TPTP Problem File: ITP031^2.p
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%------------------------------------------------------------------------------
% File : ITP031^2 : TPTP v8.2.0. Released v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer BinaryTree problem prob_301__3253080_1
% Version : Especial.
% English :
% Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% : [Des21] Desharnais (2021), Email to Geoff Sutcliffe
% Source : [Des21]
% Names : BinaryTree/prob_301__3253080_1 [Des21]
% Status : Theorem
% Rating : 0.00 v7.5.0
% Syntax : Number of formulae : 355 ( 139 unt; 52 typ; 0 def)
% Number of atoms : 778 ( 300 equ; 0 cnn)
% Maximal formula atoms : 7 ( 2 avg)
% Number of connectives : 3436 ( 90 ~; 17 |; 66 &;2943 @)
% ( 0 <=>; 320 =>; 0 <=; 0 <~>)
% Maximal formula depth : 17 ( 7 avg)
% Number of types : 3 ( 2 usr)
% Number of type conns : 147 ( 147 >; 0 *; 0 +; 0 <<)
% Number of symbols : 53 ( 50 usr; 6 con; 0-5 aty)
% Number of variables : 1009 ( 92 ^; 856 !; 18 ?;1009 :)
% ( 43 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Sledgehammer 2021-02-23 16:16:01.044
%------------------------------------------------------------------------------
% Could-be-implicit typings (4)
thf(ty_t_BinaryTree__Mirabelle__pchhvghoao_OTree,type,
binary1291135688e_Tree: $tType > $tType ).
thf(ty_t_Set_Oset,type,
set: $tType > $tType ).
thf(ty_t_Int_Oint,type,
int: $tType ).
thf(ty_tf_a,type,
a: $tType ).
% Explicit typings (48)
thf(sy_cl_Lattices_Obounded__lattice,type,
bounded_lattice:
!>[A: $tType] : $o ).
thf(sy_cl_HOL_Otype,type,
type:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ominus,type,
minus:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Obot,type,
bot:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Oord,type,
ord:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Oorder,type,
order:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ogroup__add,type,
group_add:
!>[A: $tType] : $o ).
thf(sy_cl_Lattices_Olattice,type,
lattice:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Ono__bot,type,
no_bot:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Ono__top,type,
no_top:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Olinorder,type,
linorder:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Opreorder,type,
preorder:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Oorder__bot,type,
order_bot:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Owellorder,type,
wellorder:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Odense__order,type,
dense_order:
!>[A: $tType] : $o ).
thf(sy_cl_Lattices_Osemilattice__sup,type,
semilattice_sup:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oordered__ab__group__add,type,
ordered_ab_group_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ocancel__ab__semigroup__add,type,
cancel146912293up_add:
!>[A: $tType] : $o ).
thf(sy_cl_Lattices_Obounded__semilattice__sup__bot,type,
bounde1808546759up_bot:
!>[A: $tType] : $o ).
thf(sy_c_BinaryTree__Mirabelle__pchhvghoao_OTree_OT,type,
binary210054475elle_T:
!>[A: $tType] : ( ( binary1291135688e_Tree @ A ) > A > ( binary1291135688e_Tree @ A ) > ( binary1291135688e_Tree @ A ) ) ).
thf(sy_c_BinaryTree__Mirabelle__pchhvghoao_OTree_OTip,type,
binary1746293266le_Tip:
!>[A: $tType] : ( binary1291135688e_Tree @ A ) ).
thf(sy_c_BinaryTree__Mirabelle__pchhvghoao_OTree_Oset__Tree,type,
binary2130109271t_Tree:
!>[A: $tType] : ( ( binary1291135688e_Tree @ A ) > ( set @ A ) ) ).
thf(sy_c_BinaryTree__Mirabelle__pchhvghoao_Obinsert,type,
binary1830089824insert:
!>[A: $tType] : ( ( A > int ) > A > ( binary1291135688e_Tree @ A ) > ( binary1291135688e_Tree @ A ) ) ).
thf(sy_c_BinaryTree__Mirabelle__pchhvghoao_Oeqs,type,
binary64540844le_eqs:
!>[A: $tType] : ( ( A > int ) > A > ( set @ A ) ) ).
thf(sy_c_BinaryTree__Mirabelle__pchhvghoao_Omemb,type,
binary827270440e_memb:
!>[A: $tType] : ( ( A > int ) > A > ( binary1291135688e_Tree @ A ) > $o ) ).
thf(sy_c_BinaryTree__Mirabelle__pchhvghoao_OsetOf,type,
binary1653327646_setOf:
!>[A: $tType] : ( ( binary1291135688e_Tree @ A ) > ( set @ A ) ) ).
thf(sy_c_BinaryTree__Mirabelle__pchhvghoao_OsortedTree,type,
binary1610619414edTree:
!>[A: $tType] : ( ( A > int ) > ( binary1291135688e_Tree @ A ) > $o ) ).
thf(sy_c_BinaryTree__Mirabelle__pchhvghoao_Osorted__distinct__pred,type,
binary231205461t_pred:
!>[A: $tType] : ( ( A > int ) > A > A > ( binary1291135688e_Tree @ A ) > $o ) ).
thf(sy_c_Groups_Ominus__class_Ominus,type,
minus_minus:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_HOL_OThe,type,
the:
!>[A: $tType] : ( ( A > $o ) > A ) ).
thf(sy_c_If,type,
if:
!>[A: $tType] : ( $o > A > A > A ) ).
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_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_e,type,
e: a ).
thf(sy_v_h,type,
h: a > int ).
thf(sy_v_t1____,type,
t1: binary1291135688e_Tree @ a ).
thf(sy_v_t2____,type,
t2: binary1291135688e_Tree @ a ).
thf(sy_v_x____,type,
x: a ).
% Relevant facts (256)
thf(fact_0_eqsLessX,axiom,
! [X: a] :
( ( member @ a @ X @ ( binary64540844le_eqs @ a @ h @ e ) )
=> ( ord_less @ int @ ( h @ X ) @ ( h @ x ) ) ) ).
% eqsLessX
thf(fact_1_eLess,axiom,
ord_less @ int @ ( h @ e ) @ ( h @ x ) ).
% eLess
thf(fact_2_res,axiom,
( ( binary1830089824insert @ a @ h @ e @ ( binary210054475elle_T @ a @ t1 @ x @ t2 ) )
= ( binary210054475elle_T @ a @ ( binary1830089824insert @ a @ h @ e @ t1 ) @ x @ t2 ) ) ).
% res
thf(fact_3_s1,axiom,
binary1610619414edTree @ a @ h @ t1 ).
% s1
thf(fact_4_s2,axiom,
binary1610619414edTree @ a @ h @ t2 ).
% s2
thf(fact_5_s,axiom,
binary1610619414edTree @ a @ h @ ( binary210054475elle_T @ a @ t1 @ x @ t2 ) ).
% s
thf(fact_6_sorted__distinct,axiom,
! [A: $tType,H: A > int,A2: A,B: A,T: binary1291135688e_Tree @ A] : ( binary231205461t_pred @ A @ H @ A2 @ B @ T ) ).
% sorted_distinct
thf(fact_7_c1,axiom,
( ( binary1653327646_setOf @ a @ ( binary1830089824insert @ a @ h @ e @ t1 ) )
= ( sup_sup @ ( set @ a ) @ ( minus_minus @ ( set @ a ) @ ( binary1653327646_setOf @ a @ t1 ) @ ( binary64540844le_eqs @ a @ h @ e ) ) @ ( insert @ a @ e @ ( bot_bot @ ( set @ a ) ) ) ) ) ).
% c1
thf(fact_8_c2,axiom,
( ( binary1653327646_setOf @ a @ ( binary1830089824insert @ a @ h @ e @ t2 ) )
= ( sup_sup @ ( set @ a ) @ ( minus_minus @ ( set @ a ) @ ( binary1653327646_setOf @ a @ t2 ) @ ( binary64540844le_eqs @ a @ h @ e ) ) @ ( insert @ a @ e @ ( bot_bot @ ( set @ a ) ) ) ) ) ).
% c2
thf(fact_9_eqs__def,axiom,
! [A: $tType] :
( ( binary64540844le_eqs @ A )
= ( ^ [H2: A > int,X2: A] :
( collect @ A
@ ^ [Y: A] :
( ( H2 @ Y )
= ( H2 @ X2 ) ) ) ) ) ).
% eqs_def
thf(fact_10_h1,axiom,
( ( binary1610619414edTree @ a @ h @ t1 )
=> ( ( binary1653327646_setOf @ a @ ( binary1830089824insert @ a @ h @ e @ t1 ) )
= ( sup_sup @ ( set @ a ) @ ( minus_minus @ ( set @ a ) @ ( binary1653327646_setOf @ a @ t1 ) @ ( binary64540844le_eqs @ a @ h @ e ) ) @ ( insert @ a @ e @ ( bot_bot @ ( set @ a ) ) ) ) ) ) ).
% h1
thf(fact_11_h2,axiom,
( ( binary1610619414edTree @ a @ h @ t2 )
=> ( ( binary1653327646_setOf @ a @ ( binary1830089824insert @ a @ h @ e @ t2 ) )
= ( sup_sup @ ( set @ a ) @ ( minus_minus @ ( set @ a ) @ ( binary1653327646_setOf @ a @ t2 ) @ ( binary64540844le_eqs @ a @ h @ e ) ) @ ( insert @ a @ e @ ( bot_bot @ ( set @ a ) ) ) ) ) ) ).
% h2
thf(fact_12__092_060open_062sortedTree_Ah_ATip_A_092_060longrightarrow_062_AsetOf_A_Ibinsert_Ah_Ae_ATip_J_A_061_AsetOf_ATip_A_N_Aeqs_Ah_Ae_A_092_060union_062_A_123e_125_092_060close_062,axiom,
( ( binary1610619414edTree @ a @ h @ ( binary1746293266le_Tip @ a ) )
=> ( ( binary1653327646_setOf @ a @ ( binary1830089824insert @ a @ h @ e @ ( binary1746293266le_Tip @ a ) ) )
= ( sup_sup @ ( set @ a ) @ ( minus_minus @ ( set @ a ) @ ( binary1653327646_setOf @ a @ ( binary1746293266le_Tip @ a ) ) @ ( binary64540844le_eqs @ a @ h @ e ) ) @ ( insert @ a @ e @ ( bot_bot @ ( set @ a ) ) ) ) ) ) ).
% \<open>sortedTree h Tip \<longrightarrow> setOf (binsert h e Tip) = setOf Tip - eqs h e \<union> {e}\<close>
thf(fact_13_Tree_Oinject,axiom,
! [A: $tType,X21: binary1291135688e_Tree @ A,X22: A,X23: binary1291135688e_Tree @ A,Y21: binary1291135688e_Tree @ A,Y22: A,Y23: binary1291135688e_Tree @ A] :
( ( ( binary210054475elle_T @ A @ X21 @ X22 @ X23 )
= ( binary210054475elle_T @ A @ Y21 @ Y22 @ Y23 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 )
& ( X23 = Y23 ) ) ) ).
% Tree.inject
thf(fact_14_setOf_Osimps_I2_J,axiom,
! [A: $tType,T1: binary1291135688e_Tree @ A,X3: A,T2: binary1291135688e_Tree @ A] :
( ( binary1653327646_setOf @ A @ ( binary210054475elle_T @ A @ T1 @ X3 @ T2 ) )
= ( sup_sup @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ ( binary1653327646_setOf @ A @ T1 ) @ ( binary1653327646_setOf @ A @ T2 ) ) @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% setOf.simps(2)
thf(fact_15_setOf_Osimps_I1_J,axiom,
! [A: $tType] :
( ( binary1653327646_setOf @ A @ ( binary1746293266le_Tip @ A ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% setOf.simps(1)
thf(fact_16_Tree_Odistinct_I1_J,axiom,
! [A: $tType,X21: binary1291135688e_Tree @ A,X22: A,X23: binary1291135688e_Tree @ A] :
( ( binary1746293266le_Tip @ A )
!= ( binary210054475elle_T @ A @ X21 @ X22 @ X23 ) ) ).
