TPTP Problem File: ITP030^2.p
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%------------------------------------------------------------------------------
% File : ITP030^2 : TPTP v8.2.0. Released v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer BinaryTree problem prob_220__3252356_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_220__3252356_1 [Des21]
% Status : ContradictoryAxioms
% Rating : 0.00 v8.1.0, 0.25 v7.5.0
% Syntax : Number of formulae : 362 ( 161 unt; 60 typ; 0 def)
% Number of atoms : 735 ( 355 equ; 0 cnn)
% Maximal formula atoms : 7 ( 2 avg)
% Number of connectives : 3204 ( 106 ~; 20 |; 71 &;2715 @)
% ( 0 <=>; 292 =>; 0 <=; 0 <~>)
% Maximal formula depth : 18 ( 6 avg)
% Number of types : 3 ( 2 usr)
% Number of type conns : 185 ( 185 >; 0 *; 0 +; 0 <<)
% Number of symbols : 61 ( 58 usr; 4 con; 0-5 aty)
% Number of variables : 1066 ( 103 ^; 880 !; 28 ?;1066 :)
% ( 55 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Sledgehammer 2021-02-23 16:15:42.144
%------------------------------------------------------------------------------
% Could-be-implicit typings (5)
thf(ty_t_BinaryTree__Mirabelle__pchhvghoao_OTree,type,
binary1291135688e_Tree: $tType > $tType ).
thf(ty_t_Option_Ooption,type,
option: $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 (55)
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_Otop,type,
top:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Oorder,type,
order:
!>[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_Complete__Lattices_OInf,type,
complete_Inf:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Odense__order,type,
dense_order:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Olinordered__idom,type,
linordered_idom:
!>[A: $tType] : $o ).
thf(sy_cl_Fields_Olinordered__field,type,
linordered_field:
!>[A: $tType] : $o ).
thf(sy_cl_Lattices_Osemilattice__sup,type,
semilattice_sup:
!>[A: $tType] : $o ).
thf(sy_cl_Complete__Lattices_Ocomplete__lattice,type,
comple187826305attice:
!>[A: $tType] : $o ).
thf(sy_cl_Complete__Lattices_Ocomplete__linorder,type,
comple1035589618norder:
!>[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_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_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_BinaryTree__Mirabelle__pchhvghoao_Otlookup,type,
binary1265063667lookup:
!>[A: $tType] : ( ( A > int ) > int > ( binary1291135688e_Tree @ A ) > ( option @ A ) ) ).
thf(sy_c_Complete__Lattices_OInf__class_OInf,type,
complete_Inf_Inf:
!>[A: $tType] : ( ( set @ A ) > A ) ).
thf(sy_c_HOL_OThe,type,
the:
!>[A: $tType] : ( ( A > $o ) > A ) ).
thf(sy_c_HOL_Oundefined,type,
undefined:
!>[A: $tType] : 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_Option_Ooption_ONone,type,
none:
!>[A: $tType] : ( option @ A ) ).
thf(sy_c_Option_Ooption_OSome,type,
some:
!>[A: $tType] : ( A > ( option @ A ) ) ).
thf(sy_c_Option_Ooption_Ocase__option,type,
case_option:
!>[B: $tType,A: $tType] : ( B > ( A > B ) > ( option @ A ) > B ) ).
thf(sy_c_Option_Ooption_Oset__option,type,
set_option:
!>[A: $tType] : ( ( option @ A ) > ( set @ A ) ) ).
thf(sy_c_Option_Ooption_Othe,type,
the2:
!>[A: $tType] : ( ( option @ A ) > A ) ).
thf(sy_c_Option_Othese,type,
these:
!>[A: $tType] : ( ( set @ ( option @ A ) ) > ( set @ 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_Otop__class_Otop,type,
top_top:
!>[A: $tType] : A ).
thf(sy_c_Set_OCollect,type,
collect:
!>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
thf(sy_c_Set_Obind,type,
bind:
!>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > ( set @ B ) ) > ( set @ B ) ) ).
thf(sy_c_Set_Oimage,type,
image:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ A ) > ( set @ B ) ) ).
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_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_t,type,
t: binary1291135688e_Tree @ a ).
thf(sy_v_x,type,
x: a ).
% Relevant facts (255)
thf(fact_0_h,axiom,
member @ a @ x @ ( binary1653327646_setOf @ a @ t ) ).
% h
thf(fact_1_sorted__distinct,axiom,
! [A: $tType,H: A > int,A2: A,B2: A,T: binary1291135688e_Tree @ A] : ( binary231205461t_pred @ A @ H @ A2 @ B2 @ T ) ).
% sorted_distinct
thf(fact_2_s,axiom,
binary1610619414edTree @ a @ h @ t ).
% s
thf(fact_3_res,axiom,
~ ( binary827270440e_memb @ a @ h @ x @ t ) ).
% res
thf(fact_4__092_060open_062h_Ax_A_092_060noteq_062_Ah_Ax_092_060close_062,axiom,
( ( h @ x )
!= ( h @ x ) ) ).
% \<open>h x \<noteq> h x\<close>
thf(fact_5_o1,axiom,
! [X: a] :
( ( member @ a @ X @ ( binary1653327646_setOf @ a @ t ) )
=> ( ( h @ X )
!= ( h @ x ) ) ) ).
% o1
thf(fact_6_sorted__distinct__pred__def,axiom,
! [A: $tType] :
( ( binary231205461t_pred @ A )
= ( ^ [H2: A > int,A3: A,B3: A,T2: binary1291135688e_Tree @ A] :
( ( ( binary1610619414edTree @ A @ H2 @ T2 )
& ( member @ A @ A3 @ ( binary1653327646_setOf @ A @ T2 ) )
& ( member @ A @ B3 @ ( binary1653327646_setOf @ A @ T2 ) )
& ( ( H2 @ A3 )
= ( H2 @ B3 ) ) )
=> ( A3 = B3 ) ) ) ) ).
% sorted_distinct_pred_def
thf(fact_7_tNone,axiom,
( ( binary1265063667lookup @ a @ h @ ( h @ x ) @ t )
= ( none @ a ) ) ).
% tNone
thf(fact_8_sortedTree_Osimps_I1_J,axiom,
! [A: $tType,H: A > int] : ( binary1610619414edTree @ A @ H @ ( binary1746293266le_Tip @ A ) ) ).
% sortedTree.simps(1)
thf(fact_9_sortLemmaL,axiom,
! [A: $tType,H: A > int,T1: binary1291135688e_Tree @ A,X2: A,T22: binary1291135688e_Tree @ A] :
( ( binary1610619414edTree @ A @ H @ ( binary210054475elle_T @ A @ T1 @ X2 @ T22 ) )
=> ( binary1610619414edTree @ A @ H @ T1 ) ) ).
% sortLemmaL
thf(fact_10_sortLemmaR,axiom,
! [A: $tType,H: A > int,T1: binary1291135688e_Tree @ A,X2: A,T22: binary1291135688e_Tree @ A] :
( ( binary1610619414edTree @ A @ H @ ( binary210054475elle_T @ A @ T1 @ X2 @ T22 ) )
=> ( binary1610619414edTree @ A @ H @ T22 ) ) ).
% sortLemmaR
thf(fact_11_tlookup__none,axiom,
! [A: $tType,H: A > int,T: binary1291135688e_Tree @ A,K: int] :
( ( ( binary1610619414edTree @ A @ H @ T )
& ( ( binary1265063667lookup @ A @ H @ K @ T )
= ( none @ A ) ) )
=> ! [X: A] :
( ( member @ A @ X @ ( binary1653327646_setOf @ A @ T ) )
=> ( ( H @ X )
!= K ) ) ) ).
% tlookup_none
thf(fact_12_tlookup__some,axiom,
! [A: $tType,H: A > int,T: binary1291135688e_Tree @ A,K: int,X2: A] :
( ( ( binary1610619414edTree @ A @ H @ T )
& ( ( binary1265063667lookup @ A @ H @ K @ T )
= ( some @ A @ X2 ) ) )
=> ( ( member @ A @ X2 @ ( binary1653327646_setOf @ A @ T ) )
& ( ( H @ X2 )
= K ) ) ) ).
% tlookup_some
thf(fact_13_tlookup__finds,axiom,
! [A: $tType,H: A > int,T: binary1291135688e_Tree @ A,Y: A] :
( ( ( binary1610619414edTree @ A @ H @ T )
& ( member @ A @ Y @ ( binary1653327646_setOf @ A @ T ) ) )
=> ( ( binary1265063667lookup @ A @ H @ ( H @ Y ) @ T )
= ( some @ A @ Y ) ) ) ).
% tlookup_finds
thf(fact_14_sortedTree_Osimps_I2_J,axiom,
! [A: $tType,H: A > int,T1: binary1291135688e_Tree @ A,X2: A,T22: binary1291135688e_Tree @ A] :
( ( binary1610619414edTree @ A @ H @ ( binary210054475elle_T @ A @ T1 @ X2 @ T22 ) )
= ( ( binary1610619414edTree @ A @ H @ T1 )
& ! [X3: A] :
( ( member @ A @ X3 @ ( binary1653327646_setOf @ A @ T1 ) )
=> ( ord_less @ int @ ( H @ X3 ) @ ( H @ X2 ) ) )
& ! [X3: A] :
( ( member @ A @ X3 @ ( binary1653327646_setOf @ A @ T22 ) )
=> ( ord_less @ int @ ( H @ X2 ) @ ( H @ X3 ) ) )
& ( binary1610619414edTree @ A @ H @ T22 ) ) ) ).
% sortedTree.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_memb__def,axiom,
! [A: $tType] :
( ( binary827270440e_memb @ A )
= ( ^ [H2: A > int,X3: A,T2: binary1291135688e_Tree @ A] :
( case_option @ $o @ A @ $false
@ ( ^ [Y2: A,Z: A] : Y2 = Z
@ X3 )
@ ( binary1265063667lookup @ A @ H2 @ ( H2 @ X3 ) @ T2 ) ) ) ) ).
% memb_def
thf(fact_17_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_18_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_19_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_20_tlookup_Osimps_I1_J,axiom,
! [A: $tType,H: A > int,K: int] :
( ( binary1265063667lookup @ A @ H @ K @ ( binary1746293266le_Tip @ A ) )
= ( none @ A ) ) ).
