TPTP Problem File: DAT167^1.p
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%------------------------------------------------------------------------------
% File : DAT167^1 : TPTP v8.2.0. Released v7.0.0.
% Domain : Data Structures
% Problem : Huffman 1213
% Version : [Bla16] axioms : Especial.
% English :
% Refs : [Bla08] Blanchette (2008), The Textbook Proof of Huffman's Alg
% : [RB15] Reynolds & Blanchette (2015), A Decision Procedure for
% : [Bla16] Blanchette (2016), Email to Geoff Sutcliffe
% Source : [Bla16]
% Names : huffman__1213.p [Bla16]
% Status : Theorem
% Rating : 1.00 v7.1.0
% Syntax : Number of formulae : 360 ( 128 unt; 53 typ; 0 def)
% Number of atoms : 801 ( 268 equ; 0 cnn)
% Maximal formula atoms : 13 ( 2 avg)
% Number of connectives : 3849 ( 64 ~; 11 |; 50 &;3396 @)
% ( 0 <=>; 328 =>; 0 <=; 0 <~>)
% Maximal formula depth : 18 ( 7 avg)
% Number of types : 3 ( 2 usr)
% Number of type conns : 192 ( 192 >; 0 *; 0 +; 0 <<)
% Number of symbols : 52 ( 51 usr; 5 con; 0-6 aty)
% Number of variables : 1009 ( 67 ^; 890 !; 8 ?;1009 :)
% ( 44 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2016-07-13 14:40:52.298
%------------------------------------------------------------------------------
%----Could-be-implicit typings (5)
thf(ty_t_Huffman__Mirabelle__gjololrwrm_Otree,type,
huffma16452318e_tree: $tType > $tType ).
thf(ty_t_Set_Oset,type,
set: $tType > $tType ).
thf(ty_t_Nat_Onat,type,
nat: $tType ).
thf(ty_t_itself,type,
itself: $tType > $tType ).
thf(ty_tf_a,type,
a: $tType ).
%----Explicit typings (48)
thf(sy_cl_Lattices_Obounded__lattice,type,
bounded_lattice:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
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_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_Lattices_Olattice,type,
lattice:
!>[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_Lattices_Oboolean__algebra,type,
boolean_algebra:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Lattices_Osemilattice__sup,type,
semilattice_sup:
!>[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_Lattices_Obounded__lattice__bot,type,
bounded_lattice_bot:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Ocancel__ab__semigroup__add,type,
cancel146912293up_add:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Lattices_Obounded__semilattice__sup__bot,type,
bounde1808546759up_bot:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_c_Groups_Ominus__class_Ominus,type,
minus_minus:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Groups_Omonoid,type,
monoid:
!>[A: $tType] : ( ( A > A > A ) > A > $o ) ).
thf(sy_c_Groups_Omonoid__axioms,type,
monoid_axioms:
!>[A: $tType] : ( ( A > A > A ) > A > $o ) ).
thf(sy_c_Groups_Osemigroup,type,
semigroup:
!>[A: $tType] : ( ( A > A > A ) > $o ) ).
thf(sy_c_Groups_Ouminus__class_Ouminus,type,
uminus_uminus:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Huffman__Mirabelle__gjololrwrm_Oalphabet,type,
huffma505251170phabet:
!>[A: $tType] : ( ( huffma16452318e_tree @ A ) > ( set @ A ) ) ).
thf(sy_c_Huffman__Mirabelle__gjololrwrm_OswapLeaves,type,
huffma2094459102Leaves:
!>[A: $tType] : ( ( huffma16452318e_tree @ A ) > nat > A > nat > A > ( huffma16452318e_tree @ A ) ) ).
thf(sy_c_Huffman__Mirabelle__gjololrwrm_Otree_OLeaf,type,
huffma1554276827e_Leaf:
!>[A: $tType] : ( nat > A > ( huffma16452318e_tree @ A ) ) ).
thf(sy_c_Huffman__Mirabelle__gjololrwrm_Otree_Oset__tree,type,
huffma778495363t_tree:
!>[A: $tType] : ( ( huffma16452318e_tree @ A ) > ( set @ A ) ) ).
thf(sy_c_Huffman__Mirabelle__gjololrwrm_OuniteTrees,type,
huffma453905539eTrees:
!>[A: $tType] : ( ( huffma16452318e_tree @ A ) > ( huffma16452318e_tree @ A ) > ( huffma16452318e_tree @ A ) ) ).
thf(sy_c_Lattices_Osemilattice__neutr,type,
semilattice_neutr:
!>[A: $tType] : ( ( A > A > A ) > A > $o ) ).
thf(sy_c_Lattices_Osup__class_Osup,type,
sup_sup:
!>[A: $tType] : ( A > A > 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_Pure_Otype,type,
type2:
!>[A: $tType] : ( itself @ A ) ).
thf(sy_c_Relation_Oinv__imagep,type,
inv_imagep:
!>[B: $tType,A: $tType] : ( ( B > B > $o ) > ( A > B ) > A > A > $o ) ).
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 ) ).
thf(sy_v_a,type,
a2: a ).
thf(sy_v_b,type,
b: a ).
thf(sy_v_t,type,
t: huffma16452318e_tree @ a ).
thf(sy_v_w_092_060_094sub_062a,type,
w_a: nat ).
thf(sy_v_w_092_060_094sub_062b,type,
w_b: nat ).
%----Relevant facts (256)
thf(fact_0_swapLeaves__id__when__notin__alphabet,axiom,
! [A: $tType,A2: A,T: huffma16452318e_tree @ A,W: nat,W2: nat] :
( ~ ( member @ A @ A2 @ ( huffma505251170phabet @ A @ T ) )
=> ( ( huffma2094459102Leaves @ A @ T @ W @ A2 @ W2 @ A2 )
= T ) ) ).
% swapLeaves_id_when_notin_alphabet
thf(fact_1_exists__in__alphabet,axiom,
! [A: $tType,T: huffma16452318e_tree @ A] :
? [A3: A] : ( member @ A @ A3 @ ( huffma505251170phabet @ A @ T ) ) ).
% exists_in_alphabet
thf(fact_2_insert__Diff__single,axiom,
! [A: $tType,A2: A,A4: set @ A] :
( ( insert @ A @ A2 @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( insert @ A @ A2 @ A4 ) ) ).
% insert_Diff_single
thf(fact_3_Un__Diff__cancel,axiom,
! [A: $tType,A4: set @ A,B2: set @ A] :
( ( sup_sup @ ( set @ A ) @ A4 @ ( minus_minus @ ( set @ A ) @ B2 @ A4 ) )
= ( sup_sup @ ( set @ A ) @ A4 @ B2 ) ) ).
% Un_Diff_cancel
thf(fact_4_Un__Diff__cancel2,axiom,
! [A: $tType,B2: set @ A,A4: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ B2 @ A4 ) @ A4 )
= ( sup_sup @ ( set @ A ) @ B2 @ A4 ) ) ).
% Un_Diff_cancel2
thf(fact_5_Diff__insert0,axiom,
! [A: $tType,X: A,A4: set @ A,B2: set @ A] :
( ~ ( member @ A @ X @ A4 )
=> ( ( minus_minus @ ( set @ A ) @ A4 @ ( insert @ A @ X @ B2 ) )
= ( minus_minus @ ( set @ A ) @ A4 @ B2 ) ) ) ).
% Diff_insert0
thf(fact_6_insert__Diff1,axiom,
! [A: $tType,X: A,B2: set @ A,A4: set @ A] :
( ( member @ A @ X @ B2 )
=> ( ( minus_minus @ ( set @ A ) @ ( insert @ A @ X @ A4 ) @ B2 )
= ( minus_minus @ ( set @ A ) @ A4 @ B2 ) ) ) ).
% insert_Diff1
thf(fact_7_Un__insert__left,axiom,
! [A: $tType,A2: A,B2: set @ A,C: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( insert @ A @ A2 @ B2 ) @ C )
= ( insert @ A @ A2 @ ( sup_sup @ ( set @ A ) @ B2 @ C ) ) ) ).
% Un_insert_left
thf(fact_8_Un__insert__right,axiom,
! [A: $tType,A4: set @ A,A2: A,B2: set @ A] :
( ( sup_sup @ ( set @ A ) @ A4 @ ( insert @ A @ A2 @ B2 ) )
= ( insert @ A @ A2 @ ( sup_sup @ ( set @ A ) @ A4 @ B2 ) ) ) ).
% Un_insert_right
thf(fact_9_Diff__empty,axiom,
! [A: $tType,A4: set @ A] :
( ( minus_minus @ ( set @ A ) @ A4 @ ( bot_bot @ ( set @ A ) ) )
= A4 ) ).
% Diff_empty
thf(fact_10_empty__Diff,axiom,
! [A: $tType,A4: set @ A] :
( ( minus_minus @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ A4 )
= ( bot_bot @ ( set @ A ) ) ) ).
% empty_Diff
thf(fact_11_Diff__cancel,axiom,
! [A: $tType,A4: set @ A] :
( ( minus_minus @ ( set @ A ) @ A4 @ A4 )
= ( bot_bot @ ( set @ A ) ) ) ).
% Diff_cancel
thf(fact_12_Un__empty,axiom,
! [A: $tType,A4: set @ A,B2: set @ A] :
( ( ( sup_sup @ ( set @ A ) @ A4 @ B2 )
= ( bot_bot @ ( set @ A ) ) )
= ( ( A4
= ( bot_bot @ ( set @ A ) ) )
& ( B2
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Un_empty
thf(fact_13_singletonI,axiom,
! [A: $tType,A2: A] : ( member @ A @ A2 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% singletonI
thf(fact_14_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_15_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_16_all__not__in__conv,axiom,
! [A: $tType,A4: set @ A] :
( ( ! [X2: A] :
~ ( member @ A @ X2 @ A4 ) )
= ( A4
= ( bot_bot @ ( set @ A ) ) ) ) ).
% all_not_in_conv
thf(fact_17_empty__iff,axiom,
! [A: $tType,C2: A] :
~ ( member @ A @ C2 @ ( bot_bot @ ( set @ A ) ) ) ).
% empty_iff
thf(fact_18_insert__absorb2,axiom,
! [A: $tType,X: A,A4: set @ A] :
( ( insert @ A @ X @ ( insert @ A @ X @ A4 ) )
= ( insert @ A @ X @ A4 ) ) ).
% insert_absorb2
thf(fact_19_insert__iff,axiom,
! [A: $tType,A2: A,B3: A,A4: set @ A] :
( ( member @ A @ A2 @ ( insert @ A @ B3 @ A4 ) )
= ( ( A2 = B3 )
| ( member @ A @ A2 @ A4 ) ) ) ).
% insert_iff
thf(fact_20_insertCI,axiom,
! [A: $tType,A2: A,B2: set @ A,B3: A] :
( ( ~ ( member @ A @ A2 @ B2 )
=> ( A2 = B3 ) )
=> ( member @ A @ A2 @ ( insert @ A @ B3 @ B2 ) ) ) ).
% insertCI
thf(fact_21_Un__iff,axiom,
! [A: $tType,C2: A,A4: set @ A,B2: set @ A] :
( ( member @ A @ C2 @ ( sup_sup @ ( set @ A ) @ A4 @ B2 ) )
= ( ( member @ A @ C2 @ A4 )
| ( member @ A @ C2 @ B2 ) ) ) ).