% Tree.distinct(1)
thf(fact_17_binsert_Osimps_I2_J,axiom,
! [A: $tType,H: A > int,E: A,X3: A,T1: binary1291135688e_Tree @ A,T2: binary1291135688e_Tree @ A] :
( ( ( ord_less @ int @ ( H @ E ) @ ( H @ X3 ) )
=> ( ( binary1830089824insert @ A @ H @ E @ ( binary210054475elle_T @ A @ T1 @ X3 @ T2 ) )
= ( binary210054475elle_T @ A @ ( binary1830089824insert @ A @ H @ E @ T1 ) @ X3 @ T2 ) ) )
& ( ~ ( ord_less @ int @ ( H @ E ) @ ( H @ X3 ) )
=> ( ( ( ord_less @ int @ ( H @ X3 ) @ ( H @ E ) )
=> ( ( binary1830089824insert @ A @ H @ E @ ( binary210054475elle_T @ A @ T1 @ X3 @ T2 ) )
= ( binary210054475elle_T @ A @ T1 @ X3 @ ( binary1830089824insert @ A @ H @ E @ T2 ) ) ) )
& ( ~ ( ord_less @ int @ ( H @ X3 ) @ ( H @ E ) )
=> ( ( binary1830089824insert @ A @ H @ E @ ( binary210054475elle_T @ A @ T1 @ X3 @ T2 ) )
= ( binary210054475elle_T @ A @ T1 @ E @ T2 ) ) ) ) ) ) ).
% binsert.simps(2)
thf(fact_18_binsert_Osimps_I1_J,axiom,
! [A: $tType,H: A > int,E: A] :
( ( binary1830089824insert @ A @ H @ E @ ( binary1746293266le_Tip @ A ) )
= ( binary210054475elle_T @ A @ ( binary1746293266le_Tip @ A ) @ E @ ( binary1746293266le_Tip @ A ) ) ) ).
% binsert.simps(1)
thf(fact_19_sortedTree_Osimps_I2_J,axiom,
! [A: $tType,H: A > int,T1: binary1291135688e_Tree @ A,X3: A,T2: binary1291135688e_Tree @ A] :
( ( binary1610619414edTree @ A @ H @ ( binary210054475elle_T @ A @ T1 @ X3 @ T2 ) )
= ( ( binary1610619414edTree @ A @ H @ T1 )
& ! [X2: A] :
( ( member @ A @ X2 @ ( binary1653327646_setOf @ A @ T1 ) )
=> ( ord_less @ int @ ( H @ X2 ) @ ( H @ X3 ) ) )
& ! [X2: A] :
( ( member @ A @ X2 @ ( binary1653327646_setOf @ A @ T2 ) )
=> ( ord_less @ int @ ( H @ X3 ) @ ( H @ X2 ) ) )
& ( binary1610619414edTree @ A @ H @ T2 ) ) ) ).
% sortedTree.simps(2)
thf(fact_20_sortedTree_Osimps_I1_J,axiom,
! [A: $tType,H: A > int] : ( binary1610619414edTree @ A @ H @ ( binary1746293266le_Tip @ A ) ) ).
% sortedTree.simps(1)
thf(fact_21_sortLemmaL,axiom,
! [A: $tType,H: A > int,T1: binary1291135688e_Tree @ A,X3: A,T2: binary1291135688e_Tree @ A] :
( ( binary1610619414edTree @ A @ H @ ( binary210054475elle_T @ A @ T1 @ X3 @ T2 ) )
=> ( binary1610619414edTree @ A @ H @ T1 ) ) ).
% sortLemmaL
thf(fact_22_sortLemmaR,axiom,
! [A: $tType,H: A > int,T1: binary1291135688e_Tree @ A,X3: A,T2: binary1291135688e_Tree @ A] :
( ( binary1610619414edTree @ A @ H @ ( binary210054475elle_T @ A @ T1 @ X3 @ T2 ) )
=> ( binary1610619414edTree @ A @ H @ T2 ) ) ).
% sortLemmaR
thf(fact_23_Tree_Oinduct,axiom,
! [A: $tType,P: ( binary1291135688e_Tree @ A ) > $o,Tree: binary1291135688e_Tree @ A] :
( ( P @ ( binary1746293266le_Tip @ A ) )
=> ( ! [X1: binary1291135688e_Tree @ A,X24: A,X32: binary1291135688e_Tree @ A] :
( ( P @ X1 )
=> ( ( P @ X32 )
=> ( P @ ( binary210054475elle_T @ A @ X1 @ X24 @ X32 ) ) ) )
=> ( P @ Tree ) ) ) ).
% Tree.induct
thf(fact_24_Tree_Oexhaust,axiom,
! [A: $tType,Y2: binary1291135688e_Tree @ A] :
( ( Y2
!= ( binary1746293266le_Tip @ A ) )
=> ~ ! [X212: binary1291135688e_Tree @ A,X222: A,X232: binary1291135688e_Tree @ A] :
( Y2
!= ( binary210054475elle_T @ A @ X212 @ X222 @ X232 ) ) ) ).
% Tree.exhaust
thf(fact_25_sorted__distinct__pred__def,axiom,
! [A: $tType] :
( ( binary231205461t_pred @ A )
= ( ^ [H2: A > int,A3: A,B2: A,T3: binary1291135688e_Tree @ A] :
( ( ( binary1610619414edTree @ A @ H2 @ T3 )
& ( member @ A @ A3 @ ( binary1653327646_setOf @ A @ T3 ) )
& ( member @ A @ B2 @ ( binary1653327646_setOf @ A @ T3 ) )
& ( ( H2 @ A3 )
= ( H2 @ B2 ) ) )
=> ( A3 = B2 ) ) ) ) ).
% sorted_distinct_pred_def
thf(fact_26_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_27_singleton__conv,axiom,
! [A: $tType,A2: A] :
( ( collect @ A
@ ^ [X2: A] : X2 = A2 )
= ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% singleton_conv
thf(fact_28_singleton__conv2,axiom,
! [A: $tType,A2: A] :
( ( collect @ A
@ ( ^ [Y3: A,Z: A] : Y3 = Z
@ A2 ) )
= ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% singleton_conv2
thf(fact_29_Un__Diff__cancel,axiom,
! [A: $tType,A4: set @ A,B3: set @ A] :
( ( sup_sup @ ( set @ A ) @ A4 @ ( minus_minus @ ( set @ A ) @ B3 @ A4 ) )
= ( sup_sup @ ( set @ A ) @ A4 @ B3 ) ) ).
% Un_Diff_cancel
thf(fact_30_Un__Diff__cancel2,axiom,
! [A: $tType,B3: set @ A,A4: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ B3 @ A4 ) @ A4 )
= ( sup_sup @ ( set @ A ) @ B3 @ A4 ) ) ).
% Un_Diff_cancel2
thf(fact_31_Diff__insert0,axiom,
! [A: $tType,X3: A,A4: set @ A,B3: set @ A] :
( ~ ( member @ A @ X3 @ A4 )
=> ( ( minus_minus @ ( set @ A ) @ A4 @ ( insert @ A @ X3 @ B3 ) )
= ( minus_minus @ ( set @ A ) @ A4 @ B3 ) ) ) ).
% Diff_insert0
thf(fact_32_insert__Diff1,axiom,
! [A: $tType,X3: A,B3: set @ A,A4: set @ A] :
( ( member @ A @ X3 @ B3 )
=> ( ( minus_minus @ ( set @ A ) @ ( insert @ A @ X3 @ A4 ) @ B3 )
= ( minus_minus @ ( set @ A ) @ A4 @ B3 ) ) ) ).
% insert_Diff1
thf(fact_33_Un__insert__left,axiom,
! [A: $tType,A2: A,B3: set @ A,C: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( insert @ A @ A2 @ B3 ) @ C )
= ( insert @ A @ A2 @ ( sup_sup @ ( set @ A ) @ B3 @ C ) ) ) ).
% Un_insert_left
thf(fact_34_Un__insert__right,axiom,
! [A: $tType,A4: set @ A,A2: A,B3: set @ A] :
( ( sup_sup @ ( set @ A ) @ A4 @ ( insert @ A @ A2 @ B3 ) )
= ( insert @ A @ A2 @ ( sup_sup @ ( set @ A ) @ A4 @ B3 ) ) ) ).
% Un_insert_right
thf(fact_35_Diff__empty,axiom,
! [A: $tType,A4: set @ A] :
( ( minus_minus @ ( set @ A ) @ A4 @ ( bot_bot @ ( set @ A ) ) )
= A4 ) ).
% Diff_empty
thf(fact_36_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_37_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_38_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_39_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_40_empty__iff,axiom,
! [A: $tType,C2: A] :
~ ( member @ A @ C2 @ ( bot_bot @ ( set @ A ) ) ) ).
% empty_iff
thf(fact_41_insert__absorb2,axiom,
! [A: $tType,X3: A,A4: set @ A] :
( ( insert @ A @ X3 @ ( insert @ A @ X3 @ A4 ) )
= ( insert @ A @ X3 @ A4 ) ) ).
% insert_absorb2
thf(fact_42_insert__iff,axiom,
! [A: $tType,A2: A,B: A,A4: set @ A] :
( ( member @ A @ A2 @ ( insert @ A @ B @ A4 ) )
= ( ( A2 = B )
| ( member @ A @ A2 @ A4 ) ) ) ).
% insert_iff
thf(fact_43_insertCI,axiom,
! [A: $tType,A2: A,B3: set @ A,B: A] :
( ( ~ ( member @ A @ A2 @ B3 )
=> ( A2 = B ) )
=> ( member @ A @ A2 @ ( insert @ A @ B @ B3 ) ) ) ).
% insertCI
thf(fact_44_Un__iff,axiom,
! [A: $tType,C2: A,A4: set @ A,B3: set @ A] :
( ( member @ A @ C2 @ ( sup_sup @ ( set @ A ) @ A4 @ B3 ) )
= ( ( member @ A @ C2 @ A4 )
| ( member @ A @ C2 @ B3 ) ) ) ).
% Un_iff
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] :
( ! [X4: A] :
( ( P @ X4 )
= ( Q @ X4 ) )
=> ( ( collect @ A @ P )
= ( collect @ A @ Q ) ) ) ).
% Collect_cong
thf(fact_48_ext,axiom,
! [B4: $tType,A: $tType,F: A > B4,G: A > B4] :
( ! [X4: A] :
( ( F @ X4 )
= ( G @ X4 ) )
=> ( F = G ) ) ).
% ext
thf(fact_49_UnCI,axiom,
! [A: $tType,C2: A,B3: set @ A,A4: set @ A] :
( ( ~ ( member @ A @ C2 @ B3 )
=> ( member @ A @ C2 @ A4 ) )
=> ( member @ A @ C2 @ ( sup_sup @ ( set @ A ) @ A4 @ B3 ) ) ) ).
% UnCI
thf(fact_50_Diff__idemp,axiom,
! [A: $tType,A4: set @ A,B3: set @ A] :
( ( minus_minus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A4 @ B3 ) @ B3 )
= ( minus_minus @ ( set @ A ) @ A4 @ B3 ) ) ).
% Diff_idemp
thf(fact_51_Diff__iff,axiom,
! [A: $tType,C2: A,A4: set @ A,B3: set @ A] :
( ( member @ A @ C2 @ ( minus_minus @ ( set @ A ) @ A4 @ B3 ) )
= ( ( member @ A @ C2 @ A4 )
& ~ ( member @ A @ C2 @ B3 ) ) ) ).
