TPTP Problem File: DAT174^1.p
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%------------------------------------------------------------------------------
% File : DAT174^1 : TPTP v8.2.0. Released v7.0.0.
% Domain : Data Structures
% Problem : Lazy lists II 73
% Version : [Bla16] axioms : Especial.
% English :
% Refs : [Fri04] Friedrich (2004), Lazy Lists II
% : [RB15] Reynolds & Blanchette (2015), A Decision Procedure for
% : [Bla16] Blanchette (2016), Email to Geoff Sutcliffe
% Source : [Bla16]
% Names : llist2__73.p [Bla16]
% Status : Theorem
% Rating : 1.00 v7.5.0, 0.33 v7.2.0, 0.50 v7.1.0
% Syntax : Number of formulae : 327 ( 95 unt; 45 typ; 0 def)
% Number of atoms : 853 ( 210 equ; 0 cnn)
% Maximal formula atoms : 13 ( 3 avg)
% Number of connectives : 3685 ( 98 ~; 15 |; 47 &;3112 @)
% ( 0 <=>; 413 =>; 0 <=; 0 <~>)
% Maximal formula depth : 21 ( 8 avg)
% Number of types : 2 ( 1 usr)
% Number of type conns : 135 ( 135 >; 0 *; 0 +; 0 <<)
% Number of symbols : 45 ( 44 usr; 0 con; 1-3 aty)
% Number of variables : 978 ( 57 ^; 870 !; 10 ?; 978 :)
% ( 41 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2016-07-13 14:43:52.898
%------------------------------------------------------------------------------
%----Could-be-implicit typings (4)
thf(ty_t_Coinductive__List_Ollist,type,
coinductive_llist: $tType > $tType ).
thf(ty_t_Set_Oset,type,
set: $tType > $tType ).
thf(ty_t_itself,type,
itself: $tType > $tType ).
thf(ty_tf_a,type,
a: $tType ).
%----Explicit typings (41)
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_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_Orderings_Ono__bot,type,
no_bot:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Ono__top,type,
no_top:
!>[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_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_Orderings_Owellorder,type,
wellorder:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Odense__order,type,
dense_order:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Odense__linorder,type,
dense_linorder:
!>[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_Conditionally__Complete__Lattices_Oconditionally__complete__linorder,type,
condit1037483654norder:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_c_Coinductive__List_OlSup,type,
coinductive_lSup:
!>[A: $tType] : ( ( set @ ( coinductive_llist @ A ) ) > ( coinductive_llist @ A ) ) ).
thf(sy_c_Coinductive__List_Ollist_OLNil,type,
coinductive_LNil:
!>[A: $tType] : ( coinductive_llist @ A ) ).
thf(sy_c_Groups_Ominus__class_Ominus,type,
minus_minus:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_LList2__Mirabelle__hamjzmohle_Oalllsts,type,
lList2435255213lllsts:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( coinductive_llist @ A ) ) ) ).
thf(sy_c_LList2__Mirabelle__hamjzmohle_Ofinlsts,type,
lList2236698231inlsts:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( coinductive_llist @ A ) ) ) ).
thf(sy_c_LList2__Mirabelle__hamjzmohle_Ofpslsts,type,
lList22096119349pslsts:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( coinductive_llist @ A ) ) ) ).
thf(sy_c_LList2__Mirabelle__hamjzmohle_Oinflsts,type,
lList21612149805nflsts:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( coinductive_llist @ A ) ) ) ).
thf(sy_c_LList2__Mirabelle__hamjzmohle_Oposlsts,type,
lList21148268032oslsts:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( coinductive_llist @ A ) ) ) ).
thf(sy_c_Orderings_Obot__class_Obot,type,
bot_bot:
!>[A: $tType] : A ).
thf(sy_c_Orderings_Oord__class_Oless,type,
ord_less:
!>[A: $tType] : ( A > A > $o ) ).
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_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_Oremove,type,
remove:
!>[A: $tType] : ( A > ( set @ A ) > ( set @ A ) ) ).
thf(sy_c_Set_Othe__elem,type,
the_elem:
!>[A: $tType] : ( ( set @ A ) > A ) ).
thf(sy_c_member,type,
member:
!>[A: $tType] : ( A > ( set @ A ) > $o ) ).
%----Relevant facts (256)
thf(fact_0_alllsts_OLNil__all,axiom,
! [A: $tType,A2: set @ A] : ( member @ ( coinductive_llist @ A ) @ ( coinductive_LNil @ A ) @ ( lList2435255213lllsts @ A @ A2 ) ) ).
% alllsts.LNil_all
thf(fact_1_singletonI,axiom,
! [A: $tType,A3: A] : ( member @ A @ A3 @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% singletonI
thf(fact_2_insertCI,axiom,
! [A: $tType,A3: A,B: set @ A,B2: A] :
( ( ~ ( member @ A @ A3 @ B )
=> ( A3 = B2 ) )
=> ( member @ A @ A3 @ ( insert @ A @ B2 @ B ) ) ) ).
% insertCI
thf(fact_3_insert__iff,axiom,
! [A: $tType,A3: A,B2: A,A2: set @ A] :
( ( member @ A @ A3 @ ( insert @ A @ B2 @ A2 ) )
= ( ( A3 = B2 )
| ( member @ A @ A3 @ A2 ) ) ) ).
% insert_iff
thf(fact_4_insert__absorb2,axiom,
! [A: $tType,X: A,A2: set @ A] :
( ( insert @ A @ X @ ( insert @ A @ X @ A2 ) )
= ( insert @ A @ X @ A2 ) ) ).
% insert_absorb2
thf(fact_5_empty__iff,axiom,
! [A: $tType,C: A] :
~ ( member @ A @ C @ ( bot_bot @ ( set @ A ) ) ) ).
% empty_iff
thf(fact_6_all__not__in__conv,axiom,
! [A: $tType,A2: set @ A] :
( ( ! [X2: A] :
~ ( member @ A @ X2 @ A2 ) )
= ( A2
= ( bot_bot @ ( set @ A ) ) ) ) ).
% all_not_in_conv
thf(fact_7_Collect__empty__eq,axiom,
! [A: $tType,P: A > $o] :
( ( ( collect @ A @ P )
= ( bot_bot @ ( set @ A ) ) )
= ( ! [X2: A] :
~ ( P @ X2 ) ) ) ).
% Collect_empty_eq
thf(fact_8_empty__Collect__eq,axiom,
! [A: $tType,P: A > $o] :
( ( ( bot_bot @ ( set @ A ) )
= ( collect @ A @ P ) )
= ( ! [X2: A] :
~ ( P @ X2 ) ) ) ).
% empty_Collect_eq
thf(fact_9_bot__apply,axiom,
! [C2: $tType,D: $tType] :
( ( bot @ C2 @ ( type2 @ C2 ) )
=> ( ( bot_bot @ ( D > C2 ) )
= ( ^ [X2: D] : ( bot_bot @ C2 ) ) ) ) ).
% bot_apply
thf(fact_10_singletonD,axiom,
! [A: $tType,B2: A,A3: A] :
( ( member @ A @ B2 @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) )
=> ( B2 = A3 ) ) ).
% singletonD
thf(fact_11_singleton__iff,axiom,
! [A: $tType,B2: A,A3: A] :
( ( member @ A @ B2 @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) )
= ( B2 = A3 ) ) ).
% singleton_iff
thf(fact_12_doubleton__eq__iff,axiom,
! [A: $tType,A3: A,B2: A,C: A,D2: A] :
( ( ( insert @ A @ A3 @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) )
= ( insert @ A @ C @ ( insert @ A @ D2 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( ( ( A3 = C )
& ( B2 = D2 ) )
| ( ( A3 = D2 )
& ( B2 = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_13_bot__set__def,axiom,
! [A: $tType] :
( ( bot_bot @ ( set @ A ) )
= ( collect @ A @ ( bot_bot @ ( A > $o ) ) ) ) ).
% bot_set_def
thf(fact_14_bot__fun__def,axiom,
! [B3: $tType,A: $tType] :
( ( bot @ B3 @ ( type2 @ B3 ) )
=> ( ( bot_bot @ ( A > B3 ) )
= ( ^ [X2: A] : ( bot_bot @ B3 ) ) ) ) ).
% bot_fun_def
thf(fact_15_ex__in__conv,axiom,
! [A: $tType,A2: set @ A] :
( ( ? [X2: A] : ( member @ A @ X2 @ A2 ) )
= ( A2
!= ( bot_bot @ ( set @ A ) ) ) ) ).
% ex_in_conv
thf(fact_16_equals0I,axiom,
! [A: $tType,A2: set @ A] :
( ! [Y: A] :
~ ( member @ A @ Y @ A2 )
=> ( A2
= ( bot_bot @ ( set @ A ) ) ) ) ).
% equals0I
thf(fact_17_equals0D,axiom,
! [A: $tType,A2: set @ A,A3: A] :
( ( A2
= ( bot_bot @ ( set @ A ) ) )
=> ~ ( member @ A @ A3 @ A2 ) ) ).
% equals0D
thf(fact_18_emptyE,axiom,
! [A: $tType,A3: A] :
~ ( member @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ).
% emptyE
thf(fact_19_mk__disjoint__insert,axiom,
! [A: $tType,A3: A,A2: set @ A] :
( ( member @ A @ A3 @ A2 )
=> ? [B4: set @ A] :
( ( A2
= ( insert @ A @ A3 @ B4 ) )
& ~ ( member @ A @ A3 @ B4 ) ) ) ).
% mk_disjoint_insert
thf(fact_20_insert__commute,axiom,
! [A: $tType,X: A,Y2: A,A2: set @ A] :
( ( insert @ A @ X @ ( insert @ A @ Y2 @ A2 ) )
= ( insert @ A @ Y2 @ ( insert @ A @ X @ A2 ) ) ) ).