% Un_iff
thf(fact_22_UnCI,axiom,
! [A: $tType,C2: A,B2: set @ A,A4: set @ A] :
( ( ~ ( member @ A @ C2 @ B2 )
=> ( member @ A @ C2 @ A4 ) )
=> ( member @ A @ C2 @ ( sup_sup @ ( set @ A ) @ A4 @ B2 ) ) ) ).
% UnCI
thf(fact_23_Diff__idemp,axiom,
! [A: $tType,A4: set @ A,B2: set @ A] :
( ( minus_minus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A4 @ B2 ) @ B2 )
= ( minus_minus @ ( set @ A ) @ A4 @ B2 ) ) ).
% Diff_idemp
thf(fact_24_Diff__iff,axiom,
! [A: $tType,C2: A,A4: set @ A,B2: set @ A] :
( ( member @ A @ C2 @ ( minus_minus @ ( set @ A ) @ A4 @ B2 ) )
= ( ( member @ A @ C2 @ A4 )
& ~ ( member @ A @ C2 @ B2 ) ) ) ).
% Diff_iff
thf(fact_25_DiffI,axiom,
! [A: $tType,C2: A,A4: set @ A,B2: set @ A] :
( ( member @ A @ C2 @ A4 )
=> ( ~ ( member @ A @ C2 @ B2 )
=> ( member @ A @ C2 @ ( minus_minus @ ( set @ A ) @ A4 @ B2 ) ) ) ) ).
% DiffI
thf(fact_26_ex__in__conv,axiom,
! [A: $tType,A4: set @ A] :
( ( ? [X2: A] : ( member @ A @ X2 @ A4 ) )
= ( A4
!= ( bot_bot @ ( set @ A ) ) ) ) ).
% ex_in_conv
thf(fact_27_equals0I,axiom,
! [A: $tType,A4: set @ A] :
( ! [Y: A] :
~ ( member @ A @ Y @ A4 )
=> ( A4
= ( bot_bot @ ( set @ A ) ) ) ) ).
% equals0I
thf(fact_28_equals0D,axiom,
! [A: $tType,A4: set @ A,A2: A] :
( ( A4
= ( bot_bot @ ( set @ A ) ) )
=> ~ ( member @ A @ A2 @ A4 ) ) ).
% equals0D
thf(fact_29_emptyE,axiom,
! [A: $tType,A2: A] :
~ ( member @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ).
% emptyE
thf(fact_30_mk__disjoint__insert,axiom,
! [A: $tType,A2: A,A4: set @ A] :
( ( member @ A @ A2 @ A4 )
=> ? [B4: set @ A] :
( ( A4
= ( insert @ A @ A2 @ B4 ) )
& ~ ( member @ A @ A2 @ B4 ) ) ) ).
% mk_disjoint_insert
thf(fact_31_insert__commute,axiom,
! [A: $tType,X: A,Y2: A,A4: set @ A] :
( ( insert @ A @ X @ ( insert @ A @ Y2 @ A4 ) )
= ( insert @ A @ Y2 @ ( insert @ A @ X @ A4 ) ) ) ).
% insert_commute
thf(fact_32_insert__eq__iff,axiom,
! [A: $tType,A2: A,A4: set @ A,B3: A,B2: set @ A] :
( ~ ( member @ A @ A2 @ A4 )
=> ( ~ ( member @ A @ B3 @ B2 )
=> ( ( ( insert @ A @ A2 @ A4 )
= ( insert @ A @ B3 @ B2 ) )
= ( ( ( A2 = B3 )
=> ( A4 = B2 ) )
& ( ( A2 != B3 )
=> ? [C3: set @ A] :
( ( A4
= ( insert @ A @ B3 @ C3 ) )
& ~ ( member @ A @ B3 @ C3 )
& ( B2
= ( insert @ A @ A2 @ C3 ) )
& ~ ( member @ A @ A2 @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_33_insert__absorb,axiom,
! [A: $tType,A2: A,A4: set @ A] :
( ( member @ A @ A2 @ A4 )
=> ( ( insert @ A @ A2 @ A4 )
= A4 ) ) ).
% insert_absorb
thf(fact_34_insert__ident,axiom,
! [A: $tType,X: A,A4: set @ A,B2: set @ A] :
( ~ ( member @ A @ X @ A4 )
=> ( ~ ( member @ A @ X @ B2 )
=> ( ( ( insert @ A @ X @ A4 )
= ( insert @ A @ X @ B2 ) )
= ( A4 = B2 ) ) ) ) ).
% insert_ident
thf(fact_35_Set_Oset__insert,axiom,
! [A: $tType,X: A,A4: set @ A] :
( ( member @ A @ X @ A4 )
=> ~ ! [B4: set @ A] :
( ( A4
= ( insert @ A @ X @ B4 ) )
=> ( member @ A @ X @ B4 ) ) ) ).
% Set.set_insert
thf(fact_36_insertI2,axiom,
! [A: $tType,A2: A,B2: set @ A,B3: A] :
( ( member @ A @ A2 @ B2 )
=> ( member @ A @ A2 @ ( insert @ A @ B3 @ B2 ) ) ) ).
% insertI2
thf(fact_37_insertI1,axiom,
! [A: $tType,A2: A,B2: set @ A] : ( member @ A @ A2 @ ( insert @ A @ A2 @ B2 ) ) ).
% insertI1
thf(fact_38_insertE,axiom,
! [A: $tType,A2: A,B3: A,A4: set @ A] :
( ( member @ A @ A2 @ ( insert @ A @ B3 @ A4 ) )
=> ( ( A2 != B3 )
=> ( member @ A @ A2 @ A4 ) ) ) ).
% insertE
thf(fact_39_Un__left__commute,axiom,
! [A: $tType,A4: set @ A,B2: set @ A,C: set @ A] :
( ( sup_sup @ ( set @ A ) @ A4 @ ( sup_sup @ ( set @ A ) @ B2 @ C ) )
= ( sup_sup @ ( set @ A ) @ B2 @ ( sup_sup @ ( set @ A ) @ A4 @ C ) ) ) ).
% Un_left_commute
thf(fact_40_Un__left__absorb,axiom,
! [A: $tType,A4: set @ A,B2: set @ A] :
( ( sup_sup @ ( set @ A ) @ A4 @ ( sup_sup @ ( set @ A ) @ A4 @ B2 ) )
= ( sup_sup @ ( set @ A ) @ A4 @ B2 ) ) ).
% Un_left_absorb
thf(fact_41_Un__commute,axiom,
! [A: $tType] :
( ( sup_sup @ ( set @ A ) )
= ( ^ [A5: set @ A,B5: set @ A] : ( sup_sup @ ( set @ A ) @ B5 @ A5 ) ) ) ).
% Un_commute
thf(fact_42_Un__absorb,axiom,
! [A: $tType,A4: set @ A] :
( ( sup_sup @ ( set @ A ) @ A4 @ A4 )
= A4 ) ).
% Un_absorb
thf(fact_43_Un__assoc,axiom,
! [A: $tType,A4: set @ A,B2: set @ A,C: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A4 @ B2 ) @ C )
= ( sup_sup @ ( set @ A ) @ A4 @ ( sup_sup @ ( set @ A ) @ B2 @ C ) ) ) ).
% Un_assoc
thf(fact_44_ball__Un,axiom,
! [A: $tType,A4: set @ A,B2: set @ A,P: A > $o] :
( ( ! [X2: A] :
( ( member @ A @ X2 @ ( sup_sup @ ( set @ A ) @ A4 @ B2 ) )
=> ( P @ X2 ) ) )
= ( ! [X2: A] :
( ( member @ A @ X2 @ A4 )
=> ( P @ X2 ) )
& ! [X2: A] :
( ( member @ A @ X2 @ B2 )
=> ( P @ X2 ) ) ) ) ).
% ball_Un
thf(fact_45_mem__Collect__eq,axiom,
! [A: $tType,A2: A,P: A > $o] :
( ( member @ A @ A2 @ ( collect @ A @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_46_Collect__mem__eq,axiom,
! [A: $tType,A4: set @ A] :
( ( collect @ A
@ ^ [X2: A] : ( member @ A @ X2 @ A4 ) )
= A4 ) ).
% 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,F: A > B,G: A > B] :
( ! [X3: A] :
( ( F @ X3 )
= ( G @ X3 ) )
=> ( F = G ) ) ).
% ext
thf(fact_49_bex__Un,axiom,
! [A: $tType,A4: set @ A,B2: set @ A,P: A > $o] :
( ( ? [X2: A] :
( ( member @ A @ X2 @ ( sup_sup @ ( set @ A ) @ A4 @ B2 ) )
& ( P @ X2 ) ) )
= ( ? [X2: A] :
( ( member @ A @ X2 @ A4 )
& ( P @ X2 ) )
| ? [X2: A] :
( ( member @ A @ X2 @ B2 )
& ( P @ X2 ) ) ) ) ).
% bex_Un
thf(fact_50_UnI2,axiom,
! [A: $tType,C2: A,B2: set @ A,A4: set @ A] :
( ( member @ A @ C2 @ B2 )
=> ( member @ A @ C2 @ ( sup_sup @ ( set @ A ) @ A4 @ B2 ) ) ) ).
% UnI2
thf(fact_51_UnI1,axiom,
! [A: $tType,C2: A,A4: set @ A,B2: set @ A] :
( ( member @ A @ C2 @ A4 )
=> ( member @ A @ C2 @ ( sup_sup @ ( set @ A ) @ A4 @ B2 ) ) ) ).
% UnI1
thf(fact_52_UnE,axiom,
! [A: $tType,C2: A,A4: set @ A,B2: set @ A] :
( ( member @ A @ C2 @ ( sup_sup @ ( set @ A ) @ A4 @ B2 ) )
=> ( ~ ( member @ A @ C2 @ A4 )
=> ( member @ A @ C2 @ B2 ) ) ) ).
% UnE
thf(fact_53_DiffD2,axiom,
! [A: $tType,C2: A,A4: set @ A,B2: set @ A] :
( ( member @ A @ C2 @ ( minus_minus @ ( set @ A ) @ A4 @ B2 ) )
=> ~ ( member @ A @ C2 @ B2 ) ) ).
% DiffD2
thf(fact_54_DiffD1,axiom,
! [A: $tType,C2: A,A4: set @ A,B2: set @ A] :
( ( member @ A @ C2 @ ( minus_minus @ ( set @ A ) @ A4 @ B2 ) )
=> ( member @ A @ C2 @ A4 ) ) ).
% DiffD1
thf(fact_55_DiffE,axiom,
! [A: $tType,C2: A,A4: set @ A,B2: set @ A] :
( ( member @ A @ C2 @ ( minus_minus @ ( set @ A ) @ A4 @ B2 ) )
=> ~ ( ( member @ A @ C2 @ A4 )
=> ( member @ A @ C2 @ B2 ) ) ) ).
% DiffE
thf(fact_56_singleton__inject,axiom,
! [A: $tType,A2: A,B3: A] :
( ( ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) )
= ( insert @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) )
=> ( A2 = B3 ) ) ).
% singleton_inject
thf(fact_57_insert__not__empty,axiom,
! [A: $tType,A2: A,A4: set @ A] :
( ( insert @ A @ A2 @ A4 )
!= ( bot_bot @ ( set @ A ) ) ) ).