% Diff_iff
thf(fact_52_DiffI,axiom,
! [A: $tType,C2: A,A4: set @ A,B3: set @ A] :
( ( member @ A @ C2 @ A4 )
=> ( ~ ( member @ A @ C2 @ B3 )
=> ( member @ A @ C2 @ ( minus_minus @ ( set @ A ) @ A4 @ B3 ) ) ) ) ).
% DiffI
thf(fact_53_singletonI,axiom,
! [A: $tType,A2: A] : ( member @ A @ A2 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% singletonI
thf(fact_54_Un__empty,axiom,
! [A: $tType,A4: set @ A,B3: set @ A] :
( ( ( sup_sup @ ( set @ A ) @ A4 @ B3 )
= ( bot_bot @ ( set @ A ) ) )
= ( ( A4
= ( bot_bot @ ( set @ A ) ) )
& ( B3
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Un_empty
thf(fact_55_Diff__cancel,axiom,
! [A: $tType,A4: set @ A] :
( ( minus_minus @ ( set @ A ) @ A4 @ A4 )
= ( bot_bot @ ( set @ A ) ) ) ).
% Diff_cancel
thf(fact_56_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_57_equals0I,axiom,
! [A: $tType,A4: set @ A] :
( ! [Y4: A] :
~ ( member @ A @ Y4 @ A4 )
=> ( A4
= ( bot_bot @ ( set @ A ) ) ) ) ).
% equals0I
thf(fact_58_equals0D,axiom,
! [A: $tType,A4: set @ A,A2: A] :
( ( A4
= ( bot_bot @ ( set @ A ) ) )
=> ~ ( member @ A @ A2 @ A4 ) ) ).
% equals0D
thf(fact_59_emptyE,axiom,
! [A: $tType,A2: A] :
~ ( member @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ).
% emptyE
thf(fact_60_mk__disjoint__insert,axiom,
! [A: $tType,A2: A,A4: set @ A] :
( ( member @ A @ A2 @ A4 )
=> ? [B5: set @ A] :
( ( A4
= ( insert @ A @ A2 @ B5 ) )
& ~ ( member @ A @ A2 @ B5 ) ) ) ).
% mk_disjoint_insert
thf(fact_61_insert__commute,axiom,
! [A: $tType,X3: A,Y2: A,A4: set @ A] :
( ( insert @ A @ X3 @ ( insert @ A @ Y2 @ A4 ) )
= ( insert @ A @ Y2 @ ( insert @ A @ X3 @ A4 ) ) ) ).
% insert_commute
thf(fact_62_insert__eq__iff,axiom,
! [A: $tType,A2: A,A4: set @ A,B: A,B3: set @ A] :
( ~ ( member @ A @ A2 @ A4 )
=> ( ~ ( member @ A @ B @ B3 )
=> ( ( ( insert @ A @ A2 @ A4 )
= ( insert @ A @ B @ B3 ) )
= ( ( ( A2 = B )
=> ( A4 = B3 ) )
& ( ( A2 != B )
=> ? [C3: set @ A] :
( ( A4
= ( insert @ A @ B @ C3 ) )
& ~ ( member @ A @ B @ C3 )
& ( B3
= ( insert @ A @ A2 @ C3 ) )
& ~ ( member @ A @ A2 @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_63_insert__absorb,axiom,
! [A: $tType,A2: A,A4: set @ A] :
( ( member @ A @ A2 @ A4 )
=> ( ( insert @ A @ A2 @ A4 )
= A4 ) ) ).
% insert_absorb
thf(fact_64_insert__ident,axiom,
! [A: $tType,X3: A,A4: set @ A,B3: set @ A] :
( ~ ( member @ A @ X3 @ A4 )
=> ( ~ ( member @ A @ X3 @ B3 )
=> ( ( ( insert @ A @ X3 @ A4 )
= ( insert @ A @ X3 @ B3 ) )
= ( A4 = B3 ) ) ) ) ).
% insert_ident
thf(fact_65_Set_Oset__insert,axiom,
! [A: $tType,X3: A,A4: set @ A] :
( ( member @ A @ X3 @ A4 )
=> ~ ! [B5: set @ A] :
( ( A4
= ( insert @ A @ X3 @ B5 ) )
=> ( member @ A @ X3 @ B5 ) ) ) ).
% Set.set_insert
thf(fact_66_insertI2,axiom,
! [A: $tType,A2: A,B3: set @ A,B: A] :
( ( member @ A @ A2 @ B3 )
=> ( member @ A @ A2 @ ( insert @ A @ B @ B3 ) ) ) ).
% insertI2
thf(fact_67_insertI1,axiom,
! [A: $tType,A2: A,B3: set @ A] : ( member @ A @ A2 @ ( insert @ A @ A2 @ B3 ) ) ).
% insertI1
thf(fact_68_insertE,axiom,
! [A: $tType,A2: A,B: A,A4: set @ A] :
( ( member @ A @ A2 @ ( insert @ A @ B @ A4 ) )
=> ( ( A2 != B )
=> ( member @ A @ A2 @ A4 ) ) ) ).
% insertE
thf(fact_69_Un__left__commute,axiom,
! [A: $tType,A4: set @ A,B3: set @ A,C: set @ A] :
( ( sup_sup @ ( set @ A ) @ A4 @ ( sup_sup @ ( set @ A ) @ B3 @ C ) )
= ( sup_sup @ ( set @ A ) @ B3 @ ( sup_sup @ ( set @ A ) @ A4 @ C ) ) ) ).
% Un_left_commute
thf(fact_70_Un__left__absorb,axiom,
! [A: $tType,A4: set @ A,B3: set @ A] :
( ( sup_sup @ ( set @ A ) @ A4 @ ( sup_sup @ ( set @ A ) @ A4 @ B3 ) )
= ( sup_sup @ ( set @ A ) @ A4 @ B3 ) ) ).
% Un_left_absorb
thf(fact_71_Un__commute,axiom,
! [A: $tType] :
( ( sup_sup @ ( set @ A ) )
= ( ^ [A5: set @ A,B6: set @ A] : ( sup_sup @ ( set @ A ) @ B6 @ A5 ) ) ) ).
% Un_commute
thf(fact_72_Un__absorb,axiom,
! [A: $tType,A4: set @ A] :
( ( sup_sup @ ( set @ A ) @ A4 @ A4 )
= A4 ) ).
% Un_absorb
thf(fact_73_Un__assoc,axiom,
! [A: $tType,A4: set @ A,B3: set @ A,C: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A4 @ B3 ) @ C )
= ( sup_sup @ ( set @ A ) @ A4 @ ( sup_sup @ ( set @ A ) @ B3 @ C ) ) ) ).
% Un_assoc
thf(fact_74_ball__Un,axiom,
! [A: $tType,A4: set @ A,B3: set @ A,P: A > $o] :
( ( ! [X2: A] :
( ( member @ A @ X2 @ ( sup_sup @ ( set @ A ) @ A4 @ B3 ) )
=> ( P @ X2 ) ) )
= ( ! [X2: A] :
( ( member @ A @ X2 @ A4 )
=> ( P @ X2 ) )
& ! [X2: A] :
( ( member @ A @ X2 @ B3 )
=> ( P @ X2 ) ) ) ) ).
% ball_Un
thf(fact_75_bex__Un,axiom,
! [A: $tType,A4: set @ A,B3: set @ A,P: A > $o] :
( ( ? [X2: A] :
( ( member @ A @ X2 @ ( sup_sup @ ( set @ A ) @ A4 @ B3 ) )
& ( P @ X2 ) ) )
= ( ? [X2: A] :
( ( member @ A @ X2 @ A4 )
& ( P @ X2 ) )
| ? [X2: A] :
( ( member @ A @ X2 @ B3 )
& ( P @ X2 ) ) ) ) ).
% bex_Un
thf(fact_76_UnI2,axiom,
! [A: $tType,C2: A,B3: set @ A,A4: set @ A] :
( ( member @ A @ C2 @ B3 )
=> ( member @ A @ C2 @ ( sup_sup @ ( set @ A ) @ A4 @ B3 ) ) ) ).
% UnI2
thf(fact_77_UnI1,axiom,
! [A: $tType,C2: A,A4: set @ A,B3: set @ A] :
( ( member @ A @ C2 @ A4 )
=> ( member @ A @ C2 @ ( sup_sup @ ( set @ A ) @ A4 @ B3 ) ) ) ).
% UnI1
thf(fact_78_UnE,axiom,
! [A: $tType,C2: A,A4: set @ A,B3: set @ A] :
( ( member @ A @ C2 @ ( sup_sup @ ( set @ A ) @ A4 @ B3 ) )
=> ( ~ ( member @ A @ C2 @ A4 )
=> ( member @ A @ C2 @ B3 ) ) ) ).
% UnE
thf(fact_79_DiffD2,axiom,
! [A: $tType,C2: A,A4: set @ A,B3: set @ A] :
( ( member @ A @ C2 @ ( minus_minus @ ( set @ A ) @ A4 @ B3 ) )
=> ~ ( member @ A @ C2 @ B3 ) ) ).
% DiffD2
thf(fact_80_DiffD1,axiom,
! [A: $tType,C2: A,A4: set @ A,B3: set @ A] :
( ( member @ A @ C2 @ ( minus_minus @ ( set @ A ) @ A4 @ B3 ) )
=> ( member @ A @ C2 @ A4 ) ) ).
% DiffD1
thf(fact_81_DiffE,axiom,
! [A: $tType,C2: A,A4: set @ A,B3: set @ A] :
( ( member @ A @ C2 @ ( minus_minus @ ( set @ A ) @ A4 @ B3 ) )
=> ~ ( ( member @ A @ C2 @ A4 )
=> ( member @ A @ C2 @ B3 ) ) ) ).
% DiffE
thf(fact_82_empty__def,axiom,
! [A: $tType] :
( ( bot_bot @ ( set @ A ) )
= ( collect @ A
@ ^ [X2: A] : $false ) ) ).
% empty_def
thf(fact_83_insert__Collect,axiom,
! [A: $tType,A2: A,P: A > $o] :
( ( insert @ A @ A2 @ ( collect @ A @ P ) )
= ( collect @ A
@ ^ [U: A] :
( ( U != A2 )
=> ( P @ U ) ) ) ) ).
% insert_Collect
thf(fact_84_insert__compr,axiom,
! [A: $tType] :
( ( insert @ A )
= ( ^ [A3: A,B6: set @ A] :
( collect @ A
@ ^ [X2: A] :
( ( X2 = A3 )
| ( member @ A @ X2 @ B6 ) ) ) ) ) ).
% insert_compr
thf(fact_85_Collect__disj__eq,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ( collect @ A
@ ^ [X2: A] :
( ( P @ X2 )
| ( Q @ X2 ) ) )
= ( sup_sup @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) ) ) ).
% Collect_disj_eq
thf(fact_86_Un__def,axiom,
! [A: $tType] :
( ( sup_sup @ ( set @ A ) )
= ( ^ [A5: set @ A,B6: set @ A] :
( collect @ A
@ ^ [X2: A] :
( ( member @ A @ X2 @ A5 )
| ( member @ A @ X2 @ B6 ) ) ) ) ) ).
% Un_def
thf(fact_87_set__diff__eq,axiom,
! [A: $tType] :
( ( minus_minus @ ( set @ A ) )
= ( ^ [A5: set @ A,B6: set @ A] :
( collect @ A
@ ^ [X2: A] :
( ( member @ A @ X2 @ A5 )
& ~ ( member @ A @ X2 @ B6 ) ) ) ) ) ).
% set_diff_eq
thf(fact_88_singleton__inject,axiom,
! [A: $tType,A2: A,B: A] :
( ( ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) )
= ( insert @ A @ B @ ( bot_bot @ ( set @ A ) ) ) )
=> ( A2 = B ) ) ).
% singleton_inject
thf(fact_89_insert__not__empty,axiom,
! [A: $tType,A2: A,A4: set @ A] :
( ( insert @ A @ A2 @ A4 )
!= ( bot_bot @ ( set @ A ) ) ) ).