% insert_commute
thf(fact_21_insert__eq__iff,axiom,
! [A: $tType,A3: A,A2: set @ A,B2: A,B: set @ A] :
( ~ ( member @ A @ A3 @ A2 )
=> ( ~ ( member @ A @ B2 @ B )
=> ( ( ( insert @ A @ A3 @ A2 )
= ( insert @ A @ B2 @ B ) )
= ( ( ( A3 = B2 )
=> ( A2 = B ) )
& ( ( A3 != B2 )
=> ? [C3: set @ A] :
( ( A2
= ( insert @ A @ B2 @ C3 ) )
& ~ ( member @ A @ B2 @ C3 )
& ( B
= ( insert @ A @ A3 @ C3 ) )
& ~ ( member @ A @ A3 @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_22_insert__absorb,axiom,
! [A: $tType,A3: A,A2: set @ A] :
( ( member @ A @ A3 @ A2 )
=> ( ( insert @ A @ A3 @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_23_insert__ident,axiom,
! [A: $tType,X: A,A2: set @ A,B: set @ A] :
( ~ ( member @ A @ X @ A2 )
=> ( ~ ( member @ A @ X @ B )
=> ( ( ( insert @ A @ X @ A2 )
= ( insert @ A @ X @ B ) )
= ( A2 = B ) ) ) ) ).
% insert_ident
thf(fact_24_Set_Oset__insert,axiom,
! [A: $tType,X: A,A2: set @ A] :
( ( member @ A @ X @ A2 )
=> ~ ! [B4: set @ A] :
( ( A2
= ( insert @ A @ X @ B4 ) )
=> ( member @ A @ X @ B4 ) ) ) ).
% Set.set_insert
thf(fact_25_insertI2,axiom,
! [A: $tType,A3: A,B: set @ A,B2: A] :
( ( member @ A @ A3 @ B )
=> ( member @ A @ A3 @ ( insert @ A @ B2 @ B ) ) ) ).
% insertI2
thf(fact_26_insertI1,axiom,
! [A: $tType,A3: A,B: set @ A] : ( member @ A @ A3 @ ( insert @ A @ A3 @ B ) ) ).
% insertI1
thf(fact_27_insertE,axiom,
! [A: $tType,A3: A,B2: A,A2: set @ A] :
( ( member @ A @ A3 @ ( insert @ A @ B2 @ A2 ) )
=> ( ( A3 != B2 )
=> ( member @ A @ A3 @ A2 ) ) ) ).
% insertE
thf(fact_28_singleton__inject,axiom,
! [A: $tType,A3: A,B2: A] :
( ( ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) )
= ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) )
=> ( A3 = B2 ) ) ).
% singleton_inject
thf(fact_29_insert__not__empty,axiom,
! [A: $tType,A3: A,A2: set @ A] :
( ( insert @ A @ A3 @ A2 )
!= ( bot_bot @ ( set @ A ) ) ) ).
% insert_not_empty
thf(fact_30_the__elem__eq,axiom,
! [A: $tType,X: A] :
( ( the_elem @ A @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= X ) ).
% the_elem_eq
thf(fact_31_is__singletonI,axiom,
! [A: $tType,X: A] : ( is_singleton @ A @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ).
% is_singletonI
thf(fact_32_Set_Ois__empty__def,axiom,
! [A: $tType] :
( ( is_empty @ A )
= ( ^ [A4: set @ A] :
( A4
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Set.is_empty_def
thf(fact_33_is__singleton__def,axiom,
! [A: $tType] :
( ( is_singleton @ A )
= ( ^ [A4: set @ A] :
? [X2: A] :
( A4
= ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% is_singleton_def
thf(fact_34_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_35_poslsts__def,axiom,
! [A: $tType] :
( ( lList21148268032oslsts @ A )
= ( ^ [A4: set @ A] : ( minus_minus @ ( set @ ( coinductive_llist @ A ) ) @ ( lList2435255213lllsts @ A @ A4 ) @ ( insert @ ( coinductive_llist @ A ) @ ( coinductive_LNil @ A ) @ ( bot_bot @ ( set @ ( coinductive_llist @ A ) ) ) ) ) ) ) ).
% poslsts_def
thf(fact_36_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_37_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_38_bot__empty__eq,axiom,
! [A: $tType] :
( ( bot_bot @ ( A > $o ) )
= ( ^ [X2: A] : ( member @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% bot_empty_eq
thf(fact_39_singleton__insert__inj__eq_H,axiom,
! [A: $tType,A3: A,A2: set @ A,B2: A] :
( ( ( insert @ A @ A3 @ A2 )
= ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) )
= ( ( A3 = B2 )
& ( ord_less_eq @ ( set @ A ) @ A2 @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_40_singleton__insert__inj__eq,axiom,
! [A: $tType,B2: A,A3: A,A2: set @ A] :
( ( ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) )
= ( insert @ A @ A3 @ A2 ) )
= ( ( A3 = B2 )
& ( ord_less_eq @ ( set @ A ) @ A2 @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_41_order__refl,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A] : ( ord_less_eq @ A @ X @ X ) ) ).
% order_refl
thf(fact_42_subsetI,axiom,
! [A: $tType,A2: set @ A,B: set @ A] :
( ! [X3: A] :
( ( member @ A @ X3 @ A2 )
=> ( member @ A @ X3 @ B ) )
=> ( ord_less_eq @ ( set @ A ) @ A2 @ B ) ) ).
% subsetI
thf(fact_43_subset__antisym,axiom,
! [A: $tType,A2: set @ A,B: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B )
=> ( ( ord_less_eq @ ( set @ A ) @ B @ A2 )
=> ( A2 = B ) ) ) ).
% subset_antisym
thf(fact_44_DiffI,axiom,
! [A: $tType,C: A,A2: set @ A,B: set @ A] :
( ( member @ A @ C @ A2 )
=> ( ~ ( member @ A @ C @ B )
=> ( member @ A @ C @ ( minus_minus @ ( set @ A ) @ A2 @ B ) ) ) ) ).
% DiffI
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
@ ^ [X2: A] : ( member @ A @ X2 @ 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,
! [B3: $tType,A: $tType,F: A > B3,G: A > B3] :
( ! [X3: A] :
( ( F @ X3 )
= ( G @ X3 ) )
=> ( F = G ) ) ).
% ext
thf(fact_49_Diff__iff,axiom,
! [A: $tType,C: A,A2: set @ A,B: set @ A] :
( ( member @ A @ C @ ( minus_minus @ ( set @ A ) @ A2 @ B ) )
= ( ( member @ A @ C @ A2 )
& ~ ( member @ A @ C @ B ) ) ) ).
% Diff_iff
thf(fact_50_Diff__idemp,axiom,
! [A: $tType,A2: set @ A,B: set @ A] :
( ( minus_minus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A2 @ B ) @ B )
= ( minus_minus @ ( set @ A ) @ A2 @ B ) ) ).
% Diff_idemp
thf(fact_51_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_52_empty__subsetI,axiom,
! [A: $tType,A2: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ A2 ) ).
% empty_subsetI
thf(fact_53_insert__subset,axiom,
! [A: $tType,X: A,A2: set @ A,B: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ X @ A2 ) @ B )
= ( ( member @ A @ X @ B )
& ( ord_less_eq @ ( set @ A ) @ A2 @ B ) ) ) ).
% insert_subset
thf(fact_54_Diff__empty,axiom,
! [A: $tType,A2: set @ A] :
( ( minus_minus @ ( set @ A ) @ A2 @ ( bot_bot @ ( set @ A ) ) )
= A2 ) ).
% Diff_empty
thf(fact_55_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_56_Diff__cancel,axiom,
! [A: $tType,A2: set @ A] :
( ( minus_minus @ ( set @ A ) @ A2 @ A2 )
= ( bot_bot @ ( set @ A ) ) ) ).
% Diff_cancel
thf(fact_57_Diff__insert0,axiom,
! [A: $tType,X: A,A2: set @ A,B: set @ A] :
( ~ ( member @ A @ X @ A2 )
=> ( ( minus_minus @ ( set @ A ) @ A2 @ ( insert @ A @ X @ B ) )
= ( minus_minus @ ( set @ A ) @ A2 @ B ) ) ) ).
% Diff_insert0
thf(fact_58_insert__Diff1,axiom,
! [A: $tType,X: A,B: set @ A,A2: set @ A] :
( ( member @ A @ X @ B )
=> ( ( minus_minus @ ( set @ A ) @ ( insert @ A @ X @ A2 ) @ B )
= ( minus_minus @ ( set @ A ) @ A2 @ B ) ) ) ).
% insert_Diff1
thf(fact_59_Diff__eq__empty__iff,axiom,
! [A: $tType,A2: set @ A,B: set @ A] :
( ( ( minus_minus @ ( set @ A ) @ A2 @ B )
= ( bot_bot @ ( set @ A ) ) )
= ( ord_less_eq @ ( set @ A ) @ A2 @ B ) ) ).
% Diff_eq_empty_iff
thf(fact_60_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_61_DiffE,axiom,
! [A: $tType,C: A,A2: set @ A,B: set @ A] :
( ( member @ A @ C @ ( minus_minus @ ( set @ A ) @ A2 @ B ) )
=> ~ ( ( member @ A @ C @ A2 )
=> ( member @ A @ C @ B ) ) ) ).
% DiffE
thf(fact_62_DiffD1,axiom,
! [A: $tType,C: A,A2: set @ A,B: set @ A] :
( ( member @ A @ C @ ( minus_minus @ ( set @ A ) @ A2 @ B ) )
=> ( member @ A @ C @ A2 ) ) ).
% DiffD1
thf(fact_63_DiffD2,axiom,
! [A: $tType,C: A,A2: set @ A,B: set @ A] :
( ( member @ A @ C @ ( minus_minus @ ( set @ A ) @ A2 @ B ) )
=> ~ ( member @ A @ C @ B ) ) ).
% DiffD2
thf(fact_64_set__mp,axiom,
! [A: $tType,A2: set @ A,B: set @ A,X: A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B )
=> ( ( member @ A @ X @ A2 )
=> ( member @ A @ X @ B ) ) ) ).
% set_mp
thf(fact_65_in__mono,axiom,
! [A: $tType,A2: set @ A,B: set @ A,X: A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B )
=> ( ( member @ A @ X @ A2 )
=> ( member @ A @ X @ B ) ) ) ).
% in_mono
thf(fact_66_subsetD,axiom,
! [A: $tType,A2: set @ A,B: set @ A,C: A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B )
=> ( ( member @ A @ C @ A2 )
=> ( member @ A @ C @ B ) ) ) ).