% tlookup.simps(1)
thf(fact_21_tlookup_Osimps_I2_J,axiom,
! [A: $tType,K: int,H: A > int,X2: A,T1: binary1291135688e_Tree @ A,T22: binary1291135688e_Tree @ A] :
( ( ( ord_less @ int @ K @ ( H @ X2 ) )
=> ( ( binary1265063667lookup @ A @ H @ K @ ( binary210054475elle_T @ A @ T1 @ X2 @ T22 ) )
= ( binary1265063667lookup @ A @ H @ K @ T1 ) ) )
& ( ~ ( ord_less @ int @ K @ ( H @ X2 ) )
=> ( ( ( ord_less @ int @ ( H @ X2 ) @ K )
=> ( ( binary1265063667lookup @ A @ H @ K @ ( binary210054475elle_T @ A @ T1 @ X2 @ T22 ) )
= ( binary1265063667lookup @ A @ H @ K @ T22 ) ) )
& ( ~ ( ord_less @ int @ ( H @ X2 ) @ K )
=> ( ( binary1265063667lookup @ A @ H @ K @ ( binary210054475elle_T @ A @ T1 @ X2 @ T22 ) )
= ( some @ A @ X2 ) ) ) ) ) ) ).
% tlookup.simps(2)
thf(fact_22_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_23_not__Some__eq,axiom,
! [A: $tType,X2: option @ A] :
( ( ! [Y3: A] :
( X2
!= ( some @ A @ Y3 ) ) )
= ( X2
= ( none @ A ) ) ) ).
% not_Some_eq
thf(fact_24_not__None__eq,axiom,
! [A: $tType,X2: option @ A] :
( ( X2
!= ( none @ A ) )
= ( ? [Y3: A] :
( X2
= ( some @ A @ Y3 ) ) ) ) ).
% not_None_eq
thf(fact_25_case__optionE,axiom,
! [A: $tType,P: $o,Q: A > $o,X2: option @ A] :
( ( case_option @ $o @ A @ P @ Q @ X2 )
=> ( ( ( X2
= ( none @ A ) )
=> ~ P )
=> ~ ! [Y4: A] :
( ( X2
= ( some @ A @ Y4 ) )
=> ~ ( Q @ Y4 ) ) ) ) ).
% case_optionE
thf(fact_26_disjE__realizer2,axiom,
! [B: $tType,A: $tType,P: $o,Q: A > $o,X2: option @ A,R: B > $o,F: B,G: A > B] :
( ( case_option @ $o @ A @ P @ Q @ X2 )
=> ( ( P
=> ( R @ F ) )
=> ( ! [Q2: A] :
( ( Q @ Q2 )
=> ( R @ ( G @ Q2 ) ) )
=> ( R @ ( case_option @ B @ A @ F @ G @ X2 ) ) ) ) ) ).
% disjE_realizer2
thf(fact_27_option_Odisc__eq__case_I1_J,axiom,
! [A: $tType,Option: option @ A] :
( ( Option
= ( none @ A ) )
= ( case_option @ $o @ A @ $true
@ ^ [Uu: A] : $false
@ Option ) ) ).
% option.disc_eq_case(1)
thf(fact_28_option_Odisc__eq__case_I2_J,axiom,
! [A: $tType,Option: option @ A] :
( ( Option
!= ( none @ A ) )
= ( case_option @ $o @ A @ $false
@ ^ [Uu: A] : $true
@ Option ) ) ).
% option.disc_eq_case(2)
thf(fact_29_option_Oinject,axiom,
! [A: $tType,X25: A,Y24: A] :
( ( ( some @ A @ X25 )
= ( some @ A @ Y24 ) )
= ( X25 = Y24 ) ) ).
% option.inject
thf(fact_30_empty__Collect__eq,axiom,
! [A: $tType,P: A > $o] :
( ( ( bot_bot @ ( set @ A ) )
= ( collect @ A @ P ) )
= ( ! [X3: A] :
~ ( P @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_31_Collect__empty__eq,axiom,
! [A: $tType,P: A > $o] :
( ( ( collect @ A @ P )
= ( bot_bot @ ( set @ A ) ) )
= ( ! [X3: A] :
~ ( P @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_32_all__not__in__conv,axiom,
! [A: $tType,A4: set @ A] :
( ( ! [X3: A] :
~ ( member @ A @ X3 @ A4 ) )
= ( A4
= ( bot_bot @ ( set @ A ) ) ) ) ).
% all_not_in_conv
thf(fact_33_empty__iff,axiom,
! [A: $tType,C: A] :
~ ( member @ A @ C @ ( bot_bot @ ( set @ A ) ) ) ).
% empty_iff
thf(fact_34_emptyE,axiom,
! [A: $tType,A2: A] :
~ ( member @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ).
% emptyE
thf(fact_35_equals0D,axiom,
! [A: $tType,A4: set @ A,A2: A] :
( ( A4
= ( bot_bot @ ( set @ A ) ) )
=> ~ ( member @ A @ A2 @ A4 ) ) ).
% equals0D
thf(fact_36_equals0I,axiom,
! [A: $tType,A4: set @ A] :
( ! [Y4: A] :
~ ( member @ A @ Y4 @ A4 )
=> ( A4
= ( bot_bot @ ( set @ A ) ) ) ) ).
% equals0I
thf(fact_37_ex__in__conv,axiom,
! [A: $tType,A4: set @ A] :
( ( ? [X3: A] : ( member @ A @ X3 @ A4 ) )
= ( A4
!= ( bot_bot @ ( set @ A ) ) ) ) ).
% ex_in_conv
thf(fact_38_empty__def,axiom,
! [A: $tType] :
( ( bot_bot @ ( set @ A ) )
= ( collect @ A
@ ^ [X3: A] : $false ) ) ).
% empty_def
thf(fact_39_option_Ocase__distrib,axiom,
! [C2: $tType,B: $tType,A: $tType,H: B > C2,F1: B,F2: A > B,Option: option @ A] :
( ( H @ ( case_option @ B @ A @ F1 @ F2 @ Option ) )
= ( case_option @ C2 @ A @ ( H @ F1 )
@ ^ [X3: A] : ( H @ ( F2 @ X3 ) )
@ Option ) ) ).
% option.case_distrib
thf(fact_40_option_Odistinct_I1_J,axiom,
! [A: $tType,X25: A] :
( ( none @ A )
!= ( some @ A @ X25 ) ) ).
% option.distinct(1)
thf(fact_41_option_OdiscI,axiom,
! [A: $tType,Option: option @ A,X25: A] :
( ( Option
= ( some @ A @ X25 ) )
=> ( Option
!= ( none @ A ) ) ) ).
% option.discI
thf(fact_42_option_Oexhaust,axiom,
! [A: $tType,Y: option @ A] :
( ( Y
!= ( none @ A ) )
=> ~ ! [X24: A] :
( Y
!= ( some @ A @ X24 ) ) ) ).
% option.exhaust
thf(fact_43_option_Oinducts,axiom,
! [A: $tType,P: ( option @ A ) > $o,Option: option @ A] :
( ( P @ ( none @ A ) )
=> ( ! [X4: A] : ( P @ ( some @ A @ X4 ) )
=> ( P @ Option ) ) ) ).
% option.inducts
thf(fact_44_mem__Collect__eq,axiom,
! [A: $tType,A2: A,P: A > $o] :
( ( member @ A @ A2 @ ( collect @ A @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_45_Collect__mem__eq,axiom,
! [A: $tType,A4: set @ A] :
( ( collect @ A
@ ^ [X3: A] : ( member @ A @ X3 @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_46_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_47_ext,axiom,
! [B: $tType,A: $tType,F: A > B,G: A > B] :
( ! [X4: A] :
( ( F @ X4 )
= ( G @ X4 ) )
=> ( F = G ) ) ).
% ext
thf(fact_48_split__option__ex,axiom,
! [A: $tType] :
( ( ^ [P2: ( option @ A ) > $o] :
? [X5: option @ A] : ( P2 @ X5 ) )
= ( ^ [P3: ( option @ A ) > $o] :
( ( P3 @ ( none @ A ) )
| ? [X3: A] : ( P3 @ ( some @ A @ X3 ) ) ) ) ) ).
% split_option_ex
thf(fact_49_split__option__all,axiom,
! [A: $tType] :
( ( ^ [P2: ( option @ A ) > $o] :
! [X5: option @ A] : ( P2 @ X5 ) )
= ( ^ [P3: ( option @ A ) > $o] :
( ( P3 @ ( none @ A ) )
& ! [X3: A] : ( P3 @ ( some @ A @ X3 ) ) ) ) ) ).
% split_option_all
thf(fact_50_combine__options__cases,axiom,
! [A: $tType,B: $tType,X2: option @ A,P: ( option @ A ) > ( option @ B ) > $o,Y: option @ B] :
( ( ( X2
= ( none @ A ) )
=> ( P @ X2 @ Y ) )
=> ( ( ( Y
= ( none @ B ) )
=> ( P @ X2 @ Y ) )
=> ( ! [A5: A,B4: B] :
( ( X2
= ( some @ A @ A5 ) )
=> ( ( Y
= ( some @ B @ B4 ) )
=> ( P @ X2 @ Y ) ) )
=> ( P @ X2 @ Y ) ) ) ) ).
% combine_options_cases
thf(fact_51_option_Osimps_I4_J,axiom,
! [A: $tType,B: $tType,F1: B,F2: A > B] :
( ( case_option @ B @ A @ F1 @ F2 @ ( none @ A ) )
= F1 ) ).
% option.simps(4)
thf(fact_52_option_Osimps_I5_J,axiom,
! [B: $tType,A: $tType,F1: B,F2: A > B,X25: A] :
( ( case_option @ B @ A @ F1 @ F2 @ ( some @ A @ X25 ) )
= ( F2 @ X25 ) ) ).
% option.simps(5)
thf(fact_53_bot__apply,axiom,
! [C2: $tType,D: $tType] :
( ( bot @ C2 )
=> ( ( bot_bot @ ( D > C2 ) )
= ( ^ [X3: D] : ( bot_bot @ C2 ) ) ) ) ).