% insert_not_empty
thf(fact_90_doubleton__eq__iff,axiom,
! [A: $tType,A2: A,B: A,C2: A,D: A] :
( ( ( insert @ A @ A2 @ ( insert @ A @ B @ ( bot_bot @ ( set @ A ) ) ) )
= ( insert @ A @ C2 @ ( insert @ A @ D @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( ( ( A2 = C2 )
& ( B = D ) )
| ( ( A2 = D )
& ( B = C2 ) ) ) ) ).
% doubleton_eq_iff
thf(fact_91_singleton__iff,axiom,
! [A: $tType,B: A,A2: A] :
( ( member @ A @ B @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) )
= ( B = A2 ) ) ).
% singleton_iff
thf(fact_92_singletonD,axiom,
! [A: $tType,B: A,A2: A] :
( ( member @ A @ B @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) )
=> ( B = A2 ) ) ).
% singletonD
thf(fact_93_Un__empty__right,axiom,
! [A: $tType,A4: set @ A] :
( ( sup_sup @ ( set @ A ) @ A4 @ ( bot_bot @ ( set @ A ) ) )
= A4 ) ).
% Un_empty_right
thf(fact_94_Un__empty__left,axiom,
! [A: $tType,B3: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ B3 )
= B3 ) ).
% Un_empty_left
thf(fact_95_insert__Diff__if,axiom,
! [A: $tType,X3: A,B3: set @ A,A4: set @ A] :
( ( ( member @ A @ X3 @ B3 )
=> ( ( minus_minus @ ( set @ A ) @ ( insert @ A @ X3 @ A4 ) @ B3 )
= ( minus_minus @ ( set @ A ) @ A4 @ B3 ) ) )
& ( ~ ( member @ A @ X3 @ B3 )
=> ( ( minus_minus @ ( set @ A ) @ ( insert @ A @ X3 @ A4 ) @ B3 )
= ( insert @ A @ X3 @ ( minus_minus @ ( set @ A ) @ A4 @ B3 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_96_Un__Diff,axiom,
! [A: $tType,A4: set @ A,B3: set @ A,C: set @ A] :
( ( minus_minus @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A4 @ B3 ) @ C )
= ( sup_sup @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A4 @ C ) @ ( minus_minus @ ( set @ A ) @ B3 @ C ) ) ) ).
% Un_Diff
thf(fact_97_Collect__conv__if2,axiom,
! [A: $tType,P: A > $o,A2: A] :
( ( ( P @ A2 )
=> ( ( collect @ A
@ ^ [X2: A] :
( ( A2 = X2 )
& ( P @ X2 ) ) )
= ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) )
& ( ~ ( P @ A2 )
=> ( ( collect @ A
@ ^ [X2: A] :
( ( A2 = X2 )
& ( P @ X2 ) ) )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Collect_conv_if2
thf(fact_98_Collect__conv__if,axiom,
! [A: $tType,P: A > $o,A2: A] :
( ( ( P @ A2 )
=> ( ( collect @ A
@ ^ [X2: A] :
( ( X2 = A2 )
& ( P @ X2 ) ) )
= ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) )
& ( ~ ( P @ A2 )
=> ( ( collect @ A
@ ^ [X2: A] :
( ( X2 = A2 )
& ( P @ X2 ) ) )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Collect_conv_if
thf(fact_99_insert__def,axiom,
! [A: $tType] :
( ( insert @ A )
= ( ^ [A3: A] :
( sup_sup @ ( set @ A )
@ ( collect @ A
@ ^ [X2: A] : X2 = A3 ) ) ) ) ).
% insert_def
thf(fact_100_singleton__Un__iff,axiom,
! [A: $tType,X3: A,A4: set @ A,B3: set @ A] :
( ( ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) )
= ( sup_sup @ ( set @ A ) @ A4 @ B3 ) )
= ( ( ( A4
= ( bot_bot @ ( set @ A ) ) )
& ( B3
= ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) )
| ( ( A4
= ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
& ( B3
= ( bot_bot @ ( set @ A ) ) ) )
| ( ( A4
= ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
& ( B3
= ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_101_Un__singleton__iff,axiom,
! [A: $tType,A4: set @ A,B3: set @ A,X3: A] :
( ( ( sup_sup @ ( set @ A ) @ A4 @ B3 )
= ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
= ( ( ( A4
= ( bot_bot @ ( set @ A ) ) )
& ( B3
= ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) )
| ( ( A4
= ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
& ( B3
= ( bot_bot @ ( set @ A ) ) ) )
| ( ( A4
= ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
& ( B3
= ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_102_insert__is__Un,axiom,
! [A: $tType] :
( ( insert @ A )
= ( ^ [A3: A] : ( sup_sup @ ( set @ A ) @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% insert_is_Un
thf(fact_103_Diff__insert__absorb,axiom,
! [A: $tType,X3: A,A4: set @ A] :
( ~ ( member @ A @ X3 @ A4 )
=> ( ( minus_minus @ ( set @ A ) @ ( insert @ A @ X3 @ A4 ) @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
= A4 ) ) ).
% Diff_insert_absorb
thf(fact_104_Diff__insert2,axiom,
! [A: $tType,A4: set @ A,A2: A,B3: set @ A] :
( ( minus_minus @ ( set @ A ) @ A4 @ ( insert @ A @ A2 @ B3 ) )
= ( minus_minus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) @ B3 ) ) ).
% Diff_insert2
thf(fact_105_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_106_Diff__insert,axiom,
! [A: $tType,A4: set @ A,A2: A,B3: set @ A] :
( ( minus_minus @ ( set @ A ) @ A4 @ ( insert @ A @ A2 @ B3 ) )
= ( minus_minus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A4 @ B3 ) @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Diff_insert
thf(fact_107_sup__bot_Oright__neutral,axiom,
! [A: $tType] :
( ( bounde1808546759up_bot @ A )
=> ! [A2: A] :
( ( sup_sup @ A @ A2 @ ( bot_bot @ A ) )
= A2 ) ) ).
% sup_bot.right_neutral
thf(fact_108_sup__bot_Oneutr__eq__iff,axiom,
! [A: $tType] :
( ( bounde1808546759up_bot @ A )
=> ! [A2: A,B: A] :
( ( ( bot_bot @ A )
= ( sup_sup @ A @ A2 @ B ) )
= ( ( A2
= ( bot_bot @ A ) )
& ( B
= ( bot_bot @ A ) ) ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_109_sup__bot_Oleft__neutral,axiom,
! [A: $tType] :
( ( bounde1808546759up_bot @ A )
=> ! [A2: A] :
( ( sup_sup @ A @ ( bot_bot @ A ) @ A2 )
= A2 ) ) ).
% sup_bot.left_neutral
thf(fact_110_sup__bot_Oeq__neutr__iff,axiom,
! [A: $tType] :
( ( bounde1808546759up_bot @ A )
=> ! [A2: A,B: A] :
( ( ( sup_sup @ A @ A2 @ B )
= ( bot_bot @ A ) )
= ( ( A2
= ( bot_bot @ A ) )
& ( B
= ( bot_bot @ A ) ) ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_111_sup__eq__bot__iff,axiom,
! [A: $tType] :
( ( bounde1808546759up_bot @ A )
=> ! [X3: A,Y2: A] :
( ( ( sup_sup @ A @ X3 @ Y2 )
= ( bot_bot @ A ) )
= ( ( X3
= ( bot_bot @ A ) )
& ( Y2
= ( bot_bot @ A ) ) ) ) ) ).
% sup_eq_bot_iff
thf(fact_112_bot__eq__sup__iff,axiom,
! [A: $tType] :
( ( bounde1808546759up_bot @ A )
=> ! [X3: A,Y2: A] :
( ( ( bot_bot @ A )
= ( sup_sup @ A @ X3 @ Y2 ) )
= ( ( X3
= ( bot_bot @ A ) )
& ( Y2
= ( bot_bot @ A ) ) ) ) ) ).
% bot_eq_sup_iff
thf(fact_113_sup__bot__right,axiom,
! [A: $tType] :
( ( bounde1808546759up_bot @ A )
=> ! [X3: A] :
( ( sup_sup @ A @ X3 @ ( bot_bot @ A ) )
= X3 ) ) ).
% sup_bot_right
thf(fact_114_sup__bot__left,axiom,
! [A: $tType] :
( ( bounde1808546759up_bot @ A )
=> ! [X3: A] :
( ( sup_sup @ A @ ( bot_bot @ A ) @ X3 )
= X3 ) ) ).
% sup_bot_left
thf(fact_115_memb__spec,axiom,
! [A: $tType,H: A > int,T: binary1291135688e_Tree @ A,X3: A] :
( ( binary1610619414edTree @ A @ H @ T )
=> ( ( binary827270440e_memb @ A @ H @ X3 @ T )
= ( member @ A @ X3 @ ( binary1653327646_setOf @ A @ T ) ) ) ) ).
% memb_spec
thf(fact_116_minus__apply,axiom,
! [B4: $tType,A: $tType] :
( ( minus @ B4 )
=> ( ( minus_minus @ ( A > B4 ) )
= ( ^ [A5: A > B4,B6: A > B4,X2: A] : ( minus_minus @ B4 @ ( A5 @ X2 ) @ ( B6 @ X2 ) ) ) ) ) ).
% minus_apply
thf(fact_117_sup__apply,axiom,
! [B4: $tType,A: $tType] :
( ( semilattice_sup @ B4 )
=> ( ( sup_sup @ ( A > B4 ) )
= ( ^ [F2: A > B4,G2: A > B4,X2: A] : ( sup_sup @ B4 @ ( F2 @ X2 ) @ ( G2 @ X2 ) ) ) ) ) ).
% sup_apply
thf(fact_118_sup_Oidem,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A2: A] :
( ( sup_sup @ A @ A2 @ A2 )
= A2 ) ) ).
% sup.idem
thf(fact_119_sup__idem,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X3: A] :
( ( sup_sup @ A @ X3 @ X3 )
= X3 ) ) ).
% sup_idem
thf(fact_120_sup_Oleft__idem,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A2: A,B: A] :
( ( sup_sup @ A @ A2 @ ( sup_sup @ A @ A2 @ B ) )
= ( sup_sup @ A @ A2 @ B ) ) ) ).
% sup.left_idem
thf(fact_121_sup__left__idem,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X3: A,Y2: A] :
( ( sup_sup @ A @ X3 @ ( sup_sup @ A @ X3 @ Y2 ) )
= ( sup_sup @ A @ X3 @ Y2 ) ) ) ).
% sup_left_idem
thf(fact_122_sup_Oright__idem,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A2: A,B: A] :
( ( sup_sup @ A @ ( sup_sup @ A @ A2 @ B ) @ B )
= ( sup_sup @ A @ A2 @ B ) ) ) ).
% sup.right_idem
thf(fact_123_bot__set__def,axiom,
! [A: $tType] :
( ( bot_bot @ ( set @ A ) )
= ( collect @ A @ ( bot_bot @ ( A > $o ) ) ) ) ).
% bot_set_def
thf(fact_124_sup__set__def,axiom,
! [A: $tType] :
( ( sup_sup @ ( set @ A ) )
= ( ^ [A5: set @ A,B6: set @ A] :
( collect @ A
@ ( sup_sup @ ( A > $o )
@ ^ [X2: A] : ( member @ A @ X2 @ A5 )
@ ^ [X2: A] : ( member @ A @ X2 @ B6 ) ) ) ) ) ).
% sup_set_def
thf(fact_125_fun__diff__def,axiom,
! [B4: $tType,A: $tType] :
( ( minus @ B4 )
=> ( ( minus_minus @ ( A > B4 ) )
= ( ^ [A5: A > B4,B6: A > B4,X2: A] : ( minus_minus @ B4 @ ( A5 @ X2 ) @ ( B6 @ X2 ) ) ) ) ) ).