% insert_not_empty
thf(fact_58_doubleton__eq__iff,axiom,
! [A: $tType,A2: A,B3: A,C2: A,D: A] :
( ( ( insert @ A @ A2 @ ( insert @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) )
= ( insert @ A @ C2 @ ( insert @ A @ D @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( ( ( A2 = C2 )
& ( B3 = D ) )
| ( ( A2 = D )
& ( B3 = C2 ) ) ) ) ).
% doubleton_eq_iff
thf(fact_59_singleton__iff,axiom,
! [A: $tType,B3: A,A2: A] :
( ( member @ A @ B3 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) )
= ( B3 = A2 ) ) ).
% singleton_iff
thf(fact_60_singletonD,axiom,
! [A: $tType,B3: A,A2: A] :
( ( member @ A @ B3 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) )
=> ( B3 = A2 ) ) ).
% singletonD
thf(fact_61_Un__empty__right,axiom,
! [A: $tType,A4: set @ A] :
( ( sup_sup @ ( set @ A ) @ A4 @ ( bot_bot @ ( set @ A ) ) )
= A4 ) ).
% Un_empty_right
thf(fact_62_Un__empty__left,axiom,
! [A: $tType,B2: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ B2 )
= B2 ) ).
% Un_empty_left
thf(fact_63_insert__Diff__if,axiom,
! [A: $tType,X: A,B2: set @ A,A4: set @ A] :
( ( ( member @ A @ X @ B2 )
=> ( ( minus_minus @ ( set @ A ) @ ( insert @ A @ X @ A4 ) @ B2 )
= ( minus_minus @ ( set @ A ) @ A4 @ B2 ) ) )
& ( ~ ( member @ A @ X @ B2 )
=> ( ( minus_minus @ ( set @ A ) @ ( insert @ A @ X @ A4 ) @ B2 )
= ( insert @ A @ X @ ( minus_minus @ ( set @ A ) @ A4 @ B2 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_64_Un__Diff,axiom,
! [A: $tType,A4: set @ A,B2: set @ A,C: set @ A] :
( ( minus_minus @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A4 @ B2 ) @ C )
= ( sup_sup @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A4 @ C ) @ ( minus_minus @ ( set @ A ) @ B2 @ C ) ) ) ).
% Un_Diff
thf(fact_65_singleton__Un__iff,axiom,
! [A: $tType,X: A,A4: set @ A,B2: set @ A] :
( ( ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) )
= ( sup_sup @ ( set @ A ) @ A4 @ B2 ) )
= ( ( ( A4
= ( bot_bot @ ( set @ A ) ) )
& ( B2
= ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) )
| ( ( A4
= ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
& ( B2
= ( bot_bot @ ( set @ A ) ) ) )
| ( ( A4
= ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
& ( B2
= ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_66_Un__singleton__iff,axiom,
! [A: $tType,A4: set @ A,B2: set @ A,X: A] :
( ( ( sup_sup @ ( set @ A ) @ A4 @ B2 )
= ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= ( ( ( A4
= ( bot_bot @ ( set @ A ) ) )
& ( B2
= ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) )
| ( ( A4
= ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
& ( B2
= ( bot_bot @ ( set @ A ) ) ) )
| ( ( A4
= ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
& ( B2
= ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_67_insert__is__Un,axiom,
! [A: $tType] :
( ( insert @ A )
= ( ^ [A6: A] : ( sup_sup @ ( set @ A ) @ ( insert @ A @ A6 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% insert_is_Un
thf(fact_68_Diff__insert__absorb,axiom,
! [A: $tType,X: A,A4: set @ A] :
( ~ ( member @ A @ X @ A4 )
=> ( ( minus_minus @ ( set @ A ) @ ( insert @ A @ X @ A4 ) @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= A4 ) ) ).
% Diff_insert_absorb
thf(fact_69_Diff__insert2,axiom,
! [A: $tType,A4: set @ A,A2: A,B2: set @ A] :
( ( minus_minus @ ( set @ A ) @ A4 @ ( insert @ A @ A2 @ B2 ) )
= ( minus_minus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) @ B2 ) ) ).
% Diff_insert2
thf(fact_70_insert__Diff,axiom,
! [A: $tType,A2: A,A4: set @ A] :
( ( member @ A @ A2 @ A4 )
=> ( ( insert @ A @ A2 @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) )
= A4 ) ) ).
% insert_Diff
thf(fact_71_Diff__insert,axiom,
! [A: $tType,A4: set @ A,A2: A,B2: set @ A] :
( ( minus_minus @ ( set @ A ) @ A4 @ ( insert @ A @ A2 @ B2 ) )
= ( minus_minus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A4 @ B2 ) @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Diff_insert
thf(fact_72_sup__bot_Oright__neutral,axiom,
! [A: $tType] :
( ( bounde1808546759up_bot @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( sup_sup @ A @ A2 @ ( bot_bot @ A ) )
= A2 ) ) ).
% sup_bot.right_neutral
thf(fact_73_sup__bot_Oleft__neutral,axiom,
! [A: $tType] :
( ( bounde1808546759up_bot @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( sup_sup @ A @ ( bot_bot @ A ) @ A2 )
= A2 ) ) ).
% sup_bot.left_neutral
thf(fact_74_sup__eq__bot__iff,axiom,
! [A: $tType] :
( ( bounded_lattice_bot @ A @ ( type2 @ A ) )
=> ! [X: A,Y2: A] :
( ( ( sup_sup @ A @ X @ Y2 )
= ( bot_bot @ A ) )
= ( ( X
= ( bot_bot @ A ) )
& ( Y2
= ( bot_bot @ A ) ) ) ) ) ).
% sup_eq_bot_iff
thf(fact_75_bot__eq__sup__iff,axiom,
! [A: $tType] :
( ( bounded_lattice_bot @ A @ ( type2 @ A ) )
=> ! [X: A,Y2: A] :
( ( ( bot_bot @ A )
= ( sup_sup @ A @ X @ Y2 ) )
= ( ( X
= ( bot_bot @ A ) )
& ( Y2
= ( bot_bot @ A ) ) ) ) ) ).
% bot_eq_sup_iff
thf(fact_76_sup_Oright__idem,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [A2: A,B3: A] :
( ( sup_sup @ A @ ( sup_sup @ A @ A2 @ B3 ) @ B3 )
= ( sup_sup @ A @ A2 @ B3 ) ) ) ).
% sup.right_idem
thf(fact_77_sup__left__idem,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [X: A,Y2: A] :
( ( sup_sup @ A @ X @ ( sup_sup @ A @ X @ Y2 ) )
= ( sup_sup @ A @ X @ Y2 ) ) ) ).
% sup_left_idem
thf(fact_78_sup_Oleft__idem,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [A2: A,B3: A] :
( ( sup_sup @ A @ A2 @ ( sup_sup @ A @ A2 @ B3 ) )
= ( sup_sup @ A @ A2 @ B3 ) ) ) ).
% sup.left_idem
thf(fact_79_sup__idem,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [X: A] :
( ( sup_sup @ A @ X @ X )
= X ) ) ).
% sup_idem
thf(fact_80_sup_Oidem,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( sup_sup @ A @ A2 @ A2 )
= A2 ) ) ).
% sup.idem
thf(fact_81_sup__apply,axiom,
! [B: $tType,A: $tType] :
( ( semilattice_sup @ B @ ( type2 @ B ) )
=> ( ( sup_sup @ ( A > B ) )
= ( ^ [F2: A > B,G2: A > B,X2: A] : ( sup_sup @ B @ ( F2 @ X2 ) @ ( G2 @ X2 ) ) ) ) ) ).
% sup_apply
thf(fact_82_the__elem__eq,axiom,
! [A: $tType,X: A] :
( ( the_elem @ A @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= X ) ).
% the_elem_eq
thf(fact_83_minus__apply,axiom,
! [B: $tType,A: $tType] :
( ( minus @ B @ ( type2 @ B ) )
=> ( ( minus_minus @ ( A > B ) )
= ( ^ [A5: A > B,B5: A > B,X2: A] : ( minus_minus @ B @ ( A5 @ X2 ) @ ( B5 @ X2 ) ) ) ) ) ).
% minus_apply
thf(fact_84_bot__set__def,axiom,
! [A: $tType] :
( ( bot_bot @ ( set @ A ) )
= ( collect @ A @ ( bot_bot @ ( A > $o ) ) ) ) ).
% bot_set_def
thf(fact_85_fun__diff__def,axiom,
! [B: $tType,A: $tType] :
( ( minus @ B @ ( type2 @ B ) )
=> ( ( minus_minus @ ( A > B ) )
= ( ^ [A5: A > B,B5: A > B,X2: A] : ( minus_minus @ B @ ( A5 @ X2 ) @ ( B5 @ X2 ) ) ) ) ) ).
% fun_diff_def
thf(fact_86_inf__sup__aci_I8_J,axiom,
! [A: $tType] :
( ( lattice @ A @ ( type2 @ A ) )
=> ! [X: A,Y2: A] :
( ( sup_sup @ A @ X @ ( sup_sup @ A @ X @ Y2 ) )
= ( sup_sup @ A @ X @ Y2 ) ) ) ).
% inf_sup_aci(8)
thf(fact_87_inf__sup__aci_I7_J,axiom,
! [A: $tType] :
( ( lattice @ A @ ( type2 @ A ) )
=> ! [X: A,Y2: A,Z: A] :
( ( sup_sup @ A @ X @ ( sup_sup @ A @ Y2 @ Z ) )
= ( sup_sup @ A @ Y2 @ ( sup_sup @ A @ X @ Z ) ) ) ) ).
% inf_sup_aci(7)
thf(fact_88_inf__sup__aci_I6_J,axiom,
! [A: $tType] :
( ( lattice @ A @ ( type2 @ A ) )
=> ! [X: A,Y2: A,Z: A] :
( ( sup_sup @ A @ ( sup_sup @ A @ X @ Y2 ) @ Z )
= ( sup_sup @ A @ X @ ( sup_sup @ A @ Y2 @ Z ) ) ) ) ).
% inf_sup_aci(6)
thf(fact_89_inf__sup__aci_I5_J,axiom,
! [A: $tType] :
( ( lattice @ A @ ( type2 @ A ) )
=> ( ( sup_sup @ A )
= ( ^ [X2: A,Y3: A] : ( sup_sup @ A @ Y3 @ X2 ) ) ) ) ).
% inf_sup_aci(5)
thf(fact_90_sup__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( semilattice_sup @ B @ ( type2 @ B ) )
=> ( ( sup_sup @ ( A > B ) )
= ( ^ [F2: A > B,G2: A > B,X2: A] : ( sup_sup @ B @ ( F2 @ X2 ) @ ( G2 @ X2 ) ) ) ) ) ).
% sup_fun_def
thf(fact_91_sup_Oassoc,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [A2: A,B3: A,C2: A] :
( ( sup_sup @ A @ ( sup_sup @ A @ A2 @ B3 ) @ C2 )
= ( sup_sup @ A @ A2 @ ( sup_sup @ A @ B3 @ C2 ) ) ) ) ).
% sup.assoc
thf(fact_92_sup__assoc,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [X: A,Y2: A,Z: A] :
( ( sup_sup @ A @ ( sup_sup @ A @ X @ Y2 ) @ Z )
= ( sup_sup @ A @ X @ ( sup_sup @ A @ Y2 @ Z ) ) ) ) ).
% sup_assoc
thf(fact_93_sup_Ocommute,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ( ( sup_sup @ A )
= ( ^ [A6: A,B6: A] : ( sup_sup @ A @ B6 @ A6 ) ) ) ) ).
