TPTP Problem File: ITP034^2.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : ITP034^2 : TPTP v8.2.0. Released v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer BinaryTree problem prob_562__3255854_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_562__3255854_1 [Des21]
% Status : Theorem
% Rating : 0.00 v7.5.0
% Syntax : Number of formulae : 329 ( 113 unt; 40 typ; 0 def)
% Number of atoms : 801 ( 273 equ; 0 cnn)
% Maximal formula atoms : 13 ( 2 avg)
% Number of connectives : 3400 ( 77 ~; 20 |; 71 &;2879 @)
% ( 0 <=>; 353 =>; 0 <=; 0 <~>)
% Maximal formula depth : 21 ( 7 avg)
% Number of types : 3 ( 2 usr)
% Number of type conns : 142 ( 142 >; 0 *; 0 +; 0 <<)
% Number of symbols : 40 ( 38 usr; 5 con; 0-5 aty)
% Number of variables : 1009 ( 88 ^; 856 !; 34 ?;1009 :)
% ( 31 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Sledgehammer 2021-02-23 16:17:10.404
%------------------------------------------------------------------------------
% 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 (36)
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_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_Lattices_Olattice,type,
lattice:
!>[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_Rings_Olinordered__idom,type,
linordered_idom:
!>[A: $tType] : $o ).
thf(sy_cl_Lattices_Osemilattice__sup,type,
semilattice_sup:
!>[A: $tType] : $o ).
thf(sy_cl_Lattices_Obounded__semilattice__sup__bot,type,
bounde1808546759up_bot:
!>[A: $tType] : $o ).
thf(sy_cl_Conditionally__Complete__Lattices_Olinear__continuum,type,
condit1656338222tinuum:
!>[A: $tType] : $o ).
thf(sy_cl_Conditionally__Complete__Lattices_Oconditionally__complete__linorder,type,
condit1037483654norder:
!>[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_Obinsert,type,
binary1830089824insert:
!>[A: $tType] : ( ( A > int ) > A > ( binary1291135688e_Tree @ A ) > ( binary1291135688e_Tree @ 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_Oremove,type,
binary997842527remove:
!>[A: $tType] : ( ( A > int ) > A > ( binary1291135688e_Tree @ A ) > ( binary1291135688e_Tree @ A ) ) ).
thf(sy_c_BinaryTree__Mirabelle__pchhvghoao_Orm,type,
binary576689334lle_rm:
!>[A: $tType] : ( ( A > int ) > ( binary1291135688e_Tree @ A ) > A ) ).
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_BinaryTree__Mirabelle__pchhvghoao_Owrm,type,
binary213313527le_wrm:
!>[A: $tType] : ( ( A > int ) > ( binary1291135688e_Tree @ A ) > ( binary1291135688e_Tree @ 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_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_h,type,
h: a > int ).
thf(sy_v_l____,type,
l: a ).
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 (255)
thf(fact_0_hLess,axiom,
ord_less @ int @ ( h @ l ) @ ( h @ ( binary576689334lle_rm @ a @ h @ t2 ) ) ).
% hLess
thf(fact_1_rm__res,axiom,
( ( binary576689334lle_rm @ a @ h @ ( binary210054475elle_T @ a @ t1 @ x @ t2 ) )
= ( binary576689334lle_rm @ a @ h @ t2 ) ) ).
% rm_res
thf(fact_2_t2nTip,axiom,
( t2
!= ( binary1746293266le_Tip @ a ) ) ).
% t2nTip
thf(fact_3_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_4_s,axiom,
binary1610619414edTree @ a @ h @ ( binary210054475elle_T @ a @ t1 @ x @ t2 ) ).
% s
thf(fact_5_wrm__res,axiom,
( ( binary213313527le_wrm @ a @ h @ ( binary210054475elle_T @ a @ t1 @ x @ t2 ) )
= ( binary210054475elle_T @ a @ t1 @ x @ ( binary213313527le_wrm @ a @ h @ t2 ) ) ) ).
% wrm_res
thf(fact_6_ldef,axiom,
member @ a @ l @ ( binary1653327646_setOf @ a @ ( binary213313527le_wrm @ a @ h @ ( binary210054475elle_T @ a @ t1 @ x @ t2 ) ) ) ).
% ldef
thf(fact_7_s1,axiom,
binary1610619414edTree @ a @ h @ t1 ).
% s1
thf(fact_8_s2,axiom,
binary1610619414edTree @ a @ h @ t2 ).
% s2
thf(fact_9_binsert_Osimps_I2_J,axiom,
! [A: $tType,H: A > int,E: A,X: A,T1: binary1291135688e_Tree @ A,T2: binary1291135688e_Tree @ A] :
( ( ( ord_less @ int @ ( H @ E ) @ ( H @ X ) )
=> ( ( binary1830089824insert @ A @ H @ E @ ( binary210054475elle_T @ A @ T1 @ X @ T2 ) )
= ( binary210054475elle_T @ A @ ( binary1830089824insert @ A @ H @ E @ T1 ) @ X @ T2 ) ) )
& ( ~ ( ord_less @ int @ ( H @ E ) @ ( H @ X ) )
=> ( ( ( ord_less @ int @ ( H @ X ) @ ( H @ E ) )
=> ( ( binary1830089824insert @ A @ H @ E @ ( binary210054475elle_T @ A @ T1 @ X @ T2 ) )
= ( binary210054475elle_T @ A @ T1 @ X @ ( binary1830089824insert @ A @ H @ E @ T2 ) ) ) )
& ( ~ ( ord_less @ int @ ( H @ X ) @ ( H @ E ) )
=> ( ( binary1830089824insert @ A @ H @ E @ ( binary210054475elle_T @ A @ T1 @ X @ T2 ) )
= ( binary210054475elle_T @ A @ T1 @ E @ T2 ) ) ) ) ) ) ).
% binsert.simps(2)
thf(fact_10_rm_Osimps,axiom,
! [A: $tType,T2: binary1291135688e_Tree @ A,H: A > int,T1: binary1291135688e_Tree @ A,X: A] :
( ( ( T2
= ( binary1746293266le_Tip @ A ) )
=> ( ( binary576689334lle_rm @ A @ H @ ( binary210054475elle_T @ A @ T1 @ X @ T2 ) )
= X ) )
& ( ( T2
!= ( binary1746293266le_Tip @ A ) )
=> ( ( binary576689334lle_rm @ A @ H @ ( binary210054475elle_T @ A @ T1 @ X @ T2 ) )
= ( binary576689334lle_rm @ A @ H @ T2 ) ) ) ) ).
% rm.simps
thf(fact_11_minf_I11_J,axiom,
! [C: $tType,D: $tType] :
( ( ord @ C )
=> ! [F: D] :
? [Z: C] :
! [X2: C] :
( ( ord_less @ C @ X2 @ Z )
=> ( F = F ) ) ) ).
% minf(11)
thf(fact_12_minf_I7_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T: A] :
? [Z: A] :
! [X2: A] :
( ( ord_less @ A @ X2 @ Z )
=> ~ ( ord_less @ A @ T @ X2 ) ) ) ).
% minf(7)
thf(fact_13_minf_I5_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T: A] :
? [Z: A] :
! [X2: A] :
( ( ord_less @ A @ X2 @ Z )
=> ( ord_less @ A @ X2 @ T ) ) ) ).
% minf(5)
thf(fact_14_minf_I4_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T: A] :
? [Z: A] :
! [X2: A] :
( ( ord_less @ A @ X2 @ Z )
=> ( X2 != T ) ) ) ).
% minf(4)
thf(fact_15_minf_I3_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T: A] :
? [Z: A] :
! [X2: A] :
( ( ord_less @ A @ X2 @ Z )
=> ( X2 != T ) ) ) ).
% minf(3)
thf(fact_16_minf_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > $o,P2: A > $o,Q: A > $o,Q2: A > $o] :
( ? [Z2: A] :
! [X3: A] :
( ( ord_less @ A @ X3 @ Z2 )
=> ( ( P @ X3 )
= ( P2 @ X3 ) ) )
=> ( ? [Z2: A] :
! [X3: A] :
( ( ord_less @ A @ X3 @ Z2 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z: A] :
! [X2: A] :
( ( ord_less @ A @ X2 @ Z )
=> ( ( ( P @ X2 )
| ( Q @ X2 ) )
= ( ( P2 @ X2 )
| ( Q2 @ X2 ) ) ) ) ) ) ) ).
% minf(2)
thf(fact_17_sortedTree_Osimps_I1_J,axiom,
! [A: $tType,H: A > int] : ( binary1610619414edTree @ A @ H @ ( binary1746293266le_Tip @ A ) ) ).
% sortedTree.simps(1)
thf(fact_18_wrm__sort,axiom,
! [A: $tType,T: binary1291135688e_Tree @ A,H: A > int] :
( ( ( T
!= ( binary1746293266le_Tip @ A ) )
& ( binary1610619414edTree @ A @ H @ T ) )
=> ( binary1610619414edTree @ A @ H @ ( binary213313527le_wrm @ A @ H @ T ) ) ) ).
% wrm_sort
thf(fact_19_binsert__sorted,axiom,
! [A: $tType,H: A > int,T: binary1291135688e_Tree @ A,X: A] :
( ( binary1610619414edTree @ A @ H @ T )
=> ( binary1610619414edTree @ A @ H @ ( binary1830089824insert @ A @ H @ X @ T ) ) ) ).
% binsert_sorted
thf(fact_20_rm__set,axiom,
! [A: $tType,T: binary1291135688e_Tree @ A,H: A > int] :
( ( ( T
!= ( binary1746293266le_Tip @ A ) )
& ( binary1610619414edTree @ A @ H @ T ) )
=> ( member @ A @ ( binary576689334lle_rm @ A @ H @ T ) @ ( binary1653327646_setOf @ A @ T ) ) ) ).
% rm_set
thf(fact_21_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_22_wrm_Osimps,axiom,
! [A: $tType,T2: binary1291135688e_Tree @ A,H: A > int,T1: binary1291135688e_Tree @ A,X: A] :
( ( ( T2
= ( binary1746293266le_Tip @ A ) )
=> ( ( binary213313527le_wrm @ A @ H @ ( binary210054475elle_T @ A @ T1 @ X @ T2 ) )
= T1 ) )
& ( ( T2
!= ( binary1746293266le_Tip @ A ) )
=> ( ( binary213313527le_wrm @ A @ H @ ( binary210054475elle_T @ A @ T1 @ X @ T2 ) )
= ( binary210054475elle_T @ A @ T1 @ X @ ( binary213313527le_wrm @ A @ H @ T2 ) ) ) ) ) ).
% wrm.simps
thf(fact_23_sortedTree_Osimps_I2_J,axiom,
! [A: $tType,H: A > int,T1: binary1291135688e_Tree @ A,X: A,T2: binary1291135688e_Tree @ A] :
( ( binary1610619414edTree @ A @ H @ ( binary210054475elle_T @ A @ T1 @ X @ T2 ) )
= ( ( binary1610619414edTree @ A @ H @ T1 )
& ! [X4: A] :
( ( member @ A @ X4 @ ( binary1653327646_setOf @ A @ T1 ) )
=> ( ord_less @ int @ ( H @ X4 ) @ ( H @ X ) ) )
& ! [X4: A] :
( ( member @ A @ X4 @ ( binary1653327646_setOf @ A @ T2 ) )
=> ( ord_less @ int @ ( H @ X ) @ ( H @ X4 ) ) )
& ( binary1610619414edTree @ A @ H @ T2 ) ) ) ).