% bot_apply
thf(fact_54_bot_Oextremum__strict,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [A2: A] :
~ ( ord_less @ A @ A2 @ ( bot_bot @ A ) ) ) ).
% bot.extremum_strict
thf(fact_55_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_56_Set_Ois__empty__def,axiom,
! [A: $tType] :
( ( is_empty @ A )
= ( ^ [A6: set @ A] :
( A6
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Set.is_empty_def
thf(fact_57_eqs__def,axiom,
! [A: $tType] :
( ( binary64540844le_eqs @ A )
= ( ^ [H2: A > int,X3: A] :
( collect @ A
@ ^ [Y3: A] :
( ( H2 @ Y3 )
= ( H2 @ X3 ) ) ) ) ) ).
% eqs_def
thf(fact_58_option_Osplit__sel,axiom,
! [B: $tType,A: $tType,P: B > $o,F1: B,F2: A > B,Option: option @ A] :
( ( P @ ( case_option @ B @ A @ F1 @ F2 @ Option ) )
= ( ( ( Option
= ( none @ A ) )
=> ( P @ F1 ) )
& ( ( Option
= ( some @ A @ ( the2 @ A @ Option ) ) )
=> ( P @ ( F2 @ ( the2 @ A @ Option ) ) ) ) ) ) ).
% option.split_sel
thf(fact_59_option_Osplit__sel__asm,axiom,
! [B: $tType,A: $tType,P: B > $o,F1: B,F2: A > B,Option: option @ A] :
( ( P @ ( case_option @ B @ A @ F1 @ F2 @ Option ) )
= ( ~ ( ( ( Option
= ( none @ A ) )
& ~ ( P @ F1 ) )
| ( ( Option
= ( some @ A @ ( the2 @ A @ Option ) ) )
& ~ ( P @ ( F2 @ ( the2 @ A @ Option ) ) ) ) ) ) ) ).
% option.split_sel_asm
thf(fact_60_set__empty__eq,axiom,
! [A: $tType,Xo: option @ A] :
( ( ( set_option @ A @ Xo )
= ( bot_bot @ ( set @ A ) ) )
= ( Xo
= ( none @ A ) ) ) ).
% set_empty_eq
thf(fact_61_bot__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( bot @ B )
=> ( ( bot_bot @ ( A > B ) )
= ( ^ [X3: A] : ( bot_bot @ B ) ) ) ) ).
% bot_fun_def
thf(fact_62_elem__set,axiom,
! [A: $tType,X2: A,Xo: option @ A] :
( ( member @ A @ X2 @ ( set_option @ A @ Xo ) )
= ( Xo
= ( some @ A @ X2 ) ) ) ).
% elem_set
thf(fact_63_option_Ocollapse,axiom,
! [A: $tType,Option: option @ A] :
( ( Option
!= ( none @ A ) )
=> ( ( some @ A @ ( the2 @ A @ Option ) )
= Option ) ) ).
% option.collapse
thf(fact_64_option_Oset__sel,axiom,
! [A: $tType,A2: option @ A] :
( ( A2
!= ( none @ A ) )
=> ( member @ A @ ( the2 @ A @ A2 ) @ ( set_option @ A @ A2 ) ) ) ).
% option.set_sel
thf(fact_65_bot__set__def,axiom,
! [A: $tType] :
( ( bot_bot @ ( set @ A ) )
= ( collect @ A @ ( bot_bot @ ( A > $o ) ) ) ) ).
% bot_set_def
thf(fact_66_not__psubset__empty,axiom,
! [A: $tType,A4: set @ A] :
~ ( ord_less @ ( set @ A ) @ A4 @ ( bot_bot @ ( set @ A ) ) ) ).
% not_psubset_empty
thf(fact_67_option_Oexpand,axiom,
! [A: $tType,Option: option @ A,Option2: option @ A] :
( ( ( Option
= ( none @ A ) )
= ( Option2
= ( none @ A ) ) )
=> ( ( ( Option
!= ( none @ A ) )
=> ( ( Option2
!= ( none @ A ) )
=> ( ( the2 @ A @ Option )
= ( the2 @ A @ Option2 ) ) ) )
=> ( Option = Option2 ) ) ) ).
% option.expand
thf(fact_68_option_Osel,axiom,
! [A: $tType,X25: A] :
( ( the2 @ A @ ( some @ A @ X25 ) )
= X25 ) ).
% option.sel
thf(fact_69_option_Oset__intros,axiom,
! [A: $tType,X25: A] : ( member @ A @ X25 @ ( set_option @ A @ ( some @ A @ X25 ) ) ) ).
% option.set_intros
thf(fact_70_option_Oset__cases,axiom,
! [A: $tType,E: A,A2: option @ A] :
( ( member @ A @ E @ ( set_option @ A @ A2 ) )
=> ( A2
= ( some @ A @ E ) ) ) ).
% option.set_cases
thf(fact_71_ospec,axiom,
! [A: $tType,A4: option @ A,P: A > $o,X2: A] :
( ! [X4: A] :
( ( member @ A @ X4 @ ( set_option @ A @ A4 ) )
=> ( P @ X4 ) )
=> ( ( A4
= ( some @ A @ X2 ) )
=> ( P @ X2 ) ) ) ).
% ospec
thf(fact_72_option_Oexhaust__sel,axiom,
! [A: $tType,Option: option @ A] :
( ( Option
!= ( none @ A ) )
=> ( Option
= ( some @ A @ ( the2 @ A @ Option ) ) ) ) ).
% option.exhaust_sel
thf(fact_73_option_Osimps_I14_J,axiom,
! [A: $tType] :
( ( set_option @ A @ ( none @ A ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% option.simps(14)
thf(fact_74_option_Ocase__eq__if,axiom,
! [A: $tType,B: $tType] :
( ( case_option @ B @ A )
= ( ^ [F12: B,F22: A > B,Option3: option @ A] :
( if @ B
@ ( Option3
= ( none @ A ) )
@ F12
@ ( F22 @ ( the2 @ A @ Option3 ) ) ) ) ) ).
% option.case_eq_if
thf(fact_75_dual__order_Ostrict__implies__not__eq,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B2: A,A2: A] :
( ( ord_less @ A @ B2 @ A2 )
=> ( A2 != B2 ) ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_76_order_Ostrict__implies__not__eq,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( A2 != B2 ) ) ) ).
% order.strict_implies_not_eq
thf(fact_77_not__less__iff__gr__or__eq,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X2: A,Y: A] :
( ( ~ ( ord_less @ A @ X2 @ Y ) )
= ( ( ord_less @ A @ Y @ X2 )
| ( X2 = Y ) ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_78_dual__order_Ostrict__trans,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B2: A,A2: A,C: A] :
( ( ord_less @ A @ B2 @ A2 )
=> ( ( ord_less @ A @ C @ B2 )
=> ( ord_less @ A @ C @ A2 ) ) ) ) ).
% dual_order.strict_trans
thf(fact_79_linorder__less__wlog,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > A > $o,A2: A,B2: A] :
( ! [A5: A,B4: A] :
( ( ord_less @ A @ A5 @ B4 )
=> ( P @ A5 @ B4 ) )
=> ( ! [A5: A] : ( P @ A5 @ A5 )
=> ( ! [A5: A,B4: A] :
( ( P @ B4 @ A5 )
=> ( P @ A5 @ B4 ) )
=> ( P @ A2 @ B2 ) ) ) ) ) ).
% linorder_less_wlog
thf(fact_80_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_81_less__imp__not__less,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X2: A,Y: A] :
( ( ord_less @ A @ X2 @ Y )
=> ~ ( ord_less @ A @ Y @ X2 ) ) ) ).
% less_imp_not_less
thf(fact_82_order_Ostrict__trans,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: A,B2: A,C: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less @ A @ B2 @ C )
=> ( ord_less @ A @ A2 @ C ) ) ) ) ).
% order.strict_trans
thf(fact_83_dual__order_Oirrefl,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: A] :
~ ( ord_less @ A @ A2 @ A2 ) ) ).
% dual_order.irrefl
thf(fact_84_linorder__cases,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X2: A,Y: A] :
( ~ ( ord_less @ A @ X2 @ Y )
=> ( ( X2 != Y )
=> ( ord_less @ A @ Y @ X2 ) ) ) ) ).
% linorder_cases
thf(fact_85_less__imp__triv,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X2: A,Y: A,P: $o] :
( ( ord_less @ A @ X2 @ Y )
=> ( ( ord_less @ A @ Y @ X2 )
=> P ) ) ) ).
% less_imp_triv
thf(fact_86_less__imp__not__eq2,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X2: A,Y: A] :
( ( ord_less @ A @ X2 @ Y )
=> ( Y != X2 ) ) ) ).
% less_imp_not_eq2
thf(fact_87_antisym__conv3,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Y: A,X2: A] :
( ~ ( ord_less @ A @ Y @ X2 )
=> ( ( ~ ( ord_less @ A @ X2 @ Y ) )
= ( X2 = Y ) ) ) ) ).
% antisym_conv3
thf(fact_88_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_89_less__not__sym,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X2: A,Y: A] :
( ( ord_less @ A @ X2 @ Y )
=> ~ ( ord_less @ A @ Y @ X2 ) ) ) ).
% less_not_sym
thf(fact_90_less__imp__not__eq,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X2: A,Y: A] :
( ( ord_less @ A @ X2 @ Y )
=> ( X2 != Y ) ) ) ).
% less_imp_not_eq
thf(fact_91_dual__order_Oasym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B2: A,A2: A] :
( ( ord_less @ A @ B2 @ A2 )
=> ~ ( ord_less @ A @ A2 @ B2 ) ) ) ).
% dual_order.asym
thf(fact_92_ord__less__eq__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A2: A,B2: A,C: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( B2 = C )
=> ( ord_less @ A @ A2 @ C ) ) ) ) ).
% ord_less_eq_trans
thf(fact_93_ord__eq__less__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A2: A,B2: A,C: A] :
( ( A2 = B2 )
=> ( ( ord_less @ A @ B2 @ C )
=> ( ord_less @ A @ A2 @ C ) ) ) ) ).
% ord_eq_less_trans
thf(fact_94_less__irrefl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X2: A] :
~ ( ord_less @ A @ X2 @ X2 ) ) ).