% sup.commute
thf(fact_94_sup__commute,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ( ( sup_sup @ A )
= ( ^ [X2: A,Y3: A] : ( sup_sup @ A @ Y3 @ X2 ) ) ) ) ).
% sup_commute
thf(fact_95_sup_Oleft__commute,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [B3: A,A2: A,C2: A] :
( ( sup_sup @ A @ B3 @ ( sup_sup @ A @ A2 @ C2 ) )
= ( sup_sup @ A @ A2 @ ( sup_sup @ A @ B3 @ C2 ) ) ) ) ).
% sup.left_commute
thf(fact_96_sup__left__commute,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [X: A,Y2: A,Z: A] :
( ( sup_sup @ A @ X @ ( sup_sup @ A @ Y2 @ Z ) )
= ( sup_sup @ A @ Y2 @ ( sup_sup @ A @ X @ Z ) ) ) ) ).
% sup_left_commute
thf(fact_97_sup__bot__left,axiom,
! [A: $tType] :
( ( bounded_lattice_bot @ A @ ( type2 @ A ) )
=> ! [X: A] :
( ( sup_sup @ A @ ( bot_bot @ A ) @ X )
= X ) ) ).
% sup_bot_left
thf(fact_98_sup__bot__right,axiom,
! [A: $tType] :
( ( bounded_lattice_bot @ A @ ( type2 @ A ) )
=> ! [X: A] :
( ( sup_sup @ A @ X @ ( bot_bot @ A ) )
= X ) ) ).
% sup_bot_right
thf(fact_99_bot__apply,axiom,
! [C4: $tType,D2: $tType] :
( ( bot @ C4 @ ( type2 @ C4 ) )
=> ( ( bot_bot @ ( D2 > C4 ) )
= ( ^ [X2: D2] : ( bot_bot @ C4 ) ) ) ) ).
% bot_apply
thf(fact_100_is__singleton__the__elem,axiom,
! [A: $tType] :
( ( is_singleton @ A )
= ( ^ [A5: set @ A] :
( A5
= ( insert @ A @ ( the_elem @ A @ A5 ) @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% is_singleton_the_elem
thf(fact_101_is__singletonI,axiom,
! [A: $tType,X: A] : ( is_singleton @ A @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ).
% is_singletonI
thf(fact_102_alphabet__uniteTrees,axiom,
! [A: $tType,T_1: huffma16452318e_tree @ A,T_2: huffma16452318e_tree @ A] :
( ( huffma505251170phabet @ A @ ( huffma453905539eTrees @ A @ T_1 @ T_2 ) )
= ( sup_sup @ ( set @ A ) @ ( huffma505251170phabet @ A @ T_1 ) @ ( huffma505251170phabet @ A @ T_2 ) ) ) ).
% alphabet_uniteTrees
thf(fact_103_remove__def,axiom,
! [A: $tType] :
( ( remove @ A )
= ( ^ [X2: A,A5: set @ A] : ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% remove_def
thf(fact_104_Set_Ois__empty__def,axiom,
! [A: $tType] :
( ( is_empty @ A )
= ( ^ [A5: set @ A] :
( A5
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Set.is_empty_def
thf(fact_105_member__remove,axiom,
! [A: $tType,X: A,Y2: A,A4: set @ A] :
( ( member @ A @ X @ ( remove @ A @ Y2 @ A4 ) )
= ( ( member @ A @ X @ A4 )
& ( X != Y2 ) ) ) ).
% member_remove
thf(fact_106_is__singletonI_H,axiom,
! [A: $tType,A4: set @ A] :
( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X3: A,Y: A] :
( ( member @ A @ X3 @ A4 )
=> ( ( member @ A @ Y @ A4 )
=> ( X3 = Y ) ) )
=> ( is_singleton @ A @ A4 ) ) ) ).
% is_singletonI'
thf(fact_107_bot__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( bot @ B @ ( type2 @ B ) )
=> ( ( bot_bot @ ( A > B ) )
= ( ^ [X2: A] : ( bot_bot @ B ) ) ) ) ).
% bot_fun_def
thf(fact_108_is__singleton__def,axiom,
! [A: $tType] :
( ( is_singleton @ A )
= ( ^ [A5: set @ A] :
? [X2: A] :
( A5
= ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% is_singleton_def
thf(fact_109_is__singletonE,axiom,
! [A: $tType,A4: set @ A] :
( ( is_singleton @ A @ A4 )
=> ~ ! [X3: A] :
( A4
!= ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% is_singletonE
thf(fact_110_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_111_bot__empty__eq,axiom,
! [A: $tType] :
( ( bot_bot @ ( A > $o ) )
= ( ^ [X2: A] : ( member @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% bot_empty_eq
thf(fact_112_pairwise__singleton,axiom,
! [A: $tType,P: A > A > $o,A4: A] : ( pairwise @ A @ P @ ( insert @ A @ A4 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% pairwise_singleton
thf(fact_113_alphabet_Osimps_I1_J,axiom,
! [A: $tType,W: nat,A2: A] :
( ( huffma505251170phabet @ A @ ( huffma1554276827e_Leaf @ A @ W @ A2 ) )
= ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% alphabet.simps(1)
thf(fact_114_sup__bot_Osemilattice__neutr__axioms,axiom,
! [A: $tType] :
( ( bounde1808546759up_bot @ A @ ( type2 @ A ) )
=> ( semilattice_neutr @ A @ ( sup_sup @ A ) @ ( bot_bot @ A ) ) ) ).
% sup_bot.semilattice_neutr_axioms
thf(fact_115_tree_Oinject_I1_J,axiom,
! [A: $tType,X11: nat,X12: A,Y11: nat,Y12: A] :
( ( ( huffma1554276827e_Leaf @ A @ X11 @ X12 )
= ( huffma1554276827e_Leaf @ A @ Y11 @ Y12 ) )
= ( ( X11 = Y11 )
& ( X12 = Y12 ) ) ) ).
% tree.inject(1)
thf(fact_116_pairwise__def,axiom,
! [A: $tType] :
( ( pairwise @ A )
= ( ^ [R: A > A > $o,S: set @ A] :
! [X2: A] :
( ( member @ A @ X2 @ S )
=> ! [Y3: A] :
( ( member @ A @ Y3 @ S )
=> ( ( X2 != Y3 )
=> ( R @ X2 @ Y3 ) ) ) ) ) ) ).
% pairwise_def
thf(fact_117_swapLeaves_Osimps_I1_J,axiom,
! [A: $tType,C2: A,A2: A,W_c: nat,W_a: nat,W_b: nat,B3: A] :
( ( ( C2 = A2 )
=> ( ( huffma2094459102Leaves @ A @ ( huffma1554276827e_Leaf @ A @ W_c @ C2 ) @ W_a @ A2 @ W_b @ B3 )
= ( huffma1554276827e_Leaf @ A @ W_b @ B3 ) ) )
& ( ( C2 != A2 )
=> ( ( ( C2 = B3 )
=> ( ( huffma2094459102Leaves @ A @ ( huffma1554276827e_Leaf @ A @ W_c @ C2 ) @ W_a @ A2 @ W_b @ B3 )
= ( huffma1554276827e_Leaf @ A @ W_a @ A2 ) ) )
& ( ( C2 != B3 )
=> ( ( huffma2094459102Leaves @ A @ ( huffma1554276827e_Leaf @ A @ W_c @ C2 ) @ W_a @ A2 @ W_b @ B3 )
= ( huffma1554276827e_Leaf @ A @ W_c @ C2 ) ) ) ) ) ) ).
% swapLeaves.simps(1)
thf(fact_118_pairwise__empty,axiom,
! [A: $tType,P: A > A > $o] : ( pairwise @ A @ P @ ( bot_bot @ ( set @ A ) ) ) ).
% pairwise_empty
thf(fact_119_pairwise__insert,axiom,
! [A: $tType,R2: A > A > $o,X: A,S2: set @ A] :
( ( pairwise @ A @ R2 @ ( insert @ A @ X @ S2 ) )
= ( ! [Y3: A] :
( ( ( member @ A @ Y3 @ S2 )
& ( Y3 != X ) )
=> ( ( R2 @ X @ Y3 )
& ( R2 @ Y3 @ X ) ) )
& ( pairwise @ A @ R2 @ S2 ) ) ) ).
% pairwise_insert
thf(fact_120_tree_Osimps_I15_J,axiom,
! [A: $tType,X11: nat,X12: A] :
( ( huffma778495363t_tree @ A @ ( huffma1554276827e_Leaf @ A @ X11 @ X12 ) )
= ( insert @ A @ X12 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% tree.simps(15)
thf(fact_121_diff__eq__diff__eq,axiom,
! [A: $tType] :
( ( group_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B3: A,C2: A,D: A] :
( ( ( minus_minus @ A @ A2 @ B3 )
= ( minus_minus @ A @ C2 @ D ) )
=> ( ( A2 = B3 )
= ( C2 = D ) ) ) ) ).
% diff_eq_diff_eq
thf(fact_122_diff__right__commute,axiom,
! [A: $tType] :
( ( cancel146912293up_add @ A @ ( type2 @ A ) )
=> ! [A2: A,C2: A,B3: A] :
( ( minus_minus @ A @ ( minus_minus @ A @ A2 @ C2 ) @ B3 )
= ( minus_minus @ A @ ( minus_minus @ A @ A2 @ B3 ) @ C2 ) ) ) ).
% diff_right_commute
thf(fact_123_sup__bot_Omonoid__axioms,axiom,
! [A: $tType] :
( ( bounde1808546759up_bot @ A @ ( type2 @ A ) )
=> ( monoid @ A @ ( sup_sup @ A ) @ ( bot_bot @ A ) ) ) ).
% sup_bot.monoid_axioms
thf(fact_124_monoid_Oleft__neutral,axiom,
! [A: $tType,F: A > A > A,Z: A,A2: A] :
( ( monoid @ A @ F @ Z )
=> ( ( F @ Z @ A2 )
= A2 ) ) ).
% monoid.left_neutral
thf(fact_125_monoid_Oright__neutral,axiom,
! [A: $tType,F: A > A > A,Z: A,A2: A] :
( ( monoid @ A @ F @ Z )
=> ( ( F @ A2 @ Z )
= A2 ) ) ).
% monoid.right_neutral
thf(fact_126_tree_Oset__intros_I1_J,axiom,
! [A: $tType,A22: A,A1: nat] : ( member @ A @ A22 @ ( huffma778495363t_tree @ A @ ( huffma1554276827e_Leaf @ A @ A1 @ A22 ) ) ) ).
% tree.set_intros(1)
thf(fact_127_monoid_Oaxioms_I2_J,axiom,
! [A: $tType,F: A > A > A,Z: A] :
( ( monoid @ A @ F @ Z )
=> ( monoid_axioms @ A @ F @ Z ) ) ).
% monoid.axioms(2)
thf(fact_128_Diff__single__insert,axiom,
! [A: $tType,A4: set @ A,X: A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) @ B2 )
=> ( ord_less_eq @ ( set @ A ) @ A4 @ ( insert @ A @ X @ B2 ) ) ) ).
% Diff_single_insert
thf(fact_129_subset__insert__iff,axiom,
! [A: $tType,A4: set @ A,X: A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ ( insert @ A @ X @ B2 ) )
= ( ( ( member @ A @ X @ A4 )
=> ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) @ B2 ) )
& ( ~ ( member @ A @ X @ A4 )
=> ( ord_less_eq @ ( set @ A ) @ A4 @ B2 ) ) ) ) ).
