ) ) ) ) ).
% all_not_in_conv
thf(fact_25_empty__iff,axiom,
! [A: $tType,C2: A] :
~ ( member @ A @ C2 @ ( bot_bot @ ( set @ A ) ) ) ).
% empty_iff
thf(fact_26_UNIV__I,axiom,
! [A: $tType,X2: A] : ( member @ A @ X2 @ ( top_top @ ( set @ A ) ) ) ).
% UNIV_I
thf(fact_27_finite__code,axiom,
! [A: $tType] :
( ( finite_finite @ A @ ( type2 @ A ) )
=> ( ( finite_finite2 @ A )
= ( ^ [A7: set @ A] : $true ) ) ) ).
% finite_code
thf(fact_28_insert__absorb2,axiom,
! [A: $tType,X2: A,A2: set @ A] :
( ( insert @ A @ X2 @ ( insert @ A @ X2 @ A2 ) )
= ( insert @ A @ X2 @ A2 ) ) ).
% insert_absorb2
thf(fact_29_insert__iff,axiom,
! [A: $tType,A3: A,B3: A,A2: set @ A] :
( ( member @ A @ A3 @ ( insert @ A @ B3 @ A2 ) )
= ( ( A3 = B3 )
| ( member @ A @ A3 @ A2 ) ) ) ).
% insert_iff
thf(fact_30_insertCI,axiom,
! [A: $tType,A3: A,B4: set @ A,B3: A] :
( ( ~ ( member @ A @ A3 @ B4 )
=> ( A3 = B3 ) )
=> ( member @ A @ A3 @ ( insert @ A @ B3 @ B4 ) ) ) ).
% insertCI
thf(fact_31_Compl__eq__Compl__iff,axiom,
! [A: $tType,A2: set @ A,B4: set @ A] :
( ( ( uminus_uminus @ ( set @ A ) @ A2 )
= ( uminus_uminus @ ( set @ A ) @ B4 ) )
= ( A2 = B4 ) ) ).
% Compl_eq_Compl_iff
thf(fact_32_Compl__iff,axiom,
! [A: $tType,C2: A,A2: set @ A] :
( ( member @ A @ C2 @ ( uminus_uminus @ ( set @ A ) @ A2 ) )
= ( ~ ( member @ A @ C2 @ A2 ) ) ) ).
% Compl_iff
thf(fact_33_ComplI,axiom,
! [A: $tType,C2: A,A2: set @ A] :
( ~ ( member @ A @ C2 @ A2 )
=> ( member @ A @ C2 @ ( uminus_uminus @ ( set @ A ) @ A2 ) ) ) ).
% ComplI
thf(fact_34_bot__set__def,axiom,
! [A: $tType] :
( ( bot_bot @ ( set @ A ) )
= ( collect @ A @ ( bot_bot @ ( A > $o ) ) ) ) ).
% bot_set_def
thf(fact_35_top__set__def,axiom,
! [A: $tType] :
( ( top_top @ ( set @ A ) )
= ( collect @ A @ ( top_top @ ( A > $o ) ) ) ) ).
% top_set_def
thf(fact_36_fun__Compl__def,axiom,
! [B: $tType,A: $tType] :
( ( uminus @ B @ ( type2 @ B ) )
=> ( ( uminus_uminus @ ( A > B ) )
= ( ^ [A7: A > B,X: A] : ( uminus_uminus @ B @ ( A7 @ X ) ) ) ) ) ).
% fun_Compl_def
thf(fact_37_ex__in__conv,axiom,
! [A: $tType,A2: set @ A] :
( ( ? [X: A] : ( member @ A @ X @ A2 ) )
= ( A2
!= ( bot_bot @ ( set @ A ) ) ) ) ).
% ex_in_conv
thf(fact_38_equals0I,axiom,
! [A: $tType,A2: set @ A] :
( ! [Y2: A] :
~ ( member @ A @ Y2 @ A2 )
=> ( A2
= ( bot_bot @ ( set @ A ) ) ) ) ).
% equals0I
thf(fact_39_equals0D,axiom,
! [A: $tType,A2: set @ A,A3: A] :
( ( A2
= ( bot_bot @ ( set @ A ) ) )
=> ~ ( member @ A @ A3 @ A2 ) ) ).
% equals0D
thf(fact_40_emptyE,axiom,
! [A: $tType,A3: A] :
~ ( member @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ).
% emptyE
thf(fact_41_UNIV__witness,axiom,
! [A: $tType] :
? [X3: A] : ( member @ A @ X3 @ ( top_top @ ( set @ A ) ) ) ).
% UNIV_witness
thf(fact_42_UNIV__eq__I,axiom,
! [A: $tType,A2: set @ A] :
( ! [X3: A] : ( member @ A @ X3 @ A2 )
=> ( ( top_top @ ( set @ A ) )
= A2 ) ) ).
% UNIV_eq_I
thf(fact_43_finite,axiom,
! [A: $tType] :
( ( finite_finite @ A @ ( type2 @ A ) )
=> ! [A2: set @ A] : ( finite_finite2 @ A @ A2 ) ) ).
% finite
thf(fact_44_finite__set__choice,axiom,
! [B: $tType,A: $tType,A2: set @ A,P: A > B > $o] :
( ( finite_finite2 @ A @ A2 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ A2 )
=> ? [X1: B] : ( P @ X3 @ X1 ) )
=> ? [F: A > B] :
! [X4: A] :
( ( member @ A @ X4 @ A2 )
=> ( P @ X4 @ ( F @ X4 ) ) ) ) ) ).
% finite_set_choice
thf(fact_45_mem__Collect__eq,axiom,
! [A: $tType,A3: A,P: A > $o] :
( ( member @ A @ A3 @ ( collect @ A @ P ) )
= ( P @ A3 ) ) ).
% mem_Collect_eq
thf(fact_46_Collect__mem__eq,axiom,
! [A: $tType,A2: set @ A] :
( ( collect @ A
@ ^ [X: A] : ( member @ A @ X @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_47_Collect__cong,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ! [X3: A] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collect @ A @ P )
= ( collect @ A @ Q ) ) ) ).
% Collect_cong
thf(fact_48_ext,axiom,
! [B: $tType,A: $tType,F2: A > B,G: A > B] :
( ! [X3: A] :
( ( F2 @ X3 )
= ( G @ X3 ) )
=> ( F2 = G ) ) ).
% ext
thf(fact_49_mk__disjoint__insert,axiom,
! [A: $tType,A3: A,A2: set @ A] :
( ( member @ A @ A3 @ A2 )
=> ? [B5: set @ A] :
( ( A2
= ( insert @ A @ A3 @ B5 ) )
& ~ ( member @ A @ A3 @ B5 ) ) ) ).
% mk_disjoint_insert
thf(fact_50_insert__commute,axiom,
! [A: $tType,X2: A,Y: A,A2: set @ A] :
( ( insert @ A @ X2 @ ( insert @ A @ Y @ A2 ) )
= ( insert @ A @ Y @ ( insert @ A @ X2 @ A2 ) ) ) ).
% insert_commute
thf(fact_51_insert__eq__iff,axiom,
! [A: $tType,A3: A,A2: set @ A,B3: A,B4: set @ A] :
( ~ ( member @ A @ A3 @ A2 )
=> ( ~ ( member @ A @ B3 @ B4 )
=> ( ( ( insert @ A @ A3 @ A2 )
= ( insert @ A @ B3 @ B4 ) )
= ( ( ( A3 = B3 )
=> ( A2 = B4 ) )
& ( ( A3 != B3 )
=> ? [C3: set @ A] :
( ( A2
= ( insert @ A @ B3 @ C3 ) )
& ~ ( member @ A @ B3 @ C3 )
& ( B4
= ( insert @ A @ A3 @ C3 ) )
& ~ ( member @ A @ A3 @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_52_insert__absorb,axiom,
! [A: $tType,A3: A,A2: set @ A] :
( ( member @ A @ A3 @ A2 )
=> ( ( insert @ A @ A3 @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_53_insert__ident,axiom,
! [A: $tType,X2: A,A2: set @ A,B4: set @ A] :
( ~ ( member @ A @ X2 @ A2 )
=> ( ~ ( member @ A @ X2 @ B4 )
=> ( ( ( insert @ A @ X2 @ A2 )
= ( insert @ A @ X2 @ B4 ) )
= ( A2 = B4 ) ) ) ) ).
% insert_ident
thf(fact_54_Set_Oset__insert,axiom,
! [A: $tType,X2: A,A2: set @ A] :
( ( member @ A @ X2 @ A2 )
=> ~ ! [B5: set @ A] :
( ( A2
= ( insert @ A @ X2 @ B5 ) )
=> ( member @ A @ X2 @ B5 ) ) ) ).
% Set.set_insert
thf(fact_55_insertI2,axiom,
! [A: $tType,A3: A,B4: set @ A,B3: A] :
( ( member @ A @ A3 @ B4 )
=> ( member @ A @ A3 @ ( insert @ A @ B3 @ B4 ) ) ) ).
% insertI2
thf(fact_56_insertI1,axiom,
! [A: $tType,A3: A,B4: set @ A] : ( member @ A @ A3 @ ( insert @ A @ A3 @ B4 ) ) ).
% insertI1
thf(fact_57_insertE,axiom,
! [A: $tType,A3: A,B3: A,A2: set @ A] :
( ( member @ A @ A3 @ ( insert @ A @ B3 @ A2 ) )
=> ( ( A3 != B3 )
=> ( member @ A @ A3 @ A2 ) ) ) ).
% insertE
thf(fact_58_double__complement,axiom,
! [A: $tType,A2: set @ A] :
( ( uminus_uminus @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A2 ) )
= A2 ) ).
% double_complement
thf(fact_59_ComplD,axiom,
! [A: $tType,C2: A,A2: set @ A] :
( ( member @ A @ C2 @ ( uminus_uminus @ ( set @ A ) @ A2 ) )
=> ~ ( member @ A @ C2 @ A2 ) ) ).
% ComplD
thf(fact_60_empty__not__UNIV,axiom,
! [A: $tType] :
( ( bot_bot @ ( set @ A ) )
!= ( top_top @ ( set @ A ) ) ) ).
% empty_not_UNIV
thf(fact_61_infinite__imp__nonempty,axiom,
! [A: $tType,S: set @ A] :
( ~ ( finite_finite2 @ A @ S )
=> ( S
!= ( bot_bot @ ( set @ A ) ) ) ) ).