% less_irrefl
thf(fact_95_less__linear,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X2: A,Y: A] :
( ( ord_less @ A @ X2 @ Y )
| ( X2 = Y )
| ( ord_less @ A @ Y @ X2 ) ) ) ).
% less_linear
thf(fact_96_less__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X2: A,Y: A,Z2: A] :
( ( ord_less @ A @ X2 @ Y )
=> ( ( ord_less @ A @ Y @ Z2 )
=> ( ord_less @ A @ X2 @ Z2 ) ) ) ) ).
% less_trans
thf(fact_97_less__asym_H,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ~ ( ord_less @ A @ B2 @ A2 ) ) ) ).
% less_asym'
thf(fact_98_less__asym,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X2: A,Y: A] :
( ( ord_less @ A @ X2 @ Y )
=> ~ ( ord_less @ A @ Y @ X2 ) ) ) ).
% less_asym
thf(fact_99_less__imp__neq,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X2: A,Y: A] :
( ( ord_less @ A @ X2 @ Y )
=> ( X2 != Y ) ) ) ).
% less_imp_neq
thf(fact_100_dense,axiom,
! [A: $tType] :
( ( dense_order @ A )
=> ! [X2: A,Y: A] :
( ( ord_less @ A @ X2 @ Y )
=> ? [Z3: A] :
( ( ord_less @ A @ X2 @ Z3 )
& ( ord_less @ A @ Z3 @ Y ) ) ) ) ).
% dense
thf(fact_101_order_Oasym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ~ ( ord_less @ A @ B2 @ A2 ) ) ) ).
% order.asym
thf(fact_102_neq__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X2: A,Y: A] :
( ( X2 != Y )
= ( ( ord_less @ A @ X2 @ Y )
| ( ord_less @ A @ Y @ X2 ) ) ) ) ).
% neq_iff
thf(fact_103_neqE,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X2: A,Y: A] :
( ( X2 != Y )
=> ( ~ ( ord_less @ A @ X2 @ Y )
=> ( ord_less @ A @ Y @ X2 ) ) ) ) ).
% neqE
thf(fact_104_gt__ex,axiom,
! [A: $tType] :
( ( no_top @ A )
=> ! [X2: A] :
? [X_1: A] : ( ord_less @ A @ X2 @ X_1 ) ) ).
% gt_ex
thf(fact_105_lt__ex,axiom,
! [A: $tType] :
( ( no_bot @ A )
=> ! [X2: A] :
? [Y4: A] : ( ord_less @ A @ Y4 @ X2 ) ) ).
% lt_ex
thf(fact_106_order__less__subst2,axiom,
! [A: $tType,C2: $tType] :
( ( ( order @ C2 )
& ( order @ A ) )
=> ! [A2: A,B2: A,F: A > C2,C: C2] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less @ C2 @ ( F @ B2 ) @ C )
=> ( ! [X4: A,Y4: A] :
( ( ord_less @ A @ X4 @ Y4 )
=> ( ord_less @ C2 @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less @ C2 @ ( F @ A2 ) @ C ) ) ) ) ) ).
% order_less_subst2
thf(fact_107_order__less__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B )
& ( order @ A ) )
=> ! [A2: A,F: B > A,B2: B,C: B] :
( ( ord_less @ A @ A2 @ ( F @ B2 ) )
=> ( ( ord_less @ B @ B2 @ C )
=> ( ! [X4: B,Y4: B] :
( ( ord_less @ B @ X4 @ Y4 )
=> ( ord_less @ A @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% order_less_subst1
thf(fact_108_ord__less__eq__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( ord @ A ) )
=> ! [A2: A,B2: A,F: A > B,C: B] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X4: A,Y4: A] :
( ( ord_less @ A @ X4 @ Y4 )
=> ( ord_less @ B @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less @ B @ ( F @ A2 ) @ C ) ) ) ) ) ).
% ord_less_eq_subst
thf(fact_109_ord__eq__less__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( ord @ A ) )
=> ! [A2: A,F: B > A,B2: B,C: B] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less @ B @ B2 @ C )
=> ( ! [X4: B,Y4: B] :
( ( ord_less @ B @ X4 @ Y4 )
=> ( ord_less @ A @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_110_option_Osimps_I15_J,axiom,
! [A: $tType,X25: A] :
( ( set_option @ A @ ( some @ A @ X25 ) )
= ( insert @ A @ X25 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% option.simps(15)
thf(fact_111_option_Othe__def,axiom,
! [A: $tType] :
( ( the2 @ A )
= ( case_option @ A @ A @ ( undefined @ A )
@ ^ [X26: A] : X26 ) ) ).
% option.the_def
thf(fact_112_bot__empty__eq,axiom,
! [A: $tType] :
( ( bot_bot @ ( A > $o ) )
= ( ^ [X3: A] : ( member @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% bot_empty_eq
thf(fact_113_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_114_ex__gt__or__lt,axiom,
! [A: $tType] :
( ( condit1656338222tinuum @ A )
=> ! [A2: A] :
? [B4: A] :
( ( ord_less @ A @ A2 @ B4 )
| ( ord_less @ A @ B4 @ A2 ) ) ) ).
% ex_gt_or_lt
thf(fact_115_linorder__neqE__linordered__idom,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X2: A,Y: A] :
( ( X2 != Y )
=> ( ~ ( ord_less @ A @ X2 @ Y )
=> ( ord_less @ A @ Y @ X2 ) ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_116_linordered__field__no__ub,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A] :
? [X_1: A] : ( ord_less @ A @ X @ X_1 ) ) ).
% linordered_field_no_ub
thf(fact_117_insert__absorb2,axiom,
! [A: $tType,X2: A,A4: set @ A] :
( ( insert @ A @ X2 @ ( insert @ A @ X2 @ A4 ) )
= ( insert @ A @ X2 @ A4 ) ) ).
% insert_absorb2
thf(fact_118_insert__iff,axiom,
! [A: $tType,A2: A,B2: A,A4: set @ A] :
( ( member @ A @ A2 @ ( insert @ A @ B2 @ A4 ) )
= ( ( A2 = B2 )
| ( member @ A @ A2 @ A4 ) ) ) ).
% insert_iff
thf(fact_119_insertCI,axiom,
! [A: $tType,A2: A,B5: set @ A,B2: A] :
( ( ~ ( member @ A @ A2 @ B5 )
=> ( A2 = B2 ) )
=> ( member @ A @ A2 @ ( insert @ A @ B2 @ B5 ) ) ) ).
% insertCI
thf(fact_120_singletonI,axiom,
! [A: $tType,A2: A] : ( member @ A @ A2 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% singletonI
thf(fact_121_singleton__conv,axiom,
! [A: $tType,A2: A] :
( ( collect @ A
@ ^ [X3: A] : X3 = A2 )
= ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% singleton_conv
thf(fact_122_singleton__conv2,axiom,
! [A: $tType,A2: A] :
( ( collect @ A
@ ( ^ [Y2: A,Z: A] : Y2 = Z
@ A2 ) )
= ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% singleton_conv2
thf(fact_123_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_124_less__set__def,axiom,
! [A: $tType] :
( ( ord_less @ ( set @ A ) )
= ( ^ [A6: set @ A,B6: set @ A] :
( ord_less @ ( A > $o )
@ ^ [X3: A] : ( member @ A @ X3 @ A6 )
@ ^ [X3: A] : ( member @ A @ X3 @ B6 ) ) ) ) ).
% less_set_def
thf(fact_125_psubsetD,axiom,
! [A: $tType,A4: set @ A,B5: set @ A,C: A] :
( ( ord_less @ ( set @ A ) @ A4 @ B5 )
=> ( ( member @ A @ C @ A4 )
=> ( member @ A @ C @ B5 ) ) ) ).
% psubsetD
thf(fact_126_mk__disjoint__insert,axiom,
! [A: $tType,A2: A,A4: set @ A] :
( ( member @ A @ A2 @ A4 )
=> ? [B7: set @ A] :
( ( A4
= ( insert @ A @ A2 @ B7 ) )
& ~ ( member @ A @ A2 @ B7 ) ) ) ).
% mk_disjoint_insert
thf(fact_127_insert__commute,axiom,
! [A: $tType,X2: A,Y: A,A4: set @ A] :
( ( insert @ A @ X2 @ ( insert @ A @ Y @ A4 ) )
= ( insert @ A @ Y @ ( insert @ A @ X2 @ A4 ) ) ) ).
% insert_commute
thf(fact_128_insert__eq__iff,axiom,
! [A: $tType,A2: A,A4: set @ A,B2: A,B5: set @ A] :
( ~ ( member @ A @ A2 @ A4 )
=> ( ~ ( member @ A @ B2 @ B5 )
=> ( ( ( insert @ A @ A2 @ A4 )
= ( insert @ A @ B2 @ B5 ) )
= ( ( ( A2 = B2 )
=> ( A4 = B5 ) )
& ( ( A2 != B2 )
=> ? [C4: set @ A] :
( ( A4
= ( insert @ A @ B2 @ C4 ) )
& ~ ( member @ A @ B2 @ C4 )
& ( B5
= ( insert @ A @ A2 @ C4 ) )
& ~ ( member @ A @ A2 @ C4 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_129_insert__absorb,axiom,
! [A: $tType,A2: A,A4: set @ A] :
( ( member @ A @ A2 @ A4 )
=> ( ( insert @ A @ A2 @ A4 )
= A4 ) ) ).
% insert_absorb
thf(fact_130_insert__ident,axiom,
! [A: $tType,X2: A,A4: set @ A,B5: set @ A] :
( ~ ( member @ A @ X2 @ A4 )
=> ( ~ ( member @ A @ X2 @ B5 )
=> ( ( ( insert @ A @ X2 @ A4 )
= ( insert @ A @ X2 @ B5 ) )
= ( A4 = B5 ) ) ) ) ).
% insert_ident
thf(fact_131_Set_Oset__insert,axiom,
! [A: $tType,X2: A,A4: set @ A] :
( ( member @ A @ X2 @ A4 )
=> ~ ! [B7: set @ A] :
( ( A4
= ( insert @ A @ X2 @ B7 ) )
=> ( member @ A @ X2 @ B7 ) ) ) ).