% sortedTree.simps(2)
thf(fact_24_Tree_Oexhaust,axiom,
! [A: $tType,Y: binary1291135688e_Tree @ A] :
( ( Y
!= ( binary1746293266le_Tip @ A ) )
=> ~ ! [X212: binary1291135688e_Tree @ A,X222: A,X232: binary1291135688e_Tree @ A] :
( Y
!= ( binary210054475elle_T @ A @ X212 @ X222 @ X232 ) ) ) ).
% Tree.exhaust
thf(fact_25_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_26_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_27_sortLemmaR,axiom,
! [A: $tType,H: A > int,T1: binary1291135688e_Tree @ A,X: A,T2: binary1291135688e_Tree @ A] :
( ( binary1610619414edTree @ A @ H @ ( binary210054475elle_T @ A @ T1 @ X @ T2 ) )
=> ( binary1610619414edTree @ A @ H @ T2 ) ) ).
% sortLemmaR
thf(fact_28_sortLemmaL,axiom,
! [A: $tType,H: A > int,T1: binary1291135688e_Tree @ A,X: A,T2: binary1291135688e_Tree @ A] :
( ( binary1610619414edTree @ A @ H @ ( binary210054475elle_T @ A @ T1 @ X @ T2 ) )
=> ( binary1610619414edTree @ A @ H @ T1 ) ) ).
% sortLemmaL
thf(fact_29_pinf_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > $o,P2: A > $o,Q: A > $o,Q2: A > $o] :
( ? [Z2: A] :
! [X3: A] :
( ( ord_less @ A @ Z2 @ X3 )
=> ( ( P @ X3 )
= ( P2 @ X3 ) ) )
=> ( ? [Z2: A] :
! [X3: A] :
( ( ord_less @ A @ Z2 @ X3 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z: A] :
! [X2: A] :
( ( ord_less @ A @ Z @ X2 )
=> ( ( ( P @ X2 )
& ( Q @ X2 ) )
= ( ( P2 @ X2 )
& ( Q2 @ X2 ) ) ) ) ) ) ) ).
% pinf(1)
thf(fact_30_pinf_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > $o,P2: A > $o,Q: A > $o,Q2: A > $o] :
( ? [Z2: A] :
! [X3: A] :
( ( ord_less @ A @ Z2 @ X3 )
=> ( ( P @ X3 )
= ( P2 @ X3 ) ) )
=> ( ? [Z2: A] :
! [X3: A] :
( ( ord_less @ A @ Z2 @ X3 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z: A] :
! [X2: A] :
( ( ord_less @ A @ Z @ X2 )
=> ( ( ( P @ X2 )
| ( Q @ X2 ) )
= ( ( P2 @ X2 )
| ( Q2 @ X2 ) ) ) ) ) ) ) ).
% pinf(2)
thf(fact_31_pinf_I3_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T: A] :
? [Z: A] :
! [X2: A] :
( ( ord_less @ A @ Z @ X2 )
=> ( X2 != T ) ) ) ).
% pinf(3)
thf(fact_32_pinf_I4_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T: A] :
? [Z: A] :
! [X2: A] :
( ( ord_less @ A @ Z @ X2 )
=> ( X2 != T ) ) ) ).
% pinf(4)
thf(fact_33_pinf_I5_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T: A] :
? [Z: A] :
! [X2: A] :
( ( ord_less @ A @ Z @ X2 )
=> ~ ( ord_less @ A @ X2 @ T ) ) ) ).
% pinf(5)
thf(fact_34_pinf_I7_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T: A] :
? [Z: A] :
! [X2: A] :
( ( ord_less @ A @ Z @ X2 )
=> ( ord_less @ A @ T @ X2 ) ) ) ).
% pinf(7)
thf(fact_35_pinf_I11_J,axiom,
! [C: $tType,D: $tType] :
( ( ord @ C )
=> ! [F: D] :
? [Z: C] :
! [X2: C] :
( ( ord_less @ C @ Z @ X2 )
=> ( F = F ) ) ) ).
% pinf(11)
thf(fact_36_minf_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > $o,P2: A > $o,Q: A > $o,Q2: A > $o] :
( ? [Z2: A] :
! [X3: A] :
( ( ord_less @ A @ X3 @ Z2 )
=> ( ( P @ X3 )
= ( P2 @ X3 ) ) )
=> ( ? [Z2: A] :
! [X3: A] :
( ( ord_less @ A @ X3 @ Z2 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z: A] :
! [X2: A] :
( ( ord_less @ A @ X2 @ Z )
=> ( ( ( P @ X2 )
& ( Q @ X2 ) )
= ( ( P2 @ X2 )
& ( Q2 @ X2 ) ) ) ) ) ) ) ).
% minf(1)
thf(fact_37_h1,axiom,
( ( ( t1
!= ( binary1746293266le_Tip @ a ) )
& ( binary1610619414edTree @ a @ h @ t1 ) )
=> ! [X2: a] :
( ( member @ a @ X2 @ ( binary1653327646_setOf @ a @ ( binary213313527le_wrm @ a @ h @ t1 ) ) )
=> ( ord_less @ int @ ( h @ X2 ) @ ( h @ ( binary576689334lle_rm @ a @ h @ t1 ) ) ) ) ) ).
% h1
thf(fact_38_h2,axiom,
( ( ( t2
!= ( binary1746293266le_Tip @ a ) )
& ( binary1610619414edTree @ a @ h @ t2 ) )
=> ! [X2: a] :
( ( member @ a @ X2 @ ( binary1653327646_setOf @ a @ ( binary213313527le_wrm @ a @ h @ t2 ) ) )
=> ( ord_less @ int @ ( h @ X2 ) @ ( h @ ( binary576689334lle_rm @ a @ h @ t2 ) ) ) ) ) ).
% h2
thf(fact_39_memb__spec,axiom,
! [A: $tType,H: A > int,T: binary1291135688e_Tree @ A,X: A] :
( ( binary1610619414edTree @ A @ H @ T )
=> ( ( binary827270440e_memb @ A @ H @ X @ T )
= ( member @ A @ X @ ( binary1653327646_setOf @ A @ T ) ) ) ) ).
% memb_spec
thf(fact_40_sorted__distinct__pred__def,axiom,
! [A: $tType] :
( ( binary231205461t_pred @ A )
= ( ^ [H2: A > int,A2: A,B: A,T3: binary1291135688e_Tree @ A] :
( ( ( binary1610619414edTree @ A @ H2 @ T3 )
& ( member @ A @ A2 @ ( binary1653327646_setOf @ A @ T3 ) )
& ( member @ A @ B @ ( binary1653327646_setOf @ A @ T3 ) )
& ( ( H2 @ A2 )
= ( H2 @ B ) ) )
=> ( A2 = B ) ) ) ) ).
% sorted_distinct_pred_def
thf(fact_41_l__scope,axiom,
member @ a @ l @ ( sup_sup @ ( set @ a ) @ ( sup_sup @ ( set @ a ) @ ( insert @ a @ x @ ( bot_bot @ ( set @ a ) ) ) @ ( binary1653327646_setOf @ a @ t1 ) ) @ ( binary1653327646_setOf @ a @ ( binary213313527le_wrm @ a @ h @ t2 ) ) ) ).
% l_scope
thf(fact_42_wrm__set1,axiom,
! [A: $tType,T: binary1291135688e_Tree @ A,H: A > int] :
( ( ( T
!= ( binary1746293266le_Tip @ A ) )
& ( binary1610619414edTree @ A @ H @ T ) )
=> ( ord_less_eq @ ( set @ A ) @ ( binary1653327646_setOf @ A @ ( binary213313527le_wrm @ A @ H @ T ) ) @ ( binary1653327646_setOf @ A @ T ) ) ) ).
% wrm_set1
thf(fact_43_remove_Osimps_I1_J,axiom,
! [A: $tType,H: A > int,E: A] :
( ( binary997842527remove @ A @ H @ E @ ( binary1746293266le_Tip @ A ) )
= ( binary1746293266le_Tip @ A ) ) ).
% remove.simps(1)
thf(fact_44_mem__Collect__eq,axiom,
! [A: $tType,A3: A,P: A > $o] :
( ( member @ A @ A3 @ ( collect @ A @ P ) )
= ( P @ A3 ) ) ).
% mem_Collect_eq
thf(fact_45_Collect__mem__eq,axiom,
! [A: $tType,A4: set @ A] :
( ( collect @ A
@ ^ [X4: A] : ( member @ A @ X4 @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_46_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_47_ext,axiom,
! [B2: $tType,A: $tType,F2: A > B2,G: A > B2] :
( ! [X3: A] :
( ( F2 @ X3 )
= ( G @ X3 ) )
=> ( F2 = G ) ) ).
% ext
thf(fact_48_ex__gt__or__lt,axiom,
! [A: $tType] :
( ( condit1656338222tinuum @ A )
=> ! [A3: A] :
? [B3: A] :
( ( ord_less @ A @ A3 @ B3 )
| ( ord_less @ A @ B3 @ A3 ) ) ) ).
% ex_gt_or_lt
thf(fact_49_linorder__neqE__linordered__idom,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: A,Y: A] :
( ( X != Y )
=> ( ~ ( ord_less @ A @ X @ Y )
=> ( ord_less @ A @ Y @ X ) ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_50_complete__interval,axiom,
! [A: $tType] :
( ( condit1037483654norder @ A )
=> ! [A3: A,B4: A,P: A > $o] :
( ( ord_less @ A @ A3 @ B4 )
=> ( ( P @ A3 )
=> ( ~ ( P @ B4 )
=> ? [C2: A] :
( ( ord_less_eq @ A @ A3 @ C2 )
& ( ord_less_eq @ A @ C2 @ B4 )
& ! [X2: A] :
( ( ( ord_less_eq @ A @ A3 @ X2 )
& ( ord_less @ A @ X2 @ C2 ) )
=> ( P @ X2 ) )
& ! [D2: A] :
( ! [X3: A] :
( ( ( ord_less_eq @ A @ A3 @ X3 )
& ( ord_less @ A @ X3 @ D2 ) )
=> ( P @ X3 ) )
=> ( ord_less_eq @ A @ D2 @ C2 ) ) ) ) ) ) ) ).
% complete_interval
thf(fact_51_setOf_Osimps_I2_J,axiom,
! [A: $tType,T1: binary1291135688e_Tree @ A,X: A,T2: binary1291135688e_Tree @ A] :
( ( binary1653327646_setOf @ A @ ( binary210054475elle_T @ A @ T1 @ X @ T2 ) )
= ( sup_sup @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ ( binary1653327646_setOf @ A @ T1 ) @ ( binary1653327646_setOf @ A @ T2 ) ) @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% setOf.simps(2)
thf(fact_52_minf_I8_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T: A] :
? [Z: A] :
! [X2: A] :
( ( ord_less @ A @ X2 @ Z )
=> ~ ( ord_less_eq @ A @ T @ X2 ) ) ) ).