% subset_insert_iff
thf(fact_130_in__inv__imagep,axiom,
! [B: $tType,A: $tType] :
( ( inv_imagep @ A @ B )
= ( ^ [R3: A > A > $o,F2: B > A,X2: B,Y3: B] : ( R3 @ ( F2 @ X2 ) @ ( F2 @ Y3 ) ) ) ) ).
% in_inv_imagep
thf(fact_131_order__refl,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A] : ( ord_less_eq @ A @ X @ X ) ) ).
% order_refl
thf(fact_132_subsetI,axiom,
! [A: $tType,A4: set @ A,B2: set @ A] :
( ! [X3: A] :
( ( member @ A @ X3 @ A4 )
=> ( member @ A @ X3 @ B2 ) )
=> ( ord_less_eq @ ( set @ A ) @ A4 @ B2 ) ) ).
% subsetI
thf(fact_133_subset__antisym,axiom,
! [A: $tType,A4: set @ A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B2 )
=> ( ( ord_less_eq @ ( set @ A ) @ B2 @ A4 )
=> ( A4 = B2 ) ) ) ).
% subset_antisym
thf(fact_134_le__sup__iff,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [X: A,Y2: A,Z: A] :
( ( ord_less_eq @ A @ ( sup_sup @ A @ X @ Y2 ) @ Z )
= ( ( ord_less_eq @ A @ X @ Z )
& ( ord_less_eq @ A @ Y2 @ Z ) ) ) ) ).
% le_sup_iff
thf(fact_135_sup_Obounded__iff,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [B3: A,C2: A,A2: A] :
( ( ord_less_eq @ A @ ( sup_sup @ A @ B3 @ C2 ) @ A2 )
= ( ( ord_less_eq @ A @ B3 @ A2 )
& ( ord_less_eq @ A @ C2 @ A2 ) ) ) ) ).
% sup.bounded_iff
thf(fact_136_subset__empty,axiom,
! [A: $tType,A4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ ( bot_bot @ ( set @ A ) ) )
= ( A4
= ( bot_bot @ ( set @ A ) ) ) ) ).
% subset_empty
thf(fact_137_empty__subsetI,axiom,
! [A: $tType,A4: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ A4 ) ).
% empty_subsetI
thf(fact_138_insert__subset,axiom,
! [A: $tType,X: A,A4: set @ A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ X @ A4 ) @ B2 )
= ( ( member @ A @ X @ B2 )
& ( ord_less_eq @ ( set @ A ) @ A4 @ B2 ) ) ) ).
% insert_subset
thf(fact_139_Un__subset__iff,axiom,
! [A: $tType,A4: set @ A,B2: set @ A,C: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A4 @ B2 ) @ C )
= ( ( ord_less_eq @ ( set @ A ) @ A4 @ C )
& ( ord_less_eq @ ( set @ A ) @ B2 @ C ) ) ) ).
% Un_subset_iff
thf(fact_140_singleton__insert__inj__eq,axiom,
! [A: $tType,B3: A,A2: A,A4: set @ A] :
( ( ( insert @ A @ B3 @ ( bot_bot @ ( set @ A ) ) )
= ( insert @ A @ A2 @ A4 ) )
= ( ( A2 = B3 )
& ( ord_less_eq @ ( set @ A ) @ A4 @ ( insert @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_141_singleton__insert__inj__eq_H,axiom,
! [A: $tType,A2: A,A4: set @ A,B3: A] :
( ( ( insert @ A @ A2 @ A4 )
= ( insert @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) )
= ( ( A2 = B3 )
& ( ord_less_eq @ ( set @ A ) @ A4 @ ( insert @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_142_Diff__eq__empty__iff,axiom,
! [A: $tType,A4: set @ A,B2: set @ A] :
( ( ( minus_minus @ ( set @ A ) @ A4 @ B2 )
= ( bot_bot @ ( set @ A ) ) )
= ( ord_less_eq @ ( set @ A ) @ A4 @ B2 ) ) ).
% Diff_eq_empty_iff
thf(fact_143_monoid__axioms__def,axiom,
! [A: $tType] :
( ( monoid_axioms @ A )
= ( ^ [F2: A > A > A,Z2: A] :
( ! [A6: A] :
( ( F2 @ Z2 @ A6 )
= A6 )
& ! [A6: A] :
( ( F2 @ A6 @ Z2 )
= A6 ) ) ) ) ).
% monoid_axioms_def
thf(fact_144_monoid__axioms_Ointro,axiom,
! [A: $tType,F: A > A > A,Z: A] :
( ! [A3: A] :
( ( F @ Z @ A3 )
= A3 )
=> ( ! [A3: A] :
( ( F @ A3 @ Z )
= A3 )
=> ( monoid_axioms @ A @ F @ Z ) ) ) ).
% monoid_axioms.intro
thf(fact_145_diff__eq__diff__less__eq,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B3: A,C2: A,D: A] :
( ( ( minus_minus @ A @ A2 @ B3 )
= ( minus_minus @ A @ C2 @ D ) )
=> ( ( ord_less_eq @ A @ A2 @ B3 )
= ( ord_less_eq @ A @ C2 @ D ) ) ) ) ).
% diff_eq_diff_less_eq
thf(fact_146_diff__right__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B3: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ B3 )
=> ( ord_less_eq @ A @ ( minus_minus @ A @ A2 @ C2 ) @ ( minus_minus @ A @ B3 @ C2 ) ) ) ) ).
% diff_right_mono
thf(fact_147_diff__left__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
=> ! [B3: A,A2: A,C2: A] :
( ( ord_less_eq @ A @ B3 @ A2 )
=> ( ord_less_eq @ A @ ( minus_minus @ A @ C2 @ A2 ) @ ( minus_minus @ A @ C2 @ B3 ) ) ) ) ).
% diff_left_mono
thf(fact_148_diff__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B3: A,D: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ B3 )
=> ( ( ord_less_eq @ A @ D @ C2 )
=> ( ord_less_eq @ A @ ( minus_minus @ A @ A2 @ C2 ) @ ( minus_minus @ A @ B3 @ D ) ) ) ) ) ).
% diff_mono
thf(fact_149_bot_Oextremum,axiom,
! [A: $tType] :
( ( order_bot @ A @ ( type2 @ A ) )
=> ! [A2: A] : ( ord_less_eq @ A @ ( bot_bot @ A ) @ A2 ) ) ).
% bot.extremum
thf(fact_150_bot_Oextremum__unique,axiom,
! [A: $tType] :
( ( order_bot @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( ord_less_eq @ A @ A2 @ ( bot_bot @ A ) )
= ( A2
= ( bot_bot @ A ) ) ) ) ).
% bot.extremum_unique
thf(fact_151_bot_Oextremum__uniqueI,axiom,
! [A: $tType] :
( ( order_bot @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( ord_less_eq @ A @ A2 @ ( bot_bot @ A ) )
=> ( A2
= ( bot_bot @ A ) ) ) ) ).
% bot.extremum_uniqueI
thf(fact_152_le__funD,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B @ ( type2 @ B ) )
=> ! [F: A > B,G: A > B,X: A] :
( ( ord_less_eq @ ( A > B ) @ F @ G )
=> ( ord_less_eq @ B @ ( F @ X ) @ ( G @ X ) ) ) ) ).
% le_funD
thf(fact_153_le__funE,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B @ ( type2 @ B ) )
=> ! [F: A > B,G: A > B,X: A] :
( ( ord_less_eq @ ( A > B ) @ F @ G )
=> ( ord_less_eq @ B @ ( F @ X ) @ ( G @ X ) ) ) ) ).
% le_funE
thf(fact_154_le__funI,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B @ ( type2 @ B ) )
=> ! [F: A > B,G: A > B] :
( ! [X3: A] : ( ord_less_eq @ B @ ( F @ X3 ) @ ( G @ X3 ) )
=> ( ord_less_eq @ ( A > B ) @ F @ G ) ) ) ).
% le_funI
thf(fact_155_le__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B @ ( type2 @ B ) )
=> ( ( ord_less_eq @ ( A > B ) )
= ( ^ [F2: A > B,G2: A > B] :
! [X2: A] : ( ord_less_eq @ B @ ( F2 @ X2 ) @ ( G2 @ X2 ) ) ) ) ) ).
% le_fun_def
thf(fact_156_order__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B @ ( type2 @ B ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A2: A,F: B > A,B3: B,C2: B] :
( ( ord_less_eq @ A @ A2 @ ( F @ B3 ) )
=> ( ( ord_less_eq @ B @ B3 @ C2 )
=> ( ! [X3: B,Y: B] :
( ( ord_less_eq @ B @ X3 @ Y )
=> ( ord_less_eq @ A @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_eq @ A @ A2 @ ( F @ C2 ) ) ) ) ) ) ).
% order_subst1
thf(fact_157_order__subst2,axiom,
! [A: $tType,C4: $tType] :
( ( ( order @ C4 @ ( type2 @ C4 ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A2: A,B3: A,F: A > C4,C2: C4] :
( ( ord_less_eq @ A @ A2 @ B3 )
=> ( ( ord_less_eq @ C4 @ ( F @ B3 ) @ C2 )
=> ( ! [X3: A,Y: A] :
( ( ord_less_eq @ A @ X3 @ Y )
=> ( ord_less_eq @ C4 @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_eq @ C4 @ ( F @ A2 ) @ C2 ) ) ) ) ) ).
% order_subst2
thf(fact_158_ord__eq__le__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B @ ( type2 @ B ) )
& ( ord @ A @ ( type2 @ A ) ) )
=> ! [A2: A,F: B > A,B3: B,C2: B] :
( ( A2
= ( F @ B3 ) )
=> ( ( ord_less_eq @ B @ B3 @ C2 )
=> ( ! [X3: B,Y: B] :
( ( ord_less_eq @ B @ X3 @ Y )
=> ( ord_less_eq @ A @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_eq @ A @ A2 @ ( F @ C2 ) ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_159_ord__le__eq__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B @ ( type2 @ B ) )
& ( ord @ A @ ( type2 @ A ) ) )
=> ! [A2: A,B3: A,F: A > B,C2: B] :
( ( ord_less_eq @ A @ A2 @ B3 )
=> ( ( ( F @ B3 )
= C2 )
=> ( ! [X3: A,Y: A] :
( ( ord_less_eq @ A @ X3 @ Y )
=> ( ord_less_eq @ B @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_eq @ B @ ( F @ A2 ) @ C2 ) ) ) ) ) ).
% ord_le_eq_subst
thf(fact_160_eq__iff,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ( ( ^ [Y4: A,Z3: A] : Y4 = Z3 )
= ( ^ [X2: A,Y3: A] :
( ( ord_less_eq @ A @ X2 @ Y3 )
& ( ord_less_eq @ A @ Y3 @ X2 ) ) ) ) ) ).
% eq_iff
thf(fact_161_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_162_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_163_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_164_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_165_order_Otrans,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A2: A,B3: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ B3 )
=> ( ( ord_less_eq @ A @ B3 @ C2 )
=> ( ord_less_eq @ A @ A2 @ C2 ) ) ) ) ).