% infinite_imp_nonempty
thf(fact_62_finite_OemptyI,axiom,
! [A: $tType] : ( finite_finite2 @ A @ ( bot_bot @ ( set @ A ) ) ) ).
% finite.emptyI
thf(fact_63_finite__UNIV,axiom,
! [A: $tType] :
( ( finite_finite @ A @ ( type2 @ A ) )
=> ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) ) ) ).
% finite_UNIV
thf(fact_64_infinite__UNIV__char__0,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A @ ( type2 @ A ) )
=> ~ ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) ) ) ).
% infinite_UNIV_char_0
thf(fact_65_finite__fun__UNIVD2,axiom,
! [A: $tType,B: $tType] :
( ( finite_finite2 @ ( A > B ) @ ( top_top @ ( set @ ( A > B ) ) ) )
=> ( finite_finite2 @ B @ ( top_top @ ( set @ B ) ) ) ) ).
% finite_fun_UNIVD2
thf(fact_66_finite__Prod__UNIV,axiom,
! [B: $tType,A: $tType] :
( ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) )
=> ( ( finite_finite2 @ B @ ( top_top @ ( set @ B ) ) )
=> ( finite_finite2 @ ( product_prod @ A @ B ) @ ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) ) ) ) ).
% finite_Prod_UNIV
thf(fact_67_ex__new__if__finite,axiom,
! [A: $tType,A2: set @ A] :
( ~ ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) )
=> ( ( finite_finite2 @ A @ A2 )
=> ? [A5: A] :
~ ( member @ A @ A5 @ A2 ) ) ) ).
% ex_new_if_finite
thf(fact_68_finite__prod,axiom,
! [A: $tType,B: $tType] :
( ( finite_finite2 @ ( product_prod @ A @ B ) @ ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) )
= ( ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) )
& ( finite_finite2 @ B @ ( top_top @ ( set @ B ) ) ) ) ) ).
% finite_prod
thf(fact_69_Finite__Set_Ofinite__set,axiom,
! [A: $tType] :
( ( finite_finite2 @ ( set @ A ) @ ( top_top @ ( set @ ( set @ A ) ) ) )
= ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) ) ) ).
% Finite_Set.finite_set
thf(fact_70_singleton__inject,axiom,
! [A: $tType,A3: A,B3: A] :
( ( ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) )
= ( insert @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) )
=> ( A3 = B3 ) ) ).
% singleton_inject
thf(fact_71_insert__not__empty,axiom,
! [A: $tType,A3: A,A2: set @ A] :
( ( insert @ A @ A3 @ A2 )
!= ( bot_bot @ ( set @ A ) ) ) ).
% insert_not_empty
thf(fact_72_doubleton__eq__iff,axiom,
! [A: $tType,A3: A,B3: A,C2: A,D: A] :
( ( ( insert @ A @ A3 @ ( insert @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) )
= ( insert @ A @ C2 @ ( insert @ A @ D @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( ( ( A3 = C2 )
& ( B3 = D ) )
| ( ( A3 = D )
& ( B3 = C2 ) ) ) ) ).
% doubleton_eq_iff
thf(fact_73_singleton__iff,axiom,
! [A: $tType,B3: A,A3: A] :
( ( member @ A @ B3 @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) )
= ( B3 = A3 ) ) ).
% singleton_iff
thf(fact_74_singletonD,axiom,
! [A: $tType,B3: A,A3: A] :
( ( member @ A @ B3 @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) )
=> ( B3 = A3 ) ) ).
% singletonD
thf(fact_75_insert__UNIV,axiom,
! [A: $tType,X2: A] :
( ( insert @ A @ X2 @ ( top_top @ ( set @ A ) ) )
= ( top_top @ ( set @ A ) ) ) ).
% insert_UNIV
thf(fact_76_finite_OinsertI,axiom,
! [A: $tType,A2: set @ A,A3: A] :
( ( finite_finite2 @ A @ A2 )
=> ( finite_finite2 @ A @ ( insert @ A @ A3 @ A2 ) ) ) ).
% finite.insertI
thf(fact_77_infinite__finite__induct,axiom,
! [A: $tType,P: ( set @ A ) > $o,A2: set @ A] :
( ! [A4: set @ A] :
( ~ ( finite_finite2 @ A @ A4 )
=> ( P @ A4 ) )
=> ( ( P @ ( bot_bot @ ( set @ A ) ) )
=> ( ! [X3: A,F3: set @ A] :
( ( finite_finite2 @ A @ F3 )
=> ( ~ ( member @ A @ X3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert @ A @ X3 @ F3 ) ) ) ) )
=> ( P @ A2 ) ) ) ) ).
% infinite_finite_induct
thf(fact_78_finite__ne__induct,axiom,
! [A: $tType,F4: set @ A,P: ( set @ A ) > $o] :
( ( finite_finite2 @ A @ F4 )
=> ( ( F4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X3: A] : ( P @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
=> ( ! [X3: A,F3: set @ A] :
( ( finite_finite2 @ A @ F3 )
=> ( ( F3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ~ ( member @ A @ X3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert @ A @ X3 @ F3 ) ) ) ) ) )
=> ( P @ F4 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_79_finite_Oinducts,axiom,
! [A: $tType,X2: set @ A,P: ( set @ A ) > $o] :
( ( finite_finite2 @ A @ X2 )
=> ( ( P @ ( bot_bot @ ( set @ A ) ) )
=> ( ! [A4: set @ A,A5: A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( P @ A4 )
=> ( P @ ( insert @ A @ A5 @ A4 ) ) ) )
=> ( P @ X2 ) ) ) ) ).
% finite.inducts
thf(fact_80_finite__induct,axiom,
! [A: $tType,F4: set @ A,P: ( set @ A ) > $o] :
( ( finite_finite2 @ A @ F4 )
=> ( ( P @ ( bot_bot @ ( set @ A ) ) )
=> ( ! [X3: A,F3: set @ A] :
( ( finite_finite2 @ A @ F3 )
=> ( ~ ( member @ A @ X3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert @ A @ X3 @ F3 ) ) ) ) )
=> ( P @ F4 ) ) ) ) ).
% finite_induct
thf(fact_81_Topological__Spaces_OUNIV__not__singleton,axiom,
! [A: $tType] :
( ( topolo890362671_space @ A @ ( type2 @ A ) )
=> ! [X2: A] :
( ( top_top @ ( set @ A ) )
!= ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Topological_Spaces.UNIV_not_singleton
thf(fact_82_f__simps_I1_J,axiom,
! [Na: node,Nsa: set @ node] :
~ ( coinductive_lnull @ node @ ( f @ ( product_Pair @ node @ ( set @ node ) @ Na @ Nsa ) ) ) ).
% f_simps(1)
thf(fact_83_the__elem__eq,axiom,
! [A: $tType,X2: A] :
( ( the_elem @ A @ ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) )
= X2 ) ).
% the_elem_eq
thf(fact_84_iso__tuple__UNIV__I,axiom,
! [A: $tType,X2: A] : ( member @ A @ X2 @ ( top_top @ ( set @ A ) ) ) ).
% iso_tuple_UNIV_I
thf(fact_85_neg__equal__iff__equal,axiom,
! [A: $tType] :
( ( group_add @ A @ ( type2 @ A ) )
=> ! [A3: A,B3: A] :
( ( ( uminus_uminus @ A @ A3 )
= ( uminus_uminus @ A @ B3 ) )
= ( A3 = B3 ) ) ) ).
% neg_equal_iff_equal
thf(fact_86_add_Oinverse__inverse,axiom,
! [A: $tType] :
( ( group_add @ A @ ( type2 @ A ) )
=> ! [A3: A] :
( ( uminus_uminus @ A @ ( uminus_uminus @ A @ A3 ) )
= A3 ) ) ).
% add.inverse_inverse
thf(fact_87_top__apply,axiom,
! [C: $tType,D2: $tType] :
( ( top @ C @ ( type2 @ C ) )
=> ( ( top_top @ ( D2 > C ) )
= ( ^ [X: D2] : ( top_top @ C ) ) ) ) ).
% top_apply
thf(fact_88_bot__apply,axiom,
! [C: $tType,D2: $tType] :
( ( bot @ C @ ( type2 @ C ) )
=> ( ( bot_bot @ ( D2 > C ) )
= ( ^ [X: D2] : ( bot_bot @ C ) ) ) ) ).
% bot_apply
thf(fact_89_old_Oprod_Oinject,axiom,
! [A: $tType,B: $tType,A3: A,B3: B,A8: A,B6: B] :
( ( ( product_Pair @ A @ B @ A3 @ B3 )
= ( product_Pair @ A @ B @ A8 @ B6 ) )
= ( ( A3 = A8 )
& ( B3 = B6 ) ) ) ).
% old.prod.inject
thf(fact_90_prod_Oinject,axiom,
! [A: $tType,B: $tType,X12: A,X22: B,Y1: A,Y22: B] :
( ( ( product_Pair @ A @ B @ X12 @ X22 )
= ( product_Pair @ A @ B @ Y1 @ Y22 ) )
= ( ( X12 = Y1 )
& ( X22 = Y22 ) ) ) ).
% prod.inject
thf(fact_91_top1I,axiom,
! [A: $tType,X2: A] : ( top_top @ ( A > $o ) @ X2 ) ).
% top1I
thf(fact_92_surj__pair,axiom,
! [A: $tType,B: $tType,P2: product_prod @ A @ B] :
? [X3: A,Y2: B] :
( P2
= ( product_Pair @ A @ B @ X3 @ Y2 ) ) ).
% surj_pair
thf(fact_93_prod__cases,axiom,
! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o,P2: product_prod @ A @ B] :
( ! [A5: A,B7: B] : ( P @ ( product_Pair @ A @ B @ A5 @ B7 ) )
=> ( P @ P2 ) ) ).
% prod_cases
thf(fact_94_Pair__inject,axiom,
! [A: $tType,B: $tType,A3: A,B3: B,A8: A,B6: B] :
( ( ( product_Pair @ A @ B @ A3 @ B3 )
= ( product_Pair @ A @ B @ A8 @ B6 ) )
=> ~ ( ( A3 = A8 )
=> ( B3 != B6 ) ) ) ).