% Set.set_insert
thf(fact_132_insertI2,axiom,
! [A: $tType,A2: A,B5: set @ A,B2: A] :
( ( member @ A @ A2 @ B5 )
=> ( member @ A @ A2 @ ( insert @ A @ B2 @ B5 ) ) ) ).
% insertI2
thf(fact_133_insertI1,axiom,
! [A: $tType,A2: A,B5: set @ A] : ( member @ A @ A2 @ ( insert @ A @ A2 @ B5 ) ) ).
% insertI1
thf(fact_134_insertE,axiom,
! [A: $tType,A2: A,B2: A,A4: set @ A] :
( ( member @ A @ A2 @ ( insert @ A @ B2 @ A4 ) )
=> ( ( A2 != B2 )
=> ( member @ A @ A2 @ A4 ) ) ) ).
% insertE
thf(fact_135_insert__compr,axiom,
! [A: $tType] :
( ( insert @ A )
= ( ^ [A3: A,B6: set @ A] :
( collect @ A
@ ^ [X3: A] :
( ( X3 = A3 )
| ( member @ A @ X3 @ B6 ) ) ) ) ) ).
% insert_compr
thf(fact_136_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_137_singletonD,axiom,
! [A: $tType,B2: A,A2: A] :
( ( member @ A @ B2 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) )
=> ( B2 = A2 ) ) ).
% singletonD
thf(fact_138_singleton__iff,axiom,
! [A: $tType,B2: A,A2: A] :
( ( member @ A @ B2 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) )
= ( B2 = A2 ) ) ).
% singleton_iff
thf(fact_139_doubleton__eq__iff,axiom,
! [A: $tType,A2: A,B2: A,C: A,D2: A] :
( ( ( insert @ A @ A2 @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) )
= ( insert @ A @ C @ ( insert @ A @ D2 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( ( ( A2 = C )
& ( B2 = D2 ) )
| ( ( A2 = D2 )
& ( B2 = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_140_insert__not__empty,axiom,
! [A: $tType,A2: A,A4: set @ A] :
( ( insert @ A @ A2 @ A4 )
!= ( bot_bot @ ( set @ A ) ) ) ).
% insert_not_empty
thf(fact_141_singleton__inject,axiom,
! [A: $tType,A2: A,B2: A] :
( ( ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) )
= ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) )
=> ( A2 = B2 ) ) ).
% singleton_inject
thf(fact_142_Collect__conv__if,axiom,
! [A: $tType,P: A > $o,A2: A] :
( ( ( P @ A2 )
=> ( ( collect @ A
@ ^ [X3: A] :
( ( X3 = A2 )
& ( P @ X3 ) ) )
= ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) )
& ( ~ ( P @ A2 )
=> ( ( collect @ A
@ ^ [X3: A] :
( ( X3 = A2 )
& ( P @ X3 ) ) )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Collect_conv_if
thf(fact_143_Collect__conv__if2,axiom,
! [A: $tType,P: A > $o,A2: A] :
( ( ( P @ A2 )
=> ( ( collect @ A
@ ^ [X3: A] :
( ( A2 = X3 )
& ( P @ X3 ) ) )
= ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) )
& ( ~ ( P @ A2 )
=> ( ( collect @ A
@ ^ [X3: A] :
( ( A2 = X3 )
& ( P @ X3 ) ) )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Collect_conv_if2
thf(fact_144_linordered__field__no__lb,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A] :
? [Y4: A] : ( ord_less @ A @ Y4 @ X ) ) ).
% linordered_field_no_lb
thf(fact_145_the__elem__eq,axiom,
! [A: $tType,X2: A] :
( ( the_elem @ A @ ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) )
= X2 ) ).
% the_elem_eq
thf(fact_146_is__singletonI,axiom,
! [A: $tType,X2: A] : ( is_singleton @ A @ ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% is_singletonI
thf(fact_147_these__insert__Some,axiom,
! [A: $tType,X2: A,A4: set @ ( option @ A )] :
( ( these @ A @ ( insert @ ( option @ A ) @ ( some @ A @ X2 ) @ A4 ) )
= ( insert @ A @ X2 @ ( these @ A @ A4 ) ) ) ).
% these_insert_Some
thf(fact_148_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_149_these__empty,axiom,
! [A: $tType] :
( ( these @ A @ ( bot_bot @ ( set @ ( option @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% these_empty
thf(fact_150_these__insert__None,axiom,
! [A: $tType,A4: set @ ( option @ A )] :
( ( these @ A @ ( insert @ ( option @ A ) @ ( none @ A ) @ A4 ) )
= ( these @ A @ A4 ) ) ).
% these_insert_None
thf(fact_151_in__these__eq,axiom,
! [A: $tType,X2: A,A4: set @ ( option @ A )] :
( ( member @ A @ X2 @ ( these @ A @ A4 ) )
= ( member @ ( option @ A ) @ ( some @ A @ X2 ) @ A4 ) ) ).
% in_these_eq
thf(fact_152_is__singleton__the__elem,axiom,
! [A: $tType] :
( ( is_singleton @ A )
= ( ^ [A6: set @ A] :
( A6
= ( insert @ A @ ( the_elem @ A @ A6 ) @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% is_singleton_the_elem
thf(fact_153_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_154_these__not__empty__eq,axiom,
! [A: $tType,B5: set @ ( option @ A )] :
( ( ( these @ A @ B5 )
!= ( bot_bot @ ( set @ A ) ) )
= ( ( B5
!= ( bot_bot @ ( set @ ( option @ A ) ) ) )
& ( B5
!= ( insert @ ( option @ A ) @ ( none @ A ) @ ( bot_bot @ ( set @ ( option @ A ) ) ) ) ) ) ) ).
% these_not_empty_eq
thf(fact_155_these__empty__eq,axiom,
! [A: $tType,B5: set @ ( option @ A )] :
( ( ( these @ A @ B5 )
= ( bot_bot @ ( set @ A ) ) )
= ( ( B5
= ( bot_bot @ ( set @ ( option @ A ) ) ) )
| ( B5
= ( insert @ ( option @ A ) @ ( none @ A ) @ ( bot_bot @ ( set @ ( option @ A ) ) ) ) ) ) ) ).
% these_empty_eq
thf(fact_156_is__singleton__def,axiom,
! [A: $tType] :
( ( is_singleton @ A )
= ( ^ [A6: set @ A] :
? [X3: A] :
( A6
= ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% is_singleton_def
thf(fact_157_the__elem__def,axiom,
! [A: $tType] :
( ( the_elem @ A )
= ( ^ [X6: set @ A] :
( the @ A
@ ^ [X3: A] :
( X6
= ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% the_elem_def
thf(fact_158_setOf_Osimps_I2_J,axiom,
! [A: $tType,T1: binary1291135688e_Tree @ A,X2: A,T22: binary1291135688e_Tree @ A] :
( ( binary1653327646_setOf @ A @ ( binary210054475elle_T @ A @ T1 @ X2 @ T22 ) )
= ( sup_sup @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ ( binary1653327646_setOf @ A @ T1 ) @ ( binary1653327646_setOf @ A @ T22 ) ) @ ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% setOf.simps(2)
thf(fact_159_Option_Othese__def,axiom,
! [A: $tType] :
( ( these @ A )
= ( ^ [A6: set @ ( option @ A )] :
( image @ ( option @ A ) @ A @ ( the2 @ A )
@ ( collect @ ( option @ A )
@ ^ [X3: option @ A] :
( ( member @ ( option @ A ) @ X3 @ A6 )
& ( X3
!= ( none @ A ) ) ) ) ) ) ) ).
% Option.these_def
thf(fact_160_image__eqI,axiom,
! [A: $tType,B: $tType,B2: A,F: B > A,X2: B,A4: set @ B] :
( ( B2
= ( F @ X2 ) )
=> ( ( member @ B @ X2 @ A4 )
=> ( member @ A @ B2 @ ( image @ B @ A @ F @ A4 ) ) ) ) ).
% image_eqI
thf(fact_161_Un__iff,axiom,
! [A: $tType,C: A,A4: set @ A,B5: set @ A] :
( ( member @ A @ C @ ( sup_sup @ ( set @ A ) @ A4 @ B5 ) )
= ( ( member @ A @ C @ A4 )
| ( member @ A @ C @ B5 ) ) ) ).
% Un_iff
thf(fact_162_UnCI,axiom,
! [A: $tType,C: A,B5: set @ A,A4: set @ A] :
( ( ~ ( member @ A @ C @ B5 )
=> ( member @ A @ C @ A4 ) )
=> ( member @ A @ C @ ( sup_sup @ ( set @ A ) @ A4 @ B5 ) ) ) ).
% UnCI
thf(fact_163_image__ident,axiom,
! [A: $tType,Y6: set @ A] :
( ( image @ A @ A
@ ^ [X3: A] : X3
@ Y6 )
= Y6 ) ).
% image_ident
thf(fact_164_image__is__empty,axiom,
! [A: $tType,B: $tType,F: B > A,A4: set @ B] :
( ( ( image @ B @ A @ F @ A4 )
= ( bot_bot @ ( set @ A ) ) )
= ( A4
= ( bot_bot @ ( set @ B ) ) ) ) ).
% image_is_empty
thf(fact_165_empty__is__image,axiom,
! [A: $tType,B: $tType,F: B > A,A4: set @ B] :
( ( ( bot_bot @ ( set @ A ) )
= ( image @ B @ A @ F @ A4 ) )
= ( A4
= ( bot_bot @ ( set @ B ) ) ) ) ).
% empty_is_image
thf(fact_166_image__empty,axiom,
! [B: $tType,A: $tType,F: B > A] :
( ( image @ B @ A @ F @ ( bot_bot @ ( set @ B ) ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% image_empty
thf(fact_167_image__insert,axiom,
! [A: $tType,B: $tType,F: B > A,A2: B,B5: set @ B] :
( ( image @ B @ A @ F @ ( insert @ B @ A2 @ B5 ) )
= ( insert @ A @ ( F @ A2 ) @ ( image @ B @ A @ F @ B5 ) ) ) ).