% order.trans
thf(fact_166_le__cases3,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y2: A,Z: A] :
( ( ( ord_less_eq @ A @ X @ Y2 )
=> ~ ( ord_less_eq @ A @ Y2 @ Z ) )
=> ( ( ( ord_less_eq @ A @ Y2 @ X )
=> ~ ( ord_less_eq @ A @ X @ Z ) )
=> ( ( ( ord_less_eq @ A @ X @ Z )
=> ~ ( ord_less_eq @ A @ Z @ Y2 ) )
=> ( ( ( ord_less_eq @ A @ Z @ Y2 )
=> ~ ( ord_less_eq @ A @ Y2 @ X ) )
=> ( ( ( ord_less_eq @ A @ Y2 @ Z )
=> ~ ( ord_less_eq @ A @ Z @ X ) )
=> ~ ( ( ord_less_eq @ A @ Z @ X )
=> ~ ( ord_less_eq @ A @ X @ Y2 ) ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_167_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_168_ord__eq__le__trans,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [A2: A,B3: A,C2: A] :
( ( A2 = B3 )
=> ( ( ord_less_eq @ A @ B3 @ C2 )
=> ( ord_less_eq @ A @ A2 @ C2 ) ) ) ) ).
% ord_eq_le_trans
thf(fact_169_ord__le__eq__trans,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [A2: A,B3: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ B3 )
=> ( ( B3 = C2 )
=> ( ord_less_eq @ A @ A2 @ C2 ) ) ) ) ).
% ord_le_eq_trans
thf(fact_170_order__class_Oorder_Oantisym,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A2: A,B3: A] :
( ( ord_less_eq @ A @ A2 @ B3 )
=> ( ( ord_less_eq @ A @ B3 @ A2 )
=> ( A2 = B3 ) ) ) ) ).
% order_class.order.antisym
thf(fact_171_order__trans,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y2: A,Z: A] :
( ( ord_less_eq @ A @ X @ Y2 )
=> ( ( ord_less_eq @ A @ Y2 @ Z )
=> ( ord_less_eq @ A @ X @ Z ) ) ) ) ).
% order_trans
thf(fact_172_dual__order_Orefl,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A2: A] : ( ord_less_eq @ A @ A2 @ A2 ) ) ).
% dual_order.refl
thf(fact_173_linorder__wlog,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [P: A > A > $o,A2: A,B3: A] :
( ! [A3: A,B7: A] :
( ( ord_less_eq @ A @ A3 @ B7 )
=> ( P @ A3 @ B7 ) )
=> ( ! [A3: A,B7: A] :
( ( P @ B7 @ A3 )
=> ( P @ A3 @ B7 ) )
=> ( P @ A2 @ B3 ) ) ) ) ).
% linorder_wlog
thf(fact_174_dual__order_Otrans,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [B3: A,A2: A,C2: A] :
( ( ord_less_eq @ A @ B3 @ A2 )
=> ( ( ord_less_eq @ A @ C2 @ B3 )
=> ( ord_less_eq @ A @ C2 @ A2 ) ) ) ) ).
% dual_order.trans
thf(fact_175_dual__order_Oantisym,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [B3: A,A2: A] :
( ( ord_less_eq @ A @ B3 @ A2 )
=> ( ( ord_less_eq @ A @ A2 @ B3 )
=> ( A2 = B3 ) ) ) ) ).
% dual_order.antisym
thf(fact_176_set__mp,axiom,
! [A: $tType,A4: set @ A,B2: set @ A,X: A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B2 )
=> ( ( member @ A @ X @ A4 )
=> ( member @ A @ X @ B2 ) ) ) ).
% set_mp
thf(fact_177_in__mono,axiom,
! [A: $tType,A4: set @ A,B2: set @ A,X: A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B2 )
=> ( ( member @ A @ X @ A4 )
=> ( member @ A @ X @ B2 ) ) ) ).
% in_mono
thf(fact_178_subsetD,axiom,
! [A: $tType,A4: set @ A,B2: set @ A,C2: A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B2 )
=> ( ( member @ A @ C2 @ A4 )
=> ( member @ A @ C2 @ B2 ) ) ) ).
% subsetD
thf(fact_179_subsetCE,axiom,
! [A: $tType,A4: set @ A,B2: set @ A,C2: A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B2 )
=> ( ( member @ A @ C2 @ A4 )
=> ( member @ A @ C2 @ B2 ) ) ) ).
% subsetCE
thf(fact_180_equalityE,axiom,
! [A: $tType,A4: set @ A,B2: set @ A] :
( ( A4 = B2 )
=> ~ ( ( ord_less_eq @ ( set @ A ) @ A4 @ B2 )
=> ~ ( ord_less_eq @ ( set @ A ) @ B2 @ A4 ) ) ) ).
% equalityE
thf(fact_181_subset__eq,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A5: set @ A,B5: set @ A] :
! [X2: A] :
( ( member @ A @ X2 @ A5 )
=> ( member @ A @ X2 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_182_equalityD1,axiom,
! [A: $tType,A4: set @ A,B2: set @ A] :
( ( A4 = B2 )
=> ( ord_less_eq @ ( set @ A ) @ A4 @ B2 ) ) ).
% equalityD1
thf(fact_183_equalityD2,axiom,
! [A: $tType,A4: set @ A,B2: set @ A] :
( ( A4 = B2 )
=> ( ord_less_eq @ ( set @ A ) @ B2 @ A4 ) ) ).
% equalityD2
thf(fact_184_set__rev__mp,axiom,
! [A: $tType,X: A,A4: set @ A,B2: set @ A] :
( ( member @ A @ X @ A4 )
=> ( ( ord_less_eq @ ( set @ A ) @ A4 @ B2 )
=> ( member @ A @ X @ B2 ) ) ) ).
% set_rev_mp
thf(fact_185_subset__iff,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A5: set @ A,B5: set @ A] :
! [T2: A] :
( ( member @ A @ T2 @ A5 )
=> ( member @ A @ T2 @ B5 ) ) ) ) ).
% subset_iff
thf(fact_186_rev__subsetD,axiom,
! [A: $tType,C2: A,A4: set @ A,B2: set @ A] :
( ( member @ A @ C2 @ A4 )
=> ( ( ord_less_eq @ ( set @ A ) @ A4 @ B2 )
=> ( member @ A @ C2 @ B2 ) ) ) ).
% rev_subsetD
thf(fact_187_subset__refl,axiom,
! [A: $tType,A4: set @ A] : ( ord_less_eq @ ( set @ A ) @ A4 @ A4 ) ).
% subset_refl
thf(fact_188_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_189_subset__trans,axiom,
! [A: $tType,A4: set @ A,B2: set @ A,C: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B2 )
=> ( ( ord_less_eq @ ( set @ A ) @ B2 @ C )
=> ( ord_less_eq @ ( set @ A ) @ A4 @ C ) ) ) ).
% subset_trans
thf(fact_190_set__eq__subset,axiom,
! [A: $tType] :
( ( ^ [Y4: set @ A,Z3: set @ A] : Y4 = Z3 )
= ( ^ [A5: set @ A,B5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ B5 )
& ( ord_less_eq @ ( set @ A ) @ B5 @ A5 ) ) ) ) ).
% set_eq_subset
thf(fact_191_contra__subsetD,axiom,
! [A: $tType,A4: set @ A,B2: set @ A,C2: A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B2 )
=> ( ~ ( member @ A @ C2 @ B2 )
=> ~ ( member @ A @ C2 @ A4 ) ) ) ).
% contra_subsetD
thf(fact_192_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_193_double__diff,axiom,
! [A: $tType,A4: set @ A,B2: set @ A,C: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B2 )
=> ( ( ord_less_eq @ ( set @ A ) @ B2 @ C )
=> ( ( minus_minus @ ( set @ A ) @ B2 @ ( minus_minus @ ( set @ A ) @ C @ A4 ) )
= A4 ) ) ) ).
% double_diff
thf(fact_194_Diff__subset,axiom,
! [A: $tType,A4: set @ A,B2: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A4 @ B2 ) @ A4 ) ).
% Diff_subset
thf(fact_195_Diff__mono,axiom,
! [A: $tType,A4: set @ A,C: set @ A,D3: set @ A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ C )
=> ( ( ord_less_eq @ ( set @ A ) @ D3 @ B2 )
=> ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A4 @ B2 ) @ ( minus_minus @ ( set @ A ) @ C @ D3 ) ) ) ) ).
% Diff_mono
thf(fact_196_subset__Un__eq,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A5: set @ A,B5: set @ A] :
( ( sup_sup @ ( set @ A ) @ A5 @ B5 )
= B5 ) ) ) ).
% subset_Un_eq
thf(fact_197_Un__absorb2,axiom,
! [A: $tType,B2: set @ A,A4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ B2 @ A4 )
=> ( ( sup_sup @ ( set @ A ) @ A4 @ B2 )
= A4 ) ) ).
% Un_absorb2
thf(fact_198_Un__absorb1,axiom,
! [A: $tType,A4: set @ A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B2 )
=> ( ( sup_sup @ ( set @ A ) @ A4 @ B2 )
= B2 ) ) ).
% Un_absorb1
thf(fact_199_Un__upper2,axiom,
! [A: $tType,B2: set @ A,A4: set @ A] : ( ord_less_eq @ ( set @ A ) @ B2 @ ( sup_sup @ ( set @ A ) @ A4 @ B2 ) ) ).
% Un_upper2
thf(fact_200_Un__upper1,axiom,
! [A: $tType,A4: set @ A,B2: set @ A] : ( ord_less_eq @ ( set @ A ) @ A4 @ ( sup_sup @ ( set @ A ) @ A4 @ B2 ) ) ).
% Un_upper1
thf(fact_201_Un__least,axiom,
! [A: $tType,A4: set @ A,C: set @ A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ C )
=> ( ( ord_less_eq @ ( set @ A ) @ B2 @ C )
=> ( ord_less_eq @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A4 @ B2 ) @ C ) ) ) ).
% Un_least
thf(fact_202_Un__mono,axiom,
! [A: $tType,A4: set @ A,C: set @ A,B2: set @ A,D3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ C )
=> ( ( ord_less_eq @ ( set @ A ) @ B2 @ D3 )
=> ( ord_less_eq @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A4 @ B2 ) @ ( sup_sup @ ( set @ A ) @ C @ D3 ) ) ) ) ).
% Un_mono
thf(fact_203_subset__insertI2,axiom,
! [A: $tType,A4: set @ A,B2: set @ A,B3: A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B2 )
=> ( ord_less_eq @ ( set @ A ) @ A4 @ ( insert @ A @ B3 @ B2 ) ) ) ).
% subset_insertI2
thf(fact_204_subset__insertI,axiom,
! [A: $tType,B2: set @ A,A2: A] : ( ord_less_eq @ ( set @ A ) @ B2 @ ( insert @ A @ A2 @ B2 ) ) ).
% subset_insertI
thf(fact_205_subset__insert,axiom,
! [A: $tType,X: A,A4: set @ A,B2: set @ A] :
( ~ ( member @ A @ X @ A4 )
=> ( ( ord_less_eq @ ( set @ A ) @ A4 @ ( insert @ A @ X @ B2 ) )
= ( ord_less_eq @ ( set @ A ) @ A4 @ B2 ) ) ) ).
% subset_insert
thf(fact_206_insert__mono,axiom,
! [A: $tType,C: set @ A,D3: set @ A,A2: A] :
( ( ord_less_eq @ ( set @ A ) @ C @ D3 )
=> ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ A2 @ C ) @ ( insert @ A @ A2 @ D3 ) ) ) ).