% Pair_inject
thf(fact_95_prod__cases3,axiom,
! [A: $tType,B: $tType,C: $tType,Y: product_prod @ A @ ( product_prod @ B @ C )] :
~ ! [A5: A,B7: B,C4: C] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ C ) @ A5 @ ( product_Pair @ B @ C @ B7 @ C4 ) ) ) ).
% prod_cases3
thf(fact_96_prod__cases4,axiom,
! [A: $tType,B: $tType,C: $tType,D2: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D2 ) )] :
~ ! [A5: A,B7: B,C4: C,D3: D2] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D2 ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ D2 ) @ B7 @ ( product_Pair @ C @ D2 @ C4 @ D3 ) ) ) ) ).
% prod_cases4
thf(fact_97_prod__cases5,axiom,
! [A: $tType,B: $tType,C: $tType,D2: $tType,E: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ E ) ) )] :
~ ! [A5: A,B7: B,C4: C,D3: D2,E2: E] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ E ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D2 @ E ) ) @ B7 @ ( product_Pair @ C @ ( product_prod @ D2 @ E ) @ C4 @ ( product_Pair @ D2 @ E @ D3 @ E2 ) ) ) ) ) ).
% prod_cases5
thf(fact_98_prod__cases6,axiom,
! [A: $tType,B: $tType,C: $tType,D2: $tType,E: $tType,F5: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E @ F5 ) ) ) )] :
~ ! [A5: A,B7: B,C4: C,D3: D2,E2: E,F: F5] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E @ F5 ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E @ F5 ) ) ) @ B7 @ ( product_Pair @ C @ ( product_prod @ D2 @ ( product_prod @ E @ F5 ) ) @ C4 @ ( product_Pair @ D2 @ ( product_prod @ E @ F5 ) @ D3 @ ( product_Pair @ E @ F5 @ E2 @ F ) ) ) ) ) ) ).
% prod_cases6
thf(fact_99_prod__cases7,axiom,
! [A: $tType,B: $tType,C: $tType,D2: $tType,E: $tType,F5: $tType,G2: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E @ ( product_prod @ F5 @ G2 ) ) ) ) )] :
~ ! [A5: A,B7: B,C4: C,D3: D2,E2: E,F: F5,G3: G2] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E @ ( product_prod @ F5 @ G2 ) ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E @ ( product_prod @ F5 @ G2 ) ) ) ) @ B7 @ ( product_Pair @ C @ ( product_prod @ D2 @ ( product_prod @ E @ ( product_prod @ F5 @ G2 ) ) ) @ C4 @ ( product_Pair @ D2 @ ( product_prod @ E @ ( product_prod @ F5 @ G2 ) ) @ D3 @ ( product_Pair @ E @ ( product_prod @ F5 @ G2 ) @ E2 @ ( product_Pair @ F5 @ G2 @ F @ G3 ) ) ) ) ) ) ) ).
% prod_cases7
thf(fact_100_prod__induct3,axiom,
! [C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ C ) ) > $o,X2: product_prod @ A @ ( product_prod @ B @ C )] :
( ! [A5: A,B7: B,C4: C] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ C ) @ A5 @ ( product_Pair @ B @ C @ B7 @ C4 ) ) )
=> ( P @ X2 ) ) ).
% prod_induct3
thf(fact_101_prod__induct4,axiom,
! [D2: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D2 ) ) ) > $o,X2: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D2 ) )] :
( ! [A5: A,B7: B,C4: C,D3: D2] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D2 ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ D2 ) @ B7 @ ( product_Pair @ C @ D2 @ C4 @ D3 ) ) ) )
=> ( P @ X2 ) ) ).
% prod_induct4
thf(fact_102_prod__induct5,axiom,
! [E: $tType,D2: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ E ) ) ) ) > $o,X2: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ E ) ) )] :
( ! [A5: A,B7: B,C4: C,D3: D2,E2: E] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ E ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D2 @ E ) ) @ B7 @ ( product_Pair @ C @ ( product_prod @ D2 @ E ) @ C4 @ ( product_Pair @ D2 @ E @ D3 @ E2 ) ) ) ) )
=> ( P @ X2 ) ) ).
% prod_induct5
thf(fact_103_prod__induct6,axiom,
! [F5: $tType,E: $tType,D2: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E @ F5 ) ) ) ) ) > $o,X2: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E @ F5 ) ) ) )] :
( ! [A5: A,B7: B,C4: C,D3: D2,E2: E,F: F5] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E @ F5 ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E @ F5 ) ) ) @ B7 @ ( product_Pair @ C @ ( product_prod @ D2 @ ( product_prod @ E @ F5 ) ) @ C4 @ ( product_Pair @ D2 @ ( product_prod @ E @ F5 ) @ D3 @ ( product_Pair @ E @ F5 @ E2 @ F ) ) ) ) ) )
=> ( P @ X2 ) ) ).
% prod_induct6
thf(fact_104_prod__induct7,axiom,
! [G2: $tType,F5: $tType,E: $tType,D2: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E @ ( product_prod @ F5 @ G2 ) ) ) ) ) ) > $o,X2: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E @ ( product_prod @ F5 @ G2 ) ) ) ) )] :
( ! [A5: A,B7: B,C4: C,D3: D2,E2: E,F: F5,G3: G2] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E @ ( product_prod @ F5 @ G2 ) ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E @ ( product_prod @ F5 @ G2 ) ) ) ) @ B7 @ ( product_Pair @ C @ ( product_prod @ D2 @ ( product_prod @ E @ ( product_prod @ F5 @ G2 ) ) ) @ C4 @ ( product_Pair @ D2 @ ( product_prod @ E @ ( product_prod @ F5 @ G2 ) ) @ D3 @ ( product_Pair @ E @ ( product_prod @ F5 @ G2 ) @ E2 @ ( product_Pair @ F5 @ G2 @ F @ G3 ) ) ) ) ) ) )
=> ( P @ X2 ) ) ).
% prod_induct7
thf(fact_105_old_Oprod_Oexhaust,axiom,
! [A: $tType,B: $tType,Y: product_prod @ A @ B] :
~ ! [A5: A,B7: B] :
( Y
!= ( product_Pair @ A @ B @ A5 @ B7 ) ) ).
% old.prod.exhaust
thf(fact_106_old_Oprod_Oinducts,axiom,
! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o,Prod: product_prod @ A @ B] :
( ! [A5: A,B7: B] : ( P @ ( product_Pair @ A @ B @ A5 @ B7 ) )
=> ( P @ Prod ) ) ).
% old.prod.inducts
thf(fact_107_bot__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( bot @ B @ ( type2 @ B ) )
=> ( ( bot_bot @ ( A > B ) )
= ( ^ [X: A] : ( bot_bot @ B ) ) ) ) ).
% bot_fun_def
thf(fact_108_equation__minus__iff,axiom,
! [A: $tType] :
( ( group_add @ A @ ( type2 @ A ) )
=> ! [A3: A,B3: A] :
( ( A3
= ( uminus_uminus @ A @ B3 ) )
= ( B3
= ( uminus_uminus @ A @ A3 ) ) ) ) ).
% equation_minus_iff
thf(fact_109_minus__equation__iff,axiom,
! [A: $tType] :
( ( group_add @ A @ ( type2 @ A ) )
=> ! [A3: A,B3: A] :
( ( ( uminus_uminus @ A @ A3 )
= B3 )
= ( ( uminus_uminus @ A @ B3 )
= A3 ) ) ) ).
% minus_equation_iff
thf(fact_110_lset__eq__empty,axiom,
! [A: $tType,Xs: coinductive_llist @ A] :
( ( ( coinductive_lset @ A @ Xs )
= ( bot_bot @ ( set @ A ) ) )
= ( coinductive_lnull @ A @ Xs ) ) ).
% lset_eq_empty
thf(fact_111_old_Oprod_Orec,axiom,
! [A: $tType,T: $tType,B: $tType,F1: A > B > T,A3: A,B3: B] :
( ( product_rec_prod @ A @ B @ T @ F1 @ ( product_Pair @ A @ B @ A3 @ B3 ) )
= ( F1 @ A3 @ B3 ) ) ).
% old.prod.rec
thf(fact_112_lset__lnull,axiom,
! [A: $tType,Xs: coinductive_llist @ A] :
( ( coinductive_lnull @ A @ Xs )
=> ( ( coinductive_lset @ A @ Xs )
= ( bot_bot @ ( set @ A ) ) ) ) ).
% lset_lnull
thf(fact_113_top__empty__eq,axiom,
! [A: $tType] :
( ( top_top @ ( A > $o ) )
= ( ^ [X: A] : ( member @ A @ X @ ( top_top @ ( set @ A ) ) ) ) ) ).
% top_empty_eq
thf(fact_114_bot__empty__eq,axiom,
! [A: $tType] :
( ( bot_bot @ ( A > $o ) )
= ( ^ [X: A] : ( member @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% bot_empty_eq
thf(fact_115_Collect__empty__eq__bot,axiom,
! [A: $tType,P: A > $o] :
( ( ( collect @ A @ P )
= ( bot_bot @ ( set @ A ) ) )
= ( P
= ( bot_bot @ ( A > $o ) ) ) ) ).
% Collect_empty_eq_bot
thf(fact_116_lappend_Oexhaust,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( ( coinductive_lnull @ A @ Xs )
=> ~ ( coinductive_lnull @ A @ Ys ) )
=> ( ~ ( coinductive_lnull @ A @ Xs )
| ~ ( coinductive_lnull @ A @ Ys ) ) ) ).
% lappend.exhaust
thf(fact_117_lzip_Oexhaust,axiom,
! [A: $tType,B: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ B] :
( ~ ( ( coinductive_lnull @ A @ Xs )
| ( coinductive_lnull @ B @ Ys ) )
=> ~ ( ~ ( coinductive_lnull @ A @ Xs )
=> ( coinductive_lnull @ B @ Ys ) ) ) ).
% lzip.exhaust
thf(fact_118_is__singleton__the__elem,axiom,
! [A: $tType] :
( ( is_singleton @ A )
= ( ^ [A7: set @ A] :
( A7
= ( insert @ A @ ( the_elem @ A @ A7 ) @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% is_singleton_the_elem
thf(fact_119_is__singletonI,axiom,
! [A: $tType,X2: A] : ( is_singleton @ A @ ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% is_singletonI
thf(fact_120_lset__code,axiom,
! [A: $tType] :
( ( coinductive_lset @ A )
= ( coinductive_gen_lset @ A @ ( bot_bot @ ( set @ A ) ) ) ) ).