% image_insert
thf(fact_168_insert__image,axiom,
! [B: $tType,A: $tType,X2: A,A4: set @ A,F: A > B] :
( ( member @ A @ X2 @ A4 )
=> ( ( insert @ B @ ( F @ X2 ) @ ( image @ A @ B @ F @ A4 ) )
= ( image @ A @ B @ F @ A4 ) ) ) ).
% insert_image
thf(fact_169_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_170_Un__insert__left,axiom,
! [A: $tType,A2: A,B5: set @ A,C3: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( insert @ A @ A2 @ B5 ) @ C3 )
= ( insert @ A @ A2 @ ( sup_sup @ ( set @ A ) @ B5 @ C3 ) ) ) ).
% Un_insert_left
thf(fact_171_Un__insert__right,axiom,
! [A: $tType,A4: set @ A,A2: A,B5: set @ A] :
( ( sup_sup @ ( set @ A ) @ A4 @ ( insert @ A @ A2 @ B5 ) )
= ( insert @ A @ A2 @ ( sup_sup @ ( set @ A ) @ A4 @ B5 ) ) ) ).
% Un_insert_right
thf(fact_172_insert__def,axiom,
! [A: $tType] :
( ( insert @ A )
= ( ^ [A3: A] :
( sup_sup @ ( set @ A )
@ ( collect @ A
@ ^ [X3: A] : X3 = A3 ) ) ) ) ).
% insert_def
thf(fact_173_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_174_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_175_rev__image__eqI,axiom,
! [B: $tType,A: $tType,X2: A,A4: set @ A,B2: B,F: A > B] :
( ( member @ A @ X2 @ A4 )
=> ( ( B2
= ( F @ X2 ) )
=> ( member @ B @ B2 @ ( image @ A @ B @ F @ A4 ) ) ) ) ).
% rev_image_eqI
thf(fact_176_ball__imageD,axiom,
! [A: $tType,B: $tType,F: B > A,A4: set @ B,P: A > $o] :
( ! [X4: A] :
( ( member @ A @ X4 @ ( image @ B @ A @ F @ A4 ) )
=> ( P @ X4 ) )
=> ! [X: B] :
( ( member @ B @ X @ A4 )
=> ( P @ ( F @ X ) ) ) ) ).
% ball_imageD
thf(fact_177_image__cong,axiom,
! [B: $tType,A: $tType,M2: set @ A,N2: set @ A,F: A > B,G: A > B] :
( ( M2 = N2 )
=> ( ! [X4: A] :
( ( member @ A @ X4 @ N2 )
=> ( ( F @ X4 )
= ( G @ X4 ) ) )
=> ( ( image @ A @ B @ F @ M2 )
= ( image @ A @ B @ G @ N2 ) ) ) ) ).
% image_cong
thf(fact_178_bex__imageD,axiom,
! [A: $tType,B: $tType,F: B > A,A4: set @ B,P: A > $o] :
( ? [X: A] :
( ( member @ A @ X @ ( image @ B @ A @ F @ A4 ) )
& ( P @ X ) )
=> ? [X4: B] :
( ( member @ B @ X4 @ A4 )
& ( P @ ( F @ X4 ) ) ) ) ).
% bex_imageD
thf(fact_179_Un__commute,axiom,
! [A: $tType] :
( ( sup_sup @ ( set @ A ) )
= ( ^ [A6: set @ A,B6: set @ A] : ( sup_sup @ ( set @ A ) @ B6 @ A6 ) ) ) ).
% Un_commute
thf(fact_180_image__iff,axiom,
! [A: $tType,B: $tType,Z2: A,F: B > A,A4: set @ B] :
( ( member @ A @ Z2 @ ( image @ B @ A @ F @ A4 ) )
= ( ? [X3: B] :
( ( member @ B @ X3 @ A4 )
& ( Z2
= ( F @ X3 ) ) ) ) ) ).
% image_iff
thf(fact_181_Un__absorb,axiom,
! [A: $tType,A4: set @ A] :
( ( sup_sup @ ( set @ A ) @ A4 @ A4 )
= A4 ) ).
% Un_absorb
thf(fact_182_image__Un,axiom,
! [A: $tType,B: $tType,F: B > A,A4: set @ B,B5: set @ B] :
( ( image @ B @ A @ F @ ( sup_sup @ ( set @ B ) @ A4 @ B5 ) )
= ( sup_sup @ ( set @ A ) @ ( image @ B @ A @ F @ A4 ) @ ( image @ B @ A @ F @ B5 ) ) ) ).
% image_Un
thf(fact_183_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_184_ball__Un,axiom,
! [A: $tType,A4: set @ A,B5: set @ A,P: A > $o] :
( ( ! [X3: A] :
( ( member @ A @ X3 @ ( sup_sup @ ( set @ A ) @ A4 @ B5 ) )
=> ( P @ X3 ) ) )
= ( ! [X3: A] :
( ( member @ A @ X3 @ A4 )
=> ( P @ X3 ) )
& ! [X3: A] :
( ( member @ A @ X3 @ B5 )
=> ( P @ X3 ) ) ) ) ).
% ball_Un
thf(fact_185_imageI,axiom,
! [B: $tType,A: $tType,X2: A,A4: set @ A,F: A > B] :
( ( member @ A @ X2 @ A4 )
=> ( member @ B @ ( F @ X2 ) @ ( image @ A @ B @ F @ A4 ) ) ) ).
% imageI
thf(fact_186_bex__Un,axiom,
! [A: $tType,A4: set @ A,B5: set @ A,P: A > $o] :
( ( ? [X3: A] :
( ( member @ A @ X3 @ ( sup_sup @ ( set @ A ) @ A4 @ B5 ) )
& ( P @ X3 ) ) )
= ( ? [X3: A] :
( ( member @ A @ X3 @ A4 )
& ( P @ X3 ) )
| ? [X3: A] :
( ( member @ A @ X3 @ B5 )
& ( P @ X3 ) ) ) ) ).
% bex_Un
thf(fact_187_UnI2,axiom,
! [A: $tType,C: A,B5: set @ A,A4: set @ A] :
( ( member @ A @ C @ B5 )
=> ( member @ A @ C @ ( sup_sup @ ( set @ A ) @ A4 @ B5 ) ) ) ).
% UnI2
thf(fact_188_UnI1,axiom,
! [A: $tType,C: A,A4: set @ A,B5: set @ A] :
( ( member @ A @ C @ A4 )
=> ( member @ A @ C @ ( sup_sup @ ( set @ A ) @ A4 @ B5 ) ) ) ).
% UnI1
thf(fact_189_UnE,axiom,
! [A: $tType,C: A,A4: set @ A,B5: set @ A] :
( ( member @ A @ C @ ( sup_sup @ ( set @ A ) @ A4 @ B5 ) )
=> ( ~ ( member @ A @ C @ A4 )
=> ( member @ A @ C @ B5 ) ) ) ).
% UnE
thf(fact_190_Un__def,axiom,
! [A: $tType] :
( ( sup_sup @ ( set @ A ) )
= ( ^ [A6: set @ A,B6: set @ A] :
( collect @ A
@ ^ [X3: A] :
( ( member @ A @ X3 @ A6 )
| ( member @ A @ X3 @ B6 ) ) ) ) ) ).
% Un_def
thf(fact_191_imageE,axiom,
! [A: $tType,B: $tType,B2: A,F: B > A,A4: set @ B] :
( ( member @ A @ B2 @ ( image @ B @ A @ F @ A4 ) )
=> ~ ! [X4: B] :
( ( B2
= ( F @ X4 ) )
=> ~ ( member @ B @ X4 @ A4 ) ) ) ).
% imageE
thf(fact_192_image__image,axiom,
! [A: $tType,B: $tType,C2: $tType,F: B > A,G: C2 > B,A4: set @ C2] :
( ( image @ B @ A @ F @ ( image @ C2 @ B @ G @ A4 ) )
= ( image @ C2 @ A
@ ^ [X3: C2] : ( F @ ( G @ X3 ) )
@ A4 ) ) ).
% image_image
thf(fact_193_Compr__image__eq,axiom,
! [A: $tType,B: $tType,F: B > A,A4: set @ B,P: A > $o] :
( ( collect @ A
@ ^ [X3: A] :
( ( member @ A @ X3 @ ( image @ B @ A @ F @ A4 ) )
& ( P @ X3 ) ) )
= ( image @ B @ A @ F
@ ( collect @ B
@ ^ [X3: B] :
( ( member @ B @ X3 @ A4 )
& ( P @ ( F @ X3 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_194_Collect__disj__eq,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ( collect @ A
@ ^ [X3: A] :
( ( P @ X3 )
| ( Q @ X3 ) ) )
= ( sup_sup @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) ) ) ).
% Collect_disj_eq
thf(fact_195_Un__empty__right,axiom,
! [A: $tType,A4: set @ A] :
( ( sup_sup @ ( set @ A ) @ A4 @ ( bot_bot @ ( set @ A ) ) )
= A4 ) ).
% Un_empty_right
thf(fact_196_Un__empty__left,axiom,
! [A: $tType,B5: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ B5 )
= B5 ) ).
% Un_empty_left
thf(fact_197_singleton__Un__iff,axiom,
! [A: $tType,X2: A,A4: set @ A,B5: set @ A] :
( ( ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) )
= ( sup_sup @ ( set @ A ) @ A4 @ B5 ) )
= ( ( ( A4
= ( bot_bot @ ( set @ A ) ) )
& ( B5
= ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) )
| ( ( A4
= ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) )
& ( B5
= ( bot_bot @ ( set @ A ) ) ) )
| ( ( A4
= ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) )
& ( B5
= ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_198_Un__singleton__iff,axiom,
! [A: $tType,A4: set @ A,B5: set @ A,X2: A] :
( ( ( sup_sup @ ( set @ A ) @ A4 @ B5 )
= ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) )
= ( ( ( A4
= ( bot_bot @ ( set @ A ) ) )
& ( B5
= ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) )
| ( ( A4
= ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) )
& ( B5
= ( bot_bot @ ( set @ A ) ) ) )
| ( ( A4
= ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) )
& ( B5
= ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_199_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_200_image__constant,axiom,
! [A: $tType,B: $tType,X2: A,A4: set @ A,C: B] :
( ( member @ A @ X2 @ A4 )
=> ( ( image @ A @ B
@ ^ [X3: A] : C
@ A4 )
= ( insert @ B @ C @ ( bot_bot @ ( set @ B ) ) ) ) ) ).