% minf(8)
thf(fact_53_minf_I6_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T: A] :
? [Z: A] :
! [X2: A] :
( ( ord_less @ A @ X2 @ Z )
=> ( ord_less_eq @ A @ X2 @ T ) ) ) ).
% minf(6)
thf(fact_54_pinf_I8_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T: A] :
? [Z: A] :
! [X2: A] :
( ( ord_less @ A @ Z @ X2 )
=> ( ord_less_eq @ A @ T @ X2 ) ) ) ).
% pinf(8)
thf(fact_55_pinf_I6_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T: A] :
? [Z: A] :
! [X2: A] :
( ( ord_less @ A @ Z @ X2 )
=> ~ ( ord_less_eq @ A @ X2 @ T ) ) ) ).
% pinf(6)
thf(fact_56_sorted__distinct,axiom,
! [A: $tType,H: A > int,A3: A,B4: A,T: binary1291135688e_Tree @ A] : ( binary231205461t_pred @ A @ H @ A3 @ B4 @ T ) ).
% sorted_distinct
thf(fact_57_setOf_Osimps_I1_J,axiom,
! [A: $tType] :
( ( binary1653327646_setOf @ A @ ( binary1746293266le_Tip @ A ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% setOf.simps(1)
thf(fact_58_singleton__insert__inj__eq_H,axiom,
! [A: $tType,A3: A,A4: set @ A,B4: A] :
( ( ( insert @ A @ A3 @ A4 )
= ( insert @ A @ B4 @ ( bot_bot @ ( set @ A ) ) ) )
= ( ( A3 = B4 )
& ( ord_less_eq @ ( set @ A ) @ A4 @ ( insert @ A @ B4 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_59_singleton__insert__inj__eq,axiom,
! [A: $tType,B4: A,A3: A,A4: set @ A] :
( ( ( insert @ A @ B4 @ ( bot_bot @ ( set @ A ) ) )
= ( insert @ A @ A3 @ A4 ) )
= ( ( A3 = B4 )
& ( ord_less_eq @ ( set @ A ) @ A4 @ ( insert @ A @ B4 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_60_singleton__conv2,axiom,
! [A: $tType,A3: A] :
( ( collect @ A
@ ( ^ [Y2: A,Z3: A] : Y2 = Z3
@ A3 ) )
= ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% singleton_conv2
thf(fact_61_singleton__conv,axiom,
! [A: $tType,A3: A] :
( ( collect @ A
@ ^ [X4: A] : X4 = A3 )
= ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% singleton_conv
thf(fact_62_Un__insert__right,axiom,
! [A: $tType,A4: set @ A,A3: A,B5: set @ A] :
( ( sup_sup @ ( set @ A ) @ A4 @ ( insert @ A @ A3 @ B5 ) )
= ( insert @ A @ A3 @ ( sup_sup @ ( set @ A ) @ A4 @ B5 ) ) ) ).
% Un_insert_right
thf(fact_63_Un__insert__left,axiom,
! [A: $tType,A3: A,B5: set @ A,C3: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( insert @ A @ A3 @ B5 ) @ C3 )
= ( insert @ A @ A3 @ ( sup_sup @ ( set @ A ) @ B5 @ C3 ) ) ) ).
% Un_insert_left
thf(fact_64_Un__subset__iff,axiom,
! [A: $tType,A4: set @ A,B5: set @ A,C3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A4 @ B5 ) @ C3 )
= ( ( ord_less_eq @ ( set @ A ) @ A4 @ C3 )
& ( ord_less_eq @ ( set @ A ) @ B5 @ C3 ) ) ) ).
% Un_subset_iff
thf(fact_65_Un__empty,axiom,
! [A: $tType,A4: set @ A,B5: set @ A] :
( ( ( sup_sup @ ( set @ A ) @ A4 @ B5 )
= ( bot_bot @ ( set @ A ) ) )
= ( ( A4
= ( bot_bot @ ( set @ A ) ) )
& ( B5
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Un_empty
thf(fact_66_empty__iff,axiom,
! [A: $tType,C4: A] :
~ ( member @ A @ C4 @ ( bot_bot @ ( set @ A ) ) ) ).
% empty_iff
thf(fact_67_all__not__in__conv,axiom,
! [A: $tType,A4: set @ A] :
( ( ! [X4: A] :
~ ( member @ A @ X4 @ A4 ) )
= ( A4
= ( bot_bot @ ( set @ A ) ) ) ) ).
% all_not_in_conv
thf(fact_68_Collect__empty__eq,axiom,
! [A: $tType,P: A > $o] :
( ( ( collect @ A @ P )
= ( bot_bot @ ( set @ A ) ) )
= ( ! [X4: A] :
~ ( P @ X4 ) ) ) ).
% Collect_empty_eq
thf(fact_69_empty__Collect__eq,axiom,
! [A: $tType,P: A > $o] :
( ( ( bot_bot @ ( set @ A ) )
= ( collect @ A @ P ) )
= ( ! [X4: A] :
~ ( P @ X4 ) ) ) ).
% empty_Collect_eq
thf(fact_70_subsetI,axiom,
! [A: $tType,A4: set @ A,B5: set @ A] :
( ! [X3: A] :
( ( member @ A @ X3 @ A4 )
=> ( member @ A @ X3 @ B5 ) )
=> ( ord_less_eq @ ( set @ A ) @ A4 @ B5 ) ) ).
% subsetI
thf(fact_71_subset__antisym,axiom,
! [A: $tType,A4: set @ A,B5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B5 )
=> ( ( ord_less_eq @ ( set @ A ) @ B5 @ A4 )
=> ( A4 = B5 ) ) ) ).
% subset_antisym
thf(fact_72_psubsetI,axiom,
! [A: $tType,A4: set @ A,B5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B5 )
=> ( ( A4 != B5 )
=> ( ord_less @ ( set @ A ) @ A4 @ B5 ) ) ) ).
% psubsetI
thf(fact_73_insertCI,axiom,
! [A: $tType,A3: A,B5: set @ A,B4: A] :
( ( ~ ( member @ A @ A3 @ B5 )
=> ( A3 = B4 ) )
=> ( member @ A @ A3 @ ( insert @ A @ B4 @ B5 ) ) ) ).
% insertCI
thf(fact_74_insert__iff,axiom,
! [A: $tType,A3: A,B4: A,A4: set @ A] :
( ( member @ A @ A3 @ ( insert @ A @ B4 @ A4 ) )
= ( ( A3 = B4 )
| ( member @ A @ A3 @ A4 ) ) ) ).
% insert_iff
thf(fact_75_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_76_UnCI,axiom,
! [A: $tType,C4: A,B5: set @ A,A4: set @ A] :
( ( ~ ( member @ A @ C4 @ B5 )
=> ( member @ A @ C4 @ A4 ) )
=> ( member @ A @ C4 @ ( sup_sup @ ( set @ A ) @ A4 @ B5 ) ) ) ).
% UnCI
thf(fact_77_Un__iff,axiom,
! [A: $tType,C4: A,A4: set @ A,B5: set @ A] :
( ( member @ A @ C4 @ ( sup_sup @ ( set @ A ) @ A4 @ B5 ) )
= ( ( member @ A @ C4 @ A4 )
| ( member @ A @ C4 @ B5 ) ) ) ).
% Un_iff
thf(fact_78_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_79_empty__subsetI,axiom,
! [A: $tType,A4: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ A4 ) ).
% empty_subsetI
thf(fact_80_singletonI,axiom,
! [A: $tType,A3: A] : ( member @ A @ A3 @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% singletonI
thf(fact_81_insert__subset,axiom,
! [A: $tType,X: A,A4: set @ A,B5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ X @ A4 ) @ B5 )
= ( ( member @ A @ X @ B5 )
& ( ord_less_eq @ ( set @ A ) @ A4 @ B5 ) ) ) ).
% insert_subset
thf(fact_82_less__eq__set__def,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A5: set @ A,B6: set @ A] :
( ord_less_eq @ ( A > $o )
@ ^ [X4: A] : ( member @ A @ X4 @ A5 )
@ ^ [X4: A] : ( member @ A @ X4 @ B6 ) ) ) ) ).
% less_eq_set_def
thf(fact_83_emptyE,axiom,
! [A: $tType,A3: A] :
~ ( member @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ).
% emptyE
thf(fact_84_equals0D,axiom,
! [A: $tType,A4: set @ A,A3: A] :
( ( A4
= ( bot_bot @ ( set @ A ) ) )
=> ~ ( member @ A @ A3 @ A4 ) ) ).
% equals0D
thf(fact_85_equals0I,axiom,
! [A: $tType,A4: set @ A] :
( ! [Y3: A] :
~ ( member @ A @ Y3 @ A4 )
=> ( A4
= ( bot_bot @ ( set @ A ) ) ) ) ).
% equals0I
thf(fact_86_ex__in__conv,axiom,
! [A: $tType,A4: set @ A] :
( ( ? [X4: A] : ( member @ A @ X4 @ A4 ) )
= ( A4
!= ( bot_bot @ ( set @ A ) ) ) ) ).
% ex_in_conv
thf(fact_87_not__psubset__empty,axiom,
! [A: $tType,A4: set @ A] :
~ ( ord_less @ ( set @ A ) @ A4 @ ( bot_bot @ ( set @ A ) ) ) ).
% not_psubset_empty
thf(fact_88_in__mono,axiom,
! [A: $tType,A4: set @ A,B5: set @ A,X: A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B5 )
=> ( ( member @ A @ X @ A4 )
=> ( member @ A @ X @ B5 ) ) ) ).
% in_mono
thf(fact_89_subsetD,axiom,
! [A: $tType,A4: set @ A,B5: set @ A,C4: A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B5 )
=> ( ( member @ A @ C4 @ A4 )
=> ( member @ A @ C4 @ B5 ) ) ) ).
% subsetD
thf(fact_90_equalityE,axiom,
! [A: $tType,A4: set @ A,B5: set @ A] :
( ( A4 = B5 )
=> ~ ( ( ord_less_eq @ ( set @ A ) @ A4 @ B5 )
=> ~ ( ord_less_eq @ ( set @ A ) @ B5 @ A4 ) ) ) ).
% equalityE
thf(fact_91_subset__eq,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A5: set @ A,B6: set @ A] :
! [X4: A] :
( ( member @ A @ X4 @ A5 )
=> ( member @ A @ X4 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_92_equalityD1,axiom,
! [A: $tType,A4: set @ A,B5: set @ A] :
( ( A4 = B5 )
=> ( ord_less_eq @ ( set @ A ) @ A4 @ B5 ) ) ).
% equalityD1
thf(fact_93_equalityD2,axiom,
! [A: $tType,A4: set @ A,B5: set @ A] :
( ( A4 = B5 )
=> ( ord_less_eq @ ( set @ A ) @ B5 @ A4 ) ) ).