% lset_code
thf(fact_121_top__conj_I1_J,axiom,
! [A: $tType,X2: A,P: $o] :
( ( ( top_top @ ( A > $o ) @ X2 )
& P )
= P ) ).
% top_conj(1)
thf(fact_122_is__singletonI_H,axiom,
! [A: $tType,A2: set @ A] :
( ( A2
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X3: A,Y2: A] :
( ( member @ A @ X3 @ A2 )
=> ( ( member @ A @ Y2 @ A2 )
=> ( X3 = Y2 ) ) )
=> ( is_singleton @ A @ A2 ) ) ) ).
% is_singletonI'
thf(fact_123_is__singletonE,axiom,
! [A: $tType,A2: set @ A] :
( ( is_singleton @ A @ A2 )
=> ~ ! [X3: A] :
( A2
!= ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% is_singletonE
thf(fact_124_is__singleton__def,axiom,
! [A: $tType] :
( ( is_singleton @ A )
= ( ^ [A7: set @ A] :
? [X: A] :
( A7
= ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% is_singleton_def
thf(fact_125_top__conj_I2_J,axiom,
! [A: $tType,P: $o,X2: A] :
( ( P
& ( top_top @ ( A > $o ) @ X2 ) )
= P ) ).
% top_conj(2)
thf(fact_126_prod__set__simps_I2_J,axiom,
! [A: $tType,B: $tType,X2: A,Y: B] :
( ( basic_snds @ A @ B @ ( product_Pair @ A @ B @ X2 @ Y ) )
= ( insert @ B @ Y @ ( bot_bot @ ( set @ B ) ) ) ) ).
% prod_set_simps(2)
thf(fact_127_prod__set__simps_I1_J,axiom,
! [B: $tType,A: $tType,X2: A,Y: B] :
( ( basic_fsts @ A @ B @ ( product_Pair @ A @ B @ X2 @ Y ) )
= ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% prod_set_simps(1)
thf(fact_128_Set_Ois__empty__def,axiom,
! [A: $tType] :
( ( is_empty @ A )
= ( ^ [A7: set @ A] :
( A7
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Set.is_empty_def
thf(fact_129_internal__case__prod__conv,axiom,
! [B: $tType,A: $tType,C: $tType,C2: B > C > A,A3: B,B3: C] :
( ( produc2004651681e_prod @ B @ C @ A @ C2 @ ( product_Pair @ B @ C @ A3 @ B3 ) )
= ( C2 @ A3 @ B3 ) ) ).
% internal_case_prod_conv
thf(fact_130_f__simps_I2_J,axiom,
! [Na: node,Nsa: set @ node] :
( ( coinductive_lhd @ node @ ( f @ ( product_Pair @ node @ ( set @ node ) @ Na @ Nsa ) ) )
= Na ) ).
% f_simps(2)
thf(fact_131_subset__Compl__singleton,axiom,
! [A: $tType,A2: set @ A,B3: A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ ( uminus_uminus @ ( set @ A ) @ ( insert @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( ~ ( member @ A @ B3 @ A2 ) ) ) ).
% subset_Compl_singleton
thf(fact_132_totally__bounded__empty,axiom,
! [A: $tType] :
( ( topolo47006728_space @ A @ ( type2 @ A ) )
=> ( topolo406746546ounded @ A @ ( bot_bot @ ( set @ A ) ) ) ) ).
% totally_bounded_empty
thf(fact_133_pairwise__singleton,axiom,
! [A: $tType,P: A > A > $o,A2: A] : ( pairwise @ A @ P @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% pairwise_singleton
thf(fact_134_order__refl,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X2: A] : ( ord_less_eq @ A @ X2 @ X2 ) ) ).
% order_refl
thf(fact_135_subset__antisym,axiom,
! [A: $tType,A2: set @ A,B4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B4 )
=> ( ( ord_less_eq @ ( set @ A ) @ B4 @ A2 )
=> ( A2 = B4 ) ) ) ).
% subset_antisym
thf(fact_136_subsetI,axiom,
! [A: $tType,A2: set @ A,B4: set @ A] :
( ! [X3: A] :
( ( member @ A @ X3 @ A2 )
=> ( member @ A @ X3 @ B4 ) )
=> ( ord_less_eq @ ( set @ A ) @ A2 @ B4 ) ) ).
% subsetI
thf(fact_137_compl__le__compl__iff,axiom,
! [A: $tType] :
( ( boolean_algebra @ A @ ( type2 @ A ) )
=> ! [X2: A,Y: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ X2 ) @ ( uminus_uminus @ A @ Y ) )
= ( ord_less_eq @ A @ Y @ X2 ) ) ) ).
% compl_le_compl_iff
thf(fact_138_neg__le__iff__le,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
=> ! [B3: A,A3: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ B3 ) @ ( uminus_uminus @ A @ A3 ) )
= ( ord_less_eq @ A @ A3 @ B3 ) ) ) ).
% neg_le_iff_le
thf(fact_139_empty__subsetI,axiom,
! [A: $tType,A2: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ A2 ) ).
% empty_subsetI
thf(fact_140_subset__empty,axiom,
! [A: $tType,A2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ ( bot_bot @ ( set @ A ) ) )
= ( A2
= ( bot_bot @ ( set @ A ) ) ) ) ).
% subset_empty
thf(fact_141_insert__subset,axiom,
! [A: $tType,X2: A,A2: set @ A,B4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ X2 @ A2 ) @ B4 )
= ( ( member @ A @ X2 @ B4 )
& ( ord_less_eq @ ( set @ A ) @ A2 @ B4 ) ) ) ).
% insert_subset
thf(fact_142_Compl__anti__mono,axiom,
! [A: $tType,A2: set @ A,B4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B4 )
=> ( ord_less_eq @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ B4 ) @ ( uminus_uminus @ ( set @ A ) @ A2 ) ) ) ).
% Compl_anti_mono
thf(fact_143_Compl__subset__Compl__iff,axiom,
! [A: $tType,A2: set @ A,B4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A2 ) @ ( uminus_uminus @ ( set @ A ) @ B4 ) )
= ( ord_less_eq @ ( set @ A ) @ B4 @ A2 ) ) ).
% Compl_subset_Compl_iff
thf(fact_144_singleton__insert__inj__eq,axiom,
! [A: $tType,B3: A,A3: A,A2: set @ A] :
( ( ( insert @ A @ B3 @ ( bot_bot @ ( set @ A ) ) )
= ( insert @ A @ A3 @ A2 ) )
= ( ( A3 = B3 )
& ( ord_less_eq @ ( set @ A ) @ A2 @ ( insert @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_145_singleton__insert__inj__eq_H,axiom,
! [A: $tType,A3: A,A2: set @ A,B3: A] :
( ( ( insert @ A @ A3 @ A2 )
= ( insert @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) )
= ( ( A3 = B3 )
& ( ord_less_eq @ ( set @ A ) @ A2 @ ( insert @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_146_ltakeWhile_Oexhaust,axiom,
! [A: $tType,Xs: coinductive_llist @ A,P: A > $o] :
( ~ ( ( coinductive_lnull @ A @ Xs )
| ~ ( P @ ( coinductive_lhd @ A @ Xs ) ) )
=> ~ ( ~ ( coinductive_lnull @ A @ Xs )
=> ~ ( P @ ( coinductive_lhd @ A @ Xs ) ) ) ) ).
% ltakeWhile.exhaust
thf(fact_147_subrelI,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ B ),S2: set @ ( product_prod @ A @ B )] :
( ! [X3: A,Y2: B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y2 ) @ R )
=> ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y2 ) @ S2 ) )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R @ S2 ) ) ).
% subrelI
thf(fact_148_wlog__linorder__le,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [P: A > A > $o,B3: A,A3: A] :
( ! [A5: A,B7: A] :
( ( ord_less_eq @ A @ A5 @ B7 )
=> ( P @ A5 @ B7 ) )
=> ( ( ( P @ B3 @ A3 )
=> ( P @ A3 @ B3 ) )
=> ( P @ A3 @ B3 ) ) ) ) ).
% wlog_linorder_le
thf(fact_149_bot_Oextremum__uniqueI,axiom,
! [A: $tType] :
( ( order_bot @ A @ ( type2 @ A ) )
=> ! [A3: A] :
( ( ord_less_eq @ A @ A3 @ ( bot_bot @ A ) )
=> ( A3
= ( bot_bot @ A ) ) ) ) ).
% bot.extremum_uniqueI
thf(fact_150_bot_Oextremum__unique,axiom,
! [A: $tType] :
( ( order_bot @ A @ ( type2 @ A ) )
=> ! [A3: A] :
( ( ord_less_eq @ A @ A3 @ ( bot_bot @ A ) )
= ( A3
= ( bot_bot @ A ) ) ) ) ).
% bot.extremum_unique
thf(fact_151_bot_Oextremum,axiom,
! [A: $tType] :
( ( order_bot @ A @ ( type2 @ A ) )
=> ! [A3: A] : ( ord_less_eq @ A @ ( bot_bot @ A ) @ A3 ) ) ).
% bot.extremum
thf(fact_152_top_Oextremum__uniqueI,axiom,
! [A: $tType] :
( ( order_top @ A @ ( type2 @ A ) )
=> ! [A3: A] :
( ( ord_less_eq @ A @ ( top_top @ A ) @ A3 )
=> ( A3
= ( top_top @ A ) ) ) ) ).
% top.extremum_uniqueI
thf(fact_153_top_Oextremum__unique,axiom,
! [A: $tType] :
( ( order_top @ A @ ( type2 @ A ) )
=> ! [A3: A] :
( ( ord_less_eq @ A @ ( top_top @ A ) @ A3 )
= ( A3
= ( top_top @ A ) ) ) ) ).
% top.extremum_unique
thf(fact_154_top__greatest,axiom,
! [A: $tType] :
( ( order_top @ A @ ( type2 @ A ) )
=> ! [A3: A] : ( ord_less_eq @ A @ A3 @ ( top_top @ A ) ) ) ).
% top_greatest
thf(fact_155_compl__le__swap2,axiom,
! [A: $tType] :
( ( boolean_algebra @ A @ ( type2 @ A ) )
=> ! [Y: A,X2: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ Y ) @ X2 )
=> ( ord_less_eq @ A @ ( uminus_uminus @ A @ X2 ) @ Y ) ) ) ).