% subsetD
thf(fact_67_subsetCE,axiom,
! [A: $tType,A2: set @ A,B: set @ A,C: A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B )
=> ( ( member @ A @ C @ A2 )
=> ( member @ A @ C @ B ) ) ) ).
% subsetCE
thf(fact_68_Diff__mono,axiom,
! [A: $tType,A2: set @ A,C4: set @ A,D3: set @ A,B: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ C4 )
=> ( ( ord_less_eq @ ( set @ A ) @ D3 @ B )
=> ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A2 @ B ) @ ( minus_minus @ ( set @ A ) @ C4 @ D3 ) ) ) ) ).
% Diff_mono
thf(fact_69_equalityE,axiom,
! [A: $tType,A2: set @ A,B: set @ A] :
( ( A2 = B )
=> ~ ( ( ord_less_eq @ ( set @ A ) @ A2 @ B )
=> ~ ( ord_less_eq @ ( set @ A ) @ B @ A2 ) ) ) ).
% equalityE
thf(fact_70_subset__eq,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A4: set @ A,B5: set @ A] :
! [X2: A] :
( ( member @ A @ X2 @ A4 )
=> ( member @ A @ X2 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_71_equalityD1,axiom,
! [A: $tType,A2: set @ A,B: set @ A] :
( ( A2 = B )
=> ( ord_less_eq @ ( set @ A ) @ A2 @ B ) ) ).
% equalityD1
thf(fact_72_equalityD2,axiom,
! [A: $tType,A2: set @ A,B: set @ A] :
( ( A2 = B )
=> ( ord_less_eq @ ( set @ A ) @ B @ A2 ) ) ).
% equalityD2
thf(fact_73_set__rev__mp,axiom,
! [A: $tType,X: A,A2: set @ A,B: set @ A] :
( ( member @ A @ X @ A2 )
=> ( ( ord_less_eq @ ( set @ A ) @ A2 @ B )
=> ( member @ A @ X @ B ) ) ) ).
% set_rev_mp
thf(fact_74_subset__iff,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A4: set @ A,B5: set @ A] :
! [T: A] :
( ( member @ A @ T @ A4 )
=> ( member @ A @ T @ B5 ) ) ) ) ).
% subset_iff
thf(fact_75_Diff__subset,axiom,
! [A: $tType,A2: set @ A,B: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A2 @ B ) @ A2 ) ).
% Diff_subset
thf(fact_76_double__diff,axiom,
! [A: $tType,A2: set @ A,B: set @ A,C4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B )
=> ( ( ord_less_eq @ ( set @ A ) @ B @ C4 )
=> ( ( minus_minus @ ( set @ A ) @ B @ ( minus_minus @ ( set @ A ) @ C4 @ A2 ) )
= A2 ) ) ) ).
% double_diff
thf(fact_77_rev__subsetD,axiom,
! [A: $tType,C: A,A2: set @ A,B: set @ A] :
( ( member @ A @ C @ A2 )
=> ( ( ord_less_eq @ ( set @ A ) @ A2 @ B )
=> ( member @ A @ C @ B ) ) ) ).
% rev_subsetD
thf(fact_78_subset__refl,axiom,
! [A: $tType,A2: set @ A] : ( ord_less_eq @ ( set @ A ) @ A2 @ A2 ) ).
% subset_refl
thf(fact_79_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_80_subset__trans,axiom,
! [A: $tType,A2: set @ A,B: set @ A,C4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B )
=> ( ( ord_less_eq @ ( set @ A ) @ B @ C4 )
=> ( ord_less_eq @ ( set @ A ) @ A2 @ C4 ) ) ) ).
% subset_trans
thf(fact_81_le__funD,axiom,
! [B3: $tType,A: $tType] :
( ( ord @ B3 @ ( type2 @ B3 ) )
=> ! [F: A > B3,G: A > B3,X: A] :
( ( ord_less_eq @ ( A > B3 ) @ F @ G )
=> ( ord_less_eq @ B3 @ ( F @ X ) @ ( G @ X ) ) ) ) ).
% le_funD
thf(fact_82_le__funE,axiom,
! [B3: $tType,A: $tType] :
( ( ord @ B3 @ ( type2 @ B3 ) )
=> ! [F: A > B3,G: A > B3,X: A] :
( ( ord_less_eq @ ( A > B3 ) @ F @ G )
=> ( ord_less_eq @ B3 @ ( F @ X ) @ ( G @ X ) ) ) ) ).
% le_funE
thf(fact_83_le__funI,axiom,
! [B3: $tType,A: $tType] :
( ( ord @ B3 @ ( type2 @ B3 ) )
=> ! [F: A > B3,G: A > B3] :
( ! [X3: A] : ( ord_less_eq @ B3 @ ( F @ X3 ) @ ( G @ X3 ) )
=> ( ord_less_eq @ ( A > B3 ) @ F @ G ) ) ) ).
% le_funI
thf(fact_84_set__eq__subset,axiom,
! [A: $tType] :
( ( ^ [Y3: set @ A,Z: set @ A] : Y3 = Z )
= ( ^ [A4: set @ A,B5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B5 )
& ( ord_less_eq @ ( set @ A ) @ B5 @ A4 ) ) ) ) ).
% set_eq_subset
thf(fact_85_contra__subsetD,axiom,
! [A: $tType,A2: set @ A,B: set @ A,C: A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B )
=> ( ~ ( member @ A @ C @ B )
=> ~ ( member @ A @ C @ A2 ) ) ) ).
% contra_subsetD
thf(fact_86_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_87_le__fun__def,axiom,
! [B3: $tType,A: $tType] :
( ( ord @ B3 @ ( type2 @ B3 ) )
=> ( ( ord_less_eq @ ( A > B3 ) )
= ( ^ [F2: A > B3,G2: A > B3] :
! [X2: A] : ( ord_less_eq @ B3 @ ( F2 @ X2 ) @ ( G2 @ X2 ) ) ) ) ) ).
% le_fun_def
thf(fact_88_Collect__mono__iff,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) )
= ( ! [X2: A] :
( ( P @ X2 )
=> ( Q @ X2 ) ) ) ) ).
% Collect_mono_iff
thf(fact_89_order__subst1,axiom,
! [A: $tType,B3: $tType] :
( ( ( order @ B3 @ ( type2 @ B3 ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A3: A,F: B3 > A,B2: B3,C: B3] :
( ( ord_less_eq @ A @ A3 @ ( F @ B2 ) )
=> ( ( ord_less_eq @ B3 @ B2 @ C )
=> ( ! [X3: B3,Y: B3] :
( ( ord_less_eq @ B3 @ X3 @ Y )
=> ( ord_less_eq @ A @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_eq @ A @ A3 @ ( F @ C ) ) ) ) ) ) ).
% order_subst1
thf(fact_90_order__subst2,axiom,
! [A: $tType,C2: $tType] :
( ( ( order @ C2 @ ( type2 @ C2 ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A3: A,B2: A,F: A > C2,C: C2] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( ord_less_eq @ C2 @ ( F @ B2 ) @ C )
=> ( ! [X3: A,Y: A] :
( ( ord_less_eq @ A @ X3 @ Y )
=> ( ord_less_eq @ C2 @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_eq @ C2 @ ( F @ A3 ) @ C ) ) ) ) ) ).
% order_subst2
thf(fact_91_ord__eq__le__subst,axiom,
! [A: $tType,B3: $tType] :
( ( ( ord @ B3 @ ( type2 @ B3 ) )
& ( ord @ A @ ( type2 @ A ) ) )
=> ! [A3: A,F: B3 > A,B2: B3,C: B3] :
( ( A3
= ( F @ B2 ) )
=> ( ( ord_less_eq @ B3 @ B2 @ C )
=> ( ! [X3: B3,Y: B3] :
( ( ord_less_eq @ B3 @ X3 @ Y )
=> ( ord_less_eq @ A @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_eq @ A @ A3 @ ( F @ C ) ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_92_ord__le__eq__subst,axiom,
! [A: $tType,B3: $tType] :
( ( ( ord @ B3 @ ( type2 @ B3 ) )
& ( ord @ A @ ( type2 @ A ) ) )
=> ! [A3: A,B2: A,F: A > B3,C: B3] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: A,Y: A] :
( ( ord_less_eq @ A @ X3 @ Y )
=> ( ord_less_eq @ B3 @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_eq @ B3 @ ( F @ A3 ) @ C ) ) ) ) ) ).
% ord_le_eq_subst
thf(fact_93_eq__iff,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ( ( ^ [Y3: A,Z: A] : Y3 = Z )
= ( ^ [X2: A,Y4: A] :
( ( ord_less_eq @ A @ X2 @ Y4 )
& ( ord_less_eq @ A @ Y4 @ X2 ) ) ) ) ) ).
% eq_iff
thf(fact_94_antisym,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [X: A,Y2: A] :
( ( ord_less_eq @ A @ X @ Y2 )
=> ( ( ord_less_eq @ A @ Y2 @ X )
=> ( X = Y2 ) ) ) ) ).
% antisym
thf(fact_95_linear,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y2: A] :
( ( ord_less_eq @ A @ X @ Y2 )
| ( ord_less_eq @ A @ Y2 @ X ) ) ) ).
% linear
thf(fact_96_eq__refl,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y2: A] :
( ( X = Y2 )
=> ( ord_less_eq @ A @ X @ Y2 ) ) ) ).
% eq_refl
thf(fact_97_le__cases,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y2: A] :
( ~ ( ord_less_eq @ A @ X @ Y2 )
=> ( ord_less_eq @ A @ Y2 @ X ) ) ) ).
% le_cases
thf(fact_98_order_Otrans,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A3: A,B2: A,C: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( ord_less_eq @ A @ B2 @ C )
=> ( ord_less_eq @ A @ A3 @ C ) ) ) ) ).