% equalityD2
thf(fact_94_subset__iff,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A5: set @ A,B6: set @ A] :
! [T3: A] :
( ( member @ A @ T3 @ A5 )
=> ( member @ A @ T3 @ B6 ) ) ) ) ).
% subset_iff
thf(fact_95_subset__refl,axiom,
! [A: $tType,A4: set @ A] : ( ord_less_eq @ ( set @ A ) @ A4 @ A4 ) ).
% subset_refl
thf(fact_96_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_97_subset__trans,axiom,
! [A: $tType,A4: set @ A,B5: set @ A,C3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B5 )
=> ( ( ord_less_eq @ ( set @ A ) @ B5 @ C3 )
=> ( ord_less_eq @ ( set @ A ) @ A4 @ C3 ) ) ) ).
% subset_trans
thf(fact_98_set__eq__subset,axiom,
! [A: $tType] :
( ( ^ [Y2: set @ A,Z3: set @ A] : Y2 = Z3 )
= ( ^ [A5: set @ A,B6: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ B6 )
& ( ord_less_eq @ ( set @ A ) @ B6 @ A5 ) ) ) ) ).
% set_eq_subset
thf(fact_99_Collect__mono__iff,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) )
= ( ! [X4: A] :
( ( P @ X4 )
=> ( Q @ X4 ) ) ) ) ).
% Collect_mono_iff
thf(fact_100_subset__iff__psubset__eq,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A5: set @ A,B6: set @ A] :
( ( ord_less @ ( set @ A ) @ A5 @ B6 )
| ( A5 = B6 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_101_subset__psubset__trans,axiom,
! [A: $tType,A4: set @ A,B5: set @ A,C3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B5 )
=> ( ( ord_less @ ( set @ A ) @ B5 @ C3 )
=> ( ord_less @ ( set @ A ) @ A4 @ C3 ) ) ) ).
% subset_psubset_trans
thf(fact_102_subset__not__subset__eq,axiom,
! [A: $tType] :
( ( ord_less @ ( set @ A ) )
= ( ^ [A5: set @ A,B6: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ B6 )
& ~ ( ord_less_eq @ ( set @ A ) @ B6 @ A5 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_103_psubset__subset__trans,axiom,
! [A: $tType,A4: set @ A,B5: set @ A,C3: set @ A] :
( ( ord_less @ ( set @ A ) @ A4 @ B5 )
=> ( ( ord_less_eq @ ( set @ A ) @ B5 @ C3 )
=> ( ord_less @ ( set @ A ) @ A4 @ C3 ) ) ) ).
% psubset_subset_trans
thf(fact_104_psubset__imp__subset,axiom,
! [A: $tType,A4: set @ A,B5: set @ A] :
( ( ord_less @ ( set @ A ) @ A4 @ B5 )
=> ( ord_less_eq @ ( set @ A ) @ A4 @ B5 ) ) ).
% psubset_imp_subset
thf(fact_105_psubset__eq,axiom,
! [A: $tType] :
( ( ord_less @ ( set @ A ) )
= ( ^ [A5: set @ A,B6: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ B6 )
& ( A5 != B6 ) ) ) ) ).
% psubset_eq
thf(fact_106_psubsetE,axiom,
! [A: $tType,A4: set @ A,B5: set @ A] :
( ( ord_less @ ( set @ A ) @ A4 @ B5 )
=> ~ ( ( ord_less_eq @ ( set @ A ) @ A4 @ B5 )
=> ( ord_less_eq @ ( set @ A ) @ B5 @ A4 ) ) ) ).
% psubsetE
thf(fact_107_insertE,axiom,
! [A: $tType,A3: A,B4: A,A4: set @ A] :
( ( member @ A @ A3 @ ( insert @ A @ B4 @ A4 ) )
=> ( ( A3 != B4 )
=> ( member @ A @ A3 @ A4 ) ) ) ).
% insertE
thf(fact_108_insertI1,axiom,
! [A: $tType,A3: A,B5: set @ A] : ( member @ A @ A3 @ ( insert @ A @ A3 @ B5 ) ) ).
% insertI1
thf(fact_109_insertI2,axiom,
! [A: $tType,A3: A,B5: set @ A,B4: A] :
( ( member @ A @ A3 @ B5 )
=> ( member @ A @ A3 @ ( insert @ A @ B4 @ B5 ) ) ) ).
% insertI2
thf(fact_110_Set_Oset__insert,axiom,
! [A: $tType,X: A,A4: set @ A] :
( ( member @ A @ X @ A4 )
=> ~ ! [B7: set @ A] :
( ( A4
= ( insert @ A @ X @ B7 ) )
=> ( member @ A @ X @ B7 ) ) ) ).
% Set.set_insert
thf(fact_111_insert__ident,axiom,
! [A: $tType,X: A,A4: set @ A,B5: set @ A] :
( ~ ( member @ A @ X @ A4 )
=> ( ~ ( member @ A @ X @ B5 )
=> ( ( ( insert @ A @ X @ A4 )
= ( insert @ A @ X @ B5 ) )
= ( A4 = B5 ) ) ) ) ).
% insert_ident
thf(fact_112_insert__absorb,axiom,
! [A: $tType,A3: A,A4: set @ A] :
( ( member @ A @ A3 @ A4 )
=> ( ( insert @ A @ A3 @ A4 )
= A4 ) ) ).
% insert_absorb
thf(fact_113_insert__eq__iff,axiom,
! [A: $tType,A3: A,A4: set @ A,B4: A,B5: set @ A] :
( ~ ( member @ A @ A3 @ A4 )
=> ( ~ ( member @ A @ B4 @ B5 )
=> ( ( ( insert @ A @ A3 @ A4 )
= ( insert @ A @ B4 @ B5 ) )
= ( ( ( A3 = B4 )
=> ( A4 = B5 ) )
& ( ( A3 != B4 )
=> ? [C5: set @ A] :
( ( A4
= ( insert @ A @ B4 @ C5 ) )
& ~ ( member @ A @ B4 @ C5 )
& ( B5
= ( insert @ A @ A3 @ C5 ) )
& ~ ( member @ A @ A3 @ C5 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_114_insert__commute,axiom,
! [A: $tType,X: A,Y: A,A4: set @ A] :
( ( insert @ A @ X @ ( insert @ A @ Y @ A4 ) )
= ( insert @ A @ Y @ ( insert @ A @ X @ A4 ) ) ) ).
% insert_commute
thf(fact_115_mk__disjoint__insert,axiom,
! [A: $tType,A3: A,A4: set @ A] :
( ( member @ A @ A3 @ A4 )
=> ? [B7: set @ A] :
( ( A4
= ( insert @ A @ A3 @ B7 ) )
& ~ ( member @ A @ A3 @ B7 ) ) ) ).
% mk_disjoint_insert
thf(fact_116_UnE,axiom,
! [A: $tType,C4: A,A4: set @ A,B5: set @ A] :
( ( member @ A @ C4 @ ( sup_sup @ ( set @ A ) @ A4 @ B5 ) )
=> ( ~ ( member @ A @ C4 @ A4 )
=> ( member @ A @ C4 @ B5 ) ) ) ).
% UnE
thf(fact_117_UnI1,axiom,
! [A: $tType,C4: A,A4: set @ A,B5: set @ A] :
( ( member @ A @ C4 @ A4 )
=> ( member @ A @ C4 @ ( sup_sup @ ( set @ A ) @ A4 @ B5 ) ) ) ).
% UnI1
thf(fact_118_UnI2,axiom,
! [A: $tType,C4: A,B5: set @ A,A4: set @ A] :
( ( member @ A @ C4 @ B5 )
=> ( member @ A @ C4 @ ( sup_sup @ ( set @ A ) @ A4 @ B5 ) ) ) ).
% UnI2
thf(fact_119_bex__Un,axiom,
! [A: $tType,A4: set @ A,B5: set @ A,P: A > $o] :
( ( ? [X4: A] :
( ( member @ A @ X4 @ ( sup_sup @ ( set @ A ) @ A4 @ B5 ) )
& ( P @ X4 ) ) )
= ( ? [X4: A] :
( ( member @ A @ X4 @ A4 )
& ( P @ X4 ) )
| ? [X4: A] :
( ( member @ A @ X4 @ B5 )
& ( P @ X4 ) ) ) ) ).
% bex_Un
thf(fact_120_ball__Un,axiom,
! [A: $tType,A4: set @ A,B5: set @ A,P: A > $o] :
( ( ! [X4: A] :
( ( member @ A @ X4 @ ( sup_sup @ ( set @ A ) @ A4 @ B5 ) )
=> ( P @ X4 ) ) )
= ( ! [X4: A] :
( ( member @ A @ X4 @ A4 )
=> ( P @ X4 ) )
& ! [X4: A] :
( ( member @ A @ X4 @ B5 )
=> ( P @ X4 ) ) ) ) ).
% ball_Un
thf(fact_121_Un__assoc,axiom,
! [A: $tType,A4: set @ A,B5: set @ A,C3: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A4 @ B5 ) @ C3 )
= ( sup_sup @ ( set @ A ) @ A4 @ ( sup_sup @ ( set @ A ) @ B5 @ C3 ) ) ) ).
% Un_assoc
thf(fact_122_Un__absorb,axiom,
! [A: $tType,A4: set @ A] :
( ( sup_sup @ ( set @ A ) @ A4 @ A4 )
= A4 ) ).
% Un_absorb
thf(fact_123_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_124_Un__left__absorb,axiom,
! [A: $tType,A4: set @ A,B5: set @ A] :
( ( sup_sup @ ( set @ A ) @ A4 @ ( sup_sup @ ( set @ A ) @ A4 @ B5 ) )
= ( sup_sup @ ( set @ A ) @ A4 @ B5 ) ) ).
% Un_left_absorb
thf(fact_125_Un__left__commute,axiom,
! [A: $tType,A4: set @ A,B5: set @ A,C3: set @ A] :
( ( sup_sup @ ( set @ A ) @ A4 @ ( sup_sup @ ( set @ A ) @ B5 @ C3 ) )
= ( sup_sup @ ( set @ A ) @ B5 @ ( sup_sup @ ( set @ A ) @ A4 @ C3 ) ) ) ).
% Un_left_commute
thf(fact_126_empty__def,axiom,
! [A: $tType] :
( ( bot_bot @ ( set @ A ) )
= ( collect @ A
@ ^ [X4: A] : $false ) ) ).
% empty_def
thf(fact_127_Collect__subset,axiom,
! [A: $tType,A4: set @ A,P: A > $o] :
( ord_less_eq @ ( set @ A )
@ ( collect @ A
@ ^ [X4: A] :
( ( member @ A @ X4 @ A4 )
& ( P @ X4 ) ) )
@ A4 ) ).
% Collect_subset
thf(fact_128_insert__compr,axiom,
! [A: $tType] :
( ( insert @ A )
= ( ^ [A2: A,B6: set @ A] :
( collect @ A
@ ^ [X4: A] :
( ( X4 = A2 )
| ( member @ A @ X4 @ B6 ) ) ) ) ) ).