% fun_diff_def
thf(fact_126_inf__sup__aci_I8_J,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [X3: A,Y2: A] :
( ( sup_sup @ A @ X3 @ ( sup_sup @ A @ X3 @ Y2 ) )
= ( sup_sup @ A @ X3 @ Y2 ) ) ) ).
% inf_sup_aci(8)
thf(fact_127_inf__sup__aci_I7_J,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [X3: A,Y2: A,Z2: A] :
( ( sup_sup @ A @ X3 @ ( sup_sup @ A @ Y2 @ Z2 ) )
= ( sup_sup @ A @ Y2 @ ( sup_sup @ A @ X3 @ Z2 ) ) ) ) ).
% inf_sup_aci(7)
thf(fact_128_inf__sup__aci_I6_J,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [X3: A,Y2: A,Z2: A] :
( ( sup_sup @ A @ ( sup_sup @ A @ X3 @ Y2 ) @ Z2 )
= ( sup_sup @ A @ X3 @ ( sup_sup @ A @ Y2 @ Z2 ) ) ) ) ).
% inf_sup_aci(6)
thf(fact_129_inf__sup__aci_I5_J,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ( ( sup_sup @ A )
= ( ^ [X2: A,Y: A] : ( sup_sup @ A @ Y @ X2 ) ) ) ) ).
% inf_sup_aci(5)
thf(fact_130_sup__fun__def,axiom,
! [B4: $tType,A: $tType] :
( ( semilattice_sup @ B4 )
=> ( ( sup_sup @ ( A > B4 ) )
= ( ^ [F2: A > B4,G2: A > B4,X2: A] : ( sup_sup @ B4 @ ( F2 @ X2 ) @ ( G2 @ X2 ) ) ) ) ) ).
% sup_fun_def
thf(fact_131_boolean__algebra__cancel_Osup1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A4: A,K: A,A2: A,B: A] :
( ( A4
= ( sup_sup @ A @ K @ A2 ) )
=> ( ( sup_sup @ A @ A4 @ B )
= ( sup_sup @ A @ K @ ( sup_sup @ A @ A2 @ B ) ) ) ) ) ).
% boolean_algebra_cancel.sup1
thf(fact_132_boolean__algebra__cancel_Osup2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [B3: A,K: A,B: A,A2: A] :
( ( B3
= ( sup_sup @ A @ K @ B ) )
=> ( ( sup_sup @ A @ A2 @ B3 )
= ( sup_sup @ A @ K @ ( sup_sup @ A @ A2 @ B ) ) ) ) ) ).
% boolean_algebra_cancel.sup2
thf(fact_133_sup_Oassoc,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A2: A,B: A,C2: A] :
( ( sup_sup @ A @ ( sup_sup @ A @ A2 @ B ) @ C2 )
= ( sup_sup @ A @ A2 @ ( sup_sup @ A @ B @ C2 ) ) ) ) ).
% sup.assoc
thf(fact_134_sup__assoc,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X3: A,Y2: A,Z2: A] :
( ( sup_sup @ A @ ( sup_sup @ A @ X3 @ Y2 ) @ Z2 )
= ( sup_sup @ A @ X3 @ ( sup_sup @ A @ Y2 @ Z2 ) ) ) ) ).
% sup_assoc
thf(fact_135_sup_Ocommute,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ( ( sup_sup @ A )
= ( ^ [A3: A,B2: A] : ( sup_sup @ A @ B2 @ A3 ) ) ) ) ).
% sup.commute
thf(fact_136_sup__commute,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ( ( sup_sup @ A )
= ( ^ [X2: A,Y: A] : ( sup_sup @ A @ Y @ X2 ) ) ) ) ).
% sup_commute
thf(fact_137_sup_Oleft__commute,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [B: A,A2: A,C2: A] :
( ( sup_sup @ A @ B @ ( sup_sup @ A @ A2 @ C2 ) )
= ( sup_sup @ A @ A2 @ ( sup_sup @ A @ B @ C2 ) ) ) ) ).
% sup.left_commute
thf(fact_138_sup__left__commute,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X3: A,Y2: A,Z2: A] :
( ( sup_sup @ A @ X3 @ ( sup_sup @ A @ Y2 @ Z2 ) )
= ( sup_sup @ A @ Y2 @ ( sup_sup @ A @ X3 @ Z2 ) ) ) ) ).
% sup_left_commute
thf(fact_139_less__supI1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X3: A,A2: A,B: A] :
( ( ord_less @ A @ X3 @ A2 )
=> ( ord_less @ A @ X3 @ ( sup_sup @ A @ A2 @ B ) ) ) ) ).
% less_supI1
thf(fact_140_less__supI2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X3: A,B: A,A2: A] :
( ( ord_less @ A @ X3 @ B )
=> ( ord_less @ A @ X3 @ ( sup_sup @ A @ A2 @ B ) ) ) ) ).
% less_supI2
thf(fact_141_sup_Ostrict__boundedE,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [B: A,C2: A,A2: A] :
( ( ord_less @ A @ ( sup_sup @ A @ B @ C2 ) @ A2 )
=> ~ ( ( ord_less @ A @ B @ A2 )
=> ~ ( ord_less @ A @ C2 @ A2 ) ) ) ) ).
% sup.strict_boundedE
thf(fact_142_sup_Ostrict__order__iff,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ( ( ord_less @ A )
= ( ^ [B2: A,A3: A] :
( ( A3
= ( sup_sup @ A @ A3 @ B2 ) )
& ( A3 != B2 ) ) ) ) ) ).
% sup.strict_order_iff
thf(fact_143_sup_Ostrict__coboundedI1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [C2: A,A2: A,B: A] :
( ( ord_less @ A @ C2 @ A2 )
=> ( ord_less @ A @ C2 @ ( sup_sup @ A @ A2 @ B ) ) ) ) ).
% sup.strict_coboundedI1
thf(fact_144_sup_Ostrict__coboundedI2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [C2: A,B: A,A2: A] :
( ( ord_less @ A @ C2 @ B )
=> ( ord_less @ A @ C2 @ ( sup_sup @ A @ A2 @ B ) ) ) ) ).
% sup.strict_coboundedI2
thf(fact_145_the__elem__eq,axiom,
! [A: $tType,X3: A] :
( ( the_elem @ A @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
= X3 ) ).
% the_elem_eq
thf(fact_146_bot__apply,axiom,
! [C4: $tType,D2: $tType] :
( ( bot @ C4 )
=> ( ( bot_bot @ ( D2 > C4 ) )
= ( ^ [X2: D2] : ( bot_bot @ C4 ) ) ) ) ).
% bot_apply
thf(fact_147_is__singletonI,axiom,
! [A: $tType,X3: A] : ( is_singleton @ A @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% is_singletonI
thf(fact_148_diff__strict__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: A,B: A,D: A,C2: A] :
( ( ord_less @ A @ A2 @ B )
=> ( ( ord_less @ A @ D @ C2 )
=> ( ord_less @ A @ ( minus_minus @ A @ A2 @ C2 ) @ ( minus_minus @ A @ B @ D ) ) ) ) ) ).
% diff_strict_mono
thf(fact_149_sup1CI,axiom,
! [A: $tType,B3: A > $o,X3: A,A4: A > $o] :
( ( ~ ( B3 @ X3 )
=> ( A4 @ X3 ) )
=> ( sup_sup @ ( A > $o ) @ A4 @ B3 @ X3 ) ) ).
% sup1CI
thf(fact_150_sup1E,axiom,
! [A: $tType,A4: A > $o,B3: A > $o,X3: A] :
( ( sup_sup @ ( A > $o ) @ A4 @ B3 @ X3 )
=> ( ~ ( A4 @ X3 )
=> ( B3 @ X3 ) ) ) ).
% sup1E
thf(fact_151_sup1I1,axiom,
! [A: $tType,A4: A > $o,X3: A,B3: A > $o] :
( ( A4 @ X3 )
=> ( sup_sup @ ( A > $o ) @ A4 @ B3 @ X3 ) ) ).
% sup1I1
thf(fact_152_sup1I2,axiom,
! [A: $tType,B3: A > $o,X3: A,A4: A > $o] :
( ( B3 @ X3 )
=> ( sup_sup @ ( A > $o ) @ A4 @ B3 @ X3 ) ) ).
% sup1I2
thf(fact_153_not__psubset__empty,axiom,
! [A: $tType,A4: set @ A] :
~ ( ord_less @ ( set @ A ) @ A4 @ ( bot_bot @ ( set @ A ) ) ) ).
% not_psubset_empty
thf(fact_154_psubset__imp__ex__mem,axiom,
! [A: $tType,A4: set @ A,B3: set @ A] :
( ( ord_less @ ( set @ A ) @ A4 @ B3 )
=> ? [B7: A] : ( member @ A @ B7 @ ( minus_minus @ ( set @ A ) @ B3 @ A4 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_155_minus__set__def,axiom,
! [A: $tType] :
( ( minus_minus @ ( set @ A ) )
= ( ^ [A5: set @ A,B6: set @ A] :
( collect @ A
@ ( minus_minus @ ( A > $o )
@ ^ [X2: A] : ( member @ A @ X2 @ A5 )
@ ^ [X2: A] : ( member @ A @ X2 @ B6 ) ) ) ) ) ).
% minus_set_def
thf(fact_156_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_157_is__singletonI_H,axiom,
! [A: $tType,A4: set @ A] :
( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X4: A,Y4: A] :
( ( member @ A @ X4 @ A4 )
=> ( ( member @ A @ Y4 @ A4 )
=> ( X4 = Y4 ) ) )
=> ( is_singleton @ A @ A4 ) ) ) ).
% is_singletonI'
thf(fact_158_dual__order_Ostrict__implies__not__eq,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B: A,A2: A] :
( ( ord_less @ A @ B @ A2 )
=> ( A2 != B ) ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_159_order_Ostrict__implies__not__eq,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: A,B: A] :
( ( ord_less @ A @ A2 @ B )
=> ( A2 != B ) ) ) ).
% order.strict_implies_not_eq
thf(fact_160_not__less__iff__gr__or__eq,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X3: A,Y2: A] :
( ( ~ ( ord_less @ A @ X3 @ Y2 ) )
= ( ( ord_less @ A @ Y2 @ X3 )
| ( X3 = Y2 ) ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_161_dual__order_Ostrict__trans,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B: A,A2: A,C2: A] :
( ( ord_less @ A @ B @ A2 )
=> ( ( ord_less @ A @ C2 @ B )
=> ( ord_less @ A @ C2 @ A2 ) ) ) ) ).
% dual_order.strict_trans
thf(fact_162_linorder__less__wlog,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > A > $o,A2: A,B: A] :
( ! [A6: A,B7: A] :
( ( ord_less @ A @ A6 @ B7 )
=> ( P @ A6 @ B7 ) )
=> ( ! [A6: A] : ( P @ A6 @ A6 )
=> ( ! [A6: A,B7: A] :
( ( P @ B7 @ A6 )
=> ( P @ A6 @ B7 ) )
=> ( P @ A2 @ B ) ) ) ) ) ).
% linorder_less_wlog
thf(fact_163_exists__least__iff,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ( ( ^ [P2: A > $o] :
? [X5: A] : ( P2 @ X5 ) )
= ( ^ [P3: A > $o] :
? [N: A] :
( ( P3 @ N )
& ! [M: A] :
( ( ord_less @ A @ M @ N )
=> ~ ( P3 @ M ) ) ) ) ) ) ).
% exists_least_iff
thf(fact_164_less__imp__not__less,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X3: A,Y2: A] :
( ( ord_less @ A @ X3 @ Y2 )
=> ~ ( ord_less @ A @ Y2 @ X3 ) ) ) ).
% less_imp_not_less
thf(fact_165_order_Ostrict__trans,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: A,B: A,C2: A] :
( ( ord_less @ A @ A2 @ B )
=> ( ( ord_less @ A @ B @ C2 )
=> ( ord_less @ A @ A2 @ C2 ) ) ) ) ).