% order.trans
thf(fact_99_le__cases3,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y2: A,Z2: A] :
( ( ( ord_less_eq @ A @ X @ Y2 )
=> ~ ( ord_less_eq @ A @ Y2 @ Z2 ) )
=> ( ( ( ord_less_eq @ A @ Y2 @ X )
=> ~ ( ord_less_eq @ A @ X @ Z2 ) )
=> ( ( ( ord_less_eq @ A @ X @ Z2 )
=> ~ ( ord_less_eq @ A @ Z2 @ Y2 ) )
=> ( ( ( ord_less_eq @ A @ Z2 @ Y2 )
=> ~ ( ord_less_eq @ A @ Y2 @ X ) )
=> ( ( ( ord_less_eq @ A @ Y2 @ Z2 )
=> ~ ( ord_less_eq @ A @ Z2 @ X ) )
=> ~ ( ( ord_less_eq @ A @ Z2 @ X )
=> ~ ( ord_less_eq @ A @ X @ Y2 ) ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_100_antisym__conv,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [Y2: A,X: A] :
( ( ord_less_eq @ A @ Y2 @ X )
=> ( ( ord_less_eq @ A @ X @ Y2 )
= ( X = Y2 ) ) ) ) ).
% antisym_conv
thf(fact_101_ord__eq__le__trans,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [A3: A,B2: A,C: A] :
( ( A3 = B2 )
=> ( ( ord_less_eq @ A @ B2 @ C )
=> ( ord_less_eq @ A @ A3 @ C ) ) ) ) ).
% ord_eq_le_trans
thf(fact_102_ord__le__eq__trans,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [A3: A,B2: A,C: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( B2 = C )
=> ( ord_less_eq @ A @ A3 @ C ) ) ) ) ).
% ord_le_eq_trans
thf(fact_103_order__class_Oorder_Oantisym,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( ord_less_eq @ A @ B2 @ A3 )
=> ( A3 = B2 ) ) ) ) ).
% order_class.order.antisym
thf(fact_104_order__trans,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y2: A,Z2: A] :
( ( ord_less_eq @ A @ X @ Y2 )
=> ( ( ord_less_eq @ A @ Y2 @ Z2 )
=> ( ord_less_eq @ A @ X @ Z2 ) ) ) ) ).
% order_trans
thf(fact_105_dual__order_Orefl,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A3: A] : ( ord_less_eq @ A @ A3 @ A3 ) ) ).
% dual_order.refl
thf(fact_106_linorder__wlog,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [P: A > A > $o,A3: A,B2: A] :
( ! [A5: A,B6: A] :
( ( ord_less_eq @ A @ A5 @ B6 )
=> ( P @ A5 @ B6 ) )
=> ( ! [A5: A,B6: A] :
( ( P @ B6 @ A5 )
=> ( P @ A5 @ B6 ) )
=> ( P @ A3 @ B2 ) ) ) ) ).
% linorder_wlog
thf(fact_107_dual__order_Otrans,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [B2: A,A3: A,C: A] :
( ( ord_less_eq @ A @ B2 @ A3 )
=> ( ( ord_less_eq @ A @ C @ B2 )
=> ( ord_less_eq @ A @ C @ A3 ) ) ) ) ).
% dual_order.trans
thf(fact_108_dual__order_Oantisym,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [B2: A,A3: A] :
( ( ord_less_eq @ A @ B2 @ A3 )
=> ( ( ord_less_eq @ A @ A3 @ B2 )
=> ( A3 = B2 ) ) ) ) ).
% dual_order.antisym
thf(fact_109_pairwise__def,axiom,
! [A: $tType] :
( ( pairwise @ A )
= ( ^ [R: A > A > $o,S2: set @ A] :
! [X2: A] :
( ( member @ A @ X2 @ S2 )
=> ! [Y4: A] :
( ( member @ A @ Y4 @ S2 )
=> ( ( X2 != Y4 )
=> ( R @ X2 @ Y4 ) ) ) ) ) ) ).
% pairwise_def
thf(fact_110_subset__Diff__insert,axiom,
! [A: $tType,A2: set @ A,B: set @ A,X: A,C4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ ( minus_minus @ ( set @ A ) @ B @ ( insert @ A @ X @ C4 ) ) )
= ( ( ord_less_eq @ ( set @ A ) @ A2 @ ( minus_minus @ ( set @ A ) @ B @ C4 ) )
& ~ ( member @ A @ X @ A2 ) ) ) ).
% subset_Diff_insert
thf(fact_111_subset__insert__iff,axiom,
! [A: $tType,A2: set @ A,X: A,B: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ ( insert @ A @ X @ B ) )
= ( ( ( member @ A @ X @ A2 )
=> ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A2 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) @ B ) )
& ( ~ ( member @ A @ X @ A2 )
=> ( ord_less_eq @ ( set @ A ) @ A2 @ B ) ) ) ) ).
% subset_insert_iff
thf(fact_112_Diff__single__insert,axiom,
! [A: $tType,A2: set @ A,X: A,B: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A2 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) @ B )
=> ( ord_less_eq @ ( set @ A ) @ A2 @ ( insert @ A @ X @ B ) ) ) ).
% Diff_single_insert
thf(fact_113_insert__Diff__if,axiom,
! [A: $tType,X: A,B: set @ A,A2: set @ A] :
( ( ( member @ A @ X @ B )
=> ( ( minus_minus @ ( set @ A ) @ ( insert @ A @ X @ A2 ) @ B )
= ( minus_minus @ ( set @ A ) @ A2 @ B ) ) )
& ( ~ ( member @ A @ X @ B )
=> ( ( minus_minus @ ( set @ A ) @ ( insert @ A @ X @ A2 ) @ B )
= ( insert @ A @ X @ ( minus_minus @ ( set @ A ) @ A2 @ B ) ) ) ) ) ).
% insert_Diff_if
thf(fact_114_bot_Oextremum,axiom,
! [A: $tType] :
( ( order_bot @ A @ ( type2 @ A ) )
=> ! [A3: A] : ( ord_less_eq @ A @ ( bot_bot @ A ) @ A3 ) ) ).
% bot.extremum
thf(fact_115_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_116_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_117_Set_Oinsert__mono,axiom,
! [A: $tType,C4: set @ A,D3: set @ A,A3: A] :
( ( ord_less_eq @ ( set @ A ) @ C4 @ D3 )
=> ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ A3 @ C4 ) @ ( insert @ A @ A3 @ D3 ) ) ) ).
% Set.insert_mono
thf(fact_118_subset__insert,axiom,
! [A: $tType,X: A,A2: set @ A,B: set @ A] :
( ~ ( member @ A @ X @ A2 )
=> ( ( ord_less_eq @ ( set @ A ) @ A2 @ ( insert @ A @ X @ B ) )
= ( ord_less_eq @ ( set @ A ) @ A2 @ B ) ) ) ).
% subset_insert
thf(fact_119_subset__insertI,axiom,
! [A: $tType,B: set @ A,A3: A] : ( ord_less_eq @ ( set @ A ) @ B @ ( insert @ A @ A3 @ B ) ) ).
% subset_insertI
thf(fact_120_subset__insertI2,axiom,
! [A: $tType,A2: set @ A,B: set @ A,B2: A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B )
=> ( ord_less_eq @ ( set @ A ) @ A2 @ ( insert @ A @ B2 @ B ) ) ) ).
% subset_insertI2
thf(fact_121_pairwise__empty,axiom,
! [A: $tType,P: A > A > $o] : ( pairwise @ A @ P @ ( bot_bot @ ( set @ A ) ) ) ).
% pairwise_empty
thf(fact_122_Diff__insert,axiom,
! [A: $tType,A2: set @ A,A3: A,B: set @ A] :
( ( minus_minus @ ( set @ A ) @ A2 @ ( insert @ A @ A3 @ B ) )
= ( minus_minus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A2 @ B ) @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Diff_insert
thf(fact_123_insert__Diff,axiom,
! [A: $tType,A3: A,A2: set @ A] :
( ( member @ A @ A3 @ A2 )
=> ( ( insert @ A @ A3 @ ( minus_minus @ ( set @ A ) @ A2 @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) )
= A2 ) ) ).
% insert_Diff
thf(fact_124_Diff__insert2,axiom,
! [A: $tType,A2: set @ A,A3: A,B: set @ A] :
( ( minus_minus @ ( set @ A ) @ A2 @ ( insert @ A @ A3 @ B ) )
= ( minus_minus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A2 @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) @ B ) ) ).
% Diff_insert2
thf(fact_125_Diff__insert__absorb,axiom,
! [A: $tType,X: A,A2: set @ A] :
( ~ ( member @ A @ X @ A2 )
=> ( ( minus_minus @ ( set @ A ) @ ( insert @ A @ X @ A2 ) @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= A2 ) ) ).
% Diff_insert_absorb
thf(fact_126_pairwise__insert,axiom,
! [A: $tType,R2: A > A > $o,X: A,S3: set @ A] :
( ( pairwise @ A @ R2 @ ( insert @ A @ X @ S3 ) )
= ( ! [Y4: A] :
( ( ( member @ A @ Y4 @ S3 )
& ( Y4 != X ) )
=> ( ( R2 @ X @ Y4 )
& ( R2 @ Y4 @ X ) ) )
& ( pairwise @ A @ R2 @ S3 ) ) ) ).
% pairwise_insert
thf(fact_127_subset__singletonD,axiom,
! [A: $tType,A2: set @ A,X: A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
=> ( ( A2
= ( bot_bot @ ( set @ A ) ) )
| ( A2
= ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% subset_singletonD
thf(fact_128_subset__singleton__iff,axiom,
! [A: $tType,X4: set @ A,A3: A] :
( ( ord_less_eq @ ( set @ A ) @ X4 @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) )
= ( ( X4
= ( bot_bot @ ( set @ A ) ) )
| ( X4
= ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% subset_singleton_iff
thf(fact_129_is__singleton__the__elem,axiom,
! [A: $tType] :
( ( is_singleton @ A )
= ( ^ [A4: set @ A] :
( A4
= ( insert @ A @ ( the_elem @ A @ A4 ) @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% is_singleton_the_elem
thf(fact_130_is__singletonI_H,axiom,
! [A: $tType,A2: set @ A] :
( ( A2
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X3: A,Y: A] :
( ( member @ A @ X3 @ A2 )
=> ( ( member @ A @ Y @ A2 )
=> ( X3 = Y ) ) )
=> ( is_singleton @ A @ A2 ) ) ) ).
% is_singletonI'
thf(fact_131_minus__apply,axiom,
! [B3: $tType,A: $tType] :
( ( minus @ B3 @ ( type2 @ B3 ) )
=> ( ( minus_minus @ ( A > B3 ) )
= ( ^ [A4: A > B3,B5: A > B3,X2: A] : ( minus_minus @ B3 @ ( A4 @ X2 ) @ ( B5 @ X2 ) ) ) ) ) ).