% insert_mono
thf(fact_207_sup_OcoboundedI2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [C2: A,B3: A,A2: A] :
( ( ord_less_eq @ A @ C2 @ B3 )
=> ( ord_less_eq @ A @ C2 @ ( sup_sup @ A @ A2 @ B3 ) ) ) ) ).
% sup.coboundedI2
thf(fact_208_sup_OcoboundedI1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [C2: A,A2: A,B3: A] :
( ( ord_less_eq @ A @ C2 @ A2 )
=> ( ord_less_eq @ A @ C2 @ ( sup_sup @ A @ A2 @ B3 ) ) ) ) ).
% sup.coboundedI1
thf(fact_209_sup_Oabsorb__iff2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ( ( ord_less_eq @ A )
= ( ^ [A6: A,B6: A] :
( ( sup_sup @ A @ A6 @ B6 )
= B6 ) ) ) ) ).
% sup.absorb_iff2
thf(fact_210_sup_Oabsorb__iff1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ( ( ord_less_eq @ A )
= ( ^ [B6: A,A6: A] :
( ( sup_sup @ A @ A6 @ B6 )
= A6 ) ) ) ) ).
% sup.absorb_iff1
thf(fact_211_sup_Ocobounded2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [B3: A,A2: A] : ( ord_less_eq @ A @ B3 @ ( sup_sup @ A @ A2 @ B3 ) ) ) ).
% sup.cobounded2
thf(fact_212_sup_Ocobounded1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [A2: A,B3: A] : ( ord_less_eq @ A @ A2 @ ( sup_sup @ A @ A2 @ B3 ) ) ) ).
% sup.cobounded1
thf(fact_213_sup_Oorder__iff,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ( ( ord_less_eq @ A )
= ( ^ [B6: A,A6: A] :
( A6
= ( sup_sup @ A @ A6 @ B6 ) ) ) ) ) ).
% sup.order_iff
thf(fact_214_sup_OboundedI,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [B3: A,A2: A,C2: A] :
( ( ord_less_eq @ A @ B3 @ A2 )
=> ( ( ord_less_eq @ A @ C2 @ A2 )
=> ( ord_less_eq @ A @ ( sup_sup @ A @ B3 @ C2 ) @ A2 ) ) ) ) ).
% sup.boundedI
thf(fact_215_sup_OboundedE,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [B3: A,C2: A,A2: A] :
( ( ord_less_eq @ A @ ( sup_sup @ A @ B3 @ C2 ) @ A2 )
=> ~ ( ( ord_less_eq @ A @ B3 @ A2 )
=> ~ ( ord_less_eq @ A @ C2 @ A2 ) ) ) ) ).
% sup.boundedE
thf(fact_216_sup__absorb2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [X: A,Y2: A] :
( ( ord_less_eq @ A @ X @ Y2 )
=> ( ( sup_sup @ A @ X @ Y2 )
= Y2 ) ) ) ).
% sup_absorb2
thf(fact_217_sup__absorb1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [Y2: A,X: A] :
( ( ord_less_eq @ A @ Y2 @ X )
=> ( ( sup_sup @ A @ X @ Y2 )
= X ) ) ) ).
% sup_absorb1
thf(fact_218_sup_Oabsorb2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [A2: A,B3: A] :
( ( ord_less_eq @ A @ A2 @ B3 )
=> ( ( sup_sup @ A @ A2 @ B3 )
= B3 ) ) ) ).
% sup.absorb2
thf(fact_219_sup_Oabsorb1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [B3: A,A2: A] :
( ( ord_less_eq @ A @ B3 @ A2 )
=> ( ( sup_sup @ A @ A2 @ B3 )
= A2 ) ) ) ).
% sup.absorb1
thf(fact_220_sup__unique,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [F: A > A > A,X: A,Y2: A] :
( ! [X3: A,Y: A] : ( ord_less_eq @ A @ X3 @ ( F @ X3 @ Y ) )
=> ( ! [X3: A,Y: A] : ( ord_less_eq @ A @ Y @ ( F @ X3 @ Y ) )
=> ( ! [X3: A,Y: A,Z4: A] :
( ( ord_less_eq @ A @ Y @ X3 )
=> ( ( ord_less_eq @ A @ Z4 @ X3 )
=> ( ord_less_eq @ A @ ( F @ Y @ Z4 ) @ X3 ) ) )
=> ( ( sup_sup @ A @ X @ Y2 )
= ( F @ X @ Y2 ) ) ) ) ) ) ).
% sup_unique
thf(fact_221_sup_OorderI,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [A2: A,B3: A] :
( ( A2
= ( sup_sup @ A @ A2 @ B3 ) )
=> ( ord_less_eq @ A @ B3 @ A2 ) ) ) ).
% sup.orderI
thf(fact_222_sup_OorderE,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [B3: A,A2: A] :
( ( ord_less_eq @ A @ B3 @ A2 )
=> ( A2
= ( sup_sup @ A @ A2 @ B3 ) ) ) ) ).
% sup.orderE
thf(fact_223_le__iff__sup,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ( ( ord_less_eq @ A )
= ( ^ [X2: A,Y3: A] :
( ( sup_sup @ A @ X2 @ Y3 )
= Y3 ) ) ) ) ).
% le_iff_sup
thf(fact_224_sup__least,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [Y2: A,X: A,Z: A] :
( ( ord_less_eq @ A @ Y2 @ X )
=> ( ( ord_less_eq @ A @ Z @ X )
=> ( ord_less_eq @ A @ ( sup_sup @ A @ Y2 @ Z ) @ X ) ) ) ) ).
% sup_least
thf(fact_225_sup__mono,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [A2: A,C2: A,B3: A,D: A] :
( ( ord_less_eq @ A @ A2 @ C2 )
=> ( ( ord_less_eq @ A @ B3 @ D )
=> ( ord_less_eq @ A @ ( sup_sup @ A @ A2 @ B3 ) @ ( sup_sup @ A @ C2 @ D ) ) ) ) ) ).
% sup_mono
thf(fact_226_sup_Omono,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [C2: A,A2: A,D: A,B3: A] :
( ( ord_less_eq @ A @ C2 @ A2 )
=> ( ( ord_less_eq @ A @ D @ B3 )
=> ( ord_less_eq @ A @ ( sup_sup @ A @ C2 @ D ) @ ( sup_sup @ A @ A2 @ B3 ) ) ) ) ) ).
% sup.mono
thf(fact_227_le__supI2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [X: A,B3: A,A2: A] :
( ( ord_less_eq @ A @ X @ B3 )
=> ( ord_less_eq @ A @ X @ ( sup_sup @ A @ A2 @ B3 ) ) ) ) ).
% le_supI2
thf(fact_228_le__supI1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [X: A,A2: A,B3: A] :
( ( ord_less_eq @ A @ X @ A2 )
=> ( ord_less_eq @ A @ X @ ( sup_sup @ A @ A2 @ B3 ) ) ) ) ).
% le_supI1
thf(fact_229_sup__ge2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [Y2: A,X: A] : ( ord_less_eq @ A @ Y2 @ ( sup_sup @ A @ X @ Y2 ) ) ) ).
% sup_ge2
thf(fact_230_sup__ge1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [X: A,Y2: A] : ( ord_less_eq @ A @ X @ ( sup_sup @ A @ X @ Y2 ) ) ) ).
% sup_ge1
thf(fact_231_le__supI,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [A2: A,X: A,B3: A] :
( ( ord_less_eq @ A @ A2 @ X )
=> ( ( ord_less_eq @ A @ B3 @ X )
=> ( ord_less_eq @ A @ ( sup_sup @ A @ A2 @ B3 ) @ X ) ) ) ) ).
% le_supI
thf(fact_232_le__supE,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [A2: A,B3: A,X: A] :
( ( ord_less_eq @ A @ ( sup_sup @ A @ A2 @ B3 ) @ X )
=> ~ ( ( ord_less_eq @ A @ A2 @ X )
=> ~ ( ord_less_eq @ A @ B3 @ X ) ) ) ) ).
% le_supE
thf(fact_233_inf__sup__ord_I3_J,axiom,
! [A: $tType] :
( ( lattice @ A @ ( type2 @ A ) )
=> ! [X: A,Y2: A] : ( ord_less_eq @ A @ X @ ( sup_sup @ A @ X @ Y2 ) ) ) ).
% inf_sup_ord(3)
thf(fact_234_inf__sup__ord_I4_J,axiom,
! [A: $tType] :
( ( lattice @ A @ ( type2 @ A ) )
=> ! [Y2: A,X: A] : ( ord_less_eq @ A @ Y2 @ ( sup_sup @ A @ X @ Y2 ) ) ) ).
% inf_sup_ord(4)
thf(fact_235_pairwise__subset,axiom,
! [A: $tType,P: A > A > $o,S3: set @ A,T3: set @ A] :
( ( pairwise @ A @ P @ S3 )
=> ( ( ord_less_eq @ ( set @ A ) @ T3 @ S3 )
=> ( pairwise @ A @ P @ T3 ) ) ) ).
% pairwise_subset
thf(fact_236_subset__singletonD,axiom,
! [A: $tType,A4: set @ A,X: A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
=> ( ( A4
= ( bot_bot @ ( set @ A ) ) )
| ( A4
= ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% subset_singletonD
thf(fact_237_subset__singleton__iff,axiom,
! [A: $tType,X4: set @ A,A2: A] :
( ( ord_less_eq @ ( set @ A ) @ X4 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) )
= ( ( X4
= ( bot_bot @ ( set @ A ) ) )
| ( X4
= ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% subset_singleton_iff
thf(fact_238_subset__Diff__insert,axiom,
! [A: $tType,A4: set @ A,B2: set @ A,X: A,C: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ ( minus_minus @ ( set @ A ) @ B2 @ ( insert @ A @ X @ C ) ) )
= ( ( ord_less_eq @ ( set @ A ) @ A4 @ ( minus_minus @ ( set @ A ) @ B2 @ C ) )
& ~ ( member @ A @ X @ A4 ) ) ) ).
% subset_Diff_insert
thf(fact_239_Diff__partition,axiom,
! [A: $tType,A4: set @ A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B2 )
=> ( ( sup_sup @ ( set @ A ) @ A4 @ ( minus_minus @ ( set @ A ) @ B2 @ A4 ) )
= B2 ) ) ).
% Diff_partition
thf(fact_240_Diff__subset__conv,axiom,
! [A: $tType,A4: set @ A,B2: set @ A,C: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A4 @ B2 ) @ C )
= ( ord_less_eq @ ( set @ A ) @ A4 @ ( sup_sup @ ( set @ A ) @ B2 @ C ) ) ) ).
% Diff_subset_conv
thf(fact_241_insert__subsetI,axiom,
! [A: $tType,X: A,A4: set @ A,X4: set @ A] :
( ( member @ A @ X @ A4 )
=> ( ( ord_less_eq @ ( set @ A ) @ X4 @ A4 )
=> ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ X @ X4 ) @ A4 ) ) ) ).
% insert_subsetI
thf(fact_242_subset__emptyI,axiom,
! [A: $tType,A4: set @ A] :
( ! [X3: A] :
~ ( member @ A @ X3 @ A4 )
=> ( ord_less_eq @ ( set @ A ) @ A4 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% subset_emptyI
thf(fact_243_monoid__def,axiom,
! [A: $tType] :
( ( monoid @ A )
= ( ^ [F2: A > A > A,Z2: A] :
( ( semigroup @ A @ F2 )
& ( monoid_axioms @ A @ F2 @ Z2 ) ) ) ) ).