% compl_le_swap2
thf(fact_156_compl__le__swap1,axiom,
! [A: $tType] :
( ( boolean_algebra @ A @ ( type2 @ A ) )
=> ! [Y: A,X2: A] :
( ( ord_less_eq @ A @ Y @ ( uminus_uminus @ A @ X2 ) )
=> ( ord_less_eq @ A @ X2 @ ( uminus_uminus @ A @ Y ) ) ) ) ).
% compl_le_swap1
thf(fact_157_compl__mono,axiom,
! [A: $tType] :
( ( boolean_algebra @ A @ ( type2 @ A ) )
=> ! [X2: A,Y: A] :
( ( ord_less_eq @ A @ X2 @ Y )
=> ( ord_less_eq @ A @ ( uminus_uminus @ A @ Y ) @ ( uminus_uminus @ A @ X2 ) ) ) ) ).
% compl_mono
thf(fact_158_le__imp__neg__le,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
=> ! [A3: A,B3: A] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ord_less_eq @ A @ ( uminus_uminus @ A @ B3 ) @ ( uminus_uminus @ A @ A3 ) ) ) ) ).
% le_imp_neg_le
thf(fact_159_minus__le__iff,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
=> ! [A3: A,B3: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ A3 ) @ B3 )
= ( ord_less_eq @ A @ ( uminus_uminus @ A @ B3 ) @ A3 ) ) ) ).
% minus_le_iff
thf(fact_160_le__minus__iff,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
=> ! [A3: A,B3: A] :
( ( ord_less_eq @ A @ A3 @ ( uminus_uminus @ A @ B3 ) )
= ( ord_less_eq @ A @ B3 @ ( uminus_uminus @ A @ A3 ) ) ) ) ).
% le_minus_iff
thf(fact_161_subset__UNIV,axiom,
! [A: $tType,A2: set @ A] : ( ord_less_eq @ ( set @ A ) @ A2 @ ( top_top @ ( set @ A ) ) ) ).
% subset_UNIV
thf(fact_162_rev__finite__subset,axiom,
! [A: $tType,B4: set @ A,A2: set @ A] :
( ( finite_finite2 @ A @ B4 )
=> ( ( ord_less_eq @ ( set @ A ) @ A2 @ B4 )
=> ( finite_finite2 @ A @ A2 ) ) ) ).
% rev_finite_subset
thf(fact_163_infinite__super,axiom,
! [A: $tType,S: set @ A,T2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ S @ T2 )
=> ( ~ ( finite_finite2 @ A @ S )
=> ~ ( finite_finite2 @ A @ T2 ) ) ) ).
% infinite_super
thf(fact_164_finite__subset,axiom,
! [A: $tType,A2: set @ A,B4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B4 )
=> ( ( finite_finite2 @ A @ B4 )
=> ( finite_finite2 @ A @ A2 ) ) ) ).
% finite_subset
thf(fact_165_subset__insertI2,axiom,
! [A: $tType,A2: set @ A,B4: set @ A,B3: A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B4 )
=> ( ord_less_eq @ ( set @ A ) @ A2 @ ( insert @ A @ B3 @ B4 ) ) ) ).
% subset_insertI2
thf(fact_166_subset__insertI,axiom,
! [A: $tType,B4: set @ A,A3: A] : ( ord_less_eq @ ( set @ A ) @ B4 @ ( insert @ A @ A3 @ B4 ) ) ).
% subset_insertI
thf(fact_167_subset__insert,axiom,
! [A: $tType,X2: A,A2: set @ A,B4: set @ A] :
( ~ ( member @ A @ X2 @ A2 )
=> ( ( ord_less_eq @ ( set @ A ) @ A2 @ ( insert @ A @ X2 @ B4 ) )
= ( ord_less_eq @ ( set @ A ) @ A2 @ B4 ) ) ) ).
% subset_insert
thf(fact_168_Set_Oinsert__mono,axiom,
! [A: $tType,C5: set @ A,D4: set @ A,A3: A] :
( ( ord_less_eq @ ( set @ A ) @ C5 @ D4 )
=> ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ A3 @ C5 ) @ ( insert @ A @ A3 @ D4 ) ) ) ).
% Set.insert_mono
thf(fact_169_Collect__mono__iff,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) )
= ( ! [X: A] :
( ( P @ X )
=> ( Q @ X ) ) ) ) ).
% Collect_mono_iff
thf(fact_170_pairwise__subset,axiom,
! [A: $tType,P: A > A > $o,S: set @ A,T2: set @ A] :
( ( pairwise @ A @ P @ S )
=> ( ( ord_less_eq @ ( set @ A ) @ T2 @ S )
=> ( pairwise @ A @ P @ T2 ) ) ) ).
% pairwise_subset
thf(fact_171_contra__subsetD,axiom,
! [A: $tType,A2: set @ A,B4: set @ A,C2: A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B4 )
=> ( ~ ( member @ A @ C2 @ B4 )
=> ~ ( member @ A @ C2 @ A2 ) ) ) ).
% contra_subsetD
thf(fact_172_set__eq__subset,axiom,
! [A: $tType] :
( ( ^ [Y3: set @ A,Z: set @ A] : ( Y3 = Z ) )
= ( ^ [A7: set @ A,B8: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A7 @ B8 )
& ( ord_less_eq @ ( set @ A ) @ B8 @ A7 ) ) ) ) ).
% set_eq_subset
thf(fact_173_subset__trans,axiom,
! [A: $tType,A2: set @ A,B4: set @ A,C5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B4 )
=> ( ( ord_less_eq @ ( set @ A ) @ B4 @ C5 )
=> ( ord_less_eq @ ( set @ A ) @ A2 @ C5 ) ) ) ).
% subset_trans
thf(fact_174_pairwise__def,axiom,
! [A: $tType] :
( ( pairwise @ A )
= ( ^ [R2: A > A > $o,S3: set @ A] :
! [X: A] :
( ( member @ A @ X @ S3 )
=> ! [Y4: A] :
( ( member @ A @ Y4 @ S3 )
=> ( ( X != Y4 )
=> ( R2 @ X @ Y4 ) ) ) ) ) ) ).
% pairwise_def
thf(fact_175_Collect__mono,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ! [X3: A] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) ) ) ).
% Collect_mono
thf(fact_176_subset__refl,axiom,
! [A: $tType,A2: set @ A] : ( ord_less_eq @ ( set @ A ) @ A2 @ A2 ) ).
% subset_refl
thf(fact_177_rev__subsetD,axiom,
! [A: $tType,C2: A,A2: set @ A,B4: set @ A] :
( ( member @ A @ C2 @ A2 )
=> ( ( ord_less_eq @ ( set @ A ) @ A2 @ B4 )
=> ( member @ A @ C2 @ B4 ) ) ) ).
% rev_subsetD
thf(fact_178_subset__iff,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A7: set @ A,B8: set @ A] :
! [T3: A] :
( ( member @ A @ T3 @ A7 )
=> ( member @ A @ T3 @ B8 ) ) ) ) ).
% subset_iff
thf(fact_179_set__rev__mp,axiom,
! [A: $tType,X2: A,A2: set @ A,B4: set @ A] :
( ( member @ A @ X2 @ A2 )
=> ( ( ord_less_eq @ ( set @ A ) @ A2 @ B4 )
=> ( member @ A @ X2 @ B4 ) ) ) ).
% set_rev_mp
thf(fact_180_equalityD2,axiom,
! [A: $tType,A2: set @ A,B4: set @ A] :
( ( A2 = B4 )
=> ( ord_less_eq @ ( set @ A ) @ B4 @ A2 ) ) ).
% equalityD2
thf(fact_181_equalityD1,axiom,
! [A: $tType,A2: set @ A,B4: set @ A] :
( ( A2 = B4 )
=> ( ord_less_eq @ ( set @ A ) @ A2 @ B4 ) ) ).
% equalityD1
thf(fact_182_subset__eq,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A7: set @ A,B8: set @ A] :
! [X: A] :
( ( member @ A @ X @ A7 )
=> ( member @ A @ X @ B8 ) ) ) ) ).
% subset_eq
thf(fact_183_equalityE,axiom,
! [A: $tType,A2: set @ A,B4: set @ A] :
( ( A2 = B4 )
=> ~ ( ( ord_less_eq @ ( set @ A ) @ A2 @ B4 )
=> ~ ( ord_less_eq @ ( set @ A ) @ B4 @ A2 ) ) ) ).
% equalityE
thf(fact_184_subsetCE,axiom,
! [A: $tType,A2: set @ A,B4: set @ A,C2: A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B4 )
=> ( ( member @ A @ C2 @ A2 )
=> ( member @ A @ C2 @ B4 ) ) ) ).
% subsetCE
thf(fact_185_subsetD,axiom,
! [A: $tType,A2: set @ A,B4: set @ A,C2: A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B4 )
=> ( ( member @ A @ C2 @ A2 )
=> ( member @ A @ C2 @ B4 ) ) ) ).
% subsetD
thf(fact_186_in__mono,axiom,
! [A: $tType,A2: set @ A,B4: set @ A,X2: A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B4 )
=> ( ( member @ A @ X2 @ A2 )
=> ( member @ A @ X2 @ B4 ) ) ) ).
% in_mono
thf(fact_187_set__mp,axiom,
! [A: $tType,A2: set @ A,B4: set @ A,X2: A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B4 )
=> ( ( member @ A @ X2 @ A2 )
=> ( member @ A @ X2 @ B4 ) ) ) ).
% set_mp
thf(fact_188_totally__bounded__subset,axiom,
! [A: $tType] :
( ( topolo47006728_space @ A @ ( type2 @ A ) )
=> ! [S: set @ A,T2: set @ A] :
( ( topolo406746546ounded @ A @ S )
=> ( ( ord_less_eq @ ( set @ A ) @ T2 @ S )
=> ( topolo406746546ounded @ A @ T2 ) ) ) ) ).
% totally_bounded_subset
thf(fact_189_dual__order_Oantisym,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [B3: A,A3: A] :
( ( ord_less_eq @ A @ B3 @ A3 )
=> ( ( ord_less_eq @ A @ A3 @ B3 )
=> ( A3 = B3 ) ) ) ) ).