% minus_apply
thf(fact_132_fpslsts__def,axiom,
! [A: $tType] :
( ( lList22096119349pslsts @ A )
= ( ^ [A4: set @ A] : ( minus_minus @ ( set @ ( coinductive_llist @ A ) ) @ ( lList2236698231inlsts @ A @ A4 ) @ ( insert @ ( coinductive_llist @ A ) @ ( coinductive_LNil @ A ) @ ( bot_bot @ ( set @ ( coinductive_llist @ A ) ) ) ) ) ) ) ).
% fpslsts_def
thf(fact_133_insert__subsetI,axiom,
! [A: $tType,X: A,A2: set @ A,X4: set @ A] :
( ( member @ A @ X @ A2 )
=> ( ( ord_less_eq @ ( set @ A ) @ X4 @ A2 )
=> ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ X @ X4 ) @ A2 ) ) ) ).
% insert_subsetI
thf(fact_134_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_135_diff__eq__diff__less__eq,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
=> ! [A3: A,B2: A,C: A,D2: A] :
( ( ( minus_minus @ A @ A3 @ B2 )
= ( minus_minus @ A @ C @ D2 ) )
=> ( ( ord_less_eq @ A @ A3 @ B2 )
= ( ord_less_eq @ A @ C @ D2 ) ) ) ) ).
% diff_eq_diff_less_eq
thf(fact_136_diff__right__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
=> ! [A3: A,B2: A,C: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ord_less_eq @ A @ ( minus_minus @ A @ A3 @ C ) @ ( minus_minus @ A @ B2 @ C ) ) ) ) ).
% diff_right_mono
thf(fact_137_diff__left__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
=> ! [B2: A,A3: A,C: A] :
( ( ord_less_eq @ A @ B2 @ A3 )
=> ( ord_less_eq @ A @ ( minus_minus @ A @ C @ A3 ) @ ( minus_minus @ A @ C @ B2 ) ) ) ) ).
% diff_left_mono
thf(fact_138_diff__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
=> ! [A3: A,B2: A,D2: A,C: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( ord_less_eq @ A @ D2 @ C )
=> ( ord_less_eq @ A @ ( minus_minus @ A @ A3 @ C ) @ ( minus_minus @ A @ B2 @ D2 ) ) ) ) ) ).
% diff_mono
thf(fact_139_finlsts_OLNil__fin,axiom,
! [A: $tType,A2: set @ A] : ( member @ ( coinductive_llist @ A ) @ ( coinductive_LNil @ A ) @ ( lList2236698231inlsts @ A @ A2 ) ) ).
% finlsts.LNil_fin
thf(fact_140_fun__diff__def,axiom,
! [B3: $tType,A: $tType] :
( ( minus @ B3 @ ( type2 @ B3 ) )
=> ( ( minus_minus @ ( A > B3 ) )
= ( ^ [A4: A > B3,B5: A > B3,X2: A] : ( minus_minus @ B3 @ ( A4 @ X2 ) @ ( B5 @ X2 ) ) ) ) ) ).
% fun_diff_def
thf(fact_141_diff__eq__diff__eq,axiom,
! [A: $tType] :
( ( group_add @ A @ ( type2 @ A ) )
=> ! [A3: A,B2: A,C: A,D2: A] :
( ( ( minus_minus @ A @ A3 @ B2 )
= ( minus_minus @ A @ C @ D2 ) )
=> ( ( A3 = B2 )
= ( C = D2 ) ) ) ) ).
% diff_eq_diff_eq
thf(fact_142_diff__right__commute,axiom,
! [A: $tType] :
( ( cancel146912293up_add @ A @ ( type2 @ A ) )
=> ! [A3: A,C: A,B2: A] :
( ( minus_minus @ A @ ( minus_minus @ A @ A3 @ C ) @ B2 )
= ( minus_minus @ A @ ( minus_minus @ A @ A3 @ B2 ) @ C ) ) ) ).
% diff_right_commute
thf(fact_143_remove__def,axiom,
! [A: $tType] :
( ( remove @ A )
= ( ^ [X2: A,A4: set @ A] : ( minus_minus @ ( set @ A ) @ A4 @ ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% remove_def
thf(fact_144_psubset__insert__iff,axiom,
! [A: $tType,A2: set @ A,X: A,B: set @ A] :
( ( ord_less @ ( set @ A ) @ A2 @ ( insert @ A @ X @ B ) )
= ( ( ( member @ A @ X @ B )
=> ( ord_less @ ( set @ A ) @ A2 @ B ) )
& ( ~ ( member @ A @ X @ B )
=> ( ( ( member @ A @ X @ A2 )
=> ( ord_less @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A2 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) @ B ) )
& ( ~ ( member @ A @ X @ A2 )
=> ( ord_less_eq @ ( set @ A ) @ A2 @ B ) ) ) ) ) ) ).
% psubset_insert_iff
thf(fact_145_inflsts__def,axiom,
! [A: $tType] :
( ( lList21612149805nflsts @ A )
= ( ^ [A4: set @ A] : ( minus_minus @ ( set @ ( coinductive_llist @ A ) ) @ ( lList2435255213lllsts @ A @ A4 ) @ ( lList2236698231inlsts @ A @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% inflsts_def
thf(fact_146_lSup__minus__LNil,axiom,
! [A: $tType,Y5: set @ ( coinductive_llist @ A )] :
( ( coinductive_lSup @ A @ ( minus_minus @ ( set @ ( coinductive_llist @ A ) ) @ Y5 @ ( insert @ ( coinductive_llist @ A ) @ ( coinductive_LNil @ A ) @ ( bot_bot @ ( set @ ( coinductive_llist @ A ) ) ) ) ) )
= ( coinductive_lSup @ A @ Y5 ) ) ).
% lSup_minus_LNil
thf(fact_147_top__apply,axiom,
! [C2: $tType,D: $tType] :
( ( top @ C2 @ ( type2 @ C2 ) )
=> ( ( top_top @ ( D > C2 ) )
= ( ^ [X2: D] : ( top_top @ C2 ) ) ) ) ).
% top_apply
thf(fact_148_UNIV__I,axiom,
! [A: $tType,X: A] : ( member @ A @ X @ ( top_top @ ( set @ A ) ) ) ).
% UNIV_I
thf(fact_149_member__remove,axiom,
! [A: $tType,X: A,Y2: A,A2: set @ A] :
( ( member @ A @ X @ ( remove @ A @ Y2 @ A2 ) )
= ( ( member @ A @ X @ A2 )
& ( X != Y2 ) ) ) ).
% member_remove
thf(fact_150_psubsetI,axiom,
! [A: $tType,A2: set @ A,B: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B )
=> ( ( A2 != B )
=> ( ord_less @ ( set @ A ) @ A2 @ B ) ) ) ).
% psubsetI
thf(fact_151_alllsts__UNIV,axiom,
! [A: $tType,S3: coinductive_llist @ A] : ( member @ ( coinductive_llist @ A ) @ S3 @ ( lList2435255213lllsts @ A @ ( top_top @ ( set @ A ) ) ) ) ).
% alllsts_UNIV
thf(fact_152_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_153_lSup__empty,axiom,
! [A: $tType] :
( ( coinductive_lSup @ A @ ( bot_bot @ ( set @ ( coinductive_llist @ A ) ) ) )
= ( coinductive_LNil @ A ) ) ).
% lSup_empty
thf(fact_154_lSup__singleton,axiom,
! [A: $tType,Xs: coinductive_llist @ A] :
( ( coinductive_lSup @ A @ ( insert @ ( coinductive_llist @ A ) @ Xs @ ( bot_bot @ ( set @ ( coinductive_llist @ A ) ) ) ) )
= Xs ) ).
% lSup_singleton
thf(fact_155_diff__strict__right__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
=> ! [A3: A,B2: A,C: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ord_less @ A @ ( minus_minus @ A @ A3 @ C ) @ ( minus_minus @ A @ B2 @ C ) ) ) ) ).
% diff_strict_right_mono
thf(fact_156_diff__strict__left__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
=> ! [B2: A,A3: A,C: A] :
( ( ord_less @ A @ B2 @ A3 )
=> ( ord_less @ A @ ( minus_minus @ A @ C @ A3 ) @ ( minus_minus @ A @ C @ B2 ) ) ) ) ).
% diff_strict_left_mono
thf(fact_157_diff__eq__diff__less,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
=> ! [A3: A,B2: A,C: A,D2: A] :
( ( ( minus_minus @ A @ A3 @ B2 )
= ( minus_minus @ A @ C @ D2 ) )
=> ( ( ord_less @ A @ A3 @ B2 )
= ( ord_less @ A @ C @ D2 ) ) ) ) ).
% diff_eq_diff_less
thf(fact_158_diff__strict__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
=> ! [A3: A,B2: A,D2: A,C: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ( ord_less @ A @ D2 @ C )
=> ( ord_less @ A @ ( minus_minus @ A @ A3 @ C ) @ ( minus_minus @ A @ B2 @ D2 ) ) ) ) ) ).
% diff_strict_mono
thf(fact_159_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_160_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_161_top__greatest,axiom,
! [A: $tType] :
( ( order_top @ A @ ( type2 @ A ) )
=> ! [A3: A] : ( ord_less_eq @ A @ A3 @ ( top_top @ A ) ) ) ).
% top_greatest
thf(fact_162_order_Onot__eq__order__implies__strict,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A3: A,B2: A] :
( ( A3 != B2 )
=> ( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ord_less @ A @ A3 @ B2 ) ) ) ) ).
% order.not_eq_order_implies_strict
thf(fact_163_dual__order_Ostrict__implies__order,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [B2: A,A3: A] :
( ( ord_less @ A @ B2 @ A3 )
=> ( ord_less_eq @ A @ B2 @ A3 ) ) ) ).
% dual_order.strict_implies_order
thf(fact_164_dual__order_Ostrict__iff__order,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ( ( ord_less @ A )
= ( ^ [B7: A,A6: A] :
( ( ord_less_eq @ A @ B7 @ A6 )
& ( A6 != B7 ) ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_165_dual__order_Oorder__iff__strict,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ( ( ord_less_eq @ A )
= ( ^ [B7: A,A6: A] :
( ( ord_less @ A @ B7 @ A6 )
| ( A6 = B7 ) ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_166_order_Ostrict__implies__order,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ord_less_eq @ A @ A3 @ B2 ) ) ) ).