% order.strict_trans
thf(fact_166_dual__order_Oirrefl,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: A] :
~ ( ord_less @ A @ A2 @ A2 ) ) ).
% dual_order.irrefl
thf(fact_167_linorder__cases,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X3: A,Y2: A] :
( ~ ( ord_less @ A @ X3 @ Y2 )
=> ( ( X3 != Y2 )
=> ( ord_less @ A @ Y2 @ X3 ) ) ) ) ).
% linorder_cases
thf(fact_168_less__imp__triv,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X3: A,Y2: A,P: $o] :
( ( ord_less @ A @ X3 @ Y2 )
=> ( ( ord_less @ A @ Y2 @ X3 )
=> P ) ) ) ).
% less_imp_triv
thf(fact_169_less__imp__not__eq2,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X3: A,Y2: A] :
( ( ord_less @ A @ X3 @ Y2 )
=> ( Y2 != X3 ) ) ) ).
% less_imp_not_eq2
thf(fact_170_antisym__conv3,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Y2: A,X3: A] :
( ~ ( ord_less @ A @ Y2 @ X3 )
=> ( ( ~ ( ord_less @ A @ X3 @ Y2 ) )
= ( X3 = Y2 ) ) ) ) ).
% antisym_conv3
thf(fact_171_less__induct,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [P: A > $o,A2: A] :
( ! [X4: A] :
( ! [Y5: A] :
( ( ord_less @ A @ Y5 @ X4 )
=> ( P @ Y5 ) )
=> ( P @ X4 ) )
=> ( P @ A2 ) ) ) ).
% less_induct
thf(fact_172_less__not__sym,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X3: A,Y2: A] :
( ( ord_less @ A @ X3 @ Y2 )
=> ~ ( ord_less @ A @ Y2 @ X3 ) ) ) ).
% less_not_sym
thf(fact_173_less__imp__not__eq,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X3: A,Y2: A] :
( ( ord_less @ A @ X3 @ Y2 )
=> ( X3 != Y2 ) ) ) ).
% less_imp_not_eq
thf(fact_174_dual__order_Oasym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B: A,A2: A] :
( ( ord_less @ A @ B @ A2 )
=> ~ ( ord_less @ A @ A2 @ B ) ) ) ).
% dual_order.asym
thf(fact_175_ord__less__eq__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A2: A,B: A,C2: A] :
( ( ord_less @ A @ A2 @ B )
=> ( ( B = C2 )
=> ( ord_less @ A @ A2 @ C2 ) ) ) ) ).
% ord_less_eq_trans
thf(fact_176_ord__eq__less__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A2: A,B: A,C2: A] :
( ( A2 = B )
=> ( ( ord_less @ A @ B @ C2 )
=> ( ord_less @ A @ A2 @ C2 ) ) ) ) ).
% ord_eq_less_trans
thf(fact_177_less__irrefl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X3: A] :
~ ( ord_less @ A @ X3 @ X3 ) ) ).
% less_irrefl
thf(fact_178_less__linear,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X3: A,Y2: A] :
( ( ord_less @ A @ X3 @ Y2 )
| ( X3 = Y2 )
| ( ord_less @ A @ Y2 @ X3 ) ) ) ).
% less_linear
thf(fact_179_less__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X3: A,Y2: A,Z2: A] :
( ( ord_less @ A @ X3 @ Y2 )
=> ( ( ord_less @ A @ Y2 @ Z2 )
=> ( ord_less @ A @ X3 @ Z2 ) ) ) ) ).
% less_trans
thf(fact_180_less__asym_H,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A2: A,B: A] :
( ( ord_less @ A @ A2 @ B )
=> ~ ( ord_less @ A @ B @ A2 ) ) ) ).
% less_asym'
thf(fact_181_less__asym,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X3: A,Y2: A] :
( ( ord_less @ A @ X3 @ Y2 )
=> ~ ( ord_less @ A @ Y2 @ X3 ) ) ) ).
% less_asym
thf(fact_182_less__imp__neq,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X3: A,Y2: A] :
( ( ord_less @ A @ X3 @ Y2 )
=> ( X3 != Y2 ) ) ) ).
% less_imp_neq
thf(fact_183_dense,axiom,
! [A: $tType] :
( ( dense_order @ A )
=> ! [X3: A,Y2: A] :
( ( ord_less @ A @ X3 @ Y2 )
=> ? [Z3: A] :
( ( ord_less @ A @ X3 @ Z3 )
& ( ord_less @ A @ Z3 @ Y2 ) ) ) ) ).
% dense
thf(fact_184_order_Oasym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: A,B: A] :
( ( ord_less @ A @ A2 @ B )
=> ~ ( ord_less @ A @ B @ A2 ) ) ) ).
% order.asym
thf(fact_185_neq__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X3: A,Y2: A] :
( ( X3 != Y2 )
= ( ( ord_less @ A @ X3 @ Y2 )
| ( ord_less @ A @ Y2 @ X3 ) ) ) ) ).
% neq_iff
thf(fact_186_neqE,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X3: A,Y2: A] :
( ( X3 != Y2 )
=> ( ~ ( ord_less @ A @ X3 @ Y2 )
=> ( ord_less @ A @ Y2 @ X3 ) ) ) ) ).
% neqE
thf(fact_187_gt__ex,axiom,
! [A: $tType] :
( ( no_top @ A )
=> ! [X3: A] :
? [X_1: A] : ( ord_less @ A @ X3 @ X_1 ) ) ).
% gt_ex
thf(fact_188_lt__ex,axiom,
! [A: $tType] :
( ( no_bot @ A )
=> ! [X3: A] :
? [Y4: A] : ( ord_less @ A @ Y4 @ X3 ) ) ).
% lt_ex
thf(fact_189_order__less__subst2,axiom,
! [A: $tType,C4: $tType] :
( ( ( order @ C4 )
& ( order @ A ) )
=> ! [A2: A,B: A,F: A > C4,C2: C4] :
( ( ord_less @ A @ A2 @ B )
=> ( ( ord_less @ C4 @ ( F @ B ) @ C2 )
=> ( ! [X4: A,Y4: A] :
( ( ord_less @ A @ X4 @ Y4 )
=> ( ord_less @ C4 @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less @ C4 @ ( F @ A2 ) @ C2 ) ) ) ) ) ).
% order_less_subst2
thf(fact_190_order__less__subst1,axiom,
! [A: $tType,B4: $tType] :
( ( ( order @ B4 )
& ( order @ A ) )
=> ! [A2: A,F: B4 > A,B: B4,C2: B4] :
( ( ord_less @ A @ A2 @ ( F @ B ) )
=> ( ( ord_less @ B4 @ B @ C2 )
=> ( ! [X4: B4,Y4: B4] :
( ( ord_less @ B4 @ X4 @ Y4 )
=> ( ord_less @ A @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less @ A @ A2 @ ( F @ C2 ) ) ) ) ) ) ).
% order_less_subst1
thf(fact_191_ord__less__eq__subst,axiom,
! [A: $tType,B4: $tType] :
( ( ( ord @ B4 )
& ( ord @ A ) )
=> ! [A2: A,B: A,F: A > B4,C2: B4] :
( ( ord_less @ A @ A2 @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X4: A,Y4: A] :
( ( ord_less @ A @ X4 @ Y4 )
=> ( ord_less @ B4 @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less @ B4 @ ( F @ A2 ) @ C2 ) ) ) ) ) ).
% ord_less_eq_subst
thf(fact_192_ord__eq__less__subst,axiom,
! [A: $tType,B4: $tType] :
( ( ( ord @ B4 )
& ( ord @ A ) )
=> ! [A2: A,F: B4 > A,B: B4,C2: B4] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less @ B4 @ B @ C2 )
=> ( ! [X4: B4,Y4: B4] :
( ( ord_less @ B4 @ X4 @ Y4 )
=> ( ord_less @ A @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less @ A @ A2 @ ( F @ C2 ) ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_193_bot__fun__def,axiom,
! [B4: $tType,A: $tType] :
( ( bot @ B4 )
=> ( ( bot_bot @ ( A > B4 ) )
= ( ^ [X2: A] : ( bot_bot @ B4 ) ) ) ) ).
% bot_fun_def
thf(fact_194_diff__right__commute,axiom,
! [A: $tType] :
( ( cancel146912293up_add @ A )
=> ! [A2: A,C2: A,B: A] :
( ( minus_minus @ A @ ( minus_minus @ A @ A2 @ C2 ) @ B )
= ( minus_minus @ A @ ( minus_minus @ A @ A2 @ B ) @ C2 ) ) ) ).
% diff_right_commute
thf(fact_195_diff__eq__diff__eq,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A,B: A,C2: A,D: A] :
( ( ( minus_minus @ A @ A2 @ B )
= ( minus_minus @ A @ C2 @ D ) )
=> ( ( A2 = B )
= ( C2 = D ) ) ) ) ).
% diff_eq_diff_eq
thf(fact_196_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_197_is__singletonE,axiom,
! [A: $tType,A4: set @ A] :
( ( is_singleton @ A @ A4 )
=> ~ ! [X4: A] :
( A4
!= ( insert @ A @ X4 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% is_singletonE
thf(fact_198_bot_Onot__eq__extremum,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [A2: A] :
( ( A2
!= ( bot_bot @ A ) )
= ( ord_less @ A @ ( bot_bot @ A ) @ A2 ) ) ) ).
% bot.not_eq_extremum
thf(fact_199_bot_Oextremum__strict,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [A2: A] :
~ ( ord_less @ A @ A2 @ ( bot_bot @ A ) ) ) ).
% bot.extremum_strict
thf(fact_200_diff__strict__right__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: A,B: A,C2: A] :
( ( ord_less @ A @ A2 @ B )
=> ( ord_less @ A @ ( minus_minus @ A @ A2 @ C2 ) @ ( minus_minus @ A @ B @ C2 ) ) ) ) ).
% diff_strict_right_mono
thf(fact_201_diff__strict__left__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [B: A,A2: A,C2: A] :
( ( ord_less @ A @ B @ A2 )
=> ( ord_less @ A @ ( minus_minus @ A @ C2 @ A2 ) @ ( minus_minus @ A @ C2 @ B ) ) ) ) ).
% diff_strict_left_mono
thf(fact_202_diff__eq__diff__less,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: A,B: A,C2: A,D: A] :
( ( ( minus_minus @ A @ A2 @ B )
= ( minus_minus @ A @ C2 @ D ) )
=> ( ( ord_less @ A @ A2 @ B )
= ( ord_less @ A @ C2 @ D ) ) ) ) ).
% diff_eq_diff_less
thf(fact_203_sup__Un__eq,axiom,
! [A: $tType,R: set @ A,S: set @ A] :
( ( sup_sup @ ( A > $o )
@ ^ [X2: A] : ( member @ A @ X2 @ R )
@ ^ [X2: A] : ( member @ A @ X2 @ S ) )
= ( ^ [X2: A] : ( member @ A @ X2 @ ( sup_sup @ ( set @ A ) @ R @ S ) ) ) ) ).
% sup_Un_eq
thf(fact_204_bot__empty__eq,axiom,
! [A: $tType] :
( ( bot_bot @ ( A > $o ) )
= ( ^ [X2: A] : ( member @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% bot_empty_eq
thf(fact_205_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_206_psubset__trans,axiom,
! [A: $tType,A4: set @ A,B3: set @ A,C: set @ A] :
( ( ord_less @ ( set @ A ) @ A4 @ B3 )
=> ( ( ord_less @ ( set @ A ) @ B3 @ C )
=> ( ord_less @ ( set @ A ) @ A4 @ C ) ) ) ).
% psubset_trans
thf(fact_207_less__set__def,axiom,
! [A: $tType] :
( ( ord_less @ ( set @ A ) )
= ( ^ [A5: set @ A,B6: set @ A] :
( ord_less @ ( A > $o )
@ ^ [X2: A] : ( member @ A @ X2 @ A5 )
@ ^ [X2: A] : ( member @ A @ X2 @ B6 ) ) ) ) ).