% dual_order.antisym
thf(fact_190_dual__order_Otrans,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [B3: A,A3: A,C2: A] :
( ( ord_less_eq @ A @ B3 @ A3 )
=> ( ( ord_less_eq @ A @ C2 @ B3 )
=> ( ord_less_eq @ A @ C2 @ A3 ) ) ) ) ).
% dual_order.trans
thf(fact_191_linorder__wlog,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [P: A > A > $o,A3: A,B3: A] :
( ! [A5: A,B7: A] :
( ( ord_less_eq @ A @ A5 @ B7 )
=> ( P @ A5 @ B7 ) )
=> ( ! [A5: A,B7: A] :
( ( P @ B7 @ A5 )
=> ( P @ A5 @ B7 ) )
=> ( P @ A3 @ B3 ) ) ) ) ).
% linorder_wlog
thf(fact_192_dual__order_Orefl,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A3: A] : ( ord_less_eq @ A @ A3 @ A3 ) ) ).
% dual_order.refl
thf(fact_193_order__trans,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X2: A,Y: A,Z2: A] :
( ( ord_less_eq @ A @ X2 @ Y )
=> ( ( ord_less_eq @ A @ Y @ Z2 )
=> ( ord_less_eq @ A @ X2 @ Z2 ) ) ) ) ).
% order_trans
thf(fact_194_order__class_Oorder_Oantisym,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A3: A,B3: A] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( ord_less_eq @ A @ B3 @ A3 )
=> ( A3 = B3 ) ) ) ) ).
% order_class.order.antisym
thf(fact_195_ord__le__eq__trans,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [A3: A,B3: A,C2: A] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( B3 = C2 )
=> ( ord_less_eq @ A @ A3 @ C2 ) ) ) ) ).
% ord_le_eq_trans
thf(fact_196_ord__eq__le__trans,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [A3: A,B3: A,C2: A] :
( ( A3 = B3 )
=> ( ( ord_less_eq @ A @ B3 @ C2 )
=> ( ord_less_eq @ A @ A3 @ C2 ) ) ) ) ).
% ord_eq_le_trans
thf(fact_197_antisym__conv,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [Y: A,X2: A] :
( ( ord_less_eq @ A @ Y @ X2 )
=> ( ( ord_less_eq @ A @ X2 @ Y )
= ( X2 = Y ) ) ) ) ).
% antisym_conv
thf(fact_198_le__cases3,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X2: A,Y: A,Z2: A] :
( ( ( ord_less_eq @ A @ X2 @ Y )
=> ~ ( ord_less_eq @ A @ Y @ Z2 ) )
=> ( ( ( ord_less_eq @ A @ Y @ X2 )
=> ~ ( ord_less_eq @ A @ X2 @ Z2 ) )
=> ( ( ( ord_less_eq @ A @ X2 @ Z2 )
=> ~ ( ord_less_eq @ A @ Z2 @ Y ) )
=> ( ( ( ord_less_eq @ A @ Z2 @ Y )
=> ~ ( ord_less_eq @ A @ Y @ X2 ) )
=> ( ( ( ord_less_eq @ A @ Y @ Z2 )
=> ~ ( ord_less_eq @ A @ Z2 @ X2 ) )
=> ~ ( ( ord_less_eq @ A @ Z2 @ X2 )
=> ~ ( ord_less_eq @ A @ X2 @ Y ) ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_199_order_Otrans,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A3: A,B3: A,C2: A] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( ord_less_eq @ A @ B3 @ C2 )
=> ( ord_less_eq @ A @ A3 @ C2 ) ) ) ) ).
% order.trans
thf(fact_200_le__cases,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X2: A,Y: A] :
( ~ ( ord_less_eq @ A @ X2 @ Y )
=> ( ord_less_eq @ A @ Y @ X2 ) ) ) ).
% le_cases
thf(fact_201_eq__refl,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X2: A,Y: A] :
( ( X2 = Y )
=> ( ord_less_eq @ A @ X2 @ Y ) ) ) ).
% eq_refl
thf(fact_202_linear,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X2: A,Y: A] :
( ( ord_less_eq @ A @ X2 @ Y )
| ( ord_less_eq @ A @ Y @ X2 ) ) ) ).
% linear
thf(fact_203_antisym,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [X2: A,Y: A] :
( ( ord_less_eq @ A @ X2 @ Y )
=> ( ( ord_less_eq @ A @ Y @ X2 )
=> ( X2 = Y ) ) ) ) ).
% antisym
thf(fact_204_eq__iff,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ( ( ^ [Y3: A,Z: A] : ( Y3 = Z ) )
= ( ^ [X: A,Y4: A] :
( ( ord_less_eq @ A @ X @ Y4 )
& ( ord_less_eq @ A @ Y4 @ X ) ) ) ) ) ).
% eq_iff
thf(fact_205_ord__le__eq__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B @ ( type2 @ B ) )
& ( ord @ A @ ( type2 @ A ) ) )
=> ! [A3: A,B3: A,F2: A > B,C2: B] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( ( F2 @ B3 )
= C2 )
=> ( ! [X3: A,Y2: A] :
( ( ord_less_eq @ A @ X3 @ Y2 )
=> ( ord_less_eq @ B @ ( F2 @ X3 ) @ ( F2 @ Y2 ) ) )
=> ( ord_less_eq @ B @ ( F2 @ A3 ) @ C2 ) ) ) ) ) ).
% ord_le_eq_subst
thf(fact_206_ord__eq__le__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B @ ( type2 @ B ) )
& ( ord @ A @ ( type2 @ A ) ) )
=> ! [A3: A,F2: B > A,B3: B,C2: B] :
( ( A3
= ( F2 @ B3 ) )
=> ( ( ord_less_eq @ B @ B3 @ C2 )
=> ( ! [X3: B,Y2: B] :
( ( ord_less_eq @ B @ X3 @ Y2 )
=> ( ord_less_eq @ A @ ( F2 @ X3 ) @ ( F2 @ Y2 ) ) )
=> ( ord_less_eq @ A @ A3 @ ( F2 @ C2 ) ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_207_order__subst2,axiom,
! [A: $tType,C: $tType] :
( ( ( order @ C @ ( type2 @ C ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A3: A,B3: A,F2: A > C,C2: C] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( ord_less_eq @ C @ ( F2 @ B3 ) @ C2 )
=> ( ! [X3: A,Y2: A] :
( ( ord_less_eq @ A @ X3 @ Y2 )
=> ( ord_less_eq @ C @ ( F2 @ X3 ) @ ( F2 @ Y2 ) ) )
=> ( ord_less_eq @ C @ ( F2 @ A3 ) @ C2 ) ) ) ) ) ).
% order_subst2
thf(fact_208_order__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B @ ( type2 @ B ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A3: A,F2: B > A,B3: B,C2: B] :
( ( ord_less_eq @ A @ A3 @ ( F2 @ B3 ) )
=> ( ( ord_less_eq @ B @ B3 @ C2 )
=> ( ! [X3: B,Y2: B] :
( ( ord_less_eq @ B @ X3 @ Y2 )
=> ( ord_less_eq @ A @ ( F2 @ X3 ) @ ( F2 @ Y2 ) ) )
=> ( ord_less_eq @ A @ A3 @ ( F2 @ C2 ) ) ) ) ) ) ).
% order_subst1
thf(fact_209_le__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B @ ( type2 @ B ) )
=> ( ( ord_less_eq @ ( A > B ) )
= ( ^ [F6: A > B,G4: A > B] :
! [X: A] : ( ord_less_eq @ B @ ( F6 @ X ) @ ( G4 @ X ) ) ) ) ) ).
% le_fun_def
thf(fact_210_le__funI,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B @ ( type2 @ B ) )
=> ! [F2: A > B,G: A > B] :
( ! [X3: A] : ( ord_less_eq @ B @ ( F2 @ X3 ) @ ( G @ X3 ) )
=> ( ord_less_eq @ ( A > B ) @ F2 @ G ) ) ) ).
% le_funI
thf(fact_211_le__funE,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B @ ( type2 @ B ) )
=> ! [F2: A > B,G: A > B,X2: A] :
( ( ord_less_eq @ ( A > B ) @ F2 @ G )
=> ( ord_less_eq @ B @ ( F2 @ X2 ) @ ( G @ X2 ) ) ) ) ).
% le_funE
thf(fact_212_le__funD,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B @ ( type2 @ B ) )
=> ! [F2: A > B,G: A > B,X2: A] :
( ( ord_less_eq @ ( A > B ) @ F2 @ G )
=> ( ord_less_eq @ B @ ( F2 @ X2 ) @ ( G @ X2 ) ) ) ) ).
% le_funD
thf(fact_213_llist_Oset__sel_I1_J,axiom,
! [A: $tType,A3: coinductive_llist @ A] :
( ~ ( coinductive_lnull @ A @ A3 )
=> ( member @ A @ ( coinductive_lhd @ A @ A3 ) @ ( coinductive_lset @ A @ A3 ) ) ) ).
% llist.set_sel(1)
thf(fact_214_pairwise__empty,axiom,
! [A: $tType,P: A > A > $o] : ( pairwise @ A @ P @ ( bot_bot @ ( set @ A ) ) ) ).
% pairwise_empty
thf(fact_215_pairwise__insert,axiom,
! [A: $tType,R: A > A > $o,X2: A,S2: set @ A] :
( ( pairwise @ A @ R @ ( insert @ A @ X2 @ S2 ) )
= ( ! [Y4: A] :
( ( ( member @ A @ Y4 @ S2 )
& ( Y4 != X2 ) )
=> ( ( R @ X2 @ Y4 )
& ( R @ Y4 @ X2 ) ) )
& ( pairwise @ A @ R @ S2 ) ) ) ).
% pairwise_insert
thf(fact_216_subset__singletonD,axiom,
! [A: $tType,A2: set @ A,X2: A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) )
=> ( ( A2
= ( bot_bot @ ( set @ A ) ) )
| ( A2
= ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% subset_singletonD
thf(fact_217_subset__singleton__iff,axiom,
! [A: $tType,X5: set @ A,A3: A] :
( ( ord_less_eq @ ( set @ A ) @ X5 @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) )
= ( ( X5
= ( bot_bot @ ( set @ A ) ) )
| ( X5
= ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% subset_singleton_iff
thf(fact_218_subset__Compl__self__eq,axiom,
! [A: $tType,A2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ ( uminus_uminus @ ( set @ A ) @ A2 ) )
= ( A2
= ( bot_bot @ ( set @ A ) ) ) ) ).