% order.strict_implies_order
thf(fact_167_dense__le__bounded,axiom,
! [A: $tType] :
( ( dense_linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y2: A,Z2: A] :
( ( ord_less @ A @ X @ Y2 )
=> ( ! [W: A] :
( ( ord_less @ A @ X @ W )
=> ( ( ord_less @ A @ W @ Y2 )
=> ( ord_less_eq @ A @ W @ Z2 ) ) )
=> ( ord_less_eq @ A @ Y2 @ Z2 ) ) ) ) ).
% dense_le_bounded
thf(fact_168_dense__ge__bounded,axiom,
! [A: $tType] :
( ( dense_linorder @ A @ ( type2 @ A ) )
=> ! [Z2: A,X: A,Y2: A] :
( ( ord_less @ A @ Z2 @ X )
=> ( ! [W: A] :
( ( ord_less @ A @ Z2 @ W )
=> ( ( ord_less @ A @ W @ X )
=> ( ord_less_eq @ A @ Y2 @ W ) ) )
=> ( ord_less_eq @ A @ Y2 @ Z2 ) ) ) ) ).
% dense_ge_bounded
thf(fact_169_dual__order_Ostrict__trans2,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [B2: A,A3: A,C: A] :
( ( ord_less @ A @ B2 @ A3 )
=> ( ( ord_less_eq @ A @ C @ B2 )
=> ( ord_less @ A @ C @ A3 ) ) ) ) ).
% dual_order.strict_trans2
thf(fact_170_dual__order_Ostrict__trans1,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [B2: A,A3: A,C: A] :
( ( ord_less_eq @ A @ B2 @ A3 )
=> ( ( ord_less @ A @ C @ B2 )
=> ( ord_less @ A @ C @ A3 ) ) ) ) ).
% dual_order.strict_trans1
thf(fact_171_order_Ostrict__iff__order,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ( ( ord_less @ A )
= ( ^ [A6: A,B7: A] :
( ( ord_less_eq @ A @ A6 @ B7 )
& ( A6 != B7 ) ) ) ) ) ).
% order.strict_iff_order
thf(fact_172_order_Oorder__iff__strict,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ( ( ord_less_eq @ A )
= ( ^ [A6: A,B7: A] :
( ( ord_less @ A @ A6 @ B7 )
| ( A6 = B7 ) ) ) ) ) ).
% order.order_iff_strict
thf(fact_173_order_Ostrict__trans2,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A3: A,B2: A,C: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ( ord_less_eq @ A @ B2 @ C )
=> ( ord_less @ A @ A3 @ C ) ) ) ) ).
% order.strict_trans2
thf(fact_174_order_Ostrict__trans1,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A3: A,B2: A,C: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( ord_less @ A @ B2 @ C )
=> ( ord_less @ A @ A3 @ C ) ) ) ) ).
% order.strict_trans1
thf(fact_175_not__le__imp__less,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [Y2: A,X: A] :
( ~ ( ord_less_eq @ A @ Y2 @ X )
=> ( ord_less @ A @ X @ Y2 ) ) ) ).
% not_le_imp_less
thf(fact_176_less__le__not__le,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ( ( ord_less @ A )
= ( ^ [X2: A,Y4: A] :
( ( ord_less_eq @ A @ X2 @ Y4 )
& ~ ( ord_less_eq @ A @ Y4 @ X2 ) ) ) ) ) ).
% less_le_not_le
thf(fact_177_le__imp__less__or__eq,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [X: A,Y2: A] :
( ( ord_less_eq @ A @ X @ Y2 )
=> ( ( ord_less @ A @ X @ Y2 )
| ( X = Y2 ) ) ) ) ).
% le_imp_less_or_eq
thf(fact_178_le__less__linear,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y2: A] :
( ( ord_less_eq @ A @ X @ Y2 )
| ( ord_less @ A @ Y2 @ X ) ) ) ).
% le_less_linear
thf(fact_179_dense__le,axiom,
! [A: $tType] :
( ( dense_linorder @ A @ ( type2 @ A ) )
=> ! [Y2: A,Z2: A] :
( ! [X3: A] :
( ( ord_less @ A @ X3 @ Y2 )
=> ( ord_less_eq @ A @ X3 @ Z2 ) )
=> ( ord_less_eq @ A @ Y2 @ Z2 ) ) ) ).
% dense_le
thf(fact_180_dense__ge,axiom,
! [A: $tType] :
( ( dense_linorder @ A @ ( type2 @ A ) )
=> ! [Z2: A,Y2: A] :
( ! [X3: A] :
( ( ord_less @ A @ Z2 @ X3 )
=> ( ord_less_eq @ A @ Y2 @ X3 ) )
=> ( ord_less_eq @ A @ Y2 @ Z2 ) ) ) ).
% dense_ge
thf(fact_181_less__le__trans,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y2: A,Z2: A] :
( ( ord_less @ A @ X @ Y2 )
=> ( ( ord_less_eq @ A @ Y2 @ Z2 )
=> ( ord_less @ A @ X @ Z2 ) ) ) ) ).
% less_le_trans
thf(fact_182_le__less__trans,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y2: A,Z2: A] :
( ( ord_less_eq @ A @ X @ Y2 )
=> ( ( ord_less @ A @ Y2 @ Z2 )
=> ( ord_less @ A @ X @ Z2 ) ) ) ) ).
% le_less_trans
thf(fact_183_antisym__conv2,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y2: A] :
( ( ord_less_eq @ A @ X @ Y2 )
=> ( ( ~ ( ord_less @ A @ X @ Y2 ) )
= ( X = Y2 ) ) ) ) ).
% antisym_conv2
thf(fact_184_antisym__conv1,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y2: A] :
( ~ ( ord_less @ A @ X @ Y2 )
=> ( ( ord_less_eq @ A @ X @ Y2 )
= ( X = Y2 ) ) ) ) ).
% antisym_conv1
thf(fact_185_less__imp__le,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y2: A] :
( ( ord_less @ A @ X @ Y2 )
=> ( ord_less_eq @ A @ X @ Y2 ) ) ) ).
% less_imp_le
thf(fact_186_le__neq__trans,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( A3 != B2 )
=> ( ord_less @ A @ A3 @ B2 ) ) ) ) ).
% le_neq_trans
thf(fact_187_not__less,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y2: A] :
( ( ~ ( ord_less @ A @ X @ Y2 ) )
= ( ord_less_eq @ A @ Y2 @ X ) ) ) ).
% not_less
thf(fact_188_not__le,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y2: A] :
( ( ~ ( ord_less_eq @ A @ X @ Y2 ) )
= ( ord_less @ A @ Y2 @ X ) ) ) ).
% not_le
thf(fact_189_order__less__le__subst2,axiom,
! [A: $tType,C2: $tType] :
( ( ( order @ C2 @ ( type2 @ C2 ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A3: A,B2: A,F: A > C2,C: C2] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ( ord_less_eq @ C2 @ ( F @ B2 ) @ C )
=> ( ! [X3: A,Y: A] :
( ( ord_less @ A @ X3 @ Y )
=> ( ord_less @ C2 @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less @ C2 @ ( F @ A3 ) @ C ) ) ) ) ) ).
% order_less_le_subst2
thf(fact_190_order__less__le__subst1,axiom,
! [A: $tType,B3: $tType] :
( ( ( order @ B3 @ ( type2 @ B3 ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A3: A,F: B3 > A,B2: B3,C: B3] :
( ( ord_less @ A @ A3 @ ( F @ B2 ) )
=> ( ( ord_less_eq @ B3 @ B2 @ C )
=> ( ! [X3: B3,Y: B3] :
( ( ord_less_eq @ B3 @ X3 @ Y )
=> ( ord_less_eq @ A @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less @ A @ A3 @ ( F @ C ) ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_191_order__le__less__subst2,axiom,
! [A: $tType,C2: $tType] :
( ( ( order @ C2 @ ( type2 @ C2 ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A3: A,B2: A,F: A > C2,C: C2] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( ord_less @ C2 @ ( F @ B2 ) @ C )
=> ( ! [X3: A,Y: A] :
( ( ord_less_eq @ A @ X3 @ Y )
=> ( ord_less_eq @ C2 @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less @ C2 @ ( F @ A3 ) @ C ) ) ) ) ) ).
% order_le_less_subst2
thf(fact_192_order__le__less__subst1,axiom,
! [A: $tType,B3: $tType] :
( ( ( order @ B3 @ ( type2 @ B3 ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A3: A,F: B3 > A,B2: B3,C: B3] :
( ( ord_less_eq @ A @ A3 @ ( F @ B2 ) )
=> ( ( ord_less @ B3 @ B2 @ C )
=> ( ! [X3: B3,Y: B3] :
( ( ord_less @ B3 @ X3 @ Y )
=> ( ord_less @ A @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less @ A @ A3 @ ( F @ C ) ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_193_less__le,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ( ( ord_less @ A )
= ( ^ [X2: A,Y4: A] :
( ( ord_less_eq @ A @ X2 @ Y4 )
& ( X2 != Y4 ) ) ) ) ) ).
% less_le
thf(fact_194_le__less,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ( ( ord_less_eq @ A )
= ( ^ [X2: A,Y4: A] :
( ( ord_less @ A @ X2 @ Y4 )
| ( X2 = Y4 ) ) ) ) ) ).
% le_less
thf(fact_195_leI,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y2: A] :
( ~ ( ord_less @ A @ X @ Y2 )
=> ( ord_less_eq @ A @ Y2 @ X ) ) ) ).
% leI
thf(fact_196_leD,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [Y2: A,X: A] :
( ( ord_less_eq @ A @ Y2 @ X )
=> ~ ( ord_less @ A @ X @ Y2 ) ) ) ).
% leD
thf(fact_197_bot_Onot__eq__extremum,axiom,
! [A: $tType] :
( ( order_bot @ A @ ( type2 @ A ) )
=> ! [A3: A] :
( ( A3
!= ( bot_bot @ A ) )
= ( ord_less @ A @ ( bot_bot @ A ) @ A3 ) ) ) ).