% less_set_def
thf(fact_208_psubsetD,axiom,
! [A: $tType,A4: set @ A,B3: set @ A,C2: A] :
( ( ord_less @ ( set @ A ) @ A4 @ B3 )
=> ( ( member @ A @ C2 @ A4 )
=> ( member @ A @ C2 @ B3 ) ) ) ).
% psubsetD
thf(fact_209_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_210_Tree_Osimps_I15_J,axiom,
! [A: $tType,X21: binary1291135688e_Tree @ A,X22: A,X23: binary1291135688e_Tree @ A] :
( ( binary2130109271t_Tree @ A @ ( binary210054475elle_T @ A @ X21 @ X22 @ X23 ) )
= ( sup_sup @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ ( binary2130109271t_Tree @ A @ X21 ) @ ( insert @ A @ X22 @ ( bot_bot @ ( set @ A ) ) ) ) @ ( binary2130109271t_Tree @ A @ X23 ) ) ) ).
% Tree.simps(15)
thf(fact_211_member__remove,axiom,
! [A: $tType,X3: A,Y2: A,A4: set @ A] :
( ( member @ A @ X3 @ ( remove @ A @ Y2 @ A4 ) )
= ( ( member @ A @ X3 @ A4 )
& ( X3 != Y2 ) ) ) ).
% member_remove
thf(fact_212_Tree_Oset__intros_I3_J,axiom,
! [A: $tType,Ya: A,X23: binary1291135688e_Tree @ A,X21: binary1291135688e_Tree @ A,X22: A] :
( ( member @ A @ Ya @ ( binary2130109271t_Tree @ A @ X23 ) )
=> ( member @ A @ Ya @ ( binary2130109271t_Tree @ A @ ( binary210054475elle_T @ A @ X21 @ X22 @ X23 ) ) ) ) ).
% Tree.set_intros(3)
thf(fact_213_Tree_Oset__intros_I2_J,axiom,
! [A: $tType,X22: A,X21: binary1291135688e_Tree @ A,X23: binary1291135688e_Tree @ A] : ( member @ A @ X22 @ ( binary2130109271t_Tree @ A @ ( binary210054475elle_T @ A @ X21 @ X22 @ X23 ) ) ) ).
% Tree.set_intros(2)
thf(fact_214_Tree_Oset__intros_I1_J,axiom,
! [A: $tType,Y2: A,X21: binary1291135688e_Tree @ A,X22: A,X23: binary1291135688e_Tree @ A] :
( ( member @ A @ Y2 @ ( binary2130109271t_Tree @ A @ X21 ) )
=> ( member @ A @ Y2 @ ( binary2130109271t_Tree @ A @ ( binary210054475elle_T @ A @ X21 @ X22 @ X23 ) ) ) ) ).
% Tree.set_intros(1)
thf(fact_215_Tree_Oset__cases,axiom,
! [A: $tType,E: A,A2: binary1291135688e_Tree @ A] :
( ( member @ A @ E @ ( binary2130109271t_Tree @ A @ A2 ) )
=> ( ! [Z1: binary1291135688e_Tree @ A] :
( ? [Z22: A,Z32: binary1291135688e_Tree @ A] :
( A2
= ( binary210054475elle_T @ A @ Z1 @ Z22 @ Z32 ) )
=> ~ ( member @ A @ E @ ( binary2130109271t_Tree @ A @ Z1 ) ) )
=> ( ! [Z1: binary1291135688e_Tree @ A,Z32: binary1291135688e_Tree @ A] :
( A2
!= ( binary210054475elle_T @ A @ Z1 @ E @ Z32 ) )
=> ~ ! [Z1: binary1291135688e_Tree @ A,Z22: A,Z32: binary1291135688e_Tree @ A] :
( ( A2
= ( binary210054475elle_T @ A @ Z1 @ Z22 @ Z32 ) )
=> ~ ( member @ A @ E @ ( binary2130109271t_Tree @ A @ Z32 ) ) ) ) ) ) ).
% Tree.set_cases
thf(fact_216_Tree_Osimps_I14_J,axiom,
! [A: $tType] :
( ( binary2130109271t_Tree @ A @ ( binary1746293266le_Tip @ A ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% Tree.simps(14)
thf(fact_217_Set_Ois__empty__def,axiom,
! [A: $tType] :
( ( is_empty @ A )
= ( ^ [A5: set @ A] :
( A5
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Set.is_empty_def
thf(fact_218_the__elem__def,axiom,
! [A: $tType] :
( ( the_elem @ A )
= ( ^ [X6: set @ A] :
( the @ A
@ ^ [X2: A] :
( X6
= ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% the_elem_def
thf(fact_219_the__sym__eq__trivial,axiom,
! [A: $tType,X3: A] :
( ( the @ A
@ ( ^ [Y3: A,Z: A] : Y3 = Z
@ X3 ) )
= X3 ) ).
% the_sym_eq_trivial
thf(fact_220_the__eq__trivial,axiom,
! [A: $tType,A2: A] :
( ( the @ A
@ ^ [X2: A] : X2 = A2 )
= A2 ) ).
% the_eq_trivial
thf(fact_221_the__equality,axiom,
! [A: $tType,P: A > $o,A2: A] :
( ( P @ A2 )
=> ( ! [X4: A] :
( ( P @ X4 )
=> ( X4 = A2 ) )
=> ( ( the @ A @ P )
= A2 ) ) ) ).
% the_equality
thf(fact_222_theI,axiom,
! [A: $tType,P: A > $o,A2: A] :
( ( P @ A2 )
=> ( ! [X4: A] :
( ( P @ X4 )
=> ( X4 = A2 ) )
=> ( P @ ( the @ A @ P ) ) ) ) ).
% theI
thf(fact_223_theI_H,axiom,
! [A: $tType,P: A > $o] :
( ? [X: A] :
( ( P @ X )
& ! [Y4: A] :
( ( P @ Y4 )
=> ( Y4 = X ) ) )
=> ( P @ ( the @ A @ P ) ) ) ).
% theI'
thf(fact_224_theI2,axiom,
! [A: $tType,P: A > $o,A2: A,Q: A > $o] :
( ( P @ A2 )
=> ( ! [X4: A] :
( ( P @ X4 )
=> ( X4 = A2 ) )
=> ( ! [X4: A] :
( ( P @ X4 )
=> ( Q @ X4 ) )
=> ( Q @ ( the @ A @ P ) ) ) ) ) ).
% theI2
thf(fact_225_If__def,axiom,
! [A: $tType] :
( ( if @ A )
= ( ^ [P3: $o,X2: A,Y: A] :
( the @ A
@ ^ [Z4: A] :
( ( P3
=> ( Z4 = X2 ) )
& ( ~ P3
=> ( Z4 = Y ) ) ) ) ) ) ).
% If_def
thf(fact_226_the1I2,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ? [X: A] :
( ( P @ X )
& ! [Y4: A] :
( ( P @ Y4 )
=> ( Y4 = X ) ) )
=> ( ! [X4: A] :
( ( P @ X4 )
=> ( Q @ X4 ) )
=> ( Q @ ( the @ A @ P ) ) ) ) ).
% the1I2
thf(fact_227_the1__equality,axiom,
! [A: $tType,P: A > $o,A2: A] :
( ? [X: A] :
( ( P @ X )
& ! [Y4: A] :
( ( P @ Y4 )
=> ( Y4 = X ) ) )
=> ( ( P @ A2 )
=> ( ( the @ A @ P )
= A2 ) ) ) ).
% the1_equality
thf(fact_228_psubset__insert__iff,axiom,
! [A: $tType,A4: set @ A,X3: A,B3: set @ A] :
( ( ord_less @ ( set @ A ) @ A4 @ ( insert @ A @ X3 @ B3 ) )
= ( ( ( member @ A @ X3 @ B3 )
=> ( ord_less @ ( set @ A ) @ A4 @ B3 ) )
& ( ~ ( member @ A @ X3 @ B3 )
=> ( ( ( member @ A @ X3 @ A4 )
=> ( ord_less @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) @ B3 ) )
& ( ~ ( member @ A @ X3 @ A4 )
=> ( ord_less_eq @ ( set @ A ) @ A4 @ B3 ) ) ) ) ) ) ).
% psubset_insert_iff
thf(fact_229_pairwise__alt,axiom,
! [A: $tType] :
( ( pairwise @ A )
= ( ^ [R2: A > A > $o,S2: set @ A] :
! [X2: A] :
( ( member @ A @ X2 @ S2 )
=> ! [Y: A] :
( ( member @ A @ Y @ ( minus_minus @ ( set @ A ) @ S2 @ ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) )
=> ( R2 @ X2 @ Y ) ) ) ) ) ).
% pairwise_alt
thf(fact_230_order__refl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X3: A] : ( ord_less_eq @ A @ X3 @ X3 ) ) ).
% order_refl
thf(fact_231_subset__antisym,axiom,
! [A: $tType,A4: set @ A,B3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B3 )
=> ( ( ord_less_eq @ ( set @ A ) @ B3 @ A4 )
=> ( A4 = B3 ) ) ) ).
% subset_antisym
thf(fact_232_subsetI,axiom,
! [A: $tType,A4: set @ A,B3: set @ A] :
( ! [X4: A] :
( ( member @ A @ X4 @ A4 )
=> ( member @ A @ X4 @ B3 ) )
=> ( ord_less_eq @ ( set @ A ) @ A4 @ B3 ) ) ).
% subsetI
thf(fact_233_sup_Obounded__iff,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [B: A,C2: A,A2: A] :
( ( ord_less_eq @ A @ ( sup_sup @ A @ B @ C2 ) @ A2 )
= ( ( ord_less_eq @ A @ B @ A2 )
& ( ord_less_eq @ A @ C2 @ A2 ) ) ) ) ).
% sup.bounded_iff
thf(fact_234_le__sup__iff,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X3: A,Y2: A,Z2: A] :
( ( ord_less_eq @ A @ ( sup_sup @ A @ X3 @ Y2 ) @ Z2 )
= ( ( ord_less_eq @ A @ X3 @ Z2 )
& ( ord_less_eq @ A @ Y2 @ Z2 ) ) ) ) ).
% le_sup_iff
thf(fact_235_empty__subsetI,axiom,
! [A: $tType,A4: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ A4 ) ).
% empty_subsetI
thf(fact_236_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_237_insert__subset,axiom,
! [A: $tType,X3: A,A4: set @ A,B3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ X3 @ A4 ) @ B3 )
= ( ( member @ A @ X3 @ B3 )
& ( ord_less_eq @ ( set @ A ) @ A4 @ B3 ) ) ) ).
% insert_subset
thf(fact_238_Un__subset__iff,axiom,
! [A: $tType,A4: set @ A,B3: set @ A,C: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A4 @ B3 ) @ C )
= ( ( ord_less_eq @ ( set @ A ) @ A4 @ C )
& ( ord_less_eq @ ( set @ A ) @ B3 @ C ) ) ) ).
% Un_subset_iff
thf(fact_239_psubsetI,axiom,
! [A: $tType,A4: set @ A,B3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B3 )
=> ( ( A4 != B3 )
=> ( ord_less @ ( set @ A ) @ A4 @ B3 ) ) ) ).
% psubsetI
thf(fact_240_singleton__insert__inj__eq_H,axiom,
! [A: $tType,A2: A,A4: set @ A,B: A] :
( ( ( insert @ A @ A2 @ A4 )
= ( insert @ A @ B @ ( bot_bot @ ( set @ A ) ) ) )
= ( ( A2 = B )
& ( ord_less_eq @ ( set @ A ) @ A4 @ ( insert @ A @ B @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_241_singleton__insert__inj__eq,axiom,
! [A: $tType,B: A,A2: A,A4: set @ A] :
( ( ( insert @ A @ B @ ( bot_bot @ ( set @ A ) ) )
= ( insert @ A @ A2 @ A4 ) )
= ( ( A2 = B )
& ( ord_less_eq @ ( set @ A ) @ A4 @ ( insert @ A @ B @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_242_Diff__eq__empty__iff,axiom,
! [A: $tType,A4: set @ A,B3: set @ A] :
( ( ( minus_minus @ ( set @ A ) @ A4 @ B3 )
= ( bot_bot @ ( set @ A ) ) )
= ( ord_less_eq @ ( set @ A ) @ A4 @ B3 ) ) ).