% insert_compr
thf(fact_129_insert__Collect,axiom,
! [A: $tType,A3: A,P: A > $o] :
( ( insert @ A @ A3 @ ( collect @ A @ P ) )
= ( collect @ A
@ ^ [U: A] :
( ( U != A3 )
=> ( P @ U ) ) ) ) ).
% insert_Collect
thf(fact_130_Un__def,axiom,
! [A: $tType] :
( ( sup_sup @ ( set @ A ) )
= ( ^ [A5: set @ A,B6: set @ A] :
( collect @ A
@ ^ [X4: A] :
( ( member @ A @ X4 @ A5 )
| ( member @ A @ X4 @ B6 ) ) ) ) ) ).
% Un_def
thf(fact_131_Collect__disj__eq,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ( collect @ A
@ ^ [X4: A] :
( ( P @ X4 )
| ( Q @ X4 ) ) )
= ( sup_sup @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) ) ) ).
% Collect_disj_eq
thf(fact_132_singletonD,axiom,
! [A: $tType,B4: A,A3: A] :
( ( member @ A @ B4 @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) )
=> ( B4 = A3 ) ) ).
% singletonD
thf(fact_133_singleton__iff,axiom,
! [A: $tType,B4: A,A3: A] :
( ( member @ A @ B4 @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) )
= ( B4 = A3 ) ) ).
% singleton_iff
thf(fact_134_doubleton__eq__iff,axiom,
! [A: $tType,A3: A,B4: A,C4: A,D3: A] :
( ( ( insert @ A @ A3 @ ( insert @ A @ B4 @ ( bot_bot @ ( set @ A ) ) ) )
= ( insert @ A @ C4 @ ( insert @ A @ D3 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( ( ( A3 = C4 )
& ( B4 = D3 ) )
| ( ( A3 = D3 )
& ( B4 = C4 ) ) ) ) ).
% doubleton_eq_iff
thf(fact_135_insert__not__empty,axiom,
! [A: $tType,A3: A,A4: set @ A] :
( ( insert @ A @ A3 @ A4 )
!= ( bot_bot @ ( set @ A ) ) ) ).
% insert_not_empty
thf(fact_136_singleton__inject,axiom,
! [A: $tType,A3: A,B4: A] :
( ( ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) )
= ( insert @ A @ B4 @ ( bot_bot @ ( set @ A ) ) ) )
=> ( A3 = B4 ) ) ).
% singleton_inject
thf(fact_137_insert__mono,axiom,
! [A: $tType,C3: set @ A,D4: set @ A,A3: A] :
( ( ord_less_eq @ ( set @ A ) @ C3 @ D4 )
=> ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ A3 @ C3 ) @ ( insert @ A @ A3 @ D4 ) ) ) ).
% insert_mono
thf(fact_138_subset__insert,axiom,
! [A: $tType,X: A,A4: set @ A,B5: set @ A] :
( ~ ( member @ A @ X @ A4 )
=> ( ( ord_less_eq @ ( set @ A ) @ A4 @ ( insert @ A @ X @ B5 ) )
= ( ord_less_eq @ ( set @ A ) @ A4 @ B5 ) ) ) ).
% subset_insert
thf(fact_139_subset__insertI,axiom,
! [A: $tType,B5: set @ A,A3: A] : ( ord_less_eq @ ( set @ A ) @ B5 @ ( insert @ A @ A3 @ B5 ) ) ).
% subset_insertI
thf(fact_140_subset__insertI2,axiom,
! [A: $tType,A4: set @ A,B5: set @ A,B4: A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B5 )
=> ( ord_less_eq @ ( set @ A ) @ A4 @ ( insert @ A @ B4 @ B5 ) ) ) ).
% subset_insertI2
thf(fact_141_Un__empty__left,axiom,
! [A: $tType,B5: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ B5 )
= B5 ) ).
% Un_empty_left
thf(fact_142_Un__empty__right,axiom,
! [A: $tType,A4: set @ A] :
( ( sup_sup @ ( set @ A ) @ A4 @ ( bot_bot @ ( set @ A ) ) )
= A4 ) ).
% Un_empty_right
thf(fact_143_Un__mono,axiom,
! [A: $tType,A4: set @ A,C3: set @ A,B5: set @ A,D4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ C3 )
=> ( ( ord_less_eq @ ( set @ A ) @ B5 @ D4 )
=> ( ord_less_eq @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A4 @ B5 ) @ ( sup_sup @ ( set @ A ) @ C3 @ D4 ) ) ) ) ).
% Un_mono
thf(fact_144_Un__least,axiom,
! [A: $tType,A4: set @ A,C3: set @ A,B5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ C3 )
=> ( ( ord_less_eq @ ( set @ A ) @ B5 @ C3 )
=> ( ord_less_eq @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A4 @ B5 ) @ C3 ) ) ) ).
% Un_least
thf(fact_145_Un__upper1,axiom,
! [A: $tType,A4: set @ A,B5: set @ A] : ( ord_less_eq @ ( set @ A ) @ A4 @ ( sup_sup @ ( set @ A ) @ A4 @ B5 ) ) ).
% Un_upper1
thf(fact_146_Un__upper2,axiom,
! [A: $tType,B5: set @ A,A4: set @ A] : ( ord_less_eq @ ( set @ A ) @ B5 @ ( sup_sup @ ( set @ A ) @ A4 @ B5 ) ) ).
% Un_upper2
thf(fact_147_Un__absorb1,axiom,
! [A: $tType,A4: set @ A,B5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B5 )
=> ( ( sup_sup @ ( set @ A ) @ A4 @ B5 )
= B5 ) ) ).
% Un_absorb1
thf(fact_148_Un__absorb2,axiom,
! [A: $tType,B5: set @ A,A4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ B5 @ A4 )
=> ( ( sup_sup @ ( set @ A ) @ A4 @ B5 )
= A4 ) ) ).
% Un_absorb2
thf(fact_149_subset__UnE,axiom,
! [A: $tType,C3: set @ A,A4: set @ A,B5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ C3 @ ( sup_sup @ ( set @ A ) @ A4 @ B5 ) )
=> ~ ! [A6: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A6 @ A4 )
=> ! [B8: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ B8 @ B5 )
=> ( C3
!= ( sup_sup @ ( set @ A ) @ A6 @ B8 ) ) ) ) ) ).
% subset_UnE
thf(fact_150_subset__Un__eq,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A5: set @ A,B6: set @ A] :
( ( sup_sup @ ( set @ A ) @ A5 @ B6 )
= B6 ) ) ) ).
% subset_Un_eq
thf(fact_151_Collect__conv__if,axiom,
! [A: $tType,P: A > $o,A3: A] :
( ( ( P @ A3 )
=> ( ( collect @ A
@ ^ [X4: A] :
( ( X4 = A3 )
& ( P @ X4 ) ) )
= ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) )
& ( ~ ( P @ A3 )
=> ( ( collect @ A
@ ^ [X4: A] :
( ( X4 = A3 )
& ( P @ X4 ) ) )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Collect_conv_if
thf(fact_152_Collect__conv__if2,axiom,
! [A: $tType,P: A > $o,A3: A] :
( ( ( P @ A3 )
=> ( ( collect @ A
@ ^ [X4: A] :
( ( A3 = X4 )
& ( P @ X4 ) ) )
= ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) )
& ( ~ ( P @ A3 )
=> ( ( collect @ A
@ ^ [X4: A] :
( ( A3 = X4 )
& ( P @ X4 ) ) )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Collect_conv_if2
thf(fact_153_insert__def,axiom,
! [A: $tType] :
( ( insert @ A )
= ( ^ [A2: A] :
( sup_sup @ ( set @ A )
@ ( collect @ A
@ ^ [X4: A] : X4 = A2 ) ) ) ) ).
% insert_def
thf(fact_154_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_155_subset__singleton__iff,axiom,
! [A: $tType,X5: set @ A,A3: A] :
( ( ord_less_eq @ ( set @ A ) @ X5 @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) )
= ( ( X5
= ( bot_bot @ ( set @ A ) ) )
| ( X5
= ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% subset_singleton_iff
thf(fact_156_insert__is__Un,axiom,
! [A: $tType] :
( ( insert @ A )
= ( ^ [A2: A] : ( sup_sup @ ( set @ A ) @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% insert_is_Un
thf(fact_157_Un__singleton__iff,axiom,
! [A: $tType,A4: set @ A,B5: set @ A,X: A] :
( ( ( sup_sup @ ( set @ A ) @ A4 @ B5 )
= ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= ( ( ( A4
= ( bot_bot @ ( set @ A ) ) )
& ( B5
= ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) )
| ( ( A4
= ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
& ( B5
= ( bot_bot @ ( set @ A ) ) ) )
| ( ( A4
= ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
& ( B5
= ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_158_singleton__Un__iff,axiom,
! [A: $tType,X: A,A4: set @ A,B5: set @ A] :
( ( ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) )
= ( sup_sup @ ( set @ A ) @ A4 @ B5 ) )
= ( ( ( A4
= ( bot_bot @ ( set @ A ) ) )
& ( B5
= ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) )
| ( ( A4
= ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
& ( B5
= ( bot_bot @ ( set @ A ) ) ) )
| ( ( A4
= ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
& ( B5
= ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_159_sup__bot__left,axiom,
! [A: $tType] :
( ( bounde1808546759up_bot @ A )
=> ! [X: A] :
( ( sup_sup @ A @ ( bot_bot @ A ) @ X )
= X ) ) ).
% sup_bot_left
thf(fact_160_sup__bot__right,axiom,
! [A: $tType] :
( ( bounde1808546759up_bot @ A )
=> ! [X: A] :
( ( sup_sup @ A @ X @ ( bot_bot @ A ) )
= X ) ) ).
% sup_bot_right
thf(fact_161_bot__eq__sup__iff,axiom,
! [A: $tType] :
( ( bounde1808546759up_bot @ A )
=> ! [X: A,Y: A] :
( ( ( bot_bot @ A )
= ( sup_sup @ A @ X @ Y ) )
= ( ( X
= ( bot_bot @ A ) )
& ( Y
= ( bot_bot @ A ) ) ) ) ) ).
% bot_eq_sup_iff
thf(fact_162_sup__eq__bot__iff,axiom,
! [A: $tType] :
( ( bounde1808546759up_bot @ A )
=> ! [X: A,Y: A] :
( ( ( sup_sup @ A @ X @ Y )
= ( bot_bot @ A ) )
= ( ( X
= ( bot_bot @ A ) )
& ( Y
= ( bot_bot @ A ) ) ) ) ) ).
% sup_eq_bot_iff
thf(fact_163_sup__bot_Oeq__neutr__iff,axiom,
! [A: $tType] :
( ( bounde1808546759up_bot @ A )
=> ! [A3: A,B4: A] :
( ( ( sup_sup @ A @ A3 @ B4 )
= ( bot_bot @ A ) )
= ( ( A3
= ( bot_bot @ A ) )
& ( B4
= ( bot_bot @ A ) ) ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_164_sup__bot_Oleft__neutral,axiom,
! [A: $tType] :
( ( bounde1808546759up_bot @ A )
=> ! [A3: A] :
( ( sup_sup @ A @ ( bot_bot @ A ) @ A3 )
= A3 ) ) ).