% image_constant
thf(fact_201_image__constant__conv,axiom,
! [B: $tType,A: $tType,A4: set @ B,C: A] :
( ( ( A4
= ( bot_bot @ ( set @ B ) ) )
=> ( ( image @ B @ A
@ ^ [X3: B] : C
@ A4 )
= ( bot_bot @ ( set @ A ) ) ) )
& ( ( A4
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( image @ B @ A
@ ^ [X3: B] : C
@ A4 )
= ( insert @ A @ C @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% image_constant_conv
thf(fact_202_the__elem__image__unique,axiom,
! [B: $tType,A: $tType,A4: set @ A,F: A > B,X2: A] :
( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [Y4: A] :
( ( member @ A @ Y4 @ A4 )
=> ( ( F @ Y4 )
= ( F @ X2 ) ) )
=> ( ( the_elem @ B @ ( image @ A @ B @ F @ A4 ) )
= ( F @ X2 ) ) ) ) ).
% the_elem_image_unique
thf(fact_203_sup__bot__left,axiom,
! [A: $tType] :
( ( bounde1808546759up_bot @ A )
=> ! [X2: A] :
( ( sup_sup @ A @ ( bot_bot @ A ) @ X2 )
= X2 ) ) ).
% sup_bot_left
thf(fact_204_sup__bot__right,axiom,
! [A: $tType] :
( ( bounde1808546759up_bot @ A )
=> ! [X2: A] :
( ( sup_sup @ A @ X2 @ ( bot_bot @ A ) )
= X2 ) ) ).
% sup_bot_right
thf(fact_205_bot__eq__sup__iff,axiom,
! [A: $tType] :
( ( bounde1808546759up_bot @ A )
=> ! [X2: A,Y: A] :
( ( ( bot_bot @ A )
= ( sup_sup @ A @ X2 @ Y ) )
= ( ( X2
= ( bot_bot @ A ) )
& ( Y
= ( bot_bot @ A ) ) ) ) ) ).
% bot_eq_sup_iff
thf(fact_206_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_207_sup__bot_Oneutr__eq__iff,axiom,
! [A: $tType] :
( ( bounde1808546759up_bot @ A )
=> ! [A2: A,B2: A] :
( ( ( bot_bot @ A )
= ( sup_sup @ A @ A2 @ B2 ) )
= ( ( A2
= ( bot_bot @ A ) )
& ( B2
= ( bot_bot @ A ) ) ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_208_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_209_sup__bot_Oeq__neutr__iff,axiom,
! [A: $tType] :
( ( bounde1808546759up_bot @ A )
=> ! [A2: A,B2: A] :
( ( ( sup_sup @ A @ A2 @ B2 )
= ( bot_bot @ A ) )
= ( ( A2
= ( bot_bot @ A ) )
& ( B2
= ( bot_bot @ A ) ) ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_210_sup__eq__bot__iff,axiom,
! [A: $tType] :
( ( bounde1808546759up_bot @ A )
=> ! [X2: A,Y: A] :
( ( ( sup_sup @ A @ X2 @ Y )
= ( bot_bot @ A ) )
= ( ( X2
= ( bot_bot @ A ) )
& ( Y
= ( bot_bot @ A ) ) ) ) ) ).
% sup_eq_bot_iff
thf(fact_211_these__image__Some__eq,axiom,
! [A: $tType,A4: set @ A] :
( ( these @ A @ ( image @ A @ ( option @ A ) @ ( some @ A ) @ A4 ) )
= A4 ) ).
% these_image_Some_eq
thf(fact_212_sup__set__def,axiom,
! [A: $tType] :
( ( sup_sup @ ( set @ A ) )
= ( ^ [A6: set @ A,B6: set @ A] :
( collect @ A
@ ( sup_sup @ ( A > $o )
@ ^ [X3: A] : ( member @ A @ X3 @ A6 )
@ ^ [X3: A] : ( member @ A @ X3 @ B6 ) ) ) ) ) ).
% sup_set_def
thf(fact_213_sup__Un__eq,axiom,
! [A: $tType,R: set @ A,S: set @ A] :
( ( sup_sup @ ( A > $o )
@ ^ [X3: A] : ( member @ A @ X3 @ R )
@ ^ [X3: A] : ( member @ A @ X3 @ S ) )
= ( ^ [X3: A] : ( member @ A @ X3 @ ( sup_sup @ ( set @ A ) @ R @ S ) ) ) ) ).
% sup_Un_eq
thf(fact_214_None__notin__image__Some,axiom,
! [A: $tType,A4: set @ A] :
~ ( member @ ( option @ A ) @ ( none @ A ) @ ( image @ A @ ( option @ A ) @ ( some @ A ) @ A4 ) ) ).
% None_notin_image_Some
thf(fact_215_Some__image__these__eq,axiom,
! [A: $tType,A4: set @ ( option @ A )] :
( ( image @ A @ ( option @ A ) @ ( some @ A ) @ ( these @ A @ A4 ) )
= ( collect @ ( option @ A )
@ ^ [X3: option @ A] :
( ( member @ ( option @ A ) @ X3 @ A4 )
& ( X3
!= ( none @ A ) ) ) ) ) ).
% Some_image_these_eq
thf(fact_216_sup_Ostrict__coboundedI2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [C: A,B2: A,A2: A] :
( ( ord_less @ A @ C @ B2 )
=> ( ord_less @ A @ C @ ( sup_sup @ A @ A2 @ B2 ) ) ) ) ).
% sup.strict_coboundedI2
thf(fact_217_sup_Ostrict__coboundedI1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [C: A,A2: A,B2: A] :
( ( ord_less @ A @ C @ A2 )
=> ( ord_less @ A @ C @ ( sup_sup @ A @ A2 @ B2 ) ) ) ) ).
% sup.strict_coboundedI1
thf(fact_218_sup_Ostrict__order__iff,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ( ( ord_less @ A )
= ( ^ [B3: A,A3: A] :
( ( A3
= ( sup_sup @ A @ A3 @ B3 ) )
& ( A3 != B3 ) ) ) ) ) ).
% sup.strict_order_iff
thf(fact_219_sup_Ostrict__boundedE,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [B2: A,C: A,A2: A] :
( ( ord_less @ A @ ( sup_sup @ A @ B2 @ C ) @ A2 )
=> ~ ( ( ord_less @ A @ B2 @ A2 )
=> ~ ( ord_less @ A @ C @ A2 ) ) ) ) ).
% sup.strict_boundedE
thf(fact_220_less__supI2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X2: A,B2: A,A2: A] :
( ( ord_less @ A @ X2 @ B2 )
=> ( ord_less @ A @ X2 @ ( sup_sup @ A @ A2 @ B2 ) ) ) ) ).
% less_supI2
thf(fact_221_less__supI1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X2: A,A2: A,B2: A] :
( ( ord_less @ A @ X2 @ A2 )
=> ( ord_less @ A @ X2 @ ( sup_sup @ A @ A2 @ B2 ) ) ) ) ).
% less_supI1
thf(fact_222_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_223_the__eq__trivial,axiom,
! [A: $tType,A2: A] :
( ( the @ A
@ ^ [X3: A] : X3 = A2 )
= A2 ) ).
% the_eq_trivial
thf(fact_224_the__sym__eq__trivial,axiom,
! [A: $tType,X2: A] :
( ( the @ A
@ ( ^ [Y2: A,Z: A] : Y2 = Z
@ X2 ) )
= X2 ) ).
% the_sym_eq_trivial
thf(fact_225_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_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_If__def,axiom,
! [A: $tType] :
( ( if @ A )
= ( ^ [P3: $o,X3: A,Y3: A] :
( the @ A
@ ^ [Z4: A] :
( ( P3
=> ( Z4 = X3 ) )
& ( ~ P3
=> ( Z4 = Y3 ) ) ) ) ) ) ).
% If_def
thf(fact_228_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_229_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_230_theI,axiom,
! [A: $tType,P: A > $o,A2: A] :
( ( P @ A2 )
=> ( ! [X4: A] :
( ( P @ X4 )
=> ( X4 = A2 ) )
=> ( P @ ( the @ A @ P ) ) ) ) ).
% theI
thf(fact_231_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_232_bind__singleton__conv__image,axiom,
! [A: $tType,B: $tType,A4: set @ B,F: B > A] :
( ( bind @ B @ A @ A4
@ ^ [X3: B] : ( insert @ A @ ( F @ X3 ) @ ( bot_bot @ ( set @ A ) ) ) )
= ( image @ B @ A @ F @ A4 ) ) ).
% bind_singleton_conv_image
thf(fact_233_empty__bind,axiom,
! [B: $tType,A: $tType,F: B > ( set @ A )] :
( ( bind @ B @ A @ ( bot_bot @ ( set @ B ) ) @ F )
= ( bot_bot @ ( set @ A ) ) ) ).
% empty_bind
thf(fact_234_Set_Obind__bind,axiom,
! [C2: $tType,B: $tType,A: $tType,A4: set @ A,B5: A > ( set @ C2 ),C3: C2 > ( set @ B )] :
( ( bind @ C2 @ B @ ( bind @ A @ C2 @ A4 @ B5 ) @ C3 )
= ( bind @ A @ B @ A4
@ ^ [X3: A] : ( bind @ C2 @ B @ ( B5 @ X3 ) @ C3 ) ) ) ).
% Set.bind_bind
thf(fact_235_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_236_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_237_Tree_Oset__intros_I1_J,axiom,
! [A: $tType,Y: A,X21: binary1291135688e_Tree @ A,X22: A,X23: binary1291135688e_Tree @ A] :
( ( member @ A @ Y @ ( binary2130109271t_Tree @ A @ X21 ) )
=> ( member @ A @ Y @ ( binary2130109271t_Tree @ A @ ( binary210054475elle_T @ A @ X21 @ X22 @ X23 ) ) ) ) ).