% Diff_eq_empty_iff
thf(fact_243_leD,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [Y2: A,X3: A] :
( ( ord_less_eq @ A @ Y2 @ X3 )
=> ~ ( ord_less @ A @ X3 @ Y2 ) ) ) ).
% leD
thf(fact_244_leI,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X3: A,Y2: A] :
( ~ ( ord_less @ A @ X3 @ Y2 )
=> ( ord_less_eq @ A @ Y2 @ X3 ) ) ) ).
% leI
thf(fact_245_le__less,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [X2: A,Y: A] :
( ( ord_less @ A @ X2 @ Y )
| ( X2 = Y ) ) ) ) ) ).
% le_less
thf(fact_246_less__le,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less @ A )
= ( ^ [X2: A,Y: A] :
( ( ord_less_eq @ A @ X2 @ Y )
& ( X2 != Y ) ) ) ) ) ).
% less_le
thf(fact_247_order__le__less__subst1,axiom,
! [A: $tType,B4: $tType] :
( ( ( order @ B4 )
& ( order @ A ) )
=> ! [A2: A,F: B4 > A,B: B4,C2: B4] :
( ( ord_less_eq @ A @ A2 @ ( F @ B ) )
=> ( ( ord_less @ B4 @ B @ C2 )
=> ( ! [X4: B4,Y4: B4] :
( ( ord_less @ B4 @ X4 @ Y4 )
=> ( ord_less @ A @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less @ A @ A2 @ ( F @ C2 ) ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_248_order__le__less__subst2,axiom,
! [A: $tType,C4: $tType] :
( ( ( order @ C4 )
& ( order @ A ) )
=> ! [A2: A,B: A,F: A > C4,C2: C4] :
( ( ord_less_eq @ A @ A2 @ B )
=> ( ( ord_less @ C4 @ ( F @ B ) @ C2 )
=> ( ! [X4: A,Y4: A] :
( ( ord_less_eq @ A @ X4 @ Y4 )
=> ( ord_less_eq @ C4 @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less @ C4 @ ( F @ A2 ) @ C2 ) ) ) ) ) ).
% order_le_less_subst2
thf(fact_249_order__less__le__subst1,axiom,
! [A: $tType,B4: $tType] :
( ( ( order @ B4 )
& ( order @ A ) )
=> ! [A2: A,F: B4 > A,B: B4,C2: B4] :
( ( ord_less @ A @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq @ B4 @ B @ C2 )
=> ( ! [X4: B4,Y4: B4] :
( ( ord_less_eq @ B4 @ X4 @ Y4 )
=> ( ord_less_eq @ A @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less @ A @ A2 @ ( F @ C2 ) ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_250_order__less__le__subst2,axiom,
! [A: $tType,C4: $tType] :
( ( ( order @ C4 )
& ( order @ A ) )
=> ! [A2: A,B: A,F: A > C4,C2: C4] :
( ( ord_less @ A @ A2 @ B )
=> ( ( ord_less_eq @ C4 @ ( F @ B ) @ C2 )
=> ( ! [X4: A,Y4: A] :
( ( ord_less @ A @ X4 @ Y4 )
=> ( ord_less @ C4 @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less @ C4 @ ( F @ A2 ) @ C2 ) ) ) ) ) ).
% order_less_le_subst2
thf(fact_251_not__le,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X3: A,Y2: A] :
( ( ~ ( ord_less_eq @ A @ X3 @ Y2 ) )
= ( ord_less @ A @ Y2 @ X3 ) ) ) ).
% not_le
thf(fact_252_not__less,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X3: A,Y2: A] :
( ( ~ ( ord_less @ A @ X3 @ Y2 ) )
= ( ord_less_eq @ A @ Y2 @ X3 ) ) ) ).
% not_less
thf(fact_253_le__neq__trans,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: A,B: A] :
( ( ord_less_eq @ A @ A2 @ B )
=> ( ( A2 != B )
=> ( ord_less @ A @ A2 @ B ) ) ) ) ).
% le_neq_trans
thf(fact_254_antisym__conv1,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X3: A,Y2: A] :
( ~ ( ord_less @ A @ X3 @ Y2 )
=> ( ( ord_less_eq @ A @ X3 @ Y2 )
= ( X3 = Y2 ) ) ) ) ).
% antisym_conv1
thf(fact_255_antisym__conv2,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X3: A,Y2: A] :
( ( ord_less_eq @ A @ X3 @ Y2 )
=> ( ( ~ ( ord_less @ A @ X3 @ Y2 ) )
= ( X3 = Y2 ) ) ) ) ).
% antisym_conv2
% Type constructors (43)
thf(tcon_HOL_Obool___Lattices_Obounded__lattice,axiom,
bounded_lattice @ $o ).
thf(tcon_Set_Oset___Lattices_Obounded__lattice_1,axiom,
! [A7: $tType] : ( bounded_lattice @ ( set @ A7 ) ) ).
thf(tcon_fun___Lattices_Obounded__lattice_2,axiom,
! [A7: $tType,A8: $tType] :
( ( bounded_lattice @ A8 )
=> ( bounded_lattice @ ( A7 > A8 ) ) ) ).
thf(tcon_fun___Lattices_Obounded__semilattice__sup__bot,axiom,
! [A7: $tType,A8: $tType] :
( ( bounded_lattice @ A8 )
=> ( bounde1808546759up_bot @ ( A7 > A8 ) ) ) ).
thf(tcon_fun___Lattices_Osemilattice__sup,axiom,
! [A7: $tType,A8: $tType] :
( ( semilattice_sup @ A8 )
=> ( semilattice_sup @ ( A7 > A8 ) ) ) ).
thf(tcon_fun___Orderings_Oorder__bot,axiom,
! [A7: $tType,A8: $tType] :
( ( order_bot @ A8 )
=> ( order_bot @ ( A7 > A8 ) ) ) ).
thf(tcon_fun___Orderings_Opreorder,axiom,
! [A7: $tType,A8: $tType] :
( ( preorder @ A8 )
=> ( preorder @ ( A7 > A8 ) ) ) ).
thf(tcon_fun___Lattices_Olattice,axiom,
! [A7: $tType,A8: $tType] :
( ( lattice @ A8 )
=> ( lattice @ ( A7 > A8 ) ) ) ).
thf(tcon_fun___Orderings_Oorder,axiom,
! [A7: $tType,A8: $tType] :
( ( order @ A8 )
=> ( order @ ( A7 > A8 ) ) ) ).
thf(tcon_fun___Orderings_Oord,axiom,
! [A7: $tType,A8: $tType] :
( ( ord @ A8 )
=> ( ord @ ( A7 > A8 ) ) ) ).
thf(tcon_fun___Orderings_Obot,axiom,
! [A7: $tType,A8: $tType] :
( ( bot @ A8 )
=> ( bot @ ( A7 > A8 ) ) ) ).
thf(tcon_fun___Groups_Ominus,axiom,
! [A7: $tType,A8: $tType] :
( ( minus @ A8 )
=> ( minus @ ( A7 > A8 ) ) ) ).
thf(tcon_Int_Oint___Groups_Ocancel__ab__semigroup__add,axiom,
cancel146912293up_add @ int ).
thf(tcon_Int_Oint___Groups_Oordered__ab__group__add,axiom,
ordered_ab_group_add @ int ).
thf(tcon_Int_Oint___Lattices_Osemilattice__sup_3,axiom,
semilattice_sup @ int ).
thf(tcon_Int_Oint___Orderings_Opreorder_4,axiom,
preorder @ int ).
thf(tcon_Int_Oint___Orderings_Olinorder,axiom,
linorder @ int ).
thf(tcon_Int_Oint___Orderings_Ono__top,axiom,
no_top @ int ).
thf(tcon_Int_Oint___Orderings_Ono__bot,axiom,
no_bot @ int ).
thf(tcon_Int_Oint___Lattices_Olattice_5,axiom,
lattice @ int ).
thf(tcon_Int_Oint___Groups_Ogroup__add,axiom,
group_add @ int ).
thf(tcon_Int_Oint___Orderings_Oorder_6,axiom,
order @ int ).
thf(tcon_Int_Oint___Orderings_Oord_7,axiom,
ord @ int ).
thf(tcon_Int_Oint___Groups_Ominus_8,axiom,
minus @ int ).
thf(tcon_Set_Oset___Lattices_Obounded__semilattice__sup__bot_9,axiom,
! [A7: $tType] : ( bounde1808546759up_bot @ ( set @ A7 ) ) ).
thf(tcon_Set_Oset___Lattices_Osemilattice__sup_10,axiom,
! [A7: $tType] : ( semilattice_sup @ ( set @ A7 ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder__bot_11,axiom,
! [A7: $tType] : ( order_bot @ ( set @ A7 ) ) ).
thf(tcon_Set_Oset___Orderings_Opreorder_12,axiom,
! [A7: $tType] : ( preorder @ ( set @ A7 ) ) ).
thf(tcon_Set_Oset___Lattices_Olattice_13,axiom,
! [A7: $tType] : ( lattice @ ( set @ A7 ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder_14,axiom,
! [A7: $tType] : ( order @ ( set @ A7 ) ) ).
thf(tcon_Set_Oset___Orderings_Oord_15,axiom,
! [A7: $tType] : ( ord @ ( set @ A7 ) ) ).
thf(tcon_Set_Oset___Orderings_Obot_16,axiom,
! [A7: $tType] : ( bot @ ( set @ A7 ) ) ).
thf(tcon_Set_Oset___Groups_Ominus_17,axiom,
! [A7: $tType] : ( minus @ ( set @ A7 ) ) ).
thf(tcon_HOL_Obool___Lattices_Obounded__semilattice__sup__bot_18,axiom,
bounde1808546759up_bot @ $o ).
thf(tcon_HOL_Obool___Lattices_Osemilattice__sup_19,axiom,
semilattice_sup @ $o ).
thf(tcon_HOL_Obool___Orderings_Oorder__bot_20,axiom,
order_bot @ $o ).
thf(tcon_HOL_Obool___Orderings_Opreorder_21,axiom,
preorder @ $o ).
thf(tcon_HOL_Obool___Orderings_Olinorder_22,axiom,
linorder @ $o ).
thf(tcon_HOL_Obool___Lattices_Olattice_23,axiom,
lattice @ $o ).
thf(tcon_HOL_Obool___Orderings_Oorder_24,axiom,
order @ $o ).
thf(tcon_HOL_Obool___Orderings_Oord_25,axiom,
ord @ $o ).
thf(tcon_HOL_Obool___Orderings_Obot_26,axiom,
bot @ $o ).
thf(tcon_HOL_Obool___Groups_Ominus_27,axiom,
minus @ $o ).
% Helper facts (3)
thf(help_If_3_1_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_T,axiom,
! [A: $tType,X3: A,Y2: A] :
( ( if @ A @ $false @ X3 @ Y2 )
= Y2 ) ).
thf(help_If_1_1_T,axiom,
! [A: $tType,X3: A,Y2: A] :
( ( if @ A @ $true @ X3 @ Y2 )
= X3 ) ).
% Conjectures (1)
thf(conj_0,conjecture,
! [X4: a] :
( ( member @ a @ X4 @ ( binary64540844le_eqs @ a @ h @ e ) )
=> ( ( h @ X4 )
!= ( h @ x ) ) ) ).
%------------------------------------------------------------------------------