% subset_Compl_self_eq
thf(fact_219_finite__subset__induct,axiom,
! [A: $tType,F4: set @ A,A2: set @ A,P: ( set @ A ) > $o] :
( ( finite_finite2 @ A @ F4 )
=> ( ( ord_less_eq @ ( set @ A ) @ F4 @ A2 )
=> ( ( P @ ( bot_bot @ ( set @ A ) ) )
=> ( ! [A5: A,F3: set @ A] :
( ( finite_finite2 @ A @ F3 )
=> ( ( member @ A @ A5 @ A2 )
=> ( ~ ( member @ A @ A5 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert @ A @ A5 @ F3 ) ) ) ) ) )
=> ( P @ F4 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_220_insert__subsetI,axiom,
! [A: $tType,X2: A,A2: set @ A,X5: set @ A] :
( ( member @ A @ X2 @ A2 )
=> ( ( ord_less_eq @ ( set @ A ) @ X5 @ A2 )
=> ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ X2 @ X5 ) @ A2 ) ) ) ).
% insert_subsetI
thf(fact_221_subset__emptyI,axiom,
! [A: $tType,A2: set @ A] :
( ! [X3: A] :
~ ( member @ A @ X3 @ A2 )
=> ( ord_less_eq @ ( set @ A ) @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% subset_emptyI
thf(fact_222_finite__remove__induct,axiom,
! [A: $tType,B4: set @ A,P: ( set @ A ) > $o] :
( ( finite_finite2 @ A @ B4 )
=> ( ( P @ ( bot_bot @ ( set @ A ) ) )
=> ( ! [A4: set @ A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
=> ( ! [X4: A] :
( ( member @ A @ X4 @ A4 )
=> ( P @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert @ A @ X4 @ ( bot_bot @ ( set @ A ) ) ) ) ) )
=> ( P @ A4 ) ) ) ) )
=> ( P @ B4 ) ) ) ) ).
% finite_remove_induct
thf(fact_223_remove__induct,axiom,
! [A: $tType,P: ( set @ A ) > $o,B4: set @ A] :
( ( P @ ( bot_bot @ ( set @ A ) ) )
=> ( ( ~ ( finite_finite2 @ A @ B4 )
=> ( P @ B4 ) )
=> ( ! [A4: set @ A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
=> ( ! [X4: A] :
( ( member @ A @ X4 @ A4 )
=> ( P @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert @ A @ X4 @ ( bot_bot @ ( set @ A ) ) ) ) ) )
=> ( P @ A4 ) ) ) ) )
=> ( P @ B4 ) ) ) ) ).
% remove_induct
thf(fact_224_minus__apply,axiom,
! [B: $tType,A: $tType] :
( ( minus @ B @ ( type2 @ B ) )
=> ( ( minus_minus @ ( A > B ) )
= ( ^ [A7: A > B,B8: A > B,X: A] : ( minus_minus @ B @ ( A7 @ X ) @ ( B8 @ X ) ) ) ) ) ).
% minus_apply
thf(fact_225_DiffI,axiom,
! [A: $tType,C2: A,A2: set @ A,B4: set @ A] :
( ( member @ A @ C2 @ A2 )
=> ( ~ ( member @ A @ C2 @ B4 )
=> ( member @ A @ C2 @ ( minus_minus @ ( set @ A ) @ A2 @ B4 ) ) ) ) ).
% DiffI
thf(fact_226_Diff__iff,axiom,
! [A: $tType,C2: A,A2: set @ A,B4: set @ A] :
( ( member @ A @ C2 @ ( minus_minus @ ( set @ A ) @ A2 @ B4 ) )
= ( ( member @ A @ C2 @ A2 )
& ~ ( member @ A @ C2 @ B4 ) ) ) ).
% Diff_iff
thf(fact_227_Diff__idemp,axiom,
! [A: $tType,A2: set @ A,B4: set @ A] :
( ( minus_minus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A2 @ B4 ) @ B4 )
= ( minus_minus @ ( set @ A ) @ A2 @ B4 ) ) ).
% Diff_idemp
thf(fact_228_minus__diff__eq,axiom,
! [A: $tType] :
( ( group_add @ A @ ( type2 @ A ) )
=> ! [A3: A,B3: A] :
( ( uminus_uminus @ A @ ( minus_minus @ A @ A3 @ B3 ) )
= ( minus_minus @ A @ B3 @ A3 ) ) ) ).
% minus_diff_eq
thf(fact_229_Diff__empty,axiom,
! [A: $tType,A2: set @ A] :
( ( minus_minus @ ( set @ A ) @ A2 @ ( bot_bot @ ( set @ A ) ) )
= A2 ) ).
% Diff_empty
thf(fact_230_empty__Diff,axiom,
! [A: $tType,A2: set @ A] :
( ( minus_minus @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ A2 )
= ( bot_bot @ ( set @ A ) ) ) ).
% empty_Diff
thf(fact_231_Diff__cancel,axiom,
! [A: $tType,A2: set @ A] :
( ( minus_minus @ ( set @ A ) @ A2 @ A2 )
= ( bot_bot @ ( set @ A ) ) ) ).
% Diff_cancel
thf(fact_232_finite__Diff,axiom,
! [A: $tType,A2: set @ A,B4: set @ A] :
( ( finite_finite2 @ A @ A2 )
=> ( finite_finite2 @ A @ ( minus_minus @ ( set @ A ) @ A2 @ B4 ) ) ) ).
% finite_Diff
thf(fact_233_finite__Diff2,axiom,
! [A: $tType,B4: set @ A,A2: set @ A] :
( ( finite_finite2 @ A @ B4 )
=> ( ( finite_finite2 @ A @ ( minus_minus @ ( set @ A ) @ A2 @ B4 ) )
= ( finite_finite2 @ A @ A2 ) ) ) ).
% finite_Diff2
thf(fact_234_Diff__insert0,axiom,
! [A: $tType,X2: A,A2: set @ A,B4: set @ A] :
( ~ ( member @ A @ X2 @ A2 )
=> ( ( minus_minus @ ( set @ A ) @ A2 @ ( insert @ A @ X2 @ B4 ) )
= ( minus_minus @ ( set @ A ) @ A2 @ B4 ) ) ) ).
% Diff_insert0
thf(fact_235_insert__Diff1,axiom,
! [A: $tType,X2: A,B4: set @ A,A2: set @ A] :
( ( member @ A @ X2 @ B4 )
=> ( ( minus_minus @ ( set @ A ) @ ( insert @ A @ X2 @ A2 ) @ B4 )
= ( minus_minus @ ( set @ A ) @ A2 @ B4 ) ) ) ).
% insert_Diff1
thf(fact_236_Diff__UNIV,axiom,
! [A: $tType,A2: set @ A] :
( ( minus_minus @ ( set @ A ) @ A2 @ ( top_top @ ( set @ A ) ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% Diff_UNIV
thf(fact_237_Diff__eq__empty__iff,axiom,
! [A: $tType,A2: set @ A,B4: set @ A] :
( ( ( minus_minus @ ( set @ A ) @ A2 @ B4 )
= ( bot_bot @ ( set @ A ) ) )
= ( ord_less_eq @ ( set @ A ) @ A2 @ B4 ) ) ).
% Diff_eq_empty_iff
thf(fact_238_insert__Diff__single,axiom,
! [A: $tType,A3: A,A2: set @ A] :
( ( insert @ A @ A3 @ ( minus_minus @ ( set @ A ) @ A2 @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( insert @ A @ A3 @ A2 ) ) ).
% insert_Diff_single
thf(fact_239_finite__Diff__insert,axiom,
! [A: $tType,A2: set @ A,A3: A,B4: set @ A] :
( ( finite_finite2 @ A @ ( minus_minus @ ( set @ A ) @ A2 @ ( insert @ A @ A3 @ B4 ) ) )
= ( finite_finite2 @ A @ ( minus_minus @ ( set @ A ) @ A2 @ B4 ) ) ) ).
% finite_Diff_insert
thf(fact_240_double__diff,axiom,
! [A: $tType,A2: set @ A,B4: set @ A,C5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B4 )
=> ( ( ord_less_eq @ ( set @ A ) @ B4 @ C5 )
=> ( ( minus_minus @ ( set @ A ) @ B4 @ ( minus_minus @ ( set @ A ) @ C5 @ A2 ) )
= A2 ) ) ) ).
% double_diff
thf(fact_241_Diff__subset,axiom,
! [A: $tType,A2: set @ A,B4: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A2 @ B4 ) @ A2 ) ).
% Diff_subset
thf(fact_242_Diff__mono,axiom,
! [A: $tType,A2: set @ A,C5: set @ A,D4: set @ A,B4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ C5 )
=> ( ( ord_less_eq @ ( set @ A ) @ D4 @ B4 )
=> ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A2 @ B4 ) @ ( minus_minus @ ( set @ A ) @ C5 @ D4 ) ) ) ) ).
% Diff_mono
thf(fact_243_diff__eq__diff__less__eq,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
=> ! [A3: A,B3: A,C2: A,D: A] :
( ( ( minus_minus @ A @ A3 @ B3 )
= ( minus_minus @ A @ C2 @ D ) )
=> ( ( ord_less_eq @ A @ A3 @ B3 )
= ( ord_less_eq @ A @ C2 @ D ) ) ) ) ).
% diff_eq_diff_less_eq
thf(fact_244_diff__right__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
=> ! [A3: A,B3: A,C2: A] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ord_less_eq @ A @ ( minus_minus @ A @ A3 @ C2 ) @ ( minus_minus @ A @ B3 @ C2 ) ) ) ) ).
% diff_right_mono
thf(fact_245_diff__left__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
=> ! [B3: A,A3: A,C2: A] :
( ( ord_less_eq @ A @ B3 @ A3 )
=> ( ord_less_eq @ A @ ( minus_minus @ A @ C2 @ A3 ) @ ( minus_minus @ A @ C2 @ B3 ) ) ) ) ).
% diff_left_mono
thf(fact_246_diff__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
=> ! [A3: A,B3: A,D: A,C2: A] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( ord_less_eq @ A @ D @ C2 )
=> ( ord_less_eq @ A @ ( minus_minus @ A @ A3 @ C2 ) @ ( minus_minus @ A @ B3 @ D ) ) ) ) ) ).