% Tree.set_intros(1)
thf(fact_238_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_239_bind__const,axiom,
! [B: $tType,A: $tType,A4: set @ B,B5: set @ A] :
( ( ( A4
= ( bot_bot @ ( set @ B ) ) )
=> ( ( bind @ B @ A @ A4
@ ^ [Uu: B] : B5 )
= ( bot_bot @ ( set @ A ) ) ) )
& ( ( A4
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( bind @ B @ A @ A4
@ ^ [Uu: B] : B5 )
= B5 ) ) ) ).
% bind_const
thf(fact_240_nonempty__bind__const,axiom,
! [A: $tType,B: $tType,A4: set @ A,B5: set @ B] :
( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( bind @ A @ B @ A4
@ ^ [Uu: A] : B5 )
= B5 ) ) ).
% nonempty_bind_const
thf(fact_241_Tree_Osimps_I14_J,axiom,
! [A: $tType] :
( ( binary2130109271t_Tree @ A @ ( binary1746293266le_Tip @ A ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% Tree.simps(14)
thf(fact_242_Inf_OINF__identity__eq,axiom,
! [A: $tType,Inf: ( set @ A ) > A,A4: set @ A] :
( ( Inf
@ ( image @ A @ A
@ ^ [X3: A] : X3
@ A4 ) )
= ( Inf @ A4 ) ) ).
% Inf.INF_identity_eq
thf(fact_243_Sup_OSUP__identity__eq,axiom,
! [A: $tType,Sup: ( set @ A ) > A,A4: set @ A] :
( ( Sup
@ ( image @ A @ A
@ ^ [X3: A] : X3
@ A4 ) )
= ( Sup @ A4 ) ) ).
% Sup.SUP_identity_eq
thf(fact_244_range__constant,axiom,
! [B: $tType,A: $tType,X2: A] :
( ( image @ B @ A
@ ^ [Uu: B] : X2
@ ( top_top @ ( set @ B ) ) )
= ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% range_constant
thf(fact_245_INF__eq__bot__iff,axiom,
! [B: $tType,A: $tType] :
( ( comple1035589618norder @ A )
=> ! [F: B > A,A4: set @ B] :
( ( ( complete_Inf_Inf @ A @ ( image @ B @ A @ F @ A4 ) )
= ( bot_bot @ A ) )
= ( ! [X3: A] :
( ( ord_less @ A @ ( bot_bot @ A ) @ X3 )
=> ? [Y3: B] :
( ( member @ B @ Y3 @ A4 )
& ( ord_less @ A @ ( F @ Y3 ) @ X3 ) ) ) ) ) ) ).
% INF_eq_bot_iff
thf(fact_246_top__apply,axiom,
! [C2: $tType,D: $tType] :
( ( top @ C2 )
=> ( ( top_top @ ( D > C2 ) )
= ( ^ [X3: D] : ( top_top @ C2 ) ) ) ) ).
% top_apply
thf(fact_247_UNIV__I,axiom,
! [A: $tType,X2: A] : ( member @ A @ X2 @ ( top_top @ ( set @ A ) ) ) ).
% UNIV_I
thf(fact_248_Inf__apply,axiom,
! [B: $tType,A: $tType] :
( ( complete_Inf @ B )
=> ( ( complete_Inf_Inf @ ( A > B ) )
= ( ^ [A6: set @ ( A > B ),X3: A] :
( complete_Inf_Inf @ B
@ ( image @ ( A > B ) @ B
@ ^ [F3: A > B] : ( F3 @ X3 )
@ A6 ) ) ) ) ) ).
% Inf_apply
thf(fact_249_INT__constant,axiom,
! [B: $tType,A: $tType,A4: set @ B,C: set @ A] :
( ( ( A4
= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Inf_Inf @ ( set @ A )
@ ( image @ B @ ( set @ A )
@ ^ [Y3: B] : C
@ A4 ) )
= ( top_top @ ( set @ A ) ) ) )
& ( ( A4
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Inf_Inf @ ( set @ A )
@ ( image @ B @ ( set @ A )
@ ^ [Y3: B] : C
@ A4 ) )
= C ) ) ) ).
% INT_constant
thf(fact_250_Collect__const,axiom,
! [A: $tType,P: $o] :
( ( P
=> ( ( collect @ A
@ ^ [S2: A] : P )
= ( top_top @ ( set @ A ) ) ) )
& ( ~ P
=> ( ( collect @ A
@ ^ [S2: A] : P )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Collect_const
thf(fact_251_INF__apply,axiom,
! [A: $tType,B: $tType,C2: $tType] :
( ( complete_Inf @ A )
=> ! [F: C2 > B > A,A4: set @ C2,X2: B] :
( ( complete_Inf_Inf @ ( B > A ) @ ( image @ C2 @ ( B > A ) @ F @ A4 ) @ X2 )
= ( complete_Inf_Inf @ A
@ ( image @ C2 @ A
@ ^ [Y3: C2] : ( F @ Y3 @ X2 )
@ A4 ) ) ) ) ).
% INF_apply
thf(fact_252_INF__identity__eq,axiom,
! [A: $tType] :
( ( complete_Inf @ A )
=> ! [A4: set @ A] :
( ( complete_Inf_Inf @ A
@ ( image @ A @ A
@ ^ [X3: A] : X3
@ A4 ) )
= ( complete_Inf_Inf @ A @ A4 ) ) ) ).
% INF_identity_eq
thf(fact_253_Inf__eq__bot__iff,axiom,
! [A: $tType] :
( ( comple1035589618norder @ A )
=> ! [A4: set @ A] :
( ( ( complete_Inf_Inf @ A @ A4 )
= ( bot_bot @ A ) )
= ( ! [X3: A] :
( ( ord_less @ A @ ( bot_bot @ A ) @ X3 )
=> ? [Y3: A] :
( ( member @ A @ Y3 @ A4 )
& ( ord_less @ A @ Y3 @ X3 ) ) ) ) ) ) ).
% Inf_eq_bot_iff
thf(fact_254_Inf__empty,axiom,
! [A: $tType] :
( ( comple187826305attice @ A )
=> ( ( complete_Inf_Inf @ A @ ( bot_bot @ ( set @ A ) ) )
= ( top_top @ A ) ) ) ).
% Inf_empty
% 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___Complete__Lattices_Ocomplete__lattice,axiom,
! [A7: $tType,A8: $tType] :
( ( comple187826305attice @ A8 )
=> ( comple187826305attice @ ( A7 > A8 ) ) ) ).
thf(tcon_fun___Lattices_Osemilattice__sup,axiom,
! [A7: $tType,A8: $tType] :
( ( semilattice_sup @ A8 )
=> ( semilattice_sup @ ( A7 > A8 ) ) ) ).
thf(tcon_fun___Complete__Lattices_OInf,axiom,
! [A7: $tType,A8: $tType] :
( ( complete_Inf @ A8 )
=> ( complete_Inf @ ( 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___Orderings_Oorder,axiom,
! [A7: $tType,A8: $tType] :
( ( order @ A8 )
=> ( order @ ( A7 > A8 ) ) ) ).
thf(tcon_fun___Orderings_Otop,axiom,
! [A7: $tType,A8: $tType] :
( ( top @ A8 )
=> ( top @ ( 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_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___Complete__Lattices_OInf_4,axiom,
complete_Inf @ int ).
thf(tcon_Int_Oint___Orderings_Opreorder_5,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___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,
! [A7: $tType] : ( bounde1808546759up_bot @ ( set @ A7 ) ) ).
thf(tcon_Set_Oset___Complete__Lattices_Ocomplete__lattice_9,axiom,
! [A7: $tType] : ( comple187826305attice @ ( set @ A7 ) ) ).
thf(tcon_Set_Oset___Lattices_Osemilattice__sup_10,axiom,
! [A7: $tType] : ( semilattice_sup @ ( set @ A7 ) ) ).
thf(tcon_Set_Oset___Complete__Lattices_OInf_11,axiom,
! [A7: $tType] : ( complete_Inf @ ( set @ A7 ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder__bot_12,axiom,
! [A7: $tType] : ( order_bot @ ( set @ A7 ) ) ).
thf(tcon_Set_Oset___Orderings_Opreorder_13,axiom,
! [A7: $tType] : ( preorder @ ( set @ A7 ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder_14,axiom,
! [A7: $tType] : ( order @ ( set @ A7 ) ) ).
thf(tcon_Set_Oset___Orderings_Otop_15,axiom,
! [A7: $tType] : ( top @ ( set @ A7 ) ) ).
thf(tcon_Set_Oset___Orderings_Oord_16,axiom,
! [A7: $tType] : ( ord @ ( set @ A7 ) ) ).
thf(tcon_Set_Oset___Orderings_Obot_17,axiom,
! [A7: $tType] : ( bot @ ( set @ A7 ) ) ).
thf(tcon_HOL_Obool___Lattices_Obounded__semilattice__sup__bot_18,axiom,
bounde1808546759up_bot @ $o ).
thf(tcon_HOL_Obool___Complete__Lattices_Ocomplete__lattice_19,axiom,
comple187826305attice @ $o ).
thf(tcon_HOL_Obool___Lattices_Osemilattice__sup_20,axiom,
semilattice_sup @ $o ).
thf(tcon_HOL_Obool___Complete__Lattices_OInf_21,axiom,
complete_Inf @ $o ).
thf(tcon_HOL_Obool___Orderings_Oorder__bot_22,axiom,
order_bot @ $o ).
thf(tcon_HOL_Obool___Orderings_Opreorder_23,axiom,
preorder @ $o ).
thf(tcon_HOL_Obool___Orderings_Olinorder_24,axiom,
linorder @ $o ).
thf(tcon_HOL_Obool___Orderings_Oorder_25,axiom,
order @ $o ).
thf(tcon_HOL_Obool___Orderings_Otop_26,axiom,
top @ $o ).
thf(tcon_HOL_Obool___Orderings_Oord_27,axiom,
ord @ $o ).
thf(tcon_HOL_Obool___Orderings_Obot_28,axiom,
bot @ $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,X2: A,Y: A] :
( ( if @ A @ $false @ X2 @ Y )
= Y ) ).
thf(help_If_1_1_T,axiom,
! [A: $tType,X2: A,Y: A] :
( ( if @ A @ $true @ X2 @ Y )
= X2 ) ).
% Conjectures (1)
thf(conj_0,conjecture,
$false ).
%------------------------------------------------------------------------------