% bot.not_eq_extremum
thf(fact_198_bot_Oextremum__strict,axiom,
! [A: $tType] :
( ( order_bot @ A @ ( type2 @ A ) )
=> ! [A3: A] :
~ ( ord_less @ A @ A3 @ ( bot_bot @ A ) ) ) ).
% bot.extremum_strict
thf(fact_199_not__psubset__empty,axiom,
! [A: $tType,A2: set @ A] :
~ ( ord_less @ ( set @ A ) @ A2 @ ( bot_bot @ ( set @ A ) ) ) ).
% not_psubset_empty
thf(fact_200_subset__iff__psubset__eq,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A4: set @ A,B5: set @ A] :
( ( ord_less @ ( set @ A ) @ A4 @ B5 )
| ( A4 = B5 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_201_subset__psubset__trans,axiom,
! [A: $tType,A2: set @ A,B: set @ A,C4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B )
=> ( ( ord_less @ ( set @ A ) @ B @ C4 )
=> ( ord_less @ ( set @ A ) @ A2 @ C4 ) ) ) ).
% subset_psubset_trans
thf(fact_202_subset__not__subset__eq,axiom,
! [A: $tType] :
( ( ord_less @ ( set @ A ) )
= ( ^ [A4: set @ A,B5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B5 )
& ~ ( ord_less_eq @ ( set @ A ) @ B5 @ A4 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_203_psubset__subset__trans,axiom,
! [A: $tType,A2: set @ A,B: set @ A,C4: set @ A] :
( ( ord_less @ ( set @ A ) @ A2 @ B )
=> ( ( ord_less_eq @ ( set @ A ) @ B @ C4 )
=> ( ord_less @ ( set @ A ) @ A2 @ C4 ) ) ) ).
% psubset_subset_trans
thf(fact_204_psubset__imp__subset,axiom,
! [A: $tType,A2: set @ A,B: set @ A] :
( ( ord_less @ ( set @ A ) @ A2 @ B )
=> ( ord_less_eq @ ( set @ A ) @ A2 @ B ) ) ).
% psubset_imp_subset
thf(fact_205_psubset__eq,axiom,
! [A: $tType] :
( ( ord_less @ ( set @ A ) )
= ( ^ [A4: set @ A,B5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B5 )
& ( A4 != B5 ) ) ) ) ).
% psubset_eq
thf(fact_206_psubsetE,axiom,
! [A: $tType,A2: set @ A,B: set @ A] :
( ( ord_less @ ( set @ A ) @ A2 @ B )
=> ~ ( ( ord_less_eq @ ( set @ A ) @ A2 @ B )
=> ( ord_less_eq @ ( set @ A ) @ B @ A2 ) ) ) ).
% psubsetE
thf(fact_207_empty__not__UNIV,axiom,
! [A: $tType] :
( ( bot_bot @ ( set @ A ) )
!= ( top_top @ ( set @ A ) ) ) ).
% empty_not_UNIV
thf(fact_208_psubset__imp__ex__mem,axiom,
! [A: $tType,A2: set @ A,B: set @ A] :
( ( ord_less @ ( set @ A ) @ A2 @ B )
=> ? [B6: A] : ( member @ A @ B6 @ ( minus_minus @ ( set @ A ) @ B @ A2 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_209_subset__UNIV,axiom,
! [A: $tType,A2: set @ A] : ( ord_less_eq @ ( set @ A ) @ A2 @ ( top_top @ ( set @ A ) ) ) ).
% subset_UNIV
thf(fact_210_insert__UNIV,axiom,
! [A: $tType,X: A] :
( ( insert @ A @ X @ ( top_top @ ( set @ A ) ) )
= ( top_top @ ( set @ A ) ) ) ).
% insert_UNIV
thf(fact_211_psubsetD,axiom,
! [A: $tType,A2: set @ A,B: set @ A,C: A] :
( ( ord_less @ ( set @ A ) @ A2 @ B )
=> ( ( member @ A @ C @ A2 )
=> ( member @ A @ C @ B ) ) ) ).
% psubsetD
thf(fact_212_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_213_UNIV__witness,axiom,
! [A: $tType] :
? [X3: A] : ( member @ A @ X3 @ ( top_top @ ( set @ A ) ) ) ).
% UNIV_witness
thf(fact_214_psubset__trans,axiom,
! [A: $tType,A2: set @ A,B: set @ A,C4: set @ A] :
( ( ord_less @ ( set @ A ) @ A2 @ B )
=> ( ( ord_less @ ( set @ A ) @ B @ C4 )
=> ( ord_less @ ( set @ A ) @ A2 @ C4 ) ) ) ).
% psubset_trans
thf(fact_215_ord__eq__less__subst,axiom,
! [A: $tType,B3: $tType] :
( ( ( ord @ B3 @ ( type2 @ B3 ) )
& ( ord @ A @ ( type2 @ A ) ) )
=> ! [A3: A,F: B3 > A,B2: B3,C: B3] :
( ( A3
= ( F @ B2 ) )
=> ( ( ord_less @ B3 @ B2 @ C )
=> ( ! [X3: B3,Y: B3] :
( ( ord_less @ B3 @ X3 @ Y )
=> ( ord_less @ A @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less @ A @ A3 @ ( F @ C ) ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_216_ord__less__eq__subst,axiom,
! [A: $tType,B3: $tType] :
( ( ( ord @ B3 @ ( type2 @ B3 ) )
& ( ord @ A @ ( type2 @ A ) ) )
=> ! [A3: A,B2: A,F: A > B3,C: B3] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: A,Y: A] :
( ( ord_less @ A @ X3 @ Y )
=> ( ord_less @ B3 @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less @ B3 @ ( F @ A3 ) @ C ) ) ) ) ) ).
% ord_less_eq_subst
thf(fact_217_order__less__subst1,axiom,
! [A: $tType,B3: $tType] :
( ( ( order @ B3 @ ( type2 @ B3 ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A3: A,F: B3 > A,B2: B3,C: B3] :
( ( ord_less @ A @ A3 @ ( F @ B2 ) )
=> ( ( ord_less @ B3 @ B2 @ C )
=> ( ! [X3: B3,Y: B3] :
( ( ord_less @ B3 @ X3 @ Y )
=> ( ord_less @ A @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less @ A @ A3 @ ( F @ C ) ) ) ) ) ) ).
% order_less_subst1
thf(fact_218_order__less__subst2,axiom,
! [A: $tType,C2: $tType] :
( ( ( order @ C2 @ ( type2 @ C2 ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A3: A,B2: A,F: A > C2,C: C2] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ( ord_less @ C2 @ ( F @ B2 ) @ C )
=> ( ! [X3: A,Y: A] :
( ( ord_less @ A @ X3 @ Y )
=> ( ord_less @ C2 @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less @ C2 @ ( F @ A3 ) @ C ) ) ) ) ) ).
% order_less_subst2
thf(fact_219_lt__ex,axiom,
! [A: $tType] :
( ( no_bot @ A @ ( type2 @ A ) )
=> ! [X: A] :
? [Y: A] : ( ord_less @ A @ Y @ X ) ) ).
% lt_ex
thf(fact_220_gt__ex,axiom,
! [A: $tType] :
( ( no_top @ A @ ( type2 @ A ) )
=> ! [X: A] :
? [X1: A] : ( ord_less @ A @ X @ X1 ) ) ).
% gt_ex
thf(fact_221_neqE,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y2: A] :
( ( X != Y2 )
=> ( ~ ( ord_less @ A @ X @ Y2 )
=> ( ord_less @ A @ Y2 @ X ) ) ) ) ).
% neqE
thf(fact_222_neq__iff,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y2: A] :
( ( X != Y2 )
= ( ( ord_less @ A @ X @ Y2 )
| ( ord_less @ A @ Y2 @ X ) ) ) ) ).
% neq_iff
thf(fact_223_order_Oasym,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ~ ( ord_less @ A @ B2 @ A3 ) ) ) ).
% order.asym
thf(fact_224_dense,axiom,
! [A: $tType] :
( ( dense_order @ A @ ( type2 @ A ) )
=> ! [X: A,Y2: A] :
( ( ord_less @ A @ X @ Y2 )
=> ? [Z3: A] :
( ( ord_less @ A @ X @ Z3 )
& ( ord_less @ A @ Z3 @ Y2 ) ) ) ) ).
% dense
thf(fact_225_less__imp__neq,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [X: A,Y2: A] :
( ( ord_less @ A @ X @ Y2 )
=> ( X != Y2 ) ) ) ).
% less_imp_neq
thf(fact_226_less__asym,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y2: A] :
( ( ord_less @ A @ X @ Y2 )
=> ~ ( ord_less @ A @ Y2 @ X ) ) ) ).
% less_asym
thf(fact_227_less__asym_H,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ~ ( ord_less @ A @ B2 @ A3 ) ) ) ).
% less_asym'
thf(fact_228_less__trans,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y2: A,Z2: A] :
( ( ord_less @ A @ X @ Y2 )
=> ( ( ord_less @ A @ Y2 @ Z2 )
=> ( ord_less @ A @ X @ Z2 ) ) ) ) ).
% less_trans
thf(fact_229_less__linear,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y2: A] :
( ( ord_less @ A @ X @ Y2 )
| ( X = Y2 )
| ( ord_less @ A @ Y2 @ X ) ) ) ).
% less_linear
thf(fact_230_less__irrefl,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A] :
~ ( ord_less @ A @ X @ X ) ) ).
% less_irrefl
thf(fact_231_ord__eq__less__trans,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [A3: A,B2: A,C: A] :
( ( A3 = B2 )
=> ( ( ord_less @ A @ B2 @ C )
=> ( ord_less @ A @ A3 @ C ) ) ) ) ).
% ord_eq_less_trans
thf(fact_232_ord__less__eq__trans,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [A3: A,B2: A,C: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ( B2 = C )
=> ( ord_less @ A @ A3 @ C ) ) ) ) ).
% ord_less_eq_trans
thf(fact_233_dual__order_Oasym,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [B2: A,A3: A] :
( ( ord_less @ A @ B2 @ A3 )
=> ~ ( ord_less @ A @ A3 @ B2 ) ) ) ).
% dual_order.asym
thf(fact_234_less__imp__not__eq,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [X: A,Y2: A] :
( ( ord_less @ A @ X @ Y2 )
=> ( X != Y2 ) ) ) ).