% sup_bot.left_neutral
thf(fact_165_sup__bot_Oneutr__eq__iff,axiom,
! [A: $tType] :
( ( bounde1808546759up_bot @ A )
=> ! [A3: A,B4: A] :
( ( ( bot_bot @ A )
= ( sup_sup @ A @ A3 @ B4 ) )
= ( ( A3
= ( bot_bot @ A ) )
& ( B4
= ( bot_bot @ A ) ) ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_166_sup_Oright__idem,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A3: A,B4: A] :
( ( sup_sup @ A @ ( sup_sup @ A @ A3 @ B4 ) @ B4 )
= ( sup_sup @ A @ A3 @ B4 ) ) ) ).
% sup.right_idem
thf(fact_167_sup__left__idem,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X: A,Y: A] :
( ( sup_sup @ A @ X @ ( sup_sup @ A @ X @ Y ) )
= ( sup_sup @ A @ X @ Y ) ) ) ).
% sup_left_idem
thf(fact_168_sup_Oleft__idem,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A3: A,B4: A] :
( ( sup_sup @ A @ A3 @ ( sup_sup @ A @ A3 @ B4 ) )
= ( sup_sup @ A @ A3 @ B4 ) ) ) ).
% sup.left_idem
thf(fact_169_sup__idem,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X: A] :
( ( sup_sup @ A @ X @ X )
= X ) ) ).
% sup_idem
thf(fact_170_sup_Oidem,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A3: A] :
( ( sup_sup @ A @ A3 @ A3 )
= A3 ) ) ).
% sup.idem
thf(fact_171_sup__apply,axiom,
! [B2: $tType,A: $tType] :
( ( semilattice_sup @ B2 )
=> ( ( sup_sup @ ( A > B2 ) )
= ( ^ [F3: A > B2,G2: A > B2,X4: A] : ( sup_sup @ B2 @ ( F3 @ X4 ) @ ( G2 @ X4 ) ) ) ) ) ).
% sup_apply
thf(fact_172_sup_Obounded__iff,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [B4: A,C4: A,A3: A] :
( ( ord_less_eq @ A @ ( sup_sup @ A @ B4 @ C4 ) @ A3 )
= ( ( ord_less_eq @ A @ B4 @ A3 )
& ( ord_less_eq @ A @ C4 @ A3 ) ) ) ) ).
% sup.bounded_iff
thf(fact_173_le__sup__iff,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X: A,Y: A,Z4: A] :
( ( ord_less_eq @ A @ ( sup_sup @ A @ X @ Y ) @ Z4 )
= ( ( ord_less_eq @ A @ X @ Z4 )
& ( ord_less_eq @ A @ Y @ Z4 ) ) ) ) ).
% le_sup_iff
thf(fact_174_sup__bot_Oright__neutral,axiom,
! [A: $tType] :
( ( bounde1808546759up_bot @ A )
=> ! [A3: A] :
( ( sup_sup @ A @ A3 @ ( bot_bot @ A ) )
= A3 ) ) ).
% sup_bot.right_neutral
thf(fact_175_less__set__def,axiom,
! [A: $tType] :
( ( ord_less @ ( set @ A ) )
= ( ^ [A5: set @ A,B6: set @ A] :
( ord_less @ ( A > $o )
@ ^ [X4: A] : ( member @ A @ X4 @ A5 )
@ ^ [X4: A] : ( member @ A @ X4 @ B6 ) ) ) ) ).
% less_set_def
thf(fact_176_psubset__trans,axiom,
! [A: $tType,A4: set @ A,B5: set @ A,C3: set @ A] :
( ( ord_less @ ( set @ A ) @ A4 @ B5 )
=> ( ( ord_less @ ( set @ A ) @ B5 @ C3 )
=> ( ord_less @ ( set @ A ) @ A4 @ C3 ) ) ) ).
% psubset_trans
thf(fact_177_psubsetD,axiom,
! [A: $tType,A4: set @ A,B5: set @ A,C4: A] :
( ( ord_less @ ( set @ A ) @ A4 @ B5 )
=> ( ( member @ A @ C4 @ A4 )
=> ( member @ A @ C4 @ B5 ) ) ) ).
% psubsetD
thf(fact_178_bot__set__def,axiom,
! [A: $tType] :
( ( bot_bot @ ( set @ A ) )
= ( collect @ A @ ( bot_bot @ ( A > $o ) ) ) ) ).
% bot_set_def
thf(fact_179_sup__set__def,axiom,
! [A: $tType] :
( ( sup_sup @ ( set @ A ) )
= ( ^ [A5: set @ A,B6: set @ A] :
( collect @ A
@ ( sup_sup @ ( A > $o )
@ ^ [X4: A] : ( member @ A @ X4 @ A5 )
@ ^ [X4: A] : ( member @ A @ X4 @ B6 ) ) ) ) ) ).
% sup_set_def
thf(fact_180_sup__left__commute,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X: A,Y: A,Z4: A] :
( ( sup_sup @ A @ X @ ( sup_sup @ A @ Y @ Z4 ) )
= ( sup_sup @ A @ Y @ ( sup_sup @ A @ X @ Z4 ) ) ) ) ).
% sup_left_commute
thf(fact_181_sup_Oleft__commute,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [B4: A,A3: A,C4: A] :
( ( sup_sup @ A @ B4 @ ( sup_sup @ A @ A3 @ C4 ) )
= ( sup_sup @ A @ A3 @ ( sup_sup @ A @ B4 @ C4 ) ) ) ) ).
% sup.left_commute
thf(fact_182_sup__commute,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ( ( sup_sup @ A )
= ( ^ [X4: A,Y4: A] : ( sup_sup @ A @ Y4 @ X4 ) ) ) ) ).
% sup_commute
thf(fact_183_sup_Ocommute,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ( ( sup_sup @ A )
= ( ^ [A2: A,B: A] : ( sup_sup @ A @ B @ A2 ) ) ) ) ).
% sup.commute
thf(fact_184_sup__assoc,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X: A,Y: A,Z4: A] :
( ( sup_sup @ A @ ( sup_sup @ A @ X @ Y ) @ Z4 )
= ( sup_sup @ A @ X @ ( sup_sup @ A @ Y @ Z4 ) ) ) ) ).
% sup_assoc
thf(fact_185_sup_Oassoc,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A3: A,B4: A,C4: A] :
( ( sup_sup @ A @ ( sup_sup @ A @ A3 @ B4 ) @ C4 )
= ( sup_sup @ A @ A3 @ ( sup_sup @ A @ B4 @ C4 ) ) ) ) ).
% sup.assoc
thf(fact_186_boolean__algebra__cancel_Osup2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [B5: A,K: A,B4: A,A3: A] :
( ( B5
= ( sup_sup @ A @ K @ B4 ) )
=> ( ( sup_sup @ A @ A3 @ B5 )
= ( sup_sup @ A @ K @ ( sup_sup @ A @ A3 @ B4 ) ) ) ) ) ).
% boolean_algebra_cancel.sup2
thf(fact_187_boolean__algebra__cancel_Osup1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A4: A,K: A,A3: A,B4: A] :
( ( A4
= ( sup_sup @ A @ K @ A3 ) )
=> ( ( sup_sup @ A @ A4 @ B4 )
= ( sup_sup @ A @ K @ ( sup_sup @ A @ A3 @ B4 ) ) ) ) ) ).
% boolean_algebra_cancel.sup1
thf(fact_188_sup__fun__def,axiom,
! [B2: $tType,A: $tType] :
( ( semilattice_sup @ B2 )
=> ( ( sup_sup @ ( A > B2 ) )
= ( ^ [F3: A > B2,G2: A > B2,X4: A] : ( sup_sup @ B2 @ ( F3 @ X4 ) @ ( G2 @ X4 ) ) ) ) ) ).
% sup_fun_def
thf(fact_189_inf__sup__aci_I5_J,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ( ( sup_sup @ A )
= ( ^ [X4: A,Y4: A] : ( sup_sup @ A @ Y4 @ X4 ) ) ) ) ).
% inf_sup_aci(5)
thf(fact_190_inf__sup__aci_I6_J,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [X: A,Y: A,Z4: A] :
( ( sup_sup @ A @ ( sup_sup @ A @ X @ Y ) @ Z4 )
= ( sup_sup @ A @ X @ ( sup_sup @ A @ Y @ Z4 ) ) ) ) ).
% inf_sup_aci(6)
thf(fact_191_inf__sup__aci_I7_J,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [X: A,Y: A,Z4: A] :
( ( sup_sup @ A @ X @ ( sup_sup @ A @ Y @ Z4 ) )
= ( sup_sup @ A @ Y @ ( sup_sup @ A @ X @ Z4 ) ) ) ) ).
% inf_sup_aci(7)
thf(fact_192_inf__sup__aci_I8_J,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [X: A,Y: A] :
( ( sup_sup @ A @ X @ ( sup_sup @ A @ X @ Y ) )
= ( sup_sup @ A @ X @ Y ) ) ) ).
% inf_sup_aci(8)
thf(fact_193_sup_OcoboundedI2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [C4: A,B4: A,A3: A] :
( ( ord_less_eq @ A @ C4 @ B4 )
=> ( ord_less_eq @ A @ C4 @ ( sup_sup @ A @ A3 @ B4 ) ) ) ) ).
% sup.coboundedI2
thf(fact_194_sup_OcoboundedI1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [C4: A,A3: A,B4: A] :
( ( ord_less_eq @ A @ C4 @ A3 )
=> ( ord_less_eq @ A @ C4 @ ( sup_sup @ A @ A3 @ B4 ) ) ) ) ).
% sup.coboundedI1
thf(fact_195_sup_Oabsorb__iff2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A2: A,B: A] :
( ( sup_sup @ A @ A2 @ B )
= B ) ) ) ) ).
% sup.absorb_iff2
thf(fact_196_sup_Oabsorb__iff1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [B: A,A2: A] :
( ( sup_sup @ A @ A2 @ B )
= A2 ) ) ) ) ).
% sup.absorb_iff1
thf(fact_197_sup_Ocobounded2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [B4: A,A3: A] : ( ord_less_eq @ A @ B4 @ ( sup_sup @ A @ A3 @ B4 ) ) ) ).
% sup.cobounded2
thf(fact_198_sup_Ocobounded1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A3: A,B4: A] : ( ord_less_eq @ A @ A3 @ ( sup_sup @ A @ A3 @ B4 ) ) ) ).
% sup.cobounded1
thf(fact_199_sup_Oorder__iff,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [B: A,A2: A] :
( A2
= ( sup_sup @ A @ A2 @ B ) ) ) ) ) ).
% sup.order_iff
thf(fact_200_sup_OboundedI,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [B4: A,A3: A,C4: A] :
( ( ord_less_eq @ A @ B4 @ A3 )
=> ( ( ord_less_eq @ A @ C4 @ A3 )
=> ( ord_less_eq @ A @ ( sup_sup @ A @ B4 @ C4 ) @ A3 ) ) ) ) ).