% monoid_def
thf(fact_244_monoid_Ointro,axiom,
! [A: $tType,F: A > A > A,Z: A] :
( ( semigroup @ A @ F )
=> ( ( monoid_axioms @ A @ F @ Z )
=> ( monoid @ A @ F @ Z ) ) ) ).
% monoid.intro
thf(fact_245_semigroup_Ointro,axiom,
! [A: $tType,F: A > A > A] :
( ! [A3: A,B7: A,C5: A] :
( ( F @ ( F @ A3 @ B7 ) @ C5 )
= ( F @ A3 @ ( F @ B7 @ C5 ) ) )
=> ( semigroup @ A @ F ) ) ).
% semigroup.intro
thf(fact_246_semigroup_Oassoc,axiom,
! [A: $tType,F: A > A > A,A2: A,B3: A,C2: A] :
( ( semigroup @ A @ F )
=> ( ( F @ ( F @ A2 @ B3 ) @ C2 )
= ( F @ A2 @ ( F @ B3 @ C2 ) ) ) ) ).
% semigroup.assoc
thf(fact_247_semigroup__def,axiom,
! [A: $tType] :
( ( semigroup @ A )
= ( ^ [F2: A > A > A] :
! [A6: A,B6: A,C6: A] :
( ( F2 @ ( F2 @ A6 @ B6 ) @ C6 )
= ( F2 @ A6 @ ( F2 @ B6 @ C6 ) ) ) ) ) ).
% semigroup_def
thf(fact_248_sup_Osemigroup__axioms,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ( semigroup @ A @ ( sup_sup @ A ) ) ) ).
% sup.semigroup_axioms
thf(fact_249_monoid_Oaxioms_I1_J,axiom,
! [A: $tType,F: A > A > A,Z: A] :
( ( monoid @ A @ F @ Z )
=> ( semigroup @ A @ F ) ) ).
% monoid.axioms(1)
thf(fact_250_psubset__insert__iff,axiom,
! [A: $tType,A4: set @ A,X: A,B2: set @ A] :
( ( ord_less @ ( set @ A ) @ A4 @ ( insert @ A @ X @ B2 ) )
= ( ( ( member @ A @ X @ B2 )
=> ( ord_less @ ( set @ A ) @ A4 @ B2 ) )
& ( ~ ( member @ A @ X @ B2 )
=> ( ( ( member @ A @ X @ A4 )
=> ( ord_less @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) @ B2 ) )
& ( ~ ( member @ A @ X @ A4 )
=> ( ord_less_eq @ ( set @ A ) @ A4 @ B2 ) ) ) ) ) ) ).
% psubset_insert_iff
thf(fact_251_subset__Compl__singleton,axiom,
! [A: $tType,A4: set @ A,B3: A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ ( uminus_uminus @ ( set @ A ) @ ( insert @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( ~ ( member @ A @ B3 @ A4 ) ) ) ).
% subset_Compl_singleton
thf(fact_252_add_Oinverse__inverse,axiom,
! [A: $tType] :
( ( group_add @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( uminus_uminus @ A @ ( uminus_uminus @ A @ A2 ) )
= A2 ) ) ).
% add.inverse_inverse
thf(fact_253_neg__equal__iff__equal,axiom,
! [A: $tType] :
( ( group_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B3: A] :
( ( ( uminus_uminus @ A @ A2 )
= ( uminus_uminus @ A @ B3 ) )
= ( A2 = B3 ) ) ) ).
% neg_equal_iff_equal
thf(fact_254_uminus__apply,axiom,
! [B: $tType,A: $tType] :
( ( uminus @ B @ ( type2 @ B ) )
=> ( ( uminus_uminus @ ( A > B ) )
= ( ^ [A5: A > B,X2: A] : ( uminus_uminus @ B @ ( A5 @ X2 ) ) ) ) ) ).
% uminus_apply
thf(fact_255_double__compl,axiom,
! [A: $tType] :
( ( boolean_algebra @ A @ ( type2 @ A ) )
=> ! [X: A] :
( ( uminus_uminus @ A @ ( uminus_uminus @ A @ X ) )
= X ) ) ).
% double_compl
%----Type constructors (50)
thf(tcon_HOL_Obool___Lattices_Obounded__lattice,axiom,
bounded_lattice @ $o @ ( type2 @ $o ) ).
thf(tcon_Set_Oset___Lattices_Obounded__lattice_1,axiom,
! [A7: $tType] : ( bounded_lattice @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
thf(tcon_fun___Lattices_Obounded__lattice_2,axiom,
! [A7: $tType,A8: $tType] :
( ( bounded_lattice @ A8 @ ( type2 @ A8 ) )
=> ( bounded_lattice @ ( A7 > A8 ) @ ( type2 @ ( A7 > A8 ) ) ) ) ).
thf(tcon_fun___Lattices_Obounded__semilattice__sup__bot,axiom,
! [A7: $tType,A8: $tType] :
( ( bounded_lattice @ A8 @ ( type2 @ A8 ) )
=> ( bounde1808546759up_bot @ ( A7 > A8 ) @ ( type2 @ ( A7 > A8 ) ) ) ) ).
thf(tcon_fun___Lattices_Obounded__lattice__bot,axiom,
! [A7: $tType,A8: $tType] :
( ( bounded_lattice @ A8 @ ( type2 @ A8 ) )
=> ( bounded_lattice_bot @ ( A7 > A8 ) @ ( type2 @ ( A7 > A8 ) ) ) ) ).
thf(tcon_fun___Lattices_Osemilattice__sup,axiom,
! [A7: $tType,A8: $tType] :
( ( semilattice_sup @ A8 @ ( type2 @ A8 ) )
=> ( semilattice_sup @ ( A7 > A8 ) @ ( type2 @ ( A7 > A8 ) ) ) ) ).
thf(tcon_fun___Lattices_Oboolean__algebra,axiom,
! [A7: $tType,A8: $tType] :
( ( boolean_algebra @ A8 @ ( type2 @ A8 ) )
=> ( boolean_algebra @ ( 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___Lattices_Olattice,axiom,
! [A7: $tType,A8: $tType] :
( ( lattice @ A8 @ ( type2 @ A8 ) )
=> ( lattice @ ( 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_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_Ouminus,axiom,
! [A7: $tType,A8: $tType] :
( ( uminus @ A8 @ ( type2 @ A8 ) )
=> ( uminus @ ( 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_Nat_Onat___Groups_Ocancel__ab__semigroup__add,axiom,
cancel146912293up_add @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Lattices_Osemilattice__sup_3,axiom,
semilattice_sup @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Orderings_Oorder__bot_4,axiom,
order_bot @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Orderings_Opreorder_5,axiom,
preorder @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Orderings_Olinorder,axiom,
linorder @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Lattices_Olattice_6,axiom,
lattice @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Orderings_Oorder_7,axiom,
order @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Orderings_Oord_8,axiom,
ord @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Orderings_Obot_9,axiom,
bot @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Groups_Ominus_10,axiom,
minus @ nat @ ( type2 @ nat ) ).
thf(tcon_Set_Oset___Lattices_Obounded__semilattice__sup__bot_11,axiom,
! [A7: $tType] : ( bounde1808546759up_bot @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
thf(tcon_Set_Oset___Lattices_Obounded__lattice__bot_12,axiom,
! [A7: $tType] : ( bounded_lattice_bot @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
thf(tcon_Set_Oset___Lattices_Osemilattice__sup_13,axiom,
! [A7: $tType] : ( semilattice_sup @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
thf(tcon_Set_Oset___Lattices_Oboolean__algebra_14,axiom,
! [A7: $tType] : ( boolean_algebra @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder__bot_15,axiom,
! [A7: $tType] : ( order_bot @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Opreorder_16,axiom,
! [A7: $tType] : ( preorder @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
thf(tcon_Set_Oset___Lattices_Olattice_17,axiom,
! [A7: $tType] : ( lattice @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder_18,axiom,
! [A7: $tType] : ( order @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Oord_19,axiom,
! [A7: $tType] : ( ord @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Obot_20,axiom,
! [A7: $tType] : ( bot @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
thf(tcon_Set_Oset___Groups_Ouminus_21,axiom,
! [A7: $tType] : ( uminus @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
thf(tcon_Set_Oset___Groups_Ominus_22,axiom,
! [A7: $tType] : ( minus @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
thf(tcon_HOL_Obool___Lattices_Obounded__semilattice__sup__bot_23,axiom,
bounde1808546759up_bot @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Lattices_Obounded__lattice__bot_24,axiom,
bounded_lattice_bot @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Lattices_Osemilattice__sup_25,axiom,
semilattice_sup @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Lattices_Oboolean__algebra_26,axiom,
boolean_algebra @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Oorder__bot_27,axiom,
order_bot @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Opreorder_28,axiom,
preorder @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Olinorder_29,axiom,
linorder @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Lattices_Olattice_30,axiom,
lattice @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Oorder_31,axiom,
order @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Oord_32,axiom,
ord @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Obot_33,axiom,
bot @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Groups_Ouminus_34,axiom,
uminus @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Groups_Ominus_35,axiom,
minus @ $o @ ( type2 @ $o ) ).
%----Conjectures (1)
thf(conj_0,conjecture,
( ( ( member @ a @ a2 @ ( huffma505251170phabet @ a @ t ) )
=> ( ( ( member @ a @ b @ ( huffma505251170phabet @ a @ t ) )
=> ( ( huffma505251170phabet @ a @ ( huffma2094459102Leaves @ a @ t @ w_a @ a2 @ w_b @ b ) )
= ( huffma505251170phabet @ a @ t ) ) )
& ( ~ ( member @ a @ b @ ( huffma505251170phabet @ a @ t ) )
=> ( ( huffma505251170phabet @ a @ ( huffma2094459102Leaves @ a @ t @ w_a @ a2 @ w_b @ b ) )
= ( sup_sup @ ( set @ a ) @ ( minus_minus @ ( set @ a ) @ ( huffma505251170phabet @ a @ t ) @ ( insert @ a @ a2 @ ( bot_bot @ ( set @ a ) ) ) ) @ ( insert @ a @ b @ ( bot_bot @ ( set @ a ) ) ) ) ) ) ) )
& ( ~ ( member @ a @ a2 @ ( huffma505251170phabet @ a @ t ) )
=> ( ( ( member @ a @ b @ ( huffma505251170phabet @ a @ t ) )
=> ( ( huffma505251170phabet @ a @ ( huffma2094459102Leaves @ a @ t @ w_a @ a2 @ w_b @ b ) )
= ( sup_sup @ ( set @ a ) @ ( minus_minus @ ( set @ a ) @ ( huffma505251170phabet @ a @ t ) @ ( insert @ a @ b @ ( bot_bot @ ( set @ a ) ) ) ) @ ( insert @ a @ a2 @ ( bot_bot @ ( set @ a ) ) ) ) ) )
& ( ~ ( member @ a @ b @ ( huffma505251170phabet @ a @ t ) )
=> ( ( huffma505251170phabet @ a @ ( huffma2094459102Leaves @ a @ t @ w_a @ a2 @ w_b @ b ) )
= ( huffma505251170phabet @ a @ t ) ) ) ) ) ) ).
%------------------------------------------------------------------------------