% less_imp_not_eq
thf(fact_235_less__not__sym,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y2: A] :
( ( ord_less @ A @ X @ Y2 )
=> ~ ( ord_less @ A @ Y2 @ X ) ) ) ).
% less_not_sym
thf(fact_236_less__induct,axiom,
! [A: $tType] :
( ( wellorder @ A @ ( type2 @ A ) )
=> ! [P: A > $o,A3: A] :
( ! [X3: A] :
( ! [Y6: A] :
( ( ord_less @ A @ Y6 @ X3 )
=> ( P @ Y6 ) )
=> ( P @ X3 ) )
=> ( P @ A3 ) ) ) ).
% less_induct
thf(fact_237_antisym__conv3,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [Y2: A,X: A] :
( ~ ( ord_less @ A @ Y2 @ X )
=> ( ( ~ ( ord_less @ A @ X @ Y2 ) )
= ( X = Y2 ) ) ) ) ).
% antisym_conv3
thf(fact_238_less__imp__not__eq2,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [X: A,Y2: A] :
( ( ord_less @ A @ X @ Y2 )
=> ( Y2 != X ) ) ) ).
% less_imp_not_eq2
thf(fact_239_less__imp__triv,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y2: A,P: $o] :
( ( ord_less @ A @ X @ Y2 )
=> ( ( ord_less @ A @ Y2 @ X )
=> P ) ) ) ).
% less_imp_triv
thf(fact_240_linorder__cases,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y2: A] :
( ~ ( ord_less @ A @ X @ Y2 )
=> ( ( X != Y2 )
=> ( ord_less @ A @ Y2 @ X ) ) ) ) ).
% linorder_cases
thf(fact_241_dual__order_Oirrefl,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A3: A] :
~ ( ord_less @ A @ A3 @ A3 ) ) ).
% dual_order.irrefl
thf(fact_242_order_Ostrict__trans,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A3: A,B2: A,C: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ( ord_less @ A @ B2 @ C )
=> ( ord_less @ A @ A3 @ C ) ) ) ) ).
% order.strict_trans
thf(fact_243_less__imp__not__less,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y2: A] :
( ( ord_less @ A @ X @ Y2 )
=> ~ ( ord_less @ A @ Y2 @ X ) ) ) ).
% less_imp_not_less
thf(fact_244_dual__order_Ostrict__trans,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [B2: A,A3: A,C: A] :
( ( ord_less @ A @ B2 @ A3 )
=> ( ( ord_less @ A @ C @ B2 )
=> ( ord_less @ A @ C @ A3 ) ) ) ) ).
% dual_order.strict_trans
thf(fact_245_top_Oextremum__strict,axiom,
! [A: $tType] :
( ( order_top @ A @ ( type2 @ A ) )
=> ! [A3: A] :
~ ( ord_less @ A @ ( top_top @ A ) @ A3 ) ) ).
% top.extremum_strict
thf(fact_246_top_Onot__eq__extremum,axiom,
! [A: $tType] :
( ( order_top @ A @ ( type2 @ A ) )
=> ! [A3: A] :
( ( A3
!= ( top_top @ A ) )
= ( ord_less @ A @ A3 @ ( top_top @ A ) ) ) ) ).
% top.not_eq_extremum
thf(fact_247_not__less__iff__gr__or__eq,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y2: A] :
( ( ~ ( ord_less @ A @ X @ Y2 ) )
= ( ( ord_less @ A @ Y2 @ X )
| ( X = Y2 ) ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_248_order_Ostrict__implies__not__eq,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ( A3 != B2 ) ) ) ).
% order.strict_implies_not_eq
thf(fact_249_dual__order_Ostrict__implies__not__eq,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [B2: A,A3: A] :
( ( ord_less @ A @ B2 @ A3 )
=> ( A3 != B2 ) ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_250_lSup__insert__LNil,axiom,
! [A: $tType,Y5: set @ ( coinductive_llist @ A )] :
( ( coinductive_lSup @ A @ ( insert @ ( coinductive_llist @ A ) @ ( coinductive_LNil @ A ) @ Y5 ) )
= ( coinductive_lSup @ A @ Y5 ) ) ).
% lSup_insert_LNil
thf(fact_251_wlog__linorder__le,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [P: A > A > $o,B2: A,A3: A] :
( ! [A5: A,B6: A] :
( ( ord_less_eq @ A @ A5 @ B6 )
=> ( P @ A5 @ B6 ) )
=> ( ( ( P @ B2 @ A3 )
=> ( P @ A3 @ B2 ) )
=> ( P @ A3 @ B2 ) ) ) ) ).
% wlog_linorder_le
thf(fact_252_Topological__Spaces_OUNIV__not__singleton,axiom,
! [A: $tType] :
( ( topolo890362671_space @ A @ ( type2 @ A ) )
=> ! [X: A] :
( ( top_top @ ( set @ A ) )
!= ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Topological_Spaces.UNIV_not_singleton
thf(fact_253_iso__tuple__UNIV__I,axiom,
! [A: $tType,X: A] : ( member @ A @ X @ ( top_top @ ( set @ A ) ) ) ).
% iso_tuple_UNIV_I
thf(fact_254_complete__interval,axiom,
! [A: $tType] :
( ( condit1037483654norder @ A @ ( type2 @ A ) )
=> ! [A3: A,B2: A,P: A > $o] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ( P @ A3 )
=> ( ~ ( P @ B2 )
=> ? [C5: A] :
( ( ord_less_eq @ A @ A3 @ C5 )
& ( ord_less_eq @ A @ C5 @ B2 )
& ! [X5: A] :
( ( ( ord_less_eq @ A @ A3 @ X5 )
& ( ord_less @ A @ X5 @ C5 ) )
=> ( P @ X5 ) )
& ! [D4: A] :
( ! [X3: A] :
( ( ( ord_less_eq @ A @ A3 @ X3 )
& ( ord_less @ A @ X3 @ D4 ) )
=> ( P @ X3 ) )
=> ( ord_less_eq @ A @ D4 @ C5 ) ) ) ) ) ) ) ).
% complete_interval
thf(fact_255_less__fun__def,axiom,
! [B3: $tType,A: $tType] :
( ( ord @ B3 @ ( type2 @ B3 ) )
=> ( ( ord_less @ ( A > B3 ) )
= ( ^ [F2: A > B3,G2: A > B3] :
( ( ord_less_eq @ ( A > B3 ) @ F2 @ G2 )
& ~ ( ord_less_eq @ ( A > B3 ) @ G2 @ F2 ) ) ) ) ) ).
% less_fun_def
%----Type constructors (25)
thf(tcon_fun___Orderings_Oorder__top,axiom,
! [A7: $tType,A8: $tType] :
( ( order_top @ A8 @ ( type2 @ A8 ) )
=> ( order_top @ ( A7 > A8 ) @ ( type2 @ ( A7 > A8 ) ) ) ) ).
thf(tcon_fun___Orderings_Oorder__bot,axiom,
! [A7: $tType,A8: $tType] :
( ( order_bot @ A8 @ ( type2 @ A8 ) )
=> ( order_bot @ ( A7 > A8 ) @ ( type2 @ ( A7 > A8 ) ) ) ) ).
thf(tcon_fun___Orderings_Opreorder,axiom,
! [A7: $tType,A8: $tType] :
( ( preorder @ A8 @ ( type2 @ A8 ) )
=> ( preorder @ ( A7 > A8 ) @ ( type2 @ ( A7 > A8 ) ) ) ) ).
thf(tcon_fun___Orderings_Oorder,axiom,
! [A7: $tType,A8: $tType] :
( ( order @ A8 @ ( type2 @ A8 ) )
=> ( order @ ( A7 > A8 ) @ ( type2 @ ( A7 > A8 ) ) ) ) ).
thf(tcon_fun___Orderings_Otop,axiom,
! [A7: $tType,A8: $tType] :
( ( top @ A8 @ ( type2 @ A8 ) )
=> ( top @ ( A7 > A8 ) @ ( type2 @ ( A7 > A8 ) ) ) ) ).
thf(tcon_fun___Orderings_Oord,axiom,
! [A7: $tType,A8: $tType] :
( ( ord @ A8 @ ( type2 @ A8 ) )
=> ( ord @ ( A7 > A8 ) @ ( type2 @ ( A7 > A8 ) ) ) ) ).
thf(tcon_fun___Orderings_Obot,axiom,
! [A7: $tType,A8: $tType] :
( ( bot @ A8 @ ( type2 @ A8 ) )
=> ( bot @ ( A7 > A8 ) @ ( type2 @ ( A7 > A8 ) ) ) ) ).
thf(tcon_fun___Groups_Ominus,axiom,
! [A7: $tType,A8: $tType] :
( ( minus @ A8 @ ( type2 @ A8 ) )
=> ( minus @ ( A7 > A8 ) @ ( type2 @ ( A7 > A8 ) ) ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder__top_1,axiom,
! [A7: $tType] : ( order_top @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder__bot_2,axiom,
! [A7: $tType] : ( order_bot @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Opreorder_3,axiom,
! [A7: $tType] : ( preorder @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder_4,axiom,
! [A7: $tType] : ( order @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Otop_5,axiom,
! [A7: $tType] : ( top @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Oord_6,axiom,
! [A7: $tType] : ( ord @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Obot_7,axiom,
! [A7: $tType] : ( bot @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
thf(tcon_Set_Oset___Groups_Ominus_8,axiom,
! [A7: $tType] : ( minus @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
thf(tcon_HOL_Obool___Orderings_Oorder__top_9,axiom,
order_top @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Oorder__bot_10,axiom,
order_bot @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Opreorder_11,axiom,
preorder @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Olinorder,axiom,
linorder @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Oorder_12,axiom,
order @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Otop_13,axiom,
top @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Oord_14,axiom,
ord @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Obot_15,axiom,
bot @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Groups_Ominus_16,axiom,
minus @ $o @ ( type2 @ $o ) ).
%----Conjectures (1)
thf(conj_0,conjecture,
( ( lList2435255213lllsts @ a @ ( bot_bot @ ( set @ a ) ) )
= ( insert @ ( coinductive_llist @ a ) @ ( coinductive_LNil @ a ) @ ( bot_bot @ ( set @ ( coinductive_llist @ a ) ) ) ) ) ).
%------------------------------------------------------------------------------