% sup.boundedI
thf(fact_201_sup_OboundedE,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [B4: A,C4: A,A3: A] :
( ( ord_less_eq @ A @ ( sup_sup @ A @ B4 @ C4 ) @ A3 )
=> ~ ( ( ord_less_eq @ A @ B4 @ A3 )
=> ~ ( ord_less_eq @ A @ C4 @ A3 ) ) ) ) ).
% sup.boundedE
thf(fact_202_sup__absorb2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( sup_sup @ A @ X @ Y )
= Y ) ) ) ).
% sup_absorb2
thf(fact_203_sup__absorb1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [Y: A,X: A] :
( ( ord_less_eq @ A @ Y @ X )
=> ( ( sup_sup @ A @ X @ Y )
= X ) ) ) ).
% sup_absorb1
thf(fact_204_sup_Oabsorb2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A3: A,B4: A] :
( ( ord_less_eq @ A @ A3 @ B4 )
=> ( ( sup_sup @ A @ A3 @ B4 )
= B4 ) ) ) ).
% sup.absorb2
thf(fact_205_sup_Oabsorb1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [B4: A,A3: A] :
( ( ord_less_eq @ A @ B4 @ A3 )
=> ( ( sup_sup @ A @ A3 @ B4 )
= A3 ) ) ) ).
% sup.absorb1
thf(fact_206_sup__unique,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [F2: A > A > A,X: A,Y: A] :
( ! [X3: A,Y3: A] : ( ord_less_eq @ A @ X3 @ ( F2 @ X3 @ Y3 ) )
=> ( ! [X3: A,Y3: A] : ( ord_less_eq @ A @ Y3 @ ( F2 @ X3 @ Y3 ) )
=> ( ! [X3: A,Y3: A,Z: A] :
( ( ord_less_eq @ A @ Y3 @ X3 )
=> ( ( ord_less_eq @ A @ Z @ X3 )
=> ( ord_less_eq @ A @ ( F2 @ Y3 @ Z ) @ X3 ) ) )
=> ( ( sup_sup @ A @ X @ Y )
= ( F2 @ X @ Y ) ) ) ) ) ) ).
% sup_unique
thf(fact_207_sup_OorderI,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A3: A,B4: A] :
( ( A3
= ( sup_sup @ A @ A3 @ B4 ) )
=> ( ord_less_eq @ A @ B4 @ A3 ) ) ) ).
% sup.orderI
thf(fact_208_sup_OorderE,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [B4: A,A3: A] :
( ( ord_less_eq @ A @ B4 @ A3 )
=> ( A3
= ( sup_sup @ A @ A3 @ B4 ) ) ) ) ).
% sup.orderE
thf(fact_209_le__iff__sup,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [X4: A,Y4: A] :
( ( sup_sup @ A @ X4 @ Y4 )
= Y4 ) ) ) ) ).
% le_iff_sup
thf(fact_210_sup__least,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [Y: A,X: A,Z4: A] :
( ( ord_less_eq @ A @ Y @ X )
=> ( ( ord_less_eq @ A @ Z4 @ X )
=> ( ord_less_eq @ A @ ( sup_sup @ A @ Y @ Z4 ) @ X ) ) ) ) ).
% sup_least
thf(fact_211_sup__mono,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A3: A,C4: A,B4: A,D3: A] :
( ( ord_less_eq @ A @ A3 @ C4 )
=> ( ( ord_less_eq @ A @ B4 @ D3 )
=> ( ord_less_eq @ A @ ( sup_sup @ A @ A3 @ B4 ) @ ( sup_sup @ A @ C4 @ D3 ) ) ) ) ) ).
% sup_mono
thf(fact_212_sup_Omono,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [C4: A,A3: A,D3: A,B4: A] :
( ( ord_less_eq @ A @ C4 @ A3 )
=> ( ( ord_less_eq @ A @ D3 @ B4 )
=> ( ord_less_eq @ A @ ( sup_sup @ A @ C4 @ D3 ) @ ( sup_sup @ A @ A3 @ B4 ) ) ) ) ) ).
% sup.mono
thf(fact_213_le__supI2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X: A,B4: A,A3: A] :
( ( ord_less_eq @ A @ X @ B4 )
=> ( ord_less_eq @ A @ X @ ( sup_sup @ A @ A3 @ B4 ) ) ) ) ).
% le_supI2
thf(fact_214_le__supI1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X: A,A3: A,B4: A] :
( ( ord_less_eq @ A @ X @ A3 )
=> ( ord_less_eq @ A @ X @ ( sup_sup @ A @ A3 @ B4 ) ) ) ) ).
% le_supI1
thf(fact_215_sup__ge2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [Y: A,X: A] : ( ord_less_eq @ A @ Y @ ( sup_sup @ A @ X @ Y ) ) ) ).
% sup_ge2
thf(fact_216_sup__ge1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X: A,Y: A] : ( ord_less_eq @ A @ X @ ( sup_sup @ A @ X @ Y ) ) ) ).
% sup_ge1
thf(fact_217_le__supI,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A3: A,X: A,B4: A] :
( ( ord_less_eq @ A @ A3 @ X )
=> ( ( ord_less_eq @ A @ B4 @ X )
=> ( ord_less_eq @ A @ ( sup_sup @ A @ A3 @ B4 ) @ X ) ) ) ) ).
% le_supI
thf(fact_218_le__supE,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A3: A,B4: A,X: A] :
( ( ord_less_eq @ A @ ( sup_sup @ A @ A3 @ B4 ) @ X )
=> ~ ( ( ord_less_eq @ A @ A3 @ X )
=> ~ ( ord_less_eq @ A @ B4 @ X ) ) ) ) ).
% le_supE
thf(fact_219_inf__sup__ord_I3_J,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [X: A,Y: A] : ( ord_less_eq @ A @ X @ ( sup_sup @ A @ X @ Y ) ) ) ).
% inf_sup_ord(3)
thf(fact_220_inf__sup__ord_I4_J,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [Y: A,X: A] : ( ord_less_eq @ A @ Y @ ( sup_sup @ A @ X @ Y ) ) ) ).
% inf_sup_ord(4)
thf(fact_221_less__supI1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X: A,A3: A,B4: A] :
( ( ord_less @ A @ X @ A3 )
=> ( ord_less @ A @ X @ ( sup_sup @ A @ A3 @ B4 ) ) ) ) ).
% less_supI1
thf(fact_222_less__supI2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X: A,B4: A,A3: A] :
( ( ord_less @ A @ X @ B4 )
=> ( ord_less @ A @ X @ ( sup_sup @ A @ A3 @ B4 ) ) ) ) ).
% less_supI2
thf(fact_223_sup_Ostrict__boundedE,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [B4: A,C4: A,A3: A] :
( ( ord_less @ A @ ( sup_sup @ A @ B4 @ C4 ) @ A3 )
=> ~ ( ( ord_less @ A @ B4 @ A3 )
=> ~ ( ord_less @ A @ C4 @ A3 ) ) ) ) ).
% sup.strict_boundedE
thf(fact_224_sup_Ostrict__order__iff,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ( ( ord_less @ A )
= ( ^ [B: A,A2: A] :
( ( A2
= ( sup_sup @ A @ A2 @ B ) )
& ( A2 != B ) ) ) ) ) ).
% sup.strict_order_iff
thf(fact_225_sup_Ostrict__coboundedI1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [C4: A,A3: A,B4: A] :
( ( ord_less @ A @ C4 @ A3 )
=> ( ord_less @ A @ C4 @ ( sup_sup @ A @ A3 @ B4 ) ) ) ) ).
% sup.strict_coboundedI1
thf(fact_226_sup_Ostrict__coboundedI2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [C4: A,B4: A,A3: A] :
( ( ord_less @ A @ C4 @ B4 )
=> ( ord_less @ A @ C4 @ ( sup_sup @ A @ A3 @ B4 ) ) ) ) ).
% sup.strict_coboundedI2
thf(fact_227_predicate1I,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ! [X3: A] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( ord_less_eq @ ( A > $o ) @ P @ Q ) ) ).
% predicate1I
thf(fact_228_pred__subset__eq,axiom,
! [A: $tType,R: set @ A,S: set @ A] :
( ( ord_less_eq @ ( A > $o )
@ ^ [X4: A] : ( member @ A @ X4 @ R )
@ ^ [X4: A] : ( member @ A @ X4 @ S ) )
= ( ord_less_eq @ ( set @ A ) @ R @ S ) ) ).
% pred_subset_eq
thf(fact_229_the__elem__eq,axiom,
! [A: $tType,X: A] :
( ( the_elem @ A @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= X ) ).
% the_elem_eq
thf(fact_230_order__refl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A] : ( ord_less_eq @ A @ X @ X ) ) ).
% order_refl
thf(fact_231_sup1CI,axiom,
! [A: $tType,B5: A > $o,X: A,A4: A > $o] :
( ( ~ ( B5 @ X )
=> ( A4 @ X ) )
=> ( sup_sup @ ( A > $o ) @ A4 @ B5 @ X ) ) ).
% sup1CI
thf(fact_232_bot__apply,axiom,
! [C: $tType,D: $tType] :
( ( bot @ C )
=> ( ( bot_bot @ ( D > C ) )
= ( ^ [X4: D] : ( bot_bot @ C ) ) ) ) ).
% bot_apply
thf(fact_233_sup1I2,axiom,
! [A: $tType,B5: A > $o,X: A,A4: A > $o] :
( ( B5 @ X )
=> ( sup_sup @ ( A > $o ) @ A4 @ B5 @ X ) ) ).
% sup1I2
thf(fact_234_sup1I1,axiom,
! [A: $tType,A4: A > $o,X: A,B5: A > $o] :
( ( A4 @ X )
=> ( sup_sup @ ( A > $o ) @ A4 @ B5 @ X ) ) ).
% sup1I1
thf(fact_235_sup1E,axiom,
! [A: $tType,A4: A > $o,B5: A > $o,X: A] :
( ( sup_sup @ ( A > $o ) @ A4 @ B5 @ X )
=> ( ~ ( A4 @ X )
=> ( B5 @ X ) ) ) ).
% sup1E
thf(fact_236_order__subst1,axiom,
! [A: $tType,B2: $tType] :
( ( ( order @ B2 )
& ( order @ A ) )
=> ! [A3: A,F2: B2 > A,B4: B2,C4: B2] :
( ( ord_less_eq @ A @ A3 @ ( F2 @ B4 ) )
=> ( ( ord_less_eq @ B2 @ B4 @ C4 )
=> ( ! [X3: B2,Y3: B2] :
( ( ord_less_eq @ B2 @ X3 @ Y3 )
=> ( ord_less_eq @ A @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq @ A @ A3 @ ( F2 @ C4 ) ) ) ) ) ) ).
% order_subst1
thf(fact_237_order__subst2,axiom,
! [A: $tType,C: $tType] :
( ( ( order @ C )
& ( order @ A ) )
=> ! [A3: A,B4: A,F2: A > C,C4: C] :
( ( ord_less_eq @ A @ A3 @ B4 )
=> ( ( ord_less_eq @ C @ ( F2 @ B4 ) @ C4 )
=> ( ! [X3: A,Y3: A] :
( ( ord_less_eq @ A @ X3 @ Y3 )
=> ( ord_less_eq @ C @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq @ C @ ( F2 @ A3 ) @ C4 ) ) ) ) ) ).