% diff_mono
thf(fact_247_diff__eq__diff__eq,axiom,
! [A: $tType] :
( ( group_add @ A @ ( type2 @ A ) )
=> ! [A3: A,B3: A,C2: A,D: A] :
( ( ( minus_minus @ A @ A3 @ B3 )
= ( minus_minus @ A @ C2 @ D ) )
=> ( ( A3 = B3 )
= ( C2 = D ) ) ) ) ).
% diff_eq_diff_eq
thf(fact_248_diff__right__commute,axiom,
! [A: $tType] :
( ( cancel146912293up_add @ A @ ( type2 @ A ) )
=> ! [A3: A,C2: A,B3: A] :
( ( minus_minus @ A @ ( minus_minus @ A @ A3 @ C2 ) @ B3 )
= ( minus_minus @ A @ ( minus_minus @ A @ A3 @ B3 ) @ C2 ) ) ) ).
% diff_right_commute
thf(fact_249_DiffE,axiom,
! [A: $tType,C2: A,A2: set @ A,B4: set @ A] :
( ( member @ A @ C2 @ ( minus_minus @ ( set @ A ) @ A2 @ B4 ) )
=> ~ ( ( member @ A @ C2 @ A2 )
=> ( member @ A @ C2 @ B4 ) ) ) ).
% DiffE
thf(fact_250_DiffD1,axiom,
! [A: $tType,C2: A,A2: set @ A,B4: set @ A] :
( ( member @ A @ C2 @ ( minus_minus @ ( set @ A ) @ A2 @ B4 ) )
=> ( member @ A @ C2 @ A2 ) ) ).
% DiffD1
thf(fact_251_DiffD2,axiom,
! [A: $tType,C2: A,A2: set @ A,B4: set @ A] :
( ( member @ A @ C2 @ ( minus_minus @ ( set @ A ) @ A2 @ B4 ) )
=> ~ ( member @ A @ C2 @ B4 ) ) ).
% DiffD2
thf(fact_252_fun__diff__def,axiom,
! [B: $tType,A: $tType] :
( ( minus @ B @ ( type2 @ B ) )
=> ( ( minus_minus @ ( A > B ) )
= ( ^ [A7: A > B,B8: A > B,X: A] : ( minus_minus @ B @ ( A7 @ X ) @ ( B8 @ X ) ) ) ) ) ).
% fun_diff_def
thf(fact_253_insert__Diff__if,axiom,
! [A: $tType,X2: A,B4: set @ A,A2: set @ A] :
( ( ( member @ A @ X2 @ B4 )
=> ( ( minus_minus @ ( set @ A ) @ ( insert @ A @ X2 @ A2 ) @ B4 )
= ( minus_minus @ ( set @ A ) @ A2 @ B4 ) ) )
& ( ~ ( member @ A @ X2 @ B4 )
=> ( ( minus_minus @ ( set @ A ) @ ( insert @ A @ X2 @ A2 ) @ B4 )
= ( insert @ A @ X2 @ ( minus_minus @ ( set @ A ) @ A2 @ B4 ) ) ) ) ) ).
% insert_Diff_if
%----Type constructors (37)
thf(tcon_fun___Lattices_Oboolean__algebra,axiom,
! [A9: $tType,A10: $tType] :
( ( boolean_algebra @ A10 @ ( type2 @ A10 ) )
=> ( boolean_algebra @ ( A9 > A10 ) @ ( type2 @ ( A9 > A10 ) ) ) ) ).
thf(tcon_fun___Orderings_Oorder__top,axiom,
! [A9: $tType,A10: $tType] :
( ( order_top @ A10 @ ( type2 @ A10 ) )
=> ( order_top @ ( A9 > A10 ) @ ( type2 @ ( A9 > A10 ) ) ) ) ).
thf(tcon_fun___Orderings_Oorder__bot,axiom,
! [A9: $tType,A10: $tType] :
( ( order_bot @ A10 @ ( type2 @ A10 ) )
=> ( order_bot @ ( A9 > A10 ) @ ( type2 @ ( A9 > A10 ) ) ) ) ).
thf(tcon_fun___Orderings_Opreorder,axiom,
! [A9: $tType,A10: $tType] :
( ( preorder @ A10 @ ( type2 @ A10 ) )
=> ( preorder @ ( A9 > A10 ) @ ( type2 @ ( A9 > A10 ) ) ) ) ).
thf(tcon_fun___Finite__Set_Ofinite,axiom,
! [A9: $tType,A10: $tType] :
( ( ( finite_finite @ A9 @ ( type2 @ A9 ) )
& ( finite_finite @ A10 @ ( type2 @ A10 ) ) )
=> ( finite_finite @ ( A9 > A10 ) @ ( type2 @ ( A9 > A10 ) ) ) ) ).
thf(tcon_fun___Orderings_Oorder,axiom,
! [A9: $tType,A10: $tType] :
( ( order @ A10 @ ( type2 @ A10 ) )
=> ( order @ ( A9 > A10 ) @ ( type2 @ ( A9 > A10 ) ) ) ) ).
thf(tcon_fun___Orderings_Otop,axiom,
! [A9: $tType,A10: $tType] :
( ( top @ A10 @ ( type2 @ A10 ) )
=> ( top @ ( A9 > A10 ) @ ( type2 @ ( A9 > A10 ) ) ) ) ).
thf(tcon_fun___Orderings_Oord,axiom,
! [A9: $tType,A10: $tType] :
( ( ord @ A10 @ ( type2 @ A10 ) )
=> ( ord @ ( A9 > A10 ) @ ( type2 @ ( A9 > A10 ) ) ) ) ).
thf(tcon_fun___Orderings_Obot,axiom,
! [A9: $tType,A10: $tType] :
( ( bot @ A10 @ ( type2 @ A10 ) )
=> ( bot @ ( A9 > A10 ) @ ( type2 @ ( A9 > A10 ) ) ) ) ).
thf(tcon_fun___Groups_Ouminus,axiom,
! [A9: $tType,A10: $tType] :
( ( uminus @ A10 @ ( type2 @ A10 ) )
=> ( uminus @ ( A9 > A10 ) @ ( type2 @ ( A9 > A10 ) ) ) ) ).
thf(tcon_fun___Groups_Ominus,axiom,
! [A9: $tType,A10: $tType] :
( ( minus @ A10 @ ( type2 @ A10 ) )
=> ( minus @ ( A9 > A10 ) @ ( type2 @ ( A9 > A10 ) ) ) ) ).
thf(tcon_Set_Oset___Lattices_Oboolean__algebra_1,axiom,
! [A9: $tType] : ( boolean_algebra @ ( set @ A9 ) @ ( type2 @ ( set @ A9 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder__top_2,axiom,
! [A9: $tType] : ( order_top @ ( set @ A9 ) @ ( type2 @ ( set @ A9 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder__bot_3,axiom,
! [A9: $tType] : ( order_bot @ ( set @ A9 ) @ ( type2 @ ( set @ A9 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Opreorder_4,axiom,
! [A9: $tType] : ( preorder @ ( set @ A9 ) @ ( type2 @ ( set @ A9 ) ) ) ).
thf(tcon_Set_Oset___Finite__Set_Ofinite_5,axiom,
! [A9: $tType] :
( ( finite_finite @ A9 @ ( type2 @ A9 ) )
=> ( finite_finite @ ( set @ A9 ) @ ( type2 @ ( set @ A9 ) ) ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder_6,axiom,
! [A9: $tType] : ( order @ ( set @ A9 ) @ ( type2 @ ( set @ A9 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Otop_7,axiom,
! [A9: $tType] : ( top @ ( set @ A9 ) @ ( type2 @ ( set @ A9 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Oord_8,axiom,
! [A9: $tType] : ( ord @ ( set @ A9 ) @ ( type2 @ ( set @ A9 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Obot_9,axiom,
! [A9: $tType] : ( bot @ ( set @ A9 ) @ ( type2 @ ( set @ A9 ) ) ) ).
thf(tcon_Set_Oset___Groups_Ouminus_10,axiom,
! [A9: $tType] : ( uminus @ ( set @ A9 ) @ ( type2 @ ( set @ A9 ) ) ) ).
thf(tcon_Set_Oset___Groups_Ominus_11,axiom,
! [A9: $tType] : ( minus @ ( set @ A9 ) @ ( type2 @ ( set @ A9 ) ) ) ).
thf(tcon_HOL_Obool___Lattices_Oboolean__algebra_12,axiom,
boolean_algebra @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Oorder__top_13,axiom,
order_top @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Oorder__bot_14,axiom,
order_bot @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Opreorder_15,axiom,
preorder @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Olinorder,axiom,
linorder @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Finite__Set_Ofinite_16,axiom,
finite_finite @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Oorder_17,axiom,
order @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Otop_18,axiom,
top @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Oord_19,axiom,
ord @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Obot_20,axiom,
bot @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Groups_Ouminus_21,axiom,
uminus @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Groups_Ominus_22,axiom,
minus @ $o @ ( type2 @ $o ) ).
thf(tcon_Sum__Type_Osum___Finite__Set_Ofinite_23,axiom,
! [A9: $tType,A10: $tType] :
( ( ( finite_finite @ A9 @ ( type2 @ A9 ) )
& ( finite_finite @ A10 @ ( type2 @ A10 ) ) )
=> ( finite_finite @ ( sum_sum @ A9 @ A10 ) @ ( type2 @ ( sum_sum @ A9 @ A10 ) ) ) ) ).
thf(tcon_Option_Ooption___Finite__Set_Ofinite_24,axiom,
! [A9: $tType] :
( ( finite_finite @ A9 @ ( type2 @ A9 ) )
=> ( finite_finite @ ( option @ A9 ) @ ( type2 @ ( option @ A9 ) ) ) ) ).
thf(tcon_Product__Type_Oprod___Finite__Set_Ofinite_25,axiom,
! [A9: $tType,A10: $tType] :
( ( ( finite_finite @ A9 @ ( type2 @ A9 ) )
& ( finite_finite @ A10 @ ( type2 @ A10 ) ) )
=> ( finite_finite @ ( product_prod @ A9 @ A10 ) @ ( type2 @ ( product_prod @ A9 @ A10 ) ) ) ) ).
%----Conjectures (1)
thf(conj_0,conjecture,
~ ( member @ node @ x @ nsa ) ).
%------------------------------------------------------------------------------