% order_subst2
thf(fact_238_ord__eq__le__subst,axiom,
! [A: $tType,B2: $tType] :
( ( ( ord @ B2 )
& ( ord @ A ) )
=> ! [A3: A,F2: B2 > A,B4: B2,C4: B2] :
( ( A3
= ( F2 @ B4 ) )
=> ( ( ord_less_eq @ B2 @ B4 @ C4 )
=> ( ! [X3: B2,Y3: B2] :
( ( ord_less_eq @ B2 @ X3 @ Y3 )
=> ( ord_less_eq @ A @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq @ A @ A3 @ ( F2 @ C4 ) ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_239_ord__le__eq__subst,axiom,
! [A: $tType,B2: $tType] :
( ( ( ord @ B2 )
& ( ord @ A ) )
=> ! [A3: A,B4: A,F2: A > B2,C4: B2] :
( ( ord_less_eq @ A @ A3 @ B4 )
=> ( ( ( F2 @ B4 )
= C4 )
=> ( ! [X3: A,Y3: A] :
( ( ord_less_eq @ A @ X3 @ Y3 )
=> ( ord_less_eq @ B2 @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq @ B2 @ ( F2 @ A3 ) @ C4 ) ) ) ) ) ).
% ord_le_eq_subst
thf(fact_240_eq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y2: A,Z3: A] : Y2 = Z3 )
= ( ^ [X4: A,Y4: A] :
( ( ord_less_eq @ A @ X4 @ Y4 )
& ( ord_less_eq @ A @ Y4 @ X4 ) ) ) ) ) ).
% eq_iff
thf(fact_241_antisym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( ord_less_eq @ A @ Y @ X )
=> ( X = Y ) ) ) ) ).
% antisym
thf(fact_242_linear,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
| ( ord_less_eq @ A @ Y @ X ) ) ) ).
% linear
thf(fact_243_eq__refl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A,Y: A] :
( ( X = Y )
=> ( ord_less_eq @ A @ X @ Y ) ) ) ).
% eq_refl
thf(fact_244_le__cases,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A] :
( ~ ( ord_less_eq @ A @ X @ Y )
=> ( ord_less_eq @ A @ Y @ X ) ) ) ).
% le_cases
thf(fact_245_order_Otrans,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: A,B4: A,C4: A] :
( ( ord_less_eq @ A @ A3 @ B4 )
=> ( ( ord_less_eq @ A @ B4 @ C4 )
=> ( ord_less_eq @ A @ A3 @ C4 ) ) ) ) ).
% order.trans
thf(fact_246_le__cases3,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A,Z4: A] :
( ( ( ord_less_eq @ A @ X @ Y )
=> ~ ( ord_less_eq @ A @ Y @ Z4 ) )
=> ( ( ( ord_less_eq @ A @ Y @ X )
=> ~ ( ord_less_eq @ A @ X @ Z4 ) )
=> ( ( ( ord_less_eq @ A @ X @ Z4 )
=> ~ ( ord_less_eq @ A @ Z4 @ Y ) )
=> ( ( ( ord_less_eq @ A @ Z4 @ Y )
=> ~ ( ord_less_eq @ A @ Y @ X ) )
=> ( ( ( ord_less_eq @ A @ Y @ Z4 )
=> ~ ( ord_less_eq @ A @ Z4 @ X ) )
=> ~ ( ( ord_less_eq @ A @ Z4 @ X )
=> ~ ( ord_less_eq @ A @ X @ Y ) ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_247_antisym__conv,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [Y: A,X: A] :
( ( ord_less_eq @ A @ Y @ X )
=> ( ( ord_less_eq @ A @ X @ Y )
= ( X = Y ) ) ) ) ).
% antisym_conv
thf(fact_248_order__class_Oorder_Oeq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y2: A,Z3: A] : Y2 = Z3 )
= ( ^ [A2: A,B: A] :
( ( ord_less_eq @ A @ A2 @ B )
& ( ord_less_eq @ A @ B @ A2 ) ) ) ) ) ).
% order_class.order.eq_iff
thf(fact_249_ord__eq__le__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A3: A,B4: A,C4: A] :
( ( A3 = B4 )
=> ( ( ord_less_eq @ A @ B4 @ C4 )
=> ( ord_less_eq @ A @ A3 @ C4 ) ) ) ) ).
% ord_eq_le_trans
thf(fact_250_ord__le__eq__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A3: A,B4: A,C4: A] :
( ( ord_less_eq @ A @ A3 @ B4 )
=> ( ( B4 = C4 )
=> ( ord_less_eq @ A @ A3 @ C4 ) ) ) ) ).
% ord_le_eq_trans
thf(fact_251_order__class_Oorder_Oantisym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: A,B4: A] :
( ( ord_less_eq @ A @ A3 @ B4 )
=> ( ( ord_less_eq @ A @ B4 @ A3 )
=> ( A3 = B4 ) ) ) ) ).
% order_class.order.antisym
thf(fact_252_order__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A,Y: A,Z4: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( ord_less_eq @ A @ Y @ Z4 )
=> ( ord_less_eq @ A @ X @ Z4 ) ) ) ) ).
% order_trans
thf(fact_253_dual__order_Orefl,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: A] : ( ord_less_eq @ A @ A3 @ A3 ) ) ).
% dual_order.refl
thf(fact_254_linorder__wlog,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > A > $o,A3: A,B4: A] :
( ! [A7: A,B3: A] :
( ( ord_less_eq @ A @ A7 @ B3 )
=> ( P @ A7 @ B3 ) )
=> ( ! [A7: A,B3: A] :
( ( P @ B3 @ A7 )
=> ( P @ A7 @ B3 ) )
=> ( P @ A3 @ B4 ) ) ) ) ).
% linorder_wlog
% Type constructors (33)
thf(tcon_HOL_Obool___Lattices_Obounded__lattice,axiom,
bounded_lattice @ $o ).
thf(tcon_Set_Oset___Lattices_Obounded__lattice_1,axiom,
! [A8: $tType] : ( bounded_lattice @ ( set @ A8 ) ) ).
thf(tcon_fun___Lattices_Obounded__lattice_2,axiom,
! [A8: $tType,A9: $tType] :
( ( bounded_lattice @ A9 )
=> ( bounded_lattice @ ( A8 > A9 ) ) ) ).
thf(tcon_fun___Lattices_Obounded__semilattice__sup__bot,axiom,
! [A8: $tType,A9: $tType] :
( ( bounded_lattice @ A9 )
=> ( bounde1808546759up_bot @ ( A8 > A9 ) ) ) ).
thf(tcon_fun___Lattices_Osemilattice__sup,axiom,
! [A8: $tType,A9: $tType] :
( ( semilattice_sup @ A9 )
=> ( semilattice_sup @ ( A8 > A9 ) ) ) ).
thf(tcon_fun___Orderings_Opreorder,axiom,
! [A8: $tType,A9: $tType] :
( ( preorder @ A9 )
=> ( preorder @ ( A8 > A9 ) ) ) ).
thf(tcon_fun___Lattices_Olattice,axiom,
! [A8: $tType,A9: $tType] :
( ( lattice @ A9 )
=> ( lattice @ ( A8 > A9 ) ) ) ).
thf(tcon_fun___Orderings_Oorder,axiom,
! [A8: $tType,A9: $tType] :
( ( order @ A9 )
=> ( order @ ( A8 > A9 ) ) ) ).
thf(tcon_fun___Orderings_Oord,axiom,
! [A8: $tType,A9: $tType] :
( ( ord @ A9 )
=> ( ord @ ( A8 > A9 ) ) ) ).
thf(tcon_fun___Orderings_Obot,axiom,
! [A8: $tType,A9: $tType] :
( ( bot @ A9 )
=> ( bot @ ( A8 > A9 ) ) ) ).
thf(tcon_Int_Oint___Conditionally__Complete__Lattices_Oconditionally__complete__linorder,axiom,
condit1037483654norder @ int ).
thf(tcon_Int_Oint___Lattices_Osemilattice__sup_3,axiom,
semilattice_sup @ int ).
thf(tcon_Int_Oint___Rings_Olinordered__idom,axiom,
linordered_idom @ int ).
thf(tcon_Int_Oint___Orderings_Opreorder_4,axiom,
preorder @ int ).
thf(tcon_Int_Oint___Orderings_Olinorder,axiom,
linorder @ int ).
thf(tcon_Int_Oint___Lattices_Olattice_5,axiom,
lattice @ int ).
thf(tcon_Int_Oint___Orderings_Oorder_6,axiom,
order @ int ).
thf(tcon_Int_Oint___Orderings_Oord_7,axiom,
ord @ int ).
thf(tcon_Set_Oset___Lattices_Obounded__semilattice__sup__bot_8,axiom,
! [A8: $tType] : ( bounde1808546759up_bot @ ( set @ A8 ) ) ).
thf(tcon_Set_Oset___Lattices_Osemilattice__sup_9,axiom,
! [A8: $tType] : ( semilattice_sup @ ( set @ A8 ) ) ).
thf(tcon_Set_Oset___Orderings_Opreorder_10,axiom,
! [A8: $tType] : ( preorder @ ( set @ A8 ) ) ).
thf(tcon_Set_Oset___Lattices_Olattice_11,axiom,
! [A8: $tType] : ( lattice @ ( set @ A8 ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder_12,axiom,
! [A8: $tType] : ( order @ ( set @ A8 ) ) ).
thf(tcon_Set_Oset___Orderings_Oord_13,axiom,
! [A8: $tType] : ( ord @ ( set @ A8 ) ) ).
thf(tcon_Set_Oset___Orderings_Obot_14,axiom,
! [A8: $tType] : ( bot @ ( set @ A8 ) ) ).
thf(tcon_HOL_Obool___Lattices_Obounded__semilattice__sup__bot_15,axiom,
bounde1808546759up_bot @ $o ).
thf(tcon_HOL_Obool___Lattices_Osemilattice__sup_16,axiom,
semilattice_sup @ $o ).
thf(tcon_HOL_Obool___Orderings_Opreorder_17,axiom,
preorder @ $o ).
thf(tcon_HOL_Obool___Orderings_Olinorder_18,axiom,
linorder @ $o ).
thf(tcon_HOL_Obool___Lattices_Olattice_19,axiom,
lattice @ $o ).
thf(tcon_HOL_Obool___Orderings_Oorder_20,axiom,
order @ $o ).
thf(tcon_HOL_Obool___Orderings_Oord_21,axiom,
ord @ $o ).
thf(tcon_HOL_Obool___Orderings_Obot_22,axiom,
bot @ $o ).
% Conjectures (1)
thf(conj_0,conjecture,
ord_less @ int @ ( h @ l ) @ ( h @ ( binary576689334lle_rm @ a @ h @ ( binary210054475elle_T @ a @ t1 @ x @ t2 ) ) ) ).
%------------------------------------------------------------------------------